appl. gen. topol. 21, no. 1 (2020), 81-85 doi:10.4995/agt.2020.12065 c© agt, upv, 2020 a note on rank 2 diagonals angelo bella and santi spadaro dipartimento di matematica e informatica, university of catania, città universitaria, viale a. doria 6, 95125 catania, italy (bella@dmi.unict.it, santidspadaro@gmail.com) communicated by s. garćıa-ferreira abstract we solve two questions regarding spaces with a (gδ)-diagonal of rank 2. one is a question of basile, bella and ridderbos about weakly lindelöf spaces with a gδ-diagonal of rank 2 and the other is a question of arhangel’skii and bella asking whether every space with a diagonal of rank 2 and cellularity continuum has cardinality at most continuum. 2010 msc: 54d10; 54a25. keywords: cardinality bounds; weakly lindelöf; gδ-diagonal; neighbourhood assignment; dual properties. 1. introduction a space is said to have a gδ-diagonal if its diagonal can be written as the intersection of a countable family of open subsets in the square. this notion is of central importance in metrization theory, ever since sneider’s 1945 theorem [14] stating that every compact hausdorff space with a gδ-diagonal is metrizable. sneider’s result was later improved by chaber [8] who proved that every countably compact space with a gδ-diagonal is compact and hence metrizable. around the same time, ginsburg and woods [10] showed the influence of gδdiagonals in the theory of cardinal invariants for topological spaces by proving that every space with a gδ-diagonal without uncountable closed discrete sets has cardinality at most continuum. their result led them to conjecture that every ccc space with a gδ-diagonal must have cardinality at most continuum. received 09 july 2019 – accepted 22 october 2019 http://dx.doi.org/10.4995/agt.2020.12065 a. bella and s. spadaro shakhmatov [13] and uspenskii [15] gave a pretty strong disproof to this conjecture by constructing tychonoff ccc spaces with a gδ-diagonal of arbitrarily large cardinality. however, in the meanwhile, several strengthenings of the notion of a gδ-diagonal had been introduced, leading several researchers to test ginsburg and woods’s conjecture against these stronger diagonal properties. that culminated in buzyakova’s surprising result [7] that a ccc space with a regular gδ-diagonal has cardinality at most continuum. a space has a regular gδ-diagonal if there is a countable family of neighbourhoods of the diagonal in the square such that the diagonal is the intersection of their closures. another way of strengthening the property of having a gδ-diagonal is by considering the notion of rank. recall that given a family u of subsets of a topological space and a point x ∈ x, st(x,u) := ⋃ {u ∈ u : x ∈ u}. the set stn(x,u) is defined by induction as follows: st1(x,u) = st(x,u) and stn(x,u) = ⋃ {u ∈u : u ∩stn−1(x,u) 6= ∅} for every n > 1. a space is said to have a diagonal of rank n if there is a sequence {uk : k < ω} of open covers of x such that ⋂ {stn(x,uk) : k < ω} = {x}, for every x ∈ x. by a wellknown characterization, having a diagonal of rank 1 is equivalent to having a gδ-diagonal. note also that a space with a gδ-diagonal of rank 2 is necessarily t2. zenor [17] observed that every space with a diagonal of rank 3 also has a regular gδ-diagonal so by buzyakova’s result, every ccc space with a diagonal of rank 3 has cardinality at most continuum. in [3], the first author proved the stronger result that every ccc space with a gδ-diagonal of rank 2 has cardinality at most 2ω. the following question is still open though: question 1.1 (arhangel’skii and bella [1]). is every regular gδ-diagonal always of rank 2? a positive answer would lead to a far-reaching generalization of buzyakova’s cardinal bound. arhangel’skii and the first-named author proved in [1] that every space with a diagonal of rank 4 and cellularity ≤ c has cardinality at most continuum, and leave open whether this is also true for spaces with a diagonal of rank 2 or 3. question 1.2. let x be a space with a diagonal of rank 2 or 3 and cellularity at most c. is it true that |x| ≤ c. from proposition 4.7 of [4] it follows that |x| ≤ c(x)ω for every space x with a diagonal of rank 3, which in turn that the answer to arhangel’skii and bella’s question is yes for spaces with a diagonal of rank 3. we show that the answer to their question is no for spaces with a diagonal of rank 2, by constructing a space with a diagonal of rank 2, cellularity ≤ c and cardinality larger than the continuum. that leads to a complete solution to arhangel’skii and bella’s question. recall that space x is weakly lindelöf provided that every open cover has a countable subfamily whose union is dense in x. this notion is a common generalisation of the lindelöf property and the countable chain condition (ccc). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 82 a note on rank 2 diagonals in view of the results by ginsburg-woods and bella mentioned above it is natural to consider the following question: question 1.3 ([4]). let x be a weakly lindelöf space with a gδ-diagonal of rank 2. is it true that |x| ≤ 2ω? the above question was explicitly formulated in [4] and two partial positive answers were obtained there under the assumptions that the space is either baire or has a rank 3 diagonal. here we will prove that question 1.3 has a positive answer assuming that the space is normal. all undefined notions can be found in [12]. 2. spaces with a diagonal of rank 2 recall that a neighbourhood assignment for a space x is a function φ from x to its topology such that x ∈ φ(x) for every x ∈ x. a set y ⊆ x is a kernel for φ if x = ⋃ {φ(y) : y ∈ y}. following [11], we say that a space x is dually p if every neighbourhood assignment in x has a kernel y satisfying the property p. of course, p implies dually p. a dually ccc space may fail to be even weakly lindelöf. here we need the countable version of a well-known result of erdös and rado: lemma 2.1. let x be a set with |x| > 2ω. if [x]2 = ⋃ {pn : n < ω}, then there exist an uncountable set s ⊆ x and an integer n0 ∈ ω such that [s]2 ⊆ pn0. theorem 2.2. if x is a dually weakly lindelöf normal space with a gδdiagonal of rank 2, then |x| ≤ 2ω. proof. let {un : n < ω} be a sequence of open covers of x such that {x} =⋂ {st2(x,un) : n < ω} for each x ∈ x. assume by contradiction that |x| > 2ω and for any n < ω put pn = {{x,y} ∈ [x]2 : st(x,un) ∩ st(y,un) = ∅}. the assumption that the sequence {un : n < ω} has rank 2 implies that [x]2 = ⋃ {pn : n < ω}. by lemma 2.1 there exists an uncountable set s ⊆ x and an integer n0 such that [s] 2 ⊆ pn0 . the collection {st(x,un0 ) : x ∈ s} consists of pairwise disjoint open sets. from that it follows that, for any z ∈ x, the set st(z,un0 ) cannot meet s in two distinct points, which implies that the set s is closed and discrete. we define a neighbourhood assignment φ for x as follows: if x ∈ s let φ(x) = st(x,un0 ) and if x ∈ x \s let φ(x) = x \s. since x is dually weakly lindelöf, there exists a weakly lindelöf subspace y such that x = ⋃ {φ(y) : y ∈ y}. by the way φ is defined, it follows that s ⊆ ⋃ {φ(y) : y ∈ y ∩ s} and hence s ⊆ y . as x is normal, we may pick an open set v such that s ⊆ v and v ⊆ ⋃ {st(x,un0 ) : x ∈ s}. the trace on y of the open cover {st(x,un0 ) : x ∈ x} ∪ {x \ v} witnesses the failure of the weak lindelöf property on y . this is a contradiction and we are done. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 83 a. bella and s. spadaro related results for the classes of dually ccc spaces and for that of cellularlindelöf spaces were proved in [16] and [6]. finally we will construct a space with a diagonal of rank 2, cellularity at most continuum and cardinality larger than the continuum, thus solving problem 2 from [1]. recall that a κ-suslin line l is a continuous linear order (endowed with the order topology) such that c(l) ≤ κ < d(l). the existence of a κsuslin line for every κ ≥ ω is consistent with zfc (jensen proved that it follows from v = l). theorem 2.3. (v = l) there is a space x with a diagonal of rank 2 such that c(x) ≤ c and |x| ≥ c+. proof. let t be an ω1-suslin line. let s be the set of all points of l which have countable cofinality. since t is a continuous linear order the set s is dense in t and hence d(s) > ℵ1. in particular, |s| > ℵ1. let τ be the topology on s generated by intervals of the form (x,y], where x < y ∈ s. note that c((s,τ)) = ℵ1 and that the space (s,τ) is first-countable and regular. so applying mike reed’s moore machine (see, for example [9]) to (s,τ) we obtain a moore space m(s) such that |m(s)| = |s| > ℵ1 and c(m(s)) ≤ℵ1. recalling that moore spaces have a diagonal of rank 2 (see proposition 1.1 of [2]) and that c = ℵ1 under v = l, we see that x = m(s) satisfies the statement of the theorem. � the above theorem also shows that the assumption that the space is baire is essential in proposition 4.5 from [4], thus solving a question asked by the authors of [4] (see the paragraph after the proof of lemma 4.6). 3. acknowledgements the authors acknowledge support from indam-gnsaga. references [1] a. v. arhangel’skii and a. bella, the diagonal of a first-countable paratopological groups, submetrizability and related results, appl. gen. topol. 8 (2007), 207–212. [2] a. v. arhangel’skii and r. z. buzyakova, the rank of the diagonal and submetrizability, comment. math. univ. carolinae 47 (2006), 585–597. [3] a. bella, remarks on the metrizability degree, boll. union. mat. ital. 1-3 (1987), 391–396. [4] d. basile, a. bella and g. j. ridderbos, weak extent, submetrizability and diagonal degrees, houston j. math. 40 (2014), 255–266. [5] m. bell, j. ginsburg and g. woods, cardinal inequalities for topological spaces involving the weak lindelöf number, pacific j. math. 79 (1978), no. 1, 37–45. [6] a. bella and s. spadaro, cardinal invariants of cellular lindelöf spaces, rev. r. acad. cienc. exactas f́ıs. nat. ser. a mat. racsam 113 (2019), 2805–2811. [7] r. buzyakova, cardinalities of ccc spaces with regular gδ-diagonals, topology appl. 153 (2006), 1696–1698. [8] j. chaber, conditions which imply compactness in countably compact spaces, bull. acad. pol. sci. ser. math. 24 (1976), 993–998. [9] e. k. van douwen and m. reed, on chain conditions in moore spaces ii, topology appl. 39 (1991), 65–69. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 84 a note on rank 2 diagonals [10] j. ginsburg and r. g. woods, a cardinal inequality for topological spaces involving closed discrete sets, proc. amer. math. soc. 64 (1977), 357–360. [11] j. van mill, v. v. tkachuk and r. g. wilson, classes defined by stars and neighbourhood assignments, topology appl. 154, no. 10 (2007), 2127–2134. [12] r. engelking, general topology, heldermann verlag, berlin, second ed., 1989. [13] d. shakhmatov, no upper bound for cardinalities of tychonoff c.c.c. spaces with a gδ diagonal exist (an answer to j. ginsburg and r.g. woods’ question), comment. math. univ. carolinae 25 (1984), 731–746. [14] v. sneider, continuous images of souslin and borel sets: metrization theorems, dokl. acad. nauk ussr, 50 (1945), 77–79. [15] v. uspenskij, a large fσ-discrete fréchet space having the souslin property, comment. math. univ. carolinae 25 (1984), 257–260. [16] w.-f. xuan and y.-k. song, dually properties and cardinal inequalities, topology appl. 234 (2018), 1–6. [17] p. zenor, on spaces with regular gδ-diagonals, pacific j. math. 40 (1972), 959–963. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 85 @ appl. gen. topol. 21, no. 2 (2020), 195-200 doi:10.4995/agt.2020.11903 c© agt, upv, 2020 closure formula for ideals in intermediate rings john paul jala kharbhih and sanghita dutta department of mathematics, north eastern hill university, mawkynroh, umshing, shillong 22, india (jpkharbhih@gmail.com, sanghita22@gmail.com) communicated by f. mynard abstract in this paper, we prove that the closure formula for ideals in c(x) under m topology holds in intermediate rings also. i.e. for any ideal i in an intermediate ring with m topology, its closure is the intersection of all the maximal ideals containing i. 2010 msc: 46e25; 54c30; 54c35; 54c40. keywords: m topology; rings of continuous functions; β-ideals. 1. introduction the m topology on c(x) was defined by hewitt in [9]. let cm(x) denote the ring c(x) equipped with m topology. cm(x) was shown to be a topological ring. in any topological ring, the closure of a proper ideal is either a proper ideal or the whole ring [8, 2m1]. amongst other results, hewitt in [9] showed that every maximal ideal in c(x) under m topology is closed. he conjectured that every m closed ideal of c(x) is an intersection of maximal ideals of c(x). this conjecture was settled by gillman, henriksen and jerison [7]. it was also settled independently by t.shirota [12]. in [7](also [8, 7q.3]), it was further shown that the closed ideals in c∗(x) (under subspace m topology) coincide with the intersections of maximal ideals in c∗(x) if and only if x is pseudocompact. intermediate rings denoted by a(x), are rings of continuous functions which lie in between c∗(x) and c(x). these rings were studied by donald plank as βsubalgebras in [10]. subsequently, a number of researchers generated renewed interests in these intermediate rings as can be seen in [11], [5], [2], [4], [3] and [1]. received 27 may 2019 – accepted 26 april 2020 http://dx.doi.org/10.4995/agt.2020.11903 j. p. j. kharbhih and s. dutta given a real number " > 0 and g ∈ a(x), let e!(g) [8, 2l] denote the set {x ∈ x : |g(x)| ≤ "}. given " > 0, f ∈ a(x), it is not difficult to construct a function t satisfying ft = 1 on the complement of e!(f). i.e. e!(f) ∈ za(f) ∀ " > 0. given an ideal i in a(x), let i′ denote the intersection of all the maximal ideals of am(x) that contain i. evidently i ′ is closed. let f ∈ a(x) and e ∈ z(x). then, f is said to be ec-regular, if ∃ g ∈ a(x) such that fg|ec = 1. for each f ∈ a(x), let za(f) denote the set {e ∈ z(x): f is ec − regular}. for an ideal i of a(x), za[i] denote the set ! f∈i za(f). the set of cluster points of a z-filter f is denoted by s[f ]. an ideal i in a(x) is said to be a β-ideal if za(f) ⊂ za[i] =⇒ f ∈ i. we shall denote intermediate rings a(x) with m topology by am(x). for undefined terms and references, we refer the reader to [8]. in this paper, we ask if hewitt’s formula for closure of an ideal holds for the case of am(x) also. we answer this question in the affirmative, and as an outcome we obtain the result that an ideal in an intermediate ring is closed iff the ideal is a β-ideal. theorem 1.1 ([5, theorem 3.3]). let m p a be the maximal ideal of a(x) corresponding to the point p of βx. then m p a = {f ∈ a(x): p ∈ s[za(f)]}. 2. closure formula in intermediate rings let ua(x) denote the set of positive units of a(x). for each f ∈ a(x) and each u ∈ ua(x), let ba(f, u) denote the collection {g ∈ a(x): |f − g| < u}. for each f ∈ a(x), the set bf = {ba(f, u) : u ∈ ua(x)} forms a base for the neighborhood system at f and the topology so formed is the m topology in a(x). definition 2.1. let a(x) be an intermediate subring. for an ideal i in a(x), let ∆a(i) = {p ∈ βx : m p a ⊃ i}. theorem 2.2. let i be an ideal in a(x) and p ∈ βx − ∆a(i). then, ∃ f ∈ i ∩ c∗(x) such that fβ(p) = 1. proof. since p ∕∈ ∆a(i), so m p a ∕⊃ i. therefore, ∃ g ∈ i, such that g ∕∈ m p a. so, ∃ a neighborhood u of p (in βx) which does not meet e, for some e ∈ za(g). now e ∈ za(g) =⇒ gl|ec = 1 for some l ∈ a(x). let f ∈ c ∗(x) be such that 0 ≤ f ≤ 1, fβ(p) = 1 and (2.1) fβ(uc) = 0. we define h: x → r by h(x) = " f(x) (|f(x)|+1)l(x)g(x), if x ∈ clβxu ∩ x 0, if x ∈ (βx − u) ∩ x. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 196 closure formula for ideals in intermediate rings then, h is well-defined and continuous. in fact h ∈ a(x) since h ∈ c∗(x). moreover the definition of h shows that f is a multiple of g so that f ∈ i, which completes the proof. □ theorem 2.3. let ω be an open subset of βx such that ω ⊃ ∆a(i) for some ideal i in a(x). then, given " with 0 < " < 1, ∃ g ∈ i with 0 ≤ g ≤ 1 such that ω ∩ x ⊃ e!(g). proof. let p ∈ βx − ω. then, p ∕∈ ∆a(i). by theorem 2.2 we see that ∃ gp ∈ i ∩ c∗(x) such that gβp (p) = 1. we choose an " ∈ r with 0 < " < 1. let σp = {q ∈ βx : gβp (q) > √ "0}. then, σp is open in βx and non-empty as p ∈ σp. now, the collection {σp : p ∈ βx − ω} forms an open cover for the compact set βx − ω. let {σp1, σp2, . . . , σpn} be a finite subcover of this open cover. let g = g2p1 + g 2 p2 + . . . + g2pn. for any p ∈ βx − ω, we then have g β(p) = (gβp1(p)) 2 + (gβp2(p)) 2 + . . . + (gβpn(p)) 2 > ". therefore, if |gβ(p)| ≤ ", then p ∕∈ βx − ω. i.e. p ∈ ω. hence, e!(g) ⊂ ω ∩ x. □ definition 2.4. let f ∈ a(x). we say that f is zc-related to i, if ∃ " > 0, such that z(f) ⊃ c ⊃ e!(g) for some cozero-set c and some g ∈ i. definition 2.5. for an ideal i of a(x), we define ka(i) = {f ∈ a(x) : f is zc-related to i}. theorem 2.6. for every ideal i of an intermediate subring am(x), we have ka(i) ⊂ i and clm(ka(i)) = clm(i). proof. let f ∈ ka(i). then, ∃ " > 0 such that z(f) ⊃ c ⊃ e!(g) for some cozero-set c and some g ∈ i. let us denote e!(g) by e. since e ∈ za(g), ∃ l ∈ am(x) such that (gl)|ec = 1. now, we define h by h(x) = " 0, if x ∈ clxc f (|f|+1)lg if x ∕∈ c. then, h is a well-defined bounded function. moreover, h is continuous. i.e. h ∈ c∗(x) ⊂ a(x). also, we get f = h(|f| + 1)lg, which shows that f ∈ i. thus ka(i) ⊂ i and hence clm(ka(i)) ⊂ clm(i). to prove that clm(i) ⊂ clm(ka(i)), it is enough to prove that i ⊂ clm(ka(i)). so, we take a g ∈ i. let π ∈ ua(x). we define f by f(x) = # $% $& 0, if − π(x) 2 ≤ g(x) ≤ π(x) 2 g(x) − π(x) 2 , if g(x) > π(x) 2 g(x) + π(x) 2 , if g(x) < −π(x) 2 . then, f lies in the π neighborhood of g. we also notice that f ∈ am(x) since f may be rewritten as follows : f(x) = '( g(x) − π(x) 2 ) ∨ 0 * + '( g(x) + π(x) 2 ) ∧ 0 * . c© agt, upv, 2020 appl. gen. topol. 21, no. 2 197 j. p. j. kharbhih and s. dutta we shall now show that f ∈ ka(i). let c = {x ∈ x : − π(x) 2 < g(x) < π(x) 2 }. then z(f) ⊃ c. moreover, c is the cozero-set of the function h ∈ a(x) defined by: h(x) = ( |g(x)| − π(x) 2 ) ∧ 0. we choose any real number " > 0 and define a function θ by θ(x) = 4!g(x) π(x) . clearly, θ ∈ i. moreover |θ(x)| ≤ " ⇐⇒ |g(x)| ≤ π(x) 4 . in otherwords, x ∈ e!(θ) ⇐⇒ |g(x)| ≤ π(x) 4 . but, |g(x)| ≤ π(x) 4 =⇒ x ∈ z(f). hence z(f) ⊃ c ⊃ e!(θ) which completes the proof. □ example 2.7. now, we will give an example of an ideal i such that ka(i) ⊊ i. let x = r and a(x) = c(x). let i = m0. we will show that ka(i) = o0. firstly, if f ∈ o0, then ∃ an open set c such that 0 ∈ c ⊂ z(f). now, ∃ " > 0 such that e = [−", "] ⊂ c. then e = e!(g), where g is the identity map on r. moreover, c is a cozero-set as x is a metric space. hence we have f ∈ ka(i). secondly, if f ∈ ka(i), then ∃ g ∈ i, " > 0 such that z(f) ⊃ c ⊃ e!(g) for some cozero-set c. since 0 ∈ e!(g), this gives that z(f) is a neighborhood of 0 i.e. f ∈ o0. theorem 2.8. k ∈ i′ ⇐⇒ s[za(k)] ⊃ ∆a(i). proof. (⇒) we assume that k ∈ i′. let p ∈ ∆a[i]. then, m p a ⊃ i and so k ∈ mpa. by definition of m p a, p ∈ s[za(k)]. (⇐) let mpa be a maximal ideal which contains i. so, p ∈ ∆a(i) and thus, p ∈ s[za(k)]. therefore, k ∈ m p a and hence k ∈ i ′. □ we now prove the main result. theorem 2.9. the m closure of any ideal i in am(x) is the intersection of all the maximal ideals containing i. proof. we have clm(i) ⊂ i′ as i′ is closed. to prove i′ ⊂ clm(i), it is sufficient to prove that ka(i ′) ⊂ ka(i). then, by theorem 2.6, we will get i′ ⊂ clmi. let f ∈ ka(i′). then, ∃ a cozero-set c, a real number " > 0 and θ ∈ i′ such that z(f) ⊃ c ⊃ e!(θ) = e(say).(2.2) let z = x − c. then, z and e are completely separated being disjoint zerosets. therefore, ∃ h ∈ c∗(x), 0 ≤ h ≤ 1 such that h(e) = 0 and h(z) = 1. let ω = {p ∈ βx : hβ(p) < 1}. we observe that x = c ∪ z, so βx = clβxc ∪ clβxz. if p ∈ ω, i.e. hβ(p) < 1, then p ∕∈ clβxz as hβ(clβxz) = 1. so p ∈ clβxc. i.e. clβxc ⊃ ω.(2.3) since e ∈ za(θ), therefore ω ⊃ s[za(θ)] because p ∈ s[za(θ)] gives hβ(p) = 0. hence by theorem 2.8, we see that ω ⊃ ∆a(i). theorem 2.3 now gives a g ∈ i with 0 ≤ g ≤ 1 and some " with 0 < " < 1 such that ω ∩ x ⊃ e!(g).(2.4) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 198 closure formula for ideals in intermediate rings from (2.2) and (2.3), we get, clβxz(f) ⊃ clβxc ⊃ ω. then clβxz(f) ∩ x ⊃ ω ∩ x. thus z(f) ⊃ ω ∩ x. therefore, by (2.4) z(f) ⊃ ω ∩ x ⊃ e!(g). finally, we have ω ∩ x is a co-zero-set as ω ∩ x = {p ∈ x : h(p) < 1}. □ corollary 2.10. every closed ideal is a β-ideal. proof. first we claim that an arbitrary intersection of β-ideals is also a β-ideal. let {iα : α ∈ λ} be a collection of β-ideals. let za(f) ⊂ za[ + α∈λ iα]. since each iα is a β-ideal, it is enough to prove that za(f) ⊂ za[iα] ∀ α ∈ λ, for this would imply that f ∈ iα ∀ α ∈ λ. so take e ∈ za(f). therefore e ∈ za(g) for some g ∈ + α∈λ iα. this then gives e ∈ za[iα] ∀ α ∈ λ. now, let i be a closed ideal in am(x). therefore, i is an intersection of maximal ideals. but, as every maximal ideal is a β-ideal, therefore i is an intersection of β-ideals and hence a β-ideal. □ remark 2.11. in [6, theorem 3.13], it was shown that the β-ideals of an intermediate ring are just the intersections of maximal ideals of the ring. this says that β-ideals are closed, since maximal ideals are closed. hence the class of β-ideals and the class of closed ideals in intermediate rings coincide. this coincidence also occurs in the case of the subring c∗(x) with m topology. here, the class of e-ideals is the same as the class of closed ideals [8, 2m]. however, this coincidence does not extend to z-ideals in cm(x) since the ideal o p is a z-ideal which is not closed. remark 2.12. in [1], it was proven that if an intermediate ring a(x) is different from c(x), then there exists at least one non-maximal prime ideal p in a(x). thus, p is not closed in am(x). on the other hand if a(x) = c(x) and x is a p space then each ideal in am(x) is closed [8, 7q4]. thus within the class of p spaces x, for an intermediate ring a(x), each ideal in am(x) is closed ⇐⇒ a(x) = c(x) this is a special property of c(x) which distinguishes c(x) amongst all the intermediate rings (in the category of p spaces x). acknowledgements. the authors would like to thank the referee for the valuable comments and suggestions towards the improvement of the paper. in particular, remark 2.12 is due to the referee. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 199 j. p. j. kharbhih and s. dutta references [1] s. k. acharyya and b. bose, a correspondence between ideals and z-filters for certain rings of continuous functions some remarks, topology and its applications 160, no. 13 (2013), 1603–1605. 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[8] l. gillman and m. jerison, rings of continuous functions, univ. ser. higher math, d. van nostrand company, inc., princeton, n. j., 1960. [9] e. hewitt, rings of real-valued continuous functions i, trans. amer. math. soc. 64, no. 1 (1948), 45–99. [10] d. plank, on a class of subalgebras of c(x) with applications to βx \ x, fund. math. 64 (1969), 41–54. [11] l. redlin and s. watson, maximal ideals in subalgebras of c(x), proc. amer. math. soc. 100, no. 4 (1987), 763–766. [12] t. shirota, on ideals in rings of continuous functions, proc. japan acad. 30, no. 2 (1954), 85–89. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 200 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 33-40 range-preserving ae(0)-spaces w. w. comfort and a. w. hager abstract all spaces here are tychonoff spaces. the class ae(0) consists of those spaces which are absolute extensors for compact zero-dimensional spaces. we define and study here the subclass ae(0)rp, consisting of those spaces for which extensions of continuous functions can be chosen to have the same range. we prove these results. if each point of t ∈ ae(0) is a gδ-point of t , then t ∈ ae(0) rp. these are equivalent: (a) t ∈ ae(0)rp; (b) every compact subspace of t is metrizable; (c) every compact subspace of t is dyadic; and (d) every subspace of t is ae(0). thus in particular, every metrizable space is an ae(0)rpspace. 2010 msc: primary 54c55. secondary 06f20, 46e10, 54e18 keywords: absolute extensor; retraction; zero-dimensional space; rangepreserving function; dugundji space; dyadic space; countable chain condition 1. preliminaries all spaces here are assumed tychonoff. for spaces x and y , the symbol c(x, y ) denotes the set of continuous functions from x into y . we write y ⊆h x to indicates that x contains a homeomorph of y . let x be a homeomorphism-closed class of spaces. then ae(x) [resp., ae(x)rp], the class of absolute extensors [resp., range-preserving absolute extensors] for x, consists of those spaces t for which, whenever x ∈ x and f is a closed subset of x, every f ∈ c(f, t ) extends to f ∈ c(x, t ) [resp., and with f[x] = f[f ]]. 34 w. w. comfort and a. w. hager for x a class of spaces, we write px := {πi∈i xi : i ∈ i ⇒ xi ∈ x}. it is clear for arbitrary x, since πi ◦ f ∈ c(f, ti) for each space t = πi∈i ti and f ∈ c(x, t ), that (1.1) pae(x) = ae(x) for every class x. we note below in theorem 1.5((a) and (d)) that the relation pae(x)rp = ae(x)rp can fail—indeed it fails when x = 0, the class of compact zerodimensional spaces. the class ae(0) has been much studied; see [1] for information and extensive bibliographic citations. in this paper we focus on its subclass ae(0)rp, which so far as we know is defined and studied for the first time here. the class of compact spaces in ae(0) has been intensively studied. according to haydon [10], it coincides with the class of dugundji spaces as defined by pe lczyński [14], and the subclass 0 ∩ ae(0) of ae(0) coincides with the class of stone spaces of projective boolean algebras ([13]). let 2 denote the two-point discrete space. we begin with a simple basic observation. theorem 1.1. 2 ∈ ae(0)rp. proof. let f ∈ c(f, 2) with f closed in x ∈ 0. if f is a constant function then surely f extends to f ∈ c(x, 2) with f[x] = f[f ], so we assume fi := f−1(i) 6= ∅ for i ∈ 2. for x ∈ f0 there is a clopen neighborhood ux of x ∈ x such that ux ∩ f1 = ∅, and since f0 is compact some finitely many of the sets ux (x ∈ f0) cover f0. the union u of those sets covers f0, is clopen in x, and is disjoint from f1, and then the function f ∈ c(x, 2) defined by f ≡ 0 on u, f ≡ 1 on x\u extends f as required. � from [4](6.2.16) we have for each space x that x is zero-dimensional if and only if there is a cardinal κ such that x ⊆h 2 κ. it follows then quickly from (1.1) that (1.2) ae(0) = ae(p({2}), hence ae(0)rp = ae(p{2})rp. for a qualitative distinction between the class ae(0) and its subclass ae(0)rp, one may compare the equivalence (a) ⇔ (b) of the following theorem with the fact that every (tychonoff) space y embeds as a subspace of a (compact) space t ∈ ae(0). (to see that, recall that [0, 1] ∈ ae(0) by the classical tietzeurysohn extension theorem, and that y embeds into some t ∈ p{[0, 1]}; then, use (1.1) with x = {[0, 1]}.) theorem 1.2. for each space t , these conditions are equivalent. (a) t ∈ ae(0)rp; (b) s ⊆ t ⇒ s ∈ ae(0)rp; and (c) s ⊆ t , s compact ⇒ s ∈ ae(0)rp. range-preserving ae(0)-spaces 35 proof. (a) ⇒ (b). given such s and t and f ∈ c(f, s) with f closed in x ∈ 0, there is f ∈ c(x, t ) such that f ⊆ f and f[x] = f[f ] ⊆ s. (b) ⇒ (c). this is obvious. (c) ⇒ (a). given f ∈ c(f, t ) with closed f ⊆ x ∈ 0, the space s := f[f ] is compact. thus there is f ∈ c(x, t ) such that f ⊆ f and f[f ] = f[f ] = s. this shows t ∈ ae(0)rp. � in theorem 1.5 we make additional simple observations which highlight differences between the classes ae(0) and ae(0)rp. for that, these definitions will be useful. definition 1.3. let t be a space. (a) t is a countable chain condition space (briefly, a c.c.c. space) if every family of pairwise disjoint open subsets of t is countable. (b) t is dyadic if for some cardinal κ there is a continuous surjection from 2κ onto t . lemma 1.4. (a) every compact t ∈ ae(0) is dyadic. (b) every dyadic space is a c.c.c. space. proof. (a) as with every compact space, there exist a cardinal κ, a closed subspace f of 2κ, and a continuous surjection f : f ։ t ([4](3.2.2)). since 2κ ∈ 0 and t ∈ ae(0) there is f ∈ c(2κ, t ) such that f ⊇ f. then f[2κ] = t . (b) it is well known ([4](2.3.18)) that every product of separable spaces—in particular, the space 2κ—is a c.c.c. space; and the c.c.c. property is preserved under continuous surjections. � here and later we denote by αd the one-point compactification of the discrete space d of cardinality ℵ1. theorem 1.5. let κ be a cardinal. (a) 2 ∈ ae(0)rp; (b) 2κ ∈ ae(0); (c) if κ > ℵ0 then there is compact t ⊆ 2 κ such that t /∈ ae(0); (d) if κ > ℵ0 then 2 κ /∈ ae(0)rp. proof. (a) was noted in theorem 1.1, and (b) follows from (1.1) since ae(0)rp ⊆ ae(0). it is easily seen, as in [4](6.2.16), that αd ⊆h 2 ℵ1 . clearly the (compact) space αd is not a c.c.c. space, so αd /∈ ae(0) by lemma 1.4. that shows (c), and (d) follows from theorem 1.2. � the gist of theorem 1.5 is that while the class ae(0)rp is “completely hereditary” (theorem 1.2), the class ae(0) is not even compact-hereditary; and ae(0), like every class ae(x), is completely productive (1.1), while the class ae(0)rp is not even ℵ1-productive. we will see in corollary 2.5 below that ae(0)rp is (exactly) countably productive. 36 w. w. comfort and a. w. hager 2. characterizing the spaces in ae(0)rp our principal results about the class ae(0)rp are given in theorems 2.1 and 2.2 and its corollaries. theorem 2.1. if t ∈ ae(0) and each point of t is a gδ-point, then t ∈ ae(0)rp. proof. given f ∈ c(f, t ) with f closed in x ∈ 0 and t ∈ ae(0), we must find f ∈ c(x, t ) such that f ⊆ f and f[x] = f[f ]. since x ∈ 0 there is κ ≥ ω such that x ⊆h 2 κ, and since 2κ ∈ 0 and t ∈ ae(0) there is f∗ ∈ c(2κ, t ) such that f ⊆ f∗. then, since 2 is a separable space and points of t are gδ-points, the function f ∗ factors through a countable subproduct of 2κ in the sense that there exist countable c ⊆ κ and g ∈ c(2c, t ) such that f∗ = g ◦ πc (with πc the usual projection πc : 2 κ ։ 2c). (the theorem just used, due to a. gleason, is stated and proved in detail by isbell [12](p. 132).) since 2c is compact metrizable, its continuous image g[2c] is compact metrizable ([4](3.1.28)). then since f[f ] is closed in the separable, zero-dimensional, metrizable space g[2c ], there is a (continuous) retraction r : g[2c] ։ f[f ] (see [4](6.2.b) for a proof of this assertion, credited by engelking to sierpiński [15]). then f := r ◦ f∗|x = r ◦ g ◦ πc|x is as required. in detail: (1) f∗ is defined on 2κ, so f is well-defined on x; (2) x ∈ x ⇒ f(x) = (r ◦ g ◦ πc )(x) ∈ r[g[2 c ]] ∈ f[x]; and (3) x ∈ f ⇒ f(x) ∈ f[f ], so f(x) = (r ◦ g ◦ πc )(x) = r(f(x)) = f(x). � theorem 2.2. for each space t , these conditions are equivalent. (a) t ∈ ae(0)rp; (b) each compact subspace of t is dyadic; (c) each compact subspace of t is metrizable. proof. (a) ⇒ (b). if compact s ⊆ t , then s ∈ ae(0)rp ⊆ ae(0) by theorems 1.2 and 1.5, so (b) holds by lemma 1.4. (b) ⇒ (c). suppose that some compact s ⊆ t is nonmetrizable, so that w(s) = κ > ℵ0. then, since s is dyadic, some point of s has local weight (character) κ (by a theorem of esenin-vol′pin [5], cited in [4](3.12.12(e))). then s contains a copy of the one-point compactification of the discrete space of cardinality κ (by a theorem of engelking [3], cited in [4](3.12.12(i))). then s contains the (compact, non-c.c.c.) space αd. since αd is not dyadic (by lemma 1.4(b)), the assumption w(s) > ℵ0 is false so s is metrizable. (c) ⇒ (a). according to theorem 1.2((a) ⇒ (b)), it suffices to show for each compact s ⊆ t that s ∈ ae(0)rp. given such s, from (c) we have s ⊆h [0, 1] ω with [0, 1]ω ∈ ae(0) by (1.1) (since surely [0, 1] ∈ ae(0)), so s ⊆ [0, 1]ω ∈ ae(0)rp by theorem 2.1. then s ∈ ae(0)rp, as required. � it is immediate from theorem 2.2 that a compact space is closed-hereditarily dyadic if and only if it is metrizable. that is a result of efimov [2], reproved in [3](p. 300). corollary 2.3. every metrizable space is an ae(0)rp-space. range-preserving ae(0)-spaces 37 corollary 2.4. for each space t , these conditions are equivalent. (a) t ∈ ae(0)rp; (b) s ⊆ t ⇒ s ∈ ae(0); (c) s ⊆ t , s closed ⇒ s ∈ ae(0); and (d) s ⊆ t , s compact ⇒ s ∈ ae(0). proof. that (a) ⇒ (b) is clear, since ae(0)rp ⊆ ae(0) and the class ae(0)rp is hereditary. that (b) ⇒ (c) and (c) ⇒ (d) are obvious. if (d) holds then by lemma 1.4(a) every compact s ⊆ t is dyadic and theorem 2.2((b) ⇒ (a)) gives (a). � corollary 2.5. let {ti : i ∈ i} be a set of nonempty spaces and set t := πi∈i ti. then t ∈ ae(0) rp if and only if (i) each ti ∈ ae(0) rp, and (ii) |{i ∈ i : |ti| > 1}| ≤ ℵ0. proof. “only if”. each ti ⊆h t , so theorem 1.2((a) ⇒ (b)) shows (i). if (ii) fails then 2ℵ1 ⊆h t , and then from 2 ℵ1 /∈ ae(0)rp (theorem 1.5(d)) would follow the contradiction t /∈ ae(0)rp (from theorem 1.2). “if”. we assume without loss of generality that |i| ≤ ℵ0. by theorem 2.2, it suffices to show that each compact s ⊆ t is metrizable. given such s we have for each i ∈ i that the (compact) space πi[s] is metrizable, so πi∈i πi[s] (and hence its subspace s) is metrizable. � we continue with additional corollaries of the foregoing theorems. in corollary 2.7 we note that a number of familiar spaces are in the class ae(0)rp, and in corollary 2.8 we show that spaces which are “locally in ae(0)rp” are in fact in ae(0)rp. (that result is in parallel with the theorem from [14] that “locally dugundji” implies dugundji; the converse to that result is given by hoffmann [11].) we first remind the reader of the relevant definitions. definition 2.6. let t = (t, t ) be a space. (a) a network in t is a family n of subsets of t such that if x ∈ u ∈ t then there is n ∈ n such that x ∈ n ⊆ u; (b) t is a σ-space if it has a σ-discrete network; (c) t is a p -space if every gδ-subset of t is open. we refer the reader to [7], especially (§4), for a useful introduction to σspaces. it is noted there, for example, that every moore space (in particular, every metrizable space and every countable space), is a σ-space; further, every (countably) compact subspace of a σ-space is metrizable ([7](p. 447)). every compact subspace of a p -space, being finite ([6](4k)), is metrizable. using those facts, or otherwise, we have the following corollary to theorem 2.2((c) ⇒ (a)). corollary 2.7. every σ-space, and every p-space, and every countable space, is in the class ae(0)rp. 38 w. w. comfort and a. w. hager (this shows that the converse to theorem 2.1 fails: in a p -space, each gδ-point is isolated.) corollary 2.8. let t be a space. (a) if each x ∈ t has a neighborhood ux ∈ ae(0) rp, then t ∈ ae(0)rp; and (b) if t is the topological sum (the “disjoint union”) of spaces in ae(0)rp, then t ∈ ae(0)rp. proof. it suffices to prove (a), since (b) is then immediate. by theorem 2.1, it suffices to show that every compact s ⊆ t is metrizable. let {ux : x ∈ t } be a cover of t as indicated (with each ux ∈ ae(0) rp), and for x ∈ t choose open vx such that x ∈ vx ⊆ vx ⊆ ux. there is finite f ⊆ s such that s ⊆ ⋃ x∈f vx ⊆ ⋃ x∈f vx ⊆ ⋃ x∈f ux and hence s = ⋃ x∈f (s∩vx). each space s∩vx is compact (being closed in s) and is in ae(0)rp (being a subset of ux ∈ ae(0) rp). so by theorem 2.2, each space s ∩ vx is metrizable. thus s, the union of finitely many of its closed, metrizable subspaces, is itself metrizable ([4](4.19)). � 3. an application to lattice-ordered groups we consider the category w∗ of archimedean lattice-ordered groups g with distinguished strong order unit eg (that means: for each g ∈ g there is n ∈ n such that |g| ≤ neg), together with groupand lattice-homomorphisms which preserve unit. the notation g ≤ h indicates that g ∈ w∗ is a subobject of h ∈ w∗. the yosida representation theorem, as exposed in [9], tells us that each g ∈ w∗ has an essentially unique representation g ≃ ĝ ≤ c(y g, r) with y g compact (hausdorff) and with ĝ separating points of y g; and, for each φ : g → h ∈ w∗ there corresponds a unique continuous τ : y h ։ y g such that φ̂(g) = ĝ ◦τ for each g ∈ g, and with τ an injection (hence an embedding) if φ is a surjection. we identity each g ∈ w∗ with its ĝ. thus, a surjection φ : g ։ h becomes the restriction to y h of the functions in g. now let e ≤ r (that is, e is a subgroup of r, and 1 ∈ e), and set ce := {c(x, e) : x is compact} ⊆ w ∗. theorem 3.1. ce is closed under surjections in w ∗. proof. [we sketch.] first consider the case e = r. then y c(x, r) = x, and each surjection φ : c(x, r) ։ h is induced by the restriction g → g|y h to the subspace y h ⊆ x. each f ∈ c(y h, r) has an extension g ∈ c(x, r) (tietze-urysohn), so f = g|y h and h = c(y h, r). now if e 6= r then e is zero-dimensional, and y := y c(x, e) is the zerodimensional reflection of x: y ∈ 0. so for a surjection φ : c(x, e) ։ h the “dual” topological inclusion y h ⊆ y lives in 0. then, each f ∈ c(y h, e) has an extension g ∈ c(y, e), because e ∈ ae(0)rp (e.g., by theorem 2.2), so f = g|y h and again h = c(y h, e), as required. � range-preserving ae(0)-spaces 39 we note that when e 6= r in the preceding theorem, either e is cyclic (and thus discrete) or e is dense in r. in the former case, an extension g of f ∈ c(y h, e) is easily manufactured, using the fact that |f[y h]| < ω, by extending the resulting finite clopen partition of y h to one of y (much as in the proof of theorem 1.1). in the (proof of the) dense case, however, the relation e ∈ ae(0)rp is crucial; the proof of that appears to require much of the argumentation we have given above in theorem 2.2. more issues of the sort addressed in this section are considered in the work [9]. references [1] a. b laszczyk, compactness, in: encyclopedia of general topology (k. hart, j. nagata, and j. vaughan, eds.), pp. 169–173. elsevier, amsterdam, 2004. [2] b. efimov, dyadic bicompacta, soviet math. doklady 4 (1963), 496–500, russian original in: doklady akad. nauk sssr 149 (1963), 1011-1014. [3] r. engelking, cartesian products and dyadic spaces, fund. math. 57 (1965), 287–304. [4] ryszard engelking, general topology, heldermann verlag, berlin, 1989. [5] a. s. esenin-vol′pin, on the relation between the local and integral weight in dyadic bicompacta, doklady akad. nauk sssr n.s. 68 (1949), 441–444. [in russian.] [6] l. gillman and m.r jerison, rings of continuous functions, d. van nostrand co., new york, 1960. [7] g. gruenhage, generalized metric spaces, in: handbook of set-theoretic topology (kenneth kunen and jerry e. vaughan, eds.), pp. 423–501. north-holland, amsterdam, 1984. [8] a. w. hager and l. c. robertson, representing and ringifying a riesz space, in: symposia mathematica vol. xxi, indam, rome, 1975, pp. 411–431. academic press, london, 1977. [9] a. w. hager, j. martinez and c. monaco, some basics of the category of archimedean ℓ-groups with strong unit. manuscript in preparation. [10] r. haydon, on a problem of pe lczyński: milutin spaces, dugunjdi spaces, and ae(0 − dim), studia math. 52 (1974), 23–31. [11] b. hoffmann, a surjective characterization of dugundji spaces, proc. amer. math. soc. 76 (1979), 151–156. [12] j. r. isbell, uniform spaces, math. surveys #12, american mathematical society, providence, rhode island, 1964. [13] s. koppelberg, projective boolen algebras, in: handbook of boolean algebras (j. donald monk and robert bennett, eds.), chapter 20. north-holland publ. co., amsterdam, 1989. [14] a. pe lczyński, linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, dissertationes math. 58 (1968), 92 pages. rozprawy mat. polish scientific publishers, warszawa, 1958. [15] w. sierpiński, sur les projections des ensembles complémentaire aux ensembles (a), fund. math. 11 (1928), 117–122. (received november 2011 – accepted november 2012) 40 w. w. comfort and a. w. hager w. w. comfort (wcomfort@wesleyan.edu) department of mathematics and computer science, wesleyan university, middletown, ct 06459, usa a. w. hager (ahager@wesleyan.edu) department of mathematics and computer science, wesleyan university, middletown, ct 06459, usa range-preserving ae(0)-spaces. by w. w. comfort and a. w. hager @ appl. gen. topol. 22, no. 2 (2021), 295-302doi:10.4995/agt.2021.13696 © agt, upv, 2021 intrinsic characterizations of c-realcompact spaces sudip kumar acharyya, rakesh bharati and a. deb ray department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata 700019, india (sdpacharyya@gmail.com, bharti.rakesh292@gmail.com, debrayatasi@gmail.com) communicated by o. valero abstract c-realcompact spaces are introduced by karamzadeh and keshtkar in quaest. math. 41, no. 8 (2018), 1135–1167. we offer a characterization of these spaces x via c-stable family of closed sets in x by showing that x is c-realcompact if and only if each c-stable family of closed sets in x with finite intersection property has nonempty intersection. this last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. we show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. an allied class of spaces viz cp-compact spaces akin to that of c-realcompact spaces are introduced. the paper ends after examining how far a known class of c-realcompact spaces could be realized as cp-compact for appropriately chosen ideal p of closed sets in x. 2010 msc: 54c40. keywords: c-realcompact spaces; banaschewski compactification; c-stable family of closed sets; ideals of closed sets; initially θ-compact spaces. 1. introduction in what follows x stands for a completely regular hausdorff topological space. as usual c(x) and c∗(x) denote respectively the ring of all real valued continuous functions on x and that of all bounded real valued continuous functions received 16 may 2020 – accepted 10 may 2021 http://dx.doi.org/10.4995/agt.2021.13696 s. k. acharyya, r. bharati and a. deb ray on x. suppose cc(x) is the subring of c(x) containing those functions f for which f(x) is a countable set and c∗c (x) = cc(x) ∩ c ∗(x). formal investigations of these two rings vis-a-vis the topological structure of x are being carried on only in the recent times. it turns out that there is an interplay between the topological structure of x and the ring and lattice structure of cc(x) and c ∗ c (x), which incidentally sheds much light on the topology of x. the articles [3], [4], [7], [8], [11] may be referred in this context. the notion of c-realcompact spaces is the fruit of one such endeavours in the study of x versus cc(x) or c ∗ c (x). a space x is declared c-realcompact in [8] if each real maximal ideal m in cc(x) is fixed in the sense that there exits a point x ∈ x such that for each f ∈ m, f(x) = 0. m is called real when the residue class field cc(x)/m is isomorphic to the field r. a number of interesting facts concerning these spaces is discovered in [8]. these may be called countable analogues of the corresponding properties of real compact spaces as developed in [6], chapter 8. in the present article we offer a new characterization of crealcompact spaces on using the notion, c-stable family of closed sets in x. a family f of subsets of x is called c-stable if given f ∈ c(x,z), there exists f ∈ f such that f is bounded on f . we define a topological space x (not necessarily completely regular ) to be cc-realcompact if each c-stable family of closed sets in x with finite intersection property has nonempty intersection. we check that this new notion of ccrealcompactness agrees with the already introduced notion of c-realcompactness in [8], within the class of zero-dimensional hausdorff spaces (theorem 2.3). we re-establish a modified version of a few known properties of c-realcompact spaces using our new definition of cc-realcompactness (theorem 2.4). furthermore we realize that any topological space x can be extended as a dense subspace to a cc-realcompact space υ0x enjoying some desired extension properties (theorem 2.5). while constructing this extension of x, we follow closely the technique adopted in [9]. the results mentioned above constitute the first technical section viz §2 of this article. a family p of closed sets in x is called an ideal of closed sets if a ∈ p, b ∈ p and c is a closed subset of a imply that a ∪ b ∈ p and c ∈ p. let ω(x) stand for the aggregate of all ideals of closed sets in x. for any p ∈ ω(x) let cp(x) = {f ∈ c(x) : clx(x \ z(f)) ∈ p}, here z(f) = {x ∈ x : f(x) = 0} is the zero set of f in x. it is well known that cp(x) is an ideal in the ring c(x), see [1] and [2] for more information on these ideals. with reference to any such p ∈ ω(x), we call a family f of subsets of x cp-stable if given f ∈ c(x,z) ∩ cp(x) there exists f ∈ f such that f is bounded on f . we define a space x to be cp-compact if any cp-stable family of closed sets in x with finite intersection property has non-empty intersection. it is clear that a zero-dimensional space x is cc-realcompact if it is already cp-compact. we have shown that if x is a noncompact zero-dimensional space and p ∈ ω(x) such that x is cp-compact, then there exists an r ∈ ω(x) such that r & p and x is cr-compact. thus within the class of zero-dimensional noncompact spaces x, there is no minimal member p ∈ ω(x) in the set inclusion sense of © agt, upv, 2021 appl. gen. topol. 22, no. 2 296 intrinsic characterizations of c-realcompact spaces the term for which x becomes cp-compact (theorem 3.2). in the concluding portion of §3 of this article we have examined, how far the known classes of c-realcompact spaces could be achieved as cp-compact spaces for appropriately chosen p ∈ ω(x). for any infinite cardinal number θ, x is called finally θ-compact if each open cover of x has a subcover with cardinality < θ (see [10]). in this terminology finally ω1-compact spaces are lindelöf and finally ω0-compact spaces are compact. it is realized that a c-realcompact space x is finally θ-compact if and only if it is cq-compact, where q is the ideal of all closed finally θ-compact subsets of x (theorem 3.4). a special case of this result reads: x is lindelöf when and only when x is cα-compact where α is the ideal of all closed lindelöf subsets of x. 2. properties of cc-realcompact spaces and cc-realcompactifications before stating the first technical result of this section, we need to recall a few terminologies and results from [4] and [8]. our intention is to make the present article self contained as far as possible. an element α on a totally ordered field f is called infinitely large if α > n for each n ∈ n. it is clear that f is archimedean if and only if it does not contain any infinitely large element. if m is maximal ideal in cc(x) then the residue class field cc(x)/m is totally ordered according to the following definition: for f ∈ c(x), m(f) ≧ 0 if and only if there exists g ∈ m such that f ≧ 0 on z(g). here m(f) stands for the residue class in c(x)/m, which contains the function f. theorem 2.1 (proposition 2.3 in [8]). for a maximal ideal m in cc(x) and for f ∈ cc(x), |m(f)| is infinitely large in cc(x)/m if and only if f is unbounded on every zero set of zc(m) = {z(g) : g ∈ m}. it is proved in [4], remark 3.6 that if x is a zero-dimensional space, then the set of all maximal ideals of cc(x) equipped with hull-kernel topology, also called the structure space of cc(x) is homeomorphic to the banaschewski compactification β0x of x. thus the maximal ideals of cc(x) can be indexed by virtue of the points of β0x. indeed a complete description of all these maximal ideals is given by the list {mpc : p ∈ β0x}, where m p c = {f ∈ cc(x) : p ∈ clβ0xz(f)} with m p c is a fixed maximal ideal if and only if p ∈ x (see theorem 4.2 in [4]). it is well known that any continuous map f : x → y , where x and y are both zero-dimensional spaces with y compact also, has an extension to a continuous map f̄ : β0x → y (we call this property, the c-extension property of β0x) (see remark 3.6 in [4]). it follows that for a zero-dimensional space x, any continuous map f : x → z (also written as f ∈ c(x,z)), has an extension to a continuous map f∗ : β0x → z ∗ = z∪{ω}, the one point compactification of z. we also write f∗ ∈ c(β0x,z ∗). a slightly variant form of the next result is proved in [8], theorem 2.17 and theorem 2.18. theorem 2.2. let x be zero-dimensional and p ∈ β0x, then the maximal ideal mpc in cc(x) is real if and only if for each f ∈ c(x,z), f ∗(p) 6= ω if and only if |mpc (f)| is not infinitely large in cc(x)/m p c . © agt, upv, 2021 appl. gen. topol. 22, no. 2 297 s. k. acharyya, r. bharati and a. deb ray theorem 2.3. a zero-dimensional space x is cc-realcompact if and only if it is c-realcompact. proof. let x be a c-realcompact space and f be a family of closed subsets of x with finite intersection property but with ⋂ f=∅. to show that x is cc-realcompact we shall prove that f is not a c-stable family. indeed {clβ0xf : f ∈ f} is a family of closed subsets of β0x with finite intersection property. since β0x is compact, there exists a point p ∈ ⋂ f ∈f clβ0xf and of course p ∈ β0x \ x. here m p c is a free maximal ideal in cc(x). since x is crealcompact this implies that mpc is a hyperreal maximal ideal (meaning that it is not a real maximal ideal of cc(x)). it follows from theorem 2.2 that there exists f ∈ c(x,z) with f∗(p) = ω. since p ∈ clβ0xf for each f ∈ f, it is therefore clear that ‘f’ is unbounded on each set in the family f. therefore f is not a c-stable family. conversely let x be not c-realcompact. then there exists a real maximal ideal m in cc(x), which is not fixed. this means that there is a point p ∈ β0x \ x for which m = m p c . since p ∈ clβ0xz(f) for each f ∈ m p c , it follows that {z(f) : f ∈ mpc } is a family of closed sets in x with finite intersection property but with empty intersection. to show that x is not cc-realcompact, it suffices to show that {z(f) : f ∈ mpc } is a c-stable family. so let g ∈ c(x,z). since mpc is real, this implies in view of theorem 2.2 that g ∗(p) 6= ω and hence |mpc (g)| is not infinitely large. it follows therefore from theorem 2.1 that g is bounded on some z(f) for an f ∈ mpc . this settles that {z(f) : f ∈ m p c } is a c-stable family. � by adapting the arguments of theorem 5.2, theorem 5.3 and theorem 5.4 in [9] appropriately, we can establish the following facts about cc-realcompact spaces without difficulty: theorem 2.4. (1) a compact space is cc-realcompact. (2) a pseudocompact cc-realcompact space is compact. (3) a closed subspace of a cc-realcompact space is cc-realcompact. (4) the product of any set of cc-realcompact spaces is cc-realcompact. (5) if a topological space x = e ∪ f where e is a compact subset of x and f is a z-embedded cc-realcompact subset of x, meaning that each function in c(f,z) can be extended to a function in c(x,z), then x is cc-realcompact. (6) a z-embedded cc-realcompact subset of a hausdorff space x is a closed subset of x. we now show that any topological space x can be extended to a cc-realcompact space containing the original space x as a c-embedded dense subspace and enjoying a desirable extension property. the proof can be accomplished by closely following the arguments adopted to prove theorem 6.1 in [9]. nevertheless we give a brief outline of the main points of proof in our theorem. © agt, upv, 2021 appl. gen. topol. 22, no. 2 298 intrinsic characterizations of c-realcompact spaces theorem 2.5. every topological space x can be extended to a cc-realcompact space υcx as a dense subspace with the following extension property: each continuous map from x into a regular cc-realcompact space y can be extended to a continuous map from υcx into y . x is cc-realcompact if and only if x = υcx. proof. for each x ∈ x let gx be the aggregate of all closed sets in x which contain the point x. then gx is a c-stable family of closed sets in x with finite intersection property and with the prime condition: a∪b ∈ gx =⇒ a ∈ gx or b ∈ gx, a,b ⊆ x. we extend the set x to a bigger set υcx, so that υcx \ x becomes an index set for the collection of all maximal c-stable families of closed subsets of x with finite intersection property but with empty intersection. for each p ∈ υcx\x, let gp designate the corresponding maximal c-stable family of closed sets in x with finite intersection property and with empty intersection. for each closed set f in x, we write f̄ = {p ∈ υcx : f ∈ gp}. then {f̄: f is closed in x} forms a base for closed sets of some topology on υcx and in this topology for any closed set f in x f̄ = clυcxf . since x belongs to each g p, it is clear that x is dense in υcx. let t : x → y be a continuous map with y , a regular cc-realcompact space. choose p ∈ υ cx. let hp = {g ⊆ y : g is closed in y and t−1(g) ∈ gp}. then hp is a c-stable family of closed sets in y with finite intersection property. we select a point y ∈ ⋂ hp and we set t0(p) = y with the aggrement that t0(p) = t(p) in case p ∈ x. thus t0 : υcx → y is a well defined map which is further continuous. the remaining parts of the theorem can be proved by making arguments closely as in the proof of theorem 6.1 of [9]. � 3. cp-compact spaces in this section all the topological spaces x that will appear will be assumed to be zero-dimensional. we define for any p ∈ ω(x), υp0 (x) = {p ∈ β0x : f∗(p) 6= ω for each f ∈ cp(x) ∩ c(x,z)}. it is clear that if p=e≡ the ideal of all closed sets in x then υe0 (x) = υ0x ≡ {p ∈ β0x : f ∗(p) 6= ω for each f ∈ c(x,z)} the set defined in the begining of the proof of theorem 3.8 in [8]. the next theorem puts theorem 2.3 in a more general setting. theorem 3.1. for a p ∈ ω, x is cp-compact if and only if x = υ p 0 (x). we omit the proof of this theorem because it can be done by making some appropriate modification in the arguments adopted in the proof of theorem 2.3. it is clear that if p, q ∈ ω(x) with p ⊂ q, then any cq-stable family of closed sets in x is also cp-stable, consequently if x is cp-compact then x is cq-compact also. in particular every cp-compact space is cc-realcompact and hence c-realcompact in view of theorem 2.3. the following question therefore seems to be natural. if x is a zero-dimensional non-compat c-realcompact space, then does there exist a minimal ideal p of closed sets in x (minimal in some sense of the term) for which x becomes cp-compact? © agt, upv, 2021 appl. gen. topol. 22, no. 2 299 s. k. acharyya, r. bharati and a. deb ray no possible answer to this question is known to us, however the following proposition shows that the answer to this question is in the negative if the phrase ‘minimal’ is interpreted in the set inclusion sense of the term. theorem 3.2. let x be a non compact zero-dimensional space. suppose p ∈ ω(x) is such that x is cp-compact. then there exists r ∈ ω(x) such that r & p and x is cr-compact. proof. we get from theorem 3.1 that x = υp0 x. as x is non compact we can choose a point p ∈ β0x \ x. then p /∈ υ p 0 x. accordingly there exists f ∈ cp(x) ∩ c(x,z) such that f ∗(p) = ω. we select a point x ∈ x such that f(x) 6= 0. set r = {d ∈ p : x /∈ d}. it is easy to check that r is an ideal of closed sets in x, i.e., r ∈ ω(x). furthermore, clx(x − z(f)) is a member of p containing the point x. this implies that clx(x − z(f)) /∈ r. thus r p. to show that x is cr-compact. we shall show that x = υ r 0 x (see theorem 3.1). so choose a point q ∈ β0x \x then q /∈ υ p 0 x, consequently there exists g ∈ cp(x) ∩ c(x,z) such that g ∗(q) = ω. for the distinct points q,x in β0x there exist disjoint open sets u, v in this space such that x ∈ u, q ∈ v . since β0x is zero-dimensional there exists therefore a clopen set w in β0x such that q ∈ w ⊂ v . the map h : β0x → {0,1} given by h(w) = {1} and h(β0x \ w) = {0} is continuous. we note that h(u) = {0} and h(q) = 1. let ψ = h|x. then ψ ∈ c(x,z). take l = g.ψ. since g ∈ cp(x) and cp(x) is an ideal of c(x), it follows that l ∈ cp(x). furthermore the fact that g and ψ are both functions in c(x,z) implies that l ∈ c(x,z). also the function h ∈ c(β0x,z) is the unique continuous extension of ψ ∈ c(x,z), hence we can write h = ψ∗. this implies that l∗(q) = g∗(q)ψ∗(q) = g∗(q)h(q) = ω, because g∗(q) = ω and h(q) 6= 0. on the other hand if y ∈ u∩x then h(y) = 0 and hence l(y) = 0. since u∩x is an open neighbourhood of x in the space x, this implies that x /∈ clx(x\z(l)). since clx(x \ z(l)) ∈ p, already verified, it follows that clx(x \ z(l)) ∈ r. thus l ∈ cr(x) ∩ c(x,z). since l ∗(q) = ω, this further implies that q /∈ υr0 (x). � it is trivial that a (zero-dimensional) compact space is c-realcompact. it is also observed that a lindelöf space is c-realcompact (corollary 3.6, [8]). but for an infinite cardinal number θ, a finally θ-compact space may not be crealcompact. indeed the space [0,ω1) of all countable ordinals is a celebrated example of a zero-dimensional space which is not realcompact (see 8.1, [6]). since a zero-dimensional c-realcompact space is necessarily realcompact (vide proposition 5.8, [8]) it follows therefore that [0,ω1) is not a c-realcompact space. but it is easy to show that [0,ω1) is finally ω2-compact. for the same reason, the tychonoff plank t ≡[0,ω1)×[0,ω0)-{(ω1,ω0)} of 8.20 in [6], is finally ω2compact without being c-realcompact. it can be easily shown that a closed subset of a finally θ-compact space is finally θ-compact. furthermore, the following characterization of finally θ-compactness of a topological space can be established by routine arguments. © agt, upv, 2021 appl. gen. topol. 22, no. 2 300 intrinsic characterizations of c-realcompact spaces theorem 3.3. the following two statements are equivalent for an infinite cardinal number θ. (1) x is finally θ-compact. (2) if b is a family of closed sets in x, such that for any subfamily b0 of b with |b0| < θ, ⋂ b0 6= ∅, then ⋂ b 6= ∅. theorem 3.4. let x be c-realcompact and pθ the ideal of all closed finally θ-compact subsets of x. then x is finally θ-compact if and only if it is cpθ compact. proof. let x be finally θ-compact and p ∈ β0x \ x. to show that x is cpθ compact, it suffices to show in view of theorem 3.1 that p /∈ υpθ0 (x). indeed x is c-realcompact implies that the maximal ideal mpc of cc(x) is not real. consequently by theorem 2.2, there exists f ∈ c(x,z) such that f∗(p) = ω. now clx(x \ z(f)), like any closed subsets of x is finally θ-compact. thus f ∈ cpθ (x) ∩ c(x,z), hence p /∈ υ pθ 0 (x). to prove the converse, let x be not finally θ-compact. it follows from theorem 3.3 that there exists a family b = {bα : α ∈ λ} of closed sets in x with the following properties: for any subfamily b1 of b with |b1| < θ, ⋂ b1 6= ∅ but⋂ b = ∅. let d = {dα : α ∈ λ ∗} be the aggregate of all sets d′αs, which are intersections of < θ many sets in the family b. then b ⊆ d and hence ⋂ d = ∅. also d has finite intersection property. we shall show that d is a cpθ-stable family and hence x is not pθ-compact. towards such a proof choose f ∈ cpθ (x) ∩ c(x,z), then clx(x \ z(f) is a finally θ-compact subset of x. since {x \ bα : α ∈ λ} is an open cover of x, there exists a subset λ0 of λ with |λ0| < θ such that clx(x \ z(f)) ⊆ ⋃ α∈λ0 (x \ bα). this implies that ⋂ α∈λ0 bα ⊆ z(f) and we note that ⋂ α∈λ0 bα ∈ d. thus f becomes bounded on a set lying in the family d. hence d becomes a cpθ-stable family. � acknowledgements. the second author acknowledges financial support from university grand commission, new delhi, for the award of research fellowship (f. no. 16-9(june 2018)/2019 (net/csir)). references [1] s. k. acharyya and s. k. ghosh, a note on functions in c(x) with support lying on an ideal of closed subsets of x, topology proc. 40 (2012), 297–301. [2] s. k. acharyya and s. k. ghosh, functions in c(x) with support lying on a class of subsets of x, topology proc. 35 (2010), 127–148. [3] s. k. acharyya, r. bharati and a. deb ray, rings and subrings of continuous functions with countable range, queast. math., to appear. [4] f. azarpanah, o. a. s. karamzadeh, z. keshtkar and a. r. olfati, on maximal ideals of cc(x) and the uniformity of its localizations, rocky mountain j. math. 48, no. 2 (2018), 345–384. © agt, upv, 2021 appl. gen. topol. 22, no. 2 301 s. k. acharyya, r. bharati and a. deb ray [5] p. bhattacherjee, m. l. knox and w. w. mcgovern, the classical ring of quotients of cc(x), appl. gen. topol. 15, no. 2 (2014), 147–154. [6] l. gillman and m. jerison, rings of continuous functions, van nostrand reinhold co., new york, 1960. [7] m. ghadermazi, o. a. s. karamzadeh and m. namdari, on the functionally countable subalgebras of c(x), rend. sem. mat. univ. padova. 129 (2013), 47–69. [8] o. a. s. karamzadeh and z. keshtkar, on c-realcompact spaces, queast. math. 41, no. 8 (2018), 1135–1167. [9] m. mandelkar, supports of continuous functions, trans. amer. math. soc. 156 (1971), 73–83. [10] r. m. stephenson jr, initially k-compact and related spaces, in: handbook of settheoretic topology, ed. kenneth kunen and jerry e. vaughan. amsterdam, northholland, (1984) 603–632. [11] a. veisi, ec-filters and ec-ideals in the functionally countable subalgebra of c ∗(x), appl. gen. topol. 20, no. 2 (2019), 395–405. © agt, upv, 2021 appl. gen. topol. 22, no. 2 302 @ appl. gen. topol. 21, no. 2 (2020), 295-304 doi:10.4995/agt.2020.13156 c© agt, upv, 2020 rough action on topological rough groups alaa altassan a, nof alharbi a, hassen aydi b,c, cenap özel a a department of mathematics, king abdulaziz university, jeddah, kingdom of saudi arabia (aaltassan@kau.edu.sa,nof20081900@hotmail.com,cozel@kau.edu.sa) b université de sousse, institut supérieur d’informatique et des techniques de communication, h. sousse 4000, tunisia. ( hassen.aydi@isima.rnu.tn) c china medical university hospital, china medical university, taichung 40402, taiwan. ( hassen.aydi@isima.rnu.tn) communicated by j. galindo abstract in this paper we explore the interrelations between rough set theory and group theory. to this end, we first define a topological rough group homomorphism and its kernel. moreover, we introduce rough action and topological rough group homeomorphisms, providing several examples. next, we combine these two notions in order to define topological rough homogeneous spaces, discussing results concerning open subsets in topological rough groups. 2010 msc: 22a05; 54a05; 03e25. keywords: rough groups; topological rough groups; topological rough subgroups; product of topological rough groups; topological rough group homomorphisms; topological rough group homeomorphisms; topologically rough homogeneous spaces; rough kernel. 1. introduction rough set theory has many applications in economic, medicine and engeneering [13, 14, 16, 17]. such a theory was introduced by pawlak [21] and deals with uncertainty, impression and vagueness in information systems.the starting point of his analysis is the notion of approximation space, namely a pair(u, r), where u is any arbitrary non-empty set, called universe, and r is received 18 february 2020 – accepted 28 august 2020 http://dx.doi.org/10.4995/agt.2020.13156 a. altassan, n. alharbi, h. aydi and c. özel an equivalence relaon u. the set u/r of all equivalence classes [x]r forms a partition of u . moreover, for any x ⊂ u, he introduced the notions of lower and upper approximations of x as follows: x = {[x]r : [x]r ∩ x ∕= ∅} and x = {[x]r : [x]r ⊂ x}. next, he defined the rough set to be the orderd pair x = (x, x). recently, the interrelations between rough set theory and various branches of mathematics, such as combinatorics [12], monoids [10], matroids [15, 23, 24, 25], groups [7, 18], integral domains [11] and modules [9] has been deeply studied and constitute a research field which is developing rapidly. in our perspective, we are interested in the interrelations between rough set theory and groups. to this regard, let us first recall that in [7] and [18] the notions of rough groups, rough subgroups, rough homomorphisms and rough antihomomorphisms have been analyzed in detail. moreover, the notion of topological rough groups was introduced by bagirmaz et al in [6]. in this paper, we present rough actions and rough homogenous spaces, and discuss some of their properties. we also define a rough kernel. we organise the paper as follows. in section 2, we collect the needed material about rough groups and rough homomorphisms. then the definition of topological rough groups and important properties have been recalled in section 3. section 4 presents our main results where we introduce rough action and homogenous spaces. this paper is produced from the phd thesis of ms. nof alharbi registered in king abdulaziz university. 2. rough groups and rough homomorphisms in this section, we recall rough groups, rough homomorphisms and some of their properties. let (u, r) be an approximation space, where u is a non-empty set and r is an equivalence relation on u. let (∗) be a binary operation defined on u. for all x, y ∈ u, we write xy instead of x∗y. in 1994, biswas and nanda introduced the definition of rough groups which is given in the following definition. definition 2.1 ([6]). let (u, r) be an approximation space. suppose that g is a subset of u and g and g are respectively its upper and lower approximations. then the rough set g = (g, g) is called a rough group if the following conditions are satisfied: (1) ∀x, y ∈ g, xy ∈ g. (2) (xy)z = x(yz), ∀x, y, z ∈ g. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 296 rough action on topological rough groups (3) ∀x ∈ g, ∃e ∈ g such that xe = ex = x. (4) ∀x ∈ g, ∃y ∈ g such that xy = yx = e. definition 2.2 ([6]). a non-empty rough subset h = (h, h) of a rough group g = (g, g) is called a rough subgroup if it is a rough group itself. note that g = (g, g) is a trivial rough subgroup of itself. moreover, if e ∈ g, then e = (e, e) is a trivial rough subgroup of the rough group g. theorem 2.3 ([6]). suppose that g is a subset of u and g and g are respectively its upper and lower approximations. then a rough subset h is a rough subgroup of a rough group g if (1) ∀x, y ∈ h, xy ∈ h; (2) ∀y ∈ h, y−1 ∈ h. let h be a rough subgroup of a rough group g. then h is said to be a rough normal subgroup of g if xh = hx, ∀x ∈ g definition 2.4 ([18]). let (u1, r1) and (u2, r2) be approximation spaces and ∗, ∗ ′ be binary operations on u1 and u2, respectively. let g1 ⊆ u1 and g2 ⊆ u2 be two rough groups. if the mapping ϕ : g1 → g2 satisfies that ϕ(x ∗ y) = ϕ(x) ∗ ′ ϕ(y), for all x, y ∈ g1, then ϕ is called a rough homomorphism. definition 2.5 ([18]). let g1 and g2 be two rough groups. a rough homomorphism ϕ : g1 → g2 is said to be : (1) a rough epimorphism (or surjective) if ϕ : g1 → g2 is onto. (2) a rough monomorphism if ϕ : g1 → g2 is one-to-one. (3) a rough isomorphism if ϕ : g1 → g2 is both onto and one-to-one. example 2.6. let (r, r) be an approximation space, where r is the set of real numbers under addition. consider the partition r/r = {q, qc}, where q is the set of rational numbers and qc is the set of irrational numbers. let g1 = q, and g2 = r∗ = r − 0. then g1 = q and g2 = r. it is clear that g1 and g2 are rough groups. define ϕ : q → r as follow: for every x ∈ q, ϕ(x) = x. it is not difficult to see that ϕ is a rough monomorphism. example 2.7. let u = z4 and consider the partition u/r = {{1, 2}, {0, 3}}. let g1 = {0, 2}, and g2 = {1, 2, 3}. then g1 = z4, and g2 = z4. it is clear that g1 and g2 are rough groups. define ϕ : g1 → g2 as follows: ϕ(x) = x, ∀x ∈ g1. it is not difficult to see that ϕ is a rough epimorphism and a rough monomorphism. thus ϕ is a rough isomorphism. 3. topological rough groups throughout this section, we recall the definition of topological rough groups and we give some examples. for more details and properties of these structures, we refer the reader to [6]. definition 3.1 ([6]). a rough group g with a topology τ on g is called a topological rough group if the following hold. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 297 a. altassan, n. alharbi, h. aydi and c. özel (1) f : g×g → g which defined by f(x, y) = xy is continuous with respect to a product topology on g × g and the topology τg on g induced by τ; (2) ι : g → g which defined by ι(x) = x−1 is continuous with respect to the topology τg on g induced by τ. now, we present three different examples of topological rough groups. example 3.2. let u = z4 be the group of integers modulo 4. let u/r = {{0, 2, 3}, {1}} be a classification of an equivalence relation and g = {1, 2, 3}. then g = {1} and g = {0, 1, 2, 3} = z4. given a topology τ = {∅, g, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} on g. then the relative topology on g is τg = {∅, g, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}. the two conditions in definition 3.1 are satisfied as follows: (1) the product mapping f : g × g → g = z4 is continuous with respect to product topology on g × g and the topology τg on g induced by the topology τ on g = z4. for instance, the open set {1, 2} in τg has inverse {{3} × {2} ∪ {3} × {3}} which is open in the product topology. (2) the inverse mapping i : g → g is continuous with respect to the topology τg on g induced by the topology τ. for instance the open set {1} has inverse {3} which is open in τg. hence g is a topological rough group. example 3.3. let u = r and u/r = {{x : x ≥ 0}, {x : x < 0}} be a partition of u. consider g = r∗ = r − 0. then g = {x : x < 0}, g = r. and g is a rough additive group. let d be the discrete topology on g = r, then (1) the product mapping f : r∗ × r∗ → r is continuous with respect to product topology on r∗ × r∗ and the topology dg on r∗ induced by the discrete topology d on r. (2) the inverse mapping i : r∗ → r∗ is continuous with respect to topology dg on r∗ induced by the discrete topology d. therefore the rough group g is a topological rough group with the discrete topology d on g = r. example 3.4 ([6]). let u = s4 be the set of all permutations of four objects and (∗) be the multiplication operation of permutations. consider u/r = {e1, e2, e3, e4}, to be a partition of u, where e1 = {1, (12), (13), (14), (23), (24), (34)} e2 = {(123), (132), (142), (124), (134), (143), (234), (243)} e3 = {(1234), (1243), (1342), (1324), (1423), (1432)} e4 = {(12)(34), (13)(24), (14)(23)}. for g = {(12), (123), (132)}, g = e1 ∪ e2. it is not difficult to see that g is a rough group. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 298 rough action on topological rough groups for a given topology τ = {∅, g, {(12)}, {1, (132), (123)}, {1, (12), (132), (123)}} on g, the relative topology on g is τg = {∅, g, {(12)}, {(132), (123)}}. moreover by examine the two conditions in definition 3.1, we can see that g is a topological rough group. definition 3.5 ([6]). let g be a topological rough group. for a fixed element a in g, we define the following: (1) a mapping la : g → g which is defined by la(x) = ax, is called a left transformation from g into g. (2) a mapping ra : g → g which is defined by ra(x) = xa, is called a right transformation from g into g. proposition 3.6 ([6]). let g be a topological rough group. then (1) the left transformation map la : g → g is continuous and one-to-one. (2) the right transformation map ra : g → g is continuous and one-toone. (3) the inverse mapping ι : g → g is a homeomorphism for all x ∈ g. 4. rough action and rough homogenous spaces in classical set topology in this section, we discuss our main results. we introduce rough action and rough homogenous spaces in classical set topology using rough groups. first, we recall cartesian product of topological rough groups. let (u, r1) and (v, r2) be approximation spaces with binary operations ∗1 and ∗2, respectively. for x, x ′ ∈ u and y, y ′ ∈ v, we have (x, y), (x ′ , y ′ ) ∈ u × v . define ∗ as (x, y) ∗ (x ′ , y ′ ) = (x ∗1 x ′ , y ∗2 y ′ ). then ∗ is a binary operation on u × v . in deed, that the product of equivalence relations r1 and r2 is also an equivalence relation on u × v (see [3] ). moreover, we have the following result. theorem 4.1 ([4]). let g1 ⊆ u and g2 ⊆ v be two rough groups. then the cartesian product g1 × g2 is also a rough group. now, let (u, r1) and (v, r2) be approximation spaces. let g1 ⊆ u and g2 ⊆ v be topological rough groups such that τ1 and τ2 are topologies on g1 and g2, respectively inducing τg1 and τg2 on g1 and g2, respectively. a mapping ϕ : g1 → g2 is called a topological rough group homomorphism, if ϕ is a rough homomorphism and continuous with respect to the topology τ2 on g2 inducing τg2 on g2 and the topology τ1 on g1 inducing τg1 on g1. a topological rough group homomorphism ϕ : g1 → g2 is called a topological rough group homeomorphism, if there exists a topological rough group homomorphism ϕ−1 : g2 → g1 such that ϕ−1 ◦ ϕ = 1g1. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 299 a. altassan, n. alharbi, h. aydi and c. özel the next definition is equivalent to the definition of rough kernel in rough groups that is given in [18]. definition 4.2. let g1 and g2 be topological rough groups, ϕ : g1 → g2 be a topological rough group homomorphism and let e2 be the rough identity element in g2. then ker(ϕ) = {g ∈ g1 : ϕ(g) = e2}. is called the rough kernel associated with the map ϕ. in the next theorem, we prove that, the kernel in definition 4.2 is a rough normal subgroup of g1. theorem 4.3. let ϕ be a topological rough group homomorphism from g1 to g2. then the rough kernel is a rough normal subgroup of g1. proof. for every x, y ∈ ker(ϕ), we have ϕ(x) = e2, and ϕ(y) = e2. (1) since ϕ(x ∗ y) = ϕ(x) ∗ ′ ϕ(y) = e2, we have x ∗ y ∈ ker(ϕ). (2) we have ϕ(x−1) = (ϕ(x))−1 = (e2) −1. hence ker(ϕ) is a rough subgroup of g1. (3) for every x ∈ g1 and r ∈ ker(ϕ), we have ϕ(x ∗ r ∗ x−1) = ϕ(x) ∗ ′ ϕ(r) ∗ ′ ϕ(x−1) = e2. therefore, x ∗ r ∗ x−1 ∈ ker(ϕ). thus ker(ϕ) is a rough normal subgroup of g1. □ note that, the rough kernel is always a subset of the upper approximation of g1. indeed, if g1 is a group then the kernel is a normal subgroup of g1. example 4.4. consider the map ϕ : z4 → r, where g = {1, 2, 3} and r∗ are the rough groups in example 3.2 and example 3.3, respectively. define ϕ as follows: ϕ(0) = 0, ϕ(1) = 0, ϕ(2) = 0, ϕ(3) = 0. clearly, ϕ is continuous and homomorphism. hence ϕ is a topological rough group homomorphism. from definition 4.2, it is easy to see that ker(ϕ) = {1, 2, 3} is a subset of g. moreover, ker(ϕ) is a rough normal subgroup of g. let (u, r) be an approximation space. assume moreover that g is a topological rough group and x is a subset of u. denote by x† the topological space inducing the topological space x; where x is a rough set with ordinary topology. now, we are ready to give the definition of the action of a rough group g on a rough space. definition 4.5. a continuous map ϕ : g×x† → x† (resp. ϕ : x† ×g → x†) is called a left (resp. right) rough action of g on x, if it satisfies the following conditions: (1) g(g ′ x) = (gg ′ )x (resp. ((xg)g ′ = x(gg ′ )), for all g, g ′ ∈ g and x ∈ x†. (2) ex = x (resp. xe = x), for every x ∈ x†, where e ∈ g is the rough identity. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 300 rough action on topological rough groups then the rough set x is called a rough g-space. the action ϕ is said to be effective if gx = g ′ x, for every x ∈ x† implies g = g ′ . in addition, the action ϕ is said to be transitive, if for every x, x ′ ∈ x†, there exists g ∈ g such that gx = x ′ . definition 4.6. let x be a rough g-space. then x is said to be topologically rough homogeneous if for all x, y ∈ x†, there is a topological homeomorphism ϕ : x† → x† such that ϕ(x) = y. the action of a topological rough group on itself is discussed in the following proposition. proposition 4.7. let h be a rough subgroup of the topological rough group g, and let g be a group. then h acts on g. moreover, g acts roughly on itself. proof. since h is a rough subgroup of g, h is a subset of the group g. therefore the continuous map ϕ : h × g → g is a left rough action of h on g. also since g is a group, the continuous map ϕ : g × g → g is a left rough action of g on g. □ theorem 4.8. let g be a topological rough group and x be a rough gspace. then for every g ∈ g, the left transformation map lg : x† → x†, (resp. right transformation map rg : x † → x†) which is defined by lg(x) = gx (rg(x) = xg), is a topological homeomorphism. proof. indeed, the continuity of the action ϕ implies the continuity of lg. the conditions 1 and 2 in definition 4.5 are respectively equivalent to (1) lg ◦ lg′ = lgg′ . (2) le = 1x. therefore, the maps lg and lg−1 are inverses of each other. thus, lg is a topological homeomorphism from x to x†. □ note that, the left (resp. right) transformation map lg(rg) from x † into x†, is not in general a topological homeomorphism for every g ∈ g. indeed, this is only true in the case where g is a group. from now on, we will focus on studying open subsets in topological rough groups. corollary 4.9. let g be topological rough group. then for every open set o in x† and g ∈ g, lg(o) = go is open in x†. proof. by theorem 4.8, lg(o) = x † → x† is a topological homeomorphism. thus go is open set in x†. □ theorem 4.10. let g be a topological rough group such that g is a group. for any open subset o of g, if a is a subset of g, then ao (respectively oa) is open in g. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 301 a. altassan, n. alharbi, h. aydi and c. özel proof. the fact that g is a group implies that g acts on itself. thus for every g ∈ g, lg is a topological homeomorphism. the rest of proof follows immediately from left transformation definition. therefore ao = ∪a∈ala(o). similarly oa = ∪a∈ara(o) is open in g. □ theorem 4.11. let g be a topological rough group such that g is a group. let h be a rough subgroup of g such that h is closed under multiplication. if there is an open set o in g such that e ∈ o and o ⊆ h, then h is open set in g. proof. let o be a non-empty open set in g such that o ⊆ h and e ∈ o. then for every h ∈ h, lh(o) = ho is open in g. hence h = ! h∈h ho is open in g. □ theorem 4.12. let g be a topological rough group such that g is a group and let h be a rough subgroup of g. let o be an open set in g such that o ⊆ h. then for every h ∈ h, ho is an open set in h. proof. since h ⊆ g, and g is a group, lh is a topological homeomorphism. by the definition of left transformation, lh(o) = ho is open in g. the fact that o ⊆ h implies ho ⊆ h. hence, ho is open in h. □ using the notion of open subsets in topological rough groups, we define the following set. definition 4.13. let g be a topological rough group and let b ⊆ τ be a base for τ. for g ∈ g, the family bg = {o ∩ g : o ∈ b, g ∈ o} ⊆ b is called a base at g in τg. example 4.14. in example 3.3 the family b = {{x} : x ∈ r} is a base for d. for every g ∈ g the collection b = {{g} : g ∈ r∗} is a base for τg. theorem 4.15. let g be a topological rough group such that the identity element e ∈ g and g is closed under multiplication. let g be an open set in g. for g ∈ g the base of g in g is equal to bg = {go : o ∈ be}, where be is the base of the identity e in τg. proof. since g ∈ g, we have g ∈ g. let o1 be an open set in g and let g ∈ u. since e ∈ g, and g is a topological rough group, there are two open sets o2 and o3 such that g ∈ o2, e ∈ o3 and ϕ(o2 ×o3) ⊆ o1. we have g is an open set in τ. then o3 is a neighbourhood of e in τ. then there is a basic open set o ∈ be such that e ∈ o ⊆ o3. hence lg(o) = go ⊆ ϕ(o2×o) ⊆ ϕ(o2×o3) ⊆ o1. □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 302 rough action on topological rough groups acknowledgement the authors wish to thank the deanship for scientific research (dsr) at king abdulaziz university for financially funding this project under grant no. kep-phd-2-130-39. also, we would like to thank the editor and referees for their valuable suggestions which have improved the presentation of the paper. references [1] s. akduman, e. zeliha, a. zemci and s. narli, rough topology on covering based rough sets, 1st international eurasian conference on mathematical sciences and applications (iecmsa), prishtine, kosovo, 3ôçô7 september 2012. 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[22] j. pomykala, the stone algebra of rough sets, bulletin of the polish academy of sciences, mathematics 36, no. 7-8 (1988), 495–508. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 303 a. altassan, n. alharbi, h. aydi and c. özel [23] j. tanga, k. shea, f. min and w. zhu, a matroidal approach to rough set theory, theoretical computer science 471 (2013), 1–11. [24] s. wang, q. zhu, w. zhu and f. min, graph and matrix approaches to rough sets through matroids, information sciences 288 (2014), 1–11. [25] s. wang, q. zhu, w. zhu and f. min, rough set characterization for 2-circuit matroid, fundamenta informaticae 129 (2014), 377–393. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 304 () @ appl. gen. topol. 14, no. 2 (2013), 195-203doi:10.4995/agt.2013.1588 c© agt, upv, 2013 on functions between generalized topological spaces s. bayhan a, a. kanibir b and i. l. reilly c,∗ a department of mathematics, mehmet akif ersoy university, 15030, istiklal campus, burdur, turkey. (bayhan@mehmetakif.edu.tr) b department of mathematics, hacettepe university, 06532, beytepe, ankara, turkey. (kanibir@hacettepe.edu.tr) c department of mathematics, university of auckland, p. b. 92019, auckland, new zealand. (i.reilly@auckland.ac.nz) abstract this paper investigates generalized topological spaces and functions between such spaces from the perspective of change of generalized topology. in particular, it considers the preservation of generalized connectedness properties by various classes of functions between generalized topological spaces. 2010 msc: 54a05, 54c05, 54c08. keywords: generalized topology; generalized continuity; change of generalized topology; generalized connectedness. 0. introduction the notion of continuity is one of the main ideas in the whole of mathematics. so much so, that in recent decades there has been the growing trend of speaking of two sorts of mathematics continuous mathematics and discrete mathematics. in the topological category the morphisms between the basic objects, the topological spaces, are continuous functions. so any attempt to generalize these basic objects must also provide a discussion of a property of functions that ∗ the authors gratefully acknowledge financial support of this research by the scientific and technological council of turkey (tubitak). received february 2013 – accepted july 2013 http://dx.doi.org/10.4995/agt.2013.1575 s. bayhan, a. kanibir and i. l. reilly corresponds to continuity. in the past decade, császár [4] and others have been considering generalized topological spaces, and developing a theory for them. in the title of his foundational paper, császár [4] has given equal prominence to these two aspects “generalized topology, generalized continuity”. this paper is concerned with the adaptation of the change of topology approach from topological topics to aspects of the theory of generalized topological spaces. it shows that ” change of generalized topology ” exhibits some characterictics analogous to change of topology in the topological category. a common application of the change of generalized topology approach occurs where the spaces are (ordinary) topological spaces. in this case, the generalized topologies are families of distinguished subsets of a topological space which are not topologies but are generalized topologies. it appears that this case was one of the main motivations for császár [4] to introduce and develop the concepts of generalized topology. sections 1 and 2 provide the necessary preliminary ideas, while section 3 uses three variations of generalized continuity between generalized topological spaces, defined by min [12,13,14], to illustrate the change of generalized topology procedure in its most general form. section 4 considers the preservation of generalized connectedness properties by appropriate classes of functions. 1. preliminaries if expx is the power set of a nonempty set x, then a subset g of expx is defined to be a generalized topology (briefly a gt ) on x if ∅ ∈ g and any union of elements of g belongs to g, (császár [4]) . if g is a gt on x, the members of g are called generalized open sets, or g-open sets, and their complements are called generalized closed sets, or g-closed sets. if a is a subset of x, then the generalized interior of a in (x, g), denoted by ig(a), is the union of the collection of all generalized open sets contained in a. it is the largest g-open set contained in a. the generalized closure of a in (x, g), denoted by cg(a) is the intersection of the collection of all generalized closed sets containing a. it is the smallest g-closed set containing a. császár [4] has pointed out some common examples of generalized topologies that are associated with a given topological space. if τ is a topology on x, we denote by ia the τ-interior of a and by ca the τ-closure of a. then a is semiopen if a ⊂ c(ia), a is preopen if a ⊂ i(ca), a is β-open if a ⊂ c(i(c(a)), and a is α-open if a ⊂ i(c(ia)). we denote by sτ (pτ, βτ and ατ respectively) the collection of all semi-open sets (preopen sets, β-open sets and α-open sets respectively) of the topological space (x, τ). each of these collections is a gt on x, and, in fact, ατ is a topology. in general, the other three collections are not topologies on x. a generalized topological space (x, g) is said to be connected (called γconnected in [5]) if there are no nonempty disjoint sets u and v in g such that x = u ∪ v. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 196 on functions between generalized topological spaces let gx and gy be generalized topologies on x and y, respectively. then the function f : (x, gx) → (y, gy ) is defined to be generalized continuous or more properly (gx, gy )-continuous if f −1(v ) ∈ gx for each v ∈ gy [4]. if β is an arbitrary subset of expx, then the family ϑ ⊂ expx composed of ∅ and all sets n ⊂ x of the form n = ∪i∈ibi where bi ∈ β and i is a nonempty index set, is a gt on x. we say that β is a base of ϑ, and that β generates ϑ. it is enough to check generalized continuity for each member of a base for the co-domain generalized topology, just as is the case for continuity of functions between (ordinary) topological spaces. lemma 1.1 (6, lemma 3.2). a function f : (x, gx) → (y, gy ) is generalized continuous if and only if f−1(v ) ∈ gx for each member v of a base β of gy . 2. six classes of generalized topologies if g is a gt on x, then császár [7] has shown that δ(g) and θ(g) are also generalized topologies on x, where 1) a ∈ δ(g) if and only if a ⊂ x and, if x ∈ a then there is a g-closed set q such that x ∈ igq ⊂ a, and 2) a ∈ θ(g) if and only if a ⊂ x and x ∈ a implies the existence of m ∈ g such that x ∈ m ⊂ cgm ⊂ a. furthermore, θ(g) ⊂ δ(g) ⊂ g [7, theorem 2.3]. a subset r of (x, g) is defined to be g-regular open (or gr-open) if r = ig(cg(r)). then the elements of δ(g) coincide with the unions of gr-open sets [7, theorem 3.3]. to put it another way, the collection of all gr-open sets of (x, g) is a base for the generalized topology δ(g) on x. therefore, the relationship between a gt g on a set x and the gt δ(g) on x is analogous to that between a topological space (x, τ) and its semiregularization (x, τs), where τs has as a base the collection of all τ-regular open subsets of x. recall that a subset b of (x, τ) is defined to be τ-regular open if b = i(c(b)). mršević, reilly and vamanamurthy [15] have given a systematic discussion of this relationship, especially from the point of view of change of topology. on the other hand, veličko [18] provided the basic discussion of the properties of θ-topologies. császár [3] has introduced four classes of generalized open sets in a generalized topological space with the following definition. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 197 s. bayhan, a. kanibir and i. l. reilly definition 2.1 ([3]). let (x, gx) be a generalized topological space and a ⊂ x. then a is said to be (1) g-semi-open if a ⊂ cg(ig(a)), (2) g-pre-open if a ⊂ ig(cg(a)), (3) g-α-open if a ⊂ ig(cg(ig(a))), (4) g-β-open if a ⊂ cg(ig(cg(a))). the collection of all g-semi-open sets (resp. g-pre-open sets, g-α-open sets, g-β-open sets) in (x, gx) is denoted by σ(g) (resp. π(g), α(g), β(g)). each of these families is a gt [3]. furthermore these families satisfy the following relationships g ⊂ α(g) ⊂ σ(g) ⊂ β(g) and α(g) ⊂ π(g) ⊂ β(g), see [3]. 3. change of generalized topology three variants of generalized continuity have been recently defined and studied by min [12, 13, 14]. if (x, gx) and (y, gy ) are generalized topological spaces, then a function f : (x, gx) → (y, gy ) is defined to be almost (gx, gy )-continuous at x ∈ x if for each gy -open set v containing f(x) there exists a gx-open set u containing x such that f(u) ⊂ igy (cgy (v )) [12, definition 3.1]. furthermore, f is defined to be almost (gx, gy )-continuous if it is almost (gx, gy )-continuous at each point of x [12, definition 3.3]. a function f : (x, gx) → (y, gy ) is defined to be weakly θ(gx, gy )continuous if for x ∈ x and each v ∈ θ(gy ) containing f(x), there exists u ∈ gx such that x ∈ u and f(u) ⊂ v [13, definition 3.12]. more recently, min [14] has introduced and studied another property of functions between generalized topological spaces. let (x, gx) and (y, gy ) be generalized topological spaces and δ = δ(gx), δ ′ = δ(gy ). then a function f : (x, gx) → (y, gy ) is defined to be (δ, δ ′)-continuous at the point x ∈ x if for each gy -open set v containing f(x), there exists a gx-open set u containing x such that f(igx (cgx u)) ⊂ igy (cgy (v )). moreover, the function f is defined to be (δ, δ′)-continuous if it is (δ, δ′)-continuous at every point of x [14, definition 3.4]. min has shown [12, example 3.5] that a function f : (x, gx) → (y, gy ) can be almost (gx, gy )-continuous but not (gx, gy )-continuous. it is clear from the definitions that (gx, gy )-continuity implies almost (gx, gy )-continuity. however, this distinction between these two concepts must be interpreted very carefully. our next result shows that almost (gx, gy )-continuity is really (gx, gy )continuity in disguise. the notion of almost generalized continuity coincides with the concept of generalized continuity when a suitable change of generalized topology is made to the co-domain of the function in question. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 198 on functions between generalized topological spaces proposition 3.1. a function f : (x, gx) → (y, gy ) between generalized topological space is almost (gx, gy )-continuous if and only if f : (x, gx) → (y, δ(gy )) is generalized continuous. proof. min [12, theorem 3.6 (1) and (6)] has shown that f : (x, gx) → (y, gy ) is almost (gx, gy )-continuous if and only if for every gy r-open subset v in y , f−1(v ) is gx-open. by lemma 1.1, this equivalent to f : (x, gx) → (y, δ(gy )) is generalized continuous. � note that the topological analogue of the result above is proposition 12 (1) of mršević, reilly and vamanamurthy [15]. an exactly similar situation obtains for the concept of weakly θ(gx, gy )continuity defined and studied by min [13]. here, a different change of generalized topology on the co-domain of the function reduces the property of weak θ(gx, gy )-continuity to generalized continuity. min [13, example 3.14] has shown that a function can be weakly θ(gx, gy )continuous but not be weakly (gx, gy )-continuous, and hence not be (gx, gy )continuous. however, if we make a change of generalized topology on the co-domain space of f, then weak θ(gx, gy )-continuity reduces to generalized continuity. indeed, we have the following result. proposition 3.2. a function f : (x, gx) → (y, gy ) between generalized topological space is weakly θ(gx, gy )-continuous if and only if f : (x, gx) → (y, θ(gy )) is generalized continuous. proof. [13, theorem 3.15 (1) and (2)]. � in a more recent paper min [14] has introduced the notion of (δ, δ′)-continuity of functions between generalized topological spaces. his examples 3.5 and 3.6, his remark 3.7 and its diagram are designed to distinguish between the generalized continuity of császár [4] and min’s (δ, δ′)-continuity. but, once again, it is necessary to be very careful in making claims about one concept being “independent” of another concept. in this case, this independence holds only if the generalized topologies on the domain and co-domain of the function in question are regarded as fixed. however, our next result shows that if appropriate changes of generalized topology are made on both the domain and co-domain of the function, then (δ, δ′)-continuity of the given function is just generalized continuity in disguise. the new concept of (δ, δ′)-continuity coincides with the old notion of generalized continuity when we make suitable changes of generalized topology. proposition 3.3. a function f : (x, gx) → (y, gy ) between generalized topological spaces is (δ, δ′)-continuous if and only if f : (x, δ(gx)) → (y, δ(gy )) is generalized continuous. proof. min [14] theorem 3.9 (1) and (5). � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 199 s. bayhan, a. kanibir and i. l. reilly we have just considered three examples of the change of generalized topology technique for functions between generalized topological spaces. this is “change of generalized topology” in its full manifestation. this is the true analogue of the “change of topology” approach in the topological category. the fundamental paper on the change of topology technique in the topological category is gauld, mršević, reilly and vamanamurthy [9]. more recently, gauld, greenwood and reilly [8] have classified more than one hundred properties of functions between (ordinary) topological spaces from this perspective. however, there is another way of thinking about the change of generalized topology approach. in this application of the approach we do not have full generality. rather, we consider functions between (ordinary) topological spaces. can some properties of such functions be considered in the setting of generalized continuity? to do so, it may be necessary to consider the (ordinary) topological spaces as examples of generalized topological spaces. it is clear from the work of császár [4] that this is one of the main reasons for introducing the notion of generalized topology. now let gx and gy be generalized topologies on x and y, respectively. császár [4] has pointed out how many variations of continuity in the topological case can be regarded as examples of generalized continuity of functions between generalized topological spaces. specifically, if τx and τy are (ordinary) topologies on x and y , then (τx, τy )-continuity is classical topological continuity. furthermore, (sτx, τy )-continuity is the semi-continuity of [10], (pτx, τy )-continuity is precontinuity in the sense of [11], (sτx, sτy )-continuous functions are defined to be irresolute in [2], while (pτx, pτy )-continuous functions are called preirresolute in [16]. so these are all examples of the use of the change of generalized topology technique to reduce a particular property of functions between ordinary topological spaces to generalized continuity. the corresponding change of topology technique has a much more limited breadth of application (it is restricted to the topological category). an interesting question can be posed. question 3.4. is there a variant of continuity between (ordinary) topological spaces that cannot be reduced to generalized continuity by applying the change of generalized topology technique ? we recall that a subset a of a topological space (x, τx) is defined to be locally closed if a = u ∩ f where u is τx-open and f is τx-closed. moreover, a function f : (x, τx) → (y, τy ) is defined to be lc-continuous if f −1(u) is locally closed in (x, τx) for each τy -open u in y. let lc(x, τx) denote the collection of all locally closed subsets of (x, τx). the next example shows that lc(x, τx) is not always a gt . example 3.5. let x be the set r of real numbers, and take τx = {(a, +∞) : a ∈ r}∪ {∅}. then (x, τx) is a topological space, and the τx-closed sets c© agt, upv, 2013 appl. gen. topol. 14, no. 2 200 on functions between generalized topological spaces are {(−∞, b] : b ∈ r} ∪ {x}. if a < b then (a, b] ∈ lc(x, τx). hence, for each positive integer n, (a, b − 1 n ] ∈ lc(x, τx). but ∞⋃ n=1 (a, b − 1 n ] = (a, b) /∈ lc(x, τx). thus lc(x, τx) is not a gt . it is clear from example 3.5 that lc-continuity cannot be characterized as an example of generalized continuity, thereby providing an affirmative answer to the question above. 4. preservation of generalized connectedness in a recent paper [1], bai and zuo have introduced the class of g-α-irresolute functions between generalized topological spaces. bai and zuo [1, definition 3.1] provide the following definition. definition 4.1. let (x, gx) and (y, gy ) be generalized topological spaces. then a function f : (x, gx) → (y, gy ) is g-α-irresolute if f −1(v ) is g-α-open in x for each g-α-open subset v of y . the change of generalized topology approach has an immediate application here. the next result follows from definition 4.1. proposition 4.2. f : (x, gx) → (y, gy ) is g-α-irresolute if and only if f : (x, α(gx)) → (y, α(gy )) is generalized continuous. bai and zuo [1, examples 3.9.1 and 3.9.2] distinguish between g-α-irresolute functions and (gx, gy )-continuous functions. indeed, they claim “g-α-irresolute and (gx, gy )-continuous are independent of each other in generalized topological spaces”. this distinction between these two concepts requires very careful interpretation. from proposition 4.2 it is clear that the concept of g-α-irresoluteness coincides with the standard notion of generalized continuity when an appropriate change of generalized topology is made to each of the domain and co-domain of the function. the independence of these concepts, claimed by bai and zou [1], holds only when the generalized topologies on the domain and co-domain spaces are taken as fixed. in fact, the concept of g-α-irresoluteness is a disguised form of generalized continuity. to see that this observation is more than a philosophical point, consider the discussion of α-connectedness in [1, page 387]. we recall that a generalized topological space (x, gx) is α-connected if (x, α(gx)) is connected [17, definition 2.1]. now our proposition 4.2, together with the preservation of connectedness by generalized continuous functions (see császár [5], theorem 2.2]), provides an elegant alternative proof of one of the main results of [1]. proposition 4.3 ([1, theorem 3.16]). let f be a g-α-irresolute function from (x, gx) onto (y, gy ). if (x, gx) is α-connected then (y, gy ) is α-connected. proof. we have that f : (x, α(gx)) → (y, α(gy )) is generalized continuous, by proposition 4.2, and that (x, α(gx)) is connected. hence (y, α(gy )) is connected, and thus (y, gy ) is α-connected. � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 201 s. bayhan, a. kanibir and i. l. reilly a similar discussion can be provided for the theorem 3.18 of [1] after noting that (x, gx) is α-compact if and only if (x, α(gx)) is compact. császár [5, page 276] has given the following definitions. let (x, gx) and (y, gy ) be generalized topological spaces. then a function f : (x, gx) → (y, gy ) is said to be (1) (α, gy )-continuous if f −1(v ) is g-α-open in x for every g-open set v in y . (2) (σ, gy )-continuous if f −1(v ) is g-semi-open in x for every g-open set v in y . (3) (π, gy )-continuous if f −1(v ) is g-pre-open in x for every g-open set v in y . (4) (β, gy )-continuous if f −1(v ) is g-β-open in x for every g-open set v in y . now it is clear that the change of generalized topology approach can be applied to each of these variants of generalized continuity. in each case, a change of generalized topology on the domain of the function reduces the variant of generalized continuity to generalized continuity. proposition 4.4. (1) f : (x, gx) → (y, gy ) is (α, gy )-continuous if and only if f : (x, α(gx)) → (y, gy ) is generalized continuous. (2) f : (x, gx) → (y, gy ) is (σ, gy )-continuous if and only if f : (x, σ(gx)) → (y, gy ) is generalized continuous. (3) f : (x, gx) → (y, gy ) is (π, gy )-continuous if and only if f : (x, π(gx)) → (y, gy ) is generalized continuous. (4) f : (x, gx) → (y, gy ) is (β, gy )-continuous if and only if f : (x, β(gx)) → (y, gy ) is generalized continuous. the change of generalized topology approach now provides proofs of four results that are similar to proposition 4.3. shen [17, definition 2.1] defined a generalized topological space (x, gx) to be σ-connected (π-connected, βconnected) if (x, σ(gx)) (respectively (x, π(gx)), (x, β(gx))) is connected. proposition 4.5. (1) let f be (α, gy )-continuous from (x, gx) onto (y, gy ). if (x, gx) is α-connected then (y, gy ) is connected. (2) let f be (σ, gy )-continuous from (x, gx) onto (y, gy ). if (x, gx) is σ-connected then (y, gy ) is connected. (3) let f be (π, gy )-continuous from (x, gx) onto (y, gy ). if (x, gx) is π-connected then (y, gy ) is connected. (4) let f be (β, gy )-continuous from (x, gx) onto (y, gy ). if (x, gx) is β-connected then (y, gy ) is connected. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 202 on functions between generalized topological spaces references [1] s.-z. bai and y.-p. zuo, on g-α-irresolute functions, acta math. hungar. 130, no. 4 (2011), 382–389. [2] s. g. crossley and s. k. hildebrand, semi-topological properties, fund. math. 74 (1972), 233–254. [3] a.császár, generalized open sets in generalized topologies, acta math. hungar. 106, no. 1-2 (2005), 53–66. [4] a.császár, generalized topology, generalized continuity, acta math. hungar. 96 (2002), 351–357. [5] a.császár, γ-connected sets, acta math. hungar. 101 (2003), 273–279. [6] a.császár, normal generalized topologies, acta math. hungar. 115, no. 4 (2007), 309– 313. [7] a.császár, δand θ-modifications of generalized topologies, acta math. hungar. 120 (2008), 275–279. [8] d. b. gauld, s. greenwood and i. l. reilly, on variations of continuity, topology atlas, invited contributions 4 (1) (1999), 1–54 (http://at.yorku.ca/t/a/i/c/32.htm). [9] d. b. gauld, m. mršević, i. l. reilly and m. k. vamanamurthy, continuity properties of functions, colloquia math. soc. janos bolyai, 41 (1983), 311–322. [10] n. levine, semi-open sets and semicontinuity in topological spaces, amer. math. monthly 70 (1963), 36–41. [11] a. mashhour, m. abd. el-monsef and s. el-deeb, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. egypt 53 (1982), 47–53. [12] w. k. min, almost continuity on generalized topological spaces, acta math. hungar. 125, no. 1-2 (2009), 121–125. [13] w. k. min, a note on θ(g, g′)-continuity in generalized topological spaces, acta math. hungar., 125 (4) (2009), 387-393. [14] w. k. min, (δ, δ′)-continuity on generalized topological spaces, acta math. hungar. 129, no. 4 (2010), 350–356. [15] m. mršević, i. l. reilly and m. k. vamanamurthy, on semi-regularization topologies, j. austral. math. soc. (series a) 38 (1985), 40–54. [16] i. l. reilly and m. k. vamanamurthy, on α-continuity in topological spaces, acta math. hungar. 45 (1985), 27–32. [17] r.-x. shen, a note on generalized connectedness, acta math. hungar. 122, no. 3 (2009), 231–235. [18] n. v. veličko, h-closed topological spaces, mat. sbornik 70 (112) (1966), 98–112. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 203 @ appl. gen. topol. 22, no. 2 (2021), 311-319doi:10.4995/agt.2021.14449 © agt, upv, 2021 the periodic points of ε-contractive maps in fuzzy metric spaces taixiang sun a, caihong han a,∗, guangwang su a, bin qin b and lue li a a college of information and statistics, guangxi university of finance and economics, nanning, 530003, china (stx1963@163.com,h198204c@163.com, s1g6w3@163.com, li1982lue@163.com) b guangxi (asean) research center of finance and economics nanning, 530003, china (q3009b@163.com) communicated by h. dutta abstract in this paper, we introduce the notion of ε-contractive maps in fuzzy metric space (x, m, ∗) and study the periodicities of ε-contractive maps. we show that if (x, m, ∗) is compact and f : x −→ x is ε-contractive, then p(f) = ∩∞ n=1f n(x) and each connected component of x contains at most one periodic point of f, where p(f) is the set of periodic points of f. furthermore, we present two examples to illustrate the applicability of the obtained results. 2010 msc: 54e35; 54h25. keywords: fuzzy metric space; ε-contractive map; periodic point. 1. introduction the notion of fuzzy metric spaces was introduced by kramosil and michalek [10] and later was modified by george and veeramani [3] in order to obtain a hausdorff topology in a fuzzy metric space. recently there has been a great interest in discussing some properties on discrete dynamical systems in fuzzy project supported by nnsf of china (11761011) and nsf of guangxi (2020gxnsfaa297010) and pymrbap for guangxi cu(2021ky0651). ∗corresponding author. received 07 october 2020 – accepted 25 april 2021 http://dx.doi.org/10.4995/agt.2021.14449 t. sun, c. han, g. su, b. qin and l. li metric spaces. many authors introduced and investigated the different types of fuzzy contractive maps and obtained a lot of fixed point theorems (see [1, 2, 5, 6, 8, 11, 12, 14, 15, 16, 17, 18, 19]). until now, there are very little of works that investigates the periodic points of discrete dynamical systems in fuzzy metric spaces because it is much more difficult to find various conditions to obtain the periodic points of discrete dynamical systems. in the present paper, we introduce the notion of ε-contractive map in fuzzy metric space (x, m, ∗) and obtain the following results: (1) if (x, m, ∗) is compact and f : x −→ x is ε-contractive, then p(f) = ∩∞n=1f n(x), where p(f) is the set of periodic points of f. (2) if (x, m, ∗) is compact and f : x −→ x is ε-contractive, then each connected component of x contains at most one periodic point of f. furthermore, if (x, m, ∗) is also connected, then f has at most one fixed point. 2. preliminaries throughout the paper, let n be the set of all positive integers. firstly, we recall the basic definitions and the properties about fuzzy metric spaces. definition 2.1 (schweizer and sklar [13]). a binary operation t : [0, 1]2 −→ [0, 1] is a continuous t-norm if it satisfies the following conditions (i)-(v): (i) t (a, b) = t (b, a); (ii) t (a, b) ≤ t (c, d) for a ≤ c and b ≤ d; (iii) t (t (a, b), c) = t (a, t (b, c)); (iv) t (a, 0) = 0 and t (a, 1) = a; (v) t is continuous on [0, 1]2, where a, b, c, d ∈ [0, 1]. for any a, b ∈ [0, 1], we will use the notation a∗b instead of t (a, b). t (a, b) = min{a, b}, t (a, b) = ab and t (a, b) = max{a+ b− 1, 0} are the most commonly used t-norms. in the present paper, we also use the following definition of a fuzzy metric space. definition 2.2 (george and veeramani [3]). a triple (x, m, ∗) is called a fuzzy metric space if x is a nonempty set, ∗ is a continuous t-norm and m is a map defined on x2 × (0, +∞) into [0, 1] satisfying the following conditions (i)-(v) for any x, y, z ∈ x and s, t ∈ (0, +∞): (i) m(x, y, t) > 0; (ii) m(x, y, t) = 1 (for any t > 0) ⇐⇒ x = y; (iii) m(x, y, t) = m(y, x, t); (iv) m(x, z, t + s) ≥ m(x, y, t) ∗ m(y, z, s); (v) mxy : (0, +∞) −→ [0, 1] is a continuous ( where mxy(t) = m(x, y, t)). remark 2.3 (grabiec [4]). mxy is non-decreasing for all x, y ∈ x. if (x, m, ∗) is a fuzzy metric space, then (m, ∗), or simply m, is called a fuzzy metric on x. in [3], george and veeramani showed that every fuzzy © agt, upv, 2021 appl. gen. topol. 22, no. 2 312 the periodic points of contractive maps metric m on x generates a topology τm on x which has as a base the family of open sets of the form {bm(x, ε, t) : x ∈ x, ε ∈ (0, 1), t > 0} , where bm(x, ε, t) = {y ∈ x : m(x, y, t) > 1 − ε} for any x ∈ x, ε ∈ (0, 1) and t > 0. definition 2.4 (gregori and sapena [7]). let (x, m, ∗) be a fuzzy metric space. a sequence of points xn ∈ x is called to converge to x ( denoted by xn −→ x) ⇐⇒ limn−→+∞ m(xn, x, t) = 1 (for any t > 0), i.e. for each δ ∈ (0, 1) and t > 0, there exists n ∈ n such that m(xn, x, t) > 1 − δ for all n ≥ n. definition 2.5. (1) a fuzzy metric space (x, m, ∗) is said to be compact if each sequence of points in x has a convergent subsequence. a subset a of x is said to be compact if a as a fuzzy metric subspace is compact. (2) a fuzzy metric space (x, m, ∗) is said to be connected if there exist two nonempty closed sets u, v of x with u ∩v = ∅ such that x = u ∪v . a subset a of x is said to be connected if a as a fuzzy metric subspace is connected. definition 2.6. let (x, m, ∗) be a fuzzy metric space and ε ∈ (0, 1). a map f : x −→ x is said to be ε-contractive if m(x, y, t) > 1 − ε for any x, y ∈ x with x 6= y and t > 0, then m(f(x), f(y), t) > m(x, y, t). denote by c(x, ε) the set of all ε-contractive maps in x. remark 2.7. (1) let (x, m, ∗) be compact and a be a subset of x. then a is compact ⇐⇒ a is closed (see [9]). (2) if f ∈ c(x, ε), then fn ∈ c(x, ε) for any n ∈ n. (3) if a is a connected component of x, then a is closed. (4) if f ∈ c(x, ε) and a is a closed subset of x, then f−1(a) is also closed. indeed, let a sequence of points xn ∈ f −1(a) with xn −→ x. then there exists n ∈ n such that 1 ≥ m(f(xn), f(x), t) ≥ m(xn, x, t) > 1 − ε for any n ≥ n, which implies 1 ≥ limn−→∞ m(f(xn), f(x), t) ≥ limn−→∞ m(xn, x, t) = 1 and f(xn) −→ f(x). since f(xn) ∈ a and a is closed, we see f(x) ∈ a. thus x ∈ f−1(a) , which implies that f−1(a) is closed. (5) by (4) we see that if f ∈ c(x, ε) and a is a connected subset of x, then f(a) is also connected. let (x, m, ∗) be a fuzzy metric space and f : x −→ x. write f0(x) = x and fn = f ◦ fn−1 for any x ∈ x and n ∈ n. we write p(f) = {x : there exists some n ∈ n such that fn(x) = x}. for any x ∈ x, we write ω(x, f) = {y : there exists a sequence of positive integers n1 < n2 < · · · such that lim k−→∞ m(fnk(x), y, t) = 1 for any t > 0}. p(f) is called the set of periodic points of f. ω(x, f) is called the set of ω-limit points of x under f. if f(x) = x, then x is called the fixed point of f. © agt, upv, 2021 appl. gen. topol. 22, no. 2 313 t. sun, c. han, g. su, b. qin and l. li 3. main results in this section, we study the set of periodic points of ε-contractive maps in fuzzy metric spaces. lemma 3.1. let (x, m, ∗) be a compact fuzzy metric space and f ∈ c(x, ε). then f(ω(x, f)) = ω(x, f) for any x ∈ x and ω(x, f) = ω(y, f) for any y ∈ ω(x, f). proof. let u ∈ ω(x, f). then there exists a sequence of positive integers n1 < n2 < · · · < nk < · · · such that limk−→∞ m(f nk(x), u, t) = 1 for any t > 0. thus for any 0 < δ < ε and t > 0, there exists n ∈ n such that m(fnk(x), u, t) > 1 − δ for all n ≥ n. from which it follows that m(fnk+1(x), f(u), t) ≥ m(fnk(x), u, t) > 1 − δ. therefore f(u) ∈ ω(x, f) and f(ω(x, f)) ⊂ ω(x, f). on the other hand, by taking subsequence we may assume that fnk−1(x) −→ v for some v ∈ ω(x, f) since (x, m, ∗) is a compact. then for any 0 < δ < ε and t > 0, there exist 0 < δ1 < δ and n ∈ n such that (1−δ1)∗(1−δ1) > 1−δ and m(fnk(x), u, t/2) > 1 − δ1 and m(f nk−1(x), v, t/2) > 1 − δ1 for all n ≥ n. note that f ∈ c(x, ε), we see that when n ≥ n, one has m(u, f(v), t) ≥ m(u, fnk(x), t 2 ) ∗ m(fnk(x), f(v), t 2 ) ≥ m(u, fnk(x), t 2 ) ∗ m(fnk−1(x), v, t 2 ) ≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ. thus we obtain m(u, f(v), t) = 1 for any t > 0 and u = f(v) ∈ f(ω(x, f)), which implies ω(x, f) ⊂ f(ω(x, f)). now we show that ω(y, f) ⊂ ω(x, f) for any y ∈ ω(x, f). let z ∈ ω(y, f). then there exist two sequences of positive integers k1 < k2 < · · · and r1 < r2 < · · · such that f kn(x) −→ y and frn(y) −→ z. thus for any 0 < δ < ε and t > 0, there exist 0 < δ1 < δ and n ∈ n such that (1 − δ1) ∗ (1 − δ1) > 1 − δ and m(y, fkn(x), t/2) > 1 − δ1 and m(z, f rn(y), t/2) > 1 − δ1 for any n > n. note that f ∈ c(x, ε), we see that when n > n, one has m(z, fkn+rn(x), t) ≥ m(z, frn(y), t 2 ) ∗ m(frn(y), fkn+rn(x), t 2 ) ≥ m(z, frn(y), t 2 ) ∗ m(y, fkn(x), t 2 ) ≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ. therefore we obtain z ∈ ω(x, f), which implies ω(y, f) ⊂ ω(x, f). finally we show that ω(y, f) = ω(x, f) for any y ∈ ω(x, f). let z ∈ ω(x, f). then there exist two sequences of positive integers k1 < k2 < · · · and r1 < r2 < · · · such that f kn(x) −→ y and frn(x) −→ z with rn − kn > n. thus for any 0 < δ < ε and t > 0, there exist 0 < δ1 < δ and n ∈ n such that (1−δ1)∗(1−δ1) > 1−δ and m(y, f kn(x), t/2) > 1−δ1 and m(z, f rn(x), t/2) > © agt, upv, 2021 appl. gen. topol. 22, no. 2 314 the periodic points of contractive maps 1 − δ1 for any n > n. note that f ∈ c(x, ε), we see that when n > n, one has m(z, frn−kn(y), t) ≥ m(z, frn(x), t 2 ) ∗ m(frn(x), frn−kn(y), t 2 ) ≥ m(z, frn(x), t 2 ) ∗ m(y, fkn(x), t 2 ) ≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ. therefore we obtain z ∈ ω(y, f), which implies ω(y, f) = ω(x, f). the proof is completed. � theorem 3.2. let (x, m, ∗) be a compact fuzzy metric space and f ∈ c(x, ε). then p(f) = ∩∞n=1f n(x). proof. let y = ∩∞n=1f n(x). then f(y ) = y . it is easy to see that p(f) ⊂ y . in the following we show that y ⊂ p(f). let x0 ∈ y . then there exists a sequence of points xn ∈ y such that f(xn) = xn−1 for every n ∈ n. since x is compact, there exist a subsequence {xnk} ∞ k=1 of {xn} ∞ n=1 and a point u ∈ y satisfying xnk −→ u . thus, for any 0 < δ < ε and t > 0, there exits n ∈ n such that m(xnk , u, t) ≥ 1 − δ for all k ≥ n. note that f ∈ c(x, ε), we have m(x0, f nk(u), t) = m(fnk(xnk ), f nk (u), t) > · · · > m(xnk , u, t) ≥ 1 − δ for all k ≥ n. therefore x0 ∈ ω(u, f), which with lemma 3.1 implies x0 ∈ ω(u, f) = ω(x0, f). by x0 ∈ ω(x0, f) we see that there exist 1 < n1 < n2 < · · · < nk < · · · such that limk−→∞ m(f nk(x0), x0, t) = 1. choose nr with m(f nr(x0), x0, t) > 1−ε. in the following we show fnr(x0) = x0, which implies x0 ∈ p(f). indeed, if fnr(x0) 6= x0, then m(f nr(x0), x0, t0) < 1 for some t0 > 0. since m(fnr(x0), x0, t) is continuous and non-decreasing in (0, +∞) and f ∈ c(x, ε), we see that there exists γ > 0 such that m(fnr(x0), x0, t0) ≤ m(f nr (x0), x0, t0+ γ) < m(fnr+1(x0), f(x0), t0). thus m(fnr(x0), x0, t0 + γ) ≥ m(fnr(x0), f nk+nr (x0), γ 2 ) ∗ m(fnk+nr (x0), f nk(x0), t0) ∗ m(f nk(x0), x0, γ 2 ) ≥ m(x0, f nk(x0), γ 2 ) ∗ m(f1+nr(x0), f(x0), t0) ∗ m(f nk(x0), x0, γ 2 ). taking the limit on both sides in the above as k −→ ∞ we obtain m(fnr(x0), x0, t0+γ) ≥ 1∗m(f 1+nr(x0), f(x0), t)∗1 = m(f 1+nr(x0), f(x0), t0). this leads a contradiction. the proof is completed. � remark 3.3. it is easy to see that if (x, m, ∗) is a compact fuzzy metric space and f ∈ c(x, ε), then ∪x∈xω(x, f) = p(f) since f(ω(x, f)) = ω(x, f) and p(f) ⊂ ∪x∈xω(x, f) ⊂ ∩ ∞ n=1f n(x) = p(f). © agt, upv, 2021 appl. gen. topol. 22, no. 2 315 t. sun, c. han, g. su, b. qin and l. li lemma 3.4. let (x, m, ∗) be a fuzzy metric space, u ∈ x and f ∈ c(x, ε). write j(u, f) = {x ∈ x : u ∈ ω(x, f)}. then j(u, f) is a closed subset of x. furthermore, if (x, m, ∗) is compact, then j(u, f) is a open subset of x. proof. let x1, x2, · · · , xn, · · · ∈ j(u, f) and x ∈ x such that limn−→∞ m(xn, x, t) = 1 for any t > 0. for any n ∈ n, there exist 1 < k1n < k2n < · · · < krn < · · · such that limr−→∞ m(f krn(xn), u, t) = 1 for any t > 0. then for any 0 < δ < ε and t > 0, we can choose 0 < δ1 < δ with (1 − δ1) ∗ (1 − δ1) > 1 − δ and n ∈ n such that (by taking subsequence ) m(fknn(xn), u, t/2) ≥ 1 − δ1 and m(xn, x, t/2) > 1 − δ1 for any n ≥ n. then m(fknn(x), u, t) ≥ m(fknn(x), fknn(xn), t 2 ) ∗ m(fknn(xn), u, t 2 ) ≥ m(x, xn, t 2 ) ∗ m(fknn(xn), u, t 2 ) ≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ. thus u ∈ ω(x, f) and j(u, f) is closed. now we prove the second part of the lemma. assume that x is compact. then for any 0 < δ < ε, there exists 0 < δ1 < δ < ε such that (1−δ1)∗(1−δ1) > 1 − δ. let x ∈ j(u, f). we prove that b(x, δ1, t/2) ⊂ j(u, f). let y ∈ b(x, δ1, t/2). since x ∈ j(u, f), there exist n1 < n2 < · · · < nk < · · · such that limk−→∞ m(f nk(x), u, t/2) = 1 for any t > 0. thus for any t > 0, there exists n ∈ n such that m(fnk(x), u, t/2) ≥ 1 − δ1 for any k > n. therefore we have that for k > n and t > 0, m(fnk(y), u, t) ≥ m(fnk(y), fnk (x), t 2 ) ∗ m(fnk(x), u, t 2 ) ≥ m(y, x, t 2 ) ∗ m(fnk(x), u, t 2 ) ≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ. by taking subsequence we may assume that fnk(y) −→ v. taking the limit on both sides in the above as k −→ ∞ we obtain m(v, u, t) ≥ 1 − δ. if j(u, f) = ∅, then j(u, f) is open. if j(u, f) 6= ∅, then by remark 3.3 we see that v, u ∈ p(f). let m and n be the periods of u and v, respectively. note that f ∈ c(x, ε), we have m(u, v, t) = m(fmn(u), fmn(v), t) > m(u, v, t), which is impossible unless u = v. hence u ∈ ω(y, f) and b(x, δ1, t/2) ⊂ j(u, f). since x is an arbitrarily chosen point of j(u, f), we see that j(u, f) is an open subset of x. the proof is completed. � theorem 3.5. if (x, m, ∗) is a compact fuzzy metric space and f ∈ c(x, ε), then each connected component of x contains at most one periodic point of f. furthermore, if (x, m, ∗) is also connected, then f has at most one fixed point. © agt, upv, 2021 appl. gen. topol. 22, no. 2 316 the periodic points of contractive maps proof. let p ∈ p(f) and y (p) be the connected component of x containing p and r be the period of p. write h = fr. by remark 2.7 we see that h ∈ c(x, ε) and h(y (p)) is connected and y (p) is a compact. since p ∈ y (p) ∩ h(y (p)), we have h(y (p)) ⊂ y (p). replace x and f of lemma 3.4 by y (p) and h, respectively, and write j(p, h) ⊂ y (p) to be as in lemma 3.4. by lemma 3.4 we see that j(p, h) is both closed and open in y (p). also, since p ∈ ω(p, h) and y (p) is connected, we have that j(p, h) = y (p) in the following, we show p(f) ∩ y (p) = {p}. indeed, if p(f) ∩ y (p) 6= {p} and q ∈ p(f) ∩ y (p) − {p}, then q ∈ p(h) ∩ y (p) = p(h) ∩ j(p, h) and p ∈ ω(q, h). let n1 < n2 < · · · < nk < · · · such that h nk(q) −→ p. then for any 0 < δ < ε and t > 0, there exists n ∈ n such that m(hnk (q), p, t) > 1 − δ for any k ≥ n. let s be the period of q. we have m(p, hnk(q), t) = m(hsnk (p), h(s+1)nk (q), t) > m(p, hnk(q), t). this will lead a contradiction. the proof is completed. � in the following we present two examples to illustrate the applicability of the obtained results. example 3.6. let x = [0, 1/3] ∪ [2/3, 1] ⊂ (−∞, +∞). define s ∗ t = st for any s, t ∈ [0, 1], and let m : x × x × (0, ∞) −→ [0, 1] by, for any x, y ∈ x and t > 0, m(x, y, t) = { 1 1+|x−y| , if t ≥ 1, t t+|x−y| , if 0 < t < 1. then (x, m, ∗) is a compact fuzzy metric space. take k ∈ (0, 1) and define f : x −→ x by, for any x ∈ x, f(x) = { kx + 2 3 , if x ∈ [0, 1 3 ], k(1 − x), if x ∈ [2 3 , 1]. we claim that f ∈ c(x, 1/4). indeed, for any x, y ∈ x and t > 0 with m(x, y, t) > 1−1/4, we have |x−y| < 1/3. then x, y ∈ [0, 1/3] or x, y ∈ [2/3, 1], from which it follows that |f(x) − f(x)| = k|x − y| < |x − y|. thus m(f(x), f(y), t) > m(x, y, t) and f ∈ c(x, 1/4). by theorem 3.2 and theorem 3.5 we see that p(f) = ∩∞n=1f n(x) contains at most 2 points, and [0, 1/3] ∩ p(f) contains at most one point, and [2/3, 1] ∩ p(f) contains at most one point. in fact, we have p(f) = {k/3(k2 + 1), (3k2 + 2)/3(k2 + 1)} with f(k/3(k2 + 1)) = (3k2 + 2)/3(k2 + 1) and f((3k2 + 2)/3(k2 + 1)) = k/3(k2 + 1). example 3.7. let x = [0, 1/3] ∪ [2/3, 1] ⊂ (−∞, +∞). define s ∗ t = st for any s, t ∈ [0, 1], and let m : x × x × (0, ∞) −→ [0, 1] by, for any x, y ∈ x and t > 0, m(x, y, t) = t t + |x − y| . © agt, upv, 2021 appl. gen. topol. 22, no. 2 317 t. sun, c. han, g. su, b. qin and l. li then (x, m, ∗) is a compact fuzzy metric space. take k ∈ (0, 1) and define f : x −→ x by, for any x ∈ x, f(x) = { kx + 2 3 , if x ∈ [0, 1 3 ], k(1 − x) + 2 3 , if x ∈ [2 3 , 1]. we claim that f ∈ c(x, 1/4). indeed, if x, y ∈ [0, 1/3] or x, y ∈ [2/3, 1], then |f(x) − f(y)| = k|x − y| < |x − y|, which follows that m(f(x), f(y), t) > m(x, y, t). if x ∈ [0, 1/3] and y ∈ [2/3, 1], then |f(x) − f(y)| = 1/3 < |x − y|, which also follow that m(f(x), f(y), t) > m(x, y, t). by theorem 3.2 and theorem 3.5 we see that p(f) = ∩∞n=1f n(x) contains at most 2 points, and [0, 1/3]∩p(f) contains at most one point, and [2/3, 1]∩p(f) contains at most one point. in fact, we have p(f) = {3k+2/(3k+3)} ⊂ [2/3, 1]. 4. conclusion in this paper, we introduce the notion of ε-contractive maps in a fuzzy metric space, and study the periodicities of ε-contractive maps, and obtain the following result: if f is a ε-contractive map in compact fuzzy metric space (x, m, ∗), then p(f) = ∩∞n=1f n(x) and each connected component of x contains at most one periodic point of f. furthermore, we present two examples to illustrate the applicability of the obtained results. acknowledgements. the authors thanks the referee for his/her valuable suggestions which improved the paper. references [1] m. abbas, m. imdad and d. gopal, ψ-weak contractions in fuzzy metric spaces, iranian j. fuzzy syst. 8 (2011), 141–148. 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[19] d. zheng and p. wang, meir-keeler theorems in fuzzy metric spaces, fuzzy sets syst. 370 (2019), 120–128. © agt, upv, 2021 appl. gen. topol. 22, no. 2 319 @ appl. gen. topol. 23, no. 1 (2022), 157-167 doi:10.4995/agt.2022.15356 © agt, upv, 2022 boyd-wong contractions in f-metric spaces and applications ashis bera a, lakshmi kanta dey a, sumit som b, hiranmoy garai a and wutiphol sintunavarat c a department of mathematics, national institute of technology durgapur 713209, india (beraashis.math@gmail.com,lakshmikdey@yahoo.co.in,hiran.garai24@gmail.com) b department of mathematics, school of basic and applied sciences, adamas university, barasat700126, west bengal, india (somkakdwip@gmail.com) c department of mathematics and statistics, faculty of science and technology, thammasat university rangsit center, pathum thani 12120, thailand. (wutiphol@mathstat.sci.tu.ac.th) communicated by i. altun abstract the main aim of this paper is to study the boyd-wong type fixed point result in the f-metric context and apply it to obtain some existence and uniqueness criteria of solution(s) to a second order initial value problem and a caputo fractional differential equation. we substantiate our obtained result by finding a suitable non-trivial example. 2020 msc: 47h10; 54a20; 54e50. keywords: f-metric space; fractional differential equation; boyd-wong fixed point theorem. 1. introduction and preliminaries after the invention of metric spaces by frèchet, many mathematicians have generalized the metric structure by making some changes in the original definition of a metric given by frèchet. most of the generalizations are made by making some changes in the triangle inequality of the original definition. some well-known metrics of such generalizations are b-metric due to czerwik [10], rectangular metric due to branciari [7], bv(s)metric due to mitrović and received 06 april 2021 – accepted 15 december 2021 http://dx.doi.org/10.4995/agt.2022.15356 https://orcid.org/0000-0002-0932-1332 a. bera, l. k. dey, s. som, h. garai and w. sintunavarat radenović [20], js-metric due to jleli and samet [16] etc. after all such types of generalizations, recently in 2018, jleli and samet [15] introduced another such abstraction, which they denominate as f-metric. they defined this metric structure by means of a certain class f, which contains the set of functions f : (0,∞) → r satisfying the following conditions: (f1) f is non-decreasing, i.e., 0 < s < t ⇒ f(s) ≤ f(t); (f2) for every sequence (tn) ⊆ (0,∞), lim n→∞ tn = 0 ⇐⇒ lim n→+∞ f(tn) = −∞. the definition of an f-metric based on the idea of a class f is as follows: definition 1.1 ([15]). a function d : x ×x → [0,∞), x being a nonempty set, is called an f-metric on x if there exists (f,α) ∈f× [0,∞) such that the following conditions hold: (d1) for (x,y) ∈ x ×x, d(x,y) = 0 ⇐⇒ x = y; (d2) d(x,y) = d(y,x) for all (x,y) ∈ x ×x; (d3) for every (x,y) ∈ x × x, for each n ∈ n,n ≥ 2, and for every (ui) n i=1 ⊆ x with (u1,un ) = (x,y), we have d(x,y) > 0 =⇒ f(d(x,y)) ≤ f ( n−1∑ i=1 d(ui,ui+1) ) + α. furthermore, jleli and samet [15] also introduced the concepts of f-openness, f-convergence, f-cauchyness and f-completeness as follows: definition 1.2 ([15]). a subset o of an f-metric space (x,d) is said to be f-open if for every x ∈o, there is some r > 0 such that bd(x,r) ⊆o, where bd(x,r) = {y ∈ x : d(x,y) < r}. from the above definition, it is easy to see that the family of all f-open subsets of an f-metric space (x,d) is a topology on x. definition 1.3 ([15]). let (x,d) be an f-metric space and (xn) be a sequence in x. (1) we say that (xn) is f-convergent to x ∈ x if for every f-open subset ox of x containing x, there exists some n ∈ n such that xn ∈ox for all n ≥ n. (2) we say that (xn) is an f-cauchy sequence if lim n,m→∞ d(xn,xm) = 0. (3) we say that x is f-complete if every f-cauchy sequence in x is fconvergent to some point in x. remark 1.4. if (xn) is a sequence in an f-metric space (x,d), then (xn) is f-convergent to x ∈ x if and only if lim n→∞ d(xn,x) = 0. in addition, the limit of an f-convergent sequence is unique. among all the mathematical theories, which make all such generalized structures interesting and important, fixed point theory is one of these. throughout © agt, upv, 2022 appl. gen. topol. 23, no. 1 158 boyd-wong contractions in f-metric spaces and applications the last few decades, many renowned mathematicians have achieved a lot of well-known metric fixed point theorems in these structures (see in [1, 3, 23, 25, 26] and references therein). however, if a distance space is metrizable, then sometimes it may happen that some metric fixed point theorems directly follow from the metrizability result of the spaces. still, it is essential to note that some well-known fixed point results can’t be obtained from the fact that the space is metrizable. in this paper, we deal with one such structure, f-metric space, which is metrizable, and at the same time, some fixed point theorems like the banach contraction principle follows from its metric counterpart using metrizability result. however, the general non-linear contractions like boyd-wong contraction [8] can’t be obtained from the metrizability result. som et al. proved the above facts in [24]. thus the study of boyd-wong fixed point theorem in the context of f-metric spaces seems to be interesting. we could not establish an analogous result in this setting to that of usual metric spaces, i.e., it is not known whether a mapping satisfying the boyd-wong contraction condition in an f-complete f-metric space possesses a fixed point or not. we leave it as an open question in section 2. so it is challenging work to have boyd-wong type result in fmetric spaces assuming some mild additional hypotheses. we succeed in this direction and propose boyd-wong type result in f-metric spaces. moreover, we apply our result in the context of special types of ordinary and caputo fractional differential equations. we present the above-mentioned results in section 3. 2. boyd-wong type results in the f-metric structure in the previous section, we have already mentioned that we need some additional hypotheses either on an f-metric space x or on a mapping t : x → x to get a fixed point of t satisfying the boyd-wong contractive condition. in this section, we present such additional hypotheses via the following theorem. theorem 2.1. let (x,d) be an f-complete f-metric space with (f,α) ∈ f×[0,∞) such that f is continuous and ψ : [0,∞) → [0,∞) be a nondecreasing upper semi-continuous mapping from right such that ψ(t) < t for all t > 0. suppose that t : x → x is a mapping such that (2.1) d(tx,ty) ≤ ψ(d(x,y)) for all x,y ∈ x. further, assume that (2.2) f(t) > f(ψ(t)) + α for all t ∈ (0,∞). then t has a unique fixed point and (tnx) converges to that fixed point for all x ∈ x. proof. let x0 be an arbitrary point in x. define a sequence (xn) by setting xn = t nx0 for all n ∈ n. if xn∗−1 = xn∗ for some n∗ ∈ n, then the proof is done. so, we now assume that xn−1 6= xn for all n ∈ n. then from the given © agt, upv, 2022 appl. gen. topol. 23, no. 1 159 a. bera, l. k. dey, s. som, h. garai and w. sintunavarat condition, we have d(xn,xn+1) = d(txn−1,txn) ≤ ψ (d(xn−1,xn)) < d(xn−1,xn) for all n ∈ n. hence, the sequence (d(xn,xn+1)) is a strictly decreasing sequence and also this sequence is bounded below. therefore, lim n→∞ d(xn,xn+1) exists. let lim n→∞ d(xn,xn+1) =: l ≥ 0. if l > 0, then by the property of ψ, we get l = lim n→∞ d(xn,xn+1) ≤ lim sup n→∞ ψ(d(xn−1,xn)) ≤ ψ(l) < l, which is a contradiction. therefore, we obtain (2.3) l = lim n→∞ d(xn,xn+1) = 0. next, we will show that (tnx) is an f-cauchy sequence. we will prove it by contradiction. if possible let (tnx) be not an f-cauchy sequence. then there exist ε > 0 and subsequences (xmk ) and (xnk ) of (xn) with mk > nk ≥ k such that (2.4) d(xmk,xnk ) ≥ ε for each k ∈ n. we choose nk as the smallest number not exceeding mk for which the equation (2.4) holds. then we have d(xmk,xnk−1) < ε.(2.5) now, from (d3) and (2.5), we have f(ε) ≤ f(d(xmk,xnk )) ≤ f (d(xmk,xmk+1) + d(xmk+1,xnk )) + α ≤ f (d(xmk,xmk+1) + ψ(d(xmk,xnk−1)) + α ≤ f (d(xmk,xmk+1) + ψ(ε)) + α. now, letting k →∞ in both sides of the above equation, we get f(ε) ≤ f(ψ(ε)) + α, which is a contradiction. this shows that the sequence (xn) is an f-cauchy sequence. since x is f-complete, there is a point x∗ ∈ x such that lim n→∞ d(xn,x ∗) = 0.(2.6) next, we prove that x∗ is a fixed point of t. we prove it by contradiction. let tx∗ 6= x∗. then by (d3), we have (2.7) f(d(tx∗,x∗)) ≤ f ( d(tx∗,txn) + d(txn,x ∗) ) + α for all n ∈ n. if there are two natural numbers n1 and n2 such that d(xn1,x∗) = 0 and d(xn2,x ∗) = 0, we obtain xn1 = x ∗ = xn2 , which is a contradiction. so © agt, upv, 2022 appl. gen. topol. 23, no. 1 160 boyd-wong contractions in f-metric spaces and applications we may choose a subsequence {xnl} of {xn} such that d(xnl,x ∗) 6= 0 for all l ∈ n. using the given condition, (2.7) and (f1), we get f(d(tx∗,x∗)) ≤ f ( d(tx∗,txnl ) + d(txnl,x ∗) ) + α ≤ f ( ψ(d(x∗,xnl )) + d(xnl+1,x ∗) ) + α ≤ f ( d(x∗,xnl ) + d(xnl+1,x ∗) ) + α.(2.8) therefore, by the condition (f2) and the equation (2.6), we have lim l→∞ f ( d(x∗,xnl ) + d(xnl+1,x ∗) ) + α = −∞, which is a contradiction. hence, tx∗ = x∗. for the uniqueness, let t has two fixed points, say x∗ and y∗ such that x∗ 6= y∗. then d(x∗,y∗) = d(tx∗,ty∗) ≤ ψ(d(x∗,y∗)) < d(x∗,y∗), which is impossible. hence, this theorem is proved. � next, we provide an example to validate our obtained result. example 2.2. let x = n and consider the mapping d : x × x → [0,∞), defined by d(x,y) = { |x−y|, if x and y both are even or both are odd; 3|x−y| + 5, if any one of x and y is even and the other is odd. then (x,d) is an f-metric space with f(t) = ln t and α = ln 3. also, x is f-complete. let us define a mapping t : x → x by tx = { 2, if x is even; 4, if x is odd. we define a mapping ψ : [0,∞) → [0,∞) by ψ(t) = 1 4 t for all t ∈ [0,∞). then ψ is a nondecreasing upper semicontinuous from right on [0,∞) and ψ(t) < t for all t > 0. in this case it is clear that f is continuous and satisfies f(t) > f(ψ(t)) + α for all t ∈ (0,∞). let x,y ∈ x be arbitrary. if x,y both are even or both are odd, then it is obvious that d(tx,ty) ≤ ψ(d(x,y)). if x is even and y is odd, then d(tx,ty) = 2 and d(x,y) = 3|x−y|+ 5 ≥ 2, which implies that ψ(d(x,y)) ≥ 2. hence, d(tx,ty) ≤ ψ(d(x,y)) for all x,y ∈ x. thus, all conditions of theorem 2.1 hold. so by this theorem, t has a unique fixed point in x. note that z = 2 is the unique fixed point of t. now we give the open question, which we mentioned in the previous section, as follows: © agt, upv, 2022 appl. gen. topol. 23, no. 1 161 a. bera, l. k. dey, s. som, h. garai and w. sintunavarat question 2.3. does there exist a fixed point free self mapping t which satisfies the boyd-wong contractive condition in an f-complete f-metric space? 3. applications of boyd-wong contractions the aims of this section is to give applications of the fixed result for boydwong contractions in the previous section. 3.1. application to a second-order ivp. in this part, we apply our result to the following initial value problem of the second-order differential equation: (3.1)   d2x d2t + ω2x = g(t,x(t)) x(0) = a,x′(0) = b, where x ∈ c([0,t],r) is an unknown function, ω( 6= 0),a,b ∈ r and g is a continuous function from [0,t] ×r+ into r. the above differential equation plays a crucial role in different engineering problems of activation of a spring governed by an exterior force. it can be easily shown that the given differential equation (3.1) is equivalent to the integral equation x(t) = ∫ t 0 g(t,u)g(u,x(u))du + a cos(wt) + b sin(wt), t ∈ [0,t], where g(t,u) is the green’s function defined by g(x,t) = 1 ω sin(ω(t−u))h(t−u), where h is the heaviside unit function. we like to study the existence of solution(s) of the differential equation (3.1) (studying the above equivalent integral equation) using our obtained result (theorem 2.1). for this, we need to consider an underlying f-metric space as (x,d), where x is the set of all real-valued continuous functions defined on [0,t], and d is defined by (3.2) d(x,y) = ‖x−y‖∞ = max t∈[0,t] |x(t) −y(t)| for all x,y ∈ x. then clearly (x,d) is an f-metric space with f(t) = ln t and α = 0. now we have the following theorem. theorem 3.1. consider the following differential equation (3.1) under the assumptions: (1) g is a continuous function; (2) there exists a nondecreasing function ψ : [0,∞) → [0,∞) such that ψ is upper semi-continuous from right and ψ(t) < t for all t > 0, and |g(t,r) −g(t,s)| ≤ ω2ψ(|r −s|) for all t ∈ [0,t] and r,s ∈ r. © agt, upv, 2022 appl. gen. topol. 23, no. 1 162 boyd-wong contractions in f-metric spaces and applications then the differential equation (3.1) has a unique solution in x. proof. consider the f-metric space (x,d) as in (3.2). then by assumption (2), it is clear that f(t) > f(ψ(t)) + α for all t ∈ (0,∞). let us now define a mapping t : x → x for each x ∈ x by (tx)(t) = a cos(wt) + b sin(wt) + ∫ t 0 g(t,u)g(u,x(u))du for all t ∈ [0,t]. then the existence of fixed point(s) of the mapping t is equivalent to the existence of the solution(s) of the ivp (3.1). now, for each x,y ∈ x and each t ∈ [0,t], by applying the conditions (1) and (2), we have |(tx)(t) − (ty)(t)| = |a cos(wt) + b sin(wt) + ∫ t 0 g(t,u)g(u,x(u))du −a cos(wt) − b sin(wt) − ∫ t 0 g(t,u)g(u,y(u))du| = ∣∣∣∣ ∫ t 0 g(t,u)g(u,x(u))du− ∫ t 0 g(t,u)g(u,y(u))du ∣∣∣∣ ≤ ∫ t 0 g(t,u)|g(u,x(u)) −g(u,y(u))|du ≤ ∫ t 0 g(t,u)ω2ψ(|x(u) −y(u)|)du ≤ ω2ψ(‖x−y‖∞) sup t∈[0,t] ∫ t 0 g(t,u)du = ω2ψ(‖x(u) −y(u)‖∞) sup t∈[0,t] ∫ t 0 1 ω sin(ω(t−u))du ≤ ψ(‖x−y‖∞). therefore, we get ‖(tx)(t) − (ty)(t)‖∞ ≤ ψ(‖x−y‖∞). this yields that d(tx,ty) ≤ ψ(d(x,y)). by theorem 2.1, t has a unique fixed point, say x. therefore, x is the unique solution of the second-order differential equation (3.1) in x. � 3.2. application to caputo fractional differential equations. in this part, we discuss on caputo fractional differential equation and apply our result to this equation. before going further, we first recall some basic definitions of fractional derivatives. we first start with the definition of fractional integral operators as follows. © agt, upv, 2022 appl. gen. topol. 23, no. 1 163 a. bera, l. k. dey, s. som, h. garai and w. sintunavarat definition 3.2. • the fractional integral operator of order q ∈ (0,∞) (denoted by iq0 ) is defined as follows: i q 0f(t) = 1 γ(q) ∫ t 0 f(u) (t−u)1−q du.(3.3) • the riemann-liouville fractional derivative of order q > 0 is defined as follows: d q 0f(t) = { 1 γ(m−q) dm dtm ∫ t 0 f(u) (t−u)q−m+1 du, if m− 1 < q < m; dmf(t) dtm , if q = m, where m is a positive integer and m− 1 < q < m. • the caputo fractional derivative of order q > 0 with m−1 < q < m is defined as follows: d q 0f(t) = { 1 γ(m−q) dm dtm ∫ t 0 f(m)(u) (t−u)q−m+1 du, if m− 1 < q < m; dmf(t) dtm , if q = m. now, we recall the following lemma due to [18], which will be needed for the application. lemma 3.3 ([18]). for q > 0, the homogeneous fractional differential equation d q 0g(t) = 0 has a solution g(t) = c1 + c2t + · · · + cntn−1, where ci ∈ r for i = 1, 2, 3, . . . ,n and n = [q] + 1. for more informations concerning the fractional calculus, one can see [12, 13, 14, 22] and the references therein. now, we consider the boundary value problem (bvp) as follows: (3.4)   cd q 0x(t) −g(t,x(t)) = 0, 0 ≤ t ≤ 1, 1 < q ≤ 2 x′(0) = 0,x(0) −βx(1) = ∫ r 0 h(u,x(u))du,r ∈ (0, 1),β 6= 1, where x ∈ c([0, 1],r) is an unknown function and g,h : [0, 1] × r+ → r are continuous functions. using lemma 3.3 and the above bvp, we get x(t) = i q 0g(t,x(t)) + c1 + c2t. © agt, upv, 2022 appl. gen. topol. 23, no. 1 164 boyd-wong contractions in f-metric spaces and applications using the boundary conditions, we have x(t) = i q 0g(t,x(t)) + β 1 −β i q 0g(1,x(1)) + 1 1 −β ∫ r 0 h(u,x(u))du = 1 γ(q) ∫ t 0 g(u,x(u)) (t−u)1−q du + β 1 −β 1 γ(q) ∫ 1 0 g(u,x(u)) (1 −u)1−q du + 1 1 −β ∫ r 0 h(u,x(u))du = 1 γ(q) [∫ t 0 g(u,x(u)) (t−u)1−q du + β 1 −β ∫ 1 0 g(u,x(u)) (1 −u)1−q du ] + 1 1 −β ∫ r 0 h(u,x(u))du = 1 γ(q) ∫ 1 0 g(t,u)g(u,x(u))du + 1 1 −β ∫ r 0 h(u,x(u))du, where g(t,u) = { 1 (t−u)1−q + β 1−β 1 (1−u)1−q , 0 ≤ u ≤ t ≤ 1; β 1−β 1 (1−u)1−q , 0 ≤ t ≤ u ≤ 1. thus, we see that the bvp (3.4) is equivalent to the integral equation (3.5) x(t) = 1 γ(q) ∫ 1 0 g(t,u)g(u,x(u))du + 1 1 −β ∫ r 0 h(u,x(u))du. now we are in a position to present a result concerning the existence of a solution of the above bvp. theorem 3.4. consider the bvp (3.4) under the following assumptions: (1) g,h are continuous functions; (2) there exists a nondecreasing function ψ : [0,∞) → [0,∞) such that ψ is upper semi-continuous mapping from right and ψ(t) < t for all t > 0 max{|g(t,a) −g(t,b)|, |h(t,a) −h(t,b)|}≤ kψ(|a− b|) for all t ∈ [0, 1] and a,b ∈ r, where k := γ(q)−βγ(q) 1+γ(q)−β . then the equation (3.4) has a unique solution in x. proof. consider the f-metric space (x,d) considered in (3.2). then by assumption (2), it is clear that f(t) > f(ψ(t)) + α for all t ∈ (0,∞). note that α is zero in this case. let us now define a mapping t : x → x for each x ∈ x by (tx)(t) = 1 γ(q) ∫ 1 0 g(t,u)g(u,x(u))du + 1 1 −β ∫ r 0 h(u,x(u))du for all t ∈ [0, 1]. then a point x ∈ x is a solution of the bvp (3.4) if and only if x is a fixed point of t . © agt, upv, 2022 appl. gen. topol. 23, no. 1 165 a. bera, l. k. dey, s. som, h. garai and w. sintunavarat let x,y ∈ c[0, 1] and t ∈ [0, 1]. then we have |tx(t) −ty(t)| = ∣∣∣ 1 γ(q) ∫ 1 0 g(t,u)g(u,y(u))du + 1 1 −β ∫ r 0 h(u,y(u))du − 1 γ(q) ∫ 1 0 g(t,u)g(u,y(u))du + 1 1 −β ∫ r 0 h(u,y(u))du ∣∣∣ = 1 γ(q) ∫ 1 0 g(t,u)|g(u,x(u)) −g(u,y(u))|du + 1 1 −β ∫ r 0 |h(u,x(u)) −h(u,y(u))|du ≤ 1 γ(q) ∫ 1 0 g(t,u)kψ(|x−y|)du + 1 1 −β ∫ r 0 kψ(|x−y|)du ≤ kψ(‖x−y‖∞) sup t∈[0,1] { 1 γ(q) ∫ 1 0 g(t,u)du + r 1 −β } ≤ kψ(‖x−y‖∞) ( 1 γ(q) + 1 1 −β ) ≤ ψ(‖x−y‖∞) and so ‖tx(t) −ty(t)‖∞ ≤ ψ(‖x−y‖∞). then d(tx,ty) ≤ ψ(d(x,y)) for all x,y ∈ x. hence, from theorem 2.1, t has a unique fixed point, and hence the caputo fractional differential equation (3.4) has a unique solution. � acknowledgements. the research is funded by the ministry of human resource and development, government of india and by the council of scientific and industrial research (csir), government of india under the grant number: 25(0285)/18/emr-ii. this project is funded by national research council of thailand (nrct) n41a640092. references [1] i. altun, m. aslantas and h. sahin, kw-type nonlinear contractions and their best proximity points, numerical functional analysis and optimization 42, no. 8 (2021), 935–954. 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[26] o. yamaod and w. sintunavarat, discussion of hybrid js-contractions in b-metric spaces with applications to the existence of solution for nonlinear integral equations, fixed point theory 22, no. 2 (2021), 899–912. © agt, upv, 2022 appl. gen. topol. 23, no. 1 167 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 61-72 a notion of continuity in discrete spaces and applications valerio capraro∗ abstract we propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of a-theory for locally finite connected graphs. we use this notion of continuity to derive an analogue in z2 of the jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the ℓp-distortion of a metric space that has been recently proved by pierre-nicolas jolissaint and alain valette for finite connected graphs. 2010 msc: primary 52a01; secondary 46l36. keywords: a-homotopy theory, ℓp-distortion, digital jordan curve theorem. 1. introduction, main results and motivations the a-theory is a homotopy theory for locally finite connected graphs that has been developed by barcelo et alii in a recent series of three papers [4],[5],and [6]. this theory is based on ideas that go back to the work of atkin[2],[3] indeed, the letter a is in honour of atkin and that were already re-explored in [12]. this theory is based on a notion of continuous path that makes sense for all locally finite metric spaces1. ∗supported by swiss snf sinergia project crsi22-130435. 1a metric space is called locally finite if every bounded set is finite. 62 v. capraro definition 1.1. let (x, d) be a locally finite metric space. given x ∈ x, denote by dn1(x) the smallest closed ball with the center x which contains at least two points. a finite sequence of points in x, say x0, x1, . . . , xn−1, xn, is called a continuous path if xi ∈ dn1(xi−1) and xi−1 ∈ dn1(xi) for all i = 1 . . . n a locally finite metric space is called path-connected if any pair of points can be joined by a continuous path. as we will show in the next sections, the main interest of this definition is that it allows, on one hand, to bring down results from topology of manifolds to locally finite spaces (as the jordan curve theorem); on the other hand, it allows to bring up results from graph theory to locally finite metric spaces (as the p.n.jolissaint-valette inequality). we now state our main results. for the exact definitions, we refer the reader to the next sections. theorem 1.2. let γ be a simple circuit in z2. then z2 \ γ has two path connected components, one finite and one infinite, and γ is the boundary of each of them. the reason behind the choice of this application is that the jordan curve theorem in z2 is of interest in digital geometry, a branch of theoretical computer science that studies, roughly speaking, the geometry of the screen of a computer. the basic idea is that the screen of a computer is modeled by the grid z2, whose points are pixels, and one has to turn on some pixels in order to form an image. in this context, the importance of a jordan curve theorem is clear. for an introduction to digital geometry, see the beautiful introduction and sec. 1.2 of melin’s ph.d. thesis[15]; references on digital version of the jordan curve theorem include, for instance, [10],[16],[7] and references therein. our proposal of a jordan curve theorem is different from those ones, since it is based on a new definition of a simple curve and on a different notion of connectivity. the same notion of connectivity has been considered in [11] and [17], where also versions of the jordan curve theorem have been presented. in particular, in [11] a jordan curve theorem has been proved for the so-called strict curves. we will see that every strict curve is also simple and that there are simple curves that are not strict (see remark 2.2). our second main result is the following theorem 1.3. let x be a locally finite metric space such that every pathconnected component xi is finite. the ℓ p-distortion of x has the following lower bound cp(x) ≥ sup i d(xi) 2d(xi) ( |xi| |e(xi)|λ (p) 1 ) 1 p (1.1) this inequality was recently proved for finite connected graphs by pierrenicolas jolissaint and alain valette (see [9], theorem 1). here we propose a a notion of continuity in discrete spaces and applications 63 generalization that holds for the class of locally finite spaces such that each path-connected component is finite. of course, this class contains all finite connected graphs. 2. the jordan curve theorem in z2 the classical jordan curve theorem states that a simple closed curve γ in r 2 separates r2 in two path-connected components, one bounded and one unbounded and γ is the boundary of each of these components. in this section we want to prove an analogous result in z2 with the euclidean distance. one is tempted to define a simple circuit in z2 as a continuous circuit x0x1 . . . xn−1xn such that the xi’s are pairwise distinct for i ∈ {0, 1, . . . , n − 1} and x0 = xn. with this definition the jordan curve theorem would be false: consider the following continuous circuit: (0, 0)(0, −1)(1, −1)(2, −1)(2, 0)(2, 1)(1, 1)(1, 2)(0, 2)(−1, 2)(−1, 1)(−1, 0)(0, 0). this circuit is simple in the previous sense, but it separates the grid z2 in three path-connected components: in some sense, this circuit behaves like the 8-shape curve in r2, which is not simple. a discrete analogue of a simple circuit is, in our opinion, something different. first of all, we need to introduce some notation. let (x, y) ∈ z2, • db1(x, y) denotes the set {(x + 1, y), (x − 1, y), (x, y − 1), (x, y + 1)}, • db2(x, y) denotes the set {(x−1, y−1), (x−1, y+1), (x+1, y+1), (x+ 1, y − 1)}. definition 2.1. a simple circuit in z2 is a continuous path x0x1 . . . xn−1xn such that • the xi’s are pairwise distinct for i ∈ {0, 1, . . . , n − 1} and x0 = xn. • whenever xi ∈ db2(xj), for j > i, then j = i + 2 and xi+1 ∈ db1(xi) ∩ db1(xj) remark 2.2. a jordan curve theorem in z2 with the same notion of connectivity as ours has been proved in [11] for the so-called strict circuits. they are circuits γ = x0x1 . . . xn−1xn such that |db2(xi)| = 2, for all i. consequently, every strict circuit is also simple. the converse is not true, since the circuit γ = (1, 0)(1, 1)(0, 1)(−1, 1)(−1, 0)(−1, −1)(0, −1)(1, −1)(1, 0) is simple but not strict, since |db2(1, 0)| = 3. for simplicity, we divide the proof of the jordan curve theorem in z2 in two parts: in theorem 2.3, we prove that z2\γ has two path-connected components, one finite and one infinite; in proposition 2.4, after defining a good notion of boundary, we prove that γ is the boundary of each of these path-connected components. 64 v. capraro theorem 2.3. let γ be a simple circuit in z2 not containing squares2. then z 2 \ γ has two path connected components, one finite and one infinite. proof. embed canonically z2 into r2 and construct the following subset γ̃ in r 2: • γ̃ contains all points contained by γ. • if two points of γ are adjacent vertices of a square, then γ̃ contains the segment connecting these two points. it is easy to see that γ̃ can be realized as a closed curve in r2 which is simple in the standard sense. indeed it is made by sides of squares, without repetitions. by the classical jordan curve theorem, let int(γ̃) and ext(γ̃) be the two path connected components of r2 \ γ̃. define int(γ) = int(γ̃) ∩ z2 and ext(γ) = ext(γ̃) ∩ z2 and let us prove that these are exactly the two path connected components of z2 \ γ. of course, it is enough to show that both int(γ) and ext(γ) are path-connected, since the sets int(γ), ext(γ), γ form a partition of z2. so, let us start proving that int(γ) is path connected. since int(γ̃) is bounded, we need first to prove that int(γ) is not empty. let (x0, y0) ∈ int(γ̃). by continuity, we may suppose that (x0, y0) is in the interior of some square with vertices in z2. let (x1, y1), (x1 + 1, y1), (x1 + 1, y1 + 1), (x1, y1 + 1) be such vertices. observe that at least one of these points must belong outside of γ, since γ does not contain squares. this point has to belong also in int(γ̃), since the segment line connecting this point to (x0, y0) does not intersect γ̃. therefore, we have found a point in int(γ). now we prove that int(γ) is pathconnected. let (w0, z0), (w1, z1) ∈ int(γ) and let δ̃ be a continuous path in int(γ̃) connecting them. let ii, i = 0, . . . n, be a covering of the interval [0, 1], made of intervals ii = [ai, bi], such that • a0 = 0 and bn = 1, • ai < ai+1, for all i, • ai = bi−1, for all i = 1, . . . n, • for all i = 0 . . . , n, γ̃|ii belongs to some closed squares with vertices in z2, • for all i = 0, . . . , n both γ̃(ai) and γ̃(bi) belongs to the side of some square with vertices in z2. it is easy to construct explicitly such a covering, making use of continuity of γ̃ and compactness of the interval [0, 1]. now, lift δ̃ to a discrete path {(ck, dk)} as follows: • of course, define (c0, d0) = δ̃(0) = (w0, z0), • if δ̃( 1 n ) ∈ z2, we have two sub-cases: 2a square in z2 is just a set of four points of the shape (x0, y0), (x0 + 1, y0), (x0 + 1, y0 + 1), (x0, y0 + 1). this hypothesis has an explanation in terms of a-theory. in this theory, squares are homotopic equivalent to one point and therefore they do not contribute in disconnecting the grid z2. a notion of continuity in discrete spaces and applications 65 – if δ̃( 1 n ) is adjacent3 to (w0, z0), define (c1, d1) = δ̃( 1 n ), – if if δ̃( 1 n ) is opposite to (w0, z0), first define (c2, d2) = δ̃( 1 n ) and then observe that the two points of db1(c0, d0) ∩ db1(c2, d2) cannot belong both to γ, since γ is simple. let, (c1, d1) be the one that does not belong to γ. to see that it is the right choice, it suffices to observe that (c1, d1) /∈ ext(γ̃). to see this, just observe that the segment line connecting (c1, d1) to (c2, d2) does not intersect γ̃ and therefore (c1, d1) /∈ ext(γ̃). now suppose that δ̃( 1 n ) /∈ z2. since δ̃( 1 n ) ∈ int(γ̃), in particular δ̃( 1 n ) /∈ γ̃. it follows that there is at least one extremal vertex of the (unique) side of z2 containing δ̃( 1 n ) which does not belong to γ (otherwise, by construction of γ̃, we would have δ̃( 1 n ) ∈ γ̃). this vertex, now denoted by (c2, d2) has to belong to int(γ), since the segment line connecting (c2, d2) to δ̃( 1 n ) does not intersect γ̃. now, if (c2, d2) is adjacent to (c0, d0), we can just define (c1, d1) := (c2, d2); otherwise, observe that one of the two points in db1(c0, d0) ∩ db1(c2, d2) has to belong to int(γ) and we can pick (c1, d1) to be one of them. • and so on, for all i up to n. in a similar way one shows that ext(γ) is path-connected. � now, in order to have a complete analogue of the classical jordan curve theorem we need to prove a discrete analogue of the fact that γ is the boundary of each of the path-connected components of r2 \ γ. in order to do that, we use a property that in the classical setting is implied by injectivity. in order to describe this property, let γ be a simple circuit in r2 in classical sense. let int(γ) be the bounded path-connected component of r2 \ γ. for any point (x0, y0) ∈ int(γ), define four points as follows • (x+0 , y0) is the first point where the horizontal half-line x ≥ x0, y = y0, intersects γ, • (x−0 , y0) is the first point where the horizontal half-line x ≤ x0, y = y0, intersects γ, • (x0, y + 0 ) is the first point where the vertical half-line x = x0, y ≥ y0, intersects γ, • (x0, y − 0 ) is the first point where the vertical half-line x = x0, y ≤ y0, intersects γ. making this procedure for each point belonging to the path-connected component containing (x0, y0), we get the whole γ. observe that this would have been false if γ were not simple. our definition of simplicity is exactly the one that makes this procedure working in our discrete world of z2. indeed, if now 3recall that we are working inside a square and therefore adjacent vertices are exactly the extremal point of a side of the square and opposite vertices are the extremal points of a diagonal of the square. 66 v. capraro γ is a simple circuit in z2, the previous construction gives a set which is in general smaller than γ, because of angles, but, if γ is simple, it can be completed without ambiguity. formally, given a point (x0, y0) ∈ int(γ), construct four points as before and denote by a the set of points obtained by making this procedure for all points belonging to the path-connected component containing (x0, y0). now, if (x1, y1), (x2, y2) ∈ a are opposite vertices of some square, our hypothesis that γ is simple and does not contain squares, tells us that there is a unique common adjacent point to both (x1, y1), (x2, y2) belonging to γ. denote by int(γ) the set obtained adding to a all these angle points. by an analogous construction starting from (x0, y0) ∈ ext(γ), we may define ext(γ). the following proposition is now straightforward and concludes our proposal of a discrete analogue of the jordan curve theorem in z2. proposition 2.4. let γ be a simple circuit in z2 that does not contain squares. then int(γ) = ext(γ) = γ proof. just observe that int(γ) and ext(γ) are respectively the boundary of int(γ̃) and ext(γ̃), where γ̃ is constructed as in the proof of theorem 2.3. � 3. p. n. jolissaint-valette’s inequality for finite metric spaces the theory of (approximate) embedding of metric spaces in some other well-understood metric space, as a banach space or a hilbert space, is now a widely explored field of research, after the breakthrough papers of liniallondon-rabinovich[13] and yu[18], that found relations among it, theoretical computer science and k-theory of c∗-algebras. one of the most basic notions in this theory is the notion of ℓp-distortion, which measures how badly a metric space can be embedded in an ℓp-space in a bi-lipschitz way. for the convenience of the reader we recall that a bi-lipschitz embedding of a metric space (x, d), in this context, is a mapping f : x → ℓp such that there are constants c1, c2 such that for all x, y ∈ x one has c1d(x, y) ≤ dp(f(x), f(y)) ≤ c2d(x, y) where dp stands for the ℓ p-distance. it is clear that f is injective and so we can consider f −1 : f(x) → x. therefore, the following notation makes sense, ||f ||lip = sup x 6=y dp(f(x), f(y)) d(x, y) and ||f −1||lip = sup x 6=y d(x, y) dp(f(x), f(y)) the product ||f ||lip||f −1||lip is called distortion of f and denoted by dist(f). a notion of continuity in discrete spaces and applications 67 definition 3.1. the ℓp-distortion of a metric space x is the following number cp(x) = inf {dist(f) : f is a bi-lipschitz embedding}(3.1) in [9] (theorem 1 and proposition 3), pierre-nicolas jolissaint and alain valette proved that for finite graphs the following inequality holds: cp(x) ≥ d(x) 2 ( |x| |e|λ (p) 1 ) 1 p (3.2) where e denotes the edge set and d(x) = max α∈sym(x) min x∈x d(x, α(x))(3.3) we want to extend this inequality to at least all finite metric spaces. let (x, d) be a locally finite metric space and let x1, . . . xn be the partition of x in path-connected components. the basic idea is clearly to apply p.n.jolissaintvalette’s inequality on each of them, but unfortunately this application is not straightforward, since a path-connected component might not look like a graph (think, for instance, of the space [−n, n]2 \ {(0, 0)} ⊆ z2 equipped with the metric induced by the standard embedding into r2). so we have to be a bit careful to apply p.n.jolissaint-valette’s argument. remark 3.2. since the ℓp-distortion does not depend on rescaling the metric and since we are going to work on each path-connected component separately, we can suppose that each xi is in normal form 4. since we are going to work on a fixed path-connected component, let us simplify the notation assuming directly that x is finite path-connected metric space in normal form. at the end of this section it will be easy to put together all path-connected components. let x, y ∈ x, x 6= y and let x0x1 . . . xn−1xn be a continuous path joining x and y of minimal length n. denote by s the floor of d(x, y), i.e. s is the greatest positive integer smaller than or equal d(x, y). notice that s ≥ 1, since x is in normal form. denote by p(x, y) the set of coverings p = {p1, . . . , ps} of the set {0, 1, . . . , n} such that5 • if a ∈ pi and b ∈ pi+1, then a ≤ b, • the greatest element of pi is equal to the smallest element of pi+1. we denote by p−i and p + i respectively the smallest and the greatest element of pi. 4let (x, d) be a locally finite path-connected metric space. given x ∈ x, let rx be the radius of the smallest closed ball with the center x containing at least two points. it is straightforward to prove that rx does not depend on x and it is called step of the space. we say that a locally finite path-connected metric space is in normal form if the metric is normalized in such a way that the step is 1. 5observe that each pi is a subset of {0, 1, . . . , n}. 68 v. capraro now we introduce the following set e(x) = { (e−, e+) ∈ x × x : ∃x, y ∈ x, p ∈ p(x, y) : e− = p−i , e + = p+i } (3.4) remark 3.3. if x = (v, e) is a finite connected graph equipped with the shortest path metric, then e(x) = e. indeed in this case s = n and so the only coverings belonging to p(x, y) have the shape pi = {xi−1, xi}, where the xi’s are taken along a shortest path joining x and y. we define a metric analogue of the p-spectral gap: for 1 ≤ p < ∞, we set λ (p) 1 = inf { ∑ e∈e(x) |f(e +) − f(e−)|p infα∈r ∑ x∈x |f(x) − α|p } (3.5) where the infimum is taken over all functions f ∈ ℓp(x) which are not constant. lemma 3.4. let (x, d) be a finite path-connected metric space in normal form. (1) for any permutation α ∈ sym(x) and f : x → ℓp(n), one has ∑ x∈x ||f(x) − f(α(x))||pp ≤ 2 p ∑ x∈x ||f(x)||pp (2) for any bi-lipschitz embedding f : x → ℓp(n), there is another bilipschitz embedding g : x → ℓp(n) such that ||f ||lip||f −1||lip = ||g||lip||g −1||lip and ∑ x∈x ||g(x)||pp ≤ 1 λ (p) 1 ∑ e∈e(x) ||g(e+) − g(e−)||pp proof. (1) this proof is absolutely the same as the proof of lemma 1 in [9]. (2) observe that the construction of g made in [8] is purely algebraic and so we can apply it. the inequality just follows from our definition of λ (p) 1 . � now we have to prove a version for metric spaces of a useful lemma already proved by linial and magen for finite connected graph (see [14], claim 3.2). we need to introduce a number that measures how far is the metric space to be a graph. we set d(x) = max e∈e(x) d(e−, e+)(3.6) we have told that d(x) measures how far x is far from being a graph. indeed, the following proposition holds: a notion of continuity in discrete spaces and applications 69 proposition 3.5. the followings are equivalent: (1) d(x) = 1, (2) the distance of any two points x, y ∈ x is exactly the length of the shortest path connecting x, y proof. in one sense the thesis is trivial: if x is a finite graph, then e(x) = e (by remark 3.3) and d(x) = 1, since the space is supposed to be in normal form. conversely, suppose that d(x) = 1, choose two distinct points x, y ∈ x and let x0x1 . . . xn−1xn be a continuous path of minimal length such that x0 = x and xn = y. of, course s ≤ d(x, y) ≤ n. so, it suffices to prove that s = n. in order to do that, suppose that s < n and observe that every p ∈ p(x, y) would contain some pi containing at least three points p−i = xi−1, xi, xi+1 = p + i (we suppose that they are exactly three, since the general case is similar). since x is in normal form, it follows that d(xi−1, xi) = d(xi, xi+1) = 1. now, suppose that d(x) = 1, it follows that also d(xi−1, xi+1) = 1 and then the path x0 . . . xi−1xi+1 . . . xn is still a continuous path connecting x with y, contradicting the minimality of the length of the previous path. � we are now able to prove the generalization of linial-magen’s lemma that we need. lemma 3.6. let (x, d) be a finite path-connected metric space in normal form and f : x → r. then max x 6=y |f(x) − f(y)| d(x, y) ≤ max e∈e(x) |f(e+) − f(e−)| ≤ d(x) max x 6=y |f(x) − f(y)| d(x, y) (3.7) proof. let us prove only the first inequality, since the second will be trivial a posteriori. let x, y ∈ x where the maximum in the left hand side is attained and let x0x1 . . . xn−1xn be a continuous path of minimal length connecting x with y. let p ∈ p(x, y) and let k be an integer such that |f(p+ k ) − f(p− k )| ≥ |f(p+i ) − f(p − i )| for all i. it follows |f(p+ k ) − f(p− k )| = s|f(p+ k ) − f(p− k )| s ≥ ∑s i=1 |f(p + i ) − f(p− i )| s ≥ now we use the fact that the covering p is made exactly by s sets. it follows that x and y belong to the union of the pi’s and we can use the triangle inequality and obtain ≥ |f(x) − f(y)| s ≥ |f(x) − f(y)| d(x, y) � we are now ready to prove the main result of this section. 70 v. capraro theorem 3.7. let (x, d) be a finite path-connected metric space in normal form. for all 1 ≤ p < ∞, one has cp(x) ≥ d(x) 2d(x) ( |x| |e(x)|λ (p) 1 ) 1 p (3.8) where d(x) = max α∈sym(x) min x∈x d(x, α(x))(3.9) proof. let g be a bi-lipschitz embedding which verifies the second condition in lemma 3.4 and let α be a permutation of x without fixed points. let ρ(α) = minx∈x d(x, α(x)). one has 1 ||g−1|| p lip = min x 6=y ||g(x) − g(y)||pp d(x, y)p ≤ min x∈x ||g(x) − g(α(x))||pp d(x, α(x))p ≤ 1 ρ(α)p min x∈x ||g(x) − g(α(x))||pp ≤ 1 ρ(α)p|x| ∑ x∈x ||g(x) − g(α(x))||pp now apply the first statement of lemma 3.4: ≤ 2p ρ(α)p|x| ∑ x∈x ||g(x)||pp now apply the second statement of lemma 3.4: ≤ 2p ρ(α)p|x|λ (p) 1 ∑ e∈e(x) ||g(e+) − g(e−)||pp ≤ 2p|e(x)| ρ(α)p|x|λ (p) 1 max e∈e(x) ||g(e+) − g(e−)||pp now apply lemma 3.6: ≤ 2pd(x)p|e(x)|||g|| p lip ρ(α)p|x|λ (p) 1 now recall the definitions in equation 3.1 and 3.9 and just re-arrange the terms to get the desired inequality. � notice that d(x) ≥ 1 and so the inequality gets worse when the metric space is not a graph. a notion of continuity in discrete spaces and applications 71 corollary 3.8. let x be a metric space such that every path-connected component xi is finite. one has cp(x) ≥ sup i d(xi) 2d(xi) ( |xi| |e(xi)|λ (p) 1 ) 1 p (3.10) proof. just observe that if xi is a partition of x then cp(x) ≥ supi cp(xi). � acknowledgements. we would like to thank pierre-nicolas jolissaint for useful comments on sec. 3 and a referee for helpful comments to improve the exposition of the paper. references [1] r. ayala, e. domı́nguez, a. r. francés and a. quintero, determining the components of the complement of a digital (n − 1)-manifold in zn, discrete geometry for computer imagery; lecture notes on computer science, vol. 11 76/1996 (1996), 163–176. 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(received march 2012 – accepted october 2012) v. capraro (valerio.capraro@unine.ch) university of neuchatel, switzerland. a notion of continuity in discrete spaces and[6pt] applications. by v. capraro @ appl. gen. topol. 23, no. 1 (2022), 225-234 doi:10.4995/agt.2022.15586 © agt, upv, 2022 alexandroff duplicate and βκ andrzej szymanski department of mathematics and statistics, slippery rock university of pennsylvania, u.s.a. (andrzej.szymanski@sru.edu) communicated by s. garćıa-ferreira abstract we discuss spaces and the alexandroff duplicates of those spaces that admit a č-s embedding into the čech-stone compactification of a discrete space. 2020 msc: 54g05; 03e75; 54b99. keywords: extremally disconnected; ultrafilter space. 1. introduction the alexandroff duplicate of a topological space x, denoted by a (x), is the topological space defined as follows. the underlying set consists of two disjoint copies of the set x, say x ×{0} and x ×{1}; for the sake of some technical simplicity, elements and subsets of the first copy are going to be denoted the same as it was for the original set, whereas the elements and subsets of the second copy are going to be denoted by priming the corresponding symbols, i.e., x′, y ′, etc. if y ⊆ x, d (y ) stands for the duplicate of y , i.e., for the set d (y ) = y ∪y ′. the topology of the space a (x) is generated by the subsets of the form d (u) − f ′ = u ∪ (u −f)′, where u is open in x and f ⊆ x is finite, and by {x′} where x ∈ x. thus each point of x′ is an isolated point of the space a (x). the original space x is contained in a (x) as its closed subset and also as its exactly two-to-one retract. the concept itself originated with p.s. alexandroff and p.s. urysohn in 1929 [1] for x = the unit circle; the generalization for arbitrary x, as defined above, is due to r. engelking [12] it has been extensively utilized and studied received 08 may 2021 – accepted 10 december 2021 http://dx.doi.org/10.4995/agt.2022.15586 https://orcid.org/0000-0002-6992-8953 a. szymanski since (cf. [16], [7], [6], [14], [4], and [5]). of the other possible generalizations, the most general generalization of engelking’s one was accomplished by r.p. chandler et al. [8], but it didn’t get as much traction as engelking’s. if x is a crowded (i.e., without isolated points) space, then x′ is a dense subset of isolated points and x is the remainder of a (x). in this regard, the structure of a (x) is akin to that of βκ, so the following question arises: when is a (x) embeddable into the čech-stone compactification of the discrete space x′? specifically, we say that a (x) is č-s embeddable if there exists an embedding h : a (x) → β (x′) such that h (x′) = x′ for each x ∈ x. suppose that a (x) is č-s embeddable. there are some obvious necessary conditions a (x) has to satisfy. to list a few. (0) x has to be a completely regular space; (1) a (x) has to be extremally disconnected (since x′ ⊆ a (x) ⊆ β (x′); (2) any bounded real function on x′ has to have a continuous extension to a (x) (since it has a continuous extension to β (x′)); (3) any two disjoint subsets of x cannot have a common accumulation point. finally, considering a (x) as a subspace of β (x′), one can also observe the following phenomenon: let p ∈ x be a non-isolated point of x. considering p as a free ultrafilter on x′, if a ⊆ x′ is such that a ∈ p, then there is an open neighborhood u of p in x and a finite subset f of x such that u −f ⊆ a. it means that (4) for each non-isolated point p ∈ x, the family {u −f : u is an open neighborhood of p and f is a finite subset of x} is a base for the ultrafilter p. the main goal of this note is to show that any of the properties (1) − (4) stated above constitutes also a sufficient (thus, equivalent) condition for any (completely) regular crowded space and its alexandroff duplicate to be čs embedabble. additional mutual relationships between those properties are discussed in section 1 and section 2. spaces satisfying condition (4), above, are defined and studied in section 1 under the name ultrafilter spaces. they play a crucial role in establishing the aforementioned equivalences. spaces satisfying condition (3), above, are called perfectly disconnected. they were first defined and studied by e. van douwen in [9]. the equivalence between perfectly disconnected space and extremal disconnectedness of alexandroff duplicate, i.e., the equivalence of the conditions (1) and (3), above, was first established by p. bhattacharjee, m. knox, and w. mcgovern, [3]. we refer to r. engelking’s book [13] for all undefined topological notions. 2. topological ultrafilter spaces all considered topological spaces are t1 and let x be a topological space. © agt, upv, 2022 appl. gen. topol. 23, no. 1 226 alexandroff duplicate and βκ for a ⊆ x, ∂ (a) denotes the set of all accumulation points of the set a, i.e., ∂ (a) = {x ∈ x : |u ∩a| ≥ ω for each open neighborhood u of x} an ultrafilter on a non-empty set x is a family ξ of non-empty subsets of x closed under finite intersections and maximal with respect to that property, i.e., if x = a∪b, then a ∈ ξ or b ∈ ξ. we say that the space x is an ultrafilter space at p if the family ξp = {a ⊆ x : u −f ⊆ a, where p ∈ u is open and f ⊆ x is finite} is an ultrafilter on the set x. proposition 2.1. a space x is an ultrafilter space at p if and only if the following condition holds true: (∂) if y and z are disjoint subsets of x, then p /∈ ∂ (y ) ∩∂ (z). proof. (⇒) let x be an ultrafilter space at p and let y and z be disjoint subsets of x. we may assume that they cover x. there exists an open neighborhood u of p and a finite subset f of x such that u −f ⊆ y or u −f ⊆ z; say the former holds true. hence u ∩z ⊆ f, which means that p /∈ θ (z). (⇐) suppose to the contrary that x is not an ultrafilter space at p. thus x = a∪b, where a∩b = ∅, and for each open neighborhood u of p and a finite subset of x, u −f " a and u −f " b. thus both sets (u −f) ∩a and (u −f) ∩b are infinite, i.e., p ∈ ∂ (a) ∩∂ (b); a contradiction. � we say that the space x is an ultrafilter space if x is an ultrafilter space at each point p ∈ x. thus. proposition 2.2. a space x is an ultrafilter space if and only if the following condition holds true (∂∂) if y and z are disjoint subsets of x, then ∂ (y ) ∩∂ (z) = ∅. condition (∂∂) implies: corollary 2.3. let x be an ultrafilter space. then (a) x is hereditarily extremally disconnected and nodec (= each nowhere dense subset of x is closed and, consequently, discrete). (b) (see also van douwen [9]) ∂ (e) = cl (inte) for arbitrary e ⊆ x. proof. part (a) follows immediately from (∂∂). part (b) needs some additional argument. inclusion ∂ (e) ⊇ cl (inte) is obvious. to show the converse inclusion, let x /∈ cl (inte) and let u be an open neighborhood of x disjoint from inte. it means that u ∩e is boundary which, in turn, implies that u ∩e is nowhere dense. hence, by (a), x is not an accumulation point of e. � topological spaces satisfying condition (∂∂) were introduced by e. van douwen under the name perfectly disconnected space. he also gave the following characterization of such spaces. for the sake of completeness, we provide (a different) proof. theorem 2.4 (van douwen [9]). x is a perfectly disconnected space if and only if x is extremally disconnected and each dense subset of x is open, i.e., x is submaximal. © agt, upv, 2022 appl. gen. topol. 23, no. 1 227 a. szymanski proof. if x is perfectly disconnected, then, x is (hereditarily) extremally disconnected. it is submaximal as well for if d is dense in x, then x −d cannot have any accumulation point. thus x −d is closed and discrete and so d is open. let x be extremally disconnected and submaximal and assume, to the contrary, that x does not satisfy the condition (∂∂). so let y , z be disjoint subsets of x such that there is a p ∈ ∂ (y ) ∩ ∂ (z). hence p ∈ cly ∩ clz. set u = int (cl (y )) and v = int (cl (z)). it follows from submaximality that u ∩v = ∅ for no non-empty open subset of x can contain two disjoint dense subsets. the space x, being submaximal, is also nodec. hence y − u and z −v are two disjoint closed discrete subsets of x. hence cl (y ) ∩ cl (z) = [clu ∩ (z −v )] ∪ [clv ∩ cl (y −u)]. so if p ∈ cly ∩ clz, then p ∈ clu ∩ (z −v ) or p ∈ clv ∩ cl (y −u); assume the former holds (similar argument applies when the latter holds true). there exists an open neighborhood w of p such that w ∩ (z −v ) = {p} and w ⊆ clu. hence w ∩ clz = {p}, thus p /∈ ∂ (z); a contradiction. � it is well known that the two aforementioned properties, i.e., that of submaximality and extremal disconnectedness, characterize maximal crowded topological spaces (cf. e. hewitt [15] and m. katĕtov [17]). by the kuratowski-zorn lemma, maximal crowded topology majorizes any given t1 or t2 crowded topology. however examples of maximal crowded topologies which are t3 are unknown with the exception of van douwen’s example from 1993 which happens to be countable (see [9] ). uncountable examples can be obtained by taking a disjoint union of the van douwen’s example. the next characterization of ultrafilter spaces entails a new setting. lemma 2.5. a set w is an open set in the space a (x) if and only if w = [d (u) −f ′] ∪ e′, where u = w ∩ x, f ⊆ u is discrete in u, and e is a subset of x −u. proof. let w ⊆ a (x) be open. by setting u = x ∩ w we have: w = (w ∩d (u)) ∪ [w −d (u)]. set e = w − d (u). since w ∩ d (u) = u ∪ [w ∩u′], we need to show that u′ − w = f ′ for some discrete in u subset f of u. indeed, if x ∈ u = w ∩ x, then there exists an open neighborhood v of x and a finite subset s of x such that d (v ) −s′ ⊆ w . hence v ∩f ⊆ s, which shows that f is discrete in u. the converse is obvious. � theorem 2.6. a space x is an ultrafilter space viz. perfectly disconnected space if and only if a (x) is extremally disconnected. proof. necessity is obvious. to prove sufficiency, pick any open subset w of a (x) and let us show that clw is open in a (x). by lemma 2.5, w = [d (u) −f ′]∪e′, where u is an open subset of x, f ⊆ u is discrete in u, and © agt, upv, 2022 appl. gen. topol. 23, no. 1 228 alexandroff duplicate and βκ e is a subset of x −u. thus clw = cl [d (u) −f ′] ∪cle′. one can easily verify that cl [d (u) −f ′] = cl (u) ∪ u′ − f ′ and that cle′ = e′ ∪ ∂ (e). the perfect disconnectedness of x yields that cle ∩ clu = ∅ and that clu is clopen. subsequently, cl [d (u) −f ′] = d (clu) − (clu −u)′ − f ′ is an open subset of a (x). using part (b) of corollary 2.3, we get: cle′ = e′ ∪ cl (inte) = [ d (cl (inte)) − (cl (inte) − inte)′ ] ∪ (e − cl (inte))′ is an open subset of a (x) too. � the characterization of the extremal disconnectedness of the alexandroff duplicate in terms of perfect disconnectedness (= theorem 2.6) was first established by p. bhattacharjee, m. l. knox, and w. mcgovern, [3], in 2020. the aforementioned van douwen’s example of a regular countable perfectly disconnected space provides an affirmative answer to a problem posed by k. almontashery and l. kalantan, [2] the following corollary recaps the main results of this section, corollary 2.7. for arbitrary t1 crowded space x, the following conditions are pairwise equivalent. (j) x is an ultrafilter space; (jj) x is a perfectly disconnected space; (jjj) x is a maximal crowded space; (ij) a (x) is an extremally disconnected space. remark 2.8. the ultrafilters ξp induced by the topology on an ultrafilters space x may not be uniform, i.e., that any member of ξp has to be of cardinality |x|. however there exists an open subset g of x such that if g is considered as an ultrafilter space, then the ultrafilters ξp are going to be uniform. to see this, let’s recall some (known) definitions. if z is an arbitrary topological space, y ⊆ z, and p ∈ z is its non-isolated point, then: ∆ (p,z) = min{|u| : u is an open neighborhood of p}− the dispersion character of z at p; and ∆ (z) = min{∆ (p,z) : p ∈ z and p is non-isolated}− the dispersion character of z. thus any open subset g of x such that |g| = ∆ (x) will yield an ultrafilter space with all the ultrafilters ξp to be uniform. there may be a variety of maximal filters of open sets, the question arises whether all types of maximal filters of open sets on an ultrafilter space x are ultrafilters on the set x. it turns out, it depends on the separation axioms of x. a space x is said to be a strong ultrafilter space if any maximal filter of open sets on the space x generates an ultrafilter on the set x. proposition 2.9. if x is normal ultrafilter space, then x is a strong ultrafilter space. proof. let x be an ultrafilter space with an underlying set being a cardinal number κ. let ξ be a maximal filter of open subsets of x, and let us show that ξ is an ultrafilter on κ. © agt, upv, 2022 appl. gen. topol. 23, no. 1 229 a. szymanski suppose to the contrary that κ = a∪b, where a,b are disjoint and u∩a 6= ∅ 6= u ∩b for each u ∈ ξ. by proposition 2.2, int (cla) ∩ int (clb) = ∅. let us assume int (cla) ∈ ξ. since b ∩ int (cla) is a nowhere dense subset of x, it is closed and nowhere dense by corollary 2.3. hence int (cla) − b ∈ ξ and (int (cla) −b) ∩ b = ∅, which contradicts the initial assumption. by the similar argument, int (clb) /∈ ξ. by maximality of ξ, there exists v ∈ ξ such that v ∩ int (cla) = ∅ = v ∩ int (clb). hence u ∩ (a− int (cla)) 6= ∅ 6= u ∩ (b − int (clb)) for each u ∈ ξ. the sets e = a − int (cla) and f = b − int (clb) are nowhere dense, thus closed, and disjoint. by normality of x, there are disjoint open subsets w1, w2 of x such that e ⊆ w1 and f ⊆ w2. but then w1, w2 ∈ ξ; a contradiction. � 3. ultrafilter spaces vs. spaces of ultrafilters let κ be an infinite cardinal number. βκ stands, as usual, for the set of all ultrafilters (free or principal) on the set κ endowed with the topology generated by the sets â = {ξ ∈ βκ : a ∈ ξ}, where ∅ 6= a ⊆ κ. in what follows, all considered ultrafilters spaces are assumed to be crowded. let x be an ultrafilter space with the underlying set κ, where κ is a cardinal number. for each α ∈ κ, let ξα = {u −f : u is an open neighborhood of α and f is a finite subset of κ}. thus ξα ∈ βκ−κ for each α ∈ κ. we can define a function ϕ : x → βκ−κ by setting: ϕ (α) = ξα for each α ∈ κ = x. thus ϕ (x) can be thought off as a pointless copy of x. let e (x) = κ∪ϕ (κ) ⊆ βκ. in what follows, both ϕ (x) and e (x) are considered to be subspaces of the space βκ. proposition 3.1. (a) the function ϕ : x → βκ−κ is continuous; (b) ϕ is one-to-one if and only if x is t2; (c) ϕ is an embedding if and only if x is t3. proof. (a) let v be an open set in βκ − κ and let α ∈ κ = x be such that ξα ∈ v . there exists a ⊆ κ such that ξα ∈ {ξ ∈ βκ−κ : a ∈ ξ} ⊆ v . since a ∈ ξα, there exists an open set u of x such that α ∈ u ⊆ a. thus α ∈ u ⊆ ϕ−1 (v ). (b) assume that ϕ is one-to-one and let α 6= β ∈ κ = x. since ξα 6= ξβ, there exist disjoint sets a and b such that ξα ∈ {ξ ∈ βκ−κ : a ∈ ξ} and ξβ ∈ {ξ ∈ βκ−κ : b ∈ ξ}. hence there exist open sets u, v in x such that α ∈ u ⊆ a and β ∈ v ⊆ b. thus α ∈ u and β ∈ v and u ∩ v = ∅. the converse implication is obvious. (c) assume that ϕ is an embedding and let α ∈ u ⊆ x, where u is open. thus ξα ∈ ϕ (u) and since ϕ (u) is open, there exists a ⊆ κ such that ξα ∈ {ξ ∈ βκ−κ : a ∈ ξ} ⊆ ϕ (u). since the set {ξ ∈ βκ−κ : a ∈ ξ} is clopen and since a ∈ ξα, there exists an open set v of x such that α ∈ v ⊆ a. thus α ∈ v ⊆ clv ⊆ ϕ−1 (u) = u. conversely, let u be open in x and let ξα ∈ ϕ (u). there exists an open set v in x such that α ∈ v ⊆ clv ⊆ u. © agt, upv, 2022 appl. gen. topol. 23, no. 1 230 alexandroff duplicate and βκ hence clv is clopen and such that ξα ∈{ξ ∈ βκ−κ : clv ∈ ξ}⊆ ϕ (u). thus ξα ∈ intϕ (u). � theorem 3.2. let x be an ultrafilter space with an underlying set being a cardinal number κ. if x is regular, then e (x) is homeomorphic to a (x). proof. let h : a (x) → e (x) be defined as follows: h (α′) = α for each α′ ∈ x′ and h (α) = ξα for each α ∈ x. let’s show that h is a homeomorphism. clearly, h is a bijection. to see that h is continuous, let a ⊆ κ and α ∈ x satisfy ξα ∈ â (i.e., h (α) = ξα ∈ â ). there exists an open neighborhood u of α and a finite set f ⊆ x such that u − f ⊆ a. it is obvious that h (d (u) −f ′) ⊆ â∩e (x). to see that h−1 is continuous, let w ⊆ a (x) be open and let ξα ∈ h (w). since h and ϕ coincide on x, h|x is a homeomorphism (cf. proposition 3.1 (c) ). there exists a ⊆ κ such that ξα ∈ â∩h (x) ⊆ h (w). by lemma 2.5, w = [d (u) −h′]∪g′, where h is a discrete and closed (in u) subset of u and g ⊆ x−u is arbitrary. there exists an open set v ⊆ x and a finite set f ⊆ x such that α ∈ v , v ∩h = ∅, and u −f ⊆ a. hence ξα ∈ û ⊆ h (w). � corollary 3.3. x is a regular ultrafilter space if and only if a (x) is č-s embeddable. let us consider a function ψ : e (x) → ϕ (κ) ⊆ βκ−κ, given by: ψ (α) = ϕ (α) = ξα for each α ∈ κ, and ψ (ξα) = ξα for each α ∈ κ. clearly, ψ is continuous (see also proposition 3.1 (a)): ψ−1 (u∗) = û ∩e (x) for each open subset u of x. thus ψ is an exactly two-to-one retraction onto ϕ (κ). there exists a continuous extension ψ̂ of ψ to βκ into clϕ (κ) ⊆ βκ − κ. hence ψ̂ is a retraction from the space βκ onto its subspace clϕ (κ). let us note the following. lemma 3.4. let x be a regular ultrafilter space and a ⊆ κ, â∩ clϕ (κ) = ∅ if and only if a is a discrete and closed (in the topology on x). consequently, if ξ ∈ a∗, then ψ̂ (ξ) ∈ clϕ (a) −ϕ (κ). proof. since ϕ (κ) consists only of those ultrafilters that contain an open subset of x, â∩clϕ (κ) = ∅ means that a (considered as a subspace of x) is boundary and so it is an infinite closed and nowhere dense subset of x. since ξ ∈ â = cla (when considered in βκ), ψ̂ (ξ) ∈ clψ̂ (a) = clϕ (a). since a is closed in x, ϕ (a) is closed in ϕ (κ) (by proposition 3.1). hence ψ̂ (ξ) ∈ clϕ (a) −ϕ (κ). � theorem 3.5. let x be a regular ultrafilter space., the retraction ψ̂ is oneto-one on the subspace βκ− clϕ (κ) if and only if x is a normal space. proof. assume that x is normal and let ξ 6= ζ ∈ βκ−clϕ (κ). the three cases when at least one of the two points ξ, ζ belongs to κ are obvious. so assume that ξ, ζ ∈ βκ−κ. there exist disjoint sets a,b ⊆ κ such that a ∈ ξ, b ∈ ζ, and â ∩ clϕ (κ) = ∅ = b̂ ∩ clϕ (κ). by lemma 3.4, ψ̂ (ξ) ∈ clϕ (a) − ϕ (κ) © agt, upv, 2022 appl. gen. topol. 23, no. 1 231 a. szymanski and ψ̂ (ς) ∈ clϕ (b) −ϕ (κ). since x is normal, clϕ (a) ∩ clϕ (b) = ∅. hence ψ̂ (ξ) 6= ψ̂ (ζ). assume that the retraction ψ is a bijection outside of ϕ (κ). to show that then x is a normal space (given that x is perfectly disconnected space), it suffices to show that any two disjoint discrete subsets of x can be separated. for suppose that a,b ⊆ x are disjoint and discrete but they cannot be separated. it follows that clϕ (a) ∩ clϕ (b) 6= ∅; let p ∈ clϕ (a) ∩ clϕ (b). by lemma 3.4, ψ restricted to â is a homeomorphism between â and clϕ (a). similarly, ψ restricted to b̂ is a homeomorphism between b̂ and clϕ (b). there exist ξ ∈ â and ζ ∈ b̂ such that ψ̂ (ξ) = p = ψ̂ (ζ). since â∩ b̂ = ∅ and ξ,ζ /∈ clϕ (κ), the retraction ψ is not a bijection outside of clϕ (κ); a contradiction. � remark 3.6. the retraction ψ̂ in theorem 3.5 is an instance of a ≤ two-to-one continuous maps on βκ onto a compact space of density κ. an extensive and deep study of such maps was done by e. van douwen in [9]. a. dow, among others, has published several papers on related subjects (cf. [10]). the retraction ψ̂ is also an instance of a one-to-one retraction. a. dow did a thorough and profound study of one-to-one retractions on βω in [11]. let f : x → y be a function. the duplicate of f, d (f), is a function d (f) : a (x) → y defined as follows: d (f) (x) = f (x) and d (f) (x′) = f (x). it is easy to see that the duplicate of a continuous function on x yields a continuous function on a (x), regardless of the separation axioms or other properties of its domain. in particular, x is c/c∗− embedded into a (x). can x′ be c∗− embedded into a (x)? in this regard, we have the following. corollary 3.7. if x′ is c∗− embedded into a (x), then x is an ultrafilter space. conversely, if x is an ultrafilter space, then x′ is c∗− embedded into a (x) provided that x is a regular space. proof. let a and b disjoint subsets of x. without loss of generality, we may also assume that a ∪ b = x. define f : x′ → r setting f (x′) = 0 if x ∈ a and f (x′) = 1 if x ∈ b. if f̃ is a continuous extension of f to a (x), then f̃ (x) = 0 if x ∈ ∂ (a) and f̃ (x) = 1 if x ∈ ∂ (b). hence ∂ (a) ∩ ∂ (b) = ∅, which means, x is perfectly disconnected, viz. x is an ultrafilter space. the converse part follows immediately from theorem 3.2. � proposition 3.8. let x be a regular ultrafilter space with the underlying set κ. then βx is homeomorphic to cl (ϕ (κ)). in particular, cl (ϕ (κ)) is extremally disconnected. proof. let f : ϕ (κ) → r be a continuous bounded function. take the duplicate d (f) of f, i.e., d (f) (ϕ (α)) = f (ϕ (α)) and d (f) (α) = f (ϕ (α)) for each α ∈ κ. since d (f) is continuos on e (x), it has a continuous extension onto βκ since e (x) = κ∪ϕ (κ) ⊆ βκ. since d (f) |ϕ (κ) = f, f has a continuous extension to cl (ϕ (x)). thus cl (ϕ (κ)) = β (ϕ (κ)) = βx since ϕ (κ) and x are © agt, upv, 2022 appl. gen. topol. 23, no. 1 232 alexandroff duplicate and βκ homeomorphic (see proposition 3.1(c)). since the čech-stone compactification of an extremally disconnected space is extremally disconnected, the proof is finished. � corollary 3.9. let x be a regular ultrafilter space. (a) if h ⊆ βx −x is a discrete set of remote points such that x ∩clh = ∅, then y = x ∪h is an ultrafilter space. (b) if x is normal and h ⊆ βx − x is a countable discrete set of remote points such that x ∩ clh = ∅, then y = x ∪h is a strong ultrafilter space. proof. (a) y is a perfectly disconnected space. indeed, let a and b be disjoint subsets of y , and suppose that p ∈ ∂ (a) ∩ ∂ (b). if p ∈ h, then p ∈ ∂ (a∩x)∩∂ (b ∩x). since p is remote, p /∈ cl (a∩x − int (a∩x)) and p /∈ cl (b ∩x − int (x ∩x)). hence p ∈ cl (int (a∩x))∩ cl (int (b ∩x)), which is impossible since y is extremally disconnected. suppose that p ∈ x. then p /∈ clh and therefore p ∈ ∂ (a∩x)∩∂ (b ∩x). this is impossible since x is perfectly normal. 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[13] r. engelking, general topology, sigma series in pure mathematics, vol. 6, berlin, heldermann, 1989. [14] s. garćıa-ferreira and c. yescas-aparicio, families of retractions and families of closed subsets on compact spaces, topology and its applications 301 (2021): 107504. [15] e. hewitt, a problem of set-theoretic topology, duke j. math. 10 (1943), 309–333. [16] i. juhasz and s. mrowka, e-compactness and the alexandroff duplicate, indagationes math. 73 (1970), 26–29. [17] m. katĕtov, on spaces which do not have disjoint dense subspaces, mat sb. 21 (1947), 3–12. © agt, upv, 2022 appl. gen. topol. 23, no. 1 234 @ appl. gen. topol. 22, no. 1 (2021), 67-77doi:10.4995/agt.2021.13446 © agt, upv, 2021 equicontinuous local dendrite maps aymen haj salem a, hawete hattab b,a, tarek rejeiba a,∗ a laboratory of dynamical systems and combinatorics, sfax university, tunisia (hajsalem@hotmail.com, tarekhamzah28@gmail.com) b umm alqura university makkah, saudi arabia. (hshattab@uqu.edu.sa) communicated by f. balibrea abstract let x be a local dendrite, and f : x → x be a map. denote by e(x) the set of endpoints of x. we show that if e(x) is countable, then the following are equivalent: (1) f is equicontinuous; (2) ∞⋂ n=1 f n (x) = r(f); (3) f| ∞⋂ n=1 f n (x) is equicontinuous; (4) f| ∞⋂ n=1 f n (x) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of s 1 ; (5) ω(x, f) = ω(x, f) for all x ∈ x. this result generalizes [17, theorem 5.2], [24, theorem 2] and [11, theorem 2.8]. 2010 msc: 37b05; 37b20; 37b45. keywords: dendrite; equicontinuity; local dendrite; recurrent point. ∗this work is supported by the laboratory of research ”dynamical systems and combinatorics” university of sfax, tunisia received 11 april 2020 – accepted 26 september 2020 http://dx.doi.org/10.4995/agt.2021.13446 a. h. salem, h. hattab and t. rejeiba 1. introduction a map is a continuous function between two topological spaces. a topological dynamical system is a pair (x, f), where x is a compact metric space (d is a metric on x) and f is a map from x to itself. let n be the set of positive integers. let f0 be the identity map of x. define, inductively, fn = f ◦ fn−1 for any non-zero positive integer n. for x ∈ x, {fn(x) : n ∈ n} is called the orbit of x and is denoted by o(x, f). for any x ∈ x, write ω(x, f) = {y : ∃ nk ∈ n, nk → ∞, lim k→∞ fnk(x) = y} called the ω-limit set of x under f, and write ω(x, f) = {y : ∃ xk ∈ x and nk ∈ n, nk → ∞, lim k→∞ xk = x and lim k→∞ fnk(xk) = y}. x is a periodic point if fn(x) = x for some non-zero positive integer n. note that, if n = 1, then x is a fixed pint. also, x is called a recurrent point of f if for any neighborhood u of x and any m ∈ n there exists n > m such that fn(x) ∈ u. note that, x is a recurrent point of f if and only if x ∈ ω(x, f). let fix(f), p(f) and r(f) denote the set of fixed points, periodic points and recurrent points, respectively. we say that the map f is pointwise periodic if p(f) = x. also, f is said to be pointwise recurrent if r(f) = x. a subset a of x is called f-invariant if f(a) is a subset of a. it is called a minimal set of f if it is non-empty, closed, f-invariant and minimal (in the sense of inclusion) for these properties. if x is a minimal set, then f is called a minimal map. f is said to be equicontinuous (with respect to d) if for each ε > 0, there exists 0 < α < ε such that for any non zero-integer n and any x, y ∈ x with d(x, y) < α, one has d(fn(x), fn(y)) < ε. it is interesting to give some characterizations of equicontinuous maps [12, 13, 16, 17, 24]. in [3, theorem 2.1], by means of the orbit map of : x → c(n, x) and the metric df , akin, auslander and berg gave some necessary and sufficient conditions for a map f of a compact metric space (x, d) to be equicontinuous. in [6, proposition 2.2], blanchard, host and maass discussed the topological complexity, and showed that a surjective map f of a compact metric space x is equicontinuous if and only if any finite open cover of x under f has bounded complexity. on one-dimensional spaces, one has some still finer results. in [12], cano proved that if f is an equicontinuous map from the interval i = [0, 1] to itself, then fix(f) is connected, and furthermore, if fix(f) is non-degenerate, then f has no periodic points except fixed points. bruckner and hu (only if) and boyce (if) proved that a map f : i → i is equicontinuous if and only if ∞⋂ n=1 fn(i) = fix(f2); see [9, 10]. this result was also proved by blokh in [8]. valaristos [25] described the characters of equicontinuous circle maps: a map f of the unit circle s1 to itself is equicontinuous if and only if one of the following four statements holds: (1) f is topologically conjugate to a rotation; (2) fix(f) contains exactly two points and fix(f2) = s1; © agt, upv, 2021 appl. gen. topol. 22, no. 1 68 equicontinuous local dendrite maps (3) fix(f) contains exactly one point and fix(f2) = ∞⋂ n=1 fn(s1); (4) fix(f) = ∞⋂ n=1 fn(s1). in [21], sun obtained some necessary and sufficient conditions of equicontinuous σ-maps. when x is a graph, mai proved in [17, theorem 5.2] that the following properties are equivalent: (1) f is equicontinuous; (2) r(f) = ∞⋂ n=1 fn(x); (3) the restriction of f on ∞⋂ n=1 fn(x) is equicontinuous; (4) the restriction of f on ∞⋂ n=1 fn(x) is a periodic homeomorphism, or is topologically conjugate to the irrational rotation of the unit circle s1. in [11, 22, 23], the authors studied equicontinuous dendrite maps. for a dendrite x with countable set of endpoints, in [22, theorem 2.8] it is shown that f is equicontinuous if and only if ω(x, fn) = ω(x, fn) for any x ∈ x and each n ∈ n. for a dendrite x with finite branch points, in [23, theorem 28] it is proved that the following statements are equivalent: (1) f is equicontinuous; (2) ω(x, f) = ω(x, f) for any x ∈ x; (3) p(f) = ∞⋂ n=1 fn(x), and ω(x, f) is a periodic orbit for every x ∈ x and the function ωf : x → ω(x, f) (x ∈ x) is continuous; (4) ω(x, f) is a periodic orbit for any x ∈ t . recently, for general dendrites, in [11, theorem 4.12] it is shown that the following statements are equivalent: (1) f is equicontinuous; (2) ω(x, f) = ω(x, f) for any x ∈ x; (3) ωf is continuous and p(f) = ∞⋂ n=1 fn(x). in this paper we will give some equivalent conditions of equicontinuity for local dendrite maps, whose dynamical behavior is both important and interesting in the study of discrete dynamical systems and continuum theory. our main results are the following: theorem 1.1. let x be a compact metric space and f : x → x be a map. consider the following statement: (1) f is equicontinuous; (2) ω(x, fn) = ω(x, fn) for all x ∈ x and n ∈ n. © agt, upv, 2021 appl. gen. topol. 22, no. 1 69 a. h. salem, h. hattab and t. rejeiba then (1) implies (2) and, if ∞⋂ n=1 fn(x) = p(f), then (2) implies (1). theorem 1.1 generalizes [22, theorem 2.8]. proposition 1.2. every pointwise recurrent local dendrite map is equicontinuous. the following result generalizes [17, theorem 5.2], [24, theorem 2] and [11, theorem 2.8]. theorem 1.3. let f : x → x be a local dendrite map. if e(x) is countable, then the following are equivalent: (1) f is equicontinuous; (2) ∞⋂ n=1 fn(x) = r(f); (3) f| ∞⋂ n=1 fn(x) is equicontinuous and surjective; (4) f| ∞⋂ n=1 fn(x) is either a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of s1; (5) ω(x, f) = ω(x, f) for all x ∈ x. corollary 1.4. under the assumptions of theorem 1.3, if ω(x, f) = ω(x, f) for all x ∈ x, then ω(x, fn) = ω(x, fn) for all x ∈ x and every n ∈ n. recently, several authors have been interested in studying local dendrite maps (for example one can see [1, 2, 4]). 2. preliminaries a compact connected metric space is called a continuum. a peano continuum is a locally connected continuum. an arc is any space homeomorphic to the interval i = [0, 1]. we mean by a simple closed cure every continuum homeomorphic to the circle s1. recall that a graph is a continuum which can be written as the union of finitely many arcs any two of which are either disjoint or intersect only in one or both of their endpoints (i.e., it is a one-dimensional compact connected polyhedron). a tree is a graph which contains no simple closed curves. a dendrite is a peano continuum which contains no simple closed curves. not that every dendrite is uniquely arcwise connected continuum, that is any two distinct points in a dendrite d can be joined by a unique arc [x, y]. we define by (x, y) = [x, y] \ {x, y}. for more properties of dendrites (see [19, chapter x]). let d be a dendrite and x ∈ d, x is called anendpoint of d if d \ {x} is connected. the set of all endpoints of d is denoted by e(d). we say that x is a cut point of d if x ∈ d \ e(d) [19, theorem 10.7, p. 168]. © agt, upv, 2021 appl. gen. topol. 22, no. 1 70 equicontinuous local dendrite maps by a local dendrite, we mean a continuum every point of which has a dendrite neighborhood. each local dendrite is a peano continuum which has a finite number of circles [15]. since every subcontinuum of a dendrite is a dendrite [15, º51, vi, theorem 4], a subcontinuum of a local dendrite is a local dendrite. the simple examples of local dendrites are graphs and dendrites. let x be a local dendrite. a point e ∈ x is called an endpoint of x if it admits a neighborhood u in x such that u is an arc and u \ {e} is connected. the set of endpoints of x is denoted by e(x). a point x ∈ x is called a branch point of x if there exists a closed neighborhood d of x which is a dendrite such that x is a branch point of d (i.e. d \{x} has more than two connected components). we denoted by b(x) the set of branch points of x. by [15, theorem 6,304 and theorem 7, 302], b(x) is at most countable. a local dendrite map is a map from a local dendrite into itself. lemma 2.1 ([1]). let x be a local dendrite and y be a sub-local dendrite of x distinct of x such that any arc j of x joining two distinct points of y is included in y . then for any connected component c of x \ y , c ∩ y is degenerate (i.e. reduced to a point). if s1, s2, . . ., sr are the circles in a local dendrite x, then γ(x) is the intersection of all subgraphs in x containing the union of s′is. therefore, γ(x) is the smallest graph containing all circles of the local dendrite x. define x \ γ(x) = ⋃ i∈a ci where ci are the connected components of x \ γ(x). since b(x) is at most countable, the set a is at most countable. by lemma 2.1, for any i ∈ a, ci ∩ γ(x) is reduced to a point zi. let ak be a subset of a such that, for each i ∈ ak, ci ∩ γ(x) = {zk}. put c k = ⊔ i∈ak ci. since γ(x) contains all circles of x, by [1] and [4, lemma 2.4], we obtain the following lemma. lemma 2.2. under the notation above, the (ck)k are pairwise disjoint subdendrites of x. lemma 2.3. if x is a local dendrite with e(x) is countable then every sublocal dendrite of x has a countable set of endpoints. proof. let y be a sub-local dendrite of x. if e(y ) is uncountable, then there exists a connected component c of y \ γ(y ) such that c is a dendrite with uncountable set of endpoints. since y ∩ γ(x) = γ(y ), y \ γ(y ) = y \ γ(x) ⊂ x \ γ(x). therefore, there exists a connected component l of x \ γ(x) such that c ⊂ l. now we define a one-to-one function f : e(c) → e(l) by: let e ∈ e(c). if e ∈ e(l), then f(e) = e. if e /∈ e(l), let ce be a connected component of l \ c such that ce ∩ c = {e}. we take f(e) ∈ e(ce) \ {e}. it is easy to see that f is a one-to-one function. thus the cardinality of e(c) is less than or equal to the cardinality of e(l). consequently, e(l) is uncountable which implies that e(x) is uncountable, a contradiction. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 71 a. h. salem, h. hattab and t. rejeiba lemma 2.4 ([2, lemma 4.3]). let (x,d) be a local dendrite. thus, for every ε > 0, there exists δ = δ(ε) > 0 such that, for any x and y in x with d(x, y) < δ, the diameter diam([x, y]) < ε. 3. proof of theorem 1.1 to prove theorem 1.1 we will use the following two results. the first one is proved in [11, lemma 3.5]. lemma 3.1. let x be a compact metric space and f : x → x be an equicontinuous map. then ω(x, fn) = ω(x, fn) for all x ∈ x and for all n ∈ n. part (2) of the following result is shown in [17, proposition 2.4], so we prove only part (1). lemma 3.2. let x be a compact metric space and f : x → x be a map. then the following assertions hold: (1) if ω(x, f) = ω(x, f) for all x ∈ x, then ∞⋂ n=1 fn(x) = r(f); (2) if f is equicontinuous, then ∞⋂ n=1 fn(x) = r(f). proof. since x is compact and f(r(f)) = r(f), ⋂ n∈n fn(x) ⊇ r(f). conversely, given x ∈ ⋂ n∈n fn(x), there is a sequence xk ∈ x and x0 ∈ x with xk → x0 and nk → +∞ such that f nk(xk) = x for all k ∈ n. hence x ∈ ω(x0, f) = ω(x0, f). since x ∈ ω(x0, f), there exists a sequence (pk)k ∈ n, pk → ∞ such that f pk(x0) → x. by choosing a subsequence, we can suppose that pk+1 − pk > k for all k ∈ n. then f pk+1−pk (fpk (x0)) = f pk+1(x0) → x. therefore, x ∈ ω(x, f) = ω(x, f) which implies that x ∈ r(f). consequently, ∞⋂ n=0 fn(x) = r(f). � remark that, in the above lemma, if we suppose that f is an onto map, we infer that f is pointwise recurrent that is, r(f) = x. proof of theorem 1.1. by lemma 3.1 we have (1) ⇒ (2). to show that (2) ⇒ (1), assume that ∞⋂ n=1 fn(x) = p(f) and that (2) holds. note that, in [22, lemma 2.4] it is proved that (2) ⇒ (1) if x is a dendrite. we will extend this result for every compact metric space x. since x is compact then ω(x, fn) = ω(x, fn) ⊂ ∞⋂ n=1 fn(x) = p(f) for all x ∈ x and every n ∈ n. if f is not equicontinuous, then there exist x ∈ x, a sequence (xn)n in x with xn → x and a sequence (pn)n in n, © agt, upv, 2021 appl. gen. topol. 22, no. 1 72 equicontinuous local dendrite maps pn → ∞ such that f pn(xn) → a ∈ x and f pn(x) → b ∈ x with a 6= b. then a ∈ ω(x, f) ⊂ p(f) and b ∈ ω(x, f) ⊂ p(f). since a and b are in p(f), there exists k ∈ n such that a, b ∈ fix(fk). we can assume, without loss of generality, that modulus k, pn is congruent with r (pn = kqn + r), for every n ∈ n. note that fr(xn) → f r(x), kqn → ∞ and f kqn(fr(xn)) → a. thus a ∈ ω(fr(x), fk) = ω(fr(x), fk). consequently, there is a sequence sn → ∞ (by choosing a subsequence, we can suppose sn−qn > n) such that (f k)sn(fr(x)) → a. since r = pn − kqn, f ksn(fpn−kqn (x)) = fksn−kqn(fpn(x)) → a. since fpn(x) → b, a ∈ ω(b, fk) = ω(b, fk) = {b}. therefore, a = b which leads to a contradiction. then f is equicontinuous. � the following examples show that the condition ∞⋂ n=1 fn(x) = p(f) of theorem 1.1 is essential. example 3.3 is inspired from [14, page 92]. example 3.3. there exist a compact metric space x and a homeomorphism fx → x such that ∞⋂ n=1 fn(x) 6= p(f), ω((x, y), fn) = ω(x, fn) for all x ∈ x and n ∈ n, and f is not equicontinuous. let yn ∈ r \ q be a sequence which converges to y∞ ∈ r \ q, s 1 = r/z be a unit circle, and x = s1 × {yn : n ∈ n ∪ {∞}}. then x is a compact metric space with the metric d induced from one of the euclidean space r2. let f : x → x by f(x, y) = (x + y, y). note that, f is a homeomorphism. thus ∞⋂ n=1 fn(x) = x. since {yn : n ∈ n∪{∞}} ⊂ r\q, p(f) = ∅. therefore, ∞⋂ n=1 fn(x) 6= p(f). let (x, y) ∈ x and n ∈ n. since fn(x, y) = (x + ny, y), on s1, x 7→ x + ny is an irrational rotation. then ω((x, y), fn) = s1 × {y} and ω((x, y), fn) = s1 × {y}. consequently, ω((x, y), fn) = ω(x, fn) for all x ∈ x and n ∈ n. f is not equicontinuous; indeed, fix a small ε > 0 (e.g. ε = 1 4 ) and arbitrary δ > 0. choose a pair p = (x, y); q = (x, y∞) with 0 < |y − y∞| < ε and d(q, p) < δ. fix an integer n > 1 such that k|y − y∞| < ε < n|y − y∞| < 1 for any integer k < n. then d(fn(p), fn(q)) > n|y − y∞| > ε. consequently, f is not equicontinuous. remark that the action of f is distal; indeed, let (x, y) 6= (x′, y′) be two points of x. if limfki(x, y) = limfki(x′, y′), then lim(x + kiy, y) = lim(x ′ + kiy ′, y′) which implies that y = y′ and x = x′ which is impossible. example 3.4. there exist a compact metric space x and a non surjective map f : x → x such that ∞⋂ n=1 fn(x) 6= p(f), ω((x, y), fn) = ω(x, fn) for all x ∈ x and n ∈ n, and f is not equicontinuous. © agt, upv, 2021 appl. gen. topol. 22, no. 1 73 a. h. salem, h. hattab and t. rejeiba as in the example 3.3, we put x = s1 × {yn : n ∈ n ∪ {∞}}. let f : x → x by f(x, y) = (x + y, y∞). note that, f is a non surjective map and ∞⋂ n=1 fn(x) = s1 × {y∞}. since {yn : n ∈ n ∪ {∞}} ⊂ r \ q, p(f) = ∅. therefore, ∞⋂ n=1 fn(x) 6= p(f). let (x, y) ∈ x and n ∈ n. since fn(x, y) = (x + ny, y∞), on s 1, x 7→ x + ny is an irrational rotation. then ω((x, y), fn) = s1 × {y∞} and ω((x, y), f n) = s1 × {y∞}. consequently, ω((x, y), f n) = ω(x, fn) for all x ∈ x and n ∈ n. as in the example 3.3, f is not equicontinuous. 4. proofs of proposition 1.2 and theorem 1.3 lemma 4.1. let x be a compact metric space and f : x → x be a map. consider the following statements: (1) r(f) = x; (2) f is an equicontinuous homeomorphism. then (2) implies (1) and if x is a dendrite, then (1) implies (2). proof. by [18, lemma 2.4], a homeomorphism of a compact metric space is equicontinuous if and only if it is distal and locally almost periodic. the last condition implies that r(f) = x. hence (2) implies (1). (1) ⇒ (2) from [7, theorem 1.18] we have f is a homeomorphism and x \ e(x) ⊂ p(f) hence, by the equivalence between clauses (1) and (8) of [20, theorem 3.8], f is equicontinuous. � according to [4, lemma 2.8], we get the following lemma. lemma 4.2. let f : x → x be a local dendrite map. if r(f) = x, then f(γ(x)) = γ(x). note that, by [1, proposition 3.6], if f is a monotone onto local dendrite map, then f(γ(x)) = γ(x). now we introduce some notations used in the proof of lemma 4.3. a space x is said to be almost totally disconnected if the set of its degenerate components, considered as a subset of x, is dense in x. a compact metric space x is called a cantoroid if it is almost totally disconnected and has no isolated point. a generalized brain is a cantoroid whose nondegenerate components form a null family, they are local dendrites and only finitely many of them contain circles. lemma 4.3. let f : x → x be a local dendrite map. if x 6= s1 then f is not minimal. proof. if x 6= s1, by [16, theorem 3.2] and [19, theorems 10.31], the result holds whenever x is either a graph or a dendrite. if x is neither a dendrite nor a graph, then x contains at least an attached dendrite. according to [5, theorem c], a minimal set on local dendrites is either a finite set or a finite © agt, upv, 2021 appl. gen. topol. 22, no. 1 74 equicontinuous local dendrite maps union of disjoint circles or a generalized brain. since a generalized brain is not connected, a local dendrite can not be a minimal set. this ends the proof of lemma 4.3. � according to [17, proposition 2.5], we get the following lemma. lemma 4.4. let f : x → x be a local dendrite map and let p ∈ n. then f is equicontinuous if and only if fp is equicontinuous. according to [4, theorem 2.1], we have the following result. theorem 4.5. let f : x → x be a local dendrite map. then f is pointwise recurrent if and only if f is a homeomorphism such that one of the following statements holds: (1) if f is not minimal, then every non endpoint has a finite orbit; (2) if f is minimal, then x = s1 and f is topologically conjugate to an irrational rotation. proof of proposition 1.2. assume that r(f) = x. by [17, corollary 5.1] and lemma 4.1, the result holds whenever x is either a graph or a dendrite. assume that x is neither a dendrite nor a graph. assume that x is neither a dendrite nor a graph. if f is minimal, then, by theorem 4.5 f is topologically conjugate to an irrational rotation. it is well known that every rotation is an isometric, consequently, it is equicontinuous. thus, by [17, lemma 3.1] f is equicontinuous. if f is not minimal, then, by theorem 4.5, f is a homeomorphism and every non endpoint has finite orbit. by lemma 4.2, γ(x) is invariant. then, by [17, corollary 5.1], f|γ(x) is equicontinuous. we further have x \ γ(x) is the union of pairwise disjoint dendrites (ck) such that ck ∩ γ(x) = {zk} and as fn0(zk) = zk hence by [1, lemma 3.8] one has f n0(ck) = ck and since r(f) = r(fn0) = x, fn0 : ck → ck is pointwise recurrent. thus, by lemma 4.1 and lemma 4.4, f|ck is equicontinuous for each k which implies f|x\γ(x) is equicontinuous ((ck) are pairwise disjoint dendrites). consequently, f is equicontinuous on x. � according to [4, corollary 2.2] we obtain the following result. proposition 4.6. let x be a local dendrite and f : x → x be a local dendrite map. assume that e(x) is countable. then f is pointwise recurrent if and only if one the following statements holds: (1) f is a pointwise periodic homeomorphism; (2) x = s1 and f is topologically conjugate to an irrational rotation. © agt, upv, 2021 appl. gen. topol. 22, no. 1 75 a. h. salem, h. hattab and t. rejeiba proof of theorem 1.3. since x is compact and connected, for every n ∈ n, fn(x) is also compact and connected. let m(f) = ∞⋂ n=1 fn(x). it follows from x ⊃ f(x) ⊃ f2(x) ⊃ · · · that m(f) is a nonempty sub-local dendrite of x. by [11, lemma 3.1], f(m(f)) = m(f). since m(f) is an f-invariant sub-local dendrite, by proposition 4.6, (2) is equivalent to (4). if r(f) = m(f), then f|m(f) : m(f) → m(f) is pointwise recurrent. consequently, by proposition 1.2, f|m(f) is equicontinuous. therefore, (2) ⇒ (3). by lemma 3.2, (1) implies (2). (3) ⇒ (1). if u = m(f) ∩ x \ m(f), then u ⊂ e(m(f)). by lemma 2.3, e(m(f)) is countable. since u is compact, it is a finite set. obviously, for every k ∈ n, there exists pk ∈ n such that f pk(x) ⊂ m(f) ∪ b(u, 2−k). since f|m(f) is equicontinuous, by [17, theorem 5.1], f is equicontinuous. by lemma 3.1, (1) ⇒ (5). by lemma 3.2, (5) ⇒ (2). � the following example shows that e(x) is countable cannot be removed from the hypothesis of theorem 1.3. example 4.7. by [11, example 5.4], there exist a dendrite x with uncountable set of endpoints and a homeomorphism f : x → x such that f satisfies (1) and does not satisfies (4). by applying theorem 1.3 and lemma 3.1, we obtain corollary 1.4. references 1. h. abdelli, ω-limit sets for monotone local dendrite maps. chaos, solitons and fractals, 71 (2015), 66–72. 2. h. abdelli and h. marzougui, invariant sets for monotone local dendrite maps, internat. j. bifur. chaos appl. sci. engrg. 26, no. 9 (2016), 1650150 (10 pages). 3. e. akin, j. auslander and k. berg, when is a transitive map chaotic?, in: convergence in ergodic theory and probability, walter de gruyter and co., berlin, 1996, pp. 25-40. 4. g. askri and i. naghmouchi, pointwise recurrence on local dendrites, qual. theory dyn syst 19, 6 (2020). 5. f. balibrea, t. downarowicz, r. hric, l. snoha and v. spitalsky, almost totally disconnected minimal systems, ergodic th. & dynam sys. 29, no. 3 (2009), 737–766. 6. f. blanchard, b. host and a. maass, topological complexity, ergodic th. & dynam sys. 20 (2000), 641–662. 7. a. m. blokh, pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces, proc. amer. math. soc. 143 (2015), 3985–4000. 8. a. m. blokh, the set of all iterates is nowhere dense in c([0,1],[0,1]), trans. amer. math. soc. 333, no. 2 (1992), 787–798. 9. w. boyce, γ-compact maps on an interval and fixed points, trans. amer. math. soc. 160 (1971), 87–102. 10. a. m. bruckner and t. hu, equicontinuity of iterates of an interval map, tamkang j. math. 21, no. 3 (1990), 287–294. © agt, upv, 2021 appl. gen. topol. 22, no. 1 76 equicontinuous local dendrite maps 11. j. camargo, m. rincón and c. uzcátegui, equicontinuity of maps on dendrites, chaos, solitons and fractals 126 (2019), 1–6. 12. j. cano, common fixed points for a class of commuting mappings on an interval, trans. amer. math. soc. 86, no. 2 (1982), 336–338. 13. r. gu and z. qiao, equicontinuity of maps on figure-eight space, southeast asian bull. math. 25 (2001), 413–419. 14. a. haj salem and h. hattab, group action on local dendrites, topology appl. 247, no. 15 (2018), 91–99. 15. k. kuratowski, topology, vol. 2. new york: academic press; 1968. 16. j. mai, pointwise-recurrent graph maps, ergodic th. & dynam sys. 25 (2005), 629–637. 17. j. mai, the structure of equicontinuous maps, trans. amer. math. soc. 355, no. 10 (2003), 4125–4136. 18. c. a. morales, equicontinuity on semi-locally connected spaces, topology appl. 198 (2016), 101–106. 19. s. nadler, continuum theory. inc., new york: marcel dekker; 1992. 20. g. su and b. qin, equicontinuous dendrites flows, journal of difference equations and applications 25, no. 12 (2019), 1744–1754. 21. t. sun, equicontinuity of σ-maps, pure and applied math. 16, no. 3 (2000), 9–14. 22. t. sun, z. chen, x. liu and h. g. xi, equicontinuity of dendrite maps, chaos, solitons and fractals 69 (2014), 10–13. 23. t. sun, g. wang and h. j. xi, equicontinuity of maps on a dendrite with finite branch points. acta mat. sin. 33, no. 8 (2017), 1125–1130. 24. t. sun, y. zhang and x. zhang, equicontinuity of a graph map, bull. austral math. soc. 71 (2005), 61–67. 25. a. valaristos, equicontinuity of iterates of circle maps, internat. j. math. and math. sci. 21 (1998), 453–458. © agt, upv, 2021 appl. gen. topol. 22, no. 1 77 () @ appl. gen. topol. 14, no. 2 (2013), 135-145doi:10.4995/agt.2013.1583 c© agt, upv, 2013 concerning nearly metrizable spaces m. n. mukherjee and dhananjoy mandal department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata– 700019. india (mukherjeemn@yahoo.co.in, dmandal.cu@gmail.com ) abstract the purpose of this paper is to introduce the notion of near metrizability for topological spaces, which is strictly weaker than the concept of metrizability. a number of characterizations of nearly metrizable spaces is achieved here as analogues of the corresponding ones for metrizable spaces. it is seen that near metrizability is a natural idea vis-a-vis near paracompactness, playing the similar role as played by paracompactness with regard to metrizability. 2010 msc: 54d20, 54e99. keywords: regular open set, semiregularization space, almost regular space, nearly metrizable space, near paracompactness. 1. introduction and preliminary results the notion of nearly paracompact space was introduced by singal and arya [12], and such spaces have so far been studied by many researchers with keen interest (e.g. see [3], [5], [6], [8], [10]). now, there are well known interrelations between paracompactness and metrizability; for instance, every metrizable space is always paracompact. it is thus natural to search for the kind of spaces which take the corresponding role of metrizability vis-a-vis near paracompactness. in this paper, we like to introduce and study a class of spaces, called nearly metrizable, which properly contains all metrizable spaces and which bears a similar kind of relationship with the class of all nearly paracompact spaces [12] as the family of all metrizable spaces has with the class of paracompact spaces. we shall show that within the class of paracompact spaces the concept received april 2012 – accepted october 2012 http://dx.doi.org/10.4995/agt.2013.1583 m. n. mukherjee and d. mandal of near metrizability and that of metrizability coincide. thus there exist nearly metrizable, non-metrizable spaces which are not paracompact. we recall that a subset a of a topological space x is called regular open, if a = intcla (as usual, ‘int’ and ‘cl’ stand for interior and closure operators respectively). to simplify notation, we shall write a∗ instead of intcla. it is easy to see that a subset of a space x is regular open iff it is of the form a∗, for some a ⊆ x. we denote by ro(x) the family of all regular open subsets of x, i.e., ro(x) = {a∗ : a ⊆ x}. it is well known [1] that ro(x) is an open base for some topology τs on a space (x, τ). the set x endowed with this topology τs will be denoted by (x)s and is called the semiregularization space of x. in particular, x is called semiregular if these two topologies on x coincide. as rightly observed by mrševic et al.[7], semiregularization topologies and the associated techniques are found quite important in the study of hclosed, minimal hausdorff and s-closed spaces. any subset a of x, which is open in (x)s, is called δ-open [15]. a subset a of a space x is called regular closed if x \a ∈ ro(x). we denote by rc(x) the family of all regular closed subsets of x. for any family a of subsets of x, we denote by a# the family given by a# = {a∗ : a ∈ a}. we now mention some simple facts which will be needed for our discussion. lemma 1.1. if a and b are two open sets in x, then a ⋂ b 6= φ ⇔ a∗ ⋂ b 6= φ ⇔ a∗ ⋂ b∗ 6= φ. lemma 1.2. a space x is t2 iff (x)s is t2. in [11], singal and arya called a space x almost regular if for any a ∈ rc(x) and any x ∈ x \ a, there exist disjoint open sets u and v in x such that x ∈ u and a ⊆ v . the same authors in [12] called a space x nearly paracompact if every regular open cover of x has a locally finite open refinement. we now state the following well known results (one may find them in [7] ): theorem 1.3. let x be a topological space. then (a) x is nearly paracompact iff (x)s is paracompact. (b) x is almost regular iff (x)s is regular. in the next section we introduce pseudo-embedding and thereby near metrizability. certain characterizations of near metrizability and its study vis-a-vis paracompactness and near paracompactness are taken up in this section. in section 3, two other notions viz. pseudo-bases and local pseudo-bases are defined to facilitate further investigations of near metrizability, where we will show, among other things, that a space x is nearly metrizable iff (x)s is metrizable. the last section consists of just two characterizations of near metrizability deduced from perspectives different from those in the earlier sections. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 136 concerning nearly metrizable spaces 2. pseudo-embedding and near metrizability we begin by introducing the idea of pseudo-embedding as a generalized concept of an embedding. definition 2.1. if x and y are two topological spaces, then a continuous, injective map f : x → y is called a pseudo-embedding of x into y , if for any a ∈ ro(x), f(a) is open. if there is a pseudo-embedding f of x into y , then we say that x is pseudo-embeddable in y . if a pseudo-embedding f : x → y is surjective, we say that f is a pseudo-embedding of x onto y . it is easy to see that every embedding is a pseudo-embedding; but the converse is false as is shown in the following example. example 2.2. let r be the set of all real numbers; and, τf and τc be respectively the co-finite topology and the co-countable topology on r. since τf ⊆ τc, the identity map i : (r, τc) → (r, τf ) is a continuous bijection which maps every regular open subsets of (r, τc) onto an open subset of (r, τf ) (note that r and φ are the only regular open sets in (r, τc)). hence i is a pseudoembedding; but it is not an embedding, because r \ q ∈ τc whereas r \ q 6∈ τf (q denoting the set of all rational numbers). remark 2.3. if x is semiregular, then ro(x) makes an open base for the topology of x and hence any pseudo-embedding of a semi-regular space x into any space y is an embedding. definition 2.4. a space x is called nearly metrizable if it is pseudo-embeddable in a metric space y . remark 2.5. it is obvious that every metrizable space is nearly metrizable; we show below that the converse does not hold, in general. example 2.6. let τ1 and τ2 respectively denote the euclidean and co-countable topologies on the set r of all real numbers, and let τ be the smallest topology on r generated by τ1 ⋃ τ2. since τ1 ⊆ τ, the identity map i : (r, τ) → (r, τ1) is a continuous bijection. since the regular open subsets of (r, τ) are precisely the sets which are regular open in (r, τ1) (see example 63 of steen and seeback [14]), i maps regular open subsets of (r, τ) onto open subsets of (r, τ1) and hence i becomes a pseudo-embedding. since (r, τ1) is a metrizable space, it follows that (r, τ) is nearly metrizable; but (r, τ) is not metrizable as it is not regular. remark 2.7. (a) in view of remark 2.3 it follows that a semiregular, nearly metrizable space is metrizable. (b) since metrizability is a hereditary property, it follows that a space x is nearly metrizable iff there is a pseudo-embedding f from x onto a metrizable space y . we now prove a few properties of nearly metrizable spaces. theorem 2.8. every nearly metrizable space is hausdorff and almost regular. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 137 m. n. mukherjee and d. mandal proof. we only prove that a nearly metrizable space x is almost regular, the hausdorffness of x can similarly be proved. by near metrizability of x, there is a pseudo-embedding f of x onto a metric space y . let a ∈ rc(x) and x ∈ x \ a. since f : x → y is a bijection and f maps regular open subsets of x onto open subsets of y , it follows that f(a) is closed in y and f(x) ∈ y \ f(a). since y is regular, there exist disjoint open sets u and v in y such that f(x) ∈ u and f(a) ⊆ v . it is now easy to see that f−1(u) and f−1(v ) are two disjoint open sets in x with x ∈ f−1(u) and a ⊆ f−1(v ). thus x is almost regular. � remark 2.9. we shall call an almost regular, hausdorff space an almost t3space. so, what we have proved in the above theorem is that a nearly metrizable space is an almost t3-space. it is well known that every metrizable space is paracompact. but we give here an example to show that a nearly metrizable space may not be paracompact. example 2.10. consider the nearly metrizable space (r, τ) of example 2.6. since every hausdorff, paracompact space is regular and (r, τ) is not regular, it cannot be paracompact. it is then a natural question: when does a nearly metrizable space become paracompact ? the following result answers it. theorem 2.11. a nearly metrizable space x is paracompact iff it is metrizable. proof. for the necessity, let x be a nearly metrizable, paracompact space. since x is hausdorff (by theorem 2.8) and paracompact, it is regular. as a semiregular, nearly metrizable space is metrizable (see remark 2.7), x becomes metrizable. the sufficiency part is clear. � it then follows from example 2.10 and the above theorem that a nearly metrizable and non-metrizable space is never paracompact. however, as expected, we show below that every nearly metrizable space is nearly paracompact. theorem 2.12. every nearly metrizable space x is nearly paracompact. proof. let x be a nearly metrizable space and u ⊆ ro(x) be a cover of x. then there is a pseudo-embedding f of x onto a metrizable space y . since f maps regular open subsets of x onto open subsets of y , it follows that {f(a) : a ∈ u} is an open cover of y . since y is paracompact (being a metric space), there is an open locally finite refinement v of {f(a) : a ∈ u} in y . it is then easy to check that {f−1(v ) : v ∈ v} is a locally finite open refinement of u in x and hence x is nearly paracompact. � we conclude this section by giving a sufficient condition for near metrizability. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 138 concerning nearly metrizable spaces theorem 2.13. if the semiregularization space (x)s of a space x is metrizable, then x is nearly metrizable. proof. if (x)s is metrizable, then as the topology of x is finer than that of (x)s, the identity map i : x → (x)s is a pseudo-embedding of x onto (x)s. hence x is nearly metrizable. � 3. (local) pseudo-bases and near metrizability in this section we shall give some characterizations of nearly metrizable spaces by introducing the ideas of pseudo-bases and local pseudo-bases. we shall, in addition, prove the converse of theorem 2.13. definition 3.1. suppose b is a family of open subsets of x. we say that b is a pseudo-base in x if for any a ∈ ro(x), there is a subfamily b0 of b such that a = ⋃ {b : b ∈ b0}. we call a pseudo-base b σ-locally finite if b can be expressed as b = ∞⋃ n=1 bn, where bn is locally finite, for each n ∈ n. it is obvious that every base is a pseudo-base; but the converse is false as is shown in the following example. example 3.2. let r be the set of reals and τ be the co-countable topology on r. then b = {r, φ} is a pseudo-base for (r, τ), but is not a base for it. remark 3.3. for a semiregular space x, ro(x) makes an open base for x so that every pseudo-base in a semiregular space is a base. we now prove a lemma which will be very useful for the rest of the paper. lemma 3.4. suppose b is a family of open subsets of a space x. if b is a pseudo-base in x then b# = {b∗ : b ∈ b} is a base for the topology of (x)s. proof. it is enough to show that each member of ro(x) can be expressed as a union of some members of b#, as ro(x) is an open base for the topology of (x)s. so, let a ∈ ro(x). since b is a pseudo-base in x, there exists a subfamily b0 of b such that a = ⋃ {b : b ∈ b0}. therefore, b ⊆ a, for all b ∈ b0, i.e., b ∗ ⊆ a, for all b ∈ b0 (since a ∈ ro(x), a ∗ = a). also, b ⊆ b∗, for all b ∈ b0. thus b ⊆ b ∗ ⊆ a, for all b ∈ b0, i.e.,⋃ {b : b ∈ b0} ⊆ ⋃ {b∗ : b ∈ b0} ⊆ a, i.e., a = ⋃ {b∗ : b ∈ b0} and hence b# becomes a base for the topology of (x)s. � remark 3.5. in the above lemma, if b# is a base for (x)s, then clearly b # is a pseudo-base of x, but b is not necessarily a pseudo-base for x. in fact, for the space r with co-countable topology τ we have τs = {φ, r}. if we take b = {r \ {1}, r \ {1, 2}, ...} then clearly b# = {r, φ} is a base for (r)s, but b is not a pseudo-base in r. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 139 m. n. mukherjee and d. mandal the first characterization which we now give for nearly metrizable spaces is quite similar to the celebrated nagata-smirnov metrization theorem which states that ‘a t3-space x is metrizable iff it has a σ-locally finite open base’ (see [9]). one part of the proposed result goes as follows. theorem 3.6. an almost t3-space x possessing a σ-locally finite pseudo-base is nearly metrizable. proof. let b be a σ-locally finite pseudo-base in x. then b can be expressed as b = ∞⋃ n=1 bn, where each bn is locally finite in x. we claim that b # = ∞⋃ n=1 b#n is a σ-locally finite base for (x)s. that b # is a base for (x)s follows from lemma 3.4. we now show that b# is a σ-locally finite family i.e., we show that b#n is a locally finite family in (x)s for each n ∈ n. for this, let x ∈ x. since for each n ∈ n, bn is locally finite in x, there exists an open neighbourhood u of x in x which intersects at most finitely many members of bn for each n ∈ n. now, u∗ is an open neighbourhood of x in (x)s and, in view of lemma 1.1, u∗ can intersect at most finitely many members of b#n , for each n ∈ n. thus b# is a σ-locally finite base for (x)s. since x is almost t3, by lemma 1.2 and theorem 1.3(b), (x)s is a t3-space. thus by nagata-smirnov metrization theorem, (x)s is metrizable and hence, in view of theorem 2.13, x is nearly metrizable. � corollary 3.7. an almost t3-space with a countable pseudo-base is nearly metrizable. proof. follows directly from theorem 3.6, since every countable family is a σ-locally finite family. � theorem 3.8. every nearly metrizable space admits a σ-locally finite pseudobase. proof. if x is nearly metrizable then there is a pseudo-embedding f of x onto a metric space y . by nagata-smirnov metrization theorem, there is a σ-locally finite open base b in y . let a = {f−1(b) : b ∈ b}. then a is a family of open sets in x. we show that a is a pseudo-base in x. for that, let a ∈ ro(x). since f(a) is open in y , there is a subfamily b0 of b such that f(a) = ⋃ {b : b ∈ b0} which, in turn, implies that a = ⋃ {f−1(b) : b ∈ b0}. thus a becomes a pseudo-base in x. that a is σ-locally finite is clear and hence the result follows. � as a consequence of theorem 2.8, 3.6 and 3.8 we have : theorem 3.9. a space x is nearly metrizable iff it is almost t3 and possesses a σ-locally finite pseudo-base. we now prove the converse of theorem 2.13. theorem 3.10. if a space x is nearly metrizable, then (x)s is metrizable. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 140 concerning nearly metrizable spaces proof. if x is nearly metrizable, then by theorem 2.8 and 3.8 it follows that x is almost t3 and has a σ-locally finite pseudo-base b. then b # is a σ-locally finite open base for (x)s (see the proof of theorem 3.6). since x is almost t3, by lemma 1.2 and theorem 1.3(b), (x)s is t3 and hence by nagata-smirnov metrization theorem, (x)s is metrizable. � question: does there exist any direct proof of the above theorem without using nagata-smirnov metrization theorem? combining theorems 2.13 and 3.10, we obtain : theorem 3.11. a space x is nearly metrizable iff (x)s is metrizable. remark 3.12. as observed in corollary 1 of [7] , the half-disc topology (counterexample 78 of [14]) gives an example of a space x which is not regular, but such that (x)s is metrizable. then by theorem 3.11 it follows that x is another example of a space which is nearly metrizable but is not metrizable. we now introduce the concept of local pseudo-bases which will also be used to obtain some further characterizations of near metrizability. definition 3.13. suppose x is a topological space and x ∈ x. a family a of open subsets of x each of which contains x, is called a local pseudo-base at x, if for any b ∈ ro(x) with x ∈ b, there exists an a ∈ a such that a ⊆ b. it is obvious that any local base at x in a space x is a local pseudo-base at x; but in the following example we show that the converse may not be true. example 3.14. let (r, τ) be the space of real numbers endowed with the co-countable topology τ. then a = {r, φ} is a local pseudo-base at x for each x ∈ r; but a is not a local base at any point of r. we shall use the following result for our discussion in the sequel. lemma 3.15. let a be a family of open subsets of x each of which contains x(∈ x). then a is a local pseudo-base at x in x iff a# is a local base at x in (x)s. proof. obviously, a# is a family of open subsets of (x)s each of which contains x. let b ∈ ro(x) and x ∈ b. since a is a local pseudo-base at x in x, there exists some a ∈ a such that a ⊆ b which gives a∗ ⊆ b (as b ∈ ro(x), b∗ = b), where a∗ ∈ a# and hence a# becomes a local base at x in (x)s ( since ro(x) is an open base for the topology of (x)s). conversely, let x ∈ b ∈ ro(x). as a# is a local base at x in (x)s, there exists some a ∈ a such that x ∈ a ⊆ b. then x ∈ a ⊆int(cl(a)) ⊆int(cl(b)) = b. thus a is a local pseudo-base at x in x. � the following is a well known characterization for metrizable spaces, which may be found in page 192 of [9]. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 141 m. n. mukherjee and d. mandal theorem 3.16. a t1space x is metrizable iff there exists a countable local base {bn(x) : n ∈ n} at x, for each x ∈ x such that for every n ∈ n, there exists an m ∈ n for which bm(x) ⋂ bm(y) 6= φ implies that bm(y) ⊆ bn(x), for any y ∈ x. we now prove an analogous version of the above result for nearly metrizable spaces: theorem 3.17. a t2-space x is nearly metrizable iff there exists a countable local pseudo-base {bn(x) : n ∈ n} at x, for each x ∈ x such that for every x ∈ x and every n ∈ n, there exists an m ∈ n for which bm(x) ⋂ bm(y) 6= φ implies that bm(y) ⊆ bn(x), for any y ∈ x. proof. since x is t2, (x)s is t1 (in fact, t2). first, let x be nearly metrizable. then by theorem 3.10, (x)s is metrizable. thus by theorem 3.16, there exists a countable local base bx at each x in (x)s satisfying the condition of the hypothesis and hence the necessity follows. conversely, let bx = {bn(x) : n ∈ n} be a local pseudo-base at x, for each x ∈ x such that the given condition holds. by lemma 3.15, b#x is a countable local base at x in (x)s, for each x ∈ x. we now show that these b # x ’s satisfy the hypothesis of theorem 3.16. for that, let x ∈ x and n ∈ n. then there exists m ∈ n such that (bm(x) ⋂ bm(y) 6= φ ⇒ bm(y) ⊆ bn(x), for each y ∈ x). now b∗m(x) ⋂ b∗m(y) 6= φ ⇒ bm(x) ⋂ bm(y) 6= φ ⇒ bm(y) ⊆ bn(x) ⇒ b∗m(y) ⊆ b ∗ n(x) for each y in x. then by theorem 3.16, (x)s is metrizable and hence, in view of theorem 2.13, x is nearly metrizable. � another nice characterization of metrizable spaces can be found in [9]. it goes as follows: theorem 3.18. a t1-space x is metrizable iff for each x ∈ x, there exist two sequences {an(x) : n ∈ n} and {bn(x) : n ∈ n} of open neighbourhoods of x in x such that (i) {an(x) : n ∈ n} is a local base at x, for each x ∈ x, (ii) y 6∈ an(x) ⇒ bn(x) ⋂ bn(y) = φ, and (iii) y ∈ bn(x) ⇒ bn(y) ⊆ an(x). remark 3.19. it is easy to see that in the above theorem we can assume that bn(x) ∈ b (for all n ∈ n) for a given open base b for x. an analogue of the above theorem for near metrizability is now proved. theorem 3.20. a t2-space x is nearly metrizable iff for each x ∈ x, there exist two sequences {an(x) : n ∈ n} and {bn(x) : n ∈ n} of open neighbourhoods of x in x with bn(x) ∈ ro(x), for all n ∈ n satisfying the following conditions: (i) {an(x) : n ∈ n} is a local pseudo-base at x, for each x ∈ x, (ii) y 6∈ an(x) ⇒ bn(x) ⋂ bn(y) = φ, and (iii) y ∈ bn(x) ⇒ bn(y) ⊆ an(x). c© agt, upv, 2013 appl. gen. topol. 14, no. 2 142 concerning nearly metrizable spaces proof. first let x be nearly metrizable. then (x)s is metrizable and also t2 (as x is t2). thus by theorem 3.18, there exist two sequences {an(x) : n ∈ n} and {bn(x) : n ∈ n} of open neighbourhoods of x in (x)s (and hence in x) satisfying the conditions of the above theorem, where, in view of remark 3.19, we can assume that bn(x) ∈ ro(x) for all n ∈ n as ro(x) is an open base for (x)s. conversely, let for each x ∈ x, there exist two sequences {an(x) : n ∈ n} and {bn(x) : n ∈ n} of open neighbourhoods of x in x with bn(x) ∈ ro(x), for all n ∈ n such that (i) {an(x) : n ∈ n} is a local pseudo-base at x, for each x ∈ x; (ii) y 6∈ an(x) ⇒ bn(x) ⋂ bn(y) = φ, and (iii) y ∈ bn(x) ⇒ bn(y) ⊆ an(x). then {a∗n(x) : n ∈ n} and {bn(x) : n ∈ n} are two sequences of open neighbourhoods of x in (x)s such that (a) {a∗n(x) : n ∈ n} is a local base at x in (x)s, for each x ∈ x (see lemma 3.15), (b) y 6∈ a∗n(x) ⇒ y 6∈ an(x) (since an(x) ⊆ a ∗ n(x)) ⇒ bn(x) ⋂ bn(y) = φ, and (c) y ∈ bn(x) ⇒ bn(y) ⊆ an(x) ⇒ bn(y) ⊆ a ∗ n(x) (since an(x) ⊆ a ∗ n(x)). this shows that {a∗n(x) : n ∈ n} and {bn(x) : n ∈ n} satisfy all the conditions of the hypothesis of theorem 3.18 for (x)s and hence (x)s becomes metrizable which, in turn, implies that x is nearly metrizable. � 4. two more characterizations of nearly metrizable spaces this section is meant for deriving two more characterizations of nearly metrizable spaces from two different perspectives, which are similar versions of two well known characterizations of metrizable spaces. the first such latter characterization, which may be found in [9], goes as follows: theorem 4.1. a compact, t2-space x is metrizable iff the diagonal ∆x = {(x, x) : x ∈ x} is gδ-set in x × x. we need a few definitions to arrive at an analogous version of the above theorem. definition 4.2 ([13]). a space x is called nearly compact if every regular open cover of x has a finite subcover. remark 4.3. since ro(x) is an open base for (x)s, it follows that x is nearly compact iff (x)s is compact. definition 4.4. a subset b of a space x will be called a regular gδ-set in x if there is a sequence {bn : n ∈ n} of δ-open sets such that b = ∞⋂ n=1 bn. obviously, every regular gδ-set in x is a gδ-set in x; but the converse fails as we see below: c© agt, upv, 2013 appl. gen. topol. 14, no. 2 143 m. n. mukherjee and d. mandal example 4.5. let x = (r, τ), where r is the set of all real numbers and τ is the co-countable topology on r. then (r \ q) is a gδ-set in x but is not regular gδ. remark 4.6. it is easy to see that a subset b of a space x is a regular gδ-set in x iff it is a gδ-set in (x)s. theorem 4.7. a nearly compact, t2 space x is nearly metrizable iff the diagonal ∆x = {(x, x) : x ∈ x} is a regular gδ-set in x × x. proof. if x is nearly compact and t2, then (x)s is compact and t2. thus x is nearly metrizable ⇔ (x)s is metrizable (by theorem 3.11) ⇔ the diagonal ∆x is a gδ-set in (x)s × (x)s (by theorem 4.1) ⇔ the diagonal ∆x is a gδ-set in (x × x)s (as (x)s × (x)s = (x × x)s [4]) ⇔ the diagonal ∆x is a regular gδ-set in x × x (by remark 4.6). � definition 4.8 ([2]). suppose a is a cover of x by means of subsets of x. a sequence {an : n ∈ n} of open covers of x is called locally starring for a if for any x ∈ x, there are an open neighbourhood u of x in x and an n ∈ n such that st(u, an) ⊆ a, for some a ∈ a (where st(u, an) = ⋃ {a ∈ an : a ⋂ u 6= φ}). the following characterization of metrizability is due to arhangelskii (see [2]). theorem 4.9. a t1-space x is metrizable iff there is a sequence {an : n ∈ n} of open covers of x that is locally starring for every open cover of x. our last characterization of near metrizability in this paper is an offshoot of the above theorem. theorem 4.10. a t2 space (x, τ) is nearly metrizable iff there is a sequence {an : n ∈ n} of τ-open covers of x that is locally starring in (x, τ) for every regular open cover of x. proof. since x is t2, (x)s is t2. if x is nearly metrizable, then (x)s is metrizable and hence by theorem 4.9, there is a sequence {an : n ∈ n} of τs-open covers of x that is locally starring in (x)s for every regular open cover of x. then clearly {an : n ∈ n} is a sequence of τ-open cover of x that is locally starring in x for every regular open cover of x. conversely, let {an : n ∈ n} be a sequence of τ-open covers of x, that is locally starring in (x, τ) for every regular open cover of x. then {a#n : n ∈ n} is a sequence of τs-open covers of x with a # n ⊆ ro(x), for all n ∈ n, which can be checked (by use of lemma 1.1) to be also a locally starring in (x)s for every regular open cover of x. since ro(x) is an open base for (x)s, in view of theorem 4.9, (x)s becomes metrizable so that x is nearly metrizable. � acknowledgements. the authors are thankful to the referee for some suggestions towards certain improvement of the paper. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 144 concerning nearly metrizable spaces references [1] d. e. cameron, maximal qhc-space, rocky mountain jour. math. 7, no. 2 (1977), 313–322. [2] j. dugundji, topology , allyn and bacon, boston (1966). [3] n. ergun, a note on nearly paracompactness, yokahama math. jour. 31 (1983), 21–25. [4] l. l. herrington, properties of nearly compact spaces, proc. amer. math. soc. 45 (1974), 431–436. [5] i. kovačević, almost regularity as a relaxation of nearly paracompactness, glasnik mat. 13(33) (1978), 339–341. [6] i. kovačević, on nearly paracomapct spaces, publ. inst. math. 25 (1979), 63–69. [7] m. mršević, i. l. reilly and m. k. vamanamurthy, on semi-regularization topologies, jour. austral. math. soc. (series a) 38 (1985), 40–54. [8] m. n. mukherjee and d. mandal, on some new characterizations of near paracompactness and associated results, mat. vesnik 65 (3) (2013), 334–345. [9] jun-iti nagata, modern general topology, elsevier sciences publishings second revised edition b.v. (1985). [10] t. noiri, a note on nearly paracompact spaces, mat. vesnik 5 (18) (33) (1981), 103– 108. [11] m. k. singal and s. p. arya, on almost regular spaces, glasnik mat. 4 (24) (1969), 89–99. [12] m. k. singal and s. p. arya, on nearly paracompact spaces, mat. vesnik 6 (21) (1969), 3–16. [13] m. k. singal and a. mathur, on nearly compact spaces, boll. un. mat. ital. 4 (1969), 702–710. [14] l. a. steen and j. a. seebach, counterexamples in topology, spinger-verlag, new york (1970). [15] n. v. veličko, h-closed topological spaces, amer. math. soc. transl. 78 (1968), 103–118. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 145 @ appl. gen. topol. 20, no. 1 (2019), 97-108doi:10.4995/agt.2019.10036 c© agt, upv, 2019 f-n-resolvable spaces and compactifications intissar dahane a, lobna dridi b and sami lazaar a a faculty of sciences of tunis, university of tunis el manar, tunisia. (intissardahane@gmail.com, salazaar72@yahoo.fr) b department of mathematics, tunis preparatory engineering institute. university of tunis, 1089 tunis, tunisia. (lobna dridi 2006@yahoo.fr) communicated by s. garćıa-ferreira abstract a topological space is said to be resolvable if it is a union of two disjoint dense subsets. more generally it is called n-resolvable if it is a union of n pairwise disjoint dense subsets. in this paper, we characterize topological spaces such that their reflections (resp., compactifications) are n-resolvable (resp., exactly-nresolvable, strongly-exactly-n-resolvable), for some particular cases of reflections and compactifications. 2010 msc: 54b30; 54d10; 46m15. keywords: categories; functors; resolvable spaces; compactifications. introduction let n > 1 be an integer. generalizing the concept of resolvable spaces introduced by hewitt in [16], ceder in [6] defined a topological space x to be n-resolvable space if it has a family of n pairwise disjoints dense subsets. the latter is called exactly n-resolvable if it is n-resolvable but not (n + 1)resolvable and it is called strongly exactly n-resolvable denoted by senr if it is n-resolvable and no empty subset of x is (n + 1)-resolvable. se1r space is commonly said strongly irresolvable space (abbreviated as si-space) or hereditarily irresolvable (see [7] and [13]). received 25 april 2018 – accepted 01 february 2019 http://dx.doi.org/10.4995/agt.2019.10036 i. dahane, l. dridi and s. lazaar the theory of categories and functors play an enigmatic role in topology, specially the notion of reflective subcategories. recently, some authors have been interested by particular functors like t0, s, ρ and fh. in [10], [11] and [8], the authors have characterized topological spaces whose f-reflections are door, submaximal, nodec and resolvable. some papers, as [5] and [3] were interested in spaces such that their compactifications are submaximal, door and nodec. specially in [2], k. belaid and m. al-hajri have characterized topological spaces such that their one point compactifications (resp., wallman compactifications) are resolvable. in the first section of this paper, we characterize topological spaces such that their t0-reflections are n-resolvable (resp., exactly n-resolvable, strongly exactly n-resolvable). in the second section, topological spaces, such that their tychonoff reflections and functionally hausdorff reflections are n-resolvable (resp., exactly nresolvable), have been characterized. the third section of this paper is devoted to a characterization of topological spaces such that their one point compactifications (resp., wallman compactifications) are n-resolvable (resp., exactly n-resolvable, strongly exactly n-resolvable). 1. t0-n-resolvable spaces, t0-exactly-n-resolvable spaces and t0-strongly-exactly-n-resolvable spaces. let x be a topological space. the t0-reflection of x denoted by t0(x) is defined as follow. consider the equivalence relation ∼ on x by: x ∼ y if and only if {x} = {y}, for x, y ∈ x. then the resulting quotient space t0(x) := x/ ∼ is a kolmogroff space called the t0-reflection of x. recall that a continuous map q : x −→ y is said to be a quasihomeomorphism if u 7−→ q−1(u) (resp., c 7−→ q−1(c) ) defines a bijection o(y ) −→ o(x) (resp., f(y ) −→ f(x)), where o(x) (resp., f(x)) is the collection of all open sets (resp., closed sets) of x )[15]. in particular the canonical surjection µx : x −→ t0(x) is an onto quasihomeomorphism and consequently a closed (resp., open) map, (see [4]). in order to give the main result of this section we recall the following results introduced in [10]. notation 1.1 ([10, notations 2.2]). let x be a topological space, a ∈ x and a ⊆ x. we denote by: (1) d0(a) := {x ∈ x : {x} = {a}}. (2) d0(a) = ∪[d0(a); a ∈ a]. remark 1.2 ([10, remarks 2.3]). let x be a topological space and a be a subset of x. the following properties hold. (i) d0(a) = µ −1 x (µx(a)). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 98 f -n-resolvable spaces and compactifications (ii) d0(d0(a)) = d0(a). (iii) a ⊆ d0(a) ⊆ a and consequently d0(a) = a. (iv) in particular if a is open (resp., closed ), then d0(a) = a. the following definitions are natural. definition 1.3. a topological space x is called t0-n-resolvable (resp., t0exactly-n-resolvable, t0-strongly-exactly-n-resolvable) if its t0-reflection is nresolvable (resp., exactly-n-resolvable, strongly-exactly-n-resolvable). before giving the characterization of t0-n-resolvable spaces, let us introduce the following definition. definition 1.4. a family {ai : i ∈ i} of subsets of a topological space x is called pairwise d0-disjoint if and only if d0(ai)∩d0(aj) = ∅, for any i 6= j ∈ i. by remarks 1.2 (iii), a pairwise d0-disjoint family is a pairwise disjoint family. the following result characterise t0-n-resolvable spaces. theorem 1.5. let x be a topological space. then the following statements are equivalent: (1) x is a t0-n-resolvable space; (2) x have a dense pairwise d0-disjoint family with cardinality n. proof. (1)=⇒(2) suppose that x is a t0-n-resolvable space. then t0(x) has a dense pairwise disjoint family {µx(ai); 1 ≤ i ≤ n}, where a1,..., an are subsets in x. so applying µ−1 x , one can see easily that {d0(ai) : 1 ≤ i ≤ n} is a family of pairwise disjoint subsets of x. now since µ x is an onto quasihomeomorphism then, by [10, lemma 2.16], we have: ∀1 ≤ i ≤ n x = µ−1 x (t0(x)) = µ −1 x ( µ x (ai) ) = µ−1 x (µ x (ai)) = d0(ai). therefore {ai; 1 ≤ i ≤ n} is a dense pairwise d0-disjoint family of x. (2)=⇒(1) suppose that x has a dense pairwise d0-disjoint family {ai; 1 ≤ i ≤ n} with cardinality n. then, for any 1 ≤ i 6= j ≤ n, the condition d0(ai) ∩ d0(aj) = ∅ implies immediately that µ x (ai) ∩ µx (aj)) = ∅. now, let 1 ≤ i ≤ n. the density of d0(ai) in x shows that: t0(x) = µx(x) = µx(d0(ai)) = µx(µ−1 x (µ x (ai))) = µx(µ −1 x ( µ x (ai) ) ) = µx(ai). therefore, {µx(ai) : 1 ≤ i ≤ n} is a dense pairwise disjoint family of subsets of t0(x). � remark 1.6. clearly every t0-n-resolvable space is a n-resolvable space. the converse does not hold, indeed: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 99 i. dahane, l. dridi and s. lazaar let x be a subset of cardinality n (n > 1) equipped with the indiscreet topology. clearly the family {{x}; x ∈ x} is composed by disjoint dense subsets of x and thus x is n-resolvable. but t0(x) is a one point which is not 2-resolvable. remark that in this case d0({x}) = x, for any x ∈ x and consequently, d0(a) = x for any subset a of x, therefore there is no d0-disjoint family of x with cardinality greater or equal to 2. the following result is an immediate consequence of the previous theorem. corollary 1.7. let x be a topological space. x is a t0-exactly-n-resolvable space if and only if max{| f | f is a dense d0 − disjoint family of x} = n. before giving a characterization of a t0-strongly-exactly-n-resolvable space we need the following lemma. lemma 1.8. let x be a topological space and s a subset of x. then µx(s) ≃ µs(s). proof. s is a subset of x then, the following diagram is commutative. s � i // x µx �� t0(s) �� µs t0(i) // t0(x) t0(i) : t0(s) −→ t0(i)(t0(s)) is bijective. in fact it is enough to show that t0(i) is one-to-one. let x, y two elements of s such that t0(i)(µs(x)) = t0(i)(µs(y)). then, µx(i(x)) = µx(i(y)) and thus µx(x) = µx(y). hence, we get {x} s = {x}∩s = {y} ∩ s = {y} s , as desired. t0(i) is an open map. indeed, let ũ be an open set of t0(s). then, there exists v an open set of x such that µ−1 s (ũ) = v ∩ s. thus t0(i)(ũ) = t0(i)(µs(v ∩ s)) = µx(i(v ∩ s)) = µx(v ∩ s) so, let us show that µx(v ∩ s) = µx(v ) ∩ t0(i)(t0(s)). indeed: µx(v ∩ s) ⊆ µx(v ) ∩ µx(s) = µx(v ) ∩ µx(i(s)) = µx(v ) ∩ t0(i)(µs(s)) = µx(v ) ∩ t0(i)(t0(s)) which gives the first inclusion. conversely, let x ∈ µx(v ) ∩ t0(i)(t0(s)). then there exist y ∈ v and t ∈ s such that µx(y) = x = t0(i)(µs(t)) = µx(i(t)) = µx(t). thus, {y} = {t}. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 100 f -n-resolvable spaces and compactifications since y ∈ v , then v ∩ {t} 6= ∅. so, x = µx(t) ∈ µx(v ∩ s) which proves that µx(v )∩t0(i)(t0(s)) ⊆ µx(v ∩s) which gives the second inclusion as desired. µx(s) ≃ µs(s). according to the above, we conclude that t0(i) is an homeomorphism from t0(s) to t0(i)(t0(s)). then, µx(s) = µx(i(s)) = t0(i)(µs(s)) = t0(i)(t0(s)) ≃ t0(s) = µs(s). � theorem 1.9. let x be a topological space. then the following statements are equivalent: (1) x is a t0-strongly-exactly-n-resolvable space. (2) x is t0-n-resolvable and for any subset s of x, s is not t0-(n + 1)resolvable. proof. (1)=⇒(2) let s be a subset of x. since x is t0-strongly-exactly-n-resolvable, µx(s) is not (n + 1)-resolvable. then, by lemma 1.8, µs(s) = t0(s) is not (n + 1)resolvable. therefore, x is a t0-n-resolvable space in which every subset s of x, is not t0-(n + 1)-resolvable. (2)=⇒(1) let µx(s) be a subset of t0(x), where s be a subset of x. by hypothesis, s is not t0 − (n + 1)-resolvable that is t0(s) = µs(s) is not (n + 1)resolvable. using lemma 1.8, µx(s) is not (n + 1)-resolvable. so that every subset µx(s) of t0(x) is not (n + 1)-resolvable and thus t0(x) is stronglyexactly-n-resolvable. � 2. ρ-n-resolvable spaces and f h-n-resolvable spaces recall that a t1 topological space x is called tychonoff if for any closed subset f of x and for any x ∈ x not in f there exists a real continuous map f from x to r ( we write f ∈ c(x) ) such that f(x) = 0 and f(f) = {1}. we say that f and x are completely separated. in particular two distinct points in a given tychonoff space x are said to be completely separated if x and {y} are completely separated. a t1 topological space in which every two distinct points are completely separated, is called functionally hausdorff space. give a topological space x. we define the equivalence relation ∼ on x by x ∼ y if and only if f(x) = f(y) for all f ∈ c(x). on the one hand, the set of equivalence classes x/ ∼ equipped with the quotient topology, is a functionally hausdorff space called the fh-reflection of x. on the other hand, consider ρx the canonical surjection map from x to x/ ∼. then for any continuous map fα from x to r, there exists a unique map ρ(fα) from x/ ∼ to r satisfying ρ(fα)(ρx(x)) = f(x), for any x ∈ x. so, x/ ∼ equipped with the the topology whose closed sets are of the form ∩[ρ(fα) −1(fα) : α ∈ i], where fα : x −→ r (resp., fα) is a continuous map (resp., a closed subset of r), is a a tychonoff space (see for instance [22]) called the ρ-reflection of x. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 101 i. dahane, l. dridi and s. lazaar we need to introduce and recall some definitions, notations and results. notation 2.1 ([10, notation 3.1]). let x be a topological space, a ∈ x and a a subset of x. we denote by: (1) dρ(a) := ∩[f −1(f({a})) : f ∈ c(x)]. (2) dρ(a) := ∪[dρ(a) : a ∈ a]. definition 2.2. let x be a topological space. x is called: (1) ρ-n-resolvable (resp., fh-n-resolvable) space if its ρ-reflection (resp., fh-reflection) is a n-resolvable space. (2) ρ-exactly-n-resolvable (resp., fh-exactly-n-resolvable) space if its ρreflection (resp., fh-reflection) is an exactly-n-resolvable space. (3) ρ-strongly-exactly-n-resolvable (resp., fh-strongly-exactly-n-resolvable) space if its ρ-reflection (resp., fh-reflection ) is a strongly-exactly-nresolvable space. recall that for a given topological space x and a ⊆ x, a is called a zeroset if there exists f ∈ c(x) such that a = f−1({0}). the complement of a zero-set is called a cozero-set. a space is tychonoff if and only if the family of zero-sets of the space is a base for the closed sets (equivalently, the family of cozero-sets of the space is a base for the open sets)(see [22, proposition 1.7]). in [10] it is showen that a closed (resp., open) subset of ρ(x) is of the form ∩[ρ(f)−1({0}) : f ∈ h] (resp., ∪[ρ(f)−1(r⋆) : f ∈ h]) , where h is a collection of continuous maps from x to r. definition 2.3 ([10, definition 3.14]). let x be a topological space, a subset v of x is called: (i) a functionally open subset of x (for short f-open ) if and only if dρ(v ) is open in x. (ii) a functionally dense subset of x (for short f-dense) if and only if for any f-open subset w of x, dρ(v ) meets dρ(w). (iii) ρ-dense, if g(v ) 6= {0} for every nonzero continuous map g from x to r. definition 2.4. let x be a topological space and {ai : i ∈ i} be a family of subsets of x. we say that this family is pairwise dρ-disjoint if and only if dρ(ai) ∩ dρ(aj) = ∅, for any i 6= j ∈ i. theorem 2.5. let x be a topological space. then the following statements are equivalent: (i) x is fh-n-resolvable. (ii) x have a f-dense pairwise dρ-disjoint family with cardinality n. proof. (i) =⇒ (ii) suppose that x is an fh-n-resolvable space. then, there exists a family {ρx(a1), ..., ρx(an)} of dense pairwise disjoint subsets of fh(x). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 102 f -n-resolvable spaces and compactifications now, applying ρ−1 x , we see easily that the family {a1, ......, an} is pairwise dρ-disjoint. finally, the equality ρx(ai) = fh(x) means that ai is a f-dense subset of x. therefore, {a1, ......, an} is pairwise dρ-disjoint family of x with cardinality n. (ii) =⇒ (i) conversely, let {ai : 1 ≤ i ≤ n} be a family of f-dense pairwise dρ-disjoint subsets of x. then on the one hand, for every 1 ≤ i ≤ n, ρx(ai) is a dense subset of fh(x) and on the other hand, ∀ 1 ≤ i 6= j ≤ n, we have dρ(ai) ∩ dρ(aj)) = ρ −1 x (ρx(a1)) ∩ ρ −1 x (ρx(aj))) = ρ−1 x (ρx(ai) ∩ ρx(aj))) = ∅ therefore, {ρx(a1), ..., ρx(an)} is a family of dense pairwise disjoint subsets of fh(x). � by the same way as in theorem 2.5, the following result is immediate. theorem 2.6. let x be a topological space. then the following statements are equivalent: (i) x is ρ-n-resolvable. (ii) x have a ρ-dense and pairwise dρ-disjoint family of cardinality n. remark 2.7. since every f-dense subset is a ρ-dense subset ( see [10, remarks 3.15] ), then by theorem 2.6, every fh-n-resolvable space is ρ-n-resolvable. the following results are immediate. corollary 2.8. let x be a topological space. x is a fh-exactly-n-resolvable space if and only if max{| f | f is f-dense and dρ−disjoint family of x} = n. corollary 2.9. let x be a topological space. x is a ρ-exactly-n-resolvable space if and only if max{| f | f is ρ-dense and dρ−disjoint family of x} = n. remark 2.10. regarding lemma 1.8, this result does not subsist in the case of the functors fh and ρ as showing by the following example. consider the alexandroff space x = z ∪ {∞} such that {n} = {n}, for every n ∈ z and {∞} = x. it is clear that every real continuous map from x is constant and thus fh(x) = ρ(x) is a one point space. now, consider s = z, then fh(s) = ρ(s) = s, but ρx(s) is a one point. one can illustrates this situation by the following picture. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 103 i. dahane, l. dridi and s. lazaar .. . -4 -3 -2 -1 0 1 2 3 4 . . . ∞ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚✚ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩❩ question 2.11. the theorem 1.9 is an immediate consequence of lemma 1.8 which is not valuable in the case of the functors fh and ρ as showing by remark 2.10. hence the following question is immediate. are fh-stronglyexactly-n-resolvable (resp., ρ-strongly-exactly-n-resolvable) spaces equivalent to fh-n-resolvable (resp., ρ-n-resolvable) in which every subset s of x, is not fh-(n + 1)-resolvable (resp., ρ-(n + 1)-resolvable)? 3. n-resolvable spaces and compactifications definition 3.1. a compactification of a topological space x is a pair (k(x), e) where k(x) is a compact space and e an embedding of x as a dense subset of k(x). remark 3.2. in many cases, e will be an inclusion map, so that x ⊆ k(x). in other cases , we can agree to write x when mean e(x), so that we can again write x ⊆ k(x). whenever one of this situations occurs we say simply that k(x) is a compactification of x, and think of k(x) as containing x as a dense subspace. lemma 3.3 ([2, lemma 2.1]). let x be a topological space and k(x) be a compactification of x and a be a subset of k(x). if x is an open set of k(x) then the following statements are equivalent: (1) a is a dense subset of k(x). (2) a ∩ x is a dense subset of x. using lemma 3.3, the following proposition is immediate. proposition 3.4. let x be a topological space and k(x) be a compactification of x. if x is an open set of k(x) then the following statements are equivalent: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 104 f -n-resolvable spaces and compactifications (1) x is n-resolvable. (2) k(x) is n-resolvable. recall that for a topological space x, the set x̃ = x∪{∞} with the topology whose members are the open sets of x and all subsets u of x̃ such that x̃ \ u is a closed compact subset of x, is called the alexandroff extension of x ( or the one-point compactification of x ). now, regarding proposition 3.4, we get immediately the following result. corollary 3.5. let x be a non compact topological space then the following statements are equivalent: (1) the one point compactification x̃ of x is n-resolvable. (2) x is n-resolvable. we turn our attention to spaces such that their wallman compactifications are n-resolvable spaces. first, let us recall the construction of wallman compactification of t1-space (a concept introduced, in 1938, by wallman [23]). let p be a class of subsets of a topological space x wich is closed under fnite intersections and finite unions. a p-filter on x is a collection f of nonempty elements of p with the properties: (a) p1, p2 ∈ f implies p1 ∩ p2 ∈ f. (b) p1 ∈ f p1 ⊆ p2 implies p2 ∈ f. a p-ultrafilter is a maximal p-filter. when p is the class of closed sets of x, then the p-filters are called closed filters. the points of the wallman compactification wx of a space x are the closed ultrafilters on x. for each closed set d ⊆ x, define d∗ to be the set d∗ = {a ∈ wx : d ∈ a}, if d 6= ∅ and ∅∗ = ∅. thus c = {d∗ : d is a closed set of x} is a base for the closed sets of a topology on wx. let u be an open subset of x. we define u∗ = {a ∈ wx : f ⊆ u for some f in a}. it is easily seen that the class {u∗ : u is an open set of x} is a base for the open sets of the topology of wx. the following properties of wx are frequently useful: proposition 3.6. for x ∈ x, let w x (x) = {a | a is a closed set of x and x ∈ a}. then wx is an embedding of x into wx. thus, if x ∈ x, then wx (x) will be identified to x. proposition 3.7. if u ⊂ x is open, then wx \ u∗ = (x \ u)∗. proposition 3.8. if d ⊂ x is closed, then wx \ d∗ = (x \ d)∗. proposition 3.9. if u1 and u2 are open in x, then (u1 ∩ u2) ∗ = u∗1 ∩ u ∗ 2 and (u1 ∪ u2) ∗ = u∗1 ∪ u ∗ 2 . in [19], kovar has characterized space with finite wallman compactification remainder as following: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 105 i. dahane, l. dridi and s. lazaar proposition 3.10. let x be a t1-space, wx the wallman compactification of x and k a finite number. then the following statements are equivalent: (1) card(wx − x) = k. (2) there exists a collection of k pairwise disjoint non compact closed sets of x and every family of non compact pairwise disjoint closed sets of x contain at most k elements. the following proposition follows immediately from proposition 3.10. proposition 3.11. let x be a t1-space and k ∈ n such that every family of non compact pairwise disjoint closed sets of x contain at most k elements. then x is n-resolvable if and only if wx is n-resolvable. corollary 3.12 ([2, corollary 3.5]). let x be a t1-space, wx be the wallman compactification of x and u be an open set of x. then the following statements are equivalent: (1) u u∗. (2) there exists a non compact closed set f of x such that f ⊆ u. definition 3.13. let x be a t1-topological space. then x is said to be w-n-resolvable, if its wallman compactification is n-resolvable. before characterizing w-n-resolvable spaces, let us introduce the useful definition. definition 3.14. we said that a finite family of subsets {di, i ∈ i} of a topological space (x, o(x)) satisfies the property (p) if: for every (i, o) ∈ j = i × {o ∈ o(x) : o ∩ di = ∅}, there exists a non compact closed subset fo,i ⊂ o with {fo,i : (i, o) ∈ j} is a family of pairwise disjoint subsets of x. now, let us give one of the main result of this section. theorem 3.15. let x be a t1topological space, then the following statements are equivalent: (1) x is w-n-resolvable. (2) x is a partition of a family of n subsets satisfying (p). proof. let x be a w-n-resolvable space. then there exist n pairwise disjoint dense subsets a1,a2,..., an of wx such that wx = a1 ∪ a2 ∪ ... ∪ an. we denote di = ai∩x. it is clear that the family {di; 1 ≤ i ≤ n} is a partition of x. let o be a nonempty open subset of x such that o ∩ di = ∅. the density of ai in wx gives an element fi ∈ o ∗ ∩ ai. by corollary 3.12, there exists a non compact closed subset g(i,o) ⊂ o such that g(i,o) ∈ fi. now, if i′ is distinct from i and o′ is a given nonempty open subset of x such that o′ ∩di′ = ∅, by the same way, there exists an element fi′ ∈ o ′∗ ∩ai′ and consequently there exists a non compact closed subset g(i′,o′) ⊂ o ′ such that g(i′,o′) ∈ fi′. since ai ∩ ai′ = ∅, then fi 6= fi′. thus, there exist a c© agt, upv, 2019 appl. gen. topol. 20, no. 1 106 f -n-resolvable spaces and compactifications closed subsets fi ∈ fi and fi′ ∈ fi′ such that fi ∩ fi′ = ∅. let f(i,o) = g(i,o) ∩ fi and f(i′,o′) = g(i′,o′) ∩ f ′ i . it is clear that f(i,o) ∈ fi ∈ wx \ x and f(i′,o′) ∈ fi′ ∈ wx \ x. hence, f(i,o) and f(i′,o′) are non compact closed subsets ( see [2, lemma 3.4]), which are disjoint. conversely, let {di; 1 ≤ i ≤ n} be a partition of x by n subsets satisfying (p). for every 1 ≤ i ≤ n, set ai = di ∪ {f ∈ wx − x : f(i,o) ∈ f}, where o is an open subset of x such that o ∩di = ∅ (it is clearly seen that if ai = di, then di is dense in wx). clearly, by construction, ai is a dense subset of wx for every 1 ≤ i ≤ n. to finish, let us show that the family {ai : 1 ≤ i ≤ n} are pairwise disjoint. so, suppose the existence of 1 ≤ i 6= j ≤ n such that ai ∩ aj 6= ∅. since di ∩ dj = ∅, then ai ∩ aj ∩ (wx − x) 6= ∅. by construction of ai and aj, there exist an ultrafilter fi ∈ ai and fj ∈ aj such that fi = fj. furthermore, there exist open subsets o, o′ and non compact closed subsets f(i,o) ∈ fi and f(j,o′) ∈ fj such that o ∩ di = ∅, o ′ ∩ dj = ∅, f(i,o) ⊂ o and f(j,o′) ⊂ o ′. hence, by the property (p), f(i,o) ∩ f(j,o′) = ∅ and consequently fi 6= fj, which leads to a contradiction. � as an immediate consequence of theorem 3.15, for the particular case when n = 2, we have the following corollary. corollary 3.16 ([2, theorem 3.6]). let x be a t1topological space, then the following statements are equivalent: (1) x is w-resolvable. (2) x is a partition of two subsets {d1, d2} and for each nonempty open subset o ⊆ di (i ∈ {1, 2}), there exists a non compact closed subset f such that f ⊆ o. to close this section the following result is immediate. corollary 3.17. let x be a t1-topological space. x is w-exactly-n-resolvable if and only if max{| f | ; f is a partition of x, of n dense subsets satisfying (p)} = n. acknowledgements. the authors gratefully acknoweledge helpful corrections, comments and suggestions of the referee. this paper is supported by the latao lr11es12. references [1] a. v. arhangel’skii and a. j. collins, on submaximal spaces, topology appl. 64 (1995), 219–241. [2] m. al-hajri and k. belaid, resolvable spaces and compactifications, advances in pure mathematics 3 (2013), 365–367. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 107 i. dahane, l. dridi and s. lazaar [3] k. belaid and l. dridi, i-spaces, nodec spaces and compactifications, topology appl. 161 (2014), 196–205. 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[23] h. wallman, lattices and topological spaces, ann. math. 39 (1938), 112–126. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 108 @ appl. gen. topol. 22, no. 2 (2021), 355-366doi:10.4995/agt.2021.14868 © agt, upv, 2021 further aspects of ik-convergence in topological spaces ankur sharmah and debajit hazarika department of mathematical sciences, tezpur university, napam 784028, assam, india (ankurs@tezu.ernet.in, debajit@tezu.ernet.in) communicated by d. georgiou abstract in this paper, we obtain some results on the relationships between different ideal convergence modes namely, i k , i k ∗ , i, k, i ∪ k and (i ∪k)∗. we introduce a topological space namely ik-sequential space and show that the class of i k -sequential spaces contain the sequential spaces. further i k -notions of cluster points and limit points of a function are also introduced here. for a given sequence in a topological space x, we characterize the set of i k -cluster points of the sequence as closed subsets of x. 2010 msc: 54a20; 40a05; 40a35. keywords: i-convergence; ik-convergence; ik ∗ -convergence; i k -sequential space; i k -cluster point. 1. introduction for basic general topological terminologies and results we refer to [5]. the ideal convergence of a sequence of real numbers was introduced by kostyrko et al. [11], as a natural generalization of existing convergence notions such as usual convergence [5], statistical convergence [4]. it was further introduced in arbitrary topological spaces accordingly for sequences [3] and nets [2] by das et al. the main goal of this article is to study ik-convergence which arose as a generalization of a type of ideal convergence. in this continuation we begin with a prior mentioning of ideals and ideal convergence in topological spaces. received 01 january 2020 – accepted 26 march 2021 http://dx.doi.org/10.4995/agt.2021.14868 a. sharmah and d. hazarika an ideal i on a arbitrary set s is a family i ⊂ 2s (the power set of s) that is closed under finite unions and taking subsets. fin and i0 are two basic ideals on ω, the set of all natural numbers, defined as fin:= collection of all finite subsets of ω and i0:= subsets of ω with density 0, we say a(⊂ ω) ∈ i0 if and only if lim supn→∞ |a∩{1,2,...,n}| n = 0. for an ideal i in p(ω), we have two additional subsets of p(ω) namely i⋆ and i+, where i⋆ := {a ⊂ ω : ac ∈ i}, the filter dual of i and i+:= collection of all subsets not in i. clearly, i⋆ ⊆i+. a sequence x = {xn}n∈ω is said to be i-convergent [3] to ξ, denoted by xn →i ξ, if {n : xn /∈ u} ∈ i, for all neighborhood u of ξ. a sequence x = {xn}n∈ω of elements of x is said to be i ⋆-convergent to ξ if there exists a set m := {m1 < m2 < ... < mk < ...} ∈ i ⋆ such that limk→∞ xmk = ξ. lahiri and das [3] found an equivalence between i and i⋆-convergences under certain assumptions. in 2011, macaj and sleziak [6] introduced the ik-convergence of function in a topological space, which was derived from i∗-convergence [3] by simply replacing fin by an arbitrary ideal k. interestingly, ik-convergence arose as an independent mode of convergence. comparisons of ik-convergence with i-convergence [11] can be found in [1, 6, 8]. a few articles for example [9, 7] contributed to the study of ik-convergence of sequence of functions. some of the definitions and results of [3, 6] that are used in subsequent sections are listed below. here x is a topological space and s is a set where ideals are defined. we say that a function f : s → x is im-convergent to a point x ∈ x if ∃m ∈i∗ such that the function g : s → x given by g(s) = { f(s), s ∈ m x, s /∈ m is m-convergent to x, where m is a convergence mode via ideal. if m = k∗, then f : s → x is said to be ik ∗ -convergent [6] to a point x ∈ x. also, if m = k, then f : s → x is said to be ik-convergent [6] to a point x ∈ x. in particular, if x is a discrete space, our immediate observation is that only the i-constant functions are i-convergent, for a given ideal i, f : s → x is an i-constant function if it attains a constant value except for a set in i. it follows that i and i∗ convergence coincide for x. thus, ik and ik ∗ -convergence modes also coincide on discrete spaces. lemma 1.1 ([6, lemma 2.1]). if i and k are two ideals on a set s and f : s → x is a function such that k− lim f = x, then ik − lim f = x. remark 1.2. we say two ideals i and k satisfy ideality condition if i ∪k is an proper ideal [10]. again, i and k satisfy ideality condition if and only if s 6= i ∪ k, for all i ∈i, k ∈k. the main results of this article are divided into 3 sections. section 2 is devoted to a comparative study of different convergence modes for example ik, ik ∗ , i, k, i ∪k, (i ∪k)∗ etc. we justify the existence of an ideal j , © agt, upv, 2021 appl. gen. topol. 22, no. 2 356 further aspects of ik-convergence in topological spaces such that the behavior of ik and j -convergence coincides in hausdorff spaces. then in section 3, we introduce ik-sequential space and study its properties. in section 4 we basically define ik-cluster point and ik-limit point of a function in a topological space. here we observe that the ideality condition of i and k in ik-convergence allows to get some effective conclusions. moreover, we characterize the set of ik-cluster points of a function as closed sets. throughout this paper we focus on the proper ideals [10] containing fin (s /∈i). 2. ik-convergence and several comparisons in this section, we study some more relations among different convergence modes ik, ik ∗ , i∪k, (i∪k)∗ etc. we mainly focus on ik-convergence where i∪k forms an ideal. proposition 2.1. let x be a topological space and f : s → x be a function. let i,k be two ideals on s such that i∪k is an ideal. then (i) ik ∗ − lim f = x if and only if (i∪k)∗ − lim f = x. (ii) ik − lim f = x implies i∪k− lim f = x. proof. (i) let f : s → x be ik ∗ -convergent to x. so, there exists a set m ∈i∗ for which the function g : s → x such that g(s) = { f(s), s ∈ m x, s /∈ m is k∗-convergent to x. so, there further exists a set n ∈k∗ for which we can consider the function h : s → x such that h(s) = { f(s), s ∈ m, s ∈ n x, s /∈ m or s /∈ n is fin-convergent to x. now, let k = n∁ ∈ k, i = m∁ ∈ i (say). then h(s) = { f(s), s ∈ (i ∪k)∁ x, s /∈ (i ∪k)∁. in essence, we can conclude f is (i∪k)∗-convergent to x. conversely, the function f : s → x is (i ∪k)∗-convergent to x. so, there exists a set p = (i ∪ k)∁ ∈ (i ∪k)∗ for which the function h : s → x such that h(s) = { f(s), s ∈ p x, s /∈ p h(s) = { f(s), s ∈ (i ∪k)∁ x, s /∈ (i ∪k)∁ © agt, upv, 2021 appl. gen. topol. 22, no. 2 357 a. sharmah and d. hazarika is fin-convergent to x. lets consider the function g : s → x defined as g(s) = { f(s), s ∈ i∁ x, s /∈ i∁ for which the function h : s → x such that h(s) = { f(s), s ∈ i∁, s ∈ k∁ x, s /∈ (i ∪k)∁ is fin-convergent to x. consequently, f is ik ∗ -convergent to x. (ii) let f : s → x be ik-convergent to x. so, there exists a set m ∈ i∗ for which the function g : s → x such that g(s) = { f(s), s ∈ m x, s /∈ m is k-convergent to x. then for each ux, neighborhood of x, we have {s : g(s) /∈ ux} ∈ k. accordingly, the set given by {s : f(s) /∈ ux, s ∈ m} ∈ k. further {s : f(s) /∈ ux} ⊆ {s : f(s) /∈ ux, s ∈ m}∪{s : s /∈ m}. hence, {s : f(s) /∈ux}∈i∪k. � following are immediate corollaries of the above proposition provided i∪k is an ideal. corollary 2.2. ik ∗ -convergence implies i− convergence. corollary 2.3. ik ∗ -convergence implies k− convergence. following results in [1] are corollaries of the above proposition. corollary 2.4. ik-convergence implies i− convergence provided k⊆i. corollary 2.5. ik-convergence implies k− convergence provided i ⊆k. following diagram shows the connections between different convergence modes. i∪k←ik ←i∗ → (i∪k)∗ ≡ik ∗ →ik j in this segment we are interested to find whether there exists an ideal j such that the behavior of ik and j -convergence coincides. recalling that a filter-base is a non empty collection closed under finite intersection, we have the following result for a given function f in x by taking an ideal-base to be complement of a filter-base. lemma 2.6. let i and k be two ideals on s satisfying ideality condition. f : s → x be a function on a topological space x. if j = ideal generated by (k∪j), for any j ∈i. then f is j-convergent to x =⇒ f is ik-convergent to x. © agt, upv, 2021 appl. gen. topol. 22, no. 2 358 further aspects of ik-convergence in topological spaces proof. let f be j -convergent to x, where j= ideal generated by the ideal base (k∪ j), for any j ∈ i. now for j = mc, consider the function g : s → x defined as g(s) = { f(s), s ∈ m x, s /∈ m. then, for any open set v containing x, we have {s ∈ s : g(s) /∈ v} = {s ∈ s : f(s) /∈ v, s ∈ m} ⊆{s ∈ s : f(s) /∈ v}\{s ∈ s : s /∈ m}. since, f be j -convergent to x, that implies {s ∈ s : f(s) /∈ v}∈j . therefore, there exists k ∈ k such that {s ∈ s : f(s) /∈ v} \ j ⊆ (k ∪ j) \ j ∈ k. subsequently, g is k− convergent to x. hence, f is ik-convergent to x. � theorem 2.7 ([1, theorem 3.1]). in a hausdorff space x, each function f : s → x possess a unique ik-limit provided i∪k is an ideal. theorem 2.8. let x be a hausdorff space. let f : s → x be ik-convergent to x. then ∃ an ideal j such that x ∈ x is an ik-limit of the function f if and only if x is also a j -limit of f provided i∪k is an ideal. proof. let f : s → x is ik-convergent to x. so, there exists a set m ∈ i∗ such that g : s → x with g(s) = { f(s), s ∈ m x, s /∈ m is k-convergent to x. consequently, for each neigfhborhood ux of x. we have {s ∈ s : g(s) /∈ux}∈k. =⇒ {s ∈ s : f(s) /∈ux, s ∈ m}∈k. now, let j = mc and (k∪ j) is an ideal base provided (i ∪k) is an ideal. now we consider j , the ideal generated by (k∪ j). then {s ∈ s : f(s) /∈ux}⊆{s ∈ s : f(s) /∈ux, s ∈ m}∪{s ∈ s : s /∈ m}. therefore, {s ∈ s : f(s) /∈ux}∈ (k∪ j). converse part of the proof is immediate by lemma 2.6. � the following arrow diagram exhibit the equivalence shown in theorem 2.8. k for any j∈i −−−−−−−−→j →ik fixed j∈i −−−−−−→j →i∪k comprehensively, we may ask the following question. problem. whether there exists an ideal j for ik-convergence in a given non-hausdorff topological space x such that ik ≡j -convergence? © agt, upv, 2021 appl. gen. topol. 22, no. 2 359 a. sharmah and d. hazarika 3. ik-sequential space recently, i-sequential space were defined by s.k. pal [12] for an ideal i on ω. an equivalent definition was suggested by zhou et al. [13] and further obtain that class of i-sequential spaces includes sequential spaces [5]. first, recall the notion of i-sequential spaces. let x be a topological space and o ⊆ x is i-open if no sequence in x\o has an i-limit in o. equivalently, for each sequence {xn : n ∈ ω}⊆ x\o with i−lim xn = x ∈ x, then x ∈ x\o. now x is said to be an i-sequential space if and only if each i-open subset of x is open. here we introduce a topological space namely ik-sequential space for given ideals i and k on ω. definition 3.1. let x be a topological space and o, a ⊆ x. then (1) o is said to be ik-open if no sequence in x \o has an ik-limit in o. otherwise, for each sequence {xn : n ∈ ω}⊆ x\o with i k − lim xn = x ∈ x, then x ∈ x \o. (2) a subset f ⊆ x is said to be ik-closed if x \a is ik-open in x. remark 3.2. the following are obvious for a topological space x and ideals i and k on ω. 1. each open(closed) set of x is ik-open(closed). 2. if a and b are ik-open (closed), then a ∪ b is ik-open (closed). 3. a topological space x is said to be an ik-sequential space if and only if each ik-open set of x is open. for i = k, each ik-sequential space coincides with a i-sequential space. lemma 3.3. let m1,m2 be two convergence modes in a topological space x such that m1-convergence implies m2-convergence. then o ⊆ x is m2-open implies that o is m1-open. proof. let o be not m1-open in x, then ∃{xn} in (x \ o) which is m1convergent in x. so, {xn} is (x \ o) is m2-convergent in x and hence o is not m2-open. � corollary 3.4. let m1,m2 be two convergence modes in x such that m1convergence implies m2-convergence in x. then x is a m1-sequential space implies that x is m2-sequential space. the following is an example of a topological space which is not ik-sequential space. example 3.5. let s = [a, b] be a closed interval with the countable complement topology τcc, where a, b ∈ r. let a be any subset of s and xn be a sequence in a, ik-convergent to x, provided i,k and i ∪k is an ideal i.e, i ∪k− lim xn = x. consider the neighborhood u of x, be the complement of the set {xn : xn 6= x} in s. then xn = x for all n except for a set in the ideal i ∪k. therefore, a sequence in any set a can only i ∪k-convergent to © agt, upv, 2021 appl. gen. topol. 22, no. 2 360 further aspects of ik-convergence in topological spaces an element of a i.e c is i∪k-open. thus every subset of c is ik-sequentially open. but not every subset of s is open. hence ([a, b], τcc) is not i k-sequential. proposition 3.6. let x be a topological space and i1,i2,k1,k2 be ideals on s. then the following implications hold: (1) for k1 ⊆k2 whenever u ⊆ x is i k2-open, then it is ik1-open. (2) for i1 ⊆i2 whenever u ⊆ x is i k 2 -open, then it is i k 1 -open. proof. let f : s → x be a function. then by proposition 3.6 in [6], ik1 − lim f = x =⇒ i k 2 − lim f = x. ik1 − lim f = x =⇒ ik2 − lim f = x. by lemma 3.3 we have the required results correspondingly. � corollary 3.7. for x be topological space and i1,i2,k1,k2 be ideals on ω where i1 ⊆i2, k1 ⊆k2. then the following observations are valid: (1) if x is ik1 -sequential, then it is i k 2 -sequential. (2) if x is ik1-sequential, then it is ik2-sequential. theorem 3.8. in a topological space x, if o is open then o is ik-open. proof. let o be open and {xn} be a sequence in x\o. let y ∈ o. then there is a neighborhood u of y which contained in o. hence u can not contain any term of {xn}. so y is not an i k-limit of the sequence and o is ik-open. � theorem 3.9. in a metric space x, the notions of open and ik-open coincide. proof. forward implication is obvious from theorem 3.8. conversely, let o be not open i.e., ∃y ∈ o such that for all neighborhood of y intersect (x \ o). let xn ∈ (x \ o) ∩ b(y, 1 n+1 ). then xn → y. hence xn is ik-convergent to y. thus o is not ik-open. � theorem 3.10. every first countable space is ik-sequential space. proof. we need to prove the reverse implication. if a ⊂ x be not open. then ∃y ∈ a such that every neighborhood of y intersects x \ a. let {un : n ∈ ω} be a decreasing countable basis at y (say). consider xn ∈ (x \a)∩un. then for each neighborhood v of y, ∃n ∈ ω with un ⊂ v . so, xm ∈ v,∀m ≥ n i.e xn → y. hence k− lim xn = y. therefore, a is not ik-open. � the following theorem about continuous mapping was also proved by banerjee et al. [1]. however, we have given here an alternative approach to prove. theorem 3.11. every continuous function preserves ik-convergence. proof. let x and y be two topological spaces and c : x → y be a continuous function. let f : s → x be ik-convergent. so ∃m ∈i∗ such that g : s → x given by © agt, upv, 2021 appl. gen. topol. 22, no. 2 361 a. sharmah and d. hazarika g(s)= { f(s), s ∈ m x, s /∈ m is k-convergent to x. now the function c ◦ f : s → y , the image function on y is k-convergent to x by theorem 3 in [3]. hence c◦f is ik-convergent. � we now recall the definition of a quotient space. let (x,∼) be a topological space with an equivalence relation ∼ on x. consider the projection mapping ∏ : x → x/ ∼ (the set of equivalence classes) and taking a ⊂ x/ ∼ to be open if and only if ∏−1 (a) is open in x, we have the quotient space x/ ∼ induced by ∼ on x. theorem 3.12. every quotient space of an ik-sequential space x is iksequential. proof. let a ⊂ x/ ∼ be not open. let x/ ∼ is a quotient space with an equivalence relation ∼, ∏−1 (a) is not open in x i.e., ∃ a sequence {xn} in x \ ∏−1 (a) which is ik-convergent to y ∈ ∏−1 (a). also ∏ is continuous, hence preserves ik-convergence by theorem 3.11. therefore, ∏ (xn) ∈ (x/ ∼) \ a with ik-limit ∏ (y) ∈ a. so, a is not ik-open i.e., x/ ∼ is iksequential. � following result is immediate via proposition 3.4. theorem 3.13. every sequential space is an ik-sequential space. recall that a topological space x is said to be of countable tightness, if for a ⊆ x and x ∈ ā, then x ∈ c̄ for some countable subset c ⊆ a. every sequential and i-sequential space is of countable tightness [13]. proposition 3.14. every ik-sequential space x is of countable tightness. proof. let x be an ik-sequential space and a ⊆ x. consider [a]ω = ⋃ {b : b is a countable subset of a}. clearly, a ⊆ [a]ω ⊆ a. we claim that, [a]ω is ik-closed in x. consider {xn} be a sequence in [a]ω, i k-convergent to x ∈ x. since xn ∈ [a]ω, then we can find a countable subset b of a such that xn ∈ b for all n ∈ ω. since x be an ik-sequential space, so b is ik-closed, thus x ∈ b ⊆ [a]ω, and further [a]ω is i k-closed in x. now, let x be an ik-sequential space and a be a subset of x. since the set [a]ω is closed in x, and [a]ω ⊆ a ⊆ [a]ω, thus a = [a]ω. if x ∈ a, then x ∈ [a]ω, and further, there exists a countable subset c of a such that x ∈ c, i.e., x is of countable tightness. � now we show that every ik-sequential space is hereditary with respect to ik-open (ik-closed) subspaces. first we have the following lemma. lemma 3.15 ([13, lemma 2.4]). let i be an ideal on ω and xn, yn be two sequences in a topological space x such that {n ∈ ω : xn 6= yn} ∈ i. then i− lim xn = x if and only if i− lim yn = x. © agt, upv, 2021 appl. gen. topol. 22, no. 2 362 further aspects of ik-convergence in topological spaces theorem 3.16. if x is an ik-sequential space then every ik-open (ik-closed) subspaces of x is ik-sequential. proof. let x be an ik-sequential space. suppose that y is an ik-open subset of x. then y is also open in x. we anticipate y to be ik-sequential space. consider u to be ik-open in y . here y is open, so we claim that u is open in x. since x is ik-sequential space, we need to show that u is ik-open in x. contra-positively, take u be not ik-open in x. then, ∃{xn} in x \ u such that ik − lim xn = x (∈ u.) i.e. ∃m ∈i ∗ such that xnk →k x, where nk ∈ m and xnk ∈ x \ u. now {nk : xnk /∈ y } ∈k. for a point y ∈ y \ u (assume), now consider a sequence {yn} such that yn = xn for n ∈ m and yn = ynk for n /∈ m where {ynk} is defined as ynk = xnk for xnk ∈ y and ynk = y for xnk /∈ y . then by lemma 3.15, {ynk} is k-convergent to x. hence {yn} is ik-convergent to x. so u is not ik-open in y . that is a contradiction to our assumption. let y be an ik-closed subset of x. then y is closed in x. for any ikclosed subset f of y , it is sufficient to show that f is closed in x. since x is an ik-sequential space, it is enough to show that f is ik-closed in x. therefore, let {xn : n ∈ ω} be an arbitrary sequence in f with i k − lim xn = x in x. we claim that x ∈ f . indeed, since y is closed, we have x ∈ y , and then it is also clear that x ∈ f since f is an ik-closed subset of y . � proposition 3.17. the disjoint topological sum of any family of ik-sequential spaces is ik-sequential. proof. let (xα)α∈∆ be a family of i k-sequential space and x = ⊕α∈∆xα. we claim that x is ik-sequential space. let f be ik-closed in x. for each α ∈ ∆, xα is closed in x i.e., xα is i k-closed in x. hence, f∩xα is i k-closed in x by remark 3.2. as (f ∩ xα) ⊆ xα i.e. f ∩ xα is closed in xα. now f is closed in x ≡ x \f is open in x ≡∪α(xα \f) is open in x if and only if xα \f is open in xα ≡ f ∩ xα is closed in xα. hence f is closed in x. � 4. ik-cluster point and ik-limit point the notions i-cluster point and i-limit point in a topological space x were defined by das et al. [3] and also characterized cx(i), the collection of all i-cluster points of a given sequence x = {xn} in x, as closed subsets of x (theorem 10, [3]). here we define ik-notions of cluster point and limit points for a function in x. for i∗-convergence, i∪fin is an ideal, thereupon i and fin satisfy ideality condition. moreover we assume ideality condition of i and k in ik-convergence to investigate some results. definition 4.1. let f : s → x be a function and i, k be two ideals on s. then x ∈ x is called an ik-cluster point of f if there exists m ∈i∗ such that the function g : s → x defined by g(s)= { f(s), s ∈ m x, s /∈ m © agt, upv, 2021 appl. gen. topol. 22, no. 2 363 a. sharmah and d. hazarika has a k-cluster point x, i.e., {s ∈ s : g(s) ∈ ux} /∈k. definition 4.2. let f : s → x be a function and i, k be two ideals on s. then x ∈ x is called an ik-limit point of f if there exists m ∈ i∗ such that for the function g : s → x defined by g(s)= { f(s), s ∈ m x, s /∈ m has a k-limit point x. for i = k, we know the convergence modes ik ≡ i ≡ k. hence definitions 4.1 and 4.2 generalizes the definitions of i or k-(limit point and cluster point) correspondingly. again, for nets in a topological space i-limit points and icluster points coincide [2]. therefore, ik-cluster points and ik-limit points of nets also coincide. following the notation in [11], we denote the collection of all ik-limit points and ik-cluster points of a function f in a topological space x by lf(i k) and cf (i k) respectively. we observe that cf (i k) ⊆ cf (k) and lf(i k) ⊆ lf(k). we also observe that lf (i ∗) = l(i∗), where l(i∗) denote the collection of i∗-limits of f. lemma 4.3. if i and k be two ideal then lf (i k) ⊆ cf (i k). proof. since lf (k) ⊆ cf (k) for an ideal k, hence the result is immediate. � we have the following lemma provided the ideals i and k satisfy ideality condition. lemma 4.4. cf (i∪k) ⊆ cf (i k). proof. let y be not a ik-cluster point of x = {xn}n∈ω. then for all m ∈ i ∗ such that for the function g : s → x defined by g(s)= { f(s), s ∈ m x, s /∈ m, the set {s ∈ s : g(s) ∈ ux}∈k. since {s : f(s) ∈ ux}⊆{s : g(s) ∈ ux}∈k. i.e. {s : f(s) ∈ ux}∈i∪k. hence y is not a (i∪k)-cluster point of x. � since above set inequalities signify the implication k → ik → i ∪k, we expect the following conclusion. conjecture 4.5. lf(i∪k) ⊆ lf(i k). for sequential criteria in [11], we observe the following result. theorem 4.6. let i, k be two ideals on ω and x be a topological space. then (i) for x = {xn}n∈ω, a sequence in x; cx(i k) is a closed set. (ii) if (x, τ) is closed hereditary separable and there exists a disjoint sequence of sets {pn} such that pn ⊂ ω, pn /∈ i,k for all n, then for every non empty closed subset f of x, there exists a sequence x in x such that f = cx(i k) provided i∪k is an ideal. © agt, upv, 2021 appl. gen. topol. 22, no. 2 364 further aspects of ik-convergence in topological spaces proof. consider the sequence x = {xn} in x and i, k be the two ideals on ω. (i) let y ∈ cx(ik); the derived set of cx(i k). let u be an open set containing y. it is clear that u ∩ cx(i k) 6= φ. let p ∈ (u ∩ cx(i k)) i.e., p ∈ u and p ∈ cx(i k). now there exist a set m ∈ i∗, such that {yn}n∈ω given by yn = xn if n ∈ m and p, otherwise; we have {n ∈ ω : yn ∈ u} /∈ k. consider the sequence {zn}n∈ω given by zn = xn if n ∈ m and y, otherwise; then {n ∈ ω : zn ∈ u} = {n ∈ ω : yn ∈ u} /∈k. hence y ∈ cx(i k). (ii) being a closed subset of x, f is separable. let s = {s1, s2, ...}⊂ f be a countable set such that s = f . consider xn = si for n ∈ pi. thus we have the subsequence {kn} of {n} for which assume the sequence x = {xnk}. let y ∈ cx(k) (taking y 6= si otherwise if y = si for some i, then y is eventually in f). we claim cx(k) ⊂ f . let u be any open set containing y. then {n : xnk ∈ u} /∈ k and hence non empty i.e., si ∈ u for some i. therefore f ∩u is non empty, so y is a limit point of f and closedness of f gives y ∈ f . hence cx(k) ⊂ f . further cx(i k) ⊆ cx(k) ⊂ f . conversely, for a ∈ f and u be an open set containing a, then there exists si ∈ s such that si ∈ u. then {n : xnk ∈ u} ⊃ pi (/∈ k, i). thus {n : xnk ∈ u} /∈ (i ∪k) i.e., a ∈ cx(i ∪k). on the otherhand, by lemma 4.4, cf (i∪k) ⊆ cf (i k). so we get the reverse implication. � remark 4.7. theorem 4.6 generalizes theorem 10 in [3], it follows by letting i = k in the above theorem. acknowledgements. the first author would like to thank the university grants comission (ugc) for awarding the junior research fellowship vide ugcref. no.: 1115/(csir-ugc net dec. 2017), india. references [1] a. k. banerjee and m. paul, a note on ik and ik ∗ -convergence in topological spaces, arxiv:1807.11772v1 [math.gn], 2018. [2] b. k. lahiri and p. das, i and i⋆-convergence of nets, real anal. exchange 33 (2007), 431–442. [3] b. k. lahiri and p. das, i and i⋆-convergence in topological spaces, math. bohemica 130 (2005), 153–160. [4] h. fast, sur la convergence statistique, colloq. math, 2(1951), 241–244. [5] k. p. hart, j. nagata and j. e. vaughan. encyclopedia of general topology, elsevier science publications, amsterdam-netherlands, 2004. [6] m. macaj and m. sleziak, ik-convergence, real anal. exchange 36 (2011), 177–194. © agt, upv, 2021 appl. gen. topol. 22, no. 2 365 a. sharmah and d. hazarika [7] p. das, m. sleziak and v. tomac, ik-cauchy functions, topology appl. 173 (2014), 9–27. [8] p. das, s. dasgupta, s. glab and m. bienias, certain aspects of ideal convergence in topological spaces, topology appl. 275 (2020), 107005. [9] p. das, s. sengupta, j and supina, ik-convergence of sequence of functions, math. slovaca 69, no. 5(2019), 1137–1148. [10] p. halmos and s. givant, introduction to boolean algebras, undergraduate texts in mathematics, springer, new york, 2009. [11] p. kostyrko, t. salat and w. wilczynski, i-convergence, real anal. exchange 26 (2001), 669–685. [12] s. k. pal, i-sequential topological spaces, appl. math. e-notes 14 (2014), 236–241. [13] x. zhou, l. liu and l. shou, on topological space defined by i-convergence, bull. iranian math. soc. 46 (2020), 675–692. © agt, upv, 2021 appl. gen. topol. 22, no. 2 366 @ appl. gen. topol. 21, no. 1 (2020), 159-169 doi:10.4995/agt.2020.12238 c© agt, upv, 2020 selection principles and covering properties in bitopological spaces moiz ud din khan and amani sabah department of mathematics, comsats university islamabad, chack shahzad, park road, islamabad 45550, pakistan (moiz@comsats.edu.pk, amaniusssabah@gmail.com) communicated by s. özçăg abstract our main focus in this paper is to introduce and study various selection principles in bitopological spaces. in particular, menger type, and hurewicz type covering properties like: almost p-menger, star p-menger, strongly star p-menger, weakly p-hurewicz, almost phurewicz, star p-hurewicz and strongly star p-hurewicz spaces are defined and corresponding properties are investigated. relations between some of these spaces are established. 2010 msc: 54d20; 54e55. keywords: bitopological space; selection principles; (star) p-menger bispace; (star) p-hurewicz bispace. 1. introduction our main focus in this paper is to introduce and study various selection principles, by using p-open covers in bitopological spaces. we will deal with variations of the following classical selection principles originaly studied in topological spaces: let a and b be sets whose elements are families of subsets of an infinite set x and o denotes the family of all open covers of a topological space (x,τ). then: s1(a,b) denotes the selection hypothesis: received 21 august 2019 – accepted 20 november 2019 http://dx.doi.org/10.4995/agt.2020.12238 moiz ud din khan and amani sabah for each sequence (un : n ∈ n) of elements of a there is a sequence (un : n ∈ n) such that for each n ∈ n, un is a member of un, and {un : n ∈ n} is an element of b (see [12]). the covering property s1(o,o) is called the rothberger (covering) property, and topological spaces with the rothberger property are called rothberger spaces. sfin(a,b) denotes the selection hypothesis: for each sequence (un : n ∈ n) of elements of a there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un, and ⋃ n∈n vn is an element of b. the property sfin(o,o) is called the menger (covering) property. ufin(a,b) denotes the selection hypothesis: for each sequence (un : n ∈ n) of elements of a there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un, and the family {∪vn : n ∈ n} is a γ-cover of x. the property ufin(o,o) is called the hurewicz (covering) property. an indexed family {an : n ∈ n} is a γ-cover of x if for every x ∈ x the set {n ∈ n : x /∈ an} is finite. the properties of menger and hurewicz were defined in [3]. the concept of bitopological spaces was introduced by kelly [4] in 1969. for details on the topic we refer the reader to see [2]. according to kelly, a bitopological space is a set endowed with two topologies which may be independent of each other. some mathematicians studied bitopological spaces with some relation between the two topologies, but here we consider bitopological spaces in the sense of kelly. in 2011, kočinac and özçağ introduced and studied in [8] the selective versions of separability in bitopological spaces. in particular, they investegated these properties in function spaces endowed with two topologies with one topology of pointwise convergence and the other with compact-open topology. in 2012, kočinac and özçağ [9], reviewed some known results of selection principles in the context of bitopology. they defined three versions of the menger property in a bitopological space (x,τ1,τ2), namely, δ2−menger, (1, 2)-almost menger, and (1, 2)−weakly menger. these results are mainly related to function spaces and hyperspaces endowed with two arbitrary topologies. they proposed some possible lines of investigation in the areas. in 2016, özçağ and eysen in [11] introduced the notion of almost menger property and almost γ-set in bitopological spaces. our focus in this paper is to continue study of selection principles in bitopological spaces. 2. preliminaries throughout this paper a space (x,τ1,τ2) is an infinite bitopological space (called here bispace x) in the sense of kelly. for a subset u of x, cli(u) (resp. inti(u)) will denote the closure (resp. interior) of u in (x,τi), i = 1, 2, c© agt, upv, 2020 appl. gen. topol. 21, no. 1 160 selection principles and covering properties in bitopological spaces respectively. we use the standard bitopological notion and terminology as in [2]. a subset f of a bispace x is said to be: (i) i-open if f is open with respect to τi in x, f is called open in x if it is both 1-open and 2-open in x, or equivalently, f ∈ (τ1 ∩ τ2); (ii) i-closed if f is closed with respect τi in x, f is called closed in x if it is both 1-closed and 2-closed in x, or equivalently, x\f ∈ (τ1 ∩ τ2); (iii) i-clopen if f is both i-closed and i-open set in x, f is called clopen in x if it is both 1-clopen and 2-clopen. (iv) τi-regular open if f is regular open set with respect to τi. (v) τi-regular closed if f is regular closed set with respect to τi. a bitopological space x is said to be (i,j)-regular (i,j = 1, 2, i 6= j) if, for each point x ∈ x and each τi-open (i-open) set v of x containing x, there exists an i-open set u such that x ∈ u ⊆ clj(u) ⊆ v . x is said to be pairwise regular if it is both (1, 2)-regular and (2, 1)-regular. a cover u of a bispace x is said to be a p-open cover if it is τ1τ2-open and u ∩ (τ1\φ) 6= φ and u ∩ (τ2\φ) 6= φ, where u is τ1τ2-open if u ⊂τ1 ∪ τ2. p−o denotes the family of all p-open covers of x. a p-open cover u of a bispace x is a p-ω-cover [9] if x /∈ u and each finite subset of x is contained in a member of u. u is a p-γ-cover if it is infinite and each x ∈ x belongs to all but finitely many elements of u. the symbols p-ω and p-γ denote the family of all p-ω-covers and p-γ-covers of a bispace respectively. definition 2.1 ([9]). a bispace x is called: (1) p-lindelöf if every p-open cover has a countable subcover. (2) d-paracompct if every dense family of subsets of x has a locally finite refinement. (3) p-metacompact if every p-open cover u of x has a point-finite p-open refinement v (that is, every point of x belongs to at most finitely many members of v). (4) p-metalindelöf if every p-open cover u of x has a point-countable, p-open refinement v. (5) p-menger if for each sequence (un : n ∈ n) of p-open covers of x, there is a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and ⋃ n∈n vn is a p-open cover of x. a ⊂ x is p-menger in x if for each sequence (un : n ∈ n) of covers of a by p-open sets in x, there is a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and a ⊂ ⋃ n∈n vn. (6) p-rothberger if for each sequence (un : n ∈ n) of p-open covers of x, there is a sequence (un : n ∈ n) such that for every n ∈ n, un∈un and {un : n ∈ n} is a p-open cover of x. (7) p-hurewicz (or simply pairwise hurewicz), if it satisfies: for each sequence (un : n ∈ n) of elements of p-o, there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un, and for each x ∈ x, for all but finitely many n, x ∈∪vn. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 161 moiz ud din khan and amani sabah a bispace x is called a p-space, if every countable intersection of open sets is open in x. 3. p-menger and related bispaces following the fact that every p-ω-cover of x is a p-open cover of x, we state the following theorem: theorem 3.1. (1) if a bispace x is p-menger then x satisfies sfin(p−ω,p−o). (2) if a bispace x is p-rothberger then x satisfies s1(p−ω,p−o). (3) if a bispace x is p-hurewicz then x satisfies ufin(p−ω,p−o). in [7], the notion of almost menger topological space was introduced, and in [5] kocev studied this class of spaces. we make use of this concept and define almost p-menger and almost p-rothberger bispaces with the help of p-open covers. definition 3.2. a bitopological space (x,τ1,τ2) is almost p-menger if for each sequence (un : n ∈ n) of p-open covers of x there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and⋃ n∈n {cli(v) : v ∈vn; i = { 1 if v ∈ τ1 2 if v ∈ τ2 } } = x definition 3.3. a bitopological space (x,τ1,τ2) is almost p-rothberger if for each sequence (un : n ∈ n) of p-open covers of x there exists a sequence (un : n ∈ n) such that for every n ∈ n, un∈un and⋃ n∈n {cli(un) ; i = { 1 if un ∈ τ1 2 if un ∈ τ2 } } = x we note that every p-menger (resp. p-rothberger) bispace is almost pmenger (resp. almost p-rothberger). a subset of a bitopological space is said to be dense if it is dense with respect to both topologies. proposition 3.4. if a bispace x contains a dense subset which is p-menger in x, then x is almost p-menger. proof. let a be a p-menger dense subset of a bispace x and let (un : n ∈ n) be a sequence of p-open covers of x. since a is p-menger in x therefore there exist finite sets vn, n ∈ n such that a ⊂ ⋃ n∈n{v : v ∈vn}⊂ ⋃ n∈n{cli(v ) : v ∈vn ; i = { 1 if v ∈ τ1 2 if v ∈ τ2 } }. since a is dense in x, we have x = ⋃ n∈n {cli(v ) : v ∈vn; i = { 1 if v ∈ τ1 2 if v ∈ τ2 } } � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 162 selection principles and covering properties in bitopological spaces the following theorem shows: when an almost p-menger bispace becomes p-menger? theorem 3.5. let x be a pairwise regular bispace. if x is an almost pmenger, then x is a p-menger bispace. proof. let (un : n ∈ n) be a sequence of p-open covers of x. since x is a pairwise regular bispace, by definition there exists for each n a p-open cover vn of x such that v′n = {cli(v ) : v ∈ vn; i = { 1 if v ∈ τ1 2 if v ∈ τ2 } } form a refinement of un. by assumption, there exists a sequence (wn : n ∈ n) such that for each n, wn is a finite subset of vn and ⋃ (w′n : n ∈ n) is a cover of x, where w′n = {cli(w) : w ∈ wn; i = { 1 if w ∈ τ1 2 if w ∈ τ2 } }. for every n ∈ n and every w ∈ wn we can choose uw ∈ un such that cli(w) ⊂ uw . let u′n = {uw : w ∈ wn}. we shall prove that u′n is a p-open cover of x. let x ∈ x. there exists n ∈ n and cli(w) ∈ w′n such that x ∈ cli(w). by construction, there exists uw ∈ u′n such that cli(w) ⊂ uw . hence, x ∈ uw . � theorem 3.6. a bispace x is almost p-menger if and only if for each sequence (un : n ∈ n) of covers of x by τi-regular closed sets (i =1 or i = 2), there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and ⋃ n∈n vn is a cover of x. proof. let x be an almost p-menger bispace. let (un : n ∈ n) be a sequence of covers of x by τi-regular closed sets (i =1 or i = 2), (un : n ∈ n) is a sequence of p-open covers of x. by assumption, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and ⋃ n∈n vn is a cover of x, where cli(v ) = v for all v ∈vn; i= { 1 if v ∈ τ1 2 if v ∈ τ2 } . conversely, let (un : n ∈ n) be a sequence of p-open covers of x. let (u′n : n ∈ n) be a sequence defined by u′n = {cli(u) : u ∈ un}. then each u′n is a cover of x by τi-regular closed sets. thus there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of u′n and ⋃ n∈n vn is a cover of x. by construction, for each n ∈ n and v ∈ vn there exists uv ∈ un such that v = cli(uv ). hence, ⋃ n∈n{cli(uv ) : v ∈ vn} = x. so, x is an almost p-menger bispace. � 3.1. star p-menger bispaces. a number of results in the literature shows that many topological properties can be defined and studied in terms of star covering properties. in particular, such a method is also used in investigation of selection principles for topological spaces. this investigation was initiated by kočinac in [6] and then studied in many papers. we extend this idea for bitopological spaces. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 163 moiz ud din khan and amani sabah let a be a subset of a topological space (x, τ) and u be a collection of subsets of x. then st(a,u) = ∪{u ∈u : u ∩a 6= ∅}, stn+1(a,u) = ∪{u ∈u : u ∩ stn(a,u) 6= ∅}. we usually write st(x,u) for st({x},u). definition 3.7 ([6]). in a topological space (x,τ), (1) sfin ∗(a,b) denotes the selection hypothesis: for each sequence (un : n ∈ n) of elements of a there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un, and ⋃ n∈n{st(v,un) : v ∈vn} is an element of b. (2) ssfin ∗(a,b) denotes the selection hypothesis: for each sequence (un : n ∈ n) of elements of a there is a sequence (fn : n ∈ n) of finite subsets of x such that {st(fn,un) : n ∈ n} is an element of b. the symbols sfin ∗(o,o) and ssfin∗(o,o) denotes the star-menger property and strongly star-menger property, respectively in topological spaces. in a similar way we introduce the following definition for bitopological spaces. definition 3.8. a bitopological space (x,τ1,τ2) is said to have: (1) the star p-menger property if it satisfies sfin ∗(p−o,p−o). (2) the strongly star p-menger property if it satisfies ssfin ∗(p−o,p−o). theorem 3.9. every strongly star p-menger, p-metacompact bispace is pmenger bispace. proof. let x be a strongly star p-menger p-metacompact bispace. let (un : n ∈ n) be a sequence of p-open covers of x. for each n ∈ n, let vn be a pointfinite p-open refinement of un. since x is strongly star p-menger, there is a sequence (fn : n ∈ n) of finite subsets of x such that ⋃ n∈n st(fn,vn) = x. as vn is a point-finite refinement and each fn is finite, elements of each fn belongs to finitely many members of vn say vn1,vn2,vn3, . . . ,vnk . let v′n = {vn1,vn2,vn3, . . . ,vnk}. then st(fn,vn) = ⋃ v′n for each n ∈ n. we have that ⋃ n∈n( ⋃ v′n) = x. for every v ∈ v′n choose uv ∈ un such that v ⊂ uv . then, for every n, wn := {uv : v ∈ v′n} is a finite subfamily of un and ⋃ n∈n ⋃ wn = x, that is x is p-menger bispace. � theorem 3.10. every strongly star p-menger, p-metalindelöf bispace is lindelöf bispace. proof. let x be a strongly star p-menger p-metalindelöf bispace. let u be a p-open cover of x and let v be a point-countable, p-open refinement of u. since x is strongly star p-menger, there is a sequence (fn : n ∈ n) of finite subsets of x such that ⋃ n∈n st(fn,vn) = x. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 164 selection principles and covering properties in bitopological spaces for every n ∈ n, denote by wn the collection of all members of v which intersects fn. since v is point-countable and fn is finite, wn is countable. so, the collection w = ⋃ n∈n wn is a countable subfamily of v and is a cover of x. for every w ∈ w pick a member uw ∈ u such that w ∈ uw . then {uw : w ∈w} is a countable subcover of u. hence, x is lindelöf bispace. � definition 3.11. a bispace x is an almost star p-menger if for each sequence (un : n ∈ n) of p-open covers of x, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and {cli(st(∪vn,un)) : n ∈ n; i = 1 or i = 2} is a cover of x. theorem 3.12. a bispace x is an almost star p-menger if and only if for each sequence (un : n ∈ n) of covers of x by τi−regular open sets there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and {cli(st(∪vn,un)) : n ∈ n} is a cover of x. proof. since every cover by τi−regular open sets is p-open, necessity follows. conversely, let (un : n ∈ n) be a sequence of p-open covers of x. let u′n = {cli(u) : u ∈ un and i = { 1 if u ∈ τ1 2 if u ∈ τ2 } }. then u′n is a cover of x by τi−regular open sets. then by assumption there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of u′n and {cli(st(∪vn,u′n)) : n ∈ n}a cover of x. first we shall prove that st(u,un) = st(cli(u),un) for all u ∈un. it is obvious that st(u,un) ⊂ st(cli(u),un) since u ⊂ cli(u). let x ∈ st(cli(u),un). then there exists some u′ ∈un such that x ∈ u′ and u′ ∩ cli(u) 6= ∅. then u′∩cli(u) 6= ∅ implies that x ∈ st(u,un). hence, st(cli(u),un) ⊂ st(u,un). for each v ∈ vn we can find uv ∈ un such that v = cli(uv ). let v′n = {uv : v ∈vn}. let x ∈ x.then there exists n ∈ n such that x ∈ cli(st(∪vn,u′n)). for each p-open set v , we have v ∩ st(∪vn,u′n) 6= ∅. then there exists u ∈ un such that (v ∩ cli(u) 6= ∅ and ∪vn ∩ cli(u) 6= ∅) imply that (v ∩ u 6= ∅ and ∪vn ∩ cli(u) 6= ∅). we have that ∪v′n ∩u 6= ∅, so x ∈ cli(st(∪v′n,un)). hence, {cli(st(∪v′n,un)) : n ∈ n} is a cover of x. � 4. p-hurewicz and related bispaces definition 4.1. call a bitopological space (x,τ1,τ2): (1) weakly p-hurewicz if for every sequence (un : n ∈ n) of p-open covers of x, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and each non-empty set u ∈ τ1∪τ2, u∩(∪vn) 6= φ for all but finitely many n. (2) almost p-hurewicz if for every sequence (un : n ∈ n) of p-open covers of x, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and each x ∈ x belongs to ∪{cli(v ) : v ∈vn; i = { 1 if v ∈ τ1 2 if v ∈ τ2 } } for all but finitely many n. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 165 moiz ud din khan and amani sabah theorem 4.2. let x be a pairwise regular bispace. if x is an almost phurewicz, then x is p-hurewicz bispace. proof. let (un : n ∈ n) be a sequence of p-open covers of x. since x is a pairwise regular bispace, using the definition, there exists for each n a p-open cover vn of x such that v′n = {cli(v ) : v ∈vn,v ∈ τi; i = 1 or i = 2} forms a refinement of un. by assumption, there exists a sequence (wn : n ∈ n) such that for each n, wn is a finite subset of vn and each x ∈ x belongs to ∪w′n for all but finitely many n, where w′n = {cli(w) : w ∈wn,w ∈ τi; i = 1, 2}. for every n ∈ n and every w ∈ wn we can choose uw ∈ un such that cli(w) ⊂ uw . let u′n = {uw : w ∈wn}. we shall prove that each x ∈∪u′n for all but finitely many n. let x ∈ x. there exists n0 ∈ n and cli(w) ∈w′n such that x ∈ cli(w) for all n > n0. by construction, there exists uw ∈ u′n such that cli(w) ⊂ uw . hence, x ∈ uw for all n > n0. � theorem 4.3. a bispace x is almost p-hurewicz if and only if for each sequence (un : n ∈ n) of covers of x by τi−regular closed sets, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and each x ∈ x belongs to ∪vn for all but finitely many n ∈ n. proof. let x be an almost p-hurewicz bispace. let (un : n ∈ n) be a sequence of covers of x by τi-regular closed sets; i =1 or i = 2. this implies that (un : n ∈ n) is a sequence of p-open covers of x. by assumption, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and each x ∈ x belongs to ∪vn for all but finitely many n, where cli(v ) = v for all v ∈vn; i = { 1 if v ∈ τ1 2 if v ∈ τ2 } . conversely, let (un : n ∈ n) be a sequence of p-open covers of x. let (u′n : n ∈ n) be a sequence defined by u′n = {cli(u) : u ∈ un}. then each x ∈ x belongs to ∪u′n for all but finitely many n and elements of u′n are τiregular closed sets.then there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of u′n and each x ∈ x belongs to ∪vn for all but finitely many n. by construction, for each n ∈ n and v ∈vn there exists uv ∈ un such that v = cli(uv ). hence, x ∈ cli(uv ) : v ∈ vn for all but finitely many n. so x is almost p-hurewicz bispace. � theorem 4.4. if a bispace x is weakly p-hurewicz and d-paracompact, then x is almost p-hurewicz. proof. let (un : n ∈ n) be a sequence of p-open covers of a bispace x. since x is weakly p-hurewicz, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and every non-empty set u ∈ τ1 ∪ τ2, u ∩ (∪vn) 6= φ for all but finitely many n. let x ∈ x. by the assumption {vn : n ∈ n} has a locally finite refinement say w. then ∪w = ∪n∈n ∪vn and therefore cli(∪w) =cli(∪n∈n ∪vn). as w is locally finite family, cli(∪w) = ∪w∈wcli(w). since for every w ∈ w there exists vw ∈ vn, so that w ⊂ vw , we have that each x ∈ cli(v ) where v ∈ vn, for all but finitely many n. hence, it is shown that x is almost p-hurewicz. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 166 selection principles and covering properties in bitopological spaces theorem 4.5. if a p-space x is weakly p-hurewicz, then x is almost phurewicz. proof. let (un : n ∈ n) be a sequence of p-open covers of x. since x is weakly p-hurewicz, there exists a sequence (vn : n ∈ n) such that for every n ∈ n, vn is a finite subset of un and every non-empty set u ∈ τ1 ∪ τ2, u ∩ (∪vn) 6= φ for all but finitely many n. let x ∈ x and u contains x. by the condition x is p-space, the intersection of every countable family of open subsets of x is open and hence, every countable union of closed sets is closed. so, cli(∪n∈n ∪vn) = ∪n∈n{cli(v ) : v ∈ vn} implies that x ∈ cli(v ) for all but finitely many n where v ∈vn, which shows that x is an almost p-hurewicz space. � theorem 4.6. every i−clopen subset of an almost p-hurewicz bispace is almost p-hurewicz; i =1 or i = 2. proof. let f be an i−clopen subset of an almost p-hurewicz bispace x and let (un : n ∈ n) be a sequence of p-open covers of f . then vn = un∪{x−f} is a p-open cover for x for every n ∈ n. since x is an almost p-hurewicz bispace, there exist finite subsets wn of vn for which x ∈ x belongs to cli(w) : w ∈ wn for all but finitely many n ∈ n. since,cli(x−f) = x−f and each a ∈ f belongs to cli(w) : w ∈wn,w 6= x −f for all but finitely many n. � theorem 4.7. every i−closed subset of a weakly p-hurewicz bispace is weakly p-hurewicz. i =1 or i = 2. proof. let f be an i−closed subset of a weakly p-hurewicz space and let (un : n ∈ n) be a sequence of p-open covers of f . then vn = un ∪ {x − f} is a p-open cover of x for every n ∈ n. since x is a weakly p-hurewicz space, there exists finite subsets wn of vn for each n ∈ n such that every non-empty i-open set u ⊂ x and u ∩ (∪vn) 6= φ for all but finitely many n. put w = ∪n∈n{w : w ∈ wn,w 6= x − f}. then every non-empty i-open set u ⊂ x, u ∩ (w ∪ (x −f)) 6= φ for all but finitely many n. since f = cli(inti(f)) we have inti(f)∩cli(x −f) = φ. so, inti(f) ⊂ cli(∪w) and f = cli(inti(f)) ⊂ cli(∪w). every non-empty i-open set a ⊂ f, a∩ (∪w) 6= φ for all but finitely many n. � 4.1. star p-hurewicz bispaces. the method of stars is one of classical popular topological methods. it has been used, for example, to study the problem of metrization of topological spaces, and for definitions and investigations of several important classical topological notions [1],[10]. definition 4.8. a bitopological space (x,τ1,τ2) is said to have: • star p-hurewicz property, if it satisfies: for each sequence (un : n ∈ n) of elements of p-o there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un, and each x ∈ x belongs to st(∪vn,un) for all but finitely many n. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 167 moiz ud din khan and amani sabah • strongly star p-hurewicz property, if it satisfies: for each sequence (un : n ∈ n) of elements of p-o there is a sequence (an : n ∈ n) of finite subsets of x, and each x ∈ x belongs to st(an,un) for all but finitely many n. every strongly star p-hurewicz bispace is star p-hurewicz. the implications among the mentioned covering properties are as follows: p − hurewicz ⇒ star p − hurewicz ⇓ ⇓ p − menger ⇒ star p − menger a bitopological space x is called strongly star pairwise-compact if for each p-open cover u of x there is a finite set f ⊂ x such that st(f,u) = x. call a space x strongly star pairwise σ-compact if it is union of countably many strongly star pairwise-compact bispaces. clearly, every strongly star pairwisecompact bispace is stongly star p-hurewicz. a bitopological space x is called star-p-lindelöf if for every p-open cover u of x there is a countable set f ⊂u such that st(f,u) = x. theorem 4.9. every star p-hurewicz bispace is star-p-lindelöf. proof. let x be a star p-hurewicz bispace. let u be a p-open cover of x. let (un : n ∈ n) be a sequence such that each un = u. then, by definition, there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un, and each x ∈ x belongs to ∪n∈n(st(∪vn,u)) for all but finitely many n. let v = ∪n∈n vn. now ∪n∈n(st(∪vn,un)) = st(∪v,u). then v = ∪n∈n vn is a countable subfamily of u satisfying st(∪v,u) = ∪n∈n (st(∪vn,un)) = x, that is x is star-p-lindelöf. � theorem 4.10. every strongly star p-hurewicz bispace is strongly star plindelöf. proof. let x be a strongly star p-hurewicz bispace. let u be a p-open cover of x. let f be the collection of all finite subsets of x. then, by definition, there is a sequence (fn : n ∈ n) of elements of f such that each x ∈ x belongs to st(fn,un) for all but finitely many n. let a = ∪n∈nfn; then a is a countable set being countable union of finite sets. also, ∪n∈nst(fn,un)=∪n∈n (st(∪n∈nfn,un)) = st(a,un) = x. hence, x is strongly star-p-lindelöf bispace. � theorem 4.11. every strongly star p-hurewicz, p-metacompact bispace is phurewicz bispace. proof. let x be a strongly star p-hurewicz metacompact bispace. let (un : n ∈ n) be a sequence of p-open covers of x. for each n ∈ n, let vn be a pointfinite p-open refinement of un. since x is strongly star p-hurewicz, there is a sequence (fn : n ∈ n) of finite subsets of x such that each x ∈ x belongs to st(fn,vn) for all but finitely many n. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 168 selection principles and covering properties in bitopological spaces since vn is a point-finite refinement and each fn is finite, elements of each fn belongs to finitely many members of vn say vn1,vn2,vn3, ...,vnk . let v ′ n = {vn1,vn2,vn3, ...,vnk}. then st(fn,vn) = ∪v ′ n for each n ∈ n. we have that each x ∈ x belongs to ∪v′n for all but finitely many n. for every v ∈ v′n choose uv ∈ un such that v ⊂ uv . then, for every n, {uv : v ∈ v′n} = wn is a finite subfamily of un and each x ∈ x belongs to ∪wn for all but finitely many n, that is x is p-hurewicz bispace. � theorem 4.12. every strongly star p-hurewicz, p-metalindelöf bispace is plindelöf. proof. let x be a strongly star p-hurewicz, pmetalindelof bispace. let u be a p-open cover of x then there exists v, a point-countable p-open refinement of u. since x is strongly star p-hurewicz, there exists a sequence (fn : n ∈ n) of finite subsets of x such that for each x ∈ x,x ∈ st(fn,vn) for all but finitely many n. for every n ∈ n denote by wn the collection of all members of v which intersects with fn. since v is point-countable and fn is finite, wn is countable. so, the collection w = ∪n∈nwn is countable subfamily of v and is a cover of x. for every w ∈ w pick a member uw ∈ u such that w ⊂ uw . then {uw : w ∈ w} is a countable subcover of u. hence, x is a p-lindelof bispace. � references [1] o. t. alas, l. r. junqueira and r. g. wilson, countability and star covering properties, topology appl. 158 (2011), 620–626. [2] b. p. dvalishvili, bitopological spaces: theory, relations with generalized algebraic structures, and applications, north-holand math. stud. (2005). [3] w. hurewicz, über die verallgemeinerung des borelschen theorems, math. z. 24 (1925), 401–425. [4] j. c. kelly bitopological spaces, proc. london math. soc. 13, no. 3 (1963), 71–89. [5] d. kocev, almost menger and related spaces, mat. vesnik 61 (2009), 173–180. [6] lj. d. r. kočinac , star-menger and related spaces, publ. math. debrecen 55 (1999), 421–431. [7] lj. d. r. kočinac, star-menger and related spaces ii, filomat 13 (1999), 129–140. [8] lj. d. r. kočinac and s. özçağ, versions of separability in bitopological spaces, topology appl. 158 (2011), 1471–1477. [9] lj. d. r. kočinac and s. özçağ, bitopological spaces and selection principles, proceedings icta2011, cambridge scientific publishers, (2012), 243–255. [10] m. v. matveev, a survey on star covering properties, topology atlas, preprint no. 330 (1998). [11] s. özçağ and a. e. eysen, almost menger property in bitopological spaces, ukranian math. j. 68, no. 6 (2016), 950–958. [12] f. rothberger, eine ver schörfung der eigenschaftc, fundamenta mathematicae 30 (1938), 50–55. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 169 @ appl. gen. topol. 20, no. 2 (2019), 395-405 doi:10.4995/agt.2019.11524 c© agt, upv, 2019 ec-filters and ec-ideals in the functionally countable subalgebra of c∗(x) amir veisi faculty of petroleum and gas, yasouj university, gachsaran, iran (aveisi@yu.ac.ir) communicated by a. tamariz-mascarúa abstract the purpose of this article is to study and investigate ec-filters on x and ec-ideals in c ∗ c (x) in which they are in fact the counterparts of zc-filters on x and zc-ideals in cc(x) respectively. we show that the maximal ideals of c∗c (x) are in one-to-one correspondence with the ec-ultrafilters on x. in addition, the sets of ec-ultrafilters and zcultrafilters are in one-to-one correspondence. it is also shown that the sets of maximal ideals of cc(x) and c ∗ c (x) have the same cardinality. as another application of the new concepts, we characterized maximal ideals of c∗c (x). finally, we show that whether the space x is compact, a proper ideal i of cc(x) is an ec-ideal if and only if it is a closed ideal in cc(x) if and only if it is an intersection of maximal ideals of cc(x). 2010 msc: 54c30; 54c40; 54c05; 54g12; 13c11; 16h20. keywords: c-completely regular space; closed ideal; functionally countable space; ec-filter; ec-ideal; zero-dimensional space. 1. introduction all topological spaces are completely regular hausdorff spaces and we shall assume that the reader is familiar with the terminology and basic results of [6]. given a topological space x, we let c(x) denote the ring of all real-valued continuous functions defined on x. cc(x) is the subalgebra of c(x) consisting of functions with countable image and c∗c (x) is its subalgebra consisting of bounded functions. in fact, c∗c (x) = cc(x)∩c∗(x), where elements of c∗(x) received 19 march 2019 – accepted 30 july 2019 http://dx.doi.org/10.4995/agt.2019.11524 a. veisi are bounded functions of c(x). recall that for f ∈ c(x),z(f) denotes its zero-set: z(f) = {x ∈ x : f(x) = 0}. the set-theoretic complement of a zero-set is known as a cozero-set and we denote this set by coz(f). let us put zc(x) = {z(f) : f ∈ cc(x)} and z∗c (x) = {z(g) : g ∈ c∗c (x)}. these two latter sets are in fact equal, since z(f) = z( f 1+|f|), where f ∈ cc(x). a nonempty subfamily f of zc(x) is called a zc-filter if it is a filter on x. if i is an ideal in cc(x) and f is a zc-filter on x then, we denote zc[i] = {z(f) : f ∈ i}, ∩zc[i] = ∩{z(f) : f ∈ i} and z−1c [f] = {f : z(f) ∈f}. we see that zc[i] is a zc-filter and z−1c [zc[i]] ⊇ i. if the equality holds, then i is called a zc-ideal. moreover, z −1 c [f] is a zc-ideal and we always have zc[z −1 c [f]] = f. so maximal ideals in cc(x) are zc-ideals. in [5], a huasdorff space x is called countably completely regular (briefly, ccompletely regular) if whenever f is a closed subset of x and x /∈ f, there exists f ∈ cc(x) such that f(x) = 0 and f(f) = 1. in addition, two closed sets a and b of x are also called countably separated (in brief, c-separated) if there exists f ∈ cc(x) with f(a) = 0 and f(b) = 1. c-completely regular and zero-dimensional spaces are the same, see theorem 1.1. if we let mcp = {f ∈ cc(x) : f(p) = 0} (p ∈ x), then the ring isomorphism cc(x) mcp ∼= r gives that mcp is a maximal ideal, in fact, mcp is a fixed maximal ideal. moreover, ∩zc[mcp] = {p}. our concentration is on the zero-dimensional spaces since in [5] the authors proved that for any space x there is a zero-dimensional hausdorff space y such that cc(x) and cc(y ) are isomorphic as rings, see theorem 1.2. in section 2, we study and investigate the ec-filters on x and ec-ideals in c∗c (x) which they are in fact the counterpart of [6, 2l]. we show that the maximal ideals of c∗c (x) are in one-to-one correspondence with the ec-ultrafilters on x. moreover, the sets of ec-ultrafilters and zc-ultrafilters are in one-to-one correspondence. by using the latter facts, it is shown that the sets of maximal ideals of cc(x) and c ∗ c (x) have the same cardinality. finally, maximal ideals of c∗c (x) are characterized based on these concepts. in section 3, our concentration is on the uniform norm topology on c∗c (x) which is the restriction of the uniform norm topology on c∗(x). it is shown that whenever the space x is compact, a proper ideal i of cc(x) is an ec-ideal if and only if it is a closed ideal in cc(x) if and only if it is an intersection of maximal ideals of cc(x). we recite the following results from [5]. theorem 1.1 ([5, proposition 4.4]). let x be a topological space. then, x is a zero-dimensional space (i.e., a t1-space with a base consisting of clopen sets) if and only if x is c-completely regular space. theorem 1.2 ([5, theorem 4.6]). let x be any topological space (not necessarily completely regular). then, there is a zero-dimensional space y which is a continuous image of x with cc(x) ∼= cc(y ) and cf (x) ∼= cf (y ). c© agt, upv, 2019 appl. gen. topol. 20, no. 2 396 ec-filters and ec-ideals in the functionally countable subalgebra of c ∗(x) remark 1.3 ([5, remark 7.5]). there is a topological space x, such that there is no space y with cc(x) ∼= c(y ). the following results are the known facts about cc(x) and we are seeking to get similar results for c∗c (x). proposition 1.4. let i be a proper ideal in cc(x) and f a zc-filter on x. then: (i) zc[i] is a zc-filter and z −1 c [f] is a zc-ideal of cc(x). (ii) if i is maximal then zc[i] is a zc-ultrafilter, and the converse holds if i is a zc-ideal. (iii) f is a zc-ultrafilter if and only if z−1c [f] is a maximal ideal. (iv) if f is a zc-ultrafilter and z ∈ zc(x) meets each element of f, then z ∈f. corollary 1.5. there is a one-to-one correspondence ψ between the sets of zc-ideals of cc(x) and zc-filters on x, defined by ψ(i) = zc[i]. in particular, the restriction of ψ to the set of maximal ideals is a one-to-one correspondence between the sets of maximal ideals of cc(x) and zc-ultrafilters on x. 2. ec-filters on x and ec-ideals in c ∗ c (x) for f ∈ c∗c (x) and � > 0, we define ec� (f) = f −1([−�,�]) = {x ∈ x : |f(x)| ≤ �}. each such set is a zero set, since it is equal to z((|f|−�)∨0). conversely, every zero set is also of this form, since for g ∈ c∗c (x) we have z(g) = ec� (|g| + �). for a nonempty subset i of c∗c (x) we denote e c � [i] = {ec� (f) : f ∈ i}, and ec(i) = ⋃ � e c � [i]. moreover, if f is a nonempty subfamily of z∗c (x), then we define ec� −1[f] = {f ∈ c∗c (x) : ec� (f) ∈f} and e−1c (f) = ⋂ � e c � −1[f]. so we have ec(i) = {ec� (f) : f ∈ i and � > 0}, and e−1c (f) = {f ∈ c∗c (x) : ec� (f) ∈ f, for all �}. moreover, e−1c (ec(i)) = {g ∈ c∗c (x) : ecδ(g) ∈ ec(i), for all δ > 0} and ec(e−1c (f)) = {ec� (f) : ecδ(f) ∈f, for all δ > 0}. the next result is now immediate. corollary 2.1. the following statements hold. (i) i ⊆ e−1c (ec(i)) and ec(e−1c (f)) ⊆f. (ii) the mappings ec and e −1 c preserve the inclusion. (iii) if f ∈ i then for each positive integer n, ec� (f) = ec�n(fn). (iv) if i is an ideal, then ec(i) is a zc-filter. proof. the proofs of (i), (ii) and (iii) are clear. (iv). this is presented in the proof of proposition 2.5. � examples 2.2 and 2.3 below show that the inclusions in (i) of the above corollary may be strict even when i is an ideal and f is a zc-filter. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 397 a. veisi example 2.2. let x be the discrete space n×n, f(m,n) = 1 mn and i the ideal in c∗c (x)(= c ∗(x)) generated by f2. obviously f /∈ i. since {x ∈ x : f(x) ≤ �} = {x ∈ x : f2(x) ≤ �2}, we have ec� (f) ∈ ec(i). so i $ e−1c (ec(i)). example 2.3. let x be the zero-dimensional space q×q, where q is the set of rational numbers, and f = {z ∈ zc(x) : (0, 0) ∈ z}. then f is a zc-filter on x. now, if we define f(x,y) = |x|+|y| 1+|x|+|y| then f ∈ c ∗ c (x)(= c ∗(x)) and z(f) = {(0, 0)}. given � > 0 and g = f + �, so we have ec� (g) = {(0, 0)}. if we take 0 < δ < � then ecδ(g) = ∅. hence e c � (g) is not contained in ec(e −1 c (f)). therefore the latter set is contained in f properly, which gives the result. definition 2.4. a zc-filter f is called an ec-filter if f = ec(e−1c (f)), or equivalently, whenever z ∈f then there exist f ∈ c∗c (x) and � > 0 such that z = ec� (f) and e c δ(f) ∈f, for each δ > 0. proposition 2.5. if i is a proper ideal in c∗c (x), then ec(i) is an ec-filter. proof. first, we show that ec(i) is a zc-filter, i.e., it satisfies the following conditions. (i) ∅ /∈ ec(i). (ii) ec� (f),e c δ(g) ∈ ec(i), then e c � (f) ∩ecδ(g) ∈ ec(i). (iii) ec� (f) ∈ ec(i), z ∈ zc(x) with z ⊇ ec� (f), then z ∈ ec(i). (i). suppose that for some � > 0 and f ∈ i, ec� (f) = ∅. so � < |f|, which yields f is a bounded away from zero. hence i contains the unit f, which is impossible. (ii). this is equivalent to say that if ec� (f),e c δ(g) ∈ ec(i), then ec� (f) ∩ ecδ(g) contains a member of ec(i). suppose that f ′,g′ ∈ i and �′,δ′ > 0 such that ec� (f) = e c �′(f ′) and ecδ(g) = e c δ′(g ′). without loss of generality, we may suppose that δ′ < �′ < 1. hence f′2 + g′2 ∈ i and ec δ′2 (f′2 + g′2) ⊆ ec� (f) ∩ ecδ(g), which gives the result. (iii). assume that ec� (f) ⊆ z(f′), where f ∈ i and f′ ∈ c∗c (x). since ec� (f) = ec�2 (f 2) and z(f′) = z(|f′|), we can suppose that f ≥ 0 and f′ ≥ 0. now, define g(x) = { 1, if x ∈ ec� (f) f′(x) + � f(x) , if x ∈ x r intec� (f). so g is continuous, since it is continuous on two closed sets whose union is x, in fact, g ∈ c∗c (x). note that fg ∈ i and (fg)(x) = { f(x), if x ∈ ec� (f) (ff′)(x) + �, if x ∈ x r intec� (f). it is easily seen that z(f′) = ec� (fg). so z(f ′) ∈ ec(i). this shows that ec(i) is a zc-filter. now, apply (i) and (ii) of corollary 2.1 for the ideal i and the zc-filter ec(i), to get the inclusions ec(i) ⊆ ec(e−1c (ec(i))) and ec(e −1 c (ec(i))) ⊆ ec(i), which yields ec(i) is an ec-filter. � definition 2.6. an ideal i in c∗c (x) is called ec-ideal if i = e −1 c (ec(i)), or equivalently, if f ∈ c∗c (x) and ec� (f) ∈ ec(i) for all �, then f ∈ i. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 398 ec-filters and ec-ideals in the functionally countable subalgebra of c ∗(x) proposition 2.7. if f is a zc-filter, then e−1c (f) is an ec-ideal in c∗c (x). proof. let f,g ∈ e−1c (f), h ∈ c∗c (x) and let u be an upper bound for h and � > 0. then ec� 2 (f), ec� 2 (g) and hence ec� 2 (f) ∩ ec� 2 (g) belong to f. hence ec� 2 (f)∩ec� 2 (g) ⊆ ec� (f +g) implies that ec� (f +g) ∈f, or equivalently, f +g ∈ e−1c (f). moreover, ec� u (f) ⊆ ec� (fh) implies fh ∈ e−1c (f). therefore e−1c (f) is ideal. in view of corollary 2.1, we have e−1c (f) ⊆ e−1c (ec(e−1c (f))) ⊆ e−1c (f) and so the equality holds, i.e., e−1c (f) is an ec-ideal. � corollary 2.8. maximal ideals of c∗c (x) and an arbitrary intersection of them are ec-ideals. proof. let m be a maximal ideal of c∗c (x). if e −1 c (ec(m)) is not a proper ec-ideal, then it contains the constant function 1 and e c � (1) = ∅ ∈ ec(m) (0 < � < 1) which is impossible, see propositions 2.5 and 2.7. hence m = e−1c (ec(m)), i.e., m is an ec-ideal. the second part is obtained by this fact, the fact that the intersection of a family of maximal ideals is an ideal contained in each of them and (ii) of corollary 2.1. � the next corollary is an immediate result of propositions 2.5 and 2.7. corollary 2.9. the correspondence i 7→ ec(i) is one-one from the set of ec-ideals in c ∗ c (x) onto the set of ec-filters on x. lemma 2.10. (i) let i and j be ideals in c∗c (x) and j an ec-ideal. then i ⊆ j if and only if ec(i) ⊆ ec(j). (ii) let f1 and f2 be zc-filters on x and f1 an ec-filter. then f1 ⊆f2 if and only if e−1c (f1) ⊆ e−1c (f2). proof. it is straightforward. � proposition 2.11. let i be an ideal in c∗c (x) and f a zc-filter on x. then: (i) e−1c (ec(i)) is the smallest ec-ideal containing i. (ii) ec(e −1 c (f)) is the largest ec-filter contained in f. proof. (i). propositions 2.5 and 2.7 respectively show that ec(i) is an ec-filter and e−1c (ec(i)) is an ec-ideal. now, suppose that k is an ec-ideal containing i. so e−1c (ec(i)) ⊆ e−1c (ec(k)) = k. hence we are done. (ii). this is proved similarly. � the next theorem plays an important role in many of the following results. theorem 2.12. let a be a zc-ultrafilter. then a zero set z meets every element of ec(e −1 c (a)) if and only if z ∈a. proof. since a is a filter and ec(e−1c (a)) ⊆ a, the sufficient condition is evident. for the necessary condition, it is recalled at first that if z meets every element of a then z ∈a, see (iv) of proposition 1.4. now, we claim that if z meets every element of ec(e −1 c (a)) as a particular subfamily of a, then also z ∈a. otherwise, for some z′ ∈a, z ∩z′ = ∅. since the closed sets z and c© agt, upv, 2019 appl. gen. topol. 20, no. 2 399 a. veisi z′ are c-completely separated, there is f ∈ c∗c (x) (in fact 0 ≤ f ≤ 1) such that f(z) = 1 and f(z′) = 0. notice that z′ ⊆ z(f) ⊆ ec� (f), for all �, and; ec� (f) ∈a, since z′ ∈a. so ec� (f) ∈ ec(e−1c (a)). now, if � is taken less than 1, then z ∩ec� (f) = ∅ which contradicts with our assumption of z. so z ∈a and the proof is complete. � the following proposition shows that, as z−1c (a) is a maximal ideal in cc(x), e −1 c (a) is also a maximal ideal in c∗c (x), where a is a zc-ultrafilter on x. proposition 2.13. let a be a zc-ultrafilter on x. then: (i) e−1c (a) is a maximal ideal. (ii) e−1c (a) is an ec-ideal. (iii) e−1c (a) = e−1c (ec(e−1c (a))). proof. (i). let m be a maximal ideal of c∗c (x) containing e −1 c (a). hence ec(e −1 c (a)) ⊆ ec(m). since every element of ec(m) meets every element of ec(e −1 c (a)), theorem 2.12 gives ec(m) ⊆ a. so m = e−1c (ec(m)) ⊆ e−1c (a) and hence m = e−1c (a). (ii). it follows by (i). (iii). since the maximal ideal e−1c (a) is contained in the proper ideal e−1c (ec(e−1c (a))), the result now holds. � an ec-ultrafilter on x is meant a maximal ec-filter, i.e., one not contained in any other ec-filter. as usual, every ec-filter f is contained in an ec-ultrafilter. this is obtained by considering the collection of all ec-filters containing f and the use of the zorn’s lemma, where the partially ordered relation on f is inclusion. proposition 2.14. let m be an ideal in c∗c (x) and f a zc-filter on x. then: (i) if m is a maximal ideal then ec(m) is an ec-ultrafilter. (ii) if f is an ec-ultrafilter then e−1c (f) is a maximal ideal. (iii) if m is an ec-ideal, then m is maximal if and only if ec(m) is an ec-ultrafilter. (iv) if f is an ec-filter, then f is ec-ultrafilter if and only if e−1c (f) is a maximal ideal. proof. (i). note that m = e−1c (ec(m)). let f′ be an ec-ultrafilter containing ec(m), then m ⊆ e−1c (f′) and hence m = e−1c (f′). therefore ec(m) = ec(e −1 c (f′)) = f′, which yields the result. (ii). let m be a maximal ideal of c∗c (x) containing e −1 c (f). then f ⊆ ec(m). hence f = ec(m) and so e−1c (f) = m. the proofs of (iii) and (iv) are similarly done and further details are omitted. � corollary 2.15. there is a one-to-one correspondence ψ between the sets of maximal ideals of c∗c (x) and ec-ultrafilters on x, defined by ψ(m) = ec(m). proposition 2.16. let a be a zc-ultrafilter. then it is the unique zc-ultrafilter containing ec(e −1 c (a)), and also ec(e−1c (a)) is the unique ec-ultrafilter contained in a. hence every ec-ultrafilter is contained in unique zc-ultrafilter. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 400 ec-filters and ec-ideals in the functionally countable subalgebra of c ∗(x) proof. let b be a zc-ultrafilter containing ec(e−1c (a)) and z ∈ b. since z meets every element of ec(e −1 c (a)), theorem 2.12 gives b ⊆ a and hence b = a. so the first part of the proposition holds. now, let k be an ecultrafilter contained in a. then k = ec(e−1c (k)) ⊆ ec(e−1c (a)). since the latter set is an ec-filter, the inclusion cannot be proper, i.e., k = ec(e−1c (a)). hence the result is obtained. � corollary 2.17. the zc-ultrafilters are in one-to-one correspondence with the ec-ultrafilters. proof. consider the mapping ψ from the set of zc-ultrafilters into the set of ec-ultrafilters defined by ψ(a) = ec(e−1c (a)). if ψ(a) = ψ(b), then we have that ec(e −1 c (a)) = ec(e−1c (b)) and it is contained in both a and b. so each element of b meets each element of ec(e−1c (a)). now, theorem 2.12 gives b ⊆ a. similarly, a ⊆ b. therefore ψ is one-one. let k be an ecultrafilter and a the unique zc-ultrafilter containing it (proposition 2.16). then k = ec(e−1c (a)) and hence ψ(a) = k. therefore ψ is onto. � our next two theorems are applications that are based on the concepts of ec-filters and ec-ideals. in the first result (theorem 2.18) we show that the maximal ideals of cc(x) are in one-to-one correspondence with those ones of c∗c (x) and the second result (theorem 2.20) involves characterization of maximal ideals of c∗c (x). theorem 2.18. let m (resp. m∗) be the set of maximal ideals of cc(x) (resp. c∗c (x)). then m and m∗ have the same cardinality. proof. if m ∈m then zc[m] is a zc-ultrafilter and hence e−1c (zc[m]) ∈m∗, see propositions 1.4 and 2.13. define ϕ : m→m∗ which m 7→ e−1c (zc[m]). if ϕ(m) = ϕ(m′) then ec(e −1 c (zc[m])) = ec(e −1 c (zc[m ′])) and it is contained in both zc[m] and zc[m ′]. since each element of zc[m ′] meets each element of ec(e −1 c (zc[m])), theorem 2.12 yields zc[m ′] ⊆ zc[m]. similarly, zc[m] ⊆ zc[m′]. therefore m = m′. this verifies ϕ is one-one. to show that ϕ is onto, suppose that m∗ ∈ m∗. hence ec(m∗) is an ec-ultrafilter (proposition 2.14). now, let a be the unique zc-ultrafilter containing ec(m∗) (proposition 2.16), then z−1c [a] is a maximal ideal in cc(x) (proposition 1.4) and m∗ = e−1c (a). recall that if f is a zc-filter, then we always have zc[z −1 c [f]] = f and ec(e−1c (f)) ⊆ f, but the equality occurs if f is an ecfilter. now, if we let m = z−1c [a] then ϕ(m) = e−1c (zc[m]) = e−1c (a) = m∗. hence ϕ is onto, which it completes the proof. � remark 2.19. combining corollaries 1.5, 2.15 and 2.17 gives another proof of the above theorem. theorem 2.20. let m be an ideal in c∗c (x). then m is maximal if and only if whenever f ∈ c∗c (x) and each ec� (f) meets every element of ec(m), then f ∈ m. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 401 a. veisi proof. necessity: suppose that f /∈ m. so (m,f) = c∗c (x). hence h+fg = 1, for some h ∈ m and g ∈ c∗c (x). let u be an upper bound for g and 0 < � < 1. then ∅ = ec� (1) = e c � (h + fg) ⊇ e c � 2 (h) ∩ec� 2 (fg) ⊇ ec� 2 (h) ∩ec� 2u (f), which contradicts with our assumption, since ec� 2 (h) ∩ec� 2u (f) 6= ∅. so we are done. sufficiency: let m′ be a maximal ideal of c∗c (x) containing m and f ∈ m′. then ec(m) ⊆ ec(m′) and ec� (f) ∈ ec(m′), for all �. since ec(m′) is an ecfilter, ec� (f) meets every element of ec(m ′). hence it also meets each element of ec(m). now, by hypothesis f ∈ m. therefore m = m′, which gives the result. � 3. uniform norm topology on c∗c (x) and related closed ideals consider the supremum-norm on c∗(x), i.e., ‖f‖ = supx∈x |f(x)|, where f ∈ c∗(x). so its restriction on c∗c (x) is also the supremum-norm. this defines a metric d as usual, d(f,g) = ‖f −g‖. the resulting metric topology is called the uniform norm topology on c∗c (x). convergence in this topology is uniform convergence of the functions. a base for the neighborhood system at g consists of all sets of the form {f : ‖f −g‖≤ �} (� > 0). equivalently, a base at g is given by all sets {f : |f(x) −g(x)| ≤ u(x) for every x ∈ x}, where u is a positive unit of c∗c (x). if i is an ideal in c∗c (x) then its closure in c ∗ c (x) is denoted by cli. proposition 3.1. let i be an ideal in c∗c (x). then: (i) cli is ideal. (ii) if i is a proper ideal then cli is also a proper ideal. (iii) if i is an ec-ideal then it is a closed ideal. proof. (i). let f,g ∈ cli, h ∈ c∗c (x) and let u be an upper bound for h and � > 0 is fixed. then for some f′ ∈ n� 2 (f) ∩ i and g′ ∈ n� 2 (g) ∩ i we have f′ + g′ ∈ n�(f + g) ∩ i. moreover, there exists f1 ∈ n � u (f) ∩ i and hence f1h ∈ n�(fh) ∩i. so cli contains f + g and fh. hence cli is ideal. (ii). if cli is not a proper ideal then 1 ∈ cli and hence n�(1) ∩i contains a unit element of c∗c (x) such as f, since 1−� < f < 1 + � gives f is bounded away from zero (of course, when 0 < � < 1). but this is impossible since f ∈ i. thus cli is a proper ideal. (iii). let g ∈ cli and � > 0 arbitrary. then for some f ∈ n� 2 (g) ∩ i and all x ∈ ec� 2 (f), we have |g(x)| = |g(x) −f(x) + f(x)| ≤ |g(x) −f(x)| + |f(x)| ≤ � 2 + � 2 = �. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 402 ec-filters and ec-ideals in the functionally countable subalgebra of c ∗(x) hence ec� 2 (f) ⊆ ec� (g). since the zc-filter ec(i) contains ec� 2 (f), it also contains ec� (g), for all �. so g ∈ e−1c (ec(i)) = i and therefore cli ⊆ i. this proves that i is closed and hence the proof is complete. � immediately, we find there is no proper dense ideal in c∗c (x), and further maximal ideals of c∗c (x) and hence every intersection of them are closed, see corollary 2.8 and (iii) of the above proposition. we recall that [6, 1d(1)] plays a useful role in the context of c(x). the following is the counterpart for cc(x). proposition 3.2. if f,g ∈ cc(x) and z(f) is a neighborhood of z(g), then f = gh for some h ∈ cc(x). in the remainder of this section, the zero-dimensional topological space x will be assumed to be compact. hence it is c-pseudo-compact, i.e, cc(x) = c∗c (x). lemma 3.3. let x be a compact space, i an ideal in cc(x), f ∈ cc(x) and z(f) a neighborhood of ∩zc[i]. then f ∈ i. proof. first, we recall that x is compact if and only if the intersection of members of any collection consisting of nonempty closed subsets of x with the finite intersection property (i.e., the intersection of each of a finite number of them is nonempty) is nonempty. the lemma is obvious when i = cc(x). now, if i is a proper ideal in cc(x) then zc[i] satisfies the finite intersection property and hence ∩zc[i] 6= ∅. by assumption ∩zc[i] ⊆ intz(f). hence x rintz(f) ⊆ ⋃ g∈i coz(g) and so x = ⋃ g∈i coz(g)∪intz(f). by compactness of x, there are a finite number of elements of i, say g1,g2, . . . ,gn, such that x = n⋃ i=1 coz(gi) ∪ intz(f). now, if we let g = ∑n i=1 g 2 i then g ∈ i and ∅ 6= z(g) = ⋂n i=1 z(gi) ⊆ intz(f). in view of proposition 3.2, f is a multiple of g and hence it is contained in i. so the proof is complete. � proposition 3.4. if g ∈ cc(x) and � > 0 is fixed, then there exists f ∈ cc(x) such that ‖g −f‖≤ � and z(f) is a neighborhood of z(g). proof. the trivial solution is f = g, of course when z(g) is open. in general, it suffices to define f(x) =   g(x) − �, if x ∈ g−1([�, +∞)) 0, if x ∈ ec� (g) g(x) + �, if x ∈ g−1((−∞,−�]). we note that x is the union of three closed sets g−1([�, +∞)), ec� (g) and g−1((−∞,−�]) and further f is continuous on each of them. therefore f is continuous on x, i.e., f ∈ c(x). notice that the definition of f makes the c© agt, upv, 2019 appl. gen. topol. 20, no. 2 403 a. veisi cardinality of the range of f the same cardinality of the range of g. hence this leads us f ∈ cc(x). moreover, ‖g −f‖≤ �. evidently, z(g) ⊆ g−1((−�,�)) ⊆ intz(f) which yields z(f) is a neighborhood of z(g). � theorem 3.5. let i be a proper ideal in cc(x), i = ∩{mcp : mcp ⊇ i} and j = {g ∈ cc(x) : z(g) ⊇∩zc[i]}. then: (i) i = j. (ii) ∩zc[i] = ∩zc[i]. proof. (i). let g ∈ j and mcp be a fixed maximal ideal of cc(x) containing i. then z(g) ⊇ ∩zc[i] ⊇ ∩zc[mcp] = {p}. so g(p) = 0 and hence g ∈ mcp. therefore g ∈ i. for the reverse inclusion, we show that if g /∈ j then g /∈ i. if g /∈ j then there exists x ∈ ∩zc[i] r z(g). so i ⊆ mcx but g /∈ mcx. this means that g /∈ i. the proof of (i) is now complete. (ii). by (i), we have ∩zc[i] = ∩zc[j] ⊇ ∩zc[i]. on the other hand, i ⊆ i implies zc[i] ⊆ zc[i] and therefore ∩zc[i] ⊇∩zc[i]. so it gives the result. � corollary 3.6. let i be a proper ideal in cc(x) and i as defined in theorem 3.5. then i = cli. proof. since maximal ideals are closed, ⋂ i⊆m m is also closed, where m is a maximal ideal in cc(x). therefore cli ⊆ ⋂ i⊆m m ⊆ i. let g ∈ i and n�(g) is a neighborhood of g. by proposition 3.4, there is f such that z(f) is a neighborhood of z(g) and ‖g −f‖≤ �. hence, by theorem 3.5, ∩zc[i] ⊆ z(g) ⊆ intz(f) and therefore lemma 3.3 implies f ∈ i. now, since f ∈ n�(g) ∩ i, it gives g ∈ cli. so i ⊆ cli and we are done. � we conclude the article with the following results for the proper ideals of cc(x). corollary 3.7 is a consequence of corollary 2.8, proposition 3.1 (iii) and corollary 3.6; by the same results, plus corollary 3.7 we obtain corollary 3.8; finally corollary 3.9 is the combination of corollaries 3.7 and 3.8. corollary 3.7. an ideal i of cc(x) is closed in cc(x) if and only if it is an intersection of maximal ideals of cc(x). corollary 3.8. an ideal i of cc(x) is an ec-ideal if and only if it is closed in cc(x). corollary 3.9. an ideal i of cc(x) is an ec-ideal if and only if it is an intersection of maximal ideals of cc(x). acknowledgements. the author would like to thank the referee for the careful reading of the manuscript and for pointing out some very useful suggestions toward the improvement of the paper. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 404 ec-filters and ec-ideals in the functionally countable subalgebra of c ∗(x) references [1] f. azarpanah, intersection of essential ideals in c(x), proc. amer. math. soc. 125 (1997), 2149–2154. 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[18] s. willard, general topology, addison-wesley, 1970. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 405 @ appl. gen. topol. 23, no. 1 (2022), 107-119 doi:10.4995/agt.2022.15669 © agt, upv, 2022 fixed point index computations for multivalued mapping and application to the problem of positive eigenvalues in ordered space vo viet tri division of applied mathematics, thu dau mot university, binh duong province, vietnam (trivv@tdmu.edu.vn) communicated by e. karapinar abstract in this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue. 2020 msc: 47h07; 47h08; 47h10; 35p30. keywords: multivalued operator; multivalued mapping; fixed point index; eigenvalue; eigenvector. 1. introduction the theory of the fixed point index for the compact single-valued mapping has achieved many brilliant achievements in studying the existence of solutions of equations, through the existence results on fixed points of operators (see e.g. [1, 9, 14–17]). this concept has been extended to multivalued mappings very early in [10] and the references therein. up to now, this topic has always been interested by many mathematicians (see e.g. [2–8,12,18–23,28,30,31]). an impressive achievement when extending this theory to multivalued mappings can be found in [12], in which the authors established the concept of a fixed point index with respect to the cone for multivalued operators acting on convex fréchet spaces with convex and closed values. this concept has remarkable received 22 may 2021 – accepted 10 july 2021 http://dx.doi.org/10.4995/agt.2022.15669 v. v. tri properties including the fixed point index concept for compact single-valued mappings. when studying multivalued equations, we may face several difficulties due to some strict properties appearing on the multivalued operator including possessing the value of a closed, convex set (which the single-valued mapping obviously owns). a natural problem generating is to find an alternative one, which is still feasible for the study on the existence of solutions. for example, we can find a selection function f satisfying the condition f(x) ∈ t(x), where t(x) is the multivalued operator. in this article, the strategy of finding alternative functions can be described as follows. we choose functions which can act as upper/lower bounds thanks to several relations between the two sets and possess natural properties including continuous linear. in addition, we also consider the condition which allows us to use a map with better properties in the case the neighborhood of the origin is sufficiently small or large enough. let us recall a well-known result on the relationship between the concept of spectral radius and the eigenvalues of a linear mapping, which is known as the krein-rutman theorem [24]. theorem 1.1. let e be a banach space with the ordered by cone k and ϕ : e → e be a positive completely continuous with spectral radius r(ϕ) > 0. then, r(ϕ) is eigenvalue of ϕ with respect to eigenvector x0. further, if ϕ is strongly positive and intk 6= ∅, then 1. x0 ∈ intk, 2. r(ϕ) is geometrically simple, 3. if λ 6= r(ϕ) is the eigenvalue of ϕ, |λ| ≤ r(ϕ). the above results have been extended to some non-strong positive mapping classes such as u0-positive [31], non-decomposable maps, etc, in the works of krasnoselskii and his students [25]. recently, in the papers of nussbaum [27], k.chang [11], mahadevan [26], krein’s theorem has been extended to the increasing, positively 1-homogeneous mapping class. following these works, in [14], we have extended these concepts to positively 1-homogeneous positivehomogeneous multivalued mappings. in [29], we evaluate the range of eigenvalues for multivalued operators, find a sufficient condition for existence of eigenvalues for the dual operator of the multivalued mapping [30]. in this paper, we continue to demonstrate a result that looks like the spectral radius of a linear mapping. we have structured our paper as follows. in the next section, we briefly recall some useful preliminaries. section 3 is divided in to two subsections with two separate results. in subsection 3.1, some results on the fixed point index of the multivalued operator are established. in subsection 3.2, some existence results for the positive eigen-pair are stated. 2. preliminaries let x be a banach space and k be a cone in x, i.e, k is a closed convex subset of x such that k + k ⊂ k, λk ⊂ k for λ ≥ 0 and k ∩−k = {θ} (θ © agt, upv, 2022 appl. gen. topol. 23, no. 1 108 fixed point index computations for multivalued mapping is the zero element of x). a partial order in x is defined by a ≤ b (or, b ≥ a) if and only if a− b ∈−k. for nonempty subsets a,b of x, we write a �1 b (or, b �1 a) iff for every a ∈ a, there is b ∈ b satisfying a ≤ b (or, a ≥ b), and write a �2 b (or, b �2 a) iff for every b ∈ b, there is a ∈ a satisfying a ≤ b (or, a ≥ b). a mapping t : x → 2x\{∅} is said to be positive if t(k) ⊂ k. throughout this paper, we use the following notations if there is no appearance of special cases. let (x,k,‖.‖) be an ordered banach space with cone k, x∗ be the dual topology space of x, ω ⊂ x be a convex neighbourhood of the origin θ, cc(k) be the all nonempty closed convex subset of k, � k = k\{θ}, ∂kω = k ∩∂ω, where ∂ω is boundary of ω in x, 〈x〉+ = {αx : α > 0}, where x ∈ x, b(x,r) = {y ∈ x : ‖x−y‖ < r}, where x ∈ x,r > 0; k∗ = {f ∈ x∗ : f(x) ≥ 0 ∀x ∈ k} , s∗+ = k ∗ ∩{p ∈ x∗ : ‖p‖ = 1} r+ = {x ∈ r : x ≥ 0}; � r+ = r+\{0}. a multivalued mapping t : k ∩ ω → 2k\{∅} is said to be compact if t(e) is relatively compact for any bounded subset e of k∩ω, where t(e) = ∪x∈et(x) and ω is the closure of ω in x. t is called an upper semi-continuous (in short, u.s.c.) if {x ∈ k∩ω : t(x) ⊂ w} is open in k∩ω for every open subset w of k. further, if x /∈ t(x) for all x ∈ ∂kω, the fixed point index of t in ω with respect to k is defined and we denote this integer index by ik(t, ω) (see e.g. [12]). t is said to be convex if its graph is convex subset in (x×x). clearly, t is convex iff (1 −λ)t(x) + λt(y) ⊂ t((1 −λ)x + λy) for all λ ∈ [0, 1] and x,y ∈ x. in what follows, we present some useful properties, which are of importance in constructing the main results in the next section. proposition 2.1 ( [12]). let ω be a bounded open and t : k ∩ω :→ cc(k) be an u.s.c compact satisfying x /∈ ∂kω. then 1. if ik(t, ω) 6= 0, then t has a fixed point, 2. if x0 ∈ ω, then ik(x̂0, ω) = 1, where x̂0 is a constant mapping with x̂0(x) = x0 ,∀x ∈ k ∩ ω. 3. if ω1, ω2 ⊂ ω are onpen with ω1 ∩ ω2 = ∅ and x /∈ t(x) for all x ∈ k ∩ (ω\(ω1 ∪ ω2)), then ik(t, ω1) + ik(t, ω2) = ik(t, ω). 4. if h : [0, 1] × (k ∩ ω) :→ cc(k) is an u.s.c compact satisfying x /∈ h(α,x) for all (α,x) ∈ [0, 1] ×∂kω, then ik(h(1, .), ω) = ik(h(0, .), ω) proposition 2.2 ( [12, 14]). let ω be a bounded open subset of x, and t : k∩ω → cc(k) be an u.s.c compact such that x /∈ t(x) for all x ∈ ∂kω. then © agt, upv, 2022 appl. gen. topol. 23, no. 1 109 v. v. tri 1. ik(t, ω) = 0 if there is u ∈ � k such that x /∈ t(x) + λu for all (λ,x) ∈ (0,∞) ×∂kω. 2. ik(t, ω) = 1 if λx /∈ t(x) for all (λ,x) ∈ (1,∞) ×∂kω. let l : x → x be positive continuous linear operator, and u0 ∈ � k. l is said to be u0-positive if for every x ∈ � k, there are α > 0, β > 0 and n,m ∈ n satisfying αu0 ≤ lnx and lmx ≤ βu0. proposition 2.3 ( [31]). let l1,l2 : x → x be positive continuous linear operators, and one of them is u0-positive. assume that l1u ≤ l2u for all u ∈ k and there exists (λ,x) ∈ � r+ × � k, (µ,y) ∈ � r+ × � k such that λx ≤ l1x and l2y ≤ µy. then, the following properties hold 1. λ ≤ µ, 2. 〈x〉 = 〈y〉 if λ = µ. proposition 2.4. 1. x ∈ k iff 〈f,x〉≥ 0 ,∀f ∈ k∗. 2. for x ∈ � k, there exists f ∈ k∗ such that 〈f,x〉 > 0. proposition 2.5 ( [13]). let x,y be banach spaces, t : ω ⊂ x → 2y\{∅} be u.s.c. assume that {(xn,yn)} is a consequence in graph(t) satisfying limn→∞(xn,yn) = (x,y). then, we have (x,y) ∈ graph(t) if t(x) is closed subset of y , where graph(t) = {(a,b) : a ∈ ω,b ∈ t(a)}. 3. abstract results 3.1. the fixed point index of the multivalued operator. in this subsection, we present several results on the fixed point index for multivalued mappings by using some useful tools including some continuous linear operators and approximate mappings at the origin and the infinity. theorem 3.1. let ω be a bounded open subset of x, a : k ∩ ω → cc(k) be an u.s.c and compact operator. 1. ik(a, ω) = 1 if there exists a continuous linear operator l with the spectral radius r(l) ≤ 1 such that (3.1) a(u) �1 lu and u /∈ a(u) ∀u ∈ ∂kω. 2. assume that x = k −k and there exists a continuous linear mapping and u0-positive l with the spectral radius r(l) ≥ 1 such that (3.2) lu �2 a(u) and u /∈ a(u) ∀u ∈ k ∩∂ω, then ik(a, ω) = 0. © agt, upv, 2022 appl. gen. topol. 23, no. 1 110 fixed point index computations for multivalued mapping proof. 1. to use proposition 2.2, we aim at showing that (3.3) λu /∈ t(u) for all (λ,u) ∈ (1,∞) ×∂kω. assume that it is not true, then we can find (λ,u) ∈ (1,∞) ×∂kω satisfying λu ∈ t(u). from (3.1), we have λu ≤ lu, which implies that (i −λ−1l)−1 is a positive continuous linear operaor. this gives u ≤ θ, which leads to u = θ. a contradiction can be seen here obviously. 2. let us choose x0 ∈ � k, we will prove that (3.4) u /∈ t(u) + λx0 ,∀(λ,u) ∈ (0,∞) ×∂kω. indeed, assume that (3.4) is not true, then u ∈ t(u) + λx0, for some (λ,u) ∈ (0,∞) × ∂kω. then, from (3.2), one obtain u ≥ lu. by the krein-rutman theorem, we have r(l) is the eigen vallue of l, i.e, there exists y ∈ � k such that ly = r(l)y. using proposition 2.3, we have r(l) = 1 and u ∈ 〈y〉+. by setting u = αy (α > 0), one can see lu = u which implies that u ≥ lu + λx0 = u + λx0. this is impossible. by proposition 2.2, we obtain ik(t, ω) = 0. the proof is complete. � theorem 3.2. let ω be a bounded open subset of x, t : k → cc(k) is an u.s.c compact convex satisfying x /∈ t(x) for all x ∈ k. then 1. ik(t, ω) = 0 if there exists (λ0,x0) ∈ (1,∞) × � k such that λ0x0 ∈ t(x0). 2. ik(t, ω) = 1 if λx ∈ t(x) for all (λ,x) ∈ (1,∞) × � k. proof. the second assertion can be seen as a consequence of proposition 2.2. to prove the first assertion, we will show that (3.5) x ∈ t(x) + λx0 ∀(λ,x) ∈ (0,∞) ×∂kω. indeed, assume the contrary, namely, x /∈ t(x)+λx0, for some (λ,x) ∈ (0,∞)× ∂kω. then, there exists y ∈ t(x),x = y + λx0. for arbitrary positive numbers α,β we have αλ0x + βx0 = αλ0y + ( β λ0 + αλ ) λ0x0. therefore, the following identity holds (3.6) αλ0x + βx0 ∈ αλ0t(x) + ( β λ0 + αλ ) t(x0). let us choose α as follows α = ( λ0 + λλ0 λ0 − 1 )−1 and β = αλλ0 λ0 − 1 . then, it is clear that β satisfies β = β λ0 + αλ and αλ0 + β λ0 + αλ = 1. © agt, upv, 2022 appl. gen. topol. 23, no. 1 111 v. v. tri now, we set v = αλ0x + βx0, v ∈ � k. since t is convex, we have αλ0t(x) + ( β λ0 + αλ ) t(x0) ⊂ t ( αλ0x + ( β λ0 + αλ )) this together with (3.6) yields v ∈ t(v). this is a contradiction, hence ik(t, ω) = 0. � let f,ϕ : k → 2k\{∅}. for every x ∈ k, we denote d (f(x),ϕ(x)) = sup{‖y −y′‖ : y ∈ f(x),y′ ∈ ϕ(x)} . we consider the following conditions for the pair (f,ϕ). (c0) : lim x∈ � k,‖x‖→0 d(f(x),ϕ(x)) ‖x‖ = 0. (c∞) : lim x∈ � k,‖x‖→∞ d(f(x),ϕ(x)) ‖x‖ = 0. in what follows, we aim at giving several relations between the aforementioned conditions and the results on the fixed point index for multivalued mappings. first of all, we are interested in giving an answer for the natural question “when does the two aforementioned conditions for the pair (f,ϕ) are guaranteed? ” by presenting some simple illustrations for such pair. example 3.3. let x = r, k = r+,b = [0, 1]. 1. let f,ϕ : k → 2k with f(x) = x + x2b and ϕ(x) = x. then, d (f(x),ϕ(x)) = x2; hence, the pair (f,ϕ) satisfies the condition (c0). 2. we define f,ϕ : k → 2k by f(x) = { {0}, x = 0, x + b, x ∈ (0,∞) and ϕ(x) = x. then, for x 6= 0 we have d (f(x),ϕ(x)) = sup{|y −x| : y = x + α,α ∈ b} = 1. therefore the pair (f,ϕ) satisfies the condition (c∞). 3. let f : r → r be a fréchet differentiable function with f(0) = 0, ϕ be the fréchet differentiable of f with recpect to k at 0 (at ∞, resp.). then, the pair (f,ϕ) satisfies the condition (c0) ((c∞), resp.) theorem 3.4. let f,ϕ : k → cc(k) be u.s.c and compact with x /∈ ϕ(x), for all x ∈ k. assume that ϕ(λx) = λϕ(x), for all λ > 0,x ∈ k (in order words, a is positively 1-homogeneous). then, it holds ik(f,b(θ,r)) = ik(ϕ,b(θ,r)), in the case the following conditions hold © agt, upv, 2022 appl. gen. topol. 23, no. 1 112 fixed point index computations for multivalued mapping 1. (f,ϕ) satisfies the condition (c0) if r is sufficiently small. 2. (f,ϕ) satisfies the condition (c∞) if r is sufficiently large. proof. for the sake of convenience, we denote h(α,x) = αf(x) + (1 −α)ϕ(x), x ∈ � k,α ∈ [0, 1]. the operator h(α,.) is u.s.c compact with closed convex values. for y ∈ f(x) and y′ ∈ ϕ(x) we have ‖x−αy − (1 −α)y′‖ = ‖x−y′ −α(y −y′)‖ ≥‖x−y′‖−α‖y −y′‖ ≥‖x−y′‖−‖y −y′‖.(3.7) set b = inf{‖x−y‖ : x ∈ k,‖x‖ = 1,y ∈ ϕ(x)}. if b = 0, we can find sequences {xn}⊂ k, {yn}⊂ k such that ‖xn‖ = 1, yn ∈ ϕ(xn) and ‖xn −yn‖→ 0. thanks to the compactness of ϕ(x), we can assume that limn→∞yn = y0 ∈ k, hence ‖y0‖ = 1. since ϕ is u.s.c, we have y0 ∈ ϕ(y0) which is contradictory with the assumption. thus, b > 0. now, fix x ∈ � k, write x = λx′ with λ = ‖x‖, then x′ ∈ � k and ‖x′‖ = 1. it follows from the positively 1-homogeneous properties of ϕ that infw∈ϕ(x) ‖x−w‖ ‖x‖ = inf 1 λ w∈ϕ(x′) ‖x′ − 1 λ w‖ ‖x‖ ≥ b. this implies that (3.8) inf w∈ϕ(x) ‖x−w‖≥ b‖x‖. from (3.8) and (3.7), we have (3.9) ‖x−αy − (1 −α)y′‖ ‖x‖ ≥ b− d (f(x),ϕ(x)) ‖x‖ . if (f,ϕ) satisfies (c0), there exists r > 0 such that b− d(f(x),ϕ(x)) ‖x‖ > 0 for all x ∈ � k with ‖x‖≤ r. from (3.9), it follows that x ∈ h(α,x) for all x ∈ ∂kb(θ, 0). hence, we deduce that ik(f,b(0,r)) = ik(ϕ,b(0,r)). if (f,ϕ) satisfies (c∞), we make the same argument as above. the proof is complete. � 3.2. existence of a positive eigen-pair. in this section we present results on the existence of the eigenvalue for multivalued operator. theorem 3.5. let a : k → cc(k) be u.s.c compact. assume that x = k−k, ω1, ω2 are bounded open subsets of x, θ ∈ ω1 ( ω2 satisfy the following conditions © agt, upv, 2022 appl. gen. topol. 23, no. 1 113 v. v. tri 1. there exist completely continuous linear maps p,q : k → k with spectral radius r(p),r(q), respectively, and p is u0-positive such that either (3.10) px �2 a(x) ∀x ∈ ∂kω1, a(x) �1 qx ∀x ∈ ∂kω2 or (3.11) px �2 a(x) ∀x ∈ ∂kω2, a(x) �1 qx ∀x ∈ ∂kω1 2. 0 < r(q) < r(p). then, for λ ∈ (r(q),r(p)), the inclusion λx ∈ a(x) has a positive solution. proof. we assume that (3.10) is satisfied and x /∈ µa(x), for all x ∈ ∂ω1 ∪ ω2. denote by µ = λ−1. then, we have µa(u) �1 µqu ,∀u ∈ ∂kω2. since r(µq) ≤ 1 by theorem 3.1, we obtain ik(µa, ω2) = 1. similarly, ik(µa, ω1) = 0. it follows that ik(µa, ω2\ω1) = 1 by proposition 2.1. hence, µa has a fixed point in ω2\ω1. by a similar argument as in the previous one with the condition (3.11). � let ϕ : k → 2k\{∅}, we denote r∗(ϕ) = sup { λ > 0 : ∃x ∈ � k,λx ∈ ϕ(x) } , define sup ∅ = 0; r∗(ϕ) = inf { λ > 0 : ∃x ∈ � k,λx ∈ ϕ(x) } , define inf ∅ = ∞; theorem 3.6. let a : k → cc(k) be u.s.c compact. assume that there exist positively 1-homogeneous convex operators p,q : k → cc(k0) satisfying the following conditions 1. (a,p) satisfies (c0) and (a,q) satisfies (c∞), 2. 0 < r∗(p) < r∗(q) < ∞ (or 0 < r∗(q) < r∗(p) < ∞, resp.) then, if λ ∈ (r∗(p),r∗(q)) (or λ ∈ (r∗(q),r∗(p)) resp.), the equation λx ∈ a(x) has a solution in � k. proof. denote by µ = λ−1, f = µa, ϕ1 = µp, ϕ2 = µq. we first prove that there are r1 > 0,r2 > 0 (r1 < r2) such that (3.12) ik(f, ω1) = ik(ϕ1, ω1) and ik(f, ω2) = ik(ϕ2, ω2), where ω1 = b(θ,r1), ω2 = b(θ,r2). indeed, applying theorem 3.4 for the pair (f,ϕ1) we can find r1 > 0 (small enough) such that ik(f,b(θ,r1)) = ik(ϕ1,b(θ,r1)). similarly, there exists r2 > 0 (large enough) satisfying ik(f,b(θ,r2)) = ik(ϕ2,b(θ,r2)). now, we assume that 0 < r ∗(p) < r∗(q). by theorem 3.2, ik(f, ω2) = 0 and ik(f, ω1) = 1, this leads to the assertion that needs to be proved. in the case 0 < r∗(q) < r∗(p) < ∞ the proof is analogous to the one above. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 114 fixed point index computations for multivalued mapping let a in a nonempty subset of k, for ervery p ∈ k∗ we define σ(a,p) = {〈p,x〉 : x ∈ a} , where 〈p,x〉 is value of p at x. for u ∈ � k we denote s = u+k. in the following lemma, we present the eigenvalue for the bounded multivalued operator. lemma 3.7. assume f : s → 2k\{∅} satisfying the conditions following (i) σ(f(x),p) for all (p,x) ∈ s∗+ ×s, (ii) f(s) is relatively compact in x, (iii) there is (α,v) ∈ (0,∞) ×s such that αv �1 f(v). then (3.13) 0 0. for any x ∈ s with x = u + y,y ∈ k, we have 〈p0,x〉 = 〈p0,u〉 + 〈p0,y〉≥ µ0. hence, 〈p0,x〉 σ(f(x),p0) ≥ µ0 m ∀x ∈ s, which gives inf x∈s 〈p0,x〉 σ(f(x),p0) ≥ µ0 m > 0. this implies that (3.14) sup p∈s∗ + ( inf p∈s 〈p,x〉 σ(f(x),p) ) ≥ inf x∈s 〈p0,x〉 σ(f(x),p0) ≥ µ0 m > 0. from the condition (iii), we can find z ∈ f(v) such that αz ≤ z. therefore 〈p,αv〉≤ 〈p,αz〉≤ σ(f(v),p), so 〈p,v〉 σ(f(v),p) ≤ 1 α for all p ∈ s∗+. it follows that inf y∈s 〈p,y〉 σ(f(y),p) ≤ 1 α for all p ∈ s∗+. this implies that (3.15) sup p∈s∗ + ( inf p∈s 〈p,x〉 σ(f(x),p) ) ≤ 1 α . the proof is complete. � theorem 3.8. let f : s → cc(k) be an u.s.c convex multivalued operator satisfying the conditions in lemma 3.7. then © agt, upv, 2022 appl. gen. topol. 23, no. 1 115 v. v. tri 1. if λ0 is defined by 1 λ0 = sup p∈s∗ + ( inf p∈s 〈p,x〉 σ(f(x),p) ) , there exists x0 ∈ s such that λ0x0 ∈ f(x0) and 1 λ0 = sup p∈s∗ + 〈p,x0〉 σ(f(x0),p) . 2. further, if (λ,x) ∈ (0,∞) ×s with λx ∈ f(x), we have λ ≤ λ0. proof. we first see that λ0 is fine defined by lemma 3.7. we will prove the first assertion by steps following step 1. (showing that (f − λ0i)(s) is convex subset of k). assmue that z,z′ ∈ (f −λ0i)(s) and α,β ∈ [0, 1] with α + β = 1. there is (x,x′) ∈ s ×s such that z ∈ f(x) −λ0x and z′ ∈ f(x′) −λ0x′. we have αz + βx ∈ αf(x) + βf(x′) −λ0(αx + βx′). by the convexity of f, αf(x) + βf(x′) ⊂ f(αx + βx′). since s is convex, αx + βx′ ∈ s, hence αz + βz′ ∈ (f −λ0i)(s). step 2. (showing that (f −λ0i)(s) is close subset of k). assume {zn}n=1,2,... is a sequence in (f − λ0i)(s) with limn→∞zn = z. we can find a sequence {xn}⊂ s and {yn} satisfying zn ∈ f(xn) −λ0xn, yn ∈ f(xn) and (3.16) zn = yn −λ0xn. since f(s) is relatively compact, we can assume limn→∞yn = y. therefore, there exists x ∈ s, limn→∞xn = x. on the other hand, f is u.s.c and f(x) is closed set, by proposition 2.5 it follows that y ∈ f(x). letting n → ∞ in (3.16) we obtain z = y −λ0x, thus z ∈ (f −λ0i)(s). step 3. (proving θ ∈ (f − λ0i)(s)). assume the contrary, that θ /∈ (f − λ0i)(s). by applying separation theorem for two sets {θ} and (f −λ0i)(s) we can fine a number � > 0 and p1 ∈ x∗ with ‖p1‖ = 1 (for if not, we replace p1 by 1 ‖p1 p1) such that 〈p1,z〉 < −� ∀z ∈ (f −λ0i)(s), i.e, 〈p1,y〉−λ0〈p1,x〉 < −� ∀(x,y) ∈ s ×f(x). this implies that (3.17) σ(f(x),p1) −λ0〈p1,x〉≤−� for all x ∈ s. we now will show that p1 ∈ s∗+. indeed, if there exists y ∈ k such that 〈p1,y〉 < 0, using (3.17) for x = u + ny, (n = 1, 2, ...) we have (3.18) σ(f(x),p1) −λ0〈p1,u〉−nλ0〈p1,y〉≤−�. set c = sup{σ(f(x),p1) : x ∈ s}. since f(s) is relatively compact, c ∈ (−∞,∞). letting n →∞ in (3.17) we obtain a contradiction, hence p1 ∈ s∗+. from (3.17) it follows that 1 λ0 ≤ 〈p1,x〉 σ(f(x),p1) − � λ0c for all x ∈ s. © agt, upv, 2022 appl. gen. topol. 23, no. 1 116 fixed point index computations for multivalued mapping thus 1 λ0 ≤ inf x∈s 〈p1,x〉 σ(f(x),p1) − � λ0c . on the other hand, inf x∈s 〈p1,x〉 σ(f(x),p1) ≤ sup p∈s∗ + ( inf x∈s 〈p,x〉 σ(f(x),p) ) = 1 λ0 . this implies 1 λ0 ≤ 1 λ0 − � λ0c . we have a contradiction. hence θ ∈ (f −λ0i)(s). therefore, there exists x0 ∈ s such that λ0x0 ∈ f(x0). step 4. (showing that 1 λ0 = sup p∈s∗ + ( 〈p,x0〉 σ(f(x0),p) ) ). for every p ∈ s∗+, we have σ(f(x0),p) ≥〈p,λ0x0〉 = λ0〈p,x0〉. thus, 1 λ0 ≥ 〈p,x0〉 σ(f(x0),p) for all p ∈ s∗+. on the other hand, we have 1 λ0 ≥ 〈p,x0〉 σ(f(x0),p) ≥ inf x∈s 〈p,x〉 σ(f(x),p) for all p ∈ s∗+. this implies that 1 λ0 ≥ sup p∈s∗ + 〈p,x0〉 σ(f(x0),p) ≥ sup p∈s∗ + ( inf x∈s 〈p,x〉 σ(f(x),p) ) = 1 λ0 . we deduce 1 λ0 = sup p∈s∗ + ( 〈p,x0〉 σ(f(x0),p) ) . now, we prove the second assertion. assume that λx ∈ f(x) for some (λ,x) ∈ (0,∞) ×s. then, we have σ(f(x),p) ≥〈p,λx〉 = λ〈p,x〉. thus, 1 λ ≥ 〈p,x〉 σ(f(x),p) ≥ inf y∈s 〈p,y〉 σ(f(y),p) . it follows that 1 λ ≥ sup p∈s∗ + ( inf x∈s 〈p,x〉 σ(f(x),p) ) = 1 λ0 . hence λ ≤ λ0. the proof is complete. � remark 3.9. 1. in the proofs of our results, we have not used the cone condition with nonempty interior (which is called the solid cone).therefore, the case int(k) = ∅ is just a special case of the results in this work. in theorem 3.1 and theorem 3.5 , we have used the condition that k is a reproducing cone. a solid cone is a reproducing cone. however, the opposite is not true. © agt, upv, 2022 appl. gen. topol. 23, no. 1 117 v. v. tri for example, let ω be a bouned subset of rn, x = lp(ω). the set of nonnegative functions k in x is a reproducing cone. however, it has empty interior. 2. same as above, the normal cone condition has not been used. 4. conclusion this paper is a continuation of the series works [14, 29, 30] of extending the well-known result of krein-rutman theorem. initially, we investigate the fixed point index for multivalued mappings by using some useful tools including some continuous linear operators and approximate mappings at the origin and the infinity. lastly, distinct results on the existence of solutions to the multivalued equations are constructed flexibly. acknowledgements. this paper was supported by thu dau mot university under grant number dt.21.1-014. references [1] t. abdeljawad, e. karapinar and k. tas, existence and uniqueness of a common fixed point on partial metric spaces. applied mathematics letters 24, no. 11 (2011), 1900– 1904. 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[31] j. r. l. webb, remarks on u0-positive operators, j. fixed point theory appl. 5 (2009), 37–45. © agt, upv, 2022 appl. gen. topol. 23, no. 1 119 @ appl. gen. topol. 21, no. 2 (2020), 235-246 doi:10.4995/agt.2020.12967 c© agt, upv, 2020 on i-quotient mappings and i-cs′-networks under a maximal ideal xiangeng zhou∗ department of mathematics, ningde normal university, fujian, 352100, p.r. china (56667400@qq.com) communicated by p. das abstract let i be an ideal on n and f : x → y be a mapping. f is said to be an i-quotient mapping provided f−1(u) is i-open in x, then u is i-open in y . p is called an i-cs′-network of x if whenever {xn}n∈n is a sequence i-converging to a point x ∈ u with u open in x, then there is p ∈ p and some n0 ∈ n such that {x, xn0} ⊆ p ⊆ u. in this paper, we introduce the concepts of i-quotient mappings and i-cs′-networks, and study some characterizations of i-quotient mappings and i-cs′networks, especially j -quotient mappings and j -cs′-networks under a maximal ideal j of n. with those concepts, we obtain that if x is an j -fu space with a point-countable j -cs′-network, then x is a meta-lindelöf space. 2010 msc: 54a20; 54b15; 54c08; 54d55; 40a05; 26a03. keywords: ideal convergence; maximal ideal; i-sequential neighborhood; i-quotient mappings; i-cs′-networks; i-fu spaces. 1. introduction statistical convergence was introduced by h. fast [9] and h. steinhaus [16], which is a generalization of the usual notion of convergence. it is doubtless that the study of statistical convergence and its various generalizations has become an active research area [2, 3, 7, 17, 18]. in particular, p. kostyrko, t. šalát ∗this research is supported by nsfc (no. 11801254) and ningde normal university (no. 2017t01; 2018zdk11; 2019zdk11). received 10 january 2020 – accepted 11 april 2020 http://dx.doi.org/10.4995/agt.2020.12967 x. zhou and w. wilczynski [11] introduced two interesting generalizations of statistical convergence by using the notion of ideals of subsets of positive integers, which were named as i and i∗-convergence, and studied some properties of i and i∗-convergence in metric spaces. later, b.k. lahiri and p. das [12] discussed i and i∗-convergence in topological spaces. some further results connected with i and i∗-convergence can be found in [4, 5, 6]. as we know, mappings and networks are important tools of investigating topological spaces. continuous mappings, quotient mappings , pseudo-open mappings, cs-networks, sn-networks, k-networks and so on are the most important tools for studying convergence, sequential spaces, fréchet-urysohn spaces [14] and generalized metric spaces. for this reason, this paper draws into iquotient mappings and i-cs′-networks for an ideal i on n and discusses some basic properties of them. recently, the researches on i-convergence are mainly focused on aspects of i∗-convergence [12], i-limit points [11], i-cauchy sequence [5], ideal-convergence classes [4], selection principles [6], ideal sequence covering mappings [15, 19] and so on. it is expected that i-quotient mappings and i-cs′-networks will also play active roles in the topological spaces. in this paper, the letter x always denote a topological space. the cardinality of a set b is denoted by |b|. the set of all positive integers, the first infinite ordinal, and the first uncountable ordinal are denoted by n, ω and ω1, respectively. the reader may refer to [8, 14] for notation and terminology not explicitly given here. 2. preliminaries recall the notion of statistical convergence in topological spaces. for each subset a of n the asymptotic density of a, denoted δ(a), is given by δ(a) = lim n→∞ 1 n |{k ∈ a : k ≤ n}|, if this limit exists. let x be a topological space. a sequence {xn}n∈n in x is said to converge statistically to a point x ∈ x [7], if δ({n ∈ n : xn ∕∈ u}) = 0, i.e., δ({n ∈ n : xn ∈ u}) = 1 for each neighborhood u of x in x, which is denoted by slim n→∞ xn = x or xn s−→ x. the concept of i-convergence of sequences in a topological space is a generalization of statistical convergence which is based on the ideal of subsets of the set n of all positive integers. let a = 2n be the family of all subsets of n. an ideal i ⊆ a is a hereditary family of subsets of n which is stable under finite unions [11], i.e., the following are satisfied: if b ⊆ a ∈ i, then b ∈ i; if a, b ∈ i, then a ∪ b ∈ i. an ideal i is said to be non-trivial, if i ∕= ∅ and n /∈ i. a non-trivial ideal i ⊆ a is called admissible if i ⊇ {{n} : n ∈ n}. clearly, every non-trivial ideal i defines a dual filter fi = {a ⊆ n : n\a ∈ i} on n. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 236 on i-quotient mappings and i-cs′-networks under a maximal ideal let if be the family of all finite subsets of n. then if is an admissible ideal. let iδ [11] be the family of subsets a ⊆ n with δ(a) = 0. then iδ is an admissible ideal, and the dual filter fiδ = {a ⊆ n : δ(a) = 1}. definition 2.1 ([11]). a sequence {xn}n∈n in a topological space x is said to be i-convergent to a point x ∈ x provided for any neighborhood u of x, we have {n ∈ n : xn /∈ u} ∈ i, which is denoted by ilim n→∞ xn = x or xn i−→ x, and the point x is called the i-limit of the sequence {xn}n∈n. definition 2.2 ([20]). let i be an ideal on n and x be a topological space. (1) a subset f ⊆ x is said to be i-closed if for each sequence {xn}n∈n ⊆ f with xn i−→ x ∈ x, we have x ∈ f . (2) a subset u ⊆ x is said to be i-open if x \ u is i-closed. (3) x is called an i-sequential space if each i-closed subset of x is closed. obviously, each sequential space is an i-sequential space [20]. definition 2.3 ([20]). let i be an ideal on n, x, y be topological spaces and f : x → y be a mapping. (1) f is called preserving i-convergence provided for each sequence {xn}n∈n in x with xn i−→ x, the sequence {f(xn)}n∈n i-converges to f(x) [12]. (2) f is called i-continuous provided u is i-open in y , then f−1(u) is i-open in x. it is easy to verify that a mapping f : x → y is i-continuous if and only if, whenever f is i-closed in y , then f−1(f) is i-closed in x. lemma 2.4 ([20]). let i be an ideal on n and x be a topological space. if a sequence {xn}n∈n i-converges to a point x ∈ x, and {yn}n∈n is a sequence in x with {n ∈ n : xn ∕= yn} ∈ i, then the sequence {yn}n∈n i-converges to x ∈ x. lemma 2.5 ([20]). let i be an ideal on n. the following are equivalent for a topological space x and a subset a ⊆ x. (1) a is i-open. (2) {n ∈ n : xn ∈ a} /∈ i for each sequence {xn}n∈n in x with xn i−→ x ∈ a. (3) |{n ∈ n : xn ∈ a}| = ω for each sequence {xn}n∈n in x with xn i−→ x ∈ a. lemma 2.6 ([20]). let x, y be topological spaces and f : x → y be a mapping. (1) if f is continuous, then f preserves i-convergence [12]. (2) if f preserves i-convergence, then f is i-continuous. definition 2.7 ([20]). let a ⊆ x and {xn}n∈n be a sequence in x. if i is an ideal on n, then {xn}n∈n is i-eventually in a if there is e ∈ i such that for all n ∈ n \ e, xn ∈ a. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 237 x. zhou if a is a subset of x with the property that every sequence i-converging to a point in a is i-eventually in a, then a is i-open. when we assume j to be a maximal ideal, the following proposition shows that such sets must coincide with j -open sets. proposition 2.8 ([20]). if j is a maximal ideal of n, then a ⊆ x is j -open if and only if for each j -converging sequence {xn}n∈n with xn j−→ x ∈ a, then {xn}n∈n is j -eventually in a. by definition 2.2, the union of a family of i-open sets in a topological space is i-open. whenever j is a maximal ideal, the intersection of two j -open sets is an j -open set. proposition 2.9 ([20]). if j is a maximal ideal of n and u, v are two j -open subsets of x, then u ∩ v is j -open in x. it is well known that the sequential coreflection sx of a space x is the set x endowed with the topology consisting of sequentially open subsets of x. let j be a maximal ideal of n and x be a topological space. by definition 2.2 and proposition 2.9, the family of all j -open subsets of x forms a topology of the set x. the j -sequential coreflection of a space x is the set x endowed with the topology consisting of j -open subsets of x, which is denoted by j sx. the spaces x and j -sx have the same j -convergent sequences; j -sx is an j -sequential space; a space x is an j -sequential space if and only if j -sx = x [20]. if no otherwise specified, we consider ideal i is always an admissible ideal on n, all mappings are continuous and surjection, all spaces are hausdorff. 3. i-quotient mappings in this section, we introduce the concept of i-quotient mappings, and obtain some characterizations of i-quotient mappings, especially j -quotient mappings under a maximal ideal of n. let x, y be arbitrary topological spaces, and f : x → y be a mapping. f is said to be quotient provided f−1(u) is open in x, then u is open in y ; f is said to be sequentially quotient provided f−1(u) is sequentially open in x, then u is sequentially open in y [1]. definition 3.1. let i be an ideal on n and f : x → y be a mapping. (1) f is said to be an i-quotient mapping (or shortly, i-quotient) provided f−1(u) is i-open in x, then u is i-open in y . (2) f is said to be an i-covering mapping (or shortly, i-covering) if, whenever {yn}n∈n is a sequence in y i-converging to y in y , there exist a sequence {xn}n∈n of points xn ∈ f−1(yn) for all n ∈ n and x ∈ f−1(y) such that xn i−→ x. in [20], it was showed that each i-covering mapping is i-quotient. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 238 on i-quotient mappings and i-cs′-networks under a maximal ideal definition 3.2. let i be an ideal on n, x be a topological space and p ⊂ x. p is called an i-sequential neighborhood of x, if each sequence {xn}n∈n iconverges to a point x ∈ p , then {xn}n∈n is i-eventually in p , i.e., there is i ∈ i such that {n ∈ n : xn /∈ p} = i. remark 3.3. let j be a maximal ideal of n and a ⊆ x. by proposition 2.8, a is j -open in x if and only if a is an j -sequential neighborhood of x for each x ∈ a. proposition 3.4. let j be a maximal ideal of n and a ⊆ x. if a is not an j -sequential neighborhood of x, then there is a sequence {xn}n∈n in x \ a such that xn j−→ x. proof. if a is not an j -sequential neighborhood of x, then there is a sequence {yn}n∈n in x such that yn j−→ x, but {n ∈ n : yn /∈ a} /∈ j . since j is a maximal ideal of n, this means that {n ∈ n : yn ∈ a} ∈ j . let {n ∈ n : yn ∈ a} = j ∈ j . and since j is a non-trivial ideal, it follows that a ∕= x. take a point a ∈ x \ a. define a sequence {xn}n∈n by xn = a if n ∈ j; xn = yn if n ∈ n \ j. then the sequence {xn}n∈n in x \ a and xn j−→ x from lemma 2.5. □ theorem 3.5. let i be an ideal on n. if f : x → y is an i-quotient mapping, then for each i-convergent sequence {yn}n∈n in y with yn i−→ y, there is a sequence {xi}i∈n in x such that {xi : i ∈ n} ⊆ f−1({yn : n ∈ n}) and xi i−→ x /∈ f−1({yn : n ∈ n}). proof. suppose that f : x → y is an i-quotient mapping and {yn}n∈n is a sequence in y with yn i−→ y. without loss of generality, we can assume that yn ∕= y for each n ∈ n. let u = y \ {yn : n ∈ n}. then u is not i-open in y . since f is an i-quotient mapping, f−1(u) = f−1(y \ {yn : n ∈ n}) = x \ f−1({yn : n ∈ n}) is not i-open in x. thus there is a sequence {xi}i∈n in x \ f−1(u) = f−1({yn : n ∈ n}) such that xi i−→ x /∈ f−1({yn : n ∈ n}). □ in [20], it was discussed that quotient mappings, sequentially quotient mappings and i-quotient mappings are mutually independent; and the following two theorems are useful and can be seen in it. theorem 3.6. let f : x → y be a mapping. (1) if x is an i-sequential space and f is quotient, then y is an isequential space and f is i-quotient. (2) if y is an i-sequential space and f is i-quotient, then f is quotient. (3) x is an i-sequential space if and only if for an arbitrary topological space y , if f is quotient, then f is i-quotient. theorem 3.7. let j be a maximal ideal of n and x be a topological space. then x is an j -sequential space if and only if each j -quotient mapping onto x is quotient. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 239 x. zhou let j be a maximal ideal of n and a ⊆ x. denote [a]j -s = {x ∈ x : there is a sequence {xn}n∈n in a such that xn j−→ x}; (a)j -s = {x ∈ x : a is an j -sequential neighborhood of x}. a subset u ⊆ x is said to be an j -sequential neighborhood of a if a ⊆ (u)j -s. proposition 3.8. let j be a maximal ideal of n and a ⊆ x. then [a]j -s = x \ (x \ a)j -s. proof. suppose that x ∈ [a]j -s, then there is a sequence {xn}n∈n in a such that xn j−→ x. thus x \ a is not an j -sequential neighborhood of x in x. in fact, if x \ a is an j -sequential neighborhood of x in x, then {xn}n∈n is j eventually in x \a, i.e., there is e ∈ j such that for all n ∈ n\e, xn ∈ x \a. since j is an admissible ideal, this contradicts to {xn}n∈n in a. therefore x /∈ (x \ a)j -s, and further x ∈ x \ (x \ a)j -s. on the other hand, assume that x ∈ x \ (x \ a)j -s, then x /∈ (x \ a)j -s, and hence x\a is not an j -sequential neighborhood of x in x. by proposition 3.4, there is a sequence {xn}n∈n in a such that xn j−→ x. thus x ∈ [a]j -s. □ by definition 2.2 and proposition 3.8, the following proposition is correct. proposition 3.9. let j be a maximal ideal of n and a, b ⊆ x. then (1) [∅]j -s = ∅, a◦ ⊆ (a)j -s ⊆ a ⊆ [a]j -s ⊆ a. (2) a is j -open in x if and only if a = (a)j -s. (3) a is j -closed in x if and only if a = [a]j -s. (4) if b ⊆ a, then (b)j -s ⊆ (a)j -s and [b]j -s ⊆ [a]j -s. (5) (a ∩ b)j -s = (a)j -s ∩ (b)j -s and [a ∪ b]j -s = [a]j -s ∪ [b]j -s. proof. we only prove that (5) is true. since a ∩ b ⊆ a, a ∩ b ⊆ b, it follows that (a ∩ b)j -s ⊆ (a)j -s, (a ∩ b)j -s ⊆ (b)j -s. hence (a ∩ b)j -s ⊆ (a)j -s ∩ (b)j -s. on the other hand, assume that x ∈ (a)j -s ∩ (b)j -s. then for each sequence {xn}n∈n in x with xn j−→ x, there is e, f ∈ j , such that for each n ∈ n \ e, xn ∈ a and for each n ∈ n \ f , xn ∈ b. since e ∪ f ∈ j and for each n ∈ n\(e ∪f), xn ∈ a∩b. this means that a∩b is an j -sequential neighborhood of x in x. thus x ∈ (a ∩ b)j -s. now replace x \ a with a and x \ b with b, it follows that ((x \ a) ∩ (x \ b))j -s = (x \a)j -s ∩(x \b)j -s. hence [a∪b]j -s = x \(x \(a∪b))j -s = x \ ((x \ a) ∩ (x \ b))j -s = x \ ((x \ a))j -s ∩ (x \ b))j -s) = (x \ (x \ a)j -s) ∪ (x \ (x \ b)j -s) = [a]j -s ∪ [b]j -s. □ theorem 3.10. let j be a maximal ideal of n and f : x → y be a mapping. then the following conditions are equivalent. (1) for each j -convergent sequence {yn}n∈n in y with yn j−→ y, there is a sequence {xi}i∈n in x with xi j−→ x ∈ f−1(y) and {xi : i ∈ n} ⊆ f−1({yn : n ∈ n}). (2) for each a ⊆ y , it has f([f−1(a)]j -s) = [a]j -s. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 240 on i-quotient mappings and i-cs′-networks under a maximal ideal (3) if y ∈ [a]j -s ⊆ y , then f−1(y) ∩ [f−1(a)]j -s ∕= ∅. (4) if y ∈ [a]j -s ⊆ y , then there is a point x ∈ f−1(y) such that whenever v is an j -sequential neighborhood of x, y ∈ [f(v ) ∩ a]j -s. (5) if y ∈ [a]j -s ⊆ y , then there is a point x ∈ f−1(y) such that whenever v is an j -sequential neighborhood of x, f(v ) ∩ a ∕= ∅. (6) for each y ∈ y , if u is an j -sequential neighborhood of f−1(y), then f(u) is an j -sequential neighborhood of y. proof. (1) ⇒ (2) suppose that x ∈ [f−1(a)]j -s. then there is a sequence {xn}n∈n in f−1(a) such that xn j−→ x. hence {f(xn) : n ∈ n} ⊆ a and f(xn) j−→ f(x). this means that f(x) ∈ [a]j -s. hence f([f−1(a)]j -s) ⊆ [a]j -s. on the other hand, assume that y ∈ [a]j -s. then there is a sequence {yn}n∈n in a such that yn j−→ y. by the condition (1), there is a sequence {xi}i∈n in x with {xi : i ∈ n} ⊆ f−1({yn : n ∈ n}) ⊆ f−1(a) and xi j−→ x ∈ f−1(y). thus x ∈ [f−1(a)]j -s, hence y = f(x) ∈ f([f−1(a)]j -s), and further [a]j -s ⊆ f([f−1(a)]j -s). (2) ⇒ (3) let y ∈ [a]j -s for each a ⊆ y . by the condition (2), it follows that y ∈ f([f−1(a)]j -s). thus f−1(y) ∩ [f−1(a)]j -s ∕= ∅. (3) ⇒ (4) let y ∈ [a]j -s ⊆ y . by the condition (3), assume that x ∈ f−1(y) ∩ [f−1(a)]j -s. then there is a sequence {xn}n∈n in f−1(a) such that xn j−→ x. if v is an j -sequential neighborhood of x, then there is e ∈ j such that xn ∈ v for all n ∈ n \ e. hence f(xn) ∈ f(v ) ∩ a for all n ∈ n \ e and f(xn) j−→ f(x). take a point a ∈ f(v ) ∩ a. define a sequence {yn}n∈n by yn = f(xn) if n ∈ n \ e; yn = a if n ∈ e. then {yn : n ∈ n} ⊆ f(v ) ∩ a and yn j−→ f(x) = y from lemma 2.4. thus y ∈ [f(v ) ∩ a]j -s. (4) ⇒ (5) it is clear. (5) ⇒ (6) let y ∈ y and u be an j -sequential neighborhood of f−1(y). if f(u) is not an j -sequential neighborhood of y, then y ∈ y \ (f(u))j -s = [y \ f(u)]j -s. by the condition (5), it follows that f(u) ∩ (y \ f(u)) = ∅, a contradiction. (6) ⇒ (3) let y ∈ [a]j -s ⊆ y . suppose that f−1(y) ∩ [f−1(a)]j -s = ∅. then f−1(y) ⊆ x \ [f−1(a)]j -s = (x \ f−1(a))j -s. this means that x \ f−1(a) is an j -sequential neighborhood of f−1(y). by the condition (6), y ∈ (f(x \ f−1(a)))j -s = (y \ a)j -s = y \ [a]j -s, a contradiction. (3) ⇒ (1) let {yn}n∈n be an j -convergent sequence in y with yn j−→ y. put a = {yn : n ∈ n}, then y ∈ [a]j -s. by the condition (3), there is x ∈ f−1(y) ∩ [f−1(a)]j -s. hence there is a sequence {xi}i∈n in x with {xi : i ∈ n} ⊆ f−1(a) ⊆ f−1({yn : n ∈ n}) and xi j−→ x ∈ f−1(y). □ remark 3.11. (1) theorem 3.5 is different from lemma 3.10 (1). in lemma 3.10 (1), xi j−→ x ∈ f−1(y). but we don’t know whether the i-limit point x in f−1(y) or not in theorem 3.5. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 241 x. zhou (2) one of the above six conditions can deduce that f is an j -quotient mapping. in fact, let u be non-i-closed in y . then there is a sequence {yn}n∈n in u j -converging to y ∈ y \u. thus y ∕= yn for each n ∈ n. by the assumption of the condition (1), there is a sequence {xi}i∈n in x such that {xi : i ∈ n} ⊆ f−1({yn : n ∈ n}) ⊆ f−1(u) and xi j−→ x ∈ f−1(y) /∈ f−1(u). this implies that f−1(u) is non-j -closed in x. hence, f is an j -quotient mapping. (3) if the maximal ideal j is replaced by if in theorem3.10, then (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇔ (5) ⇔ (6) ⇔ f is an if -quotient mapping. but the following example shows that there exist a t1 space x, an ideal i of n and an i-quotient mapping f such that f does not satisfy the condition (6) of theorem 3.10. example 3.12. there exist a t1 space x, an ideal i of n and an i-quotient mapping f, but f does not satisfy the condition (6) of theorem 3.10. proof. let i = {a ⊆ n : a contains at most only finite odd positive integers}. then i is an admissible ideal of n . let y be the set ω which is endowed with the finite complement topology. then y is a first-countable t1-space. put x0 = y \ {0} and x1 = {2k : k ∈ ω} as the subspaces of the space y , and x = x0 ! x1. a mapping f : x → y is defined by the natural mapping. it is easy to see that the mapping f is a continuous quotient mapping. since x0 and x1 are first-countable space, x is a first-countable space. thus, x is an i-sequential space. by theorem 3.6, it follows that f is an i-quotient mapping. note that the set x1 is open in x and f −1(0) ⊆ x1, and hence x1 is an i-sequential neighborhood of f−1(0). for each open neighborhood u of 0 in y , {n ∈ n : n /∈ u} is a finite subset, hence {n ∈ n : n /∈ u} ∈ i. this means that the sequence {n}n∈n in y satisfies n i−→ 0. but {n ∈ n : n /∈ f(x1)} = {2k + 1, k ∈ ω} /∈ i. thus f(x1) is not an i-sequential neighborhood of 0 in y . □ problem 3.13. for some maximal ideal j of n and an j -quotient mapping f, does it satisfy the condition (6) of theorem 3.10? 4. on spaces with i-cs′-networks in this section, we introduce the concepts of i-cs-networks, i-cs′-networks and i-wcs′-networks for a space x; and obtain that if x is an j -fu space with a point-countable j -cs′-network, then x is a meta-lindelöf space, for a maximal ideal j of n. definition 4.1 ([13]). let i be an ideal on n, x be a topological space and p be a cover of x. (1) p is a network of x if whenever x ∈ u with u open in x, then x ∈ p ⊆ u for some p ∈ p. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 242 on i-quotient mappings and i-cs′-networks under a maximal ideal (2) p is called an i-cs-network of x if whenever {xn}n∈n is a sequence in x i-converging to a point x ∈ u with u open in x, then {xn}n∈n is i-eventually in p and x ∈ p ⊆ u for some p ∈ p. (3) p is called an i-cs′-network of x if whenever {xn}n∈n is a sequence in x i-converging to a point x ∈ u with u open in x, then there is p ∈ p and some n0 ∈ n such that {x, xn0} ⊆ p ⊆ u. (4) p is called an i-wcs′-network of x if whenever {xn}n∈n is a sequence in x i-converging to a point x ∈ u with u open in x, then there is p ∈ p and some n0 ∈ n such that {xn0} ⊆ p ⊆ u. obviously, i-cs-networks ⇒ i-cs′-networks ⇒ i-wcs′-networks ⇒ networks. definition 4.2. let j be a maximal ideal of n and x be a topological space. u is said to be j -sn-cover of x, if {(u)j -s : u ∈ u} is a cover of x. theorem 4.3. each i-cs-network is preserved by an i-covering mapping. proof. let f : x → y be an i-covering mapping and p be an i-cs-network of x. suppose that {yn}n∈n is a sequence i-converging to a point y ∈ u with u open in y . since f is an i-covering mapping, there exist a sequence {xn}n∈n of points xn ∈ f−1(yn) for all n ∈ n and x ∈ f−1(y) such that xn i−→ x. since p is an i-cs-network of x, there is some p ∈ p such that {xn}n∈n is i-eventually in p and x ∈ p ⊆ f−1(u). thus there is e ∈ i such that {n ∈ n : xn /∈ p} ⊆ e. note that {n ∈ n : yn /∈ f(p)} ⊆ {n ∈ n : xn /∈ p} ⊆ e, hence yn ∈ f(p) for all n ∈ n \ e, i.e. {yn}n∈n is i-eventually in f(p) and y ∈ f(p) ⊆ u. this means that f(p) = {f(p) : p ∈ p} is an i-cs-network of y . □ corollary 4.4. each i-cs′-network is preserved by an i-covering mapping. theorem 4.5. each i-wcs′-network is preserved by an i-quotient mapping. proof. let f : x → y be an i-quotient mapping and p be an i-wcs′-network of x. suppose that {yn}n∈n is a sequence i-converging to a point y ∈ u with u open in y . since f is an i-quotient mapping, there is a sequence {xi}i∈n in x such that {xi : i ∈ n} ⊆ f−1({yn : n ∈ n}) and xi i−→ x /∈ f−1({yn : n ∈ n}). and because p is an i-wcs′-network of x, there is some p0 ∈ p and i0 ∈ n such that {xi0} ⊆ p0 ⊆ f−1(u). and hence {f(xi0)} = {yn0} ⊆ f(p0) ⊆ u for some n0 ∈ n. this implies that f(p) = {f(p) : p ∈ p} is an i-wcs′-network of y . □ lemma 4.6. let j be a maximal ideal of n and p be a family of subsets of x. then p is an j -cs′-network of x if and only if, whenever u is an open neighborhood of x, " {p ∈ p : x ∈ p ⊆ u} is an j -sequential neighborhood of x. proof. necessity: let u be an open neighborhood of x. if " {p ∈ p : x ∈ p ⊆ u} is not an j -sequential neighborhood of x, then there is a sequence {xn}n∈n such that xn j−→ x and xn /∈ " {p ∈ p : x ∈ p ⊆ u} for each n ∈ n. since p is an j -cs′-network of x, there is p0 ∈ p and n0 ∈ n such that {x, xn0} ⊆ p0 ⊆ u, a contradiction. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 243 x. zhou sufficiency: suppose that xn j−→ x ∈ u ∈ τx and " {p ∈ p : x ∈ p ⊆ u} is an j -sequential neighborhood of x. then {xn}n∈n is j -eventually in " {p ∈ p : x ∈ p ⊆ u}. hence there exists n0 ∈ n such that xn0 ∈ " {p ∈ p : x ∈ p ⊆ u}. and hence there is p0 ∈ p such that xn0 ∈ p0 and x ∈ p0 ⊆ u. thus {x, xn0} ⊆ p0 ⊆ u. this means that p is an j -cs′-network of x. □ theorem 4.7. let j be a maximal ideal of n and a space x be of a pointcountable j -cs′-network. then each open cover of x has a point-countable j -sn refinement. proof. suppose that p is a point-countable j -cs′-network for a space x. let u = {uα}α<γ be an open cover of x, where γ is an ordinal. for each α < γ, put vα = # {p ∈ p : p ⊆ uα, p ∕⊆ uβ if β < α}. clearly, vα ⊆ uα. next we shall show that the family v = {vα}α<γ is a pointcountable j -sn-cover of x. for each x ∈ x, let α(x) = min{α < γ : x ∈ uα}. then x ∈ uα(x) and # {p ∈ p : x ∈ p ⊆ uα(x)} ⊆ # {p ∈ p : p ⊆ uα(x), p ∕⊆ uβ if β < α(x)}. since p is an j -cs′-network for a space x, it follows from lemma 4.6 that x ∈ ( # {p ∈ p : x ∈ p ⊆ uα(x)})j -s ⊆ ( # {p ∈ p : p ⊆ uα(x), p ∕⊆ uβ if β < α(x)})j -s = (vα(x))j -s. this means that v = {vα}α<γ is an j -sn-cover of x. we claim that v is point-countable. suppose, to the contrary, that there exist a point x ∈ x and an uncountable subset γ of γ such that x ∈ vα for each α ∈ γ. hence there is pα ∈ p such that x ∈ pα ⊆ uα and pα ∕⊆ uβ for β < α. since p is a point-countable family and γ is an uncountable set, there are α, β ∈ γ, α ∕= β such that pα = pβ. assume that β < α, then uβ ⊇ pβ = pα ∕⊆ uβ, a contradiction. □ definition 4.8. (1) a space x is called i-fréchet-urysohn (or shortly, i-fu) space, if for each a ⊂ x and each x ∈ a, there exists a sequence in a i-converging to the point x in x [20]. (2) a space x is called a meta-lindelöf space if each open cover of x has a point-countable open refinement [13]. corollary 4.9. let j be a maximal ideal of n. if x is an j -fu space with a point-countable j -cs′-network, then x is a meta-lindelöf space. proof. x is an j -fu space ⇔ a = [a]j -s for each a ⊆ x ⇔ inta = (a)j -s for each a ⊆ x. □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 244 on i-quotient mappings and i-cs′-networks under a maximal ideal theorem 4.10. let j be a maximal ideal of n. the following are equivalent for a space x. (1) j -sx is an j -fréchet-urysohn space. (2) clj -sx(a) = [a]j -s, for each a ⊆ x. (3) [a]j -s is j -closed in x, for each a ⊆ x. (4) (a)j -s is j -open in x, for each a ⊆ x. proof. since the spaces x and j -sx have the same j -convergent sequences, by the definition 4.8 and proposition 3.8, it follows that (1) ⇔ (2) and (3) ⇔ (4). hence, it suffices to show that (2) ⇔ (3). if clj -sx(a) = [a]j -s, then [a]j -s is closed in j -sx, and hence [a]j -s is j -closed in x, for each a ⊆ x. on the other hand, if [a]j -s is j -closed in x, then [a]j -s is closed in j -sx, and further clj -sx(a) = [a]j -s, for each a ⊆ x. □ acknowledgements. my gratitude goes to professor shou lin, for his friendly encouragement and inspiring suggestions. references [1] j. r. boone and f. siwiec, sequentially quotient mappings, czech. math. j. 26 (1976), 174–182. [2] l. x. cheng, g. c. lin, y. y. lan and h. liu, measure theory of statistical convergence, sci. china ser. a 51 (2008), 2285–2303. 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[11] p. kostyrko, t. šalát and w. wilczynski, i-convergence, real anal. exch. 26 (2000/2001), 669–686. [12] b. k. lahiri and p. das, i and i∗-convergence in topological spaces, math. bohemica 130, no. 2 (2005), 153–160. [13] s. lin, point-countable covers and sequence-covering mappings, science press, beijing, 2015 (in chinese). [14] s. lin and z.q. yun, generalized metric spaces and mapping, atlantis studies in mathematics 6, atlantis press, paris, 2016. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 245 x. zhou [15] s. k. pal, n. adhikary and u. samanta, on ideal sequence covering maps, appl. gen. topol. 20, no. 2 (2019), 363–377. [16] h. steinhaus, sur la convergence ordinaire et la convergence asymptotique, colloq. math. 2 (1951), 73–74. [17] z. tang and f. lin, statistical versions of sequential and fréchet-urysohn spaces, adv. math. (china) 44 (2015), 945–954. [18] x. g. zhou and m. zhang, more about the kernel convergence and the ideal convergence, acta math. sinica, english series 29 (2013), 2367–2372. [19] x. g. zhou and l. liu, on i-covering mappings and 1-i-covering mappings, j. math. res. appl. (china) 40, no. 1 (2020) 47–56. [20] x. g. zhou, l. liu and s. lin, on topological spaces defined by i-convergence, bull. iran. math. soc. 46 (2020), 675–692. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 246 @ appl. gen. topol. 23, no. 1 (2022), 121-133 doi:10.4995/agt.2022.11368 © agt, upv, 2022 fixed point results with respect to a wt-distance in partially ordered b-metric spaces and its application to nonlinear fourth-order differential equation reza babaei, hamidreza rahimi* and ghasem soleimani rad department of mathematics, faculty of science, central tehran branch, islamic azad university, tehran, iran (rez.babaei.sci@iauctb.ac.ir, rahimi@iauctb.ac.ir, gha.soleimani.sci@iauctb.ac.ir) communicated by m. abbas abstract in this paper we study the existence of the fixed points for hardy-rogers type mappings with respect to a wt-distance in partially ordered metric spaces. our results provide a more general statement, since we replace a w-distance with a wt-distance and ordered metric spaces with ordered b-metric spaces. some examples are presented to validate our obtained results and an application to nonlinear fourth-order differential equation are given to support the main results. 2020 msc: 54h25; 47h10; 05c20. keywords: partially ordered set; b-metric space; wt-distance; fixed point. 1. introduction and preliminaries fixed point theory is an important and useful tool for different branches of mathematical analysis and it has many applications in mathematics and sciences. in 1922, banach proved the famous contraction mapping principle [3]. afterward, other authors considered various definitions of contractive mappings *corresponding author received 7 february 2019 – accepted 24 november 2021 http://dx.doi.org/10.4995/agt.2022.11368 https://orcid.org/0000-0003-4883-4250 https://orcid.org/0000-0002-0758-2758 r. babaei, h. rahimi and g. soleimani rad and proved several fixed and common fixed point theorems (see rhoades survey [22] and references therein). on the other hand, the symmetric spaces as metriclike spaces lacking the triangle inequality was introduced in 1931 by wilson [24]. in the sequel, a new type of spaces which they called b-metric spaces (or metric type spaces) are defined by bakhtin [2] and czerwik [6]. after that, several papers have dealt with fixed point theory for single-valued and multi-valued operators in b-metric spaces (for example, see [4, 5, 15, 17]). definition 1.1 ([2, 6]). let x be a nonempty set and s ≥ 1 be a real number. suppose that the mapping d : x ×x → [0,∞) satisfies (d1) d(x,y) = 0 if and only if x = y; (d2) d(x,y) = d(y,x) for all x,y ∈ x; (d3) d(x,z) ≤ s[d(x,y) + d(y,z)] for all x,y,z ∈ x. then d is called a b-metric and (x,d) is called a b-metric space (or metric type space). obviously, for s = 1, a b-metric space is a metric space. also, for notions such as convergent and cauchy sequences, completeness, continuity, etc in bmetric spaces, we refer to [1, 5, 17]. in 1996, kada et al. [16] introduced the concept of w-distance in metric spaces, where nonconvex minimization problems were treated. then some fixed point results and common fixed point theorem with respect to w-distance in metric spaces were proved by ilić and rakočević [12] and shioji et al. [23]. in 2014, hussain et al. [11] introduced the concept of wt-distance on a bmetric space and proved some fixed point theorems under wt-distance in a partially ordered b-metric space. then demma et al. [7] considered multivalued operators with respect to a wt-distance on b-metric spaces and proved some results on fixed point theory. definition 1.2 ([11]). let (x,d) be a b-metric space and s ≥ 1 be a given real number. a function ρ : x ×x → [0, +∞) is called a wt-distance on x if the following properties are satisfied: (ρ1) ρ(x,z) ≤ s[ρ(x,y) + ρ(y,z)] for all x,y,z ∈ x; (ρ2) ρ is b-lower semi-continuous in its second variable i.e., if x ∈ x and yn → y in x, then ρ(x,y) ≤ s lim infn ρ(x,yn); (ρ3) for each ε > 0 there exists δ > 0 such that ρ(z,x) ≤ δ and ρ(z,y) ≤ δ imply d(x,y) ≤ ε. let us recall that a real-valued function f defined on a metric space x is said to be b-lower semi-continuous at a point x ∈ x if either lim inf xn→x f(xn) = ∞ or f(x) ≤ lim inf xn→x sf(xn), whenever xn ∈ x and xn → x for each n ∈ n [12]. note that each b-metric d is a wt-distance, but the converse is not hold. thus, wt-distance is a generalization of b-metric d. obviously, for s = 1, every wtdistance is a w-distance. but, a w-distance is not necessary a wt-distance. thus, each wt-distance is a generalization of w-distance. © agt, upv, 2022 appl. gen. topol. 23, no. 1 122 fixed point results with respect to a wt-distance example 1.3 ([11]). let x = r and define a mapping d : x × x → r by d(x,y) = (x−y)2 for all x,y ∈ x. then (x,d) is a b-metric with s = 2. define a mapping ρ : x × x → [0,∞) by ρ(x,y) = y2 or ρ(x,y) = x2 + y2 for all x,y ∈ x. then ρ is a wt-distance. from example 1.3, we have two important results: (1) for any wt-distance ρ, ρ(x,y) = 0 is not necessarily equivalent to x = y for all x,y ∈ x. (2) for any wt-distance ρ, ρ(x,y) = ρ(y,x) does not necessarily hold for all x,y ∈ x. lemma 1.4 ([11]). let (x,d) be a b-metric space with parameter s ≥ 1 and ρ be a wt-distance on x. also, let {xn} and {yn} be sequences in x, and {αn} and {βn} be a sequences in [0, +∞) converging to zero and x,y,z ∈ x. then the following conditions hold: (i) if ρ(xn,y) ≤ αn and ρ(xn,z) ≤ βn for all n ∈ n, then y = z. in particular, if ρ(x,y) = 0 and ρ(x,z) = 0, then y = z; (ii) if ρ(xn,yn) ≤ αn and ρ(xn,z) ≤ βn for n ∈ n, then {yn} converges to z; (iii) if ρ(xn,xm) ≤ αn for all m,n ∈ n with m > n, then {xn} is a cauchy sequence in x; (iv) if ρ(y,xn) ≤ αn for all n ∈ n, then {xn} is a cauchy sequence in x. existence of fixed points in ordered metric spaces has been applied by ran and reurings [21]. the key feature in this fixed point theorem is that the contractivity condition on the nonlinear map is only assumed to hold on elements that are comparable in the partial order. however, the map is assumed to be monotone. they showed that under such conditions the conclusions of banach’s fixed point theorem still hold. this fixed point theorem was extended by nieto and rodŕıguez-lópez [18] and was applied to the periodic boundary value problem (see, e.g., [8, 9, 13, 14, 19, 20]). a relation v on x is called (i) reflexive if x v x for all x ∈ x; (ii) transitive if x v y and y v z imply x v z for all x,y,z ∈ x; (iii) antisymmetric if x v y and y v x imply x = y for all x,y ∈ x; (iv) pre-order if it is reflexive and transitive. a pre-order v is called partial order or an order relation if it is antisymmetric. given a partially ordered set (x,v); that is, the set x equipped with a partial order v, the notation x < y stands for x v y and x 6= y. also, let (x,v) be a partially ordered set. a mapping f : x → x is said to be nondecreasing if x v y implies that fx v fy for all x,y ∈ x. 2. main results our main result is the following theorem for mappings satisfying hardyrogers type conditions [10] with respect to a given wt-distance in a complete partially ordered b-metric space. © agt, upv, 2022 appl. gen. topol. 23, no. 1 123 r. babaei, h. rahimi and g. soleimani rad theorem 2.1. let (x,v) be a partially ordered set, (x,d) be a complete bmetric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exist mappings αi : x → [0, 1) such that αi(fx) ≤ αi(x)(2.1) for all x ∈ x and i = 1, 2, · · · , 5, where f : x → x be a continuous and nondecreasing mapping with respect to v satisfying the following conditions: ρ(fx,fy) ≤ α1(x)ρ(x,y) + α2(x)ρ(x,fx) + α3(x)ρ(y,fy) + α4(x)ρ(x,fy) + α5(x)ρ(y,fx),(2.2) ρ(fy,fx) ≤ α1(x)ρ(y,x) + α2(x)ρ(fx,x) + α3(x)ρ(fy,y) + α4(x)ρ(fy,x) + α5(x)ρ(fx,y)(2.3) for all x,y ∈ x with y v x such that (s(α1 + α3 + 2α4) + α2 + (s 2 + s)α5)(x) < 1.(2.4) if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. proof. if fx0 = x0, then x0 is a fixed point of f and the proof is finished. now, suppose that fx0 6= x0. since f is nondecreasing with respect to v and x0 v fx0, we obtain by induction that x0 v fx0 v f2x0 v ···v fnx0 v fn+1x0 v ··· , where xn = fxn−1 = f nx0. first we shall prove that {xn} is a cauchy sequence. now, setting x = xn and y = xn−1 in (2.2) and applying (2.1) and (ρ1), we have ρ(xn+1,xn) = ρ(fxn,fxn−1) ≤ α1(xn)ρ(xn,xn−1) + α2(xn)ρ(xn,fxn) + α3(xn)ρ(xn−1,fxn−1) + α4(xn)ρ(xn,fxn−1) + α5(xn)ρ(xn−1,fxn) = α1(fxn−1)ρ(xn,xn−1) + α2(fxn−1)ρ(xn,xn+1) + α3(fxn−1)ρ(xn−1,xn) + α4(fxn−1)ρ(xn,xn) + α5(fxn−1)ρ(xn−1,xn+1) ≤ α1(xn−1)ρ(xn,xn−1) + (α3 + sα5)(xn−1)ρ(xn−1,xn) + sα4(xn−1)ρ(xn+1,xn) + (α2 + sα4 + sα5)(xn−1)ρ(xn,xn+1) ... ≤ α1(x0)ρ(xn,xn−1) + (α3 + sα5)(x0)ρ(xn−1,xn) + sα4(x0)ρ(xn+1,xn) + (α2 + sα4 + sα5)(x0)ρ(xn,xn+1).(2.5) © agt, upv, 2022 appl. gen. topol. 23, no. 1 124 fixed point results with respect to a wt-distance similarly, setting x = xn and y = xn−1 in (2.3) and applying (2.1) and (ρ1), we have ρ(xn,xn+1) ≤ α1(x0)ρ(xn−1,xn) + (α3 + sα5)(x0)ρ(xn,xn−1) + sα4(x0)ρ(xn,xn+1) + (α2 + sα4 + sα5)(x0)ρ(xn+1,xn).(2.6) now, adding up (2.5) and (2.6), we obtain ρ(xn+1,xn) + ρ(xn,xn+1) ≤ (α1 + α3 + sα5)(x0)[ρ(xn,xn−1) + ρ(xn−1,xn)] + (α2 + 2sα4 + sα5)(x0)[ρ(xn+1,xn) + ρ(xn,xn+1)]. let an = ρ(xn+1,xn) + ρ(xn,xn+1). then we get an ≤ (α1 + α3 + sα5)(x0)an−1 + (α2 + 2sα4 + sα5)(x0)an. therefore, an ≤ kan−1 for all n ∈ n, where 0 ≤ k = (α1 + α3 + sα5)(x0) 1 − (α2 + 2sα4 + sα5)(x0) < 1 s by (2.4) and since (α1 + α3 + sα5)(x0) ≥ 0. by repeating the procedure, we obtain an ≤ kna0 for all n ∈ n. it follows that ρ(xn,xn+1) ≤ an ≤ kn[ρ(x1,x0) + ρ(x0,x1)].(2.7) let m > n. it follows from (2.7) and 0 ≤ sk < 1 that ρ(xn,xm) ≤ s[ρ(xn,xn+1) + ρ(xn+1,xm)] ≤ sρ(xn,xn+1) + s[sρ(xn+1,xn+2) + ρ(xn+2,xm)] ... ≤ sρ(xn,xn+1) + s2ρ(xn+1,xn+2) + · · · + sm−nρ(xm−1,xm)] ≤ (skn + s2kn+1 · · · + sm−nkm−1)[ρ(x1,x0) + ρ(x0,x1)] ≤ skn 1 −sk [ρ(x1,x0) + ρ(x0,x1)]. now, lemma 1.4 (iii) implies that {xn} is a cauchy sequence in x. since x is complete, there exists a point x′ ∈ x such that xn → x′ as n → ∞. the continuity of f implies that xn+1 = fxn → fx′ as n →∞, and since the limit of a sequence is unique, we get that fx′ = x′. thus, x′ is a fixed point of f. further, suppose that fz = z. then, by using (2.2), we have ρ(z,z) = ρ(fz,fz) ≤ α1(z)ρ(z,z) + α2(z)ρ(z,fz) + α3(z)ρ(z,fz) + α4(z)ρ(z,fz) + α5(z)ρ(z,fz) ≤ (α1 + α2 + α3 + α4 + α5)(z)ρ(z,z). since 5∑ i=1 αi(z) < s(α1 + α3 + 2α4)(z) + α2 + (s 2 + s)α5)(z) < 1, we obtain that ρ(z,z) = 0 by using lemma 1.4 (i). this completes the proof. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 125 r. babaei, h. rahimi and g. soleimani rad example 2.2. let x = [0, 1] and define a mapping d : x × x → r by d(x,y) = (x − y)2 for all x,y ∈ x. then (x,d) is a complete b-metric space with s = 2. define a function ρ : x × x → [0,∞) by ρ(x,y) = d(x,y) for all x,y ∈ x. then ρ is a wt-distance. let an order relation v be defined by x v y if and only if x ≤ y also, let a mapping f : x → x be defined by fx = x 2 5 for all x ∈ x. then f is a continuous and nondecreasing mapping with respect to v and there exists a 0 ∈ x such that 0 v f0. define the mappings α1(x) = (x+1)2 25 and αi(x) = 0 for all x ∈ x and i = 2, 3, 4, 5. observe that s(α1 + α3 + 2α4)(x) + α2 + (s 2 + s)α5)(x) = 2 (x + 1)2 25 < 1. also, α1(fx) = 1 25 (x2 5 + 1 )2 ≤ 1 25 ( x2 + 1 )2 ≤ (x + 1)2 25 = α1(x) for all x ∈ x and αi(fx) = 0 = αi(x) for all x ∈ x and i = 2, 3, 4, 5. moreover, for all x,y ∈ x with y v x, we get ρ(fx,fy) = ( x2 5 − y2 5 )2 = (x + y)2(x−y)2 25 ≤ (x + 1)2 25 (x−y)2 ≤ α1(x)ρ(x,y) + α2(x)ρ(x,fx) + α3(x)ρ(y,fy) + α4(x)ρ(x,fy) + α5(x)ρ(y,fx). similarly, for all x,y ∈ x with y v x, we get ρ(fy,fx) ≤ α1(x)ρ(y,x) + α2(x)ρ(fx,x) + α3(x)ρ(fy,y) + α4(x)ρ(fy,x) + α5(x)ρ(fx,y). therefore, all the conditions of theorem 2.1 are satisfied. hence, f has a fixed point x = 0 with ρ(0, 0) = 0. several consequences of theorem 2.1 follow now for particular choices of the contractions. corollary 2.3. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exist mappings α,β,γ : x → [0, 1) such that α(fx) ≤ α(x), β(fx) ≤ β(x), γ(fx) ≤ γ(x) for all x ∈ x, where f : x → x be a continuous and nondecreasing mapping with respect to v satisfying the following conditions: ρ(fx,fy) ≤ α(x)ρ(x,y) + β(x)ρ(x,fy) + γ(x)ρ(y,fx), ρ(fy,fx) ≤ α(x)ρ(y,x) + β(x)ρ(fy,x) + γ(x)ρ(fx,y) © agt, upv, 2022 appl. gen. topol. 23, no. 1 126 fixed point results with respect to a wt-distance for all x,y ∈ x with y v x such that (s(α + 2β) + (s2 + s)γ)(x) < 1. if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. proof. we obtain this result by applying theorem 2.1 with α1(x) = α(x), α2(x) = α3(x) = 0, α4(x) = β(x) and α5(x) = γ(x). � in the process of proving theorem 2.1, consider x = xn−1 and y = xn with x v y (instead of x = xn and y = xn−1 with y v x). then, we only need one condition for some following types of the contractions. corollary 2.4. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exist mappings αi : x → [0, 1) such that αi(fx) ≤ αi(x) for all x ∈ x and i = 1, 2, 3, 4, where f : x → x be a continuous and nondecreasing mapping with respect to v satisfying the following condition: ρ(fx,fy) ≤ α1(x)ρ(x,y) + α2(x)ρ(x,fx) + α3(x)ρ(y,fy) + α4(x)ρ(x,fy) for all x,y ∈ x with x v y such that (s(α1 + α2) + α3 + (s 2 + s)α4)(x) < 1. if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. corollary 2.5. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exist mappings α,β,γ : x → [0, 1) such that α(fx) ≤ α(x), β(fx) ≤ β(x), γ(fx) ≤ γ(x) for all x ∈ x, where f : x → x be a continuous and nondecreasing mapping with respect to v satisfying the following condition: ρ(fx,fy) ≤ α(x)ρ(x,y) + β(x)ρ(x,fx) + γ(x)ρ(y,fy) for all x,y ∈ x with x v y such that (s(α + β) + γ)(x) < 1. if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. theorem 2.6. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exists a continuous and nondecreasing mapping f : x → x with respect to v such that the following conditions hold: ρ(fx,fy) ≤ α1ρ(x,y) + α2ρ(x,fx) + α3ρ(y,fy) + α4ρ(x,fy) + α5ρ(y,fx), ρ(fy,fx) ≤ α1ρ(y,x) + α2ρ(fx,x) + α3ρ(fy,y) + α4ρ(fy,x) + α5ρ(fx,y) © agt, upv, 2022 appl. gen. topol. 23, no. 1 127 r. babaei, h. rahimi and g. soleimani rad for all x,y ∈ x with y v x, where αi are nonnegative coefficients for i = 1, 2, · · · , 5 with s(α1 + α3 + 2α4) + α2 + (s 2 + s)α5 < 1. if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. proof. we can prove this result by applying theorem 2.1 with αi(x) = αi for i = 1, 2, · · · , 5. � several consequences of theorem 2.6 follow now for particular choices of the contractions. corollary 2.7. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exists a continuous and nondecreasing mapping f : x → x with respect to v such that the following conditions hold: ρ(fx,fy) ≤ αρ(x,y) + βρ(x,fy) + γρ(y,fx), ρ(fy,fx) ≤ αρ(y,x) + βρ(fy,x) + γρ(fx,y) for all x,y ∈ x with y v x, where α,β,γ are nonnegative coefficients with s(α + 2β) + (s2 + s)γ < 1. if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. proof. we obtain this result by applying theorem 2.6 with α1 = α, α2 = α3 = 0, α4 = β and α5 = γ. � in the process of proving theorem 2.6, consider x = xn−1 and y = xn with x v y (instead of x = xn and y = xn−1 with y v x). then, we only need one condition for some types of the contractions. corollary 2.8. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exists a continuous and nondecreasing mapping f : x → x with respect to v such that the following condition hold: ρ(fx,fy) ≤ α1ρ(x,y) + α2ρ(x,fx) + α3ρ(y,fy) + α4ρ(x,fy) for all x,y ∈ x with x v y, where αi for i = 1, 2, 3, 4 are nonnegative coefficients with s(α1 + α2) + α3 + (s 2 + s)α4 < 1. if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. corollary 2.9. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. © agt, upv, 2022 appl. gen. topol. 23, no. 1 128 fixed point results with respect to a wt-distance suppose that there exists a continuous and nondecreasing mapping f : x → x with respect to v such that the following condition hold: ρ(fx,fy) ≤ αρ(x,y) + βρ(x,fx) + γρ(y,fy) for all x,y ∈ x with x v y, where α,β,γ are nonnegative coefficients with s(α + β) + γ < 1. if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. example 2.10. consider x, d, s and order relation v as in example 2.2. define a function ρ : x ×x → [0,∞) by ρ(x,y) = y2 for all x,y ∈ x. then ρ is a wt-distance. also, let a mapping f : x → x be defined by fx = x 2 3 for all x ∈ x. then f is a continuous and nondecreasing mapping with respect to v and there exists a 0 ∈ x such that 0 v f0. take α = 1 9 , β = 1 8 and γ = 1 4 . then we obtain ρ(fx,fy) = (fy)2 = ( y2 3 )2 = y4 9 ≤ 1 9 y2 = 1 9 ρ(x,y) ≤ αρ(x,y) + βρ(x,fx) + γρ(y,fy). also, we have s(α + β) + γ = 2( 1 9 + 1 8 ) + 1 4 = 26 36 < 1. hence, all the conditions of corollary 2.9 are satisfied. therefore, f has a fixed point x = 0. moreover, ρ(0, 0) = 0. corollary 2.11. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exists a continuous and nondecreasing mapping f : x → x with respect to v such that (2.8) ρ(fx,fy) ≤ λρ(x,y) for all x,y ∈ x with x v y, where λ ∈ [0, 1 s ). if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. example 2.12. let x = [0, 2] and consider d, s and order relation v as in example 2.2. also, let a mapping f : x → x be defined by fx = x 2 2 for all x ∈ x. since d(f0,f2) = d(0, 2), there is not 0 ≤ α < 1 s such that d(fx,fy) ≤ αd(x,y) for all x,y ∈ x. hence, banach-type result on b-metric space cannot be applied for this example. now, let x = [0, 1] and define a function ρ : x ×x → [0,∞) by ρ(x,y) = y2 + x2 for all x,y ∈ x. then ρ is a © agt, upv, 2022 appl. gen. topol. 23, no. 1 129 r. babaei, h. rahimi and g. soleimani rad wt-distance. ρ(fx,fy) = (fy)2 + (fx)2 = ( y2 2 )2 + ( x2 2 )2 = y4 + x4 4 ≤ y2 + x2 4 = 1 4 ρ(x,y). thus, (2.8) is hold with λ = 1 4 ∈ [0, 1 2 ). hence, all conditions of banach-type fixed point results (or same corollary 2.11) with respect to the wt-distance on b-metric spaces are satisfied. note that f has a (trivial) fixed point 0 ∈ [0, 1] ⊆ [0, 2] and ρ(0, 0) = 0. corollary 2.13. let (x,v) be a partially ordered set, (x,d) be a complete b-metric space with given real number s ≥ 1 and ρ be a wt-distance on x. suppose that there exists a continuous and nondecreasing mapping f : x → x with respect to v such that ρ(fx,fy) ≤ δ(ρ(x,fx) + ρ(y,fy)) for all x,y ∈ x with x v y, where δ ∈ [0, 1 s+1 ). if there exists x0 ∈ x such that x0 v fx0, then f has a fixed point. moreover, if fz = z, then ρ(z,z) = 0. 3. an application it is well-known that fourth-order differential equations are important and useful tools for modeling the elastic beam deformation. precisely, we refer to beams in equilibrium state, whose two ends are simply supported. consequently, this study has many applications in engineering and physical science. now, we establish the existence of solutions of fourth-order boundary value problems as a consequence of theorem 2.1. in particular, the focus is on the equivalent integral formulation of the boundary value problem below and the use of green’s functions. at the first, we introduce the mathematical background as follows (also, see [14]). let x = c([0, 1],r) be the set of all non-negative real-valued continuous functions on the interval [0, 1]. also, let x be endowed with the supremum norm ‖x‖∞ = supt∈[0,1] |x(t)| and define a mapping d : x×x → r by d(x,y) = supt∈[0,1](x(t) −y(t))2 for all x,y ∈ x. also, consider the partial order (x,y) ∈ x ×x, x v y ⇐⇒ x(t) ≤ y(t) for all t ∈ [0, 1]. clearly, (x,v) is a partially ordered set and (x,d) is a complete b-metric space with s = 2. finally, consider the wt-distance ρ : x × x → r given by ρ(x,y) = d(x,y) for all x,y ∈ x. thus, we study the following fourth-order two-point boundary value problem  xiv(t) = k(t,x(t)), 0 < t < 1, x(0) = x′(0) = x′′(1) = x′′′(1) = 0, (3.1) with k ∈ c([0, 1] ×r,r). © agt, upv, 2022 appl. gen. topol. 23, no. 1 130 fixed point results with respect to a wt-distance it is well-known that the problem (3.1) may be equivalently expressed in integral form: find x∗ ∈ x solution of (3.2) x(t) = ∫ 1 0 g(t,τ)k(τ,x(τ)) dτ, t ∈ [0, 1], where the green function g(t,τ) is given by g(t,τ) = 1 6   τ2(3t− τ), 0 ≤ τ ≤ t ≤ 1, t2(3τ − t), 0 ≤ t ≤ τ ≤ 1. also, it is immediate to show that (3.3) 0 ≤ g(t,τ) ≤ 1 2 t2τ for all t,τ ∈ [0, 1]. next, we consider the following hypotheses: (i) there exists α1 : x → [0, 12 ) such that (3.4) 0 ≤ k(t,y(t)) −k(t,x(t)) ≤ 4 √ α1(x)ρ(x,y) for all x,y ∈ x with y v x and for all t ∈ [0, 1] and (3.5) α1 (∫ 1 0 g(t,τ)k(τ,x(τ)) dτ ) ≤ α1(x) for all x ∈ x. (ii) there exists x0 ∈ x such that x0(t) ≤ ∫ 1 0 g(t,τ)k(τ,x0(τ)) dτ, t ∈ [0, 1]; that is, the integral equation (3.2) admits a lower solution in x. now, we prove the existence of at least a solution of (3.1) in x. theorem 3.1. the existence of at least a solution of problem (3.1) in x is established, provided that the function k ∈ c([0, 1]×r,r) satisfies the hypotheses (i) and (ii). proof. the problem in study is equivalent to the fixed point problem obtained by introducing the continuous integral operator f : x → x given as (fx)(t) = ∫ 1 0 g(t,τ)k(τ,x(τ)) dτ, t ∈ [0, 1] and x ∈ x. now, we show that the operator f satisfies all the conditions in theorem 2.1 to conclude that there exists a fixed point of f in x. by using the inequality (3.4) in hypothesis (i), we deduce that f is a nondecreasing mapping with respect to v. also, by using (3.4), for all t ∈ [0, 1] and for all x,y ∈ x with y v x, we © agt, upv, 2022 appl. gen. topol. 23, no. 1 131 r. babaei, h. rahimi and g. soleimani rad get |(fy)(t) − (fx)(t)| = ∫ 1 0 g(t,τ)[k(τ,y(τ)) −k(τ,x(τ))] dτ ≤ ∫ 1 0 g(t,τ)4 √ α1(x)ρ(x,y) dτ ≤ (∫ 1 0 g(t,τ) dτ ) 4 √ α1(x)ρ(x,y) ≤ √ α1(x)ρ(x,y) (from (3.3)). since ρ(x,y) = d(x,y) for all x,y ∈ x, by passing to square and taking the supremum with respect to t, we get ρ(fx,fy) = d(fx,fy) = sup t∈[0,1] ((fy)(t) − (fx)(t))2 ≤ α1(x)ρ(x,y) for all x,y ∈ x with y v x. it follows that the conditions (2.2) and (2.3) of theorem 2.1 hold αi(x) = 0 for all x ∈ x and i = 2, 3, 4, 5. by hypothesis (ii), we get that there exists x0 ∈ x such that x0 v fx0. also, from (3.5) and the fact that the function α1 assumes values in the interval [0, 1 2 ), we have α1(fx) ≤ α1(x) < 1 2 for all x ∈ x; that is, the condition (2.1) of theorem 2.1 hold with αi(x) = 0 for all x ∈ x and i = 2, 3, 4, 5. we conclude that all the conditions of theorem 2.1 hold and so we deduce the existence of a fixed point of f; that is, there exists a solution of problem (3.1) in x. � acknowledgements. the authors thank the editorial board and the referees for their valuable comments. references [1] r. p. agarwal, e. karapinar, d. o’regan and a. f. roldan-lopez-de-hierro, fixed point theory in metric type spaces, springer-international publishing, switzerland, 2015. 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[24] w. a. wilson, on semi-metric spaces, amer. jour. math. 53, no. 2 (1931), 361–373. © agt, upv, 2022 appl. gen. topol. 23, no. 1 133 @ appl. gen. topol. 20, no. 1 (2019), 237-249doi:10.4995/agt.2019.10776 c© agt, upv, 2019 on rings of baire one functions a. deb ray and atanu mondal department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata 700019, india (debrayatasi@gmail.com, mail.atanu12@yahoo.com) communicated by f. lin abstract this paper introduces the ring of all real valued baire one functions, denoted by b1(x) and also the ring of all real valued bounded baire one functions, denoted by b∗1(x). though the resemblance between c(x) and b1(x) is the focal theme of this paper, it is observed that unlike c(x) and c∗(x) (real valued bounded continuous functions), b∗1 (x) is a proper subclass of b1(x) in almost every non-trivial situation. introducing b1-embedding and b ∗ 1 -embedding, several analogous results, especially, an analogue of urysohn’s extension theorem is established. 2010 msc: 26a21; 54c30; 54c45; 54c50. keywords: b1(x); b ∗ 1 (x); zero set of a baire one function; completely separated by b1(x); b1-embedded; b ∗ 1 -embedded. 1. introduction the real valued baire class one functions on a real variable is very important and intensively studied in real analysis. several characterizations of baire one functions defined on metric spaces were obtained by different mathematicians in [1], [8] etc. lee, tang and zhao [5] characterized baire class one functions in terms of the usual ǫ-δ formulation as in the case of continuous functions under the assumptions that x and y are complete separable metric spaces. in [7], baire one functions on normal topological spaces have been investigated and characterized by observing that pull-backs of open sets are fσ. the study of rings of continuous functions was initiated long back and the received 03 november 2018 – accepted 10 january 2019 http://dx.doi.org/10.4995/agt.2019.10776 a. deb ray and a. mondal theory was enriched by publication of several outstanding results, cited in [2], [3], [4], [6], etc., established by various well known mathematicians around the first half of the twentieth century. the class of all real valued baire one functions defined on any topological space is of course a superset of the class of all real valued continuous functions on the same space and therefore, it is a natural query whether one can extend the study on the class of baire one functions. this paper puts forward some basic results which came out of such investigation. in section 2, we introduce two rings, the ring b1(x) of all the real valued baire one functions and the ring b∗1(x) of all the real valued bounded baire one functions. we observe that b1(x) is a lattice ordered commutative ring with unity and the other one, i.e., b∗1(x) is a commutative subring with unity (and a sublattice) of b1(x). the rest of this section is devoted to build the basics that are required to carry on research in this field. though the focal theme of this work is to observe the similarities of b1(x) with c(x), we find in section 3 that the scenario is quite different. it is well known that x is pseudocompact if c(x) = c∗(x) and there are plenty of examples, including compact spaces, which are pseudocompact. but in case of b1(x) and b ∗ 1(x), the equality occurs very rarely. section 4 introduces zero sets of b1(x) and discusses several algebraic equalities involving union and intersection of zero sets and zero sets of modulus and power of baire one functions. the sets which are completely separated by b1(x) is characterized via zero sets of functions from b1(x). however, the well known result that two sets a and b are completely separated by c(x) if and only if a and b are completely separated by c(x) becomes one sided in the context of b1(x). that it is indeed one-sided is supported by an example. the final section of this paper is devoted for developing the idea of b1-embedding and b∗1-embedding of a set and thereby obtaining an analogue of urysohn’s extension theorem in this new context. 2. b1(x) and b ∗ 1(x) let x be any arbitrary topological space and c(x) be the collection of all real valued continuous functions from x to r. we define b1(x) as the collection of all pointwise limit functions of sequnces in c(x). so, b1(x) = {f : x → r : ∃ {fn} ⊆ c(x), for which {fn(x)} pointwise converges to f(x) for all x ∈ x }, is called the set of baire class one functions or baire one functions. it is clear that, c(x) ⊆ b1(x). let f and g be two functions in b1(x). there exist two sequences of continuous functions {fn} and {gn} such that, {fn} converges pointwise to f and {gn} converges pointwise to g on x. then c© agt, upv, 2019 appl. gen. topol. 20, no. 1 238 on rings of baire one functions • {fn + gn} pointwise converges to f + g. • {−fn} pointwise converges to −f. • {fn.gn} converges pointwise to f.g. • {|fn|} converges pointwise to |f|. in view of the above observations, it is easy to see that (b1(x), +, .) is a commutative ring with unity 1 (where 1 : x → r is given by 1(x) = 1, ∀x ∈ x) with respect to usual pointwise addition and multiplication. in [7], baire one functions are described in terms of pull-backs of open sets imposing conditions on domain and co-domain of functions. theorem 2.1 ([7]). (i) for any topological space x and any metric space y , b1(x, y ) ⊆ fσ(x, y ) where b1(x, y ) denotes the collection of baire one functions from x to y and fσ(x, y ) = {f : x → y : f −1(g) is an fσ set, for any open set g ⊆ y }. (ii) for a normal topological space x, b1(x, r) = fσ(x, r), where b1(x, r) denotes the collection of baire one functions from x to r and fσ(x, r) = {f : x → r : f−1(g) is an fσ set, for any open set g ⊆ r}. define a partial order ‘≤’ on b1(x) by f ≤ g iff f(x) ≤ g(x), ∀x ∈ x. it is clear that (b1(x), ≤) is a lattice, where sup{f, g} = f ∨g = 1 2 {(f +g)+|f −g|} and inf{f, g} = f ∧ g = 1 2 {(f + g) − |f − g|} both are in b1(x). also for any f, g, h ∈ b1(x) • f ≤ g =⇒ f + h ≤ g + h. • f ≥ 0 and g ≥ 0 =⇒ f.g ≥ 0. so, b1(x) is a commutative lattice ordered ring with unity. moreover c(x) is a commutative subring with unity and also a sublattice of b1(x). if a baire one function f on x is a unit in the ring b1(x) then {x ∈ x : f(x) = 0} = ∅. the following result shows that in case of a normal space, this condition is also sufficient. theorem 2.2. for a normal space x, f ∈ b1(x) is a unit in b1(x) if and only if z(f) = {x ∈ x : f(x) = 0} = ∅. proof. if f ∈ b1(x) is a unit then clearly the condition holds. let f ∈ b1(x) be such that z(f) = ∅. define 1 f (x) = 1 f(x) , for all x ∈ x. to show that 1 f ∈ b1(x). let u = (a, b) be any open interval in r. it is enough to show that ( 1 f )−1 (u) is fσ. ( 1 f )−1 (u) = {x ∈ x : a < 1 f(x) < b}. case 1 : suppose 0 /∈ u. then {x ∈ x : a < 1 f(x) < b} = {x ∈ x : 1 b < f(x) < 1 a } = f−1 ( 1 b , 1 a ) , when a 6= 0, b 6= 0 and ( 1 f )−1 (u) = f−1 ( 1 b , ∞ ) or f−1 ( −∞, 1 a ) according as a = 0 or b = 0 . in any case the resultant set is an fσ set. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 239 a. deb ray and a. mondal case 2 : let 0 ∈ u. since z(f) = ∅, f(x) 6= 0 and hence 1 f(x) 6= 0, ( 1 f )−1 (u) = ( 1 f )−1 ((a, 0)) ∪ ( 1 f )−1 ((0, b)). then ( 1 f )−1 (u) = {x ∈ x : a < 1 f(x) < 0} ⋃ {x ∈ x : 0 < 1 f(x) < b} = {x ∈ x : −∞ < f(x) < 1 a } ⋃ {x ∈ x : 1 b < f(x) < ∞} = f−1 ( −∞, 1 a ) ∪ f−1 ( 1 b , ∞ ) , which is an fσ set. hence, 1 f ∈ b1(x). � however, the following theorem provides an useful sufficient criterion to determine units of b1(x), where x is any topological space. theorem 2.3. let x be a topological space and f ∈ b1(x) be such that f(x) > 0 (or f(x) < 0), ∀x ∈ x, then 1 f exists and belongs to b1(x). proof. without loss of generality let, f ∈ b1(x) and f(x) > 0, ∀x ∈ x. then there exists a sequence of continuous functions {fn} such that {fn} converges pointwise to f. now we construct a sequence of continuous functions {gn}, where gn(x) = |fn(x)| + 1 n , ∀n ∈ n and ∀x ∈ x. clearly gn(x) > 0, ∀n ∈ n and ∀x ∈ x. we now show that, {gn(x)} converges to f(x), for each x ∈ x. for each x ∈ x, {fn(x)} converges to f(x) and { 1 n } converges to 0 imply that {gn(x)} converges to |f(x)| = f(x), ∀x ∈ x. consider the function g : r − {0} → r defined by g(y) = 1 y . g is continuous and {g ◦ gn} is a sequence of continuous functions from x to r. our claim is {g ◦ gn} converges to g ◦ f on x. let, ǫ > 0. then there exists a δ > 0 such that |(g ◦ gn)(x) − (g ◦ f)(x)| = |g(gn(x)) − g(f(x))| < ǫ, whenever |gn(x) − f(x)| < δ (by using continuity of g). since {gn} converges pointwise to f, for δ > 0, ∃ k ∈ n such that, |gn(x) − f(x)| < δ, whenever n ≥ k. so, |(g ◦ gn)(x) − (g ◦ f)(x)| < ǫ whenever n ≥ k. hence, {(g ◦ gn)} converges pointwise to g ◦ f, i.e. g ◦ f ∈ b1(x). now (g ◦ f)(x) = 1 f(x) shows that 1 f belongs to b1(x). � in the last theorem, we have shown that composition of a typical continuous function g : r − {0} → r given by g(x) = 1 x and a typical baire one function f : x → r produces a baire one function g ◦f. in the next theorem we further generalize this. theorem 2.4. let f be any baire one function on x and g : r → r be a continuous function. then the composition function g ◦ f is also a baire one function. proof. since f ∈ b1(x), there exists a sequence of continuous functions {fn} which converges pointwise to f. the functions g ◦fn are all defined and continuous functions on x, ∀n ∈ n. let ǫ > 0 be any arbitrary positive real number and x ∈ x. by continuity of g there exists a positive δ depending on ǫ such that, |(g◦fn)(x)−(g◦f)(x)| = |g(fn(x))−g(f(x))| < ǫ, whenever |fn(x)−f(x)| < δ. again by using pointwise convergence of {fn} we can find a natural number c© agt, upv, 2019 appl. gen. topol. 20, no. 1 240 on rings of baire one functions k such that, |fn(x) − f(x)| < δ, ∀n ≥ k. so, |(g ◦ fn)(x) − (g ◦ f)(x)| < ǫ, ∀ n ≥ k. hence, g ◦ f is a baire one function on x. � we introduce another subcollection of b1(x), called bounded baire one functions, denoted by b∗1(x) consisting of all real valued bounded baire one functions on x. i.e., b∗1(x) = {f ∈ b1(x) : f is bounded on x}. b ∗ 1(x) also forms a commutative lattice ordered ring with unity 1, which is a subring and sublattice of b1(x). 3. is b∗1(x) always a proper subring of b1(x)? in case of rings of continuous functions we have seen that there are spaces for which c∗(x) coincides with c(x), where c(x) and c∗(x) denote respectively the collection of all real valued continuous functions and the collection of all real valued bounded continuous functions on x. for example, if x is compact then c(x) = c∗(x). in fact, the spaces for which c(x) = c∗(x) are known as pseudocompact spaces. but for baire one functions, the scenario is quite different. we show in the next theorem that for most of the spaces unbounded baire one functions do exist. theorem 3.1. let x be any topological space. if f ∈ c(x) is such that 0 ≤ f(x) ≤ 1, ∀x ∈ x and 0 is a limit point of the range set f(x), then there exists an unbounded baire one function on x (i.e., b∗1(x) is a proper subset of b1(x)). proof. for each n ∈ n, define gn : x → r by gn(x) = { n2f(x) if x ∈ f−1([0, 1 n ]) 1 f(x) if x ∈ f−1([ 1 n , 1]) each gn is continuous and it is clear that {gn(x)} converges pointwise to the function g : x → r defined by g(x) = { 0 if x ∈ z(f) 1 f(x) if x /∈ z(f) so, g is a baire one function on x. since 0 is a limit point of f(x), g is unbounded baire one function. � remark 3.2. if b1(x) = b ∗ 1(x), then for every f ∈ c(x), 0 cannot be a limit point of f(x). in fact, we can say more, if b1(x) = b ∗ 1(x), then for every f ∈ c(x), r is not a limit point of f(x), where r is any real number. this follows from the fact that if r is a limit point of the range set of the continuous function f, then 0 becomes a limit point of the set g(x), where g = f − r and we can apply the previous theorem to the function (0 ∨ |g|) ∧ 1 to get an unbounded function. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 241 a. deb ray and a. mondal remark 3.3. one may observe that whenever b1(x) = b ∗ 1(x) and x possesses at least one non-constant continuous function then x cannot be connected, because in that case f(x) must be an interval and hence possesses a limit point. also, it follows easily from b1(x) = b ∗ 1(x) that x is pseudocompact. therefore, it is natural to ask, which class of spaces is precisely determined by b1(x) = b ∗ 1(x)? as a consequence of remark 3.2 we obtain, theorem 3.4. a completely hausdorff space x (i.e., where any two distinct points are completely separated by continuous function) is totally disconnected if b1(x) = b ∗ 1(x). proof. since x is completely hausdorff, it possesses a non-constant continuous function. so, by previous remark b1(x) = b ∗ 1(x) implies x is disconnected. now assume that c be any component of x and x, y be two distinct points in c. since x is completely hausdorff, there exists a continuous function f on x such that f(x) = 0 and f(y) = 1. then the function g = f ∣ ∣ c is a continuous function on c and c being connected g(c) must be an interval. so, f(x) has a limit point as g is the restriction function of f on c. this contradicts to the fact that b1(x) = b ∗ 1(x). so, c cannot contain more than one point. since c is arbitrary component of x, every component of x contains single point. hence, x is totally disconnected. � remark 3.5. converse of the theorem is not true. q, the set of all rational numbers is both completely hausdorff and totally disconnected but q possesses an unbounded real valued continuous function, hence an unbounded baire one function. in the context of ring homomorphism we may further get the following results similar to the known results about homomorphism from c(y ) (or c∗(y )) to c(x) [3]. theorem 3.6. every ring homomorphism t from b1(y ) (or b ∗ 1(y )) to b1(x) is a lattice homomorphism. theorem 3.7. every ring homomorphism t from b1(y ) (or b ∗ 1(y )) to b1(x) takes bounded functions to bounded functions. corollary 3.8. if a completely hausdorff space x is not totally disconnected, then b1(x) cannot be a homomorphic image of b ∗ 1(y ), for any y . corollary 3.9. b1(x) and b ∗ 1(x) are isomorphic if and only if they are identical. theorem 3.10. let, t be a homomorphism from b1(y ) into b1(x), such that b∗1(x) ⊆ t(b1(y )). then t maps b ∗ 1(y ) onto b ∗ 1(x). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 242 on rings of baire one functions 4. zero sets in b1(x) the zero set z(f) of a function f ∈ b1(x) is defined by z(f) = {x ∈ x : f(x) = 0} and the collection of all zero sets in b1(x) is denoted by z(b1(x)). we say a subset e of x is a zero set in b1(x) if e = z(f), for some f ∈ b1(x). we call a set to be a cozero set in b1(x) if it is the complement of a zero set in b1(x). z(b1(x)) is closed under finite union and finite intersection as z(f) ∪ z(g) = z(f.g) and z(f) ∩ z(g) = z(f2 + g2) = z(|f| + |g|). it is evident that, z(f) = z(|f|) = z(fn), for all f ∈ b1(x) and for all n ∈ n, z(0) = x and z(1) = ∅. here 0 and 1 denote the constant functions whose values are 0 and 1 on x. examples of zero and cozero sets in b1(x) : • every zero set of a continuous function is also a zero set of a baire one function. every clopen set k of x is in z(b1(x)), as it is a zero set of a continuous function. • for any f ∈ b1(x), {x ∈ x : f(x) ≥ 0} and {x ∈ x : f(x) ≤ 0} belong to z(b1(x)) and they are the zero sets of the functions f − |f| and f + |f| respectively. • for any f ∈ b1(x) and any real number r ∈ r, {x ∈ x : f(x) ≤ r} and {x ∈ x : f(x) ≥ r} are also in z(b1(x)). • for any f ∈ b1(x) and any real number r ∈ r, {x ∈ x : f(x) < r} and {x ∈ x : f(x) > r} are cozero sets in b1(x). it is easy to observe that, for any f ∈ b1(x) there exists g ∈ b ∗ 1(x) given by g = f ∧ 1 such that z(f) = z(g). hence z(b1(x)) and z(b ∗ 1(x)) produce same family of zero sets. theorem 4.1. for any f ∈ b1(x), z(f) is a gδ set. proof. since r is a metric space, by theorem 2.1, f−1(g) is an fσ set, for every open set g of r. therefore in particular f−1(r − {0}) is an fσ set. i.e. z(f) = f−1({0}) is a gδ set. � corollary 4.2. every cozero set in z(b1(x)) is fσ. observation: countable union of zero sets in b1(x) need not be a zero set in b1(x). for example, q can be written as a countable union of singleton sets and each singleton set is a zero set in b1(r), but q is not a zero set in b1(r), as it is not a gδ set in r. however it can be proved that, z(b1(x)) is closed under countable intersection. to establish this result we need two important lemmas, which are already proved for the functions of a real variable. we generalize these results here for any arbitrary topological spaces. lemma 4.3. if f : x → r is a bounded baire one function with bound m, where x is any topological space, then there exists a sequence of continuous function {fn} such that, each fn has the same bound m and {fn} converges pointwise to f on x. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 243 a. deb ray and a. mondal proof. let {gn} be a sequence of continuous functions that converges pointwise to f on x. suppose also that |f(x)| ≤ m, ∀x ∈ x. define {fn} by fn(x) =      −m if gn(x) ≤ −m gn(x) if − m ≤ gn(x) ≤ m m if gn(x) ≥ m. . each fn of the sequence of functions {fn(x)} is continuous and has bound m. also {fn(x)} converges pointwise to f on x with bound m. � lemma 4.4. let {fk} be a sequence of baire one functions defined on a topological space x and let ∞ ∑ k=1 mk < ∞, where each mk > 0. if |fk(x)| ≤ mk, ∀k ∈ n and ∀x ∈ x, then the function f(x) = ∞ ∑ k=1 fk(x) is also a baire one function on x. proof. since each fk is a baire one function, for each positive integer k there exists a sequence of continuous functions {gki} ∞ i=1 on x such that {gki} ∞ i=1 converges pointwise to the function fk on x and by lemma 4.3 we can choose {gki} ∞ i=1 in such a way that, |gki| ≤ mk, ∀i ∈ n. for each n ∈ n, let hn = g1n + g2n + . . . + gnn. it is easy to verify that each hn is continuous on x. we will show that {hn} converges pointwise to f on x. fix a point x ∈ x and let ǫ > 0 be an arbitrary positive real number. since ∞ ∑ k=1 mk < ∞, we can find k ∈ n so that ∞ ∑ k=k+1 mk < ǫ. now choose an integer n > k such that |gki(x) − fk(x)| < ǫ k for 1 ≤ k ≤ k and ∀i ≥ n. for any n ≥ n we have, |hn(x)−f(x)| = | n ∑ k=1 gkn(x)− ∞ ∑ k=1 fk(x)| ≤ | n ∑ k=1 (gkn(x)−fk(x))|+| ∞ ∑ k=n+1 fk(x)| ≤ | k ∑ k=1 (gkn(x)−fk(x))|+ n ∑ k=k+1 |gkn(x)|+ ∞ ∑ k=k+1 |fk(x)| < k ∑ k=1 ǫ k +2 ∞ ∑ k=k+1 mk < 3ǫ. since ǫ is arbitrary, it follows that {hn(x)} converges pointwise to f on x. hence f(x) = ∞ ∑ k=1 fk(x) is a baire one function on x. � theorem 4.5. z((b1(x))) is closed under countable intersection. proof. let z(fn) ∈ z(b1(x)), ∀n ∈ n. we define gn(x) = |fn(x)| ∧ 1 2n , ∀x ∈ x and ∀n ∈ n and let g(x) = ∞ ∑ n=1 gn(x). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 244 on rings of baire one functions here, |gn(x)| ≤ 1 2n , ∀n ∈ n and ∀x ∈ x. also ∞ ∑ n=1 1 2n < ∞. so, by lemma 4.4 g ∈ b1(x) and z(g) = ∞ ⋂ n=1 z(gn) = ∞ ⋂ n=1 z(fn). � definition 4.6. two subsets a and b are said to be completely separated in x by b1(x), if there exists a function f ∈ b ∗ 1(x) such that f(a) = r and f(b) = s with r < s and r ≤ f ≤ s, ∀x ∈ x. it is enough to say that a and b are completely separated by b1(x), if we get a function g in b1(x) satisfying g(x) ≤ r, ∀x ∈ a and g(x) ≥ s, ∀x ∈ b, for then, (r ∨ g) ∧ s has the required property. moreover we can replace r and s by 0 and 1, as there always exists a continuous bijection h from [r, s] to [0, 1] and h ◦ g is a baire one function with the desired property. it is well known that two sets a and b are completely separated by c(x) if and only if a and b are completely separated by c(x). but in case of completely separated by b1(x), this result is one-sided. certainly, a and b are completely separated by b1(x) implies that a and b are completely separated by b1(x). but the converse is not true, as seen in the following example. example 4.7. in [0, 1], the sets a = [0, 1) and b = {1} are completely separated by b1([0, 1]), because {fn} ⊆ c[0, 1] defined by fn(x) = x n, ∀x ∈ [0, 1] and ∀n ∈ n converges to the function f(x) defined by f(x) = { 0 if 0 ≤ x < 1 1 if x = 1. so, f belongs to b∗1[0, 1] and f(a) = 0, f(b) = 1. but a, b are not disjoint and therefore are not completely separated by b1([0, 1]). theorem 4.8. two subsets of x are completely separated by b1(x) if and only if they are contained in disjoint zero sets in z(b1(x)). proof. let z(f) and z(g) are two members of z(b1(x)) such that a ⊆ z(f) and b ⊆ z(g) with z(f) ∩ z(g) = ∅. clearly, the zero set of |f| + |g| is empty and we may define h(x) = |f(x)| |f(x)|+|g(x)| . here (|f(x)| + |g(x)|) > 0 on x and |f| + |g| ∈ b1(x), so by theorem 2.3 1 |f(x)|+|g(x)| ∈ b1(x) which implies h(x) = |f(x)| |f(x)|+|g(x)| ∈ b1(x). now, h(z(f)) = 0 and h(z(g)) = 1. so, z(f) and z(g) are completely separated by b1(x). therefore a and b are completely separated by b1(x). conversely, let a and a′ be completely separated by b1(x). so, there exists f ∈ b1(x) such that f(a) = 0 and f(a ′) = 1. the disjoint sets f = {x : f(x) ≤ 1 3 } and f ′ = {x : f(x) ≥ 2 3 } belong to z(b1(x)) and a ⊆ f, a ′ ⊆ f ′. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 245 a. deb ray and a. mondal 5. b1-embedded and b ∗ 1-embedded in this section we introduce the analogues of c-embedding and c∗-embedding, called b1-embedding and b ∗ 1-embedding in connection with the extensions of baire one functions. definition 5.1. a subset y of a topological space x is called b1-embedded in x, if each f ∈ b1(y ) has an entension to a g ∈ b1(x). i.e. ∃ g ∈ b1(x) such that g ∣ ∣ y = f. similarly, y is called b∗1-embedded in x, if each f ∈ b ∗ 1(y ) has an entension to a g ∈ b∗1(x). theorem 5.2. any b1-embedded subset is b ∗ 1-embedded. proof. let y be any b1-embedded subset of x and f ∈ b ∗ 1(y ). there is a natural number n such that |f(y)| ≤ n, ∀y ∈ y . since y is b1-embedded, it has an extension to g ∈ b1(x). taking h = (−n ∨ g) ∧ n. we get h ∈ b ∗ 1(x) and g(y) = h(y), ∀ y ∈ y . so, h is an extension of f and h is bounded. hence y is b∗1-embedded in x. � we establish an analogue of urysohn’s extension theorem for continuous functions which gives a necessary and sufficient condition for a subspace to be b∗1-embedded in a topological space x. before proving this theorem we need a lemma that ensures that the uniform limit of a sequence of baire one functions is a baire one function. it is to be noted that a special case of this particular result when functions are on real line has already been established. we prove the result in a more general setting. lemma 5.3. let {fn} be a sequence of baire one functions on x. if {fn} converges uniformly to f on x then f is a baire one function on x. proof. let {fn} be a sequence of baire one functions converging uniformly to f on x. by definition of uniform convergence, for each k ∈ n, there exists a subsequence {fnk} such that |fnk(x) − f(x)| < 1 2k , ∀x ∈ x. consider the sequence {fnk+1 − fnk}. then |fnk+1(x)−fnk(x)| ≤ |fnk+1(x)−f(x)|+|fnk(x)−f(x)| ≤ 1 2k+1 + 1 2k = ( 3 2 )2−k. let mk = ( 3 2 )2−k and note that, |fnk+1(x) − fnk(x)| ≤ mk, ∀x ∈ x and ∞ ∑ k=1 mk < ∞, where mk > 0, ∀ k ∈ n. so, ∞ ∑ k=1 [fnk+1(x) − fnk(x)] is a convergent series and so by lemma 4.4 the sum function ∞ ∑ k=1 [fnk+1(x) − fnk(x)] is a baire one function on x. now, ∞ ∑ k=1 [fnk+1(x) − fnk(x)] = lim n→∞ n ∑ k=1 [fnk+1(x) − fnk(x)] = f(x) − fn1(x). since, fn1 is a baire one function, f is also a baire one function. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 246 on rings of baire one functions theorem 5.4. a subset y of a topological space x is b∗1-embedded in x iff any pair of subsets of y which are completely separated in y by b1(y ) are also completely separated in x by b1(x). proof. let y be a b∗1-embedded subspace of x and p , q be subsets of y completely separated in y by b1(y ). so, there exists f ∈ b ∗ 1(y ) such that, f(p) = 0 and f(q) = 1. since y is b∗1-embedded in x, f has an extension to g ∈ b∗1(x) such that g(p) = 0 and g(q) = 1. hence, p , q are completely separated in x by b1(x). conversely, assume that any two completely separated sets in y by b1(y ) are also completely separated in x by b1(x). let f1 ∈ b ∗ 1(y ). then there exists a natural number m such that, |f1(y)| ≤ m, ∀y ∈ y . for n ∈ n, let rn = ( m 2 )(2 3 )n. then 3rn+1 = 2rn, ∀n and |f1| ≤ 3r1. let a1 = {y ∈ y : f1(y) ≤ −r1} and b1 = {y ∈ y : f1(y) ≥ r1}. clearly, a1 and b1 are disjoint members in z(b1(y )) and therefore, are completely separated in y by b1(y ). by our hypothesis a1, b1 are completely separated in x by b1(x). so, there exists g1 ∈ b ∗ 1(x) such that, g1(a1) = −r1 and g1(b1) = r1 also −r1 ≤ g1 ≤ r1. let f2 = f1 − g1 ∣ ∣ y , then f2 ∈ b ∗ 1(y ) and |f2| ≤ 2r1 on y . so, |f2| ≤ 3r2. let a2 = {y ∈ y : f2(y) ≤ −r2} and b2 = {y ∈ y : f2(y) ≥ r2}. then a2 and b2 are completely separated in x by member of b1(x). so, there exists g2 : x → [−r2, r2], such that, g2 ∈ b ∗ 1(x) and g2(a2) = −r2, g2(b2) = r2. taking f3 = f2 − g2 ∣ ∣ y we get f3 ∈ b ∗ 1(y ) with |f3| ≤ 2r2 = 3r3. if we continue this process and use principle of mathematical induction then we get two sequences {fn} ⊆ b ∗ 1(y ) and {gn} ⊆ b ∗ 1(x), with the following properties: |fn| ≤ 3rn and fn+1 = fn − gn ∣ ∣ y , ∀n ∈ n. let g(x) = ∞ ∑ n=1 gn(x), ∀x ∈ x. it is clear that, |gn(x)| ≤ rn, ∀n and ∀x ∈ x, gn ∈ b1(x), ∀n ∈ n. also, ∞ ∑ n=1 rn < ∞, where rn > 0, ∀n ∈ n. so, by lemma 4.4 g is a baire one function on x. we claim that, g(y) = f1(y), ∀y ∈ y . g(y) = lim n→∞ [g1(y) + g2(y) + ... + gn(y)] = lim n→∞ [f1(y) − f2(y) + f2(y) − f3(y) + ... + fn(y) − fn+1(y)] = f1(y) − lim n→∞ fn+1(y) = f1(y) (as |fn| ≤ 3rn and ∞ ∑ n=1 rn is a geometric series with common ratio less than 1). so, f ∈ b∗1(y ) has an extension g ∈ b1(x) and we know that if a baire one function f ∈ b ∗ 1(y ) has an extension in b1(x), then it has an extension in b ∗ 1(x). hence, y is b∗1-embedded. � theorem 5.5. if y is any b1-embedded subspace of a topological space x then y is completely separated in x by b1(x), from any zero set of z(b1(x)) disjoint from y . c© agt, upv, 2019 appl. gen. topol. 20, no. 1 247 a. deb ray and a. mondal proof. let y be a b1-embedded subspace of x and z(f) a zero set in z(b1(x)), disjoint from y . let h : y → r be a map defined as, h(y) = 1 |f(y)| . here, |f| ∣ ∣ ∣ ∣ y ∈ b1(y ) and |f(y)| > 0 on y as z(f) ∩ y = ∅. so, by theorem 2.3, h ∈ b1(y ). as y is b1-embedded, h has an extension g ∈ b1(x). then |f|.g ∈ b1(x) and (|f|.g)(z(f)) = 0, (|f|.g)(y) = 1, ∀ y ∈ y . it follows that y and z(f) are completely separated in x by b1(x). � theorem 5.6. a b∗1-embedded subset y of x is a b1-embedded subset in x if and only if it is completely separated in x by b1(x) from any disjoint zero set in z(b1(x)). proof. let b∗1-embedded subset y of x be b1-embedded. then the result follows from theorem 5.5. conversely, let y be completely separated in x by b1(x) from any zero set in z(b1(x)) disjoint from y . let f ∈ b1(y ). we consider the function tan−1 : r → (−π 2 , π 2 ). by theorem 2.4 the function tan−1 ◦ f is a bounded baire one function. as y is b∗1-embedded in x, tan −1 ◦ f has an extension to a function g ∈ b1(x). let z = {x ∈ x : |g(x)| ≥ π 2 }, then z ∈ z(b1(x)) and z ∩ y = ∅. so, by hypothesis, there exists a baire one function h : x → [0, 1] such that h(z) = 0 and h(y ) = 1. we see that, g.h ∈ b1(x) and ∀x ∈ x, |(g.h)(x)| < π 2 . so, tan(g.h) ∈ b1(x) and ∀ y ∈ y , tan(g.h)(y) = f(y). hence, y is b1embedded. � corollary 5.7. for any topological space x, a zero set z ∈ z(b1(x)) is b∗1-embedded if and only if it is b1-embedded. acknowledgements. the authors are thankful to professor s. k. acharyya, retired professor, department of pure mathematics, university of calcutta for his continuous support and encouragement. we also acknowledge the learned referee for his valuable comments. the second author is supported by council of scientific and industrial research, hrdg, india. sanction no.09/028(0998)/2017-emr-1. references [1] j. p. fenecios and e. a. cabral, on some properties of baire-1 functions, int. journal of math. analysis 7, no. 8 (2013), 393–402. 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[8] d. zhao, functions whose composition with baire class one functions are baire class one, soochow journal of mathematics 33, no. 4 (2007), 543–551. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 249 @ appl. gen. topol. 23, no. 1 (2022), 55-68 doi:10.4995/agt.2022.16126 © agt, upv, 2022 on certain new notion of order cauchy sequences, continuity in (l)-group sudip kumar pal a,1 and sagar chakraborty b a department of mathematics, diamond harbour women’s university, diamond harbour-743368, west bengal, india. (sudipkmpal@yahoo.co.in) b jadavpur university, kolkata-700032, west bengal, india. (sagarchakraborty55@gmail.com) communicated by m. a. sánchez-granero abstract in this paper, we introduce the notions of order quasi-cauchy sequences, downward and upward order quasi-cauchy sequences, order half cauchy sequences. next we consider an associated idea of continuity namely, ward order continuous functions [2] and investigate certain interesting results. the entire investigation is performed in (l)-group setting to extend the recent results in [5, 6]. 2020 msc: 54d20; 54c30; 54a25. keywords: (l)-group; order quasi-cauchy sequences; statistical ward continuity; uniform order continuity; statistical ward compact set; downward order continuity; upward order continuity. 1. introduction the concept of continuity and any concept involving continuity play a very important role not only in pure mathematics but also in other branches of sciences especially in computer science, information theory, biological science. in 2010, burton and coleman first introduce the term quasi-cauchy sequence which is weaker than cauchy sequence but interesting in their own right. they defined the term quasi-cauchy sequence as: any sequence of real numbers (xn) is quasi-cauchy if given any � > 0 there exists an integer k > 0 such that 1corresponding author. received 25 august 2021 – accepted 22 december 2021 http://dx.doi.org/10.4995/agt.2022.16126 s. k. pal and s. chakraborty n ≥ k implies |xn+1 −xn| < �. evidently cauchy sequences are quasi-cauchy but the converse is not true in general as the counter example is provided by the sequence of partial sums of the harmonic series. this and several such examples establish the important fact that the class of quasi-cauchy sequences is much bigger than the class of cauchy sequences, taking in the process more sequences under the preview. understandably mathematical consequence are not analogous to the already existing notions bases on cauchy sequences, like the usual idea of compactness. in a current development of the study of generalized metric space, the term ward continuity comes remembering the definition of continuity in sequential sense. the concepts of ward continuity of real valued function and ward compactness of subsets of r are introduced by cakalli [5]. a real valued function f is called ward continuous on e if for every quasi-cauchy sequence in e, the corresponding f-image sequence is also quasi-cauchy. the aim of this paper is to introduce the notion of order quasi-cauchy sequences and some weaker versions of it [2]. we primarily investigate several features of this new notion. finally, a new concept, namely the concept of order statistical ward continuity of a function is introduced and investigated[7]. in this investigation we have obtained theorems related to order statistical ward continuity, order statistical ward compactness, compactness, and uniform order continuity. we also introduced and studied some other continuities involving statistical order quasi-cauchy sequences and order convergent sequences of points in l-group[8]. throughout r and n stand for the sets of all real numbers and natural numbers respectively and our topological terminologies and notations are as in the book [9] from where the notions (undefined inside the article) can be found. all spaces in the sequel are l-group. 2. preliminaries first we recall the concept of ‘natural density’ [9] of a set a of positive integers, which is defined by δ(a) = lim n→∞ d(k ≤ n : k ∈ a) n , where d denotes the cardinality of the concerned sets. the notion of statistical convergence which is an extension of the idea of usual convergence, was introduced by h. fast [10] and i. j. schoenberg [13]. any sequence (xn) in r is statistical convergent to the number l provided that for each � > 0, lim n→∞ d(k ≤ n : |xk −l| ≥ �) n = 0. equivalently, |xk − l| < � for almost all k. topological consequences of statistical convergence were studied by fridy [11] and šalát [12]. the study of statistical convergence and its numerous extensions and, in particular, of the ideal convergence and its applications has been one of the most active areas of research in the summability theory over the last 15 years. © agt, upv, 2022 appl. gen. topol. 23, no. 1 56 on certain new notion of order cauchy sequences, continuity in (l)-group now we recall some concepts related to lattice, order convergence and lattice order group. a nonempty set l is said to be a lattice with respect to the partial order ≤ if for each pair of elements x,y ∈ l, both the supremum and infimum of the set {x,y} exists in l. we shall write x∨y = sup{x,y} and x∧y = inf{x,y}. definition 2.1. an abelian group (l, +) is said to be an (l)-group if it is lattice and a ≤ b implies a + c ≤ b + c for all a,b,c ∈ l. from now throughout this paper we will write l for (l)-group (l, +) and θ denotes the identity element of the (l)-group (l, +). let x ∈ l be any element, we define |x| = x∨ (−x) where −x denotes the additive inverse of x. also we use the notation a ≥ b equivalent as b ≤ a, and a > b as equivalent to b ≤ a with b 6= a. a sequence (xn) in l( i.e. a map : n → l) is said to be increasing(or decreasing) if x1 ≤ x2 ≤ ... (or x1 ≥ x2 ≥ ...) and we write it symbolically as xn ↑ (or xn ↓). a sequence (pn) is called an order sequence if pn ↓ and inf pn = θ. in this case we write pn ↓ θ. some author use the term monotone sequences instead of order sequence. it is easy to observe that if (an) and (bn) are two order sequences then the sequence (an + bn) is also an order sequence. a sequence (xn) in l is said to be order bounded if there exists an order interval [a,b] such that a ≤ xn ≤ b for all n ∈ n. a sequence (an) in l is said to be convergent in order(or order convergence)to a ∈ l if there exists an order sequence (pn) such that |an −a| ≤ pn holds for all n. we write it symbolically as an ord−−−−→ a. in the literature, there are two ways to define order convergence. other than the above way one can define order convergence as: a sequence (an) in l is said to be order convergence to a ∈ l if there exists an order sequence (pn) such that for each n0 ∈ n, there exists some m ∈ n satisfying |an −a| ≤ pn0 for all n ≥ m. the later definition is useful for defining order convergence in filter. the first definition is called 1-converging and the second one is called 2-converging. if the lattice is dedekind complete then the two definitions are equivalent[1] throughout the paper we use the second type definition of order convergence. if a sequence (xn) is order convergent to x0 then we call the sequence (xn− x0) as a null order sequence which converges to θ. 3. main results 3.1. ward continuity in (l)-group. in this section, we introduce the concept of order quasi-cauchy sequences and ward continuity in (l)-group l, and we study some results related to it. we know that any sequence of real numbers (xn) is said to be cauchy if given any � > 0 there exists an integer k > 0 such that m,n ≥ k implies |xm −xn| < �. in 2010, burton and coleman first use the term quasi-cauchy sequence. they defined the term quasi-cauchy sequence as: any sequence of real numbers © agt, upv, 2022 appl. gen. topol. 23, no. 1 57 s. k. pal and s. chakraborty (xn) is quasi-cauchy if given any � > 0 there exists an integer k > 0 such that n ≥ k implies |xn+1−xn| < �. using this idea we first introduce two definitions. definition 3.1. any sequence (xn) in a (l)-group l is said to be an ordercauchy sequence if for any order sequence (pn) and for each n0 ∈ n there exists m ∈ n such that |xi −xj| ≤ pn0 for all i,j ≥ m. definition 3.2. any sequence (xn) in a (l)-group l is said to be an order quasi-cauchy sequence if for any order sequence (pn) and for each n0 ∈ n there exists m ∈ n such that |xn+1 −xn| ≤ pn0 for all n ≥ m. remark 3.3. it is easy to verify that every order-cauchy sequence is order quasicauchy but the converse is not true in general. for counter example we take (l)-group (r, +) and consider the sequence (xn) where xn = 1 + 12 + 1 3 + ... + 1 n . clearly this sequence is order quasi-cauchy but not order-cauchy. remark 3.4. every order convergent sequence is also order quasi-cauchy. remark 3.5. also every subsequence of order-cauchy sequence is order-cauchy. but the analogous property fails for quasi-cauchy sequences. for instance we take the sequence (xn) in r, xn = √ n. (xn) is order quasi-cauchy but the subsequence (xn2 ) is not order quasi-cauchy. we know that if a function preserves cauchy sequences, then it is called cauchy continuous function. similarly we define quasi-cauchy continuous function in (l)-group. some author rename it as ‘ward continuous function’. throughout this paper we use the name ward continuous. definition 3.6. let l be a (l)-group. a function f : l → l is said to be ward continuous on l if the sequence (f(xn)) is order quasi-cauchy whenever (xn) is order quasi-cauchy in l. definition 3.7. a subset e of l is said to be ward compact if any sequence in e has an order quasi-cauchy subsequence. the following theorems are obvious. theorem 3.8. (1) every finite subset of l is ward compact. (2) union of any two ward compact subsets of l is ward compact. (3) intersection of any family of ward compact sets is ward compact. (4) any subset of ward compact set is ward compact. we see that for any metric space, continuity can be described by using sequence. remembering this idea, we introduce continuity in (l)-group l. we call it order continuity. definition 3.9. a function f : l → l is said to be order continuous at x0 if for any sequence (xn) in l, which is order convergent to x0, the corresponding image sequence (f(xn)) is order convergent to f(x0). in the next theorem we will investigate the relationship between ward continuity and order continuity [3]. © agt, upv, 2022 appl. gen. topol. 23, no. 1 58 on certain new notion of order cauchy sequences, continuity in (l)-group theorem 3.10. let f : r → l be a ward continuous function, where r ⊂ l then it is order continuous on r. proof. suppose that f : r → l is ward continuous on r ⊂ l. let (xn) be a sequence in r such that xn ord−−−−→ x0. now we define a new sequence (yn) as : yn = { x0, if n is even xk, if n = 2k − 1, k ∈ n. so, yn −x0 = { θ, if n is even xk −x0, if n = 2k − 1, k ∈ n. as (xn) is order convergent to x0 so for any order sequence (pn) and n0 ∈ n there exists m ∈ n such that |xn − x0| ≤ pn0 for all n ≥ m. now using this (pn) and n0 ∈ n with some suitable changes of m, from the construction of yn − x0, we can easily conclude that (yn) is order convergent. hence it is an order quasi-cauchy sequence. as f is ward continuous so it preserves order quasi cauchy sequence. hence (f(yn)) is also order quasi-cauchy, which is given by: f(yn) = { f(x0), if n is even f(xk), if n = 2k − 1, k ∈ n. now for any order sequence (qn) and n ′ 0 ∈ n, there exists m′ ∈ n such that |f(yn+1)−f(yn)| ≤ pn′0 for all n ≥ m ′ which implies |f(xk)−f(x0)| ≤ pn′0 , for all k ≥ m, where m(∈ n) depends on m′. so f(xn) ord−−−−→ f(x0). hence f is order continuous. � converse of the above theorem is not true in general, which follows from the next example. example 3.11. conisder the function f : r → r, given by f(x) = x2 and consider the sequence (xn) given by xn = √ n. theorem 3.12. ward continuous function preserves ward compact set. proof. let f : l → l be a ward continuous map and e ⊆ l be ward compact set. let (xn) be any sequence in e, as e is ward compact so we get subsequence (yn) of (xn) such that (yn) is order quasi-cauchy sequence. now as f is ward continuous function, (f(yn)) is order quasi-cauchy subsequence of the sequence (f(xn)) in f(e). this completes the proof of the theorem. � definition 3.13. a function f : l → l is said to be uniformly order continuous on a subset e of l if for any order sequence (�n) and n0 ∈ n, depending on this we get another order sequence (δn) and m ∈ n, such that |f(x)−f(y)| < �n0 whenever |x−y| < δm. theorem 3.14. if f : l → l is an uniform order continuous map on e ⊂ l then it is ward continuous on e. © agt, upv, 2022 appl. gen. topol. 23, no. 1 59 s. k. pal and s. chakraborty proof. let (xn) be any order quasi-cauchy sequence of points in e. as f is uniform order continuous so any order sequence (�n) and n0 ∈ n, depending on this we get another order sequence (δn) and m ∈ n, such that |f(x) − f(y)| < �n0 whenever |x−y| < δm. this implies for this δn and m,n0, we get suitably n, depends on δn,m,n0 such that |xn+1 −xn| < δn for all n > n. so |f(xn) −f(xn+1)| < �no, for all n > n. � from the above theorem we can easily conclude that uniform order continuous functions are also order continuous. remark 3.15. uniform order continuous image of ward compact set is ward compact. we use the following notations : c[l,l] = set of all order continuous functions on l. wc[l,l] = set of all ward continuous functions on l. uc[l,l] = set of all uniform order continuous functions on l. now from the above discussion we can easily conclude that, remark 3.16. uc[l,l] ⊆ wc[l,l] ⊆ c[l,l]. we see that, a sequence (xn) in l is said to be order convergent to x0 ∈ l if there exists an order sequence (pn) such that for each n0 ∈ n, there exists some m ∈ n satisfying |xn − x0| ≤ pn0 for all n ≥ m. in this case choice of m depends on x0. to deal this type of situation we introduce the concept of uniform order convergence in (l)-group. definition 3.17. a sequence (xn) in e ⊂ l is said to be uniform order convergent to x ∈ e if there exists an order sequence (pn) in l such that for each n0 ∈ n, there exists some m ∈ n satisfying |xn −x| ≤ pn0 for all n ≥ m and for all x ∈ e. theorem 3.18. let (fn) be a sequence of uniform order continuous functions on e ⊂ l and (fn) is uniformly order convergent to a function f then f is uniform order continuous on e. proof. suppose that (fn) is uniformly order convergent to some function f. then for a given order sequence (pn) in l and for each n0 ∈ n, there exists some m ∈ n satisfying |fn(x) − f(x)| ≤ pn0 for all n ≥ m and for all x ∈ e. now each fn : l → l is uniform order continuous on e. hence for this order sequence (pn), n ′ ∈ n, there exists order sequence qn and m′ ∈ n such that |fn(x) − fn(y)| < pn′ , for all n ≥ m′, whenever |x − y| ≤ qn. now f(x) −f(y) = f(x) −fn(x) + fn(x) −fn(y) + fn(y) −f(y), sum of three null sequences, hence f(x) is uniform order continuous on e. � theorem 3.19. let (fn) be a sequence of ward continuous functions defined on e ⊂ l and (fn) is uniformly order convergent to a function f then f is ward continuous on e. © agt, upv, 2022 appl. gen. topol. 23, no. 1 60 on certain new notion of order cauchy sequences, continuity in (l)-group proof. let (xn) be an order quasi-cauchy sequence of points on e. as (fn) is uniformly order convergent to f, given order sequence (pn) in l such that for each n0 ∈ n, there exists some m ∈ n satisfying |fn(x) − f(x)| ≤ pn0 for all n ≥ m and for all x ∈ e. now each fn is ward continuous on e. so for this (pn) and n ′ ∈ n there exists m′ ∈ n with |fm(xn+1) − fm(xn)| < pn′ for all n ≥ m′. now f(xn+1)−f(xn) = f(xn+1)−fm(xn+1) +fm(xn+1)−fm(xn) +fm(xn)−f(xn). this implies f(xn+1) −f(xn) is sum of three null sequences, so we easily conclude that f is ward continuous on e. � 3.2. statistical ward continuity in (l)-group. recently, it has been proved that a real-valued function defined on an interval a of the set of real numbers, is uniformly continuous on a if and only if it preserves quasi-cauchy sequences of points in a. in this section we call a real-valued function order statistically ward continuous if it preserves statistical order quasi-cauchy sequences. it turns out that any order statistically ward continuous function on a statistically ward order compact subset a of an l -group is uniformly order continuous on a. we prove theorems related to order statistical ward compactness, order statistical compactness, order continuity, statistical order continuity, ward order continuity, and uniform order continuity. definition 3.20. we call a sequence (xn) of points in (l)-group l statistically order quasi-cauchy if for any order sequence (pn) and n0 ∈ n, lim n→∞ d(k ≤ n : |xk+1 −xk| ≥ pn0 ) n = 0. where d(a) denotes the cardinality of the set a. it is clear that, any order quasi cauchy sequence is statistically order quasicauchy. definition 3.21. a subset e of l is said to be statistically ward compact if any sequence of points in e has a statistically order quasi-cauchy subsequence. definition 3.22. a function f : e → l is said to be statistically order continuous on e ⊆ l if it preserves statistically order convergent sequences. definition 3.23. let e ⊆ l. a function f : e → l is said to be statistically ward continuous if it preserves statistically order quasi-cauchy sequences. theorem 3.24. every statistically ward continuous functions are also statistically order continuous. proof. let f : e → l be a statistically ward continuous function and (xn) be any statistically order convergent sequence which converges to x0. for any order sequence (pn), n0 ∈ n, limn→∞ d(k≤n:|xk−x0|≥pn0 ) n = 0. hence the sequence (x1,x0,x2,x0, ...,xn−1,x0,xn,x0, ...) is also statistically order convergent to x0. hence it is statistically order quasi-cauchy. as f is statistically © agt, upv, 2022 appl. gen. topol. 23, no. 1 61 s. k. pal and s. chakraborty ward continuous so (f(x1),f(x0),f(x2),f(x0),f(x3), ...) is also statistically order quasi-cauchy. as the even terms of the sequence are f(x0), odd terms are nothing but the sequence (f(xn)), we can easily conclude that (f(xn)) is statistically order convergent to f(x0). this completes the proof. � the converse is not true in general. for counter example we take the function f : r → r given by f(x) = x2 and consider the sequence ( √ n). we know that any continuous function on a compact set is uniformly continuous. similarly for statistically ward continuous function defined on a statistically ward compact subset of an (l)-group, we have the following result : theorem 3.25. let e be a statistically ward compact subset of an (l)-group l and f : e → l be a statistically ward continuous function on e. then it is uniform order continuous. proof. if possible suppose that f is not uniformly order continuous on e. then there exists order sequence (�n) and n0 ∈ n such that for any δn and m ∈ n with |x − y| ≤ δm, |f(x) − f(y)| > �n0 . now for each n ∈ n fix |xn − yn| < δn and |f(xn) − f(yn)| ≥ �n0 . since e is statistically ward compact so (xn) has a subsequence (xnk ) which is statistically order quasi-cauchy. now ynk+1 − ynk = (ynk+1 − xnk+1 ) + (xnk+1 − xnk ) + (xnk − ynk ) which is clearly sum of three null sequences hence (ynk ) is statistically order quasi-cauchy subsequence of (yn). as xnk+1 −ynk = (xnk+1 −ynk+1 ) + (ynk+1 −ynk ) so the sequence (xnk+1−ynk ) is statistically order convergent to θ. hence the sequence (xn1,yn1,xn2,yn2, ...,xnk,ynk, ...) is statistically order quasi-cauchy. which implies that the sequence (f(xn1 ),f(yn1 ),f(xn2 ),f(yn2 ), ...,f(xnk ),f(ynk ), ...) is also statistically order quasi-cauchy in f(e). but this contradicts the fact that |f(x) −f(y)| > �n0 . thus f is uniformly order continuous on e. � theorem 3.26. statistically ward continuous image of any statistically ward compact subset of l is statistically ward compact. proof. let f : a → l be a statistically ward continuous function defined on a subset a of l and e be a statistically ward compact subset of a. we want to show f(e) is statistically ward compact subset of l. let (yn) be any sequence of points in f(e). then clearly yn = f(xn) for some sequence (xn) of points in e. as e is statistically ward compact set so there exists subsequence (zk) = (xnk ) of (xn) such that (zk) is statistically ward compact. now as f is ward continuous function so f(zk) is statistically order quasi-cauchy sequence. thus we get a order quasi-cauchy subsequence (f(zk)) of the sequence (yn) of f(e). hence f(e) is a ward compact set. � theorem 3.27. if (fn) is a sequence of statistically ward continuous functions on a subset e of l and (fn) is uniformly order convergent to a function f then f is statistically order ward continuous on e. proof. suppose that (fn) is uniform order convergence to f. for any order sequence (�n) and n0 ∈ n there exists m ∈ n such that |fn(x) − f(x)| < �n0 , © agt, upv, 2022 appl. gen. topol. 23, no. 1 62 on certain new notion of order cauchy sequences, continuity in (l)-group for all n ≥ m and for all x ∈ e. consider any statistically order quasicauchy sequence (xn) of points in e. as fm is statistically ward continuous on e so it preserves statistically order quasi-cauchy sequence. hence limn→∞ d(k≤n:|fm(xk+1)−fm(xk)|≥�n0 ) n = 0. now f(xk+1)−f(xk) = [f(xk+1)−fm(xk+1)] + [fm(xk+1)−fm(xk)] + [fm(xk)− f(xk)]. hence using the fact that fm is statistically ward continuous and fm is uniform convergent to f we can easily conclude that lim n→∞ d(k ≤ n : |f(xk+1) −f(xk)| ≥ �n0 ) n = 0. this completes the proof. � the following result follows immediately: theorem 3.28. the set swc[e,l], the set of all statistical ward continuous functions is a closed set. from the above discussion we see that uc[l,l] ⊆ wc[l,l] ⊆ c[l,l]. now the obvious question is when these sets are equal. in [2], burton and coleman gives some partial idea about the equality of uc[l,l] ⊆ wc[l,l]. theorem 3.29. let i ⊆ r be any interval. then uc[i,r] = wc[i,r]. now we introduce another type of convergence in (l) − group called slowly oscillating order convergence. definition 3.30. a sequence (xn) of points in (l)-group l is called slowly oscillating order convergence if for any order sequence (�n) and n0 ∈ n there exists m ∈ n such that |xi −xj| < �n0 for all i ≥ m and 1 ≤ i j and i j → 1 as i,j →∞. from definition it is clear that order cauchy sequences are obviously slowly oscillating and every slowly oscillating sequence is order quasi-cauchy. definition 3.31 ([4, 14]). a function f : e → l is said to be slowly oscillating continuous if it preserves slowly oscillating order sequences. by soc[e, l] we denote set of all slowly oscillating continuous functions defined on e. now we introduce the concept of connectedness in (l)-group l. definition 3.32. let l be a (l)-group. suppose that x ∈ u ⊆ l. u is called order sequential neighborhood of x ∈ l if any sequence (xn) which is order convergent to x then {xn : n ≥ m}⊂ u for some m ∈ n. definition 3.33. u is said to be an order sequential open subset of l if for each x ∈ u, u is order sequential neighborhood of x. definition 3.34. a is said to be an order sequential closed subset of l if l\a is an order sequential open subset of l. definition 3.35. let a ⊆ l, by ā we denote closure of a, defined as intersection of all sequentially closed sets containing a. © agt, upv, 2022 appl. gen. topol. 23, no. 1 63 s. k. pal and s. chakraborty definition 3.36. an (l)-group l is said to be order connected if there do not exists any non empty subsets a,b such that x = a∪b with ā∩ b̄ = φ. now we modify the lemma 1 in [2] which is given in metric space setting. lemma 3.37. let ((an,bn)) be a sequence of ordered pair of points in a connected subset e ⊆ l such that given any ordered sequence (�n) there exists n0,m ∈ n, we have |an − bn| ≤ �n0 for all n ≥ m. then there exists an order quasi-cauchy sequence (tn) with the property that for any positive integer i there exists a positive integer k such that (ai,bi) = (tj−1, tj). it turns out that a function defined on a connected subset e of a metric space is uniformly continuous if and only if it preserves either quasi-cauchy sequences or slowly oscillating sequences of points in e. now we are in the position of most desired result: theorem 3.38. let e be an order connected subset of a (l)-group l then the three sets uc[e,l],wc[e,l] and soc[e,l] are equivalent. proof. uc[e,l] ⊆ wc[e,l] : let f : e → l be any uniformly order continuous function on e. let (xn) be any order quasi-cauchy sequence of points in e. as f is uniform order continuous so any order sequence (�n) and n0 ∈ n, depending on this we get another order sequence (δn) and m ∈ n such that |f(x) −f(y)| < �n0 whenever |x−y| < δm. this implies for this δn and m,n0, we get suitably n, depends on δn,m,n0 such that |xn+1 − xn| < δn for all n > n. so, |f(xn) −f(xn+1)| < �n0, for all n > n. uc[e,l] ⊆ soc[e,l] : let f : e → l be uniform order continuous. we take slowly oscillating sequence (xn) of points on e. let (�n) be any order sequence. we get another order sequence (δn) and n0,m0 ∈ n such that |f(xi) − f(xj)| ≤ �n0 whenever xi,xj ∈ e and |xi − xj| ≤ δm0 . as (xn) is slowly oscillating so |xi − xj| ≤ δk, for all i ≥ m and 1 ≤ ij and i j → 1 as i,j → ∞. now using uniform continuity of f, |f(xi) − f(xj)| ≤ �k, for all i ≥ m and 1 ≤ i j and i j → 1 as i,j → ∞. this implies (f(xn)) is slowly oscillating. hence f ∈ soc[e,l]. soc[e,l] ⊆ uc[e,l] : let f : e → l be not uniformly continuous on e. now each n ∈ n, we fixed |xn −yn| < δn then as f is not uniformly order continuous so there exists order sequence �n such that |f(xn) −f(yn) ≥ �(k)|. now as it is given that e is order connected so by the lemma 3.1, from (xn) we can construct a slowly oscillating sequence (tn) but as f is not uniformly continuous, so the transformed sequence (f(tn)) is not slowly oscillating. hence f /∈ soc[e,l]. this implies soc[e,l] ⊆ uc[e,l]. wc[e,l] ⊆ uc[e,l] : suppose f is not uniformly order continuous on e. since we know that slowly oscillating sequences are also order quasi-cauchy hence the sequence (tn) constructed on previous case is also order quasi-cauchy, but as f is not uniformly order continuous so f(tn) is not order quasi-cauchy. so wc[e,l] ⊆ uc[e,l]. this completes the proof of the theorem. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 64 on certain new notion of order cauchy sequences, continuity in (l)-group 4. downward and upward order continuity in (l)-group in this section, we introduce and investigate the concepts of down order continuity and down order compactness. a real valued function f on a subset e of the set of real numbers is down continuous if it preserves downward half cauchy sequences, i.e. the sequence (f(an)) is downward half cauchy whenever (an) is a downward half cauchy sequence of points in e. a sequence (ak) of points in r is called downward half cauchy if for every � > 0 there exists an n0 ∈ n such that am − an < � for m ≥ n ≥ n0. it turns out that the set of all down continuous functions is a proper subset of the set of all continuous functions. first we introduce the following definition: definition 4.1. let (xn) be a sequence in (l)-group l. then (xn) is called downward order quasi-cauchy if for any order sequence (pn) and for each n0 ∈ n there exists m ∈ n such that xn+1 −xn ≤ pn0 , for all n ≥ m. it is clear that every order quasi-cauchy sequence is also downward order quasi-cauchy but the converse is not true in general. for example we take the lattice order group (r, +) and the sequence (xn), where xn = −n. this sequence is downward order quasi-cauchy but not order quasi-cauchy. any order cauchy sequence is obviously order quasi-cauchy and hence downward order quasi-cauchy. definition 4.2. a sequence (xn) of points in l is said to be downward order half cauchy if for any order sequence (pn) and for each n0 ∈ n there exists k ∈ n such that xm −xn ≤ pn0 where m,n ∈ n with m ≥ n > k. it is obvious that downward order half cauchy sequences are also downward order quasi-cauchy and any subsequence of downward order half cauchy sequence are same type. but for the downward order quasi cauchy sequences the situation is different. we take the sequence (xn) in (r, +) such that xn = √ n. clearly (xn) is downward order half cauchy but one of it’s subsequence, namely (xnk ) is not downward order half cauchy. in [2], authors proved that a sequence of real numbers is cauchy if and only if every subsequence is quasi-cauchy. in the next theorem we present similar type result for downward order half cauchy sequences in (l)-group. theorem 4.3. a sequence (xn) in l is downward order half cauchy if and only if every subsequence of (xn) is downward order quasi-cauchy. proof. if (xn) is downward order half cauchy then every subsequence of (xn) is downward half cauchy so is downward order quasi-cauchy. to prove the converse part, we use contrapositive statement. let (xn) be not downward order half cauchy. then there exists order sequence (pn) and n0 ∈ n such that for every positive integer m, xni −xnj > pn0 , ni > nj ≥ m. now for m = 1 we get such ni,nj, we rename it as k1,k2. so k2 > k1 > 1 and xk2−xk1 > pn0 . similarly for m = 2, 3, .... inductively we get xkn+1−xkn > pn0 , where kn+1 > kn > kn−1 > .... which shows that the subsequence (xkn ) is not downward order quasi-cauchy. hence the proof. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 65 s. k. pal and s. chakraborty now we study the sequential compactness like property. first of all we introducing the idea of downward order compact set in a lattice order group as: definition 4.4. a subset e of l is called downward order compact if any sequence of points in e has a downward order quasi-cauchy subsequence. we know that a real valued function of real variables is continuous if it preserves convergent sequences. in a similar way we already defined order continuity. now if a function preserves downward order quasi-cauchy sequences then we get a new type of continuity, we call it downward order continuity. definition 4.5. a function f : e → l is called downward order continuous on a subgroup e of l if it preserves downward order quasi-cauchy sequences. theorem 4.6. sum of two downward order continuous functions is downward order continuous. proof. suppose that f : e → l and g : e → l are two downward order continuous functions on e, a subgroup of l. let (xn) be a downward order quasi-cauchy sequence in e. as f,g both are downward order continuous so the sequences (f(xn)) and (g(xn)) both are downward order quasi-cauchy. we know that sum of two order sequences is also an order sequence. take any order sequence (pn) then (2pn) is also order sequence. so for order sequence 2pn and for n0 ∈ n there exists positive integers m1 and m2 such that f(xn+1)− f(xn) ≤ pn0 for all n ≥ m1 and g(xn+1) − g(xn) ≤ pn0 for all n ≥ m2. take m = max{m1,m2}. then f(xn+1) + g(xn+1) − g(xn) − f(xn) ≤ 2pn0 for all n ≥ m. this proves the theorem. � from definition it is quite obvious that every ward order continuous function is downward order continuous. the following theorem make the link between ward order continuity and order continuity. theorem 4.7. every downward order continuous function is order continuous. proof. let f : e → l be downward order continuous on e and (xn) be a sequence which order converges to x0. now we construct a sequence (x1,x0,x1,x0,x2,x0,x2,x0, ...). as (xn) is order converges to x0 so the new sequence is also order converges to x0. also from the construction it is clear that the new sequence is also downward order quasi-cauchy. as f is downward order continuous so (f(x1),f(x0),f(x2),f(x0),f(x2),f(x0), ...) is downward order quasi-cauchy. from here we can easily conclude that f(xn) order converges to f(x0). hence f is order continuous. � theorem 4.8. let e be a downward compact subgroup of l and f : e → l be downward order continuous function. then f(e) is also downward order compact. © agt, upv, 2022 appl. gen. topol. 23, no. 1 66 on certain new notion of order cauchy sequences, continuity in (l)-group proof. suppose that e is a downward compact subgroup of l. let us take any sequence (yn) in f(e). so yn = f(xn) where xn ∈ a for each n. as e is downward order compact and (xn) is any sequence in e so (xn) has a downward order quasi-cauchy sub-sequence, say, (zk). as f : e → l is a downward order continuous function, f(zk) is a downward order quasi-cauchy sequence. hence we get f(zk) is a downward order quasi-cauchy sub-sequence of (yn). this completes the proof. � now the question arise that does the downward continuous function preserves uniform limit? the following theorem gives the answer. the technique of the proof is almost same as word continuous function. so we just state the theorem. theorem 4.9. if (fn) be a sequence of downward continuous functions defined on a subgroup e of l and (fn) is uniform order convergent to a function f then f is downward order continuous on e. if we change the position of xn+1 and xn in the definition of downward order quasi-cauchy sequence we get a new type of sequence, we call it upward order quasi-cauchy sequence. the results are similar. we just state the result. definition 4.10. let (xn) be a sequence in (l)-group l. then (xn) is called upward order quasi-cauchy if for any order sequence (pn) and for each n0 ∈ n there exists m ∈ n such that xn −xn+1 ≤ pn0 . it is clear that every order quasi-cauchy sequence is also upward order quasicauchy. but the converse is not true in general. for example we take the lattice order group (r, +) and the sequence (xn), where xn = n. this sequence is upward order quasi-cauchy but not order quasi-cauchy. any order cauchy sequence is obviously order quasi-cauchy and hence upward order quasi-cauchy. acknowledgements. the second author is thankful to the council of scientific and industrial research, hrdg, india for granting the senior research fellowship during the tenure of which this work was done. the authors are grateful to prof. pratulananda das for his valuable suggestions to improve the quality of the paper. references [1] y. abramovich and g. sirotkin, on order convergence of nets, positivity 9 (2005), 287– 292. [2] d. burton and j. coleman, quasi-cauchy sequences, amer. math. monthly 117, no. 4 (2010), 328–333. [3] i. canak and d. mik, new types of continuity, abstr. appl. anal. 2010 (2010), article id: 258980. © agt, upv, 2022 appl. gen. topol. 23, no. 1 67 s. k. pal and s. chakraborty [4] h. çakalli, slowly oscillating continuity, abstr. appl. anal. 2008 (2008), article id 485706. [5] h. çakalli, forward continuity, j. comput anal. appl. 13, no. 2 (2011), 225–230. [6] h. çakalli, statistical ward continuity, applied mathematics letters 24 (2011), 1724– 1728. [7] h. çakalli, statistical quasi-cauchy sequences, mathematical and computer modelling 54 (2011), 1620–1624. [8] h. çakalli and b. haarika, ideal quasi-cauchy sequences, j. inequal. appl. 234 (2012), 1–11. [9] r. engelking, general topology, sigma ser. pure math., heldermann, berlin vol. 6 2nd edition (1989). [10] h. fast, sur la convergence statistique, colloq. math. 2 (1951) 241–244. [11] j. a. fridy, on stastistical convergence, analysis 5 (1985), 301–313. [12] p. kostyrko, t. šalát and w. wilczyński, i-convergence, real anal. exchange. 26, no. 2 (2000/2001), 669–685. [13] i. j. schoenberg, the integrability methods, amer. math. monthly 66 (1959), 361–375. [14] r. w. vallin, creating slowly oscillating sequences and slowly oscilating continuous functions, acta. math. univ. comenianae 25 (2011), 71–78. © agt, upv, 2022 appl. gen. topol. 23, no. 1 68 @ appl. gen. topol. 20, no. 1 (2019), 119-133doi:10.4995/agt.2019.10360 c© agt, upv, 2019 existence of fixed points for pointwise eventually asymptotically nonexpansive mappings m. radhakrishnan a and s. rajesh b a ramanujan institute for advanced study in mathematics, university of madras, chennai 600 005, india. (radhariasm@gmail.com ) b department of mathematics, indian institute of technology, tirupati 517 506, india. (srajeshiitmdt@gmail.com) communicated by e. a. sánchez-pérez abstract kirk introduced the notion of pointwise eventually asymptotically nonexpansive mappings and proved that uniformly convex banach spaces have the fixed point property for pointwise eventually asymptotically nonexpansive maps. further, kirk raised the following question: “does a banach space x have the fixed point property for pointwise eventually asymptotically nonexpansive mappings whenever x has the fixed point property for nonexpansive mappings?”. in this paper, we prove that a banach space x has the fixed point property for pointwise eventually asymptotically nonexpansive maps if x has uniform normal sturcture or x is uniformly convex in every direction with the maluta constant d(x) < 1. also, we study the asymptotic behavior of the sequence {t nx} for a pointwise eventually asymptotically nonexpansive map t defined on a nonempty weakly compact convex subset k of a banach space x whenever x satisfies the uniform opial condition or x has a weakly continuous duality map. 2010 msc: 47h10; 47h09. keywords: fixed points; pointwise eventually asymptotically nonexpansive mappings; uniform normal structure; uniform opial condition; duality mappings. received 06 june 2018 – accepted 12 december 2018 http://dx.doi.org/10.4995/agt.2019.10360 m. radhakrishnan and s. rajesh 1. introduction let k be a nonempty weakly compact convex subset of a banach space x. a mapping t : k → k is said to be asymptotically nonexpansive if there exists a sequence {αn} ⊆ [1, ∞) with lim n→∞ αn = 1 such that for each integer n ≥ 1, (1.1) ‖t nx − t ny‖ ≤ αn‖x − y‖, for all x, y ∈ k. if αn = 1 in (1.1) for all n ∈ n, then t is said to be nonexpansive. if for each x ∈ k, the following inequality holds lim sup n→∞ ( sup y∈k {‖t nx − t ny‖ − ‖x − y‖} ) ≤ 0, then t is said to be asymptotically nonexpansive type. kirk [13] proved that if k is a nonempty weakly compact convex set in a banach space x with normal structure, then every nonexpansive map t on k has a fixed point. goebel and kirk [9] further proved that if x is a uniformly convex banach space, then every asymptotically nonexpansive map t on k has a fixed point. later, this result was extended to mappings of asymptotically nonexpansive type by kirk [14]. however, it remains open that whether normal structure condition on a banach space x guarantees the existence of fixed points of asymptotically nonexpansive mapping. kim and xu [12] proved that if x is a banach space with uniform normal structure, then every asymptotically nonexpansive map t on k has a fixed point. li and sims [8] proved the existence of fixed points of asymptotically nonexpansive type mappings in the setting of banach spaces having uniform normal structure. gossez and lami dozo [11] studied the class of spaces which satisfies opial’s condition and observed that all such spaces have normal structure. hence, if x is a banach space satisfying opial’s condition, then every nonexpansive map t on k has a fixed point. however, it is not clear whether opial’s condition implies the existence of fixed points for asymptotically nonexpansive mappings. in this direction, lin et. al [18] proved that every asymptotically nonexpansive map t on k has a fixed point whenever x is a banach space that satisfies the uniform opial condition. also, lim and xu [17] proved the existence of fixed points of an asymptotically nonexpansive map t on k in a banach space x whenever the maluta constant d(x) < 1 and t is weakly asymptotically regular on k. in 2008, kirk and xu [16] introduced the notion of pointwise asymptotically nonexpansive mappings and studied the existence of fixed points in the setting of uniformly convex banach spaces. definition 1.1 ([16]). a mapping t : k → k is said to be pointwise asymptotically nonexpansive if for each x ∈ k there exists a sequence {αn(x)} ⊆ [1, ∞) with lim n→∞ αn(x) = 1 such that for each integer n ≥ 1, ‖t nx − t ny‖ ≤ αn(x)‖x − y‖, for all y ∈ k. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 120 existence of fixed points for pointwise eventually asymptotically nonexpansive mappings theorem 1.2 ([16]). let k be a nonempty closed bounded convex subset of a uniformly convex banach space x and t : k → k be a pointwise asymptotically nonexpansive map. then t has a fixed point in k. recently, kirk [15] introduced the notion of pointwise eventually asymptotically nonexpansive mappings as follows: definition 1.3 ([15]). a mapping t : k → k is said to be pointwise eventually asymptotically nonexpansive if for each x ∈ k there exists a sequence {αn(x)} ⊆ [1, ∞) with lim n→∞ αn(x) = 1 and an integer n(x) ∈ n such that for n ≥ n(x), ‖t nx − t ny‖ ≤ αn(x)‖x − y‖, for all y ∈ k. though the definition of pointwise asymptotically nonexpansive mappings and pointwise eventually asymptotically nonexpansive mappings are quite simple to understand, examples of such mappings are rare. in [15] kirk proved that theorem 1.2 holds for pointwise eventually asymptotically nonexpansive mappings. further, kirk [15] raised the following question: does a banach space x have the fixed point property for pointwise eventually asymptotically nonexpansive mappings whenever x has the fixed point property for nonexpansive mappings? in this paper we give a partial answer to the above question. in section 3, we prove that a banach space x have the fixed point property for pointwise eventually asymptotically nonexpansive mappings if x has uniform normal structure or x is uniformly convex in every direction with the maluta constant d(x) < 1. in section 4, we study the asymptotic behavior of the sequence {t nx} for a pointwise eventually asymptotically nonexpansive map t defined on a nonempty weakly compact convex set k in a banach space x whenever x satisfies the uniform opial condition or x has a weakly continuous duality mapping. for results about asymptotic behavior of nonexpansive mappings and asymptotically nonexpansive mappings, one may refer to [2, 7, 17, 18]. 2. preliminaries let x be a banach space and c be the collection of all closed bounded convex sets in x. for k ∈ c, define (1) for x ∈ x, δ(x, k) = sup{‖x − y‖ : y ∈ k}; (2) r(k) = inf{δ(x, k) : x ∈ k} and (3) δ(k) = diam(k) = sup{δ(x, k) : x ∈ k}. definition 2.1 ([3]). a banach space x is said to have normal structure if every nonempty closed bounded convex subset k of x with diam(k) > 0 has a point x0 ∈ k such that δ(x0, k) < diam(k). also, brodskii and milman [3] gave a characterization for normal structure in terms of sequences as follows: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 121 m. radhakrishnan and s. rajesh theorem 2.2 ([3]). a banach space x does not have normal structure if and only if there exists a nonconstant bounded sequence {xn} in x such that lim n→∞ d (xn+1, co{x1, . . . , xn}) = diam({xn}). using this characterization of normal structure, maluta [19] defined the constant d(x) of a given banach space as follows: definition 2.3 ([19]). let x be a banach space. the maluta constant d(x) of x is defined as d(x) = sup    lim sup n→∞ d (xn+1, co{x1, . . . , xn}) diam({xn})    where the supremum is taken over all non-constant bounded sequences in x. it is known from [19] that a banach space x with d(x) < 1 has normal structure. in [5] bynum defined the concept of uniform normal structure as follows: definition 2.4 ([5]). a banach space x is said to have uniform normal structure if n(x) < 1, where n(x) = sup { r(k) δ(k) : k ∈ c with δ(k) > 0 } . remark 2.5. the following facts are known from [19]. (1) for a banach space x, 0 ≤ d(x) ≤ 1 and d(x) ≤ n(x). (2) if x is nonreflexive banach space, then d(x) = 1. thus if d(x) < 1 then x is reflexive. definition 2.6 ([20]). a banach space x is said to satisfy opial’s condition if for each weakly convergent sequence {xn} in x with limit x0 ∈ x, lim sup n→∞ ‖xn − x0‖ < lim sup n→∞ ‖xn − x‖, for all x ∈ x with x 6= x0. it is known that every hilbert space, finite dimensional banach spaces and the banach space lp(n) for 1 0, there exists an r > 0 such that 1 + r ≤ lim sup n→∞ ‖xn + x‖ for each x ∈ x with ‖x‖ ≥ c and each sequence {xn} in x such that w − lim n→∞ xn = 0 and lim sup n→∞ ‖xn‖ ≥ 1. further, prus [21] defined the opial modulus of x denoted by rx, as follows rx(c) = inf { lim sup n→∞ ‖xn + x‖ − 1 } , c© agt, upv, 2019 appl. gen. topol. 20, no. 1 122 existence of fixed points for pointwise eventually asymptotically nonexpansive mappings where c ≥ 0 and the infimum is taken over all x ∈ x with ‖x‖ ≥ c and sequences {xn} in x such that w − lim n→∞ xn = 0 and lim sup n→∞ ‖xn‖ ≥ 1. it is easy to see that x satisfies the uniform opial condition if and only if rx(c) > 0 for all c > 0. definition 2.8 ([10]). a continuous strictly increasing function ϕ : [0, ∞) → [0, ∞) is said to be gauge if ϕ(0) = 0 and lim t→∞ ϕ(t) = ∞. definition 2.9 ([10]). let x be a banach space and ϕ be a gauge function. then we associate with x a generalized duality map jϕ : x → 2 x ∗ defined by jϕ(x) = {x ∗ ∈ x∗ : 〈x, x∗〉 = ‖x‖ϕ(‖x‖) and ‖x∗‖ = ϕ(‖x‖)} , where 〈., .〉 is the pairing between x and x∗. define φ(t) = t ∫ 0 ϕ(s)ds for t ≥ 0. then it is easy to see that φ is a convex and gauge function on [0, ∞). also, it is known from [4] that jϕ(x) is the subdifferential of the convex function φ(‖.‖) at x ∈ x. definition 2.10 ([10]). a banach space x is said to have a weakly continuous duality map if there exists a gauge function ϕ such that the duality map jϕ is single-valued and continuous from x with the weak topology to x∗ with the weak∗ topology. it is clear from [10] that the banach space lp(n) for 1 < p < ∞ has a weakly continuous duality map with the gauge function ϕ(t) = tp−1. also, browder [4] proved that a banach space x with a weakly continuous duality map satisfies opial’s condition. 3. existence result in this section, we prove that a pointwise eventually asymptotically nonexpansive mappings defined on a weakly compact convex subset k of a banach space x always has a fixed point if x has uniform normal structure or x is uniformly convex in every direction with the maluta constant d(x) < 1. we use the following two lemmas in the sequel. lemma 3.1 ([6]). let x be a banach space with uniform normal structure. then for every bounded sequence {xn} in x there exists a point z ∈ ∞ ⋂ k=1 co{xn : n ≥ k} such that (1) lim sup n→∞ ‖xn − z‖ ≤ n(x)δ(co({xn})) (2) ‖z − y‖ ≤ lim sup n→∞ ‖xn − y‖, for all y ∈ x. lemma 3.2 ([22]). let k be a nonempty weakly compact convex subset of a banach space x and t : k → k be an asymptotically nonexpansive type mapping. let k0 be minimal with respect to being a nonempty closed convex subset of k such that for every x ∈ k0 we have ωw(x) ⊆ k0, where ωw(x) is c© agt, upv, 2019 appl. gen. topol. 20, no. 1 123 m. radhakrishnan and s. rajesh the set of all weak subsequential limit points of the sequence {t nx : n ∈ n}. then there exists a constant ρ0 ≥ 0 such that lim sup n→∞ ‖t nx − y‖ = ρ0 for all x, y ∈ k0. remark 3.3. we infer that if k is bounded, then a pointwise eventually asymptotically nonexpansive mapping on k is of asymptotically nonexpansive type. theorem 3.4. let k be a nonempty weakly compact convex subset of a banach space x with uniform normal structure and t : k → k be a pointwise eventually asymptotically nonexpansive map. then t has a fixed point in k. proof. assume that k0 is minimal with respect to being a nonempty closed convex subset of k with the property that for every x ∈ k0, ωw(x) ⊆ k0. by lemma 3.2, there is a constant ρ0 ≥ 0 such that lim sup n→∞ ‖t nx − y‖ = ρ0, for all x, y ∈ k0. first note that if ρ0 = 0, then lim n→∞ ‖t nx − x‖ = 0 and lim n→∞ ‖t nx − t x‖ = 0 for x ∈ k0, and it follows that k0 = {x} with t x = x. to see that ρ0 = 0 we break the proof into two assertions. assertion i : if {t nx} has a convergent subsequence for some x ∈ k0, then ρ0 = 0. proof. assume ρ0 > 0, and suppose that there exists a x ∈ k0 such that lim i→∞ t nix = y for some y ∈ k0, and choose r1 > 0 so that (1 + r1)n(x) < 1. since αn(y) → 1, there exists a natural number n1 ≥ n(y) such that αn(y) < 1 + r1, for all n ≥ n1. define f = co{t ny : n ≥ n1}. for l, m ∈ n with l > m ≥ n1, ‖t l(y) − t m(y)‖ = lim i→∞ ‖t l+ni(x) − t my‖ ≤ lim sup n→∞ ‖t nx − t my‖ ≤ lim sup n→∞ αm(y)‖t n−mx − y‖ = αm(y)ρ0 < (1 + r1)ρ0. this gives that δ(f) ≤ (1 + r1)ρ0. now by lemma 3.1, there exists a z ∈ f ∩ k0 such that ρ0 = lim sup n→∞ ‖t ny − z‖ ≤ n(x)δ(f) ≤ n(x)(1 + r1)ρ0 < ρ0. hence assertion i is proved. assertion ii : there exists a x ∈ k0 such that {t nx} has a convergent subsequence. proof. let x0 ∈ k0 and define d1 = co{t nx0 : n = 0, 1, 2, . . .}. by lemma 3.1, there exists a x1 ∈ d1 ∩ k0 such that 0 ≤ β1 = lim sup n→∞ ‖x1 − t n x0‖ ≤ n(x)δ(d1). choose r0 > 0 so that r = (1 + r0) 2n(x) < 1. since αn(x1) → 1, there exists a natural number n1 ≥ n(x1) such that αn(x1) < 1 + r0, for all n ≥ n1. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 124 existence of fixed points for pointwise eventually asymptotically nonexpansive mappings define d2 = co{t kn1(x1) : k = 0, 1, 2, . . .}. for k ≥ 1, we have ‖t kn1(x1) − t n(x0)‖ ≤ αkn1(x1)‖x1 − t n−kn1(x0)‖. letting lim sup n→∞ on both sides, we get lim sup n→∞ ‖t kn1(x1) − t n(x0)‖ ≤ αkn1 (x1) lim sup n→∞ ‖x1 − t n−kn1(x0)‖ = αkn1 (x1) lim sup n→∞ ‖x1 − t n(x0)‖ = αkn1 (x1)β1 and for l, m ∈ n with l > m, ‖t mn1(x1) − t ln1(x1)‖ ≤ αmn1(x1)‖x1 − t (l−m)n1(x1)‖ ≤ αmn1(x1) lim sup n→∞ ‖t (l−m)n1(x1) − t nx0‖ = αmn1(x1)α(l−m)n1(x1)β1 < (1 + r0) 2 β1 ≤ (1 + r0) 2n(x)δ(d1) = rδ(d1). this gives that δ(d2) ≤ rδ(d1). now by lemma 3.1, there exists a x2 ∈ d2 ∩ k0 such that 0 ≤ β2 = lim sup k→∞ ‖x2 − t kn1(x1)‖ ≤ n(x)δ(d2). since αkn1(x2) → 1 as k → ∞, we can choose k1 ∈ n such that n2 = k1n1 ≥ n(x2) and αkn1(x2) < 1 + r0, for all k ≥ k1. define d3 = co{t kn2(x2) : k = 0, 1, 2, . . .}. for l ≥ 1, we have ‖t ln2(x2) − t kn1(x1)‖ ≤ αln2(x2)‖x2 − t kn1−ln2(x1)‖ = αln2(x2)‖x2 − t (k−lk1)n1(x1)‖. this implies that lim sup k→∞ ‖t ln2(x2) − t kn1(x1)‖ ≤ αln2(x2)β2. so for l, m ∈ n with l > m, ‖t mn2(x2) − t ln2(x2)‖ ≤ αmn2(x2)‖x2 − t (l−m)n2(x2)‖ ≤ αmn2(x2) lim sup k→∞ ‖t (l−m)n2(x2) − t kn1(x1)‖ ≤ αmn2(x2)α(l−m)n2(x2)β2 < (1 + r0) 2β2 ≤ (1 + r0) 2 n(x)δ(d2) ≤ r2δ(d1). this gives that δ(d3) ≤ r 2δ(d1). by continuing the above process, we obtain a sequence {xm} and a sequence of sets {dm} with the following properties: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 125 m. radhakrishnan and s. rajesh (1) there exists a xm ∈ dm ∩ k0 such that 0 ≤ βm = lim sup k→∞ ‖xm − t knm−1(xm−1)‖ ≤ n(x)δ(dm) where dm = co{t knm−1(xm−1) : k = 0, 1, 2, . . .} and nm−1 = km−2nm−2 ≥ n(xm−1) for some km−2 ∈ n. (2) δ(dm) ≤ rδ(dm−1) and hence δ(dm) ≤ r m−1δ(d1). note that xm−1, xm ∈ dm and ‖xm − xm−1‖ ≤ δ(dm) ≤ r m−1δ(d1). this implies that {xm} is a cauchy sequence in k0. thus, there exists a x ∈ k0 such that x = lim m→∞ xm. now, for k ≥ n(x), ‖t kx − x‖ ≤ ‖t kx − t kxm‖ + ‖t kxm − x‖ ≤ αk(x)‖x − xm‖ + ‖t kxm − xm+1‖ + ‖xm+1 − x‖ letting lim inf k→∞ on both sides, we get lim inf k→∞ ‖t kx − x‖ ≤ ‖x − xm‖ + lim inf k→∞ ‖t kxm − xm+1‖ + ‖xm+1 − x‖ ≤ ‖x − xm‖ + lim inf k→∞ ‖t knmxm − xm+1‖ + ‖xm+1 − x‖ ≤ ‖x − xm‖ + βm+1 + ‖xm+1 − x‖. as βm → 0 and xm → x, we have lim inf k→∞ ‖t kx − x‖ = 0. thus, there exists a subsequence {t nix} of {t nx} such that lim i→∞ t nix = x. hence assertion ii is proved. therefore ρ0 = 0 and k0 is singleton. hence t has a fixed point in k. � next, we use the ultrafilter techniques to prove the existence of fixed points for a pointwise eventually asymptotically nonexpansive mapping t on k in a banach space x with the maluta constant d(x) < 1. this result (theorem 3.6) is motivated by the following result of lim and xu [17]: if t is an asymptotically nonexpansive map defined on a nonempty weakly compact convex set k in a banach space with the maluta constant d(x) < 1 and t is weakly asymptotically regular on k, then t has a fixed point in k. in the following remark, we recall some facts about ultrafilter which are used in the proof of theorem 3.6. for more information about ultrafilters, one may refer to [1, 10]. remark 3.5 ([1]). a filter f on n is a nonempty collection of subsets of n satisfying (1) if a, b ∈ f then a ∩ b ∈ f; (2) if a ∈ f and a ⊆ b then b ∈ f. if f is a filter on n and ∅ ∈ f, then f = 2n and is called an improper filter. by usual set inclusion, the family p of all proper filters on n is a partially ordered set and every chain in p has an upper bound. by zorn’s lemma, we get a maximal proper filter in p. a maximal filter in p is called an ultrafilter on n. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 126 existence of fixed points for pointwise eventually asymptotically nonexpansive mappings a sequence {xn} in a banach space x converges to x ∈ x over the filter f if for every neighbourhood v of x, the set {n ∈ n : xn ∈ v } belongs to f and it is denoted by lim f xn = x. a trivial ultrafilter fn0 on n is the collection of subsets of n which contains an element n0 ∈ n, where n0 ∈ n is fixed and all other ultrafilters on n are said to be non-trivial. it is known that nontrivial ultrafilters always exist (zorn’s lemma), and a sequence {xn} in a compact set always converges relative to any nontrivial ultrafilter over n. theorem 3.6. let k be a nonempty weakly compact convex subset of a banach space x with the maluta constant d(x) < 1 and t : k → k be a pointwise eventually asymptotically nonexpansive map. further, assume that t is weakly asymptotically regular on k ( i.e., w − lim n→∞ (t nx − t n+1x) = 0 for all x ∈ k ) . then t has a fixed point in k. proof. let u be a non-trivial ultrafilter on n. define a mapping s on k by s(x) = w−lim u t nx, for x ∈ k. since k is weakly compact, s(x) is well defined and there exists a subsequence {t nix} of {t nx} such that {t nix} converges weakly to s(x). for x, y ∈ k, we have ‖sx − sy‖ ≤ lim inf k→∞ ‖t nkx − t nky‖ ≤ lim sup n→∞ ‖t nx − t ny‖ ≤ lim sup n→∞ αn(x)‖x − y‖ = ‖x − y‖. hence s is a nonexpansive map on k. since d(x) < 1, x has normal structure. therefore, the nonexpansive map s : k → k has a fixed point in k. that is, there exists a x ∈ k such that w − lim u t nx = x. this implies that there exists a subsequence {t n ′ k(x)} of {t nx} converges weakly to x. without loss of generality, we may assume that n′ k ≥ n(x) for all k ∈ n. choose r > 0 so that (1+r)2d(x) < 1. since αn(x) → 1 as n → ∞, we can find a subsequence {ni} of {n ′ k} such that αni(x) < 1 + r and αni+1−ni(x) < 1 + r for all i ∈ n. from the definition of d(x), lim sup i→∞ ‖t nix − x‖ ≤ d(x)δ({t nix}). since t is weakly asymptotically regular at x ∈ k, for fixed i > j it must be the case that t nt+(ni−nj)(x) converges weakly to x as t → ∞. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 127 m. radhakrishnan and s. rajesh now observe that ‖t nix − t nj x‖ ≤ αnj (x)‖t ni−nj (x) − x‖ ≤ (1 + r) lim sup t→∞ ‖t ni−nj (x) − t nt+(ni−nj)(x)‖ ≤ (1 + r)αni−nj (x) lim sup t→∞ ‖x − t nt(x)‖ ≤ (1 + r)2 lim sup t→∞ ‖x − t nt(x)‖ therefore lim sup i→∞ ‖x − t ni(x)‖ ≤ (1 + r)2d(x) lim sup t→∞ ‖x − t nt(x)‖ and since (1 + r)2d(x) < 1, we conclude lim sup i→∞ ‖x − t ni(x)‖ = 0. as t m is continuous at x for all m ≥ n(x), {t ni+m(x)} converges weakly to t m(x) and since t is weakly asymptotically regular at x this in turn implies x = t m(x) for all m ≥ n(x). therefore x = t m+1x = t (t mx) = t x. � theorem 3.7. let k be a nonempty weakly compact convex subset of a banach space x, which is uniformly convex in every direction and t : k → k be a pointwise eventually asymptotically nonexpansive map. further, assume that the maluta constant d(x) < 1. then t has a fixed point in k. proof. assume that k0 is minimal with respect to being a nonempty closed convex subset of k with the property that for every x ∈ k0, ωw(x) ⊆ k0. by lemma 3.2, there is a constant ρ0 ≥ 0 such that lim sup n→∞ ‖t nx − y‖ = ρ0, for all x, y ∈ k0. since x is uniformly convex in every direction, k0 is singleton, say k0 = {x} and so the sequence {t nx} converges weakly to x. now choose r > 0 so that (1+r)2d(x) < 1. since αn(x) → 1 as n → ∞, we can find a subsequence {ni} of {n(x), n(x) + 1, . . . } such that αni(x) < 1 + r and αni+1−ni(x) < 1 + r for all i ∈ n. from the definition of d(x), lim sup i→∞ ‖x − t nix‖ ≤ d(x)δ({t nix}). now for i > j, we have ‖t nix − t nj x‖ ≤ αnj (x)‖t ni−nj (x) − x‖ ≤ (1 + r) lim sup t→∞ ‖t ni−nj (x) − t nt+(ni−nj)(x)‖ ≤ (1 + r)αni−nj (x) lim sup t→∞ ‖x − t nt(x)‖ ≤ (1 + r)2 lim sup t→∞ ‖x − t nt(x)‖ thus, lim sup i→∞ ‖x − t ni(x)‖ ≤ (1 + r)2d(x) lim sup t→∞ ‖x − t nt(x)‖. since (1 + r)2d(x) < 1, we have lim sup i→∞ ‖x − t ni(x)‖ = 0. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 128 existence of fixed points for pointwise eventually asymptotically nonexpansive mappings finally, by the definition of t, we have {t ni+m(x)} converging weakly to t m(x) for all m ≥ n(x). but, we know that the sequence {t nx} converges weakly to x. this implies that x = t m(x) for all m ≥ n(x). hence x is a fixed point of t. � 4. asymptotic behavior in this section, we investigate the asymptotic behavior of the sequence {t nx} for a pointwise eventually asymptotically nonexpansive map t defined on a nonempty weakly compact convex subset k of a banach space x whenever x satisfies the uniform opial condition or x has a weakly continuous duality map. lemma 4.1. let k be a nonempty weakly compact convex subset of a banach space x satisfying opial’s condition and t : k → k be a pointwise eventually asymptotically nonexpansive map. further, assume that t is weakly asymptotically regular at some x ∈ k. for m ∈ n, define bm = lim sup i→∞ ‖t ni+m(x) − y‖ where y = w − lim i→∞ t nix. then the sequence {bm} converges. proof. let y = w − lim i→∞ t nix. since t is weakly asymptotically regular at x ∈ k, we have y = w − lim i→∞ t ni+m(x) for m ∈ n. for any j ≥ n(y), we have bm+j = lim sup i→∞ ‖t ni+m+j(x) − y‖ ≤ lim sup i→∞ ‖t ni+m+j(x) − t jy‖ ≤ αj(y) lim sup i→∞ ‖t ni+m(x) − y‖ = αj(y)bm. we claim that lim m→∞ bm exists. note that there exists two subsequences {mi} and {m′i} of n such that lim i→∞ bmi = lim sup m→∞ bm and lim i→∞ bm′ i = lim inf m→∞ bm. this gives that there exists k0 ∈ n such that mj > m ′ 1 +n(y), for all j ≥ k0. i.e., mj = m ′ 1 + n(y) + nj, for some nj ∈ n. for any j ≥ k0, we have bmj = bm′1+n(y)+nj ≤ αn(y)+nj (y)bm′1 = αmj−m′1(y)bm ′ 1 so lim sup m→∞ bm = lim j→∞ bmj ≤ bm′1. similarly, for each i ∈ n, we have lim sup m→∞ bm ≤ bm′ i and we get lim sup m→∞ bm ≤ lim inf m→∞ bm. hence the sequence {bm} converges. � theorem 4.2. let k be a nonempty weakly compact convex subset of a banach space x satisfying the uniform opial condition and t : k → k be a pointwise c© agt, upv, 2019 appl. gen. topol. 20, no. 1 129 m. radhakrishnan and s. rajesh eventually asymptotically nonexpansive map. then given an x ∈ k, {t nx} converges weakly to a fixed point of t if and only if t is weakly asymptotically regular at x. proof. if {t nx} converges weakly to a fixed point of t, then {t nx} is obviously weakly asymptotically regular at x. conversely, assume that t is weakly asymptotically regular at x ∈ k. then we claim that ωw(x) ⊆ f(t ), where f(t ) is the set of all fixed point of t and ωw(x) is singleton. to see that ωw(x) ⊆ f(t ), let y ∈ ωw(x). then there exists a subsequence {t nix} of {t nx} such that y = w− lim i→∞ t nix. since t is weakly asymptotically regular at x, we have y = w − lim i→∞ t ni+m(x) for m ∈ n. now, let bm = lim sup i→∞ ‖t ni+m(x) − y‖. by lemma 4.1, the sequence {bm} converges to b ≥ 0. assume b = 0. for m ≥ n(y), we have ‖t my − y‖ ≤ ‖t my − t ni+2m(x)‖ + ‖t ni+2m(x) − y‖ ≤ lim sup i→∞ ‖t my − t ni+2m(x)‖ + lim sup i→∞ ‖t ni+2m(x) − y‖ ≤ αm(y)bm + b2m. letting m → ∞ on both sides, we get t my → y. now, for m ≥ n ≥ n(y), we have ‖t my − t ny‖ ≤ αn(y)‖y − t m−ny‖. it follows that if n ≥ n(y), then t my → t ny as m → ∞. this implies that y = t ny, for n ≥ n(y) and hence t y = y. now, suppose b > 0 and let z (m) i = t ni+m(x) − y bm . then for each m ≥ 0, w − lim i→∞ z (m) i = 0 and lim sup i→∞ ‖z (m) i ‖ = 1. by the uniform opial condition of x, we have 1 + rx(c) ≤ lim sup i→∞ ‖z (m) i + z‖, for all z ∈ x with ‖z‖ ≥ c. take z = y − t my b2m . then for m ≥ n(y), 1 + rx ( ‖y − t my‖ b2m ) ≤ lim sup i→∞ ∥ ∥ ∥ ∥ t ni+2m(x) − t my b2m ∥ ∥ ∥ ∥ and hence b2m ( 1 + rx ( ‖y − t my‖ b2m )) ≤ lim sup i→∞ ‖t ni+2m(x) − t my‖ ≤ αm(y)bm. letting m → ∞, we get b  1 + rx   lim sup m→∞ ‖y − t my‖ b     ≤ b; rx   lim sup m→∞ ‖y − t my‖ b   ≤ 0. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 130 existence of fixed points for pointwise eventually asymptotically nonexpansive mappings since x satisfies the uniform opial condition, we have lim m→∞ ‖t my − y‖ = 0. this implies that t y = y and thus ωw(x) ⊆ f(t ). now, it remains to prove that ωw(x) is a singleton. first to observe that lim n→∞ ‖t nx−p‖ exists for every p ∈ f(t ). for this, let p ∈ f(t ). for m ≥ n(p) and n ≥ 1, we have ‖t n+mx − p‖ = ‖t n+mx − t mp‖ ≤ αm(p)‖t nx − p‖ letting lim sup m→∞ on both sides, we get lim sup m→∞ ‖t mx − p‖ ≤ ‖t nx − p‖ for all n ≥ 1. thus, we have lim n→∞ ‖t nx − p‖ exists for all p ∈ f(t ). suppose that there exists two subsequences {ni} and {mi} of n such that w − lim i→∞ t ni(x) = p1 and w − lim i→∞ t mi(x) = p2 for p1 6= p2. lim n→∞ ‖t nx − p1‖ = lim i→∞ ‖t ni(x) − p1‖ < lim i→∞ ‖t ni(x) − p2‖ = lim i→∞ ‖t mi(x) − p2‖ < lim i→∞ ‖t mi(x) − p1‖ = lim n→∞ ‖t nx − p1‖ which is a contradiction. hence ωw(x) is singleton and the sequence {t nx} converges weakly to a fixed point of t. � theorem 4.3. let k be a nonempty weakly compact convex subset of a banach space x with a weakly continuous duality map jϕ and t : k → k be a pointwise eventually asymptotically nonexpansive map. then given an x ∈ k, {t nx} converges weakly to a fixed point of t if and only if t is weakly asymptotically regular at x. proof. if {t nx} converges weakly to a fixed point of t, then {t nx} is obviously weakly asymptotically regular at x. conversely, assume that t is weakly asymptotically regular at x ∈ k. then we claim that ωw(x) ⊆ f(t ) and ωw(x) is singleton. since x has weakly continuous duality map jϕ, so it satisfies opial’s condition. by theorem 4.2, it is enough to show that wω(x) ⊆ f(t ). let y ∈ ωw(x). then there exists a subsequence {t nix} of {t nx} such that y = w − lim i→∞ t nix. since t is weakly asymptotically regular at x, we have y = w − lim i→∞ t ni+m(x) for m ∈ n. let bm = lim sup i→∞ ‖t ni+m(x) − y‖. by lemma 4.1, the sequence {bm} converges to b. for m ≥ n(y), we have c© agt, upv, 2019 appl. gen. topol. 20, no. 1 131 m. radhakrishnan and s. rajesh φ ( ‖t ni+2m(x) − y‖ ) = φ ( ‖t ni+2m(x) − t my + t my − y‖ ) = φ ( ‖t ni+2m(x) − t my‖ ) + 1 ∫ 0 〈 t my − y, jϕ(t ni+2m(x) − t my + t(t my − y)) 〉 dt ≤ φ ( αm(y)‖t ni+m(x) − y‖ ) + 1 ∫ 0 〈 t my − y, jϕ(t ni+2m(x) − t my + t(t my − y)) 〉 dt letting lim sup i→∞ on both sides, we get φ (b2m) ≤ φ (αm(y)bm) + 1 ∫ 0 〈t my − y, jϕ(y − t my + t(t my − y))〉 dt = φ (αm(y)bm) − 1 ∫ 0 ‖y − t my‖ϕ ((1 − t)‖y − t my‖) dt = φ (αm(y)bm) − φ (‖y − t m y‖) letting lim sup m→∞ on both sides, we get φ ( lim sup m→∞ ‖y − t my‖ ) ≤ φ(b) − φ(b). thus, t my → y as m → ∞ and t y = y. hence ωw(x) ⊆ f(t ) and the sequence {t nx} converges weakly to a fixed point of t. � acknowledgements. the authors would like to thank the anonymous referee for the comments and suggestions. the first author acknowledges the university grants commission, new delhi, for providing financial support in the form of project fellow through ramanujan institute for advanced study in mathematics, university of madras, chennai. c© agt, upv, 2019 appl. gen. topol. 20, 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[22] h. k. xu, existence and convergence for fixed points of mappings of asymptotically nonexpansive type, nonlinear anal. 16 (1991), 1139–1146. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 133 @ appl. gen. topol. 22, no. 2 (2021), 331-344doi:10.4995/agt.2021.14602 © agt, upv, 2021 disconnection in the alexandroff duplicate papiya bhattacharjee a, michelle l. knoxb and warren wm. mcgovern c,1 a florida atlantic university, boca raton, fl 33431, usa (pbhattacharjee@fau.edu) b midwestern state university, wichita falls, tx 76308, usa (michelle.knox@msutexas.edu) c h. l. wilkes honors college, florida atlantic university, jupiter, fl 33458, usa (warren.mcgovern@fau.edu) communicated by m. a. sánchez-granero abstract it was demonstrated in [2] that the alexandroff duplicate of the čechstone compactification of the naturals is not extremally disconnected. the question was raised as to whether the alexandroff duplicate of a non-discrete extremally disconnected space can ever be extremally disconnected. we answer this question in the affirmative; an example of van douwen is significant. in a slightly different direction we also characterize when the alexandroff duplicate of a space is a p-space as well as when it is an almost p-space. 2010 msc: 54f65; 54d15. keywords: extremally disconnected space; alexandroff duplicate; p-space. 1. introduction in 1929, alexandroff [1] constructed a non-metrizable first countable compact hausdorff space. alexandroff’s construction was generalized by engelking [6] in 1966, and since then the construction has been a constant force in the supply of counter-examples to topological questions. our aim here is to consider some well-known peculiar topological properties and classify when the alexandroff duplicate of a space x has such properties. throughout, we assume that x is a tychonoff space, that is, completely regular and hausdorff. for such 1corresponding author received 10 november 2020 – accepted 08 july 2021 http://dx.doi.org/10.4995/agt.2021.14602 p. bhattacharjee, m. l. knox and w. wm. mcgovern a space x, c(x) denotes the collection of real-valued continuous functions on x. since x is tychonoff, it is also the case that a(x) is tychonoff (see [5]). definition 1.1. let x be a topological space. the alexandroff duplicate of x is the space created by taking two (disjoint) copies of x, say a(x) = x ∪ x′. for any subset t ⊆ x, we let t ′ be its copy in x′. then we define basic open sets of a(x) to be the singletons of x′ as well as any subset of the form o ∪ o′ r {x′} for some open subset o ⊆ x and x ∈ x. (the space a(x) is also known as the alexandroff double.) observe that x′ is open in a(x), so x is a closed subspace of a(x). the space a(x) always has a dense set of isolated points. also, x has no isolated points if and only if x′ is dense in a(x). moreover, every f ∈ c(x) has a continuous extension to all of a(x), that is, x is c-embedded in a(x). simply extend f to x′ by assigning to each x′ ∈ x′ the value of its original x ∈ x. since there will be a lot of movement between the spaces x and a(x), we use clt to denote the closure of t in x and cl z to denote the closure of a subset z ⊆ a(x). given a subset t of x, we use acc(t) to denote the derived set of t , that is, the set of accumulation points of t (i.e. points x ∈ x such that x ∈ cl(t r {x})). if needed, we shall add a subscript, as in accx(t). recall that t is closed if and only if acc(t) ⊆ t , and furthermore, for any set t , clt = t ∪ acc(t). the condition acc(t) = ∅ is equivalent to saying that t is closed and discrete. lemma 1.2. let x be a space. for any subset t ⊆ x. cl t ′ = t ′ ∪accx(t). subsets of x which are closed and discrete are of utmost importance in the study of clopen subsets of a(x). for the sake of convenience, we call a set that is both closed and discrete disclosed. in this article we shall use the term σ-disclosed to describe a countable union of disclosed subsets. it is straightforward to show that a σ-disclosed subset is a countable union of disclosed subsets which are pairwise disjoint. clearly, every countable set is σ-disclosed, and hence it is obvious that σ-disclosed sets need not be closed. in a lindelöf space, the σ-disclosed subsets are precisely the countable subsets. in the second section, we answer the question raised in [2] by characterizing those x for which a(x) is extremally disconnected. an example of van douwen [8] is used to show that there are crowded such x, see example 2.5. in the last section we consider other peculiar properties such as basically disconnected spaces, (weak) p-spaces, and almost p-spaces. we end this section with a characterization of open (and clopen) subsets of a(x), which will be helpful in later sections. proposition 1.3. a subset of a(x) is clopen if and only if it is of the form k ∪ (k′ r c′) ∪ d′ where k is a clopen subset of x, c ⊆ k, d ∩ k = ∅, and both c and d are disclosed. © agt, upv, 2021 appl. gen. topol. 22, no. 2 332 disconnection in the alexandroff duplicate proof. first, let y = k ∪ (k′ r c′) ∪ d′ where k is a clopen subset of x, c ⊆ k, and d ∩ k = ∅, and both satisfy accx(c) = ∅ = accx(d). then d ′ and c′ are clopen in a(x) by lemma 1.2 since accx(c) = ∅ = accx(d), and k ∪k′ is clopen in a(x) by lemma 1.2. hence k ∪(k′ rc′) = (k ∪k′)rc′ is clopen in a(x) and so is y . now, let y be a clopen subset of a(x). if y ⊆ x′, then simply let k = c = ∅ and d′ = y . otherwise, let k = y ∩ x, so k is clopen in x. it follows that k ∪ k′ is clopen in a(x). let d′ = (y ∩ x′) r k′, then d∩k = ∅ since d′ ∩k′ = ∅. we need to show that accx(d) = ∅, so consider cl d′ = d′ ∪ accx(d). if there is some x ∈ accx(d), then x ∈ acca(x) y ∩ x implies that x ∈ k. but k ∪ k′ is an open neighborhood of x disjoint from d′, a contradiction. thus accx(d) = ∅. next, let c′ = k′ r y , and observe that c′ is open in a(x) with c ⊆ k since c′ ⊆ k′. we also want to show that accx(c) = ∅. note that cl c′ = c′ ∪ accx(c), and suppose there exists x ∈ accx(c). but x /∈ cl c ′ because y is an open neighborhood of x disjoint from c′ in a(x). therefore accx(c) = ∅. finally, it is clear that k ∪ (k′ r c′) ∪ d′ = k ∪ (k′ r (k′ r y )) ∪ d′ = k ∪ (k′ ∩ y ) ∪ [(y ∩ x′) r k′] = y. � in generalizing from clopen subsets to open subsets the proof of the following should be evident. lemma 1.4. let u ⊆ a(x) be an open subset of a(x) and set v = (u ∩x)∪ (u ∩ x)′. then v r u = c′ for some c ⊆ u ∩ x which is discrete in x and closed in u ∩ x. furthermore, u r v = d′ for some d ⊆ x with the property that for each subset e ⊆ u which is closed in a(x), the set {x ∈ d : x′ ∈ e} is disclosed in x. 2. extremally disconnected spaces recall that x is extremally disconnected (e.d. for short) if the closure of each open set is clopen. this is equivalent to saying that no point is in the simultaneous closure of two disjoint open subsets. the compact e.d. spaces are precisely the stone duals of complete boolean algebras. a space x is e.d. if and only if βx, the čech-stone compactification of x, is e.d. locally, a point x ∈ x is said to be an extremally disconnected point if it is not in the closure of two disjoint open sets. in [2] the authors demonstrate that a(βn) is not an extremally disconnected space. the authors raise the question of whether there is a non-discrete space whose alexandroff duplicate is extremally disconnected. in our attempt to © agt, upv, 2021 appl. gen. topol. 22, no. 2 333 p. bhattacharjee, m. l. knox and w. wm. mcgovern answer this question, we created a condition on a topological space that characterized the situation: definition 2.1. we then set off to find a non-discrete example. in looking for such a space we happened upon a particular piece of work of van douwen ([8]). not only was this article invaluable to our efforts, moreover, we realized that our “new” condition had already been discovered and studied by van douwen. (the reader should be aware that in [8] it is not assumed that spaces are tychonoff.) definition 2.1. we call a point x ∈ x perfectly disconnected in x if it is not simultaneously an accumulation point of two disjoint subsets. if every point of x is perfectly disconnected, then in line with definition 1.3 of [8], we say that x is perfectly disconnected. notice that every perfectly disconnected point is extremally disconnected. recall that a space is hereditarily extremally disconnected if every subspace is extremally disconnected. we prove that a perfectly disconnected space is hereditarily extremally disconnected. proposition 2.2. suppose x is perfectly disconnected. let t ⊆ x. if x ∈ x is an accumulation point of t, then it belongs to the interior of clt. it follows that for all t ⊆ x, clt r t ⊆ int(clt). moreover, a perfectly disconnected space is hereditarily extremally disconnected. proof. suppose x is perfectly disconnected and x ∈ acc(t). notice that x /∈ acc(x rt), so there exists an open neighborhood o of x such that (or{x})∩ (x r t) = ∅. it follows that o ⊆ clt . as for the claim that a perfectly disconnected space is hereditarily extremally disconnected, consider that in [4, proposition 3.1], the authors prove that a topological space is hereditarily extremally disconnected if and only if for all a,b ⊆ x cl(a r clb) ∩ cl(b r cla) = ∅. notice that if x ∈ (a r clb), then x /∈ (b r cla). furthermore, it is always the case that (a r clb) ∩ (b r cla) = ∅. therefore, if x is perfectly disconnected, then the above display must be empty. � example 2.3. it is not true that every hereditarily extremally disconnected is perfectly disconnected. in [4], the authors construct countable hereditarily extremally disconnected crowded spaces as follows. there are countable dense subspaces of e([0,1]), where (e([0,1]),π) is the absolute of the unit interval. (the absolute is also known as the gleason cover of x. in particular, ex denotes the stone space of the complete boolean algebra of regular open subsets of x.) these subspaces are dense and hence both extremally disconnected and crowded. for example, if {qn} = q = q ∩ [0,1] then by choosing xn ∈ π −1(qn) produces such a space. let q1 and q2 be disjoint countable dense subspaces of © agt, upv, 2021 appl. gen. topol. 22, no. 2 334 disconnection in the alexandroff duplicate [0,1] and construct e1 and e2 in this manner. let x = e1 ∪ e2, a countable hereditarily extremally disconnected crowded space. notice that each point x ∈ x satisfies x ∈ cle1 r {x} ∩ cle2 r {x}, whence x is not perfectly disconnected. we are now able to prove the main theorem of the section. theorem 2.4. the following statements are equivalent. (1) the space a(x) is extremally disconnected. (2) x is perfectly disconnected and the set of isolated points of x is clopen. (3) x is the topological sum of a discrete space and a crowded perfectly disconnected space. proof. (1) ⇒ (2). clearly, x must be e.d. if a(x) is e.d. since x is cembedded in x. the space x must have the even stronger property that no point of x can be in the closure of two disjoint subsets of x since if it were, say x ∈ clt1 ∩ clt2 and t1 ∩ t2 = ∅, then x ∈ cl t ′ 1 ∩ cl t ′ 2 with both t ′ 1 and t ′2 open in a(x) and disjoint. to show that the set is(x) of isolated points of x is clopen, let p ∈ cl is(x)r is(x) then p ∈ cl is(x). but p ∈ cl x′ ∩ cl is(x) contradicts the fact that a(x) is e.d. therefore, is(x) is clopen in x. (2) ⇔ (3). note that if is(x) is clopen, then we can separate x into two clopen subsets: x ris(x) and is(x). this separation induces a clopen separation of a(x) each of which is e.d. furthermore, it follows that is(x) is discrete and x r is(x) is crowded. thus (2) ⇔ (3). (2) ⇒ (1). without loss of generality, we assume that x is a perfectly disconnected crowded space. suppose p ∈ a(x) is not an e.d. point of a(x). clearly, p ∈ x. then there are disjoint open subsets of a(x), say o1,o2, such that p ∈ cl o1 ∩ cl o2. there are collections u1,u2 of basic open subsets of a(x) such that both u1,u2 are pairwise disjoint and ∪ ui ⊆ oi is dense. since p ∈ cl ∪ u1 ∩ cl ∪ u2, it follows that without loss of generality we can assume that oi = ∪ ui. now for appropriate index sets i1 and i2, basic open sets oα ∪ o ′ α r {x ′ α} and oβ ∪ o ′ β r {x′ β }, and disjoint subsets t1,t2 ⊆ x o1 = ⋃ α∈i1 (oα ∪ o ′ α r {x ′ α}) ∪ t ′ 1 and o2 = ⋃ β∈i2 (oβ ∪ o ′ β r {x ′ β}) ∪ t ′ 2. let o1 = ⋃ α∈i1 oα and o2 = ⋃ β∈i2 oβ, which are clearly are disjoint open subsets of x. that p ∈ cl o1 means that either p ∈ clo1 or p ∈ clt1. similarly, either p ∈ clo2 or p ∈ clt2. since p is not simultaneously in the closure of two disjoint subsets of x it follows that either p ∈ clo1 and p ∈ clt2, or p ∈ clo2 and p ∈ clt1. we consider the case that p ∈ clo1 and p ∈ clt2. observe that not only is it entirely possible that t2 ∩ o1 6= ∅, but in fact this must be the case. let t3 = t2 ∩ {xα}α∈i1 and observe that p ∈ clt3 since otherwise would © agt, upv, 2021 appl. gen. topol. 22, no. 2 335 p. bhattacharjee, m. l. knox and w. wm. mcgovern lead us to conclude that p is in the closure of two disjoint subsets of x. so, we can replace t2 with t3. since u1 is pairwise disjoint it follows that t3 is discrete; this fact is pivotal. let j = {α ∈ i1 : xα ∈ t3}. define vα = oα r {xα} and let v = ∪α∈jvα. the crucial piece is that v ∩ t3 = ∅. recall that x is crowded, therefore each xα ∈ clvα. finally, we can now see that p ∈ clv and p ∈ clt3 even though v ∩ t3 = ∅, the desired contradiction. � example 2.5. e. van douwen [8] constructed a perfectly disconnected crowded tychonoff space. (this construction was generalized in [10].) the construction starts with a countable crowded tychonoff space, say the rationals, and then enlarges the topology to a maximal regular topology; this means maximal amongst the regular topologies. call this new space x; this space happens to also be hausdorff as it is larger than a hausdorff topology. he then takes θ to be the set of points of x which do not lie in the closure of any nowhere dense subset. he shows this set is non-empty, crowded, and perfectly disconnected. consequently, a(θ) is extremally disconnected. e. van douwen’s examples serves as an affirmative answer to the question raised in [2] of whether there is a non-discrete space x for which a(x) is extremally disconnected. it is worth pointing out that applying the alexandroff duplicate twice to any non-discrete space never yields an extremally disconnected space. proposition 2.6. for a space x, the following are equivalent. (1) a(a(x)) is extremally disconnected. (2) a(x) is perfectly disconnected. (3) x is discrete. proof. if a(a(x)) is e.d., then a(x) is perfectly disconnected by theorem 2.4. if x ∈ a(x) is not isolated then x ∈ x is not isolated in x. in this case x ∈ cl a(x)(x r {x}) and x ∈ cl a(x)x ′, contradicting that a(x) is perfectly disconnected. so every point of a(x) and thus every point of x is isolated. � 3. basically disconnected spaces as the title of this section suggests, we are interested in determining which spaces x have basically disconnected duplicates. recall that a space x is basically disconnected if the closure of every cozero-set is clopen. a quick recap is in order. for f ∈ c(x), z(f) = {x ∈ x : f(x) = 0} and coz(f) = x r z(f) denote the zero-set and cozero-set of f, respectively. a subset v of x is a zero-set if v = z(f) for some f ∈ c(x), and similarly for cozero-sets. each zeroset (cozero-set) is closed (open), and since x is tychonoff, the collection of zero-sets (cozero-sets) forms a base for the topology of closed (open) sets. it is well known that x is basically disconnected if and only if βx is basically disconnected. compact basically disconnected spaces are precisely the stone © agt, upv, 2021 appl. gen. topol. 22, no. 2 336 disconnection in the alexandroff duplicate duals of σ-complete boolean algebras. locally, a point p ∈ x is called a basically disconnected point (or a b.d. point for short) if whenever o and c are disjoint open sets and c is a cozero, then p /∈ clo ∩ clc. a nice source of basically disconnected spaces are p-spaces. recall that a tychonoff space is a p-space when every cozero-set is clopen. this is equivalent to saying that the topology of open subsets of x is closed under countable intersections. this equivalent condition lends itself to generalization. a space x is said to be a pκ-space whenever its topology of open subsets is closed under the intersection of fewer than κ-many open set; p-space and pℵ1-space mean the same thing. it follows that any pκ-space has the property that every subset of size less than κ is closed. spaces with this latter property are called weak pκ-spaces. it ought to be clear that x is a weak pκ-space if and only if a(x) is a weak pκ-space. to see this note that any subspace of a (weak) pκ-space if again a (weak) pκ-space. conversely, any subset of a(x) of size less than κ must be of the form t ∪ s′ where both t and s are of size less than κ. we leave it to the interested reader to check that if x is pκ-space, then any intersection in a(x) of fewer than κ-many open sets is again open. we summarize as follows. proposition 3.1. the space x is a (weak) pκ-space if and only if a(x) is a (weak) pκ-space. we turn to characterizing when a(x) is basically disconnected. to aid us, the next lemma is useful in that it describes what a cozero-set of a(x) looks like in terms of a cozero-set of x and certain subsets of x′. lemma 3.2. let f ∈ c(a(x)). then there exist some f ∈ c(x), some discrete subset c ⊆ coz(f) such that (clc r c) ∩ coz(f) = ∅, and some d ⊆ z(f) which is σ-disclosed in x, such that coz(f) = coz(f) ∪ [coz(f)′ r c′] ∪ d′. proof. we apply lemma 1.4 to the open set coz(f). in this case, coz(f) ∩ x = coz(f) where f is the restriction of f to x; so f ∈ c(x). thus, there exists a subset c ⊆ coz(f) which is discrete and closed in x and a subset d ⊆ x which is disjoint from coz(f) and has the property that for each subset e ⊆ coz(f) which is closed in a(x), the set {x ∈ d : x′ ∈ e} is disclosed in x. furthermore, coz(f) = coz(f) ∪ [coz(f)′ r c′] ∪ d′. obviously, d ⊆ z(f). so all that is left to be shown is that d is σ-disclosed in x. let t ′n = [f −1([ 1 n ,∞)) ∩ x′] r coz(f)′ for each n ∈ n, and observe that d′ = ∞ ⋃ n=1 t ′n = [coz(f) ∩ x ′] r coz(f)′. this implies that d is the union of the tn. we claim accx tn = ∅ for all n. suppose x ∈ accx tn, then x ∈ coz(f) since x ∈ cl t ′ n and f(y ′) ≥ 1 n for all © agt, upv, 2021 appl. gen. topol. 22, no. 2 337 p. bhattacharjee, m. l. knox and w. wm. mcgovern y′ ∈ t ′n. moreover, f(y ′) /∈ coz(f)′ for every y′ ∈ t ′n, so f(y) = 0 implies x ∈ z(f), a contradiction. thus accx tn = ∅ and so tn is disclosed in x. � corollary 3.3. if f ∈ c(a(x)) with coz(f) ⊆ x′, then the set t = {x ∈ x : x′ ∈ coz(f)} is a σ-disclosed subset of x. moreover, x is a zero-set of a(x) if and only if x is σ-disclosed in x. in particular, if the extent of x is countable (e.g. lindelöf), then x is a zero-set of a(x) if and only if x is countable. remark 3.4. a couple of remarks are in order. first, recall that the extent of a space is the supremum of cardinalities of closed discrete subsets. to say that a space has countable extent means that every disclosed subset is countable. second, the sets cl coz(f) r coz(f) play a pivotal role in the description of the cozero-sets of a(x). we shall call the set clv r v the residue of v . theorem 3.5. let x be tychonoff space. the following statements are equivalent. (1) a(x) is basically disconnected. (2) the space x satisfies the following three conditions: i) x is basically disconnected, ii) the set of accumulation points of any σ-disclosed subset of x is open in x, and iii) the residue of any cozero-set of x is discrete. proof. (1) ⇒ (2). suppose that a(x) is basically disconnected. let coz(f) be an arbitrary cozero-set of x. then coz(f) = coz(f) ∪ coz(f)′ is a cozero-set of a(x) for f ∈ c(a(x)) defined by f(x) = f(x′) = f(x). by hypothesis, cl[coz(f) ∪ coz(f)′] is clopen. notice that x ∩ cl[coz(f) ∪ coz(f)′] = cl coz(f) from which we gather that cl coz(f) is clopen in x, whence x is basically disconnected. next we will show (iii). it is straightforward to check that cl coz(f) = cl[coz(f) ∪ coz(f)′] = cl coz(f) ∪ coz(f)′. let p ∈ cl coz(f) r coz(f) and choose a basic open set around p in cl coz(f), say p is an element of o ∪ o′ r {p′} ⊆ cl coz(f) = cl coz(f) ∪ coz(f)′. it follows that for all q ∈ cl coz(f) r coz(f) (q 6= p), then q′ /∈ coz(f) so that q /∈ o. therefore, the residue of coz(f) is discrete. finally, to show (ii), let {tn} be a σ-disclosed subset of x and set t = ⋃ tn. without loss of generality, we can assume that the tn are pairwise disjoint. let f : a(x) → r be defined by f(x′) = 1 n if x′ ∈ t ′n and zero otherwise. since each t ′n is clopen, it follows that f ∈ c(a(x)) and coz(f) = t ′. hence, by hypothesis, cl t ′ = accx(t)∪t ′ is clopen. as a result, then accx(t) is clopen in x. © agt, upv, 2021 appl. gen. topol. 22, no. 2 338 disconnection in the alexandroff duplicate (2) ⇒ (1). suppose that x is basically disconnected, the set of accumulation points of any σ-disclosed subset of x is open in x, and the residue of any cozero-set of x is discrete. let f ∈ c(a(x)), and without loss of generality we may assume that f ≥ 0. let f be the continuous restriction of f to x. by lemma 3.2, we can write coz(f) as coz(f) = coz(f) ∪ (coz(f)′ r c′) ∪ d′ where c ⊆ coz(f) is discrete in x and closed in coz(f), and d ⊆ z(f) is a countable union of disclosed subsets of x. thus, by hypothesis accx(d) is clopen in x which means cl d′ = accx(d) ∪ d ′ is clopen in a(x). we also have that accx c = ∅. we will apply corollary 1.3 to prove that cl coz(f) is clopen in a(x). let k = cl coz(f), then k is clopen in x. by hypothesis, cl coz(f)r coz(f) is discrete and so it has no accumulation points in x. thus, k ∪ k′ r [c ∪ (cl coz(f) r coz(f))]′ is clopen in a(x). observe that cl coz(f) = cl ( coz(f) ∪ (coz(f)′ r c′) ∪ d′ ) = cl coz(f) ∪ cl(coz(f)′ r c′) ∪ cl d′ = cl coz(f) ∪ (cl (coz(f)′) r c′ ∪ cl d′ = cl coz(f) ∪ (cl (coz(f)′) r c′ ∪ cl d′ = k ∪ k′ r [c ∪ (cl coz(f) r coz(f))]′ ∪ cl d′. is hence a clopen set in a(x). consequently, a(x) is basically disconnected. � we now give two examples of basically disconnected spaces that are not pspaces. the first is an example for which a(x) is basically disconnected and the second is for which a(x) is not basically disconnected. example 3.6. as a result of theorem 0.1 of [9], given a crowded p-space of π-weight ω1, call it z, there is a non-principal z-ultrafilter on z, say v, which is a far point in βz. set z∗ = z ∪ {v} equipped with the subspace topology inherited from βz. next, let d be the disjoint union of countably many copies of z together with one additional point σ. let z∗n denote the n-th copy of z ∗ and zn = v. fix a non-principal ultrafilter u ∈ βn − n and then define a topology on d as follows. take the usual topology on the disjoint union of copies of z together with a set o containing σ to be open if {n ∈ n|(o ∩ z∗n) ∪ {zn} is an open set in z ∗ n} ∈ u. now, σ has the property that it is not in the closure of any closed discrete subset of zn. this is because for any disclosed subset of zn there is an open subset of z∗n containing zn and disjoint from said disclosed subset. then the disjoint union of copies of the open set together with σ is an open set. consequently, σ is not in the closure of any disclosed subset of d. it follows that a(d) is basically disconnected. however, σ is not a p-point of d. © agt, upv, 2021 appl. gen. topol. 22, no. 2 339 p. bhattacharjee, m. l. knox and w. wm. mcgovern let d be a discrete space of size ω1 and then let z = cδ(αd,z), that is, the space of z-valued continuous functions equipped with the p-ification of the topology of pointwise convergence, then z is a p-space of weight ω1. example 3.7. consider the space σ from [7] defined as follows. let u be a free ultrafilter on n, and let σ = n ∪ {σ} (where σ /∈ n). points of n are isolated, and a neighborhood of σ is of the form u ∪{σ} for u ∈ u. it is known that this space is extremally disconnected (and thus basically disconnected). in fact, σ is perfectly disconnected. note that n is a σ-disclosed subset of σ, but the set of accumulation points of n is {σ}, which is not open in σ. so by theorem 3.5, a(σ) is not basically disconnected. we generalize this to arbitrary discrete spaces. proposition 3.8. let x = d ∪ {p} where d is an uncountable discrete space and p ∈ βd r d is a non-principal ultrafilter. the following statements are equivalent. (1) a(x) is p-space. (2) a(x) is basically disconnected. (3) d is of measurable cardinality and p is a measurable ultrafilter. (4) x is a p-space. proof. all we need to note is that the set d is σ-disclosed set in x if and only if p is a p-point of x. � proposition 3.9. suppose a(x) is basically disconnected and no non-isolated point of x is a gδ-point of x. then x is a p-space. proof. let t be a cozero-set of x, and without loss of generality assume that t is proper. if t is not closed, then we show that each point in res(t) is a non-isolated gδ-point of x. let p ∈ res(t), then clearly, p is not isolated. by iii) of theorem 3.5, there is a clopen subset of the clopen set clt , say k, such that p ∈ k and {p} = k ∩ res(t), which as we point out is an intersection of two zero-sets, whence a zero-set. hence p is a gδ-point, contradicting the hypothesis. consequently, every cozero-set in x is clopen, i.e. x is a pspace. � proposition 3.10. suppose a(x) is basically disconnected. then every crowded σ-disclosed subset of x is clopen. in particular, every countable discrete set is closed. proof. let t be a crowded σ-disclosed subset of x. since t is crowded, it follows that t ⊆ acc(t), whence clt = t ∪ acc(t) = acc(t). by ii) of theorem 3.5, acc(t) is open. as to the last statement suppose t = {xn} is a countable discrete set. since t is countable it is a σ-disclosed set and hence acc(t) is clopen. but, by discreteness, t ∩ acc(t) = ∅ and since acc(t) is open it also follows that clt ∩acc(t) = ∅. putting everything together means that acc(t) = ∅, whence t is closed. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 340 disconnection in the alexandroff duplicate corollary 3.11. suppose x is compact. then a(x) is basically disconnected if and only if x is finite. proof. the sufficiency is obvious, so suppose that a(x) is basically disconnected. we suppose that x is infinite and compact and then choose an infinite discrete set in x, say n. since x is compact, n is not a closed set, and therefore a(x) is not basically disconnected. � remark 3.12. corollary 3.11 does not generalize to lindelöf spaces since there are common examples of infinite lindelöf p-spaces. we end this section by looking at almost p-spaces. recall that a space x is an almost p-space if every nonempty gδ-set has dense interior. this is equivalent to the condition that every nonempty zero-set has nonempty interior. clearly, every p-space is an almost p-space, but the converse is not true. for example, the space n∗ = βn r n is a compact almost p-space for which it is consistent that n∗ has no p-points. our next theorem gives necessary and sufficient conditions for when a(x) is an almost p-space. but first we need a characterization of when x possesses a non-isolated gδ-point as this is essential to the result. lemma 3.13. let x be a tychonoff space. x has a non-isolated gδ-point if and only if x has a non-empty σ-disclosed zero-set containing no isolated points. proof. first, notice that for any gδ-point p ∈ x, {p} is a zero-set that is also closed and discrete. therefore, if x has a non-isolated gδ-point, then it has a non-empty σ-disclosed zero-set containing no isolated points. conversely, suppose that x has a (non-empty) σ-disclosed zero-set z which contains no isolated points. there is a sequence of pairwise disjoint disclosed sets, say {tn}, such that z = ⋃ tn. choose any p ∈ z, and by renumbering, we assume that p ∈ t1. for each n ≥ 2, p is not in the closed set tn and hence there is a zero-set zn such that p ∈ zn and zn ∩ tn = ∅. furthermore, since t1 is discrete there is some zero-set z1 such that p ∈ z1 and z1 ∩ t1 = {p}. a quick check reveals that z ∩ ⋂ zn = {p}. since the countable intersection of zero-sets of x is again a zero-set of x it follows that p is a gδ-point. � theorem 3.14. let x be a tychonoff space. the following statements are equivalent. (1) the alexandroff duplicate a(x) is an almost p-space. (2) x has the property that there is no non-empty zero-set of x which is simultaneously σ-disclosed and does not contain any isolated points. © agt, upv, 2021 appl. gen. topol. 22, no. 2 341 p. bhattacharjee, m. l. knox and w. wm. mcgovern (3) x has no non-isolated gδ-points. proof. (1) ⇒ (2). suppose that x has a non-empty zero-set which is a countable union of disclosed subsets, say z(f) = ⋃ n∈n tn where each tn is disclosed, and that z(f) contains no isolated points. without loss of generality, we can assume that the collection is pairwise disjoint. define f : a(x) → r by f(x) =      f(x), if x ∈ x f(x), if x ∈ coz(f)′ 1 n , if x ∈ t ′n. since each tn is closed and discrete we know that t ′ n is a clopen subset of a(x). therefore, f ∈ c(a(x)). notice that z(f) = z(f), and also that this set is a co-dense subset of a(x), whence it is nowhere dense. it follows that coz(f) is a proper dense cozero-set of a(x) and consequently, a(x) is not an almost p-space. (2) ⇒ (1). suppose that a(x) is not an almost p-space and choose f ∈ c(a(x)) so that f ≥ 0 and coz(f) is a proper dense cozero-set of a(x). notice that z(f) ⊆ x, since otherwise z(f) would contain an isolated point in x′, forcing z(f) to have nonempty interior. similarly, z(f) contains no isolated points of x. for each n ∈ n, let tn = { x ∈ z(f) : f(x′) ≥ 1 n } , if p ∈ cltn, then p ∈ z(f) and so there is a basic open centered at p which misses f−1([ 1 n ,∞)) and hence t ′n. therefore, p ∈ tn, whence tn is closed in x. notice that cl t ′n ⊆ z(f), but due to continuity of f , cl t ′ n = t ′ n, whence t ′n is a clopen subset of a(x). therefore, tn is a closed and discrete subset of x. furthermore, for any x ∈ z(f) we know that f(x′) > 0 and therefore, x ∈ tn. consequently, z(f) is a countable union of closed and discrete subsets of x. that (2) are (3) are equivalent follows from lemma 3.13. � corollary 3.15. if x is an almost p-space, then a(x) is an almost p-space. in particular, a(n∗) is an almost p-space. proof. in an almost p-space, any gδ-point must be isolated. � remark 3.16. recall that the pseudo-character of a point, notated ψ(p,x), is the minimum (infinite) cardinality needed to write {p} as an intersection of open sets. thus, ψ(p,x) = ℵ0 if and only if p is a gδ-point. we leave it to the interested reader to modify the above proofs to show that for any p ∈ x, ψ(p,x) = ψ(p,a(x)). furthermore, a non-isolated p ∈ x is an almost p-point of a(x) if and only if it is not a gδ-point of x; we find this result to be rather striking. the result corroborates the fact that a(βn) is not an extremally disconnected space as it is an almost p-space, and it is well-known that if x © agt, upv, 2021 appl. gen. topol. 22, no. 2 342 disconnection in the alexandroff duplicate has non-measurable cardinality, then the only points that are simultaneously e.d. points and almost p-points are the isolated points. example 3.17. we would like to share more examples of spaces x which are not almost p-spaces yet a(x) is an almost p-space. it suffices to simply find a space that has no non-isolated gδ points and is not an almost p-space. examples are everywhere but we take one from cp-theory. start with a topological space x and consider c(x) with the subspace topology inherited from the product rx. this is known as the topology of point-wise convergence denote this space by cp(x). this topology on c(x) makes c(x) into a topological ring. in particular, the topology is homogeneous. a base of open sets centered at 0 are sets of the form o(f,ǫ) = {f ∈ c(x) : |f(xi)| ≤ ǫ for each i = 1, . . . ,n} for some finite subset f = {x1, . . . ,xn} ⊆ x. it is known that cp(x) has gδ-points if and only if x is separable; see [3, theorem i.1.4.]. our claim is that cp(x) is never an almost p-space. notice that the set mx = {f ∈ c(x) : f(x) = 0} = ⋂ n∈n o({x}, 1 n ) is a gδ-set, and clearly, this set has empty interior. thus, for example if x = αd, the one-point compactification of an uncountable discrete set, then a(cp(x)) is an almost p-space even though cp(x) is not. next, recall that no compact extremally disconnected space has non-isolated gδ-points. this is because if x is compact extremally disconnected and p ∈ x is not isolated, then x r {p} is c∗-embedded in x and thus β(x r {p}) = x and no point of βx r x can be a gδ-point of βx. [thanks to a.w. hager for reminding us of this result.] acknowledgements. we would like to thank the referee for making some suggestions that improved the quality of the paper. references [1] p. alexandrov and p. urysohn, memoire sur les espaces topologiques compacts, verh. akad. wetensch. amsterdam, 14 (1929), 1–96. [2] k. almontashery and l. kalantan, results about the alexandroff duplicate space, appl. gen. topol. 17, no. 2 (2016), 117–122. [3] a. v. arkhangel’skii, topological function spaces, mathematics and its applications, 78, springer, netherlands, 1992. © agt, upv, 2021 appl. gen. topol. 22, no. 2 343 p. bhattacharjee, m. l. knox and w. wm. mcgovern [4] g. bezhanishvili, n. bezhanishvili, j. lucero-bryan and j. van mill, s4.3 and hereditarily extremally disconnected spaces, georgian mathematical journal 22, no. 4 (2015), 469– 475. [5] a. caserta and s. watson, the alexandroff duplicate and its subspaces, appl. gen. topol. 8, no. 2 (2007), 187–205. [6] r. engelking, on functions defined on cartesian products, fund. math. 59 (1966), 221– 231. [7] l. gillman and m. jerison, rings of continuous functions, graduate texts in mathametics, vol. 43, springer verlag, berlin-heidelberg-new york, 1976. [8] e. van douwen, applications of maximal topologies, topology appl. 51 (1993), 125–139. [9] j. van mill, weak p -points in čech-stone compactifications, trans. amer. math. soc. 273 (1982), 657–678. [10] j. l. verner, lonely points revisited, comment. math. univ. carolin. 54, no. 1 (2013), 105–110. © agt, upv, 2021 appl. gen. topol. 22, no. 2 344 () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 151-166 common fixed points for generalized (ψ,φ)-weak contractions in ordered cone metric spaces hemant kumar nashine and hassen aydi abstract the purpose of this paper is to establish coincidence point and common fixed point results for four maps satisfying generalized (ψ,φ)-weak contractions in partially ordered cone metric spaces. also, some illustrative examples are presented. 2010 msc: 54h25, 47h10. keywords: coincidence point, common fixed point, weakly contractive condition, dominating map, dominated map, ordered set, cone metric space. 1. introduction one of the simplest and useful results in the fixed point theory is the banach– caccioppoli contraction mapping principle. in the last years, this principal has been generalized in many directions to generalized structures as cone metrics, partial metric spaces and quasi-metric spaces has received a lot of attention. fixed point theory in k-metric and k-normed spaces was developed by perov et al. [24], mukhamadijev and stetsenko [16], vandergraft [33]. for more details on fixed point theory in k-metric and k-normed spaces, we refer the reader to fine survey paper of zabrejko [34]. the main idea was to use an ordered banach space instead of the set of real numbers, as the codomain for a metric. in 2007, huang and zhang [13] reintroduced such spaces under the name of cone metric spaces and reintroduced definition of convergent and cauchy sequences in the terms of interior points of the underlying cone. they also proved some fixed point theorems in such spaces in the same work. after that, fixed point points in k-metric spaces have been the subject of intensive research (see, e.g., 152 h. k. nashine and h. aydi [1, 3, 7, 11, 13, 14, 15, 16, 23, 25, 30]). the main motivation for such research is a point raised by agarwal [4], that the domain of existence of a solution to a system of first-order differential equations may be increased by considering generalized distances. recently, wei-shih du [12] used the scalarization function and investigated the equivalence of vectorial versions of fixed point theorems in k-metric spaces and scalar versions of fixed point theorems in metric spaces. he showed that many of the fixed point theorems for mappings satisfying contractive conditions of a linear type in k-metric spaces can be considered as the corollaries of corresponding theorems in metric spaces. nevertheless, the fixed point theory in k-metric spaces proceeds to be actual, since the method of scalarization cannot be applied for a wide class of mappings satisfying contractive conditions more general than contractive conditions of a linear type. on the other hand, fixed point theory has developed rapidly in metric spaces endowed with a partial ordering. one of results in this direction was given by ran and reurings [26] who presented its applications to matrix equations. subsequently, nieto and rodŕıguez-lópez [22] extended the result of ran and reurings for nondecreasing mappings and applied it to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. thereafter, many authors obtained many fixed point theorems in ordered metric spaces. for more details, see [5, 6, 8, 10, 17, 19, 20, 21, 22, 27, 29, 31] and the references cited therein. in this paper, an attempt has been made to derive some common fixed point theorems for four maps involving generalized (ψ,φ)-weak contractions in ordered cone metric spaces. the presented theorems generalize, extend and improve some recent fixed point results in k-metric spaces. 2. preliminaries in what follows, we recall some notations and definitions that will be utilized in our subsequent discussion. let e be always a banach space. definition 2.1. a non-empty subset k of e is called a cone if and only if (i) k = k, k 6= 0e where k is the closure of k, (ii) a,b ∈ r, a,b ≥ 0, x,y ∈ k ⇒ ax + by ∈ k, (iii) k ∩ (−k) = {0e}. a cone k defines a partial ordering ≤e in e by x ≤e y if and only if y − x ∈ k. we shall write x m > n. we say that {xn} converges to x ∈ x if for every c ∈ e with 0e ≪ c, there exists n ∈ n such that d(xn,x) ≪ c for all n > n. in this case, we denote xn → x as n → ∞. a cone metric space (x,d) is said to be complete if every cauchy sequence in x is convergent in x. definition 2.4. let f : e → e be a given mapping. we say that f is a monotone non-decreasing mapping with respect to ≤e if for every x,y ∈ e, x ≤e y implies fx ≤e fy. definition 2.5 ([9]). let ψ : k → k be a given function. (i) we say that ψ is strongly monotone increasing if for x,y ∈ k, we have x ≤e y ⇐⇒ ψ(x) ≤e ψ(y). (ii) ψ is said to be continuous at x0 ∈ k if for any sequence {xn} in k, we have ‖xn − x0‖e → 0 =⇒ ‖ψ(xn) − ψ(x0)‖e → 0. definition 2.6. let (x,d) be a cone metric space and f,g : x → x. if w = fx = gx, for some x ∈ x, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. if w = x, then x is a common fixed point of f and g. the pair {f,g} is said to be compatible if and only if lim n→+∞ d(fgxn,gfxn) = 0, whenever {xn} is a sequence in x such that lim n→+∞ fxn = lim n→+∞ gxn = t for some t ∈ x. definition 2.7 ([2]). let f and g be two self-maps defined on a set x. then f and g are said to be weakly compatible if they commute at every coincidence point. definition 2.8. let x be a nonempty set. then (x,d,�) is called an ordered cone metric space if and only if (i) (x,d) is a metric space, (ii) (x,�) is a partial order. definition 2.9. let (x,�) be a partial ordered set. then x,y ∈ x are called comparable if x � y or y � x holds. definition 2.10 ([2]). let (x,�) be a partially ordered set. a mapping f is called dominating if x � fx for each x in x. 154 h. k. nashine and h. aydi example 2.11 ([2]). let x = [0,1] be endowed with usual ordering and f : x → x be defined by fx = n √ x. since x ≤ x13 = fx for all x ∈ x. therefore f is a dominating map. definition 2.12 ([18]). let (x,�) be a partially ordered set. a mapping f is called dominated if fx � x for each x in x. example 2.13 ([18]). let x = [0,1] be endowed with usual ordering and f : x → x be defined by fx = xn for all n ≥ 1. since fx = xn ≤ x for all x ∈ x. therefore f is a dominated map. 3. common fixed point results first, let ψ be the set of functions ψ : k → k such that (i) ψ is continuous; (ii) ψ(t) = 0e if and only if t = 0e; (iii) ψ is strongly monotone increasing. also, let φ be the set of functions φ : int(k)∪{0e} → int(k)∪{0e} such that (i’) φ is continuous; (ii’) φ(t) = 0e if and only if t = 0e; (iii’) φ(t) ≪e t for all t ∈ int(k); (iv’) either φ(t) ≤e d(x,y) or d(x,y) ≤e φ(t) for t ∈ int(k) ∪ {0e} and x,y ∈ x. the following lemma will be useful later. lemma 3.1. [30]. let e be a banach space, {an}, {bn} and {cn} are sequences in e such that bn → b ∈ e, cn → c ∈ e as n → +∞. suppose also that an ∈ {bn,cn} for all n ∈ n. then there exists a subsequence {an(p)} of {an} such that an(p) → a ∈ {b,c} as p → +∞. our first result is the following. theorem 3.2. let (x,d,�) be an ordered complete cone metric space over a solid cone k. let t,s,i,j : x → x be given mappings satisfying for every pair (x,y) ∈ x × x such that x and y are comparable, ψ(d(sx,ty)) ≤e ψ(θ(x,y)) − φ(θ(x,y)),(3.1) where θ(x,y) ∈ {d(ix,jy), 1 2 [d(ix,sx)+d(jy,ty)], 1 2 [d(ix,ty)+d(jy,sx)]}, ψ ∈ ψ and φ ∈ φ. suppose that (i) tx ⊆ ix and sx ⊆ jx; (ii) i and j are dominating maps and s and t are dominated maps; (iii) if for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. also, assume either (a) {s,i} are compatible, s or i is continuous and {t,j} are weakly compatible or common fixed points in ordered cone metric spaces 155 (b) {t,j} are compatible, t or j is continuous and {s,i} are weakly compatible. then s,t,i and j have a common fixed point. proof. let x0 be an arbitrary point in x. since tx ⊆ ix and sx ⊆ jx, we can define the sequences {xn} and {yn} in x by (3.2) y2n−1 = sx2n−2 = jx2n−1, y2n = tx2n−1 = ix2n, ∀n ∈ n. by given assumptions x2n+1 � jx2n+1 = sx2n � x2n and x2n � ix2n = tx2n−1 � x2n−1. thus, for all n ≥ 0, we have (3.3) xn+1 � xn. putting x = x2n+1 and y = x2n, from (3.3) and the considered contraction (3.1), we have ψ(d(y2n+1,y2n+2)) = ψ(d(sx2n,tx2n+1))(3.4) ≤e ψ(θ(x2n,x2n+1)) − φ(θ(x2n,x2n+1)) ≤e ψ(θ(x2n,x2n+1)). the function ψ is strongly increasing, so we get that (3.5) d(y2n+1,y2n+2) ≤e θ(x2n,x2n+1). note that θ(x2n,x2n+1) ∈ {d(ix2n,jx2n+1), 1 2 [d(ix2n,sx2n) + d(jx2n+1,tx2n+1)], 1 2 [d(ix2n,tx2n+1) + d(sx2n,jx2n+1)]} = {d(y2n,y2n+1), 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], 1 2 [d(y2n,y2n+2) + d(y2n+1,y2n+1)]} = {d(y2n,y2n+1), 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], 1 2 d(y2n,y2n+2)}. if θ(x2n,x2n+1) = d(y2n,y2n+1), (3.5) becomes d(y2n+1,y2n+2) ≤e d(y2n,y2n+1). if θ(x2n,x2n+1) = 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], then (3.5) becomes d(y2n+1,y2n+2) ≤e 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], so d(y2n+1,y2n+2) ≤e d(y2n,y2n+1). if θ(x2n,x2n+1) = 1 2 d(y2n,y2n+2), by (3.4) and a triangular inequality, we find that d(y2n+1,y2n+2) ≤e 1 2 d(y2n,y2n+2) ≤e 1 2 d(y2n,y2n+1) + 1 2 d(y2n+1,y2n+2), so d(y2n+1,y2n+2) ≤e d(y2n,y2n+1). in all cases, we obtained that (3.6) d(y2n+1,y2n+2) ≤e θ(x2n,x2n+1) ≤e d(y2n,y2n+1). 156 h. k. nashine and h. aydi similarly, we have (3.7) d(y2n+1,y2n) ≤e θ(x2n,x2n−1) ≤e d(y2n,y2n−1). by (3.6) and (3.7), we get that (3.8) d(yn+1,yn) ≤e d(yn,yn−1) for all n ≥ 1. it follows that the sequence {d(yn,yn+1)} is monotone non-increasing. since k is a regular cone and 0e ≤e d(yn,yn+1) for all n ≥ 0, there exists r ≥e 0e such that d(yn,yn+1) → r as n → +∞. by (3.6) and (3.7), we have lim n→+∞ θ(x2n,x2n+1) = lim n→+∞ θ(x2n,x2n−1) = r. now, letting n → +∞ in (3.4) and using the continuity property of ψ and φ, we get ψ(r) ≤ ψ(r) − φ(r), which yields that φ(r) = 0e. since φ(t) = 0e ⇐⇒ t = 0e, then r = 0e. therefore, (3.9) lim n→+∞ d(yn,yn+1) = 0. now, we will show that {yn} is a cauchy sequence in the cone metric space (x,d). we proceed by negation and suppose that {y2n} is not a cauchy sequence. then, there exists ε > 0 for which we can find two sequences of positive integers {m(i)} and {n(i)} such that for all positive integer i, (3.10) n(i) > m(i) > i, d(y2m(i),y2n(i)) ≥e ε, d(y2m(i),y2n(i)−2) 0. if lim i→+∞ θ(x2n(i),x2m(i)−1) = ε, then using similar arguments, we obtain that ψ(ε) ≤e ψ(ε) − φ(ε), so φ(ε) = 0e, which is a contradiction. thus {y2n} is a cauchy sequence in x, so {yn} is also a cauchy sequence in x. finally, we shall prove existence of a common fixed point of the four mappings i,j,s and t . since x is complete, there exists a point z in x, such that {y2n} converges to z. therefore, (3.17) y2n+1 = jx2n+1 = sx2n → z as n → ∞ and (3.18) y2n+2 = ix2n+2 = tx2n+1 → z as n → ∞. 158 h. k. nashine and h. aydi assume that (a) holds. suppose that i is continuous. since the pair {s,i} is compatible, we have (3.19) lim n→∞ six2n+2 = lim n→∞ isx2n+2 = iz. also, ix2n+2 = tx2n+1 � x2n+1. now, by (3.1) (3.20) ψ(d(six2n+2,tx2n+1)) ≤e ψ(θ(ix2n+2,x2n+1)) − φ(θ(ix2n+2,x2n+1)), where θ(ix2n+2,x2n+1)) ∈ {d(iix2n+2,jx2n+1), 1 2 [d(iix2n+2,six2n+2)+ d(jx2n+1,tx2n+1)], 1 2 [d(iix2n+2,tx2n+1) + d(six2n+2,jx2n+1)]}. by (3.9), (3.17), (3.18) and (3.19), we get that lim n→∞ d(iix2n+2,jx2n+1) = lim n→∞ 1 2 [d(iix2n+2,tx2n+1)+d(six2n+2,jx2n+1)] = d(iz,z), lim n→∞ 1 2 [d(iix2n+2,six2n+2) + d(jx2n+1,tx2n+1)] = 0e. by lemma 3.1, there exists a subsequence of {θ(ix2n+2,x2n+1)} still denoted θ(ix2n+2,x2n+1) such that from the above limits (3.21) lim n→+∞ θ(ix2n+2,x2n+1) ∈ {0e,d(iz,z)}. if lim n→+∞ θ(ix2n+2,x2n+1) = 0e, then then letting n → +∞ in (3.20) and using the fact that lim n→∞ d(six2n+2,tx2n+1) = d(iz,z), and the continuities of ψ and φ, we obtain ψ(d(iz,z)) ≤e ψ(0e) − φ(0e), so ψ(d(iz,z)) = 0e, which yields that d(iz,z) = 0e, so iz = z. if lim n→+∞ θ(ix2n+2,x2n+1) = d(iz,z), using the similar arguments we get that ψ(d(iz,z)) − ψ(d(iz,z)) − φ(d(iz,z)), so similarly, iz = z. in each case, we obtained (3.22) iz = z. now, tx2n+1 � x2n+1 and tx2n+1 → z as n → ∞, so by assumption [(iii)] we have z � x2n+1. from (3.1), (3.23) ψ(d(sz,tx2n+1)) ≤e ψ(d(θ(z,x2n+1))) − φ(d(θ(z,x2n+1))), common fixed points in ordered cone metric spaces 159 where θ(z,x2n+1) ∈ {d(iz,jx2n+1), 1 2 [d(iz,sz) + d(jx2n+1,tx2n+1)], 1 2 [d(iz,tx2n+1) + d(sz,jx2n+1)]} = {d(z,jx2n+1), 1 2 [d(z,sz) + d(jx2n+1,tx2n+1)], 1 2 [d(z,tx2n+1) + d(sz,jx2n+1)]}. by (3.9), (3.17), (3.18) and (3.19), we get that lim n→∞ 1 2 [d(z,sz)+d(jx2n+1,tx2n+1)] = 1 2 d(z,sz) = lim n→∞ 1 2 [d(iix2n+2,tx2n+1) +d(six2n+2,jx2n+1)]}, lim n→∞ d(z,jx2n+1) = 0e. by lemma 3.1, there exists a subsequence of {θ(z,x2n+1)} still denoted θ(ix2n+2,x2n+1) such that from the above limits (3.24) lim n→+∞ θ(ix2n+2,x2n+1) ∈ {0e, 1 2 d(sz,z)}. if lim n→+∞ θ(ix2n+2,x2n+1) = 0e, then then letting n → +∞ in (3.24) and using the fact that lim n→∞ d(sz,tx2n+1) = d(sz,z), and the continuities of ψ and φ, we obtain ψ(d(sz,z)) ≤e ψ(0e) − φ(0e), so ψ(d(iz,z)) = 0e, which yields that sz = z. if lim n→+∞ θ(ix2n+2,x2n+1) = 1 2 d(sz,z) and using the similar arguments, we get that ψ(d(sz,z)) ≤e ψ( 1 2 d(sz,z)) − φ(1 2 d(sz,z)) ≤e ψ( 1 2 d(sz,z)), so d(sz,z) ≤e 12d(sz,z), which holds unless d(sz,z) = 0e, so (3.25) sz = z. since s(x) ⊆ j(x), there exists a point w ∈ x such that sz = jw. suppose that tw 6= jw. since w � jw = sz � z implies w � z. from (3.1), we obtain (3.26) ψ(d(jw,tw)) = ψ(d(sz,tw)) ≤e ψ(θ(z,w)) − φ(θ(z,w)), where θ(z,w) ∈ {d(iz,jw), 1 2 [d(iz,sz) + d(jw,tw)], 1 2 [d(iz,tw) + d(sz,jw)]} = {0e, 1 2 d(jw,tw)}. 160 h. k. nashine and h. aydi if θ(z,w) = 0e, we easily deduce from (3.26) that d(jw,tw) = 0e. if θ(z,w) = d(jw,tw), similarly we get that d(jw,tw) = 0e. thus, we obtained (3.27) jw = tw. since t and j are weakly compatible, tz = tsz = tjw = jtw = jsz = jz. thus, z is a coincidence point of t and j. now, since sx2n � x2n and sx2n → z as n → ∞, so by assumption [(iii)], z � x2n. then, from (3.1) (3.28) ψ(d(sx2n,tz)) ≤e ψ(θ(x2n,z)) − φ(θ(x2n,z)), where θ(x2n,z) ∈ {d(ix2n,jz), 1 2 [d(ix2n,sx2n) + d(jz,tz)], 1 2 [d(ix2n+1,tz) + d(sx2n,jz)]} = {d(ix2n,tz), 1 2 d(ix2n,sx2n), 1 2 [d(ix2n+1,tz) + d(sx2n,tz)]} we have lim n→∞ d(ix2n,tz) = lim n→∞ 1 2 [d(ix2n+1,tz) + d(sx2n,tz)] = d(z,tz), and lim n→∞ d(ix2n,sx2n) = 0, lim n→∞ d(sx2n,tz) = d(z,tz). by lemma 3.1, there exists a subsequence of {θ(x2n,z))} still denoted θ(x2n,z) such that from the above limits (3.29) lim n→+∞ θ(x2n,z) ∈ {0e,d(z,tz)}. similarly, letting n → ∞ in (3.28) and having in mind (3.29), we get that (3.30) z = tz. therefore sz = tz = iz = jz = z, so z is a common fixed point of i, j, s and t . the proof is similar when s is continuous. similarly, the result follows when (b) holds. � now, it is easy to state a corollary of theorem 3.2 involving a contraction of integral type. corollary 3.3. let t,s,i and j satisfy the conditions of theorem 3.2, except that condition (3.1) is replaced by the following: there exists a positive lebesgue integrable function u on r+ such that ∫ ε 0 u(t)dt > 0 for each ε > 0 and that (3.31) ∫ ψ(d(sx,ty)) 0 u(t)dt ≤ ∫ ψ(θ(x,y)) 0 u(t)dt − ∫ φ(θ(x,y)) 0 u(t)dt. then, s,t,i and j have a common fixed point. common fixed points in ordered cone metric spaces 161 corollary 3.4. let (x,d,�) be an ordered complete cone metric space over a solid cone k. let t,s,i : x → x be given mappings satisfying for every pair (x,y) ∈ x × x such that x and y are comparable, ψ(d(sx,ty)) ≤e ψ(θ1(x,y)) − φ(θ1(x,y)),(3.32) where θ1(x,y) ∈ {d(ix,iy), 12[d(ix,sx)+d(iy,ty)], 1 2 [d(ix,ty)+d(iy,sx)]}, ψ ∈ ψ and φ ∈ φ. suppose that (i) tx ⊆ ix and sx ⊆ ix; (ii) i is a dominating map and s and t are dominated maps; (iii) if for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. also, assume either (a) {s,i} are compatible, s or i is continuous and {t,i} are weakly compatible or (b) {t,i} are compatible, t or i is continuous and {s,i} are weakly compatible, then s,t and i have a common fixed point. proof. it follows by taking i = j in theorem 3.2. � corollary 3.5. let (x,d,�) be an ordered complete cone metric space over a solid cone k. let s,i : x → x be given mappings satisfying for every pair (x,y) ∈ x × x such that x and y are comparable, ψ(d(sx,sy)) ≤e ψ(θ2(x,y)) − φ(θ2(x,y)),(3.33) where θ2(x,y) ∈ {d(ix,iy), 12[d(ix,sx)+d(iy,sy)], 1 2 [d(ix,sy)+d(iy,sx)]}, ψ ∈ ψ and φ ∈ φ. suppose that (i) sx ⊆ ix; (ii) i is a dominating map and s is dominated maps; (iii) if for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. also, assume {s,i} are compatible and s or i is continuous, then s and i have a common fixed point. proof. it follows by taking s = t in corollary 3.4. � corollary 3.6. let (x,d,�) be an ordered complete cone metric space over a solid cone k. let t,s : x → x be given mappings satisfying for every pair (x,y) ∈ x × x such that x and y are comparable, ψ(d(sx,ty)) ≤e ψ(θ3(x,y)) − φ(θ3(x,y)),(3.34) where θ3(x,y) ∈ {d(x,y), 12[d(x,sx)+d(y,ty)], 1 2 [d(x,ty)+d(y,sx)]}, ψ ∈ ψ and φ ∈ φ. suppose that (i) s and t are dominated maps; (ii) if for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. 162 h. k. nashine and h. aydi also, assume either s or t is continuous, then s and t have a common fixed point. proof. it follows by taking i = idx, the identity on x, in corollary 3.4. � corollary 3.7. let (x,d,�) be an ordered complete cone metric space over a solid cone k. let t,s,i,j : x → x be given mappings satisfying for every pair (x,y) ∈ x × x such that x and y are comparable, d(sx,ty) ≤e θ(x,y) − φ(θ(x,y)), where θ(x,y) ∈ {d(ix,jy), 1 2 [d(ix,sx) + d(jy,ty)], 1 2 [d(ix,ty) + d(jy,sx)]} and φ ∈ φ. suppose that (i) tx ⊆ ix and sx ⊆ jx; (ii) i and j are dominating maps and s and t are dominated maps; (iii) if for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. also, assume either (a) {s,i} are compatible, s or i is continuous and {t,j} are weakly compatible or (b) {t,j} are compatible, t or j is continuous and {s,i} are weakly compatible, then s,t,i and j have a common fixed point. proof. it suffices to take ψ(t) = t in theorem 3.2. � remark 3.8. theorem 3.2 extends theorem 2.1 of shatanawi and samet [32] to cone metric spaces. now, we state the following illustrative examples. example 3.9 (the case of a non-normal cone). let x = [0, 1 4 ] be equipped with the usual order. take e = c1 r ([0,1]) and k = {ϕ ∈ e, ϕ(t) ≥ 0, t ∈ [0,1]}. define d : x × x → e by d(x,y)(t) = |x − y|ϕ where ϕ ∈ k is a fixed function, for example ϕ(t) = et. then, (x,d) is a complete cone metric space with a nonnormal solid cone. also, define s,t,i,j : x → x by sx = tx = x2 and ix = jx = x. for all comparable x,y ∈ x, we have d(sx,ty)(t) = d(sx,sy)(t) = |x2−y2|et = |x−y||x+y|et ≤ 1 2 |x−y|et = 1 2 d(ix,jy)(t), that is, (3.1) holds for ψ(t) = t and φ(t) = 1 2 t. on the other hand, x ≤ ix = jx and sx = tx ≤ x for all x ∈ x. also, sx = tx ⊆ ix = jx and the pairs {s,i} = {t,j} are compatible. all hypotheses of theorem 3.2 are verified and x = 0 is a common fixed point of s,t,i and j. example 3.10. (the case of a normal cone). let x = [0,∞] be equipped with the usual order. take e = r2 and k = {(x,y), x ≥ 0, y ≥ 0}. define common fixed points in ordered cone metric spaces 163 d : x × x → e by d(x,y) = (|x − y|,α|x − y|) where α ≥ 0 a constant. then, (x,d) is a complete cone metric space with a normal solid cone. also, define s,t,i,j : x → x by sx = tx = ax and ix = jx = bx where 0 < a < 1 and b > 1. for all comparable x,y ∈ x, we have d(sx,ty) = d(sx,sy) = (a|x−y|,aα|x−y|) = (a b b|x−y|, a b bα|x−y|) = a b d(ix,jy), that is, (3.1) holds for ψ(t) = t and φ(t) = (1 − a b )t. also, it is clear that all other hypotheses of theorem 3.2 are verified and x = 0 is a common fixed point of s,t,i and j. the following example (which is inspired by [18]) demonstrates the validity of theorem 3.2. example 3.11 (the case of a non-normal cone). let x = [0,1] be equipped with the usual order. take e = c1 r ([0,1]) and k = {ϕ ∈ e, ϕ(t) ≥ 0, t ∈ [0,1]}. define d : x × x → e by d(x,y)(t) = |x − y|ϕ where ϕ ∈ k is a fixed function, for example ϕ(t) = et. then, (x,d) is a complete cone metric space with a nonnormal solid cone. define the self maps i, j, s and t on x by s(x) = { 0, if x ≤ 1 3 1 2 (x − 1 3 ), if x ∈ (1 3 ,1] , tx = { 0, if x ≤ 1 3 1 3 , if x ∈ (1 3 ,1] , j(x) =    0, if x = 0 x, if x ∈ (0, 1 3 ] 1, if x ∈ (1 3 ,1] , ix =    0, if x = 0 1 3 , if x ∈ (0, 1 3 ] 1, if x ∈ (1 3 ,1] . then i and j are dominating maps and s and t are dominated maps with s(x) ⊆ j(x) and t(x) ⊆ i(x),i.e. s is dominated map t is dominated map i is dominating map j is dominating map for each x in x sx ≤ x tx ≤ x x ≤ ix x ≤ jx x = 0 s (0) = 0 t (0) = 0 0 = i(0) 0 = j(0) x ∈ (0, 1 3 ] sx = 0 < x tx = 0 < x x ≤ 1 3 = i(x) x = j(x) x ∈ ( 1 3 , 1] sx = 1 2 (x − 1 3 ) < x tx = 1 3 < x x ≤ 1 = i(x) x ≤ 1 = j(x) also, {s,i} are compatible, s is continuous and {t,j} are weakly compatible. define ψ : k → k and φ : int(k) ∪ {0e} → int(k) ∪ {0e} by ψ(t) = t and φ(t) = 1 2 t. the inequality (3.1) holds for all comparable x,y ∈ x. without loss of generality, take x ≤ y. we consider the following cases: (i) if x = y = 0, then d(s0,t0)(t) = 0 and (3.1) is satisfied. (ii) for x = 0 and y ∈ (0, 1 3 ], then again d(sx,ty)(t) = 0 and (3.1) is satisfied. 164 h. k. nashine and h. aydi (iii) for x = 0 and y ∈ (1 3 ,1], d(sx,ty)(t) = 1 3 e t < 1 2 e t = 1 2 d(ix,jy)(t). (iv) for x,y ∈ (0, 1 3 ], then d(sx,ty) = 0 and hence (3.1) is satisfied. (v) for x = (0, 1 3 ] and y ∈ (1 3 ,1], d(sx,ty)(t) = 1 3 et < 1 2 et = 1 2 d(ix,jy)(t). (vi) for x,y ∈ (1 3 ,1], d(sx,ty)(t) = 1 2 (1 − x)et ≤ 1 3 et ≤ 1 2 d(jy,ty)(t). all hypotheses of theorem 3.2 are verified and x = 0 is a common fixed point of s,t,i and j. acknowledgements. the authors are grateful to the editor and referee for their valuable remarks for improving this paper. references [1] m. abbas and g. jungck, common fixed point results for noncommuting mappings without continuity in cone metric spaces, j. math. anal. appl. 341 (2008), 416–420. [2] m. abbas, t. nazir and s. radenović, common fixed point of four maps in partially ordered metric spaces, appl. math. lett. 24 (2011), 1520–1526. 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(received december 2011 – accepted august 2012) 166 h. k. nashine and h. aydi h. aydi (hassen.aydi@isima.rnu.tn) université de sousse, institut supérieur d’informatique et des technologies de communication de hammam sousse, route gp1-4011, h. sousse, tunisie. h. k. nashine (drhknashine@gmail.com,hemantnashine@rediffmail.com) department of mathematics, disha institute of management and technology, satya vihar, vidhansabha-chandrakhuri marg, mandir hasaud, raipur492101(chhattisgarh), india. common fixed points for generalized (,)-weak contractions in ordered cone metric spaces. by h. k. nashine and h. aydi @ appl. gen. topol. 22, no. 1 (2021), 17-30doi:10.4995/agt.2021.13084 © agt, upv, 2021 on soft quasi-pseudometric spaces hope sabao a and olivier olela otafudu b a school of mathematics, university of the witwatersrand johannesburg 2050 , south africa. (hope.sabao@wits.ac.za) b department of mathematics and applied mathematics, university of the western cape, bellville 7535, south africa. (olmaolela@gmail.com) communicated by s. romaguera abstract in this article, we introduce the concept of a soft quasi-pseudometric space. we show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. we then introduce the concept of soft isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft isbell convex fixed point set. 2010 msc: 03e72; 08a72; 47h10; 54e35; 54e15. keywords: soft-metric; soft-quasi-pseudometric; soft isbell convexity. 1. introduction soft set theory has several applications in solving practical problems in economics, engineering, social sciences and medical science e.t.c. the study of soft sets was first initiated by molodtsov [8] in 1999. since then, many other scholars have taken interest in soft set theory (see [1], [2], [4], [15] ). the study of soft metric spaces was initiated by das and samanta in [16]. using the concept of a soft point in a soft set, they introduced a soft metric and some of their basic properties. thereafter, they investigated some topological structures such as soft open sets, soft closed sets and soft closures of soft sets e.t.c. furthermore, they investigated the notion of completeness of soft metric received 31 january 2020 – accepted 19 october 2020 http://dx.doi.org/10.4995/agt.2021.13084 h. sabao and o. olela otafudu spaces and the cantor’s intersection theorem. recently, abbas et al. [5] studied the concept of fixed point theory of soft metric spaces. they showed that a soft metric induces a compatible metric on the collection of all soft points of the absolute soft set, whenever the set of parameters is finite. thereafter, they used this concept to show that several fixed point theorems for metric spaces can be directly deduced from comparable existing results. until recently, most studies in topology has been based on spaces arising from a collection of metrics, which like the euclidean distance, are symmetric. this was very natural since many problems it was used for were based on the euclidean topology on the real numbers, which arises from the usual distance on reals numbers. but most topologies are not distance based, they are based on things like “effort” which have many properties of metrics but lack symmetry. a quasi-metric space is an example of a space which lack symmetry. it is also well known that quasi-metric spaces constitute an efficient tool to discuss and solve several problems in topological algebra, approximation theory, theoretical computer science, etc. (see [10]). on the other hand, t -theory is a theory that involves trees, injective envelopes of metric spaces (hyperconvex hull), and all of the areas that are connected with these topics. these topics have been used in the development of mathematical tools for reconstructing phylogenetic trees (see [6]). these are our motivations for generalising soft metric spaces to the asymmetric setting and introducing the concept of hyperconvexity in our new space. in this article, we introduce soft quasi-pseudometric spaces, a concept that generalise soft metric spaces to the asymmetric setting. we then show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. we then introduce the concept of hyperconvexity in soft quasipseudometric spaces, which we call soft isbell convexity, and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft isbell convex fixed point set. 2. preliminaries the letters u, e and p(u) will denote the universal set, the set of parameters and the power set of u respectively. according to [8], if f is a set valued mapping on a ⊂ e taking values in p(u), then the pair (f, a) is called a soft set over u. we will denote the collection of soft sets over a common universe u by s(u). a soft set (f, a) over u is said to be a soft point if there is exactly one λ ∈ a such that f(λ) = {x} and f(e) = ∅ for all e ∈ a \ {λ}. we shall denote such © agt, upv, 2021 appl. gen. topol. 22, no. 1 18 on soft quasi-pseudometric spaces a point by (f xλ , a) or simply f x λ . a soft point f x λ is said to belong to a soft set (f, a), denoted by f xλ ∼ ∈ (f, a), if f xλ (λ) = {x} ⊂ f(λ). the collection of soft points of (f, a) is denoted by sp(f, a). a soft set (f, e) is said to be a null soft set, denoted by φ if for all e ∈ a, f(e) = ∅. a soft set which is not null is said to be a non-null soft set. f is a soft mapping from the soft set (f, a) to a soft set (g, b), denoted by f : (f, a) ∼ −→ (g, b), if for each soft point f xλ ∼ ∈ (f, a) there exists only one soft point gyµ ∼ ∈ (g, b) such that f(f xλ ) = g y µ. now let r be the set of real numbers. we denote the collection of all nonempty bounded subsets of r by b(r). a soft real set, denoted by (f̂, a) or simply f̂ is a mapping f̂ : a → b(r). if f̂ is a single valued mapping on a ⊂ e taking values in r, then the pair (f̂, a) or simply f̂ is called a soft element of r or a soft real number. if f̂ is a single valued mapping on a ⊂ e taking values in the set r+ of nonnegative real numbers, then the pair (f̂, a), or simply f̂, is called a nonnegative soft real number. we shall denote the set of nonnegative soft real numbers by r(a)∗. a constant soft real number c is a soft real number such that for each e ∈ a, we have c(e) = c, where c is some real number. definition 2.1 ([8]). for two soft real numbers f̂, ĝ we say that (i) f̂ ∼ ≤ ĝ if f̂(e) ≤ ĝ(e) for all e ∈ a (ii) f̂ ∼ ≥ ĝ if f̂(e) ≥ ĝ(e) for all e ∈ a (iii) f̂ ∼ < ĝ if f̂(e) < ĝ(e) for all e ∈ a (iv) f̂ ∼ > ĝ if f̂(e) > ĝ(e) for all e ∈ a definition 2.2. let u be a universal set, a be a nonempty subset of parameters and ∼ u be the absolute soft set, i.e f(λ) = u for all λ ∈ a, where (f, a) = ∼ u. a mapping d : sp( ∼ u) × sp( ∼ u) → r(a)∗ is said to be a soft pseudometric on ∼ u if for any uxλ , u y µ, u z λ ∈ sp( ∼ u) (equivalently uxλ , u y µ, u z λ ∼ ∈ ∼ u), the following hold: (i) d(uxλ , u x λ ) = 0 (ii) d(uxλ , u y µ) = d(u y µ, u x λ ) (iii) d(uxλ , u z λ) ∼ ≤ d(uxλ , u y µ) + d(u y µ, u z λ) the soft set ∼ u endowed with a soft pseudometric d is called a soft pseudometric space and is denoted by ( ∼ u, d, a), or simply by ( ∼ u, d) if no confusion arises. definition 2.3. let u be a universal set, a be a nonempty subset of parameters and ∼ u be the absolute soft set, i.e f(λ) = u for all λ ∈ a, where (f, a) = ∼ u. a mapping d : sp( ∼ u) × sp( ∼ u) → r(a)∗ is said to be a soft metric on ∼ u if for any uxλ , u y µ, u z λ ∈ sp( ∼ u) (equivalently uxλ , u y µ, u z λ ∼ ∈ ∼ u), the following hold: © agt, upv, 2021 appl. gen. topol. 22, no. 1 19 h. sabao and o. olela otafudu (i) d(uxλ , u y µ) = 0 iff u x λ = u y µ (ii) d(uxλ , u y µ) = d(u y µ, u x λ ) (iii) d(uxλ , u z λ) ∼ ≤ d(uxλ , u y µ) + d(u y µ, u z λ) the soft set ∼ u endowed with a soft metric d is called a soft metric space and is denoted by ( ∼ u, d, a), or simply by ( ∼ u, d) if no confusion arises. 3. soft quasi-pseudometric spaces in this section, we introduce the concept of a soft quasi-pseudometric space. we show that the symmetrised soft pseudometric coincides with the soft pseudometric in the sense of [16]. definition 3.1. let u be a universal set, a be a nonempty subset of parameters and ∼ u be the absolute soft set. a mapping q : sp( ∼ u) × sp( ∼ u) → r(a)∗ is said to be a soft quasi-pseudometric on ∼ u if for any uxλ , u y µ, u z λ ∈ sp( ∼ u) (equivalently uxλ , u y µ, u z λ ∼ ∈ ∼ u), the following hold: (i) q(uxλ , u x λ ) = 0 (ii) q(uxλ , u z λ) ∼ ≤ q(uxλ , u y µ) + q(u y µ, u z λ). we say q is a soft quasi-metric provided that q also satisfies the following condition: q(uxλ , u y µ) = 0 = q(u y µ, u x λ ) implies u x λ = u y µ. the soft set ∼ u endowed with a soft quasi-pseudometric is called a soft quasi pseudometric space denoted by ( ∼ u, q, a) or simply by ( ∼ u, q) if no confusion arises. remark 3.2. if q is a soft quasi-pseudometric (soft quasi-metric) on ∼ u, then qt : sp( ∼ u) × sp( ∼ u) → r(a)∗ and qs : sp( ∼ u) × sp( ∼ u) → r(a)∗ defined by qt(uxλ , u y µ) = q(u y µ, u x λ ) and q s(uxλ , u y µ) = max{q(u x λ , u y µ), q t(uxλ , u y µ)} are also a soft quasi-pseudometric (soft quasi-metric) and soft pseudometric (soft metric) on ∼ u respectively. note that qs is a soft (pseudometric) metric in the sense of [16]. furthermore, qt is called the conjugate of q. furthermore, we have q(uxλ , u y µ) ≤ q s(uxλ , u y µ) and q t(uxλ , u y µ) ≤ q s(uxλ , u y µ). definition 3.3. let ( ∼ u, q) be a soft quasi-pseudometric space and r̂ be a nonnegative soft real number. for any uxλ ∼ ∈ ∼ u, we define the open and closed balls with radius r̂ and center uxλ respectively as follows: bq(u x λ , r̂) = {u y µ ∼ ∈ ∼ u : q(uxλ , u y µ) ∼ < r̂} and cq(u x λ , r̂) = {u y µ ∼ ∈ ∼ u : q(uxλ , u y µ) ∼ ≤ r̂}. © agt, upv, 2021 appl. gen. topol. 22, no. 1 20 on soft quasi-pseudometric spaces ss(bq(u x λ , r̂)) is called the soft open ball with center u x λ and radius r̂ while ss(cq(u x λ , r̂)) is called the soft closed ball with center u x λ and radius r̂. example 3.4. let u ⊆ r be a non-empty set and a ⊂ r be the non-empty set of parameters. let ∼ u be the absolute soft set, that is, f(λ) = u ∀λ ∈ a, where (f, a) = ∼ u. let x denote the soft real number such that x(λ) = x for all λ ∈ a. furthermore, for constant soft real numbers x and y, put x−̇y = max{x−y, 0}. then q : sp( ∼ u) × sp( ∼ u) −→ r(a)∗, defined by q(uxλ , u y µ) = x−̇y + λ−̇µ, is a soft quasi-metric. proof. (i) q(uxλ , u x λ ) = x−̇x + λ−̇λ = 0 (ii) q(uxλ , u y µ) = q(u y µ, u x λ ) = 0 =⇒ x−̇y + λ−̇µ = 0 =⇒ x−̇y = 0 and λ−̇µ = 0 =⇒ x = y and λ = µ =⇒ uxλ = u y µ. (iii) q(uxλ , u y µ) = x−̇y + λ−̇µ = x−̇y + z−̇z + λ−̇µ + γ−̇γ ∼ ≤ x−̇z + z−̇y + λ−̇γ + γ−̇µ ∼ ≤ q(uxλ , u z γ ) + q(u z γ , u y µ). therefore, ( ∼ u, q, a) is a soft quasi-metric space. � remark 3.5. notice in the example above that qt : sp( ∼ u)×sp( ∼ u) −→ r(a)∗ defined by qt(uxλ , u y µ) = q(u y µ, u x λ ) is also a soft quasi-metric on ∼ u. furthermore, qs : sp( ∼ u) × sp( ∼ u) −→ r(a)∗ defined by qs(uxλ , u y µ) = max{q(u x λ , u y µ), q t(uxλ , u y µ)} = |x − y| + |λ − µ| is a soft metric on ∼ u in the sense of [16]. proposition 3.6. let ( ∼ u, q) be a soft quasi-pseudometric space and uxλ ∼ ∈ ∼ u. then we have the following: (i) bqs(u x λ , r̂) ⊆ bq(u x λ , r̂) (ii) cqs(u x λ , r̂) ⊆ cq(u x λ , r̂) definition 3.7. a soft subset (y, a) in a soft quasi-pseudometric space ( ∼ u, q, a) is said to be τ(q)-soft open if for any soft point uxλ of (y, a), there exists a positive soft real number r̂ such that uxλ ∈ bq(u x λ , r̂) ⊂ sp(y, a). © agt, upv, 2021 appl. gen. topol. 22, no. 1 21 h. sabao and o. olela otafudu remark 3.8. the collection τ(q) of all τ(q)-soft open sets in a soft quasipseudometric space ( ∼ u, q) form a τ(q)-soft topology on ∼ u. similarly, the collection τ(qt) of all τ(qt)-soft open sets in a soft quasi-pseudometric space ( ∼ u, q) form a τ(qt)-soft topology on ∼ u. furthermore, the collection τ(qs) of all τ(qs)soft open sets in a soft quasi-pseudometric space ( ∼ u, q) form a τ(qs)-soft topology on ∼ u. the soft topology τ(qs) is finer than the soft topologies τ(q) and τ(qt). finally the triple (x, τ(q), τ(qt)) is a soft bitopological space. definition 3.9. let ( ∼ u, q) be a soft quasi-pseudometric space. a sequence (uxn λn )n∈n of soft points in ∼ u is said to be τ(q)-convergent in ( ∼ u, q) if there is a soft point uyµ ∼ ∈ ∼ u such that q(uxn λn , u y µ) → 0 as n → ∞. definition 3.10. let ( ∼ u, q) be a soft quasi-pseudometric space. a sequence {uxi λi }i∈n of soft points in ∼ u is said to be τ(qt)-convergent in ( ∼ u, q) if there is a soft point uyµ ∼ ∈ ∼ u such that qt(uxn λn , uyµ) = q(u y µ, u xn λn ) → 0 as n → ∞. proposition 3.11. let (uxn λn )n∈n be a sequence in a soft quasi-pseudometric space ( ∼ u, q). then (i) if (uxn λn )n∈n is τ(q)-convergent to u x λ and τ(q t)-convergent to uyµ, then q(uxλ , u y µ) = 0. (ii) if (uxn λn )n∈n is τ(q)-convergent to u x λ and q(u y µ, u x λ ) = 0, then (u xn λn )n∈n is τ(q)-convergent to uyµ. proof. (i) by letting n → ∞ in the inequality q(uxλ , u y µ) ∼ ≤ q(uxλ , u xn λn )+q(uxn λn , uzµ), we get q(uxλ , u y µ) = 0. (ii) follows from the relation q(uyµ, u xn λn ) ∼ ≤ q(uyµ, u x λ ) + q(u x λ , u xn λn ) → 0 as n → ∞. � definition 3.12. a sequence (uxn λn )n∈n of soft points in a soft metric space ( ∼ u, d) is said to be cauchy in ( ∼ u, d) if for each ǫ̂ ∼ ≥ 0, there exists an m ∈ n such that d(uxi λi , u xj λj ) ∼ < ǫ̂ for all i, j ≥ m. definition 3.13. a soft metric space ( ∼ u, d) is said to be complete if every cauchy sequence in ( ∼ u, d) converges to some soft point of ∼ u. © agt, upv, 2021 appl. gen. topol. 22, no. 1 22 on soft quasi-pseudometric spaces definition 3.14. a soft quasi-metric space ( ∼ u, q) is said to be bicomplete provided that ( ∼ u, qs) is a complete soft metric space. 4. the compatible quasi-pseudometric in [5], abbas et al. introduced the concept of a compatible soft metric and used this concept to prove some fixed point theorems. in this section, we introduce the concept of a compatible soft quasi-pseudometric metric whose symmetrised (pseudo) metric coincides with the compatible metric in the sense of [5]. theorem 4.1. let ( ∼ u, q, a) be a soft quasi-pseudometric with a a finite set. define a function mq : sp( ∼ u) × sp( ∼ u) → r+ as mq(u x λ , u y µ) = max η∈a q(uxλ , u y µ)(η) for all uxλ , u y µ ∈ sp( ∼ u). then the following holds: (i) mq is a quasi-pseudometric on sp( ∼ u) (ii) mq is a quasi-metric on sp( ∼ u) if and only if q is a soft quasi-metric on ∼ u (iii) ( ∼ u, q, a) is bicomplete if and only if (sp( ∼ u), mq) is bicomplete. proof. let uxλ , u y µ, u z γ ∈ sp( ∼ u). then we have (i) we first show that mq satisfies the conditions of a quasi-pseudometric. (i) mq(u x λ , u x λ ) = 0 by condition (i) of definition 3.1. (ii) mq(u x λ , u z γ )+mq(u z γ , u z µ) by condition (ii) of definition 3.1. this is because mq(u x λ , u y µ) = max η∈a q(uxλ , u y µ) ≤ max η∈a q(uxλ , u z γ ) + max η∈a q(uzγ , u y µ) = mq(u x λ , u z γ ) + mq(u z γ , u z µ). therefore, ( ∼ u, mq) is a quasi-pseudometric space. (ii) if q is a quasi-metric on ∼ u, then mq(u x λ , u y µ) = mq(u y µ, u x λ ) = 0 =⇒ u x λ = u y µ. (iii) suppose ( ∼ u, q, a) is bicomplete. then ( ∼ u, qs, a) is complete. then by [5, theorem 1], (sp( ∼ u), (mq) s) is complete and so (sp( ∼ u), mq) is bicomplete. conversely, suppose (sp( ∼ u), mq) is bicomplete. then (sp( ∼ u), (mq) s) is complete. thus by [5, theorem 1], ( ∼ u, qs, a) is complete. therefore, ( ∼ u, q, a) is bicomplete. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 23 h. sabao and o. olela otafudu proposition 4.2. let ( ∼ u, d, a) be a soft quasi-pseudometric space. then the function (mq) t : sp( ∼ u) × sp( ∼ u) → r+ defined by (mq) t(uxλ , u y µ) = mq(u y µ, u x λ ) is also a quasi-pseudometric on sp( ∼ u). moreover, (mq) t = mqt. proof. one can easily check that (mq) satisfies the axioms of a quasi-pseudometric. we only show that (mq) t = mqt. observe that (mq) t(uxλ , u y µ) = mq(u y µ, u x λ ) = max η∈a q(uyµ, u x λ )(η) = max η∈a qt(uxλ , u y µ)(η) = mqt � proposition 4.3. let ( ∼ u, q, a) be a soft quasi-pseudometric space. then for any soft point uxλ of ∼ u the following holds: (i) bq(u x λ , r) = bmq (u x λ , r) and bqt(u x λ , r) = bmqt (u x λ , r) (ii) cq(u x λ , r) = cmq (u x λ , r) and cqt (u x λ , r) = cmqt (u x λ , r) proof. we show that bq(u x λ , r) ⊆ bmq (u x λ , r), the rest follows the same arguments. suppose uyµ ∈ bq(u x λ , r). then q(u x λ , u y µ) ∼ < r. this implies that q(uxλ , u y µ)(η) < r(η) for all η ∈ a. thus max η∈a q(uxλ , u y µ)(η) < r. therefore, mq(u x λ , u y µ) < r and so bq(u x λ , r) ⊆ bmq (u x λ , r). conversely, suppose u y µ ∈ bmq (u x λ , r), then mq(u x λ , u y µ) < r this implies that max η∈a q(uxλ , u y µ)(η) < r. therefore, q(uxλ , u y µ)(η) < r(η) for all η ∈ a. hence q(u x λ , u y µ) ∼ < r and so bmq (u x λ , r) ⊆ bq(u x λ , r) � 5. soft isbell convexity in this section, we extend the concept of isbell convexity, introduced in [13] to soft quasi-pseudometric spaces. definition 5.1. a soft quasi-pseudometric space ( ∼ u, q) is said to be soft isbell convex provided that for each family (uxi λi )i∈i of soft points of ∼ u and families (ri)i∈i and (si)i∈i of constant non-negative soft real numbers satisfying q(uxi λi , u xj λj ) ∼ ≤ ri + sj whenever i, j ∈ i, the following holds: ∼⋂ i∈i ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt(u xi λi , si)) 6= φ. or equivalently ⋂ i∈i cq(u xi λi , ri) ∩ cqt(u xi λi , si) 6= ∅. lemma 5.2. suppose ( ∼ u, q, a), where a is finite, is a soft quasi-pseudometric space. then ( ∼ u, q, a) is soft isbell convex if and only if (sp( ∼ u), mq) is isbell convex. © agt, upv, 2021 appl. gen. topol. 22, no. 1 24 on soft quasi-pseudometric spaces proof. suppose ( ∼ u, q, a) is soft isbell convex. let (uxi λi )i∈i be a family of soft points of ∼ u and (ri)i∈i and (si)i∈i be families of non-negative real numbers satisfying mq(u xj λi , u xj λi ) ≤ ri + sj whenever i, j ∈ i. then max η∈a q(u xj λi , u xj λi )(η) ≤ ri + sj whenever i, j ∈ i. thus q(u xj λi , u xj λi )(η) ≤ ri + sj for all η ∈ a and i, j ∈ i. thus q(u xj λi , u xj λi ) ∼ ≤ ri + sj whenever i, j ∈ i. since ( ∼ u, q, a) is soft isbell convex, we have ∼⋂ i∈i ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt(u xi λi , si)) 6= φ. or equivalently ⋂ i∈i cq(u xi λi , ri) ∩ cqt(u xi λi , si) 6= ∅. then by proposition 4.3, we have ⋂ i∈i cmq (u xi λi , ri) ∩ cm qt (uxi λi , si) = ⋂ i∈i cq(u xi λi , ri) ∩ cqt(u xi λi , si) 6= ∅. therefore, (sp( ∼ u), mq) is isbell convex. conversely, suppose (sp( ∼ u), mq) is isbell convex. let (u xi λi )i∈i be a family of soft points of ∼ u and (ri)i∈i and (si)i∈i be families of non-negative soft real numbers satisfying q(u xj λi , u xj λi ) ∼ ≤ ri + sj whenever i, j ∈ i. then q(u xj λi , u xj λi )(η) ≤ (ri + sj)(η) for all η ∈ a and i, j ∈ i. thus max η q(uxi λi , u xj λj )(η) ≤ ri + sj whenever i, j ∈ i. this implies that mq(u xj λi , u xj λi ) ≤ ri + sj whenever i, j ∈ i. by isbell convexity of (sp( ∼ u, mq), we have ⋂ i∈i cmq (u xi λi , ri) ∩ cm qt (uxi λi , si) 6= ∅. by proposition 4.3, we have ⋂ i∈i cq(u xi λi , ri) ∩ cqt (u xi λi , si) = ⋂ i∈i cmq (u xi λi , ri) ∩ cm qt (uxi λi , si) 6= ∅. hence ∼⋂ i∈i ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt(u xi λi , si)) 6= φ. therefore, ( ∼ u, q, a) is soft isbell convex. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 25 h. sabao and o. olela otafudu definition 5.3. let ( ∼ u, q, a) be a soft quasi-pseudometric space. a family of soft double balls [(ss(cq(u xi λi , ri)))i∈i, (ss(cqt(u xi λi , si)))i∈i], where ri and si are non-negative constant soft real numbers and u xi λi is a soft point of ∼ u whenever i ∈ i, is said to have a mixed binary intersection property if for all indices i, j ∈ i, ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt (u xj λj , sj)) 6= φ or equivalently, cq(u xi λi , ri) ∩ cqt(u xj λj , sj) 6= ∅. definition 5.4. a soft quasi-pseudometric space ( ∼ u, q, a) is said to be soft isbell complete if for every family of soft double balls [(ss(cq(u xi λi , ri)))i∈i , (ss(cqt(u xi λi , si)))i∈i], where ri and si are non-negative constant soft real numbers and u xi λi is a soft point of ∼ u whenever i ∈ i, having a mixed binary intersection property satisfy ∼⋂ i∈i ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt (u xi λi , si)) 6= φ or equivalently ⋂ i∈i cq(u xi λi , ri) ∩ cqt(u xi λi , si) 6= ∅. lemma 5.5. a soft quasi-pseudometric space ( ∼ u, q, a), where a is finite, is soft isbell complete if and only if (sp( ∼ u), mq) is an isbell complete quasipseudometric space. proof. suppose ( ∼ u, q, a) is soft isbell complete. let [(cmq (u xi λi , ri))i∈i, (cm qt (uxi λi , si))i∈i], where ri and si are non-negative real numbers and u xi λi is a soft point of ∼ u whenever i ∈ i, have a mixed binary intersection property. then cmq (u xi λi , ri) ∩ cm qt (u xj λj , sj) 6= ∅. by proposition 4.3, cq(u xi λi , ri) ∩ cqt(u xj λj , sj) 6= ∅. whenever i, j ∈ i. hence the family of soft double balls [(ss(cq(u xi λi , ri)))i∈i , (ss(cqt(u xi λi , si)))i∈i], where ri and si are non-negative constant soft real numbers and u xi λi is a soft point of ∼ u whenever i ∈ i, have a mixed binary intersection property. by isbell © agt, upv, 2021 appl. gen. topol. 22, no. 1 26 on soft quasi-pseudometric spaces completeness of ( ∼ u, q, a), we have ∼⋂ i∈i ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt (u xi λi , si)) 6= φ or equivalently ⋂ i∈i cq(u xi λi , ri) ∩ cqt(u xi λi , si) 6= ∅. therefore by proposition 4.3, ⋂ i∈i cmq (u xi λi , ri) ∩ cm qt (uxi λi , si) = ⋂ i∈i cq(u xi λi , ri) ∩ cqt(u xi λi , si) 6= ∅. conversely, suppose (sp( ∼ u), q, a) is isbell complete. let [(ss(cq(u xi λi , ri)))i∈i , (ss(cqt(u xi λi , si)))i∈i], where ri and si are non-negative soft real numbers and u xi λi is a soft point of ∼ u whenever i ∈ i, be a family of soft double balls having a mixed binary intersection property. then ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt (u xj λj , sj)) 6= φ or equivalently, cq(u xi λi , ri) ∩ cqt (u xj λj , sj) 6= ∅ whenever i, j ∈ i. by proposition 4.3, we have cmq (u xi λi , ri) ∩ cm qt (u xj λj , sj) 6= ∅ whenever i, j ∈ i. since (sp( ∼ u), q, a) is isbell complete, it follows that ⋂ i∈i cmq (u xi λi , ri) ∩ cm qt (uxi λi , si) 6= ∅. by proposition 4.3, we have ⋂ i∈i cq(u xi λi , ri) ∩ cqt (u xi λi , si) = ⋂ i∈i cmq (u xi λi , ri) ∩ cm qt (uxi λi , si) 6= ∅. therefore, ∼⋂ i∈i ss(cq(u xi λi , ri)) ∼ ∩ ss(cqt(u xi λi , si)) 6= φ. hence, ( ∼ u, q, a) is soft isbell complete. � definition 5.6. a soft quasi-pseudometric space ( ∼ u, q, a) is said to be soft metrically convex if for any soft points uxλ and u y µ of ∼ u and non-negative constant soft real numbers r and s, such that q(uxλ , u y µ) ∼ ≤ r + s there exists a © agt, upv, 2021 appl. gen. topol. 22, no. 1 27 h. sabao and o. olela otafudu soft point uzλ of ∼ u such that q(uxλ , u z λ) ∼ ≤ r and q(uzλ, u y µ) ∼ ≤ s or equivalently cq(u x λ , r) ∩ cqt(u y µ, s) 6= ∅. remark 5.7. notice that cq(u x λ , r)∩cqt (u y µ, s) 6= ∅ is equivalent to ss(cq(u x λ , r))∩ ss(cqt(u y µ, s)) 6= φ. lemma 5.8. a soft quasi-pseudometric space ( ∼ u, q, a), where a is finite, is soft metrically convex if and if (sp( ∼ u), mq) is metrically convex. proof. suppose ( ∼ u, q, a) is soft metrically convex. let uxλ and u y µ be soft points of ∼ u and r and s are non-negative real numbers such that mq(u x λ , u y µ) ≤ r + s. then max η∈a q(uxλ , u y µ)(η) ≤ r + s. thus q(u x λ , u y µ)(η) ≤ r + s for all η ∈ a. therefore, q(uxλ , u y µ) ∼ ≤ r + s. by soft metric convexity of ( ∼ u, q, a), we have cq(u x λ , r) ∩ cqt (u y µ, s) 6= ∅. by proposition 4.3, we have cmq (u x λ , r) ∩ cm qt (uyµ, s) 6= ∅. therefore, (sp( ∼ u), mq) is metrically convex. conversely, suppose (sp( ∼ u), mq) is metrically convex. let u x λ and u y µ be soft points of ∼ u and r and s are non-negative soft real numbers such that q(uxλ , u y µ) ≤ r + s. then q(u x λ , u y µ) ≤ r(η) + s(η) for all η ∈ a. then max η∈a q(uxλ , u y µ)(η) ≤ r + s and by soft metric convexity of (sp( ∼ u), mq), we have cmq (u x λ , r)∩cmqt (u y µ, s) 6= ∅. therefore, ( ∼ u, q, a) is soft metrically convex. � lemma 5.9. a soft quasi-pseudometric space ( ∼ u, q, a), where a is finite, is soft isbell convex if and if ( ∼ u, q, a) is soft-isbell complete and soft metrically convex. proof. suppose ( ∼ u, q, a) is soft isbell convex. then by lemma 5.2, (sp( ∼ u), mq) is isbell convex. by [13, lemma 3.1.1], (sp( ∼ u), mq) is isbell complete and metrically convex. by lemma 5.5 and lemma 5.8, ( ∼ u, q, a) is soft metrically convex and soft isbell complete. conversely, suppose ( ∼ u, q, a) is soft metrically convex and soft isbell complete. then by lemma 5.5 and lemma 5.8, (sp( ∼ u), mq) is isbell complete and metrically convex. therefore, by [13, lemma 3.1.1] (sp( ∼ u), mq) is isbell convex. therefore, by lemma 5.2, ( ∼ u, q, a) is soft metrically convex. � proposition 5.10. suppose ( ∼ u, q, a) is a soft isbell convex soft quasi-metric space. then ( ∼ u, q, a) is bicomplete. © agt, upv, 2021 appl. gen. topol. 22, no. 1 28 on soft quasi-pseudometric spaces proof. since ( ∼ u, q, a) is soft isbell convex. then (sp( ∼ u), mq) is isbell convex. by [13, corollary 3.1.3] (sp( ∼ u), mq) is bicomplete. this implies that ( ∼ u, q, a) is bicomplete by theorem 4.1. � definition 5.11. a soft quasi-pseudometric space ( ∼ u, q, a) is said to be bounded if for each soft points uxλ and u y µ of ∼ u, there exists a positive soft real number k̂ such that q(uxλ , u y µ) ∼ ≤ k̂. remark 5.12. notice that if ( ∼ u, q, a) is a soft quasi-pseudometric space, where a is finite, then boundedness of ( ∼ u, q, a) implies boundedness of (sp( ∼ u), mq). theorem 5.13. if ( ∼ u, q, a), where a is finite, is a bounded isbell convex soft quasi-metric space and t : ( ∼ u, q, a) −→ ( ∼ u, q, a) is a non-expansive map, then the fixed point set fix(t ) of t in ( ∼ u, q, a) is nonempty and soft isbell convex. proof. since ( ∼ u, q, a) is soft isbell convex, then (sp( ∼ u), mq) is isbell convex. also, since ( ∼ u, q, a) is bounded, then (sp( ∼ u), mq) is bounded by remark 5.12. furthermore, since f : ( ∼ u, q, a) −→ ( ∼ u, q, a) satisfies q(f(uxλ ), f(u y µ)) ∼ ≤ q(uxλ , u y µ) for all uxλ , u y µ ∈ sp( ∼ u). then we have mq(f(u x λ ), f(u y µ)) = max η∈a q(f(uxλ ), f(u y µ))(η) ≤ max η∈a q(uxλ , u y µ)(η) = mq(u x λ , u y µ) therefore, f : (sp( ∼ u), mq) −→ (sp( ∼ u), mq) is a non-expansive map and by [11, theorem 3.3] fix(f) is nonemepty and isbell convex. by lemma 5.2, fix(f) is soft isbell convex. � acknowledgements. the authors would like to thank the anonymous referee for the suggestions that have improved the presentation of this paper references [1] h. aktas and n. cagman, soft sets and soft groups, inform. sci. 177 (2007), 2226–2735. 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[16] s. das and s. k. samanta, soft metric, ann fuzzy math. inform. 6 (2013), 77–94. © agt, upv, 2021 appl. gen. topol. 22, no. 1 30 @ appl. gen. topol. 21, no. 2 (2020), 331-347 doi:10.4995/agt.2020.13926 c© agt, upv, 2020 weak proximal normal structure and coincidence quasi-best proximity points farhad fouladi a, ali abkar a and erdal karapınar b,c a department of pure mathemathics, faculty of science, imam khomeini international university, qazvin 34149, iran (fa folade@yahoo.com; abkar@sci.ikiu.ac.ir) b etsi division of applied mathematics, thu dau mot university, binh duong province, vietnam (erdalkarapinar@tdmu.edu.vn, erdalkarapinar@yahoo.com) c department of mathematics, çankaya university, 06790, etimesgut, ankara, turkey (erdal.karapinar@cankaya.edu.tr) communicated by s. romaguera abstract we introduce the notion of pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. we study the best proximity point problem for this class of mappings. we also study the same problem for the class of pointwise noncyclic-noncyclic relatively nonexpansive pairs involving orbits. finally, under the assumption of weak proximal normal structure, we prove a coincidence quasi-best proximity point theorem for pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. examples are provided to illustrate the observed results. 2010 msc: 47h09; 46b20;46t99. keywords: pointwise cyclic-noncyclic pairs; weak proximal normal structure; coincidence quasi-best proximity point. 1. introduction let a, b be nonempty subsets of banach space x. a mapping t : a ∪ b → a ∪ b is said to be cyclic provided that t(a) ⊆ b and t(b) ⊆ a. on the other hand, a mapping s : a ∪ b → a ∪ b is said to be noncyclic if s(a) ⊆ a and s(b) ⊆ b. received 26 june 2020 – accepted 15 july 2020 http://dx.doi.org/10.4995/agt.2020.13926 f. fouladi, a. abkar and e. karapınar for a cyclic mapping t : a ∪ b → a ∪ b, a point p ∈ a ∪ b is said to be a best proximity point provided that d(p, tp) = dist(a, b). furthermore, we say that a pair (a, b) of subsets in a banach space satisfies a property if each of the sets a and b has that property. similarly, the pair (a, b) is called convex if both a and b are convex; moreover we write (a, b) ⊆ (e, f) ⇔ a ⊆ e, b ⊆ f. in addition, we will use the following notations: δ(a, b) = sup{‖x − y‖ : x ∈ a, y ∈ b}; δ(x, b) = sup{‖x − y‖ : y ∈ b}. for a nonempty, bounded and convex subset f of a banach space x, we write rx(f) = sup{‖x − y‖ : y ∈ f}; r(f) = inf{rx(f) : x ∈ f}; fc = {x ∈ f : rx(f) = r(f)}. in 2017, m. gabeleh introduced the notion of a pointwise cyclic relatively nonexpansive mapping involving orbits, and proved a theorem on the existence of best proximity points. definition 1.1 ([11]). let (a, b) be a nonempty pair of subsets of a banach space x. a mapping t : a∪b → a∪b is said to be pointwise cyclic relatively nonexpansive involving orbits if t is cyclic and for any (x, y) ∈ a × b, if ‖x − y‖ = dist(a, b), then ‖tx − ty‖ = dist(a, b), and otherwise, there exists a function α : a × b → [0, 1] such that ‖tx − ty‖ ≤ α(x, y)‖x − y‖ + (1 − α(x, y)) min{δx[o2(y; ∞)], δy[o2(x; ∞)]}, where, for any (x, y) ∈ a × b δx[o2(y; ∞)] = sup n∈n ‖x − t 2ny‖, δy[o2(x; ∞)] = sup n∈n ‖t 2nx − y‖. note that, if a = b, then we say that t is a pointwise nonexpansive mapping involving orbits. in [12], m. gabeleh, o. olela otafudu, and n. shahzad considered a pair of mappings t and s. according to [12], for a nonempty pair of subsets (a, b) in a metric space (x, d), and a cyclic-noncyclic pair (t ; s) on a∪b (that is, t : a∪b → a∪b is cyclic and s : a∪b → a∪b is noncyclic); they called a point p ∈ a ∪ b a coincidence best proximity point for (t ; s) if d(sp, tp) = dist(a, b). note that if s = i, the identity map on a ∪ b, then p ∈ a ∪ b is a best proximity point for t . c© agt, upv, 2020 appl. gen. topol. 21, no. 2 332 weak proximal normal structure in 2019, a. abkar and m. norouzian introduced the concept of coincidence quasi-best proximity point and proved the existence of such points for quasicyclic-noncyclic contraction pairs. we remark that the coincidence quasi-best proximity point theory is more general and includes both the best proximity point theory and the coincidence best proximity point theory. definition 1.2 ([2]). let (a, b) be a nonempty pair of subsets of a metric space (x, d) and t, s : x → x be a quasi-cyclic-noncyclic pair on a ∪ b; that is, t(a) ⊆ s(b) and t(b) ⊆ s(a). a point p ∈ a ∪ b is said to be a coincidence quasi-best proximity point for (t ; s) if d(sp, tp) = dist(s(a), s(b)). in case that s = i, the point p reduces to a best proximity point for t . in this article, we will focus on the coincidence quasi-best proximity point problem for pointwise cyclic-noncyclic and noncyclic-noncyclic relatively nonexpansive pairs. to do this, we need to recall some definitions and theorems. we begin with the following definition which is a modification of a concept in [8]. definition 1.3. let (a, b) be a nonempty pair of subsets of a banach space x and s : a ∪ b → a ∪ b be a noncyclic mapping on a ∪ b. a convex pair (s(a), s(b)) is called a proximal pair if for each (a1, b1) ∈ a × b, there exists (a2, b2) ∈ a × b such that for each i, j ∈ {1, 2} with i ∕= j we have ‖sai − sbj‖ = dist(s(a), s(b)). given (a, b) a pair of nonempty subsets of a banach space x, the associated proximal pair of (s(a), s(b)) is the pair (s(as0), s(b s 0)) given by as0 := {a ∈ a : ‖sa − sb‖ = dist(s(a), s(b)) for some b ∈ b}, bs0 := {b ∈ b : ‖sa − sb‖ = dist(s(a), s(b)) for some a ∈ a}, in fact, if the pair (s(a), s(b)) is nonempty, weakly compact and convex, then its associated pair (s(as0), s(b s 0)) is also nonempty, weakly compact and convex. furthermore, we have dist(s(as0), s(b s 0)) = dist(s(a), s(b)). the proof of the above statements goes in the same lines as in the case for the pair (a, b); see for instance [21]. here’s a definition we derive from [8] and we’ve made some changes to meet our needs. definition 1.4. let (k1, k2) be a nonempty pair of subsets of a banach space x and s : k1 ∪k2 → k1 ∪k2 be a noncyclic mapping on k1 ∪k2. we say that a convex pair (s(k1), s(k2)) has proximal normal structure (pns) if for any closed, bounded, convex and proximal pair (s(h1), s(h2)) ⊆ (s(k1), s(k2)) which dist(s(h1), s(h2)) = dist(s(k1), s(k2)), δ(s(h1), s(h2)) > dist(s(h1), s(h2)), c© agt, upv, 2020 appl. gen. topol. 21, no. 2 333 f. fouladi, a. abkar and e. karapınar there exists (x, y) ∈ h1 × h2 such that δ(sx, s(h2)) < δ(s(h1), s(h2)), δ(sy, s(h1)) < δ(s(h1), s(h2)). note that the pair (k, k) has proximal normal structure if and only if k has normal structure in the sense of brodskii and milman (see [4] and [20]). theorem 1.5 ([8]). every bounded, closed and convex pair in a uniformly convex banach space x has proximal normal structure. the following definition is a modification of what already appeared in [11]. definition 1.6. let (k1, k2) be a nonempty pair of subsets of a banach space x and s : k1 ∪ k2 → k1 ∪ k2 be a noncyclic mapping on k1 ∪ k2. we say that a convex pair (s(k1), s(k2)) has weak proximal normal structure (wpns) if for each nonempty, weakly compact and convex proximal pair (s(h1), s(h2)) ⊆ (s(k1), s(k2)) for which dist(s(h1), s(h2)) = dist(s(k1), s(k2)), δ(s(h1), s(h2)) > dist(s(h1), s(h2)), there exists (x, y) ∈ h1 × h2 such that δ(sx, s(h2)) < δ(s(h1), s(h2)), δ(sy, s(h1)) < δ(s(h1), s(h2)). in this article, we intend to generalize some results of [8] and [11]. our results have the following advantages: first, we introduce the class of the pointwise cyclic-noncyclic and noncyclic-noncyclic relatively nonexpansive pairs involving orbits, that in particular, includes the class of pointwise cyclic-noncyclic and noncyclic-noncyclic relatively nonexpansive mappings involving orbits. second, we consider a pair of mappings while the previous articles are concerned with one single mapping, and finally, we study the coincidence quasi-best proximity point problem, which in particular, includes the best proximity point problem as a special case. 2. cyclic-noncyclic pairs we begin this section by introducing the new concept of a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits. definition 2.1. assume that (a, b) is a nonempty pair of subsets of a banach space x and t, s : a ∪ b → a ∪ b are two mappings. a pair (t ; s) is said to be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits if (t ; s) is a cyclic-noncyclic pair and for any (x, y) ∈ a × b, if ‖x − y‖ = dist(s(a), s(b)), then ‖tx − ty‖ = dist(s(a), s(b)), ‖sx − sy‖ = dist(s(a), s(b)) and otherwise, there exists a function α : a × b → [0, 1] such that ‖tx−ty‖ ≤ α(x, y)‖sx−sy‖+(1−α(x, y)) max{δx[o2(y; ∞)], δy[o2(x; ∞)]}, where, for any (x, y) ∈ a × b δx[o2(y; ∞)] = sup n∈n ‖x − t 2ny‖, δy[o2(x; ∞)] = sup n∈n ‖t 2nx − y‖. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 334 weak proximal normal structure we note that if s = i, then the class of pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits reduces to the class of pointwise cyclic relatively nonexpansive mappings involving orbits introduced in [11]. definition 2.2 ([20]). we say that a banach space x has the property (c) if every bounded decreasing sequence of nonempty, closed and convex subsets of x have a nonempty intersection. for c ⊆ x, we denote the diameter of c by δ(c). a point x ∈ c is a diametral point of c provided that sup{‖x − y‖ : y ∈ c} = δ(c). a convex set k ⊆ x is said to have normal structure if for each bounded convex subset h of k which contains at least two points, there is some point x ∈ h which is not a diametral point of h. lemma 2.3 ([20]). assume that x is a banach space with the property (c), then fc is nonempty, closed and convex. lemma 2.4 ([20]). assume that f is a closed and convex subset of a banach space x which contains at least two points. if f has normal structure, then δ(fc) < δ(f). theorem 2.5. assume that k is a nonempty, bounded, closed and convex subset of a banach space x with property (c). suppose that k has normal structure. let (t, s) be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits on k. then there exists a point p ∈ k such that ‖tp−sp‖ = 0. proof. suppose γ denotes the collection of all nonempty, closed and convex subsets of k such that (t, s) is a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits on k. by zorn’s lemma, γ has a minimal member, say f . we complete the proof by verifying that f consists of a single point. assume that x ∈ fc. in this case, for any y ∈ fc we have ‖sx − y‖ ≤ sup{‖z − y‖ : z ∈ f} = ry(f) = r(f), therefore, sup{‖sx − y‖ : x ∈ fc} ≤ r(f). then, rsx(f) = sup{‖sx − y‖ : y ∈ f} ≤ sup{‖sx − y‖ : x ∈ fc, y ∈ f} ≤ sup{r(f), y ∈ f} = r(f). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 335 f. fouladi, a. abkar and e. karapınar then, for any x ∈ fc we have rsx(f) = r(f); that is, s : fc → fc. moreover, for any x, y ∈ fc we have ‖sx−sy‖ ≤ r(f). on other hand, for any x, y ∈ fc, δx[o2(y; ∞)] = sup n∈n ‖x − t 2ny‖ ≤ sup{‖x − z‖ : z ∈ f} = rx(f) = r(f). similarly, for any x, y ∈ fc we have δy[o2(x; ∞)] ≤ r(f). in particular, for each x, y ∈ fc, ‖tx − ty‖ ≤ α(x, y)‖sx − sy‖ + (1 − α(x, y)) max{δx[o2(y; ∞)], δy[o2(x; ∞)]} ≤ α(x, y)r(f) + (1 − α(x, y))r(f) = r(f); that is, rt x(f) = r(f). then, t : fc → fc. by lemma 2.3, we have fc ∈ γ. if δ(f) > 0, then by lemma 2.4, fc is properly contained in f which contradicts the minimality of f . hence δ(f) = 0 and f consists of a single point; this is, there exists a point p ∈ k such that tp = p and sp = p. so, there exists a p ∈ k such that ‖tp − p‖ = 0. □ theorem 2.6. assume that (a, b) is a nonempty pair of subsets in a banach space x with pns. let t, s : a ∪ b → a ∪ b be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits, and such that t(a) ⊆ s(b) and t(b) ⊆ s(a). suppose that (s(a), s(b)) is a weakly compact and convex pair of subsets in x. then there exists (x, y) ∈ a × b such that for p ∈ {x, y} we have ‖tp − sp‖ = dist(s(a), s(b)). proof. the result follows from theorem 2.5 if dist(s(a), s(b)) = 0, so we assume that dist(s(a), s(b)) > 0. let (s(as0), s(b s 0)) be the associated proximal pair of (s(a), s(b)). we have already observed that s(as0) and s(b s 0) are nonempty, weakly compact and convex, moreover dist(s(as0), s(b s 0)) = dist(s(a), s(b)). assume that x ∈ as0, then there exists y ∈ bs0 such that ‖sx − sy‖ = dist(s(a), s(b)). on other hand, (t ; s) is a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits. thus, ‖t(sx) − t(sy)‖ = dist(s(a), s(b)), ‖s(sx) − s(sy)‖ = dist(s(a), s(b)). this implies that ‖s(sx) − s(sy)‖ = dist(s(as0), s(b s 0)), and ‖t(sx) − t(sy)‖ = dist(s(as0), s(b s 0)). therefore, we have t(sx) ∈ s(bs0), t(sy) ∈ s(a s 0); c© agt, upv, 2020 appl. gen. topol. 21, no. 2 336 weak proximal normal structure that is, t(s(as0)) ⊆ s(b s 0), t(s(b s 0)) ⊆ s(a s 0). similarly, s(s(as0)) ⊆ s(a s 0), s(s(b s 0)) ⊆ s(b s 0). so, for each x ∈ as0 and y ∈ bs0 we have ‖t(sx) − t(sy)‖ = dist(s(as0), s(b s 0)), and ‖s(sx) − s(sy)‖ = dist(s(as0), s(b s 0)). clearly (s(as0), s(b s 0)) also has proximal normal structure. now, assume that ω denotes the collection of all nonempty subsets s(f) of s(as0) ∪ s(bs0) for which s(f) ∩ s(as0) and s(f) ∩ s(bs0) are nonempty, closed, convex, and such that t(s(f) ∩ s(as0)) ⊆ s(f) ∩ s(b s 0), t(s(f) ∩ s(b s 0)) ⊆ s(f) ∩ s(a s 0), and s(s(f) ∩ s(as0)) ⊆ s(f) ∩ s(a s 0), s(s(f) ∩ s(b s 0)) ⊆ s(f) ∩ s(b s 0), and so dist(s(f) ∩ s(as0), s(f) ∩ s(b s 0)) = dist(s(a), s(b)). since, s(as0)∪s(bs0) ∈ ω and ω is nonempty, we may assume that {s(fα)}α∈ω is a decreasing chain in ω such that s(f0) = ∩α∈ωs(fα). then s(f0) ∩ s(as0) = ∩α∈ω(s(fα) ∩ s(as0)), so s(f0) ∩ s(as0) is nonempty, closed and convex. similarly, s(f0) ∩ s(bs0) is nonempty, closed and convex. also, t(s(f0) ∩ s(as0)) ⊆ s(f0) ∩ s(b s 0), t(s(f0) ∩ s(b s 0)) ⊆ s(f0) ∩ s(a s 0) and s(s(f0) ∩ s(as0)) ⊆ s(f0) ∩ s(a s 0), s(s(f0) ∩ s(b s 0)) ⊆ s(f0) ∩ s(b s 0). to show that s(f0) ∈ ω we only need to verify that dist(s(f0) ∩ s(as0), s(f0) ∩ s(b s 0)) = dist(s(a), s(b)). note that for each α ∈ j it is possible to select sxα ∈ s(fα) ∩ s(as0), syα ∈ s(fα) ∩ s(b s 0) such that ‖sxα − syα‖ = dist(s(a), s(b)). it is also possible to choose convergent subnets {sxα′} and {syα′} (with the same indices), say lim α′ sxα′ = sx, lim α′ syα′ = sy. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 337 f. fouladi, a. abkar and e. karapınar then clearly sx ∈ s(f0) ∩ s(as0) and sy ∈ s(f0) ∩ s(bs0). by weak lower semicontinuity of the norm, we have ‖sx − sy‖ ≤ dist(s(a), s(b)); hence, dist(s(a), s(b)) ≤ dist(s(f0) ∩ s(as0), s(f0) ∩ s(b s 0)) ≤ ‖sx − sy‖ ≤ dist(s(a), s(b)). therefore, dist(s(f0) ∩ s(as0), s(f0) ∩ s(b s 0)) = dist(s(a), s(b)). since, every chain in ω is bounded below by a member of ω, zorn’s lemma implies that ω has a minimal element, say s(k). assume that s(k1) = s(k)∩ s(as0) and s(k2) = s(k) ∩ s(bs0). observe that if δ(s(k1), s(k2)) = dist(s(k1), s(k2)), then for any x ∈ s(k1), we have ‖tx − sx‖ = dist(s(k1), s(k2)) = dist(s(a), s(b)). similarly, for any y ∈ s(k2), we have ‖ty − sy‖ = dist(s(k1), s(k2)) = dist(s(a), s(b)). now, we assume that δ(s(k1), s(k2)) > dist(s(k1), s(k2)). we complete the proof by showing that this leads to a contradiction. since s(k) is minimal, it follows that (s(k1), s(k2)) is a proximal pair in (s(a s 0), s(b s 0)). by the pns property of x, there exist (x1, y1) ∈ k1 × k2 and β ∈ (0, 1) such that δ(sx1, s(k2)) ≤ βδ(s(k1), s(k2)) and δ(sy1, s(k1)) ≤ βδ(s(k1), s(k2)). since, (s(k1), s(k2)) is a proximal pair, there exists (x2, y2) ∈ k1 × k2 such that for each distinct i, j ∈ {1, 2}, we have ‖sxi − syj‖ = dist(s(k1), s(k2)). so, for each u ∈ s(k2) we have ‖ sx1 + sx2 2 − u‖ ≤ ‖ sx1 − u 2 ‖ + ‖ sx2 − u 2 ‖ ≤ βδ(s(k1), s(k2)) 2 + δ(s(k1), s(k2)) 2 = αδ(s(k1), s(k2)), where α = 1+β 2 ∈ (0, 1). assume that sw1 = (sx1+sx2) 2 and sw2 = (sy1+sy2) 2 . then δ(sw1, s(k2)) ≤ αδ(s(k1), s(k2)) and δ(sw2, s(k1)) ≤ αδ(s(k1), s(k2)). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 338 weak proximal normal structure since, dist(s(k1), s(k2)) ≤ ‖sw1 − sw2‖ = ‖ (sx1 + sx2) 2 − (sy1 + sy2) 2 ‖ ≤ 1 2 [‖sx1 − sy2‖ + ‖sx2 − sy1‖] = dist(s(k1), s(k2)), we have ‖sw1 − sw2‖ = dist(s(k1), s(k2)). put s(l1) = {sx ∈ s(k1) : δ(sx, s(k2)) ≤ αδ(s(k1), s(k2))}, s(l2) = {sy ∈ s(k2) : δ(sy, s(k1)) ≤ αδ(s(k1), s(k2))}. then for each i = 1, 2, s(li) is a nonempty, closed and convex subset of s(ki) and since sw1 ∈ s(l1) and sw2 ∈ s(l2), we have dist(s(k1), s(k2)) ≤ dist(s(l1), s(l2)) ≤ ‖sw1 − sw2‖ = dist(s(k1), s(k2)). therefore, dist(s(l1), s(l2)) = dist(s(k1), s(k2)) = dist(s(a), s(b)). now, assume that sx ∈ s(l1) and sy ∈ s(k2). then sx ∈ s(as0) and sy ∈ s(bs0); that is, x ∈ as0 and y ∈ bs0. thus, ‖t(sx) − t(sy)‖ = dist(s(a), s(b)) ≤ δ(sx, s(k2)) ≤ αδ(s(k1), s(k2)). so, t(sy) ∈ b(t(sx); αδ(s(k1), s(k2))) ∩ s(k1); that is, t(s(k2)) ⊆ b(t(sx); αδ(s(k1), s(k2))) ∩ s(k1) := s(k′1). clearly s(k′1) is closed and convex. also, if sy ∈ s(k2) satisfies ‖sx − sy‖ = dist(s(a), s(b)), then ‖t(sx) − t(sy)‖ = dist(s(k1), s(k2)). since, t(sy) ∈ s(k′1), we conclude that dist(s(k′1), s(k2)) = dist(s(a), s(b)). therefore, s(k′1) ∪ s(k2) ∈ ω and by the minimality of s(k) we must have s(k′1) = s(k1). hence, s(k1) ⊆ b(t(sx); αδ(s(k1), s(k2))); that is, δ(t(sx), s(k1)) ≤ αδ(s(k1), s(k2)) and since sx ∈ s(l1) was arbitrary, we obtain t(s(l1)) ⊆ s(l2). similarly, t(s(l2)) ⊆ s(l1), s(s(l1)) ⊆ s(l1) and s(s(l2)) ⊆ s(l2). thus, s(l1)∪s(l2) ∈ ω, but δ(s(l1), s(l2)) ≤ αδ(s(k1), s(k2)), contradicting the minimality of s(k). □ corollary 2.7. assume that (a, b) is a nonempty pair of subsets in a uniformly convex banach space x. let t, s : a ∪ b → a ∪ b be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits, and such that t(a) ⊆ s(b) and t(b) ⊆ s(a). suppose that (s(a), s(b)) is a bounded, c© agt, upv, 2020 appl. gen. topol. 21, no. 2 339 f. fouladi, a. abkar and e. karapınar closed and convex pair of subsets in x. then there exists (x, y) ∈ a × b such that for p ∈ {x, y} we have ‖tp − sp‖ = dist(s(a), s(b)). 3. noncyclic-noncyclic pairs in this section we study the case in which both mappings are noncyclic. indeed, we first introduce a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits, and proceed to study its best proximity points. definition 3.1. assume that (a, b) is a nonempty pair of subsets of a banach space x and t, s : a ∪ b → a ∪ b are two mappings. a pair (t ; s) is said to be a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits if (t ; s) is a noncyclic-noncyclic pair and for any (x, y) ∈ a × b, if ‖x − y‖ = dist(s(a), s(b)), then ‖tx − ty‖ = dist(s(a), s(b)), ‖sx − sy‖ = dist(s(a), s(b)) and otherwise, there exists a function α : a × b → [0, 1] such that ‖tx − ty‖ ≤ α(x, y)‖sx − sy‖ + (1 − α(x, y)) max{δx[o(y; ∞)], δy[o(x; ∞)]}, where, for any (x, y) ∈ a × b δx[o(y; ∞)] = sup n∈n ‖x − t ny‖, δy[o(x; ∞)] = sup n∈n ‖t nx − y‖. theorem 3.2. assume that (a, b) is a nonempty pair of subsets in a strictly convex banach space x with pns, and t, s : a ∪ b → a ∪ b is a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits such that t(a) ⊆ s(a) and t(b) ⊆ s(b). suppose that (s(a), s(b)) is a weakly compact and convex pair of subsets in x. then, there exists x0 ∈ a and y0 ∈ b such that tx0 = x0, ty0 = y0 and ‖x0 − y0‖ = dist(s(a), s(b)). proof. suppose that (s(as0), s(b s 0)) is the associated proximal pair of (s(a), s(b)), and choose x ∈ as0. then there exists y ∈ bs0 such that ‖sx − sy‖ = dist(s(a), s(b)), and furthermore ‖t(sx) − t(sy)‖ = dist(s(a), s(b)) = dist(s(as0), s(b s 0)). thus, t : s(as0) → s(as0) and similarly, t : s(bs0) → s(bs0). now let ω denote the collection of nonempty subsets s(f) of s(as0) ∪ s(bs0) for which s(f) ∩ s(as0) and s(f) ∩ s(bs0) are nonempty, closed and convex, t(s(f) ∩ s(as0)) ⊆ s(f) ∩ s(a s 0), t(s(f) ∩ s(b s 0)) ⊆ s(f) ∩ s(b s 0), s(s(f) ∩ s(as0)) ⊆ s(f) ∩ s(a s 0), s(s(f) ∩ s(b s 0)) ⊆ s(f) ∩ s(b s 0) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 340 weak proximal normal structure and dist(s(f) ∩ s(as0), s(f) ∩ s(b s 0)) = dist(s(a), s(b)). since, s(as0) ∪ s(bs0) ∈ ω, ω is nonempty. we proceed as in the proof of theorem 2.6 to show that ω has a minimal element s(k). assume that s(k1) = s(k) ∩ s(as0), and s(k2) = s(k) ∩ s(bs0). first, assume that one of the sets is a singleton, say s(k1) = {x}. then tx = x and if y is the unique point of s(k2) for which ‖x − y‖ = dist(s(k1), s(k2)), it must be the case that ty = y. since, ‖y − x‖ = dist(s(a), s(b)), we are finished. so, we may assume that s(k1) and s(k2) have positive diameter and because the space is strictly convex, this in turn implies that δ(s(k1), s(k2)) > dist(s(k1), s(k2)). we shall see that this leads to a contradiction. since (s(as0), s(b s 0)) has proximal normal structure, we may define s(l1) and s(l2) as in the proof of theorem 2.6. choose sx ∈ s(l1). for any sy ∈ s(k2), we have sx ∈ s(as0) and sy ∈ s(bs0); that is, x ∈ as0 and y ∈ bs0. thus, ‖sx − sy‖ = dist(s(a), s(b)) and so, ‖t(sx) − t(sy)‖ = dist(s(a), s(b)) ≤ δ(sx, s(k2)) ≤ αδ(s(k1), s(k2)). this implies that t(sy) ∈ b(t(sx); αδ(s(k1), s(k2))) ∩ s(k2), thus, t(s(k2)) ⊆ b(t(sx); αδ(s(k1), s(k2))) ∩ s(k2). it follows from the minimality of s(k) that s(k2) ⊆ b(t(sx); αδ(s(k1), s(k2))) and this in turn implies that δ(t(sx), s(k2)) ≤ αδ(s(k1), s(k2)). therefore, t(sx) ∈ s(l1); in fact t(s(l1)) ⊆ s(l1). similarly, t(s(l2)) ⊆ s(l2), s(s(l1)) ⊆ s(l1) and s(s(l2)) ⊆ s(l2). since, s(l1) and s(l2) are, respectively, nonempty, closed and convex subsets of s(k1) and s(k2) and since for α < 1 we have δ(s(l1), s(l2)) ≤ αδ(s(k1), s(k2)), which contradicts the minimality of s(k). □ corollary 3.3. assume that (a, b) is a nonempty pair of subsets in a uniformly convex banach space x and t, s : a ∪ b → a ∪ b is a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits such that t(a) ⊆ s(a) and t(b) ⊆ s(b). suppose that (s(a), s(b)) is a bounded, closed and convex pair of subsets in x. then, there exists x0 ∈ a and y0 ∈ b such that tx0 = x0, ty0 = y0 and ‖x0 − y0‖ = dist(s(a), s(b)). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 341 f. fouladi, a. abkar and e. karapınar 4. wpns and cyclic-noncyclic pairs in this section, and under weak proximal normal structure, we discuss the coincidence quasi-best proximity point problem for pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. lemma 4.1. assume that (a, b) is a nonempty pair of subsets in a banach space x, and t, s : a ∪ b → a ∪ b is a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits such that t(a) ⊆ s(b) and t(b) ⊆ s(a). suppose that (s(a), s(b)) is a weakly compact and convex pair of subsets in x. then, there exists (s(k1), s(k2)) ⊆ (s(as0), s(bs0)) ⊆ (s(a), s(b)) which is minimal with respect to being nonempty, closed, convex and t and s-invariant pair of subsets of (s(a), s(b)), such that dist(s(k1), s(k2)) = dist(s(a), s(b)). moreover, the pair (s(k1), s(k2)) is proximal. proof. the proof essentially goes in the same lines as in the proof of theorem 2.6. we omit the details. □ theorem 4.2. assume that (a, b) is a nonempty pair of subsets in a banach space x with wpns, and t, s : a ∪ b → a ∪ b is a pointwise cyclicnoncyclic relatively nonexpansive pair involving orbits such that t(a) ⊆ s(b) and t(b) ⊆ s(a). suppose that (s(a), s(b)) is a weakly compact and convex pair of subsets in x. then (t ; s) has a coincidence quasi-best proximity point. proof. by lemma 4.1, assume that (s(k1), s(k2)) is a minimal, weakly compact, convex and proximal pair which is t and s-invariant, and such that dist(s(k1), s(k2)) = dist(s(a), s(b)). notice that con(t(s(k1))) ⊆ s(k2) and so, t(con(t(s(k1)))) ⊆ t(s(k2)) ⊆ con(t(s(k2))). similarly, t(con(t(s(k2)))) ⊆ con(t(s(k1))); that is, t is cyclic on con(t(s(k1))) ∪ con(t(s(k2))). on other hand, s is noncyclic on con(s(s(k1))) ∪ con(s(s(k2))). the minimality of (s(k1), s(k2)) implies that con(t(s(k1))) = s(k2) and con(t(s(k2))) = s(k1). besides, con(s(s(k1))) = s(k1) and con(s(s(k2))) = s(k2). we note that if δ(s(k1), s(k2)) = dist(s(k1), s(k2)) = dist(s(a), s(b)), then every point of s(k1) ∪ s(k2) is a coincidence quasi-best proximity point c© agt, upv, 2020 appl. gen. topol. 21, no. 2 342 weak proximal normal structure of (t ; s) and we are finished. otherwise, since (s(a), s(b)) has wpns, there exists a point (x1, y1) ∈ k1 × k2 and c ∈ (0, 1), so that δ(sx1, s(k2)) ≤ c δ(s(k1), s(k2)), δ(sy1, s(k1)) ≤ c δ(s(k1), s(k2)). since (s(k1), s(k2)) is a proximal pair, there exists (x2, y2) ∈ k1 × k2 such that ‖sx1 − sy2‖ = ‖sx2 − sy1‖ = dist(s(a), s(b)). put su := sx1+sx2 2 and sv := sy1+sy2 2 . then, (su, sv) ∈ s(k1) × s(k2) and ‖su − sv‖ = dist(s(k1), s(k2)). moreover, for each z ∈ k2, we have ‖su − sz‖ = ‖ sx1 + sx2 2 − sz‖ ≤ 1 2 [‖sx1 − sz‖ + ‖sx2 − sz‖] ≤ c + 1 2 δ(s(k1), s(k2)). now, if r := c+1 2 , then r ∈ (0, 1) and δ(su, (s(k2)) ≤ rδ(s(k1), s(k2)). similarly, we can see that δ(sv, (s(k1)) ≤ rδ(s(k1), s(k2)). assume that s(l1) = {sx ∈ s(k1) : δ(sx, s(k2)) ≤ rδ(s(k1), s(k2))}, s(l2) = {sy ∈ s(k2) : δ(sy, s(k1)) ≤ rδ(s(k1), s(k2))}. thus, (su, sv) ∈ s(l1)×s(l2) and so, dist(s(l1), s(l2)) = dist(s(k1), s(k2)). moreover, (s(l1), s(l2)) is a weakly compact and convex pair in x. we show that t is cyclic on s(l1)∪s(l2). suppose sx ∈ s(l1) and sy ∈ s(k2). then, similar to proof of theorem 2.6, sx ∈ s(as0) and sy ∈ s(bs0); that is, x ∈ as0 and y ∈ bs0. thus, ‖t(sx) − t(sy)‖ = dist(s(a), s(b)) ≤ δ(sx, s(k2)) ≤ rδ(s(k1), s(k2)). so, t(sy) ∈ b(t(sx); rδ(s(k1), s(k2))); that is, t(s(k2)) ⊆ b(t(sx); rδ(s(k1), s(k2))) and s(k1) = cont(s(k2)) ⊆ b(t(sx); rδ(s(k1), s(k2))). therefore, δ(t(sx), s(k1)) ≤ rδ(s(k1), s(k2)); that is, t(sx) ∈ s(l2). thus, t(s(l1)) ⊆ s(l2). similarly, t(s(l2)) ⊆ s(l1), s(s(l1)) ⊆ s(l1) and s(s(l2)) ⊆ s(l2). hence, t is cyclic and s is noncyclic on s(l1)∪s(l2). the minimality of (s(k1), s(k2)) now implies that s(l1) = s(k1) and s(l2) = s(k2). now, we have δ(s(k1), s(k2)) = sup x∈k1 δ(sx, s(k2)) ≤ rδ(s(k1), s(k2)), which is a contradiction. □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 343 f. fouladi, a. abkar and e. karapınar 5. examples we clarify the above results with some examples. example 5.1. let a = [−4, 0] and b = [0, 4] be subsets of the uniformly convex banach space (r, |.|). for any x ∈ a ∪ b we define tx = − 1 4 x, sx = 1 2 x. then, t(a) = [0, 1] ⊆ [0, 2] = s(b), t(b) = [−1, 0] ⊆ [−2, 0] = s(a). moreover, for any (x, y) ∈ a × b, we define α(x, y) = ! 0, if x = y 1, if x ∕= y. if (x, y) ∈ a × b such that ‖x − y‖ = dist(s(a), s(b)) = 0, then x = y and ‖tx − ty‖ = dist(s(a), s(b)), ‖sx − sy‖ = dist(s(a), s(b)). otherwise, ‖tx − ty‖ = ‖ 1 4 y − 1 4 x‖ = 1 2 ‖ 1 2 y − 1 2 x‖ = 1 2 ‖sy − sx‖ = 1 2 ‖sx − sy‖ ≤ ‖sx − sy‖ = α(x, y)‖sx − sy‖ + (1 − α(x, y)) max{δx[o2(y; ∞)], δy[o2(x; ∞)]}. thus, (t ; s) is a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits, and by corollary 2.7, there exists (x, y) ∈ a × b such that ‖tx − sx‖ = dist(s(a), s(b)), ‖ty − sy‖ = dist(s(a), s(b)). example 5.2. let a = [−4, −1] and b = [1, 4] be subsets in (r, |.|). let k1 = [−4, −2], k2 = [2, 4] and sx = " ###$ ###% − √ −x − 2, if x ∈ a \ k1√ x + 2, if x ∈ b \ k2 −3, if x ∈ k1 3, if x ∈ k2. therefore, s is a noncyclic mapping. moreover, s(a) = [−4, −3] ⊆ a, s(b) = [3, 4] ⊆ b. so, (s(a), s(b)) is a closed, convex and bounded pair and we have dist(s(a), s(b)) = 6. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 344 weak proximal normal structure suppose that tx = " ###$ ###% √ −x + 2, if x ∈ a \ k1 − √ x − 2, if x ∈ b \ k2 3, if x ∈ k1 −3, if x ∈ k2. therefore, t is a cyclic mapping. besides, t(a) = [3, 4] = s(b) ⊆ b, t(b) = [−4, −3] = s(a) ⊆ a. moreover, we suppose that for any (x, y) ∈ a × b, α(x, y) = ! 1, if (x, y) ∈ (a \ k1) × (b \ k2) 0, otherwise. if ‖x − y‖ = dist(s(a), s(b)), then (x, y) ∈ k1 × k2 and we have ‖sx − sy‖ = ‖ − 3 − 3‖ = 6 = dist(s(a), s(b)) and ‖tx − ty‖ = ‖3 − (−3)‖ = 6 = dist(s(a), s(b)). onherwise, for any (x, y) ∈ (a \ k1) × (b \ k2), we have ‖tx − ty‖ = ‖ √ −x + 2 − (− √ y − 2)‖ = ‖ √ −x + √ y + 4‖ = ‖ √ y + 2 − (− √ −x − 2)‖ = ‖sy − sx‖ = ‖sx − sy‖ ≤ α(x, y)‖sx − sy‖ + (1 − α(x, y)) max{δx[o(y; ∞)], δy[o(x; ∞)]}. thus, (t ; s) is a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits, and by corollary 2.7, there exists (x, y) ∈ a × b such that ‖tx − sx‖ = dist(s(a), s(b)), ‖ty − sy‖ = dist(s(a), s(b)). in fact, for any (x, y) ∈ k1 × k2, we have ‖tx − sx‖ = 6 = dist(s(a), s(b)), ‖ty − sy‖ = 6 = dist(s(a), s(b)). we clarify the above result with an example. example 5.3. assume that a = [−4, 0] and b = [0, 4] are subsets of (r, |.|). for any x ∈ a ∪ b, we set tx = 1 4 x, sx = 1 2 x. then, t(a) = [−1, 0] ⊆ [−2, 0] = s(a), t(b) = [0, 1] ⊆ [0, 2] = s(b). moreover, we suppose that for any (x, y) ∈ a × b, α(x, y) = ! 0, if x = y 1, if x ∕= y. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 345 f. fouladi, a. abkar and e. karapınar if (x, y) ∈ a × b such that ‖x − y‖ = dist(s(a), s(b)) = 0, then x = y and ‖tx − ty‖ = dist(s(a), s(b)), ‖sx − sy‖ = dist(s(a), s(b)). otherwise, ‖tx − ty‖ = ‖ 1 4 x − 1 4 y‖ = 1 2 ‖ 1 2 x − 1 2 y‖ = 1 2 ‖sx − sy‖ ≤ ‖sx − sy‖ = α(x, y)‖sx − sy‖ + (1 − α(x, y)) max{δx[o(y; ∞)], δy[o(x; ∞)]}. thus, (t ; s) is a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits, and by corollary 3.3, there exists (x0, y0) ∈ a × b such that ‖x0 − y0‖ = dist(s(a), s(b)). in fact, for x0 = 0 and y0 = 0, we have tx0 = x0, ty0 = y0 and ‖x0 − y0‖ = dist(s(a), s(b)). acknowledgements. the authors express their gratitude to colleagues at china medical university for their sincere hospitality during the visit of china medical university, taichung, taiwan. references [1] a. abkar and m. gabeleh, best proximity points for cyclic mappings in ordered metric spaces, j. optim. theorey. appl. 150 (2011), 188–193. 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[8] a. a. eldred, w. a. kirk and p. veeramani, proximal normal structure and relatively nonexpansive mappings, studia math. 171 (2005), 283–293. [9] r. espinola, m. gabeleh and p. veeramani, on the structure of minimal sets of relatively nonexpansive mappings, numer. funct. anal. optim. 34 (2013), 845–860. [10] a. f. leon and m. gabeleh, best proximity pair theorems for noncyclic mappings in banach and metric spaces, fixed point theory 17 (2016), 63–84. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 346 weak proximal normal structure [11] m. gabeleh, a characterization of proximal normal structure via proximal diametral sequences, j. fixed point theory appl. 19 (2017), 2909–2925. [12] m. gabeleh, o. olela otafudu and n. shahzad, coincidence best proximity points in convex metric spaces, filomat 32 (2018), 1–12. [13] m. gabeleh, h. lakzian and n.shahzad, best proximity points for asymptotic pointwise contractions, j. nonlinear convex anal. 16 (2015), 83–93. 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[25] t. suzuki, m. kikkawa and c. vetro, the existence of best proximity points in metric spaces with to property uc, nonlinear anal. 71 (2009), 2918–2926. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 347 @ appl. gen. topol. 22, no. 1 (2021), 121-137doi:10.4995/agt.2021.14185 © agt, upv, 2021 convexity and freezing sets in digital topology laurence boxer department of computer and information sciences, niagara university, ny 14109, usa; and department of computer science and engineering, state university of new york at buffalo, usa. (boxer@niagara.edu) communicated by v. gregori abstract we continue the study of freezing sets in digital topology, introduced in [4]. we show how to find a minimal freezing set for a “thick” convex disk x in the digital plane z 2 . we give examples showing the significance of the assumption that x is convex. 2010 msc: 54h25. keywords: digital topology; freezing set; convexity; digital disk. 1. introduction we often use a digital image as a mathematical model of an object or a set of objects “pictured” by the image. methods inspired by classical topology are used to determine whether a digital image has properties analogous to the topological properties of a “real world” object represented by the image. the literature now contains considerable success in adapting to digital topology notions from classical topology such as connectedness, continuous function, homotopy, fundamental group, homology, automorphism group, etc. the fixed point properties (including “approximate fixed point” properties) of a digital image are in some ways similar and in other ways very different from those of the euclidean object modeled by the image. this claim is borne out in such papers as [8, 6, 7]. a large number of papers have been written to study fixed point properties of digital images considered as digital metric spaces [12]. most of assertions of these papers have been shown [9, 2, 3, 5] to be incorrect or trivial; this leads received 12 august 2020 – accepted 04 december 2020 http://dx.doi.org/10.4995/agt.2021.14185 l. boxer to the conclusion that the digital metric space is not an idea worth developing further. knowledge of the fixed point set fix(f) of a continuous self-map on a nontrivial topological space x rarely tells us much about f|x\fix(f). by contrast, it was shown in [9, 4] that knowledge of the fixed point set fix(f) of a digitally continuous self-map on a nontrivial digital image (x, κ) may tell us a great deal about f|x\fix(f). indeed, if a is a subset of x that is a “freezing set” and a ⊂ fix(f), then f is constrained to be the identity function idx. some results concerning freezing sets were presented in [4]. in this paper, we continue the study of freezing sets. in particular, we show how to find minimal freezing sets for “thick” convex disks in the digital plane, and we give examples showing the importance of the assumption of convexity in our theorems. 2. preliminaries we use z to indicate the set of integers and r for the set of real numbers. for a finite set x, we denote by #x the number of distinct members of x. 2.1. adjacencies. material in this section is quoted or paraphrased from [4]. the cu-adjacencies are commonly used in digital topology. let x, y ∈ z n, x 6= y, where we consider these points as n-tuples of integers: x = (x1, . . . , xn), y = (y1, . . . , yn). let u ∈ z, 1 ≤ u ≤ n. we say x and y are cu-adjacent if • there are at most u indices i for which |xi − yi| = 1, and • for all indices j such that |xj − yj| 6= 1 we have xj = yj. often, a cu-adjacency is denoted by the number of points adjacent to a given point in zn using this adjacency. e.g., • in z1, c1-adjacency is 2-adjacency. • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency. • in z3, c1-adjacency is 6-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. for κ-adjacent x, y, we write x ↔κ y or x ↔ y when κ is understood. we write x -κ y or x y to mean that either x ↔κ y or x = y. we say subsets a, b of a digital image x are (κ-)adjacent, a -κ b or a b when κ is understood, if there exist a ∈ a and b ∈ b such that a -κ b. we say {xn} k n=0 ⊂ (x, κ) is a κ-path (or a path if κ is understood) from x0 to xk if xi -κ xi+1 for i ∈ {0, . . . , k − 1}, and k is the length of the path. a subset y of a digital image (x, κ) is κ-connected [15], or connected when κ is understood, if for every pair of points a, b ∈ y there exists a κ-path in y from a to b. we define n(x, κ, x) = {y ∈ x | x ↔κ y}, n∗(x, κ, x) = {y ∈ x | x -κ y} = n(x, κ, x) ∪ {x}. © agt, upv, 2021 appl. gen. topol. 22, no. 1 122 convexity and freezing sets in digital topology definition 2.1. let x ⊂ zn. • the boundary of x with respect to the ci adjacency, i ∈ {1, 2}, is bdi(x) = {x ∈ x | there exists y ∈ z n \ x such that y ↔ci x}. note bd1(x) is what is called the boundary of x in [14]. however, for this paper, bd2(x) offers certain advantages. • the interior of x with respect to the ci adjacency is inti(x) = x \ bdi(x). 2.2. digitally continuous functions. material in this section is quoted or paraphrased from [4]. the following generalizes a definition of [15]. definition 2.2 ([1]). let (x, κ) and (y, λ) be digital images. a function f : x → y is (κ, λ)-continuous if for every κ-connected a ⊂ x we have that f(a) is a λ-connected subset of y . if (x, κ) = (y, λ), we say such a function is κ-continuous, denoted f ∈ c(x, κ). ✷ when the adjacency relations are understood, we may simply say that f is continuous. continuity can be expressed in terms of adjacency of points: theorem 2.3 ([15, 1]). a function f : (x, κ) → (y, λ) is continuous if and only if x ↔κ x ′ in x implies f(x) -λ f(x ′). similar notions are referred to as immersions, gradually varied operators, and gradually varied mappings in [10, 11]. composition preserves continuity, in the sense of the following. theorem 2.4 ([1]). let (x, κ), (y, λ), and (z, µ) be digital images. let f : x → y be (κ, λ)-continuous and let g : y → z be (λ, µ)-continuous. then g ◦ f : x → z is (κ, µ)-continuous. given x = πvi=1xi, we denote throughout this paper the projection onto the ith factor by pi; i.e., pi : x → xi is defined by pi(x1, . . . , xv) = xi, where xj ∈ xj. given a function f : x → x, we say x ∈ x is a fixed point of f if f(x) = x. the set of points {x ∈ x | f(x) = x} we denote as fix(f). we use the notation idx to denote the identity function: idx : x → x is the function idx(x) = x for all x ∈ x. definition 2.5 ([4]). let (x, κ) be a digital image. we say a ⊂ x is a freezing set for x if given f ∈ c(x, κ), a ⊂ fix(f) implies f = idx. 2.3. digital disks and bounding curves. let κ ∈ {c1, c2}, n > 1. we say a κ-connected set s = {xi} n i=1 ⊂ z 2 is a (digital) line segment if the members of s are collinear. remark 2.6. a digital line segment must be vertical, horizontal, or have slope of ±1. we say a segment with slope of ±1 is slanted. © agt, upv, 2021 appl. gen. topol. 22, no. 1 123 l. boxer a (digital) κ-closed curve is a path s = {si} m i=0 such that s0 = sm, and 0 < |i − j| < m implies si 6= sj. if i, j < m and si ↔κ sj implies |i − j| mod m = 1, s is a (digital) κ-simple closed curve. for a simple closed curve s ⊂ z2 we generally assume • m ≥ 8 if κ = c1, and • m ≥ 4 if κ = c2. these requirements are necessary for the jordan curve theorem of digital topology, below, as a c1-simple closed curve in z 2 needs at least 8 points to have a nonempty finite complementary c2-component, and a c2-simple closed curve in z2 needs at least 4 points to have a nonempty finite complementary c1-component. examples in [14] show why it is desirable to consider s and z 2 \ s with different adjacencies. theorem 2.7 ([14], jordan curve theorem for digital topology). let {κ, κ′} = {c1, c2}. let s ⊂ z 2 be a simple closed κ-curve such that s has at least 8 points if κ = c1 and such that s has at least 4 points if κ = c2. then z 2 \s has exactly 2 κ′-connected components. one of the κ′-components of z2 \ s is finite and the other is infinite. this suggests the following. definition 2.8. let s ⊂ z2 be a c2-closed curve such that z 2 \ s has two c1-components, one finite and the other infinite. the union d of s and the finite c1-component of z 2 \ s is a (digital) disk. s is a bounding curve of d. the finite c1-component of z 2 \ s is the interior of s, denoted int(s), and the infinite c1-component of z 2 \ s is the exterior of s, denoted ext(s). notes: • if d is a digital disk determined as above by a bounding c2-closed curve s, then (s, c1) can be disconnected. see figure 1. • there may be more than one closed curve s bounding a given disk d. see figure 2. since we are interested in finding minimal freezing sets and since it turns out we often compute these from bounding curves, we may prefer those that are minimal bounding curves. a bounding curve s for a disk d is minimal if there is no bounding curve s′ for d such that #s′ < #s. • in particular, a bounding curve need not be contained in bd1(d). e.g., in the disk d shown in figure 2(i), (2, 2) is a point of the bounding curve; however, all of the points c1-adjacent to (2, 2) are members of d, so by definition 2.1, (2, 2) 6∈ bd1(d). however, a bounding curve for d must be contained in bd2(d). • in definition 2.8, we use c2 adjacency for s and we do not require s to be simple. figure 2 shows why these seem appropriate. – the use of c2 adjacency allows slanted segments in bounding curves and makes possible a bounding curve in subfigure (ii) with fewer points than the bounding curve in subfigure (i) in which adjacent pairs of the bounding curve are restricted to c1 adjacency. © agt, upv, 2021 appl. gen. topol. 22, no. 1 124 convexity and freezing sets in digital topology figure 1. the c1-disk d = {(x, y) ∈ z 2 | |x| + |y| < 2}. the bounding curve s = {(x, y) ∈ z2 | |x| + |y| = 1} = d \ {(0, 0)} is not c1-connected. figure 2. two views of [0, 3]2 z \ {(3, 3)}, which can be regarded as a c1-disk with either of the closed curves shown in dark as a bounding curve. (i) the dark line segments show a c1-simple closed curve s that is a bounding curve for d. (ii) the dark line segments show a c2-closed curve s that is a minimal bounding curve for d. note the point (2, 2) in the bounding curve shown in (i). by definition 2.1, (2, 2) 6∈ bd1(d); however, (2, 2) ∈ bd2(d). – neither of the bounding curves shown in figure 2 is a c2-simple closed curve. e.g., non-consecutive points of each of the bounding curves, (0, 1) and (1, 0), are c2-adjacent. the bounding curve shown in figure 2(ii) is clearly also not a c1-simple closed curve. • a closed curve that is not simple may be the boundary bd2 of a digital image that is not a disk. this is illustrated in figure 3. more generally, we have the following. definition 2.9. let x ⊂ z2 be a finite, ci-connected set, i ∈ {1, 2}. suppose there are pairwise disjoint c2-closed curves sj ⊂ x, 1 ≤ j ≤ n, such that • x ⊂ s1 ∪ int(s1); • for j > 1, dj = sj ∪ int(sj) is a digital disk; © agt, upv, 2021 appl. gen. topol. 22, no. 1 125 l. boxer figure 3. d = [0, 6]z×[0, 2]z\{(3, 2)} shown with a bounding curve s in dark segments. d is not a disk with either the c1 or the c2 adjacency, since with either of these adjacencies, z 2 \ s has two bounded components, {(1, 1), (2, 1)} and {(4, 1), (5, 1)}. • no two of s1 ∪ ext(s1), d2, . . . , dn are c1-adjacent or c2-adjacent; and • we have z 2 \ x = ext(s1) ∪ n⋃ j=2 int(sj). then {sj} n j=1 is a set of bounding curves of x. note: as above, a digital image x ⊂ z2 may have more than one set of bounding curves. a set x in a euclidean space rn is convex if for every pair of distinct points x, y ∈ x, the line segment xy from x to y is contained in x. the convex hull of y ⊂ rn, denoted hull(y ), is the smallest convex subset of rn that contains y . if y ⊂ r2 is a finite set, then hull(y ) is a single point if y is a singleton; a line segment if y has at least 2 members and all are collinear; otherwise, hull(y ) is a polygonal disk, and the endpoints of the edges of hull(y ) are its vertices. a digital version of convexity can be stated for subsets of the digital plane z2 as follows. a finite set y ⊂ z2 is (digitally) convex if either • y is a single point, or • y is a digital line segment, or • y is a digital disk with a bounding curve s such that the endpoints of the maximal line segments of s are the vertices of hull(y ) ⊂ r2. let s1 and s2 be sides of a digital disk x ⊂ z 2, i.e., maximal digital line segments in a bounding curve s of x, such that s1 ∩ s2 = {p} ⊂ x. the interior angle of x at p is the angle formed by s1, s2, and int(x). remark 2.10. let (x, κ) be a digital disk in z2, κ ∈ {c1, c2}. let s1 and s2 be sides of x such that s1 ∩ s2 = {p} ⊂ x. then the interior angle of x at p is well defined. proof. if there exists q ∈ x \ (s1 ∪ s2) such that q ↔c2 p, then the interior angle of x at p is the angle obtained by rotating s1 about p through q to reach s2. © agt, upv, 2021 appl. gen. topol. 22, no. 1 126 convexity and freezing sets in digital topology figure 4. illustration of lemma 2.13. arrows show the images of q, q′ under f ∈ c(x, c2). since f(q) is to the right of q and q′ ↔c1,c2 q with q ′ to the left of q, f pulls q′ to the right so that f(q′) is to the right of q′. otherwise, the angles formed by s1 and s2 measure 45 ◦ (π/4 radians) and 315◦ (7π/4 radians). since x is convex, the 45◦ angle determined by s1 and s2 is the interior angle of x at p. � remark 2.11. it follows from remark 2.6 that every interior angle measures as a multiple of 45◦ (π/4 radians). for a convex disk, an interior angle must be 45◦ (π/4 radians), 90◦ (π/2 radians), or 135◦ (3π/4 radians). 2.4. tools for determining fixed point sets. the following assertions will be useful in determining fixed point and freezing sets. proposition 2.12 (corollary 8.4 of [9]). let (x, κ) be a digital image and f ∈ c(x, κ). suppose x, x′ ∈ fix(f) are such that there is a unique shortest κ-path p in x from x to x′. then p ⊂ fix(f). lemma 2.13 below is in the spirit of “pulling” as introduced in [13]. we quote [4]: the following assertion can be interpreted to say that in a cu-adjacency, a continuous function that moves a point p also [pulls along] a point that is “behind” p. e.g., in z2, if q and q′ are c1or c2-adjacent with q left, right, above, or below q ′, and a continuous function f moves q to the left, right, higher, or lower, respectively, then f also moves q′ to the left, right, higher, or lower, respectively. lemma 2.13 ([4]). let (x, cu) ⊂ z n be a digital image, 1 ≤ u ≤ n. let q, q′ ∈ x be such that q ↔cu q ′. let f ∈ c(x, cu). (1) if pi(f(q)) > pi(q) > pi(q ′) then pi(f(q ′)) > pi(q ′). (2) if pi(f(q)) < pi(q) < pi(q ′) then pi(f(q ′)) < pi(q ′). figure 4 illustrates lemma 2.13. theorem 2.14 ([4]). let x ⊂ zn be finite. then for 1 ≤ u ≤ n, bd1(x) is a freezing set for (x, cu). remark 2.15. a similar proof can be used to show that if x ⊂ z2 is finite, then a set of bounding curves for x is a freezing set for (x, ci), i ∈ {1, 2}. © agt, upv, 2021 appl. gen. topol. 22, no. 1 127 l. boxer theorem 2.16. let d be a digital disk in z2. let s be a bounding curve for d. then s is a freezing set for (d, c1) and for (d, c2). proof. this is like the proof of theorem 2.14 in [4]. let κ ∈ {c1, c2}. let f ∈ c(d, κ) such that s ∈ fix(f). suppose there exists x ∈ d such that f(x) 6= x. then x lies on a horizontal segment ab and on a vertical segment cd such that {a, b, c, d} ⊂ s, p1(a) < p1(b), and p2(c) < p2(d). • if p1(f(x)) > p1(x) then by lemma 2.13, p1(f(a)) > p1(a), contrary to a ∈ s ⊂ fix(f). • if p1(f(x)) < p1(x) then by lemma 2.13, p1(f(b)) < p1(b), contrary to b ∈ s ⊂ fix(f). • if p2(f(x)) > p2(x) then by lemma 2.13, p1(f(c)) > p1(c), contrary to c ∈ s ⊂ fix(f). • if p2(f(x)) < p2(x) then by lemma 2.13, p1(f(d)) < p1(d), contrary to d ∈ s ⊂ fix(f). in all cases, we have a contradiction brought on by assuming x 6∈ fix(f). therefore, f = idd, so s is a freezing set for (d, κ). � we will use the following. definition 2.17. let (x, κ) be a digital image. let p, q ∈ x such that n(x, p, κ) ⊂ n∗(x, q, κ). then q is a close κ-neighbor of p. lemma 2.18. let (x, κ) be a digital image. let p, q ∈ x such that q is a close κ-neighbor of p. then there is a κ-retraction r : x → x of x to x \ {p}. proof. this was shown in the proof of lemma 4.8 of [9]. � lemma 2.19. let (x, κ) be a digital image. let p, q ∈ x such that q is a close κ-neighbor of p. then p belongs to every freezing set of (x, κ). proof. let a be a freezing set of (x, κ). it follows from lemma 2.18 that a \ {p} is not a freezing set of (x, κ). the assertion follows. � 3. c1-freezing sets for disks in z 2 the following can be interpreted as stating that the set of “corner points” form a freezing set for a digital cube with the c1 adjacency. theorem 3.1 ([4]). let x = πni=1[0, mi]z. let a = π n i=1{0, mi}. then a is a freezing set for (x, c1); minimal for n ∈ {1, 2}. remark 3.2. example 5.16 of [4] shows that the freezing set of theorem 3.1 is not minimal for n = 3. the argument used to prove theorem 3.1 may lead one to ask if this theorem can be generalized for n = 2 as follows: © agt, upv, 2021 appl. gen. topol. 22, no. 1 128 convexity and freezing sets in digital topology given a digital disk d ⊂ z2 such that all of the maximal segments of a bounding curve s of d are horizontal or vertical, is the set of the endpoints of the maximal segments of s a minimal freezing set for (d, c1)? the following provides a negative answer to this question. example 3.3. let d = [0, 3]z × [0, 6]z \ {(3, 3)}). then a = {(0, 0), (3, 0), (3, 2), (3, 4), (3, 6), (0, 6)} (see figure 5(i)) is a minimal freezing set for (d, c1). note (2, 2) and (2, 4) are endpoints of maximal horizontal and vertical bounding segments of d and are not members of a. while (2, 2) and (2, 4) are members of a bounding curve for d, they are not members of a minimal bounding curve, which includes edges from (3, 4) to (2, 3) and from (2, 3) to (3, 2) (see figure 5(ii)). proof. let f ∈ c(d, c1) such that a ⊂ fix(f). it follows from proposition 2.12 that the vertical segments {0} × [0, 6]z, {3} × [0, 2]z, and {3} × [4, 6]z, the horizontal segments [0, 3]z × {0} and [0, 3]z × {6}, and the path {(3, 2), (2, 2), (2, 3), (2, 4), (3, 4)} are all subsets of fix(f). since the union of these paths is a bounding curve s for d, we have s ⊂ fix(f). that a is a freezing set follows from theorem 2.16. to show a is a minimal freezing set, we observe that the following are pairs (p, q) such that p ∈ a and q is a close c1-neighbor of p (see figure 5): ((0, 0), (1, 1)), ((3, 0), (2, 1)), ((3, 2), (2, 1)), ((3, 4), (2, 5)), ((3, 6), (2, 5)), and ((0, 6), (1, 5)). by lemma 2.19, every member of a must belong to every freezing set of (x, c1). it follows that a is a minimal freezing set. � definition 3.4. let x ⊂ z2 be a digital disk. we say x is thick if the following are satisfied. for some bounding curve s of x, • for every slanted segment s of bd2(x), if p ∈ s is not an endpoint of s, then there exists c ∈ x such that (see figure 6) (3.1) c ↔c2 p 6↔c1 c, and • if p is the vertex of a 90◦ (π/2 radians) interior angle θ of s, then there exists q ∈ int(x) such that – if θ has horizontal and vertical sides then q ↔c2 p 6↔c1 q (see figure 7); – if θ has slanted sides then q ↔c1 p (see figure 8); and • if p is the vertex of a 135◦ (3π/4 radians) interior angle θ of s, there exist b, b′ ∈ x such that b and b′ are in the interior of θ and (see figure 9) b ↔c2 p 6↔c1 b and b ′ ↔c1 p. © agt, upv, 2021 appl. gen. topol. 22, no. 1 129 l. boxer figure 5. there are distinct boundary curves for the disk d that contain the horizontal segments from (0, 0) to (3, 0) and from (0, 6) to (3, 6); and vertical segments from (0, 0) to (0, 6), from (3, 0) to (3, 2), and from (3, 4) to (3, 6). (i) we can complete a boundary curve by using the horizontal segments from (2, 2) to (3, 2) and from (2, 4) to (3, 4) and the vertical segment from (2, 2) to (2, 4), as shown in dark. this lets us view d as a disk with horizontal and vertical sides. members of the minimal freezing set a for (d, c1), determined in example 3.3, are marked “a”. note {(2, 2), (2, 4)} ∩ a = ∅. (2, 2) and (2, 4) are endpoints of a maximal horizontal segment of a bounding curve, but not of the minimal bounding curve s; the latter is shown in (ii). indeed, by definition 2.8, {(2, 2), (2, 4)} ⊂ int(d). (ii) alternately, we can complete a boundary curve by using the slanted line segments from (2, 3) to (3, 4) and from (2, 3) to (3, 2). this is a minimal boundary curve s that lets us view d as in example 4.1. a minimal freezing set for (d, c2) is s \ {(2, 3)}. examples of digital images that fail to be thick are shown in figure 10. the following expands on the dimension 2 case of theorem 3.1 to give a subset of bd(x) that is a freezing set. theorem 3.5. let x be a finite digital image in z2 with a set of bounding curves si, 1 ≤ i ≤ n, as in definition 2.9. let a1 be the set of points x ∈ ⋃n i=1 si such that x is an endpoint of a maximal horizontal or a maximal vertical edge of ⋃n i=1 si. let a2 be the union of slanted line segments in⋃n i=1 si. then a = a1 ∪ a2 is a freezing set for (x, c1). © agt, upv, 2021 appl. gen. topol. 22, no. 1 130 convexity and freezing sets in digital topology figure 6. p ∈ uv in a bounding curve, with uv slanted. note u 6↔c1 p 6↔c1 v, p ↔c2 c 6↔c1 p, {p, c} ⊂ n(z 2, c1, b) ∩ n(z2, c1, d). if x is thick then c ∈ x. (not meant to be understood as showing all of x.) figure 7. ∠apb is a 90◦ (π/2 radians) angle of a bounding curve of x at p ∈ a1, with horizontal and vertical sides. if x is thick then q ∈ int(x). (not meant to be understood as showing all of x.) figure 8. ∠apb is a 90◦ (π/2 radians) angle between slanted segments of a bounding curve. if x is thick then q ∈ int(x). (not meant to be understood as showing all of x). figure 9. ∠apq is an angle of 135◦ degrees (3π/4 radians) of a bounding curve of x at p, with ap ∪ pq a subset of the bounding curve. if x is thick then b, b′ ∈ x. (not meant to be understood as showing all of x.) © agt, upv, 2021 appl. gen. topol. 22, no. 1 131 l. boxer figure 10. digital disks that are not thick. (i) (1, 2) is a non-endpoint of a slanted boundary segment for which there is no point corresponding to c of figure 6. (ii) (0, 1) is the vertex of a 90◦-degree (π/2 radians) interior angle but is not c2-adjacent to any member of the interior of the disk. note the segment from (1,1) to (2,0) belongs to a minimal bounding curve, so (2,1) is an interior point. therefore, this image really is a disk. (iii) (0, 2) is the vertex of a 135◦ interior angle of a bounding curve for which there is no point corresponding to b of figure 9. proof. let f ∈ c(x, c1) such that a ⊂ fix(f). let x, x ′ be distinct members of a1 that are endpoints of the same maximal horizontal or vertical edge e in some si. then e is the unique shortest c1-path in x from x to x ′. by proposition 2.12, e ⊂ fix(f). it follows that every horizontal and every vertical side of si belongs to fix(f). by hypothesis we also have that a2 ⊂ fix(f), so si ⊂ fix(f). therefore, bd2(x) ⊂ fix(f). by remark 2.15, f = idx. thus a is a freezing set for (x, c1). � remark 3.6. the set a of theorem 3.5 need not be minimal. this is shown in example 3.3, where (2, 3), as a member of a slanted edge of a minimal bounding curve (see figure 5), is a member of the set a of theorem 3.5, but is not a member of the minimal freezing set. theorem 3.7. let x be a thick convex disk with a bounding curve s. let a1 be the set of points x ∈ s such that x is an endpoint of a maximal horizontal or a maximal vertical edge of s. let a2 be the union of slanted line segments in s. then a = a1 ∪ a2 is a minimal freezing set for (x, c1) (see figure 11(ii)). proof. that a is a freezing set follows as in the proof of theorem 3.5. to show a is minimal, we must show that if we remove a point p from a, the remaining set a \ {p} is not a freezing set. we start by considering p ∈ a1. since x is convex, the interior angle of s at p must be 45◦ (π/4 radians), 90◦ (π/2 radians), or 135◦ (3π/4 radians). • suppose the interior angle of s at p is 45◦ (π/4 radians). let b be a point of s that is c1-adjacent to p on the horizontal or vertical edge of this angle (see figure 12). then b is a close c1-neighbor of p in x. by lemma 2.19, x \ {p} is not a freezing set for (x, c2). © agt, upv, 2021 appl. gen. topol. 22, no. 1 132 convexity and freezing sets in digital topology figure 11. the convex disk d = [0, 4]2 z \{(0, 3), (0, 4), (1, 4)}. the dashed segment from (0, 2) to (2, 4) shown in (i) and (ii) indicates part of the bounding curve and not c1-adjacencies. (i) d with a c2 bounding curve. (ii) (d, c1) with members of a minimal freezing set a marked “a” these are the endpoints of the maximal horizontal and vertical segments of the bounding curve, and all points of the slanted segment of the bounding curve, per theorem 3.5. (iii) (d, c2) with members of a minimal freezing set b marked “b” these are the endpoints of the maximal slanted edge and all the points of the horizontal and vertical edges of the bounding curve, per theorem 4.2. figure 12. ∠apb is a 45◦ (π/4 radians) interior angle of a bounding curve at p ∈ a1. (not meant to be understood as showing all of x.) • suppose the interior angle of s at p is 90◦ (π/2 radians). let a, b be the points of s that are c1-adjacent to p on the horizontal and vertical edges of this angle and let q be the point of int(x) that is c1-adjacent to each of a and b (see figure 7). then q is a close c1-neighbor of p in x. thus, a \ {p} is not a freezing set for (x, c1). • suppose the interior angle of s at p is 135◦ (3π/4 radians). let a, q ∈ s be such that a and q are the members of this angle that are c2-adjacent to p, where ap is slanted and pq is horizontal or vertical. since x is thick, definition 3.4 yields that there exists b ∈ x such that b ↔c2 p (as © agt, upv, 2021 appl. gen. topol. 22, no. 1 133 l. boxer in figure 9). then b is a close c1-neighbor of p in x. by lemma 2.19, a \ {p} is not a freezing set for (x, c1). thus we have shown that if p ∈ a1 then a\{p} is not a freezing set for (x, c1). now we wish to show if p ∈ a2 then a \ {p} is not a freezing set for (x, c1). let s be a slanted segment of bd2(x) containing p. if p is not an endpoint of s, then from the assumption (3.1) there exist b, c, d ∈ x such that p ↔c2 c, p 6↔c1 c, and b ↔c1 c ↔c1 d (see figure 6). then c is a close c1-neighbor of p in x. by lemma 2.19, a \ {p} is not a freezing set for (x, c1). if p is an endpoint of s, let s′ be the other maximal segment of bd2(x) for which p is an endpoint. if s′ is horizontal or vertical, then p ∈ a1, hence, as discussed above, a \ {p} is not a freezing set for (x, c1). therefore, we assume s′ is slanted. since x is convex and both s and s′ are slanted, the interior angle of s at p must be 90◦ (π/2 radians). there exists q ∈ int(x) such that q ↔c1 p (see figure 8). then q is a close c1-neighbor of p in x. by lemma 2.19, a\{p} is not a freezing set for (x, c1). � 4. c2-freezing sets for disks in z 2 for disks in z2, we obtain results for the c2 adjacency that are dual to those obtained for the c1 adjacency in the previous section. as was true of the c1 adjacency and theorem 3.5, we see, by comparing example 4.1 and theorem 4.3 below, that with c2 adjacency, convexity can affect determination of a minimal freezing set for a digital image in z2. example 4.1. let d = [0, 3]z × [0, 6]z \ {(3, 3)}). (this is the set used in example 3.3. see figure 5.) let s be the bounding curve shown in figure 5(ii), i.e., s is the union of the vertical segments {0} × [0, 6]z, {3} × [0, 2]z, and {3} × [4, 6]z; the horizontal segments [0, 3]z × {0} and [0, 3]z × {6}; and the slanted segments (2, 3)(3, 4) and (2, 3)(3, 2). let b = s \ {(2, 3)}. then b is a minimal freezing set for (d, c2). proof. let f ∈ c(d, c2) be such that (4.1) f|b = idb . let p = (2, 3), q = (3, 2) ∈ b, s = (3, 4) ∈ b. note the following: • if p1(f(p)) > p1(p) then by lemma 2.13, p1(f(1, 3)) > 1 and therefore p1(f(0, 3)) > 0, contrary to (4.1). • if p1(f(p)) < p1(p) then by lemma 2.13, p1(f(q)) < 3, contrary to (4.1). • if p2(f(p)) > p2(p) then by lemma 2.13, p2(f(q)) > 2, contrary to (4.1). • if p2(f(p)) < p2(p) then by lemma 2.13, p1(f(s)) < 4, contrary to (4.1). © agt, upv, 2021 appl. gen. topol. 22, no. 1 134 convexity and freezing sets in digital topology it follows that p ∈ fix(f). thus b∪{p} = bd1(d) ⊂ fix(f). by theorem 2.14, f = idd. this establishes that b is a freezing set. to show b is minimal, for b ∈ b notice that there is a c2-close neighbor q ∈ x: (1, 1) is a c2-close neighbor of (0, 0) ∈ b; (i, 1) is a c2-close neighbor of (i, 0) ∈ b, i ∈ {1, 2}; (1, j) is a c2-close neighbor of (0, j) ∈ b, 1 < j < 5; (1, 5) is a c2-close neighbor of (0, 6) ∈ b; (i, 5) is a c2-close neighbor of (i, 6) ∈ b, i ∈ {1, 2}; (2, 5) is a c2-close neighbor of (3, 6) ∈ b; (2, j) is a c2-close neighbor of (3, j) ∈ b, j ∈ {1, 2, 4, 5}. by lemma 2.19, b \ {b} is not a freezing set for (d, c2). the assertion follows. � theorem 4.2. let x be a finite digital image in z2 with a set of bounding curves si, 1 ≤ i ≤ n, as in definition 2.9. let b1 be the set of points x ∈⋃n i=1 si such that x is an endpoint of a maximal slanted edge in ⋃n i=1 si. let b2 be the union of maximal horizontal and maximal vertical line segments in⋃n i=1 si. let b = b1 ∪ b2. then b is a freezing set for (x, c2). proof. let f ∈ c(x, c2) such that f|b = idb. let p be a point of a slanted edge e of ⋃n i=1 si such that p 6∈ b1. let s and s′ be the endpoints of e. if f(p) 6= p, it follows from lemma 2.13 that either f(s) 6= s or f(s′) 6= s′, a contradiction since by hypothesis we have {s, s′} ⊂ fix(f). therefore, p ∈ fix(f); hence, every slanted edge of ⋃n i=1 si is a subset of fix(f). since by hypothesis all horizontal and vertical edges of⋃n i=1 si belong to fix(f), we conclude that ⋃n i=1 si ⊂ fix(f). it follows from remark 2.15 that f = idx. thus, b is a freezing set for (x, c2). � theorem 4.3. let x be a thick convex disk with a bounding curve s. let b1 be the set of points x ∈ s such that x is an endpoint of a maximal slanted edge in s. let b2 be the union of maximal horizontal and maximal vertical line segments in s. let b = b1 ∪ b2. then b is a minimal freezing set for (x, c2) (see figure 11(iii)). proof. that b is a freezing set follows as in the proof of theorem 4.2. to show b is a minimal freezing set, we must show that b \ {p} is not a freezing set for every p ∈ b. we start with p ∈ b1. since x is a convex disk, we only have the following possibilities to consider. • x has an interior angle θ at p of 45◦ (π/4 radians). let a ∈ x be such that a ↔c2 p and a is adjacent to p on, say, the slanted edge of θ (see figure 12). then a is a c2-close neighbor of p in x. by lemma 2.19, b \ {p} is not a freezing set for (x, c2). • x has an interior angle at p of 90◦ (π/2 radians). then there is a point q ∈ int(x) such that p ↔c1 q as in figure 8. then q is a c2-close © agt, upv, 2021 appl. gen. topol. 22, no. 1 135 l. boxer figure 13. p ∈ ab, a segment of the bounding curve s. q ∈ int(x). p ↔c1 q. (not meant to be understood as showing all of x.) neighbor of p in x. by lemma 2.19, b \ {p} is not a freezing set for (x, c2). • x has an interior angle at p of 135◦ (3π/4 radians). since x is thick, there is a point b′ as in figure 9 that is a close c2-neighbor of p. by lemma 2.19, b \ {p} is not a freezing set for (x, c2). now consider p as a member of b2. since x is convex, this leaves only the following possibilities. • x has an interior angle at p of 45◦ (π/4 radians). then p ∈ b1 ∩ b2 ⊂ b1. as discussed above, b \ {p} is not a freezing set for (x, c2). • x has an interior angle at p of 90◦ (π/2 radians). let a and b be the points of the horizontal and vertical segments of bd(x) such that a ↔c1 p ↔c1 b and let q ∈ int(x) be the point such that a ↔c1 q ↔c1 b (see figure 7). then q is a close c2-neighbor of p in x. by lemma 2.19, b \ {p} is not a freezing set for (x, c2). • x has an interior angle at p of 135◦ (3π/4 radians). then p ∈ b1∩b2 ⊂ b1. as shown above, b \ {p} is not a freezing set for (x, c2). • p is not an endpoint of its segment of bd(x). then p has a c1-close neighbor q ∈ x (see figure 13). by lemma 2.19, b \ {p} is not a freezing set for (x, c2). we have shown that for all p ∈ b, b \ {p} is not a freezing set for (x, c2). therefore, b is a minimal freezing set for (x, c2). � 5. further remarks let x be a thick convex digital disk in z2 . we have shown how to find minimal freezing sets for (x, c1) and for (x, c2). we have given examples showing that our assertions do not extend to non-convex disks in z2. however, for non-convex disks in z2 we have shown how to obtain smaller freezing sets than were previously known. we have also obtained results for freezing sets of c1and c2-connected finite subsets of z 2 bounded by multiple bounding curves. we have left unanswered the following. question 5.1. is every convex disk in z2 thick? © agt, upv, 2021 appl. gen. topol. 22, no. 1 136 convexity and freezing sets in digital topology acknowledgements. the suggestions and corrections of the anonymous reviewers are gratefully acknowledged. references [1] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. 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[15] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. © agt, upv, 2021 appl. gen. topol. 22, no. 1 137 @ appl. gen. topol. 21, no. 1 (2020), 53-56 doi:10.4995/agt.2020.11976 c© agt, upv, 2020 counterexample to theorems on star versions of hurewicz property manoj bhardwaj department of mathematics, university of delhi, new delhi-110007, india (manojmnj27@gmail.com) communicated by s. romaguera abstract in this paper, an example contradicting theorem 4.5 and theorem 5.3 [1] is provided and these theorems are proved under some extra hypothesis. 2010 msc: 54d20; 54b20. keywords: hurewicz spac; star-hurewicz space; strongly star-hurewicz space. 1. introduction in covering properties, hurewicz property is one of the most important property. in 1925, hurewicz [4] (see also [5]) introduced hurewicz property in topological spaces and studied it. this property is stronger than lindelöf and weaker than σ-compactness. in 2004, the authors m. bonanzinga, f. cammaroto, lj.d.r. kočinac [1] introduced the star version of hurewicz property. for the terms and symbols that we do not define follow [2]. the basic definitions are given. let a and b be collections of open covers of a topological space x. in [6], kočinac introduced star selection principles in the following way. the symbol s?1 (a,b) denotes the selection hypothesis that for each sequence < un : n ∈ ω > of elements of a there exists a sequence < un : n ∈ ω > such that for each n, un ∈un and {st(un,un) : n ∈ ω}∈b. the symbol s?fin(a,b) denotes the selection hypothesis that for each sequence < un : n ∈ ω > of elements of a there exists a sequence < vn : n ∈ ω > received 12 june 2019 – accepted 24 november 2019 http://dx.doi.org/10.4995/agt.2020.11976 m. bhardwaj such that for each n, vn is a finite subset of un and ⋃ n∈ω{st(v,un) : v ∈vn} is an element of b the symbol u?fin(a,b) denotes the selection hypothesis that for each sequence < un : n ∈ ω > of elements of a there exists a sequence < vn : n ∈ ω > such that for each n, vn is a finite subset of un and {st( ⋃ vn,un) : n ∈ ω}∈b or there is some n such that st( ⋃ vn,un) = x. let k be a family of subsets of x. then we say that x belongs to the class ss?k(a,b) if x satisfies the following selection hypothesis that for every sequence < un : n ∈ ω > of elements of a there exists a sequence < kn : n ∈ ω > of elements of k such that {st(kn,un) : n ∈ ω}∈b. when k is the collection of all one-point [resp., finite, compact] subspaces of x we write ss?1 (a,b) [resp., ss?fin(a,b), ss ? comp(a,b)] instead of ss?k(a,b). in this paper a and b will be collections of the following open covers of a space x: o : the collection of all open covers of x. ω : the collection of ω-covers of x. an open cover u of x is an ω-cover [3] if x does not belong to u and every finite subset of x is contained in an element of u. γ : the collection of γ-covers of x. an open cover u of x is a γ-cover [3] if it is infinite and each x ∈ x belongs to all but finitely many elements of u. ogp : the collection of groupable open covers. an open cover u of x is groupable [7] if it can be expressed as a countable union of finite, pairwise disjoint subfamilies un, such that each x ∈ x belongs to ⋃ un for all but finitely many n. 2. main results in [1], bonanzinga, cammaroto and kočinac introduced the notion of sh≤n in topological spaces. a space x is said to have sh≤n if for each sequence < un : n ∈ ω > of open covers of x there is a sequence < vn : n ∈ ω > such that for each n, vn is a finite subset of un of cardinalilty atmost n and {st( ⋃ vn,un) : n ∈ ω} is a γ-cover of x. theorem 2.1 ([1]). let a space x satisfies sh≤n. then x satisfies s ? 1 (o,ogp). according to definition of sh≤n, there does not exist any topological space which satisfies sh≤n, take any topological space x and un = {x} for each n. then for any finite subset vn of un, st( ⋃ vn,un) = x for each n. so {x} is not a γ-cover of x since it is finite. to avoid this possibility, without loss of generality if we consider infinite open covers for the theorem then following example shows that the above theorem is not correct. example 2.2. there is a space which satisfies sh≤n but does not satisfy s?1 (o,ogp). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 54 counterexample to theorems on star versions of hurewicz property proof. let n be set of natural numbers with discrete topology on it. since n is countable, it satisfies sh≤n. now consider a sequence < un = {{n} : n ∈ ω} > of open covers for each n. then it does not satisfy s?1 (o,ogp) since {{n} : n ∈ ω} is not groupable. � for the existence of sh≤n, we consider only infinite covers such that x does not belong to each cover. in order to prove above theorem we need more hypothesis on a space x, that is, we define cdr?sub(a,b). definition 2.3 ([8]). let a and b be families of subsets of the infinite set s. then cdrsub(a,b) denotes the statement that for each sequence < an : n ∈ ω > of elements of a there is a sequence < bn : n ∈ ω > such that for each n, bn ⊆ an, for m 6= n, bm ∩bn = ∅, and each bn is a member of b. definition 2.4. let a and b be families of subsets of the infinite set s. then cdr?sub(a,b) denotes the statement that for each sequence < an : n ∈ ω > of elements of a there is a sequence < bn : n ∈ ω > such that for each n, bn ⊆ an, for m 6= n, {st(b,am) : b ∈ bm}∩{st(b,an) : b ∈ bn} = ∅, and each bn is a member of b. theorem 2.5. let a space x satisfies sh≤n and cdr ? sub(o,o). then x satisfies s?1 (o,ogp). proof. the proof is similer as given in [1] with necessary modifications. � in [1], bonanzinga, cammaroto and kočinac consider the hypothesis : for each sequence < un : n ∈ ω > of open covers of x there is a sequence < vn : n ∈ ω > of finite subsets of x such that for each n, vn has atmost n elements and {st(vn,un) : n ∈ ω} is a γ-cover of x. theorem 2.6 ([1]). let a space x satisfies the following condition : for each sequence < un : n ∈ ω > of open covers of x there is a sequence < vn : n ∈ ω > of finite subsets of x such that for each n, vn has atmost n elements and {st(vn,un) : n ∈ ω} is a γ-cover of x. then x satisfies ss?1 (o,ogp). according to hypothesis, there does not exist any topological space which satisfies the hypothesis considered in above theorem, because take any topological space x and un = {x} for each n. then for any finite subset vn of un, st( ⋃ vn,un) = x for each n. so {x} is not a γ-cover of x since it is finite. to avoid this possibility, without loss of generality if we consider infinite open covers for the theorem then the following example shows that the above theorem is not correct. example 2.7. there is a space which satisfies following condition : for each sequence < un : n ∈ ω > of open covers of x there is a sequence < vn : n ∈ ω > of finite subsets of x such that for each n, vn has atmost n elements and {st(vn,un) : n ∈ ω} is a γ-cover of x but does not satisfy ss?1 (o,ogp). proof. let n be set of natural numbers with discrete topology on it. since n is countable, it satisfies following condition : for each sequence < un : n ∈ ω > c© agt, upv, 2020 appl. gen. topol. 21, no. 1 55 m. bhardwaj of open covers of x there is a sequence < vn : n ∈ ω > of finite subsets of x such that for each n, vn has at most n elements and {st(vn,un) : n ∈ ω} is a γ-cover of x. now consider a sequence < un = {{n} : n ∈ ω} > of open covers for each n. then it does not satisfy ss?1 (o,ogp) since {{n} : n ∈ ω} is not groupable. � for the existence of above hypothesis, we consider only infinite covers such that x does not belong to each cover. in order to prove above theorem, we need more hypothesis on a space x, that is, we define cdrf?sub(a,b). definition 2.8. let a and b be families of subsets of the infinite set s. then cdrf?sub(a,b) denotes the statement that for each sequence < an : n ∈ ω > of elements of a there is a sequence < bn : n ∈ ω > such that for each n, bn ⊆ an, for m 6= n and for each finite subset f of s, {st(x,bm) : x ∈ f}∩{st(x,bn) : x ∈ f} = ∅, and each bn is a member of b. theorem 2.9. let a space x satisfies cdrf?sub(o,o) and the following condition : for each sequence < un : n ∈ ω > of open covers of x there is a sequence < vn : n ∈ ω > of finite subsets of x such that for each n, vn has atmost n elements and {st(vn,un) : n ∈ ω} is a γ-cover of x. then x satisfies ss?1 (o,ogp). proof. the proof is similer as given in [1] with necessary modifications. � acknowledgements. the author acknowledges the fellowship grant of university grants commission, india. references [1] m. bonanzinga, f. cammaroto and lj. d. r. kočinac, star-hurewicz and related properties, appl. gen. topol. 5, no. 1 (2004), 79–89. [2] r. engelking, general topology, revised and completed edition, heldermann verlag berlin (1989). [3] j. gerlits and zs. nagy, some properties of c(x), i, topology appl. 14 (1982), 151–161. [4] w. hurewicz, über eine verallgemeinerung des borelschen theorems, math. z. 24 (1925), 401–421. [5] w. hurewicz, über folgen stetiger funktionen, fund. math. 9 (1927), 193–204. [6] lj. d. kočinac, star-menger and related spaces, publ. math. debrecen 55 (1999), 421– 431. [7] lj. d. kočinac and m. scheepers, combinatorics of open covers (vii): groupability, fund. math. 179 (2003), 131–155. [8] m. scheepers, combinatorics of open covers (i) : ramsey theory, topology appl. 69 (1996), 31–62. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 56 @ appl. gen. topol. 21, no. 2 (2020), 326-329 doi:10.4995/agt.2020.13836 c© agt, upv, 2020 a disjointly tight irresolvable space angelo bella a and michael hrušák b a department of mathematics and computer science, university of catania, città universitaria, viale a. doria 6, 95125 catania, italy (bella@dmi.unict.it) b centro de ciencias matemáticas, universidad nacional autónoma de méxico, campus morelia, morelia, michoacán, 58089, méxico. (michael@matmor.unam.mx) communicated by s. garćıa-ferreira abstract in this short note we prove the existence (in zfc) of a completely regular countable disjointly tight irresolvable space by showing that every sub-maximal countable dense subset of 2c is disjointly tight. 2010 msc: 54g15. keywords: irresolvable; disjointly tight; empty interior tightness. in [2] the first author and v. malykhin constructed using the continuum hypothesis a completely regular countable disjointly tight irresolvable space, and asked if it exists in zfc. in this note we show how to combine their ideas with those of juhász, soukup and szentmiklóssy [6] to obtain such a space. our notation is standard and follows [7, 4]. all topological spaces considered are completely regular. recall that a topological space x is irresolvable if it has no isolated points and no disjoint dense sets. following [2] we say that a space x is disjointly tight (or it has disjoint tightness) if whenever x ∈ x is an accumulation point of a set a there are disjoint a0, a1 ⊆ a each containing x in its closure. a space x is sub-maximal if it has no isolated points and every dense subset of x is open, or equivalently (see [1]), if every open subspace of x is irresolvable and every nowhere dense subset of x is closed. these notions were introduced by hewitt [5] and have bee extensively studied in the past fifty years. received 11 june 2020 – accepted 25 august 2020 http://dx.doi.org/10.4995/agt.2020.13836 a disjointly tight irresolvable space answering a question in [1], juhász, soukup and szentmiklóssy [6] showed that theorem 1.1 ([6]). 2c contains a countable dense subspace which is submaximal. the content of the paper is essentially reduced to proving the following: lemma 1.2. let x be a countable dense subspace of 2κ, for some infinite cardinal κ. if u ⊆ x is open, and x ∈ x is an accumulation point of u, then there are disjoint open u0, u1 ⊆ u such that x is an accumulation point of both u0 and u1. proof. as 2κ is c.c.c., we can write u as a countable union of disjoint basic clopen sets u = ! n∈ω wn ∩ x. for each n ∈ ω there is a finite fn ⊆ κ and σn : fn → 2 such that wn = {y ∈ 2κ : σn ⊆ y}. let j ⊇ " n∈ω fn be a countable subset of κ such that for every y ∕= y ′ ∈ x there is an α ∈ j such that y(α) ∕= y′(α). then the projection function h : 2κ → 2j defined by h(y) = y ↾ j is one-to-one on x. let z = h[x]. then z is a dense subset of 2j, and the sets h[wn] are disjoint clopen subsets of 2j having h(x) = x ↾ j as an accumulation point of their union. as j is countable, 2j is a compact metrizable space, hence there are disjoint infinite sets a0, a1 ⊆ ω such that every neighborhood of h(x) in 2j contains all but finitely many of the sets h[wn], n ∈ a0 ∪ a1. let, for i = 0, 1 ui = ! n∈ai wn ∩ x. both u0 and u1 are then open in x and we claim that x is in their closure. to see this let w ⊆ 2κ be a basic open neighborhood of x, i.e. there is a finite set f ⊆ κ such that w = {y ∈ 2κ : y ↾ f = x ↾ f}. then there is a k ∈ ω such that h[w ]∩h[wn] ∕= ∅, for every n ∈ (a0 ∪a1)\k. this implies that w ∩ wn is a non-empty clopen subset of 2κ for every n ∈ (a0 ∪ a1) \ k. by density of x, w ∩ wn ∩ x ∕= ∅ for n ∈ (a0 ∪ a1) \ k and wn ∩ x ⊆ ui, for n ∈ ai \ k and i = 0, 1. hence, w ∩ u0 ∕= ∅ and w ∩ u1 ∕= ∅. □ proposition 1.3. every countable dense sub-maximal subset of 2κ is disjointly tight. proof. note (or recall) that a subset with empty interior of a sub-maximal space is closed discrete. consequently, if x is an accumulation point of a subset a of a submaximal space, then x is an accumulation point of int(a). now, if c© agt, upv, 2020 appl. gen. topol. 21, no. 2 327 a. bella and m. hrušák x is a countable dense sub-maximal subset of 2κ, then by lemma 1.2 there are disjoint open u0, u1 ⊆ int(a) such that x is an accumulation point of both u0 and u1. □ the main result of the note now easily follows. corollary 1.4. there is a countable irresolvable space which is disjointly tight. proof. follows immediately from proposition 1.3, theorem 1.1 and the trivial fact that every sub-maximal space is irresolvable. □ another corollary of proposition 1.3 is the result of [1] that no countable dense subset of 2κ is maximal (i.e. every strictly stronger topology has an isolated point). proposition 1.3 actually shows that the countable sub-maximal spaces which are densely embedable in some cantor cube are quite far from being maximal (see [3], theorems 2.1b and 2.2c). however, every countable maximal space is homeomorphic to a subspace of 2c and, consistently, there are maximal spaces which are embedable into 2κ for some κ < c. problem 1.5. is it consistent that there is a countable irresolvable disjointly tight space of weight strictly less than c? another variation on tightness was also considered in [2]. definition 1.6. a space x has empty interior tightness if for any set a ⊆ x and any point x ∈ a there exists a set b ⊆ a, with empty interior in x, such that x ∈ b. this notion is relevant here because it fully characterizes resolvability of a countable space. theorem 1.7 ([2]). a countable space x is resolvable if and only if it has empty interior tightness. theorem 1.7 and corollary 1.4 provide a regular space with disjoint tightness but not empty interior tightness. the next example shows that these two notions are actually mutually independent. example 1.8. a countable regular space with empty interior tightness but not disjoint tightness. proof. let x = (q × ω) ∪ {p}, where q is the set of rationals and p a free ultrafilter on ω. we topologize x by taking the collection {u×{n} : u open in the euclidean topology on q, n ∈ ω}∪{{p}∪(q×a) : a ∈ p} as a basis. it is easily seen that x is a countable regular resolvable space. hence, theorem 1.7 shows that x has empty interior tightness. on the other hand, the set a = {0} × ω and p ∈ a witness that x does not have disjoint tightness. □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 328 a disjointly tight irresolvable space acknowledgements. the research of the first author was supported by the grand piaceri 2020/22 (linea 2) from the university of catania. the research of the second author was supported by papiit grant in104220, and by a conacyt grant a1-s-16164. references [1] o. t. alas, m. sanchis, m. g. tkac̆enko, v. v. tkachuk and r. g. wilson, irresolvable and submaximal spaces: homogeneity versus σ-discreteness and new zfc examples, topol. appl. 107 (2000), 259–273. [2] a. bella and v. i. malykhin, tightness and resolvability, comment. math. univ. carolinae 39, no. 1 (1998), 177–184. [3] e. k. van douwen, applications of maximal topologies, topol. appl. 51 (1993), 125–139. [4] r. engelking, general topology, heldermann verlag, berlin, 1989. [5] e. hewitt, a problem of set-theoretic topology, duke math. j. 10 (1943), 309–333. [6] i. juhász, l. soukup and z. szentmiklóssy, d-forced spaces: a new approach to resolvability, topol. appl. 153, no. 11 (2006), 1800–1824. [7] k. kunen, set theory an introduction to independence proofs, north-holland, amsterdam, 1980. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 329 @ appl. gen. topol. 22, no. 2 (2021), 399-415doi:10.4995/agt.2021.15101 © agt, upv, 2021 geometrical properties of the space of idempotent probability measures kholsaid fayzullayevich kholturaev tashkent institute of irrigation and agricultural mechanization engineers, 39, kori niyoziy str., 100000, tashkent, uzbekistan (xolsaid 81@mail.ru) communicated by s. romaguera abstract although traditional and idempotent mathematics are “parallel”, by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be “parallel”. at first we establish for a compact metric space x the spaces p(x) of probability measures and i(x) idempotent probability measures are homeomorphic (“parallelism”). then we construct an example which shows that the constructions p and i form distinguished functors from each other (“parallelism” negation). further for a compact hausdorff space x we establish that the hereditary normality of i3(x)\x implies the metrizability of x. 2010 msc: 52a30; 54e35; 54b30; 18b30. keywords: category; functor; compact hausdorff space; idempotent measure. 1. introduction idempotent mathematics is a new branch of mathematical sciences, rapidly developing and gaining popularity over the last two decades. it is closely related to mathematical physics. the literature on the subject is vast and includes numerous books and an all but innumerable body of journal papers. an important stage of development of the subject was presented in the book received 13 february 2021 – accepted 06 may 2021 http://dx.doi.org/10.4995/agt.2021.15101 k. f. kholturaev “idempotency” edited by j. gunawardena [6]. this book arose out of the wellknown international workshop that was held in bristol, england, in october 1994. the next stage of development of idempotent and tropical mathematics was presented in the book idempotent mathematics and mathematical physics edited by g. l. litvinov and v. p. maslov [13]. the book arose out of the international workshop that was held in vienna, austria, in february 2003. in [14] it was delivered the proceedings of the international workshop on idempotent and tropical mathematics and problems of mathematical physics, held at the independent university of moscow, russia, on august 25-30, 2007. idempotent mathematics is based on replacing the usual arithmetic operations with a new set of basic operations, i. e., on replacing numerical fields by idempotent semirings and semifields. typical example is the so-called max-plus algebra rmax [8], [19]. the modern idempotent analysis (or idempotent calculus, or idempotent mathematics) was founded by v. p. maslov and his collaborators [11], [12], [10], [14]. some preliminary results are due to e. hopf and g. choquet, see [1], [7]. idempotent mathematics can be treated as the result of a dequantization of the traditional mathematics over numerical fields as the planck constant h tends to zero taking imaginary values. this point of view was presented in [13]. in other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the fields of real and complex numbers. the basic paradigm is expressed in terms of an idempotent correspondence principle. this principle is closely related to the well-known correspondence principle of n. bohr in quantum theory. actually, there exists a heuristic correspondence between important, interesting, and useful constructions and results of the traditional mathematics over fields and analogous constructions and results over idempotent semirings and semifields (i. e., semirings and semifields with idempotent addition). a systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. as a result, in parallel with the traditional mathematics over fields, its “shadow,” idempotent mathematics, appears. this “shadow” stands approximately in the same relation to traditional mathematics as classical physics does to quantum theory. the notion of idempotent (maslov) measure finds important applications in different parts of mathematics, mathematical physics and economics (see the survey article [14] and the bibliography therein). topological and categorical properties of the functor of idempotent measures were studied in [20], [21]. although idempotent measures are not additive and the corresponding functionals are not linear, there are some parallels between topological properties of the functor of probability measures and the functor of idempotent measures (see, for example [16], [17], [20]) which are based on existence of natural equiconnectedness structure on both functors. © agt, upv, 2021 appl. gen. topol. 22, no. 2 400 geometrical properties of idempotent measures a notion of central importance in categorical topology is that of topological functor. various applications of topological functors described in [5]. in the present paper we show that for a compact metric space x the spaces p(x) of probability measures and i(x) idempotent probability measures are homeomorphic. further we construct an example which shows that the constructions p and i form distinguished functors from each other. this phenomenon shows that the category theory finds out such subtle moments of relations between topological spaces which against common sense. in other words, we get such a conclusion: although traditional and idempotent mathematics are parallel, an application of the category theory shows, objects obtained the similar rules over traditional and idempotent mathematics must not be “parallel”. further, for a compact hausdorff space x we establish that the hereditary normality of i3(x)\x implies the metrizability of x. 2. preliminaries recall [14] that a set s equipped with two algebraic operations: addition ⊕ and multiplication ⊙, is said to be a semiring if the following conditions are satisfied: • the addition ⊕ and the multiplication ⊙ are associative; • the addition ⊕ is commutative; • the multiplication ⊙ is distributive with respect to the addition ⊕: x ⊙ (y ⊕ z) = x ⊙ y ⊕ x ⊙ z and (x ⊕ y) ⊙ z = x ⊙ z ⊕ y ⊙ z for all x, y, z ∈ s. a unit of a semiring s is an element 1 ∈ s such that 1 ⊙ x = x ⊙ 1 = x for all x ∈ s. a zero of the semiring s is an element 0 ∈ s such that 0 6= 1 and 0 ⊕ x = x ⊕ 0 = x for all x ∈ s. a semiring s with neutral elements 0 and 1 is called a semifield if every nonzero element of s is invertible. a semiring s is called an idempotent semiring if x ⊕ x = x for all x ∈ s. an idempotent semiring s is called an idempotent semifield if it is a semifield. note that diöıds, quantales and inclines are examples of idempotent semirings [14]. many authors (s. c. kleene, s. n. n. pandit, n. n. vorobjev, b. a. carré, r. a. cuninghame-green, k. zimmermann, u. zimmermann, m. gondran, f. l. baccelli, g. cohen, s. gaubert, g. j. olsder, j.-p. quadrat, v. n. kolokoltsov and others) used idempotent semirings and matrices over these semirings for solving some applied problems in computer science and discrete mathematics. let r = (−∞, +∞) be the field of real numbers and r+ = [0, +∞) be the semiring of all nonnegative real numbers (with respect to the usual addition “+” and multiplication “·”). consider a map φh : r+ → r (h) = r ∪ {−∞} © agt, upv, 2021 appl. gen. topol. 22, no. 2 401 k. f. kholturaev defined by the equality φh(x) = h ln x, h > 0. let x, y ∈ x and u = φh(x), v = φh(y). put u ⊕h v = φh(x + y) and u ⊙ v = φh(xy). the imagine φh(0) = −∞ of the usual zero 0 is a zero 0 and the imagine φh(1) = 0 of the usual unit 1 is a unit 1 in s with respect to these new operations. the convention −∞ ⊙ x = −∞ allows us to extend ⊕h and ⊙ over r(h). thus we obtained the structure of a semiring (r(h), ⊕h, ⊙) which is isomorphic to (r+, +, ·). a direct check shows that u ⊕h v → max{u, v} as h → 0. it can easily be checked that r ∪ {−∞} forms a semiring with respect to the addition u ⊕ v = max{u, v} and the multiplication u ⊙ v = u + v with zero 0 = −∞ and unit 1 = 0. denote this semiring by rmax; it is idempotent, i. e., u⊕u = u for all its elements u. the semiring (rmax, ⊕, ⊙) generates the semifield (rmax, ⊕, ⊙, 0, 1). the analogy with quantization is obvious; the parameter h plays the role of the planck constant, so r+ can be viewed as a “quantum object” and rmax as the result of its “dequantization”. the described passage (r+, +, ·, 0, 1) φh ≃ (r(h), ⊕h, ⊙, −∞, 0) h→0 −−−→ (rmax, ⊕, ⊙, 0, 1) is called the maslov dequantization. let x be a compact hausdorff space, c(x) be the algebra of continuous functions on x with the usual algebraic operations (i. e. with the addition “+” and the multiplication “·”). on c(x) the operations ⊕ and ⊙ are determined by ϕ ⊕ ψ = max{ϕ,ψ} and ϕ ⊙ ψ = ϕ + ψ where ϕ, ψ ∈ c(x). recall [21] that a functional µ: c(x) → r is said to be an idempotent probability measure on x if it has the following properties: (i1) µ(λx) = λ for all λ ∈ r, where λx is a constant function; (i2) µ(λ ⊙ ϕ) = λ ⊙ µ(ϕ) for all λ ∈ r and ϕ ∈ c(x); (i3) µ(ϕ ⊕ ψ) = µ(ϕ) ⊕ µ(ψ) for all ϕ, ψ ∈ c(x). let i(x) denote the set of all idempotent probability measures on a compact hausdorff space x, and rc(x) be a set of all maps c(x) → r. obviously i(x) ⊂ rc(x). one can treat rc(x) = ∏ ϕ∈c(x) rϕ where rϕ = r, ϕ ∈ c(x). we consider rc(x) with the product topology and consider i(x) as its subspace. a family of sets of the form 〈µ; ϕ1, . . . , ϕn; ε〉 = {ν ∈ i(x) : |ν(ϕi) − µ(ϕi)| < ε, i = 1, . . . , n} is a base of open neighbourhoods of a given idempotent probability measure µ ∈ i(x) according to the induced topology, where ϕi ∈ c(x), i = 1, . . . , n, and ε > 0. it is obvious that the induced topology and the pointwise convergence topology on i(x) coincide. so, we get a topological space i(x), equipped with the pointwise convergence topology. in [21] it was shown that for each compact hausdorff space x the space i(x) is also a compact hausdorff space. let x, y be compact hausdorff spaces and f : x → y be a continuous map. it is easy to check that the map i(f): i(x) → i(y ) determined by the formula i(f)(µ)(ψ) = µ(ψ ◦ f) is continuous. the construction i is a normal functor © agt, upv, 2021 appl. gen. topol. 22, no. 2 402 geometrical properties of idempotent measures acting in the category comp of compact hausdorff spaces and their continuous maps. remind that a functor f : comp → comp on the category of compact hausdorff spaces and continuous maps is said to be normal (see [15], definition 14) if it satisfies the following conditions: (f1) f is continuous (i. e., f(lim s) = lim f(s)); (f2) f preserves weight (i. e., wx = wf(x)); (f3) f is monomorphic (i. e., preserves the injectivity of maps); (f4) f is epimorphic (i. e., preserves the surjectivity of maps); (f5) f preserves intersections (i. e., f(∩ α xα) = ∩ α f(xα)); (f6) f preserves preimages (i. e., f(f−1) = f(f)−1); (f7) f preserves singletons and the empty space (i. e., f(1) = 1 and f(∅) = ∅). let us decipher this definition. let s = {xα, p β α; a} be an inverse system of compact hausdorff spaces, and let lim s = lim ← s be its limit. according to the kurosh theorem, the limit of any inverse system of nonempty compact hausdorff spaces is nonempty (see [2], theorem 3.13) and compact hausdorff space (see [2], proposition 3.12). the action of the functor f on the compact hausdorff spaces xα and the maps p β α, where α, β ∈ a and α ≺ β, produces the inverse system f(s) = {f(xα), f(p β α); a}. let lim f(s) be the limit of this system. by virtue of condition (f1), we have f(lim s) = lim f(s). given a topological space x, let wx denote its weight, i. e., the minimum cardinality of a base of x. by condition (f2), the weights of the compact spaces x and f(x) are equal. since the functor f is monomorphic (by condition (f3)), we can assume f(a) to be a subspace of f(x) for a closed a ⊂ x. the space f(a) is identified with a subspace of f(x) by means of the embedding f(ia), where ia : a → x is the identity embedding. according to condition (f4), if f : x → y is a continuous map “onto” then so is f(f): f(x) → f(y ). for a monomorphic functor f , conditions (f5) and (f6) mean that, for any family {aα} of closed subsets of a compact hausdorff space x, we have f(∩ α aα) = ∩ α f(aα) (this is condition (f5)), and for any continuous map f : x → y and any closed b in y , we have f(f−1(b)) = f(f)−1(f(b)) (this is condition (f6)). the singleton preservation condition means that f takes any one-point space to a one-point space. the intersection preservation condition makes it possible to define an important notion of the support of a monomorphic functor f . the support of a point x ∈ f(x) is a closed set supp x ⊂ x such that, for any closed a ⊂ x, © agt, upv, 2021 appl. gen. topol. 22, no. 2 403 k. f. kholturaev we have a ⊃ supp x if and only if x ∈ f(a) ([15], definition 18). given an intersection-preserving functor f , each point x ∈ f(x) has support, which is defined by supp x = ∩{a ⊂ x : a = a, x ∈ f(a)}, where a denotes the closure of a. as it was mentioned above, the functor i is normal; therefore, for each compact hausdorff space x and any idempotent probability measure µ ∈ i(x), the support of µ is defined as: suppµ = ∩ { a ⊂ x : a = a, µ ∈ i(a) } . for a positive integer n we define the following set in(x) = {µ ∈ i(x) : |suppµ| ≤ n} . put iω(x) = ∞ ∪ n=1 in(x). the set iω(x) is everywhere dense in i(x) [18], [21]. an idempotent probability measure µ ∈ iω(x) is called an idempotent probability measure with finite support. note that if µ is an idempotent probability measure with the finite support suppµ = {x1, x2, . . . , xk} then µ can be represented as µ = λ1 ⊙ δx1 ⊕ λ2 ⊙ δx2 ⊕ . . . , ⊕λk ⊙ δxk uniquely, where −∞ < λi ≤ 0, i = 1, . . . k, λ1 ⊕λ2 ⊕ . . . , ⊕λk = 0. here, as usual, for x ∈ x by δx we denote a functional on c(x) defined by the formula δx(ϕ) = ϕ(x), ϕ ∈ c(x), and called the dirac measure. it is supported at the point x. let x be a compact hausdorff space. a continuous linear functional µ: c(x) → r is said to be a measure on x. the riesz theorem about isomorphism between the normalized space (c(x))∗ dual to c(x) (i. e. the space of all continuous functional on c(x)) and the space m(x) of all finite regular measures on x is substantiation of the above definition (see [4], page 192, paragraph 3.1). a measure µ ∈ m(x) is positive (µ ≥ 0) if µ(ϕ) ≥ 0 for each ϕ ∈ c(x), ϕ ≥ 0. a measure µ is positive if and only if ‖µ‖ = µ(1x). really, let µ ≥ 0 and ‖ϕ‖ ≤ 1. then µ(1x − ϕ) ≥ 0. consequently, µ(1x) ≥ µ(ϕ). from here ‖µ‖ = sup{|µ(ϕ)| : ϕ ∈ c(x), ‖ϕ‖ ≤ 1} = µ(1x). contrary, let now µ(1x) = ‖µ‖ and ϕ ≥ 0. put ψ = 1x − ϕ ‖ϕ‖ . since ‖ψ‖ ≤ 1 we have µ(ψ) ≤ ‖µ‖ = µ(1x), i. e. µ(1x)− 1 ‖ϕ‖ ·µ(ϕ) ≤ µ(1x). hence µ(ϕ) ≥ 0. a measure µ is normed, if ‖µ‖ = 1. a positive, normed measure is said to be a probability measure. thus, we can define the notion of probability measure as the following. a probability measure on a given compact hausdorff space x is a functional µ: c(x) → r satisfying the conditions: (p1) µ(λx) = λ for all λ ∈ r, where λx – constant function; (p2) µ(λϕ) = λµ(ϕ) for all λ ∈ r and ϕ ∈ c(x); © agt, upv, 2021 appl. gen. topol. 22, no. 2 404 geometrical properties of idempotent measures (p3) µ(ϕ + ψ) = µ(ϕ) + µ(ψ) for all ϕ, ψ ∈ c(x). the set of all probability measures on a compact hausdorff space x is denoted by p(x). the set p(x) is endowed with the pointwise convergence topology, i. e. we consider p(x) as a subspace of rc(x). it is well known the topological spaces p(x) and i(x) equipped with the pointwise convergence topology are compact hausdorff spaces. it is easy to sea that, the conditions of normality (p1) and (i1) are the same, and the conditions of homogeneity (p2) and (i2), and the conditions of additivity (p3) and (i3) are mutually similar, just operations are different. in other words, the definition of the idempotent probability measure is “parallel” to the traditional one. in section 3 we will show that the spaces p(x) and i(x) are homeomorphic, i. e. constructions p and i generate “parallel” objects. at the same time, in section 4 we will show that functors p and i are not isomorphic, i. e. the constructions p and i themselves are not “parallel”. note that idempotent probability measures were investigated in [21]. unlike this work in the present paper we establish our results constructively, while in [21] the results were gotten descriptively. 3. spaces p(x) and i(x) are homeomorphic theorem 3.1. for an arbitrary hausdorff finite space x the spaces p(x) and i(x) are homeomorphic. proof. we determine the map z p i : p(x) −→ i(x), by the following equality z p i ( n∑ i=1 αiδxi ) = n⊕ i=1    ln αi − n⊕ j=1 ln αj   ⊙ δxi   , n∑ i=1 αiδxi ∈ p(x), here n∑ i=1 αi = 1, αi > 0 for all i = 1, . . . , n, and the map z i p : i(x) −→ p(x), by the rule z i p ( n⊕ i=1 λi ⊙ δxi ) = n∑ i=1 eλi n∑ j=1 eλj · δxi, n⊕ i=1 λi ⊙ δxi ∈ i(x), here n ⊕ i=1 λi = 0, λi > −∞, i = 1, . . . , n. we will show that the maps zpi and z i p are continuous and mutually inverse. © agt, upv, 2021 appl. gen. topol. 22, no. 2 405 k. f. kholturaev 1) for each probability measure n∑ i=1 αiδxi ∈ p(x) the following equalities hold z i p ( z p i ( n∑ i=1 αiδxi )) = zip   n⊕ i=1  ln αi − n⊕ j=1 ln αj   ⊙ δxi   = = n∑ i=1 e ln αi− n⊕ j=1 ln αj n∑ l=1 e ln αl− n⊕ j=1 ln αj · δxi = n∑ i=1 eln αi : e n⊕ j=1 ln αj ( n∑ l=1 eln αl ) : e n⊕ j=1 ln αj · δxi = = n∑ i=1 αiδxi n∑ l=1 αl = n∑ i=1 αiδxi; 2) for each idempotent probability measure ⊕n i=1 λi ⊙ δxi ∈ i(x) we have zpi ( zip ( n⊕ i=1 λi ⊙ δxi )) = zpi   n∑ i=1 eλi n∑ j=1 eλj δxi   = = n⊕ i=1  ln eλi n∑ j=1 eλj − n⊕ l=1 ln eλl n∑ j=1 eλj   ⊙ δxi = = n⊕ i=1  ln eλi − ln n∑ j=1 eλj − n⊕ l=1  lneλl − ln n∑ j=1 eλj     ⊙ δxi = = n⊕ i=1  λi − ln n∑ j=1 eλj − n⊕ l=1 λl + ln n∑ j=1 eλj   ⊙ δxi = n⊕ i=1 λi ⊙ δxi. consequently, the compositions zpi z i p : i(x) → i(x) and z i p z p i : p(x) → p(x) are the identical maps. now we will show that the maps zpi and z i p are continuous. since they are mutually inverse maps between compact hausdorff spaces, it suffices to show the continuity only of one of them. we show that the map zpi : p(x) → i(x) is continuous. let µ0 = n0∑ i(0)=1 α i(0) 0 δxi(0)0 ∈ p(x) be a probability measure, {µt} ∞ t=1 = { nt∑ i(t)=1 α i(t) t δxi(t) t }∞ t=1 ⊂ p(x) be a sequence converging to µ0 in the pointwise convergence topology (symbolically lim t→∞ µt = µ0). it means that lim t→∞ µt(ϕ) = µ0(ϕ) for all ϕ ∈ c(x). © agt, upv, 2021 appl. gen. topol. 22, no. 2 406 geometrical properties of idempotent measures for each point x i(0) 0 ∈ supp µ0 = {x 1 0, x 2 0, . . . , x n0 0 } consider a characteristic function χi(0) = χ{ x i(0) 0 } : x → r, i(0) = 1, . . . , n0. these functions are continuous, i. e. χi(0) ∈ c(x) since x is provided with the discrete topology. evidently, lim t→∞ µt(χi(0)) = µ0(χi(0)) = α i(0) 0 , i(0) = 1, . . . , n0.(⋆) (⋆) implies the following two conclusions: (output1) since each α i(0) 0 > 0, we have x i(0) 0 ∈ supp µt for all t greater than or equals to some ti(0). hence supp µ0 ⊂ supp µt for all t ≥ max{t1, . . . , tn0}; (output2) let α i(0) t be the barycentre mass of µt at x i(0) 0 ∈ supp µt, t = 1, 2 . . . . then lim t→∞ α i(0) t = α i(0) 0 . on the other hand, the continuity of the logarithm function ln and the operation ⊕ implies the equality lim t→∞ ln α i(0) t = ln α i(0) 0 . hence, lim t→∞  ln αi(0)t − n0⊕ j(0)=1 ln α j(0) t   = ln αi(0)0 − n0⊕ j(0)=1 ln α j(0) 0 . therefore, lim t→∞ zpi (µt) = z p i (µ0), i. e. the map z p i is continuous. theorem 3.1 is proved. � corollary 3.2. for an arbitrary metrizable compact space x the spaces p(x) and i(x) are homomorphic. proof. as well-known that a metrizable compact space has a dense countable subset. let m be a dense countable set in x. for each n let mn be a npoint subset of m, n = 1, 2, . . . , such that m1 ⊂ · · · ⊂ mn ⊂ mn+1 ⊂ . . . , and ∞ ∪ n=1 mn = m. one can directly verified that ∞ ∪ n=1 p(mn) and ∞ ∪ n=1 i(mn) are dense in p(x) and i(x) respectively. let z∞ : ∞ ∪ n=1 p(mn) → ∞ ∪ n=1 i(mn) be such a map that z∞|p (mn) = z p i for each n = 1, 2, . . . . then z∞ is a homeomorphism and continued over all p(x) uniquely. let z : p(x) → i(x) be this continuation. it is clear z is a homeomorphism. corollary 3.2 is proved. � 4. functors p and i are not isomorphic a subset l of the space c(x) is called [21] a max-plus-linear subspace in c(x), if: 1) λx ∈ l for each λ ∈ r; 2) λ ⊙ ϕ ∈ l for each λ ∈ r and ϕ ∈ l; 3) ϕ ⊕ ψ ∈ l for each ϕ,ψ ∈ l. © agt, upv, 2021 appl. gen. topol. 22, no. 2 407 k. f. kholturaev lemma 4.1 ([21] the max-plus variant of the hahn-banach theorem). let l be a max-plus-linear subspace in c(x). let µ : l −→ r be a functional satisfying the conditions of normality (i1), homogeneity (i2) and additivity (i3) (with c(x) replaced by l). for an arbitrary ϕ0 ∈ c(x)\l there exists an extension of the functional µ satisfying the conditions of normality, homogeneity and additivity on the minimal max-plus-linear subspace l ′ containing l∪{ϕ0}. consider the following subset in c(x × y ): c0 = { n⊕ i=1 ϕi ⊙ ψi : ϕi ∈ c(x) and ψi ∈ c(y ), i = 1, . . . , n; n ∈ n } . it is obvious that c0 is a max-plus-linear subspace in c(x). for every pair (µ,ν) ∈ i(x) × i(y ) we put (µ⊗̃ν) ( n⊕ i=1 ϕi ⊙ ψi ) = n⊕ i=1 µ(ϕi) ⊙ ν(ψi). proposition 4.2. the constructed functional µ⊗̃ν satisfies the conditions of normality, homogeneity and additivity on c0. proof. each c ∈ r can be represented as cx×y = ax ⊙by , where a, b ∈ r and a+b = c. therefore, (µ⊗̃ν)(cx×y ) = (µ⊗̃ν)(ax ⊙by ) = µ(a)⊙ν(b) = a⊙b = c. let λ ∈ r and ⊕n i=1 ϕi ⊙ ψi ∈ c0. then (µ⊗̃ν) ( λ ⊙ n⊕ i=1 ϕi ⊙ ψi ) = (µ⊗̃ν) ( n⊕ i=1 (λ ⊙ ϕi) ⊙ ψi ) = = n⊕ i=1 µ(λ ⊙ ϕi) ⊙ ν(ψi) = n⊕ i=1 λ ⊙ µ(ϕi) ⊙ ν(ψi) = = λ ⊙ n⊕ i=1 µ(ϕi) ⊙ ν(ψi) = λ ⊙ (µ⊗̃ν) ( n⊕ i=1 ϕi ⊙ ψi ) . finally, let ⊕n i=1 ϕ1 i ⊙ ψ1 i ∈ c0 and ⊕m j=1 ϕ2 j ⊙ ψ2 j ∈ c0. then (µ⊗̃ν)   n⊕ i=1 ϕ1 i ⊙ ψ1 i ⊕ m⊕ j=1 ϕ2 j ⊙ ψ2 j   = = (µ⊗̃ν) (⊕ ϕk l ⊙ ψk l ) = ⊕ µ(ϕk l) ⊙ ν(ψk l) = = n⊕ i=1 µ(ϕ1 i) ⊙ ν(ψ1 i) ⊕ m⊕ j=1 µ(ϕ2 j) ⊙ ν(ψ2 j) = = (µ⊗̃ν) ( n⊕ i=1 ϕ1 i ⊙ ψ1 i ) ⊕ (µ⊗̃ν)   m⊕ j=1 ϕ2 j ⊙ ψ2 j   . © agt, upv, 2021 appl. gen. topol. 22, no. 2 408 geometrical properties of idempotent measures proposition 4.2 is proved. � since c0 is a max-plus-linear subspace in c(x × y ), according to lemma 1 for the idempotent probability measure µ⊗̃ν there exists its extension ξ all over c(x × y ) which satisfies the conditions of normality, homogeneity and additivity on c(x × y ). so, we have proved the following max-plus variant of the fubini theorem. theorem 4.3. for every pair (µ, ν) ∈ i(x)×i(y ) there exists an idempotent probability measure ξ ∈ i(x ×y ) such that ξ(ϕ⊙ψ) = µ(ϕ)⊙ν(ψ), ϕ ∈ c(x), ψ ∈ c(y ). from the results of work [18] (see section 3) it follows that if |x| ≥ 2, |y | ≥ 2, then µ⊗̃ν has uncountable many extensions on c(x × y ). put µ ⊗ ν = ⊕{ ξ ∈ i(x × y ) : ξ|c0 = µ⊗̃ν } . similarly to the traditional case, µ ⊗ ν we call as a “tensor” product of idempotent probability measure µ and ν. further, to distinguish the tensor products we will use symbols ⊗i and ⊗p for the idempotent and traditional cases, respectively. let us give the classical option of the fubini theorem. theorem 4.4 ([4]). for every pair (µ, ν) ∈ p(x)×p(y ) there exists a unique probability measure µ⊗p ν ∈ p(x ×y ) such that (µ⊗p ν)(ϕ·ψ) = µ(ϕ)·ν(ψ), ϕ ∈ c(x), ψ ∈ c(y ). now we need some concepts from the category theory [4], [15]. let fi : c → c ′, i = 1, 2, be to functors from the category c = (o, m) to the category c ′ = (o ′, m ′). a family of morphisms φ = {ϕx : f1(x) → f2(x)|x ∈ o} ⊂ m ′ is said to be a natural transformation of the functor f1 to the functor f2, if for each morphism f : x → y of the category c a diagram f1(x) f1(f) −−−−→ f1(y ) ϕx y ϕy y f2(x) f2(f) −−−−→ f2(y ) is commutative, i. e. f2(f) ◦ ϕx = ϕy ◦ f1(f). if, for every object x in c, the morphism fx is an isomorphism in c ′, then φ = {fx} is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). two functors fi : c → c ′, i = 1, 2, are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from f1 to f2. our goal is to show that the functors p and i are not isomorphic. © agt, upv, 2021 appl. gen. topol. 22, no. 2 409 k. f. kholturaev example 4.5. consider the sets x = {a,b,c}, y = {a,b}, z = {a,c}, where a, b, c are different points (these sets are supplied with discrete topologies). define the following maps: f : x −→ y, f(a) = f(c) = a, f(b) = b, g : x −→ z, g(a) = g(b) = a, g(c) = c. consider compact hausdorff spaces x, y × z, and the map (f, g): x −→ y × z. it suffices to show the map (p(f), p(g)) has a property that the map (i(f), i(g)) does not possess it. at first we show the map (p(f), p(g)): p(x) −→ p(y ) × p(z) is an embedding. in fact, for any pair of probability measures µ = α1δa + α2δb + α3δc, ν = β1δa + β2δb + β3δc, with positive α1, α2, α3, β1, β2, β3, α1 + α2 + α3 = 1, β1 + β2 + β3 = 1, the following equalities take place p(f)(µ) = (α1 + α3)δa + α2δb, p(f)(ν) = (β1 + β3)δa + β2δb, p(g)(µ) = (α1 + α2)δa + α3δc, p(g)(ν) = (β1 + β2)δa + β3δc. therefore, (p(f), p(g))(µ) = (p(f), p(g))(ν) if and only if    α1 + α3 = β1 + β3, α2 = β2, α1 + α2 = β1 + β2, α3 = β3. (1.p) system (1.p) has a unique solution α1 = β1, α2 = β2 and α3 = β3. hence, µ = ν. thus, (p(f), p(g))(µ) = (p(f), p(g))(ν) if and only if µ = ν, i. e. (p(f), p(g)): p(x) → p(y ) × p(z) is an embedding. consequently, the diagram p(x) (p (f), p (g)) // p ((f, g)) �� p(y ) × p(z) ⊗p ww♦♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ p(y × z) holds, i. e. p((f, g)) = ⊗p ◦ (p(f), p(g)),(2.p) where the map ⊗p : p(y ) × p(z) → p(y × z) acts as ⊗p (µ, ν) = µ ⊗p ν. remind, the uniqueness of the solution of system (1.p) and equality (2.p) may be considered as corollaries of theorem 4.4. © agt, upv, 2021 appl. gen. topol. 22, no. 2 410 geometrical properties of idempotent measures now we show that the map (i(f), i(g)): i(x) → i(y ) × i(z) is not an embedding. really, for idempotent probability measures µ = λ1 ⊙ δa ⊕ λ2 ⊙ δb ⊕ λ3 ⊙ δc, ν = γ1 ⊙ δa ⊕ γ2 ⊙ δb ⊕ γ3 ⊙ δc, with −∞ < λ1, λ2, λ3, γ1, γ2, γ3 ≤ 0 and λ1 ⊕ λ2 ⊕ λ3 = γ1 ⊕ γ2 ⊕ γ3 = 0, the following equalities hold i(f)(µ) = (λ1 ⊕ λ3) ⊙ δa ⊕ λ2 ⊙ δb, i(f)(ν) = (γ1 ⊕ γ3) ⊙ δa ⊕ γ2 ⊙ δb, i(g)(µ) = (λ1 ⊕ λ2) ⊙ δa ⊕ λ3 ⊙ δc, i(g)(ν) = (γ1 ⊕ γ2) ⊙ δa ⊕ γ3 ⊙ δc. the equality (i(f), i(g))(µ) = (i(f), i(g))(ν) is true if and only if    λ1 ⊕ λ3 = γ1 ⊕ γ3, λ2 = γ2, λ1 ⊕ λ2 = γ1 ⊕ γ2, λ3 = γ3. (1.i) system (1.i) has infinitely many solutions. for example, for every pair of λ1 and γ1 with −∞ < λ1 ≤ 0, −∞ < γ1 ≤ 0 a 6-tuple (λ1, γ1, 0, 0, 0, 0) is its solution. the equality (i(f), i(g))(µ) = (i(f), i(g))(ν) is true for this 6-tuple although λ1 6= γ1. this means that the map (i(f), i(g)) is not an embedding. that means that the following diagram i(x) (i(f), i(g)) // i((f, g)) �� i(y ) × i(z) ⊗i ww♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ i(y × z) does not hold, i. e. the equality i((f, g)) = ⊗i ◦ (i(f), i(g))(2.i) is wrong. to present the existence of infinitely many solutions of system (1.i), or relation (2.i) consider idempotent probability measures µ = −1 ⊙ δa ⊕ 0 ⊙ δb ⊕ 0 ⊙ δc, ν = −2 ⊙ δa ⊕ 0 ⊙ δb ⊕ 0 ⊙ δc. then i(f)(µ) = 0 ⊙ δa ⊕ 0 ⊙ δb, i(f)(ν) = 0 ⊙ δa ⊕ 0 ⊙ δb, i(g)(µ) = 0 ⊙ δa ⊕ 0 ⊙ δc, i(g)(ν) = 0 ⊙ δa ⊕ 0 ⊙ δc, and (i(f), i(g))(µ) = (0 ⊙ δa ⊕ 0 ⊙ δb, 0 ⊙ δa ⊕ 0 ⊙ δc) = (i(f), i(g))(ν), which yields i(f)(µ) ⊗i i(g)(µ) = 0 ⊙ δ(a, a) ⊕ 0 ⊙ δ(b, c) = i(f)(ν) ⊗i i(g)(ν). © agt, upv, 2021 appl. gen. topol. 22, no. 2 411 k. f. kholturaev on the other hand since (f, g)(a) = (a, a), (f, g)(b) = (b, a), (f, g)(c) = (a, c), we have i((f, g))(µ) = −1 ⊙ δ(a, a) ⊕ 0 ⊙ δ(b, a) ⊕ 0 ⊙ δ(a, c), i((f, g))(ν) = −2 ⊙ δ(a, a) ⊕ 0 ⊙ δ(b, a) ⊕ 0 ⊙ δ(a, c). so, i((f, g))(µ) 6= i((f, g))(ν). moreover, i((f, g))(µ) 6= ⊗i ◦ (i(f), i(g))(µ) and i((f, g))(ν) 6= ⊗i ◦ (i(f), i(g))(ν). thus, (1) the map (p(f), p(g)): p(x) → p(y )×p(z) is an embedding, while the map (i(f), i(g)): i(x) → i(y ) × i(z) is not an embedding. (2) for the embedding (f, g): x → y ×z the embedding p((f, g)): px) → p(y × z) might be defined by the rule (2.p), while the embedding i((f, g)): i(x) → i(y × z) does not have such a decomposition. (3) regardless of the natural transformation φ = {ϕx : p(x) → i(x) : x ∈ comp} a diagram p(x) (p (f), p (g)) −−−−−−−−→ p(y ) × p(z) ϕx y yϕy ×ϕz i(x) (i(f), i(g)) −−−−−−−→ i(y ) × i(z) can not be commutative. so, we came to the following important conclusion: although for a metrizable hausdorff compact x the spaces i(x) and p(x) are homeomorphic, example 4.5 shows, the difference between the constructions p and i appears even on finite sets. thus, the functors p and i are not isomorphic. note this conclusion was proclaimed in [21] as proposition 2.15. but it was not equipped with detailed proof. 5. on a metricise criterion of the compact hausdorff spaces the well-known m. katětov’s theorem states [9] that the hereditary normality of the cube x3 of a hausdorff compact space x follows metrizability of x. in 1989, v. v. fedorchuk generalized [3] katětov’s theorem for a normal functors of the degree ≥ 3, acting in the category comp. many publications in the field of general topology are devoted to the issues of katětovæs theorem and problem. © agt, upv, 2021 appl. gen. topol. 22, no. 2 412 geometrical properties of idempotent measures the set of all nonempty closed subsets of the topological space x is denoted by exp x. for open subsets u1, . . . , un ⊂ x a family of the sets of the type o〈u1, . . . , un〉 = {f : f ∈ exp x, f ⊂ n⋃ i=1 un, f ∩ ui 6= ∅, i = . . . , n} forms a base of a topology on exp x. this topology is called the vietoris topology, the set exp x equipped with the vietoris topology is called a hyperspace of the topological space x. for a compact x its hyperspace exp x is a compact. for the compact x, the natural number n, the functor f we put fn = {a ∈ f(x) : | supp a| ≤ n}, f0n = fn(x) \ fn−1(x), where supp a = ⋂{ a ⊂ x : a = a, a ∈ f(a) } is a support of the element a ∈ f(x). in particular, expn x = {k ∈ exp x : |k| ≤ n}, exp0n = expn x \ expn−1 x, and in = {µ ∈ i(x) : | supp µ| ≤ n}, i0n = in(x) \ in−1(x). let τ be an uncountable cardinal number. put nτ = {x : x < τ}. provide nτ with the discrete topology. then it becomes a local compact hausdorff space (which is not compact, since |nτ| = τ > ℵ0). by αnτ = nτ ∪ {p} we denote its one-point compactification, where p 6∈ nτ. in [3] (proposition 1) it was shown that if τ is an uncountable cardinal then exp2 αnτ is not hereditary normal. we claim the following statement. proposition 5.1. for every uncountable cardinal number τ the space exp03 αnτ is not normal. proof. obviously, there exist disjoint subsets f1 and f2 of nτ such that f1 is uncountable and f2 is countable. take a point x0 ∈ f1 ∪ f2. choose subsets a1 and a2 of the space exp 0 3 αnτ, assuming a1 = { {p,x,x0} : x ∈ f1 \ {x0} } , a2 = { {p,x ′ ,x0} : x ′ ∈ f2 \ {x0} } , obviously, a1 ∩ a2 = ∅. let f = {x1,x2,x3} ∈ exp 0 3 αnτ \ a1. the set o({x1},{x2},{x3}) is an open neighbourhood of the set f wich does not intersect a1. hence, the set a1 is closed in exp03 αnτ. similarly one can check that a2 is closed in exp 0 3 αnτ. for each x ∈ nτ we put ux = o〈αnτ \ {x0,x},{x0},{x}〉 ∩ exp 0 3 αnτ. it is easy to see that the smallest by inclusion neighbourhoods of the sets a1 and a2 in exp 0 3 (αnτ) are the sets oa1 = ⋃ x∈f1 ux and oa2 = ⋃ x∈f2 ux, respectively. for the set {a,b,x0}, where a ∈ f1, b ∈ f2, we have {a,b,x0} ∈ o (αnτ \ {x0,a},{x0},{a}) ⊂ oa1, {a,b,x0} ∈ o (αnτ \ {x0,b},{x0},{b}) ⊂ oa2. © agt, upv, 2021 appl. gen. topol. 22, no. 2 413 k. f. kholturaev this means, oa1 ∩ oa2 6= ∅. proposition 5.1 is proved. � proposition 5.2. let τ is an uncountable cardinal number. then i03 (αnτ) is not normal space. proof. for each compact x the set exp03 x is closed in i 0 3 (x). really, a correspondence {a, b, c} 7→ 0⊙δa ⊕0⊙δb ⊕0⊙δc establishes an identical embedding of exp03 x into i 0 3 (x). this embedding is continuous and it is closed map. on the other hand the normality is a hereditary property for the closed subsets of the space. therefore according to proposition 5.1 the space i03 (αnτ) is not normal. proposition 5.2 is proved. � in [3] (theorem 2) it was established that if for a normal functor f of degree ≥ 3 the hausdorff compact space f(x) is hereditary normal then a hausdorff compact space x is metrizable. we get modified shape of this result for the functor i, and it might be considered as a metricize criterion of compact hausdorff spaces. theorem 5.3. let x be a compact hausdorff space. if i3(x) \ x is a hereditarily normal space then x is metrizable. proof. suppose the compact x is non-metrizable. if x has a unique nonisolated point then x is homeomorphic to αnτ for τ = |x| > ω. proposition 5.2 implies that i03 (x) is not normal. but according to the condition i 0 3 (x) must be normal as a subset of the hereditarily normal space i3(x) \ x. we get a contradiction. now let a and b be distinguished nonisolated points of the compact x. there are open neighbourhoods u and v of points a and b, respectively, such that u ⋂ v = ∅. we consider set z = u × exp2 v and by the formula λ(x,y,z) = 0 ⊙ δx ⊕ 0 ⊙ δy ⊕ 0 ⊙ δz we define the topological embedding λ : z → i3(x) \ x. the result of m. katetov [9, corollary 1] (which asserts the perfectly normality of the factor x under the condition of hereditarily normality of the product of x × y ) implies that the factor exp2 v of the product z = u × exp2 v is perfectly normal. further, applying the result of v. v. fedorchuk [3] (which asserts the metrizability of the compact x if for a normal functor f of degree ≤ 2 the space f(x) is perfectly normal) we conclude that v is metrizable. similarly, one can show that u is metrizable. therefore each nonisolated point of the compact x has a metrizable closed neighbourhood. hence the compact x is locally metrizable. therefore it is metrizable. theorem 5.3 is proved. � corollary 5.4. let x be a compact hausdorff space and n ≥ 3. if in(x) \ x is a hereditarily normal space then x is metrizable. © agt, upv, 2021 appl. gen. topol. 22, no. 2 414 geometrical properties of idempotent measures acknowledgements. the author expresses deep gratitude to the editor, professor salvador romaguera and the referee for suggestions and useful advice. also the author would like to express thanks to professor adilbek zaitov for the revealed shortcomings, the specified remarks. references [1] g. choquet, theory of capacities, ann. inst. fourier 5 (1955), 131–295. 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[21] m. m. zarichnyi, spaces and maps of idempotent measures, izv. math. 74, no. 3 (2010), 481–499. © agt, upv, 2021 appl. gen. topol. 22, no. 2 415 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 115-134 pseudo perfectly continuous functions and closedness/compactness of their function spaces j. k. kohli, d. singh, jeetendra aggarwal and manoj rana abstract a new class of functions called ‘pseudo perfectly continuous’ functions is introduced. their place in the hierarchy of variants of continuity which already exist in the literature is highlighted. the interplay between topological properties and pseudo perfect continuity is investigated. function spaces of pseudo perfectly continuous functions are considered and sufficient conditions for their closedness and compactness in the topology of pointwise convergence are formulated. 2010 msc: 54c05, 54c10, 54c35, 54g15, 54g20,54d10 keywords: (quasi) perfectly continuous function, dδ-supercontinuous function, dδ-map, slightly continuous function, pseudo-partition topology, dδt0-space, δ-completely regular space, alexandroff space (≡ saturated space). 1. introduction the class of pseudo perfectly continuous functions properly contains the class of quasi perfectly continuous functions [36] which in its turn strictly contains the class of δ-perfectly continuous functions [27] and so includes all perfectly continuous functions due to noiri [48] and hence contains all strongly continuous functions of levine [37]. it is well known that the set c(x, y ) of all continuous functions from a space x into a space y is not closed in y x in the topology of pointwise convergence. however, naimpally [46] showed that in contrast to continuous functions the set s(x, y ) of strongly continuous functions is closed in y x in the topology 116 j. k. kohli, d. singh, j. aggarwal and m. rana of pointwise convergence if x is locally connected and y is hausdorff. naimpally’s result is extended in ([25], [27], [36], [55]) for larger classes of functions and spaces. the main purpose of this paper is to further strengthen these results to show that if x is sum connected and y is a dδt0-space, then several classes of functions are identical and closed in y x in the topology of pointwise convergence. moreover, conditions are formulated for these classes of functions to be compact hausdorff subspaces of y x in the topology of pointwise convergence. the organization of the paper is as follows: section 2 is devoted to preliminaries and basic definitions. in section 3 examples are included to ascertain the distinctiveness of the notion so defined from the existing notions in the mathematical literature. section 4 is devoted to the study of basic properties of pseudo perfectly continuous functions wherein, in particular, it is shown that (i) pseudo perfect continuity is preserved under compositions and expansion of range (ii) sufficient conditions are formulated for the preservation of pseudo perfect continuity in the passage to the graph function and under restriction of range. the notion of pseudo partition topology is introduced and sufficient conditions are given for its direct and inverse preservation under mappings. a sum theorem is proved showing when the presence of pseudo perfect continuity on parts of a function implies pseudo perfect continuity on the whole space. in section 5 we discuss the interplay between topological properties and pseudo perfectly continuous functions. section 6 is devoted to function spaces wherein it is shown that if x is sum connected [16] (e.g. connected or locally connected) and y is a dδt0-space, then the function space pp(x, y ) of all pseudo perfectly continuous functions from x to y is closed in y x in the topology of pointwise convergence. moreover, if y is a compact dδt0-space, then pp(x, y ) and several other function spaces are shown to be compact hausdorff in the topology of pointwise convergence. 2. preliminaries and basic definitions a collection β of subsets of a space x is called an open complementary system [12] if β consists of open sets such that for every b ∈ β, there exist b1, b2, . . . ∈ β with b = ∪{x \ bi : i ∈ n}. a subset a of a space x is called a strongly open fσ-set [12] if there exists a countable open complementary system β(a) with a ∈ β(a). the complement of a strongly open fσ-set is called strongly closed gδ-set. a subset a of a space x is called a regular gδ-set [42] if a is an intersection of a sequence of closed sets whose interiors contain a, i.e., if a = ∞⋂ n=1 fn = ∞⋂ n=1 f 0n, where each fn is a closed subset of x (here f 0n denotes the interior of fn). the complement of a regular gδ-set is called a regular fσ-set. a point x ∈ x is called a θ-adherent point [59] of a ⊂ x if every closed neigbourhood of x intersects a. let clθa denote the set of all θ-adherent points of a. the set a is called θ-closed if a = clθa. the complement of a θ-closed set is referred to as a θ-open set. a subset a of a pseudo perfectly continuous functions and closedness/compactness of their function spaces117 space x is said to be regular open if it is the interior of its closure, i.e., a = a 0 . the complement of a regular open set is referred to as a regular closed set. any union of regular open sets is called δ-open [59]. the complement of a δ-open set is referred to as a δ-closed set. an open subset u of a space x is said to be r-open [30] if for each x ∈ u there exists a closed set b such that x ∈ b ⊂ u or equivalently, if u is expressible as a union of closed sets.the complement of an r-open set is called r-closed set. a subset a of a space x is said to be cl-open [54] if for each x ∈ a there exists a clopen set h such that x ∈ h ⊂ a ; or equivalently a is expressible as a union of clopen sets. the complement of a cl-open set is referred to as a cl-closed set. definitions 2.1. a function f : x → y from a topological space x into a topological space y is said to be (a) strongly continuous [37] if f(ā) ⊂ f(a) for each subset a of x. (b) perfectly continuous ([32], [48]) if f−1(v ) is clopen in x for every open set v ⊂ y . (c) cl-supercontinuous [54] (≡ clopen continuous [51])if for each x ∈ x and each open set v containing f(x) there is a clopen set u containing x such that f(u) ⊂ v . (d) z-supercontinuous [20] (respectively dδ-supercontinuous [21], respectively supercontinuous [45]) if for each x ∈ x and for each open set v containing f(x), there exists a cozero set (respectively regular fσ-set, respectively regular open set) u containing x such that f(u) ⊂ v . (e) strongly θ-continuous ([39], [47]) if for each x ∈ x and for each open set v containing f(x), there exists an open set u containing x such that f(u) ⊂ v . definitions 2.2. a function f : x → y from a topological space x into a topological space y is said to be (a) dδ-continuous [22] (respectively d-continuous [18], respectively z-continuous [52]) if for each point x ∈ x and each regular fσ-set (respectively open fσ-set, respectively cozero set) v containing f(x) there is an open set u containing x such that f(u) ⊂ v . (b) almost continuous [53] (respectively faintly continuous [40], respectively r-continuous [35]) if for each x ∈ x and each regular open set (respectively θ-open set, respectively r-open set) v containing f(x) there is an open set u containing x such that f(u) ⊂ v . (c) dδ-map [23] if for each regular fσ-set u in y , f −1(u) is a regular fσ-set in x. (d) θ-continuous [10] if for each x ∈ x and each open set v containing f(x) there is an open set u containing x such that f(u) ⊂ v . (e) weakly continuous [38] if for each x ∈ x and each open set v containing f(x) there exists an open set u containing x such that f(u) ⊂ v . (f) quasi θ-continuous function [50] if for each x ∈ x and each θ-open set v containing f(x) there exists a θ-open set u containing x such that f(u) ⊂ v . 118 j. k. kohli, d. singh, j. aggarwal and m. rana (g) slightly continuous1[13] if f−1(v ) is open in x for every clopen set v ⊂ y . definitions 2.3. a function f : x → y from a topological space x into a topological space y is said to be (a) δ-perfectly continuous [27] if for each δ-open set v in y , f−1(v ) is a clopen set in x. (b) almost perfectly continuous [55] (≡ regular set connected [7]) if f−1(v ) is clopen for every regular open set v in y . (c) almost cl-supercontinuous [26] (≡ almost clopen continuous [9]) if for each x ∈ x and each regular open set v containing f(x), there is a clopen set u containing x such that f(u) ⊂ v . (d) almost z-supercontinuous [34] (almost dδ-supercontinuous) if for each x ∈ x and for each regular open set v containing f(x), there exists a cozero set (regular fσ-set) u containing x such that f(u) ⊂ v . (e) almost strongly θ-continuous [49] if for each x ∈ x and for each regular open set v containing f(x), there exists an open set u containing x such that f(u) ⊂ v . (f) quasi perfectly continuous [36] if f−1(v ) is clopen in x for every θ-open set v in y . (g) quasi z-supercontinuous [33] (quasi cl-supercontinuos [19]) if for each x ∈ x and each θ-open set v containing f(x), there exists a cozero (clopen) set u containing x such that f(u) ⊂ v . (h) pseudo z-supercontinuous [33] (pseudo cl-supercontinuos [31]) if for each x ∈ x and each regular fσ-set v containing f(x), there exists a cozero (clopen) set u containing x such that f(u) ⊂ v . (i) δ-continuous [47] if for each x ∈ x and for each regular open set v containing f(x), there exists a regular open set u containing x such that f(u) ⊂ v . 3. pseudo perfectly continuous functions we call a function f : x → y from a topological space x into a topological space y pseudo perfectly continuous if f−1(v ) is clopen in x for every regular fσ-set v in y . the adjoining diagram (figure 1) well exhibits the interrelations that exist among pseudo perfect continuity and other variants of continuity that already exist in the literature and are related to the theme of the present paper and thus well reflects the place of pseudo perfect continuity in the hierarchy of known variants of continuity. examples. 3.1. let x denote the real line with usual topology and let y be the real line with cofinite (or cocountable) topology. then the identity function f : x → y is quasi perfectly continuous but not continuous. 1slightly continuous functions have been referred to as cl-continuous in ([22], [35]). p se u d o p e rfe c tly c o n tin u o u s fu n c tio n s a n d c lo se d n e ss/ c o m p a c tn e ss o f th e ir fu n c tio n sp a c e s1 1 9 strongly continuous perfectly continuous cl-supercontinuous z-supercontinuous -supercontinuous strongly -continuous supercontinuous continuous -perfectly continuous almost perfectly continuous almost cl-supercontinuous almost z-supercontinuous almost -supercontinuous almost strongly -continuous -continuous almost continuous -continuous weakly continuous quasi perfectly continuous quasi cl-supercontinuous quasi z-supercontinuous quasi -supercontinuous quasi -continuous quasi faintly continuous supercontinuous [28] -continuous pseudo perfectly continuous pseudo cl-supercontinuous pseudo z-supercontinuous pseudo -supercontinuous pseudo pseudo supercontinuous [28] strongly -continuous [29] z-continuous d d d d d d d d d d q d q d q q q slightly continuous figure 1. 120 j. k. kohli, d. singh, j. aggarwal and m. rana 3.2. let x be the real line endowed with usual topology. then the identity function defined on x is continuous as well as z-supercontinuous but not pseudo perfectly continuous. the space e0 in the following example is due to misra [44, example 3.1, p. 352]. 3.3. let w1 be the first uncountable ordinal. let the space e0 be the union of disjoint sets {a, b}, {aαβ : 0 ≤ α, β < w1}, {bαβ : 0 ≤ α, β < w1} and {cγ : 0 ≤ γ < w1}. the basic neighbourhoods of various points be as follows: all the points aαβ and bαβ, 0 ≤ α, β < w1 are isolated; for each fixed γ, a typical basic neighbourhood of the point cγ contains the points aγβ and bγβ for all but countably many indices β, 0 ≤ β < w1; a typical basic neighbourhood of a (respectively b) contains for every α greater than some ordinal δ < w1, all but countably many points aαβ (respectively bαβ). then e0 is a hausdorff, non urysohn p-space and every real valued continuous function defined on e0 takes the same value at the points a and b. it is easily verified that in the space e0 every fσ-set and hence every regular fσ-set is clopen. thus the identity mapping ie0 : e0 → e0 is pseudo perfectly continuous. however, it is not a quasi perfectly continuous function, since the inverse image of θ-closed set {a} is not clopen. 4. basic properties of pseudo perfectly continuous functions proposition 4.1. if f : x → y is a pseudo perfectly continuous function and g : y → z is a dδ-map, then g ◦ f is a pseudo perfectly continuous function. in particular, composition of two pseudo perfectly continuous functions is pseudo perfectly continuous. corollary 4.2. if f : x → y is a pseudo perfectly continuous function and g : y → z is a continuous function, then g ◦ f is a pseudo perfectly continuous function. proof. every continuous map is a dδ-map. � proposition 4.3. let f : x → y be a slightly continuous function and let g : y → z be a pseudo perfectly continuous function. then g ◦ f is pseudo perfectly continuous. remark 4.4. the hypothesis of ‘slightly continuity’ in proposition 4.3 can be traded of by any one of the weak variants of continuity in the following diagram, since each one of them implies slight continuity. pseudo perfectly continuous functions and closedness/compactness of their function spaces121 continuous almost continuous θ -continuous r-continuous -continuous quasi -continuous weakly continuous slightly continuous z-continuous continuous faintly continuous q d q d d figure 2. definition 4.5. a space x is said to be endowed with a (a) pseudo partition topology if every regular fσ-set in x is closed; or equivalently every regular gδ-set in x is open. (b) partition topology [58] if every open set in x is closed. (c) δ-partition topology [27] if every δ-open set in x is closed. (d) almost partition topology [55] (≡extremally disconnected topology [58]) if every regular open set in x is closed. (e) quasi partition topology [36] if every θ-open set in x is closed. the following implications are immediate from definitions. partition topology -partition topology quasi partition topology almost partition topology ( extremally disconnected topology) pseudo partition topology ! º figure 3. however, none of the above implications is reversible as shown in ([27], [55]) and the following examples. example 4.6 ([43, example 3.20]). consider the space r of reals with countable complement extension topology τ [58, example 63, p. 85]. let r \ q be the quotient space obtained from (r, τ) by identifying the set q of rationals to a point. since q is closed in (r, τ), the space r \ q is t1 and the quotient map p : (r, τ) → r \ q is a closed map. let z be an irrational and let a and b be any rationals such that a < z < b. then g = {[x] : x ∈ (a, b), x irrational} is a regular open set containing [z] in r \ q. the quotient topology on r \ q is not an almost partition topology and hence not a δ-partition topology since g is a regular open set but not closed in r \ q. on the other hand r \ q is the only θ-open set in r \ q so it is equipped with a quasi partition topology. example 4.7. the hausdorff space e0 also discussed in example 3.3 which is due to misra [44, example 3.1] has a pseudo partition topology since every regular fσ-set is clopen in e0 but it is not endowed with a quasi partition topology. 122 j. k. kohli, d. singh, j. aggarwal and m. rana theorem 4.8. let f : x → y be a function and g : x → x × y , defined by g(x) = (x, f(x)) for each x ∈ x, be the graph function. if g is pseudo perfectly continuous, then so is f and the space x is endowed with a pseudo partition topology. further, if x has a pseudo partition topology and f is pseudo perfectly continuous, then g is pseudo cl-supercontinuous. proof. suppose that the graph function g : x → x × y is pseudo perfectly continuous. consider the projection map py : x × y → y . since it is continuous, it is a dδ-map. hence in view of proposition 4.1, the function f = py ◦ g is pseudo perfectly continuous. to prove that the space x possesses a pseudo partition topology, let u be a regular fσ-set in x. then u × y is a regular fσ-set in x × y . since g is pseudo perfectly continuous, g −1(u × y ) = u is clopen in x and so the topology of x is a pseudo partition topology. finally, suppose that x has pseudo partition topology and f is a pseudo perfectly continuous function. to show that g is pseudo cl-supercontinuous, let u × v be a basic regular fσ-set in x × y . then g −1(u × v ) = u ∩ f−1(v ) is a clopen set in x and so g is pseudo cl-supercontinuous. � the following result gives sufficient conditions on mappings for domain or range of the mapping to be endowed with pseudo partition topology. theorem 4.9. let f : x → y be a pseudo perfectly continuous surjection which maps clopen sets to closed (open) sets. then y is endowed with a pseudo partition topology. moreover, if f is a bijection which maps regular fσ-sets (regular gδ-sets) to regular fσ-sets (regular gδ-sets), then x is also equipped with a pseudo partition topology. proof. suppose f maps clopen sets to closed (open) sets. let v be a regular fσ-set (regular gδ-set) in y . in view of pseudo perfect continuity of f, f −1(v ) is a clopen set in x. again, since f is a surjection which maps clopen sets to closed (open) sets, the set f(f−1(v )) = v is closed (open) in y and hence clopen in y . thus y is endowed with a pseudo partition topology. to prove the last part of the theorem assume that f is a bijection which maps regular fσ-sets (regular gδ-sets) to regular fσ-sets (regular gδ-sets) and let u be a regular fσ-set (regular gδ-set) in x. then f(u) is a regular fσset (regular gδ-set) in y . since f is a pseudo perfectly continuous bijection, f−1(f(u)) = u is a clopen set in x and so x is endowed with a pseudo partition topology. � remark 4.10. a space x is endowed with a pseudo partition topology if and only if every dδ-map f : x → y is pseudo perfectly continuous. necessity is obvious in view of definitions. to prove sufficiency, assume contrapositive and let v be a regular fσ-set in x which is not clopen. then the identity mapping defined on x is a dδ-map but not pseudo perfectly continuous. pseudo perfectly continuous functions and closedness/compactness of their function spaces123 proposition 4.11. if f : x → y is a surjection which maps clopen sets to open sets and g : y → z is a function such that g ◦ f is pseudo perfectly continuous, then g is a dδ-continuous function. moreover, if f maps clopen sets to clopen sets, then g is a pseudo perfectly continuous function. proof. let v be a regular fσ-set in z. since g◦f is pseudo perfectly continuous, (g ◦ f)−1(v ) = f−1(g−1(v )) is clopen set in x. again, since f is a surjection which maps clopen sets to open sets, f(f−1(g−1(v ))) = g−1(v ) is open in y and so g is a dδ-continuous function. the last assertion is immediate, since in this case g−1(v ) is a clopen set in y . � proposition 4.12. if f : x → y is a pseudo perfectly continuous function and g : y → z is a dδ-supercontinuous function, then their composition is cl-supercontinuous. proof. let v be an open set in z. in view of dδ-supercontinuity of g, g −1(v ) is a dδ-open set in y and so g −1(v ) = ⋃ α vα, where each vα is a regular fσ-set. since f is pseudo perfectly continuous, each f−1(vα) is a clopen set. hence (g ◦ f)−1(v ) = f−1(g−1(v )) = f−1( ⋃ α vα) = ⋃ α f−1(vα) is cl-open. so g ◦ f is cl-supercontinuous. � proposition 4.13. if f : x → y is a pseudo perfectly continuous function and g : y → z is an almost dδ-supercontinuous function, then their composition g ◦ f is almost cl-supercontinuous. proposition 4.14. if f : x → y is a pseudo perfectly continuous function and g : y → z is quasi dδ-supercontinuous, then their composition g ◦ f is quasi cl-supercontinuous. theorem 4.15. let f : x → y be a function and let q = {xα : α ∈ λ} be a locally finite clopen cover of x. for each α ∈ λ, let fα = f|xα : xα → y denote the restriction map. then f is pseudo perfectly continuous if and only if each fα is pseudo perfectly continuous. proof. necessity is immediate in view of the fact that quasi perfect continuity is preserved under the restriction of domain. to prove sufficiency, let v be a regular fσ-set in y . then f −1(v ) = ⋃ α∈λ (f|xα) −1(v ) = ⋃ α∈λ (f−1(v ) ∩ xα). since each f−1(v ) ∩ xα is clopen in xα and hence in x. thus f −1(v ) is open being the union of clopen sets. moreover, since the collection q is locally finite, the collection {f−1(v )∩xα : α ∈ λ} is a locally finite collection of clopen sets. since the union of a locally finite collection of closed sets is closed, f−1(v ) is also closed and hence clopen. � definition 4.16. a subset s of a space x is said to be regular gδ-embedded [6] in x if every regular gδ-set in s is the intersection of a regular gδ-set in x with s; or equivalently every regular fσ-set in s is the intersection of a regular fσ-set in x with s. 124 j. k. kohli, d. singh, j. aggarwal and m. rana proposition 4.17. let f : x → y be a pseudo perfectly continuous function. if f(x) is regular gδ-embedded in y , then f : x → f(x) is pseudo perfectly continuous. proof. let v1 be a regular fσ-set in f(x). since f(x) is regular gδ-embedded in y , there exists a regular fσ-set v in y such that v1 = v ∩ f(x). in view of pseudo perfect continuity of f, f−1(v ) is clopen in x. now f−1(v ∩ f(x)) = f−1(v ) ∩ f−1(f(x)) = f−1(v ) and hence the result. � definition 4.18. a topological space x is called an alexandroff space [2] if any intersection of open sets in x is itself an open in x, or equivalently any union of closed sets in x is closed in x. alexandroff spaces have been referred to as saturated spaces by lorrain in [41]. theorem 4.19. for each α ∈ λ, let fα : x → xα be a function and let f : x → ∏ α∈λ xα be defined by f(x) = (fα(x)) for each x ∈ x. if f is pseudo perfectly continuous, then each fα is pseudo perfectly continuous. further, if x is an alexandroff space and each fα is pseudo perfectly continuous, then f is pseudo perfectly continuous. proof. let f be pseudo perfectly continuous. now for each α, fα = pα ◦ f, where pα : ∏ α∈λ xα → x denotes the projection map. since each projection map pα is continuous and hence a dδ-map, in view of proposition 4.1 it follows that each fα is pseudo perfectly continuous. conversely, suppose that x is an alexandroff space and each fα is a pseudo perfectly continuous function. since x is alexandroff, to show that the function f is pseudo perfectly continuous, it is sufficient to show that f−1(s) is clopen for every subbasic regular fσ-set s in the product space ∏ α∈λ xα. let uβ × ∏ α6=β xα be a subbasic regular fσ-set in ∏ α∈λ xα, where uβ is a regular fσ-set in xβ. then f−1(uβ × ∏ α6=β xα) = f −1(p−1 β (uβ)) = f −1 β (uβ) is clopen in x and so f is pseudo perfectly continuous. � 5. interplay between topological properties and pseudo perfectly continuous functions definition 5.1. a space x is called a dδt0-space if for each pair of distinct points x, y in x, there is a regular fσ-set u containing one of the points x and y but not the other. definition 5.2. a space x is said to be dδ-hausdorff [22] (ultra hausdorff [57]) if every pair of distinct points in x are contained in disjoint regular fσ-sets (clopen sets). pseudo perfectly continuous functions and closedness/compactness of their function spaces125 in particular, every dδt0-space is hausdorff. the following diagram illustrates the relationships that exist among dδt0spaces and other strong variants of hausdorffness. ultra hausdorff functionally hausdorff q-hausdorff hausdorff d t δ 0-space dδ-hausdorff urysohn figure 4. the following example shows that even a hausdorff regular space need not be a dδt0-space. example 5.3. let x be the skyline space due to heldermann [12, example 7.7]. the space x is a hausdorff regular space. it is not a dδt0-space since x is the only regular fσ-set containing the points p − and p+. so there exists no regular fσ-set containing one of the points p − and p+ and missing other. proposition 5.4. let f : x → y be a pseudo perfectly continuous injection into a dδt0-space y . then x is an ultra hausdorff space. proof. let x, y ∈ x, x 6= y. then f(x) 6= f(y). since y is a dδt0-space, there exists a regular fσ-sets v containing one of the points f(x) and f(y) but not both. to be precise, suppose that f(x) ∈ v . since f is pseudo perfectly continuous, f−1(v ) is a clopen set containing x but not y. then f−1(v ) and x \ f−1(v ) are disjoint clopen sets containing x and y, respectively. hence x is an ultra hausdorff space. � the following theorem is related to a class of spaces, important in the theories studying the p-adic topologies and the stone duality for boolean algebras, namely spaces having large inductive dimension zero or ultranormal spaces [57]. these are precisely the spaces in which each pair of nonempty disjoint closed sets can be separated by disjoint clopen sets. theorem 5.5. let f : x → y be a closed, pseudo perfectly continuous injection into a normal space y . then x is an ultranormal space. proof. let a and b be any two disjoint closed sets in x. since the function f is closed and injective, f(a) and f(b) are disjoint closed subsets of y . again, since y is normal, by urysohn’s lemma there exists a continuous function ϕ : y → [0, 1] such that ϕ(f(a)) = 0 and ϕ(f(b)) = 1. then v = ϕ−1([0, 1/2)) and w = ϕ−1((1/2, 1]) are disjoint cozero sets in y containing f(a) and f(b), 126 j. k. kohli, d. singh, j. aggarwal and m. rana respectively. since every cozero set is a regular fσ-set, f −1(v ) and f−1(w) are disjoint clopen sets containing a and b, respectively and so x is an ultranormal space. � definition 5.6. a space x is said to be dδ-compact [23] (mildly compact [57]) if every cover of x by regular fσ-sets (clopen sets) has a finite subcover. proposition 5.7. let f : x → y be a pseudo perfectly continuous function from a mildly compact space x onto a space y . then y is dδ-compact. proof. let ω = {uα : α ∈ λ} be a cover of y by regular fσ-sets. since f is pseudo perfectly continuous, the collection β = {f−1(uα) : α ∈ λ} is a clopen cover of x. since x is mildly compact, let {f−1(uα1), . . . , f −1(uαn)} be a finite subcollection of β which covers x. then {uα1, . . . , uαn} is a finite subcollection of ω which covers y . hence y is dδ-compact. � proposition 5.8. let f : x → y be a pseudo perfectly continuous function from a space x onto a space y . if (i) f is an open bijection; or (ii) f is a closed surjection, then any pair of disjoint regular gδ-sets in y are clopen in y . proof. let a and b be disjoint regular gδ-subsets of y . since f is pseudo perfectly continuous f−1(a) and f−1(b) are disjoint clopen subsets of x. (i) in case f is an open bijection, f(f−1(a)) = a and f(f−1(b)) = b are disjoint open sets and hence clopen sets in y . (ii) in case f is a closed surjection, the sets a = y \ f(x \ f−1(a)) and b = y \ f(x \ f−1(b)) are disjoint clopen sets in y . � definition 5.9. a space x is said to be δ-completely regular space ([22], [24]) if for each regular gδ-set f and a point x /∈ f , there exists a continuous function f : x → [0, 1] such that f(x) = 0 and f(f) = 1. theorem 5.10. let f : x → y be an open closed pseudo perfectly continuous surjection. then y is a δ-completely regular space. proof. let k ⊂ y be a regular gδ-set and let z /∈ k. since f is pseudo perfectly continuous, f−1(k) is clopen. let x0 ∈ f −1(z). then x0 /∈ f −1(k). since f−1(k) is clopen, its characteristic function φ : x → [0, 1] is continuous and φ(x0) = 0 and φ(f −1(k)) = 1. define ϕ̂ : y → [0, 1] by taking ϕ̂(y) = sup{φ(x) : x ∈ f−1(y)}. then ϕ̂(z) = 0, ϕ̂(k) = 1 and by [8, exercise 16] ϕ̂ is continuous. hence y is a δ-completely regular space. � remark 5.11. there exists no open closed pseudo perfectly continuous surjection from a space onto a non δ-completely regular space. proposition 5.12. let f, g : x → y be pseudo perfectly continuous functions from a space x into a dδ-hausdorff space y . then the set a = {x : f(x) = g(x)} is cl-closed in x. pseudo perfectly continuous functions and closedness/compactness of their function spaces127 proof. let x ∈ x \ a. then f(x) 6= g(x), and so by hypothesis on y , there are disjoint regular fσ-sets u and v containing f(x) and g(x), respectively. since f and g are pseudo perfectly continuous, the sets f−1(u) and g−1(v ) are clopen and containing the point x. let g = f−1(u) ∩ g−1(v ). then g is a clopen set contain x and g ∩ a = ∅. thus a is cl-closed in x. � proposition 5.13. let f : x → y be a pseudo perfectly continuous function from a space x into a dδ-hausdorff space y . then the set a = {(x1, x2) ∈ x × x : f(x1) = f(x2)} is cl-closed in x × x. proof. let (x1, x2) ∈ x × x \ a. then f(x1) 6= f(x2). since y is a dδhausdorff space, there exist disjoint regular fσ-sets u and v containing f(x1) and f(x2), respectively. since f is pseudo perfectly continuous, f −1(u) and f−1(v ) are disjoint clopen sets in x containing x1 and x2, respectively. let g = f−1(u)×f−1(v ). then g is a clopen subset of x ×x containing (x1, x2) and g ∩ a = ∅. thus a is cl-closed in x × x. � definition 5.14. a space x is said to be (i) pseudo hyperconnected if there exists no nonempty proper regular gδset in x or equivalently there exists no nonempty proper regular fσ-set in x (≡ x is the only regular fσ-set in x). (ii) hyperconnected ([1], [58]) if every nonempty open subset of x is dense in x (≡ x is the only regular open set in x). (iii) quasi hyperconnected [19] if there exists no nonempty proper θ-open set in x or equivalently there exists no nonempty proper θ-closed set in x (≡ x is the only θ-open set in x). following implications are immediate from definitions. hyperconnected quasi hyperconnected pseudo hyperconnected figure 5. example 5.15. the space r \ q given by mancuso [43, example 3.20] and also discussed in example 4.6 is quasi hyperconnected but not hyperconnected. proposition 5.16. let f : x → y be a pseudo perfectly continuous surjection from a connected space x onto y . then y is pseudo hyperconnected. proof. suppose y is not pseudo hyperconnected and let v be a nonempty proper regular fσ-set in y . since f is pseudo perfectly continuous, f −1(v ) is a nonempty proper clopen subset of x contradicting the fact that x is connected. � remark 5.17. there exists no pseudo perfectly continuous surjection from a connected space onto a non pseudo hyperconnected space. 128 j. k. kohli, d. singh, j. aggarwal and m. rana definition 5.18. the graph g(f) of a function f : x → y is said to be (i) clopen θ-closed [19] if for each (x, y) /∈ g(f) there exists a clopen set u containing x and a θ-open set v containing y such that (u ×v ) ∩ g(f)=∅. (ii) clopen dδ-closed if for each (x, y) /∈ g(f) there exists a clopen set u containing x and a regular fσ-set v containing y such that (u × v ) ∩ g(f) = ∅. proposition 5.19. let f : x → y be a pseudo perfectly continuous function into a dδ-hausdorff space y . then the graph g(f) of f is a clopen dδ-closed set in x × y . proof. suppose (x, y) /∈ g(f). then f(x) 6= y. since y is dδ-hausdorff, there exist disjoint regular fσ-sets v and w containing f(x) and y, respectively. since f is pseudo perfectly continuous, f−1(v ) is a clopen set containing x. clearly (f−1(v ) × w) ∩ g(f) = ∅ and so the graph g(f) of f is clopen dδclosed in x × y . � corollary 5.20. if f : x → y is a pseudo perfectly continuous function into a dδ-hausdorff space y , then the graph g(f) of f is clopen θ-closed in x × y . 6. function spaces and pseudo perfectly continuous functions it is of fundamental importance in topology, analysis and other branches of mathematics to know whether a given function space is closed/compact in y x in the topology of pointwise convergence. so it is of considerable significance both from intrinsic interest as well as from applications viewpoint to formulate conditions on the spaces x, y and subsets of c(x, y ) or y x to be closed/compact in the topology of pointwise convergence. results of this type and ascoli type theorems abound in the literature (see [3], [14]). naimpally’s result [46] that in contrast to continuous functions, the set s(x, y ) of strongly continuous functions is closed in y x in the topology of pointwise convergence if x is locally connected and y is hausdorff; is extended to a larger framework by kohli and singh [25] wherein it is shown that if x is sum connected and y is hausdorff, then the function space p(x, y ) of all perfectly continuous functions as well as the function space l(x, y ) of all cl-supercontinuous functions is closed in y x in the topology of pointwise convergence. this result is further extended in ([27], [55], [36]) for the set p∆(x, y ) of all δ-perfectly continuous functions as well as for the set pδ(x, y ) of all almost perfectly continuous (≡ regular set connected) functions and the set pq(x, y ) of all quasi perfectly continuous functions under the same hypotheses on x and y . herein we further strengthen these results to show that if x is a sum connected space and y is a dδt0-space, then all the seven classes of functions are identical, i.e. s(x, y ) = p(x, y ) = l(x, y ) = p∆(x, y ) = pδ(x, y ) = pq(x, y ) = pp(x, y ) and are closed in y x in the topology of pointwise convergence. pseudo perfectly continuous functions and closedness/compactness of their function spaces129 proposition 6.1. let f : x → y be a pseudo perfectly continuous function into a dδt0-space y . then f is constant on each connected subset of x. in particular, if x is connected, then f is constant on x and hence strongly continuous. proof. assume contrapositive and let c be the connected subset of x such that f(c) is not a singleton. let f(x), f(y) ∈ f(c), f(x) 6= f(y). since y is a dδt0-space, there exists a regular fσ-set v containing one of the points f(x) and f(y) but not other. for definiteness assume f(x) ∈ v . since f is a pseudo perfectly continuous, f−1(v ) ∩ c is a non empty proper clopen subset of c, contradicting the fact that c is connected. the last part of the theorem is immediate, since every constant function is strongly continuous. � remark 6.2. the hypothesis of ‘dδt0-space’ in proposition 6.1 cannot be omitted. for let x be the real line with usual topology and let y denote the real line endowed with cofinite topology. let f denote the identity mapping from x onto y . then f is a nonconstant pseudo perfectly continuous function. corollary 6.3. let f : x → y be a pseudo perfectly continuous function from a sum connected space x into a dδt0-space y . then f is constant on each component of x and hence strongly continuous. proof. clearly, in view of proposition 6.1 f is constant on each component of x. since x is a sum connected space, each component of x is clopen in x. hence it follows that any union of components of x and the complement of this union are complementary clopen sets in x. thus f is constant on each component on x. therefore, for every subset a of y , f−1(a) and x \ f−1(a) are complementary clopen sets in x being the union of component of x. so f is strongly continuous. � we may recall that a space x is a δt0-space [26] if for each pair of distinct points x and y in x there exists a regular open set containing one of the points x and y but not the other. in particular, every hausdorff space is a δt0-space. next, we quote the following results from ([27], [36], [55]). theorem 6.4 ([27, theorem 5.3]). let f : x → y be a function from a sum connected space x into a δt0-space y . then the following statements are equivalent. (a) f is strongly continuous (b) f is perfectly continuous (c) f is cl-supercontinuous (d) f is δ-perfectly continuous. theorem 6.5 ([55, theorem 4.5]). let f : x → y be a function from a sum connected space x into a δt0-space y . then the following statements are equivalent. (a) f is strongly continuous (b) f is perfectly continuous 130 j. k. kohli, d. singh, j. aggarwal and m. rana (c) f is cl-supercontinuous (d) f is δ-perfectly continuous (e) f is almost perfectly continuous. theorem 6.6 ([36, theorem 5.6]). let f : x → y be a function from a sum connected space x into a hausdorff space y . then the following statements are equivalent. (a) f is strongly continuous (b) f is perfectly continuous (c) f is cl-supercontinuous (d) f is δ-perfectly continuous (e) f is almost perfectly continuous (f) f is quasi perfectly continuous. theorem 6.7. let f : x → y be a function from a sum connected space x into a dδt0-space y . then the following statements are equivalent. (a) f is strongly continuous (b) f is perfectly continuous (c) f is cl-supercontinuous (d) f is δ-perfectly continuous (e) f is almost perfectly continuous (f) f is quasi perfectly continuous (g) f is pseudo perfectly continuous. proof. since every dδt0-space is hausdorff, the equivalence of the assertions (a)-(f) is a consequence of theorem 6.6. the implications (a)⇒(b)⇒(d)⇒(f) ⇒(g) are trivial and the implication (g)⇒(a) is immediate in view of corollary 6.3. � theorem 6.8. let x be a sum connected space and let y be a dδt0-space. then s(x, y ) = p(x, y ) = l(x, y ) = p∆(x, y ) = pδ(x, y ) = pq(x, y ) = pp(x, y ) is closed in y x in the topology of pointwise convergence. proof. it is immediate from theorem 6.7 that the above seven classes of functions are identical and its closedness in y x in the topology of pointwise convergence follows either from [27, theorem 5.4] or [55, theorem 4.6] or [36, theorem 5.7]. the above results are important from applications view point since in particular it follows that if x is sum connected (e.g. connected or locally connected) and y is dδt0-space, then the pointwise limit of a sequence {fn : x → y : n ∈ n} of pseudo perfectly continuous functions is pseudo perfectly continuous. � we conclude this section with the following result which seems to be of considerable significance from applications view point. theorem 6.9. if x is a sum connected space and y is a compact dδt0-space, then the spaces s(x, y ) = p(x, y ) = l(x, y ) = p∆(x, y ) = pδ(x, y ) = pq(x, y ) = pp(x, y ) are compact hausdorff subspaces of y x in the topology of pointwise convergence. pseudo perfectly continuous functions and closedness/compactness of their function spaces131 7. change of topology the technique of change of topology of a space is prevalent all through mathematics and is of considerable significance and widely used in topology, functional analysis and several other branches of mathematics. for example, weak and weak∗ topology of a banach space, hull kernel topology and the multitude of other topologies on id(a) the space of all closed two sided ideals of a banach algebra a ([4], [5], [56]). moreover, to taste the flavour of applications of the technique in topology see ([11], [15], [17], [30], [60]). here we show that if the range of a pseudo perfectly continuous function is retopologized in an appropriate way, then it is simply a cl-supercontinuous function. let (x, τ) be a topological space and let bdδ denote the collection of all regular fσ-subsets of (x, τ). since the intersection of two regular fσ-sets is a regular fσ-set, the collection bdδ is a base for a topology τdδ on x which is coarser than τ (see [21], [22]). for interrelations and interplay among various other coarser topologies obtained in this way for a given topology we refer the interested reader to [30]. finally, we conclude with the following result. proposition 7.1. if f : (x, τ) → (y, ϑ) is a pseudo perfectly continuous function, then f : (x, τ) → (y, ϑdδ ) is cl-supercontinuous. references [1] n. ajmal and j. k. kohli, properties of hyperconnected spaces, their mappings into hausdorff spaces and embeddings into hyperconnected spaces, acta math. hungar. 60, no. 1-2 (1992), 41–49. 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(received june 2012 – accepted november 2012) j. k. kohli (jk kohli@yahoo.co.in) department of mathematics, hindu college, university of delhi, delhi 110 007, india 134 j. k. kohli, d. singh, j. aggarwal and m. rana d. singh (dstopology@rediffmail.com) department of mathematics, sri aurobindo college, university of delhi-south campus, delhi 110 017, india jeetendra aggarwal (jitenaggarwal@gmail.com) department of mathematics, university of delhi, delhi 110 007, india manoj rana (krana 71@yahoo.co.uk) department of mathematics, university of delhi, delhi 110 007, india pseudo perfectly continuous functions and[8pt] closedness/compactness of their function spaces. by j. k. kohli, d. singh, j. aggarwal and m. rana @ appl. gen. topol. 20, no. 1 (2019), 155-175doi:10.4995/agt.2019.10667 c© agt, upv, 2019 remarks on fixed point assertions in digital topology, 2 laurence boxer department of computer and information sciences, niagara university, ny 14109, usa; and department of computer science and engineering, state university of new york at buffalo (boxer@niagara.edu) communicated by s. romaguera abstract several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. we show that in many cases, researchers using these tools have derived conclusions that are incorrect, trivial, or limited. 2010 msc: 54h25. keywords: digital topology; fixed point; metric space. 1. introduction this paper continues the work of [5] and quotes or paraphrases from it. recent papers have attempted to apply to digital images ideas from euclidean topology and real analysis concerning metrics and fixed points. while the underlying motivation of digital topology comes from euclidean topology and real analysis, some applications of fixed point theory recently featured in the literature of digital topology seem of doubtful worth. although papers including [19, 4] have valid and interesting results for fixed points and for “almost” or “approximate” fixed points in digital topology, many other published assertions concerning fixed points in digital topology are incorrect, trivial (e.g., received 30 august 2018 – accepted 25 january 2019 http://dx.doi.org/10.4995/agt.2019.10667 l. boxer applicable only to singletons, or only to constant functions), or limited, as discussed in [5]. after submitting [5], we learned of several additional publications with assertions characterized as above; these are discussed in the current paper. the less-than-ideal papers we discuss have in common a definition of a digital metric space (x,d,κ), where x is a set of lattice points, d is a metric (typically, the euclidean), and κ is an adjacency relation on x, making (x,κ) a graph; and then these papers never make use of κ. functions considered usually are all continuous in the topological sense, since the metric d usually imposes a discrete topology on the digital image; but are often discontinuous in the digital sense of preserving graph connectedness. many of these papers’ assertions are modifications of results known for the euclidean topology of rn that tell us little or nothing about digital images as graphs. some of these papers’ assertions are of interest if we regard the functions investigated as defined on subsets of rn. in many cases, we offer corrections, notes on their limitations, or improvements. 2. preliminaries we let z denote the set of integers, and r, the real line. we consider a digital image as a graph (x,κ), where x ⊂ zn for some positive integer n and κ is an adjacency relation on x. we will often assume that x is a finite set, as in the “real world.” a digital metric space is [8] a triple (x,d,κ) where (x,κ) is a digital image and d is a metric for x. in [8], d was taken to be the euclidean metric, as was the case in many subsequent papers, but we will not limit our discussion to the euclidean metric. often, however, we will assume d is an ℓp metric (see section 2.2). the diameter of a metric space (x,d) is diamx = max{d(x,y) |x,y ∈ x}. 2.1. adjacencies. the most commonly used adjacencies for digital images are the cu-adjacencies, defined as follows. definition 2.1. let p,q ∈ zn, p = (p1, . . . ,pn), q = (q1, . . . ,qn), p 6= q. let 1 ≤ u ≤ n. we say p and q are cu-adjacent, denoted p ↔cu q or p ↔ q when the adjacency is understood, if • for at most u distinct indices i, |pi − qi| = 1, and • for all other indices j, pj = qj. often, a cu-adjacency is denoted by the number of points in z n that are cu-adjacent to a given point. e.g., • in z1, c1-adjacency is 2-adjacency; • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency; • in z3, c1-adjacency is 8-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 156 fixed point assertions in digital topology, 2 other adjacencies for digital images are discussed in papers such as [9, 2, 3]. a digital interval is a digital image of the form ([a,b]z,2), where a < b and [a,b]z = {z ∈ z |a ≤ z ≤ b}. 2.2. ℓp metric. let x ⊂ rn and let x = (x1, . . . ,xn) and y = (y1, . . . ,yn) be points of x. let 1 ≤ p ≤ ∞. the ℓp metric d for x is defined by d(x,y) = { ( ∑n i=1 |xi − yi|p) 1/p for 1 ≤ p < ∞; max{|xi − yi|}ni=1 for p = ∞. for p = 1, this gives us the manhattan metric d(x,y) = ∑n i=1 |xi − yi|; for p = 2, we have the euclidean metric d(x,y) = ( ∑n i=1 |xi − yi|2)1/2. the following are easily proved. proposition 2.2. let x,y ∈ zn and let d be any ℓp metric. then • if d(x,y) < 1, then x = y; • if 1 ≤ u ≤ n and x ↔cu y, then d(x,y) ≤ u1/p. 2.3. digital continuity. definition 2.3 ([19, 1]). a function f : (x,κ) → (y,λ) between digital images is (κ,λ)-continuous (or just continuous when κ and λ are understood) if for every κ-connected subset x′ of x, f(x′) is a λ-connected subset of y . theorem 2.4 ([1]). a function f : (x,κ) → (y,λ) between digital images is (κ,λ)-continuous if and only if x ↔κ x′ in x implies either f(x) = f(x′) or f(x) ↔λ f(x′) in y . 2.4. cauchy sequences and complete metric spaces. the papers [6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22] apply to digital images the notions of cauchy sequence and complete metric space. since if the digital image x is finite or uses a common metric d such as an ℓp metric, the digital metric space (x,d,κ) is a discrete topological space, the digital versions of these notions are quite limited. recall that a sequence of points {xn} in a metric space (x,d) is a cauchy sequence if for all ε > 0 there exists n0 ∈ n such that m,n > n0 implies d(xm,xn) < ε. if every cauchy sequence in x has a limit, then (x,d) is a complete metric space. it has been shown that under a mild additional assumption, a digital cauchy sequence is eventually constant. theorem 2.5 ([10, 5]). let a > 0. if d is a metric on a digital image (x,κ) such that for all distinct x,y ∈ x we have d(x,y) > a, then for any cauchy sequence {xi}∞i=1 ⊂ x there exists n0 ∈ n such that m,n > n0 implies xm = xn. an immediate consequence of theorem 2.5 is the following. corollary 2.6 ([10]). let (x,d,κ) be a digital metric space. if d is a metric on (x,κ) such that for all distinct x,y ∈ x we have d(x,y) > a for some constant a > 0, then any cauchy sequence in x is eventually constant, and (x,d) is a complete metric space. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 157 l. boxer remarks 2.7 ([5]). it is easily seen that the hypotheses of theorem 2.5 and corollary 2.6 are satisfied for any finite digital metric space, or for a digital metric space (x,d,κ) for which the metric d is any ℓp metric. thus, a cauchy sequence that is not eventually constant can only occur in an infinite digital metric space with an unusual metric. such an example is given below. example 2.8 ([5]). let d be the metric on (n,c1) defined by d(i,j) = |1/i − 1/j|. then {i}∞i=1 is a cauchy sequence for this metric that does not have a limit. 2.5. function sets ψ, φ. below, we define sets of functions ψ,φ that will be used in the following. definition 2.9 ([15]). let ψ0 be a set of functions ψ : [0,∞) → [0,∞) such that for each ψ ∈ ψ0 we have • ψ is nondecreasing, and • there exists k0 ∈ n, a ∈ (0,1), and a convergent series ∑ ∞ k=1 vk of non-negative terms such that k ≥ k0 implies ψk+1(t) ≤ aψk(t) + vk for all t ∈ [0,∞), where ψk represents the k-fold composition of ψ. the following will be used later in the paper. example 2.10 ([5]). the constant function with value 0 is a member of ψ0. definition 2.11 ([13]). let ψ be the set of functions ψ : [0,∞) → [0,∞) such that for each ψ ∈ ψ we have • ψ is nondecreasing, and • ∑ ∞ n=1 ψ n(t) < ∞ for all t > 0, where ψn represents the n-fold composition of ψ. definition 2.12 ([21]). let φ be the set of functions φ : [0,∞) → [0,∞) such that φ is increasing, φ(t) = 0 if and only if t = 0, and φ(t) < t for t > 0. proposition 2.13. let ψ ∈ ψ. then for all t > 0, ψ(t) < t. proof. suppose there exists t0 > 0 such that ψ(t0) ≥ t0. since ψ is nondecreasing, an easy induction yields that ψn+1(t0) ≥ ψn(t0) for all n ∈ n. therefore, ∑ ∞ n=1 ψ n(t0) = ∞, contrary to definition 2.11. the contradiction establishes the assertion. � a notion often used with the set ψ is given by the following. definition 2.14 ([20]). let t : x → x and α : x × x → [0,∞). we say t is α-admissible if α(x,y) ≥ 1 implies α(t(x),t(y)) ≥ 1. 2.6. an example. example 2.15 ([5]). let x = {p1 = (0,0,0,0,0), p2 = (2,0,0,0,0), p3 = (1,1,1,1,1)} ⊂ z5. let f : x → x be defined by f(p1) = f(p2) = p1, f(p3) = p2. then f is not (c5,c5)-continuous. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 158 fixed point assertions in digital topology, 2 despite its discontinuity, the function of example 2.15 was shown in [5] to exemplify many different types of functions studied in [8, 10, 11, 12, 14, 15, 16, 17, 18]. in the current paper, this example is also used to show that functions of the type studied need not be digitally continuous. 3. compatible functions 3.1. equivalence of compatibilities. in this section, we examine the equivalence of several different types of “compatible” functions discussed in [6]. definition 3.1 ([6]). suppose s and t are self-maps on a digital metric space (x,d,κ). suppose {xn}∞n=1 is a sequence in x such that (3.1) limn→∞s(xn) = limn→∞t(xn) = t for some t ∈ x. we have the following. • s and t are called compatible if limn→∞d(s(t(xn)),t(s(xn))) = 0 for all sequences {xn}∞n=1 ⊂ x that satisfy statement (3.1). • s and t are called compatible of type (a) if limn→∞d(s(t(xn)),t(t(xn))) = 0 = limn→∞d(t(s(xn)),s(s(xn))) for all sequences {xn}∞n=1 ⊂ x that satisfy statement (3.1). • s and t are called compatible of type (p) if limn→∞d(s(s(xn)),t(t(xn))) = 0 for all sequences {xn}∞n=1 ⊂ x that satisfy statement (3.1). because digital metric spaces are typically discrete, these properties can be simplified as shown in the remainder of this section. proposition 3.2 ([6]). let s and t be compatible maps of type (a) on a digital metric space (x,d,κ). if one of s and t is continuous, then s and t are compatible. the proof given for proposition 3.2 clarifies that the continuity expected of s or t is in the sense of the classical “ε−δ definition”, not in the sense of digital continuity. however, it turns out that this assumption is usually unnecessary, as shown below. theorem 3.3. let (x,d,κ) be a digital metric space, where either x is finite or d is an ℓp metric. let s and t be self-maps on x. then the following are equivalent. • s and t are compatible. • s and t are compatible of type (a). • s and t are compatible of type (p). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 159 l. boxer proof. throughout this proof, let {xn}∞n=1 ⊂ x satisfy statement (3.1). suppose s and t are compatible. then, by the triangle inequality, (3.2) d(s(s(xn)),t(t(xn))) ≤ d(s(s(xn)),s(t(xn))) + d(s(t(xn)),t(s(xn))) + d(t(s(xn)),t(t(xn))). by (3.1) and corollary 2.6, the first term on the right side of (3.2) is, for sufficiently large n, d(s(s(xn)),s(t(xn))) = d(s(t),s(t)) = 0. similarly, the third term on the right side of (3.2) is, for sufficiently large n, d(t(s(xn)),t(t(xn))) = d(t(t),t(t)) = 0. the middle term on the right side of (3.2) is d(s(t(xn)),t(s(xn))), which, by compatibility, tends to 0 as n → ∞. since all the terms on the right side of (3.2) tend to 0, s and t are compatible of type (p). suppose s and t are compatible of type (p). then, using corollary 2.6, limn→∞d(s(t(xn)),t(t(xn))) = limn→∞d(s(t),t(t(xn))) = limn→∞d(s(s(xn)),t(t(xn))) = (because compatible of type (p)) 0. similarly, limn→∞d(t(s(xn)),s(s(xn))) = 0. therefore, s and t are compatible of type (a). if s and t are compatible of type (a), then, using the triangle inequality and definition 3.1, d(s(t(xn)),t(s(xn))) ≤ d(s(t(xn)),t(t(xn))) + d(t(t(xn)),t(s(xn))) →n→∞ 0 + 0 = 0. hence, s and t are compatible. � 3.2. compatible functions’ common fixed points. the assertions stated as theorem 23 and theorem 24 of [6] are concerned with the existence of a common fixed point of four self-maps of a digital metric space. however, these assertions are incorrect. we give a counterexample below. stated as theorem 23 of [6] is the following. let a,b,s, and t be self-maps on a complete digital metric space (x,d,κ) such that (a) s(x) ⊂ b(x) and t(x) ⊂ a(x); (b) the pairs (a,s) and (b,t) are compatible; (c) one of s,t,a, and b is continuous; and (d) we have f [d(a(x),b(y)),d(s(x),t(y)),d(a(x),s(x)),d(b(y),t(y)), d(a(x),t(y)),d(b(y),s(x))] ≤ 0 for all x,y ∈ x. then a,b,s, and t have a unique common fixed point. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 160 fixed point assertions in digital topology, 2 condition (d) above gives the only assumption stated in [6] about the function f . perhaps the authors intended to say more, but their statement permits f to be any function that satisfies (d). stated as theorem 24 of [6] is the following. let a,b,s, and t be self-maps on a complete digital metric space (x,d,κ) satisfying conditions (a), (c), and (d). if the pairs (a,s) and (b,t) are compatible of type (a) or of type (p) then a,b,s, and t have a unique common fixed point. by theorem 3.3, if d is an ℓp metric then these assertions are equivalent. a counterexample to these assertions: example 3.4. let x = z, d(x,y) = |x − y|, κ = c1, a(x) = b(x) = x − 1, s(x) = t(x) = x + 1, f(x1,x2,x3,x4,x5,x6) = 0. proof. that this is a counterexample to the assertion stated as theorem 23 of [6] is shown as follows. clearly s(x) = b(x) = z = t(x) = a(x). the pair (a,s) is compatible, since a(s(x)) = x = s(a(x)) for all x ∈ x; similarly, (b,t) is a compatible pair. all of s,t,a, and b are continuous in both the “ε−δ” sense and in the digital sense with respect to the c1 adjacency. trivially, f(d(ax,by),d(sx,ty),d(ax,sx),d(by,ty),d(ax,ty),d(by,sx)) ≤ 0, for all x,y ∈ x. however, none of s,t,a, and b has a fixed point. � 4. expansive mappings the paper [13] is concerned with fixed points for expansive maps on digital metric spaces. definition 4.1 ([13]). let t be a self map on a complete metric space (x,d) such that t is onto, and for some k ≥ 1 and all x,y ∈ x, (4.1) d(t(x),t(y)) ≥ kd(x,y). then t is called an expansive map. remarks 4.2. in [15], the constant k of (4.1) was restricted to k > 1. theorem 4.3 below shows there is no such map if x is finite. the following shows a limitation on the application of expansive maps in digital topology. the result seems contrary to the spirit of definition 4.1. theorem 4.3. if t is an expansive map on a finite digital image (x,d,κ), then for all x,y ∈ x, d(t(x),t(y)) = d(x,y). proof. it is shown at theorem 4.9 of [5] that t cannot satisfy (4.1) for k > 1. thus, we have k = 1, so (4.2) d(t(x),t(y)) ≥ d(x,y) for all x,y ∈ x. since x is finite, there exists a maximal finite set {di}mi=1 ∈ (0,∞) such that 0 < d1 < d2 < ... < dm and sets si = {(u,v) ∈ x2 |d(u,v) = di} 6= ∅. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 161 l. boxer suppose that there exist x,y ∈ x such that d(t(x),t(y)) > d(x,y). then there exists j such that j = min{i ∈ {1, . . . ,m} |d(t(x0),t(y0)) > d(x0,y0) for some {x0,y0} ∈ si}. thus {{t(x),t(y)} | {x,y} ∈ sj} 6⊂ sj. but t is onto and x is finite, so there exist x1,y1 ∈ x such that {x1,y1} 6∈ sj and {t(x1),t(y1)} ∈ sj. by our choice of j, there exists an index k > j such that {x1,y1} ∈ sk. therefore, d(t(x1),t(y1)) = dj < dk = d(x1,y1), a contradiction of (4.2). this establishes the assertion. � even being an isomorphism need not make a self-map expansive, as shown in the following. example 4.4. let x = {p0 = (0,0),p1 = (1,0),p2 = (1,1)} ⊂ z2. let f : (x,c2) → (x,c2) be the rotation defined by f(pi) = p(i+1) mod 3. then it is easily seen that f is a (c2,c2)-isomorphism. however, if we let d be any ℓp metric, then by theorem 4.3, f is not an expansive mapping, since d(p1,p2) = 1 < 2 1/p = d(p2,p0) = d(f(p1),f(p2)). definition 4.5 ([13]). let (x,d,κ) be a digital metric space and let t : x → x be a mapping. we say that t is a generalised α-ψ-expansive mapping if there exist functions α : x × x → [0,∞) and ψ ∈ ψ such that for all x,y ∈ x we have (4.3) ψ(d(t(x),t(y))) ≥ α(x,y)m(x,y) where m(x,y) = max{d(x,y), d(x,t(x)) + d(y,t(y)) 2 , d(x,t(y)) + d(y,t(x)) 2 }. remarks 4.6. any function t : x → x on a digital metric space is a generalized α-ψ-expansive mapping if α is the constant function with value 0. therefore, a generalised α-ψ-expansive mapping need not be digitally continuous. if one desires an example of a discontinuous generalised α-ψ-expansive mapping for which α is not the constant function with value 0, consider the following. this example also shows that the status of a map as an expansive map depends on the metric used, and that an expansive map need not be digitally continuous. example 4.7. let x = {p0,p1,p2} ⊂ z2, where p0 = (0,0), p1 = (1,1), p2 = (2,0). let t : x → x be the circular rotation t(pi) = p(i+1) mod 3. then • t is an expansive map with respect to the manhattan metric, but not with respect to the euclidean metric; • t is not (c2,c2)-continuous; c© agt, upv, 2019 appl. gen. topol. 20, no. 1 162 fixed point assertions in digital topology, 2 • t is a generalised α-ψ-expansive mapping for ψ(t) = t/2 and α(x,y) = 1/3. proof. with respect to the manhattan metric, (4.4) i 6= j implies d(pi,pj) = 2. since t is a bijection, we have d(t(pi),t(pj)) = d(pi,pj), so t is an expansive map. with respect to the euclidean metric, d(p0,p2) = 2 > √ 2 = d(p1,p0) = d(t(p0),t(p2)), so t is not an expansive map. since p1 ↔c2 p2 but t(p1) = p2 and t(p2) = p0 are neither equal nor c2-adjacent, t is not c2-continuous. using the manhattan metric, we have from (4.4) that i 6= j implies d(f(xi),f(xj)) = 2 = d(xi,xj). therefore, from definition 4.5 we have m(xi,xi) = 0 and i 6= j implies m(xi,xj) = 2. then one sees easily that t is a generalised α-ψ-expansive mapping for ψ(t) = t/2 and α(x,y) = 1/3. � the assertion that appears as theorem 3.4 of [13] can be corrected and improved as discussed below. the assertion is let (x,d,κ) be a complete digital metric space and let t : x → x be a bijective and generalised α-ψ-expansive mapping that satisfies the following conditions: (1) t −1 is α-admissible; (2) there exists x0 ∈ x such that α(x0,t −1(x0)) ≥ 1; and (3) t is digitally continuous. then t has a fixed point. remarks 4.8. there are several errors in the argument given in [13] as a proof of the assertion above, including confusion of digital continuity with “ε − δ continuity.” the following is a corrected, somewhat modified, version of the assertion above. note we do not require t to be continuous. theorem 4.9. let (x,d,κ) be a complete digital metric space and let t : x → x be a bijective and generalized α-ψ-expansive mapping that satisfies the following conditions: (1) t −1 is α-admissible; (2) there exists x0 ∈ x such that α(x0,t −1(x0)) ≥ 1. assume also that either x is finite or d is an ℓp metric. then t has a fixed point. proof. our argument is based on its analog in [13]. we have hypothesized the existence of x0 ∈ x such that α(x0,t −1(x0)) ≥ 1. define the sequence {xn}∞n=0 ∈ x by xn+1 = t −1(xn) for n > 0. if xm+1 = xm c© agt, upv, 2019 appl. gen. topol. 20, no. 1 163 l. boxer for some m, then xm is a fixed point of t −1, hence of t . otherwise, xn+1 6= xn for all n. since t −1 is α-admissible, we have that α(x0,x1) = α(x0,t −1(x0)) ≥ 1 implies α(x1,x2) = α(t −1(x0),t −1(x1)) ≥ 1, and, by induction, α(xn,xn+1) = α(t −1(xn−1),t −1(xn)) ≥ 1 for all n. from definition 4.5, we have (4.5) ψ(d(xn−1,xn)) = ψ(d(t(xn),t(xn+1)) ≥ α(xn,xn+1)m(xn,xn+1) ≥ m(xn,xn+1) where m(xn,xn+1) = max{d(xn,xn+1), d(xn,t(xn)) + d(xn+1,t(xn+1)) 2 , d(xn,t(xn+1)) + d(t(xn),xn+1) 2 } = max{d(xn,xn+1), d(xn,xn−1) + d(xn+1,xn) 2 , d(xn,xn) + d(xn−1,xn+1) 2 } = max{d(xn,xn+1), d(xn,xn−1) + d(xn+1,xn) 2 , d(xn−1,xn+1) 2 }. since by the triangle inequality, we have d(xn−1,xn+1) 2 ≤ d(xn,xn−1) + d(xn+1,xn) 2 , it follows that (4.6) m(xn,xn+1) = max{d(xn,xn+1), d(xn,xn−1) + d(xn+1,xn) 2 }. it follows from proposition 2.13 and inequalities (4.5) and (4.6) that d(xn−1,xn) > ψ(d(xn−1,xn)) ≥ max{d(xn,xn+1), d(xn,xn−1) + d(xn+1,xn) 2 }, so (4.7) d(xn,xn+1) < d(xn−1,xn). by theorem 2.5, it follows that limn→∞d(xn,xn+1) = 0. from corollary 2.6, we conclude that for large n we have t(xn+1) = xn = xn+1. thus, xn+1 is a fixed point of t . � theorem 3.5 of [13] is concerned with the existence of a fixed point for a generalised α-ψ-expansive mapping satisfying conditions somewhat like those of theorem 4.9. the assertion is incorrectly stated (corrections noted below) as follows. let (x,d,κ) be a complete digital metric space. suppose t : x → x is a generalized α-ψ expansive mapping such that c© agt, upv, 2019 appl. gen. topol. 20, no. 1 164 fixed point assertions in digital topology, 2 (1) t −1 is α-admissible; (2) there exists x0 ∈ x such that α(x0,t −1(x0)) ≥ 1; and (3) if {xn}∞n=0 ⊂ x such that α(xn,xn+1) ≥ 1 for all n and {xn}∞n=0 is digitally convergent to x′ ∈ x, then α(t −1(xn),t −1(x)) ≥ 1 for all n. then t has a fixed point. remarks 4.10. theorem 3.5 of [13] can be improved as follows. • since the existence of the function t −1 is assumed, it should be stated that t is assumed to be a bijection. • the term “digitally convergent” is undefined. what the proof actually uses is metric convergence. • if we add the hypothesis that x is finite or d is an ℓp metric, then the assertion, as amended by these observations, follows from our theorem 4.9. remarks 4.11. examples 3.6 and 3.7 of [13] claim [0,∞) as a digital metric space. clearly, it is not. remarks 4.12. in the proof of theorem 3.8 of [13], the inequality d(u,t nv) ≤ ψ(d(u,v)) for all n = 1,2,3, . . . should be d(u,t nv) ≤ ψn(d(u,v)) for all n = 1,2,3, . . . 5. common fixed points 5.1. commuting maps. the paper [18] is concerned with common fixed points of pairs of commuting self-maps on digital metric spaces. theorem 5.1 ([18]). let t be a continuous mapping of a complete digital metric space (x,d,κ) into itself. then t has a fixed point if and only if there exists α ∈ (0,1) and a function s : x → x that commutes with t such that (5.1) s(x) ⊂ t(x) and d(s(x),s(y)) ≤ αd(t(x),t(y)) for all x,y ∈ x. remarks 5.2. let s and t be as in theorem 5.1, where “continuous” is interpreted as (cu,cu)-continuous, x is cu-connected, d is an ℓp metric, and 0 < α < 1 u1/p . then s must be a constant function. proof. let x ↔cu x′ in x. since t is (cu,cu)-continuous, either t(x) = t(x′) or t(x) ↔cu t(x′), so d(t(x),t(x′)) ≤ u1/p. by the inequality (5.1), d(s(x),s(x′)) ≤ αd(t(x),t(x′)) < 1. since d is an ℓp metric, d(s(x),s(x′)) = 0, so s(x) = s(x′). it follows from the cu-connectedness of x that s is constant. � below, we show that if we assume common conditions, the requirement that t be continuous in theorem 5.1 is unnecessary. our proof is similar to its analog in [18]. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 165 l. boxer theorem 5.3. let t be a mapping of a digital metric space (x,d,κ) into itself, where x is finite or d is an ℓp metric. then t has a fixed point if and only if there exists α ∈ (0,1) and a function s : x → x that commutes with t such that (5.2) s(x) ⊂ t(x) and d(s(x),s(y)) ≤ αd(t(x),t(y)) for all x,y ∈ x. proof. suppose t has a fixed point, say, t(a) = a for some a ∈ x. let s : x → x be the constant function s(x) = a. then clearly s ◦ t = t ◦ s and the condition (5.2) is satisfied. suppose there exist a function s : x → x and α ∈ (0,1) such that s ◦ t = t ◦ s and the condition (5.2) is satisfied. let x0 ∈ x. since s(x) ⊂ t(x), there exists x1 ∈ x such that t(x1) = s(x0), and, inductively, for all n ∈ n there exists xn ∈ x such that t(xn) = s(xn−1). it follows from (5.2) that d(t(xn+1),t(xn)) = d(s(xn),s(xn−1)) ≤ αd(t(xn),t(xn−1)). an easy induction yields that d(t(xn+1),t(xn)) ≤ αnd(t(x1),t(x0)). since the right side of the latter inequality tends to 0 as n → ∞, it follows from theorem 2.5 that for sufficiently large n, t(xn+1) = t(xn) = t for some t ∈ x. but t(xn+1) = s(xn), so for sufficiently large n, s(xn) = t, and since s and t commute, (5.3) t(t) = t(t(xn)) = t(s(xn)) = s(t(xn)) = s(t). thus, t(t(t)) = t(s(t)) = s(t(t)), so d(s(t),s(s(t))) ≤ αd(t(t),t(s(t))) = αd(s(t),s(s(t))), or 0 ≤ (α − 1)d(s(t),s(s(t))). since the factor α − 1 < 0, we must have d(s(t),s(s(t))) = 0, so s(t) = s(s(t)). from (5.3) it follows that s(t) = s(s(t)) = s(t(t)) = t(s(t)), so s(t) is a common fixed point of s and t . suppose x and y are common fixed points of s and t . then d(x,y) = d(s(x),s(y)) ≤ αd(t(x),t(y)) = αd(x,y). since 0 < α < 1, we must have d(x,y) = 0, so x = y. � similarly, under common circumstances we can omit the assumption of continuity that is in the version of the following that appears in [18] . corollary 5.4. let t and s be commuting mappings of a complete digital metric space (x,d,κ) into itself, where d is an ℓp metric. suppose that s(x) ⊂ t(x). if there exists α ∈ (0,1) and a positive integer k such that d(sk(x),sk(y)) ≤ αd(t(x),t(y)) for all x,y ∈ x, then t and s have a common fixed point. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 166 fixed point assertions in digital topology, 2 proof. our proof is much like that of its analog in [18]. since s and t commute, sk and t commute. further, sk(x) ⊂ s(x) ⊂ t(x). therefore, we can apply theorem 5.3 to the maps sk and t to conclude that there is a unique a ∈ x that is a common fixed point of sk and t , i.e., sk(a) = t(a) = a. therefore, t(s(a)) = s(t(a)) = s(a) = s(sk(a)) = sk(s(a)), so s(a) is a common fixed point of t and sk. but a is the unique common fixed point of t and sk, so a = s(a). � remarks 5.5. the assertion given as corollary 3.2.5 of [18] has errors in its statement and in the argument given as its proof. in order for the assertion to be a corollary of the preceding theorem 3.2.4, the former should be stated as corollary. let n be a positive integer and let 0 < k < 1. if s is a self-map of a digital metric space (x,d,κ) such that d(sn(x),sn(y)) ≤ kd(x,y) for all x,y ∈ x, then s has a unique fixed point. remarks 5.6. the assertion at example 3.3.8 of [18] considers the maps s,t : (z,d,c1) → z given by s(x) = 2 − x2, t(x) = x2 for all x ∈ z, where d(x,y) = |x − y|. it is claimed that d(s(x),s(y)) ≤ 1 2 d(t(x),t(y)) for all x,y ∈ z, but this is clearly incorrect, since d(s(x),s(y)) = d(t(x),t(y)). 5.2. other common fixed point assertions. the paper [22] is another that is concerned with common fixed points of pairs of self-maps on digital metric spaces. we have the following. definition 5.7 ([22]). let (x,d,κ) be a digital metric space. let s,t : x → x. then s and t are weakly compatible if s(x) = t(x) implies s(t(x)) = t(s(x)). i.e., s and t are weakly compatible if they commute at all coincidence points. note this definition does not require that coincidence points exist. example 5.8. the maps s,t : [0,1]z → [0,1]z given by s(x) = 0, t(x) = 1, are weakly compatible, despite having no coincidence points, as they vacuously satisfy the requirement of commuting at all coincidence points. theorem 3.6 and corollaries 3.7 and 3.8 of [22] are concerned with common fixed points of self-maps s and t on a digital metric space (x,d,κ) for which d(s(x),s(y)) < d(t(x),t(y)) for all x,y ∈ x such that x 6= y. but such results may be quite limited under common conditions, as in the following propositions 5.9 and 5.10. proposition 5.9. let (x,d,κ) be a digital metric space with |x| > 1. let s,t : x → x be such that x 6= y implies d(s(x),s(y)) < d(t(x),t(y)). then t is one-to-one. if, further, x is finite, then t is a bijection and s is neither one-to-one nor onto. proof. since x 6= y implies 0 ≤ d(s(x),s(y)) < d(t(x),t(y)), we have that x 6= y implies t(x) 6= t(y). therefore, t is one-to-one. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 167 l. boxer suppose x is finite. since t is one-to-one, it follows that t is a bijection. further, there exist x0,y0 ∈ x such that d(x0,y0) = diamx. if s were onto, there would exist x′,y′ ∈ x such that s(x′) = x0 and s(y′) = y0. by hypothesis, we would then have d(x0,y0) = d(s(x ′),s(y′)) < d(t(x′),t(y′)), contrary to our choice of x0,y0. therefore, s is not onto. since x is finite, it follows that s is not one-to-one. � proposition 5.10. let (x,d,c1) be a connected digital metric space, where d is an ℓp metric. let s,t : z → z be such that x 6= y implies d(s(x),s(y)) < d(t(x),t(y)). if t is c1-continuous, then s is a constant function. proof. since t is c1-continuous, for x ↔c1 x′, we have t(x) ↔c1 t(x′) or t(x) = t(x′), so, since d is an ℓp metric, d(t(x),t(x ′)) ∈ {0,1}. since d(s(x),s(y)) < d(t(x),t(y)), we must have s(x) = s(x′). since x is c1connected, it follows that s is a constant function. � 6. contractive mappings 6.1. φ-contractive, contraction, α-φ-ψ-contraction maps. the paper [21] discusses fixed point assertions for contractive-type mappings on digital metric spaces. definition 6.1 ([21]). suppose (x,d,κ) is a digital metric space, t : x → x, and φ ∈ φ. if d(t(x),t(y)) ≤ φ(d(x,y)) for all x,y ∈ x, then t is called a digital φ-contraction. remarks 6.2. the function of example 2.15 is a digital φ-contraction for φ(t) = t/2. this shows that a digital φ-contraction need not be digitally continuous. a limitation of such functions is given in the following. proposition 6.3. let (x,d,κ) be a digital metric space and let t : x → x be a digital φ-contraction for some φ ∈ φ. suppose d is an ℓp metric and φ(t) < 1 for all t ∈ r. then t is a constant function. proof. let x,x′ ∈ x. then d(t(x),t(x′)) ≤ φ(d(x,x′)) < 1. therefore t(x) = t(x′). it follows that t is a constant function. � definition 6.4. let (x,d,κ) be a digital metric space, t : x → x, and φ ∈ φ. we say that • t is φ-contractive [21] if φ(d(t(x),t(y))) < φ(d(x,y)) for all x,y ∈ x, x 6= y. • t is a digital contraction map [8] if for some α ∈ (0,1), d(t(x),t(y)) ≤ αd(x,y) for all t ∈ r. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 168 fixed point assertions in digital topology, 2 the similarity of these definitions yields the following. proposition 6.5. let (x,d,κ) be a digital metric space, |x| > 1, and let t : x → x. if t is a non-constant digital contraction map then for some φ ∈ φ, t is φ-contractive. the converse is true when x is finite. proof. let t be a digital contraction map. then for some α ∈ (0,1), d(t(x),t(y)) < αd(x,y) for all x,y ∈ x. therefore, αd(t(x),t(y)) < α2d(x,y). note the function φ(t) = αt is a member of φ. then x 6= y implies φ(d(t(x),t(y))) = αd(t(x),t(y)) ≤ α2d(x,y) < αd(x,y) = φ(d(x,y)), so t is φ-contractive. suppose x is finite and t is φ-contractive for some φ ∈ φ. then x 6= y implies φ(d(t(x),t(y))) < φ(d(x,y)). since φ is monotone increasing, x 6= y implies d(t(x),t(y)) < d(x,y). since x is finite and t is non-constant, we can have α ∈ (0,1) well defined by α = max { d(t(x),t(y)) d(x,y) |x,y ∈ x,x 6= y } . since x is finite, x 6= y implies d(t(x),t(y)) d(x,y) ≤ α, or d(t(x),t(y)) ≤ αd(x,y). since the latter inequality also holds when x = y, it follows that t is a digital contraction map. � limitations on digital contraction maps are discussed in [5]. fixed point results for such maps appear in [8, 21]. definition 6.6 ([21]). let (x,d,κ) be a digital metric space. the function t : x → x is an α-ψ-φ-contractive type mapping if there exist three functions α : x × x → [0,∞) and ψ,φ ∈ φ such that α(x,y)ψ(d(tx,ty)) ≤ ψ(d(x,y)) − φ(d(x,y)) for all x,y ∈ x. remarks 6.7. one sees easily that every map t : x → x is an α-ψ-φcontractive type mapping, for α(x,y) = 0 and ψ(t) ≥ φ(t). hence such a function t need not be digitally continuous. the assertion stated as theorem 3.12 of [21] is (after correction) as follows. let (x,d,κ) be a digital metric space, t : x → x, and α : x2 → [0,∞). suppose (1) t is α-admissible; (2) there exists x0 ∈ x such that α(x0,t(x0)) ≥ 1; (3) t is digitally continuous; and c© agt, upv, 2019 appl. gen. topol. 20, no. 1 169 l. boxer (4) α(x,y)ψ(d(t(x),t(y))) ≤ ψ(m(x,y))−φ(m(x,y)), where m(x,y) = max{d(x,y), d(x,t(x)), d(y,t(y)), [d(x,t(y)) + d(y,t(x))]/2} for all x,y ∈ x. then t has a fixed point. further, if u and v are fixed points of t such that α(u,v) ≥ 1, then u = v. this assertion is incorrect without additional hypotheses. for example, if we take x = [0,1]z, t(x) = 1 − x, α(x,y) = 1, ψ(x) = x, φ(x) = −1, then all the hypotheses above are satisfied, but t has no fixed points. if x is finite or d is an ℓp metric, then the argument given as a proof for this assertion in [21] does not require the continuity assumption, but does require that ψ and φ have properties of φ. thus, a correct, somewhat modified version is as follows. theorem 6.8. let (x,d,κ) be a digital metric space where x is finite or d is an ℓp metric. let t : x → x, and α : x2 → [0,∞). suppose (1) t is α-admissible; (2) there exists x0 ∈ x such that α(x0,t(x0)) ≥ 1; and (3) there exist ψ,φ ∈ φ such that α(x,y)ψ(d(t(x),t(y))) ≤ ψ(m(x,y)) − φ(m(x,y)), where for all x,y ∈ x, m(x,y) = max{d(x,y), d(x,t(x)), d(y,t(y)), [d(x,t(y)) + d(y,t(x))]/2}. then t has a fixed point. further, if u and v are fixed points of t such that α(u,v) ≥ 1, then u = v. proof. our argument is similar to its analog in [21]. let x0 ∈ x be such that α(x0,t(x0)) ≥ 1. let xn be defined inductively by xn+1 = t(xn) for n ≥ 0. therefore, α(x0,x1) = α(x0,t(x0)) ≥ 1, so since t is α-admissible, we have α(x1,x2) = α(t(x0),t(x1)) ≥ 1, and, inductively, α(xn,xn+1) = α(t(xn−1),t(xn)) ≥ 1 for all n. then ψ(d(xn,xn+1)) = ψ(d(t(xn−1),t(xn))) ≤ α(xn−1,xn)ψ(d(t(xn−1),t(xn)) ≤ ψ(m(xn−1,xn)) − φ(m(xn−1,xn)) < ψ(m(xn−1,xn)). since ψ is increasing, d(xn,xn+1) < m(xn−1,xn) = max{d(xn−1,xn),d(xn−1,t(xn−1)),d(xn,t(xn)), [d(xn−1,t(xn)) + d(xn,t(xn−1))]/2} = max{d(xn−1,xn),d(xn−1,xn),d(xn,xn+1), [d(xn−1,xn+1) + d(xn,xn)]/2} = c© agt, upv, 2019 appl. gen. topol. 20, no. 1 170 fixed point assertions in digital topology, 2 max{d(xn−1,xn),d(xn,xn+1), [d(xn−1,xn+1) + 0]/2} = (6.1) max{d(xn−1,xn),d(xn,xn+1),d(xn−1,xn+1)/2} by the triangle inequality, d(xn−1,xn+1)/2 ≤ [d(xn−1,xn) + d(xn,xn+1)]/2 (6.2) ≤ max{d(xn−1,xn),d(xn,xn+1)} combining (6.1) and (6.2), d(xn,xn+1) < max{d(xn−1,xn),d(xn,xn+1)}, so d(xn,xn+1) < d(xn−1,xn). since x is finite or d is an ℓp metric, it follows that for sufficiently large n, d(xn,xn+1) = 0, or xn = xn+1 = t(xn), so xn is a fixed point of t . suppose u and v are fixed points of t with α(u,v) ≥ 1. we have ψ(d(u,v)) = ψ(d(t(u),t(v)) ≤ α(u,v)ψ(d(t(u),t(v))) ≤ ψ(m(u,v)) − φ(m(u,v)) = ψ(d(u,v)) − φ(d(u,v)), or 0 ≤ −φ(d(u,v)), which implies d(u,v) = 0, or u = v. � 6.2. weakly uniformly strict contractions. in [5], we discussed the paper [7], including mention of the fact that the author of the current work was identified as a reviewer of [7] and that the latter work was published without correction of several flaws mentioned in the review. here, we present additional discussion of [7]. definition 6.9 ([7]). let (x,d,κ) be a digital metric space. a self map t : x → x is a weakly uniformly strict digital contraction if given ε > 0, there exists δ > 0 such that ε ≤ d(x,y) < ε + δ implies d(t(x),t(y)) < ε for all x,y ∈ x. lemma 6.10. let (x,d,κ) be a digital metric space. let t : x → x be a weakly uniformly strict digital contraction. then for all x,y ∈ x such that x 6= y, d(t(x),t(y)) < d(x,y). proof. this is shown in the first paragraph of the argument given as a proof of theorem 3.3 in [7]. � corollary 6.11. let (x,d,κ) be a digital metric space, such that x is finite. let t : x → x be a weakly uniformly strict digital contraction. then t is a digital contraction map. proof. without loss of generality, t is not constant. since x is finite, α = max { d(t(x),t(y)) d(x,y) | x,y ∈ x,x 6= y } c© agt, upv, 2019 appl. gen. topol. 20, no. 1 171 l. boxer is well defined. since t is not constant, α > 0; and, since x is finite, by lemma 6.10, α < 1. then for x,y ∈ x such that x 6= y, d(t(x),t(y)) = d(t(x),t(y)) d(x,y) d(x,y) ≤ αd(x,y). by definition 6.1, t is a digital contraction map. � proposition 6.12. let (x,d,κ) be a digital metric space such that 1 < |x| < ∞. let t : x → x be a weakly uniformly strict digital contraction. then t is neither one-to-one nor onto. proof. since 1 < |x| < ∞, m = min{d(x,y) |x,y ∈ x,x 6= y} and m = max{d(x,y) |x,y ∈ x,x 6= y} are well defined positive numbers, and there exist x0,y0 ∈ x such that x0 6= y0 and d(x0,y0) = m, and x1,y1 ∈ x such that x1 6= y1 and d(x1,y1) = m. by lemma 6.10, d(t(x0),t(y0)) < d(x0,y0) = m. by definition of m, this implies t(x0) = t(y0), so t is not one-to-one. if t is onto, then there exist u,v ∈ x such that t(u) = x1 and t(v) = y1. thus, by lemma 6.10 we conclude that m = d(x1,y1) = d(t(u),t(v)) < d(u,v), which contradicts our choice of m. therefore, t is not onto. � a limitation on weakly uniformly strict digital contractions is shown in the following. proposition 6.13. let (x,d,cu) be a digital metric space, where d is an ℓp metric. let t : x → x be a self map such that for some α > 0, 1 ≤ d(x,y) < u1/p + α implies d(t(x),t(y)) < 1. if x is cu-connected, then t is constant. proof. let x ↔cu y in x. then for every α > 0, 1 ≤ d(x,y) < u1/p + α, so d(t(x),t(y)) < 1. therefore, t(x) = t(y). since x is cu-connected, it follows that t is constant. � as an immediate consequence, we have the following. corollary 6.14. let (x,d,c1) be a digital metric space, where d is an ℓp metric. let t : x → x be a weakly uniformly strict digital contraction. if x is c1-connected, then t is constant. proof. since t is a weakly uniformly strict digital contraction, for some α > 0, 1 ≤ d(x,y) < 1 + α implies d(t(x),t(y)) < 1. since d is an ℓp metric, c1adjacent x,y ∈ x satisfy d(x,y) = 1, hence d(t(x),t(y)) < 1. since x is c1-connected, it follows from theorem 6.13 that t is constant. � the arguments given as proof for theorems 3.1 and 3.3 in [7] are marred by confusion of digital and metric continuity. below, we make minor modifications of the stated assumptions in both theorems, we replace the assumption of a complete metric space with the assumption of an ℓp metric and give corrected c© agt, upv, 2019 appl. gen. topol. 20, no. 1 172 fixed point assertions in digital topology, 2 proofs that are much shorter than the arguments given in [7], in part because discussion of continuity is unnecessary. the following is theorem 3.1 of [7], modified as discussed above. theorem 6.15. let (x,d,κ) be a digital metric space, where d is an ℓp metric, and suppose that t : x → x satisfies d(t(x),t(y)) ≤ ψ(d(x,y)) for all x,y ∈ x, where ψ ∈ φ. then t has a unique fixed point. proof. let x0 ∈ x. inductively, let xn+1 = t(xn) for n ≥ 0. then d(xn,xn+1) = d(t(xn−1),t(xn)) ≤ ψ(d(xn−1,xn)) and a simple induction argument leads to the conclusion that for n ≥ 0, d(xn,xn+1) ≤ ψn(d(x0,x1)) →n→∞ 0. by corollary 2.6, for sufficiently large n we have xn = xn+1 = t(xn), so xn is a fixed point. suppose x and x′ are fixed points of t . then d(x,x′) = d(t(x),t(x′)) ≤ ψ(d(x,x′)), since ψ ∈ φ, where equality occurs only for d(x,x′) = 0, i.e., x = x′. thus t has a unique fixed point. � remarks 6.16. if, in theorem 6.15, d is an ℓp metric and x is c1-connected, then t is a constant function. proof. given x ↔c1 x′ in x, we have, since d is an ℓp metric, d(x,x′) = 1, and d(t(x),t(x′)) ≤ ψ(d(x,x′)) = ψ(1) < 1, so t(x) = t(x′). since x is c1-connected, it follows that t is constant. � the following is theorem 3.3 of [7], modified as discussed above. theorem 6.17. let (x,d,κ) be a complete digital metric space, where d is an ℓp metric, and let t : x → x be a weakly uniformly strict digital contraction mapping. then t has a unique fixed point z. moreover, for any x ∈ x, limn→∞t n(x) = z. proof. by lemma 6.10, for all x,y ∈ x, d(t(x),t(y)) < d(x,y). let x0 ∈ x and, for n ≥ 0, inductively define xn+1 = t(xn). we may assume x1 6= x0, since, otherwise, x0 is a fixed point of t . we have, for n > 0, d(xn,xn+1) = d(t(xn−1),t(xn)) < d(xn−1,xn). since d is an ℓp metric, for sufficiently large n, xn = xn+1 = t(xn); thus, xn is a fixed point of t . suppose x and y are fixed points of t . if x 6= y then, by lemma 6.10, d(x,y) = d(t(x),t(y)) < d(x,y), which is impossible. therefore x = y. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 173 l. boxer 7. concluding remarks we have corrected or noted limitations of published assertions concerning fixed points for self-maps on digital metric spaces. this continues the work of [5]. in many cases, flaws we have noted were so obvious that reviewers and/or editors of these papers share responsibility with the authors. we have also offered improvements to some of the assertions discussed. acknowledgements. we are grateful to the anonymous referee for many corrections and excellent suggestions. references [1] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. 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[19] a. rosenfeld, ‘continuous’ functions on digital images, pattern recognition letters 4 (1986), 177–184. [20] b. samet, c. vetro, and p. vetro, fixed point theorems for contractive mappings, nonlinear analysis: theory, methods & applications 75, no. 4 (2012), 2154–2165. [21] k. sridevi, m. v. r. kameswari and d. m. k. kiran, fixed point theorems for digital contractive type mappings in digital metric spaces, international journal of mathematics trends and technology 48, no. 3 (2017), 159–167. [22] k. sridevi, m. v. r. kameswari and d. m. k. kiran, common fixed points for commuting and weakly compatible self-maps on digital metric spaces, international advanced research journal in science, engineering and technology 4, no. 9 (2017), 21–27. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 175 @ appl. gen. topol. 20, no. 2 (2019), 449-469 doi:10.4995/agt.2019.11683 c© agt, upv, 2019 existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle rehana tabassum a, akbar azam a and mohammed shehu shagari b a department of mathematics, comsats university, chak shahzad, islamabad, 44000, pakistan (reha alvi@yahoo.com,akbarazam@yahoo.com) b department of mathematics, faculty of physical sciences, ahmadu bello university,zaria, nigeria (ssmohammed@abu.edu.ng) communicated by s. romaguera abstract the purpose of this article is to extend the results derived through former articles with respect to the notion of weak contraction into intuitionistic fuzzy weak contraction in the context of (t ,n ,α)−cut set of an intuitionistic fuzzy set. we intend to prove common fixed point theorem for a pair of intuitionistic fuzzy mappings satisfying weakly contractive condition in a complete metric space which generalizes many results existing in the literature. moreover, concrete results on existence of the solution of a delay differential equation and a system of riemann-liouville cauchy type problems have been derived. in addition, we also present illustrative examples to substantiate the usability of our main result. 2010 msc: 46s40; 47h10; 54h25. keywords: common fixed point; intuitionistic fuzzy set-valued maps; (t ,n ,α) − cut set; weakly contractive condition; delay differential equation; riemann-liouville fractional differential equations. received 15 april 2019 – accepted 29 july 2019 http://dx.doi.org/10.4995/agt.2019.11683 r. tabassum, a. azam and m. s. shagari 1. introduction metric fixed point theory is generally based on the banach contraction principle, which has been used to study the existence and uniqueness of fixed points. this principle has been extensively studied in different directions. in 1997, alber and guerr-delabriere [3] proposed the notion of weak contractive mappings on hilbert spaces and studied the existence of fixed point results in the context of weakly contractive single valued maps on hilbert spaces as a generalization of banach contraction principle. however, in 2001, rhoades [30] presented some results of [3] to arbitrary banach spaces. later on, bae [10] established the fixed points of weakly contractive multivalued mappings and beg and abbas [19] demonstrated the fixed point results for a pair of single valued mappings one is weakly contractive relative to the other. on the other hand, there are many complicated practical problems in the domain of real world such as engineering, economics, social sciences, medical science and many other fields that involve data which are not always precise. to overcome these difficulties, classical mathematical notions may not be applied effectively, because there are numerous types of vagueness appear in these domains. however, in response to this fact, zadeh [39] developed the concept of fuzzy set as an extension of conventional set theory. over the years, several mathematicians extended this notion in different directions, for instance, l-fuzzy set, intuitionstic fuzzy set, fuzzy soft set and hesitant fuzzy soft set. consequently, in 1981, heilpern [20] initiated the idea of fuzzy mapping and proved a fixed point theorem for fuzzy contraction mappings as an extension of multivalued mappings of nadler’s contraction principle. thus, this result motivated several researchers to study and establish the fixed point results satisfying a fuzzy contractive inequalities ( see, [1, 6, 7]). one of the generalizations of fuzzy set theory [39] is the notion of intuitionistic fuzzy set (if-set) introduced by atanassov [5]. moreover, if-sets create a valuable mathematical structure to deal with inaccuracy and hesitancy originating from insufficient decision information and as a consequence ,it has remarkable applications in various fields like image proccessing [21], medical diagnosis [18], drug selection [22], decision analysis [26], etc. until now, research on if-set has been very active and many results have been proved with different aspects. recently, azam et al. [8] developed new approach to discuss the fixed point theorems using the idea of intuitionistic fuzzy mappings [38] on a complete metric space. later on, azam and rehana [9] presented existence of common coincidence point for three intuitionistic fuzzy set valued maps and they also studied existence results for a system of integral equations. in this manuscript, the idea of weakly contraction is used for intuitionistic fuzzy mappings in association with (t ,n ,α)−cut set of an if-set [27]. on the basis of this concept, an existence result of common fixed point on complete metric space is presented. from an application perspective, we apply our main c© agt, upv, 2019 appl. gen. topol. 20, no. 2 450 existence results of delay and fractional differential equations result to establish existence theorems of the solution of a delay differential equation and a system of riemann-liouville cauchy type problems. 2. preliminaries throughout this paper, (u,ρ), (w,ρw ) and v (w) denote a metric space, a metric linear space and a subcollection of all approximate quantities in w , respectively. let cb (u) = {a∗ : a∗ is nonempty, closed and bounded subset of u}, c(u) = { a∗ : a∗ is nonempty compact subset of u}. for a∗, b∗ ∈ cb(u), define ρ(u,a∗) = inf ρ v∈a∗ (u,v), ρ(a∗,b∗) = inf u∈a∗, v∈b∗ ρ(u,v). the hausdorff metric ρh on cb(u) induced by ρ is defined as ρh (a ∗,b∗) = max { sup u∈a∗ ρ (u,b∗) , sup v∈b∗ ρ (a∗,v) } . definition 2.1 ([3, 17]). let (u,ρ) be a metric space and a mapping f : u → u is called a weakly contractive mapping if for u, v ∈ u, ρ (f (u) ,f (v)) ≤ ρ (u,v) −φ (ρ (u,v)) , where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) = 0 if and only if t = 0. definition 2.2 ([10]). let (u,ρ) be a metric space. a mapping f : u → c (u) is said to be a weakly contractive multivalued mapping, if there exists a continuous non-decreasing function φ : [0,∞) → [0,∞) with φ (0) = 0 and φ (t) > 0 for all t > 0, such that ρh (f (u) ,f (v)) ≤ ρ (u,v) −φ (ρ (u,v)) , for all u,v ∈ u. definition 2.3 ([20, 39]). let z be a universal set. a fuzzy set in u is an object of the form a∗ = {(z,a∗(z)) : z ∈ z}, where a∗(z) denotes the membership values of z in a∗. definition 2.4 ([20, 39]). let a∗ be a fuzzy set of universe z. the α − cut set of a∗ denoted by [a∗]α is a crisp subset of z whose membership value in a∗ is greater than or equal to some specific value of α, i.e. [a∗]α = {z ∈ z : a ∗(z) ≥ α} if α ∈ (0, 1] . definition 2.5 ([20]). a fuzzy set a∗ in a metric linear space w is said to be an approximate quantity if and only if only if [a∗]α is compact and convex in w for each α ∈ (0, 1] with sup a∗ (w) w∈w = 1. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 451 r. tabassum, a. azam and m. s. shagari definition 2.6 ([20]). let z be an arbitrary set and u be a metric space. a mapping from z into iu is called a fuzzy mapping. definition 2.7 ([7]). an element u∗ ∈ u is called a fuzzy fixed point of a fuzzy mapping s : u → iu if there exists α ∈ (0, 1] such that u∗ ∈ [s (u∗)]α. let iu be the collection of all fuzzy sets in u and f (u) = { a∗ ∈ iu : [a∗]α ∈ c (u) for all α ∈ [0, 1] } . for a∗, b∗ ∈ iu , if there exists an α ∈ [0, 1] such that [a∗]α, [b ∗]α ∈ c (u), then define dα (a ∗,b∗) = ρh ([a ∗]α , [b ∗]α) , d (a∗,b∗) = sup α dα ([a ∗]α , [a ∗]α) , where d is a metric on f (u) and the completness of (u,ρ) implies (c (u) ,hρ) and (f (u) ,d) are complete. lemma 2.8 ([28]). let (u,ρ) be a metric space and a∗, b∗ ∈ c(u), then for each u ∈ a∗, there exists an element v ∈ b∗ such that ρ(u,v) ≤ ρh(a∗,b∗). lemma 2.9 ([28]). let (u,ρ) be a metric space and a∗, b∗ ∈ cb(u). if u ∈ a∗, then ρ(u,b∗) ≤ ρh(a∗,b∗). definition 2.10 ([5]). let z be a fixed set. then an if-set e in z is a set of ordered triples given by e = {〈z,µe (z) ,υe (z)〉 : z ∈ z} , where µe : z → [0, 1] and υe : z → [0, 1] define the degree of membership and the degree of non-membership respectively, of the elements z in e and satisfying 0 ≤ µe (z) + υe (z) ≤ 1, for each element z ∈ z. in addition, the degree of hesitancy of z to e is defined by πe (z) = 1 −µe (z) −υe (z) . particularly, if πe (z) = 0, for all z ∈ z, then an if-set e is reduced to a fuzzy set a∗. example 2.11. consider an if-set e of high-experienced and low-experienced employees of a company z, whose degrees of membership µe(z) and nonmembership υe (z) are depicted in fig. 1. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 452 existence results of delay and fractional differential equations definition 2.12 ([5]). let e be an if-set of universe z. the α − cut set of e is a classical subset of elements of z denoted by [e]α and is defined by [e]α = {z ∈ z : µe(z) ≥ α and υe (z) ≤ 1 −α} if α ∈ [0, 1] . definition 2.13 ([27]). a mapping t : [0, 1]2 → [0, 1] is called a triangular norm (t-norm), if the following conditions are satisfied: (i) . t (z1,t (z2,z3)) = t (t (z1,z2) ,z3) for all z1, z2, z3 ∈ z. (ii) . t (z1,z2) = t (z2,z1) for all z1, z2 ∈ z. (iii) . if z1, z2, z3 ∈ [0, 1] and z1 ≤ z2, then t (z1,z3) ≤t (z2,z3) . (iv) . t (z1, 1) = z1 for all z1 ∈ z. minium t-norm denoted by tm and is defined by tm (z1,z2) = min (z1,z2) for all z1, z2 ∈ [0, 1] . definition 2.14 ([27]). fuzzy negation is a non-increasing mapping n : [0, 1] → [0, 1] such that n (0) = 1, n (1) = 0. if n is continuous and strictly decreasing, then it is called strict. fuzzy negations with n (n (z)) = z, for all c© agt, upv, 2019 appl. gen. topol. 20, no. 2 453 r. tabassum, a. azam and m. s. shagari z ∈ [0, 1] , are called strong fuzzy negations. the example of fuzzy negation is a standard negation defined by ns (z) = 1 −z, for all z ∈ z. definition 2.15 ([27]). let e be an if-set of u, t and n a triangular norm and a fuzzy negation, respectively. then (t ,n ,α) −cut set of e is a crisp set denoted by [e](t ,n,α) and is defined by [e](t ,n,α) = {z ∈ z : t (µe(z),n (υe (z))) ≥ α} if α ∈ [0, 1] . remark 2.16. if we take t = tm and n = ns, then (t ,n ,α) − cut set is reduced into original definition of a cut set by atanassov [5]. definition 2.17 ([38]). let z be an arbitrary set, u a metric space. a mapping s is called intuitionistic fuzzy mapping if s is a mapping from z into (ifs) u . definition 2.18. a point u∗ ∈ u is said to be an intuitionistic fuzzy fixed point of an intuitionistic fuzzy mapping s : u → (ifs)u if there exists α ∈ [0, 1] such that u∗ ∈ [s (u∗)](t ,n,α) . let (ifs) u be the collection of all intuitionistic fuzzy subsets of u and define fif (u) = { e ∈ (ifs)u : [e](t ,n,α) ∈ c (u) for all α ∈ [0, 1] } . for e1, e2 ∈ (ifs) u and α ∈ [0, 1] such that [e1](t ,n,α) , [e2](t ,n,α) ∈ c (u) , the following notations are defined by dα (e1,e2) = ρh ( [e1](t ,n,α) , [e2](t ,n,α) ) , dif (e1,e2) = sup α dα ( [e1](t ,n,α) , [e2](t ,n,α) ) , where dif is a metric on fif (u) . 3. main results in what follows hereafter, we present our main results. theorem 3.1. let (u,ρ) be a complete metric space and f, g be a pair of intuitionistic fuzzy mappings from u into (ifs) u . for u ∈ u, there exist αf (ξ) , αg (ξ) ∈ [0, 1] such that [f (ξ)](t ,n,αf (ξ)) , [g (ξ)](t ,n,αg(ξ)) ∈ c (u) . if for all u,v ∈ u, (3.1) ρh ( [f (u)](t ,n,αf (u)) , [g (v)](t ,n,αg(v)) ) ≤ ρ (u,v) −φ (ρ (u,v)) , where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) = 0 if and only if t = 0 and lim t→∞ φ (t) = ∞. thus, there exists ω ∈ u such that ω ∈ [f (ω)](t ,n,αf(ω)) ∩ [g (ω)](t ,n,αg(ω)). c© agt, upv, 2019 appl. gen. topol. 20, no. 2 454 existence results of delay and fractional differential equations proof. let u0 be an arbitrary but fixed element of u, then by assumptions, there exists αf(u0) ∈ [0, 1] such that [f (u0)](t ,n,αf (u0)) ∈ c (u). choose u1 ∈ [f (u0)](t ,n,αf (u0)). it follows from lemma 2.8, there exists u2 ∈ [g (u1)](t ,n,αg(u1)) such that ρ (u1,u2) ≤ ρh ( [f (u0)](t ,n,αf (u0)) , [g (u1)](t ,n,αg(u1)) ) ≤ ρ (u0,u1) −φ (ρ (u0,u1)) . again by lemma 2.8, for u2 ∈ [g (u1)](t ,n,αg(u1)) there exists u3 ∈ [f (u2)](t ,n,αf (u2)) such that ρ (u2,u3) ≤ ρh ( [g (u1)](t ,n,αg(u1)) , [f (u2)](t ,n,αf (u2)) ) ≤ ρ (u1,u2) −φ (ρ (u1,u2)) . continuing this process, for un ∈ u we obtain un+1 ∈ u such that un+1 ∈ [f (un)](t ,n,αf(un)) , n = 0, 1, 2, · · ·, un+2 ∈ [g (un+1)](t ,n,αg(un+1)) , n = 0, 1, 2, · · ·, where, ρ (un+1,un+2) ≤ ρh ( [f (un)](t ,n,αf (un)) , [g (un+1)](t ,n,αg(un+1)) ) ≤ ρ (un,un+1) −φ (ρ (un,un+1)) ≤ ρ (un,un+1) , n = 0, 1, 2, · · ·. it follows that {ρ (un,un+1)} is a non-increasing sequence of positive real numbers and hence tends to limit r ≥ 0. if r > 0, then we obtain (3.2) ρ (un+1,un+2) ≤ ρ (un,un+1) −φ (r) . therefore, (3.3) ρ (un+n,un+n+1) ≤ ρ (un,un+1) −nφ (r) , which is a contradiction for large enough n. hence, ρ (un,un+1) → 0. therefore, by a similar argument of [11], it follows {un} is a cauchy sequence in u. as u is complete, therefore there exists ω ∈ u such that un → ω. then by lemma 2.9, we get ρ ( [f (ω)](t ,n,αf (ω)) ,un+2 ) ≤ ρh ( [f (ω)](t ,n,αf (ω)) , [g (un+1)](t ,n,αg(un+1)) ) ≤ ρ (ω,un+1) −φ (ρ (ω,un+1)) .(3.4) letting n →∞ and using the fact that φ (0) = 0, we obtain ρ ( [f (ω)](t ,n,αf (ω)) ,ω ) ≤ 0. this implies ω ∈ [f (ω)](t ,n,αf (ω)) . c© agt, upv, 2019 appl. gen. topol. 20, no. 2 455 r. tabassum, a. azam and m. s. shagari similarly, ω ∈ [g (ω)](t ,n,αg(ω)) . hence, there exists ω ∈ u such that ω ∈ [f (ω)](t ,n,αf (ω))∩[g (ω)](t ,n,αg(ω)) . � corollary 3.2. let (u,ρ) be a complete metric space and f : u → (ifs)u be a intuitionistic fuzzy mapping. for u ∈ u, there exists αf (u) ∈ [0, 1] such that [f (u)](t ,n,αf (u)) ∈ c (u) . if for all u, v ∈ u, ρh ( [f (u)](t ,n,αf (u)) , [f (v)](t ,n,αf (v)) ) ≤ ρ (u,v) −φ (ρ (u,v)) , where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) = 0 if and only if t = 0 and lim t→∞ φ (t) = ∞. thus, there exists ω ∈ u such that ω ∈ [f (ω)](t ,n,αf (ω)). if we take φ (t) = (1 −q) (t) , where 0 < q < 1, then corollary 3.2 reduces to the following result. corollary 3.3. let (u,ρ) be a complete metric space and f : u → (ifs)u be intuitionistic fuzzy mapping. for u ∈ u there exists αf (u) ∈ [0, 1] such that [f (u)](t ,n,αf (u)) ∈ c (u) . if 0 < q < 1 and for all u, v ∈ u, ρh ( [f (u)](t ,n,αf (u)) , [f (v)](t ,n,αf (v)) ) ≤ qρ (u,v) . thus, there exists ω ∈ u such that ω ∈ [f (ω)](t ,n,αf (ω)). corollary 3.4. let (u,ρ) be a complete metric space and f, g : u → iu be a pair of fuzzy mappings. for u ∈ u there exists αf (u) , αg (u) ∈ (0, 1] such that [f (u)]αf (ξ) , [g (u)]αg(ξ) ∈ c (u) . if for all u,v ∈ u, ρh ( [f (u)]αf (u) , [g (v)]αg(v) ) ≤ ρ (u,v) −φ (ρ (u,v)) , where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) = 0 if and only if t = 0 and lim t→∞ φ (t) = ∞. thus, there exists ω ∈ u such that ω ∈ [f (ω)]αf (ω) ∩ [g (ω)]αg(ω). corollary 3.5. let (w,ρw ) be a complete metric linear space and f, g : w → v (w) be a pair of fuzzy mappings satisfying the following condition d (f (u) ,g (v)) ≤ αρ (u,v) , for each u,v ∈ w, where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) = 0 if and only if t = 0 and lim t→∞ φ (t) = ∞. thus, there exists a ω ∈ w such that {ω}⊂ f (ω) and {ω}⊂ g (ω) . corollary 3.6. let (w,ρw ) be a complete metric linear space and f : w → v (w) be a fuzzy mapping satisfying d (f (u) ,f (v)) ≤ ρ (u,v) −φ (ρ (u,v)) , for each u,v ∈ w. thus, there exists ω ∈ w such that {ω}⊂ f (ω) . c© agt, upv, 2019 appl. gen. topol. 20, no. 2 456 existence results of delay and fractional differential equations example 3.7. let u = r+, ρ (u,v) = |u−v| , whenever u, v ∈ u and γ, δ ∈ [0, 1]. consider a pair of intuitionistic fuzzy mappings f = 〈µf ,υf〉 , g = 〈µg,υg〉 : u → (ifs) u as follow: case (i) : if u = v = 0, then we have µf(0) (t) = µg(0) (t) =   1 if t = 0 2 5 if 0 < t ≤ 1002 0 if t > 1002 and υf(0) (t) = υg(0) (t) =   0 if t = 0 3 5 if 0 < t ≤ 1003 1 if t > 1003 . if we take αf(0) = 1 = αg(0), then we obtain [f (0)](t ,n,1) = {0} = [g (0)](t ,n,1) . moreover, ρh ( [f (0)](t ,n,α f (0) ) , [g (0)](t ,n,αg(0)) ) = ρ (u,v) −φ (ρ (u,v)) . case (ii) : if u 6= 0, v 6= 0 then we have µf(u) (t) =   γ if 0 ≤ t ≤ u− u 2 2 γ 3 if u− u 2 2 < t ≤ u− u 2 4 γ 7 if u− u 2 4 < t < u 0 if u ≤ t < ∞ , υf(u) (t) =   0 if 0 ≤ t ≤ u− u 2 2 γ 4 if u− u 2 4 < t ≤ u− u 2 6 γ 2 if u− u 2 6 < t < u 1 if u ≤ t < ∞ . and µg(u) (t) =   δ if 0 ≤ t ≤ u− u 2 2 δ 2 if u− u 2 2 < t ≤ ξ − u 2 5 δ 3 if u− u 2 5 < t < u 0 if u ≤ t < ∞ , υg(u) (t) =   0 if 0 ≤ t ≤ u− u 2 2 δ 8 if u− u 2 2 < t ≤ u− u 2 8 δ 6 if u− u 2 8 < t < u 1 if u ≤ t < ∞ . if αf(u) = γ and αg(u) = δ, then we have [f (u)] (t ,n,γ) = { t ∈ u : t ( µf(u) (t) ,n ( υf(u) (t) )) = γ } = [ 0,u− u2 2 ] and [g (u)](t ,n,δ) = { t ∈ u : t ( µg(u) (t) ,n ( υg(u) (t) )) = δ } = [ 0,u− u2 2 ] . c© agt, upv, 2019 appl. gen. topol. 20, no. 2 457 r. tabassum, a. azam and m. s. shagari however, ρh ( [f (u)](t ,n,γ) , [g (u)](t ,n,δ) ) = ∣∣∣∣u− u22 −v + v 2 2 ∣∣∣∣ = ∣∣∣∣(u−v) ( 1 − u + v 2 )∣∣∣∣ ≤ |u−v| ∣∣∣∣1 − |u−v|2 ∣∣∣∣ ≤ |u−v|− |u−v|2 2 = |u−v|−ϕ (|u−v|) ≤ ρ (u,v) −ϕ (ρ (u,v)) . thus, in both the cases, for ϕ (t) = 1 2 t2, all the assumptions of theorem 3.1 are satisfied to obtain ω ∈ [f (ω)](t ,n,α f (ω) ) ∩ [g (ω)](t ,n,α g(ω) ) . 3.1. application to delay differential equations. in this section, we will establish an existence result of delay differential equation with constant delay, where the only independent variable is the time variable. delay differential equations appear naturally in modelling the numerous biological systems. for instance, primary infection [14], drug therapy [29] and immune response [15]. they have also been used in the study of epidemiology [16], the respiratory system [36] and tumor growth [37]. moreover, statistical analysis of ecological data [34], indicates the delay effects in many classes of population dynamics. general form of delay differential equation consider the general form of equation with delay u· (t) = g (t,ut) , where ut : [−τ, 0] → rn is a function such that ut (λ) = u (t + λ) for λ ∈ [−τ, 0] , as shown in fig. 2. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 458 existence results of delay and fractional differential equations for an ordinary differential system, a unique solution is obtained using an initial point in euclidean space at an initial time t0. on the other hand, one needs information on the entire interval [t0 − τ,t0], for a delay differential equations. delay differential equations are solved by considering previous values of dependent variable u at every time step. for this, one requires initial function or initial history, the value of u (t) for the interval [−τ, 0] is used to demonstrate the behavior of the system prior to the starting time. theorem 3.8. let u = c ([a,b] ,r) be the space of all continuous real valued functions on [a,b] with a metric ρ : x ×x → r defined by ρ (u,v) = max t∈[a,b] |u (t) −v (t)| , for all u, v ∈ c [a,b] . assume that g : [t0,b] × r2 → r and ψ : [t0 − τ,b] → r are continuous mappings, where t0, b ∈ r and τ > 0. if there exists λg > 0 such that λg < 1 2(b−t0) and (3.5) |g (t,u1,u2) −g (t,v1,v2)| ≤ λg 2∑ i=1 |ui −vi|, for all ui, vi ∈ r, i = 1, 2, t ∈ [t0,b] . thus, the delay differential equation (3.6) u′ (t) = g (t,u (t) ,u (t− τ)) , t ∈ [t0,b] with initial condition (3.7) u (t) = ψ (t) , t ∈ [t0 − τ,t0] has a solution u ∈ c ([t0 − τ,b] ,r) ∩c1 ([t0,b] ,r) . proof. let f : u → (ifs)u be intuitionistic fuzzy mapping and define an arbitrary mapping h from u into (0, 1]. the integral reformulation of problem (3.5)-(3.7) is given by (3.8) u (t) =   ψ (t) , t ∈ [t0 − τ,t0] ψ (t0) + t∫ t0 g (s,u (s) ,u (s− τ)) ds, t ∈ [t0,b] . define an intuitionistic fuzzy mapping f = 〈µf ,υf〉 : u → (ifs) u as follows: µf(u) (e) = { h (u) if e (t) = u (t) for all t ∈ [t0,b] 0 otherwise , υf(u) (e) = { 0 if e (t) = u (t) for all t ∈ [t0,b] h (u) otherwise . c© agt, upv, 2019 appl. gen. topol. 20, no. 2 459 r. tabassum, a. azam and m. s. shagari if αf(u) = h (u) , then we have [f (u)](t ,n,αf (u)) = { e ∈ u : t ( µf(u) (e) ,n ( υf(u) (e) )) = h (u) } = {u} . however, ρh ( [f (u)](t ,n,αf (u)) , [f (v)](t ,n,αf (v)) ) = max t∈[t0−τ,b] |u (t) −v (t)| . therefore, by assumptions, we obtain max t∈[t0−τ,b] |u (t) −v (t)| = max t∈[t0−τ,b] ∣∣∣∣∣∣ t∫ t0 g (s,u (s) ,u (s− τ)) ds− t∫ t0 g (s,v (s) ,v (s− τ)) ds ∣∣∣∣∣∣ ≤ max t∈[t0−τ,b] t∫ t0 |g (s,u (s) ,u (s− τ)) −g (s,v (s) ,v (s− τ))|ds ≤ max t∈[t0−τ,b] t∫ t0 λg (|u (s) −v (s)| + |u (s− τ) −v (s− τ)|) ds ≤ t∫ t0 λg ( max s∈[t0−τ,b] |u (s) −v (s)| + max s∈[t0−τ,b] |u (s− τ) −v (s− τ)| ) ds ≤ t∫ t0 λg (ρ (u,v) + ρ (u,v)) ds ≤ 2λgd (u,v) t∫ t0 ds ≤ 2λg (b− t0) ρ (u,v) ≤ ρ (u,v) − (1 −q) ρ (u,v) ≤ ρ (u,v) −ϕ (ρ (u,v)) . where, q = 2λg (b− t0) and ϕ (u) = (1 −q) (u) . thus, all the assumptions of theorem 3.1 are satisfied for f = g to obtain ω ∈ u such that ω ∈ [ [f (ω)](t ,n,αf (ω)) ] . hence, ω is a solution of (3.5) and (3.6). � example 3.9. consider the delay differential equation (3.9) u′ (t) = t3 + 1 10 u5 (t) + 1 10 u5 ( t− 1 2 ) , t ∈ [0, 1] c© agt, upv, 2019 appl. gen. topol. 20, no. 2 460 existence results of delay and fractional differential equations with initial condition (3.10) u (t) = t + 1, t ∈ [ − 1 2 , 0 ] , where τ = 1 2 and ψ (t) = t + 1, g (t,u (t) ,u (t− τ)) = t3 + 1 10 u5 (t) + 1 10 u5 ( t− 1 2 ) . the associated integral equation of problem (3.9)-(3.10) is given by u (t) =   t + 1, t ∈ [ −1 2 , 0 ] 1 + t∫ 0 ( s3 + 1 10 u5 (s) + 1 10 u5 ( s− 1 2 )) ds, t ∈ [0, 1] . if ui, vi ∈ r, i = 1, 2, and t ∈ [0, 1] then we obtain |g (t,u1,u2) −g (t,v1,v2)| = ∣∣∣∣t3 + 110u51 + 110u52 − t3 − 110v51 − 110v52 ∣∣∣∣ = ∣∣∣∣ 110 (u51 −v51) + 110 (u52 −v52) ∣∣∣∣ ≤ 1 10 ∣∣(u51 −v51)∣∣ + 110 ∣∣(u52 −v52)∣∣ ≤ 1 10 2∑ i=1 ∣∣u5i −v5i ∣∣ . hence, for λg = 1 10 , all the conditions of theorem 3.8 are satisfied to obtain a solution of the given delay differential equation. 4. application to a system of riemann-liouville fractional differential equations in recent time, fractional calculus has drawn the interests of researchers due to its wide range of applications in solving problems in diverse areas such as viscoelasticity, biological science, aerodynamics, statistical physics, etc. for some noted applications and developmental history of fractional calculus, the interested reader may see [24, 35]. undoubtably, the first problem of every fractional differential equation is the conditions for the existence of its solution. thus, this section is devoted to providing existence conditions of solutions to riemann-liouville cauchy type problem on a finite interval of the real line in a space of summable and continuous functions. our investigations are based on reformulating the problem to volterra integral equation of the second kind and using intuitionistic fuzzy maps. the nonlinear riemann-liouville fractional derivative (d ξ a+ v)(u) of order ξ, defined for re(ξ) > 0 on a finite interval [a,b] is given by (4.1) (d ξ a+ v)(u) = g(u,v(u)), c© agt, upv, 2019 appl. gen. topol. 20, no. 2 461 r. tabassum, a. azam and m. s. shagari with initial conditions (4.2) (d ξ−i a+ v)(a+) = di, di ∈ c (i = 1, 2, 3, · · ·n), where n = re(ξ) + 1 for ξ /∈ n and ξ = n for ξ ∈ n. notice that for ξ = n ∈ n, problems (4.1)-(4.2) are reduced to classical cauchy problem for the ordinary differential equation. the cauchy type problem (4.1)-(4.2) with complex ξ ∈ c was first studied by kilbas [23] in the space of summable functions l(a,b). al-bassam [12] studied problem (4.1)-(4.2) for a real 0 < ξ ≤ 1 in the space of continuous functions c[a,b], provided that g(u,v) is a real-valued continuous function in a domain h ⊂ r2. most likely, he was the first to show that the method of contraction mapping could be employed to prove the existence of solution to (4.1)-(4.2). it was however observed by kilbas [23] that the condition given by al-bassam [12] was not suitable for solving the problem. afterwards, delbosco and rodino [19] studied the nonlinear riemann-liouville cauchy problem: (4.3) (d ξ a+ v)(u) = g(u,v(u)), v(i)(0) = vi ∈ r (i = 1, 2, 3, · · ·n) with 0 ≤ u ≤ 1, λ > 0 and g(u,v) is a continuous function on [0, 1] × r. they showed the equivalence to the corresponding volterra integral equation and applied schauder’s fixed point theorem to prove that problem (4.3) has at least one solution v(u) defined on [0,τ] provided that uκg(u,v) is continuous on [0, 1] × r for some κ ∈ [0, 1). later on, problems (4.1)-(4.2) and (4.3) were studied by several authors (see, [2, 4, 32]). but the above investigations were not complete due to the missing of some techniques of nonlinear functional analysis [23]. for details in this observation, the interested readers may go through the survey paper by kilbas and trujillo [33]. as far as we know, no contribution exists in the literature concerning with the study of existence conditions of the riemann-liouville cauchy type problem (4.1)-(4.2) in the setting of intuitionistic fuzzy mappings and even then for fuzzy and multivalued mappings. thus, in this section, we establish existence conditions for the solution of problem (4.1)-(4.2) in the space l1(a,b) = l(a,b) of summable functions on a finite interval [a,b] of r by appealing to intuitionistic fuzzy mappings defined on a complete metric space. for our convenience, we recall the definitions of riemann-liouville fractional integrals and fractional derivatives on a finite interval of the real line and present specific results. for these basic concepts and notations, we follow the books of kilbas et al. [23] and samko et al. [31]. the riemann-liouville fractional integrals i ξ a+ g and i ξ b− g of order ξ ∈ c where re(ξ) > 0 are defined by (4.4) (i ξ a+ g)(u) = 1 γ(ξ) ∫ u a g(t) (u− t)1−ξ dt and (4.5) (i ξ b− g)(u) = 1 γ(ξ) ∫ b u g(t) (u− t)1−ξ dt, c© agt, upv, 2019 appl. gen. topol. 20, no. 2 462 existence results of delay and fractional differential equations where γ(.) is the gamma function. the integrals (4.4) and (4.5) are called the left-sided and right-sided fractional integrals, respectively. the riemann-liouville fractional derivatives d ξ a+ v, d ξ b− v of order ξ ∈ c are defined by (d ξ a+ v)(u) = ( d du )n ( i n−ξ a+ ) (u) = 1 γ(n− ξ) ( d du )n ∫ u a v(t) (u− t)ξ−n+1 dt (n = [re(ξ)] + 1) and (d ξ b− v)(u) = ( − d du )n ( i n−ξ b− ) (u) = 1 γ(n− ξ) ( − d du )n ∫ b u v(t) (u− t)ξ−n+1 dt (n = [re(ξ)] + 1), respectively, where [re(ξ)] means the integral part of re(ξ). we denote by lp(a,b), where 1 ≤ p ≤∞, the set of all lebesgue complex-valued measurable functions g on ω for which ‖g‖p < ∞ with ‖g‖p = (∫ ω |f(t)|p dt )1 p and ‖g‖∞ = esssup a≤u≤b |g(u)|. the following result shows that fractional differentiation is an operator inverse to the fractional integral operator from the left. lemma 4.1 ([31]). if re(ξ) > 0 and g(u) ∈ lp(a,b), where 1 ≤ p ≤∞, then the following equalities( d ξ a+ i ξ a+ g ) (u) = g(u) and ( d ξ b− i ξ b− g ) (u) = g(u), hold almost everywhere on [a,b]. lemma 4.2 ([31]). the fractional integral operator i ξ a+ with ξ > 0 is bounded in l(a,b) satisfying ‖iξ a+ z‖1 ≤ (b−a)ξ γ(ξ + 1) ‖z‖1. lemma 4.3 ([23]). let ξ ∈ c and n− 1 < re(ξ) < n (n ∈ n). let h be an open set in c and g : [a,b] ×h −→ c be a function such that g(u,v) ∈ l(a,b) for any v ∈ h. if v(u) ∈ l(a,b), then v(u) satisfies almost every where the riemann-liouville cauchy type problem (4.1)-(4.2) if and only if v(u) satisfies the integral equation (4.6) v(u) = n∑ i=1 di γ(ξ − i + 1) (u−a)ξ−i + 1 γ(ξ) ∫ u a g(t,v(t)) (u− t)1−ξ dt c© agt, upv, 2019 appl. gen. topol. 20, no. 2 463 r. tabassum, a. azam and m. s. shagari our main result of this section runs as follows: theorem 4.4. let h be an open set in c and g1,g2 : [a,b] × h −→ c be functions such that g1(u,v), g2 (u,v) ∈ l(a,b) = u, where l(a,b) is the set of all lebesgue complex-valued measurable functions on [a,b] endowed with the metric ρ : u ×u −→ r defined as ρ(v1,v2) = ‖v1 −v2‖ = ∫ u a |v1(u) −v2(u)|du, for all v1,v2 ∈ u and a 0 such that ‖g1(u,v1) −g2(u,v2)‖≤ %‖v1 −v2‖. thus, the system of riemann-liouville cauchy type problems (srlctps) given by (4.7) (d ξ a+ v)(u) = g1(u,v(u)), with initial conditions (4.8) (d ξ−i a+ v)(a+) = di, di ∈ c (i = 1, 2, 3, · · ·n) and (4.9) (d ξ a+ v)(u) = g2(u,v(u)), with initial conditions (4.10) (d ξ−i a+ v)(a+) = di, di ∈ c (i = 1, 2, 3, · · ·n), have a common solution in l(a,b). proof. by lemma 4.3, the common solution of (4.7)-(4.10) is also the common solution of their integral reformulation, respectively given as: (4.11) v(u) = n∑ i=1 di γ(ξ − i + 1) (u−a)λ−i + 1 γ(ξ) ∫ u a g1(t,v(t)) (u− t)1−ξ dt (4.12) v(u) = n∑ i=1 di γ(ξ − i + 1) (u−a)λ−i + 1 γ(ξ) ∫ u a g2(t,v(t)) (u− t)1−ξ dt clearly, the set u equipped with the given metric ρ is a complete metric space. let r,s : u −→ (0, 1] be any two arbitrary mappings and φ : [0,∞) −→ [0,∞) be a continuous non-decreasing function. choose u1 ∈ (a,b) such that % (u1 −a)ξ γ(ξ + 1) ≤ ρ(v1,v2) −ϕ(ρ(v1,v2)) 1 + ρ(v1,v2) . for v ∈ u, we have ωv(t) = v0(t) + 1 γ(ξ) ∫ u a g1(t,v(t)) (u− t)1−ξ dt c© agt, upv, 2019 appl. gen. topol. 20, no. 2 464 existence results of delay and fractional differential equations and τv(t) = v0(t) + 1 γ(ξ) ∫ u a g2(t,v(t)) (u− t)1−ξ dt, where v0(t) = n∑ i=1 di γ(ξ − i + 1) (u−a)ξ−i. consider a pair of intuitionistic fuzzy mappings f,g : u −→ (ifs)u defined as follows: µf(v)(q) = { r(v), q(t) = ωv(t), t ∈ [a,b] 0, q(t) 6= ωv(t), υf(v)(q) = { 0, q(t) = ωv(t), t ∈ [a,b] r(v), q(t) 6= ωv(t) and µg(v)(q) = { s(v), q(t) = τv(t), t ∈ [a,b] 0, q(t) 6= τv(t), υg(v)(q) = { 0, q(t) = τv(t), t ∈ [a,b] s(v), q(t) 6= τv(t). if we take αf(v) = r(v) and αg(v) = s(v), then we have [f(v)](t ,n,αf (v)) = { q ∈ u : t ( µf(v)(q),n ( υf(v)(q) )) = r(v) } = {ωv} and [g(v)](t ,n,αg(v)) = { q ∈ u : t ( µg(v)(q),n ( υg(v)(q) )) = s(v) } = {τv}. therefore, for v1,v2 ∈ u, we obtain [f(v1)](t ,n,αf (v1)) = {ωv1} and [g(v2)](t ,n,αg(v2)) = {τv2}. consequently, ρh ( [f(v1)](t ,n,αf (v1)) , [g(v2)](t ,n,αg(v2)) ) = ‖ωv1 − τv2‖1. for the remaining steps, we employ a standard method for nonlinear volterra integral equations of the proof of result on a subinterval of [a,b], (see, [25, 23]). notice that equations (4.11)-(4.12) are valid in any interval [a,u1] ⊂ [a,b] for a < u1 < b. thus, for an interval [a,u1], a metric ρ : l(a,u1) ×l(a,u1) −→ r is defined by ρ(v1,v2) = ‖v1 −v2‖1 = ∫ u1 a |v1(u) −v2(u)|du. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 465 r. tabassum, a. azam and m. s. shagari since f, g ∈ l(a,b), therefore, by lemma 4.2, u1 = b and z = g1(u,v1) − g2(u,v2), we have ρh ( [f(v1)](t ,n,αf (v1)) , [g(v2)](t ,n,αg(v2)) ) = ‖ωv1 − τv2‖1 = ∥∥∥∥ 1γ(ξ) ∫ u1 a g1(t,v1(t)) (u− t)1−ξ dt− ∫ u1 a g2(t,v2(t)) (u− t)1−ξ dt ∥∥∥∥ 1 ≤ ∥∥∥∥ 1γ(ξ) ∫ u1 a [g1(t,v1) −g2(t,v2)] (u− t)1−ξ dt ∥∥∥∥ 1 ≤ ∥∥∥iξa+ [g1(t,v1) −g2(t,v2)]∥∥∥ 1 ≤ (u1 −a)ξ γ(ξ + 1) ‖g1(t,v1) −g2(t,v2)‖1 ≤ % (u1 −a)ξ γ(ξ + 1) ρ(v1,v2) ≤ % (u1 −a)ξ γ(ξ + 1) (1 + ρ(v1,v2)) ≤ ρ(v1,v2) −ϕ(ρ(v1,v2)). hence, by theorem 3.1, there exists a common solution v∗ ∈ l(a,u1) to the volterra integral equations (4.11)-(4.12) in the interval [a,u1]. next, consider the interval [u1,u2], where u2 = u1 + ζ1 and ζ1 > 0 are such that u2 < b. rewrite equations (4.11)-(4.12) as follows: v(u) = 1 γ(ξ) ∫ u u1 g1(t,v(t)) (u− t)1−ξ dt + n∑ i=1 di γ(ξ − i + 1) (u−a)ξ−i + 1 γ(ξ) ∫ u1 a g1(t,v(t)) (u− t)1−ξ dt.(4.13) v(u) = 1 γ(ξ) ∫ u u1 g2(t,v(t)) (u− t)1−ξ dt + n∑ i=1 di γ(ξ − i + 1) (u−a)ξ−i + 1 γ(ξ) ∫ u1 a g2(t,v(t)) (u− t)1−ξ dt.(4.14) again, equations (4.13)-(4.14) can be rewrriten as v(u) = v01(u) + 1 γ(ξ) ∫ u u1 g1(t,v(t)) (u− t)1−ξ dt, v(u) = v01(u) + 1 γ(ξ) ∫ u u1 g2(t,v(t)) (u− t)1−ξ dt, c© agt, upv, 2019 appl. gen. topol. 20, no. 2 466 existence results of delay and fractional differential equations where v01(u) = n∑ i=1 di γ(ξ − i + 1) (u−a)ξ−i + 1 γ(ξ) ∫ u1 a g1(t,v(t)) (u− t)1−ξ = n∑ i=1 di γ(ξ − i + 1) (u−a)ξ−i + 1 γ(ξ) ∫ u1 a g2(t,v(t)) (u− t)1−ξ (4.15) are the known functions. the idea of (4.15) is to ignore the previous interval [a,u1] for which a solution is known. next, by re-considering any two arbitrary mappings r, s : u −→ (0, 1], a pair of intuitionistic fuzzy mappings f,g : u −→ (ifs)u and a nondecreasing continuous function φ : [0,∞) −→ [0,∞) such that % (u2 −a)ξ γ(ξ + 1) ≤ ρ(v2,v3) −φ(ρ(v2,v3)) 1 + ρ(v2,v3) (v2,v3 ∈ l(a,b), a < u2 < b) . so, one can obtain ρh ( [f(v2)](t ,n,αf (v2)) , [g(v3)](t ,n,αg(v3)) ) ≤ ρ(v2,v3) −ϕ(ρ(v2,v3)). again, theorem 3.1 can be applied to find a solution v∗(u) ∈ l(u1,u2) to the integral equations (4.11)-(4.12) on the interval [u1,u2]. by repeating this procedure inductively on the intervals [u2,u3], · · · , [un,un+1], where un+1 = un+ζn and ζn > 0 are such that un+1 < b, therefore, we can conclude according to theorem 3.1 that there exists a common solution v(u) = v∗(u) ∈ l(a,b) to the riemann-liouville cauchy type problems (4.7)-(4.10) on the interval [a,b]. � remark 4.5. the result of theorem 4.4 only gives existence conditions for the riemann-liouville cauchy type problem (4.7)-(4.8) and its equivalent integral equation (4.11) in the space l(a,b) for ξ ∈ c and n− 1 < re(ξ) < n (n ∈ n). the case of the problem (4.7)-(4.8) for order ξ = n+im, (n ∈ n,m ∈ r,m 6= 0) may be considered in due course. conclusion in the framework of if-sets, we have established a common fixed point theorem using weakly contractive condition for a pair of intuitionistic fuzzy mappings in the context of (t ,n ,α)−cut set of an if-set in a complete metric space. moreover, in our research work, we have constructed the iterations to establish the fixed point of intuitionistic fuzzy mappings. by building on the constructive approach, one will be able to define a procedure for obtaining the solution of certain functional equations arising in dynamical systems. on other hand, there is a rich variety of dynamics with multifaceted mathematical structures such as industrial control devices and systems handling imprecise information. therefore, the knowledge of cut sets of an if-set is beneficial to handle such uncertain and imprecise informations and processes, because these sets can transform an if-set into a crisp set. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 467 r. tabassum, a. azam and m. s. shagari as an application, we have investigated the existence of solution of time dependent delay differential equations with constant delay and riemann-liouville cauchy type fractional differential equations, which involve completeness property of function spaces. moreover, an example has been given to support the validity of existence theorem of the considered delay 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[39] l. a. zadeh, fuzzy sets, information and control 8, no. 3 (1965), 338–353. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 469 @ appl. gen. topol. 21, no. 2 (2020), 171-176 doi:10.4995/agt.2020.10129 c© agt, upv, 2020 structure of symmetry group of some composite links and some applications yang liu∗ shenzhen technology university, shenzhen, china (liuyang2@sztu.edu.cn) communicated by f. lin abstract in this paper, we study the symmetry group of a type of composite topological links, such as 221m!2 2 1. we have done a complete analysis on the elements of the symmetric group of this link and show the structure of the group. the results can be generalized to the study of the symmetry group of any composite topological link, and therefore it can be used for the classification of composite topological links, which can also be potentially used to identify synthetics molecules. 2010 msc: 57q45; 57m25; 20b30; 20b35; 51h05. keywords: knot; link; geometric topology; symmetry group; classification of links. 1. introduction in topology, geometry, and physics, various knots, which are mathematically various embeddings of a circle in the 3-dimensional euclidean space, have been interesting objects, which have been studied in recent decades (see for instance, [12], [3], and [10]). in particular, knots have been used to construct examples for the study of low-dimensional topology (see for instance, [9]). two or more knots can make up of a link, which have appeared to be somewhat more interesting as a single knot, because of the combinatorial structure involved. the theory of knots and links has applications in many areas such as physics, biochemistry, ∗this work is partially supported by shenzhen municipal finance for research. received 13 september 2019 – accepted 22 november 2019 http://dx.doi.org/10.4995/agt.2020.10129 y. liu and biology, in particular, dna and enzyme action (see for instance, [19], [14], and [7]). the structure of the symmetry group has become important information to understand the geometrical, physical properties of knots and links, as well as the enumeration of knots and links (see, for instance, [5] and [6]). as shown in [1], the knots are algebraic, and the symmetry of knots has been one of the interesting topics presented in [18]. in this article, we show heuristically that all composite topological links are actually also algebraic, in particular, the composite link, which is the knot sum of the hopf link and its mirror image, denoted as 221m!2 2 1, has a symmetry group. the main contribution of this paper is that we show that the symmetry group of the composite link 221m!2 2 1 is (1.1) z2 × z2 × z2 = {1, α} × {1, β} × {1, γ} , where (1.2) α = (1, 1, −1, −1, (2, 3)) , (1.3) β = (1, −1, −1, −1, e), and (1.4) γ = (−1, −1, 1, 1, e). the results can be generalized in the study of the symmetry group of any composite topological link, and so it can be used for the classification of composite topological links. this paper is structured as follows: in section 2, we show the classification of symmetries of the link 221m!2 2 1; and in section 3, we analyze the structure of the symmetry group from the perspective of algebraic group and prove our main theorem. 2. classifications let us compute the compatible (p, r) permutations first. since the compatible permutations just determine which component of each link is connected, the compatible permutations are the same with the case 221!2 2 1. so the compatible permutations are p = (2, 3), r = (1, 2) and p = e, r = e. now, let’s take the next step, by which we can find p̄1 and p̄2, and indeed, p̄1 = p̄2 = e. knowing the fact that 221m and 2 2 1 are in different cosets and that 2 2 1 has the symmetry group (2.1) 〈 (1, −1, −1, e), (−1, 1, −1, e), (1, 1, 1, (1, 2) 〉 , we can see that there are 16 cases, as follows, to to considered: (1) γ = (1, 1, 1, 1, (2, 3)); γ1 = (1, 1, 1, e); γ2 = (1, 1, 1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 172 structure of symmetry group of some composite links and some applications (2) γ = (1, 1, 1, −1, (2, 3)); γ1 = (1, 1, 1, e); γ2 = (1, 1, −1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. (3) γ = (1, 1, −1, 1, (2, 3)); γ1 = (1, 1, −1, e); γ2 = (1, 1, 1, e). since l γ1 2 is in the coset of l1, but l γ2 1 is not in the coset of l2, this case is not in the symmetry group of 221m!2 2 1. (4) γ = (1, 1, −1, −1, (2, 3)); γ1 = (1, 1, −1, e); γ2 = (1, 1, −1, e). since l γ1 2 is in the coset of l1, and l γ2 1 is in the coset of l2, this case is in the symmetry group of 221m!2 2 1. (5) γ = (1, −1, 1, 1, (2, 3)); γ1 = (1, −1, 1, e); γ2 = (1, −1, 1, e). since l γ1 2 is in the coset of l1, and l γ2 1 is in the coset of l2, this case is in the symmetry group of 221m!2 2 1. (6) γ = (1, −1, 1, −1, (2, 3)); γ1 = (1, −1, 1, e); γ2 = (1, −1, −1, e). since l γ2 1 is not in the coset of l2, this case is not in the symmetry group of 221m!2 2 1. (7) γ = (1, −1, −1, 1, (2, 3)); γ1 = (1, −1, −1, e); γ2 = (1, −1, 1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. (8) γ = (1, −1, −1, −1, (2, 3)); γ1 = (1, −1, −1, e); γ2 = (1, −1, −1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. the next 8 cases of mirror image are the followings: (9) γ = (−1, 1, 1, 1, (2, 3)); γ1 = (−1, 1, 1, e); γ2 = (−1, 1, 1, e). since l γ1 2 is in the coset of l1, and and l γ2 1 is in the coset of l2, this case is in the symmetry group of 221m!2 2 1. (10) γ = (−1, 1, 1, −1, (2, 3)); γ1 = (−1, 1, 1, e); γ2 = (−1, 1, −1, e). since l γ2 1 is not in the coset of l2, this case is not in the symmetry group of 221m!2 2 1. (11) γ = (−1, 1, −1, 1, (2, 3)); γ1 = (−1, 1, −1, e); γ2 = (−1, 1, 1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. (12) γ = (−1, 1, −1, −1, (2, 3)); γ1 = (−1, 1, −1, e); γ2 = (−1, 1, −1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. (13) γ = (−1, −1, 1, 1, (2, 3)); γ1 = (−1, −1, 1, e); γ2 = (−1, −1, 1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. (14) γ = (−1, −1, 1, −1, (2, 3)); γ1 = (−1, −1, 1, e); γ2 = (−1, −1, −1, e). since l γ1 2 is not in the coset of l1, this case is not in the symmetry group of 221m!2 2 1. (15) γ = (−1, −1, −1, 1, (2, 3)); γ1 = (−1, −1, −1, e); γ2 = (−1, −1, 1, e). since l γ2 1 is not in the coset of l2, this case is not in the symmetry group of 221m!2 2 1. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 173 y. liu (16) γ = (−1, −1, −1, −1, (2, 3)); γ1 = (−1, −1, −1, e); γ2 = (−1, −1, −1, e). since l γ1 2 is in the coset of l1, and and l γ2 1 is in the coset of l2, this case is in the symmetry group of 221m!2 2 1. in summary, the set of elements involving (2, 3) as the permutation in the symmetry group is (2.2) s1 = {(1, 1, −1, −1, (2, 3)), (1, −1, 1, 1, (2, 3)), (−1, 1, 1, 1, (2, 3)), (−1, −1, −1, −1, (2, 3))} . for the other compatible permutation p = r = e, we can compare l γ1 1 and l1, as well as l γ2 2 and l2 , then we obtain the following set of elements in the symmetry group, (2.3) s2 = {(1, 1, 1, 1, e), (1, −1, −1, −1, e), (−1, −1, 1, 1, e), (−1, 1, −1, −1, e)} . 3. structure analysis and theorem in this section, we analyze the structure of the symmetry group with the multiplication operation of the group and have the following theorem. theorem 3.1. the symmetry group of the composite link 221m!2 2 1 is isomorphic to z2 × z2 × z2. proof. let α = (1, 1, −1, −1, (2, 3)), β = (1, −1, −1, −1, e), γ = (−1, −1, 1, 1, e), δ = (−1, 1, −1, −1, e), and the unit element 1 = (1, 1, 1, 1, e), we have (3.1) (1, −1, 1, 1, (2, 3)) = αβ, (3.2) (−1, 1, 1, 1, (2, 3)) = αδ, and (3.3) (−1, −1, −1, −1, (2, 3)) = αγ. noticing that δ = βγ, we now have the symmetry group (3.4) g = 〈 1, α, β, γ 〉 . since any other element in g than the identity is of order 2, then we know that g is abelian. therefore, g is an abelian group of order 8. by the fundamental theorem of finitely generated abelian group (see for instance [8]), the structure of g is z8, z2 × z4, or z2 × z2 × z2. but since any other element in g than the identity has an order 2, the structure of g must be z2 × z2 × z2. hence, (3.5) g = {1, α} × {1, β} × {1, γ} ∼= z2 × z2 × z2 where α = (1, 1, −1, −1, (2, 3), β = (1, −1, −1, −1, e), and γ = (−1, −1, 1, 1, e). □ so it turns out that the structure of the symmetry group of the composite link 221m!2 2 1 is the same as the composite link 2 2 1!2 2 1, but the symmetry group of 221m!2 2 1has different elements in its symmetry group. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 174 structure of symmetry group of some composite links and some applications remark 3.2. topological knots and links were studied by using integral geometry, on which one can refer to [13], but other theoretic work of integral geometry such as [2], [16], [4], [15], and [17], may also be used to study knots and links. on the other hand, the fundamental theorem of finitely generated abelian group, in the case if the group is abelian, the classification of finite simple groups (see for instance [11]), and other algebraic theories, can be used to determine the structure of the symmetry group. another remark about the applications of the symmetry groups of knots we would like to make is remark 3.3. in some physical movements or processes of dna, the group structure of the double helix strands of dna is invariant, and therefore, it can be used to track these movements or processes. furthermore, the chirality of synthetics molecules (see for instance, [20]), which should be induced by the symmetry groups, can be used to identify synthetics molecules and can be potentially applied to the testing on virus infections, which might potentially help with the control on diseases, in particular, the infectious disease, covid19, in the recent pandemic. figure 1. discovered dna knot (c.f. 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[21] s. a. wasserman, j. m. dungan and n. r. cozzarelli, discovery of a predicted dna knot substantiates a model for site-specific recombination, science 229, no. 4709 (1985), 171–174. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 176 @ appl. gen. topol. 23, no. 1 (2022), 69-77 doi:10.4995/agt.2022.15893 © agt, upv, 2022 beyond the hausdorff metric in digital topology laurence boxer department of computer and information sciences, niagara university, usa; and department of computer science and engineering, state university of new york at buffalo, usa (boxer@niagara.edu) communicated by j. rodŕıguez-lópez abstract two objects may be close in the hausdorff metric, yet have very different geometric and topological properties. we examine other methods of comparing digital images such that objects close in each of these measures have some similar geometric or topological property. such measures may be combined with the hausdorff metric to yield a metric in which close images are similar with respect to multiple properties. 2020 msc: 54b20. keywords: digital topology; digital image; hausdorff metric. 1. introduction a key question in digital image processing is whether two digital images a and b represent the same object. if, after magnification or shrinking and translation, copies a′ and b′ of the respective images have been scaled to approximately the same size and are located in approximately the same position, a hausdorff metric h may be employed: if h(a′,b′) is small, then perhaps a and b represent the same object; if h(a′,b′) is large, then a and b probably do not represent the same object. however, the hausdorff metric is very crude as a measure of similarity. in this paper, we consider other comparisons of digital images. received 06 july 2021 – accepted 27 september 2021 http://dx.doi.org/10.4995/agt.2022.15893 l. boxer 2. preliminaries much of this section is quoted or paraphrased from [10]. we use n to indicate the set of natural numbers, z for the set of integers, and r for the set of real numbers. 2.1. adjacencies. a digital image is a graph (x,κ), where x is a subset of zn for some positive integer n, and κ is an adjacency relation for the points of x. the cu-adjacencies are commonly used. let x,y ∈ zn, x 6= y, where we consider these points as n-tuples of integers: x = (x1, . . . ,xn), y = (y1, . . . ,yn). let u ∈ z, 1 ≤ u ≤ n. we say x and y are cu-adjacent if • there are at most u indices i for which |xi −yi| = 1. • for all indices j such that |xj −yj| 6= 1 we have xj = yj. often, a cu-adjacency is denoted by the number of points adjacent to a given point in zn using this adjacency. e.g., • in z1, c1-adjacency is 2-adjacency. • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency. • in z3, c1-adjacency is 6-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. we write x ↔κ x′, or x ↔ x′ when κ is understood, to indicate that x and x′ are κ-adjacent. similarly, we write x -κ x′, or x x′ when κ is understood, to indicate that x and x′ are κ-adjacent or equal. a subset y of a digital image (x,κ) is κ-connected [17], or connected when κ is understood, if for every pair of points a,b ∈ y there exists a sequence {yi}mi=0 ⊂ y such that a = y0, b = ym, and yi ↔κ yi+1 for 0 ≤ i < m. 2.2. digitally continuous functions. the following generalizes a definition of [17]. definition 2.1 ([5]). let (x,κ) and (y,λ) be digital images. a single-valued function f : x → y is (κ,λ)-continuous if for every κ-connected a ⊂ x we have that f(a) is a λ-connected subset of y . when the adjacency relations are understood, we will simply say that f is continuous. continuity can be expressed in terms of adjacency of points: theorem 2.2 ([17, 5]). a function f : x → y is continuous if and only if x ↔ x′ in x implies f(x) f(x′). see also [11, 12], where similar notions are referred to as immersions, gradually varied operators, and gradually varied mappings. 2.3. pseudometrics and metrics. definition 2.3 ([13]). let x be a nonempty set. let d : x2 → [0,∞) be a function such that for all x,y,z ∈ x, • d(x,y) ≥ 0; © agt, upv, 2022 appl. gen. topol. 23, no. 1 70 beyond the hausdorff metric in digital topology • d(x,x) = 0; • d(x,y) = d(y,x); and • d(x,z) ≤ d(x,y) + d(y,z). then d is a pseudometric for x. if, further, d(x,y) = 0 implies x = y then d is a metric for x. pseudometrics that can be applied to pairs (a,b) of nonempty subsets of a digital image x include the absolute values of the differences in their • deviations from convexity. several such deviations are discussed in [19, 4], for each of which it was shown that two objects can be “close” in the hausdorff metric yet quite different with respect to the deviation from convexity. these can be adapted to digital images with respect to digital convexity as defined in [7]. • euler characteristics. i.e., the function sχ(a,b) = |χ(a) −χ(b)|, where χ(x) is the euler characteristic of (x,κ), is a pseudometric for digital images in zn. an improper definition of the euler characteristic for digital images was given in [15]. an appropriate definition is given in [8]. • lusternik-schnirelman category catκ(x) [1]. i.e., the function sls,κ(a,b) = |catκ(a) − catκ(b)|, where catκ(x) is the lusternik-schnirelman category of (x,κ), is a pseudometric for digital images in zn. • diameters. this is discussed below. the following is easily verified and extends an assertion of [4]. lemma 2.4. let ∆i : x 2 → [0,∞) be a pseudometric, 1 ≤ i ≤ n. then d = ∑n i=1 ∆i : x 2 → [0,∞) is a pseudometric. further, if at least one of the ∆i is a metric, then d is a metric. here we mention metrics we use in this paper for rn or zn. let x = (x1, . . . ,xn), y = (y1, . . . ,yn). • let p ≥ 1. the `p metric for rn is given by dp(x,y) = ( n∑ i=1 |xi −yi|p )1/p . the special case p = 1 gives the manhattan or city block metric d1 : (rn)2 → [0,∞), given by d1(x,y) = n∑ i=1 |xi −yi|. © agt, upv, 2022 appl. gen. topol. 23, no. 1 71 l. boxer the special case p = 2 gives the euclidean metric d2 : (rn)2 → [0,∞), given by d2(x,y) = ( n∑ i=1 (xi −yi)2 )1/2 . • the shortest path metric [14]: let (x,κ) be a connected digital image. for x,y ∈ x, let dκ(x,y) = min{n | there is a κ-path of length n in x from x to y.} • the hausdorff metric based on a metric d [16]: let d : x2 → [0,∞) be a metric where x ⊂ rn. the hausdorff metric for nonempty bounded and closed subsets a and b of x (hence, in the case x ⊂ zn, finite subsets of x) based on d is h(a,b) = min { ε > 0 | ∀(a,b) ∈ a×b, ∃ (a′,b′) ∈ a×b such that ε ≥ d(a,b′) and ε ≥ d(a′,b) } . we make the following modification of the hausdorff metric based on dκ as presented in [20]. definition 2.5. let x ⊂ zn, ∅ 6= a ⊂ x, ∅ 6= b ⊂ x. let κ be an adjacency on x. then h(x,κ)(a,b) = min   ε ≥ 0 | ∀(a,b) ∈ a×b, ∃(a′,b′) ∈ a×b such that there are κ-paths in x of length ≤ ε from a to b′ and from b to a′   . in the version of the hausdorff metric based on dκ in [20], x = zn. we show below that we can get very different results for the more general situation ∅ 6= x ⊂ zn. we use the notations hd for the hausdorff metric based on the metric d, hp for the hausdorff metric based on the `p metric dp (i.e., hp = hdp), and h(x,κ) for the hausdorff metric based on dκ for subsets of x (i.e., hκ = hdκ). another metric from classical topology that is easily adapted to digital topology is borsuk’s metric of continuity [2, 3] based on a metric d which is typically, but not necessarily, the euclidean metric. for digital images (x,κ) and (y,κ) in zn, define the metric of continuity δd(x,y ) as the greatest lower bound of numbers t > 0 such that there are κ-continuous f : x → y and g : y → x with d(x,f(x)) ≤ t for all x ∈ x and d(y,g(y)) ≤ t for all y ∈ y. proposition 2.6. given finite digital images (x,κ) and (y,κ) in zn and a metric d for zn, hd(x,y ) ≤ δd(x,y ). proof. this is largely the argument of the analogous assertion in [2]. let u = hd(x,y ). since x and y are finite, without loss of generality, there exists x0 ∈ x such that u = min{y ∈ y | d(x0,y)}. then for all κ-continuous f : x → y , d(x0,f(x0)) ≥ u. therefore, δd(x,y ) ≥ u. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 72 beyond the hausdorff metric in digital topology an example for which the inequality of proposition 2.6 is strict is given in the following. theorem 2.7. let x = {(x,y) ∈ z2 | |x| = n or |y| = n}. let y = x \ {(n,n)}. then, using the manhattan metric for d and κ = c1, we have h1(x,y ) = 1 and δd(x,y ) ≥ 2n− 1. proof. it is clear that h1(x,y ) = 1. notice there is an isomorphism f : (y,c1) to a subset of (z,c1). let f : x → y be c1-continuous. by [6], there is a pair of antipodal points p,−p ∈ x such that |f ◦ f(p) − f ◦ f(−p)| ≤ 1. since f is an isomorphism, we must have f(p) -c1 f(−p). we will show that either d(p,f(p)) ≥ 2n − 1 or d(−p,f(−p)) ≥ 2n− 1, as follows. if p = (n,u) then −p = (−n,−u). then: • if f(p) = (n,−n) then f(−p) ∈{(n−1,−n), (n,−n), (n,−n + 1)}, so d(−p,f(−p)) ≥ 2n− 1. • note (n,n) 6∈ y so f(p) cannot equal (n,n). • if f(p) = (n,v) for |v| < n then f(−p) ∈{(n,v−1), (n,v), (n,v + 1)}, so d(−p,f(−p)) ≥ 2n. the cases p = (−n,u), p = (w,n), and p = (w,−n) are similar. thus δd(x,y ) ≥ 2n− 1. � we say the diameter of a nonempty bounded set a ⊂ rn with respect to a metric d is diamd(a) = max{d(a,b) |a,b ∈ a}. we will use the notations diamp for diamdp, and diamκ for diamdκ. we define a function sd for pairs of nonempty bounded sets in rn by sd(a,b) = |diamd(a) −diamd(b)|. we use notations sp for sdp, and sκ for sdκ. the following is easily verified. lemma 2.8. the function sd is a pseudometric. 3. comparing (pseudo)metrics on digital images in this section, we compare the use of some of the (pseudo)metrics discussed above. theorem 3.1. let a and b be nonempty, bounded subsets of rn. let hp be the hausdorff metric based on the `p metric dp and suppose hp(a,b) ≤ m. then sp(a,b) ≤ 2m. proof. there exist a,a′ ∈ a such that dp(a,a′) = diamp(a). there exist b,b′ ∈ b such that dp(a,b) ≤ m and dp(a′,b′) ≤ m. so diamp(a) = dp(a,a ′) ≤ dp(a,b) + dp(b,b′) + dp(b′,a′) ≤ m + diamp(b) + m = diamp(b) + 2m. similarly, diamp(b) ≤ diamp(a) + 2m. the assertion follows. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 73 l. boxer figure 1. left: q = [0,n]2z (here, n = 6). right: s = q\ [ ( ⋃ k∈z{4k + 1}× [1,n]z) ∪ ( ⋃ k∈z{4k + 3}× [0,n− 1]z) ] (here, n = 6). q and s are within 1 with respect to the hausdorff metric based on the manhattan metric; however, they differ considerably with respect to diameter in the shortest path metric. by contrast, we have the following. example 3.2. let n ∈ n such that n is even. let q = [0,n]2z. let s = q\ [ ( ⋃ k∈z {4k + 1}× [1,n]z) ∪ ( ⋃ k∈z {4k + 3}× [0,n− 1]z) ] . (see figure 1.) then s1(q,s) = 0, but while diamc1 (q) = 2n, we have diamc1 (s) = n + n(1 + n/2). thus sc1 (q,s) = n 2/2. proof. it is easy to see that both q and s have diagonally opposed points that are maximally distant in the d1 metric. therefore, diam1(s) = diam1(q) = 2n, so s1(q,s) = 0. diagonally opposed points of q are maximally separated with respect to dc1 , so diamc1 (q) = 2n. maximally separated points of s with respect to dc1 are (0,n) and (n,n) if n = 4k + 2 for some k ∈ z; (0,n) and (n, 0) if n = 4k for some k ∈ z. in either case, the unique shortest c1-path between maximally separated points requires n horizontal steps. the number of vertical steps is computed as follows. there are 1 + n/2 vertical line segments that must be traversed, each of length n, so the number of vertical steps is n(1 + n/2). thus the number of steps between maximally separated members of s is diamc1 (s) = n + n(1 + n/2). hence for κ = c1 we have sκ(q,s) = |n + n(1 + n/2) − 2n| = n2/2. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 74 beyond the hausdorff metric in digital topology figure 2. digital images a (left) and b (right) for example 3.3, using n = 5. using the shortest path metric and either κ = c1 or κ = c2, maximally distant points in a are (0, 0) and (n, 2), and maximally distant points in b are (n, 0) and (n, 2). we do not get an analog of theorem 3.1 by using the hausdorff metric based on an adjacency κ instead of hp. this is shown in the following example. example 3.3. let a = [0,n]z × [0, 2]z. let b = a \ ([1,n]z ×{1}). (see figure 2.) then h1(a,b) = 1. however, we have the following. • for κ = c1, diamκ(a) = n+2 and diamκ(b) = 2n+2, so sκ(a,b) = n. • for κ = c2, diamκ(a) = n and diamκ(b) = 2n, so sκ(a,b) = n. theorem 3.4. let a and b be finite, nonempty cu-connected subsets of a cuconnected subset x of zn, where 1 ≤ u ≤ n. suppose we have h(x,cu)(a,b) ≤ m for some m ∈ n. then hp(a,b) ≤ mu1/p. proof. by hypothesis, given x ∈ a and y ∈ b, there exist x′ ∈ a, y′ ∈ b, and cu-paths p from x to y ′ and q from y to x′ in x such that each of p and q has length of at most m. since each cu-adjacency corresponds to a euclidean distance of at most u1/p, it follows that dp(x,y ′) ≤ mu1/p and dp(y,x ′) ≤ mu1/p. it follows that hp(x,y ) ≤ mu1/p. � we do not get a converse for theorem 3.4, as the following shows. example 3.5. let b = [0,n]z × [0, 2]z \ ([1,n]z ×{1}) as in example 3.3. (see figure 2.) let c = [0,n]z ×{0} ⊂ b. then h1(b,c) = h2(b,c) = 2. however, h(b,c1)(b,c) = n + 2 and h(b,c2)(b,c) = n + 1. proof. it is easy to see that h1(b,c) = h2(b,c) = 2. since c ⊂ b, finding a hausdorff distance between b and c comes down to considering a furthest point of b from c. with respect to κ = c1 and also with respect to κ = c2, the furthest point of b from c in the shortest path metric is b = (n, 2) and its closest point of c is c = (0, 0). since dc1 (b,c) = n + 2 and dc2 (b,c) = n + 1, the assertion follows. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 75 l. boxer roughly, it appears that the great differences found in examples 3.3 and 3.5, between measures based in `p metrics and measures based on the shortest path metric, are due to significant deviations from convexity. if we consider h(x,ci)(a,b) for a set x such as a digital cube, we may find hp and h(x,cp) are more alike, as we see below. proposition 3.6. let a 6= ∅ 6= b, a∪b ⊂ j = [0,m]2z. then h1(a,b) = h(j,c1)(a,b). proof. let n = h1(a,b). let x ∈ a. then there exists y ∈ b such that d1(x,y) ≤ n. by definition of d1, it follows that there is a c1-path in j of length at most n from x to y. similarly, given u in b, there is a c1-path in j of length at most n from u to a point v ∈ a. therefore, h(j,c1)(a,b) ≤ n = h1(a,b). now let n = h(j,c1)(a,b). then given x ∈ a, there is a c1-path in j of length at most n from x to some y ∈ b. similarly, given u ∈ b, there is a c1path in j of length at most n from u to some v ∈ a. since every c1 adjacency represents a d1 distance of 1, it follows that d1(x,y) ≤ n and d1(u,v) ≤ n. thus h1(a,b) ≤ n = h(j,c1)(a,b). the assertion follows. � using the observation that a cu-adjacency in zr, 1 ≤ u ≤ r, represents a dp distance between the adjacent points that is between 1 and u 1/p, we can generalize the argument used to prove proposition 3.6, getting the following. theorem 3.7. let a 6= ∅ 6= b, a ∪ b ⊂ j = [0,m]vz. then for 1 ≤ u ≤ v, h(j,cu)(a,b) ≤ u 1/p ·h(j,cu)(a,b). 4. further remarks the hausdorff metric is often used to compare objects a and b. it is easy to compute efficiently [18, 9] and gives a good indication of how well each of its arguments approximates the other with respect to location. however, two objects may be close in the hausdorff metric and yet have very different geometric or topological properties. lemma 2.4 tells us that by adding other pseudometrics or metrics, such as those we have discussed, to the hausdorff metric, we can get another metric in which closeness is more likely to validate the parameters as digital images representing the same physical object. acknowledgements. the suggestions and corrections of an anonymous reviewer are gratefully acknowledged. references [1] a. borat and t. vergili, digital lusternik-schnirelmann category, turkish journal of mathematics 42 (2018), 1845–1852. [2] k. borsuk, on some metrizations of the hyperspace of compact sets, fundamenta mathematicae 41 (1954), 168–202. © agt, upv, 2022 appl. gen. topol. 23, no. 1 76 beyond the hausdorff metric in digital topology [3] k. borsuk, theory of retracts, polish scientific publishers, warsaw, 1967. [4] l. boxer, computing deviations from convexity in polygons, pattern recognition letters 14 (1993), 163–167. [5] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62 . [6] l. boxer, continuous maps on digital simple closed curves, applied mathematics 1 (2010), 377–386. [7] l. boxer, convexity and freezing sets in digital topology, applied general topology 22, no. 1 (2021), 121–137. [8] l. boxer, i. karaca and a. oztel, topological invariants in digital images, journal of mathematical sciences: advances and applications 11, no. 2 (2011), 109–140. 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[20] t. vergili, digital hausdorff distance on a connected digital image, communications faculty of sciences university of ankara series a1 mathematics and statistics 69, no. 2 (2020), 76–88. © agt, upv, 2022 appl. gen. topol. 23, no. 1 77 @ appl. gen. topol. 22, no. 1 (2021), 183-192doi:10.4995/agt.2021.14542 © agt, upv, 2021 digital homotopic distance between digital functions ayşe borat bursa technical university, faculty of engineering and natural scieces, department of mathematics, bursa, turkey (ayse.borat@btu.edu.tr) communicated by v. gregori abstract in this paper, we define digital homotopic distance and give its relation with ls category of a digital function and of a digital image. moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance. 2010 msc: 55m30; 68u10. keywords: homotopic distance; lusternik schnirelmann category; digital topology. 1. introduction macias-virgos and mosquera-lois introduced homotopic distance between maps in [16]. one of the benefits of homotopic distance is to cover the concepts of lusternik-schnirelmann category (denoted by cat, see [8]) and topological complexity (denoted by tc, see [9]). if one computes the homotopic distance between some specific maps, then they end up with cat or tc of the domain of these maps. so there is a well-defined relation between homotopic distance, tc, cat and even the sectional category (secat) of some specific fibrations. we investigate an analog of this relationship in the digital setting. after constructing the digital homotopic distance between digital functions, one of our aims is to show the relation between digital homotopic distance and digital received 27 october 2020 – accepted 14 december 2020 http://dx.doi.org/10.4995/agt.2021.14542 a. borat ls category of a digital image as defined in [2] and digital ls cat of a digital function as defined in [18]. our another aim is to investigate how the adjacency relation effects the digital homotopic distance; see theorem 3.5 and theorem 3.6. 2. background in this section, we recall some definitions and theorems from digital topology. a digital image x is a subset of zn with an adjacency relation which is defined as follows. definition 2.1 ([6]). let p = (p1, . . . , pn) and q = (q1, . . . , qn) in zn. then for 1 ≤ ℓ ≤ n, p and q are said to be cℓ-adjacent if (i) there are at most ℓ indices i which satisfies |pi − qi| = 1 we would like to call attention to that digital ls category is also introduced with a different point of view using subdivisions by lupton, oprea, and scoville in [14] and [15]. (ii) pj = qj for all other indices j satisfying |pi − qi| 6= 1 here cℓ indicates the number of adjacent points in zn. for example, c1 = 2 in z; c1 = 4 and c2 = 8 in z2. also notice that adjacency relations are often denoted by greek letters. definition 2.2 ([4]). a digital interval which is a subset of z can be defined as follows [a, b]z = {n ∈ z|a ≤ n ≤ b} where 2-adjacency is assumed. definition 2.3 ([5]). let (x, κ) and (y, λ) be digital images. given a function f : x → y , if f(x) and f(y) are λ-adjacent or f(x) = f(y) in y whenever x and y are κ-adjacent in x, then f is called (κ, λ)-continuous. definition 2.4 ([5, 13]). let f, g : x → y be (κ, λ)-continuous functions. if there exist m ∈ z+ and a function f : x × [0, m]z → y with the following conditions, then f is called a (κ, λ)-homotopy, and f and g are called (κ, λ)-homotopic in y (denoted by f ≃κ,λ g). (i) for all x ∈ x, f(x, 0) = f(x) and f(x, m) = g(x). (ii) for all x ∈ x, the induced function fx : [0, m]z → y , fx(t) = f(x, t) is (2, λ)-continuous. (iii) for all t ∈ [0, m]z, the induced function ft : x → y , ft(x) = f(x, t) is (κ, λ)-continuous. proposition 2.5 ([5]). if f : x → y and g : y → z are (κ, λ)-continuous and (λ, γ)-continuous functions, respectively, then g ◦ f : x → z is (κ, γ)continuous. © agt, upv, 2021 appl. gen. topol. 22, no. 1 184 digital homotopic distance between digital functions definition 2.6 ([7]). if κ and λ are two adjacency relations on x, then we say κ dominates λ (denoted by κ ≥d λ) if for x, y ∈ x and if x, y are κ-adjacent imply x, y are λ-adjacent. proposition 2.7 ([7]). let κ, κ1, κ2 be adjacency relations on x and λ, λ1, λ2 be adjacency relations on y . (a) if f : x → y is (κ, λ1)-continuous and λ1 ≥d λ2, then f is (κ, λ2)continuous. (b) if f : x → y is (κ1, λ)-continuous and κ2 ≥d κ1, then f is (κ2, λ)continuous. definition 2.8 ([1, 17]). let (x, κ) and (y, λ) be digital images. two elements (x1, y1), (x2, y2) ∈ x × y are called np(κ, λ)-adjacent if either (i) x1 = x2 and y1, y2 are λ-adjacent or (ii) y1 = y2 and x1, x2 are κ-adjacent or (iii) x1, x2 are κ-adjacent and y1, y2 are λ-adjacent. definition 2.9. if a (κ, λ)-continuous function f : x → y is (κ, λ)-homotopic to a constant map c : x → y , c(x) = y0, then f is said to be (κ, λ)nullhomotopic. if f : x → x is (κ, κ)-nullhomotopic, then we omit one of the adjacency relations and simply write “κ-nullhomotopic”. 3. digital homotopic distance recall that a covering of a space x is a collection of subsets of x whose union is x. definition 3.1. let f, g : x → y be (κ, λ)-continuous functions. the (κ, λ) homotopic distance (so-called digital homotopic distance) between f and g is the least non-negative integer n such that there exists a covering u0, u1, . . . , un of the digital image x with the property f|ui ≃κ,λ g|ui for each i. it is denoted by dκ,λ(f, g). if there is no such covering, we define dκ,λ(f, g) = ∞. proposition 3.2. let f : (x, κ) → (y, λ) be continuous. if x is finite and κ-connected, then dκ,λ(f, g) < ∞. proof. let ux = {x}. then {ux |x ∈ x} is a finite covering of x. since x is κconnected, for each x ∈ x there is a (c1, κ)-continuous fx : [0, mx]z → x such that fx(0) = x and fx(mx) ∈ f−1(g(x)). then the function h : ux×[0, mx]z → y defined by h(x, t) = f(fx(t)) is a homotopy between f|ux and g|ux. it follows that dκ,λ(f, g) < ∞. � proposition 3.3. let f, g : x → y be (κ, λ)-continuous. the following properties hold. (a) dκ,λ(f, g) = dκ,λ(g, f). (b) dκ,λ(f, g) = 0 iff f ≃κ,λ g. © agt, upv, 2021 appl. gen. topol. 22, no. 1 185 a. borat proposition 3.4. let (x, κ) and (y, λ) be digital images. let f, f′, g, g′ : x → y be (κ, λ)-continuous functions. if f ≃κ,λ f′ and g ≃κ,λ g′ then dκ,λ(f, g) = dκ,λ(f ′, g′). proof. suppose dκ,λ(f′, g′) = n. then there exist subsets u0, u1, . . . , un covering (x, κ) such that f′|ui ≃κ,λ g ′|ui for all i. from the assumption f ≃κ,λ f′ and g ≃κ,λ g′, we have f|ui ≃κ,λ f ′|ui and g|ui ≃κ,λ g ′|ui for all i. therefore, for all i, we have f|ui ≃κ,λ f ′|ui ≃κ,λ g ′|ui ≃κ,λ g|ui. hence dκ,λ(f, g) ≤ n. the other way around can be proved similarly. thus we conclude that dκ,λ(f ′, g′) = dκ,λ(f, g). � the following theorems state how the adjacency relations in the domain and in the image affect the digital homotopic distance. theorem 3.5. let (x, κ), (y, λ) and (y, λ′) be digital images. let f, f′ : x → y be (κ, λ)-continuous and g, g′ : x → y (κ, λ′)-continuous functions. if f ≃κ,λ f ′, g ≃κ,λ′ g′ and λ′ ≥d λ, then dκ,λ(f, g) ≤ dκ,λ′(f′, g′). proof. suppose dκ,λ′(f′, g′) = n. then there exist subsets u0, u1, . . . , un covering (x, κ) such that f′|ui ≃κ,λ′ g ′|ui for all i. from the assumption f ≃κ,λ f′ and g ≃κ,λ′ g′, we have f|ui ≃κ,λ f ′|ui and g|ui ≃κ,λ′ g ′|ui for all i. on the other hand, since λ′ ≥d λ and g, g′ are (κ, λ′)-continuous, by proposition 2.7(a) g, g′ are (κ, λ)-continuous. moreover, (κ, λ′)-homotopies are also (κ, λ)-homotopies. hence, since g|ui ≃κ,λ′ g ′|ui and f ′|ui ≃κ,λ′ g ′|ui, thus g|ui ≃κ,λ g ′|ui and f ′|ui ≃κ,λ g ′|ui for all i. therefore, for all i, we have f|ui ≃κ,λ f ′|ui ≃κ,λ g ′|ui ≃κ,λ g|ui. it follows that f|ui ≃κ,λ g|ui for all i. so we have dκ,λ(f, g) ≤ n. thus we conclude that dκ,λ(f, g) ≤ dκ,λ′(f′, g′). � theorem 3.6. let (x, κ), (x, κ′) and (y, λ) be digital images. let f, f′ : x → y be (κ, λ)-continuous and g, g′ : x → y (κ′, λ)-continuous functions. if f ≃κ,λ f ′, g ≃κ′,λ g′ and κ′ ≥d κ, then dκ′,λ(f′, g′) ≤ dκ,λ(f, g). proof. suppose dκ,λ(f, g) = n. then there exist subsets u0, u1, . . . , un covering (x, κ) such that f|ui ≃κ,λ g|ui for all i. from the assumption f ≃κ,λ f′ and g ≃κ′,λ g′, we have f|ui ≃κ,λ f ′|ui and g|ui ≃κ′,λ g ′|ui for all i. on the other hand, since κ′ ≥d κ and f, f′ are (κ, λ)-continuous, by proposition 2.7(b) f, f′ are (κ′, λ)-continuous. moreover, (κ, λ)-homotopies are also (κ′, λ)-homotopies. hence, since f|ui ≃κ,λ f ′|ui and f|ui ≃κ,λ g|ui, it follows that f|ui ≃κ′,λ f ′|ui and f|ui ≃κ′,λ g|ui for all i. © agt, upv, 2021 appl. gen. topol. 22, no. 1 186 digital homotopic distance between digital functions therefore, for all i, we have f ′|ui ≃κ′,λ f|ui ≃κ′,λ g|ui ≃κ′,λ g ′|ui. it follows that f′|ui ≃κ,λ g ′|ui for all i. so we have dκ′,λ(f ′, g′) ≤ n. notice that ui’s also admit κ′-adjacency. thus we conclude that dκ′,λ(f′, g′) ≤ dκ,λ(f, g). � proposition 3.7. if f, g : x → y are (κ, λ)-continuous maps and if {u0, u1, . . . , un} is a finite covering of x, then we have dκ,λ(f, g) ≤ n ∑ i=0 dκ,λ(f|ui, g|ui) + n. proof. suppose dκ,λ(f|ui, g|ui) = mi for each i = 0, 1, . . . , n. so there exist u0i , u 1 i , . . . , u mi i covering (ui, κ) such that f|uj i ≃κ,λ g|uj i for all i, j. consider the collection u = {u00 , u 1 0 , . . . , u m0 0 , u 0 1 , u 1 1 , . . . , u m1 1 , . . . , u 0 n, u 1 n, . . . , u mn n }. notice that ⋃ v ∈u v = x. moreover, f|v ≃κ,λ g|v for all v ∈ u. hence dκ,λ(f, g) ≤ (m0 + 1) + (m1 + 1) + . . . + (mn + 1) = (m0 + m1 + . . . + mn) + n = n ∑ i=0 dκ,λ(f|ui, g|ui) + n. � 4. relations between the digital analogs of homotopic distance and cat in this section we introduce the relation between digital homotopic distance and digital ls category both of a digital image and a digital function. let us first recall the definition of digital ls categories. definition 4.1 ([2]). the digital ls category of a digital image (x, κ) is the least non-negative integer k such that there is a covering u0, u1, · · · , uk of x such that inclusion map ιi : ui →֒ x is digitally κ-nullhomotopic in x for each i = 0, 1, . . . , k. it is denoted by catκ(x) = k. definition 4.2 ([18]). let f : x → y be a (κ, λ)-continuous function. the digital ls category of f is the least non-negative integer k such that there is a covering {u0, . . . , uk} of x such that f|uj is (κ, λ)-nullhomotopic for each j = 0, 1, . . . , k. it is denoted by catκ,λ(f). by definition 4.2, if f : x → y is (κ, λ)-continuous and c : x → y is a digital constant function, then catκ,λ(f) = dκ,λ(f, c) provided x and y are κand λ-connected, respectively. by definition 4.1, catκ(x) = dκ,κ(id, c) where c : x → x is a digital constant map and x is κ-connected. © agt, upv, 2021 appl. gen. topol. 22, no. 1 187 a. borat example 4.3. consider the digital image ({0, 2}, c1). catc1({0, 2}) = 1 but dc1,c1(id, c) = ∞. digital lusternik-schnirelmann category can be written in terms of digital homotopic distance as follows. theorem 4.4. for a fixed x0 ∈ x, let i1 : x → x × x, i1(x) = (x, x0) and i2 : x → x × x, i2(x) = (x0, x). then catκ(x) = dκ,np (κ,κ)(i1, i2). proof. let us first show that dκ,np (κ,κ)(i1, i2) ≤ catκ(x). let catκ(x) = k. then there are u0, u1, . . . , uk covering x such that ji : ui →֒ x is κnullhomotopic for each i = 0, 1, . . . , k. let f i : ui × [0, mi]z → x be (κ, κ)-homotopy such that (a1) f i(x, 0) = ji(x) = x and f i(x, mi) = c(x) = x0 where c : x → x, c(x) = x0 is a constant map. (a2) for all x ∈ ui, the induced function f ix : [0, mi]z → x, f i x(t) = f i(x, t) is (2, κ)-continuous. (a3) for all t ∈ [0, mi]z, the induced function f it : ui → x, f i t (x) = f i(x, t) is (κ, κ)-continuous. define hi : ui × [0, 2mi]z → x × x as follows. hi(x, t) = { (f i(x, t), x0), t ∈ [0, mi]z (x0, f i(2mi − t)), t ∈ [mi, 2mi]z then we have (b1) hi(x, 0) = i1(x) and hi(x, 2mi) = i2(x). (b2) for all x ∈ ui, the induced function hix : [0, mi]z → x × x, h i x(t) = hi(x, t) is (2, np(κ, κ))-continuous: suppose t1, t2 are 2-adjacent. case i: if t1, t2 ∈ [0, mi]z, then hix(t1) = (f i(x, t1), x0) and hix(t2) = (f i(x, t2), x0) are np(κ, κ)-continuous since the first components are κ-adjacent or equal from (a2) and the second components are equal. case ii: if t1, t2 ∈ [mi, 2mi]z, then hix(t1) = (x0, f i(x, 2mi − t1)) and hix(t2) = (x0, f i(x, 2mi − t2)) are np(κ, κ)-continuous since the first components are equal and the second components are κ-adjacent or equal from (a2). case iii: if mi ∈ {t1, t2} then either case i or case ii applies. (b3) for all t ∈ [0, mi]z, the induced function hit : ui → x, h i t(x) = hi(x, t) is (κ, np(κ, κ))-continuous: suppose x, y ∈ ui are κ-adjacent. case i: if t ∈ [0, mi]z, then hit(x) = h i(x, t) = (f(x, t), x0) and hit(y) = h i(y, t) = (f(y, t), x0). so the first components are κadjacent or equal due to (a3) and the second components are equal. case ii: if t ∈ [mi, 2mi]z, then hit(x) = h i(x, t) = (x0, f(x, 2mi−t)) and hit(y) = h i(y, t) = (x0, f(y, 2mi − t)). so the first components © agt, upv, 2021 appl. gen. topol. 22, no. 1 188 digital homotopic distance between digital functions are equal and the second components are κ-adjacent or equal due to (a3). hence i1|ui ≃κ,np (κ,κ) i2|ui. this establishes the desired inequality. now let us show that catκ(x) ≤ dκ,np (κ,κ)(i1, i2). let dκ,np (κ,κ)(i1, i2) = k. then there are u0, u1, . . . , uk covering x × x such that i1|uj ≃κ,np (κ,κ) i2|uj for each j = 0, 1, . . . , k. let gj : uj × [0, mj]z → x × x be (κ, np(κ, κ))-homotopy such that (c1) gj(x, 0) = i1(x) and gj(x, mj) = i2(x). (c2) for all x ∈ uj, the induced function gjx : [0, mj]z → x × x, g j x(t) = gj(x, t) is (2, np(κ, κ))-continuous. (c3) for all t ∈ [0, mj]z, the induced function g j t : uj → x × x, g j t(x) = gj(x, t) is (κ, np(kappa, κ))-continuous. let ιj : uj →֒ x be the inclusion function and c : uj → x, c(x) = x0 be a constant function. define kj : uj × [0, mj]z → x by kj(x, t) = pr1 ◦ g j(x, t) where pr1 : x×x → x is the projection to the first factor. notice that pr1 is (np(κ, κ), κ)continuous, [10]. then we have (d1) kj(x, 0) = x = ιj(x) and kj(x, mj) = x0 = c(x). (d2) for all x ∈ uj, the induced function kjx : [0, mj]z → x, k j(x, t) =: kjx(t) = pr1 ◦g j x(t) is (2, np(κ, κ))-continuous due to proposition 2.5, the (np(κ, κ), κ)-continuity of pr1 and the (2, np(κ, κ))-continuity of gjx. (d3) for all t ∈ [0, mj]z, the induced function k j t : uj → x ×x, k j(x, t) =: k j t (x) = pr1 ◦ g j t(x) is (κ, κ)-continuous due to due to proposition 2.5, the (np(κ, κ), κ)-continuity of pr1 and the (2, np(κ, κ))-continuity of gjx. thus we have ιj|uj ≃κ,κ c for each j. � theorem 4.5 ([18]). catκ,np (κ,κ)(∆x) = catκ(x) where ∆x : x → x × x, ∆x(x) = (x, x) is (κ, np(κ, κ))-continuous. corollary 4.6. let i1, i2 be inclusions as defined in theorem 4.4 and ∆x : x → x × x be (κ, np(κ, κ))-continuous digital diagonal function. then dκ,np (κ,κ)(∆x, c) = dκ(i1, i2). proof. this follows from theorems 4.4 and 4.5. � 5. digitally homotopy invariance of digital homotopic distance theorem 5.1 which states that the digital homotopic distance is homotopy invariant, is the main theorem of this section. before we mention the theorem, let us recall right and left digital homotopy equivalences. a (κ, λ)-continuous function f : x → y is left digital homotopy inverse if there exists a (λ, κ)-continuous function g : y → x such that g ◦ f ≃κ,κ idx. © agt, upv, 2021 appl. gen. topol. 22, no. 1 189 a. borat a (κ, λ)-continuous function f : x → y is right digital homotopy inverse if there exists a (λ, κ)-continuous function h : y → x such that f ◦ g ≃λ,λ idy . theorem 5.1. let f, g : x → y be (κ, λ)-continuous and f′, g′ : x′ → y ′ be (κ′, λ′)-continuous functions. if h2 : x′ → x and h1 : y → y ′ have left and right digital homotopy equivalences respectively such that the following diagram is commutative both with respect to f and g in the following sense: h1 ◦ f ◦ h2 ≃κ′,λ′ f ′ and h1 ◦ g ◦ h2 ≃κ′,λ′ g′. then dκ,λ(f, g) = dκ′,λ′(f′, g′). x y x′ y ′ f g h1h2 f ′ g ′ an immediate consequence of theorem 5.1 is the following. corollary 5.2 ([2]). digital ls category is digitally homotopy invariant. for a proof of theorem 5.1 we need the following lemmas. lemma 5.3. let f, g : x → y be (κ, λ)-continuous and h : y → z be (λ, γ)continuous functions. then dκ,γ(h ◦ f, h ◦ g) ≤ dκ,λ(f, g). proof. let dκ,λ(f, g) = k. then there exists a covering u0, . . . , uk of x such that f|ui ≃κ,λ g|ui for each i = 0, 1, . . . , k. then for each i, we have (h ◦ f)|ui = h ◦ f|ui ≃κ,γ h ◦ g|ui = (h ◦ g)|ui where the (κ, γ)-homotopy follows from proposition 2.5. hence dκ,γ(h ◦ f, h ◦ g) ≤ dκ,λ(f, g). � lemma 5.4. let f, g : x → y be (κ, λ)-continuous and h : z → x be (γ, κ)continuous functions. then dγ,λ(f ◦ h, g ◦ h) ≤ dκ,λ(f, g). proof. let dκ,λ(f, g) = k. then there exists a covering u0, . . . , uk of x such that f|ui ≃κ,λ g|ui for each i = 0, 1, . . . , k. consider h−1(uj) ⊆ z. notice that {h−1(uj)}kj=0 is a covering of x and the restriction map hj : h−1(uj) → z can be written in terms of h as hj : h−1(uj) h −→ uj ι −֒→ z, hj = ι ◦ h. then we have (f ◦ h)|h−1(uj) = fh−1(uj) ◦ hj ≃γ,λ g|h−1(uj ) ◦ hj = g|h−1(uj ) ◦ (ι ◦ h)|h−1(uj) = g ◦ (ι ◦ h)|h−1(uj) = (g ◦ h)|h−1(uj) so (f ◦ h)|h−1(uj ) ≃γ,λ (g ◦ h)|h−1(uj) for each i. notice that the (γ, λ)homotopy on above line follows from proposition 2.5. hence dγ,λ(f ◦h, g◦h) ≤ k. � by using these lemmas we prove the following propositions. © agt, upv, 2021 appl. gen. topol. 22, no. 1 190 digital homotopic distance between digital functions proposition 5.5. let f, g : x → y be (κ, λ)-continuous and h1 : y → y ′ be (λ, λ′)-continuous function with a left digital homotopy inverse. then dκ,λ′(h1 ◦ f, h1 ◦ g) = dκ,λ(f, g). proof. by proposition 3.4 and lemma 5.3, it follows that dκ,λ(f, g) = dκ,λ(h◦ h1 ◦ f, h ◦ h1 ◦ g) ≤ dκ,λ′(h1 ◦ f, h1 ◦ g) ≤ dκ,λ(f, g). � proposition 5.6. let f, g : x → y be (κ, λ)-continuous and h2 : x′ → x be (κ′, κ)-continuous function with a right digital homotopy inverse. then dκ′,λ(f ◦ h2, g ◦ h2) = dκ,λ(f, g). proof. by proposition 3.4 and lemma 5.4, it follows that dκ,λ(f, g) = dκ,λ(f ◦ h2 ◦ h, g ◦ h2 ◦ h) ≤ dκ′,λ(f ◦ h2, g ◦ h2) ≤ dκ,λ(f, g). � proof of theorem 5.1. by proposition 5.5 and proposition 5.6, we have d(f′, g′) = d(h1 ◦ f ◦ h2, h1 ◦ g ◦ h2) = d(f ◦ h2, g ◦ h2) = dκ,λ(f, g). 6. future work there is a relation between usual homotopic distance and tc. a similar relation can be found between digital homotopic distance and digital tc (as defined in [12]). higher digital topological complexity is studied by is and karaca in [11]. motivated from the fact that if digital homotopic distance is a generalization of digital tc, it can be predicted that a higher analog of homotopic distance (defined in a similar way as in [16]; see also [3]) can be realized as a generalization of higher digital tc. acknowledgements. the author would like to thank tane vergili and the referees for their helpful suggestions. in particular, the author would like to thank the referee who contributed proposition 3.2 and example 4.3. references [1] c. berge, graphs and hypergraphs, 2nd edition, north-holland, amsterdam, 1976. [2] a. borat and t. vergili, digital lusternik-schnirelmann category, turkish journal of mathematics 42, no 1 (2018), 1845–1852. [3] a. borat and t. vergili, higher homotopic distance, topological methods in nonlinear analysis, to appear. [4] l. boxer, digitally continuous functions, pattern recognit. lett. 15 (1994), 883–839. [5] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. [6] l. boxer, homotopy properties of sphere-like digital images, j. math. imaging vision, 24 (2006), 167–175. © agt, upv, 2021 appl. gen. topol. 22, no. 1 191 a. borat [7] l. boxer, alternate product adjacencies in digital topology, applied general topology 19, no. 1 (2018), 21–53. [8] o. cornea, g. lupton, j. oprea and d. tanre, lusternik-schnirelmann category, mathematical surveys and monographs, vol. 103, american mathematical society 2003. [9] m. farber, topological complexity of motion planning, discrete and computational geometry 29 (2003), 211–221. [10] s. e. han, non-product property of the digital fundamental group, information sciences 171 (2005), 73–91. [11] m. is and i. karaca, the higher topological complexity in digital images, applied general topology 21, no. 2 (2020), 305–325. [12] i. karaca and m. is, digital topological complexity numbers, turkish journal of mathematics 42, no. 6 (2018), 3173–3181. [13] e. khalimsky, motion, deformation, and homotopy in finite spaces, proceedings ieee international conference on systems, man, and cybernetics (1987), 227–234. [14] g. lupton, j. oprea and n. a. scoville, homotopy theory in digital topology, arxiv: 1905.07783. [15] g. lupton, j. oprea and n. a. scoville, subdivisions of maps of digital images, arxiv: 1906.03170. [16] e. macias-virgos and d. mosquera-lois, homotopic distance between maps, math. proc. cambridge philos. soc., to appear. [17] g. sabidussi, graph multiplication, math. z. 72 (1960), 446–457. [18] t. vergili and a. borat, digital lusternik-schnirelmann category of digital functions, hacettepe journal of mathematics and statistics 49, no. 4 (2020), 1414–1422. © agt, upv, 2021 appl. gen. topol. 22, no. 1 192 @ appl. gen. topol. 23, no. 1 (2022), 31-43 doi:10.4995/agt.2022.16165 © agt, upv, 2022 some fixed point results for enriched nonexpansive type mappings in banach spaces rahul shukla and rajendra pant department of mathematics & applied mathematics, university of johannesburg kingsway campus, auckland park 2006, south africa (rshukla.vnit@gmail.com, pant.rajendra@gmail.com) communicated by s. romaguera abstract in this paper, we introduce two new classes of nonlinear mappings and present some new existence and convergence theorems for these mappings in banach spaces. more precisely, we employ the krasnosel’skĭı iterative method to obtain fixed points of suzuki-enriched nonexpansive mappings under different conditions. moreover, we approximate the fixed point of enriched-quasinonexpansive mappings via ishikawa iterative method. 2020 msc: 47h10; 47h09. keywords: nonexpansive mapping; enriched nonexpansive mapping; banach space. 1. introduction let c be a nonempty subset of a banach space (b,‖.‖). a mapping ξ : c →c is said to be nonexpansive if ‖ξ(ϑ) − ξ(ν)‖≤‖ϑ−ν‖ for all ϑ,ν ∈ c. bruck [5] observed that apart from being an obvious generalization of the contraction mapping, nonexpansive mappings are important due to their connection with the monotonicity methods. perhaps, nonexpansive mappings belong to the first class of nonlinear mappings for which fixed point theorems were obtained by using the geometric properties of the underlying banach spaces rather than the compactness assumptions (see fixed point theorems received 31 august 2021 – accepted 24 november 2021 http://dx.doi.org/10.4995/agt.2022.16165 https://orcid.org/0000-0002-9835-0935 https://orcid.org/0000-0001-9990-2298 r. shukla and r. pant due to browder [3], göhde [12] kirk [15]). this class of mappings also appears in applications as transition operators for initial value problems (of differential inclusion), accretive operators, monotone operators, variational inequality problems and equilibrium problems. a number of extensions and generalizations of nonexpansive mappings in different directions have been considered by many mathematicians in the literature to enlarge the class of nonexpansive mappings, see [11, 17, 8, 25, 20, 21, 24] (see also the references therein). in 2008, suzuki [25] introduced a new type of mapping which is more general than nonexpansive mapping, as follows. definition 1.1 ([25]). let c be a nonempty subset of a banach space (b,‖.‖). a mapping ξ : c →c is said to satisfy condition (c) if for all ϑ,ν ∈c 1 2 ‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies ‖ξ(ϑ) − ξ(ν)‖≤‖ϑ−ν‖. a mapping satisfying condition (c) is also known as suzuki type generalized nonexpansive mapping. recently, berinde [1] introduced the following class of nonlinear mappings. definition 1.2. let (b,‖.‖) be a banach space. a mapping ξ : b →b is said to be b-enriched nonexpansive mapping if there exists b ∈ [0,∞) such that for all ϑ,ν ∈b (1.1) ‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖≤ (b + 1)‖ϑ−ν‖. it is shown that every nonexpansive mapping ξ is a 0-enriched mapping. it is interesting to note that both these classes of mappings, suzuki type nonexpansive and b-enriched nonexpansive mappings are independent. a couple of examples below illustrate these facts. example 1.3 ([25]). let c = [0, 3] be a subset of r endowed with the usual norm. define ξ : c →c by ξ(ϑ) = { 0, if ϑ 6= 3, 1, if ϑ = 3. then ξ satisfies condition (c). however at ϑ = 2.5 and ν = 3 ‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖ = ‖b(2.5 − 3) + (0 − 1)‖ = b(0.5) + 1 > b(0.5) + 0.5 = (b + 1)|ϑ−ν| and ξ is not b-enriched nonexpansive mapping for any b ∈ [0,∞). example 1.4 ([1]). let c = [ 1 2 , 2 ] ⊂ r and ξ : c → c be a mapping defined as ξ(ϑ) = 1 ϑ . then f(ξ) = {1} and ξ is a 3 2 -enriched nonexpansive mapping. on the other hand at ϑ = 1 and ν = 1 2 , we have 1 2 ‖1 − ξ(1)‖ = 0 ≤ 1 2 = ∥∥∥∥1 − 12 ∥∥∥∥ © agt, upv, 2022 appl. gen. topol. 23, no. 1 32 fixed point results for enriched nonexpansive type mappings and ∥∥∥∥ξ ( 1 2 ) − ξ(1) ∥∥∥∥ = |2 − 1| = 1 > 12 = ∥∥∥∥12 − 1 ∥∥∥∥ . thus ξ is not a mapping satisfying condition (c). now an interesting question arises that does there exists a class of mappings, which contains both the b-enriched nonexpansive mappings and suzuki-type generalized nonexpansive mappings? herein, we answer this question, affirmatively. indeed, we introduce a new class of mappings, namely, suzuki-enriched nonexpansive mapping. on the other hand, to check that a given mapping belongs to any of the classes of nonexpansive type mappings can not be an easy task. keeping this point in mind to make task easier, diaz and metcalf [7] considered the following class of mappings known as quasinonexpansive mapping definition 1.5. a mapping ξ : c → c is said to be quasinonexpansive if for all ϑ ∈c and ϑ† ∈ f(ξ) 6= ∅, ‖ξ(ϑ) −ϑ†‖≤‖ϑ−ϑ†‖ where f(ξ) is the set of all fixed points of ξ. it is well known that a nonexpansive mapping with a fixed point is quasinonexpansive. however the converse need not to be true. again, it is interesting to see that the classes of b-enriched nonexpansive mappings and that of quasi-nonexpansive mappings are independent in nature, see [23]. keeping this in mind, we generalize the class of quasinonexpansive mappings in the sense of b-enriched nonexpansive mappings. in particular, we introduce a new class of mappings namely enriched-quasinonexpansive mappings. this class of mappings properly contains both quasinonexpansive mappings and b-enriched nonexpansive mappings. motivated by berinde [1, 2], suzuki [25], diaz and metcalf [7] and others, we introduce two new nonlinear classes of mappings in the setting of banach spaces and establish some existence and convergence theorems for these classes of mappings. we ensure the existence of fixed points for suzuki-enriched nonexpansive mappings in banach spaces under certain assuptions. we employ ishikawa iterative method to approximate the fixed points of enriched-quasinonexpansive mappings and obtain some weak and strong convergence theorems. our results complement, extend, and generalize certain results from [1, 2, 25, 7, 18, 9]. 2. preliminaries definition 2.1 ([10]). a banach space b is said to be uniformly convex if for every ε ∈ (0, 2] there is some δ > 0 so that, for any ϑ,ν ∈b with ‖ϑ‖ = ‖ν‖ = 1, the condition ‖ϑ−ν‖≥ ε implies that ∥∥ϑ+ν 2 ∥∥ ≤ 1 − δ. © agt, upv, 2022 appl. gen. topol. 23, no. 1 33 r. shukla and r. pant definition 2.2 ([19]). let (b,‖ · ‖) be a banach space. a space b satisfies opial property if, for every weakly convergent sequence {ϑn} with weak limit ϑ ∈b it holds: lim inf n→∞ ‖ϑn −ϑ‖ < lim inf n→∞ ‖ϑn −ν‖ for all ν ∈b with ϑ 6= ν. all finite dimensional banach spaces and all hilbert spaces satisfy the weakopial property. spaces `p (p ∈ (1,∞)) are opial spaces but lp(∈ (1,∞), p 6= 2) spaces are not. [10]. definition 2.3 ([22]). the mapping ξ : c → c with f(ξ) 6= ∅ satisfies condition (i) if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0,f(r) > 0 for r ∈ (0,∞) such that ‖ϑ− ξ(ϑ)‖ ≥ f(d(ϑ,f(ξ))) for all ϑ ∈c, where d(ϑ,f(ξ)) = inf{‖ϑ−ν‖ : ν ∈ f(ξ)}. let c be a convex subset of a banach space b and ξ : c → c a mapping. the following iterative method is known as the krasnosel’skĭı iterative method (see [16]): (2.1) { ϑ1 ∈c ϑn+1 = αϑn + (1 −α)ξ(ϑn) where α ∈ (0, 1). lemma 2.4. let c be a nonempty convex subset a banach space b. let ξ : c →c be a mapping, define s : c →c as follows: s(ϑ) = (1 −λ)ϑ + λξ(ϑ) for all ϑ ∈c and λ ∈ (0, 1). then f(s) = f(ξ). definition 2.5. let c be a nonempty subset of a banach space b. a mapping ξ : c →c is said to be compact if ξ(c) has a compact closure. lemma 2.6 ([27, p. 484]). let b be a uniformly convex banach space. if two sequences {ϑn}, {νn} in b such that lim sup n→∞ ‖ϑn‖ ≤ θ, lim sup n→∞ ‖νn‖ ≤ θ, lim n→∞ ‖αnϑn + (1 −αn)νn‖ = θ, where {αn}⊆ [η1,η2] ⊂ [0, 1] and θ ≥ 0. then lim n→∞ ‖ϑn −νn‖ = 0. lemma 2.7. let b be a uniformly convex banach space and c a nonempty closed convex subset of b. let s : c →c be a quasinonexpansive mapping with f(s) 6= ∅. for given ϑ1 ∈c, for all n ∈ n, γn,δn ∈ [c,d] with c,d ∈ (0, 1), we can define a sequence {ϑn} (ishikawa iterative method [13]) as follows: (2.2) { νn = (1 −γn)ϑn + γns(ϑn) ϑn+1 = (1 − δn)ϑn + δns(νn), then we have the followings: (1) lim n→∞ ‖ϑn −z‖ exists for all z ∈ f(s). © agt, upv, 2022 appl. gen. topol. 23, no. 1 34 fixed point results for enriched nonexpansive type mappings (2) lim n→∞ ‖ϑn −s(ϑn)‖ = 0. proof. from (2.2) ‖ϑn+1 −z‖ ≤ (1 −δn)‖ϑn −z‖ + δn‖s(νn) −z‖ ≤ (1 −δn)‖ϑn −z‖ + δn‖νn −z‖ ≤ (1 −δn)‖ϑn −z‖ + δn{(1 −γn)‖ϑn −z‖ + γn‖s(ϑn) −z‖} ≤ (1 −δn)‖ϑn −z‖ + δn‖ϑn −z‖ = ‖ϑn −z‖. hence the sequence {‖ϑn −z‖} is monotone nonincreasing and lim n→∞ ‖ϑn −z‖ exists. let (2.3) lim n→∞ ‖ϑn −z‖ = r > 0. since, s is a quasinonexpansive mapping (2.4) lim sup n→∞ ‖s(νn) −z‖≤ lim sup n→∞ ‖νn −z‖≤ lim n→∞ ‖ϑn −z‖ = r and (2.5) lim n→∞ ‖(1 − δn)(ϑn −z) + δn(s(νn) −z)‖ = lim n→∞ ‖ϑn+1 −z‖ = r. from (2.3), (2.4), (2.5) and lemma 2.6, we have (2.6) lim n→∞ ‖ϑn −s(νn)‖ = 0 again ‖ϑn+1 −z‖ ≤ (1 − δn)‖ϑn −z‖ + δn‖s(νn) −z‖ ≤ (1 − δn)‖ϑn −z‖ + δn‖νn −z‖ which implies ‖ϑn+1 −z‖−‖ϑn −z‖ δn ≤‖νn −z‖−‖ϑn −z‖. since δn ∈ [c,d] ‖ϑn+1 −z‖−‖ϑn −z‖ d ≤‖νn −z‖−‖ϑn −z‖. thus r ≤ lim inf n→∞ ‖νn −z‖ from (2.4), we get (2.7) lim n→∞ ‖νn −z‖ = r = lim n→∞ ‖(1 −γn)(ϑn −z) + γn(s(ϑn) −z)‖ from (2.3), (2.7) and lemma 2.6, we get lim n→∞ ‖ϑn −s(ϑn)‖ = 0. � lemma 2.8. let b be a uniformly convex banach space and c a nonempty closed convex subset of b. let s : c →c be a quasinonexpansive mapping with f(s) 6= ∅. then f(s) is closed. © agt, upv, 2022 appl. gen. topol. 23, no. 1 35 r. shukla and r. pant lemma 2.9 (demiclosedness principle, [4]). let b be a uniformly convex banach space, c a closed convex subset of b and ξ : c →c a mapping with a fixed point. suppose {ϑn} is a sequence in b such that {ϑn} converges weakly to ϑ and lim n→∞ ‖ϑn − ξ(ϑn)‖ = 0. then ξ(ϑ) = ϑ. that is, i − ξ is demiclosed at zero. 3. suzuki-enriched nonexpansive mapping in this section, we introduce the following new class of mappings: definition 3.1. let (b,‖.‖) be a banach space and c a nonempty subset of b. a mapping ξ : c → c is said to be suzuki-enriched nonexpansive mapping if there exists b ∈ [0,∞) such that for all ϑ,ν ∈c 1 2(b + 1) ‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies(3.1) ‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖≤ (b + 1)‖ϑ−ν‖. it can be seen that every suzuki-nonexpansive mapping ξ is a suzukienriched nonexpansive mapping with b = 0. theorem 3.2. let b be a banach space and c a nonempty compact convex subset of b. let ξ : c → c be a mapping satisfying (3.1). for given ϑ1 ∈ c, define a sequence {ϑn} in c by (3.2) ϑn+1 = (1 −λ)ϑn + λξ(ϑn) for all n ∈ n, where λ ∈ [ 1 2(b+1) , 1 b+1 ) . then f(ξ) 6= ∅ and {ϑn} strongly converges to a point in f(ξ). proof. by the definition of mapping ξ, we have 1 2(b + 1) ‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies(3.3) ‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖≤ (b + 1)‖ϑ−ν‖. for all ϑ,ν ∈c. take µ = 1 b+1 ∈ (0, 1) and put b = 1−µ µ in (3.3) then the above inequality is equivalent to 1 2 µ‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies(3.4) ‖(1 −µ)(ϑ−ν) + µ(ξ(ϑ) − ξ(ν))‖≤‖ϑ−ν‖. define the mapping s as follows: s(ϑ) = (1 −µ)ϑ + µξ(ϑ) for all ϑ ∈c. thus (3.5) ‖s(ϑ) −ϑ‖ = µ‖ξ(ϑ) −ϑ‖. then from (3.4), we get 1 2 ‖ϑ−s(ϑ)‖≤‖ϑ−ν‖ implies ‖s(ϑ) −s(ν)‖≤‖ϑ−ν‖ © agt, upv, 2022 appl. gen. topol. 23, no. 1 36 fixed point results for enriched nonexpansive type mappings for all ϑ,ν ∈ c. thus s is a mapping satisfying condition (c). thus all the assumptions of [25, theorem 2] are satisfied and s has a fixed point in c. from lemma 2.4, f(s) = f(ξ) 6= ∅. next, for given ϑ1 ∈ c and any λ ∈ [ 1 2 , 1 ) consider the sequence (3.6) ϑn+1 = (1 −λ)ϑn + λs(ϑn). from [25, theorem 2], {ϑn} strongly converges to a fixed point of s. but f(s) = f(ξ) and (1 −λ)ϑ + λs(ϑ) = (1 −λµ)ϑ + λµξ(ϑ) for all ϑ ∈c. since λ ∈ [ 1 2 , 1 ) and µ = 1 b+1 . this implies that λµ ∈ [ 1 2(b+1) , 1 b+1 ) . therefore for any λ ∈ [ 1 2(b+1) , 1 b+1 ) , the sequence {ϑn} defined by (3.8) strongly converges to a point in f(ξ). � theorem 3.3. let b be a banach space with the opial property. let c be a nonempty weakly compact convex subset of b and ξ : c → c a mapping satisfying (3.1). for given ϑ1 ∈c, define a sequence {ϑn} in c by (3.7) ϑn+1 = (1 −λ)ϑn + λξ(ϑn) for all n ∈ n, where λ ∈ [ 1 2(b+1) , 1 b+1 ) . then f(ξ) 6= ∅ and {ϑn} weakly converges to a point in f(ξ). proof. following the same proof technique as in theorem 3.2, we can define a mapping s : c →c as follows: s(ϑ) = ( 1 − 1 b + 1 ) ϑ + 1 b + 1 ξ(ϑ) for all ϑ ∈c and s is a mapping satisfying condition (c). then all the assumptions of [23, theorem 5] are satisfied, hence {ϑn} weakly converges to a fixed point of s. but f(s) = f(ξ). this completes the proof. � theorem 3.4. let b be a uniformly convex in every direction (or uced) banach space and c a nonempty weakly compact convex subset of b. let ξ : c →c be a a mapping satisfying (3.1). then ξ admits a fixed point in c. proof. following largely the proof of theorem 3.2, we can define a mapping s satisfying condition (c). thus all the assumptions of [25, theorem 5] are satisfied and it is guaranteed that s has at least one fixed point. from lemma 2.4, f(s) = f(ξ) 6= ∅. � theorem 3.5. let b be a uced banach space and c a nonempty weakly compact convex subset of b. let g be a family of commuting mappings on c satisfying (3.1). then g has a common fixed point. proof. following the same proof technique of [25, theorem 6] one can get the desired result. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 37 r. shukla and r. pant theorem 3.6. let b be a uniformly convex banach space whose dual b∗ has the kadec-klee property. let c be a nonempty bounded closed convex subset of b and ξ : c → c a mapping satisfying (3.1). for given ϑ1 ∈ c, define a sequence {ϑn} in c by (3.8) ϑn+1 = (1 −λ)ϑn + λξ(ϑn) for all n ∈ n, where λ ∈ [ 1 2(b+1) , 1 b+1 ) . then f(ξ) 6= ∅ and {ϑn} weakly converges to a point in f(ξ). proof. following the same proof technique as in theorem 3.2, we can define a mapping s : c →c as follows: s(ϑ) = ( 1 − 1 b + 1 ) ϑ + 1 b + 1 ξ(ϑ) for all ϑ ∈c and s is a mapping satisfying condition (c). then all the assumptions of [14, theorem 11] are satisfied, hence {ϑn} weakly converges to a fixed point of s. but f(s) = f(ξ). this completes the proof. � we can obtain the following results due to consequence of theorem 3.6. corollary 3.7. let b be a uniformly convex banach space having fréchet differentiable norm. let c, ξ and {ϑn} be same as in theorem 3.6. then f(ξ) 6= ∅ and {ϑn} weakly converges to a point in f(ξ). 4. enriched-quasinonexpansive mapping now, we introduce the following new class of mappings: definition 4.1. let (b,‖.‖) be a banach space and c a nonempty subset of b. a mapping ξ : c → c is said to be b-enriched quasinonexpansive mapping if there exists b ∈ [0,∞) such that for all ϑ ∈c and ν ∈ f(ξ) 6= ∅ (4.1) ‖b(ϑ−ν) + ξ(ϑ) −ν‖≤ (b + 1)‖ϑ−ν‖. remark 4.2. • it can be seen that every quasinonexpansive mapping is a 0-enriched quasinonexpansive mapping. • every b-enriched nonexpansive mapping with a fixed point is b-enriched quasinonexpansive mapping but the converse need not be true. we consider the following examples, see [6, example 6.23]. example 4.3. let b = `∞ and c := {ϑ ∈ `∞ : ‖ϑ‖∞ ≤ 1}. define ξ : c → c by ξ(ϑ) = (0,ϑ21,ϑ 2 2,ϑ 2 3, . . . ) for ϑ = (ϑ1,ϑ2,ϑ3, . . . ) ∈ c. then it can be seen that ξ is continuous from c into c with p = (0, 0, . . . ) and f(ξ) = {p}. furthermore, ‖ξ(ϑ) −p‖∞ = ‖ξ(ϑ)‖∞ = ‖(0,ϑ21,ϑ 2 2,ϑ 2 3, . . . )‖∞ ≤ ‖(ϑ1,ϑ2,ϑ3, . . . )‖∞ = ‖ϑ‖∞ = ‖ϑ−p‖∞ © agt, upv, 2022 appl. gen. topol. 23, no. 1 38 fixed point results for enriched nonexpansive type mappings for all ϑ ∈ c. thus, ξ is quasi-nonexpansive mapping and hence 0-enriched quasinonexpansive mapping. however, ξ is not enriched-nonexpansive for any b ∈ [0,∞). for, if ϑ = ( 3 4 , 3 4 , 3 4 , . . . ) and ν = ( 1 2 , 1 2 , 1 2 , . . . ), it is clear that ϑ,ν ∈c. furthermore, for any b ∈ [0,∞) ‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖∞ = ∥∥∥∥ ( b 4 , 4b + 5 16 , 4b + 5 16 , . . . )∥∥∥∥ ∞ = 4b + 5 16 > b + 1 4 = (b + 1) ∥∥∥∥14, 14, 14, . . . ∥∥∥∥ ∞ = (b + 1)‖ϑ−ν‖∞. proposition 4.4. let ξ : c → c be a suzuki-enriched nonexpansive mapping with any b ∈ [0,∞) and f(ξ) 6= ∅. then ξ is a b-enriched quasinonexpansive mapping for any b ∈ [0,∞). proof. let z ∈ f(ξ) and ϑ ∈c. since 1 2(b+1) ‖z −ξ(z)‖ = 0 ≤‖ϑ−z‖, we have ‖b(ϑ−ν) + ξ(ϑ) −z‖≤ (b + 1)‖ϑ−z‖. � in the above proposition the inclusion is strict, the following illustrative example [25, example 2] verifies this fact. example 4.5. let c = [0, 3] ⊂ r and ξ : c →c be a mapping defined as ξ(ϑ) = { 0, if ϑ 6= 3 2, if ϑ = 3. then f(ξ) = {0} and ξ is a b-enriched quasinonexpansive mapping for any b ∈ [0,∞). on the other hand at ϑ = 3 and ν = 4, 1 2(b+1) ‖3−ξ(3)‖ = 1 2(b+1) ≤ 1 = ‖3 − 2‖, we have ‖b(ϑ−ν)+ξ(ϑ)−ξ(ν)‖ = ‖b(3−2)+ξ(3)−ξ(2)‖ = (b+2) > (b+1) = (b+1)‖ϑ−ν‖ and ξ is not a suzuki-enriched nonexpansive mapping for any b ∈ [0,∞). for some fix ϑ1 ∈c, the ishikawa iterative method can be defined as follows [13]: (4.2) { νn = (1 −βn)ϑn + βnξ(ϑn) ϑn+1 = (1 −αn)ϑn + αnξ(νn), where {βn} and {αn} are sequences in [0, 1]. theorem 4.6. let b be a uniformly convex banach space and c a nonempty closed convex subset of b. let ξ : c → c be a b-enriched quasinonexpansive mapping and ξ satisfies condition i. for given ϑ1 ∈ c, for all n ∈ n, αn ∈ (c,d), βn ∈ ( c b+1 , d b+1 ) with c,d ∈ (0, 1), define a sequence {ϑn} as follows:{ νn = (1 −βn)ϑn + βnξ(ϑn) ϑn+1 = (1 −αn)ϑn + αn [( 1 − 1 b+1 ) νn + 1 b+1 ξ(νn) ] . © agt, upv, 2022 appl. gen. topol. 23, no. 1 39 r. shukla and r. pant then {ϑn} strongly converges to a point in f(ξ). proof. by the definition of b-enriched quasinonexpansive mapping, we have ‖b(ϑ−ν) + ξ(ϑ) −ν‖≤ (b + 1)‖ϑ−ν‖(4.3) for all ϑ ∈ c and ν ∈ f(ξ). take µ = 1 b+1 ∈ (0, 1) and put b = 1−µ µ in (4.3), then the above inequality is equivalent to ‖(1 −µ)(ϑ−ν) + µ(ξ(ϑ) −ν)‖≤‖ϑ−ν‖.(4.4) define the mapping s as follows: (4.5) s(ϑ) = (1 −µ)ϑ + µξ(ϑ) for all ϑ ∈c. from lemma 2.4, f(s) = f(ξ). then from (4.4), we get ‖s(ϑ) −ν‖≤‖ϑ−ν‖ for all ϑ ∈ c and ν ∈ f(s). thus s : c → c is a quasinonexpansive mapping. for given ϑ1 ∈ c, for all n ∈ n, γn,αn ∈ [c,d] with c,d ∈ (0, 1), we can define a sequence {ϑn} as follows: (4.6) { νn = (1 −γn)ϑn + γns(ϑn) ϑn+1 = (1 −αn)ϑn + αns(νn), from lemma 2.7, lim n→∞ ‖ϑn−z‖ exists for all p ∈ f(s). thus lim n→∞ d(ϑn,f(s)) exists. since ξ satisfies condition i ‖ϑn − ξ(ϑn)‖ = (b + 1)‖ϑn −s(ϑn)‖≥ f(d(ϑn,f(ξ))) = f(d(ϑn,f(s))). from lemma 2.7, lim n→∞ ‖ϑn −s(νn)‖ = 0. thus lim n→∞ d(ϑn,f(s)) = 0. following largely the proof of [26, theorem 3], we can choose a subsequence {ϑnj} of {ϑn} such that ‖ϑnj −pj‖≤ 1 2j for all j ∈ n, where {pj} ⊆ f(ξ). it can be easily seen that {pj} is a cauchy sequence and strongly converges to a point p in f(ξ), since f(ξ) is closed. therefore {ϑn} strongly converges to p ∈ f(ξ). using the definition of s, we have { νn = (1 −βn)ϑn + βnξ(ϑn) ϑn+1 = (1 −αn)ϑn + αn [( 1 − 1 b+1 ) νn + 1 b+1 ξ(νn) ] where βn = γn b+1 . this completes the proof. � theorem 4.7. let b be a uniformly convex banach space and c a nonempty closed convex subset of b. let ξ : c → c be a continuous and b-enriched quasinonexpansive mapping with f(ξ) 6= ∅. for given ϑ1 ∈c, define a sequence {ϑn} in c by ϑn = ξ n α,β(ϑ1), ξα,β = (1−α)i + α [( 1 − 1 b + 1 ) i + 1 b + 1 ξ ] [(1−β)i + βξ] © agt, upv, 2022 appl. gen. topol. 23, no. 1 40 fixed point results for enriched nonexpansive type mappings for all n ∈ n, where α ∈ (0, 1), β ∈ [ 0, 1 b+1 ) and i is an identity mapping. then {ϑn} strongly converges to a point in f(ξ) if and only if d(ϑn,f(ξ))) →∞ as n →∞. proof. following the same proof technique as in theorem 3.2, we can define a mapping s : c →c as follows: s(ϑ) = ( 1 − 1 b + 1 ) ϑ + 1 b + 1 ξ(ϑ) for all ϑ ∈c and s is a quasinonexpansive mapping. for given ϑ1 ∈c, α ∈ (0, 1), γ ∈ [0, 1), we can define a sequence {ϑn} as follows: ϑn = s n α,γ(ϑ1), sα,γ = (1 −α)i + αs[(1 −γ)i + γs] using the definition of s, we have ϑn = ξ n α,β(ϑ1), ξα,β = (1−α)i + α [( 1 − 1 b + 1 ) i + 1 b + 1 ξ ] [(1−β)i + βξ] where β = γ b+1 ∈ [ 0, 1 b+1 ) . since ξ is continuous, s is continuous. then all the assumptions of [9, theorem 3.1] are satisfied, hence {ϑn} strongly converges to a fixed point of s. but f(s) = f(ξ). this completes the proof. � theorem 4.8. let b be a uniformly convex banach space and c a nonempty closed convex subset of b. let ξ : c → c be a b-enriched quasinonexpansive mapping and i − ξ is demiclosed at zero. for given ϑ1 ∈ c, for all n ∈ n, αn ∈ (c,d), βn ∈ ( c b+1 , d b+1 ) with c,d ∈ (0, 1), define a sequence {ϑn} as follows: { νn = (1 −βn)ϑn + βnξ(ϑn) ϑn+1 = (1 −αn)ϑn + αn [( 1 − 1 b+1 ) νn + 1 b+1 ξ(νn) ] . then {ϑn} weakly converges to a point in f(ξ). proof. we can define a sequence {ϑn} as in (4.6). since space b is uniformly convex, b is reflexive. by the reflexiveness of b there exists a subsequence {ϑnj} of {ϑn} such that {ϑnj} weakly converges to some p ∈ c. by lemma 2.7, lim n→∞ ‖ϑn −s(νn)‖ = 0 and lim n→∞ ‖ϑn − ξ(ϑn)‖ = 0. from the demiclosedness principle of i − ξ we have p ∈ ωw(ϑn) ⊂ f(ξ). thus, to prove that {ϑn} weakly converges to a fixed point of ξ, it suffices to show the unique weak limit for each subsequences of {ϑn}, that is, ωw(ϑn) (cluster points (ω-limit) set of a sequence {ϑn}) is a singleton. arguing by contradiction, assume that {ϑn} does not converge weakly to p, i.e., take p,q ∈ ωw(ϑn) and let {ϑnj} and {ϑmj} be subsequences of {ϑn} such that ϑnj ⇀ p © agt, upv, 2022 appl. gen. topol. 23, no. 1 41 r. shukla and r. pant and ϑmj ⇀ q, respectively. if p 6= q, and standard application of opial’s property gives us the following contradiction: lim n→∞ ‖ϑn −p‖ = lim j→∞ ‖ϑnj −p‖ < lim j→∞ ‖ϑnj −q‖ = lim n→∞ ‖ϑn −q‖ = lim j→∞ ‖ϑnj −q‖ < lim j→∞ ‖ϑnj −p‖ = lim n→∞ ‖ϑn −p‖. this completes the proof � theorem 4.9. let b be a uniformly convex banach space and c a nonempty closed convex subset of b. let ξ : c → c be a compact b-enriched quasinonexpansive mapping and i−ξ is demiclosed at zero. for given ϑ1 ∈c, αn ∈ (c,d), βn ∈ ( c b+1 , d b+1 ) with c,d ∈ (0, 1), define a sequence {ϑn} as follows:{ νn = (1 −βn)ϑn + βnξ(ϑn) ϑn+1 = (1 −αn)ϑn + αn [( 1 − 1 b+1 ) νn + 1 b+1 ξ(νn) ] . then {ϑn} strongly converges to a point in f(ξ). proof. from the proof of theorem 3.2, we can define a quasinonexpansive mapping s (as in (4.5)). we can define a sequence as in (2.2). from lemma 2.7, lim n→∞ ‖ϑn −s(νn)‖ = 0 and (4.7) lim n→∞ ‖ϑn − ξ(ϑn)‖ = 0. since the range of c under ξ is contained in a compact set, there exists a subsequence {ξ(ϑnj )} of {ξ(ϑn)} strongly converges to ϑ† ∈ c. by (4.7), the subsequence {ϑnj} strongly converges ϑ†. by the demiclosedness principle ξ(ϑ†) = ϑ†, and {ϑn} strongly converges to a point ϑ† in f(ξ). � acknowledgements. the authors are thankful to the reviewer and the editor for their constructive comments. the first author acknowledges the support from the ges 4.0 fellowship, university of johannesburg, south africa. references [1] v. berinde, approximating fixed points of enriched nonexpansive mappings by krasnoselskij iteration in hilbert spaces, carpathian j. math. 35 (2019), 293–304. 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[27] e. zeidler, nonlinear functional analysis and its applications. i. fixed-point theorems, springer-verlag, new york, 1986. © agt, upv, 2022 appl. gen. topol. 23, no. 1 43 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 73-84 the hyperspaces cn(x) for finite ray-graphs norah esty abstract in this paper we consider the hyperspace cn(x) of non-empty and closed subsets of a base space x with up to n connected components. the class of base spaces we consider we call finite ray-graphs, and are a noncompact variation on finite graphs. we prove two results about the structure of these hyperspaces under different topologies (hausdorff metric topology and vietoris topology). 2010 msc: 54b20 keywords: hyperspaces, finite graphs 1. introduction the last thirty years have produced a large amount of research in the area of hyperspaces. a hyperspace is a topological space whose points are subsets of a given base space. there are several topologies available for such spaces. if the base space is compact, two of the most popular topologies, the hausdorff metric topology and the vietoris topology, agree. however, when the base space is not compact, they differ, and in fact the vietoris topology is non-metrizable. in contrast, by using a bounded metric on the space, or allowing for infinitevalued metrics, the hausdorff topology arises from a metric. most of the study of hyperspaces has been done in the case where the base space x is a continuum. in 1968, duda did an examination of the hyperspace of subcontinua of finite connected graphs, and under some minor conditions was able to give a description of the hyperspace as a polyhedron, decomposable into balls of various dimensions [1], [2]. a single hyperspace may consist of several sections of different dimension: a two-dimensional disc glued to a three dimensional ball, 74 n. esty etc. in particular, for x a finite graph, the hyperspace of subcontinua is known to be compact and connected. in this paper we are interested in the situation where the base space is not compact. we look at a natural generalization of finite graphs which we call finite ray-graphs, which consist of vertices, edges, and rays. because the graphs are not compact we must always specify which topology we are using, and in section 5 of this paper we will use first the hausdorff metric topology, arising from the hausdorff metric, which we allow to be infinite-valued; we will call this simply the hausdorff topology. in section 6 we will use the vietoris topology. to assist the reader, in sections 3 and 4 we present several models of the hyperspace c(x) of closed, connected subsets of the base space x. we begin with a few known examples on compact base spaces, in order to provide an analog for our non-compact examples. in section 4 we state a theorem about hyperspace c(x ∨p y ) of a wedge product at a point, when the hyperspaces of the c(x) and c(y ) are known. we state this theorem without proof, as it seems to be well-known in the folk-lore (although we have been unable to find a reference). this theorem gives a nice algorithm for drawing hyperspaces. in sections 5 and 6 we prove two main results about the number of connected components of the hyperspace cn(x) (closed subsets of x with up to n connected components) of a finite, connected ray-graph x: once in the hausdorff topology and once in the vietoris. in particular, we show that when allowing for an infinite-valued hausdorff metric, a finite, connected ray-graph with k rays will have a hyperspace cn(x) with 2 k connected components for all n, and will not be compact. in contrast, under the vietoris topology cn(x) is connected for all n. 2. preliminaries and notation 2.1. notation. there is not always consistent notation used for the different hyperspaces of a given base space x. we attempt to use those notations from the literature which are least ambiguous. given a metric space x, we define the following notation for the hyperspaces we will discuss: • cl(x) = {a ⊂ x : a closed and a 6= ∅} • cn(x) = {a ∈ cl(x) : a has at most n connected components } • c(x) = c1(x) • ca(x) = {b ∈ c(x) : a ⊂ b}. the final example is called the containment hyperspace. this concept is especially useful to us when a is a vertex of the graph. if a = {p} we may write cp(x) rather than c{p}(x). it should be pointed out that much of the literature on hyperspaces assumes that the base space x is compact, in effect making c(x) the hyperspace of the hyperspaces cn(x) for finite ray-graphs 75 subcontinua, but we are not assuming that here. this is also why we write cl(x) rather than 2x, which is more common, but to many readers may mean bounded closed subsets, which we do not mean. when we wish to refer to a general hyperspace, we will write h(x). initially we will endow our hyperspaces with the hausdorff topology (τh). the hausdorff topology has the virtue that it arises from a metric in our approach, although since we are interested in unbounded base spaces, we allow the metric to be infinite-valued. let h(y ) be a hyperspace over a metric base space y . if a, b ⊂ y , and if ny (a, ǫ) indicates the ǫ-neighborhood in the space y around the subset a, then the hausdorff distance in the hyperspace is given by dh(a, b) = inf{ǫ : a ⊂ ny (b, ǫ) and b ⊂ ny (a, ǫ)} if the elements of the hyperspace are not closed subsets, then it is possible to have the distance between two non-equal sets be zero. however we will deal exclusively with closed sets. one can see from this definition that if a is bounded and b is not, the hausdorff distance between a and b is infinite. it is worth noting that using this infinite-valued metric results in a different picture of the hyperspaces than one gets from putting a bounded metric on the space x, which is why we do it. later in the paper we will use the vietoris topology. this topology is usually given by a basis definition, which we will recall in section 6. 2.2. the class of base spaces: finite ray-graphs. for our base spaces, we will consider a variation on finite graphs, which we will call finite raygraphs. these graphs will consist of a finite number of vertices (points), edges (homeomorphic to [0, 1] and attached at two vertices, or at one vertex twice) and rays (homeomorphic to [0, ∞) and attached at one vertex). we will restrict our attention to finite connected ray-graphs. we give some simple examples of models for c(x) in sections 3 and 4. the metric on these graphs will be that of arc-length, and we will consider all edges as having length one. we shall call the class of all such ray-graphs x and elements of that class x. 3. some basic examples of (c(x), τh) here we give a few geometric models of the hyperspace (c(x), τh) for specific x ∈ x . we will provide a brief explanation but no full proofs for the examples in sections 3.1, 3.2 and 4.1 as they are well-known. the purpose of outlining the examples is to use them to compare to their analogous noncompact counterparts (sections 3.3, 4.2 and 4.3). for more detail or proofs on the known hyperspaces, see [7]. 76 n. esty 3.1. x ≈ [0, 1]. if x is a segment homeomorphic to [0, 1] then any element a ∈ c(x) is of the form [a, b], with 0 ≤ a ≤ b ≤ 1. clearly there is a homeomorphism from the hyperspace c(x) to the solid triangle in r2 with vertices at (0, 0), (0, 1) and (1, 1) which takes an interval [a, b] to the point (a, b). (here we are abusing the notation to say that [a, a] = {a}.) see figure 1. notice that the left edge of the triangle corresponds to subsets of x which contain 0, i.e. the containment hyperspace c{0}(x). the top edge corresponds to subsets which contain 1, c{1}(x), and the hypotenuse corresponds to the single-element sets. we will refer to this triangle as t . 0 1 x (0, 0) (0, 1) (1, 1) c(x) a a figure 1. x = [0, 1] and c(x), as well as an element a ∈ c(x) 3.2. x ≈ s1. if x ≈ s1, then elements of c(x) can be categorized by their midpoint and their length. we can make a homeomorphism from c(x) to the solid unit disc by mapping an arc with length l and midpoint p to the point which sits on the radial line through p, and whose distance from the origin is 1− l 2π . see figure 2. notice that points on the boundary of the disc correspond to single-element sets, and the center point of the disc corresponds to the full circle. we will refer to this disc as d. although [0, 1] 6≈ s1, their two hyperspaces d and t are homeomorphic. it is known that for finite graphs this is the only such example [1]. 3.3. x ≈ [0, ∞). here we give our first non-compact example, as an analogy to the compact closed interval. if x is a ray homeomorphic to [0, ∞), then elements of c(x) are either bounded intervals of the form [a, b] or unbounded intervals of the form [a, ∞). we can make a homeomorphism from c(x) to the space t ∞ ⊔ [0, ∞), where t ∞ = {(a, b) ∈ r2 : 0 ≤ a ≤ b} is an “infinite triangle.” this is done by mapping [a, b] to (a, b) ∈ t ∞ and [a, ∞) to a ∈ [0, ∞). see figure 3. notice that although t ∞ is itself unbounded, elements of t ∞ correspond to bounded subsets of x, and in particular, the left edge of t ∞ corresponds to the hyperspaces cn(x) for finite ray-graphs 77 x a b c(x) a b figure 2. x = s1 and c(x), as well as two elements a, b ∈ c(x) 0 x (0, 0) c(x) 0 a b figure 3. x = [0, ∞) and c(x), as well as the elements a = [.25, .5] and b = [.25, ∞), both in c(x) bounded elements which contain 0, and the hypotenuse corresponds to singleelement sets. for a fixed horizontal value a, increasing the vertical value b corresponds to longer bounded intervals. since the second component, [0, ∞), corresponds to unbounded intervals, it can be loosely thought of as the “top” of the infinite triangle. in this example, unlike before, the containment hyperspace c{0}(x) has two components: the left edge of the triangle and the leftmost point 0 ∈ [0, ∞). clearly this c(x) is not connected and not compact. with these three examples we can form several more examples by understanding what happens to the hyperspace when you form the wedge product of two graphs. 78 n. esty 4. the wedge product of graphs let x = x1 ∨p x2 be the wedge product of two ray-graphs, where p is a vertex of both. it is clear that the hyperspace c(x) will contain all the elements which are in c(x1) and c(x2). it will also contain elements which correspond to subsets of x that contain the joining point p and part of x1 and x2. therefore c(x) will contain a cross product of cp(x1) and cp(x2). theorem 4.1. for x1, x2 ∈ x and the wedge produce x = x1 ∨p x2 (where p is a vertex of x1 and x2), then c(x) ≈ c(x1) ⊔ cp(x1) × cp(x2) ⊔ c(x2) (cp(x1) ∼ cp(x1) × {p} and {p} × cp(x2) ∼ cp(x2)) this theorem, which seems to be well-known in the folklore (and certainly applies to a larger class of spaces than graphs), gives us the following nice algorithm for drawing c(x): (1) draw cp(x1) × cp(x2). (2) attach the rest of c(x1) to the figure by identifying its subset cp(x1) with the slice cp(x1) × {p} in the cross product. (3) attach the rest of c(x2) to the figure by identifying its subset cp(x2) with the slice {p} × cp(x2) in the cross product. we shall use this algorithm in the following examples. 4.1. the noose. the space is the wedge-product of a circle and an interval. by following the steps outlined above, we get for the hyperspace a solid cylinder with a disc attached along the bottom and a fin attached along the side. (the cross-section of the cylinder is in fact a cardiod, but we represent it here as a circular disc.) see figure 4. p x p c(x1) c(x2) cp(x1) × cp(x2) c(x) figure 4. x a noose, and c(x) the hyperspaces cn(x) for finite ray-graphs 79 4.2. the infinite noose. let x be the infinite noose, made up of x1 ≈ s 1 and x2 ≈ [0, ∞), joined at the point p. cp(x1), as we have already noted in section 4.1, is a cardiod inside the unit disc, which we will draw as a subdisc. recall from section 3.3 that cp(x2) has two components: the left edge of the infinite triangle t ∞, and the left-most point of the ray. the point of c(x2) which corresponds to the single-point set {p} ⊂ x2 is at the bottom of the triangle. crossing cp(x1) with cp(x2), we get an infinite cylinder and a disc. we attach c(x1) to the slice cp(x1) × {p}, along the bottom of the cylinder. we attach c(x2) along {p} × cp(x2), producing an infinite fin off the side of the cylinder and a ray off the side of the disc. see figure 5. p x p c(x) figure 5. x is the infinite noose. c(x) has two components. 4.3. the real line. let x1 = x2 ≈ [0, ∞), both just a single vertex with a ray attached. then we can think of x ≈ r as the result of attaching these two subgraphs along their vertex. both subgraphs have hyperspaces which consist of t ∞ ⊔ [0, ∞), and the containment hyperspace for the vertex is the union of the left edge of the triangle and the leftmost point of the ray. following the algorithm, cp(x1) × cp(x2) gives us four components: an infinite square, two rays, and a point. when we attach the rest of c(x1) along the correct slice, it attaches the rest of the triangle along one side of the infinite square, and the rest of a ray along one of the rays. similarly when we attach the rest of c(x2) it attaches the rest of an infinite triangle along the other side of the infinite square, and another ray along the second ray. the end result is four components: a half-plane, two real lines, and a point. 80 n. esty 5. the connected components of (cn(x), τh) the last two examples of section 4 show that under the hausdorff topology, there is a relationship between the number of rays in a given graph, and the number of connected components of its hyperspace. that relationship is what we explore in this section. let r = {r1, . . . rk} denote the set of rays in a given ray-graph. if #r = k, we will call x an k-legged graph. we will denote by xg = x − ∪ k i=1ri. if a ⊂ x, and a ∩ ri is an unbounded interval, we say that a is unbounded in direction i. in this way we can talk about the unbounded direction set of a, which is the set of indices between 1 and k for which a is unbounded in direction i. clearly there are 2k possible unbounded direction sets, in one-to-one correspondence with the power set of {1, 2, . . . , k}. let pk be the power set of {1, 2, . . . , k}. define a function φ : cn(x) → pk by φ(a) = ∆, where ∆ ∈ pk is the unbounded direction set of a. recall that we denote the hausdorff distance between two elements a and b by dh(a, b). lemma 5.1. let a, b ∈ cn(x) under the hausdorff topology. (1) if dh(a, b) < ∞, then a and b have the same unbounded direction set. (2) if a and b have distinct unbounded direction sets, e.g. there exists a ray ri ∈ x such that a is unbounded in direction i but b is not, then there does not exist any path through cn(x) from a to b. proof. if a is unbounded in direction i and b is not, then clearly dh(a, b) = ∞. since any path is a continuous image of a compact set, it must have a compact image which contains a and b. if dh(a, b) = ∞, this is impossible. � lemma 5.2. if x ∈ x is a k-legged graph, then for all n, the hyperspace (cn(x), τh) has at least 2 k connected components. proof. let x ∈ x be a graph with k distinct rays, labelled r1, . . . , rk. consider the power set pk of {1, . . . , k}. each element of the power set corresponds to an unbounded direction set. we will use the map φ : cn(x) → pk from above, given by φ(a) = ∆ if a is unbounded in the direction set ∆. we will show that φ is continuous, and therefore cn(x) has at least 2 k connected components. because (cn(x), τh) is first countable, it is enough to show convergent sequences are mapped to convergent sequences. let am → a be a convergent sequence of elements of cn(x), meaning that dh(am, a) → 0 as m → ∞. if a is unbounded in direction ri, and am is not (or vice versa) we know the hyperspaces cn(x) for finite ray-graphs 81 dh(a, am) = ∞, so for all m greater than some m ∗ we must have am unbounded in the same set of directions as a. therefore φ(am) = φ(a) for all m > m∗ and φ is continuous. � theorem 5.3. if x ∈ x is a k-legged graph, then for all n, the hyperspace (cn(x), τh) has exactly 2 k path-connected components. the previous lemma showed that cn(x) has at least 2 k connected components. we will now show that it has no more than that, by showing that for all ∆ ∈ pk, {a ∈ cn(x) : φ(a) = ∆} is a path-connected set. this will be done by taking any element in a given component and constructing a path from it to a designated “default” element of that component. a note on notation: a subinterval of a inside the edge ei will be denoted [a, b]ei . a subinterval of a inside the ray ri will be denoted [a, b] r i . for rays, the vertex is 0. for edges which only have one ramification point x, the endpoint 0 is the ramification point. proof. fix n. we begin by choosing for each ∆ ∈ pk a particular element a∆ of {a ∈ cn(x) : φ(a) = ∆}. the element a∆ will consist of the complete finite-graph xg, and all the rays which are in the unbounded direction set ∆, but no part of the other rays. it will have one connected component. precisely, a∆ = xg ∪ ⋃ i∈∆ ri given an element a ∈ cn(x) with φ(a) = ∆, we will construct a path from a to a∆. there are three steps. first, any sections of a contained completely in rays ri where i ∈ ∆ we will grow until they touch xg at the vertex. secondly, any sections of a contained in rays rj where j 6∈ ∆, we will shrink down until they are gone. finally, we grow the remaining subset out so that it includes all of xg. the first two steps of this process will either keep constant or decrease the number of components of a; the last step will produce an element with one component. so the path will stay in cn(x) at all times. by definition, if i ∈ ∆ then a ∩ ri 6= ∅. consider those i ∈ ∆ for which a ∩ ri 6= ri. for each such i, that intersection will be a finite number of intervals, one of which is unbounded. call the unbounded one [ai, ∞) r i , and call the vertex where that ray is attached vi. we will grow this interval out so that it encompasses all of ri. we define the first step in the path as follows. f0 : [0, 1] → cn(x) is given by f0(t) = a ∪ ⋃ i∈∆ [tvi + (1 − t)ai, ∞) r i if a had several intervals contained in that ray, this process will consume them, reducing the number of components of a. if a had empty unbounded 82 n. esty direction set, this will do nothing to a. let a1 = f0(1). now consider those j 6∈ ∆ with a1 ∩ rj 6= ∅. that intersection will consist of a finite number of bounded intervals [aij, b i j] r j (where i = 1, . . . , lj for some lj ≤ k). we wish to shrink and slide each of those intersections down to the vertex vj. to do that we define a path f1 : [0, 1] → cn(x) by f1(t) = a1 ∪ ⋃ j 6∈∆ lj ⋃ i=1 [tvj + (1 − t)a i j, tvj + (t − 1)b i j] r j this is a path from a1 to the set which agrees with a1 in xg, contains all of the rays in the unbounded direction set, but does not contain any section of any rays which are not in the unbounded direction set (apart from possibly the vertices). call this second intermediate set a2 = f1(1). the final step will grow the subset a2 out until it includes all of xg. fix an element a ∈ a2 ∩xg. because xg is a graph, it is path connected, so there exists a path γ : [0, 1] → xg which starts at a and whose image contains all of xg. define f2(t) = a2 ∪ ⋃ x∈[0,t] γ(x) clearly f2(0) = a2 and f2(1) = a∆. the continuity of γ makes f2 continuous, and because components may merge together, but never split apart, the construction ensures f2(t) ∈ cn(x) at all times. following f0 with f1 and f2, we have a path from a to a∆. hence the set {a ∈ cn(x) : φ(a) = ∆} is path-connected. this completes the proof. � 6. connectedness of (cn(x), τv ) in this section we will explore some of the distinctions between a hyperspace of a finite ray-graph under the hausdorff topology with that same hyperspace under the vietoris topology. we begin by recalling the definition. 6.1. the vietoris topology. let x be the base space, and let u1, . . . , un be a finite number of open subsets of x. for any hyperspace h(x) over x, we define the open set u∗ =< u1, . . . , un > in the following way: a ∈ u∗ iff (1) a ⊂ ⋃n i=1 ui (2) a ∩ ui 6= ∅ for all i = 1, . . . , n such open sets make a basis for the vietoris topology on h(x). it is sometimes more useful to treat the topology as the supremum of the upper and lower vietoris topologies. the upper vietoris topology is generated by sets of the form u+ = {a ∈ h(x) : a ⊂ u} the hyperspaces cn(x) for finite ray-graphs 83 where u is open in x. the lower vietoris topology is generated by sets of the form v − = {a ∈ h(x) : a ∩ v 6= ∅} where v is open in x. subbase elements of the vietoris topology are then of the form u+ and v −1 ∩ v − 2 ∩ · · · ∩ v − n . 6.2. path-connectedness. in [4] we proved that cl(m) was contractible for any borel compact space m having the property that the closure of open balls is closed balls. since ray-graphs satisfy those conditions, we know that cl(x) is contractible: theorem 6.1. (cl(x), τv ) is contractible. we now prove a companion theorem to theorem 5.3. theorem 6.2. (cn(x), τv ) is path-connected. the proof is similar in flavor to the proof of theorem 5.3. in fact, the first part is identical: take an element a ∈ c(x) and construct a path from it to the element a∆, ∆ = φ(a). as it was in the hausdorff topology, this construction is continuous in the vietoris topology. the distinction comes when we then form a path from a∆ to the element x. proof. recall that if φ(a) = ∆ ∈ pk, we define the element a∆ as a∆ = xg ∪ ⋃ i∈∆ ri start with the same path f = f2 ◦ f1 ◦ f0 from a to a∆ as given in the proof of theorem 5.3. as before, the path remains in cn(x) at all times, and results in a∆ ∈ c(x). the proof that this is continuous under the vietoris topology is quite similar to the proof that the second path is continuous, so we omit it. for the second path we will connect a∆ to x. let f(t) = t 1−t , and define a path γ : [0, 1] → c(x) by: γ(t) = { a∆ ∪ ⋃ i6∈∆[0, f(t)] r i t ∈ [0, 1) x t = 1 obviously γ(0) = a∆ and by construction, γ(t) ∈ c(x) for all t. to show γ is continuous, it is enough to show that it is continuous with respect to the upper and lower vietoris topologies. note that s < t implies γ(s) ⊂ γ(t). this means for the upper topology, which is concerned with containment, we only have to worry about t increasing, and for the lower topology, which is concerned with intersection, we only have to worry about t decreasing. we begin by checking continuity with respect to the upper vietoris topology. fix t0 ∈ [0, 1] and suppose γ(t0) ∈ u +. if t0 = 1 then γ(t0) = x and as x ∈ u+ we have u = x, which clearly implies γ(t) ∈ u+ for all t. so we 84 n. esty assume that t0 ∈ [0, 1) and that u 6= x. since u 6= x and γ(t0) ⊂ u, there exists ǫ = d(γ(t0), u c) > 0. continuity of f(t) implies that there exists some δ, 0 < δ ≤ ǫ such that if |t − t0| < δ then |f(t) − f(t0)| < ǫ. then the total growth from γ(t0) to γ(t) is small, i.e. d(γ(t), γ(t0)) < ǫ and hence γ(t) ∈ u +. so γ is continuous with respect to the upper vietoris topology. now we check continuity with respect to the lower vietoris topology. let γ(t0) ∈ v − 1 ∩ v − 2 ∩ · · · ∩ v − n , meaning that γ(t0) ∩ vi 6= ∅ for each i. pick a point xi ∈ γ(t0) ∩ vi. if xi ∈ a∆ then xi ∈ γ(t) for all t. if not, then xi ∈ [0, f(t0)] r ℓ for some ℓ. possibly xi is in the interior of that interval, or possibly it is the endpoint f(t0). if xi is in the interior, i.e. xi < f(t0), then pick δi > 0 such that |t − t0| < δi implies |f(t0) − f(t)| < f(t0) − xi, and then xi ∈ f(t) also. if xi = f(t0), then let di = d(xi, v c i ) > 0 and choose δi such that |t − t0| < δi implies |f(t) − f(t0)| < di. then [0, f(t)] r ℓ ∩ vi 6= ∅. all together, let δ = min{δi : i = 1, . . . , n}. then for each i, γ(t) ∩ vi 6= ∅. so γ is continuous with respect to the lower vietoris topology. combining with the path f, we have constructed a path from any a ∈ cn(x) to x. � references [1] r. duda, on the hyperspace of subcontinua of a finite graph i, fund. math. 62 (1968), 265–286. [2] r. duda, on the hyperspace of subcontinua of a finite graph ii, fund. math. 63 (1968), 225–255. [3] c. eberhart and s. nadler, hyperspaces of cones and fans, proc. amer. math. soc. 77 (1979), no. 2, 279–288. [4] n. esty, on the contractibility of certain hyperspaces, top. proc. 32 (2008), 291–300. [5] a. illanes, the hyperspace c2(x) for a finte graph is unique, glasnik mat. 37 (2002), 347–363. [6] a. illanes, finite graphs x have unique hyperspaces cn(x), top. proc. 27 (2003), 179–188. [7] a. illanes and s. nadler, hyperspaces: fundamentals and recent advances, marcel dekker, inc., new york, 1999. (received april 2012 – accepted november 2012) norah esty (nesty@stonehill.edu) department of mathematics, stonehill college, easton, massachusetts 02357 the hyperspaces cn(x) for finite ray-graphs. by n. esty @ appl. gen. topol. 23, no. 1 (2022), 45-54 doi:10.4995/agt.2022.16128 © agt, upv, 2022 investigation of topological spaces using relators gergely pataki department of analysis, budapest university of technology and economics, hungary (pataki@math.bme.hu) department of mathematics and modelling, hungarian university of agriculture and life sciences, hungary communicated by h. dutta abstract in this paper, we define uniformities and topologies as relators and show the equivalences of these definitions with the classical ones. for this, we summarize the essential properties of relators, using their theory from earlier works of á. száz. moreover, we prove implications between important topological properties of relators and disprove others. finally, we show that our earlier analogous definition [g. pataki, investigation of proximal spaces using relators, axioms 10, no. 3 (2021): 143.] for uniformly and proximally filtered property is equivalent to the topological one. at the end of this paper, uniformities and topologies are defined in the same way. this will give us new possibilities to compare these and other topological structures. 2020 msc: 54e15; 54a05; 54g15; 54g20; 54d10. keywords: (generalized) uniformities; (generalized) topologies; relators. 1. introduction at the beginning of the 20th century some mathematicians tried to define abstract topological structures. the most relevant results due to poincaré 1895, fréchet 1906, hausdorff 1914, and kuratowski 1922. uniform spaces in terms of relations were introduced by weil in 1937 [12]. uniform and topological spaces formulated the recently usable form by bourbaki in 1953 [1]. received 26 august 2021 – accepted 23 november 2021 http://dx.doi.org/10.4995/agt.2022.16128 https://orcid.org/0000-0002-4630-9949 g. pataki after the works of davis, pervin, and nakano [2], [8], and [4], in 1987 száz [9] introduced the notion of relators and relator spaces in the following way. definition 1.1. a nonvoid family r of relations on a nonvoid set x is called a relator on x, and the ordered pair (x,r) is called a relator space. in the last decades, a few authors investigated the interpretation of wellknown topological properties in terms of relators. for more details, see, for instance, [7] and [6], but for the readers’ convenience, we summarize the necessary notions and notations. remark 1.2. with the usual notations, the statement r is a relator on x means that x 6= ∅, ∅ 6= r⊂ exp(x2), where exp(x) is the power set of x, and x2 = x ×x. if r is a relation on x, x ∈ x, and a ⊂ x, then the sets r(x) = {y ∈ x : (x,y) ∈ r}, and r[a] = ⋃ x∈a r(x) are called the images of x and a under r, respectively. 2. preliminary concepts definition 2.1. if r and s are relations on x, then the composition of r and s can be defined, such that (r◦s)(x) = r[s(x)] for all x ∈ x. moreover, let r−1 = {(y,x) : (x,y) ∈ r}, r0 = ∆x = {(x,x) : x ∈ x} and rn = r◦rn−1, for all n = 1, 2, . . . . finally, we say that r is • reflexive if r0 ⊂ r, • symmetric if r−1 ⊂ r, • transitive if r2 ⊂ r. lemma 2.2. if r is a relation on x, and a,b ⊂ x, then r[a] ⊂ b ⇐⇒ r−1[x \b] ⊂ x \a. definition 2.3. if r is a relator on x, then the relators r∗ = {s ⊂ x2 : ∃r ∈r : r ⊂ s}, and r∧ = {s ⊂ x2 : ∀x ∈ x : ∃r ∈r : r(x) ⊂ s(x)}, are called the uniform and the topological refinements of r, respectively. for more details, see [7]. moreover, for all n = −1, 0, 1, 2, . . . , we define rn = {rn : r ∈r} . © agt, upv, 2022 appl. gen. topol. 23, no. 1 46 investigation of topological spaces using relators remark 2.4. ∗ and ∧ are really refinements as we defined in [7], that is, they are self-increasing in the sense that r⊂s∗ ⇐⇒ r∗ ⊂s∗ and r⊂s∧ ⇐⇒ r∧ ⊂s∧, or equivalently, they are expansive, increasing, and idempotent, in the sense that r⊂r∗, r⊂s =⇒ r∗ ⊂s∗, r∗∗ = r∗ and r⊂r∧, r⊂s =⇒ r∧ ⊂s∧, r∧∧ = r∧, for all r and s relators on x. moreover, ∧ is ∗-dominating, ∗-invariant, ∗-absorbing and ∗-compatible, that is if r is a relator on x, then r∗ ⊂r∧, r∧ = r∧∗, r∧ = r∗∧, r∧∗ = r∗∧. for all n = −1, 0, 1, 2, . . . the mapping r 7→rn of relators on x is increasing. finally, ∗ is inversion-compatible, that is, for all r relators on x r∗−1 = r−1∗. and we have that for all r relators on x r2∗ = r∗2∗. the following examples show, that the analog assertions are not true for ∧. example 2.5. let x = {1, 2, 3}, and r = {∆x ∪{(1, 2)}, ∆x ∪{(3, 2)} is an elementwise reflexive and transitive relator on x. now, r∧−1 6⊂ r−1∧, since ∆x ∈ r∧−1 however ∆x /∈ r−1∧. moreover, if s = r−1, then ∆x ∈ s−1∧ \s∧−1. note, that r = { �� , �� } . example 2.6. let x = {1, 2, 3, 4}, and r = {∆x ∪{(1, 2), (4, 2), (2, 1), (2, 4)}, ∆x ∪{(1, 3), (4, 3), (3, 1), (3, 4)}} is an elementwise reflexive and symmetric relator on x. now, r∧2 6⊂ r2∧, since r = x2 \{(1, 4), (4, 1)}∈r∧2 however r /∈r2∧. note, that r = { �� , �� } , and r = �� . definition 2.7. if r is a relator on x, then for any x ∈ x and a ⊂ x, we write: x ∈ intr(a) if r(x) ⊂ a for some r ∈r, and a ∈tr if a ⊂ intr(a). © agt, upv, 2022 appl. gen. topol. 23, no. 1 47 g. pataki the relation intr is called the topological interior, and the elements of tr are called the topologically open subsets induced by r on x. theorem 2.8. int : exp(exp(x2)) \{∅}→ exp(exp(x) ×x) is a ∧–increasing set-valued function for relators on x in the sense that s ⊂r∧ ⇐⇒ ints ⊂ intr for any two relators r and s on x. moreover, it follows that if r is a relator on x, then r∧ is the largest relator on x such that intr = intr∧ . theorem 2.9. t : exp(exp(x2)) \{∅}→ exp(exp(x)) is an increasing set-valued function for relators on x in the sense that s ⊂r =⇒ ts ⊂tr for any two relators r and s on x. moreover, if r is a relator on x, then tr = tr∧. 3. a new form of generalized topologies definition 3.1. let r be a relator on x, and let � ∈{∗,∧} be a refinement for relators on x. we define the followings. • r is �-reflexive, if r⊂r0�; • r is �-symmetric, if r⊂r−1�; • r is �-transitive, if r⊂r2�; • r is �-fine, if r = r�. for instance, we say that r is uniformly symmetric or topologically transitive instead of ∗-symmetric or ∧-transitive, respectively. following weil, we say that the relator r on x is a generalized uniformity on x, and the relator space (x,r) is a generalized uniform space if it is • uniformly reflexive; • uniformly symmetric; • uniformly transitive; • uniformly fine. moreover, we say that the relator r on x, is a generalized topology on x, and the relator space (x,r) is a generalized topological space if it is • topologically reflexive; • topologically transitive; • topologically fine. © agt, upv, 2022 appl. gen. topol. 23, no. 1 48 investigation of topological spaces using relators definition 3.2. if x is an arbitrary set, and t ⊂ exp(x) satisfies the following axioms, then we say that it is a generalized set-topology on x. (1) t is closed under arbitrary union, that is ⋃ a∈t for any a⊂t ; (2) x ∈t . the following relations were investigated by davis, pervin, and száz. definition 3.3. let a be a subset of x. then, the relation da = a 2 ∪ (x \a) ×x is called the davis–pervin relation on x generated by a. some parts of the following theorem were proved in [10] and [11]. proposition 3.4. if r is a relator on x, and ψ : p(p(x)) →p(p(x2)), ψ(t ) = {da : a ∈t}, then (1) tr is a generalized set-topology for all r relators on x; (2) ψ(t )∧ is a generalized topology for all ∅ 6= t ⊂p(x); (3) if t is a generalized set-topology on x, then t = tψ(t ); (4) if r is a generalized topology on x, then r = ψ(tr)∧. proof. (1) if a ⊂ tr and x ∈ ⋃ a, then there exists an a ∈ a such that x ∈ a. it follows that r(x) ⊂ a ⊂ ⋃ a for some r ∈r, since a ∈tr. therefore, ⋃ a∈tr. x ∈tr means only that the triviality r(x) ⊂ x for all x ∈ x and r ∈r. note that r 6= ∅. (2) ψ(t )∧ is obviously reflexive and topologically fine. for proving topologically transitivity of ψ(t )∧, that is ψ(t )∧ ⊂ ψ(t )∧2∧ or equivalently ψ(t ) ⊂ ψ(t )∧2∧ let r ∈ ψ(t ) be arbitrary. by the definition of the davis–pervin relations, we have that r2 = r and by using the expansivity of ∧ the proof is complete. (3) at first, we prove that t ⊂ tψ(t ) for all t ⊂ p(x). if a ∈ t , then da ∈ ψ(t ). since da[a] = a, we have that a ∈tψ(t ). on the other hand, let a ∈ tψ(t ). if a = x, then a ∈ t , since t is a generalized set-topology. if a 6= x, and x ∈ a, then there exists a ux ∈ t such that x ∈ ux and ux = dux (x) ⊂ a. a = ⋃ x∈a {x}⊂ ⋃ x∈a ux ⊂ ⋃ x∈a a = a implies that a ∈t since t is closed under arbitrary union. (4) let r ∈ ψ(tr)∧ and x ∈ x. there exists an a ∈ tr such that da ∈ ψ(tr) and da(x) ⊂ r(x). if x ∈ a then s(x) ⊂ a for some s ∈ r since a is topologically open, and da(x) = a implies s(x) ⊂ r(x). if x /∈ a, then da(x) = x therefore obviously s(x) ⊂ x = r(x) for © agt, upv, 2022 appl. gen. topol. 23, no. 1 49 g. pataki some s ∈r. we proved that ψ(tr)∧ ⊂r∧ for an arbitrary r relator on x, that is ψ(tr)∧ ⊂r if r is topologically fine. for the converse inclusion let r ∈ r, x ∈ x and a = intr(r(x)). by the definition of interior, for all y ∈ a there exists an s ∈ r such that s(y) ⊂ r(x). topological transitivity of r implies that there exists a q ∈ r such that q[q(y)] = q2(y) ⊂ s(y) ⊂ r(x), that is q(y) ⊂ a. it follows that a ∈ tr, and then da ∈ ψ(tr). since x ∈ a and r is reflexive we have that da(x) = a ⊂ r(x). we proved that r ∈ ψ(tr)∧ for an arbitrary r ∈r. � several papers including [10] used {r ◦ s : r,s ⊂ r} instead of our r2. because of the definition of uniformities, we need our one, but later in theorem 4.8 we will see that in some cases, these definitions are equivalent. theorem 3.5. if r is a relator on x, then r 7→ tr is a bijection of the set of generalized topologies on x onto the set of generalized set-topologies on x. proof. the injectivity and surjectivity are followed by proposition 3.4 (4) and (3), respectively. � 4. a new form of topologies definition 4.1. let a be a family of sets, or equivalently a ⊂ exp(x) for some set x. we call φ(a) = {⋂ b : ∅ 6= b ⊂a, and b is finite } the filtered family of sets generated by a. moreover, we say that a is filtered if φ(a) = a. remark 4.2. since φ is a refinement for relators on x, we write rφ instead of φ(r), if r is a relator on x. note also that r is filtered iff rφ ⊂r. moreover, note that φ is an inversion compatible refinement for relators on x, that is, if r is a relator on x, then r−1φ = rφ−1. finally, if r is finite, then rφ∗ = { ⋂ r}∗. remark 4.3. by [6] we know that if r is a relator on x, then r∗φ = rφ∗, r2φ ⊂rφ2∗. now, we can state a similar assertion for topological refinement. lemma 4.4. if r is a relator on x, then r∧φ ⊂rφ∧. proof. it is easy to see that ∃∅ 6= s ⊂r∧ finite : ⋂ s = r =⇒ =⇒ ∀x ∈ x : ∃∅ 6= q⊂r finite : ⋂ q(x) ⊂ r(x). � © agt, upv, 2022 appl. gen. topol. 23, no. 1 50 investigation of topological spaces using relators proposition 4.5. if r is a relator on x and � ∈ {∗,∧}, then the following assertions are equivalent. (1) rφ ⊂r�; (2) there exists an s relator on x, such that sφ ⊂s� = r�. proof. we need only to prove the (2) =⇒ (1) implication. for this, we note that if (2) is true, then rφ ⊂r�φ = s�φ ⊂sφ� ⊂s�� = s� = r�. � because of the above proposition, the following definition seems unduly overcomplicated, but from [6] we know, that this is reasonable. definition 4.6. if � is a refinement for relators on x, then we say that the r relator on x is �-filtered if there exists an s relator on x such that sφ ⊂s� = r�. we use the uniformly filtered and topologically filtered notions instead of ∗-filtered and ∧-filtered. note that by proposition 4.5 we have that r is a uniformly (topologically) filtered relator on x iff rφ ⊂ r∗ (rφ ⊂ r∧), which can be found in the definition of uniformities by weil. definition 4.7. if the generalized uniformity/generalized topology r on x is also uniformly/topologically filtered, then we say that r is a uniformity/topology on x, and (x,r) is a uniform space/topological space. moreover, if t is a generalized set-topology on x, such that t is filtered, then we say that t is a set-topology on x. several papers including [7] investigated the following properties. theorem 4.8. if r is topologically fine and topologically reflexive (or topologically filtered), then the following assertions are equivalent. (1) r is topologically transitive; (2) ∀x ∈ x,r ∈r : ∃p,q ∈r : p [q(x)] ⊂ r(x); (3) ∀x ∈ x,r ∈r : x ∈ intr(intr(r(x))); (4) ∀a ⊂ x : intr(a) ∈tr; (5) ∀x ∈ x,r ∈r : intr(r(x)) ∈tr. proof. (1) =⇒ (2) =⇒ (3) =⇒ (4) =⇒ (5) are quite obvious (without extra conditions). note only for proving (3) =⇒ (4), that if x ∈ intr(a), that is there exists an r ∈ r such that r(x) ⊂ a, then by (3) x ∈ intr(intr(r(x))) ⊂ intr(intr(a)) and it follows that there exists an s ∈ r such that s(x) ⊂ intr(a). therefore we prove only the (5) =⇒ (1) implication. for this, let r ∈ r and x ∈ x be fixed and use (5). we have x ∈ intr(r(x)) ∈ tr, that is there exists a q ∈ r such that q(x) ⊂ intr(r(x)). © agt, upv, 2022 appl. gen. topol. 23, no. 1 51 g. pataki for such a q, and for all y ∈ q(x) there exists a py ∈r such that py(y) ⊂ r(x). now define the s relation on x by the following. s(y) =   q(y), if y = x, py(y), if y ∈ q(x) \{x}, x, else. note that because of r is topologically fine, we have that s ∈r∧ = r. if r is topologically reflexive, then intr(r(x)) ⊂ r(x), therefore s2(x) = s[q(x)] = ⋃ y∈q(x)\{x} py(y) ∪q(x) ⊂ r(x). if r is topologically filtered, then write q ∩ r in place of q. note that q∩r ∈rφ ⊂r∧ = r. in this case s2(x) = s[q(x)] ⊂ ⋃ y∈q(x)\{x} py(y) ∪q(x) ⊂ r(x). � proposition 4.9. let r be a relator on x. (1) if r is topologically filtered, then tr is filtered. (2) if r is topologically reflexive and topologically transitive, moreover tr is filtered, then r is topologically filtered. proof. (1) if a ∈ φ(tr), then there exists a nonvoid finite subset b of tr such that a = ⋂ b. let x ∈ a be an arbitrary fixed point. for all b ∈ b we have that x ∈ b ∈ tr, therefore there exists sb ∈ r such that sb(x) ⊂ b. now, with r = ⋂ b∈b sb ∈ r φ ⊂ r∧ we can see that r(x) ⊂ ⋂ b = a, that is a ∈tr∧ = tr. (2) if r ∈rφ, then let s ⊂r nonvoid and finite such that r = ⋂ s. let x ∈ x be an arbitrary fixed point. we need to show that there exists a p ∈r such that p(x) ⊂ r(x). topologically transitivity of r, the filtered property of tr and theorem 4.8 give that u = ⋂ s∈s intr(s(x)) ∈ tr. it is easy to see, that x ∈ intr(s(x)) for all s ∈ r, that is x ∈ u, therefore there exists a p ∈r such that p(x) ⊂ u. on the other hand, since r is topologically reflexive, we have that intr(s(x)) ⊂ s(x), and it follows that u ⊂ ( ⋂ s) (x) = r(x). � it gives the following. theorem 4.10. if r is a relator on x, then r 7→tr is a bijection of the set of topologies on x onto the set of set-topologies on x. proof. proposition 4.9 yields that the range of the bijection in theorem 3.5 restricted to the set of topologies on x is the set of set-topologies on x. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 52 investigation of topological spaces using relators 5. a new form of s0-topologies following notations of [3], (x,r) is called quasi-uniformities, iff r is a uniformly reflexive, uniformly transitive, uniformly filtered, and uniformly fine relator on x. by definition 4.7, we have that (x,r) is a topology, iff r is a topologically reflexive, topologically transitive, topologically filtered and topologically fine relator on x. uniformities have the symmetric property. let us see topologies with this. definition 5.1. if t is a (generalized) set-topology on x, such that for all x,y ∈ x (∃u ∈t : x ∈ u ⊂ x \{y}) =⇒ (∃u ∈t : y ∈ u ⊂ x \{x}) , then we say that t is a (generalized) s0-set-topology on x. lemma 5.2. if r is a relator on x, then ⋂ r⊂ ⋂ r∧. proof. on the contrary, assume that there exist an (x,y) ∈ ⋂ r and an r ∈r∧ such that (x,y) /∈ r. in this case, y /∈ r(x) that is r(x) ⊂ x \{y}. it follows that there exists an s ∈r such that s(x) ⊂ r(x), and this is a contradiction because y /∈ s(x) means y /∈ ( ⋂ r) (x). � note that by the above lemma, we have that ⋂ r = ⋂ r∧ for all r relators on x. proposition 5.3. if r is a generalized topology on x, then the following assertions are equivalent. (1) ⋂ r⊂ ⋂ r−1; (2) ⋂ r⊃ ⋂ r−1; (3) ⋂ r = ⋂ r−1; (4) r is topologically symmetric; (5) tr is a generalized s0-set-topology on x. proof. (1) ⇐⇒ (2) ⇐⇒ (3) is quite obvious since ⋂ r−1 = ( ⋂ r)−1. (4) =⇒ (2): by lemma 5.2 and (4), we have that ⋂ r−1 ⊂ ⋂ r−1∧ ⊂⋂ r. (1) =⇒ (4): by (1) ⋂ r ⊂ ⋂ r−1 ⊂ r−1 for all r ∈ r, that is r−1 ⊂ { ⋂ r}∗ = r∧−1∧−1 since [5] definition 3.1. (3) and 4.1., remark 4.2. and theorem 5.3. it follows that r⊂r∧−1∧ = r−1∧ because of r is topologically fine. (3) =⇒ (5): let x,y ∈ x be fixed, and assume that there exists a u ∈tr such that x ∈ u ⊂ x \ {y}. since u ∈ tr hence y /∈ r(x) for some r ∈ r, and hence y /∈ ( ⋂ r) (x) = (⋂ r−1 ) (x). in this case, x /∈ ( ⋂ r) (y), that is there exists an r ∈ r such that x /∈ r(y). by theorem 4.8 (5), we have that intr(r(y)) ∈tr. it is easy to see that y ∈ intr(r(y)). because of the topologically reflexivity of r it is also easy to see that x /∈ intr(r(y)) that is intr(r(y)) ⊂ x \{x}. © agt, upv, 2022 appl. gen. topol. 23, no. 1 53 g. pataki (5) =⇒ (1): let (x,y) ∈ ⋂ r be arbitrary. if the u topologically open subset contains x, then there exists an r ∈r such that y ∈ ( ⋂ r)(x) ⊂ r(x) ⊂ u. now, by using (5), we have that for all u ∈tr y ∈ u =⇒ x ∈ u. if r ∈ r, then y ∈ intr(r(y)) ∈ tr and hence x ∈ intr(r(y)) ⊂ r(y) since r is topologically reflexive. it follows that (y,x) ∈ ⋂ r that is (x,y) ∈ ⋂ r−1. it holds for an arbitrary (x,y) ∈ ⋂ r, therefore (1) is true. � the previous proposition shows the appropriateness of the following. definition 5.4. if r is a topologically symmetric (generalized) topology on x, then we say that r is a (generalized) s0-topology on x, and the ordered pair (x,r) is called a (generalized) s0-topological space. the previous proposition gives the following. theorem 5.5. if r is a relator on x, then r 7→tr is a bijection of the set of (generalized) s0-topologies on x onto the set of (generalized) s0-set-topologies on x. proof. proposition 5.3 yields that the range of the bijection in theorem 3.5 restricted to the set of generalized s0-topologies on x is the set of generalized s0-set-topologies on x. on the other hand, proposition 5.3 yields that the range of the bijection in theorem 4.10 restricted to the set of s0-topologies on x is the set of s0-settopologies on x. � references [1] n. bourbaki, topologie générale, herman, paris (1953). [2] s. a. davis, indexed systems of neighborhoods for general topological spaces, amer. math. monthly 68 (1961), 886–893. [3] l. nachbin, topology and order, d. van nostrand (princetown, 1965). [4] h. nakano and k. nakano, connector theory, pacific j. math. 56 (1975), 195–213. [5] g. pataki, on the extensions, refinements and modifications of relators, math. balk. 15 (2001), 155–186. [6] g. pataki, investigation of proximal spaces using relators, axioms 10, no. 3 (2021): 143. [7] g. pataki and a. száz, a unified treatment of well-chainedness and connectedness properties, acta math. acad. paedagog. nyházi. (n.s.) 19 (2003), 101–166. [8] w. j. pervin, quasi-uniformization of topological spaces, math. ann. 147 (1962), 316– 317. [9] á. száz, basic tools and mild continuities in relator spaces, acta math. hungar. 50 (1987), 177–201. [10] á. száz, directed, topological and transitive relators, publ. math. debrecen 35 (1988), 179–196. [11] á. száz, relators, nets and integrals, unfinished doctoral thesis (1991). [12] a. weil, sur les espaces a structure uniforme at sur la topologie générale, actualités sci. ind. 551, herman and cie, paris, 1937. © agt, upv, 2022 appl. gen. topol. 23, no. 1 54 @ appl. gen. topol. 20, no. 2 (2019), 325-347 doi:10.4995/agt.2019.11065 c© agt, upv, 2019 the function ωf on simple n-ods ivon vidal-escobar and salvador garcia-ferreira centro de ciencias matemáticas, universidad nacional autónoma de méxico, campus morelia, apartado postal 61-3, santa maŕıa, 58089, morelia, michoacán, méxico. (jpaula@matmor.unam.mx; sgarcia@matmor.unam.mx) communicated by j. galindo abstract given a discrete dynamical system (x, f), we consider the function ωf -limit set from x to 2 x as ωf (x) = {y ∈ x : there exists a sequence of positive integers n1 < n2 < . . . such that lim k→∞ f nk (x) = y}, for each x ∈ x. in the article [1], a. m. bruckner and j. ceder established several conditions which are equivalent to the continuity of the function ωf where f : [0, 1] → [0, 1] is continuous surjection. it is natural to ask whether or not some results of [1] can be extended to finite graphs. in this direction, we study the function ωf when the phase space is a n-od simple t . we prove that if ωf is a continuous map, then fix(f2) and fix(f3) are connected sets. we will provide examples to show that the inverse implication fails when the phase space is a simple triod. however, we will prove that: theorem a 2. if f : t → t is a continuous function where t is a simple triod, then ωf is a continuous set valued function iff the family {f0, f1, f2, . . . } is equicontinuous. as a consequence of our results concerning the ωf function on the simple triod, we obtain the following characterization of the unit interval. theorem a 1. let g be a finite graph. then g is an arc iff for each continuous function f : g → g the following conditions are equivalent: (1) the function ωf is continuous. (2) the set of all fixed points of f2 is nonempty and connected. 2010 msc: 54h20; 54e40; 37b45. keywords: simple triod; equicontinuity; ω-limit set; fixed points; discrete dynamical system. received 28 november 2018 – accepted 15 april 2019 http://dx.doi.org/10.4995/agt.2019.11065 i. vidal-escobar and s. garcia-ferreira 1. introduction in this article, a continuum is a nonempty compact connected metric space. we shall consider discrete dynamical systems (x,f) where the space x is a continuum and f : x → x is a continuous map. given a dynamical system (x,f), we define: f0 as the identity map of x, fn = f◦fn−1 for every positive n ∈ n, and the function ωf -limit set from x to 2x as ωf (x) = {y ∈ x : there is a sequence of positive integers n1 < n2 < ... such that lim k→∞ fnk (x) = y}, for each x ∈ x. we remark that ωf (x) = ⋂ m≥0 {fn(x) : n ≥ m} for every x ∈ x. given ([0, 1],f) a discrete dynamical system, the authors of [1] proved that the following conditions are equivalence: (1) ωf is a continuous function. (2) the set of fixed points of f2, fix(f2), is connected and nonempty. (3) f is equicontinuous. we wonder whether or not this result can be extended to discrete dynamical systems (g,f), where g is finite graph. we answer this question in negative form, actually, we prove that the unique finite graph that satisfies the equivalence (1) ⇔ (2) is the arc. to obtain this result, we start proving some properties that satisfies ωf , when the phase space is a dendroid, a fan and finally a simple triod t, the latter continuum is the union of 3 arcs emanating from a point v such that the intersection of any two of the arcs is v. for a simple triod, we prove that if ωf is a continuous function, then fix(f 2) and fix(f3) are connected and nonempty. moreover, we give examples to show that the connectivity of fix(f2) and fix(f3) does not imply the continuity of the map ωf . the proofs of these assertions will be given in the third section. in the fourth section, we will prove the equivalence (1) ⇔ (3) when the phase space is a simple triod, t. this assertion requires some properties of fix(f) when f is a surjective map and ωf is a continuous function. more precisely, we prove that f−1(fix(f)) = fix(f) and fix(f) coincides with one of the following sets: t , the vertex of t and some edge of t . we also show that each point of t is a periodic point of any continuous surjection f : t → t , with period at most 3. in general is interesting to find conditions equivalent to the equicontinuity of a map (the papers [1], [2], [7], and [9] contain results in this direction). 2. preliminaries given a discrete dynamical system (x,f). for a point x ∈ x, the orbit of x under f is the set of (x) = {fn(x) : n ∈ n}; x is said to be a fixed point of f if f(x) = x, x is said to be n-periodic point, if fn(x) = x and fi(x) 6= x for every 1 ≤ i < n with n ≥ 1; x is said to be periodic point if there exists an n ∈ n such that x is an n-periodic point. the sets of fixed points, n-periodic c© agt, upv, 2019 appl. gen. topol. 20, no. 2 326 the function ωf on simple n-ods points and periodic points of f are denoted by fix(f), pern(f) and per(f), respectively. a function f : x → x is said to be equicontinuous (relative to the metric d) if for each ε > 0, there exists δ > 0 such that d(fn(x),fn(y)) < ε for each x,y ∈ x with d(x,y) < δ and all n ∈ n. a function f : x → x is said to be topological transitivity if for every pair of nonempty open sets u and v in x, there exists n ∈ n such that fn(u) ∩v 6= ∅. for a continuum x we denote the collection of all nonempty compact subsets of x by 2x, we consider 2x equipped with the hausdorff metric. an arc is a continuum homeomorphic to [0, 1]. given an arc a and a homeomorphism h : [0, 1] → a, the points h(0) and h(1) are called the end points of a. a simple closed curve is a continuum homeomorphic to a circle. given a point x ∈ x, x is said to be an end point of x, if for each arc a in x such that x ∈ a, then x is an end point of the arc a. let n ∈ n ∪{ω}, x is said to be a point of order n in the classical sense, or here briefly a point of order n, ordx(x) = n, if x is a unique common end point of every two of exactly n arcs contained in x. if ordx(x) ≥ 3, then x is said to be a ramification point of x. the sets of end points and ramification points of x are denoted by e(x) and r(x), respectively. a continuum x is said to be arc-wise connected provided that for every two points a,b ∈ x there exists an arc in x with end points a and b. x is said to be unicoherent provided that for every two proper subcontinua a and b of x such that x = a∪b, we have that a∩b is connected. a continuum x is irreducible between a and b if no proper subcontinuum of x contains a and b. given a, b and c subontinua of x, we said that c is irreducible from a to b if a ∩ c 6= ∅ 6= b ∩ c and no proper subcontinuum of c intersects both a and b. given a property p, a continuum x is said to be hereditarily p , provided that every non degenerate subcontinuum of x has the property p. a dendroid is a hereditarily arc-wise connected and hereditarily unicoherent continuum. it is well know that a dendroid is unique arc-wise connected. if x is a dendroid we denote by [a,b] the arc in x with end points a and b. a fan is a dendroid with exactly one ramification point. a finite graph is a continuum which can be written as the union of finitely many arcs any two of which are either disjoint or intersect only in one or both of their end points. a tree is finite graph without simple closed curves. given n ∈ n, n > 2 a simple n−od with vertex v is the union of n arcs emanating from the point v and such that v is the intersection of any two of the arcs, v is called de vertex of the simple n-od. throughout this paper, t will be denote a simple n-od with vertex v and set of end points e(t) = {e1,e2, . . . ,en}, we consider t with convex metric d. in order to define some specific functions on a simple triod we will consider the special case when t = ([−1, 1] ×{0}) ∪ ({0}× [0, 1]), t is a subspace of the euclidian plane r2, and its vertex will be v = (0, 0) and its end points will be e1 = (−1, 0), e2 = (1, 0), e3 = (0, 1). the following result is well known and will very useful throughout this work, we present a proof for the convenience of the reader. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 327 i. vidal-escobar and s. garcia-ferreira proposition 2.1. let x be a dendroid. if f be a closed non-connected subset of x, then there exist points a,b ∈ x such that [a,b] ∩f = {a,b}. proof. let a and b components of f, and let y be the irreducible continuum from a to b. by [8, 11.37(a)] there exist a0 ∈ a and b0 ∈ b such that y be the irreducible continuum between a0 and b0. since x is a dendroid, we have that y = [a0,b0]. if [a0,b0] ∩ f = {a0,b0}, then the proposition follows. suppose that [a0,b0] ∩ f 6= {a0,b0}. let p ∈ [a0,b0] − f and let u ⊂ [a0,b0] − f be the component of [a0,b0] − f that contains p. we observe that u = (a,b). by [8, 5.6], we have that f ∩bd(u) 6= ∅ and then f ∩{a,b} = {a,b}, hence [a,b] ∩f = {a,b}. � the implicit properties of continua that we will use in this article may be found in the book [8]. 3. connectivity of the set of fixed points of an iterate to start this section, we shall study those discrete dynamical systems (x,f) where x is a fan and the function ωf is continuous. the next, four lemmas will provide the basic information that we need for our purposes. lemma 3.1. let (x,f) be a discrete dynamical system where x is a dendroid. if n ∈ n and a,x ∈ x such that [a,x] ∩fix(fn) = {a} and fn(z) ∈ [a,z) for each z ∈ (a,x], then we have that a ∈ ωf (z) for each z ∈ [a,x]. proof. let z ∈ [a,x]. if z = a the result is immediately. assume that z 6= a, it follows from the hypothesis that fnk(z) ∈ [a,fn(k−1)(z)) for all k ∈ n. then we have that {fnk(z)}∞k=1 is a monotone sequence and hence this sequence converges. set c := limk→∞f nk(z) and notice that limk→∞f n(fnk(z)) = c. then we have that fn(c) = c. since fnk(z) ∈ [a,x] for each k ∈ n, we must have that c ∈ [a,x] and since [a,x]∩fix(fn) = {a}, c = a. thus, a ∈ ωf (z). � lemma 3.2. let (x,f) be a discrete dynamical system where x is a dendroid such that ωf is a continuous function. if n ∈ n and a,b,c ∈ x such that b ∈ (a,c), [a,c] ∩ fix(fn) = {a,c}, fn(b) = a, then ωf (y) = ωf (c) for each y ∈ [a,c]. proof. fix an arbitrary point y0 ∈ [a,c]. since fn(b) = a and fn(c) = c, we have that y0 ∈ [a,c] ⊂ fn([b,c]). hence, there exists y1 ∈ [b,c] such that fn(y1) = y0. notice that y1 ∈ (y0,c]; otherwise, y0 ∈ [y1,c] and since fn(b) = a, [b,y1] ∩ fix(fn) 6= ∅ which contradicts the hypothesis. as y1 ∈ [y0,c] ⊂ fn([y1,c]), then there exists y2 ∈ [y1,c] such that f(y2) = y1, and since y2 ∈ [y1,c] ⊂ fn([y2,c]), we can find y3 ∈ [y2,c] so that f(y3) = y2. by continuing with this process, we may construct a sequence {yk}∞k=1 such that yk+1 ∈ [yk,c] and fn(yk) = yk−1 for each k ∈ n. since {yk}∞k=1 is a monotone sequence, it converges and ωf (yk) = ωf (y0), for each k ∈ n. put limk→∞yk = d and notice that limk→∞f n(yk) = d ∈ [b,c]. then fn(d) = d and since [b,c] ∩ fix(fn) = {c}, we have that d = c. by the continuity of ωf , we obtain c© agt, upv, 2019 appl. gen. topol. 20, no. 2 328 the function ωf on simple n-ods that limk→∞ωf (yk) = ωf (c), and as ωf (yk) = ωf (y0) for each k ∈ n, then ωf (y0) = ωf (c). � lemma 3.3. let (x,f) be a discrete dynamical system where x is a dendroid such that ωf is a continuous function. if n ∈ n and a,b,c ∈ x such that b ∈ (a,c) and [a,b] ∩fix(fn) = {a}, fn(b) = c, then ωf (y) = ωf (a) for each y ∈ [a,c]. proof. by a procedure similar to the one used in the proof of lemma 3.2, we may construct a sequence {yk}∞k=1 such that limk→∞yk = d exists, yk+1 ∈ [a,yk] and f n(yk) = yk−1 for every k ∈ n. notice that limk→∞fn(yk) = d ∈ [a,b]. then fn(d) = d and since [a,c] ∩ fix(fn) = {a} we have that d = a. since ωf is a continuous function we have that limk→∞ωf (yk) = ωf (a), and as ωf (yk) = ωf (y0) for each k ∈ n, then ωf (y0) = ωf (a). � lemma 3.4. let (x,f) be a discrete dynamical system where x is a dendroid such that ωf is a continuous function. if b,c,d ∈ x such that b 6= d, c ∈ (b,d), f(c) = d, f(d) = b, then ωf (c) ⊂ [c,d]. proof. as f(c) = d and f(d) = b, then [b,d] ⊂ f([c,d]). hence, there exist r1,s1 ∈ [c,b] and r1 6= s1 such that f(r1) = c, f(s1) = d and f((r1,s1)) = (c,d). it is easy to proof that r1 6= c and s1 6= d. following with this procedure, we may construct two sequences {rn}∞n=1 and {sn}∞n=1 such that [rn+1,sn+1] ⊂ [rn,sn], rn 6= rn+1 6= sn, sn+1 6= sn, fn(rn) = c, fn(sn) = b, fn((rn,sn)) = (c,b) for each n ∈ n. then, we have that ωf (rn) = ωf (c) and ωf (sn) = ωf (b) for each n ∈ n. without loss of generality, suppose that limn→∞rn = r. observe that r ∈ ⋂∞ n=1[rn,sn]. since ωf is a continuous function limn→∞ωf (rn) = ωf (r). on the other hand, since ωf (rn) = ωf (c) for each n ∈ n, ωf (r) = ωf (c). since fn(r) ∈ fn([rn,sn]) = [c,d] for each n ∈ n, then we obtain that ωf (r) ⊂ [c,d]. therefore, ωf (c) ⊂ [c,d]. � theorem 3.5. let (x,f) be a discrete dynamical system where x is a fan such that ωf is a continuous function. if n ∈ n and a,b ∈ fix(fn) such that a 6= b and fix(fn) ∩ [a,b] = {a,b} then ωf (a) = ωf (b). proof. let v be the ramification point of x. we shall consider the following cases: case 1. fn(x) ∈ [a,b] for every x ∈ (a,b). since fix(fn) ∩ [a,b] = {a,b}, we have that either fn(x) ∈ [a,x) for every x ∈ (a,b) or fn(x) ∈ (x,b] for every x ∈ (a,b). without loss of generality suppose that fn(x) ∈ [a,x) for every x ∈ (a,b). it follows from lemma 3.1 that a ∈ ωf (x), for each x ∈ (a,b). as ωf is a continuous function, we obtain that a ∈ ωf (b). since fn(b) = b and fn(a) = a, and a ∈ ωf (b) = {f(b), . . . ,fn(b)}, we must have that ωf (b) = ωf (a). case 2. there exists x ∈ (a,b) such that fn(x) /∈ [a,b]. choose e1,e2 ∈ e(x) such that [a,b] ⊂ [e1,e2] and a ∈ [e1,v]. case 2.1. fn(x) ∈ [e1,e2]. since fn(x) ∈ [e1,e2] − [a,b], without loss of generality, we may assume that fn(x) ∈ [e1,a). then, it follows from the continuity of fn that there exists c ∈ [a,b] such that fn(c) = a. thus, by c© agt, upv, 2019 appl. gen. topol. 20, no. 2 329 i. vidal-escobar and s. garcia-ferreira lemma 3.2, we obtain that ωf (y) = ωf (b) for each y ∈ [a,b]. as a consequence, ωf (a) = ωf (b). case 2.2. fn(x) ∈ x − [e1,e2]. without loss of generality, we suppose that x ∈ [a,fn(x)]. it follows from lemma 3.3 that ωf (y) = ωf (a) for each y ∈ [a,fn(x)]. if [a,b] ⊂ [a,v], then ωf (b) = ωf (a) since [a,v] ⊂ [a,fn(x)]. next, assume that [a,b] 6⊂ [a,v]. then, b 6= v and so either fn((v,b]) 6⊂ [v,e2] or fn((v,b]) ⊂ [v,e2]. case 2.2.1. fn((v,b]) 6⊂ [v,e2]. then there exists y ∈ [v,b] so that fn(y) = v. according to lemma 3.3, we have that ωf (y) = ωf (b) for each y ∈ [v,b]. since ωf (v) = ωf (a) and ωf (v) = ωf (b), we obtain that ωf (a) = ωf (b). case 2.2.2. fn((v,b]) ⊂ [v,e2]. • if fn(y) ∈ (y,b], for each y ∈ [v,b], by lemma 3.1, then we have that b ∈ ωf (y) for each y ∈ [v,b]. in particular, b ∈ ωf (v) = ωf (a). therefore, ωf (a) = ωf (b). • there exists c ∈ [v,b] ⊂ [a,b] such that fn(c) = b. by lemma 3.2, we know that ωf (y) = ωf (a) for each y ∈ [a,b]. hence, ωf (b) = ωf (a). in both subcases, we conclude that ωf (a) = ωf (b). � in [1] the authors studied the relation between the continuity of ωf and the connectivity of the sets fix(f) and fix(f2), where f is a continuous map from [0, 1] to [0, 1] . the corollary 3.6 is a generalization of [1, lemma 1.1] when f is a continuous function from a fan to itself. further the theorem 3.8 generalizes [1, theorem 1.2 (1) → (6)], in the case where f is a continuous function from a simple triod to itself. corollary 3.6. let (x,f) be a discrete dynamical system where x is a fan such that ωf is a continuous function. then fix(f) is connected. proof. suppose to the contrary, fix(f) is not connected. then, by proposition 2.1, there are points a,b ∈ fix(f), a 6= b such that [a,b] ∩fix(f) = {a,b}. it follows from theorem 3.5 that ωf (a) = ωf (b), but this is impossible because of ωf (a) = {a} and ωf (b) = {b}. � remark 3.7. let (x,f) be a discrete dynamical system where x is a fan. if a,b ∈ fix(fn) are distinct, for some 1 < n ∈ n, and ωf (b) = ωf (a), then the following statements a 6= f(a) and b 6= f(b). from now on we will consider discrete dynamical systems on a simple triod. our next task is to analyze the consequences when ωf is a continuous function. theorem 3.8. let (t,f) be a discrete dynamical system where t is a simple n-od and such that ωf is a continuous function, then fix(f 2) is connected. proof. let v ∈ t the vertex of t. suppose to the contrary that fix(f2) is nonconnected. by proposition 2.1, there exist two points a,b ∈ fix(f2) a 6= b such that [a,b]∩fix(f2) = {a,b}. theorem 3.5 asserts we have that ωf (a) = ωf (b). so, f(a) = b, and f(b) = a. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 330 the function ωf on simple n-ods case 1. v /∈ (a,b). since [a,b] ⊂ f([a,b]) and v /∈ (a,b), then f would have a fixed point on (a,b), and hence (a,b) also would have a fixed point of f2, this is a contradiction. case 2. v ∈ (a,b). without loss of generality, we may assume that a ∈ [e1,v] and b ∈ [v,e2]. notice that [a,b] ⊂ [e1,e2]. case 2.1. f(v) ∈ [e1,e2]. suppose that f(v) ∈ (v,e2]. since [v,f(v)] ⊂ [a,f(v)] ⊂ f([v,b]), we have that [v,b) has a fixed point of f. therefore [v,b) has a fixed point of f2, this contradicts our supposition. similarly, we analyze the case when f(v) ∈ (v,e1]. case 2.2. f(v) /∈ [e1,e2]. since v ∈ [a,f(v)] = [f(b),f(v)] ⊂ f([v,b]) there exists v1 ∈ [v,b] such that f(v1) = v. notice that v 6= v1, in other case v ∈ fix(f2) which is impossible because of fix(f2) ∩ [a,b] = {a,b}. as v1 ∈ [v,b] ⊂ [b,f(v)] and [v1,b] ∩ fix(f2) = {b} it follows from lemma 3.3 that ωf (y) = ωf (b), for each y ∈ [b,f(v)], in particular ωf (v) = ωf (a) = {a,b}. let ε > 0 such that bε(b) ⊂ (v1,e2] and bε(a) ⊂ (v,e1], by continuity there exists 0 < δ < ε such that if d(x,y) < δ then d(f(x),f(y)) < ε. since ωf (v) = ωf (b) = {a,b} there exists m ∈ n such that fm(v) ∈ bδ(a). so, fm+1(v) ∈ bε(b). since [v,v1] ⊂ [fm(v),fm+1(v)] = [fm+1(v1),fm+1(v)] ⊂ fm+1([v,v1]) we have that [v,v1] ∩fix(fm+1) 6= ∅. given x ∈ [v,v1] ∩fix(fm+1), we obtain that ωf (x) = of (x). on the other hand as x ∈ [v,v1] ⊂ [b,f(v)], we have that ωf (x) = ωf (b) = {a,b}, then of (x) = {a,b} which is impossible because of a 6= x 6= b and x ∈of (x). � the following example show that the converse of corollary 3.6 and theorem 3.8 are not true. therefore, we have that [1, theorem 1.2 (6) ⇔ (1)] is not satisfied, when f is a continuous function from a simple triod to itself. example 3.9. we will give a function f from t to itself such that the sets fix(f) and fix(f2) are connected but the function ωf is not continuous and fix(f3) is not connected. in order to have this done, we will assume that t = ([−1, 1] ×{0}) ∪ ({0}× [0, 1]). define f : t → t given by f((x,y)) =   (2x + 3 2 , 0) (x,y) ∈ [−1,−3 4 ] ×{0}, (3 + 4x, 0) (x,y) ∈ [−3 4 ,−1 2 ] ×{0}, (−2x, 0) (x,y) ∈ [−1 2 , 0] ×{0}, (0, 2x) (x,y) ∈ [0, 1 2 ] ×{0}, (0, 3 − 4x) (x,y) ∈ [ 1 2 ,−3 4 ] ×{0}, (2x− 3 2 , 0) (x,y) ∈ [−3 4 , 1] ×{0}, (−2y, 0) (x,y) ∈{0}× [0, 1 2 ], (3 − 4y, 0) (x,y) ∈{0}× [ 1 2 , 3 4 ], (0, 2y − 3 2 ) (x,y) ∈{0}× [ 3 4 , 1]. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 331 i. vidal-escobar and s. garcia-ferreira in the figure 1 we have the graph of f. figure 1. in the figure 2 we have the graph of f2. figure 2. we will see that fix(f) and fix(f2) are connected sets however ωf is not a continuous function. indeed, it follows from the definition that fix(f) = {v} = fix(f2). further, if a is a subset of t , with t 6= a 6= {v}, then we have that f(a) 6= a. by [5, theorem 2.2.2 (1) ⇔ (9)], we know that f is a transitive continuous function. hence, according to [5, theorem 2.2.2 (1) ⇔ (16)], we have that there exists x0 ∈ t such that of (x0) = t . so, for every ε > 0, we can find y0 ∈ of (x0) so that d(y0,v) < ε, but ωf (v) = {v} and ωf (y0) = ωf (x0) = t , which shows that ωf is not a continuous function at v. finally, its easy to see that fix(f3) is not a connected set (compare with theorem 3.11 below). next let see how example 3.9 gives a new characterization of the arc in terms of the continuity of ωf and the connectivity of fix(f 2). theorem a 1. let g be a finite graph. then g is an arc if and only if for each continuous function f : g → g, the following conditions are equivalent: (1) ωf is a continuous function, (2) fix(f2) is connected and nonempty. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 332 the function ωf on simple n-ods proof. necessity. suppose that g is an arc and that f : g → g is a continuous function, since an arc has the fixed point property, we have that fix(f2) is nonempty. in accordance with [1, theorem 1.2 (1) ↔ (6)], we have that the conditions (1) and (2) are equivalents. sufficiency. assume that conditions (1) and (2) are equivalents for each continuous function f : g → g. suppose that g is not an arc, we need analyze the following two cases: case 1. g has a simple closed curve. let s ⊂ g be a simple closed curve. since g is 1-dimensional, by [3, theorem vi 4], there is retraction r : g → s and we consider an irrational rotation g : s → s. then, the function f = g◦r is a continuous and satisfies that ωf (x) = s for each x ∈ g. that is, ωf is a continuous function. on the other hand, since g is irrational rotation we have that fix(f2) = ∅, this is a contradiction. case 2. g has not a simple closed curve. in this case, g contains a simple triod t with vertex v. according to [6, theorem 2.1], there is a retraction r : g → t. if f : t → t is as in example 3.9, then h = f ◦ r is a continuous function such that fix(h2) = {v} and ωh is not a continuous function, but this is impossible. in any case we obtain a contradiction. therefore, g is an arc. � the following theorem shows that the statement of theorem 3.8 is valid for f3. lemma 3.10. let (x,f) be a discrete dynamical system. if a,b ∈ fix(f3), a 6= b and ωf (a) = ωf (b). then the following conditions hold: (1) a 6= fi(a), with i ∈{1, 2}, (2) b 6= fi(b), with i ∈{1, 2}. proof. as ωf (a) = ωf (b), we know that either a = f(b) or a = f 2(b). (1). suppose that a = f(a). if a = f(b), then a = f2(a) = f3(b) = b, but this is a contradiction. now, if a = f2(b), then a = f(a) = f3(b) = b and so we obtain a contradiction. assume that f2(a) = a. if a = f(b), then a = f2(a) = f3(b) = b, which is impossible. if a = f2(b), then f(a) = f3(b) = b and a = f2(a) = f(b), then b = f2(b), hence a = b and we have a contradiction. clause (2) is established in a similar way. � the function f : [0, 1] → [0, 1], define by f(x) = 1 − x for each x ∈ [0, 1], witnesses that lemma 3.10 does not work for n = 4. theorem 3.11. let (t,f) be a discrete dynamical system where t is a simple n-od and such that ωf is a continuous function, then fix(f 3) is connected. proof. let v ∈ t the vertex of t. suppose that fix(f3) is not connected. by proposition 2.1, there are points a,b ∈ fix(f3) such that [a,b] ∩ fix(f3) = {a,b}. as a consequence of theorem 3.5, we obtain that ωf (a) = ωf (b). by lemma 3.10, we can suppose that ωf (b) = {b,f(b),a}. case 1. there exists an arc a ⊂ t such that ωf (b) ⊂ a. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 333 i. vidal-escobar and s. garcia-ferreira case 1.1. a ∈ (b,f(b)). being that f(a) = b and f(b) = f(b), by lemma 3.4, we obtain that ωf (a) ⊂ [b,a], which contradicts f(b) 6∈ [b,a]. case 1.2. f(b) ∈ (a,b). since f(f(b)) = a and f(a) = b, we obtain of lemma 3.4 that ωf (f(b)) ⊂ [a,f(b)], this is impossible because of b 6∈ [a,f(b)]. case 1.3. b ∈ (a,f(b)). as f(b) = f(b) and f(f(b)) = a, by lemma 3.4 we have that ωf (b) ⊂ [b,f(b)], this is a contradiction to a 6∈ [b,f(b)]. case 2. ωf (b) 6⊂ a for any arc a ⊂ t. without loss of generality, suppose that a ∈ [v,e1], b ∈ [v,e2] and f(b) ∈ [v,e3]. case 2.1. f(v) ∈ [e1,e2]. assume that f(v) ∈ (v,e2]. as [f(v),v] ⊂ [f(v),f(b)] ⊂ f([v,b]), we must have that [v,b) has a fixed point of f; hence, (a,b) has a fixed point of f3 which is a contradiction. similarly, we obtain a contradiction if f(v) ∈ (v,e1]. case 2.2. f(v) /∈ [e1,e2]. case 2.2.1. f(v) ∈ (v,e3]. since [v,f(v)] ⊂ [a,f(v)] = [f(f(b)),f(v)] ⊂ f([v,f(b)]), we have that [v,f(b)] has a fixed point of f. let z ∈ [v,f(b)] be a fixed point of f. as [v,f(b)] ⊂ [f(v),b] = [f(v),f(a)] ⊂ f([a,v]), there exits y ∈ [a,v] for which f(y) = z. then, ωf (y) = {z}. on the other hand, as v ∈ [a,f(v)] = [f(f(b)),f(v)] ⊂ f([f(b),v]) there is v1 ∈ (v,f(b)) such that f(v1) = v. as v1 ∈ [b,f(b)] = [f(a),f(b)] ⊂ f([a,b]) there is v2 ∈ (a,b) such that f(v2) = v1 and so f 3(v2) = f(v). as [a,b]∩fix(f3) = {a,b}, if v2 ∈ [a,v], then [a,v2] ∩fix(f3) = {a} and v2 ∈ [a,f(v)] it follows from lemma 3.3 that ωf (x) = ωf (a) for each x ∈ [a,f(v)] in particular ωf (y) = ωf (a). hence, ωf (a) = {z} but this is impossible because of ωf (a) = {b,f(b),a}. similarly in the case that v2 ∈ [v,b] we obtain a contradiction. case 2.2.2. f(v) /∈ (v,e3]. since v ∈ [f(b),f(v)] ⊂ f([v,b]) there exists v1 ∈ [v,b] such that f(v1) = v. since [v,b] ∩ fix(f) = ∅ we have that v 6= v1. as v1 ∈ [v,b] ⊂ [f(v),b] and [v1,b] ∩ fix(f2) = {b} it follows from lemma 3.3 that ωf (y) = ωf (b), for each y ∈ [b,f(v)]. thus, ωf (v) = ωf (b) = {b,f(b),a}. let ε > 0 such that bε(b) ⊂ (v1,e2] and bε(a) ⊂ (v,e1], since f is a continuous function there exists 0 < δ < ε such that if d(x,y) < δ then d(f(x),f(y)) < ε. since ωf (v) = ωf (b) = {b,f(b),a} there exists m ∈ n such that fm(v) ∈ bδ(a). hence, as f(a) = b we have that fm+1(v) ∈ bε(b). on the other hand [v,v1] ⊂ [fm(v),fm+1(v)] = [fm+1(v1),fm+1(v)] ⊂ fm+1([v,v1]) we have that [v,v1]∩fix(fm+1) 6= ∅. let x ∈ [v,v1]∩fix(fm+1). we have that ωf (x) = of (x). as x ∈ [v,v1] ⊂ [b,f(v)], we obtain that ωf (x) = ωf (b) = {b,f(b),a} which is not possible because of a 6= x 6= b, f(b) 6= x and x ∈of (x) � the following example shows that the converse of theorem 3.11 is not true. example 3.12. we will give a continuous function f from t to itself such that the sets fix(f) and fix(f3) are connected but the function ωf is not continuous and fix(f2) is not connected. define f : t → t given by c© agt, upv, 2019 appl. gen. topol. 20, no. 2 334 the function ωf on simple n-ods f((x,y)) =   (−3 2 (x + 1 3 ), 0) (x,y) ∈ [−1,−1 3 ] ×{0}, (0, 1 3 + x) (x,y) ∈ [−1 3 , 0] ×{0}, (0, 1 3 −x) (x,y) ∈ [0, 1 3 ] ×{0}, ( 3 2 ( 1 3 −x), 0) (x,y) ∈ [ 1 3 , 1] ×{0}, (0, 1 3 ) (x,y) ∈{0}× [0, 1]. the graph of f, f2 and f3 appear in the figure 3. figure 3. it follows from the definition that fix(f) = {(0, 1 3 )} = fix(f3) which are both connected. since fix(f2) = {(0, 1 3 ),e1,e2}, then fix(f2) cannot be connected. hence, by theorem 3.8 we conclude that the function ωf is not continuous. by a procedure similar to the one used in the construction of example 3.9, we can define a function f from a simple 4-od to itself, for which the sets fix(f), fix(f2) and fix(f3) are connected, but fix(f4) is not connected and ωf is not continuous. question 3.13. let t be a simple n-od and f : t → t be a continuous map. if there is m ∈ n such that fix(fn) is connected, for all n > m, must ωf be a continuous map? in relation with theorem 3.8 and theorem 3.11 we have the following questions. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 335 i. vidal-escobar and s. garcia-ferreira question 3.14. let t be a simple n-od and let f : t → t be a continuous map. if ωf is a continuous map, must fix(f m) be connected for all m ≥ 4? question 3.15. let x be a fan and let f : x → x be a continuous map. if ωf is a continuous map, must fix(f m) be connected for all m ≥ 2? in connection with question 3.13, the following example shows that, there is a fan x and a continuous function f : x → x that satisfy: fix(fn) is a connected set, for all n ∈ n and ωf is not continuous. example 3.16. let l be the segment of the line that join (0, 0) with (1, 0) and for each n ∈ n let ln be the segment of the line that join (0, 0) with (1, 1n). define x = ⋃∞ n=1 ln ∪ l. for each n ∈ n define fn : [0, 1] → ln given by fn(t) = (t, t n ), fn is an homeomorphism. define f : x → x given by f((x,y)) = { fn( n n+1 (f−1n (x,y))) (x,y) ∈ ln, (x,y) (x,y) ∈ l. figure 4. then, we have that ωf ((x,y)) = { (0, 0) (x,y) ∈ ln, (x,y) (x,y) ∈ l. hence, ωf is not continuous. on the other hand fix(f n) = l, so fix(fn) is connected, for each n ∈ n. 4. equicontinuity of functions on the simple triod now, we address our attention to discrete dynamical systems on a simple triod. through this section t will be denote a simple triod with vertex v and set of end points e(t) = {e1,e2,e3}, we consider t with convex metric d . mainly we prove that ωf is a continuous function if and only if f is equicontinuous. to have this done we need the following preliminary results. lemma 4.1. if (t,f) is a discrete dynamical system such that f is a surjective map and ωf is a continuous function. if x0 ∈ t −fix(f) satisfies that f(x0) ∈ fix(f), then the following conditions hold: c© agt, upv, 2019 appl. gen. topol. 20, no. 2 336 the function ωf on simple n-ods (1) there exists a sequence {xn}∞n=1 with the following properties: (a) xn ∈ t −fix(f) for each n ∈ n, (b) xi 6= xj for each i,j ∈ n such that i 6= j, (c) f(xn) = xn−1 for each n ∈ n, and (d) limn→∞xn = f(x0). (2) f(x0) ∈ fr(fix(f)), (3) [f(x0),x0] ∩fix(f) = {f(x0)}. even more, when fix(f) = {f(x0)} we have stronger properties: (4) if f(x0) ∈ (v,ei) for some i{1, 2, 3}, then there exists a strictly growing sequence {kn}∞n=1, when kn ∈ n, that satisfies the following conditions: (i) if n is odd xkn ∈ (ei,f(x0)) and xkj+1 ∈ (ei,xkj ) for each j ∈ {kn, . . . ,kn+1 − 2}, (ii) if n is even xkn ∈ (f(x0),x0) and xkj+1 ∈ (xkj,x0) for all j ∈ {kn, . . . ,kn+1 − 2}. (5) if f(x0) = v and the sequence {xn}∞n=1 satisfies that {xn : n ∈ n}∩ (v,ei] is infinite, for each i ∈{1, 2, 3}, then there are strictly increasing sequences {kn}∞n=1 and {ln}∞n=1 that satisfy kn ≤ ln < kn+1, for every n ∈ n, and the following conditions: (i) if n ∈ n∪{0} and i ∈{1, 2, 3}, then xk3n+i ∈ (ei,v), (ii) if n ∈ n, i ∈ {1, 2, 3}, ln > kn and xkn ∈ (ei,v), then xkj+1 ∈ (xkj,ei], for each j ∈{kn, . . . , ln − 1}, (iii) if n ∈ n and i ∈ {1, 2, 3}, then xl3n+i ∈ (v,ei) and xl3n+i+1 /∈ (v,ei). proof. (1). since f is surjective there exists x1 ∈ t such that f(x1) = x0, notice that x1 6= x0 and x1 ∈ t − fix(f). assume that we constructed x1,x2, . . . ,xn−1 that satisfy (a) − (c). being as f is surjective there exists xn ∈ t such that f(xn) = xn−1, it is clear that xn satisfy (a) − (c). thus, we construct our sequence {xn}∞n=1. it follows from (c) that fn+1(xn) = f(x0) for each n ∈ n. hence, ωf (xn) = {f(x0)} for each n ∈ n. now, we procedure to prove (d). without loss of generality, suppose that limn→∞xn = z, then f(z) = limn→∞f(xn) = limn→∞xn−1 = z. on the other hand, since ωf is a continuous function, we have that limn→∞ωf (xn) = {f(x0)} for each n ∈ n, then ωf (z) = {f(x0)}. then, z = f(x0) and (d) is proved. (2). by (a) and (d), we have that xn ∈ t − fix(f) for each n ∈ n, and limn→∞xn = f(x0), further, by hypothesis f(x0) ∈ fix(f). therefore, f(x0) ∈ fr(fix(f)) and this shows (2). to prove (3) suppose that [f(x0),x0] ∩ fix(f) 6= {f(x0)}. since fix(f) is connected, there is z ∈ fix(f) −{f(x0)}, for which [f(x0),x0] ∩fix(f) = [f(x0),z]. let y ∈ (f(x0),z) ⊂ fix(f). since [f(x0),z] ⊂ f([z,x0]), there exists x ∈ (z,x0) such that f(x) = y and x /∈ fix(f). according to (d) and (2), we have that there exists a sequence {yn}∞n=1 such that limn→∞yn = u, and y ∈ fr(fix(f)), but this is a contradiction because of u ∈ (f(x0),z). so, (3) is proved. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 337 i. vidal-escobar and s. garcia-ferreira (4) without loss of generality, we assume that f(x0) ∈ (e1,v). notice that [e1,f(x0)) ∩fix(f) = ∅ = (f(x0),v] ∩fix(f). hence, we must have that • f(y) ∈ (y,v] ∪ [v,e2] ∪ [v,e3] for each y ∈ [e1,f(x0)) and • f(y) ∈ [e1,y), for each y ∈ (f(x0),v]. let ε > 0 such that bε(f(x0)) ⊂ (e1,v). since limn→∞xn = f(x0), there is m ∈ n that satisfies that xn ∈ bε(f(x0)) for each n > m. fix k1 > m so that xk1 ∈ (e1,f(x0)). since f(xk1+1) = xk1 we have that xk1+1 ∈ [e1,xk1 ) ∪ (f(x0),x0). if xk1+1 ∈ (f(x0),x0), then put k2 = k1 + 1. if not, then xk1+1 ∈ [e1,xk1 ), and since limn→∞xn = f(x0), there is k2 > k1 such that xk2 ∈ (f(x0),v) and xkj+1 ∈ (e1,xkj ) for each j ∈ {k1, . . . ,k2 − 2}. as f(xk2+1) = xk2 , we must have that xk2+1 ∈ [e1,f(x0)) ∪ (xk2,x0). if xk2+1 ∈ [e1,f(x0)), then we define k3 = k2 + 1. assume that xk2+1 ∈ [e1,f(x0)). since limn→∞xn = f(x0), there exists k3 > k1 such that xk3 ∈ (e1,f(x0)) and xkj+1 ∈ (xkj,x0) for all j ∈{k2, . . . ,k3 −2}. by following with this process we obtain our desired sequence. (5) without loss of generality, we suppose that x0 ∈ [e1,v). since fix(f) = {v}, we have that f(y) ∈ (y,v]∪[v,e2]∪[v,e3] for each y ∈ (x0,v). choose ε > 0 so that bε(v) is connected and bε(v) ⊂ t −{x0}. since limn→∞xn = f(x0), there exist m ∈ n such that xn ∈ bε(f(x0)) for each n > m. fix k1 > m such that xk1 ∈ (x0,v). since f(xk1+1) = xk1 , we obtain that xk1+1 ∈ (x0,xk1 ) ∪ (v,e2] ∪ (v,e3]. if xk1+1 ∈ (v,e2], then put l1 = k1 and k2 = k1 + 1. then assume that xk1+1 ∈ (x0,xk1 ). since limn→∞xn = f(x0), there is l1 > k1 so that xkj+1 ∈ (xkj,x0) ⊂ (xkj,e1] and xl1+1 /∈ (v,e1) for j ∈ {k1, . . . , l1 − 1}. if xl1+1 ∈ (v,e2], we set k2 = l1 + 1. suppose that xl1+1 ∈ (v,e3]. since {xn : n ∈ n}∩ (v,e2] is infinite and limn→∞xn = v, there is k2 > l1 + 1 such that xk2 ∈ (v,e2) and xj /∈ (v,e2) for each j ∈ {l1 + 1, . . . ,k2−1}. thus, f(xk2+1) = xk2 and then xk2+1 ∈ (xk2,e2)∪(v,e1]∪(v,e3]. we have arrived to the conditions from the beginning. by this way we define our required sequences {kn}∞n=1 and {ln}∞n=1. � theorem 4.2. let (t,f) be a discrete dynamical system such that f is a surjective map and ωf is a continuous function. if fix(f) 6= t , then f(t − fix(f)) ⊂ t −fix(f). proof. suppose that there is x0 ∈ t − fix(f) such that f(x0) ∈ fix(f). without of generality, suppose that x0 ∈ [e1,v]. by (1) of lemma 4.1, we obtain that there exists a sequence {xn}∞n=1 with the following properties: (a) xn ∈ t −fix(f) for each n ∈ n, (b) xi 6= xj for each i,j ∈ n such that i 6= j, (c) f(xn) = xn−1 for each n ∈ n, (d) limn→∞xn = f(x0). we consider the following cases: case 1. either fix(f) is not degenerate or fix(f) ∈ e(t). since ωf is a continuous function, by corollary 3.6, we have that fix(f) is connected and, then we have that either fix(f) ⊂ [e1,x0) or fix(f) ⊂ (x0,v]∪ [v,e2]∪ [v,e3]. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 338 the function ωf on simple n-ods case 1.1. fix(f) ⊂ [e1,x0). suppose that a,b ∈ [e1,x0), such that b ∈ [a,x0) and fix(f) = [a,b]. by (2) and (3) of lemma 4.1 we have that f(x0) ∈ fr(fix(f)) and [f(x0),x0] ∩ fix(f) = {f(x0)}, then b = f(x0). let ε > 0 such that bε(f(x0)) ⊂ (a,x0). since limn→∞xn = f(x0), then there exists k ∈ n such that xn ∈ bε(f(x0)) ∩ (f(x0),x0) for each n > k, and there exists l > k such that xl+1 ∈ (f(x0),xl). since [b,xl] ⊂ f([xl+1,x0]), [xl+1,x0] has a fixed point of f, but this is a contradiction. case 1.2. fix(f) ⊂ (x0,v] ∪ [v,e2] ∪ [v,e3]. choose ε > 0 such that bε(f(x0)) ∩ [e1,f(x0)] ⊂ (x0,f(x0)]. since limn→∞xn = f(x0), we can find k ∈ n such that xn ∈ bε(f(x0))∩ [x0,f(x0)] for each n > k, and there is l > k such that xl+1 ∈ (xl,f(x0)). since [xl,f(x0)] ⊂ f([x0,xl+1]), we obtain that [x0,xl+1] has a fixed point of f, but this is impossible. case 2. fix(f) = {f(x0)}, f(x0) /∈ e(t) and f(x0) 6= v. suppose that f(x0) ∈ (ei,v) for some i ∈ {1, 2, 3}, we obtain from (4) of lemma 4.1 that there exists a strictly increasing sequence {kn}∞n=1, when kn ∈ n, that satisfies the following conditions: (i) if n is odd xkn ∈ (ei,f(x0)) and xkj+1 ∈ (ei,xkj ) for each j ∈{kn, . . . ,kn+1 − 2}, (ii) if n is even xkn ∈ (f(x0),x0) and xkj+1 ∈ (xkj,x0) for all j ∈{kn, . . . ,kn+1 − 2}. pick an odd integer m so that xkm ∈ [ei,f(x0)) and xkm+2 ∈ (ei,xkm ). for definition, we have that f(xkm+1 ) = xkm+1−1 and xkm+1−1 ∈ (ei,xkm ) ⊂ [e1,f(x0)). then [xkm+1−1,f(x0)] ⊂ f([f(x0),xkm+1 ]). on the other hand, since xkm+2 ∈ (ei,xkm ) and xkm+1−1 ∈ (ei,xkm ) ⊂ [ei,f(x0)), then xkm+2 ∈ [xkm+1−1,f(x0)] ⊂ f([f(x0),xkm+1 ]). hence, there is z ∈ [f(x0),xkm+1 ] such that f(z) = xkm+2 . observe that f 2(z) = f(xkm+2 ) = xkm+2−1. since [f(x0),xkm+2−1] ⊂ f2([z,x0]), we must have that f2([z,x0]) has a fixed point of f2. that is, there is y ∈ [z,x0], such that f2(y) = y. by theorem 3.8, we have that fix(f2) is connected and, so [f(x0),y] ⊂ fix(f2). hence, f2(z) = z which is impossible because of f2(z) = xkm+2−1. case 3. fix(f) = {f(x0)} and f(x0) = v. if there exist i,j ∈{1, 2, 3} and k ∈ n such that xn ∈ [ei,ej] for each n > k, then the proof follows as in the case 2. thus, we may suppose that {xn : n ∈ n,n > k}∩ (v,ei] is infinite for each i ∈ {1, 2, 3}. it follows from (5) of lemma 4.1 that there are strictly increasing sequences {kn}∞n=1 and {ln}∞n=1, that satisfy kn ≤ ln < kn+1, for every n ∈ n, and the following conditions: (i) for each n ∈ n∪{0} and i ∈{1, 2, 3}, xk3n+i ∈ (ei,v), (ii) for each n ∈ n and i ∈ {1, 2, 3}, if ln > kn and xkn ∈ (ei,v), then xkj+1 ∈ (xkj,ei], for each j ∈{kn, . . . , ln − 1}, (iii) for each n ∈ n and i ∈{1, 2, 3}, xl3n+i ∈ (v,ei) and xl3n+i+1 /∈ (v,ei). if there exists r ∈ n, such that lr + 1 6= kr+1 the proof follows as in the case 2. suppose that ln + 1 = kn+1 for each n ∈ n. let m ∈ n such that xkm ∈ [x0,v], xkm+3 ∈ (xkm,v). then we have that xkm+2 ∈ [e3,v) and xkm+1 ∈ [e2,v). by definition we obtain that f(xkm+3 ) = xkm+3−1, c© agt, upv, 2019 appl. gen. topol. 20, no. 2 339 i. vidal-escobar and s. garcia-ferreira then [xkm+3−1,v]) ⊂ f([xkm+3,v], by (ii) xkm+3 ∈ (xkm+3−1,v) ⊂ f([xkm+3,v]. then there is z ∈ [xkm+1,v] such that f(z) = xkm+2 . moreover f(xkm+2 ) = xkm+2−1 then [xkm+2−1,v] ⊂ f([xkm+2,v]) ⊂ f2([z,v]). by (ii) we have that xkm+1 ∈ (xkm+2−1,v), then there exists w ∈ [z,v] such that f2(w) = xkm+1 . so [xkm+1−1,v] ⊂ f([xkm+1,v]) ⊂ f3([w,v]), since f(xkm+1 ) = xkm+1−1. by (ii), we obtain that xk1 ∈ (xkm+1−1,v) and xkm+3 ∈ (xk1,v). then, there exists y ∈ [w,v] such that f3(y) = xkm+3 . hence, we obtain that [v,xkm+3 ] ⊂ f3([x0,y]). thus, [x0,y] has a fixed point of f 3, says y′ ∈ [x0,y]. by theorem 3.11, we know that fix(f3) is connected and, then [y′,v] ⊂ fix(f3), which implies that f3(y) = y, but this contradicts the equality f3(y) = xkm+3 . � by the corollary 3.6, we know that fix(f) is connected, when ωf is a continuous map and f is a surjective map of a simple triod to itself. so, it can be a point, an arc or a simple triod. the following result limits these possibilities and clarifies that set it is. theorem 4.3. let (t,f) be a discrete dynamical system such that f is a surjective map and ωf is a continuous function. then, one of the following conditions holds: (1) fix(f) = t , (2) fix(f) = {v}, or (3) there exists i ∈{1, 2, 3}, such that fix(t) = [v,ei]. proof. suppose that neither of the conditions, (1)-(3), is true. since ωf is a continuous function, we obtain from corollary 3.6 that fix(f) is connected. we consider the following cases: case 1. v ∈ fix(f). by assumption, for each i ∈{1, 2, 3} there is ci ∈ [v,ei] so that fix(f) = [c1,v]∪[v,c2]∪[v,c3]. as (2) and (3) fail, we have that ci 6= v and cj 6= ej for some i,j ∈ {1, 2, 3}. suppose, without loss of generality, that e1 6= c1. since f is a surjective map, we can find e ∈ t such that f(e) = e1. we may assume that e /∈ (e1,c1); otherwise, (e1,c1) would have a fixed point of f. suppose, without loss of generality, that e ∈ (c2,e2]. case 1.1. c1 6= v. since [e1,c1] ⊂ [e1,v] ⊂ f([v,e]), there exists c ∈ [v,e] such that f(c) = c1. as c ∈ (v,e] ⊂ (v,e2], then c /∈ fix(f). it follows from theorem 4.2 that f(c) /∈ fix(f). this contradicts the fact f(c) = c1 ∈ fix(f). case 1.2. c2 6= v. as v ∈ [e1,c2] ⊂ f([c2,e]), there is c ∈ [c2,e] such that f(c) = v. since c ∈ [c2,e] and c2 6= v, we have that c 6= v. hence, c /∈ fix(f), but this contradicts theorem 4.2. case 1.3. c1 = v = c2. since neither (2) nor (3) hold, we obtain that v 6= c3 6= e3. as f is a surjective function there exits d ∈ t such that f(d) = e3. then we proceed as in case 1.1. thus, this case is impossible. case 2. v /∈ fix(f). suppose, without of generality, that fix(f) ⊂ [e1,v). pick a,b ∈ [e1,v) such that b ∈ [a,v) and fix(f) = [a,b]. notice that f(v) ∈ (b,v); in other case, f(v) ∈ [e2,e3] and then [e2,e3] has a fixed point of f, but this impossible because of fix(f) ⊂ [e1,v). since f(v) ∈ (b,v), f(y) ∈ (b,y) for each y ∈ (b,v], we obtain from lemma 3.1 that b ∈ ωf (v). c© agt, upv, 2019 appl. gen. topol. 20, no. 2 340 the function ωf on simple n-ods case 2.1. a = e1. as f is a surjective map, there exists di ∈ t such that f(di) = ei for each i ∈ {2, 3}. note that di /∈ (v,ei); otherwise, (v,ei) has a fixed point of f. thus, d2 ∈ (v,e3) and d3 ∈ (v,e2). since [e2,v] ⊂ f([v,d2]), we can choose c2 ∈ [v,d2] for which f(c3) = d3. so f2(c3) = e3 and, then (v,e3] has a fixed point of f 2, says y ∈ (v,e3]. we know from theorem 3.5 that fix(f2) is connected, and so [b,y] ⊂ fix(f2) which implies that v ∈ fix(f2). therefore, ωf (v) = {v,f(v)}. as b ∈ ωf (v), we have that f(v) = b, but this contradicts theorem 4.2 since v /∈ fix(f). case 2.2. a 6= e1. since f is a surjective function, there is e ∈ t so that f(e) = e1. notice that e /∈ [e1,a); if not, [e1,a) has a fixed point of f. remember that f(y) ∈ (b,y) for each y ∈ (b,v]. hence, we have that e ∈ [e2,e3]. as [e1,f(v)] ⊂ f([v,e]), there exists c ∈ [v,e] such that f(c) = a. since c ∈ [v,e] ⊂ [e2,e3], c /∈ [a,b] ⊂ [e1,v). but this is impossible by the theorem 4.2. in each case we obtain a contradiction. thus, we obtain that one of the conditions either (1), (2) or (3) holds. � proposition 4.4. let (t,f) be a discrete dynamical system such that f is a surjective map and ωf is a continuous function. then for each i ∈ {1, 2, 3} there exists j ∈{1, 2, 3} such that f([v,ei]) = [v,ej]. proof. fix i ∈{1, 2, 3} and assume that f([v,ei]) 6= [v,ej] for each j ∈{1, 2, 3}. it follows from theorem 4.3 that v ∈ fix(f). we consider the following cases. case 1. there exist distinct j,k ∈{1, 2, 3} so that f([v,ei])∩(v,ej] 6= ∅ 6= f([v,ei]) ∩ (v,ek]. it follows from assumption that there exists c,d ∈ (v,ei] such that f(c) ∈ (v,ej] and f(d) ∈ (v,ek]. then, v ∈ f([c,e]) ⊂ f((v,ei]) and, hence there is v′ ∈ [c,e] such that f(v′) = v. by theorem 4.2, we have that v = f(v′) /∈ fix(f), which is a contradiction. case 2. there exists j ∈ {1, 2, 3} such that f([v,ei]) ⊂ [v,ej). as f is a surjective function there exists e ∈ t such that f(e) = ej. since f([v,ei]) ⊂ [v,ej), e /∈ [v,ei]. suppose that e ∈ [v,ek] for k ∈{1, 2, 3}\{i}. • if f([v,ek]) 6= [v,ej]. then, there exits c ∈ (v,ek] such that f(c) /∈ (v,ej]. assume that f(c) ∈ [v,el], where l 6= j. then, f([v,ek]) ∩ (v,ej] 6= ∅ 6= f([v,ek]) ∩ (v,el]. • suppose that f([v,ek]) = [v,ej]. then, f([v,ei])∪f([v,ek]) = [v,ej]. first, if i 6= j 6= k, then [v,ek] ∪ [v,ei] ⊂ f([v,ej]) and so f([v,ej]) ∩ (v,ek] 6= ∅ 6= f([v,ej]) ∩ (v,ei]. now, if i = j and l ∈{1, 2, 3}\{i,k}, then [v,ek] ∪ [v,el] ⊂ f([v,el]) and hence f([v,el]) ∩ (v,ek] 6= ∅ 6= f([v,el]) ∩ (v,el]. finally if j = k and l ∈ {1, 2, 3} \ {i,k}, then [v,ei] ∪ [v,el] ⊂ f([v,el]) which implies that f([v,el]) ∩ (v,ei] 6= ∅ 6= f([v,el]) ∩ (v,el]. in each one of the previous case, we arrived to the conditions of case 1. so we can obtain a contradiction. � theorem 4.5. let (t,f) be a discrete dynamical system such that f is a surjective map and ωf is a continuous function, then f|e(t) is a permutation. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 341 i. vidal-escobar and s. garcia-ferreira proof. suppose that fix(f) 6= t. by proposition 4.4, we have that for each i ∈{1, 2, 3} there exists ji ∈{1, 2, 3} such that f([v,ei]) = [v,eji ]. by theorem 4.3 we need consider the following cases. case 1. there exists i ∈ {1, 2, 3} satisfying fix(f) = [v,ei]. assume, without of generality, that i = 1. then, we have that f([v,e2]) = [v,e3] and f([v,e3]) = [v,e2]. we will prove, that f(e2) = e3 and f(e3) = e2. suppose that f(e2) 6= e3, since f([v,e2]) = [v,e3], we have that there exists b2 ∈ (v,e2) such that f(b2) = e3. case 1.1. f(e3) = e2. since f(b2) = e3 we have that [v,e2] ⊂ f2([v,b2]). then, (b2,e2] has a fixed point of f 2, says c ∈ (b2,e2]. by theorem 3.8, we have that fix(f2) is connected. thus, [v,c] ⊂ fix(f2) for which f2(b2) = b2, this is a contradiction, because of f2(b2) = e2. case 1.2. f(e3) 6= e2. as f([v,e2]) = [v,e3] we have that, there exists b3 ∈ (v,e3) such that f(b3) = e2. since [v,e3] ⊂ f([v,b2]), there is c2 ∈ [v,b2] such that f(c2) = b3. then [v,e2] ⊂ f2([v,b3]) = and so f((c2,e2]) has a fixed point of f2, says c ∈ (c2,e2]. we know from the theorem 3.8 that fix(f2) is connected. hence, [v,c] ⊂ fix(f2) and then f2(c2) = c2. this contradicts the fact f2(c2) = e2. case 2. fix(f) = {v}. by proposition 4.4, we can assume, without of generality, that f([v,e1]) = [v,e2], f([v,e2]) = [v,e3], f([v,e3]) = [v,e1]. we will show, that f(e1) = e2, f(e2) = e3 and f(e3) = e1. suppose that f(e1) 6= e2. since f([v,e1]) = [v,e2], there exists b1 ∈ (v,e1) that satisfies f(b1) = e2. case 2.1. f(e2) = e3 and f(e3) = e1. since f(b1) = e2, we obtain that [v,e1] = f([v,e3]) = f 2([v,e2]) = f 3([v,b1]). hence, (b1,e1] has a fixed point of f3, says c ∈ (b1,e1]. it follows from theorem 3.11 that fix(f3) is connected. thus, [v,c] ⊂ fix(f3) for which f3(b1) = b1, this is a contradiction, because of f3(b1) = e1. case 2.2. f(e2) 6= e3 and f(e3) = e1. by assumption f([v,e2]) = [v,e3], for which, there is b2 ∈ [v,e2] such that f(b2) = e3. by other hand as [v,e2] = f([v,b1]), we can find c1 ∈ [v,e1] such that f(c1) = b2. hence, [v,e1] = f([v,e3]) = f 2([v,b2]) ⊂ f3([v,c1]) and then (c1,e1] has a fixed point of f3, says c ∈ (c1,e1]. we know from the theorem 3.11 that fix(f3) is connected. hence, [v,c] ⊂ fix(f3) and so f3(c1) = c1. this contradicts the fact f3(c1) = e1. case 2.3. f(e2) = e3 and f(e3) 6= e1. since f([v,e3]) = [v,e1]. so, we can choose b3 ∈ [v,e3] such that f(b3) = e1. by other hand as f(b1) = e2, then [v,e3] = f([v,e2]) = f 2([v,b1]), and so, we can pick c1 ∈ [v,b1] such that f2(c1) = b3. hence, [v,e1] = f([v,b3]) ⊂ f3([v,c1]) (c1,e1] has a fixed point of f3, says c ∈ (c1,e1]. by theorem 3.11, fix(f3) is connected. thus, [v,c] ⊂ fix(f3) and then f3(c1) = c1, but this is impossible since f3(c1) = e1. case 2.4. f(e2) 6= e3 and f(e3) 6= e1. by supposition f([v,e2]) = [v,e3] and f([v,e3]) = [v,e1], then we can find b2 ∈ (v,e2) and b3 ∈ (v,e3) such that f(b2) = e3 and f(b3) = e1. since [v,e2] ⊂ f([v,b1]), there is c1 ∈ (v,b1) such that f(c1) = b2. so, [v,e3] ⊂ f2([v,c1]), then choose d1 ∈ [v,c1] such that f2(d1) = b3. hence, [v,e1] ⊂ f3([v,d1]) for which (d1,e1] has a fixed point of f3, says c ∈ (d1,e1]. we know from the theorem 3.11 that fix(f3) is connected. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 342 the function ωf on simple n-ods hence, [v,c] ⊂ fix(f3) and then f3(d1) = d1. this is a contradiction because of f3(d1) = e1. to proof f(e2) = e3 and f(e3) = e1, we proceed in the similar way. in each case we have that f|e(t) is a permutation. � corollary 4.6. let (t,f) be a discrete dynamical system such that f is a surjective map and ωf is a continuous function. then t = per(f) and each point of t is n−periodic, where n ∈{1, 2, 3}. proof. it follows from theorem 4.3 that one of the following conditions holds. (1) fix(f) = t, (2) fix(f) = {v}, (3) there exists i ∈{1, 2, 3}, such that fix(t) = [v,ei]. if fix(f) = t , then each point of t is a fixed point, so 1-periodic and t = per(f). case 1. there exists i ∈ {1, 2, 3} such that fix(f) = [v,ei]. suppose that i = 1. we obtain from proposition 4.4 that f([v,e2]) = [v,e3] and f([v,e3]) = [v,e2]. it follows from theorem 4.5 that f(e2) = e3 and f(e3) = e2. thus, f2(e2) = e2, f 2(e3) = e3 and f 2(e1) = e1. by theorem 3.8 we have that fix(f2) is connected, hence fix(f2) = t. therefore, each point of t is at most 2-periodic and t = per(f). case 2. fix(f) = {v}. by proposition 4.4, we have that for each i ∈ {1, 2, 3} there exists ji ∈ {1, 2, 3} such that f([v,ei]) = [v,eji ]. without of generality, suppose that f([v,e1]) = [v,e2], f([v,e2]) = [v,e3] and f([v,e3]) = [v,e1]. we obtain from the theorem 4.5 that f(e1) = e2, f(e2) = e3 and f(e3) = e1. so, f 3(e1) = e1, f 3(e2) = e2 and f 3(e3) = e3. we know from the theorem 3.11 that fix(f3) is connected. hence, fix(f3) = t and so each point of t is 3-periodic and t = per(f). � the following result is the version of the theorem 4.5, when the phase space is the arc. following the same way, of the proof, of theorem 4.5 it is easy to prove it. theorem 4.7. let f : [0, 1] → [0, 1] be a surjective continuous function such that ωf is a continuous function. then f(0) = 0 and f(1) = 1 or f(0) = 1 and f(1) = 0. corollary 4.8. let f : [0, 1] → [0, 1] be a surjective continuous function such that ωf is a continuous function. then f = f 0 or f2 = f0 corollary 4.9. let f : [0, 1] → [0, 1] be a surjective continuous function such that ωf is a continuous function. then each point of [0, 1] is n−periodic with n ∈{1, 2}. the following result shows that it is sufficient to request the equicontinuity of a function f, so that, the function ωf will be a continuous map, when the phase space is a compact metric space. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 343 i. vidal-escobar and s. garcia-ferreira proposition 4.10. let (x,f) be a discrete dynamical system, where x is a metric compact space with metric d. if f is equicontinuous, then ωf is a continuous function. proof. let ε > 0. since f is equicontinuous there is 0 < δ ≤ ε 2 such that d(fn(x),fn(y)) < ε 2 , for each x,y ∈ x with d(x,y) ≤ δ, and every n ∈ n. fix x,y ∈ x such that d(x,y) < δ. choose z ∈ ωf (x), by definition, there is a sequence of positive integers n1 < n2 < ..., such that limk→∞f nk (x) = z. as x is compact, we can assume that limk→∞f nk (y) = y′. hence, y′ ∈ ωf (y). since d(fnk (x),fnk (y)) < ε 2 , for each k ∈ n, we obtain that d(z,y′) ≤ ε 2 < ε. thus, ωf (x) ⊂ nε(ωf (y)). similarly we obtain that ωf (y) ⊂ nε(ωf (x)). it follows from [4, 2.9] that h(ωf (x),ωf (y)) < ε. therefore, ωf is a continuous function. � now we will prove the main result of this work, in which we show that the equicontinuity of a function f is equivalent to the continuity of ωf , when the phase space is the simple triod, this result was proved in [1], when the phases space is the arc. theorem a 2. let (t,f) be a discrete dynamical system. then, ωf is a continuous function if and only if f is equicontinuous. proof. if f is equicontinuous by proposition 4.10 we have that ωf is a continuous function. suppose that ωf is a continuous function, we consider the following cases. case 1. f is a surjective function. we know from theorem 4.6 that t = per(f). thus, each point of t has periodic orbit and so ωf is a periodic orbit for every point of x. moreover ωf is a continuous function, then follows of [9, theorem 3.8] that f is equicontinuous. case 2. f is not a surjective function. define r = ⋂∞ n=1 f n(t). notice that f(r) = r 6= ∅, then f|r : r → r is a surjective continuous function. • if r is a point, then f|r is equicontinuous. • if r is an arc, it follows from [1, theorem 1.2] that f|r is equicontinuous. • if r is a simple triod, we know from case 1 that f|r is equicontinuous. so, f|r is equicontinuous. case 2.1. r is degenerate. pick a ∈ t such that r = {a}. let ε > 0, since {a} = ⋂∞ n=1 f n(t) = limn→∞f n(t), then there exists m ∈ n such that h(fn(t),{a}) < ε 2 for each n > m. for which d(fn(x),a) < ε 2 for each x ∈ t and n ∈ n. by other hand as fn is a continuous function for each n ∈ {1, . . . ,m}, there exists δ > 0 such that d(fn(x),fn(y)) < ε, for each x,y ∈ t with d(x,y) ≤ δ, and all n ∈ {1, . . . ,m}. it follows from the above that d(fn(x),fn(y)) < ε, for each x,y ∈ t with d(x,y) ≤ δ, and every n ∈ n. therefore f is equicontiuous. case 2.2. r is not degenerate. by assumption, either r is an arc or r a a simple triod. we suppose that r is a simple triod, the proof when r is an arc it follows similarly. since r is a simple triod, by corollary 4.6, we have that r = per(f|r) and each point of r is n− periodic for some n ∈ {1, 2, 3}. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 344 the function ωf on simple n-ods thus, f6(x) = x for each x ∈ r. for i ∈ {1, 2, 3}, choose ci ∈ (v,ei] such that r = [c1,v] ∪ [v,c2] ∪ [v,c3]. set si = diam([v,ci]) for each i ∈ {1, 2, 3}, and s = min{si : i ∈ {1, 2, 3}}. let 0 < ε < s such that nε(r) ⊆ t . now, we will find positive numbers δ3 < δ2 < δ1 < ε 2 as follows: • since f, f2, . . . ,f6 are continuous maps, there exists 0 < δ1 < ε2 such that, if x,y ∈ t and d(x,y) < δ1, then d(fn(x),fn(y)) < ε2 for each n ∈{1, 2, . . . , 6}. • f|r is equicontinuous, we can find 0 < δ2 < δ1 such that d(fn(x),fn(y)) < ε 2 for each x,y ∈ t with d(x,y) ≤ δ2, and every n ∈ n. • since r = limn→∞fn(t) = r, there is m ∈ n, m = 6k for some k ∈ n, such that h(fn(t),r) < δ2 for each n > m. • since f, f2, . . . ,fm+1 are continuous functions, then there exists 0 < δ3 < δ2 such that, if x,y ∈ t , d(x,y) < δ3 and n ∈ {1, 2, . . . ,m + 1}; then d(fn(x),fn(y)) < δ2. fix x,y ∈ t such that d(x,y) < δ3. we have that d(fm+1(x),fm+1(y)) < δ2 and fm+1(x),fm+1(y) ∈nδ2 (r). case 2.2.1. fm+1(x),fm+1(y) ∈ r. since d(fm+1(x),fm+1(y)) < δ2, we have that d(fn(fm+1(x)),fn(fm+1(y))) < ε 2 for each n ∈ n. moreover, since d(x,y) < δ3, we know that d(f n(x),fn(y)) < δ2 < ε 2 for each n ∈ {1, 2, . . . ,m + 1}. therefore, d(fn(x),fn(y)) < ε for each n ∈ n. case 2.2.2. fm+1(x) ∈ r and fm+1(y) /∈ r. suppose, without of generality, that c1 ∈ [fm+1(y),fm+1(x)]. then we have that d(fm+1(y),c1) < δ2 and d(fm+1(x),c1) < δ2. for which d(f n(fm+1(x)),fn(c1)) < ε 2 , for each n ∈ n. since d(fm+1(y),c1) < δ2 < δ1, we have that d(f m+1+i(y),fi(c1)) < ε 2 for each i ∈ {1, 2, . . . , 6}, but we know that h(fn(t),r) < δ2 for each n > m, then d(fm+1+i(y),fi(c1)) < δ2 for each i ∈ {1, 2, . . . , 6}. now, if there is i ∈ {1, 2, . . . , 6} such that fm+1+i(y) ∈ r, then d(fn(fm+1+i(y)),fn(fi(c1))) < ε2 , for each n ∈ n, so d(fn(fm+1(x)),fn(fm+1(y))) < ε. further, since d(x,y) < δ3, we have that d(f n(x),fn(y)) < ε 2 for each n ∈{1, 2, . . . ,m + 1}. therefore, d(fn(x),fn(y)) < ε for each n ∈ n. assume that fm+1+i(y) /∈ r for each i ∈ {1, 2, . . . , 6}. we know that d(fm+7(y),f6(c1)) = d(f m+7(y),c1) < δ2, then we can proceed in a way similar to what was done previously, and so d(fn(fm+1(y),fn(c1))) < ε 2 for each n ∈ n. further, since d(fn(fm+1(x)),fn(c1))) < ε 2 and d(fn(fm+1(y)),fn(c1))) < ε 2 , we obtain d(fn(fm+1(x)),fn(fm+1(y))) < ε. moreover, as d(x,y) < δ3, we have that d(fn(x),fn(y)) < δ2 < ε 2 for each n ∈ {1, 2, . . . ,m + 1}. therefore, d(fn(x),fn(y)) < ε for each n ∈ n. case 2.2.3. fm+1(x),fm+1(y) /∈ r. since d(fm+1(x),fm+1(y)) < δ2 < s, and h(fm+1(t),r) < δ2 we can suppose, without loss of generality that d(fm+1(x),c1) < δ2, and d(f m+1(x),c1) < δ2. following similarly to case 2.2.2, we obtain the following inequality d(fn(fm+1(x),fn(c1))) < ε 2 and d(fn(fm+1(y),fn(c1))) < ε 2 for each n ∈ n, so d(fn(fm+1(x)),fn(fm+1(y))) < ε. as d(x,y) < δ3, d(f n(x),fn(y)) < δ2 < ε 2 for each n ∈ {1, 2, . . . ,m + 1}. therefore, d(fn(x),fn(y)) < ε for each n ∈ n. � c© agt, upv, 2019 appl. gen. topol. 20, no. 2 345 i. vidal-escobar and s. garcia-ferreira corollary 4.11. let (t,f) be a discrete dynamical system such that ωf is a continuous function. then ωfn is a continuous function and fix(f n) is connected, for each n ∈ n. proof. by theorem a 2 we have that f is equicontinuos, notice that if f es equicontinuos then fn is equicontinuos then by theorem a 2 we have that ωfn is a continuous function, and so it follows from corollary 3.6 that fix(fn) is connected. � question 4.12. can theorem a 2 be extended when the phase space is a n-od with n ≥ 4? to finish this paper, we give an example of a function f that is equicontinuous, however the function ωf is not continuous. when the phase space is a harmonic fan. example 4.13. consider x, l, ln and fn : [0, 1] → ln as in example 3.16. define gn : [0, 1 2n ] → [0, n+1 2n ] given by gn(t) = (n + 1)(t); hn : [ 1 2n , 1 2n−1 ] → [0, n+1 2n ] given by hn(t) = (n + 1)( 1 2n−1 − t) and f((x,y)) =   fn+1(gn(f −1 n (x,y))) (x,y) ∈ ln and f−1n (x,y) ∈ [0, 1 2n ], fn+1(hn(f −1 n (x,y))) (x,y) ∈ ln and f−1n (x,y) ∈ [ 1 2n , 1 2n−1 ], (0, 0) (x,y) ∈ ln and f−1n (x,y) ∈ [ 1 2n−1 , 1], (0, 0) (x,y) ∈ l. we have that for each x ∈ x, ωf (x) = {(0, 0)}, hence ωf is a continuous function. we will show that f is not equicontinuous. we consider x with the maximum metric, dm . let ε > 0, given m ∈ n such that 1 2m < ε < 1 2m−1 . we consider the point (x,y) = ( 1 2mm! , 1 2mm! ) ∈ l1, then we have that fm−1((x,y)) = fm( 1 2m ) ∈ lm, so fm((x,y)) = fm+1(m+12m ), then fm+1((x,y)) = (0, 0) because of m+1 2m > 1 2m−1 . now, we have that dm ((x,y), (0, 0)) = 1 2mm! < 1 2m < ε, and we have that dm (f m(x,y),fm(0, 0)) = m+1 2m > 1 2m−1 > ε. therefore, f is not equicontinuous. acknowledgements. the authors would like to thank the anonymous referee for careful reading and very useful suggestions and comments that help to improve the presentation of the paper. references [1] a. m. bruckner and j. ceder, chaos in terms of the map x → ω(x, f), pacific j. math. 156 (1992), 63–96. [2] r. gu, equicontinuity of maps on figure-eight space, southeast asian bull. math. 25 (2001), 413–419. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 346 the function ωf on simple n-ods [3] w. hurewicz and h. wallman, dimension theory, princeton university press, princeton (1941). [4] a. illanes and s. b. nadler, jr., hyperspaces: fundamental and recent advances, a series of monographs and textbooks pure and applied mathematics 216, marcel decker inc. new york (1998). [5] s. kolyada and l. snoha, some aspects of topological transitivity a survey, grazer math. ber. 334 (1997), 3–35. [6] l. lum, a characterization of local connectivity in dendroids, studies in topology (proc. conf., univ. north carolina, charlotte nc 1974); academic press (1975) 331– 338. [7] j. mai, the structure of equicontinuous maps, trans. amer. math. soc. 355 (2003), 4125–4136. [8] s. b. nadler, jr., continuum theory: an introduction, a series of monographs and textbooks pure and applied mathematics 158, marcel decker inc. new york (1992). [9] t. x. sun, g. w. su, h. j. xi and x. kong, equicontinuity of maps on a dendrite with finite branch points, acta math. sin. (engl. ser.) 33 (2017), 1125–1130. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 347 @ appl. gen. topol. 22, no. 2 (2021), 435-446doi:10.4995/agt.2021.15561 © agt, upv, 2021 on a probabilistic version of meir-keeler type fixed point theorem for a family of discontinuous operators ravindra k. bisht a and vladimir rakočević b a department of mathematics, national defence academy, khadakwasla-411023, pune, india (ravindra.bisht@yahoo.com) b university of nǐs, faculty of sciences and mathematics, vǐsegradska 33, 18000 nǐs, serbia. (vrakoc@sbb.rs) communicated by s. romaguera abstract a meir-keeler type fixed point theorem for a family of mappings is proved in menger probabilistic metric space (menger pm-space). we establish that completeness of the space is equivalent to fixed point property for a larger class of mappings that includes continuous as well as discontinuous mappings. in addition to it, a probabilistic fixed point theorem satisfying (ǫ − δ) type non-expansive mappings is established. 2010 msc: 47h09; 47h10. keywords: menger pm-spaces; fixed point; almost orbital continuity; nonexpansive mapping. 1. introduction and preliminaries the idea of statistical metric space or probabilistic menger space can be traced back to menger [10], who extended the concept of metric space (x, d), by replacing the notion of distance d(x, y) (x, y ∈ x) by a distributive function fx,y : x × x → r, where fx,y(t) represents the probability that the distance between x and y is less than t. schweizer and sklar [22, 23] studied various properties, e.g., topology, convergence of sequences, continuity of mappings, received 04 may 2021 – accepted 15 june 2021 http://dx.doi.org/10.4995/agt.2021.15561 r. k. bisht and v. rakočević completeness, etc., of these spaces. in 1972, sehgal and bharucha–reid [24] showed the role of distributive functions in metric fixed point theory and established the probabilistic metric version of the classical banach contraction mapping principle. since then the study of fixed point theorems in pm-space has emerged as an active area of research. let g be a selfmapping which satisfy some contractive condition on a complete menger pm-space (x, f, t ). then there exists a cauchy sequence of successive iterates {gnx}n∈n for each x in x which converges to some point, say z ∈ x, and the limiting point z of the sequence of iterates is nothing but a fixed point of g. however, there exist various contractive definitions which ensure the existence of the cauchy sequence of iterates converging to some limit point, but the limit point may not be a fixed point. pant et al. [17] (see also bisht [2]) proved the following theorem where the meir-keeler [9] type operator ensures the convergence of sequence of iterates but does not ensure the existence of a fixed point. lemma 1.1. let (x, f, t ) be a complete menger pm-space, and let f be selfmapping of x satisfying one of the following conditions (i) for every ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that ǫ − δ < min { fx,gx(t), fy,gy(t) } < ǫ ⇒ fgx,gy(t) ≥ ǫ, (ii) fgx,gy(t) > min { fx,gx(t), fy,gy(t) } , or (i’) for every ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that ǫ − δ ≤ min { fx,gx(t), fy,gy(t) } < ǫ ⇒ fgx,gy(t) > ǫ, for all x, y ∈ x. then for any x in x the sequence of iterates {gnx}n∈n is a cauchy sequence and there exists a point z in x such that lim n→∞ gnx = z for each x in x. the triple (x, f, tmin) is a complete menger pm-space, for x ⊆ r (see remark 2.3). the following example illustrates lemma 1.1, but does not possess a fixed point. example 1.2. let x = [1, 2] ∪ { 1 − 1 3n : n = 0, 1, 2, · · · } and d be the usual metric. define g : x → x by gx = { 0 if 1 ≤ x ≤ 2. 1 − 1 3n+1 if x = 1 − 1 3n , n = 1, 2, · · · . then g(x) = { 1 − 1 3n : n = 1, 2, · · · } © agt, upv, 2021 appl. gen. topol. 22, no. 2 436 on a probabilistic version of meir-keeler type fixed point theorem and g is fixed point free. the mapping g satisfies the contractive condition (i′) of lemma 1.1 with δ(ε) = { 1 3n − ε if 1 3n+1 ≤ ε < 1 3n , n = 1, 2, · · · ε if ε ≥ 1. therefore, to ensure the existence of a fixed point under such contractive definitions, one needs to assume some additional hypotheses on the mappings. ćirić [5] introduced the notion of orbital continuity. if g is a self-mapping of a metric space (x, d) then the set og(x) = {g nx | n = 0, 1, 2, . . .} is called the orbit of g at x and g is called orbitally continuous if u = limi g mix implies gu = limi gg mix. every continuous self-mapping is orbitally continuous but not conversely. in 1977, jaggi [7] introduced the concept of x0-orbital continuity which is weaker than orbital continuity of the mapping. a self-mapping g of a metric space (x, d) is called x0-orbitally continuous for some x0 ∈ x if its restriction to the set o(g, x0), is continuous, i.e., g : o(g, x0) → x, is continuous, here o(g, x) represents closure of the orbit of g at x0. the mapping g is said to be orbitally continuous if it is x0-orbitally continuous for all x0 ∈ x. in 2011, jungck [8] gave a generalized notion of orbital continuity, namely, almost orbital continuity. a self-mapping g of a metric space (x, d) is called almost orbitally continuous at x0 ∈ x if whenever limn g in(x) = x0 for some x ∈ x and subsequence {gin(x)} of gn(x), there exists a subsequence {gjn(x)} of gn(x) such that limn g jn(x) = g(x0). orbital continuity implies almost orbital continuity, but the implication is not reversible. in 2017, pant and pant [11] introduced the notion of k−continuity. a self-mapping g of a metric space x is called k-continuous, k = 1, 2, 3, . . . , if gkxn → gt, whenever {xn}n∈n is a sequence in x such that g k−1xn → t. it may be observed that 1-continuity is equivalent to continuity and continuity implies 2-continuity, 2continuity implies 3-continuity and so on but not conversely. it is important to note that k−continuity of the mapping implies orbital continuity but not conversely. more recently, pant et al. [12] introduced the notion of weak orbital continuity, which is weaker than orbital continuity of the mapping. a self-mapping g of a metric space (x, d) is called weakly orbitally continuous [12] if the set {y ∈ x : limi g miy = u implies limi gg miy = gu} is nonempty, whenever the set {x ∈ x : limi g mix = u} is nonempty. example 1.3. let x = [0, 2] and d be the usual metric. define g : x → x by gx = (1 + x) 2 if 0 ≤ x < 1, gx = 0 if 1 ≤ x < 2, g2 = 2. then [12]: (i) g is not orbitally continuous. since gn0 → 1 and g(gn0) → 1 6= g1. (ii) g is weakly orbitally continuous. if we take x = 2 then gn2 → 2 and g(gn2) → 2 = g2. © agt, upv, 2021 appl. gen. topol. 22, no. 2 437 r. k. bisht and v. rakočević (iii) g is not k−continuous. if we consider the sequence {gn0}, then for any positive integer k, we have gk−1(gn0) → 1 and gk(gn0) → 1 6= g1. example 1.4. let x = [0, +∞) and d be the usual metric. define g : x → x by gx = 1 if 0 ≤ x ≤ 1, gx = x 5 if x > 1. then g is orbitally continuous. let k ≥ 1 be any integer. consider the sequence {xn} given by xn = 5 k−1 + 1 n . then gk−1xn = 1+ 1 n5k−1 , gkxn = 1 5 + 1 n5k . this implies gk−1xn → 1, g kxn → 1 5 6= g1 as n → +∞. hence g is not k−continuous. the above examples show that orbital continuity implies weak orbital continuity but the converse need not be true. also, every k-continuous mapping is orbitally continuous, but the converse is not true. the question of continuity of contractive definitions at their fixed point in metric space was studied by rhoades [20] (see also, hicks and rhoades [6]). all the contractive definitions studied by them forced the mappings to be continuous at the fixed point. rhoades [20] also listed the question of the existence of a contractive condition that intromits discontinuity at the fixed point as an open problem. pant [15] gave the first affirmative answer to this problem in the setting of metric space. various other distinct answers to this problem and their possible applications to neural networks having discontinuities in activation functions can be found in bisht and pant [1], bisht and rakočević [3], pant and pant [11], pant et al. [12, 16, 17, 18], taş and özgür [27]. bisht and rakočević [4] presented some new solutions to rhoades’ open problem on the existence of contractive mappings that admit discontinuity at the fixed point. this was done via new fixed point theorems for a generalized class of meir-keeler type mappings which were proved by the authors. rhoade’s question was related, in part, to the important problem of characterizing metric completeness in terms of fixed point results; in this direction solutions to that problem were deduced. in 2020. romaguera [21] introduced and studied the notion of w-distance for fuzzy metric spaces and he obtained a characterization of complete fuzzy metric spaces via a suitable fixed point theorem. in this paper, we prove a meir-keeler type fixed point theorem for a family of mappings in menger pmspace. a probabilistic fixed point theorem satisfying (ǫ − δ) type non-expansive mappings is also established. we assume the notions of weak continuity which may imply discontinuity at the fixed point but characterize completeness of the space. 2. preliminaries we start with some standard definitions and notations of a probabilistic metric space. let d+ be the set of all distribution functions f : r → [0, 1] such that f is a non-decreasing, left-continuous mapping satisfying f(0) = 0 and supx∈r f(x) = © agt, upv, 2021 appl. gen. topol. 22, no. 2 438 on a probabilistic version of meir-keeler type fixed point theorem 1. the space d+ is partially ordered by the usual point-wise ordering of functions, i.e., f ≤ g if and only if f(t) ≤ g(t) for all t ∈ r. the maximal element for d+ in this order is the distribution function given by ε0(t) = { 0, t ≤ 0, 1, t > 0. definition 2.1 ([23]). a binary operation t : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if t satisfies the following conditions: (a) t is commutative and associative; (b) t is continuous; (c) t (a, 1) = a for all a ∈ [0, 1]; (d) t (a, b) ≤ t (c, d), whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1]. some of the simple examples of t-norm are t (a, b) = max{a+b−1, 0}, t (a, b) = min{a, b}, t (a, b) = ab and t (a, b) = { ab a+b−ab , ab 6= 0, 0, ab = 0. the t-norms are defined recursively by t 1 = t and t n(x1, . . . , xn+1) = t (t n−1(x1, . . . , xn), xn+1), for n ≥ 2 and xi ∈ [0, 1] for all i ∈ {1, . . . , n + 1}. definition 2.2. a menger probabilistic metric space (briefly, menger pmspace) is a triple (x, f, t ) where x is a non-void set, t is a continuous tnorm, and f is a mapping from x × x into d+ such that, if fx,y denotes the value of f at the pair (x, y), then the following conditions hold: (pm1) fx,y(t) = ε0(t) if and only if x = y; (pm2) fx,y(t) = fy,x(t); (pm3) fx,z(t + s) ≥ t (fx,y(t), fy,z(s)) for all x, y, z ∈ x and s, t ≥ 0. remark 2.3 ([24]). every metric space is a pm-space. let (x, d) be a metric space and t (a, b) = min{a, b} is a continuous t-norm. define fx,y(t) = ε0(t − d(x, y)) for all x, y ∈ x and t > 0. the triple (x, f, t ) is a pm-space induced by the metric d. definition 2.4. let (x, f, t ) be a menger pm-space. (1) a sequence {xn}n=1,2,... in x is said to be convergent to x in x if, for every ε > 0 and λ > 0 there exists positive integer n such that fxn,x(ε) > 1 − λ whenever n ≥ n. (2) a sequence {xn}n=1,2,... in x is called cauchy sequence if, for every ε > 0 and λ > 0 there exists positive integer n such that fxn,xm(ε) > 1 − λ whenever n, m ≥ n. (3) a menger pm-space is said to be complete if every cauchy sequence in x is convergent to a point in x. © agt, upv, 2021 appl. gen. topol. 22, no. 2 439 r. k. bisht and v. rakočević the following lemma was given in [22, 23]. lemma 2.5 ([23]). let (x, f, t ) be a menger pm-space. then the function f is lower semi-continuous for every fixed t > 0, i.e., for every fixed t > 0 and every two convergent sequences {xn}, {yn} ⊆ x such that xn → x, yn → y it follows that lim inf n→+∞ fxn,yn(t) = fx,y(t). 3. main results 3.1. fixed points of a family of meir-keeler type mappings in menger pm-space. the meir-keeler type contractive condition employed in the next theorem for a family of self-mappings ensures the convergence of sequence of iterates as well as the existence of fixed points under some weaker notion of continuity assumption. theorem 3.1. let (x, f, t ) be a complete menger pm-space, and let {fj : 0 ≤ j ≤ 1} be a family of self-mappings of x such that for any given fj the following conditions are satisfied: (i) ffj x,fjy(t) ≥ min { fx,fjx(t), fy,fjy(t) } for all x, y ∈ x; (ii) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ x with ǫ − δ ≤ min { fx,fjx(t), fy,fjy(t) } < ǫ implies ffj x,fjy(t) > ǫ. if fj is weakly orbitally continuous, then fj has a unique fixed point, say z, and lim n→+∞ fnj x0 = z for each x in x. moreover, if every pair of mappings (fr, fs) satisfies the condition (iii) ffj x,fsy(t) ≥ min { fx,fjx(t), fy,fsy(t) } ; then the mappings {fj} have a unique common fixed point which is also the unique fixed point of each fr. proof. consider any mapping fj. by virtue of (ii), it is obvious that fj satisfies the following condition: (3.1) ffj x,fjy(t) > min { fx,fjx(t), fy,fjy(t) } . let x0 be any point in x. define a sequence {xn} in x recursively by xn = fjxn−1, n = 1, 2, . . .. if xp = xp+1 for some p ∈ n, then xp is a fixed point of fj. suppose xn 6= xn+1 for all n ≥ 0. then using (3.1) we have fxn,xn+1(t) = ffj xn−1,fjxn(t) > min { fxn−1,fjxn−1(t), fxn,fjxn(t) } = min { fxn−1,xn(t), fxn,xn+1(t) } = fxn−1,xn(t). thus {fxn,xn+1(t)} is a strictly increasing sequence of positive real numbers in [0, 1] and, hence, tends to a limit r ≤ 1. suppose r < 1. then there exists a positive integer n with n ≥ n such that (3.2) r − δ(r) < fxn,xn+1(t) < r. © agt, upv, 2021 appl. gen. topol. 22, no. 2 440 on a probabilistic version of meir-keeler type fixed point theorem this further implies r − δ(r) < min { fxn,xn+1(t), fxn+1,xn+2(t) } < r, that is, r − δ(r) < min { fxn,fjxn(t), fxn+1,fjxn+1(t) } < r. by virtue of (ii), this yields ffj xn,fjxn+1(t) = fxn+1,xn+2(t) > r. this contradicts (3.2). hence lim inf n→+∞ fxn,xn+1(t) = 1. further, if q is any positive integer then for each t > 0, we have fxn,xn+q(t) = ffj xn−1,fjxn+q−1(t) > > min { fxn−1,fjxn−1(t), fxn+q−1,fjxn+q−1(t) } = min { fxn−1,xn(t), fxn+q−1,xn+q(t) } . since lim inf n→+∞ fxn,xn+1(t) = 1, making limit as n → +∞, the above inequality yields lim inf n→+∞ fxn,xn+q(t) = 1. therefore, {xn} is a cauchy sequence. since x is complete, there exists a point z in x such that lim n→+∞ xn = lim n→+∞ fnj x0 = z. moreover, if y0 is any other point in x and yn = fjyn−1 = f n j y0, then (3.1) yields fxn,yn(t) = ffj xn−1,fjyn−1(t) > min { fxn−1,fjxn−1(t), fyn−1,fyn−1(t) } = min { fxn−1,xn(t), fyn−1,yn(t) } . letting n → +∞, we get lim inf n→+∞ fz,yn(t) = 1 for each t > 0. therefore, lim n→+∞ yn = lim n→+∞ fnj y0 = z. suppose that fj is weakly orbitally continuous. since fnj x0 → z for each x0, by virtue of weak orbital continuity of fj we get, fnj y0 → z and f n+1 j y0 → fjz for some y0 ∈ x. this implies that z = fjz since fn+1j y0 → z. therefore z is a fixed point of fj. uniqueness of the fixed point follows from (i). moreover, if v and w are the fixed points of fj and fs respectively, then by (iii) we have fv,w(t) = ffj v,fsw(t) ≥ min { fv,fj v(t), fw,fsw(t) } . in view of lim inf n→+∞ fv,w(t) = 1 for each t > 0, we get v = w and each mapping {fj} has a unique fixed point which is also the unique common fixed point of the family of mappings. � the following result is an easy consequence of theorem 3.1: corollary 3.2. let (x, f, t ) be a complete menger pm-space, and let {fj : 0 ≤ j ≤ 1} be a family of self-mappings of x such that for any given fj satisfying conditions (i)-(ii) of theorem 3.1. if fj is either k−continuous or fkj is continuous for some positive integer k ≥ 1 or fj is orbitally continuous, then fj has a unique fixed point moreover, if every pair of mappings (fr, fs) © agt, upv, 2021 appl. gen. topol. 22, no. 2 441 r. k. bisht and v. rakočević satisfies the condition (iii) of theorem 3.1, then the mappings {fj} have a unique common fixed point which is also the unique fixed point of each fr. the triple (x, f, tmin) is a complete menger pm-space, for x ⊆ r (see remark 2.3). the following example illustrates theorem 3.1. example 3.3. let x = [0, 2] and d be the usual metric. for each 0 ≤ j ≤ 1}, we define fj : x 7→ x by fjx = { 1, if 0 ≤ x ≤ 1, j(x − 1), if 1 < x ≤ 2. then the mappings fj satisfy all the conditions of theorem 3.1 and have a unique common fixed point x = 1 which is also the unique fixed point of each mapping. the mapping fj is discontinuous at the fixed point. the mapping fj satisfies condition (ii) with δ(ǫ) = 1 − ǫ, if ǫ < 1, and δ(ǫ) = ǫ, for ǫ ≥ 1. it is also easy to see that the mapping fj is orbitally continuous and, hence, weak orbitally continuous [14]. taking fj = g in theorm 3.1, we get the following result as a corollary which is a probabilistic version of theorem 2.1 of pant et al. [12]: theorem 3.4. let (x, f, t ) be a complete menger pm-space, and let g be a self-mapping of x such that fgx,gy(t) ≥ min { fx,gx(t), fy,gy(t) } for all x, y ∈ x; (iv) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ x with ǫ − δ ≤ min { fx,gx(t), fy,gy(t) } < ǫ implies fgx,gy(t) > ǫ. then (a) g possesses a unique fixed point if and only if g is weakly orbitally continuous. (b) g possesses a unique fixed point provided g is either orbitally continuous or k−continuous or gk is continuous for some positive integer k ≥ 1. (c) g possesses a unique fixed point provided g is either x0-orbitally continuous or almost orbitally continuous. in the next result, we show that theorem 3.4 characterizes metric completeness of x. various workers have proved fixed point theorems that characterize metric completeness [4, 17, 19, 25, 26]. in the next theorem, we show that completeness of the space is equivalent to fixed point property for a large class of mappings including both continuous and discontinuous mappings. in what follows we use the notation a ≫ b (or a ≪ b) to show that the positive number a is much greater (smaller) than the positive number b. theorem 3.5. let (x, f, t ) be a menger pm-space. if every k−continuous or almost orbitally continuous self-mapping of x satisfying the condition (iv) of theorem 3.4 has a fixed point, then x is complete. © agt, upv, 2021 appl. gen. topol. 22, no. 2 442 on a probabilistic version of meir-keeler type fixed point theorem proof. suppose that every k−continuous self-mapping of x satisfying condition (iv) of theorem 3.4 possesses a fixed point. we will prove that x is complete. if possible, suppose x is not complete. then there exists a cauchy sequence in x, say m = {u1, u2, u3, . . .}, consisting of distinct points which does not converge. let x ∈ x be given. then, since x is not a limit point of the cauchy sequence m, there exists a least positive integer n(x) such that x 6= un(x) and for each m ≥ n(x) and t > 0 we have (3.3) 1 − fx,un(x)(t) ≫ 1 − fun(x),um(t). consider a mapping g : x 7→ x by g(x) = un(x). then, g(x) 6= x for each x and, using (3.3), for any x, y in x and t > 0 we get 1 − fgx,gy(t) = 1 − fun(x),un(y)(t) ≪ 1 − fx,un(x)(t) = 1 − fx,gx(t) if n(x) ≤ n(y), or 1 − fgx,gy(t) = 1 − fun(x),un(y)(t) ≪ 1 − fy,un(y)(t) = 1 − fy,gy(t) if n(x) > n(y). this implies that (3.4) fgx,gy(t) > min { fx,gx(t), fy,gy(t) } . in other words, given ǫ > 0 we can select δ(ǫ) = ǫ such that (3.5) ǫ − δ ≤ min { fx,gx(t), fy,gy(t) } < ǫ implies fgx,gy(t) > ǫ. it is clear from (3.4) and (3.5) that the mapping g satisfies condition (iv) of theorem 3.4. moreover, g is a fixed point free mapping whose range is contained in the non-convergent cauchy sequence m = {un}n∈n. hence, there exists no sequence {xn}n∈n in x for which {gxn}n∈n converges, that is, there exists no sequence {xn}n∈n in x for which the condition gxn → t implies g2xn → gt is violated. therefore, g is a 2-continuous mapping. in a similar manner it follows that g is almost orbitally continuous. thus, we have a selfmapping g of x which satisfies condition (iv) of theorems 3.4 but does not possess a fixed point. this contradicts the hypothesis of the theorem. hence x is complete. � we now give a weaker version of theorem 3.4 which extends theorem 3.2 of [17]. theorem 3.6. let (x, f, t ) be a complete menger pm-space, and let g be a self-mapping of x such that (v) fgx,gy(t) > min { fx,gx(t), fy,gy(t) } for all x, y ∈ x; (vi) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ x with ǫ − δ < min { fx,gx(t), fy,gy(t) } < ǫ implies fgx,gy(t) ≥ ǫ. then g possesses a unique fixed point if g is either weakly orbitally continuous or x0-orbitally continuous or almost orbitally continuous. proof. the proof follows on the similar lines as the proof of theorem 3.5. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 443 r. k. bisht and v. rakočević 3.2. fixed points of a family of (ǫ − δ) non-expansive mappings in menger pm-space. we now prove a fixed point theorem for a family of (ǫ − δ) non-expansive mappings in menger pm-space. theorem 3.7. let (x, f, t ) be a complete menger pm-space, and let {fj : 0 ≤ j ≤ 1} be a family of self-mappings of x such that for any given fj the following conditions are satisfied: (i’) ffj x,fjy(t) ≥ min { fx,fjx(t), fy,fjy(t) } for all x, y ∈ x; (ii’) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ x with ǫ − δ < min { fx,fjx(t), fy,fjy(t) } < ǫ implies ffj x,fjy(t) ≤ ǫ; if fj is continuous, then fj has a unique fixed point, say z, moreover, if every pair of mappings (fr, fs) satisfies the condition (iii’) ffj x,fsy(t) ≥ min { fx,fjx(t), fy,fsy(t) } ; then the mappings {fj} have a unique common fixed point which is also the unique fixed point of each fr. proof. let x0 be any point in x. define a sequence {xn} in x recursively by xn = fjxn−1, n = 1, 2, . . .. then following the lines of theorem 3.1, it can be shown that {xn} is a cauchy sequence. continuity of fj now implies that fjz = z and z is a fixed point of fj. rest of the proof follows from theorem 3.1. � taking fj = g in theorm 3.7, we get the following result as a corollary: theorem 3.8. let (x, f, t ) be a complete menger pm-space, and let g be a continuous self-mapping of x such that (iv’) fgx,gy(t) ≥ min { fx,gx(t), fy,gy(t) } for all x, y ∈ x; (v’) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ x with ǫ − δ < min { fx,gx(t), fy,gy(t) } < ǫ implies fgx,gy(t) ≤ ǫ. then g possesses a unique fixed point. the triple (x, f, tmin) is a complete menger pm-space, for x ⊆ r (remark 2.3). the following example [13] illustrates theorem 3.8. example 3.9. let x = [−1, 1] and d be the usual metric. define g : x 7→ x by gx = −|x|x, for each x ∈ x. then the mapping g satisfies all the conditions of theorem 3.8 and has a unique fixed point x = 0. also, g possesses two periodic points x = 1 and x = −1. the mapping g satisfies condition (v′) with δ(ǫ) = ( √ (ǫ/2)) − (ǫ/2), if ǫ < 2, and δ(2) = 2. remark 3.10. it is pertinent to mention here that uniqueness of the fixed point in theorem 3.8 is because of the particular form (iv′). if we change (iv′) by the following fgx,gy(t) ≥ min { fx,y(t), fx,gx(t), fy,gy(t) } for all x, y ∈ x, © agt, upv, 2021 appl. gen. topol. 22, no. 2 444 on a probabilistic version of meir-keeler type fixed point theorem then the fixed point need not be unique. remark 3.11. theorem 3.1 provides a new answer to the once open question (see rhoades [20], p. 242) on the existence of contractive mappings which admit discontinuity at the fixed point in the setting of menger pm-space. references [1] r. k. bisht and r. p. pant, a remark on discontinuity at fixed point, j. math. anal. appl. 445 (2017), 1239–1242. 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[27] n. taş and n. y. özgür, a new contribution to discontinuity at fixed point, fixed point theory 20, no. 2 (2019), 715–728. © agt, upv, 2021 appl. gen. topol. 22, no. 2 446 @ appl. gen. topol. 23, no. 1 (2022), 179-187 doi:10.4995/agt.2022.15187 © agt, upv, 2022 topologically mixing extensions of endomorphisms on polish groups john burke and leonardo pinheiro department of mathematical sciences, rhode island college, 600 mt pleasant ave, providence, ri 02908 (jburke@ric.edu, lpinheiro@ric.edu) communicated by j. galindo abstract in this paper we study the dynamics of continuous endomorphisms on polish groups. we offer necessary and sufficient conditions for a continuous endomorphism on a polish group to be weakly mixing. we prove that any continuous endomorphism of an abelian polish group can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said group. 2020 msc: 37b99. keywords: weak mixing; polish group; hypercyclicity criterion. 1. introduction the theory of discrete dynamical systems is concerned with the behavior of the iterates of a continuous map on a (usually compact) metric space. the most interesting and studied examples include maps that, in some sense, ‘mix’ the space. for a nice survey on the subject see the article by kolyada and snoha [7]. formally, let x be a topological space and f : x → x a continuous map. let fn(x) = f ◦f ◦ · · · ◦f︸︷︷︸ n−fold denote the n− th iteration of the map f. received 28 february 2021 – accepted 08 january 2022 http://dx.doi.org/10.4995/agt.2022.15187 https://orcid.org/0000-0002-1633-3606 j. burke and l.pinheiro we say f is topologically transitive if given any two non-empty open subsets u and v of x there exists a natural number n ≥ 1 such that fn(u) ∩v 6= ∅. a continuous map f : x → x is said to be topologically mixing if given any two non-empty open subsets u and v of x there exists a natural number n such that fn(u)∩v 6= ∅, whenever n > n. a map f such that f×f is transitive in x ×x is called a weakly mixing map. topological mixing is a stronger condition than weak mixing which is generally stronger than topological transitivity. for example, the irrational rotation of the circle is a topologically transitive map that is not weakly mixing. in the setting of linear operators acting on a banach space, a very celebrated result is the set of sufficient conditions for topological mixing known as the hypercyclicity criterion. the result first appeared in kitai [6] and was later independently rediscovered by gethner and shapiro [5]. in [1], bes and peris showed that satisfying slightly laxer hypothesis than those in the hypercyclicity criterion is equivalent to the weak mixing of the operator in question. in [8] reed and de la rosa constructed a topologically transitive operator on a banach space that is not weak mixing and hence does not satisfy the hypercyclicity criterion, settling the question of whether the conditions were necessary. notice that the general theory of discrete dynamical systems is usually not concerned with any underlying algebraic structure of the space x. operator theorists studying linear dynamics, on the other hand, only consider maps preserving the linear structure of the underlying space. in this note we will revisit this theme in a more general setting; we will study the dynamics of continuous endomorphisms on polish groups. in particular, we conclude that every topologically mixing and weakly mixing continuous endomorphism on a polish group satisfies conditions analogous to those in the hypercylicity criterion. 2. main results in what follows g will denote a metric, complete, separable topological group: a polish group for short. if g has no isolated points, then a result by g. d. birkhoff [2] tells us that a continuous map ϕ : g → g is topologically transitive if and only if there exists an element x0 such that the orbit orb(x0,ϕ) = {x0,ϕ(x0),ϕ2(x0),ϕ3(x0), · · ·} is dense in x. in other words, any element of g can be arbitrarily approximated by a sequence of elements in the orbit of x0. in this setting, ϕ is weakly mixing if there exists (h,f) in g×g whose orbit under ϕ×ϕ is dense. recall that for a group g, with the group operation written as multiplication, an endomorphism is a map ϕ : g → g such that ϕ(gh) = ϕ(g)ϕ(h) for all g and h in g. before we continue with our discussion, we introduce the following definition: definition 2.1 (weak mixing criterion). let g be a polish semigroup with identity e. we say that a continuous endomorphism ϕ : g → g satisfies the weak mixing criterion if there exists an increasing sequence {nk} of natural © agt, upv, 2022 appl. gen. topol. 23, no. 1 180 topologically mixing extensions of endomorphisms on polish groups numbers, dense sets f,h ⊂ g, and maps ψnk : f → g such that for any f ∈ f and h ∈ h: (i) ϕnk (h) → e as k →∞. (ii) ψnk (f) → e as k →∞. (iii) ϕnkψnk (f) → f as k →∞. chan [3] and moothatu [9] independently remarked that if ϕ is a continuous endomorphism on a polish group satisfying the weak mixing criterion in the particular case nk = k,k ≥ 0, then ϕ is topologically mixing. in the same paper moothatu showed that if g is a compact hausdorff group and ϕ is a topologically transitive endomorphism, then ϕ is weakly mixing. in view of chan and moothatu’s observation we ask if every weak mixing continuous endomorphism on a polish group must satisfy the weak mixing criterion. we answer the question in the affirmative for compact polish groups in contrast to de la rosa and reed result for banach spaces presented in [8]. the connection between the two settings comes from the fact that continuous linear operators are endomorphisms on the additive abelian group of a topological vector space. indeed, we have: theorem 2.2. let g be a polish group with identity e and f : g → g a continuous endomorphism. then ϕ satisfies the weak mixing criterion if and only if ϕ is weakly mixing. proof. assume ϕ satisfies the the weak mixing criterion for a sequence {nk} and dense sets f and h. let u1,u2,v1 and v2 be nonempty open subsets of g. in order to show that ϕ is weakly mixing, it suffices to exhibit n ∈ n such that (ϕn ×ϕn) (u1 ×u2) ∩ (v1 ×v2) 6= ∅. since f and h are dense in g, we can choose f1 ∈ v1 ∩f, f2 ∈ v2 ∩f , h1 ∈ u1 ∩h, and h2 ∈ u2 ∩h. then h1ψnk (f1) → h1e = h1 ∈ u1 as k →∞ similarly, h2ψnk (f2) → h2e = h2 ∈ u2 as k →∞, so for sufficiently large k (h1ψnk (f1),h2ψnk (f2)) ∈ u1 ×u2. now, notice that ϕnk (h1ψnk (f1)) = ϕ nk (h1)ϕ nk (ψnk (f1)) → ef1 = f1 ∈ v1 as k →∞ and also ϕnk (h2ψnk (f2)) = ϕ nk (h2)ϕ nk (ψnk (f2)) → ef2 = f2 ∈ v2 as k →∞. again, for k large enough, (ϕnk (u1) ×ϕnk (u2)) ∩ (v1 ×v2) 6= ∅, and we can conclude ϕ is weakly mixing. now, we assume that is ϕ weakly mixing with (h,f) an element whose orbit is dense in g×g. let id be the identity map on g. since id×ϕn commutes © agt, upv, 2022 appl. gen. topol. 23, no. 1 181 j. burke and l.pinheiro with ϕ×ϕ and its image is dense in g×g, we have that for any n ∈ n, the orbit of (h,ϕn(f)) is also dense. this is implies that for all u ⊂ g open, there is u ∈ u such that (h,u) has dense orbit under ϕ×ϕ. we will now exhibit the sets h and f and construct the sequence {nk} in the statement of the weak mixing criterion. for each natural number k > 1, consider bk be the open ball in g centered at e of radius 1/k. since bk is open in g, and left multiplication is continuous, the set bk × hbk is open in g × g. by the observation above, we can pick wk ∈ bk such that, the orbit of (h,wk) under ϕ × ϕ is dense in g × g. it follows that there exists an integer nk > k such that (ϕnk ×ϕnk )(h,wk) ∈ (bk ×hbk). that is, we can inductively construct an increasing sequence of positive integer {nk} such that: (i) ϕnk (h) ∈ bk and (ii) ϕnk (wk) ∈ hbk. letting k →∞, we have that: wk → e because wk ∈ bk, ϕnk (h) → e by (i), and ϕnk (wk) → h by (ii) . now, let f = h = {h,ϕ(h),ϕ2(h), · · ·} and define ψnk : f → g by ψnk (ϕ j(h)) = ϕj(wk). we have: (i) ϕnk (ϕj(h)) = ϕj(ϕnk (h)) → ϕj(e) = e, (ii) ψnk (ϕ j(h)) = ϕj(wk) → ϕj(e) = e, and (iii) ϕnk (ψnk (ϕ j(h))) = ϕnk (ϕj(wk)) = ϕ j(ϕnk (wk)) → ϕj(h), which is what we needed to show. � it is worth noting again, that if the subsequence {nk} is given by nk = k then we can conclude that ϕ is topologically mixing. we can now state: corollary 2.3. let g be a compact polish group and let ϕ be a continuous homomorphism on g. then ϕ is topologically transitive if and only if it satisfies the weak mixing criterion. proof. if ϕ is topologically transitive, then by a result of moothatu [9] ϕ must be weak mixing and by theorem 2.2 it satisfies the weak mixing criterion. conversely, if ϕ satisfies that weak mixing criterion then its transitivity follows immediately from theorem 2.2. � a natural question is whether any polish group supports a topologically mixing endomorphism. we show that if the group in question is an abelian group that can be written as a countable infinite product of isomorphic copies of one of its closed subgroups, then the answer is affirmative. more precisely, we show that any continuous endomorphism on an abelian polish group can be extended in a natural way to a topologically mixing continuous endomorphism of the infinite direct product of said group. the result is analogous to that of chan [4]. © agt, upv, 2022 appl. gen. topol. 23, no. 1 182 topologically mixing extensions of endomorphisms on polish groups theorem 2.4. let g be an abelian polish group, and ϕ : g → g a continuous endomorphism. there exists a continuous endomorphism φ : ∏∞ i=1 g →∏∞ i=1 g such that: (i) φ is topologically mixing, and (ii) j ◦ϕ = φ ◦ j where j : g → ∏∞ i=1 g is defined by j(g) = (g,e,e, ...) proof. we will construct the endomorphism φ and check it is topologically mixing by verifying it satisfies the weak mixing criterion. consider a bounded metric d on g, we can define a metric ρ on ∏∞ i=1 g by ρ(g,h) = ∞∑ i=1 d(gi,hi) 2i , forf,g ∈ g which induces the product topology on ∏∞ i=1 g. for each g ∈ ∏∞ i=1 g , write g = (g1,g2,g3, · · ·), gi ∈ g, and define the maps φ, ψ : ∏∞ i=1 g → ∏∞ i=1 g by φ(g) = (ϕ(g1)g2,g3,g4, · · ·) and ψ(g) = (e,g1,g2,g3, · · ·) where e is the identity in g. first we verify that φ is an endomorphism in ∏∞ i=1 g. indeed, let f,g,∈∏∞ i=1 g and we have φ(fg) = φ((f1,f2,f3, ...)(g1,g2,g3, ...)) = φ(f1g1,f2g2,f3g3, ...) = (ϕ(f1g1)f2g2,f3g3, ...) = (ϕ(f1)f2ϕ(g1)g2,f3g3, ...) = (ϕ(f1)f2,f3, ...)(ϕ(g1)g2,g3, ...) = φ(f)φ(g). notice that we have used the fact that g is abelian on the third equality from the bottom. the continuity of φ follows from the fact that the group operation is continuous. now, we will check that j ◦ϕ = φ ◦ j. indeed, for any g ∈ g, (j ◦ϕ)(g) = j(ϕ(g)) = (ϕ(g),e,e, ...) = (ϕ(g)e,e, ...) = φ((ϕ(g),e,e, ...)) = φ(j(g) = (φ ◦ j)(g) © agt, upv, 2022 appl. gen. topol. 23, no. 1 183 j. burke and l.pinheiro now, note that for all g ∈ ∏∞ i=1 g, φ(ψ(g)) = g and ψ ng → ẽ as n → ∞ where ẽ = (e,e,e, · · ·) is the identity element of ∏∞ i=1 g. we will now verify that φ is mixing. let h be a dense set in g and consider the subgroup d̃ in∏∞ i=1 g whose elements are of the form (h1,h2,h3, · · · ,hk,e,e,e, . . . ) for some natural k and hi ∈ h. this is clearly dense in ∏∞ i=1 g and for any element h = (h1,h2,h3, · · · ,hk,e,e,e, . . . ) ∈ d̃, we have φk(h) = ( ϕk(h1)ϕ k−1(h2) · · ·ϕ(hk−1)hk,e,e, · · · ) and ψk(φk(h)) = (e,e, · · · ,e︸︷︷︸ k positions ,ϕk(h1)ϕ k−1(h2) · · ·ϕ(hk−1)hk,e,e, · · ·). notice that: ρ(ψk(φk(h)), ẽ) = d(ϕik(h0)ϕ k−1(h1) · · ·ϕ(hk−1)hk,e) 2k ≤ 1 2k+1 so we conclude that as n →∞, ρ(ψk(φk(h)), ẽ) → 0. now, consider the set: f̃ = {g(ψn(φn(g))−1 : g ∈ d̃,n ∈ n}, and note that lim n→∞ g(ψn(φn(g))−1) = g lim n→∞ (ψn(φn(g))−1) = g( lim n→∞ (ψn(φn(g−1))) = gẽ = g which shows that f̃ is dense in ∏∞ i=1 g. also, let f ∈ f̃, so f = g(ψn(φn(g))−1 for some g ∈ d̃ and we have φn(f) = φn(g(ψn(φn(g))−1)) = φn(g)φn(ψn(φn(g))−1) = φn(g)φn(ψn(φn(g−1))) = φn(g)(φn(g))−1 = ẽ. the third equality follows from the fact that ψ is a left inverse for φ. we can then conclude that φn → ẽ on f̃ and by the results of chan [3] and moothatu [9], we conclude φ is mixing. � now, we define a weakly mixing continuous endomorphism on the infinite product in an analogous fashion to theorem 2.4. © agt, upv, 2022 appl. gen. topol. 23, no. 1 184 topologically mixing extensions of endomorphisms on polish groups theorem 2.5. let g be an abelian polish group and let ϕ : g → g be a continuous endomorphism. then the map φ : ∏∞ i=1 g → ∏∞ i=1 g defined by: φ(a1,a2,a3, . . . ) = (ϕ(a1 −1)a−12 ,a3 −1,a4 −1, . . . ) is weakly mixing. proof. this proof will be a slight modification of the proof of theorem 2.4. we follow the notation in the proof of theorem 2.4 and let ψ and d̃ be defined in the same manner. it is not hard to verify that φ is a continuous endomorphism. as in the proof of theorem 2.4, we have ψk(g) → ẽ as k → ∞ and thus ψ2k(g) → ẽ as k → ∞. also observe that φ2k(ψ2k(g)) = g. next note that since g is an abelian group we have, φ2(a1,a2,a3, . . . ) = (ϕ 2(a1)ϕ(a2)a3,a4,a5 . . . ). one can thus verify that ψ2k(φ2k(h)) = (e,e, · · · ,e︸︷︷︸ 2k positions ,ϕ2k(h0)ϕ 2k−1(h1) · · ·ϕ(h2k−1)h2k,e,e, · · ·). and ρ(ψ2k(φ2k(h)), ẽ) = d̂(ϕ2k(h0)ϕ 2k−1(h1) · · ·ϕ(h2k−1)h2k,e) 22k ≤ 1 22k+1 we conclude that as n →∞, ρ(ψ2k(φ2k(h)), ẽ) → 0. now, let d be a dense set in g and define f̃ as follows, f̃ = {g(ψ2n(φ2n(g))−1 : g ∈ d,n ∈ n}. notice that lim n→∞ g(ψ2n(φ2n(g))−1 = g( lim n→∞ (ψ2n(φ2n(g))−1) = g( lim n→∞ (ψ2n(φ2n(g))−1 = gẽ = g which shows that f̃ is dense in ∏∞ i=1 g. also, let f ∈ f̃, so f = g(ψ2n(φ2n(g))−1 for some g ∈ ∏∞ i=1 g and we have φ2n(f) = φ2n(g(ψ2n(φ2n(g))−1) = φ2n(g)φ2n(ψ2n(φ2n(g))−1) = φ2n(g)(φ2n(g))−1 = ẽ. the third equality follows from the fact that ψ is a left inverse for φ. hence, φ2n(f) → ẽ on f̃ and by theorem 2.2, we can conclude φ is weakly mixing. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 185 j. burke and l.pinheiro 3. some general results the first result in this section is concerned with the size of the orbit of an endomorphism; the second result gives an idea of how common are group elements with a dense orbit. before we do so, we will fix some notation. for a topological space x and a a subset of x, we denote the closure of a by ā and its interior by a◦. we say that a subset of a topological space is somewhere dense when a ◦ 6= ∅. proposition 3.1. let t be a continuous endomorphism on a polish group. if t has an element g with a somewhere dense orbit then the subgroup generated by g is clopen in g. proof. it is a well-known fact that if a subgroup of a topological group has non-empty interior then the subgroup is open. since orb(t,g) ◦ ⊂〈orb(t,x)〉 6= ∅ we have that 〈orb(t,x)〉 is open but being the closure of the orbit, it is closed. � we immediately get the following. corollary 3.2. if g is a connected polish group, then the subgroup generated by an element with a somewhere dense orbit is dense. proof. the result follows from the basic fact that the only clopen subsets of a connected space are the empty set and the space itself. � now, we turn our attention to the algebraic structure of the set of group elements whose orbit is dense. we begin with the observation made by kitai in [6] that a continuous self-map f on a complete metrizable space x without isolated points is topologically transitive if and only if the set of transitive elements tr(f) = {x ∈ x; orb(x,f) is dense} is a dense gδ set. we then have : proposition 3.3. let g be a polish group that admits a topologically transitive endomorphism, then every element x ∈ g is the product of two elements whose orbits are dense under t . proof. let t be a topologically transitive endomorphism on g and let x ∈ g. we have that both tr(t) and tr(t)x−1 are dense gδ sets, by baire’s theorem their intersection is not empty. let h be an element in the intersection, so h = gx−1 with g and h with dense orbit. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 186 topologically mixing extensions of endomorphisms on polish groups references [1] j. bès and a. peris, hereditarily hypercyclic operators, journal of functional analysis 167 (1999), 94–112. [2] g. d. birkhoff, surface transformations and their dynamical applications, acta. math. 43 (1922), 1–119. [3] k. chan, universal meromorphic functions, complex variables, theory and applications 46 (2001), 307-314. [4] c. chan and g. turcu, chaotic extensions of operators on hilbert subspaces, revista de la real academia de ciencias exactas, fisicas y naturales. serie a. matemáticas 105 (2011), 415–421. [5] m. gethner and j. shapiro, universal vectors for operators on spaces of holomorphic functions, proceedings of the american mathematical society 100 (1987), 281–288. [6] c. kitai, invariant closed sets for linear operators, university of toronto thesis (1982). [7] s. kolyada and l’. snoha, topological transitivity a survey, grazer math. ber. 334 (1997), 3–35. [8] m. de la rosa and c. reed, invariant closed sets for linear operators, journal of operator theory 61 (2009), 369–380. [9] t. k. subrahmonian moothathu, weak mixing and mixing of a single transformation of a topological (semi)group, aequationes mathematicae 78 (2009), 147–155. © agt, upv, 2022 appl. gen. topol. 23, no. 1 187 @ appl. gen. topol. 22, no. 2 (2021), 303-309doi:10.4995/agt.2021.13874 © agt, upv, 2021 orbitally discrete coarse spaces igor protasov taras shevchenko national university of kyiv, department of computer science and cybernetics, academic glushkov pr. 4d, 03680 kyiv, ukraine (i.v.protasov@gmail.com) communicated by f. mynard abstract given a coarse space (x, e), we endow x with the discrete topology and denote x ♯ = {p ∈ βg : each member p ∈ p is unbounded }. for p, q ∈ x♯, p||q means that there exists an entourage e ∈ e such that e[p ] ∈ q for each p ∈ p. we say that (x, e) is orbitally discrete if, for every p ∈ x♯, the orbit p = {q ∈ x♯ : p||q} is discrete in βg. we prove that every orbitally discrete space is almost finitary and scattered. 2010 msc: 54d80; 20b35; 20f69. keywords: coarse space; ultrafilter; orbitally discrete space; almost finitary space; scattered space. 1. introduction and preiminaries given a set x, a family e of subsets of x × x is called a coarse structure on x if • each e ∈ e contains the diagonal △x = {(x, x) ∈ x : x ∈ x}; • if e, e′ ∈ e then e ◦ e′ ∈ e and e−1 ∈ e, where e ◦ e′ = {(x, y) : ∃z((x, z) ∈ e, (z, y) ∈ e′)}, e−1 = {(y, x) : (x, y) ∈ e}; • if e ∈ e and △x ⊆ e ′ ⊆ e then e′ ∈ e; • ⋃ e = x × x. received 16 june 2020 – accepted 15 april 2021 http://dx.doi.org/10.4995/agt.2021.13874 i. protasov a subfamily e′ ⊆ e is called a base for e if, for every e ∈ e, there exists e′ ∈ e′ such that e ⊆ e′. for x ∈ x, a ⊆ x and e ∈ e, we denote e[x] = {y ∈ x : (x, y) ∈ e}, e[a] = ⋃ a∈a e[a], ea[x] = e[x] ∩ a and say that e[x] and e[a] are balls of radius e around x and a. the pair (x, e) is called a coarse space [19] or a ballean [12], [18]. for a coarse space (x, e), a subset b ⊆ x is called bounded if b ⊆ e[x] for some e ∈ e and x ∈ x. the family b(x,e) of all bounded subsets of (x, e) is called the bornology of (x, e). we recall that a family b of subsets of a set x is a bornology if b is closed under taking subsets and finite unions, and b contains all finite subsets of x. a coarse space (x, e) is called finitary, if for each e ∈ e there exists a natural number n such that |e[x]| < n for each x ∈ x. let g be a transitive group of permutations of a set x. we denote by xg the set x endowed with the coarse structure with the base {{(x, gx) : g ∈ f}} : f ∈ [g]<ω, id ∈ f}. by [8, theorem 1], for every finitary coarse structure (x, e), there exists a transitive group g of permutations of x such that (x, e) = xg. for more general results, see [10]. let x be a discrete space and let βx denote the stone-čech compactification of x. we take the points of βx to be the ultrafilters on x, with the points of x identified with the principal ultrafilters, so x∗ = βx \ x is the set of all free ultrafilters. the topology of βx is generated by the base consisting of the sets ā = {p ∈ βx : a ∈ p}, where a ⊆ x. the universal property of βx states that every mapping f : x −→ y to a compact hausdorff space y can be extended to a continuous mapping fβ : βx −→ x. given a coarse space (x, e), we endow x with the discrete topology and denote by x♯ the set of all ultrafilters p on x such that each member p ∈ p is unbounded. clearly, x♯ is a closed subset of x∗ and x♯ = x∗ if (x, e) is finitary. following [7], we say that two ultrafilters p, q ∈ x♯ are parallel (and write p||q) if there exists e ∈ e such that e[p ] ∈ q for each p ∈ p. then || is an equivalence on x♯. we denote p = {q ∈ x♯ : q||p} and say that p is the orbit of p. if (x, e) is finitary and (x, e) = xg then p = gp. a coarse space (x, e) is called orbitally discrete if, for every p ∈ x♯, the orbit p is discrete. every discrete coarse space is orbitally discrete. we recall that (x, e) is discrete if, for each e ∈ e, there exists a bounded subset b such that e[x] = {x} for each x ∈ x \ b. in this case, p = {p} for each p ∈ x♯. © agt, upv, 2021 appl. gen. topol. 22, no. 2 304 orbitally discrete coarse spaces every bornology b on a set x defines the discrete coarse structure on x with the base {eb : b ∈ b}, eb[x] = b if x ∈ b, and eb[x] = {x} if x ∈ x \ b. by [15, theorem 5.4], for a finitary coarse space (x, e), the following conditions are equivalent: xg is orbitally discrete, xg is scattered, xg has no piecewise shifted fp-sets. a coarse space (x, e) is called scattered if, for every unbounded subset a of x, there exists e ∈ e such that a has asymptotically e-isolated balls: for each e′ ∈ e, there is a ∈ a such that e′a[a] \ ea[a] = ∅. this notion arouse in the characterization of the cantor macrocube [3] and, in the case of finitary coarse groups, was explored in [2]. let g be a group of permutations of a set x. let (gn)n∈ω be a sequence in g and let (xn)n∈ω be a sequence in x such that (1) {gǫ00 . . . g ǫn n xn : (ǫi) n i=0 ∈ {0, 1} n+1} ∩ {gǫ00 . . . g ǫm m xm : (ǫi) n i=0 ∈ {0, 1}n+1} = ∅ for all distinct n, m ∈ ω; (2) {gǫ00 . . . g ǫn n xn : (ǫi) n i=0 ∈ {0, 1} n+1}| = 2n+1 for every n ∈ ω. following [15], we say that a subset y of x is a piecewise shifted fp-set if there exist (gn)n∈ω, (xn)n∈ω satisfying (1), (2) and such that y = {gǫ00 . . . g ǫn n xn : ǫi ∈ {0, 1}}, n ∈ ω}. after exposition of results in section 2, we survey some known classes of orbitally discrete spaces in section 3. 2. resuts a coarse space (x, e) is called almost finitary if, for every e ∈ e , there exists a bounded subset b and a natural number n such that |e[x]| < n for each x ∈ x \ b. every discrete space and every finitary space are almost finitary. theorem 2.1. every orbitally discrete coarse space is almost finitary. proof. we suppose the contrary and choose e ∈ e, e = e−1 such that, for any bounded subset b and a natural number n, there exists x ∈ x \ b such that |e[x]| > n. we claim that there exists p ∈ x♯ such that, for every p ∈ p, {x ∈ p : |e2[x] ∩ p | > 1} ∈ p. otherwise, for every p ∈ x♯, there exists qp ∈ p such that {x ∈ qp : e 2[x] ∩ qp| = 1} ∈ p. we consider the open covering {q♯p : p ∈ x ♯} of x♯ and choose its finite subcovering q♯p1, . . . , q ♯ pm . then the set b = x\(qp1∪, . . . , ∪qpm) is bounded and |e[x]| ≤ m for each x ∈ x\e[b], but this contradicts the choice of e. we show that the orbit p is not discrete. given any p ∈ p, we choose q ∈ p, q ⊆ p such that |e2[x] ∩ p | > 1 for each x ∈ q. for every x ∈ q, we take f(x) ∈ e2[x] ∩ p such that x 6= f(x). then we extend the mapping x 7→ f(x) © agt, upv, 2021 appl. gen. topol. 22, no. 2 305 i. protasov from q to x by f(x) = x for each x ∈ x \ q. clearly, fβ(p) 6= p, p ∈ fβ(p) and fβ(p)||p because (x, f(x)) ∈ e2 for each x ∈ x. � to clarify the structure of an almost finitary coarse space, we use the following construction from [6]. a bornology b on a coarse space (x, e) is called e-compatible if e[b] ∈ b for all b ∈ b, e ∈ e. every e-compatible bornology b defines the b-strengthening (x, h) of (x, e), where h has the base {hb,e : b ∈ b, e ∈ e}, hb,e[x] = { e[b], if x ∈ b, e[x], if x ∈ x \ b. for description of the upper bound e ∨ e′ of coarse structures, see [13]. theorem 2.2. for a coarse space (x, e), the following statements are equivalent (i) (x, e) is almost finitary; (ii) (x, e) is the b-stregthening of some finitary coarse space (x, e′) by the bornology b of bounded subspaces of (x, e); (iii) e is the upper bound of a discrete and a finitary coarse structures on x. proof. (i) =⇒ (ii). for b ∈ b and e ∈ e, we pick b′b,e ∈ b and a natural number n such that b ⊆ b′b,e and |e[x]| < n for each x ∈ x \ b ′ b,e. we note that {b′b,e : b ∈ b, e ∈ e} is a base for b. for b ∈ b, e ∈ e we put e′b,e = { x if x ∈ b′b,e, e[x] if x ∈ x \ b′b,e, denote by e′ the smallest coarse structure on x containing all entourages {hb,e : b ∈ b, e ∈ e}, observe that e ′ is finitary and (x, e) is the bstrengthening of (x, e′). (ii) =⇒ (iii). if (x, e) is the b-strengthening of (x, e′) then e is the upper bounded of e′ and the discrete coarse structure on x defined by the bornology b. (iii) =⇒ (i). we assume that e is the upper bound of finitary coarse structure e′ and discrete coarse structure on x defined by some bornology b. we choose the smallest bornology b′ on x such that b ⊆ b′ and e′(b′) ∈ b′ for all e′ ∈ e′. then b′ is the bornology of bounded subsets of (x, e) and (x, e) is the b′-strengthening of (x, e′), so (x, e) is almost finitary. � remark. let (x, e) be the b-strengthening of a finitary coarse space (x, e′). if (x, e′) is orbitally discrete then (x, e) is orbitally discrete, but the converse statement needs not to be true. let x be the disjoint union of © agt, upv, 2021 appl. gen. topol. 22, no. 2 306 orbitally discrete coarse spaces two infinite subsets y, z. we endow y with the finitary coarse structure ey such that (y, ey ) is not orbitaly discrete, and denote by ez the discrete coarse structure on z defined by the bornology of finite subset. we take the smallest coarse structure e′ on x such that e′|y = ey , e ′|z = ez. clearly, e ′ is finitary but not orbitally discrete. we denote by b the smallest bornology on x such that y ∈ b. then the b-strengthening of (x, e′) is discrete. theorem 2.3. for almost finitary coarse space (x, e) and p, q ∈ x♯, we have p||q if and only if there exist e ∈ e and a permutation g of x such that gp = q, gp = {gp : p ∈ p} and (x, gx) ∈ e for each x ∈ x. proof. let p||q. we take e ∈ e such that e = e−1 and e[p ] ∈ q for each p ∈ p. since (x, e) is almost finitary, there exist a bounded subset b of x and a natural number n such that |e[x]| < n for each x ∈ x \ b. we put y = x \ e[b], note that y ∈ p and define a set-valued mapping f : x −→ [x]<ω. f(x) = e[x] if x ∈ y and f(x) = {x} if x = x \ y . by theorem 1 from [10], there exists bijection f1, . . . , fm of x such that fi(x) ∈ f(x) and f1(x)∪· · · ∪fn(x) = f(x). we take i ∈ {1, . . . , m} such that fi(p) ∈ q for each p ∈ p and put g = fi. the converse statement follows directly from the definition of the parallelity relation ||. � corollary 2.4. if (x, e) is almost finitary, p ∈ x♯ and p is an isolated point of p then p is discrete. proof. we assume that some point q ∈ p is not isolated in p, use theorem 2.3 to choose a permutation g of x such that gq = p and note that p is not isolated in p. � for a subset a of (x, e) and p ∈ x♯, we denote ∆p(a) = p ∩ a ♯. theorem 2.5. an almost finitary coarse space (x, e) is scattered if and only if, for every unbounded subset a of x, there exists p ∈ a such that ∆p(a) is finite. proof. we suppose that x is scattered and choose e ∈ e such that a has an asymptotically isolated e-balls. for each h ∈ e, we denote ph = {x ∈ a : ha[x] \ ea[x] = ∅} and take p ∈ a ♯ such that ph ∈ p for each h ∈ e. if q ∈ a♯ and q||p then e[p ] ∈ q for each p ∈ p. we take the bijections f1, . . . , fm from the proof of theorem 2.3. since q = gp for some g ∈ {f1, . . . , fm}, we have ∆p(a) ≤ m. let ∆p(a) = {p1, . . . , pm}. for each i ∈ {1, . . . , m}, we pick ei ∈ e such that ei[p ] ∈ pi for each p ∈ p. then we take e ∈ e such that ei ⊆ e for each i ∈ {1, . . . , m}, and observe that a has an asymptotically isolated e-balls. � theorem 2.6. every orbitally discrete space is scattered. © agt, upv, 2021 appl. gen. topol. 22, no. 2 307 i. protasov proof. to apply theorem 2.5, we take an arbitrary unbounded subset a of x and find p ∈ a♯ such that ∆p(a) is finite. we use the zorn lemma to choose a minimal (by inclusion) closed subset s of a♯ such that ∆q(a) ⊆ s for each q ∈ s. let p ∈ s but ∆p(a) is infinite. we take the limit point q of ∆p(a). by the minimality of s, we have p ∈ cl∆q(a). applying theorem 2.3, we conclude that p is not isolated in p. � question. let x be an almost finitary scattered space. is x orbitally discrete? 3. comments 1. for a natural number n, a coarse space (x, e) is called n-thin if, for every e ∈ e, there exists a bounded subset b of x such that |e[x]| ≤ n, for every x ∈ x \ b. a space (x, e) is n-thin if and only if |p| ≤ n for each p ∈ x♯. for finite partitions of an n-thin space into discrete subspaces, see [5], [14], [17], [1, section 6]. 2. a coarse space (x, e) is called sparse if each orbit p, p ∈ x♯ is finite. sparse subsets of groups are studied in [4], [16]. for sparse metric spaces, see [9]. 3. a coarse space (x, e) is called indiscrete if each discrete subspace of x is bounded. by theorem 3.15 from [11], a finitary indiscrete space has no unbounded orbitally discrete subspaces. we do not know whether this statement holds for any almost finitary indiscrete spaces. references [1] t. banakh and i. protasov, set-theoretical problems in asymptology, arxiv: 2004.01979. [2] t. banakh, i. protasov and s. slobodianiuk, scattered subsets of groups, ukr. math. j. 67 (2015), 347–356. [3] t. banakh and i. zarichnyi, characterizing the cantor bi-cube in asymptotic categories, groups geom. dyn. 5, no. 4 (2011), 691–728. [4] ie. lutsenko and i. protasov, space, thin and other subsets of groups, intern. j. algebra comput. 19 (2009), 491–510. [5] ie. lutsenko and i. v. protasov, thin subset of balleans, appl. gen. topology 11 (2010), 89–93. [6] o. petrenko and i. v. protasov, balleans and filters, mat. stud. 38 (2012), 3–11. [7] i. v. protasov, normal ball structures, mat. stud. 20 (2003), 3–16. [8] i. v. protasov, balleans of bounded geometry and g-spaces, algebra dicrete math. 7, no. 2 (2008), 101–108. [9] i. v. protasov, sparse and thin metric spaces, mat. stud. 41 (2014), 92–100. [10] i. protasov, decompositions of set-valued mappings, algebra discrete math. 30, no. 2 (2020), 235–238. [11] i. protasov, coarse spaces, ultrafilters and dynamical systems, topol. proc. 57 (2021), 137–148. [12] i. protasov and t. banakh, ball structures and colorings of groups and graphs, mat. stud. monogr. ser. vol. 11, vntl, lviv, 2003. [13] i. protasov and k. protasova, lattices of coarse structures, math. stud. 48 (2017), 115–123. © agt, upv, 2021 appl. gen. topol. 22, no. 2 308 orbitally discrete coarse spaces [14] i. v. protasov and s. slobodianiuk, thin subsets of groups, ukrain. math. j. 65 (2013), 1245–1253. [15] i. protasov and s. slobodianiuk, on the subset combinatorics of g-spaces, algebra dicrete math. 17, no. 1 (2014), 98–109. [16] i. protasov and s. slobodianiuk, ultracompanions of subsets of a group, comment. math. univ. carolin. 55, no. 1 (2014), 257–265. [17] i. protasov and s. slobodianiuk, the dynamical look at the subsets of groups, appl. gen. topology 16 (2015), 217–224. [18] i. protasov and m. zarichnyi, general asymptology, math. stud. monogr. ser., vol. 12, vntl, lviv, 2007. [19] j. roe, lectures on coarse geometry, univ. lecture ser., vol. 31, american mathematical society, providence ri, 2003. © agt, upv, 2021 appl. gen. topol. 22, no. 2 309 @ appl. gen. topol. 22, no. 1 (2021), 139-147doi:10.4995/agt.2021.14332 © agt, upv, 2021 remarks on the rings of functions which have a finite number of discontinuities m. r. ahmadi zand and zahra khosravi department of mathematics, yazd university, iran (mahmadi@yazd.ac.ir, zahra.khosravi@stu.yazd.ac.ir) communicated by d. georgiou abstract let x be an arbitrary topological space. f(x) denotes the set of all real-valued functions on x and c(x)f denotes the set of all f ∈ f(x) such that f is discontinuous at most on a finite set. it is proved that if r is a positive real number, then for any f ∈ c(x)f which is not a unit of c(x)f there exists g ∈ c(x)f such that g 6= 1 and f = g r f. we show that every member of c(x)f is continuous on a dense open subset of x if and only if every non-isolated point of x is nowhere dense. it is shown that c(x)f is an artinian ring if and only if the space x is finite. we also provide examples to illustrate the results presented herein. 2010 msc: 54c40; 13c99. keywords: c(x)f ; z-ultrafilter; completely separated; c(x)f -embedded; z-filter; over-rings of c(x); artinian ring. 1. introduction let x be a nonempty topological space and i(x) denote the set of isolated points of x. the ring of all real-valued functions on x with pointwise addition and multiplication is denoted by f(x), continuous members of f(x) is denoted by c(x), the set of points at which f ∈ f(x) is continuous is denoted by c(f) and bounded members of c(x) is denoted by c∗(x). for any f ∈ f(x), it is well known and easy to prove that x \ c(f) is a countable union of closed sets. the set of all f ∈ f(x) such that x \ c(f) is finite is a subring of f(x) received 12 september 2020 – accepted 25 december 2020 http://dx.doi.org/10.4995/agt.2021.14332 m. r. ahmadi zand and z. khosravi and it is denoted by c(x)f . the ring c(x)f where x is t1 was introduced and studied in [5]. in this paper, topological spaces don’t have to satisfy any separation axioms unless otherwise stated. recall that lattice-ordered rings are subdirect sums of totally ordered rings. let f, g ∈ c(x)f , then f ∨ g = 1 2 (f + g + |f − g|) ∈ c(x)f , and f ∧ g = −(−f ∨ −g) [5]. thus, for any topological space x we observe that c(x)f is a lattice-ordered subring of f(x) and c(x) is a lattice-ordered subring of c(x)f . also c ∗(x)f , consisting of all functions in c(x)f which are bounded, is a lattice-ordered subring of c(x)f . let s be a subset of x. the characteristic function of s is denoted by χs. let r be a commutative ring. a nonzero ideal i in r is an essential ideal if i intersects every nonzero ideal of r nontrivially. the socle of r denoted by soc(r) is the sum of all minimal ideals of r, or the intersection of all essential ideals of r. in [5], it is shown that soc(c(x)f ) consists of all functions which vanish everywhere except on a finite subset of x. a subset of x is called a gδ-set if it is a countable intersection of open sets. as usual, cla and inta will denote the closure and interior of a subset a in a space x, respectively. in section 2, we show that for any f ∈ c(x)f which is not a unit of c(x)f there exists g ∈ c(x)f such that g 6= 1 and f = g rf where r is a positive number. also, it is proved that if x is a t1-space, f ∈ c(x)f and z(f) ⊆ c(f), then z(f) is gδ. in section 3, it is shown that c(x)f is an artinian ring if and only if the space x is finite. we prove that every point of x \ i(x) is nowhere dense and every dense open subset of x has finite complement if and only if c(x)f = {f ∈ f(x)|f|d ∈ c(d) for some dense open subset of x }. for undefined notations, the reader is referred to [4] and [6]. 2. cf -embedded subsets of x let f ∈ c(x)f . the set f −1(0) = {x ∈ x|f(x) = 0} denoted by z(f) is called the zero-set of f and coz(f) = x \ z(f) is called a co-zero-set in x [5]. the collection of all zero-sets in x is denoted by z[c(x)f ] or z(x). proposition 2.1 ([5]). for a topological space x, f ∈ c(x)f is a unit element if and only if z(f) = ∅. units in c∗(x)f is characterized by the following result. lemma 2.2. a function g in c∗(x)f is a unit of c ∗(x)f if and only if g is bounded away from zero. proof. necessity. if there is a function h in c∗(x)f such that gh = 1, then |h| < n for some n ∈ n and so |g| > 1 n . hence g is bounded away from zero. sufficiency. suppose that g is bounded away from zero. so there is r > 0 such that |g(x)| > r for every x ∈ x. thus for any x ∈ x, 0 < 1 |g(x)| < 1 r and so 1 g ∈ c∗(x)f , i.e., g is a unit of c ∗(x)f . � © agt, upv, 2021 appl. gen. topol. 22, no. 1 140 rings of functions which have a finite number of discontinuities proposition 2.3. if x is an arbitrary topological space and r is a positive real number, then, for any f ∈ c(x)f which is not a unit of c(x)f there exists g ∈ c(x)f such that g 6= 1 and f = g rf. proof. since f is not a unit of c(x)f , proposition 2.1 implies that there exists a ∈ z(f). if g = 1 − χ{a}, then 1 6= g ∈ c(x)f and f = g rf. � definition 2.4 ([5]). a nonempty subset f of z(x) is said to be a z-filter on x, if it satisfies the following conditions: (1) ∅ /∈ f; (2) if z1, z2 ∈ f, then z1 ∩ z2 ∈ f; and (3) if z ∈ f, z′ ∈ z(x) and z′ ⊇ z, then z′ ∈ f. let i be a proper ideal in c(x)f . then z[i] = {z(f)|f ∈ i} is a z-filter on x [5]. definition 2.5. a z-filter u on x is called a z-ultrafilter on x, if there isn’t any zfilter f on x, such that u ( f. definition 2.6 ([5]). an ideal i in a subring of c(x)f is called fixed ( resp., free), if the intersection of all members of z[i] is non-empty (resp., empty). in [5], it is noted that every z-filter f is of the form z[if] for some ideal if in c(x)f and f is called fixed ( resp., free), if if is fixed ( resp., free). definition 2.7. a z-filter f on x is called a prime z-filter, if the union of two zero-sets belongs to f, then at least one of them belongs to f. theorem 2.8. the following statements are correct. (1) if i is a prime ideal in c(x)f , then z[i] = {z(f)|f ∈ i} is a prime z-filter on x. (2) if f is a prime z-filter on x, then z−1[f] = {f ∈ c(x)f |z(f) ∈ f} is a prime ideal in c(x)f . proof. the proof is similar to [6, theorem 2.12]. � lemma 2.9. let f ∈ c(x)f . then there is a positive unit u of c(x)f such that (−1 ∨ f) ∧ 1 = uf. proof. it’s straightforward. � theorem 2.10. let x be a topological space which is t1. then the following statements are equivalent. (1) for any f ∈ c(x)f , there is a unit u of c(x)f such that f = u|f|. (2) for any g ∈ c∗(x)f , there exists a unit v of c ∗(x)f such that g = v|g|. proof. (1) ⇒ (2). let g ∈ c∗(x)f . then by (1), there exists a unit u of c(x)f such that g = u|g|, and so |u(x)| = 1 for any x ∈ x such that g(x) 6= 0. now, proposition 2.1 implies that z(u) = ∅. let a = {x ∈ x|u(x) > 0} and b = {x ∈ x|u(x) < 0}. then x is a disjoint union of a and b. if u = 1 or © agt, upv, 2021 appl. gen. topol. 22, no. 1 141 m. r. ahmadi zand and z. khosravi u = −1, then u ∈ c∗(x). let u 6= 1 and u 6= −1, then there is a finite subset f of x such that c(u) = x \ f . since x is a t1-space, c(u) is open in x. a \ f and b \ f are disjoint open sets in c(u) and so a \ f and b \ f are disjoint open sets in x. therefore, there is a unite v of c∗(x)f such that v|a = 1 and v|b = −1. clearly, g = v|g| and so (1) implies (2). (2) ⇒ (1). let f ∈ c(x)f . then, there is g ∈ c ∗(x)f such that g = (f ∧1)∨−1, so according to (2) there is a unit v of c∗(x)f such that v|g| = g. from z(f) = z(g) it follows that v|f| = f and hence, (2) implies (1). � theorem 2.11. let x be an arbitrary topological space, f be in f(x) and h be a finite closed subset of x such that f ∈ c(x \ h). if h ∩f−1(0) = ∅, then f−1(0) is a gδ-set in x. proof. since by our assumption f−1(0) ∩ h = ∅, there is m ∈ n such that f−1(− 1 m , 1 m )∩h = ∅. thus for every n ≥ m, f−1(− 1 n , 1 n ) is open in x \h and so it is open in x. hence, f−1(0) = ⋂∞ n=m f −1(− 1 n , 1 n ) is a gδ-set in x. � by the above theorem we note that if x is a t1-space, f ∈ c(x)f and z(f) ⊆ c(f), then z(f) is gδ. the following example shows that theorem 2.11 is not true in general case. example 2.12. let βn be the stone-čech compactification of n and y ∈ βn\ n. it is well known that {y} is not a gδ-set in βn. thus, f = 1−χ{y} ∈ c(βn)f and z(f) = {y} is not a gδ-set. definition 2.13 ([5]). two nonempty subsets a and b of a topological space x are said to be f-completely separated in x if there is a function f ∈ c∗(x)f such that f[a] = {0} and f[b] = {1}. theorem 2.14 ([5]). let x be a topological space. then two nonempty subsets a and b are f-completely separated in x if and only if they are contained in disjoint zero-sets. definition 2.15. let s be a nonempty subset of x. we say that s is cf embedded in x if for any function f ∈ c(s)f , there exists g ∈ c(x)f called an extension of f such that g|s = f. in the same way, s is called c∗f -embedded if every f ∈ c ∗(s)f can be extended to g ∈ c∗(x)f , i.e., ∃g ∈ c∗(x)f such that g|s = f. example 2.16. let x = r and s1 = r \ {0}. then, f ∈ c ∗(s1) with value 1 for all positive r, and -1 for negative r, has no continuous extension, i.e., s1 is not c ∗-embedded in x. clearly, s1 is c ∗ f -embedded and cf -embedded. let s = r \ z and h be the restriction of the bracket function to s. then, g ∈ c(s)f since it is in c(s). since there is no g ∈ c(r)f such that g|s = h, s is not cf -embedded in x. we observe that if s is a subset of a topological space x with finite complement, then s is c∗f -embedded and cf -embedded in x. © agt, upv, 2021 appl. gen. topol. 22, no. 1 142 rings of functions which have a finite number of discontinuities proposition 2.17. let η : s → x be a one-one mapping and k be the set of discontinuity points of η. if k is a finite set , then φ : c(x)f → c(s)f defined by φ(g) = g ◦ η is a ring homomorphism. proof. if g ∈ c(x)f , then h = x \ c(g) is a finite set. thus, s \ c(g ◦ η) is a finite set since it is a subset of η−1(h) ∪ k and η is continuous at every point of x \ k and one-one so φ(g) = g ◦ η ∈ c(s)f , i.e., φ is well defined. clearly, φ is a ring homomorphism. � the following result is an immediate consequence of proposition 2.17. corollary 2.18. let t be a nonempty subset of a topological space x. then, the restriction function φ : c(x)f → c(t )f (resp. φ : c ∗(x)f → c ∗(t )f ) defined by φ(f) = f|t is an onto ring homomorphism if and only if t is cf embedded (resp. c∗f -embedded). theorem 2.19. each cf -embedded subset is a c ∗ f -embedded subset of x. proof. let x be a topological space and s ⊆ x be a cf -embedded subset of x. now suppose that f ∈ c∗(s)f . since c ∗(s)f ⊆ c(s)f , there exists an extension g ∈ c(x)f such that g|s = f. thus there is a positive integer m such that for every x ∈ s, |f(x)| ≤ m. we put h := (g ∨ −m) ∧ m, obviously h ∈ c∗(x)f and for each s ∈ s, we have h(s) = (g(s) ∨ −m) ∧ m = (f(s) ∨ −m) ∧ m = f(s). hence, s is a c∗f -embedded subset of x. � theorem 2.20. let s and x be two subsets of a topological space y such that ∅ 6= s ⊆ x ⊆ y . if x is a cf -embedded in y then s is a cf -embedded in x if and only if s is cf -embedded in y . proof. necessity, assume that s is a cf -embedded in x and f ∈ c(s). so f has an extension g ∈ c(x)f such that g|s = f and according to assumption, x is cf -embedded in y , so there is h ∈ c(y )f such that h|x = g. hence, s is cf -embedded in y . sufficiency, assume that s is cf -embedded in y and put f ∈ c(s)f . since s is cf -embedded in y , there is an extension g ∈ c(y )f such that g|s = f and so g|x ∈ c(x)f is an extension of f which completes the proof. � theorem 2.21. a c∗f -embedded subset s of x is cf -embedded in x if and only if it is f-completely separated from every zero-set disjoint from it. proof. let s be a c∗f -embedded in x. assume that h ∈ c(x)f and z(h)∩s = ∅ and let f(s) = 1 h(s) for any s ∈ s, then f ∈ c(s)f . thus by the hypothesis, there is g ∈ c(x)f , such that g|s = f. therefore, gh ∈ c(x)f . from gh[s] = {1} and gh[z(h)] = {0} it follows that z(h) and s are f-completely separated in x. conversely, suppose that f ∈ c(s)f . so arctan◦f ∈ c ∗(x)f and there is © agt, upv, 2021 appl. gen. topol. 22, no. 1 143 m. r. ahmadi zand and z. khosravi g ∈ c(x)f such that g|s = arctan◦f. therefore z = {x ∈ x : |g(x)| ≥ π 2 } belongs to z(x) and z ∩ s = ∅. so according to the assumption there exists h ∈ c∗f (x) such that h[s] = {1} and h[z] = {0}. it’s obvious that for every x ∈ x, |g(x)h(x)| < π 2 and gh|s = arctan ◦f. hence tan ◦(gh) ∈ c(x)f is an extension of f and so s is cf -embedded in x. � by the above theorem and theorem 2.14 we have the following result. theorem 2.22. each c∗f -embedded zero-set is cf -embedded. proof. it is straigthforward. � 3. some algebraic aspects of c(x)f recall that t ′ (x) is the ring of all f ∈ f(x) where for each f there is an open dense subset d of x such that f|d ∈ c(d) [1]. let x be a topological space which is t1. then c(x)f is a subring of t ′ (x). the following example shows that it is not true in general. example 3.1. let x = {0, 1, 2} and τ = {∅, {0}, {1, 2}, x}. then, c(x)f = f(x) and (x, τ) is not a discrete space and so c(x) 6= f(x). since x is the only dense open subset of the space, c(x) = t ′ (x). thus, c(x)f is not a subring of t ′ (x). in the following result we show that every non-isolated point of a topological space x is nowhere dense if and only if c(x)f is a subring of t ′ (x). lemma 3.2. let x be an arbitrary topological space. then, c(x)f is a subring of t ′ (x) if and only if for any x ∈ x \ i(x), intcl{x} = ∅, i.e., {x} is nowhere dense. proof. let c(x)f be a subring of t ′ (x). if x ∈ x \ i(x), then χ{x} ∈ t ′ (x) since χ{x} ∈ c(x)f . thus, there is a dense open subset d of x such that χ{x} ∈ c(d). clearly, x /∈ d and so int(cl{x}) = ∅. conversely, let int(cl{x}) = ∅ for every x ∈ x \ i(x). if f ∈ c(x)f , then f = x \ c(f) is a finite subset of x \ i(x) and x \ clf = x \ (∪x∈f cl{x}) = ∩x∈f (x \ cl{x}) is an open dense subset of x contained in c(f). thus, f ∈ t ′(x), i.e., c(x)f ⊆ t ′ (x). � remark 3.3. recall that a subset a of a topological space x is said to be a generalized closed (briefly g-closed) set if cla ⊆ u whenever a ⊆ u and u is open in x. if the set of all closed sets and g-closed sets in x are coincide, then x is called a t 1 2 -space [7]. it is well known that x is a t 1 2 -space if and only if for each x ∈ x , {x} is either closed or open [3]. thus, if x is a t 1 2 space then by lemma 3.2, c(x)f is a subring of t ′(x). it is noted that all topological spaces in the paper [5] is assume to be t1 but some of its results hold in the class of t 1 2 -topological spaces, for example if we borrow the proof of [5, proposition 3.1], word-by-word, then we observe that proposition 3.1 in [5] holds for t 1 2 -spaces. © agt, upv, 2021 appl. gen. topol. 22, no. 1 144 rings of functions which have a finite number of discontinuities lemma 3.4. let x be an arbitrary topological space. then, t ′(x) ⊆ c(x)f if and only if every dense open subset of x has finite complement. proof. ⇒. let d be a dense open subset of x and t ′(x) ⊆ c(x)f . then, χd ∈ c(x)f since χd ∈ t ′(x). if x \ d is infinite, then χd is not continuous at x for any x ∈ x \ d and so x \ d ⊂ x \ c(χd). thus, χd /∈ c(x)f which is a contradiction. ⇐. it is obvious. � the following results is a direct consequence of the above lemma and lemma 3.2. theorem 3.5. let x be an arbitrary topological space. then, t ′(x) = c(x)f if and only if every point of x \ i(x) is nowhere dense and every dense open subset of x has finite complement. corollary 3.6. let x be a topological space which is t 1 2 . then, t ′(x) = c(x)f if and only if every dense open subset of x has finite complement. the above corollary shows that [5, proposition 5.4 ] holds for topological spaces which are t 1 2 . remark 3.7. recall that if the intersection of any two nonempty open sets in a topological space x is nonempty, then x is called hyperconnected. thus, if f ∈ c(x)f and x is t1, then there is a finite closed subset h of x such that f ∈ c(x \ h). since x is a hyperconnected t1-space and h is a finite subset of x, we have x \ h is hyperconnected. therefore c(x \ h) consists of constant functions. hence, the image of the function f is a finite set. also, we have the following: r ≃ c∗(x) = c(x) c(x)f ⊆ t ′(x) ( f(x). in particular if the topology of x is cofinite topology and y is the direct sum of x with itself, then by theorem 3.5 we have the following: c(y ) c(y )f = t ′(y ) ( f(y ). let f be a free ultrafilter on n. the set n with topology f ∪ {∅} become a topological space denoted by x. from [1, example 3.4] it follows that c(x) t ′(x) f(x). clearly, c(x) 6= c(x)f and there is a dense open subset d in x such that x \ d is infinite and so χd ∈ t ′(x) \ c(x)f . let x be an arbitrary topological space and r be a subring of f(x) such that χ{x} ∈ r for some x ∈ x. if i is the ideal generated by χ{x} in r, then i is ring isomorphic with the field of real numbers and so i is a minimal ideal of r. note that for any x ∈ x we have χ{x} ∈ c(x)f . if an ideal of c(x)f contains a nonzero function f and r = f(x) 6= 0 for some x ∈ x, then it contains χ{x} = 1 r fχ{x} and so we have the following well known result. © agt, upv, 2021 appl. gen. topol. 22, no. 1 145 m. r. ahmadi zand and z. khosravi lemma 3.8 ([5]). let x be an arbitrary topological space, then soc(c(x)f ) which is equal to the ideal generated by χ{x}’s is a free ideal both in c(x)f and in c∗(x)f . proposition 3.9. let x be an arbitrary topological space. then, soc(c(x)f ) is the intersection of all the free ideals in c(x)f , and of all the free ideals in c∗(x)f . proof. if i is a free ideal in c(x)f or c ∗(x)f and x ∈ x, then there exists g ∈ i such that a = g(x) 6= 0 and so by the comment before lemma 3.8, χ{x} ∈ i. thus, soc(c(x)f ) ⊆ i and so by lemma 3.8 the proof is complete. � an algebraic characterization of finite topological spaces is given in the following result. theorem 3.10. let x be a topological space. then the following are equivalent. (1) x is a finite set. (2) c(x)f = soc(c(x)f ). proof. (1) ⇒ (2). let x be a finite set. then 1 = ∑ x∈x χ{x} ∈ soc(c(x)f ) by lemma 3.8 . thus, c(x)f = soc(c(x)f ). (2) ⇒ (1). if c(x)f = soc(c(x)f ), then 1 ∈ soc(c(x)f ) and so by lemma 3.8 , there are xi ∈ x and ci ∈ r where 1 ≤ i ≤ n and n ∈ n such that 1 = ∑n i=1 ciχ{xi}. thus, x is a finite set. � recall that a commutative ring is called artinian if it satisfies the descending chain condition on ideals and it is called semisimple if the intersection of all the maximal ideals called jacobson radical is zero. it is well known that a ring is artinian semisimple if and only if it is equal to the sum of its minimal ideals [2]. let x be a topological space, then c(x)f is semisimple since by [5] its jacobson radical is zero. thus by the above theorem we have the following result. corollary 3.11. a topological space x is finite if and only if c(x)f is an artinian ring. theorem 3.12. let φ : c(x)f → c(x)f be a c(x)f -module homomorphism which is one to one. then φ is a c(x)f -module isomorphism. proof. first we claim that φ(1) is a unit of c(x)f . assume to the contrary that φ(1) is not a unit of c(x)f . thus by proposition 2.3, there is 1 6= g ∈ c(x)f such that φ(1) = gφ(1) and so φ(1 − g) = (1 − g)φ(1) = 0, which is a contradiction since φ is one to one. therefore, there exists h ∈ c(x)f such that hφ(1) = 1. now for any f ∈ c(x)f , we have f = fφ(h) = φ(fh) ∈ φ(c(x)f ) which completes the proof. � corollary 3.13. let f : x → x be a bijection. if f is continuous, then φ : c(x)f → c(x)f defined by φ(g) = g◦f is a c(x)f -module isomorphism. © agt, upv, 2021 appl. gen. topol. 22, no. 1 146 rings of functions which have a finite number of discontinuities proof. it is straightforward. � acknowledgements. we record our pleasure to the anonymous referee for his or her constructive report and many helpful suggestions on the main results of the earlier version of the manuscript which improved the presentation of the paper. references [1] m. r. ahmadi zand, an algebraic characterization of blumberg spaces, quaest. math. 33, no. 2 (2010), 223–230. [2] a. j. berrick and m. e. keating, an introduction to rings and modules, cambridge university press, 2000. [3] w. dunham, t 1 2 -spaces, kyungpook math. j. 17, no. 2 (1977), 161–169. [4] r. engelking, general topology, sigma ser. pure math. 6, heldermann-verlag, berlin, 1989. [5] z. gharabaghi, m. ghirati and a. taherifar, on the rings of functions which are discontinuous on a finite set, houston j. math. 44, no. 2 (2018), 721–739. [6] l. gillman and m. jerison, rings of continuous functions, springer-verlag, new yorkheidelberg, 1976. [7] n. levine, generalized closed sets in topology. rend. circ. mat. palermo. 19, no. 2 (1970), 89–96. © agt, upv, 2021 appl. gen. topol. 22, no. 1 147 @ appl. gen. topol. 20, no. 2 (2019), 407-418 doi:10.4995/agt.2019.11541 c© agt, upv, 2019 matrix characterization of multidimensional subshifts of finite type puneet sharma and dileep kumar department of mathematics, indian institute of technology jodhpur, india (puneet.iitd@yahoo.com, pg201383502@iitj.ac.in) communicated by f. balibrea abstract let x ⊂ az d be a 2-dimensional subshift of finite type. we prove that any 2-dimensional subshift of finite type can be characterized by a square matrix of infinite dimension. we extend our result to a general d-dimensional case. we prove that the multidimensional shift space is non-empty if and only if the matrix obtained is of positive dimension. in the process, we give an alternative view of the necessary and sufficient conditions obtained for the non-emptiness of the multidimensional shift space. we also give sufficient conditions for the shift space x to exhibit periodic points. 2010 msc: 37b10; 37b20; 37b50. keywords: multidimensional shift spaces; shifts of finite type; periodicity in multidimensional shifts of finite type. 1. introduction the study of dynamical systems originated to facilitate the study of natural processes and phenomenon. many natural phenomenon can be modeled as discrete dynamical systems and their long term behavior can be approximated using the modeled system. however, investigating a general discrete dynamical system is complex in nature and the long term behavior of the system cannot always be determined accurately. the uncertainty in predicting long term behavior introduces dynamical complexity in the system which in turn results in received 21 march 2019 – accepted 07 may 2019 http://dx.doi.org/10.4995/agt.2019.11541 p. sharma and d. kumar erroneous behavior of the modeled system. thus, there is a need to develop tools to facilitate the study of a general dynamical system which are not erroneous and can model the physical system with the sufficient accuracy. symbolic dynamics is one of such tools which are structurally simpler and can be used to model the physical system with desired accuracy. in one of the early studies, jacques hadamard used symbolic dynamics to study the geodesic flows on surfaces of negative curvature [6]. claude shennon applied symbolic dynamics to the field of communication to develop the mathematical theory of communication systems [14]. since then the topic has found applications in areas like data storage, data transmission and planetary motion to name a few. the area has also found significant applications in various branches of science and engineering [9, 11]. its simpler structure and easy computability can be used to investigate any general dynamical system. infact, it is known that every discrete dynamical system can be embedded in a symbolic dynamical system with appropriate number of symbols [5]. thus, to investigate a general discrete dynamical system, it is sufficient to study the shift spaces and its subsystems. multidimensional shift spaces has been a topic of interest to many researchers. in one of the early works, berger investigated multidimensional subshifts of finite type over finite number of symbols. he proved that for a multidimensional subshift, it is algorithmically undecidable whether an allowed partial configuration can be extended to a point in the multidimensional shift space [3]. consequently, he observed that it is algorithmically undecidable to verify the non-emptiness of a multidimensional shift defined by a set of finite forbidden patterns. in [13], the author gives examples to show that a multidimensional shift space may or may not contain any periodic points. these results unraveled the uncertainty associated with a multidimensional shift space and attracted attention of several researchers around the globe. as a result, several researchers have explored the field and a lot of work has been done [1, 2, 4, 7, 8, 10, 12]. in [12], authors proved that multidimensional shifts of finite type with positive topological entropy cannot be minimal. in fact, if x is subshift of finite type with positive topological entropy, then x contains a subshift which is not of finite type, and hence contains infinitely many subshifts of finite type. in the same paper, the authors proved that every shift space x contains an entropy minimal subshift y , i.e., a subshift y of x such that h(y ) = h(x). while [1] investigated mixing properties of multidimensional shift of finite type, [2] investigated minimal forbidden patterns for multidimensional shift spaces. in [4], authors exhibit mixing zd shifts of finite type and sofic shifts with large entropy. however, they establish that such systems exhibit poorly separated subsystems. they give examples to show that while there exists zd mixing systems such that no non-trivial full shift is a factor for such systems, they provide examples of sofic systems where the only minimal subsystem is a single point. in [8], for multidimensional shifts with d ≥ 2, authors proved that a real number h ≥ 0 is the entropy of a zd shift of finite type if and only if it is the infimum of a recursive sequence of rational numbers. in [7], hochman c© agt, upv, 2019 appl. gen. topol. 20, no. 2 408 matrix characterization of multidimensional subshifts of finite type improved the result and showed that h ≥ 0 is the entropy of a zd effective dynamical system if and only if it is the lim inf of a recursive sequence of rational numbers. the problem of determining which class of shifts have a dense set of periodic points is still open. for two-dimensional shifts, lightwood proved that strongly irreducible shifts of finite type have dense set of periodic points [10]. however, the problem is still open for shifts of dimension greater than two. let a = {ai : i ∈ i} be a finite set and let d be a positive integer. let the set a be equipped with the discrete metric and let az d , the collection of all functions c : zd → a be equipped with the product topology. any such function c is called a configuration over a. any configuration c is called periodic if there exists u ∈ zd (u 6= 0) such that c(v + u) = c(v) ∀v ∈ zd. the set γc = {w ∈ zd : c(v + w) = c(v) ∀v ∈ zd} is called the lattice of periods for the configuration c. the function d : az d ×az d → r+ be defined as d(x,y) = 1 n+1 , where n is the least non-negative integer such that x 6= y in rn = [−n,n]d, is a metric on az d and generates the product topology. for any a ∈ zd, the map σa : a zd → az d defined as (σa(x))(k) = x(k + a) is a d-dimensional shift and is a homeomorphism. for any a,b ∈ zd, σa ◦ σb = σb ◦σa and hence zd acts on az d through commuting homeomorphisms. a set x ⊆ az d is σa-invariant if σa(x) ⊆ x. any set x ⊆ az d is shift-invariant if it is invariant under σa for all a ∈ zd. a non-empty, closed shift invariant subset of az d is called a shift space. if y ⊆ x is a closed, nonempty shift invariant subset of x, then y is called a subshift of x. for any nonempty s ⊂ zd, the projection map πs : a zd → as defined as πs = az d |s projects the elements of az d to as. any element in as is called a pattern over s. a pattern is said to be finite if it is defined over a finite subset of zd. a pattern q over s is said to be extension of the pattern p over t if t ⊂ s and q|t = p. the extension q is said to be proper extension if t ∩bd(s) = φ, where bd(s) denotes the boundary of s. let f be a given set of finite patterns (possibly over different subsets of zd) and let x = {x ∈ azd : any pattern from f does not appear in x}. the set x defines a subshift of zd generated by set of forbidden patterns f. if the set f is a finite set of finite patterns, we say that the shift space x is a shift of finite type. we say that a pattern is allowed if it is not an extension of any forbidden pattern. we denote the shift space generated by the set of forbidden patterns f by xf. two forbidden sets f1 and f2 are said to be equivalent if they generate the same shift space, i.e. xf1 = xf2 . a forbidden set f of patterns is called minimal for the shift space x if f is the set with least cardinality such that x = xf. it is worth mentioning that a shift space x is of finite type if and only if its minimal forbidden set is a finite set of finite patterns. it may be noted that the shift space can equivalently be defined in terms of the allowed patterns. for a shift space x and any set s ⊂ zd, let as = {x ∈ as : x = πs(y), for some y ∈ x}. then, as is the set of allowed patterns (for x) over s. the set a = ⋃ s⊂zd as is called the language c© agt, upv, 2019 appl. gen. topol. 20, no. 2 409 p. sharma and d. kumar for the shift space x. given a set s ⊂ zd and a set of patterns p in as, the set x = x(s,p) = {x ∈ azd : πs ◦σn(x) ∈p for every n ∈ zd} is a subshift generated by the (allowed) patterns p. refer [2, 12] for details. let m be a square 0 − 1 matrix (possibly infinite) with indices {i : i ∈ i}. we say that the index i is u-related to j if mji = 1. let the collection of indices u-related to j be denoted by ruj . we say that the indices j is d-related to i if mji = 1. let the collection of indices d-related to i be denoted by r d i . it may be noted that i is u-related to j if and only if j is d-related to i. the non-empty subset k of the index set i is said to be complementary if for each i ∈ k, there exists j,k ∈ k such that j is u-related to i and k is d-related to i. in this paper we investigate some of the questions raised in [3]. in the process we address the problem of non-emptiness and existence of periodic points for a multidimensional shift of finite type. we prove that the any 2-dimensional shift of finite type can be characterized by an infinite square matrix (possibly of infinite dimension). we prove that the elements of a shift of finite type can equivalently be characterized by limits of periodic configurations arising from allowed cubes for the shift space x. we investigate the non-emptiness problem using complementary set of indices. we extend our results to a general d-dimensional case. we also give sufficient condition ensuring existence of periodic points for a shift space x. 2. main results proposition 2.1. x is a d-dimensional shift of finite type =⇒ there exists a set c of d-dimensional cubes such that x = xc. proof. let x be a shift of finite type and let f be the minimal forbidden set of patterns for the shift space x. it may be noted that f contains finitely many patterns defined over finite subsets of zd. for any pattern p in f, let lip be the length of the pattern p in the i-th direction. let lp = max{lip : i = 1, 2, . . . ,d} denote the width of the pattern p and let l = max{lp : p ∈ f}. let cl be the collection of d-dimensional cubes of length l and let ef denote the set of extensions of patterns in f. let c = cl ∩ef. it may be observed that if p is a pattern with width l, forbidding a pattern p for x is equivalent to forbidding all extensions q of p in cl. thus, each pattern in the forbidden set of width l can be replaced by an equivalent forbidden set of cubes of length l and c is an equivalent forbidden set for the shift space x. consequently, x = xc and the proof is complete. � remark 2.2. the above result proves that every d-dimensional shift of finite type is generated by a set of cubes of fixed finite length. such a consideration leads to an equivalent forbidden set which in general is not minimal. the above result constructs an equivalent forbidden set by considering all the cubes which are extension of the set of patterns in f. however, the cardinality of the new set can be reduced by considering only those cubes which are not c© agt, upv, 2019 appl. gen. topol. 20, no. 2 410 matrix characterization of multidimensional subshifts of finite type proper extensions of patterns in f (but are of same size l). such a construction reduces the cardinality of the forbidden set considerably and hence reduces the complexity of the system. it may be noted that the forbidden set obtained on reduction is still not minimal. however, the d-dimensional cubes generating the elements of x are of same size and can be used for generating the shift space x. we say that a shift of finite type x is generated by cubes of length l if there exists a set of cubes c of length l such that x = xc. proposition 2.3. every 2-dimensional shift of finite type x can be characterized by an infinite square matrix. proof. let x be a 2-dimensional shift of finite type and let f be the equivalent set of forbidden cubes (of fixed length, say l) for the space x. let a be the generating set of cubes (of length l) for the space x. it may be noted that as cubes of length l form a generating set for the shift space x, to verify whether any x ∈ az d belongs to x, it is sufficient to examine strips of height l in x. let a2 = { ( s1 s2 ) : s1,s2 ∈a, ( s1 s2 ) is allowed in x}. by construction, a2 is a finite set of 2l×l allowed rectangles, say {a1,a2, . . . ,ak}, generating the shift space x. define a k ×k matrix m as mij = { 0, (aiaj) is forbidden in x; 1 (aiaj) is allowed in x; then, the sequence space corresponding to the matrix m, σm = {(xn) : mxixi+1 = 1, ∀i} generates all allowed infinite strips(of height 2l) in x. it may be noted that any element in σm is element of the form ( p q ) , where p and q are allowed infinite strips of height l. generate an infinite matrix m, indexed by allowed infinite strips of height l, using the following algorithm: (1) pick any ( p q ) ∈ σm and index first two rows and columns of the matrix by p and q. set mqp = 1. (2) for each ( p q ) ∈ σm , if the rows and columns indexed p and q exist, set mqp = 1. else, label next row and/or column as p and/or q (whichever required) and set mqp = 1. (3) in the infinite matrix generated in step 2, set mqp = 0, if mqp has so far not been assigned a value. (4) in the infinite matrix obtained, if there exists an index p such that the p-th row or column is zero, delete the p-th row and column from the matrix generated. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 411 p. sharma and d. kumar the above algorithm generates an infinite 0-1 matrix where mqp = 1 if and only if ( p q ) is allowed in x, where p and q are allowed infinite strips (of height l) in x. let σm be the sequence space associated with the matrix m. consequently, any sequence in σm gives a vertical arrangement of infinite allowed strips (of height l) such that the arrangement is allowed in x and hence generates an element in x. conversely, any element in x is a sequential (vertical) arrangement of infinitestrips of height l and hence is generated by a sequence in σm. consequently, x = σm and the proof is complete. � remark 2.4. the above result characterizes elements of the shift space x by a infinite square matrix m. it may be noted that if row/column for an index a is zero, the algorithm deletes the row and column with index a. such a criteria reduces the size of the matrix and will result in a matrix of dimension 0, if the shift space is empty. further, the characterization of the space may yield a matrix of infinite (uncountable) dimension. consequently, it is undecidable whether a shift of finite type generated by set of cubes a is non-empty. it may be noted that although the algorithm does not guarantee a positive dimensional matrix, if the shift space x is non-empty the matrix generated is definitely of positive dimension and characterizes the elements in x. further, as each row/column of the matrix generated has atleast one non-zero entry, each block indexing the matrix can be extended to an element of x. consequently, any submatrix of the matrix m cannot generate the shift space x. in light of the remark stated, we get the following result. corollary 2.5. a 2-dimensional shift of finite type is non-empty if and only if the characterizing matrix m is of positive dimension. further, any proper submatrix of the matrix m generates a proper subshift and hence the matrix m is minimal. remark 2.6. it may be noted that although the above algorithm characterizes the elements of the shift space using (possibly) a matrix of infinite dimension, the same can be achieved by approximating each point of x by a sequence of periodic points (which may not lie in the shift space x). to illustrate, let a is the collection of generating cubes (of size l) of x and let ar be the collection of all allowed cubes of rl×rl obtained by r×r arrangement of elements of a. let xr denote the all periodic configurations arising from the collection ar. then x = ∞⋂ k=1 xk and hence any element of the shift space can be obtained by approximation through periodic points (which may not lie in x themselves). hence we get the following result. proposition 2.7. any point in a 2-dimensional shift of finite type can be approximated by a sequence of periodic points. proof. let a denote the collection of generating cubes (of size l) of x and ar be the collection of all allowed cubes of rl×rl obtained by r ×r arrangement of elements of a. note that as all central blocks of an element in x are c© agt, upv, 2019 appl. gen. topol. 20, no. 2 412 matrix characterization of multidimensional subshifts of finite type allowed, any element is a limit of periodic configurations (generated by its central blocks). also, if x is a limit of periodic configurations arising from the collection ar, then any central block of x is allowed and hence x is an element of the shift space x (proof follows from the fact that any element belongs to x if and only if all central blocks of x are allowed in x). consequently x = ∞⋂ k=1 xk and the proof is complete. � remark 2.8. for a shift space x, with generating set of cubes of height l, let l denote set of all allowed infinite strips of height l. recalling the notions of u-related indices for a square matrix m, for any two infinite strips p,q of height l, we say that p is u-related (d-related) to q if p and q are indices of m such that mqp = 1 (mp q = 1). further, generalizing the definition, a family of allowed infinite strips of height l is complementary if for each p in l there exists infinite strips q,r ∈l such that q is u-related to p and r is d-related to p . thus, the algorithm generates u-related (d-related) infinite strips for the shift space x which in turn generates an arbitrary element of x. as any element of the shift space is a sequential arrangement of u-related (d-related) infinite strips, the characterization of the elements of the space x by a matrix m is equivalent to finding all the u-related (d-related) pairs of infinite strips for the space x. as any infinite strip of height l (say p) can be extended to an element of x only if there exists infinite strips q,r of height l such that q is u-related to p and r is d-related to p , only members of complementary family can form the building blocks for an element of x. as a result, we get the following corollary. corollary 2.9. let x be a multidimensional shift space generated by cubes of length l and let b be the infinite strips of height l allowed in x. then, the shift space x is non-empty if and only if there exists non-empty set of indices b0 ⊆ b such that b0 is complementary. remark 2.10. the above result provides an alternate view of the criteria established for the non-emptiness of the space x. although the matrix generated characterizes the elements of the shift space x, one does not require the matrix m for establishing the non-emptiness for the shift space. the set of indices of the matrix may be observed at each iteration and existence of a complementary subfamily can be used to establish the non-emptiness of the space x. however, as the algorithm does not provide any optimal technique for picking the block( p q ) at each iteration, such a consideration does not reduce the time complexity of the problem. however, algorithms for optimal selection of the infinite blocks ( p q ) may be proposed which in turn may reduce the time complexity of the algorithm. as an extension of the proposed algorithm characterizes the elements of a d-dimensional shift space, similar results are true for a general d-dimensional shift of finite type. for the sake of completion, we include the proof of the result below. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 413 p. sharma and d. kumar proposition 2.11. if x is a d-dimensional shift of finite type, then the elements of x can be determined by an infinite square matrix. proof. let x be a d-dimensional shift of finite type and let f be the equivalent set of forbidden cubes (of fixed length, say l) for the space x. let a be the generating set of cuboids of size 2l× 2l× . . . 2l︸︷︷︸ d−1times ×l for the space x. by construction, a is a finite set of allowed rectangles, say {a1,a2, . . . ,ak}. define a k ×k matrix m0 as m0ij = { 0, (aiaj) is forbidden in x; 1 (aiaj) is allowed in x; where (aiaj) denotes adjacent placement of aj with ai in the positive d-th direction. then, the sequence space corresponding to the matrix m0, σm0 = {(xn) : m0xixi+1 = 1, ∀i} generates all allowed one directional (in d-th direction) infinite strips in x. it may be noted that any element in σm0 is element of the form ( p q ) 0 , where p and q are allowed infinite strips (in direction d) of dimension 2l× 2l× . . . 2l︸︷︷︸ d−2times ×l ×∞ and ( p q ) 0 denotes adjacent placement of q with p in the negative d− 1-th direction. generate an infinite matrix m1, indexed by allowed infinite strips of dimension 2l× 2l× . . . 2l︸︷︷︸ d−2times ×l×∞ , using the following algorithm: (1) pick any ( p q ) 0 ∈ σm0 and index first two rows and columns of the matrix by p and q. set mqp = 1. (2) for each ( p q ) 0 ∈ σm0 , if the rows and columns indexed p and q exist, set mqp = 1. else, label next row and/or column as p and/or q (whichever required) and set mqp = 1. (3) in the infinite matrix generated in step 2, set mqp = 0, if mqp has so far not been assigned a value. (4) in the infinite matrix obtained, if there exists an index p such that the p-th row or column is zero, delete the p-th row and column from the matrix. the above algorithm generates an infinite 0-1 matrix where mqp = 1 if and only if ( p q ) 0 is allowed in x, where p and q are of dimension 2l× 2l× . . . 2l︸︷︷︸ d−2times ×l ×∞. let σm1 denote the sequence space corresponding to c© agt, upv, 2019 appl. gen. topol. 20, no. 2 414 matrix characterization of multidimensional subshifts of finite type the matrix generated above. it can be seen that the space σm1 precisely is the collection of allowed bi-infinite strips (in direction d and d − 1). further, as any element in σm1 is of the form ( p q ) 1 , where p and q are allowed infinite strips (in direction d and d− 1) of dimension 2l× 2l× . . . 2l︸︷︷︸ d−3times ×l ×∞×∞ and ( p q ) 1 denotes adjacent placement of q with p in the negative d− 2-th direction, a repeated application of the algorithm generates a matrix m2 which extends the infinite patterns in σm1 along the direction d− 3 to generate the space σm2 . consequently, repeated application of the above algorithm extends the allowed patterns infinitely in all the d directions (one direction at each step) to obtain a point in x. further, as any point in x can be visualized as such an extension of allowed cubes in the d directions, the matrix obtained (at the final step) characterizes the elements of the space x. � remark 2.12. the above result characterizes the multidimensional shift space by a infinite matrix m. the characterization is obtained by repeated application of the 2-dimensional case, extending the allowed blocks in each of the d directions. in the process, at each step we obtain an infinite matrix characterizing the extension of an allowed block in the i-th direction. although the rows and columns of the characterizing matrix m are indexed by infinite blocks allowed in x, their existence is guaranteed as they are procured from the allowed blocks obtained in the previous step. it may be noted that extension in any of the directions (at i-th step) does not guarantee an extension to the element of x. in particular, a block extendable in a direction i (or in a few directions i1, i2, . . . , ir) need not necessarily extend to an element in x. in particular if the shift space is empty, positive dimension of matrix at i-th step does not guarantee a matrix of positive dimension at the final step. consequently, once again, the shift space is non-empty if and only if the matrix generated (at the final step) is of positive dimension. thus, we obtain the following corollary. corollary 2.13. a multidimensional shift of finite type is non-empty if and only if the characterizing matrix m is of positive dimension. further, any proper submatrix of m generates a proper subshift and hence the matrix m is minimal. remark 2.14. it may be noted that the matrix characterizing the elements of the multidimensional shift space is once again (possibly) infinite. however, such a construction helps in better visualization of the problem and can help in better understanding of the subsystems of the shift space under consideration. it may be noted that the elements of the shift space can be obtained as sequential limits of the periodic points generated using allowed cubes of finite size (which may not lie in the shift space itself). consequently, the points of the multidimensional shift space can be obtained by approximations through c© agt, upv, 2019 appl. gen. topol. 20, no. 2 415 p. sharma and d. kumar periodic points (which may not lie in the shift space x). hence we get the following result. proposition 2.15. any point in a d-dimensional shift of finite type can be approximated by a sequence of periodic points. proof. let a denote the collection of generating cubes (of size l) of x and ar be the collection of all allowed cubes of side rl. it may be noted that any element of ar is an r ×r × . . .×r︸︷︷︸ d times arrangement of elements of a. let xr denote the collection of all periodic configurations (periodic of same period in all the d-directions) generated by elements of ar. as all central blocks of an element in x are allowed, any element of x is a limit of periodic configurations (generated by its central blocks). also, if x is a limit of periodic configurations arising from the collection ar, then any central block of x is allowed and hence x is an element of the shift space x (proof follows from the fact that any element belongs to x if and only if all central blocks of x are allowed in x). consequently x = ∞⋂ k=1 xk and the proof is complete. � remark 2.16. the above proof characterizes the points of the shift space as limits of periodic points generated by the allowed cubes for the shift space. note that although the periodic points generated are periodic in all the d-directions (with the same period), the construction of periodic points can be further simplified by constructing them as adjacent tiling of a single element (of ar) throughout the zd domain. as the arguments given in the proof hold good in this setting too, elements of the shift space can be realized as limits of periodic points constructed in this manner (note that as periodicity in one direction need not imply periodicity in the other, periodic points in general have infinite orbits in the multidimensional shift space). once again, the construction of elements of the shift space can be captured through the notion of complimentary sets. as any element of the shift space can be visualized as an alignment of elements of a complimentary set, the shift space is non-empty if and only if the exists a subset b0 of indices (of matrix obtained at the final step) which forms a complimentary set. the result is an analogous extension of the result obtained for the two dimensional case and hence characterize the elements of the shift space x. hence we get the following corollary. corollary 2.17. let x be a multidimensional shift space and let b be the infinite strips of height l allowed in x. then, the shift space x is non-empty if and only if there exists b0 ⊆ b such that b0 is complementary. we now discuss the periodicity for a given multidimensional shift space. proposition 2.18. let x be a multidimensional shift space and let b be the infinite strips of height l allowed in x. if there exists a finite complementary set b0 ⊂ b, then the set of periodic points is non-empty. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 416 matrix characterization of multidimensional subshifts of finite type proof. let b be the infinite strips of height l allowed in x and let b0 ⊂ b be a finite complementary set. by definition, elements of b0 form indices (not all) for the matrix m. let nbe the submatrix of m indexed by elements of b0. as the set b0 is complementary, the shift generated by b0 (say σb0 ) is non-empty. further, as shift defined by a finite dimensional matrix contains periodic points, there exists periodic points for σb0 (and hence for the shift space x). � remark 2.19. the above result establishes a sufficient condition for existence of periodic points in a multidimensional shift space. however, the condition derived is sufficient in nature and the shift space may exhibit periodic points without exhibiting the derived condition. note that a point is periodicity of a point in a direction dk ensures (and is equivalent to) existence of a finite complementary set in the direction dk. consequently, a point in the shift space is periodic in all the d directions if and only if there exists a finite complementary set for the shift space under consideration. thus we get the following corollary. corollary 2.20. a shift space x contains a point periodic in all the directions if and only if it there exists a finite set of finite patterns complementary for the shift space x. 3. conclusion in this paper, we investigate the non-emptiness problem and existence of periodic points for a multidimensional shift space of finite type. in the process, we prove that any multidimensional shift of finite type can be characterized by an infinite square matrix (possibly of infinite dimension). we prove that the multidimensional shift space is non-empty if and only if the characterizing matrix is of positive dimension. we prove that the elements of the shifts space can equivalently be characterized as limits of the periodic points generated by the cubes allowed for the shift space x. we also investigate the existence of periodic points for a multidimensional shift space. we address the nonemptiness problem and existence of periodic points using complementary set of indices. we prove that a shift space exhibits a point periodic in all the directions if and only if there exists a finite set of finite cubes complimentary for the shift space x. acknowledgements. the authors thank the referee for his/her useful suggestions and remarks. the first author thanks national board for higher mathematics (nbhm) grant no. 2/48(39)/2016/nbhm(r.p)/r&d ii/4519 for financial support. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 417 p. sharma and d. kumar references [1] j. c. ban, w. g. hu, s. s. lin and y. h. lin, verification of mixing properties in two-dimensional shifts of finite type, arxiv:1112.2471v2. 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[15] h. h. wicke and j. m. worrell, jr., open continuous mappings of spaces having bases of countable order, duke math. j. 34 (1967), 255–271. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 418 @ appl. gen. topol. 20, no. 1 (2019), 251-264doi:10.4995/agt.2019.10928 c© agt, upv, 2019 exact computation for existence of a knot counterexample k. marinelli and t. j. peters department of computer science & engineering, university of connecticut, usa (kevin.marinelli@uconn.edu,tpeters@cse.uconn.edu) communicated by m. schellekens abstract previously, numerical evidence was presented of a self-intersecting bézier curve having the unknot for its control polygon. this numerical demonstration resolved open questions in scientific visualization, but did not provide a formal proof of self-intersection. an example with a formal existence proof is given, even while the exact self-intersection point remains undetermined. 2010 msc: 68u05. keywords: knot theory; isotopy; parametric curve. 1. introduction to existence condition for self-intersection the formal proof of an unknotted control polygon with a self-intersecting bézier curve (see definition 3.3.) appears in theorem 8.4. the progression in figure 1, from left to right, shows ‘snap shots’ of a piecewise linear (pl) curve being linearly perturbed, generating new bézier curves at each instant. the top-most point is the only one perturbed. the initial (left) and final (right) bézier curves are shown to have differing prime knot types, so there must be an intermediate bézier curve with a self-intersection (as shown in the middle image by a black dot). relevant definitions follow. received 25 april 2018 – accepted 06 november 2018 http://dx.doi.org/10.4995/agt.2019.10928 k. marinelli and t. j. peters figure 1. existence of self-intersection 2. data for the example let (0, 9, 20), (−15, −95, −50), (40, 80, −20), (−10, −60, 58), (−60, 30, 20), (40, −60, −60), (0, 9, 20) be the ordered list of vertices for the initial pl knot, denoted by k0. integer values are chosen for exact computation. in figure 1, the vertex (0, 9, 20) is the initial and final point of the bézier curves (shown as the highest point on the polynomial curve, colored green). the remaining vertices are arranged counterclockwise. a single reidemeister move of vertex (40, -60, -60) shows that k0 is the unknot. the vertex in bold font is perturbed linearly to (10, −60, 58) to obtain the final pl knot, denoted by k1, with all other vertices remaining fixed. the perturbation generates uncountably many new control polygons and associated bézier curves. the bézier curves corresponding to k0 and k1 are denoted by β0 and β1, respectively. both k0 and k1 resulted from visual experiments. 3. mathematical preliminaries some relevant mathematical definitions are summarized. definition 3.1. a subspace of r3 is called a knot [7] if it is homeomorphic to a circle. the stronger equivalence of ambient isotopy is fundamental in knot theory. definition 3.2. let x and y be two subspaces of r3. a continuous function h : r3 × [0, 1] → r3 is an ambient isotopy between x and y if h satisfies the following conditions: (1) h(·, 0) is the identity, (2) h(x, 1) = y , and (3) ∀t ∈ [0, 1], h(·, t) is a homeomorphism from r3 onto r3. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 252 computational knot counterexample definition 3.3. denote c(t) as the parameterized bézier curve of degree n with control points pi ∈ r 3, defined by c(t) = n ∑ i=0 bi,n(t)pi, t ∈ [0, 1], where bi,n(t) = ( n i ) ti(1 − t)n−i. the curve p formed by pl interpolation on the ordered list of points p0, p1, . . . , pn is called the control polygon and is a pl approximation of c. the work presented was partially motivated by development of wireless data gloves. the smooth bézier representations are manipulated by the gloves, while the illustrative computer graphics are created by pl approximations derived from the control polygon. topological fidelity between these representations is of interest to provide reliable visual feedback to the user. 4. related work this article arose from a question posed by a referee1 from a previous numerical example [22]. the mathematical objects of study here are bézier curves. the canonical references [13] (any edition) and [27] focus on approximation and modeling, with less attention to associated topological properties. relations between a smooth curve and its pl approximation are dominant in computer graphics [14], but the topological aspects are often ignored. one prominent property is that any bézier curve is contained in the convex hull of its control points [13], while recent enhancements have been shown [26], where the containing set is a subset of the convex hull. the push receives prominent attention by r. h. bing [8] as a fundamental tool in developing ambient isotopies in ℜ2. the perturbation in this article is the trivial extension of a push to ℜ3. the preservation of topological characteristics in geometric modeling and graphics has become of contemporary interest [4, 5, 6, 10, 12, 16, 19, 21]. sufficient conditions for a homeomorphism between a bézier curve and its control polygon have been studied [25], while topological differences have also been shown [9, 22, 27]. topological aspects are relevant in ‘molecular movies’ [23]. sufficient conditions were established for preservation of knot type [17] during dynamic visualization of ongoing molecular simulations [29]. interest by bio-chemists in using hand gestures to interactively manipulate these complex molecular images prompts related topological considerations for data gloves [1, 2, 3]. 1author t. j. peters thanks that referee of the previous numerical work [22] for a suggestion that led to the example shown here. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 253 k. marinelli and t. j. peters the beautiful visual studies of knot symmetry [28] provided an example of the 74 knot that was helpful in the initial visual studies performed to create the example presented here. exact computation has some advantages over floating point arithmetic [11, 18], particularly regarding loss of precision in finite bit strings, with alternative views expressed [15]. some ad hoc techniques are often invoked to provide adequate bit length for full precision relative to a particular computation, as has been done here. 5. exact computation for subdivision the de casteljau algorithm [13] is a subdivision method for bézier curves2. the algorithm recursively generates control polygons that more closely approximate the curve [13] under hausdorff distance [24]. the algorithm is presented here, for a bézier curve denoted by α, for the input value of t = 1/2, as is customary3. the point α(1/2), is an endpoint output by the algorithm. for t = 1/2 subdivision proceeds by selecting the midpoint of each edge of the control polygon and these midpoints are connected to create new edges, as shown in the left image of figure 2. recursive creation and connection of midpoints continues until an edge is created that is tangent to the bézier curve [13]. termination is guaranteed since there are only finitely many edges. this splits the original curve into two pieces, at α(1/2), designated here as the ‘left’ and ‘right’ pieces of α. the left piece has a control polygon with a final point of α(1/2) and the right piece has a control polygon with initial point of α(1/2), as shown in the right image of figure 2, where α(1/2), is the point of tangency for terminating the algorithm. the union of the edges from the final step then forms two new pl curves (each described as a sub-control polygon4) as shown in blue and green in the right image of figure 2. figure 2. a subdivision of a cubic bézier curve at parameter 1 2 for each iteration, m, the subdivision process generates 2m pl sub-curves, each being a sub-control polygon [13], as denoted by pmj for j = 1, 2, 3, . . . , 2 m. for a bézier curve initially defined by n + 1 control points, each pmj has n + 1 points and their union ⋃ j pmj forms a new pl curve that converges in hausdorff distance to the original bézier curve. the bézier curve defined by ⋃ j pmj is exactly the same as the original bézier curve [20]. 2an illustrative image appears [13, figure 3.2]. 3any parameter t from (0, 1) can be substituted. 4note that each sub-control polygon of a simple bézier curve is open. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 254 computational knot counterexample remark 5.1. a bézier curve is contained within the convex hull of its control points [13]. remark 5.1 also applies to each sub-control polygon, as will be extensively used. the stick knots of section 2 are refined via the decasteljau subdivision algorithm, repeatedly computing averages for the coordinate vertices in the form (a+b) 2 . in order to retain integer valued vertices throughout all computed subdivisions, the vertices will be scaled by 2k, for an appropriately chosen k. the existence of self-intersections in the bézier curves is preserved under scaling. each subdivision iteration has one division by 2 for each degree d. in our examples, we invoke ℓ levels of subdivision, so that k = ℓ ∗ (d + 1) + 1. 6. ambient isotopy between stick knots the two curves, k0 and k1 will be shown to be the unknot. lemma 6.1. the curve k0 is the unknot. proof. the demonstration that k0 is simple proceeds directly over the six edges. the non-consecutive pairs of edges have separating hyperplanes. � lemma 6.2. the curves k0 and k1 are ambient isotopic. proof. the single perturbation to create k1 is similar to a push [8] and incurs no self-intersections with other segments. it is easily extended to an ambient isotopy having compact support. � corollary 6.3. the curve k1 is the unknot. 7. subdivision analysis the points from the 4th iteration of the decasteljau algorithm, using a subdivision value of 1/2 are listed in appendix a. these can be verified simply by executing the algorithm on the initial data for k0 and k1, with the provision that the implementation relies upon exact arithmetic. it was observed that each sub-control polygon in appendix a had the property that its control points were strictly monotone in one of its coordinate values. this is annotated in the appendix by abbreviations noting the coordinates in which strict monotonicity was observed. in some cases, this occurs for all three coordinates. this observation greatly simplified the proofs presented. let k0,4 denote the pl curve formed from the 4th subdivision on k0 and similarly denote k1,4 for k1. within each of k0,4 and k1,4, there are 16 subcontrol polygons. the rest of the paper proceeds by showing that k0,4 is the unknot and is ambient isotopic to β0 and that k1,4 is the trefoil and is ambient isotopic to β1. as the convex hulls of the sub-control polygons become central, their images are shown in figure 3. lemma 7.1. both k0,4 and k1,4 are simple. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 255 k. marinelli and t. j. peters (a) k0 (b) k1 figure 3. convex hulls – fourth subdivision (parameter 1 2 ) proof. first consider k0,4. let k0,4,i, for i = 0, 1, . . .15, denote the 16 subcontrol polygons for k0,4. each k0,4,i, for i = 0, 1, . . . 15, is simple, because of the strict monotonicity in at least one coordinate. the k0,4,i, for i = 0, 1, . . .15, are pairwise disjoint (except at shared endpoints), as established by computing the convex hull for each. again, any ambiguity that could arise from floating point arithmetic is avoided by exact computations. the argument for k1,4 follows the same pattern. � the derivative of a bézier curve c is expressed as c ′ (t) = n ∑ i=0 ( n i ) ti(1 − t)n−i(pi+1 − pi), i ∈ {1, . . . , n − 1}. lemma 7.2. each of β0 and β1 are strictly monotonic in the same coordinate as their corresponding control polygons and are simple. proof. the strict monotonicity in some coordinate for each sub-control polygon implies that the partial derivative in the x, y or z variable will be positive, implying simplicity for that segment of the bézier curve. since each bézier segment is contained in the convex hull of its sub-control polygon and since those convex hulls are pairwise disjoint (except at shared endpoints), the simplicity follows. � corollary 7.3. if a control polygon is strictly monotonic in some coordinate, then its associated bézier curve is also. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 256 computational knot counterexample subdivision need not preserve knot type, so we consider k0,4 and k1,4. lemma 7.4. the curve k0,4 is the unknot. proof. project k0,4 onto the x − y plane, where 3 intersections are computed exactly, listed according to the oriented direction of k0,4, with the z coordinates also computed exactly. overcrossing at (x, y) = (70600874219532518400/90998355737, −331936500725210686464/90998355737), since z: 346821834258400839680/90998355737 > −179855360, overcrossing at (x, y) = (−2606810091676070400/2228701007, −141298996572704411136/15600907049), since z: −695629074309606400/821100371 > −2734161920, undercrossing at (x, y) = (70736463317966883840/46441845451, −2183313901438239630336/232209227255), since z: −312474348049921062912/46441845451 < −5234410880. this crossing information permits the conclusion that k0,4 is the unknot. � lemma 7.5. the curve k1,4 is the trefoil. proof. project k1,4 onto the x − y plane, as done, similarly, in the proof of lemma 7.4, with, again, 3 intersections computed exactly. undercrossing at (x, y) = (−2321511588927897600/2778711187, −12860064917297823744/2778711187), since z: = 8352902508791726080/2778711187 < 4026531840, overcrossing at (x, y) = (−8217249783839948800/7079707793, −58835463893254373376/7079707793), since z: = −24693447116718080/7079707793 > −1547779858, undercrossing at (x, y) = (2736710937161450496/968030497, −222045650166834551808/24200762425), since z: = −6443291820066594816/968030497 < −5348303360. these three pairs of alternating crossings permit conclusion of the trefoil. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 257 k. marinelli and t. j. peters 8. ambient isotopy for bézier curves we show that β0 is ambient isotopic to k0,4 and that β1 is ambient isotopic to k1,4. both proofs proceed by showing that each sub-control polygon is ambient isotopic to its associated bézier curve. we show this for one subcontrol polygon, as representative. let k0,4,i be one such control polygon and let β0,i be its corresponding bézier curve segment. theorem 8.1. each k0,4,i is ambient isotopic to its corresponding bézier curve, denoted as β0,i. proof. without loss of generality assume k0,4,i is strictly monotonic in its x coordinate, which implies that β0,i is also strictly monotonic in its x coordinate by corollary 7.3. for each p ∈ k0,4,i, let πp be the plane at p that is parallel to the y z-plane. by the indicated strict monotonicity, πp ∩ k0,4,i = {p}, is a unique point. similarly, by the same strict monotonicity on β0,i, the intersection πp ∩β0,i has at most one point, and the connectivity of β0,i (as the continuous image of a connected subset of [0, 1]) implies that this intersection is non-empty. for each p ∈ k0,4,i, let qp = πp ∩ β0,i. to construct the ambient isotopy, consider the line segment between each p and qp. the interiors of these line segments will not intersect, due to the strict monotonicity. the remaining details to construct an ambient isotopy of compact support are standard and left to the reader. � corollary 8.2. the bézier curve β0 is ambient isotopic to its control polygon k0,4, the unknot. proof. the existence of an ambient isotopy within each convex hull of k0,4,i, the lack of intersection between convex hulls k0,4,i and k0,4,j for i 6= j (except at the end points when j = i + 1, where those end points remain fixed) and composition of the ambient isotopies on each sub-control polygon conclude this proof. � corollary 8.3. the bézier curve β1 is ambient isotopic to its control polygon k1,4, the trefoil. theorem 8.4. the ambient isotopy between k0 and k1 generates a bézier curve with a self-intersection. proof. the perturbation of the control point (−10, 60, 58) to (10, 60, 58) creates a homotopy on the bézier curve β0 to β1, defined over the interval [0, 1]. if for all values of t ∈ [0, 1], the perturbed bézier curve was simple, then the homotopy would be an isotopy between β0 and β1. however, since β0 and β1 have differing prime knot types, this cannot be. so, for some value t̃ ∈ [0, 1], the corresponding bézier curve must have a self-intersection. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 258 computational knot counterexample 9. conclusion and future work an example is shown of a closed bézier curve and its control polygon which are both the unknot. a perturbation of one vertex is shown to cause the perturbed bézier curve to be the trefoil, while its control polygon remains the unknot. this transition demonstrates that there must have been an intermediate state where a self-intersection occurred in the transforming bézier curves. these topological differences between the mathematical representation and its pl approximation are of interest in computer graphics and animation. visual evidence previously existed, but the only supportive mathematics relied on floating point arithmetic which left open the question of whether an intermediate intersection could be rigorously proven. the formal proof presented relies on the implementation of exact, integer computations to resolve that open question. comparisons between the exact techniques and certifiable numeric methods merit further consideration, as it may often be of interest to specify a neighborhood in which a self-intersection is known to exist. the present example was quite carefully constructed to show the self-intersection and robust numerical methods are likely to have broader scope for applications. acknowledgements. the authors acknowledge, with appreciation, the contributions of • d. marsh, for software that generated experimental visualizations and related computations, • the reviewers, for singularly comprehensive and constructive comments, and • the editors, for their keen insight and informed perspective in selecting those reviewers. references [1] cybergloves. http://www.cyberglovesystems.com/cyberglove-iii/. [2] manusvr. https://manus-vr.com/. [3] virtual motion labs. http://www.virtualmotionlabs.com/. [4] n. amenta, t. j. peters and a. c. russell, computational topology: ambient isotopic approximation of 2-manifolds, theoretical computer science 305 (2003), 3–15. [5] l. e. andersson, s. m. dorney, t. j. peters and n. f. stewart, polyhedral perturbations that preserve topological form, cagd 12, no. 8 (1995), 785–799. [6] l. e. andersson, t. j. peters and n. f. stewart, selfintersection of composite curves and surfaces, cagd 15 (1998), 507–527. [7] m. a. armstrong, basic topology, springer, new york, 1983. [8] r. h. bing, the geometric topology of 3-manifolds, american mathematical society, providence, ri, 1983. 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[26] j. peters and x. wu, on the optimality of piecewise linear max-norm enclosures based on slefes, international conference on curves and surfaces, saint-malo, france, 2002. [27] l. piegl and w. tiller, the nurbs book, springer, new york, 1997. [28] c. h. sequin, spline knots and their control polygons with differing knottedness, http://www.eecs.berkeley.edu/pubs/techrpts/2009/eecs-2009-152.html. [29] m. wertheim and k. millett, where the wild things are: an interview with ken millett, cabinet 20, 2006. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 260 http://www.eecs.berkeley.edu/pubs/techrpts/2009/eecs-2009-152.html computational knot counterexample appendix a. subdivision data the following two tables list the vertices for both the original and perturbed control polygons, each after 4 subdivisions. the strict monotonicity of specific coordinates is indicated for each subdivided control polygon. in table i for subdivision of k0, the sixteen control polygons are denoted by p[0], . . . , p[15] and all consecutive control polygons are separated by halfplanes. between p[0] and p[1] the separating plane is the xz plane at the y-value of the last point of p[0]. between p[1] and p[2] the separating plane is the xz plane at the y-value of the last point of p[1]. for all others, it is a y z plane at the x value of the shared subdivision point. in table ii for subdivision of k1, the sixteen control polygons are denoted by q[0], . . . , q[15] and all consecutive control polygons are separated by halfplanes. between q[9] and q[10] the separating plane is the xz plane at the y-value of the last point of p[0]. between q[14] and q[15] the separating plane is the xz plane at the y-value of the last point of p[14]. for all others, it is a y z plane at the x value of the shared subdivision point. a notational change for points occurs in appendix a. in the preceding narrative, points are denoted by parentheses in the format of (x, y, z). within appendix a, points are bounded by the set brackets in the format of {x, y, z}, as the required syntax within mathematica, where many of the computations were performed. for integrity of the data, it was judged best to report the points using this alternative formatting. the code used and its output is posted for public access at https://github.com/kmarinelli/agt 2018 tp km table i: subdivision points from 4 iterations on k0 p[0]: y,z { 0, 4831838208,10737418240 }, { -503316480, 1342177280,8388608000 }, { -859832320, -1562378240,6249512960 }, { -1092485120, -3959685120,4313317376 }, { -1221918720, -5918269440,2572288000 }, { -1266603520, -7498398720,1017953280 }, { -1242964800, -8753032512,-358732160 } p[1]: x,y,z { -1242964800, -8753032512,-358732160}, { -1219326080, -10007666304,-1735417600}, { -1127363840, -10936804608,-2934453760}, { -983503360, -11593406976,-3964886016}, { -802124800, -12023124992,-4836333568}, { -595691520, -12265252864,-5558824960}, { -374886400, -12353556480,-6142648320} p[2]: x { -374886400, -12353556480,-6142648320}, { -154081280, -12441860096,-6726471680}, { 81095680, -12376339456,-7171627008}, { 319961600, -12190760448,-7488402432}, { 553614080, -11913360640,-7687345664}, { 774794880, -11567678336,-7779114240}, { 977745600, -11173260096,-7774340480} p[3]; x,y,z { 977745600, -11173260096,-7774340480}, { 1180696320, -10778841856,-7769566720 }, { 1365416960, -10335687680,-7668250624 }, { 1526149120, -9863344128,-7481024512 }, { 1658634240, -9377366016,-7218495488 }, { 1759969280, -8890023936,-6891110400 }, { 1828454400, -8410890240,-6509035520 } p[4]: y,z { 1828454400, -8410890240,-6509035520 }, { 1896939520, -7931756544,-6126960640 }, { 1932574720, -7460831232,-5690195968 }, { 1933660160, -7007686656,-5208907776 }, { 1899699200, -6579196928,-4692981760 }, { 1831246080, -6180123904,-4151902720 }, { 1729745600, -5813581632,-3594648960 } c© agt, upv, 2019 appl. gen. topol. 20, no. 1 261 k. marinelli and t. j. peters p[5]: x,y,z { 1729745600, -5813581632,-3594648960 }, { 1628245120, -5447039360,-3037395200 }, { 1493697280, -5113027840,-2463966720 }, { 1327546880, -4814661120,-1883341824 }, { 1132129280, -4553405440,-1303992320 }, { 910510080, -4329543680,-733777920 }, { 666316800, -4142518272,-179855360 } p[6]: x,y,z { 666316800, -4142518272,-179855360 }, { 422123520, -3955492864, 374067200 }, { 155356160, -3805303808, 911697920 }, { -130357760, -3691393536,1425880064 }, { -430828800, -3612364032,1910159872 }, { -741473920, -3566319744,2358877440 }, { -1057492800, -3551088960,2767242880 } p[7]: x,z { -1057492800, -3551088960,2767242880 }, { -1373511680, -3535858176,3175608320 }, { -1694904320, -3551440896,3543621632 }, { -2016870400, -3595665408,3866492928 }, { -2334392320, -3666083840,4140302336 }, { -2642411520, -3760193536,4362076160 }, { -2936012800, -3875536896,4529848320 } p[8]: x,y { -2936012800, -3875536896,4529848320 }, { -3229614080, -3990880256,4697620480 }, { -3508797440, -4127457280,4811390976 }, { -3768647680, -4282810368,4869193728 }, { -4004392960, -4454526976,4870070272 }, { -4211589120, -4640339456,4814131200 }, { -4386312000, -4838103360,4702602880 } p[9]: y,z { -4386312000, -4838103360,4702602880 }, { -4561034880, -5035867264,4591074560 }, { -4703284480, -5245582592,4423956992 }, { -4809136640, -5465104896,4202476544 }, { -4875187200, -5692412928,3928975360 }, { -4898744320, -5925586944,3606958080 }, { -4878028800, -6162665472,3241123840 } p[10]: x,y,z { -4878028800, -6162665472,3241123840 }, { -4857313280, -6399744000,2875289600 }, { -4792325120, -6640727040,2465638400 }, { -4681285120, -6883653120,2016869376 }, { -4523326720, -7126519040,1534876160 }, { -4318696320, -7367136640,1026778880 }, { -4068961600, -7602868032, 500941440 } p[11]: x,y,z { -4068961600, -7602868032, 500941440}, { -3819226880, -7838599424, -24896000}, { -3524387840, -8069444608,-568473600}, { -3186012160, -8292765696,-1121427456}, { -2806988800, -8505475072,-1674149888}, { -2391736320, -8703770624,-2215772160}, { -1946419200, -8882749440,-2734161920} p[12]: x,y,z { -1946419200, -8882749440,-2734161920}, { -1501102080, -9061728256,-3252551680 }, { -1025720320, -9221390336,-3747708928 }, { -526438400, -9356832768,-4207501312 }, { -11166720, -9462051840,-4618532864 }, { 510222080, -9529556736,-4966141440 }, { 1025668800, -9549861696,-5234410880 } p[13]: x { 1025668800, -9549861696,-5234410880 }, { 1541115520, -9570166656,-5502680320 }, { 2050620160, -9543271680,-5691610624 }, { 2542123520, -9459691008,-5785285632 }, { 3001379840, -9307943936,-5766535168 }, { 3411732480, -9074046976,-5616947200 }, { 3753881600, -8740884480,-5316894720 } p[14]: y,z { 3753881600, -8740884480,-5316894720 }, { 4096030720, -8407721984,-5016842240 }, { 4369976320, -7975293952,-4566325248 }, { 4556418560, -7426484736,-3945716736 }, { 4633414400, -6741046528,-3134174720 }, { 4576145280, -5894969984,-2109669120 }, { 4356676800, -4859733312,-849023360 } p[15]: x,y,z { 4356676800, -4859733312,-849023360 }, { 4137208320, -3824496640, 411622400 }, { 3755540480, -2600099840,1908408320 }, { 3183738880, -1158021120,3664510976 }, { 2390753280, 534773760,5704253440 }, { 1342177280, 2516582400,8053063680 }, { 0, 4831838208,10737418240 } c© agt, upv, 2019 appl. gen. topol. 20, no. 1 262 computational knot counterexample table ii: subdivision points from 4 iterations on k1 q[0]: y,z { 0, 4831838208,10737418240 }, { -503316480, 1342177280,8388608000 }, { -859832320, -1562378240,6249512960 }, { -1089863680, -3959685120,4312924160 }, { -1212088320, -5918269440,2570813440 }, { -1243563520, -7498398720,1014497280 }, { -1199764800, -8753032512,-365212160 } q[1]: x, y, z { -1199764800, -8753032512,-365212160 }, {-1155966080, -10007666304,-1744921600}, { -1036893440, -10936804608,-2948024320 }, { -858022400, -11593406976,-3983708160 }, { -633246720, -12023124992,-4861665280 }, { -374917120, -12265252864,-5591941120 }, { -93900800, -12353556480,-6184796160 } q[2]: x { -93900800, -12353556480,-6184796160 }, { 187115520, -12441860096,-6777651200 }, { 490818560, -12376339456,-7233085440 }, { 806341120, -12190760448,-7561359360 }, { 1124299520, -11913360640,-7772948480 }, { 1436734080, -11567678336,-7878405120 }, { 1737028800, -11173260096,-7888232960 } q[3]: x, y, z { 1737028800, -11173260096,-7888232960 }, { 2037323520, -10778841856,-7898060800 }, { 2325478400, -10335687680,-7812259840 }, { 2594877440, -9863344128,-7641333760 }, { 2840248320, -9377366016,-7395737600 }, { 3057582080, -8890023936,-7085752320 }, { 3244032000, -8410890240,-6721372160 } q[4]: x, y, z { 3244032000, -8410890240,-6721372160 }, { 3430481920, -7931756544,-6356992000 }, { 3586048000, -7460831232,-5938216960 }, { 3707883520, -7007686656,-5475041280 }, { 3794304000, -6579196928,-4977172480 }, { 3844686080, -6180123904,-4453918720 }, { 3859345600, -5813581632,-3914088960 } q[5]: y, z { 3859345600, -5813581632,-3914088960 }, { 3874005120, -5447039360,-3374259200 }, { 3852942080, -5113027840,-2817853440 }, { 3796472320, -4814661120,-2253680640 }, { 3705850880, -4553405440,-1690050560 }, { 3583150080, -4329543680,-1134673920 }, { 3431116800, -4142518272,-594575360 } q[6]: x, y, z { 3431116800, -4142518272,-594575360 }, { 3279083520, -3955492864, -54476800 }, { 3097717760, -3805303808, 470343680 }, { 2889766400, -3691393536, 972861440 }, { 2658650880, -3612364032,1446737920 }, { 2408324480, -3566319744,1886407680 }, { 2143108800, -3551088960,2287152640 } q[7]: x, z { 2143108800, -3551088960,2287152640 }, { 1877893120, -3535858176,2687897600 }, { 1597788160, -3551440896,3049717760 }, { 1307115520, -3595665408,3367895040 }, { 1010565120, -3666083840,3638558720 }, { 713031680, -3760193536,3858759680 }, { 419430400, -3875536896,4026531840 } q[8]: x, y { 419430400, -3875536896,4026531840 }, { 125829120, -3990880256,4194304000 }, { -163840000, -4127457280,4309647360 }, { -444661760, -4282810368,4370595840 }, { -711700480, -4454526976,4376166400 }, { -960184320, -4640339456,4326420480 }, { -1185710400, -4838103360,4222512640 } q[9]: x, y, z { -1185710400, -4838103360,4222512640 }, { -1411236480, -5035867264,4118604800 }, { -1613804800, -5245582592,3960535040 }, { -1789012480, -5465104896,3749457920 }, { -1932825600, -5692412928,3487621120 }, { -2041784320, -5925586944,3178414080 }, { -2113228800, -6162665472,2826403840 } q[10]: y, z { -2113228800, -6162665472,2826403840 }, { -2184673280, -6399744000,2474393600 }, { -2218603520, -6640727040,2079580160 }, { -2212359680, -6883653120,1646530560 }, { -2164081920, -7126519040,1180989440 }, { -2072936320, -7367136640, 689914880 }, { -1939361600, -7602868032, 181501440 } q[11]: x, y, z { -1939361600, -7602868032, 181501440 }, { -1805786880, -7838599424,-326912000 }, c© agt, upv, 2019 appl. gen. topol. 20, no. 1 263 k. marinelli and t. j. peters { -1629783040, -8069444608,-852664320 }, { -1411788800, -8292765696,-1387560960 }, { -1153515520, -8505475072,-1922170880 }, { -858193920, -8703770624,-2445803520 }, { -530841600, -8882749440,-2946498560 } q[12]: x, y, z { -530841600, -8882749440,-2946498560 }, { -203489280, -9061728256,-3447193600 }, { 155893760, -9221390336,-3924951040 }, { 542289920, -9356832768,-4367810560 }, { 948894720, -9462051840,-4762542080 }, { 1366849280, -9529556736,-5094635520 }, { 1784952000, -9549861696,-5348303360 } q[13]: x { 1784952000, -9549861696,-5348303360 }, { 2203054720, -9570166656,-5601971200 }, { 2621305600, -9543271680,-5777213440 }, { 3028503040, -9459691008,-5858242560 }, { 3411102720, -9307943936,-5827993600 }, { 3752929280, -9074046976,-5668126720 }, { 4034867200, -8740884480,-5359042560 } q[14]: y, z { 4034867200, -8740884480,-5359042560 }, { 4316805120, -8407721984,-5049958400 }, { 4538854400, -7975293952,-4591656960 }, { 4681899520, -7426484736,-3964538880 }, { 4723884800, -6741046528,-3147745280 }, { 4639505280, -5894969984,-2119173120 }, { 4399876800, -4859733312,-855503360 } q[15]: x, y, z { 4399876800, -4859733312,-855503360 }, { 4160248320, -3824496640, 408166400 }, { 3765370880, -2600099840,1906933760 }, { 3186360320, -1158021120,3664117760 }, { 2390753280, 534773760,5704253440 }, { 1342177280, 2516582400,8053063680 }, { 0, 4831838208,10737418240 } c© agt, upv, 2019 appl. gen. topol. 20, no. 1 264 @ appl. gen. topol. 23, no. 1 (2022), 79-90 doi:10.4995/agt.2022.15844 © agt, upv, 2022 closed ideals in the functionally countable subalgebra of c(x) amir veisi faculty of petroleum and gas, yasouj university, gachsaran, iran (aveisi@yu.ac.ir) communicated by a. tamariz-mascarúa abstract in this paper, closed ideals in cc(x), the functionally countable subalgebra of c(x), with the mc-topology are studied. we show that if x is a cuc-space, then c∗c (x) with the uniform norm-topology is a banach algebra. closed ideals in cc(x) as a modified countable analogue of closed ideals in c(x) with the m-topology, are characterized. for a zero-dimensional space x, we show that a proper ideal in cc(x) is closed if and only if it is an intersection of maximal ideals of cc(x). it is also shown that every ideal in cc(x) with the mc-topology is closed if and only if x is a p -space if and only if every ideal in c(x) with the m-topology is closed. also, for a strongly zero-dimensional space x, it is proved that every properly closed ideal in c∗c (x) is an intersection of maximal ideals of c∗c (x) if and only if x is pseudocompact if and only if every properly closed ideal in c∗(x) is an intersection of maximal ideals of c∗(x). finally, we show that if x is a p -space, then the family of ec-ultrafilters and zc-ultrafilter coincide. 2020 msc: 54c30; 54c40; 13c11. keywords: zero-dimensional space; functionally countable subalgebra; mtopology; closed ideal; ec-filter; ec-ideal; p -space. 1. introduction in what follows x stands for an infinite completely regular hausdorff topological space (i.e., infinite tychonoff space) and c(x) as usual denotes the ring of all real-valued continuous functions on x. c∗(x) designates the subring of c(x) containing all those members which are bounded over x. for received 30 june 2021 – accepted 09 october 2021 http://dx.doi.org/10.4995/agt.2022.15844 a. veisi each f ∈ c(x), the zero-set of f, denoted by z(f), is the set of zeros of f and x \z(f) is the cozero-set of f and the set of all zero-sets in x is denoted by z(x). an ideal i in c(x) is called a z-ideal if f ∈ i, g ∈ c(x) and z(f) ⊆ z(g), then g ∈ i. the space βx is the stone-c̆ech compactification of x and for any p ∈ βx, the maximal ideal mp of c(x) is the set of all f ∈ c(x) for which p ∈ clβxz(f). moreover, mp is fixed if and only if p ∈ x (in which case, we put mp = mp = {f ∈ c(x) : p ∈ z(f)}). whenever c(x) mp ∼= r, then mp is called real, else hyper-real, see [5, chapter 8]. we recall that a zero-dimensional space is a hausdorff space with a base consisting of clopen (closed-open) sets. a tychonoff space x is called strongly zero-dimensional if for every finite cover {ui}ki=1 of x by cozero-sets there exists a finite refinement {vi}mi=1 of mutually disjoint open sets. a tychonoff space x is strongly zero-dimensional if and only if βx is zero-dimensional, see [2]. the subring of c(x) consisting of those functions with countable (resp. finite) image, which is denoted by cc(x) (resp. c f (x)) is an r-subalgebra of c(x). the subring c∗c (x) of cc(x) consists of bounded elements of cc(x). so c∗c (x) = c ∗(x) ∩ cc(x). the rings cc(x) and cf (x) are introduced and investigated in [3] and more studied in [1], [4], [9], [10] and [12]. a topological space x is called countably pseudocompact, briefly, c-pseudocompact if cc(x) = c ∗ c (x). a nonempty subfamily f of zc(x) := {z(f) : f ∈ cc(x)} is called a zc-filter if it is a filter on x. for an ideal i in cc(x) and a zcfilter f, we define zc[i] = {z(f) : f ∈ i}, ∩zc[i] = ∩{z(f) : f ∈ i} and z−1c [f] = {f ∈ cc(x) : z(f) ∈f}. it is observed that f = zc[z−1c [f]]. also, zc[i] is a zc-filter on x and z −1 c [zc[i]] ⊇ i. if the equality holds, then i is called a zc-ideal. this means that if f ∈ i, g ∈ cc(x) and z(f) ⊆ z(g), then g ∈ i. so maximal ideals in cc(x) are zc-ideals. in the same way, for an ideal i of c∗c (x) and a zc-filter f on x, ec(i) is an ec-filter and e−1c (f) is an ecideal. the counterpart notions are e−1c (ec(i)) ⊇ i and ec(e−1c (f)) = f, see [14]. by β0x, we mean the banaschewski compactification of a zerodimensional space x. if βx is zero-dimensional, then βx = β0x, see [13, section 4.7] for more details. according to [1, theorems 4.2, 4.8], for any p ∈ β0x, the maximal ideal mpc of cc(x) is the set of all f ∈ cc(x) for which p ∈ clβ0xz(f), or equivalently, it is the set of all f ∈ cc(x) for which πp ∈ clβxz(f). moreover, mpc is fixed if and only if p ∈ x (in which case, we put mpc = mcp = {f ∈ cc(x) : p ∈ z(f)}). let s be a subring of c(x) and a topological space. an ideal i of s is called a closed ideal if i = clsi, briefly, i = cli. the paper is organized as follows. in section 2, we introduce the mc-topology on cc(x) and derive some corollaries on the ideals of cc(x) and c∗c (x). we show that if x is a cuc-space, then c ∗ c (x) with the uniform-norm topology is a banach algebra. it is shown that an ideal in cc(x) is a z-ideal if and only if it is a zc-ideal. in [5], closed ideals in c(x) with the m-topology are characterized. in section 3, the countable analogue of this characterization is given. we show that a proper ideal in cc(x) is closed if and only if it is an intersection of maximal ideals in cc(x). it is also shown that every ideal © agt, upv, 2022 appl. gen. topol. 23, no. 1 80 closed ideals in the functionally countable subalgebra of c(x) in cc(x) is closed if and only if x is a p-space if and only if every ideal in c(x) is closed. for a strongly zero-dimensional space x, we prove that every properly closed ideal in c∗c (x) is an intersection of maximal ideals of c ∗ c (x) if and only if x is pseudocompact if and only if every properly closed ideal in c∗(x) is an intersection of maximal ideals of c∗(x). finally, we show that if x is a p-space, then the family of ec-ultrafilters and zc-ultrafilter coincide. 2. some properties of ideals in cc(x) the m-topology on c(x) was first introduced and studied by hewitt [8], the generalizing work of e. h. moore. in his article, he demonstrated that certain classes of topological spaces x can be characterized by topological properties of c(x) with the m-topology. for example, he showed that x is pseudocompact if and only if c(x) with the m-topology is first countable. several authors have investigated the topological properties of x via properties of c(x), for more information, one can refer to [6] and [11]. the m-topology on c(x) is defined by taking the sets of the form b(f,u) = {g ∈ c(x) : |f(x) −g(x)| < u(x) for all x ∈ x}, as a base for the neighborhood system at f, for each f ∈ c(x) and each positive unit u of c(x). the mc-topology (in brief, mc) on cc(x) is determined by considering the sets of the form b(f,u) = {g ∈ cc(x) : |f(x) −g(x)| < u(x) for all x ∈ x}, as a base for the neighborhood system at f, for each f ∈ cc(x) and each positive unit u of cc(x). the uniform topology, or the uc-topology (in brief, uc) on cc(x) is defined by taking the sets of the form b(f,ε) = {g ∈ cc(x) : |f(x) −g(x)| < ε for all x ∈ x}, as a base for the neighborhood system at f, for each f ∈ cc(x) and each ε > 0. equivalently, a base at f is given by all sets b(f,u) = {g ∈ cc(x) : |f(x) −g(x)| < u(x) for all x ∈ x}, where u is a positive unit of c∗c (x). we observe that uc ⊆ mc. it is shown in [15] that uc = mc if and only if x is countably pseudocompact. the uctopology turns cc(x) into a metric space with d(f,g) = ‖f−g‖ = sup{|f(x)− g(x)| : x ∈ x}. also, the mc-topology is contained in the relative m-topology. we remind a well-known result that due to rudin, pelczynski and semadeni which asserts that a compact hausdorff space x is functionally countable (i.e., c(x) = cc(x)) if and only if x is scattered. so if x is a compact scattered space or a countable space, then c(x) = cc(x), and thus the mc-topology and the m-topology coincide. proposition 2.1. let i be an ideal in cc(x) (resp. c ∗ c (x)) and the topology on cc(x) be the mc-topology. then: (i) cl i is an ideal in cc(x) (resp. c ∗ c (x)) and hence i is contained in a closed ideal. (ii) if i is a proper ideal, then cl i is also a proper ideal and hence there is no proper dense ideal in cc(x) (resp. c ∗ c (x)). © agt, upv, 2022 appl. gen. topol. 23, no. 1 81 a. veisi proof. we provide the proof for which case i is an ideal in cc(x). in the same way, the proof holds for the ideal i in c∗c (x). (i). clearly, the result holds if i = cc(x). suppose that i $ cc(x). let f,g ∈ cli, h ∈ cc(x) and u be a positive unit of cc(x). then for some f ′ ∈ b(f, u 2 ) ∩ i, and g′ ∈ b(g, u 2 ) ∩ i, we have f′ + g′ ∈ b(f + g,u) ∩ i. to show that fh ∈ cli, we consider the positive unit u1 = u (|h| + 1)(u + 1) ∈ cc(x). therefore, for some f1 ∈ b(f,u1) ∩ i we have that |fh−f1h| < u1|h| < u. so f1h ∈ b(fh,u)∩i. moreover, if f ∈ cli, then also −f ∈ cli. thus, cli contains both f + g and fh. so cli is ideal. (ii). suppose that i is a proper ideal in cc(x) and cli = cc(x). consider the constant function 1 ∈ cli and 0 < ε < 1. hence, the nonempty set b(1,ε) ∩ i contains a nonzero element of cc(x), f say. since 1 −ε < f(x) < 1 + ε for each x ∈ x, we have z(f) = ∅, i.e., f is a unit of cc(x), which is impossible (because f ∈ i). thus, cli $ cc(x), and we are done. � the next result is now immediate. corollary 2.2. any maximal ideal in cc(x) (resp. c ∗ c (x)) and hence any intersection of maximal ideals in cc(x) (resp. c ∗ c (x)) is closed. definition 2.3. an ideal i in a commutative ring with unity r is called a z-ideal in r if for each a ∈ i, we have ma ⊆ i, here ma is the intersection of all maximal ideals in r containing a. evidently, each maximal ideal in r is a z-ideal. this notion of z-ideal is consistent with the notion of z-ideals in c(x), see [5, 4a(5)]. proposition 2.4. let x be zero-dimensional and i be an ideal in c∗c (x). then i is a z-ideal if and only if g ∈ i whenever z(fβ) ⊆ z(gβ) with f ∈ i and g ∈ c∗c (x), where fβ is the extension of f to βx. proof. (⇒) : let f ∈ i, g ∈ c∗c (x) and z(fβ) ⊆ z(gβ) and let mf be the intersection of all the maximal ideals in c∗c (x) containing f. by the assumption, mf ⊆ i. let m be a maximal ideal in c∗c (x) containing f. according to [9, corollary 2.11], m has a form of m∗pc = {h ∈ c∗c (x) : hβ(p) = 0}, for some p ∈ βx. now, z(fβ) ⊆ z(gβ) implies that g ∈ m. hence, g ∈ i. (⇐) : let f ∈ i and g ∈ mf . then f ∈ m∗pc implies that g ∈ m∗pc , i.e., z(fβ) ⊆ z(gβ). therefore, by the hypothesis, g ∈ i. � lemma 2.5. let x be zero-dimensional and i be an ideal in cc(x). then i is a z-ideal if and only if it is a zc-ideal. proof. (⇒) : let i be a z-ideal in cc(x), f ∈ i and z(f) ⊆ z(g) with g ∈ cc(x). we have to show that g ∈ i. since i is a z-ideal, we have mf ⊆ i, where mf is the intersection of all the maximal ideals in cc(x) containing f. it suffices to show that g ∈ mf . so let mpc (p ∈ β0x) be any maximal ideal in cc(x) which contains f, we have to show that g ∈ mpc (see [1, theorem © agt, upv, 2022 appl. gen. topol. 23, no. 1 82 closed ideals in the functionally countable subalgebra of c(x) 4.2]). indeed f ∈ mpc implies that p ∈ clβ0xz(f) which further implies that p ∈ clβ0xz(g), by the assumption, z(f) ⊆ z(g). hence, g ∈ mpc . thus, i becomes a zc-ideal in cc(x). (⇐) : let i be a zc-ideal in cc(x) and f ∈ i. we must show mf ⊆ i. let g ∈ mf . then f ∈ mpc gives g ∈ mpc , where p ∈ β0x. equivalently, clβ0xz(f) ⊆ clβ0xz(g). so z(f) = clβ0xz(f) ∩x ⊆ clβ0xz(g) ∩x = z(g). now, the assumption yields that g ∈ i. � proposition 2.6. if i is a closed ideal in cc(x), then i is a zc-ideal. proof. suppose that z(f) ⊆ z(g), f ∈ i and g ∈ cc(x). to show that g ∈ i, we show that g ∈ cli because i = cli. let u ∈ cc(x) be a positive unit and let us define a function h : x → r as follows: h(x) =   g(x)−u(x) 2 f(x) where g(x) ≥ u(x) 2 , 0 where |g(x)| ≤ u(x) 2 , g(x)+ u(x) 2 f(x) where g(x) ≤−u(x) 2 . from the continuity of h on the three closed sets (g − u 2 )−1([0,∞)), (g + u 2 )−1([0,∞)) ∩(g− u 2 )−1((−∞, 0]), and (g + u 2 )−1((−∞, 0]), which their union is x, we infer that h ∈ c(x). moreover, since the ranges of g,u and f are countable, the range of h is also countable, i.e., h ∈ cc(x). thus, fh ∈ i. furthermore, it is easy to see that |g(x) −f(x)h(x)| < u(x) for every x ∈ x, i.e., fh ∈ b(g,u) ∩ i and thus g ∈ cli, which completes the proof. � the next example shows that the converse of the above proposition is not true in general. example 2.7. consider the zero-dimensional space x = q×q, p = (0, 0) ∈ x, and put op = {f ∈ c(x) : p ∈ intxz(f)} (note, cc(x) = c(x) because x is countable). recall that op is a zc-ideal. we now claim that op is not a closed ideal in c(x). to see this, consider f(x,y) = |x|+|y| 1+|x|+|y| ∈ c(x) and let u be a fixed positive unit of c(x). define a function g by g(x,y) = { 0 where f(x,y) ≤ u(x,y) 2 , f(x,y) − u(x,y) 2 where f(x,y) ≥ u(x,y) 2 . obviously, g ∈ c(x). let g = {(x,y) ∈ x : f(x,y) < u(x,y) 2 }. then p ∈ g ⊆ z(g) and therefore g ∈ op, in fact, g ∈ b(f,u) ∩ op. it follows that f ∈ clc(x)op. on the other hand, the set z(f) = {p} is not open in x. hence, f ∈ clc(x)op \op. i.e., op is not a closed ideal in c(x). a banach algebra b is an algebra that is a banach space with a norm that satisfies ‖xy‖≤ ‖x‖‖y‖ for all x,y ∈ b, and there exists a unit element e ∈ b such that ex = xe = x, ‖e‖ = 1. in [7, definition 2.2], a topological space x is called a countably uniform closed-space, briefly, a cuc-space, if whenever {fn}n∈n is a sequence of functions of cc(x) and fn → f uniformly, then f belongs to cc(x). © agt, upv, 2022 appl. gen. topol. 23, no. 1 83 a. veisi theorem 2.8. if x is a cuc-space, then c∗c (x) with the supremum-norm topology is a banach algebra. proof. let {fn}n∈n be a cauchy sequence of functions in c∗c (x). given ε > 0, we can find a natural number n such that ‖fn −fm‖≤ ε for every m,n > n. thus, |fn(x) −fm(x)| ≤ ε for all x ∈ x and all m,n > n. let x ∈ x be fixed and ax be the limit of the numerical sequence {fn(x)}n∈n in r (note, r is a banach space). now, define f : x → r by f(x) = ax. let n be fixed, then |fn(x)−limm→∞fm(x)| ≤ ε for each x ∈ x and each m > n. so ‖fn−f‖≤ ε. since n is arbitrary, we get fn → f in the norm, uniformly. consequently, f ∈ c(x). furthermore, our assumption implies that f ∈ cc(x). moreover, ‖f‖ ≤ ‖f −fn‖ + ‖fn‖ gives f is bounded. hence, c∗c (x) is a banach space. the proof is completed by the fact that ‖fg‖≤‖f‖‖g‖ for all f,g ∈ c∗c (x). � 3. closed ideals in cc(x) and c ∗ c (x) (with the mc-topology) we need the next statement which is the counterpart of [5, 1d(1)] for cc(x). proposition 3.1. if f,g ∈ cc(x) and z(f) is a neighborhood of z(g), then f = gh for some h ∈ cc(x). proposition 3.2. let x be a zero-dimensional space, f ∈ cc(β0x) and let f0 be the restriction of f on x. then intβ0xz(f) ⊆ clβ0xz(f0) ⊆ z(f). proof. let p ∈ intβ0xz(f) and v be an open set in β0x containing p. since x is dense in β0x, we have ∅ 6= v ∩ intβ0xz(f) ∩ x ⊆ v ∩ z(f0). so p ∈ clβ0xz(f0). for the second inclusion, since z(f0) ⊆ z(f), we have that clβ0xz(f0) ⊆ clβ0xz(f) = z(f). � corollary 3.3. let x be zero-dimensional and p ∈ β0x. then (i) ⋂ f∈mpc clβ0xz(f) = {p}. (ii) if p ∈ x, then ⋂ f∈mcp z(f) = {p}, i.e., mcp is fixed. proof. (i). recall that f ∈ mpc if and only if p ∈ clβ0xz(f) (see [1, theorem 4.2]). therefore, p ∈ ⋂ f∈mpc clβ0xz(f). now, we claim that the latter intersection is the singleton set {p}. on the contrary, suppose that this set contains an element q ∈ β0x distinct from p. since β0x is zero-dimensional, by [3, proposition 4.4], there exists g ∈ cc(β0x) such that p ∈ intβ0xz(g) and g(q) = 1. let g0 be the restriction of g on x. then by proposition 3.2, clβ0xz(g0) contains p but not q. this means that g0 ∈ mpc \mqc which is a contradiction, so (i) holds. (ii). clearly, ⋂ f∈mcp z(f) = ⋂ f∈mcp clβ0xz(f) ∩x = {p}. � in a similar way to proposition 3.2 and corollary 3.3, we get: proposition 3.4. for a tychonoff space x and f ∈ c∗(x), we have that intβxz(f β) ⊆ clβxz(f) ⊆ z(fβ), where fβ is the extension of f to βx. moreover, if p ∈ βx, then ⋂ f∈mp clβxz(f) = {p}. in particular, if p ∈ x, then ⋂ f∈mp z(f) = {p}, i.e., mp is fixed. © agt, upv, 2022 appl. gen. topol. 23, no. 1 84 closed ideals in the functionally countable subalgebra of c(x) proposition 3.5. let x be zero-dimensional, p ∈ β0x and πp be its corresponding point of βx in characterizing of maximal ideals in cc(x). then mpc ∩c∗c (x) ⊆ m∗πp ∩c∗c (x). particularly, if x is strongly zero-dimensional, then mpc ∩c∗c (x) ⊆ m∗p ∩c∗c (x). proof. in view of [1, theorems 4.2, 4.8], we have mpc = {f ∈ cc(x) : p ∈ clβ0xz(f)} = {f ∈ cc(x) : πp ∈ clβxz(f)}. let f ∈ mpc ∩c∗c (x). then πp ∈ clβxz(f) and hence fβ(πp) = 0, by proposition 3.4. therefore, f ∈ m∗πp ∩ c∗c (x). the second part follows from the assumption, i.e., β0x = βx and so πp = p. � remark 3.6. replacing t with β0x in [1, proposition 3.2] implies that for any two zero-sets z1 and z2 in zc(x), we get clβ0x(z1 ∩z2) = clβ0xz1 ∩clβ0xz2. remark 3.7. ([1, remark 4.12]) if x is zero-dimensional and f,g ∈ cc(x), then clβ0xz(f) is a neighborhood of clβ0xz(g) if and only if there exists h ∈ cc(x) such that z(g) ⊆ coz(h) ⊆ z(f). proposition 3.8. let x be zero-dimensional and i a proper ideal in cc(x) and let vc(i) = {p ∈ β0x : mpc ⊇ i}. then: (i) vc(i) = ⋂ g∈i clβ0xz(g). (ii) if f ∈ cc(x) and clβ0xz(f) is a neighborhood of vc(i), then f ∈ i. proof. (i). this is easily obtained from the fact that g ∈ mpc if and only if p ∈ clβ0xz(g). (ii). suppose that vc(i) = ⋂ g∈i clβ0xz(g) ⊆ intβ0xclβ0xz(f). then we have ⋃ g∈i ( β0x \ clβ0xz(g) ) ⊇ β0x \ intβ0xclβ0xz(f). hence, the collection c = {intβ0xclβ0xz(f), β0x \ clβ0xz(g) : g ∈ i} is an open cover for the compact set β0x. therefore, there is a finite number of elements of i; g1,g2, . . . ,gn say, such that β0x = intβ0xclβ0xz(f) ∪ ( β0x \ intβ0xclβ0xz(f) ) = intβ0xclβ0xz(f) ∪ ( n⋃ i=1 (β0x \ clβ0xz(gi)) ) . now, we have that(⋂n i=1 clβ0xz(gi) ) ∩ ( β0x \ intβ0xclβ0xz(f) ) = ∅. thus, ⋂n i=1 clβ0xz(gi) ⊆ intβ0xclβ0xz(f). since i is a proper ideal, the element g = ∑n i=1 g 2 i of i is not a unit of cc(x) and hence z(g) = ⋂n i=1 z(gi) 6= ∅. from remark 3.6 we conclude that clβ0xz(g) = clβ0x (⋂n i=1 z(gi) ) = ⋂n i=1 clβ0xz(gi) ⊆ intβ0xclβ0xz(f). © agt, upv, 2022 appl. gen. topol. 23, no. 1 85 a. veisi this leads us clβ0xz(f) is a neighborhood of clβ0xz(g). in view of remark 3.7, there exists h ∈ cc(x) such that z(g) ⊆ coz(h) ⊆ z(f). so z(f) is a neighborhood of z(g). by proposition 3.1, we get f ∈ i. � lemma 3.9. let x be zero-dimensional and g ∈ cc(x). then for any neighborhood b(g,u) of g in the mc-topology, there exists some fu ∈ b(g,u) such that clβ0xz(fu) is a neighborhood of clβ0xz(g). proof. if clβ0xz(g) is an open set in β0x, then we set fu = g. in general, we define a function fu : x → r by fu(x) =   g(x) − u(x) 2 where g(x) ≥ u(x) 2 , 0 where |g(x)| ≤ u(x) 2 , g(x) + u(x) 2 where g(x) ≤−u(x) 2 . it is clear that fu ∈ c(x) and further since the range of g and u is countable, we get fu ∈ cc(x). moreover, fu ∈ b(g,u). to establish the conclusion, consider the function h below h(x) = { ( g(x) + u(x) 2 )( g(x) − u(x) 2 ) where |g(x)| ≤ u(x) 2 , 0 where |g(x)| ≥ u(x) 2 . we observe that h ∈ cc(x). furthermore, z(g) ⊆ coz (h) ⊆ z(fu). now, remark 3.7 implies that clβ0xz(fu) is a neighborhood of clβ0xz(g), and we are through. � theorem 3.10. let x be zero-dimensional and i a proper ideal in cc(x) and let vc(i) be the same as the set in proposition 3.8 ( vc(i) = ⋂ g∈i clβ0xz(g) ) . let j = {f ∈ cc(x) : clβ0xz(f) ⊇ vc(i)}, and ī = ∩{m p c : m p c ⊇ i}. then: (i) ī is a closed ideal in cc(x) containing i. (ii) j = ī, in other words, j is the kernel of the hull of i in the structure space of cc(x). (iii) vc(i) = vc(ī). (iv) cl i = ī. proof. (i). it follows from corollary 2.2. (ii). let f ∈ j and mpc (p ∈ β0x) be a maximal ideal in cc(x) containing i. then (3.1) vc(i) ⊇ vc(mpc ) and so clβ0xz(f) ⊇ vc(i) ⊇ vc(m p c ) = {p} (note, the last equality follows from corollary 3.3). therefore, f ∈ mpc and thus f ∈ ī, i.e., j ⊆ ī. for the reverse inclusion, we show that if f /∈ j, then f /∈ ī. since f /∈ j, there exists q ∈ β0x such that q ∈ vc(i) \ clβ0xz(f). therefore, g ∈ mqc for every g ∈ i and hence i ⊆ mqc . but f /∈ mqc . thus, mqc is a maximal ideal containing i but not f. this yields that f /∈ ī. (iii). using (ii) and the definition of j, we have vc(ī) = vc(j) ⊇ vc(i). on the other hand, the inclusion i ⊆ ī implies that vc(ī) ⊆ vc(i). so (iii) holds. © agt, upv, 2022 appl. gen. topol. 23, no. 1 86 closed ideals in the functionally countable subalgebra of c(x) (iv). by (i), cli ⊆ ī. now, suppose that g ∈ ī and u is a positive unit of cc(x). we claim that b(g,u) ∩ i 6= ∅. according to lemma 3.9, there exists fu ∈ cc(x) such that fu ∈ b(g,u), and clβ0xz(fu) is a neighborhood of clβ0xz(g). now, it remains to show that fu ∈ i. from (iii), we infer that vc(i) = vc(ī) ⊆ clβ0xz(g) ⊆ intβ0xclβ0xz(fu). proposition 3.8(ii) now yields that fu ∈ i. therefore, fu ∈ b(g,u) ∩ i and so g ∈ cli, i.e., ī ⊆ cli. � it is known that a proper ideal in c(x) with the m-topology is closed if and only if it is an intersection of maximal ideals in c(x) (see [5, 7q(2)]). the next theorem involves the countable analogue characterization of closed ideals in cc(x). using theorem 3.10(iv) and corollary 2.2, we obtain: theorem 3.11. let x be zero-dimensional and the topology on cc(x) be the mc-topology. then a proper ideal in cc(x) is closed if and only if it is an intersection of maximal ideals of cc(x). theorem 3.12. let x be zero-dimensional and the topology on cc(x) (resp. c(x)) be the mc-topology (resp. the m-topology). then the following statements are equivalent. (i) every ideal in c(x) is closed. (ii) x is a p -space. (iii) every ideal in cc(x) is closed. (iv) every prime ideal in cc(x) is closed. proof. (i) ⇔ (ii). it follows from [5, 4j(9), 7q(2)]. (ii) ⇒ (iii). by [3, proposition 5.3], x is a cp-space. now, the result is obtained by [3, theorem 5.8(7)] and corollary 2.2. (iii) ⇒ (iv). it is evident. (iv) ⇒ (ii). according to [3, corollary 5.7], it is enough to show that x is a cp-space. let p be a prime ideal in cc(x), then by [1, lemma 4.11(4)], p is contained in a unique maximal ideal mpc of cc(x), where p ∈ β0x. now, by the assumption and theorem 3.11, we get p = mpc , i.e., x is a cp-space. � theorem 3.13. let x be strongly zero-dimensional and the topology on c∗c (x) (resp. c∗(x)) be the mc-topology (resp. the m-topology). then the following statements are equivalent. (i) every properly closed ideal in c∗c (x) is an intersection of maximal ideals of c∗c (x). (ii) x is pseudocompact. (iii) every properly closed ideal in c∗(x) is an intersection of maximal ideals of c∗(x). proof. a maximal ideal in c∗c (x) is of the form m ∗p c = {f ∈ c∗c (x) : fβ(p) = 0}, where p ∈ βx. also, m∗pc = m∗p ∩c∗c (x), see [9, corollaries 2.10, 2.11]. (i) ⇒ (ii). suppose that x is not pseudocompact, so c∗c (x) $ cc(x), by [9, theorem 6.3]. hence, cc(x) contains an unbounded element, f say. so for some p ∈ βx and the maximal ideal mpc of cc(x), we have |mpc (f)| is infinitely © agt, upv, 2022 appl. gen. topol. 23, no. 1 87 a. veisi large ([9, proposition 2.4]). in other words, mpc is hyper-real, i.e., r $ cc(x) m p c . hence, by [9, corollary 2.13], mpc ∩ c∗c (x) is not a maximal ideal in c∗c (x). using proposition 3.5, we infer that (3.2) mpc ∩c ∗ c (x) $ m ∗p ∩c∗c (x). furthermore, since the maximal ideal mpc is closed in cc(x) (corollary 2.2), the ideal mpc ∩c∗c (x) is also closed in c∗c (x). we now claim that the latter closed ideal cannot be an intersection of maximal ideals of c∗c (x). otherwise, (3.3) mpc ∩c ∗ c (x) = ⋂ q∈a⊆βx ( m∗q ∩c∗c (x) ) , for a subset a of βx. notice that by (3.2), a 6= ∅ since p ∈ a. now, we claim that a = {p}. on the contrary, suppose that a contains an element q distinct from p. we can take f ∈ cc(βx) such that z(f) is a neighborhood of p and f(q) = 1 (note, by the assumption, βx is zero-dimensional). let f0 be the restriction of f on x. then the compactness of βx gives f and hence f0 are bounded, i.e., f0 ∈ c∗c (x). by density of x in βx, we get f = f β 0 , where f β 0 is the extension of f0 to βx. due to proposition 3.2, we infer that p ∈ clβxz(f0), since p ∈ intβxz(f). hence, f0 ∈ mpc ∩ c∗c (x). on the other hand, since q /∈ z(f), we have that f0 /∈ m∗q. therefore, f0 ∈ mpc ∩c∗c (x) \ (m∗q ∩c∗c (x)), which contradicts the equation in (3.3). so a = {p} and hence mpc ∩c∗c (x) = m∗p ∩c∗c (x). but this also contradicts (3.2). thus, if x is not pseudocompact, then there exists a closed ideal in c∗c (x) which is not an intersection of maximal ideals of c ∗ c (x), and we are done. (ii) ⇒ (i). since x is pseudocompact, c(x) = c∗(x) gives cc(x) = c∗c (x). now, it follows from theorem 3.11. (ii) ⇔ (iii). it follows from [5, 7q(3)]. � we end the article with some results on ec-filters on x and ec-ideals in c∗c (x), for more details, see [14, section 2]. let p ∈ βx and fβ be the extension of f ∈ c∗(x) to βx. let us recall that m∗pc = {f ∈ c ∗ c (x) : f β(p) = 0} = m∗p ∩c∗c (x), and o ∗p c = o p c ∩c ∗ c (x), where m∗p = {f ∈ c∗(x) : fβ(p) = 0}, and opc = {f ∈ cc(x) : p ∈ intβxclβxz(f)}. lemma 3.14. let x be strongly zero-dimensional and p ∈ βx. then ec(m ∗p c ) = zc[o p c ] = zc[o ∗p c ] = ec(o ∗p c ). proof. by the hypothesis, βx = β0x. to get the result, we show the following chain of inclusions holds. (3.4) ec(m ∗p c ) ⊆ zc[o p c ] ⊆ zc[o ∗p c ] ⊆ ec(o ∗p c ) ⊆ ec(m ∗p c ). to establish the first inclusion, let ecε(f) := {x ∈ x : |f(x)| ≤ ε} ∈ ec(m∗pc ), where f ∈ m∗pc and ε > 0. then fβ(p) = 0. notice that ecε(f) = z((|f|−ε)∨0) © agt, upv, 2022 appl. gen. topol. 23, no. 1 88 closed ideals in the functionally countable subalgebra of c(x) and (3.5) clβxz((|f|−ε) ∨ 0) = clβxecε(f) = {q ∈ βx : |f β(q)| ≤ ε}. hence, p ∈ intβxclβxz((|f|−ε) ∨ 0), in other words, (|f|−ε) ∨ 0 ∈ opc . here, we are going to show the last equality in (3.5). let q ∈ βx such that |fβ(q)| ≤ ε. since x is dense in βx, there exists a net (xλ)λ∈λ ⊆ x converging to q and so f(xλ) = f β(xλ) → fβ(q). moreover, |f(xλ)| → |fβ(q)|. now, let v be an open set in βx containing q. then for some λ0 ∈ λ and each λ ≥ λ0, we have xλ ∈ v . furthermore, |fβ(q)| ≤ ε yields that |f(xλ)| ≤ ε. hence, v ∩ecε(f) 6= ∅, i.e., q ∈ clβxec� (f). the second inclusion in (3.4) follows from the fact that z(f) = z( f 1+|f|), where f ∈ opc (and thus f 1+|f| ∈ o ∗p c ). to verify the third inclusion, we let f ∈ o∗pc and show that z(f) ∈ ec(o∗pc ). since p does not belong to the closed set f := βx \ intβxclβxz(f) and βx is zero-dimensional, by [3, proposition 4.4], there is some g ∈ cc(βx) = c∗c (βx) such that p ∈ intβxz(g) and g(f) = {1}. let g0 be the restriction of g on x. then by proposition 3.2, p ∈ intβxclβxz(g0). so g0 ∈ o∗pc and hence ecε(g0) ∈ ec(o∗pc ) for all ε > 0. let 0 < ε < 1 be fixed. since x is dense in βx, the open set {q ∈ βx : |g(q)| < ε} intersects x nontrivially (since it contains p). therefore, ∅ 6= {q ∈ βx : |g(q)| ≤ ε}∩x = {x ∈ x : |g0(x)| ≤ ε} = ecε(g0) ⊆ (βx \f) ∩x ⊆ z(f). now, since the zc-filter (in fact, the ec-filter) ec(o ∗p c ) contains e c ε(g0) and ecε(g0) ⊆ z(f), we infer that z(f) ∈ ec(o∗pc ), and we are done. finally, the last inclusion in (3.4) follows from the inclusion o∗pc ⊆ m∗pc and the fact that ec preserves the order, see [14, corollary 2.1]. � theorem 3.15. let x be a p -space and f, an ec-filter on x. then f is an ec-ultrafilter if and only if it is a zc-ultrafilter. proof. (⇒) : by [5, 4k(7), 6m(1), 16o], every p-space is strongly zerodimensional (see also [15, proposition 2.12]). by [5, 7l], we have op = mp for every p ∈ βx. therefore, opc = op∩cc(x) = mp∩cc(x) = mpc (note, βx = β0x). let f be an ec-ultrafilter on x. then e−1c (f) is a maximal ideal in c∗c (x), see [14, proposition 2.14]. therefore, e−1c (f) = m∗pc for some p ∈ βx. by lemma 3.14, we have f = ec(e−1c (f)) = ec(m ∗p c ) = zc[o p c ] = zc[m p c ]. since mpc is a maximal ideal in cc(x), f is a zc-ultrafilter. (⇐) : suppose that f is a zc-ultrafilter. then z−1c [f] is a maximal ideal in cc(x). so z −1 c [f] = mpc for some p ∈ βx. therefore, f = zc[z−1c [f]] = zc[m p c ] = ec(m ∗p c ). since m∗pc is a maximal ideal in c ∗ c (x), f is an ec-ultrafilter. � corollary 3.16. for a strongly zero-dimensional space x and p ∈ βx, m∗pc is the only ec-ideal in c ∗ c (x) containing o ∗p c . © agt, upv, 2022 appl. gen. topol. 23, no. 1 89 a. veisi proof. let j be an ec-ideal in c ∗ c (x) which contains o ∗p c . then e−1c (ec(o ∗p c )) ⊆ e−1c (ec(j)) = j. by lemma 3.14, ec(m∗pc ) = ec(o∗pc ) and therefore m∗pc = e −1 c (ec(m ∗p c )) = e −1 c (ec(o ∗p c )) ⊆ j. so m∗pc = j, and we are through. � acknowledgements. the author is grateful to the referee for useful comments and recommendations towards the improvement of the paper. references [1] f. azarpanah, o. a. s. karamzadeh, z. keshtkar and a. r. olfati, on maximal ideals of cc(x) and uniformity its localizations, rocky mountain journal of mathematics 48, no. 2 (2018), 345–382. 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[16] a. veisi and a. delbaznasab, metric spaces related to abelian groups, appl. gen. topol. 22, no. 1 (2021), 169–181. © agt, upv, 2022 appl. gen. topol. 23, no. 1 90 @ appl. gen. topol. 23, no. 1 (2022), 201-212 doi:10.4995/agt.2022.14778 © agt, upv, 2022 topological krasner hyperrings with special emphasis on isomorphism theorems manoranjan singha and kousik das department of mathematics, university of north bengal, darjeeling-734013, india (manoranjan.math@nbu.ac.in, das.kousik1991@nbu.ac.in) communicated by a. tamariz-mascarúa abstract krasner hyperring is studied in topological flavor. it is seen that krasner hyperring endowed with topology, when the topology is compatible with the hyperoperations in some sense, fruits new results comprising algebraic as well as topological properties such as topological isomorphism theorems. 2020 msc: 16y20; 22a30. keywords: topological hyperring; quotient hyperring; topological isomorphism. 1. introduction and relevant literature the theory of hyperalgebra which is extended in this article gets birth in the year 1934 but it gets acquaintance during the last two decades and so far it is wide in various branches of mathematics including physics and chemistry: geometry [25, 26], graph theory [5, 27], codes [31, 10], cryptography [4], probability [20], automata [19], artificial intelligence [17], lattice theory [14, 15], chemistry [1, 12], physics [8, 21], and all credits for these go to the hyperoperations. it is investigated that how the difference between hyperoperation and binary operation affects on the theory of topological krasner hyperring, especially on the topological isomorphisms. hyperring, introduced by krasner [16] is one of the most general structures so far in the literature that satisfies the ring-like axioms. later, many mathematicians, like ameri [3, 2], massouros received 11 december 2020 – accepted 10 december 2021 http://dx.doi.org/10.4995/agt.2022.14778 https://orcid.org/0000-0003-4875-4330 https://orcid.org/0000-0001-8011-3635 m. singha and k. das [18], spartalis [29], davvaz [7], stratigopoulos [30], kemprasit [24] extended this field of study. in literature, a topological ring is a combination of two structures, namely a topological space and a ring. these two structures are connected in such a way that one affects another. in this paper, we generalize this concept as topological krasner hyperring, supported by illustrative examples. we also present an example that makes difference between the classical and the new concept. in the later part, we use the notion of complete parts to study isomorphism theorems on hyperrings. let’s begin with some basic definitions and results which will be used as ready references in the sequel. on a nonempty set h, a hyperoperation is a function + : h × h → p∗(h), where p∗(h) is the collection of nonempty subsets of h. for nonempty subsets a,b of h and x ∈h, consider a + b = ⋃ a∈a, b∈b a + b, x + a = {x} + a and a + x = a + {x}. a krasner hyperring is an algebraic structure (h, +, ·) satisfying the following axioms: (1) (h, +) is a canonical hypergroup, i.e., + is a hyperoperation on h such that (a) for every x,y,z ∈h, x + (y + z) = (x + y) + z, (b) for every x,y ∈h, x + y = y + x, (c) there exists 0 ∈h such that 0 + x = {x} for every x ∈h, (d) for every x ∈h, there exists a unique x′ ∈h such that 0 ∈ x+x′, (write −x instead of such x′), (e) z ∈ x + y implies y ∈−x + z and x ∈ z −y; (2) (h, ·) is a semigroup having zero as a bilaterally absorbing element, i.e., 0 ·x = x · 0 = 0 for all x ∈h. (3) the multiplication, ‘·’ is distributive with respect to the hyperoperation +. throughout this context, hyperring stands for krasner hyperring. the following elementary facts are the consequences of the above axioms: −(−x) = x, for any nonempty subset x of h, −x = {−x : x ∈ x} and −(x+y) = −x−y. also, for all a,b,c,d ∈h, (a+b)·(c+d) ⊆ a·c+b·c+a·d+b·d. a nonempty subset k of the hyperring h is said to be a subhyperring of h if (k, +, ·) is itself a hyperring. the subhyperring k is a hyperideal of h if h ·k ∈ k and k ·h ∈ k for all h ∈ h and k ∈ k. the subhyperring k is said to be normal in h if and only if h + k−h ⊆ k for all h ∈ h. for a normal hyperideal k of a hyperring h the following results hold: (1) x + k = k + x for all x ∈h, (2) (x + k) + (y + k) = x + y + k for all x,y ∈h, (3) if x,y ∈h, x + y + k = z + k for all z ∈ x + y, (4) x + k = y + k for all y ∈ x + k. let k1,k2 be two hyperideals of a hyperring h such that k2 is normal in h. then, (1) k1 ∩k2 is a normal hyperideal of k1, © agt, upv, 2022 appl. gen. topol. 23, no. 1 202 topological krasner hyperrings with special emphasis on isomorphism theorems (2) k2 is a normal hyperideal of k1 + k2. for a normal hyperideal k of a hyperring h, define an equivalence relation k∗ as follows: x ≡ y(modk) if and only if (x−y) ∩k 6= φ. then, for all x ∈h, k∗(x) = x+k. the collection [h : k∗] = {k∗(x) : x ∈h} of all equivalence classes forms a hyperring together with the hyperoperations ⊕ and multiplication � defined as follows: k∗(x) ⊕k∗(y) = {k∗(z) : z ∈k∗(x) + k∗(y)}, k∗(x) �k∗(y) = k∗(x ·y). a homomorphism from a hyperring (h, +, ·) into another hyperring (h′, +′, ·′) is a map f : h→h′ such that f(x+y) ⊆ f(x)+′f(y) and f(x·y) = f(x)·′f(y), for all x,y ∈ h. a homomorphism f from (h, +, ·) into (h′, +′, ·′) is said to be a good homomorphism if f(x + y) = f(x) +′ f(y), for all x,y ∈ h. an onto homomorphism is called epimorphism. an isomorphism from (h, +, ·) onto (h′, +′, ·′) is a bijective good homomorphism and if such map exists, then write h ∼= h′. if f is an isomorphism from h onto h′, then f−1 is an isomorphism from h′ onto h. for a homomorphism f : h → h′, the kernel of the homomorphism is defined as ker f = {x ∈ h : f(x) = 0h′}. it is seen (example 1.2 [24]) that the kernel of a homomorphism may be empty, but, if it is nonempty (i.e., ker f 6= φ), then the following results ([24]) hold: (1) f(0h) = 0h′ ; (2) f(−x) = −f(x), for all x ∈h; (3) ker f is a hyperideal of h; (4) if f is injective, then ker f = {0h}; (5) if f is a good homomorphism and ker f = {0h}, then f is injective; (6) if f is a good homomorphism, f(h) is a subhyperring of h′. note that, if f is onto, then f(x) = 0h′ for some x ∈h, i.e., ker f 6= φ. a nonempty subset c of a hyperring h is said to be a complete part of h if for any nonzero natural number n and for all x1,x2, · · · ,xn of h, the following implication holds: c ∩ n σ i=1 xi 6= φ ⇒ n σ i=1 xi ⊆ c. let a and b be two nonempty subsets of the hyperring h such that a is a complete part of h and x ∈h. then, (1) −x + x + a = x−x + a = a; (2) −a is a complete part of h; (3) x + a and a + x are complete parts; (4) b ⊆−x + a if and only if x + b ⊆ a. for more details about hyperring we refer to [16, 9, 24, 7]. © agt, upv, 2022 appl. gen. topol. 23, no. 1 203 m. singha and k. das 2. topological hyperring and isomorphism theorems to define topological hyperring, the codomain of the hyperoperation is to be topologized, but there is no straightforward way to obtain such topology. so, let’s consider the following. lemma 2.1 ([13]). for a topological space (h,τ), the family b consisting of the sets sv = {u ∈ p∗(h) : u ⊆ v}, where v ∈ τ is a base for a topology τ∗ on p∗(h). now, we are in a situation to define topological hyperring. definition 2.2. let (h, +, ·) be a krasner hyperring endowed with some topology τ. then, h is said to be a topological krasner hyperring, denoted by (h, +, ·,τ), if with respect to the product topology on h×h and the topology τ∗ on p∗(h), the following maps (th1) (x,y) 7→ x + y from h×h to p∗(h); (th2) x 7→−x from h to h; (th3) (x,y) 7→ x ·y from h×h to h; are continuous. for the ease of writing throughout this context, let’s write topological hyperring instead of topological krasner hyperring. every topological ring is a topological hyperring. here, we consider some other examples. example 2.3. consider the hyperring (r, +, ·), where r = {0, 1, 2}, the hyperoperation + and the binary operation ‘·’ are defined as follows + 0 1 2 0 {0} {1} {2} 1 {1} {0,2} {1} 2 {2} {1} {0} · 0 1 2 0 0 0 0 1 0 1 2 2 0 2 0 let r be endowed with the topology τ = {φ,{1},{0, 2},r}. then, (r, +, ·,τ) is a topological hyperring. example 2.4. consider the unit interval [0, 1] as a subspace of r with standard topology. for x,y ∈ [0, 1], let + be the hyperoperation defined as follows x + y = { {max{x,y}}, if x 6= y; [0,x], if x = y. then ([0, 1], +, ·) is a topological hyperring, where ‘·’ is the usual multiplication on r. remark 2.5. unlike in topological rings, some results may fail to hold in the new setting. for, in the above example 2.4, 1 2 ∈ [0, 1] and [0, 1 2 ) is open in [0, 1], but 1 2 ⊕ [0, 1 2 ) = {1 2 }, which is not open in [0, 1]. lemma 2.6. in a topological hyperring (h, +, ·,τ), the following results hold. (1) for a ∈h, the map ta(x) = a + x from h to p∗(h) is continuous. © agt, upv, 2022 appl. gen. topol. 23, no. 1 204 topological krasner hyperrings with special emphasis on isomorphism theorems (2) let u be open and a complete part of h. then, for a ∈h, a + u is an open subset of h. moreover, if a is any subset of h, then a + u is an open subset of h. (3) the map i(x) = −x from h to h is a homeomorphism. proof. (1) being a restriction of the map + : h×h→p∗(h), ta is continuous for any a ∈h. (2) consider the basic open set su of p∗(h) and a ∈h. then, t−1−a (su ) = {x ∈h : t−a(x) ∈ su} = {x ∈h : −a + x ⊆ u} suppose s = {x ∈ h : −a + x ⊆ u}. take y ∈ s, then −a + y ⊆ u. so, y ∈ 0 + y ⊆ (a + (−a)) + y ⊆ a + (−a + y) ⊆ a + u. again, if z ∈ a + u, then −a + z ⊆ −a + a + u = u, which implies z ∈ s. hence, t−1−a (su ) = a + u, which is open in h. for a ⊆h, a + u = ⋃ a∈a (a + u), which is also open for being arbitrary union of open sets. (3) the inverse of an element in the canonical hypergroup (h, +) is unique. so, the map i on h is a homeomorphism. � remark 2.7. the open subsets in the above example 2.3 are complete parts. theorem 2.8. in a topological hyperring (h, +, ·,τ), the following straightforward results easily hold. (1) for a neighborhood v of zero, there exist a neighborhood u of zero and a neighborhood w of x, where x ∈ h such that u + u ⊆ v and u ·w ⊆ v . (2) if v is any neighborhood of zero, then −v is also a neighborhood of zero. (3) every neighborhood u of zero contains a symmetric neighborhood of zero (i.e., u ∩ (−u) ). (4) if u is a neighborhood of zero and n > 1, then there exists a symmetric neighborhood v of zero such that v + v + · · · + v︸︷︷︸ n terms ⊆ u. proof. the proofs are straightforward. � any subhyperring of a topological hyperring is also a topological hyperring when considering the relative topology on it, such subhyperrings are called topological subhyperrings. let i be a hyperideal of a hyperring (h, +, ·) and h/i = {x + i : x ∈ h}. then, (h/i,⊕,�) is a hyperring, called quotient hyperring of h by i, where (x + i) ⊕ (y + i) = {z + i : z ∈ x + y} and (x + i) � (y + i) = (x ·y) + i for x,y ∈h. let i be a normal hyperideal of a topological hyperring (h, +, ·,τ) and φi be the canonical map of h onto h/i, defined by φi(x) = x + i for x ∈ h. let’s topologize h/i by declaring the map φi to be quotient, i.e., a subset a © agt, upv, 2022 appl. gen. topol. 23, no. 1 205 m. singha and k. das of h/i is open in h/i if and only if φ−1i (a) is open in h. this topology is called the quotient topology on h/i and denoted by τφ. theorem 2.9. let i be a normal hyperideal of a topological hyperring (h, +, ·,τ) such that the members of τ are complete parts and φi be the above mentioned map, then the following results hold. (1) φi is continuous, open good epimorphism. (2) (h/i,⊕,�,τφ) is a topological hyperring. (3) if v is a fundamental system of neighborhoods of zero (i.e., 0) in h, then {φi(v ) : v ∈ v} is a fundamental system of neighborhoods of zero (i.e., 0 + i = i) for the quotient topology τφ of h/i. proof. (1) for x,y ∈h, φi(x + y) = {z +i : z ∈ x + y} = (x +i)⊕(y +i) = φi(x)⊕φi(y). φi is continuous as per the definition of the quotient topology. let o be an open subset of h. claim that φ−1i (φi(o)) = o + i. to prove o + i ⊆ φ−1i (φi(o)), take y ∈ o + i. then, y ∈ p + i for some p ∈ o, which implies y + i = p + i. thus, y ∈ φ−1i (φi(o)). now, take x ∈ φ −1 i (φi(o)), then x + i ∈ φi(o), which implies x + i = r + i for some r ∈ o. thus, x ∈ r + i ⊆ o + i. hence, φi is an open map (by (2) of lemma 2.6). (2) φi×φi is the map from h×h to h/i×h/i defined by (φi×φi)(x,y) = (φi(x), φi(y)) for all (x,y) ∈ h×h. as φi is a continuous open surjection, so is φi × φi. then, ⊕◦ (φi × φi) = φi ◦ f and �◦ (φi × φi) = φi ◦ f, where f = + and · respectively. so, the continuity of f implies both ⊕ and � are continuous (by theorem 5.3 of [33, p. 33]). let i : h/i →h/i be defined by i(x + i) = (−x) + i for x ∈h. also, i ◦ φi = φi ◦− is continuous, as − is continuous; hence i is continuous (by theorem 5.3 of [33, p. 33]). (3) for every neighborhood u of zero in h/i, φ−1i (u) is a neighborhood of zero in h, so there exists v ∈ v such that v ⊆ φ−1i (u). then, φi(v ) ⊆ φi(φ −1 i (u)) = u. � remark 2.10. it is clear from the above theorem 2.9 that if a is some open subset of h/i, then there exists open subset a of h such that a = a/i. theorem 2.11. let b be a subhyperring and i be a normal hyperideal of a topological hyperring (h, +, ·,τ) such that i ⊆ b. if the members of τ are complete parts, then the quotient topology of b/i is identical with the topology induced on the subhyperring b/i of h/i by the quotient topology of h/i. proof. let φb,i : b →b/i and φh,i : h→h/i be the canonical surjections. let o be open for the quotient topology of b/i. then, φ−1b,i(o) is open in b and φ−1b,i(o) = b∩q for some open subset q of h. claim that o = b/i∩φh,i(q). for, clearly o ⊆ b/i ∩ φh,i(q). for the converse, take α ∈ b/i ∩ φh,i(q). then, α = b + i for some b ∈b and α = q + i for some q ∈q, which implies q ∈ b + i ⊆ b + i = b as i ⊆ b. consequently, q ∈ b∩q = φ−1b,i(o), so α = q + i ∈ o. now suppose r be open in b/i for the topology on b/i induced by the quotient topology on h/i. then, r = b/i ∩ s for some open subset s of © agt, upv, 2022 appl. gen. topol. 23, no. 1 206 topological krasner hyperrings with special emphasis on isomorphism theorems h/i. so, φ−1b,i(r) = b∩ φ −1 h,i(s), which is open in b. hence, r is open for the quotient topology of b/i. � corollary 2.12. let b be a subhyperring and i be a normal hyperideal of a topological hyperring (h, +, ·,τ). if the members of τ are complete parts, then the quotient topology on (b +i)/i is identical with the topology induced by the quotient topology of h/i. let’s define topological isomorphism and prove some topological isomorphism theorems. definition 2.13. a homomorphism f between two topological hyperrings satisfying ker f 6= φ is said to be a topological homomorphism if it is a continuous open mapping. if f is a good, one to one and onto topological homomorphism, then it is called a topological isomorphism and in this case the hyperrings are topologically isomorphic. example 2.14. consider the topological hyperring ([0, 1], +, ·,τu) as in example 2.4, where τu is the subspace topology induced from r with standard topology. [0, 1) being a normal hyperideal of [0, 1], the quotient hyperring [0, 1]/[0, 1) is a topological hyperring with respect to the quotient topology induced by canonical projection φ : [0, 1] → [0, 1]/[0, 1). now, consider x = {0, 1} together with the hyperoperation ⊕ and the binary operation � defined as follows: ⊕ 0 1 0 {0} {1} 1 {1} {0,1} � 0 1 0 0 0 1 0 1 then, (x,⊕,�,τ′) is a topological hyperring, where τ′ = {φ,{0},x}. if we define ψ : [0, 1]/[0, 1) →{0, 1} by ψ(x + [0, 1)) = [x] = the greatest integer less than or equal to x, then, ψ is a topological isomorphism. example 2.15. for a hyperring (r, +, ·) and for some positive integer n, the collection mn(r) of all n × n matrices over r forms a krasner hyperring with respect to the hyperaddition ⊕ and multiplication � defined as, for a = (aij),b = (bij) ∈ mn(r), a⊕b = {c ∈ mn(r) : c = (cij),cij ∈ aij +bij} and a � b = (aij · bij). now, replace (r, +, ·) by the topological hyperring ([0, 1], +, ·,τu) of example 2.4, where τu is the subspace topology induced from r with standard topology. if we topologize mn([0, 1]) by identifying it with [0, 1]n 2 , then, mn([0, 1]) is a topological krasner hyperring. in a similar manner, we can also obtain the topological krasner hyperring mn(mn([0, 1])). now, let a =   a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44   be an element of m4([0, 1]) and consider the map f : m4([0, 1]) → m2(m2([0, 1])) defined by © agt, upv, 2022 appl. gen. topol. 23, no. 1 207 m. singha and k. das f(a) =   ( a11 a12 a21 a22 ) ( a13 a14 a23 a24 ) ( a31 a32 a41 a42 ) ( a33 a34 a43 a44 )  . then, f is a topological isomorphism. theorem 2.16. let h1 and h2 be two topological hyperrings such that the open subsets of h1 are complete parts. let f be an open, continuous and good topological homomorphism from h1 onto h2 such that ker f is normal in h1. then, h1/ ker f and h2 are topologically isomorphic. proof. clearly, ψ : h1/ ker f → h2 defined by ψ(x + ker f) = f(x), for all x ∈ h1 is the required isomorphism[24, 7]. so, it is only required to prove that ψ is open and continuous. consider an open subset u of h1/ ker f. then, by remark 2.10 there exists an open subset u of h1 such that u = u/ ker f. so, ψ(u) = ψ(u/ ker f) = f(u), which is open in h2. to prove ψ continuous, consider an open subset v of h2. then, ψ−1(v ) = {x + ker f : ψ(x + ker f) ∈ v} = {x + ker f : f(x) ∈ v} = f−1(v )/ ker f, which is open in h1/ ker f as f−1(v ) is open in h1. � theorem 2.17. let f be a homomorphism from a topological hyperring (h, +, ·,τ) into a topological hyperring (h′, +′, ·′,τ′) such that ker f(6= φ) is normal in h and members of τ are complete parts. let j be a normal hyperideal of h such that j ⊆ ker f. then, the homomorphism g from h/j to h′ satisfying g ◦ φj = f is continuous (open, a topological homomorphism) if and only if f is. in particular, if j = ker f, g is a topological isomorphism (good monomorphism) if and only if f is a topological good epimorphism (good homomorphism). proof. continuity and openness are consequences of theorem 5.3 of [33, p. 33] and theorem 2.9. to prove ker g 6= φ, take x ∈ ker f. then, f(x) = 0h′ , which implies g(x + j) = 0h′ . so, x + j ∈ ker g. for the second part, suppose j = ker f. let f be a topological good homomorphism. then, for x,y ∈h, g((x + j) ⊕ (y + j)) = g({z + j : z ∈ x + y}) = {(g ◦ φj )(z) : z ∈ x + y} = f(x + y) = f(x) +′ f(y) = (g ◦ φj )(x) +′ (g ◦ φj )(y) = g(x + j) +′ g(y + j) and g((x + j) � (y + j)) = g((x · y) + j) = (g ◦ φj )(x · y) = f(x · y) = f(x) ·′ f(y) = (g ◦ φj )(x) ·′ (g ◦ φj )(y) = g(x + j) ·′ g(y + j). now, g(0h + j) = (g ◦ φj )(0h) = f(0h) = 0h′ . therefore, 0h + j ∈ ker g. © agt, upv, 2022 appl. gen. topol. 23, no. 1 208 topological krasner hyperrings with special emphasis on isomorphism theorems let x ∈h be such that g(x+j) = 0h′ . then, f(x) = 0h′ and hence x ∈ ker f, which implies x + ker f = 0h + ker f. clearly, g is surjective if and only if f is. for the converse, suppose g is a topological good monomorphism. then, for x,y ∈h, f(x + y) = (g ◦ φj )(x + y) = g(φj (x) ⊕ φj (y)) = g((x + j) ⊕ (y + j)) = g(x + j) +′ g(y + j) = (g ◦ φj )(x) +′ (g ◦ φj )(y) = f(x) +′ f(y). and f(x · y) = (g ◦ φj )(x · y) = g((x · y) + j) = g((x + j) � (y + j)) = g(x + j) ·′ g(y + j) = (g ◦ φj )(x) ·′ (g ◦ φj )(y) = f(x) ·′ f(y). � corollary 2.18. let i,j be normal hyperideals of a topological hyperring h such that j ⊆ i and the open subsets of h are complete parts. then, the following results hold: (1) the map f : h/j →h/i defined by f(x + j) = x + i is a topological good epimorphism. (2) (h/j)/(i/j) and h/i are topologically isomorphic. proof. (1) here, f ◦ φj = φi is an open, continuous good epimorphism from h → h/i by theorem 2.9. then, by theorem 2.17, f is a topological good epimorphism. (2) as f is surjective, so, ker f 6= φ and ker f = {x + j : f(x + j) = 0 + i} = {x + j : f ◦ φj (x) = i} = {x + j : φi(x) = i} = {x + j : x + i = i} = {x + j : x ∈i} = i/j . then, by theorem 2.17, g : (h/j)/(i/j) → h/i satisfying g ◦ φi/j = f is the required topological isomorphism. � corollary 2.19. let i,j be hyperideals of a topological hyperring h such that • j is normal and i is open in h; • open subsets of h are complete parts. then, i/(i∩j) is topologically isomorphic to (i + j)/j . © agt, upv, 2022 appl. gen. topol. 23, no. 1 209 m. singha and k. das proof. here, f : i → (i + j)/j defined by f(x) = x + j , for all x ∈ i is an epimorphism such that ker f = {x ∈i : f(x) = 0 + j} = {x ∈i : x + j = j} = {x ∈i : x ∈j} = i∩j 6= φ. as i is open in h, by theorem 22.1 [22, p. 140] f is a quotient map and hence, f is a topological good epimorphism (by theorem 2.9). so, by theorem 2.17, g : i/(i∩j) → (i +j)/j satisfying g◦φi∩j = f is the required topological isomorphism. � theorem 2.20. let k be a complete part dense subhyperring of a topological hyperring h, and let i be a closed normal hyperideal of k, i its closure in h. if the open subsets of h are complete parts, then g(x +i) = x +i is a topological isomorphism from k/i to the dense subhyperring (k + i)/i of h/i. proof. as the open subsets of h are complete parts, lemma 3.2 ([11]), continuity of multiplication map and lemma 3.9 ([28]) imply i is a normal hyperideal of h. here, the kernel of the restriction φi|k : k→ (k + i)/i is ker φi|k = {x ∈k : φi(x) = i} = {x ∈k : x + i = i} = k∩i = i 6= φ and g satisfies g ◦ φi = φi|k. φi|k is a continuous good epimorphism, so, by theorem 2.17, g is a continuous isomorphism from k/i to (k +i)/i. as k is dense in h and φi is continuous, φi(k) = (k + i)/i is dense in h/i. consider an open subset o of k/i and let φ−1i (o) = p . then, remark 2.10 and (1) of theorem 2.9 imply g(o) = φi(p) and p + i = p . p being open in k, there exists an open subset u of h such that p = u ∩k. claim that (u + i) ∩ k = p . to prove (u + i) ∩ k ⊆ p , let k ∈ k such that k ∈ (u + i). then, there exist u ∈ u and h ∈ i such that k ∈ u + h. u being a neighborhood of u, there exists a symmetric neighborhood v of zero such that u + v ⊆ u. then, (h + v ) ∩i 6= φ, so there exists v ∈ v such that (h+v)∩i 6= φ. as i is a closed subset of h, i is also a complete part of h and hence (h+v) ⊆i. consequently, u+h ⊆ (u−v)+(v+h) ⊆ (u+v )+i ⊆ u +i. as k is a complete part, u + h ⊆ k and hence, by proposition 2.3 ([28]) u + h ⊆ (u + i) ∩k = (u + i) ∩ (k + i) = (u ∩k) + i = p + i = p . the reverse inclusion also follows from proposition 2.3 ([28]). therefore, g(o) = φi(p) = φi(u)∩((k+i)/i), an open subset of (k+i)/i, for if x+i ∈ φi(u) where x ∈ k, then x ∈ φ−1 i (φi(u)) ∩k = (u + i) ∩k = p , which implies x + i ∈ φi(p) = g(o). � conclusion. as in algebra, distinguishing and classifying topological hyperrings is of great importance in the theory of topological krasner hyperring too. © agt, upv, 2022 appl. gen. topol. 23, no. 1 210 topological krasner hyperrings with special emphasis on isomorphism theorems this article provides basic tools for this task too. in short, the essence of this article is to show identities in difference between binary and hyper operations. it is hoped that our investigations in the present article will throw much light towards extension of the theory of hyperalgebra and pave the way for further research in this direction. acknowledgements. the authors are thankful to the referee for suggesting a few revisions and also pointing out some issues that enabled us to complete this paper. references [1] m. al tahan and b. davvaz, electrochemical cells as experimental verifications of n-ary hyperstructures, matematika 35, no. 1 (2019), 13–24. 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[33] s. warner, topological rings, north-holland, 1993. © agt, upv, 2022 appl. gen. topol. 23, no. 1 212 @ appl. gen. topol. 21, no. 2 (2020), 285-294 doi:10.4995/agt.2020.13126 c© agt, upv, 2020 the depth and the attracting centre for a continuous map on a fuzzy metric interval taixiang sun a, lue li a, guangwang su a, caihong han a,∗ and guoen xia b a college of information and statistics, guangxi university of finance and economics, nanning, 530003, china. (stx1963@163.com,li1982lue@163.com,s1g6w3@163.com,h198204c@163.com) b college of of business administration, guangxi university of finance and economics, nanning, 530003, china. (x3009h@163.com) communicated by d. georgiou abstract let i be a fuzzy metric interval and f be a continuous map from i to i. denote by r(f), ω(f) and ω(x, f) the set of recurrent points of f, the set of non-wandering points of f and the set of ωlimit points of x under f, respectively. write ω(f) = ∪x∈iω(x, f), ωn+1(f) = ∪x∈ωn(f)ω(x, f) and ωn+1(f) = ω(f|ωn(f)) for any positive integer n. in this paper, we show that ω2(f) = r(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2. 2010 msc: 54e35; 54h25. keywords: fuzzy metric interval; attracting centre; depth. 1. introduction by extending the notion of menger space to the fuzzy setting, kramosil and michalek [7] obtained the notion of fuzzy metric space with the help of continuous t-norms. in order to obtain a hausdorff topology in fuzzy metric ∗corresponding author. project supported by nnsf of china (11761011, 71862003) and nsf of guangxi (2018gxnsfaa294010) and sf of guangxi university of finance and economics (2019qnb10). received 13 february 2020 – accepted 11 may 2020 http://dx.doi.org/10.4995/agt.2020.13126 t. sun, l. li, g. su, c. han and g. xia spaces, george and veeramani [1] modified the notion given by kramosil and michalek in a slight but appealing way. recently many authors studied several properties of the hausdorff fuzzy metric spaces (see [6, 8, 11]) and introduced and investigated the different types of fuzzy contractive maps and obtained a lot of fixed point theorems (see [3, 4, 9, 10, 13, 14, 15, 16]). until now, there are little of works that investigates some properties of discrete dynamical systems on fuzzy metric spaces. in this paper, we introduce the notion of fuzzy metric interval and study the depth and the attracting centre for a continuous map on a fuzzy metric interval. the rest of this paper is organized as follows. in section 2 we give some definitions and notations. in section 3 we study the depth for a continuous map on a fuzzy metric interval. in section 4 we study the depth of the attracting centre for a continuous map on a fuzzy metric interval. 2. preliminaries throughout the paper, let n be the set of all positive integers and n! = n ∪ {0}. firstly, we recall the basic definitions and the properties about fuzzy metric spaces. definition 2.1 (see [12]). we say that a continuous map ξ : [0, 1]2 −→ [0, 1] is a continuous t-norm if for any a, b, c, d ∈ [0, 1], the following conditions hold: (1) ξ(a, b) = ξ(b, a). (2) ξ(a, b) ≤ ξ(c, d) for a ≤ c and b ≤ d. (3) ξ(ξ(a, b), c) = ξ(a, ξ(b, c)). (4) ξ(a, 0) = 0 and ξ(a, 1) = a. for a, b ∈ [0, 1], we will use the notation a∗b instead of ξ(a, b). for example, ξ(a, b) = min{a, b}, ξ(a, b) = ab and ξ(a, b) = max{a + b − 1, 0} are the most commonly used t-norms. in the present paper, we also use the following definition of the fuzzy metric space. definition 2.2 (see [1]). we say that a triple (x, m, ∗) is a fuzzy metric space if x is a nonempty set, ∗ is a continuous t-norm and m is a map defined on x2×(0, +∞) into [0, 1] and for any x, y, z ∈ x and s, t ∈ (0, +∞), the following conditions hold: (1) m(x, y, t) > 0. (2) m(x, y, t) = 1 (for any t > 0) ⇐⇒ x = y. (3) m(x, y, t) = m(y, x, t). (4) m(x, z, t + s) ≥ m(x, y, t) ∗ m(y, z, s). (5) mxy : (0, +∞) −→ [0, 1] is a continuous mapping ( where mxy(t) = m(x, y, t)). remark 2.3. (1) mxy is a non-decreasing function on (0, ∞) for all x, y ∈ x (see [2]). (2) m is a continuous function on x × x × (0, +∞) (see [11]). if (x, m, ∗) is a fuzzy metric space, then we will say that (m, ∗), or simply m, is a fuzzy metric on x. in [1], george and veeramani showed that every c© agt, upv, 2020 appl. gen. topol. 21, no. 2 286 the depth and the attracting centre for map fuzzy metric m on x generates a topology τm on x which has as a base the family of open sets of the form {bm(x, ε, t) : x ∈ x, 0 < ε < 1, t > 0} , where bm(x, ε, t) = {y ∈ x : m(x, y, t) > 1 − ε} for all x ∈ x, ε ∈ (0, 1) and t > 0, and (x, τm) is a hausdorff space. definition 2.4 (see [5]). let (x, m, ∗) be a fuzzy metric space. we say that a sequence of points xn ∈ x converges to x ( denoted by xn −→ x) ⇐⇒ limn−→+∞ m(xn, x, t) = 1 (for any t > 0), i.e. for each δ ∈ (0, 1) and t > 0, there exists n ∈ n such that m(xn, x, t) > 1 − δ for all n ≥ n. definition 2.5. (1) we say that a fuzzy metric space (x, m, ∗) is compact if (x, τm) is compact. we say that a subset a of x is compact if a as a fuzzy metric subspace is compact. (2) we say that a fuzzy metric space (x, m, ∗) is connected if (x, τm) is connected. we say that a subset a of x is connected if a as a fuzzy metric subspace is connected. by [6] we know that: (1) (x, m, ∗) is a compact fuzzy metric space ⇐⇒ each sequence of points in x has a convergent subsequence. (2) if (x, m, ∗) is a compact fuzzy metric space and a is a subset of x, then a is compact ⇐⇒ a is closed. (3) if (x, m, ∗) is a compact fuzzy metric space, then for any x, y ∈ x with x ∕= y, there exist b(x, ε1, t1) and b(y, ε2, t2) with ε1, ε1 ∈ (0, 1) and t1, t2 ∈ (0, +∞) such that b(x, ε1, t1) ∩ b(y, ε2, t2) = ∅. definition 2.6. let (x, m, ∗) be a compact fuzzy metric space and a, b ∈ x. we say that x is a fuzzy metric interval with ends a and b if the following conditions hold: (1) m(a, x, t) ≥ m(a, b, t) for any x ∈ x and t > 0. (2) for any x, y ∈ x with x ∕= y, we have m(a, x, t) < m(a, y, t) for any t > 0, which is denoted by x > y, or m(a, x, t) > m(a, y, t) for any t > 0, which is denoted by x < y . (3) for any x, y ∈ x with m(a, x, t) ≥ m(a, y, t) for any t > 0, set {z ∈ x : m(a, x, t) ≥ m(a, z, t) ≥ m(a, y, t) for any t > 0}, which is denoted by [x; y], is a connected subset of x. (4) if y ∈ b(x, ε, t) for some x ∈ x and some ε ∈ (0, 1) and some t > 0, then [y; x] ⊂ b(x, ε, t) if y ≤ x or [x; y] ⊂ b(x, ε, t) if y ≥ x. write [x; y) = [x; y] − {y} ≡ {z ∈ x : m(a, x, t) ≥ m(a, z, t) > m(a, y, t) and (x; y) = [x; y) − {x} ≡ {z ∈ x : m(a, x, t) > m(a, z, t) ≥ m(a, y, t). remark 2.7. (1) [a; b] = x. (2) let x, y ∈ [a; b]. if m(a, x, t) = m(a, y, t) for some t > 0, then x = y. example 2.8. let i = [a, b] be a compact interval of r= (−∞, +∞). define s ∗ t = st for any s; t ∈ [0, 1], and let md : i × i × (0, ∞) −→ [0, 1] such that for any x, y ∈ i and t > 0, md(x, y, t) = t t + |x − y| . c© agt, upv, 2020 appl. gen. topol. 21, no. 2 287 t. sun, l. li, g. su, c. han and g. xia then (i, md, ∗) is a fuzzy metric interval. further, it was proven [1] that the topologies induced by (i, d) with d(x, y) = |x−y| for any x, y ∈ i and (i, md, ∗) are the same. let (i, m, ∗) be a fuzzy metric interval and a ⊂ i. denote by a the closure of a in (i, τm). let c 0(i) denote the set of all continuous maps on i. for any f ∈ c0(i) and x ∈ i, write f0(x) = x and fn = f ◦ fn−1 for any n ∈ n. we also introduce a number of notations: o(x, f) = {fn(x) : n ∈ n!}. λ(x, f) = ∪∞n=1f −n(x). fix(f) = {x : f(x) = x}. p(f) = {x : there exists some n ∈ n such that fn(x) = x}. ω(x, f) = {y : there exists a sequence of positive integers k1 < k2 < · · · such that lim n−→∞ m(fkn(x), y, t) = 1 (for any t > 0)}. r(f) = {x : x ∈ ω(x, f)}. uγ(f) = {y : there exist a connected component j of i − {y}, a point x ∈ j, a sequence of points x1, x2, · · · ∈ λ(x, f) ∩ j and a sequence of positive integers k1 < k2 < · · · such that fkn(x) ∈ j for any n ∈ n and lim n−→∞ m(xn, y, t) = lim n−→∞ m(fkn(x), y, t) = 1(for any t > 0)}. ω(f) = {y : there exist a sequence of points x1, x2, · · · ∈ i and a sequence of positive integers k1 ≤ k2 ≤ · · · such that lim n−→∞ m(xn, y, t) = lim n−→∞ m(fkn(xn), y, t) = 1 (for any t > 0)}. p(f), r(f), uγ(f) and ω(f) are called the set of periodic points, the set of recurrent points, the set of unilateral γlimit points and the set of nonwandering points of f, respectively. o(x, f), λ(x, f) and ω(x, f) are called the orbit of x under f, the reverse orbit of x under f and the set of ω-limit points of x under f, respectively. remark 2.9. let (i, m, ∗) be a fuzzy metric interval and f ∈ c0(i). then the following statements hold: (1) f(ω(f)) ⊂ ω(f) and ω(f) is closed. (2) fix(f) ⊂ p(f) ⊂ r(f) ⊂ ω(f) ⊂ ω(f). (3) aγ(f) ⊂ ω(f) and r(f) ⊂ ωn(f) for any ∈ n. definition 2.10. let (i, m, ∗) be a fuzzy metric interval and f ∈ c0(i). (1) for any a ⊂ i, write ω(a) = ∪x∈aω(x, f) and ω1(f) = ω(f) = ω(i) and ωn+1(f) = ∪x∈ωn(f)ω(x, f) for any n ∈ n. the minimal m ∈ n ∪ {∞} such that ωm(f) = ωm+1(f) is called the depth of the attracting centre of f. (2) write ω1(f) = ω(f) and ωn+1(f) = ω(f|ωn(f)) for any n ∈ n. the minimal m ∈ n ∪ {∞} such that ωm(f) = ωm+1(f) is called the depth of f. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 288 the depth and the attracting centre for map in this paper, we show that if f is a continuous map on a fuzzy metric interval, then ω2(f) = r(f) and the depth of f is at most 2, and ω 3(f) = ω2(f) and the depth of the attracting centre of f is at most 2. 3. the depth for a continuous map on a fuzzy metric interval in this section, we study the depth of a continuous map on a fuzzy metric interval i = [a, b] with a ∕= b. proposition 3.1. the following two statements hold: (1) let xn ∈ i for any n ∈ n. then xn −→ x ⇐⇒ m(a, xn, t) −→ m(a, x, t) for any t > 0. (2) for any x, y ∈ i with x ≤ y, [x; y] is closed. (3) for any x, y, z ∈ x with x < y < z, [x; z] − {y} is not connected. proof. (1) =⇒ is obvious since m is continuous. ⇐= assume on the contrary that xn ∕−→ x. then there exist an open neighbourhood u of x and a sequence of positive integers k1 < k2 < · · · such that fkn(x) ∈ i − u for any n ∈ n. by taking subsequence we let xkn −→ u ∈ i − u since i − u is closed. then by the above (=⇒) we have m(a, xkn, t) −→ m(a, u, t) for any t > 0. thus m(a, x, t) = m(a, u, t). by remark 2.7 we have x = u. this is a contradiction. (2) let xn ∈ [x; y] and xn −→ u ∈ i since i is compact. then m(a, x, t) ≥ m(a, xn, t) ≥ m(a, y, t) with m(a, xn, t) −→ m(a, u, t) ∈ [m(a, y, t), m(a, x, t)], which implies u ∈ [x; y]. (3) we claim that [x; y) is an open subset of [x; z]. indeed, for any w ∈ [x; y), there exists an open neighbourhood u = b(w, ε, t0) of w such that y ∕∈ u. by definition 2.6 (4) we see that u ∩ [x; z] ⊂ [x; y), which implies that [x; y) is an open subset of [x; z]. in a similar fashion we can also show that (y; z] is an open subset of [x; z]. since [x; y) ∩ (y; z] = ∅ and [x; z] − {y} = [x; y) ∩ (y; z], from which it follows that [x; z] − {y} is not connected. the proof is completed. □ lemma 3.2. let f ∈ c(i). if there exist x, y ∈ i with x ≤ y such that f(x) ≤ x ≤ y ≤ f(y) or x ≤ f(x) and f(y) ≤ y, then [x; y] ∩ fix(f) ∕= ∅. proof. we can assume that f(x) ∕= x and f(y) ∕= y. define f : [x; y] −→ r such that for any z ∈ [x; y], f(z) = m(a, z, t) − m(a, f(z), t). by remark 2.3 we see that f is continuous. since f(x)f(y) < 0 and [x; y] is connected, we have 0 ∈ f([x; y]) and there exists p ∈ [x; y] such that f(p) = 0. thus m(a, p, t) = m(a, f(p), t) for t > 0. by remark 2.7 we see f(p) = p. the proof is completed. □ corollary 3.3. let f ∈ c(i). if there exist x ∈ i and n ∈ n such that fn(x) < x and [fn(x); x] ∩ p(f) = ∅, then fkn(x) < fn(x) for any k ≥ 2. if there exist x ∈ i and n ∈ n such that fn(x) > x and [x; fn(x)] ∩ p(f) = ∅, then fkn(x) > fn(x) for any k ≥ 2. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 289 t. sun, l. li, g. su, c. han and g. xia proof. let fn(x) < x. assume on the contrary that fn(x) ≤ fkn(x) for some k ≥ 2. then it follows from lemma 3.2 that [fn(x); x] ∩ fix(f(k−1)n) ∕= ∅. this contradicts [fn(x); x] ∩ p(f) = ∅. for the another case, the proof is similar. the proof is completed. □ by lemma 3.2 and corollary 3.3 we obtain the following corollary. corollary 3.4. let f ∈ c(i) and x, y ∈ i with x < y and [x; y] ∩ p(f) = ∅. if u, v, fm(u), fn(v) ∈ [x; y] for some m, n ∈ n, then u ∈ [x; fm(u)) and v ∈ [x; fn(v)), or u ∈ (fm(u); y] and v ∈ (fn(v); y]. theorem 3.5. let f ∈ c(i). then p(f) = r(f). proof. by remark 2.9 it is suffice to show that p(f) ⊃ r(f). let x ∈ r(f) − p(f). take an open neighbourhood u = b(x, ε, t0) of x. then there exists a sequence of positive integers k1 < k2 < · · · such that fkn(x) −→ x and fkn(x) ∈ u. choose fkm(x) ∈ u. without loss of generality we may assume that fkm(x) < x. then by remark 2.7 we see that there exists r > m such that fkm(x) < fkr (x). by corollary 3.4 we obtain [fkm(x); fkr (x)]∩p(f) ∕= ∅ if fkr (x) ≥ x or [fkm(x); x] ∩ p(f) ∕= ∅ if fkr (x) ≤ x. thus u ∩ p(f) ∕= ∅ (definition 2.6 (4)), which implies x ∈ p(f) and r(f) ⊂ p(f). the proof is completed. □ theorem 3.6. let f ∈ c(i). then ω(f|ω(f)) = r(f) and the depth of f is at most 2. proof. by remark 2.9 it is suffice to show that ω(f|ω(f)) − r(f) ⊂ r(f). let x ∈ ω(f|ω(f)) − r(f). take an open neighbourhood u = b(x, ε, t0) of x. then there exist a sequence of positive integer k1 ≤ k2 ≤ · · · and a sequence of points xn ∈ ω(f) such that fkn(xn) −→ x, xn −→ x and fkn(xn), xn ∈ u for any n ∈ n. without loss of generality we may assume that xn ∕∈ p(f) for any n ∈ n. choose xm, fkm(xm) ∈ u. without loss of generality we may assume that xm < f km(xm). we can choose an open neighbourhood w = b(xn, δ, t1) of xm such that w, f km(w) ⊂ u and w ∩ fkm(w) = ∅ since i is a compact hausdorff space. note that xm ∈ ω(f) and xm ∕∈ p(f). then there exist a sequence of positive integers r1 < r2 < · · · and a sequence of points yn ∈ i such that frn(yn) −→ xm, yn −→ xm and frn(yn), yn ∈ w for any n ∈ n. by proposition 3.1 we see that there exists rn > km such that m(a, frn(yn), t) > m(a, f km(yn), t) and m(a, yn, t) > m(a, f km(yn), t). thus frn(yn) < f km(yn) and yn < f km(yn). by corollary 3.4 we obtain [yn; f km(yn)] ∩ p(f) ∕= ∅ if yn ≤ frn(yn), or [frn(yn); fkm(yn)] ∩ p(f) ∕= ∅ if yn ≥ frn(yn), which implies u ∩ p(f) ∕= ∅ (definition 2.6 (4)). thus x ∈ p(f) = r(f). the proof is completed. □ 4. the attracting centre for a continuous map on a fuzzy metric interval in this section, we study the attracting centre of a continuous map on a fuzzy metric interval i = [a; b] with a ∕= b. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 290 the depth and the attracting centre for map lemma 4.1. let f ∈ c0(i) and x, w, y ∈ i with w ∈ [x; y] and [x; y] ∩ (r(f) ∪ λ(w, f)) = ∅. if there exist x1, x2, · · · ∈ λ(w, f) with xn < x and y1, y2, · · · ∈ λ(w, f) with y < yn for any n ∈ n such that limn−→∞ m(xn, x, t) = limn−→∞ m(yn, y, t) = 1 (for any t > 0), then {x, y} ∩ aγ(f) ∕= ∅. proof. we claim that fn([x; y]) ∩ [x; y] = ∅ for any n ∈ n. indeed, if fm([x; y]) ∩ [x; y] ∕= ∅ for some m ∈ n, then fm([x; y]) ∕⊂ [x; y] since [x; y] ∩ p(f) = ∅. thus there exists n ∈ n such that xn ∈ fm([x; y]) or yn ∈ fm([x; y]) for any n ≥ n (proposition 3.1), which implies that [x; y]∩λ(w, f) ∕= ∅. this is a contradiction. since x, y ∕∈ r(f), we see that there exist δ0 ∈ (0, 1), t1 > 0 and t2 > 0 such that m(fn(x), x, t1) ≤ 1−δ0 and m(fn(y), y, t2) ≤ 1−δ0 for any n ∈ n. thus by taking subsequence we can assume that fn([x; y]) ∩ [x1; y1] = ∅ for any n ∈ n and choose two sequences of positive integers µ1, µ2, · · · and λ1, λ2, · · · such that (1) fµn(xn) = f λn(yn) = w for all n ∈ n. (2) (i) fµn([xn; x]) ⊃ [x1; x] for any n ∈ n or (ii) fµn([xn; x]) ⊃ [y; y1] for any n ∈ n. (3) (i) fλn([y; yn]) ⊃ [x1; x] for any n ∈ n or (ii) fλn([y; yn]) ⊃ [y; y1] for any n ∈ n. if (2(i)) holds, then write e1 = [x1; x] and for n ≥ 2, write en = en−1 ∩ f−µ1−···−µn−1([xn; x]). thus en+1 ⊂ en and en ∕= ∅ for all n ∈ n, which implies that ∩∞n=1en ∕= ∅. let u ∈ ∩∞n=1en. then one has fµ1+···+µn−1(u) ∈ [xn; x] and u ∈ fµn([xn; x]) for all n ∈ n, from which we see that x ∈ aγ(f). if (3(i)) holds, then using arguments similar to the ones developed in the proof of above case, also we can show that y ∈ aγ(f). if (2(ii)) and (3(ii)) hold, then fλn+µn([yn; y]) ⊃ [y1; y] and fλn+µn([x; xn]) ⊃ [x; x1] for any n ∈ n. in a similar fashion, also we can show that x, y ∈ aγ(f). the proof is completed. □ lemma 4.2. let f ∈ c0(i). then r(f) ∪ aγ(f) ⊂ ω(r(f) ∪ aγ(f)). proof. if x ∈ r(f), then x ∈ ω(x, f) ⊂ ω(r(f) ∪ aγ(f)). in the following we show that x ∈ ω(r(f) ∪ aγ(f)) if x ∈ aγ(f) − r(f). since x ∈ aγ(f) − r(f), there exist δ0 ∈ (0, 1) and t1 > 0 such that m(fn(x), x, t1) ≤ 1 − δ0 for any n ∈ n and without loss of generality we may assume that there exist a sequence of points x0, x1, · · · , xn, · · · ∈ (x; b], and two sequences of positive integers λ1 ≤ λ2 ≤ · · · and µ1 < µ2 < · · · such that: (1) m(x, xn, t1) > 1 − δ0/2 for every n ∈ n!. (2) fλn(xn) = x0 and xn ∈ (x; xn−1) for every n ∈ n and limn−→∞ m(xn, x, t) = 1 (for any t > 0). (3) fµn(x0) ∈ (x; x0) and fµn+1(x0) ∈ (x; fµn(x0)) for every n ∈ n and limn−→∞ m(f µn(x0), x, t) = 1 (for any t > 0). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 291 t. sun, l. li, g. su, c. han and g. xia by corollary 3.4 we see that there exists a sequence of periodic points p1, p2, · · · such that pn+1 ∈ (x; pn) for any n ∈ n and limn−→∞ m(pn, x, t) = 1 (for any t > 0). without loss of generality we may assume that x < · · · < x2n < x2n−1 < fµn(x0) < · · · < fµ3(x0) < x4 < x3 < f µ2(x0) < x2 < x1 < p2 < f µ1(x0) < p1 < x0. let en be the connected component of i − λ(fµ1(x0), f) containing fn+µ1(x0) and e0 = [u; v]. then f n([u; v]) ⊂ en. we claim that [u; v] ∩ (aγ(f) ∪ r(f)) ∕= ∅. without loss of generality we may assume that [u; v]∩r(f) = ∅. then [u; v] ⊂ [p2, p1] and fn([u; v]) ∕⊂ [u; v] for all n ∈ n and the following two statements hold: (4) (i) u ∈ λ(fµ1(x0), f), or (ii) there exist u1, u2, · · · ∈ λ(fµ1(x0), f) with p2 < u1 < u2 < · · · 0). (5) (i) v ∈ λ(fµ1(x0), f), or (ii) there exist v1, v2, · · · ∈ λ(fµ1(x0), f) with v < · · · < v2 < v1 < p1 such that limn−→∞ m(vn, v, t) = 1 (for any t > 0). now we show that u ∕∈ λ(fµ1(x0), f). otherwise, if u ∈ λ(fµ1(x0), f), then there exists some n ∈ n such that fn(u) = fµ1(x0) ∕= u. by lemma 3.2 and corollary 3.4 we have fn(v) > v and v ∕∈ λ(fµ1(x0), f) since [u; v] ∩ p(f) = ∅. thus by (5(ii)) we see that there exists n ∈ n such that vn ∈ fn((u; v)) for any n ≥ n, which contradicts the definition of e0. in a similar fashion we can also show that v ∕∈ λ(fµ1(x0), f). since u, v ∕∈ λ(fµ1(x0), f), we know that (4(ii)) and (5(ii)) hold. by lemma 4.1 we see that {u, v} ∩ aγ(f) ∕= ∅. the claim is proven it follows from above claim that x ∈ ω(u, f) ∩ ω(v, f). the proof is completed. □ lemma 4.3. let f ∈ c0(i). then ω(ω(f)) ⊂ r(f) ∪ aγ(f). proof. we may assume that y ∈ ω(ω(f)) − r(f). then there exist δ0 ∈ (0, 1) and t1 > 0 such that m(f n(y), y, t1) ≤ 1 − δ0 for any n ∈ n and without loss of generality we may assume that there exists some z ∈ ω(f) such that y ∈ ω(z, f) and there exist two sequences of positive integers µ1 < µ2 < · · · and λ1 < λ2 < · · · , a sequence of points z1, z2, · · · such that (1) limn−→∞ m(f µn(z), y, t) = limn−→∞ m(f λn(zn), z, t) = limn−→∞ m(zn, z, t) = 1 (for any t > 0). (2) y < · · · < fµn(z) < · · · < fµ2(z) < fµ1(z) with m(fµn(z), y, t1) > 1 − δ0/2. for every n ≥ 2, there exists a point zi with sn = λi +µn −µn+4 > 0 such that fµn+5(z) < un+4 = f µn+4(zi) < f µn+3(z) < fµn+2(z) < fµn+1(z) < vn = f λi+µn(zi) = f sn(un+4) < f µn−1(z). if there exist k1 < k2 < · · · such that fskn (y) > y, then fskn (y) > fµ1(z) and fskn ([y; ukn+4]) ⊃ [vkn; fµ1(z)], then there exists wn ∈ (y; ukn+4] such that fskn (wn) = f µ1(z) for any n ≥ 2 with wn −→ y, which implies that y ∈ aγ(f). in the following we may assume that fsn(y) < y for any n ≥ 2. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 292 the depth and the attracting centre for map since fsn([y; un+4]) ⊃ [y; vn] for any n ≥ 2, there exists wn ∈ [y; un+4] such that fsn(wn) = wn−1 for any n ≥ 2, w1 = fµ3(z) and wn −→ y, which implies that y ∈ aγ(f). the proof is completed. □ theorem 4.4. let f ∈ c0(i). then for any n ∈ n, ωn+2(f) = ω2(f) = ω(ω(f)) = ω(r(f) ∪ aγ(f)) and the depth of the attracting centre of f is at most 2. proof. it follows from lemma 4.2 and lemma 4.3 that ω(r(f) ∪ (aγ(f))) ⊂ ω2(f) ⊂ ω(ω(f)) ⊂ r(f) ∪ (aγ(f) ⊂ ω(r(f) ∪ aγ(f)). the last implies that ω(r(f) ∪ (aγ(f))) = ω2(f) = ω(ω(f)) = r(f) ∪ (aγ(f) = ω(r(f) ∪ aγ(f)). thus we know that for any n ∈ n, ωn+2(f) = ω2(f) = ω(ω(f)) = ω(r(f) ∪ aγ(f)) = ω(r(f) ∪ aγ(f)). the proof is completed. □ 5. conclusion in this paper, we introduce the notion of fuzzy metric interval, and study the depth and the attracting centre for a continuous map f on a fuzzy metric interval, and show that ω2(f) = r(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2. acknowledgements. the authors thank the referee for his/her valuable suggestions which improved the paper. references [1] a. george and p. veeramani, on some results in fuzzy metric spaces, fuzzy sets sys. 64 (1994), 395–399. 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[16] d. zheng and p. wang, meir-keeler theorems in fuzzy metric spaces, fuzzy sets sys. 370 (2019), 120–128. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 294 @ appl. gen. topol. 23, no. 1 (2022), 91-105 doi:10.4995/agt.2022.15739 © agt, upv, 2022 on w-isbell-convexity olivier olela otafudu a and katlego sebogodi b dedicated to the memory of prof. hans-peter künzi a school of mathematical and statistical sciences north-west university, potchefstroom campus, potchefstroom 2520, south africa. (olivier.olelaotafudu@nwu.ac.za) b department of mathematics and applied mathematics, university of johannesburg, auckland park, 2006, south africa. (7katlego3@gmail.com) communicated by m. a. sánchez-granero abstract chistyakov introduced and developed the concept of modular metric for an arbitrary set in order to generalise the classical notion of modular on a linear space. in this article, we introduce the theory of hyperconvexity in the setting of modular pseudometric that is herein called w-isbell-convexity. we show that on a modular set, w-isbell-convexity is equivalent to hyperconvexity whenever the modular pseudometric is continuous from the right on the set of positive numbers. 2020 msc: 45e35; 46a80. keywords: modular pseudometric; isbell-convexity; w-isbell-convexity. 1. introduction modular metric spaces were introduced by chistyakov [8] in 2010. he developed the theory of modular metric on an arbitrary set and investigated the theory of metric spaces induced by a modular metric. he defined a modular metric in the following way. let x be a nonempty set, then the function w : (0,∞) × x × x −→ [0,∞] is called a modular metric if it satisfies (a) w(λ,x,y) = 0 if and only if x = y whenever λ > 0, (b) w(λ,x,y) = w(λ,y,x) whenever x,y ∈ x and λ > 0 and (c) w(λ + µ,x,y) ≤ w(λ,x,z) + w(µ,z,y) whenever x,z,y ∈ x and λ,µ > 0. received 06 june 2021 – accepted 10 december 2021 http://dx.doi.org/10.4995/agt.2022.15739 o. olela otafudu and k. sebogodi furthermore, the function w is said to be modular pseudometric on x if instead of (a), the function satisfies (d) w(λ,x,x) = 0 for all λ > 0 and x ∈ x. for a ∈ x, the modular set xw(a) is defined by xw = xw(a) = {x ∈ x : limλ→∞w(λ,a,x) = 0}. chistyakov equipped xw with the metric qw, where qw(x,y) = inf{λ > 0 : w(λ,x,y) ≤ λ} whenever x,y ∈ xw. in [6], chistyakov introduced a topology τ(w) on xw in the following sense. a subset v of xw is τ(w)-open if for any λ > 0 and x ∈ v , there exists µ > 0 such that the entourage set bλ,µ(x) = {y ∈ xw : w(λ,x,y) ≤ µ} ⊂ v . he studied the hausdorff modular pseudometric on a power set of a nonempty set equipped with a modular pseudometric. in addition, chistyakov provided an application of modular metrics which consists of an extended kind of helly’s theorem on the pointwise selection principle. this was obtained by building a special modular space, the set of all bounded and regulated mappings on an interval. furthermore, chistyakov considered the description of superposition operators acting in modular spaces, the existence of regular selections of setvalued maps, the new interpretation of lipschitzian and absolutely continuous maps, and the existence of solutions to the carathéodory-type ordinary differential equations in banach spaces with the right-hand side from the orlicz space. since chistyakov developed the theory of metric spaces generated by a modular metric defined on an arbitrary set, the interest of this concept has grown up in the community of mathematicians, especially in operator theory where many authors are applying modular metrics to study the existence and uniquenesse of fixed points of self-maps on a modular set that satisfy some particular properties (see for instance [4, 5, 20, 28]). let mention some important and popular recent works on different type of modular metric spaces [1, 3, 10, 11, 12, 14, 15]. in addition, the well-known and important concept of hyperconvexity in a metric space has been successfully investigated and applied in many areas of mathematics and other fields. for instance the theory of hyperconvexity has been introduced in the framework of quasi-pseudometric spaces which is called isbell-convexity (or q-hyperconvexity) (see for instance [16, 19, 21, 23, 25]). naturally this has led us to the believe that isbell-convexity on a set equipped with a odular pseudometric should be investigated. the aim of this article is to introduce and study the theory of isbell-convexity in the setting of modular pseudometrics and we call it w-isbell-convexity. for instance, we study connections between hyperconvexity in a metric space and w-isbell-convexity on a modular set. in addition, we discuss the boundedness (w-boundedness) of a set endowed with a modular pseudometric. we eventually show that a nonexpansive self-map (w-nonexpansive map) on a w-isbell-convex modular set has a fixed point. in addition, its fixed points set is w-isbell convex whenever its modular pseudometric is continuous from the right on the set of positive numbers. © agt, upv, 2022 appl. gen. topol. 23, no. 1 92 on w-isbell-convexity 2. preliminaries for the comfort of the reader and in preparation of the terminology that we are going through this article, we recall the following concepts that can be found in [5, 6, 7, 8]. definition 2.1. consider a nonempty set x. a function w : (0,∞)×x×x → [0,∞] is said to be a modular pseudometric on x if it satisfies the following conditions: (a) w(λ,x,x) = 0 for all x ∈ x and λ ∈ (0,∞), (b) w(λ,x,y) = w(λ,y,x) for all x,y ∈ x and λ ∈ (0,∞), (c) w(λ + µ,x,y) ≤ w(λ,x,z) + w(µ,z,y) for all x,y,z ∈ x and λ,µ ∈ (0,∞). we shall say that w is a modular metric provided that w satisfies also the following condition: for all x,y ∈ x and λ ∈ (0,∞), (d) w(λ,x,y) = 0 for all λ > 0 imply x = y. let w be a modular pseudometric on a nonempty set x. for any x ∈ x, we denote by xw = xw(x) := {y ∈ x : lim λ→∞ w(λ,x,y) = 0}. let us fix an element x0 ∈ x. then set x∗w(x0) defined by x∗w = x ∗ w(x0) = {x ∈ x : w(λ,x,x0) < ∞ for some λ > 0} is called a modular set (around x0) and x0 is called the center of x ∗ w. it has been observed in [8] that it is easy to see that xw(x0) ⊆ x∗w(x0). for further details on modular sets and examples of these sets, we refer the reader to [8]. the function qw defined by qw(x,y) = inf{λ > 0 : w(λ,x,y) ≤ λ} whenever x,y ∈ xw is a (pseudo) metric on xw whenever w is modular (pseudo) metric on x. if x,y ∈ x, then qw is an extended (pseudo) metric on x. for any x ∈ xw and λ and µ > 0, we define the sets bwλ,µ(x) and c w λ,µ(x) by bwλ,µ(x) := {z ∈ xw : w(λ,x,z) < µ} and cwλ,µ(x) := {z ∈ xw : w(λ,x,z) ≤ µ}. then the set bwλ,µ(x) is called a w<-entourage about x relative to λ and µ and the set cwλ,µ(x) is called a w≤-entourage about x relative to λ and µ. we need the following two examples in the sequel. © agt, upv, 2022 appl. gen. topol. 23, no. 1 93 o. olela otafudu and k. sebogodi example 2.2 ([8, example 2.4 (a)]). let r be equipped with the modular metric w defined by w(λ,x,y) = { ∞ if x 6= y 0 if x = y whenever x,y ∈ r and λ > 0. it is readily checked that xw = xw(x0) = {x0}, where x0 ∈ r and qw(x,y) = 0 for all x,y ∈ xw. moreover, for any r > 0, we have bqw (x0,r) = {y ∈ xw : 0 = qw(x0,y) < r} = {x0} and bwr,r(x0) = {y ∈ xw : w(r,x0,y) < r} = {x0} = c w r,r(x0). example 2.3 ([8, example 2.7 (b)]). consider a pseudometric space (x,q). if we equip x with the modular pseudometric w(λ,x,y) = q(x,y) λp whenever x,y ∈ x and λ > 0, where p is a strictly positive constant. then it follows that xw = x and qw(x,y) = [q(x,y)] 1 p+1 . furthermore, for any λ > 0, we have bqw (x,λ) = { y ∈ xw : [q(x,y)] 1 p+1 < λ } = { y ∈ xw : q(x,y) λp < λ } = {y ∈ xw : w(λ,x,y) < λ} = bwλ,λ(x). if w is a modular pseudometric on a nonempty set x, then the topology induced by w (denoted by τ(w)) is defined by the following: a subset a of xw is said to be τ(w)-open (or w-open) if for any x ∈ a and λ > 0, there exists µ := µ(x,λ) > 0 such that bwλ,µ(x) ⊂ a. note that b w λ,µ(x) is not τ(w)-open, in general. lemma 2.4 ([5]). let w be a modular pseudometric on x and x ∈ xw. then, whenever λ > 0 we have (a) bqw (x,λ) ⊆ bwλ,λ(x), (b) cqw (x,λ) ⊆ cwλ,λ(x), where bqw (x,λ) and cqw (x,λ) are respectively open and closed balls with centre x and radius λ with respect to the pseudometric qw. the following important concept of continuity of a modular pseudometric space was introduced in [6]. definition 2.5. let w be a modular pseudometric on a set x. given x,y ∈ x, (a) the limit from the right w+0(λ,x,y) of w at a point λ > 0 is defined by w+0(λ,x,y) := lim µ→λ+ w(µ,x,y) = sup{w(µ,x,y) : µ > λ}. © agt, upv, 2022 appl. gen. topol. 23, no. 1 94 on w-isbell-convexity (b) the limit from the left w−0(λ,x,y) of w at a point λ > 0 is defined by w−0(λ,x,y) := lim µ→λ− w(µ,x,y) = inf{w(µ,x,y) : 0 < µ < λ}. moreover, (c) we say that w is continuous from the right on (0,∞) if for any λ > 0 we have w(λ,x,y) = w+0(λ,x,y). (d) we say that w is continuous from the left on (0,∞) if for any λ > 0 we have w(λ,x,y) = w−0(λ,x,y). (e) we say that w is continuous on (0,∞) if w is continuous from the right and continuous from the left on (0,∞). lemma 2.6 ([6]). let w be a modular pseudometric on a set x. if x,y ∈ xw and 0 < inf{λ > 0 : w(λ,x,y) ≤ λ} < ∞, then qw(x,y) = inf{λ > 0 : w(λ,x,y) ≤ λ} > λ if and only if w+0(λ,x,y) > λ. the following remark is a consequence of definition 2.5. remark 2.7. if w is continuous from the right on (0,∞), then for any x,y ∈ xw and λ > 0, we have that qw(x,y) ≤ λ if and only if w(λ,x,y) ≤ λ. let us recall the well-known concept of hyperconvexity. definition 2.8. a pseudometric space (x,q) is called hyperconvex provided that for any (xi)i∈i and family (ri)i∈i of nonnegative real numbers satisfying q(xi,xj) ≤ ri + rj whenever i,j ∈ i, the following condition holds:⋂ i∈i [ cq(xi,ri) ] 6= ∅. let (x,q) be a pseudometric space. then x is said to be metrically convex if for any points x,y ∈ x and positive numbers r and s such that q(x,y) ≤ r + s, there exists z ∈ x such that q(x,z) ≤ r and q(z,y) ≤ s. furthermore, a family of balls (cq(xi,ri))i∈i is said to have the mixed binary intersection property if for all indices i ∈ i, cq(xi,ri) 6= ∅. definition 2.9. a pseudometric space (x,q) is called hypercomplete if every family (cq(xi,ri))i∈i of balls having the mixed binary intersection property satisfies ⋂ i∈i [ cq(xi,ri) ] 6= ∅. the following is a well-known characterisation of hyperconvexity in terms of metric convexity and hypercompleteness. proposition 2.10 ([26]). a pseudometric space (x,q) is hyperconvex if and only if it is metrically convex and hypercomplete. © agt, upv, 2022 appl. gen. topol. 23, no. 1 95 o. olela otafudu and k. sebogodi 3. isbell-convexity in this section, we introduce the concept of hyperconvexity in modular metric spaces. definition 3.1. let w be a modular pseudometric on a nonempty set x. we say that xw is w-isbell-convex if for any family of points (xi)i∈i in xw and family of points (λi)i∈i in (0,∞) such that w(λi + λj,xi,xj) ≤ λi + λj, whenever i,j ∈ i, then ⋂ i∈i [ cwλi,λi(xi) ] 6= ∅. example 3.2. let (x,d) be a metric space. for any x,y ∈ x and λ > 0, it is well-known from [8, example 2.4] that the function w defined by w(λ,x,y) = d(x,y) ϕ(λ) , where ϕ : (0,∞) −→ (0,∞) is a bounded nondecreasing function is a modular metric on x. furthermore, whenever x0 ∈ x the set xw(x0) = {x0} is w-isbell-convex. indeed, for any λ,µ > 0 such that 0 = w(λ + µ,x0,x0) < λ + µ, then we have x0 ∈ bλ,λ(x0) ∩bµ,µ(x0) ⊆ cλ,λ(x0) ∩cµ,µ(x0). definition 3.3. let w be a modular pseudometric on a nonempty set x. we say that xw is w-metrically convex if for any points x,y ∈ xw so that w(λ + µ,x,y) ≤ λ + µ whenever λ,µ > 0, there exists z ∈ xw such that w(λ,x,z) ≤ λ and w(µ,z,y) ≤ µ. example 3.4. let x = r be equipped with the modular metric w(λ,x,y) = d(x,y) λ for any x,y ∈ x, where d is the discrete metric on r. then it is obvious that xw = r. moreover xw is not w-metrically convex since w ( 1, 1 2 , 1 ) = d( 1 2 , 1) 1 = 1 ≤ 1 2 + 1 2 but there is no z ∈ r such that w ( 1 2 , 1 2 ,z ) = d( 1 2 ,z) 1 2 ≤ 1 2 and w ( 1 2 ,z, 1 ) = d(z, 1) 1 2 ≤ 1 2 . if such z exits it would satisfy z = 1 and z = 1 2 . remark 3.5. let w be a modular metric on a set x. it is easy to see that if (xw,qw) is metrically convex, then xw is w-metrically convex too. © agt, upv, 2022 appl. gen. topol. 23, no. 1 96 on w-isbell-convexity proof. let x,y ∈ xw and λ,µ > 0 such that w(λ + µ,x,y) ≤ λ + µ. it follows that qw(x,y) ≤ λ + µ. by metric convexity of (xw,qw), there exists z ∈ xw such that z ∈ cqw (x,λ) ∩cqw (y,µ) ⊆ cwλ,λ(x) ∩c w µ,µ(y). hence there exists z ∈ xw such that w(λ,x,z) ≤ λ and w(µ,z,y) ≤ µ. therefore, xw is w-metrically convex. � definition 3.6. let w be a modular pseudometric on a nonempty set x. a family (cwλi,λi(xi))i∈i of w≤-entourages with λi > 0 and xi ∈ xw for all i ∈ i is said to have the mixed binary intersection property provided that cwλi,λi(xi) 6= ∅ whenever i ∈ i. definition 3.7. let w be a modular pseudometric on a nonempty set x. the modular set xw is called w-isbell-complete if every family (c w λi,λi (xi))i∈i of w≤-entourages, where λi > 0 and xi ∈ xw for all i ∈ i, having the mixed binary intersection property satisfies⋂ i∈i [ cwλi,λi(xi) ] 6= ∅. proposition 3.8. if w is a modular pseudometric on x, then xw is w-isbellconvex if and only if xw is w-metrically-convex and w-isbell-complete. proof. (=⇒) suppose that xw be w-isbell-convex. let x1,x2 ∈ xw and λ1,µ2 > 0 such that w(λ1 + µ2,x1,x2) ≤ λ1 + µ2. then, we set λ2 = µ1 = w(λ2 + µ1,x2,x1). by w-isbell-convexity of xw, there exists a ∈ cwλ1,λ1 (x1) ∩c w µ2,µ2 (x2). it follows that w(λ1,x1,a) ≤ λ1 and w(µ2,a,x2) ≤ µ2. hence xw is wmetrically convex. consider the family (cwλi,λi(xi))i∈i of w≤-entourages having mixed binary intersection property. then there exists z ∈ cwλi,λi(xi) ∩c w λj,λj (xj) whenever i,j ∈ i. furthermore, whenever i,j ∈ i we have w(λi + λj,xi,xj) ≤ w(λi,xi,z) + w(λj,z,xj) ≤ λi + λj. thus ⋂ i∈i [ cwλi,λi(xi) ] 6= ∅ by w-isbell-convexity of xw. therefore, xw is w-isbell-complete. (⇐=) let xw be w-metrically convex and w-isbell-complete. let (xi)i∈i be a family of points in xw and (λi)i∈i be a family of points in (0,∞) such that w(λi + λj,xi,xj) ≤ λi + λj whenever i,j ∈ i. by w-metrical convexity of xw there exists z ∈ xw such that w(λi,xi,z) ≤ λi and w(λj,z,xj) ≤ λj whenever i,j ∈ i. © agt, upv, 2022 appl. gen. topol. 23, no. 1 97 o. olela otafudu and k. sebogodi then the family (cwλi,λi(xi))i∈i has the mixed binary intersection property. since xw is w-isbell-complete, we have⋂ i∈i [ cwλi,λi(xi) ] 6= ∅. � proposition 3.9. let w be a modular pseudometric on a nonempty set x. if (xw,qw) be a hyperconvex pseudometric space, then xw is w-isbell-convex. proof. suppose that (xw,qw) be a hyperconvex pseudometric space. let (xi)i∈i be a family of points in xw and (λi)i∈i be a family of points in (0,∞) such that w(λi + λj,xi,xj) ≤ λi + λj whenever i,j ∈ i. then qw(xi,xj) = inf{λ > 0 : w(λ,xi,xj) ≤ λ}≤ λi + λj whenever i,j ∈ i. by hyperconvexity of (xw,qw), we have⋂ i∈i [ cqw (xi,λi) ] 6= ∅. since by lemma 2.4, cqw (xi,λi) ⊆ cwλi,λi(xi), it follows that ∅ 6= ⋂ i∈i [ cqw (xi,λi) ] ⊆ ⋂ i∈i [ cwλi,λi(xi) ] . therefore, xw is w-isbell-convex. � remark 3.10. let w be a modular pseudometric on a set x. if w be continuous from the right on (0,∞), then qw(x,y) ≤ λ if and only if w(λ,x,y) ≤ λ for any x,y ∈ xw and positive number λ. it is natural to wonder about the converse of proposition 3.9. lemma 3.11. let w be a modular pseudometric on x. if w be continuous from the right on (0,∞), then xw is w-metrically convex if and only if (xw,qw) is a metrically convex pseudometric space. proof. (=⇒) suppose that xw be w-metrically convex. let x,y ∈ xw and λ,µ > 0 such that qw(x,y) ≤ λ + µ. since w is continuous from the right on (0,∞) we have w(λ+µ,x,y) ≤ λ+µ. it follows that there exists a ∈ xw such that w(λ,x,a) ≤ λ and w(µ,a,z) ≤ µ. then qw(x,a) ≤ λ and qw(a,y) ≤ µ by the right continuity of w. so (xw,qw) is a metrically convex. (⇐=) follows from remark 3.5. � we leave the proof of the following lemma to the reader. © agt, upv, 2022 appl. gen. topol. 23, no. 1 98 on w-isbell-convexity lemma 3.12. let w be a modular pseudometric on x. if w be continuous from the right on (0,∞), then xw is w-isbell-complete if and only if (xw,qw) is an isbell-complete pseudometric space. theorem 3.13. let w be a modular pseudometric on x. if w be continuous from the right on (0,∞), then xw is w-isbell-convex if and only if (xw,qw) is a hyperconvex pseudometric space. proof. suppose that xw be w-isbell-convex. then, by proposition 3.8 xw is w-isbell-complete and w-metrically convex. thus (xw,qw) is an hypercomplete and metrically convex pseudometric space by lemma 3.11 and lemma 3.12. therefore, (xw,qw) is a hyperconvex pseudometric space. the converse follows by a similar argument. � 4. nonexpansive maps we are aware of [2], where the concept of boundedness of a subset of modular set was introduced in order to study the existence of fixed point of modular contractive maps in modular metric spaces. this is a motivation for our next definitions. let w be a modular pseudometric on x. we say that a nonempty subset a of xw is w-bounded if there exists x ∈ xw such that a ⊆ cwλ,λ(x) for some λ > 0. remark 4.1. if w be a modular pseudometric on a set x, then boundeness on (xw,qw) implies w-boundeness. this observation follows from the fact that cqw (x,λ) ⊆ cwλ,λ(x) whenever λ > 0 and x ∈ xw. definition 4.2. let a be a w-bounded subset of xw. then we denote by diamw(a) the w-diameter of a and it is defined by diamw(a) := sup{w(λ,x,y) : x,y ∈ a} for some λ > 0. lemma 4.3. let w be a modular pseudometric on x. if a is a w-bounded subset of xw, then diamw(a) < ∞. proof. suppose that a is w-bounded. then for some x ∈ xw, we have a ⊆ cwλ,λ(x) for some λ > 0. if z,y ∈ a then w(λ,x,z) ≤ λ and w(λ,x,y) ≤ λ. it follows that w(λ + λ,z,y) ≤ w(λ,z,x) + w(λ,x,y) ≤ λ + λ. hence for λ′ = λ + λ > 0, we have sup{w(λ′,z,y) : z,y ∈ a}≤ λ′. therefore, diamw(a) < ∞. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 99 o. olela otafudu and k. sebogodi lemma 4.4. let w be a modular pseudometric on x. if w be continuous from the right on (0,∞), then boundeness on (xw,qw) is equivalent to w-boundeness. proof. we only prove the sufficient condition since the necessary condition follows from remark 4.1. suppose that a is a w-bounded subset of xw. then there exists x ∈ xw such that a ⊆ cwλ,λ(x) for some λ > 0. let y ∈ a. then w(λ,x,y) ≤ λ. by the right continuity of w on (0,∞), we have qw(x,y) ≤ λ for some x ∈ xw and λ > 0. thus a ⊆ cqw (x,λ). therefore, a is bounded in (xw,qw). � proposition 4.5. let w be a modular pseudometric on x. let xw be wisbell-convex and (xi)i∈i is a family of points in xw and (λi)i∈i is a family of positive real numbers such that w(λi + λj,xi,xj) ≤ λi + λj whenever i,j ∈ i. then the set a := ⋂ i∈i[c w λi,λi (xi)] is nonempty and w-isbell-convex. proof. observe that the set a is nonempty by w-isbell-convexity of xw. now we show that the set a is w-isbell-convex. let (xα)α∈γ be a family of points in a and (λα)α∈γ be a family of positive real numbers such that w(λα + λβ,xα,xβ) ≤ λα + λβ whenever α,β ∈ γ. we have to show that the family of w≤-entourages ((cwλα,λα(xα))α∈γ; (c w λi,λi (xi))i∈i) satisfies the hypothesis of w-isbell-convexity. then, in particular, for all α ∈ γ and i ∈ i, we have w(λi,xi,xα) ≤ λi < λi + λα since xα ∈ a. by w-isbell-convexity of xw, it follows that ∅ 6= ⋂ α∈γ [ cwλα,λα(xα) ] ∩ ⋂ i∈i [ cwλi,λi(xi) ] = a∩ ⋂ α∈γ [ cwλα,λα(xα) ] . hence a is w-isbell-convex. � for a w-bounded subset a of xα, we set (4.1) covw(a) := ⋂{ cwλ,λ(x) : a ⊆ c w λ,λ(x),x ∈ xw,λ > 0 } . furthermore, we set: rwx,λ(a) := sup y∈a {w(λ,x,y) : for some λ > 0}, where,x ∈ xw. proposition 4.6 (compare [13, lemma 3.3]). let w be a modular pseudometric on x and a be a w-bounded subset of xw. then we have: © agt, upv, 2022 appl. gen. topol. 23, no. 1 100 on w-isbell-convexity (a) covw(a) = ⋂ x∈x [ cwrw x,λ (a),rw x,λ (a)(x) ] . (b) rx(covw(a)) = rx(a). definition 4.7 (compare [13, definition 3.4]). let w be a modular pseudometric on x. a nonempty and w-bounded subset a of xw is called w-admissible if a = covw(a). in the sequel, we will denote by aw(xw), the set of all w-admissible subsets of xw. remark 4.8. observe that a w-admissible subset of xw can be written as the intersection of a family of the form cwλ,λ(x), where x ∈ xw and λ > 0. lemma 4.9. let w be a modular pseudometric on x which is continuous from the right on (0,∞). then cqw (x,λ) = c w λ,λ(x) whenever λ > 0 and x ∈ xw. proof. since we know that cqw (x,λ) ⊆ cwλ,λ(x), then we only prove the reverse inclusion. if a ∈ cwλ,λ(x), then w(λ,x,a) ≤ λ which is equivalent to qw(x,a) ≤ λ by the right continuity of w. thus a ∈ cqw (x,λ). � corollary 4.10. let w be a modular pseudometric on x which is continuous from the right on (0,∞) and a ⊆ xw. then a is w-admissible if and only if a is qw-admissible. definition 4.11. let w be a modular pseudometric on x. given a subset a of xw, we define for λ > 0 the λ-parallel set of a as pλ(a) = ⋃ a∈a [ cwλ,λ(a) ] . proposition 4.12 (compare [17, lemma 4.2]). let w be a modular pseudometric on x. if xw is w-isbell-convex and a is a w-admissible subset of xw, that is a = ⋂ i∈i c w λi,λi (xi) with xi ∈ xw and λi > 0 for each i ∈ i 6= ∅, then pλ(a) = ⋂ i∈i [ cwλi+λ,λi+λ(xi) ] whenever λ > 0. proof. let y ∈ pλ(a). then, for some a ∈ a we have w(λ,a,y) ≤ λ. furthermore, for each i ∈ i, w(λi + λ,xi,y) ≤ w(λi,xi,a) + w(λ,a,y) ≤ λi + λ. it follows that y ∈ cwλi+λ,λi+λ(xi) whenever ∈ i. hence pλ(a) ⊆ ⋂ i∈i [ cwλi+λ,λi+λ(xi) ] . © agt, upv, 2022 appl. gen. topol. 23, no. 1 101 o. olela otafudu and k. sebogodi we now suppose that y ∈ ⋂ i∈i [ cwλi+λ,λi+λ(xi) ] . hence, for any i ∈ i, w(λi + λ,xi,y) ≤ λi + λ. thus the family of w≤-entourages [(c w λi,λi (xi))i∈i,c w λ,λ(y)] satisfies the hypothesis of w-isbell-convexity of xw. then ∅ 6= [⋂ i∈i cwλi,λi(xi) ]⋂ cwλ,λ(y) = a ⋂ cwλ,λ(y). it follows that w(λ,y,a) ≤ λ for some a ∈ a. therefore, y ∈ pλ(a). � definition 4.13. let w be a modular pseudometric on a set x. then we say that a map t : xw → xw is w-lipschitz if there exists a k > 0 such that w(kλ,t(x),t(y)) ≤ w(λ,x,y) for all λ > 0 and x,y ∈ xw. if k = 1, then the map t is called a w-nonexpansive map. remark 4.14. let w be a modular pseudometric on x. in light of [5, theorem 5.1] and [5, theorem 5.2], one can easily prove that for any w-lipschitz map t : xw → xw, we have that w(kλ,t(x),t(y)) ≤ w(λ,x,y) implies qw(t(x),t(y)) ≤ kqw(x,y) for some k > 0 and whenever λ > 0 and x,y ∈ xw. theorem 4.15 (compare [17, lemma 4.3]). let w be a modular pseudometric on x which is continuous from the right on (0,∞) and xw be w-isbell-convex. if a is a w-admissible subset of xw, then there is a w-nonexpansive retraction r of pλ(a) onto a such that w(λ,x,r(x)) ≤ λ whenever x ∈ pλ(a) and λ > 0. proof. suppose that a is w-admissible. then a = ⋂ i∈i cwλi,λi(xi) with i 6= ∅. since pλ(a) is an intersection of w≤-entourages from proposition 4.12, then pλ(a) is w-admissible in xw. moreover, from proposition 4.5 we have pλ(a) is w-isbell-convex. let us consider the family ω defined by ω = {(d,rd) : a ⊆ d ⊆ pλ(a) and rd : d → a is a w-nonexpansive retraction such that w(λ,rd(x),x) ≤ λ whenever x ∈ d}. if the map ia be the identity on a, then (a,ia) ∈ ω. hence ω 6= ∅. furthermore, one can ordered the family ω by the partial order (d,rd) ≤ (e,re) if and only if d ⊆ e and the w-nonexpansive map rd is an extension of the w-nonexpansive map re. it follows that every chain in (ω,≤) is bounded above. thus by zorn’s lemma, ω has a maximal element. suppose that (d,rd) is the maximal element of (ω,≤). © agt, upv, 2022 appl. gen. topol. 23, no. 1 102 on w-isbell-convexity let us prove that d = pλ(a). suppose that there exists an element x ∈ pλ(a)\d such that w(λd,d,x) = λd whenever d ∈ d and λd > 0. we consider the set c := [ ⋂ d∈d cww(λd,d,x),w(λd,d,x)(rd(d)) ]⋂[⋂ i∈i cwλi,λi(xi) ]⋂ [cwλ,λ(x)]. it is easy to see that c 6= ∅ since the family [(cwλi,λi(xi))i∈i, (c w λ,λ(x)), (c w w(λd,d,x),w(λd,d,x) (rd(d)))d∈d] of ≤-entourages has the mixed binary intersection property in light of proposition 3.8. since ∅ 6= c ⊆ a. we now suppose that z ∈ c and we define the map r′ : d ∪{x} → a by r′(d) = rd(d) if d ∈ d and r′(x) = z. then, we have for any d ∈ d w(λd,r ′(d),r′(x)) = w(λd,rd(d),z) ≤ w(λd,d,x). so r′ is w-nonexpansive. moreover, w(λ,r′(x),x) = w(λ,z,x) ≤ λ since z ∈c. hence (d∪{x},r′) ∈ ω which contradicts the maximality of d. therefore, d = pλ(a). � theorem 4.16. let w be a modular pseudometric on x. if xw be w-bounded w-isbell-convex and w be continuous from the right on (0,∞) and t : xw → xw be a w-nonexpansive map, then the fixed point set fix(t) is nonempty and wisbell-convex. proof. since t : xw → xw is a w-nonexpansive map, then t : (xw,qw) → (xw,qw) is a nonexpansive map. moreover, xw is qw-bounded by lemma 4.4. hence for any λ > 0, we have w(t(x),t(y)) ≤ w(λ,x,y) whenever x,y ∈ xw. then from remark 4.14 when k = 1, we have qw(t(x),t(y)) ≤ qw(x,y) whenever x,y ∈ xw. observe that (xw,qw) is a hyperconvex pseudometric space by theorem 3.13. furthermore, we have fix(t) is nonempty and isbell-convex by [13, theorem 6.1]. therefore, fix(t) is w-isbell-convex by theorem 3.13. � definition 4.17 (compare [19, theorem 5.5]). let w be a modular pseudometric on x and t : xw → xw be a map. for λ1,λ2 > 0, we define the set fλ1,λ2 (t) by fλ1,λ2 (t) = {x ∈ xw : w(λ2,x,t(x)) ≤ λ2 and w(λ1,t(x),x) ≤ λ1}. corollary 4.18 (compare [17, theorem 4.11]). let w be a modular pseudometric on x and xw be w-isbell-convex. if w be continuous from the right on (0,∞) and t : xw → xw be a w-nonexpansive map, then the set fλ1,λ2 (t) is w-isbell-convex whenever fλ1,λ2 (t) is nonempty. © agt, upv, 2022 appl. gen. topol. 23, no. 1 103 o. olela otafudu and k. sebogodi the conclusion leads us to list some of the open problems that can be studied in future for the continuation of this work. 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[28] c. i. zhu, j. chen, x. j. huang and j. h. chen, fixed point theorems in modular spaces with simulation functions and altering distance functions with applications, j. nonlinear convex anal. 21 (2020), 1403–1424. © agt, upv, 2022 appl. gen. topol. 23, no. 1 105 () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 103-113 supersymmetry and the hopf fibration simon davis abstract the serre spectral sequence of the hopf fibration s 15 s 7 → s 8 is computed. it is used in a study of supersymmetry and actions based on this fibration. 2010 msc: 14d21, 55f20, 81q60 keywords: spectral sequence, hopf fibration, supersymmetry 1. introduction there are compactifications of eleven-dimensional supergravity with an su(3)× su(2) × u(1) isometry group of the compact space that are known to yield the particle spectrum of the standard model [1], [2]. the fermion multiplets can be included in a spinor space represented by a tensor product of division algebras for each generation. the automorphism group of this product would be g2 × su(2) × u(1) and it may be demonstrated that there are coset spaces g2×su(2)×u(1) su(3)×u(1)′×u(1)′′ yielding particles and antiparticles with the known quantum numbers [3]. the dimensions of the normed real alternative division algebras correspond to the parallelizability of the spheres. the spheres s1, s3 and s7 in the reduction sequence of the unified field theory represent submanifolds of the higherdimensional coset space. the representation of unit elements in the components of the spinor space could be related to the fermion bilinears arising in the set of light-like lines in two larger dimensions, yielding s2, s4 and s8. the unit fermions can exist in a fibre of a bundle over the space of light-like lines. amongst the s7 bundles over s8 is the hopf fibration s15 s 7 → s8. a classification of physical states described by the hopf fibrations is given in §2. 104 s. davis it has been demonstrated previously that an s7 transformation rule cannot be constructed for a pure yang-mills theory with the connection taking values only on the four-dimensional base space [4]. since twistor variables that transform under sp(4; o) can be combined to transform parameterize s7 [5], the problem of constructing a model with this invariance may be considered. this can be done only if the space s8 of lightlike lines of octonionic superparticles is interpreted in terms of fundamental variables in the theory. if the fermion field is allowed to take values in a one-point compactification of the space identified with the division algebra, an equivalence with the bosonic sector given by the light-like lines can provide a basis for a supersymmetry. this approach can be compared to an algebraic description of the supersymmetric hopf fibration. when the base space super-sphere s2∗, a supersymmetric version of the u(1) theory is found [6]. the spectral sequences of the hopf fibrations of the superspheres and the homology groups are found to unaltered by the introduction of supersymmetry in §3. the effect of an s7 transformation on fields in the twistor formalism can be elevated to an invariant action directly because there are anomalous terms in the commutators. although various spinor bilinears and combinations of supertwistors are found to be invariant, there is a associator term with a spinor field, which must be cancelled for invariance under the composition of these transformations. a method for eliminating the additional terms through the commutator of brst and gauge transformations is suggested in §4. 2. spectral sequences and hopf fibrations it may be recalled that the homology group of the total space of a fibre bundle may be determined from the serre spectral sequence. for a filtration x0 ⊂ x1 ⊂ x2 ⊂ ... ⊂ x, let d = ⊕m,nhm(xn) and e = ⊕m,nhm(xn,xn−1) define an exact couple such that im j = ker k where j : d → e and k : e → d. let d′ = i(d) and d = jk : e → e with d2 = 0. suppose that e′ = h(e;d), i′ = i|′d and the map j ′ : d′ → e′ is defined by j′(x) is the coset of j(y) in z(e), where x = i(y) ∈ d′ for y ∈ d. the map k′ completes the exact sequence and (d′,e′, i′,j′,k′) is an exact couple. further iterations give dn,en;in,jn,kn) such that dn = jnkn : e n → en, where en+1 = h(en;dn), and d = ⊕p,qdp,q and e = ⊕p,qep,q. then i(dp,q) ⊂ dp+1,q−1, j(dp,q) ⊂ ep,q and k(ep,q) ⊂ dp−1,q. if d n p,q = d n ∩ dp,q and e n p,q = h(e n−1 p,q ;dn), i(dnp,q) ⊂ d n p+1,q−1, jn(d n p,q) ⊂ e n p−n+1,q+n−1 and kn(e n p,q) ⊂ d n p−1,q, while dn : e n p,q → e n p−n,q+n−1. for a fibre bundle, e f → b, e2p,q = h(e 1 p,q;d1) = hp(b;hq(f)) [7]. consider the hopf fibration s7 s 3 → s4. e2p,q = hp(s 4;hq(s 3)) = { hp(s 4) q = 0,3 0 otherwise (2.1) = { z p = 0,4, q = 0,3 0 otherwise supersymmetry and the hopf fibration 105 the boundary mapping d4 is injective as there would exist an element y 6= d4x mapped to zero otherwise, implying that e54,0 = h 4(e44,0;d4) 6= 0. this latter statement would imply e54,0 ≃ ... ≃ e ∞ 4,0 6≃ 0 (2.2) contrary to h4(s7) ≃ 0. also, d4 is surjective, because e 5 0,3 ≃ ... ≃ e ∞ 0,3 ≃ 0 since h3(s 7) ≃ 0. it follows that d4 is an isomorphism and d4 : e 4 4,3 → e 4 0,6 is surjective. since e54,0 ≃ 0, e 5 3,0 ≃ d4(e 5 4,0) ≃ 0, removing the (0,4) and (3,0) elements in the sequence for e5p,q. the remaining non-zero entries in the e 5 p,q sequence may be deduced from exact sequences derived for filtrations of the total space. given that dp,q = 0 for p < 0 and ep,q = 0 for p < 0 or q < 0, e n p,q = 0 for p < 0, q < 0 and p+q < 0. for large n, there are the following exact sequences: →enp+n−1,q−n+2 kn → dnp+n,q−n+2 in → dnp+n−1,q−n+1 jn → enp,q kn → dnp−1,q → ... i ↓ i ↓ (2.3) 0 → dn+1p+n−2,q−n+2 in+1 → dn+1p+n−1,q−n+1 jn+1 → en+1p−1,q+1 → ... . there is a related sequence 0 → dn+1p+n−2,q−n+2 in+1 → dn+1p+n−1,q−n+1 jn → en+1p,q → ... (2.4) which holds if the domain of jn can be chosen such that jn(d n+1 p+n−1,q−n+1) = en+1p,q . since d n+1 p,q = i(d n p,q) ⊆ d n p,q, this is feasible, although jn has not been defined to have the range en+1p,q . this could be ensured, however, if en+1p,q = e n p,q. from the sequence (2.4), it follows that the exactness of the sequence enp+n,q−n+1 dn → enp,q dn → enp−n,q+n−1 (2.5) which is equivalent to 0 dn → enp,q dn → 0 (2.6) for n > p and n > q + 1, implying the constancy of enp,q for large n. from the complex of sequences 0 → dn+1p+n−2,q−n+2 in+1 → dn+1p+n−1,q−n+1 jn → en+1p,q kn → dn+1p−1,q → ... i ↓ i ↓ 0 → dn+2p+n−2,q−n+2 in+1 → dn+2p+n−1,q−n+1 jn → en+2p,q kn+2 → dn+2p−1,q → ... (2.7) ... 0 → d∞p+n−2,q−n+2 i∞ → d∞p+n−1,q−n+1 j∞ → e∞p,q k∞ → d∞p−1,q → ... 106 s. davis the last exact sequence does not end. it implies 0 → d∞p−1,q+1 i∞ → d∞p,q j1 → e∞p,q k∞ → d∞p−1,q → ... (2.8) when n = 1 is substituted in the final sequence of eq.(2.7). for the sequences, dn+1p+n−1,q−n+1 jn+1 → en+1p−1,q+1 kn+1 → dn+1p−2,q+1 → ... i ↓ dn+2p+n−1,q−n+1 jn+2 → en+2p−2,q+2 kn+2 → dn+2p−3,q+2 → ... i ↓ ... (2.9) d n+p−1 p+n−1,q−n+1 jn+p−1 → e n+p−1 1,p+q−1 kn+p−1 → d n+p−1 0,p+q−1 → ... i ↓ d n+p p+n−1,q−n+1 jn+p → e n+p 0,p+q kn+p → d n+p −1,p+q ≃ 0 and 0 → d n+p p+n−2,q−n+2 in+p → d n+p p+n−1,q−n+1 jn+p → e n+p 0,p+q kn+p → 0 (2.10) yielding eventually the sequence 0 → d∞p+n−2,q−n+2 i∞ → d∞p+n−1,q−n+1 j∞ → e∞0,p+q k∞ → 0. (2.11) with n = 1 in the indices 0 → d∞p−1,q+1 i∞ → d∞p,q j∞ → e∞0,p+q k∞ → 0 (2.12) implying e∞0,p+q ≃ d ∞ p,q/i∞(d ∞ p−1,q+1). since hp+q(xp−1) = d ∞ p−1,q+1 ⊂ hp+q(xp) = d ∞ p,q, d ∞ p−1,q+1 ⊂ d ∞ p,q ⊂ d ∞ p+1,q−1 ⊂ ... ⊂ d ∞ p+n,q−n ⊂ .... for n sufficiently large, d∞p+n,q−n = d ∞ ∩ dp+n,q−n = d ∞ ∩ hp,q(xp+n) = d∞ ∩hp,q(x) is constant, when the exhaustion of x contains a finite sequence of proper subspaces. the sequences supersymmetry and the hopf fibration 107 0 → d n+p p+n−2,q−n+2 in+p → d n+p p+n−1,q−n+1 jn+p → e n+p 0,p+q kn+p → ...0 i ↓ i ↓ 0 → d n+p+1 p+n−2,q−n+2 in+p+1 → d n+p p+n−1,q−n+1 jn+p+1 → e n+p+1 0,p+q kn+p+1 → 0 i ↓ i ↓ ... (2.13) 0 → dn+n ′ p+n−2,q−n+2 in+n′ → dn+n ′ p+n−1,q−n+1 jn+n′ → en+n ′ 0,p+q kn+n′ → 0 ... 0 → d∞p+n−2,q−n+2 i∞ → d∞p+n−1,q−n+1 j∞ → e∞0,p+q k∞ → 0 yield the isomorphisms e n+p 0,p+q ≃ d n+p p+n−1,q−n+1/in+p(d n+p p++n−2,q−n+2) ... en+n ′ 0,p+q ≃ d n+n′ p+n−1,q−n+1/in+n′(d n+n′ p+n−2,q−n+2) ... e∞0,p+q ≃ d ∞ p+n−1,q−n+1/i∞(d ∞ p+n−2,q−n+2). (2.14) it is apparent that dn+n ′ p+n−1,q−n+1 ≃ d ′ ∩dp+n−1,q−n+1 = d ′ ∩hp+q(xp+n−1) dn+n ′ p+n−2,q−n+2 ≃ d ′∩dp+n−2,q−n+2 = d ′∩hp+q(xp+n−2). (2.15) for sufficiently large n, hp+q(xp+n−1) ≃ hp+q(xp+n−2) ≃ hp+q(x) and e n+p 0,p+q ≃ [d ′ ∩hp+q(x)]/in+p(d ′ ∩hp+q(x)) ... en+n ′ 0,p+q ≃ [d ′∩hp+q(x)]/in+n′(d ′∩hp+q(x)) ... e∞0,p+q ≃ [d ′ ∩ hp+q(x)]/i∞(d ′ ∩ hp+q(x)) (2.16) are the quotient groups related to hp+q(x), and d ′ = i(d) = i(⊕p,qdp,q) = i(⊕p,qhp+q(xp)). 108 s. davis fixing p + q, ⊕p+q=constanthp+q(xp) = hp+q(x0) ⊕ hp+q(x1) ⊕ ... ⊕ hp+q(xm) ⊕ ... ⊕ hp+q(x) ≃ hp+q(x) ⊕p,qhp+q(xp) ≃ ⊕ ∞ p+q=−∞hp+q(x) ⊕∞p′+q′=0hp′+q′(x) ∩ hp+q(x) ≃ hp+q(x) (2.17) and the following isomorphisms hold: e n+p 0,p+q ≃ hp+q(x)/in+p(hp+q(x)) ... e∞0,p+q ≃ hp+q(x)/i∞(hp+q(x)). (2.18) from the sequences (2.8) and 0 → d∞p−1,q+1 ι∞ → d∞p,q j2 → e∞p−1,q+1 k∞ → d∞p−2,q+1 → .. (2.19) isomorphisms of the form e∞p−1,q+1 ≃ e ∞ p,q ≃ ... may be deduced, and e∞p,q ≃ d ∞ p,q/i∞(d ∞ p−1,q+1). by the exact sequence (2.12), i∞(d ∞ p−1,q+1) = ker j∞(d ∞ p,q) consists of the identity element, because j∞ must be an injective homomorphism as j∞(d ∞ p,q) ⊂ e ∞ p,q = e ∞ 0,p+q, and e ∞ p,q = hp+q(x). it follows that, for the hopf fibration s7 s 3 → s4, e∞p,q ≃ hp+q(s 7) and hr(s 7) ≃ { z r = 0,7 0 otherwise . (2.20) for the hopf fibration s3 s 1 → s2, there exist multi-soliton solutions parameterized by the homotopy group π3(s 2) [8]. similarly the homotopy group π7(s 4) could be used to parameterize the hopf number of soliton solutions to theories based on the next hopf fibration, as the one-soliton solutions can be combined to give n-soliton solutions. from the hurewicz theorem [9], the kth homology and homotopy groups of the sphere sk are isomorphic to z. by the exact sequence s3 → s7 → s4, it follows that π7(s 4) = π7(s 7) ⊕ π6(s 3) = z ⊕ z12 [10] [11], implying that there would be twelve varieties of each n-soliton. for the hopf fibration s15 s 7 → s8, e2p,q = hp(s 8;hq(s 7)) = { z p = 0,8, q = 0,7, p + q = 15 0 otherwise . (2.21) since every element in e87,0 is d8x, x ∈ e 8 0,8, and e 9 7,0 ≃ h7(e 8 7,0;d8) ≃ 0. similarly, d8 : e 7,0 8 → e 0,8 8 . therefore, e 0,8 9 ≃ h 8(e 0,8 8 ;d8) ≃ 0 (2.22) supersymmetry and the hopf fibration 109 therefore, e90,8 ≃ 0. again e 9 p,q ≃ ... ≃ e ∞ p,q and e∞p,q ≃ hp+q(s 15) ≃ { z p = q = 0, p = 8, q = 7, p + q = 15 0 otherwise , (2.23) which is consistent with hr(s 15) ≃ z r = 0, 15 and hr(s 15) ≃ 0 otherwise. an instanton solution to the yang-mills equations related to the last hopf fibration has been found with the euler number equal to n ∫ s8 (f ∧ f ∧ f ∧ f ∧ f)dv [12]. however, there is no reference to the gauge instanton in the euler number, which is a topological invariant that is entirely characteristic of the spheres in the fibration. this result has been explained through the equivalence of this integral with that of the pfaffian of 1 2π f̂ , where f̂µν is the field of the spinor connection [13]. the expression for the euler number is derived from the curvature form. however, the formula for the curvature form of the spinor connection is given by ωµν = eµ ∧eν [13], and it would appear that equivalence with a volume form would follow. the hopf invariant, given by the integral ∫ s15 α∧dα, where α is a volume form on s7, can be projected to ∫ s8 dαs, where αs is a singular form as a result of the intersections of the seven-spheres, which has integer values. while the hopf invariant is equal to the number of links of seven-spheres in s15, its integrality is similar, therefore, to that of the euler class, which is a generator of a homology group isomorphic to z. upon deriving an n-soliton configuration from an n-instanton solution, the classification would be given by the homotopy group π15(s 8) = π15(s 15) ⊕ π14(s 7) = z ⊕ z120 [11]. 3. supersymmetric hopf fibrations it has been shown that there is a generalization of the hopf fibration s3 s 1 → s2 to su(2)∗ u(1)∗ → s2∗, where each of the spaces is a supersphere [14]. the analogue of an element of su(2) represented by s(t) = exp(it aǫa(t)), t a = σa 2 , is s∗(t,θ) = exp(it aηa(t,θ)) ηa(t,θ) = ǫa(t) − 2θiξa(t) (3.1) from eq.(3.1), s∗(t,θ) = 1 + it a(ǫa(t) − 2θiξa(t)) − 1 2! t a(ǫa(t) − 2θiξa(t))t b(ǫb(t) − 2θiξb(t)) + ... = (1 + θσaξa(t))(1 + it aǫa(t) − 1 2! t at bǫa(t)ǫb(t) + ...) (3.2) = (1 + θξ)s(t) ξ = σaξa(t) and s † ∗(t,θ)s∗(t,θ) = s(t) †(1 − θξ)(1 + θξ)s(t) = s(t)†(1 − θξ + θξ + θξθξ)s(t) = s(t)†s(t) = 1 (3.3) 110 s. davis the action of u(1)∗ on su(2)∗ is s∗ → s∗e iσ3α and the projection from su(2) to s2, s(t) → s(t)σ3s(t) † is generalized such that x̂∗ = x̂∗aσ = s∗σ3σ † ∗ (3.4) parameterizes s2∗. this space may be compared with the supersphere s2,2 defined as osp(1|2)/u(1), which has even coordinates xi and odd coordinates θα satisfying ∑ i xixi + ∑ α,β cαβθαθβ (3.5) where c = ( 0 1 −1 0 ) [15]. the coordinates of s2 are given by yi = ( 1 + θcθ 2r2 ) (3.6) and ∑ i yiyi = r 2. the action of the u(1)∗ is right multiplication by an unitary group element and therefore identified with the action of u(1). since the supersymmetry algebra has the form {d∗,d} = 0, where d = ∂ ∂θ − iθ ∂ ∂t (3.7) is similar to the exterior derivative operator, it might be considered useful to determine de rham cohomology for the supersymmetric hopf fibration. the graded differential calculus on a supersphere can be constructed such that (ω ∧ ω′) = dω ∧ ω′ + (−1)pω ∧ dω′ ω ∈ ωp(sm,n), ω′ ∈ ωp ′ (sm,n) (3.8) where ωp(sm,n) and ωp ′ (sm,n) are exterior form algebra. and d2 = 0. as the dimension of a supermanifold belongs to z[ǫ]/(ǫ2 − 1) = z ⊕ zǫ, there is an isomorphism of the de rham cohomology of a supermanifold with that of the underlying manifold [16][17]. the de rham cohomology groups of the spheres have given hkdr(s n) ∼ { r if k = 0,n 0 if k 6= 0,n (3.9) whereas, hkdr(s n; z) ∼ { z if k = 0,n 0 if k 6= 0,n (3.10) by de rham’s theorem, there is an isomorphism between the de rham cohomology group hkdr(m) and the cohomology groups h k(m; r) for any smooth manifold. from the commutative diagram of isomorphisms, it follows that the spectral sequences based on the homology groups of spheres could be adapted to the superspheres after specializing to a specific coefficient field. consequently, the results of §2 may be used for the superspheres and each of the supersymmetry and the hopf fibration 111 supersymmetric hopf fibrations, based on the exact sequences. 0 → s3∗ → s 7 ∗ → s 4 ∗ → 0 0 → s7∗ → s 15 ∗ → s 8 ∗ → 0 (3.11) 4. the action of s7 and its supersymmetric generalization although it has been demonstrated that the principal bundle structure of gauge theories is dependent on a lie group structure, the action of s7 has been developed for twistor variables. in ten dimensions, the momentum vector of a massless particle can be expressed as p = ψψ†, where ψ is a spinor that traces out s8 . by the action of s7 on s8, consistent with that of the hopf fibration, there exists a transformation δψα = tψα = ψαo(α) such that [t,t ′]ψα = o(α)(o′(α)ψα) − o′(α)(oαψα) = ([o(α),o′(α)] − 2[o(α),o′(α),e(α)]ē(α))ψα e(α) = |ψα|−1ψα (4.1) the action is not covariant, and this prevents the construction of an entirely invariant action [18]. for a supersymmetric particle, the variables ξ = ψθ† + θψ† ω = xψ + iξθ za = (ψα,ωα̇) (4.2) may be used to construct invariants under the action of s7, ψ → ψo, ω → ωo, |o| = 1, jmn = [ 1 2 z†γmnz ] (4.3) where the square brackets refer to the selection of the e0 component. while the components of jmn, ψ1ψ̄2, ω1ω̄2, ψ1ω̄2 and ψ2ω̄1 are separately invariant, the repeated action of the s7 transformations generates additional terms through the nonvanishing associator. . one method for eliminating the extra term may be based on the construction of a charge which could cancel the associator containing either e(1) or e(2). if this is included in the s7 transformation, it would cause the transformations to be covariant. the brst charge, for example, is typically constructed such that an exact term does not affect the invariance of the lagrangian under local gauge transformations. however, this would depend on the associativity of the operations of the gauge transformation and brst transformation. through the variables transforming under sl(2; o), an additional term derived from an associator containing the two transformations would be introduced. since d = jk in the exact couple, a correspondence between spectral sequences and the theories with an operator satisfying d2 = 0 can be established. the brst cohomology of quantum field theories has been calculated previously with spectral sequences [19]. similarly, in supersymmetric models, the anticommutator of the supercharge satisfies {qα,q † α̇} = σ µ αα̇pµ, and the trace will be non-zero except for a vacuum with zero energy. the existence of a 112 s. davis spectral sequence only for the ground state is indicative of a connection with the index tr (−1)f . the necessity of the brst charge in the study of the action of s7 transformations on components of the supertwistor and the brst cohomology of ten-dimensional supergravity and superstring theories implies that the role of the anomaly, which can be determined through spectral sequences, is similar to that of the quantum terms representing lack of closure of the classical s7 algebra on spinor fields. 5. conclusion the homology groups of the total spaces in the hopf fibrations are calculated with the spectral sequence. the conditions for the equality of e∞p,q = hp+q(x) will be satisfied if enp,q is constant for sufficiently large n. the derived homology groups enp,q are found to be constant if n ≥ 5 for the fibration s 7 s 3 → s4. and n ≥ 9 for the fibration s15 s 7 → s8. the classification of solitons resulting from these fibrations are given by the homotopy groups π7(s 4) and π15(s 8) respectively. with the introduction of supersymmetry, the spectral sequences for the superspheres would have the same form. the homotopy groups determining the classification of the soliton states would follow. the description of the spaces in the third and fourth supersymmetric fibrations may be given together with a derivation of the homology and homotopy groups of the superspheres. it is suggested that an invariant action under s7 transformations can be developed if a brst charge is added to the theory. this brst charge generates another cohomology, which may be evaluated for a general quantum field theory. the consistency of the theory through a vanishing anomaly provides conditions on the matter content. references [1] l. castellani, r. d’auria and p. fre, su(3) × su(2) × u(1) from d = 11 supergravity, nucl. phys. b239 (1984), 610–652. [2] a. f. zerrouk, standard model gauge group and realistic fermions from most symmetric coset space mklm, il nuovo cim. b106 (1991), 457–500. [3] s. davis, the coset space of the unified field theory, rfsc-07-01. [4] s. davis, a constraint on the geometry of yang-mills theory, j. geom. phys. 4 (1987), 405–415. [5] m. cederwall, introduction to division algebras, sphere algebras and twistors, talk presented at the theoretical physics network meeting at nordita, copenhagen, september, 1993, hep-th/9310115. [6] g. landi and g. marmo, extensions of lie superalgebras and supersymmetric abelian gauge fields, phys. lett. b193 (1987), 61–66. [7] j.-p. serre, homologie singuliere des espaces, ann. math. ser. 2 54 (1951), 425–505. [8] r. s. ward, hopf solitons on s3 and r3, nonlinearity 12 (1999), 241–246. [9] w. hurewicz, beiträge zur topologie der deformationen. iv., neder. akad. wetensch. 39 (1936), 215–224. [10] w. hurewicz and n. steenrod, homotopy relations in fibre spaces, proc. nat. acad. sci. u.s.a. 27 (1941), 60–64. supersymmetry and the hopf fibration 113 [11] h. toda, composition methods in homotopy groups of spheres, ann. of math. studies no. 49, princeton university press, princeton, 1962. [12] b. grossman, t. w. kephart and j. d. stasheff, solutions to yang-mills field equations in eight dimensions and the last hopf map, commun. math. phys. 96 (1984), 431–437. [13] g. landi, the natural spinor connection on s8 is a gauge field, lett. math. phys. 11 (1986), 171–175. [14] a. p. balachandran, g. marmo, b.-s. skagerstam and a. stern, gauge symmetries and fibre bundles, lect. notes in physics 188, springer-verlag, berlin, 1983. [15] k. hasebe and y. kimura, fuzzy supersphere and supermonopole, nucl. phys. b709 (2005), 94–114. [16] h. grosse and g. reiter, the fuzzy supersphere, j. geom. phys. 28 (1998), 349–383. [17] d. leites, the index theorem for homogeneous differential operators on supermanifolds, math-ph/0202024. [18] m. cederwall, octonionic particles and the s7 symmetry, j. math. phys. 33 (1992), 388–393. [19] j. a. dixon, calculation of brs cohomology with spectral sequences, commun. math. phys. 139 (1991), 495–526. (received october 2011 – accepted september 2012) simon davis (sbdavis@resfdnsca.org) research foundation of southern california, 8837 villa la jolla drive #13595, la jolla, ca 92039, usa. supersymmetry and the hopf fibration. by s. davis () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 167-178 classification of separately continuous mappings with values in σ-metrizable spaces olena karlova abstract we prove that every vertically nearly separately continuous mapping defined on a product of a strong pp-space and a topological space and with values in a strongly σ-metrizable space with a special stratification, is a pointwise limit of continuous mappings. 2010 msc: 54c08, 54c05, 54e20 keywords: separately continuous mapping, σ-metrizable space, strong ppspace, baire classification, lebesgue classification 1. introduction let x, y and z be topological spaces. by c(x, y ) we denote the collection of all continuous mappings from x to y . for a mapping f : x × y → z and a point (x, y) ∈ x × y we write f x(y) = fy(x) = f(x, y). we say that a mapping f : x × y → z is separately continuous, f ∈ cc(x×y, z), if fx ∈ c(y, z) and fy ∈ c(x, z) for every point (x, y) ∈ x×y . a mapping f : x ×y → z is said to be vertically nearly separately continuous, f ∈ cc(x × y, z), if fy ∈ c(x, z) for every y ∈ y and there exists a dense set d ⊆ x such that fx ∈ c(y, z) for all x ∈ d. let b0(x, y ) = c(x, y ). assume that the classes bξ(x, y ) are already defined for all ξ < α, where α < ω1. then f : x → y is said to be of the α-th baire class, f ∈ bα(x, y ), if f is a pointwise limit of a sequence of mappings fn ∈ bξn(x, y ), where ξn < α. in particular, f ∈ b1(x, y ) if it is a pointwise limit of a sequence of continuous mappings. 168 o. karlova in 1898 h. lebesgue [12] proved that every real-valued separately continuous function of two real variables is of the first baire class. lebesgue’s theorem was generalized by many mathematicians (see [4, 15, 17, 19, 18, 1, 2, 5, 6, 16] and the references given there). w. rudin[17] showed that cc(x × y, z) ⊆ b1(x × y, z) if x is a metrizable space, y a topological space and z a locally convex topological vector space. naturally the following question has been arose, which is still unanswered. problem 1.1. let x be a metrizable space, y a topological space and z a topological vector space. does every separately continuous mapping f : x×y → z belong to the first baire class? v. maslyuchenko and a. kalancha [5] showed that the answer is positive, when x is a metrizable space with finite čech-lebesgue dimension. t. banakh [1] gave a positive answer in the case that x is a metrically quarter-stratifiable paracompact strongly countably dimensional space and z is an equiconnected space. in [8] it was shown that the answer to problem 1.1 is positive for metrizable spaces x and y and a metrizable arcwise connected and locally arcwise connected space z. it was pointed out in [9] that cc(x × y, z) ⊆ b1(x × y, z) if x is a metrizable space, y is a topological space and z is an equiconnected strongly σ-metrizable space with a stratification (zn) ∞ n=1 (see the definitions below), where zn is a metrizable arcwise connected and locally arcwise connected space for every n ∈ n. in this paper we generalize the above-mentioned result from [9] to the case of vertically nearly separately continuous mappings. to do this, we introduce the class of strong pp-spaces which includes the class of all metrizable spaces. in section 3 we investigate some properties of strong pp-spaces. in section 4 we establish an auxiliary result which generalizes the famous kuratowskimontgomery theorem (see [11] and [14]). finally, in section 5 we prove that the inclusion cc(x ×y, z) ⊆ b1(x ×y, z) holds if x is a strongly pp-space, y is a topological space and z is a contractible space with a stratification (zn) ∞ n=1, where zn is a metrizable arcwise connected and locally arcwise connected space for every n ∈ n. 2. preliminary observations a subset a of a topological space x is a zero (co-zero) set if a = f−1(0) (a = f−1((0, 1])) for some continuous function f : x → [0, 1]. let g∗0 and f ∗ 0 be collections of all co-zero and zero subsets of x, respectively. assume that the classes g∗ξ and f ∗ ξ are defined for all ξ < α, where 0 < α < ω1. then, if α is odd, the class g∗α (f ∗ α) is consists of all countable intersections (unions) of sets of lower classes, and, if α is even, the class g∗α (f ∗ α) is consists of all countable unions (intersections) of sets of lower classes. the classes f∗α for odd α and g∗α for even α are said to be functionally additive, and the classes f∗α for even α and g ∗ α for odd α are called functionally multiplicative. if a set belongs to the α’th functionally additive and functionally multiplicative class, classification of separately continuous mappings 169 then it is called functionally ambiguous of the α’th class. note that a ∈ f∗α if and only if x \ a ∈ g∗α. if a set a is of the first functionally additive (multiplicative) class, we say that a is an f ∗σ (g ∗ δ) set. let us observe that if x is a perfectly normal space (i.e. a normal space in which every closed subset is gδ), then functionally additive and functionally multiplicative classes coincide with ordinary additive and multiplicative classes respectively, since every open set in x is functionally open. lemma 2.1. let α ≥ 0, x be a topological space and let a ⊆ x be of the α’th functionally multiplicative class. then there exists a function f ∈ bα(x, [0, 1]) such that a = f−1(0). proof. the hypothesis of the lemma is obvious if α = 0. suppose the assertion of the lemma is true for all ξ < α and let a be a set of the α’th functionally multiplicative class. then a = ∞ ⋂ n=1 an, where an belong to the αn’th functionally additive class with αn < α for all n ∈ n. by assumption, there exists a sequence of functions fn ∈ bαn(x, [0, 1]) such that an = f −1 n ((0, 1]). notice that for every n the characteristic function χan of an belongs to the α-th baire class. indeed, setting hn,m(x) = m √ fn(x), we obtain a sequence of functions hn,m ∈ bαn(x, [0, 1]) which is pointwise convergent to χan. now let f(x) = 1 − ∞ ∑ n=1 1 2n χan(x). for all x ∈ x. then f ∈ bα(x, [0, 1]) as a sum of a uniform convergent series of functions of the α’th class. moreover, it is easy to see that a = f−1(0). � a topological space x is called • equiconnected if there exists a continuous function λ : x × x × [0, 1] → x such that (1) λ(x, y, 0) = x; (2) λ(x, y, 1) = y; (3) λ(x, x, t) = x for all x, y ∈ x and t ∈ [0, 1]. • contractible if there exist x∗ ∈ x and a continuous mapping γ : x × [0, 1] → x such that γ(x, 0) = x and γ(x, 1) = x∗. a contractible space x with such a point x∗ and such a mapping γ is denoted by (x, x∗, γ). remark that every convex subset x of a topological vector space is equiconnected, where λ : x × x × [0, 1] → x is defined by the formula λ(x, y, t) = (1 − t)x + ty, x, y ∈ x, t ∈ [0, 1]. it is easily seen that a topological space x is contractible if and only if there exists a continuous mapping λ : x × x × [0, 1] → x such that λ(x, y, 0) = x and λ(x, y, 1) = y for all x, y ∈ x. indeed, if (x, x∗, γ) is a contractible space, 170 o. karlova then the formula λ(x, y, t) = { γ(x, 2t), 0 ≤ t ≤ 1 2 , γ(y, −2t + 2), 1 2 < t ≤ 1. defines a continuous mapping λ : x ×x ×[0, 1] → x with the required properties. conversely, if x is equiconnected, then fixing a point x∗ ∈ x and setting γ(x, t) = λ(x, x∗, t), we obtain that the space (x, x∗, γ) is contractible. lemma 2.2. let 0 ≤ α < ω1, x a topological space, y a contractible space, a1, . . . , an be disjoint sets of the α’th functionally multiplicative class in x and fi ∈ bα(x, y ) for each 1 ≤ i ≤ n. then there exists a mapping f ∈ bα(x, y ) such that f|ai = fi for each 1 ≤ i ≤ n. proof. let n = 2. in view of lemma 2.1 there exist functions hi ∈ bα(x, [0, 1]) such that ai = h −1 i (0) for i = 1, 2. we set h(x) = h1(x) h1(x) + h2(x) for all x ∈ x. it is easy to verify that h ∈ bα(x, [0, 1]) and ai = h −1(i − 1), i = 1, 2. consider a continuous mapping λ : y ×y ×[0, 1] → y such that λ(y, z, 0) = y and λ(y, z, 1) = z for all y, z ∈ y . let f(x) = λ(f1(x), f2(x), h(x)) for every x ∈ x. clearly, f ∈ bα(x, y ). if x ∈ a1, then f(x) = λ(f1(x), f2(x), 0) = f1(x). if x ∈ a2, then f(x) = λ(f1(x), f2(x), 1) = f2(x). assume that the lemma is true for all 2 ≤ k < n and let k = n. according to our assumption, there exists a mapping g ∈ bα(x, y ) such that g|ai = fi for all 1 ≤ i < n. since a = n−1 ⋃ i=1 ai and an are disjoint sets which belong to the α’th functionally multiplicative class in x, by the assumption, there is a mapping f ∈ bα(x, y ) with f|a = g and f|fn = fn. then f|fi = fi for every 1 ≤ i ≤ n. � let 0 ≤ α < ω1. we say that a mapping f : x → y is of the (functional) α-th lebesgue class, f ∈ hα(x, y ) (f ∈ h ∗ α(x, y )), if the preimage f −1(v ) belongs to the α’th (functionally) additive class in x for any open set v ⊆ y . clearly, hα(x, y ) = h ∗ α(x, y ) for any perfectly normal space x. the following statement is well-known, but we present a proof here for convenience of the reader. lemma 2.3. let x and y be topological spaces, (fk) ∞ k=1 a sequence of mappings fk : x → y which is pointwise convergent to a mapping f : x → y , f ⊆ y be a closed set such that f = ∞ ⋂ n=1 v n, where (vn) ∞ n=1 is a sequence of open sets in y such that vn+1 ⊆ vn for all n ∈ n. then (2.1) f−1(f) = ∞ ⋂ n=1 ∞ ⋃ k=n f −1 k (vn). classification of separately continuous mappings 171 proof. let x ∈ f−1(f) and n ∈ n. taking into account that vn is an open neighborhood of f(x) and lim k→∞ fk(x) = f(x), we obtain that there is k ≥ n such that fk(x) ∈ vn. now let x belong to the right-hand side of (2.1), i.e. for every n ∈ n there exists a number k ≥ n such that fk(x) ∈ vn. suppose f(x) 6∈ f . then there exists n ∈ n such that f(x) 6∈ vn. since u = x \ vn is a neighborhood of f(x), there exists k0 such that fk(x) ∈ u for all k ≥ k0. in particular, fk(x) ∈ u for k = max{k0, n}. but then fk(x) 6∈ vn, a contradiction. hence, x ∈ f −1(f). � lemma 2.4. let x be a topological space, y a perfectly normal space and 0 ≤ α < ω1. then bα(x, y ) ⊆ h ∗ α(x, y ) if α is finite, and bα(x, y ) ⊆ h∗α+1(x, y ) if α is infinite. proof. let f ∈ bα(x, y ). fix an arbitrary closed set f ⊆ y . since y is perfectly normal, there exists a sequence of open sets vn ⊆ y such that vn+1 ⊆ vn and f = ∞ ⋂ n=1 v n. moreover, there exists a sequence of mappings fk : x → y of baire classes < α which is pointwise convergent to f on x. by lemma 2.3, equality (2.1) holds. now put an = ∞ ⋃ k=n f −1 k (vn). if α = 0, then f is continuous and f−1(f) is a zero set in x, since f is a zero set in y . suppose the assertion of the lemma is true for all finite ordinals 1 ≤ ξ < α. we show that it is true for α. remark that fk ∈ bα−1(x, y ) for every k ≥ 1. by assumption, fk ∈ h ∗ α−1(x, y ) for every k ∈ n. then an is of the functionally additive class α − 1. therefore, f−1(f) belongs to the α’th functionally multiplicative class. assume the assertion of the lemma is true for all ordinals ω0 ≤ ξ < α. for all k ∈ n we choose αk < α such that fk ∈ bαk (x, y ) for every k ≥ 1. the preimage f−1 k (vn), being of the (αk + 1)’th functionally additive class, belongs to the α’th functionally additive class for all k, n ∈ n, provided αk + 1 ≤ α. then an is of the α’th functionally additive class, hence, f −1(f) belongs to the (α + 1)’th functionally multiplicative class. � recall that a family a = (ai : i ∈ i) of sets ai refines a family b = (bj : j ∈ j) of sets bj if for every i ∈ i there exists j ∈ j such that ai ⊆ bj. we write in this case a � b. 3. pp-spaces and their properties definition 3.1. a topological space x is said to be a (strong) pp-space if (for every dense set d in x) there exist a sequence ((ϕi,n : i ∈ in)) ∞ n=1 of locally finite partitions of unity on x and a sequence ((xi,n : i ∈ in)) ∞ n=1 of families of points of x (of d) such that (3.1) (∀x ∈ x)((∀n ∈ n x ∈ suppϕin,n) =⇒ (xin,n → x)) remark that definition 3.1 is equivalent to the following one. 172 o. karlova definition 3.2. a topological space x is a (strong) pp-space if (for every dense set d in x) there exist a sequence ((ui,n : i ∈ in)) ∞ n=1 of locally finite covers of x by co-zero sets ui,n and a sequence ((xi,n : i ∈ in)) ∞ n=1 of families of points of x (of d) such that (3.2) (∀x ∈ x)((∀n ∈ n x ∈ uin,n) =⇒ (xin,n → x)) clearly, every strong pp-space is a pp-space. proposition 3.3. every metrizable space is a strong pp-space. proof. let x be a metrizable space and d a metric on x which generates its topology. fix an arbitrary dense set d in x. for every n ∈ n let bn be a cover of x by open balls of diameter 1 n . since x is paracompact, for every n there exists a locally finite cover un = (ui,n : i ∈ in) of x by open sets ui,n such that un � bn. notice that each ui,n is a co-zero set. choose a point xi,n ∈ d ∩ui,n for all n ∈ n and i ∈ in. let x ∈ x and let u be an arbitrary neighborhood of x. then there is n0 ∈ n such that b(x, 1 n ) ⊆ u for all n ≥ n0. fix n ≥ n0 and take i ∈ in such that x ∈ ui,n. since diam ui,n ≤ 1 n , d(x, xi,n) ≤ 1 n , consequently, xi,n ∈ u. � example 3.4. the sorgenfrey line l is a strong pp-space which is not metrizable. proof. recall that the sorgenfrey line is the real line r endowed with the topology generated by the base consisting of all semi-intervals [a, b), where a < b (see [3, example 1.2.2]). let d ⊆ l be a dense set. for any n ∈ n and i ∈ z by ϕi,n we denote the characteristic function of [i−1 n , i n ) and choose a point xi,n ∈ [ i n , i+1 n )∩d. then the sequences ( (ϕi,n : i ∈ in) )∞ n=1 and ( (xi,n : i ∈ in) )∞ n=1 satisfy (3.1). � proposition 3.5. every σ-metrizable paracompact space is a pp-space. proof. let x = ∞ ⋃ n=1 xn, where (xn) ∞ n=1 is an increasing sequence of closed metrizable subspaces, and let d1 be a metric on x1 which generates its topology. according to hausdorff’s theorem [3, p. 297] we can extend the metric d1 to a metric d2 on x2. further, we extend the metric d2 to a metric d3 on x3. repeating this process, we obtain a sequence (dn) ∞ n=1 of metrics dn on xn such that dn+1|xn = dn for every n ∈ n. we define a function d : x 2 → r by setting d(x, y) = dn(x, y) for (x, y) ∈ x 2 n. fix n ∈ n and m ≥ n. let bn,m be a cover of xm by d-open balls of diameter 1 n . for every b ∈ bn,m there exists an open set vb in x such that vb ∩ xm = b. let vn,m = {vb : b ∈ bn,m} and un = ∞ ⋃ m=1 vn,m. then un is an open cover of x for every n ∈ n. since x is paracompact, for every n ∈ n there exists a locally finite partition of unity (hi,n : i ∈ in) on x subordinated to un. for every n ∈ n and i ∈ in we choose xi,n ∈ xk(i,n) ∩ supp hi,n, where k(i, n) = min{m ∈ n : xm ∩ supp hi,n 6= ø}. classification of separately continuous mappings 173 now fix x ∈ x. let (in)n=1 be a sequence of indexes in ∈ in such that x ∈ supp hin,n. we choose m ∈ n such that x ∈ xm. it is easy to see that k(in, n) ≤ m for every n ∈ n. then xin,n ∈ xm. since dm(xin,n, x) ≤ diam supp hin,n ≤ 1 n , xin,n → x in xm. therefore, xin,n → x in x. � denote by r∞ the collection of all sequences with a finite support, i.e. sequences of the form (ξ1, ξ2, . . . , ξn, 0, 0, . . . ), where ξ1, ξ2, . . . , ξn ∈ r. clearly, r ∞ is a linear subspace of the space rn of all sequences. denote by e the set of all sequences e = (εn) ∞ n=1 of positive reals εn and let ue = {x = (ξn) ∞ n=1 ∈ r ∞ : (∀n ∈ n)(|ξn| ≤ εn)}. we consider on r∞ the topology in which the system u0 = {ue : e ∈ e} forms the base of neighborhoods of zero. then r∞ equipped with this topology is a locally convex σ-metrizable paracompact space which is not a first countable space, consequently, non-metrizable. example 3.6. the space r∞ is a pp-space which is not a strong pp-space. proof. remark that r∞ is a pp-space by proposition 3.5. we show that r∞ is not a strong pp-space. indeed, let an = {(ξ1, ξ2, . . . , ξn, 0, 0, . . . ) : |ξk| ≤ 1 n (∀1 ≤ k ≤ n)}, d = ∞ ⋃ m=1 m ⋂ n=1 (r∞ \ an). then d is dense in r∞, but there is no sequence in d which converges to x = (0, 0, 0, . . .) ∈ r∞. hence, r∞ is not a strong pp-space. � 4. the lebesgue classification the following result is an analog of theorems of k. kuratowski [11] and d. montgomery [14] who proved that every separately continuous function, defined on a product of a metrizable space and a topological space and with values in a metrizable space, belongs to the first baire class. theorem 4.1. let x be a strong pp-space, y a topological space, z a perfectly normal space and 0 ≤ α < ω1. then ch∗α(x × y, z) ⊆ h ∗ α+1(x × y, z). proof. let f ∈ ch∗α(x ×y, z). then for the set xh∗α(f) there exist a sequence (un) ∞ n=1 of locally finite covers un = (ui,n : i ∈ in) of x by co-zero sets ui,n and a sequence ((xi,n : i ∈ in)) ∞ n=1 of families of points of the set xh∗α(f) satisfying condition (3.2). we choose an arbitrary closed set f ⊆ z. since z is perfectly normal, f = ∞ ⋂ m=1 gm, where gm are open sets in z such that gm+1 ⊆ gm for every 174 o. karlova m ∈ n. let us verify that the equality (4.1) f−1(f) = ∞ ⋂ m=1 ∞ ⋃ n≥m ⋃ i∈in ui,n × (f xi,n)−1(gm). holds. indeed, let (x0, y0) ∈ f −1(f). then f(x0, y0) ∈ gm for every m ∈ n. fix any m ∈ n. since vm = f −1 y0 (gm) is an open neighborhood of x0, there exists a number n0 ≥ m such that for all n ≥ n0 and i ∈ in the inclusion xi,n ∈ vm holds whenever x0 ∈ ui,n. we choose i0 ∈ in0 such that x0 ∈ ui0,n0. then f(xi0,n0, y0) ∈ gm. hence, (x0, y0) belongs to the right-hand side of (4.1). conversely, let (x0, y0) belong to the right-hand side of (4.1). fix m ∈ n. we choose sequences (nk) ∞ k=1, (mk) ∞ k=1 of numbers nk, mk ∈ n and a sequence (ik) ∞ k=1 of indexes ik ∈ ink such that m = m1 ≤ n1 < m2 ≤ n2 < · · · < mk ≤ nk < . . . , x0 ∈ uik,nk and f(xik,nk, y0) ∈ gmk ⊆ gm for every k ∈ n. since lim k→∞ xik,nk = x0 and the mapping f is continuous with respect to the first variable, lim k→∞ f(xik,nk, y0) = f(x0, y0). therefore, f(x0, y0) ∈ gm for every m ∈ n. hence, (x0, y0) belongs to the left-hand side of (4.1). since fxi,n ∈ h∗α(y, z), the sets (f xi,n)−1(gm) are of the functionally additive class α in y . moreover, all ui,n are co-zero sets in x, consequently, by [6, theorem 1.5] the set en = ⋃ i∈in ui,n × (f xi,n)−1(gm) belongs to the α’th functionally additive class for every n. therefore, ⋃ n≥m en is of the α’th functionally additive class. hence, f−1(f) is of the (α + 1)’th functionally multiplicative class in x × y . � definition 4.2. we say that a topological space x has the (strong) l-property or is a (strong) l-space, if for every topological space y every (nearly vertically) separately continuous function f : x × y → r is of the first lebesgue class. according to theorem 4.1 every strong pp-space has the strong l-property. proposition 4.3. let x be a completely regular strong l-space. then for any dense set a ⊆ x and a point x0 ∈ x there exists a countable dense set a0 ⊆ a such that x0 ∈ a0. proof. fix an arbitrary everywhere dense set a ⊆ x and a point x0 ∈ a. let y be the space of all real-valued continuous functions on x, endowed with the topology of pointwise convergence on a. since the evaluation function e : x × y → r, e(x, y) = y(x), is nearly vertically separately continuous, e ∈ h1(x × y, r). then b = e −1(0) is gδ-set in x × y . hence, b0 = {y ∈ y : y(x0) = 0} is a gδ-set in y . we set y0 ≡ 0 and choose a sequence (vn) ∞ n=1 of basic neighborhoods of y0 in y such that ∞ ⋂ n=1 vn ⊆ b0. for every n there classification of separately continuous mappings 175 exist a finite set {xi,n : i ∈ in} of x and εn > 0 such that vn = {y ∈ y : max i∈in |y(xi,n)| < εn}. let a0 = ⋃ n∈n ⋃ i∈in {xi,n}. take an open neighborhood u of x0 in x and suppose that u ∩ a0 = ø. since x is completely regular and x0 6∈ x \ u, there exists a continuous function y : x → r such that y(x0) = 1 and y(x \ u) ⊆ {0}. then y ∈ ∞ ⋂ n=1 vn, but y 6∈ b0, a contradiction. therefore, u ∩ a0 6= ø, and x0 ∈ a0. � 5. baire classification and σ-metrizable spaces we recall that a topological space y is b-favorable for a space x, if h1(x, y ) ⊆ b1(x, y ) (see [10]). definition 5.1. let 0 ≤ α < ω1. a topological space y is called weakly bα-favorable for a space x, if h ∗ α(x, y ) ⊆ bα(x, y ). clearly, every b-favorable space is weakly b1-favorable. proposition 5.2. let 0 ≤ α < ω1, x a topological space, y = ∞ ⋃ n=1 yn a contractible space, f : x → y a mapping, (xn) ∞ n=1 a sequence of sets of the α’th functionally additive class such that x = ∞ ⋃ n=1 xn and f(xn) ⊆ yn for every n ∈ n. if one of the following conditions holds (i) yn is a nonempty weakly bα-favorable space for x for all n and f ∈ h∗α(x, y ), or (ii) α > 0 and for every n there exists a mapping fn ∈ bα(x, yn) such that fn|xn = f|xn, then f ∈ bα(x, y ). proof. if α = 0 then the statement is obvious in case (i). let α > 0. by [6, lemma 2.1] there exists a sequence (en) ∞ n=1 of disjoint functionally ambiguous sets of the α’th class such that en ⊆ xn and x = ∞ ⋃ n=1 en. in case (i) for every n we choose a point yn ∈ yn and let fn(x) = { f(x), if x ∈ en, yn, if x ∈ x \ en. since f ∈ h∗α(x, y ) and en is functionally ambiguous set of the α’th class, fn ∈ h ∗ α(x, yn). then fn ∈ bα(x, yn) provided yn is weakly bα-favorable for x. for every n there exists a sequence of mappings gn,m : x → yn of classes < α such that gn,m(x) → m→∞ fn(x) for every x ∈ x. in particular, lim m→∞ gn,m(x) = 176 o. karlova f(x) on en. since en is of the α-th functionally additive class, en = ∞ ⋃ m=1 bn,m, where (bn,m) ∞ m=1 is an increasing sequence of sets of functionally additive classes < α. let fn,m = ø if n > m, and let fn,m = bn,m if n ≤ m. according to lemma 2.2, for every m ∈ n there exists a mapping gm : x → y of a class < α such that gm|fn,m = gn,m, since the system {fn,m : n ∈ n} is finite for every m ∈ n. it remains to prove that gm(x) → f(x) on x. let x ∈ x. we choose a number n ∈ n such that x ∈ en. since the sequence (fn,m) ∞ m=1 is increasing, there exists a number m0 such that x ∈ fn,m for all m ≥ m0. then gm(x) = gn,m(x) for all m ≥ m0. hence, lim m→∞ gm(x) = lim m→∞ gn,m(x) = f(x). therefore, f ∈ bα(x, y ). � definition 5.3. let {xn : n ∈ n} be a cover of a topological space x. we say that (x, (xn) ∞ n=1) has the property (∗) if for every convergent sequence (xk) ∞ k=1 in x there exists a number n such that {xk : k ∈ n} ⊆ xn. proposition 5.4. let 0 ≤ α < ω1, x a strong pp-space, y a topological space, (z, (zn) ∞ n=1) have the property (∗), let zn be closed in z (and let zn be a zero-set in z if α = 0) for every n ∈ n, and f ∈ cbα(x × y, z). then there exists a sequence (bn) ∞ n=1 of sets of the α’th /(α + 1)’th/ functionally multiplicative class in x × y , if α is finite /infinite/, such that ∞ ⋃ n=1 bn = x × y and f(bn) ⊆ zn for every n ∈ n. proof. since xbα(f) is dense in x, there exists a sequence (um = (ui,m : i ∈ im)) ∞ m=1 of locally finite co-zero covers of x and a sequence ((xi,m : i ∈ im)) ∞ m=1 of families of points of xbα(f) such that condition (3.2) holds. in accordance with [16, proposition 3.2] there exists a pseudo-metric on x such that all the set ui,m are co-zero with respect to this pseudo-metric. denote by t the topology on x generated by the pseudo-metric. obviously, the topology t is weaker than the initial one. using the paracompactness of (x, t ), for every m we choose a locally finite open cover vm = (vs,m : s ∈ sm) which refines um. by [3, lemma 1.5.6], for every m there exists a locally finite closed cover (fs,m : s ∈ sm) of (x, t ) such that fs,m ⊆ vs,m for every s ∈ sm. now for every s ∈ sm we choose i(s) ∈ im such that fs,m ⊆ ui(s),m. for all m, n ∈ n and s ∈ sm let as,m,n = (f xi(s),m)−1(zn), bm,n = ⋃ s∈sm (fs,m × as,m,n), bn = ∞ ⋂ m=1 bm,n. since f is of the α’th baire class with respect to the second variable, for every n the set as,m,n belongs to the α’th functionally multiplicative class /α+1/ in y for all m ∈ n and s ∈ sm, if α is finite /infinite/ by lemma 2.4. according to [6, proposition 1.4] the set bm,n is of the α’th /(α + 1)’th/ functionally classification of separately continuous mappings 177 multiplicative class in (x, t ) × y . then the set bn is of the α’th /(α + 1)’th/ functionally multiplicative class in (x, t )× y , and, consequently, in x × y for every n. we prove that f(bn) ⊆ zn for every n. to do this, fix n ∈ n and (x, y) ∈ bn. we choose a sequence (sm) ∞ m=1 such that x ∈ fm,sm ⊆ um,i(sm) and f(xm,i(sm), y) ∈ zn. then xm,i(sm) → m→∞ x. since f is continuous with respect to the first variable, f(xm,i(sm), y) → m→∞ f(x, y). the set zn is closed, then f(x, y) ∈ zn. now we show that ∞ ⋃ n=1 bn = x ×y . let (x, y) ∈ x ×y . then there exists a sequence (sm) ∞ m=1 such that x ∈ fm,sm ⊆ um,i(sm) and f(xm,i(sm), y) → m→∞ f(x, y). since (z, (zn) ∞ n=1) satisfies (∗), there is a number n such that {f(xm,im, y) : m ∈ n} is contained in zn, i.e. y ∈ am,n,i for every m ∈ n. hence, (x, y) ∈ bn. � theorem 5.5. let x be a strong pp-space, y a topological space, {zn : n ∈ n} a closed cover of a contractible perfectly normal space z, let (z, (zn) ∞ n=1) satisfy (∗) and zn be weakly b1-favorable for x × y for every n ∈ n. then cc(x × y, z) ⊆ b1(x × y, z). proof. let f ∈ cc(x × y, z). in accordance with theorem 4.1, f ∈ h∗1 (x × y, z). moreover, proposition 5.4 implies that there exists a sequence of zerosets bn ⊆ x × y such that ∞ ⋃ n=1 bn = x × y and f(bn) ⊆ zn for every n ∈ n. since for every n the set bn is an f ∗ σ -set and h ∗ 1 (x × y, zn) ⊆ b1(x × y, zn), f ∈ b1(x × y, z) by proposition 5.2. � definition 5.6. a topological space x is called strongly σ-metrizable, if it is σmetrizable with a stratification (xn) ∞ n=1 and (x, (xn) ∞ n=1) has the property (∗). taking into account that every regular strongly σ-metrizable space with metrizable separable stratification is perfectly normal (see [13, corollary 4.1.6]) and every metrizable separable arcwise connected and locally arcwise connected space is weakly bα-favorable for any topological space x for all 0 ≤ α < ω1 [7, theorem 3.3.5], we immediately obtain the following corollary of theorem 5.5. corollary 5.7. let x be a strong pp-space, y a topological space and z a contractible regular strongly σ-metrizable space with a stratification (zn) ∞ n=1, where zn is a metrizable separable arcwise connected and locally arcwise connected space for every n ∈ n. then cc(x × y, z) ⊆ b1(x × y, z). 178 o. karlova references [1] t. banakh, (metrically) quarter-stratifiable spaces and their applications, math. stud. 18, no. 1 (2002), 10–28. [2] m. burke, borel measurability of separately continuous functions, topology appl. 129, no. 1 (2003), 29–65. [3] r. engelking, general topology. revised and completed edition. heldermann verlag, berlin (1989). [4] h. hahn, reelle funktionen.1.teil. punktfunktionen., leipzig: academische verlagsgesellscheft m.b.h. (1932). [5] a. kalancha and v. maslyuchenko, čech-lebesgue dimension and baire classification of vector-valued separately continuous mappings, ukr. math. j. 55, no. 11 (2003), 1596– 1599 (in ukrainian). [6] o. karlova, baire classification of mappings which are continuous in the first variable and of the functional class α in the second one, math. bull. ntsh. 2 (2005), 98–114 (in ukrainian). [7] o. karlova, baire and lebesgue classification of vector-values and multi-valued mappings, phd thesis (2006) (in ukrainian). [8] o. karlova, separately continuous σ-discrete mappings, bull. of chernivtsi nat. univ., mathematics 314–315 (2006), 77–79 (in ukrainian). [9] o. karlova and v. maslyuchenko, separately continuous mappings with values in non locally convex spaces, ukr. math. j. 59, no. 12 (2007), 1639–1646 (in ukrainian). [10] o. karlova and v. mykhaylyuk, weak local homeomorphisms and b-favorable spaces, ukr. math. j. 60, no. 9 (2008), 1189–1195 (in ukrainian). [11] k. kuratowski, quelques probémes concernant les espaces métriques non-séparables, fund. math. 25 (1935), 534–545. [12] h. lebesgue, sur l’approximation des fonctions, bull. sci. math. 22 (1898), 278–287. [13] v. maslyuchenko, separately continuous mappings and köthe spaces, doctoral thesis (1999) (in ukrainian). [14] d. montgomery, non-separable metric spaces, fund. math. 25 (1935), 527–533. [15] w. moran, separate continuity and supports of measures, j. london math. soc. 44 (1969), 320–324. [16] v. mykhaylyuk, baire classification of separately continuous functions and namioka property, ukr. math. bull. 5, no. 2 (2008), 203–218 (in ukrainian). 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(received december 2011 – accepted may 2012) o. karlova (maslenizza.ua@gmail.com) chernivtsi national university, department of mathematical analysis, kotsjubyns’koho 2, chernivtsi 58012, ukraine classification of separately continuous mappings[4pt] with values in -metrizable spaces. by o. karlova @ appl. gen. topol. 21, no. 1 (2020), 87-110 doi:10.4995/agt.2020.12091 c© agt, upv, 2020 fixed point sets in digital topology, 1 laurence boxer a and p. christopher staecker b a department of computer and information sciences, niagara university, niagara university, ny 14109, usa; and department of computer science and engineering, suny at buffalo (boxer@niagara.edu) b department of mathematics, fairfield university, fairfield, ct 06823-5195,usa. (cstaecker@fairfield.edu) communicated by v. gregori abstract in this paper, we examine some properties of the fixed point set of a digitally continuous function. the digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology. we introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. perhaps the most important of these is the fixed point spectrum f(x) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. we give a complete computation of f(cn) where cn is the digital cycle of n points. for other digital images, we show that, if x has at least 4 points, then f(x) always contains the numbers 0, 1, 2, 3, and the cardinality of x. we give several examples, including cn, in which f(x) does not equal {0, 1, . . . , #x}. we examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and cartesian products. we also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected. 2010 msc: 54h25. keywords: digital image; fixed point; retraction. received 16 july 2019 – accepted 08 october 2019 http://dx.doi.org/10.4995/agt.2020.12091 l. boxer and p. christopher staecker 1. introduction digital images are often used as mathematical models of real-world objects. a digital model of the notion of a continuous function, borrowed from the study of topology, is often useful for the study of digital images. however, a digital image is typically a finite, discrete point set. thus, it is often necessary to study digital images using methods not directly derived from topology. in this paper, we introduce several such methods to study properties of the fixed point set of a continuous self-map on a digital image. many of our results have elementary proofs. their importance is, in part, due to the following. digital topology has been successful in showing that digital images resemble the euclidean objects they model with respect to topological properties such as connectedness, homotopy, covering maps, fundamental groups, retractions, and homology; however, we see in this paper that the fixed point properties of digital images and the euclidean objects they model can be very different. some of the results of this paper were presented in [7]. 2. preliminaries let n denote the set of natural numbers; and z, the set of integers. #x will be used for the number of elements of a set x. 2.1. adjacencies. a digital image is a pair (x,κ) where x ⊂ zn for some n and κ is an adjacency on x. thus, (x,κ) is a graph for which x is the vertex set and κ determines the edge set. usually, x is finite, although there are papers that consider infinite x. usually, adjacency reflects some type of “closeness” in zn of the adjacent points. when these “usual” conditions are satisfied, one may consider the digital image as a model of a black-and-white “real world” digital image in which the black points (foreground) are the members of x and the white points (background) are members of zn \x. we write x ↔κ y, or x ↔ y when κ is understood or when it is unnecessary to mention κ, to indicate that x and y are κ-adjacent. notations x -κ y, or x y when κ is understood, indicate that x and y are κ-adjacent or are equal. the most commonly used adjacencies are the cu adjacencies, defined as follows. let x ⊂ zn and let u ∈ z, 1 ≤ u ≤ n. then for points x = (x1, . . . ,xn) 6= (y1, . . . ,yn) = y we have x ↔cu y if and only if • for at most u indices i we have |xi −yi| = 1, and • for all indices j, |xj −yj| 6= 1 implies xj = yj. the cu-adjacencies are often denoted by the number of adjacent points a point can have in the adjacency. e.g., • in z, c1-adjacency is 2-adjacency; • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency; c© agt, upv, 2020 appl. gen. topol. 21, no. 1 88 fixed point sets in digital topology, 1 • in z3, c1-adjacency is 8-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. we discuss the digital n-cycle, the n-point image cn = {x0, . . . ,xn−1} in which each xi is adjacent only to xi+1 and xi−1, and subscripts are always read modulo n. the literature also contains several adjacencies to exploit properties of cartesian products of digital images. these include the following. definition 2.1 ([1]). let (x,κ) and (y,λ) be digital images. the normal product adjacency or strong adjacency on x ×y , denoted np(κ,λ), is defined as follows. given x0,x1 ∈ x, y0,y1 ∈ y such that p0 = (x0,y0) 6= (x1,y1) = p1, we have p0 ↔np(κ,λ) p1 if and only if one of the following is valid: • x0 ↔κ x1 and y0 = y1, or • x0 = x1 and y0 ↔λ y1, or • x0 ↔κ x1 and y0 ↔λ y1. theorem 2.2 ([9]). let x ⊂ zm, y ⊂ zn. then (x ×y,np(cm,cn)) = (x ×y,cm+n), i.e., the cm+n-adjacency on x ×y ⊂ zm+n coincides with the normal product adjacency based on cm and cn. building on the normal product adjacency, we have the following. definition 2.3 ([5]). given u,v ∈ n, 1 ≤ u ≤ v, and digital images (xi,κi), 1 ≤ i ≤ v, let x = πvi=1xi. the adjacency npu(κ1, . . . ,κv) for x is defined as follows. given xi,x ′ i ∈ xi, let p = (x1, . . . ,xv) 6= (x′1, . . . ,x ′ v) = q. then p ↔npu(κ1,...,κv) q if for at least 1 and at most u indices i we have xi ↔κi x′i and for all other indices j we have xj = x ′ j. notice np(κ,λ) = np2(κ,λ) [5]. when (x,κ) is understood to be a digital image under discussion, we use the following notations. for x ∈ x, n(x) = {y ∈ x |y ↔κ x}, n∗(x) = {y ∈ x |y -κ x} = n(x) ∪{x}. 2.2. digitally continuous functions. we denote by id or idx the identity map id(x) = x for all x ∈ x. definition 2.4 ([14, 3]). let (x,κ) and (y,λ) be digital images. a function f : x → y is (κ,λ)-continuous, or digitally continuous or just continuous when κ and λ are understood, if for every κ-connected subset x′ of x, f(x′) is a λ-connected subset of y . if (x,κ) = (y,λ), we say a function is κ-continuous to abbreviate “(κ,κ)-continuous.” c© agt, upv, 2020 appl. gen. topol. 21, no. 1 89 l. boxer and p. christopher staecker theorem 2.5 ([3]). a function f : x → y between digital images (x,κ) and (y,λ) is (κ,λ)-continuous if and only if for every x,y ∈ x, if x ↔κ y then f(x) -λ f(y). theorem 2.6 ([3]). let f : (x,κ) → (y,λ) and g : (y,λ) → (z,µ) be continuous functions between digital images. then g ◦ f : (x,κ) → (z,µ) is continuous. a path is a continuous function r : [0,m]z → x. we use the following notation. for a digital image (x,κ), c(x,κ) = {f : x → x |f is continuous}. definition 2.7 ([3]; see also [13]). let x and y be digital images. let f,g : x → y be (κ,κ′)-continuous functions. suppose there is a positive integer m and a function h : x × [0,m]z → y such that • for all x ∈ x, h(x, 0) = f(x) and h(x,m) = g(x); • for all x ∈ x, the induced function hx : [0,m]z → y defined by hx(t) = h(x,t) for all t ∈ [0,m]z is (c1,κ ′)−continuous. that is, hx(t) is a path in y . • for all t ∈ [0,m]z, the induced function ht : x → y defined by ht(x) = h(x,t) for all x ∈ x is (κ,κ′)−continuous. then h is a digital (κ,κ′)−homotopy between f and g, and f and g are digitally (κ,κ′)−homotopic in y , denoted f 'κ,κ′ g or f ' g when κ and κ′ are understood. if (x,κ) = (y,κ′), we say f and g are κ-homotopic to abbreviate “(κ,κ)-homotopic” and write f 'κ g to abbreviate “f 'κ,κ g”. if there is a κ-homotopy between idx and a constant map, we say x is κ-contractible, or just contractible when κ is understood. definition 2.8. let a ⊆ x. a κ-continuous function r : x → a is a retraction, and a is a retract of x, if r(a) = a for all a ∈ a. if such a map r satisfies i◦ r 'κ idx where i : a → x is the inclusion map, then r is a κ-deformation retraction and a is a κ-deformation retract of x. a topological space x has the fixed point property (fpp) if every continuous f : x → x has a fixed point. a similar definition has appeared in digital topology: a digital image (x,κ) has the fixed point property (fpp) if every κ-continuous f : x → x has a fixed point. however, this property turns out to be trivial, in the sense of the following. theorem 2.9 ([8]). a digital image (x,κ) has the fpp if and only if #x = 1. the proof of theorem 2.9 was due to the establishment of the following. lemma 2.10 ([8]). let (x,κ) be a digital image, where #x > 1. let x0,x1 ∈ x be such that x0 ↔κ x1. then the function f : x → x given by f(x0) = x1 and f(x) = x0 for x 6= x0 is κ-continuous and has 0 fixed points. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 90 fixed point sets in digital topology, 1 figure 1. a rigid image in z2 with 8-adjacency a function f : (x,κ) → (y,λ) is an isomorphism (called a homeomorphism in [2]) if f is a continuous bijection such that f−1 is continuous. 3. rigidity we will say a function f : x → y is rigid when no continuous map is homotopic to f except f itself. this generalizes a definition in [11]. when the identity map id : x → x is rigid, we say x is rigid. many digital images are rigid, though it can be difficult to show directly that a given example is rigid. a computer search described in [15] has shown that no rigid images in z2 with 4-adjacency exist having fewer than 13 points, and no rigid images in z2 with 8-adjacency exist having fewer than 10 points. we will demonstrate some methods for showing that a given image is rigid. for example, the digital image in figure 1 is rigid, as shown below in example 3.11. an immediate consequence of the definition of rigidity is the following. proposition 3.1. let (x,κ) be a rigid digital image such that #x > 1. then x is not κ-contractible. rigidity of functions is preserved when composing with an isomorphism, as the following theorems demonstrate: theorem 3.2. let f : x → y be rigid and g : y → z be an isomorphism. then g ◦f : x → z is rigid. proof. suppose otherwise. then there is a homotopy h : x × [0,m]z → z from g ◦f to a map g : x → z such that g ◦f 6= g. then by theorem 2.6, g−1 ◦h : x × [0,m]z → x is a homotopy from f to g−1 ◦g, and since g−1 is one-to-one, f 6= g−1 ◦g. this contradiction of the assumption that f is rigid completes the proof. � theorem 3.3. let f : x → y be rigid. let g : w → x be an isomorphism. then f ◦g is rigid. proof. suppose otherwise. then there is a homotopy h : w × [0,m]z → y from f ◦ g to some g : w → y such that g 6= f ◦ g. thus, for some w ∈ w , g(w) 6= f ◦g(w). now consider the function h′ : x × [0,m]z → y defined by c© agt, upv, 2020 appl. gen. topol. 21, no. 1 91 l. boxer and p. christopher staecker h′(x,t) = h(g−1(x), t). by theorem 2.6, h′ is a homotopy from f ◦g◦g−1 = f to g◦g−1. since f ◦g(w) 6= g(w) = (g◦g−1)(g(w)), the homotopic functions f and g◦g−1 differ at g(w), contrary to the assumption that f is rigid. the assertion follows. � as an immediate corollary, we obtain: corollary 3.4. if f : x → y is an isomorphism and one of x and y are rigid, then f is rigid. proof. in the case where x is rigid, the identity map idx is rigid. then by theorem 3.3 we have f◦idx = f is rigid. in the case where y is rigid, similarly by theorem 3.2 we have idy ◦f = f is rigid. � the corollary above can be stated equivalently as follows: corollary 3.5. a digital image x is rigid if and only if every digital image y that is isomorphic to x is rigid. it is easy to see that no digital image in z is rigid: proposition 3.6. if x ⊂ z is a connected digital image with c1 adjacency and #x > 1, then x is not rigid. proof. a connected subset of z having more than one point takes one of the forms [a,b]z, {z ∈ z |z ≥ a}, {z ∈ z |z ≤ b}, z. in all of these cases, it is easily seen that there is a deformation retraction of x to a proper subset of x. therefore, x is not rigid. � we also show that a normal product of images is rigid if and only if all of its factors are rigid. theorem 3.7. let (xi,κi) be digital images for each 1 ≤ i ≤ v, and (x,κ) = ( v∏ i=1 xi,npu(κ1, . . . ,κv)) for some u, 1 ≤ u ≤ v. then x is rigid if and only if xi is rigid for each i. proof. first we assume x is rigid, and we will show that xi is rigid for each i. for some i, let hi : xi × [0,m]z → xi be a κi-homotopy from idxi to fi : xi → xi. without loss of generality we may assume m = 1, and we will show that hi(xi, 1) = xi, and thus fi = idxi . the function h : x × [0, 1]z → x defined by h(x1, . . . ,xv, t) = (x1, . . . ,xi−1,hi(xi, t),xi+1, . . . ,xv), is a homotopy. since x is rigid we must have h(x1, . . . ,xv, 1) = idx, and this means hi(xi, 1) = xi as desired. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 92 fixed point sets in digital topology, 1 now we prove the converse: assume xi is rigid, and we will show x is rigid. let ji : xi → x be the function ji(x) = (x1, . . . ,xi−1,x,xi+1, . . . ,xn). let pi : x → xi be the projection function, pi(x1, . . . ,xv) = xi. then ji is (κi,κ)-continuous, where κ = npu(κ1, . . . ,κv), and pi is (κ,κi)continuous [5, 6]. for the sake of a contradiction, suppose x is not rigid. then there is a homotopy h : (x,κ)× [0,m]z → x between idx and a function g such that for some y = (y1, . . . ,yv) ∈ x, g(y) 6= y. then for some index j, pj(y) 6= pj(g(y)). then the function h′ : xj × [0,m]z → xj defined by h′(x,t) = pj(h(jj(x), t)), is a homotopy from idxj to a function gj, with gj(yj) = pj(h(jj(yj),m)) = pj(h(y,m)) = pj(g(y)) 6= pj(y) = yj, contrary to the assumption that xi is rigid. we conclude that x is rigid. � we have a similar result when x is a disjoint union of digital images. let x be a digital image of the form x = a∪b where a and b are disjoint and no point of a is adjacent to any point of b. we say x is the disjoint union of a and b, and we write x = atb. theorem 3.8. let x = atb. then x is rigid if and only if a and b are rigid. proof. first we assume that x is rigid, and we will show that a is rigid. (it will follow from a similar argument that b is rigid.) let f : a → a be any self-map homotopic to ida, and we will show that f = ida. define g : x → x by g(x) = { f(x) if x ∈ a; x if x ∈ b. then g is continuous and homotopic to idx, and since x is rigid we must have g = idx, which means that f = ida. now for the converse, assume that a and b are both rigid. take some selfmap f : x → x homotopic to idx, and we will show that f = idx. since f is homotopic to the identity, we must have f(a) ⊆ a and f(b) ⊆ b. this is because there will always be a path from any point x to f(x) given by the homotopy from idx to f(x). thus if x ∈ a we must also have f(x) ∈ a since there are no paths from points of a to points of b. since f(a) ⊆ a and f(b) ⊆ b, there are well-defined restrictions fa : a → a and fb : b → b, and the homotopy from idx to f induces homotopies from ida to fa and idb to fb. since a and b are rigid we must have fa = ida and fb = idb, and thus f = idx as desired. � since every digital image is a disjoint union of its connected components, we have: c© agt, upv, 2020 appl. gen. topol. 21, no. 1 93 l. boxer and p. christopher staecker corollary 3.9. a digital image x is rigid if and only if every connected component of x is rigid. let x be some digital image of the form x = a∪b, where a∩b is a single point x0, and no point of a is adjacent to any point of b except x0. we say x = a∪b is the wedge of a and b, denoted x = a∨b, and x0 is called the wedge point of a∨b. we have the following. theorem 3.10. if x = a∨b and a and b are rigid, then x is rigid. proof. let x0 be the wedge point of a∨b, and let a0 and b0 be the components of a and b that include x0. if #a0 = 1 or #b0 = 1, then the components of a∨b are in direct correspondence to the components of a and b and the result follows by corollary 3.9. thus we assume #a0 > 1 and #b0 > 1. let h : a∨b × [0,m]z → a∨b be a homotopy such that h(x, 0) = x for all x ∈ a∨b. without loss of generality, m = 1. if the induced map h1 is not idx then there is a point x ′ ∈ x such that h1(x′) = h(x′, 1) 6= x′. without loss of generality, x′ ∈ a. let pa : x → a be the projection pa(x) = { x for x ∈ a; x0 for x ∈ b. since pa ◦h is a homotopy from ida to pa ◦h1, and a is rigid, we have (3.1) pa ◦h1 = ida . were h1(x ′) ∈ a then it would follow that h1(x ′) = pa ◦h1(x′) = x′, contrary to our choice of x′. therefore we have h1(x ′) ∈ b \ {x0}. but x′ ↔ h1(x′), so x′ = x0. since a0 is connected and has more than 1 point, there exists x1 ∈ a such that x1 ↔κ x0. by the continuity of h1 and choice of x0, we must therefore have h1(x1) = x0, and therefore pa ◦h1(x1) = pa(x0) = x0. this contradicts statement (3.1), so the assumption that h1 is not idx is incorrect, and the assertion follows. � a loop is a continuous function p : cm → x. the converse of theorem 3.10 is not generally true. in [11] it was mentioned (without proof) that a wedge of two long cycles is in general rigid. we give a specific example: example 3.11. let a and b be non-contractible simple closed curves. then a and b are non-rigid [11]. however, x = a∨b is rigid. e.g., using c2 = 8adjacency in z2, let a = {a0 = (0, 0),a1 = (1,−1),a2 = (2,−1),a3 = (3, 0),a4 = (2, 1),a5 = (1, 1)} and let b = {b0 = a0,b1 = (−1,−1),b2 = (−2,−1),b3 = (−3, 0),b4 = (−2, 1),b5 = (−1, 1)}. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 94 fixed point sets in digital topology, 1 by continuity, if there is a homotopy h : x × [0,m]z → x without loss of generality, m = 1 such that h0 = idx and h(x, 1) 6= x, then h “pulls” [11] every point of a or of b and therefore “breaks” one of the loops of x, a contradiction since “breaking” a loop is a discontinuity. thus no such homotopy exists. 2 4. homotopy fixed point spectrum the paper [10] gave a brief treatment of homotopy-invariant fixed point theory, defining two quantities m(f) and x(f), respectively the minimum and maximum possible number of fixed points among all maps homotopic to f. when f : x → x, clearly we will have: 0 ≤ m(f) ≤ x(f) ≤ #x. we will see in the examples below that any one of these inequalities can be strict in some cases, or equality in some cases. more generally, for some map f : x → x, let fix(f) denote the set of fixed points of f. we consider the following set s(f), which we call the homotopy fixed point spectrum of f: s(f) = {# fix(g) | g ' f}⊆{0, . . . , #x}. an immediate consequence of lemma 2.10: corollary 4.1. let (x,κ) be a connected digital image, where #x > 1. then 0 ∈ s(c), where c ∈ c(x,κ) is a constant map. we can also consider the fixed point spectrum of x, defined as: f(x) = {# fix(f) | f : x → x is continuous} remark 4.2. the following assertions are immediate consequences of the relevant definitions. • if x is a digital image of only one point, then f(x) = {1}. • if f : x → x is rigid, then s(f) = {# fix(f)}. if x is rigid, then s(id) = {#x}. since every image x has a constant map and an identity map, we always have: {1, #x}⊆ f(x). the number of fixed points is always preserved by isomorphism: lemma 4.3. let x and y be isomorphic digital images. let f : x → x be continuous. then there is a continuous g : y → y such that # fix(f) = # fix(g). proof. let g : x → y be an isomorphism. let a = fix(f). since g is oneto-one, #g(a) = #a. let g : y → y be defined by g = g ◦ f ◦ g−1. for y0 ∈ g(a), let x0 = g−1(y0). then g(y0) = g◦f ◦g−1(y0) = g◦f(x0) = g(x0) = y0. let b = fix(g) it follows that g(a) ⊆ b, so #a ≤ #b. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 95 l. boxer and p. christopher staecker since g−1 is an isomorphism, it similarly follows that #b ≤ #a. thus, # fix(f) = # fix(g). � as an immediate consequence, we have the following. corollary 4.4. let x and y be isomorphic digital images. then f(x) = f(y ). there is a certain regularity to the fixed point spectrum for connected digital images. when x has only a single point, we have already remarked that f(x) = {1}. for images of more than 1 point, we will show that f(x) always includes 0, 1, and #x, and, provided the image is large enough, the set f(x) also includes 2 and 3. the following statements hold for connected images. we discuss the fixed point spectrum of disconnected images in terms of their connected components in theorem 7.2 and its corollary. we begin with a simple lemma: lemma 4.5. let x be any connected digital image with #x > 1. let x0 ∈ x, and let 0 ≤ k ≤ #n∗(x0). then k ∈ s(c) ⊆ f(x), where c is the map with image {x0}. proof. by corollary 4.1, a constant map is homotopic to a map with no fixed points, so 0 ∈ s(c) as desired. for k > 0, let n = #n∗(x0) and write n∗(x0) = {x0,x1, . . . ,xn−1}. then define f : x → x by: f(x) = { x if x = xi for some i < k, x0 otherwise. then f is continuous with fix(f) = {x0, . . . ,xk−1} and thus k ∈ f(x). furthermore, f is homotopic to the constant map at x0, and so in fact k ∈ s(c). � theorem 4.6. let x be a connected digital image, and let c : x → x be any constant map. if #x ≥ 2 then {0, 1, 2}⊆ s(c). if #x ≥ 3, then {0, 1, 2, 3}⊆ s(c). proof. if #x = 2, then x consists simply of two adjacent points. thus #n∗(x) = 2 for each x ∈ x, and so lemma 4.5 implies that {0, 1, 2}⊆ s(c). when #x ≥ 3, there must be some x ∈ x with #n∗(x) ≥ 3. (otherwise the image would consist only of disjoint pairs of adjacent points, which would not be connected.) thus by lemma 4.5 we have {0, 1, 2, 3}⊆ s(c). � since we always have #x ∈ s(id) and s(c) ∪s(id) ⊆ f(x) ⊆{0, 1, . . . , #x}, the theorem above directly gives: c© agt, upv, 2020 appl. gen. topol. 21, no. 1 96 fixed point sets in digital topology, 1 corollary 4.7. let x be a connected digital image. if #x = 2 then f(x) = {0, 1, 2}. if #x > 2, then {0, 1, 2, 3, #x}⊆ f(x). we have already seen that #x ∈ f(x) in all cases. there is an easy condition that determines whether or not #x − 1 ∈ f(x). lemma 4.8. let x be connected with n = #x > 1. then n − 1 ∈ f(x) if and only if there are distinct points x1,x2 ∈ x with n(x1) ⊆ n∗(x2). proof. suppose there are points x1,x2 ∈ x, x1 6= x2, such that n(x1) ⊆ n∗(x2). then the map f(x1) = x2, f(x) = x for all x 6= x1, is a self-map on x with exactly n−1 fixed points. that f is continuous is seen as follows. suppose x,x′ ∈ x with x ↔ x′. • if x1 6∈ {x,x′}, then f(x) = x ↔ x′ = f(x′). • if, say, x = x1, then x′ ∈ n(x1) ⊆ n∗(x2), so f(x′) = x′ x2 = f(x1). thus f is continuous, and we conclude n− 1 ∈ f(x). now assume that n− 1 ∈ f(x). thus there is some continuous self-map f with exactly n− 1 fixed points. let x1 be the single point not fixed by f, and let x2 = f(x1). then let x ∈ x with x ↔ x1. then x = f(x) f(x1) = x2, so n(x1) ⊆ n∗(x2). � lemma 4.8 can be used to show that a large class of digital images will satisfy n − 1 6∈ f(x). for example when x = cn for n > 4, no n(xi) is contained in n∗(xj) for j 6= i. thus we have: corollary 4.9. let n > 4. then n− 1 6∈ f(cn). in particular this means that 4 6∈ f(c5), so the result of theorem 4.7 cannot in general be improved to state that 4 ∈ f(x) for all images of more than 4 points. 5. pull indices let fix(f) be the complement of the fixed point set, that is, fix(f) = {x ∈ x | f(x) 6= x}. when f(x) 6= x, we say f moves x. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 97 l. boxer and p. christopher staecker definition 5.1. let (x,κ) be a digital image with #x > 1 and let x ∈ x. the pull index of x, p(x) or p(x,x) or p(x,x,κ), is p(x) = min{#fix(f) | f : x → x is continuous and f(x) 6= x}. when f(x) 6= x, the set fix(f) always contains at least the point x, and so p(x) ≥ 1 for any x that is moved by some f. example 5.2. let x = [1, 3]z with c1-adjacency. to compute p(3), consider the function f(x) = min{x, 2}. this is continuous, not the identity, and fix(f) = {1, 2}, and thus p(3) = 1. similarly we can show that p(1) = 1. but we have p(2) = 2, since any continuous self-map f on x that moves 2 must also move at least one other point: if f(2) = 1 we must have f(3) ∈{1, 2}, and if f(2) = 3 we must have f(1) ∈{2, 3}. proposition 5.3. let (x,κ) be a connected digital image with n = #x > 1. let m ∈ n, 1 ≤ m ≤ n. suppose, for all x ∈ x, we have p(x) ≥ m . then f(x) ∩{i}n−1i=n−m+1 = ∅. proof. by hypothesis, f ∈ c(x,κ) \{idx} implies f moves at least m points, hence # fix(f) ≤ n−m. the assertion follows. � theorem 5.4. let (x,κ) be a connected digital image with n = #x > 1. the following are equivalent. 1) n− 1 ∈ f(x). 2) there are distinct x1,x2 ∈ x such that n(x1) ⊆ n∗(x2). 3) there exists x ∈ x such that p(x) = 1. proof. 1) ⇔ 2) is shown in lemma 4.8. 1) ⇔ 3): we have n−1 ∈ f(x) ⇔ there exists f ∈ c(x) with exactly n−1 fixed points, i.e., the only x ∈ x not fixed by f has p(x) = 1. � the following generalizes 1) ⇒ 3) of theorem 5.4. proposition 5.5. let (x,κ) be a connected digital image with n = #x > 1. let k ∈ [1,n−1]z. then k ∈ f(x) implies there exist distinct x1, . . . ,xn−k ∈ x such that p(xi) ≤ n−k. proof. k ∈ f(x) implies there exists f ∈ c(x) with exactly k fixed points, hence distinct x1, . . . ,xn−k ∈ x such that xi 6∈ fix(f). thus for each i, the members of fix(f) are not pulled by f and xi. thus p(xi) ≤ n−k. � 6. retracts in this section, we study how retractions interact with fixed point spectra. theorem 6.1 ([2]). let (x,κ) be a digital image and let a ⊆ x. then a is a retract of x if and only if for every continuous f : (a,κ) → (y,λ) there is an extension of f to a continuous g : (x,κ) → (y,λ). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 98 fixed point sets in digital topology, 1 in the proof of theorem 6.1, an extension of f is obtained by using g = f◦r, where r : x → a is a retraction. we use this in the proof of the next assertion. theorem 6.2. let a be a retract of (x,κ). then f(a) ⊆ f(x). proof. let f : a → a be κ-continuous. let r : x → a be a κ-retraction. let i : a → x be the inclusion function. by theorem 2.6, g = i◦f ◦ r : x → x is continuous. further, g(x) = f(x) if and only if x ∈ a, so fix(g) = fix(f). since f was taken arbitrarily, the assertion follows. � remark 6.3. we do not have an analog to theorem 6.2 by replacing fixed point spectra by spectra of identity maps. e.g, in example 3.11 we have {0, #a}⊆ s(ida), and a is a retract of x, but x is rigid, so s(idx) = {#x}. however, we have the following corollaries 6.4 and 6.5. corollary 6.4. let a be a deformation retract of x. then s(ida) ⊆ s(idx) ⊆ f(x). in particular, #a ∈ s(idx). corollary 6.5. let a,b ∈ z, a < b. then s(id[a,b]z,c1) = f([a,b]z,c1) = {0, 1, . . . ,b−a + 1}. proof. since a < b and [a,b]z is c1-contractible, it follows from theorem 2.9 that 0 ∈ s(id[a,b]z,c1). since for each d ∈ [a,b]z there is a c1-deformation of [a,b]z to [a,d]z, it follows from corollary 6.4 that #[a,d]z ∈ s(id[a,b]z,c1). thus, f([a,b]z,c1) ⊆{i}b−a+1i=0 = s(id[a,b]z,c1) ⊆ f([a,b]z,c1). the assertion follows. � we can generalize this result about intervals to a two-dimensional box in z2. theorem 6.6. let x = [1,a]z × [1,b]z, with adjacency κ ∈{c1,c2}. then s(idx) = f(x) = {0, 1, . . . ,ab} proof. all self-maps on [1,a]z × [1,b]z are homotopic to the identity, so it suffices only to show that f(x) = {0, 1, . . . ,ab}. the proof is by induction on a. for a = 1, our image x is isomorphic to the one-dimensional image ([1,b]z,c1). thus by theorem 6.5 we have f(x) = {0, 1, . . . ,b} = {0, 1, . . . ,ab} as desired. for the inductive step, first note that [1,a − 1]z × [1,b]z is a retract of x (using either κ = c1 or c2). thus by induction and theorem 6.2 we have {0, 1, . . . , (a− 1)b}⊆ f(x). it remains only to show that {(a− 1)b + 1, (a− 1)b + 2, . . . ,ab}⊆ f(x). we do this by exhibiting a family of self-maps of x having these numbers of fixed points. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 99 l. boxer and p. christopher staecker figure 2. the map ft from theorem 6.6, pictured in the case t = 2. all points are fixed except those with arrows indicating where they map to. let t ∈{0, . . . ,b− 1}, and define ft : x → x as follows: ft(x,y) = { (x,y) if x > 1 or y > t, (x + 1,y + 1) if x = 1 and y ≤ t see figure 2 for a pictorial depiction of ft. this ft is well-defined and both c1and c2-continuous for each t ∈{0, . . . ,b− 1} and has ab− t fixed points. thus we have {ab,ab− 1, . . . ,ab− (b− 1) = (a− 1)b + 1}⊆ f(x) as desired. � 7. cartesian products and disjoint unions in the following, assume ai ⊂ n, 1 ≤ i ≤ v. define v⊗ i=1 ai = { v∏ i=1 ai | ai ∈ ai } and v⊕ i=1 ai = { v∑ i=1 ai | ai ∈ ai } . if fi : xi → yi, let πvi=1fi : π v i=1xi → π v i=1yi be the product function defined by πvi=1fi(x1, . . .xv) = (f1(x1), . . . ,fv(xv)) for xi ∈ xi. theorem 7.1. suppose (xi,κi) is a digital image, 1 ≤ i ≤ v. let x = πvi=1xi. then ⊗v i=1 f(xi,κi) ⊆ f(x,npv(κ1, . . . ,κv)). proof. let fi : xi → xi be κi-continuous. let x = πvi=1xi. then the product function f = πvi=1fi(x1, . . .xv) : x → x is npv(κ1, . . . ,κv)-continuous [5]. if ai = {yi,ji} pi ji=1 is the set of distinct fixed points of fi, then each point (y1,j1, . . . ,yv,jv ), for 1 ≤ ji ≤ pi, is a fixed point of f. the assertion follows. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 100 fixed point sets in digital topology, 1 we note that the conclusion of theorem 7.1 cannot in general be strengthened to say that ⊗v i=1 f(xi) = f(x). for example, if x = [1, 3]z × [1, 3]z, we have f(x) = {0, 1, . . . , 9} by theorem 6.6, but f([1, 3]z) ⊗f([1, 3]z) = {0, 1, 2, 3}⊗{0, 1, 2, 3} = {0, 1, 2, 3, 4, 6, 9}. we do have a similar result, this time with equality, for a disjoint union of digital images. theorem 7.2. let x = atb. if a and b both have at least 2 points, then f(x) = f(a) ⊕f(b). proof. first we show that f(a)⊕f(b) ⊆ f(x). take some k ∈ f(a)⊕f(b), say k = m + n with m ∈ f(a) and n ∈ f(b). that means there are two selfmaps f : a → a and g : b → b with # fix(f) = m and # fix(g) = n. let h : x → x be defined by: h(x) = { f(x) if x ∈ a g(x) if x ∈ b then # fix(h) = # fix(f) + # fix(g) = m + n = k and so k ∈ f(x) as desired. next we show f(x) ⊆ f(a) ⊕ f(b). take some k ∈ f(x), so there is some self-map f with # fix(f) = k. let fa : a → x and fb : b → x be the restrictions of f to a and b. since x = a∪b, we have fix(f) = fix(fa) ∪ fix(fb), and fix(fa) = fix(f) ∩ a and fix(fb) = fix(f) ∩ b. since a and b are disjoint, the union of the fixed point sets above is disjoint. thus we have k = # fix(fa) + # fix(fb). since continuous functions preserve connectedness, we must have fa(a) ⊆ a or fa(a) ⊆ b. similarly fb(b) ⊆ a or fb(b) ⊆ b. we show that k ∈ f(a) ⊕f(b) in several cases. in the case where fa(a) ⊆ b and fb(b) ⊆ a, there are no fixed points of fa or fb, and thus no fixed points of f. thus k = 0, and it is true that k ∈ f(a) ⊕f(b) since 0 ∈ f(a) and 0 ∈ f(b) by theorem 4.6. in the case where fa(a) ⊆ b and fb(b) ⊆ b, there are no fixed points of fa, and thus fix(f) = fix(fb). in this case in fact fb is a self-map of b, and so k = # fix(f) = 0 + # fix(fb) ∈ f(a) ⊕f(b) since 0 ∈ f(a) by theorem 4.6 and # fix(fb) ∈ f(b) since fb is a self-map on b. the case where fa(a) ⊆ a and fb(b) ⊆ a is similar. the final case is when fa(a) ⊆ a and fb(b) ⊆ b. in this case fa is a self-map of a and fb is a self-map of b. since fix(f) = fix(fa) ∪ fix(fb), the k fixed points of f must partition into m fixed points of fa and n fixed points of fb, where m + n = k. thus m ∈ f(a) and n ∈ f(b), and so k = m + n ∈ f(a) ⊕f(b). � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 101 l. boxer and p. christopher staecker the assumption above that a and b have at least 2 points is necessary. for example if a and b are each a single point, then f(x) = {0, 1, 2} while f(a) = f(b) = {1} and thus f(a) ⊕f(b) = {2}. since any digital image is a disjoint union of its connected components, we have: corollary 7.3. let x1, . . . ,xk be the connected components of a digital image x, and assume that #xi > 1 for all i. then we have: f(x) = k⊕ i=1 f(xi) 8. locations of fixed points in many cases, the existence of two fixed points will imply that other fixed points must exist in certain locations. in some cases we will show that fix(f) must be connected. we do not have fix(f) connected in general, as shown by the following. example 8.1. let x = {p0 = (0, 0),p1 = (1, 0),p2 = (2, 0),p3 = (1, 1)}. let f : x → x be defined by f(p0) = p0, f(p1) = p3, f(p2) = p2, f(p3) = p1. then x is c2-connected, f ∈ c(x,c2), and fix(f) = {p0,p2} is c2-disconnected. lemma 8.2. let (x,κ) be a digital image and f : x → x be continuous. suppose that x,x′ ∈ fix(f) and that y ∈ x lies on every path of minimal length between x and x′. then y ∈ fix(f). proof. let k be the minimal length of a path from x to x′. first we show that y must occur at the same location along any minimal path from x to x′. that is, we show that there is some i ∈ [0,k]z with p(i) = y for every minimal path p from x to x′. this we prove by contradiction: assume we have two minimal paths p and q with p(i) = y = q(j) for some j < i. then construct a new path r by traveling from x to y along q, and then from y to x′ along p. then this path r has length less than the length of p, contradicting the minimality of p. thus we have some i ∈ [0,k]z with p(i) = y for every minimal path p from x to x′. let p be some minimal path from x to x′, and since the endpoints of p are fixed, the path f(p) is also a path from x to x′. furthermore the length of f(p) must be at most k, and thus must equal k since this is the minimal possible length of a path from x to x′. since both p and f(p) are minimal paths from x to x′, we have p(i) = f(p(i)) = y, and thus y = f(y) as desired. � a vertex v of a connected graph (x,κ) is an articulation point of x if (x\{v},κ) is disconnected. we have the following immediate consequences of lemma 8.2. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 102 fixed point sets in digital topology, 1 corollary 8.3. let (x,κ) be a connected digital image. let v be an articulation point of x. suppose f ∈ c(x,κ) has fixed points in distinct components of x \{v}. then v is a fixed point of f. corollary 8.4. let (x,κ) be a digital image and f ∈ c(x,κ). suppose x,x′ ∈ fix(f) are such that there is a unique shortest κ-path p in x from x to x′. then p ⊆ fix(f). proof. this follows immediately from lemma 8.2. � corollary 8.5. let (x,κ) be a digital image that is a tree. then f ∈ c(x,κ) implies fix(f) is κ-connected. proof. this follows from corollary 8.4, since given x,x′ in a tree x, there is a unique shortest path in x from x to x′. � for a digital cycle, the fixed point set is typically connected. the only exception is in a very particular case, as we see below. theorem 8.6. let f : cn → cn be any continuous map. then fix(f) is connected, or is a set of 2 nonadjacent points. the latter case occurs only when n is even and the two fixed points are at opposite positions in the cycle. proof. if # fix(f) ∈{0, 1}, then fix(f) is connected. when # fix(f) > 1, we show that if xi,xj ∈ fix(f) are two distinct fixed points, then either there is a path from xi to xj through other fixed points, or that no other points are fixed. there are two canonical paths p and q from xi to xj: the two injective paths going in either “direction” around the cycle. without loss of generality assume |p| ≥ |q|. this means that |q| is the shortest possible length of a path from xi to xj. consider the case in which |p| > |q|. in this case |q| is the unique shortest path from xi to xj, and by lemma 8.4, q ⊆ fix(f), and so xi and xj are connected by a path of fixed points as desired. now consider the case in which |p| = |q|. in this case again |q| is the shortest possible length of a path from xi to xj, and p and q are the only two paths from xi to xj having this length. then f(q) is a path from xi to xj of length |q|, and so we must have either f(q) = q or f(q) = p. in the former case, q is a path of fixed points connecting xi and xj as desired. in the latter case, fix(f) ∩q = {xi,xj}. similarly considering the path f(p), we must have either f(p) = p (in which case p is a path of fixed points connecting xi and xj); or f(p) = f(q), in which case fix(f) ∩p = {xi,xj}. considering all cases, either a minimal-length path from xi to xj is contained in fix(f), or fix(f) = {xi,xj}. the second sentence of the theorem follows from our analysis of the various cases. the only case which gives 2 nonadjacent fixed points requires xi and xj to be opposite points on the cycle, which requires n to be even. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 103 l. boxer and p. christopher staecker x3 x2 x7 x6 x0 x1 x4 x5 figure 3. a contractible image for which f(x) 6= {0, 1, . . . , #x}. 9. remarks and examples in classical topology m(f) is the only interesting homotopy invariant count of the number of fixed points. s(f) is not studied in classical topology, since in all typical cases (all continuous maps on polyhedra) we would have s(f) = [m(f),∞)z. in classical topology the value of m(f) is generally hard to compute. the lefschetz number gives a very rough indication of homotopy invariant fixed point information, and the more sophisticated nielsen number is a homotopy invariant lower bound for m(f). see [12]. when x is contractible, all self-maps are homotopic, so s(f) = f(x) for any self-map f. it is natural to suspect that when x is contractible with #x > 1, we will always have f(x) = {0, 1, . . . , #x}. this is false, however, as the following example shows: example 9.1. let x ⊂ z3 be the unit cube of 8 points with c1 adjacency, shown in figure 3. then x is contractible, so s(f) = f(x) for any self-map f. by projecting the cube into one of its faces, we see that x retracts to c4, and since f(c4) = {0, 1, 2, 3, 4}, we have {0, 1, 2, 3, 4}⊆ f(x) by theorem 6.2. in fact there are also continuous maps having 5 or 6 fixed points: let: g(x5) = x0, g(x6) = x3, g(xi) = xi for i 6∈ {5, 6} then g is continuous with 6 fixed points. let: h(x5) = h(x7) = x0, h(x6) = x3, h(xi) = xi for i 6∈ {5, 6, 7} then h is continuous with 5 fixed points. since of course the identity map has 8 fixed points, we have so far shown that {0, 1, 2, 3, 4, 5, 6, 8}⊆ f(x). in fact 7 6∈ f(x). this follows from lemma 4.8. we have shown that: f(x) = {0, 1, 2, 3, 4, 5, 6, 8}. the computation of s(f) in general seems to be a difficult and interesting problem. even in the case of self-maps on the cycle cn, the results are interesting. first we show that in fact there are exactly 3 homotopy classes of self-maps on cn: the identity map id(xi) = xi, the constant map c(xi) = x0, and the flip map l(xi) = x−i. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 104 fixed point sets in digital topology, 1 theorem 9.2. given f ∈ c(cn), f is homotopic to one of: a constant map, the identity map, or the flip map. proof. we have noted above that if n ≤ 4 then cn is contractible, so every f ∈ c(cn) is homotopic to a constant map. thus, in the following, we assume n > 4. we can compose f by some rotation r to obtain g = r ◦ f ' f such that g(x0) = x0. we will show that g is either the identity, the flip map, or homotopic to a constant map. if g is not a surjection, then its continuity implies g(cn) is a connected proper subset of cn, hence is contractible. therefore, g is homotopic to a constant map. if g is a surjection, then g is a bijection because the domain and codomain of g both have cardinality n. by continuity, g(x1) ↔ g(x0) = x0. therefore, either g(x1) = xn−1 or g(x1) = x1. if g(x1) = x−1, then continuity and the fact that g is a bijection yield an easy induction showing that g(xi) = x−i, 0 ≤ i < n. therefore, g is the flip map. if g(x1) = x1, a similar argument shows that g is the identity. � in fact the proof of theorem 9.2 demonstrates the following stronger statement. let rd : cn → cn be the rotation map rd(xi) = xi+d. the following generalizes theorem 3.4 of [4], which states that any map homotopic to the identity must be a rotation. theorem 9.3. let f : cn → cn be continuous. then one of the following is true: • f is homotopic to a constant map • f is homotopic to the identity, and f = rd for some d • f is homotopic to the flip map l, and f = rd ◦ l for some d the proof of theorem 9.2 also demonstrated that all non-isomorphisms on cn must be nullhomotopic. thus all isomorphisms on cn fall into the second and third categories above, and in fact all maps in those two categories are isomorphisms. thus we obtain: corollary 9.4. let n > 4, and f : cn → cn be an isomorphism with f ' g for some g. then g is an isomorphism. now we are ready to compute the values of s(f) for our three classes of self-maps on cn. theorem 9.5. we have s(f) = {1} for every f : c1 → c1. when 1 < n ≤ 4, we have s(f) = {0, . . . ,n} for any f : cn → cn. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 105 l. boxer and p. christopher staecker x0 x1 x4 x2 x3 x0 x1 x5 x2 x3 x4 figure 4. the map f from theorem 9.5 pictured in the cases n = 5 and n = 6. all points are fixed except those with arrows indicating where they map to. the path f(cn) is in bold. when n > 4, let c be any constant map, id be the identity map, and l be the flip map on cn. we have: s(id) = {0,n} s(c) = {0, 1, . . . ,bn/2c + 1} s(l) = { {0, 1} if n is odd {0, 2} if n is even proof. when n = 1, our image is a single point, and the constant map (which is also the identity map) is the only continuous self-map. thus s(f) = f(x) = {1} for every f : c1 → c1. when 1 < n ≤ 4, again all maps are homotopic, and we have s(f) = f(x) = {0, . . . ,n} for any f by theorem 4.7. now we consider cn with n > 4, which is the interesting case. by theorem 9.3 the only maps homotopic to id are rotation maps rd. since # fix(r0) = n and # fix(rd) = 0 for d 6= 0, we have s(id) = {0,n}. now we consider the constant map c(xi) = x0. let f ∈ c(cn) be defined as follows. f(xi) = { x−i for 0 ≤ i ≤bn/2c; xi for bn/2c < i < n. this map “folds” the cycle onto a path that is “about half” of the cycle, with bn/2c + 1 fixed points. see figure 4. this can be taken as the first step of a homotopy, in which successive steps shrink the path and the number of fixed points by one per step, until a constant map is reached at the end of the homotopy. thus {1, . . . ,bn/2c + 1} ⊆ s(c), and of course 0 ∈ s(c) also by theorem 4.6. thus we have shown there is a fixed path p between fixed points of f, xi,xj, of length at least bn/2c + 1. we wish to show that in fact s(c) = {0, . . . ,bn/2c + 1}. we show this by contradiction: take some nullhomotopic f, assume that k ∈ s(f) with k > bn/2c + 1, and we will show in fact that c© agt, upv, 2020 appl. gen. topol. 21, no. 1 106 fixed point sets in digital topology, 1 figure 5. an image having a self-map with x(f) = 0 all points are fixed; this would be a contradiction since f 6= id. choose any x ∈ cn\p. then x lies on the unique shortest path in cn from xi to xj. then x ∈ fix(f) by lemma 8.4; this gives our desired contradiction. finally we consider the flip map l(xi) = x−i. by theorem 9.3, all maps homotopic to l have the form f(xi) = rd◦l(xi) = xd−i. such a map has a fixed point at xi if and only if d = 2i (mod n). when d is odd there are no solutions, and so # fix(f) = 0. when d is even, say d = 2a, and n is odd, there is one solution: i = a. when d is even and n is also even, there are two solutions: i = a and i = a + n/2. thus we have some maps with no fixed points, and when k is odd we have some with one fixed point, and when k is even we have some with two. we conclude: s(l) = { {0, 1} if n is odd {0, 2} if n is even � by theorem 9.2, any self-map on cn is homotopic to the constant, identity, or flip. thus by taking unions of the sets above, we have: corollary 9.6. f(cn) =   {1} if n = 1, {0, . . . ,n} if 1 < n ≤ 4, {0, 1, . . . ,bn/2c + 1,n} if n > 4. from the corollary above we see that f(c5) = {0, 1, 2, 3, 5}, and thus the formula of corollary 4.7 is exact for x = c5. this is the only example known to the authors in which this occurs. question 9.7. is there any digital image x 6= c5 with #x > 4 and f(x) = {0, 1, 2, 3, #x}? we conclude this section with two interesting examples showing the wide variety of fixed point sets that can be exhibited for other digital images. tools we use in our discussion include the following. a path r : [0,m]z → x that is an isomorphism onto r([0,m]z) is a simple path. if a loop p is an isomorphism onto p(cm), p is a simple loop. definition 9.8 ([11]). a simple path or a simple loop in a digital image x has no right angles if no pair of consecutive edges of the path or loop belong to a loop of length 4 in x. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 107 l. boxer and p. christopher staecker x x′ figure 6. a lasso for the points x = (0, 1) and x′ = (0, 0) figure 7. an image with many different values for s(f) definition 9.9 ([11]). a lasso in x is a simple loop p : cm → x and a path r : [0,k]z → x such that k > 0, m ≥ 5, r(k) = p(x0), and neither p(x1) nor p(xm−1) is adjacent to r(k − 1). the lasso has no right angles if neither p nor r has a right angle, and no right angle is formed where r meets p, i.e., the final edge of r and neither of the edges of p at p(x0) form 2 edges of a loop of length 4 in x. theorem 9.10 ([11]). let x be an image in which, for any two adjacent points x ↔ x′ ∈ x, there is a lasso with no right angles having path r : [0,k]z → x with r(0) = x and r(1) = x′. then x is rigid. example 9.11. let x be the digital image x = ([0, 6]z ×{0, 2}) ∪{(0, 1), (2, 1), (4, 1), (6, 1)} (see figure 5), with 4-adjacency. by theorem 9.10, this image is rigid. it is easy, though a bit tedious, to verify that the hypothesis of theorem 9.10 is satisfied by x. for example, in figure 6 we exhibit a lasso with no right angles for two adjacent points. it is easy to construct such lassos for any pair of adjacent points. since x is rigid, we have s(id) = {#x} = {18}. let f : x → x be the 180-degree rotation of x. then f is an isomorphism, and so by theorem 3.2, f = f ◦ idx is rigid. thus s(f) = {# fix(f)} = {0}. in particular this provides an example for the question posed in [10] if x(f) could ever equal 0 for a connected image. the following example demonstrates an image which has many different possible sets which can occur as s(f) for various self-maps f. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 108 fixed point sets in digital topology, 1 example 9.12. let x = ([0, 5]z ×{0, 2}) ∪{(0, 1), (2, 1), (5, 1)} (see figure 7), with 4-adjacency. in this image we have several different homotopy classes of maps. we will derive sufficient information about s for some of these to compute f(x). by theorem 9.10, x is rigid, so s(id) = {#x} = {15}. let f be a vertical reflection. then f is rigid by corollary 3.4, and has 3 fixed points, so s(f) = {3}. let g be the function that maps the bottom horizontal bar onto the top one, and fixes all other points. then g has 9 fixed points, and is homotopic to a constant map. we can retract the image of g down to a point one point at a time, and so {0, 1, . . . , 9}⊆ s(g). let h be the function which maps the left vertical bar into the middle vertical bar and fixes all other points. then h has 12 fixed points. we can additionally map one or both of the next two points into the middle vertical bar to obtain maps homotopic to h with 11 or 10 fixed points. we can do these retractions followed by a rotation around the 10-cycle on the right to obtain a map homotopic to h with no fixed points. thus {0, 10, 11, 12}⊆ s(h). we have f(x)∩{13, 14} = ∅ by proposition 5.3, since for all x ∈ x we see easily that p(x) ≥ 3. we therefore have f(x) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15}. 10. further remarks we have studied several measures concerning the fixed point set of a continuous self-map on a digital image. we anticipate further research in this area. acknowledgements. we are grateful for the suggestions of an anonymous reviewer, and some careful corrections by muhammad sirajo abdullahi and jamilu abubakar. references [1] c. berge, graphs and hypergraphs, 2nd edition, north-holland, amsterdam, 1976. [2] l. boxer, digitally continuous functions, pattern recognition letters 15 (1994), 833– 839. [3] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. [4] l. boxer, continuous maps on digital simple closed curves, applied mathematics 1 (2010), 377–386. [5] l. boxer, generalized normal product adjacency in digital topology, applied general topology 18, no. 2 (2017), 401–427. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 109 l. boxer and p. christopher staecker [6] l. boxer, alternate product adjacencies in digital topology, applied general topology 19, no. 1 (2018), 21–53. [7] l. boxer, fixed points and freezing sets in digital topology, proceedings, 2019 interdisciplinary colloquium in topology and its applications, in vigo, spain; 55–61. [8] l. boxer, o. ege, i. karaca, j. lopez, and j. louwsma, digital fixed points, approximate fixed points, and universal functions, applied general topology 17, no. 2 (2016), 159– 172. [9] l. boxer and i. karaca, fundamental groups for digital products, advances and applications in mathematical sciences 11, no. 4 (2012), 161–180. [10] l. boxer and p. c. staecker, remarks on fixed point assertions in digital topology, applied general topology 20, no. 1 (2019), 135–153. [11] j. haarmann, m. p. murphy, c. s. peters and p. c. staecker, homotopy equivalence in finite digital images, journal of mathematical imaging and vision 53 (2015), 288–302. [12] b. jiang, lectures on nielsen fixed point theory, contemporary mathematics 18 (1983). [13] e. khalimsky, motion, deformation, and homotopy in finite spaces, in proceedings ieee intl. conf. on systems, man, and cybernetics (1987), 227–234. [14] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. [15] p. c. staecker, some enumerations of binary digital images, arxiv:1502.06236, 2015. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 110 @ appl. gen. topol. 21, no. 2 (2020), 201-214 doi:10.4995/agt.2020.12258 c© agt, upv, 2020 closed subsets of compact-like topological spaces serhii bardyla a and alex ravsky b a institute of mathematics, university of vienna, austria. (sbardyla@yahoo.com) b pidstryhach institute for applied problems of mechanics and mathematics, nat. acad. sciences of ukraine, lviv, ukraine. (alexander.ravsky@uni-wuerzburg.de) communicated by j. galindo abstract we investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). we show that each hausdorff topological space is a closed subspace of some hausdorff ω-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. this example provides an affirmative answer to a question posed by banakh, dimitrova, and gutik in [4]. also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a hausdorff topological semigroup whose space is weakly h-closed. 2010 msc: 54d30; 22a15. keywords: pseudocompact space; h-closed space; semigroup of matrix units; bicyclic monoid. 1. preliminaries in this paper all topological spaces are assumed to be hausdorff. by ω we denote the first infinite cardinal. for ordinals α, β put α ≤ β, (α < β, resp.) iff α ⊂ β (α ⊂ β and α ∕= β , resp.). by [α, β] ([α, β), (α, β], (α, β), resp.) we denote the set of all ordinals γ such that α ≤ γ ≤ β (α ≤ γ < β, α < γ ≤ β, α < γ < β, resp.). the cardinality of a set x is denoted by |x|. received 26 august 2019 – accepted 29 march 2020 http://dx.doi.org/10.4995/agt.2020.12258 s. bardyla and a. ravsky for a subset a of a topological space x by a we denote the closure of the set a in x. a family f of subsets of a set x is called a filter if it satisfies the following conditions: (1) ∅ /∈ f; (2) if a ∈ f and a ⊂ b then b ∈ f; (3) if a, b ∈ f then a ∩ b ∈ f. a family b is called a base of a filter f if for each element a ∈ f there exists an element b ∈ b such that b ⊂ a. a filter on a topological space x is called an ω-filter if it has a countable base. a filter f is called free if ! f = ∅. a filter on a topological space x is called open if it has a base which consists of open subsets. a point x is called an accumulation point (θ-accumulation point, resp.) of a filter f if for each open neighborhood u of x and for each f ∈ f the set u ∩ f (u ∩ f , resp.) is non-empty. a topological space x is said to be • compact, if each filter has an accumulation point; • sequentially compact, if each sequence {xn}n∈ω of points of x has a convergent subsequence; • ω-bounded, if each countable subset of x has compact closure; • totally countably compact, if each sequence of x contains a subsequence with compact closure; • countably compact, if each infinite subset a ⊆ x has an accumulation point; • ω-bounded pracompact, if there exists a dense subset d of x such that each countable subset of the set d has compact closure in x; • totally countably pracompact, if there exists a dense subset d of x such that each sequence of points of the set d has a subsequence with compact closure in x; • countably pracompact, if there exists a dense subset d of x such that every infinite subset a ⊆ d has an accumulation point in x; • pseudocompact, if x is tychonoff and each real-valued function on x is bounded; • h-closed, if each filter on x has a θ-accumulation point; • feebly ω-bounded, if for each sequence {un}n∈ω of non-empty open subsets of x there is a compact subset k of x such that k ∩ un ∕= ∅ for each n ∈ ω; • totally feebly compact, if for each sequence {un}n∈ω of non-empty open subsets of x there is a compact subset k of x such that k ∩ un ∕= ∅ for infinitely many n ∈ ω; • selectively feebly compact, if for each sequence {un}n∈ω of non-empty open subsets of x, for each n ∈ ω we can choose a point xn ∈ un such that the sequence {xn : n ∈ ω} has an accumulation point. • feebly compact, if each open ω-filter on x has an accumulation point. the interplay between some of the above properties is shown in the diagram at page 3 in [13]. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 202 closed subsets of compact-like topological spaces remark 1.1. h-closed topological spaces have few different equivalent definitions. for a topological space x the following conditions are equivalent: • x is h-closed; • if x is a subspace of a hausdorff topological space y , then x is closed in y ; • each open filter on x has an accumulation point; • for each open cover f = {fα}α∈a of x there exists a finite subset b ⊂ a such that ∪α∈bfα = x. h-closed topological spaces in terms of θ-accumulation points were investigated in [7, 16, 17, 19, 21, 22, 28, 29]. also observe that each h-closed space is feebly compact. in this paper we investigate closed subsets (subsemigroups, resp.) of compactlike topological spaces (semigroups, resp.). we prove that each topological space can be embedded as a closed subspace into an h-closed topological space. however, the semigroup of ω×ω-matrix units cannot be embedded into a topological semigroup which is a weakly h-closed topological space. we show that each topological space is a closed subspace of some ω-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. also, we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup, providing a positive solution of [4, problem 7.2]. 2. closed subspaces of compact-like topological spaces the productivity of compact-like properties is a known topic in general topology. according to tychonoff’s theorem, a (tychonoff) product of a family of compact spaces is compact, on the other hand, there are two countably compact spaces whose product is not feebly compact (see [10], the paragraph before theorem 3.10.16). the product of a countable family of sequentially compact spaces is sequentially compact [10, theorem 3.10.35]. but already the cantor cube dc is not sequentially compact (see [10], the paragraph after example 3.10.38). on the other hand some compact-like properties are also preserved by products, see [27, § 3-4] (especially theorem 3.3, proposition 3,4, example 3.15, theorem 4.7, and example 4.15), [26, § 5], and [13, sec. 2.3]. proposition 2.1. a product of any family of feebly ω-bounded spaces is feebly ω-bounded. proof. let x = " {xα : α ∈ a} be a product of a family of feebly ω-bounded spaces and let {un}n∈ω be a family of non-empty open subsets of the space x. for each n ∈ ω let vn be a basic open set in x which is contained in un. for each n ∈ ω and α ∈ a let vn,α = πα(vn) where by πα we denote the projection on xα. for each α ∈ a there exists a compact subset kα of xα, intersecting each vn,α. then the set k = " {kα : α ∈ a} is a compact subset of x intersecting each vn ⊂ un. □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 203 s. bardyla and a. ravsky a non-productive compact-like properties still can be preserved by products with more strong compact-like spaces. for instance, a product of a countably compact space and a countably compact k-space or a sequentially compact space is countably compact, and a product of a pseudocompact space and a pseudocompact k-space or a sequentially compact tychonoff space is pseudocompact (see [10, sec. 3.10]). proposition 2.2. a product x × y of a countably pracompact space x and a totally countably pracompact space y is countably pracompact. proof. let d be a dense subset of x such that each infinite subset of d has an accumulation point in x and f be a dense subset of y such that each sequence of points of the set f has a subsequence contained in a compact set. then d×f is a dense subset of x×y . so to prove that the space x×y is countably pracompact it suffices to show that each sequence {(xn, yn)}n∈ω of points of d×f has an accumulation point. taking a subsequence, if needed, we can assume that a sequence {yn}n∈ω is contained in a compact set k. let x ∈ x be an accumulation point of a sequence {xn}n∈ω and b(x) be the family of neighborhoods of the point x. for each u ∈ b(x) put yu = {yn | xn ∈ u}. then {yu | u ∈ b(x)} is a centered family of closed subsets of a compact space k, so there exists a point y ∈ ! {yu | u ∈ b(x)}. clearly, (x, y) is an accumulation point of the sequence {(xn, yn)}n∈ω. □ proposition 2.3. a product x × y of a selectively feebly compact space x and a totally feebly compact space y is selectively feebly compact. proof. let {un}n∈ω be a sequence of open subsets of x×y . for each n ∈ ω pick a non-empty open subsets u1n of x and u 2 n of y such that u 1 n×u2n ⊂ un. taking a subsequence, if needed, we can assume that that there exists a compact subset k of the space y intersecting each set u2n, n ∈ ω. since x is selectively feebly compact, for each n ∈ ω we can choose a point xn ∈ u1n such that a sequence {xn}n∈ω has an accumulation point x ∈ x. for each n ∈ ω pick a point yn ∈ u2n ∩ k. then (xn, yn) ∈ u1n×u2n ⊂ un. let b(x) be the family of neighborhoods of the point x in x. for each u ∈ b(x) put yu = {yn | xn ∈ u}. then {yu | u ∈ b(x)} is a centered family of closed subsets of the compact space k, so there exists a point y ∈ ! {yu | u ∈ b(x)}. clearly, (x, y) is an accumulation point of the sequence {(xn, yn)}n∈ω. □ an extension of a space x is a hausdorff space y containing x as a dense subspace. extensions of topological spaces were investigated in [8, 18, 23, 24, 25]. a class c of spaces is called extension closed provided each extension of each space of c belongs to c. if y is a space, a class c of spaces is y productive provided x×y ∈ c for each space x ∈ c. it is well-known or easy to check that each of the following classes of spaces is extension closed: countably pracompact, ω-bounded pracompact, totally countably pracompact, feebly compact, selectively feebly compact, and feebly ω-bounded. since [0, ω1) endowed with the order topology is ω-bounded and sequentially compact, each c© agt, upv, 2020 appl. gen. topol. 21, no. 2 204 closed subsets of compact-like topological spaces of these classes is [0, ω1)-productive by proposition 2.2, [13, proposition 3], [13, proposition 2], [9, lemma 4.2], proposition 2.3, and proposition 2.1, respectively. next we introduce a construction which helps us to construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup providing a positive answer to [4, problem 7.2]. let x and y be topological spaces such that there exists a continuous injection f : x → y . then by efy (x) we denote the subset [0, ω1]×y \ {(ω1, y) | y ∈ y \ f(x)} of a product [0, ω1]×y endowed with a topology τ which is defined as follows. a subset u ⊂ ey (x) is open if it satisfies the following conditions: • for each α < ω1, if (α, y) ∈ u then there exist β < α and an open neighborhood vy of y in y such that (β, α]×vy ⊂ u; • if (ω1, f(x)) ∈ u then there exist α < ω1, an open neighborhood vf(x) of f(x) in y and an open neighborhood wx of x in x, such that f(wx) ⊂ vf(x) and (α, ω1)×vf(x) ∪ {ω1}×f(wx) ⊂ u. remark that {ω1}×f(x) is a closed subset of e f y (x) homeomorphic to x. proposition 2.4. let x be a topological space which admits a continuous injection f into a space y and c be any extension closed, [0, ω1)-productive class of spaces. if y ∈ c then efy (x) ∈ c. proof. let y ∈ c. since c is [0, ω1)-productive, [0, ω1) × y ∈ c. a space e f y (x) is an extension of the space [0, ω1) × y providing that e f y (x) ∈ c. □ if a space x is a subspace of a topological space y and id is the identity embedding of x into y , then by ey (x) we denote the space e id y (x). it is easy to see that ey (x) is a subspace of a product [0, ω1]×y which implies that if y is tychonoff then so is ey (x). proposition 2.5. let x be a subspace of a pseudocompact space y . then ey (x) is pseudocompact and contains a closed copy of x. proof. the argument above implies that ey (x) is tychonoff. fix any continuous real valued function f on ey (x). observe that the dense subspace [0, ω1)×y of ey (x) is pseudocompact. then the restriction of f on the subset [0, ω1)×y is bounded, i.e., there exist reals a, b such that f([0, ω1)×y ) ⊂ [a, b]. then f−1[a, b] is closed in ey (x) and contains the dense subset [0, ω1) × y witnessing that f−1[a, b] = ey (x). hence the space ey (x) is pseudocompact. □ embeddings into countable compact and ω-bounded topological spaces were investigated in [2, 3]. a family a of countable subsets of a set x is called almost disjoint if for each a, b ∈ a the set a ∩ b is finite. given a property p , an almost disjoint family a is called p-maximal if each element of a has the property p and for c© agt, upv, 2020 appl. gen. topol. 21, no. 2 205 s. bardyla and a. ravsky each countable subset f ⊂ x which has the property p there exists a ∈ a such that the set a ∩ f is infinite. let f be a family of closed subsets of a topological space x. the topological space x is called • f-regular, if for any set f ∈ f and point x ∈ x \f there exist disjoint open sets u, v ⊂ x such that f ⊂ u and x ∈ v ; • f-normal, if for any disjoint sets a, b ∈ f there exist disjoint open sets u, v ⊂ x such that a ⊂ u and b ⊂ v . given a topological space x, by dω we denote the family of countable closed discrete subsets of x. we say that a subset a of x satisfies a property dω iff a ∈ dω. theorem 2.6. each dω-regular topological space x can be embedded as an open dense subset into a hausdorff countably pracompact topological space. proof. by zorn’s lemma, there exists a dω-maximal almost disjoint family a in x. let y = x ∪ a. we endow y with the topology τ defined as follows. a subset u ⊂ y belongs to τ iff it satisfies the following conditions: • if x ∈ u ∩x, then there exists an open neighborhood v of x in x such that v ⊂ u; • if a ∈ u ∩ a, then there exists a cofinite subset a ′ ⊂ a and an open set v in x such that a ′ ⊂ v ⊂ u. observe that x is an open dense subset of y and a is a discrete and closed subspace of y . since x is dω-regular for each distinct points x ∈ x and y ∈ y there exist disjoint open neighborhoods ux and uy in y . by proposition 2.1 from [2], each dω-regular topological space is dω-normal. fix any distinct a, b ∈ a. put a ′ = a \ (a ∩ b) and b ′ = b \ (a ∩ b). by the dω-normality of x, there exist disjoint open neighborhoods ua′ and ub′ of a ′ and b ′ , respectively. then the sets ua = {a} ∪ ua′ and ub = {b} ∪ ub′ are disjoint open neighborhoods of a and b, respectively, in y . hence the space y is hausdorff. observe that the maximality of the family a implies that there exists no countable discrete subset d ⊂ x which is closed in y . hence each infinite subset a in x has an accumulation point in y , that is, y is countably pracompact. □ however, there exists a hausdorff topological space which cannot be embedded as a dense open subset into any hausdorff countably pracompact topological space. example 2.7. let τ be the usual topology on the real line r and c = {a ⊂ r : |r\a| ≤ ω}. by τ∗ we denote the topology on r which is generated by the subbase τ ∪ c. obviously, the space r∗ = (r, τ∗) is hausdorff. we claim that r∗ cannot be embedded as a dense open subset into any hausdorff countably pracompact topological space. assuming the contrary, let x be a hausdorff countably pracompact topological space which contains r∗ as a dense open c© agt, upv, 2020 appl. gen. topol. 21, no. 2 206 closed subsets of compact-like topological spaces subspace. since x is countably pracompact there exists a dense subset y of x such that each infinite subset of y has an accumulation point in x. since r∗ is open and dense in x the set z = r∗ ∩ y is dense in x. moreover, it is dense in (r, τ). fix any point z ∈ z and a sequence {zn}n∈ω of distinct points of z \ {z} converging to z in (r, τ). since z is dense in (r, τ) such a sequence exists. observe that {zn}n∈ω is closed and discrete in r∗. so, its accumulation point x belongs to x \ r∗. note that for each open neighborhood u of z in r∗ all but finitely many zn belongs to the closure of u. hence x ∈ u for each open neighborhood u of z which contradicts to the hausdorffness of x. theorem 2.8. each topological space can be embedded as a closed subset into a hausdorff ω-bounded pracompact topological space. proof. let x be a topological space. by xd we denote the set x endowed with a discrete topology. let x∗ be the one point compactification of the space xd. the unique non-isolated point of x ∗ is denoted by ∞. put y = [0, ω1]×x∗ \ {(ω1, ∞)}. we endow y with a topology τ defined as follows. a subset u is open in (y, τ) if it satisfies the following conditions: • if (α, ∞) ∈ u, then there exist β < α and a cofinite subset a of x∗ which contains ∞ such that (β, α]×a ⊂ u; • if (ω1, x) ∈ u, then there exist α < ω1 and an open (in x) neighborhood v of x such that (α, ω1]×v ⊂ u. it is easy to check that the space (y, τ) is hausdorff. observe that the subset [0, ω1)×x∗ ⊂ y is open, dense and ω-bounded. hence y is ω-bounded pracompact. finally, note that the subset {ω1}×x ⊂ y is closed and homeomorphic to x. □ next we introduce a construction which helps us to prove that any space can be embedded as a closed subspace into an h-closed topological space. denote the subspace {1 − 1/n | n ∈ n} ∪ {1} of the real line by j. let x be a dense open subset of a topological space y . by z we denote the set (j×y )\{(t, y) | y ∈ y \x and t > 0}. by hy (x) we denote the set z endowed with a topology defined as follows. a subset u ⊂ z is open in hy (x) if it satisfies the following conditions: • for each x ∈ x if (t, x) ∈ u, then there exist open neighborhoods vt of t in j and vx of x in x such that vt×vx ⊂ u; • for each y ∈ y \x if (0, y) ∈ u, then there exists an open neighborhood vy of y in y such that {0}×(vy \ x) ∪ (j \ {1})×(vy ∩ x) ⊂ u. obviously, the space hy (x) is hausdorff and the subset {(1, x) | x ∈ x} ⊂ hy (x) is closed and homeomorphic to x. proposition 2.9. if y is an h-closed topological space, then hy (x) is hclosed. proof. fix an arbitrary filter f on hy (x). one of the following three cases holds: c© agt, upv, 2020 appl. gen. topol. 21, no. 2 207 s. bardyla and a. ravsky (1) there exists t ∈ j \ {1} such that for each f ∈ f there exists y ∈ y such that (t, y) ∈ f ; (2) for each f ∈ f there exists x ∈ x such that (1, x) ∈ f ; (3) for every t ∈ j there exists f ∈ f such that (t, y) /∈ f for each y ∈ y . consider case (1). for each f ∈ f put ft = f ∩ ({t}×x ∪ {0}×(y \ x)). clearly, a family ft = {ft | f ∈ f} is a filter on {t}×x∪{0}×(y \x). observe that for each t ∈ j \ {1} the subspace {t}×x ∪ {0}×(y \ x) is homeomorphic to y and hence is h-closed. then there exists a θ-accumulation point z ∈ {t}×x ∪ {0}×(y \ x) of the filter ft. obviously, z is a θ-accumulation point of the filter f. consider case (2). for each f ∈ f put f0 = {(0, x) | (1, x) ∈ f}. clearly, the family f0 = {f0 | f ∈ f} is a filter on the h-closed space {0}×y . hence there exists y ∈ y such that (0, y) is a θ-accumulation point of the filter f0. if y ∈ x, then (1, y) is a θ-accumulation point of the filter f. if y ∈ y \x, then we claim that (0, y) is a θ-accumulation point of the filter f. indeed, let u be any open neighborhood of the point (y, 0). there exists an open neighborhood vy of y in y such that v = {0}×(vy \x)∪(j \{1})×(vy ∩x) ⊂ u. since (0, y) is a θaccumulation point of the filter f0, v ∩f0 ∕= ∅ for each f0 ∈ f0. fix any f ∈ f and (0, z) ∈ v ∩f0. the definition of the topology on hy (x) yields that the set {(t, z) | t ∈ j \ {1}} is contained in v . then (1, z) ∈ {(t, z) | t ∈ j \ {1}} ⊂ v . hence for each f ∈ f the set u ∩ f is non-empty providing that (0, y) is a θ-accumulation point of the filter f. consider case (3). for each f ∈ f denote f ∗ = {(0, x) | there exists t ∈ i such that (t, x) ∈ f}. let (0, y) be a θ-accumulation point of the filter f∗ = {f ∗ | f ∈ f}. if y ∈ x, then we claim that (1, y) is a θ-accumulation point of the filter f. indeed fix any f ∈ f and an open neighborhood v of (1, y). then there exist a positive integer n and an open neighborhood u of y in x such that {t ∈ j | t > 1−1/n}×u ⊂ v . by the assumption, there exist sets f0, . . . , fn ∈ f such that fi ∩ {(1 − 1/i, x) | x ∈ y } = ∅ for every i ≤ n. then the set h = ∩i≤nfi ∩ f belongs to f and for each (t, x) ∈ h, t > 1 − 1/n. since (0, y) is a θ-accumulation point of the filter f∗ the set {0} × u ∩ h∗ is nonempty. fix any (0, x) ∈ {0} × u ∩ h∗. then there exists k > n such that (1 − 1/k, x) ∈ h ⊂ f . the definition of the topology of hy (x) implies that (1 − 1/k, x) ∈ v ∩ h ⊂ v ∩ f which implies that (1, y) is a θ-accumulation point of the filter f. if y ∈ y \ x, then even more simple arguments show that (0, y) is a θaccumulation point of the filter f. hence the space hy (x) is h-closed. □ theorem 2.10. for any hausdorff topological space x there exists an h-closed space z which contains x as a closed subspace. proof. for each hausdorff topological space x there exists an h-closed space y which contains x as a dense open subspace (see [10, problem 3.12.6]). by c© agt, upv, 2020 appl. gen. topol. 21, no. 2 208 closed subsets of compact-like topological spaces proposition 2.9, the space hy (x) is h-closed. it remains to note that the set {(1, x) | x ∈ x} ⊂ hy (x) is closed and homeomorphic to x. □ 3. applications for topological semigroups a set endowed with an associative binary operation is called a semigroup. a semigroup s is called an inverse semigroup, if for each element a ∈ s there exists a unique element a−1 ∈ s such that aa−1a = a and a−1aa−1 = a−1. the map which associates every element of an inverse semigroup to its inverse is called an inversion. a topological (inverse) semigroup is a hausdorff topological space endowed with a continuous semigroup operation (and a continuous inversion, resp.). in this case the topology of the space is called (inverse, resp.) semigroup topology. a semitopological semigroup is a hausdorff topological space endowed with a separately continuous semigroup operation. it this case the topology of the space is called shift-continuous. let x be a non-empty set. by bx we denote the set x×x ∪ {0} where 0 /∈ x×x endowed with the following semigroup operation: (a, b) · (c, d) = # (a, d), if b = c; 0, if b ∕= c, and (a, b) · 0 = 0 · (a, b) = 0 · 0 = 0, for each a, b, c, d ∈ x. the semigroup bx is called the semigroup of x×x-matrix units. observe that semigroups bx and by are isomorphic iff |x| = |y |. if a set x is infinite then the semigroup of x×x-matrix units cannot be embedded into a compact topological semigroup (see [11, theorem 3]). in [12, theorem 5] this result was generalized for countably compact topological semigroups. moreover, in [6, theorem 4.4] it was shown that for an infinite set x the semigroup bx cannot be embedded densely into a feebly compact topological semigroup. a bicyclic monoid c(p, q) is the semigroup with the identity 1 generated by two elements p and q subject to the condition pq = 1. the bicyclic monoid is isomorphic to the set ω×ω endowed with the following semigroup operation: (a, b) · (c, d) = # (a + c − b, d), if b ≤ c; (a, d + b − c), if b > c. neither stable nor γ-compact topological semigroups can contain a copy of the bicyclic monoid (see [1, 15]). in [14] it was proved that the bicyclic monoid does not embed into a countably compact topological inverse semigroup. also, a topological semigroup with a feebly compact square cannot contain the bicyclic monoid [4]. on the other hand, in [4, theorem 6.1] it was proved that there exists a tychonoff countably pracompact topological semigroup s densely containing the bicyclic monoid. moreover, under martin’s axiom the semigroup s is countably compact (see [4, theorem 6.6 and corollary 6.7]). however, it is still unknown whether there exists under zfc a countably c© agt, upv, 2020 appl. gen. topol. 21, no. 2 209 s. bardyla and a. ravsky compact topological semigroup containing the bicyclic monoid (see [4, problem 7.1]). also, in [4] the following problem was posed: problem 3.1 ([4, problem 7.2]). is there a pseudocompact topological semigroup s that contains a closed copy of the bicyclic monoid? embeddings of semigroups which are generalizations of the bicyclic monoid into compact-like topological semigroups were investigated in [5, 6]. namely, in [6] it was proved that for each cardinal λ > 1 a polycyclic monoid pλ does not embed as a dense subsemigroup into a feebly compact topological semigroup. in [5] embeddings of graph inverse semigroups into clp-compact topological semigroups were described. observe that the space [0, ω1] endowed with a semigroup operation of taking minimum becomes a topological semilattice and therefore a topological inverse semigroup. lemma 3.2. let x and y be semitopological (topological, topological inverse, resp.) semigroups such that there exists a continuous injective homomorphism f : x → y . then efy (x) is a semitopological (topological, topological inverse, resp.) semigroup with respect to the semigroup operation inherited from a direct product of semigroups (ω1, min) and y . proof. we prove this lemma for the case of topological semigroups x and y . proofs in other cases are similar. fix any elements (α, x), (β, y) of e f y (x). also, assume that β ≤ α. in the other case the proof will be similar. fix any open neighborhood u of (β, xy) = (α, x) · (β, y). there are three cases to consider: (1) β ≤ α < ω1; (2) β < α = ω1; (3) α = β = ω1. in case (1) there exist γ < β and an open neighborhood vxy of xy in y such that (γ, β]×vxy ⊂ u. since y is a topological semigroup there exist open neighborhoods vx and vy of x and y, respectively, such that vx ·vy ⊂ vxy. put u(α,x) = (γ, α]×vx and u(β,y) = (γ, β]×vy. it is easy to check that u(α,x) · u(β,y) ⊂ (γ, β]×vxy ⊂ u. consider case (2). similarly as in case (1) there exist an ordinal γ < β and open neighborhoods vx, vy and vxy of x, y and xy, respectively, such that (γ, β]×vxy ⊂ u and vx · vy ⊂ vxy. since the map f is continuous there exists an open neighborhood vf−1(x) of f −1(x) in x such that f(vf−1(x)) ⊂ vx. put u(ω1,x) = (β, ω1)×vx ∪ {ω1}×f(vf−1(x)) and u(β,y) = (γ, β]×vy. it is easy to check that u(ω1,x) · u(β,y) ⊂ (γ, β]×vxy ⊂ u. consider case (3). there exist ordinal γ < ω1, an open neighborhood vxy of xy in y and an open neighborhood wf−1(xy) of f −1(xy) in x such that (γ, ω1)×vxy ∪ {ω1}×f(wf−1(xy)) ⊂ u. since y is a topological semigroup there exist open (in y ) neighborhoods vx and vy of x and y, respectively, such that vx · vy ⊂ vxy. since the map f is continuous and x is a topological semigroup there exist open (in x) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 210 closed subsets of compact-like topological spaces neighborhoods wf−1(x) and wf−1(y) of f −1(x) and f−1(y), respectively, such that wf−1(x) · wf−1(y) ⊂ wf−1(xy), f(wf−1(x)) ⊂ vx and f(wf−1(y)) ⊂ vy. put u(ω1,x) = (γ, ω1)×vx ∪ {ω1}×f(wf−1(x)) and u(ω1,y) = (γ, ω1)×vy ∪ {ω1}×f(wf−1(y)). it is easy to check that u(ω1,x) · u(ω1,y) ⊂ u. hence the semigroup operation in e f y (x, τx) is continuous. □ remark 3.3. the subsemigroup {(ω1, f(x)) | x ∈ x} ⊂ e f y (x) is closed and topologically isomorphic to x. proposition 2.4, lemma 3.2 and remark 3.3 imply the following: proposition 3.4. let x be a (semi)topological semigroup which admits a continuous injective homomorphism f into a (semi)topological semigroup y and c be any [0, ω1)-productive, extension closed class of spaces. if y ∈ c then the (semi)topological semigroup e f y (x) ∈ c and contains a closed copy of a (semi)topological semigroup x. proposition 2.5, lemma 3.2 and remark 3.3 imply the following: proposition 3.5. let x be a subsemigroup of a pseudocompact (semi)topological semigroup y . then the (semi)topological semigroup ey (x) is pseudocompact and contains a closed copy of the (semi)topological semigroup x. by [4, theorem 6.1], there exists a tychonoff countably pracompact (and hence pseudocompact) topological semigroup s containing densely the bicyclic monoid. hence proposition 3.5 implies the following corollary which gives an affirmative answer to problem 3.1. corollary 3.6. there exists a pseudocompact topological semigroup which contains a closed copy of the bicyclic monoid. further we will need the following definitions. a subset a of a topological space is called θ-closed if for each element x ∈ x \ a there exists an open neighborhood u of x such that u ∩ a = ∅. observe that if a topological space x is regular then each closed subset a of x is θ-closed. a topological space x is called weakly h-closed if each ω-filter f has a θ-accumulation point in x. generalizations of h-closed spaces were investigated by osipov in [19, 20]. obviously, for a topological space x the following implications hold: x is hclosed ⇒ x is weakly h-closed ⇒ x is feebly compact. neither of the above implications can be inverted. indeed, an arbitrary pseudocompact but not countably compact space will be an example of feebly compact space which is not weakly h-closed. the space [0, ω1) with an order topology is an example of weakly h-closed but not h-closed space. the following theorem shows that theorem 2.10 cannot be generalized for topological semigroups. theorem 3.7. the semigroup bω of ω×ω-matrix units does not embed into a weakly h-closed topological semigroup. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 211 s. bardyla and a. ravsky proof. suppose to the contrary that bω is a subsemigroup of a weakly h-closed topological semigroup s. by e(bω) we denote the semilattice of idempotents of bω. observe that e(bω) = {(n, n) | n ∈ ω} ∪ {0} and a · b = 0 for each distinct elements a, b ∈ e(bω). let f be an arbitrary free ω-filter on the set {(n, n) | n ∈ ω}. since s is weakly h-closed, there exists a θ-accumulation point s ∈ s of the filter f. fix any open neighborhood up of the point p = s · s. the continuity of the semigroup operation in s yields an open neighborhood vs of s such that vs · vs ⊂ up. since s is a θ-accumulation point of the filter f there exist distinct elements (n, n), (m, m) ∈ vs ∩ e(bω). hence 0 = (n, n) · (m, m) ∈ vs · vs ⊂ up which implies that 0 ∈ up for each open neighborhood up of p witnessing that p = 0. we claim that 0 is a θ-accumulation point of the filter f. indeed, fix any open neighborhood u of 0. since s · s = 0 and s is a topological semigroup there exists an open neighborhood vs of s such that vs · vs ⊂ u. observe that (n, n) = (n, n) · (n, n) ∈ vs · vs ⊂ u for each (n, n) ∈ vs. hence 0 is a θ-accumulation point of the filter f. since the filter f was selected arbitrarily we have that 0 is a θ-accumulation point of any free ω-filter on the set {(n, n) | n ∈ ω}. thus for each open neighborhood u of 0 the set au = {(n | (n, n) /∈ u} is finite, because if there exists an open neighborhood u of 0 such that the set au is infinite, then 0 is not a θ-accumulation point of the ω-filter f which has a base consisting of cofinite subsets of au. let f be an arbitrary free ω-filter on the set {(1, n) | n ∈ ω}. since s is weakly h-closed there exists a θ-accumulation point s ∈ s of the filter f. we claim that s · 0 = 0. indeed, fix any open neighborhood w of s · 0. the continuity of the semigroup operation in s yields open neighborhoods vs of s and v0 of 0 such that vs·v0 ⊂ w . since the set av0 = {(n | (n, n) /∈ v0} is finite and s is a θ-accumulation point of the filter f there exist distinct n, m ∈ ω such that (1, n) ∈ vs and (m, m) ∈ v0. then 0 = (1, n) · (m, m) ∈ vs · v0 ⊂ w . hence 0 ∈ w for each open neighborhood w of s · 0 witnessing that s · 0 = 0. fix an arbitrary open neighborhood u of 0. since s · 0 = 0 and s is a topological semigroup, there exist open neighborhoods vs of s and v0 of 0 such that vs · v0 ⊂ u. recall that the set {n | (n, n) /∈ v0} is finite. then (1, n) = (1, n)·(n, n) ∈ vs·v0 ⊂ u for all but finitely many elements (1, n) ∈ vs. hence 0 is a θ-accumulation point of the ω-filter f. since the filter f was selected arbitrarily, 0 is a θ-accumulation point of any free ω-filter on the set {(1, n) | n ∈ ω}. as a consequence, for each open neighborhood u of 0 the set bu = {n | (1, n) /∈ u} is finite. similarly it can be shown that for each open neighborhood u of 0 the set cu = {n | (n, 1) /∈ u} is finite. fix an open neighborhood u of 0 such that (1, 1) /∈ u. since 0 = 0 · 0 the continuity of the semigroup operation implies that there exists an open neighborhood v of 0 such that v · v ⊂ u. the finiteness of the sets bv c© agt, upv, 2020 appl. gen. topol. 21, no. 2 212 closed subsets of compact-like topological spaces and cv implies that there exists n ∈ ω such that {(1, n), (n, 1)} ⊂ v . hence (1, 1) = (1, n) · (n, 1) ∈ v · v ⊂ u, which contradicts to the choice of u. □ corollary 3.8. the semigroup of ω×ω-matrix units does not embed into a topological semigroup s whose space is h-closed. however, we have the following questions: problem 3.9. does there exist a feebly compact topological semigroup s which contains the semigroup of ω×ω-matrix units? problem 3.10. does there exist a topological semigroup s which cannot be embedded into a feebly compact topological semigroup t? we remark that these 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(pranav15851@gmail.com) communicated by f. mynard abstract in the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the pontryagin duality is the continuous duality. we prove that local quasi-convexity is a necessary condition for a convergence group to be creflexive. further, we prove that every character group of a convergence group is locally quasi-convex. 2010 msc: 43a40; 54a20; 54h11. keywords: continuous duality; convergence groups; local quasi-convexity; pontryagin duality. 1. introduction and preliminaries the character group ĝ of an abelian topological group g is the group of all continuous characters equipped with the compact-open topology. pontryagin duality theorem states that for any locally compact abelian (lca) topological group, the canonical map αg : g → ˆ̂ g (from the group to the double dual group) is a topological isomorphism. mart́ın-peinador [10] proves that if g is reflexive and the evaluation map e : ĝ × g → t defined as e(χ, x) = χ(x) is continuous, then the group g is locally compact. this result explains the role of local compactness in the pontryagin duality theory. this result is further generalized in [11, theorem 1.1] where the same statement with the condition “reflexive” replaced by the “quasi-convex compactness property” is proved. received 06 november 2020 – accepted 07 january 2021 http://dx.doi.org/10.4995/agt.2021.14585 p. sharma in the last seventy years, extending the pontryagin duality theory beyond local compactness has gained the attention of several researchers [8, 13]. one of these approaches is the continuous duality theory [5]. the compact-open topology on the continuous character group is replaced with the continuous convergence structure, and as a consequence of this, the evaluation map is always continuous. a comparison of this approach with the pontryagin duality is presented in [7]. on the other hand, there are several situations in analysis (like convergence in measure) where non-topological convergence originates, and the corresponding generalisation of a topological space is the convergence space. in the realm of the convergence spaces, the generalisation of topological groups is the convergence groups. for details regarding the continuous duality for convergence groups refer [4, 6]. further, the notion of locally quasi-convex convergence groups is introduced in [12]. here, we prove that local quasi-convexity is a necessary condition for a convergence group to be c-reflexive, and then we prove that every character group of a convergence group is locally quasi-convex. before proceeding further, we present certain terms and notations required for the rest of the article. a filter f is a non-empty family of non-empty subsets of a set x which is closed under supersets and finite intersection. we denote the set of all filters on a set x by fx. further, a subset h of a filter f is called basis of the filter if each set in f contains a set in h. if f and g are filters on a set x then, f is called coarser than g if f ⊂ g. let λ be an arbitrary relation between x and fx. the relation is called convergence (and the pair (x, λ) a convergence space) on x if for f1, f2 in fx and x in x the following conditions hold: (i) centred: x↑ ∈ λ(x), (ii) isotone: if f1 ∈ λ(x) and f1 ≤ f2, then f2 ∈ λ(x) , and (iii) finitely deep: if f1, f2 ∈ λ(x), then f1 ∩ f2 ∈ λ(x). this relation is also denoted by f −→ λ x. a convergence space x is hausdorff if every filter on x converges to at most one point. further, a map f : (x, λ1) → (y, λ2) between two convergence spaces is said to be continuous if (f −→ λ1 x ⇒ f(f) −→ λ2 f(x)). for any convergence spaces (x, λ1) and (y, λ2) let c(x, y ) denote the set of all continuous functions from x to y . the evaluation mapping e : c(x, y ) × x → y is defined as e(f, x) = f(x) ∀ f ∈ c(x, y ) and x ∈ x. the continuous convergence structure on c(x, y ) is defined as: a filter g converge to f in c(x, y ) iff e(g × f) converge to f(x) in y , whenever f converge to x in x. the space c(x, y ) equipped with continuous convergence structure is denoted by cc(x, y ). a convergence group is a group with a compatible (the group operations are continuous in the sense of convergence) convergence structure. the class © agt, upv, 2021 appl. gen. topol. 22, no. 1 194 duality of locally quasi-convex convergence groups of convergence groups contains the class of topological groups. some other examples include the underlying groups of the convergence vector spaces. for details about convergence spaces and convergence groups, we refer the reader to [3, 9]. for a convergence abelian group (g, λ) denote by γg the set of all continuous homomorphisms of g into the circle group t. the set of all continuous homomorphisms with the structure of continuous convergence is defined as convergence dual of g and is denoted as (γg, λc). the evaluation map κ : g → γγg defined as κ(g)(χ) = χ(g) ∀ g ∈ g, χ ∈ γg is a continuous group homomorphism. the convergence group is c-reflexive if this evaluation map κ is a bicontinuous isomorphism. as proved in [3, example 8.5.14], there exists a locally compact non-reflexive convergence group. therefore, the analog of pontryagin duality theorem, valid in the context of abelian topological groups, cannot be directly extended to abelian convergence groups. motivated from the notion of local quasi-convexity [2] in topological groups, the notion of local quasi-convexity for convergence groups is defined in [12]. for a convergence abelian group g we define the polar and the inverse polar of subsets of g and of γg as follows: definition 1.1 (polar and inverse polar). for any subset h of g and l of γg the polar and the inverse polar of h and l respectively are subsets defined as: h✄ = {χ ∈ γg : χ(h) ⊂ t+}; l ✁ = {g ∈ g : χ(g) ⊂ t+, ∀ χ ∈ l}, here t+ = {z ∈ t : re (z) ≥ 0}. the local quasi-convexity for convergence groups is defined as: definition 1.2 (quasi-convex set [3]). a subset a of a convergence abelian group g is quasi-convex if for each point g in g\a, there is a character χ in the polar set of a such that reχ(g) < 0, that is a✄✁ = a. proposition 1.3. let g be a convergence group, h ⊂ g and l ⊂ γg. the polar h⊲ and the inverse polar l⊳ are quasi convex subsets of γcg and g respectively. definition 1.4 (locally quasi-convex convergence group [12]). a convergence group g is locally quasi-convex if for each filter f −→ g 0 , there exists another filter g coarser than f such that g −→ g 0 and g has a filter base composed of quasi-convex sets. 2. main results proposition 2.1. for a filter u on a convergence group g and for all u, v ∈ u we have (u ∩ v )⊲⊲ ⊆ u⊲⊲ ∩ v ⊲⊲. proof. the proof is trivial. � proposition 2.2. {u⊲⊲ : u ∈ u} is a basis of a filter in γcγcg. © agt, upv, 2021 appl. gen. topol. 22, no. 1 195 p. sharma proof. the proof is similar to [3, theorem 8.4.3]. � we denote the filter generated by {u⊲⊲ : u ∈ u} as u⊲⊲. similarly, {u⊲⊳ : u ∈ u} is a basis of a filter in g which we denote by u⊲⊳. lemma 2.3. for a convergence abelian group g, the following statements hold: (1) if a filter φ −−−→ γcg 0, then a✄ ∈ φ for every finite subset a of g. (2) for every filter f −→ g 0, there is b ∈ f such that b✄ ∈ φ. proof. the proof follows from [3, proposition 8.1.8]. � theorem 2.4. for a convergence abelian group g, if u −→ g 0, then u✄✄ −−−−→ γcγcg 0. proof. applying lemma 2.3 to γcg we need to prove that a. a⊲ ∈ u⊲⊲ for each finite set a ⊆ γcg, and b. for each filter φ which converges to 0 in γcg, there is some p ∈ φ such that p ⊲ ∈ u⊲⊲. a) let a = {χ1, ..., χn}, then χi(u) → 0 for all i = 1, . . . , n. further, there are ui ∈ u such that χi(ui) ⊆ t+. if u = u1 ∩ ... ∩ un, then χi(u) ⊆ t+ for all i which implies a ⊆ u⊲. finally we have, u⊲⊲ ⊆ a⊲ and hence, a⊲ ∈ u⊲⊲. b) if φ → 0 in γcg, then φ(u) → 0 in t. further, there are p ∈ φ and u ∈ u such that p(u) ⊆ t+. this gives p ⊆ u ⊲ which implies u⊲⊲ ⊆ p ⊲ and hence, p ⊲ ∈ u⊲⊲. � theorem 2.5. if a convergence group g is c-reflexive then it must be locally quasi-convex. proof. to prove this result it is sufficient to prove that if a convergence group is embedded then it must be locally quasi-convex. in a convergence abelian group g let, κg : g → γcγcg be an embedding. if u → 0 in g then by theorem 2.4, we have, u⊲⊲ → 0 in γcγcg and so κ −1(u⊲⊲) → 0 in g. now we have κ−1(u⊲⊲) = u⊲⊳ ⊆ u. in view of proposition 1.3 the proof follows. � next we prove that the continuous character group of a convergence group is locally quasi-convex. © agt, upv, 2021 appl. gen. topol. 22, no. 1 196 duality of locally quasi-convex convergence groups lemma 2.6. let x be a convergence space and a is a subset of x. furthermore, let g be a convergence group and m ⊆ g a quasi-convex set. then p(a, m) = {g ∈ cc(x, g) : g(a) ⊂ m} is quasi-convex in the group cc(x, g). proof. take any g0 6∈ p(a, m). then there is a point x0 ∈ a such that g0(x0) 6∈ m. since m is quasi-convex there is some χ ∈ γ(g) such that χ(m) ⊂ t+ while χ(g0(x0)) 6∈ t+. define φ : cc(x, g) → t by φ(g) = χ(g(x0)). then φ is a continuous character, φ(g0) = χ(g0(x0)) 6∈ t+ while φ(g) = χ(g(x0)) ∈ χ(m) ⊆ m for all g ∈ p(a, m). � the next lemma is an extension of [1, proposition 6.2, (ii)] to the realm of convergence groups. lemma 2.7. let g be a convergence group and b be a family of quasi-convex subsets of g. then b0 = ⋂ {b : b ∈ b} is a quasi-convex subset of g. proof. a direct proof is easy, so omitted. � theorem 2.8. let x be a convergence space and g be a locally quasi-convex topological group. then cc(x, g) is locally quasi-convex. proof. choose a zero neighbourhood basis b in g consisting of quasi-convex sets and take a filter g on cc(x, g) which converges to 0. then g(φ) → 0 ∈ g for each x ∈ x, and each filter φ on x which converges to x. so for each b ∈ b there are gx,φ,b ∈ g and fx,φ,b ∈ φ such that gx,φ,b(fx,φ,b) ⊆ b. set hx,φ,b = p(fx,φ,b, b) then {hx,φ,b : φ → x, b ∈ b} is the subbasis of a filter h which converges to 0 in cc(x, g). by the lemma 2.6 and lemma 2.7, h has a base of quasi-convex sets and h ⊆ g since gx,φ,b ⊆ hx,φ,b for all x, φ, b. � corollary 2.9. for each convergence group g, γcg is locally quasi-convex. proof. since t and cc(g, t) are locally quasi-convex so is γc(g) as a subgroup. � acknowledgements. we thank prof. h.-p. butzmann and the anonymous reviewers for their many insightful comments and suggestions. © agt, upv, 2021 appl. gen. topol. 22, no. 1 197 p. sharma references [1] l. außenhofer, contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, dissertationes mathematicae. institute of mathematics, polish academy of sciences, 1999. [2] w. banaszczyk, additive subgroups of topological vector spaces, lecture notes in mathematics, springer berlin heidelberg, 1991. [3] r. beattie and h.-p. butzmann, convergence structures and applications to functional analysis, bücher, springer netherlands, 2013. [4] m. bruguera, topological groups and convergence groups: study of the pontryagin duality, thesis, 1999. [5] h.-p. butzmann, über diec-reflexivität von cc(x), comment. math. helv. 47, no. 1 (1972), 92–101. [6] h.-p. butzmann, duality theory for convergence groups, topology appl. 111, no. 1 (2000), 95–104. [7] m. j. chasco and e. mart́ın-peinador, binz-butzmann duality versus pontryagin duality, arch. math. (basel) 63, no. 3 (1994), 264–270. [8] m. j. chasco, d. dikranjan and e. mart́ın-peinador, a survey on reflexivity of abelian topological groups, topology appl. 159, no. 9 (2012), 2290–2309. [9] s. dolecki and f. mynard, convergence foundations of topology, world scientific publishing company, 2016. [10] e. mart́ın-peinador, a reflexive admissible topological group must be locally compact, proc. amer. math. soc. 123, no. 11 (1995), 3563–3566. [11] e. mart́ın-peinador and v. tarieladze, a property of dunford-pettis type in topological groups, proc. amer. math. soc. 132, no. 6 (2004), 1827–1837. [12] p. sharma, locally quasi-convex convergence groups, topology appl. 285 (2020), 107384. [13] p. sharma and s. mishra, duality in topological and convergence groups, top. proc., to appear. © agt, upv, 2021 appl. gen. topol. 22, no. 1 198 @ appl. gen. topol. 21, no. 2 (2020), 247-264 doi:10.4995/agt.2020.13049 c© agt, upv, 2020 topological distances and geometry over the symmetrized omega algebra mesfer alqahtani a, cenap özel a and hanifa zekraoui b a department of mathematics, king abdulaziz university, jeddah, kingdom of saudi arabia (mesfer ¯ alqhtani@hotmail.com,cozel@kau.edu.sa) b department of mathematics, larbi ben m’hidi university, oum el bouaghi, algeria (hzekraoui421@gmail.com) communicated by f. lin abstract the aim of this paper is to study some topological distances properties, semidendrites and convexity on the symmetrized omega algebra. furthermore, some properties and exponents on the symmetrized omega algebra are introduced. 2010 msc: 15a80; 16y60; 54f65. keywords: omega algebra; symmetrized omega algebra; semidendrite; exponents; convex and topology. 1. introduction omega algebra unifies the min and the max plus algebras and introduces an original structure which in fact is an ”abstract tropical algebra”. termed it as ”omega algebra” or in short just, ”ω− algebra”. the r−∞ and r∞ and their nearby structures, like min − max and max − times algebras, etc., are all subsumed under omega algebra. all these are idempotent semirings which sometimes also termed as dioids. in the previous studies, for the construction of all such semirings, an ordered infinite abelian group is mandatory, see [4], [5], [7] and [8]. in ω− algebra, the definition is extended to cyclically ordered abelian groups and also for finite sets under some suitable ordering, for more details one can refer to [3]. received 23 january 2020 – accepted 29 july 2020 http://dx.doi.org/10.4995/agt.2020.13049 m. alqahtani, c. özel and h. zekraoui the aim of this paper is to define some topological distances, semidendrites, convexity, some properties and exponents on the symmetrized omega algebra. our paper is organized as follows. in section 2, we review some basic facts for omega algebra and a brief of the symmetrized omega algebra, furthermore the rules of calculation and the absolute value on the symmetrized omega algebra. in section 3, some properties and exponents on the symmetrized omega algebra are introduced. we study some properties of the given topological distances and semidendrite on the symmetrized omega algebra in section 4. in section 5, we generalize the notion of convex sets in paper [6] over sω. this paper is produced from the phd thesis of mr. mesfer hayyan alqahtani in king abdulaziz university.. 2. some preliminaries in abstract omega algebra in this section, we recall some basic facts for omega algebra, the symmetrized omega algebra, rules of calculation and ω−absolute value. for more details see [3]. 2.1. omega algebra. let (g, ◦, e) be an abelian group. let a be a closed subset of g and e ∈ a. then (a, ◦, e) is a submonoid of g. assume that ω is an indeterminate (may belong to a or g, as we will see in examples 1 and 2. obviously, in this case ω is no longer an indeterminate). because the terms are generated from tropical geometry, so such an indeterminate may be termed as a tropical indeterminate. definition 2.1 ([3]). we say that aω = a∪{ω} is an omega algebra (in short ω− algebra) over the group g in case aω is closed under two binary operations, ⊕, ⊗ : aω × aω −→ aω, such that ∀a1, a2, a3 ∈ a, the following axioms are satisfied: 1) a1 ⊕ a2 = a1 or a2; 2) a1 ⊕ ω = a1 = ω ⊕ a1; 3) ω ⊕ ω = ω; 4) a1 ⊗ a2 = a2 ⊗ a1 ∈ a; 5) (a1 ⊗ a2) ⊗ a3 = a1 ⊗ (a2 ⊗ a3); 6) a1 ⊗ e = a1; 7) a1 ⊗ ω = ω ⊗ a1 = ! ω if ω ∕= e a1 if ω = e ; 8) ω ⊗ ω = ω; 9) a1 ⊗ (a2 ⊕ a3) = (a1 ⊗ a2) ⊕ (a1 ⊗ a3). remark 2.2 ([3]). 1) ⊕ is a pairwise comparison operation, such as, max, min, inf, sup, up, down, lexicographic ordering, or any thing else that compairs two elements of aω. obviously, it is associative and commutative and the tropical indeterminate ω plays the role of the identity. hence (aω, ⊕, ω) is a commutative monoid. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 248 topological distances and geometry over the symmetrized omega algebra 2) ⊗ is also associative and commutative on aω and e plays the role of the multiplicative identity of aω. hence (aω, ⊗, e) is also a commutative monoid. 3) the left distributive law (9) also gives the right distributive law. 4) every element of aω is an idempotent under ⊕. 5) altogether, we write both structures as: aω = (aω, ⊕, ⊗, ω, e). this is an idempotent semiring also called ”dioid” in literature. remark 2.3 ([3]). in this note, we confined ourselves to only ω− algebras over abelian groups and rings. 2.2. the symmetrized omega algebra. in this subsection, we give a brief of the symmetrized omega algebra, for more details see [3]. let (g, ◦, e) be an abelian group and (aω, ⊕, ⊗, ω, e) an ω−algebra over the group g. we consider the set of ordered pairs pω = a2ω. let ≤ be the ordering defined on aω by the relation (2.1) a ≤ b ⇐⇒ a ⊕ b = b which gives a total order on aω and for all a ∈ aω, we have ω ≤ a. for a ∕= b, such that a ⊕ b = b, we denote by a < b. let ▽ be the relation defined on pω as follows: for all (a, b), (c, d) ∈ pω (a, b) ▽ (c, d) ⇐⇒ a ⊕ d = b ⊕ c. definition 2.4 ([3]). let ∼ be the equivalence relation close to ▽ defined as follows: for all (a, b), (c, d) ∈ pω, (a, b) ∼ (c, d) ⇐⇒ ! (a, b) ▽ (c, d) if a ∕= b and c ∕= d (a, b) = (c, d) otherwise in addition to the class element ω = (ω, ω); for all a ∈ aω, with a ∕= ω, we have three kinds of equivalence classes: (a) (a, ω) = {(a, b) ∈ pω, b < a}, called positive ω−element. (b) (ω, a) = {(b, a) ∈ pω, b < a}, called negative ω−element. (c) (a, a) called balenced ω−element. proposition 2.5 ([3]). the addition operation ⊕ defined by (a, b)⊕(c, d) = (a ⊕ c, b ⊕ d) on the quotient set pω∼ is well defined and satisfies the axioms (1), (2) and (3) of definition 2.1. proposition 2.6 ([3]). the set pω∼ is closed under the binary multiplication operation ⊗ defined as follows: for all (a, b), (c, d) ∈ pω∼ ; (a, b)⊗(c, d) = ((a ⊗ c) ⊕ (b ⊗ d), (a ⊗ d) ⊕ (b ⊗ c)) and satisfies axioms from (4) to (9) of definition 2.1 with the unit class element e = (e, ω). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 249 m. alqahtani, c. özel and h. zekraoui definition 2.7 ([3]). the structure "pω ∼ , ⊕, ⊗, ω, e # is called the symmetrized ω−algebra over the abelian group g × g and denoted by sω. in the coming sections just for simplicity we will use ⊕, ⊗, ω and e instead of ⊕, ⊗, ω and e respectively. remark 2.8 ([3]). 1) despite the nature of the positive and the negative ω−elements, they are not the inverses of each others for the additive operation ⊕, 2) we have three symmetrized ω−subalgebras of sω, s(+)ω = $ (a, ω), a ∈ aω % , s(−)ω = $ (ω, a), a ∈ aω % , s(0)ω = $ (a, a), a ∈ aω % . 3) the three symmetrized ω−subalgebras of sω are connected by the zero class element ω. 4) the positive ω−elements and the negative ω−elements are called signed and denoted by s∨ω = s (+) ω ∪ s (−) ω , where the zero class (ω, ω) corresponds to ω. 2.3. rules of calculation in omega. let a ∈ aω. then we admit the following notations: +a = (a, ω), − a = (ω, a), · a = (a, a). by results in proposition 2.5, proposition 2.6 and the above notation, it is easy to verify the rules of calculation in the following proposition: proposition 2.9 ([3]). for all a, b ∈ aω, we have (i) (+a) ⊕ (+b) = + (a ⊕ b) ; (ii) (+a) ⊕ (−b) = & ' ( +a if b < a −b if b > a ·a if b = a ; (iii) (±a) ⊕ (·b) = ! ±a if b < a ·b if b > a : (iv) (−a) ⊕ (−b) = − (a ⊕ b) ; (v) (+a) ⊗ (+b) = + (a ⊗ b) ; (vi) (+a) ⊗ (−b) = − (a ⊗ b) ; (vii) (±a) ⊗ (·b) = · (a ⊗ b) ; (viii) (−a) ⊗ (−b) = + (a ⊗ b) . from the previous rules, we can notice that the sign of the result in the addition operation follows the greater element in aω. while in the multiplication operation, the balance sign is the strong one (has priority). from proposition 2.9, we can deduce the following : c© agt, upv, 2020 appl. gen. topol. 21, no. 2 250 topological distances and geometry over the symmetrized omega algebra proposition 2.10 ([3]). the map |.|ω : sω −→ aω, such that for all a ∈ aω, |+a|ω = |−a|ω = |·a|ω = a is an absolute value on sω. we call it the ω−absolute value. 3. some properties and exponents on sω in this section, we give some properties and exponents on sω. 3.1. some properties on sω. as aω is isomorphic to the subdioid of pairs (a, ω), a ∈ aω, then aω itself can be considered as a subdioid of pω. if u = (a, b) ∈ sω, then we have two unary operators: ⊖ (the ω-algebraic minus operator) and (·)• (the balance operator) such that ⊖u = (b, a) and u• = u ⊕ (⊖u). proposition 3.1. for all u = (a, b), v = (c, d) ∈ sω, we have: (i) u• = (⊖u)• = (u•)•; (ii) u ⊗ v• = (u ⊗ v)• = u• ⊗ v = u• ⊗ v•; (iii) ⊖ (⊖u) = u, ⊖ (u ⊕ v) = (⊖u) ⊕ (⊖v) and (⊖u) ⊗ v = ⊖ (u ⊗ v) = u ⊗ (⊖v). proof. a direct calculations gives the desired results. □ proposition 3.2. let a, b ∈ aω be arbitrary. then 1) ⊖(+a) = −a; 2) ⊖(−a) = +a; 3) ⊖(·a) = ·a; 4) +a ⊖ +b = (a, b). proof. 1) ⊖(+a) = ⊖(a, ω) = ⊖ (a, ω) = (ω, a) = −a; 2) ⊖(−a) = ⊖(ω, a) = ⊖ (ω, a) = (a, ω) = +a; 3) ⊖(·a) = ⊖(a, a) = ⊖ (a, a) = (a, a) = ·a; 4) +a ⊖ +b = (a, ω) ⊖ (b, ω) = (a, ω) ⊕ (⊖(b, ω)) = (a, ω) ⊕ (ω, b) = (a, b). □ one can easily prove that the ordering ≤ defined on aω by the relation (3.1) a ≤ b ⇐⇒ a ⊕ b = b is a total order on aω and for all a ∈ aω, we have ω ≤ a. for a ∕= b, such that a ⊕ b = b, we denote by a < b. by the construction of the symmetrized omega algebra, it is easy to extend the total order ≤ on aω to the ordering: definition 3.3. for any u = (a, b), v = (c, d) ∈ sω, we have u ≤ v ⇐⇒ u ⊕ v = v corollary 3.4. for any (a, b) ∈ sω, we have (ω, ω) ≤ (a, b). proof. (a, b) ⊕ (ω, ω) = (a ⊕ ω, b ⊕ ω) = (a, b) ⇐⇒ ω = (ω, ω) ≤ (a, b). □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 251 m. alqahtani, c. özel and h. zekraoui proposition 3.5. let (a, b) ∈ sω, where a, b ∈ aω, then we have: 1) if a > b then (a, b) = (a, ω). 2) if a b (a ∕= ω, because ω > b impossible). since (a, ω) = {(a, b) ∈ pω, b < a}, then (a, b) ∈ (a, ω). also (a, b) ∈ (a, b), because (a, b) ∼ (a, b). then we have (a, b) ∈ (a, ω)∩(a, b). hence (a, b) = (a, ω), because any two equivalence classes are disjoint or equal. 2) let a a}, then (a, b) ∈ (ω, b). also (a, b) ∈ (a, b), because (a, b) ∼ (a, b). then we have (a, b) ∈ (ω, b) ∩ (a, b). hence (a, b) = (ω, b), because any two equivalence classes are disjoint or equal. 3) if a = b, then we have (a, b) = (a, a) or (a, b) = (b, b). □ proposition 3.6. let u = (a, ω), v = (ω, b), z = (c, c) ∈ sω, where a, b, c ∈ aω are arbitrary. then we have 1) if a ≥ b, then u ⊕ v = u ⊕ (⊖v); 2) if a < b, then u ⊕ (⊖v) = (⊖u) ⊕ (⊖v); 3) if b ≤ c, then v ⊕ z = (⊖v) ⊕ z = v ⊕ (⊖z) = (⊖v) ⊕ (⊖z); 4) if b > c, then (⊖v) ⊕ (⊖z) = (⊖v) ⊕ z and v ⊕ z = v ⊕ (⊖z). proof. 1) (3.2) u ⊕ v = (a, ω) ⊕ (ω, b) = (a ⊕ ω, ω ⊕ b) = (a, b) = (a, ω) and (3.3) u ⊕ (⊖v) = (a, ω) ⊕ (b, ω) = (a ⊕ b, ω) = (a, ω) by equations 3.2 and 3.3, we get u ⊕ v = u ⊕ (⊖u). 2) (3.4) u ⊕ (⊖u) = (a, ω) ⊕ (b, ω) = (a ⊕ b, ω) = (b, ω) and (3.5) (⊖u) ⊕ (⊖v) = (ω, a) ⊕ (b, ω) = (b, a) = (b, ω) by equations 3.4 and 3.5, we get u ⊕ (⊖v) = (⊖u) ⊕ (⊖v). 3) (3.6) v ⊕ z = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, c) (3.7) (⊖v) ⊕ z = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (c, c) (3.8) v ⊕ (⊖z) = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, c) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 252 topological distances and geometry over the symmetrized omega algebra (3.9) (⊖v) ⊕ (⊖z) = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (c, c) by equations 3.6, 3.7, 3.8 and 3.9, we get v⊕z = (⊖v)⊕z = v⊕(⊖z) = (⊖v) ⊕ (⊖z). 4) (3.10) (⊖v) ⊕ (⊖z) = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (b, c) = (b, ω) and (3.11) (⊖v) ⊕ z) = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (b, c) = (b, ω) by equations 3.10 and 3.11, we get (⊖v) ⊕ (⊖z) = (⊖v) ⊕ z. also (3.12) v ⊕ z = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, b) = (ω, b) and (3.13) v ⊕ (⊖z) = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, b) = (ω, b) by equations 3.12 and 3.13, we get v ⊕ z = v ⊕ (⊖z). □ 3.2. exponents on sω. in this subsection, we assume that ⊗|a = ◦. let sign (.) ∈ {+, −, ·}. definition 3.7. by proposition 2.9, we can define the ω−power of an element in sω as following: let sign(a)a ∈ sω be arbitrary, where a ∈ aω and n ∈ z+, then (1): if a ∕= ω. (sign(a)a)⊗n = (sign(a)a) ⊗ ... ⊗ (sign(a)a) ) *+ , n times = sign(.)(a ⊗ ... ⊗ a) ) *+ , n times = sign(.)(a ◦ ... ◦ a) ) *+ , n times , where sign (.) = ! sign(a) if sign(a) ∈ {+, ·} or sign(a) = − and n is odd + if sign(a) = − and n is even ; (2): if a = ω, then (sign(ω)ω)⊗n = sign(ω)ω. example 3.8. let −a ∈ sω, where a ∕= ω, then c© agt, upv, 2020 appl. gen. topol. 21, no. 2 253 m. alqahtani, c. özel and h. zekraoui 1) (−a)⊗2 = (ω, a) ⊗2 = (ω, a) ⊗ (ω, a) =((ω ⊗ ω) ⊕ (a ⊗ a), (ω ⊗ a) ⊕ (a ⊗ ω)) = (a ⊗ a, ω) = (a ◦ a, ω) = + (a ◦ a) 2) (−a)⊗3 = (ω, a) ⊗3 =(ω, a) ⊗ (ω, a) ⊗ (ω, a) = ((ω ⊗ ω) ⊕ (a ⊗ a), (ω ⊗ a) ⊕ (a ⊗ ω)) ⊗ (ω, a) = (a ⊗ a, ω) ⊗ (ω, a) = ((a ⊗ a ⊗ ω) ⊕ (ω ⊗ a), (a ⊗ a ⊗ a) ⊕ (ω ⊗ ω)) = (ω, a ⊗ a ⊗ a) = (ω, a ◦ a ◦ a) = − (a ◦ a ◦ a) theorem 3.9. let aω = (aω, ⊕, ⊗, ω, e) be an ω− algebra over an abelian group g = (g, ◦, e) and sω = " aω×aω ∼ , ⊕, ⊗, ω, e # is the symmetrized ω−algebra over the abelian group g×g. let ⊗|a = ◦, ω ∕= e and a ∕= ω. then the following are equivalent: 1) if a ∈ (a, ◦, e), has an inverse in a, then 2) a ∈ (aω \ {ω}, ⊗, e) has a multiplicativ inverse, and then 3) +a, −a ∈ sω \ s (0) ω have a multiplicative inverse, but ·a ∈ s (0) ω has no multiplicative inverse. proof. (1 ⇒ 2) let a ∈ (a, ◦, e) be arbitrary, which has an inverse and denoted by a−1, then a ⊗ a−1 = a ◦ a−1 = e. hence a⊗−1 = a−1 is the multiplicative inverse of a in (aω \ {ω}, ⊗, e). (2 ⇒ 3) let a ∈ aω be arbitrary, where a ∕= ω and ω ∕= e, which has a multiplicative inverse and denoted by a⊗−1, then (+a) ⊗ (+a⊗−1) = (a, ω) ⊗ (a⊗−1, ω) =((a ⊗ a⊗−1) ⊕ (ω ⊗ ω), (a ⊗ ω) ⊕ (ω ⊗ a⊗−1)) = (a ⊗ a⊗−1, ω) = (a ◦ a⊗−1, ω) = (e, ω) = e then +a⊗−1 is a multiplicative inverse of +a in sω and (−a) ⊗ (−a⊗−1) = (ω, a) ⊗ (ω, a⊗−1) =((ω ⊗ ω) ⊕ (a ⊗ a⊗−1), (ω ⊗ a⊗−1) ⊕ (a ⊗ ω)) = (a ⊗ a⊗−1, ω) = (a ◦ a⊗−1, ω) = (e, ω) = e c© agt, upv, 2020 appl. gen. topol. 21, no. 2 254 topological distances and geometry over the symmetrized omega algebra then −a⊗−1 is a multiplicative inverse of −a in sω. on the other hand suppose that ·a ∈ s(0)ω has a multiplicative inverse (x, y), where x, y ∈ aω, then (·a) ⊗ (x, y) = (a, a) ⊗ (x, y) =((a ⊗ x) ⊕ (a ⊗ y), (a ⊗ y) ⊕ (a ⊗ x)) = ((a ◦ x) ⊕ (a ◦ y), (a ◦ y) ⊕ (a ◦ x)) = (e, ω) hence (a ◦ x) ⊕ (a ◦ y) = e and (a ◦ y) ⊕ (a ◦ x) = ω, thus contradiction (note that if (x, y) = ·a⊗−1, then we have (a ◦ a⊗−1) ⊕ (a ◦ a⊗−1) = e and (a ◦ a⊗−1) ⊕ (a ◦ a⊗−1) = ω, thus condradiction). (3 ⇒ 1) let +a ∈ sω be arbitrary, where a ∕= ω, ω ∕= e and the multiplicative inverse of +a in sω is (x, y), where x, y ∈ aω, then we have: (+a) ⊗ (x, y) = (a, ω) ⊗ (x, y) =((a ⊗ x) ⊕ (ω ⊗ y), (a ⊗ y) ⊕ (ω ⊗ x)) = (a ⊗ x, a ⊗ y) = (a ◦ x, a ◦ y) = (e, ω) = e then a ◦ x = e and a ◦ y = ω. hence x = a−1 is the multiplicative of a in (a, ◦, e). let −a ∈ sω be arbitrary, where a ∕= ω, ω ∕= e and the multiplicative inverse of −a is (x, y), where x, y ∈ aω, then we have: (−a) ⊗ (x, y) = (ω, a) ⊗ (x, y) =((ω ⊗ x) ⊕ (a ⊗ y), (ω ⊗ y) ⊕ (a ⊗ x)) = (a ⊗ y, a ⊗ x) = (a ◦ y, a ◦ x) = (e, ω) = e then a ◦ y = e and a ◦ x = ω. hence y = a−1 is the multiplicative inverse of a in (a, ◦, e). □ corollary 3.10. (aω \ {ω}, ⊗, e) is a group if and only if for any +a or −a ∈ sω \ s (0) ω , where ⊗|a = ◦, ω ∕= e and a ∕= ω has a multiplicative inverse. theorem 3.11. in sω, with n, m ∈ z+, a ∈ aω be arbitrary and ⊗|a = ◦, the following exponents hold: 1) (+a)⊗n ⊗ (+a)⊗m = (+a)⊗(m+n) 2) (·a)⊗n ⊗ (·a)⊗m = (·a)⊗(m+n) 3) (−a)⊗n ⊗ (−a)⊗m = (−a)⊗(m+n) 4) ((+a)⊗n)⊗m = (+a)⊗(m×n) 5) ((·a)⊗n)⊗m = (·a)⊗(m×n) 6) ((−a)⊗n)⊗m = (−a)⊗(m×n) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 255 m. alqahtani, c. özel and h. zekraoui proof. 1) let a ∕= ω, then (+a)⊗n ⊗ (+a)⊗m = +(a ◦ ... ◦ a) ) *+ , n times ⊗ +(a ◦ ... ◦ a) ) *+ , m times = +((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n+m times = (+a)⊗(m+n). if a = ω, then (+ω)⊗n ⊗ (+ω)⊗m = +ω ⊗ (+ω) = +ω = (+ω)⊗(m+n). 2) by direct calculation similar above, we obtain to the desired result. 3) we have three cases: case i) let a ∕= ω and n, m are odd (note that n + m is even), then (−a)⊗n ⊗ (−a)⊗m = −(a ◦ ... ◦ a) ) *+ , n times ⊗ −(a ◦ ... ◦ a) ) *+ , m times = +((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n+m times = (−a)⊗(m+n) if a = ω, then (−ω)⊗n ⊗ (−ω)⊗m = −ω ⊗ (−ω) = +ω = (−ω)⊗(m+n). case ii) let a ∕= ω and n, m are even (note that n + m is even), then (−a)⊗n ⊗ (−a)⊗m = +(a ◦ ... ◦ a) ) *+ , n times ⊗ +(a ◦ ... ◦ a) ) *+ , m times = +((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n+m times = (−a)⊗(m+n) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 256 topological distances and geometry over the symmetrized omega algebra if a = ω, then (−ω)⊗n ⊗ (−ω)⊗m = +ω ⊗ (+ω) = +ω = (−ω)⊗(m+n). case iii) let a ∕= ω, n is odd and m is even (note that n + m is odd and we get to the same result if n is even and m is odd), then (−a)⊗n ⊗ (−a)⊗m = −(a ◦ ... ◦ a) ) *+ , n times ⊗ +(a ◦ ... ◦ a) ) *+ , m times = −((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = −((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = −(a ◦ ... ◦ a) ) *+ , n+m times = (−a)⊗(m+n) if a = ω, then (−ω)⊗n ⊗ (−ω)⊗m = −ω ⊗ (+ω) = −ω = (−ω)⊗(m+n). 4) let a ∕= ω, then ((+a)⊗n)⊗m = (+(a ◦ ... ◦ a) ) *+ , n times )⊗m = (+(a ◦ ... ◦ a) ) *+ , n times ⊗ ... ⊗ + (a ◦ ... ◦ a) ) *+ , n times) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ⊗ ... ⊗ (a ◦ ... ◦ a) ) *+ , n times) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ ... ◦ (a ◦ ... ◦ a) ) *+ , n times) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n×m times = (+a)⊗(m×n) if a = ω, then ((+ω)⊗n)⊗m = (+ω)⊗m = +ω = (+ω)⊗(m×n). 5) by direct calculation similar above, we obtain to the desired result. 6) by direct calculation similar above with respect to n and m which is even or odd, we obtain to the desired result. □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 257 m. alqahtani, c. özel and h. zekraoui 4. some topological distances properties on sω by the euclidean and frobenius distances definitions on papers [6] and [3] respectively, we give some topological distances properties over sω. we denote the euclidean metric in rn by de,n or by de when it clear what n is. the euclidean metric in cn is the same euclidean metric in r2n. as we have already mentioned, rn will be usually considered with its euclidean metric de,n and with the natural topology induced by de,n. from paper [3] we recall that, some metrics were first time introduced and some algebraic and topological properties were studied in the symmetrized max-plus-algebra [6]. let (g, ◦) be a finitely generated (or finite) abelian group and aω be an ω−algebra over g such that the restriction ⊗|a = ◦. by the fundamental theorem of finitely generated (or finite) abelian groups, the group g is a direct sum (direct product) of its cyclic groups, which make it isomorphic to a direct sum of finite copies of the cyclic groups z of integers with finite summands of quotients of z or isomorphic to a direct sum of cyclic groups of the quotients zqi of z (in other words, according to the group is infinite or finite, there exist some natural numbers m, r, q1, ...,qr such that g ∼= zm or g ∼= zm ⊕zq1 ⊕...⊕zqr or g ∼= zq1 ⊕...⊕zqr ). in all cases we can represent an element of g by n-tuple of elements of z for some natural number n via that isomorphism. from this point of view, we will define metrics (or semimetrics) on our ω−algebra via the distance in z. let φ be a such isomorphism. for any a ∈ g, there exists n ∈ n and there exist α1, . . ., αn ∈ z (also they can be the representatives of classes in z), such that φ (a) = (α1, ..., αn). let us extend φ on aω as follows: (4.1) φ (a) = ! (α1, ..., αn) for a ∈ a, n ∈ n 0 for a = ω 4.1. the euclidean distance on sω. in this section, we suppose that the positive integer n is the number of the direct sum (direct product) of copies of the cyclic groups z of integers and/or the direct sum of the cyclic groups of quotients of z, which is isomorphic to g, where g be a finitely generated (or finite) abelian group and aω be an ω−algebra over g such that the restriction ⊗|a = ◦. definition 4.1. the embedding of sω into cn is the mapping ϕ : sω → cn defined in the following way: let θ be a cube root of unity, say, θ = −1+ √ 3i 2 . as elements of sω are defined by three signs, then we can emerge from sω into cn by the map ϕ defined by: for all a ∈ a, and by (4.1) we have φ (a) = (α1, α2, ..., αn), such that n ∈ n and α1, α2, ..., αn ∈ z, then c© agt, upv, 2020 appl. gen. topol. 21, no. 2 258 topological distances and geometry over the symmetrized omega algebra ϕ (sign(a)a) = & --' --( θ (exp(α1), ..., exp(αn)), if a ∈ a and sign(a) = +, θ2 (exp(α1), ..., exp(αn)), if a ∈ a and sign(a) = −, (exp(α1), ..., exp(αn)), if a ∈ a and sign(a) = ·, (0, ..., 0) , if a = ω. we shall consider the natural metric in sω that is defined below: definition 4.2. let d1 : sω × sω−→r be defined as follows: d1(sign (a) a, sign (b) b) = de,2n(ϕ(sign (a) a), ϕ(sign (b) b)) for all sign(a)a, sign(b)b ∈ sω, where de,2n is the euclidean distance of r2n. the metric d1 in sω will be called the euclidean distance in sω. remark 4.3. (1) 1) it is clear that the euclidean distances in sω is induce the natural topology of sω. as a topological space, we shall consider sω only with its natural topology. 2) we have (cn, de,2n) is a metric space and let ϕ(sω) ⊆ cn. then the subspace metric on ϕ(sω) is defined by simply restricting the metric on cn to points in ϕ(sω). in other words, for any ϕ(sign(a)a), ϕ(sign(b)b) ∈ ϕ(sω), we define ρ(ϕ(sign(a)a), ϕ(sign(b)b)) = de,2n(ϕ(sign(a)a), ϕ(sign(b)b)). hence (ϕ(sω), ρ) is a metric subspace of a metric space (cn, de,2n). proposition 4.4. let (sω, d1) and (ϕ(sω), ρ) be two metrics spaces, then 1) the mapping ϕ is an isometry from (sω, d1) onto (ϕ(sω), ρ). 2) (sω, d1) is homeomorphic to (ϕ(sω), ρ). proof. 1) let sign(a)a, sign(b)b ∈ sω be arbitrary, then ρ(ϕ(sign(a)a), ϕ(sign(b)b)) = de,2n(ϕ(sign (a) a), ϕ(sign (b) b)) = d1(sign (a) a, sign (b) b). hence ϕ is an isometry map. 2) since the map ϕ from (sω, d1) onto (ϕ(sω), ρ) is an isometry, then ϕ is homeomorphism map. hence (sω, d1) ∼= (ϕ(sω), ρ). □ corollary 4.5. since the mapping ϕ from (sω, d1) onto (ϕ(sω), ρ) is an isometry, then (sω, τd1) ∼= (ϕ(sω), τρ) , where τd1 and τρ are the natural topologies induced by the usual metrics d1 and ρ respectively. let x be a topological space. we recall that it is said that a point c ∈ x disconnects x between points a, b ∈ x if there exists a pair u, v of disjoint open sets in x such that x \ {c} = u ∪ v and a ∈ u, while b ∈ v . the space x is a dendrite if it is a continuum such that, for each pair a, b of distinct points of x, there exists a point c ∈ x which disconnects x between points a and b or the space x is continuum if x is compact, connected and metric c© agt, upv, 2020 appl. gen. topol. 21, no. 2 259 m. alqahtani, c. özel and h. zekraoui space. it is said that x is a semicontinuum if, for each pair a, b of points of x, there is a continuum c in x such that a, b ∈ c. let us say that x is a semidendrite if it is a semicontinuum. proposition 4.6. the topological space (sω, τd1) is a semidendrite. proof. since ϕ(sω) ⊂ cn, then we have ϕ(sω) is a semidendrite. by corollary 4.5, we have (sω, τd1) is a semidendrite. □ 4.2. the frobenius distance on sω. we suppose that the group g is a direct sum of its n cyclic subgroups (then g is a finitely generated group). then for every a ∈ a, we have φ (a) = (α1, α2, ..., αn), such that n ∈ n. let ca be a right circulant matrix, such that the first row is (α1, α2..., αn), then ca = . /////// 0 α1 α2 . . . αn−1 αn αn α1 . . . αn−2 αn−1 αn−1 αn . . . αn−3 αn−2 ... ... ... ... ... α3 α4 . . . α1 α2 α2 α3 . . . αn α1 1 2222222 3 each row of this matrix is right cyclic shift of the row above it. as right circulant matrices are diagonalizable in a same basis, we can benefit from this property to define a metric on sω by using frobenius norm ‖‖f (it is a matrix norm). let θ be a cube root of unity, say, θ = −1+ √ 3i 2 . as elements of sω are defined by three signs, then we can emerge from sω into the algebra of right circulant matrices by the map φ defined by: for all a ∈ aω, φ (sign(a)a) = & --' --( θ exp ((ca)) , if a ∈ a and sign(a) = +, θ2 exp ((ca)) , if a ∈ a and sign(a) = −, exp ((ca)) , if a ∈ a and sign(a) = ·, (0)n×n , if a = ω, where (0)n×n is the zero right circulant matrix. let us recall the frobenius norm, sometimes also called the euclidean norm is a matrix norm of an m × n matrix a, defined as the square root of the sum of the absolute squares of its elements, ‖a‖f = 4556 m7 i=1 n7 j=1 | aij |2 let cn×n(c), be the space of all right circulant matrices over c, the definition of distance over cn×n(c) give by df (a, b) = ‖a − b‖f , where a, b ∈ cn×n(c). hence (cn×n(c), df ) is a metric space. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 260 topological distances and geometry over the symmetrized omega algebra corollary 4.7. the exponential function of right circulant matrix is right circulant matrix [1] and the multiplication scalar of right circulant matrix is right circulant matrix. hence, we have φ(sω) ⊂ cn×n(c). corollary 4.8. the map φ : sω → cn×n(c) is an embedding. definition 4.9. let d2 : sω × sω −→ r be defined by d2 (sign(a)a, sign(b)b) = df (φ (sign(a)a) , φ (sign(b)b)) = ‖φ (sign(a)a) − φ (sign(b)b)‖f for all sign(a)a, sign(b)b ∈ sω, where df is the frobenius distance of the right circulant matrices cn×n(c). the metric d2 in sω will be called the omega frobenius distance in sω. remark 4.10. let (cn×n(c), df ) be a metric space. let φ(sω) be a non-empty subset of the right circulant matrices space cn×n(c). then df |φ(sω)×φ(sω), the restriction of df to φ(sω) × φ(sω) is a metric on φ(sω). let us denoted for df |φ(sω)×φ(sω) by ρ. in other words, for any φ(sign(a)a), φ(sign(b)b) ∈ φ(sω), we define ρ(φ(sign(a)a), φ(sign(b)b)) = df (φ(sign(a)a), φ(sign(b)b)). hence (φ(sω), ρ) is a metric subspace of a metric space (cn×n(c), df ). proposition 4.11. let (sω, d2) and (φ(sω), ρ) be two metrics spaces, then 1) the mapping φ is an isometry from (sω, d2) onto (φ(sω), ρ). 2) (sω, d2) is homeomorphic to (φ(sω), ρ). proof. 1) let sign(a)a, sign(b)b ∈ sω be arbitrary, then ρ(φ(sign(a)a), φ(sign(b)b)) = df (φ(sign(a)a), φ(sign(b)b)) = d2(sign (a) a, sign (b) b). hence φ is an isometry map. 2) since the map φ from (sω, d2) onto (φ(sω), ρ) is an isometry, then φ is homeomorphism map. hence (sω, d2) ∼= (φ(sω), ρ). □ corollary 4.12. since the mapping φ from (sω, d2) onto (φ(sω), ρ) is an isometry, then (sω, τd2) ∼= (φ(sω), τρ) , where τd2 and τρ are the natural topologies induced by the omega frobenius distance d2 and frobenius distance ρ respectively. proposition 4.13. let (sω, τd2) and (cn×n(c), τdf ) be a topological space, then φ : (sω, τd2) → (cn×n(c), τdf ) is an embedding map. proof. by corollary 4.12, we have the space sω is homeomorphic with the subspace φ(sω) of cn×n(c). hence the map φ is embedding. □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 261 m. alqahtani, c. özel and h. zekraoui 5. convexity in symmetrized omega algebra in this section, we generalize the notion of convex sets in paper [6] over sω. we suppose that n ∈ z+ is the number of the direct sum (direct product) of copies of the cyclic groups z of integers and/or the direct sum of the cyclic groups of quotients of z, which is isomorphic to g, where g be a finitely generated (or finite) abelian group and aω be an ω−algebra over g such that the restriction ⊗|a = ◦ and ω /∈ a. by ω− absolute value, we have the following corollaries: corollary 5.1. each sign(a)a ∈ sω can be written as sign(a)a = sign(a) |sign(a)a|ω where sign(a) ∈ {+. − .·}. corollary 5.2. one can write the formulas for actions in sω: (5.1) sign(a)a ⊕ sign(b)b = sign(a ⊕ b)( |sign(a)a|ω ⊕ |sign(b)b|ω) (5.2) sign(a)a ⊗ sign(b)b = (sign(a)) ⊗ (sign(b)) (|sign(a)a|ω ⊗ |sign(b)b|ω) where, sign(a ⊕ b) = & --' --( sign(a), a > b, sign(b), b > a, sign(a), a = b, sign(a) = sign(b), · , a = b, sign(a) ∕= sign(b), and the multiplication table for signs as follow. (sign(a))⊗(sign(b)) = & ' ( +, if sign(a) = sign(b) = + or−, −, if sign(a) ∕= sign(b), sign(a) ∕= · and sign(b) ∕= ·, ·, if either sign(a) = · or sign(b) = ·, for any sign(a)a, sign(b)b ∈ sω, we can define sign(a)a ⊕ sign(b)b ∈ sω and sign(a)a ⊗ sign(b)b ∈ sω as in equations 5.1 and 5.2 respectively. then (sω, ⊕, ⊗) is a semiring. by a scalar we mean an element of aω. for sign(a)a ∈ sω, µ ∈ aω, we define µ ⊗ sign(a)a = sign(a)(µ ◦ |sign(a)a|ω).(5.3) the operation ⊗ : aω × sω → sω is well defined, and we can consider it as an outer operation from aω ×sω to sω or the restriction to aω ×sω of the inner operation ⊗ in sω because aω ⊂ sω. considering ⊗ as the outer operation, we can look at sω as at a semimodule(”vector space over semiring”) over the semiring aω. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 262 topological distances and geometry over the symmetrized omega algebra we recall that, for each pair of points x1, x2 of a space x, a segment [x1, x2] with end-points x1 and x2 is defined, then a set a ⊆ x is called convex if, for all x1, x2 ∈ a, we have [x1, x2] ⊆ a see [2]. in particular, for sign(a)a, sign(b)b ∈ sω, we can consider distinct kinds of segments with endpoints sign(a)a and sign(b)b, between them,traditional segments and semimodule segments. these kinds of segments in sω lead to distinct kinds of convexity in sω. let us turn our attention to traditional and semimodule convexity in sω. a convex set in cn is a set b ⊆ cn such that, for each pair x1, x2 ∈ b, the traditional segment [x1, x2] in the space cn is contained in b. definition 5.3. a set b ⊆ sω will be called traditionally convex if ϕ(b) is convex in cn. remark 5.4. 1) let us notice that a nonempty set b ⊆ sω is simultaneously compact and traditionally convex in sω if and only if b is a traditional segment in sω. 2) all traditionally convex subsets of sω are connected. 5.1. semimodule convex sets in sω. the semimodule structure of sω allows us to think about semimodule convexity of subsets of sω. in analogy to conventional algebra, we define a semimodule segment [sign(a)a, sign(b)b]s with end-points sign(a)a ∈ sω and sign(b)b ∈ sω as follows: [sign(a)a, sign(b)b]s ={(µ ⊗ sign(a)a) ⊕ (γ ⊗ sign(a)b) : µ, γ ∈ aω, with µ ⊕ γ = e}. definition 5.5. a set b ⊆ sω is said to be semimodule convex if, for each pair sign(a)a, sign(b)b of points of b, the semimodule segment [sign(a)a, sign(b)b]s is contained in b. acknowledgements. the authors would thank the referees for the valuable remarks and advices. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 263 m. alqahtani, c. özel and h. zekraoui references [1] a. c. f. bueno, on the exponential function of right circulant matrices, international journal of mathematics and scientific computing 3, no. 2 (2013). [2] l. hörmander, notions of convexity, progress in mathematics 127, birkhäuser, bostonbasel-berlin (1994). [3] s. khalid nauman, c. ozel and h. zekraoui, abstract omega algebra that subsumes min and max plus algebras, turkish journal of mathematics and computer science 11 (2019) 1–10. [4] g. l. litvinov, the maslov dequantization, idempotent and tropical mathematics: a brief introduction, journal of mathematical sciences 140, no. 3 (2007), 426–444. [5] d. maclagan and b. sturmfels, introduction to tropical geometry, graduate studies in mathematics, vol. 161, american mathematical society, 2015. [6] c. ozel, a. piekosz, e. wajch and h. zekraoui, the minimizing vector theorem in symmetrized max-plus algebra, journal of convex analysis 26, no. 2 (2019), 661–686. [7] j.-e. pin, tropical semirings, idempotency (bristol, 1994), 50–69, publ. newton inst., vol. 11, cambridge univ. press, cambridge, 1998. [8] i. simon, recognizable sets with multiplicities in the tropical semiring, in: mathematical foundations of computer science (carlsbad, 1988), lecture notes in computer science, vol. 324, springer, berlin, 1988, pp. 107–120. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 264 @ appl. gen. topol. 21, no. 2 (2020), 349-362 doi:10.4995/agt.2020.13943 c© agt, upv, 2020 discontinuity at fixed point and metric completeness ravindra k. bisht a and vladimir rakočević b a department of mathematics, national defence academy, khadakwasla-411023, pune, india (ravindra.bisht@yahoo.com) b university of nǐs, faculty of sciences and mathematics, vǐsegradska 33, 18000 nǐs, serbia (vrakoc@sbb.rs) communicated by s. romaguera abstract in this paper, we prove some new fixed point theorems for a generalized class of meir-keeler type mappings, which give some new solutions to the rhoades open problem regarding the existence of contractive mappings that admit discontinuity at the fixed point. in addition to it, we prove that our theorems characterize completeness of the metric space as well as cantor’s intersection property. 2010 msc: 47h09; 47h10. keywords: fixed point; completeness; discontinuity; cantor’s intersection property. 1. introduction and preliminaries if f is a self-mapping on a complete metric space (x, d) satisfying a contractive condition, then the contractive mapping, in general, ascertains that: for a point x ∈ x, the sequence of iterates is a cauchy sequence, {fnx} → z, and z is the fixed point of f. however, there exists meir-keeler type contractive mapping which ensures the existence of sequence of iterates which is cauchy and {fnx} → z, but z may not be a fixed point of f. meanwhile, a more complete study (data dependence, well-posedness, ulam-hyers stability, ostrowski received 30 june 2020 – accepted 30 july 2020 http://dx.doi.org/10.4995/agt.2020.13943 r. k. bisht and v. rakočević property) was recently proposed in [34]. fixed point theorems studied by kannan are considered as the genesis of the question on continuity of contractive mappings at the fixed point (see [15, 16]). subsequently, the question whether there exists a contractive definition which is strong enough to generate a fixed point but which does not force the mapping to be continuous at the fixed point was ingeminated as an open problem by rhoades[33]. in 1999, pant [27] proved two fixed point theorems in which the considered mappings were discontinuous at the fixed points, hence gave affirmative solutions to the rhoades problem for both the single and a pair of self-mappings. some new solutions to this problem with applications to neural networks have been reported in [2, 3, 4, 5, 6, 12, 23, 24, 25, 26, 29, 30, 31, 32, 37, 39]. fixed point theorems for discontinuous mappings have found a variety of applications, e.g., neural networks are generally used in character recognition, image compression, stock market prediction and to solve non-negative sparse approximation problems ([10, 11, 20, 21, 22, 38]). here, we list various classes of meir-keeler (m-k) type conditions which ensure convergence of the successive approximations but the limiting point may or may not be a fixed point for the given mappings. for i ∈ {1, ..., 11}, we consider: [m-k] for a given ! > 0 there exists a δ(!) > 0 such that, for any x, y ∈ x, ! ≤ mi(x, y) < ! + δ implies d(fx, fy) < !. it is obvious that [m-k] satisfies the following contractive condition: d(tx, ty) < mi(x, y), for any x, y ∈ x with mi(x, y) > 0, where m1(x, y) = d(x, y), (meir-keeler [18]) m2(x, y) = d(x, fx) + d(y, fy) 2 , (kannan [15]) m3(x, y) = max{d(x, fx), d(y, fy)}, (bianchini [1]) m4(x, y) = d(x, fy) + d(y, fx) 2 , (chatterjea [7]) m5(x, y) = max ! ad(x, fx) + (1 − a)d(y, fy), (1 − a)d(x, fx) + ad(y, fy) " , 0 ≤ a < 1, (pant [28]) m6(x, y) = max{d(x, y), d(x, fx), d(y, fy)}, (maiti and pal [17]) m7(x, y) = max ! d(x, y), d(x, fx), d(y, fy), d(x, fy), d(y, fx) " , (ćirić [9]) m8(x, y) = max ! d(x, y), d(x, fx), d(y, fy), d(x, fy) + d(y, fx) 2 " , (jachymski[14]) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 350 discontinuity at fixed point and metric completeness m9(x, y) = max ! d(x, y), k[d(x, fx) + d(y, fy)] 2 , k[d(x, fy) + d(y, fx)] 2 " , 0 ≤ k < 1, m10(x, y) = max ! d(x, y), ad(x, fx) + (1 − a)d(y, fy), (1 − a)d(x, fx) + ad(y, fy), b[d(x, fy) + d(y, fx)] 2 " , 0 ≤ a < 1 and 0 ≤ b < 1, ( bisht and rakočević [5]) m11(x, y) = max ! d(x, y), ad(x, fx) + (1 − a)d(y, fy), (1 − a)d(x, fx) + ad(y, fy), [d(x, fy) + d(y, fx)] 2 " , 0 ≤ a < 1. now, we recall some notions of weaker forms of continuity conditions. definition 1.1 ([8, 9]). if f is a self-mapping of a metric space (x, d), then the set o (x, f) = {fnx : n = 1, 2, ...} is called the orbit of f at x and f is called orbitally continuous if u = limi f mix implies fu = limi ff mix. definition 1.2 ([25]). a self-mapping f of a metric space x is called kcontinuous, k = 1, 2, 3, ..., if fkxn → ft whenever, {xn} is a sequence in x such that fk−1xn → t. definition 1.3 ([26]). a self-mapping f of a metric space (x, d) is called weakly orbitally continuous if the set {y ∈ x : limifmiy = u =⇒ limiffmiy = fu} is nonempty, whenever the set {x ∈ x : limifmix = u} is nonempty. remark 1.4. the following observations are now well-established (see [25, 26]): (i) continuity implies orbital continuity but not conversely. (ii) 1-continuity is equivalent to continuity and continuity =⇒ 2 − continuity =⇒ 3 − continuity =⇒ . . . , but not conversely. (iii) orbital continuity implies weak orbital continuity but the converse need not be true. (iv) k-continuous mappings are orbitally continuous but the converse need not be true. the notion of f-orbitally lower semi-continuity was given by hicks and rhoades [13]. definition 1.5. let (x, d) be a metric space and f : x → x. a mapping g : x → r is said to be f-orbitally lower semi-continuous at a point z ∈ x if {xn} is a sequence in o(x, f) for some x ∈ x, limn→∞ xn = z implies lim infn→∞ g(xn) ≥ g(z). in [19], the author has shown that the f-orbital lower semi-continuity of x → d(x, fx) is weaker than orbital continuity and k-continuity of f. the following example illustrates this fact: c© agt, upv, 2020 appl. gen. topol. 21, no. 2 351 r. k. bisht and v. rakočević example 1.6. let x = {0, 1}∪ ! 1 3n : n = 1, 2, · · · " ∪ ! 1 + 1 3n : n = 1, 2, · · · " and d be the usual metric. define f : x → x by fx = # $$$$$$% $$$$$$& 0 if x = 0 4 3 if x = 1 1 + 1 3n+1 if x = 1 3n , n = 1, 2, · · · 1 3n if x = 1 + 1 3n , n = 1, 2, · · · then o(1, f) = ! 1, 1 + 1 3 , 1 3 , 1 + 1 32 , 1 32 , · · · , 1 + 1 3n , 1 3n , · · · " . let {xn} be a sequence in o(1, f) with xn = 1 3n for n ≥ 1. then, xn → 0 as n → ∞. however, fxn = 1 + 13n+1 → 1 ∕= f0. thus, f is not orbitally continuous. now, let {zn} be a sequence in x with zn = 1 3n for n ≥ 1. then we have fzn = 1 + 1 3n+1 and f2zn = 1 3n+1 . since limn→∞ fzn = 1 and limn→∞ f 2zn = 0 ∕= 4/3 = f1, the mapping f is not 2-continuous. also, g(x) = d(x, fx) = # $$$$$$% $$$$$$& 0 if x = 0 1 3 if x = 1 1 − 2 3n+1 if x = 1 3n , n = 1, 2, · · · 1 if x = 1 + 1 3n , n = 1, 2, · · · let x ∈ x and {xn} ⊂ o(x, f). if {xn} converges, then it converges to 0 or 1. if limn→∞ xn = 0, then lim inf n→∞ g(xn) = # % & 0 or 1 ≥ g(0) = 0 . if limn→∞ xn = 1, then lim infn→∞ g(xn) = 1 > 1/3 = g(1). thus, x → d(x, fx) is f-orbitally lower semi-continuous. in this paper, we prove some new fixed point theorems for a generalized class of meir-keeler type mappings, which give some new solutions to rhoades open problem regarding the existence of contractive mappings that admit discontinuity at the fixed point. further, we prove that our theorems characterize completeness of the metric space as well as cantor’s intersection property. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 352 discontinuity at fixed point and metric completeness 2. sequence of successive approximations here we show that there exists a large class of contractive mappings which ensure that the sequences of iterates are cauchy and {fnx} → z, but z may not be a fixed point of f. we begin with the following result: proposition 2.1. let f be a self-mapping of a complete metric space (x, d) such that for all x, y in x we have (i) given ε > 0 there exists a δ > 0 such that ε ≤ m10(x, y) < ε + δ ⇒ d(fx, fy) < ε. then given x in x, the sequence of iterates {fnx} is a cauchy sequence and lim n→∞ fnx = z for some z in x. proof. condition (i) implies that if m10(x, y) > 0, then (2.1) d(fx, fy) < m10(x, y). let x0 be any point in x. define a sequence {xn} in x recursively by xn+1 = fxn = f nx0 and γn = d(xn, xn+1) for each n ∈ n ∪ {0}. we assume that xn ∕= xn+1 for each n. using (2.1), we get γn = d(xn, xn+1) = d(fxn−1, fxn) < max ' d(xn−1, xn), ad(xn−1, fxn−1) + (1 − a) d(xn, fxn), (1 − a)d(xn−1, fxn−1) + ad(xn, fxn), b[d(xn−1, fxn) + d(xn, fxn−1)] 2 ( = max ' γn−1, aγn−1 + (1 − a) γn, (1 − a)γn−1 + aγn, b[γn−1 + γn] 2 ( = γn−1. this shows that {γn} is a strictly decreasing sequence of positive real numbers and, hence, tends to a limit γ ≥ 0. suppose γ > 0. then there exists a positive integer n such that (2.2) n ≥ n ⇒ γ < γn < γ + δ(γ). by virtue of (i) the above inequality yields d(fxn, fxn+1) = d(xn+1, xn+2) < r. this contradicts with (2.2). hence, γn → 0 as n → ∞. we now prove that {xn} is a cauchy sequence. arguing by contradiction, suppose that {xn} is not a cauchy sequence. then there exist an ε > 0 and a sub-sequence {xni} of {xn} such that (2.3) d(xni, xni+1) > 2ε. select δ in (i) such a way that 0 < δ ≤ ε. since lim n→∞ γn = 0, there exists a positive integer n such that (2.4) γn < δ 6 , whenever n ≥ n. let ni > n. then there exist integers mi satisfying ni < mi < ni+1 such that (2.5) d(xni, xmi) ≥ ε + δ 3 . c© agt, upv, 2020 appl. gen. topol. 21, no. 2 353 r. k. bisht and v. rakočević if not, then using (2.4) and (2.5), we have d(xni, xni+1) ≤ d(xni, xni+1−1) + d(xni+1−1, xni+1) < ε + δ 3 + δ 6 = ε + δ 2 < 2ε, a contradiction with (2.3). let m∗i be the smallest integer such that ni < m ∗ i < ni+1 and (2.6) d(xni, xm∗i ) ≥ ε + δ 3 . then d(xni, xm∗i −1) < ε + δ 3 . in view of (2.1) and (2.2), we get ε < ε + δ 3 ≤ d(xni, xm∗i ) = d(fxni−1, fxm∗i −1) < m10(xni−1, xm∗i −1) = max # $% $& d ) xni−1, xm∗i −1 * , ad(xni−1, fxni−1) + (1 − a)d(xm∗i −1, fxm∗i −1), (1 − a)d(xni−1, fxni−1) + ad(xm∗i −1, fxm∗i −1), b[d(xni−1, fxm∗i −1) + d(xm ∗ i −1, fxni−1)] 2 + $, $≤ max . d(xni−1, xm∗i −1), d(xni−1, xni), d(xm ∗ i −1, xni) + d(xni−1, xni) / ≤ d(xni, xm∗i −1) + d(xni−1, xni) < ε + δ 3 + δ 6 = ε + δ 2 , that is, ε + δ 3 ≤ m(xni−1, xm∗i −1) < ε + δ 2 . by virtue of (i), the last inequality yields d(fxni−1, fxm∗i −1) < ε, i.e., d(xni, xm∗i ) < ε. this contradicts (2.6) and, hence, {xn} is a cauchy sequence. since x is complete, there exists a point z ∈ x such that lim n→∞ xn = lim n→∞ fnx = z. □ we now present two examples which satisfy the above proposition but f is fixed point free. example 2.2. let x = [1, 2] ∪ ! 1 − 1 3n : n = 0, 1, 2, · · · " and d be the usual metric. define f : x → x by fx = ' 0 if 1 ≤ x ≤ 2. 1 − 1 3n+1 if x = 1 − 1 3n , n = 0, 1, 2, · · · . then f(x) = ! 1 − 1 3n : n = 0, 1, 2, · · · " and f is fixed point free. the mapping f satisfies the contractive condition (i) with δ(ε) = ! 1 3n − ε if 1 3n+1 ≤ ε < 1 3n , n = 0, 1, 2, · · · ε if ε ≥ 1. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 354 discontinuity at fixed point and metric completeness example 2.3. let x = [0, 2] equipped with the euclidean metric d. define f : x → x by fx = ! 1+3x 4 if 0 ≤ x < 1 0 if x ≥ 1 . then x is a complete metric space and f satisfies the contractive condition (i) with δ(ε) = # % & ε 3 if 0 ≤ ε ≤ 3 4 1 − ε if 3 4 < ε < 1 ε if ε ≥ 1 but does not posses a fixed point. it is easy to verify that for each x in x, the sequence of iterates {fnx} is a cauchy sequence and fnx → 1 (see [29]). replacing m10(x, y) given in the condition (i) of proposition 2.1 by m11(x, y) and using similar arguments, proof of the following proposition follows easily. proposition 2.4. let (x, d) be a complete metric space and f : x → x be a self-mapping such that for all x, y ∈ x we have (i)′ given ε > 0 there exists a δ > 0 such that ε ≤ m11(x, y) < ε + δ =⇒ d(fx, fy) < ε. then the sequence of iterates {fnx} is a cauchy sequence for a given x ∈ x and lim n→∞ fnx = z for some z ∈ x. 3. discontinuity at fixed point we give a new solution to the problem of continuity at the fixed point for the number m10(x, y). theorem 3.1. let f be a self-mapping of a complete metric space (x, d) such that (i) given ε > 0 there exists a δ = δ(ε) > 0 such that ε ≤ m10(x, y) < ε + δ ⇒ d(fx, fy) < ε, for all x, y in x. if x → d(x, fx) is f-orbitally lower semi-continuous, then f has a unique fixed point z ∈ x and fnx → z as n → ∞. moreover, f is continuous at z if and only if lim x→z max {d (x, fx) , d (z, fz)} = 0 or, equivalently, lim x→z sup d (fz, fx) = 0. proof. let x be any point in x. define a sequence {xn} in x recursively by xn = f nx, n = 0, 1, 2, 3.... then following the proof of proposition 2.1 above, we get that {xn} is a cauchy sequence. since x is complete, there exists a point z ∈ x such that xn → z. since xn → z satisfying d(xn, fxn) = d(fnx, fn+1x) → 0 as n → ∞, by f-orbital lower semi-continuity of x → d(x, fx), one gets d(z, fz) ≤ lim inf n→∞ d(xn, fxn) = 0, which implies that z = fz, i.e., z is a fixed point of f. uniqueness of the fixed point follows easily. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 355 r. k. bisht and v. rakočević let f be continuous at the fixed point z. for lim x→z max {d (x, fx) , d (z, fz)} = 0; let yn → z. then fyn → fz = z, d (yn, fyn) = 0 and lim n→∞ max {d (yn, fyn) , d (z, fz)} = 0. conversely, let lim x→z max {d (x, fx) , d (z, fz)} = 0. to show that f is continuous at the fixed point z, let yn → z. then, we have lim n→∞ max {d (yn, fyn) , d (z, fz)} = 0. this implies that lim n→∞ d (yn, fyn) = 0 and, hence, lim n→∞ fyn = lim n→∞ yn = z = fz. therefore, f is continuous at the fixed point z. this completes the proof. □ theorem 3.2. let f be a self-mapping of a complete metric space (x, d) such that (i) given ε > 0 there exists a δ = δ(ε) > 0 such that ε ≤ m10(x, y) < ε + δ ⇒ d(fx, fy) < ε, for all x, y in x. then f possesses a fixed point if and only if f is weakly orbitally continuous. moreover, the fixed point is unique and f is continuous at z if and only if lim x→z max {d (x, fx) , d (z, fz)} = 0 or, equivalently, lim x→z sup d (fz, fx) = 0. proof. let x be any point in x. define a sequence {xn} in x recursively by xn = f nx, n = 0, 1, 2, 3.... then following the proof of theorem 3.1 above, we get that {xn} is a cauchy sequence. since x is complete, there exists a point z ∈ x such that xn → z. suppose that f is weakly orbitally continuous. since fnx0 → z for each x0, by virtue of weak orbital continuity of f we get, fny0 → z and fn+1y0 → fz for some y0 ∈ x. this implies that z = fz since fn+1y0 → z. therefore z is a fixed point of f. conversely, suppose that the mapping f has a fixed point, say z. then {fnz = z} is a constant sequence such that limn fnz = z and limn fn+1z = z = fz. hence, f is weak orbitally continuous. rest of the the proof follows from the proof of theorem 3.1. □ remark 3.3. the last part of theorem 3.1 or theorem 3.2 can alternatively be stated as: f is discontinuous at z if and only if lim x→z max {d (x, fx) , d (z, fz)} ∕= 0 or equivalently, lim x→z sup d (fz, fx) > 0. remark 3.4. theorems 3.1 and 3.2 hold true if we replace m10(x, y) given in (i) by m11(x, y). the following theorem is a consequence of above theorems. theorem 3.5. let f be a self-mapping of a complete metric space (x, d) such that c© agt, upv, 2020 appl. gen. topol. 21, no. 2 356 discontinuity at fixed point and metric completeness (i) given ε > 0 there exists a δ = δ(ε) > 0 such that ε ≤ m3(x, y) < ε + δ ⇒ d(fx, fy) < ε, for all x, y in x. then f possesses a fixed point if and only if f is weakly orbitally continuous or if x → d(x, fx) is f-orbitally lower semi-continuous, then f has a fixed point z ∈ x and fnx → z as n → ∞. moreover, the fixed point is unique and f is continuous at z if and only if lim x→z sup d (fz, fx) = 0. the following result is an easy consequence of theorem 3.1. corollary 3.6. let f be a self-mapping of a complete metric space (x, d) satisfying (i) of theorem 3.1 for all x, y in x. if f is k-continuous for some k ≥ 1 or if f is orbitally continuous then f has a unique fixed point, say z. moreover, f is continuous at z if and only if lim x→z sup d (fz, fx) = 0. w now give an example to illustrate the above result. example 3.7. let x = [0, 2] and d be the usual metric. define f : x → x by fx = ! 1+x 2 if 0 ≤ x ≤ 1 x−1 2 if 1 < x ≤ 2 . then f satisfies all the conditions of theorem 3.5 and has a unique fixed point z = 1 at which f is discontinuous [29]. one can compute that g(x) = d(x, fx) = ! x−1 2 if 0 ≤ x ≤ 1 1+x 2 if 1 < x ≤ 2 . the mapping g(x) = x → d(x, fx) is f-orbitally lower semi-continuous and satisfies the condition (i) with δ(ε) = # % & ε if ε ≤ 1 2 1 − ε if 1 2 < ε < 1 ε if ε ≥ 1 . here, lim x→z max {d (x, fx) , d (z, fz)} does not exist. also, lim x→z sup d (fz, fx) = 1. theorem 3.8. let f be a self-mapping of a complete metric space (x, d) such that (ii) given ε > 0 there exists a δ = δ(ε) > 0 such that ε ≤ m10(x, y) < ε + δ ⇒ d(fx, fy) < ε, for all x, y in x. if a = b = 0, then f has a unique fixed point whenever x → d(x, fx) is f-orbitally lower semi-continuous or f is weakly orbitally continuous. if a, b > 0, then f possesses a unique fixed point at which f is continuous. proof. the proof follows on the same lines of the proof of theorem 3.1 above. as seen in theorem 3.1 above, f need not be continuous at the fixed point if a = 0 = b. we now show that f is continuous at the fixed point when a, b > 0. suppose a > 0 and z is the fixed point of f. let {xn} be any sequence in x c© agt, upv, 2020 appl. gen. topol. 21, no. 2 357 r. k. bisht and v. rakočević such that xn → z as n → ∞. then using (ii), for sufficiently large values of n, we get d(z, fxn) = d(fz, fxn) < max ! d(z, xn), ad(z, fz) + (1 − a) d(xn, fxn), (1 − a)d(z, fz) + ad(xn, fxn), b[d(z, fxn) + d(xn, fz)] 2 " = max ! d(z, xn), (1 − a) d(xn, fxn), ad(xn, fxn), b[d(z, fxn) + d(xn, fz)] 2 " ≤ max ! !1, !2 + (1 − a) d(z, fxn), !3 + ad(z, fxn), b[d(z, fxn) + !4] 2 " , where !i(i = 1, 2, 3, 4) → 0 as n → ∞. this yields d(z, fxn) < !1, ad(z, fxn) < !2, (1 − a)d(z, fxn) < !3 or bd(z, fxn) 2 < !4. on letting n → ∞, we get limn→∞ fxn = z = fz. hence f is continuous at the fixed point. □ 4. characterization of metric completeness in a complete metric space, the following well-known variant of cantor’s intersection theorem holds. theorem 4.1. suppose that x is a complete metric space, and cn is a sequence of non-empty closed nested subsets of x whose diameters tend to zero: limn→∞ diam(cn) = 0, where diam(cn) is defined by diam(cn) = sup{d(x, y)| x, y ∈ cn}. then the intersection of the cn contains exactly one point:0∞ n=1 cn = {x} for some x ∈ x. a converse to this theorem is also true: if x is a metric space with the property that the intersection of any nested family of non-empty closed subsets whose diameters tend to zero is non-empty, then x is a complete metric space. in the next result, we show that theorem 3.5 characterizes metric completeness of x. however, there is a substantive difference between the next theorem and similar theorems (e. g., subrahmanyam [35], suzuki [36]) giving characterization of completeness in terms of fixed point property for contractive type mappings. subrahmanyam [35] and suzuki [36] have shown that the contractive condition implies continuity at the fixed point; and completeness of the metric space x is equivalent to the existence of fixed point. in the next theorem motivated by ([25, 26]) we prove that completeness of the space is equivalent to fixed point property for a larger class of mappings including continuous as well as discontinuous mappings. theorem 4.2. if every x → g(x) = d(x, fx) is f-orbitally lower semi-continuous or weak orbitally continuous self-mapping f of a metric space (x, d) satisfying the condition (i) of theorem 3.5 has a fixed point, then x is complete. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 358 discontinuity at fixed point and metric completeness proof. suppose that every x → d(x, fx) is f-orbitally lower semi-continuous or weak orbitally continuous self-mapping f of a metric space (x, d) satisfying the condition (i) of theorem 3.5 possesses a fixed point. we will prove that x is complete. arguing by contradiction, suppose that x is not complete. then there exists a cauchy sequence in x, say m = {un}n∈n having distinct points which does not converge in x. let x ∈ x be any arbitrary point. then, since x is not a limit point of the cauchy sequence m, and we have d(x, m − {x}) > 0 and there exists an integer nx ∈ n such that x ∕= unx and for each m ≥ nx (4.1) d(unx, um) < 1 2 d(x, unx). consider a mapping f : x +→ x by f(x) = unx. then, f(x) ∕= x for each x and, using (4.1), for any x, y ∈ x we get d(fx, fy) = d(unx, unx=y) < 1 2 d(x, unx) = d(x, fx), if nx ≤ ny, or d(fx, fy) = d(unx, uny ) < 1 2 d(y, uny ) = d(y, fy), if nx > ny. this implies that (4.2) d(fx, fy) < 1 2 max{d(x, fx), d(y, fy)}. or equivalently, given ! > 0 we can select δ(!) = ! such that (4.3) ! ≤ max{d(x, fx), d(y, fy)} < ! + δ ⇒ d(fx, fy) < !. it is clear from (4.2) and (4.3) that the mapping f satisfies condition (i) of theorem 3.5. moreover, f is a fixed point free mapping whose range is contained in the non-convergent cauchy sequence m. hence, there exists no sequence {xn}n∈n in x for which {fxn}n∈n converges, i.e., there exists no sequence {xn}n∈n in x for which the condition fk−1xn → t ⇒ fkxn → ft for k > 1 is violated. therefore, f is k-continuous mapping. since xn ∈ o(x, f), it is of the form xn = f inx with in ∈ n. set yn = fin+1−kx when in ≥ k − 1. then, fk−1yn = xn → t as n → ∞. by k-continuity of f, we get fxn = fkyn → ft as n → ∞. thus, limn→∞ g(xn) = limn→∞ d(xn, fxn) = d(t, ft) = g(t), which implies that g is f-orbitally lower semi-continuous at t. in a similar manner it follows that f is weak orbitally continuous. thus, we have a self-mapping f of x which satisfies all the conditions of theorems 3.5 but does not possess a fixed point. this contradicts the hypothesis of the theorem. hence x is complete. □ we now show that theorem 3.5 characterizes cantor’s intersection property. theorem 4.3. let (x, d) be a metric space and f a self-mapping of x satisfying condition (i) of theorem 3.5. suppose x satisfies cantor’s intersection property and x → d(x, fx) is f-orbitally lower semi-continuous or f is weakly orbitally continuous. then f has a fixed point and f is continuous at z if and only if limx→z max{d(x, fx), d(z, fz)} = 0. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 359 r. k. bisht and v. rakočević proof. let x be any point in x. define a sequence {xn} in x recursively by xn = f nx, n = 0, 1, 2, 3.... then following the proof of proposition 2.1 above, we get that {xn} is a cauchy sequence. define a sequence {tn : n = 1, 2, 3, ...} of nonempty subsets of x by tn = {xi : i ≥ n}. let cn denote the closure of tn ∈ x. then for each n it is obvious that cn is a nonempty closed subset of x, cn+1 ⊆ cn and limn→∞ diam(cn) = 0. since cantor’s intersection property holds in x, ∩{cn} consists of exactly one point, say z, which is nothing but the limit of the cauchy sequence {xn}. rest of the proof follows from theorem 3.5. □ theorem 4.4. let (x, d) be a metric space. if every x → d(x, fx) is forbitally lower semi-continuous or weak orbitally continuous self-mapping f of a metric space (x, d) satisfying the condition (i) of theorem 3.5 has a fixed point, then x satisfies cantor’s intersection property. proof. suppose that every x → d(x, fx) is f-orbitally lower semi-continuous or weak orbitally continuous self-mapping f of a metric space (x, d) satisfying the condition (i) of theorem 3.5 possesses a fixed point. we assert that x satisfies cantor’s intersection property. if possible, suppose x does not satisfy cantor’s intersection property, then there exists a sequence {cn}of nonempty closed subsets of x satisfying cn+1 ⊆ cn and limn→∞ diam(cn) = 0 and having empty intersection. construct a sequence t = {xn} ∈ x such that xi ∈ ai. limn→∞ diam(cn) = 0, given ! > 0 there exists a positive integer n such that n, m ≥ n implies d(xn, xm) < !. therefore {xn} is a cauchy sequence. however, t = {xn} is a non-convergent cauchy sequence since the sequence {cn} has empty intersection. as done in the proof of theorem 4.2 we can now define f-orbitally lower semi-continuous or weakly orbitally continuous self-mapping on a metric space (x, d) satisfying condition (i) which does not possess a fixed point. this contradicts our hypothesis. therefore, cantor’s intersection property holds in x. □ combining theorem 4.2 and theorem 4.4, we get the following theorem: theorem 4.5. for a metric space (x, d), the following are equivalent: (a) x is complete. 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[39] d. zheng and p. wang, weak θ-φ-contractions and discontinuity, j. nonlinear sci. appl. 10 (2017), 2318–2323. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 362 @ appl. gen. topol. 21, no. 1 (2020), 57-70 doi:10.4995/agt.2020.11992 c© agt, upv, 2020 existence of picard operator and iterated function system medha garg and sumit chandok school of mathematics, thapar institute of engineering & technology, patiala-147004, punjab, india. (sumit.chandok@thapar.edu) communicated by e. a. sánchez-pérez abstract in this paper, we define weak θm− contraction mappings and give a new class of picard operators for such class of mappings on a complete metric space. also, we obtain some new results on the existence and uniqueness of attractor for a weak θm− iterated multifunction system. moreover, we introduce (α,β,θm)− contractions using cyclic (α,β)− admissible mappings and obtain some results for such class of mappings without the continuity of the operator. we also provide an illustrative example to support the concepts and results proved herein. 2010 msc: 47h10; 54h25; 46j10; 46j15. keywords: picard operator; fixed point; weak θm-contraction; iterated function system. 1. introduction the iterated function system (ifs) is the main generator of fractals. it is introduced by hutchinson [7] and generalized by barnsley [2]. an ifs is a finite family of contractions {fi}ni=1 on a complete metric space (m,d). for an ifs there is always a non-empty set a ⊂ m such that a = ⋃n i=1 fi(a), such a is known as attractor of the respective ifs. in this paper, we study the concept of weak θ -contraction used by imdad and alfaqih [8] which is an extension of θ -contraction (or js contraction) introduced by jleli and samet [9]. we consider the family θ1,2,4 and introduce received 17 june 2019 – accepted 11 october 2019 http://dx.doi.org/10.4995/agt.2020.11992 m. garg and s. chandok weak θm-contraction and prove that every (continuous) weak θm -contraction is a picard operator in section 3. in section 4, we study about iterated multifunction system (ims) and obtain some results on the existence and uniqueness of attractor for a weak θm− ims. also, we obtain some results on (α,β,θm)− contractions using cyclic (α,β)− admissible mappings without the continuity of the operator in the last section. 2. preliminaries in this section, we recall some notations, basic definitions and results to be used in the sequel. definition 2.1 (see [12, 13]). let (m,d) be a metric space and f : m → m be a self mapping. a sequence {un} defined by un = fnu0 is called a picard sequence based at the point u0 ∈ m. a self-mapping f is said to be a picard operator if it has a unique fixed point z ∈ m and z = lim n→∞ fnu for all u ∈ m. definition 2.2 (see [12, 13]). let (m,d) be a metric space, and let k(m) be the class of all non-empty compact sets of m. the function η : k(m) × k(m) → [0,∞) define by η(a,b) = max{d(a,b),d(b,a)} where d(a,b) = supa∈a infb∈b d(a,b) for all a,b ∈ k(m) is a metric known as hausdorffpompeiu metric. it is well known that if (m,d) is complete then (k(m),η) is also complete. alizadeh et al. [1] introduced the notion of cyclic (α,β)-admissible mapping which is defined as follows: definition 2.3. let m be a nonempty set, f be a self-mapping on m and α,β : m → [0,∞) be two mappings. we say that f is a cyclic (α,β)-admissible mapping if x ∈ m with α(x) ≥ 1 implies β(fx) ≥ 1 and β(x) ≥ 1 implies α(fx) ≥ 1. the following results will be needed in the proof of our main results. lemma 2.4 ([10]). let (m,d) be a metric space and let {xn} be a sequence in m such that (2.1) lim n→∞ d (xn,xn+1) = 0. if {xn} is not a cauchy sequence in m, then there exist ε > 0 and two sequences {m (k)} and {n (k)} of positive integers such that n (k) > m (k) > k and the following sequences tend to ε+ when k → +∞: (2.2) d ( xm(k),xn(k) ) , d ( xm(k),xn(k)+1 ) , d ( xm(k)−1,xn(k) ) , d ( xm(k)−1,xn(k)+1 ) , d ( xm(k)+1,xn(k)+1 ) . remark 2.5. let {xn}n∈n be a sequence in a metric space (x,d) . if for all n ∈ n holds d (xn+1,xn) < d (xn,xn−1), then n 6= m implies xn 6= xm whenever n,m ∈ n. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 58 existence of picard operator and iterated function system lemma 2.6 ([14]). let a,b,c ∈ k(m). then we have the following: (i) a ⊂ b if and only if d(a,b) = 0; (ii) d(a,b) ≤ d(a,c) + d(c,b). lemma 2.7 ([15]). if {ei}i∈τ and {fi}i∈τ are finite collection of elements in k(m), then η( ⋃ i∈τ ei, ⋃ i∈τ fi) ≤ sup i∈τ η(ei,fi). 3. weak θm-contraction now we use the definition of an auxiliary function and utilize the same to introduce weak θm-contraction. definition 3.1 (see [6, 8, 9]). let θ : (0,∞) → (1,∞) be a function and consider the following conditions: θ1 : θ is non-decreasing. θ2: for each sequence {αn} in (0,∞), lim n→∞ θ(αn) = 1 ⇔ lim n→∞ (αn) = 0. θ3: there exist r ∈ (0, 1) and l ∈ (0,∞) such that lim α→0+ θ(α)−1 αr = l; θ4: θ is continuous. the following notations to be used in the sequel. • θ1,2,3 the family of all θ that satisfy θ1 − θ3. • θ1,2,4 the family of all θ that satisfy θ1, θ2 and θ4. • θ2,3 the family of all θ that satisfy θ2 and θ3. • θ2,4 the family of all θ that satisfy θ2 and θ4. • θ2 the family of all θ that satisfy θ2. example 3.2 ([6]). define θ : (0,∞) → (1,∞) by θ(α) = e √ α, for all α ∈ (0,∞).then θ ∈ θ1,2,3,4. example 3.3 ([6]). define θ : (0,∞) → (1,∞) by θ(α) = eα, for all α ∈ (0,∞).then θ ∈ θ1,2,3. example 3.4 ([8]). the following function θ : (0,∞) → (1,∞) are in θ2,4: (1) θ(α) = e α 2 +sinα; (2) θ(α) = αr + 1,r ∈ (0,∞). example 3.5. define θ : (0,∞) → (1,∞) by θ(α) = 2 √ α2 − 1√ α , for all α ∈ (0,∞).then θ ∈ θ1,2,4. now, we define weak θm-contraction mapping. definition 3.6. let (m,d) be a metric space and f : m → m is a selfmapping. a mapping f is called a weak θm-contraction if there exist a θ ∈ θ2,4 (or θ ∈ θ1,2,4) and h ∈ (0, 1), such that for all u,v ∈ m, we have d(fu,fv) > 0 ⇒ θ(d(fu,fv)) ≤ [θ(m(u,v))]h,(3.1) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 59 m. garg and s. chandok where m(u,v) = max{d(u,fu),d(v,fv),d(u,v)}. remark 3.7. here, we note that weak θm-contraction mapping has at most one fixed point. assume that f has another fixed point say v ∈ m, d(u,v) > 0. using (3.1) we have θ(d(u,v)) = θ(d(fu,fv)) ≤ [θ(max{d(u,fu),d(v,fv),d(u,v)})]h = [θ(d(u,v))]h, which is a contradiction. lemma 3.8. let (m,d) be a metric space and f : m → m is a weak θmcontraction. suppose that there exists a picard sequence {un}⊆ m defined by un+1 = f nu0 = fun for all n ∈ n ∪{0}. then d(un,un+1) → 0 as n → ∞, where un 6= un+1 (here θ ∈ θ2,4 or θ1,2,4). proof. let u0 ∈ m be an arbitrary point. define the picard sequence as {un}⊆ m by un+1 = fnu0 = fun for all n ∈ n∪{0}. assume that un 6= un+1 for all n ∈ n∪{0}. applying (3.1) we have, for all n ∈ n∪{0}, θ(d(un,un+1)) = θ(d(fun−1,fun)) ≤ [θ(max{d(un−1,fun−1),d(un,fun),d(un−1,un)})]h = [θ(max{d(un,un+1),d(un,un−1)})]h case 1: when d(un,un+1) > d(un,un−1), then we have θ(d(fun−1,fun)) = θ(d(un,un+1)) ≤ [θ(d(un,un+1)]h, but α ≥ αh,∀α ∈ r+,h ∈ (0, 1). thus we get contradiction. case 2: when d(un,un−1) > d(un,un+1), we have θ(d(fun−1,fun)) ≤ [θ(d(un,un−1)]h. hence on the same lines, we have [θ(d(fun−1,fun−2))] h ≤ [θ(max{d(un−1,fun−1),d(un−2,fun−2),d(un−1,un−2)})]h 2 = [θ(max{d(un−1,un),d(un−1,un−2)})]h 2 ≤ [θ(d(un−1,un−2))]h 2 . proceeding on these lines, we get θ(d(fun,fun−1)) ≤ [θ(d(fun−1,fun−2))]h ≤ [θ(d(fun−2,fun−3))]h 2 ≤ ... ≤ [θ(d(fu0,u0))]h n . thus, we have θ(d(un,un+1)) ≤ [θ(d(u1,u0))]h n . now, taking n →∞ we have, lim n→∞ θ(d(un,un+1)) = 1. using θ2, we have lim n→∞ d(un,un+1) = 0. � lemma 3.9. let (m,d) be a metric space and f : m → m is a weak θmcontraction. suppose that there exists a picard sequence {un}⊆ m defined by un+1 = f nu0 = fun for all n ∈ n ∪ {0}. then picard sequence {un} is a cauchy sequence (here θ ∈ θ2,4 or θ1,2,4). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 60 existence of picard operator and iterated function system proof. let u0 ∈ m be an arbitrary point. define the picard sequence as {un}⊆ m by un+1 = fnu0 = fun for all n ∈ n∪{0}. assume that un 6= un+1 for all n ∈ n ∪{0}. using lemma 3.8, we have lim n→∞ d(un,un+1) = 0. now we have to prove that {un} is a cauchy sequence. we’ll prove this by contradiction. assume that {un} is not a cauchy sequence. now, since the sequence {un} is not a cauchy sequence, then by lemma 2.4, we have d ( um(k),un(k) ) and d ( um(k)+1,un(k)+1 ) tend to ε > 0, as k →∞. using (3.1), we have θ(d(um(k),un(k))) = θ(d(fum(k)−1,fun(k)−1)) ≤ [θ(max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1), d(um(k)−1,un(k)−1)})]h. case 1: if max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1),d(um(k)−1,un(k)−1)} = d(um(k)−1,fum(k)−1), then we have θ(d(um(k),un(k))) ≤ [θ(d(um(k)−1,fum(k)−1)]h. letting k →∞, from lemma 2.4 and θ4, we have θ(�) ≤ [θ(0)]h, which is a contradiction. case 2: if max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1),d(um(k)−1,un(k)−1)} = d(un(k)−1,fun(k)−1), then proceeding the same way as in case 1 we again get a contradiction. case 3: if max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1),d(um(k)−1,un(k)−1)} = d(um(k)−1,un(k)−1), then we have θ(d(um(k),un(k))) ≤ [θ(d(um(k)−1,un(k)−1)]h. letting k →∞ and using lemma 2.4 and θ4, we obtain θ(�) ≤ [θ(�)]h, which is again a contradiction. hence picard sequence {un} is a cauchy sequence. � theorem 3.10. every weak θm-contraction on a complete metric space is a picard operator. [here, we consider θ ∈ θ1,2,4.] proof. let u0 ∈ m be an arbitrary point. define the picard sequence as {un}⊆ m by un+1 = fnu0 = fun for all n ∈ n∪{0}. if there exist n0 ∈ n∪{0} such that un0 = fun0 , then we are done. assume that un 6= un+1 for all n ∈ n ∪{0}. using lemma 3.9, we have {un} is a cauchy sequence. now as (m,d) is a complete metric space so there exist u ∈ m such that {un} converges to u. from (θ1) and (3.1), it is easy to conclude that θ(d(fu,fv)) ≤ [θ(max{d(u,fu),d(v,fv),d(u,v)})]h ≤ θ(max{d(u,fu),d(v,fv),d(u,v)}) for all u,v ∈ m with d(fu,fv) > 0. using (θ1) and above inequality, we have d(fu,fv) ≤ max{d(u,fu),d(v,fv),d(u,v)}. suppose that u 6= fu. therefore, we have d(un+1,fu) = d(fun,fu) ≤ max{d(un,fun),d(u,fu),d(un,u)} = max{d(un,un+1),d(u,fu),d(un,u)}. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 61 m. garg and s. chandok taking n → ∞, using lemma 3.8 we have d(u,fu) ≤ d(u,fu), which is a contradiction. hence fu = u, thus we get a fixed point. further, now we prove the uniqueness of the fixed point. assume that f has another fixed point say v ∈ m, v 6= u. using (3.1) we have θ(d(fu,fv)) ≤ [θ(max{d(u,fu),d(v,fv),d(u,v)})]h = [θ(d(u,v))]h, which is a contradiction. hence the result. � theorem 3.11. every continuous weak θm-contraction on a complete metric space is a picard operator. [here, we consider θ ∈ θ2,4.] proof. let u0 ∈ m be an arbitrary point. define the picard sequence as {un}⊆ m by un+1 = fnu0 = fun for all n ∈ n∪{0}. if there exist n0 ∈ n∪{0} such that un0 = fun0 , then we are done. assume that un 6= un+1 for all n ∈ n ∪{0}. proceeding as in theorem 3.10, we have picard sequence {un} is a cauchy sequence. now as (m,d) is a complete metric space so there exist u ∈ m such that {un} converges to u. the continuity of f and uniqueness of limit implies fu = u, thus we get a fixed point. hence every continuous weak θm-contraction on a complete metric space is a picard operator. � example 3.12. let m = {1, 2, 3}. define the metric d : m × m → [0,∞) by d(x,y) = |x − y|, for all x,y ∈ m. define a function f : m → m as f(1) = 2,f(2) = 2,f(3) = 1. define a function θ : (0,∞) → (1,∞) by θ(t) = e √ t. so θ ∈ θ1,2,3,4. case 1. consider (u,v) = (1, 3). we have θ(d(f1,f3)) = θ(d(2, 1)) = θ(1) = e. also, [θ(max{d(1,f1),d(3,f3),d(1, 3)})]h = [θ(d(1, 2),d(1, 3))]h = [θ(2)]h = [e √ 2]h. therefore θ(d(f1,f3)) ≤ [θ(max{d(1,f1),d(3,f3),d(1, 3)})]h, for all h ∈ [ 1√ 2 , 1). case 2. consider (u,v) = (2, 3). we have θ(d(f2,f3)) = θ(d(2, 1)) = θ(1) = e. also, [θ(max{d(2,f2),d(3,f3),d(2, 3)})]h = [θ(d(3, 1),d(2, 3))]h = [θ(2)]h = [e √ 2]h. therefore θ(d(f1,f3)) ≤ [θ(max{d(2,f2),d(3,f3),d(2, 3)})]h, for all h ∈ [ 1√ 2 , 1). thus all the conditions of theorem 3.10 are satisfied and 2 is a unique fixed point of f. here is to note that when (u,v) = (2, 3) in the above example, then (a) f is not banach contraction; (b) f is not weak θ-contraction of imdad et al. [8]; (c) f is weak θm-contraction. (d) f is a picard operator. theorem 3.13. let (m,d) be a complete metric space and let f : m → m be a self mapping. if there exist n ∈ n such that fn is a weak θm -contraction, then f is a picard operator. proof. from theorem 3.10, it is obvious that fn is a picard operator, thus there exists a unique z ∈ m such that fnz = z and lim m→∞ tm+1 = (f n) m u = z, c© agt, upv, 2020 appl. gen. topol. 21, no. 1 62 existence of picard operator and iterated function system for all u ∈ m. also, we observe that fn+1z = fnz, that is fn(fz) = fz, thus fz is also a fixed point of fn. thus fz = z. further, if z∗ is another fixed point of f, then it must be a fixed point of fn. hence z = z∗. therefore f has a unique fixed point. now, let m be a positive integer greater than n. then there exist l ≥ 1 and s ∈{0, 1, 2, ...,n−1} such that m = nl + s. here, we notice that for all u ∈ m, we have lim m→∞ um+1 = lim m→∞ fmu = lim l→∞ fnl(fsu) = lim l→∞ (fn)l(fsu) = z. hence the result. � haghi et al. [5], in 2011, proved a lemma by using the axiom of choice as follows: lemma 3.14. let m be a nonempty set and f : m → m a function. then there exist a set e ⊆ m such that f(e) = f(m) and f : e → m is one-to-one. by using above lemma, we prove common fixed point theorems for two self mappings on m as follows: theorem 3.15. let (m,d) be a complete metric space and f,g be two self maps on m satisfying d(fu,fv) > 0 ⇒ θ(d(fu,fv)) ≤ [θ(max{d(gu,fu),d(gv,fv),d(gu,gv)})]h.(3.2) for all u,v ∈ m and θ ∈ θ2,4 (or θ ∈ θ1,2,4). if f(m) ⊆ g(m) and g(m) is a complete subset of m then f and g have a unique common fixed point in m. proof. by using lemma 3.14, there exist e ⊆ m such that g(e) = g(m) and g : e → m is one-to-one. define h : g(e) → g(e) by h(gu) = fu. clearly, h is well defined as g is one-to-one on e. also, θ(d(h(gu),h(gv))) ≤ [θ(max{d(gu,fu),d(gv,fv),d(gu,gv)})]h, for all gx,gy ∈ g(e). since g(e) = g(m) is complete, then by using theorem 3.10, we can easily prove that f and g have a unique common fixed point in m. � 4. weak θm iterated multifunction system as application of results proved in the last section, we obtain some results on the existence and uniqueness of attractor of iterated multifunction system composed by weak θm-contraction in the setting of complete metric space in this section. in the following section, we consider (m,d) is a complete metric space, n ∈ n and θ ∈ θ1,2,4. definition 4.1. let {fi}ni=1 be a finite family of self mappings on m. if fi : m → m is a weak θm−contraction (for each i), then the family {fi}ni=1 is called a weak θm− iterated function system (weak θm−ifs). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 63 m. garg and s. chandok the set function g : k(m) → k(m) define by g(b) = ⋃n i=1 fi(y ) (for all y ∈ k(m)) is said to be associated hutchinson operator. a set a ∈ k(m) is called an attractor of the weak θm− ifs if g(a) = a. let (m,d) be a metric space and f1,f2, ...fn : m → k(m) be multivalued operator. then the system f = (f1,f2, ...,fn) is called an iterated multifunction system (abbreviated as ims). definition 4.2. let {fi}ni=1 be a finite family of iterated multifunction system. if fi : m → k(m) is a weak θm−contraction (for each i), then the family {fi}ni=1 is called a weak θm− iterated multifunction system (weak θm−ims). define p(m) = {y ⊂ m : y is nonempty}. if t : m → p(m) is a multivalued operator then t(y ) := ⋃ x∈y t(x),y ∈ p(m). let f1,f2, ...fm : m → k(m) be a finite family of multivalued operators, we define multifractal operator tf generated by the iterated multifunction system f = (f1,f2, ...fm) by gf : k(m) → k(m), gf (y ) = ⋃m i=1 fi(y ). in this framework, a nonempty compact subset a∗ of m is said to be a multivalued fractal with respect to the iterated multifunctions system f = (f1,f2, ...fm) if and only if it is a fixed point for the associated multifractal operator. in particular, if the operators fi = fi are singlevalued, then a fixed point for the fractal operator gf : k(m) → k(m), gf (y ) = ∪mi=1fi(y ) generated by generated by iterated function system f = (f1,f2, ...fm) is said to be a self similar set or a fractal. throughout, fix(f) denotes the set of fixed points of f (see [2, 4, 7]). definition 4.3. if {fi}ni=1 is weak θm− ims such that fi : m → k(m) is continuous for i = 1, 2, . . . ,n then the operator gf : k(m) → k(m), gf (y ) = ⋃n i=1 fi(y ) is well defined and is called weak θm− multi-fractal operator. a fixed point of gf is called a multivalued fractal. now we will use the following lemma to show that a weak θm− multi-fractal operator has a unique multivalued fractal. lemma 4.4. let f : m → k(m) is a continuous weak θmmultivalued operator. then the mapping a 7→ f(a) is also a weak θm-multivalued operator from k(m) into itself. proof. let a,b ∈ k(m) be such that η(f(a),f(b)) > 0. assume that η(f(a),f(b)) = d(f(a),f(b)) = sup u∈a inf v∈b d(fu,fv), for all a,b ∈ k(m).(4.1) as f is a continuous weak θm-multivalued operator so there exist h ∈ (0, 1) such that θ(d(fu,fv)) ≤ [θ(max{d(u,fu),d(v,fv),d(u,v)})]h, for all u,v ∈ m. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 64 existence of picard operator and iterated function system now using (4.1), compactness of a, and continuity of f, we can find a ∈ a such that d(f(a),f(b)) = infv∈b d(fa,fv) > 0, so that d(fa,fv) > 0, for all v ∈ b. hence, for all v ∈ b, we have θ( inf v∈b d(fa,fv)) ≤ θ(d(fa,fv) ≤ [θ(max{d(a,fa),d(v,fv),d(a,v)})]h. therefore, for all v ∈ b we get (4.2) θ(η(f(a),f(b)) ≤ [θ(max{d(a,fa),d(v,fv),d(a,v)})]h. case 1: if max{d(a,fa),d(v,fv),d(a,v)} = d(a,fa), then we have: θ(infv∈b d(fa,fv)) ≤ [θ(d(a,fa))]h, now from (4.2) we have θ(η(f(a),f(b))) ≤ [θ(d(a,fa′))]h ≤ [θ(sup a∈a inf fa∈f(a) d(a,fa))]h = [θ(d(a,a))]h, which is a contradiction. case 2: if max{d(a,fa),d(v,fv),d(a,v)} = d(v,fv), then proceeding in the same way as in case 1 we again get a contradiction. case 3: if max{d(a,fa),d(v,fv),d(a,v)} = d(a,v), then for all v ∈ b we have θ(η(f(a),f(b))) ≤ [θ(d(a,v))]h. now let v ∈ b be such that d(a,v) = infv∈b d(a,v). from (4.2) we have, θ(η(f(a),f(b))) ≤ [θ(d(a,v))]h, = [θ( inf b∈b d(a,v))]h, ≤ [θ(sup a∈a inf v∈b d(a,v))]h = [θ(d(a,b))]h ≤ [η(a,b)]h. hence we get the result. � theorem 4.5. let (m,d) be a complete metric space and fi : m → k(m), i = {1, 2, ...,m} be continuous multivalued operator satisfying θ(η(fiu,fiv)) ≤ [θ(max{d(u,fiu),d(v,fiv),d(u,v)})]h, for all u,v ∈ m and h ∈ (0, 1). then there exists a unique multivalued fractal with respect to the iterated multifunction system f = (f1,f2, ...fm), that is, fix(gf ) = {a∗} and {gnf (a)}n∈n converges to a ∗, for each a ∈ k(m). proof. first we prove that the operator gf : k(m) → k(m), gf (y ) = ∪mi=1fi(y ) satisfies the conditions of theorem 3.10. let b,c ∈ k(m) such that c© agt, upv, 2020 appl. gen. topol. 21, no. 1 65 m. garg and s. chandok 0 < η(gf (b),gf (c)) = η( ⋃m i=1 fi(b), ⋃m i=1 fi(c)). now lemma 2.7 implies that η(gf (b),gf (c)) = η( m⋃ i=1 fi(b), m⋃ i=1 fi(c)) ≤ sup 1≤i≤n η(fi(b),fi(c)) = η(fi0 (b),fi0 (c)), for some i0 ∈{1, 2, 3, . . . ,n}. using θ1 and lemma 4.4, we have θ(η(gf (b),gf (c)) ≤ θ(η(fi0 (b),fi0 (c))) ≤ [θ(η(b,c))]hi0 . therefore gf is also a continuous weak θm contraction on the complete metric space (k(m),η). theorem 3.11 ensures the existence and uniqueness of a∗ ∈ k(m) such that gf (a ∗) = a∗ and a∗ = lim n→∞ gnf (b) for all b ∈ k(m). this completes the proof. � in particular, when the operators are single valued, we have the following result. theorem 4.6. if {fi}ni=1 is a continuous weak θm -ifs, then it has unique attractor. moreover, a = lim n→∞ gn(b) for all b ∈ k(m), the limit being taken with respect to the hutchinson-pompeiu metric. proof. for each i ∈{1, 2, . . .n}, let hi be constant such that hi ∈ (0, 1) and is associated with fi. let b,c ∈ k(m) such that 0 < η(g(b),g(c)) = η( ⋃n i=1 fi(b), ⋃n i=1 fi(c)). now lemma 2.7 implies that η(g(b),g(c)) = η( n⋃ i=1 fi(b), n⋃ i=1 fi(c)) ≤ sup 1≤i≤n η(fi(b),fi(c)) = η(fi0 (b),fi0 (c)), for some i0 ∈{1, 2, 3, . . . ,n}. using θ1 and lemma 4.4, we have θ(η(g(b),g(c)) ≤ θ(η(fi0 (b),fi0 (c))) ≤ [θ(η(b,c))]hi0 . therefore g is also a continuous weak θm contraction on the complete metric space (k(m),η). theorem 3.11 ensures the existence and uniqueness of a ∈ k(m) such that g(a) = a and a = lim n→∞ gn(b) for all b ∈ k(m). this completes the proof. � example 4.7. let m=[0, 1] ⊂ r, with the metric given by the usual metric. we define, f : k(m) → k(m) by f(a) = f1(a) ∪f2(a), where f1(x) = 1 3 x, f2(x) = 1 3 x + 2 3 , 0≤x≤1. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 66 existence of picard operator and iterated function system first we verify that f1 and f2 are weak θm contraction. take θ = ex and d(x,y) = |x−y|, thus d(x,f1x) = ∣∣x− x 3 ∣∣ = ∣∣2x 3 ∣∣ for all x ∈ m. therefore, max{d(x,f1x),d(y,f1y),d(x,y)} = max{2x3 , 2y 3 , |x−y| 3 }. case 1: if x > y, max{d(x,f1x),d(y,f1y),d(x,y)} = 2x3 . we know that (4.3) x−y 3 ≤ 2xy 3 for all x,y ∈ m. therefore, e x−y 3 ≤ e 2xy 3 = [e 2x 3 ]y = [e 2x 3 ]h, where h = y ∈ (0, 1). hence we have θ(d(f1x.f1y)) = e x−y 3 ≤ [θd(x,f1x)]h, h = y ∈ (0, 1). now choose f2(x) = 1 3 x + 2 3 , d(x,f2x) = ∣∣x−(1 3 x + 2 3 )∣∣ = ∣∣2x 3 − 2 3 ∣∣ for all x ∈ m. as we know that (4.4) x−y 3 − 2 3 ≤ 2xy 3 − 2 3 for all x,y ∈ m. therefore, e x−y 3 −2 3 ≤ e 2xy 3 −2 3 ≤ e 2xy 3 = [e 2x 3 ]y = [e 2x 3 ]h, we have h = y ∈ (0, 1). thus we have θ(d(f2x.f2y)) = e x−y 3 −2 3 ≤ [θd(x,f2x)]h,h = y ∈ (0, 1). case 2: now take y > x, we have max{d(x,f1x),d(y,f1y),d(x,y)} = d(y,f1y). in this case we also obtain same conclusion as in case 1. therefore, d(x,y) 6= max{d(x,f1x),d(y,f1y),d(x,y)}, for any value of x,y ∈ [0, 1]. hence from both cases we can say that f1 is a weak θm-contraction for θ = ex. in the similar way, we can prove that f2 is also a weak θm-contraction for θ = ex. thus f = (f1,f2) is iterated multifunction system. the unique fixed point of f must satisfy a = f(a) = f1(a) ∪f2(a). considering the nature of the two transformations, we get a unique fractal a ⊂ k(m) which is cantor subset of [0, 1]. 5. cyclic (α,β)-admissible mappings definition 5.1. let (m,d) be a complete metric space, f : m → m be a mapping and α,β : r → [0,∞) be two functions. then s is said to be a generalized (α,β,θm)− contraction mapping if f satisfies the following conditions: (1) f is cyclic (α,β)-admissible; (2) there exits a θ ∈ θ2,4 and h ∈ (0, 1) such that for all u,v ∈ m, we have α(u)β(v) ≥ 1,d(fu,fv) > 0 ⇒ θ(d(fu,fv)) ≤ [θ(m(u,v))]h,(5.1) where m(u,v) = max{d(u,fu),d(v,fv),d(u,v)}. theorem 5.2. let (m,d) be a complete metric space, f : m → m be a mapping and α,β : m → [0, 1) be two functions. suppose that the following conditions hold. (1) f is a generalized (α,β,θm)− contraction mapping; c© agt, upv, 2020 appl. gen. topol. 21, no. 1 67 m. garg and s. chandok (2) there exists an element x0 ∈ m such that α(x0) ≥ 1 and β(x0) ≥ 1; (3) f is continuous; or if sequence {xn} in m converges to x ∈ m with the property α(xn) ≥ 1 (or β(xn) ≥ 1) for all n ∈ n, then α(x) ≥ 1 (or β(x) ≥ 1). then f is a picard operator. proof. assume that there exist x0 ∈ m such that α(x0) ≥ 1. define a picard sequence {xn} by xn+1 = fxn = fnx0, for all n ∈ n ∪ {0}. if there exist n0 ∈ n∪{0} such that un0 = fun0 , then we are done. assume that un 6= un+1 for all n ∈ n ∪{0}. assume that there exist x0,x1 ∈ m such that α(x0) ≥ 1 =⇒ β(fx0) = β(x1) ≥ 1 and β(x0) ≥ 1 =⇒ α(fx0) = α(x1) ≥ 1. by continuing above process, we have α(xn) ≥ 1 =⇒ β(fxn) = β(xn+1) ≥ 1 and β(xn) ≥ 1 =⇒ α(fxn) = α(xn+1) ≥ 1. since α(xm) ≥ 1 =⇒ β(fxm) = β(xm+1) ≥ 1 and β(xm) ≥ 1 =⇒ α(fxm) = α(xm+1) ≥ 1, for all m,n ∈ n with n < m. moreover, since α(xm) ≥ 1 =⇒ β(xm+2) ≥ 1 and β(xm) ≥ 1 =⇒ α(xm+2) ≥ 1, for all m,n ∈ n with n < m. by continuing this process, we have α(xn) ≥ 1 =⇒ β(xm) ≥ 1 and β(xn) ≥ 1 =⇒ α(xm) ≥ 1, for all m,n ∈ n. thus α(xn)β(xn+1) ≥ 1, for all n ∈ n∪{0}. therefore, using (5.1) we have θ(d(fxn,fxn+1)) ≤ [θ(max{d(xn,fxn),d(xn+1,fxn+1),d(xn,xn+1)})]h = [θ(max{d(xn,xn+1),d(xn+1,xn+2),d(xn,xn+1)})]h = [θ(max{d(xn,xn+1),d(xn+1,xn+2)})]h(5.2) analysis similar to that in the proof of theorem 3.11 shows that d (xn,xn+1) → 0, as n →∞. now, we prove that {xn} is a cauchy sequence. on the contrary, suppose that {xn} is not a cauchy sequence. by lemma 2.4, there exist ε > 0 and two sequences {n (k)} and {m(k)} of positive integers such that n(k) > m(k) > k and the sequences {d(xm(k),xn(k))} and {d(xm(k)+1,xn(k)+1)} tend to ε+ > 0 as k →∞. substituting x = xm(k) and y = xn(k) into the inequality (5.1), we obtain α ( xm(k) ) β ( xn(k) ) ≥ 1 ⇒ θ ( d ( fxm(k),fxn(k) )) ≤ [θ ( m ( xm(k),xn(k) )) ]h,(5.3) where m ( xm(k),xn(k) ) = max { d ( xm(k),xn(k) ) ,d ( xm(k),xm(k)+1 ) ,d ( xn(k),xn(k)+1 )} . since d ( xm(k),xm(k)+1 ) → 0 and d ( xn(k),xn(k)+1 ) → 0 as k →∞. then using the fact that α ( xm(k) ) β ( xn(k) ) ≥ 1 holds and that d ( xm(k)+1,xn(k)+1 ) and d ( xm(k),xn(k) ) are both positive numbers, by using the property θ4, lemma c© agt, upv, 2020 appl. gen. topol. 21, no. 1 68 existence of picard operator and iterated function system 2.4 and similar arguments as in theorem 3.11, we obtain α ( xm(k) ) β ( xn(k) ) ≥ 1 ⇒ θ ( d ( fxm(k),fxn(k) )) ≤ [θ ( d ( xm(k),xn(k) )) ]h. for sufficiently large k, k →∞, we get θ(ε) ≤ [θ(ε)]h, which is a contradiction. hence, {xn}n∈n∪{0} is a cauchy sequence. now as (m,d) is a complete metric space so there exist x ∈ m such that {xn} converges to x. the continuity of f and uniqueness of limit implies fx = x, thus we get a fixed point. now, suppose that the sequence {xn} in m converges to x ∈ m with the property α(xn) ≥ 1 (or β(xn) ≥ 1) for all n ∈ n, then α(x) ≥ 1 (or β(x) ≥ 1). hence α(x)β(x) ≥ 1 further, we claim that fx = x. suppose not, that is, fx 6= x. so d(fx,x) > 0 and lim n→∞ d(xn+1,fx) 6= 0. using (5.1) we have θ(d(xn+1,fx)) = θ(d(fxn,fx)) ≤ [θ(max{d(xn,fxn),d(x,fx),d(xn,x)})]h = [θ(max{d(xn,xn+1),d(x,fx),d(xn,x)})]h.(5.4) taking n →∞ and using property θ4, we have θ(d(x,fx)) ≤ [d(x,fx)]h, which is a contradiction. we, thus, obtain that f has a fixed point fx = x. it is easy to prove the uniqueness of fixed point. � remark 5.3. • note that, throughout this paper, lemma 2.4 and the contractive conditions imply that the iterative sequence, i.e. picard sequence is a cauchy. • for different variants of inequality (3.1), we have many interesting results. for example, when, we replace m(u,v) in (2.1) and (5.1) with m(u,v) = max{d(u,f(u)),d(v,f(v)))} (type of bianchini [3]), we may extend theorem 3.11, theorem 3.13, theorem 3.15, theorem 4.6 and theorem 5.2 to these different variants of inequality. also when, we replace m(u,v) in (2.1) with m(u,v) = d(u,v), we have the corresponding results of imdad et al. [8]. acknowledgements. the authors are thankful to the learned referee for valuable suggestions. the second author is also thankful to aistdf, dst for the research grant vide project no. crd/2018/000017. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 69 m. garg and s. chandok references [1] s. alizadeh, f. moradlou and p. salimi, some fixed point results for (α,β) − (ψ,φ)contractive mappings, filomat 28 (2014), 635–647. [2] m. f. barnsley, fractals everywhere, revised with the assistance of and with a foreword by hawley rising, iii. academic press professional, boston (1993). [3] r. m. t. bianchini, su un problema di s. reich riguardante la teoria dei punti fissi, boll. un. mat. ital. 5 (1972), 103–108. [4] e. l. fuster, a. petrusel and j. c. yao, iterated function system and well-posedness, chaos sol. fract. 41 (2009), 1561–1568. [5] r. h. haghi, sh. rezapour and n. shahzad, some fixed point generalizations are not real generalizations, nonlinear anal. 74 (2011), 1799–1803. 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[13] i. a. rus, a. petrusel and g. petrusel, fixed point theory, cluj university press, cluj-napoca, 2008. [14] n. a. secelean, countable iterated function systems, lap lambert academic publishing (2013). [15] n. a. secelean, iterated function systems consisting of f-contractions, fixed point theory appl. 2013, 277 (2013). [16] v. m. sehgal, on fixed and periodic points for a class of mappings, j. london math. soc. 5 (1972), 571–576. [17] s.-a. urziceanu, alternative charaterizations of agifss having attactors, fixed point theory 20 (2019), 729–740. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 70 @ appl. gen. topol. 22, no. 1 (2021), 47-65doi:10.4995/agt.2021.13165 © agt, upv, 2021 intermediate rings of complex-valued continuous functions amrita acharyya a, sudip kumar acharyya b, sagarmoy bag b and joshua sack c a department of mathematics and statistics, university of toledo, main campus, toledo, oh 43606-3390. (amrita.acharyya@utoledo.edu) b department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata 700019, west bengal, india (sdpacharyya@gmail.com, sagarmoy.bag01@gmail.com) c department of mathematics and statistics, california state university long beach, 1250 bellflower blvd, long beach, ca 90840, usa (joshua.sack@csulb.edu) communicated by a. tamariz-mascarúa abstract for a completely regular hausdorff topological space x, let c(x, c) be the ring of complex-valued continuous functions on x, let c∗(x, c) be its subring of bounded functions, and let σ(x, c) denote the collection of all the rings that lie between c∗(x, c) and c(x, c). we show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z◦-ideals in the rings p(x, c) in σ(x, c) and in their real-valued counterparts p(x, c) ∩ c(x). these correlations culminate to the fact that the structure space of any such p(x, c) is βx. for any ideal i in c(x, c), we observe that c∗(x, c)+i is a member of σ(x, c), which is further isomorphic to a ring of the type c(y, c). incidentally these are the only c-type intermediate rings in σ(x, c) if and only if x is pseudocompact. we show that for any maximal ideal m in c(x, c), c(x, c)/m is an algebraically closed field, which is furthermore the algebraic closure of c(x)/m ∩c(x). we give a necessary and sufficient condition for the ideal cp(x, c) of c(x, c), which consists of all those functions whose support lie on an ideal p of closed sets in x, to be a prime ideal, and we examine a few special cases thereafter. at the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a p(x, c) in σ(x, c). 2010 msc: 54c40; 46e25. keywords: z-ideals; z◦-ideals; algebraically closed field; c-type rings; zero divisor graph. received 20 february 2020 – accepted 06 october 2020 http://dx.doi.org/10.4995/agt.2021.13165 a. acharyya, s. k. acharyya, s. bag and j. sack 1. introduction in what follows, x stands for a completely regular hausdorff topological space and c(x,c) denotes the ring of all complex-valued continuous functions on x. c∗(x,c) is the subring of c(x,c) containing those functions which are bounded over x. as usual c(x) designates the ring of all real-valued continuous functions on x and c∗(x) consists of those functions in c(x) which are bounded over x. an intermediate ring of real-valued continuous functions on x is a ring that lies between c∗(x) and c(x). let σ(x) be the aggregate of all such rings. likewise an intermediate ring of complex-valued continuous functions on x is a ring lying between c∗(x,c) and c(x,c). let σ(x,c) be the family of all such intermediate rings. it turns out that each member p(x,c) of σ(x,c) is absolutely convex in the sense that |f| ≤ |g|,g ∈ p(x,c),f ∈ c(x,c) implies f ∈ p(x,c). it follows that each such p(x,c) is conjugate-closed in the sense that if whenever f + ig ∈ p(x,c) where f,g ∈ c(x), then f −ig ∈ p(x,c). it is realised that there is a natural correlation between the prime ideals/ maximal ideals/ z-ideals/ z◦-ideals in the rings p(x,c) and the prime ideals/ maximal ideals/ z-ideals/ z◦-ideals in the ring p(x,c) ∩ c(x). in the second and third sections of this article, we examine these correlations in some detail. incidentally an interconnection between prime ideals in the two rings c(x,c) and c(x) is already observed in corollary 1.2[7]. as a follow up of our investigations on the ideals in these two rings, we establish that the structure spaces of the two rings p(x,c) and p(x,c)∩c(x) are homeomorphic. the structure space of a commutative ring r with unity stands for the set of all maximal ideals of r equipped with the wellknown hull-kernel topology. it was established in [21] and [22], independently that the structure space of all the intermediate rings of real-valued continuous functions on x are one and the same viz the stone-čech compactification βx of x. it follows therefore that the structure space of each intermediate ring of complex-valued continuous functions on x is also βx. this is one of the main technical results in our article. we like to mention in this context that a special case of this result telling that the structure space of c(x,c) is βx is quite well known, see [19]. we call a ring p(x,c) in the family σ(x,c) a c-type ring if it is isomorphic to a ring of the form c(y,c) for tychonoff space y . we establish that if i is any ideal of c(x,c), then the linear sum c∗(x,c) + i is a c-type ring. this is the complex analogue of the corresponding result in the intermediate rings of real-valued continuous functions on x as proved in [16]. we further realise that these are the only c-type intermediate rings in the family σ(x,c) when and only when x is pseudocompact i.e. c(x,c) = c∗(x,c). it is well-known that if m is a maximal ideal in c(x), then the residue class field c(x)/m is real closed in the sense that every positive element in this field is a square and each odd degree polynomial over this field has a root in the same field [17, theorem 13.4]. the complex analogue of this result as we realise © agt, upv, 2021 appl. gen. topol. 22, no. 1 48 intermediate rings of complex-valued continuous functions is that for a maximal ideal m in c(x,c),c(x,c)/m is an algebraically closed field and furthermore this field is the algebraic closure of c(x)/m ∩ c(x). in section 4 of this article, we deal with a few special problems originating from an ideal p of closed sets in x and a certain class of ideals in the ring c(x,c). a family p of closed sets in x is called an ideal of closed sets in x if for any two sets a,b in p,a ∪ b ∈ p and for any closed set c contained in a,c is also a member of p. we let cp(x,c) be the set of all those functions f in c(x,c) whose support clx(x \ z(f)) is a member of p; here z(f) = {x ∈ x : f(x) = 0} is the zero set of f in x. we determine a necessary and sufficient condition for cp(x,c) to become a prime ideal in the ring c(x,c) and examine a few special cases corresponding to some specific choices of the ideal p. the ring c∞(x,c) = {f ∈ c(x,c) : f vanishes at infinity in the sense that for each n ∈ n,{x ∈ x : |f(x)| ≥ 1 n } is compact} is an ideal of c∗(x,c) but not necessarily an ideal of c(x,c). on the assumption that x is locally compact, we determine a necessary and sufficient condition for c∞(x,c) to become an ideal of c(x,c). the fifth section of this article is devoted to finding out the estimates of a few standard parameters concerning zero divisor graphs of a few rings of complex-valued continuous functions on x. thus for instance we have checked that if γ(p(x,c)) is the zero divisor graph of an intermediate ring p(x,c) belonging to the family σ(x,c), then each cycle of this graph has length 3, 4 or 6 and each edge is an edge of a cycle with length 3 or 4. these are the complex analogues of the corresponding results in the zero divisor graph of c(x) as obtained in [9]. 2. ideals in intermediate rings notation: for any subset a(x) of c(x) such that 0 ∈ a(x), we set [a(x)]c = {f + ig : f,g ∈ a(x)} and call it the extension of a(x). then it is easy to see that [a(x)]c ∩ c(x) = a(x) = [a(x)]c ∩ a(x). from now on, unless otherwise stated, we assume that a(x) is an intermediate ring of real-valued continuous functions on x, i.e. a(x) is a member of the family σ(x). it follows at once that [a(x)]c is an intermediate ring of complex-valued continuous functions and it is not hard to verify that [a(x)]c is the smallest intermediate ring in σ(x,c) which contains a(x) and the constant function i. furthermore [a(x)]c is conjugate-closed meaning that if f +ig ∈ [a(x)]c with f,g ∈ a(x), then f −ig ∈ [a(x)]c. the following result tells that intermediate rings in the family σ(x,c) are the extensions of intermediate rings in σ(x). theorem 2.1. let p(x,c) be an intermediate ring of c(x,c). then p(x,c) is absolutely convex. proof. let |f| ≤ |g|, f ∈ c(x,c),g ∈ p(x,c). then f = f 1+g2 (1 + g2) ∈ p(x,c). hence p(x,c) is absolutely convex. � theorem 2.2. an intermediate ring p(x,c) of c(x,c) is conjugate closed. © agt, upv, 2021 appl. gen. topol. 22, no. 1 49 a. acharyya, s. k. acharyya, s. bag and j. sack proof. let f + ig ∈ p(x,c). we have |f| ≤ |f + ig|, |g| ≤ |f + ig| and f + ig ∈ p(x,c). since p(x,c) is absolutely convex, then f,g ∈ p(x,c). this implies f,ig ∈ p(x,c) as i ∈ p(x,c). thus f − ig ∈ p(x,c). hence p(x,c) is conjugate closed. � theorem 2.3. a ring p(x,c) of complex valued continuous functions on x is a member of σ(x,c) if and only if there exists a ring a(x) in the family σ(x) such that p(x,c) = [a(x)]c. proof. assume that p(x,c) ∈ σ(x,c) and let a(x) = p(x,c)∩c(x). then it is clear that a(x) ∈ σ(x) and [a(x)]c ⊆ p(x,c). to prove the reverse containment, let f + ig ∈ p(x,c). here f,g ∈ c(x). since p(x,c) is conjugate closed, f − ig ∈ p(x,c), and hence 2f and 2ig both belong to p(x,c). since constant functions are bounded and hence in p(x,c), both the constant functions 1 2 and 1 2i are in p(x,c). it follows that both f and g are in p(x,c) ∩ c(x), and hence in a(x). consequently, f + ig ∈ [a(x)]c. thus, p(x,c) ⊆ [a(x)]c. � the following facts involving convex sets will be useful. a subset s of c(x) is called absolutely convex if whenever |f| ≤ |g| with g ∈ s and f ∈ c(x), then f ∈ s. theorem 2.4. let a(x) ∈ σ(x). then (a) a(x) is an absolutely convex subring of c(x) (in the sense that if |f| ≤ |g| with g ∈ a(x) and f ∈ c(x), then f ∈ a(x)) ([16, proposition 3.3]). (b) a prime ideal p in a(x) is an absolutely convex subset of a(x) ([13, theorem 2.5]). the following convenient formula for [a(x)]c with a(x) ∈ σ(x) will often be helpful to us. theorem 2.5. for any a(x) ∈ σ(x), [a(x)]c = {h ∈ c(x,c) : |h| ∈ a(x)}. proof. first assume that h = f + ig ∈ [a(x)]c with f,g ∈ a(x). then |h| ≤ |f| + |g|. this implies, in view of theorem 2.4(a), that h ∈ a(x) and also |h| ∈ a(x). conversely, let h = f+ig ∈ c(x,c) with f,g ∈ c(x), be such that |h| ∈ a(x). this means that (f2 + g2) 1 2 ∈ a(x). since |f| ≤ (f2 + g2) 1 2 , this implies in view of theorem 2.4(a) that f ∈ a(x). analogously g ∈ a(x). thus h ∈ [a(x)]c. � theorem 2.6. if i is an ideal in a(x) ∈ σ(x), then ic = {f + ig : f,g ∈ i} is the smallest ideal in [a(x)]c containing i. furthermore ic ∩ a(x) = i = ic ∩ c(x). proof. it is easy to show that ic is an ideal in [a(x)]c containing i. let k be an ideal of [a(x)]c containing i. to show ic ⊆ k. let f + ig ∈ k, where f,g ∈ i. since i ⊆ k, then f,g ∈ k. now k is an ideal of [a(x)]c, f,g ∈ k © agt, upv, 2021 appl. gen. topol. 22, no. 1 50 intermediate rings of complex-valued continuous functions implies f +ig ∈ k. therefore ic ⊆ k. hence ic is the smallest ideal of [a(x)]c containing i. proof of the second part is trivial. � theorem 2.7. if i and j are ideals in a(x) ∈ σ(x), then i ⊆ j if and only if ic ⊆ jc. also i ( j when and only when ic ( jc. proof. if i ⊆ j, then clearly ic ⊆ jc. conversely, let ic ⊆ jc. let f ∈ i. since i ⊂ ic, we have f ∈ ic ⊆ jc. now f = f + i0 and jc = {g + ih : g,h ∈ j}. therefore f ∈ j. hence i ⊆ j. for the second part we consider i ( j and f ∈ j \ i. then f ∈ jc \ ic. thus ic ( jc. conversely, let ic ( jc and f + ig ∈ jc \ ic. then either f or g is outside i. let f /∈ i. then f ∈ j \ i. hence i ( j. this completes the proof. � we have the following convenient formula for ic when i is an absolutely convex ideal of a(x). theorem 2.8. if i is an absolutely convex ideal of a(x) (in particular if i is a prime ideal or a maximal ideal of a(x)), then ic = {h ∈ [a(x)]c : |h| ∈ i}. proof. let h = f + ig ∈ ic. then f,g ∈ i. since |h| ≤ |f| + |g|, the absolute convexity of i implies that |h| ∈ i. conversely, let h = f +ig ∈ [a(x)]c be such that |h| ∈ i. here f,g ∈ a(x). since |f| ≤ (f2 + g2) 1 2 = |h|, it follows from the absolute convexity of i that f ∈ i. analogously g ∈ i. hence h ∈ ic. � the above theorem prompts us to define the notion of an absolutely convex ideal in p(x,c) ∈ σ(x,c) as follows: definition 2.9. an ideal j in p(x,c) in σ(x,c) is called absolutely convex if for g,h in c(x,c) with |g| ≤ |h| and h ∈ j, it follows that g ∈ j. the first part of the following proposition is immediate, while the second part follows from theorem 2.3 and theorem 2.8. theorem 2.10. let p(x,c) ∈ σ(x,c). (i) if j is an absolutely convex ideal of p(x,c), then j ∩ c(x) is an absolutely convex ideal of the intermediate ring p(x,c) ∩c(x) ∈ σ(x). (ii) an ideal i in p(x,c) ∩ c(x) is absolutely convex in this ring if and only if ic is an absolutely convex ideal of p(x,c). (iii) if j is an absolutely convex ideal of p(x,c), then j = [j ∩ c(x)]c. proof. (iii) it is trivial that [j ∩ c(x)]c ⊆ j. to prove the reverse implication relation let h = f + ig ∈ j, with f,g ∈ c(x). the absolute convexity of j implies that |h| ∈ j. consequently |h| ∈ j∩c(x). but since |f| ≤ (f2+g2) 1 2 = |h|, it follows again due to the absolute convexity of p(x,c) as a subring of c(x,c) that f ∈ p(x,c). we further use absolute convexity of j in p(x,c) to assert that f ∈ j. analogously g ∈ j. thus h = f + ig ∈ [j ∩ c(x)]c. therefore j ⊆ [j ∩ c(x)]c. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 51 a. acharyya, s. k. acharyya, s. bag and j. sack remark 2.11. for any p(x,c) ∈ σ(x,c), the assignment i 7→ ic provides a one-to-one correspondence between the absolutely convex ideals of p(x,c) ∩ c(x) and those of p(x,c). the following theorem gives a one-to-one correspondence between the prime ideals of p(x,c) and those of p(x,c) ∩ c(x). theorem 2.12. let p(x,c) be member of σ(x,c). an ideal j of p(x,c) is prime if and only if there exists a prime ideal q in p(x,c) ∩ c(x) such that j = qc. proof. let j be a prime ideal in p(x,c) and let q = j ∩ c(x) and a(x) = p(x,c) ∩ c(x). then q is a prime ideal in the ring a(x). it is easy to see that qc ⊆ j. to prove the reverse containment, let h = f + ig ∈ j, where f,g ∈ p(x,c). note that p(x,c) = [a(x)]c by theorem 2.3. hence f,g ∈ a(x) and therefore f − ig ∈ p(x,c). as j is an ideal of p(x,c), it follows that (f + ig)(f − ig) ∈ j i.e, f2 + g2 ∈ j ∩ c(x) = q. since q is a prime ideal in a(x), we can apply theorem 2.4(b), yielding f2 ∈ q and hence f ∈ q. analogously g ∈ q. thus h ∈ qc. therefore j ⊆ qc. to prove the converse of this theorem, let q be a prime ideal in a(x). it follows from theorem 2.8 that qc = {h ∈ p(x,c) : |h| ∈ q} and therefore qc is a prime ideal in p(x,c). finally we note that qc ∩ c(x) = q. � remark 2.13. for any p(x,c) ∈ σ(x,c), the collection of all prime ideals in p(x,c) is precisely {qc : q is a prime ideal in p(x,c) ∩ c(x)}. remark 2.14. the collection of all minimal prime ideals in p(x,c) is precisely {qc : q is a minimal prime ideal in p(x, c) ∩ c(x)}. [this follows from remark 2.13 and theorem 2.7]. theorem 2.15. for any p(x,c) ∈ σ(x,c), the collection of all maximal ideals in p(x,c) is {mc : m is a maximal ideal of p(x,c) ∩ c(x)}. proof. let m be a maximal ideal in p(x,c) ∩ c(x) = a(x). then by theorem 2.12, mc is a prime ideal in p(x,c). suppose that mc is not a maximal ideal in p(x,c), then there exists a prime ideal t in p(x,c) such that mc ( t . by remark 2.11, there exists a prime ideal p in a(x) such that j = pc. so mc ( pc. this implies in view of theorem 2.5 that m ( p , a contradiction to the maximality of m in a(x). conversely, let j be a maximal ideal of p(x,c). in particular j is a prime ideal in this ring. by remark 2.13, j = qc for some prime ideal q in a(x). we claim that q is a maximal ideal in a(x). suppose not; then q ( k for some proper ideal k in a(x). then by theorem 2.7, qc ( kc and kc a proper ideal in p(x,c); this contradicts the maximality of j = qc. � we next prove analogoues of remark 2.13 and theorem 2.15 for two important classes of ideals viz z-ideals and z◦-ideals in p(x,c) ∈ σ(x,c). these ideals are defined as follows. © agt, upv, 2021 appl. gen. topol. 22, no. 1 52 intermediate rings of complex-valued continuous functions definition 2.16. let r be a commutative ring with unity. for each a ∈ r, let ma (respectively pa) stand for the intersection of all maximal ideals (respectively all minimal prime ideals) which contain a. an ideal i in r is called a z-ideal (respectively z◦-ideal) if for each a ∈ i,ma ⊆ i (respectively pa ⊆ i). this notion of z-ideals is consistent with the notion of z-ideal in c(x) (see [17, 4a5]). since each prime ideal in an intermediate ring a(x) ∈ σ(x) is absolutely convex (theorem 2.4(b)), it follows from theorem 2.10(ii) and remark 2.13 that each prime ideal in p(x,c) ∈ σ(x,c) is absolutely convex. in particular each maximal ideal is absolutely convex. now if i is a z-ideal in p(x,c) ∈ σ(x,c) and |f| ≤ |g|,g ∈ i,f ∈ p(x,c), then mg ⊆ i. let m be a maximal ideal in p(x,c) containing g. it follows due to the absolute convexity of m that f ∈ m. therefore f ∈ mg ⊂ i. thus each z-ideal in p(x,c) is absolutely convex. analogously it can be proved that each z◦-ideal in p(x,c) is absolutely convex. the following subsidiary result can be proved using routine arguments. lemma 2.17. for any family {iα : α ∈ λ} of ideals in an intermediate ring a(x) ∈ σ(x), ( ⋂ α∈λ iα)c = ⋂ α∈λ(iα)c. theorem 2.18. an ideal j in a ring p(x,c) ∈ σ(x,c) is a z-ideal in p(x,c) if and only if there exists a z-ideal i in p(x,c) ∩ c(x) such that j = ic. proof. first assume that j is a z-ideal in p(x,c). let i = j ∩ c(x). since j is absolutely convex, it follows from theorem 2.10(iii) that j = ic. we show that i is a z-ideal in p(x,c) ∩ c(x). choose f ∈ i. suppose {mα : α ∈ λ} is the set of all maximal ideals in the ring p(x,c) ∩ c(x) which contain f. it follows from theorem 2.15 that {(mα)c : α ∈ λ} is the set of all maximal ideals in p(x,c) containing f. since f ∈ j and j is a z-ideal in p(x,c), it follows that ⋂ α∈λ(mα)c ⊆ j. this implies in the view of lemma 2.17 that ( ⋂ α∈λ mα)c ∩ c(x) ⊆ i, and hence f ∈ ⋂ α∈λ mα ⊆ i. thus it is proved that i is a z-ideal in p(x,c) ∩ c(x). conversely, let i be a z-ideal in the ring p(x,c)∩c(x). we shall prove that ic is a z-ideal in p(x,c). we recall from theorem 2.3 that [p(x,c)∩c(x)]c = p(x,c). choose f from ic. from theorem 2.8, it follows that (taking care of the fact that each z-ideal in p(x,c) is absolutely convex) |f| ∈ i. let {nβ : β ∈ λ ∗} be the set of all maximal ideals in p(x,c)∩c(x) which contain the function |f|. the hypothesis that i is a z-ideal in p(x,c)∩c(x) therefore implies that ⋂ β∈λ∗ nβ ⊆ i. this further implies in view of lemma 2.17 that⋂ β∈λ∗(nβ)c ⊆ ic. again it follows from theorem 2.8 that, for any maximal ideal m in p(x,c) ∩ c(x) and any g ∈ p(x,c), g ∈ mc if and only if |g| ∈ m. thus for any β ∈ λ∗, |f| ∈ nβ if and only if f ∈ (nβ)c. this means that {(nβ)c}β∈λ∗ is the collection of maximal ideals in p(x,c) which contain f, and we have already observed that f ∈ ∩β∈λ∗(nβ)c ⊆ ic. consequently ic is a z-ideal in p(x,c). � © agt, upv, 2021 appl. gen. topol. 22, no. 1 53 a. acharyya, s. k. acharyya, s. bag and j. sack if we use the result embodied in remark 2.14 and take note of the fact that each minimal prime ideal in p(x,c) is absolutely convex and argue as in the proof of theorem 2.18, we get the following proposition: theorem 2.19. an ideal j in a ring p(x,c) ∈ σ(x,c) is a z◦-ideal in p(x,c) if and only if there exists a z◦-ideal i in p(x,c) ∩ c(x) such that j = ic. an ideal j in p(x,c) ∈ σ(x,c) is called fixed if ⋂ f∈j z(f) 6= ∅. the following proposition is a straightforward consequence of theorem 2.6. theorem 2.20. an ideal j in a ring p(x,c) ∈ σ(x,c) is a fixed ideal in p(x,c) if and only if j ∩ c(x) is a fixed ideal in p(x,c) ∩ c(x). we recall that a space x is called an almost p space if every non-empty gδ subset of x has non-empty interior. these spaces have been characterized via z-ideals and z◦-ideals in the ring c(x) in [8]. we would like to mention that the same class of spaces have witnessed a very recent characterization in terms of fixed maximal ideals in a given intermediate ring a(x) ∈ σ(x). we reproduce below these two results to make the paper self-contained. theorem 2.21 ([8]). x is an almost p space if and only if each maximal ideal in c(x) is a z◦-ideal if and only if each z-ideal in c(x) is a z◦-ideal. theorem 2.22 ([12]). let a(x) ∈ σ(x) be an intermediate ring of real-valued continuous functions on x. then x is an almost p space if and only if each fixed maximal ideal m p a = {g ∈ a(x) : g(p) = 0} of a(x) is a z◦-ideal. it is further realised in [12] that if x is an almost p space, then the statement of theorem 2.21 cannot be improved by replacing c(x) by an intermediate ring a(x), different from c(x). indeed it is shown in [12, theorem 2.4] that if an intermediate ring a(x) 6= c(x), then there exists a maximal ideal in a(x) (which is incidentally also a z-ideal in a(x)), which is not a z◦-ideal in a(x). we record below the complex analogue of the above results. theorem 2.23. x is an almost p space if and only if each maximal ideal of c(x,c) is a z◦-ideal if and only if each z-ideal in c(x,c) is a z◦-ideal. proof. this follows from combining theorems 2.15, 2.18, 2.19, and 2.21. � theorem 2.24. let p(x,c) ∈ σ(x,c). then x is almost p if and only if each fixed maximal ideal m p p = {g ∈ p(x,c) : g(p) = 0} of p(x,c) is a z◦-ideal. proof. this follows from combining theorems 2.15, 2.20, and 2.22. � theorem 2.25. let x be an almost p space and let p(x,c) be a member of σ(x,c) such that p(x,c) ( c(x,c). then there exists a maximal ideal in p(x,c), which is not a z◦-ideal in p(x,c). © agt, upv, 2021 appl. gen. topol. 22, no. 1 54 intermediate rings of complex-valued continuous functions thus, within the class of almost p-spaces x, c(x,c) is characterized amongst all the intermediate rings p(x,c) of σ(x,c) by the requirement that z-ideals and z◦-ideals (equivalently maximal ideals and z◦-ideals) in p(x,c) are one and the same. proof. this follows from combining theorems 2.15, 2.18, and 2.19 of this article together with [12, theorem 2.4]. � we recall the classical result that x is a p space if and only if c(x) is a von-neumann regular ring meaning that each prime ideal in c(x) is maximal. incidentally the following fact was rather recently established: theorem 2.26 ([3, 20, 12]). if a(x) ∈ σ(x) is different from c(x), then a(x) is never a regular ring. theorems 2.12, 2.15, and 2.26 yield in a straight forward manner the following result: theorem 2.27. if p(x,c) ∈ σ(x,c) is a proper subring of c(x,c), then p(x,c) is not a von-neumann regular ring. it is well-known that if p is a non maximal prime ideal in c(x) and m is the unique maximal ideal containing p , then the set of all prime ideals in c(x) that lie between p and m makes a dedekind complete chain containing no fewer than 2ℵ1 many members (see [17, theorem 14.19]). if we use this standard result and combine with theorems 2.7, 2.12, and 2.15, we obtain the complex-version of this fact: theorem 2.28. suppose p is a non maximal prime ideal in the ring c(x,c). then there exists a unique maximal ideal m containing p in this ring. furthermore, the collection of all prime ideals that are situated between p and m constitutes a dedekind complete chain containing at least 2α1 many members. thus for all practical purposes (say for example when x is not a p space), c(x,c) is far from being a noetherian ring. incidentally we shall decide the noetherianness condition of c(x,c) by deducing it from a result in section 4; in particular, we show that c(x,c) is noetherian if and only if x is a finite set. 3. structure spaces of intermediate rings we need to recall a few technicalities associated with the hull-kernel topology on the set of all maximal ideals m(a) of a commutative ring a with unity. if we set for any element a of a, m(a)a = {m ∈ m(a) : a ∈ m}, then the family {m(a)a : a ∈ a} constitutes a base for closed sets of the hull-kernel topology on m(a). we may write ma for m(a)a when context is clear. the set m(a) equipped with this hull-kernel topology is called the structure space of the ring a. © agt, upv, 2021 appl. gen. topol. 22, no. 1 55 a. acharyya, s. k. acharyya, s. bag and j. sack for any subset m◦ of m(a), its closure m◦ in this topology is given by: m◦ = {m ∈ m(a) : m ⊇ ⋂ m◦}. for further information on this topology, see [17, 7m]. following the terminology of [14], by a (hausdorff) compactification of a tychonoff space x we mean a pair (α,αx), where αx is a compact hausdorff space and α : x → αx a topological embedding with α(x) dense in αx. for simplicity, we often designate such a pair by the notation αx. two compactifications αx and γx of x are called topologically equivalent if there exists a homeomorphism ψ : αx → γx with the property ψ ◦ α = γ. a compactification αx of x is said to possess the extension property if given a compact hausdorff space y and a continuous map f : x → y , there exists a continuous map fα : αx → y with the property fα ◦ α = f. it is well known that the stone-čech compactification βx of x or more formally the pair (e,βx), where e is the evaluation map on x induced by c∗(x) defined by the formula: e(x) = (f(x) : f ∈ c∗(x)) such that e : x 7→ rc ∗(x) , enjoys the extension property. furthermore this extension property characterizes βx amongst all the compactifications of x in the sense that whenever a compactification αx of x has extension property, it is topologically equivalent to βx. for more information on these topic, see [14, chapter 1]. the structure space m(a(x)) of an arbitrary intermediate ring a(x) ∈ σ(x) has been proved to be homeomorphic to βx, independently by the authors in [21] and [22]. nevertheless we offer yet another independent technique to establish a modified version of the same fact by using the above terminology of [14]. theorem 3.1. let ηa : x → m(a(x)) be the map defined by ηa(x) = mxa = {g ∈ a(x) : g(x) = 0} (a fixed maximal ideal in a(x)). then the pair (ηa,m(a(x))) is a (hausdorff) compactification of x, which further satisfies the extension property. hence the pair (ηa,m(a(x))) is topologically equivalent to the stone-čech compactification βx of x. proof. since x is tychonoff, ηa is one-to-one. also clm(a(x))ηa(x) = {m ∈ m(a(x)) : m ⊇ ⋂ x∈x m x a} = {m ∈ m(a(x)) : m ⊇ {0}} = m(a(x)). it follows from a result proved in theorem 3.3 and theorem 3.4 [23] that m(a(x)) is a compact hausdorff space and ηa is an embedding. thus (ηa,m(a(x))) is a compactification of x. to prove that this compactification of x possesses the extension property we take a compact hausdorff space y and a continuous map f : x → y . it suffices to define a continuous map fβa : m(a(x)) → y with the property that fβa ◦ ηa = f. let m be any member of m(a(x)) i.e. m is a maximal ideal of the ring a(x). define m̂ = {g ∈ c(y ) : g ◦ f ∈ m}. note that if g ∈ c(y ) then g ◦ f ∈ c(x). further note that since y is compact and g ∈ c(y ), g is bounded i.e. g(y ) is a bounded subset of r. it follows that (g ◦ f)(x) is a bounded subset of r and hence g ◦ f ∈ c∗(x). consequently g ◦ f ∈ a(x). thus the definition of m̂ is without any ambiguity. it is easy to see that m̂ is an ideal of c(y ). it follows, since m is a maximal ideal and therefore a prime ideal of a(x), that © agt, upv, 2021 appl. gen. topol. 22, no. 1 56 intermediate rings of complex-valued continuous functions m̂ is a prime ideal of c(y ). since c(y ) is a gelfand ring, m̂ can be extended to a unique maximal ideal n in c(y ). since y is compact, n is fixed (see [17, theorem 4.11]). thus we can write: n = ny = {g ∈ c(y ) : g(y) = 0} for some y ∈ y . we observe that y ∈ ⋂ g∈m̂ z(g). indeed ⋂ g∈m̂ z(g) = {y} for if y1,y2 ∈ ⋂ g∈m̂ z(g), for y1 6= y2, then m̂ ⊆ ny1 and m̂ ⊆ ny2 which is impossible as ny1 6= ny2 and c(y ) is a gelfand ring. we then set f βa(m) = y. note that {fβa(m)} = ⋂ g∈m̂ z(g). thus f βa : m(a(x)) → y is a well defined map. now choose x ∈ x and then g ∈ m̂x a ; then g ◦ f ∈ mxa, which implies that (g ◦ f)(x) = 0. consequently f(x) ∈ z(g) for each g ∈ m̂xa. on the other hand {fβa(mxa)} = ⋂ g∈m̂x a z(g). this implies that fβa(mxa) = f(x); in other words (fβa ◦ ηa)(x) = f(x) and this relation is true for each x ∈ x. hence fβa ◦ ηa = f. now towards the proof of the continuity of the map fβa, choose m ∈ m(a(x)) and a neighbourhood w of fβa(m) in the space y . in a tychonoff space every neighbourhood of a point x contains a zero set neighbourhood of x, which contains, a co-zero set neighbourhood of x. so there exist some g1,g2 ∈ c(y ), such that f βa(m) ∈ y \ z(g1) ⊆ z(g2) ⊆ w . it follows that g1g2 = 0 as z(g1) ∪ z(g2) = y which means that z(g1g2) = y . furthermore fβa(m) /∈ z(g1). since {f βa(m)} = ⋂ g∈m̂ z(g), as observed earlier, we then have g1 /∈ m̂. this means that g1 ◦ f /∈ m. in other words m ∈ m(a(x)) \ mg1◦f, which is an open neighbourhood of m in m(a(x)). we shall check that fβa(m(a(x)) \ mg1◦f) ⊆ w and that settles the continuity of fβa at m. towards that end, choose a maximal ideal n ∈ m(a(x)) \ mg1◦f. this means that n /∈ mg1◦f, i.e. g1 ◦ f /∈ n. thus g1 /∈ n̂. but as g1g2 = 0 and n̂ is prime ideal in c(y ), it must be that g2 ∈ n̂. since {fβa(n)} = ⋂ g∈n̂ z(g), it follows that fβa(n) ∈ z(g2) ⊆ w . � to achieve the complex analogue of the above mentioned theorem, we need to prove the following proposition, which is by itself a result of independent interest. theorem 3.2. let a(x) ∈ σ(x). then the map ψa : m([a(x)]c) → m(a(x)) mapping m → m ∩ a(x) is a homeomorphism from the structure space of [a(x)]c onto the structure space of a(x). proof. that the above map ψa is a bijection between the structure spaces of [a(x)]c and a(x) follows from theorems 2.3, 2.6, 2.7, and 2.15. recall (same notation as before) that m([a(x)]c)f is the set of maximal ideals in the ring [a(x)]c containing the function f ∈ [a(x)]c. a typical basic closed set in the structure space m([a(x)]c) is given by m([a(x)]c)h where h ∈ [a(x)]c. note that m([a(x)]c)h = {j ∈ m([a(x)]c) : h ∈ j}. so for h ∈ [a(x)]c, j ∈ m([a(x)]c)h if and only if h ∈ j, and this is true in view of theorem 2.8 and the absolute convexity of maximal ideals (see theorem 2.4(b) of the present article) if and only if |h| ∈ j ∩ a(x), and this holds when and only when j ∩ a(x) ∈ m(a(x))|h|, which is a basic closed set in the structure space © agt, upv, 2021 appl. gen. topol. 22, no. 1 57 a. acharyya, s. k. acharyya, s. bag and j. sack m(a(x)) of the ring a(x). thus (3.1) ψa[m([a(x)]c)h] = m(a(x))|h| therefore ψa carries a basic closed set in the domain space onto a basic closed set in the range space. now for a maximal ideal n in a(x) and a function g ∈ a(x),g belongs to n if and only if |g| ∈ n, because of the absolutely convexity of a maximal ideal in an intermediate ring. consequently m(a(x))g = m(a(x))|g| for any g ∈ a(x). hence from relation (3.1), we get: ψa[m([a(x)]c)g] = m(a(x))g which implies that ψ −1 a [m(a(x))g] = m([a(x)]c)g. thus ψ −1 a carries a basic closed set in the structure space m(a(x)) onto a basic closed in the structure space m([a(x)]c). altogether ψa becomes a homeomorphism. � for any x ∈ x and a(x) ∈ σ(x), set mx a[c] = {h ∈ [a(x)]c : h(x) = 0}. it is easy to check by using standard arguments, such as those employed to prove the textbook theorem [17, theorem 4.1], that mx a[c] is a fixed maximal in [a(x)]c and m x a[c] ∩ a(x) = mxa = {g ∈ a(x) : g(x) = 0}. let ζa : x 7→ m([a(x)]c) be the map defined by: ζa(x) = m x a[c] . then we have the following results. theorem 3.3. (ζa,m([a(x)]c)) is a hausdorff compactification of x. furthermore (ψa ◦ ζa)(x) = ηa(x) for all x in x. hence (ζa,m([a(x)]c)) is topologically equivalent to the hausdorff compactification (ηa,m(a(x))) as considered in theorem 3.1. consequently (ζa,m([a(x)]c)) turns out to be topologically equivalent to the stone-čech compactification βx of x. proof. since m(a(x)) is hausdorff [23], it follows from theorem 3.2 that m([a(x)]c) is a hausdorff space. now by following closely the arguments made at the very beginning of the proof of theorem 3.1, one can easily see that (ζa,m([a(x)]c)) is a hausdorff compactification of x. the second part of the theorem is already realised in theorem 3.2. the third part of the present theorem also follows from theorem 3.2. � definition 3.4. an intermediate ring a(x) ∈ σ(x) is called c-type in [16], if it is isomorphic to c(y ) for some tychonoff space y . in [16], the authors have shown that if i is an ideal of the ring c(x), then the linear sum c∗(x)+i is a c-type ring and of course c∗(x)+i ∈ σ(x). recently the authors in [1] have realised that these are the only c-type intermediate rings of real-valued continuous functions on x if and only if x is pseudocompact. we now show that the complex analogues of all these results are also true. we reproduce the following result established in [15], which will be needed for this purpose. theorem 3.5. a ring a(x) ∈ σ(x) is c-type if and only if a(x) is isomorphic to the ring c(υax), where υax = {p ∈ βx : f ∗(p) ∈ r for each f ∈ a(x)} and f∗ : βx 7→ r ∪ {∞} is the stone extension of the function f. © agt, upv, 2021 appl. gen. topol. 22, no. 1 58 intermediate rings of complex-valued continuous functions we extend the notion of c-type ring to rings of complex-valued continuous functions: a ring p(x,c) ∈ σ(x,c) is a c-type ring if it is isomorphic to a ring c(y,c) for some tychonoff space y . theorem 3.6. suppose a(x) ∈ σ(x) is a c-type intermediate ring of realvalued continuous functions on x. then [a(x)]c is a c-type intermediate ring of complex-valued continuous functions on x. proof. since a(x) is a c-type intermediate ring by theorem 3.5, there exists an isomorphism ψ : a(x) 7→ c(υax). let ψ̂ : [a(x)]c 7→ c(υax,c) be defined as follows: ψ̂(f +ig) = ψ(f) +iψ(g), where f,g ∈ a(x). it is not hard to check that ψ̂ is an isomorphism from [a(x)]c onto c(υax,c). � theorem 3.7. let i be a z-ideal in c(x,c). then c∗(x,c) + i is a c-type intermediate ring of complex-valued continuous functions on x. furthermore these are the only c-type rings lying between c∗(x,c) and c(x,c) if and only if x is pseudocompact. proof. as mentioned above, it is proved in [16] that for any ideal j in c(x), c∗(x) + j is a c-type intermediate ring of real-valued continuous functions on x. in light of this and theorem 3.6, it is sufficient to prove for the first part of this theorem that c∗(x,c) + i = [c∗(x) + i ∩ c(x)]c. towards proving that, let f,g ∈ c∗(x) + i ∩ c(x). we can write g = g1 + g2 where g1 ∈ c ∗(x) and g2 ∈ i ∩ c(x). it follows that ig1 ∈ c ∗(x,c) and ig2 ∈ i and this implies that i(g1 + g2) ∈ c ∗(x,c) + i. thus f + ig ∈ c∗(x) + i. hence [c∗(x) + i ∩ c(x)]c ⊆ c ∗(x,c) + i. to prove the reverse inclusion relation, let h1 + h2 ∈ c ∗(x,c) + i, where h1 ∈ c ∗(x,c) and h2 ∈ i. we can write h1 = f1 + ig1,h2 = f2 + ig2, where f1,f2,g1,g2 ∈ c(x). since h1 ∈ c ∗(x,c), it follows that f1,g1 ∈ c ∗(x). thus |f2| ≤ |h2| and h2 ∈ i. this implies, because of the absolute convexity of the z-ideal i in c(x,c), that f2 ∈ i. analogously g2 ∈ i. it is now clear that f1 + f2 ∈ c ∗(x) + i ∩ c(x) and g1 + g2 ∈ c ∗(x) + i ∩ c(x). thus h1 + h2 = (f1 + f2) + i(g1 + g2) ∈ [c∗(x) + i ∩ c(x)]c. hence c ∗(x,c) + i ⊆ [c∗(x) + i ∩ c(x)]c. to prove the second part of the theorem, we first observe that if x is pseudocompact, then there is practically nothing to prove. assume therefore that x is not pseudocompact. hence by [1], there exists an a(x) ∈ σ(x) such that a(x) is a c-type ring but a(x) 6= c∗(x) + j for any ideal j in c(x). it follows from theorem 3.6 that [a(x)]c is a c-type intermediate ring of complex-valued continuous functions belonging to the family σ(x,c). we assert that there does not exist any z-ideal i in c(x,c) with the relation: c∗(x,c) + i = [a(x)]c and that finishes the present theorem. suppose towards a contradiction, there exists a z-ideal i in c(x,c) such that c∗(x,c) +i = [a(x)]c. now from the proof of the first part of this theorem, we have already settled that c∗(x,c)+i = [c∗(x)+i∩c(x)]c. consequently [c∗(x) + i ∩ c(x)]c = [a(x)]c which yields [c ∗(x) + i ∩ c(x)]c ∩ c(x) = [a(x)]c ∩ c(x), and hence c ∗(x) + i ∩ c(x) = a(x), a contradiction. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 59 a. acharyya, s. k. acharyya, s. bag and j. sack we shall conclude this section after incorporating a purely algebraic result pertaining to the residue class field of c(x,c) modulo a maximal ideal in the same field. for each a = (a1,a2, . . . ,an) ∈ c n if p1a,p2a,. . . ,pna are the zeroes of the polynomial pa(λ) = λ n + a1λ n−1 + · · · + an, ordered so that |p1a| ≤ |p2a| ≤ · · · ≤ |pna|, then by following closely the arguments of [17, 13.3(a)], the following result can be obtained. theorem 3.8. for each k, the function pk : c n 7→ c, described above, is continuous. by employing the main argument of [17, theorem 13.4], we obtain the following proposition as a consequence of theorem 3.8. theorem 3.9. for any maximal ideal n in c(x,c), the residue class field c(x,c)/n is algebraically closed. we recall from theorem 2.15 that the assignment m 7→ mc establishes a one-to-one correspondence between maximal ideals in c(x) and those in c(x,c). let φ : c(x)/m 7→ c(x,c)/mc be the induced assignment between the corresponding residue class fields, explicitly φ(f + m) = f + mc for each f ∈ c(x). it is easy to check that φ is a ring homomorphism and is one-to-one because if f + mc = g + mc with f,g ∈ c(x), then f − g ∈ mc ∩ c(x) = m and hence f + m = g + m. furthermore, if we choose an element f + ig + mc from c(x,c)/mc, with f,g ∈ c(x), then one can verify easily that it is a root of the polynomial λ2 − 2(f +mc)λ+ (f 2 +g2 +mc) over the field φ(c(x)/m). identifying c(x)/m with φ(c(x)/m), and taking note of theorem 3.9 we get the following result. theorem 3.10. for any maximal ideal m in c(x), the residue class field c(x,c)/mc is the algebraic closure of c(x)/m. 4. ideals of the form cp(x,c) and c p ∞(x,c) let p be an ideal of closed sets in x. we set cp(x,c) = {f ∈ c(x,c) : clx(x \ z(f)) ∈ p} and c p ∞(x,c) = {f ∈ c(x,c) : for each ǫ > 0 in r,{x ∈ x : |f(x)| ≥ ǫ} ∈ p}. these are the complex analogues of the rings, cp(x) = {f ∈ c(x) : clx(x\z(f)) ∈ p} and c p ∞(x) = {f ∈ c(x) : for each ǫ > 0,{x ∈ x : |f(x)| ≥ ǫ} ∈ p} already introduced in [4] and investigated subsequently in [5], [12]. as in the real case, it is easy to check that cp(x,c) is a z-ideal in c(x,c) with cp∞(x,c) just a subring of c(x,c). plainly we have: cp(x,c) ∩ c(x) = cp(x) and c p ∞(x,c) ∩ c(x) = c p ∞(x). the following results need only routine verifications. theorem 4.1. for any ideal p of closed sets in x, [cp(x)]c = {f +ig : f,g ∈ cp(x)} = cp(x,c) and [c p ∞(x)]c = c p ∞(x,c). theorem 4.2. a) if i is an ideal of the ring cp(x), then ic = {f + ig : f,g ∈ i} is an ideal of cp(x,c) and ic ∩ cp(x) = i. © agt, upv, 2021 appl. gen. topol. 22, no. 1 60 intermediate rings of complex-valued continuous functions b) if i is an ideal of the ring cp∞(x), then ic is an ideal of c p ∞(x,c) and ic ∩ c p ∞(x) = i. we record below the following consequence of the above theorem. theorem 4.3. if i1 ( i2 ( ... is a strictly ascending sequence of ideals in cp(x)(respectively c p ∞(x)), then i1c ( i2c ( · · · becomes a strictly ascending sequence of ideals in cp(x,c)(respectively c p ∞(x,c)). the analogous results for a strictly descending sequence of ideals in both the rings cp(x) and c p ∞(x) are also valid. definition 4.4. a space x is called locally p if each point of x has an open neighbourhood w such that clxw ∈ p. observe that if p is the ideal of all compact sets in x, then x is locally p if and only if x is locally compact. towards finding a condition for which cp(x,c) and c p ∞(x,c) are noetherian ring/artinian rings, we reproduce a special version of a fact proved in [6]: theorem 4.5 (from [6, theorem 1.1]). let p be an ideal of closed sets in x and suppose x is locally p. then the following statements are equivalent: 1) cp(x) is a noetherian ring. 2) cp(x) is an artinian ring. 3) cp∞(x) is a noetherian ring. 4) cp∞(x) is an artinian ring. 5) x is finite set. we also note the following standard result of algebra. theorem 4.6. let {r1,r2, ...,rn} be a finite family of commutative rings with identity. the ideals of the direct product r1 × r2 × · · · × rn are exactly of the form i1 × i2 × · · · × in, where for k = 1,2, . . . ,n, ik is an ideal of rk. now if x is a finite set, with say n elements, then as it is tychonoff, it is discrete space. furthermore if x is locally p, then clearly p is the power set of x. consequently cp(x,c) = c p ∞(x,c) = c(x,c) = c n, which is equal to the direct product of c with itself ‘n’ times. since c is a field, it has just 2 ideals, hence by theorem 4.6 there are exactly 2n many ideals in the ring cn. hence cp(x,c) and c p ∞(x,c) are both noetherian rings and artinian rings. on the other hand if x is an infinite space and is locally p space then it follows from the theorem 4.3 and theorem 4.5 that neither of the two rings cp(x,c) and c p ∞(x,c) is either noetherian or artinian. this leads to the following proposition as the complex analogue of theorem 4.5. theorem 4.7. let p be an ideal of closed sets in x and suppose x is locally p. then the following statements are equivalent: 1) cp(x,c) is a noetherian ring. 2) cp(x,c) is an artinian ring. © agt, upv, 2021 appl. gen. topol. 22, no. 1 61 a. acharyya, s. k. acharyya, s. bag and j. sack 3) cp∞(x,c) is a noetherian ring. 4) cp∞(x,c) is an artinian ring. 5) x is finite set. a special case of this theorem, choosing p to be the ideal of all closed sets in x reads: c(x,c) is a noetherian ring if and only if x is finite set. the following gives a necessary and sufficient condition for the ideal cp(x,c) in c(x,c) to be prime. theorem 4.8. let p be an ideal of closed sets in x and suppose x is locally p. then the following statements are equivalent: (1) cp(x,c) is a prime ideal in c(x,c). (2) cp(x) is a prime ideal in c(x). (3) x /∈ p and for any two disjoint co-zero sets in x, one has its closure lying in p. proof. the equivalence of (1) and (2) follows from theorem 2.12 and theorem 4.1. towards the equivalence (2) and (3), assume that cp(x) is a prime ideal in c(x). if x ∈ p, then for each f ∈ c(x), clx(x \z(f)) ∈ p meaning that f ∈ cp(x) and hence cp(x) = c(x), a contradiction to the assumption that cp(x) is a prime ideal and in particular a proper ideal of c(x). thus x /∈ p. now consider two disjoint co-zero sets x \ z(f) and x \ z(g) in x, with f,g ∈ c(x). it follows that z(f) ∪ z(g) = x, i.e. fg = 0. since cp(x) is prime, this implies that f ∈ cp(x) or g ∈ cp(x), i.e. clx(x \ z(f)) ∈ p or clx(x \ z(g)) ∈ p. conversely let the statement (3) be true. since a z-ideal i in c(x) is prime if and only if for each f,g ∈ c(x), fg = 0 implies f ∈ i or g ∈ i (see [17, theorem 2.9]) and since cp(x) is a z-ideal in c(x), it is sufficient to show that for each f,g ∈ c(x), if fg = 0 then f ∈ cp(x) or g ∈ cp(x). indeed fg = 0 implies that x\z(f) and x\z(g) are disjoint co-zero sets in x. hence by supposition (3), either clx(x \ z(f))p or clx(x \ z(g)) ∈ p meaning that f ∈ cp(x) or g ∈ cp(x). � a special case of theorem 4.8, with p equal to the ideal of all compact sets in x, is proved in [10]. we examine a second special case of theorem 4.8. a subset y of x is called a bounded subset of x if each f ∈ c(x) is bounded on y . let β denote the family of all closed bounded subsets of x. then β is an ideal of closed sets in x. it is plain that a pseudocompact subset of x is bounded but a bounded subset of x may not be pseudocompact. here is a counterexample: the open interval (0,1) in r is a bounded subset of r without being a pseudocompact subset of r. however for a certain class of subsets of x, the two notions of boundedness and pseudocompactness coincide. the following well-known proposition substantiates this fact: theorem 4.9 (mandelkar [18]). a support of x, i.e. a subset of x of the form clx(x \ z(f)) for some f ∈ c(x), is a bounded subset of x if and only if it is a pseudocompact subset of x. © agt, upv, 2021 appl. gen. topol. 22, no. 1 62 intermediate rings of complex-valued continuous functions it is clear that the conclusion of theorem 4.9 remains unchanged if we replace c(x) by c(x,c). let cψ(x) = {f ∈ c(x) : f has pseudocompact support} and recall that cβ(x) = {f ∈ c(x) : f has bounded support}. we would like to mention here that the closed pseudocompact subsets of a pseudocompact space x might not constitute an ideal of closed sets in x. indeed a closed subset of a pseudocompact space may not be pseucdocompact. the celebrated example of a tychonoff plank in [17, 8.20]: [0,ω1] × [0,ω] \ {(ω1,ω)}, where ω1 is the 1st uncountable ordinal and ω is the first infinite ordinal, demonstrates this fact. nevertheless cψ(x) is an ideal of the ring c(x). indeed it follows directly from theorem 4.9 that cψ(x) = cβ(x). a tychonoff space x is called locally pseudocompact if each point on x has an open neighbourhood with its closure pseudocompact. on the other hand, x is called locally bounded (or locally β) if each point in x has an open neighbourhood with its closure bounded. since each open neighbourhooad of a point x in a tychonoff space x contains a co-zero set neighbourhood of x, it follows from theorem 4.9 that x is locally bounded if and only if x is locally pseudocompact. this combined with theorem 2.12 leads to the following special case of theorem 4.8. theorem 4.10. let x be locally pseudocompact. then the following statements are equivalent: (1) cψ(x) is a prime ideal of c(x). (2) cψ(x,c) = {f ∈ c(x,c) : f has pseudocompact support} is a prime ideal of c(x,c). (3) x is not pseudocompact and for any two disjoint co-zero sets in x, the closure of one of them is pseudocompact. since for f ∈ c(x,c), f ∈ c∞(x,c) if and only if |f| ∈ c∞(x), it follows that c∞(x,c) is an ideal of c(x,c) if and only if c∞(x) is an ideal of c(x). in general however c∞(x) need not be an ideal of c(x). if x is assumed to be locally compact, then it is proved in [2] and [11] that c∞(x) is an ideal of c(x) when and only when x is pseudocompact. therefore the following theorem holds. theorem 4.11. let x be locally compact. then the following three statements are equivalent: 1) c∞(x,c) is an ideal of c(x,c). 2) c∞(x) is an ideal of c(x). 3) x is pseudocompact. 5. zero divisor graphs of rings in the family σ(x,c) we fix any intermediate ring p(x,c) in the family σ(x,c). suppose g = g(p(x,c)) designates the graph whose vertices are zero divisors of p(x,c) and there is an edge between vertices f and g if and only if fg = 0. for any two vertices f,g in g, let d(f,g) be the length of the shortest path between f and © agt, upv, 2021 appl. gen. topol. 22, no. 1 63 a. acharyya, s. k. acharyya, s. bag and j. sack g and diam g = sup{d(f,g) : f,g ∈ g}. suppose gr g designates the length of the shortest cycle in g, often called the girth of g. it is easy to check that a vertex f in g is a divisor of zero in p(x,c) if and only if intxz(f) 6= ∅. this parallels the statement that a vertex f in the zero-divisor graph γc(x) of c(x) considered in [9] is a divisor of zero in c(x) if and only if intxz(f) 6= ∅. we would like to point out in this connection that a close scrutiny into the proof of various results in [9] reveal that several facts related to the nature of the vertices and the length of the cycles related to γc(x) have been established in [9] by employing skillfully the last mentioned simple characterization of divisors of zero in c(x). it is expected that the anlogous facts pertaining to the various parameters of the graph g(p(x,c)) = g should also hold. we therefore just record the following results related to the graph g, without any proof. theorem 5.1. let f,g be vertices of the graph g. then d(f,g) = 1 if and only if z(f) ∪ z(g) = x; d(f,g) = 2 if and only if z(f) ∪ z(g) ( x and intxz(f) ∩ intxz(g) 6= φ; d(f,g) = 3 if and only if z(f) ∪ z(g) ( x and intxz(f)∩intxz(g) = ∅. consequently on assuming that x contains at least 3 points, diam g and gr g are both equal to 3 (compare with [9, corollary 1.3]). theorem 5.2. each cycle in g has length 3,4 or 6. furthermore every edge of g is an edge of a cycle with length 3 or 4 (compare with [9, corollary 2.3]). theorem 5.3. suppose x contains at least 2 points. then a) each vertex of g is a 4 cycle vertex. b) g is a triangulated graph meaning that each vertex of g is a vertex of a triangle if and only if x is devoid of any isolated point. c) g is a hypertriangulated graph in the sense that each edge of g is edge of a triangle if and only if x is a connected middle p space (compare with the analogous facts in [9, proposition 2.1]). acknowledgements. the authors wish to thank the referee for his/her remarks which improved the paper. references [1] s. k. acharyya, s. bag, g. bhunia and p. rooj, some new results on functions in c(x) having their support on ideals of closed sets, quest. math. 42 (2019), 1017–1090. 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[22] l. redlin and s. watson, maximal ideals in subalgebras of c(x), proc. amer. math. soc. 100, no. 4 (1987), 763–766. [23] l. redlin and s. watson, structure spaces for rings of continuous functions with applications to real compactifications, fundamenta mathematicae 152 (1997), 151–163. © agt, upv, 2021 appl. gen. topol. 22, no. 1 65 @ appl. gen. topol. 20, no. 1 (2019), 109-117doi:10.4995/agt.2019.10171 c© agt, upv, 2019 a class of ideals in intermediate rings of continuous functions sagarmoy bag, sudip kumar acharyya and dhananjoy mandal department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata 700019, west bengal, india (sagarmoy.bag01@gmail.com, sdpacharyya@gmail.com, dmandal.cu@gmail.com) communicated by a. tamariz-mascarúa abstract for any completely regular hausdorff topological space x, an intermediate ring a(x) of continuous functions stands for any ring lying between c ∗(x) and c(x). it is a rather recently established fact that if a(x) 6= c(x), then there exist non maximal prime ideals in a(x). we offer an alternative proof of it on using the notion of z ◦ -ideals. it is realized that a p-space x is discrete if and only if c(x) is identical to the ring of real valued measurable functions defined on the σ-algebra β(x) of all borel sets in x. interrelation between z-ideals, z◦-ideal and za-ideals in a(x) are examined. it is proved that within the family of almost p-spaces x, each za-ideal in a(x) is a z ◦ -ideal if and only if each z-ideal in a(x) is a z◦-ideal if and only if a(x) = c(x). 2010 msc: primary 54c40; secondary 46e25. keywords: p-space; almost p-space; ump-space; z-ideal; z◦-ideal; zaideal. 1. introduction let c(x) be the ring of all real valued continuous functions on a completely regular hausdorff topological space x. c∗(x) is the subring of c(x) consisting of those functions which are bounded over x. a ring a(x) lying between c∗(x) and c(x) is called an intermediate ring. these intermediate rings have an important common property which says that the structure space of all these rings are one and the same and is the stone-čech compactification βx of x received 17 may 2018 – accepted 10 september 2018 http://dx.doi.org/10.4995/agt.2019.10171 s. bag, s. k. acharyya and d. mandal (see [7]). the structure space of a commutative ring r with unity stands for the set of all maximal ideals of r equipped with hull kernel topology. in the present paper our purpose is to point out a few dissimilarities existing between the ambient ring c(x) and its proper intermediate subrings. to achieve that we have chosen three special classes of ideals z-ideals, z◦-ideals and za-ideals in a typical intermediate ring a(x). an ideal i unmodified in a commutative ring r with unity will also designate a proper ideal of r. for each a in r, let ma(pa) be the intersection of all maximal ideals (minimal prime ideals) of r containing a. an ideal i in r is called a z-ideal (respectively a z◦-ideal) if for each a in i, ma ⊆ i( respectively pa ⊆ i). it is well known that if r is a reduced ring meaning that ′0′ is the only nilpotent element of r, then each z◦-ideal of r is a z-ideal also (see [4]). in particular therefore each z◦-ideal in an intermediate ring a(x) is a z-ideal. the notion z-ideals and z◦-ideals in commutative rings are quite well known and are being investigated since around 1970’s. however the concept of za-ideals in an intermediate ring is rather recent and is initiated in [15]. given e ⊆ x an f ∈ a(x) is called e-regular if there exist g ∈ a(x) such that f(x)g(x) = 1 for each x ∈ e. for any non-invertible f ∈ a(x), za(f) = {e ∈ z[x] : f is x \ e-regular} and za(f) = {e ∈ z[x] : f is h-regular for each zero set h ⊆ x \ e} are z-filters on x, here z[x] stands for the family of all zero sets in x. for any ideal i in a(x) , za[i] = ∪f∈iza(f) and za[i] = ∪f∈iza(f) are z-filters on x. for any z-filter f on x, z−1a [f] = {f ∈ a(x) : za(f) ⊆ f} and z −1 a [f] = {f ∈ a(x) : za(f) ⊆ f} are easily seen to be ideals in a(x). it is easily verified that for any ideal i in a(x), z−1a za[i] ⊇ i. i is called za-ideal if z −1 a za[i] = i. it is plain that zc-ideals and z-ideals in c(x) are same. for an arbitrary a(x), we check that every za-ideal is a z-ideal (theorem 3.4). we establish a partial converse of this theorem that, within the class of p spaces x if every z-ideal is za-ideal in a(x) then a(x) = c(x) (theorem 3.5). a za-ideal in a(x) need not be a z ◦-ideal. indeed a zc-ideal in c(x) is not necessarily a z◦-ideal, a fact which is not hard to realize on choosing x = r. we prove that such things do not happen if and only if x is an almost p-space (theorem 3.6). this further yields that if x is almost p and a(x) 6= c(x), then there does exist a za ideal in a(x) which is not a z◦-ideal (theorem 3.7). relations between maximal ideals and z◦-ideals in a(x) have also been investigated by the present authors, indeed an improved version of such interrelations have already been pondered upon in a recently communicated paper [5] however to make the present article self contained and also for the benefit of the readers we reproduce a few of these relevant facts from this paper in the technical section §2. it is a rather recently established fact that if an intermediate ring a(x) is different from c(x), then there exist non maximal prime ideals in a(x) [1], [11]. we give an alternative proof of it in the concluding section §4 of this article. in this last section we find out two conditions each necessary and sufficient for a p-space x to be discrete (theorem 4.3). on using certain c© agt, upv, 2019 appl. gen. topol. 20, no. 1 110 a class of ideals in intermediate rings of continuous functions facts about z◦-ideals we prove a second special result which tells that certain important subspaces of an ump-space are also ump-spaces. a space x is called an ump-space if every maximal ideal in c(x) is a union of minimal prime ideals contained in it (see [6] for results about these spaces). 2. z◦-ideals in intermediate rings versus almost pspaces we start with the following characterization of minimal prime ideals in a commutative ring r with unity ([12]), which is also recorded in ([10], lemma 1.1). theorem 2.1. a prime ideal p in r is a minimal prime ideal if and only if given a ∈ p, there is a b ∈ r \ p such that a.b is a nilpotent element of r, in particular ab = 0, if r is assumed to be a reduced ring. it follows from this theorem that each non zero element of a minimal prime ideal in a reduced ring r is a divisor of zero in r. therefore non zero elements of a z◦-ideal in such a ring r are all divisors of zero. we shall use this singularly important fact in the proof of several theorems that follow. incidentally it is not hard to prove on using theorem 3.1 that if ann(a) is the annihilator of an element a in a reduced ring r, then pa = {b ∈ r : ann(a) ⊆ ann(b)}. this formula together with the fact that each bounded function in c(x) belongs to each intermediate ring a(x) yields the following theorem: theorem 2.2. for f ∈ a(x), pf = {g ∈ a(x) : intxz(f) ⊆ intxz(g)}, here z(f) stands for the zero set of f in x. as recorded in [9], p-spaces are fairly rare. interesting examples of non discrete p-spaces are rather pathological (see examples 7.3,7.4,7.5,7.6 in [9]). a larger family of spaces, the so-called almost p-spaces x viz those for which non empty zero sets in x have non empty interior (equivalently non empty gδ sets in x have non empty interior) have been introduced in [13]. it turns out that almost p-spaces are far more abundant than p-spaces. those spaces have already been characterized viz z◦-ideals and z-ideals in c(x) in [3]. we have offered a some what improved characterization of those spaces via z◦-ideals in intermediate rings. the following proposition attests to this fact: theorem 2.3. suppose a(x) is an intermediate ring. then x is an almost p-space if and only if each fixed maximal ideal m p a = {f ∈ a(x) : f(p) = 0}, p ∈ x of a(x) is a z◦-ideal. proof. let x be almost p-space and p ∈ x. choose f ∈ m p a and g ∈ pf ≡ the intersection of all minimal prime ideals of a(x) which contain f. then from theorem 3.2, intxz(f) ⊆ intxz(g). therefore the hypothesis that x is almost p implies that z(f) = clxintxz(f) ⊆ clxintxz(g) = z(g) and therefore g ∈ m p a. thus pf ⊆ m p a and hence m p a is a z ◦-ideal of a(x). to prove the other containment le x be not almost p . so there exists f ∈ c∗(x) such that z(f) 6= φ but intxz(f) = φ. hence f ∈ m p a yet f is not a divisor of zero in a(x). as members of z◦-ideals are necessarily divisors of zero in the ambient ring, if follows that m p a is not a z◦-ideal in a(x). � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 111 s. bag, s. k. acharyya and d. mandal we would like to mention in this context that in the paper [3] a space x was realized as almost p when and only when each maximal ideal of c(x) is a z◦ideal. we show in the next result that, the last characterization can not be improved to say that x is almost p if and only if each maximal ideal of an intermediate ring a(x) is a z◦-ideal. theorem 2.4. let x be almost p. then for an intermediate ring a(x), each maximal ideal is a z◦-ideal in a(x) if and only if each z-ideal in a(x) is a z◦-ideal if and only if a(x) = c(x). proof. if a(x) = c(x), then it follows from [3], that each z ideal of a(x), in particular each maximal ideal of a(x) is a z◦-ideal. to prove the converse let a(x) 6= c(x). then there exists f ∈ c(x) such that f /∈ a(x). since a(x) is an absolutely convex subring of c(x) (see [7]), it follows that |f| /∈ a(x). it follows that g = 1 1+|f| is an element of a(x) which is not invertible in this ring. accordingly there exists a maximal ideal m of a(x) (which is incidentally a z ideal of a(x) also) such that g ∈ m. as z(g) = φ, g is not a divisor of zero, hence m can not be a z◦-ideal of a(x). � remark 2.5. since the converse part of theorem 2.4 does not use the almost p hypothesis on x, we can say that for any space x (not necessarily almost p), if a(x) is an intermediate subring of c(x) properly contained in c(x), then there exists a free maximal ideal of a(x), which is not a z◦-ideal. 3. za-ideals in a(x) versus p-spaces/ almost p-spaces x for any z-filter f on x, the hull hf of f is the set of all z-ultrafilters containing f and for every set u of z-ultrafilters on x, the kernel ku of u is the intersection of all z-ultrafilters belonging to u. we reproduce the following theorem established in [16]. theorem 3.1. for any f in a(x), za(f) = khza(f). this further yields the following result: theorem 3.2. for f, g ∈ a(x), hza(f) ⊆ hza(g) if and only if za(g) ⊆ za(f). proof. let hza(f) ⊆ hza(g). then khza(g) ⊆ khza(f) and hence from theorem 3.1, za(g) ⊆ za(f). to prove the other containment let za(g) ⊆ za(f). choose p from the set hza(f), then za(f) ⊆ u p (here we are identifying the point p in βx with the z-ultrafilter up on x associated with p). this means that up ∈ hza(f) and therefore za(g) ⊆ khza(g) = za(g) ⊆ za(f) = khza(f) ⊆ u p. consequently, up ∈ hza(g). thus hza(f) ⊆ hza(g). � the next proposition furnishes us with a convenient formula for the intersection of all maximal ideals containing a function f in a(x). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 112 a class of ideals in intermediate rings of continuous functions theorem 3.3. let f ∈ a(x). then the intersection of all maximal ideals of a(x) containing f is given by: mf = {g ∈ a(x) : hza(f) ⊆ hza(g)} = {g ∈ a(x) : za(g) ⊆ za(f)}. proof. it follows from theorem 3.3 of [7] that the maximal ideal in a(x) corresponding to a point p ∈ βx is given by: m p a = {f ∈ a(x) : za(f) ⊆ up} ≡ z−1a [u p] = {f ∈ a(x) : p ∈ hza(f)}. the desired result therefore follows in view of theorem 3.2. � the following result comes out as a consequence of the last one: theorem 3.4. every za ideal in a(x) is a z-ideal. proof. let i be a za-ideal in a(x) and f ∈ i. let g ∈ mf . then from theorem 3.3, we have za(g) ⊆ za(f). as f ∈ i, it is plain that za(f) ⊆ za[i] and hence za(g) ⊆ za[i]. since i is a za-ideal in a(x), it follows that g ∈ i. thus mf ⊆ i and hence i is a z-ideal in a(x). � it is recently established in [11], theorem 3.7 that if x is a p-space and each ideal in a(x) is a za ideal then a(x) = c(x). the following theorem is a somewhat improved version of this fact. theorem 3.5. let x be a p-space. then a(x) = c(x) if and only if every z-ideal in a(x) is a za-ideal. proof. if a(x) = c(x), then z-ideals and zc-ideals in c(x) are the same. to prove the converse let a(x) ( c(x). then by theorem 3.10 in [11], it follows that there is a point p ∈ βx for which o p a = {f ∈ a(x) : p ∈ intβxhza(f)} ( m p a = {f ∈ a(x) : p ∈ hza(f)}. it is not hard to verify that o p a is a z-ideal in a(x) indeed let f ∈ o p a and g ∈ mf . then from theorem 3.3, we have hza(f) ⊆ hza(g). hence p ∈ intβxhza(f) ⊆ intβxhza(g), this implies that g ∈ o p a. we assert that o p a is not a za ideal in a(x). we argue by contradiction and let o p a be a za-ideal. since x is a p-space it follows from corollary 2.4 of [11] that for any ideal i in a(x), we have za[i] = za[i]. again from [7], theorem 4.1 it follows that za[o p a] = za[m p a]. thus we can write za[o p a ] = za[m p a ], which implies in view of the assumption that o p a is a za ideal in a(x) that m p a = o p a, a contradiction. � a za-ideal in a(x) need not be a z ◦-ideal, indeed a maximal ideal in c(x) and therefore a zc-ideal in c(x) is not necessarily a z ◦-ideal. an easy example is produced by m0 = {f ∈ c(r) : f(0) = 0}, we only note that the function i ∈ c(r) defined by i(r) = r, r ∈ r is a member of m0 without being a divisor of zero in the ring c(r). the following theorem settles the exact class of spaces x for which zc-ideals in c(x) are z ◦-ideals. theorem 3.6. x is an almost p-space if and only if every zc-ideal in c(x) is a z◦-ideal. proof. let x be almost p and i a zc-ideal of c(x). then from theorem 3.4, i is a z ideal of c(x). the hypothesis, x is almost p implies in view of theorem 2.14 in [3] that i is a z◦-ideal of c(x). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 113 s. bag, s. k. acharyya and d. mandal to prove the converse let every zc-ideal in c(x) be a z ◦-ideal. since maximal ideals in c(x) are always zc-ideals (see [15]), it follows from the theorem 2.3 that, x is almost p-space. � the next proposition shows that, the last result characterizes c(x) amongst all the intermediate rings within the class of almost p-spaces. theorem 3.7. let x be an almost p-space. then a(x) = c(x) if and only if each za ideal in a(x) is a z ◦-ideal. proof. let a(x) = c(x). then from theorem 3.6, it follows that every zaideal in a(x) is a z◦-ideal. to prove the converse let a(x) ( c(x). then as in the proof of the converse part of theorem 2.4, we can ensure the existence of a maximal ideal m of a(x), which is not a z◦-ideal. surely m is a za-ideal in a(x). � 4. two special results it has been established recently by the authors in [1] and [11], independently that if a(x) is an intermediate ring, properly contained in c(x), then a(x) is never regular in the sense of von-neumann, which means that there exist non maximal prime ideals in a(x). we offer yet another proof of the above mentioned fact by using the notion of z◦-ideals. we will need the following general result for commutative rings. theorem 4.1. let r be a commutative reduced regular ring with unity. then each ideal in r is a z◦-ideal. proof. let i be an ideal in r. let a ∈ i and b ∈ pa, then ann(a) ⊆ ann(b). since r is regular there exists x ∈ r such that a = a2x. therefore a(1−ax) = 0 and hence 1 − ax ∈ ann(a) ⊆ ann(b). this implies that (1 − ax)b = 0, hence b = abx and therefore b ∈ i as a ∈ i. thus pa ⊆ i. hence i is a z ◦-ideal in r. � theorem 4.2. let a(x) 6= c(x). then a(x) is not von-neumann regular, equivalently if a(x) is a regular ring, then a(x) = c(x). proof. assume that a(x) is a regular ring and choose f ∈ c(x). to show that f lies in a(x) it is sufficient to show in view of the absolute convexity of a(x) that g = 1 1+|f| is a multiplicative unit of the ring a(x). if possible let g be not a unit in a(x). then there exists a maximal ideal m in a(x) such that g ∈ m. surely g is not a divisor of zero in a(x) and therefore m cannot be a z◦-ideal in a(x). on the other hand it follows from theorem 4.1 that each ideal of a(x) is a z◦-ideal, a contradiction. � before using the above theorem, we let b(x) be the set of all borel sets in the space x. thus b(x) is the smallest σ-algebra on x containing all the open sets in x. we call a function f : x 7→ r, b measurable if for any open set v in r, f−1(v ) is a member of b(x). it is quite well known that the family b(x) of all b measurable functions on x constitutes a commutative lattice ordered c© agt, upv, 2019 appl. gen. topol. 20, no. 1 114 a class of ideals in intermediate rings of continuous functions ring with unity if the relevant operations are defined point wise on x and of course c(x) ⊆ b(x) (see [2]). the following theorem gives three conditions involving c(x) and b(x) for a p-space x to become a discrete one. theorem 4.3. let x be a p-space. then the following three statements are equivalent: (1) x is discrete. (2) z[x] = b(x), we recall that z[x] is the family of all zero sets in x. (3) c(x) = b(x). proof. it is trivial that the truth of the statement (1) implies the truth of each of the statements (2), (3). (2) ⇒ (1) : let x be not discrete, then there exists x ∈ x such that {x} is not open in x. but {x} is closed in x implies that {x} ∈ b(x). on the other hand the fact that each zero set in a p-space x is open implies that {x} /∈ z(x). thus z[x] 6= b(x). hence the statements (1) and (2) are equivalent. we make the further observation that the characteristic function χ{x} : x 7→ r defined by χ{x}(y) = { 1, if y = x 0, if y 6= x is not continuous as {x} is not open in x. but χ{x} ∈ b(x) because {x} is a borel set in x. thus c(x) 6= b(x). so (3) ⇒ (1) is also proved. � we shall now prove the last principal theorem of this paper. theorem 4.4. every dense c∗-embedded subspace y of an ump-space x is an ump-space. proof. define a map φ : c(x) 7→ c(y ) as follows φ(f) = f|y . then φ is an injective homomorphism. since y is c∗-embedded in x, it follows that φ(c∗(x)) = c∗(y ). consequently φ(c(x)) becomes an intermediate subring of c(y ), say φ(c(x)) = a(y ). the hypothesis x is an ump-space, therefore ensures that a(y ) is an ump-ring meaning that each maximal ideal is union of minimal prime ideals contained in it. in particular each maximal ideal of a(x) consists of divisor of zero. but as we have observed in the proof of the second part of theorem 2.4 and also in remark 2.5 that if a(y ) ( c(y ), then there exist a maximal ideal m of a(y ) and g ∈ m such that g is not a divisor of zero. hence we should necessarily have a(y ) = c(y ). thus φ(c(x)) = c(y ). hence y is an ump-space. � corollary 4.5. if x is ump-space, then every subspace of υx containing x is ump-space. in this context we record the following result proved in [6], corollary 1.11. theorem 4.6. no dense c∗-embedded proper realcompact subspace of a compact ump-space is a ump-space. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 115 s. bag, s. k. acharyya and d. mandal this above two theorems 4.4 and 4.6 shows that a compact ump-space does not contains any proper dense c∗-embedded realcompact subspace. we conclude this article after raising the following open questions: question 4.7. does an isomorphism between the rings c(x) and b(x) of a p-space x imply that c(x) = b(x)? question 4.8. is o p a , p ∈ βx necessarily a z◦-ideal of a(x)? question 4.9. if a(x) ( c(x), then what is the least cardinal number of the set of all free maximal ideals of a(x), which are not z◦-ideal? acknowledgements. the authors would like to thank the referee for suggesting a few revisions and also pointing out some errors appearing in the original version of this paper. references [1] s. k. acharyya and b. bose, a correspondence between ideals and z-filters for certain rings of continuous functions-some remarks, topology appl. 160 (2013), 1603–1605. 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[15] p. panman, j. sack and s. watson, correspondences between ideals and z-filters for rings of continuous functions between c∗ and c, commentationes mathematicae 52, no. 1, (2012) 11–20. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 116 a class of ideals in intermediate rings of continuous functions [16] j. sack and s. watson, c and c∗ among intermediate rings,topology proceedings 43 (2014), 69–82. [17] j. sack and s. watson, characterizing c(x) among intermediate c-rings on x, topology proceedings 45 (2015), 301–313. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 117 @ appl. gen. topol. 20, no. 1 (2019), 135-153doi:10.4995/agt.2019.10474 c© agt, upv, 2019 remarks on fixed point assertions in digital topology laurence boxer a and p. christopher staecker b a department of computer and information sciences, niagara university, ny 14109, usa; and department of computer science and engineering, state university of new york at buffalo. (boxer@niagara.edu) b department of mathematics, fairfield university, fairfield, ct 06823-5195, usa. (cstaecker@fairfield.edu) communicated by s. romaguera abstract several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. we show that some of the published assertions based on these tools are incorrect or trivial; we offer improvements on others. 2010 msc: 54h25. keywords: digital topology; fixed point; metric space. 1. introduction recent papers have attempted to apply to digital images ideas from euclidean topology and real analysis concerning metrics and fixed points. while the underlying motivation of digital topology comes from euclidean topology and real analysis, some applications recently featured in the literature seem of doubtful worth. although papers including [30, 7] have valid and interesting results for fixed points and for “almost” or “approximate” fixed points in digital topology, other published assertions concerning fixed points in digital topology are incorrect or trivial (e.g., applicable only to singletons, or functions studied forced to be constant), as we will discuss in the current paper. received 02 july 2018 – accepted 19 november 2018 http://dx.doi.org/10.4995/agt.2019.10474 l. boxer and p. c. staecker 2. preliminaries we let z denote the set of integers, and r, the real line. we consider a digital image as a graph (x,κ), where x ⊂ zn for some positive integer n and κ is an adjacency relation on x. a digital metric space is [12] a triple (x,d,κ) where (x,κ) is a digital image and d is a metric for x. in [12], d was taken to be the euclidean metric, but we will not limit our discussion to the euclidean metric. the diameter of a metric space (x,d) is diamx = sup{d(x,y) |x,y ∈ x}. 2.1. adjacencies. the most commonly used adjacencies for digital images are the cu-adjacencies, defined as follows. definition 2.1. let p,q ∈ zn, p = (p1, . . . ,pn), q = (q1, . . . ,qn), p 6= q. let 1 ≤ u ≤ n. we say p and q are cu-adjacent, denoted p ↔cu q or p ↔ q when the adjacency is understood, if • for at most u distinct indices i, |pi − qi| = 1, and • for all other indices j, pj = qj. often, a cu-adjacency is denoted by the number of points in z n that are cu-adjacent to a given point. e.g., • in z1, c1-adjacency is 2-adjacency; • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency; • in z3, c1-adjacency is 8-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. an adjacency often used for cartesian products of digital images is the normal product adjacency, denoted in the following by κ∗ and defined [2] as follows. given digital images (x,κ) and (y,λ) and points x,x′ ∈ x, y,y′ ∈ y , we have (x,y) ↔κ∗ (x ′,y′) in x × y if and only if one of the following holds. • x ↔κ x ′ and y = y′, or • x = x′ and y ↔λ y ′, or • x ↔κ x ′ and y ↔λ y ′. other adjacencies for digital images are discussed in papers such as [16, 5, 6]. a digital interval is a digital image of the form ([a,b]z,2), where a < b and [a,b]z = {z ∈ z |a ≤ z ≤ b}. digital connectedness is defined in terms of adjacency. definition 2.2 ([30]). a digital image (x,κ) is κ-connected (or connected when κ is understood) if given distinct x,y ∈ x there is a sequence {xi} n i=0 ⊂ x such that x = x0, xn = y, and xi ↔κ xi+1 for 0 ≤ i < n. 2.2. ℓp metric. let x ⊂ r n and let x = (x1, . . . ,xn) and y = (y1, . . . ,yn) be points of x. let 1 ≤ p ≤ ∞. the ℓp metric d for x is defined by d(x,y) = { ( ∑n i=1 |xi − yi| p) 1/p for 1 ≤ p < ∞; max{|xi − yi|} n i=1 for p = ∞. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 136 remarks on fixed point assertions in digital topology for p = 1, this gives us the manhattan metric d(x,y) = ∑n i=1 |xi − yi|; for p = 2, we have the euclidean metric d(x,y) = ( ∑n i=1 |xi − yi| 2)1/2. notice that for any ℓp metric d, if x,y ∈ z n and d(x,y) < 1, then x = y. we note that digital images in the literature are commonly, although not exclusively, finite; also, the cu adjacencies or adjacencies based on them (e.g., the normal product adjacency for cartesian products with cu adjacencies) are common. many papers simply state an assumption that every digital image is a finite subset of zn with some cu adjacency. digital metric spaces typically use the euclidean or some other ℓp metric. the development of classical topology has often placed emphasis on “wild” counterexamples, and in that setting finiteness and focus on ℓp metrics is very restrictive. but digital topology is motivated primarily by “real-world” digital images, which are represented in the real world as subsets of z2 with either the c1 or c2 adjacency. when a metric is used in a real world digital image, it’s usually ℓ1 or ℓ2. thus if in some results we assume that our images are finite and use the ℓp metric, these should be regarded as light assumptions. 2.3. digital continuity and homotopy. definition 2.3 ([30, 4]). a function f : (x,κ) → (y,λ) between digital images is (κ,λ)-continuous (or just continuous when κ and λ are understood) if for every κ-connected subset x′ of x, f(x′) is a λ-connected subset of y . theorem 2.4 ([4]). a function f : (x,κ) → (y,λ) between digital images is (κ,λ)-continuous if and only if x ↔κ x ′ in x implies either f(x) = f(x′) or f(x) ↔λ f(x ′) in y . as in topology, the digital topology notion of homotopy can be understood as one function deforming in a continuous fashion into another. precisely, we have the following. definition 2.5 ([4]). let f,g : (x,κ) → (y,λ). we say f and g are homotopic, denoted f ≃(κ,λ) g or f ≃ g when κ and λ are understood, if there is a function f : x × [0,m]z → y for some m ∈ n such that • f(x,0) = f(x) and f(x,m) = g(x) for all x ∈ x. • the induced function ft : x → y defined by ft(x) = f(x,t) is (κ,λ)continuous for all t ∈ [0,m]z. • the induced function fx : [0,m]z → y defined by fx(t) = f(x,t) is (2,λ)-continuous for all x ∈ x. 2.4. cauchy sequences and complete metric spaces. the papers [12, 18, 20, 21, 23, 24, 26, 27] apply to digital images the notions of cauchy sequence and complete metric space. since for common metrics such as an ℓp metric, a digital metric space is discrete, the digital versions of these notions are quite limited. recall that a sequence of points {xn} in a metric space (x,d) is a cauchy sequence if for all ε > 0 there exists n0 ∈ n such that m,n > n0 implies c© agt, upv, 2019 appl. gen. topol. 20, no. 1 137 l. boxer and p. c. staecker d(xm,xn) < ε. if every cauchy sequence in x has a limit, then (x,d) is a complete metric space. it has been shown that under a mild additional assumption, a digital cauchy sequence is eventually constant. the following is an easy generalization of proposition 3.6 of [18], where only the euclidean metric was considered. the proof given in [18] is easily modified to give the following. theorem 2.6. let a > 0. if d is a metric on a digital image (x,κ) such that for all distinct x,y ∈ x we have d(x,y) > a, then for any cauchy sequence {xi} ∞ i=1 ⊂ x there exists n0 ∈ n such that m,n > n0 implies xm = xn. an immediate consequence of theorem 2.6 is the following. corollary 2.7 ([18]). let (x,d,κ) be a digital metric space. if d is a metric on (x,κ) such that for all distinct x,y ∈ x we have d(x,y) > a for some constant a > 0, then any cauchy sequence in x is eventually constant, and (x,d) is a complete metric space. remark 2.8. it is easily seen that the hypotheses of theorem 2.6 and corollary 2.7 are satisfied for any finite digital metric space, or for a digital metric space (x,d,κ) for which the metric d is any ℓp metric. thus, a cauchy sequence that is not eventually constant can only occur in an infinite digital metric space with an unusual metric. such an example is given below. example 2.9. let d be the metric on (n,c1) defined by d(i,j) = |1/i − 1/j|. then {i}∞i=1 is a cauchy sequence for this metric that does not have a limit. 2.5. digital fixed points and approximate fixed points. the study of the fixed points of continuous self-maps is prominent in many areas of mathematics. we say a topological space x or a digital image (x,κ) has the fixed point property if every continuous (respectively, (κ,κ)-continuous) f : x → x has a fixed point, i.e., a point p ∈ x such that f(p) = p. a version of theorem 2.10 below was proved by rosenfeld in [30] for the case when x is a digital picture, that is, a digital image of the form πni=1[ai,bi]z ⊂ z n with the cn-adjacency. for general digital images, theorem 2.10 was proved in [7]. theorem 2.10. a digital image (x,κ) has the fixed point property if and only if x is a singleton. this theorem led to the study in [7] of the approximate fixed point property, an idea suggested by results of [30]. an approximate fixed point [7], called an almost fixed point in [30], of a (κ,κ)-continuous function f : (x,κ) → (x,κ) is a point p ∈ x such that f(p) = p or f(p) ↔κ p. a digital image (x,κ) has the approximate fixed point property (afpp) [7] if for every continuous f : x → x there is an approximate fixed point of f. we have rephrased the following to conform with terminology used in this paper. theorem 2.11 (theorem 4.1 of [30]). every digital picture (πni=1[ai,bi]z,cn) has the afpp. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 138 remarks on fixed point assertions in digital topology in remark 6.2 (2) of [19], the author incorrectly attributes to [30] the claim that “every digital image (y,8) has the afpp.” the attribution is incorrect, since, as we have shown above, the citation should be about digital pictures, not the more general digital images. further, the claim is false, as the following example shows. example 2.12. let n ≥ 4 and let y = ({yi} n−1 i=0 ,8) ⊂ z 2 be a digital simple closed curve with the points yi indexed circularly. then (y,8) does not have the afpp. proof. the function f : y → y defined by f(yi) = y(i+2) mod n is easily seen to be (8,8)-continuous and free of approximate fixed points. � 2.6. contraction and expansion functions. we cite several fixed point theorems for digital topology that are modeled on analogs for the topology of rn. in section 4 below, we explore limitations on many of the types of functions introduced in this section; in many cases, their inspirations in topology are not similarly limited. in the following definitions, we assume (x,d,κ) is a digital metric space and f : x → x is a function. several of these definitions are unmodified from their inspirations in the topology of metric spaces. definition 2.13 ([12]). if for some α ∈ (0,1) and all x,y ∈ x, d(f(x),f(y)) < αd(x,y), then f is a digital contraction map. we say α is the multiplier. note such a function should not be confused with a digital contraction [3], a homotopy between an identity map and a constant function. definition 2.14. if d(f(x),f(y)) ≤ α[d(x,f(x)) + d(y,f(y))] for all x,y ∈ x, where 0 < α < 1/2, we say f is a kannan contraction map. definition 2.15. if d(f(x),f(y)) ≤ α[d(x,f(y)) + d(y,f(x))] for all x,y ∈ x, where 0 < α < 1/2, we say f is a chatterjea contraction map. definition 2.16. if d(f(x),f(y)) ≤ ad(x,f(x)) + bd(y,f(y)) + cd(x,y) for all x,y ∈ x and all nonnegative a,b,c such that a + b + c < 1, then f is a reich contraction map. proposition 2.17. a reich contraction map is a digital contraction map and is a kannan contraction map. proof. let f be a reich contraction map. that f is a digital contraction map follows from the observation of [28] that in definition 2.16, we can take a = b = 0 and obtain the conclusion from definition 2.13. that f is a kannan contraction map follows from the observation that in definition 2.16, we can c© agt, upv, 2019 appl. gen. topol. 20, no. 1 139 l. boxer and p. c. staecker take a = b ∈ (0,1/2) and c = 0 to obtain the conclusion from definition 2.14. � definition 2.18 ([26]). let (x,d,κ) be a digital metric space and let f : x → x be a function. if there exists α ∈ (0,1) such that for all x,y ∈ x we have d(f(x),f(y)) ≤ α max{d(x,y), d(x,f(x)) + d(y,f(y)) 2 , d(x,f(y)) + d(y,f(x)) 2 } then f is called a zamfirescu digital contraction. definition 2.19 ([26]). let (x,d,κ) be a digital metric space and let f : x → x be a function. if there exists α ∈ (0,1) such that for all x,y ∈ x we have d(f(x),f(y)) ≤ α max{d(x,y), d(x,f(x)) + d(y,f(y)) 2 ,d(x,f(x)),d(y,f(y))} then f is called a rhoades digital contraction. proposition 2.20. let (x,d,κ) be a digital metric space and let f : x → x be a function. if f is a digital contraction map, then f is a zamfirescu digital contraction and a rhoades digital contraction. proof. if f is a digital contraction map, then d(f(x),f(y)) ≤ αd(x,y) for all x,y ∈ x. the assertion follows from definitions 2.18 and 2.19,. � the following is a minor generalization of a definition in [20]. therefore, results we derive in this paper for digitally (α,κ)-uniformly locally contractive functions apply to the version in [20]. definition 2.21. suppose 0 ≤ α < 1. let (x,d,κ) be a digital metric space. let f : x → x be a function such that d(x,y) ≤ 1 implies d(f(x),f(y)) ≤ αd(x,y). then f is called digitally (α,κ)-uniformly locally contractive. proposition 2.22. a digital contraction map with multiplier α is a digitally (α,κ)-uniformly locally contractive map. proof. this is obvious from definition 2.13 and definition 2.21. � below, we define a set of functions ψ that will be used in the following. definition 2.23 ([24]). let ψ be a set of functions ψ : [0,∞) → [0,∞) such that for each ψ ∈ ψ we have • ψ is nondecreasing, and • there exists k0 ∈ n, a ∈ (0,1), and a convergent series ∑ ∞ k=1 vk of non-negative terms such that k ≥ k0 implies ψ k+1(t) ≤ aψk(t) + vk for all t ∈ [0,∞), where ψk represents the k-fold composition of ψ. the following will be used later in the paper. example 2.24. the constant function with value 0 is a member of ψ. definition 2.25 ([31]). let t : x → x and α : x × x → [0,∞). we say t is α-admissible if α(x,y) ≥ 1 implies α(t(x),t(y)) ≥ 1. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 140 remarks on fixed point assertions in digital topology definition 2.26 ([31]). let (x,d) be a metric space, t : x → x, α : x → x, and ψ ∈ ψ. we say t is an α−ψ-contractive mapping if α(x,y)d(t(x),t(y)) ≤ ψ(d(x,y)) for all x,y ∈ x. remark 2.27 ([24]). a digital contraction map f : (x,d,κ) → (x,d,κ) is an α − ψ-contractive mapping for α(x,y) = 1 and ψ(t) = λt for λ ∈ (0,1). definition 2.28 ([24]). let (x,d,κ) be a digital metric space, t : x → x, β : x × x → [0,∞), and ψ,φ ∈ ψ such that ψ(d(t(x),t(y))) ≥ β(x,y)ψ(d(x,y)) + φ(d(x,y)) for all x,y ∈ x. then t is a β − ψ − φ-expansive mapping. depending on the choice of functions β,φ in definition 2.28, the definition may not be very discriminating, as we see in the following. remark 2.29. every function t : x → x is a β − ψ − φ-expansive mapping if we take β and φ to be constant functions with value 0. proof. the assertion follows from example 2.24 and definition 2.28. � definition 2.30 ([9]). let (x,d,κ) be a digital metric space. let t : x → x. then t is a weakly uniformly strict digital contraction if given ε > 0 there exists δ > 0 such that ε < d(x,y) < ε + δ implies d(t(x),t(y)) < ε for all x,y ∈ x. definition 2.31 ([24]). let (x,d,κ) be a complete digital metric space. let t : x → x. if t satisfies the condition d(t(x),t(y)) ≥ kd(x,y) for all x,y ∈ x and some k > 1, then t is a digital expansive mapping. example 2.32. the function t : n → n defined by t(n) = 2n is a digital expansive mapping, using the usual euclidean metric. this map is not c1continuous [30]. the literature contains the following theorems concerning fixed points for such functions. the following is a digital version of the banach contraction principle [1]. theorem 2.33 ([12]). let (x,d,κ) be a complete digital metric space, where d is the euclidean metric in zn. let f : x → x be a digital contraction map. then f has a unique fixed point. the following is a digital version of the kannan fixed point theorem [25]. theorem 2.34 ([27]). let f : x → x be a kannan contraction map on a digital metric space (x,d,κ). then f has a unique fixed point in x. the following is a digital version of the chatterjea fixed point theorem [8]. theorem 2.35 ([27]). if f : x → x is a chatterjea contraction map on a digital metric space (x,d,κ), then f has a unique fixed point. the following gives digital versions of zamfirescu [32] and rhoades [29] fixed point theorems. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 141 l. boxer and p. c. staecker theorem 2.36 ([26]). let d be the euclidean metric on zn and let (x,d,κ) be a digital metric space. if f : x → x is a zamfirescu digital contraction or a rhoades contraction map, then f has a unique fixed point. the following is a digital version of the reich fixed point theorem [28]. we give a simpler proof of the digital version than appeared in [27]. theorem 2.37. let f : (x,d,κ) → (x,d,κ) be a reich contraction map on a digital metric space. then f has a unique fixed point in x. proof. this follows immediately from proposition 2.17 and theorem 2.33. � the following is a version of the edelstein fixed point theorem [10] for digital images. theorem 2.38 ([20]). a digitally (α,κ)-uniformly locally contractive function on a connected complete digital metric space has a unique fixed point. 3. digital homotopy fixed point theory the paper [13] defines a digital fixed point property as follows. the digital image (x,κ) has the digital fixed point property with respect to the digital interval [0,m]z if for all (κ∗,κ)-continuous functions f : (x × [0,m]z,κ∗) → (x,κ), where κ∗ is the normal product adjacency for (x,κ)×([0,m]z,2), there is a κ-path p : [0,m]z → x of fixed points, i.e., p(t) is a fixed point of the induced function ft : x → x defined by ft(x) = f(x,t). also, [13] defines a digital homotopy fixed point property and states that this is equivalent to the following: a digital image (x,κ) has the digital homotopy fixed point property if for each digital homotopy f : x × [0,m]z → x there is a κ-path p : [0,m]z → x such that for all t ∈ [0,m]z, p(t) is a fixed point of the induced function ft. these imply triviality, as follows. theorem 3.1. a digital image (x,κ) has the digital fixed point property and the digital homotopy fixed point property if and only if x is a singleton. proof. clearly a singleton has the digital homotopy fixed point property and the digital homotopy fixed point property. conversely, if x is not a singleton, then by theorem 2.10 there is a continuous function f : x → x that does not have a fixed point. let f : x × [0,m]z → x be defined by f(x,t) = f(x). then f is both (κ∗,κ)-continuous (where κ∗ = κ∗(κ,2) is the normal product adjacency) and a homotopy, and fails to have a fixed point for any of the induced functions ft = f. thus, (x,κ) does not have the digital fixed point property or the digital homotopy fixed point property. � 4. results for various contraction and expansion maps 4.1. digital contraction maps. in both of the papers [12, 20], arguments are given for the incorrect assertion that every digital contraction map is digitally continuous. both papers present an error of confusing (topological) continuity c© agt, upv, 2019 appl. gen. topol. 20, no. 1 142 remarks on fixed point assertions in digital topology of a map between metric spaces with (digital) continuity of a map between digital images. we present a counterexample to this assertion; our example is also used to show that kannan, chatterjea, zamfirescu, and rhoades contraction maps need not be digitally continuous. we use the manhattan metric for its ease of computation, but the euclidean or other ℓp metrics could be used to obtain similar conclusions. example 4.1. let x = {p1 = (0,0,0,0,0), p2 = (2,0,0,0,0), p3 = (1,1,1,1,1)} ⊂ z 5. let f : x → x be defined by f(p1) = f(p2) = p1, f(p3) = p2. then f is not (c5,c5)-continuous. however, with respect to the manhattan metric d, f is • a digital contraction map, • a kannan contraction map, • a chatterjea contraction map, • a zamfirescu contraction map, • a rhoades contraction map, • a (0.45,c5)-uniformly locally contractive function, • an α − ψ-contractive mapping for α(x,y) = 1 and ψ(t) = λt for λ ∈ (0,1), • a β − ψ − φ-expansive mapping, where ψ and φ are constant functions with the value 0, • a weakly uniformly strict digital contraction. proof. note f is not (c5,c5)-continuous, since p1 ↔c5 p3 ↔c5 p2, but f(x) = {p1,p2} is not c5-connected. observe that d(p1,p2) = 2, d(f(p1),f(p2)) = 0, d(p1,p3) = 5, d(f(p1),f(p3)) = 2, d(p2,p3) = 5, d(f(p2),f(p3)) = 2. therefore, we have, for all x,y ∈ x such that x 6= y, d(f(x),f(y)) ≤ 2/5 d(x,y) < 0.45d(x,y). therefore, f is a digital contraction map, a zamfirescu contraction map, a rhoades contraction map, and a (0.45,c5)-uniformly locally contractive function. since d(f(x),f(y)) ≤ 2/5[d(x,f(x)) + d(y,f(y))] < 0.45[d(x,f(x)) + d(y,f(y))] for all x,y ∈ x such that x 6= y, f is a kannan contraction map. note d(f(p1),f(p2)) = 0 < 0.45[d(p1,f(p2)) + d(p2,f(p1))], d(f(p1,f(p3))) = 2 < 0.45(2 + 5) = 0.45[d(p1,f(p3)) + d(p3,f(p1)), ] d(f(p2),f(p3)) = 2 < 0.45(0 + 5) = 0.45(d(p2,f(p3)) + d(p3,f(p2))). therefore, f is a chatterjea contraction map. that f is an α − ψ-contractive mapping for α(x,y) = 1 and ψ(t) = λt for λ ∈ (0,1) follows from remark 2.27. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 143 l. boxer and p. c. staecker since example 2.24 notes that the constant function with value 0 is in ψ, it follows from definition 2.28 that f is a β − ψ − φ-expansive mapping. it follows easily from definition 2.30 that f is a weakly uniformly strict digital contraction. � the following generalizes theorem 4.7(1) of [18]. we give a proof, essentially that of [18], so we can refer to it below. theorem 4.2. let (x,d,c1) be a digital metric space that is c1-connected, where d is any ℓp metric in z n. let f : x → x be a digital contraction map. then f is a constant function. proof. let α ∈ (0,1) satisfy d(f(x),f(y)) ≤ αd(x,y) for all x,y ∈ x. if x ↔c1 y in x, then d(x,y) = 1, so d(f(x),f(y)) ≤ α, which implies f(x) = f(y), since every distinct pair of points in zn has distance of at least 1. given x0 ∈ x, for any x ∈ x there is a path p = {x0,x1, . . . ,xm = x} ⊂ x from x0 to x such that xi ↔c1 xi+1, 0 ≤ i < m. it follows from the above that f is the constant function with value f(x0). � theorem 4.7(2) of [18] gives examples of c2-connected images with digital contraction maps that are continuous and not constant. however, modification of theorem 4.2 yields the following. theorem 4.3. let (x,d,κ) be a digital metric space that is κ-connected, where, for some m1 ≥ m2 > 0 we have that x 6= y implies d(x,y) ≥ m2 and x ↔κ y implies d(x,y) ≤ m1 let f : x → x be a digital contraction map with multiplier α such that α < m2/m1. then f is a constant function. proof. let x,y ∈ x such that x ↔κ y. then d(x,y) ≤ m1 and d(f(x),f(y)) < αd(x,y) < (m2/m1)m1 = m2. by choice of m2, f(x) = f(y). it follows as in the proof of theorem 4.2 that f is constant. � remark 4.4. notice that theorem 4.3 applies to all connected digital images (x,d,cu), where x ⊂ z n, 1 ≤ u ≤ n, and d is any ℓp metric. 4.2. kannan and chatterjea contractions. example 4.1 shows that neither a kannan contraction map nor a chatterjea contraction map must be constant. however, we have the following. theorem 4.5. let (x,d,κ) be a digital metric space of finite diameter, where d is any ℓp metric. let f : x → x be a function. if f is a kannan contraction map or a chatterjea contraction map with α as in definition 2.14 or definition 2.15, respectively, satisfying 0 < α < 1 2 diamx , then f is a constant function. proof. we have d(f(x),f(y)) < 1 for all x,y ∈ x, by definition 2.14 in the case of a kannan contraction map, and by definition 2.15 in the case of a chatterjea contraction map. since d is an ℓp metric, it follows that f(x) = f(y) for all x,y ∈ x. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 144 remarks on fixed point assertions in digital topology 4.3. reich contractions. theorem 4.6. let (x,d,κ) be a digital metric space of finite diameter, where d is any ℓp metric. let f : x → x be a function. if f is a reich contraction map with a,b,c as in definition 2.16 satisfying a,b,c ∈ (0, 1 3 diamx ), then f is a constant function. proof. we have d(f(x),f(y)) < 1 for all x,y ∈ x. since d is an ℓp metric, it follows that f(x) = f(y) for all x,y ∈ x. � also, it follows from proposition 2.17 that theorem 4.2 and theorem 4.3 apply to a reich contraction map. 4.4. (α,κ)-uniformly locally contractive functions. theorem 2.38 turns out to be a trivial result for connected digital metric spaces that use an ℓp metric, as we see below. theorem 4.7. let (x,d,κ) be a κ-connected digital metric space, where d is any ℓp metric. let f : x → x be an (α,κ)-uniformly locally contractive function. then f is a constant function. proof. let x0,x ∈ x. since x is connected, there is a κ-path in x, {xi} m i=0, from x0 to x such that xm = x and xi ↔c1 xi+1 for 0 ≤ i < m. if d(xi,xi+1) ≤ 1, then d(f(xi),f(xi+1)) ≤ αd(xi,xi+1) < 1. since d is an ℓp metric, f(xi) = f(xi+1). it follows that f is a constant function. � 4.5. digital expansive mappings. we saw in example 2.32 that a digital expansive mapping need not be digitally continuous. theorem 3.2 and corollary 3.3 of [23] hypothesize a digital expansive mapping t : x → x that is onto. but in “real world” image processing, a digital image is finite, and therefore cannot support such a map, as shown by the following theorems 4.8 and 4.9. theorem 4.8. let (x,d,κ) be a digital metric space. if x has points x0,y0 such that d(x0,y0) = diamx > 0, then there is no self-map t : x → x that is onto and a digital expansive mapping. proof. suppose there is a digital expansive mapping t : x → x. let x0,y0 ∈ x be such that d(x0,y0) = diamx > 0. then for some k > 1, (4.1) d(t(x0),t(y0)) ≥ kd(x0,y0) = k diamx > diamx. since statement (4.1) is contradictory, the assertion follows. � theorem 4.9. let (x,d,κ) be a digital metric space of more than one point. if there exist x0,y0 ∈ x such that (4.2) d(x0,y0) = min{d(x,y) |x,y ∈ x,x 6= y} then there is no self-map t : x → x that is onto and a digital expansive mapping. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 145 l. boxer and p. c. staecker proof. suppose there exists a map t : x → x that is a digital expansive mapping. let x0,y0 ∈ x be as in equation (4.2). let k be the expansive constant of t . since t is onto, there exist distinct x′,y′ ∈ x such that t(x′) = x0 and t(y ′) = y0. then d(x0,y0) = d(t(x ′),t(y′)) ≥ kd(x′,y′), which contradicts our choice of x0,y0. the assertion follows. � note theorem 4.9 is applicable when x is finite or when d is any ℓp metric. remark 4.10. example 3.8 of [23] claims that the self-map t : x → x of some subset of z given by t(n) = 2n − 1 is onto. it is easy to see that this claim is only true for x = {1}. 4.6. β − ψ − φ-expansive mappings. an analog of theorem 2.1 of [31] is asserted as theorem 3.2 of [24]: let (x,d,κ) be a complete digital metric space, β : x × x → [0,∞), and let t : x → x be a β − ψ − φ-expansive mapping for some ψ,φ ∈ ψ. if there exist functions and such that • t −1 is β-admissible; • there exists x0 ∈ x such that β(x0,t −1(x0)) ≥ 1; and • t is digitally continuous, then t has a fixed point. however, this assertion is false, as we see in the following. example 4.11. let x = [−1,1]2 z \{(0,0)} ⊂ z2. let β = d be the manhattan metric on z2. let φ and ψ be constant functions with value 0. let t : x → x be the map defined by t(x,y) = (−x,−y). by remark 2.29, t is a β − ψ − φexpansive mapping. clearly t is β-admissible. for every p ∈ x we have β(p,t −1(p)) = d(p,−p) > 1. also, t is both c1-continuous and c2-continuous. however, t has no fixed point. proof. it was observed in example 2.24 that the constant function with value 0 is a member of ψ. it is easy to see that the assertion follows. � the following is given as theorem 3.3 (and, with another hypothesis, as theorem 3.7) of [24]. theorem 4.12. let (x,d,κ) be a complete digital metric space and let t : x → x be a β − ψ − φ-expansive mapping such that for some sequence {xn} ∞ n=1 ∈ x we have β(xn,xn+1) ≥ 1 for all n and xn → y ∈ x as n → ∞, then there is a subsequence {xnk} of {xn} ∞ n=1 such that β(xnk,y) ≥ 1 for all k. then t has a fixed point. however, we have the following. example 4.13. if x is finite or d is an ℓp metric, and β = d, then theorem 4.12 is vacuously true, as no such sequence {xn} ∞ n=1 ∈ x exists. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 146 remarks on fixed point assertions in digital topology proof. by corollary 2.7, xn → y implies that for some k0, k > k0 implies xn = y and therefore β(xn,y) = d(xn,y) = 0. thus, no sequence {xn} ∞ n=1 satisfies the hypotheses of theorem 4.12. � remark 4.14. the following assertion is stated as theorem 3.6 of [24]. let (x,d,κ) be a complete digital metric space. let t : x → x be a β − ψ − φ-expansive mapping such that (4.3) ψ(d(t(x),t(y))) ≥ β(x,y)ψ(m(x,y)) + φ(m(x,y)) for all x,y ∈ x, where m(x,y) = max{d(x,y),d(x,t(x)),d(y,t(y)), d(x,t(y)) + d(y,t(x)) 2 }. then t has a fixed point. but this assertion is false. proof. consider the choices of x,d,κ,t,β,ψ,φ of example 4.11, where we saw that t is a β −ψ−φ-expansive mapping. since ψ and φ are constant functions with value 0, (4.3) is satisfied. however, as noted at example 4.11, t has no fixed point. � 4.7. remarks on [9]. the publisher of [9] identified one of the authors of the current paper, l. boxer, as a reviewer. in fact, errors and other shortcomings mentioned in boxer’s review remain in the published version of [9]. the assertion stated as theorem 3.1 of [9] is the following. let (x,d,κ) be a complete metric space such that t : x → x satisfies d(t(x),t(y)) ≤ ψ(d(x,y)) for all x,y ∈ x, where ψ : [0,∞) → [0,∞) is monotone nondecreasing and ψn(t) → 0 as n → ∞. then t has a unique fixed point. the argument offered in proof of this assertion confuses topological continuity (the “ε − δ definition”) and digital continuity (preservation of connectedness) in order to conclude that t is continuous. however, in example 4.1, using ψ(t) = t/2, we have a function that satisfies the hypotheses above and is not digitally continuous. further, if we add hypotheses that are often satisfied to theorem 3.1 of [9], then t is forced to be a constant function, as seen in the following. proposition 4.15. let (x,d,cu) be a digital metric space, x ⊂ z n, such that t : x → x is (cu,cu)-continuous and satisfies d(t(x),t(y)) ≤ ψ(d(x,y)) for all x,y ∈ x, where ψ : [0,∞) → [0,∞) is monotone nondecreasing and ψn(t) → 0 as n → ∞. if x is cu-connected, d is an ℓp metric, and ψ(t) < 1/u1/p for all t ∈ [0,∞), then t is a constant function. proof. this follows easily from theorem 4.3. � the argument given in proof of the assertion stated as theorem 3.3 of [9] is similarly flawed. the assertion is the following. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 147 l. boxer and p. c. staecker let (x,d,κ) be a complete digital metric space and t : x → x a weakly uniformly strict digital contraction mapping. then t has a unique fixed point z. moreover, for any x ∈ x, limn→∞t n(x) = z. as above, example 4.1 provides a counterexample to the claim appearing in the “proof” of this assertion that a weakly uniformly strict digital contraction mapping is digitally continuous. therefore, we must regard the assertions stated as theorems 3.1 and 3.3 of [9] as unproven. since these and assertions dependent on these make up all of the new assertions of the paper, we conclude that nothing new is correctly established in [9]. 5. common fixed points of intimate maps the paper [21] obtains a result for common fixed points of intimate maps. we show in this section that the characterization of intimate maps given in [21] can be simplified, and that the primary result of [21] is rather limited. definition 5.1 ([21]). let (x,d,κ) be a digital metric space. let f,g : x → x. let α be either the liminf or the limsup operation. if for every {xn} ∞ n=1 ⊂ x such that (5.1) lim n→∞ f(xn) = lim n→∞ g(xn) = t for some t ∈ x we have, for n sufficiently large, (5.2) αd(g(f(xn)),g(xn)) ≤ αd(f(f(xn)),f(xn)) then we say f is g-intimate. proposition 5.2. let (x,d,κ) be a digital metric space, where d is any ℓp metric. let f,g : x → x. then f is g-intimate if and only if for every sequence {xn} ∞ n=1 ⊂ x satisfying statement (5.1) we have d(g(t), t) ≤ d(f(t), t). proof. from theorem 2.6, a sequence {xn} ∞ n=1 ⊂ x satisfying statement (5.1) has, for n sufficiently large, f(xn) = g(xn) = t. the assertion follows easily. � theorem 5.3 ([21]). if (x,d,κ) is a digital metric space and a,b,s,t : x → x are such that a) s(x) ⊂ b(x) and t(x) ⊂ a(x); b) for some α ∈ (0,1) and all x,y ∈ x, (5.3) d(s(x),t(y)) ≤ αf(x,y) where f(x,y) = max{d(a(x),b(y)),d(a(x),s(x)),d(b(y),t(y)), d(s(x),b(y)),d(a(x),t(y))}; c) a(x) is complete; d) s is a-intimate and t is b-intimate, c© agt, upv, 2019 appl. gen. topol. 20, no. 1 148 remarks on fixed point assertions in digital topology then a, b, s, and t have a unique common fixed point. but theorem 5.3 is limited, as shown by the following. proposition 5.4. suppose we assume the hypotheses of theorem 5.3, with d being an ℓp metric. suppose (5.4) 0 < α < inf{1/f(x,y) |x,y ∈ x} or (5.5) diam(s(x) ∪ t(x)) = diamx < ∞. then s and t are constant functions that have the same value, and, in case (5.5), x is a singleton. proof. inequality (5.4) and hypothesis b) of theorem 5.3 imply that d(s(x),t(y)) < 1, hence s(x) = t(y) for all x,y ∈ x. since d is an ℓp metric, we have, for some x0,y0 ∈ x, s(x0) = t(y) and s(x) = t(y0) for all x,y ∈ x. hence s and t are constant functions with the same values. statement (5.5) implies there exist x0,y0 ∈ x such that d(s(x0),t(y0)) = diamx. since f(x0,y0) ≤ diamx, inequality (5.3) becomes diamx ≤ αdiamx, which implies diamx = 0. thus, x is a singleton, so s and t are constant functions with the same values. � 6. homotopy invariant fixed point theory it is natural in topology to consider the behavior of the fixed point set when a continuous function is changed by homotopy. in classical topology for nice spaces (for example the geometric realization of any finite simplicial complex), when fixed points of a certain function exist, then there is a standard construction to change the function by homotopy to increase the number of fixed points. the more interesting question is whether or not the number of fixed points can be decreased by homotopy. the following proposition 6.1 was the key to the proof of theorem 2.10. proposition 6.1 ([7]). let (x,κ) be a connected digital image of more than one point. let x0 ↔κ x1 in x. then the function g : x → x defined by g(x) = { x0 if x 6= x0; x1 if x = x0, is continuous and has no fixed points. proposition 6.2. let (x,κ) be a connected digital image of more than one point. then any constant map f : x → x is homotopic to a map without fixed points. proof. let x0,x1 ∈ x with x0 ↔κ x1. let f : x → x be the constant map with image {x0}. let h : x × [0,1]z → x be defined by h(x,0) = x0; h(x,1) = x0 for x 6= x0; h(x0,1) = x1. it is easy to see that h is a homotopy from f to a function g as in proposition 6.1 without fixed points. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 149 l. boxer and p. c. staecker let mf(f) be the minimal number of fixed points among all continuous functions homotopic to f. for example, if x has only 1 point then clearly mf(f) = 1. if x has more than 1 point and (x,κ) is contractible, then any continuous function f : x → x is homotopic to a constant function, and it follows from proposition 6.2 that mf(f) = 0. several examples are given in [15] of digital images (x,κ) for which no function on x is homotopic to the identity except for the identity itself. such images are called rigid. for example, a wedge product of two loops, each having at least 5 points, is rigid. clearly if x is a rigid digital image having n points and id denotes the identity function, then mf(id) = |x|. in classical topology, mf(f) can often be computed by nielsen fixed point theory; see [22]. each fixed point is assigned an integer-valued fixed point index, which can be computed homologically. when x is an isolated fixed point of f, the fixed point index of x is denoted ind(f,x). the general definition of the index is complicated, but if the space is a smooth manifold, then f can be smoothed by homotopy so that ind(f,x) = sign(det |i − dfx|) where dfx is the derivative map and i is the identity matrix. this index is a sort of multiplicity count for the fixed point: when ind(f,x) = 0 then the fixed point at x can be removed by a homotopy. the fixed point index is homotopy invariant in the following sense: if, during some homotopy f ≃ g, the fixed point x of f moves into a fixed point y of g, then ind(f,x) = ind(g,y). furthermore, when the fixed point set of f is finite, then the sum of all the fixed point indices equals the lefschetz number l(f), which is the alternating sum of traces of the induced maps of f in the homology groups: l(f) = ∞ ∑ i=1 (−1)i trace(f∗i : hi(x,q) → hi(x,q)). since ind(f,x) sums to l(f), it is often said that ind(f,x) is localized version of the lefschetz number. in nielsen fixed point theory, the fixed points are grouped into nielsen classes, and the number of such classes having nonzero index sum is the nielsen number n(f). this number is a homotopy invariant satisfying n(f) ≤ mf(f), and in many cases (for example when x is a manifold of dimension different from 2), n(f) = mf(f). the 2012 paper [11] by ege & karaca attempts to develop a lefschetz fixed point theorem for digital images, but the main result is incorrect, and was retracted in the 2016 paper [7]. the same authors attempted to develop a nielsen theory in the 2017 paper [14] based on their faulty lefschetz theory. the theory developed in [14] is also incorrect. the main problem in [14] is inherited from problems in [11], and concerns the definition of the fixed point index. definition 3.2 of [14] states the following. let (x,κ) be a digital image, a ⊂ x, and f : a → x a digital map. we define the fixed point index of f as ind(f) = deg(f) where f(x) = x − f(x) and x ∈ x. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 150 remarks on fixed point assertions in digital topology this does not give a satisfactory definition of the function f . it seems motivated by the classical fact that ind(f,x) = sign(det |i − dfx|), but the domain and range of f are never specified. the subtraction x − f(x) is apparently performed in x, but x − f(x) need not be a member of x. furthermore, the degree deg(f) used is inadequate because the appropriate homology groups, as defined earlier in [14], are not necessarily isomorphic to z, which is a requirement for the definition of the degree of a function. ege & karaca claim to define an integer valued nielsen number n(f) which is a homotopy invariant (theorem 3.6) and a lower bound for mf(f) (theorem 3.7). their example 3.4, claiming that if f is a constant then n(f) = 1, yields a contradiction for connected images x such that |x| > 1, since our proposition 6.2 implies that mf(f) = 0 for such functions f. the errors in this work are not merely mistakes but indicate fundamental flaws in the theory. anything resembling the standard homological definitions of l(f) and the fixed point index will require that the lefschetz number and fixed point index of the constant map equal 1. this cannot be reconciled with the fact that, when x has more than 1 point, the constant map can be changed by homotopy to have no fixed points. the authors believe that any successful theory for computing mf(f) will involve techniques very different from classical lefschetz and nielsen theory. the setting of digital images also allows the study of the quantity xf(f), the maximum number of fixed points among all functions homotopic to f. in classical topological fixed point theory this number is typically infinite, but for a digital image with n points, clearly xf(f) ≤ n for any f. in fact our definition of xf(f) implies that f is homotopic to the identity if and only if xf(f) = n. when f is a constant, then 1 ≤ xf(f) ≤ n, and for many choices of the image x we will have xf(f) < n. we do not know if it is possible for xf(f) = 0 for any function on a connected digital image. although many of the concepts discussed in this paper turn out to be trivial or otherwise uninteresting, the questions of computing mf(f) and xf(f) seem to be difficult and interesting, and present opportunities for further work. variations that count approximate fixed points would also be interesting objects of study. 7. concluding remarks although the study of fixed points, or approximate fixed points, is important in digital topology as in other branches of mathematics, it does not appear that the use of metric spaces yields useful knowledge in this area. we have seen that metric space functions introduced to study fixed points in digital topology digital contraction maps, kannan contraction maps, chatterjea contraction maps, zamfirescu contraction maps, rhoades contraction maps, reich contraction maps, uniformly locally contractive functions, intimate functions often turn out to be either discontinuous or constant hence, arguably uninteresting when the image considered is finite or when common metrics are used. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 151 l. boxer and p. c. staecker it appears to us that the most natural metric function to use for a connected digital image (x,κ) is the path length metric [17]: d(x,y) is the length of a shortest κ-path from x to y. since this metric reflects κ, it seems far superior to an ℓp metric on a digital image. however, even this metric gives us little new information. since it is integer-valued, its cauchy sequences are also eventually constant. we have also corrected errors and pointed out trivialities in other papers concerned with fixed points or approximate fixed points of continuous self-maps of digital images. acknowledgements. the authors are grateful to the anonymous reviewer for many suggestions and corrections. references [1] s. banach, sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, fundamenta mathematicae 3 (1922), 133–181. 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[32] t. zamfirescu, fixed point theorems in metric spaces, archiv der mathematik 23 (1972), 292–298. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 153 @ appl. gen. topol. 22, no. 1 (2021), 199-221doi:10.4995/agt.2021.14915 © agt, upv, 2021 on the menger and almost menger properties in locales tilahun bayih, themba dube and oghenetega ighedo department of mathematical sciences, university of south africa, p.o. box 392, 0003 pretoria, south africa (tileesa24@gmail.com,dubeta@unisa.ac.za,ighedo@unisa.ac.za) communicated by f. mynard abstract the menger and the almost menger properties are extended to locales. regarding the former, the extension is conservative (meaning that a space is menger if and only if it is menger as a locale), and the latter is conservative for sober td-spaces. non-spatial menger (and hence almost menger) locales do exist, so that the extensions genuinely transcend the topological notions. we also consider projectively menger locales, and show that, as in spaces, a locale is menger precisely when it is lindelöf and projectively menger. transference of these properties along localic maps (via direct image or pullback) is considered. 2010 msc: 06d22; 54c05; 54d20. keywords: menger; almost menger; frame; locale; sublocale; spectrum of a frame. introduction recall that a topological space x is menger if for every sequence (un)n∈n of open covers of x we can select, for each n, a finite vn ⊆ un such that ⋃ n∈nvn is a cover of x. this definition is purely in terms of the lattice of open subsets, and can thus be extended to frames almost verbatim. that is exactly what we do. it then turns that the extension of the menger property to frames is conservative. on the other hand, a topological space is called almost menger if for every sequence (un)n∈n of open covers of x we can select, for each n, a finite vn ⊆ un received 11 january 2021 – accepted 11 february 2021 http://dx.doi.org/10.4995/agt.2021.14915 t. bayih, t. dube and o. ighedo such that ⋃ {v | v ∈ ⋃ n∈nvn} = x. although this definition is not solely in terms of the lattice of open sets (because of the appearance of closures), it can be adapted to frames by working within the lattice of sublocales, with the union replaced by the join. that is precisely what we do to define almost menger frames. our aim in this paper is to initiate the study of the menger-type properties in pointfree topology. some of the results we obtain not only extend the known topological ones to frames, but also sharpen the topological ones. there are various weaker forms of the menger property in spaces, but we restrict ourselves to extensions of the menger property and the almost menger property. here is a brief overview of the paper. since the theory of frames and locales has by now come of age, the preliminaries in section 1 are written tersely; the main purpose being just to fix notation and recall the concepts that are used most throughout the paper. in section 2 we study some properties of menger frames. we start by observing that (as already been mentioned) a topological space x is menger if and only if the frame ω(x) is menger, and that non-spatial menger frames do exist, so that our extension to frames of this property is a genuine extension covering more objects than topologies of menger spaces. since the contravariant functor ω: top → frm preserves and reflects the menger property, one may ask about its right adjoint σ: frm → top. a frame whose spatial reflection is a codense sublocale is menger if and only if its spectrum is menger (proposition 2.6). it is perhaps worth underscoring that a frame whose spatial reflection is a codense sublocale is not necessarily spatial. defining a frame to be projectively menger if every subframe with a countable base is menger, we have that a frame is menger precisely when it is lindelöf and projectively menger (corollary 2.14). a completely regular normal countably paracompact frame is projectively menger if and only if its lindelöf coreflection is menger (corollary 2.16). in section 3 we consider almost menger frames. our definition, adapted from spaces as indicated above, turns out to be conservative for sober td-spaces (theorem 3.3). although our definition invokes the lattice of sublocales, we have a characterisation (proposition 3.6) solely in terms of elements. 1. preliminaries we assume familiarity with frames and locales. our references are [12] and [15]. in this section we recall just a few of the concepts that we shall need. our notation is standard, and is, by and large, that of our references. 1.1. frames and spatiality. throughout this section, l denotes a frame. we denote by ω(x) the frame of open subsets of a topological space x. an element p ∈ l is called a point (or a prime) if it satisfies the property that p < 1 and (∀x, y ∈ l)(x ∧ y ≤ p =⇒ x ≤ p or y ≤ p). © agt, upv, 2021 appl. gen. topol. 22, no. 1 200 on the menger and almost menger properties in locales we write pt(l) for the set of points of l. a frame is spatial if it is isomorphic to ω(x) for some space x. this is the case precisely when every element is a meet of primes. we view the spectrum of l as the topological space whose underlying set is pt(l) with the topology ω(σl) = {σa | a ∈ l} where, for each a ∈ l, σa = {p ∈ pt(l) | a � p}. the map ηl : l → ω(σl) given by ηl(a) = σa is an onto frame homomorphism, and is the reflection map from l to spatial frames. as usual, we shall write ≺ and ≺≺, respectively, for the rather below and the completely below relations, and recall that l is called regular (resp. completely regular) if every element of l is the join of the elements that are rather below (resp. completely below) it. 1.2. sublocales and localic maps. the lattice of sublocales of l, ordered by inclusion, is a coframe denoted by s(l). for later use, we recall that joins in s(l) are given by ∨ i∈i si = {∧ m | m ⊆ ⋃ i∈i si } . since s(l) is a coframe, when turned upside down it is a frame, denoted s(l)op, whose top element is the void sublocale o = {1}. a sublocale of l is called a one-point sublocale if it is of the form {p, 1} for some p ∈ pt(l). spatial frames are precisely those that are joins of their one-point sublocales. the open sublocale associated with a ∈ l is denoted by ol(a), and the closed one by cl(a). we shall drop the subscript if no confusion may result from that. the closure of a sublocale s of l, denoted s or cll s, is the sublocale s = cl (∧ s ) . in particular, ol(a) = cl(a ∗). a sublocale s of l is dense if s = l. if s and t are sublocales of l and s ⊆ t , then s is a sublocale of t . the closure of s in t will be denoted by clt s, and s (unadorned) will be understood to be the closure in l. a localic map f : l → m gives rise to two maps f[−]: s(l) → s(m) and f−1[−]: s(m) → s(l) given by f[s] = {f(x) | x ∈ s} and f−1[t ] = ∨ {a ∈ s(l) | a ⊆ f−1[t ]}. the map f[−] preserves all joins and f−1[−] preserves all meets (recall that they are intersections) and all binary joins. for any s ∈ s(l) and t ∈ s(m), f[s] ⊆ t ⇐⇒ s ⊆ f−1[t ]. writing h for the left adjoint of f, we have that, for any b ∈ m, f−1[om(b)] = ol(h(b)) and f−1[cm (b)] = cl(h(b)). © agt, upv, 2021 appl. gen. topol. 22, no. 1 201 t. bayih, t. dube and o. ighedo this then shows that the map f−1[−] also preserves arbitrary joins of open sublocales. for, if {bi | i ∈ i} ⊆ m, then f−1 [∨ i∈i om(bi) ] = f−1 [ om (∨ i∈i bi )] = ol ( h (∨ i∈i bi )) = ol (∨ i∈i h(bi) ) = ∨ i∈i ol ( h(bi) ) = ∨ i∈i f−1[om (bi)]. 1.3. covers and coverings. by a cover of l we mean a set c ⊆ l such that∨ c = 1. on the other hand, to avoid possible confusion, we say a collection c of sublocales of l is a covering of l if ∨ {c | c ∈ c } = l, where the join is calculated in s(l). this terminology is not standard. a cover consists of elements of l, whereas a covering consists of sublocales of l. if every sublocale in a covering c of l is open, then c is an open covering of l. there is a bijection between covers and open coverings given by c 7→ c c = {ol(c) | c ∈ c} and c 7→ c c = {x ∈ l | ol(x) ∈ c }. a cover c of l is said to refine a cover d if for every c ∈ c there is a d ∈ d such that c ≤ d. in this case, c is called a refinement of d. 2. menger locales we aim to define menger locales in such a way that a space x is menger precisely when the frame ω(x) is menger. our definition will be localic, and we will then cast it in frame terms, which will enable us to show easier that the definition is conservative. throughout, every sequence is indexed by n. definition 2.1. a frame l is menger if for every sequence (cn) of open coverings of l, there exists, for each n, a finite dn ⊆ cn such that ⋃ n∈ndn is a covering of l. in this case, we say the sequence (dn) is a menger witness for (cn). from the bijection between covers and coverings, this definition could equivalently have been stated in terms of covers. the reason is that if c is a cover of l and d is a finite subset of c, then, in the notation of subsection 1.3, c d is a finite subset of c c. conversely, if c is an open covering of l and d is a finite subset of c , then cd is a finite subset of cc because the mapping u 7→ ol(u) is one-one. proposition 2.2. a frame l is menger iff for every sequence (cn) of covers of l, there exists, for each n, a finite dn ⊆ cn such that ⋃ n∈ndn is a cover of l. as with coverings, we shall say such a sequence (dn) is a menger witness for the sequence (cn). this proposition makes it most apparent that every menger frame is lindelöf, and every compact frame (in fact, every σ-compact one – meaning one that is a join of countably many compact sublocales) is menger. since there are non-spatial compact frames (see [12, p. 89]), it follows that: © agt, upv, 2021 appl. gen. topol. 22, no. 1 202 on the menger and almost menger properties in locales a menger frame need not be spatial. since every cover of a subframe is a cover of the ambient frame, we deduce that: every subframe of a menger frame is menger. thus, a localic image of a menger frame is menger. since a collection of open subsets of a space x is a cover of the frame ω(x) if and only if it is an open cover of the space x, we deduce the following from proposition 2.2. corollary 2.3. a topological space x is menger iff ω(x) is menger. recall that the sobrification of a topological space is the spectrum of its frame of open sets. since a space and its sobrification have isomorphic frames of open sets, we have the following result. corollary 2.4. a topological space is menger iff its sobrification is menger. in light of the dual adjunction top ω // frm σ oo and the result in corollary 2.3, one may ask if it is the case that a frame is menger if and only if its spectrum is menger. we address this for some types of frames. as is well known, a frame l is spatial if and only if the frame homomorphism ηl : l → ω(σl) is one-one. we will show that for frames l for which ηl is codense (meaning that ηl(a) = 1ω(σl) implies a = 1l) the spectrum analogue of corollary 2.3 holds. we reiterate that such frames need not be spatial, as the following example shows. example 2.5. let l be a frame with no points, such as the smallest dense sublocale of ω(r). let l̃ be the frame obtained from l by adjoining a new top element 1l̃ > 1l. then l̃ is not spatial and pt(l̃) = {1l}. from the latter, it is not hard to see that ηl̃ is codense. proposition 2.6. a frame whose spatial reflection is a codense sublocale is menger iff its spectrum is menger. proof. let l be such a frame. for any a ⊆ l we set σa = {σa | a ∈ a}. since σ∨ i∈i ai = ⋃ i∈iσai for any collection {ai | i ∈ i} of elements of l, it follows that σc is an open cover of σl whenever c is a cover of l. on the other hand, the part of the hypothesis that says ηl is codense ensures that every open cover of σl is of the form σc for some cover c of l. now assume that σl is menger. we apply proposition 2.2 to show that l is menger. let (cn) be a sequence of covers of l. then (σcn) is a sequence of open covers of σl. so for each n there exists a finite dn ⊆ cn such that⋃ n∈nσdn = σl. since ⋃ n∈nσdn = σ ⋃ n∈n dn, we deduce from the codensity part of the hypothesis that ⋃ n∈ndn is a cover of l. therefore l is menger. the converse is shown similarly. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 203 t. bayih, t. dube and o. ighedo when working with covers or coverings, it is at times convenient to deal with directed ones. when we say a subset of a poset is directed, we mean that it is up-directed. if c is a cover of l and c is an open covering of l, we set c× = {∨ f | f is a finite subset of c } and c × = {∨ f | f is a finite subset of c } , and observe that c× is a directed cover of l and c × is directed (open) covering of l. proposition 2.7. the following are equivalent for a frame l. (1) l is menger. (2) for every sequence (cn) of directed covers of l, there exists, for each n, an element cn ∈ cn such that {cn | n ∈ n} is a cover of l. (3) for every sequence (un) of directed open coverings of l, there exists, for each n, a sublocale un ∈ un such that {un | n ∈ n} is a covering of l. proof. (2) ⇔ (3): this equivalence comes from the bijection between covers and open coverings, together with the observation that this bijection induces a bijection between directed covers and directed open coverings. (1) ⇔ (2): assume that l is menger, and let (cn) be a sequence of directed covers of l. let (dn) be a menger witness for (cn). since each dn is a finite subset of cn and cn is directed, there exists an element cn ∈ cn such that∨ dn ≤ cn. therefore the cover ⋃ n∈ndn refines {cn | n ∈ n}, showing that the latter is a cover of l. conversely, let (cn) be a sequence of covers of l, and consider the sequence (c×n ) of directed covers of l. the current hypothesis furnishes, for each n, a finite bn ⊆ cn such that the set { ∨ bn | n ∈ n} is a cover of l. clearly, this makes the sequence (bn) a menger witness for (cn), and hence l is a menger frame. � now, let us observe that the menger property is preserved under finite joins. in the proof we shall use the fact that families of open sublocales are distributive in the coframe of sublocales, that is, if (uα)α∈a is a family of open sublocales of l and t is any sublocale of l, then t ∩ ∨ α∈a uα = ∨ α∈a (t ∩ uα). recall that open sublocales of a sublocale a of a frame l are precisely the intersections with a of the open sublocales of l. proposition 2.8. the join of finitely many menger sublocales of a given frame is menger. proof. let us first show that a sublocale a of l is menger if and only if whenever (un) is a sequence of families of open sublocales of l such that a ⊆ ∨ un for © agt, upv, 2021 appl. gen. topol. 22, no. 1 204 on the menger and almost menger properties in locales every n, then there exists, for each n, a finite vn ⊆ un such that a ⊆ ∨⋃ n∈nvn. to see this, let us introduce the ad hoc notation that if f is a family of sublocales of l, we write a ∩ f = {a ∩ f | f ∈ f }. now, for the “if” part, the equality a = ∨ {a ∩ u | u ∈ un} implies that a ∩ un is an open covering of a for each n. since a is menger, we can find, for each n, a finite vn ⊆ un such that a ∩ ⋃ n∈nvn is a covering of a. thus, a = ∨{ a ∩ s | s ∈ ⋃ n∈n vn } = a ∩ ∨{ s | s ∈ ⋃ n∈n vn } , which implies a ⊆ ∨⋃ n∈nvn, as desired. the “only if” part is proved similarly, taking into account the fact every open covering of a is of the form a ∩ u , where u is a family of open sublocales of l with a ⊆ ∨ u . now let a and b be menger sublocales of l, and consider a sequence (cn) of families of open sublocales of l with a ∨ b ⊆ ∨ cn for each n. then a ⊆ ∨ cn and b ⊆ ∨ cn for each n. so we can find a finite d a n ⊆ cn and a finite d b n ⊆ cn such that a ⊆ ∨⋃ n∈n d a n and b ⊆ ∨⋃ n∈n d b n . consequently, for each n, dan ∪ d b n is a finite subset of cn such that a ∨ b ⊆ ∨{ s | s ∈ dan ∪ d b n } , which proves that a ∨ b is menger. the general case follows by induction because the binary join is an associative operation on s(l). � the sublocales that inherit the menger property include the closed ones because for any frame l and a ∈ l, a cover of c(a) is a cover of l. in fact, we have a stronger result. recall that a frame homomorphism is called perfect if its right adjoint preserves directed joins. in [8], a frame homomorphism is called weakly perfect if its right adjoint preserves directed covers. perfect homomorphisms are weakly perfect. weak perfectness is strictly weaker than perfectness. indeed, as observed in [7, example 3.11], if l is a compact frame which is not boolean, then the right adjoint of the join map jl → l (where jl denotes the frame of ideals of l) takes covers to covers (and hence is weakly perfect), but it is not perfect. on the other hand, weak perfectness does not imply that the right adjoint takes covers to covers. a counterexample (also sourced from [7]) is the embedding of the two-element chain in the four-element boolean algebra. we can summarise these “pictorially” using the acronyms that a frame homomorphism satisfies: • (dj) if its right adjoint preserves directed joins; • (dc) if its right adjoint sends directed covers to covers; and • (cc) if its right adjoint sends covers to covers. © agt, upv, 2021 appl. gen. topol. 22, no. 1 205 t. bayih, t. dube and o. ighedo then (dj) 6=⇒ (cc) 6=⇒ (dj); (cc) =⇒ (dc) 6=⇒ (cc) and (dj) =⇒ (dc) 6=⇒ (dj). thus, weak perfectness is the least restricted of these properties of homomorphisms. so a property of frames that is preserved or reflected by weakly perfect homomorphisms is also done so by the other types of homomorphisms. in [6], a cover b of a frame l is called a strong refinement of a cover c if for every b ∈ b there is a c ∈ c such that b ≺ c. then l is called cover regular if every cover of l has a strong refinement. regular frames are cover regular. any finite chain with more than two elements is a cover regular frame which is not regular. note that if a directed cover has a strong refinement, then it has a strong refinement which is directed because whenever bi ≺ ci for each i ∈ {1, . . . , n}, with n ∈ n, then (b1 ∨ · · · ∨ bn) ≺ (c1 ∨ · · · ∨ cn). recall that a frame homomorphism h: l → m is called dense if the zero of its domain is the only element mapped to the zero of its codomain. this is precisely when h∗(0) = 0, where h∗ denotes the right adjoint of h. it is well known that if h is a dense frame homomorphism, then h∗(h(x)) ≤ a whenever x ≺ a in the domain of h. corollary 2.9. let h: l → m be a frame homomorphism. (a) if h is weakly perfect and l is menger, then m is menger. (b) if h is dense and weakly perfect, l is cover regular, and m is menger, then l is menger. proof. (a) let (cn) be a sequence of directed covers of m. since h is weakly perfect, h∗[cn] is a directed cover of l for each n. since l is menger, by proposition 2.7 there exists, for each n, an element cn ∈ cn such that {h∗(cn) | n ∈ n} is a cover of l. therefore {hh∗(cn) | n ∈ n} is a cover of m, and hence {cn | n ∈ n} is a cover of m because hh∗(x) ≤ x for every x ∈ m. proposition 2.7 again shows that m is menger. (b) let (cn) be a sequence of directed covers of l, and, for each n, let bn be a directed strong refinement of cn. then h[bn] is a directed cover of m for each n. since m is menger, proposition 2.7 enables us to find, for each n ∈ n, an element bn ∈ bn such that {h(bn) | n ∈ n} is a cover of m. since h∗ takes directed covers to covers, the set c = {h∗h(bn) | n ∈ n} is a cover of l. since bn is a strong refinement of cn, for each n, there exists some cn ∈ cn such that bn ≺ cn, and since h is dense, we have h∗h(bn) ≤ cn. so the cover c refines {cn | n ∈ n}, whence we deduce that l is menger. � part (a) of this corollary enables us to say a word about coproducts. we do not need to recall the construction of coproducts. corollary 2.10. if l is compact and m is menger, then l ⊕ m is menger. proof. it is shown in [11, lemma 2] that if l is compact, then the coproduct injection m → l ⊕ m is a perfect map (actually, it is a proper map – we © agt, upv, 2021 appl. gen. topol. 22, no. 1 206 on the menger and almost menger properties in locales will recall the definition later). since any perfect homomorphism is weakly perfect, the result follows from corollary 2.9(a) because compact frames are menger. � we recall that a topological space x is called “projectively menger” if every continuous second countable image of x is menger. in [4], it is shown that a tychonoff space is menger if and only if it is lindelöf and projectively menger. we extend this result to frames without restricting to completely regular ones. let us first recall some frame-theoretic notions. a subset s of a frame l is said to be a generating set if l is the smallest (under inclusion) subframe containing s. this is the case precisely when each a ∈ l is of the form a = ∨ {x ∈ l | x is the meet of some finite f ⊆ s}. a base of a frame l is a subset b with the property that every element of l is a join of some elements of b. definition 2.11. a frame l is projectively menger if every subframe of l with a countable base is menger. this definition can clearly be rephrased to say l is projectively menger in case whenever h: m → l is a one-one frame homomorphism and m has a countable base, then m is menger. let us record the following easy observation, which should certainly be known, but for which we provide a proof as we do not have a reference. lemma 2.12. every frame with a countable base is lindelöf. proof. let b be a countable base of a frame l. let c be a cover of l. for each c ∈ c, let b(c) be a subset of b such that c = ∨ b(c). then ⋃ c∈cb (c) is a countable cover of l refining c. therefore c has a countable subcover. � in what follows, we say a countable cover c = {c1, c2, . . . } is increasing if cn ≤ cn+1 for each n. this is standard terminology. of course, an increasing cover is directed. for any given countable cover c = {c1, c2, . . . }, we denote by c+ = {c1, c1 ∨ c2, c1 ∨ c2 ∨ c3, . . . } the increasing cover constructed from c as indicated. proposition 2.13. the following are equivalent for a frame l. (1) l is projectively menger. (2) every lindelöf subframe of l is menger. (3) for every sequence (cn) of countable covers of l, there exists, for each n, a finite dn ⊆ cn such that ⋃ n∈ndn is a cover of l. (4) for every sequence (cn) of increasing countable covers of l, there exists, for each n, an element cn ∈ cn such that {cn | n ∈ n} is a cover of l. (5) for every sequence (cn) of countable open coverings of l, there exists, for each n, a finite dn ⊆ cn such that ⋃ n∈ndn is a covering of l. © agt, upv, 2021 appl. gen. topol. 22, no. 1 207 t. bayih, t. dube and o. ighedo proof. (1) ⇒ (4): assume that l is projectively menger, and let (cn) be a sequence of increasing countable covers of l. put c = ⋃ n∈ncn. then c is a countable subset of l. let m be the subframe of l generated by c. then the set cf whose elements are the meets of finite subsets of c is a base for m. since c is countable, cf is countable, and so m has a countable base. therefore m is menger, by hypothesis. since each cn ⊆ m, (cn) is a sequence of increasing covers of m, so for each n there exists some cn ∈ cn such that {cn | n ∈ n} is a cover of m, and hence of l. therefore (1) implies (4). (4) ⇒ (3): assume that condition (4) holds, and let (cn) be a sequence of countable covers of l. consider the sequence (c+n ) of increasing covers of l. by (4), there exists, for each n, some dn ∈ c + n such that {dn | n ∈ n} is a cover of l. since each dn is a join of some finite dn ⊆ cn, ⋃ n∈ndn is a cover of l. (3) ⇒ (2): assume condition (3), and let m be a lindelöf subframe of l. let (un) be a sequence of covers of m. since m is lindelöf, we can find, for each n, a countable cn ⊆ un such that cn is a cover of m. then of course cn is a cover of l, and so, by (3), there exists a finite dn ⊆ cn such that⋃ n∈ndn is a cover of l, and hence of m. thus, m is menger, and therefore l is projectively menger. (2) ⇒ (1): this follows from the fact that every frame with a countable base is lindelöf. (4) ⇔ (5): this follows from the bijection between covers and open coverings. � since every menger frame is lindelöf and every subframe of a menger frame is menger, we deduce the following characterisation of menger frames from this proposition. corollary 2.14. a frame is menger iff it is lindelöf and projectively menger. we now turn to some subclass of completely regular frames. recall from [6] that a frame l is countably paracompact if for every countable cover c of l, there is a cover w of l such that each element of w misses all but a finite number of elements of c. let us also recall that a countable cover d = {dn | n ∈ n} is called a shrinking of a countable cover c = {cn | n ∈ n} if dn ≺ cn for every n ∈ n. by a cozero cover of a frame we mean a cover consisting entirely of cozero elements. theorem 2.15. the following are equivalent for a normal countably paracompact completely regular frame l. (1) l is projectively menger. (2) for every sequence (cn) of countable cozero covers of l, there exists, for each n, a finite dn ⊆ cn such that ⋃ n∈ndn is a cover of l. (3) for every sequence (cn) of increasing countable cozero covers of l, there exists, for each n, an element cn ∈ cn such that {cn | n ∈ n} is a cover of l. © agt, upv, 2021 appl. gen. topol. 22, no. 1 208 on the menger and almost menger properties in locales proof. the implication (1) ⇒ (2) follows from proposition 2.13, and the implication (2) ⇒ (3) is almost immediate. (3) ⇒ (1): let (un) be a sequence of increasing countable covers of l. now fix n ∈ n, and write un = {un1, un2, . . . } with un1 ≤ un2 ≤ · · · . since l is countably paracompact, un has a shrinking [5, corollary to proposition 1]. since x ∨ y ≺ a whenever x ≺ a and y ≺ a, and since x ≤ a ≺ b implies x ≺ b, the cover un actually has an increasing shrinking vn = {vn1, vn2, . . . } with vnk ≺ unk for each k ∈ n. by normality, vnk ≺≺ unk for each k, and hence, by [2, corollary 1], there is a cozero element znk such that vnk ≺≺ znk ≺≺ unk. since the join of finitely many (actually, countably many) cozero elements is a cozero element, the sequence (znk)k∈n can be chosen so that it is increasing. thus, if for each n ∈ n we let zn be the set zn = {znk | k ∈ n}, then (zn) is a sequence of countable increasing cozero covers of l, and so by (3) there exists, for each n, an element zn ∈ zn such that {zn | n ∈ n} is a cover of l. since each zn refines un, there exists, for each n, an element un ∈ un such that {un | n ∈ n} is cover of l. therefore l is projectively menger by proposition 2.13. � let l be a completely regular frame and λl be its lindelöf coreflection (see [14] for details). as shown in [1, corollary 8.2.13], if we let h: λl → l be the coreflection map to l from completely regular lindelöf frames, then for any countable cozero cover c of l, h∗[c] is a (countable) cozero cover of λl. since, as is well known, coz(λl) = {h∗(c) | c ∈ coz l}, the countable cozero covers of λl are precisely the covers h∗[c], for c a countable cozero cover of l. furthermore, h∗[c] is increasing if and only if c is increasing. in all, this yields the following corollary. corollary 2.16. a normal countably paracompact completely regular frame l is projectively menger iff λl is menger. proof. suppose that l is projectively menger, and denote by h: λl → l the coreflection map to l from completely regular lindelöf frames. let (h∗[cn]) be a sequence of increasing countable cozero covers of λl. then (cn) is a sequence of increasing countable cozero covers of l. by theorem 2.15, there exists, for each n, an element cn ∈ cn such that {cn | n ∈ n} is a cover of l. thus, for each n, there exists dn ∈ h∗[cn] such that {dn | n ∈ n} is a cover of λl. therefore λl is projectively menger by theorem 2.15, and hence it is menger by corollary 2.14 because it is lindelöf. the converse is proved similarly. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 209 t. bayih, t. dube and o. ighedo 3. almost menger frames apart from the projective menger property that we discussed in the previous section (solely for purposes of characterising menger frames), we shall also consider a weaker form of the menger property in frames. as with the case of the menger property, we will define it by a condition lifted straight from spaces, mutatis mutandis. we will show that the localic extension is conservative for sober td-spaces. recall that a space is sober if it is a t0-space and the complements of the closures of its singletons are exactly its meet-irreducible open sets. that is, a space x is sober if and only if it is a t0-space and pt(ω(x)) = {x r {x} | x ∈ x}. on the other hand, x is a td-space if each x ∈ x has an open neighbourhood u such that u r {x} is open. hausdorff spaces are sober td-spaces. let us recall that a space x is called almost menger if for every sequence (cn) of open covers of x, there exists, for each n, a finite dn ⊆ cn such that⋃{ d | d ∈ ⋃ n∈ndn } = x. definition 3.1. a frame l is almost menger if for every sequence (cn) of open coverings of l, there exists, for each n, a finite dn ⊆ cn such that∨ {d | d ∈ ⋃ n∈ndn } = l. in this case, we say the sequence (dn) is an almost menger witness for the sequence (cn). it is clear that every menger frame is almost menger. before investigating some of the properties of almost menger frames, we show that, among sober td-spaces, the definition of almost menger frames is conservative. we need some background. recall (from [16], for instance) that a prime element p of a frame l is said to be a covered prime if whenever p = ∧ s for some s ⊆ l, then p ∈ s. since for any sober space x the prime elements of ω(x) are precisely the open sets x r {x}, for x ∈ x, we deduce from [16, proposition 1.6.2] that if x is a sober td-space, then all its prime elements are covered. we recalled in the preliminaries that l is spatial iff l = ∨{ {p, 1} | p ∈ pt(l) } . if x is sober, then the one-point sublocales of ω(x) are exactly the sublocales {x r {x}, 1ω(x)}, with x ∈ x. hence, if x is sober, then ω(x) = ∨{ {x r {x}, 1ω(x)} | x ∈ x } . in fact, this equality holds without sobriety, as observed in [17]. the argument goes as follows. if x is any topological space and u ⊆ x is open, then u =⋂{ x r {w} | w /∈ u } , so that, calculating in ω(x) and s(ω(x)), u = int ( ⋂ w /∈u ( x r {w} ) ) = ∧ w /∈u ( x r {w} ) ∈ ∨{ {x r {x}, 1ω(x)} | x ∈ x } . © agt, upv, 2021 appl. gen. topol. 22, no. 1 210 on the menger and almost menger properties in locales lemma 3.2. let {uα | α ∈ a} be a family of open subsets of a topological space x. (a) if ⋃ α∈auα = x, then ∨{ oω(x)(uα) | α ∈ a } = ω(x). (b) if x is a sober td-space and ∨{ oω(x)(uα) | α ∈ a } = ω(x), then⋃ α∈auα = x. proof. (a) note that oω(x)(uα) = cω(x)(u ∗ α) = cω(x)(x r uα). now, given p ∈ x, since ⋃ α∈auα = x, there is an index γ ∈ a such that p ∈ uγ, which implies {p} ⊆ uγ, and hence x r uγ ⊆ x r {p}. consequently, the element x r{p} of the frame ω(x) belongs to the sublocale cω(x)(x ruγ), whence we deduce that {x r {p}, 1ω(x)} ⊆ cω(x)(x r uγ) and therefore, ω(x) = ∨{ {x r {x}, 1ω(x)} | x ∈ x } ⊆ ∨{ cω(x)(x r uα) | α ∈ a } = ∨{ oω(x)(uα) | α ∈ a } ⊆ ω(x), which proves the claimed equality. (b) assume that x is a sober td-space and ∨{ oω(x)(uα) | α ∈ a } = ω(x). then, for any x ∈ x, the element x r{x} of ω(x) belongs to this join, so that x r {x} ∈ ∨{ cω(x)(x r uα) | α ∈ a } . by the way joins of sublocales are computed, for each α there is an open subset vα of x such that x r uα ⊆ vα and x r {x} = ∧ αvα. since primes are covered here, there is an index γ ∈ a such that x r {x} = vγ. therefore x ∈ x r vγ ⊆ uγ, which then shows that ⋃ α∈auα = x. � theorem 3.3. let x be a topological space. (a) if x is almost menger, then ω(x) is almost menger. (b) if x is a sober td-space, then ω(x) is almost menger iff x is almost menger. proof. (a) let (cn) be a sequence of open coverings of ω(x). for each n, there is an open cover un of x such that cn = {oω(x)(u) | u ∈ un}. now, since x is almost menger, there exists, for each n, a finite vn ⊆ un such that⋃ {v | v ∈ ⋃ n∈nvn} = x. for each n, put dn = {oω(x)(v ) | v ∈ vn}. © agt, upv, 2021 appl. gen. topol. 22, no. 1 211 t. bayih, t. dube and o. ighedo we will show that (dn) is an almost menger witness for (cn). clearly, each dn is finite and dn ⊆ cn. let us check the final condition. since ⋃ {v | v ∈⋃ n∈nvn} = x, we deduce from lemma 3.2(a) that ω(x) = ∨{ oω(x)(v ) | v ∈ ⋃ n∈n vn } = ∨{ d | d ∈ ⋃ n∈n dn } , which shows that the final condition to make (dn) an almost menger witness for (cn) is satisfied. therefore ω(x) is almost menger. (b) assume that x is a sober td-space and the frame ω(x) is almost menger. let (un) be a sequence of open covers of x. for each n ∈ n, put u ′ n = {oω(x)(u) | u ∈ un}, and observe that u ′n is an open covering of ω(x); and so we have a sequence (u ′n) of open coverings of the almost menger frame ω(x). in accordance with the definition, for each n, there is a finite v ′n ⊆ u ′ n such that the sequence (v ′ n) is an almost menger witness for (u ′n), and hence (†) ∨{ s | s ∈ ⋃ n∈n v ′ n } = ω(x). for each n, put vn = {v ∈ ω(x) | oω(x)(v ) ∈ v ′ n}. since the mapping oω(x) : ω(x) → s(ω(x)) is injective and v ′ n is a finite set, it follows that vn is a finite set. furthermore, vn ⊆ un because if v ∈ vn, then oω(x)(v ) ∈ v ′ n ⊆ u ′ n, so that oω(x)(v ) = oω(x)(u) for some u ∈ un, whence v = u. observe that, for any sublocale t of l, t ∈ ⋃ n∈n v ′ n ⇐⇒ t = oω(x)(v ) for some v ∈ ⋃ n∈n vn, and so, by lemma 3.2(b), the equality in (†) implies ⋃{ v | v ∈ ⋃ n∈n vn } = x, which then shows that x is almost menger. � next, we show that in the definition of almost menger frames, “open coverings” can be replaced with “directed open coverings” without violating the concept. in the proof we are going to use the fact that s1 ∨ · · · ∨ sn = s1 ∨ · · · ∨ sn for any collection of finitely many sublocales [15, proposition iii.8.1]. proposition 3.4. a frame l is almost menger iff for every sequence (cn) of directed open coverings of l, there exists, for each n, a sublocale cn ∈ cn such that ∨ {cn | n ∈ n} = l. © agt, upv, 2021 appl. gen. topol. 22, no. 1 212 on the menger and almost menger properties in locales proof. suppose, first, that l is almost menger. let (cn) be a sequence of directed open coverings of l, and let (dn) be an almost menger witness for (cn). since each dn is a finite subset of cn and cn is directed, there exists some cn ∈ cn such that d ⊆ cn for every d ∈ dn. it follows therefore that l = ∨{ d | d ∈ ⋃ n∈n dn} ⊆ ∨{ cn | n ∈ n } ⊆ l, which proves the left-to-right implication. conversely, suppose that the stated condition holds, and let (cn) be a sequence of open coverings of l. for each n, consider the collection c × n = {∨ f | f is a finite subset of cn } . then (c ×n ) is a sequence of directed open coverings of l. by hypothesis, there exists, for each n, some wn ∈ c × n such that ∨ {wn | n ∈ n} = l. now, for each n ∈ n, there exists some k(n) ∈ n and elements f (1) n , . . . , f k(n) n of cn such that wn = f (1) n ∨ · · · ∨ f k(n) n and hence wn = f (1) n ∨ · · · ∨ f k(n) n . consequently, if for each n we let c ′n = { f (1) n , . . . , f k(n) n } , then each c ′n is a finite subset of cn such that l = ∨{ wn | n ∈ n } ⊆ ∨{ t | t ∈ ⋃ n∈n c ′ n } ⊆ l, which then shows that (c ′n) is an almost menger witness for (cn), and hence l is almost menger. � although we could have proved directly from the definition that a localic image of an almost menger frame is almost menger, we shall use this result. if f : l → m is a localic map and u is a collection of sublocales of m, we write f−1[u ] for the set {f−1[u] | u ∈ u }. proposition 3.5. a localic image of any almost menger frame is itself almost menger. proof. let f : l → m be an onto localic map with l almost menger. let (un) be a sequence of directed open coverings of m. since f−1[−] preserves joins of open sublocales, (f−1[un])n∈n is a sequence of open coverings of l, and since f−1[−] preserves order, this sequence is directed. since l is almost menger, the foregoing proposition furnishes, for each n, an element un ∈ un such that∨ {f−1[un] | n ∈ n} = l. since f[−] preserves joins and f[−]◦f−1[−] ≤ ids(m), © agt, upv, 2021 appl. gen. topol. 22, no. 1 213 t. bayih, t. dube and o. ighedo we have (in light of f being onto) m = f[l] = f [∨ n∈n f−1[un] ] = ∨ n∈n f [ f−1[un] ] ⊆ ∨ n∈n f [ f−1 [ un ]] ⊆ ∨ n∈n un ⊆ m, which shows that l is almost menger by proposition 3.4. � as with the case of menger frames, the almost menger ones can also be characterised frame-theoretically without invoking sublocales. to do this, given a collection {ol(aα) | α ∈ a} of open sublocales of l, we note that ∨ α∈a ol(aα) = l ⇐⇒ ∨ α∈a cl(a ∗ α) = l ⇐⇒ (∀a ∈ l) ( a = ∧ α∈a tα, for some tα ≥ a ∗ α ) ; where we have surreptitiously used the fact that l = {∧ m | m ⊆ ⋃ a∈a cl(a ∗ α) } and that if m ⊆ l is expressible as a union m = ⋃ i∈imi of some subsets, then, setting mi = ∧ mi for each i, we have ∧ m = ∧ i∈i mi. given a sequence (cn) of covers of l, suppose that, for each n, there is a finite dn ⊆ cn such that every element a of l is expressible as a = ∧ αtα where each tα ≥ d ∗ α for some dα ∈ ⋃ n∈ndn. we then say the sequence (dn) is an almost menger witness for the sequence (cn). recall the bijection between covers and coverings. the calculation above shows that a sequence of covers has an almost menger witness if and only if the corresponding sequence of open coverings has an almost menger witness. consequently we have the following result. proposition 3.6. a frame l is almost menger iff for every sequence (cn) of covers of l, there exists, for each n, a finite dn ⊆ cn such that every element a of l is expressible as a = ∧ αtα where each tα ≥ d ∗ α for some dα ∈ ⋃ n∈ndn. we have seen that working with directed covers (or coverings) is often neater. after all, informally speaking, selecting an element is easier and quicker than selecting a finite subset. the frame-theoretic characterisation just stated can be couched in terms of directed covers. corollary 3.7. a frame l is almost menger iff for every sequence (cn) of directed covers of l, we can select, for each n, an element cn ∈ cn such that any a ∈ l is expressible as a = ∧ n∈ntn for some elements tn ∈ l with each tn ≥ c ∗ n. © agt, upv, 2021 appl. gen. topol. 22, no. 1 214 on the menger and almost menger properties in locales proof. this follows from proposition 3.6, the equivalences displayed in the paragraph following the proof of proposition 3.5, and the fact that if (cn) is a sequence of directed covers of l, then (c cn)n∈n is a sequence of directed open coverings of l, and, conversely, if (cn) is a sequence of directed open coverings of l, then (ccn )n∈n is a sequence of directed covers of l. � it is known that regular-closed subspaces of almost menger spaces need not be almost menger [20, example 3.1], but clopen subspaces inherit the almost menger property [13, proposition 3.3]. in frames we present a formally stronger result. we are going to impose conditions on an onto frame homomorphism which we first show by an example not to be so stringent as to make the homomorphism an isomorphism. recall that an element a of a frame l is colinear in case a ∨ ∧ i∈ixi = ∧ i∈i(a ∨ xi) for all families {xi}i∈i of elements of l. example 3.8. let a be a co-linear element of l and let κa : l → ↑a be the map given by κa(x) = a ∨ x. then κa is an onto, weakly perfect (actually, perfect) frame homomorphism preserving meets (since a is co-linear). note though that κa is not an isomorphism if a 6= 0. now let us recall from [10, remark 7.1] that: if h: l → m is a perfect frame homomorphism, then h∗(a ∗) ≤ h∗(a) ∗ for every a ∈ m. proposition 3.9. let h: l → m be a meet-preserving perfect onto frame homomorphism. if l is an almost menger frame, then so is m. proof. let (cn) be a sequence of directed covers of m. then (h∗[cn]) is a sequence of directed covers of l. since l is almost menger, there exists, for each n, an element un ∈ h∗[cn] such that the set {un | n ∈ n} has the property stated in corollary 3.7. each un is of the form h∗(cn) for some cn ∈ cn. we show that the set {cn | n ∈ n} has the desired property as per corollary 3.7. let a ∈ m. for each n, we can select tn ∈ l such that tn ≥ h∗(cn) ∗ and h∗(a) = ∧ n∈ntn. by the result cited above from [10], for each n we have h(tn) ≥ h ( h∗(cn) ∗ ) ≥ h ( h∗(c ∗ n) ) = c∗n because h is onto. now, using the fact that h preserves meets, we see that the elements h(tn) of m, for n ∈ n have the property that a = h(h∗(a)) = ∧ n∈n h(tn) and h(tn) ≥ c ∗ n for each n , so it follows that m is almost menger. � recall that a coframe is a complete lattice in which binary joins distributive over meets. a frame which is simultaneously a coframe need not be boolean. corollary 3.10. if l is almost menger, then cl(a) is almost menger for every co-linear a ∈ l. in particular, every closed sublocale of an almost menger frame which is also a coframe is almost menger. © agt, upv, 2021 appl. gen. topol. 22, no. 1 215 t. bayih, t. dube and o. ighedo we saw in the proof of proposition 2.8 that if a is a sublocale of l, then a is menger if and only if whenever (un) is a sequence of families of open sublocales of l such that a ⊆ ∨ un for every n, then there exists, for each n, a finite vn ⊆ un such that a ⊆ ∨⋃ n∈nvn. we present an almost similar result for the almost menger property, but only for dense complemented sublocales. in the proof we shall use the fact that if s is a dense sublocale of l and u is an open sublocale of l, then s ∩ u = u [15, xiii.1.2.3]. let us also recall that if t ⊆ s are sublocales of l, then the closure of t in s is given by cls t = s ∩ t [15, iii.8.5]. we also recall that if s is a sublocale of l and (ti)i∈i is a family of sublocales of s, then s(s)∨ {ti | i ∈ i} = s(l)∨ {ti | i ∈ i}. in the upcoming proof, the unadorned joins will be in s(l). theorem 3.11. the following are equivalent for a complemented dense sublocale a of l. (1) a is almost menger. (2) whenever (un) is a sequence of families of open sublocales of l with a ⊆ ∨ un for every n, then there exists, for each n, a finite vn ⊆ un such that a ⊆ ∨{ v | v ∈ ⋃ n∈nvn } . proof. assume that a is almost menger, and let (un) be a sequence of families of open sublocales of l with a ⊆ ∨ un for every n. using the notation in the proof of proposition 2.8, we have that (a∩ un) is a sequence of open coverings of a. since a is almost menger, for each n, there is a finite vn ⊆ un such that a = s(a)∨ { cla(a ∩ v ) | v ∈ ⋃ n∈n vn } = s(a)∨ { a ∩ a ∩ v | v ∈ ⋃ n∈n vn } ⊆ ∨{ v | v ∈ ⋃ n∈n vn } , which shows that (1) implies (2). conversely, assume that (2) holds. let (cn) be a sequence of open coverings of a. then, for each n, there is a family un of open sublocales of l such that cn = a ∩ un. then a ⊆ ∨ un. by hypothesis, there exists, for each n, a finite vn ⊆ un such that a ⊆ ∨{ v | v ∈ ⋃ n∈nvn } , so that (‡) a = ∨{ a ∩ v | v ∈ ⋃ n∈n vn } because a is complemented. for each n, put dn = {a ∩ v | v ∈ vn}, © agt, upv, 2021 appl. gen. topol. 22, no. 1 216 on the menger and almost menger properties in locales and observe that dn is a finite subset of cn. now, for each v ∈ ⋃ n∈nvn, a ∩ v = a ∩ a ∩ v because a ∩ v = v as a is dense and v is open. since { a ∩ v | v ∈ ⋃ n∈n vn } = { d | d ∈ ⋃ n∈n dn } , we deduce from (‡) that a = s(a)∨ { a ∩ d | d ∈ ⋃ n∈n dn } = s(a)∨ { cla d | d ∈ ⋃ n∈n dn } , which shows that a is almost menger. � as in spaces, regular almost menger frames are menger. in fact, there is a stronger result. note that if d is a strong refinement of a cover c of a frame l, then the open covering d = {o(d) | d ∈ d} of l has the property that for each u ∈ d there exists a v ∈ c , where c = {o(c) | c ∈ c}, such that u ⊆ v because d ≺ c implies o(d) ⊆ o(c). as in the case of covers, let us say an open covering u is a strong refinement of an open covering v if for each u ∈ u there is a v ∈ v such that u ⊆ v . proposition 3.12. a cover regular almost menger frame is menger. proof. let l be a cover regular almost menger frame, and let (cn) be a sequence of open coverings of l. for each n, let ĉn be a strong refinement of cn, so that we have the sequence (ĉn) of open coverings of l. since l is almost menger, we can choose, for each n, a finite vn ⊆ ĉn such that ∨{ v | v ∈ ⋃ n∈n vn } = l. since each ĉn is a strong refinement of cn, and since vn is finite, there is a finite dn ⊆ cn such that the closure of each sublocale in vn is contained in some sublocale in dn. consequently, l = ∨{ v | v ∈ ⋃ n∈n vn } ⊆ ∨{ d | d ∈ ⋃ n∈n dn } ⊆ l, showing that (dn) is a menger witness for (cn). therefore l is menger. � we shall now present a result which is, in a way, an analogue of [20, proposition 3.7]. we recall some pertinent terminology and facts. a frame is called scattered [18] just in case every sublocale of it is complemented. as observed in [18, p. 315]: if f : l → m is a localic map with m scattered, then f−1 [∨ i∈isi ] = ∨ i∈if−1[si] for every family {si}i∈i of sublocales of m. a localic map f : l → m is called nearly open if f−1[v ] = f−1[v ] for every open sublocale v of m. this is a conservative extension of pták’s [19] notion of nearly open continuous maps, and it is equivalent to saying h(a∗) = h(a)∗ © agt, upv, 2021 appl. gen. topol. 22, no. 1 217 t. bayih, t. dube and o. ighedo for every a ∈ l, where h is the left adjoint of f. incidentally, this latter condition was used by banaschewski and pultr [3] to define nearly open frame homomorphisms. recall that a localic map is called proper if it is closed and preserves directed joins. theorem 3.13. let f : l → m be a nearly open proper map of locales. suppose that m is scattered and covered by its compact sublocales. if m is almost menger, then l is almost menger. proof. let (cn) be a sequence of directed open coverings of l. write the set of the compact sublocales of m as an indexed family {kα | α ∈ a}. since f is a proper map, each f−1[kα] is a compact sublocale of l [21, corollary 4.3]. fix n ∈ n. since cn is an open covering of l, for each α ∈ a we have the containment f−1[kα] ⊆ ∨ cn, which, by compactness and the fact that cn is an increasing open covering of l, implies f−1[kα] ⊆ cnα, for some cnα ∈ cn. since cnα is an open sublocale of l, there exists an element cnα ∈ l such that cnα = ol(cnα). since m is scattered and covered by its compact sublocales, l = f−1[m] = f−1 [ ∨ α∈a kα ] = ∨ α∈a f−1[kα] ⊆ ∨ α∈a ol(cnα), which implies that the collection un = {ol(cnα) | α ∈ a} is an open covering of l. we show from this that the collection wn = {om(f(cnα)) | α ∈ a} is an open covering of m; and the idea for that is to show that, for each α ∈ a, kα ⊆ om(f(cnα)). since f−1[kα] ⊆ ol(cnα), upon taking supplements, we have cl(cnα) = l r ol(cnα) ⊆ l r f−1[kα]. taking direct images, and using the fact that f is a closed map, we obtain cm (f(cnα)) = f[cl(cnα)] ⊆ f [ l r f−1[kα] ] ⊆ m r kα; where the last containment is obtained from [9, equation (5.2)]. since every sublocale of m is complemented, taking supplements in the containment above yields kα = m r (m r kα) ⊆ m r cm (f(cnα)) = om(f(cnα)), whence we deduce that wn is an open covering of m. since m is almost menger, the sequence (wn) has an almost menger witness, (w ′ n), say. thus, for each n ∈ n, there exists some k(n) ∈ n and indices α(n,1), . . . , α(n,k(n)) in a such that w ′ n = { om ( f(cn,α(n,1)) ) , . . . , om ( f(cn,α(n,k(n))) )} and (#) ∨{ w | w ∈ ⋃ n∈n w ′ n } = m. © agt, upv, 2021 appl. gen. topol. 22, no. 1 218 on the menger and almost menger properties in locales we claim that the sequence (c ′n), where, for each n, c ′ n = { ol(cn,α(n,1)), . . . , ol(cn,α(n,k(n))) } is an almost menger witness for (cn). it is clear that each c ′ n is a finite subset of cn. since m is scattered and f is nearly open, from (#) we obtain l = f−1 [∨{ w | w ∈ ⋃ n∈n w ′ n }] = ∨{ f−1 [ w ] | w ∈ ⋃ n∈n w ′ n } . now, if w ∈ w ′m for some m ∈ n, then w = om ( f(cm,α(m,i)) ) , for some i ∈ {1, . . . , k(m)}, so that w = cm ( f(cm,α(m,i)) ∗ ) . since h(f(b)) ≤ b, so that b∗ ≤ h(f(b))∗, for any b ∈ m, and since f is nearly open, we therefore have f−1 [ w ] = cl ( h(f(cm,α(m,i)) ∗) ) = cl ( h(f(cm,α(n,i))) ∗ ) ⊆ cl(c ∗ m,α(m,i)) ⊆ ∨{ d | d ∈ ⋃ n∈n c ′ n } , because ol(cm,α(m,i)) ∈ c ′ m. it follows therefore that ∨{ d | d ∈ ⋃ n∈n c ′ n } = l, which proves that l is almost menger. � the condition that m is scattered was used, among other things, to ensure that joins (actually, only those that cover the codomain) are preserved under pullback. it is well known that coverings are generally not preserved under pullback. however, since in the first part of the proof it is a special type of a covering (by compact sublocales) that is pulled back along a special type of a localic map (a nearly open one), it is perhaps worth pointing out that we have not over hypothesised by requiring the codomain to be scattered. here is an example demonstrating the point. example 3.14. consider the frame ω(r), and let j : b(ω(r)) → ω(r) be the inclusion of its smallest dense sublocale. for any frame l, the frame homomorphism (−)∗∗ : l → bl is nearly open because the pseudocomplement of an element in any dense sublocale calculated in the sublocale is exactly its pseudocomplement calculated in the frame. thus j is a nearly open localic map. for any x ∈ r, denote by x̃ the prime element r r {x} of ω(r). by spatiality, ω(r) = ∨{ {r̃, 1ω(r)} | r ∈ r } , so that ω(r) is covered by its (compact) one-point sublocales. for any r ∈ r, the set-theoretic inverse image j−1[{r̃, 1ω(r)}] = {1ω(r)} because r̃ ∗∗ 6= r̃. hence j−1[{r̃, 1ω(r)}] = o, which then says o = ∨ r∈r j−1[{r̃, 1ω(r)}] whereas j−1 [∨ r∈r {r̃, 1ω(r)} ] = b(ω(r)), © agt, upv, 2021 appl. gen. topol. 22, no. 1 219 t. bayih, t. dube and o. ighedo showing that, generally, j−1 fails to preserve joins of (covering) compact sublocales. we close with a result that shows that we can replace open coverings with regular-open coverings in the definition of almost menger frames. to recall, a sublocale of l is called regular-open if it is of the form ol(a) with a = a ∗∗. an element of l of the form x∗∗ is called regular. clearly, the bijection between covers and open coverings restricts to a bijection between covers consisting entirely of regular elements and open coverings consisting entirely of regularopen sublocales. for any cover c we will write c∗∗ = {c∗∗ | c ∈ c}, and observe that c∗∗ is also a cover, consisting of regular elements. proposition 3.15. a frame is almost menger iff every sequence of open coverings consisting entirely of regular-open sublocales has an almost menger witness. proof. only one implication needs proving. we do it via covers. so, suppose that every sequence of covers of l consisting entirely of regular elements has an almost menger witness. let (cn) be a sequence of covers of l, and then consider the sequence (c∗∗n ). by our supposition, (c ∗∗ n ) has an almost menger witness; so, for each n, there exists a finite un ⊆ c ∗∗ n such that any a ∈ l is expressible as a = ∧ α tα, where each tα ≥ u ∗ α for some uα ∈ ⋃ n∈n un. for each n, there exists a positive integer kn and finitely many elements cn1, . . . , cnkn in cn such that un = {c ∗∗ n1, . . . , c ∗∗ nkn }. let dn = {cn1, . . . , cnkn}. since (x∗∗)∗ = x∗ always, it follows that (dn) is an almost menger witness for (cn). therefore l is almost menger. � acknowledgements. the second-named author acknowledges funding from the national research foundation of south africa under grant 113829. references [1] r. n. ball and j. walters-wayland, cand c∗-quotients in pointfree topology, dissert. math. 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[21] j. j. c. vermeulen, proper maps of locales, j. pure appl. algebra 92 (1994), 79–107. © agt, upv, 2021 appl. gen. topol. 22, no. 1 221 @ appl. gen. topol. 20, no. 1 (2019), 211-222doi:10.4995/agt.2019.10682 c© agt, upv, 2019 generic theorems in the theory of cardinal invariants of topological spaces1 alejandro raḿırez-páramo a and jesús f. tenorio b a facultad de ciencias de la electrónica, benemérita universidad autónoma de puebla, ŕıo verde y av. san claudio s.n., puebla, puebla, méxico (alejandro.ramirez@correo.buap.mx) b instituto de f́ısica y matemáticas, universidad tecnológica de la mixteca, carretera a acatlima, km 2.5, huajuapan de león, oaxaca, méxico (jtenorio@mixteco.utm.mx) communicated by d. georgiou abstract the main aim of this paper is to present a technical result, which provides an algorithm to prove several cardinal inequalities and relative versions of cardinal inequalities related. moreover, we use this result and the weak hausdorff number, h∗, introduced by bonanzinga in [houston j. math. 39 (3) (2013), 1013–1030], to generalize some upper bounds on the cardinality of topological spaces. 2010 msc: 54a25; 54d10. keywords: cardinal functions; compact spaces; lindelöf spaces; weak hausdorff number of a space. 1. introduction among the best known theorems concerning cardinal functions are those which give an upper bound on the cardinality of a space in terms of other cardinal invariants. of course, one of these results is the famous arhangel’skǐı inequality, answering a 50 years old question posed by p.s. alexandroff and p. urysohn, namely: for each hausdorff space x, |x| ≤ 2l(x)χ(x). the previous inequality generated a great development in the theory of topological cardinal functions, as well as new questions and open problems. the 1in memory of j. r. e. arrazola-ramı́rez. received 04 september 2018 – accepted 27 december 2018 http://dx.doi.org/10.4995/agt.2019.10682 a. ramı́rez-páramo and j. f. tenorio ideas employed in their proof became proving technique, which is now a line of research. at present, there is a wide range of results that attempt to capture the central ideas of arhangel’skǐı’s proof, in order to obtain generic theorems (see [11]). the reader interested in knowing about arhangel’skǐı’s inequality can consult the work of hodel [11]. on the other hand, since the appearance of arhangel’skǐı’s inequality, in 1969, to date, more results have been obtained that are either a generalization or a variation of arhangel’skǐı’s result. in [6], bonanzinga introduced the hausdorff number and the weak hausdorff number of a space x, denoted by h(x) and h∗(x), respectively, to analyze, among others, problems related to arhangel’skǐı’s inequality. bonanzinga’s ideas have given new impetus to the theory of the topological cardinal invariants and today many authors have returned to the problems in this field. in this paper we prove in theorem 2.2 a general technical result, closely related to [2, theorem 1], which provides an algorithm for proving a wide range of cardinal inequalities in absolute and relative versions. moreover, we use theorem 2.2 to prove other well-known generic theorems and we establish upper bounds on the cardinality of topological spaces which generalize some recently presented. we recall the following. let x be a topological space and let a be a subset of x. we denote by a or clx(a) the closure of a in x. if x is a set and κ is an infinite cardinal, then [x]≤κ (respectively, [x]<κ, [x]κ and [x]≥κ) denotes the collection of all subsets of x with cardinality ≤ κ (respectively, < κ, = κ and ≥ κ). also, if x is a topological space and y ⊆ x, then the κ-closure of y in x, denoted by clκ(y ) or [y ]κ, is the set:⋃ {d : d ∈ [y ]≤κ}. we say that y is a κ-closed subset of x if y = [y ]κ. we refer the reader to [13] and [14] for definitions and terminology on cardinal functions not explicitly given. let l, χ, ψ, ψc, c, t, nw, f and d denote the following standard cardinal functions: lindelöf degree, character, pseudocharacter, closed pseudocharacter, cellularity, tightness, networkweight, free sequence number and density, respectively. the following definitions are known (see e.g. [7], compare also [12]). for y ⊆ x, the almost lindelöf degree of y relative to x, denoted by al(y,x), is the smallest infinite cardinal κ such that for every open cover u of y , by open subsets of x, there is a subcollection v ∈ [u]≤κ such that y ⊆ ⋃ v = ⋃ {v | v ∈ v}. the almost lindelöf degree of x, denoted by al(x) is al(x,x). the almost lindelöf degree relative to closed subsets of x is alc(x) = sup{al(c,x) | c is a closed subset of x}. the κ-almost lindelöf degree of x [7] is alκ(x) = sup{al(c,x) | c is a κ-closed subset of x}. for definitions of weak lindelöf degree of x, wl(x), and weak lindelöf degree relative to closed subsets, wlc(x), see [12] (compare with [1, 5]). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 212 generic theorems in the theory of cardinal invariants 2. generic theorems in what follows, τ and κ are infinite cardinals such that κ < cf(τ). let x be a nonempty set. a κ-sensor in x is a pair s = (a,f), where a is a family of subsets of x and f is a collection of families of subsets of x such that: for every a ∈ a, |a| ≤ κ and |a| ≤ κ; and, for every c ∈ f, |f| ≤ κ and |c| ≤ κ. given h ⊆ x and g ⊆ p(x), we say that a κ-sensor s = (a,f) in x is generated by the pair (h,g), if a ⊆ h, for each a ∈ a, and c ⊆ g, for each c ∈ f. the proof of the following proposition is easy. proposition 2.1. let x be a set. if h ⊆ x and g ⊆ p(x), then the collection of κ-sensors in x generated by the pair (h,g) has cardinality less than or equal to |h|κ · |g|κ. let θ denote a function such that each κ-sensor s in x, is associated with a subset θ(s) of x, called the θ-closure of s, and we say that the function θ is a θ-closure. let y be a nonempty subset of x. if s a κ-sensor in x, we say that s is small for y if y \ θ(s) 6= ∅. when y = x, we only say that s is a small κ-sensor. let τ and κ be infinite cardinals. an operator ρ : p(x) → p(x) will be called (τ,κ)-closing if whenever a ⊆ x such that |a| ≤ τκ, then |ρ(a)| ≤ τκ and a ⊆ ρ(a). it is clear that if τ = κ+, then the condition |a| ≤ τκ implies |ρ(a)| ≤ τκ in this definition is equivalent to |a| ≤ 2κ implies |ρ(a)| ≤ 2κ. throughout this paper, we put l = [x]≤τ κ and q = [p(x)]≤τ κ . moreover, if g : l → q is a function and e ⊆ l, we put ug(e) = ⋃ {g(f) | f ∈ e}. when ρ is a (τ,κ)-closing operator we denote by ρ(e) the set {ρ(e) | e ∈ e}. let ρ be a (τ,κ)-closing operator and let θ be a θ-closure operator. considering e ⊆ l and a function g : l → q we say that a κ-sensor s in x is θ-good for e with respect to y if s is generated by the pair ( ⋃ ρ(e),ug(e)) and y ∩ [ ⋃ ρ(e)] ⊆ θ(s). when y = x, we only say that s is θ-good for e. finally, a (g,ρ,θ)-quasi-propeller for y is a family e = {eα | α < τ} ⊆ l such that no small κ-sensor for y in x is θ-good for e with respect to y . when y = x, we only say that e = {eα | α < τ} ⊆ l is a (g,ρ,θ)-quasi-propeller if no small κ-sensor is θ-good for e. now we are ready to prove our main result in theorem 2.2. we mention that currently there are several results that are adapted to prove cardinal inequalities, but generally related to the inequality of arhangel’skǐı. arhangel’skǐı has a much more general result, an algorithm, for proving relative versions of cardinal inequalities (main theorem from [2]). however, arhangel’skǐı mentions in [2] that he does not know a proof of gryzlov’s result [10] (see theorem 2.6, here). following the ideas of arhangel’skǐı in [2], we obtain in theorem 2.2 a general technical result, which can be used to prove several well-known cardinal inequalities in relative and absolute version. among others, the results given in [2], and gryzlov’s inequality. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 213 a. ramı́rez-páramo and j. f. tenorio theorem 2.2. let x be a set, let y be a nonempty subset of x, and let τ and κ be such that κ < cf(τ). if g : l → q is a function, ρ : p(x) → p(x) is a (τ,κ)-closing operator, and θ is a θ-closure, then there exists a family {eα | α < τ} ⊆ l, such that: (1) for each 0 < α < τ, ⋃ {ρ(eβ) ∩ y | β < α} ⊆ eα, ⋃ {ρ(e) ∩ y | e ∈ [ ⋃ β<α eβ] ≤κ} ⊆ eα, and (2) e = {eα | α < τ} is a (g,ρ,θ)-quasi-propeller for y . proof. we construct a sequence {eα | α < τ} ⊆ l and a collection of families of subsets of x, {uα | 0 < α < τ}, such that: (a) for each 0 ≤ α < τ,⋃ {ρ(eβ) ∩ y | β < α} ⊆ eα and ⋃ {ρ(e) ∩ y | e ∈ [ ⋃ β<α eβ] ≤κ} ⊆ eα; (b) for each 0 < α < τ, uα = ⋃ {g(eβ) | β < α}; (c) for each 0 < α < τ, if s is a κ-sensor such that is small for y and is generated by the pair ( ⋃ {ρ(eβ) | β < α},uα), then (y ∩ eα) \ θ(s) 6= ∅. fix 0 < α < τ and assume that eβ and uβ are already defined such that (a)-(c) hold for each β < α. note that uα has been defined by (b). we put hα = ⋃ {ρ(eβ) ∩ y | β < α}. it is not difficult to prove that |hα| ≤ τ κ. for each small for y κ-sensor s generated by the pair ( ⋃ {ρ(eβ) | β < α},uα), we choose one point m(s) ∈ y \ θ(s), and let fα be the set of points chosen in this way. from proposition 2.1, |fα| ≤ τ κ. we put h′α = ⋃ {ρ(e) ∩ y | e ∈ [ ⋃ β<α eβ] ≤κ}. note that |h′α| ≤ τ κ. let eα = hα ∪ h ′ α ∪ fα. clearly, eα ∈ l and eα satisfies (c). this completes the construction. on the other hand, it is clear that the collection {eα | α < τ} satisfies (1). finally, the proof will be complete if we prove that e = {eα | α < τ} is a (g,ρ,θ)-quasi-propeller for y . to see this, suppose there is a κ-sensor s0 = (a,f), which is small for y , and s0 is θ-good for e with respect to y . thus, y \ θ(s0) 6= ∅, s0 is generated by the pair ( ⋃ ρ(e),ug(e)) and y ∩ [ ⋃ ρ(e)] ⊆ θ(s0). since κ < cf(τ), there exists α0 < τ such that for each a ∈ a, a ⊆ ⋃ {ρ(eβ) | β < α0}, and for each b ∈ f, b ⊆ uα0. hence, s0 is generated by the pair ( ⋃ {ρ(eβ) | β < α0},uα0) and satisfies y \ θ(s0) 6= ∅. thus, by (c) there exists m(s0) ∈ eα0 \ θ(s0), a contradiction. � we apply theorem 2.2 in section 3 to obtain some results on cardinal invariants of topological spaces. however, theorem 2.2 also can be used to obtain other generic theorems. for example, the next technical result due to hodel captures the common core of several cardinal inequalities which are either a generalization or a variation of arhangel’skǐı’s inequality: for each hausdorff space x, |x| ≤ 2l(x)χ(x). corollary 2.3 ([12]). let x be a set, let κ and λ be infinite cardinals with λ ≤ 2κ, let c′,d : p(x) → p(x) be operators on x, and for each x ∈ x, let bx = {v (γ,x) | γ < λ} a collection of subsets of x. assume the following: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 214 generic theorems in the theory of cardinal invariants (t) if x ∈ c′(h), then there exists a ∈ [h]≤κ, such that x ∈ c′(a); (c) if a ∈ [x]≤κ, then |c′(a)| ≤ 2κ; and (c-s) if h 6= ∅, c′(h) ⊆ h, and q 6∈ h, then there exist a ∈ [h]≤κ and a function f : a → λ such that h ⊆ d( ⋃ {v (f(x),x) | x ∈ a}) and q /∈ d( ⋃ {v (f(x),x) | x ∈ a}). then |x| ≤ 2κ. proof. let τ = κ+. let l = [x]≤2 κ , q = [p(x)]≤2 κ , and g : l → q given by g(f) = ⋃ {bx | x ∈ f}, for every f ∈ l. it is easy to see that ρ : p(x) → p(x) given by ρ(a) = c′(a) is a (τ,κ)-closing operator. for each κ-sensor s = (a,f), we put θ(s) = d( ⋃ { ⋃ c | c ∈ f}). then, there exists a family {eα | α < τ} ⊆ l such that parts (1) and (2) of theorem 2.2, hold. let p = ⋃ e. clearly p 6= ∅ and |p | ≤ 2κ. moreover, c′(p) ⊆ p . the proof will be complete once we show that x ⊆ p . suppose the contrary. then, there exists p ∈ x \ p . hence, by (c-s) there exist a ∈ [p ]≤κ and a function f : a → λ such that p ⊆ d( ⋃ {v (f(x),x) | x ∈ a}) and p /∈ d( ⋃ {v (f(x),x) | x ∈ a}). let s0 = (∅,{{v (f(x),x) | x ∈ a}}) and θ(s0) =⋃ {v (f(x),x) | x ∈ a}. then s0 is a small κ-sensor in x which is generated by the pair ( ⋃ ρ(e),ug(e)) and p ⊆ θ(s0), a contradiction. hence, x ⊆ p . therefore, |x| ≤ 2κ. � we observe that cammaroto, catalioto and porter [7] use corollary 2.3 to generalize the inequalities: |x| ≤ 2l(x)f(x)ψ(x) due to spadaro-juhász [16], and |x| ≤ 2l(x)fc(x)ψ(x) due to bella [4]. on the other hand, the next result improves a result by cammaroto et al. [8, main theorem], which is also a unified approach to prove several cardinal inequalities. corollary 2.4. let x be a set, κ and τ infinite cardinals such that κ < cf(τ), ρ : [x]≤τ κ → p(x) a function, and for x ∈ x, bx = {v (x,α) | α ∈ κ} a collection of subsets of x such that: (i) for a,b ∈ [x]≤τ κ , a ⊆ ρ(a) and if a ⊆ b, then ρ(a) ⊆ ρ(b). (ii) for a ∈ [x]≤τ κ , |ρ(a)| ≤ τκ. (iii) if h 6= ∅, ρ(h) ⊆ h, and q /∈ h, then there exist a ∈ [h]≤κ and a function f : a → κ such that h ⊆ ⋃ x∈a v (x,f(x)) and q /∈⋃ x∈a v (x,f(x)). then |x| ≤ τκ. proof. let l = [x]≤τ κ , q = [p(x)]≤τ κ , and g : l → q given for every f ∈ l by g(f) = ⋃ {bx | x ∈ f}. clearly, ρ is a (τ,κ)-closing operator. now, for every κ-sensor s = (a,f), we put θ(s) = ⋃ { ⋃ c | c ∈ f}. hence, there exists a family {eα | α < τ} ⊆ l such that parts (1) and (2) of theorem 2.2 hold. we see ρ(p) ⊆ p . for this end, it suffices to note that if b ∈ [p ]≤κ, then ρ(b) ⊆ p . indeed, if b ∈ [p ]≤κ, then by regularity of τ, there exists α0 < τ such that b ⊆ ⋃ {eβ | β < α0}. thus, by second contention of part (1) in theorem 2.2 ρ(b) ⊆ p . we show that x ⊆ p . suppose the contrary and fix q ∈ x\p . by (iii) there exist a ∈ [p ]≤κ and a function f : a → κ such that p ⊆ ⋃ x∈a {v (x,f(x))} c© agt, upv, 2019 appl. gen. topol. 20, no. 1 215 a. ramı́rez-páramo and j. f. tenorio and q /∈ ⋃ x∈a {v (x,f(x))}. we consider the κ-sensor s0 = (∅,{{v (x,f(x)) | x ∈ a}}) and θ(s0) = ⋃ x∈a {v (x,f(x))}. we note that q ∈ x \ θ(s0). thus, s0 is a small κ-sensor which is θ-good for e, a contradiction. it follows that x ⊆ p . therefore, |x| ≤ τκ. � we conclude this section with a proof of gryzlov’s theorem using theorem 2.2. lemma 2.5 ([10]). let x be a t1 compact space with ψ(x) ≤ κ. let h be a subset of x such that every infinity subset of h of cardinality ≤ κ has a complete accumulation point in h. then h is compact. theorem 2.6. if x is a t1 compact space, then |x| ≤ 2 ψ(x). proof. let κ = ψ(x) and τ = κ+. for each x ∈ x, let bx be a local pseudobase of x in x with |bx| ≤ κ. we consider the operator ρ : p(x) → p(x) defined by ρ(a) = a ∪ a′, where a′ is the set defined as follows: for each infinite subset, b ⊆ a with |b| ≤ κ, we take a complete accumulation point of b in x and a′ is the set formed by such points. clearly ρ is a (τ,κ)-closing operator. for each κ-sensor s = (a,f) in x, we put θ(s) = ⋃ { ⋃ c | c ∈ f}. consider g : l → q defined by g(f) = ⋃ {bx | x ∈ ρ(f)}, for f ∈ l. by theorem 2.2, there is a family e = {eα | α ∈ κ +}, which is a (g,ρ,θ)-quasi-propeller in l. let h = ⋃ {ρ(eα) | α ∈ κ +}. it is not difficult to show, using lemma 2.5, that h is compact. moreover, |h| ≤ 2κ. let us show that x ⊆ h. suppose not and let p ∈ x \ h. for each x ∈ h, let vx ∈ bx such that p /∈ vx. clearly the collection {vx | x ∈ h} cover h. hence, there exist x1, . . . ,xn ∈ h such that h ⊆ ⋃ {vxi | i ∈ {1, . . . ,n}}. let f = {vxi | i ∈ {1, . . . ,n}}. then, we have that s0 = (∅,{f}) is a small κ-sensor in x, which is θ-good for e. this is a contradiction, since e is a (g,ρ,θ)-quasi-propeller. thus, x ⊆ h. therefore, |x| ≤ 2ψ(x). � 3. some applications in cardinal functions in 1969, arhangel’skǐı [3] proved his famous result: for each hausdorff space x, |x| ≤ 2l(x)χ(x). this inequality has been generalized by some authors as sapadaro [16], bella [4], cammaroto catalioto and porter [7], among others. another generalization from arhangel’skǐı’s result is due to bonanzinga in 2013, namely she proved: for each t1 space x with h ∗(x) ≤ ω, |x| ≤ 2l(x)χ(x), the which is a positive partial answer to a question posed by arhangel’skǐı, that is: is it true that if x is a t1-space, then |x| ≤ 2 l(x)χ(x)? like all the important results, bonanzinga’s inequality, in addition to solving a long-standing problem, introduces new techniques and generates new questions. among the new concepts presented by bonanzinga we find the hausdorff number and the weak hausdorff number of a space x, denoted by h(x) and h∗(x), respectively. next, we use these cardinal functions and theorem 2.2 in order to present some generalizations of bounds to the cardinality of topological spaces. before this, we recall the notion of the weak hausdorff number of a space. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 216 generic theorems in the theory of cardinal invariants definition 3.1 ([6]). let x be a topological space. the weak hausdorff number of x is h∗(x) = min{τ | for each a ∈ [x]≥τ, there is b ∈ [a]<τ, and for every b ∈ b,there exists an open subset ub such that b ∈ ub and ⋂ b∈b ub = ∅}. the following concepts were introduced in [5]. if x is a t1 topological space then for each x ∈ x we put hw(x) = ⋂ {u | x ∈ u and u is open in x}. the hausdorff width is hw(x) = sup{|hw(x)| | x ∈ x} (see [5]). moreover, for every x ∈ x, ψw(x) = min{|ux| | ux is a family of open neighbourhoods of x and ⋂ {u | u ∈ ux} = hw(x)}. thus, we have that ψw(x) = sup{ψw(x) | x ∈ x}. in the following results, if y ⊆ x, we denote by y ∗ the set ⋃ {hw(x) : x ∈ y }. the next application of theorem 2.2, generalizes [17, theorem 2.2], [5, theorem 2.22] and [6, theorem 31] (see corollary 3.3 parts (a), (b) and (c), respectively). theorem 3.2. let x be a t1-space and for every infinite cardinal κ assume that: (i) alκ(x)ψw(x) ≤ κ; (ii) for each a ∈ [x]≤κ, |a| ≤ 2κ. then |x| ≤ hw(x)2κ. proof. let τ = κ+. for every x ∈ x we fix a collection bx of open subsets of x containing x such that |bx| ≤ κ and ⋂ bx = hw(x). we consider the operator ρ : p(x) → p(x) defined by ρ(a) = [a]κ. by (ii), we have that ρ is a (τ,κ)-closing operator. for each κ-sensor s = (a,f) in x, we put θ(s) =⋃ { ⋃ c | c ∈ f} and we take g : l → q defined by g(f) = ⋃ {bx | x ∈ ρ(f)}, for f ∈ l. by theorem 2.2, there is a family e = {eα | α < κ +}, which is a (g,ρ,θ)-quasi-propeller in l. let h = ⋃ e. we have that ρ(h) = ⋃ {ρ(eα) | α ∈ κ +}. indeed, if p ∈ ρ(h), then there exists c ∈ [h]≤κ such that p ∈ c. because h = ⋃ e, there exists αp < κ + such that c ⊆ ⋃ {eβ | β < αp}. by hypothesis in (i), [c]κ = ρ(c) ⊆ eαp. thus, p ∈ ⋃ {ρ(eα) | α ∈ κ +}. moreover, it is clear that |h| ≤ 2κ. hence |ρ(h)| ≤ 2κ. then |ρ(h)∗| ≤ hw(x)2κ. let us show that x ⊆ ρ(h)∗. assume the contrary, and fix p ∈ x \ ρ(h)∗. for each x ∈ ρ(h), choose ux ∈ bx such that p /∈ ux. hence v = {ux | x ∈ ρ(h)} is a collection of open subsets of x which cover ρ(h). thus, there exists a ∈ [ρ(h)]≤κ such that ρ(h) ⊆ ⋃ {ux | x ∈ a}. it is clear that p ∈ x \ ⋃ {ux | x ∈ a}. let f0 = {ux | x ∈ a} and we put s0 = (∅,{f0}). then, we have that s0 is a small κ-sensor in x, which is θ-good for e. this is a contradiction, since e is a (g,ρ,θ)-quasi-propeller. thus, x = ρ(h)∗ and therefore, |x| ≤ hw(x)2κ. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 217 a. ramı́rez-páramo and j. f. tenorio corollary 3.3. let x be a t1-space with h ∗(x) ≤ ω. then (a) |x| ≤ hw(x)2alκ(x)χ(x). (b) ([5]) |x| ≤ hw(x)2alc(x)χ(x). (c) ([6]) |x| ≤ hw(x)2l(x)χ(x). proof. (a) by [5, note 2.21], we have that alκ(x)ψw(x) ≤ alκ(x)χ(x). thus, part (i) from theorem 3.2 holds. moreover, by [6, proposition 2.8], we obtain that, for every a ∈ [x]≤alκ(x)χ(x), |a| ≤ 2alκ(x)χ(x). thus, part (ii) from theorem 3.2 holds. we have shown that |x| ≤ hw(x)2alκ(x)χ(x). (b) let κ = alc(x)χ(x). since t(x) ≤ χ(x) ≤ κ, we have that each κ-closed subset is a closed subset; hence, alκ(x) ≤ κ. moreover, ψw(x) ≤ χ(x) ≤ κ; thus, alκ(x)ψw(x) ≤ κ. it is easy to see that part (ii) from theorem 3.2 holds. therefore, |x| ≤ hw(x)2alc(x)χ(x). � the following definition is from [2]. let x be a t1 space. a subspace y of x is said to be lindelöf in x if for each open cover u of x, there is a subcollection v ∈ [u]≤ω such that y ⊆ ⋃ v. for a cardinal number κ, we say that y is initially κ-lindelöf in x, if for every open cover u of x of cardinality less than κ, there is a subcolection v ∈ [u]≤ω such that y ⊆ ⋃ v. theorem 3.4. let x be a t1-space with h ∗(x) ≤ ω, and let y be a subspace dense in x and initially 2χ(x)-lindelöf in x, then |x| ≤ 2χ(x) and y is lindelöf in x. proof. let κ = χ(x) and τ = κ+. for every x ∈ x fix bx a local base of x in x such that |bx| ≤ κ. we consider the operator ρ : p(x) → p(x) defined by ρ(a) = a. note that, by [6, proposition 28], ρ is a (κ+,κ)-closing operator. for each κ-sensor s = (a,f) in x, we put θ(s) = ⋃ { ⋃ c | c ∈ f}. let g : l → q be given by g(f) = ⋃ {bx | x ∈ ρ(f)}, for f ∈ l. by theorem 2.2, there is a family e = {eα | α < κ +}, which is a (g,ρ,θ)-quasi-propeller in l. let h = ⋃ e. since χ(x) ≤ κ, h = ⋃ {eα | α ∈ κ +}. thus |ρ(h)| ≤ 2κ. we show that y ⊆ ρ(h). assume the contrary, and fix p ∈ y \ ρ(h). for each x ∈ ρ(h), choose ux ∈ bx such that p /∈ ux. hence, v = {ux | x ∈ ρ(h)} ∪ {x \ ρ(h)} is an open cover of x such that |v| ≤ 2κ. since y is 2κ-lindelöf in x there exists a ∈ [ρ(h)]≤ω such that y ⊆ ⋃ {ux | x ∈ a} ∪ (x \ ρ(h)). clearly, if x ∈ y ∩ ρ(h), then x 6∈ x \ ρ(h). on the other hand, p ∈ y \ ⋃ {ux | x ∈ a}. we put f0 = {ux | x ∈ a} and we put s0 = (∅,{f0}). then, we have that s0 is a κ-sensor in x, s0 is generated by the pair ( ⋃ ρ(e),ug(e)) and y ∩ ⋃ ρ(e) ⊆ θ(s0). thus, we conclude that s0 is a κ-sensor in x, which is small for y and is θ-good for e with respect to y , a contradiction, because e is a (g,ρ,θ)-quasi-propeller in l. hence, y ⊆ ρ(h). therefore, |y | ≤ 2κ. finally, since y is dense in x, from [6, proposition 28], we conclude that |x| ≤ 2κ. in consequence, and since y is 2κ-lindelöf in x, it follows that y is lindelöf in x. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 218 generic theorems in the theory of cardinal invariants from theorem 3.4, we obtain the following result due to arhangel’skǐı (see [2, corollary 1]). corollary 3.5 ([2]). let x be a hausdorff space, and let y is a subspace dense in x and initially 2χ(x)-lindelöf in x, then |x| ≤ 2χ(x) and y is lindelöf in x. for the third application of theorem 2.2, we recall that given a topological space and κ an infinite cardinal, we say that a subset a ∈ [x]≤2 κ is κ-quasi-dense in x if for every open cover u of x, there exist b ∈ [a]≤κ and v ∈ [u]κ such that x = clx(b) ∪ ⋃ v. moreover, ql(x) = min{κ | there is a κ-quasi-dense subset a in x}. theorem 3.6. if x is a t1-space with h ∗(x) ≤ ω, then |x| ≤ 2ql(x)χ(x). proof. let κ = ql(x)χ(x), τ = κ+, for each x ∈ x, let bx be a local pseudobase of x in x with |bx| ≤ κ. since ql(x) ≤ κ, there exist a ∈ [x] ≤2κ which is a κ-quasi-dense in x. let g(f) = ⋃ {bx | x ∈ f}, for every f ∈ l. we consider ρ(a) = a. then, since h∗(x) ≤ ω, we obtain by [6, proposition 28] that ρ is a (κ+,κ)-closing operator. for every sensor s = (a,f), we put θ(s) = clx( ⋃ a)∪ ⋃ { ⋃ c | c ∈ f}. by theorem 2.2, there is a family e = {eα | α < κ +}, which is a (g,ρ,θ)-quasipropeller in l. let h = ⋃ e. note that |h| ≤ 2κ. hence, |ρ(h)| ≤ 2κ. observe that, ρ(h) = ⋃ {ρ(eα) | α ∈ κ +}. we prove that x ⊆ ρ(h). suppose that there exists p ∈ x\ρ(h). for every x ∈ ρ(h), let vx ∈ bx such that p /∈ vx. it is clear that the collection {vx | x ∈ ρ(h)} ∪ {x\ρ(p)} cover x. since ql(x) ≤ κ, there exist d ∈ [a] ≤κ and b ∈ [ρ(h)]≤κ such that x = clx(d) ∪ ⋃ {vx | x ∈ b} ∪ x\ρ(h). let a = {d}, f = {vx | x ∈ b} and let s0 = (a,f). clearly p /∈ θ(s0) and h ⊆ ρ(h) ⊆ θ(s0). then, we conclude that s0 is a small κ-sensor in x, which is θ-good for e, a contradiction, because e is a (g,ρ,θ)-quasi-propeller in l. � corollary 3.7. for every hausdorff space x, (1) |x| ≤ 2ql(x)χ(x). (2) ([15]) |x| ≤ 2ql(x)ψ(x)t(x). problem 3.8. if x is a t1-space with h ∗(x) ≤ ω, then |x| ≤ 2ql(x)ψ(x)t(x). for another application of theorem 2.2 we consider the following notion introduced by arhangel’skǐı in [2] for κ = ω. given a topological space x and κ an infinite cardinal, we say that x is strictly quasi-κ-lindelöf if for every closed subset p of x and every collection {uα | α ∈ κ} of families of open subsets in x sucht that p ⊆ ⋃ { ⋃ uα | α ∈ κ}, there exists, for each α ∈ κ, vα ∈ [uα] ω such that p ⊆ ⋃ { ⋃ vα | α ∈ κ}. it is easy to see that for every κ, if x is lindelöf, then x is strictly quasi-κ-lindelöf. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 219 a. ramı́rez-páramo and j. f. tenorio theorem 3.9. let x be a t1-space with h ∗(x) ≤ ω. let κ be an infinite cardinal such that: (i) χ(x) ≤ κ, (ii) x is strictly quasi-κ-lindelöf. then |x| ≤ hw(x)2κ. proof. let τ = κ+. for every x ∈ x, we fix a collection bx of open subset of x containing x such that |bx| ≤ κ and ⋂ bx = hw(x). let g(f) = ⋃ {bx | x ∈ f}, for every f ∈ l. we consider the operator ρ : p(x) → p(x) defined by ρ(a) = a. note that, from [6, proposition 28], ρ is a (κ+,κ)-closing operator. for each κ-sensor s = (a,f) in x, we put θ(s) = ⋃ { ⋃ c : c ∈ f}. by theorem 2.2, there is a family e = {eα | α < κ +}, which is a (g,ρ,θ)-quasipropeller in l. let h = ⋃ e. note that |h| ≤ 2κ. moreover, since ρ(h) = ⋃ {ρ(eα) | α ∈ κ+}, then |ρ(h)| ≤ 2κ. hence, |ρ(h)∗| ≤ hw(x)2κ. because χ(x) ≤ κ, we obtain that ρ(h) is a closed subset of x. we prove that x ⊆ ρ(h)∗. for this end, suppose that there exists p ∈ x \ ρ(h)∗ and let up = {uα | α ∈ κ} a local base of p in x. for every α ∈ κ, let uα = {v ∈ ug(e) | v ∩ uα = ∅}. we claim that ρ(h) ⊆ ⋃ { ⋃ uα | α ∈ κ}. indeed, let x ∈ ρ(h). since p 6∈ ρ(h)∗ = ⋃ {hw(h) : h ∈ ρ(h)}, then p 6∈ hw(x). thus, since hw(x) =⋂ {u | u ∈ bx}, there exists v ∈ bx such that p /∈ v . observe that there is uβ ∈ up such that uβ ∩ v = ∅. thus, x ∈ ⋃ uβ. hence, x ∈ ⋃ { ⋃ uα | α ∈ κ}, therefore, ρ(h) ⊆ ⋃ { ⋃ uα | α ∈ κ}. now, since x is strictly quasi-κ-lindelöf y ρ(h) is a closed subset, for every α ∈ κ, there is vα ∈ [uα] ≤ω such that ρ(h) ⊆ ⋃ { ⋃ vα | α ∈ κ}. let f = {vα | α ∈ κ} and s0 = (∅,f). by construction, we note that s is generated by ( ⋃ e,ug(e)). moreover, p /∈ ⋃ { ⋃ vα | α ∈ κ}; that is, p ∈ x \ θ(s0). hence, we conclude that s0 is a small κ-sensor in x which is θ-good for e, a contradiction, because e is a (g,ρ,θ)-quasi-propeller. thus, we obtain that x ⊆ ρ(h)∗. therefore, |x| ≤ hw(x)2κ. � from theorem 3.9, we obtain the following result due to arhangel’skii (see [2, corollary 22]). corollary 3.10 ([2]). let x be a t2-space strictly quasi-χ(x)-lindelöf. then |x| ≤ 2χ(x). it is easy to see that if κ = c(x) or κ = l(x), then x is strictly quasi-κlindelöf. hence, theorem 3.9 is a common generalization of the inequalities |x| ≤ 2c(x)χ(x) and |x| ≤ 2l(x)χ(x), where x is a hausdorff space. problem 3.11. let x be a t1-space with h ∗(x) ≤ ω and strictly quasi-κlindelöf. (1) if ψ(x)t(x) ≤ κ, then |x| ≤ 2κ? (2) if ψ(x)f(x) ≤ κ, then |x| ≤ 2κ? c© agt, upv, 2019 appl. gen. topol. 20, no. 1 220 generic theorems in the theory of cardinal invariants finally, we present the last application of theorem 2.2 to prove some upper bounds to density, netweight and cardinality of hausdorff spaces, which are inspired by the bounds obtained by charlesworth in [9]. before this, we recall that if x is a hausdorff space, then a collection of open subsets of x, u, is called closed separating cover of x, if x = ⋃ u and ⋂ {u | u ∈ u and x ∈ u} = {x}, for each x ∈ x. moreover, the closed point separating weight of x, denoted pswc(x), is the smallest infinite cardinal κ such that x has a closed separating cover u, such that |ux| ≤ κ, where ux = {u ∈ u | x ∈ u}. we have the following result. theorem 3.12. if x is a hausdorff space, then d(x) ≤ pswc(x) alc(x). proof. let κ = alc(x) and τ = pswc(x). let u be a closed separating cover of x with |ux| ≤ κ, where ux = {u ∈ u | x ∈ u}. let l = [x] ≤τ κ , q = [p(x)]≤τ κ , and g : l → q given by g(f) = ⋃ {ux | x ∈ f}. we consider the operator identity ρ. clearly, ρ is (τ,κ)-closing. for each κ-sensor s = (a,f), we put θ(s) = ⋃ ( ⋃ f). thus, there exists a family e = {eα | α < τ} ⊆ l such that (1) and (2) of theorem 2.2 hold. clearly |p | ≤ τκ, where p = ⋃ e. we show that x ⊆ p . indeed, suppose p ∈ x \ p . since u is closed separating cover of x, for each x ∈ p , there exists ux ∈ ux such that p /∈ ux. clearly, the collection u = {ux | x ∈ p} covers p . hence, there exists a ∈ [p]≤κ such that p ⊆ ⋃ {ux | x ∈ a}. note that each x ∈ a may be replaced by an x′ ∈ p . indeed, since x ∈ a, then x ∈ p. hence, ux ∩ p 6= ∅. thus, there exists x ′ ∈ ux ∩ p . hence, ux ∈ ux′. it follows p ⊆ ⋃ {ux ′ x | x ′ ∈ a′}, where ux ′ x = ux. then, s0 = (∅,{{ux ′ x | x ′ ∈ a′}}) is a small κ-sensor which is θ-good for e, a contradiction. thus, x ⊆ p . therefore, d(x) ≤ pswc(x) alc(x). � corollary 3.13. if x is a hausdorff space, then nw(x) ≤ pswc(x) alc(x). proof. let κ = alc(x) and let u be a closed separating cover of x with |u| ≤ pswc(x). by theorem 3.12, there exists a dense subset d of x with |d| ≤ pswc(x) κ. let n = {x\ ⋃ v | v ∈ [u]≤κ}. notice that |n| ≤ |[d]≤κ| ≤ pswc(x) κ. thus |n| ≤ pswc(x) κ. we claim that n is a network on x. indeed, let p ∈ x and let u be an open subset of x such that p ∈ u. for each x ∈ x \ u, we fix ux ∈ u such that p /∈ ux. clearly {ux | x ∈ x \ u} covers x \ u. hence, there exists a ∈ [x \ u] ≤κ such that x \ u ⊆ ⋃ {ux | x ∈ a}. then p ∈ x \ ⋃ {ux | x ∈ a} ⊆ u. thus, n is a network on x. the proof is complete. � corollary 3.14. if x is a hausdorff space, then |x| ≤ pswc(x) alc(x)ψ(x). proof. it follows from [13, theorem 4.1] that |x| ≤ nw(x)ψ(x). since nw(x)ψ(x) ≤ (pswc(x) alc(x))ψ(x), then, by corollary 3.13, we obtain that |x| ≤ pswc(x) alc(x)ψ(x). � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 221 a. ramı́rez-páramo and j. f. tenorio acknowledgements. the authors thanks the referee for his/her valuable suggestions which improved the paper. references [1] o. t. alas, more topological cardinal inequalities, colloq. math. 65, no. 2 (1993), 165– 168. 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[17] s. willard and u. n. b. dissanayake, the almost lindelöf degree, canad. math. bull. 27, no. 4 (1984), 452–455. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 222 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 85-95 strongly path-confluent mappings abdo qahis and mohd. salmi md. noorani abstract in this paper, we introduce a new class of path-confluent mapping, called strongly path-confluent maps. we discuss and study some characterizations and some basic properties of this class of mappings. relations between this class and some other existing classes of mappings are also obtained. also we study some operations on this class of mappings, such as: composition property, composition factor property, component restriction property and path-component restriction property. 2010 msc: 54c05, 54c08, 54c10. keywords: continuum, connectedness, components, path-components, quasi-components, confluent maps, path-confluent maps, quasiconfluent maps, strongly confluent maps, strongly pathconfluent maps. 1. introduction and preliminaries throughout this work a space will always mean a topological space on which no separation axiom is assumed unless explicitly stated and all mappings are assumed to be continuous. in this paper, we obtain a new kind of path-confluent mapping called, strongly path-confluent. a subset k of a space x is said to be a continuum if k is connected and compact. using this idea of a continuum, charatonik introduced and studied the concept of confluent mapping between topological spaces [1] as follows: a mapping f : x −→ y is said to be confluent provided that for each subcontinuum k of y and for each component c of f−1(k) we have f(c) = k. motivated by charatonik’s work, we have introduced the notions of quasi-confluent and path-confluent mappings in [9, 10] and studied their basic properties. recall that space x is said to be connected between two subsets a and b if there is no closed-open set f such that a ⊂ f and b ∩ f = φ. the connectedness of a space x between points 86 a. qahis and m. s. m. noorani is an equivalence relation on x. the equivalence classes of this relation are called quasi-components of the space x, that is, a quasi-component of a space x containing a point p ∈ x is the set of all points x ∈ x such that the space x is connected between {p} and {x}. in other words, a quasi-component of a space x containing a point p ∈ x is the intersection of all closed and open subsets of x containing p (see [7]). moreover a mapping f : x −→ y is said to be quasi-confluent provided that for each continuum k subset of y and for each quasi-component qc of f−1(k) we have f(qc) = k. the notion of path-component of a point p ∈ x is a maximal path-connected subset of x containing p and it is denoted by pc(x, p) (see [5]). by using this notion we introduced the appropriate definition for path-confluent mapping as follows: a mapping f : x −→ y is said to be path-confluent provided that for each continuum k subset of y and for each path-component pc of f−1(k) we have f(pc) = k. in this paper we are interested in further generalization of the work of charatonik in the context of path-components and connectedness. in section 2 we introduce the notion of strongly path-confluent mapping and study some characterizations and some basic properties of this class of mappings. also we study its relation with other known classes of generalized confluent mappings, namely the classes of confluent, quasi-confluent, path-confluent, and strongly confluent mappings. in section 3 we study the composition property and composition factor property for this class. in section 4 we study the notion of path-component restriction property for the class of strongly path-confluent mappings. we denote by c, qc(or qt ), and pc(or pt ) the components, quasicomponents, and path-components of any topological spaces x at any point x ∈ x, respectively, and the symbol n is used for the set of natural numbers. now we recall some known notions, definitions which will be used in this work. definition 1.1 ([4]). a mapping f : x −→ y is said to be strongly confluent provided for each connected non-empty subset k of y and for each component c of f−1(k) we have f(c) = k. definition 1.2 ([11]). a mapping f : x −→ y is a local homeomorphism if for each point x ∈ x there exists an open neighborhood u of x such that f(u) is an open neighborhood of f(x) and the restriction mapping f | u : u −→ f(u) is a homeomorphism. definition 1.3 ([2]). a class m of mappings between topological spaces is said to have the component restriction property provided that for each mapping f : x −→ y ∈ m and for each b subset of y , if a ⊂ x is the union of some components of f−1(b), the restriction mapping f | a : a −→ f(a) ∈ m. strongly path-confluent mappings 87 2. strongly path-confluent mappings in this section we introduce the concept of strongly path-confluent mapping and discuss some of its interesting properties and its relations with other known mappings. definition 2.1. a mapping f : x −→ y is said to be strongly path-confluent provided that for each connected k subset of y and for each path-component pc of f−1 (k) we have f(pc) = k. proposition 2.2. the following statements are true: (1) every strongly path-confluent mapping is path-confluent; (2) every strongly path-confluent mapping is confluent; (3) every strongly path-confluent mapping is strongly confluent; (4) every strongly path-confluent mapping is quasi-confluent. proof. the proof comes directly from the definitions. � the following diagram follows immediately from the definitions in which none of these implications is reversible as shown by the following example. ? ? � � � � � �� h h h h h hj � � � � ��* h h h h h hyquasi-confluent path-confluent strongly confluent confluent strongly path-confluent diagram 2.1 example 2.3. let x = {(x, y) : x = 0, and y ∈ [−1, 1]} ∪ {(x, y) : y = sin π x : x ∈ (0, 1]} ⊂ r2 with usual topology and y = x/r, where r the equivalence relation in x, given by r = {((0, y), (0−y)) : y ∈ [−1, 1]}∪{(p, p) : p ∈ x}. then the natural projection f : x −→ y is: (1) confluent; (2) quasi-confluent; (3) strongly confluent. but, it is neither path-confluent nor strongly path-confluent mappings. proposition 2.4. let f : x −→ y be a mapping. if y is totally disconnected space, then the following items are equivalent: (1) f is strongly path-confluent; (2) f is path-confluent. proof. (1) =⇒ (2): follows immediately from the proposition 2.2. (2) =⇒ (1): the proof comes directly from the fact that in totally disconnected space the classes of connected and continua are the same. � 88 a. qahis and m. s. m. noorani recall that a space x is called almost discrete if every open subset of x is closed; equivalently, if every closed subset of x is open (see [6, 7]). proposition 2.5. let f : x −→ y be a mapping of locally path-connected space x into totally disconnected space y . if x is almost discrete and y is hausdorff, then the following items are equivalent: (1) f is strongly path-confluent; (2) f is strongly confluent; (3) f is path-confluent; (4) f is confluent; (5) f is quasi-confluent. proof. the proof of the implication (1) =⇒ (2) is obvious. (2) =⇒ (3): suppose that mapping f : x −→ y is a strongly confluent. let k be any continuum subset of y and pc be any path-component of f−1(k). then, k is connected subset of y . since y is totally disconnected and hausdorff space, then k is closed singleton set. since, x is locally path-connected and almost discrete space, then the set f−1(k) is locally path-connected. hence, its components and path-component are the same. thus, f(pc) = k by assumption. therefore f is path-confluent mapping. the proofs of the implications (3) =⇒ (4) and (4) =⇒ (5) are obvious. (5) =⇒ (1): assume that mapping f : x −→ y is a quasi-confluent. let k be any connected subset of y and pc be any path-component of f−1(k). since y is totally disconnected and hausdorff space, then k is closed singleton set. thus, k is continuum in y . since, x is locally path-connected and almost discrete space, then the set f−1(k) is locally path-connected. hence, its quasi-components and path-components are the same. thus f(pc) = k by assumption. therefore f is path-confluent mapping. � proposition 2.6. let f : x −→ y be a mapping of space x into compact space y . if every connected subset of y is closed, then the following items are equivalent: (1) f is strongly path-confluent; (2) f is path-confluent. proof. (1) =⇒ (2) :obvious. (2) =⇒ (1): let k ⊆ y be an arbitrary connected, and pc be an arbitrary path-component of f−1(k). then k ⊆ y is closed by the assumption. so, k is compact subset of y . thus, k is continuum subset of y . then f(k) = pc by assumption. therefore, f is strongly path-confluent. � proposition 2.7. if f : x −→ y is a mapping of space x into compact hausdorff space y , then the following statements are equivalent: (1) f is path-confluent; (2) for each closed connected k ⊆ y , the path-components of f−1(k) are mapped into k under f. strongly path-confluent mappings 89 proof. (1) =⇒ (2): let k ⊆ y be any closed connected, and pc be any pathcomponent f−1(k). since, y is compact space, then k is compact. thus, k is continuum in y . so, f(pc) = k by the path-confluence of f. (2) =⇒ (1): let k be any continuum in y , and pc be any path-component of f−1(k). since, y is hausdorff space, then k is closed. so, k is closed connected subset of y . thus, f(pc) = k. hence, f is path-confluent mapping. � recall that a connected space x is said to be σ-connected provided that it can not be decomposed into countably many mutually separated non-empty subsets, (see[3, 8]). also a space x is said to be hereditarily σ-connected provided that it is connected and each connected subset of it is σ-connected (see[4]). the following theorem shows the hereditarily σ-connected property is strongly path-confluent property. theorem 2.8. a surjective strongly path-confluent mapping preserves hereditarily σ-connected spaces. proof. assume that mapping f : x −→ y be a surjective strongly pathconfluent such that x is a hereditarily σ-connected space, and suppose on the contrary that y is not hereditarily σ-connected. let k be any connected subset of y such that k = ∞⋃ i=1 ai, where ai and aj are non-empty mutually separated sets for i 6= j and i, j ∈ n. then f−1(ai) and f −1(aj) are non-empty mutually separated for i 6= j and i, j ∈ n. let pc be a path-component of f−1(k). since, f is strongly pathconfluent, then we have f(pc) = k. so, we infer that pc ∩ f−1(ai) 6= φ for i ∈ n since, pc ⊂ f−1(k) = f−1( ∞⋃ i=1 ai) = ∞⋃ i=1 f−1(ai) then, pc = ∞⋃ i=1 (f−1(ai) ∩ pc), where f−1(ai)∩pc and f −1(aj)∩pc are non-empty mutually separated sets for i 6= j and i, j ∈ n. but this contradicts the fact that pc is σ-connected set. thus, k is σ-connected. hence, y is hereditarily σ-connected space. � theorem 2.9. let f : x −→ y be a mapping between topological spaces x and y such that y = y1 ∪ y2 is a decomposition of y into connected subsets. if the following properties hold: (1) either y1 ∩ y2 6= φ or y1, and y2 are separated; (2) the intersection of any two connected subsets of y is connected; (3) f |f−1(yi) is strongly path-confluent mapping for i = 1, 2. then f is strongly path-confluent. 90 a. qahis and m. s. m. noorani proof. let k be an arbitrary connected subset of y , and pc be any pathcomponent of f−1(k). assume that y = y1 ∪ y2, if y1 ∩ cl(y2) = φ = cl(y1) ∩ y2, then either k ⊆ y1 or k ⊆ y2. so that by condition (3) we infer that f(pc) = k. therefore, f is strongly path-confluent mapping. suppose that y1 ∩ y2 6= φ and that k − y1 6= φ 6= k − y2. let x ∈ pc such that f(x) ∈ y1. since, k∩y1 is a connected subset of y1, if pc1 is a path-component of x in f−1(k ∩ y1), then by the condition (3), we have (2.1) f(pc1) = k ∩ y1, and pc1 ⊆ pc. also, let x′ ∈ pc1 ∩ f −1(y) such that y ∈ k ∩ y1 ∩ y2 6= φ. since, k ∩ y2 is a connected subset of y2, if pc2 is a path-component of x ′ in f−1(k ∩ y2), then by the condition (3), we have (2.2) f(pc2) = k ∩ y2, and pc2 ⊆ pc. by (2.1) and (2.2) we obtain k = k ∩ (y1 ∪ y2) = (k ∩ y1) ∪ (k ∩ y2) = f(pc1) ∪ f(pc2) ⊆ f(pc). but, we always have f(pc) ⊆ k. so, f(pc) = k. hence, f is strongly path-confluent mapping. � the following corollary is a generalization of the theorem 2.9. corollary 2.10. if f : x −→ y be a mapping, and y1, y2, ..., yk are connected subsets of y such that y = y1 ∪ ... ∪ yk. if the following properties hold: (1) either yi ∩ yj 6= φ or yi, yj are separated, for each i 6= j and i, j ∈ {1, ..., k}; (2) the intersection of any two connected subsets of y is connected; (3) f |f−1(yi) is strongly path-confluent mapping for i ∈ {1, ..., k}. then f is strongly path-confluent. 3. the composition properties we say that a class m of mappings has the composition property provided that for any two mappings f : x −→ y and g : y −→ z belonging to m then their composition gof belongs to m. also, we say that a class m of mappings has the composition factor property provided that for any two mappings f : x −→ y and g : y −→ z such that gof belongs to m, then g belongs to m. before we prove the main results in this section, we need to introduce the following lemma. lemma 3.1. let f : x −→ y and g : y −→ z be two mappings. if f is a strongly path-confluent and h = gof, then for each connected k subset of z, and each path-component pc of h−1(k), we have f(pc) is a path-component of g−1(k). strongly path-confluent mappings 91 proof. let k be any connected in z, and pc be any path-component of h−1(k). since pc ⊆ h−1(k) = (gof)−1(k) = f−1(g−1(k)), then pc ⊆ f−1(g−1(k)). so, f(pc) ⊆ g−1(k). it is obviously that f(pc) ⊆ pt for some path-component pt of g−1(k). then, pc ⊆ f−1(pt ). since, f(pc) ⊆ pt ⊆ g−1(k), then pc ⊆ f−1(pt ) ⊆ f−1(g−1(k)) = (gof)−1(k) = h−1(k). thus, pc is the path-component of f−1(pt ). now, let q be any connected in pt . so, f−1(q) ⊆ f−1(pt ). since pc is a path-component of f−1(pt ), then, pc ∩ f−1(q) is the path-component of f−1(q). but, f is strongly pathconfluent mapping. thus, f(pc ∩ f−1(q)) = q = f(pc) ∩ q. which implies that q ⊆ f(pc) ⊆ g−1(k). hence, f(pc) = pt . therefore, f(pc) is a path-component of g−1(k). � the following theorem shows that the class of strongly path-confluent mappings has the composition property. theorem 3.2. let f : x −→ y and g : y −→ z be two strongly path-confluent mappings. then h = gof is strongly path-confluent mapping. proof. let k ⊆ z be a connected and pc be any path-component of h−1(k). since, f is a strongly path-confluent mapping, then f(pc) is a path-component of g−1(k) by the lemma 3.1. then by the strongly path-confluence of g, we infer that h(pc) = g(f(pc)) = k. hence, h = gof is strongly path-confluent mapping. � proposition 3.3. let f : x −→ y and g : y −→ z be two mappings. if f is a homeomorphism and g is a strongly path-confluent mapping, then gof is a strongly path-confluent mapping. proof. let k be a connected subset of z, and pc be the path-component of the inverse image (gof)−1(k). we want to prove that gof(pc) = k. obviously pc ⊆ (gof)−1(k) = f−1g−1(k). so, f(pc) ⊆ g−1(k). since, f is a homeomorphism, then f(pc) is the path-component of g−1(k). since, g is strongly path-confluent, then g(f(pc)) = k. therefore, gof is strongly path-confluent mapping. � the following theorem shows that the class of strongly path-confluent mappings has the composition factor property. theorem 3.4. let f : x −→ y and g : y −→ z be two mappings. if h = gof is strongly path-confluent mapping, then g is also strongly path-confluent mapping. proof. let k be a connected subset of z and pc be any path-component of g−1(k). then (3.1) g(pc) ⊆ k. 92 a. qahis and m. s. m. noorani on the other hand, since, h is a strongly path-confluent we have that for each x ∈ f−1(pc) h(pc(h−1(k), x)) = k. then f(pc(h−1(k), x)) ⊆ pc. so, we get (3.2) k = gf(pc(h−1(k), x)) ⊆ g(pc). then from (3.1) and (3.2), we get g(pc) = k. hence, g is strongly pathconfluent. � remark 3.5. if h = gof is strongly path-confluent mapping, then f is not necessarily strongly path-confluent mapping as shown by the following example. example 3.6. let x = {a, b, c, d, e}, y = {ℓ, m, n, o} and z = {p, q, r} with topologies τ = {φ, x, {a}, {c, d}, {a, c, d}, {c, d, e}, {c, d, a, e}}, σ = {φ, y, {n}, {n, o}}, and γ = {φ, z}} defined on x, y and z, respectively. let f : x −→ y be a mapping defined by: f(a) = ℓ, f(b) = m, f(c) = f(d) = n, and f(e) = o. also let g : y −→ z be a mapping defined by: g(ℓ) = g(m) = p, g(n) = r, and g(o) = q. assume that h = g ◦ f : x −→ z which is defined by: h(a) = h(b) = p, h(c) = h(d) = r, and h(e) = q. then the mappings h and g are strongly path-confluent, but f is not strongly path-confluent. note that, if we take the subcontinuum k = {ℓ, n} of y . then the path-components of f−1(k) = {a, c, d} are pc = {a} and pt = {c, d} and f(pc) = ℓ 6= k and f(pt ) = n 6= k. hence, f is not strongly path-confluent mapping. now, the following theorem clarifies under certain conditions, the mapping f will be strongly path-confluent. let a mapping f : x −→ y be given. recall that a subset a ⊂ x is said to be an inverse set under f provided that a = f−1(f(a)), (see [1]). theorem 3.7. let f : x −→ y and g : y −→ z be two mappings. if h = g ◦f is strongly path-confluent mapping, and if every set in y is an inverse set, then f is strongly path-confluent mapping. proof. let k be any connected set in y , and let pc be any path-component of f−1(k). then, g(k) will be connected in z. since g−1(g(k)) ⊂ y , and since, every set in y is an inverse set, then k is an inverse set and, thus g−1(g(k)) = k. then f−1g−1(g(k)) = f−1(k), which implies that h−1(g(k)) = (gof)−1(g(k)) = f−1(k). that is, pc is a path-component of h−1(g(k)). since h is a strongly path-confluent mapping. so, h(pc) = g(k). therefore, f(pc) = k. hence, f is a strongly path-confluent mapping. � strongly path-confluent mappings 93 corollary 3.8. let f : x −→ y and g : y −→ z be two mappings. if h = g ◦ f is strongly path-confluent mapping, and g is a homeomorphism then f is strongly path-confluent mapping. proof. since, g is a homeomorphism, then every set in y is an inverse set. then, by theorem 3.7, the mapping f is strongly path-confluent. � 4. the path-component restriction property in this section we study the path-component restriction property for the class of strongly path-confluent mappings. definition 4.1 ([10]). a class m of mappings between topological spaces is said to have the path-component restriction property provided that for each mapping f : x −→ y ∈ m and for each b subset of y , if a ⊂ x is the union of some path-components of f−1(b), the restriction mapping f | a : a −→ f(a) ∈ m. the following theorem shows that the class of strongly path-confluent mappings has the path-component restriction property. theorem 4.2. let f : x −→ y be a strongly path-confluent mapping and let b ⊆ y and a ⊂ x is the union of some path-components of f−1(b). then the restriction mapping f | a : a −→ f(a) is a strongly path-confluent. proof. assume that the mapping f : x −→ y is a strongly path-confluent. take b ⊆ y , and a is the union of some path-components of f−1(b). suppose that k be any connected subset of f(a), and let pc and pt be the pathcomponents of (f |a) −1(k) and f−1(k), respectively. since, (f |a) −1(k) = a ∩ f−1(k), then (4.1) pc ⊂ pt. since, pc ⊂ a, then φ 6= pc = a ⋂ pc ⊂ a ⋂ pt . but, k ⊂ f(a) ⊂ b. so, f−1(k) ⊂ f−1(f(a)) ⊂ f−1(b). according to the assumption on a, we get pt ⊂ f−1(k) ⊂ a, whence pt ⊂ (f |a) −1(k) ⊂ f−1(k). which implies that (4.2) pt ⊂ pc. then from (4.1) and (4.2), we get pc = pt , and f | a(pc) = f(pc) = f(pt ) = k. therefore the restriction mapping f | a is a strongly pathconfluent. � the following corollary is a particular case of the theorem 4.2. corollary 4.3. let f : x −→ y be a strongly path-confluent mapping. let a ⊂ x be an inverse set under f, then the restriction mapping f | a : a −→ f(a) is also strongly path-confluent. 94 a. qahis and m. s. m. noorani proposition 4.4. let f : x −→ y be a strongly path-confluent mapping. if f is local homeomorphism, then restriction mapping f | u : u −→ f(u) is also strongly path-confluent for every open subset u of x. proof. since, f is local homeomorphism, then the restriction mapping f | u : u −→ f(u) is a homeomorphism. which means that u = f−1(f(u)). hence, u ⊆ x is an inverse set. by the corollary 4.3, we infer that f | u is strongly path-confluent mapping. � the following proposition shows that if the mapping f : x −→ y has the component restriction property, then it also has the path-component restriction property, which means that the path-component restriction property is weaker notion than the notion of component restriction property. proposition 4.5. let m denote the class of mappings. if m has the component restriction property, then it has also the path-component restriction property. proof. assume that class m of mappings has the component restriction property and let f : x −→ y ∈ m. take b be a subset of y , and a ⊂ x is the union of some components of f−1(b). then the restriction mapping f | a : a −→ f(a) ∈ m. we need to show that a ⊂ x is the union of some path-components of the set f−1(b). now, let a = ∪ α∈∆ cα, for some components cα of the inverse set f −1(b), where ∆ be the index set. since, each component is a disjoint union of path-components, then we can put cα = ∪ β∈i pcβ with ∩ β∈i pcβ = φ, for some path-components pcβ of the inverse set f −1(b), where i be the index set. hence, we get a = ∪ α∈∆ cα = ∪ α∈∆ ( ∪ β∈i pcβ). then by the definition 4.1, the class m of mappings has the path-component restriction property. � proposition 4.6. the classes of strongly path-confluent mappings has the component restriction property. proof. let m be the class of strongly path-confluent mappings, and let f : x −→ y ∈ m. take b ⊆ y , and a is the union of some components of f−1(b). let k ⊂ f(a) be a subcontinuum, and c and pc be the component and path-component of (f | a) −1(k) = a ∩ f−1(k). thus, c is contained in a component t of f−1(k). let pt be the path-component of f−1(k). obviously, pc ⊂ c and pt ⊂ t . so, pc ⊂ c ⊂ t . since c ⊂ a, it follows that φ 6= c = a ∩ c ⊂ a ∩ t . further k ⊂ f(a) ⊂ b, implies that t ⊂ f−1(b). according to the assumptions on a, we infer that t ⊂ a, whence pt ⊂ t ⊆ (f | a) −1(k). which implies that t = c. thus pc = pt and consequently (f | a)(pc) = f(pc) = f(pt ) = k by theorem 4.2. therefore, f | a is strongly path-confluent mapping. � strongly path-confluent mappings 95 acknowledgements. the authors would like to acknowledge the financial support received from universiti kebangsaan malaysia under the research grant ukm topdown-st-06-frgs00012012. the authors also wish to gratefully acknowledge all those who have generously given of their time to referee our paper. references [1] j. j. charatonik, confluent mappings and unicoherence of continua, fund. math. 56 (1964), 213–220. [2] j. j. charatonik, component restriction property for classes of mappings, mathematica pannonica 14, no. 1. (2003), 135–143. [3] j. grispolakis, a. lelek and e. d. tymchatyn, connected subsets of nitely suslinian continua, colloq. math. 35 (1976), 31–44. [4] j. grispolakis, confluent and related mappings defined by means of quasi-components, can. j. math. 30, no. 1, (1978), 112–132. [5] f. c. helen, introduction to general topology, boston: university of massachusetts, 1968. [6] k. kuratowski, topology, volume i, pwn polish scientific publishers, academic press, warsaw, new york and london, 1966. [7] k. kuratowski, topology, vol. 2, academic press and pwn, 1968. [8] a. lelek, ensembles-connexes et le thoereme de gehman, fund. math. 47 (1959), 265– 276. [9] a. qahis and m. s. m. noorani, on quasi-ω-confluent mappings, internat. j. math. math. sci. 2011, article id 270704, 9 pages. [10] a. qahis and m. s. m. noorani, a strong form of confluent mappings, questions and answers in general topology 30, no. 2 (2012), 139–152. [11] g. t. whyburn, analytic topology, amer. math. soc. colloq. publ.28, providence, 1942. (received may 2012 – accepted october 2012) abdo qahis (cahis82@gmail.com) school of mathematical sciences, faculty of science and technology,university kebangsaan malaysia, 43600 ukm, selangor darul ehsan, malaysia mohd. salmi md. noorani (msn@ukm.my) school of mathematical sciences, faculty of science and technology,university kebangsaan malaysia, 43600 ukm, selangor darul ehsan, malaysia strongly path-confluent mappings. by a. qahis and m. s. m. noorani @ appl. gen. topol. 21, no. 2 (2020), 177-194 doi:10.4995/agt.2020.11369 c© agt, upv, 2020 fixed points for fuzzy quasi-contractions in fuzzy metric spaces endowed with a graph mina dinarvand department of mathematics, faculty of mathematical sciences and computer, kharazmi university, 50 taleghani avenue, 15618, tehran, iran (dinarvand mina@yahoo.com) communicated by i. altun abstract in this paper, we introduce the notion of g-fuzzy h-quasi-contractions using directed graphs in the setting of fuzzy metric spaces endowed with a graph and we show that this new type of contraction generalizes a large number of different types of contractions. subsequently, we investigate some results concerning the existence of fixed points for this class of contractions under two different conditions in m-complete fuzzy metric spaces endowed with a graph. our main results of the work significantly generalize many known comparable results in the existing literature. examples are given to support the usability of our results and to show that they are improvements of some known ones. 2010 msc: 54h25; 47h10; 05c40. keywords: fuzzy metric space; (!c)-graph; !g-fuzzy quasi-contraction; fixed point. 1. introduction and preliminaries in attempt to model the real world problems, we have to deal with uncertainties and vagueness of the data, tools or conditions in the form of constraints. to deal with uncertainty, we need techniques other than classical ones wherein some specific logic is required. fuzzy set theory is one of the uncertainty approaches wherein topological structures are basic tools to develop mathematical models compatible to concrete real life situations. zadeh [21] considered the received 08 february 2019 – accepted 24 june 2020 http://dx.doi.org/10.4995/agt.2020.11369 m. dinarvand nature of uncertainty in the behaviour of systems possessing fuzzy nature by means of a fuzzy set. in 1994, george and veeramani [11] modified the concept of fuzzy metric space introduced by kramosil and michálek [15]. definition 1.1 (george and veeramani [11]). a fuzzy metric space is a triple (x, m, !) such that x is a nonempty set, ! is a continuous t-norm and m is a fuzzy set on x × x × (0, ∞) satisfying the following conditions: (fm1) m(x, y, t) > 0 for all x, y ∈ x and each t > 0; (fm2) m(x, y, t) = 1 for all x, y ∈ x and each t > 0 if and only if x = y; (fm3) m(x, y, t) = m(y, x, t) for all x, y ∈ x and each t > 0; (fm4) m(x, z, t + s) ≥ m(x, y, t) ! m(y, z, s) for all x, y, z ∈ x and each t, s > 0; (fm5) m(x, y, ·) : (0, ∞) → [0, 1] is continuous. if we replace (fm4) by (na) m(x, z, max{t, s}) ≥ m(x, y, t) ! m(y, z, s) for all x, y, z ∈ x and each t, s > 0, then the triple (x, m, !) is called a non-archimedean fuzzy metric space. it should be noted that any non-archimedean fuzzy metric space is a fuzzy metric space. example 1.2 ([11]). let (x, d) be a metric space. then the triple (x, md, !) is a fuzzy metric space, where a ! b = ab for all a, b ∈ [0, 1] and md(x, y, t) = t t+d(x,y) for all x, y ∈ x and each t > 0. we call this md as the standard fuzzy metric induced by the metric d. even if we define a ! b = min{a, b} for all a, b ∈ [0, 1], then the triple (x, md, !) will be a fuzzy metric space. let (x, md, !) be a fuzzy metric space. for t > 0, the open ball b(x, r, t) with a center x ∈ x and a radius 0 < r < 1 is defined by b(x, r, t) = ! y ∈ x : m(x, y, t) > 1 − r " . the collection {b(x, r, t) : x ∈ x, 0 < " < 1, t > 0} is a neighbourhood system for the topology τ on x induced by the fuzzy metric m. lemma 1.3 ([9]). let (x, m, !) be a fuzzy metric space. then m is a continuous function on x × x × (0, ∞). definition 1.4 (george and veeramani [11]). let (x, m, !) be a fuzzy metric space. (1) a sequence {xn} in x is said to be convergent to a point x ∈ x, denoted by xn → x as n → ∞, if and only if limn→+∞ m(xn, x, t) = 1 for all t > 0, i.e. for each r ∈ (0, 1) and t > 0, there exists n0 ∈ n such that m(xn, x, t) > 1 − r for all n ≥ n0. (2) a sequence {xn} in x is a m-cauchy sequence if and only if for all " ∈ (0, 1) and t > 0, there exists n0 ∈ n such that m(xn, xm, t) > 1 − " for all m, n ≥ n0. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 178 quasi-contractions in fuzzy metric spaces with a graph (3) an m-complete fuzzy metric space is a fuzzy metric space in which every m-cauchy sequence is convergent. definition 1.5 ([10]). let (x, m, !) be a fuzzy metric space. a mapping t : x → x is called t-uniformly continuous if for all r ∈ (0, 1), there exists s ∈ (0, 1) such that m(x, y, t) ≥ 1 − s implies m(tx, ty, t) ≥ 1 − r for all x, y ∈ x and each t > 0. remark 1.6. if t is t-uniformly continuous, then it is uniformly continuous for the uniformity generated by m and so it is continuous for the topology deduced from m. for the details concerning a uniform structure in a fuzzy metric space, the reader is directed to [10]. fixed point theory is one of the most fruitful and effective tools in mathematics which has enormous applications within as well as outside the mathematics. the paper of grabiec [9] started the investigations concerning fixed point theory in fuzzy metric spaces. afterwards, gregori and sapena [10] introduced the notion of fuzzy contractive mappings and gave some fixed point results in fuzzy metric spaces. recently, wardowski [20] introduced the following class of functions which will be used densely in the sequel. denote by h the family of all the mappings η : (0, 1] → [0, ∞) satisfying the following properties: (h1) η transforms (0, 1] onto [0, ∞); (h2) η is strictly decreasing (i.e. s < t implies η(s) > η(t) for all t, s ∈ (0, 1]). it is worth mentioning that if η ∈ h, then η(1) = 0 and η is continuous. theorem 1.7 (wardowski [20]). let (x, m, !) be an m-complete fuzzy metric space and suppose that t : x → x be a self-mapping satisfying η # m(tx, ty, t) $ ≤ kη # m(x, y, t) $ (1.1) for all x, y ∈ x and each t > 0, where k ∈ (0, 1). assume also that the following assertions hold: (a) %k i=1 m(x, tx, ti) ∕= 0 for all x ∈ x, k ∈ n and any sequence {tn} ⊆ (0, ∞), tn ↓ 0; (b) r ! s > 0 implies η(r ! s) ≤ η(r) + η(s) for all r, s ∈ ! m(x, tx, t) : x ∈ x, t > 0 " ; (c) ! η(m(x, tx, ti)) : i ∈ n " is bounded for all x ∈ x and any sequence {tn} ⊆ (0, ∞), tn ↓ 0. then t has a unique fixed point in x∗ ∈ x and for each x0 ∈ x the sequence {t nx0} converges to x∗. by considering a mapping η ∈ h of the form η(t) = 1 t − 1 where t ∈ (0, 1], the fuzzy contraction condition (1.1) reduces to the class of fuzzy contractive mappings introduced by gregori and sapena [10]. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 179 m. dinarvand on the other hand, the most important graph theory approach to metric fixed point theory introduced so far is attributed to jachymski [13]. in this new approach, the underlying metric space is equipped with a directed graph and the banach contraction is formulated in a graph language. using this simple but very interesting idea, jachymski generalized several well known versions of banach contraction principle in metric spaces simultaneously and from various aspects. we commence by reviewing some basic notions in graph theory which will be used throughout this paper. the readers interested in this topic are referred to [5, 8, 19] and references cited therein. in an arbitrary (not necessarily simple) graph g, a link is an edge of g with distinct ends and a loop is an edge of g with identical ends. two or more links of g with the same pairs of ends are called parallel edges of g. let (x, m, !) be a fuzzy metric space and ∆(x) denotes the diagonal of the cartesian product x ×x. consider a directed graph g such that the set v (g) of the vertices of g coincides with x, i.e. v (g) = x, and the set e(g) of the edges of g contains all loops, i.e. e(g) ⊇ ∆(x) (note that in general, g can have uncountably many vertices). suppose further that g has no parallel edges. in this case, the graph g can be simply denoted by the ordered pair g = (v (g), e(g)) = (x, e(g)). if g is such a graph, then it is said that the fuzzy metric space (x, m, !) is endowed with the graph g. by the notation g−1, it is meant the conversion of g as usual, i.e. a directed graph obtained from g by reversing the directions of the edges of g, and by the notation &g, it is always meant the undirected graph obtained from g by ignoring the directions of the edges of g. thus, it is clear that v (g−1) = v ( &g) = v (g) = x and so e(g−1) = ! (x, y) ∈ x × x : (y, x) ∈ e(g) " and e( &g) = e(g) ∪ e(g−1). it should be remarked that if both (x, y) and (y, x) belong to e(g), then we will face with parallel edges in the graph &g. to avoid this problem, we delete either the edge (x, y) or the edge (y, x) (but not both of them) from g and consider the graph &g obtained from the remaining graph. if (x, ≼) is a partially ordered set, then by comparable elements of (x, ≼), it is meant two elements x, y ∈ x satisfying either x ≼ y or y ≼ x, and a mapping t : x → x is called order-preserving whenever x ≼ y implies tx ≼ ty for all x, y ∈ x. in 1974, ćirić [7] introduced quasi-contractions in metric spaces and gave an example to show that this new contraction is a real generalization of some well known linear contraction. the main purpose of the present work is to formulate a g-fuzzy h-quasi-contraction which generalizes a large number of contractions in fuzzy metric spaces endowed with a graph. we then investigate some sufficient conditions which ensure the existence of fixed points for such mappings on m-complete fuzzy metric spaces in the sense of george and veeramani endowed with a graph. the obtained results generalize many known results in the recent literature. some examples are given which illustrate the c© agt, upv, 2020 appl. gen. topol. 21, no. 2 180 quasi-contractions in fuzzy metric spaces with a graph value of the obtained results in comparison to some of the existing ones in literature. 2. main results suppose that (x, m, !) be a fuzzy metric space endowed with a graph g and t : x → x be an arbitrary mapping. throughout this section, we use the letter ct to denote the set of all points x ∈ x such that (t mx, t nx) ∈ e( &g) for all m, n ∈ n ∪ {0}. remark 2.1. let (r, d) be the usual (euclidean) metric space of all real numbers and (r, md, !) be the standard fuzzy metric space induced by d. consider a graph g given by v (g) = r and e(g) = {(x, x) : x ∈ r}. define a mapping t : r → r by the rule tx = x + 2 for all x ∈ r. now one can see easily that ct = ∅. given x ∈ x and n ∈ n ∪ {0}, the n-th orbit of x under t is denoted by o(x; n), i.e. o(x; n) = ! x, tx, . . . , t nx " . if a is a subset of x, then by δt(a), it is meant the diameter of a in x, i.e. δt(a) = sup ! η # m(x, y, t) $ : x, y ∈ a " . motivated by aleomraninejad et al. [1], we say that g is a (&c)-graph whenever the triple (x, d, g) has the following property: if {xn} is a sequence in (x, d, g) such that xn → x ∈ x and (xn, xn+1) ∈ e( &g) for all n ∈ n, then there exists a subsequence {xnk} of {xn} such that (xnk, x) ∈ e( &g) for all k ∈ n. now, we are ready to introduce the concept of g-fuzzy h-quasi-contractions with respect to η ∈ h in fuzzy metric spaces endowed with a graph which is inspired by [2, definition 2.2] and [13, definition 2.1]. definition 2.2. let (x, m, !) be a fuzzy metric space endowed with a graph g and t : x → x be a mapping. we say t is a g-fuzzy h-quasi-contraction with respect to η ∈ h if (fq1) t preserves the edges of g, i.e. (x, y) ∈ e(g) implies (tx, ty) ∈ e(g) for all x, y ∈ x; (fq2) there exists λ ∈ (0, 1) such that η # m(tx, ty, t) $ ≤ λ max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ η # m(x, ty, t) $ , η # m(y, tx, t) $" for all x, y ∈ x with (x, y) ∈ e(g) and each t > 0. if t is a g-fuzzy h-quasi-contraction with respect to η ∈ h, then we call λ in (fq2) a quasi-contractive constant of t . we now give some examples of g-fuzzy h-quasi-contractions with respect to η ∈ h in fuzzy metric spaces endowed with a graph. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 181 m. dinarvand example 2.3. suppose that (x, m, !) is a fuzzy metric space endowed with a graph g and x0 be a point in x. it is elementary to check that the constant mapping x t.→ x0 is a g-fuzzy h-quasi-contraction with respect to η ∈ h with arbitrary quasi-contractive constant λ ∈ (0, 1) since e(g) contains all the loops. so the cardinality of the set of all g-fuzzy h-quasi-contractions defined on a fuzzy metric space (x, m, !) endowed with a graph g is no less than the cardinality of x. example 2.4. suppose that (x, m, !) is a fuzzy metric space and t : x → x is a fuzzy h-quasi-contraction with respect to η ∈ h in the sense that there exists λ ∈ (0, 1) such that η # m(tx, ty, t) $ ≤ λ max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" (2.1) for all x, y ∈ x and any t > 0. define a graph g0 by v (g0) = x and e(g0) = x × x, i.e. g0 is the complete graph whose vertex set coincides with x. obviously, t preserves the edges of g0 and (2.1) guarantees that t satisfies (fq2) for the complete graph g0. thus, t is a g0-fuzzy h-quasi-contraction with respect to η ∈ h with the quasi-contractive constant λ ∈ (0, 1). hence, g0-fuzzy h-quasi-contractions with respect to η ∈ h on fuzzy metric spaces endowed with the graph g0 are precisely the fuzzy h-quasi-contractions with respect to η ∈ h on fuzzy metric spaces. therefore, g-fuzzy h-quasi-contractions with respect to η ∈ h are a generalization of fuzzy h-quasi-contractions with respect to η ∈ h from fuzzy metric spaces to fuzzy metric spaces endowed with a graph. as stated before, the concept of quasi-contractions in metric spaces initiated by ćirić [7] in 1974. moreover, rhoades [18] showed that ćirić’s contractive condition is one of the most general contractive definitions in metric spaces and includes a large number of different types of contractions. example 2.5. let (x, ≼) be a partially ordered set and (x, m, !) be a fuzzy metric space. consider the poset graphs g1 and g2 by v (g1) = x and e(g1) = ! (x, y) ∈ x × x : x ≼ y " , and v (g2) = x and e(g2) = ! (x, y) ∈ x × x : x ≼ y ∨ y ≼ x " . a mapping t : x → x preserves the edges of g1 if and only if t is orderpreserving, and t satisfies (fq2) for the graph g1 if and only if there exists λ ∈ (0, 1) such that η # m(tx, ty, t) $ ≤ λ max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" (2.2) for all comparable elements x, y ∈ x and any t > 0, where η ∈ h. moreover, t preserves the edges of g2 if and only if t maps comparable elements of (x, ≼) onto comparable elements, and t satisfies (fq2) for the graph g2 if and only if (2.2) holds for all comparable elements x, y ∈ x and any t > 0. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 182 quasi-contractions in fuzzy metric spaces with a graph hence, if t is a g1-fuzzy h-quasi-contraction with respect to η ∈ h, then t is a g2-fuzzy h-quasi-contraction with respect to η ∈ h. therefore, g-fuzzy hquasi-contractions with respect to η ∈ h are a generalization of ordered fuzzy h-quasi-contractions with respect to η ∈ h from fuzzy metric spaces equipped with a partial order to fuzzy metric spaces endowed with a graph. from now on, we assume that the graphs g0, g1 and g2 are as defined in examples 2.4 and 2.5. remark 2.6. in the definitions of (&c)-graph and the set ct , let’s set g the special graphs g0, g1 and g2. then we obtain the following special cases: • the set ct related to the complete graph g0 coincides with x and g0 is a (&c)-graph. • if ≼ is a partial order on x, then the set ct related to the graph g1 (and also g2) consists of all points x ∈ x whose every two iterates under t are comparable elements of x. moreover, g1 (and also g2) is a (&c)-graph whenever the quadruple (x, m, !, ≼) has the following property: (∗): if {xn} is a sequence in (x, m, !) converging to a point x ∈ x whose successive terms are pairwise comparable elements of (x, ≼), then there exists a subsequence of {xn} whose terms and x are comparable elements of (x, ≼). example 2.7. let x = [0, 1] and ! be the usual product. then (x, m, !) is a fuzzy metric space, where m(x, y, t) = ' t t + 1 (|x−y| for all x, y ∈ x and each t > 0. define a self-mapping t : x → x by the formula tx = ) 1 9 , x = 0, 1 3 , 0 < x ≤ 1. we show that t is not a g0-fuzzy h-banach contraction with respect to η ∈ h on x. arguing by contradiction, we suppose that there exists η ∈ h such that η # m(tx, ty, t) $ ≤ λη # m(x, y, t) $ for all x, y ∈ x and each t > 0, where λ ∈ (0, 1) is a constant. now, by taking the points x = 0, 0 < y ≤ 1 and t = 1 in the above inequality, we get η((1 2 ) 2 9 ) ≤ λη((1 2 )y) and so η ##1 2 $ 2 9 $ ≤ λ lim y→0+ η ##1 2 $y$ = λη(1) = 0, which gives a contradiction. on the other hand, from the equality ln m( 1 9 , 1 3 , t) = 2 3 ln m(0, 1 3 , t) for all t > 0, c© agt, upv, 2020 appl. gen. topol. 21, no. 2 183 m. dinarvand it immediately follows that for all x, y ∈ x and each t > 0, η # m(tx, ty, t) $ ≤ 2 3 max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" , by considering a mapping η ∈ h of the form η(s) = ln(1 s ) for s ∈ (0, 1]. therefore, t is a g0-fuzzy h-quasi-contraction with respect to η ∈ h with the quasi-contractive constant λ = 2 3 . remark 2.8. suppose that (x, m, !) is a fuzzy metric space endowed with a graph g and t : x → x is a g-fuzzy h-banach contraction with respect to η ∈ h in the sense that t preserves the edges of g and there exists α ∈ (0, 1) such that η # m(tx, ty, t) $ ≤ αη # m(x, y, t) $ for all x, y ∈ x with (x, y) ∈ e(g) and any t > 0. if (x, y) ∈ e(g), then η # m(tx, ty, t) $ ≤ αη # m(x, y, t) $ ≤ α max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" . therefore, t satisfies (fq2) and so t is a g-fuzzy h-quasi-contraction with respect to η ∈ h. hence every g-fuzzy h-contraction with respect to η ∈ h is a g-fuzzy h-quasi-contraction with respect to η ∈ h. remark 2.9. suppose that (x, m, !) is a fuzzy metric space endowed with a graph g and t : x → x is a g-fuzzy h-kannan contraction with respect to η ∈ h in the sense that t preserves the edges of g and there exists α ∈ (0, 1 2 ) such that η # m(tx, ty, t) $ ≤ α # η # m(x, tx, t) $ + η # m(y, ty, t) $$ for all x, y ∈ x with (x, y) ∈ e(g) and any t > 0 (see [14] for the definition in metric spaces). if (x, y) ∈ e(g), then η # m(tx, ty, t) $ ≤ α # η # m(x, tx, t) $ + η # m(y, ty, t) $$ ≤ 2α max ! η # m(x, tx, t) $ , η # m(y, ty, t) $" ≤ 2α max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" . therefore, t satisfies (fq2) and so t is a g-fuzzy h-quasi-contraction with respect to η ∈ h. hence every g-fuzzy h-kannan contraction with respect to η ∈ h is a g-fuzzy h-quasi-contraction with respect to η ∈ h. remark 2.10. suppose that (x, m, !) is a fuzzy metric space endowed with a graph g and t : x → x is a g-fuzzy h-chatterjea contraction with respect to c© agt, upv, 2020 appl. gen. topol. 21, no. 2 184 quasi-contractions in fuzzy metric spaces with a graph η ∈ h in the sense that t preserves the edges of g and there exists α ∈ (0, 1 2 ) such that η # m(tx, ty, t) $ ≤ α # η # m(x, ty, t) $ + η # m(y, tx, t) $$ for all x, y ∈ x with (x, y) ∈ e(g) and any t > 0 (see [6] for the definition in metric spaces). if (x, y) ∈ e(g), then an argument similar to that appeared in remark 2.9 establishes that η # m(tx, ty, t) $ ≤ 2α max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" . therefore, t satisfies (fq2) and so t is a g-fuzzy h-quasi-contraction with respect to η ∈ h. hence every g-fuzzy h-chatterjea contraction with respect to η ∈ h is a g-fuzzy h-quasi-contraction with respect to η ∈ h. remark 2.11. suppose that (x, m, !) is a fuzzy metric space endowed with a graph g and t : x → x is a g-fuzzy h-ćirić-reich-rus contraction with respect to η ∈ h in the sense that t preserves the edges of g and there exist α, β, γ ≥ 0 with α + β + γ < 1 such that η # m(tx, ty, t) $ ≤ αη # m(x, y, t) $ + βη # m(x, tx, t) $ + γη # m(y, ty, t) $ for all x, y ∈ x with (x, y) ∈ e(g) and any t > 0 (see [17] for the definition in metric spaces). if (x, y) ∈ e(g), then an argument similar to that appeared in remark 2.9 establishes that η # m(tx, ty, t) $ ≤ (α + β + γ) max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" . therefore, t satisfies (fq2) and so t is a g-fuzzy h-quasi-contraction with respect to η ∈ h. hence every g-fuzzy h-ćirić-reich-rus contraction with respect to η ∈ h is a g-fuzzy h-quasi-contraction with respect to η ∈ h. remark 2.12. suppose that (x, m, !) is a fuzzy metric space endowed with a graph g and t : x → x is a g-fuzzy h-hardy-rogers contraction with respect to η ∈ h in the sense that t preserves the edges of g and there exist α, β, γ, δ, θ ≥ 0 with α + β + γ + δ + θ < 1 such that η # m(tx, ty, t) $ ≤ αη # m(x, y, t) $ + βη # m(x, tx, t) $ + γη # m(y, ty, t) $ + δη # m(x, ty, t) $ + θη # m(y, tx, t) $ for all x, y ∈ x with (x, y) ∈ e(g) and any t > 0 (see [12] for the definition in metric spaces). if (x, y) ∈ e(g), then an argument similar to that appeared in remark 2.9 establishes that η # m(tx, ty, t) $ ≤ (α + β + γ + δ + θ) max ! η # m(x, y, t) $ , η # m(x, tx, t) $ , η # m(y, ty, t) $ , η # m(x, ty, t) $ , η # m(y, tx, t) $" . therefore, t satisfies (fq2) and so t is a g-fuzzy h-quasi-contraction with respect to η ∈ h. hence every g-fuzzy h-hardy-rogers contraction with respect to η ∈ h is a g-fuzzy h-quasi-contraction with respect to η ∈ h. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 185 m. dinarvand note that every fuzzy h-contractive mapping (1.1) due to wardowski [20] is t-uniformly continuous. in the next example, we see that a g-fuzzy h-quasicontraction with respect to η ∈ h need not be even continuous. example 2.13. let (r+, d) be the usual (euclidean) metric space of all nonnegative real numbers and (r+, md, !) be the standard fuzzy metric space induced by d. consider a graph g given by v (g) = r+ and e(g) = ∆ ∪ ! (x, y) ∈ x × x : x, y ∈ q ∩ r+ with x ≤ y " , where q is the set of all rational numbers. define a mapping t : r+ → r+ by the rule tx = ) x 2 , x ∈ r+ ∩ q, 0, otherwise. then t is a g-fuzzy h-quasi-contraction with respect to η ∈ h with the quasi-contractive constant λ = 1 2 . obviously, t is only continuous at zero. in particular, t is not continuous on the whole r+. the following proposition is an immediate consequence of the definition of gfuzzy h-quasi-contractions with respect to η ∈ h and gives a simple procedure to construct new g-fuzzy h-quasi-contractions with respect to η ∈ h from older ones. proposition 2.14. let (x, m, !) be a fuzzy metric space endowed with a graph g and t : x → x be a mapping. (1) if t preserves the edges of g, then t preserves the edges of g−1 (resp. &g); (2) if t satisfies (fq2) for the graph g, then t satisfies (fq2) for the graph g−1 (resp. &g); (3) if t is a g-fuzzy h-quasi-contraction with respect to η ∈ h with a quasi-contractive constant λ ∈ (0, 1), then t is a g−1-fuzzy h-quasicontraction (resp. &g-fuzzy h-quasi-contraction) with respect to η ∈ h with a quasi-contractive constant λ. to prove the existence of a fixed point for a g-fuzzy h-quasi-contraction with respect to η ∈ h, we make use of the following useful lemmas. lemma 2.15. let (x, m, !) be a fuzzy metric space endowed with a graph g and t : x → x be a g-fuzzy h-quasi-contraction with respect to η ∈ h with a quasi-contractive constant λ ∈ (0, 1). then η # m(t ix, t jx, t) $ ≤ λδt # o(x; n) $ i, j = 1, . . . , n for all x ∈ ct and each t > 0 and any n ∈ n. proof. suppose that x ∈ ct and n ∈ n be given. if i and j are arbitrary positive integers no more than n, then (t i−1x, t j−1x) ∈ e( &g). according to proposition 2.14, t is also a &g-fuzzy h-quasi-contraction with respect to η ∈ h c© agt, upv, 2020 appl. gen. topol. 21, no. 2 186 quasi-contractions in fuzzy metric spaces with a graph with a quasi-contractive constant λ ∈ (0, 1). in particular, t satisfies (fq2) for the graph &g. hence, we have η # m(t ix, t jx, t) $ = η # m(tt i−1x, tt j−1x, t) $ ≤ λ max ! η # m(t i−1x, t j−1x, t) $ , η # m(t i−1x, t ix, t) $ , η # m(t j−1x, t jx, t) $ , η # m(t i−1x, t jx, t) $ , η # m(t j−1x, t ix, t) $" ≤ λδt # o(x; n) $ for all t > 0. □ example 2.16. consider the set r of real numbers with the usual (euclidean) metric and the standard fuzzy metric space (r, m, !). let g0 be a the complete graph and define a mapping t : r → r as tx = x 2 for all x ∈ r. then t is a g0-quasi-contraction with respect to η ∈ h with respect to η ∈ h with a quasi-contractive constant λ = 1 2 . meanwhile, t nx = x 2n and δt(o(x; n)) = |x| t (1 − 1 2n ) for all x ∈ r and all n ∈ n ∪ {0}. now, let x0 be a positive real numbers. take a mapping η ∈ h of the form η(t) = 1 t − 1 for t ∈ (0, 1] and put n = 2, i = 0 and j = 1 in lemma 2.14. hence, we have η # m(x, tx, t) $ = |x0 − tx0| t = x0 2t > x0 2t * 1 − 1 22 + = λδt # o(x0; 2) $ . lemma 2.17. let (x, m, !) be a fuzzy metric space endowed with a graph g and t : x → x be a g-fuzzy h-quasi-contraction with respect to η ∈ h. then for all x ∈ ct and each n ∈ n, there exists a positive integer k no more than n such that δt # o(x; n) $ = η # m(x, t kx, t) $ for all t > 0. proof. suppose that x ∈ ct and n ∈ n be given. on the one hand, if δt # o(x; n) $ = 0, then o(x; n) is singleton. in particular, x is a fixed point for t and η # m(t ix, t jx, t) $ = 0 for all i, j = 0, 1, . . . , n and each t > 0. hence, the assertion holds trivially for any positive integer k no more than n. on the other hand, if since o(x; n) is a finite set, it follows that there exist distinct nonnegative integers i and j no more that n such that δt # o(x; n) $ = η # m(t ix, t jx, t) $ for all t > 0. if both the integers i and j are assumed to be positive, then from lemma 2.15, we have δt # o(x; n) $ = η # m(t ix, t jx, t) $ ≤ λδt # o(x; n) $ for all t > 0, where λ ∈ (0, 1) is a quasi-contraction constant of t , which is a contradiction. therefore, either i or j must be zero. □ remark 2.18. combining lemmas 2.15 and 2.17, one can easily obtain that if (x, m, !) is a fuzzy metric space endowed with a graph g and t : x → x be a g-fuzzy h-quasi-contraction with respect to η ∈ h with a quasi-contraction c© agt, upv, 2020 appl. gen. topol. 21, no. 2 187 m. dinarvand constant λ, then for all x ∈ ct and each n ∈ n, there exists a positive integer k no more than n such that η # m(t ix, t jx, t) $ ≤ λδt # o(x; n) $ = λη # m(x, t kx, t) $ i, j = 1, . . . , n for all t > 0. lemma 2.19. let (x, m, !) be a fuzzy metric space endowed with a graph g and t : x → x be a g-fuzzy h-quasi-contraction with respect to η ∈ h such that (i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈ ! m(t ix, t jx, t) : x ∈ x, t > 0, i, j ∈ n " ; (ii) ! η(m(x, tx, ti)) : i ∈ n " is bounded for all x ∈ x and any sequence {tn} ⊆ (0, ∞), tn ↓ 0. then the sequence {t nx} is cauchy for all x ∈ ct . proof. suppose that x ∈ ct and n ∈ n ∪ {0} be given. if n = 0, then there remains nothing to prove since δt # o(x; 0) $ = 0. otherwise, from lemma 2.17, there exists a positive integer k no more than n such that δt # o(x; n) $ = η # m(x, t kx, t) $ for all t > 0.(2.3) now, we can choose a strictly decreasing sequence of positive numbers {ai} with ,∞ i=1 ai = 1 and so thanks to (i) and (2.3), we obtain δt # o(x; n) $ = η # m(x, t kx, t) $ = η # m(x, t kx, ∞i=1 ait) $ ≤ η # m(x, tx, ∞i=j+1 ait) $ + η # m(tx, t kx, ji=1 ait) $ for all t > 0 and any j. hence, by substituting i = 1 and j = k in lemma 2.15, we get δt # o(x; n) $ ≤ lim sup j→∞ η # m(x, tx, ∞i=j+1 ait) $ + η # m(tx, t kx, t) $ ≤ lim sup j→∞ η # m(x, tx, ∞i=j+1 ait) $ + λδt # o(x; n) $ , from which it follows that δt # o(x; n) $ ≤ 1 1 − λ lim sup j→∞ η # m(x, tx, ∞i=j+1 ait) $ .(2.4) c© agt, upv, 2020 appl. gen. topol. 21, no. 2 188 quasi-contractions in fuzzy metric spaces with a graph if m, n ∈ n with m ≥ n ≥ 2, since t n−1x ∈ ct , then by putting i = m − n + 1 and j = 1 in lemma 2.15, we obtain η # m(t mx, t nx, t) $ = η # m(t m−n+1t n−1x, tt n−1x, t) $ ≤ λδt # o(t n−1x; m − n + 1) $ ,(2.5) for all t > 0, where λ ∈ (0, 1) is a quasi-contractive constant of t . moreover, due to (2.3), there exists a positive integer k no more than m − n + 1 such that δt # o(t n−1x; m − n + 1) $ = η # m(t n−1x, t k+n−1x, t) $ (2.6) for all t > 0. because n ≥ 2, it follows that t n−2x ∈ ct and so putting i = 1 and j = k + 1 in lemma 2.15, we obtain η # m(t n−1x, t k+n−1x, t) $ = η # m(tt n−2x, t k+1t n−2x, t) $ ≤ λδt # o(t n−2x; m − n + 2) $ (2.7) for all t > 0. combining (2.5), (2.6) and (2.7) together with (2.4) and using induction, we get η # m(t mx, t nx, t) $ ≤ λδt # o(t n−1x; m − n + 1) $ = λη # m(t n−1x, t k+n−1x, t) $ ≤ λ2δt # o(t n−2x; m − n + 2) $ ... ≤ λnδt # o(x; m) $ ≤ λn 1 − λ lim sup j→∞ η # m(x, tx, ∞i=j+1 ait) $ , which implies from (ii) that lim m,n→∞ η # m(t mx, t nx, t) $ = 0. hence, limm,n→∞ m(t nx, t mx, t) = 1. this means that {t nx} is a cauchy sequence. □ following petruşel and rus [16], we introduce the concept of a picard and weakly picard operator in fuzzy metric spaces as follows. definition 2.20. let (x, m, !) be a fuzzy metric space and t : x → x be a mapping. (i) t is called a picard operator if t has a unique fixed point x∗ ∈ x and limn→∞ m(t nx, x∗, t) = 1 for all x ∈ x and each t > 0. (ii) t is called a weakly picard operator if {t nx} is a convergent sequence and its limit (which depends on x) is a fixed point of t for all x ∈ x. it is clear that each picard operator is weakly picard operator but the identity mapping of any fuzzy metric space with more than one point shows that the converse is not generally true. in fact, the set of fixed points of a weakly c© agt, upv, 2020 appl. gen. topol. 21, no. 2 189 m. dinarvand picard operator can have any arbitrary cardinality. nevertheless, one can easily see that a weakly picard operator is picard one if and only if it has a unique fixed point. motivated by jachymski [13], we define a weaker type of continuity of selfmaps in fuzzy metric spaces endowed with a graph as follows. definition 2.21. let (x, m, !) be a fuzzy metric space endowed with a graph g and t : x → x be a mapping. we say that t is orbitally g-continuous on x if limn→∞ m # t anx, y, t $ = 1 implies limn→∞ m # t(t anx), ty, t $ = 1 for all x, y ∈ x and each t > 0 and all sequences {an} of positive integers such that (t anx, t an+1x) ∈ e(g) for all n ∈ n. it is clear that a continuous mapping on a fuzzy metric space is orbitally g-continuous for all graphs g but the converse is not true in general as the following example shows. example 2.22. let (r+, d) be the usual (euclidean) metric space of all nonnegative real numbers and (r+, md, !) be the standard fuzzy metric space induced by d. consider a mapping t : r+ → r+ defined by the rule tx = ) x 3 , x ∕= 0, 1, x = 0. then it is clear that t is not continuous at x = 0 and in particular, t is not continuous on the whole r+. now, suppose that r+ is endowed with a graph g = (v (g), e(g)), where v (g) = r+ and e(g) = {(x, x) : x ∈ r+}, i.e. e(g) contains all loops. if x, y ∈ r+ and {an} is a sequence of positive integers with (t anx, t an+1x) ∈ e(g) for all n ∈ n such that limn→∞ m # t anx, y, t $ = 1 for any t > 0, then {t anx} is necessarily a constant sequence. hence, t anx = y for all n ∈ n and so limn→∞ m # t(t anx), ty, t $ = 1 for each t > 0. therefore, t is orbitally g-continuous on r+. now, we are ready to prove our main theorem on the existence of a fixed point for a g-fuzzy h-quasi-contraction in the setup of m-complete fuzzy metric spaces endowed with a graph. theorem 2.23. let (x, m, !) be an m-complete fuzzy metric space endowed with a graph g and t : x → x be a g-fuzzy h-quasi-contraction with respect to η ∈ h such that (i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈ ! m(t ix, t jx, t) : x ∈ x, t > 0, i, j ∈ n " ; (ii) ! η(m(x, tx, ti)) : i ∈ n " is bounded for all x ∈ x and any sequence {tn} ⊆ (0, ∞), tn ↓ 0. then the restriction of t to ct is a weakly picard operator if either t is orbitally &g-continuous on x or g is a (&c)-graph. in particular, whenever either t is orbitally &g-continuous on x or g is a (&c)-graph, t has a fixed point in x if and only if ct ∕= ∅. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 190 quasi-contractions in fuzzy metric spaces with a graph proof. if ct = ∅, then there remains nothing to prove. so assume that ct is nonempty. note that since t preserves the edges of &g, it follows immediately that ct is t-invariant, i.e. t maps ct into itself. now, suppose that x ∈ ct is given. by virtue of lemma 2.19, {t nx} is a cauchy sequence. as (x, m, !) is m-complete, there exists x∗ ∈ x (depending on x) such that limn→∞ m(t nx, x∗, t) = 1 for all t > 0. we shall show that x∗ is a fixed point for t . to this end, note that from x ∈ ct , we have (t nx, t n+1x) ∈ e( &g) for all n ∈ n ∪ {0}. on the one hand, if t is orbitally &g-continuous on x, then limn→∞ m(t nx, x∗, t) = 1 for all t > 0 implies limn→∞ m(t n+1x, tx∗, t) = limn→∞ m(t(t nx), tx∗, t) = 1 for all t > 0. by the uniqueness of the limit, we get m(x∗, tx∗, t) = 1 for all t > 0, i.e. tx∗ = x∗. on the other hand, if g is a (&c)-graph, then there exists a strictly increasing sequence {nk} of positive integers such that (t nkx, x∗) ∈ e( &g) for all k ∈ n. due to proposition 2.14, if λ ∈ (0, 1) is a quasi-contractive constant of t , then t is a &g-fuzzy h-quasi-contraction with respect to η ∈ h with a quasicontractive constant λ. let t > 0 is given. hence, for all " > 0 and k ∈ n, we have m(tx∗, x∗, t + ") ≥ m(tx∗, t nk+1x, ") ! m(t nk+1x, x∗, t) which together with (i) and (fq2) yields η # m(x∗, tx∗, t + ") $ ≤ η # m(x∗, t nk+1x, ") $ + η # m(tx∗, t nk+1x, t) $ ≤ η # m(tx∗, t nk+1x, t) $ + λ max ! η # m(x∗, t nkx, t) $ , η # m(x∗, tx∗, t) $ , η # m(t nkx∗, t nk+1x∗, t) $ , η # m(x∗, t nk+1x∗, t) $ , η # m(t nkx, tx∗, t) $" . on taking the limit as k → ∞ in the above inequality, we obtain η # m(x∗, tx∗, t + ") $ ≤ λη # m(x∗, tx∗, t) $ , which implies that η # m(x∗, tx∗, t) $ = lim !→0+ η # m(x∗, tx∗, t + ") $ ≤ λη # m(x∗, tx∗, t) $ . as λ ∈ (0, 1), it then follows that η # m(x∗, tx∗, t) $ = 0 for all t > 0. thus m(x∗, tx∗, t) = 1 for all t > 0 or equivalently, tx∗ = x∗. finally, since ct contains all fixed points of t , it follows that x ∗ ∈ ct . consequently, the restriction of t .. ct : ct → ct is a weakly picard operator. □ by putting g = g0 in theorem 2.23, we obtain the following generalization of ćirić’s fixed point theorem [7] on m-complete fuzzy metric spaces in the sense of george and veeramani. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 191 m. dinarvand corollary 2.24. let (x, m, !) be an m-complete fuzzy metric space and t : x → x be a fuzzy h-quasi-contraction with respect to η ∈ h such that (i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈ ! m(fix, fjx, t) : x ∈ x, t > 0, i, j ∈ n " ; (ii) ! η(m(x, fx, ti)) : i ∈ n " is bounded for all x ∈ x and any sequence {tn} ⊆ (0, ∞), tn ↓ 0. then t is a picard operator. proof. the set ct is nonempty because ct = x. therefore, due to theorem 2.23, the mapping t = t .. ct is a weakly picard operator. in particular, t has a fixed point in x. to see that t is a picard operator, it sufficies to show that t has a unique fixed point in x. to this end, suppose that x∗ and y∗ are two fixed points for t in x. thus, from (2.1), we have η # m(x∗, y∗, t) $ = η # m(tx∗, ty∗, t) $ ≤ λ max ! η # m(x∗, y∗, t) $ , η # m(x∗, tx∗, t) $ , η # m(y∗, ty∗, t) $ , η # m(x∗, ty∗, t) $ , η # m(y∗, tx∗, t) $" = λη # m(x∗, y∗, t) $ , for all t > 0, where λ ∈ (0, 1) is the quasi-contractive constant. therefore, η # m(x∗, y∗, t) $ = 0 which by our assumptions about η ∈ h implies that m(x∗, y∗, t) = 1 for all t > 0. hence, x∗ = y∗. □ remark 2.25. by a subtle look at the proof of corollary 2.24 and use an argument similar to that appeared there, we see that both the ends of any link of g can not be fixed points for a g-fuzzy h-quasi-contraction with respect to η ∈ h, i.e. if x ∕= y, tx = x and ty = y, then (x, y) ∕∈ e(g). roughly speaking, no g-fuzzy h-quasi-contraction with respect to η ∈ h can keep both the ends of a link of g fixed. in particular, • if g = g0, then no fuzzy h-quasi-contraction with respect to η ∈ h can have two distinct fixed points; • if ≼ is a partial order on x, then neither a g1-fuzzy h-quasi-contraction with respect to η ∈ h nor a g2-fuzzy h-quasi-contraction with respect to η ∈ h can have two distinct fixed points which are comparable elements of (x, ≼). by taking g = g1 or g = g2 in theorem 2.23, we obtain the ordered version of ćirić’s fixed point theorem on ordered fuzzy h-quasi-contractions with respect to η ∈ h in m-complete fuzzy metric spaces equipped with a partial order as follows. corollary 2.26. let (x, ≼) be a partially ordered set and (x, m, !) be an mcomplete fuzzy metric space. suppose that t : x → x be a mapping which maps comparable elements of (x, ≼) onto comparable elements and satisfies (2.2) such that for η ∈ h, c© agt, upv, 2020 appl. gen. topol. 21, no. 2 192 quasi-contractions in fuzzy metric spaces with a graph (i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈ ! m(fix, fjx, t) : x ∈ x, t > 0, i, j ∈ n " ; (ii) ! η(m(x, fx, ti)) : i ∈ n " is bounded for all x ∈ x and any sequence {tn} ⊆ (0, ∞), tn ↓ 0. then the restriction of t to the set of all points x ∈ x whose every two iterates under t are comparable elements of (x, ≼) is a weakly picard operator if either t is orbitally g2-continuous on x or the quadruple (x, m, !, ≼) satisfies (∗). in particular, whenever either t is orbitally g2-continuous on x or the quadruple (x, m, !, ≼) satisfies (∗), t has a fixed point in x if and only if there exists x ∈ x such that t mx and t nx are comparable elements of (x, ≼) for all m, n ∈ n ∪ {0}. because g-fuzzy h-banach contractions with respect to η ∈ h, g-fuzzy hkannan contractions with respect to η ∈ h, g-fuzzy h-chatterjea contractions with respect to η ∈ h, g-fuzzy h-ćirić-reich-rus contractions with respect to η ∈ h and g-fuzzy h-hardy-rogers contractions with respect to η ∈ h are all a g-fuzzy h-quasi-contraction with respect to η ∈ h, we have also the following fixed point theorem for these contractions as a consequence of theorem 2.23. corollary 2.27. let (x, m, !) be an m-complete fuzzy metric space endowed with a graph g and t : x → x be a g-fuzzy h-banach contraction (a gfuzzy h-kannan contraction, a g-fuzzy h-chatterjea contraction, a g-fuzzy h-ćirić-reich-rus contraction, or a g-fuzzy h-hardy-rogers contraction) with respect to η ∈ h such that (i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈ ! m(fix, fjx, t) : x ∈ x, t > 0, i, j ∈ n " ; (ii) ! η(m(x, fx, ti)) : i ∈ n " is bounded for all x ∈ x and any sequence {tn} ⊆ (0, ∞), tn ↓ 0. then the restriction of t to ct is a weakly picard operator if either t is orbitally &g-continuous on x or g is a (&c)-graph. in particular, whenever either t is orbitally &g-continuous on x or g is a (&c)-graph, t has a fixed point in x if and only if ct ∕= ∅. remark 2.28. by comparing corollary 2.27 as a version of theorem 2.23 for several types of contractions with some recent results in graph metric fixed point theory, one can see easily that our results can be viewed as the improvement and generalization of corresponding results in [3, 4, 6, 12, 13, 14, 18, 17] and several other comparable results. 3. conclusion despite noted improvements in computer skill and its remarkable success in facilitating many areas of research, computers are not designed to handle situations wherein uncertainties are involved. fuzzy set theory has provided many important tools in mathematics and related disciplines to resolve the issues of c© agt, upv, 2020 appl. gen. topol. 21, no. 2 193 m. dinarvand uncertainty and ambiguity. in the present work, we investigated sufficient conditions which guarantee the existence of a fixed point for a new notion called g-fuzzy h-quasi-contraction using directed graphs in the setting of fuzzy metric spaces endowed with a graph. a large number of different types of contractive mappings formulated using directed graphs satisfy the presented contractive condition and our main result is a natural generalization of [2, definition 2.3] from fuzzy metric spaces to fuzzy metric spaces with a graph and enriches our knowledge of fixed points in such spaces. as a new work, it will be interesting to study common fixed point results for two or more than two mappings on fuzzy metric spaces endowed with a graph g by considering the function η ∈ h. references [1] s. m. a. aleomraninejad, sh. rezapour and n. shahzad, some fixed point results on a metric space with a graph, topology appl. 159, no. 3 (2012), 659–663. 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[21] l. a. zadeh, fuzzy sets, inform. control, 10, no. 1 (1960), 385–389. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 194 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 41-52 epimorphisms and maximal covers in categories of compact spaces b. banaschewski and a. w. hager abstract the category c is ”projective complete” if each object has a projective cover (which is then a maximal cover). this property inherits from c to an epireflective full subcategory r provided the epimorphisms in r are also epi in c. when this condition fails, there still may be some maximal covers in r. the main point of this paper is illustration of this in compact hausdorff spaces with a class of examples, each providing quite strange epimorphisms and maximal covers. these examples are then dualized to a category of algebras providing likewise strange monics and maximal essential extensions. 2010 msc: primary 06f20, 18g05, 54b30; secondary 18a20, 18b30, 54c10, 54g05 keywords: epimorphism, cover, projective, essential extension, compact, strongly rigid. 1. introduction in a category, an essential extension of an object a is a monomorphism a b m // for which km monic implies k monic. in recent work [3], the authors have considered the inheritance from a category c to a monocoreflecive subcategory v of the property that each object has a unique maximal essential extension. the hypothesis ”each monic in v is also monic in c” was crucial. (the property was deployed to similar ends in [9].) this paper is largely directed at exhibiting in a concrete setting some pathology which can occur in the absence of these hypotheses. but we shall operate ”in dual”, as we now describe briefly, and sketch a return to essential extensions in the final §5. 42 b. banaschewski and a. w. hager in a category, a cover of the object x is an epimorphism xy g oo for which gf epi implies that f is epi. (this definition is dual to ”essential extension”). any projective cover is also a unique maximal cover (2.3). but there are categories with no projectives, and still every object has a unique maximal cover ([3], in dual.) in compact hausdorff spaces, comp, epis are onto and every object has a projective cover (the gleason cover). for an epireflective subcategory r of comp, r has a non-void projective if and only if epis in r are onto (3.5) and then the projective covers from comp are projective covers in r (3.2). we begin with a necessary discussion of simple categorical preliminaries, proceed to comp and two specific epireflective subcategories, then extract what little can be said for an epireflective r in general. penultimately, we consider a strongly rigid e ∈ comp and the epireflective subcategory r(e) which e generates. there are epis not onto, and any nonconstant {0, 1}e oo is a maximal cover. finally, we sketch the dualization of this to a category of algebras, in which any proper c(e) r2// is a maximal essential extension. we thank horst herrlich and miroslav hušek for alerting us to strongly rigid spaces. 2. preliminaries the context for 2.1 2.7 is a fixed category with no hypotheses at all before 2.4. in the following, g, h, k, . . . are assumed to be morphisms. the terms ”morphism” and ”map” will be interchangeable. definition 2.1. (a) a morphism g is an epimorphism (epi) if hg = kg implies h = k. (b) the map g is ”covering” if epi, and gf epi implies f epi. (such g could also be called essential epi (or perhaps co-essential epi).) a cover of object x is a pair (x, g) with xy g oo covering. covers of x, (y, g) and (y ′, g′) are equivalent if there is an isomorphism h with g′h = g. (c) object y is cover-complete if, (z, k) a cover of y implies k is an isomorphism. a maximal cover of x is a cover (y, g) with y cover complete. a unique maximal cover of x is a maximal cover which is equivalent to any other maximal cover of x. (d) object p is projective if whenever px h oo and yx g oo is an epi, then there is py f oo with gf = h. a projective cover is a cover (p, p) with p projective. (e) the category is called projectively complete if every object has a projective cover, and (weaker) is said to have enough projectives if for each object x there is px f oo , f epi and p projective. the following two elementary propositions are, except for 2.2 (d) and perhaps 2.3 (b), proved (in dual) in [1], 9.14, 9.19, 9.20. epimorphisms and maximal covers in categories of compact spaces 43 proposition 2.2. (a) an isomorphism is covering. (b) the composition of two covering maps is covering. (c) if g and gf are covering, then f is covering. (d) if gf is covering and f is epi, then f is covering. proof. (d) given such gf and f, suppose fh is epi. note that g is epi (because gf is). so, g(fh) is epi, and g(fh) = (gf)h shows h is epi, since gf is covering. � proposition 2.3. (a) a projective object is cover-complete. (b) a projective cover is a unique maximal cover. proof. (b) suppose (p, p) is a projective cover of x. it is a maximal cover by (a). if (y, g) is another cover of x, there is k with gk = p (since p is projective and g is epi). by 2.2 (c), k is covering, thus an isomorphism if y is cover-complete. � to proceed further, we require assumptions. two hypotheses 2.4. (to be invoked selectively). let c be a category, and r a subcategory (always assumed full and isomorphism-closed). the first condition is on c alone, and is ”the other face” of 2.2 (d): (f ◦) if gf is covering and f is epi, then g is covering the second condition is on r ⊂ c, and is the (frequently invalid) converse to the obvious truth ”any c-epi between r-objects is r-epi”: (s◦) any r-epi is c-epi. the point of this paper is, in the presence of (f ◦), what happens when (s◦) holds (2.7 and §3), and especially what can happen when it fails (§5 , §6). proposition 2.5. if c has enough projectives (in particular if c is projectively complete), then c satisfies (f ◦). proof. consider x y g oo z f oo with gf covering and f epi. since gf is epi, so is g. suppose y t t oo has gt epi; we want t epi. take t p e oo epi with p projective. there is z p h oo with fh = te (since f is epi).then, (gf)h = g(fh) = g(te) = (gt)e. the last term is epi, and so also the first term. thus h is epi (since gf is covering), and so also fh. since fh = te, t is epi. � 44 b. banaschewski and a. w. hager proposition 2.6. suppose (s◦). if x, y ∈ r, and x y g oo is c-covering, then g is r-covering. proof. suppose given x y g oo as stated, and y z f oo with z ∈ r and gf r-epi. then gf is c-epi (by (s◦)), so f is c-epi (since g is c-covering), thus also r-epi (as desired). � we say (r, r) is epireflective in c if r is a subcategory of c, and for each y ∈ c there is ry ∈ r (the reflection) and epi ry y ry oo (the reflection map) for which, whenever x y f oo with x ∈ r, there is f̄ with f̄ry = f. (see [11] for a full account of the theory of epireflective subcategories). proposition 2.7. suppose that (r, r) is epireflective in c, and satisfies (s◦). (a) if p is projective in c, then rp is projective in r. (b) suppose further that c satisfies (f ◦). if x ∈ r, and (p, p) is a projective cover in c of x, then (rp, p̄) is a projective cover in r of x. (c) if c is projectively complete, then so is r (with projective covers as in (b)). proof. (c) (from (b)). 2.5 says c satisfies (f ◦), so (b) applies. (a) suppose given r-epi x y g oo and any x rp f oo . by (s◦), g is cepi, so there is f1 with gf1 = frp (since p is c-projective). next, there is f2 with f2rp = f1, and we have frp = gf1 = g(f2rp ) = (gf2)rp . since rp is c-epi, f = gf2. (b) by (a), rp is r-projective. we need that the p̄ in p̄rp = p is rcovering. since rp is epi, (f ◦) says that p̄ is c-covering, and thus r-covering by 2.5. � remark 2.8. (a) [3], 1.2 shows (in dual) that if c has unique maximal covers, so does epireflective r, assuming the conditions (s◦) and (f ◦). the proofs above of 2.7 (a) and (b) are simplified versions of those in [3]. for 2.7 (c), the present 2.7 (new here) allows suppression of the hypothesis (f ◦). (b) if in 2.7, r already contains every c-projective, then 2.7 (a) and (b) simplify in the obvious way. this is the case for c = comp, with r having (s◦); see 3.2 below. 3. compact hausdorff spaces comp is the category of compact hausdorff spaces with continuous functions as maps. a map x y f oo in comp is called irreducible if f(y ) = x, but when f ( y (f closed), f(f) 6= x. the following is mostly due to gleason epimorphisms and maximal covers in categories of compact spaces 45 [6]. ((a) is a folk item. (e) follows from (d) and 2.4; it has a short direct proof, and is noted in [8], 2.5.) proposition 3.1. in comp: (a) epis are onto. (see comment after 3.3 below.) (b) a map is covering iff it is irreducible. (c) a space is projective iff it is extremally disconnected (every open set has open closure). (d) any object x has a projective cover (px, px); comp is projectively complete. (e) (f ◦) holds. the notation (px, px) is reserved for the rest of the paper; this will always denote the projective cover in comp of x ∈ comp. also, for brevity, we shall let ed stand for the class of extremally disconnected spaces in comp. (considerable literature developed from gleason’s [6], with various new proofs, generalizations, and variants of the theory. see [2], [8], [14] and their bibliographies.) now consider a subcategory a of comp (which can be identified with its object class). the family of all subobjects (resp., products) of spaces in a is denoted sa (resp., pa). (note that subobjects are closed subspaces.) kennison [13] has shown that r is epireflective in comp iff r is neither ∅ nor {∅} and r = spr. for ∅ 6= x ∈ comp, let r(x) = sp{x}; this is the smallest epireflective subcategory containing x. let {0} (resp., {0, 1}) denote the space with one (resp. two) points. the smallest epireflective is r({0}) = {∅, {0}}; here, {0} ∅oo is epi, so epis are not onto. we comment further on this shortly. the next largest is r({0, 1}): if r is epireflective and not r({0}), there is x ∈ r with |x| ≥ 2, thus {0, 1} ∈ r, so r({0, 1}) ⊂ r. note that r({0, 1}) = comp◦, the class of compact zerodimensional spaces [5], and ed ⊂ comp◦ [7]. thus, if r is epireflective and not r({0}), ed⊂ r. corollary 3.2. suppose r is epireflective and r-epis are onto (i.e., r ⊂ comp satisfies (s◦)). then r is projectively complete. in fact, for any x ∈ r, the r-(projective cover) is (px, px). proof. apply 3.1, 2.4, and the discussion above. � proposition 3.3. comp◦-epis are onto. 3.2 applies to comp◦. proof. the following takes place in comp◦ the only ∅ yoo has y = ∅ and the map is the identity, which is epi, and technically onto. if x 6= ∅ then x ∅oo is not epi (since there are different {0, 1} x h oo k oo ). suppose x 6= ∅, and x y g oo is epi. were g not onto, there would be p ∈ x − g(y ), and clopen u with p /∈ u ⊇ g(y ). then h constantly 1 and k the characteristic function of u has h 6= k but hg = kg. � 46 b. banaschewski and a. w. hager (to show comp-epis are onto, argue similarly using [0, 1] instead of {0, 1}, and using complete regularity of x (i.e. the tietze-urysohn theorem).) remark 3.4. we do not know if there is epireflective r different from comp◦ and comp, for which epis are onto. the following (closely related to [3], 4.1) shows that, failing ”epis are onto”, there are no 6= ∅ projectives. but there still may be some maximal covers, of at least two sorts, as the examples in §5 show. proposition 3.5. suppose (only) {0} ∈ r. the following statements in r are equivalent. (a) epis are onto. (b) {0} is projective. (c) there is a non-void projective. proof. (b) ⇒ (c) obviously, and (c) ⇒ (b) because {0} is a retract of any x 6= ∅, and a retract of a projective is projective. (a) ⇒ (b) because {0} is projective in comp, and if (a) holds, projective in r. (b) ⇒ (a). if x y g oo is an epi which is not onto, then there is p ∈ x −g(y ), and for x {0} h oo defined as h(0) = p, there can be no y {0} f oo with gf = h. � finally, we clarify the situation for ∅ and for r({0}). note the following for any r ⊂ comp with ∅ ∈ r. (i) ∅ is the initial object of r, i.e., for any x ∈ r, there is unique x ∅oo (namely, the empty map). (ii) ∅ is projective in r. (iii) if x ∅oo is epi in r, then this is a projective cover. proposition 3.6. suppose (only) {0} ∈ r = sr. the following statements in r are equivalent. (a) r = r({0}) (b) {0} ∅oo is epi (and thus a projective cover). (c) for any x ∈ r, x ∅oo is epi (and thus a projective cover). proof. the parenthetical remarks follow from the comments above. (a) ⇒ (c): ∅ ∅oo is epi, and {0} ∅oo also, since the only map out of {0} is the identity. (c) ⇒ (b): since {0} ∈ r. (b) ⇒ (a): if r 6= r({0}), then there is x ∈ r with |x| ≥ 2, so {0, 1} ∈ r (since sr = r). then there are different {0, 1} {0} h oo k oo which compose equally with {0} ∅oo , so the latter is not epi. � epimorphisms and maximal covers in categories of compact spaces 47 corollary 3.7. r({0}) is projectively complete, with epis not onto, and is the only epireflective subcategory with these two properties. proof. 3.6, (a) ⇒ (c) yields the first statement. if epireflective r has epis not onto, then by 3.5, the only projective is ∅. if r is projectively complete, then the projective cover must be x ∅oo . so these are epi, and 3.6 (c) ⇒ (a) says r = r({0}). � 4. when epis may not be onto consider r ⊂ comp. we localize the condition ”r-epis are onto”. keep in mind that r might have no projectives (but any y ∈ r∩ed is still compprojective). definition 4.1. ”x has e(r)” means x ∈ r, and whenever x · g oo is epi in r, then g is onto. proposition 4.2. suppose r = sr. (a) if x ∈ r∩comp◦, then x has e(r). (b) if y ∈ r∩ed, then y is cover-complete in r. proof. (a) identical to the proof of 3.3. (b) if (z, g) is an r-cover of y , then g is onto by (a), so there is f with gf = idy , since y is comp-projective, and f ∈ r (since y , z ∈ r). so f is an r-section. also, by 2.2, f is an r-covering map, thus r-epi. so f is an r-isomorphism, so is g, and therefore g is a comp-isomorphism, thus a homeomorphism. � (the converse to 4.2 (a) fails, with r = comp. but see 4.5 below.) proposition 4.3. suppose x has e(r). (a) if y ∈ r and x y g oo is irreducible, then (y, g) is an r-cover of x. (b) if also px ∈ r, and supposing r = sr, then (px, px) is the unique maximal r-cover of x. proof. (a) as in the proof of 2.6, mutatis mutandis (b) by (a), (px, px) is an r-cover, and px is cover-complete. if (y, g) is another r-cover of x, then g is onto (by e(r)), and there is y px f oo with gf = px (since px is comp-projective). if y is cover-complete, f is a homeomorphism. � corollary 4.4. suppose ed⊂ r = sr. if x ∈ r∩comp◦, then (px, px) is the unique maximal r-cover of x. proof. 4.2 (a) and 4.3 (b). � 48 b. banaschewski and a. w. hager the following is a qualified converse to 4.2 (a). corollary 4.5. suppose that ed⊂ r = sr. for y ∈ r, the following are equivalent. (a) y is ed. (b) y is cover-complete and y ∈comp◦. (c) y is cover-complete and y has e(r). proof. (a) ⇒ (b): 4.2 (b) and ed⊂comp◦. (b) ⇒ (c): 4.2 (a). (c) ⇒ (a): by 4.3 (b) (using ed⊂ r now), (py, py ) is the unique maximal r-cover of y , so if y is cover-complete, py is a homeomorphism. � here is one (more) triviality valid in (almost) any r. proposition 4.6. suppose {0} ∈ r. for any x ∈ r, with |x| ≥ 1, there are maps x {0} e oo (in r). such an e is r-epi iff |x| = 1. proof. given such e, there is (the retraction) x r //{0} with re = id{0}, so e is a section. if |x| = 1, then e is onto, thus r − epi. if e is r-epi, it becomes an r-isomorphism, thus a homeomorphism, so |x| = 1. � 5. epireflectives with epis not onto, and some maximal covers first, in summary so far of the situation for r epireflective in comp: if in r, there are epis not onto, then there are no non-void projectives ( 3.5). that is the case for r = {∅, {0}}, but here we have the projective (thus unique maximal) covers ∅ ∅oo and {0} ∅oo ( 3.6). if r contains the two-point space {0, 1} then r ⊇ comp◦ and at least has unique maximal covers for x ∈ comp◦, namely the (px, px) ( 4.4). we now display a large class of such r with some very strange epis, and non-unique maximal covers. this will be the r(e) = sp{e}, for e as follows. a space e in comp will be called strongly rigid if |e| ≥ 2 and the only continuous e //e are ide and constants. cook [4] has several of these, including a metric one m1. note that if e is strongly rigid, then {0, 1} ⊆ e (since |e| ≥ 2), e is connected (since a clopen u 6= ∅, e would yield e // {0, 1} � � // e ), |e| ≥ c (since there are non-constant e // // [0, 1] , using the tietze-urysohn theorem), and [0, 1] * e (since [0, 1] ⊆ e would yield non-constant e // [0, 1] � � // e , and [0, 1] is not strongly rigid). from cook’s examples, trnková [15] and isbell [12] have shown first, that if n is any cardinal, there is strongly rigid e with |e| ≥ n, and second, that if there is no measurable cardinal, there is a proper class e of strongly rigid spaces for which, whenever e1 6= e2 in e, the only continuous e1 //e2 are constants (and thus, for e1 6= e2 in e, neither of r(e1) and r(e2) contains the other). epimorphisms and maximal covers in categories of compact spaces 49 now let e be any strongly rigid space. in the following, terms epi, cover, . . . refer to r(e). of course 4.4 and 4.5 apply here. on the other hand, proposition 5.1. let f be a closed subspace of e. label the inclusion e f if oo . (a) if is epi iff |f | > 1. (b) if |f | = 2, then (f, if ) is a cover of e. if f ∈comp◦ and (f, if ) is a cover of e, then |f | = 2. (in the second part of (b), the supposition ”f ∈comp◦” cannot be dropped, because e e id oo is a cover.) corollary 5.2. any nonconstant e {0, 1} g oo is a maximal cover of e. any maximal cover of e is equivalent to one of these. two of these, with g and g′, are equivalent covers of e iff g({0, 1}) = g′({0, 1}). in particular, (pe, pe) is not a cover of e, and there are at least |e| ≥ c non-equivalent maximal covers of e; proof. (of 5.1) (a) by 4.6, if |f | = 1, then if is not epi. now suppose |f | > 1 . suppose f, g ∈ r(e) have common codomain which might as well be supposed of the form ei and fif = gif , i.e., f|f = g|f . then, for any projection ei πi //e , we have πif|f = πig|f . we want f = g, which is equivalent to πif = πig ∀ i ∈ i. let i ∈ i. then each of πif and πig is ide or constant. if πif = ide, then πif|f is not constant (since |f | ≥ 2), so πig|f is not constant, so πig = ide also. if πif is constant, say c, then πif|f = c also. so πig|f = c, and since |f | ≥ 2, πig = c. (b) suppose |f | = 2. by (a), if is epi. suppose if f is epi. then f is onto f (since if not, |range(f)| = 1, since |f | = 2, but then |range(if f)| = 1 and if f is not epi, by 4.6. so f is epi. suppose f ∈comp◦.if there are different p0, p1, p2 ∈ f , let f {0, 1} f oo be f(i) = pi. then f is not epi (by 4.2 (a)), but if f is epi by (a) above. � proof. (of 5.2) if e {0, 1} g oo is nonconstant, it is a cover because f ≡ {g(0), g(1)} {0, 1} g oo is a homeomorphism, and thus a maximal cover because f is cover-complete (being ed 4.2). suppose e y h oo is a maximal cover. then h is epi, thus nonconstant ( 6.1). so there are p0, p1 ∈ y with h(p0) 6= h(p1). define y {0, 1} f oo as f(i) = pi. so hf is a covering-map (by the preceding paragraph), thus f is a 50 b. banaschewski and a. w. hager covering map ( 2.2 (d)). since y is cover-complete, f is a homeomorphism, so (y, h) and ({0, 1}, hf) are equivalent. now suppose e {0, 1} g,g ′ oo are non-constant. there are two homeomorphisms h of {0, 1}, the identity and ”interchange 0 and 1”. and, range(g) = range(g′) iff g′ = gh for one of these h. � remark 5.3. cook’s specific strongly rigid m1 has these further features: m1 has a countable infinity of disjoint subcontinua; if k is any proper subcontinuum of m1, the only maps m1 koo are inclusion and constants. (see [4]). then in the category r(m1), in 5.1 and 5.2, e = m1 may be replaced by any proper subcontinuum k of m1 (as the proofs there show). 6. an application to lattice-ordered groups we now convert the situations of maximal covers in r ⊂comp to situations of maximal essential extensions in subcategories of a category of algebras. we use terminology categorically dual to the items in 2.1 (a) (e), respectively, namely (a) monic, (b) essential extension, (c) essentially complete, maximal essential extension (or, essential completion), (d) injective, injective hull, (e) injectively complete. the category of algebras is w ∗, the category of archimedean lattice-ordered groups with distinguished strong order unit, and ℓ-group homomorphisms carrying unit to unit. w ∗ has monics one-to-one, and is injectively complete; see [3]. consequently, the dual of 2.7 applies to w ∗. for x ∈comp, the continuous real-valued functions c(x), with unit the constant function 1, is a w ∗-object, and we have the functor w ∗ comp c oo : for x z τ oo in comp, c(x) cτ //c(z) is cτ(f) = f ◦ τ. this has a left adjoint, the yosida functor: for each g ∈ w ∗, there is y g ∈ comp and g //c(y g) monic in w ∗; for each g ϕ //h in w ∗, there is unique y g y h y ϕ oo in comp ”realizing ϕ” as ϕ(g) = g◦y ϕ. note that y c(x) ≃ x, and that ϕ is one-to-one iff y ϕ is onto. (see [10]). basic features of (y, c), and some diagram-chasing, convert the situations in comp discussed in previous sections to ”dual” situations in w ∗, as follows. (we omit the calculations). suppose r is epireflective in comp, and {0, 1} ∈ r (so comp◦ ∈ r). for brevity, set ∗r = {g ∈ w ∗|y g ∈ r}. proposition 6.1. (a) ∗r is monocoreflective in w ∗. (b) c(x) ϕ //h is monic in ∗r iff x y h y ϕ oo is epi in r. (c) ∗r has an injective other than {0} iff monics in ∗r are one-to-one iff r-epis are onto. when this occurs, ∗r is injective-complete, with injective hulls g //c(y g) //c(p(y g)). (d) if x is ed, then c(x) is essentially complete in ∗r. epimorphisms and maximal covers in categories of compact spaces 51 (e) if x ∈ comp◦, then c(d) cpx //c(px) is the unique maximal essential extension of c(x) in ∗r. now consider, as in §5, strongly rigid e ∈ comp and its generated epireflective r(e). by 5.1 and 6.1 (b), ∗r(e) has monics which are not one-to-one, and thus no 6= {0} injectives. 6.1 (d) and (e) hold in ∗r(e). note that {0, 1} ∈ comp has c({0, 1}) = r2 ∈ w ∗, the self-homeomorphisms of {0, 1} are the identity and ”interchange points”, and these correspond to the only self-isomorphisms of r2, which are the identity, and h(x, y) = (y, x). from 5.2 we obtain corollary 6.2. in ∗r(e), the maximal essential extensions of c(e) are exactly the w ∗-surjections c(e) ϕ //r2 . two of these, ϕ and ϕ′, are equivalent iff either ϕ = ϕ′, or ϕ′ = ϕh. references [1] j. adamek, h. herrlich and g. strecker, abstract and concrete categories, dover 2009. 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[11] h. herrlich and g. strecker, category theory, allyn and bacon 1973. [12] j. isbell, a closed non-reflective subcategory of compact spaces, manuscript c. 1971. [13] j. kennison, reflective functors in general topology and elsewhere, trans. amer. math. soc. 118 (1965), 303–315. [14] j. porter and r. g. woods, extensions and absolutes of hausdorff spaces, springerverlag 1988. [15] v. trnková, non-constant continuous mappings of metric or compact hausdorff spaces, comm. math. univ. carol. 13 (1972), 283–295. (received december 2011 – accepted november 2012) 52 b. banaschewski and a. w. hager b. banaschewski department of mathematics and statistics, mcmaster university, hamilton, on l8s4k1, canada. a. w. hager (ahager@wesleyan.edu) department of mathematics and computer science, wesleyan university, middletown, ct 06459 usa epimorphisms and maximal covers in categories[7pt] of compact spaces. by b. banaschewski and a. w. hager @ appl. gen. topol. 23, no. 1 (2022), 169-178 doi:10.4995/agt.2022.15214 © agt, upv, 2022 some generalizations for mixed multivalued mappings mustafa aslantaş a , hakan sahin b and ugur sadullah a a department of mathematics, faculty of science, çankırı karatekin university, çankırı, turkey (maslantas@karatekin.edu.tr, ugur s 037@hotmail.com) b department of mathematics, faculty of science and arts, amasya university, amasya, turkey (hakan.sahin@amasya.edu.tr) communicated by i. altun abstract in this paper, we first introduce a new concept of kw -type mcontraction mapping. then, we obtain some fixed point results for these mappings on m-metric spaces. thus, we extend many well-known results for both single valued mappings and multivalued mappings such as the main results of klim and wardowski [13] and altun et al. [4]. also, we provide an interesting example to show the effectiveness of our result. 2020 msc: 54h25; 47h10. keywords: fixed point; mixed multivalued mapping; m-metric space; pompeiu-hausdorff metric. 1. introduction in 1922, banach [7] proved an important theorem which is known as banach contraction principle. this principle is an important tool in the fixed point theory and has been accepted as starting of the fixed point theory in metric spaces. due to its applicability, many authors have studied to generalize this principle by considering different kinds of contractions or abstract spaces [2, 10, 11, 12, 19]. taking into account multivalued mappings, nadler [17] proved received 05 march 2021 – accepted 14 january 2022 http://dx.doi.org/10.4995/agt.2022.15214 https://orcid.org/0000-0003-4338-3518 https://orcid.org/0000-0002-4671-7950 https://orcid.org/0000-0001-8463-2862 m. aslantaş, h. sahin and u. sadullah one of the interesting and famous generalizations of this result in metric spaces as follows: theorem 1.1. let t : x → cb(x) be a multivalued mapping on a complete metric space (x,d) where cb(x) is the family of all nonempty bounded and closed subsets of x. suppose that there exists k in [0, 1) satisfying hd(tx,ty) ≤ kd(x,y) for all x,y ∈ x where hd : cb(x) × cb(x) → r is a pompei-hausdorff metric defined as hd(a,b) = max { sup x∈a d(x,b), sup y∈b d(a,y) } for all a,b ∈ cb(x). then, t has a fixed point in x. then, a lot of fixed point theorems for multivalued mappings have been obtained [15, 20]. in this sense, nadler’s result has been extended by feng and liu [9] by taking into account c(x), which is the family of all nonempty closed subsets of a metric space (x,d) valued mappings instead of cb(x) as follows: theorem 1.2. let t : x → c(x) be a multivalued mapping on a complete metric space (x,d). suppose that for all x ∈ x there exists y ∈ ixλ = {z ∈ tx : λd(x,z) ≤ d(x,tx)} such that d(y,ty) ≤ γd(x,y). if the function g(x) = d(x,tx) is lower semicontinuous (briefly l.s.c.) on x and 0 < γ < λ < 1, then t has a fixed point in x. later, klim and wardowski [13] generalized theorem 1.2 by taking into account a nonlinear contraction: theorem 1.3. let t : x → c(x) be a multivalued mapping on a complete metric space (x,d). if there exist λ in (0, 1) and ϕ : [0,∞) → [0,λ) such that (1.1) lim s→u+ sup ϕ(s) < λ for all u ∈ [0,∞) and there is y ∈ ixλ for all x ∈ x satisfying d(y,ty) ≤ ϕ (d(x,y)) d(x,y), then t has a fixed point provided that g(x) = d(x,tx) is lower-semicontinuous function on x. on the other hand, introducing the concept of partial metric, matthews [14] obtained another generalization of the banach contraction principle. now, we give the definition of the partial metric space. definition 1.4 ([14]). let x be a nonempty set and p : x ×x → [0,∞) be a function satisfying following conditions for all x,y,z ∈ x. p1) p(x,x) = p(x,y) = p(y,y) if and only if x = y p2) p(x,x) ≤ p(x,y) © agt, upv, 2022 appl. gen. topol. 23, no. 1 170 some generalizations for mixed multivalued mappings p3) p(x,y) = p(y,x) p4) p(x,z) ≤ p(x,y) + p(y,z) −p(y,y) then, p is said to be a partial metric. also, the pair (x,p) is called partial metric space. it is clear that every metric space is a partial metric space, but the converse may not be true. for some examples of partial metric space, we refer to [1, 5, 8, 21]. let (x,p) be a partial metric space. recently, asadi et al. [6] introduced a nice concept of m-metric which includes the notion of the partial metric. then, they proved a version of the banach contraction principle on these spaces. after that, many authors have proved many fixed point results for multivalued and single valued mappings [3, 16, 18] on m-metric spaces. now, we recall some notations and properties of an m-metric space. definition 1.5. let x be a nonempty set, and m : x × x → [0,∞) be a function. then, m is said to be an m-metric if the following conditions hold for all x,y,z ∈ x: m1) m(x,y) = m(x,x) = m(y,y) if and only if x = y, m2) mxy = min{m(x,x),m(y,y)}≤ m(x,y), m3) m(x,y) = m(y,x), m4) m(x,y) −mxy ≤ (m(x,z) −mxz) + (m(z,y) −mzy) . also, (x,m) is called an m-metric space. it is obvius that every standard metric and partial metric space is an m-metric space but the converse may not be true. indeed, let x = [0,∞) and m : x × x → [0,∞) be a function defined by m(x,y) = x+y 2 . hence, (x,m) is an m-metric space, but neither a partial metric space nor a standard metric. let (x,m) be an m-metric space. then, the m-metric m generates a t0 topology τm on x which has as a base the family open balls {bm(x,r) : x ∈ x, r > 0} where bm(x,r) = {y ∈ x : m(x,y) < mxy + r} for all x ∈ x and r > 0. let {xn} be a sequence in x and x ∈ x. it can be seen that the sequence {xn} m-converges to x with respect to τm if and only if lim n→∞ (m(xn,x) −mxnx) = 0. if limn,m→∞m(xn,xm) exists and is finite, then {xn} is said to be an mcauchy sequence. if every m-cauchy sequence {xn} converges to a point x in m such that lim n,k→∞ m(xn,xk) = m(x,x), then (x,m) is said to be m-complete. the following proposition is important for our main results. © agt, upv, 2022 appl. gen. topol. 23, no. 1 171 m. aslantaş, h. sahin and u. sadullah proposition 1.6. let (x,m) be an m-metric space, a ⊆ x and x ∈ x. m(x,a) = 0 =⇒ x ∈ a m where a m is the closure of a with respect to τm. the converse of proposition 1.6 may not be true. indeed, let x = [2,∞) and m : x × x → r be a function defined by m(x,y) = min{x,y}. then, (x,m) is an m-metric space. let a = (3, 5) and x = 2. it can be seen that x ∈ a m , but m(x,a) = 2 > 0. note that, since every metric space is a t1-space, every singleton is a closed set. therefore, theorem 1.2 is an extension of some fixed point result for single-valued mappings. however, τm may not be a t1-space, and thus each singleton does not have to be closed. therefore, the fixed point results obtained for multivalued mappings on an m-metric space may not be valid for singlevalued mappings unlike in the settings of metric spaces. to overcome this problem, we will use the notion of mixed multivalued mapping introduced by romaguera [22]. in the current paper, we first introduce a new concept of kw-type mcontraction for the mixed multivalued mapping. then, we obtain some fixed point results on m-metric spaces for these mappings. hence, we extend some well known results in the literature such as theorem 1.3. also, we provide a noteworth example to show the effectiveness of our results. 2. main results we start this section with the definition of kw-type m-contraction for mixed multivalued mapping. definition 2.1. let t : x → x ∪ cm(x) be a mixed multivalued mapping on an m-metric space (x,m) where cm(x) is the family of all closed subsets of x w.r.t. τm. then, t is called kw-type m-contraction mapping if there exists λ,α ∈ (0, 1) and ϕ : [0,∞) → [0,λ) satisfying lims→u+ sup ϕ(s) < λ for all u ∈ [0,∞) and for all x ∈ x with m(x,tx) > 0 there is y ∈ txλ (m) = {z ∈ tx : λm(x,z) ≤ m(x,tx)} such that m(y,ty) ≤ ϕ(m(x,y))m(x,y) and αm(y,y) ≤ m(x,y). theorem 2.2. let t : x → x ∪ cm(x) be a kw -type m-contraction on an m-complete m-metric space (x,m). if the function g : x → r defined by g(x) = m(x,tx) is l.s.c., then t has a fixed point in x. proof. let x0 ∈ x be an arbitrary point. if there exists n0 ∈ n such that m(xn0,txn0 ) = 0, then xn0 ∈ txn0 = txn0 (or xn0 = txn0 ), and so xn0 is a fixed point of t. assume that m(xn,txn) > 0 for all n ≥ 1. now, we consider the following cases: © agt, upv, 2022 appl. gen. topol. 23, no. 1 172 some generalizations for mixed multivalued mappings case 1. let |tx0| = 1. since t is a kw-type m-contraction mapping, there exists x1 = tx0 such that m(x1,tx1) ≤ ϕ(m(x0,x1))m(x0,x1) and αm(x1,x1) ≤ m(x0,x1). now, if |tx1| = 1, since t is a kw-type m-contraction mapping, there exists x2 = tx1 such that m(x2,tx2) ≤ ϕ(m(x1,x2))m(x1,x2) and αm(x2,x2) ≤ m(x1,x2), if |tx1| > 1, since t is a kw-type m-contraction mapping, there exists x2 ∈ tx1λ (m) such that m(x2,tx2) ≤ ϕ(m(x1,x2))m(x1,x2) and αm(x2,x2) ≤ m(x1,x2). case 2. let |tx0| > 1. since t is a kw-type m-contraction mapping, there exists x1 ∈ tx0λ (m) such that m(x1,tx1) ≤ ϕ(m(x0,x1))m(x0,x1) and αm(x1,x1) ≤ m(x0,x1). if |tx1| = 1. since t is a kw-type m-contraction mapping, there exists x2 = tx1 such that m(x2,tx2) ≤ ϕ(m(x1,x2))m(x1,x2) and αm(x2,x2) ≤ m(x1,x2). now, if |tx1| > 1. since t is a kw-type m-contraction mapping, there exists x2 ∈ tx1λ (m) such that m(x2,tx2) ≤ ϕ(m(x1,x2))m(x1,x2) and αm(x2,x2) ≤ m(x1,x2). repeating this process, we can construct a sequence {xn} such that for xn+1 ∈ txnλ (m), m(xn+1,txn+1) ≤ ϕ(m(xn,xn+1))m(xn,xn+1) and (2.1) αm(xn+1,xn+1) ≤ m(xn,xn+1) for all n ≥ 1. since xn+1 ∈ txnλ (m) for all n ≥ 1, we have (2.2) λm(xn,xn+1) ≤ m(xn,txn) © agt, upv, 2022 appl. gen. topol. 23, no. 1 173 m. aslantaş, h. sahin and u. sadullah for all n ≥ 1. hence, we get m(xn,xn+1) ≤ m(xn,txn) λ ≤ ϕ(m(xn−1,xn))m(xn−1,xn) λ < m(xn−1,xn)(2.3) and m(xn,txn) −m(xn+1,txn+1) ≥ λm(xn,xn+1) −ϕ(m(xn,xn+1))m(xn,xn+1) = (λ−ϕ(m(xn,xn+1)))m(xn,xn+1) > 0(2.4) for all n ≥ 1. from inequalities (2.3) and (2.4), (m(xn,xn+1)) and (m(xn,txn)) are decreasing sequences in r, and so they are convergent. because of the fact that lim n→∞ m(xn,xn+1) = r ≥ 0 and lim s→u+ sup ϕ(s) < λ, we can find q ∈ [0,λ) satisfying lim n→∞ sup ϕ(m(xn,xn+1)) = q. therefore, for any λ0 ∈ (q,λ), there exists n0 ∈ n such that ϕ(m(xn,xn+1)) < λ0 for all n ≥ n0. from (2.4), we have m(xn,txn) −m(xn+1,txn+1) ≥ (λ−ϕ(m(xn,xn+1)))m(xn,xn+1) ≥ (λ−λ0)m(xn,xn+1)(2.5) © agt, upv, 2022 appl. gen. topol. 23, no. 1 174 some generalizations for mixed multivalued mappings for all n ≥ n0. hence, for all n ≥ n0, we have m(xn,txn) ≤ ϕ (m(xn−1,xn)) m(xn−1,xn) ≤ ϕ(m(xn−1,xn)) λ m(xn−1,txn−1) ≤ ϕ(m(xn−1,xn))ϕ(m(xn−2,xn−1)) λ m(xn−2,xn−1) ≤ ϕ(m(xn−1,xn))ϕ(m(xn−2,xn−1)) λ2 m(xn−2,txn−2) ... ≤ ϕ(m(xn−1,xn)) · · ·ϕ(m(x0,x1)) λn m(x0,tx0) = ϕ(m(xn−1,xn)) · · ·ϕ(m(xn0,xn0+1)) λn−n0 × ϕ(m(xn0−1,xn0 )) · · ·ϕ(m(x0,x1)) λn0 m(x0,tx0) < ( λ0 λ )n−n0 ϕ(m(xn0−1,xn0 )) · · ·ϕ(m(x0,x1)) λn0 m(x0,tx0). then, since limn→∞ ( λ0 λ )n−n0 = 0, we have (2.6) lim n→∞ m(xn,txn) = 0. hence, for all k > n ≥ n0, from (2.5), we get m(xn,xk) −mxnxk ≤ ( m(xn,xn+1) −mxnxn+1 ) + ( m(xn+1,xn+2) −mxn+1xn+2 ) + · · · + ( m(xk−1,xk) −mxk−1xk ) ≤ m(xn,xn+1) + · · · + m(xk−1,xk) = k−1∑ j=n m(xj,xj+1) ≤ 1 λ−λ0 k−1∑ j=n (m(xj,txj) −m(xj+1,txj+1)) = 1 λ−λ0 (m(xn,txn) −m(xk,txk)) from (2.6), we have lim n,k→∞ (m(xn,xk) −mxnxk ) = 0 also, from (2.1), (2.2) and (2.6) we have limn→∞m(xn,xn) = 0, and so lim n,k→∞ m(xn,xk) = 0. © agt, upv, 2022 appl. gen. topol. 23, no. 1 175 m. aslantaş, h. sahin and u. sadullah then, {xn} is an m-cauchy sequence in (x,m). since (x,m) is an m-complete m-metric space, there exists x∗ ∈ m such that lim n→∞ (m(xn,x ∗) −mxnx∗ ) = 0 and lim n,k→∞ m(xn,xk) = m(x ∗,x∗) now, we shall show that x∗ is a fixed point of t. since g(x) = m(x,tx) is a l.w.s.c. function and limn→∞m(xn,txn) = 0, we have 0 ≤ m(x∗,tx∗) = g(x∗) ≤ lim n→∞ inf g(xn) = lim n→∞ inf m(xn,txn) = 0 hence, m(x∗,tx∗) = 0, and so we have x∗ ∈ tx∗ m = tx∗. therefore, x∗ is a fixed point of t. � the following example is important to show the effectiveness of our result. example 2.3. let x = [0, 4] and m : x ×x → [0,∞) be a function defined by m(x,y) = x + y 2 then, (x,m) is an m-complete m-metric space. define mappings t : x → x ∪cm(x) and ϕ : [0,∞) → [ 0, 3 4 ) by tx = [ 0, x2 16 ] , and ϕ(u) = { 3 4 u , u < 1 1 2 , u ≥ 1 , respectively. then, we have g(x) = m(x,tx) = x 2 for all x ∈ x. it can be seen that g is l.s.c. with respect to τm. now, we shall show that kw -type m-contraction mapping. let x be an arbitrary point in x with m(x,tx) > 0. also, we have tx3 4 = { y ∈ tx : 3 4 m(x,y) ≤ m(x,tx) } = { y ∈ tx : y ≤ x 3 } choose α = 1 2 . then, for y ∈ tx3 4 , we have m(y,ty) = y 2 ≤ x 6 ≤ 3 8 (x + y)2 = ϕ (m(x,y)) m(x,y), © agt, upv, 2022 appl. gen. topol. 23, no. 1 176 some generalizations for mixed multivalued mappings and 1 2 m(y,y) ≤ m(x,y). hence, all hypotheses of theorem 2.2 hold, and so t has a fixed point in x. if we take ϕ(u) = γ ∈ (0,λ) for all u ∈ [0,∞) in definition 2.1, we obtain the following fixed point result which is a generalization result of [4]. corollary 2.4. let t : x → x ∪cm(x) be a multivalued mapping on a mcomplete m-metric space (x,m). assume that the following conditions hold: (i) the function g : x → r defined by g(x) = m(x,tx) is l.s.c. (ii) there exist λ,γ,α ∈ (0, 1) with γ < λ such that for any x ∈ x with m(x,tx) > 0, m(y,ty) ≤ γm(x,y) and αm(y,y) ≤ m(x,y) for some y ∈ txλ (m). then, t has a fixed point in x. 3. conclusion in this paper, we extend the result given by klim and wardowski [13] to m-metric spaces. also, we generalize the main result of altun et al. [4]. since an m-metric space may not be a t1-space, we first introduce a new concept of kw-type m-contraction mapping to obtain a real generalization of the results obtained for the single valued mappings. then we obtain some fixed point results for these mappings in m-metric spaces. moreover, we provide an interesting example to show the effectiveness of our results. acknowledgements. the authors are thankful to the referees for making valuable suggestions leading to the better presentations of the paper. references [1] m. abbas and t. nazir, fixed point of generalized weakly contractive mappings in ordered partial metric spaces, fixed point theory and applications 2012, no. 1 (2012), 1-19. [2] n. alamgir, q. kiran, h. aydi and y. u. gaba, fuzzy fixed point results of generalized almost f -contractions in controlled metric spaces, adv. differ. equ. 2021 (2021): 476. 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[22] s. romaguera, on nadler’s fixed point theorem for partial metric spaces, mathematical sciences and applications e-notes 1, no. 1 (2013), 1–8. © agt, upv, 2022 appl. gen. topol. 23, no. 1 178 @ appl. gen. topol. 21, no. 2 (2020), 305-325 doi:10.4995/agt.2020.13553 c© agt, upv, 2020 the higher topological complexity in digital images melih i̇s and i̇smet karaca department of mathematics, ege university, i̇zmir, turkey (melih.is@ege.edu.tr, ismet.karaca@ege.edu.tr) communicated by s. romaguera abstract y. rudyak develops the concept of the topological complexity tc(x) defined by m. farber. we study this notion in digital images by using the fundamental properties of the digital homotopy. these properties can also be useful for the future works in some applications of algebraic topology besides topological robotics. moreover, we show that the cohomological lower bounds for the digital topological complexity tc(x, κ) do not hold. 2010 msc: 46m20; 68u05; 68u10; 68t40; 62h35. keywords: topological complexity; digital topology; homotopy theory; digital topological complexity; image analysis. 1. introduction one of the most popular topics in the field of applications of algebraic topology in recent years arises from the interpretation of algebraic topological elements in the area of robotics. after m. farber put it in the literature for the first time [17], this subject has been studied by many researchers [13], [18], [19], [20], [23] and [28] since that time. this is very important because these studies can shed light on further engineering and navigation problems. for this purpose, it is necessary to understand how tools of algebraic topology take a place in the problem of robot motion planning. the answer is configuration spaces, that is explained in [18]. one of the main consequence of this innovative approach to the robot motion planning problem in any configuration space x received 18 april 2020 – accepted 11 july 2020 http://dx.doi.org/10.4995/agt.2020.13553 m. i̇s and i̇. karaca is the calculation of the topological complexity tc(x), which depends only on the homotopy type of x [17]. this numerical invariant plays a crucial role in the selection of the best suited route for the robot by measuring the navigational complexity of a system’s configuration space. the concept of schwarz genus of a fibration [29] and some properties of fibrations are united and it follows that a new definition of the topological complexity tc(x) coincides with the first one. farber develops remarkable methods to compute lower or upper bounds for the topological complexity tc(x) [17]. using cohomological cup-product is one of the most precious one from the view of topology. farber and grant extend this method to any cohomology operations [19]. farber states that the topological complexity tc(x) is greater than the zero-divisor-cup-length of the cohomology h∗(x; k), where k is a field [theorem 7,[17]]. in addition, farber uses künneth formulae to prove this theorem. in digital setting, ege and karaca introduce the digital explanation of cup-product and also show that künneth formulae does not hold for digital images [15]. we examine in this study that whether it is valid or not for the digital cup-product or any other digital cohomology operations. it is important to consider the generalization of the topological complexity. rudyak introduced the higher topological complexity tcn(x) in [28], where x is a path-connected topological space and n is a positive integer. it is also noted that the higher topological complexity tcn(x) is more significant for n > 1. the subject of robotics can gain even more values in the future because the main areas in which digital topology is influenced are the issues directly related to the technology of our time such as robot designs, image processing, computer graphics algorithms and computer images. so karaca and is give the digital version of the topological complexity tc(x) [23]. we will show that it is also possible to generalize this digital version tc(x, κ), where (x, κ) is a digital image. to construct the digital higher topological complexity, we need a good knowledge of digital homotopy theory. then we sometimes refer to many articles or books from algebraic topology and digital topology such as [10], [11], [12], [27], [16], [24], [30], and [31]. we introduce the different version of some definitions. for example, we do not prefer using subdivisions in the definition of a cofibration defined in [27]. in this paper, we try to reveal the simple background of digital topology in the first part. after that we present some new results or reinterpretations about function spaces, evaluation maps, exponential law, fibrations, and cofibrations in digital topology using several definitions, in which take part in the same section, for reaching our main goal that is to give a new definition of the digital topological complexity. in topological setting, these are facts in the study of homotopy theory but in digital setting, they must be proved. we sometimes reinterpret some existing definitions from our own perspective. we all do it because in section 4, we want to introduce digital higher topological complexity via this new definition includes digital fibrations. also, we show some properties about the digital higher topological complexity tcn(x, κ). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 306 the higher topological complexity in digital images for example, we address when the topological complexity tc coincides with the higher topological complexity tcn in digital images. finally, in section 5, we go back to the study of the digital topological complexity for a short time. we are interested in a very important tool often used in topological robotics. it is a cohomological lower bound for the topological complexity tc. we show that it is invalid for digital images. we give a counter example to show our assertion in the last. 2. preliminaries in this section, we present fundamental notions and some basic facts for digital topology and topological robotics. let zn be the set of lattice points in the n−dimensional euclidean space. then (x, κ) is called a digital image, where x ⊂ zn and κ is an adjacency relation for the elements of x, [4]. two distinct points p and q in zn are cl−adjacent for a positive integer l with 1 ≤ l ≤ n, if there are at most l indices i such that |pi −qi| = 1 and for all other indices i such that |pi −qi| ∕= 1, pi = qi, [4]. for instance, consider the situations n = 1, n = 2 and n = 3, respectively. then we have 2−adjacency in z since c1 = 2 in the first case; 4 and 8 adjacencies in z2 since c1 = 4 and c2 = 8 in the second case and last 6, 18, and 26 adjacencies in z3 since c1 = 6, c2 = 18 and c3 = 26 as well. given the adjacency relation κ on zn, a κ−neighbor of p ∈ zn is a point of zn that is κ−adjacent to p, [22]. a digital image x ⊂ zn is κ−connected if and only if for every pair of different points x, y ∈ x, there is a set {x0, x1, ..., xr} of points of the digital image x such that x = x0, y = xr and xi and xi+1 are κ−neighbors, where i = 0, 1, ..., r − 1, [22]. let x ⊂ zn0 and y ⊂ zn1. assume that f : x −→ y is a function and κi is an adjacency relation defined on zni, for i = {0, 1}. f is called digitally (κ0, κ1)−continuous if, when any κ0−connected subset of x is taken, its image under f is also κ1−connected, [4]. proposition 2.1 ([4]). assume that two digital images (x, κ) and (x ′ , λ) are given. then the map h : x −→ x ′ is (κ, λ)−continuous if and only if for any x and x ′ ∈ x such that x and x ′ are κ−adjacent, either f(x) = f(x ′ ) or f(x) and f(x ′ ) are λ−adjacent. proposition 2.2 ([4]). if f : (x, κ1) −→ (y, κ2) and g : (y, κ2) −→ (z, κ3) are digitally continuous maps, then the composite map g ◦ f : (x, κ1) −→ (z, κ3) is digitally continuous as well. let x ⊂ zn0 and y ⊂ zn1 be digital images with κ0−adjacency and κ1−adjacency, respectively. f : x −→ y is a digital (κ0, κ1)−isomorphism if f is bijective and digital (κ0, κ1)−continuous and f−1 : y −→ x is digital (κ1, κ0)−continuous, [7]. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 307 m. i̇s and i̇. karaca proposition 2.3 ([3]). a digital isomorphism relation is equivalence on digital images. a digital interval is defined by [a, b]z = {z ∈ z : a ≤ z ≤ b}, [6]. let (x, κ) be a digital image. we say that f is a digital path from x to y in (x, κ) if f : [0, m]z −→ x is a (2, κ)−continuous function such that f(0) = x, and f(m) = y, [6]. if f(0) = f(m), then the κ−path is said to be closed, and the function f is called a κ−loop. let (x, κ0) ∈ zn0 and (y, κ1) ∈ zn1 be two digital images. for two (κ0, κ1)−continuous functions f, g : x −→ y , if there is a positive integer m and a function h : x × [0, m]z −→ y such that • for all x ∈ x, h(x, 0) = f(x) and h(x, m) = g(x); • for all x ∈ x, hx : [0, m]z −→ y , defined by hx(t) = h(x, t) for all t ∈ [0, m]z, is (2, κ1)−continuous; • for all t ∈ [0, m]z, ht : x −→ y , defined by ht(x) = h(x, t) for all x ∈ x, is (κ0, κ1)−continuous, they are said to be digitally (κ0, κ1)−homotopic in y and this is denoted by f ≃(κ0,κ1) g, [4]. the function h is called a digital (κ0, κ1)−homotopy between f and g. proposition 2.4 ([4]). a digital homotopy relation is equivalence on digitally continuous functions. a digitally continuous function f : x −→ y is digitally nullhomotopic in y if f is digitally homotopic in y to a constant function, [4]. a (κ0, κ1)−continuous function f from a digital image x to another image y is a (κ0, κ1)−homotopy equivalence if there exists a (κ1, κ0)−continuous function g from y to x such that g ◦ f is (κ0, κ0)−homotopic to the identity function 1x and f ◦ g is (κ1, κ1)−homotopic to the identity function 1y , [5]. a digital image (x, κ) is said to be κ−contractible if its identity map is (κ, κ)−homotopic to a constant function c for some c0 ∈ x, where the constant function c : x −→ x is defined by c(x) = c0 for all x ∈ x, [4]. let (x, κ) and (a, λ) be two digital images with the inclusion map i : (a, λ) −→ (x, κ). a is called a digitally κ−retract of the image x if and only if there is a digitally continuous map r : (x, κ) −→ (a, λ) such that r(a) = a for all a ∈ a, [4]. then the function r is called a digital retraction of x onto a. the product of two digital paths defined in [25]: if f : [0, m1]z −→ x and g : [0, m2]z −→ x are digital κ−paths with the condition f(m1) = g(0), then define the product (f ∗ g) : [0, m1 + m2]z −→ x by (f ∗ g)(t) = ! f(t), t ∈ [0, m1]z g(t − m1), t ∈ [m1, m1 + m2]. adjacency for the cartesian product of any two digital images is defined in [27]. let (x, κ) and (y, λ) be two digital images and consider the cartesian product x ×y of these sets. then x ×y has the following adjacency relation: for all x1, x2 ∈ x and y1, y2 ∈ y , we say that (x1, y1) and (x2, y2) are adjacent c© agt, upv, 2020 appl. gen. topol. 21, no. 2 308 the higher topological complexity in digital images on x × y if x1 and x2 are κ−adjacent and y1 and y2 are λ−adjacent. this adjacency relation on x × y is generally denoted by κ∗. definition 2.5 ([24]). let (e, κ1), (b, κ2) and (f, κ3) be any digital images, where b is a κ2−connected space. let p : (e, κ1) −→ (b, κ2) be a (κ1, κ2)−continuous surjection map. then • the triple (e, p, b) is called a digital bundle. the digital set b is called a digital base set, the digital set e is called a digital total set and the digital map p is called a digital projection of the bundle. • the quadruple ξ = (e, p, b, f) is called a digital fiber bundle with a digital base set b, a digital total set e, a digital fiber set f if p satisfies the following two conditions: 1) for all b ∈ b, the map p−1(b) −→ f is a (κ1, κ3)−isomorphism. 2) for every point b ∈ b, there exists a κ2−connected subset u of b such that ϕ : p−1(u) −→ u × f is (κ1, κ∗)−isomorphism making following diagram commute: p−1(u) ϕ !! p "" ❋❋ ❋❋ ❋❋ ❋❋ ❋ u × f. ##①① ①① ①① ①① ① u definition 2.6 ([16]). let (e, κ1) and (b, κ2) be two digital images. a digital map p : (e, κ1) −→ (b, κ2) is said to have the homotopy lifting property with respect to a digital image (x, κ3) if for any digital map "f : x −→ e and any digital homotopy g : x×[0, m]z −→ b such that p◦ "f = g◦i, where i is an inclusion map and m is a positive integer, there exists a (κ∗, κ0)−continuous map "g : x × [0, m]z −→ e making both triangles below commute: x !f !! i $$ e p $$ x × [0, m]z g !! !g %% b. definition 2.7 ([16]). a digital map p : (e, κ1) −→ (b, κ2) is a digital fibration if it has the homotopy lifting property with respect to every digital image. if b0 ∈ b, then p−1(b0) = f is called the digital fiber. ege and karaca prove that the composition of two digital fibrations is again a digital fibration, [16]. they also show that if the digital maps p1 : (e1, κ1) −→ (b1, κ2) and p2 : (e2, κ ′ 1) −→ (b2, κ ′ 2) are digital fibrations, then p1 × p2 : (e1 × e2, κ∗) −→ (b1 × b2, κ ′ ∗) is a digital fibration, where κ∗ and κ ′ ∗ are adjancency relations on cartesian products (e1 × e2) and (b1 × b2), respectively. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 309 m. i̇s and i̇. karaca let (x, κ) and (y, λ) be any digital images. the digital function space y x is defined as the set of all digitally continuous functions x −→ y and has an adjacency relation as follows, [27]: for all f, g ∈ y x, the statement x and x ′ are κ−adjacent on x implies that f(x) and g(x ′ ) are λ−adjacent. given two digital paths f, g ∈ (x, κ1)([0,m]z,2), we say that the paths f and g are γ−connected if for all t times, they are γ−connected, where γ is an adjacency relation on (x, κ1) ([0,m]z,2), [23]. suppose now (y, κ2) and (z, κ3) are digital images. digital function space map (z, κ3) (y,κ2) is the set of all maps from (y, κ2) to (z, κ3) with an adjacency as follows, [27]: for f, g ∈ (z, κ3)(y,κ2), f and g are adjacent if f(y) and g(y ′ ) are κ3−adjacent whenever y and y ′ are κ2−adjacent, for any y, y ′ ∈ (y, κ2). we recall something about a digital image px of all digital paths from [23]. to construct a digitally continuous digital motion planning algorithm s : x × x −→ px, it is needed to be define the digital connectedness on px by the definition of the digital continuity. let χ be an adjacency relation on the digital image px. let also α and β be two digital paths in the digital image px. the paths α and β are χ−connected if for all t times, they are χ−connected, where t ∈ [0, m]z. it is also noted that if we have different steps (t times) for the digital paths, then we equalize the number of steps by increasing the less one of the paths with the endpoint of it. for example, two paths α = xyzw and β = xy are given in px. then we consider β as xyyy and now we can have an idea that whether α and β are adjacent or not. definition 2.8 ([23]). the digital topological complexity tc(x, κ) is the minimal number k such that x × x = u1 ∪ u2 ∪ ... ∪ uk with the property that π admits a digitally continuous map sj : uj −→ px such that π ◦ sj = 1 over each uj ⊂ x × x. if no such k exists, then we will set tc(x, κ) = ∞. karaca and is state that when the digital image (x, κ) is κ−contractible, tc(x, κ1) is equal to 1 and the converse of this statement is also true, [23]. in addition, they show that the digital image x with two different adjacencies κ1 and κ2 yields the result that if κ1 < κ2, then tc(x, κ1) ≥ tc(x, κ2). another outstanding statement in [23] is that the digital topological complexity tc(x, κ) of a digital image (x, κ) is an invariant up to the digital homotopy. 3. some definitions and properties in the digital homotopy theory in this section, we give some useful definitions and properties that we need in the next section. these are commonly related to the homotopy theory in digital topology. definition 3.1. let f : (x, κ1) −→ (y, κ2) be a map in digital images with digitally connected spaces (x, κ1) and (y, κ2). a digital fibrational substitute c© agt, upv, 2020 appl. gen. topol. 21, no. 2 310 the higher topological complexity in digital images of f is defined as a digital fibration #f : (z, κ3) −→ (y, κ2) such that there exists a commutative diagram x h !! f $$ z "f $$ y 1y y, where h is a digital homotopy equivalence. example 3.2. let x ⊂ z be a one-point digital image and y be a digital image [0, 1]z × [0, 1]z ⊂ z2 with 4−adjacency such that f : x −→ y is a digital map. consider another digital map g : mss ′ 6 −→ y , where mss ′ 6 = [0, 1]z × [0, 1]z × [0, 1]z ⊂ z3 with 6−adjacency. we know that g is a digital fibration, [16]. moreover, there is a digital homotopy equivalence h : x −→ mss ′ 6 because mss ′ 6 is 6−contractible, [21]. then the commutativity of the following diagram holds: x h !! f && ❊❊ ❊❊ ❊❊ ❊❊ ❊ mss ′ 6 g $$ y. thus, we conclude that g is a digital fibrational substitute of f. lemma 3.3. any digital map f : (x, κ1) −→ (y, κ2) has a digital fibrational substitute. proof. let f : x −→ y be a digital map. for any m ∈ n, set the digital image z = {(x, α) : x ∈ x, α : [0, m]z −→ y, α(0) = f(x)}. let α∗ be the adjacency relation on the cartesian product digital image x × y [0,m]z. α∗ is defined as follows: for any (x1, α1), (x2, α2) ∈ x × y [0,m]z, if x1 and x2 are adjacent points and α1 and α2 are adjacent paths, then (x1, α1) and (x2, α2) are adjacent pairs of the elements in the cartesian product image. z is a subset of the cartesian product image so the adjacency relation on z is the same with the cartesian product. consider the digital map g : z −→ y (x, α) .−→ α(1). we also define h : x −→ z with h(x) = (x, αx), where αx(t) = f(x) for all t ∈ [0, m]z. we first get g ◦ h(x) = g(x, αx) = gx(1) = f(x). in order to show that g is a digital fibration, we need to build a new digital map "g : x × [0, m]z −→ z. let g : x × [0, m]z −→ y be a digital homotopy. if we take "g(x, t) = h(x) and i(x) = (x, t) for t ∈ [0, m]z, then we have that "g ◦ i(x) = "g(x, t) = h(x), c© agt, upv, 2020 appl. gen. topol. 21, no. 2 311 m. i̇s and i̇. karaca and g ◦ "g(x, t) = g ◦ h(x) = g ◦ i(x) = g(x, t) from g ◦ h = g ◦ i using the definition of a digital fibration. this shows that g is a digital fibration. finally, we say that h : x −→ z is a digital homotopy equivalence with considering the digital map k : z −→ x, defined by k(x, α) = x for any (x, α) ∈ z. as a consequence, g is a digital fibrational substitute of f. □ definition 3.4. let (e, λ1), (e ′ , λ ′ 1), (b, λ2) and (b ′ , λ ′ 2) be digital images. then a digital map of digital fibrations p : (e, λ1) −→ (b, λ2) to p ′ : (e ′ , λ ′ 1) −→ (b ′ , λ ′ 2) is a commutative diagram: e !f !! p $$ e ′ p ′ $$ b f !! b ′ . definition 3.5. let p : (e, λ1) −→ (b, λ2) be a digital fibration. two digital maps f0, f1 : (x, λ3) −→ (e, λ1) are said to be digitally fiber homotopic provided that there is a digital homotopy f : x × [0, m]z −→ e between f0 and f1 such that p ◦ f(x, t) = p ◦ f0(x) for x ∈ x and t ∈ [0, m]z. definition 3.6. let p1 : (e1, λ1) −→ (b, λ) and p2 : (e2, λ2) −→ (b, λ) be two digital fibrations. then they are said to digital fiber homotopy equivalent if there exist digital maps f : (e1, λ1) −→ (e2, λ2) and g : (e2, λ2) −→ (e1, λ1) for which g ◦f is digitally fiber homotopic to 1(e1,λ1) and f ◦g is digitally fiber homotopic to 1(e2,λ2). lemma 3.7. any two digital fibrational substitutes of a digital map f : (x, κ1) −→ (y, κ2) are digitally fiber homotopy equivalent fibrations. proof. let f : (x, κ1) −→ (y, κ2) be a digital map and assume that #f and #g are two digital fibrational substitutes of it. then we have the following diagram consisting of the union of two commutative diagrams; z ′ "g '' ❅❅ ❅❅ ❅❅ ❅❅ x k (( h !! f $$ z "f))%% %% %% %% y, i.e. #f ◦ h = f and #g ◦ k = f. since both the digital maps h and k on (x, κ1) are digitally homotopy equivalences, there exist two digitally continuous maps h1 : (z, κ3) −→ (x, κ1) and k1 : (z ′ , κ ′ 3) −→ (x, κ1) such that h ◦ h1 ≃(κ3,κ3) 1(z,κ3), h1 ◦h ≃(κ1,κ1) 1(x,κ1), k1 ◦k ≃(κ1,κ1) 1(x,κ1) and k◦k1 ≃(κ′3,κ′3) 1(z′ ,κ′3). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 312 the higher topological complexity in digital images we want to show that #f and #g are digitally fiber homotopy equivalent fibrations. so we have to guarantee the existence of two digital maps z −→ z ′ and z ′ −→ z such that their compositions are digitally fiber homotopic to idendities on z and z ′ , respectively. now, we consider the digital maps k ◦ h1 : (z, κ3) −→ (z ′ , κ ′ 3) and h ◦ k1 : (z ′ , κ ′ 3) −→ (z, κ3). it is easy to see that (h ◦ k1) ◦ (k ◦ h1) ≃(κ3,κ3) 1(z,κ3) and (k ◦ h1) ◦ (h ◦ k1) ≃(κ′3,κ′3) 1(z,κ′3). we conclude that the two digital fibrational substitutes of f is digitally fiber homotopy equivalent because #g ◦ (k ◦ h1) = #f and #f ◦ (h ◦ k1) = #g hold by definition 3.4. □ definition 3.8. the digital schwarz genus of a digital fibration p : (e, λ1) −→ (b, λ2) is defined as a minimum number k such that b = u1 ∪ u2 ∪ ... ∪ uk with the property that there is a digitally continuous map si : (ui, λ1) −→ (e, λ2) for all 1 ≤ i ≤ k, satisfies p ◦ si = 1ui over each ui ⊂ b. the digital schwarz genus of a digital map f is defined as the digital schwarz genus of the digital fibrational substitute of f. lemma 3.9. let f : (x, κ1) −→ (y, κ2) and g : (y, κ2) −→ (z, κ3) be two maps. then the digital schwarz genus of the map g ◦ f is not less than the digital schwarz genus of the map g. proof. let f and g be digital fibrations. then g ◦ f is a digital fibration. now assume that the digital schwarz genus of the digital map g◦f is k. then we have z = u1∪u2∪...∪uk such that there exists a digital map sj : (uj, κ3) −→ (x, κ1) with (g ◦ f) ◦ sj = 1uj over each uj ⊂ z, where j = 1, 2, ..., k. for each uj, we obtain a new digital map tj : (uj, κ3) −→ (y, κ2) with tj = f ◦ sj. this implies that g ◦ tj = g ◦ (f ◦ sj) = 1uj . it shows that the digital schwarz genus of the digital map g does not exceed k because the digital maps tj are constructed by the digital maps sj. if f and g are not digital fibrations, then we consider the digital fibrational substitutes of them. similarly, the digital schwarz genus of the digital map g is less than or equal to the the digital schwarz genus of the map g ◦ f. □ definition 3.10. let (x, κ1) and (y, κ2) be digital images. then the digital map e κ1,κ2 x,y : (y x × x, κ∗) −→ (y, κ2), defined by e κ1,κ2 x,y (f, x) = f(x), is called a digital evaluation map. proposition 3.11. the digital evaluation map is a digitally continuous map. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 313 m. i̇s and i̇. karaca proof. let (f, x) and (g, x ′ ) be two points in the digital image y x × x such that they are κ∗−connected. let λ be an adjacency relation on the set y x. by the definition of adjacency for the cartesian product, we have that f and g are λ−adjacent and x and x ′ are κ1−adjacent. these give us that f(x) and g(x ′ ) are κ2−adjacent. □ definition 3.12. let (x, κ1), (y, κ2) and (z, κ3) be any three digital images such that f : (x × y, κ∗) −→ (z, κ3) is a digitally continuous map. then the digital map f : (x, κ1) −→ (zy , λ), defined by f(x)(y) = f(x, y), is called a digital adjoint map. proposition 3.13. if (x, κ1), (y, κ2) and (z, κ3) are any three digital images such that f : (x × y, κ∗) −→ (z, κ3) is a digitally continuous map, then the digital adjoint map f : (x, κ1) −→ (zy , λ) x .−→ f(x) is also digitally continuous. proof. let x and x ′ be two points in x such that x and x ′ are κ1−adjacent. then (x, y) and (x ′ , y) are κ∗−adjacent for any y ∈ y . since f is digitally continuous, we have that f(x, y) and f(x ′ , y) are κ3−adjacent. hence, f(x)(y) and f(x ′ )(y) are κ3−adjacent. consequently, f(x) and f(x ′ ) are λ−adjacent. □ proposition 3.14. let (x, κ1), (y, κ2) and (z, κ3) be digital images. then the map α : (zx×y , κ∗) −→ ((zy )x, κ) f .−→ α(f) = f is digitally continuous, where κ∗ and κ are adjacency relations on z x×y and (zy )x, respectively. proof. let f and g be given in zx×y such that f and g are κ∗-adjacent and let λ∗ be an adjacency relation on x × y . we show that α(f) and α(g) are κ−adjacent with considering α(f) = f and α(g) = g. now let x and x ′ be two κ1−adjacent points in x. then (x, y) and (x ′ , y) are λ∗−adjacent in x ×y . moreover, we see that f(x, y) and g(x ′ , y) are κ3−adjacent. this means that f(x)(y) and g(x ′ )(y) are κ3−adjacent. therefore, the result holds from proposition 3.13. □ remark 3.15. a digitally continuous map ϕ : x −→ zy induces a digital map ϕ : x × y −→ z, defined by ϕ = e κ2,κ3 y,z ◦ (ϕ × 1y ); c© agt, upv, 2020 appl. gen. topol. 21, no. 2 314 the higher topological complexity in digital images x × y ϕ×1y−→ zy × y e κ2,κ3 y,z−→ z. the composite map e κ2,κ3 y,z ◦ (ϕ × 1y ) is digitally continuous since e κ2,κ3 y,z , ϕ and 1y are digitally continuous. proposition 3.16. let (x, κ1), (y, κ2) and (z, κ3) be any digital images. then the map β : ((zy )x, κ) −→ (zx×y , λ) ϕ .−→ β(ϕ) = ϕ is digitally continuous. proof. let ϕ and χ be two κ−adjacent digital maps in (zy )x, κ∗ an adjacency relation on x × y , τ an adjacency relation on zy and µ an adjaceny relation on zy ×y . we show that ϕ and χ are λ−adjacent in zx×y . now assume that (x, y) and (x ′ , y ′ ) ∈ x ×y are κ∗−adjacent. then x and x ′ are κ1−adjacent in x, so ϕ(x) and χ(x ′ ) are τ−adjacent in zy . since y and y ′ are κ2−adjacent, (ϕ × 1y )(x, y) and (χ × 1y )(x ′ , y ′ ) are µ−adjacent in zy × y . by the digital continuity of the evaluation map e κ2,κ3 y,z and remark 3.15, the result holds. □ proposition 3.17. let (x, κ1), (y, κ2) and (z, κ3) be any digital images. then the maps α : (zx×y , κ∗) → ((zy )x, κ) and β : ((zy )x, κ) → (zx×y , λ) are bijective. proof. consider the digital composition maps α◦β : ((zy )x, κ) → ((zy )x, κ) and β ◦ α : (zx×y , κ∗) → (zx×y , κ∗). then we have that α ◦ β(ϕ) = α(β(ϕ)) = α(ϕ) = (ϕ) = ϕ and β ◦ α(f) = β(α(f)) = β(f) = (f) = f for all digitally continuous maps ϕ ∈ (zy )x and f ∈ zx×y . □ we refer to [31] for more details on the topological setting. theorem 3.18. if (x, κ1), (y, κ2) and (z, κ3) are three digital images, then the map α : (zx×y , λ) −→ ((zy )x, κ), defined by α(g) = g, is a digital isomorphism, where λ is an adjacency relation on the digital image zx×y and κ is an adjacency relation on the digital image (zy )x. proof. from proposition 3.11, e λ∗,κ3 x×y,z is digitally continuous, where λ∗ is an adjacency relation on x × y . we can express another version of the digital image e λ∗,κ3 x×y,z as e κ2,κ3 y,z ◦ (α1 × 1y ), where α1 = e κ1,µ x,zy ◦ (α × 1x) such that µ is an adjacency relation on zy . as a result of proposition 3.14, proposition 3.11 and the digital continuity of the identity map, α1 is digitally continuous. finally, we conclude that α is digitally continuous. conversely, the digitally continuous map e κ2,κ3 y,z ◦(e κ1,µ x,zy ×1y ) equals the digital map e λ∗,κ3 x×y,z ◦(α −1 ×1x×y ). as a result of proposition 3.17, proposition 3.16, c© agt, upv, 2020 appl. gen. topol. 21, no. 2 315 m. i̇s and i̇. karaca proposition 3.11 and the digital continuity of the identity map, α−1 is digitally continuous as well. □ corollary 3.19. let (y, λ) be a digital image. then µ : (y [0,n]z)[0,m]z −→ (y [0,m]z)[0,n]z, for m, n ∈ n, is a digital isomorphism. proof. let κ∗ and λ∗ be adjacency relations on the digital images [0, n]z × [0, m]z and [0, m]z × [0, n]z, respectively. first, we show that the digital map γ : ([0, n]z × [0, m]z, κ∗) −→ ([0, m]z × [0, n]z, λ∗) (s, t) .−→ γ(s, t) = (t, s) is a digital isomorphism for any s ∈ [0, n]z and t ∈ [0, m]z. • to show that γ is injective, let’s define the digital map δ : ([0, m]z × [0, n]z, λ∗) −→ ([0, n]z × [0, m]z, κ∗) with δ(t, s) = (s, t). since γ ◦ δ(t, s) = γ(s, t) = (t, s) = 1[0,m]z×[0,n]z and δ ◦ γ(s, t) = δ(s, x) = (s, t) = 1[0,n]z×[0,m]z, γ has left and right inverses. • let (s, t) and (s ′ , t ′ ) be two points in [0, n]z × [0, m]z such that they are κ∗−adjacent. then s and s ′ are 2−adjacent and t and t ′ are 2−adjacent. hence, we have that (t, s) and (t ′ , s ′ ) are λ∗−adjacent. since γ(s, t) = (t, s) and γ(s ′ , t ′ ) = (t ′ , s ′ ), γ is digitally continuous. • given two λ∗−adjacent points (t, s) and (t ′ , s ′ ) in the digital image [0, m]z ×[0, n]z. this implies that t and t ′ are 2−adjacent and s and s ′ are 2−adjacent at the same time. thus, we have that (s, t) and (s ′ , t ′ ) are κ∗−adjacent. it shows that γ−1 is digitally continuous because of that γ−1(t, s) = (s, t) and γ−1(t ′ , s ′ ) = (s ′ , t ′ ). after all, we now have that γ is a digital isomorphism and γ defines a digital map φ : y [0,m]z×[0,n]z −→ y [0,n]z×[0,m]z f .−→ φ(f) = f ◦ γ for any digitally continuous map f in the image y [0,n]z×[0,m]z. second, we show that φ is a digital isomorphism. • to show that φ is bijective, we define a new digital map ψ : y [0,n]z×[0,m]z −→ y [0,m]z×[0,n]z with ψ(g) = g ◦ γ−1 for any g ∈ y [0,n]z×[0,m]z. it is easy to see that ψ ◦ φ(f) = ψ(f ◦ γ) = (f ◦ γ) ◦ γ−1 = f ◦ 1y [0,n]z×[0,m]z = f c© agt, upv, 2020 appl. gen. topol. 21, no. 2 316 the higher topological complexity in digital images and φ ◦ ψ(g) = φ(g ◦ γ−1) = (g ◦ γ−1) ◦ γ = g ◦ 1y [0,m]z×[0,n]z = g. so φ has left and right inverses. • the digital continuity of f, g, γ and γ−1 shows that φ and φ−1 are digitally continuous maps. by theorem 3.18, the digital image (y [0,n]z)[0,m]z is digitally isomorphic to y [0,m]z×[0,n]z and the digital image (y [0,m]z)[0,n]z is digitally isomorphic to y [0,n]z×[0,m]z. finally, we conclude that µ : (y [0,n]z)[0,m]z −→ (y [0,m]z)[0,n]z is a digital isomorphism because the digital isomorphism relation is an equivalence relation on digital images. □ definition 3.20. let (a, τ) and (x, κ) be digital images such that a ⊂ x. we say that a pair of digital images (x, a) has a digital homotopy extension property with respect to the digital image (y, χ) on condition that the digital map g : x −→ y and the digital homotopy g : a × [0, m]z −→ y , where m ∈ n, for which g(x) = g(x, 0) for x ∈ a, satisfy that there exists a digital homotopy f : x × [0, m]z −→ y with f(x, 0) = g(x) and f |a×[0,m]z = g for x ∈ x. definition 3.21. let (x ′ , λ) and (x, κ) be two digital images. then a digital map f : x ′ −→ x is called a digital cofibration if given two digital maps g : x −→ y and g : x ′ × [0, n]z −→ y for abritrary digital image y such that g ◦ f(x ′ ) = g(x ′ , 0) for x ′ ∈ x ′ and n ∈ n, then there exists a digital map f : x × [0, n]z −→ y satisfying f(x, 0) = g(x) and f(f(x ′ ), t) = g(x ′ , t) for x ∈ x, x ′ ∈ x ′ and t ∈ [0, n]z. x ′ × 0 ! " !! f×10 $$ x ′ × [0, n]z f×1[0,n]z $$ g **&& && && && && y x × 0 ! " !! g ++①①①①①①①①① x × [0, n]z f ,, note that the digital cofibration i : a ↩→ x corresponds to that (x, a) has the digital homotopy extension property with respect to any digital image. proposition 3.22. let i̇m = {0, m} and [0, m]z be two digital images for m ∈ n. then the pair ([0, m]z, i̇m) has the digital homotopy extension property if and only if the digital image i̇m × [0, n]z ∪ [0, m]z × 0 is a digital retract of [0, m]z × [0, n]z for n ∈ n. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 317 m. i̇s and i̇. karaca proof. (⇒) : suppose that f : i̇m −→ [0, m]z is a digital cofibration and choose the arbitrary digital image y as i̇m × [0, n]z ∪ [0, m]z ×0 in the definition 3.21. then we have a digital map f : [0, m]z × [0, n]z −→ y . hence, we can rewrite f as a retraction r : [0, m]z × [0, n]z −→ i̇m × [0, n]z ∪ [0, m]z × 0. (⇐) : let r : [0, m]z ×[0, n]z −→ i̇m ×[0, n]z ∪[0, m]z ×0 be a digital retraction. assume that g : [0, m]z −→ y and g : i̇m × [0, n]z −→ y are digital maps such that g(a) = g(a, 0) for a ∈ i̇. define a digital map k : i̇m × [0, n]z ∪ [0, m]z × 0 −→ y combining with the maps g and g. then for all a ∈ i̇m, s ∈ [0, m]z and t ∈ [0, n]z, the digital homotopy f : [0, m]z × [0, n]z −→ y (s, t) .−→ f(s, t) = k(r(s, t)) gives us f(s, 0) = k(r(s, 0)) = k(s, 0) = g(s), f(a, t) = k(r(a, t)) = k(a, t) = g(a, t). hence, f implies that f : i̇m −→ [0, m]z is a digital cofibration. □ definition 3.23. let p : (e, λ1) −→ (b, λ2) be a digital map. for n ∈ z, define a digital image b = {(e, w) ∈ e × b[o,n]z : w(0) = p(e)}. there is a digital map p ′ : e[0,n]z −→ b defined by p ′ (w) = (w(0), p ◦ w) for any w : ([0, n]z, 2) −→ (e, λ1). a digital lifting function of p is a digital map σ : b −→ e[0,n]z for which p ′ ◦ σ = 1b holds. the next three propositions and a theorem now guide us to reach our goal in this section. we first take the advantage of the relation between the digital lifting function and the digital fibration, and then we charactarize the digital cofibration. next, we obtain a new digital fibration using the characterization of the digital cofibration. as a consequence, we state the final theorem of this section with the help of these propositions. proposition 3.24. given two digital images (e, λ1) and (b, λ2). then a digital map p : (e, λ1) −→ (b, λ2) is a digital fibration if and only if there exists a digital lifting function σ of p. proof. suppose that p is a digital fibration. let f : b × [0, m]z −→ b and f ′ : b −→ e be digital maps such that f((e, w), t) = w(t) and f ′ (e, w) = e. therefore, we have f((e, w), 0) = w(0) = p(e) = p ◦ f ′ (e, w). there exists a digital homotopy f ′ : b × [0, m]z −→ e because p is a digital fibration. the equalities f ′ ((e, w), 0) = f ′ (e, w) and p ◦ f ′ = f hold. by theorem 3.18, the map σ : b −→ e[0,m]z, defined by σ(e, w)(t) = f ′ ((e, w), t), c© agt, upv, 2020 appl. gen. topol. 21, no. 2 318 the higher topological complexity in digital images is a digital lifting function for the map p. conversely, let σ be a digital lifting function for the map p. we take the digital homotopy f : x × [0, m]z −→ e and the digital map f ′ : x −→ e such that f(x, 0) = p ◦ f ′ (x). we define g : x −→ b[0,m]z by g(x)(t) = f(x, t). so we can construct a map f ′ : x × [0, m]z −→ e (x, t) .−→ f ′ (x, t) = σ(f ′ (x), g(x))(t). hence, the equalities f ′ (x, 0) = f ′ (x) and p ◦ f ′ = f hold. as a conclusion, the map p has the digital homotopy lifting property. □ consider two digital images i̇m = {0, m} and [0, m]z. let f : i̇m → [0, m]z be a digital map defined by f(0) = 0 and f(m) = m. define a digital image [0, m]z which is the quotient space of the sum (i̇m × [0, n]z) ∨ ([0, m]z × 0) obtained by identifying (x ′ , 0) ∈ i̇m × [0, n]z with (f(x ′ ), 0) ∈ [0, m]z × 0, for all x ′ ∈ i̇m. then there exists a digital map i ′ : ([0, m]z, τ) −→ ([0, m]z × [0, n]z, τ∗), defined by i ′ [x ′ , t] = (f(x ′ ), t), for all t ∈ [0, n]z and x ′ ∈ i̇m and i ′ [x, 0] = (x, 0) for all x ∈ [0, m]z, where τ and τ∗ are adjacency relations for the digital images [0, m]z and [0, m]z × [0, n]z, respectively. proposition 3.25. let i̇m = {0, m} and [0, m]z be two digital images. assume that f : i̇m −→ [0, m]z is a digital map. then f is a digital cofibration if and only if there exists a digital map ρ : ([0, m]z × [0, n]z, τ∗) −→ ([0, m]z, λ) such that ρ is a left inverse of the map i ′ . proof. (⇒) : suppose that the digital map f is a digital cofibration and consider the digital maps g : [0, m]z −→ [0, m]z and g : i̇ × [0, n]z −→ [0, m]z with g(x) = [x, 0] and g(x ′ , t) = [x ′ , t]. we now have that g(x ′ , 0) = [x ′ , 0] = [f(x ′ ), 0] = g ◦ f(x ′ ). since f is a digital cofibration, we define a digital map ρ : [0, m]z × [0, n]z −→ [0, m]z such that ρ(x, 0) = g(x) and ρ(f(x ′ ), t) = g(x ′ , t) hold for x ∈ [0, m]z and x ′ ∈ i̇. due to ρ ◦ i ′ [x, 0] = ρ(x, 0) = g(x) = [x, 0], and ρ ◦ i ′ [x ′ , t] = ρ(f(x ′ ), t) = g(f(x ′ ), t) = [x ′ , t], ρ is a left inverse of the map i ′ . (⇐) : suppose that ρ : ([0, m]z × [0, n]z, κ∗) −→ [0, m]z is a digital map with ρ ◦ i ′ = 1 [0,m]z and let g : [0, m]z −→ y and g : i̇m × [0, n]z −→ y be two c© agt, upv, 2020 appl. gen. topol. 21, no. 2 319 m. i̇s and i̇. karaca digital maps such that g(x ′ , 0) = g ◦ f(x ′ ) for x ′ ∈ i̇m. then there is a digital map g : [0, m]z −→ y defined by g[x ′ , t] = g(x ′ , t) and g[x, 0] = g(x) for any t ∈ [0, n]z. for all x ∈ [0, m]z and x ′ ∈ i̇m, we have that f(x, 0) = g ◦ ρ(x, 0) = g ◦ (ρ ◦ i ′ ([x, 0])) = g[x, 0] = g(x), and f(f(x ′ ), t) = g ◦ ρ(f(x ′ ), t) = g ◦ (ρ ◦ i ′ ([x, t])) = g[x ′ , t] = g[x ′ , t]. thus, we obtain that f is a cofibration. □ proposition 3.26. if the map f : i̇m −→ [0, m]z is a digital cofibration, then the digital map p : y [0,m]z −→ y i̇m g .−→ p(g) = g ◦ f is a digital fibration. proof. assume that the map f : i̇m −→ [0, m]z is a digital cofibration. by proposition 3.25, we have that the digital map i ′ : [0, m]z −→ [0, m]z × [0, n]z has a left inverse σ : [0, m]z × [0, n]z −→ [0, m]z. define a new digital map σ ′ : y [0,m]z −→ y [0,m]z×[0,n]z with σ ′ (g) = g ◦ σ, where g ∈ y [0,m]z. by using corollary 3.19, we see that y [0,m]z×[0,n]z is digital isomorphic to the digital image (y [0,m]z)[0,n]z. since y [0,m]z is digital isomorphic to the digital image {(g, g) ∈ y [0,m]z × (y i̇m)[0,n]z : g ◦ f = g(0)}, we conclude that σ ′ is a digital lifting function of p from proposition 3.24. □ we now give a fundamental theorem of this section which is a key for defining the digital higher topological complexity in the next section. theorem 3.27. for a digital image (x, κ1), the digital map p : x[0,m]z −→ x × x w .−→ p(w) = (w(0), w(m)) is a digital fibration. proof. for m, n ∈ z, it is easy to see that {0, m} × [0, n]z ∪ [0, m]z × 0 is a digital retract of [0, m]z × [0, n]z. by proposition 3.22, the pair of digital images ([0, m]z, {0, m}) has the digital homotopy extension property, i.e. i : {0, m} −→ [0, m]z is a digital cofibration for m ∈ n. proposition 3.26 also implies that x[0,m]z −→ x{0,m} w .−→ w ◦ i is a digital fibration. on the other hand, x{0,m} is digital isomorphic to x ×x c© agt, upv, 2020 appl. gen. topol. 21, no. 2 320 the higher topological complexity in digital images via g .→ (g(0), g(m)) for any digital map g : {0, m} −→ x. therefore, the digital map p : x[0,m]z −→ x × x w .−→ p(w) = (w(0), w(m)) is a digital fibration for any digital image (x, κ1). □ 4. the digital higher topological complexity in this section we will define higher digital topological complexity and introduce its properties. digital topological complexity can be equivalently introduced as follows. definition 4.1. let (x, κ1) be a connected digital image and p : x[0,m]z −→ x × x a digital fibration, defined by p(w) = (w(0), w(1)) for any w ∈ x[0,m]z. the digital schwarz genus of p is called the digital topological complexity of (x, κ1). it is clear that definiton 4.1 is equivalent to definition 2.8. definition 4.2. let x be any κ-connected digital image. let jn be the wedge of n−digital intervals [0, m1]z, ..., [0, mn]z for a positive integer n, where 0i ∈ [0, mi], i = 1, ..., n, are identified. then the digital higher topological complexity tcn(x, κ) is defined by the digital schwarz genus of the digital fibration en : x jn → xn f .−→ (f(m1)1, ..., f(mn)n), where (mi)k, k = 1, ..., n, denotes the endpoints of the i−th interval for each i. theorem 4.3. let (x, κ) be a digital κ−connected image and n be a positive integer. then the following hold: (1) tc1(x, κ) = 1. (2) tcn(x, κ) is a homotopy invariant in digital images. (3) tc2(x, κ) coincides with tc(x, κ). (4) tcn(x, κ) ≤ tcn+1(x, κ). proof. (1) consider a digital map e1 : x j1 → x defined by e1(f) = f(11). since there is a digitally continuous map s1 : x → xj1 such that e1 ◦ s1 = 1x, tc(x, κ) = 1. (2) the proof is a repeated use of the proof of theorem 4.1 in [23]. let x and y be digitally homotopy equivalent images with the maps f : x → y and g : y → x. let f ◦ g be digitally homotopic to 1y . we first assume that u is a subset of x × x for which s : u → px is a digitally continuous motion planning. then we define v = (g × g)−1(u) and find a digitally continuous motion planning α : v → py . for the tcn version of the proof, we have to define v as n−copies of (g×g×...×g)−1(u), where u belongs to x×x×...×x because the range of the digital map en is n−copies of x. we also take digitally continuous maps s : u −→ xjn and α : v −→ y jn since the domain of the c© agt, upv, 2020 appl. gen. topol. 21, no. 2 321 m. i̇s and i̇. karaca digital map en includes the digitally continuous maps defined from any digitally connected space to jn. (3) we first note that en : xjn −→ xn is a digital fibrational substitute of the diagonal map of digital images ∆n : x −→ xn. by definition 3.1, we must show that en ◦ h = 1xn ◦ ∆n. here we define h : x −→ xjn as a digital map taking any point x in (x, κ) and turning it into the wedge of digital loops starting and finishing at x. to say h is a digital homotopy equivalence, we consider that there exists a digital map g : xjn −→ x, defined by g(β) = β(0) for any β ∈ xjn, such that g◦h is digitally homotopy equivalent to 1x and h◦g is digitally homotopy equivalent to 1xjn . hence, we have en ◦ h = 1xn ◦ ∆n. on the other hand, the digital fibration en is digitally homotopy equivalent to the digital fibration fn : x [0,m]z −→ xn α .−→ (α(0), α(1), α(2), ..., α(n − 1)), where m is a positive integer such that n ≤ m − 1. since we take the digital homotopy equivalence h as the digital constant map at x in the digital image (x, κ), we have fn ◦h = ∆n. as a conclusion, tcn(x, κ) is the digital schwarz genus of fn and tc(x, κ) is the digital schwarz genus of p in definition 4.1. moreover, for n = 2, the digital fibration f2 coincides with p. it shows that tc2(x, κ) = tc(x, κ). (4) assume that tcn+1(x, κ) = r and examine the digital higher topological complexity tcn(x, κ). then the digital schwarz genus of the digital map en+1 equals r. so we have that x n+1 = u1 ∪ u2 ∪ ... ∪ ur and there exists si : ui −→ xjn+1 such that en+1 ◦ si = 1ui for each i ∈ [1, r]. consider the digital map an : x jn+1 −→ xjn that takes the set of n + 1 paths and deletes the last one into the set of first n paths. now choose a point a ∈ x and establish a new digital map bn : x n −→ xn+1 that assigns n points of x to the n + 1 points of it by adding a to the end of the n points. last, we set vi = {(x1, x2, ..., xn) ∈ xn : (x1, x2, ..., xn, a) ∈ ui} ⊂ xn. therefore, we obtain ti = an ◦ si ◦ bn : vi −→ xjn with the help of the digital map si such that en ◦ ti = 1vi for i = 1, 2, ..., r. it shows that the digital schwarz genus of en can be at most r and so tcn(x, κ) ≤ r. □ we currently finalize this section with an example about frequently used the digital surface mss18 and the digital curve msc8 in digital topology, [21]. example 4.4. consider the minimal simple surface mss18 = {ai}9i=0 in z3, where a0 = {0, 0, 0}, a1 = {1, 1, 0}, a2 = {0, 1, −1}, a3 = {0, 2, −1}, a4 = {1, 2, 0}, a5 = {0, 3, 0}, a6 = {−1, 2, 0}, a7 = {0, 2, 1}, a8 = {0, 1, 1}, a9 = {−1, 1, 0}. it is shown that any loop is 18−contractible in mss18, [9] and cat18(mss18) = 1, [2]. by [23, theorem 5.1], we have tc(mss18, 18) ≥ 1 c© agt, upv, 2020 appl. gen. topol. 21, no. 2 322 the higher topological complexity in digital images but [23, corollary 3.3]] says that tc(mss18, 18) cannot be greater than 1. therefore, we conclude that tc1(mss18, 18) = 1, tc2(mss18, 18) = 1 and tcn(mss18) ≥ 1, for n ≥ 3, from theorem 4.3. on the other hand, if we consider the minimal simple closed curve msc8 = {bi}5i=0 in z2, where b0 = {0, 0}, b1 = {−1, 1}, b2 = {−1, 2}, b3 = {0, 3}, b4 = {1, 2}, b5 = {1, 1}, then we have tc(msc8, 8) = 2 by [23, theorem 3.5]]. it shows that tc1(msc8, 8) = 1, tc2(msc8, 8) = 2 and tcn(msc8, 8) ≥ 2, for n ≥ 3, from theorem 4.3. 5. a different result for the digital topological complexity cohomological lower bounds are not valid for the digital topological complexity tc(x, κ), where (x, κ) is a digital image. let us explain it with an example. example 5.1. let f be a field. consider the two same digital images x = y = [0, m]z with 2−adjacency for any m ∈ n and assume that cohomological lower bounds are holds in digital images. we know h∗,2(x; f) = h∗,2(y ; f) = ! f , n = 0 0 , n ∕= 0, and h∗,4(x × y ; f) = ! f , n = 0, 1 0 , n ∕= 0, 1, from [14] and [1], respectively. since x and y are 2−contractible, we have that tc(x, 2) = 1 = tc(y, 2). so the zero-divisor-cup-lengths of h∗,2(x; f) and h∗,2(y ; f) must be equal to zero. moreover, tc(x×y, 4) = 1 because x×y is 4−contractible. this implies that the zero-divisor-cup-length of h∗,4(x×y ; f) is equal to zero as well. on the other hand, the zero-divisor-cup-lengths of h1,2(x; f) and h1,2(y ; f) are equal to zero by h1,2(x; f) = h1,2(y ; f) = 0. in addition to this, the zero-divisor-cup-length of h1,4(x×y ; f) is greater than or equal to 1. we can choose a nonzero fundamental class u ∈ h1,4(x × y ; f) such that u = 1 ⊗ u − u ⊗ 1 is a zero-divisor by h1,4(x × y ; f) = f . we focus on the first dimension of digital cohomologies with considering that the minimum adjacency is 4 for x × y because it is a digital image in z2. then we get a contradiction! 6. conclusion this study introduces the theory of higher topological complexity in digital images. we first start with establishing the homotopy background for this. as a continuation, we deal with the number of digital topological complexities. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 323 m. i̇s and i̇. karaca furthermore, we answer the open problem given in the conclusion section of [23]. we offer a clear solution in digital setting to the question of what relationship between the digital cohomological cup-product and the digital topological complexity tc(x, κ) will have. our answer is negative, i.e. it is not suitable to build such a cohomological structure in digital topology. we can study on the various areas with different examples. one open problem is to find a relation between tcn and digital lusternik-schnirelmann category cat which is introduced in [2]. acknowledgements. the authors are thankful to the anonymous referees for their valuable comments and suggestions. the first author is granted as fellowship by the scientific and technological research council of turkey tubitak2211-a. in addition, this work was partially supported by research fund of the ege university (project number: fdk-2020-21123) references [1] h. arslan, i. karaca and a. oztel, homology groups of n− dimensional digital images, xxi turkish national mathematics symposium (2008); b1-13. 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[31] t. tom dieck, algebraic topology, zurich, switzerland: ems textbooks in mathematics, ems, 2008. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 325 @ appl. gen. topol. 23, no. 2 (2022), 463-480 doi:10.4995/agt.2022.15963 © agt, upv, 2022 classical solutions for the euler equations of compressible fluid dynamics: a new topological approach dalila boureni a , svetlin georgiev b , arezki kheloufi a and karima mebarki c a laboratory of applied mathematics, bejaia university, 06000 bejaia, algeria. (dalila.boureni@univ-bejaia.dz; arezki.kheloufi@univ-bejaia.dz; arezkinet2000@yahoo.fr) b department of differential equations, faculty of mathematics and informatics, university of sofia, sofia, bulgaria. (svetlingeorgiev1@gmail.com) c laboratory of applied mathematics, faculty of exact sciences, bejaia university, 06000 bejaia, algeria. (karima.mebarki@univ-bejaia.dz; mebarqi karima@hotmail.fr) communicated by e. a. sánchez-pérez abstract in this article we study a class of euler equations of compressible fluid dynamics. we give conditions under which the considered equations have at least one and at least two classical solutions. to prove our main results we propose a new approach based upon recent theoretical results. 2020 msc: 35q31; 35a09; 35e15. keywords: euler equations; classical solution; fixed point; initial value problem. 1. introduction in this paper, we investigate an initial value problem for euler equations of compressible fluid dynamics, see [6], [10], [21]. namely, we are concerned with received 22 july 2021 – accepted 26 june 2022 http://dx.doi.org/10.4995/agt.2022.15963 https://orcid.org/0000-0003-3227-0745 https://orcid.org/0000-0001-8015-4226 https://orcid.org/0000-0001-5584-1454 https://orcid.org/0000-0002-6679-5059 d. boureni, s. georgiev, a. kheloufi and k. mebarki the following system: (1.1) ∂tρ + ∂x(ρu) = 0, ∂t(ρu) + ∂x ( ρu2 + p(ρ) ) = 0, t > 0, x ∈ r, ρ(0,x) = ρ0(x), x ∈ r, u(0,x) = u0(x), x ∈ r, where (h1): ρ0,u0 ∈ c1(r), 0 ≤ ρ0(x),u0(x) ≤ b, x ∈ r, with b is a given positive constant. here the unknowns ρ = ρ(t,x) ≥ 0 and u = u(t,x) denote respectively, the density and the velocity of the gas, while the pressure p = p(ρ) is a given function so that (h2): p ∈ c(r) is a nonnegative function for which p(z) ≤ czq, z ≥ 0, c is a positive constant, q ≥ 0. note that if p(ρ) = cρq, ρ ≥ 0, c > 0, q ≥ 1, then, the fluid is called isentropic and isothermal when q > 1 and q = 1, respectively. for other possibilities of the pressure function, readers may refer to [5] and the references therein. cauchy problem with bounded measurable initial data for (1.1): (ρ0,u0) ∈ l∞ ×l∞ where u0(x) and ρ0(x) ≥ 0 (6≡ 0) is studied in [4]. the authors established the convergence of a second-order shock-capturing scheme. in [7], a convergence result for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established. in [20], some properties for solutions of (1.1) containing a portion of the t − x plane in which ρ = 0 called vacuum state, were investigated. conservation laws of the one-dimensional isentropic gas dynamics equations in lagrangian coordinates are obtained in [16]. in [19], a 2 × 2 hyperbolic system of isentropic gas dynamics, in both eulerian or lagrangian variables is considered. whereas local existence results for problems of type (1.1) were obtained, see, for example, [2], [3], [13],[17], [18], [22], [23], the literature concerning global existence of solutions for such kind of problems does not seem to be very rich. the problem of the global in time existence of solutions of the equations of fluid mechanics in one space dimension was treated by glimm in 1965 [12]. the equations (1.1) was investigated in [11] for existence of global periodic solutions. for euler equations with damping, the global existence of solutions can be found in [24], [27], [30] and the references therein. in [5], a class of conditions for non-existence of global classical solutions is established for the initial-boundary value problem of a three-dimensional compressible euler © agt, upv, 2022 appl. gen. topol. 23, no. 2 464 classical solutions for the euler equations equations with (or without) time-dependent damping. we mention also the works [15], [26] and [28]. the aim of this paper is to investigate the ivp (1.1) for existence of global classical solutions. we call a solution a classical solution if it, along with its derivatives that appear in the equations, is of class c([0,∞) ×r). our main result for existence of classical solutions of the ivp (1.1) is as follows. theorem 1.1. suppose (h1)-(h2). then, the ivp (1.1) has at least one nonnegative solution (ρ,u) ∈c1([0,∞) ×r) ×c1([0,∞) ×r). theorem 1.2. suppose (h1)-(h2). then, the ivp (1.1) has at least two nonnegative solutions (ρ1,u1), (ρ2,u2) ∈c1([0,∞) ×r) ×c1([0,∞) ×r). the strategy for the proof of theorem 1.1 and theorem 1.2 which we develop in section 2 uses the abstract theory of the sum of two operators. this basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics. the paper is organized as follows. in the next section, we give some auxiliary results. in section 3 we prove theorem 1.1. in section 4, we prove theorem 1.2. in section 5, we give an example to illustrate our main results. 2. preliminaries and auxiliary results 2.1. preliminaries. to prove our existence results we will use theorem 2.1 and theorem 2.8, that we will present and demonstrate in the sequel. theorem 2.1. let � > 0, r > 0, e be a banach space and x = {x ∈ e : ‖x‖≤ r}. let also, tx = −�x, x ∈ x, s : x → e is a continuous, (i −s)(x) resides in a compact subset of e and (2.1) {x ∈ e : x = λ(i −s)x, ‖x‖ = r} = ∅ for any λ ∈ ( 0, 1 � ) . then, there exists x∗ ∈ x so that tx∗ + sx∗ = x∗. proof. define r ( − 1 � x ) =   −1 � x if ‖x‖≤ r� rx ‖x‖ if ‖x‖ > r�. then, r ( −1 � (i −s) ) : x → x is continuous and compact. hence and the schauder fixed point theorem, it follows that there exists x∗ ∈ x so that r ( − 1 � (i −s)x∗ ) = x∗. assume that −1 � (i −s)x∗ 6∈ x. then,∥∥∥(i −s)x∗∥∥∥ > r�, r ‖(i −s)x∗‖ < 1 � © agt, upv, 2022 appl. gen. topol. 23, no. 2 465 d. boureni, s. georgiev, a. kheloufi and k. mebarki and x∗ = r ‖(i −s)x∗‖ (i −s)x∗ = r ( − 1 � (i −s)x∗ ) and hence, ‖x∗‖ = r. this contradicts with (2.1). therefore, −1 � (i−s)x∗ ∈ x and x∗ = r ( − 1 � (i −s)x∗ ) = − 1 � (i −s)x∗ or −�x∗ + sx∗ = x∗, or tx∗ + sx∗ = x∗. this completes the proof. � let e be a real banach space. definition 2.2. a closed, convex set p in e is said to be cone if (1) αx ∈p for any α ≥ 0 and for any x ∈p, (2) x,−x ∈p implies x = 0. every cone p defines a partial ordering ≤ in e defined by : x ≤ y if and only if y −x ∈p. denote p∗ = p\{0}. definition 2.3. a mapping k : e → e is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets. in what follows, we give some results about the fixed point index theory for perturbation of a completely continuous mapping by expansive one. first, we recall the definition of an expansive mapping. definition 2.4. let x and y be real banach spaces. a mapping k : x → y is said to be expansive if there exists a constant h > 1 such that ‖kx−ky‖y ≥ h‖x−y‖x for any x,y ∈ x. in the following lemma, we present the key property of the expansive mappings which allows to extend the notion of the fixed point index in the case of a completely continuous mapping perturbed by an expansive one. lemma 2.5. [29, lemma 2.1] let (x,‖.‖) be a linear normed space and d ⊂ x. assume that the mapping t : d → x is expansive with constant h > 1. then, the inverse of i −t : d → (i −t)(d) exists and ‖(i −t)−1x− (i −t)−1y‖≤ 1 h− 1 ‖x−y‖, ∀x,y ∈ (i −t)(d). © agt, upv, 2022 appl. gen. topol. 23, no. 2 466 classical solutions for the euler equations in the sequel, p will refer to a cone in a banach space (e,‖.‖), ω is a subset of p, and u is a bounded open subset of p. assume that s : u → e is a completely continuous mapping and t : ω → e is a expansive one with constant h > 1. by lemma 2.5, the operator (i−t)−1 is (h− 1)−1-lipschtzian on (i −t)(ω). suppose that (2.2) s(u) ⊂ (i −t)(ω), and (2.3) x 6= tx + sx, for all x ∈ ∂u ∩ ω. then, x 6= (i −t)−1sx, for all x ∈ ∂u and the mapping (i −t)−1s : u →p is a completely continuous. from [14, theorem 2.3.1], the fixed point index i ((i −t)−1s,u,p) is well defined. thus we put (2.4) i∗(t + s,u ∩ ω,p) = { i (i −t)−1s,u,p), if u ∩ ω 6= ∅ 0, if u ∩ ω = ∅. this integer is called the generalized fixed point index of the sum t + s on u ∩ ω with respect to the cone p. the basic properties of the index i∗ are collected in the following lemma lemma 2.6 ([8, theorem 2.3]). the fixed point index defined in (2.4) satisfies the following properties: (a) (normalization). if u = pr, 0 ∈ ω, and sx = z0 ∈b(−t0, (h−1)r)∩p for all x ∈pr, then, i∗ (t + s,pr ∩ ω,p) = 1. (b) (additivity). for any pair of disjoint open subsets u1,u2 in u such that t + s has no fixed point on (u \(u1 ∪u2)) ∩ ω, we have i∗ (t + s,u ∩ ω,p) = i∗ (t + s,u1 ∩ ω,p) + i∗ (t + s,u2 ∩ ω,p), where i∗ (t + s,uj ∩ ω,x) : = i∗ (t + s|uj,uj ∩ ω,p), j = 1, 2. (c) (homotopy invariance). the fixed point index i∗ (t + h(t, .),u ∩ω,p) does not depend on the parameter t ∈ [0, 1] whenever (i) h : [0, 1] ×u → e be a completely continuous mapping, (ii) h([0, 1] ×u) ⊂ (i −t)(ω), (iii) tx + h(t,x) 6= x, for all t ∈ [0, 1] and x ∈ ∂u ∩ ω. (d) (solvability). if i∗ (t + s,u ∩ ω,p) 6= 0, then, t + s has a fixed point in u ∩ ω. several considerations allowing computation of the index i∗ are shown in [8]. the following result is an extension of [8, proposition 2.11] in the case of a completely continuous mapping perturbed by an expansive one. proposition 2.7. let u be a bounded open subset of p with 0 ∈ u. assume that t : ω ⊂ p → e is an expansive mapping, s : u → e is a completely continuous one and s(u) ⊂ (i −t)(ω). © agt, upv, 2022 appl. gen. topol. 23, no. 2 467 d. boureni, s. georgiev, a. kheloufi and k. mebarki if t + s has no fixed point on ∂u ∩ ω and there exists ε > 0 small enough such that sx 6= (i −t)(λx) for all λ ≥ 1 + ε, x ∈ ∂u and λx ∈ ω, then, the fixed point index i∗ (t + s,u ∩ ω,p) = 1. proof. the mapping (i − t)−1s : u → p is a completely continuous without fixed point in the boundary ∂u and it is readily seen that the following condition is satisfied (i −t)−1sx 6= λx for all x ∈ ∂u and λ ≥ 1 + ε. our claim then follows from the definition of i∗ and [1, lemma 2.3]. � now we are able to present a multiple fixed point theorem. the proof rely on proposition 2.7 and [8, proposition 2.16] producing the computation of the index i∗. this result will be used to prove theorem 1.2. theorem 2.8. let u1,u2 and u3 three open bounded subsets of p such that u1 ⊂ u2 ⊂ u3 and 0 ∈ u1. assume that t : ω ⊂ p → e is an expansive mapping, s : u3 → e is a completely continuous one and s(u3) ⊂ (i−t)(ω). suppose that (u2 \u1) ∩ ω 6= ∅, (u3 \u2) ∩ ω 6= ∅, and there exists v0 ∈ p∗ such that the following conditions hold: (i): sx 6= (i −t)(x−λv0), for all λ > 0 and x ∈ ∂u1 ∩ (ω + λv0), (ii): there exists ε > 0 small enough such that sx 6= (i−t)(λx), for all λ ≥ 1 + ε, x ∈ ∂u2, and λx ∈ ω, (iii): sx 6= (i −t)(x−λv0), for all λ > 0 and x ∈ ∂u3 ∩ (ω + λv0). then,t + s has at least two non-zero fixed points x1,x2 ∈p such that x1 ∈ ∂u2 ∩ ω and x2 ∈ (u3 \u2) ∩ ω or x1 ∈ (u2 \u1) ∩ ω and x2 ∈ (u3 \u2) ∩ ω. proof. if sx = (i−t)x for x ∈ ∂u2∩ω, then we get a fixed point x1 ∈ ∂u2∩ω of the operator t + s. suppose that sx 6= (i − t)x for any x ∈ ∂u2 ∩ ω. without loss of generality, assume that tx+sx 6= x on ∂u1∩ω and tx+sx 6= x on ∂u3 ∩ ω, otherwise the conclusion has been proved. by proposition 2.7 and [8, proposition 2.16], we have i∗ (t + s,u1 ∩ω,p) = i∗ (t + s,u3 ∩ω,p) = 0 and i∗ (t + s,u2 ∩ω,p) = 1. the additivity property of the index i∗ yields i∗ (t + s, (u2 \u1) ∩ ω,p) = 1 and i∗ (t + s, (u3 \u2) ∩ ω,p) = −1. consequently, by the existence property of the index i∗, t + s has at least two fixed points x1 ∈ (u2 \u1) ∩ ω and x2 ∈ (u3 \u2) ∩ ω. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 468 classical solutions for the euler equations 2.2. auxiliary results. in this subsection, we give some properties of solutions of ivp (1.1). let x1 = c1([0,∞) ×r) be endowed with the norm ‖u‖x1 = max { sup (t,x) ∈ [0,∞) ×r |u(t,x)|, sup (t,x) ∈ [0,∞) ×r |ut(t,x)|, sup (t,x) ∈ [0,∞) ×r |ux(t,x)| } , provided it exists. let x2 = x1 ×x1 be endowed with the norm ‖(ρ,u)‖x2 = max{‖ρ‖x1, ‖u‖x1}, (ρ,u) ∈ x2, provided it exists. for (ρ,u) ∈ x2, we will write (ρ,u) ≥ 0 if ρ(t,x) ≥ 0, u(t,x) ≥ 0 for any (t,x) ∈ [0,∞) ×r. for (ρ,u) ∈ x2, define the operators s11 (ρ,u)(t,x) = ∫ x 0 (ρ(t,x1) −ρ0(x1)) dx1 + ∫ t 0 ρ(t1,x)u(t1,x)dt1, s21 (ρ,u)(t,x) = ∫ x 0 (ρ(t,x1)u(t,x1) −ρ0(x1)u0(x1)) dx1 + ∫ t 0 ( ρ(t1,x)(u(t1,x)) 2 + p(ρ(t1,x)) ) dt1, s1(ρ,u)(t,x) = ( s11 (ρ,u)(t,x),s 2 1 (ρ,u)(t,x) ) , (t,x) ∈ [0,∞) ×r. lemma 2.9. suppose (h1) and p ∈c(r). if (ρ,u) ∈ x2 satisfies the equation (2.5) s1(ρ,u)(t,x) = 0, (t,x) ∈ [0,∞) ×r, then it is a solution of the ivp (1.1). proof. let (ρ,u) ∈ x2 is a solution to the equation (2.5). then (2.6) s11 (ρ,u)(t,x) = 0, s 2 1 (ρ,u)(t,x) = 0, (t,x) ∈ [0,∞) ×r. we differentiate the first equation of (2.6) with respect to t and x and we find ρt(t,x) + (ρu)x(t,x) = 0, (t,x) ∈ [0,∞) ×r. we put t = 0 in the first equation of (2.6) and we arrive at∫ x 0 (ρ(0,x1) −ρ0(x1)) dx1 = 0, x ∈ r, which we differentiate with respect to x and we find ρ(0,x) = ρ0(x), x ∈ r. now, we differentiate the second equation of (2.6) with respect to t and x and we find (ρu)t(t,x) + (ρu 2 + p(ρ))x(t,x) = 0, (t,x) ∈ [0,∞) ×r. © agt, upv, 2022 appl. gen. topol. 23, no. 2 469 d. boureni, s. georgiev, a. kheloufi and k. mebarki we put t = 0 in the second equation of (2.6) and we get∫ x 0 (ρ(0,x1)u(0,x1) −ρ0(x1)u0(x1)) dx1 = 0, x ∈ r, which we differentiate with respect to x and we obtain ρ(0,x)u(0,x) −ρ0(x)u0(x) = 0, x ∈ r, whereupon u(0,x) = u0(x), x ∈ r. thus, (ρ,u) is a solution to the ivp (1.1). this completes the proof. � lemma 2.10. suppose (h1) and let h ∈c([0,∞) × r) be a positive function almost everywhere on [0,∞) ×r. if (ρ,u) ∈ x2 satisfies the following integral equations:∫ t 0 ∫ x 0 (t− t1)2(x−x1)2h(t1,x1)s11 (ρ,u)(t1,x1)dx1dt1 = 0, (t,x) ∈ [0,∞) ×r and∫ t 0 ∫ x 0 (t− t1)2(x−x1)2h(t1,x1)s21 (ρ,u)(t1,x1)dx1dt1 = 0, (t,x) ∈ [0,∞) ×r, then, (ρ,u) is a solution to the ivp (1.1). proof. we differentiate three times with respect to t and three times with respect to x the integral equations of lemma 2.10 and we find h(t,x)s1(ρ,u)(t,x) = 0, (t,x) ∈ [0,∞) ×r, whereupon s1(ρ,u)(t,x) = 0, (t,x) ∈ [0,∞) ×r. hence and lemma 2.9, we conclude that (ρ,u) is a solution to the ivp (1.1). this completes the proof. � let b1 = max{b,b2,b3,cbq}. lemma 2.11. suppose (h1) and (h2). for (ρ,u) ∈ x2 with ‖(ρ,u)‖x2 ≤ b, we have |s11 (ρ,u)(t,x)| ≤ 2b1(1 + t)(1 + |x|), |s21 (ρ,u)(t,x)| ≤ 2b1(1 + t)(1 + |x|), (t,x) ∈ [0,∞) ×r. © agt, upv, 2022 appl. gen. topol. 23, no. 2 470 classical solutions for the euler equations proof. we have |s11 (ρ,u)(t,x)| = ∣∣∣∫ x 0 (ρ(t,x1) −ρ0(x1)) dx1 + ∫ t 0 ρ(t1,x)u(t1,x)dt1 ∣∣∣ ≤ ∣∣∣∫ x 0 (ρ(t,x1) −ρ0(x1)) dx1 ∣∣∣ + ∣∣∣∫ t 0 ρ(t1,x)u(t1,x)dt1 ∣∣∣ ≤ ∣∣∣∫ x 0 (|ρ(t,x1)| + ρ0(x1)) dx1 ∣∣∣ + ∫ t 0 |ρ(t1,x)||u(t1,x)|dt1 ≤ 2b|x| + b2t ≤ 2b1|x| + b1t ≤ 2b1(1 + |x|)(1 + t), (t,x) ∈ [0,∞) ×r, and ∣∣∣s21 (ρ,u)(t,x)∣∣∣ = ∣∣∣∫ x 0 (ρ(t,x1)u(t,x1) −ρ0(x1)u0(x1)) dx1 + ∫ t 0 ( ρ(t1,x)(u(t1,x)) 2 + p(ρ(t1,x)) ) dt1 ∣∣∣ ≤ ∣∣∣∫ x 0 (ρ(t,x1)u(t,x1) −ρ0(x1)u0(x1)) dx1 ∣∣∣ + ∣∣∣∫ t 0 ( ρ(t1,x)(u(t1,x)) 2 + p(ρ(t1,x)) ) dt1 ∣∣∣ ≤ ∣∣∣∫ x 0 (|ρ(t,x1)||u(t,x1)| + ρ0(x1)u0(x1)) dx1 ∣∣∣ + ∫ t 0 ( |ρ(t1,x)|(u(t1,x))2 + c(ρ(t1,x))q ) dt1 ≤ 2b2|x| + b3t + cbqt ≤ 2b1|x| + 2b1t ≤ 2b1(1 + |x|)(1 + t), (t,x) ∈ [0,∞) ×r. this completes the proof. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 471 d. boureni, s. georgiev, a. kheloufi and k. mebarki 3. proof of theorem 1.1 (a1): let a be a positive constant such that a ≤ 1 and g ∈c([0,∞)×r) is a nonnegative function such that 16(1 + t) ( 1 + t + t2 ) (1 + |x|) ( 1 + |x| + x2 )∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣∣dt1 ≤ a, (t,x) ∈ [0,∞) ×r. in the last section, we will give an example for a function g that satisfies (a1). for (ρ,u) ∈ x2, define the operators s12 (ρ,u)(t,x) = ∫ t 0 ∫ x 0 (t− t1)2(x−x1)2g(t1,x1)s11 (ρ,u)(t1,x1)dx1dt1, s22 (ρ,u)(t,x) = ∫ t 0 ∫ x 0 (t− t1)2(x−x1)2g(t1,x1)s21 (ρ,u)(t1,x1)dx1dt1, s2(ρ,u)(t,x) = ( s12 (ρ,u)(t,x),s 2 2 (ρ,u)(t,x) ) , (t,x) ∈ [0,∞) ×r. lemma 3.1. suppose (h1)-(h2). for (ρ,u) ∈ x2, ‖(ρ,u)‖x2 ≤ b, we have ‖s2(ρ,u)‖x2 ≤ ab1, where b1 = max{b,b2,b3,cbq}. proof. we have |s12 (ρ,u)(t,x)| = ∣∣∣∣ ∫ t 0 ∫ x 0 (t− t1)2(x−x1)2g(t1,x1)s11 (ρ,u)(t1,x1)dx1dt1 ∣∣∣∣ ≤ ∫ t 0 ∣∣∣∣ ∫ x 0 (t− t1)2(x−x1)2g(t1,x1)|s11 (ρ,u)(t1,x1)|dx1 ∣∣∣∣dt1 ≤ 2b1 ∫ t 0 ∣∣∣∣ ∫ x 0 (t− t1)2(x−x1)2g(t1,x1)(1 + t1)(1 + |x1|)dx1 ∣∣∣∣dt1 ≤ 8b1(1 + t)t2(1 + |x|)x2 ∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣dt1 ≤ 16b1(1 + t)(1 + t + t2)(1 + |x|)(1 + |x| + x2) ∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣dt1 ≤ ab1, (t,x) ∈ [0,∞) ×r, © agt, upv, 2022 appl. gen. topol. 23, no. 2 472 classical solutions for the euler equations and∣∣∣∣ ∂∂ts12 (ρ,u)(t,x) ∣∣∣∣ = 2 ∣∣∣∣ ∫ t 0 ∫ x 0 (t− t1)(x−x1)2g(t1,x1)s11 (ρ,u)(t1,x1)dx1dt1 ∣∣∣∣ ≤ 2 ∫ t 0 ∣∣∣∣ ∫ x 0 (t− t1)(x−x1)2g(t1,x1)|s11 (ρ,u)(t1,x1)|dx1 ∣∣∣∣dt1 ≤ 4b1 ∫ t 0 ∣∣∣∣ ∫ x 0 (t− t1)(x−x1)2g(t1,x1)(1 + t1)(1 + |x1|)dx1 ∣∣∣∣dt1 ≤ 16b1(1 + t)t(1 + |x|)x2 ∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣dt1 ≤ 16b1(1 + t)(1 + t + t2)(1 + |x|)(1 + |x| + x2) ∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣dt1 ≤ ab1, (t,x) ∈ [0,∞) ×r, and∣∣∣∣ ∂∂xs12 (ρ,u)(t,x) ∣∣∣∣ = 2 ∣∣∣∣ ∫ t 0 ∫ x 0 (t− t1)2(x−x1)g(t1,x1)s11 (ρ,u)(t1,x1)dx1dt1 ∣∣∣∣ ≤ 2 ∫ t 0 ∣∣∣∣ ∫ x 0 (t− t1)2|x−x1|g(t1,x1)|s11 (ρ,u)(t1,x1)|dx1 ∣∣∣∣dt1 ≤ 4b1 ∫ t 0 ∣∣∣∣ ∫ x 0 (t− t1)2|x−x1|g(t1,x1)(1 + t1)(1 + |x1|)dx1 ∣∣∣∣dt1 ≤ 8b1(1 + t)t2(1 + |x|)|x| ∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣dt1 ≤ 16b1(1 + t)(1 + t + t2)(1 + |x|)(1 + |x| + x2) ∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣dt1 ≤ ab1, (t,x) ∈ [0,∞) ×r. as above,∣∣s22 (ρ,u)(t,x)∣∣ , ∣∣∣∣ ∂∂ts22 (ρ,u)(t,x) ∣∣∣∣ , ∣∣∣∣ ∂∂xs22 (ρ,u)(t,x) ∣∣∣∣ ≤ ab1, (t,x) ∈ [0,∞) ×r. therefore, ‖s2(ρ,u)‖x2 ≤ ab1. this completes the proof. � below, let (a2): � ∈ (0, 1), a, b, b1 and q satisfy the inequalities �b1(1 + a) < 1 and ab1 < b. © agt, upv, 2022 appl. gen. topol. 23, no. 2 473 d. boureni, s. georgiev, a. kheloufi and k. mebarki let ỹ denotes the union of the set {(ρ0,u0)} and the closure of the set of all equi-continuous families in x2 with respect to the norm ‖ ·‖x2 . let also, y = {(ρ,u) ∈ ỹ : (ρ,u) ≥ 0, ‖(ρ,u)‖x2 ≤ b}. note that y is a compact set in x2. for (ρ,u) ∈ x2, define the operators t(ρ,u)(t,x) = −�(ρ,u)(t,x), s(ρ,u)(t,x) = (ρ,u)(t,x) + �(ρ,u)(t,x) + �s2(ρ,u)(t,x), (t,x) ∈ [0,∞) ×r. for (ρ,u) ∈ y , using lemma 3.1, we have ‖(i −s)(ρ,u)‖x2 = ‖�(ρ,u) + �s2(ρ,u)‖x2 ≤ �‖(ρ,u)‖x2 + �‖s2(ρ,u)‖x2 ≤ �b1 + �ab1 = �b1(1 + a) < b. thus, s : y → x2 is continuous and (i −s)(y ) resides in a compact subset of x2. now, suppose that there is a (ρ,u) ∈ x2 so that ‖(ρ,u)‖x2 = b and (ρ,u) = λ(i −s)(ρ,u) or 1 λ (ρ,u) = (i −s)(ρ,u) = −�(ρ,u) − �s2(ρ,u), or ( 1 λ + � ) (ρ,u) = −�s2(ρ,u) for some λ ∈ ( 0, 1 � ) . hence, ‖s2(ρ,u)‖x2 ≤ ab1 < b, �b < ( 1 λ + � ) b = ( 1 λ + � ) ‖(ρ,u)‖x2 = �‖s2(ρ,u)‖x2 < �b, which is a contradiction. hence and theorem 2.1, it follows that the operator t + s has a fixed point (ρ∗,u∗) ∈ y . therefore, (ρ∗,u∗)(t,x) = t(ρ∗,u∗)(t,x) + s(ρ∗,u∗)(t,x) = −�(ρ∗,u∗)(t,x) + (ρ∗,u∗)(t,x) + �(ρ∗,u∗)(t,x) + �s2(ρ∗,u∗)(t,x), (t,x) ∈ [0,∞) ×r, whereupon 0 = s2(ρ ∗,u∗)(t,x), (t,x) ∈ [0,∞) ×r. © agt, upv, 2022 appl. gen. topol. 23, no. 2 474 classical solutions for the euler equations from here and from lemma 2.10, it follows that (ρ∗,u∗) is a solution to the ivp (1.1). this completes the proof. 4. proof of theorem 1.2 let x2 be the space used in the previous section. let (a3): m > 0 be large enough and a, b, r, l, r1 be positive constants that satisfy the following conditions r < l < r1 ≤ b, � > 0, r1 + a m b1 + l 5m > ( 2 5m + 1 ) l, ab1 < l 5 . let p̃ = {(ρ,u) ∈ x2 : (ρ,u) ≥ 0 on [0,∞) ×r}. with p we will denote the set of all equi-continuous families in p̃ . for (ρ,v) ∈ x2, define the operators t1(ρ,v)(t,x) = (1 + m�)(ρ,v)(t,x) − ( � l 10 ,� l 10 ) , s3(ρ,v)(t,x) = −�s2(ρ,v)(t,x) −m�(ρ,v)(t,x) − ( � l 10 ,� l 10 ) , (t,x) ∈ [0,∞) × r. note that any fixed point (ρ,v) ∈ x2 of the operator t1 + s3 is a solution to the ivp (1.1). define u1 = pr = {(ρ,v) ∈p : ‖(ρ,v)‖x2 < r}, u2 = pl = {(ρ,v) ∈p : ‖(ρ,v)‖x2 < l}, u3 = pr1 = {(ρ,v) ∈p : ‖(ρ,v)‖x2 < r1}, r2 = r1 + a m b1 + l 5m , ω = pr2 = {(ρ,v) ∈p : ‖(ρ,v)‖x2 ≤ r2}. (1) for (ρ1,v1), (ρ2,v2) ∈ ω, we have ‖t1(ρ1,v1) −t1(ρ2,v2)‖x2 = (1 + m�)‖(ρ1,v1) − (ρ2,v2)‖x2, whereupon t1 : ω → x2 is an expansive operator with a constant h = 1 + m� > 1. © agt, upv, 2022 appl. gen. topol. 23, no. 2 475 d. boureni, s. georgiev, a. kheloufi and k. mebarki (2) for (ρ,v) ∈pr1 , we get ‖s3(ρ,v)‖x2 ≤ �‖s2(ρ,v)‖x2 + m�‖(ρ,v)‖x2 + � l 10 ≤ � ( ab1 + mr1 + l 10 ) . therefore, s3(pr1 ) is uniformly bounded. since s3 : pr1 → x2 is continuous, we have that s3(pr1 ) is equi-continuous. consequently, s3 : pr1 → x2 is completely continuous. (3) let (ρ1,v1) ∈pr1 . set (ρ2,v2) = (ρ1,v1) + 1 m s2(ρ1,v1) + ( l 5m , l 5m ) . note that s12 (ρ1,v1) + l 5 ≥ 0, s22 (ρ1,v1) + l 5 ≥ 0 on [0,∞) × r. we have ρ2,v2 ≥ 0 on [0,∞) ×r and ‖(ρ2,v2)‖x2 ≤ ‖(ρ1,v1)‖x2 + 1 m ‖s2(ρ1,v1)‖x2 + l 5m ≤ r1 + a m b1 + l 5m = r2. therefore, (ρ2,v2) ∈ ω and −εm(ρ2,v2) = −εm(ρ1,v1) −εs2(ρ1,v1) −ε ( l 10 , l 10 ) −ε ( l 10 , l 10 ) or (i −t1)(ρ2,v2) = −εm(ρ2,v2) + ε ( l 10 , l 10 ) = s3(ρ1,v1). consequently, s3(pr1 ) ⊂ (i −t1)(ω). (4) assume that for any (ρ1,u1) ∈ p∗ there exist λ ≥ 0 and (ρ,v) ∈ ∂pr ∩ (ω + λ(ρ1,u1)) or (ρ,v) ∈ ∂pr1 ∩ (ω + λ(ρ1,u1)) such that s3(ρ,v) = (i −t1)((ρ,v) −λ(ρ1,u1)). then −�s2(ρ,v) −m�(ρ,v) − � ( l 10 , l 10 ) = −m�((ρ,v) −λ(ρ1,u1)) + � ( l 10 , l 10 ) or −s2(ρ,v) = λm(ρ1,u1) + ( l 5 , l 5 ) . © agt, upv, 2022 appl. gen. topol. 23, no. 2 476 classical solutions for the euler equations hence, ‖s2(ρ,v)‖x2 = ∥∥∥∥λm(ρ1,u1) + ( l 5 , l 5 )∥∥∥∥ x2 > l 5 . this is a contradiction. (5) let ε1 = 2 5m . assume that there exist a (ρ1,v1) ∈ ∂pl and λ1 ≥ 1 + ε1 such that λ1(ρ1,v1) ∈pr2 and (4.1) s3(ρ1,v1) = (i −t1)(λ1(ρ1,v1)). since (ρ1,v1) ∈ ∂pl and λ1(ρ1,v1) ∈pr2 , it follows that( 2 5m + 1 ) l < λ1l = λ1‖(ρ1,v1)‖x2 ≤ r1 + a m b1 + l 5m . moreover, −�s2(ρ1,v1) −m�(ρ1,v1) − � ( l 10 , l 10 ) = −λ1m�(ρ1,v1) + � ( l 10 , l 10 ) , or s2(ρ1,v1) + ( l 5 , l 5 ) = (λ1 − 1)m(ρ1,v1). from here, 2 l 5 > ∥∥∥∥s2(ρ1,v1) + ( l 5 , l 5 )∥∥∥∥ x2 = (λ1 − 1)m‖(ρ1,v1)‖x2 = (λ1 − 1)ml and 2 5m + 1 > λ1, which is a contradiction. therefore, all conditions of theorem 2.8 hold. hence, the ivp (1.1) has at least two solutions (ρ1,u1) and (ρ2,u2) so that ‖(ρ1,u1)‖x2 = l < ‖(ρ2,u2)‖x2 ≤ r1 or r ≤‖(ρ1,u1)‖x2 < l < ‖(ρ2,u2)‖x2 ≤ r1. 5. an example below, we will illustrate our main results. let q = 2, c = 1 and r1 = b = 10, l = 5, r = 4, m = 10 50, a = � = 1 104 . then b1 = max { 10, 103 } = 103 and ab1 = 1 104 · 103 < b, �b1(1 + a) = 1 104 · 103 ( 1 + 1 104 ) < b, © agt, upv, 2022 appl. gen. topol. 23, no. 2 477 d. boureni, s. georgiev, a. kheloufi and k. mebarki i.e., (a2) holds. next, r < l < r1 ≤ b, � > 0, r1 > ( 2 5m + 1 ) l, ab1 < l 5 . i.e., (a3) holds. take h(s) = log 1 + s11 √ 2 + s22 1 −s11 √ 2 + s22 , l(s) = arctan s11 √ 2 1 −s22 , s ∈ r, s 6= ±1. then h′(s) = 22 √ 2s10(1 −s22) (1 −s11 √ 2 + s22)(1 + s11 √ 2 + s22) , l′(s) = 11 √ 2s10(1 + s22) 1 + s44 , s ∈ r, s 6= ±1. therefore, −∞ < lim s→±∞ (1 + s + s2 + s3 + s4 + s5 + s6)h(s) < ∞, −∞ < lim s→±∞ (1 + s + s2 + s3 + s4 + s5 + s6)l(s) < ∞. hence, there exists a positive constant c1 so that (1 + s + s2 + s3 + s4 + s5 + s6) ( 1 44 √ 2 log 1 + s11 √ 2 + s22 1 −s11 √ 2 + s22 + 1 22 √ 2 arctan s11 √ 2 1 −s22 ) ≤ c1, s ∈ r. note that lim s→±1 l(s) = π 2 and by [25] (pp. 707, integral 79), we have ∫ dz 1 + z4 = 1 4 √ 2 log 1 + z √ 2 + z2 1 −z √ 2 + z2 + 1 2 √ 2 arctan z √ 2 1 −z2 . let q(s) = s10 (1 + s44)(1 + s + s2)2 , s ∈ r, and g1(t,x) = q(t)q(x), t ∈ [0,∞), x ∈ r. then, there exists a constant c2 > 0 such that 16(1 + t) ( 1 + t + t2 ) (1 + |x|) ( 1 + |x| + |x|2 ) ∫ t 0 ∣∣∣∣ ∫ x 0 g1(t1,x1)dx1 ∣∣∣∣∣dt1 ≤ c2, (t,x) ∈ [0,∞) ×r. let g(t,x) = a c2 g1(t,x), (t,x) ∈ [0,∞) ×r. © agt, upv, 2022 appl. gen. topol. 23, no. 2 478 classical solutions for the euler equations then 16(1 + t) ( 1 + t + t2 ) (1 + |x|) ( 1 + |x| + |x|2 ) ∫ t 0 ∣∣∣∣ ∫ x 0 g(t1,x1)dx1 ∣∣∣∣∣dt1 ≤ a, (t,x) ∈ [0,∞) ×r, i.e., (a1) holds. therefore, for the ivp ∂tρ + ∂x(ρu) = 0, ∂t(ρu) + ∂x ( ρu2 + ρ2 ) = 0, t > 0, x ∈ r, ρ(0,x) = u(0,x) = 1 1+x8 , x ∈ r, are fulfilled all conditions of theorem 1.1 and theorem 1.2. acknowledgements. the authors d. boureni, a. kheloufi and k. mebarki acknowledge support of ”direction générale de la recherche scientifique et du développement technologique (dgrsdt)”, mesrs, algeria. references [1] s. benslimane, s. georgiev and k. mebarki, multiple nonnegative solutions for a class of fourth-order bvps via a new topological approach, advances in the theory of nonlinear analysis and its applications 6, no. 3 (2022), 390–404. 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[31] n. zabusky, fermi-pasta-ulam, solitons and the fabric of nonlinear and computational science: history, synergetics, and visiometrics, chaos 15, no. 1 (2005), 015102. © agt, upv, 2022 appl. gen. topol. 23, no. 2 480 @ appl. gen. topol. 22, no. 2 (2021), 461-481doi:10.4995/agt.2021.15610 © agt, upv, 2021 quantale-valued cauchy tower spaces and completeness gunther jäger a and t. m. g. ahsanullah b a school of mechanical engineering, university of applied sciences stralsund, 18435 stralsund, germany (gunther.jaeger@hochschule-stralsund.de) b department of mathematics, college of science, king saud university, riyadh 11451, saudi arabia (tmga1@ksu.edu.sa) communicated by j. rodŕıguez-lópez abstract generalizing the concept of a probabilistic cauchy space, we introduce quantale-valued cauchy tower spaces. these spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. for special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach cauchy spaces arise. we also study completeness and completion in this setting and establish a connection to the cauchy completeness of a quantale-valued metric space. 2010 msc: 54e15; 54a20; 54a40; 54e35; 54e70. keywords: cauchy space; quantale-valued metric space; quantale-valued uniform convergence tower space; completeness; completion; cauchy completeness. 1. introduction cauchy spaces, axiomatized in [17], are a natural setting for studying completeness and completion [32]. in the probabilistic case such spaces are introduced as certain “towers indexed by [0, 1]” by richardson and kent [33] and by nusser [27]. the connection to probabilistic metric spaces [34], however, was not clarified there. in this paper, we generalize the approaches of [33, 27] received 13 may 2021 – accepted 06 august 2021 http://dx.doi.org/10.4995/agt.2021.15610 g. jäger and t. m. g. ahsanullah by allowing the index set to be a quantale. in this way, a quantale-valued metric space, in particular a probabilistic metric space, possesses a “natural” cauchy structure, which then can in turn be used to study completeness and completion. further examples include quantale-valued uniform convergence tower spaces and quantale-valued uniform spaces as well as quantale-valued convergence groups. we study the basic categorical properties of the category of quantale-valued cauchy tower spaces and characterize those spaces that are quantale-valued metrical. finally we discuss completeness and completion and we establish a connection with the cauchy completeness [6] of a quantale-valued metric space. 2. preliminaries let l be a complete lattice with distinct top element ⊤ and bottom element ⊥. in l we can define the well-below relation α ✁ β if for all subsets d ⊆ l such that β ≤ ∨ d there is δ ∈ d such that α ≤ δ. then α ≤ β whenever α ✁ β and, for a subset b ⊆ l, we have α ✁ ∨ β∈b β iff α ✁ β for some β ∈ b. sometimes we consider also a weaker relation, the way-below relation, α ≪ β if for all directed subsets d ⊆ l such that β ≤ ∨ d there is δ ∈ d such that α ≤ δ. the properties of this relation are similar to the properties of the wellbelow relation, replacing arbitrary subsets by directed subsets. but we also have α ∨ β ≪ γ if α, β ≪ γ, [9]. a complete lattice is completely distributive, if and only if we have α = ∨ {β : β ✁ α} for any α ∈ l [31] and it is continuous if and only if we have α = ∨ {β : β ≪ α} for any α ∈ l, [9]. clearly α ✁ β implies α ≪ β and hence every completely distributive lattice is also continuous. for more results on lattices we refer to [9]. lemma 2.1. let l be a continuous lattice. then (1) ∨ δ≪α(δ ∗ δ) = α ∗ α. (2) if ǫ ≪ α ∗ α, then there is δ ≪ α such that ǫ ≪ δ ∗ δ. proof. (1) we have ∨ δ≪α(δ ∗ δ) ≤ α ∗ α = ∨ δ≪α δ ∗ ∨ γ≪α γ = ∨ δ,γ≪α δ ∗ γ ≤ ∨ δ∨γ≪α(δ ∨ γ) ∗ (δ ∨ γ) ≤ ∨ η≪α(η ∗ η). (2) follows directly from (1) as the set {δ ∈ l : δ ≪ α} is directed. � the triple l = (l, ≤, ∗), where (l, ≤) is a complete lattice, is called a commutative and integral quantale if (l, ∗) is a commutative semigroup with the top element of l as the unit, and ∗ is distributive over arbitrary joins, i.e. if we have ( ∨ i∈j αi) ∗ β = ∨ i∈j(αi ∗ β) for all αi, β ∈ l, i ∈ j. in a quantale we can define an implication operator, α → β = ∨ {γ ∈ l : α ∗ γ ≤ β}, which can be characterized by γ ≤ α → β ⇐⇒ γ ∗ α ≤ β. a quantale is called divisible [11] if for β ≤ α there exists γ ∈ l such that β = α ∗ γ. in a divisible quantale we have ∨ δ✁α(δ ∗ δ) = α ∗ α, see [30]. prominent examples of quantales are e.g. the unit interval [0, 1] with a leftcontinuous t-norm [34] or lawvere’s quantale, the interval [0, ∞] with the opposite order and addition α ∗ β = α + β (extended by α + ∞ = ∞ + α = ∞), © agt, upv, 2021 appl. gen. topol. 22, no. 2 462 quantale-valued cauchy tower spaces and completeness see e.g. [7]. a further noteworthy example is the quantale of distance distribution functions. a distance distribution function ϕ : [0, ∞] −→ [0, 1], satisfies ϕ(x) = supy 0 there is n0 ∈ in such that for all n, m ≥ n0 we have d(xn, xm)(t) > 1 − t. this definition employs the strong uniformity, i.e. the system of entourages generated by the sets u(t) = {(x, y) ∈ x × x : d(x, y)(t) > 1 − t}, t > 0. equivalently, as is pointed out in [34], we can also use the system of entourages generated by the sets u(t, ǫ) = {(x, y) ∈ x × x : d(x, y)(t) > 1 − ǫ}, t, ǫ > 0. as in [16], using the quantale of distance distribution functions (∆+, ≤, ∗), we can see that the sets bd(ϕ) = {(x, y) ∈ x × x : d(x, y) ✄ ϕ}, ϕ ✁ ǫ0 generate the same uniformity. from this we can deduce that a sequence (xn) is a strong cauchy sequence in the probabilistic metric space (x, d) if and only if the generated filter is in cdǫ0. note that for a divisible quantale l, a symmetric l-metric space (x, d) and an idempotent element α ∈ l, in proposition 4.10, (i) and (ii) are equivalent. in particular, for a frame l, we can characterize the l-cauchy tower of a symmetric l-metric space by (ii) or by the equivalent statement (i) in proposition 4.9. the following example shows that we cannot omit the idempotency here. example 4.12. let x = ir and l = ([0, ∞], ≥, +) be lawvere’s quantale and d the usual metric on ir. we consider, for α > 0, the sequence xn = α 2 + 1 n if n is even and xn = − α 2 − 1 n if n is odd and denote f the filter generated by this sequence. then ∨ f ∈f ∧ x,y∈f d(x, y) = inf k∈in sup m,n≥k ∣ ∣ ∣ ∣ α 2 + 1 n − (− α 2 − 1 m ) ∣ ∣ ∣ ∣ = inf k∈in ∣ ∣ ∣ ∣ α + 2 k ∣ ∣ ∣ ∣ = α and hence we have f ∈ cdα. however, as ∨ x∈x ∨ f ∈f ∧ y∈f d(x, y) = inf x∈ir inf k∈in sup m≥k (d(x, α 2 + 1 m ) ∨ d(x, − α 2 − 1 m )) = inf x∈ir ( ∣ ∣ ∣ x − α 2 ∣ ∣ ∣ ∨ ∣ ∣ ∣ x + α 2 ∣ ∣ ∣ ) = α 2 , we have ∨ x∈x ∨ x∈qd β (f) β = α 2 but f /∈ cdα/2. proposition 4.13. let (x, d) ∈ |l-met|. then qc d α (f) ⊆ q d α(f) and if (x, d) is symmetric, we have qdα(f) ⊆ q cd α∗α(f). proof. let first x ∈ qc d α (f). then f∧ [x] ∈ c d α. let ǫ✁ α. then there is f ∈ f such that for all u, v ∈ f ∪ {x} we have d(u, v) ≥ ǫ. we conclude for u = x that d(x, v) ≥ ǫ for all v ∈ f and hence ∨ f ∈f ∧ y∈f d(x, y) ≥ ǫ. the complete distributivity implies x ∈ qdα(f). let now (x, d) be symmetric and x ∈ qdα(f). let ǫ ≪ α. noting that the set d = { ∧ y∈f d(x, y) : f ∈ f} is directed, there is f ∈ f such that for all y ∈ f we have d(x, y) ≥ ǫ. let u, v ∈ f ∪ {x}. we distinguish four cases. case 1: u 6= x, v 6= x. then d(x, u), d(x, v) ≥ ǫ and hence d(u, v) ≥ d(u, x) ∗ d(x, v) ≥ ǫ ∗ ǫ. © agt, upv, 2021 appl. gen. topol. 22, no. 2 469 g. jäger and t. m. g. ahsanullah case 2: u = x, v 6= x. then d(u, v) = d(x, v) ≥ ǫ ≥ ǫ ∗ ǫ. case 3: u 6= x, v = x. similar to case 2. case 4: u = v = x. then d(u, v) = d(x, x) = ⊤ ≥ ǫ ∗ ǫ. hence we have ∨ f ∈f ∧ u,v∈f ∪{x} d(u, v) ≥ ǫ∗ǫ and the continuity of l yields f ∧ [x] ∈ cdα∗α, i.e. x ∈ q cd α∗α(f). � again, for idempotent α ∈ l, i.e. if α∗α = α, we have the equality qc d α (f) = qdα(f). in particular in a frame l, this equality is guaranteed for all α ∈ l. example 4.14. we use example 4.12 and show that α 4 ∈ qc d α (f). to this end, we consider the sequence (yn) = (x1, α 4 , x2, α 4 , x3, α 4 , ...), the generated filter of which is g = f ∧ [α 4 ]. we then have ∨ g∈g ∧ x,y∈g d(x, y) = inf k∈in supm,n≥k |yn − ym| = α and hence we have f ∧ [ α 4 ] ∈ cdα, i.e. α 4 ∈ qc d α (f) = q cd α/2+α/2 (f). however, we see in a similar way that inf k∈in supn≥k ∣ ∣ α 4 − xn ∣ ∣ = 3 4 α 6≤ α 2 , so that α 4 /∈ qd α/2 (f). so we have qc d α/2+α/2 (f) 6⊆ qdα(f). 5. example: l-uniform convergence tower spaces and l-uniform tower spaces definition 5.1 ([15]). a pair ( x, λ = (λα)α∈l ) , is called an l-uniform convergence tower space, if λα ⊆ f(x × x), α ∈ l, satisfy the following: (lucts1) [(x, x)] ∈ λα for all x ∈ x, α ∈ l; (lucts2) φ ∈ λα whenever φ ≤ ψ and ψ ∈ λα; (lucts3) φ, ψ ∈ λα implies φ ∧ ψ ∈ λα; (lucts4) λβ ⊆ λα whenever α ≤ β; (lucts5) φ−1 ∈ λα whenever φ ∈ λα; (lucts6) φ ◦ ψ ∈ λα∗β whenever φ ∈ λα, ψ ∈ λβ and φ ◦ ψ exists; (lucts7) λ⊥ = f(x × x). a mapping f : ( x, λ ) −→ ( x′, λ′ ) is called uniformly continuous if, for all φ ∈ f(x × x), (f × f)(φ) ∈ λ′α, whenever φ ∈ λα. the category of l-uniform convergence tower spaces and uniformly continuous mappings is denoted by l-ucts. if l = ([0, 1], ≤, ∗), then we obtain nusser’s probabilistic uniform convergence spaces [26, 27]. for l = (∆+, ≤, ∗) we obtain the probabilistic uniform convergence spaces in [2]. for (x, λ) ∈ |l-ucts|, f ∈ f(x), x ∈ x and α ∈ l we define x ∈ qλα(f) ⇐⇒ f × [x] ∈ λα. it is not difficult to show that (x, qλ = (qλα)α∈l) ∈ |l-cts| is an l-limit tower space. for (x, λ) ∈ |l-ucts| and α ∈ l, a filter f ∈ f(x) is called an α-cauchy filter, written as f ∈ cλα if and only if f × f ∈ λα. © agt, upv, 2021 appl. gen. topol. 22, no. 2 470 quantale-valued cauchy tower spaces and completeness proposition 5.2. let (x, λ) ∈ |l-ucts|. then (x, cλ) ∈ |l-chyts|. proof. (lchyts1) since for all x ∈ x and α ∈ l, [x] × [x] ∈ λα, we have [x] ∈ cλα . (lchyts2) let g ≥ f with f ∈ cλα . then f×f ∈ λα and hence, by (lucts2), g × g ∈ λα, i.e. g ∈ c λ α . (lchyts3) let α ≤ β and f ∈ cλβ . then f × f ∈ λβ, then by (lucts4), f × f ∈ λα which in turn yields f ∈ c λ α . (lchyts4) follows at once from the definition. (lchyts5) let α, β ∈ l, and let f ∈ cλα and g ∈ c λ β such that f ∨ g exists. then f × f ∈ λα and g × g ∈ λβ. then f × g = (f × g) ◦ (f × g) ∈ λα∗β. also, g × f ∈ λα∗β. since (f ∧ g) × (f ∧ g) = (f × f) ∧ (f × g) ∧ (g × f) ∧ (g × g) ∈ λα∗β, this implies f ∧ g ∈ cλα∗β. � proposition 5.3. let (x, λ) ∈ |l-ucts|. then qc λ α (f) ⊆ q λ α(f) ⊆ q cλ α∗α(f). proof. let x ∈ qc λ α (f), then f∧[x] ∈ c λ α . this implies that (f∧[x])×(f∧[x]) ∈ λα. as (f ∧ [x]) × (f ∧ [x]) ≤ f × [x], (lucts2) implies x ∈ q λ α(f). if x ∈ qλα(f), then f × [x] ∈ λα and with (lucts5) then also [x] × f = (f×[x])−1 ∈ λα. from (lucts6) we obtain f×f = (f×[x])◦([x]×f) ∈ λα∗α. this yields with (lucts3) (f ∧ [x]) × (f ∧ [x]) = (f × f) ∧ (f × [x]) ∧ ([x] × f) ∧ ([x] × [x]) ∈ λα∗α. thus, f ∧ [x] ∈ c λ α∗α which means x ∈ q cλ α∗α(f). � again, for idempotent α ∈ l, we have equality, qc λ α (f) = q λ α(f). in particular this is the case if l is a frame. definition 5.4 ([15]). let l = (l, ≤, ∗) be a quantale. a pair ( x, u ) with u = (uα)α∈l a family of filters on x × x is called an l-uniform tower space if for all α ∈ l the following holds: (luts1) uα ≤ [∆] with [∆] = ∧ x∈x[(x, x)]; (luts2) uα ≤ (uα) −1; (luts3) uα∗β ≤ uα ◦ uβ; (luts4) uα ≤ uβ whenever α ≤ β; (luts5) u⊥ = ∧ f(x × x); (luts6) u∨ a ≤ ∨ α∈a uα whenever ∅ 6= a ⊆ l. a mapping f : ( x, u ) −→ ( x′, u′ ) is called uniformly continuous if u′α ≤ (f × f)(uα) for all α ∈ l. the category with objects all l-uniform tower spaces and uniformly continuous mappings as morphisms is denoted by l-uts. if l = ([0, 1], ≤, ∗) with a t-norm ∗, then we obtain florescu’s probabilistic uniform spaces [8], for l = (∆+, ≤, ∗) we obtain the probabilistic uniform spaces in [2]. for lawvere’s quantale an l-uniform tower space is an approach uniform space [23]. © agt, upv, 2021 appl. gen. topol. 22, no. 2 471 g. jäger and t. m. g. ahsanullah for ( x, u = (uα)α∈l ) ∈ |l − uts| and α ∈ l, a filter f ∈ f(x) is called an α-cauchy filter written as f ∈ cuα if and only if f × f ≥ uα. lemma 5.5. let (x, u) ∈ |l-uts|. then (x, cu) ∈ |l-chyts|. proof. (lchyts1) clearly by (luts1), uα ≤ [x]×[x], implies [x] ∈ c u α , for all x ∈ x and α ∈ l. (lchyts2) let f, g ∈ f(x) with f ≤ g, and f ∈ cuα . then f × f ≥ uα but then g × g ≥ uα, which in turn implies g ∈ c u α . (lchyts3) let α ≤ β and f ∈ cuβ , implying f × f ≥ uβ. but then uβ ≥ uα by (luts4) for α ≤ β. so, f × f ≥ uα which gives f ∈ c u α . the condition (lchyts4) is trivially true. finally, we check the condition (lchyts5): let f ∈ cuα and g ∈ cuβ such that f ∨ g exists. then f × f ≥ uα and g × g ≥ uβ. then (f× f)◦ (g × g) ≥ uα ◦ uβ which in view of the condition (luts3) yields that f×g = (f×f)◦(g×g) ≥ uα∗β. this means that f×g ≥ uα∗β. furthermore, note that we also have: g×f ≥ uα∗β, f×f ≥ uα∗β, and g×g ≥ uα∗β; this is due to the condition (luts5) and the fact that α ∗ β ≤ α, β. hence we arrive at the following: (f ∧ g) × (f ∧ g) = (f × f) ∧ g × g) ∧ (f × g) ∧ (g × f) ≥ uα∗β, that is, (f ∧ g) × (f ∧ g) ≥ uα∗β which in turn implies that f ∧ g ∈ c u α∗β. � given an l-metric space (x, d) we define λdα ⊆ f(x × x) for α ∈ l, by [15] φ ∈ λdα ⇐⇒ ∨ f ∈φ ∧ (x,y)∈f d(x, y) ≥ α. theorem 5.6. the l-cauchy tower space (x, cd) of an l-metric space (x, d) is the same as the l-cauchy tower space (x, cλ d ) induced by the l-uniform convergence tower space (x, λd) of an l-metric space. that is, cd = cλ d . proof. this follows from ∨ h∈f×f ∧ (x,y)∈h d(x, y) = ∨ f ∈f ∧ x,y∈f d(x, y) for f ∈ f(x), as the sets f × f with f ∈ f are a basis of f × f. � for an l-metric space (x, d) we define uǫ = {(x, y) ∈ x ×x : d(x, y) ≫ ǫ}. for ǫ ≪ α, these sets form the basis of a filter udα and we have u d α = ∧ φ∈λdα φ, see [15]. we call (x, ud) the l-uniform tower space induced by the l-metric space (x, d). proposition 5.7. let (x, d) ∈ |l-met|. then cu d α = c d α for all α ∈ l. proof. let f ∈ cu d α . then f×f ∈ u d α and for all ǫ ≪ α there is f ∈ f such that f × f ⊆ uǫ. hence for all ǫ ≪ α there is f ∈ f such that ∧ x,y∈f d(x, y) ≥ ǫ. this implies ∧ x,y∈f d(x, y) ≥ α from which we conclude f ∈ c d α. conversely, we note that the set { ∧ x,y∈f d(x, y) : f ∈ f} is directed. if f ∈ cdα, then ∨ f ∈f ∧ x,y∈f d(x, y) ≥ α. hence, for ǫ ≪ α there is f ∈ f such that for all x, y ∈ f we have d(x, y) ≫ ǫ. this means f × f ⊆ uǫ and we have f × f ≥ udα. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 472 quantale-valued cauchy tower spaces and completeness 6. example: l-limit tower groups let (x, ·) be a group with identity element e ∈ x and let f, g, h ∈ f(x). we define f⊙g as the filter with filter basis {f ·g : f ∈ f, g ∈ g} and f−1 as the filter with filter basis {f −1 : f ∈ f}. then, for f, g, h ∈ f(x) and x, y ∈ x, the following properties are easily verified: f ⊙ f−1 ≤ [e]; [x] ⊙ [x]−1 = [e]; [x−1] = [x]−1; [x · y] = [x] · [y]; (f ⊙ g) ⊙ h = f ⊙ (g ⊙ h); (f−1)−1 = f; (f ⊙ g)−1 = g−1 ⊙ f−1; [e] ⊙ f = f ⊙ [e] = f; (f ∧ g)−1 = f−1 ∧ g−1 and (f ∧ g) ⊙ h = (f ⊙ h) ∧ (g ⊙ h). definition 6.1. let l = (l, ≤, ∗) be a quantale, and (x, ·) be a group with identity e. then a triple (x, ·, q = (qα)α∈l) is called an l-convergence tower group (respectively, an l-limit tower group) if the following conditions are fulfilled: (lctg) if (x, q) is an l-convergence tower space (resp. an l-limit tower space) (lctgm) x ∈ qα(f), y ∈ qβ(g) implies xy ∈ qα∗β (f ⊙ g), for all f, g ∈ f(x), for all α, β ∈ l, and x, y ∈ x; (lctgi) x ∈ qα(f) implies x −1 ∈ qα(f −1), for all f ∈ f(x), x ∈ x and α ∈ l. the category of all l-convergence tower groups and continuous group homomorphisms is denoted by l-ctgrp (respectively, the category of all l-limit tower groups and continuous group homomorphisms is denoted by l-limgrp.) if l = {0, 1}, then we obtain classical convergence groups [28]. if l = ([0, 1], ≤, ∗) with a continuous t-norm ∗, then we get probabilistic convergence group under a t-norm in the sense of [13]. if l = ([0, ∞], ≥, +), then a leftcontinuous l-convergence tower group is an approach group [24]. if l = (∆+, ≤ , ∗), we obtain a probabilistic convergence group in the definition of [3]. for (x, ·, q) ∈ |l − limtgrp|, a filter f ∈ f(x) is called an α-cauchy filter, written f ∈ cqα, if and only if e ∈ qα ( f −1 ⊙ f ) . proposition 6.2. let (x, ·, q) ∈ |l-limgrp|. then ( x, cq ) ∈ |l-chyts|. proof. (lchyts1) since e ∈ qα([x] −1 ⊙ [x]) implies [x] ∈ cqα. (lchyts2) let f ≤ g with f ∈ cqα. this implies e ∈ qα ( f −1 ⊙ f ) . but then e ∈ qα ( g −1 ⊙ g ) which gives g ∈ cqα. (lchyts3) let α, β ∈ l with α ≤ β and f ∈ c q β. then e ∈ qβ ( f −1 ⊙ f ) implies e ∈ qα ( f −1 ⊙ f ) . hence f ∈ cqα. (lchyts4) obvious. (lchyts5) let f ∈ cα and g ∈ c q β such that f ∨ g exists. it is not difficult to prove that f ⊙ g−1 ≤ [e]. as e ∈ qα ( f −1 ⊙ f ) and e ∈ qβ ( g −1 ⊙ g ) , we obtain with condition (lctgm) e = ee ∈ qα∗β ( f −1 ⊙ f ⊙ g−1 ⊙ g ) which implies that e ∈ qα∗β ( f −1 ⊙ g ) . similarly, we can get e ∈ qα∗β ( g −1 ⊙ f ) . since (f ∧ g)−1 ⊙ (f ∧ g) = (f−1 ∧ g−1) ⊙ (f ∧ g) = (f−1 ⊙ f) ∧ (f−1 ⊙ g) ∧ (g−1 ⊙ f) ∧ (g−1 ⊙ g), © agt, upv, 2021 appl. gen. topol. 22, no. 2 473 g. jäger and t. m. g. ahsanullah we conclude from condition (lcts5) e ∈ qα∗β ( (f ∧ g)−1 ⊙ (f ∧ g) ) . hence f ∧ g ∈ c q α∗β. � definition 6.3 ([14]). a triple (x, ·, d) is called an l-metric group if d is invariant, i.e., d(x, y) = d(xz, yz) = d(zx, zy) for all x, y, z ∈ x. a group homomorphism f : (x, ·) −→ (x′, ·′) between the l-metric groups (x, ·, d) and (x′, ·′, d′) is called an l-metgrp-morphism if it is an l-met-morphism between (x, d) and (x′d′). the category of l-metric groups is denoted by l-metgrp. lemma 6.4. let (x, ·, d) be a symmetric l-metric group. then (x, ·, qd) is an l-convergence tower group. proof. we need to show the conditions (lctgm) and (lctgi). for (lctgm), let x ∈ qdα(f) and y ∈ q d β(g). for ǫ ✁ α and δ ✁ β, there is f ∈ f and g ∈ g such that for all u ∈ f and for all v ∈ g we have d(x, u) ≥ ǫ and d(y, v) ≥ δ. then d(e, x−1u) ≥ ǫ and d(yv−1, e) ≥ δ and hence d(yv−1, x−1u) ≥ ǫ ∗ δ. this implies d(xy, uv) ≥ ǫ ∗ δ for all u ∈ f, v ∈ g and hence d(xy, h) ≥ ǫ ∗ δ for all h ∈ f ⊙ g. therefore ∨ h∈f⊙g ∧ h∈h d(xy, h) ≥ ǫ ∗ δ and from the complete distributivity we obtain xy ∈ qdα∗β(f ⊙ g). (lctgi) follows with the symmetry from d(x, y) = d(xy−1, e) = d(y−1, x−1). � proposition 6.5. let (x, d) ba a symmetric l-metric group. then cdα = c qd α for all α ∈ l. proof. f ∈ cq d α is equivalent to ∨ f ∈f ∧ x,y∈f d(e, x −1y) ≥ α, which is, by the invariance of the l-metric, equivalent to ∨ f ∈f ∧ x,y∈f d(x, y) ≥ α, i.e. equivalent to f ∈ cdα. � 7. completeness and completion following [33, 27] we call (x, c) ∈ |l-chyts| complete if for all α ∈ l, f ∈ cα implies the existence of x ∈ x such that f ∧ [x] ∈ cα. with this definition, the “point of convergence” x = x(f, α) not only depends on the filter f but may also depend on the “level” α ∈ l. in the left-continuous case, we can omit this extra dependency. proposition 7.1. let (x, c) ∈ |l-chyts| be left-continuous. then (x, c) is complete if for all f ∈ f(x) there is x = x(f) such that for all α ∈ l, f ∈ cα implies f ∧ [x] ∈ cα. proof. let f ∈ f(x) and define a = {α ∈ l : f ∈ cα} and δ = ∨ a. by left-continuity f ∈ cδ and hence there is x = x(f, δ) such that f ∧ [x] ∈ cδ. if f ∈ cα, then α ≤ δ and hence f ∧ [x] ∈ cδ ⊆ cα and we can choose x for each α. � the following examples show that the left-continuity is essential. © agt, upv, 2021 appl. gen. topol. 22, no. 2 474 quantale-valued cauchy tower spaces and completeness example 7.2. let l = {⊥, α, β, ⊤} with α, β incomparable, ∗ = ∧ and let x be an infinite set. there is an ultrafilter u ∈ u(x) with u 6≥ [x] for all x ∈ x. we fix x0, y0 ∈ x with x0 6= y0 and define an l-cauchy tower as follows. we let c⊥ = f(x), f ∈ cα iff f ≥ u ∧ [x0] or if f = [x] for some x ∈ x. similarly, f ∈ cβ iff f ≥ u∧ [y0] or f = [x] for some x ∈ x, and finally f ∈ c⊤ if f = [x] for some x ∈ x. we only show (lchyts5) as the other axioms are easy. let f ∈ cγ, g ∈ cδ and f ∨ g exist. we distinguish three cases. case 1: γ = α, δ = β. then α ∧ β = ⊥ and hence f ∧ g ∈ cα∧β. case 2: f, g ∈ cα. if f = [x] and g = [y] then x = y because f ∨ g exists and hence f ∧ g = [x] ∈ cα. if f = [x] and g = u ∧ [x0], then [x] ∨ (u ∧ [x0]) exists. if x 6= x0 we define f = x \ {x}. then either f ∈ u or its complement f c ∈ u and also f ∈ [x0]. if f ∈ u then f ∈ u ∧ [x0] and hence also f ∈ [x] ∨ (u ∧ [x0]), in contradiction to f c ∈ [x] and f ∩ f c = ∅. if f /∈ u then f c ∈ u and hence u = [x], again a contradiction. we conclude x = x0, i.e. f = [x0] and g = u ∧ [x0] and therefore f ∧ g = g ∈ cα = cα∧α. case 3: f, g ∈ cβ. this is similar. we note that (x, c) is complete but since we have u ∈ cα, u ∈ cβ and u /∈ cα∨β = c⊤, it is not left-continuous. clearly, u ∈ cα and u ∧ [x0] ∈ cα. likewise, u ∈ cβ and u ∧ [y0] ∈ cβ. as we have seen above u ∧ [x] ∈ cα implies x = x0 and hence in particular u ∧ [y0] /∈ cα. this shows that the point of convergence for u is different for α and β. example 7.3. we consider x = [0, ∞) and lawvere’s quantale l = ([0, ∞], ≥ , +). again we choose an ultrafilter u ∈ u(x) with u 6= [x] for all x ∈ [0, ∞). we define c∞ = f(x) and for α < ∞ we define f ∈ cα if f = [x] for some x ∈ x or if f ≥ u ∧ [x] for some 0 < x < α. then ([0, ∞), c) is an l-cauchy tower space. we again only show (lchyts5). let f ∈ cα, g ∈ cβ and let f∨g exist. the case f = [x] and g = [y] implies x = y and then f∧g = [x] ∈ cα∧β. if f = [x] and g ≥ u ∧ [y] with 0 < y < β implies again x = y and we obtain f∧g = g ∈ cβ ⊆ cα+β. it remains the case g ≥ u∧[x], g ≥ u∧[y]. if x = y, trivially f ∧ g ∈ cα+β. if x 6= y, then we choose two sets f1, f2 ⊆ [0, 1] with non-empty and finite complement and f1 ∩ f2 = ∅ such that x ∈ f1, y ∈ f2. then f2 ∈ u ∧ [x], as f2 ∈ [x] and if f2 /∈ u, then the complement f c 2 ∈ u and as f c2 is a finite set, then u = [z] for z ∈ f c 2 . similarly, f1 ∈ u ∧ [y]. as h = (u ∧ [x]) ∨ (u ∧ [y]) exists, we get the contradiction ∅ = f1 ∩ f2 ∈ h. as u ∈ cα for all 0 < α but u /∈ c0 we see that ([0, ∞), c) is not leftcontinuous. clearly, the space is complete. however, the point of convergence varies with the level α: we have u ∈ cα for all 0 < α < ∞. we fix α ∈ (0, ∞). then there is x with 0 < x < α such that u ∧ [x] ∈ cα. for β with 0 < β < x, however, u ∧ [x] /∈ cβ, because u ∧ [x] 6≥ u ∧ [y] for x 6= y. hence we cannot choose for each level the same point of convergence for u. we note that for (x, d) ∈ |l-met|, the space (x, cd) is left-continuous. © agt, upv, 2021 appl. gen. topol. 22, no. 2 475 g. jäger and t. m. g. ahsanullah proposition 7.4. let (xj, cj) be complete l-cauchy tower spaces for all j ∈ j. then the product space ( ∏ j∈j xj, π-c) is also complete. proof. let f ∈ (π-c)α. then, for all j ∈ j, prj(f) ∈ c j α. by completeness there is, for each j ∈ j, a point xj ∈ xj such that prj(f) ∧ [xj] ∈ c j α. hence with x = (xj)j∈j , then for all j ∈ j, prj(f ∧ [x]) = prj(f) ∧ [prj(x)] = prj(f) ∧ [xj] ∈ c j α which implies f ∧ [x] ∈ (π-c)α. � remark 7.5 (completeness in l-uts). we define for an l-uniform convergence tower space (x, u) the underlying l-convergence tower qu by x ∈ quα (f) ⇐⇒ f×[x] ≥ uα. noting that (f∧[x])×(f∧[x]) = (f×f)∧(f×[x])∧(f×[x]) −1 ∧ [(x, x)] and f×f = (f×[x])◦([x]×f) we immediately obtain x ∈ qc u α (f) ⇐⇒ x ∈ quα∗α(f). for the definition of completeness however, we can use either l-convergence tower, i.e. if we define (x, u) complete if (x, cu) is complete, then this is equivalent to demanding that f × f ≥ uα implies f × [x] ≥ uα for some x ∈ x. remark 7.6 (completeness in l-ucts.). similarly, if we define in l-ucts that a space (x, λ) is complete if (x, cλ) is complete, then this is equivalent to f × f ∈ λα implies f × [x] ∈ λα for some x ∈ x. let (x, c) be a non-complete l-cauchy tower space. we call a pair ((x′, c′), κ) with a complete l-cauchy tower space (x′, c′) and an initial and injective mapping κ : (x, c) −→ (x′, c′) such that κ(x) is dense in (x′, c′), a completion of (x, c). here, a set a ⊆ x is called dense in (x, c) if for all x ∈ x there is f ∈ f(x) such that a ∈ f and f ∧ [x] ∈ c⊤ and a mapping κ : (x, c) −→ (x′, c′) is initial if f ∈ cα if and only if κ(f) ∈ c ′ α. in the sequel, we describe a completion construction which goes back to [18] and [27]. the proofs in [27] can simply be adapted, replacing the quantale l = ([0, 1], ≤, ∗) with a continuous t-norm ∗ on [0, 1] by an arbitrary quantale. for this reason, they are not presented. we consider a non-complete l-cauchy tower space (x, c) and define nc = {f ∈ f(x) : f ∈ c⊤, f ∧ [x] /∈ c⊤∀x ∈ x}. furthermore, we consider the following equivalence relation on c⊤: f ∼ g ⇐⇒ f ∧ g ∈ c⊤ and we denote the equivalence class of f ∈ c⊤ by 〈f〉 = {g ∈ c⊤ : f ∼ g}. we define x∗ = {〈[x]〉 : x ∈ x} ∪ {〈f〉 : f ∈ nc} and denote the inclusion mapping ιx = ι : x −→ x ∗, x 7−→ ι(x) = 〈[x]〉. we note that if (x, c) is a t1-space, then ι is an injection. in fact, if ι(x) = ι(y), then [x] ∧ [y] ∈ c⊤ and by (t1) then x = y. let φ ∈ f(x∗) and let α 6= ⊥. we define φ ∈ c∗α if there is f ∈ cα such that φ ≥ ι(f) or if there is f ∈ cα and there are f1, ..., fn ∈ nc such that f ∨ fi exists for all i = 1, ..., n and φ ≥ ι(f) ∧ ∧n i=1[〈fi〉]. furthermore, we put c∗⊥ = f(x). we will consider the completion axiom (lca): for all f ∈ cα with f ∧ [x] /∈ cα for all x ∈ x, there is v ∈ nc such that f ∧ v ∈ cα. © agt, upv, 2021 appl. gen. topol. 22, no. 2 476 quantale-valued cauchy tower spaces and completeness then ((x∗, c∗), ι) is a completion of (x, c) if (lca) is true. we can say more. proposition 7.7 ([27, 18]). let (x, c) ∈ |l-chyts|. then (x, c) has a completion if and only if it satisfies (lca). for the completion ((x∗, c∗), ι), the following universal property is true. theorem 7.8 ([27, 18]). let (x, c), (y, d) ∈ |l-chyts| and let f : x −→ y be cauchy-continuous. then there exists a cauchy-continuous mapping f∗ : (x∗, c∗) −→ (y ∗, d∗) such that the following diagram commutes: (x, c) f −→ (y, d) ιx ↓ ↓ ιy (x∗, c∗) f∗ −→ (y ∗, d∗) corollary 7.9 ([27]). let (x, c), (y, d) ∈ |l-chyts| and let (x, c) satisfy (lca) and let (y, d) be a complete t1-space. if f : x −→ y is cauchycontinuous then there is a unique cauchy-continuous extension f∗ : (x∗, c∗) −→ (y, d) such that f∗ ◦ ι = f. it is at present not clear if for an l-metric space (x, d) the completion (x, (cd)∗) is again l-metrical, i.e. if there is an l-metric e in x∗ such that (cd)∗α = c e α for all α ∈ l. we can, however, show the one half of the axiom (lchym). proposition 7.10. let (x, d) ∈ |l-met|. if φ ∈ (cd)∗α then for all ǫ ✁ α there is φǫ ∈ φ such that for all x ∗, y∗ ∈ φǫ we have [x ∗] ∧ [y∗] ∈ (cd)∗ǫ. proof. let φ ∈ (cd)∗α. then there are f ∈ c d α and f1, ..., fn ∈ nc such that f ∨ fk exists for k = 1, ..., n, n ≥ 0, and φ ≥ ι(f) ∧n k=1[〈fk〉]. let ǫ ✁ α. by the axiom (lchym) there is fǫ ∈ f such that for all x, y ∈ fǫ we have [x] ∧ [x] ∈ cdǫ . we define φǫ = ι(fǫ) ∪ {〈f1〉, ..., 〈fn〉} ∈ φ. let x ∗, y∗ ∈ φǫ. we distinguish three cases. case 1: x∗, y∗ ∈ ι(fǫ). then x ∗ = ι(x), y∗ = ι(y) for x, y ∈ fǫ and as [x] ∧ [y] ∈ cdǫ we conclude [x ∗] ∧ [y∗] = ι([x] ∧ [y]) ∈ (cd)∗ǫ. case 2: x∗ = 〈fk〉, y ∗ = 〈fl〉. then trivially [〈fk〉] ∧ [〈fl〉] ≥ ι(f) ∧ ∧n k=1[〈fk〉] and hence [x ∗] ∧ [y∗] ∈ (cd)∗ǫ . case 3: x∗ = 〈[x]〉 ∈ ι(fǫ), y ∗ = 〈fk〉. we have [fǫ] = {g ⊆ x : fǫ ⊆ g} ≤ f and [fǫ] ≤ [x] as x ∈ fǫ. as f ∨ fk exists therefore also f ∨ [fǫ] exists. moreover, we have ∨ g∈[fǫ] ∧ u,v∈g d(u, v) ≥ ∧ u,v∈fǫ d(u, v) ≥ ǫ as d(u, v) ≥ ǫ is equivalent to [u] ∧ [v] ∈ cdǫ . this shows [fǫ] ∈ c d ǫ . we conclude [x∗] ∧ [y∗] = ι([x]) ∧ [〈fk〉] ≥ ι([fǫ]) ∧ [〈fk〉], i.e. [x ∗] ∧ [y∗] ∈ (cd)∗ǫ. � 8. the l-metric case: cauchy completeness we follow concepts and notations introduced in [6], see also [10]. let (x, d) ∈ |l-met|. a mapping φ : x −→ l is an order ideal if d(y, x) ∗ φ(x) ≤ φ(y) for all x, y ∈ x. it is called an order filter if φ(x) ∗ d(x, y) ≤ φ(y) for all x, y ∈ x. © agt, upv, 2021 appl. gen. topol. 22, no. 2 477 g. jäger and t. m. g. ahsanullah clearly, d(y, x) ≤ φ(x) → φ(y) and using the l-metric dl : l × l −→ l defined by dl(α, β) = α → β for α, β ∈ l, we see that an order ideal is an l-metric morphism from (x, dop) to (l, dl) and similarly, an l-order filter is an l-metric morphism from (x, d) to (l, dl). the following lemmas give important examples. lemma 8.1 ([6]). let (x, d) be an l-metric space and a ∈ x. then ↓ (a) defined by ↓ (a)(x) = d(x, a) is an order ideal and ↑ (a) defined by ↑ (a)(x) = d(a, x) is an order filter. lemma 8.2. let (x, d) be an l-metric space and let f be a filter on x. then φ : x −→ l, defined by φ(x) = ∨ f ∈f ∧ y∈f d(x, y) is an order ideal and ψ : x −→ l defined by ψ(x) = ∨ f ∈f ∧ y∈f d(y, x) is an order filter. proof. we only show the first case. we have for x, y ∈ x d(y, x) ∗ φ(x) = d(y, x) ∗ ∨ f ∈f ∧ z∈f d(x, z) = ∨ f ∈f d(y, x) ∗ ∧ z∈f d(x, z) ≤ ∨ f ∈f ∧ z∈f d(y, x) ∗ d(x, z) ≤ ∨ f ∈f ∧ z∈f d(y, z) = φ(y). � definition 8.3 ([6]). let (x, d) be an l-metric space and let φ : x −→ l be an order ideal and ψ : x −→ l be an order filter. the pair (φ, ψ) is called a cut on x (1) ⊤ = ∨ x∈x φ(x) ∗ ψ(x); (2) φ(x) ∗ ψ(y) ≤ d(x, y) for all x, y ∈ x. we note that for a given a ∈ x, the pair (↓ (a), ↑ (a)) is a cut on x. proposition 8.4. let (x, d) be an l-metric space and let f ∈ cd⊤. then (φ, ψ) as defined in lemma 8.2 is a cut on x. proof. let ǫ ≪ ⊤ and choose δ ≪ ⊤ such that ǫ ≪ δ ∗ δ. then δ ≪ ⊤ = ∨ f ∈f ∧ x,y∈f d(x, y) and hence there is f ∈ f such that for all x, y ∈ f we have d(x, y) ≫ δ and d(y, x) ≫ δ. we conclude for all x ∈ f ǫ ≪ δ ∗ δ ≤ ∧ y∈f d(x, y) ∗ ∧ y∈f d(y, x) ≤ ∨ f ∈f ∧ y∈f d(x, y) ∗ ∨ f ∈f ∧ y∈f d(y, x) = φ(x) ∗ ψ(x). hence ǫ ≪ ∨ x∈x φ(x) ∗ ψ(x) and taking the join for all ǫ ≪ ⊤ we obtain ⊤ = ∨ x∈x φ(x) ∗ ψ(x). moreover, we have φ(x) ∗ ψ(y) = ∨ f ∈f ∧ z∈f d(x, z) ∗ ∨ g∈f ∧ z∈g d(z, y) ≤ ∨ f,g∈f ∧ z∈f ∩g d(x, z) ∗ d(z, y) ≤ d(x, y). © agt, upv, 2021 appl. gen. topol. 22, no. 2 478 quantale-valued cauchy tower spaces and completeness hence (φ, ψ) is a cut on x. � we call a symmetric l-metric space (x, d) complete if for f ∈ cd⊤ there is a ∈ x such that a ∈ qd⊤(f). from proposition 4.3 we know that q d ⊤(f) = q cd ⊤ (f) and hence the completeness of (x, d) is the completeness of the cauchy space (x, cd⊤). clearly, the requirement that (x, c d) is complete is stronger. we shall now establish a relation to the concept of cauchy completeness of an l-metric space as defined by lawvere [19], see also [6]. definition 8.5. let (x, d) be an l-metric space. then (x, d) is called cauchy complete if for all cuts (φ, ψ) there is a ∈ x that represents the cut (φ, ψ) in the sense that φ =↓ (a) and ψ =↑ (a). theorem 8.6. let (x, d) be a symmetric l-metric space. then (x, d) is complete if and only if (x, d) is cauchy complete. proof. let (x, d) be cauchy complete and let f ∈ cd⊤. we consider the order ideal φ and the order filter ψ defined in lemma 8.2. from proposition 8.4 we see that (φ, ψ) is a cut on x. by assumption, there is a ∈ x such that φ(x) = d(x, a) for all x ∈ x and ψ(x) = d(a, x) for all x ∈ x. then ⊤ = ψ(a) ≤ ∨ f ∈f ∧ z∈f d(z, a). which means that a ∈ q d ⊤(f) and (x, d) is complete. conversely, let (x, d) be complete and let (φ, ψ) be a cut. for ǫ ≪ ⊤ we define fǫ = {x ∈ : φ(x)∗ψ(x) ≫ ǫ}. from ⊤ = ∨ x∈x φ(x)∗ψ(x) we conclude that fǫ 6= ∅ and hence {fǫ : ǫ ≪ ⊤} is a basis for a filter f. we show that f ∈ cd⊤. let ǫ ≪ ⊤ and choose δ ≪ ⊤ such that ǫ ≪ δ ∗ δ. for all x, y ∈ fδ we then have δ ∗ δ ≤ φ(x) ∗ ψ(x) ∗ φ(y) ∗ ψ(y) ≤ φ(x) ∗ ψ(y) ≤ d(x, y), and hence ǫ ≪ δ ∗ δ ≤ ∨ f ∈f ∧ x,y∈f d(x, y). taking the join over all ǫ ≪ ⊤ yields f ∈ cd⊤. hence there is a ∈ x such that a ∈ qd⊤(f). we note that this means ⊤ = ∨ ǫ≪⊤ ∧ z∈fǫ d(z, a) and, ψ : (x, d) −→ (l, dl) being an l − met-morphism, this yields ⊤ = ∨ ǫ≪⊤ ∧ z∈fǫ (ψ(z) → ψ(a)) = ∨ ǫ≪α (( ∨ z∈fǫ ψ(z)) → ψ(a)). for δ ≪ ⊤ there is ǫδ ≪ ⊤ such that δ ∗ ∨ z∈fǫδ ψ(z) ≤ ψ(a). © agt, upv, 2021 appl. gen. topol. 22, no. 2 479 g. jäger and t. m. g. ahsanullah as ǫδ ≤ ǫδ ∨ δ ≪ ⊤ we have fǫδ∨δ ⊆ fǫδ and hence δ ∗ ∨ z∈fǫδ∨δ ψ(z) ≤ ψ(a). for z ∈ fǫδ∨δ we know ψ(z) ≥ φ(z)∗ψ(z) ≥ ǫδ ∨δ ≥ δ and we conclude δ ∗δ ≤ ψ(a). taking the join over all δ ≪ ⊤ we obtain ⊤ = ψ(a). similarly we can show ⊤ = φ(a). φ being an order ideal implies d(x, a) = d(x, a) ∗ φ(a) ≤ φ(x) for all x ∈ x and from φ(x) = φ(x) ∗ ψ(a) ≤ d(x, a) we obtain φ(x) = d(x, a) for all x ∈ x. hence φ =↓ (a). similarly we can show ψ =↑ (a) and hence a ∈ x represents the cut (φ, ψ) and (x, d) is cauchy complete. � references [1] j. adámek, h. herrlich and g. e. strecker, abstract and concrete categories, wiley, new york, 1989. 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[34] b. schweizer and a. sklar, probabilistic metric spaces, north holland, new york, 1983. © agt, upv, 2021 appl. gen. topol. 22, no. 2 481 @ appl. gen. topol. 22, no. 1 (2021), 11-15doi:10.4995/agt.2021.13066 © agt, upv, 2021 metric topology on the moduli space jialong deng mathematisches institut, georg-august-universität, göttingen, germany (jialong.deng@mathematik.uni-goettingen.de) communicated by s. romaguera abstract we define the smooth lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with gromov-hausdorff topology, which offers an answer to tuschmann’s question. 2010 msc: 54e35; 54f65; 51f99. keywords: gromov-hausdorff topology; ε-topology; lipschitz-topology; smooth lipschitz-topology. 1. introduction any smooth closed manifold m can be given a smooth riemannian metric, and then one can ask: how many riemannian metrics are there, and how many different geometries of this kind does the manifold actually allow? that means one wants to understand the space of riemannian metrics on m, which is denoted by r(m), and the moduli space which is denoted by m(m). here the moduli space is the quotient space of r(m) by the action of diffeomorphism group of m. those two questions originated from riemann when he set up riemannian geometry in the nineteenth century ([1]). especially, the moduli space m(m) is the superspace in physics (see [10], [4], [3]). the cn,α-compact-open topology (n ∈ n+ and α ∈ r+) is the most common consideration [9]. tuschmann asked the following question in [8, section 3, (8)]: what can one say about the topology of moduli spaces under the gromov-hausdorff metric? what if one uses the lipschitz topology? received 28 january 2020 – accepted 28 december 2020 http://dx.doi.org/10.4995/agt.2021.13066 j. deng 2. metric topology inspired by tuschmann’s questions, we will introduce four kinds of metric topology on the moduli space, and then discuss the relationship among them. let x and y be metric spaces of finite diameter, then the gromov-hausdorff distance is defined as ρgh(x, y ) := inf z {dzh(f(x), g(y ))}, where dh is hausdorff metric and z takes all metric spaces such that f (resp. g) are isometric embeddings x (resp. y ) into z (see [5]). the gromov-hausdorff distance ρgh is a pseudo-metric in the collection of all compact metric spaces. furthermore, ρgh(x, y ) = 0 if and only if x is isometric to y . for g1 and g2 in r(m), the gromov-hausdorff distance can be defined on it by ρgh(g1, g2) = ρgh((m, d1), (m, d2)), where d1 and d2 are induced metrics on m by g1 and g2. since m is closed, the gromov-hausdorff distance is well-defined on r(m). moreover, ρgh(f ∗ 1 g1, f ∗ 2 g2) = ρgh(g1, g2), where f1 and f2 are diffeomorphism of m and f ∗ 1 g1, f ∗ 2 g2 are push-back metrics on m. then one can define ρgh on m(m) as above and then ρgh is a metric on m(m). therefore, m(m) can be endowed with the metric topology called gh-topology by the gromov-hausdorff metric ρgh. definition 2.1 (edwards [3]). a map f : x → y is called an ε-isometry between compact metric spaces x and y , if |dx(a, b) − dy (f(a), f(b))| ≤ ε for all a, b ∈ x. definition 2.2 (ε-distance). assume g1 and g2 are in r(m), then we define the ε-distance by ρε(g1, g2) := ρε(d1, d2) = inf {ε | iε(d1, d2) 6= φ 6= iε(d2, d1)} , where iε(d1, d2) are the set of ε-isometries from (m, d1) to (m, d2). remark 2.3. note that |diam(d1) − diam(d2)| ≤ ρε(d1, d2) ≤ max{diam(d1), diam(d2)}, where diam(di) is the diameter of (m, gi), i = 1, 2. thus, ρε is well-defined on r(m). moreover, ρε is the pseudo-metric and ρε(g1, g2) = 0 if and only if g1 is isometric to g2 on r(m). then ε-metric, which is also denoted by ρε, can be defined on m(m) as ρgh. thus, it induces a metric topology on m(m) called ε-topology. the conformal class dense theorem of the ε-topology on m(m) was proved by liu in [7, corollary 2.2]. theorem 2.4 (liu [7]). each conformal class is dense in m(m) that is endowed with ε-topology. lemma 2.5. if ρgh(x, y ) ≤ ε, then there is a 2ε-isometric map f : x → y . if there is an ε-isometric map f : x → y , then ρgh(x, y ) ≤ 3 2 ε. remark 2.6. the lemma can be proved by using another definition of gromovhausdorff metric, i.e. ρgh(x, y ) = 1 2 inf r {dis (r)}, where the infimum is taken © agt, upv, 2021 appl. gen. topol. 22, no. 1 12 metric topology on the moduli space over all correspondences r ⊆ x × y . a correspondence between two metric spaces x and y is a subset r of x ×y such that the projections πx : x ×y → x and πy : x × y → y remain surjective when they are restricted to r. corollary 2.7. ε-topology is equivalent to gh-topology on m(m). thus, the conformal class dense theorem is also true on gh-topology. that means the gh-topology is coarse and a finer topology is needed to define on m(m). let x and y be two compact metric spaces, the dilation of a lipschitz map f : x → y is defined by dil(f) := sup a,b∈x,a 6=b dy (f(a), f(b)) dx(a, b) . if f−1 is also a lipschitz map then it is called the bi-lipschitz homeomorphism. the lipschitz-distance ρl between x and y is defined by ρl(x, y ) := inf f:x→y log{max{dil(f), dil(f−1)}}, where the infinum is taken over all bi-lipschitz homeomorphisms between x and y . then the lipschitz-distance ρl can be defined on r(m) as the definition of gromov-hausdorff distance on r(m). moreover, ρl is pseudo-metric on r(m) and ρl(g1, g2) = 0 if and only if g1 is isometric to g2 (see [2, theorem 7.2.4]). thus, it can induce a lipschitz-metric ρl on m(m), and then ρl induces the lipschitz-topology on m(m) called ltopology. furthermore, lipschitz convergence implies gromov-hausdorff convergence, where the convergence means cauchy sequence convergence related to their metrics (see [6, proposition 3.6]). proposition 2.8. l-topology is finer than gh-topology on m(m). the gh-topology and l-topology on m(m) only catch the metric information of the basic manifold and lose much essential information of the smooth structure. so it may be useful to modify the definition of l-topology on m(m) to a finer topology on m(m). for any homorphism of metric space f : (x, dx) → (y, dy ), the lipschitz constant of f is defined by l(f) := inf{k ≥ 1 | dx(x, y) k ≤ dy (f(x), f(y)) ≤ kdx(x, y), x, y ∈ x}. if the set is empty, then let l(f) be infinity. lemma 2.9. suppose that m and n are smooth closed riemannian manifolds, then any diffeomorphism of m and n has bounded lipschitz constant. remark 2.10. the normal of tangent maps of diffeomorphism on the unit tangent bundle over closed manifold are uniform bounded, since the tangent maps are continuous and the total spaces of unit tangent bundle over compact manifold are compact. © agt, upv, 2021 appl. gen. topol. 22, no. 1 13 j. deng for the composition of diffeomorphism f ◦ g : m → n → w , we have l(f ◦ g) ≤ l(f) · l(g) by direct computation. definition 2.11. assume g1 and g2 are in r(m), we define ρsl(g1, g2) = ρsl((m, d1), (m, d2)) := inf{log l(f) | f ∈ diff}, where diff is the diffeomorphism group of m. lemma 2.12. ρsl is a pseudo-metric on r(m) and ρsl(g1, g2) = 0 if and only if g1 is isometric to g2 on m. remark 2.13. if ρsl(d1, d2) = 0, then the isometry map between (m, d1) and (m, d2) can be constructed by using the closeness of m and the arzela-ascoli theorem. continuing the game, one can define the metric topology on m(m) called sl-topology by the metric ρsl. theorem 2.14. sl-topology � l-topology � gh-topology ∼= ε-topology. usually those four metrics are not complete metrics on m(m), so m(m) is local compact topology spaces endowed with their induced metric topology in general. but if we restrict it to the subset of m(m), it may have some precompact propositions. for example, gromov precompactness theorem and other convergence theorems on the moduli space [6, chapter 5]. for the non-compact case, one can ask what is the right topology on r≥0(v ) and m≥0(v ), where v is a non-compact manifold, r≥0(v ) is the riemannian metric with non-negative sectional curvature, and m≥0(v ) is the moduli space of v with non-negative sectional curvature? acknowledgements. i thank xuchao yao for useful discussions. references [1] n. a’campo, l. ji and a. papadopoulos, on the early history of moduli and teichmüller spaces, arxiv e-prints, page arxiv:1602.07208, feb 2016. 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[6] m. gromov, metric structures for riemannian and non-riemannian spaces, modern birkhäuser classics, birkhäuser boston, inc., boston, ma (2007). © agt, upv, 2021 appl. gen. topol. 22, no. 1 14 metric topology on the moduli space [7] c.-h. li, quantum fluctuations, conformal deformations, and gromov’s topology — wheeler, dewitt, and wilson meeting gromov, arxiv e-prints, page arxiv:1509.03895, sep 2015. [8] w. tuschmann, spaces and moduli spaces of riemannian metrics, front. math. china 11, no. 5 (2016), 1335–1343. [9] w. tuschmann and d. j. wraith, moduli spaces of riemannian metrics, volume 46, oberwolfach seminars, birkhäuser verlag, basel, 2015. [10] j. a. wheeler, superspace, in: analytic methods in mathematical physics (sympos., indiana univ., bloomington, ind., 1968), pages 335–378, 1970. © agt, upv, 2021 appl. gen. topol. 22, no. 1 15 @ appl. gen. topol. 20, no. 2 (2019), 349-361 doi:10.4995/agt.2019.11117 c© agt, upv, 2019 remarks on fixed point assertions in digital topology, 3 laurence boxer department of computer and information sciences, niagara university, ny 14109, usa; and department of computer science and engineering, state university of new york at buffalo, usa. (boxer@niagara.edu) communicated by s. romaguera abstract we continue the work of [5] and [3], in which are considered papers in the literature that discuss fixed point assertions in digital topology. we discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality under “usual” conditions; or that we improve upon. 2010 msc: 54h25. keywords: digital topology; fixed point; approximate fixed point; metric space. 1. introduction the topic of fixed points in digital topology has drawn much attention in recent papers. the quality of discussion among these papers is uneven; while some assertions have been correct and interesting, others have been incorrect, incorrectly proven, or reducible to triviality. in [5] and [3], we have discussed many shortcomings in earlier papers and have offered corrections and improvements. we continue this work in the current paper. received 10 december 2018 – accepted 12 june 2019 http://dx.doi.org/10.4995/agt.2019.11117 l. boxer 2. preliminaries we use n to represent the natural numbers, z to represent the integers, and r to represent the reals. a digital image is a pair (x,κ), where x ⊂ zn for some positive integer n, and κ is an adjacency relation on x. thus, a digital image is a graph. in order to model the “real world,” we usually take x to be finite, although there are several papers that consider infinite digital images. the points of x may be thought of as the “black points” or foreground of a binary, monochrome “digital picture,” and the points of zn\x as the “white points” or background of the digital picture. 2.1. adjacencies, connectedness, continuity. in a digital image (x,κ), if x,y ∈ x, we use the notation x ↔κ y to mean x and y are κ-adjacent; we may write x ↔ y when κ can be understood. we write x -κ y, or x y when κ can be understood, to mean x ↔κ y or x = y. the most commonly used adjacencies in the study of digital images are the cu adjacencies. these are defined as follows. definition 2.1. let x ⊂ zn. let u ∈ z, 1 ≤ u ≤ n. let x = (x1, . . . ,xn), y = (y1, . . . ,yn) ∈ x. then x ↔cu y if • for at most u distinct indices i, |xi −yi| = 1, and • for all indices j such that |xj −yj| 6= 1 we have xj = yj. definition 2.2 ([13]). a digital image (x,κ) is κ-connected, or just connected when κ is understood, if given x,y ∈ x there is a set {xi}ni=0 ⊂ x such that x = x0, xi ↔κ xi+1 for 0 ≤ i < n, and xn = y. definition 2.3 ([13, 1]). let (x,κ) and (y,λ) be digital images. a function f : x → y is (κ,λ)-continuous, or κ-continuous if (x,κ) = (y,λ), or digitally continuous when κ and λ are understood, if for every κ-connected subset x′ of x, f(x′) is a λ-connected subset of y . theorem 2.4 ([1]). a function f : x → y between digital images (x,κ) and (y,λ) is (κ,λ)-continuous if and only if for every x,y ∈ x, if x ↔κ y then f(x) -λ f(y). theorem 2.5 ([1]). let f : (x,κ) → (y,λ) and g : (y,λ) → (z,µ) be continuous functions between digital images. then g ◦ f : (x,κ) → (z,µ) is continuous. 2.2. fixed, approximate fixed points. a fixed point of a function f : x → x is a point x ∈ x such that f(x) = x. if (x,κ) is a digital image, an almost fixed point [13] or approximate fixed point [4] of f : x → x is a point x ∈ x such that f(x) -κ x. 2.3. digital metric spaces. a digital metric space [8] is a triple (x,d,κ), where (x,κ) is a digital image and d is a metric on x. we are not convinced that this is a notion worth developing; under conditions in which a digital image c© agt, upv, 2019 appl. gen. topol. 20, no. 2 350 fixed point assertions in digital topology, 3 models a “real world” image, x is finite or d is (usually) an `p metric, so that (x,d,κ) is discrete as a topological space. typically, assertions in the literature do not make use of both d and κ, so that this notion has an artificial feel. e.g., for a discrete topological space, all self-maps are continuous, although on digital images, self-maps are often not digitally continuous. we say a sequence {xn}∞n=0 is eventually constant if for some m > 0, n > m implies xn = xm. proposition 2.6 ([10]). let (x,d,κ) be a digital metric space. if for some a > 0 and all distinct x,y ∈ x we have d(x,y) > a, then any cauchy sequence in x is eventually constant, and (x,d) is a complete metric space. note that the hypotheses of proposition 2.6 are satisfied if x is finite or if d is an `p metric. 3. universal functions and afpp a digital image (x,κ) has the approximate fixed point property (afpp) if every κ-continuous f : x → x has an approximate fixed point. we can paraphrase theorem 3.3 of [13] as follows. theorem 3.1. a digital interval ([a,b]z,c1) has the afpp. 2 definition 3.2 ([4]). let (x,κ) and (y,λ) be digital images. a (κ,λ)-continuous function f : x → y is universal for (x,y ) if given a (κ,λ)-continuous function g : x → y such that g 6= f, there exists x ∈ x such that f(x) ↔λ g(x). it was shown in [4] that there is a relationship between the afpp and universal functions. in this section, we show there are advantages in the study of the afpp to replacing the notion of universal function with a similar notion of a “weakly universal function.” this enables us to make several improvements on results of [4]. the following assertion, one implication of which is incorrect, appears as proposition 5.5 of [4]. let (x,κ) be a digital image. then (x,κ) has the afpp if and only if the identity function 1x is universal for (x,x). the implication of this assertion that is correct is stated in the following with its proof as given in [4]. proposition 3.3. let (x,κ) be a digital image. if the identity function 1x is universal for (x,x), then (x,κ) has the afpp. proof. if 1x is universal for (x,x), then for 1x 6= f : x → x, f being κcontinuous, there exists x ∈ x such that f(x) ↔κ 1x(x) = x. thus x has the afpp. � however, the converse of proposition 3.3 is not generally true, as shown be the following. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 351 l. boxer example 3.4. let f : ([−1, 1]z,c1) → ([−1, 1]z,c1) be the map f(z) = −z. then f is c1-continuous, and there is no z ∈ [−1, 1]z such that f(z) ↔c1 z. hence 1[−1,1]z is not a universal function for ([−1, 1]z, [−1, 1]z). however, by theorem 3.1, ([−1, 1]z,c1) has the afpp. 2 definition 3.5. let (x,κ) and (y,λ) be digital images. let f : x → y be (κ,λ)-continuous. then f is a weakly universal function for (x,y ) if for every (κ,λ)-continuous g : x → y such that g 6= f there exists x ∈ x such that f(x) -λ g(x). 2 notice the difference between definitions 3.2 and 3.5: the former requires f(x) and g(x) to be adjacent, while the latter requires f(x) and g(x) to be adjacent or equal. proposition 3.6. a universal function between digital images is weakly universal. proof. this is immediate from definitions 3.2 and 3.5. � for a graph g = (v,e) (v is the vertex set; e is the edge set), a subset d of v is called dominating if for every v ∈ v , either v ∈ d or there is a w ∈ d such that {v,w}∈ e [6]. the following generalizes a result of [4]. proposition 3.7. let (x,κ) and (y,λ) be digital images. if f : x → y is a weakly universal function, then f(x) is λ-dominating in y . proof. let y ∈ y and let ỹ : x → y be the constant function with image {y}. since f is weakly universal, there exists x ∈ x such that f(x) -λ ỹ(x) = y. since y is an arbitrary member of y , the assertion follows. � theorem 3.8. the digital image (x,κ) has the afpp if and only if 1x is weakly universal for (x,x). 2 proof. (x,κ) has the afpp if and only if given a (κ,κ)-continuous f : x → x, for some x ∈ x we have f(x) -κ 1x(x) = x; i.e., if and only if 1x is universal. � the following is suggested by theorem 5.7 of [4]. proposition 3.9. let (w,κ), (x,λ), and (y,µ) be digital images. let f : w → x be (κ,λ)-continuous and let g : x → y be (λ,µ)-continuous. if g ◦f is weakly universal, then g is weakly universal. proof. let h : x → y be (λ,µ)-continuous. since g ◦ f is weakly universal, there exists w ∈ w such that g ◦f(w) -µ h◦f(w), i.e., for x = f(w) we have g(x) -µ h(x). since h was arbitrarily chosen, the assertion follows. � the following is suggested by theorem 5.8 of [4]. theorem 3.10. let g : (u,µ) → (x,κ) and h : (y,λ) → (v,ν) be digital isomorphisms. let f : x → y be (κ,λ)-continuous. then the following are equivalent. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 352 fixed point assertions in digital topology, 3 (1) f is a weakly universal function for (x,y ). (2) f ◦g is weakly universal. (3) h◦f is weakly universal. proof. (1 implies 2): let k : u → y be (µ,λ)-continuous. since f is weakly universal, there exists x ∈ x such that (k◦g−1)(x) -λ f(x), i.e., for u = g−1(x) we have k(u) = k(g−1(x)) -λ (f ◦g)(g−1(x)) = (f ◦g)(u). since k is arbitrary, f ◦g is weakly universal. (2 implies 1): this follows from proposition 3.9. (1 implies 3): let m : x → v be (κ,ν)-continuous. since f is weakly universal, there exists x ∈ x such that (h−1 ◦m)(x) -λ f(x). by continuity, m(x) = h◦ (h−1 ◦m)(x) -ν h◦f(x). since m is arbitrary, h◦f is weakly universal. (3 implies 1): let r : x → y be (κ,λ)-continuous. since h ◦ f is weakly universal, there exists x ∈ x such that h◦f(x) -ν h◦r(x). thus, f(x) = h−1 ◦h◦f(x) -λ h−1 ◦h◦r(x) = r(x). since r is arbitrary, f must be weakly universal. � corollary 5.9 of [4] claims that an isomorphism f : (x,κ) → (y,λ) is universal for (x,y ) if and only if (x,κ) has the afpp. example 3.4 above shows that this assertion is incorrect. however, we have the following. corollary 3.11. let f : (x,κ) → (y,λ) be an isomorphism. the following are equivalent. (1) f is weakly universal for (x,y ). (2) (x,κ) has the afpp. (3) (y,λ) has the afpp. proof. (1) ⇔ (2): by theorem 3.10, f is weakly universal if and only if f◦f−1 = 1x is weakly universal, which, by theorem 3.8 is true if and only if (x,κ) has the afpp. (1) ⇔ (3): by theorem 3.10, f is weakly universal if and only if f−1◦f = 1y is weakly universal, which, by theorem 3.8 is true if and only if (y,λ) has the afpp. � the following generalizes theorem 5.10 of [4] and corrects its proof (stated in terms of proposition 5.5 of [4], which, we noted above, is incorrect). theorem 3.12. let (xi,κi) be digital images, 1 ≤ i ≤ v. let x = πvi=1xi. if (x,npv(κ1, . . . ,κv)) has the afpp, then each (xi,κi) has the afpp. proof. let fi : xi → xi be (κi,κi)-continuous, 1 ≤ i ≤ v. then the product function f = πvi=1fi : x → x is (npv(κ1, . . . ,κv),npv(κ1, . . . ,κv))continuous [2]. by theorem 3.8, 1x is weakly universal, so there exists p = (x1, . . . ,xv) ∈ x, xi ∈ xi, such that p -npv(κ1,...,κv) f(p) = (f1(x1), . . . ,fv(xv)), c© agt, upv, 2019 appl. gen. topol. 20, no. 2 353 l. boxer hence xi -κi fv(xi) for all i. since fi was taken arbitrarily, the conclusion follows. � 4. digital expansions in [12] the paper [12] contains several assertions that are incorrect or incorrectly proven, limited, or can be improved. 4.1. digital expansive mappings. definition 4.1 ([12]). let (x,d,κ) be a complete digital metric space. let t : x → x. if d(t(x),t(y)) ≥ k d(x,y) for all x,y ∈ x and some k > 1, then t is a digital expansive mapping. theorem 4.2 ([12]). if t : x → x is a digital expansive mapping on complete digital metric space (x,d,κ) and t is onto, then t has a fixed point. however, in practice, the hypotheses of theorem 4.2 often cannot be satisfied, as shown in the following, which combines theorems 4.8 and 4.9 of [5]. theorem 4.3. let (x,d,κ) be a digital metric space of more than one point. if there exist x0,y0 ∈ x such that either • d(x0,y0) = diamx > 0, or • d(x0,y0) = min{d(x,y) |x,y ∈ x,x 6= y}, then there is no self-map t : x → x that is a digital expansive mapping and is onto. in practice, a digital image (x,κ) typically consists of a finite set of more than 1 point; or, should a metric d be used, it is typically an `p metric. under such circumstances, by theorem 4.3 a digital metric space (x,d,κ) cannot have a self-map that is both a digital expansive mapping and onto. 4.2. 1st generalization of expansive mappings. theorem 3.4 of [12] states the following. theorem 4.4. let, (x,d,κ) be a complete digital metric space and t : x → x be an onto self map. let t satisfy d(t(x),t(y)) ≥ k [d(x,t(x)) + d(y,t(y))] where k ≥ 1/2, for all x,y ∈ x. then t has a fixed point. but theorem 4.4 reduces to a trivial statement, as we see in the following. proposition 4.5. a map t as in theorem 4.4 must be the identity map. proof. for such a map, we have 0 = d(t(x),t(x)) ≥ k [d(x,t(x)) + d(x,t(x))] = 2k d(x,t(x)), so d(x,t(x)) = 0 for all x ∈ x. � c© agt, upv, 2019 appl. gen. topol. 20, no. 2 354 fixed point assertions in digital topology, 3 4.3. 2nd generalization of expansive mappings. theorem 3.5 of [12] asserts the following. let (x,d,κ) be a complete digital metric space and let t : x → x be onto and continuous. let d(t(x),t(y)) ≥ kµ(x,y) for all x,y ∈ x, where k > 1 and µ(x,y) ∈ { d(x,y), d(x,t(x)) + d(y,t(y)) 2 , d(x,t(y)) + d(y,t(x)) 2 } . then t has a fixed point. the argument given as proof for this assertion has flaws, including the use in its proof of an assumption that k < 2, not stated in the hypotheses; and an incorrect application of the triangle inequality (where we need the reverse of the inequality to proceed as the authors have done) in the attempt to reduce case 3 to case 2. thus, the assertion as stated must be regarded as unproven. also, the argument given for proof clarifies that the continuity assumption is of the ε−δ type, not digital. in the following, we obtain a version of this assertion with no continuity assumption, but with an additional assumption about x or d and with greater restriction on the possible values of µ(x,y). theorem 4.6. let (x,d,κ) be a digital metric space, such that x is finite or d is an `p metric. let t : x → x be onto. suppose d(t(x),t(y)) ≥ kµ(x,y) for all x,y ∈ x, where 1 < k < 2 and µ(x,y) ∈ { d(x,y), d(x,t(x)) + d(y,t(y)) 2 } . then t has a fixed point. proof. a proof can be given via suitable modification of its analog in [12]. however, a simpler argument is as follows. without loss of generality, |x| > 1. since x is finite or d is an `p metric, there exist x0,y0 ∈ x such that m = d(x0,y0) = min{d(x,y) |x,y ∈ x,x 6= y} > 0. since t is onto, there exist x′,y′ ∈ x such that t(x′) = x0 and t(y′) = y0. suppose t has no fixed point. then for all x,y ∈ x, d(x,t(x)) ≥ m and d(y,t(y)) ≥ m; hence µ(x,y) ≥ m. therefore, m = d(x0,y0) = d(t(x ′),t(y′)) ≥ kµ(x′,y′) ≥ km, a contradiction. therefore, t must have a fixed point. � c© agt, upv, 2019 appl. gen. topol. 20, no. 2 355 l. boxer 4.4. 3rd generalization of expansive mappings. the next assertion of [12] is flawed in ways similar to the assertion discussed in section 4.3. asserted as theorem 3.6 of [12] is the following. let (x,d,κ) be a complete digital metric space. let t be an onto self-map of x that is continuous. let k > 1 and suppose t satisfies d(t(x),t(y)) ≥ kµ(x,y) for all x,y ∈ x, where µ(x,y) belongs to{ d(x,y), d(x,t(x)) + d(y,t(y)) 2 ,d(x,t(y)),d(y,t(x)) } . then t has a fixed point. observe the following. • as above, the continuity used for the proof of this assertion is of the ε− δ kind, not digital continuity. • as above, the argument given in proof for this assertion requires 1 < k < 2. • as above, the authors attempt to establish a cauchy sequence, and in doing so, they incorrectly reverse the triangle inequality in order to reduce the 3rd case considered to the 2nd case. thus, as stated, the assertion presented as theorem 3.6 of [12] must be regarded as unproven. note that theorem 4.6 above is a reasonable correct modification of this assertion. 4.5. examples of [12]. in examples 3.8, 3.9, 3.16, and 3.17 of [12], the authors lose track of the standard assumption that a digital image x is a subset of zn. in each of these examples, they write of an unspecified x using functions that clearly place x in r, but not clearly in z. 4.6. α − ψ expansive maps. in the following, we let ψ be the set of functions [14] ψ : [0,∞) → [0,∞) such that • ∑∞ n=1 ψ n(t) < ∞ for each t > 0, where ψn is the n-th iterate of ψ ([12] misquotes this requirement as ψn(t) < ∞ for each t > 0), and • ψ is non-decreasing. definition 4.7 ([12]). let (x,d,κ) be a digital metric space. let t : x → x. t is a digital α−ψ expansive mapping if α : x ×x → [0,∞), ψ ∈ ψ, and for all x,y ∈ x, ψ(d(t(x),t(y))) ≥ α(x,y)d(x,y). definition 4.8 ([12]). let t : x → x. let α : x × x → [0,∞). t is α-admissible if α(x,y) ≥ 1 implies α(t(x),t(y)) ≥ 1 theorem 4.9 ([12]). let (x,d,κ) be a complete digital metric space. let t : x → x be a bijective, digital α−ψ expansion mapping such that • t−1 is α-admissible; c© agt, upv, 2019 appl. gen. topol. 20, no. 2 356 fixed point assertions in digital topology, 3 • there exists x0 ∈ x such that α(x0,t−1(x0)) ≥ 1; and • t is digitally continuous. then t has a fixed point. despite the use of “digitally continuous” in the statement of theorem 4.9, the continuity assumption used in its proof is of the ε− δ variety. in fact, the assumption is unnecessary if we assume additional common conditions, as in the following. theorem 4.10. let (x,d,κ) be a digital metric space, where x is finite or d is an `p metric. let t : x → x be a bijective, digital α−ψ expansion mapping such that • t−1 is α-admissible; and • there exists x0 ∈ x such that α(x0,t−1(x0)) ≥ 1. then t has a fixed point. proof. our argument borrows from its analog in [12]. by hypothesis, there exists x0 ∈ x such that α(x0,t−1(x0)) ≥ 1. by induction, we obtain s = {xn}∞n=0 such that xn+1 = t−1(xn) for n > 0. since t−1 is α-admissible, by induction we have α(xn,xn+1) = α(t −1(xn−1),t −1(xn)) ≥ 1 for all n. since t is a digital α−ψ expansive mapping, for all n we have d(xn,xn+1) ≤ α(xn,xn+1)d(xn,xn+1) ≤ ψ(d(t(xn),t(xn+1)) = ψ(d(xn−1,xn)). by induction, it follows that d(xn,xn+1) ≤ ψn(d(x0,x1)). since ψ ∈ ψ, it follows that s is a cauchy sequence. by theorem 2.6, s is eventually constant, so there exists m such that t(xm+1) = xm = xm+1; thus, xm+1 is a fixed point of t . � 5. weakly commuting mappings the paper [11] presents a fixed point assertion for “weakly commuting mappings,” defined as follows. definition 5.1 ([15]). let (x,d) be a metric space and let f,g : x → x. then f and g are weakly commuting if for all x ∈ x, d(f(g(x)),g(f(x))) ≤ d(f(x),g(x)). presented as theorem 3(a) of [11] is the following. let (x,d,κ) be a complete digital metric space, x 6= ∅. let s,t : x → x such that (3.1) t(x) ⊂ s(x); (3.2) s is κ-continuous; (3.3) for some α such that 0 < α < 1 and all x,y ∈ x, d(t(x),t(y)) ≤ αd(s(x),s(y)). if s and t are weakly commuting, then they have a unique common fixed point. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 357 l. boxer the argument given in proof of this assertion is flawed by the unjustified statement (rephrased slightly), “from (3.2) the κ-continuity of s implies the κ-continuity of t.” this reasoning is incorrect, as shown in the following. example 5.2. let x = {p0 = (0, 0, 0),p1 = (1, 1, 1),p2 = (2, 0, 0)} ⊂ z3. let s = 1x : x → x and let t : x → x be defined by t(p0) = t(p2) = p2, t(p1) = p0. let κ be the c3-adjacency. clearly, (3.1) and (3.2) of the above are satisfied. let d be the `1 metric. then (3.3) above is satisfied with α = 2/3. however, t is not c3-continuous, since p0 ↔c3 p1 but f(p0) and f(p1) are not c3-adjacent. therefore, the assertion stated as theorem 3(a) of [11] must be regarded as unproven. however, we see below that replacing the assumptions of completeness and (3.2) by assumptions that are commonly realized yields a valid statement. theorem 5.3. let (x,d,κ) be a digital metric space, x 6= ∅, with x finite or d an `p metric. let s,t : x → x such that (1) t(x) ⊂ s(x); (2) for some α such that 0 < α < 1 and all x,y ∈ x, d(t(x),t(y)) ≤ αd(s(x),s(y)). if s and t are weakly commuting, then they have a unique common fixed point. proof. we use ideas from the analogous argument of [11]. let x0 ∈ x. by assumption 1, there exists x1 ∈ x such that s(x1) = t(x0). by induction we have a sequence {xn}∞n=0 such that for all n, s(xn+1) = t(xn), and we have d(s(xn),s(xn+1)) = d(t(xn−1),t(xn)) ≤ αd(s(xn−1),s(xn)). by a simple induction, this yields d(s(xn),s(xn+1)) ≤ αnd(s(x0),s(x1)). thus, {s(xn)}∞n=0 is a cauchy sequence, hence by proposition 2.6 is eventually constant, i.e., there exists z ∈ x such that for sufficiently large n, s(xn) = z. by our definition of the sequence {xn}, we also have, for sufficiently large n, t(xn) = z. so for n sufficiently large, and since s and t are weakly commuting, d(s(z),t(z)) = d(s(t(xn)),t(s(xn))) ≤ d(s(xn),t(xn)) = d(z,z) = 0, i.e., s(z) = t(z) and therefore, by the weakly commuting property, d(s(t(z)),t(s(z))) ≤ d(s(z),t(z)) = 0, i.e., s(t(z)) = t(s(z)). so d(t(z),t(t(z))) ≤ αd(s(z),s(t(z))) = αd(t(z),t(s(z))) = αd(t(z),t(t(z))). c© agt, upv, 2019 appl. gen. topol. 20, no. 2 358 fixed point assertions in digital topology, 3 thus d(t(z),t(t(z))) = 0, i.e., t(z) is a fixed point of t . further, substituting from the above gives d(s(t(z)),t(z)) = d(t(s(z)),t(z)) ≤ αd(s(s(z)),s(z)) = αd(s(t(z)),t(z)); since α > 0, it follows that d(s(t(z)),t(z)) = 0. thus, t(z) is a common fixed point of s and t. to show the common fixed point is unique, suppose y and y′ are common fixed points, i.e., s(y) = t(y) = y, s(y′) = t(y′) = y′. then d(y,y′) = d(t(y),t(y′)) ≤ αd(s(y),s(y′)) = αd(y,y′), so d(y,y′) = 0. hence y = y′. � note the following limitation on theorem 5.3 is applicable if x is finite, or if d is an `p metric. proposition 5.4. let (x,d,κ),s,t,α be as in theorem 5.3, where d0 = min{d(x,x′) |x,x′ ∈ x,x ↔κ x′} > 0, d1 = max{d(x,x′) |x,x′ ∈ x,x ↔κ x′}. if x is κ-connected, s is κ-continuous, and 0 < α < d0/d1, then t is a constant function. proof. let x ↔κ x′. since s is κ-continuous we have s(x) -κ s(x′), and therefore d(s(x),s(x′)) ≤ d1. thus, d(t(x),t(x′)) ≤ αd(s(x),s(x′)) < d0 d1 d1 = d0. our choice of d0 implies t(x) = t(x ′). since x is connected, the assertion follows. � 6. weakly compatible maps the paper [7] discusses “weakly compatible” or “coincidentally commuting” maps, defined as follows. definition 6.1. let s,t : x → x. then s and t are weakly compatible or coincidentally commuting if, for every x ∈ x such that s(x) = t(x) we have s(t(x)) = t(s(x)). the following assertion is stated as theorem 3.1 of [7]. let a,b,s,t : x → x, where (x,d,κ) is a complete digital metric space. suppose the following are satisfied. • s(x) ⊂ b(x) and t(x) ⊂ a(x). • the pairs (a,s) and (b,t) are coincidentally commuting. • one of s(x),t(x),a(x),b(x) is a complete subspace of x. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 359 l. boxer • for all x,y ∈ x, d(s(x),t(y)) ≤ φ(max{d(a(x),b(y)),d(s(x),a(x)),d(s(x),b(y)),d(b(y),t(y))}) where φ : [0,∞) → [0,∞) is continuous, monotone increasing, and satisfies φ(t) < t for all t > 0. then a,b,s, and t have a unique common fixed point. however, the argument offered as proof of this assertion is flawed as follows. a sequence {yn}∞n=0 is established and it is shown that limn→∞d(y2n,y2n+1) = 0. from this, it is claimed that {yn}∞n=0 is a cauchy sequence. but such reasoning is incorrect, as shown in the following. example 6.2. for n ∈ n, let yn = { 0 if (n mod 4) ∈{0, 1}; 1 if (n mod 4) ∈{2, 3}, for all n ∈ n, d(y2n,y2n+1) = 0, yet {yn}∞n=0 is not a cauchy sequence. thus, the assertion of [7] stated as theorem 3.1, and its dependent assertion stated as theorem 3.2, must be regarded as unproven. 7. further remarks we have discussed assertions that appeared in [4, 7, 11, 12]. we have discussed errors or corrections for some, shown some to be limited or trivial, and offered improvements for others. references [1] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. [2] l. boxer, generalized normal product adjacency in digital topology, applied general topology 18, no. 2 (2017), 401–427. [3] l. boxer, remarks on fixed point assertions in digital topology, 2, applied general topology 20, no. 1 (2019), 155–175. [4] l. boxer, o. ege, i. karaca, j. lopez and j. louwsma, digital fixed points, approximate fixed points and universal functions, applied general topology 17, no. 2 (2016), 159– 172. [5] l. boxer and p. c. staecker, remarks on fixed point assertions in digital topology, applied general topology 20, no. 1 (2019), 135–153. [6] g. chartrand and l. lesniak, graphs & digraphs, 2nd ed., wadsworth, inc., belmont, ca, 1986. [7] s. dalal, common fixed point results for weakly compatible map in digital metric spaces, scholars journal of physics, mathematics and statistics 4, no. 4 (2017), 196–201. [8] o. ege and i. karaca, digital homotopy fixed point theory, comptes rendus mathematique 353, no. 11 (2015), 1029–1033. [9] j. haarmann, m. p. murphy, c. s. peters and p. c. staecker, homotopy equivalence of finite digital images, journal of mathematical imaging and vision 53, no. 3 (2015), 288–302. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 360 fixed point assertions in digital topology, 3 [10] s.-e. han, banach fixed point theorem from the viewpoint of digital topology, journal of nonlinear science and applications 9 (2016), 895–905. [11] d. jain and a.c. upadhyaya, weakly commuting mappings in digital metric spaces, international advanced research journal in science, engineering and technology 4, no. 8 (2017), 12–16. [12] k. jyoti and a. rani, digital expansions endowed with fixed point theory, turkish journal of analysis and number theory 5, no. 5 (2017), 146–152. [13] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. [14] b. samet, c. vetro,and p. vetro, fixed point theorem for α − ψ-contractive type mappings, nonlinear analysis 75 (2012), 2154–2165. [15] s. sessa, on a weak commutative condition of mappings in fixed point considerations, publications de l’institut mathematique, nouvelle serie tome 32 (1982), 149–153. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 361 @ appl. gen. topol. 21, no. 1 (2020), 17-34 doi:10.4995/agt.2020.11807 c© agt, upv, 2020 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) franco barragán, alicia santiago-santos∗ and jesús f. tenorio† instituto de f́ısica y matemáticas, universidad tecnológica de la mixteca, carretera a acatlima, km 2.5, huajuapan de león, oaxaca, méxico. (franco@mixteco.utm.mx,alicia@mixteco.utm.mx, jtenorio@mixteco.utm.mx) communicated by f. balibrea abstract let x be a continuum and let n be a positive integer. we consider the hyperspaces fn(x) and sfn(x). if m is an integer such that n > m ≥ 1, we consider the quotient space sfnm(x). for a given map f : x → x, we consider the induced maps fn(f) : fn(x) → fn(x), sfn(f) : sfn(x) → sfn(x) and sfnm(f) : sfnm(x) → sfnm(x). in this paper, we introduce the dynamical system (sfnm(x),sfnm(f)) and we investigate some relationships between the dynamical systems (x, f), (fn(x),fn(f)), (sfn(x),sfn(f)) and (sfnm(x),sfnm(f)) when these systems are: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, irreducible, feebly open and turbulent. 2010 msc: 54b20; 37b45; 54f50; 54f15. keywords: chaotic; continuum; dynamical system; exact; feebly open; hyperspace; induced map; irreducible; mixing; strongly transitive; symmetric product; symmetric product suspension; totally transitive; transitive; turbulent; weakly mixing. ∗this paper was partially supported by the project: “propiedades dinámicas y topológicas sobre sistemas dinámicos inducidos”, (utmix-ptc-064) of prodep, 2017. †corresponding author. received 10 may 2019 – accepted 26 february 2020 http://dx.doi.org/10.4995/agt.2020.11807 f. barragán, a. santiago-santos and j. f. tenorio 1. introduction a continuum is a nonempty compact connected metric space. by a (discrete) dynamical system we mean a continuum with a continuous self-surjection. this class of dynamical systems belongs to the area of topological dynamics, which is a branch of dynamical systems and topology where the qualitative and asymptotic properties of dynamical systems are studied. in the last 30 years, dynamical systems had been greatly developed, this is because they are very useful to model problems of other sciences such as physics, biology and economics. currently, we can find several types of dynamical systems: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, minimal and sensitive, see [2, 3, 8, 10, 19, 21, 23, 24, 32]. concerning hyperspaces theory, given a continuum x, the hyperspaces of x most studied are: the hyperspace 2x which consists of all the nonempty compact subsets of x; given a natural number n, the hyperspace cn(x) consisting of the elements of 2x that have at most n components; and the hyperspace fn(x) formed by the elements of 2x which have at most n points. each of them is topologized with the hausdorff metric. these hyperspaces are extendly studied in continuum theory, see [20, 28, 30]. on the other hand, given a continuum x and a positive integer n, in 1979 [29], the study of quotient spaces of hyperspace was initiated with the introduction of the space c1(x)/f1(x). later the space cn(x)/fn(x) was defined in 2004 [27]. subsequently, the space cn(x)/f1(x) was studied [26]. in 2010 [4], the first named author of this paper defined the space fn(x)/f1(x) which is denoted by sfn(x) and is called the n-fold symmetric product suspension of the continuum x. some topological properties of sfn(x) are studied in [4, 6]. finally, in 2013 [14], the space fn(x)/fm(x) is defined (1 ≤ m < n) and is denoted by sfnm(x). in [14] are studied several properties of this quotient space. note that when m = 1, sfnm(x) = sfn(x). a map (continuous surjection) f : x → x, where x is a continuum, induces a map on the hyperspace 2x, denoted by 2f : 2x → 2x and defined by 2f (a) = f(a), for each a ∈ 2x. if n is a positive integer, the induced map to the hyperspace cn(x) is the restriction of 2f to cn(x), and is denoted by cn(f) and the induced map to the hyperspace fn(x) is simply the restriction of 2f to fn(x) which is denoted by fn(f). this last map, fn(f), induces a map on the space sfn(x) which is denoted by sfn(f) : sfn(x) → sfn(x) [5, 7]. thus, the dynamical system (x,f) induces the dynamical systems (2x, 2f ), (cn(x),cn(f)), (fn(x),fn(f)) and (sfn(x),sfn(f)). a line of research consists of analyzing the relationships between the dynamical system (x,f) (individual dynamic) and the dynamical systems on the hyperspaces (2x, 2f ), (cn(x),cn(f)), (fn(x),fn(f)) and (sfn(x),sfn(f)) (collective dynamic). in 1975 [9], the study of this line of research began, and nowadays there are a lot of results in the literature, for instance in [1, 3, 12, 13, 17, 18, 19, 25, 31, 32, 34]. it is important to note that recently, in 2016 [8], c© agt, upv, 2020 appl. gen. topol. 21, no. 1 18 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) the relationships between the dynamical systems (x,f), (fn(x),fn(f)) and (sfn(x),sfn(f)) were investigated. let n and m be two integers such that n > m ≥ 1 and let x be a continuum. note that the function fn(f) induces another map on the space sfnm(x) which is denoted by sf n m(f) : sf n m(x) → sf n m(x) [15]. in this paper, we introduce the dynamical system (sfnm(x),sf n m(f)) and we investigate some relationships between the dynamical systems (x,f), (fn(x),fn(f)), (sfn(x),sfn(f)) and (sfnm(x),sf n m(f)) when these systems are: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, irreducible, feebly open and turbulent. this paper is organized as follows: in section 2, we recall basic definitions and we introduce some notations. in section 3, we present properties related with the transitivity of the dynamical systems (x,f), (fn(x),fn(f)), (sfn(x),sfn(f)) and (sfnm(x),sf n m(f)), namely: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive and chaotic. finally, in section 4, we review others properties of these dynamical systems, namely: irreducible, feebly open and turbulent. 2. preliminaries the symbols n, q, r and c denote the set of positive integers, rational numbers, real numbers and complex numbers, respectively. a continuum is a nonempty compact connected metric space. a continuum is said to be nondegenerate if it has more than one point. a subcontinuum of a space x is a continuum contained in x. given a continuum x, a point a ∈ x and � > 0, v�(a) denotes the open ball with center a and radius �. a map is a continuous function. we denote by idx the identity map on the continuum x. given a continuum x and a positive integer n, we consider the hyperspaces of x, 2x = {a ⊆ x | a is closed and nonempty} and fn(x) = {a ∈ 2x | a has at most n points}. we topologize these sets with the hausdorff metric [30, (0.1)]. the hyperspace fn(x) is the n-fold symmetric product of x [11]. given a finite collection u1,u2, . . . ,um of nonempty subsets of x, with 〈u1,u2, . . . ,um〉 we denote the following subset of 2x:{ a ∈ 2x | a ⊆ m⋃ i=1 ui and a∩ui 6= ∅, for each i ∈{1, 2, . . . ,m} } . the family: {〈u1,u2, . . . ,ul〉 | l ∈ n and u1,u2, . . . ,ul are open subsets of x} forms a basis for a topology on 2x called the vietoris topology [30, (0.11)]. it is well known that the vietoris topology and the topology induced by the hausdorff metric coincide [30, (0.13)]. for those who are interested in learning more about these topics can see [20, 28, 30]. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 19 f. barragán, a. santiago-santos and j. f. tenorio notation 2.1. let x be a continuum, let n and m be positive integers, and let u1,u2, . . . ,um be a finite family of open subsets of x. by 〈u1,u2, . . . ,um〉n we denote the intersection 〈u1,u2, . . . ,um〉∩fn(x). given two integers n and m such that n > m ≥ 1, sfnm(x) denotes the quotient space fn(x)/fm(x) obtained by shrinking fm(x) to a point in fn(x), with the quotient topology [15]. here, we denote the quotient map by qm : fn(x) →sfnm(x) and qm(fm(x)) by fmx . thus: sfnm(x) = {{a} | a ∈fn(x) \fm(x)}∪{f m x }. note that, if m = 1, then sfn1 (x) = sfn(x) (see [4]). remark 2.2. the space sfnm(x)\{fmx } is homeomorphic to fn(x)\fm(x), using the appropriate restriction of qm. let n be a positive integer and let x be a continuum. if f : x → x is a map, we consider the induced map of f on the n-fold symmetric product of x, fn(f) : fn(x) →fn(x), defined by fn(f)(a) = f(a), for all a ∈fn(x) [28, 1.8.23]. also, given two integers n and m such that n > m ≥ 1, we consider the function sfnm(f) : sf n m(x) →sf n m(x) given by: sfnm(f)(χ) = { qm(fn(f)(q−1m (χ))), if χ 6= fmx ; fmx , if χ = f m x , for each χ ∈sfnm(x). note that, by [16, 4.3, p. 126], sfnm(f) is continuous. moreover, diagram 1 is commutative, that is, qm ◦fn(f) = sfnm(f) ◦qm. -fn(x) fn(x) fn(f) ? sfnm(x) qm ? sfnm(x) qm sfnm(f) diagram 1 note that if m = 1, then sfn1 (f) = sfn(f) (see [5]). now, by diagram (∗) from [8, p. 457] and diagram 1, the maps sfn(f) and sfnm(f) are related under the diagram 2, where q : fn(x) → sfn(x) is the quotient map. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 20 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) sfn(x) sfn(f) // sfn(x) fn(x) qm �� q oo fn(f) //fn(x) qm �� q oo sfnm(x) sfnm(f) // sfnm(x) diagram 2 observe that sfn(f) ◦q = q ◦fn(f). on the other hand, in this paper a dynamical system is a pair (x,f), where x is a nondegenerate continuum and f : x → x is a map. given a dynamical system (x,f), define f0 = idx and for each k ∈ n, let fk = f ◦fk−1. a point p ∈ x is a periodic point in (x,f) provided that there exists k ∈ n such that fk(p) = p. the set of periodic points of (x,f) is denoted by per(f). given x ∈ x, the orbit of x under f is the set o(x,f) = {fk(x) | k ∈ n ∪{0}}. finally, a subset k of x is said to be invariant under f if f(k) = k. let (x,f) be a dynamical system. we say that (x,f) is: (1) exact if for each nonempty open subset u of x, there exists k ∈ n such that fk(u) = x; (2) mixing if for every pair of nonempty open subsets u and v of x, there exists n ∈ n such that fk(u) ∩v 6= ∅, for every k ≥ n; (3) weakly mixing if for all nonempty open subsets u1,u2,v1 and v2 of x, there exists k ∈ n such that fk(ui) ∩vi 6= ∅, for each i ∈{1, 2}; (4) transitive if for every pair of nonempty open subsets u and v of x, there exists k ∈ n such that fk(u) ∩v 6= ∅; (5) totally transitive if (x,fs) is transitive, for all s ∈ n; (6) strongly transitive if for each nonempty open subset u of x, there exists s ∈ n such that x = ⋃s k=0 f k(u); (7) chaotic if it is transitive and per(f) is dense in x; (8) irreducible if the only closed subset a ⊆ x for which f(a) = x is a = x; (9) feebly open (or semi-open) if for every nonempty open subset u of x, there is a nonempty open subset v of x such that v ⊆ f(u); (10) turbulent if there are compact nondegenerate subsets c and k of x such that c ∩k has at most a point and k ∪c ⊆ f(k) ∩f(c). inclusions between some classes of dynamical systems, which are considered here, are showed in diagram 3. an arrow means inclusion; this is, the class of c© agt, upv, 2020 appl. gen. topol. 21, no. 1 21 f. barragán, a. santiago-santos and j. f. tenorio dynamical system above is contained in the class of dynamical system below. for some of these inclusions see, for instance, [21, 22]. exact // mixing �� weakly mixing �� totally transitive �� strongly transitive vv irreducible ww �� chaotic // transitive �� feebly open surjective diagram 3 by diagram 3 and [5, theorem 3.2], we have the following result (compare with [8, lemma 2.3]): lemma 2.3. let (x,f) be a dynamical system and n and m be integers such that n > m ≥ 1. let n be one of the following classes of dynamical systems: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, and irreducible. if (x,f) ∈ n, then f,fn(f), sfn(f) and sfnm(f) are surjective. let n be an integer greater than or equal to two and let (x,f) be a dynamical system. observe that f1(x) is a subcontinuum of fn(x) such that f1(x) is invariant under fn(f). in section 4 of [8] the authors defined and studied the dynamical system (sfn(x),sfn(f)). similarly, given an integer m such that n > m ≥ 1, fm(x) is also an invariant subcontinuum of fn(x) under fn(f). thus, by [8, remark 3.1], we can define the dynamical system (sfnm(x),sf n m(f)). 3. dynamical properties related to transitivity of (sfnm(x),sf n m(f)) arguing as in [8, proposition 4.1] and considering diagram 1, we have the following result. proposition 3.1. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. then, for each k,s ∈ n, the following holds: (a) (fn(f))k(a) = fk(a), for every a ∈fn(x). (b) qm ◦ (fn(f))k = (sfnm(f))k ◦qm. (c) ((fn(f))s)k = (fn(f))sk. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 22 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) (d) qm ◦ ((fn(f))s)k = ((sfnm(f))s)k ◦qm. let (x,d) be a continuum and let f : x → x be a map. recall that f is an isometry if d(x,y) = d(f(x),f(y)), for each x,y ∈ x. theorem 3.2. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. if f is an isometry, then the dynamical system (sfnm(x),sf n m(f)) is not transitive. proof. suppose that f is an isometry and that (sfnm(x),sf n m(f)) is transitive. let x1, x2, . . . ,xm+1 ∈ x be such that xi 6= xj, for each i,j ∈ {1, . . . ,m + 1} with i 6= j. let r = min{d(xi,xj) : i,j ∈{1, . . . ,m + 1}, i 6= j}, where d is the metric of x. for each i ∈ {1, . . . ,m + 1}, we put ui = vr 4 (xi). observe that u1, . . . ,um+1 are nonempty open subsets of x such that xi ∈ ui, for each i ∈{1, . . . ,m + 1} and ui∩uj = ∅, for each i,j ∈{1, . . . ,m + 1} with i 6= j. moreover, we consider v1, . . . ,vm+1 nonempty open subsets of x such that ⋃m+1 i=1 vi ⊆ u1 and vi ∩vj = ∅, for each i,j ∈{1, . . . ,m + 1} with i 6= j. it follows that 〈u1, . . . ,um+1〉n is a nonempty open subset of fn(x) such that 〈u1, . . . ,um+1〉n∩fm(x) = ∅ and 〈v1, . . . ,vm+1〉n∩fm(x) = ∅. by remark 2.2, we have that qm(〈u1, . . . ,um+1〉n) and qm(〈v1, . . . ,vm+1〉n) are nonempty open subsets of sfnm(x). since (sf n m(x),sf n m(f)) is transitive, there exists k ∈ n such that (sfnm(f))k(qm(〈u1, . . . ,um+1〉n))∩qm(〈v1, . . . ,vm+1〉n) 6= ∅. hence, by proposition 3.1-(b), we obtain that: qm((fn(f))k(〈u1, . . . ,um+1〉n)) ∩qm(〈v1, . . . ,vm+1〉n) 6= ∅. let b ∈ (fn(f))k(〈u1, . . . ,um+1〉n) with qm(b) ∈ qm(〈v1, . . . ,vm+1〉n). we consider an element a ∈ 〈v1, . . . ,vm+1〉n such that qm(a) = qm(b). by remark 2.2, we have that a = b. let c ∈ 〈u1, . . . ,um+1〉n be such that (fn(f))k(c) = b. thus, (fn(f))k(c) = a. by proposition 3.1-(a), fk(c) = a. let c1 ∈ c ∩ u1 and let c2 ∈ c ∩ u2. hence, d(x1,x2) ≤ d(x1,c1) + d(c1,c2)+d(c2,x2) < r 2 +d(c1,c2). this implies that r 2 < d(c1,c2). on the other hand, fk(c1),f k(c2) ∈ fk(c) ⊆ ⋃m+1 i=1 vi ⊆ u1. thus, d(f k(c1),f k(c2)) ≤ r2 . in consequence, d(fk(c1),f k(c2)) < d(c1,c2), which is a contradiction to [8, remark 4.2]. therefore, we conclude that (sfnm(x),sf n m(f)) is not transitive. � the proof of the following result is obtained from theorem 3.2 and diagram 3. theorem 3.3. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. let n be one of the following classes of dynamical systems: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, and chaotic. if f is an isometry, then (sfnm(x),sf n m(f)) 6∈n. we recall that s1 = { e2πiθ ∈ c | θ ∈ [0, 1] } . example 3.4. let α ∈ r \ q and let r : s1 → s1 be the map defined by r(e2πiθ) = e2πi(θ+α), for each θ ∈ [0, 1]. note that r is an isometry. let n be c© agt, upv, 2020 appl. gen. topol. 21, no. 1 23 f. barragán, a. santiago-santos and j. f. tenorio one of the following classes of dynamical systems: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, and chaotic. by theorem 3.3, we obtain that (sfnm(s1),sf n m(r)) 6∈n . on the other hand, we have that the dynamical system (s1,r) is transitive, totally transitive, and strongly transitive (see [33, p. 261]). theorem 3.5. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. then the following are equivalent: (1) (x,f) is exact; (2) (fn(x),fn(f)) is exact; (3) (sfn(x),sfn(f)) is exact; (4) (sfnm(x),sf n m(f)) is exact. proof. by [8, theorem 4.7], we have that (1), (2) and (3) are equivalent. now, if (fn(x),fn(f)) is exact, then by, [8, theorem 3.4], we obtain that (sfnm(x),sf n m(f)) is exact. that is, (2) implies (4). therefore, for complete the proof it is enough to prove that (4) implies (1). suppose that (sfnm(x),sf n m(f)) is exact, we prove that (x,f) is exact. let u be a nonempty open subset of x. we see that fk(u) = x, for some k ∈ n. we take u1, . . . ,um+1 nonempty open subsets of x such that⋃m+1 i=1 ui ⊆ u and ui ∩uj = ∅, for each i,j ∈{1, . . . ,m + 1} with i 6= j. note that 〈u1,u2, . . . ,um+1〉n is a nonempty open subset of fn(x), and moreover 〈u1,u2, . . . ,um+1〉n ∩ fm(x) = ∅. hence, by remark 2.2, we obtain that qm(〈u1,u2, . . . ,um+1〉n) is a nonempty open subset of sfnm(x). note that fmx /∈ qm(〈u1,u2, . . . ,um+1〉n). thus, by the assumption, there exists k ∈ n such that: (sfnm(f)) k(qm(〈u1,u2, . . . ,um+1〉n)) = sfnm(x). in consequence, by part (b) from proposition 3.1, we have that: qm((fn(f))k(〈u1,u2, . . . ,um+1〉n)) = sfnm(x). let x ∈ x. we take y1,y2, . . . ,ym ∈ x \ {x} such that yi 6= yj, for each i,j ∈ {1, 2, . . . ,m} with i 6= j, and we define a = {x,y1, . . . ,ym}. note that a ∈ fn(x) \fm(x). thus, qm(a) 6= fmx . since qm(a) ∈ sf n m(x), there exists b ∈ (fn(f))k(〈u1,u2, . . . ,um+1〉n) such that qm(b) = qm(a). hence, by remark 2.2, we have that b = a. let c ∈ 〈u1,u2, . . . ,um+1〉n be such that (fn(f))k(c) = b. by proposition 3.1-(a), we deduce that fk(c) = b. since a = b and c ⊆ u, it follows that a ⊆ fk(u). hence, x ∈ fk(u). thus, x ⊆ fk(u). this implies that (x,f) is exact. � as a consequence from theorem 3.5 and diagram 3, we have the following result. corollary 3.6. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. if (x,f) is exact, then (sfnm(x),sf n m(f)) is mixing, weakly mixing, totally transitive and transitive. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 24 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) corollary 3.7. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. if f is an isometry, then (x,f) is not exact. proof. suppose that f is an isometry. if the dynamical system (x,f) is exact, then, by theorem 3.5, the dynamical system (sfnm(x),sf n m(f)) is exact. however, by theorem 3.3, we know that the dynamical system (sfnm(x),sf n m(f)) is not exact. therefore, the dynamical system (x,f) is not exact. � theorem 3.8. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. then the following are equivalent: (1) (x,f) is mixing; (2) (fn(x),fn(f)) is mixing; (3) (sfn(x),sfn(f)) is mixing; (4) (sfnm(x),sf n m(f)) is mixing. proof. from [8, theorem 4.9], it follows that (1), (2) and (3) are equivalent. now, if the system (fn(x),fn(f)) is mixing, then by [8, theorem 3.4], we have that the system (sfnm(x),sf n m(f)) is mixing. thus, (2) implies (4). finally, we prove that (4) implies (1). suppose that (sfnm(x),sf n m(f)) is mixing, we prove that (x,f) is mixing. for this end, let u and v be nonempty open subsets of x. we see that there exists n ∈ n such that fk(u) ∩ v 6= ∅, for every k ≥ n. we consider nonempty open subsets u1,u2, . . . ,um+1 and v1,v2, . . . ,vm+1 of x such that⋃m+1 i=1 ui ⊆ u, ⋃m+1 i=1 vi ⊆ v , ui∩uj = ∅ for each i,j ∈{1, 2, . . . ,m + 1} with i 6= j, and vi ∩vj = ∅ for each i,j ∈ {1, 2, . . . ,m + 1} with i 6= j. it follows that 〈u1,u2, . . . ,um+1〉n and 〈v1,v2, . . . ,vm+1〉n are nonempty open subset of fn(x) such that 〈u1,u2, . . . ,um+1〉n ∩fm(x) = ∅ and 〈v1,v2, . . . ,vm+1〉n ∩ fm(x) = ∅. hence, by remark 2.2, we have that: qm(〈u1,u2, . . . ,um+1〉n) and qm(〈v1,v2, . . . ,vm+1〉n) are open subsets of sfnm(x). note that fmx /∈ qm(〈u1,u2, . . . ,um+1〉n). additionally, fmx /∈ qm(〈v1,v2, . . . ,vm+1〉n). since (sf n m(x),sf n m(f)) is mixing, there exists n ∈ n such that for each k ≥ n: (sfnm(f)) k(qm(〈u1,u2, . . . ,um+1〉n)) ∩qm(〈v1,v2, . . . ,vm+1〉n) 6= ∅. fix k ≥ n and let χ ∈ qm(〈u1, . . . ,um+1〉n) satisfying (sfnm(f))k(χ) ∈ qm(〈v1, . . . ,vm+1〉n). let a ∈ 〈u1,u2, . . . ,um+1〉n such that qm(a) = χ and let b ∈ 〈v1,v2, . . . ,vm〉n such that (sfnm(f))k(χ) = qm(b). hence, we have that (sfnm(f))k(qm(a)) = qm(b). by part (b) from proposition 3.1, we obtain that qm((fn(f))k(a)) = qm(b). from remark 2.2, it follows that (fn(f))k(a) = b. again, by part (a) from proposition 3.1, we deduce that fk(a) = b. we take a ∈ a ∩ u1. this implies that fk(a) ∈ fk(a) ∩ fk(u). moreover, fk(a) ∈ b ∩fk(u). since b ⊆ v , we have that fk(u) ∩v 6= ∅. in consequence, (x,f) is mixing. � using theorem 3.8 and diagram 3, we deduce the following result. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 25 f. barragán, a. santiago-santos and j. f. tenorio corollary 3.9. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. if (x,f) is mixing, then (sfnm(x),sf n m(f)) is weakly mixing, totally transitive and transitive. the proof of the following result is similar to the proof of the corollary 3.7. corollary 3.10. let (x,f) be a dynamical system. if f is an isometry, then (x,f) is not mixing. theorem 3.11. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. consider the following statements: (1) (x,f) is transitive; (2) (fn(x),fn(f)) is transitive; (3) (sfn(x),sfn(f)) is transitive; (4) (sfnm(x),sf n m(f)) is transitive. then (2), (3) and (4) are equivalent, (2) implies (1), (3) implies (1), (4) implies (1), (1) does not imply (2), (1) does not imply (3), and (1) does not imply (4). proof. by [8, theorem 4.10], we have that (2) and (3) are equivalent, (2) implies (1), (3) implies (1), (1) does not imply (2) and (1) does not imply (3). now, if (fn(x),fn(f)) is transitive, then by [8, theorem 3.4], (sfnm(x),sf n m(f)) is transitive. hence, we have that (2) implies (4). finally, suppose that (sfnm(x),sf n m(f)) is transitive. we prove that the system (sfn(x),sfn(f)) is transitive. let γ and λ be nonempty open subsets of sfn(x). since q−1(γ) and q−1(λ) are nonempty open subsets of fn(x), then by [19, lemma 4.2], there exist nonempty open subsets u1,u2, . . . ,un and v1,v2, . . . ,vn of x such that: 〈u1,u2, . . . ,un〉n ⊆ q−1(γ) and 〈v1,v2, . . . ,vn〉n ⊆ q−1(λ). we take, for each i ∈{1, 2, . . . ,n}, a nonempty open subset wi of x such that wi ⊆ ui and for each i,j ∈ {1, 2, . . . ,n}, wi ∩ wj = ∅ with i 6= j. also, for each i ∈ {1, 2, . . . ,n}, let oi be a nonempty open subset of x such that oi ⊆ vi and for each i,j ∈{1, 2, . . . ,n}, oi∩oj = ∅ with i 6= j. it follows that 〈u1,u2, . . . ,un〉n and 〈v1,v2, . . . ,vn〉n are nonempty open subsets of fn(x) such that: 〈w1,w2, . . . ,wn〉n ⊆〈u1,u2, . . . ,un〉n ⊆ q−1(γ) and 〈o1,o2, . . . ,on〉n ⊆〈v1,v2, . . . ,vn〉n ⊆ q−1(λ). moreover, 〈w1,w2, . . . ,wn〉n∩fm(x) = ∅ and 〈o1,o2, . . . ,on〉n∩fm(x) = ∅. hence, by remark 2.2, we have that: qm(〈w1,w2, . . . ,wn〉n) and qm(〈o1,o2, . . . ,on〉n) are nonempty open subsets of sfnm(x). note that fmx /∈ qm(〈w1, . . . ,wn〉n) and fmx /∈ qm(〈o1, . . . ,on〉n). because (sf n m(x),sf n m(f)) is transitive, there exists k ∈ n such that: (sfnm(f)) k(qm(〈w1,w2, . . . ,wn〉n)) ∩qm(〈o1,o2, . . . ,on〉n) 6= ∅. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 26 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) as a consequence of proposition 3.1-(d), it follows that: qm((fn(f))k(〈w1,w2, . . . ,wn〉n)) ∩qm(〈o1,o2, . . . ,on〉n) 6= ∅. let b ∈ (fn(f))k(〈w1,w2, . . . ,wn〉n) with qm(b) ∈ qm(〈o1,o2, . . . ,on〉n). hence, we consider a ∈ 〈o1,o2, . . . ,on〉n such that qm(a) = qm(b). by remark 2.2, we obtain that a = b. thus, it follows that: (fn(f))k(〈w1,w2, . . . ,wn〉n) ∩〈o1,o2, . . . ,on〉n 6= ∅. hence, there is an element c ∈ 〈w1,w2, . . . ,wn〉n such that (fn(f))k(c) ∈ 〈o1,o2, . . . ,on〉n. then, q(c) ∈ q(〈w1,w2, . . . ,wn〉n) and q((fn(f))k(c)) ∈ q(〈o1,o2, . . . ,on〉n). moreover, since q ◦fn(f) = sfn(f) ◦ q, we obtain that (sfn(f))k(q(c))) ∈ q(〈o1,o2, . . . ,on〉n). also, observe that: q(〈w1, . . . ,wn〉n) ⊆ q(q−1(γ)) ⊆ γ and q(〈o1,o2, . . . ,on〉n) ⊆ q(q−1(λ)) ⊆ λ. hence, (sfn(f))k(γ) ∩ λ 6= ∅. in consequence, (sfn(x),sfn(f)) is transitive. since (2) and (4) are equivalent, we obtain that (4) implies (1). by example 3.4, we note that (1) does not imply (4). � as a consequence of diagram 3 and theorem 3.11, we have the next result: corollary 3.12. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. if (sfnm(x),sf n m(f)) is strongly transitive, then the system (sfn(x),sfn(f)) is transitive. theorem 3.13. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. then the following are equivalent: (1) (x,f) is weakly mixing; (2) (fn(x),fn(f)) is weakly mixing; (3) (fn(x),fn(f)) is transitive; (4) (sfn(x),sfn(f)) is weakly mixing; (5) (sfn(x),sfn(f)) is transitive; (6) (sfnm(x),sf n m(f)) is weakly mixing; (7) (sfnm(x),sf n m(f)) is transitive. proof. by [8, theorem 4.11], we have that (1), (2), (3), (4) and (5) are equivalent. on the other hand, by theorem 3.11, we have that (5) and (7) are equivalent. it follows from diagram 3 that (6) implies (7). now, if (fn(x),fn(f)) is weakly mixing, then by [8, theorem 3.4], (sfnm(x),sf n m(f)) is weakly mixing. hence, we have that (2) implies (6). thus, (7) implies (6). therefore, (6) and (7) are equivalent. � the proof of the corollary 3.14 is similar to the proof of the corollary 3.7. corollary 3.14. let (x,f) be a dynamical system. if f is an isometry, then (x,f) is not weakly mixing. moreover, by corollary 3.12 and theorem 3.13, we obtain: c© agt, upv, 2020 appl. gen. topol. 21, no. 1 27 f. barragán, a. santiago-santos and j. f. tenorio corollary 3.15. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. if (sfnm(x),sf n m(f)) is strongly transitive, then the system (sfnm(x),sf n m(f)) is weakly mixing. theorem 3.16. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. consider the following statements: (1) (x,f) is totally transitive; (2) (fn(x),fn(f)) is totally transitive; (3) (sfn(x),sfn(f)) is totally transitive; (4) (sfnm(x),sf n m(f)) is totally transitive. then (2), (3) and (4) are equivalent, (2) implies (1), (3) implies (1), (4) implies (1), (1) does not imply (2), (1) does not imply (3) and (1) does not imply (4). proof. by [8, theorem 4.12], we have that (2) and (3) are equivalent, (3) implies (1), (2) implies (1), (1) does not imply (2) and (1) does not imply (3). now, if the system (fn(x),fn(f)) is totally transitive, then, by [8, theorem 3.4], we have that the system (sfnm(x),sf n m(f)) is totally transitive. that is, (2) implies (4). in consequence (3) implies (4). now, we prove that (4) implies (3). suppose that (sfnm(x),sf n m(f)) is totally transitive, we prove that (sfn(x),sfn(f)) is totally transitive. for this end, let s ∈ n. we see that (sfn(x), (sfn(f))s) is transitive. let γ and λ be nonempty open subsets of sfn(x). since q is continuous, q−1(γ) and q−1(λ) are nonempty open subsets of fn(x). applying [19, lemma 4.2], we can take nonempty open subsets u1,u2, . . . ,un and v1,v2, . . . ,vn of x such that 〈u1,u2, . . . ,un〉n ⊆ q−1(γ) and 〈v1,v2, . . . ,vn〉n ⊆ q−1(λ). hence, for every i ∈ {1, 2, . . . ,n}, we consider a nonempty open subset wi of x such that wi ⊆ ui and for each i,j ∈ {1, 2, . . . ,n}, wi ∩ wj = ∅, when i 6= j. moreover, for every i ∈ {1, 2, . . . ,n}, let oi be a nonempty open subset of x such that oi ⊆ vi and for each i,j ∈ {1, 2, . . . ,n}, oi ∩ oj = ∅, when i 6= j. observe that 〈u1,u2, . . . ,un〉n and 〈v1,v2, . . . ,vn〉n are nonempty open subsets of fn(x) with 〈w1,w2, . . . ,wn〉n ⊆ 〈u1,u2, . . . ,un〉n ⊆ q−1(γ) and 〈o1,o2, . . . ,on〉n ⊆ 〈v1,v2, . . . ,vn〉n ⊆ q−1(λ). moreover, 〈w1, . . . ,wn〉n ∩ fm(x) = ∅ and 〈o1,o2, . . . ,on〉n ∩ fm(x) = ∅. hence, by remark 2.2, we have that qm(〈w1,w2, . . . ,wn〉n) and qm(〈o1,o2, . . . ,on〉n) are nonempty open subsets of sfnm(x). note that: fmx /∈ qm(〈w1, . . . ,wn〉n) and f m x /∈ qm(〈o1,o2, . . . ,on〉n). since (sfnm(x),sf n m(f)) is totally transitive, (sf n m(x), (sf n m(f)) s) is transitive. it follows that there exists k ∈ n such that: ((sfnm(f)) s)k(qm(〈w1,w2, . . . ,wn〉n)) ∩qm(〈o1,o2, . . . ,on〉n) 6= ∅. using proposition 3.1-(d), we obtain that: qm(((fn(f))s)k(〈w1,w2, . . . ,wn〉n)) ∩qm(〈o1,o2, . . . ,on〉n) 6= ∅. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 28 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) by remark 2.2, we have that: (fn(f))s)k(〈w1,w2, . . . ,wn〉n) ∩〈o1,o2, . . . ,on〉n 6= ∅. in consequence, there exists c ∈ 〈w1,w2, . . . ,wn〉n such that (fn(f))s)k(c) ∈ 〈o1,o2, . . . ,on〉n. then q(c) ∈ q(〈w1,w2, . . . ,wn〉n) and q((fn(f))s)k(c)) ∈ q(〈o1,o2, . . . ,on〉n). since q ◦fn(f) = sfn(f) ◦q, we obtain that: ((sfn(f))s)k(q(c))) ∈ q(〈o1,o2, . . . ,on〉n). moreover, we note that q(〈w1,w2, . . . ,wn〉n) ⊆ γ and q(〈o1,o2, . . . ,on〉n) ⊆ λ. hence, q(c) ∈ γ and ((sfn(f))s)k(q(c))) ∈ λ. thus, it follows that ((sfn(f))s)k(γ) ∩ λ 6= ∅. therefore, (3) and (4) are equivalent. in consequence, (4) implies (2), (4) implies (1), and (1) does not imply (4). � theorem 3.17. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. consider the following statements: (1) (x,f) is strongly transitive; (2) (fn(x),fn(f)) is strongly transitive; (3) (sfn(x),sfn(f)) is strongly transitive; (4) (sfnm(x),sf n m(f)) is strongly transitive. then (2) implies (1), (2) implies (3), (2) implies (4), (3) implies (1), (4) implies (1), (1) does not imply (2), (1) does not imply (3), and (1) does not imply (4). proof. by [8, theorem 4.13], we have that (2) implies (1), (2) implies (3), (3) implies (1), (1) does not imply (2) and (1) does not imply (3). on the other hand, if (fn(x),fn(f)) is strongly transitive, then, by [8, theorem 3.4], we have that (sfnm(x),sf n m(f)) is strongly transitive. hence, (2) implies (4). also, by example 3.4, we have that (1) does not implies (4). we prove that (4) implies (1). suppose that (sfnm(x),sf n m(f)) is strongly transitive. let u be a nonempty open subset of x. let u1, . . . ,un be nonempty open subsets of x such that ⋃n i=1 ui ⊆ u and ui ∩ uj = ∅ for each i,j ∈ {1, . . . ,n} with i 6= j. it follows that 〈u1, . . . ,un〉n is a nonempty open subset of fn(x) such that 〈u1, . . . ,un〉n ∩fm(x) = ∅. using remark 2.2, we obtain that qm(〈u1, . . . ,un〉n) is a nonempty open subset of sfnm(x). note that fmx /∈ qm(〈u1, . . . ,un〉n). considering that (sf n m(x),sf n m(f)) is strongly transitive, we have that sfnm(x) = ⋃s k=0(sf n m(f)) k(qm(〈u1, . . . ,un〉n)), for some s ∈ n. as a consequence from proposition 3.1-(b), it follows that: sfnm(x) = s⋃ k=0 qm((fn(f))k(〈u1, . . . ,un〉n)). finally, we see that x = ⋃s k=0 f k(u). let x ∈ x. we take y1, . . . ,ym ∈ x\{x} such that yi 6= yj for each i,j ∈{1, . . . ,m} with i 6= j. let a = {x,y1, . . . ,ym}. we have that a ∈ fn(x) \ fm(x). in consequence, qm(a) ∈ sfnm(x) \ {fmx }. this implies that there exists j ∈ {0, 1, . . . ,s} such that qm(a) ∈ qm((fn(f))j(〈u1, . . .un〉n)). hence, there exists b ∈ (fn(f))j(〈u1, . . . ,un〉n) such that qm(b) = qm(a). note that, by remark 2.2, a = b. observe that c© agt, upv, 2020 appl. gen. topol. 21, no. 1 29 f. barragán, a. santiago-santos and j. f. tenorio there exists c ∈ 〈u1, . . . ,un〉n such that (fn(f))j(c) = b. thus, by proposition 3.1-(a), fj(c) = b. moreover, since c ⊆ u, it follows that fj(c) ⊆ fj(u). then, a ⊆ fj(u). in consequence, x ∈ fj(u). thus, x ⊆ ⋃s k=0 f k(u). hence, (x,f) is strongly transitive. � we have the following questions (compare with [8, question 4.1]). questions 3.18. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. (i) if (sfnm(x),sf n m(f)) is strongly transitive, then is (fn(x),fn(f)) strongly transitive? (ii) if (sfnm(x),sf n m(f)) is strongly transitive, then is (sfn(x),sfn(f)) strongly transitive? (iii) if (sfn(x),sfn(f)) is strongly transitive, then is (sfnm(x),sf n m(f)) strongly transitive? in order to prove the theorem 3.20, we have the next result. lemma 3.19. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. suppose that f is a surjective map. then the following are equivalent: (1) per(f) is dense in x; (2) per(fn(f)) is dense in fn(x); (3) per(sfn(f)) is dense in sfn(x); (4) per(sfnm(f)) is dense in sf n m(x). proof. by [8, theorem 4.16], we have that (1), (2) and (3) are equivalent. now, by [8, lemma 3.3], we have that (2) implies (4). therefore, for complete the proof it is enough to prove that (4) implies (2). suppose that per(sfnm(f)) is dense in sf n m(x), we prove that per(fn(f)) is dense in fn(x). for this end, let u be a nonempty open subset of fn(x). by [19, lemma 4.2], there exist nonempty open subsets u1,u2, . . . ,un of x such that 〈u1,u2, . . . ,un〉n ⊆ u. for each i ∈ {1, 2, . . . ,n}, let wi be a nonempty open subset of x such that wi ⊆ ui and for each i,j ∈{1, 2, . . . ,n}, wi ∩ wj 6= ∅, if i 6= j. it follows that 〈w1,w2, . . . ,wn〉n is a nonempty open subset of fn(x) such that 〈w1,w2, . . . ,wn〉n ⊆ 〈u1,u2, . . . ,un〉n ⊆ u and 〈w1,w2, . . . ,wn〉n ∩ fm(x) = ∅. hence, by remark 2.2, we have that qm(〈w1,w2, . . . ,wn〉n) is a nonempty open subset of sfnm(x). observe that fmx /∈ qm(〈w1,w2, . . . ,wn〉n). thus, by hypothesis, we obtain that qm(〈w1,w2, . . . ,wn〉n) ∩ per(sfnm(f)) 6= ∅. in consequence, there exist a ∈ 〈w1,w2, . . . ,wn〉n and k ∈ n such that (sfnm(f))k(qm(a)) = qm(a). this implies, by proposition 3.1-(b) that qm((fn(f))k(a)) = qm(a). furthermore, by remark 2.2, we have that (fn(f))k(a) = a. therefore, there exist a ∈ u and k ∈ n such that (fn(f))k(a) = a. hence, per(fn(f)) is dense in fn(x). � theorem 3.20. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. then the next propositions are equivalent: c© agt, upv, 2020 appl. gen. topol. 21, no. 1 30 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) (1) (x,f) is weakly mixing and chaotic; (2) (fn(x),fn(f)) is chaotic; (3) (sfn(x),sfn(f)) is chaotic; (4) (sfnm(x),sf n m(f)) is chaotic. proof. by [8, theorem 4.17], we have that (1), (2) and (3) are equivalent. now, if (fn(x),fn(f)) is chaotic, then, by [8, theorem 3.4], we have that (sfnm(x),sf n m(f)) is chaotic. thus, (2) implies (4). as a consequence of lemma 3.19 and theorem 3.11, we conclude that (4) implies (2). � arguing as in corollary 3.7, we obtain the following result. corollary 3.21. let (x,f) be a dynamical system. if f is an isometry, then (x,f) is not chaotic or (x,f) is not weakly mixing. 4. other dynamical properties of (sfnm(x),sf n m(f)) in this section we study irreducible, feebly open and turbulent dynamical systems. theorem 4.1. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. consider the following statements: (1) (x,f) is irreducible; (2) (fn(x),fn(f)) is irreducible; (3) (sfn(x),sfn(f)) is irreducible; (4) (sfnm(x),sf n m(f)) is irreducible. then (2) implies (1), (3) implies (1) and (4) implies (1). proof. by [8, theorem 5.1], we obtain that (2) implies (1) and (3) implies (1). therefore, it is enough to prove that (4) implies (1). suppose that (sfnm(x),sf n m(f)) is irreducible and we prove that (x,f) is irreducible. we take a nonempty closed subset a of x with f(a) = x. we see that a = x. note that 〈a〉n is a nonempty closed subset of fn(x) such that fn(f)(〈a〉n) = fn(x). thus, qm(fn(f)(〈a〉n)) = sfnm(x). hence, by proposition 3.1-(b), we have that sfn(f)(qm(〈a〉n)) = sfnm(x). since qm(〈a〉n) is a nonempty closed subset of sfnm(x) and (sf n m(x),sf n m(f)) is irreducible, we have that qm(〈a〉n) = sfnm(x). now, let x ∈ x and we consider y1, . . . ,ym ∈ x \ {x} such that yi 6= yj for each i,j ∈ {1, . . . ,m} with i 6= j. let b = {x,y1, . . . ,ym}. clearly, b ∈ fn(x) \fm(x). then, qm(b) ∈ sfnm(x) \ {fmx }. considering that qm(b) ∈ sf n m(x), there exists an element c ∈ 〈a〉n with qm(c) = qm(b). using remark 2.2, we obtain that c = b. thus, x ∈ a. this implies that x = a. therefore, (x,f) is irreducible. � questions 4.2. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. (i) if (x,f) is irreducible, then is (sfnm(x),sf n m(f)) irreducible? (ii) if (fn(x),fn(f)) is irreducible, then is (sfnm(x),sf n m(f)) irreducible? c© agt, upv, 2020 appl. gen. topol. 21, no. 1 31 f. barragán, a. santiago-santos and j. f. tenorio (iii) if (sfn(x),sfn(f)) is irreducible, then is (sfnm(x),sf n m(f)) irreducible? theorem 4.3. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1 with f a surjective map. then the following propositions are equivalent: (1) (x,f) is feebly open; (2) (fn(x),fn(f)) is feebly open; (3) (sfn(x),sfn(f)) is feebly open; (4) (sfnm(x),sf n m(f)) is feebly open. proof. by [7, theorem 10.1], we deduce that (1), (2) and (3) are equivalent. now, by theorem [7, theorem 3.3], it follows that (2) and (4) are equivalent. � statement (1) in corollary 4.4 is a consequence of diagram 3 and theorem 4.3. also, statement (2) in corollary 4.4 is a direct consequence of diagram 3. corollary 4.4. let (x,f) a dynamical system and n and m be integers such that n > m ≥ 1. then the following propositions hold: (1) if (x,f) is irreducible, then (sfnm(x),sf n m(f)) is feebly open. (2) if (sfnm(x),sf n m(f)) is irreducible, then (sf n m(x),sf n m(f)) is feebly open. theorem 4.5. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1, where f is a surjective map. consider the following statements: (1) (x,f) is turbulent; (2) (fn(x),fn(f)) is turbulent; (3) (sfn(x),sfn(f)) is turbulent; (4) (sfnm(x),sf n m(f)) is turbulent. then (1) implies (2), (3) and (4). proof. by [8, theorem 5.6], we have that (1) implies (2) and (3). now, suppose that (x,f) is turbulent. we see that (sfnm(x),sf n m(f)) is turbulent. let k and c be nondegenerate compact subsets of x such that k ∩c has at most one point and k ∪c ⊆ f(k) ∩f(c). observe that 〈k〉n and 〈c〉n are nondegenerate compact subsets of fn(x). let λ = qm(〈k〉n) and γ = qm(〈c〉n). this implies that λ and γ are nondegenerate compact subsets of sfnm(x). next, we see that λ ∩ γ has at most one point. we have two cases: case (1): k ∩ c = ∅. in this case, it follows that 〈k〉n ∩ 〈c〉n = ∅. moreover, since fm(k) ⊆〈k〉n and fm(c) ⊆〈c〉n, we see that fmx ∈ λ ∩ γ. case (2): k ∩ c = {a}. in this case, we have that 〈k〉n ∩〈c〉n = {{a}}. thus, fmx ∈ λ ∩ γ. now, we suppose that χ ∈ (λ ∩ γ) \ {f m x }. then, there exist a ∈ 〈k〉n\fm(x) and b ∈ 〈c〉n\fm(x) such that qm(a) = χ = qm(b). using remark 2.2, we obtain that a = b. hence, a ⊆ k∩c. thus, k∩c has at least two elements, which is a contradiction. therefore, λ ∩ γ has at most one point. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 32 dynamic properties of the dynamical system (sfnm(x),sf n m(f)) we prove that λ∪γ ⊆sfnm(f)(λ)∩sf n m(f)(γ). for this end, we consider χ ∈ λ ∪ γ. it follows that, there exists a ∈ 〈k〉n ∪〈c〉n such that qm(a) = χ. this implies that a ⊆ f(k) ∩ f(c). hence, a ∈ 〈f(k) ∩ f(c)〉n. in consequence, qm(a) ∈ qm(〈f(k)〉n) ∩qm(〈f(c)〉n). since qm(a) = χ, we have that χ ∈ qm(fn(f)(〈k〉n)) ∩ qm(fn(f)(〈c〉n)). by part (b) from proposition 3.1, we obtain that χ ∈ sfnm(f)(qm(〈k〉n)) ∩sf n m(f)(qm(〈c〉n)). thus, χ ∈ sfnm(f)(λ)∩sf n m(f)(γ). then, λ∪γ ⊆sf n m(f)(λ)∩sf n m(f)(γ). therefore, (sfnm(x),sf n m(f)) is turbulent. � finally, we have the following questions (compare with [8, questions 5.7]). questions 4.6. let (x,f) be a dynamical system and let n,m ∈ n be such that n > m ≥ 1. 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[34] y. wang and g. wei, characterizing mixing, weak mixing and transitivity on induced hyperspace dynamical systems, topology appl. 155, no. 1 (2007), 56–68. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 34 () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 179-191 weak partial metric spaces and some fixed point results i. altun and g. durmaz abstract the concept of partial metric p on a nonempty set x was introduced by matthews [8]. one of the most interesting properties of a partial metric is that p(x, x) may not be zero for x ∈ x. also, each partial metric p on a nonempty set x generates a t0 topology on x. by omitting the small self-distance axiom of partial metric, heckmann [7] defined the weak partial metric space. in the present paper, we give some fixed point results on weak partial metric spaces. 2010 msc: 54h25, 47h10. keywords: fixed point, partial metric space, weak partial metric space. 1. introduction the notion of partial metric space was introduced by matthews [8] as a part of the study of denotational semantics of data flow networks. it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation. in a partial metric spaces, the distance of a point in the self may not be zero. after the definition of partial metric space, matthews proved a partial metric version of banach’s fixed point theorem. then, valero [11], oltra and valero [9] and altun et al [1], [3] gave some generalizations of the result of matthews. recently, romaguera [10] proved the caristi type fixed point theorem on this space. first, we recall some definitions of partial metric space and some properties of theirs. see [2, 7, 8, 9, 10, 11] for details. 180 i. altun and g. durmaz a partial metric on a nonempty set x is a function p : x × x → r+ (nonnegative reals) such that for all x, y, z ∈ x : (p1) x = y ⇐⇒ p(x, x) = p(x, y) = p(y, y) (t0-separation axiom), (p2) p(x, x) ≤ p(x, y) (small self-distance axiom), (p3) p(x, y) = p(y, x) (symmetry), (p4) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z) (modified triangular inequality). a partial metric space (for short pms) is a pair (x, p) such that x is a nonempty set and p is a partial metric on x. it is clear that, if p(x, y) = 0, then, from (p1) and (p2), x = y. but if x = y, p(x, y) may not be 0. a basic example of a pms is the pair (r+, p), where p(x, y) = max{x, y} for all x, y ∈ r+. for another example, let i denote the set of all intervals [a, b] for any real numbers a ≤ b. let p : i × i → r+ be the function such that p([a, b], [c, d]) = max{b, d} − min{a, c}. then (i, p) is a pms. other examples of pms which are interesting from a computational point of view may be found in [5], [8]. each partial metric p on x generates a t0 topology τp on x which has as a base the family open p-balls {bp(x, ε) : x ∈ x, ε > 0}, where bp(x, ε) = {y ∈ x : p(x, y) < p(x, x) + ε} for all x ∈ x and ε > 0. it is easy to see that, a sequence {xn} in a pms (x, p) converges with respect to τp to a point x ∈ x if and only if p(x, x) = limn→∞ p(x, xn). if p is a partial metric on x, then the functions dp, dw : x × x → r + given by (1.1) dp(x, y) = 2p(x, y) − p(x, x) − p(y, y) and dw(x, y) = max{p(x, y) − p(x, x), p(x, y) − p(y, y)}(1.2) = p(x, y) − min{p(x, x), p(y, y)} are ordinary metrics on x. remark 1.1. let {xn} be a sequence in a pms (x, p) and x ∈ x, then lim n→∞ dw(xn, x) = 0 if and only if p(x, x) = lim n→∞ p(xn, x) = lim n,m→∞ p(xn, xm). proposition 1.2. let (x, p) be a pms, then dp and dw are equivalent metrics on x. weak partial metric spaces 181 proof. we obtain dp(x, y) = 2p(x, y) − p(x, x) − p(y, y) = p(x, y) − p(x, x) + p(x, y) − p(y, y) ≤ 2dw(x, y).(1.3) again we obtain dw(x, y) = p(x, y) − min{p(x, x), p(y, y)} ≤ p(x, y) − min{p(x, x), p(y, y)} +p(x, y) − max{p(x, x), p(y, y)} = 2p(x, y) − p(x, x) − p(y, y) = dp(x, y).(1.4) from (1.3) and (1.4) we have 1 2 dp(x, y) ≤ dw(x, y) ≤ dp(x, y). � definition 1.3. (i) a sequence {xn} in a pms (x, p) is called a cauchy sequence if there exists (and is finite) limn,m→∞ p(xn, xm). (ii) a pms (x, p) is said to be complete if every cauchy sequence {xn} in x converges, with respect to τp, to a point x ∈ x such that p(x, x) = limn,m→∞ p(xn, xm). the following lemma plays an important role to give fixed point results on a pms. lemma 1.4 ([8], [9]). let (x, p) be a pms. (a) {xn} is a cauchy sequence in (x, p) if and only if it is a cauchy sequence in the metric space (x, dw). (b) (x, p) is complete if and only if (x, dw) is complete. remark 1.5. since dp and dw are equivalent, we can take dp instead of dw in lemma 1.4. 2. weak partial metric heckmann [7] introduced the concept of weak partial metric space (for short wpms), which is a generalized version of matthews’ partial metric space by omitting the small self-distance axiom. that is, the function p : x × x → r+ is called weak partial metric on x if the conditions (p1),(p3) and (p4) are satisfied. also, heckmann shows that, if p is a weak partial metric on x, then for all x, y ∈ x, we have the following weak small self-distance property p(x, y) ≥ p(x, x) + p(y, y) 2 . weak small self-distance property shows that wpms are not far from small-self distance axiom. it is clear that every pms is a wpms, but the converse may 182 i. altun and g. durmaz not be true. a basic example of a wpms but not a pms is the pair (r+, p), where p(x, y) = x+y 2 for all x, y ∈ r+. for another example, let i denote the set of all intervals [a, b] for any real numbers a ≤ b. let p : i × i → r+ be the function such that p([a, b], [c, d]) = b+d−a−c 2 . then (i, p) is a wpms but not a pms. remark 2.1. if (x, p) be a wpms, but not a pms, then the function dp as in (1.1) may not be an ordinary metric on x. for example, let x = r+ and let p : x × x → r+ defined by p(x, y) = x+y 2 . then it is clear that dp(x, y) = 0 for all x, y ∈ x, so dp is not a metric on x. note that, in this case dw(x, y) = 1 2 |x − y|. proposition 2.2. let a, b, c ∈ r+, then we have min{a, c} + min{b, c} ≤ min{a, b} + c. proposition 2.3. let (x, p) be a wpms, then dw : x × x → r defined as in (1.2) is an ordinary metric on x. proof. since p is a weak partial metric, then we have 2p(x, y) ≥ p(x, x) + p(y, y) ≥ 2 min{p(x, x), p(y, y)}. therefore p(x, y)−min{p(x, x), p(y, y)} ≥ 0. again it is clear that, dw(x, y) = 0 if and only if x = y and dw(x, y) = dw(y, x) for all x, y ∈ x. now, let x, y, z ∈ x, then from proposition 2.2, we have dw(x, z) = p(x, z) − min{p(x, x), p(z, z)} ≤ p(x, y) + p(y, z) − p(y, y) − min{p(x, x), p(z, z)} ≤ p(x, y) − min{p(x, x), p(y, y)} +p(y, z) − min{p(y, y), p(z, z)} = dw(x, y) + dw(y, z). � in a wpms, the convergence of a sequence, cauchy sequence, completeness and continuity of a function are defined as pms. to give some fixed point results on a wpms, we need to prove lemma 1.4 by omitting the small-self distance axiom. lemma 2.4. let (x, p) be a wpms. (a) {xn} is a cauchy sequence in (x, p) if and only if it is a cauchy sequence in the metric space (x, dw). (b) (x, p) is complete if and only if (x, dw) is complete. proof. first we show that every cauchy sequence in (x, p) is a cauchy sequence in (x, dw). let {xn} be a cauchy sequence in (x, p), then there exists a ∈ r weak partial metric spaces 183 such that, given ε > 0, there is n0 ∈ n with |p(xn, xm) − a| < ε 2 for all n, m ≥ n0. hence dw(xn, xm) = p(xn, xm) − min{p(xn, xn), p(xm, xm)} = p(xn, xm) − a + a − min{p(xn, xn), p(xm, xm)} ≤ |p(xn, xm) − a| + |a − min{p(xn, xn), p(xm, xm)}| < ε 2 + ε 2 = ε for all n, m ≥ n0. therefore {xn} is a cauchy sequence in (x, dw). next we prove that completeness of (x, dw) implies completeness of (x, p). indeed, if {xn} is a cauchy sequence in (x, p), then it is also a cauchy sequence in (x, dw). since (x, dw) is complete we deduce that there exists x ∈ x such that limn→∞ dw(xn, x) = 0. now we show that limn,m→∞ p(xn, xm) = p(x, x). since {xn} is a cauchy sequence in (x, p) it is sufficient to show that limn→∞ p(xn, xn) = p(x, x). let ε > 0, then there exists n0 ∈ n such that dw(xn, x) < ε 2 for all n ≥ n0. thus |p(xn, xn) − p(x, x)| = max{p(xn, xn), p(x, x)} − min{p(xn, xn), p(x, x)} = 2 { max{p(xn, xn), p(x, x)} + min{p(xn, xn), p(x, x)} 2 − min{p(xn, xn), p(x, x)} } = 2[ p(xn, xn) + p(x, x) 2 − min{p(xn, xn), p(x, x)}] ≤ 2[p(xn, x) − min{p(xn, xn), p(x, x)}] = 2dw(xn, x) < ε whenever n ≥ n0. this shows that (x, p) is complete. now we prove that every cauchy sequence {xn} in (x, dw) is a cauchy sequence in (x, p). let ε = 1 2 . then there exists n0 ∈ n such that dw(xn, xm) < 1 2 for all m, n ≥ n0. therefore we have p(xn, xn) = p(xn, xn) − p(xn0, xn0) + p(xn0, xn0) ≤ |p(xn, xn) − p(xn0 , xn0)| + p(xn0, xn0) ≤ 2dw(xn, xn0) + p(xn0 , xn0) < 1 + p(xn0, xn0). consequently the sequence {p(xn, xn)} is bounded in r and so there exists a ∈ r such that a subsequence {p(xnk, xnk )} is convergent to a. on the other hand, since {xn} is a cauchy sequence in (x, dw), given ε > 0 there exists nε ∈ n such that dw(xn, xm) < ε 2 for all m, n ≥ nε. thus we have |p(xn, xn) − p(xm, xm)| ≤ 2dw(xn, xm) < ε. that is, the sequence {p(xn, xn)} is cauchy in r. therefore lim n→∞ p(xn, xn) = a. 184 i. altun and g. durmaz on the other hand, since |p(xn, xm) − a| ≤ |p(xn, xm) − min{p(xn, xn), p(xm, xm)}| + |min{p(xn, xn), p(xm, xm)} − a| = dw(xn, xm) + |min{p(xn, xn), p(xm, xm)} − a| , we have limn,m→∞ p(xn, xm) = a and so {xn} is a cauchy sequence in (x, p). now we prove that completeness of (x, p) implies completeness of (x, dw). indeed, if {xn} is a cauchy sequence in (x, dw), then it is also a cauchy sequence in (x, p). since (x, p) is complete we deduce that there exists x ∈ x such that limn,m→∞ p(xn, xm) = limn→∞ p(xn, x) = p(x, x). then, given ε > 0, there exists nε ∈ n such that max{|p(xn, x) − p(xn, xn)| , |p(xn, x) − p(x, x)|} < ε whenever n ≥ nε. as a consequence we have dw(xn, x) = p(xn, x) − min{p(xn, xn), p(x, x)} = |p(xn, x) − min{p(xn, xn), p(x, x)}| < ε whenever n ≥ nε. therefore (x, dw) is complete. � remark 2.5. remark 1.1 is still true for wpms. 3. fixed point results in this section we give some fixed point results on weak partial metric spaces. we begin by giving hardy and rogers type [6] fixed point theorem. theorem 3.1. let (x, p) be a complete wpms and let f : x → x be a map such that p(fx, fy) ≤ ap(x, y) + bp(x, fx) + cp(y, fy) + dp(x, fy) + ep(y, fx)(3.1) for all x, y ∈ x, where a, b, c, d, e ≥ 0 and, if d ≥ e, then a + b + c + 2d < 1, if d < e, then a + b + c + 2e < 1. then f has a unique fixed point. proof. let x0 ∈ x be an arbitrary point. define a sequence {xn} in x by xn = fxn−1 for n = 1, 2, · · · . now if xn0 = xn0+1 for some n0 = 0, 1, 2, · · · , then it is clear that xn0 is a fixed point of f . now assume xn 6= xn+1 for all n. then we have from (3.1) p(xn+1, xn) = p(fxn, fxn−1) ≤ ap(xn, xn−1) + bp(xn, fxn) + cp(xn−1, fxn−1) + dp(xn, fxn−1) + ep(xn−1, fxn) = ap(xn, xn−1) + bp(xn, xn+1) + cp(xn−1, xn) + dp(xn, xn) + ep(xn−1, xn+1) ≤ (a + c + e)p(xn, xn−1) + (b + e)p(xn, xn+1) + (d − e)p(xn, xn).(3.2) weak partial metric spaces 185 now if d ≥ e, then adding the term (d−e)p(xn+1, xn+1) or (d−e)p(xn−1, xn−1) in the right side of (3.2) and using weak small self distance axiom, we have (3.3) p(xn+1, xn) ≤ max{ a + c + e 1 − b − 2d + e , a + c + 2d − e 1 − b − e }p(xn, xn−1) for all n. if d < e, then from (3.2) by omitting the term (d − e)p(xn, xn), we have (3.4) p(xn+1, xn) ≤ a + c + e 1 − b − e p(xn, xn−1). hence from (3.3) and (3.4) we have for n = 1, 2, · · · p(xn+1, xn) ≤ λ np(x1, x0), where λ =    max{ a+c+e 1−b−2d+e , a+c+2d−e 1−b−e } , d ≥ e a+c+e 1−b−e , d < e . it is clear that λ ∈ [0, 1), thus we have (3.5) lim n→∞ p(xn+1, xn) = 0. on the other hand, since dw(xn+1, xn) = p(xn+1, xn) − min{p(xn, xn), p(xn+1, xn+1)} ≤ p(xn+1, xn) ≤ λnp(x1, x0) we have limn→∞ dw(xn, xn+1) = 0. therefore we have for k = 1, 2, · · · dw(xn+k, xn) ≤ dw(xn+k, xn+k−1) + · · · + dw(xn+1, xn) ≤ λn+k−1p(x1, x0) + · · · + λ np(x1, x0) = [λn+k−1 + · · · + λn]p(x1, x0) ≤ λn 1 − λ p(x1, x0). this shows that {xn} is a cauchy sequence in the metric space (x, dw). since (x, p) is complete then from lemma 2.4, the sequence {xn} converges in the metric space (x, dw), say limn→∞ dw(xn, x) = 0. again from lemma 2.4, we have (3.6) p(x, x) = lim n→∞ p(xn, x) = lim n,m→∞ p(xn, xm). moreover since {xn} is a cauchy sequence in the metric space (x, dw), we have limn,m→∞ dw(xn, xm) = 0. on the other hand since p(xn, xn) + p(xn+1, xn+1) ≤ 2p(xn, xn+1) we obtain by (3.5) lim n→∞ p(xn, xn) = 0. 186 i. altun and g. durmaz therefore from the definition dw we have p(xn, xm) = dw(xn, xm) + min{p(xn, xn), p(xm, xm)} and so limn,m→∞ p(xn, xm) = 0. thus from (3.6) we have p(x, x) = lim n→∞ p(xn, x) = lim n,m→∞ p(xn, xm) = 0. now we show that p(x, fx) = 0. assume this is not true, then from (3.1) we obtain p(x, fx) ≤ p(x, fxn) + p(fxn, fx) − p(fxn, fxn) ≤ p(x, xn+1) + p(fxn, fx) ≤ p(x, xn+1) + ap(x, xn) + bp(x, fx) + cp(xn, xn+1) + dp(x, xn+1) + ep(xn, fx) ≤ p(x, xn+1) + ap(x, xn) + bp(x, fx) + cp(xn, xn+1) + dp(x, xn+1) + ep(xn, x) + ep(x, fx) letting n → ∞, we have p(x, fx) ≤ (b + e)p(x, fx), which is a contradiction. thus p(x, fx) = 0 and so x = fx. moreover p(x, x) = 0. for the uniqueness, suppose y is another fixed point of f . then we have p(y, y) = p(fy, fy) ≤ (a + b + c + d + e)p(y, y). this shows that p(y, y) = 0. now, if p(x, y) > 0, then we have p(x, y) = p(fx, fy) ≤ (a + d + e)p(x, y), which is a contradiction. therefore, the fixed point is unique. � we can have the following corollaries from theorem 3.1. corollary 3.2 (banach type). let (x, p) be a complete wpms and let f : x → x be a map such that p(fx, fy) ≤ αp(x, y) for all x, y ∈ x, where 0 ≤ α < 1. then f has a unique fixed point. corollary 3.3 (kannan type). let (x, p) be a complete wpms and let f : x → x be a map such that p(fx, fy) ≤ βp(x, fx) + γp(y, fy) for all x, y ∈ x, where β, γ ≥ 0 and β + γ < 1. then f has a unique fixed point. weak partial metric spaces 187 corollary 3.4 (reich type). let (x, p) be a complete wpms and let f : x → x be a map such that p(fx, fy) ≤ αp(x, y) + βp(x, fx) + γp(y, fy) for all x, y ∈ x, where α, β, γ ≥ 0 and α + β + γ < 1. then f has a unique fixed point. next we state a nonlinear contractive type fixed point theorem. let φ : [0, ∞) → [0, ∞) be a function. in the connection with the function φ we consider the following properties: (i) φ is nondecreasing, (ii) φ(t) < t for all t > 0, (iii) φ(0) = 0, (iv) φ is continuous, (v) limn→∞ φ n(t) = 0 for all t ≥ 0, (vi) ∑ ∞ n=0 φn(t) convergent for all t > 0. it is easy to see that, (i) and (ii) imply (iii), (ii) and (iv) imply (iii), (i) and (v) imply (ii). definition 3.5 ([4]). a function φ satisfying (i) and (v) is said to be a comparison function and a function φ satisfying (i) and (vi) is said to be (c)-comparison function. it is clear that, any (c)-comparison function is a comparison function and any comparison function satisfies (iii). theorem 3.6. let (x, p) be a complete wpms and let f : x → x be a map such that p(fx, fy) ≤ φ(max{p(x, y), p(x, fx), p(y, fy), 1 2 [p(x, fy) + p(y, fx)]})(3.7) for all x, y ∈ x, where φ : [0, ∞) → [0, ∞) is a (c)-comparison function. then f has a unique fixed point. proof. let x0 ∈ x be an arbitrary point. define a sequence {xn} in x by xn = fxn−1 for n = 1, 2, · · · . now if xn0 = xn0+1 for some n0 = 0, 1, 2, · · · , then it is clear that xn0 is a fixed point of f . now assume xn 6= xn+1 for all 188 i. altun and g. durmaz n. in this case p(xn, xn+1) > 0 for all n. then we have from (3.7) p(xn+1, xn) = p(fxn, fxn−1) ≤ φ(max{p(xn, xn−1), p(xn, fxn), p(xn−1, fxn−1), 1 2 [p(xn, fxn−1) + p(xn−1, fxn)]}) ≤ φ(max{p(xn, xn−1), p(xn, xn+1), 1 2 [p(xn−1, xn) + p(xn, xn+1)]}) = φ(max{p(xn, xn−1), p(xn, xn+1)}),(3.8) since p(xn, xn) + p(xn−1, xn+1) ≤ p(xn−1, xn) + p(xn, xn+1) and φ is nondecreasing. now if max{p(xn, xn−1), p(xn, xn+1)} = p(xn, xn+1) for some n, then from (3.8) we have p(xn+1, xn) ≤ φ(p(xn, xn+1)) < p(xn+1, xn) which is a contradiction since p(xn, xn+1) > 0. thus max{p(xn, xn−1), p(xn, xn+1)} = p(xn, xn−1) for all n. then from (3.8) we have p(xn+1, xn) ≤ φ(p(xn, xn−1)) and hence (3.9) p(xn+1, xn) ≤ φ n(p(x1, x0)). this shows that (3.10) lim n→∞ p(xn, xn+1) = 0. on the other hand, since dw(xn+1, xn) = p(xn+1, xn) − min{p(xn, xn), p(xn+1, xn+1)} ≤ p(xn+1, xn) ≤ φn(p(x1, x0)) we have limn→∞ dw(xn, xn+1) = 0. therefore we have for m > n dw(xm, xn) ≤ dw(xm, xm−1) + · · · + dw(xn+1, xn) ≤ φm−1(p(x1, x0)) + · · · + φ n(p(x1, x0)) ≤ ∞ ∑ k=n φk(p(x1, x0)). since φ is (c)-comparison function, then {xn} is a cauchy sequence in the metric space (x, dw). since (x, p) is complete then from lemma 2.4, the sequence weak partial metric spaces 189 {xn} converges in the metric space (x, dw), say limn→∞ dw(xn, x) = 0. again from lemma 2.4, we have (3.11) p(x, x) = lim n→∞ p(xn, x) = lim n,m→∞ p(xn, xm). moreover since {xn} is a cauchy sequence in the metric space (x, dw), we have limn,m→∞ dw(xn, xm) = 0. on the other hand since p(xn, xn) + p(xn+1, xn+1) ≤ 2p(xn, xn+1) we obtain by (3.10) lim n→∞ p(xn, xn) = 0. therefore from the definition dw we have p(xn, xm) = dw(xn, xm) + min{p(xn, xn), p(xm, xm)} and so limn,m→∞ p(xn, xm) = 0. thus from (3.11) we have (3.12) p(x, x) = lim n→∞ p(xn, x) = lim n,m→∞ p(xn, xm) = 0. now we show that p(x, fx) = 0. suppose that p(x, fx) > 0, as limn→∞ p(xn+1, xn) = 0 and limn→∞ p(xn, x) = 0, there exists n0 ∈ n such that for n > n0, (3.13) p(xn+1, xn) < 1 3 p(x, fx) and there exist n1 ∈ n such that for n > n1, (3.14) p(xn, x) < 1 3 p(x, fx). if we take n > max{n0, n1} then, by (3.13), (3.14) and triangular inequality, we have 1 2 [p(xn, fx) + p(x, fxn)] ≤ 1 2 [p(xn, x) + p(x, fx) − p(x, x) + p(x, fxn)] ≤ 1 2 [ 1 3 p(x, fx) + p(x, fx) + 1 3 p(x, fx)] = 5 6 p(x, fx).(3.15) now for n > max{n0, n1}, then, by (3.13), (3.14) and (3.15), we have p(xn+1, fx) = p(fxn, fx) ≤ φ(max{p(xn, x), p(xn, fxn), p(x, fx), 1 2 [p(xn, fx) + p(x, fxn)]}) ≤ φ(p(x, fx)). letting n → ∞ in the last inequality, we have p(x, fx) ≤ φ(p(x, fx)), which is a contradiction. thus p(x, fx) = 0 and so x is a fixed point of f . moreover by (3.12) p(x, x) = 0. the uniqueness follows easily from (3.7). � 190 i. altun and g. durmaz example 3.7. let x = {0, 1, · · · , 10} and p(x, y) = x+y 2 , then dw(x, y) = 1 2 |x − y|. therefore, since (x, dw) is complete, then by lemma 2.4 (x, p) is complete wpms. let f : x → x, fx =    x − 1 , x 6= 0 0 x = 0 . we claim that the condition (3.7) of theorem 3.6 is satisfied with φ(t) = 9 10 t. for this, we consider the following cases. case 1. if x = y = 0, then p(fx, fy) = 0 ≤ 9 10 p(x, y). case 2. if x = y > 0, then p(fx, fy) = p(x − 1, x − 1) = x − 1 ≤ 9 10 x = 9 10 p(x, y) case 3. if x > y = 0, then p(fx, fy) = p(x − 1, 0) = x − 1 2 ≤ 9 10 x 2 = 9 10 p(x, y). case 4. if x > y > 0, then p(fx, fy) = p(x − 1, y − 1) = x + y − 2 2 ≤ 9 10 x + y 2 = 9 10 p(x, y). this shows that all conditions of theorem 3.6 are satisfied and so f has a unique fixed point in x. note that, if we use the usual metric on x, then the contractive condition is not satisfied. acknowledgements. the authors thanks to professor salvador romaguera for his valuable suggestions for improving this paper. references [1] i. altun and a. erduran, fixed point theorems for monotone mappings on partial metric spaces, fixed point theory and applications (2011), article id 508730, 10 pp. [2] i. altun and h. simsek, some fixed point theorems on dualistic partial metric spaces, j. adv. math. stud. 1 (2008), 1–8. [3] i. altun, f. sola and h. simsek, generalized contractions on partial metric spaces, topology and its applications 157 (2010), 2778–2785. [4] v. berinde, iterative approximation of fixed points, springer -verlag berlin heidelberg, 2007. weak partial metric spaces 191 [5] m. h. escardo, pcf extended with real numbers, theoretical computer sciences 162 (1996), 79–115. [6] g. e. hardy, t. d. rogers, a generalization of a fixed point theorem of reich, canad. math. bull. 16 (1973), 201–206. [7] r. heckmann, approximation of metric spaces by partial metric spaces, appl. categ. structures 7 (1999), 71–83. [8] s. g. matthews, partial metric topology, proc. 8th summer conference on general topology and applications, ann. new york acad. sci. 728 (1994), 183–197. [9] s. oltra and o. valero, banach’s fixed point theorem for partial metric spaces, rend. istid. math. univ. trieste 36 (2004), 17–26. [10] s. romaguera, a kirk type characterization of completeness for partial metric spaces, fixed point theory and applications (2010), article id 493298, 6 pp. [11] o. valero, on banach fixed point theorems for partial metric spaces, appl. general topology 6 (2005), 229–240. (received february 2012 – accepted may 2012) ishak altun (ishakaltun@yahoo.com, ialtun@kku.edu.tr) department of mathematics, faculty of science and arts, kirikkale university, 71540 yahsihan, kirikkale, turkey. gonca durmaz (gncmatematik@hotmail.com) department of mathematics, faculty of science and arts, kirikkale university, 71540 yahsihan, kirikkale, turkey. weak partial metric spaces and some fixed[5pt] point results. by i. altun and g. durmaz () @ appl. gen. topol. 14, no. 2 (2013), 159-169doi:10.4995/agt.2013.1586 c© agt, upv, 2013 zariski topology on the spectrum of graded classical prime submodules ahmad yousefian darani a and shahram motmaen b a department of mathematics and applications, faculty of mathematical sciences, university of mohaghegh ardabili, 56199-11367, ardabil, iran. (yousefian@uma.ac.ir, youseffian@gmail.com) b young researchers club, ardabil branch islamic azad university, ardabil, iran. (sh.motmaen@gmail.com) abstract let r be a g-graded commutative ring with identity and let m be a graded r-module. a proper graded submodule n of m is called graded classical prime if for every a, b ∈ h(r), m ∈ h(m), whenever abm ∈ n, then either am ∈ n or bm ∈ n. the spectrum of graded classical prime submodules of m is denoted by cl.specg(m). we topologize cl.specg(m) with the quasi-zariski topology, which is analogous to that for specg(r). 2010 msc: 13a02, 16w50. keywords: graded prime ideal, zariski topology, quasi-zariski topology. 1. introduction recently many authors have been interested in equip algebraic structures with zariski topology (cf. [4, 11, 12]). a grading on a ring and its modules usually aids computations by allowing one to focus on the homogeneous elements, which are presumably simpler or more controllable than random elements. however, for this to work one needs to know that the constructions being studied are graded. one approach to this issue is to redefine the constructions entirely in terms of the category of graded modules and thus avoid any consideration of non-graded modules or non-homogeneous elements; sharp gives such a treatment of attached primes in [15]. unfortunately, while such an approach helps to understand the graded modules themselves, it will only help received august 2012 – accepted march 2013 http://dx.doi.org/10.4995/agt.2013.1586 a. yousefian darani and s. motmaen to understand the original construction if the graded version of the concept happens to coincide with the original one. therefore, notably, the study of graded modules is very important. our main purpose is to study some new classes of graded submodules of graded modules and endow these classes of submodules with quasi-zariski topology. zariski topology on the prime spectrum of a module over a commutative ring have been already studied in [11, 12]. moreover some topologies on the spectrum of graded prime submodules of a graded module have been studied in [16]. therefore these results will be used in order to obtain the main aims of this paper. the organization of this paper is as follows: in section 2 we recollect the results concerning the topologies on the prime spectrum of a module over a commutative ring. moreover we remind the notation and the elemental properties about graded modules and rings that we will use in this paper. in section 3 we introduce and study the concept of graded classical prime submodules and define the quasi-zariski topology on the spectrum of all graded classical prime submodules of a graded module. 2. preliminaries in this section, we recall some definitions and notations used throughout. throughout this paper all rings are commutative with a nonzero identity and all modules are considered to be unitary. prime submodules play an important role in the module theory over commutative rings. let m be a module over a commutative ring r. a prime (resp. primary) submodule n of m is a proper submodule n of m with the property that for a ∈ r and m ∈ m, am ∈ n implies that m ∈ n or a ∈ (n :r m) (resp. a k ∈ (n :r m) for some positive integer k). in this case p = (n :r m) (resp. p = √ (n :r m)) is a prime ideal of r and we say that n is a p-prime (resp. p-primary) submodule of m. there are several ways to generalize the notion of prime submodules. we could restrict where am lies or we can restrict where a and/or b lie. we begin by mentioning some examples obtained by restricting where ab lies. weakly prime submodules were introduced by ebrahimi atani and farzalipour in [8]. a proper submodule n of m is weakly prime if for a ∈ r and m ∈ m with 0 6= am ∈ n, either m ∈ n or a ∈ (n :r m). behboodi and koohi in [3] defined another class of submodules and called it classical prime. a proper submodule n of m is said to be classical prime when for a, b ∈ r and m ∈ m, abm ∈ n implies that am ∈ n or bm ∈ n. recently, m. baziar and m. behboodi [2] defined a classical primary submodule in the r-module m as a proper submodule q of m such that if abm ∈ q, where a, b ∈ r and m ∈ m, then either bm ∈ q or akm ∈ q for some positive integer k. clearly, in case m = r, classical primary submodules coincide with primary ideals. let g be an arbitrary group. a commutative ring r with a non-zero identity is g-graded if it has a direct sum decomposition r = ⊕ g∈g rg such that for c© agt, upv, 2013 appl. gen. topol. 14, no. 2 160 zariski topology on the spectrum of graded classical prime submodules all g, h ∈ g, rgrh ⊆ rgh. the g-graded ring r is called a graded integral domain provided that ab = 0 implies that either a = 0 or b = 0 where a, b ∈ h(r) := ⋃ g∈g rg. if r is g-graded, then an r-module m is said to be ggraded if it has a direct sum decomposition m = ⊕ g∈g mg such that for all g, h ∈ g, rgmh ⊆ mgh . for every g ∈ g, an element of rg or mg is said to be a homogeneous element. we denote by h(m) the set of all homogeneous elements of m, that is h(m) = ⋃ g∈g mg. let m be a g-graded r-module. a submodule n of m is called graded (or homogeneous) if n = ⊕ g∈g (n ∩ mg) or equivalently n is generated by homogeneous elements. moreover, m/n becomes a g-graded r-module with g-component (m/n)g = (mg + n)/n for each g ∈ g. an ideal i of r is called a graded ideal if it is a graded submodule of r and a graded r-module. let r be a g-graded ring. a proper graded ideal i of r is said to be a graded prime ideal if whenever ab ∈ i, we have a ∈ i or b ∈ i, where a, b ∈ h(r). the graded radical of i , denoted by gr(i), is the set of all x ∈ r such that for each g ∈ g there exists ng > 0 with x ng ∈ i. a graded r-module m is called graded finitely generated if m = ∑n i=1 rxgi, where xgi ∈ h(m) for every 1 ≤ i ≤ n. it is clear that a graded module is finitely generated if and only if it is graded finitely generated. for m, consider the subset t g(m) = {m ∈ m : rm = 0 for some nonzero r ∈ h(r)}. if r is a graded integral domain, then t g(m) is a graded submodule of m. m is called graded torsion-free (g-torsion-free for short) if t g(m) = 0, and it is called graded torsion (g-torsion for short) if t g(m) = m. it is clear that if m is torsion-free, then it is g-torsion-free. moreover, if m is g-torsion, then it is torsion. let r be a g-graded ring and m a graded r-module. we recall from [8] that a proper graded submodule n of m is called graded prime (resp. graded primary) if rm ∈ n, then m ∈ n or r ∈ (n :r m) = {r ∈ r|rm ⊆ n} (resp. rk ∈ (n :r m) for some positive integer k), where r ∈ h(r), m ∈ h(m). it is shown in [8, proposition 2.7] that if n is a graded prime submodule of m, then p := (n :r m) is a graded prime ideal of r, and n is called graded p-prime submodule. let n be a graded submodule of m. then n is a graded prime submodule of m if and only if p := (n :r m) is a graded prime ideal of r and m/n is a g-torsion-free r/p-module. note that some graded r-modules m have no graded prime submodules. we call such graded modules g-primeless. a submodule s of m will be called graded semiprime if s is an intersection of graded prime submodules of m. let specg(m) denote the set of all graded prime submodules of m. let n be a graded submodule of m. the graded radical of n in m, denoted by grm (n) is defined to be the intersection of all graded prime submodules of m containing n [10]. hence grm (n) is a graded semiprime submodule. a proper graded submodule n of m is called graded weakly prime if 0 6= rm ∈ n, then m ∈ n or r ∈ (n :r m). hence every graded prime submodule is graded weakly prime. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 161 a. yousefian darani and s. motmaen from now on, r is a g-graded ring and m is a graded r-module unless otherwise stated. 3. graded classical prime submodules a proper graded submodule n of m is called graded classical prime if for every a, b ∈ h(r), m ∈ h(m), whenever abm ∈ n, then either am ∈ n or bm ∈ n. let n be a graded classical prime submodule of m. then, it is easy to see that ng is a classical prime submodule of the re-module mg for every g ∈ g. it is evident that every graded prime submodule is graded classical prime. however the next example shows that a graded classical prime submodule is not necessarily graded prime. example 3.1. assume that r is a graded integral domain and p is a non-zero graded prime ideal of r. in this case the ideal q := p ⊕ 0 is a graded classical prime submodule of the graded r-module r ⊕ r while it is not graded prime. this example shows also that a graded classical prime submodule need not be classical prime. we denote by cl.specg(m), the set of all graded classical prime submodules of m. obviously, some graded r-modules m have no graded classical prime submodules; such modules are called g-cl.primeless. for example, the zero module is clearly g-cl.primeless. a submodule s of m will be called graded classical semiprime if s is an intersection of graded classical prime submodules of m. let n be a graded submodule of m. the graded classical radical of n in m, denoted by cl.grm (n), is defined to be the intersection of m and all graded classical prime submodules of m containing n. so if cl.specg(m) = ∅, then grclm (n) = m, and if cl.specg(m) 6= ∅, then gr cl m (n) is a graded classical semiprime submodule. if n = 0, then grclm (0) is called the graded classical nil-radical of m. we know that specg(m) ⊆ cl.specg(m). as it is mentioned in example 3.1, it happens sometimes that this containment is strict. we call m a graded compatible r-module if its graded classical prime submodules and graded prime submodules coincide, that is if specg(m) = cl.specg(m). if r is a g-graded ring, then every graded classical prime ideal of r is a graded prime ideal. so, if we consider r as a graded r-module, it is graded compatible. the following lemma is obvious. lemma 3.2. let n be a proper graded submodule of m. then n is a graded classical prime submodule if and only if for each x ∈ h(m) \ n, (n :r x) is a graded prime ideal of r. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 162 zariski topology on the spectrum of graded classical prime submodules proposition 3.3. (1) let n be proper graded submodule of m. then n is a graded prime submodule of m if and only if n is graded primary and graded classical prime. (2) assume that n and k are graded submodule of m with k ⊆ n. then n is a graded classical prime submodule of m if and only if n/k is a graded classical prime submodule of the graded r-module m/k. proof. (1) if n is a graded prime submodule of m, then it clearly is both graded primary and graded classical prime. now assume that n is a graded primary and graded classical prime submodule of m. let am ∈ n but m /∈ n, where a ∈ h(r) and m ∈ h(m). since n is graded primary, there exists a positive integer k such that ak ∈ (n :r m). therefore, for every y ∈ h(m) \ n, ak ∈ (n :r y) and (n :r y) is a prime ideal of r by lemma 3.2. hence a ∈ (n :r y). it follows that a ∈ (n :r m), i.e. n is graded prime in m. (2) straightforward. � let r be a g-graded r-module and consider specg(r), the spectrum of all graded prime ideals of r. the zariski topology on specg(r) is defined in a similar way to that of spec(r). for each graded ideal i of r, the graded variety of i is the set v g r (i) = {p ∈ specg(r)|i ⊆ p}. then the set {v g r (i)|i is a graded ideal of r} satisfies the axioms for the closed sets of a topology on specg(r), called the zariski topology on specg(r) (see [14]). in [16], specg(m) has endowed with quasi-zariski topology. for each graded submodule n of m, let v g ∗ (n) = {p ∈ specg(m)|n ⊆ p}. in this case, the set ζ g ∗ (m) = {v g ∗ (n)|n is a graded submodule of m} contains the empty set and specg(m), and it is closed under arbitrary intersections, but it is not necessarily closed under finite unions. the graded r-module m is said to be a g-top module if ζ g ∗ (m) is closed under finite unions. in this case ζ g ∗ (m) satisfies the axioms for the closed sets of a unique topology τ g ∗ on specg(m). the topology τ g ∗ (m) on specg(m) is called the quasi-zariski topology. in the remainder of this section we use a similar method to define a topology on cl.specg(m). to this end, for each graded submodule n of m, set vg∗(n) = {p ∈ cl.specg(m)|n ⊆ p}. proposition 3.4. let m be a graded r-module. then (1) for each subset e ⊆ h(m), vg∗(e) = v g ∗(n) = v g ∗ (gr cl m (n)), where n is the graded submodule of m generated by e. (2) vg∗(0) = cl.specg(m), and v g ∗(m) = ∅. (3) if {nλ}λ∈λ is a family of graded submodules of m, then ⋂ λ∈λ v g ∗(nλ) = vg∗( ∑ λ∈λ nλ). (4) for every pair n and k of graded submodules of m, we have vg∗(n) ∪ vg∗(k) ⊆ v g ∗(n ∩ k). proof. the proof of (2) − (4) is easy. so we just provide a proof for part (1). assume that n is the graded submodule of m generated by e ⊆ h(m). then from e ⊆ n ⊆ grclm (n) we have c© agt, upv, 2013 appl. gen. topol. 14, no. 2 163 a. yousefian darani and s. motmaen vg∗(gr cl m (n)) ⊆ v g ∗(n) ⊆ v g ∗(e). on the other hand, n is the smallest graded submodule of m containing e, so that if p ∈ vg∗(e), then p ∈ v g ∗(n). therefore v g ∗(e) = v g ∗(n). moreover grclm (n) is the intersection of all graded classical prime submodules of m containing n; so vg∗(n) = v g ∗(gr cl m (n)). therefore v g ∗(e) = v g ∗(n) = vg∗(gr cl m (n)). � now if we set η g ∗(m) = {v g ∗(n)|n is a graded submodule of m} then η g ∗(m) contains the empty set and cl.specg(m). moreover η g ∗(m) is closed under arbitrary intersections, but it is not necessarily closed under finite unions. definition 3.5. let m be a graded r-module. (1) we shall say that m is a g-cl.top module if η g ∗(m) is closed under finite unions, i.e. for any graded submodules n and l of m there exists a graded submodule k of m such that vg∗(n) ∪ v g ∗(l) = v g ∗(k). (2) a graded classical prime submodule n of m will be called graded classical extraordinary, or g-cl.extraordinary for short, if whenever k and l are graded classical semiprime submodules of m with k ∩ l ⊆ n then k ⊆ n or l ⊆ n. note that if m is a g-cl.top module, then η g ∗(m) satisfies the axioms for the closed sets of a unique topology ̺ g ∗ on cl.specg(m). in this case, the topology ̺ g ∗(m) on cl.specg(m) is called the quasi-zariski topology. note that we are not excluding the trivial case where cl.specg(m) is empty; that is every gcl.primeless modules is a g-cl.top module. the next result is a useful tool for characterizing g-cl.top modules. theorem 3.6. let m be a graded r-module. then, the following statements are equivalent: (i) m is a g-cl.top module. (ii) every graded classical prime submodule of m is g-cl.extraordinary. (iii) vg∗(n) ∪ v g ∗(l) = v g ∗(n ∩ l) for any graded classical semiprime submodules n and l of m. proof. the result is clear when cl.specg(m) = ∅. so assume that cl.specg(m) 6= ∅. (i) ⇒ (ii) let m be a g-cl.top module. assume that p is a graded classical prime submodule of m and that n, l are graded classical semiprime submodules of m with n ∩l ⊆ p . by assumption, there exists a graded submodule k of m with vg∗(n) ∪ v g ∗(l) = v g ∗(k). since n is a graded classical semiprime submodule, n = ⋂ i∈i pi in which {pi}i∈i is a collection of graded classical prime submodules of m. for every i ∈ i, we have pi ∈ v g ∗(n) ⊆ v g ∗(k) ⇒ k ⊆ pi ⇒ k ⊆ ⋂ i∈i pi = n c© agt, upv, 2013 appl. gen. topol. 14, no. 2 164 zariski topology on the spectrum of graded classical prime submodules similarly, k ⊆ l. so k ⊆ n ∩ l. now we have vg∗(n) ∪ v g ∗(l) ⊆ v g ∗(n ∩ l) ⊆ v g ∗(k) = v g ∗(n) ∪ v g ∗(l). consequently, vg∗(n) ∪ v g ∗(l) = v g ∗(n ∩ l). now from n ∩ l ⊆ p we have p ∈ vg∗(n ∩l) = v g ∗(n)∪v g ∗(l). hence either p ∈ v g ∗(n) or p ∈ v g ∗(l), that is either n ⊆ p or l ⊆ p . so p is g-cl.extraordinary. (ii) ⇒ (iii) suppose that every graded classical prime submodule of m is g-cl.extraordinary. assume that n and l are two graded classical semiprime submodules of m. clearly vg∗(n) ∪ v g ∗(l) ⊆ v g ∗(n ∩ l). now assume that p ∈ vg∗(n∩l). then n∩l ⊆ p . since p is g-cl.extraordinary, we have n ⊆ p or l ⊆ p , that is either p ∈ vg∗(n) or p ∈ v g ∗(l). therefore v g ∗(n ∩ l) ⊆ vg∗(n) ∪ v g ∗(l), and so v g ∗(n) ∪ v g ∗(l) = v g ∗(n ∩ l). (iii) ⇒ (i) let n, l be two graded submodules of m. we can assume that vg∗(n) and v g ∗(l) are both nonempty, for otherwise v g ∗(n) ∪ v g ∗(l) = v g ∗(n) or vg∗(n) ∪ v g ∗(l) = v g ∗(l). we know that gr cl m (n) and gr cl m (l) are both graded classical semiprime submodules of m. setting k = grclm (n) ∩ gr cl m (l) we have: vg∗(n) ∪ v g ∗(l) = v g ∗(gr cl m (n)) ∪ v g ∗(gr cl m (l)) = v g ∗(gr cl m (n) ∩ gr cl m (l)) = vg∗(k) by (iii). hence m is a g-cl.top module. � corollary 3.7. every g-cl.top module is a g-top module. proof. assume that m is a g-cl.top module. let p be a graded prime submodule of m. since every graded prime l-submodule is a graded classical prime submodule, p is g-cl.extraordinary by proposition 3.6. hence it is gextraordinary. now the result follows from [16, theorem 2.3]. � theorem 3.8. let m be a g-cl.top r-module. then, (1) for every graded submodule k of m, the r-module m/k is a g-cl.top module. (2) the graded rp -module mp is a g-cl.top module for every graded prime ideal p of r. (3) if grclm (n) = n for every graded submodule n of m, then m is a graded distributive module. proof. there will be nothing to prove if m has no graded classical prime submodules. so assume that cl.specg(m) 6= ∅. (1) by proposition 3.3, the graded classical prime submodules of m/k are just the submodules n/k where n is a graded classical prime submodule of m with k ⊆ n. consequently, any graded classical semiprime submodule of m/k is of the form s/k in which s is a graded classical semiprime submodule of m with k ⊆ s. assume that s1/k and s2/k are two graded classical semiprime submodules of m/k. then, by theorem 3.6, vg∗(s1) ∪ v g ∗(s2) = v g ∗(s1 ∩ s2) since m is a g − cl.t op module. thus vg∗(s1/k) ∪ v g ∗(s2/k) = v g ∗(s1/k ∩ s2/k). it follows from theorem 3.6 that m/k is a g − cl.t op module. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 165 a. yousefian darani and s. motmaen (2) by theorem 3.6, it is enough to show that every graded classical prime submodule of mp is g-cl.extraordinary. let n be a graded classical prime submodule of mp , and let s1 ∩ s2 ⊆ n for some graded classical semiprime submodules s1, s2 of mp . clearly, n ∩m is a proper graded submodule of m. assume that a, b ∈ h(r) and m ∈ h(m) are such that abm ∈ n ∩ m. then, a/1, b/1 ∈ h(rp ) and m/1 ∈ h(mp ) with (a/1)(b/1)(m/1) = (abm)/1 ∈ n. it follows that either (a/1)(m/1) ∈ n or (b/1)(m/1) ∈ n since n is graded classical prime. therefore, either am ∈ n ∩ m or bm ∈ n ∩ m. this implies that n ∩ m is a graded classical prime submodule of m. hence n is gcl.extraordinary by theorem 3.6. as another consequence, s1 ∩m and s2 ∩m are graded classical semiprime submodules of m with (s1 ∩ m) ∩ (s2 ∩ m) ⊆ n ∩m. therefore, s1 ∩m ⊆ n ∩m or s2 ∩m ⊆ n ∩m. it follows that either s1 = (s1 ∩m)rp ⊆ (n ∩m)rp = n or s2 = (s2 ∩m)rp ⊆ (n ∩m)rp = n. hence n is a g-cl.extraordinary submodule of mp . (3) for every graded submodules n, k and l of m we have: (k + l) ∩ n = grclm ((k + l) ∩ n) = ⋂ {p |p ∈ vg∗((k + l) ∩ n)} = ⋂ {p |p ∈ vg∗(k + l) ∪ v g ∗(n)} = ⋂ {p |p ∈ (vg∗(k) ∩ v g ∗(l)) ∪ v g ∗(n)} = ⋂ {p |p ∈ (vg∗(k)∪v g ∗(n))∩(v g ∗(l)∪v g ∗(n))} = ⋂ {p |p ∈ (vg∗(k ∩ n)) ∩ (v g ∗(l ∩ n))} = ⋂ {p |p ∈ vg∗((k ∩ n) + (l ∩ n))} = grclm ((k ∩n)+(l∩n)) = (k ∩n)+(l∩n). thus m is graded distributive. � let m be a g-cl.top module and let x = cl.specg(m). we know that any closed subset of x is of the form vg∗(n) for some graded submodule n of m. but now the question arises as to what open subsets of x look like. to say that any open subset of x is of the form x − vg∗(n) for some graded prime submodule n of m, though true, is not very helpful. for every subset s of h(m), define xs = x − v g ∗(s) in particular, if s = {f}, we denote xs be xf . proposition 3.9. the set {xf|f ∈ h(m)} is a basis for the quasi-zariski topology on x. proof. let u be a non-void open subset in x. then u = x − vg∗(n) for some graded submodule n of m. assume that n is generated by some subset e ⊆ h(m). then we have u = x − vg∗(n) = x − v g ∗(e) = x − v g ∗( ⋃ f∈e {f}) = x − ⋂ f∈e vg∗(f) = ⋃ f∈e (x − vg∗(f)) = ⋃ f∈e xf therefore the set {xf |f ∈ h(m)} is a basis for x. � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 166 zariski topology on the spectrum of graded classical prime submodules a topological space x is said to be irreducible if x 6= ∅ and if every pair of non-void open sets in x intersect. let x be a topological space. a subset a ⊆ x is said to be dense in x if and only if a ∩ g 6= ∅ for every non-void open subset g ⊆ x. therefore x is irreducible if and only if every non-void open subset of x is dense. lemma 3.10. let m be a graded r-module. then, n := grclm (0) is a graded classical prime submodule of m if and only if cl.specg(m) is irreducible. proof. set x = cl.specg(m). assume first that n is a graded classical prime submodule of m. let u, v ⊆ x be non-void open subsets. pick p ∈ u. now, u = x \ vg∗(e) for some e ⊆ h(m). then p ∈ u implies that e * p . moreover, from n ⊆ p we have e * n, so that n ∈ u. similarly, n ∈ v . hence n ∈ u ∩v , and thus u ∩v 6= ∅. therefore, x is irreducible. conversely, assume that n is not a graded classical prime submodule of m. so there exist a, b ∈ h(r) and m ∈ h(m) such that am, bm /∈ n, but abm ∈ n. both xam and xbm are open in x. also am /∈ n ⇒ v g ∗(am) 6= x so xam 6= ∅. similarly, xbm 6= ∅. now we have, xam ∩ xbm = xabm = x − v g ∗ (abm) ⊆ x − vg ∗ (n) = ∅ therefore, x is not irreducible. � 4. homomorphisms and graded classical prime spectrum of modules in our discussion so far we have concerned ourselves with the graded classical prime spectrum of but one graded module at any given time. a natural question to ask is what relationships on their respective graded classical prime spectra are induced by a homomorphism between two rings. in this section, we address this question. let m and m′ be two graded r-modules and let φ : m → m′ be a graded r-homomorphism. the inverse image of a graded classical prime submodule of m′ is a graded classical prime submodule of m. for every graded submodule of m′, we write φ−1(n′) = n′ c , the contraction of n′ to m. also, if n is a graded submodule of m, then we write rφ(n) = ne, the graded submodule of m′ generated by φ(n), the extension of n to m′. let x = cl.specg(m) and y = cl.specg(m ′). thus if p ∈ y , then p c ∈ x. so we see that φ induces a map φ∗ : y → x defined by φ∗(p) = p c, for all p ∈ y . before continuing, we introduce a more explicit notation: p ∈ vg ∗m (e) means that e ⊆ h(m) and e ⊆ p ∈ x and q ∈ vg ∗m′ (f) means that f ⊆ h(m′) and f ⊆ p ∈ y c© agt, upv, 2013 appl. gen. topol. 14, no. 2 167 a. yousefian darani and s. motmaen proposition 4.1. let m and m′ be two graded r-modules and let φ : m → m′ be a graded r-homomorphism. let x = cl.specg(m), y = cl.specg(m ′), and let φ∗ : y → x be the induced map. (1) φ∗ is continuous. (2) if n is a graded submodule of m, then φ∗ −1 (vg ∗m (n)) = vg ∗m′ (ne). (3) if φ is an epimorphism, then φ∗ is a homeomorphism from y onto the closed subset vg ∗m (ker(φ)) of x. proof. (1) it is enough to show that if u is open in x, then φ∗ −1 (u) is open if y . for every subset e ⊆ h(m) and q ∈ y , we have φ(e) ⊆ q ⇔ e ⊆ φ−1(q) ⇔ φ∗(q) ∈ vg ∗m (e) ⇔ q ∈ φ∗ −1 (vg ∗m (e)) hence, if f ∈ h(m) and q ∈ y , then q ∈ yφ(f) ⇔ φ(f) /∈ q ⇔ q /∈ φ∗ −1 (vg ∗m (f)) ⇔ q ∈ φ∗ −1 (x) − φ∗ −1 (vg ∗m (f)) ⇔ q ∈ φ∗ −1 (x − vg ∗m (f)) = φ∗ −1 (xf ) therefore, φ∗ −1 (xf ) = yφ(f). in particular, if u is open in x, then φ ∗ −1 (u) is open if y . hence φ∗ is continuous. (2) assume that q ∈ y . then, q ∈ φ∗ −1 (vg ∗m (n)) ⇔ φ(n) ⊆ q ⇔ q ∈ vg ∗m′ (φ(n)) ⇔ q ∈ vg ∗m′ (ne) therefore, φ∗ −1 (vg ∗m (n)) = vg ∗m′ (ne). (3) suppose that φ is an epimorphism. then, there exists a one-to-one correspondence between graded submodules of m′ and graded submodules of m containing ker(φ). under this correspondence, graded classical prime submodules of m′ correspond to the graded classical prime submodules of m containing ker(φ). therefore, φ∗ : y → vg ∗m (ker(φ)) is bijective. as φ∗ is continuous by (1), it suffices to prove that φ∗ is an open map. assume that u is an open subset of y . then, without loss of generality, we may assume that u = yf for some f ∈ h(m′). in this case, φ∗(u) = φ∗(yf ) = {φ ∗(q)|q ∈ y and f /∈ q} = {p ∈ vg ∗m (ker(φ))|φ−1(f) /∈ p} = xφ−1(f) ∩ v g ∗m (ker(φ)) this implies that φ∗(u) is an open subset of vg ∗m (ker(φ)), that is φ∗ is an open map. consequently, φ∗ : y → vg ∗m (ker(φ)) is a homeomorphism. � corollary 4.2. let m be a graded r-module, and let n be the graded classical nil-radical of m. then cl.specg(m) and cl.specg(m/n) are naturally homeomorphic. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 168 zariski topology on the spectrum of graded classical prime submodules proof. by proposition 4.1, the canonical graded r-epimorphism f : m → m/n induces the homeomorphism f∗ : cl.specg(m/n) → v g ∗m (ker(f)). now the result follows from vg ∗m (ker(f)) = vg ∗m (n) = cl.specg(m). � references [1] m. f. atiyah and i. g. macdonald, introduction to commutative algebra, longman higher education, new york 1969. [2] m. baziar and m. behboodi, classical primary submodules and decomposition theory of modules, j. algebra appl. 8, no. 3 (2009), 351–362. [3] m. behboodi and h. koohi, weakly prime modules, vietnam j. math. 32, no. 2 (2004), 185–195. [4] m. behboodi and m. j. noori, zariski-like topology on the classical prime spectrum of a module, bull. iranian math. soc. 35, no. 1 (2009), 255–271. [5] m. behboodi and s. h. shojaee, on chains of classical prime submodules and dimension theory of modules, bulletin of the iranian mathematical society 36 (2010), 149–166. [6] j. dauns, prime modules, j. reine angew. math. 298 (1978), 156–181. [7] s. ebrahimi atani, on graded prime submodules, chiang mai j. sci. 33, no. 1 (2006), 3–7. [8] s. ebrahimi atani and f. farzalipour, on weakly prime submodules, tamkang journal of mathematics 38, no. 3 (2007), 247–252. [9] s. ebrahimi atani and f. farzalipour, on graded multiplication modules, chiang-mai journal of science, to appear. [10] s. ebrahimi atani and f. e. k. saraei, graded modules which satisfy the gr-radical formola, thai journal of mathematics 8, no. 1 (2010), 161–170. [11] c. p. lu, the zariski topology on the prime spectrum of a module, houston j. math. 25, no. 3 (1999), 417–425. [12] r. l. mccasland, m. e. moore and p. f. smith, on the spectrum of a module over a commutative ring, comm. algebra 25, no. 1 (1997), 79–103. [13] k. h. oral, u. tekir and a. g. agargun, on graded prime and primary submodules, turk. j. math. 25, no. 3 (1999), 417–425. [14] p. c. roberts, multiplicities and chern classes in local algebra, cambridge university press, 1998. [15] r. y. sharp, asymptotic behaviour of certain sets of attached prime ideals, j. london math. soc. 34, no. 2 (1986), 212–218. [16] a. yousefia darani, topologies on specg(m), buletinul academiei de stiinte a republicii moldova matematica, to appear. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 169 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 17-32 on star compactifications lorenzo acosta and i. marcela rubio abstract we study the ordered structure of the collection of star compactifications by n points and the behavior of these compactifications through quotients obtained by identification of additional points. 2010 msc: primary 54d35, 54a10. secondary 54d60, 54b15 keywords: star compactifications, quotient spaces, ordered structure. 1. introduction. when we study compactification in a general topology course, we usually only deal with three types of compactifications: (1) the alexandroff compactification, which is a one-point compactification; (2) the stone c̆ech’s compactification, and (3) some elementary examples of compactifications by a finite number of points, for instance: (1) [0, 1] is a compactification of (0, 1) by two points. (2) [0, 1] ∪ [2, 3] is a compactification of a = (0, 1) ∪ (2, 3) by four points. (3) geometrically, two disjoint circles on r2 are a compactification by two points, of the set a in the previous example. these examples of compactifications by a finite number of points are particular cases of star compactifications. star topologies were defined in [7] and we mention without proofs some results given in [7]; in particular, the necessary and sufficient conditions for such topologies to be compactifications or t2−compactifications of a non-compact space. 18 l. acosta and i. m. rubio by compactness of a topological space x, we mean that every open covering of x has a finite subcovering, and by compactification of a non-compact topological space x, not necessarily a hausdorff space, we mean a compact space containing x as a subspace, which is dense in the compactification. our intention is to illustrate the definition with simple star compactifications and present some interesting results about the ordered structure of the collection of star compactifications by n points, of a topological space. we obtain the stability of such collection by finite intersections. on the other hand, we present the behavior of the quotients of star compactifications obtained through equivalence relations in which we make some identification between the additional points. moreover we establish the relationships between the star compactifications of a topological space and other compactifications of it by a finite number of points. 2. preliminary results. 2.1. star topologies and star compactifications. in this section we present the definition and some known results concerning star topologies and star compactifications, given in [7]. for this purpose, we consider a non-compact topological space (x, τ) and xn = x ∪ {ω1, ..., ωn} , where ω1, ..., ωn are n distinct points not belonging to x, n ∈ n. proposition 2.1. let ui, i = 1, ..., n be open subsets contained in x. then b = τ ∪ {(ui \ k) ∪ {ωi} | k ⊆ x closed and compact; i = 1, ..., n} is a base for a topology µ on xn. this topology is called the star topology associated to u1, ..., un. notice that b is a closed collection under finite intersections. throughout this paper we denote the star topology over xn associated to u1, ..., un by µ = 〈〈u1, ..., un〉〉. proposition 2.2. (xn, µ) is a compactification of (x, τ) if and only if (1) x \ n ∪ i=1 ui is compact, and (2) ui * k for each k closed and compact subset of x, i = 1, ..., n. observe that the second condition implies that for each i = 1, ..., n, ui is nonempty. by the definition of µ we observe that (x, τ) is a subspace of (xn, µ) and x ∈ µ. on star compactifications 19 2.2. the alexandroff compactification. let (x, τ) be a non-compact topological space and x1 = x ∪ {ω} , where ω is a point not belonging to x. theorem 2.3. if η = τ∪{a ∪ {ω} | a ∈ τ and x \ a is compact} , then (x1, η) is a compactification of (x, τ) by one point. this compactification is called the alexandroff compactification of (x, τ) . it is the finest compactification of (x, τ) obtained by adding one point, and it is called “the” compactification by one point in the class of hausdorff spaces, because it is the only hausdorff compactification by one point when (x, τ) is a hausdorff and locally compact space. see [6]. 3. ordered structure of the star compactifications by n points. we present some results regarding the inclusion order relation in the collection of star compactifications by n points of a non-compact topological space, which enable us to conclude that this collection is stable under finite intersections. we exhibit the smallest element of the mentioned collection. this element can be seen as a generalization of the alexandroff compactification by n points, with n > 1. on the other hand, we study the relationship between the open sets that generate two star topologies when one of them is finer than the other. let (x, τ) be a non-compact topological space and w = {ω1, ω2, ω3, ...} be a set of different elements not belonging to x. we denote x0 = x, xn = xn−1 ∪ {ωn} , n ≥ 1. it is known that all compactifications of (x, τ) by n points are seen, up to homeomorphisms, as xn with a convenient topology, in such way that x is considered a subspace of the compactification. we denote en = {µ ∈ t op(xn) | (xn, µ) is a star compactification of (x, τ)} ; the collection of star compactifications of (x, τ) by n points. observe that for each µ ∈ en, we have τ ⊂ µ, i.e., x ∈ µ. notice that the inclusion order relation defined in en coincides with the order usually defined between compactifications (see [4]). proposition 3.1. let ω = 〈〈u1, ..., un〉〉 , where ui = x for each i. if µ = 〈〈v1, ..., vn〉〉 is an arbitrary star topology on xn, then ω ⊆ µ. proof. let k be a closed and compact subset of (x, τ). as vi ⊆ x for each i = 1, ..., n, then a = {ωi} ∪ (x \ k) = {ωi} ∪ (vi \ k) ∪ (x \ k) ∈ µ, since {ωi} ∪ (vi \ k) ∈ µ and x \ k ∈ τ ⊆ µ. therefore ω ⊆ µ. � this proposition asserts that ω is the smallest element of the set of star topologies on xn, ordered by inclusion. since ω satisfies the mentioned conditions in proposition 2.2, then ω is a star compactification of (x, τ) by n points and ω is the smallest element of (en, ⊆). 20 l. acosta and i. m. rubio remark 3.2. for the case n = 1, ω = 〈〈x〉〉 = b, where b = τ ∪ {{ω1} ∪ (x \ k) | k ⊆ x closed and compact} , we have that ω = b and ω is the alexandroff compactification of (x, τ). since the alexandroff compactification is the finest compactification of (x, τ) by one point, then ω is the only star compactification of (x, τ) by one point. proposition 3.3. if µ = 〈〈v1, ..., vn〉〉 is a star compactification of (x, τ) where v1 = v2 = ... = vn, then µ = ω. proof. let {ωi} ∪ (vi \ k) be a basic open set in µ for some i = 1, ..., n and some k closed and compact subset of x. x \ vi is closed and compact since n ∪ j=1 vj = vi. thus {ωi} ∪ (vi \ k) = {ωi} ∪ [x \ (k ∪ (x \ vi))] ∈ ω, and µ ⊆ ω. � remark 3.4. (1) if µ = 〈〈v1, ..., vn〉〉 is an arbitrary star topology on xn with v1 = v2 = ... = vn, it could happen that µ * ω. example: consider x = (0, 1) with the usual topology as subspace of r; v1 = ... = vn = ( 1 4 , 1 2 ) . then {ωi} ∈ µ, but {ωi} /∈ ω. (2) proposition 3.3 is a particular case of theorem 0.8 in [7]: “let a1, ..., an and b1, ..., bn be two n-tuples of open sets in x. the star topologies associated with those two n-tuples are the same if and only if the sets ai − bi and bi − ai are contained in compact and closed sets for all i = 1, ..., n”. proposition 3.5. let µ = 〈〈u1, ..., un〉〉 , β = 〈〈v1, ..., vn〉〉 be two star topologies on xn. if vi ⊆ ui for each i = 1, ..., n, then µ ⊆ β. proof. let a = {ωi} ∪ (ui \ k) ∈ µ. since vi ⊆ ui and ui \ k ∈ τ, then a = {ωi} ∪ (vi \ k) ∪ (ui \ k) ∈ β. � the next proposition asserts that the insersection of two star topologies on xn is of the same kind and it describes the open sets associated with it. proposition 3.6. let µ = 〈〈u1, ..., un〉〉 , β = 〈〈v1, ..., vn〉〉 be two star topologies on xn, then µ ∩ β = η where η = 〈〈u1 ∪ v1, ..., un ∪ vn〉〉 . proof. i) by the previous proposition we have that η ⊆ µ because ui ⊆ ui∪vi, for each i = 1, ..., n and for the same reason, η ⊆ β. ii) for the other inclusion it is enough to see that all basic open neighborhoods of ωi in µ ∩ β are open neighborhoods of ωi in η, for each i = 1, ..., n. on star compactifications 21 let m be a basic open neighborhood of ωi in µ ∩ β, for i = 1, ..., n, that is, m = {ωi} ∪ (ui \ k1) ∪ a = {ωi} ∪ (vi \ k2) ∪ b, where k1 and k2 are closed and compact subsets of x and a, b ∈ τ. since (ui ∪ vi) \ (k1 ∪ k2) ⊆ (ui \ k1) ∪ (vi \ k2) and (ui \ k1) ∪ a = (vi \ k2) ∪ b, then (ui \ k1) ∪ a = (ui \ k1) ∪ (vi \ k2) ∪ a ∪ b = [(ui ∪ vi) \ (k1 ∪ k2)] ∪ (ui \ k1) ∪ (vi \ k2) ∪ a ∪ b. so, if we call c = (ui \ k1) ∪ (vi \ k2) ∪ a ∪ b, then c is an open set of τ and therefore m = {ωi} ∪ [(ui ∪ vi) \ (k1 ∪ k2)] ∪ c ∈ η. � proposition 3.7. let µ = 〈〈u1, ..., un〉〉 , β = 〈〈v1, ..., vn〉〉 be two star topologies on xn. if µ ⊆ β then there exist mi, ni open sets of τ such that ni ⊆ mi for each i = 1, ..., n and µ = 〈〈m1, ..., mn〉〉 , β = 〈〈n1, ..., nn〉〉 . proof. let ni = vi, mi = ui ∪ vi for each i = 1, ..., n sets of τ. therefore µ = 〈〈m1, ..., mn〉〉 , β = 〈〈n1, ..., nn〉〉 and ni ⊆ mi for each i = 1, ..., n. � observe that this proposition is a weak version of the reciprocal of proposition 3.5. the next proposition guarantees that intersection is a closed operation in the collection en of star compactifications of (x, τ) by n points. proposition 3.8. if µ = 〈〈u1, ..., un〉〉 , β = 〈〈v1, ..., vn〉〉 are two star compactifications of (x, τ) by n points, then η = 〈〈u1 ∪ v1, ..., un ∪ vn〉〉 is a star compactification of (x, τ) by n points. proof. i) we have that ui ∪ vi * k for each closed and compact subset k of x, because ui * k for each i = 1, ..., n. ii) x \ n ∪ i=1 (ui ∪ vi) = x \ (( n ∪ i=1 ui ) ∪ ( n ∪ i=1 vi )) = ( x \ n ∪ i=1 ui ) ∩ ( x \ n ∪ i=1 vi ) , where x \ n ∪ i=1 ui and x \ n ∪ i=1 vi are closed and compact subsets of x, then x \ n ∪ i=1 (ui ∪ vi) is compact. therefore, η is a star compactification of (x, τ) by n points. � 4. certain quotients of star compactifications. we consider the quotients obtained by an equivalence relation ⋄ on xn; ⋄ = {(x, x) | x ∈ xn} ∪ r , where r is an equivalence relation on {ω1, ..., ωn}. the star compactifications of these quotients have an interesting behavior. definition 4.1. we say that a compactification (y, µ) of (x, τ) is of a-class if x ∈ µ. 22 l. acosta and i. m. rubio observe that star compactifications are of a-class. the next theorem asserts that a quotient of a-class compactification of (x, τ), obtained through the mentioned equivalence relation ⋄, is an a-class compactification of (x, τ). theorem 4.2. let (xn, µ) be an a-class compactification of (x, τ) by n points, n > 1. if we consider the equivalence relation ⋄ on xn defined above, then (xn/⋄, µ/⋄) is an a-class compactification of (x, τ) by m points, where m = |{ω1, ..., ωn} /r| ≤ n. proof. we have x i →֒ xn θ −→ xn/⋄, where i is the topological imbedding and θ is the standard quotient map. i) (xn/⋄, µ/⋄) is compact because θ : (xn, µ) −→ (xn/⋄, µ/⋄) is a surjective continuous map, where (xn, µ) is a compact space. ii) to see that θ◦i(x) is dense in xn/⋄ we need to see that θ ◦ i(x) µ/⋄ = xn/⋄ = {[x] | x ∈ x} ∪ {[ωi] | i = 1, ..., n} . if a = {[ωi] | i = 1, ..., n} , then |a| = |{ω1, ..., ωn} /r| = m. supose that θ ◦ i(x) µ/⋄ $ xn/⋄, that is, that there exists b ∈ p (a) \ {∅} such that b ∈ µ/⋄, then θ−1(b) ∈ p ({ω1, ..., ωn})\{∅} and θ−1(b) ∈ µ, but this is a contradiction because x is dense in xn; therefore θ ◦ i(x) µ/⋄ = xn/⋄ and θ ◦ i(x) is dense in xn/ ⋄ . iii) let us see that θ ◦ i : x → xn/⋄ is a homeomorphism between x and θ ◦ i(x), where θ ◦ i(x) = θ(x) = {[x] = {x} | x ∈ x} : i is a homeomorphism between x and i(x) = x. by the definition of ⋄, θ is a bijective map between i(x) = x and θ(x), then θ ◦ i is a bijective map between x and θ ◦ i(x) = θ(x). by the continuity of θ we have that θ ◦ i is continuous. let us see that θ ◦ i is an open map: let a be an open set of τ, we need to see that θ ◦ i(a) = θ(a) ∈ µ/ ⋄ . θ(a) = {[a] | a ∈ a} = {{a} | a ∈ a} because a ⊂ x. θ(a) ∈ µ/⋄ if and only if θ−1 (θ(a)) ∈ µ. since θ(a) ⊂ θ(x) and θ is a bijective map between x and θ(x), then θ−1 (θ(a)) = a. since i is a homeomorphism between x and i(x) = x and a ∈ τ, then i(a) = a = b ∩ x, where b ∈ µ and since x ∈ µ then a ∈ µ and θ(a) ∈ µ/ ⋄ . thus, θ ◦ i is an open map, and then, it is a homeomorphism between x and θ ◦ i(x). moreover, θ(x) ≈ x ∈ µ/⋄ because x ∈ µ. hence (xn/⋄, µ/⋄) is an a-class compactification of (x, τ) by m points, where m = |{ω1, ..., ωn} /r| ≤ n. � corollary 4.3. (xn/⋄, µ/⋄) is homeomorphic to an a-class compactification of (x, τ) by m points, seen as (xm, η) for an appropriate η. on star compactifications 23 consider (xn, µ) a star compactification of (x, τ) by n points, with µ = 〈〈u1, ..., un〉〉 . since (xn, µ) is of a-class, we know that (xn/⋄, µ/⋄) is a compactification of (x, τ) by m points, where m = |{ω1, ..., ωn} /r| ≤ n. since the quotient map θ : xn −→ xn/⋄ is bijective in x, we say θ (x) = x and θ (a) = a for each x ∈ x and for each a ⊆ x, and then xn/⋄ = x ∪ v with v = {υ1, ..., υm} = {θ (ωi) | i = 1, ..., n} , υ1 = θ (ω1) , υk = θ (ωi) where i = min {j ∈ {1, ..., n} | ωj /∈ υ1 ∪ ... ∪ υk−1} for each k = 2, ..., m. we denote ii = {j ∈ {1, ..., n} | θ (ωj) = υi} for each i = 1, ..., m and if |ii| = ni, then ii = {i1, ..., ini} . the next proposition asserts that (xn/⋄, µ/⋄) is a star compactification of (x, τ). proposition 4.4. µ/⋄ = 〈〈m1, ..., mm〉〉 where mi = ni ∪ j=1 uij for each i = 1, ..., m. proof. let a be a basic open set of 〈〈m1, ..., mm〉〉: if a ∈ τ, since θ −1(a) = a and τ ⊆ µ then a ∈ µ/ ⋄ . if a = {υi} ∪ (mi \ k) for some i = 1, ..., m and some k closed and compact subset of x, then θ−1(a) = {ωi1, ..., ωini} ∪ (mi \ k) = {ωi1, ..., ωini} ∪ (( ni ∪ j=1 uij ) \ k ) = {ωi1, ..., ωini} ∪ ( ni ∪ j=1 (uij \ k) ) = ni ∪ j=1 ({ ωij } ∪ (uij \ k) ) ∈ µ, then a ∈ µ/ ⋄ . thus 〈〈m1, ..., mm〉〉 ⊆ µ/ ⋄ . consider now a ∈ µ/⋄, we see that a ∈ 〈〈m1, ..., mm〉〉 by showing that all its points are interior in 〈〈m1, ..., mm〉〉 . we know that θ−1(a) ∈ µ and let z be an element of a : case 1: if z ∈ x then z ∈ θ−1(a) and since µ |x= τ, there exists an open set θ−1(a) ∩ x = b ∈ τ such that z ∈ b ⊆ θ−1(a) then z ∈ θ(b) = b ⊆ a. since b ∈ 〈〈m1, ..., mm〉〉 then z is an interior point of a in 〈〈m1, ..., mm〉〉 . case 2: z = υi for some i = 1, ..., m. θ−1(υi) = {ωi1, ..., ωini} ⊆ θ −1(a) and θ−1(a) is an open set of µ, then for each j ∈ ii, there exists { ωij } ∪ (uij \ kj) open set of µ and subset of θ −1(a). since ( ni ∪ j=1 uij ) \ ( ni ∪ j=1 kj ) ⊆ ni ∪ j=1 (uij \ kj) we have: e = ni ∪ j=1 [{ ωij } ∪ (uij \ kj) ] = θ−1(υi) ∪ [( ni ∪ j=1 uij ) \ ( ni ∪ j=1 kj )] ∪ ni ∪ j=1 (uij \ kj) = θ−1(υi) ∪ (mi \ k) ∪ b ⊆ θ −1(a), where k = ni ∪ j=1 kj is a closed and compact subset of x and b = ni ∪ j=1 (uij \ kj) is an open set of τ. 24 l. acosta and i. m. rubio thus, υi ∈ θ (e) = {υi} ∪ (mi \ k) ∪ b ⊆ a and since {υi} ∪ (mi \ k) ∪ b is open in 〈〈m1, ..., mm〉〉, then υi is an interior point of a in 〈〈m1, ..., mm〉〉 . by cases 1 and 2 we conclude that a ∈ 〈〈m1, ..., mm〉〉 . � therefore, the quotient of a star compactification by n points is a star compactification by m points, where n ≥ m. consequently and by remark 3.2, if ⋄ = r ∪ {(x, x) | x ∈ x} , where r is the equivalence relation on {ω1, ..., ωn} in which all these points are related, then (xn/⋄, µ/⋄) is the alexandroff compactification of (x, τ) . moreover, by observing the form of the open sets mi and in view of the proposition 3.6 we obtain the next corollary. corollary 4.5. µ/⋄ = ∩ p∈p 〈〈up1, ..., upm〉〉 where p = i1 × ... × im and p = (p1, ..., pm) . for instance, if ⋄ only identifies ω1 with ω2 on xn then, in xn−1 we have: µ/ ⋄ = 〈〈u1 ∪ u2, u3, ..., un〉〉 = 〈〈u1, u3,..., un〉〉 ∩ 〈〈u2, u3,..., un〉〉 . on the other hand, the star compactifications show an interesting behavior when, from a star compactification of (x, τ) by m points, we obtain a compactification of (x, τ) by n points, with n ≥ m, so that certain quotient on xn gives back the original compactification. in terms of theorem 4.2 we have the next proposition. proposition 4.6. if (xm, µ) with µ = 〈〈u1, ..., um〉〉 is a star compactification of (x, τ) by m points, then (xn, β) with β = µ ∪ {b ∪ a | b ⊆ {ωm+1, ..., ωn} , a ∈ µ , {ω1} ∪ a ∈ µ} is a star compactification of (x, τ) by n points, n ≥ m, such that a certain quotient of (xn, β) is (xm, µ) . proof. let see that β = 〈〈u1, u2, ..., um, u1, ..., u1〉〉 . by the definition of 〈〈u1, u2, ..., um, u1, ..., u1〉〉 we observe that µ ⊆ 〈〈u1, u2, ..., um, u1, ..., u1〉〉 . let c be a basic open of 〈〈u1, u2, ..., um, u1, ..., u1〉〉: if c ∈ τ or c = {ωi} ∪ (ui \ k) for some i = 1, ..., m and some k closed and compact subset of x, then c ∈ µ ⊆ β. if c = {ωi} ∪ (u1 \ k) for some i = m + 1, ..., n and some closed and compact subset k of x, we have that a = u1 \ k ∈ τ ⊆ µ and {ω1} ∪ a ∈ µ, thus c = b ∪ a with b = {ωi} ⊆ {ωm+1, ..., ωn} , then c ∈ β. therefore 〈〈u1, u2, ..., um, u1, ..., u1〉〉 ⊆ β. on the other hand let c be an element of β. we see that c ∈ 〈〈u1, u2, ..., um, u1, ..., u1〉〉 showing that all of its points are interior points in 〈〈u1, u2, ..., um, u1, ..., u1〉〉. on star compactifications 25 let z be a point of c : case 1: if z ∈ xm, by the definition of β we have that c ∩ xm ∈ µ, thus z ∈ c ∩ xm ⊆ c where c ∩ xm ∈ 〈〈u1, u2, ..., um, u1, ..., u1〉〉 and z is an interior point of c in 〈〈u1, u2, ..., um, u1, ..., u1〉〉 . case 2: if z = ωi, for some i = m + 1, ..., n, then c = b ∪ a where ωi ∈ b, a ∈ µ, {ω1}∪a ∈ µ, thus there exists a closed and compact subset k of x such that {ω1}∪(u1 \k) ⊆ {ω1}∪a. since u1 \k ⊆ a, then ωi ∈ {ωi}∪(u1 \k) ⊆ b ∪ a = c where {ωi} ∪ (u1 \ k) is open in 〈〈u1, u2, ..., um, u1, ..., u1〉〉 . therefore ωi is an interior point to c in 〈〈u1, u2, ..., um, u1, ..., u1〉〉 . by cases 1 and 2 we conclude that c ∈ 〈〈u1, u2, ..., um, u1, ..., u1〉〉 . finally, in the presentation of β as 〈〈u1, u2, ..., um, u1, ..., u1〉〉 it is clear that the equivalence relation on xn that identifies ω1, ωm+1, ..., ωn produces the star topology µ. � remark 4.7. from a star compactification of µ = 〈〈u1, ..., um〉〉 on xm it is quite simple to generate new compactifications by n points, n > m, by repeating the ui as the open sets associated to new points. for instance, β = 〈〈u1, u2, ..., u7, u4, u3〉〉 is a star compactification of (x, τ) by nine points obtained from the star compactification by seven points µ = 〈〈u1, ..., u7〉〉 of (x, τ). moreover, if ⋄ is the equivalence relation on x9 that identifies ω4 with ω8 and ω3 with ω9 then (x9/⋄, β/⋄) is the compactification (x7, µ) . 5. star compactifications vs magill compactifications. a magill compactification refers to the method of compactification by n points of a non-compact topological space, presented by magill in [3]. in this section we show the relation between magill compactifications and star compactifications. 5.1. the magill compactifications. using the notation that we have introduced in this paper, we present the magill compactification with his results without proofs. proposition 5.1. if x contains n non-empty open subsets gi, i = 1, ..., n; two by two disjoint such that: (1) h = x \ n ∪ i=1 gi is compact and (2) x \ ∪ j 6=i gj is not compact for each i = 1, ..., n, then the collection b∗ = τ ∪ {a ∪ {ωi} | a ∈ τ, (h ∪ gi) ∩ (x \ a) is compact in x; i = 1, ..., n} is a base for a topology ρ on xn. observe that h ∪ gi = x \ ∪ j 6=i gj for each i = 1, ..., n because gi are mutually disjoint. under these conditions we have the following propositions. 26 l. acosta and i. m. rubio proposition 5.2. (xn, ρ) is a compactification of (x, τ) by n points. proposition 5.3. if (x, τ) is locally compact and t2 then (xn, ρ) is t2. remark 5.4. although magill always considers hausdorff topological spaces in [3], it is clear that this construction still provides a compactification of x without this assumption. 5.2. relation between star compactifications and magill compactifications. in this section we obtain that each magill compactification of x by n points is a star compactification. thus the collection of magill compactifications of x by n points coincides with the collection of star compactifications with associated two by two disjoint open sets. in the case of hausdorff compactifications, when x is hausdorff and locally compact, we have that the collections of magill compactifications of x by n points and of star compactifications of x by n points coincide and are all the possible hausdorff compactifications of x by n points. proposition 5.5. each star compactification of (x, τ) by n points, with associated two by two disjoint open sets is a magill compactification of (x, τ) by n points. proof. let µ be a star compactification 〈〈u1, ..., un〉〉 of (x, τ) where the sets ui are two by two disjoint. the sets ui are open, non-empty subsets of x and h = x \ n ∪ i=1 ui is compact. moreover, x \ ∪ j 6=i uj is not compact because in the contrary case, since ui ⊆ x \ ∪ j 6=i uj, with x \ ∪ j 6=i uj a subset of x closed and compact, this contradicts that (xn, µ) is a compactification of (x, τ) . therefore, the sets ui, i = 1, ..., n produce a magill compactification that we call ρ. we assert that µ = ρ. i) let (ui \ k) ∪ {ωi} be an element of b, base of the star compactification µ, where k is a closed and compact subset of x. to see that (ui \ k) ∪ {ωi} ∈ b ∗, base of the magill compactification ρ, we show that (h ∪ ui) ∩ (x \ (ui \ k)) is compact: (h ∪ ui) ∩ (x \ (ui \ k)) = ( x \ ∪ j 6=i uj ) ∩ (x \ (ui ∩ (x \ k))) = x \ (( ∪ j 6=i uj ) ∪ (ui ∩ (x \ k)) ) = x \ (( n ∪ j=1 uj ) ∩ (( ∪ j 6=i uj ) ∪ (x \ k) )) = ( x \ n ∪ j=1 uj ) ∪ ( x \ (( ∪ j 6=i uj ) ∪ (x \ k) )) = ( x \ n ∪ j=1 uj ) ∪ (( x \ ∪ j 6=i uj ) ∩ k ) , on star compactifications 27 where x \ n ∪ j=1 uj is compact and so is ( x \ ∪ j 6=i uj ) ∩ k because k is compact and x \ ∪ j 6=i uj is closed. thus (h ∪ ui) ∩ (x \ (ui \ k)) is compact, with ui \ k ∈ τ; then b ⊆ b ∗ and µ ⊆ ρ. ii) let a ∪ {ωi} be an element of b ∗, base of ρ, then (h ∪ ui) ∩ (x \ a) is compact and closed. thus ui \ [(h ∪ ui) ∩ (x \ a)] ∪ {ωi} ∈ b. ui \ [(h ∪ ui) ∩ (x \ a)] = [ui \ (h ∪ ui)] ∪ [ui \ (x \ a)] = [ ui \ ( x \ ∪ j 6=i uj )] ∪ [ui \ (x \ a)] = ∅ ∪ [ui \ (x \ a)] = a ∩ ui. since a ∈ τ, we have that a∪{ωi} = a∪[(a ∩ ui) ∪ {ωi}] ∈ µ and ρ ⊆ µ. � proposition 5.6. each magill compactification of (x, τ) by n points is a star compactification of (x, τ) by n points. proof. let ρ be a magill compactification of (x, τ) obtained through n open, non-empty subsets of x, gi, i = 1, ..., n two by two disjoint. let µ be the star topology 〈〈g1, ..., gn〉〉. reasoning as in the previous proposition we obtain that ρ = µ. thus ρ is a star compactification of x and we have that gi * k, for each k closed and compact subset of x. � so we have that the collection of magill compactifications of x by n points: mn and the collection of star compactifications of x by n points associated to n two by two disjoint open sets: edn are the same. from this fact and proposition 4.4 we obtain the following corollary. corollary 5.7. if (xn, µ) is a magill compactification of (x, τ) by n points and ⋄ = r ∪ {(x, x) | x ∈ x}, where r is an equivalence relation on {ω1, ..., ωn}, then (xn/⋄, µ/⋄) is a magill compactification of (x, τ) by m points, where m = |{ω1, ..., ωn} /r| ≤ n. remark 5.8. there exist star compactifications that are not magill compactifications. let x be the subspace (0, 1) of r with the usual topology, and x2 = x ∪ {ω1, ω2} , µ = 〈〈u1, u2〉〉 where u1 = ( 0, 1 2 ) , u2 = (0, 1) . it is clear that (x2, µ) is a star compactification of x by two points. if µ is a magill compactification, then by proposition 5.6 we have that µ = 〈〈g1, g2〉〉 where g1 and g2 are non-empty, open and disjoint subsets of x, that satisfy the other conditions to be a compactification. but since 〈〈g1, g2〉〉 is a compactification of x by two points, with g1 and g2 disjoints, it follows that there are a, b ∈ (0, 1) such that g1 = (0, a) , g2 = (b, 1) with a < b; without loss of generality we can suppose that a < 1 2 < b then, by proposition 3.5 we have that 〈〈u1, u2〉〉 ⊆ 〈〈g1, g2〉〉 . 28 l. acosta and i. m. rubio moreover, g2 ∪ {ω2} ∈ 〈〈g1, g2〉〉 but g2 ∪ {ω2} /∈ 〈〈u1, u2〉〉 because for all a ∪ {ω2} ∈ 〈〈u1, u2〉〉 there exists c ∈ (0, 1) such that (0, c) ⊆ a and clearly (0, c) * g2. therefore 〈〈u1, u2〉〉 $ 〈〈g1, g2〉〉 and (x2, µ) is a star compactification of x that is not a magill compactification. furthermore, for each n > 1 we have that mn $ en because (xn, ω)1 is a star compactification of (x, τ) that is not a magill compactification. we mention without proof theorem 0.5 of [7]: “let x be a locally compact hausdorff space. then any hausdorff compactification for x is the star topology associated with an m− tuple of open mutually disjoint subsets of x”. we denote: hcn the collection of t2 compactifications of x by n points, hen the collection of t2 star compactifications of x by n points and hmn the collection of t2 magill compactifications of x by n points. proposition 5.9. if (x, τ) is t2, locally compact and non-compact then hcn = hen = hmn. proof. i) clearly hen ⊆ hcn. by theorem 0.5 of [7] we have hcn ⊆ hen. ii) by proposition 5.6, hmn ⊆ hen. let µ be an element of hen ⊆ hcn, by theorem 0.5 of [7], µ is a t2 star compactification of x associated with n open mutually disjoint subsets of x. then µ ∈ hmn by proposition 5.5. � remark 5.10. observe that µ = 〈〈u1, u2〉〉 is not a t2 compactification of x, by remark 5.8. 6. some examples. in this section we present simple examples of star compactifications, and examples of compactifications of a topological space by a finite number of points that are not of a-class and, therefore, are not star compactifications. example 6.1. consider the topological space (r, τ) , where τ = {(−x, x) : x > 0} ∪ {φ, r} . to find the star compactifications by two points of this space we need two open sets u1, u2 such that l = r\ (u1 ∪ u2) is compact. since the only closed and compact set of this space is φ we need that u1 ∪ u2 = r and this happens if one of them is r. thus, for this space we have basically three types of star compactifications by two points: 〈〈u, r〉〉 , 〈〈r, u〉〉 and 〈〈r, r〉〉 , where u is an open set of the form (−x, x) . on the other hand, there exist infinity star compactifications by two points of type 〈〈u, r〉〉 , that depend on the open set u = (−x, x) that we consider. all 1considered as defined in proposition 3.1. on star compactifications 29 the compactifications of this type are ordered by inclusion, so that each nonempty subcollection of them has its intersection or its union in the same type of star compactifications. that is, we again obtain a star compactification. however, in general this fact is false. example 6.2. let x be the open interval (0, 1) of r with the usual topology of subspace. in this case there exist seven star compactifications by two points, which are classified in three types: (1) the lowest compactification 〈〈x, x〉〉 ; (2) four of the type 〈〈x, u2〉〉 or 〈〈u1, x〉〉 that are obtained accordingly if ui, for i = 1, 2 is either (0, a) or (a, 1) for some a, with 0 < a < 1; and (3) two maximal compactifications of the form 〈〈u1, u2〉〉 , where u1 = (0, a) and u2 = (b, 1) or on the contrary, where a, b ∈ (0, 1) and in this case the order between a and b is irrelevant because the different possibilities produce the same compactification. the two maximal compactifications can be seen as the compactification [0, 1] of (0, 1) where the additional points 0 and 1 correspond to ω1 and ω2; ω1 = 0 if u1 = (0, a) , or ω1 = 1 if u1 = (b, 1) . example 6.3. consider x = (0, 1) ∪ (2, 3) as a subspace of r with the usual topology. in this case the star compactifications by two points are classified in twelve different types. in these twelve, we find three types of maximal compactifications in which x2 can be represented as a subset of r2, where the basic neighborhoods of ω1 and ω2 are precisely obtained by the subspace topology of r2, with the usual topology. these can be represented as in figure 1: (x2, µ) (x2, ρ) (x2, ϕ) ω1 ω2 • • .... .... .... .... .... .... ..... ..... ..... ..... ...... ...... ....... ........ .......... ................................................................................................................................................................................................................................................... .......... ........ ....... ...... ...... ..... ..... ..... ..... ..... .... .... .... .... . ω1 ω2• • ..... .... .... ..... ..... ..... ...... ....... ........ ........................................................................................................................................................................ ........ ....... ...... ..... ..... ..... .... .... .... ...................................................................................... ω1 ω2 • • .... .... .... ..... ..... ..... ..... ...... ....... ........... .................................................................................................................................................................... ........ ...... ...... ..... ..... ..... .... .... .... ..... .... .... .... ..... ..... ..... ...... ....... ......... ........................................................................................................................................................................ ........ ...... ...... ..... ..... ..... .... .... .... .. x2 is the x2 is the union of x2 is the union of circumference. the segment and the two disjoint the circumference. circumferences. figure 1: maximal compactifications. some intermediate compactifications are: • η = 〈〈u1, u2〉〉 , where u1 = (0, 1) and u2 = (b, 1)∪(2, 3) , with 0 < b < 1. we have that η ⊂ ϕ and η ⊂ ρ. • γ = 〈〈v1, v2〉〉 , where v1 = (0, 1) ∪ (2, a) and v2 = (b, 1) ∪ (2, 3) , with 2 < a < 3 and 0 < b < 1. we have that γ ⊂ η. 30 l. acosta and i. m. rubio • α = 〈〈w1, w2〉〉 , where w1 = (0, a) ∪ (2, 3) and w2 = (b, 1) ∪ (2, 3) , with 0 < a < 1 and 0 < b < 1. notice that α and γ are incomparable. we know that every star compactification is of a-class, but there exist compactifications of a-class that are not star compactifications. this is illustrated in the following example. example 6.4. consider the topological space (x, τ) of the real numbers with the usual topology and the simplest possible compactification: (xn, µ) , where µ = τ ∪{xn} , for n ≥ 1. we can easily verify that (xn, µ) is a compactification of a-class of (x, τ) that is not star. the following propositions provide examples of non a-class compactifications. proposition 6.5. if (x1, µ) is a compactification of (x, τ) by one point, where x1 = x ∪ {ω} , then τ ⊆ µ. proof. i) x ∈ µ. in fact, since (x, τ) is not compact, there exists a covering {ai | i ∈ i} of x by open sets of τ that cannot be reduced to a finite covering. for each i ∈ i there exists bi ∈ µ such that ai = bi ∩ x. x1 = ∪ i∈i bi =⇒ x1 = n ∪ k=1 bik (because x1 is compact). =⇒ x = n ∪ k=1 (bik ∩ x) = n ∪ k=1 aik. this is a contradiction and then x1 6= ∪ i∈i bi. but x1 % ∪ i∈i bi =⇒ ω /∈ bi, for all i ∈ i =⇒ bi ⊆ x, for all i ∈ i =⇒ ai = bi, for all i ∈ i =⇒ x ∈ µ. ii) a ∈ τ =⇒ a = b ∩ x for some b ∈ µ =⇒ a ∈ µ. (because x ∈ µ.) � the previous proposition asserts that x is an open set in every compactification of (x, τ) by one point. this is equivalent to saying that each element of τ is an element of µ. thus, each compactification of (x, τ) by one point is obtained with the original open sets and by adding some additional sets in an appropriate way. however, this fact cannot be generalized to compactifications by more than one point. definition 6.6. we say that a topological space (y, υ) is hyperconnected if υ \ {∅} is a collection closed for finite intersections, that is to say, if each pair of non-empty open sets has non-empty intersection. on star compactifications 31 some examples: (1) (y, ϕ) , where y is an infinite set and ϕ is the cofinite topology, is hyperconnected. (2) (r, τ), where τ is the topology with base {(−a, a) ⊂ r |a > 0}, is hyperconnected. (3) (r, µ), where µ is the topology with base {(a, +∞) ⊂ r |a ∈ r}, is hyperconnected. (4) r with the usual topology is not a hyperconnected space. the next proposition provides examples of compactifications by more than one point that are not of a-class. proposition 6.7. if (x, τ) is a hyperconnected, non-compact topological space then (x, τ) has a compactification by more than one point in which x is not an open set. proof. let x2 be the set x ∪ {ω1, ω2} µ = {a ∪ {ω1} | a ∈ τ \ {∅}} ∪ {∅, x2} . it is clear that µ is a topology on x2 and x /∈ µ. i) (x2, µ) is compact because if {bi | i ∈ i} is a covering of x2 by open sets of µ, x2 = bi, for some i ∈ i, and then this covering can be reduced to a finite one. ii) µ |x= τ. iii) x is dense in x2, because on the contrary some {ω1} , {ω2} or {ω1, ω2} will be open sets of µ. thus, (x2, µ) is a compactification of (x, τ) by more than one point, in which x is not a open set. � remark 6.8. in the previous proposition we can take y = x ∪z, where ω1 ∈ z, x ∩z = ∅ and z is a finite or infinite set that contains more than one element. following the proof we obtain that (y, µ) is a compactification of (x, τ) by more than one point that does not contain x as an open set. the next fact can be proved in a simple way. proposition 6.9. if (xn, µ) is a t1 compactification of (x, τ) then (xn, µ) is of a-class. question 6.10. if we consider other types of compactifications of (x, τ) by n points, what is the relationship between these types of compactifications and star (magill) compactifications? acknowledgements. we would like to thank professors lucimar nova and januario varela for their useful comments. 32 l. acosta and i. m. rubio references [1] j. blankespoor and j. krueger, compactifications of topological spaces, electronic journal of undergraduate mathematics, furman university. 2 (1996), 1–5. [2] n. bourbaki, elements of mathematics, general topology, addison-wesley, france, 1968. [3] k. d. magill jr., n-point compactifications, amer. math. monthly. 72 (1965), 10751081. [4] j. margalef et al. topoloǵıa, vol. 3. alhambra, madrid, 1980. [5] j. r. munkres, topology. a first course, prentice-hall, new jersey, 1975. [6] m. murdeshwar, general topology, john wiley and sons, new york, 1983. [7] t. nakassis and s. papastavridis, on compactifying a topological space, by adding a finite number of points, bull. soc. math., grece. 17 (1976), 59-65. [8] c. rúız and l. blanco, acerca del compactificado de alexandroff, bol. mat. (3) 20 (1986), 163-171. [9] s. willard, general topology, addison-wesley, publishing company, 1970. (received august 2011 – accepted november 2012) l. acosta (lmacostag@unal.edu.co) universidad nacional de colombia, facultad de ciencias, departamento de matemáticas, carrera 30 no. 45-03, bogotá, colombia. i. m. rubio (imrubiop@unal.edu.co) universidad nacional de colombia, facultad de ciencias, departamento de matemáticas, carrera 30 no. 45-03, bogotá, colombia. on star compactifications. by l. acosta and i. m. rubio @ appl. gen. topol. 22, no. 2 (2021), 345-354doi:10.4995/agt.2021.14809 © agt, upv, 2021 sum connectedness in proximity spaces beenu singh a and davinder singh b a department of mathematics, university of delhi, delhi (india) 110007. (singhbeenu47@gmail.com) b department of mathematics, sri aurobindo college (m), university of delhi, delhi (india) 110017. (dstopology@gmail.com) communicated by p. das abstract the notion of sum δ-connected proximity spaces which contain the category of δ-connected and locally δ-connected spaces is defined. several characterizations of it are substantiated. weaker forms of sum δ-connectedness are also studied. 2010 msc: 54e05; 54d05. keywords: sum δ-connected; δ-connected; δ-component; locally δconnected. 1. introduction the notion of proximity was introduced by efremovic [4, 5] as a natural generalization of metric spaces and topological groups. smirnov [10, 11] and naimpally [8, 9] did the most significant and extensive work in this area. in 2009, bezhanishvili [1] defined zero-dimensional proximities and zero-dimensional compactifications. mrówka et al. [7] introduced the theory of δ-connectedness (or equiconnectedness) in proximity spaces. consequently, dimitrijević et al. [2, 3] defined local δ-connectedness, δ-component and the treelike proximity spaces. in 1978, kohli [6] introduced the notion of sum connectedness in topological spaces. we discuss sum δ-connectedness in proximity spaces in this paper. some necessary definitions and the results which are used in further sections, are recalled in section 2. in section 3, sum δ-connectedness is defined and its relations with other kinds of connectedness are determined. several characterizations of received 16 december 2020 – accepted 18 march 2021 http://dx.doi.org/10.4995/agt.2021.14809 b. singh and d. singh it are established. it is shown that sum δ-connectedness is equivalent to local δconnectedness in a zero-dimensional proximity space. further, the stone-čech compactification of a separated proximity space x is sum δ-connected if and only if x is sum δ-connected and it has finitely many δ-components. for a sum δ-connected proximity space to be sum connected, a sufficient condition is deduced. in the last section, weaker forms of sum δ-connectedness are defined. finally, if a sum δ-connected space is δ-padded, then it is also locally δ-connected. 2. preliminaries definition 2.1 ([9]). a binary relation δ on the power set p(x) of x is said to be a proximity on x, if the following axioms are satisfied for all p , q, r in p(x): (i) (φ, p) /∈ δ; (ii) if p ∩ q 6= φ, then (p, q) ∈ δ; (iii) if (p, q) ∈ δ, then (q, p) ∈ δ; (iv) (p, q ∪ r) ∈ δ if and only if (p, q) ∈ δ or (p, r) ∈ δ; (v) if (p, q) /∈ δ, then there exists a subset r of x such that (p, r) /∈ δ and (x\r, q) /∈ δ. the pair (x, δ) is called a proximity space. throughout this paper, we simply write proximity space (x, δ) as x whenever there is no confusion of the proximity δ. definition 2.2 ([8, 9]). a proximity space x is said to be separated if x = y whenever ({x}, {y}) ∈ δ for x, y ∈ x. proposition 2.3 ([9]). let x be a proximity space and p be a subset of x. if p is δ-closed if and only if x ∈ p whenever ({x}, p) ∈ δ, then the collection of the complements of all δ-closed sets forms a topology tδ on x. proposition 2.4 ([9]). let x be a proximity space. then the closure c(p) of p with respect to tδ is given by c(p) = {x ∈ x : ({x}, p) ∈ δ}. corollary 2.5 ([9]). let x be a proximity space. then m ∈ tδ if and only if ({x}, x\m) /∈ δ for every x ∈ m. using proposition 2.4, a set f is δ-closed if c(f) = f . from corollary 2.5, a set u is δ-open, if ({x}, x\u) /∈ δ for every x ∈ u. definition 2.6 ([9]). let x be a proximity space and t be a topology on x. then δ is said to be compatible with t if the generated topology tδ and t are equal, that is, tδ = t . definition 2.7 ([9]). let x be a proximity space. then a subset n of x is said to be a δ-neighbourhood of m ⊂ x if (m, x\n) /∈ δ. it is denoted by m ≪δ n. © agt, upv, 2021 appl. gen. topol. 22, no. 2 346 sum connectedness in proximity spaces definition 2.8 ([9]). let (x, δ) and (y, δ′) be two proximity spaces. then a map f : (x, δ) −→ (y, δ′) is said to be δ-continuous ( or p-continuous ) if (f(p), f(q)) ∈ δ′ whenever (p, q) ∈ δ, for all p, q ⊂ x. definition 2.9 ([7]). let x be a proximity space. then x is said to be δconnected if every δ-continuous map from x to a discrete proximity space is constant. theorem 2.10 ([7]). let x be a proximity space. then the following statements are equivalent: (i) x is δ-connected. (ii) (p, x\p) ∈ δ for each nonempty subset p with p 6= x. (iii) for every δ-continuous real-valued function f, the image f(x) is dense in some interval of r. (iv) if x = p ∪ q and (p, q) /∈ δ, then either p = φ or q = φ. definition 2.11 ([2]). let x be a proximity space and x ∈ x. then the δ-component of a point x is defined as the union of all δ-connected subsets of x containing x. it is denoted by cδ(x). definition 2.12 ([2]). let x be a proximity space and x ∈ x. then the δ-quasi component of x is the equivalence class of x with respect to the equivalence relation ∼ defined on x as “ x ∼ y if and only if there do not exist the sets m, n such that x ∈ m and y ∈ n with x = m ∪ n and (m, n) /∈ δ”. definition 2.13 ([2]). a proximity space x is called locally δ-connected if for every point x of x and for every δ-neighbourhood n of x, there exists some δ-connected δ-neighbourhood m of x such that x ∈ m ⊂ n. definition 2.14 ([12]). let (x, δ) be a proximity space and f : x −→ y be a surjective map, where y is any set. then the quotient proximity on y is the finest proximity such that the map f is δ-continuous. when y has the quotient proximity, f is called δ-quotient map. proposition 2.15 ([12]). let (x, δ) be a proximity space and f : x −→ y be a surjective map, where y be any set. then the quotient proximity δ′ on y is given by p ≪δ′ q if and only if for each binary rational s ∈ [0, 1], there is some ps ⊆ y such that p0 = p, p1 = q and s < t implies f −1(ps) ≪δ f −1(pt). proposition 2.16 ([12]). let (x, δ) be a proximity space and f : x −→ y be a surjective map such that f−1(f(m)) = m for each δ-open set m of x, where y be any set. then the quotient proximity δ′ on y is given by (p, q) ∈ δ′ if and only if (f−1(p), f−1(q)) ∈ δ. definition 2.17 ([1]). a proximity space x is said to be zero-dimensional if the proximity δ satisfies the following axiom: if (p, q) /∈ δ, then there is a subset r of x such that (r, x\r) /∈ δ, (p, r) /∈ δ and (x\r, q) /∈ δ. © agt, upv, 2021 appl. gen. topol. 22, no. 2 347 b. singh and d. singh definition 2.18 ([6]). a topological space x is said to be sum connected at x ∈ x, if there exists an open connected neighbourhood of x. if x is sum connected at each of its points, then x is called sum connected. proposition 2.19 ([6]). let x∗ be the stone-čech compactification of a tychonoff space x. then x is sum connected and has finitely many components, if x∗ is sum connected. 3. sum δ-connectedness definition 3.1. a proximity space x is said to be sum δ-connected at x ∈ x if there exists a δ-connected δ-open δ-neighbourhood of x. if x is sum δconnected at each of its points, then it is said to be sum δ-connected. definition 3.2. let (xi, δi)i∈i be a family of proximity spaces, where i is an index set. a proximity space (x, δ) is said to be a far proximity sum of (xi)i∈i if x = ⋃ i∈i xi and (xi, xj) /∈ δ for all i 6= j in i with δ|xi = δi for all i ∈ i. note that a proximity space x is sum δ-connected if and only if each of its δ-component is δ-open. therefore, every δ-connected proximity space is sum δ-connected. example 3.3. (i) let x be any discrete proximity space with |x| ≥ 2. then x is sum δ-connected but not δ-connected. (ii) let x = (0, 1) ∪ (2, 3) with usual subspace proximity of r. then x is sum δ-connected but not δ-connected. every sum connected proximity space is sum δ-connected. but, converse may not be true. however, in compact separated proximity spaces, the notion of sum connectedness and sum δ-connectedness coincides. example 3.4. the space q of rationals with the usual proximity is sum δconnected. but, it is not sum connected. every locally δ-connected proximity space is sum δ-connected. converse may not be true. example 3.5. consider t = {(x, sin 1 x ) : 0 < x ≤ 1} ∪ {(0, y) : −1 ≤ y ≤ 1} the closed topologist’s sine curve with subspace proximity induced from r2. let x be the far proximity sum of two copies of t . then x is sum δ-connected but it is neither δ-connected nor locally δ-connected. example 3.6. let x = {0} ∪ { 1 n : n ∈ n} be a proximity space. since each { 1 n } is δ-clopen in x, there does not exists any δ-connected δ-neighbourhood of 0 in x because every δ-neighbourhood of 0 contains infinitely many members of x\{0}. thus, x is not sum δ-connected at 0. thus, we have following relationship among several connectednesses in proximity space. © agt, upv, 2021 appl. gen. topol. 22, no. 2 348 sum connectedness in proximity spaces connected =⇒ sum connected ⇐= locally connected ⇓ ⇓ ⇓ δ − connected =⇒ sum δ − connected ⇐= locally δ − connected the following theorem gives some necessary and sufficient conditions for sum δ-connectedness. theorem 3.7. for a proximity space x, the following statements are equivalent: (i) x is sum δ-connected. (ii) for each x ∈ x and each δ-clopen set u which contains x, there exists a δ-open δ-connected set w containing x such that w ⊂ u. (iii) δ-components of δ-clopen sets in x are δ-open in x. proof. (i) =⇒ (ii). let x ∈ x and u be a δ-clopen set such that x ∈ u. let cδ(x) be the δ-component of x containing x. by hypothesis, cδ(x) is δ-open. so, cδ(x) ∩ u is δ-clopen. therefore, ((cδ(x) ∩ u), cδ(x)\(cδ(x) ∩ u)) /∈ δ as cδ(x) ∩ u ⊂ cδ(x) ⊂ x. also, since cδ(x) is δ-connected, cδ(x) ∩ u = cδ(x). hence, cδ(x) is a δ-open δ-connected such that cδ(x) ⊂ u. (ii) =⇒ (iii). let u be any δ-clopen set in x and cδ be a δ-component of u. then, by hypothesis, for each x ∈ cδ there exists a δ-open δ-connected set w such that x ∈ w ⊂ u. therefore, w ⊂ cδ as cδ is δ-component. hence, cδ is δ-open. (iii) =⇒ (i). since x is δ-clopen, the result follows. � proposition 3.8. let y be a dense proximity subspace of x and x ∈ y . then x is sum δ-connected at x if y is sum δ-connected at x. proof. let w be a δ-open δ-connected δ-neighbourhood of x in y . therefore, w = u ∩ y , where u is δ-open δ-neighbourhood of x in x. thus, w ⊂ u and u ⊂ clx(u) = clx(w) as y is dense in x. note that clx(w) is δ-connected. hence, u is δ-open δ-connected δ-neighbourhood of x in x. � next example shows that the closure of sum δ-connected proximity space may not be sum δ-connected. example 3.9. let x = { 1 n : n ∈ n} be a proximity subspace of r. then each δ-component { 1 n } is δ-clopen in x. so, x is sum δ-connected. but, note that cl(x) = {0} ∪ { 1 n : n ∈ n} is not sum δ-connected at 0 by example 3.6. proposition 3.10. let x be a sum δ-connected proximity space and f : (x, δ) −→ (y, δ∗) be a δ-quotient map such that f−1(f(u)) = u for each δ-open subset u of x. then y is sum δ-connected. proof. let cδ be any δ-component of y and y ∈ cδ. we have to show that (y, y \cδ) /∈ δ ∗. by definition of δ-quotient proximity δ∗, it suffices to show that (f−1(y), x\f−1(cδ)) /∈ δ. let x ∈ f −1(y), then the δ-component cx of x in x, be δ-open in x. therefore, (z, x\cx) /∈ δ for every z ∈ cx. since © agt, upv, 2021 appl. gen. topol. 22, no. 2 349 b. singh and d. singh f is δ-continuous, f(cx) is δ-connected. thus, y = f(x) ∈ f(cx) ∩ cδ. so f(cx) ⊆ cδ, which implies cx ⊆ f −1(cδ). then, (z, x\f −1(cδ)) /∈ δ for every z ∈ cx. in particular, (f −1(y), x\f−1(cδ) /∈ δ. � corollary 3.11. let f : (x, δ) −→ (y, δ∗) be a δ-continuous, δ-closed, surjection such that f−1(f(u)) = u for each δ-open subset u of x. if x is sum δ-connected, then y is also sum δ-connected. proposition 3.12. every δ-continuous, δ-open image of a sum δ-connected proximity space is sum δ-connected. proof. let f : (x, δ) −→ (y, δ′) be a δ-continuous, δ-open, surjective map and x be sum δ-connected. let cδ be a δ-component of y and x ∈ f −1(cδ). then there is a δ-component cx in x containing x which is δ-open. since f is δ-continuous and δ-open, f(cx) ⊆ cδ and f(cx) is δ-open. therefore, (f(x), y \f(cx)) /∈ δ ′. hence, (f(x), y \cδ) /∈ δ ′. � corollary 3.13. if the product of proximity spaces is sum δ-connected, then each of its factor is also sum δ-connected. the product of sum δ-connected proximity spaces need not be sum δ-connected in general. example 3.14. let x = {0, 1}ω be infinite product of two point discrete proximity spaces. then x is not discrete proximity space. therefore, the δcomponent cδ(x) of x in x is {x} itself, which is not δ-open. hence, x is not sum δ-connected. theorem 3.15. let (x, δ) be a product of proximity spaces (xi, δi)i∈i, where i is an index set. then x = ∏ i∈i xi is sum δ-connected if and only if each xi is sum δ-connected and all but finitely many xi’s are δ-connected. proof. let x be sum δ-connected. so, by corollary 3.13, each xi is sum δconnected. now, suppose that all but finitely many xi’s are not δ-connected. then any δ-component of x is not δ-open in x, which is a contradiction. conversely, assume that each xi is sum δ-connected and all but finitely many xi’s are δ-connected. let cδ be any δ-component of x and pi be the ith projection map. then pi(cδ) is δ-connected for each i ∈ i. therefore,∏ i∈i pi(cδ) is also δ-connected. thus, cδ = ∏ i∈i pi(cδ). for each i ∈ i, suppose cδi be the δi-component of xi containing pi(cδ). put c ′ δ = ∏ i∈i cδi. if pi(cδ) ( cδi, then cδ = c ′ δ as cδ is δ-component of x. thus, pi(cδ) = cδi for each i ∈ i. since all but finitely many xi’s are δ-connected, pi(cδ) = cδi = xi for all but finitely many i ∈ i. hence, cδ is δ-open set in x. � theorem 3.16. every far proximity sum of sum δ-connected proximity spaces is sum δ-connected. it can be easily shown that a δ-closed subspace of sum δ-connected proximity space need not be sum δ-connected. © agt, upv, 2021 appl. gen. topol. 22, no. 2 350 sum connectedness in proximity spaces corollary 3.17. a proximity space x is locally δ-connected if and only if every δ-open subspace of x is sum δ-connected. theorem 3.18. let x be a pseudocompact, separated, sum δ-connected proximity space. then it has at most finitely many δ-components. proof. suppose x has infinitely many δ-components. since collection of δcomponents of x is locally finite and each δ-component of x is δ-open, we have a locally finite collection of non-empty δ-open sets which is not finite, a contradiction. � corollary 3.19. if x is compact sum δ-connected proximity space, then it has at most finitely many δ-components. corollary 3.20. if x is lindelof (or separable) sum δ-connected proximity space, then it has at most countably many δ-components. theorem 3.21. every separated, zero-dimensional, sum δ-connected proximity space is discrete. proof. let x be any separated, zero-dimensional, sum δ-connected proximity space. let s be a subset of x such that x, y ∈ s with x 6= y. therefore, ({x}, {y}) /∈ δ. then, there exists c ⊂ x such that (c, x\c) /∈ δ, ({x}, c) /∈ δ and (x\c, {y}) /∈ δ. so, (c, s\c) /∈ δ which implies s is not δ-connected. hence, every δ-component of x is singleton. as x is sum δ-connected, each singleton of x is δ-open. � next theorem shows that in a zero-dimensional proximity space, local δconnectedness and sum δ-connectedness are equivalent. proposition 3.22. a zero-dimensional proximity space x is locally δ-connected if and only if it is sum δ-connected. proof. necessity is obvious. for the sufficient part, let x be sum δ-connected. let x ∈ x and u be a δ-neighbourhood of x. therefore, there exists c ⊂ x such that (c, x\c) /∈ δ, ({x}, x\c) /∈ δ and (c, x\u) /∈ δ. thus, c is δclopen and x ∈ c ⊂ u. so, by theorem 3.7, there exists a δ-open δ-connected set w such that x ∈ w ⊂ c ⊂ u. hence, x is locally δ-connected. � now, we find the relation of sum δ-connectedness of proximity space with its stone-čech compactification. theorem 3.23. let (x∗, δ∗) be the stone-čech compactification of the separated proximity space (x, δ). then x∗ is sum δ-connected if and only if x is sum δ-connected and has finitely many δ-components. proof. let x∗ be sum δ-connected. then, by corollary 3.19, it has finitely many δ-components. so, x∗ = ⋃n i=1 ciδ, where c i δ is a δ-component of x ∗ for each 1 ≤ i ≤ n. therefore, x = ⋃n i=1 (ciδ ∩ x). as each c i δ ∩ x is δ-open in x and (ciδ ∩ x, c j δ ∩ x) /∈ δ by using hypothesis, it suffices to show that each ciδ ∩ x is δ-connected. let c i δ ∩ x = p ∪ q with (p, q) /∈ δ ∗. note that © agt, upv, 2021 appl. gen. topol. 22, no. 2 351 b. singh and d. singh clδ∗(c i δ∩x) = c i δ because c i δ is δ-open in x ∗ and x is dense in x∗. therefore, ciδ = clδ∗(c i δ ∩ x) = clδ∗(p) ∪ clδ∗(q) with (clδ∗ (p), clδ∗(q)) /∈ δ ∗. thus, ciδ is not δ-connected, a contradiction. conversely, assume x is sum δ-connected and has finitely many δ-components. therefore, x = ⋃n i=1 ciδ where c i δ is a δ-component of x for each 1 ≤ i ≤ n. thus, x∗ = clδ∗(x) = ⋃n i=1 clδ∗(c i δ). since, (c i δ, c j δ ) /∈ δ for i 6= j, (clδ∗(c i δ), clδ∗ (c j δ )) /∈ δ∗. note that each clδ∗(c i δ) is δ-connected in x ∗. thus, each clδ∗(c i δ) is a δ-component in x ∗. since δ-components in x∗ are finite, hence x∗ is sum δ-connected. � corollary 3.24. if x is pseudocompact, separated and sum δ-connected proximity space, then it’s stone-čech compactification x∗ is also sum δ-connected. every sum connected proximity space is sum δ-connected. following theorem gives the sufficient condition for a sum δ-connected proximity space to be sum connected. theorem 3.25. let (x, t ) be a tychonoff space. if x is sum δ-connected and has finitely many δ-components with respect to any proximity δ compatible with t , then x is sum connected. moreover, it has at most finitely many components. proof. let s be the collection of all proximities which are compatible with t . let δ0 = sup s, then δ0 is also compatible with t . therefore, by hypothesis, x is sum δ-connected and has finitely many δ-components with respect to δ0. since δ0 = sup s, the compactification (x ∗, δ∗) corresponding to δ0 is stonečech compactification. so, by theorem 3.23, x∗ is sum δ-connected. thus, x∗ is sum connected. by proposition 2.19, x is sum connected and has finitely many components. � 4. weaker forms of sum δ-connectedness in this section we give proximity versions of notions defined and considered in [6]. definition 4.1. let x be a proximity space which contains a point x. then x is called : (i) weakly sum δ-connected at x if there exists a δ-connected δ-neighbourhood of x. (ii) quasi sum δ-connected at x if the δ-quasi component which contains x is a δ-neighbourhood of x. (iii) δ-padded at x if for every δ-neighbourhood w of x there exist δ-open sets u and v such that x ∈ u ⊆ clδ(u) ⊆ v ⊆ w and v \clδ(u) has at most finitely many δ-components. if a proximity space x is weakly sum δ-connected (or quasi sum δ-connected) at each of its points, then the space x is called weakly sum δ-connected (or quasi sum δ-connected). for a proximity space x, © agt, upv, 2021 appl. gen. topol. 22, no. 2 352 sum connectedness in proximity spaces sum δ-connected ⇒ weakly sum δ-connected ⇒ quasi sum δ-connected example 4.2. in r2, let bn be the infinite broom containing all the closed line segments joining the point ( 1 n , 0) to the points {( 1 n+1 , 1 m ) : m = n, n + 1, · · · }, where n = 1, 2, · · · . let b = ⋃ ∞ n=1 bn and a = {(x, 0) : 0 ≤ x ≤ 2} ∪ {(y, 1 n ) : 1 ≤ y ≤ 2 and n = 1, 2, · · ·}. let x = a ∪ b. then note that x is compact. therefore, connectedness is equivalent to δ-connectedness. hence, x is weak sum δ-connected but not sum δ-connected at (0, 0). lemma 4.3. every δ-open δ-quasi component is a δ-component. proof. let u be a δ-open δ-quasi component of proximity space x and x ∈ u. let v be the δ-component of x. then v ⊂ u. let y ∈ u\v . so, x ∼ y. since v is δ-closed in x and v ⊂ u, v is δ-closed in u. so, u\v is δ-open in u. as u is δ-open in x, u\v is δ-open in x. therefore, (u\v, x\(u\v )) /∈ δ. thus, x = (u\v ) ∪ (x\(u\v )) with (u\v, x\(u\v )) /∈ δ. hence, x ≁ y which is a contradiction. � proposition 4.4. for a given proximity space x, the following statements are comparable: (i) x is quasi sum δ-connected. (ii) x is weakly sum δ-connected. (iii) x is sum δ-connected. (iv) δ-components of x are δ-open. (v) δ-quasi components of x are δ-open. proof. by lemma 4.3, δ-open δ-quasi component is a δ-component. therefore the statements (iv) and (v) are equivalent. the equivalence of (iv) with (i), (ii), (iii) follows from the fact that a set is δ-open if and only if it is a δ-neighbourhood of each of its points. � corollary 4.5. a proximity space x is sum δ-connected if and only if it is the far proximity sum of its δ-components (δ-quasi components). corollary 4.6. let x be a sum δ-connected proximity space. then the map f on x is δ-continuous if and only if it is δ-continuous on each of its δcomponent. corollary 4.7. every locally δ-connected proximity space is the far proximity sum of its δ-components (δ-quasi components). corollary 4.8. if x is sum δ-connected proximity space and u ⊂ x, then u is a δ-component if and only if it is δ-quasi component. in particular, if y is a locally δ-connected proximity space and x ⊂ y is δ-open, then u ⊂ x is δ-component if and only if it is δ-quasi component. proof. by proposition 4.4 (iv), δ-components and δ-quasi components coincide in sum δ-connected proximity space. the last statement of corollary from the fact that every locally δ-connected proximity space is sum δ-connected; © agt, upv, 2021 appl. gen. topol. 22, no. 2 353 b. singh and d. singh and every δ-open subset of a locally δ-connected proximity space is locally δ-connected. � as in example 3.5, sum δ-connected proximity space may not be locally δconneted. but, if sum δ-connected proximity space is δ-padded, then it is also locally δ-connected. proposition 4.9. let x be a sum δ-connected proximity space and x ∈ x. if x is δ-padded at x, then it is locally δ-connected at x. proof. let n be a δ-open δ-neighbourhood of x. as x is sum δ-connected, suppose that n is contained in δ-component cδ. since x is δ-padded at x, there are δ-open δ-neighbourhoods w and v of x such that clδ(w) ⊆ v ⊆ n with v \clδ(w) has only finitely many δ-components c 1 δ , c 2 δ , · · · , c n δ . now for each i, 1 ≤ i ≤ n, there exist a δ-quasi component qiδ such that c i δ ⊆ q i δ. we show that each v ∈ v is in some qiδ. if there is some v ∈ v such that v /∈ q i δ for each 1 ≤ i ≤ n, then for each i we have v = (v \qiδ)∪q i δ with (v \q i δ, q i δ) /∈ δ. let wi = v \q i δ for each 1 ≤ i ≤ n and m = ⋂ i wi. since (v \q i δ, q i δ) /∈ δ for each 1 ≤ i ≤ n, (m, qiδ) /∈ δ. note that cδ\m = ⋃ i cδ\wi and for each i, cδ\wi = (cδ\v ) ∪ q i δ. as v is δ-open in cδ, (v, cδ\v ) /∈ δ which implies (m, cδ\v ) /∈ δ. thus, (m, (cδ\v ) ∪ q i δ) /∈ δ, that is, (m, cδ\wi) /∈ δ for each i. therefore, (m, cδ\m) /∈ δ. therefore, cδ is not δ-connected, a contradiction. thus, each v ∈ v is in some qiδ. therefore, v has only finitely many δ-quasi components and each of them is δ-open. thus, each δ-quasi component is a δ-component. hence, δ-component of x in v is δ-connected δ-open neighbourhood of x contained in n. � references [1] g. bezhanishvili, zero-dimensional proximities and zero-dimensional compactifications, topology appl. 156 (2009), 1496–1504. [2] r. dimitrijević and lj. kočinac, on connectedness of proximity spaces, matem. vesnik 39 (1987), 27–35. [3] r. dimitrijević and lj. kočinac, on treelike proximity spaces, matem. vesnik 39, no. 3 (1987), 257–261. [4] v. a. efremovic, infinitesimal spaces, dokl. akad. nauk sssr 76 (1951), 341–343 (in russian). [5] v. a. efremovic, the geometry of proximity i, mat. sb. 31 (1952), 189–200 (in russian). [6] j. k. kohli, a class of spaces containing all connected and all locally connected spaces, math. nachr. 82 (1978), 121–129. [7] s. g. mrówka and w. j. pervin, on uniform connectedness, proc. amer. math. soc. 15 (1964), 446–449. [8] s. naimpally, proximity approach to problems in topology and analysis, oldenbourg verlag, münchen, 2009. [9] s. naimpally and b. d. warrack, proximity spaces, cambridge univ. press, 1970. [10] y. m. smirnov, on completeness of proximity spaces i, amer. math. soc. trans. 38 (1964), 37–73. [11] y. m. smirnov, on proximity spaces, amer. math. soc. trans. 38 (1964), 5–35. [12] s. willard, general topology, addison-wesley publishing co., reading, mass.-londondon mills, ont., 1970. © agt, upv, 2021 appl. gen. topol. 22, no. 2 354 @ appl. gen. topol. 22, no. 2 (2021), 447-459doi:10.4995/agt.2021.15566 © agt, upv, 2021 index boundedness and uniform connectedness of space of the g-permutation degree r. b. beshimov a, d. n. georgiou b and r. m. zhuraev a a national university of uzbekistan named after mirzo ulugbek, str. university, 100174 tashkent, uzbeksitan (rbeshimov@mail.ru,rmjurayev@mail.ru) b university of patras, department of mathematics, 26504 patras, greece (georgiou@math.upatras.gr) communicated by s. romaguera abstract in this paper the properties of space of the g-permutation degree, like: weight, uniform connectedness and index boundedness are studied. it is proved that: (1) if (x, u) is a uniform space, then the mapping πsn, g : (x n , u n) → (sp ngx, sp n gu) is uniformly continuous and uniformly open, moreover w (u) = w (sp ngu); (2) if the mapping f : (x, u) → (y, v) is a uniformly continuous (open), then the mapping sp ngf : (sp n gx, sp n gu) → (sp n gy, sp n gv) is also uniformly continuous (open); (3) if the uniform space (x, u) is uniformly connected, then the uniform space (sp ngx, sp n gu) is also uniformly connected. 2010 msc: 54a05; 54e15; 55s15. keywords: g-permutation degree space; uniform space; uniform connectedness; index boundedness of uniform space; uniform continuity. 1. introduction in [19], a functor o : comp → comp of weakly additive functionals acting in the category of compact and its continuous mappings is defined. it was proved that the functor o : comp → comp satisfies the normality conditions, except the preimage preservation condition. in [6], categorical and cardinal received 05 may 2021 – accepted 08 june 2021 http://dx.doi.org/10.4995/agt.2021.15566 r. b. beshimov, d. n. georgiou and r. m. zhuraev properties of hyperspaces with finite number of components are investigated. it was proved that the functor cn : comp → comp is not normal, i.e., it does not preserve epimorphisms of continuous mappings. the authors of this paper also discussed the properties of density, caliber and shanin number for the space cn(x). this space cn(x) is of great interest for researchers, since it contains the hyperspaces expn x of closed sets with cardinalities not greater than n elements. it was proved in [4] that the radon functor satisfies all the normality conditions. in [7], the topological properties of topological groups were studied. in [5], categorical and topological properties of the functor osτ of semiadditive τ-smooth functionals in the category t ych of tychonoff spaces and their continuous mappings, which extends the functors os of semiadditive functionals in the category comp of compact and their continuous mappings, were investigated. in [3], some properties of the functor oβ : t ych → t ych were considered, where β is the čech-stone compact extension in the category of tychonoff spaces and their continuous mappings. this functor is regarded as an extension of the functor o : comp → comp. the author in [3] proved that the space oβ(x) is a convex subset of the space cp(cb(x)), where cb(x) = {f ∈ c(x)| f : x → r is a bounded function} and cp(x) is the space of pointwise convergence. it was proved in [2] that if a covariant functor f : comp → comp is weakly normal, then fβ : t ych → t ych does not increase the density and weak density for any infinite tychonoff space. in [11], it was proved that the functor sp ng preserves the property of the fibers of the map to be a compact q-manifold. in [9] some classes of uniform spaces are considered. in particular, the uniformly continuous mappings and absolutes, generalizations of metrics, normed, uniform unitary spaces, topological and uniform groups, its completions and spectral characterizations are studied. in addition, the properties of uniformly continuous and uniformly open mappings between uniform spaces are studied, too. but, it should be noted here that the class of uniformly continuous and uniformly open maps itself was introduced by michael in [18]. in what follows, we present the basic notions that will be used in the rest of this article. it is known that a permutation group is the group of all permutations, that is one-to-one mappings x → x. a permutation group of a set x is usually denoted by s(x). especially, if x = {1, 2, . . . , n}, then s(x) is denoted by sn. let xn be the n-th power of a compact space x. the permutation group sn of all permutations acts on the n-th power x n as permutation of coordinates. the set of all orbits of this action with quotient topology is denoted by sp nx. thus, points of the space sp nx are finite subsets (equivalence classes) of the product xn. two points (x1, x2, . . . , xn), (y1, y2, . . . , yn) ∈ xn © agt, upv, 2021 appl. gen. topol. 22, no. 2 448 index boundedness and uniform connectedness of space of the g-permutation degree are considered to be equivalent if there exists a permutation σ ∈ sn such that yi = xσ(i). the space sp nx is called the n-permutation degree of the space x. equivalent relation by which we obtain space sp nx is called the symmetric equivalence relation. the n-th permutation degree is a quotient of xn. therefore, the quotient map is denoted by πsn : x n → sp nx, where for every x = (x1, x2, . . . , xn) ∈ xn, πsn((x1, x2, . . . , xn)) = [(x1, x2, . . . , xn)] is an orbit of the point x = (x1, x2, . . . , xn) ∈ xn [20]. let g be a subgroup of the permutation group sn and let x be a compact space. the group g acts also on the n-th power of the space x as permutation of coordinates. the set of all orbits of this action with quotient topology is denoted by sp ngx. the space sp n gx is called g-permutation degree of the space x [13]. actually, it is the quotient space of the product of xn under the g-symmetric equivalence relation. an operation sp n is the covariant functor in the category of compacts and it is said to be a functor of g-permutation degree. if g = sn, then sp n g = sp n and if the group g consists of the unique element only, then sp n = xn. let t be a set and let a and b be subsets of t × t , i.e., relations on the set t . the inverse relation of a will be denoted by a−1, that is, a−1 = {(x, y) : (y, x) ∈ a}. the composition of a and b will be denoted by ab; thus we have ab = {(x, z) : there exists a y ∈ t such that (x, y) ∈ a and (y, z) ∈ b}. for an arbitrary relation a ⊂ t × t and for a positive integer n the relation an ⊂ t × t is defined inductively by the formulas: a1 = a and an = an−1a. every set v ⊂ t × t that contains the diagonal ∆t = {(x, x) : x ∈ t } of t is called an entourage of the diagonal. definition 1.1. let t be a non-empty set. a family u of subsets of t × t is called a uniformity on t , if this family satisfies the following conditions: (u1) each u ∈ u contains the diagonal ∆t = {(x, x) : x ∈ t } of t ; (u2) if v1, v2 ∈ u, then v1 ∩ v2 ∈ u; (u3) if u ∈ u and u ⊂ v , then v ∈ u; (u4) for each u ∈ u there is a v ∈ u such that v 2 ⊂ u; (u5) for each u ∈ u we have u−1 ∈ u. the pair (t, u) is called uniform space [17]. also, the elements of the uniformity u are called entourages. for an entourage u ∈ u and a point x ∈ t the set u(x) = {y ∈ t : (x, y) ∈ u} is called the u-ball centered at x. for a subset a ⊂ t the set u(a) = ⋃ a∈a u(a) is called the u-neighborhood of a [1]. © agt, upv, 2021 appl. gen. topol. 22, no. 2 449 r. b. beshimov, d. n. georgiou and r. m. zhuraev a family b is called a base for the uniformity u, if for any v ∈ u there exists a w ∈ b with w ⊂ u. the smallest cardinal number of the form |b|, where b is a base for u, is called the weight of the uniformity u and is denoted by ω(u). every base b for a uniformity on t has the following properties: (bu1) for every v1, v2 ∈ b there exists a v ∈ b such that v ⊂ v1 ∩ v2; (bu2) for every v ∈ b there exists a w ∈ b such that w 2 ⊂ v . proposition 1.2 ([15]). suppose that a non-empty set x is given. consider a family b of entourages of the diagonal, which has the properties (bu1)–(bu2) and b−1 = b. a family u is a uniformity on x, if it consists of all entourages which contain a member of b. the family b is a base for u. the uniformity u is called the uniformity generated by the base b. let {(xs, us) : s ∈ s} be a family of uniform spaces. a family b of all entourages of the diagonal, which has the form {({xs} , {ys}) : (xsi , ysi) ∈ vsi for s1, s2, . . . , sk ∈ s, vsi ∈ usi, i = 1, 2, . . . , k} , generates a uniformity on the set ∏ s∈s xs. this uniformity is called a cartesian product of the uniformities {us : s ∈ s} and is denoted by ∏ s∈s us. if all the uniformities us are equal to each other, i.e., if xs = x and us = u for s ∈ s, then the cartesian product ∏ s∈s us is also denoted by uτ, where τ = |s| [10]. definition 1.3. a function f : (x, u) → (y, v) is called uniformly continuous, if for each v ∈ v there exists a u ∈ u such that (f × f)(u) = {(f(x1), f(x2)) : (x1, x2) ∈ u} ⊂ v. note that the condition (f × f)(u) ⊂ v is equivalent to the condition f(u(x)) ⊂ v (f(x)) or (f × f)−1(v ) ∈ u [14]. a uniformly continuous function f : (x, u) → (y, v) is uniformly open, if for any u ∈ u there exists a v ∈ v such that v (f(x)) ⊂ f(u(x)) for all x ∈ x [12]. let expc x and expc y be the hyperspaces of x and y , consisting of all nonempty compact subsets equipped with the hausdorff uniformity. in [12], it was proved that if a (continuous) surjection f : x → y between uniform spaces x and y is perfect, then f is uniformly open if and only if expc f : expc x → expc y is uniformly open. remark 1.4. the uniform continuity of the mapping f does not always imply uniform openness, i.e., there is a mapping f that can be uniformly continuous, but cannot be uniformly open. as an example, consider the mapping f : (r, u) → (r, v), where f(x) = x, x ∈ x, v = {∆, r × r}, u = {u ⊂ r × r : ∆ = {(x, x) : x ∈ r} ⊂ u} and © agt, upv, 2021 appl. gen. topol. 22, no. 2 450 index boundedness and uniform connectedness of space of the g-permutation degree r is the set of all real numbers. this mapping f is uniformly continuous, but not uniformly open. recall that a bijective mapping f : (x, u) → (y, v), acting from the uniform space (x, u) to the uniform space (y, v), is called a uniform isomorphism if the mappings f : (x, u) → (y, v) and f−1 : (y, v) → (x, u) are uniformly continuous [8]. let (x, u) be a uniform space and d ∈ u. a pair of points x, y of the uniform space (x, u) is said to be related by a d-chain, if there exists an integer k such that (x, y) ∈ dk. the uniform space x is called uniformly connected, if every entourage d of x and every pair of points of x are related by a d-chain [16]. in our paper we use the following theorem, which have been proved in [15]. theorem 1.5. the uniformly continuous image of a uniformly connected space is uniformly connected. the smallest cardinal number τ is called an index boundedness of a uniform space (x, u), if the uniformity u has a base b consisting of entourages of cardinality ≤ τ. the index boundedness is denoted by l(u). the uniform space (x, u) is called τ-bounded, if l(u) ≤ τ [8]. 2. uniformly open and uniformly continuous mappings theorem 2.1. let (x, u) be a uniform space. a family b of all subsets of sp ngx × sp ngx of the form o [ u1, u2, . . . , un ] = {([x], [y]) : there exist permutations σ, δ ∈ g such that ( xi, yσ(i) ) ∈ uδ(i), i = 1, 2, . . . , n}, where { u1, u2, . . . , un } ⊂ u, has the properties (bu1)–(bu2) and generates some uniformity on sp ngx. proof. first, we show that every set of the form o [ u1, u2, . . . , un ] is an entourage of the diagonal, where { u1, u2, . . . , un } ⊂ u. take an arbitrary point [x] = [(x1, x2, . . . , xn)] ∈ sp ngx. then ( xi, xi ) ∈ ui for all i = 1, 2, . . . , n. in this case, we have that σ = δ = e is the unit element of the group g. therefore, ([x], [x]) ∈ o [ u1, u2, . . . , un ] and ∆ = {([x], [x]) : [x] ∈ sp ngx} ⊂ o [ u1, u2, . . . , un ] . choose any two entourages o [ u1, u2, . . . , un ] and o [ v1, v2, . . . , vn ] of the family b. it is clear, that ui ∩ vi ∈ u for each i = 1, 2, . . . , n. so, it is enough to show the following relation: o [ u1 ∩ v1, u2 ∩ v2, . . . , un ∩ vn ] ⊂ o [ u1, u2, . . . , un ] ∩ o [ v1, v2, . . . , vn ] . let ([x], [y]) ∈ o [ u1 ∩ v1, u2 ∩ v2, . . . , un ∩ vn ] . © agt, upv, 2021 appl. gen. topol. 22, no. 2 451 r. b. beshimov, d. n. georgiou and r. m. zhuraev then there exist permutations θ, φ ∈ g such that ( xi, yθ(i) ) ∈ ( uφ(i) ∩ vφ(i) ) for all i = 1, 2, . . . , n. put ( xi, yθ(i) ) ∈ uφ(i) and ( xi, yθ(i) ) ∈ vφ(i) for all i = 1, 2, . . . , n. it means that ([x], [y]) ∈ o [ u1, u2, . . . , un ] ∩ o [ v1, v2, . . . , vn ] . for any entourage o [ u1, u2, . . . , un ] ∈ b there is a wi ∈ u such that w 2i ⊂ ui for each i = 1, 2, . . . , n. put w = n ⋂ i=1 wi. we prove that o [ w ′1, w ′ 2, . . . , w ′ n ]2 ⊂ o [ u1, u2, . . . , un ] , where w ′i = w for every i = 1, 2, . . . , n. let ([x], [y]) ∈ o [ w ′1, w ′ 2, . . . , w ′ n ]2 . then there exists an orbit [z] ∈ sp ngx such that ([x], [z]) ∈ o [ w ′1, w ′ 2, . . . , w ′ n ] and ([z], [y]) ∈ o [ w ′1, w ′ 2, . . . , w ′ n ] . since ([x], [z]) ∈ o [ w ′1, w ′ 2, . . . , w ′ n ] there are permutations σ1, δ1 ∈ g such that ( xi, zσ1(i) ) ∈ w ′ δ1(i) = w for all i = 1, 2, . . . , n. if ([z], [y]) ∈ o [ w ′1, w ′ 2, . . . , w ′ n ] , then there exist permutations ϕ, γ ∈ g such that ( zj, yϕ(j) ) ∈ w ′ γ(j) = w for all j = 1, 2, . . . , n. put j = σ1(i) and we obtain ( xi, zσ1(i) ) ∈ w and ( zσ1(i), yϕσ1(i) ) ∈ w . thus, ( xi, yϕσ1(i) ) ∈ w 2 ⊂ w 2i ⊂ ui for each i = 1, 2, . . . , n. consequently ([x], [y]) ∈ o[u1, u2, . . . , un], i.e. o[w ′1, w ′2, . . . , w ′n]2 ⊂ o [ u1, u2, . . . , un ] . now we prove that o [ u1, u2, . . . , un ]−1 = o [ u−11 , u −1 2 , . . . , u −1 n ] . indeed, let ([x], [y]) ∈ o [ u1, u2, . . . , un ]−1 . then ([y], [x]) ∈ o [ u1, u2, . . . , un ] and there are permutations σ2, δ2 ∈ g such that ( yi, xσ2(i) ) ∈ uδ2(i) for every i = 1, 2, . . . , n. put j = σ2(i). this implies that i = σ −1 2 (j). the relation ( y σ −1 2 (j), xj ) ∈ u δ2σ −1 2 (j) implies that ( xj, yσ−1 2 (j) ) ∈ u−1 δ2σ −1 2 (j) for all j = 1, 2, . . . , n. therefore, ([x], [y]) ∈ o [ u−11 , u −1 2 , . . . , u −1 n ] . we have o [ u1, u2, . . . , un ]−1 ⊂ o [ u−11 , u −1 2 , . . . , u −1 n ] . the reverse inclusion is similarly. by proposition 1.2, the family b generates some uniformity sp ngu on the set sp ngx. theorem 2.1 is proved. � consider a mapping πsn, g : (x n, un) → ( sp ngx, sp n gu ) defining as follows: πsn, g ( x1, x2, . . . , xn ) = [ (x1, x2, . . . , xn) ] g for each ( x1, x2, . . . , xn ) ∈ xn. theorem 2.2. let (x, u) be a uniform space. then the mapping πsn, g : ( xn, un) → (sp ngx, sp ngu ) is uniformly continuous. © agt, upv, 2021 appl. gen. topol. 22, no. 2 452 index boundedness and uniform connectedness of space of the g-permutation degree proof. let o [ u1, u2, . . . , un ] be any entourage in ( sp ngx, sp n gu ) . consider an entourage u = { (a, b) : ( ai, bi ) ∈ ui, i = 1, 2, . . . , n } in ( xn, un ) , where a = ( a1, a2, . . . , an ) and b = ( b1, b2, . . . , bn ) are points of xn. we prove that for all x = ( x1, x2, . . . , xn ) ∈ xn, πsn, g(u(x)) ⊂ o [ u1, u2, . . . , un ] ([x]). indeed, if y = ( y1, y2, . . . , yn ) ∈ u(x), then ( xi, yi ) ∈ ui for each i = 1, 2, . . . , n. from ( xi, yi ) ∈ ui we have ( xi, yσ(i) ) ∈ uδ(i), where σ = δ = e ∈ g. in this case, we have that [y] ∈ o [ u1, u2, . . . , un ] ([x]). thus, πsn, g(u(x)) ⊂ o [ u1, u2, . . . , un ] ([x]). theorem 2.2 is proved. � theorem 2.3. for a uniform space (x, u) the mapping πsn, g : ( xn, un ) → ( sp ngx, sp n gu ) is uniformly open. proof. by theorem 2.2 the mapping πsn, g is uniformly continuous. let u be an arbitrary entourage in un. then there is a trace {u1, u2, . . . , un} ⊂ u such that { (a, b) : ( ai, bi ) ∈ ui, i = 1, 2, . . . , n } ⊂ u, where a = ( a1, a2, . . . , an ) and b = ( b1, b2, . . . , bn ) are points of xn. we show that for any point x = ( x1, x2, . . . , xn ) ∈ xn we have o [ u′1, u ′ 2, . . . , u ′ n ] ([x]) ⊂ πsn, g(u(x)), where u′k = n ⋂ i=1 ui for k = 1, 2, . . . , n. indeed, if [y] ∈ o [ u′1, u ′ 2, . . . , u ′ n ] ([x]), then there exist permutations σ, δ ∈ g such that ( xi, yσ(i) ) ∈ u′ δ(i) for all i = 1, 2, . . . , n. in particular, ( xi, yσ(i) ) ∈ ui for all i = 1, 2, . . . , n, i.e., (2.1) (x, yσ) ∈ { (a, b) : ( ai, bi ) ∈ ui, i = 1, 2, . . . , n } where yσ = ( yσ(1), yσ(2), ..., yσ(n) ) . from (2.1) it follows that yσ ∈ u(x) and [y] ∈ πsn, g(u(x)). therefore, o [ u′1, u ′ 2, . . . , u ′ n ] ([x]) ⊂ πsn, g(u(x)) for a point x ∈ xn. theorem 2.3 is proved. � proposition 2.4. let f : (x, u) → (y, v) be a uniformly open mapping and f(x) = y . then w(v) ≤ w(u). proof. let w(u) = τ ≥ ℵ0. then there is a base b = {uα : α ∈ m} of uniformity u such that |m| = τ. we shall prove that the family (f × f)(b) = {(f × f)(uα) : α ∈ m} is a base of uniformity v. since the map f is uniformly open, we have that (f × f)(uα) ∈ v for each α ∈ m. for any entourage v ∈ v the relation (f × f)−1(v ) ∈ u is true. in this case, there exists an index α0 ∈ m such © agt, upv, 2021 appl. gen. topol. 22, no. 2 453 r. b. beshimov, d. n. georgiou and r. m. zhuraev that uα0 ⊂ (f × f)−1(v ), i.e. (f × f)(uα0) ⊂ v . it means that the family (f × f)(b) is a base of uniformity v. proposition 2.4 is proved. � for a uniform space (x, u) we define a mapping λ : x → sp ngx, where λ(x) = [(x, x, . . . , x)], x ∈ x. proposition 2.5. for a uniform space (x, u) the mapping λ : (x, u) → ( sp ngx, sp n gu ) is a uniform embedding. proof. let λ|∆ : x → ∆ be the restriction of the map λ : x → sp ngx, where ∆ = {[(x, x, . . . , x)] : x ∈ x}. it is known that it is bijective. let us show that the map λ|∆ is uniformly continuous. choose an arbitrary entourage o [ u1, u2, . . . , un ] ∈ sp ngu. put u = n ⋂ i=1 ui. by the definition of uniformity we have u ∈ u. it suffices to show that λ|∆(u(x)) ⊂ ( o [ u1, u2, . . . , un ] ∩ (∆ × ∆) )( λ|∆(x) ) for all x ∈ x. clearly, λ|∆(x) = λ(x) and ( o [ u1, u2, . . . , un ] ∩ (∆ × ∆) ) (λ|∆(x)) = o [ u1, u2, . . . , un ] (λ(x)) ∩ ∆ for x ∈ x. let y ∈ u(x). then (x, y) ∈ u ⊂ ui for every i = 1, 2, . . . , n. in this case, we have (λ(x), λ(y)) ∈ o [ u1, u2, . . . , un ] , i.e., λ(y) ∈ o [ u1, u2, . . . , un ] (λ(x)) ∩ ∆. now we show that the mapping (λ|∆)−1 : ∆ → x is uniformly continuous. take an arbitrary entourage v ∈ u. the following relation holds: (λ|∆)−1 ( o [ v ′1, v ′ 2, . . . , v ′ n ] (λ(x)) ∩ ∆ ) ⊂ v (x), x ∈ x, where v ′k = n ⋂ i=1 vi for each k = 1, 2, . . . , n. it means that the mapping (λ|∆)−1 is uniformly continuous. proposition 2.5 is proved. � lemma 2.6 ([8]). let (x, u) be a uniform space and (y, u|y ) be its subspace, where u|y = {u ∩ (y × y ) : u ∈ u}. then w(u|y ) ≤ w(u). theorem 2.7. let (x, u) be a uniform space. then the equality w(u) = w(sp ngu) holds. proof. let (x, u) be a uniform space. by proposition 2.5 and lemma 2.6 it follows that w(u) ≤ w(sp ngu). by the definition of uniformity sp ngu on the set sp ngx we have w(sp n gu) ≤ w(u). thus, we directly obtain w(u) = w(sp ngu). theorem 2.7 is proved. � consider an arbitrary mapping f : (x, u) → (y, v), where (x, u) and (y, v) are uniform spaces. for an equivalence class [(x1, x2, . . . , xn)] ∈ sp ngx, put sp ngf [( x1, x2, . . . , xn )] g = [( f ( x1 ) , f ( x2 ) , . . . , f ( xn ))] g . © agt, upv, 2021 appl. gen. topol. 22, no. 2 454 index boundedness and uniform connectedness of space of the g-permutation degree the following mapping is defined sp ngf : ( sp ngx, sp n gu ) → ( sp ngy, sp n gv ) . we obtained the following result. theorem 2.8. let f : (x, u) → (y, v) be a uniformly continuous mapping. then the mapping sp ngf : ( sp ngx, sp n gu ) → ( sp ngy, sp n gv ) is also uniformly continuous. proof. let f : (x, u) → (y, v) be a uniformly continuous mapping. take an arbitrary entourage o [ v1, v2, . . . , vn ] ∈ sp ngv. then there is an entourage ui ∈ u such that f ( ui(a) ) ⊂ vi ( f(a) ) for all a ∈ x and i = 1, 2, . . . , n. we show that sp ngf ( o [ u1, u2, . . . , un ] ([x]) ) ⊂ o [ v1, v2, . . . , vn ]( sp ngf([x]) ) for a point [x] ∈ sp ngx. choose an orbit [y] ∈ o [ u1, u2, . . . , un ] ([x]). then there exist permutations σ, δ ∈ g such that ( xi, yσ(i) ) ∈ uδ(i) for all i = 1, 2, . . . , n. we have yσ(i) ∈ uδ(i) ( xi ) for any i = 1, 2, . . . , n. therefore, f ( yσ(i) ) ∈ f(uδ(i) ( xi) ) ⊂ vδ(i) ( f ( xi )) . it means that (2.2) ( f ( xi ) , f ( yσ(i) )) ∈ vδ(i) for all i = 1, 2, . . . , n. put sp ngf([x]) = [( f ( x1 ) , f ( x2 ) , . . . , f ( xn ))] g and sp ngf([y]) = [( f(y1 ) , f ( y2 ) , . . . , f ( yn ))] g . from (2.2) we obtain ( sp ngf([x]), sp n gf([y]) ) ∈ o [ v1, v2, . . . , vn ] . hence, sp ngf([y]) ∈ o [ v1, v2, . . . , vn ]( sp ngf([x]) ) . theorem 2.8 is proved. � theorem 2.9. let f : (x, u) → (y, v) be a uniformly open mapping. then the mapping sp ngf : ( sp ngx, sp n gu ) → ( sp ngy, sp n gv ) is also uniformly open. proof. let f : (x, u) → (y, v) be a uniformly open mapping. take an arbitrary entourage o [ u1, u2, . . . , un ] ∈ sp ngu. in this case there exists entourage vi ∈ v such that vi(f(a)) ⊂ f(ui(a)) for each a ∈ x and i = 1, 2, . . . , n. it suffices to show that o [ v1, v2, . . . , vn ]( sp ngf([x]) ) ⊂ sp ngf ( o [ u1, u2, . . . , un ] ([x]) ) . choose an arbitrary point [y] ∈ o [ v1, v2, . . . , vn ]( sp ngf([x]) ) . then there are permutations σ, δ ∈ g such that ( f ( xi ) , yσ(i) ) ∈ vδ(i) for all i = 1, 2, . . . , n. moreover, yσ(i) ∈ vδ(i)(f(xi)) ⊂ f(uδ(i)(xi)) for each i = 1, 2, . . . , n. since © agt, upv, 2021 appl. gen. topol. 22, no. 2 455 r. b. beshimov, d. n. georgiou and r. m. zhuraev yσ(i) ∈ f ( uδ(i) ( xi )) , there exists a point zi ∈ uδ(i) ( xi ) such that yσ(i) = f ( zi ) for all i = 1, 2, . . . , n. put z = ( z1, z2, . . . , zn ) ∈ xn and we have (2.3) [y] = sp ngf([z]). the relation zi ∈ uδ(i) ( xi ) implies (2.4) [z] ∈ o [ u1, u2, . . . , un ] ([x]) for i = 1, 2, . . . , n. by relations (2.3) and (2.4) it follows that [y] ∈ sp ngf ( o [ u1, u2, . . . , un ] ([x]) ) . hence o [ v1, v2, . . . , vn ]( sp ngf([x]) ) ⊂ sp ngf ( o [ u1, u2, . . . , un ] ([x]) ) . theorem 2.9 is proved. � 3. uniformly connected spaces and index boundedness theorem 3.1. if a uniform space (x, u) is uniformly connected, then the uniform space ( sp ngx, sp n gu ) is also uniformly connected. proof. let x, y ∈ xn, where x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn). take an arbitrary entourage u = {(a, b) : (ai, bi) ∈ ui, i = 1, 2, . . . , n} ∈ un on xn, where a = (a1, a2, . . . , an) and b = (b1, b2, ..., bn). since x is uniformly connected, there is a ki ∈ z such that (xi, yi) ∈ ukii for every i = 1, 2, . . . , n. put k = max{ki : i = 1, 2, . . . , n}. therefore, (xi, yi) ∈ uki for each i = 1, 2, . . . , n. in this case there are points z1i , z 2 i , . . . , z k−1 i such that        ( xi, z 1 i ) ∈ ui, ( z1i , z 2 i ) ∈ ui, . . . . ( zk−1i , yi ) ∈ ui. for every i = 1, 2, ..., n. consider k − 1 points of xn; z1 = ( z11, z 1 2, . . . , z 1 n ) , z2 = ( z21, z 2 2, . . . , z 2 n ) , . . . , zk−1 = ( zk−11 , z k−1 2 , . . . , z k−1 n ) . by definition of entourage u, we have ( x, z1 ) ∈ u, ( z1, z2 ) ∈ u, . . . , ( zk−1, y ) ∈ u. it means that (x, y) ∈ uk. hence, ( xn, un ) is uniformly connected space. by theorem 1.1 [15] and theorem 2.2 the space ( sp ngx, sp n gu ) is uniformly connected. theorem 3.1 is proved. � we say that a uniform space (x, u) is discrete, if ∆x ∈ u [15]. example 3.2. any discrete uniform space is not uniformly connected. indeed, take points x, y ∈ x with x 6= y. then for any integer number k we have (x, y) /∈ ∆x = ∆kx. © agt, upv, 2021 appl. gen. topol. 22, no. 2 456 index boundedness and uniform connectedness of space of the g-permutation degree theorem 3.3 ([15]). for every uniformity u on a set x the family τu = {g ⊂ x : for every x ∈ g there exists a v ∈ u such that v (x) ⊂ g} is a topology on the set x, which is called the topology induced by the uniformity u. remark 3.4. the uniformly connectedness of the space (x, u) does not imply connectedness with respect to the topology induced by the uniformity u, in general. consider the family b = {uε : ε > 0} of subsets of r × r, where uε = {(x, y) ∈ r × r : |x − y| < ε} and r is the real line. the family b has the properties (bu1)–(bu2) and generates some uniformity on the real line r. this uniformity is called natural uniformity on r. the family bq = {uε ∩ (q × q) : ε > 0} is a base of a uniformity on q (the set of all rational numbers) and generates some uniformity uq on q. for any ε > 0 and r1, r2 ∈ q (r1 < r2) we have k = [r2−r1ε ] + 1. consider a sequence of points {ai}k−1i=1 defined by the formula ai = r1 + r2−r1 k i with i = 1, 2, . . . , k − 1. it is clear that ai ∈ q for all i = 1, 2, . . . , k − 1. in this case we have        ( r1, a1 ) ∈ vε, ( a1, a2 ) ∈ vε, . . . . ( ak−1, r2 ) ∈ vε, where vε = uε ∩ (q × q). thus, ( r1, r2 ) ∈ v kε . therefore, the uniform space ( q, uq ) is uniformly connected, but not connected, since q = ((−∞, √ 2) ∩ q) ∪ (( √ 2, ∞) ∩ q) and ((−∞, √ 2) ∩ q) ∩ (( √ 2, ∞) ∩ q) = ∅. theorem 3.5. let (x, u) be a uniform space. then the equality l(u) = l ( sp ngu ) holds. proof. first, we show the inequality l ( sp ngu ) ≤ l ( un ) . let l ( un ) = τ ≥ ℵ0. then there is a base bn of uniformity un such that |v | ≤ τ for any v ∈ bn. we consider the family πsn, g ( bn ) = {πsn, g(v ) : v ∈ bn} and prove that the family πsn, g ( bn ) is the base of the uniformity sp ngu. since πsn, g is a uniformly open mapping, πsn, g(v ) ∈ sp ngu for any entourage v ∈ bn. consider an arbitrary entourage o [ u1, u2, . . . , un ] ∈ sp ngu. for an entourage ( πsn, g )−1( o [ u1, u2, . . . , un ]) ∈ un there exists an entourage v ∈ bn such that v ⊂ ( πsn, g )−1( o [ u1, u2, . . . , un ]) . hence, we obtain πsn, g(v ) ⊂ o [ u1, u2, . . . , un ] , i.e., we have that l ( sp ngu ) ≤ τ. now we show the inverse inequality l ( un ) ≤ l(u). let l(u) = κ ≥ ℵ0 and let b be a base for the uniformity u, consisting of entourages of cardinality ≤ κ. © agt, upv, 2021 appl. gen. topol. 22, no. 2 457 r. b. beshimov, d. n. georgiou and r. m. zhuraev we denote by b′ the family of all entourages of the form n ⋂ i=1 pr−1i ( ui ) , where ui ∈ b and pri is the projection of xn onto xi = x for each i = 1, 2, . . . , n. then by definition of cartesian product of the uniform spaces, the family b′ is a base for the uniformity un. since | n ⋂ i=1 pr−1i (ui)| ≤ max{|ui| : i = 1, 2, . . . , n} ≤ κ it follows that l ( un ) ≤ l(u). consequently, we have l ( sp ngu ) ≤ l(u). note that a uniform subspace of a τ-bounded space is also τ-bounded. by proposition 2.5, l(u) ≤ l(sp ngu). hence, we get the equality l(u) = l ( sp ngu ) . � references [1] t. banakh, topological spaces with an ωω-base, dissertationes mathematicae, warszawa, 2019. 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[14] g. itzkowitz, s. rothman, h. strassberg and t. s. wu, characterization of equivalent uniformities in topological groups, topology and its applications 47 (1992), 9–34. [15] i. m. james, introduction to uniform spaces, london mathematical society, lecture notes series 144, cambridge university press, cambridge, 1990. [16] j. l. kelley, general topology, van nostrand reinhold, princeton, nj, 1955. © agt, upv, 2021 appl. gen. topol. 22, no. 2 458 index boundedness and uniform connectedness of space of the g-permutation degree [17] l. holá and l. d. r. kocinac, uniform boundedness in function spaces, topology and its applications 241 (2018), 242–251. [18] e. michael, topologies on spaces of subsets, trans. amer. math. soc. 71 (1951), 152–182. [19] t. n. radul, on the functor of order-preserving functionals, comment. math. univ. carol. 39, no. 3 (1998), 609–615. [20] t. k. yuldashev and f. g. mukhamadiev, the local density and the local weak density in the space of permutation degree and in hattorri space, ural mathematical journal 6, no. 2 (2020), 108–126. © agt, upv, 2021 appl. gen. topol. 22, no. 2 459 @ appl. gen. topol. 22, no. 1 (2021), 1-10doi:10.4995/agt.2021.12291 © agt, upv, 2021 a glance into the anatomy of monotonic maps in memory of my dearest friend natashka gamzina-kandaurova may your light guide me to kindness raushan buzyakova miami, florida, usa (raushan buzyakova@yahoo.com) communicated by m. a. sánchez-granero abstract given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. we note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. we observe that the latter cannot always be chosen to be order-isomorphic to the original space. also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps. 2010 msc: 26a48; 54f05; 06b30. keywords: monotonic map; ordered topological spaces; topologically equivalent maps. 1. introduction it is one of classical problems of various areas of topology if a given continuous map on a topological space with perhaps a richer structure has nice properties related to this rich structure. for clarity of exposition, let us agree on terminology. an autohomeomorphism on a topological space x is any homeomorphism of x onto itself. an open interval with end points a and b of a linearly ordered set l will be denoted by (a, b)l. if it is clear what ordered set is under consideration, we simply write (a, b). the same concerns other types received 31 august 2019 – accepted 19 october 2020 http://dx.doi.org/10.4995/agt.2021.12291 r. buzyakova of intervals. linearly ordered topological spaces are abbreviated as lots and their subspaces as go-spaces. we will mostly be concerned with go-spaces. it is due to čech ([4]) that a hausdorff space x is a go-space if and only if a family of convex sets with respect to some ordering on x is a basis for the topology of x. given a go-space x, an order ≺ on x is said to be gocompatible if some collection of ≺-convex subsets of 〈x, ≺〉 is a basis for the topology of x. note that if x is a lots, a go-compatible order on x need not witness the fact that x is a lots. some of our attention will be directed at topological groups. recall that g is a topological group if it is a topological space with a group operation · such that both · and the operation of taking the inverse (inversion) are continuous with respect to the topology of g. if g is abelian and a ∈ g, the map g 7→ ag is called a shift. we will be concerned with the following general problem. problem 1.1. let x be a go-space and let f be an autohomeomorphism on x. what conditions on x and/or f guarantee that x has a go-compatible ordering with respect to which f is monotonic? since monotonicity is an order-dependent concept, we will specify with respect to which ordering a map is monotonic. if no clarification is given, the assumed order is the original one and should be clear from the context. since our discussion will be around problem 1.1, we will isolate the target property into a definition. definition 1.2. an autohomeomorphism f on a go-space x is potentially monotonic if there exists a go-compatible order on x with respect to which f is monotonic. definition 1.2 is equivalent to the following definition: definition 1.3 (equivalent to 1.2). an autohomeomorphism f on a go-space x is potentially monotonic if there exists a go-space y , a homeomorphism h : x → y , and a monotonic autohomeomorphism m on y such that f = h−1 ◦ m ◦ h. to see why these two definitions are equivalent, let f be an autohomeomorphism on a go-space x. assume f is potentially monotonic by definition 1.2. fix a go-compatible order ≺ on x with respect to which x is monotonic. put y = 〈x, ≺〉, h = idx (the identity map), and m = f. clearly, f = id−1 x ◦ m ◦ idx. hence, f is potentially monotonic with respect to definition 1.3. we now assume that f is potentially monotonic with respect to definition 1.3. fix y, f, m as in the definition. the order on y induces an order ≺ on x as follows: a ≺ b if and only if h(a) < h(b). since h is a homeomorphism, ≺ is compatible with the go-topology of x. next, let us show that f is ≺-monotonic. we have a ≺ b is equivalent to h(a) < h(b). by the choice of m, the latter is equivalent to m◦h(a) < m◦h(b). by the definition of ≺, the latter is equivalent to h−1 ◦ m ◦ h(a) ≺ h−1 ◦ m ◦ h(b). since f = h−1 ◦ m ◦ h, we conclude that f(a) ≺ f(b). © agt, upv, 2021 appl. gen. topol. 22, no. 1 2 a glance into the anatomy of monotonic maps one may wonder if the property in definition 1.2 is equivalent to the property of being topologically equivalent to a monotonic map with respect to the existing order. recall that homeomorphisms f, g : x → x are topologically equivalent (or conjugate) if there exists a homeomorphism t : x → x such that t ◦ f = g ◦ t. a supported explanation will be given later in remark 2.9 that a map can be potentially monotonic but not topologically equivalent to a monotonic map (with respect to the existing order). it is clear, however, from definition 1.3 that a map topologically equivalent to a monotonic map is potentially monotonic. in our arguments, given a monotonic function f on a go-space l and an x ∈ l, we will make a frequent use of the set {fn(x) : n ∈ z}. in literature, similarly defined sets are often referred to as the orbit of x under f. we will also refer to this set as the f-orbit of x. similarly, the f-orbit of a set a ⊂ x is the collection {fn(a) : n ∈ z}. recall that an indexed family {si : i ∈ i} of subsets of a space x is called discrete if any point x ∈ x has a neighborhood u such that |{i ∈ i : si ∩ u 6= ∅}| ≤ 1. by looking at the behavior of monotonic maps on the reals, we quickly observe that the orbit of each point under such maps exhibits very strong properties. namely, the following holds. proposition 1.4. let f be a fixed-point free monotonic autohomeomorphism on a go-space l and x ∈ l. then there exists an open neighborhood i of x such that the family {fn(i) : n ∈ z} is discrete. proof. without loss of generality, we may assume that f is strictly increasing. to avoid repetition, we next isolate a useful statement into a claim: claim. s = {fn({x}) : n ∈ z} is a discrete family of sets for any x ∈ l. to prove the claim, we first observe that the elements of s are distinct singletons. indeed, by strict monotonicity, fn(x) 6= fm(x) for distinct integers n and m.therefore it suffices to show that s has no limit points. assume the contrary and let y be a limit point for s. by monotonicity, lim n→∞ fn(x) = y or lim n→∞ f−n(x) = y. by continuity, f(y) = y, contradicting the fact that f is fixed-point free. the claim is proved. fix x ∈ l. if x is isolated, then i = {x} is as desired by claim. assume now that x is not isolated. since f is an increasing homeomorphism, the intervals (x, f(x)) and (f−1(x), x) are not empty. pick and fix a ∈ (f−1(x), x). since f is strictly increasing, f(a) ∈ (x, f(x)). let i be an open neighborhood of x such that the closure of i is a subset of (a, f(a)). let us show that i is as desired. fix y ∈ l. we need to find an open neighborhood u of y that meets fn(i) for at most one n ∈ z. we have three cases. case (y ∈ (fn(a), fn+1(a)) for some integer n): by monotonicity of f, the interval (fn(a), fn+1(a)) contains fn(i) and misses fm(i) for every other m. therefore, u = (fn(a), fn+1(a)) is as desired. case (y = fn(a) for some integer n): then (fn−1(a), fn+1(a)) contains y and meets fm(i) only for m = n − 1 and m = n. since the closure of i is in (a, f(a)), we conclude that fn(a) is not in the closure of fn−1(i) or © agt, upv, 2021 appl. gen. topol. 22, no. 1 3 r. buzyakova fn(i). hence, there exists an neighborhood u of y that misses fm(i) for any m. case (y 6∈ [fn(a), fn+1(a)] for any integer n): if y is not in the closure of ⋃ n [fn(a), fn+1(a)], then some neighborhood of y misses [fn(a), fn+1(a)] for any n. assume y is in the closure of ⋃ n [fn(a), fn+1(a)]. by the case’s condition, y must be a limit point for {fn({a}) : n ∈ z}, which is impossible by claim. since we exhausted all cases, the proof is complete. � we will next isolate the necessary condition identified in proposition 1.4 into a property. definition 1.5. let f : x → x be a map and a ⊂ x. the f-orbit of a is strongly discrete if there exists an open neighborhood u of a such that the family {fn(u) : n ∈ z} is discrete. the f-orbit of x ∈ x is strongly discrete if the f-orbit of {x} is strongly discrete. in this note we will present partial results addressing problem 1.1. at the end of our study we will identify a few questions that may have a good chance for an affirmative resolution. in notation and terminology we will follow [3]. in particular, if ≺ is an order on l and a, b ⊂ l, by a ≺ b we denote the fact that a ≺ b for any a ∈ a and b ∈ b. 2. study one may wonder if our introduction of the concepts of strongly discrete orbits is really necessary. can we use the requirement of being ”period-point free” instead? the next example shows that a periodic-point free autohomeomorphism even on a nice space need not have strongly discrete orbits. example 2.1. there exist a periodic-point free autohomeomorphism f of the space of rationals q and a point q ∈ q such that the f-orbit of q is not strongly discrete. proof. example [1, example 2.5 ] provides a construction of a fixed point autohomeomorphism f on the rationals that satisfies the hypothesis of lemma [1, lemma 2.4]. for convenience, the cited hypotheses is copied next: hypothesis of lemma [1, lemma 2.4]: ”suppose f : q → q is not an identity map and p ∈ q satisfy the following property: (*) ∀n > 0∃m > 0 such that fm+1((p−1/n, p+1/n)q) meets f −m((p−1/n, p+ 1/n)q).” clearly an f that satisfies the above hypothesis fails having a strong f-orbit at p. � for our next affirmative result we need a technical statement that incorporates our general strategy for showing that a map is potentially monotonic. © agt, upv, 2021 appl. gen. topol. 22, no. 1 4 a glance into the anatomy of monotonic maps lemma 2.2. let l be a go-space and f : l → l an autohomeomorphism. suppose that o is a collection of clopen subsets of l with the following properties: (1) the f-orbit of each o ∈ o is strongly discrete. (2) fn(o) ∩ fm(o′) = ∅ for distinct o, o′ ∈ o and n, m ∈ z. (3) {fn(o) : n ∈ z, o ∈ o} is a cover of l . then there exists a go-compatible order ≺ on l with respect to which f is strictly increasing. proof. by < we denote some ordering with respect to which l is a generalized ordered space. enumerate elements of o as {oα : α < |o|}. we will define ≺ in three stages. stage 1: for each o ∈ o and n ∈ ω \ {0}, define ≺ on fn(o) and f−n(o) recursively as follows: step 0: put ≺ |o =< |o. assumption: assume that ≺ is defined on fk(o) and on f−k(o) for all k = 0, 1, ..., n − 1. step n: if x, y ∈ fn(o), put x ≺ y if and only if f−1(x) ≺ f−1(y). this is well defined since f−1(x), f−1(y) are in fn−1(o) and ≺ is defined on fn−1(o) by assumption. similarly, if x, y ∈ f−n(o), put x ≺ y if and only if f(x) ≺ f(y). stage 2: for any o ∈ o and any n, m ∈ z such that n < m, put fn(o) ≺ fm(o). stage 3: for any α < β < |o| and any n, m ∈ z, put fn(oα) ≺ f m(oβ). the next two claims show that ≺ is as desired. claim 1. ≺ is compatible with the go-topology of l. proof of claim. to prove the claim, for each o ∈ o, let to be the collection of all <-convex open subsets of l that are subsets of o. since < coincides with ≺ one every o ∈ o, we conclude that every element in to is ≺-convex. by the constructions at stage 1, fn(o) is ≺-convex. since f is an autohomeomorphism, the collection {fn(i) : i ∈ to, o ∈ o} is a basis for the topology of l and consists of open ≺-convex sets. the claim is proved. claim 2. f is increasing with respect to ≺. proof of claim. pick distinct x and y. if x, y ∈ ⋃ n fn(o) for some o ∈ o, then apply stages 1 and 2. otherwise, apply stage 3. � remark to lemma 2.2. note that if 〈l, <〉 is a lots and each o in the argument of the lemma has both extremities or each o has neither extremity, then 〈l, ≺〉 is a lots too. the converse of lemma 2.2 for fixed-point free autohomeomorphisms on zero-dimensional go-spaces holds too (lemma 2.4). to prove the converse, we need the following quite technical statement. recall that given a continuous self-map f : x → x, a closed set a ⊂ x is an f-color if a ∩ f(a) = ∅. for © agt, upv, 2021 appl. gen. topol. 22, no. 1 5 r. buzyakova a review of major results on colors of continuous maps, we refer the reader to [6]. proposition 2.3. let l be a zero-dimensional go-space, f : l → l a fixedpoint free monotonic homeomorphism, and x ∈ l. then there exists a maximal convex clopen set i ⊂ l containing x such that the following hold: (1) ⋃ n∈z fn(i) is clopen and convex. (2) fn(i) ∩ fm(i) = ∅ for any distinct integers n and m. (3) fn(i) is a maximal clopen convex f-color for any n ∈ z. proof. we may assume that f is strictly increasing. let dl be the largest ordered compactification of l. that is, if a and b are clopen subsets of l with the properties that a < b and a∪b = l, then cldl(a) and cldl(b) are clopen as well. therefore, dl is zero-dimensional too, and f continuously extends to f̃ : dl → dl. the map f is an autohomeomorphism too and is increasing but not necessarily strictly. since dl is a zero-dimensional compact lots, any neighborhood of x contains a ∈ dl such that a is the right member of a gap and a ≤dl x. since f̃ does not fix x we can select such an a ∈ dl with an additional property that f̃(a) >dl x. put i = [a, f̃(a))l, where the right-hand side is an abbreviation for [a, f̃(a))dl ∩ l. let us show that i is as desired. by monotonicity, {fn(i) : n ∈ z} = {..., [f̃−1(a), a)l, [a, f̃(a))l, [f̃(a), f̃ 2(a))l, ...}. enlarging any interval in this sequence would make that interval meet its image. therefore, (3) is met. visual inspection of the sequence is a convincing evidence that the union ⋃ n fn(i) is convex. the union is also open as the union of open sets. since f is fixed-point free, fn(i)’s form a discrete collection, and hence, the union is closed. by our choice, fn+1(i) = [f̃n+1(a), f̃n+2(a))l, which guarantees that (2) is met. � lemma 2.4. let l be a zero-dimensional go-space and let f : l → l be a fixed-point free monotonic autohomeomorphism. then there exists a collection o of convex clopen subsets of l with the following properties: (1) the f-orbit of each o ∈ o is strongly discrete. (2) fn(o) ∩ fm(o′) = ∅ for distinct o, o′ ∈ o and n, m ∈ z. (3) {fn(o) : n ∈ z, o ∈ o} is a cover of l . proof. without loss of generality, we may assume that f is strictly increasing. we will construct o = {oα}α recursively. assume that oβ is constructed for each β < α and the following properties hold: p1: ⋃ n∈z fn(oβ) is clopen and convex. p2: fn(oβ) ∩ f m(oβ) = ∅ for any distinct integers n and m. p3: fn(oβ) is a maximal clopen convex f-color for any n ∈ z. note that p1 and p2 imply the following: p4: the f-orbit of oβ is strongly discrete. construction of oα: let lα = l \ ⋃ {fn(oβ) : β < α, n ∈ z}. if lα is empty, then the recursive construction is complete and o = {oβ : β < α}. otherwise, © agt, upv, 2021 appl. gen. topol. 22, no. 1 6 a glance into the anatomy of monotonic maps we have f(lα) = f −1(lα) = lα. let us show that lα is clopen in l. firstly, it is closed as the complement of the union of open sets. to show that it is open, fix x ∈ lα. let i be as in proposition 2.3 for given x, f, l. if x is a limit point for l\lα, then it must contain some f n(oβ) for β < α and n ∈ z, which contradicts property p3. hence, i is an open neighborhood of x contained in lα. since properties (1)-(3) of i in the conclusion of proposition 2.3 coincide with the properties p1-p3, we can put oα = i. the family o = {oα}α is as desired by construction. � lemmas 2.2 and 2.4 form the following criterion. theorem 2.5. let f be a fixed-point free autohomeomorphism on a zerodimensional go-space x. then f is potentially monotonic if and only if there exists a collection o of convex clopen subsets of l with the following properties: (1) the f-orbit of each o ∈ o is strongly discrete. (2) fn(o) ∩ fm(o′) = ∅ for distinct o, o′ ∈ o and n, m ∈ z. (3) {fn(o) : n ∈ z, o ∈ o} is a cover of l . we next put one part (lemma 2.2) of the above criterion to a good use. theorem 2.6. let x be a zero-dimensional subspace of the reals and let f : x → x be an autohomeomorphism with strongly discrete orbits at all points. then f is potentially monotonic. proof. to prove the statement, we will construct a collection o as in the hypothesis of lemma 2.2. let x ∈ x be an arbitrary point. let u be an open neighborhood of x that witnesses the fact that x has a strongly discrete f-orbit. let v be a clopen neighborhood of x that is a subset of u. clearly, v witnesses the property too. since v is its own neighborhood, v has a strongly discrete f-orbit as well. therefore, we can fix a countable cover f = {fn : n ∈ ω} of x so that each fi is clopen and has strongly discrete f-orbit. step 0: put o0 = f0. assumption: assume that ok is defined for k < n, clopen, and has strongly discrete f-orbit. in addition, assume that ⋃ m∈z fm(oi) misses⋃ m∈z fm(oj), whenever i 6= j and i, j < n. step n: let in be the smallest index such that fin is not covered by {fm(oi) : i < n, m ∈ z}. put on = fin \ ∪{f m(oi) : i < n, m ∈ z}. construction is complete. the collection o = {on : n ∈ ω} has properties (1) and (2) in the hypothesys of lemma 2.2 by construction. to show (3), that is, the equality x = ∪{fm(oi) : i ∈ ω, m ∈ z}, fix any x ∈ x. since f is a cover of x, there exists n such that x ∈ fn. if x is not in f m(oi) for some i < n and m ∈ ω, then fn is the first element in f that meets the construction requirements at step n. therefore, x ∈ on. � corollary 2.7. every periodic-point free bijection on z is potentially monotonic. © agt, upv, 2021 appl. gen. topol. 22, no. 1 7 r. buzyakova in contrast with corollary 2.7, we next observe that not every periodic-point free bijection on z is topologically equivalent to a monotonic map. example 2.8. there exists a periodic-point free bijection on z that is not topologically equivalent to a monotonic map. proof. first observe that every monotonic bijection on z is a shift. therefore, any bijection on z that is topologically equivalent to a monotonic map is also topologically equivalent to a shift. it is observed in [2, example 1.2] that if a bijection f on z has infinitely many points with mutually disjoint orbits, then such a map is not topologically equivalent to a shift. thus, any such fixedpoint free map is an example of a potentially monotonic map on z that is not topologically equivalent to a monotonic map. � remark 2.9. corollary 2.7 and example 2.8 imply that the property of being potentially monotonic does not imply the property of being topologically equivalent to a monotonic map (with respect to the existing order). we can strengthen corollary 2.7 as follows. theorem 2.10. let f be a periodic-point free bijection on z. then there exist an ordering ≺ and a binary operation ⊕ on z such that z′ = 〈z, ⊕, ≺〉 is a discrete ordered topological group and f is a shift in z′. proof. let m be a minimal subset of z with respect to the property that the f-orbit of m covers z. if |m| = n, enumerate the elements of m by zn. clearly, zn ×l z is an ordered discrete topological group with the component-wise addition. define a bijection h : z → zn ×l z by letting g(f k(ni)) = (i, k). since any element of z is in the f-orbit of exactly one element of m, the correspondence is welldefined and is a bijection. since g is a homeomorphism, we will next abuse notation and will identify fk(xi) with (i, k). let us apply f to (i, k). we have f(fk(xi)) = f k+1(xi), and the latter is identified with (i, k + 1). therefore, f is a shift by (0, 1) in z′. if m is infinite, enumerate its elements by integers as m = {ni : i ∈ z}. define h : z → z ×l z by letting g(f k(ni)) = (i, k). argument similar to the zn case shows that the ordering on z induced by h is as desired. � note that the above statement does not hold for continuous periodic-point free bijections on the rationals. indeed, as shown in [1, example 2.5] there exists a continuous periodic-point free bijection on q with a point with nonstrongly discrete fiber. the mentioned example [1, example 2.5] is constructed to satisfy the hypothesis of [1, lemma 2.4], which is a stronger case of not having discrete fibers. nonetheless, the following takes place. theorem 2.11. a fixed-point free autohomeomorphism f : q → q is potentially monotonic if and only if f is topologically equivalent to a shift. © agt, upv, 2021 appl. gen. topol. 22, no. 1 8 a glance into the anatomy of monotonic maps proof. (⇒) since f is potentially monotonic, there exists a collection o as in the conclusion of lemma 2.4. the argument of theorem 2.3 in [1] shows that f with such a collection is topologically equivalent to a non-trivial shift. (⇐) it is proved in [2, theorem 2.8] that a periodic-point free homeomorphism h on q is topologically equivalent to a shift if and only if one can introduce a group operation ⊕ on q compatible with the topology of q so that the topological group 〈q, ⊕〉 is continuously isomorphic to q and h is a shift with respect to new operation. clearly such an 〈q, ⊕〉 is an ordered topological group, and hence, any shift is monotonic. therefore, f is potentially monotonic. � recall that given a continuous selfmap f : x → x, the chromatic number of f is the least number of f-colors needed to cover x. theorem 2.12. let f be a fixed-point free autohomeomorphism on a zerodimensional go-space l. if f is potentially monotonic, then the chromatic number of f is 2. proof. (⇒) since the chromatic number of f is a purely topological property not attached to an order, we may assume that f is strictly monotonic. let o be as in the conclusion of 2.4 for the given f and l. put a = ∪{fn(o) : n is an even integer, o ∈ o} and b = ∪{fn(o) : n is an odd integer, n ∈ o}. clearly, {a, b} is cover of l by colors. � theorem 2.12 and remark 2.9 prompt the following question. problem 2.13. let f be a periodic point free homeomorphism on a zerodimensional go-space l. let the chromatic number of f be 2. is f potentially monotonic? theorem 2.6 prompts the following question. problem 2.14. let x be a go-space and let f : x → x be an autohomeomorphism with strongly discrete orbits at all points. is f potentially monotonic? what if x is hereditarily paracompact? acknowledgements. the author would like to thank the referee for many helpful remarks and suggestions. © agt, upv, 2021 appl. gen. topol. 22, no. 1 9 r. buzyakova references [1] r. buzyakova, on monotonic fixed-point free bijections on subgroups of r, applied general topology 17, no. 2 (2016), 83–91. [2] r. buzyakova and j. west, three questions on special homeomorphisms on subgroups of r and r∞, questions and answers in general topology 36, no. 1 (2018), 1–8. [3] r. engelking, general topology, pwn, warszawa, 1977. [4] h. bennet and d. lutzer, linearly ordered and generalized ordered spaces, encyclopedia of general topology, elsevier, 2004. [5] d. lutzer, ordered topological spaces, surveys in general topology, g. m. reed., academic press, new york (1980), 247–296. [6] j. van mill, the infinite-dimensional topology of function spaces, elsevier, amsterdam, 2001. © agt, upv, 2021 appl. gen. topol. 22, no. 1 10 @ appl. gen. topol. 20, no. 2 (2019), 419-430 doi:10.4995/agt.2019.11605 c© agt, upv, 2019 on proximal fineness of topological groups in their right uniformity ahmed bouziad département de mathématiques, université de rouen, umr cnrs 6085, avenue de l’université, bp.12, f76801 saint-étienne-du-rouvray, france. (ahmed.bouziad@univ-rouen.fr) communicated by j. galindo abstract a uniform space x is said to be proximally fine if every proximally continuous function defined on x into an arbitrary uniform pace y is uniformly continuous. we supply a proof that every topological group which is functionally generated by its precompact subsets is proximally fine with respect to its right uniformity. on the other hand, we show that there are various permutation groups g on the integers n that are not proximally fine with respect to the topology generated by the sets {g ∈ g : g(a) ⊂ b}, a, b ⊂ n. 2010 msc: 22a05; 54e15. keywords: uniform space; topological group; proximal continuity; proximally fine group; symmetric group; o-radial space. 1. introduction a function f : x → y between two uniform spaces is said to be proximally continuous if for every bounded uniformly continuous function g : y → r, the composition function g ◦ f : x → r is uniformly continuous; the reals r being equipped with the usual metric. a uniform space x is said to be proximally fine if every proximally continuous function defined on x into an arbitrary uniform space y is uniformly continuous. it is well-known that metric spaces and uniform products of metric spaces are proximally fine; however, there are many uniform spaces that are not proximally fine although they are received 02 april 2019 – accepted 30 july 2019 http://dx.doi.org/10.4995/agt.2019.11605 a. bouziad topologically well-behaved (some may be locally compact and even discrete). we refer the reader to hušek’s recent paper [12] for more information about proximally fineness of general uniform spaces. we are interested here in the fine proximal condition of the topological groups when these spaces are endowed with their right uniformity. this subject seems to have no specific literature, although there are some questions which could have been naturally addressed in this setting, such as the itzkowitz problem for metric and/or locally compact groups [13] (the link between itzkowitz problem and proximity theory was made later in [6]). in view of the close relationship between the right uniformity of topological groups and the system of neighborhoods of their identity, it is reasonable to expect that the proximal fineness of a given topological group g is satisfied provided that a not too much restrictive condition is imposed on the topology of g. this feeling is heightened by corollary 2.6 in this note asserting that it is sufficient to assume that g is functionally generated (in arkhangel’skǐı’s sense) by its precompact subsets. in fact, this is still true under much less restrictive conditions (theorem 2.4 below). let us mention that one part of theorem 2.4 was asserted without proof in [5]1. the main subject of this note can be examined in the context of g-sets (or g-spaces) without significantly altering its essence, so the main result is stated and established in a somewhat more general form (theorem 2.5). some examples of non-proximally fine groups are given in section 3. to do that, we consider the topology τ on nn of uniform convergence when (the target set) n is endowed with the samuel uniformity. we show in corollary 3.4 that for any permutation group h on n for which all but finitely many orbits are finite and uniformly bounded, the only group topology on h which is finer than τ and proximally fine is the discrete topology. the question of whether there is an abelian (or at least sin) group which is non-proximally fine group is left open. 2. main result as usual, if (x,u) is a uniform space, where u is the uniform neighborhoods of the diagonal of x, then for a ⊂ x and u ∈ u, u[a] stands for the set of y ∈ x such that (x,y) ∈ u for some x ∈ a. for undefined terms we refer to the books [8] and [16]. one of the tools used here is katetov’s extension theorem of uniformly continuous bounded real-valued functions [14]. the following statement is also well-known (see [8]); its proof is outlined here for the sake of completeness and because it can be adapted to show a result in the same spirit that will be used in the proof theorem 2.5. proposition 2.1. let f : x → y , where (x,u) and (y,v) are two uniform spaces. then the following are equivalent: (1) f : x → y is proximally continuous, 1the author is indebted to professor michael d. rice for his interest in this result and its proof, which motivated the present work. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 420 on proximal fineness of topological groups (2) for every a ⊂ x and v ∈ v, there exists u ∈ u such that f(u[a]) ⊂ v [f(a)]. proof. to show that 1) implies 2), let d be a bounded uniformly continuous pseudometric on y such that (x,y) ∈ v whenever d(x,y) < 1 ([8]) and consider the uniformly continuous function φ on y defined by φ(y) = d(y,f(a)). since φ ◦ f is uniformly continuous, there is u ∈ u such that (x,y) ∈ u implies |φ(f(x)) −φ(f(y))| < 1. then f(u[a]) ⊂ v [f(a)]. for the converse, let φ : y → r be a bounded uniformly continuous function and let ε > 0. define v = {(x,y) ∈ y ×y : |φ(x) −φ(y)| < ε/3}. then v ∈v and since φ is bounded, there is a finite set f ⊂ y so that y = v [f]. let az = f −1(v [z]), z ∈ f, and choose u ∈ u such that f(u[az]) ⊂ v [f(az)] for each z ∈ f . then |φ ◦ f(x) − φ ◦ f(y)| ≤ ε for each (x,y) ∈ u. indeed, there exists z ∈ f such that x ∈ az so (f(x),z) ∈ v . since y ∈ u[x] we have f(y) ∈ f(u[az]) ⊂ v [f(az)] thus there exists t ∈ az such that (f(y),f(t)) ∈ v . as t ∈ az we also have (f(t),z) ∈ v . therefore, (f(x),f(y)) ∈ v ◦v ◦v . � let us say that a topological group g is proximally fine if the uniform space (g,ur) is proximally fine, where ur is the right uniformity of g. it is equivalent to say that (g,ul) is proximally fine, where ul is the left uniformity of g. recall that a basis of ur (respectively, ul) is given by the sets of the form {(g,h) ∈ g×g : gh−1 ∈ v} (respectively, {(g,h) ∈ g×g : g−1h ∈ v}) as v runs over the set v(e) of neighborhoods of the unit e of g. proposition 2.2. let g be a topological group, (y,u) a uniform space and let f : g → y be a function. for g ∈ g, let ψg : g → y be the function defined by ψg(h) = f(gh). then, the following are equivalent: (1) f is uniformly continuous, g being equipped with the right uniformity, (2) the function ψ : g ∈ g → ψg ∈ y g is continuous when y g is endowed with the uniformity of uniform convergence on g. proof. to show that (1) implies (2), suppose that f is right uniformly continuous and let u ∈ u. there is a neighborhood v of the unit e in g such that xy−1 ∈ v implies (f(x),f(y)) ∈ u. let g ∈ g. then v g is a neighborhood of g in g and for each h ∈ v g and x ∈ g we have hx(gx)−1 ∈ v . it follows that (f(hx),f(gx)) ∈ u, so (ψh(x),ψg(x)) ∈ u for every x ∈ g. to show that (2) implies (1), let u ∈ u and choose v ∈ v(e) such that (ψg(x),ψe(x)) ∈ u for every g ∈ v and x ∈ g. then, for every g,h ∈ g such that h ∈ v g we have (ψhg−1 (x),ψe(x)) ∈ u for each x ∈ g, equivalently, (f(hx),f(gx)) ∈ u, for every x ∈ g. in particular, (f(h),f(g)) ∈ u. � it is possible to expand substantially the framework of the starting topic of this note without major changes as follows: let g be a topological group and let x be a g-set, that is, x is a nonempty set for which there is a map ∗ : g×x → x satisfying (gh)∗x = g∗(h∗x) for every g,h ∈ g and x ∈ x. the function ∗ is called a left action of g on x. to simplify, write gx in place of g∗x and ua in place of u ∗a if u ⊂ g and a ⊂ x. no topology will be required c© agt, upv, 2019 appl. gen. topol. 20, no. 2 421 a. bouziad on x. let (y,v) be a uniform space and f : x → y a function. it is consistent with the definitions given above to say that a f is right uniformly continuous if for each v ∈v, there is u ∈v(e) such that (f(gx),f(hx)) ∈ v for each x ∈ x, whenever gh−1 ∈ u. similarly, the function f is said to be right proximally continuous if for each bounded uniformly continuous function φ : y → r, the function φ◦f is right uniformly continuous. it is easy to check (see the proof of proposition 2.2) that f is right uniformly continuous if and only if the function ψ : g ∈ g → ψ(g) ∈ y x (where ψ(g)(x) = f(gx)) is continuous when y x is endowed with the uniform convergence. similarly, a simple adaptation of the proof of proposition 2.1 shows that f is right proximally continuous if and only if for each a ⊂ gx and v ∈v, there is u ∈v(e) such that f(ua) ⊂ v [f(a)]. the following properties are required for theorem 2.5; we are formulating them separately to reduce the proof to its essential components. let x be a g-set and f : x → y a right proximally continuous, as defined above. then, for each a ⊂ g: (c1) for every x ∈ x, the function g ∈ g → f(gx) ∈ y is continuous, (c2) if ψ|a is right uniformly continuous, then ψ|a is right uniformly continuous, (c3) if a ∈ a is a point of continuity of ψ|a, then a is a point of continuity of ψ|a. we check the validity of these properties for the benefit of the reader. let v ∈ v. for every x ∈ x and g ∈ g, there is u ∈ v(e) such that f(ugx) ⊂ v [f(gx)]. since ug is a neighborhood of g in g, (c1) holds. property (c2) and (c3) follows from (c1). for, if x ∈ x, u ∈v(e) are such that (f(ax),f(bx)) ∈ v for every a,b ∈ a with a ∈ uub, then (c1) implies that f(gx),f(hx)) ∈ v 2 for each g,h ∈ a such that g ∈ uh. taking x arbitrary in x gives (c2). for (c3), let u be an open neighborhood of the unit in g such that (f(gx),f(ax)) ∈ v for ever g ∈ ua∩a and x ∈ x. since ua∩a ⊂ ua∩a, it follows from (c1) that for every g ∈ ua∩a and x ∈ x, (f(gx),f(ax)) ∈ v 2. to establish theorem 2.5 we also need the next key lemma; this is a wellknown tool in the theory of proximity spaces (most often with w 4 instead of w 3). lemma 2.3. let x be a set and w be a symmetric binary relation on x. then, for every infinite cardinal η and for every sequence (xn,yn)n<η ⊂ x×x such that (xn,yn) 6∈ w 3 for each n < η, there is a cofinal set a ⊂ η such that (xn,ym) 6∈ w for every n,m ∈ a. proof. replacing η by its cofinality cf(η), we may suppose that η is regular. let m ⊂ η be a maximal set satisfying (xn,ym) 6∈ w for every n,m ∈ m. if m is cofinal in η, the proof is finished, so suppose that m is not cofinal in η. for each j ∈ m, let aj = {n < η : (xn,yj) ∈ w}, bj = {n < η : (xj,yn) ∈ w}, cj = aj∪bj and c = ∪j∈mcj. the maximality of m implies that η ⊂ m∪c. since η is regular, there is j ∈ m such that cj is cofinal in η, therefore aj or bj is cofinal in η. we suppose that it is aj, the other case is similar. let c© agt, upv, 2019 appl. gen. topol. 20, no. 2 422 on proximal fineness of topological groups n,m ∈ aj. then (xn,yj) ∈ w and (xm,yj) ∈ w , hence (xn,ym) 6∈ w since (xm,ym) 6∈ w 3 (recall that w is symmetric). similarly, (xm,yn) 6∈ w . � we will now specify a few topological concepts that will be used in what follows. the first is a variant of herrlich’s notion of radial spaces [11]. radial spaces were characterized by a.v. arhangel’skǐı [3] as follows: a space x is radial if and only if for each x ∈ x and a ⊂ x such that x ∈ a, there is b ⊂ a of regular cardinality |b| such that x ∈ c for every c ⊂ b having the same cardinality as b. let us say that a subset a of x is relatively o-radial in x if for every collection (oi)i∈i of open sets in x and x ∈ a such that x ∈∪i∈ioi ∩a\∪i∈ioj, there is a set j ⊂ i of regular cardinality such that x ∈ ∪j∈loj whenever l ⊂ j and |l| = |j|. if the set j can always be chosen countable, then a is said to be relatively o-malykhin in x. all closures are taken in x. every almost metrizable (in particular, čech-complete) group is o-malykhin (in itself). more generally, every inframetrizable group [16] is o-malykhin, see [6]. let x be a topological space and m a collection of subspaces of x. following arhangel’skǐı [2], the space x is said to be functionally generated by the collection m if for every discontinuous function f : x → r, there exists a ∈m such that the restriction f|a : a → r of f to the subspace a of x has no continuous extension to x. the space x is said to be strongly functionally generated by m if for every discontinuous function f : x → r, there exists a ∈ m such that the restriction f|a : a → r of f to the subspace a of x is discontinuous. the following is the main result of this note. the statement corresponding to the case (2) was asserted (without proof) in [5]. theorem 2.4. let g be a topological group satisfying at least one of the following: (1) g is functionally generated by the sets a ⊂ g such that aa−1 is relatively o-radial in g, (2) g is strongly functionally generated by the sets a ⊂ g such that a is relatively o-radial in g. then g is proximally fine. according to proposition 2.2 and keeping the above notations, theorem 2.4 is obtained from the following general result by considering the left action of g on itself. theorem 2.5. let g be a topological group and suppose that for each discontinuous bounded function α : g → r, there is a set a ⊂ g having at least one of the following conditions: (1) α|a has no continuous extension to g and aa −1 is relatively o-radial in g, c© agt, upv, 2019 appl. gen. topol. 20, no. 2 423 a. bouziad (2) α|a is discontinuous at some point of a and a is relatively o-radial in g. let x be a g-set, (y,v) a uniform space and let f : x → y be a right proximally continuous. then f : x → y is right uniformly continuous. proof. we have to show that the function ψ : g ∈ g → ψ(g) ∈ y x (where ψ(g)(x) = f(gx)) is continuous. we proceed by contradiction by supposing that ψ is not continuous. then, there is a bounded uniformly continuous function θ : y x → r such that θ◦ψ is not continuous (see [8]). let a ⊂ g satisfying at least one of the conditions (1) and (2) with respect to the function θ◦ψ. in case (1), there is no compatible uniformity on g making uniformly continuous the function θ◦ψ|a; for, otherwise, katetov’s theorem would give us a continuous extension of θ ◦ψ|a. in particular, ψ|a is not right uniformly continuous. as remarked above, it follows from (c2) that ψ|a is not right uniformly continuous. there is then an open and symmetric w ∈ u such that for every v ∈ v(e), there exist av ∈ v , gv ∈ a and xv ∈ x satisfying av gv ∈ a and (f(av gv xv ),f(gv xv )) 6∈ w 6.(2.1) for each v ∈v(e), let hv = gv xv and define ov = {g ∈ g : (f(ghv ),f(av hv )) ∈ w}. since the functions g ∈ g → f(ghv ) ∈ y , v ∈ v(e), are continuous by the property (c1), each ov is open in g. since av ∈ ov ∩ ag−1v ⊂ aa −1 and av ∈ v for each v ∈v(e), it follows that e ∈ ⋃ v∈v(e) ov ∩ (aa−1).(2.2) we also have e 6∈ ov , for each v ∈ v(e). indeed, otherwise, there exist v ∈ v(e) and g ∈ ov such that (f(ghv ),f(hv )) ∈ w . therefore (f(av hv ),f(hv )) ∈ w 2 which contradicts (2.1). since aa−1 is relatively o-radial in g, in view of (2.2), there is a set γ ⊂v(e) of regular cardinal such that for each set i ⊂ γ of the same cardinal as γ, we have e ∈ ⋃ v∈i {g ∈ g : (f(ghv ),f(av hv )) ∈ w}.(2.3) by lemma 2.3 and (2.1), there is i ⊂ γ such that |i| = |γ| (since |γ| is regular) and (f(auhu ),f(hv )) 6∈ w 2 for every u,v ∈ i. since f is right proximally continuous, there exists v ∈v(e) such that f(v{hu : u ∈ i}) ⊂ w [f({hu : u ∈ i})].(2.4) by (2.3) applied to i, there is u1 ∈ i such that v ∩ ou1 6= ∅. let g ∈ v be such that (f(ghu1 ),f(au1hu1 )) ∈ w . by (2.4), there is u2 ∈ i so that (f(ghu1 ),f(hu2 )) ∈ w . it follows that (f(au1hu1 ),f(hu2 )) ∈ w 2, which is a contradiction. therefore, ψ is continuous in case (1). c© agt, upv, 2019 appl. gen. topol. 20, no. 2 424 on proximal fineness of topological groups in case (2), a is o-radial in g and θ ◦ψ|a is discontinuous at some point a ∈ a. since θ is continuous, ψ|a is necessarily discontinuous at a. it follows from the property (c3) that ψ|a is discontinuous at a. let w ∈u be symmetric and open such that for every v ∈ v(e), there exist av ∈ v and xv ∈ g, such that av a ∈ a and (f(av axv ),f(axv )) 6∈ w 6. taking hv = axv for each v ∈v(e), we have e ∈ ⋃ v∈v(e) {g ∈ g : (f(ghv ),f(hv )) ∈ w}∩ (aa−1). it is easy to see that aa−1 is relatively o-radial in g, therefore the proof can be continued and concluded in the same way as in the first case. it should be noted that katetov’s theorem was not used in this case. � let a ⊂ g, where g is a topological group. it is proved in [6] that aa−1 is relatively o-malykhin in g, provided that a is left and right precompact. thus theorem 2.4 yields: corollary 2.6. every topological group g which is functionally generated by the collection of its precompact subsets is proximally fine. in view of the role played by locally compact groups in many areas of mathematics, it is worth mentioning the following particular case of corollary 2.6. corollary 2.7. every locally compact topological group is proximally fine. recall that the topological group g is said to sin (or with small invariant neighborhoods of the identity) if the left uniformity ul and the right uniformity ul of g are equal. it is well-known and easy to see that the property of being sin for g is equivalent to the inclusion ur ⊂ul or, equivalently, to the uniform continuity of the identity map from (g,ul) to (g,ur). following protasov [15], the group g is said to be fsin (or functionally balanced) if every bounded right uniformly continuous function f : g → r is left uniformly continuous. clearly, the property of being fsin for g is equivalent to the proximal continuity of the identity map from (g,ul) to (g,ur). consequently, every proximally fine fsin group is sin. the question whether every fsin group is sin is called itzkowitz problem and is still open. we refer the reader to [5] for more information; see also [17] for a very recent contribution to this topic. since every proximally fine fsin group is sin, it follows from theorem 2.4 that every fsin group is sin provided that it is strongly functionally generated by its relatively o-radial subsets. this result has already been formulated (implicitly and without proof) in [5]. this is supplemented by the next corollary of theorem 2.4 : corollary 2.8. every fsin group g which is functionally generated by the sets a ⊂ g such that aa−1 is relatively o-radial in g is a sin group. to conclude this section, we would like to take this opportunity to comment on the parenthesized question of [5, question 6] whether every bounded topological group is fsin. the answer is of course no, since fsin is a hereditary c© agt, upv, 2019 appl. gen. topol. 20, no. 2 425 a. bouziad property (by katetov’s theorem) and every topological group is isomorphic both algebraically and topologically to a subgroup of a bounded group [10] (see also [9]). 3. examples in this section, we give some examples of non-proximally fine hausdorff topological groups and examine their behavior towards the fsin property. the set of positive integers is denoted by n and u is the samuel uniformity of the uniform discrete space n. the uniformity u is sometimes called the precompact reflection of the uniform discrete space n. a basis of u is given by the sets ∪i≤nai × ai, where a1, . . .an is a partition of the integers. let nn be endowed with the uniformity v of uniform convergence when n (the target space) is equipped with the uniformity u. let g denote the permutation group of the set n of positive integers and let s be the normal subgroup of g given by finitary permutations g ∈ g; that is, g ∈ s if and only if the set supp(g) = {x ∈ n : g(x) 6= x} is finite. it is easy to see that g (hence s) is a topological group when equipped with the topology τ induced by the uniformity v. more precisely, g is a non-archimedean group, since a basis of neighborhoods of its unit is given by the subgroups of g of the form hπ = {g ∈ g : g(ai) = ai, i = 1, . . . ,n}, where π = {a1, . . . ,an} is a finite partition of n. another approach (which we will not adopt here) is to consider g as the group of all auto-homeomorphisms of βn equipped with the compact-open topology, where βn is the stone-čechcompactification of the integers. the so-called natural polish topology τ0 on g, given by pointwise convergence, is coarser than τ and for every partition π of n the set hπ is τ0-closed. this is to say that τ0 is a cotopology for τ in the sense of [1]; in particular, (g,τ) is submetrizable and a baire space. in what follows, unless otherwise stated, the groups g and s will be systematically considered under the topology τ. for later use, we check that the quotient group g/s (with the quotient topology) is hausdorff, that is, s is closed in g. let g ∈ g \ s. a simple induction allows to construct an infinite set a ⊂ n such that g(a) ⊂ n \ a (alternatively, a is obtained from lemma 2.3 applied to the set {(x,g(x)) : x ∈ supp(g)}). let π = {a,n \ a}. then ghπ is a τ-neighborhood of g and for each h ∈ hπ, gh(a) = g(a) ⊂ n\a. hence ghπ ∩s = ∅. recall that for a topological group h, the lower uniformity ul ∧ur on h is called the roelcke uniformity and has base consisting of the sets {(x,y) ∈ h ×h : x ∈ v yv}, v ∈v(e) (see [16]). every (right) proximally fine group is proximally fine with respect to the roelcke uniformity; therefore, the following shows in a strong way that the groups g and s are not proximally fine: proposition 3.1. let k ∈ n and φ : g → n be the evaluation function φ(g) = g(k), n being equipped with the discrete uniformity. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 426 on proximal fineness of topological groups (1) the function φ : g → n is left uniformly continuous and right proximally continuous. in particular, φ is roelcke-proximally continuous. (2) if n is endowed with the uniformity u, then φ is right uniformly continuous. conversely, if v is a uniformity on n such that the restriction of φ|s : s → (n,v) is right uniformly continuous, then v ⊂u. in particular, g and s are not roelcke proximally fine. proof. 1) clearly, φ is left uniformly continuous with respect to the natural topology τ0; since τ0 ⊂ τ, φ is left uniformly continuous. to show that φ is right proximally continuous, let l ⊂ g and put π0 = {a,n \ a}, where a = {g(k) : g ∈ l}, and let us verify that φ(hπ0l) ⊂ φ(l). proposition 2.1 will then conclude the proof. let h0 ∈ hπ0 and g0 ∈ l. then h0(g0(k)) ∈ {g(k) : g ∈ l}, hence we can write h0(g0(k)) = g(k) for some g ∈ l, thus φ(h0g0) ∈ φ(l). 2) φ : g → (n,u) is right uniformly continuous, since for every partition π = {a1, . . . ,an} of n, we have (g(k),h(k)) ∈ ∪i≤nai × ai provided that hg−1 ∈ hπ. for the converse, suppose that v is a uniformity on n satisfying v 6⊂u and let v ∈v\u. we will check that for any partition π = {a1, . . . ,an} of n, there are g,h ∈ s having g ∈ hπh and (φ(g),φ(h)) 6∈ v . we may suppose that a1 = {k} and that a2 contains two elements a and b such that (a,b) 6∈ v . define g ∈ s by g(k) = a, g(a) = k and g(x) = x for x /∈ {k,a}. define also h ∈ s by h(k) = b, h(a) = k, h(b) = a and h(x) = x otherwise. then gh−1 ∈ hπ, but (φ(g),φ(h)) 6∈ v . � the next statements 3.2 and 3.4 give some extremal properties showing that the above examples provided by proposition 3.1 are somehow optimal. for a subgroup h of g and l ⊂ n, let h(l) stand for the pointwise stabilizers of l in h (that is, the set of h ∈ h such that h(x) = x for all x ∈ l). let us recall that a topological group h is said to be strongly fsin if every real-valued right uniformly continuous function on h is left uniformly continuous. proposition 3.2. let h be subgroup of g and f ⊂ n a finite set. let τ1 be a group topology on h and for each k ∈ f , let φk : g ∈ h → g(k) ∈ n, where n is endowed with the discrete uniformity. (1) if for each k ∈ f , φk : (h,τ1) → n is right proximally continuous, then τ is coarser than τ1 on h(n\hf). (2) if for each k ∈ f , φk : (h,τ1) → n is right uniformly continuous, then τ1 is discrete on h(n\hf). (3) if τ0|h ⊂ τ1 and (h,τ1) is strongly fsin, then τ1 is discrete on h(n\hf). proof. 1) we first show that for a given a ⊂ n, there is a τ1-neighborhood va of the unit (in h) such that f(a ∩ (hf)) ⊂ a for every f ∈ va. to do that, for each k ∈ f, define lk = {g ∈ h : g(k) ∈ a}. according to proposition 2.1, there is a τ1-neighborhood va of the unit such that for every k ∈ f, φk(valk) ⊂ φk(lk). let n ∈ a ∩ (hf) and f ∈ va. choose g ∈ h c© agt, upv, 2019 appl. gen. topol. 20, no. 2 427 a. bouziad and k ∈ f such that g(k) = n. then g ∈ lk, hence φk(fg) ∈ φk(lk) and thus f(n) ∈ a. this shows that f(a∩ (hf)) ⊂ a for every f ∈ va. it follows that for any partition π = {a1, . . . ,a1} of n, there is a τ1-neighborhood of the unit in h(n\hf), namely v = h(n\hf) ∩va1 ∩ . . .∩van , such that v ⊂ hπ. since τ1 is a group topology, it follows that τ is coarser than τ1 on h(n\hf). 2) suppose that τ1 is not discrete on h(n\hf) and let v be τ1-neighborhood of the unit in h. we will show that there are g,h ∈ h and k ∈ f such that gh−1 ∈ v and g(k) 6= h(k), contradicting the right uniform continuity of φl for at least one l in the finite set f . since for each k ∈ f , φk is τ1-continuous, there is f ∈ v ∩ h(n\hf) such that f(k) = k for all k ∈ f and f(a) 6= a for some a ∈ n. then a ∈ hf , hence there are h ∈ h and k ∈ f such that h(k) = a. taking g = fh, we get gh−1 ∈ v and φk(g) 6= φk(h). 3) if (h,τ1) is strongly fsin, then the functions φk, k ∈ f, are τ1-right uniformly continuous on h because they are left uniformly continuous on (g,τ0) and τ0|h ⊂ τ1. it follows form (2) that τ1 is discrete on h(n\hf). � lemma 3.3. let h be a subgroup of g, m ∈ n and l ⊂ n such that |l∩k| ≤ m for each orbit k of the action of h on n. then h(l) ∈ τ. proof. there is a finite partition a0, . . . ,am of n such that a0 = n \ l and |ai ∩k| ≤ 1 for each 1 ≤ i ≤ m and every orbit k. then hπ ⊂ h(l). indeed, if f ∈ hπ and x ∈ k ∩ ai where 1 ≤ i ≤ m, then f(x) ∈ k and f(x) ∈ ai, thus f(x) = x since |ai ∩k| ≤ 1. � corollary 3.4. let h be a subgroup of g for which all but finitely many orbits are finite and uniformly bounded. then the discrete topology is the only group topology on h that is both proximally fine and finer than τ|h. proof. if τ1 is a proximally fine group topology on h finer than τ, then the evualuation mappings φk : h → n, k ∈ n, are right uniformly continuous (with respect to τ1). it follows from proposition 3.2(2) that h(n\hf) is τ1-discrete, where f ⊂ n is a finite set such that the cardinals of all orbits hn, n 6∈ f, are finite and uniformly bounded. by lemma 3.3, h(n\hf) is τ-open hence τ1-open, consequently, τ1 is discrete. � similarly, the next result follows from proposition 3.2(3) and lemma 3.3. corollary 3.5. let h be subgroup of g for which all but finitely many orbits are finite and uniformly bounded. if (h,τ) is strongly fsin, then (h,τ) is discrete. moreover, if h has finitely many orbits and (h,τ0) is strongly fsin, then h is a (closed) discrete subgroup of (g,τ0). it is plain that strongly fsin groups are fsin, but it still isn’t known whether the converse is true or not (see question 3 in [5]). it follows from proposition 3.1 that the groups g and s are not strongly fsin, but this does not allow us to conclude that they are not fsin, because none of the functions φk, k ∈ n, is bounded. the next result shows that the groups g and s as well as their quotient g/s are not fsin. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 428 on proximal fineness of topological groups for a topological group h, let r(h), respectively u(h), stand for the real banach spaces of bounded right uniformly continuous and of bounded right and left uniformly continuous functions on h. as usual, c denotes the cardinal of r. proposition 3.6. the groups g, s and g/s are not fsin. moreover, the density character of the quotient banach space r(g/s)/u(g/s) is at least 2c. proof. we shall exhibit, in (1) below, a real-valued bounded function which is left uniformly continuous on g but not right uniformly continuous when restricted to s. it will follow that g and s are not fsin. as for g/s, our strategy is as follows: for each nonprincipal ultrafilter p on n, we shall give in (2) a bounded right uniformly continuous φp defined on g which is not left uniformly continuous. this function is in addition constant on every coset of s, so it factorizes to g/s. then, we show that for each bounded left uniformly continuous function ψ on g, we have ||ψp+ψq+ψ|| ≥ 1 for any distinct nonprincipal ultrafilters p, q. since the quotient map g → g/s is both left and right uniformly continuous, this will imply that the banach space r(g/s)/u(g/s) contains a uniformly discrete set of cardinal 2c (as βn\n, see [8]). (1) let χ : g →{0, 1} be the function defined by χ(f) = 1 if f(1) ≤ f(2) and χ(f) = 0 otherwise. then, clearly, χ is left uniformly continuous. let us show that it is not right uniformly continuous on s. let a1, . . . ,an be a partition of n. we may suppose that a1 = {a,b} with a 6= b. define f and g in s by f(1) = g(2) = a, f(2) = g(1) = b, and f(x) = g(x) for x 6∈ {1, 2}. then f−1(ai) = g −1(ai) for each i = 1, . . . ,n, but χ(f) 6= χ(g). (2) let p be a nontrivial ultrafilter on n and fix an infinite a ⊂ n such that n\a is infinite. let ψp : g →{0, 2} be the function given by ψp(f) = 2 if f−1(a) ∈ p. clearly, the function ψp is bounded and right uniformly continuous. to show that ψp is constant on every coset of s, let g ∈ g and f ∈ s. then, for every b ⊂ n, g−1(b) \ supp(f) ⊂ (gf)−1(b). thus, taking b = a if g−1(a) ∈ p or b = n\a if not, we get that ψp(gf) = ψp(g). let p and q be two distinct nonprincipal ultrafilters on n and let us verify that ||ψp +ψq +ψ|| ≥ 1 for each bounded left uniformly continuous ψ : g → r. it will follow that the quotient r(g/s)/u(g/s) contains a norm 1 discrete set of cardinal |βn\n|. let ε > 0 and π = {b1, . . . ,bn} be a partition of n such that |ψ(f) −ψ(g)| < ε for every f,g ∈ g such that g ∈ fhπ. we may suppose that b1 = c∪d with c ∈ p, d ∈ q and c∩d = ∅. write again d = d1∪d2, where d1 and d2 are infinite, disjoint and d1 ∈ q. finally, let {e,f,k} be a partition of of n \ a, with e and f infinite and |k| = |n \ b1|. there are certainly f,g ∈ g such that f(c) = a, f(d) = e ∪f , g(c) = f, g(d1) = e, g(d2) = a and f = g on n \b1. we have f−1g ∈ hπ (i.e, f(bi) = g(bi) for each i ≤ n), ψp(f) +ψq(f) = 2 and ψp(g) +ψq(g) = 0. since |ψ(f)−ψ(g)| < ε, it follows that |ψp(f) + ψq(f) + ψ(f)| ≥ 1−ε or |ψp(g) + ψq(g) + ψ(g)| ≥ 1−ε. since ε is arbitrary, we have ||ψp + ψq + ψ|| ≥ 1. � c© agt, upv, 2019 appl. gen. topol. 20, no. 2 429 a. bouziad let us mention that the corollary of 3.6 that the hausdorff group g/s is not discrete (being not fsin) was established and used by t. banakh et al. in [4] to answer a question by d. dikranjan and a. giordano bruno in [7]. knowing that the symmetric group g and its finitary subgroup s are highly nonabelian (their centers are trivial) and taking corollary 3.5 into account, we are naturally led to conclude by asking the following: question 3.7. is there a hausdorff topological group that is abelian (or at least sin) and non-proximally fine? acknowledgements. the author would like to thank the reviewer for the thoughtful comments and efforts towards improving the paper. references [1] j. m. aarts, j. de groot and r. h. mcdowell, cotopology for metrizable spaces, duke math. j. 37 (1970), 291–295. [2] a.v. arkhangel’skǐı, topological function spaces, vol. 78, kluwer academic, dordrecht, 1992. [3] a.v. arhangel’skǐı, some properties of radial spaces, math. notes russ. acad. sci. 27 (1980), 50–54. [4] t. banakh, i. guran and i. protasov, algebraically determined topologies on permutation groups, topology appl. 159 (2012), 2258–2268. [5] a. bouziad and j.-p. troallic, problems about the uniform structures of topological groups, in: open problems in topology ii. ed. elliott pearl. amsterdam: elsevier, 2007, 359–366. [6] a. bouziad and j.-p. troallic, left and right uniform structures on functionally balanced groups, topology appl. 153, no. 13 (2006), 2351–2361. [7] d. dikranjan and a. giordano bruno, arnautov’s problems on semitopological isomorphisms, appl. gen. topol. 10, no. 1 (2009), 85–119. [8] r. engelking, general topology, heldermann, berlin, 1989. [9] h. fuhr and w. roelcke, contributions to the theory of boundedness in uniform spaces and topological groups, note di matematica 16, no. 2 (1996), 189–226. [10] s. hartman and j. mycielski, on the embedding of topological groups into connected topological groups, colloq. math. 5 (1958), 167–169. [11] h. herrlich, quotienten geordneter raume und folgenkonvergenz, fund. math. 61 (1967), 79–81. [12] m. hušek, ordered sets as uniformities, topol. algebra appl. 6, no. 1 (2018), 67–76. [13] g. l. itzkowitz, continuous measures, baire category, and uniform continuity in topological groups, pacific j. math. 54 (1974), 115–125. [14] m. katětov, on real-valued functions in topological spaces, fund. math. 38 (1951), 85–91. [15] i. v. protasov, functionally balanced groups, mat. zametki 49, no. 6 (1991), 87–91, translation: math. notes 49, no. 6 (1991), 614–616. [16] w. roelcke and s. dierolof, uniform structures in topological groups and their quotients, mcgraw-hill, new york, 1981. [17] m. shlossberg, balanced and functionally balanced p-groups, topol. algebra appl. 6, no. 1 (2018), 53–59. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 430 @ appl. gen. topol. 22, no. 2 (2021), 251-257doi:10.4995/agt.2021.13225 © agt, upv, 2021 a new topology over the primary-like spectrum of a module fatemeh rashedi department of mathematics, velayat university, iranshahr, iran (f.rashedi@velayat.ac.ir) communicated by j. galindo abstract let r be a commutative ring with identity and m a unitary r-module. the primary-like spectrum specl(m) is the collection of all primarylike submodules q of m, the recent generalization of primary ideals, such that m/q is a primeful r-module. in this article, we topologies specl(m) with the patch-like topology, and show that when, specl(m) with the patch-like topology is a quasi-compact, hausdorff, totally disconnected space. 2010 msc: 13c13; 13c99. keywords: primary-like submodule; zariski topology; patch-like topology. 1. introduction throughout this paper all rings are commutative with identity and all modules are unitary. for a submodule n of m, we let (n : m) denote the ideal {r ∈ r | rm ⊆ n} and annihilator of m, denoted by ann(m), is the ideal (0 : m). by a prime submodule (or a p-prime submodule) of m, we mean a proper submodule p with p = (p : m) such that rm ∈ p for r ∈ r and m ∈ m implies that either m ∈ p or r ∈ p. the prime spectrum (or simply, the spectrum) of m, denoted by spec(m), is the set of all prime submodules of m [1, 2, 3, 5]. the intersection of all prime submodules of m containing n is called the radical of n and denoted by radn. if there is no prime submodule containing n, then we define radn = m. as a new generalization of a primary ideal on the one hand and a generalization of a prime submodule on the other received 04 march 2020 – accepted 24 may 2021 http://dx.doi.org/10.4995/agt.2021.13225 f. rashedi hand, a proper submodule q of m is said to be primary-like if rm ∈ q implies r ∈ (q : m) or m ∈ radq ([6]). we say that a submodule n of an r-module m satisfies the primeful property if for each prime ideal p of r with (n : m) ⊆ p, there exists a prime submodule p containing n such that (p : m) = p. if the zero submodule of m satisfies the primeful property, then m is called primeful. for instance finitely generated modules, projective modules over domains and (finite and infinite dimensional) vector spaces are primeful (see [9]). it is easy to see that, if q is a primary-like submodule satisfying the primeful property, then p = √ (q : m) is a prime ideal of r and so in this case, q is called a pprimary-like submodule. the primary-like spectrum specl(m) is defined to be the set of all primary-like submodules of m satisfying the primeful property. if q ∈ specl(m), since q satisfies the primeful property, there exists a maximal ideal m of r and a prime submodule p containing q such that (p : m) = m and so radq 6= m. for any submodule n of m, let ν(n) = {q ∈ specl(m)| √ (q : m) ⊇ √ (n : m)}. then we have the following lemma. lemma 1.1. let m be an r-module. let n, n′ and {ni|i ∈ i} be submodules of m. then the following hold. (1) ν(m) = ∅. (2) ν(0) = specl(m). (3) if n ⊆ n′, then ν(n′) ⊆ ν(n). (4) ⋂ i∈i ν(ni) = ν( ∑ i∈i (ni : m)m). (5) ν(n) ∪ ν(n′) = ν(n ∩ n′). (6) ν(radn) ⊆ ν(n). (7) if √ (n : m) = √ (n′ : m), then ν(n) = ν(n′). the converse is also true if both n and n′ are primary-like. (8) ν(n) = ν( √ (n : m)m). also, for each submodule n of m we denote the complement of ν(n) in specl(m) by u(n). from (1), (2), (4) and (5) above, the family η(m) = {u(n)|n ≤ m} is closed under finite intersections and arbitrary unions. moreover, we have u(m) = specl(m) and u(0) = ∅. therefore, η(m), as the family of all open sets, satisfy the axioms of a topology t on specl(m), called the zariski topology on m. in section 2, we topologies specl(m) with a patch-like topology, and show that, if m is a noetherian multiplication r-module and (n : m) is a radical ideal for every submodule n of m, then specl(m) with the patch-like topology is a quasi-compact, hausdorff, totally disconnected space (corollary 2.16). 2. main results we need to recall the patch topology (see [7, 8], for definition and more details). let x be topological space. by the patch topology on x, we mean the topology which has as a sub-basis for its closed sets the closed sets and © agt, upv, 2021 appl. gen. topol. 22, no. 2 252 a new topology over the primary-like spectrum compact open sets of the original space. by a patch we mean a set closed in the patch topology. the patch topology associated to a spectral space is compact and hausdorff (see [8]). also, the patch topology associated to the zariski topology of a ring r (not necessarily commutative) with acc on ideals is compact and hausdorff (see [7, proposition 16.1]). definition 2.1. let m be an r-module, and let ω(m) be the family of all subsets of specl(m) of the form ν(n)∪u(k) where ν(n) is any zariski-closed subset of specl(m) and u(k) is a zariski-quasi-compact subset of specl(m). clearly ω(m) is closed under finite unions and contains specl(m) and the empty set, since specl(m) equals ν(0)∪u(0) and the empty set equals ν(m)∪ u(0). therefore ω(m) is basis for the family of closed sets of a topology on specl(m), and call it patch-like topology of m. thus ω(m) = {ν(n) ∪ u(k)|n, k ≤ m, u(k) is zariski-quasi-compact}, and hence we obtain the family ω(m) = {ν(n) ∩ u(k)|n, k ≤ m, u(k) is zariski-quasi-compact}, which is a basis for the open sets of the patch-like topology, i.e., the patchlike-open subsets of specl(m) are precisely the unions of sets from ω(m). we denote the patch-like topology of specl(m) by tp(m). definition 2.2. let m be an r-module, and let ω̃(m) be the family of all subsets of specl(m) of the form ν(n)∩u(k) where n, k ≤ m. clearly ω̃(m) contains specl(m) and the empty set, since specl(m) equals ν(0)∩u(m) and the empty set equals ν(m)∩u(0). let t̃p(m) to be the collection ũ of all unions of elements of ω̃(m). then t̃p(m) is a topology on specl(m) and it is called the finer patch-like topology (in fact, ω̃(m) is a basis for the finer patch-like topology of m). we will use x to represent specl(m). lemma 2.3. let m be an r-module and q ∈ x. then for each finer patch-like-neighborhood w of q, there exists a submodule l of m such that √ (q : m) ⊆ √ (l : m) and q ∈ ν(q) ∩ u(l) ⊆ w. proof. since q ∈ w, there exists a neighborhood of the form ν(k)∩u(n) ⊆ w such that q ∈ ν(k) ∩ u(n) where √ (q : m) ⊇ √ (k : m) and √ (q : m) + √ (n : m). since q ∈ ν(q) and ν(q) ⊆ ν(k), we may replace ν(k) by ν(q). now we claim that ν(q) ∩ u(n) = ν(q) ∩ u((i + p)m), where p = √ (q : m) and i = √ (n : m). since u(im) ⊆ u((i + p)m), ν(q) ∩ u(n) = ν(q) ∩ u(im) ⊆ ν(q) ∩ u((i + p)m). suppose that q′ ∈ ν(q) ∩ u((i + p)m), then q′ /∈ u(q). on the other hand q′ ∈ u((i + p)m) = u(n) ∪ u(q). this follows that q′ ∈ u(n). thus ν(q) ∩ u(n) = ν(q) ∩ u((i + p)m). now let l = (i + p)m. then p ⊆ i + p ⊆ √ (l : m) and q ∈ ν(q) ∩ u(l) ⊆ w. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 253 f. rashedi let y be a subset of y for a module m. we will denote the closure of y in x with finer patch-like topology by y. proposition 2.4. let m be an r-module and y ⊆ x be a finite set. if q ∈ y with finer patch-like topology, then there exists a ⊆ y such that ν(q) = ν( ⋂ q′∈a q′). proof. suppose q ∈ y. if q ∈ y , then we are thorough. thus we can assume that q /∈ y. let a = {q′ ∈ y| √ (q : m) ⊂ √ (q′ : m)}. since q ∈ u(m) ∩ ν(q), there exists q′′ ∈ y such that q′′ ∈ u(m) ∩ ν(q). since q /∈ y, √ (q : m) ⊂ √ (q′′ : m) and hence a 6= ∅. since √ (q : m) ⊂ √ (q′ : m) for each q′ ∈ a, √ (q : m) ⊂ ∩q′∈a √ (q′ : m) = √ (∩q′∈aq′ : m). if ∩q′∈a √ (q′ : m) * √ (q : m), then q ∈ u(∩q′∈aq ′) ∩ ν(q). since q ∈ y, there exists q′′ ∈ y such that q′′ ∈ u(∩q′∈aq ′) ∩ ν(q). therefore q′′ ∈ ν(q) and hence q′′ ∈ a. but ⋂ q′∈a √ (q′ : m) = √ ( ⋂ q′∈a q′ : m) ⊆ √ (q′′ : m). thus q′′ /∈ u( ⋂ q′∈a q′), a contradiction. thus ⋂ q′∈a √ (q′ : m) ⊆ √ (q : m), and hence ν(q) = ν( ⋂ q′∈a q′). � a module m over a commutative ring r is called a multiplication module if each submodule of m has the form im for some ideal i of r [4]. in this case we can take i = (n : m). proposition 2.5. let m be a multiplication r-module such that (q : m) is a radical ideal for every q ∈ x. then x with the finer patch-like topology is hausdorff. moreover, x with this topology is totally disconnected. proof. assume q, q′ ∈ x are distinct points. since q 6= q′, (q : m) 6= (q′ : m). thus either (q : m) * (q′ : m) or (q′ : m) * (q : m). suppose that (q : m) * (q′ : m). by definition 2.2, u1 := u(m) ∩ ν(q) is a finer patchlike-neighborhood of q and since (q : m) * (q′ : m), u2 := u(q) ∩ ν(q ′) is a finer patch-like-neighborhood of q′. clearly u(q) ∩ ν(q) = ∅ and hence u1 ∩ u2 = ∅. thus x is a hausdorff space. on the other hand for every submodule n of m, observer that the sets u(n) and ν(n) are open in finer patch-like topology, since ν(n) = u(m)∩ν(n) and u(n) = u(n)∩ν(0). since u(n) and ν(n) are complement of each other, they are both finer both-closed as well. therefore the finer patch-like topology on x has a basis of open sets which are also closed, and hence x is totally disconnected in this topology. � the following example shows that the condition multiplication in proposition 2.5 is necessary. © agt, upv, 2021 appl. gen. topol. 22, no. 2 254 a new topology over the primary-like spectrum example 2.6. let v be a vector space over a field f with dimf v > 1. it is evident that x and spec(v ) are the set of all proper vector subspaces of v . now, √ (q : m) = √ (q′ : m) for all distinct subspaces q, q′ ∈ x . if (q : m) is a radical ideal for every q ∈ x , then x with the finer patch-like topology is not hausdorff. definition 2.7. an r-module m is called p-module if for each prime ideal p of r such that (pm : m) = p, there exists q ∈ x such that √ (q : m) = p. for example every finitely generated faithful module is a p-module. now we show that every noetherian r-module m is also a p-module. let p be a prime ideal of a ring r, m an r-module, and n ≤ m. by the saturation of n with respect to p, we mean the contraction of np in m and designate it by sp(n). it is also known that sp(n) = {m ∈ m|rm ∈ n for some r ∈ r\p}. saturations of submodules were investigated in detail in [10]. lemma 2.8. let m be a notherian r-module. then m is a p-module. proof. assume m is a notherian r-module. hence m is finitely generated. by [11, proposition 1.8], for each prime ideal p of r, sp(pm) is a prime submodule of m such that (pm : m) = p. thus sp(pm) ∈ x . � theorem 2.9. let r be a ring and m be a p-module such that r/ann(m) has acc on ideals. if (n : m) is a radical ideal for every submodule n of m, then x with the finer patch-like topology is a quasi-compact space. proof. suppose m is a p-module and r/ann(m) has acc on ideals. assume a is a family of finer patch-like-open sets covering x and suppose that no finite subfamily of a covers x . suppose s = {i|i is an ideal of r such that ann(m) ⊆ i and no finite subfamily of a covers ν(im)}. since ν(ann(m)m) = ν(0) = x , s 6= ∅. we may use the acc on ideals of r/ann(m) to choose an ideal m of r maximal with respect to the property that no finite subfamily of a covers ν(mm) (i.e., m is a maximal element of s). it is clear that mm 6= m. we claim that m is a prime ideal of r, for if not, suppose that i and j are two ideals of r properly containing m and ij ⊆ m. then ν(im) and ν(jm) covered by finite subfamily of a. suppose q ∈ ν(ijm), then ij ⊆ p := √ (q : m). since p is prime, either i ⊆ p or j ⊆ p, and hence either q ∈ ν(im) or q ∈ ν(jm). thus ν(ijm) covered by a finite subfamily of a. since ij ⊆ m, then ν(mm) ⊆ ν(ijm). thus ν(mm) covered by finite subfamily of a, a contradiction. thus m is a prime ideal of r. we claim that (mm : m) = m, for if not, then there exists an ideal m1 of r such that m1 = (mm : m) and m ⊂ m1. this follows that mm = m1m and so no finite subfamily of a covers ν(m1m), contrary to maximality of m. therefore (mm : m) = m and since m is p-module, there exists q′ ∈ x such that √ (q′ : m) = m. let u ∈ a such that q′ ∈ u. by lemma 2.3, there exists a submodule k of m such that m = √ (q′ : m) ⊆ √ (k : m) and © agt, upv, 2021 appl. gen. topol. 22, no. 2 255 f. rashedi q′ ∈ u(k) ∩ ν(q′) ⊆ u. suppose (k : m) = i. by lemma 1.1, we know that u(k) = u(im) and ν(q′) = ν(mm), and so q′ ∈ u(im) ∩ ν(mm) ⊆ u. since m ⊆ i, then ν(im) can be covered by some finite subfamily a′ of a. but ν(mm)\ν(im) = ν(mm)\[u(im)]c = ν(mm) ∩ u(im) ⊆ u and so ν(mm) can be covered by a′ ∪ {u}, contrary to our choice of q′. thus there must exist a finite subfamily of a which covers x . therefore x is quasi-compact in the finer patch-like topology of m. � it is well-known that if m is a noetherian module over a ring r, then r/ann(m) is a noetherian ring. hence we have the following result. corollary 2.10. let m be a noetherian r-module. if (n : m) is a radical ideal for every submodule n of m, then x with the finer patch-like topology is a quasi-compact space. proof. using lemma 2.8 and theorem 2.9. � we need the following evident lemma. lemma 2.11. let t , t ∗ be two topology on x such that t ⊆ t ∗. if x is quasi-compact in t ∗, then x is also quasi-compact in t . theorem 2.12. let m be an r-module. if x is quasi-compact with the finer patch-like topology, then for each submodule n of m, u(n) is a quasi-compact subset of x with the zariski topology. consequently, x with the zariski topology is quasi-compact. proof. by definition 2.2, for each submodule n of m, ν(n) = ν(n) ∩ u(m) is an open subset of x with finer patch-like topology, and hence, for each submodule n of m, u(n) is a closed subset in x with finer patch-like topology. since every closed subset of a quasi-compact space is quasi-compact, u(n) is quasi-compact in x with finer patch-like topology and so by lemma 2.11, it is quasi-compact in x with the zariski topology. now, since x = u(m), x is quasi-compact with the zariski topology. � corollary 2.13. let m be an r-module. if x is quasi-compact with finer patch-like topology, then the finer patch-like topology and the patch-like topology of m coincide. proof. by theorem 2.12, for each submodule k of m, u(k) is quasi-compact. therefore for each n, k ≤ m, ν(n) ∩ u(k) is an element of the basis ω(m) of the patch-like topology on x . � corollary 2.14. let m be an r-module such that (n : m) is a radical ideal for every submodule n of m. if m is noetherian or m is a p-module such that r/ann(m) has acc on ideals, then the finer patch-like topology and the patch-like topology of m coincide. proof. by theorem 2.9 and corollaries 2.10 and 2.13. � we conclude this section with the following results. © agt, upv, 2021 appl. gen. topol. 22, no. 2 256 a new topology over the primary-like spectrum corollary 2.15. let m be a multiplication p-module such that (n : m) is a radical ideal for every submodule n of m and r/ann(m) has acc on ideals. then x with the zariski topology is a hausdorf, quasi-compact, totally disconnected space. proof. by proposition 2.5, theorem 2.9 and corollary 2.13. � corollary 2.16. let m be a multiplication noetherian r-module such that (n : m) is a radical ideal for every submodule n of m. then x with the zariski topology is a hausdorf, quasi-compact, totally disconnected space. proof. by lemma 2.8, proposition 2.5, theorem 2.9 and corollary 2.13. � references [1] m. alkan and y. tiraş, projective modules and prime submodules, czechoslovak math. j. 56 (2006), 601–611. [2] h. ansari-toroghy and r. ovlyaee-sarmazdeh, on the prime spectrum of a module and zariski topologies, comm. algebra 38 (2010), 4461–4475. [3] a. azizi, prime submodules and flat modules, acta math. sin. (eng. ser.) 23 (2007), 47–152. [4] a. barnard, multiplication modules, j. algebra 71 (1981), 174–178. [5] j. dauns, prime modules, j. reine angew math. 298 (1978), 156–181. [6] h. fazaeli moghimi and f. rashedi, zariski-like spaces of certain modules, journal of algebraic systems 1 (2013), 101–115. [7] k. r. goodearl and r. b. warfield, an introduction to non-commutative noetherian rings (second edition), london math. soc. student texts 16, 2004. [8] m. hochster, prime ideal structure in commutative rings, trans. amer. math. soc. 137 (1969), 43–60. [9] c. p. lu, a module whose prime spectrum has the surjective natural map, houston j. math. 33 (2007), 125–143. [10] c. p. lu, saturations of submodules, comm. algebra 31 (2003), 2655–2673. [11] r. l. mccasland and p. f. smith, prime submodules of noetherian modules, rocky mountain. j. math. 23 (1993), 1041–1062. © agt, upv, 2021 appl. gen. topol. 22, no. 2 257 @ appl. gen. topol. 21, no. 1 (2020), 71-79 doi:10.4995/agt.2020.12042 c© agt, upv, 2020 new topologies between the usual and niemytzki dina abuzaid a, maha alqahtani b and lutfi kalantan a a department of mathematics, king abdulaziz university, saudi arabia. (dabuzaid@kau.edu.sa, lkalantan@kau.edu.sa) b department of mathematics, king khalid university, saudi arabia. (mjobran@kku.edu.sa) communicated by o. valero abstract we use the technique of hattori to generate new topologies on the closed upper half plane which lie between the usual metric topology and the niemytzki topology. we study some of their fundamental properties and weaker versions of normality. 2010 msc: 54a10; 54d15. keywords: generated topology; h-space; niemytzki plane; usual metric; cc-normal; l-normal; s-normal; c-paracompactness. 1. notations and basic definitions we use the technique of hattori [6, 13] to generate new topologies on the closed upper half plane which lie between the usual metric topology and the niemytzki topology. we study some of their fundamental properties and weaker versions of normality. we denote an order pair by 〈x,y〉, the set of real numbers by r, the natural numbers by n, and the rationals by q. let x = {〈x,y〉 ∈ r2 : y ≥ 0} and p = {〈x,y〉 ∈ r2 : y > 0}, so the x-axis is l = x\p . denote the usual metric topology on x by u and the niemytzki topology on x by n . for every 〈x, 0〉 ∈ l and r ∈ r, r > 0, let d(〈x, 0〉,r) be the set of all points of p inside the circle of radius r tangent to x-axis at 〈x, 0〉 and let dr(〈x, 0〉) = d(〈x, 0〉,r)∪{〈x, 0〉}. for every 〈x,y〉 ∈ x and r > 0, let ur(〈x,y〉) be the set of all points of x inside the circle of radius r and centered at 〈x,y〉. recall that the niemytzki topology n on x is generated by the following neighborhood received 03 july 2019 – accepted 28 october 2019 http://dx.doi.org/10.4995/agt.2020.12042 d. abuzaid, m. alqahtani and l. kalantan system: for every 〈x, 0〉 ∈ l, let b(〈x, 0〉) = {dr(〈x, 0〉) : r > 0}. for every 〈x,y〉 ∈ p , let b(〈x,y〉) = {ur(〈x,y〉) : r > 0}. observe that p as a subspace of x with the usual metric topology coincides with p as a subspace of x with the niemytzki topology. definition 1.1. let a be a non-empty proper subset of the x-axis l. for each 〈a, 0〉 ∈ a, let b(〈a, 0〉) = {ur(〈a, 0〉) : r > 0}, where ur(〈a, 0〉) is the set of all points of x inside the circle of radius r and centered at 〈a, 0〉. for each 〈a,b〉 ∈ p , let b(〈a,b〉) = {ur(〈a,b〉) : r > 0}. so, the points in a∪p will have the same local base as in ( x , u ). for each 〈c, 0〉 ∈ l\a, let b(〈c, 0〉) = {dr(〈c, 0〉) : r > 0}. so, the points in l\a will have the same local base as in ( x , n ). we call the topology on x generated by the neighborhood system {b(〈x,y〉) : 〈x,y〉 ∈ x} the h-generated topology on x from u and n and denote it by uan . we call x with this h-generated topology an h-space and denote it by ( x , uan ). observe that if a = ∅, then uan is the niemytzki topology, if a is the x-axis l, then uan is the usual topology. from now on, when we consider x with an h-generated topology uan we are assuming that a is a non-empty proper subset of the x-axis l. let us interchange the local bases in definition 1.1 as follows: let a be a non-empty proper subset of the x-axis l. for each 〈a, 0〉 ∈ a, let b(〈a, 0〉) = {dr(〈a, 0〉) : r > 0}. for each 〈a,b〉 ∈ p , let b(〈a,b〉) = {ur(〈a,b〉) : r > 0}. so, the points in a∪p will have the same local base as in ( x , n ). for each 〈c, 0〉 ∈ l\a, let b(〈c, 0〉) = {ur(〈c, 0〉) : r > 0}, where ur(〈c, 0〉) is the set of all points of x inside the circle of radius r and centered at 〈c, 0〉. so, the points in l \ a will have the same local base as in ( x , u ). we call the topology on x generated by the neighborhood system {b(〈x,y〉) : 〈x,y〉 ∈ x} the h-generated topology on x from n and u and denote it by nau. it is clear that uan = n(l\a)u for any subset a of the x-axis l. 2. some fundamental properties observe that for any non-empty proper subset a of the x-axis we have u ⊆uan ⊆n . thus ( x , uan ) is t0, t1, hausdorff, completely hausdorff, and connected. to show complete regularity of ( x , uan ) we use frink’s theorem [4] which is the following characterization, see also [3, 1.5.g]: theorem 2.1 (o. frink). a space x is completely regular if and only if there exists a base b for x satisfying the following two conditions: (1) for every x ∈ x and every u ∈ b that contains x there exists v ∈ b such that x 6∈ v and u ∪v = x. (2) for any u,v ∈ b satisfying u ∪v = x, there exist u′,v ′ ∈ b such that x \v ⊆ u′, x \u ⊆ v ′, and u′ ∩v ′ = ∅. theorem 2.2. every h-space ( x , uan ) is tychonoff. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 72 new topologies between the usual and niemytzki proof. denote the base of the neighborhood system {b(〈x,y〉) : 〈x,y〉 ∈ x} by b. let w = {x \g : g ∈ b}. define b = b∪w. since b ⊂ b, then b is a base. we show that b satisfies the conditions of theorem 2.1. observe that for each g,g′ ∈ b with g ⊂ g′ we have g ⊂ g ⊂ g′. let 〈x,y〉 ∈ x be arbitrary, and u ∈ b be arbitrary with 〈x,y〉 ∈ u. pick k ∈ b such that 〈x,y〉 ∈ k ⊂ u. we have two cases: (1) u ∈ b. then we can let v = x \ k. therefore 〈x,y〉 /∈ v and u ∪v = x. (2) u ∈w. put u = x \g; g ∈b. here k has two cases: (a) k ∈b. then we can let v = x \k. (b) k ∈w, where k = x\g ′ ; g′ ∈b with g ⊂ g′. then we can let v = g′. therefore in each case 〈x,y〉 /∈ v and u ∪v = x. so the first condition of theorem 2.1 is satisfied. let u,v ∈ b be arbitrary satisfying u ∪v = x. then one and only one of the following cases is satisfied: (1) u ∈b and v ∈w, where v = x \g; g ∈b with g ⊂ u. pick k ∈b with g ⊂ k ⊂ u. let u′ = k and v ′ = x\k. therefore x\v ⊂ u′, x \u ⊂ v ′, and u′ ∩v ′ = ∅. (2) u ∈w and v ∈b, where u = x \g; g ∈b with g ⊂ v . pick k ∈b with g ⊂ k ⊂ v . let v ′ = k and u′ = x\k. therefore x\v ⊂ u′, x \u ⊂ v ′, and u′ ∩v ′ = ∅. (3) u ∈ w and v ∈ w, put u = x \ k, v = x \ g where k,g ∈ b with k ∩ g = ∅. write k = br1 (〈x,y〉) and g = br2 (〈a,b〉). since x = u ∪ v , then k ∩ g = ∅. since k and g are both closed, then the distance between them are positive say δ > 0. let �1 = r1 + δ 4 and �2 = r2 + δ 4 . let v ′ = b�1 (〈x,y〉) and u′ = b�2 (〈a,b〉). therefore x \v ⊂ u′, x \u ⊂ v ′, and u′ ∩v ′ = ∅. so, in all cases, the second condition of theorem 2.1 is satisfied. thus ( x , uan ) is tychonoff. � any h-space ( x , uan ) is first countable just by taking for each 〈x,y〉 ∈ a∪p the countable local base b′(〈x,y〉) = {u 1 n (〈x,y〉) : n ∈ n} and for each 〈x, 0〉 ∈ l\a the countable local base b′(〈x, 0〉) = {d 1 n (〈x, 0〉) : n ∈ n}. any h-space ( x , uan ) is separable as (q × q) ∩p is a countable dense subset. for the second countability, we have the following theorem: theorem 2.3. let ( x , uan ) be an h-space. the following are equivalent: (1) l\a is countable. (2) ( x , uan ) is second countable. (3) ( x , uan ) is metrizable. proof. (1)⇒(2) assume that l\a is countable. since a∪p as a subspace is metrizable, let w be a countable base for a∪p . let b = {w,b′(〈x, 0〉) : 〈x, 0〉 ∈ c© agt, upv, 2020 appl. gen. topol. 21, no. 1 73 d. abuzaid, m. alqahtani and l. kalantan l \ a}, then b is a countable base for ( x , uan ) because l \ a is countable. (2)⇒(3) assume that ( x , uan ) is second countable. since ( x , uan ) is also t3, see theorem 2.2, and any t3 second countable space is metrizable [3], result follows. (3)⇒(2) assume that ( x , uan ) is metrizable. since it is separable and any metrizable separable space is second countable [3, 4.1.16], result follows. (2)⇒(1) assume that ( x , uan ) is second countable. suppose that l \ a is uncountable. since any basic open set dr(〈x, 0〉) of each element 〈x, 0〉 in l\a does not contain any element from the x-axis other than 〈x, 0〉 itself and any basic open set ur(〈x′, 0〉) of each element 〈x′, 0〉 in a cannot be contained in dr(〈x, 0〉), we conclude that ( x , uan ) cannot be second countable which is a contradiction. � ( x , uan ) need not be normal, for example, if a = {〈x, 0〉 : x < −1 or 0 < x}, then x\(a∪p) is closed uncountable discrete subspace of ( x , uan ). since ( x , uan ) is also separable, then by jones’ lemma [7], ( x , uan ) cannot be normal. since any interval c in r must contain a closed bounded interval [a,b], we conclude the following theorem: theorem 2.4. if l \ a contains a set of the form c ×{0}, where c is an interval in r, then ( x , uan ) cannot be normal. since any metrizable space is normal, theorem 2.3 gives the following: theorem 2.5. if l\a is countable, then ( x , uan ) is normal. for each n ∈ n, let wn = r × [0,n). then x ⊆ ⋃ n∈n wn and for all n ∈ n, wn ∈ uan . so the family w = {wn : n ∈ n} is a countable open cover of x which has no finite subcover. thus ( x , uan ) is neither compact nor countably compact. theorem 2.6. for a non-empty proper subset a of l, ( x , uan ) is not locally compact. proof. let 〈x, 0〉 ∈ l\a be arbitrary and let g be any open neighborhood of 〈x, 0〉. there exists an i ∈ r such that di(〈x, 0〉) ⊂ di(〈x, 0〉) = di(〈x, 0〉) u ⊆ g. observe that di(〈x, 0〉) u is closed in ( x , uan ). now, the circumference of di(〈x, 0〉) u contains a sequence of points which converges to 〈x, 0〉 in the usual topology, but this sequence can have no accumulation point in ( x , uan ) as the open neighborhood dj(〈x, 0〉), where j > i, satisfies that dj(〈x, 0〉) does not contain any member of the sequence. thus dj(〈x, 0〉) u is not countably compact, hence not compact. � the first result about the lidelöfness of an h-space is a corollary of theorem 2.3. theorem 2.7. if l\a is countable, then the h-space ( x ,uan ) is lindelöf. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 74 new topologies between the usual and niemytzki theorem 2.8. if a′ = {x : 〈x, 0〉 ∈ a} is dense in (r,u), then ( x ,uan ) is lindelöf. proof. let w = {wα : α ∈ λ} be any open cover for x. for each 〈x, 0〉 ∈ a, there exists an αx ∈ λ such that 〈x, 0〉 ∈ wαx. thus for each 〈x, 0〉 ∈ a there exists rx > 0 such that 〈x, 0〉 ∈ urx(〈x, 0〉) ⊆ wαx. since a′ is dense in (r,u) , then ∪x∈a′ urx(〈x, 0〉) covers l. now, for each 〈x,y〉 ∈ p , there exists wα〈x,y〉 ∈ w such that 〈x,y〉 ∈ wα〈x,y〉 . this means that there exists r〈x,y〉 > 0 such that 〈x,y〉 ∈ ur〈x,y〉 (〈x,y〉) ⊆ wα〈x,y〉 . thus w ′ = {ur〈x,y〉 (〈x,y〉),urx(〈x, 0〉) : 〈x, 0〉 ∈ a,〈x,y〉 ∈ p} is an open cover for (x,u). since (x,u) is lindelöf , then there exists a countable subcover from w′. then w has a countable open refinement . therefore, ( x ,uan ) is lindelöf. � theorem 2.9. ( x ,uan ) is not lindelöf if and only if l\a contains a set of the form c ×{0}, where c is an interval in r. proof. (⇒) assume that ( x ,uan ) is not lindelöf. suppose that l\a does not contain any set of the form c ×{0}, where c is an interval in r. so, for each a,b ∈ r; a < b, there exists x ∈ a such that a < x < b. this gives that a′ = {x : 〈x, 0〉 ∈ a} is dense in (r,u). then by theorem 2.8 ( x ,uan ) is lindelöf which is a contradiction. (⇐) assume that l\a contains a set of the form c ×{0}, where c is an interval in r. let j = [a,b ] ⊆ c. let � = b−a 4 . then x ⊆ (∪〈x,y〉∈(a∪p)\([a,b ]×[0, 3� 2 )) u�2 (〈x,y〉) ) ∪ (∪〈x,0〉∈l\a d�(〈x, 0〉) ). so, {u� 2 (〈x,y〉),d�(〈x, 0〉) : 〈x,y〉 ∈ (a∪p)\([a,b ]× [0, 3�2 );〈x, 0〉 ∈ l\a} is an open cover of x which has no countable subcover because j is uncountable and for each 〈x, 0〉 ∈ j ×{0} the set d� (〈x, 0〉) contains no other element of j except 〈x, 0〉 . � 3. other properties of h-spaces recall that a topological space x is called c-normal [1] (cc-normal [9], l-normal [11], s-normal [10]) if there exists a normal space y and a bijective function f : x −→ y such that the restriction f �a: a −→ f(a) is a homeomorphism for each compact (countably compact, lindelöf, separable) subspace a ⊆ x. a topological space x is called c2-paracompact [12] if there exists a t2 paracompact space y and a bijective function f : x −→ y such that the restriction f �a: a −→ f(a) is a homeomorphism for each compact subspace a ⊆ x. a topological space ( x , τ ) is called submetrizable if there exists a metric d on x such that the topology τd on x generated by d is coarser than τ , i.e., τd ⊆ τ , see [5]. since submetrizabilty implies both c-normality [1] and c2-paracompactness [12], we conclude that any h-space ( x , uan ) is both c-normal and c2-paracompact being submetrizable by the usual metric. by the theorem “if x is t3 separable l-normal and of countable tightness, then x is normal.” [11, 1.6], we obtain the following theorem: c© agt, upv, 2020 appl. gen. topol. 21, no. 1 75 d. abuzaid, m. alqahtani and l. kalantan theorem 3.1. ( x , uan ) is normal if and only if it is l-normal. as ( x , uan ) is always separable we conclude the following theorem: theorem 3.2. ( x , uan ) is normal if and only if it is s-normal. now, to study the cc-normality of an h-space, we start with the ccnormality of the niemytzki plane. we do this by three steps. lemma 3.3. a subset c of the niemytzki plane x is countably compact if and only if c ∩ l is finite and c is closed and bounded in x considered with its usual metric topology. proof. assume that c is a countably compact subspace of the niemytzki plane x. suppose that c ∩ l is infinite. pick a countably infinite subset d = {〈dn, 0〉 : n ∈ n}⊆ c∩l. for each n ∈ n, consider the basic open neighborhood d1(〈dn, 0〉) of 〈dn, 0〉. for each 〈x,y〉 ∈ c ∩ p , consider uy 2 (〈x,y〉) and let u = ( ⋃ 〈x,y〉∈c∩p uy2 (〈x,y〉)) ⋃ ( ⋃ 〈x,0〉∈(c∩l)\d d1(〈x, 0〉)). then the countable open cover {u,d1(〈dn, 0〉) : n ∈ n} of c has no finite subsover which is a contradiction. now, assume that c is countably compact and c∩l is finite. suppose that c is either not closed in x considered with its usual metric topology or not bounded. since in a metrizable space, a subspace is countably compact if and only if it compact [3, 4.1.17]. also, in the usual metric space, a subspace is compact if and only if it is closed and bounded [3, 3.2.8]. we conclude that c is not countably compact in x considered with its usual metric topology. since the usual metric topology u is coarser than the niemytzki topology n we conclude that c is not countably compact in the niemytzki plane x which is a contradiction. conversely, assume that c ∩ l is finite and c is closed and bounded in x considered with its usual metric topology. let w be any countable open cover for c. then w is a countable open cover for c ∩ p. since c is closed and bounded in x considered with its usual metric topology, then c ∩ p is closed and bounded in p considered with its usual metric topology. so, c ∩p is compact in p . pick a finite w1, ...,wn ∈ w such that c ∩ p ⊆ ⋃n i=1 wi. since c ∩ l is finite, pick for each 〈x, 0〉 ∈ c ∩ l a member wx ∈ w such that 〈x, 0〉 ∈ wx. then {w1, ...,wn,wx : x ∈ c ∩ l} is a finite subcover. therefore, c is countably compact. � lemma 3.4. let c be a subspace of the niemytzki plane x. c is countably compact if and only if c is compact. proof. let c be any countably compact subspace in the niemytzki plane x. by lemma 3.3, c ∩ l is finite and c is closed and bounded in x considered with its usual metric topology, hence c ∩ p is compact in x with its usual metric topology [3, 3.2.8]. let g be any open cover for c consisting of basic open sets. since c ∩ l is finite, pick gx ∈ g such that 〈x, 0〉 ∈ gx. since c ∩ p is compact, pick a finite subcover g′ of g which covers c ∩ p . then g′ ∪{gx : 〈x, 0〉 ∈ c ∩l} is a finite subcover of g. thus c is compact. the other direction is clear. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 76 new topologies between the usual and niemytzki theorem 3.5. the niemytzki plane is cc-normal. proof. let y = x with its usual metric topology. consider the identity function id : x −→ y . since the usual metric topology u is coarser than the niemytzki topology n , then id : x −→ y is continuous, hence any restriction function of it is continuous. let c be any countably compact subspace of x. by lemma 3.4, c is compact in the niemytzki plane, hence id|c : c −→ id(c) = c is a homeomorphism, see [3, 3.1.13]. � we use similar ideas to show that any h-space is cc-normal. lemma 3.6. a subset c of an h-space ( x , uan ) is countably compact if and only if c satisfies the following two conditions: (1) c is closed and bounded in ( x , u ). (2) any infinite subset of c ∩ (l\a) has an accumulation point in c ∩a in l with its usual metric topology. proof. assume that c is countably compact in an h-space ( x , uan ). suppose that c is either not closed in x considered with its usual metric topology or not bounded. since in a metrizable space, a subspace is countably compact if and only if it compact [3, 4.1.17]. also, in the usual metric space, a subspace is compact if and only if it is closed and bounded [3, 3.2.8]. we conclude that c is not countably compact in ( x , u ). since the usual metric topology u is coarser than the h-topology uan we conclude that c is not countably compact in the h-space ( x , uan ) which is a contradiction. now, assume that c is a countably compact subspace of an h-space ( x , uan ) and c is closed and bounded in ( x , u ). suppose that there exists a countably infinite subset d = {〈dn, 0〉 : n ∈ n} of c∩(l\a) which has no accumulation point in c∩a with respect to l with its usual metric topology. for each 〈x, 0〉 ∈ c ∩a, fix rx > 0 such that urx(〈x, 0〉) satisfies urx(〈x, 0〉) ∩d = ∅. let u = ( ⋃ 〈x,y〉∈c∩p uy 2 (〈x,y〉)) ⋃ ( ⋃ 〈x,0〉∈c∩((l\a)\d) d1(〈x, 0〉)) ⋃ ( ⋃ 〈x,0〉∈c∩a urx(〈x, 0〉)). then the countable open cover {u,d1(〈dn, 0〉) : n ∈ n} of c has no finite subsover which is a contradiction. conversely, let c be any subset of x satisfies the two conditions. let w be any countable open cover for c. by condition (1), c ∩ (p ∪a) is closed and bounded in the metrizable space p ∪a. so, c ∩ (p ∪a) is compact in p ∪a. pick a finite subcover w ′ of w for c∩(p∪a). if c∩(l\a) is finite, there exist w1, ...,wn ∈ w such that c ∩ (l\a) ⊆ ⋃n i=1 wi. then w ′ ∪{w1, ...,wn} is a finite subcover of w covers c. now, if c ∩ (l \ a) is infinite. for each 〈x, 0〉 ∈ c ∩ (l \ a) there exists wx ∈ w with 〈x, 0〉 ∈ wx. since w is countable, pick a countable subset e = {〈xn, 0〉 : n ∈ n} ⊆ c ∩ (l\a) such that c ∩ (l\a) ⊆ ⋃ n∈n wxn. by condition (2), pick 〈y, 0〉 ∈ c ∩a such that 〈y, 0〉 is an accumulation point of e in l with its usual metric. then the open set wy ∈w ′ with 〈y, 0〉 ∈ wy covers all elements of e except possibly finitely many elements say 〈xn1, 0〉, ...,〈xnm, 0〉. for each i ∈ {1, ...,m} pick wi ∈ w c© agt, upv, 2020 appl. gen. topol. 21, no. 1 77 d. abuzaid, m. alqahtani and l. kalantan such that 〈xni, 0〉 ∈ wi. then w ′ ⋃ {wni : i ∈{1, ...,m}} is a finite subcover of w covers c. therefore, c is countably compact. � lemma 3.7. let c be a subspace of an h-space ( x , uan ). c is countably compact if and only if c is compact. proof. let c be any countably compact subspace in an h-space ( x , uan ). by lemma 3.6, c is closed and bounded in x considered with its usual metric topology, hence c ∩ (p ∪ a) is compact in x with its usual metric topology [3, 3.2.8]. let g be any open cover for c consisting of basic open sets. since c ∩ (p ∪ a) is compact, pick a finite subcover g′ of g which covers c ∩ (p ∪ a). in particular, g′ covers c ∩ a. pick 〈x1, 0〉, ...,〈xn, 0〉 ∈ c ∩ a such that c ∩ a ⊆ ⋃n i=1 uri(〈xi, 0〉). by lemma 3.6, ⋃n i=1 uri(〈xi, 0〉) covers all possible accumulation points of c ∩ (l \ a) in l with its usual metric topology. thus ⋃n i=1 uri(〈xi, 0〉) covers all points of c ∩ (l \ a) except possibly finitely many points, say 〈y1, 0〉, ...,〈ym, 0〉 ∈ c ∩ (l \ a), because if (c ∩ (l \ a)) \ ( ⋃n i=1 uri(〈xi, 0〉)) is infinite, then any countably infinite subset of it will not have an accumulation point in c ∩ a in l with its usual metric and this contradicts the countable compactness of c, see lemma 3.6. for each j ∈ {1, ...,m} pick gj ∈ g such that 〈yj, 0〉 ∈ gj. then g′ ∪ {gj : j ∈ {1, ...,m}} is a finite subcover for g covers c. thus c is compact. � theorem 3.8. any h-space ( x , uan ) is cc-normal. proof. let y = x with its usual metric topology. consider the identity function id : x −→ y . since the usual metric topology u is coarser than the h-topology uan , then id : x −→ y is continuous, hence any restriction function of it is continuous. let c be any countably compact subspace in ( x , uan ). by lemma 3.7, c is compact in the h-space ( x , uan ), hence id|c : c −→ id(c) = c is a homeomorphism, see [3, 3.1.13]. � conjecture: under the cases that ( x , uan ) is not normal, is it κ-normal [8, 14]? quasi-normal [15]? references [1] s. alzahrani and l. kalantan, c-normal topological property, filomat 31 (2017), 407– 411. [2] v. a. chatyrko and y. hattori, on reversible and bijectively related topological spaces, topology and its applications 201 (2016), 432–440. [3] r. engelking, general topology, vol. 6, berlin: heldermann (sigma series in pure mathematics) poland, 1989. [4] o. frink, compactification and semi-normal spaces, amer. j. math. 86 (1964), 602–607. [5] g. gruenhage, generalized metric spaces, k. kunen, j. e. vaughan (eds.), handbook of set-theoretic topology, north-holland, amsterdam, 1984, pp. 423–510. [6] y. hattori, order and topological structures of posets of the formal balls on metric spaces, mem. fac. sci. eng. shimane univ. ser. b math sci 43 (2010), 13–26. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 78 new topologies between the usual and niemytzki [7] f. b. jones, concerning normal and completely normal spaces, bull. amer. math. soc. 43(1937), 671–677. [8] l. kalantan, results about κ-normality, topology and its applications 125 (2002) 47–62. [9] l. kalantan and m. alhomieyed, cc-normal topological spaces, turk. j. math. 41 (2017), 749–755. [10] l. kalantan and m. alhomieyed, s-normality, j. math. anal. 9, no. 5 (2018), 48–54. [11] l. kalantan and m. m. saeed, l-normality, topology proceedings 50(2017), 141–149. [12] l. kalantan, m. saeed and h. alzumi, c-paracompactness and c2-paracompactness, turk. j. math. vol. 43, no. 1 (2019), 9–20. [13] j. kulesza, results on spaces between the sorgenfrey and usual topologies, topology and its applications 231(2017), 266–275. [14] e. v. shchepin, real valued functions and spaces close to normal, sib. j. math. 13 (1972), 1182–1196. [15] v. zaitsev, on certain classes of topological spaces and their bicompactifications, dokl. akad. naur sssr 178 (1968), 778–779. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 79 @ appl. gen. topol. 20, no. 1 (2019), 265-279doi:10.4995/agt.2019.10930 c© agt, upv, 2019 a quantitative version of the arzelà-ascoli theorem based on the degree of nondensifiability and applications g. garćıa departamento de matemáticas, uned, 03202 elche, alicante (spain) (gonzalogarciamacias@gmail.com) communicated by d. werner abstract we present a novel result that, in a certain sense, generalizes the arzelàascoli theorem. our main tool will be the so called degree of nondensifiability, which is not a measure of noncompactness but can be used as an alternative tool in certain fixed problems where such measures do not work out. to justify our results, we analyze the existence of continuous solutions of certain volterra integral equations defined by vector valued functions. 2010 msc: primary 46b50; secondary 47h08; 45d05. keywords: arzelà-ascoli theorem; degree of nondensifiability; α-dense curves; measures of noncompactness; volterra integral equations. 1. introduction to set the notation, let i := [0,1], (x,‖ · ‖) a banach space and c(i,x) the banach space of continuous maps x : i −→ x, endowed the norm ‖x‖∞ := max{‖x(t)‖ : t ∈ i}, for each x ∈ c(i,x). also, bx denotes the class of non-empty and bounded subsets of x, conv(b) and b̄, as usual, denote the convex hull and closure of a subset b of x, respectively. it is well known, that one of the most useful tools to analyse the compactness of a subset of c(i,x) is the celebrated arzelà-ascoli theorem (see, for instance, received 07 november 2018 – accepted 24 january 2019 http://dx.doi.org/10.4995/agt.2019.10930 g. garćıa [29]). so, it is not surprising that this result has been extended and generalized in many ways; see, for instance, [5, 7, 25] and references therein. our main goal will be to state a novel generalization of such a result. in our context, the arzelà-ascoli theorem can be stated as follows: a set b ∈ bc(i,x) is precompact (i.e., its closure is compact) if, and only if, the following conditions hold: (1) for each t ∈ i, the set b(t) := {x(t) : x ∈ b} is precompact. (2) b is equicontinuous, that is, given ε > 0 there is δ > 0 such that ‖x(t) − x(t′)‖ ≤ ε for every x ∈ b, whenever |t − t′| ≤ δ. for a better comprehension of the manuscript, we recall the concept of measure of noncompactness. as the definition of such measures may vary according to the author (see, for instance, [1, 3]), here we consider the following one given in [13]: definition 1.1. a map µ : bx −→ r+ := [0,∞) is said to be a measure of noncompactness, mnc, if it satisfies the following properties: (i) regularity: µ(b) = 0 if and only if b ∈ bx is a precompact set. (ii) invariant under closure: µ(b) = µ(b̄), for all b ∈ bx. (iii) monotony: µ(b1 ∪ b2) = max{µ(b1),µ(b2)}, for all b1, b2 ∈ bx. (iv) semi-homogeneity: µ(λb) = |λ|µ(b), for all λ ∈ r and s ∈ bx. (v) invariance under translations: µ(x + b) = µ(b), for all x ∈ x and b ∈ bx. two widely studied mncs are these of hausdorff and kuratowski (again, [1, 3]), denoted by χ and κ respectively and defined as χ(b) := inf { ε > 0 : b can be covered by a finite number of balls of radius ≤ ε } and κ(b) := inf { ε > 0 : b can be covered by a finite number of sets of diameter ≤ ε } , for every b ∈ bx. for instance, if ux is the closed unit ball of x, we have that χ(ux) = 1, κ(ux) = 2 if x has infinite dimension and χ(ux) = κ(ux) = 0 if x has finite dimension. there are some arzelà-ascoli results type for vector-valued functions based on the mncs (see, for instance, [6, 14, 23] or [28, theorem 12.5, p. 255]), the first of them, due to ambrosetti [2], was proved in 1967: theorem 1.2. let b ∈ bc(i,x) be equicontinuous. then, κ(b) = sup { κ ( b(t) ) : t ∈ i } , where b(t) := {x(t) : x ∈ b}, for each t ∈ i. roughly speaking, the above result “quantifies” the compactness of an equicontinuous b ∈ bc(i,x). of course, if x := r then the above result holds trivially c© agt, upv, 2019 appl. gen. topol. 20, no. 1 266 a quantitative version of the arzelà-ascoli theorem from the equicontinuity of b and the arzelà-ascoli theorem. therefore, theorem 1.2 can be useful in infinite dimensional banach spaces where, unlike the finite dimensional case, a bounded set is not necessarily precompact. on the other hand, our main goal is to prove a more general result, not based on mncs, than theorem 1.2 using the so called degree of nondensifiability φd, explained in detail in section 2. we prove this result in theorem 3.2, and as a consequence we derive some bounds of the mncs of arc-connected subsets of bc(i,x). as it is shown in [10, 11, 12], we can use φd as an alternative to the mncs in certain fixed point problems, because in some cases it seems to work out under more general conditions than the mncs. so, with a suitable combination of a sadovskĭı fixed point theorem type for φd (see theorem 2.11) and theorem 3.2, we will prove in section 4 a result regarding the existence of solutions of certain volterra integral equations. 2. the degree of nondensifiability in 1997 the concepts of α-dense curve and densifiable set were introduced by cherruault and mora [18] in a metric space (y,d): definition 2.1. let α ≥ 0 and b ∈ by , the class of non-empty and bounded subsets of y . a continuous map γ : i −→ y is said to be an α-dense curve in b if the following conditions hold: (i) γ(i) ⊂ b. (ii) for any x ∈ b, there is y ∈ γ(i) such that d(x,y) ≤ α. b is said to be densifiable if for every α > 0 there is an α-dense curve in b. let us note that, given b ∈ by , the class of α-dense curves in b is nonempty for any α ≥ diam(b), the diameter of b. indeed, fixed x0 ∈ b, the map given by γ(t) := x0, for each t ∈ i, is an α-dense curve in b, for any α ≥ diam(b). also, note that for b := in, n > 1, a 0-dense curve in b is, precisely, a space-filling curve in b (see [26]). so, the α-dense curves are a generalization of these curves. it can be proved that the class of densifiable sets is strictly between the class of peano continua (i.e. those sets that are the continuous image of i) and the class of connected and precompact sets. for a detailed exposition of the above concepts, see [8, 17, 18, 19, 20] and references therein. on the other hand, from the α-dense curves we can define the following (see [13, 19]): definition 2.2. the degree of nondensifiability φd : by −→ r+ is defined as φd(b) := inf { α ≥ 0 : γα,b 6= ∅ } , for each b ∈ by , γα,b being the class of α-dense curves in b. note that φd is well defined, as we have pointed out above, γα,b 6= ∅ for every α ≥ diam(b) for each b ∈ by . so, φd(b) ≤ diam(b) for each b ∈ by . for instance, if y is a banach space, we have that φd(uy ) = 0 if y is c© agt, upv, 2019 appl. gen. topol. 20, no. 1 267 g. garćıa finite dimensional (see proposition 2.4 below) and φd(uy ) = 1 if y is infinite dimensional (see [19]), uy being the closed unit ball of y . example 2.3. let l1 be the banach space of absolute value lebesgue integrable functions defined on i, endowed its usual norm, and define the set d := { f ∈ l1 : f ≥ 0 and ∫ 1 0 f(x)dx = 1 } . then, one can check that φd(d) = 2 (see [13]). so, the inequalities 1 = φd(ul1) = φd(ul1 ∪ d) < max{φd(d),φd(ul1)} = 2, hold, where ul1 is the closed unit ball of l 1. by the above example follows that the φd is not a mnc, as the monotony condition (iii) of definition 1.1 is not satisfied. however, the degree of nondensifiability shares some properties with the mncs, as we show in the next result (see [13]). proposition 2.4. in a complete metric space (y,d), the degree of nondensifiability φd satisfies: (i) it is regular on the subclass ba,y of bounded and arc-connected sets of the class by of bounded sets of y : for a given b ∈ ba,y , φd(b) = 0 if, and only if, b is precompact. (ii) it is invariant under closure: φd(b) = φd(b), for any b ∈ by . (iii) it is semi-additive on disjoint sets of ba,y , that is to say φd(b1 ∪ b2) = max{φd(b1),φd(b2)}. whenever b1∩ b2 6= ∅. furthermore, if y is a banach space, then φd also satisfies: (i) φd ( conv(b1) ) ≤ φd(b1), φd ( conv(b1 ∪ b2) ) ≤ max{φd(b1),φd(b2)}, for every b1,b2 ∈ by . (ii) it is semi-homogeneous, that is, φd(λb) = |λ|φd(b), for λ ∈ r and b ∈ by . (iii) it is invariant under translations, that is, φd(y + b) = φd(b), for any y ∈ y and b ∈ by . the degree of nondensifiability φd and the hausdorff mnc χ are related by the following result (see [13]): proposition 2.5. for each arc-connected b ∈ bx we have χ(b) ≤ φd(b) ≤ 2χ(b), and these inequalities are the best possible if x is infinite dimensional. for the kuratowski mnc κ, we have the following result (again, [13]): proposition 2.6. for each arc-connected b ∈ bx we have 1 2 κ(b) ≤ φd(b) ≤ κ(b). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 268 a quantitative version of the arzelà-ascoli theorem another interesting property of the degree of nondensifiability φd is that it is an “universal” upper bound for any mnc. formally (see [13]): proposition 2.7. µ : bx −→ r+ a mnc. then, µ(b) ≤ µ(ux)φd(b), for each b ∈ bx, ux being the closed unit ball of x. next, we recall some well known definitions regarding with the mncs (see, for instance, [1, 3]): definition 2.8. let t : c ⊆ x −→ x continuous and µ a mnc. given k ≥ 0, we say that t is k-µ-contractive if µ ( t(b) ) ≤ kµ(b), for each b ∈ bx ∩ c. if for every non-precompact b ∈ bx ∩ c, µ ( t(b) ) < µ(b) we say that t is µ-condensing. now, we can state the following fixed point result (again, [1, 3]): theorem 2.9 (sadovskĭı fixed point theorem). let c ∈ bx be closed and convex, and t : c −→ c µ-condensing, for certain mnc µ invariant under the convex hull. then, t has some fixed point. before to state a sadovskĭı fixed point theorem for the degree of nondensifiability φd, is convenient to formalize the concepts of definition 2.8 for φd. definition 2.10. let t : c ⊆ x −→ x continuous and k ≥ 0. we will say that t is k-φd-contractive if φd ( t(b) ) ≤ kφd(b), for each convex b ∈ bx ∩ c. if for every convex and non-precompact b ∈ bx ∩ c, φd ( t(b) ) < φd(b), then t is said to be φd-condensing. now, we have the following result (we skip the proof because it follows in the same way than the given in [3], noticing proposition 2.4, or in [10, theorem 3.2]): theorem 2.11. let c ∈ bx be closed and convex, and t : c −→ c φdcondensing. then, t has some fixed point. as we have pointed out above, the degree of nondensifiability φd and the mncs (and, in particular, the hausdorff and kuratowski mncs) are essentially different. then, as expected, definitions 2.8 and 2.10 are essentially different. this fact is evidenced in the following examples. example 2.12. let ℓ2 be the banach space of the real sequences of summable square, endowed with the usual norm ‖ · ‖. fixed 2−1/2 < β < 1, let (ξn)n≥1 be a dense sequence in βuℓ2, uℓ2 being the closed unit ball of ℓ2, and define t : ℓ2 −→ ℓ2 as t(x) := ∑ n≥1 max { 0,1 − ‖x − en‖ 1 − β } ξn, ∀x := (xn)n≥1 ∈ ℓ2, c© agt, upv, 2019 appl. gen. topol. 20, no. 1 269 g. garćıa where {en : n ≥ 1} is the standard basis of ℓ2. clearly, t is continuous and t(uℓ2) ⊂ βuℓ2. then, in [9, remark 3.10] it is proved that (2.1) κ({en : n ≥ 1}) = √ 2 > 2β = κ(t({en : n ≥ 1})) χ(t(b)) ≤ β ≤ χ(b), for every non-precompact b ∈ bℓ2. that is, t is 1-χ-contractive but not 1-κ-contractive. now, let b be a non-empty, convex and non-precompact subset of uℓ2. by proposition 2.4, we can assume without loss of generality that θ ∈ b, θ being the null vector of ℓ2. then, from the definition of t , t(θ) = θ and ‖t(x)‖ ≤ β for every x ∈ uℓ2. consequently, the identically null map γ : i −→ ℓ2 is a β-dense curve in t(uℓ2). therefore, (2.2) φd(t(b)) ≤ β. finally, in view of (2.1), (2.2) and proposition 2.5 we conclude that φd(t(b)) ≤ β ≤ χ(b) ≤ φd(b), that is, t is a 1-φd-contractive map. example 2.13. let the closed and convex set c := {x ∈ c(i,r) : 0 = x(0) ≤ x(t) ≤ 1 = x(1), t ∈ i} and t : c −→ c given by t(x)(t) :=        1 2 x(2t), for 0 ≤ t ≤ 1 2 1 2 x(2t − 1) + 1 2 , for 1 2 < t ≤ 1 for each x ∈ c. then, t is continuous (in fact, affine) and t(c) ⊂ c. one can check (see [3, example 2, p. 169]) that t is 1 2 -κ-contractive and 1-χcontractive, because of χ(t(c)) = χ(c) = 1/2, whereas in [11] we show that t is 1 2 -φd-contractive. to end this section, note that the above examples show that φd and the mncs χ and κ are essentially different. moreover, if we consider the “inner” hausdorff mnc (see, for instnace, [1]) defined as χi(b) := inf { ε > 0 : b can be covered by a finite number of balls centered at points of b and radius ≤ ε } , for each b ∈ bx, in general, χi 6= φd. indeed, for the set b := { (x,sin(1/x)) : x ∈ [−1,0] ∪ (0,1] } ∪ { [0,y] : y ∈ [−1,1]} ⊂ r2, we find χi(b) = 0 < 1 = φd(b). another example that shows that φd and χi are essentially different can be found in [12, example 3.4]. despite its name, χi is not a mnc (see, for instance, [1, p. 9]). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 270 a quantitative version of the arzelà-ascoli theorem 3. the main result before to prove our main result, we need to recall the following definition (see, for instance, [28, p. 253]): definition 3.1. given b ∈ bc(i,x), the uniform modulus of equicontinuity of b is defined as ω(b) := inf δ>0 sup |t−s|<δ sup x∈b ‖x(t) − x(s)‖. is clear that ω(b) is well defined, as ω(b) < ∞ for every b ∈ bc(i,x). also, let us note that b is equicontinuous if, and only if, ω(b) = 0. now, we are ready to prove our main result: theorem 3.2. let c ∈ bc(i,x) be arc-connected. then, (3.1) max { φd(c), 1 2 ω(c) } ≤ φd(c) ≤ φd(c) + 2ω(c). where c(t) := {x(t) : x ∈ c} and φd(c) := sup{φd ( c(t) ) : t ∈ i}, for each t ∈ i. if c(t) is precompact for every t ∈ i, then (3.2) 1 2 ω(c) ≤ φd(c) ≤ ω(c). proof. the inequality (3.3) φd(c) ≤ φd(c). has been proved in [10, lemma 3.2]. let α := φd(c), ε > 0 and γ an (α + ε)dense curve in c. then, given x ∈ c there is y ∈ γ(i) such that (3.4) ‖x − y‖∞ ≤ α + ε. therefore, fixed δ > 0, in view of (3.4), for each t,s ∈ i with |t − s| < δ we have ‖x(t) − x(s)‖ ≤ ‖x(t) − y(t)‖ + ‖y(t) − y(s)‖ + ‖y(s) − x(s)‖ ≤ 2(α + ε) + ‖y(t) − y(s)‖, and, as δ > 0 can be arbitrarily small, we infer ω(c) ≤ 2(α + ε) + ω ( γ(i) ) , but, as ω ( γ(i) ) = 0 (by the arzelà-ascoli theorem, γ(i) is equicontinuous), letting ε → 0, (3.5) ω(c) ≤ 2φd(c). then, from (3.3) and (3.5), we conclude that (3.6) max { φd(c), 1 2 ω(c) } ≤ φd(c). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 271 g. garćıa on the other hand, fixed ε > 0 there is some δ > 0 such that for each t,s ∈ i with |t − s| < δ we have ‖x(t) − x(s)‖ < ω(c) + ε for each x ∈ c. by the compactness of i there is {t1, . . . , tn} ⊂ i such that i ⊂ ∪ni=1b(ti,δ) (the open balls centered at ti and radius δ). so, (3.7) c(t) ⊂ n ⋃ i=1 c(ti) + (ω(c) + ε)ux, ux being the closed unit ball of x. given α > φd(c), let γi be an αdense curve in c(ti). then, as each γi(i) is compact there is a finite set {y(i)1 (ti), . . . ,y (i) ri (ti)} ⊂ γi(i), with y (i) ri ∈ c, for each i = 1, . . . ,n, such that (3.8) c(ti) ⊂ n ⋃ i=1 γi(i) + αux ⊂ n ⋃ i=1 {y(i)1 (ti), . . . ,y(i)ri (ti)} + (α + ε)ux, and joining (3.7) and (3.8), we find (3.9) c(t) ⊂ n ⋃ i=1 {y(i)1 (ti), . . . ,y(i)ri (ti)} + (ω(c) + α + 2ε)ux. for each t ∈ i. next, we claim: (3.10) c ⊂ n ⋃ i=1 {y(i)1 , . . . ,y(i)ri } + (2ω(c) + α + 3ε)uc(i,x), where uc(i,x) is the closed unit ball of c(i,x). in other case, there is x0 ∈ c such that (3.11) ‖x0 − y(i)ri ‖∞ > 2ω(c) + α + 3ε, for each i = 1, . . . ,n. take, for each i = 1, . . . ,n, t̃i ∈ i such that ‖x0(t̃i) − y (i) ri (t̃i)‖ = ‖x0 − y (i) ri ‖∞, and let ti ∈ i be such that t̃i ∈ b(ti,δ). then, in view of (3.9) ‖x0 − y(i)ri ‖∞ = ‖x0(t̃i) − y (i) ri (t̃i)‖ ≤ ‖x0(t̃i) − y (i) ri (ti)‖ +‖y(i)ri (ti) − y (i) ri (t̃i)‖ ≤ ω(c) + α + 2ε + ω(c) + ε = 2ω(c) + α + 3ε, which, is contradictory with (3.11). therefore, (3.10) holds as claimed. now, if γ is a continuous map joining the functions y (i) 1 , . . . ,y (i) ri for i = 1, . . . ,n (this is possible because c is arc-connected), from (3.10), we find that γ is an (2ω(c)+α+3ε)-dense curve in c. letting ε → 0 and α → φd(c), we conclude that (3.12) φd(c) ≤ 2ω(c) + φd(c), and consequently, the inequality (3.1) follows from (3.6) and (3.12). to prove the inequality (3.2) of the statement, if c(t) is precompact for every t ∈ i, then φd(c(t)) = 0 by (i) of proposition 2.4. so, from (3.1), φd(c) ≥ ω(c)/2. by [23, theorem 1], κ(c) ≤ ω(c) and therefore, taking c© agt, upv, 2019 appl. gen. topol. 20, no. 1 272 a quantitative version of the arzelà-ascoli theorem into account proposition 2.6, the inequality (3.2) holds. the proof is now complete. � some comments are necessary before continuing. (a) replacing the banach space x by an arbitrary complete metric space, the above result remains true. (b) the above result has been proved in [23, theorem 1] for the kuratowski mnc κ. however, as we have pointed out in section 2 (and, in particular, in example 2.12), the degree of nondensifiability is essentially different from κ. (c) as theorem 1.2, our result “quantifies” the compactness of the bounded and arc-connected subsets of bc(i,x). indeed, if an arc-connected c ∈ bc(i,x) is precompact if, and only if, is equicontinuous (that is, ω(c) = 0) and for each t ∈ i, c(t) is precompact (that is, φd(c(t)) = 0 by virtue of proposition 2.4). (d) if c is equicontinuous then theorem 1.2 remains true for the degree of nondensifiability φd, as in such case ω(c) = 0. in the following examples we show that the estimates given in the inequality (3.2) of theorem 3.2 are the best possible. example 3.3 (see [23, example 2]). define the sequence of functions xn : i −→ r as xn(t) :=            0, for 0 ≤ t ≤ 1/2 − 1/n (t − 1/2 + 1/n)n/2, for 1/2 − 1/n ≤ t ≤ 1/2 + 1/n 1, for 1/2 + 1/n ≤ t ≤ 1 for every integer n ≥ 2. then for b := {xn : n ≥ 2} we find κ(b) = 1/2 and ω(b) = 1. putting c := conv(b), by proposition 2.6, we have φd(c) ≤ κ(c) = κ(b) = 1/2. then, noticing (3.2) of theorem 3.2, we have φd(c) = 1 2 = 1 2 ω(c). example 3.4. consider the following (closed and convex) set c := { x ∈ c(i,r) : 0 = x(0) ≤ x(t) ≤ x(1) = 1, for each t ∈ i } . then, ω(c) = 1 (see also the above example). we will prove in the following lines that φd(c) = 1. as diam(c) = 1 we have φd(c) ≤ 1. if φd(c) < 1, we can take an α-dense curve in c, put γ, with 0 < α < 1. noticing that γ(i) is a compact subset of c, by the arzelà-ascoli theorem, it is equicontinuous. so, for 0 < 3ε < 1 − α there is δ1 > 0 such that (3.13) |y(t) − y(t′)| ≤ ε for all y ∈ γ(i), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 273 g. garćıa whenever |t − t′| ≤ δ1. taking into account that y(1) = 1 for each y ∈ γ(i), again by the equicontinuity of the set γ(i), there is δ2 > 0 such that (3.14) y(t′) ≥ 1 − ε for all y ∈ γ(i), whenever 1 − t′ ≤ δ2. therefore, taking 0 < t′ < t < 1 satisfying 1 − t′ ≤ min{δ1,δ2} and an integer n ≥ 1 such that f(t) := tn ≤ ε, as γ is an α-dense curve in c we can pick y ∈ γ(i) with (3.15) ‖f − y‖∞ ≤ α. finally, from (3.13), (3.14) and (3.15) we infer α ≥ ‖f − y‖∞ ≥ |f(t) − y(t′)| − |y(t′) − y(t)| ≥ 1 − 2ε − |y(t′) − y(t)| ≥ 1 − 3ε, which is contradictory with the choice of α. so, we conclude that φd(c) = 1. on the other hand, we can derive from theorem 3.2 some bounds for the mncs of arc-connected subsets of bc(i,x). in view of proposition 2.7, our first corollary is the following: corollary 3.5. let µ : bc(i,x) −→ r+ be a mnc, and c ∈ bc(i,x) arcconnected. then, we have µ(c) ≤ µ(uc(i,x)) [ φd(c) + 2ω(c) ] , uc(i,x) being the closed unit ball of c(i,x), and φd(c) defined in theorem 3.2. in particular, if c(t) is precompact for every t ∈ i, then µ(c) ≤ µ(uc(i,x))ω(c). example 3.6. let x be an infinite dimensional banach space, b ∈ bc(i,x) convex and k : i2 −→ r, f : i × x −→ x continuous. assume that there is m > 0 with ‖f(s,x(s))‖ ≤ m for every s ∈ i and x ∈ b. consider the set c := { ∫ t 0 k(t,s)f(s,x(s))ds : x ∈ c } , meaning the above integral in the bochner sense (see, for instance, [27]). from the convexity of b and the continuity of f, it follows that c is arc-connected. also, given 0 ≤ t′ < t ≤ 1 we have ∥ ∥ ∥ ∫ t 0 k(t,s)f(s,x(s))ds − ∫ t′ 0 k(t′,s)f(s,x(s))ds ∥ ∥ ∥ ≤ ∥ ∥ ∥ ∫ t 0 (k(t,s) − k(t′,s))f(s,x(s))ds ∥ ∥ ∥ + ∥ ∥ ∥ ∫ t t′ k(t′,s)f(s,x(s))ds ∥ ∥ ∥ ≤ m ∫ t 0 |k(t,s) − k(t′,s)|ds + mmk(t − t′), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 274 a quantitative version of the arzelà-ascoli theorem and therefore, the set c is equicontinuous, mk denoting the maximum of |k| over i2. then, from corollary 3.5, we find the inequalities χ(c) ≤ φd(c) and κ(c) ≤ 2φd(c), with φd(c) as in theorem 3.2. recalling that two mncs µ and ν are said to be comparable (see, for instance, [3, definition 1.1, p. 168]) if there are a,b > 0 such that aν(b) ≤ µ(c) ≤ bν(b), for every b ∈ bx, the next result follows directly from proposition 2.5 and theorem 3.2: corollary 3.7. assume µ : bc(i,x) −→ r+ is a comparable mnc with the hausdorff mnc χ, that is to say, there are a,b > 0 such that aχ(b) ≤ µ(b) ≤ bχ(b), for every b ∈ bc(i,x). let c ∈ bc(i,x) be arc-connected. then, we have a 2 max { φd(c), 1 2 ω(c) } ≤ µ(c) ≤ b [ φd(c) + 2ω(c) ] , with φd(c) as in theorem 3.2. in particular, if c(t) is precompact for every t ∈ i, then a 4 ω(c) ≤ µ(c) ≤ bω(c). roughly speaking, the above corollaries provide us bounds for µ(c) (of an arc-connected c ∈ bc(i,x)) from the degree of nondensifiability of c(t) or from the uniform modulus of equicontinuity of c, depending if c(t) is, or not, precompact for each t ∈ i. 4. application to volterra integral equations for x := rn one of the most important example of compact operators in c(i,x) (i.e. continuous maps that transform bounded sets into precompact sets) are those defined by integral operators with sufficient regular kernels (see, for intance, [16]). however, as it is shown in [15] by several examples, if x is an infinite dimensional banach space, the situation is very different. it is due, essentially, to the fact that in infinite dimensional banach spaces bounded subsets are not necessarily precompact. so, the mncs are a useful tool to analyze the existence of solutions of differential and integral equations posed in infinite dimensional banach spaces, see, for instance, [4, 21, 22, 24] and references therein. in this section, we will use the results exposed in previous sections to prove the existence of solutions of certain volterra integral equation, which can not be solved by the mncs techniques. specifically, we consider the following volterra integral equation (4.1) x(t) = x0(t) + ∫ t 0 k(t,s)f(s,x(s))ds, c© agt, upv, 2019 appl. gen. topol. 20, no. 1 275 g. garćıa where x0 ∈ c(i,x), k : i2 −→ r and f : i × x −→ x are known, and the integral stands in the bochner sense (see, for instance, [27]). when x is a sequence banach space (such as c0, ℓp, etc.), the above integral equation is equivalent to certain infinite systems of second-order differential equations (see [21] and references therein). let the following conditions: (c1) the functions k : i2 −→ r and f : i × x −→ x are continuous. (c2) there is r > 0 such that ‖x0(t)‖ + ‖ ∫ t 0 k(t,s)f(s,x(s))ds‖ r ≤ 1 for all t ∈ i, whenever ‖x‖∞ ≤ r. (c3) for each b ∈ bc(i,x), there is c : i −→ r+, bounded and lebesgue integrable, and ψ : r+ −→ r+ continuous and increasing with ψ(r) < r for every r > 0 and ψ(0) = 0, such that φd ({ f(s,x(s)) : x ∈ b }) ≤ c(s)ψ ( φd ({ x(s) : x ∈ b })) for all s ∈ i. (c4) the functions c and k obey ∫ t 0 |k(t,s)|c(s)ds ≤ 1, for all t ∈ i. regarding the above conditions, we can do the following observations. • unlike many of the results based in mncs for integral equations of type (4.1), we do not require the uniform continuity of the family of functions (f(s,x(s))s∈i (see the above cited references). • similar conditions to (c1)-(c4) are assumed in [12] to prove the existence of solutions of certain integral equations. in fact, condition (c3) is required in [12] by taking c(s) equal to a positive constant. • from example 2.13 and [12, examples 3.2 and 3.4], not detailed here for lack of space, we can define some maps f(t,x) such that condition (c3) is satisfied by the φd but not by the hausdorff mnc χ. this fact will be also evidenced in example 4.2. • in particular, conditions (c3) and (c4) are assumed to prove that the operator which defined the integral equation (4.1) is φd-condensing and then, we can apply theorem 2.11. let us note that, as the concepts of φd-condensing and µ-condensing are essentially different (see example 2.13), sadovskĭı fixed point theorem and theorem 2.11 too. now, we can state and prove the main result of this section. theorem 4.1. assume conditions (c1)-(c4). then, equation (4.1) has some solution x∗ ∈ c(i,x). proof. clearly, the existence of solutions of (4.1) is equivalent to the existence of fixed points of the map t(x)(t) := x0(t) + ∫ t 0 k(t,s)f(s,x(s))ds for all x ∈ c(i,x), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 276 a quantitative version of the arzelà-ascoli theorem which, from condition (c1), satisfies t(x) ∈ c(i,x) for every x ∈ c(i,x). also, a routine check shows us that t is continuous. we will apply theorem 2.11 to prove the existence of fixed points of t . from condition (c2), there is r > 0 such that t(c) ⊂ c, c := ruc(i,x). also, from condition (c3) and theorem 3.2, for every s ∈ i the degree of nondensifiablity of the set {f(s,x(s)) : x ∈ c} remains bounded by sup{c(t) : t ∈ i}ψ(φd(c)) + 2ω(c) and, therefore, the set {f(s,x(s)) : x ∈ c} must be bounded. so, in view of example 3.6, we find that t(c) is equicontinuous. next, let b ⊂ c be non-empty, non-precompact and convex. by theorem 3.2 and the above considerations, we have (4.2) φd ( t(b) ) ≤ φd ( t(b) ) + 2ω ( t(b) ) = sup{φd ( t(b)(t) : t ∈ i } , meaning t(b)(t) := {t(x)(t) : x ∈ b}. on the other hand, by condition (c3), for each s ∈ i, if αs := φd(b(s)) > 0 then for an arbitrarily small ε > 0, there is an (ψ(αs) + ε)-dense curve in {f(s,x(s)) : x ∈ b}, put τ ∈ i 7−→ γτ(s). so, given x ∈ b there is τ ∈ i such that (4.3) ‖f(s,x(s)) − γτ (s)‖ ≤ c(s)ψ(αs), for each s ∈ i. now, consider the map τ ∈ i 7−→ γ̃τ (t) := x0(t) + ∫ t 0 k(t,s)γτ(s)ds, for all t ∈ i, which is continuous and γ(i) ⊂ t(b). then, noticing (4.3) and condition (c4), for every t ∈ i, we have ‖t(x)(t) − γ̃τ (t)‖ ≤ ∫ t 0 |k(t,s)|‖f(s,x(s)) − γτ(s)‖ds ≤ ∫ t 0 |k(t,s)|c(s)(ψ(αs) + ε)ds ≤ sup{ψ(αs) : s ∈ i} + ε, and letting ε → 0 ‖t(x)(t) − γ̃τ (t)‖ ≤ sup{ψ(αs) : t ∈ i} = sup{ψ(αt) : t ∈ i}, or, in other words, γ̃ is a sup{ψ ( φd(b(t)) ) : t ∈ i}-sense curve in t(b)(t). then, noticing (4.2), the properties of ψ and theorem 3.2 φd ( t(b) ) ≤ sup { ψ ( φd(b(t)) ) : t ∈ i } ≤ ψ ( sup{φd(b(t)) : t ∈ i} ) < sup { φd(b(t)) : t ∈ i } ≤ max { sup{φd(b(t)) : t ∈ i}, 12ω(b) } ≤ φd(b). so, t is a φd-condensing map and the proof is complete. � we conclude our exposition with an example. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 277 g. garćıa example 4.2. consider the following integral equation in the banach space c0 of the null sequences: (4.4) xn(t) = 4n 3n + 1 ∫ t 0 3ts3 arctan ( |xn(s)| ) ds. for all n ≥ 1. in this case, x0 ≡ 0, f(s,x) := ( 4n3n+1 arctan(|xn|))n≥1 and k(t,s) := 3ts 3. so, condition (c1) is trivially satisfied and (c2) taking, for instance, r := π. from the elementary properties of the arctan function, it is immediate to check that, if γ is an α-dense curve in b, then f(s,γ(s)) is an 4 3 arctan(α)-dense curve in the set {f(s,x(s)) : s ∈ b}, for every b ∈ bc(i,c0). so, condition (c3) is fulfilled for c(s) := 4 3 and ψ(r) := arctan(r). also, 4 3 ∫ t 0 3ts3ds = t5 ≤ 1 for all t ∈ i, and therefore condition (c4) holds. then, by theorem 4.1, the above integral equation has some continuous solution. on the other hand, from the following formula (see, for instance, [1, p. 5]) χ(b) = lim n sup x∈b { sup{|xi| : i ≥ n} } , for every b ∈ bc(i,c0), we deduce that χ({en : n ≥ 1}) = 1 < π 3 = χ({f(s,en) : n ≥ 1}), en ∈ c0 being the vector with 1 in the n-th position and zeros otherwise. in other words, condition (c3) is not satisfied for the hausdorff mnc χ and the above functions c(s) and ψ(r). moreover, sadovskĭı fixed point theorem (see theorem 2.11) can not be used in this example for χ as, by the above inequality, t is not χ-condensing. acknowledgements. to the anonymous referee, for his/her useful comments and suggestions to improve the quality of the paper. also, to my beloved loli, for her 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[29] s. willard, general topology, dover pub. inc. 2004. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 279 () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 115-134 computational topology counterexamples with 3d visualization of bézier curves j. li, t. j. peters∗, d. marsh and k. e. jordan abstract for applications in computing, bézier curves are pervasive and are defined by a piecewise linear curve l which is embedded in r 3 and yields a smooth polynomial curve c embedded in r 3 . it is of interest to understand when l and c have the same embeddings. one class of counterexamples is shown for l being unknotted, while c is knotted. another class of counterexamples is created where l is equilateral and simple, while c is self-intersecting. these counterexamples were discovered using curve visualizing software and numerical algorithms that produce general procedures to create more examples. (a) unknotted l with knotted c (b) zoomed-in view of c figure 1. unknotted l with knotted c ∗this author was partially supported by nsf grants ccf 0429477, cmmi 1053077 and cns 0923158, as well as by an ibm faculty award and ibm doctoral fellowships. all statements here are the responsibility of the author, not of the national science foundation nor of ibm. 116 j. li, t. j. peters, d. marsh and k. e. jordan (a) equilateral, simple l with self-intersecting c (b) zoomed-in self-intersection figure 2. equilateral, simple l with self-intersecting c 1. introduction computational topology lies properly within the broad scope of applied general topology and depends upon a novel integration of the ‘pure mathematics’ of topology with the ‘applied mathematics’ of numerical analysis. computational topology also blends general topology, geometric topology and knot theory with computer visualization and graphics, as presented here. for the computational cases considered here, ideas from general topology, geometric topology and knot theory are complemented by numeric arguments in a novel integration of the ‘pure’ field of topology with the ‘applied’ focus of numerical analysis. additionally, aspects of computer visualization and graphics are incorporated into the proofs. much of this work was motivated by modeling biological molecules, such as proteins, as knots for visualization synchronized to simulations of the writhing of these molecules. attention is restricted to when c is closed, implying that l is also closed. as c is created from l, it is natural to ask which topological characteristics are shared by these two curves, particularly as the control polygon often serves as an approximation to the bézier curve in many practical applications. however, topological differences between a bézier curve and its control polygon can exist and it is natural to develop counterexamples to show these topological differences. these counterexamples extend beyond related results, while we introduce new computational methods to generate additional counterexamples. in section 2, we present a counterexample of bézier curve and its control polygon being homeomorphic, yet not ambient isotopic. to develop this counterexample, we created and used a computer visualization tool to study topological relationships between a bézier curve and its control polygon. we viewed the images to motivate formal proofs, which also rely on numerical analysis and geometric arguments. in section 3, we present numerical techniques to create a class of topological counterexamples – where a bézier curve and its control polygon are not even homeomorphic, as the bézier curve is self-intersecting while the control polygon is simple. we exhibit the self-intersection by a numerical method, which finds computational topology counterexamples 117 the roots of a system of equations. we freely admit that these roots are not determined with infinite precision, but such calculations on polynomials of degree 6, as in these examples, typically elude precise calculation. we argue that two primary values of our method for these approximated solutions are (1) as a catalyst to alternative examples that may admit infinite precision calculation and rigorous topological proofs, and (2) as having specified digits of accuracy – typically crucial for acceptable approximations in computational mathematics and computer graphics. using the visualization tool described in section 4, we viewed many examples where the bézier curve was simple, while its control polygon was equilateral and simple. it is well known that a bézier curve can be self-intersecting even when its control polygon is simple, but we conjectured that the added equilateral hypothesis would imply that both curves were simple. while this visual evidence was suggestive, we present a general numerical approach in section 3 that supports a contrary interpretation. this example provides guidance for designing appropriate approximation algorithms for computer graphics. 1.1. mathematical definitions. the definitions presented are restricted to r 3, as sufficient for the purposes of this paper, but the interested reader can find appropriate generalizations in published literature. definition 1.1. a knot [4] is a curve in r3 which is homeomorphic to a circle. a knot is often described by a knot diagram [12], which is a planar projection of a knot. self-intersections in the knot diagram correspond to crossings in the knot, where each crossing has one arc that is an undercrossing and an overcrossing, relative to the direction of projection. homeomorphism is an equivalence relation over point sets and does not distinguish between different embeddings, while ambient isotopy is a stronger equivalence relation which is fundamental for classification of knots. definition 1.2. let x and y be two subspaces of r3. a continuous function h : r3 × [0, 1] → r3 is an ambient isotopy between x and y if h satisfies the following: (1) h(·, 0) is the identity, (2) h(x, 1) = y , and (3) ∀t ∈ [0, 1], h(·, t) is a homeomorphism from r3 onto r3. the sets x and y are then said to be ambient isotopic. definition 1.3. denote c(t) as the parameterized bézier curve of degree n with control points pm ∈ r 3, defined by c(t) = n ∑ i=0 bi,n(t)pi, t ∈ [0, 1] where bi,n(t) = ( n i ) ti(1 − t)n−i. 118 j. li, t. j. peters, d. marsh and k. e. jordan 1.2. related literature. a bézier curve and its control polygon may have substantial topological differences. it is well known that a bézier curve and its control polygon are not necessarily homeomorphic [14]. recently, it was shown that there exists an unknotted bézier curve with a knotted control polygon [6]. a specific dual example has also been shown [16] of a knotted bézier curve with an unknotted control polygon. however, the methodology was a visual construction without formal proof and is not easily generalized. software, knot spline vis, developed by authors marsh and peters, was used to find another example, where the methodology can more easily be generalized and knot spline vis is publicly available [13]. much of the motivation for considering these counterexamples came from applications in scientific visualization [9, 10, 11]. a primary focus was to establish tubular neighborhoods of knotted curves so that piecewise linear (pl) approximations of those curves within those neighborhoods remained ambient isotopic to the original curves. this was initially considered for approximations used in producing static images of these curves, but it became readily apparent that these same neighborhoods also provided bounds within which many perturbations of these models remained ambient isotopic under these movements. that theory is being applied to dynamic visualizations of molecular simulations, where the neighborhood boundaries permit convenient warnings as to an impending self-intersection, as of possible interest to biologists and chemists who are running the simulations. previous work [15] in knot visualization provides a rich set of data for pl knots, where each edge is of the same length. the interface of knot spline vis was designed to import this equilateral pl knot data and then generate the associated bézier curves. this matured into an empirical study of dozens of cases, where all examples examined yielded simple bézier curves for these simple equilateral control polygons. this raised the question of whether the presence of equilateral edges in the control polygon might be a sufficient additional hypothesis to ensure homeomorphic equivalence with the bézier curve, as the previously cited examples [14] did not have equilateral control polygons. 2. unknotted l with knotted c in order to produce a knotted bézier curve with an unknotted control polygon, we invoked knot spline vis with an example (figure 3) of an unknotted bézier curve, where the total curvature appeared to be larger than 4π. (the total curvature being larger than 4π is a necessary condition of knottedness.) we experimented on this example by moving control points to construct a bézier curve that visually appeared to be knotted. we then moved control points to unknot the control polygon while keeping the bézier curve knotted. in the end we obtained a bézier curve and the control polygon (of degree 10) (figure 1(a)) defined by the control points {p0, p1, · · · , p9, p0} listed below: (−5.9, 4.7, −6.2), (10.3, −1.1, 8.9), (−2.6, −12.4, −6.3), (−10, 7, −0.3), (1.9, −12, −0.6), (11.2, 7.5, −7.6), (−15.3, −1.7, −4.1), (−11.7, 20, 3.5), (17.9, −1.1, 2.9), (2.9, −13.7, 4.8), (−5.9, 4.7, −6.2). computational topology counterexamples 119 (a) unknotted c & the p (b) zoomed-in view of c figure 3. visual experiments the 3d visualization offers only suggestive evidence that the above bézier curve is knotted while the control polygon is unknotted. we provide rigorous mathematical proofs of these properties in sections 2.1 and 2.2. generally, proving knottedness or unknottedness can be difficult [8], but is accessible for the counterexample here. 2.1. proofs of the bézier curve being knotted. we prove that the bézier curve is a trefoil1. we orthogonally project the curve onto x-y plane. we then shown that there are three self-intersections in the projection and these intersections are alternating crossings in 3d. since projections preserve selfintersections, this curve can have no more than 3 self-intersections, but these self-intersections in x-y plane are shown to have pre-images that are 3d crossings, so the original curve has no self-intersections. since the 2d curve in figure 4 has degree 10, we use a numerical method implemented by matlab function ‘fminsearch’ to find the parameters where the curve intersects with itself. we provide the numerical codes in appendix a.1, and we provide the data used to find these parameters in appendix a.3. the pairs of parameters (labeled in order as t1, . . . , t6) of the self-intersections are listed below: [t1 = 0.0306, t4 = 0.5573], [t2 = 0.1573, t5 = 0.9244], [t3 = 0.3731, t6 = 0.9493]. next we prove that these 2d intersections are projections from three alternating crossings in 3d. the above parameters are substituted into the bézier curve (definition 1.3) to get pairs of points (numerical codes for this calculation 1a trefoil is a knot with three alternating crossings[12]. 120 j. li, t. j. peters, d. marsh and k. e. jordan figure 4. the 2d projection of the knot are in appendix a.2): [c(t1) = (−2.0539, 2.8001, −2.6929), c(t4) = (−2.0530, 2.7987, −2.0143)], [c(t2) = (0.4376, −2.5212, −0.0576), c(t5) = (−0.4364, −2.5206, −0.5547)], [c(t3) = (−1.3613, −1.4239, −2.2944), c(t6) = (−1.3624, −1.4232, −1.9067)]. the alternating crossings follow from comparing the z-coordinates in each pair. precisely, according to the parameters given above, the tracing of the six points in order is c(t1), c(t2), c(t3), c(t4), c(t5), c(t6), and the crossings at these points are under, over, under, over, under, over. 2.2. proof of the control polygon being unknotted. to prove that the control polygon p = (p0, p1, · · · , p9, p0) is an unknot, it is necessary to show that p is simple and unknotted. we directly tested each pair of segments of p for non-self-intersection and those calculations can be repeated by the interested reader. we prove unknottedness using a 3d push, as the obvious generalization of a 2d function from low-dimensional geometric topology [5]. we restrict attention to a median push, as defined below. the full sequence of 5 median pushes is explicated, where the first 4 median pushes are equivalently described by reidemeister moves [12]. computational topology counterexamples 121 (a) the initial control polygon (b) after the 1st push (c) after the 2nd push (d) after the 3rd push (e) after the 4th push (f) the unknot after these pushes figure 5. pushes and pl curves in 3d definition 2.1. suppose △pj−1pjpj+1 is a triangle determined by non-collinear vertices pj−1, pj and pj+1 of p . push the vertex pj, along the corresponding median of the triangle to the middle point of the side pj−1pj+1. we call this function a median push. we depict the sequence of pushes used in figure 5, showing the 3d graphs of the pl space curves after the median pushes. these graphs show, at each step, which vertex is pushed and its image. for example, in figure 5(a), the 122 j. li, t. j. peters, d. marsh and k. e. jordan vertex p3 is pushed to p ′ 3 and the resultant polygon after this push is shown by figure 5(b). figures 5(a) 5(b) 5(c) and 5(d) have corresponding reidemeister moves, as those pushes eliminate at least one crossing. using the published notation [12], figure 5(a) depicts a reidemeister move of type 2b. similarly, figure 5(b) depicts a reidemeister move of type 1b; figure 5(c) has type 2b and figure 5(d) has a move of type 2b, followed by a move of type 1b. the final push to achieve figure 5(f) does not correspond to any reidemeister move as no crossings are changed, but it is included to have a polygon with only five edges, which necessarily must be the unknot [1]. we prove that p is unknotted by showing that p is ambient isotopic to the unknotted pl curve shown in figure 5(f). as a sufficient condition, we show that the pushes do not cause intersections [2]. (we gained significant intuition for specifying the sequence of pushes by visual verification with our 3d graphics software capabilities to translate, rotate and zoom the images.) we now present the formal arguments. consider figure 5(a), where p3 is pushed to p ′ 3 (the middle point 2 of −−−→ p2p4). we show that any segments other than −−−→ p2p3 and −−−→ p3p4 of p do not intersect the triangle △p2p3p4 or the triangle interior. we parameterize each segment by: −−−−→ pipi+1 : pi + (pi+1 − pi)t, t ∈ [0, 1] for i = 0, 1, · · · , 9 and let p10 = p0. then the points given by p3 + a(p2 − p3) + b(p4 − p3), for a, b ≥ 0 and a + b ≤ 1 are on the △p2p3p4 and contained in its interior. hence pi + (pi+1 − pi)t intersects △p2p3p4 or its interior if and only if pi + (pi+1 − pi)t = p3 + a(p2 − p3) + b(p4 − p3)(2.1) has a solution for some t ∈ [0, 1] and a, b ≥ 0 and a + b ≤ 1. for each i = 0, 1, · · · , 9, we solve equation 2.1 (a system of 3 linear equations) for a, b and t with the above constraints. calculations show there is no solution for each system, and hence pi + (pi+1 − pi)t does not intersect △p2p3p4 or its interior for each i = 0, 1, · · · , 9. thus it follows that the push does not cause any intersections. similar computations verify that the other pushes do not cause any intersections, thus establishing the ambient isotopy. 3. equilateral, simple l with self-intersecting c we visualized many cases of simple, closed equilateral control polygons3 that all appeared to have unknotted bézier curves. many of these control polygons were nontrivial knots. prompted by this visual evidence, we conjectured that any simple, closed equilateral control produces a simple bézier curve, where some examples are shown in figure 6. 2it is not necessary to push p3 to the middle point. any point along −−−→ p2p4 would suffice. 3throughout this section, all control polygons are simple. computational topology counterexamples 123 we now present numerical evidence to the contrary. as noted in section 1, the degree 6 polynomials make precise computation difficult, so we do not provide a completely formal proof, independent of numerical methods. a legitimate concern is whether the numerical approximation produces coordinates for a ‘near’ intersection, subject to the accuracy of the floating point computation. we cannot refute that possibility, but we argue that in the context of graphical images, the level of approximation produced is often sufficient. in particular, we have parameters in the code to adjust the number of digits of accuracy. this level of user-defined precision is often accepted as sufficient for visualization [9]. the user can then set the graphical resolution so that points that are determined to be the same within some acceptable numerical tolerance will also appear within the same pixel. figure 6. unknotted bézier curves with equilateral control polygons 3.1. intuitive overview. we create many examples to test and retain only those that satisfy all three criteria listed in section 3.4. we begin the creation of a closed equilateral polygon by setting p0 = (0, 0, 0). we then take {q0, q1, · · · , qn−1} (equation 3.3) from the unit sphere so that p1 = p0 + q0, p2 = p1 + q1, · · · , pn = pn−1 + qn−1. we ensure that the polygon is closed in section 3.4 item (3). we consider a necessary and sufficient condition for a bézier curve being selfintersecting (equation 3.2). since we want the equilateral polygon to define a bézier curve that is self-intersecting, we not only select {q0, q1, · · · , qn−1} from the unit sphere as above, but also select them such that equation 3.2 is zero for some parameters s and t. consequently the set of control points generated 124 j. li, t. j. peters, d. marsh and k. e. jordan determines a closed equilateral control polygon and a self-intersecting bézier curve. 3.2. necessary and sufficient condition for self-intersection. we rely upon the following published equation [3] for necessary and sufficient conditions for self-intersection of a bézier curve (3.1) s(s, t) = 1 n c(1 − s) − c(t) (1 − s) − t , with the domain d = {(s, t) : s + t < 1, s, t ≥ 0, (s, t) 6= (0, 0)}. a bézier curve defined by c(t) is self-intersecting if and only if there exist s and t in the domain d such that s(s, t) = 0, , with an alternative formulation4 given by s(s, t) = 1 n n−1 ∑ i=0 n−1−i ∑ j=0 ( n − 1 − i j ) sn−1−i−j(1 − s)j i ∑ k=0 ( i k ) (1 − t)i−ktkqj+k, (3.2) where qi = pi+1 − pi for i = {0, 1, · · · , n − 1}.(3.3) 3.3. a representative counterexample generated. we present a single numerical counterexample, noting that while only one counterexample is presented, the numerical algorithm implemented can be used to find many such examples. we list six distinct control points (the seventh control point is equal to the first control point) that determine an equilateral simple l and a selfintersecting c (shown in firgure 2), as generated by the algorithm described in detail in section 3.4. (0, 0, 0), (0.0305, 0.0810, 0.9962), (−0.2074, −0.2671, 1.9030), (−0.1792, −0.3402, 0.9063), (0.0189, 0.0782, 0.0185), (0.1557, 0.2329, −0.9600). we verify that l is simple by considering all pairwise intersections of the segments of this control polygon. the self-intersection of the bézier curve occurs at a point that is numerically approximated as [s, t] = [0.2969, 0.0633] where correspondingly, s(s, t) = (−0.0003861, −0.000097, 0.0001462) ≈ (0, 0, 0). the error occurs because of numerical round off on s, t and the control points. 4one needs to write out the [3, equation (6)] to obtain equation 3.2. computational topology counterexamples 125 3.4. the numerical method for generating counterexamples. we provide the numerical codes and data used in appendix b. given a control polygon, we can determine whether a self-intersection of the bézier curve occurs by determining whether s (equation 3.2) has a root in d. we consider s as a function s(s, t, q) where q = {qi} n−1 i=0 , so that finding a self-intersecting bézier curve with an equilateral control polygon is equivalent to determining s, t and q such that the following are satisfied: (1) s(s, t, q) = 0 where s, t ∈ d; (2) ||qi|| = r for each i ∈ {0, 1, · · · , n − 1}, where || · || is the euclidean norm; (3) ∑n−1 i=0 qi = ∑n−1 i=0 pi+1 − pi = 0 since p needs to be closed. we assume r = 1 without loss of generality since the value of r can be adjusted by scaling. throughout the provided codes, n is always the degree of the bézier curve. we give the code for function s in appendix b.1, where [s, t] is labeled as u and q as [a, b, c]. the function sf (appendix b.2) takes parameters for s, t and q and outputs a floating point value. it is designed to be zero if and only if the above three conditions are satisfied simultaneously. precisely since q should be taken from the unit sphere, sf assigns a, b, c values given by a = sin(φ)cos(θ); b = sin(φ)sin(θ); c = cos(φ), where φ and θ represent input parameters. in order to satisfy condition (3) above, qn−1 is set equal to − ∑n−2 i=0 qi. but in this way, ||qn−1|| may fail to be equal to 1. so we include the function f (appendix b.2) to determine whether ||qn−1|| = 1. the function is designed to be f = ||qn−1|| − 1 such that ||qn−1|| = 1 if and only if f = 0. symbolically, sf = ||s|| + ||f ||,(3.4) where s is given by equation 3.2 and f = ||qn−1|| − 1. having the above three conditions satisfied simultaneously is equivalent to finding input values such that sf = 0. since sf ≥ 0, the minimum of sf is 0. the function sfminimizer (appendix b.3) uses fminsearch (a numerical method integrated in matlab) to search for the minimum of sf , while returning this minimum and the corresponding values of s, t and q. the user supplied initial guesses for s, t and q greatly influence the results, so we assign sfminimizer randomized initial values m times so that we get m different minimums for different initial values. but no matter which initial values we use, as long as we can get “a” minimum of 0, then we get the equilateral control polygon which determines a self-intersecting bézier curve. 126 j. li, t. j. peters, d. marsh and k. e. jordan the data of finding the counterexample of section 3.3 is included in appendix b.4. 4. visualizing bézier curves & their control polygons to study the knot types of a control polygon and its bézier curve, a knot visualization tool was developed, called knot spline vis. the tool knot spline vis takes a pl control polygon as input and displays both the pl curve and the associated bézier curve. for these studies, the input was always restricted to be a pl curve of known knot type. the functions of knot spline vis were designed to permit interactive studies of the topological relationships between a bézier curve and its control polygon. the intent is to use these examples to stimulate mathematical conjectures as a prelude to formal proofs. some of the standard graphical manipulation capabilities provided are illustrated in the following figure 7 and 8, where the rotation capabilities have been particularly useful to develop visual insights into the occurrence of selfintersections and crossings as fundamental for the study of knots. an editing window allows the user to change the coordinates of the control polygons, as shown in figure 9. the software is freely available for download at the site www.cse.uconn.edu/∼tpeters. figure 7. rotating the control polygon is an initial pl approximation to its associated bézier curve. subdivision algorithms [14] produce additional control points to further refine the original control polygon, converging in distance to the bézier curve. figure 10 illustrates performance of a standard subdivision technique, the de casteljau algorithm [7]. users can specify a subdivision parameter. figure 10 shows the case with parameter 1 2 . computational topology counterexamples 127 figure 8. scaling (a) initial display (b) some vertices moved figure 9. moving control points figure 10. subdivision 128 j. li, t. j. peters, d. marsh and k. e. jordan 5. conclusions and future work we use numerical algorithms and graphical software to generate counterexamples regarding the embeddings in r3 for a bézier curve and its control polygon. first, we present a class of counterexamples showing an unknotted control polygon for a knotted bézier curve and provide complete formal proofs of this condition. we used a spline visualization tool, knot spline vis for easy generation of many examples to gain intuitive understanding to formalize these proofs. secondly, we used knot spline vis in formulating the conjecture that any simple, closed equilateral control had a simple bézier curve. we provide contrary numerical evidence regarding this conjecture, as a useful guide for many applications in computer graphics and scientific visualization. this work also shows the importance of continuing investigation • theoretically, into an infinite precision proof for a counterexample of the simple, closed equilateral polygon conjecture, where these demands will likely require further, substantive mathematical innovation and • practically, into a user interface for knot spline vis to permit interactive 3d editing, as the current text driven interface is cumbersome. 6. web posting of supplemental materials appendices listed below are posted on this webpage: http://www.math.uconn.edu/~jili/supplemental-materials.pdf appendix a: numerical codes for knottedness of c (figure 1(a)): • a.1 codes for searching self-intersections in figure 4; • a.2 codes for determining under or over crossings; • a.3 data for searching intersections in figure 4. appendix b: numerical codes for searching the example of figure 2(a): • b.1 codes for equation 3.2; • b.2 codes for equation 3.4; • b.3 codes for searching the roots of the system of equations 3.2 and 3.4; • b.4 data for finding the example shown in figure 2(a). http://www.math.uconn.edu/~ jili/supplemental-materials.pdf computational topology counterexamples 129 a. code: knottedness of bézier curve of figure 1(a) a.1. code for self-intersections of figure 4. %the function c2d(t) defines the projection of the bézier curve. function [value] = c2d(t) n=10; p=zeros(2,11); p(1,:)= [-5.9, 10.3, -2.6, -10, 1.9, 11.2, -15.3, -11.7, 17.9, 2.9, -5.9]; p(2,:)= [4.7, -1.1, -12.4, 7, -12, 7.5, -1.7, 20, -1.1, -13.7, 4.7]; sum=0; for i=0:n sum = sum + nchoosek(n, i) ∗ ti ∗ (1 − t)(n − i) ∗ p(1, i + 1); end v1 = sum; sum=0; for i=0:n sum = sum + nchoosek(n, i) ∗ ti ∗ (1 − t)(n − i) ∗ p(2, i + 1); end v2 = sum; value = [v1, v2]; % use the function ‘fminsearch’ to find the minimums of fns(x), the zero minimums and parameters where the zeros occur are what we look for. function [value] = fns(x) value = norm(c2d(x(1)) − c2d(x(2)), 2); function [value] = smin(s0,t0) max=optimset(‘maxfunevals’,1e+19); comb=@(x)fns(x); u=[s0,t0]; [xval, fval] = fminsearch(comb, u, max) a.2. code for determining crossings. %below is the function of the bézier curve in 3d (definition 1.3). function [value] = c(t) n=10; p=zeros(3,11); p(1,:)= [-5.9, 10.3, -2.6, -10, 1.9, 11.2, -15.3, -11.7, 17.9, 2.9, -5.9]; p(2,:)= [4.7, -1.1, -12.4, 7, -12, 7.5, -1.7, 20, -1.1, -13.7, 4.7]; p(3,:)= [-6.2, 8.9, -6.3, -0.3, -0.6, -7.6, -4.1, 3.5, 2.9, 4.8, -6.2]; sum=0; 130 j. li, t. j. peters, d. marsh and k. e. jordan for i=0:n sum = sum + nchoosek(n, i) ∗ ti ∗ (1 − t)(n − i) ∗ p(1, i + 1); end v1 = sum; sum=0; for i=0:n sum = sum + nchoosek(n, i) ∗ ti ∗ (1 − t)(n − i) ∗ p(2, i + 1); end v2 = sum; sum=0; for i=0:n sum = sum + nchoosek(n, i) ∗ ti ∗ (1 − t)(n − i) ∗ p(3, i + 1); end v3 = sum; value = [v1, v2, v3]; a.3. data for searching intersections in figure 4. % the matlab commands and corresponding results: % the initial input (0.03, 0.55) is figured out by the observation and calculations on self-intersections in figure 4. similarly for the others below. smin(0.03,0.55) xval = 0.0306 0.5573 fval = 2.9567e-04 smin(0.15,0.92) xval = 0.1573 0.9244 fval = 1.5848e-04 smin(0.37,0.95) xval = 0.3731 0.9493 fval = 1.4637e-04 b. code: generating counterexamples like figure 2(a) b.1. code for equation 3.2. function [value] = s(u,a,b,c) sum=0; subsum1=0; subsum2=0; for i=0:n-1 for j=0:n-1-i % the use of adding 1 to the vectors a, b and c is required by matlab. for k=0:i subsum2 = subsum2 + nchoosek(i, k) ∗ (1 − u(2))(i−k) ∗ u(2)k ∗ a(j + k + 1); end subsum1 = subsum1 + subsum2 ∗ nchoosek(n − 1 − i, j) ∗ u(1)(n−1−i−j) ∗ (1 − computational topology counterexamples 131 u(1))j; end sum = sum + subsum1; end v1 = (1/n) ∗ sum; sum=0; subsum1=0; subsum2=0; for i=0:n-1 for j=0:n-1-i for k=0: i subsum2 = subsum2 + nchoosek(i, k) ∗ (1 − u(2))(i−k) ∗ u(2)k ∗ b(j + k + 1); end subsum1 = subsum1 + subsum2 ∗ nchoosek(n − 1 − i, j) ∗ u(1)(n−1−i−j) ∗ (1 − u(1))j; end sum = sum + subsum1; end v2 = (1/n) ∗ sum; sum=0; subsum1=0; subsum2=0; for i=0:n-1 for j=0:n-1-i for k=0:i subsum2 = subsum2 + nchoosek(i, k) ∗ (1 − u(2))(i−k) ∗ u(2)k ∗ c(j + k + 1); end subsum1 = subsum1 + subsum2 ∗ nchoosek(n − 1 − i, j) ∗ u(1)(n−1−i−j) ∗ (1 − u(1))j; end sum = sum + subsum1; end v3 = (1/n) ∗ sum; value = [v1, v2, v3]; b.2. code for equation 3.4. function [value] = sf(x,n) q=zeros(3,n); p=zeros(3,n+1); a=zeros(1,n);b=zeros(1,n);c=zeros(1,n); alpha = zeros(1, 2 ∗ n − 2); for i = 1 : 2 ∗ n − 2 alpha(i)=x(i); end for i=1:n-1 a(i) = sin(alpha(i)) ∗ cos(alpha(n − 1 + i)); b(i) = sin(alpha(i)) ∗ sin(alpha(n − 1 + i)); c(i) = cos(alpha(i)); q(:,i)=[a(i),b(i),c(i)]; 132 j. li, t. j. peters, d. marsh and k. e. jordan end a(n)=0; b(n)=0; c(n)=0; for i=1:n-1 a(n)=a(n)-a(i); b(n)=b(n)-b(i); c(n)=c(n)-c(i); q(:,n)=[a(n),b(n),c(n)]; end u=zeros(1,2); u = [x(2 ∗ n − 1), x(2 ∗ n)]; value=abs(f(alpha,n))+norm(s(u,a,b,c),2); for i=1:n for j=1:i; p(:,i+1)=p(:,i+1)+q(:,j); end end p function [value]=f(alpha,n) for i=1:n-1 a(i) = sin(alpha(i))∗cos(alpha(n−1+i)); b(i) = sin(alpha(i))∗sin(alpha(n− 1 + i)); c(i) = cos(alpha(i)); end a(n)=0; b(n)=0; c(n)=0; for i=1:n-1 a(n)=a(n)-a(i); b(n)=b(n)-b(i); c(n)=c(n)-c(i); end value = abs(a(n)2 + b(n)2 + c(n)2 − 1); b.3. codes for solving the system of equations 3.2 and 3.4. function [value] = sfminimizer(n,m) max=optimset(‘maxfunevals’ ,1e+9); comb=@(x)sf(x,n); xval=zeros(2*n,m); fval=zeros(1,m); for i=1:m phi=unifrnd(0,pi,1,n-1); theta=unifrnd(0,2*pi,1,n-1); s0=unifrnd(0,1); t0=unifrnd(0,1-s0); x0=[phi,theta,s0,t0]; [xval(:, i), fval(i)] = fminsearch(comb, x0, max); %make sure s + t = xval(2 ∗ n − 1, i) + xval(2 ∗ n, i) < 1 so that s, t ∈ d. if xval(2 ∗ n − 1, i) + xval(2 ∗ n, i) > 1 or xval(2 ∗ n − 1, i) + xval(2 ∗ n, i) = 1 computational topology counterexamples 133 fval(i)=1; else end end %display the minimums and corresponding s, t and q. xval fval %returns the optimal one. [c, i] = min(fval); value=[c,i]; b.4. data for finding the example shown in figure 2(a). the counterexample (section 3.3) uses the second column below for the input parameters. sfminimizer(6,10) xval = columns 1 through 10 2.1907 0.0867 1.0279 2.4529 0.8294 1.3029 1.0958 0.8502 1.2136 0.3308 2.1572 0.4353 1.9303 0.3841 -0.2788 0.3006 -0.2769 2.8797 3.0782 3.1699 0.0079 3.2225 3.2107 1.8739 3.1482 3.6489 1.3900 0.5959 0.1228 1.8107 0.5496 2.6633 0.2000 1.1955 0.5784 2.0316 1.6273 2.0583 3.1341 1.3276 1.8511 2.9336 1.0424 2.6615 2.3439 1.2207 2.4580 4.3850 0.1877 -0.1146 5.2200 1.2107 7.0598 3.4826 0.3696 0.5633 5.1432 4.2546 3.5523 5.5103 2.7090 4.1128 4.1941 2.3445 2.4291 2.0589 0.4295 2.9467 1.0576 5.9281 0.1954 2.0119 2.2644 7.0575 2.1168 2.3027 1.9744 -0.2264 2.5206 2.4902 0.5312 7.4240 2.2923 5.1205 1.2132 4.0211 3.4493 1.6729 1.7930 5.7145 1.3429 0.8465 1.4447 4.4661 4.0120 1.8860 0.0947 3.9222 5.1409 1.2114 0.5200 0.2969 0.7942 0.7334 0.2149 0.6042 1.3269 0.9360 0.1185 0.0320 0.0054 0.0633 0.2813 0.2543 0.6229 0.4588 0.0165 0.1667 0.5815 0.5970 fval = columns 1 through 7 0.0000 0.0000 1.0000 0.2180 0.0001 1.0000 1.0000 columns 8 through 10 1.0000 0.0001 0.0000 % the corresponding function values of s and f are given: svalue = 1.0e − 03∗ -0.3861 -0.0970 0.1462 fvalue =2.2329e − 05 134 j. li, t. j. peters, d. marsh and k. e. jordan references [1] c .c. adams, the knot book: an elementary introduction to the mathematical theory of knots, american mathematical society, 2004. [2] j. w. alexander and g. b. briggs, on types of knotted curves annals of mathematics 28 (1926-1927), 562–586. [3] l. e. andersson, t. j. peters and n. f. stewart, selfintersection of composite curves and surfaces, cagd 15 (1998), 507–527. [4] m. a. armstrong, basic topology, springer, new york, 1983. [5] r. h. bing, the geometric topology of 3-manifolds, american mathematical society, providence, ri, 1983. [6] j. bisceglio, t. j. peters, j. a. roulier and c. h. sequin, unknots with highly knotted control polygons, cagd 28, no. 3 (2011), 212–214. [7] g. farin, curves and surfaces for computer aided geometric design, academic press, san diego, ca, 1990. [8] j. hass, j. c. lagarias and n. pippenger, the computational complexity of knot and link problems, journal of the acm, 46, no, 2 (1999), 185–221. [9] k. e. jordan, r. m. kirby, c. silva and t. j. peters, through a new looking glass*: mathematically precise visualization, siam news 43, no. 5 (2010), 1–3. [10] k. e. jordan, l. e. miller, e. l. f. moore, t. j. peters and a. c. russell, modeling time and topology for animation and visualization with examples on parametric geometry, theoretical computer science 405 (2008), 41–49. [11] k. e. jordan, l. e. miller, t. j. peters and a. c. russell, geometric topology and visualizing 1-manifolds, in v. pascucci, x. tricoche, h. hagen, and j. tierny, editors, topology-based methods in visualization, pages 1 – 12, new york, 2011. springer. [12] c. livingston, knot theory, volume 24 of carus mathematical monographs, mathematical association of america, washington, dc, 1993. [13] t. j. peters and d. marsh, personal home page of t. j. peters. http://www.cse.uconn.edu/. [14] l. piegl and w. tiller, the nurbs book, springer, new york, 2nd edition, 1997. [15] r. scharein, the knotplot site, http://www.knotplot.com/. [16] c. h. sequin, spline knots and their control polygons with differing knottedness, http://www.eecs.berkeley.edu/pubs/techrpts/2009/eecs-2009-152.html. (received november 2011 – accepted may 2012) j. li (ji.li@uconn.edu) department of mathematics, university of connecticut, storrs, ct, usa. t. j. peters (tpeters@cse.uconn.edu) department of computer science and engineering, university of connecticut, storrs, ct, usa. d. marsh (david.the.marsh@gmail.com) pratt and whitney, east hartford, ct, usa. k. e. jordan (kjordan@us.ibm.com) ibm t.j. watson research, cambridge research center, cambridge, ma, usa. http://www.cse.uconn.edu/ http://www.knotplot.com/ http://www.eecs.berkeley.edu/pubs/techrpts/2009/eecs-2009-152.html computational topology counterexamples with [5pt]3d visualization of bézier curves. by j. li, t. j. peters, d. marsh and k. e. jordan @ appl. gen. topol. 23, no. 2 (2022), 255-268 doi:10.4995/agt.2022.15745 © agt, upv, 2022 partial actions of groups on hyperspaces luis mart́ınez , héctor pinedo and edwar ramirez escuela de matemáticas, universidad industrial de santander, colombia (luchomartinez9816@hotmail.com, hpinedot@uis.edu.co, edwar5119@gmail.com) communicated by f. lin abstract let x be a compact hausdorff space. in this work we translate partial actions of x to partial actions on some hyperspaces determined by x, this gives an endofunctor 2− in the category of partial actions on compact hausdorff spaces which generates a monad in this category. moreover, structural relations between partial actions θ on x and partial actions determined by 2θ as well as their corresponding globalizations are established. 2020 msc: 54h15; 54b20; 54f16. keywords: partial action; globalization; hyperspace; monad. 1. introduction given an action µ : g×y → y of a group g on a set y and an invariant subset x of y (i.e., µ(g,x) ∈ x, for all x ∈ x, and g ∈ g), the restriction of µ to g×x determines an action of g on x. however, if x is not invariant, we obtain a partial action on x. this is a collection of partially defined maps θg (g ∈ g) on x satisfying θ1 = idx and θgh is an extension of the composition θg◦θh, for all g,h ∈ g. the notion of partial action of a group was introduced by r. exel in [6, 7] motivated by problems arising from c∗-algebras. since then partial group actions have appeared in many different contexts, such as the theory of operator algebras, algebra, the theory of r-trees, tilings and model theory (see for instance [10]). in topology, partial actions on topological spaces consist of a family of homeomorphism between open subsets of the space, and have been considered in the context of polish spaces (see [13, 14]), 2-cell complexes (see received 07 june 2021 – accepted 14 december 2021 http://dx.doi.org/10.4995/agt.2022.15745 https://orcid.org/0000-0002-3957-3119 https://orcid.org/0000-0003-4432-419x https://orcid.org/0000-0003-4919-9439 j. l. mart́ınez , h. pinedo and e. ramirez [16]), topological semigroups [3] and recently in [11] where introduced in the realm of profinite spaces. it seems that when a partial action on some structure is given, one of the most relevant problems is the question of the existence and uniqueness of a globalization, that is, if a partial action can be realized as restrictions of a corresponding collection of total maps on some superspace. in the topological context, this problem was studied by abadie [1] and independently by kellendonk and lawson [10]. it was proved that for any continuous partial action θ of a topological group g on a topological space x, there is a topological space y and a continuous action µ of g on y such that x is a subspace of y and θ is the restriction of µ to x. such a space y is called a globalization of x. they also show that there is a minimal globalization xg called the enveloping space of x (see subsection 2.2 for details). recent topological advances on partial actions on (locally) compact spaces include the groupoid approach to the enveloping spaces associated to partial actions of countable discrete groups [9]. also several classes of c∗-algebras can be described as partial crossed products that correspond to partial actions of discrete groups on profinite spaces; for instance the carlsen-matsumoto c∗ -algebra ox of an arbitrary subshift x (see [5]). the interested reader may consult [4] and [8] for a detailed account in developments around partial actions. on the other hand, the study of hyperspaces has developed for more than one hundred years, topological properties in hyperspaces: dimension, shape, contractibility, admissibility, unicoherence, etc., have been topics where researchers have dedicated a lot of attention recently. furthermore, there are many papers in different areas of mathematics focused on the study of set-valued functions where hyperspaces are the natural environment to work. for instance, in [2] the authors study when a hyperspace can be embedded in a cell or when a cell can be embedded in a hyperspace. topics concerning the n-od problem, whitney properties and whitney-reversible properties have been widely considered, for a detailed account on hyperspaces the interested reader may consult [12] and the reference therein. this work is structured as follows. after the introduction in section 2 we present the preliminary notions on topological partial actions and their enveloping actions, at the end of this section we fix a compact hausdorff space x and state our conventions, notations and results on the hyperspaces h1, h2 and h3 consisting of compact, compact and connected, and finite subsets of x, respectively. in section 3 we translate partial actions θ of x to partial actions 2θ on h ∈ {h1,h2,h3} and present in theorem 3.2 and proposition 3.5 some structural properties preserved by this correspondence. separation properties relating enveloping actions of θ and 2θ are considered in corollary 3.12 and theorem 3.14. finally, section 4 has a categorical flavor, where it is considered the category g y ch whose objects are topological partial actions on compact hausdorff spaces and show in theorem 4.3 that the functor 2− generates a monad in this category. © agt, upv, 2022 appl. gen. topol. 23, no. 2 256 partial actions of groups on hyperspaces 2. the notions we present the necessary background on partial actions and hyperspaces that we use throughout the work. 2.1. preliminaries on partial actions and their enveloping actions. we start with the following. definition 2.1 ([10, p. 87-88]). let g be a group with identity element 1 and x be a set. a partially defined function θ : g × x 99k x, (g,x) 7→ g · x is called a (set theoretic) partial action of g on x if for each g,h ∈ g and x ∈ x the following assertions hold: (pa1) if ∃g ·x, then ∃g−1 · (g ·x) and g−1 · (g ·x) = x, (pa2) if ∃g · (h ·x), then ∃(gh) ·x and g · (h ·x) = (gh) ·x, (pa3) ∃1 ·x and 1 ·x = x, where ∃g ·x means that g ·x is defined. we say that θ acts (globally) on x or that θ is global if ∃g ·x, for all (g,x) ∈ g×x. given a partial action θ of g on x, g ∈ g and x ∈ x. we set: • g∗x = {(g,x) ∈ g×x | ∃g ·x} the domain of θ. • xg = {x ∈ x | ∃g−1 ·x}. then θ induces a family of bijections {θg : xg−1 3 x 7→ g · x ∈ xg}g∈g. we also denote this family by θ. the following result characterizes partial actions in terms of a family of bijections. proposition 2.2 ([15, lemma 1.2]). a partial action θ of g on x is a family θ = {θg : xg−1 → xg}g∈g, where xg ⊆ x, θg : xg−1 → xg is bijective, for all g ∈ g, and such that: (i) x1 = x and θ1 = idx; (ii) θg(xg−1 ∩xh) = xg ∩xgh; (iii) θgθh : xh−1 ∩xh−1g−1 → xg∩xgh, and θgθh = θgh in xh−1 ∩xh−1g−1 ; for all g,h ∈ g. in view of proposition 2.2 a partial action on x are frequently denoted as a family of maps (θg,xg)g∈g, between subsets of x satisfying conditions (i)-(iii) above. for the reader’s convenience we recall a characterization of partial action. proposition 2.3 ([8, proposition 2.5]). let g be a group and x a set. then a family θ = {θg : xg−1 → xg}g∈g, of bijections between subsets of x is a partial action of g on x if and only if, in addition to (i) of proposition 2.2, for all g,h ∈ g one has that: (ii’) θg(xg−1 ∩xh) ⊆ xgh, (iii’) θg(θh(x)) = θgh(x), for all x ∈ xh−1 ∩x(gh)−1. from now on in this work g will denote a topological group and x a topological space. we endow g×x with the product topology and g∗x with the topology of subspace. moreover θ : g ∗ x → x will denote a partial action. © agt, upv, 2022 appl. gen. topol. 23, no. 2 257 j. l. mart́ınez , h. pinedo and e. ramirez we say that θ is a topological partial action if xg is open and θg is a homeomorphism, for all g ∈ g. moreover, if θ is continuous, θ is called a continuous partial action. 2.2. restriction of global actions and globalization. let µ: g×y → y be a continuous action of g on a topological space y and x ⊆ y be an open set. then we can obtain by restriction a topological partial action on x by setting: (2.1) xg = x ∩µg(x), θg = µg � xg−1 and θ: g∗x 3 (g,x) 7→ θg(x) ∈ x. then θ is a topological partial action of g on x, we say that θ is the restriction of µ to x. as mentioned in the introduction, a natural problem in the study of partial actions is whether they can be restrictions of global actions. in the topological sense, this turns out to be affirmative and a proof was given in [1, theorem 1.1] and independently in [10, section 3.1]. their construction is as follows. let θ be a topological partial action of g on x and consider the following equivalence relation on g×x: (2.2) (g,x)r(h,y) ⇐⇒ x ∈ xg−1h and θh−1g(x) = y. denote by [g,x] the equivalence class of (g,x). the enveloping space or the globalization of x is the set xg = (g × x)/r endowed with the quotient topology. we have by [1, theorem 1.1] that the action (2.3) µ: g×xg 3 (g, [h,x]) → [gh,x] ∈ xg, is continuous and is the so called the enveloping action of θ. further by (ii) in [10, proposition 3.9] the map q : g×x 3 (g,x) 7→ [g,x] ∈ xg, is open. on the other hand the map (2.4) ι: x 3 x 7→ [1,x] ∈ xg satisfies g · ι(x) = xg. moreover, it follows by [10, proposition 3.12] that ι a homeomorphism onto ι(x) if and only if θ is continuous, and by [10, proposition 3.11] ι(x) is open in xg, provided that g∗x is open. we finish this section with a result that will be useful in the sequel. lemma 2.4. let µ : g × y → y be a continuous global action of g on a topological space y and let u ⊆ y be such that g ·u = y . then the following assertions hold. (i) if g and u are separable, then y is separable. (ii) if u is clopen and regular, then y is regular. proof. (i) let {un : n ∈ n} ⊆ u and {gm : m ∈ n} ⊆ g be dense subsets of u and g, respectively. then for an open nonempty set v ⊆ y we have that w := µ−1(v ) ∩ (g × u) is open in g × u. then there are n,m ∈ n © agt, upv, 2022 appl. gen. topol. 23, no. 2 258 partial actions of groups on hyperspaces such that (gm,un) ∈ w and consequently, gm · un ∈ v which implies that {gm ·un ∈ y : m,n ∈ n} is dense in y . (ii) take y ∈ y and z ⊆ y an open set such that y ∈ z. the fact that g ·u = y implies that there are g ∈ g, u ∈ u such that y = g ·u. since µ is continuous there is an open set b ⊆ y for which u ∈ b and g ·b ⊆ z. then v = u ∩b is open in u and g ·v ⊆ z. since u is regular, there is an open set w of u such that u ∈ w ⊆ clu (w) ⊆ v but u is closed then y = g ·u ∈ g ·w ⊆ g ·clu (w) = g ·w ⊆ g ·v ⊆ z, and y is regular. � 2.3. conventions on hyperspaces. from now on in this work x will denote a compact hausdorff space. the hyperspace h1 := 2x is the set consisting of non-empty compact subsets of x. for u1,u2, · · · ,un non-empty open sets of x, let 〈u1, ...,un〉h1 = { a ∈h1 : a ⊆ n⋃ i=1 ui, and a∩ui 6= ∅, 1 ≤ i ≤ n } , moreover we set 〈∅〉 := ∅. the vietoris topology on h1 is generated by collections of the form 〈u1, ...,un〉h1. we shall also work with the subspaces h2 := {c ∈h1 | c is connected} and h3 := {f ∈h1 | f is finite} that is 〈u1, · · · ,un〉hi := hi∩〈u1, · · · ,un〉h1, for u1,u2, · · · ,un open subsets of x and i = 2, 3. finally, when taking about a hyperspace h we make reference to any of the spaces h1,h2 as well as h3. we summarize some well-known properties of the space h. for more details on hyperspaces, the interested reader may consult [12]. lemma 2.5. let x be a compact hausdorff space. then the following assertions hold. (i) the map x 3 x 7→ {x}∈h is an embedding of x into h. (ii) h is a compact hausdorff space and the map u : 22 x → 2x, a 7→ ∪a is continuous. 3. from partial actions on x to partial actions on h in what follows we shall use a continuous partial action on x to construct a continuous partial action on h. the following is straightforward. lemma 3.1. let u and v be open subsets of x and f : u → v a homeomorphism, then the map 2f : 〈u〉h 3 a 7→ f(a) ∈ 〈v 〉h is a homeomorphism. theorem 3.2. let θ := (θg,xg)g∈g be a topological partial action of g on x. for g ∈ g, we set 2θg : 〈xg−1〉h 3 a 7→ θg(a) ∈ 〈xg〉h. then 2θ = (2θg,〈xg〉h)g∈g is a topological partial action of g on h and the following assertions hold. © agt, upv, 2022 appl. gen. topol. 23, no. 2 259 j. l. mart́ınez , h. pinedo and e. ramirez (i) g∗h is open provided that g∗x is open. (ii) if θ is continuous, then 2θ is continuous. (iii) if θ is global then 2θ is global. proof. we shall only deal with the case h = h2. by lemma 3.1 we have that 2θg is a homeomorphism between open subsets of h2, for any g ∈ g. we shall check conditions (i) and (ii’) (iii’) in proposition 2.2 and proposition 2.3, respectively. to see (i) notice that 2θe is the identity map of 〈x〉h2 = h2. for (ii’) take g,h ∈ g and a ∈ 〈xg−1〉h2 ∩ 〈xh〉h2 = 〈xg−1 ∩ xh〉h2 , then 2θg (a) = θg(a) ⊆ θg(xg−1 ∩xh) ⊆ xgh, and thus 2θg (〈xg−1〉h2 ∩〈xh〉h2 ) ⊆ 〈xgh〉h2 . for (iii’) take a ∈ 〈xh−1〉h2 ∩〈x(gh)−1〉h2 = 〈xh−1 ∩ x(gh)−1〉h2 , then 2θgh(a) = θgh(a) = θg(θh(a)) = 2 θg (2θh(a)), and we conclude that 2θ is a partial action of g on h2. now we check (i)−(iii). (i) suppose that g ∗ x is open in g × x. to see that g ∗ h2 is open in g × h2, take (g,a) ∈ g ∗ h2. since a ⊆ xg−1 , we have (g,a) ∈ g ∗ x for all a ∈ a. now the fact that g ∗ x is an open subset of g × x, implies that for any a ∈ a there are open sets ua ⊆ g and va ⊆ x for which (g,a) ∈ ua × va ⊆ g ∗ x. since a is compact, there exist a1, · · · ,an ∈ a with a ⊆ n⋃ i=1 vai, and a ∈ 〈va1, · · · ,van〉h2 . let u := n⋂ i=1 uai, then (g,a) ∈ u×〈va1, · · · ,van〉h2 we claim that u×〈va1, · · · ,van〉h2 ⊆ g∗h2. indeed, take (h,b) ∈ u×〈va1, · · · ,van〉h2, we shall check b ∈ 〈xh−1〉h2 . take b ∈ b. since b ⊆ n⋃ i=1 vai, there is 1 ≤ i ≤ n for which b ∈ vai and (h,b) ∈ uai×vai ⊆ g∗x, then b ∈ xh−1 . from this we get b ∈ 〈xh−1〉h2 and thus (h,b) ∈ g ∗h2. this shows that g∗h2 is open in g×h2. (ii) suppose that θ is continuous. we need to show that 2θ : g∗h2 →h2, (g,a) 7→ θg(a) is continuous. let (g,a) ∈ g ∗h2 and take v1, · · · ,vk open subsets of x such that θg(a) ∈ 〈v1, · · · ,vk〉h2 . for each a ∈ a there is 1 ≤ ia ≤ k such that θg(a) ∈ via, and since θ is continuous there are open sets uia ⊆ g and wia ⊆ x such that: (g,a) ∈ (uia ×wia) ∩ (g∗x) and θ((uia ×wia) ∩g∗x) ⊆ via. the fact that a is compact implies that there are a1, · · · ,am ∈ a such that a ⊆ m⋃ j=1 wiaj . on the other hand, since θg(a) ∩ vj 6= ∅, for any 1 ≤ j ≤ k, there are r1,r2, · · · ,rk ∈ a for which θg(rj) ∈ vj, for all j = 1, 2, · · · ,k. set irj := j, j = 1, · · · ,k. without loss of generality we may suppose {r1,r2, · · · ,rk} ⊆ {a1, · · · ,am} and ri = ai, for each i = 1, · · · ,k. let u := ⋂m j=1 uiaj . then (g,a) ∈ z := (u ×〈wia1 , · · · ,wiam〉h2 ) ∩ (g ∗h2). to finish the proof it is enough to show that 2θ(z) ⊆ 〈v1, · · · ,vk〉h2 . for this take (h,b) ∈ z and © agt, upv, 2022 appl. gen. topol. 23, no. 2 260 partial actions of groups on hyperspaces b ∈ b. since b ⊆ m⋃ j=1 wiaj , there exists 1 ≤ j ≤ m such that b ∈ wiaj and thus (h,b) ∈ uiaj ×wiaj . but b ⊆ xh−1 , then (h,b) ∈ (uiaj ×wiaj ) ∩ (g∗x) and θh(b) ∈ viaj which implies θh(b) ⊆ k⋃ i=1 vi. finally, for 1 ≤ l ≤ k we see that θh(b) ∩vl 6= ∅. indeed, take b ∈ b ∩wial , where al = rl. since h ∈ uial , we have (h,b) ∈ (uial ×wial )∩(g∗x) and θh(b) ∈ vial ∩θh(b) = virl ∩θh(b) = vl ∩θh(b) which finishes the proof of the second item. (iii) this is clear. � remark 3.3. given a partial action θ of g on x, we shall refer to 2θ as the induced partial action of θ on h. example 3.4. there is a topological partial action of z(4) on s1 given by the family {xn}n∈z(4) as it is shown below. x1x3 x2 θ0 = ids1 ; θ1 : x3 → x1 by θ1(eit) = ei(t+π); θ3 = θ−11 , θ2 : x2 → x2 is the identity. we construct the induced partial action of z(4) on h2(s1), for this we find a homeomorphism h between h2(s1), the connected sets of s1 and d = {z ∈ c : |z| ≤ 1}. o h(a) p a let p ∈s1 and take an arc center at p of length l, this arc is mapped on h(a) =( 1− l 2π ) p ∈ d. the arc {p} of length zero is mapped onto h({p}) = p ∈ d. in particular, all arcs centered at p are mapped on op. from this follows that the sets {〈xn〉h2}n∈z(4) are © agt, upv, 2022 appl. gen. topol. 23, no. 2 261 j. l. mart́ınez , h. pinedo and e. ramirez 〈x1〉h2〈x3〉h2 〈x2〉h2 we construct 2θ. the map 2θ2 is the identity on 〈x2〉h2 . notice that 2θ1 rotates each arc in x3 π radians to an arc in x1 of the same length. x1x3 x2 a 2θ1 (a) h(a) h(2θ1 (a)) then h(2θ1 (a)) is obtained by rotating h(a) π radians, from this 2θ1 in d is identified with 2θ1 : 〈x3〉h2 −→ 〈x1〉h2 , reit 7−→ rei(t+π), analogously 2θ3 : 〈x1〉h2 −→ 〈x3〉h2 , reit 7−→ rei(t+π). we finish this section with the next. proposition 3.5. let θ be a topological partial action of g on x. if g∗x is closed, then g∗h is closed. proof. take (g,a) ∈ (g ×h) \ g ∗h. then there is a ∈ a such that @g · a and (g,a) /∈ g ∗ x and there are open sets u ⊆ g and v ⊆ x such that (g,a) ∈ u × v ⊆ (g × x) \ g ∗ x. note that (g,a) ∈ u × 〈v,x〉h to finish the proof we need to show that u ×〈v,x〉h ⊆ (g×h) \g∗h. take (h,b) ∈ u ×〈v,x〉h and b ∈ b ∩v . since (h,b) ∈ u ×v , we get @h · b, then @h ·b and (h,b) /∈ g∗h as desired. � 3.1. separation properties and enveloping spaces. it is shown in [1, proposition 1.2] that a partial action has a hausdorff enveloping space if and only if the graph of the action is closed. below we show that partial actions on compact hausdorff spaces have hausdorff enveloping space, if and only if the enveloping space of the induced partial action on h is hausdorff. from now on, 2r denotes the equivalence relation associated to the enveloping action of the partial action 2θ of g on h (see equation (2.2)). that is hg = (g×h)/2r. lemma 3.6. let θ be a partial action on x and 2θ be the corresponding partial action of g on h, then the map θ : xg 3 [g,x] 7→ [g,{x}] ∈hg is an embedding. © agt, upv, 2022 appl. gen. topol. 23, no. 2 262 partial actions of groups on hyperspaces proof. first of all observe that θ is well defined. indeed, if (g,x)r(h,y), then {x} ⊆ 〈xg−1h〉h and 2 θ h−1g ({x}) = {y}, which gives (g,{x})2r(h,{y}). in an analogous way one checks that θ is injective. now we prove that θ is continuous, for this it is enough to check that β : g×x 3 (g,x) 7→ [g,{x}] ∈ hg, is continuous. for this notice that ϕ : g×x 3 (g,x) 7→ (g,{x}) ∈ g×h, is continuous because of lemma 2.5. also, β = qh◦ϕ, where qh : g×h→hg is the quotient map, form this β is continuous, and so is θ. now we need to show that θ−1 : im(θ) → xg is continuous. let u ⊆ xg be an open set and [g0,{x0}] ∈ im(θ) such that [g0,x0] ∈ u. then (g0,x0) ∈ q−1(u) and there exists open sets v ⊆ g and w ⊆ x such that (g0,x0) ∈ v × w ⊆ q−1(u). take z := qh(v × 〈w〉h) ∩ im(θ). since qh is open, then z is open in im(θ) and [g0,{x0}] ∈ z. on the other hand, take [r,{s}] ∈ z we check that θ−1([r,{s}]) = [r,s] ∈ u. for this take (v,f) ∈ v ×〈w〉h such that [v,f] = qh(v,f) = [r,{s}]. then f = {w} for some w ∈ w and θ−1([r,{s}]) = θ−1([v,{w}]) = [v,w] = q(v,w) ∈ q(v ×w) ⊆ u, this shows that θ−1 is continuous and θ is an embedding. � lemma 3.7. let θ be a partial action on x and 2θ be the induced partial action of g on h, then 2r is closed in (g×h)2 provided that r is closed in (g×x)2. proof. take ((g,a), (h,b)) ∈ (g×h)2 \ 2r, we have two cases to consider: case 1: a /∈ 〈xg−1h〉h. then there exists a ∈ a ∩ (x \ xg−1h), and ((g,a), (h,b)) ∈ (g × x)2 \ r, for any b ∈ b. since r is closed there are open sets ub,yb ⊆ g and vb,zb ⊆ x such that ((g,a), (h,b)) ∈ (ub ×vb) × (yb ×zb) ⊆ (g×x)2 \r, for any b ∈ b. the fact that b is compact implies that there are b1, · · · ,bn ∈ b for which b ⊆ n⋃ i=1 zbi. write u := n⋂ i=1 ubi, v := n⋂ i=1 vbi and y = n⋂ i=1 ybi. then a ∈ 〈x,v 〉h and ((g,a), (h,b)) ∈ (u ×〈x,v 〉h) × (y ×〈zb1, · · · ,zbn〉h). now we show that (u ×〈x,v 〉h) × (y ×〈zb1, · · · ,zbn〉h) ⊆ (g×h) 2 \ 2r. for this take ((r,c), (s,d)) ∈ (u ×〈x,v 〉h) × (y ×〈zb1, · · · ,zbn〉h). for c ∈ c∩v and d ∈ d, there is 1 ≤ j ≤ n such that d ∈ zbj , then ((r,c), (s,d)) ∈ (ubj ×vbj )×(ybj ×zbj ) ⊆ (g×x)2 \r which implies c /∈ xr−1s or c ∈ xr−1s and θs−1r(c) 6= d. if c /∈ xr−1s, then c /∈ 〈xr−1s〉h and we have done. now suppose c ∈ xr−1s and θs−1r(c) 6= d. if θs−1r(c) ∈ d, by a similar argument as above we get ((r,c), (s,θs−1r(c)) /∈ r, which leads to a contradiction. then, θs−1r(c) 6= d and ((r,c), (s,d)) /∈ 2r. case 2. a ⊆ xg−1h. then θh−1g(a) 6= b. suppose that there exists a ∈ a such that θh−1g(a) /∈ b. then ((g,a), (h,b)) /∈ r, for any b ∈ b, we argue as in case 1 to obtain b1, · · · ,bn ∈ b and families {ubi}ni=1, {ybi} n i=1 of open subsets © agt, upv, 2022 appl. gen. topol. 23, no. 2 263 j. l. mart́ınez , h. pinedo and e. ramirez of g such that g ∈ u := n⋂ i=1 ubi and h ∈ y := n⋂ i=1 ybi. also there are families {vbi}ni=1 and {zbi} n i=1 of open subsets of x such that a ∈ v := ⋂ i=1 vbi,b ∈ 〈zb1, · · · ,zbn〉h and (ubi×vbi)×(ybi×zbi) ⊆ (g×x)2\r, for any i = 1, · · ·n. as in case 1 we get ((g,a), (h,b)) ∈ (u ×〈x,v 〉h) × (y ×〈zb1, · · · ,zbn〉h) ⊆ (g×h)2 \ 2r. to finish the proof, suppose that there is b ∈ b such that θh−1g(a) 6= b, for each a ∈ a. if a ∈ a, then ((g,a), (h,b)) /∈ r and there are open sets ua,ya ⊆ g and va,za ⊆ x such that ((g,a), (h,b)) ∈ (ua×va)×(ya×za) ⊆ (g×x)2\r. the compactness of a implies that there are a1, · · · ,an ∈ a such that a ⊆ n⋃ i=1 vai. write u := n⋂ i=1 uai, y ′ := n⋂ i=1 yai and z := n⋂ i=1 zai. now ((g,a), (h,b)) ∈ (u ×〈va1, · · · ,van〉h) × (y ′ ×〈x,z〉h) ⊆ (g×h)2 \ 2r. indeed, let ((r,c), (s,d)) ∈ (u ×〈va1, · · · ,van〉h) × (y ′ ×〈x,z〉h) and d ∈ d∩z. for c ∈ c, there is 1 ≤ j ≤ n such that c ∈ vaj therefore ((r,c), (s,d)) ∈ (uaj × vaj ) × (yaj × zaj ) ⊆ (g × x)2 \ r. moreover, ((s,d), (r,c)) /∈ r and d /∈ xs−1r or d ∈ xs−1r and θr−1c(d) 6= c. in the case d /∈ xs−1r, we obtain d /∈ 〈xs−1r〉h and ((r,c), (s,d)) /∈ 2r. thus it only remains to consider the case d ∈ xs−1r and θr−1c(d) 6= c. if θr−1s(d) ∈ c, as above we get ((s,d), (r,θr−1s(d))) /∈ r, which leads to a contradiction. this shows θr−1s(d) /∈ c, and ((r,c), (s,d)) /∈ 2r. � combining [13, lemma 34] with lemma 3.6, lemma 3.7 and using that the quotient map to the globalization is open we obtain the following. theorem 3.8. let θ be a partial action of g on x. then xg is hausdorff if and only if hg is hausdorff. recall that a locally compact cantor space is a locally compact hausdorff space with a countable basis of clopen sets and no isolated points. if a locally compact cantor space x is compact, then there is a homeomorphism between x and the cantor space. we proceed with the next. proposition 3.9. let x be a metric compact cantor space, g a countable discrete group and suppose that θ = (θg,xg)g∈g is a partial action of g on x such that xg is clopen for all g ∈ g. then (2x)g is a locally compact cantor space. proof. since x is a compact hausdorff space, then 2x is a compact hausdorff space. moreover since x is metric, we get from [12, proposition 8.4] that 2x has no isolated points, and from [12, proposition 8.6] we have that 2x have a countable basis of clopen sets. therefore h1 = 2x is the cantor space. also 〈xg〉 is clopen for all g ∈ g and the result follows from [9, proposition 2.3]. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 264 partial actions of groups on hyperspaces now we shall work with the hyperspace h3 = f(x) consisting of finite subsets of x. the following result shows that the enveloping space (h3)g is t1, provided that xg is. proposition 3.10. let θ be a topological partial action of g on x and 2θ be the induced partial action of g on h3. if xg is t1, then (h3)g is t1. proof. let a = {a1, · · · ,an} ∈ h3 and g ∈ g, and qh3 : g×h3 → (h3)g be the corresponding quotient map. we need to show that q−1h3 ([g,a]) = {(h,f) ∈ g×h3 : ∃(g −1h) ·f and (g−1h) ·f = a} is closed in g×h3. take (h,f) /∈ q−1h3 ([g,a]). there are two cases to consider. case 1: @(g−1h) ·f. then there is f ∈ f such that @(g−1h) ·f. since xg is t1, for 1 ≤ i ≤ n there are open sets ui ⊆ g and vi ⊆ x for which (h,f) ∈ ui ×vi ⊆ (g×x) \q−1([g,ai]). take u := ⋂n i=1 ui and v = ⋂n i=1 vi. note that (h,f) ∈ u ×〈x,v 〉h3 ⊆ (g ×h3) \ q−1h3 ([g,a]). indeed, if (t,b) ∈ u ×〈x,v 〉h3 and b ∈ b ∩ v we have (t,b) /∈ q−1([g,ai]) for any 1 ≤ i ≤ n. if @g−1t · b, then @(g−1t) · b and (t,b) /∈ q−1h3 ([g,a]). on the other hand, if ∃(g −1t) · b, then (g−1t) · b 6= ai, for each 1 ≤ i ≤ n, then (g−1t) · b /∈ a and (t,b) /∈ q−1h3 ([g,a]). case 2: ∃(g−1h) · f and (g−1h) · f 6= a. if there is f ∈ f for which (g−1h) · f /∈ a we get (h,f) /∈ q−1([g,ai]) for 1 ≤ i ≤ n and we proceed as in case 1. if there is a ∈ a such that (g−1h) · f 6= a, for any f ∈ f write f = {f1, · · · ,fk}, then (h,fj) /∈ q−1([g,a]) for each 1 ≤ j ≤ k. hence there are open sets u ⊆ g and v ⊆ x such that (h,fj) ∈ u ×v ⊆ (g×x)\q−1([g,a]), for every 1 ≤ j ≤ k. note that (h,f) ∈ u ×〈v 〉h3 ⊆ (g×h3) \ q −1 h3 ([g,a]). indeed, if (t,b) ∈ u ×〈v 〉h3 . if @(g−1t) ·b, then (t,b) /∈ q −1 h3 ([g,a]). in the case ∃g−1t ·b, we get that for any b ∈ b the pair (t,b) belongs to u ×v and thus (g−1t) · b 6= a which gives (t,b) /∈ q−1h3 ([g,a]), as desired. � combining lemma 3.6 and proposition 3.10 we get. corollary 3.11. let θ be a topological partial action of g on x and 2θ be the induced partial action of g on h3. then (h3)g is t1 if and only if xg is t1. we proceed with the next corollary 3.12. let g be a separable group and θ be a continuous partial action of g on x such that g ∗ x is open and x is separable. take h ∈ {h1,h3} then the following assertions hold. (i) xg is separable; (ii) hg is separable; (iii) if xg is t1, then h(xg) and h(f(x)g) are separable. proof. (i) since θ is continuous with open domain then ι(x) is open in xg and (2.4) is a homeomorphism onto ι(x), in particular ι(x) is separable, moreover © agt, upv, 2022 appl. gen. topol. 23, no. 2 265 j. l. mart́ınez , h. pinedo and e. ramirez the map µ given in (2.3) acts continuously in xg and g · ι(x) = xg thus the result follows by (i) in lemma (2.4). (ii) since x is separable and t1, then h is separable. then by (i) and (ii) of theorem 3.2 and lemma (2.4) we get that hg is separable. (iii) by (i) the space xg is separable and follows that h(xg) is separable. finally, by proposition 3.10 we have that f(x)g is t1, moreover f(x)g is separable thanks to (ii), and thus h(f(x)g) is separable. � example 3.13 ([13, example 4.8]). consider the partial action of z on x = [0, 1] given by θ0 = idx and θn = id[0,1),n 6= 0 then θ is continuous with open domain and xz is t1. thus by corollary 3.12 the spaces xz,hz,h([0, 1]z) and h(f([0, 1])g) are separable, where h∈{h1,h3}. now we shall deal with the regularity condition. theorem 3.14. let θ : g∗x → x be a continuous partial action with clopen domain. then the spaces xg and hg are regular, provided that h∈{h1,h2}. proof. let ι be the embedding map defined in (2.4) then gι(x) = xg, we shall prove that ι(x) is clopen and regular. let q : g × x → xg be the quotient map, then q−1(ι(x)) = g ∗ x is clopen in g × x which shows that ι(x) is clopen in xg. now since x is a compact hausdorff space we have that ι(x) is regular and thus xg is regular thanks to item (ii) of lemma 2.4. on the other hand, we have that h is compact and hausdorff, 2θ is continuous ((ii) of theorem 2.8) and g∗h is clopen thanks to of theorem 3.2 and proposition 3.5 , then it is enough to apply (ii) of lemma 2.4. � example 3.15 ([8, p. 22]). partial bernoulli action let g be a discrete group and x := {0, 1}g. the map β : g × x 3 (g,ω) 7→ gω ∈ x, is a continuous global action. the topological partial bernoulli action is obtained by restricting β to the open set ω1 = {ω ∈ x : ω(1) = 1} (see (2.1)). it is shown in [11, example 3.4] that g ∗ ω1 is clopen. thus by theorem 3.14 we have that hg is regular where h ∈ {h1,h2}. moreover, since g · ω1 = x, then xg = {0, 1}g is regular. remark 3.16. in [13, theorem 4.6] are presented other conditions for the space xg being regular. 4. on the category g y ch we shall use some of the above results to construct a monad in the category of partial actions on compact hausdorff spaces. first recall the next. definition 4.1. let φ = (φg,xg)g∈g and ψ = (ψg,yg)g∈g, be partial actions of g on the spaces x and y , respectively. a g-map f : φ → ψ is a continuous function f : x → y such that: (i) f(xg) ⊆ yg, (ii) f(φg(x)) = ψg(f(x)), for each x ∈ xg−1, © agt, upv, 2022 appl. gen. topol. 23, no. 2 266 partial actions of groups on hyperspaces for any g ∈ g. if moreover f is a homeomorphism and f−1 is g-map, we say that φ are ψ equivalent. we denote by g y top the category whose objects are topological partial actions of g on topological spaces and morphisms are g-maps defined as above. also, we denote by g y ch the subcategory of g y top whose objects are topological partial actions of g on compact hausdorff spaces. it follows by theorem 3.2 that there is a functor 2− : g y ch → g y ch. 4.1. the monad i. recall the next. definition 4.2. let c be a category. a monad in c is a triple (t,η,µ), where t : c → c is an endofunctor, η : idc =⇒ t and µ : t 2 =⇒ t are natural transformations such that: (4.1) µ◦tη = µ◦ηt = 1t and µ◦µt = µ◦tµ. given an object α = (αg,xg)g∈g ∈ g y ch. we have by lemma 2.5 that the map ηα : x 3 x 7→ {x}∈ 2x, is a continuous function. from this it is not difficult to see that ηα : α → 2α is a morphism in g y ch. moreover, for another object β = (yg,βg)g∈g in g y ch and a morphism f : α → β the diagram: x f �� ηα // 2x 2f �� y ηβ // 2y is commutative. thus the family η = {ηα}α∈gycmet : idgycmet =⇒ 2− is a natural transformation. now set µα : 2 2x 3 a 7→ ∪a ∈ 2x, by lemma 2.5 µα is continuous. we shall check that µα : 2 2α → 2α is a morphism in g y ch . (i) take g ∈ g and a ∈ 〈〈xg〉〉. then a ⊆ 〈xg〉 and µα(a) = ∪a ⊆ xg, that is µα(a) ∈ 〈xg〉. (ii) for a ∈ 〈〈xg−1〉〉 we have 2αg [a] = {αg(f) : f ∈ a}, then µα(22 αg (a)) = µα(2 αg [a]) = ∪2αg [a] = αg(∪a) = 2αg (∪a) = 2αg (µα(a)), as desired. now we prove that µ = {µα}{α∈gych} : (2−)2 =⇒ 2− is a natural transformation. for this take β = (yg,βg)g∈g in g y ch and a morphism f : α → β in g y ch. consider the diagram (4.2) 22 x 22 f �� µα // 2x 2f �� 22 y µβ // 2y let a ∈ 22 x , then 2f [a] = {f(b) : b ∈ a} and 2f (µα(a)) = 2f (∪a) = f(∪a) = ∪2f [a] = µβ(2f [a]) thus the diagram (4.2) is commutative. © agt, upv, 2022 appl. gen. topol. 23, no. 2 267 j. l. mart́ınez , h. pinedo and e. ramirez theorem 4.3. let η and µ be as above. then the triple i = (2−,η,µ) forms a monad in the category g y ch. proof. it remains to prove that equalities in (4.1) hold. let α be an object in g y ch since(η2−)α = η2α, we have that µα◦η2α : 2x 3 a 7→ a ∈ 2x, which gives µ◦η2− = 12− . also, (2−η)α = 2−(ηα) = 2ηα, and µα ◦ 2ηα : 2x 3 a 7→ a ∈ 2x, which shows µ◦ 2−η− = µ◦η2− = 12− . finally, since (µ2−)α = µ2α and (2−µ)α = 2 −(µα), we have (µ◦µ2−)α = µα ◦µ2α = µα ◦ 2µα = (µ◦ 2−µ)α, thus µ◦µ2− = µ◦ 2−µ and (2−,η,µ) is a monad. � acknowledgements. we thank the referee for very helpful comments and suggestions that helped for the improvement of the paper. we also thank professor javier camargo for fruitful discussions. references [1] f. abadie, enveloping actions and takai duality for partial actions, j. funct. anal. 197 (2003), 14–67. [2] j. camargo, d. herrera and s. maćıas, cells and n-fold hyperspaces, colloq. math. 145, no. 2 (2016), 157–166. [3] k. choi, birget-rhodes expansions of topological groups, advanced studies in contemporary mathematics 23, no. 1 (2013), 203–211. [4] m. dokuchaev, recent developments around partial actions, são paulo j. math. sci. 13, no. 1 (2019), 195–247. [5] m. dokuchaev and r. exel, partial actions and subshifts, j. funct. analysis 272 (2017), 5038–5106. [6] r. exel, circle actions on c∗-algebras, partial automorphisms and generalized pimsnervoiculescu exact sequences, j. funct. anal. 122, no. 3 (1994), 361–401. [7] r. exel, partial actions of group and actions of inverse semigroups, proc. amer. math. soc. 126, no. 12 (1998), 3481–3494. [8] r. exel, partial dynamical systems, fell bundles and applications, mathematical surveys and monographs; volume 224, providence, rhode island: american mathematical society, 2017. 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[16] b. steinberg, partial actions of groups on cell complexes, monatsh. math. 138, no. 2 (2003), 159–170. © agt, upv, 2022 appl. gen. topol. 23, no. 2 268 @ appl. gen. topol. 22, no. 1 (2021), 91-108doi:10.4995/agt.2021.13902 © agt, upv, 2021 convexity and boundedness relaxation for fixed point theorems in modular spaces fatemeh lael a and samira shabanian b a department of mathematics, buein zahra technical university, buein zahra, qazvin, iran. (f lael@bzte.ac.ir) b microsoft research (samira.shabanian@microsoft.com) communicated by s. romaguera abstract although fixed point theorems in modular spaces have remarkably applied to a wide variety of mathematical problems, these theorems strongly depend on some assumptions which often do not hold in practice or can lead to their reformulations as particular problems in normed vector spaces. a recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further. in this paper, we focus on convexity and boundedness of modulars in fixed point results taken from the literature for contractive correspondence and single-valued mappings. to relax these two assumptions, we seek to identify the ties between modular and b-metric spaces. afterwards we present an application to a particular form of integral inclusions to support our generalized version of nadler’s theorem in modular spaces. 2010 msc: 46e30; 47h10; 54c60. keywords: modular space; fixed point; correspondences; b-metric space. 1. introduction compared to 1922 when banach fixed point theorem has been proved [8], certainly, fixed point theory now plays a significant and meaningful role in both the field of mathematics and many real-life applications due to providing received 21 june 2020 – accepted 18 january 2021 http://dx.doi.org/10.4995/agt.2021.13902 f. lael and s. shabanian a general framework which opens the door to the development of many other approaches. in one approach, fixed point theory in modular spaces has received a lot of attention after being proposed as a generalization of normed spaces [39, 40, 42, 45, 46]. a growing literature on fixed point theorems in modular spaces deals with rigorous formulations and proofs of many interesting problems which are applicable in a wide variety of settings, including quantum mechanics, machine learning and etc. fixed point theory in modular spaces has its root in [27] by using some constructive techniques for single-valued mappings. this work has been widely cited as the inspiration for a variety of fixed point work along with [25, 26]. this line of work was extended by several works in a variety of ways. in one successful approach, in 1969, nadler proposed the banach contraction principle for multivalued mappings of in modular spaces [41]. a wide range of extensions was subsequently proposed by various authors, based on different relaxations [1, 18, 19, 31, 50, 51, 53]. furthermore, authors in [34] focus on a particular case of multivalued mappings in modular spaces with a key property of modulars, additivity. then, [3] explores the existence of fixed points of a specific type of g-contraction and g-nonexpansive mappings in modular function spaces. in another approach, in 1993, czerwik in [15, 16] proposed the first banach’s fixed point theorem for both single and multivalued mapping in b-metric spaces, introduced by bourbaki and bakhtin [7, 13]. then, authors in [30] extended it for some particular types of contractions in the context of b-metric spaces. along this direction, many researchers studied the extension of various well known fixed point results for various types of contractive mapping in the framework of b-metric spaces [12, 33, 47, 48, 57]. although fixed point theory is shown to be successful in challenging problems and has contributed significantly to many real-world problems, various fixed point theorems strongly are proved under strong assumptions. in particular, in modular spaces, some of these assumptions can lead to having some induced norms. so, some assumptions that often do not hold in practice or can lead to their reformulations as a particular problem in a normed vector spaces. a recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further [2, 18, 27]. the aim of the work presented in this paper is to contribute to a deeper understanding of fixed point results in modular spaces and to improve their conditions and assumptions by addressing the open questions and challenges outlined in the literature by identifying the ties between modular spaces and b-metric spaces. in a bird-eyes view, the paper starts with section 2 which is a brief introduction to modular and b-metric spaces along with the required concepts. afterwards we describe the relation between these two particular spaces. section 3 introduces some techniques and ways of improving some current fixed point results. finally, an application to integral inclusions is provided in section 5. © agt, upv, 2021 appl. gen. topol. 22, no. 1 92 convexity and boundedness relaxation for fixed point theorems in modular spaces 2. background this section will serve as an introduction to some fundamental concepts of modular and b-metric spaces. a detailed introduction can be found, for example, in the textbooks [4, 5, 7, 32, 40, 52]. 2.1. modular spaces. a modular space is a pair (x,ρ) where x is a real linear space and ρ is a real valued functional on x which satisfies the conditions: (1) ρ(x) = 0 if and only if x = 0, (2) ρ(−x) = ρ(x), (3) ρ(αx + βy) ≤ ρ(x) + ρ(y), for any nonnegative real numbers α,β with α + β = 1. the functional ρ is called a modular on x. there are many arguably important special instances of well known spaces in which these properties are fulfilled [44, 45, 46, 54]. interestingly, it is shown that a modular induces a vector space xρ = {x ∈ x : ρ(αx) → 0 as α → 0} which is called a modular linear space. furthermore, musielak and orlicz in [39, 45, 46] naturally provide the first definitions of the following key concepts in a modular space (x,ρ): d1. a sequence xn in b ⊆ x is said to be ρ-convergent to a point x ∈ b if ρ(xn − x) → 0 as n → ∞. d2. a ρ-closed subset b ⊆ x is meant that it contains the limit of all its ρ-convergent sequences. d3. a sequence xn in b ⊆ x is said to be ρ-cauchy if ρ(xm − xn) → 0 as m,n → ∞. d4. a subset b of x is said to be ρ-complete if each ρ-cauchy sequence in b is ρ-convergent to a point of b. d5. ρ-bounded subsets: a subset b ⊆ xρ is called ρ-bounded if sup x,y∈b ρ(x− y) < ∞. d6. ρ-compact subsets: a ρ-closed subset b ⊆ x is called ρ-compact if any sequence xn ∈ b has a ρ-convergent subsequence. for a modular space (x,ρ), the function ωρ which is said growth function [17] is defined on [0,∞) as follows: ωρ(t) = inf{ω : ρ(tx) ≤ ωρ(x) : x ∈ x,0 < ρ(x)}. it is easy to show that when (x,ρ) satisfies ωρ(2) < ∞, then every ρconvergent sequence in (x,ρ) is ρ-cauchy. also, we note that in such case every ρ-compact set is ρ-bounded and ρ-complete [35]. 2.2. b-metric spaces. now, we turn our attention to another important and related space in the sense that it can be induced by a modular, namely bmetric spaces. it is shown that a modular induces some well known operators of which we are interested in b-metrics; a b-metric on a nonempty set x is a real function d : x × x → [0,∞) such that for a given real number s ≥ 1 satisfies the conditions: © agt, upv, 2021 appl. gen. topol. 22, no. 1 93 f. lael and s. shabanian (1) d(x,y) = 0 if and only if x = y, (2) d(x,y) = d(y,x), (3) d(x,z) ≤ s[d(x,y) + d(y,z)], for all x,y,z ∈ x, the pair (x,d) is called a b-metric space. as stated in [56], it is true that a b-metric space is not necessarily a metric space. with s = 1, however, a b-metric space is a metric space. furthermore, many concepts like convergent and cauchy sequences, complete spaces, closed sets and etc are easily defined in b-metric spaces [11] which are denoted by bconvergent and bcauchy sequences, bcomplete spaces, bclosed sets. moreover, it has been shown that a lot of metric fixed point theorems can be extended to b-metric spaces [11, 58], although b-metric, in the general case, is not continuous, lim n→∞ xn = x does not necessarily imply lim n→∞ d(xn,y) = d(x,y) (see [6, 21] for further details). the notion of b-metric spaces were introduced to reach the generalization of some known fixed point theorems for single valued mappings and correspondences [9, 10, 15, 16]. in the following example, we generalize some examples which are mentioned in [6]. example 2.1. suppose that (x,d) is a b-metric space with s ≥ 1. then (x,dr) is a b-metric space, for all r ∈ r+. since from the general form of holder’s inequality [55], for every x,y,z ∈ x and r ∈ r+ with 1 + 1 r ≥ 1, we get d(x,y) ≤ s(d(x,z) + d(z,y)) ≤ (2s)(dr(x,z) + dr(z,y)) 1 r , that is, dr(x,y) ≤ (2s)r(dr(x,z) + dr(z,y)). this implies that, dr is a b-metric. since every metric d is a b-metric, then dr is a b-metric. however, dr is not necessarily to be a metric. for example, if d(x,y) = |x − y| is the usual euclidean metric, d2(x,y) = |x − y|2 is not a metric on r. we note that a modular ρ induces a b-metric. in fact, for such modular, we can define d(x,y) = ρ(x − y). then, d is a b-metric with s = ωρ(2) and when (x,ρ) is a ρ-complete space, then (x,d) is a b-complete space. actually, ρ-cauchy sequence and ρ-convergent sequence are equivalent to b-cauchy sequence and b-convergent sequence respectively. we recall that for any subset c of (x,ρ) a correspondence f on a set c, denoted by f : c ։ x assigns to each a ∈ c a (nonempty) subset f(a) of x and an element x ∈ c is said to be a fixed point if x ∈ f(x). a correspondence f is called continuous if xn → x and yn → y and yn ∈ f(xn) imply y ∈ f(x). for a correspondence f we define d(a,f(b)) = inf{d(a,y) : y ∈ f(b)} and distρ(a,f(b)) = inf{ρ(a − y) : y ∈ f(b)}. also, hausdorff distance is defined as hρ(a,b) = max{sup a∈a distρ(a,b), sup b∈b distρ(a,b))} © agt, upv, 2021 appl. gen. topol. 22, no. 1 94 convexity and boundedness relaxation for fixed point theorems in modular spaces where a and b are subsets of c. 2.3. relevant literature. much work has been done on the problem of the fixed point existence for single-valued mappings and in general correspondence in modular spaces [18, 24, 29]. over the years, multiple authors have analyzed various conditions which suffice to guarantee the existence of fixed points for a board class of functions in modular spaces. arguably, the following (h1)-(h4) conditions are specified to be some of the most common and popular ones in modular spaces: (h1) ∆2-condition: a modular ρ is said to satisfy the ∆2-condition [45, 46] if ρ(2xn) → 0, whenever sup n ρ(xn) → 0 as n → ∞. (h2) ∆2-type condition: a modular ρ is said to satisfy the ∆2-type condition [45, 46] if there exists k > 0 such that ρ(2x) ≤ kρ(x) for all x ∈ xρ. (h3) s̃-convex modulars: if condition (3) in the modular definition is replaced by ρ(αx + βy) ≤ αs̃ρ(x) + βs̃ρ(y) for all α,β ∈ [0,∞) with αs̃ + βs̃ = 1 with an s̃ ∈ (0,1], the modular ρ is called an s̃-convex modular [22]. in particular, a 1-convex modular is simply called convex. (h4) fatou property: a modular ρ has the fatou property [14] if ρ(x) ≤ lim inf ρ(xn), whenever xn → x. some excellent overviews of (h1)-(h4) conditions are provided in [27, 22, 54]. it is shown that a modular ρ implies that ‖x‖ρ = inf{a > 0 : ρ( x a ) ≤ 1}, defines an f-norm on xρ. specifically, if ρ is convex, ‖ · ‖ρ is a norm and it is frequently called the luxemburg norm [23]. note that a modular space determined by a function modular ρ will be called a modular function space and will be denoted by lρ. then, it is not difficult to show that ‖ · ‖ρ is an f-norm induced by ρ. more importantly, (lρ,‖ · ‖ρ) is a complete space. being able to define such norm in a real vector space can lead to a smooth proof for many fixed point theorems in very specific modular spaces. for instance, an earlier work on this topic goes back to theorem 2-2 of [27] which was proposed in the early 1990s: theorem 2.2 ([27]). let ρ be a function modular satisfying the ∆2-condition and let b be a ‖ · ‖ρ-closed subset of lρ. let t : b → b be a single-valued mapping such that ρ(t(f) − t(g)) ≤ kρ(f − g) where f,g ∈ b and k ∈ (0,1). then t has a fixed point if supn(2t n(f0)) < 1. since then, there has been significant work on extending and improving this result further in many ways. ait taleb and hanebaly present some example illustrating that the following result (theorem i-1 of [2]) tends to be more applicable than theorem 2.2: theorem 2.3 ([2]). suppose that xρ is a ρ-complete modular space where ρ is an s̃-convex modular satisfying the ∆2-condition and has the fatou property. © agt, upv, 2021 appl. gen. topol. 22, no. 1 95 f. lael and s. shabanian moreover, assume that b is a ρ-closed subset of xρ and t : b → b is a single-valued mapping such that there are c,k ∈ r+ that c > max{1,k} and ρ(c(t(x) − t(y))) ≤ ks̃ρ(x − y) where x,y ∈ b. then t has a fixed point. however, it should be stressed that theorem 2.2 is not generalized by theorem 2.3. as mentioned in [2], we can ask what if it is mentioned that they are unable to prove whether the conclusion of theorem 2.3 is true if we have c = 1 and 0 < k < 1. we note that our main result, theorem 3.3, replied to this open question, also it generalized theorem 2.2, when wρ(2) < ∞. various extensions were subsequently proposed by various authors, based on different relaxations that require: a recent extension of theorem 2.2 to the cases of correspondence maps appeared in 2006 in theorem 3-1 of [18] as: theorem 2.4 ([18]). let ρ be a convex function modular satisfying the ∆2-type condition, b a nonempty ρ-bounded ρ-closed subset of lρ, and f : b ։ b a ρ-closed valued correspondence such that there exists a constant k ∈ [0,1) that hρ(f(f1),f(f2)) ≤ kρ(f1 − f2), where f1,f2 ∈ b. then f has a fixed point. additionally, in 2009, this result is improved to theorem 2-1 of [34]: theorem 2.5 ([34]). let ρ be a convex modular satisfying ∆2-type condition and b ⊂ lρ be a nonempty ρ-closed ρ-bounded subset of the modular space lρ. then any closed valued correspondence f : b ։ b such that for f1,f2 ∈ b and f3 ∈ f(f1), there is f4 ∈ f(f2) such that ρ(f3 − f4) ≤ kρ(f2 − f1), where k ∈ (0,1), has a fixed point. in both theorems 2.4 and 2.5, it is assumed that the correspondence defined on a ρ-bounded subset of a modular space (x,ρ) with convex modular. in [35] theorem 2-5, the correspondence has ρ-compact set values. theorem 2.6 ([35]). let b be a ρ-bounded subset of ρ-complete space (x,ρ). let f : b ։ b be a correspondence with ρ-compact values that for each x,y ∈ c and z ∈ f(x), there exists w ∈ f(y) such that ρ(z − w) ≤ kρ(x − y), where 2kwρ(2) 2 < 1. then f has a fixed point. our main result, theorem 3.3, is definitely a generalization of theorems 2.4, 2.5, 2.6. in the following sections, we provide certain conditions under which we can guarantee the existence of fixed points for myriad mappings and some strong assumptions such as the convexity of modulars and the ρ-boundedness of the domain of a correspondence are relaxed which can lead to making our theorems much stronger and more applicable. © agt, upv, 2021 appl. gen. topol. 22, no. 1 96 convexity and boundedness relaxation for fixed point theorems in modular spaces 3. main results in this section, we focus on the case of the ρ-complete modular space x and consider f : b ։ b is a correspondence with ρ-closed valued where b is a ρ-closed subset of x. further, we assume ωρ(2) < ∞. also, to ease the notation let us now denote x = (x,ρ). to prove our main results, we make use of the following lemma suggested by the reviewer. lemma 3.1 ([36]). let (x,d) be a b-metric space with s ≥ 1 and {xn} be a b-convergent sequence in x with lim n→∞ xn = x. then for all y ∈ x, s−1d(x,y) ≤ lim n→∞ inf d(xn,y) ≤ lim n→∞ supd(xn,y) ≤ sd(x,y). we also use the following lemma taken from the literature to obtain our main results. lemma 3.2 ([38]). a sequence {xn} in a b-metric space (x,d) is a b-cauchy sequence if there exists k ∈ [0,1) such that d(xn,xn+1) ≤ kd(xn−1,xn), for every n ∈ n. now we can state one of our main results which is an equivalent of nadler’s theorem in [41] in a modular space. we would like to highlight that while the convexity of ρ is required for both theorems 2.4 and 2.5, we show that it can be removed. theorem 3.3. consider k ∈ [0,1) and for every y ∈ b, there exists w ∈ f(y) such that ρ(z − w) ≤ kρ(x − y) for every x ∈ b and z ∈ f(x). then f has a fixed point. proof. take x0 ∈ b and x1 ∈ f(x0). we know from our assumption that for every n ≥ 1 there exists xn+1 ∈ b such that xn+1 ∈ f(xn) and d(xn+1,xn) ≤ kd(xn,xn−1), where d(x,y) = ρ(x − y) is the b-metric induced by the modular ρ. note that by lemma 3.2, {xn} is a b-cauchy sequence in the ρ-complete space b. which means that there exists x ∈ b such that xn → x as n → ∞. on the other hand, from our assumption, it is true that for every xn ∈ f(xn−1), there exists yn ∈ f(x) such that ρ(yn − xn) ≤ kρ(x − xn−1). it implies that limρ(yn −xn) = 0, and as a result we have limd(xn,yn) = 0. by lemma 3.1, s−1 limd(yn,x) ≤ limd(yn,xn). therefore, limd(yn,x) = 0 which means limρ(yn − x) = 0. since yn ∈ f(x) and f(x) is ρ-closed, it concludes our proof. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 97 f. lael and s. shabanian example 3.4. define the modular ρ : ℓ∞(r) → r as follows: ρ((xn)) = sup n∈n |xn|, for every (xn) ∈ ℓ∞(r) and the correspondence f : ℓ∞(r) ։ ℓ∞(r) by f((xn)) = {( 1 i , x1 2 , x2 2 , . . .) : i ∈ n}. so (ℓ∞(r),ρ) is a ρ-complete space and the correspondence f satisfies the contraction of theorem 3.3. indeed, for (xn),(yn) ∈ ℓ∞(r), and (zn) = ( 1 i , x1 2 , x2 2 , . . .) ∈ f((xn)), there is (wn) = ( 1 i , y1 2 , y2 2 , . . .) ∈ f((yn)) such that ρ((zn) − (wn)) = sup n | xn − yn 2 | ≤ 1 2 sup n |xn − yn| = 1 2 ρ((xn) − (yn)) thus all assumptions of theorem 3.3 are fulfilled and f has many fixed points such as {(1 i , 1 2i , 1 4i , . . .) : i ≥ 2}. the following result shows that theorem 3.3 can be even further generalized: theorem 3.5. consider for every x, y ∈ b and z ∈ f(x), there exists w ∈ f(y) such that ρ(z − w) ≤ k max{ρ(x − y),αρ(x − z),αρ(y − w), β 2 (ρ(x − w) + ρ(y − z))}, where α,β ∈ [0,1], and k ∈ [0,1). then f has a fixed point if one of the following assumptions satisfies: i: f is continuous. ii: ρ is continuous i.e. limρ(xn) = ρ(x) as xn → x. iii: kβωρ(2) < 1. proof. let us define a sequence {xn} with x0 ∈ b, x1 ∈ f(x0) and xn+1 ∈ f(xn) such that ρ(xn+1 − xn) ≤ k max{ρ(xn − xn−1),αρ(xn − xn−1), αρ(xn+1 − xn), β 2 ρ(xn−1 − xn+1)},(3.1) for every n ≥ 1. as becomes clear by equation (3.1), the right hand side of this equation is not αρ(xn − xn+1) or αρ(xn − xn−1). now it is easy to see that (3.2) ρ(xn+1 − xn) ≤ k max{ρ(xn − xn−1), βωρ(2) 2 (ρ(xn−1 − xn) + ρ(xn − xn+1))}, now we distinguish the two cases whether the right hand side of equation (3.2) is ρ(xn − xn−1) or βωρ(2) 2 (ρ(xn−1 − xn) + ρ(xn − xn+1)). © agt, upv, 2021 appl. gen. topol. 22, no. 1 98 convexity and boundedness relaxation for fixed point theorems in modular spaces if the former is the case, then using equation (3.2), we have ρ(xn+1 − xn) ≤ kρ(xn − xn−1). the latter leads to ρ(xn+1 − xn) ≤ kβωρ(2) 2 − kβωρ(2) ρ(xn − xn−1). now, we are ready to derive a new upper bound for equation (3.2) as: ρ(xn+1 − xn) ≤ max{k, kβωρ(2) 2 − kβωρ(2) }ρ(xn − xn−1). thus it follows from lemma 3.2 and ρ-completeness of b that there exists x ∈ b such that xn → x as n → ∞. the proof is obviously complete under assumption (i). now assume (ii) holds. we know that for xn, there exists yn ∈ f(x) such that ρ(xn − yn) ≤ k max{ρ(xn−1 − x),αρ(xn−1 − xn), αρ(yn − x), β 2 (ρ(xn−1 − yn) + ρ(x − xn))}.(3.3) by looking at the definition of the ρ-convergent sequences, it becomes clear that lim xn→x ρ(xn − x) = 0 and lim xn→x ρ(xn − xn−1) = 0. now (ii) leads to lim xn→x ρ(xn − yn) = ρ(x − yn). using this and equation (3.3), it is easy to derive that lim xn→x ρ(xn − yn) ≤ k max{ lim xn→x αρ(yn − xn), lim xn→x βρ(yn − xn) 2 } ≤ k lim xn→x ρ(yn − xn), which yields lim xn→x ρ(xn − yn) = 0. now from the facts that f(x) is ρ-closed and ρ(yn − x) ≤ ωρ(2)(ρ(xn − x) + ρ(yn − xn)), we have limn→∞ yn = x ∈ f(x). finally, if assumption (iii) is satisfied, it is possible to repeat the proof presented for assumption (ii) to get equation (3.3). therefore, it follows that ρ(xn − yn) ≤ k max{ρ(xn−1 − x),αρ(xn − xn−1),αρ(yn − x), β 2 (ρ(xn−1 − yn) + ρ(x − xn))}, ≤ k max{αωρ(2)ρ(yn − xn), β 2 (ρ(xn−1 − yn) + ρ(x − xn))}, ≤ k max{αωρ(2)ρ(yn − xn), β 2 [ωρ(2)(ρ(xn − xn−1) + ρ(xn − yn)) + ρ(x − xn)]}, ≤ max{kαωρ(2), kβωρ(2) 2 }ρ(xn − yn). this implies that limn→∞ ρ(xn − yn) = 0, by lemma 3.1, limρ(yn − x) = 0 and then x ∈ f(x). � © agt, upv, 2021 appl. gen. topol. 22, no. 1 99 f. lael and s. shabanian example 3.6. for a function p : n → [1,∞), define the vector space ℓp(·) = {(xn) ∈ r n : σ∞n=1|λxn| p(n) < ∞, for some λ > 0}, and the modular ρ : ℓp(·) → r by ρ((xn)) = σ ∞ n=1|xn| p(n), for every (xn) ∈ ℓp(·). now define the corespondence f : ℓp(·) ։ ℓp(·) by f((xn)) = {( sinx1 2 , x2 3 , · · · , xn n + 1 , . . .),( cosx1 2 , x2 3 , · · · , xn n + 1 , . . .)}, for every (xn) ∈ ℓp(·). for (xn),(yn) ∈ ℓp(·) and (zn) = ( sinx1 2 , x2 3 , · · · , xn n+1 , . . .), there is (wn) = siny1 2 , y2 3 , · · · , yn n+1 , . . .) such that ρ((zn) − (wn)) = | sinx1 − siny1 2 |p(1) + σ∞n=2| xn − yn n + 1 |p(n), ≤ 1 2 max{ρ((xn) − (yn)),ρ((xn) − (zn)),ρ((yn) − (wn)), 1 2 (ρ((xn) − (wn)) + ρ((yn) − (zn)))}. otherwise (zn) = ( cosx1 2 , x2 3 , · · · , xn n+1 , . . .), there is (wn) = cosy1 2 , y2 3 , · · · , yn n+1 , . . .) such that ρ((zn) − (wn)) = | cosx1 − cosy1 2 |p(1) + σ∞n=2| xn − yn n + 1 |p(n), ≤ 1 2 max{ρ((xn) − (yn)),ρ((xn) − (zn)),ρ((yn) − (wn)), 1 2 (ρ((xn) − (wn)) + ρ((yn) − (zn)))}. thus all assumptions of theorem 3.5 are fulfilled and f has a fixed point (0). from now on, for the sake of clearness of notation, we consider ψ to be a continuous and nondecreasing self-map on [0,∞) such that ψ(t) = 0 if and only if t = 0. this notation has been taken from the literature [28] and it was shown that, under mild assumptions, fixed point exists for many families. theorem 3.7. let k ∈ [0,1), α ≥ 0 and for every x,y ∈ b we have hx,y = distρ(x,f(y)) + distρ(y,f(x)) 2wρ(2) . furthermore, suppose that for every y ∈ b there is w ∈ f(y) such that ψ( 1 k ρ(z − w)) ≤ ψ(s(x,y)) + αψ(i(x,y)), for every x ∈ b and z ∈ f(x) where s(x,y) = max{ρ(x − y),distρ(x,f(x)),distρ(y,f(y)) 1 + distρ(x,f(x)) 1 + ρ(x − y) ,hx,y}, © agt, upv, 2021 appl. gen. topol. 22, no. 1 100 convexity and boundedness relaxation for fixed point theorems in modular spaces and i(x,y) = min{distρ(x,f(x)) + distρ(y,f(y)),distρ(x,f(y)),distρ(y,f(x))}. then f has a fixed point, provided that it satisfies one of the following condition: i: f is continuous. ii: ρ is continuous. iii: kwρ(2) < 1. proof. let x0 ∈ x and x1 ∈ f(x0). by assumption, there exists x2 ∈ f(x1) such that ψ( 1 k ρ(x2 − x1)) ≤ ψ(s(x1,x0)) + αψ(i(x1,x0)). thus, one can define a sequence of {xn} in b such that (3.4) ψ( 1 k ρ(xn − xn+1)) ≤ ψ(s(xn−1,xn)) + αψ(i(xn−1,xn)), where xn ∈ f(xn−1). on the other hand, taking into account that limn→∞ i(xn−1,xn) = 0, we have s(xn−1,xn) = max{ρ(xn−1 − xn),distρ(xn−1,f(xn−1)), distρ(xn,f(xn)) 1 + distρ(xn−1,f(xn−1)) 1 + ρ(xn−1 − xn) ,hxn−1,xn}, ≤ max{ρ(xn−1 − xn),ρ(xn − xn+1), ρ(xn−1 − xn+1) + ρ(xn − xn) 2wρ(2) }, = max{ρ(xn−1 − xn),ρ(xn − xn+1)}. the right side of this inequality can be either ρ(xn−1 − xn) or ρ(xn − xn+1). however, the latter follows that ψ( 1 k ρ(xn − xn+1)) ≤ ψ(ρ(xn − xn+1)) + αψ(0), which gives a contradiction ρ(xn − xn+1) ≤ kρ(xn − xn+1), based on the fact that ψ is nondecreasing. therefore, we have ψ( 1 k ρ(xn − xn+1)) ≤ ψ(ρ(xn−1 − xn)) + αψ(0) = ψ(ρ(xn−1 − xn)). it leads to ρ(xn − xn+1) ≤ kρ(xn−1 − xn) for every n ∈ n. lemma 3.2 implies that there exists x ∈ b such that xn → x. now, our goal is to prove x ∈ f(x). obviously, x is a fixed point of f if (i) holds. by considering assumption (ii), it becomes obvious that x ∈ f(x). for xn ∈ f(xn−1), there is qn ∈ f(x) such that ϕ( 1 k ρ(qn − xn)) ≤ ϕ(s(xn−1,x) + lϕ(i(xn−1,x)). © agt, upv, 2021 appl. gen. topol. 22, no. 1 101 f. lael and s. shabanian we have s(xn−1,x) = max { ρ(xn−1 − x),distρ(xn−1,f(xn−1)), distρ(x,f(x)) 1 + distρ(xn−1,f(xn−1)) 1 + ρ(xn−1 − x) , distρ(xn−1,f(x)) + distρ(x,f(xn−1)) 2wρ(2) }, ≤ max{ρ(xn−1 − x),ρ(xn−1 − xn), distρ(x,f(x)) 1 + ρ(xn−1 − xn) 1 + ρ(xn−1 − x) , wρ(2)(distρ(x,f(x)) + ρ(x − xn−1)) + ρ(x − xn) 2wρ(2) }. this implies that lims(xn−1,x) ≤ distρ(x,f(x)). therefore limϕ( 1 k ρ(qn − xn)) ≤ ϕ(distρ(x,f(x))) ≤ ϕ(lim ρ(x − qn)). so lim ρ(qn−xn) ≤ k limρ(x−qn). since ρ is continuous and xn → x, limρ(qn− x) ≤ kρ(x−qn). thus limρ(qn −x) = 0. this implies that, since f(x) is closed and qn ∈ f(x), we have x ∈ f(x). if condition (iii) is provided, since we have limϕ( 1 k ρ(qn − xn)) ≤ ϕ(distρ(x,f(x))), ≤ ϕ(ρ(x − qn)), ≤ ϕ(wρ(2)(ρ(x − xn) + ρ(xn − qn))). therefore lim ρ(xn − qn) ≤ kwρ(2) limρ(xn − qn). now, kwρ(2) < 1 implies limρ(xn − qn) = 0. therefore, by lemma 3.1, qn → x. thus x ∈ f(x). � 4. addressing some open questions and challenges a series of research articles addressing challenges and hitherto open questions in the context of fixed point theory in modular spaces has been presented [2, 49]. our aim is to contribute to a deeper understanding of fixed point theorems in modular spaces and to extend them to further general cases. a way of doing this is to address open problems taken from the literature. for instance, radenović et. al. in [49] considered the following open problem if t : b → b is a single valued mapping such that ρ(t(x)−t(y)) ≤ k max{ρ(x−y),ρ(x−t(x)),ρ(y−t(y)),ρ(x−t(y)),ρ(y−t(x))}, for every x,y ∈ b where b ⊆ x and k ∈ r, then under what constraints does t have a fixed point? and can answer this question under the constraints that t : b → b is a single valued mapping and k ∈ (0, 1 wρ(2)(1+wρ(2)) ). however, there is no answer to this question in the case of multi-valued t or k ≥ 1 wρ(2)(1+wρ(2)) . the open questions that have arisen address open questions outlined in [49] existence of a fixed point for some certain maps: © agt, upv, 2021 appl. gen. topol. 22, no. 1 102 convexity and boundedness relaxation for fixed point theorems in modular spaces existing of a fixed point for t has been successfully shown if k ∈ (0, 1 wρ(2)(1+wρ(2)) ) (given by radenovic et. al. in [49]). interestingly, this question can be reformulated for correspondence, and we prove it with k ∈ [0, 1 2wρ(2) ] in the next theorem. theorem 4.1. let f be a correspondence that for each x, y ∈ b and z ∈ f(x), there exists w ∈ f(y) such that ρ(z − w) ≤ k max{ρ(x − y),ρ(x − z),ρ(y − w),ρ(x − w),ρ(y − z)}, where 2kwρ(2) < 1. then f has a fixed point. proof. we first find x1 ∈ f(x0) for an arbitrary x0 ∈ b. by assumption, for every n ≥ 1 there exists xn+1 ∈ f(xn) such that (4.1) ρ(xn+1 −xn) ≤ k max{ρ(xn −xn−1),ρ(xn+1 −xn),ρ(xn−1 −xn+1),ρ(xn −xn)}, it follow that ρ(xn+1 − xn) ≤ k max{ρ(xn − xn−1),ρ(xn+1 − xn), wρ(2)(ρ(xn−1 − xn) + ρ(xn − xn+1))}, ≤ kwρ(2)(ρ(xn−1 − xn) + ρ(xn − xn+1) which implies that ρ(xn+1 − xn) ≤ k ′ ρ(xn−1 − xn). where k′ = kwρ(2) 1−kwρ(2) . note that k′ is never larger than one since k < 1 2wρ(2) . in addition, from the fact that b is a ρ-complete set and xn is a ρ-cauchy sequence by lemma 3.2, there exists x ∈ b such that xn → x. on the other hand, for every xn, there exists yn ∈ f(x) such that ρ(xn − yn) ≤ k max{ρ(xn−1 − x),ρ(xn−1 − xn),ρ(yn − x),ρ(yn − xn−1), ρ(xn − x)}, ≤ k max{ρ(xn−1 − x),wρ(2)(ρ(yn − xn) + ρ(xn − x)), ρ(xn−1 − xn),wρ(2)(ρ(yn − xn) + ρ(xn − xn−1)),ρ(xn − x)}. hence, it becomes obvious that limρ(xn −yn) = 0 by lemma 3.1, x ∈ f(x). � 5. application to integral inclusions as outlined in the introduction, a modular fixed point theorem can be used for providing sufficient (but not necessary) conditions for finding a real continuous function u defined on [a,b] such that (5.1) u(t) ∈ v(t) + γ ∫ b a g(t,s)g(s,u(s))ds, t ∈ [a,b], where γ is a constant, g : [a,b] × r ։ [a,b] is lower semicontinuous, g : [a,b] × [a,b] → [0,∞) and v : [a,b] → r are given continuous functions. © agt, upv, 2021 appl. gen. topol. 22, no. 1 103 f. lael and s. shabanian for simplicity we introduce the following shorthand notations. we use x = c[a,b] to denote all real continuous functions defined on [a,b], gu : [a,b] ։ [a,b] where gu(s) = g(s,u(s)) and a modular ρ defined on x as ρ(u) = max a≤t≤b | u(t) |2 . it is not difficult to prove that (x,ρ) is a ρ-complete modular space. now the aforementioned integral inclusion problem (5.1) can be reformulated as u is a solution of problem (5.1) if and only if it is a fixed point of f : x ։ x defined as f(u) = {x ∈ x : x(t) ∈ v(t) + γ ∫ b a g(t,s)g(s,u(s))ds, t ∈ [a,b]}. now we show under the following mild assumptions: i: for all x,y ∈ x and wx(t) ∈ gx(t), there exists hy(t) ∈ gy(t) such that |wx(t) − hy(t)| 2 ≤ 1 2s | x(t) − y(t) |2, t ∈ [a,b], ii: max a≤t≤b ∫ b a g2(t,z)dz ≤ 1 b−a , iii: | γ |≤ 1, the correspondence f has a unique fixed point. so, we assume x,y ∈ x and w ∈ f(x) by definition, we have w(t) ∈ v(t) + γ ∫ b a g(t,s)g(s,x(s))ds = v(t) + γ ∫ b a g(t,s)gx(s)ds. by michael’s selection theorem (see theorem 1 in [38]), it follows that there exists a continuous single valued mapping wx(s) ∈ gx(s) that w(t) = v(t) + γ ∫ b a g(t,s)wx(s)ds. according to assumption (i), for wx(s) ∈ gx(s), there is an hy(s) ∈ gy(s) such that |wx(s) − hy(s)| 2 ≤ 1 2s | x(s) − y(s) |2, for all s ∈ [a,b]. we define h(t) = v(t) + γ ∫ b a g(t,s)hy(s)ds which means that h(t) ∈ v(t) + γ ∫ b a g(t,s)gy(s)ds. © agt, upv, 2021 appl. gen. topol. 22, no. 1 104 convexity and boundedness relaxation for fixed point theorems in modular spaces therefore h ∈ f(y). using the cauchy-schwarz inequality and conditions (i-iii), we have ρ(w − h) = max a≤t≤b |w(t) − h(t)| 2 , = max a≤t≤b | v(t) + γ ∫ b a g(t,s)wx(s)ds − (v(t) + γ ∫ b a g(t,s)hy(s)ds) | 2 , = | γ |2 max a≤t≤b | ∫ b a g(t,s)(wx(s) − hy(s))ds | 2, ≤ | γ |2 max a≤t≤b { ∫ b a g2(t,s)ds ∫ b a | wx(s) − hy(s) | 2 ds } , = | γ |2 { max a≤t≤b ∫ b a g 2(t,s)ds } . { ∫ b a | wx(s) − hx(s) | 2 ds } , ≤ | γ |2 b − a { 1 2s ∫ b a | x(s) − y(s) |2 ds } , ≤ | γ |2 2s(b − a) ∫ b a max a≤s≤b | x(s) − y(s) |2 ds, = | γ |2 2s max a≤s≤b | x(s) − y(s) |2, = 1 2s ρ(x − y). theorem 3.3 implies that f has a unique fixed point u ∈ x, that is, the integral inclusion (5.1) has a solution which belongs to c[a,b]. 6. conclusion our main results show that strong assumptions such as convexity and boundedness of modulars in fixed point results for contractive correspondence and single-valued mappings can be relaxed by making use of some ties between modular and b-metric spaces. our approach in this work includes a unifying view on fixed point results to yield some assumptions which are more likely to hold in practice and reformulations as particular normed vector space problems are no longer required. in particular, a generalized version of nadler’s theorem along with an application 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[58] t. van an, l. quoc tuyen and n. van dung, stone-type theorem on b-metric spaces and applications, topology and its applications 185–186 (2015), 50–64. © agt, upv, 2021 appl. gen. topol. 22, no. 1 108 @ appl. gen. topol. 22, no. 2 (2021), 321-330doi:10.4995/agt.2021.14589 © agt, upv, 2021 periodic points of solenoidal automorphisms in terms of inverse limits sharan gopal and faiz imam department of mathematics, bits-pilani, hyderabad campus, india. (sharanraghu@gmail.com, mefaizy@gmail.com) communicated by f. balibrea abstract in this paper, we describe the periodic points of automorphisms of a one dimensional solenoid, considering it as the inverse limit, lim ← k (s1, γk) of a sequence (γk) of maps on the circle s 1 . the periodic points are discussed for a class of automorphisms on some higher dimensional solenoids also. 2010 msc: 54h20; 37c25; 11b50; 37p35; 22c05. keywords: solenoid; periodic points; inverse limits; pontryagin dual. 1. introduction we consider a dynamical system of the form (g,f), where g is a compact group and f is an automorphism of g. the study of dynamics involves the eventual behaviour of trajectories of its points i.e., the sequences (fn(x))∞n=0, where x ∈ g and fn = f ◦ f ◦ ... ◦ f (n times for n ∈ n) and f0 is the identity map on g. a point x ∈ g is said to be periodic with a period n if there is an n ∈ n such that fn(x) = x. the sets of periods and periodic points of a family of dynamical systems are well studied in literature. see for instance, [3], [6], [7], [8], [12], [17] and [18]. in this paper, we give a description of the sets of periodic points of automorphisms of a one dimensional solenoid i.e., we describe the set p(f) = {x ∈ σ : x is a periodic point of f}, where σ is a one received 07 november 2020 – accepted 29 may 2021 http://dx.doi.org/10.4995/agt.2021.14589 s. gopal and f. imam dimensional solenoid and f is an automorphism of σ. it is then extended to some automorphisms of certain higher dimensional solenoids. solenoids are extensively studied in literature. some of the papers consider solenoids as dual groups of subgroups of qn,n ∈ n, while others consider them as inverse limits of certain maps on the n-dimensional torus, tn. in [19], it is shown that an ergodic automorphism of a solenoid is measure theoretically isomorphic to a bernoulli shift. the papers [1], [5] and [14] discuss about the structure of a solenoid, whereas [13] describes the structure of group of automorphisms of a solenoid. the articles [2] and [11] calculate the entropy and the zeta function respectively, for an automorphism of a solenoid. the papers [9] and [10] consider the flows on higher dimensional solenoids. we use results from [9] to describe the sets of periodic points of some automorphisms on certain higher dimensional solenoids. there are articles on counting the number of periodic points of a dynamical system; this forms a crucial part in defining the zeta function. the number of periodic points of any given period for some continuous homomorphisms of a one dimensional solenoid was discussed in [15]. we show that our description of periodic points of one dimensional solenoidal automorphisms is in accordance with this result. a characterization of sets of periodic points for automorphisms of a one dimensional solenoid was given in [12], where it was described in terms of adeles, a number theoretic concept. it was based on a description of subgroups of q. it may however be noted that this characterization for one dimensional solenoids may not be extended to higher dimensions with similar ideas, as there is no neat description of subgroups of qn for n > 1. we now describe the sets of periodic points, again for automorphisms of one dimensional solenoids in a different manner, namely in terms of inverse limits and then use it to describe the periodic points of some automorphisms on certain higher dimensional solenoids that are inverse limits of sequences of maps on tn,n > 1. in all these cases, we show that the set of periodic points of a given period is the inverse limit of the same maps (that define the solenoid) restricted to a subgroup of tn. this may help in giving a characterization of periodic points for automorphisms of other higher dimensional solenoids also. 1.1. definitions and notations. a solenoid is a compact connected finite dimensional abelian group. an equivalent interpretation is that, a topological group σ is an n−dimensional solenoid if and only if its pontryagin dual σ̂ is (isomorphic to) a subgroup of the discrete additive group qn and contains zn (see [16]). thus, a one dimensional solenoid is a topological group whose dual is a subgroup of q and contains z. definition 1.2 gives an equivalent description of a one dimensional solenoid σ as the inverse limit of a sequence of maps on the circle s1, where z $ σ̂ ⊆ q. in section 3, we consider ndimensional solenoids, for n > 1, which are inverse limits of sequences of maps on tn. we use the notations z, n, n0 and p to denote the sets of integers, positive integers, non-negative integers and the prime numbers respectively. © agt, upv, 2021 appl. gen. topol. 22, no. 2 322 periodic points of solenoidal automorphisms in terms of inverse limits for a,b ∈ z, we use the notations a | b, if a divides b; else, we write a 6 | b. to represent a sequence (a1,a2, · · · ) of positive integers, besides the customary notations (ak) ∞ k=1 and (ak), we also use a single capital letter. for instance, we write a = (ak) = (ak) ∞ k=1 = (a1,a2, · · · ). we write diag [m1,m2, · · · ,mn] for an n × n diagonal matrix with m1,m2, · · · ,mn on the principal diagonal. definition 1.1. let xk be a topological space for each k ∈ n0 and fk : xk → xk−1 be a continuous map for each k ∈ n. then the subspace of ∞∏ k=0 xk defined as lim ← k (xk,fk) = {(xk) ∈ ∞∏ k=0 xk : xk−1 = fk(xk),∀k ∈ n} is called the inverse limit of the sequence of maps (fk). definition 1.2. let a = (a1,a2, · · · ) be a sequence of integers such that ak ≥ 2 for every k ∈ n. the solenoid corresponding to the sequence a, denoted by σa, is defined as σa = {(xk) ∈ (s 1)(n0) : xk−1 = akxk (mod1) for every k ∈ n}. in other words, the one dimensional solenoid σa is the inverse limit, lim ← k (s1,γk), where γk : s 1 → s1 is defined as γk(x) = akx(mod1). 2. one dimensional solenoids the descriptions of a one dimensional solenoid as an inverse limit and as the dual group of a subgroup of q are very closely related. the dual of a one dimensional solenoid σa, where a = (ak) is isomorphic to the subgroup of q generated by { 1 a1a2···ak : k ∈ n}. now, a subgroup of q is characterized by a sequence, called the height sequence, indexed by prime numbers and with values in n0 ∪ {∞}. we will now discuss about this sequence and establish a relation between the terms of this sequence and the integers ak’s. one may refer to [4] for more details about the structure of subgroups of q. let s ⊆ q and x ∈ s. for a p ∈ p , the p−height of x with respect to s, denoted by h (s) p (x) is defined as the largest non-negative integer n, if it exists, such that x pn ∈ s; otherwise, define h (s) p (x) = ∞. thus, we have a sequence (h (s) p (x)), p ranging over prime numbers in the usual order, with values in n0 ∪ {∞}. we call such sequences as height sequences. if (up) and (vp) are two height sequences such that up = vp for all but finitely many primes and up = ∞ ⇔ vp = ∞, then they are said to be equivalent. if s is a subgroup of q, then there is a unique height sequence (up to equivalence) associated to all non-zero elements of s. also, two subgroups of q are isomorphic if and only if their associated height sequences are equivalent. given a subgroup s of q, for every p ∈ p , we assign an element n(s)p of n0 ∪ {∞} as follows. let qp and zp denote the field of p−adic numbers and the ring of p−adic integers respectively and |u|p denote the p−adic norm of u ∈ qp. then define n (s) p = sup{h (s) p (x) : x ∈ s ∩ z∗p}, where z ∗ p is the multiplicative group {x ∈ zp : |x|p = 1}. now, the information whether n (s) p © agt, upv, 2021 appl. gen. topol. 22, no. 2 323 s. gopal and f. imam is finite or not, for a given p, is going to play a crucial role in our discussion. so, we define d (s) ∞ = {p ∈ p : n (s) p = ∞}. we will use the notations n (s) p and d (s) ∞ , as defined here, throughout this paper. we now have the following relation between the sequences (n (s) p ) and a, where s is the dual of σa. proposition 2.1. let σa be a one dimensional solenoid and s = σ̂a, where a = (ak). let p ∈ p and np = n (s) p . then, (1) p ∈ d (s) ∞ if and only if for every j ∈ n, there exists a k ∈ n such that pj|a1a2 · · ·ak. (2) if p /∈ d (s) ∞ , then np is the largest non-negative integer such that pnp|a1a2 · · ·ak for some k. proof. (1) suppose p ∈ d (s) ∞ . since np = ∞, for any j ∈ n, there exists an x ∈ s ∩ z∗p with h (s) p (x) > j. now, x ∈ z∗p implies that x = a b , where a,b ∈ z and (a,p) = (b,p) = 1. also, h(s)p (x) > j implies that xpj = a pj b ∈ s. but, s = { i a1a2···ak : i ∈ z,k ∈ n } . thus, a pj b = i a1a2···ak for some i ∈ z and k ∈ n. then, we have aa1a2 · · ·ak = ipjb implying that pj|a1a2 · · ·ak. for the converse, let j ∈ n. then, there exists a k ∈ n such that a1a2 · · ·ak = p ji for some i ∈ n. this implies that 1 pj = i a1a2···ak ∈ s and thus h (s) p (1) ≥ j. since j is chosen arbitrarily and 1 ∈ s ∩ z∗p, we get, np = ∞ i.e., p ∈ d (s) ∞ . (2) suppose p /∈ d (s) ∞ . then, np = max{h (s) p (x) : x ∈ s ∩ z∗p}. say np = h (s) p (x0) for some x0 ∈ s ∩ z∗p i.e., x0 pnp ∈ s. let x0 = u0 v0 , for some u0,v0 ∈ z. then (u0,p) = (v0,p) = 1. now, x0 pnp ∈ s implies that u0 pnp v0 = i a1a2···ak for some i ∈ z and k ∈ n i.e, u0a1a2 · · ·ak = ipnpv0 and hence pnp|a1a2 · · ·ak. if possible, let l > np such that p l|a1a2 · · ·aj for some j. but then, a1a2 · · ·aj = p li′ for some i′ ∈ n implying that 1 pl = i ′ a1a2···aj ∈ s and thus h (s) p (1) ≥ l > np which is a contradiction. therefore, np is the largest integer such that pnp|a1a2 · · ·ak for some k. � the following corollary follows from the above proposition. corollary 2.2. let σa, s and d (s) ∞ be defined as above. then, for a p ∈ p, p ∈ d (s) ∞ if and only if p divides infinitely many ak’s. if f is an automorphism of a one dimensional solenoid σ, then its dual is an automorphism of a subgroup of q and thus, it is multiplication by a non-zero rational number, say α β and for any (xk) ∈ σ, f((xk)) = ( α β xk(mod1)). we say that f is induced by α β . it is known that f is ergodic if and only if α β 6= ±1. further, we can assume that a = (βbk), where each bk is a positive integer © agt, upv, 2021 appl. gen. topol. 22, no. 2 324 periodic points of solenoidal automorphisms in terms of inverse limits coprime to β. in this case, we can write f((xk)) = (αb1x1,αb2x2, ...) for each (xk) ∈ σ(βbk). see [19] for all these details about automorphisms. we now state and prove our main results, namely the description of periodic points (theorem 2.5) and the number of periodic points (theorem 2.7) of an automorphism of a one dimensional solenoid. before that, the following proposition describes the elements of a one dimensional solenoid with rational coordinates, in terms of the prime factors of ak’s and the succeeding proposition shows that a periodic point should have only rational coordinates. proposition 2.3. let σa be a one dimensional solenoid where a = (ak) and (xk) = ( uk vk ) ∈ σa ∩ qn0, where uk,vk ∈ z such that (uk,vk) = 1. for a p ∈ p, denote |vk|p = 1 pck , for every k ≥ 0 and let |ak|p = 1 pdk , for every k ≥ 1. if h is the least integer such that ch > 0, then ck = ch + dh+1 + dh+2 + · · · + dk, for every k > h. proof. it follows from the definition of a one dimensional solenoid that uh vh = ah+1ah+2 · · ·ak uk vk + j for some j ∈ z. since ch > 0, it follows that (uh,p) = (uk,p) = 1. then, we can find positive integers a ′ h+1,a ′ h+2, · · · ,a ′ k,v ′ k and v ′ h, each of which is coprime to p, such that uh pch v ′ h = p dh+1+dh+2+···dk a ′ h+1a ′ h+2···a ′ kuk pck v ′ k + j ⇒ pckv ′ kuh = p ch+dh+1+···dkv ′ ha ′ h+1 · · ·a ′ kuk + jp ck+chv ′ kv ′ h ⇒ pck ( v ′ kuh − p chjv ′ kv ′ h ) = pch+dh+1+···dkv ′ ha ′ h+1 · · ·a ′ kuk now, since ch > 0, p does not divide ( v ′ kuh − p chjv ′ kv ′ h ) . thus, ck = ch + dh+1 + · · ·dk for every k > h. � proposition 2.4. let σa be a one dimensional solenoid and s = σ̂a, where a = (ak). if (xk) is periodic in (σa,φ), where φ is an automorphism of σa induced by α β , then xk ∈ q for every k ∈ n0. further, for any p ∈ d (s) ∞ , we have |xk|p ≤ 1 for every k ∈ n0. proof. say φl ((xk)) = (xk) for some l ∈ n. then, for any k ∈ n0, αl βl xk = xk + jk for some jk ∈ z and thus xk ∈ q. let xk = uk vk , where uk, vk ∈ z and (uk,vk) = 1. then, (α l − βl)uk = β lvkjk for every k ≥ 0. for a prime number p, let us now denote |vk|p = 1 pck , for every k ≥ 0 and |ak|p = 1 pdk , for every k ≥ 1. let p ∈ d (s) ∞ . then, by corollary 2.2, p|ak for infinitely many k and thus dk > 0 for infinitely many k. suppose there exists an r ∈ n0 such that p|vr. then, cr > 0 and (α l − βl)ur = β lvrjr implies that p cr|(αl − βl). now from proposition 2.3, cr+k = ch+dh+1+· · ·+dr+dr+1+· · ·+dr+k, where h is the least integer such that ch > 0. then, h ≤ r and cr+k = cr +dr+1 +dr+2 +· · ·+dr+k. again, since (αl−βl)ur+k = α lvr+kjr+k for every k ≥ 0, we get p cr+dr+1+···+dr+k| © agt, upv, 2021 appl. gen. topol. 22, no. 2 325 s. gopal and f. imam (αl − βl). this is a contradiction, as infinitely many of dr+1,dr+2, · · · are nonzero. hence, p 6 | vk for any k. therefore, |xk|p ≤ 1 for every k ≥ 0. � in the following theorem about the set of periodic points of the dynamical system(σ(ak), α β ), we assume that ak = βbk, where each bk is a positive integer coprime to β. as noted already, there is no loss of generality in assuming this (see [19]). theorem 2.5. let φ be an automorphism of a one dimensional solenoid σa induced by α β , where a = (βbk), each bk being co-prime to β. for each l ∈ n, define ul = ⋂ p∈p ( 1 p ep,l zp ∩ q ∩ s1 ) , where pep,l = 1 |αl−βl|p . if γk,l : ul → ul is the map defined as γk,l(x) = βbkx(mod 1) for each k ∈ n and l ∈ n, then p(φ) = ∞⋃ l=1 lim ← k (ul,γk,l). proof. let (xk) be a periodic point with a period l. then, xk ∈ q for every k ≥ 0; say xk = uk vk , where uk, vk ∈ z such that (uk,vk) = 1. again, for every prime p, let |vk|p = 1 pck , for every k ≥ 0. now, φl ((xk)) = (xk) implies that (αl − βl)uk = β lvkjk for some jk ∈ z. since pck|vk, it follows that pck|(αl − βl) and thus ck ≤ ep,l. we can now write xk = 1 p ep,l .p ep,l−ck .uk v ′ k , for some v ′ k ∈ z such that (v ′ k,p) = 1. it then follows that xk ∈ 1 p ep,l zp, because |p ep,l−ck .uk v ′ k |p ≤ 1 p ep,l−ck ≤ 1. since p was chosen arbitrarily, we conclude that xk ∈ ul, for every k ≥ 0. on the other hand, let (xk) ∈ lim ← k (ul,γk,l) for some l ∈ n. say xk = uk vk , where uk, vk ∈ z such that (uk,vk) = 1. write vk = ∏ p|vk pcp, for some cp ∈ n. then, for any p|vk, |xk|p = p cp. also, |xk|p ≤ p ep,l, for any p ∈ p . thus, cp ≤ ep,l and hence vk|(α l − βl). therefore, α l−βl vk ∈ z, for every k. then, φl((xk)) − (xk) = ( (αl − βl)bk+1bk+2...bk+lxk+l ) = (0) implying that (xk) is periodic. � remark 2.6. the set of periodic points of period l is equal to lim ← k (ul,γk,l). here ul is a subgroup of s 1 and the map γk,l is the restriction of γk to ul, where γk is a map on s 1 such that σ(βbk) = lim← k (s1,γk). the following theorem about the number of periodic points, which follows from the above description, is in accordance with a similar result in [15]. theorem 2.7. let φ be an automorphism of a one dimensional solenoid σa induced by α β and for every l ∈ n, let ep,l = 1 |αl−βl|p . then the number of periodic points of φ with a period l is ∏ p/∈d (s) ∞ pep,l. © agt, upv, 2021 appl. gen. topol. 22, no. 2 326 periodic points of solenoidal automorphisms in terms of inverse limits proof. since αl − βl ∈ z, ep,l is positive only for finitely many primes. thus, there is a finite subset f of p \ d (s) ∞ such that for a p /∈ d (s) ∞ , ep,l 6= 0 if and only if p ∈ f . therefore ∏ p/∈d (s) ∞ pep,l = ∏ p∈f pep,l. we first claim that (xk) is periodic with a period l if and only if for every k ∈ n0, xk = uk vk , where uk, vk ∈ z, 0 ≤ uk < vk and vk = ∏ p∈f pfp,k with 0 ≤ fp,k ≤ ep,l. if φl ((xk)) = (xk), then for every k ∈ n0, xk ∈ 1 p ep,l zp ∩ q, for every p ∈ p . let xk = uk vk for some uk, vk ∈ z such that (uk,vk) = 1. now, xk ∈ 1 p ep,l zp implies that |xk|p ≤ p ep,l, for every p. from proposition 2.4, if p ∈ d (s) ∞ , then p 6 | vk. also, for a prime p not in f , ep,l = 0 implies that p 6 | vk. thus, the prime factorisation of vk = ∏ p∈f pfp,k for some 0 ≤ fp,k ≤ ep,l. since xk ∈ [0,1), we conclude that 0 ≤ uk < vk. conversely, if xk = uk vk , where uk and vk satisfy the given conditions, then |xk|p ≤ 1, for p /∈ f and |xk|p ≤ p fp,k for p ∈ f . in any case |xk|p ≤ p ep,l and thus xk ∈ ul. hence the claim follows. for a p ∈ f , let |ak|p = 1 pdk , for every k ∈ n. as this dk depends on p we will denote dk = d (p) k . again, there are at most finitely many k ∈ n for which d (p) k > 0, as f ⊆ p \ d (s) ∞ ; let these positive integers be denoted by d (p) k1 ,d (p) k2 , ...,d (p) kα(p) , where α(p) ∈ n0. further, assume that k1 < k2 < ... < kα(p). let k = max{kα(p) : p ∈ f}, if kα(p) > 0 for at least some p ∈ f ; otherwise, define k = 0. then, d (p) k = 0 for every k > k and for every p ∈ f . let (xk) ∈ σa be periodic; say xk = uk vk , where uk, vk ∈ z such that (uk,vk) = 1. we have xk = uk vk , where 0 ≤ uk < vk and vk = ∏ p∈f pfp,k with 0 ≤ fp,k ≤ ep,l. for any k < k, the value of xk is uniquely determined by xk, as xk = ak+1ak+2...akxk (mod 1). now, let k > k. it follows from proposition 2.3 that vk = vk. also, xk = ak+1...akxk (mod 1) i.e., uk vk = ak+1...ak uk vk + j for some j ∈ z. by denoting ak+1...ak = gk and using the fact that vk = vk, we have uk vk = gk uk vk +j. since d (p) k = 0 for any k > k and every p ∈ f , it follows that p 6 | gk for any p ∈ f . having defined uk vk , the distinct possible values for uk vk are uk vk = uk gkvk − j gk , where j ∈ {0,1, ...,gk −1}. consider two such values, say u (1) k vk = uk gkvk − j1 gk and u (2) k vk = uk gkvk − j2 gk for some j1, j2 ∈ {0,1, ...gk − 1}. then, u (1) k −u (2) k vk = j2−j1 gk and thus gk ( u (1) k − u (2) k ) = vk (j2 − j1). now, if j1 6= j2, then |j2 − j1| < gk and thus gk 6 | (j2 − j1). but then, there will be a prime p such that p | gk and p | vk. on one hand, p | gk implies that p /∈ f . on the other hand, p | vk implies that p | αl − βl and also p /∈ d (s) ∞ , which means that p ∈ f leading © agt, upv, 2021 appl. gen. topol. 22, no. 2 327 s. gopal and f. imam to a contradiction. hence, j1 = j2 i.e., u (1) k = u (2) k . thus, there is only one possible value for xk. thus, a periodic point (xk) is uniquely determined by the coordinate xk. now, since 0 ≤ fp,k ≤ ep,l, the possible values of xk are i∏ p∈f p ep,l , where 0 ≤ i < ∏ p∈f pep,l. thus, the theorem follows. � 3. n-dimensional solenoids we now extend our result about periodic points to some automorphisms of certain higher dimensional solenoids. though this seems to be a small class, the reason for considering it is that the result follows immediately from what we have shown for one dimensional case. the higher dimensional solenoids that we are going to consider are isomorphic to products of one dimensional solenoids, as described in [9]. we mention here some notations, definitions and results from this paper that are needed to discuss our result. for a positive integer n > 1, let πn : rn → tn be the homomorphism defined as πn((x1,x2, ...,xn)) = (x1 (mod 1),x2 (mod 1), ...,xn (mod 1)). let m = (mk) ∞ k=1 = (m1,m2, ...) be a sequence of n × n matrices with integer entries and non-zero determinant. then, the n−dimensional solenoid ∑ m is defined as ∑ m = {(xk) ∈ (t n)n0 : πn(mkxk) = xk−1 for every k ∈ n}. in other words, ∑ m = lim← k (tn,δk), where δk : tn → tn is defined as δk(x) = πn(mkx) if φ is an automorphism of ∑ m, then there is a matrix l ∈ gl(n,q) such that φ((xk)) = (π n(lxk)). we say that φ is induced by the matrix l. now, consider n sequences of positive integers a1 = (a 1 1,a 1 2, ...), a2 = (a 2 1,a 2 2, ...), ...... an = (a n 1 ,a n 2 , ...). then define the sequence m = (mk) of matrices as mk = diag[a 1 k,a 2 k, ...,a n k]. these sequences of positive integers and matrices give rise to n one-dimensional solenoids and an n−dimensional solenoid. the following lemma from [9] gives a connection between these. lemma 3.1. the map η : ∏n i=1 ∑ ai → ∑ m given by η((x 1 k) ∞ k=1,(x 2 k) ∞ k=1, . . . . . . ,(xnk ) ∞ k=1) = ((x 1 1,x 2 1, . . . ,x n 1 ),(x 1 2,x 2 2, . . . ,x n 2 ), . . . ,(x 1 k,x 2 k, . . . ,x n k), . . .) is a topological isomorphism. we reserve these symbols ai, i = 1,2, ...,n for the sequences of positive integers and mk, k ∈ n for the corresponding diagonal matrices as described above. now, let φ be an automorphism of ∑ m induced by a diagonal matrix, say d = diag[α1 β1 , α2 β2 , ..., αn βn ]. then for each i, αi βi induces an automorphism of the one dimensional solenoid ∑ ai , say ψi. again, by following [19], we assume that ai = (βib i k) for some suitable sequence (b i k) of positive integers. then, the map ψ : ((x1k) ∞ k=1.(x 2 k) ∞ k=1, . . . ,(x n k ) ∞ k=1) 7→ (ψ1((x 1 k) ∞ k=1),ψ2((x 2 k) ∞ k=1), . . . ,ψn((x n k ) ∞ k=1)) is an automorphism of ∏n i=1 ∑ ai . it is easy to see that η ◦ ψ = φ ◦ η. thus, we have the following proposition. proposition 3.2. ( ∏n i=1 ∑ ai ,ψ) is conjugate to ( ∑ m,φ). we now state and prove a theorem regarding the periodic points. © agt, upv, 2021 appl. gen. topol. 22, no. 2 328 periodic points of solenoidal automorphisms in terms of inverse limits theorem 3.3. for each l ∈ n, define vl = ∏n i=1 ( ⋂ p∈p ( 1 p ep,l,i zp ∩ q ∩ s1 )) , where pep,l,i = 1 |αl i −βl i |p . if δk,l : vl → vl is the map defined as δk,l(x) = πn(mkx) for each k ∈ n and l ∈ n, then p(φ) = ∞⋃ l=1 lim ← k (vl,δk,l). proof. let pl(φ) and pl(ψ) be the sets of periodic points of φ and ψ respectively, with a period l ∈ n. since η is a conjugacy from ( ∏n i=1 ∑ ai ,ψ) to ( ∑ m,φ), it follows that pl(φ) = η (pl(ψ)). but pl(ψ) = ∏n i=1 pl(ψi), where ψi is the automorphism of ∑ ai induced by αi βi . thus by theorem 2.5, pl(ψ) = ∏n i=1 { (xik) ∞ k=1 ∈ ∑ ai : xik ∈ q and |x i k|p ≤ 1 p ep,l,i for every p ∈ p } . then, pl(φ) = {( (x1k,x 2 k, · · ·x n k ) )∞ k=1 ∈ ∑ m : x i k ∈ q and |x i k|p ≤ 1 p ep,l,i for every p ∈ p } = lim ← k (vl,δk,l). thus, p(φ) = ∞⋃ l=1 lim ← k (vl,δk,l). � remark 3.4. the set of periodic points of φ with a period l is equal to lim ← k (vl,δk,l). here, vl is a subgroup of tn and δk,l is the restriction of δk to vl, where each δk is a map on tn such that ∑ m = lim← k (tn,δk). 4. conclusion the periodic points of an automorphism of a one dimensional solenoid are described here. there are papers that discuss the number of periodic points or in general the zeta function of such automorphisms, whereas this paper gives an explicit description of these points. the paper [12], on the other hand, describes the sets of periodic points using adeles, but these ideas may not be useful for higher dimensional solenoids. here, we have extended this result to certain automorphisms of higher dimensional solenoids also. hence, the present description in terms of inverse limits may be helpful in more general cases. acknowledgements. both the authors thank science and engineering research board, a statutory body of department of science and technology, govt. of india for funding this work as a part of the project ecr/2017/000741. references [1] j. m. aarts and r. j. fokkink, the classification of solenoids, proc. amer. math. soc. 111 (1991), 1161–1163. [2] l. m. abramov, the entropy of an automorphism of a solenoidal group, theory of probability and its applications 4 (1959), 231–236. [3] k. ali akbar, v. kannan, s. gopal and p. chiranjeevi, the set of periods of periodic points of a linear operator, linear algebra and its applications 431 (2009), 241–246. © agt, upv, 2021 appl. gen. topol. 22, no. 2 329 s. gopal and f. imam [4] d. m. arnold, finite rank torsion free abelian groups and rings, vol. 931, lecture notes in mathematics. springer-verlag, berlin, 1982. [5] r. h. bing, a simple closed curve is the only homogeneous bounded plane continuum that contains an arc, canad. j. math. 12 (1960), 209–230. [6] l. block, periods of periodic points of maps of the circle which have a fixed point, proc. amer. math. soc. 82 (1981), 481–486. [7] r. bowen and j. franks, the periodic points of maps of the disk and the interval, topology 15 (1976), 337–342. [8] p. chiranjeevi, v. kannan and s. gopal, periodic points and periods for operators on hilbert space, discrete and continuous dynamical systems 33 (2013), 4233–4237. [9] a. clark, linear flows on κ-solenoids, topology and its applications 94 (1999), 27–49. [10] a. clark, the rotation class of a flow, topology and its applications 152 (2005), 201–208. [11] j. w. england and r. l. smith, the zeta function of automorphisms of solenoid groups, j. math. anal. appl. 39 (1972), 112–121. [12] s. gopal and c. r. e. raja, periodic points of solenoidal automorphisms, topology proceedings 50 (2017), 49–57. [13] j. keesling, the group of homeomorphisms of a solenoid, trans. amer. math. soc. 172 (1972), 119–131. [14] m. c. mccord, inverse limit sequences with covering maps, trans. amer. math. soc. 114 (1965), 197–209. [15] r. miles, periodic points of endomorphisms on solenoids and related groups, bull. lond. math. soc. 40 (2008), 696–704. [16] s. a. morris, pontryagin duality and the structure of locally compact abelian groups, london mathematical society lecture note series, no. 29, cambridge univ. press, 1977. [17] a. n. sharkovsky, coexistence of cycles of a continuous map of the line into itself, ukrain. mat. zh. 16 (1964), 61–71. (russian); english translation: international journal of bifurcation and chaos in appl. sci. engg. 5 (1995), 1263–1273. [18] t. k. subrahmonian moothathu, set of periods of additive cellular automata, theoretical computer science 352 (2006), 226–231. [19] a. m. wilson, on endomorphisms of a solenoid, proc. amer. math. soc. 55 (1976), 69–74. © agt, upv, 2021 appl. gen. topol. 22, no. 2 330 @ appl. gen. topol. 20, no. 1 (2019), 223-230doi:10.4995/agt.2019.10714 © agt, upv, 2019 a non-discrete space x with cp(x) menger at infinity angelo bellaa and rodrigo hernández-gutiérrezb a department of mathematics and computer science, university of catania, cittá universitaria, viale a. doria 6, 95125 catania, italy (bella@dmi.unict.it) b departamento de matemáticas, universidad autónoma metropolitana campus iztapalapa, av. san rafael atlixco 186, col. vicentina, iztapalapa, 09340, mexico city, mexico (rodrigo.hdz@gmail.com) communicated by á. tamariz-mascarúa abstract in a paper by bella, tokgös and zdomskyy it is asked whether there exists a tychonoff space x such that the remainder of cp(x) in some compactification is menger but not σ-compact. in this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a menger ultrafilter. 2010 msc: primary 54d20; secondary 54a35; 54c35; 54d40; 54d80; 54h11. keywords: menger spaces; non-meager p-filter; pointwise convergence topology. 1. introduction a space x is called menger if for every sequence {un ∶ n ∈ ω} of open covers of x one may choose finite sets vn ⊂ un for all n ∈ ω in such a way that ⋃{vn ∶ n ∈ ω} covers x. given a property p, a tychonoff space x will be called p at infinity if βx ∖x has p. let x be a tychonoff space. it is well-known that x is σ-compact at infinity if and only if x is čech-complete. also, henriksen and isbell proved in [7] that x is lindelöf at infinity if and only if x is of countable type. received 14 september 2018 – accepted 24 january 2019 http://dx.doi.org/10.4995/agt.2019.10714 a. bella and r. hernández-gutiérrez moreover, the menger property implies the lindelöf property and is implied by σ-compactness. so it was natural for the authors of [2] to study when x is menger at infinity. later, the authors of [4] study when a topological group is menger, hurewicz and scheepers at infinity. the hurewicz and scheepers properties are other covering properties that are stronger than the menger property and weaker than σ-compactness (see the survey [12] by boaz tsaban). essentially, [4] has two main results. theorem 1.1 ([4, theorem 1.3]). if g is a topological group and βg ∖ g is hurewicz, then βg∖g is σ-compact. theorem 1.2 ([4, theorem 1.4]). there exists a topological group g such that βg∖g is scheepers and not σ-compact if and only if there exists an ultrafilter u on ω such that, considered as a subspace of p(ω) with the cantor set topology, u is scheepers. the last section of [4] considers the specific case of the topological group cp(x) consisting of all continuous real-valued functions defined on x, with the topology of pointwise convergence. it is shown that if cp(x) is menger at infinity, then it is first countable and hereditarily baire. it is a well-known result that cp(x) is čech-complete (equivalently, σ-compact at infinity) if and only if x is countable and discrete (see [1, i.3.3]). so the authors of [4] made the natural conjecture that cp(x) is menger at infinity if and only if cp(x) is σ-compact at infinity ([4, question 6.1]). their conjecture is equivalent to the statement that if cp(x) is menger at infinity, then x is countable and discrete. in this paper we disprove this conjecture. theorem 1.3. it is consistent with zfc that there exists a regular, countable, non-discrete space x such that cp(x) is menger at infinity. our proof of theorem 1.3 uses filters with a special property that is immediately satisfied by menger ultrafilters. see theorem 3.2 for the exact property we use. according to [5, theorem 3.5], menger filters are precisely those called canjar filters. also, by [6, proposition 2], d = c implies there is a canjar (thus, menger) ultrafilter. however, a characterization of canjar ultrafilters given in [6] implies that a menger ultrafilter is a p-point. thus, menger ultrafilters consistently do not exist. however, we do not know whether there are filters in zfc that satisfy the conditions we need. see sections 3 and 4 for a more thorough explanation and concrete open questions. 2. preliminaries and notation 2.1. the menger property. the menger property has been thoroughly studied. we state some well-known facts below: (i) [9, theorem 2.2] every σ-compact set is menger. © agt, upv, 2019 appl. gen. topol. 20, no. 1 224 a non-discrete space x with cp(x) menger at infinity (ii) [9, theorem 3.1] if x is menger and y ⊂ x is closed, then y is menger. (iii) if x is menger and k is σ-compact, then x ×k is also menger. (iv) [9, theorem 3.1] the continuous image of a menger space is also menger. (v) the continuous and perfect pre-image of a menger space is also menger. (vi) [9, p. 255] if a space is the countable union of menger spaces, then it is menger as well. (vii) ωω is not menger. we also mention the following observation of aurichi and bella. lemma 2.1 ([2, corollary 1.6]). a space x is menger at infinity if and only if there exists a compactification of x with a menger remainder if and only if the remainder of every compactification of x is menger. 2.2. filters. a filter f on a non-empty set x is a subset f ⊂ p(x) such that: (a) ∅∉f, (b) if x,y ∈f then x∩y ∈f, and (c) if x ∈f and x ⊂ y ⊂ x, then y ∈ f. all filters in this paper are defined on countable sets (and most of the times, on ω). filters that contain the fréchet filter of cofinite sets are called free. maximal filters are called ultrafilters. let χ ∶ p(ω) → 2ω be the function that sends each subset to its characteristic function. using χ, a filter on a countable set can be thought of as a subspace of the cantor set. for every subset y ⊂ p(x) we may define y∗ = {a ⊂ x ∶ x ∖ a ∈ y}. if f is a filter on ω, f∗ is called its dual ideal and f+ = p(x)∖f∗ is the set of f-positive sets. moreover, the function that takes each set in p(x) to its complement is a homeomorphism. thus, a filter is always homeomorphic to its dual ideal. also, notice that the complement p(x)∖f is then homeomorphic to f+. given a ⊂ ω ×ω and n ∈ ω, define a(n) = {i ∈ ω ∶ ⟨i,n⟩ ∈ a}, and a(n) = {i ∈ ω ∶ ⟨n,i⟩ ∈ a}. for y ⊂ p(ω), let us define y(ω) ={a ⊂ ω ×ω ∶ ∀n ∈ ω (a(n) ∈y))}. there is a natural function from p(ω×ω) to p(ω)ω that takes each a ⊂ ω×ω to {⟨n,a(n)⟩ ∶ n ∈ ω}. this function is also a homeomorphism and takes y (ω) to yω. it is easy to see that if f is a filter on ω, then f(ω) is a filter on ω×ω. thus, the ω-power of a filter is always (homeomorphic to) a filter. a filter f is a p-filter if for every {fn ∶ n < ω}⊂f there exists f ∈f such that f ∖ fn is finite for all n ∈ ω. a filter is called meager if it is meager as a topological space. it is known that ultrafilters are non-meager [3, theorem 4.1.1]. a p-point is a p-filter that is also an ultrafilter. the existence of p-points is independent from zfc. for example, d = c implies there are ppoints but there are models with no p-points, see [3, section 4.4]. non-meager p-filters are a natural generalization of p-points; it is still an open question © agt, upv, 2019 appl. gen. topol. 20, no. 1 225 a. bella and r. hernández-gutiérrez whether they exist in zfc but if they don’t exist, then there is an inner model with a large cardinal, see [3, section 4.4.c]. 2.3. the hilbert cube. the hilbert cube is the countable infinite product of closed intervals of the reals; we will find it convenient to work with q= [−1,1]ω. the pseudointerior of q is s = (−1,1)ω and the pseudoboundary is b(q)=q∖s. 3. the example according to [4, proposition 6.2], if cp(x) is menger at infinity, then it is first countable and hereditarily baire. from [1, i.1.1], it follows that x is countable. in [10], witold marciszewski studied countable spaces x such that cp(x) is hereditarily baire. we will consider one specific case: when x has a unique non-isolated point. given a filter f ⊂ p(ω), consider the space ξ(f) = ω ∪{f}, where every point of ω is isolated and every neighborhood of f is of the form {f}∪a with a ∈f. all our filters will be free, that is, they contain the fréchet filter. in this case, f is not isolated. when f is the fréchet filter, ξ(f) is homeomorphic to a convergent sequence. it is easy to see that a space x is homeomorphic to a space of the form ξ(f) if and only if x is a countable space with a unique non-isolated point. theorem 3.1 ([10]). for a free filter f on ω, the following are equivalent. (a) f is a non-meager p-filter, (b) f is hereditarily baire, and (c) cp(ξ(f)) is hereditarily baire. thus, it is natural to ask when cp(ξ(f)) is menger at infinity. by lemma 2.1 it is sufficient to look at any compactification of cp(ξ(f)) and try to decide whether the remainder is menger. consider the set of functions in the hilbert cube that f-converge to 0: kf ={f ∈q ∶∀m ∈ ω {n ∈ ω ∶ ∣f(n)∣< 2 −m} ∈f}, and those that only take values in the pseudointerior cf = kf ∩ s. by [11, lemma 2.1], it easily follows that cf is homeomorphic to cp(ξ(f)). also, cf is dense in q. so our problem becomes equivalent to finding a filter f such that q∖cf is menger. in fact, we will prove the following characterization of those filters. theorem 3.2. let f be a free filter on ω. then cp(ξ(f)) is menger at infinity if and only if f+ is menger. recall that since f+ is homeomorphic to p(ω)∖f, the property in theorem 3.2 is equivalent to saying that f is menger at infinity. so the problem is reduced to the existence of such filters. as discussed in the introduction, menger ultrafilters have the desired properties. indeed, an ultrafilter coincides with its set of positive sets. thus, we conclude the following. © agt, upv, 2019 appl. gen. topol. 20, no. 1 226 a non-discrete space x with cp(x) menger at infinity corollary 3.3. if f is a menger ultrafilter on ω, then cp(ξ(f)) is menger at infinity. now we proceed to the proof of theorem 3.2. the argument is based on two classical theorems that relate the cantor set and the unit interval: [0,1] has a subspace homeomorphic to 2ω and is a continuous image of 2ω. we just have to take the filter into account and the proof will follow naturally. a family of closed, non-empty subsets {js ∶ s ∈ 2<ω} of a space x will be called a cantor scheme on x if (i) for every s ∈ 2<ω, js ⊃ js⌢0 ∪js⌢1, and (ii) for every f ∈ 2ω, jf =⋂{jf↾n ∶ n < ω} is exactly one point. let 1 = ω ×{1} and σ ={1↾n∶ n ∈ ω}. lemma 3.4. for every free filter f on ω there is a closed embedding of f(ω) into cp(ξ(f)). proof. as explained earlier we shall work on cf ⊂ q instead of the function space. recursively, construct a cantor scheme {js ∶ s ∈ 2<ω} on the interval (−1,1) such that: (i) j∅ = [−12,0], (ii) for every s ∈ 2<ω, js is a non-degenerate closed interval of length < 2 −∣s∣, (iii) for every s ∈ 2<ω, js⌢0 ∩js⌢1 =∅, and (iv) for every s ∈ σ, 0 ∈ js. now define the function ϕ ∶ p(ω ×ω)→q such that for all a ⊂ ω ×ω and n ∈ ω, ϕ(a)(n) is the unique point in jχ(a(n)). so informally speaking, the n-th row of a is used to define the value of the function ϕ(a) ∶ ω → 2 at n. from standard arguments, it is easy to see that ϕ is an embedding. now we shall prove that a ∈f(ω) if and only if ϕ(a) ∈ kf. first, assume that a ∈ f(ω), that is, a(i) ∈ f for all i ∈ ω. let m ∈ ω. by the definition of ϕ(a), for every n ∈⋂{a(i) ∶ i ≤ m} we have ϕ(a)(n) ∈ j1↾m. since j1↾m has diameter less than 2 −m and contains 0, we obtain that ⋂{a(i) ∶ i ≤ m}⊂{n ∈ ω ∶ ∣ϕ(a)(n)∣< 2 −m}. thus the set {n ∈ ω ∶ ∣ϕ(a)(n)∣< 2−m} is an element of f. since this holds for every m < ω, we obtain that ϕ(a) is f-convergent to 0. now assume that ϕ(a) ∈ kf and fix m ∈ ω. let k ∈ ω be such that the length of j1↾m is greater than 2 −k. by our hypothesis the set {n ∈ ω ∶ ∣ϕ(a)(n)∣< 2−k} is an element of f and {n ∈ ω ∶ ∣ϕ(a)(n)∣< 2−k}⊂{n ∈ ω ∶ ϕ(a)(n) ∈ j1↾m} so {n ∈ ω ∶ ϕ(a)(n) ∈ j1↾m} ∈f. finally, notice that by the definition of ϕ {n ∈ ω ∶ ϕ(a)(n) ∈ j1↾m}⊂{n ∈ ω ∶ ⟨m,n⟩ ∈ a}= a(m), so we obtain that a(m) ∈ f. since this is true for all m ∈ ω, we conclude that a ∈f(ω). © agt, upv, 2019 appl. gen. topol. 20, no. 1 227 a. bella and r. hernández-gutiérrez this concludes the proof that a ∈ f(ω) if and only if ϕ(a) ∈ kf. also, notice that the image of p(ω × ω) under ϕ is a subset of s. thus, we can even say that a ∈ f(ω) if and only if ϕ(a) ∈ cf. thus, ϕ↾f(ω) is the closed embedding we wanted. � lemma 3.5. let f be a free filter on ω. then there exists a continuous surjective function ϕ ∶ p(ω ×ω)→q such that ϕ←[kf]=f(ω). proof. as before, we work on cf ⊂ q instead of the function space. recursively, construct a cantor scheme {js ∶ s ∈ 2<ω} on the interval [−1,1] according to the following conditions: (1) j∅ = [−1,1], (2) for every s ∈ 2<ω, js is a non-degenerate closed interval of length ≤ 2 −∣s∣, (3) if s ∈ σ, there are 0 < x < y with js = [−y,y], js⌢1 = [−x,x] and as⌢0 = [−y,−x]∪ [x,y], (4) if s ∈ σ, there are 0 < x < y with js⌢⟨0,0⟩ = [−y,−x] and js⌢⟨0,1⟩ = [x,y], (5) if s ∈ 2<ω ∖ σ and there are a < b with js = [a,b], then there exists x ∈ (a,b) such that js⌢0 = [a,x] and js⌢1 = [x,b]. again define the function ϕ ∶ p(ω × ω) → q such that for all a ⊂ ω × ω and n ∈ ω, ϕ(a)(n) is the unique point in jχ(a(n)). another standard argument implies that ϕ is a continuous, surjective function. also, the equality ϕ←[kf]=f(ω) can be proved in a manner completely analogous to the corresponding equality from lemma 3.4. thus, we will leave this argument to the reader. � finally, the following allows us to simplify the characterization we will obtain. lemma 3.6. let f be a free filter. then (f(ω)) + is menger if and only if f+ is menger. proof. first, assume that f+ is menger. for each n ∈ ω, consider mn = {a ⊂ ω×ω ∶ a(n) ∈f +}, which is homeomorphic to the product f+×p(ω)ω. since the product of a menger space and a compact space is menger, it follows that mn is menger for every n ∈ ω. then notice that (f(ω))+ =⋃{mn ∶ n ∈ ω} is a countable union of menger spaces so it is menger. now assume that (f(ω))+ is menger. the diagonal in a product is always a closed subspace and the diagonal of (f(ω))+ is equal to the set {a ⊂ ω ×ω ∶ ∃f ∈f+ ∀n ∈ ω (a(n) = f)}, which is homeomorphic to f+. then f+ is menger because it is a closed subspace of a menger space. � proof of theorem 3.2. by lemma 3.6, it is sufficient to prove that cf is menger at infinity if and only if f(ω) is menger at infinity. first, assume that cf is menger at infinity. this means that q∖ cf is menger. by lemma 3.4, f(ω) can be embedded as a closed set f in cf. let © agt, upv, 2019 appl. gen. topol. 20, no. 1 228 a non-discrete space x with cp(x) menger at infinity f denote the closure of f in q. then f ∖f is a closed subset of q∖cf. then f is a compactification of f(ω) with menger remainder. now, assume that f(ω) is menger at infinity. let ϕ ∶ p(ω × ω) → q be the continuous surjection from lemma 3.5. then it follows that ϕ[p(ω×ω)∖ f(ω)] = q∖ kf. since the menger property is preserved under continuous functions, q∖kf is menger. notice that q∖cf = (q∖kf)∪b(q). since the boundary b(q) is σ-compact and the union of countably many menger spaces is menger, q∖ cf is menger. this concludes the proof of the theorem. � 4. questions let f be a free filter on ω such that f+ is menger. we have just proved that cp(ξ(f)) is menger at infinity. by the observations of aurichi and bella from [2], cp(ξ(f)) is a hereditarily baire space. then, by marciszewski’s result from [10] it follows that f is a non-meager p-filter. thus, inadvertently we proved the following, which supersedes [4, observation 3.4] (for filters only). corollary 4.1. let f be a free filter on ω. if f+ is menger, then f is a non-meager p-filter. here is another more direct proof: assume that f is a free filter that is not a non-meager p-filter. by lemmas 2.1 and 2.2 from [10], the space f has a closed subset q homeomorphic to the rationals. the closure q of q in p(ω) must be homeomorphic to the cantor set. also, q∖q is homeomorphic to ωω, contained in p(ω)∖f and closed in p(ω)∖f. since ωω is not menger and the menger property is hereditary to closed sets, p(ω)∖f is not menger. by [8, 2.7] every filter of character < d is a menger filter. however, it not hard to conclude from [3, 4.1.2] that any filter of character < d is meager. so indeed none of these filters can have its positive set menger. as discussed earlier, the existence of a non-meager p-filter in zfc is still an open question. but so far the only example of a filter f with f+ menger is a menger ultrafilter, which we know that consistently does not exist. so it is natural to ask about the consistency of filters that are menger at infinity. question 4.2. does zfc imply that there exists a free filter f on ω such that f+ is menger? question 4.3. is the existence of a free menger ultrafilter on ω equivalent to the existence of a free filter f on ω such that f+ is menger? finally, regarding the original question from [4], we could ask whether there exist examples with other properties. in [10, proposition 3.3], marciszewski gave general conditions for a countable space x to guarantee that cp(x) is hereditarily baire. so we may ask what happens in general. © agt, upv, 2019 appl. gen. topol. 20, no. 1 229 a. bella and r. hernández-gutiérrez question 4.4. does there exist a countable, regular, crowded space x such that cp(x) is menger at infinity? question 4.5. does there exist a countable, regular, maximal space x such that cp(x) is menger at infinity? question 4.6. characterize all countable regular spaces x such that cp(x) is menger at infinity. acknowledgements. the research that led to the present paper was partially supported by a grant of the group gnsaga of indam. the second-named author was also supported by the 2017 prodep grant uam-ptc-636 awarded by the mexican secretariat of public education (sep). references [1] a. v. arkhangel’skĭi, topological function spaces, mathematics and its applications (soviet series), 78. kluwer academic publishers group, dordrecht, 1992. x+205 pp. isbn: 0-7923-1531-6 [2] l. f. aurichi and a. bella, when is a space menger at infinity?, appl. gen. topol. 16, no. 1 (2015), 75–80. [3] t. bartoszyński and h. judah, set theory. on the structure of the real line, a k peters, ltd., wellesley, ma, 1995. xii+546 pp. isbn: 1-56881-044-x [4] a. bella, s. tokgöz and l. zdomskyy, menger remainders of topological groups, arch. math. logic 55, no. 5-6 (2016), 767–784. [5] d. chodounský, d. repovš and l. zdomskyy, mathias forcing and combinatorial covering properties of filters, j. symb. log. 80, no. 4 (2015), 1398–1410. [6] o. guzmán, m. hrušák and a. mart́ınez-celis, canjar filters, notre dame j. form. log. 58, no. 1 (2017), 79–95. [7] m. henriksen and j. r. isbell, some properties of compactifications, duke math. j. 25 (1958), 83–105. [8] r. hernández-gutiérrez and p. j. szeptycki, some observations on filters with properties defined by open covers, comment. math. univ. carolin. 56, no. 3 (2015), 355–364. [9] w. just, a. w. miller, m. scheepers and p. j. szeptycki, the combinatorics of open covers. ii, topology appl. 73, no. 3 (1996), 241–266. [10] w. marciszewski, p-filters and hereditary baire function spaces, topology appl. 89, no. 3 (1998), 241–247. [11] w. marciszewski, on analytic and coanalytic function spaces cp(x), topology appl. 50 (1993), 341–248. [12] b. tsaban, menger’s and hurewicz’s problems: solutions from “the book” and refinements, in: set theory and its applications, 211–226, contemp. math., 533, amer. math. soc., providence, ri, 2011. © agt, upv, 2019 appl. gen. topol. 20, no. 1 230 @ appl. gen. topol. 22, no. 1 (2021), 31-46doi:10.4995/agt.2021.13148 © agt, upv, 2021 further remarks on group-2-groupoids sedat temel department of mathematics, recep tayyip erdogan university, turkey (sedat.temel@erdogan.edu.tr) communicated by e. minguzzi abstract the aim of this paper is to obtain a group-2-groupoid as a 2-groupoid object in the category of groups and also as a special kind of an internal category in the category of group-groupoids. corresponding group2-groupoids, we obtain some categorical structures related to crossed modules and group-groupoids and prove categorical equivalences between them. these results enable us to obtain 2-dimensional notions of group-groupoids. 2010 msc: 20l05; 18d05; 18d35; 20j15. keywords: crossed module; group-groupoid; 2-groupoid. 1. introduction there are several 2-dimensional notions of groupoids such as double groupoids, 2-groupoids, and crossed modules over groupoids. the purpose of this paper is to obtain 2-dimensional notions of group-groupoids which are internal groupoids in the category of groups and widely used under the name of 2groups. the term ”categorification”, which was first used by louis crane [13] in the context of mathematical physics, is the process of replacing set-theoretic theorems by category-theoretic concepts. the aim of categorification is to develop a richer case of existing mathematics by replacing sets with categories, functions with functors and equations between functions with natural isomorphisms between functors. in this approach, the categorified version of a group is called a group-groupoid [2, 5]. group-groupoids, which are also known as g-groupoids [6] or 2-groups [4], are internal categories (hence internal groupoids) in the received 17 february 2020 – accepted 07 november 2020 http://dx.doi.org/10.4995/agt.2021.13148 s. temel category gp of groups [22, 23]. equivalently, group-groupoids can be thought as group objects in the category cat of small categories [6, 23]. another useful viewpoint of group-groupoids is to think them as crossed modules over groups. crossed modules which can be viewed as 2-dimensional groups [7] are widely used in homotopy theory [8], homological algebra [16], and algebraic k-theory [21]. the well-known categorical equivalence between crossed modules and group-groupoids is proved by brown and spencer [6]. this equivalence is introduced in [4] by obtaining a group-groupoid as a 2-category with a unique object. crossed modules, and their higher dimensional analogues, provide algebraic models for homotopy n-types; the group-2-groupoids of this paper in principle provide algebraic models for certain homotopy 3-types. in the previous paper [1], the notions of a group-2-groupoid were introduced and compared with a corresponding structure related to crossed modules over groups. on the other hand, the main objective of this paper is to obtain the structure of a group-2-groupoid as a 2-groupoid object in the category of groups and also as a special kind of internal category in the category of groupgroupoids. in section 4, we present the notion of crossed modules over groupgroupoids and prove that there is a categorical equivalence between group2-groupoids and crossed modules over group-groupoids using the categorical equivalence between 2-groupoids and crossed modules over groupoids given in [17]. in section 5, we show that group-2-groupoids are categorically equivalent to special kind of internal categories in the category of crossed modules. 2. preliminaries let c be a finitely complete category and d0, d1 are objects of the ambient category c. an internal category d = (d0, d1, s, t, ε, m) in c consists of an object d0 in c called the object of objects and an object d1 in c called the object of arrows (i.e. morphisms), together with morphisms s, t: d1 → d0, ε: d0 → d1 in c called the source, the target and the identity maps, respectively, d1 s // t // d0 εtt such that sε = tε = 1d0 and a morphism m: d1 ×d0 d1 → d1 of c called the composition map (usually expressed as m(f, g) = g ◦ f) where d1 ×d0 d1 is the pullback of s, t such that εs(f) ◦ f = f = f ◦ εs(f) [22]. an internal groupoid in c is an internal category with a morphism η : d1 → d1, η(f) = f in c called inverse such that f ◦ f = 1s(f), f ◦ f = 1t(f). we write c(x, y) for all morphisms from x to y where x, y ∈ c0. if c(x, y) = ∅ for all x, y ∈ c0 such that x 6= y, then c is called totally disconnected category. we introduce the definition of a 2-category as given in [4]. a 2-category c = (c0, c1, c2) consists of a set of objects c0, a set of 1-morphisms c1, and © agt, upv, 2021 appl. gen. topol. 22, no. 1 32 further remarks on group-2-groupoids a set of 2-morphisms c2 as follows: x f (( g 66 ✤✤ ✤✤ �� α y with maps s: c1 → c0, s(f) = x, sh : c2 → c0, sh(α) = x, sv : c2 → c1, sv(α) = f, t: c1 → c0, t(f) = y, th : c2 → c0, th(α) = y, tv : c2 → c1, tv(α) = g, called the source and the target maps, respectively, the composition of 1-morphisms as in an ordinary category, the associative horizontal composition of 2-morphisms ◦h : c2 ×c0 c2 → c2 as x f (( g 66 ✤✤ ✤✤ �� α y f1 (( g1 66 ✤✤ ✤✤ �� δ z = x f1◦f (( g1◦g 66 ✤✤ ✤✤ �� δ◦hα z , where c2 ×c0 c2 = {(α, δ) ∈ c2 × c2|sh(δ) = th(α)} and the associative vertical composition of 2-morphisms ◦v : c2 ×c1 c2 → c2 as x f !! ✤✤ ✤✤ �� α == h ✤✤ ✤✤ �� β g // y = x f (( h 66 ✤✤ ✤✤ �� β◦vα y where c2 ×c1 c2 = {(α, β) ∈ c2 × c2|sv(β) = tv(α)} such that satisfying the following interchange rule: (θ ◦v δ) ◦h (β ◦v α) = (θ ◦h β) ◦v (δ ◦h α) whenever one side makes sense, and the identity maps ε: c0 → c1, ε(x) = 1x, εh : c0 → c2, εh(x) = 11x such that α ◦h 11x = α = 11y ◦h α and εv : c1 → c2, εv(f) = 1f such that α ◦v 1f = α = 1g ◦v α. therefore, the construction of a 2-category c = (c0, c1, c2) contains compatible category structures c1 = (c0, c1, s, t, ε, ◦), c2 = (c0, c2, sh, th, εh, ◦h), and c3 = (c1, c2, sv, tv, εv, ◦v) such that the following diagram commutes. c2 sv // tv // th ��✸ ✸✸ ✸✸ ✸✸ ✸ ✸✸ ✸✸ ✸ sh ��✸ ✸ ✸✸ ✸ ✸✸ ✸✸ ✸✸ ✸ ✸ c1 εv ss c0 �� t ☞☞☞ ☞☞☞ ☞☞ ☞☞☞ ☞☞ �� s ☞ ☞☞ ☞☞☞ ☞☞ ☞☞☞ ☞☞ εh tt ε jj let c and c′ be 2-categories. a 2-functor is a map f : c → c′ sending each object of c to an object of c′, each 1-morphism of c to 1-morphism of c′ and © agt, upv, 2021 appl. gen. topol. 22, no. 1 33 s. temel 2-morphism of c to 2-morphism of c′ as follows: x f (( g 66 ✤✤ ✤✤ �� α y 7→ f(x) f (f) ** f (g) 44 ✤✤ ✤✤ �� f (α) f(y) such that f(f1 ◦ f) = f(f1) ◦ f(f), f(δ ◦h α) = f(δ) ◦h f(α), f(β ◦v α) = f(β) ◦v f(α), f(11x) = 1f (1x) = 11f (x), f(1f ) = 1f (f). hence 2-categories form a category which is denoted by 2cat [24]. a strict 2-groupoid is a 2-category all of whose 1-morphisms are invertible and in which all 2-morphisms are invertible horizontally and vertically. x f (( g 66 ✤✤ ✤✤ �� α y f̄ '' ḡ 77 ✤✤ ✤✤ �� ᾱh x = x 1x (( 1x 66 ✤✤ ✤✤ �� 11x x , x f �� ✤✤ ✤✤ �� α >> f ✤✤ ✤✤ �� ᾱ v g // y = x f (( f 66 ✤✤ ✤✤ �� 1f y let g, g′ be 2-groupoids. a morphism of 2-groupoids is a 2-functor f : g → g′ which preserves the 2-groupoid structures. thus, 2-groupoids and their morphisms form a category which is denoted by 2gpd [24]. a group-groupoid is an internal category in gp [22]. also, a group-groupoid can be obtained as a group object in the category cat of small categories (or in gpd). a morphism of group-groupoids is a morphism of groupoids which preserves group structures. hence we can define the category of group-groupoids, which is denoted by 2gp or gpgd. for further details about group-groupoids, see [24, 6, 4]. by a crossed module as defined by whitehead, it is meant a pair m, n of groups together with an action •: n × m → m of groups and a morphism ∂ : m → n of groups such that ∂(n•m) = n∂(m)n−1 and ∂(m)•m′ = mm′m−1 [28, 29]. let k = (m, n, ∂, •), k′ = (m′, n′, ∂′, •′) be crossed modules and λ1 : n → n′, λ2 : m → m ′ be morphisms of groups. if λ1, λ2 satisfies the conditions λ1∂ = ∂ ′λ2 and λ2(n • m) = λ1(n) • ′ λ2(m), then 〈λ2, λ1〉: k → k ′ is called morphism of crossed modules [6]. hence crossed modules and their morphisms form a category which we denote by cm. the following theorem was proved by brown and spencer in [6]: theorem 2.1. the category of group-groupoids and the category of crossed modules are equivalent. let g = (x, g) and h = (x, h) be groupoids over the same object set x such that h is totally disconnected. we recall from [8, 17, 11] that an action © agt, upv, 2021 appl. gen. topol. 22, no. 1 34 further remarks on group-2-groupoids of g on h is a partially defined map •: g × h → h, (g, h) 7→ g • h such that the following conditions satisfies [ag 1] g • h is defined iff t(h) = s(g), and t(g • h) = t(g), [ag 2] (g2 ◦ g1) • h = g2 • (g1 • h), [ag 3] g•(h2◦h1) = (g•h2)◦(g•h1), for h1, h2 ∈ h(x, x) and g ∈ g(x, y), [ag 4] 1x • h = h, for h ∈ h(x, x). from this conditions, it can be easily obtain that g•1x = 1y, for g ∈ g(x, y). using this action of g on h, we can obtain a groupoid which is called semidirect product of g and h denoted by g ⋉ h. let x g // y h // y are morphisms of g and h, respectively, then (g, h) is a morphism as follows x (g,h) // y where the structure maps are defined by s(g, h) = s(g), t(g, h) = t(g), ε(x) = (1x, 1x). if x g // y h // y g1 // z h1 // z then the composition of morphisms is defined by (g1, h1) ◦ (g, h) = (g1 ◦ g, h1 ◦ (g1 • h)). the notion of crossed modules over groupoids is introduced by brownhiggins [9, 10] and brown-icen [11]. let g = (x, g) and h = (x, h) be groupoids over the same object set x such that h is totally disconnected. a crossed module k = (h, g, ∂, •) over groupoids consists of a morphism ∂ = (1, ∂): h → g of groupoids which is identity on objects together with an action •: g × h → h of groupoids which satisfies ∂(g • h) = g ◦ ∂(h) ◦ g and ∂(h) • h1 = h ◦ h1 ◦ h, for h, h1 ∈ h(x, x) and g ∈ g(x, y). let k = (h, g, ∂, •) and k′ = (h′, g′, ∂′, •′) be crossed modules over groupoids. a morphism of crossed modules over groupoids is a mapping λ = 〈λ2, λ1, λ0〉: k → k′ which satisfies λ2∂ = ∂ ′λ1 and λ1(g•h) = λ2(g)• ′λ1(h) where (λ0, λ1): h → h′ and (λ0, λ2): g → g ′ are morphisms of groupoids. hence the category of crossed modules over groupoids can be defined which we denoted by cmg. the following result was proved by icen in [17]. since we need some details in section 4, we give a sketch proof in terms of our notations. theorem 2.2. the categories of 2-groupoids and of crossed module over groupoids are equivalent. proof. for any 2-groupoid g = (g0, g1, g2), we know that b = (g0, g1) is a groupoid. let a(x) = {α ∈ g2|sv(α) = ε(x)}, for x ∈ g0 and a = {a(x)}x∈g0. then a = (g0, a) is a totally disconnected groupoid. now we define a functor © agt, upv, 2021 appl. gen. topol. 22, no. 1 35 s. temel γ : 2gpd → cmg as an equivalence of categories such that γ(g) = (a, b, ∂) is a crossed module over groupoids with ∂ : a → b, ∂(α) = tv(α) and an action of groupoids such that f • α = 1f ◦h α ◦h 1f . y 1y (( ∂(f•α) 66 ✤✤ ✤✤ �� f•α y = y f (( f 66 ✤✤ ✤✤ �� 1f x 1x (( ∂(α) 66 ✤✤ ✤✤ �� α x f (( f 66 ✤✤ ✤✤ �� 1f y clearly ∂(f • α) = f ◦ ∂(α) ◦ f and ∂(α) • α1 = α ◦h α1 ◦h α h, for f ∈ g1(x, y) and α, α1 ∈ a(x). let f = (f0, f1, f2) be a morphism of 2-groupoids. then γ(f) = 〈f2 ∣ ∣ a , f1, f0〉 is a morphism of crossed modules over groupoids. now we define a functor θ : cmg → 2gpd which is an equivalence of categories. let k = (a, b, ∂) be a crossed module over groupoids a = (x, a) and b = (x, b). then 2-groupoid θ(k) = (x, b, b ⋉ a) is a 2-groupoid which is constructed as in the following way. the set of 2-morphisms is the semi-direct product b ⋉ a = {(b, a)|b ∈ b, a ∈ a, s(a) = t(a) = t(b)}. if x b // y a // y , then (b, a) is a 2-morphism as follows: x b (( ∂(a)◦b 66 ✤✤ ✤✤ �� (b,a) y where the horizontal composition of 2-morphisms is defined by (b1, a1) ◦h (b, a) = (b1 ◦ b, a1 ◦ (b1 • a)) when y b1 // z a1 // z and the vertical composition of 2-morphisms is defined by ( ∂(a) ◦ b, a2 ) ◦v (b, a) = (b, a2 ◦ a) when y a2 // y . the source and the target maps are defined by sh(b, a) = s(b), sv(b, a) = b, th(b, a) = t(b), tv(b, a) = ∂(a) ◦ b, respectively, the identity maps are defined by εh(x) = (1x, 1x), εv(b) = (b, 1y), and the inversion maps are defined by (b, a) v = (∂(a) ◦ b, a), (b, a) h = (b, b • a). let λ = 〈λ2, λ1, λ0〉 be a morphism of crossed modules over groupoids. then θ(λ) = (λ0, λ2, λ2 × λ1) is a morphism of 2-groupoids. a natural equivalence s : θγ → 12gpd is defined via the map sg : θγ(g) → g which is defined to be identity on objects and on 1-morphisms, on 2-morphisms is defined by α 7→ (f, α ◦h 1f). clearly sg is an isomorphism and preserves compositions. now, given a crossed module k = (a, b, ∂, •) over groupoids, we define a natural equivalence t : 1cmg → γθ by a map tk : k → γθ(k) which is defined to be identity on objects and on b, while on a is defined by a 7→ (s(a), a). � © agt, upv, 2021 appl. gen. topol. 22, no. 1 36 further remarks on group-2-groupoids 3. group-2-groupoids in [1], a group-2-groupoid is defined as a group object in 2cat using similar methods given in [6, 23]. in other words, a group-2-groupoid g is a small 2-groupoid equipped with the following 2-functors satisfying group axioms, written out as commutative diagrams (1) µ: g × g → g called product, (2) inv : g → g called inverse and (3) id: {∗} → g (where {∗} is a singleton) called unit or identity. then, the product of x a '' b 77 ✤✤ ✤✤ �� α y and x ′ a ′ (( b ′ 77 ✤✤ ✤✤ �� α ′ y′ is written by x · x′ a·a′ ** b·b′ 44 ✤✤ ✤✤ �� α·α ′ y · y′ , the inverse of x a '' b 77 ✤✤ ✤✤ �� α y is x −1 a −1 )) b −1 55 ✤✤ ✤✤ �� α−1 y −1 where id{∗} = e 1e '' 1e 77 ✤✤ ✤✤ �� 11e e . the condition 1 above gives us the following interchange rules (a1 ◦ a) · (a ′ 1 ◦ a ′) = (a1 · a ′ 1) ◦ (a · a ′), (δ ◦h α) · (δ ′ ◦h α ′) = (δ · δ′) ◦h (αα ′), (β ◦v α) · (β ′ ◦v α ′) = (β · β′) ◦v (α · α ′) whenever compositions are defined. we can obtain from the condition 2 that (a1 ◦ a) −1 = a−11 ◦ a −1, (δ ◦h α) −1 = δ−1 ◦h α −1, (β ◦v α) −1 = β−1 ◦v α −1, 1−1x = 1x−1, 1 −1 1x = 11 x−1 and 1−1a = 1a−1. moreover, the structure of a group-2-groupoid g = (g0, g1, g2) contains compatible group-groupoids g = (g0, g1), g ′ = (g0, g2) and g ′′ = (g1, g2) [1]. equivalently we shall describe a group-2-groupoid as a 2-groupoid object in the category gp of groups. let c0, c1 and c2 be objects of a finitely complete category c. if c1 = (c0, c1, s, t, ε, ◦), c2 = (c0, c2, sh, th, εh, ◦h), and c3 = (c1, c2, sv, tv, εv, ◦v) are internal categories in c such that the following diagram commutes whenever the usual interchange rule satisfies between ◦h and ◦v, then (c0, c1, c2) is called an internal 2-category in c. c2 sv // tv // th ��✸ ✸✸ ✸✸ ✸✸ ✸✸ ✸ ✸✸ ✸ sh ��✸ ✸ ✸✸ ✸✸ ✸ ✸✸ ✸✸ ✸✸ c1 εv ss c0 �� t ☞ ☞☞ ☞☞☞ ☞☞☞ ☞☞ ☞☞ �� s ☞ ☞☞☞ ☞☞ ☞☞☞ ☞☞ ☞☞ εh tt ε jj © agt, upv, 2021 appl. gen. topol. 22, no. 1 37 s. temel proposition 3.1. a 2-category object in gp is a group-2-groupoid. proof. let g = (g0, g1, g2) is a 2-category object in gp and µ0, µ1, µ2 be multiplications of groups g0, g1, g2, respectively. then, we can define a multiplication µ: g × g → g as a 2-functor such that µ = µ0 on objects, µ = µ1 on 1-morphisms and µ = µ2 on 2-morphisms. similarly, we can define 2-functors id: 1 → g (where 1 is the terminal object of 2cat, i.e. the oneobject discrete 2-category) which picks out an identity object, an identity 1morphism and an identity 2-morphism and inv : g → g picks out inverses for multiplications. since a = 1s(a)a −11t(a) from [6] and α v = 1sv(α)α −11tv(α), αh = 11sh(α) α−111th(α) from [1], g is a 2-groupoid. then, g is a group object in 2cat and so g is a group-2-groupoid. � example 3.2. every group-groupoid can be thought as a group-2-groupoid in which all 2-morphisms are identities as follows: x a '' a 77 ✤✤ ✤✤ �� 1a y · x ′ a ′ (( a ′ 77 ✤✤ ✤✤ �� 1 ′ a y ′ = x · x′ a·a′ ** a·a′ 44 ✤✤ ✤✤ �� 1a·a′ y · y ′ it is mentioned that a group-groupoid is a 2-category with a single object [4]. then, we shall need a different viewpoint on group-groupoids as a special kind of group-2-groupoids: proposition 3.3. a group-2-groupoid with a single object is a group-groupoid in which both groups are necessarily abelian. proof. in this approach, the composition of 1-morphisms and the horizontal composition of 2-morphisms are defined by multiplications of groups as follows: ⋆ a (( b 66 ✤✤ ✤✤ �� α ⋆ a ′ (( b ′ 66 ✤✤ ✤✤ �� α ′ ⋆ = ⋆ a ′∗a (( b ′∗b 66 ✤✤ ✤✤ �� α ′∗α ⋆ it is proved in [23] that a′ ∗ a = a′ · a = a · a′. using similar way, we get α′ ∗ α = (α′ · 1e) ∗ (1e · α) = (α ′ ∗ 1e) · (1e ∗ α) = α ′ · α and α ′ · α = (1e ∗ α ′) · (α ∗ 1e) = (1e · α) ∗ (α ′ · 1e) = α · α ′ . � a third way to understand group-2-groupoids is to view them as double group-groupoids which are defined in [26] (see also [27]). recall that a double category is a category object internal to cat. hence the structure of a double category contains four different but compatible category structures as partially © agt, upv, 2021 appl. gen. topol. 22, no. 1 38 further remarks on group-2-groupoids shown in the following diagram d2 s // t // t �� s �� dv1 t �� s �� ε tt dh1 s // t // ε ii d0 ε uu ε kk where dh1 and d v 1 are called horizontal and vertical edge categories, respectively, and d2 is called the set of squares. for further details, see [12, 14, 15, 20]. the structure of a 2-category may be regarded as a double category in which all vertical morphisms are identities (or d2 and d h 1 have the same objects) [12, 20]. therefore, a group-2-groupoid is a special kind of an internal category in the category gpgd of group-groupoids. 4. crossed modules over group-groupoids in this section, we work on crossed modules over groupoids by replacing such groupoids with group-groupoids. using the natural equivalence between crossed modules over groupoids and 2-groupoids given in [17], we will prove that there is a categorical equivalence between group-2-groupoids and crossed modules over group-groupoids. definition 4.1. let g = (x, g) and h = (x, h) are group-groupoids over the same object set, h be totally disconnected and k = (h, g, ∂) be a crossed module over g and h such that ∂ is a homomorphism of group-groupoids and the following interchange rule holds: (g • h) · (g′ • h′) = (g · g′) • (h · h′) where g, g′ ∈ g, h, h′ ∈ h. then k is called a crossed module over groupgroupoids. a morphism of crossed modules over group-groupoids is a morphism of crossed modules of groupoids which preserves group structures. then, we can construct the category of crossed modules over group-groupoids which we denote by cmg*. theorem 4.2. the categories cmg* and gp2gd are equivalent. proof. the idea of the proof is to show that the functor of 2.2 restricts to an equivalence of categories. let a = (x, a) and b = (x, b) are group-groupoids and k = (a, b, ∂) is a crossed module over a and b. then θ(k) = (x, b, b⋉a) is a group-2-groupoid via the process of the proof 2.2. the group multiplication of 2-morphisms in θ(k) is defined by (b, a) · (b′, a′) = (b · b′, a · a′). © agt, upv, 2021 appl. gen. topol. 22, no. 1 39 s. temel we draw such pairs as x b (( ∂(a)◦b 66 ✤✤ ✤✤ �� (b,a) y · x ′ b ′ )) ∂(a′)◦b′ 55 ✤✤ ✤✤ �� (b ′ ,a ′) y′ = x · x′ b·b′ ,, ∂(a·a′)◦(b·b′) 22 ✤✤ ✤✤ �� (b·b ′ ,a·a′) y · y′ now we will verify that compositions and the group multiplication satisfy the interchange rule. [ (b1, a1) ◦h (b, a) ] · [ (b′1, a ′ 1) ◦h (b ′, a′) ] = [ (b1 ◦ b, a1 ◦ (b1 • a)) ] · [ (b′1 ◦ b ′, a′1 ◦ (b ′ 1 • a ′)) ] = ( (b1 ◦ b) · (b ′ 1 ◦ b ′), (a1 ◦ (b1 • a) · (a ′ 1 ◦ (b ′ 1 • a ′)) ) = ( (b1 · b ′ 1) ◦ (b · b ′), (a1 · a ′ 1) ◦ ( (b1 • a) · (b ′ 1 • a ′) ) ) = ( (b1 · b ′ 1) ◦ (b · b ′), (a1 · a ′ 1) ◦ ( (b1 · b ′ 1) • (a · a ′) ) ) = (b1 · b ′ 1, a1 · a ′ 1) ◦h (b · b ′, a · a′) = [ (b1, a1) · (b ′ 1, a ′ 1) ] ◦h [ (b, a) · (b′, a′) ] and [ (∂(a) ◦ b, a2) ◦v (b, a) ] · [ (∂(a′) ◦ b′, a′2) ◦v (b ′, a′) ] = (b, a2 ◦ a) · (b ′, a′2 ◦ a ′) = (b · b′, (a2 · a ′ 2) ◦ (a · a ′)) = [ ∂(a · a′) ◦ (b · b′), a2 · a ′ 2 ] ◦v (b · b ′, a · a′) = [ (∂(a) ◦ b, a2) · (∂(a ′) ◦ b′, a′2) ] ◦v [ (b, a) · (b′, a′) ] whenever all above compositions are defined. now let g = (g0, g1, g2) be a group-2-groupoid. then γ(g) is a crossed module over groupoids internal to gp. we will verify that the interchange law holds: (f•α)·(f′•α′) = (1f ◦hα◦h1f )·(1f′◦hα ′◦h1f′) = 1f·f′◦h(α·α ′)◦h1f·f′ = (f·f ′)•(α·α′) now we will show that sg preserves the group multiplication: sg(α · α ′) = (f · f′, (α · α′) ◦h 1f·f′) = ( f · f′ , (α · α′) ◦h (1f · 1f′) ) = ( f · f′ , (α ◦h 1f ) · (α ′ ◦h 1f′) ) = (f, α ◦h 1f ) · (f ′, α′ ◦h 1f′) = sg(α) · sg(α ′) other details are straightforward and so are omitted. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 40 further remarks on group-2-groupoids 5. group-2-groupoids as internal categories in cm a group-2-groupoid can be also thought as a special case of an internal category in the category cm of crossed modules (see, e.g., [25] and [26] for more details about internal categories in cm). this idea comes from that the structure of a group-2-groupoid contains three compatible group-groupoid structures. given a group-2-groupoid, we can extract crossed modules as follows: g2 sv // tv // th ��✷ ✷ ✷✷ ✷ ✷✷ ✷ ✷✷ ✷✷ ✷ ✷ sh ��✷ ✷ ✷ ✷✷ ✷✷ ✷ ✷✷ ✷✷ ✷ ✷ g1 εv ss ker(sh) sv // tv // th|ker(sh) ��✼ ✼✼ ✼✼ ✼✼ ✼✼ ✼✼ ✼✼ ✼✼ ker(s) t|ker(s) ��✟✟ ✟✟ ✟✟ ✟✟ ✟✟ ✟✟ ✟✟ ✟ εv qq 7→ g0 �� t ☞ ☞☞ ☞ ☞☞ ☞☞ ☞ ☞☞ ☞☞ ☞ �� s ☞ ☞☞ ☞☞ ☞ ☞☞ ☞ ☞☞ ☞☞ ☞ εh tt ε jj g0 then, we obtain an internal groupoid in cm (ker(sh), g0) s // t // (ker(s), g0) ǫpp where the structure maps are defined by s = 〈sv, 1〉, t = 〈tv, 1〉, ǫ = 〈εv, 1〉 as morphisms of crossed modules. here s, t, ǫ are equivariant maps, since sv(x • α) = x • sv(α), tv(x • α) = x • tv(α) and εv(x • f) = x • εv(f), for all x ∈ g0 and α ∈ ker(sh). the actions of g0 on ker(sh) and on ker(s) are drawn in the following diagram: e x•f ++ x•g 33 ✤✤ ✤✤ �� x•α xyx −1 := x 1x (( 1x 66 ✤✤ ✤✤ �� 11x x · e f (( g 66 ✤✤ ✤✤ �� α y · x −1 1−1x ** 1−1x 44 ✤✤ ✤✤ �� 1 −1 1x x−1 we denote the category of such internal groupoids in cm by igcm. we know from [25, 26] that internal categories in the category cm of crossed modules are naturally equivalent to crossed squares which in turn should be viewed as a ”crossed module of crossed modules”. hence an object of the category igcm can be viewed as a special kind of crossed square. let g = (g0, g1, x, ∂0, ∂1) be an object of igcm. then, the following diagram is commutative. g1 s // t // ∂1 ❆ ❆❆ ❆❆ ❆❆ ❆ g0 ∂0~~⑥⑥ ⑥⑥ ⑥⑥ ⑥⑥ ε ss x © agt, upv, 2021 appl. gen. topol. 22, no. 1 41 s. temel let g = (g0, g1, x, ∂0, ∂1), g ′ = (g′0, g ′ 1, x ′, ∂′0, ∂ ′ 1) be objects of igcm. if (λ1, λ2) is an endomorphism of the group-groupoid g = (g0, g1), and 〈λ1, λ0〉, 〈λ2, λ0〉 are morphisms of crossed modules (g0, x, ∂0), (g1, x, ∂1), respectively, then λ = (λ2, λ1, λ0) is called a morphism of igcm. lemma 5.1. let g = (g0, g1, x, ∂0, ∂1) be an object of igcm. then x • (β ◦ α) = (x • β) ◦ (x • α) for x ∈ x, α, β ∈ g1 where s(β) = t(α). proof. let a α // b β // c . we know from [6] that β ◦α = β ·1−1 b ·α. then, we get x • (β ◦ α) = x • (β · 1−1 b · α) = (x • β) · (x • 1−1 b ) · (x • α) = (x • β) · (x • 1b) −1 · (x • α) = (x • β) · 1−1 (x•b) · (x • α) = (x • β) ◦ (x • α) � example 5.2. every crossed module k = (m, n, ∂) over groups is an object of igcm with the discrete groupoid of m where n • 1m = 1n•m and ∂1(1m) = ∂(m). theorem 5.3. there is an equivalence between igcm and gp2gd. proof. a functor γ : gp2gd → igcm is defined in the following way. let h = (h0, h1, h2) be a group-2-groupoid. then γ(h) = (g0, g1, x, ∂0, ∂1) is an object of igcm where g0 = ker(s), g1 = ker(sh), x = h0, ∂0 = t ∣ ∣ ker(s) and ∂1 = th ∣ ∣ ker(sh) h2 sv // tv // th ��✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ sh ��✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ h1 εv rr g1 s ′ // t ′ // ∂1 ��✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ g0 ∂0 ��✍✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ε ′ ss h : 7→ γ(h) : h0 �� t ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ �� s ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ εh tt ε jj x with actions x•f = 1x·f ·1 −1 x and x•α = 11x ·α·1 −1 1x , for x ∈ x, f ∈ g0, α ∈ g1. now we will verify that s′, t′, ε′ are equivariant maps. s ′(x•α) = s′(11x ·α·1 −1 1x ) = sv(11x)·sv(α)·sv(1 −1 1x ) = 1x ·sv(α)·1 −1 x = x•s ′(α), t′(x • α) = t′(11x · α · 1 −1 1x ) = tv(11x) · tv(α) · tv(1 −1 1x ) = 1x · tv(α) · 1 −1 x = x • t ′(α) © agt, upv, 2021 appl. gen. topol. 22, no. 1 42 further remarks on group-2-groupoids and ε′(x•f) = ε′(1x·f ·1x −1) = εv(1x)·εv(f)·εv(1x −1) = 11x ·εv(f)·1 −1 1x = x•ε′(f). let f = (f0, f1, f2) be a morphism of group-2-groupoids. then γ(f) = (f2|ker(sh), f1|ker(s), f0) is a morphism of igcm. next, we define a functor θ : igcm → gp2gd is an equivalence of categories. given an object g = (g0, g1, x, ∂0, ∂1) of igcm, we can obtain a group-2-groupoid θ(g) = h = (h0, h1, h2) where h0 = x, h1 = x⋉g0, h2 = x ⋉ g1 as in the following way. let a α // b be a morphism of g. then pairs x (x,a) // ∂0(a) · x and x (x,b) // ∂0(b) · x are obtained as morphisms of the group-groupoid (h0, h1), and a pair x (x,α) // ∂1(α) · x is obtained as a morphism of the group-groupoid (h0, h2). since ∂1(α) · x = ∂0s(α) · x = ∂0(a) · x, ∂1(α) · x = ∂0t(α) · x = ∂0(b) · x, then (x, α) can be considered as a 2-morphism as follows: x (x,a) ,, (x,b) 22 ✤✤ ✤✤ �� (x,α) ∂1(α) · x let a α // b β // c . then, the vertical composition of (x, α) and (x, β) is defined by (x, β) ◦v (x, α) = (x, β ◦ α) where the source and the target maps are defined by sv(x, α) = (x, s(α)) and tv(x, α) = (x, t(α)), respectively, and the identity map is defined by εv(x, a) = (x, 1a). given morphisms a α // b and a1 α1 // b1 , we obtain pairs (x, α), (∂1(α) · x, α1) and we define their horizontal composite by (∂1(α) · x, α1) ◦h (x, α) = (x, α1 · α) where the source and the target maps are defined by sh(x, α) = x, th(x, α) = ∂1(α) · x, respectively, and the identity map is defined by εh(x) = (x, 1e). clearly the vertical composition and the horizontal composition satisfy the usual interchange rule. the product of (x, α) and (x′, α′) is written by (x, α) · (x′, α′) = (x · x′, α · (x • α′)) for a α // b and a′ α ′ // b′ . if λ = (λ2, λ1, λ0) is a morphism of g, then θ(λ) = (λ0, λ0 × λ1, λ0 × λ2) is morphism of θ(g). a natural equivalence s : 1gp2gd → θγ is defined with a map sg : g → θγ(g) which is defined such that to be the identity on objects, sg(f) = © agt, upv, 2021 appl. gen. topol. 22, no. 1 43 s. temel (x, f ·1−1x ) and sg(α) = (x, α·1 −1 1x ) for f ∈ g1, α ∈ g2 where x = s(f) = sh(α). clearly sg is an isomorphism and preserves the group operations and compositions as follows: sg(α) · sg(α ′) = (x, α · 1−11x ) · (x ′ , α ′ · 1−11x′ ) = ( x · x′, α · 1−11x · (x • (α ′ · 1−11x′ )) ) = ( x · x′, α · 1−11x · 11x · α ′ · 1−11x′ · 1−11x ) = ( x · x′, α · α′ · 1−11xx′ ) = sg(α · α ′) where s(α) = x, s(α′) = x′, sg(δ◦hα) = sg(δ·1 −1 1y ·α) = (x, δ·1−11y ·α·1 −1 1x ) = (y, δ·1−11y )◦h(x, α·1 −1 1x ) = sg(δ)◦hsg(α) where t(α) = s(δ) = y and sg(β) ◦v sg(α) = (x, β · 1 −1 1x ) ◦v (x, α · 1 −1 1x ) = ( x, (β · 1−11x ) ◦v (α · 1 −1 1x ) ) = ( x, (β ◦v α) · (1 −1 1x ◦v 1 −1 1x ) ) = ( x, (β ◦v α) · 1 −1 1x ) = sg(β ◦v α) where sv(β) = tv(α). to define a natural equivalence t : 1igcm → γθ, a map tg is defined such that to be identity on x, tg(a) = (e, a) for a ∈ g0 and tg(α) = (e, α) for α ∈ g1. obviously tg is an isomorphism and preserves the 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[29] j. h. c. whitehead, note on a previous paper entitled “on adding relations to homotopy group”, ann. math. 47 (1946), 806–810. © agt, upv, 2021 appl. gen. topol. 22, no. 1 46 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 53-60 a rafu linear space uniformly dense in c [a, b] e. corbacho cortés abstract in this paper we prove that a rafu (radical functions) linear space, ∁, is uniformly dense in c [a, b] by means of a s-separation condition of certain subsets of [a, b] due to blasco-moltó. this linear space is not a lattice or an algebra. given an arbitrary function f ∈ c [a, b] we will obtain easily the sequence (cn) n of ∁ that converges uniformly to f and we will show the degree of uniform approximation to f with (cn) n . 2010 msc: 37l65 keywords: rafu; uniform density; uniform approximation; radical functions; approximation algorithm. 1. introduction let k be a compact hausdorff space. the kakutani-stone theorem [10] gives a necessary and sufficient condition for the density of a lattice of c (k) in the topology of the uniform convergence on k. the stone-weierstrass theorem [7] provides a necessary and sufficient condition under which an algebra of c (k) is uniformly dense. nevertheless, the above conditions are not sufficient to ensure the uniform density of a linear space of c (k). tietze [5], jameson [4], mrowka [11], blasco-moltó [6], garrido-montalvo [8] and recently gassóhernández-rojas [9] have studied the uniform approximation for linear spaces. in the section 2 we will construct a rafu (radical functions) linear space, ∁, in c [a, b] and we will prove that ∁ is uniformly dense in c [a, b] by using a s-separation condition according to blasco-moltó [6]. we will also see that the uniform density of ∁ in c [a, b] is not a consequence of the results given by kakutani-stone, stone-weierstrass, tietze, jameson, or mrowka. 54 e. corbacho cortés it is true that blasco-moltó showed an example of a linear space, f , uniformly dense in c [0, 1] by using the s-separation condition but some questions were not studied: the linear combinations of elements belonging to f which approximate uniformly every f ∈ c [0, 1] and the degree of uniform approximation that f provides were unknown. in the section 3 we will solve these problems by using the rafu linear space ∁. moreover, this linear space ∁ can be used as an example of approximation by series in the work of gassó-hernández-rojas. 2. a rafu linear space uniformly dense in c [a, b] for each n ∈ n we consider the partition p = {x0, x1, ..., xn} of [a, b] with xj = a + j · b−an , j = 0, ..., n and we define in [a, b] the functions (2.1) cn(x) = k1 + n ∑ i=2 (ki − ki−1) · fn(xi−1, x) where {ki}ni=1 are a family of real arbitrary numbers and (2.2) fn (xi−1, x) = 2n+1 √ xi−1 − x0 + 2n+1 √ x − xi−1 2n+1 √ xn − xi−1 + 2n+1 √ xi−1 − x0 , i = 2, ..., n we designate by ∁n the subset of c [a, b] formed by the functions cn and we also denote by ∁ the set ∁ = ∪n∈n∁n. proposition 2.1. the set ∁ is a linear space included in c [a, b]. proof. it is clear that ∁ ⊂ c [a, b]. in the first place it is easy to check that ∁n is a linear space n-dimensional because n is fixed and hence the values {xi}ni=0 are the same points. moreover, a basis of ∁n is {1, fn(x1, x), ..., fn(xn−1, x)}. ∁ is a linear space. let cp and cq be two elements belonging to ∁. then, cp ∈ ∁r·p, r ∈ n and cq ∈ ∁s·q, s ∈ n by considering zero the coefficients (ki − ki−1) of the functions fn(xi−1, x) that do not appear on the expressions of cp or cq. in particular, cp and cq belong to the linear space ∁p·q and, of course, cp + cq ∈ ∁. finally, it is inmediate to check that if cp ∈ ∁ and λ ∈ r then λ · cp ∈ ∁. � definition 2.2. a rafu linear space is a linear space whose basis is formed by radical functions of the type (2.2). we will say that ∁ is a rafu linear space. the theorems of uniform approximation in c (k) for lattices are known as kakutani-stone theorems (the interested reader can see [10], [7], [12]). the family ∁ is not a lattice. in fact, in the interval [−1, 1] the function c(x) = 3 √ x ∈ ∁ but |c(x)| /∈ ∁ because at x = 0 its side derivatives do not have the same sign. therefore, the family ∁ does not satisfy the kakutani-stone theorems. the theorems of uniform approximation in c (k) for algebras are known as stone-weierstrass theorems (the interested reader can see [7], [12]). a rafu linear space uniformly dense in c [a, b] 55 a simple count proves that ∁ is not an algebra, therefore the set ∁ does not verify the stone-weierstrass theorems. let x be a topological space and let c∗ (x) be the set consisting of all bounded continuous functions and let c (x) be the set consisting of all continuous functions. definition 2.3. let f be a family of c∗ (x). we say that (1) a zero-set in x is a set of the form z(f) = {x ∈ x : f(x) = 0} with f ∈ c∗ (x). (2) the lebesgue-sets of f ∈ c (x) are the sets lα(f) = {x ∈ x : f(x) ≤ α} and lβ(f) = {x ∈ x : f(x) ≥ β} where α and β are real numbers. (3) f s1separates the subsets a and b of x when there is f ∈ f, 0 ≤ f ≤ 1 such that f(x) = 0 if x ∈ a and f(x) = 1 if x ∈ b. (4) (blasco-moltó [?]). f s-separates the subsets a and b of x if for each δ > 0, there is f ∈ f such that 0 ≤ f ≤ 1 for every x ∈ x, f(a) ⊂ [0, δ] and f(b) ⊂ [1 − δ, 1]. (5) (garrido-montalvo [?]). f s′-separates the subsets a and b of x if for each δ > 0, there is f ∈ f such that −δ ≤ f ≤ 1 + δ for every x ∈ x, f(a) ⊂ [−δ, δ] and f(b) ⊂ [1 − δ, 1 + δ]. (6) given a series of continuous functions ∑ i∈i fi on x, the series is locally convergent, for every x ∈ x, if there is a neighborhood u of x such that the series converges uniformly on u. for e ⊂ c(x), ∑ (e) is the set of all f ∈ c (x) such that f = ∑ i∈i fi with fi ∈ e for every i ∈ i and ∑ i∈i fi is a locally convergent series. ∑ (e) denotes the uniform closure of ∑ (e). theorem 2.4 (tietze [5], mrowka [11]). let f be a linear space of c∗ (x). f is uniformly dense in c∗ (x) if and only if f s1separates every pair of disjoint zero-sets in x. theorem 2.5 (jameson [4]). let f be a linear space of c∗ (x). f is uniformly dense in c∗ (x) if and only if f s1separates every pair of disjoint closed subsets in x. by the properties of the functions of the linear space ∁ it is possible to deduce that we cannot apply to ∁ the results of tietze, mrowka or jameson. theorem 2.6 (blasco-moltó [6]). let x be a topological space. a linear space f of c∗ (x) is uniformly dense in c∗ (x) if and only if f sseparates every pair of disjoint zero-sets in x. we go to see that we can apply this theorem to prove the uniform density of ∁ in c [a, b]. let us consider in [a, b] the step function defined by f(x) = { k1 a ≤ x ≤ x1 k2 x1 < x ≤ b , k1, k2 ∈ r. if we calculate, for each n ∈ n, the expressions of the radical functions cn(x) = mn + nn · 2n+1 √ x − x1 that are obtained by the conditions cn(a) = k1 and cn(b) = k2, we obtain nn = k2−k1 2n+1 √ b−x1+ 2n+1 √ x1−a 56 e. corbacho cortés and mn = k1 + (k2−k1)· 2n+1 √ x1−a 2n+1 √ b−x1+ 2n+1 √ x1−a . in this case, a elementary count shows that the sequence (cn)n satisfies limn→+∞ cn(x) =      k1 a ≤ x < x1 k1+k2 2 x = x1 k2 x < x ≤ b now, we will consider an arbitrary step function in [a, b] (2.3) f(x) = k1 · χ[x0, x1] + m ∑ i=2 ki · χ(xi−1, xi] where ki ∈ r, i = 1, ..., m and x ∈ [x0 = a, xm = b]. by an elementary count we can write (2.3) in the form f(x) = ∑m i=1 fi(x) where f1(x) = k1 · χ[x0, xm] and fp(x) = (kp − kp−1) · χ(xp−1, xm], p = 2, ..., m. for each fp we construct its sequence of radical functions (cp,n)n. for every n ∈ n, the corresponding sequence for f1 is c1,n(x) = k1 and for fp, with p > 1, we obtain cp,n(x) = mp,n +np,n · 2n+1 √ x − xp−1 where np,n and mp,n are given by np,n = kp−kp−1 2n+1 √ xm−xp−1+ 2n+1 √ xp−1−x0 and mp,n = (kp−kp−1)· 2n+1 √ xp−1−x0 2n+1 √ xm−xp−1+ 2n+1 √ xp−1−x0 . finally, if we denote by (cm,n)n to the sequence cm,n(x) = ∑m i=1 ci,n(x) then limn→+∞ cm,n(x) = { f(x) x ∈ [x0, xm] − {x1, x2, , ..., xm−1} kp+kp+1 2 x = xp, p = 1, ..., m − 1 by elementary properties of the limits. proposition 2.7. let f be the function defined by (2.3). for any β > 0 such that (xi − β, xi + β)∩(xj − β, xj + β) = ∅ where i 6= j and i, j ∈ {1, ..., m − 1} the limit limn→+∞ cm,n = f is uniform on [x0, x1 − β] ∪ [x1 + β, x2 − β] ∪ ...∪ [xm−1 + β, xm]. proof. 1st part. it verifies that limn→∞ 2n+1 √ x = { −1 x ∈ [ −m, − 1 k ] 1 x ∈ [ 1 k , m ] where m and k are large positive real numbers. moreover, the limit becomes uniform. the function hn(x) = 2n+1 √ x is strictly increasing on r, therefore hn(−m) ≤ hn(x) ≤ hn(− 1k ) for x ∈ [ −m, − 1 k ] and fixed ǫ > 0 it is possible to find nm,k ∈ n such that if n ≥ nm,k then −1−ǫ < hn(−m) ≤ hn(x) ≤ hn(− 1k ) < −1 + ǫ. in other words, |hn(x) + 1| < ǫ. analogous, we obtain |hn(x) − 1| < ǫ on [ 1 k , m ] . 2nd part. given a partition p = {a = x0, x1, ..., xm = b} of [a, b] with a < x1 < ... < b. for each n ∈ n and p = 2, ..., m we define in [a, b] the function fn (xp−1, m, x) = 2n+1 √ xp−1 − x0 + 2n+1 √ x − xp−1 2n+1 √ xm − xp−1 + 2n+1 √ xp−1 − x0 a rafu linear space uniformly dense in c [a, b] 57 then, it follows that limn→∞ fn (xp−1, m, x) =      0 x < xp−1 1 2 x = xp−1 1 x > xp−1 , p = 2, ..., m and these limits are uniform on [x0, x1 − β]∪[x1 + β, x2 − β]∪...∪[xm−1 + β, xm] the first assertion is consequence of the elementary properties of the limits and the second is obtained by aplying the first part and take into acount that for each p = 2, ..., m only a root of fn (xp−1, m, x) depends upon x. 3rd part. by the second part and the definitions of cm,n and f we obtain the result which we want to prove. � proposition 2.8. let β > 0 be such that (xi − β, xi + β) ∩ (xj − β, xj + β) = ∅ where i 6= j and i, j ∈ {1, ..., m − 1}. then, for all ε > 0, there exists n0 ∈ n such that for n > n0 it follows that 1. | cm,n(x) − f(x) |<| kj+1 − kj | +ε 2. |cm,n(x) − (kj · (1 − α) + kj+1 · α)| < ε where x ∈ (xj − β, xj + β), j = 1, ..., m − 1and α ∈ (0, 1). proof. 1st part. let x ∈ (xj − β, xj + β) be, j = 1, ..., m − 1. by the proposition 2.7 the sequence (fn)n converges uniformly to 1 as p − 1 < j and to 0 as p − 1 > j. moreover there exists n0 ∈ n such that ∀n > n0 the function (kj+1 − kj) fn (xp−1, m, x) transforms the interval (xj − β, xj + β) into the interval (0, (kj+1 − kj)). the rest is obtained by the elementary properties of the limits and the definition of cm,n(x). 2nd part. it is analogous to the 1st part by considering ∀n > n0 the function (kj+1 − kj) fn (xp−1, m, x) attains on (xj − β, xj + β) the values (kj+1 − kj)· α, α ∈ (0, 1). � theorem 2.9. the rafu linear space ∁ is uniformly dense in c [a, b]. proof. consider the family l of all sets which are finite unions of disjoint compact intervals. first, we will prove that ∁ s-separates every pair of disjoint sets of l. clearly , it suffices to prove the following fact: given δ > 0 and the intervals [α1, βi], 1 ≤ i ≤ m, m ≥ 2, 0 ≤ αj < βj < αj+1 < 1, there is a function f in ∁ such that 0 ≤ f(x) ≤ 1 for every x ∈ [a, b], f ([αi, βi]) ⊂ [0, δ] for i odd and f ([αi, βi]) ⊂ [1 − δ, 1] for i even, 1 ≤ i ≤ m. consider a partition p = {xi}n0 of [a, b] with xj = a + j · b−a n , j = 0, ..., n such that βj < xp < αj+1 for every j and some xp. we also consider a step function h defined in [a, b] from the values xj such that h(x) = 0 or h(x) = 1 for every x ∈ [a, b] but verifying that h ([xs, xt]) = 0 when [αi, βi] ⊂ [xs, xt] and i is odd, h ([xk, xl]) = 1 when [αi, βi] ⊂ [xk, xl] and i is even, 1 ≤ i ≤ m. fixed an appropriate value β ≤ min { |xp−βj| 2 , |αj+1−xp| 2 } and given δ > 0 we can choose suitable partitions of [a, b] into 2kn intervals, if it was necessary for some k ∈ n, supporting the previous conditions and, by the propositions 2.7 and 2.8, we can obtain a function c2kn ∈ ∁ such that 0 ≤ c2kn(x) ≤ 58 e. corbacho cortés 1 and |c2kn − h| < δ, that is to say, c2kn ([αi, βi]) ⊂ [0, δ] for i odd and c2kn ([αi, βi]) ⊂ [1 − δ, 1] for i even. next, we will prove that ∁ s-separates every pair of disjoint zero-sets z1 and z2 of [a, b]. since l is a basis for the closed sets of [a, b] we have z1 = ∩ {b ∈ l : z1 ⊂ b}. as z2 is compact the family {z2} ∪ {b ∈ l : z1 ⊂ b} does not have the finite intersection property. therefore z2 ∩b1 ∩...∩bp = ø, for some bi ∈ l, zi ⊂ b, 1 ≤ i ≤ p. since l is closed under finite intersections it follows that b′ = b1 ∩ ... ∩ bp ∈ l, z1 ⊂ b′ and b′ ∩ z2 = ø. in the same way we find b′′ ∈ l, such that z2 ⊂ b′′ and b′ ∩ b′′ = ø. since ∁ s-separates b′ and b′′, by blasco-moltó’s theorem, ∁ is uniformly dense in c [a, b]. � the s-separation of subsets is equivalent to the s′-separation of subsets in linear spaces containing constant functions (garrido-montalvo [8]). clearly ∁ contains the constant functions, therefore we can also deduce the uniform density of ∁ in c [a, b] by using the s′-separation condition of every pair of disjoint zero-sets in x. 3. the degree of uniform approximation with the rafu linear space blasco-moltó [6] proved that the linear subspace f of c [0, 1] generated by the functions {exp ((x + µ)n) : µ ∈ r, x ∈ [0, 1] , n = 0, 1, 3, ..., 2k + 1, ...} is uniformly dense in c [0, 1], but the linear combinations which approximate uniformly a function f ∈ c [0, 1] and the degree of uniform approximation that f provides were not studied. the following result has been proved recently in the xxii cedya-xii cma [2] and solves these two problems by considering the linear space ∁ theorem 3.1. let f be a continuous function defined on [a, b] and let p = {x0 = a, x1, ..., xn = b} be a partition of [a, b] with xj = a + j · b−an , j = 0, ..., n. then, ‖cn − f‖ ≤ m − m√ n + ω ( b − a n ) , n ≥ 2 where ‖‖ denotes the uniform norm, m and m are the maximum and the minimum of f on [a, b] respectively, ω (δ) its modulus of continuty and cn(x) is defined for all x ∈ [a, b] and n ∈ n by cn(x) = f(a) + ∑n j=2[f(xj) − f(xj−1)] · fn (xj−1, x) let us observe that the values {ki}ni=1of (2.1) becomes {f(xi)} n i=1in this case. theorem 3.2 (gassó-hernández-rojas [9]). let a be a subset of c (x) and e a linear space of c (x) which s-separates lebesgue-sets of a. then the sublattice generated by a is contained in ∑ (e). a rafu linear space uniformly dense in c [a, b] 59 (a) approximation to f(x) (b) approximation to g(x) (c) approximation to h(x) (d) approximation to l(x) figure 1. approximation with the rafu linear space ∁ the rafu linear space ∁ satifies the theorem 3.1 when x = [a, b] because every lebesgue-set is also a zero-set since lα(f) = z((f − α) ∨ 0) and lβ(f) = z((f − β) ∧ 0) and we have proved that ∁ s-separates every pair of disjoint zero-sets z1 and z2 of [a, b]. in this case, if a = c (x) we can say that c (x) is contained in ∑ (∁). in fact, given f ∈ c [a, b], we already knew that f(x) = ∑∞ n=1 cn(x) where cn ∈ ∁, n ∈ n, and the series converges uniformly. example 3.3. given the functions f(x) = e−x 2 on [−3, 3], g(x) = 3x x2+1 on [−5, 6], h(x) = 5(x + 8)(x + 6)(x + 2)x(x − 3)(x − 5) on [−10, 6] and l(x) =| x | on [−10, 6], the figure 1 shows the graphics of these functions together with their approximations by means of its respective radical function c75 belonging to the rafu linear space ∁. 4. conclusions the rafu method is an original and unknown procedure of uniform approximation on c [a, b]. this method improves the instability of the polynomial interpolation and it is based in the use of radical functions to approximate any continuous function defined in [a, b]. we have constructed a linear space ∁ uniformly dense on c [a, b] and this linear space is not a lattice or an algebra. at the moment, the proof of this result was direct but in this work we have proved that ∁ is uniformly dense on c [a, b] by using a s-separation condition due to blasco-moltó [6] or an equivalent s′-separation condition due to garrido-montalvo [8]. we already knew another example of a linear space uniformly dense by using these separation conditions [6] but by considering the set ∁, we can know easily the linear combinations of elements belonging to 60 e. corbacho cortés ∁ which approximate uniformly every f ∈ c [a, b] and the degree of uniform approximation that ∁ provides. references [1] e. corbacho, teoŕıa general de la aproximación mediante funciones radicales, isbn 84-690-1149-9, (mérida, 2006). [2] e. corbacho, the degree of uniform approximation by radical funtions, xxii cedyaxii cma, september 5th-9th, (palma de mallorca, 2011). [3] e. corbacho, uniform approximation by means of radical functions, i jaen conference on approximation theory, july 4th-9th, (jaen, 2010). [4] g. j. o. jameson, topology and normed spaces, chapman and hall, (london, 1974). [5] h. tietze, uber functionen die anf einer abgeschlossenen menge stetig sind, journ. math. 145 (1915), 9–14. [6] j. l. blasco and a. moltó, on the uniform closure of a linear space of bounded realvalued functions, annali di matematica pura ed applicata iv vol. cxxxiv (1983), 233–239. [7] m. h. stone, applications of the theory of boolean rings to general topology, trans. amer. math. soc. 41 (1937), 375–481. [8] m. i. garrido, aproximación uniforme en espacios de funciones continuas, publicaciones del departamento de matemáticas universidad de extremadura 24, ( univ. extremadura, badajoz 1990). [9] m. t. gassó, s. hernández and s. rojas, aproximación por series en espacios de funciones continuas, (univesitat jaume i, departament de matemàtiques, 2010). [10] s. kakutani, concrete representation of abstract (m)-spaces, ann. math. 42 (1941), 994–1024. [11] s. mrowka, on some approximation theorems, nieuw archief voor wiskunde (3) xvi (1968), 94–111. [12] s. stone, a generalized weierstrass approximation theorem, math. magazine 21 (1948) 167–184, 237–254. (received january 2012 – accepted december 2012) e. corbacho cortés (ecorcor@unex.es) department of mathematics, university of extremadura, spain. a rafu linear space uniformly dense in c[a,b]. by e. corbacho cortés @ appl. gen. topol. 22, no. 1 (2021), 149-167doi:10.4995/agt.2021.14422 © agt, upv, 2021 on sheaves of abelian groups and universality s. d. iliadis ∗ and yu. v. sadovnichy moscow state university (m.v. lomonosov), moscow center of fundamental and applied mathematics (s.d.iliadis@gmail.com,sadovnichiy.yu@gmail.com) communicated by d. georgiou abstract universal elements are one of the most essential parts in research fields, investigating if there exist (or not) universal elements in different classes of objects. for example, classes of spaces and frames have been studied under the prism of this universality property. in this paper, studying classes of sheaves of abelian groups, we construct proper universal elements for these classes, giving a positive answer to the existence of such elements in these classes. 2010 msc: 14f05; 18f20; 54b40. keywords: sheaves; universal sheaves; universal spaces; containing spaces; saturated classes of spaces. 1. introduction and preliminaries the notion of “universal object” is considered in many branches of mathematics. the problem of the existence of such objects is naturally arised whenever a new category of objects is appeared. especially for the branch of topology, the problem of the existence of universal elements in different classes of topological spaces was considered at the first steps of its development. now, in the bibliography there are lots of papers concerning universal objects. many of them are indicated in the book [9]. in the paper [7] and in the above mentioned book, a method of construction of so-called containing spaces is developed. this method can be used for the ∗corresponding author. received 29 september 2020 – accepted 20 december 2020 http://dx.doi.org/10.4995/agt.2021.14422 s .d. iliadis and yu. v. sadovnichy construction of universal objects in different categories. such categories are, for example, topological spaces (with different dimension invariants)(see [3], chapter 3 of [9]), separable metric spaces (see chapter 9 of [9], [10], [12], [13], [15]), mappings (see [8], chapter 6 of [9], [10], [11]), topological groups ([11], [14], [17]), g-spaces (see chapter 7 of [9], [10], [11], [17], [18]) and frames (see [2], [4], [5], [6], [16]). in the present paper we use this method for construction of universal objects in the category of sheaves of abelian groups, which play an important role in the study of cohomology theories of general topological spaces. general notation and assumptions. an ordinal is considered as the set of all smaller ordinals. a cardinal is identified with the least ordinal of this cardinality. by τ we denote a fixed infinite cardinal. by f we denote the set of all non-empty finite subsets of τ. the symbol ≡ in a relation means that one or both sides of the relation are new notations. all spaces are assumed to be t0-spaces of weight ≤ τ. an equivalent relation on a set x is considered as a subset of x × x. 1.1 on the sheaves. we consider the notion of a sheaf according to [1]. a sheaf of abelian groups is a triad (a, π, x) satisfying the following conditions: (i) a and x are topological spaces and π is a map of a onto x. (ii) π is a local homeomorphism, that is each point a ∈ a has an open neighbourhood v in a such that the restriction of π on v is a homeomorphism of v onto an open subset of x; (iii) for each point x ∈ x the set ax ≡ π −1(x), which is called fiber of a in x, is an abelian group; (iv) the group operations are continuous. (this condition means the following. let a ⊠ a be the set of all pairs (a, b) ∈ a × a such that π(a) = π(b). then, the mapping ̟a : a ⊠ a → a for which ̟a(a, b) = a + b is continuous. similarly, the mapping ia : a → a for which ia(a) = −a is continuous.) below we give some well-known notions of sheaves and introduce some notations, which will be used in the paper. let p1 ≡ (a1, π1, x1) and p2 ≡ (a2, π2, x2) be two sheaves. a continuous mapping f of a1 into a2 is called homomorphism if the restriction of f onto each fiber of a1 is a homomorphism of this fiber into a fiber of a2. the unique mapping g of x1 into x2 satisfying the relation g ◦ π1 = π2 ◦ f is called induced by f. the homomorphism f is called isomorphism (or embedding) of p1 into p2 if f and the induced mapping g are embeddings. the isomorphism f of p1 into p2 is called proper if for each x ∈ x1 the restriction of f onto the fiber a1,x of a1 in x maps a1,x onto the fiber a2,g(x) of a2 in g(x). the sheaves p1 and p2 are called isomorphic if there exists an isomorphism of p1 onto p2. let (a, π, x) be a sheaf and u a non-empty subset of x. a continuous mapping s : u → a, for which the maping π ◦ s is the identical mapping of u, is called © agt, upv, 2021 appl. gen. topol. 22, no. 1 150 on sheaves of abelian groups and universality section of a on u. we shall consider sections on the open subsets of x and the set of all such sections will be denoted by a(sec). the set of all sections on an open subset u of x will be denoted by a(sec)(u). the set a(sec)(u) is an abelian group under the pointwise operations. for two subsets u and v such that u ⊂ v we denote by rau,v the mapping of a(sec)(v ) into a(sec)(u) by setting rau,v (s) = s|u for each s ∈ a(sec)(v ). the section s|u will be called restriction of the section s and the mapping rau,v restriction mapping. obviously, rau,v is a homomorphism of the group a(sec)(v ) into the group a(sec)(u). moreover, if u, v, w are subsets of x and u ⊂ v ⊂ w , then rau,v ◦ r a v,w = r a u,w . for each section s : u → a the set dom(s) ≡ u is called the domain of s and the set ran(s) ≡ s(u) is called the range of s. for each subset b ⊂ a(sec) and x ∈ x we put dom(b) = {dom(s) : s ∈ b}, ran(b) = {ran(s) : s ∈ b} and dom(b)(x) = {dom(s) : s ∈ b, x ∈ dom(s)}. also, for each non-empty subset u ⊂ x, we put b(u) = {s ∈ b : dom(s) = u}. we note that for each s ∈ a(sec) the set ran(s) is an open subset of a and that the set ran(a(sec)) is a base for the open subsets of a. the set dom(a(sec))(x) is directed by inclusion “ ⊂ ”. thus, for each x ∈ x we have a direct spectrum of abelian groups σax ≡ {a(sec)(u), r a u,v , dom(a(sec))(x)}, where u, v ∈ dom(a(sec))(x) with u ⊂ v and rau,v is the restriction mapping of cuts. the mapping ϑax : lim−→ σax → ax, of the limit group lim −→ σax of the spectrum σ a x into ax, defined by relation ϑax (σ) = s(x), where σ is an arbitrary element of the limit group lim−→ σax and s ∈ σ, is an isomorphism of lim −→ σax onto ax. 1.2 on the containing spaces. in this section we briefly explain the construction of the containing spaces (see [7], [9]). the spaces of the universal sheaves in the main result of the paper (see below theorem 1.3.1) will be containing spaces. a containing space is constructed for a given indexed collection s of spaces and it is uniquely determined by a base b for s (in [7] and [9] the base b is called mark and it is denoted by m): b ≡ {{uxδ : δ ∈ τ} : x ∈ s}, © agt, upv, 2021 appl. gen. topol. 22, no. 1 151 s .d. iliadis and yu. v. sadovnichy where {uxδ : δ ∈ τ} is an indexed base for the open subsets of x ∈ s, and by a family r of equivalence relations on s: r ≡ {∼t: t ∈ f}. it is required that r satisfies the following conditions: (a) for each t ∈ f the number of equivalence classes of ∼t is finite; (b) for each t1 ⊂ t2 ∈ f, ∼ t2 ⊂ ∼t1; (c) for each x, y ∈ s the condition x ∼t y for some t ∈ f, implies that the algebra of subsets of x, generated by the set {uxδ : δ ∈ t}, and the algebra of subsets of y , generated by the set {uyδ : δ ∈ t}, are isomorphic and the correspondence uxδ → u y δ , δ ∈ t, generates an isomorphism of these algebras. such a family is called b-admissible. also, for each t ∈ f we denote by c(∼t) the set of equivalence classes of ∼t and put c(r) = ∪{c(∼t) : t ∈ f}. the corresponding containing space is denoted by t ≡ t(b, r) and its construction is done as follows. let ∼br be the equivalence relation on a set of all pairs (x, x), where x ∈ x ∈ s, defined as follows: two such pairs (x, x) and (y, y ) are ∼br-equivalent if and only if: (a) x ∼t y for each t ∈ f and (b) for each δ ∈ τ, x ∈ uxδ if and only if y ∈ uyδ . then, t is the set of all equivalence classes of ∼ b r and the set bt ≡ {utδ (h) : δ ∈ τ, h ∈ c(r)}, where utδ (h) is the set consisting of all points a ∈ t such that there exists an element (x, x) ∈ a for which x ∈ h and x ∈ uxδ , is a base for a topology on t, called standard base (see corollary 2.8 of [7]). we note that if for some κ ⊂ τ and for each x ∈ s the set {uxδ : δ ∈ κ} is a base for the open subsets of x, then the set {utδ (h) : δ ∈ κ, h ∈ c(r)} is also a base for the open subsets of the space t (see corollary 2.8 of [7]). the mapping ixt : x → t, defining by the relation i x t (x) = a ∈ t, where x ∈ x ∈ s and a is the point of t containing the pair (x, x), is an embedding of x into t, which is called natural (see proposition 2.10 of [7]). in the paper, we shall use also the following notions. let b1 ≡ {{u x 1,δ : δ ∈ τ} : x ∈ s} and b2 ≡ {{u x 2,δ : δ ∈ τ} : x ∈ s}, where {ux1,δ : δ ∈ τ} and {u x 2,δ : δ ∈ τ} are indexed sets of subsets of x ∈ s (in particular, they may be indexed bases of x) and b2 is a base for s. the base b2 is an extension of b1 if there exists an one-to-one mapping ϑ : τ → τ, called extension mapping, such that ux1,δ = u y 2,ϑ(δ), δ ∈ τ. we shall also say that for a given x ∈ s, {ux2,δ : δ ∈ τ} is an extension of {u x 1,δ : δ ∈ τ} with the extension mapping ϑ. let r1 ≡ {∼ t 1: t ∈ f}, and r2 ≡ {∼ t 2: t ∈ f} © agt, upv, 2021 appl. gen. topol. 22, no. 1 152 on sheaves of abelian groups and universality be two families of equivalence relations on s. we say that r2 is a final refinement of r1 if for each t ∈ f there exists t ′ ∈ f such that ∼t ′ 2 ⊂ ∼ t 1. a class s of spaces is called saturated if for each indexed collection s of elements of s, there exists a base b0 for s such that for each extension b of b0, there exists a b-admissible family rb of equivalence relations on s with the property that for each admissible family r of equivalence relations on s being a final refinement of rb, the containing space t(b, r) belongs to s (see section 3 of [7] and chapter 2 of [9]). the base b0 is called initial base for s (corresponding to the class s) and rb initial family of equivalence relations on s corresponding to b (and the class s). below, we give some examples of saturated classes of spaces of weight ≤ τ. (1) the class of all t0-spaces (see propositions 2.9 of [7]); (2) the class of all regular spaces (see propositions 3.5 of [7]); (3) the class of all completely regular spaces (see propositions 3.8 of [7]); (4) the class of all spaces of small inductive dimension ind ≤ n ∈ n; (5) the class of all countable-dimensional spaces; (6) the class of all strongly contable-dimensional spaces; (7) the class of all locally finite-dimensional spaces; (8) the intersection of any two saturated classes of spaces. (for the above example (4) see corrolary 3.1.6 of [9], for (5), (6) and (7) see proposition 4.4.4 of [9] and for example (8) see proposition 3.3 of [7]). 1.3 the results. let s be a class of sheaves. a sheaf p̄ is called proper universal in the class s if p̄ ∈ s and for each p ∈ s there exist a proper isomorphism of p into p̄. the main result of this paper is the following theorem. theorem 1.3.1. let sd and sr be two saturated classes of spaces of weights ≤ τ. then, in the class of all sheaves (a, π, x), for which a ∈ sd and x ∈ sr, there exists a proper universal element (ā, π̄, x̄). since the class of t0-spaces of countable weight and the class of separable metric spaces are saturated classes we have the following corollary. corollary 1.3.2. in the class of all sheaves (a, π, x), where a is a t0-space of countable weight and x is a separable metrizable space there exists a proper universal element. 2. proof of the result lemma 2.1. let (a, π, x) be a sheaf of abelian groups. there exists a subset b ⊂ a(sec) such that: (a) ran(b) is a base for the open subsets of a of cardinality w(a) ≤ τ and, therefore, the set dom(b) is a base for the open subsets of x; (b) for each u ∈ dom(b), b(u) is a subgroup of the group a(sec)(u). © agt, upv, 2021 appl. gen. topol. 22, no. 1 153 s .d. iliadis and yu. v. sadovnichy (c) for each u, v ∈ dom(b) with u ⊂ v the restriction s|u of any cut s ∈ b(v ) belongs to b. proof. since a(sec) is a base for a, there exists a subset b0 ⊂ a(sec) such that ran(b0) is a base for the open subsets of a of cardinality w(a). by induction we define the subset bn ⊂ a(sec), n ∈ n, setting bn = bn−1 ∪ (∪{bn−1+ (u) ⊂ a(sec)(u) : u ∈ dom(b n−1)}) ∪{s|u : s ∈ b n−1(v ), u, v ∈ dom(bn−1), u ⊂ v }, where bn−1+ (u) is the subgroup of a(sec)(u) generated by the set b n−1(u). it is easy to see that the set b ≡ ∪{bn : n ∈ ω} is the required set. � the direct spectrum σbx . let (a, π, x) be a sheaf, b a subset of a(sec), satisfying the conditions of lemma 2.1, and x ∈ x. by property (a) of this lemma, ran(b) is a base of a and, therefore, the set dom(b) is a base for the open subsets of x. hence, the set dom(b)(x) is directed by inclusion “ ⊂ ”. by property (c) of lemma 2.1, for each u, v ∈ dom(b)(x) with u ⊂ v the restriction of rau,v onto b(v ) is an isomorphism of b(v ) into b(u). we shall denote this restriction by rbu,v . thus, for each x ∈ x we have a direct spectrum σbx of groups: σbx ≡ {b(u), r b u,v , dom(b)(x)}. (2.1.1) let σb be an arbitrary element of the limit group lim −→ σbx of the spectrum (2.1.1) and s ∈ σb. we define the mapping ϑbx : lim−→ σbx → ax setting ϑ b x (σ b) = s(x). lemma 2.2. let ϑb,ax : lim−→ σbx → lim−→ σax be the mapping defined as follows: for each σb ∈ lim −→ σbx we put ϑ b,a x (σ b) = σa, where σa is the element of lim −→ σax containing σ b. then, ϑb,ax is welldefined (that is, the element σa is uniquely determined), one-to-one, onto and preserves the group operations, that is it is an isomorphism of lim −→ σbx onto lim −→ σax . moreover, ϑ b x = ϑ a x ◦ ϑ b,a x and, therefore, ϑ b x is an isomorphism and onto mapping. proof. since the mappings rbu,v of the spectrum σ b x are the restrictions of the corresponding mappings rau,v of the spectrum σ a x each element σ b of lim −→ σbx is contained in an uniquely determined element σa of lim −→ σax , that is the mapping ϑb,ax is well-defined. we prove that ϑb,ax is one-to-one. let σ b 1 and σ b 2 be two distinct elements of lim −→ σbx and let σ a 1 , σ a 2 ∈ lim−→ σax such that σ b 1 ⊂ σ a 1 and σ b 2 ⊂ σ a 2 . suppose that σa1 = σ a 2 and let s1 ∈ σ b 1 and s2 ∈ σ b 2 and, therefore, s1, s2 ∈ σ a 1 . then, there exists s3 ∈ σ a 1 , which is a restriction of s1 and a restriction of s2. since dom(b) is a base for the open subsets of x (see property (a) of lemma 2.1) there exists s0 ∈ b such that x ∈ dom(s0) ⊂ dom(s3) and ran(s0) ⊂ ran(s3). © agt, upv, 2021 appl. gen. topol. 22, no. 1 154 on sheaves of abelian groups and universality then, s0 is the restriction of s3 and, therefore, the restriction of s1 and s2, which contradicts the fact that s1 and s2 belong to distinct elements of lim−→ σbx . thus, ϑb,ax is one-to-one. we prove that ϑb,ax is onto. let σ a be an element of lim −→ σax and s ∈ σ a x . consider an element s0 ∈ b such that x ∈ dom(s0) ⊂ dom(s) and ran(s0) ⊂ ran(s). then for the element σb containing s0 we have ϑ b,a x (σ b) = σa, proving that ϑb,ax is onto. we prove that ϑb,ax preserves the group operations. let σ b 1 , σ b 2 ∈ lim−→ σbx and let ϑb,ax (σ b 1 ) = σ a 1 and ϑ b,a x (σ b 2 ) = σ a 2 . let s1 ∈ σ b 1 and s2 ∈ σ b 2 . consider an element s0 ∈ b such that dom(s0) ⊂ dom(s1)∩dom(s2). let s ′ 1 = s1|dom(s0) and s′2 = s2|dom(s0). then, by proprety (c) of lemma 2.1, s ′ 1, s ′ 2 ∈ b and by property (b) of this lemma, s′1 + s ′ 2 ∈ b. therefore, s ′ 1 + s ′ 2 ∈ σ b 1 + σ b 2 . on the other hand, s′1 ∈ σ a 1 and s ′ 2 ∈ σ a 2 and, therefore, s ′ 1 + s ′ 2 ∈ σ a 1 + σ a 2 , proving that ϑb,ax preserves the sum operation. similarly, we can prove that ϑb,ax preserves the taking of the inverse element. thus, the mapping ϑ b,a x is an isomorphism of lim −→ σbx onto lim−→ σax .the relation ϑ b x = ϑ a x ◦ ϑ b,a x is easy to verify. � the indexed collections s, a and x. consider the saturated classes sd and sr of the theorem. by set-theoretical reasons we can suppose that there exists a collection s of sheaves (a, π, x) such that a ∈ sd, x ∈ sr and each sheaf (a′, π′, x′), for which a′ ∈ sd and x ′ ∈ sr, is isomorphic to an element of s. moreover, we can suppose that s is indexed by a set λ: s ≡ {(aλ, πλ, xλ) : λ ∈ λ}. we put a ≡ {aλ : λ ∈ λ}, x ≡ {xλ : λ ∈ λ}. and consider a and x as indexed by λ sets of topological spaces. the bases ba and bx for a and x, respectively. for each element (aλ, πλ, xλ) ∈ s we consider a subset bλ ⊂ aλ(sec) satisfying the conditions of lemma 2.1. since |bλ| = w(aλ) ≤ τ (see the property (a) of lemma 2.1), we can suppose that bλ is indexed by the set τ: bλ = {s λ η : η ∈ τ}. furthermore, we put b aλ 0 ≡ {v aλ η ≡ ran(s λ η ) : η ∈ τ}, and b xλ 0 ≡ {v xλ η ≡ dom(s λ η) : η ∈ τ}. let θ0 and θ1 be two one-to-one mappings of τ into itself such that |θ0(τ)| = |θ1(τ)|, θ0(τ) ∩ θ1(τ) = ∅ and θ0(τ) ∪ θ1(τ) = τ. © agt, upv, 2021 appl. gen. topol. 22, no. 1 155 s .d. iliadis and yu. v. sadovnichy (we note that these mappings are independed on λ ∈ λ.) for each λ ∈ λ we put w aλ ζ = v aλ θ −1 0 (ζ) if ζ ∈ θ0(τ) and w aλ ζ = π−1 λ (v xλ θ −1 1 (ζ) ) if ζ ∈ θ1(τ). therefore, the indexed set baλ1 ≡ {w aλ ζ : ζ ∈ τ} is an extension of the indexed base b aλ 0 of aλ and, simultaneously, an extension of the indexed set π−1 λ (bxλ0 ) ≡ {π −1 λ (v xλη ) : η ∈ τ} of subsets of aλ with the extension mappings θ0 and θ1, respectively. now, we consider a base b a ≡ {baλ ≡ {uaλε : ε ∈ τ} : λ ∈ λ} for a, which is an initial base corresponding to the saturated class sd and, simultaneously, is an extension of the base b a 1 ≡ {{w aλ ζ : ζ ∈ τ} : λ ∈ λ} for a with an extension mapping θa. also, we consider a base b x ≡ {bxλ ≡ {u xλ δ : δ ∈ τ} : λ ∈ λ} for x, which is an initial base corresponding to the saturated class sr and, simultaneously, is an extension of the base b x 0 ≡ {{v xλ η : η ∈ τ} : λ ∈ λ} for x with an extension mapping θx. the families ra and rx of equivalence relations. we denote by ra ≡ {∼ t a : t ∈ f} a ba-admissible family of equivalence relation on a and by rx ≡ {∼ t x : t ∈ f} a bx-admissible family of equivalence relations on x. we suppose that ra and rx satisfy the following conditions: (1) for each λ, µ ∈ λ and t ∈ f the equivalence xλ ∼ t x xµ is true if and only if the equivalence aλ ∼ t a aµ is true; (2) for each λ, µ ∈ λ, t ∈ f, and η1, η2, η ∈ t the equivalence aλ ∼ t a aµ implies that the conditions: (21) dom(s λ η1 ) = dom(sλη2) and s λ η1 + sλη2 = s λ η and (22) dom(s µ η1 ) = dom(sµη2) and s µ η1 + sµη2 = s µ η are equivalent; (3) for each λ, µ ∈ λ, t ∈ f, and η1, η2 ∈ t the equivalence aλ ∼ t a aµ implies that the conditions: © agt, upv, 2021 appl. gen. topol. 22, no. 1 156 on sheaves of abelian groups and universality (31) dom(s λ η1 ) = dom(sλη2) and s λ η1 = −sλη2 and (32) dom(s µ η1 ) = dom(sµη2) and s µ η1 = −sµη2 are equivalent ; (4) for each λ, µ ∈ λ, t ∈ f, and η1, η2 ∈ t the equivalence aλ ∼ t a aµ implies that the conditions: (41) ran(s λ η1 ) ⊂ ran(sλη2 ) and (42) ran(s µ η1 ) ⊂ ran(sµη2 ) are equivalent; (5) for each λ, µ ∈ λ, t ∈ f, and η1, η2 ∈ t the equivalence aλ ∼ t a aµ implies that the conditions: (51) dom(s λ η1 ) ⊂ dom(sλη2) and (52) dom(s µ η1 ) ⊂ dom(sµη2) are equivalent. lemma 2.3. the ba-admissible family ra and the b x-admissible family rx satisfying conditions (1) − (5) exist. proof. since the class sd is saturated there exists a b a-admissible family r0,a ≡ {∼ t 0,a: t ∈ f}, which is initial for the base b a and the class sd. similarly, there exists bx-admissible family r0,x ≡ {∼ t 0,x: t ∈ f}, which is initial for the base bx and the class sr. let t ∈ f and η1, η2, η ∈ t. we denote by ∼ t i, i ∈ {2, 3, 4, 5}, the equivalence relation on a defined as follows: aλ ∼ t i aµ, λ, µ ∈ λ, if and only if the conditions (i1) and (i2) are equivalent for all indexes η1, η2, η, which belong to t. obviously, the relations ∼ti, i ∈ {2, 3, 4, 5}, are admissible. let r1,a ≡ {∼ t 1,a: t ∈ f} be the family of equivalence relations on a, where ∼t1,a=∼ t 0,a ∩(∩{∼ t i: i ∈ {2, 3, 4, 5}}) for each t ∈ f. now, for each t ∈ f we define the equivalence relation ∼t a on a as follows: aλ ∼ t a aµ, λ, µ ∈ λ, if and only if aλ ∼ t 1,a aµ and xλ ∼ t 0,x xµ. also, we define the equivalence relation ∼t x , t ∈ f, on x as follows: xλ ∼ t x xµ if and only if aλ ∼ t a aµ. it is easy to see that ra ≡ {∼ t a : t ∈ f} and rx ≡ {∼ t x : t ∈ f} are the required families of equivalence relations. � the equivalence relations ∼a and ∼x. we put ∼a= ∩{∼ t a : t ∈ f} and ∼x= ∩{∼ t x : t ∈ f}. the following two lemmas can easily be proved. lemma 2.4. let aλ ∼a aµ, λ, µ ∈ λ. then, the algebra of subsets of aλ, generated by the set baλ, and the algebra of subsets of aµ, generated by the set baµ, are isomorphic and the correspondence uaλε → u aµ ε , ε ∈ τ, generates this isomorphism. therefore, for any κ ⊂ τ the algebra of subsets of aλ, generated by the set {uaλε : ε ∈ κ}, and the algebra of subsets of aµ, generated by the © agt, upv, 2021 appl. gen. topol. 22, no. 1 157 s .d. iliadis and yu. v. sadovnichy set {u aµ ε : ε ∈ κ}, are isomorphic and the correspondence u aλ ε → u aµ ε , ε ∈ κ, generates this isomorphism. moreover, for each η, η1, η2 ∈ τ we have: (a) the cut sλη1 is a restriction of the cut s λ η2 if and only if the cut sµη1 is the restriction of the cut sµη2; (b) the equalities dom(sλη1) = dom(s λ η2 ) and sλη = s λ η1 + sλη2 are true if and only if the equalities dom(sµη1) = dom(s µ η2 ) and sµη = s µ η1 + sµη2 are true; (c) the equalities dom(sλη1) = dom(s λ η2 ) and sλη1 = −s λ η2 are true if and only if the equalities dom(sµη1) = dom(s µ η2 ) and sµη1 = −s µ η2 are true. lemma 2.5. let xλ ∼x xµ, λ, µ ∈ λ. then, the algebra of subsets of xλ, generated by the set bxλ, and the algebra of subsets of xµ, generated by the set bxµ, are isomorphic and the correspondence u xλ δ → u xµ δ , δ ∈ τ, generates this isomorphism. therefore, for any k ⊂ τ the algebra of subsets of xλ, generated by the set {u xλ δ : δ ∈ κ}, and the algebra of subsets of xµ, generated by the set {u xµ δ : δ ∈ κ}, are isomorphic and the correspondence u xλ δ → u xµ δ , δ ∈ κ, generates this isomorphism. the triad (ā, π̄, x̄). we put ā = t(ba, ra), x̄ = t(b x, rx) and define the mapping π̄ as follows. let a ∈ ā and (aλ, aλ) ∈ a for some λ ∈ λ. then, we put π̄(a) = x, where x is the point of x̄ containing the pair (πλ(a λ), xλ). in what follows we shall prove that the triad (ā, π̄, x̄) is the required universal sheaf. lemma 2.6. the mapping π̄ is correctly defined (that is, it is independent from the element (aλ, aλ) ∈ a considered in its definition). proof. let a ∈ ā and (aλ, aλ), (b µ, aµ) ∈ a, that is (a λ, aλ) and (b µ, aµ) are ∼b a ra -equivalent. we must prove that if πλ(a λ) = xλ and πµ(b µ) = yµ, then (xλ, xλ) and (y µ, xµ) are ∼ b x rx -equivalent, that is xλ ∼x xµ and for each δ ∈ τ either xλ ∈ uxλ δ and yµ ∈ u xµ δ or xλ /∈ uxλ δ and yµ /∈ u xµ δ . since (aλ, aλ) and (bµ, aµ) are ∼ b a ra -equivalent, aλ ∼a aµ. by the condition (1) of the definitions of ra and rx we have xλ ∼x xµ. suppose that there exists δ0 ∈ τ such that, for example, x λ ∈ uxλ δ0 and yµ /∈ u xµ δ0 . then, aλ ∈ π−1 λ (uxλ δ0 ) and bµ /∈ π−1µ (u xµ δ0 ). since the set b xλ 0 is a base for the open subsets of xλ, there exists η ∈ τ such that xλ ∈ v xλη ⊂ u xλ δ0 . let δ1 = θx(η) and, therefore, v xλ η = u xλ δ1 and v xµ η = u xµ δ1 . then, u xλ δ1 ⊂ u xλ δ0 . by lemma 2.5 we have u xµ δ1 ⊂ u xµ δ0 and, therefore, yµ /∈ u xµ δ1 = v xµ η . since b a is an extension of π−1 λ (bxλ0 ) with the extension mapping θa ◦ θ1 we have u aν ε = π −1 ν (v xν η ) for each ν ∈ λ and ε = θa(θ1(η)). therefore, for λ and µ we have u aλ ε = π −1 λ (v xλη ) and u aµ ε = π −1 µ (v xµ η ), respectively, and hence a λ ∈ uaλε and b µ 6= u aµ ε , which contradicts the fact that (aλ, aλ) and (b µ, aµ) are ∼ b a ra -equivalent. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 158 on sheaves of abelian groups and universality the following lemma can easily be verified. lemma 2.7. for each λ ∈ λ the following relation is true: π̄ ◦ iaλ ā = ixλ x̄ ◦ πλ, (2.7.1) where i aλ ā is the natural embedding of aλ into ā and i xλ x̄ is the natural embedding of xλ into x̄. lemma 2.8. for each η ∈ τ and h ∈ c(ra) we have π̄(uāε (h)) = u x̄ δ (l), where ε = θa(θ0(η)), δ = θx(η) and l is the element of c(rx), consisting of all elements xλ ∈ x for which aλ ∈ h (we shall say that l and h correspond each other). proof. let η ∈ τ, ε = θa(θ0(η)), δ = θx(η) and h ∈ c(ra). then, for each λ ∈ λ, uaλε = im(s λ η). by the definition of the elements of the standard base of the containing spaces, we have uāε (h) = ∪{i aλ ā (uaλε ) : aλ ∈ h} = ∪{i aλ ā (ran(sλη )) : aλ ∈ h}. therefore, using relation (2.7.1), we have π̄(uāε (h)) = ∪{π̄(i aλ ā (ran(sλη ))) : aλ ∈ h} = ∪{i xλ x̄ (πλ(ran(s λ η))) : xλ ∈ l} = ∪{i xλ x̄ (dom(sλη)) : xλ ∈ l} = ∪{ixλ x̄ (uxλ δ )) : xλ ∈ l} = u x̄ δ (l). � proposition 2.9. the mapping π̄ is continuous. proof. since the set {ux̄δ (l) : δ ∈ θx(τ), l ∈ c(rx)} is a base of the space x̄ it suffices to prove that the set π̄−1(ūxδ (l)) is open in ā for each δ ∈ θx(τ) and l ∈ c(rx). let δ be a fixed element of θx(τ) and l a fixed element of c(rx). let η = θ−1 x (δ) and ε = θa(θ1(η)). then, for each ν ∈ λ we have π−1ν (u xν δ ) = uaνε . (2.9.1) we shall prove the following equality, which will prove the continuity of π̄: π̄−1(ux̄δ (l)) = u ā ε (h), (2.9.2) where h is the element of c(ra) corresponding to l. let a ∈ π̄ −1(ux̄δ (l)), that is π̄(a) ≡ x ∈ ux̄δ (l). let (x λ, xλ) ∈ x and (a µ, aµ) ∈ a. since π̄(a) = x, by the definition of π̄ we have (πµ(a µ), xµ) ∈ x. this means that xλ ∼x xµ, that is xµ ∈ l and, therefore, aµ ∈ h. also, πµ(a µ) ∈ u xµ δ and, therefore, aµ ∈ π−1µ (u xµ δ ) = uaµε (2.9.3) © agt, upv, 2021 appl. gen. topol. 22, no. 1 159 s .d. iliadis and yu. v. sadovnichy (see relation (2.9.1)). since aµ ∈ h, the relation (2.9.3) shows that a ∈ u ā ε (h), proving that the left side of the relation (2.9.2) is contained in the right. conversly, let a ∈ uāε (h) and (a µ, aµ) ∈ a. then, aµ ∈ h and aµ ∈ u aµ ε . therefore, xµ ∈ l and a µ ∈ π−1µ (u xµ δ ) (see relation 2.9.1) or πµ(a µ) ∈ u xµ δ . this means that π̄(a) ∈ ux̄δ (l) and, therefore, a ∈ π̄ −1(ux̄δ (l)), proving that the right side of (2.9.2) is contained in the left, completing the proof of the proposition. � the set āx, x ∈ x̄. for each x ∈ x̄ we put āx = {a ∈ ā : π̄(a) = x}. we shall prove that āx is an abelian group. first we shall prove the following lemma. lemma 2.10. let (aλ, πλ, xλ) and (aµ, πµ, xµ) be two elements of s such that aλ ∼a aµ and let x ∈ x̄. then, for each two elements (xλ, xλ), (x µ, xµ) ∈ x ∈ x̄ there exists an isomorphism ϑxλxµ of aλ,xλ onto aµ,xµ such that: (a) ϑxλxλ is the identical isomorphism; (b) ϑ xµ xν ◦ ϑ xλ xµ = ϑxλxν , where (x ν, xν) ∈ x. moreover, for each aλ ∈ aλ,xλ, we have (aλ, aλ) ∼ b a ra (ϑx λ xµ(a λ), aµ). (2.10.1) proof. let (xλ, xλ), (x µ, xµ) ∈ x ∈ x̄ for some fixed λ, µ ∈ λ. then, x λ ∈ uxλ δ for some δ ∈ θx(τ) if and only if x µ ∈ u xµ δ , that is xλ ∈ dom(sλη) for some η ∈ τ, if and only if xµ ∈ dom(sµη ). denote by κ all such η. by lemma 2.5 it follows that the mapping dom(sλη) → dom(s µ η ), η ∈ κ, is an isomorphism of the directed by inclusion set dom(bλ)(x λ) onto the directed by inclusion set dom(bµ)(x µ). let σλ ≡ {sλη : η ∈ τ(σ λ)} be an element of the limit group lim −→ σ bλ xλ , where τ(σλ) is the set of all η ∈ τ for which sλη ∈ σ λ. consider the set σµ(λ) ≡ {sµη : η ∈ τ(σ λ)} of sections of aµ. let s λ η1 , sλη2 ∈ σ λ such that ran(sλη1) ⊂ ran(s λ η2 ), that is the section sλη1 is the restriction of the section s λ η2 . by relation aλ ∼a aµ and lemma 2.4 it follows that ran(sµη1 ) ⊂ ran(s µ η2 ), that is the section sµη1 is the restriction of the section sµη2. this means that the set σ µ(λ) is a subset of an unique determined element σµ of the limit group lim −→ σ bµ xµ . similarly, the constructed element σµ of the limit group lim −→ σ bµ xµ defines a set σλ(µ) of sections of aλ. by construction, σ λ ⊂ σλ(µ) and, therefore, © agt, upv, 2021 appl. gen. topol. 22, no. 1 160 on sheaves of abelian groups and universality σλ = σλ(µ). similarly, σµ = σµ(λ). we will say that σλ ∈ lim −→ σbλ xλ and σµ ∈ lim −→ σ bµ xµ correspond each other. thus, we have defined mutually inverse, one-to-one and onto mappings ϑ(xλ, xµ) : lim −→ σ bλ xλ → lim −→ σ bµ xµ and ϑ(x µ, xλ) : lim −→ σ bµ xµ → lim−→ σ bλ xλ . we prove that the mapping ϑ(xλ, xµ) is an isomorphism, that is it preserves the group operations. let σλ1 , σ λ 2 ∈ lim−→ σbλ xλ and let sλη1 ∈ σ λ 1 and s λ η2 ∈ σλ2 . since the set dom(bλ)(x λ) is a base of xλ at the point xλ, there exists sλη ∈ bλ such that u ≡ dom(sλη) ⊂ dom(s λ η1 ) ∩ dom(sλη2). by condition (c) of lemma 2.1 the restrictions sλη1|u and s λ η2 |u belong to bλ and by condition (b), s λ η1 |u + s λ η2 |u belongs to bλ. then, there exist η ′ 1, η ′ 2, η ′ 0 ∈ τ such that sλη′ 1 = sλη1|u ∈ σ λ 1 , s λ η′ 2 = sλη2|u ∈ σ λ 2 and s λ η′ 0 = sλη′ 1 + sλη′ 2 ∈ σλ1 + σ λ 2 . let ϑ(xλ, xµ)(σλ1 ) = σ µ 1 , ϑ(x λ, xµ)(σλ2 ) = σ µ 2 , and ϑ(x λ, xµ)(σλ) = σµ. by lemmas 2.4 and 2.5 we have v ≡ dom(sµη ) ⊂ dom(s µ η1 ) ∩ dom(s µ η2), s µ η′ 1 = sµη1|v ∈ σ µ 1 , s µ η′ 2 = sµη2|v ∈ σ µ 2 , and, s µ η′ 0 = s µ η′ 1 + s µ η′ 2 ∈ σ µ 1 + σ µ 2 therefore, ϑ(xλ, xµ)(σλ1 + σ λ 2 ) = ϑ(x λ, xµ)(σλ1 ) + ϑ(x λ, xµ)(σλ2 ). similarly, we prove that ϑ(xλ, xµ)(−σλ1 ) = −ϑ(x λ, xµ)(σλ1 ). thus, the mapping ϑ(xλ, xµ) and, therefore, the mapping ϑ(xµ, xλ) is an isomorphism and onto. the required isomorphism ϑx λ xµ of aλ,xλ onto aµ,xµ is defined by setting ϑx λ xµ = ϑ bµ xµ ◦ ϑ(x λ, xµ) ◦ (ϑ bλ xλ )−1. conditions (a) and (b) of the lemma can easily be verified. we prove relation (2.10.1). let aλ ∈ aλ,xλ and σ λ 0 = (ϑ bλ xλ )−1(aλ). then, ϑx λ xµ(a λ) = ϑ bµ xµ (ϑ(x λ, xµ)(σλ0 )) = ϑ bµ xµ (σ µ 0 ), where σ µ 0 is the element lim−→ σ bµ xµ corresponding to σ λ 0 . thus, it suffices to prove that the pairs (ϑbλ xλ (σλ0 ), aλ) and (ϑ b µ xµ (σ µ 0 ), aµ) belong to the same element of ā. let uaλε be an element of b aλ containing ϑ bλ xλ (σλ). we need to prove that ϑ bµ xµ (σ µ) ∈ uaµε . (2.10.2) since the set {ran(s) : s ∈ σλ} is a base for the open subsets of aλ at the point ϑ bµ xµ (σ µ), there exists η ∈ τ such that sλη ∈ σ λ and ϑbλ xλ (σλ) ∈ ran(sλη) ⊂ u aλ ε . © agt, upv, 2021 appl. gen. topol. 22, no. 1 161 s .d. iliadis and yu. v. sadovnichy since σµ corresponds to σλ, sµη ∈ σ. by the definition of the mapping ϑ bµ xµ , ϑ bµ xµ (σ µ) = sµη(x µ) ∈ ran(sµη ). since aλ ∼a aµ, the condition ran(s λ η) ⊂ u aλ ε implies that ran(sµη ) ⊂ u aµ ε , proving relation (2.10.2). � proposition 2.11. for each x ∈ x̄ the set āx is an abelian group and for each (xλ, xλ) ∈ x, λ ∈ λ, the natural embedding i aλ ā of aλ into ā maps the fiber aλ,xλ of aλ onto the set āx. proof. let x ∈ x̄, (xλ, xλ) ∈ x for some fixed λ ∈ λ and a ∈ āx. by relation (2.7.1) it follows that iaλ ā (aλ,xλ) ⊂ āx. we must prove that i aλ ā (aλ,xλ) = āx. by the definition of the containing spaces there exists µ ∈ λ and a point aµ ∈ aµ such that (a µ, aµ) ∈ a. using relation (2.7.1) we can see that (xµ, xµ) ∈ x, where x µ = πµ(a µ). let aλ = (ϑx λ xµ) −1(aµ). then, by the relation (2.10.1), (aλ, aλ) ∼ b a ra (aµ, aµ) and, therefore, (a λ, aλ) ∈ a, that is i aλ ā (aλ) = a, proving that i aλ ā maps the fiber aλ,xλ of aλ onto the set āx. now, on the set āx, x ∈ x̄, we define the group operations. let a1, a2 ∈ āx and let (aλ1 , aλ) ∈ a1 and (a λ 2 , aλ) ∈ a2. then, we put a1 + a2 = a, where a is the element of āx, containing the pair (a λ 1 + a λ 2 , aλ). also, we consider that −a1 is the element of āx, containing the pair (−a λ 1, aλ). obviously, by these operations āx becomes an abelian group such that the restriction onto aλ,xλ of the natural embedding i aλ ā of aλ into ā, is an isomorphism of aλ,xλ onto āx. it remains to prove that the defined operations are independent of the element (xλ, xλ) ∈ x. let (x ν, xν) ∈ x for some ν ∈ λ, (a ν 1, aν) ∈ a1 and (a ν 2, aν) ∈ a2. we need to prove that the pair (a λ 1 + a λ 2, aλ) and (a ν 1 + a ν 2, aν) belong to the same element of āx. since, by lemma 2.10, ϑ x λ xν is an isomorphism we have ϑx λ xν (a λ 1 + a λ 2 ) = ϑ x λ xν (a λ 1 ) + ϑ x λ xν (a λ 2 ). on the other hand, since the pairs (aλ1 , aλ) and (ϑ x λ xν (a λ 1 ), aν) belong to the same element of ā we have (ϑx λ xν (a λ 1), aν) ∈ a1 and since (a ν 1, aν) ∈ a1 we have ϑx λ xν (a λ 1) = a ν 1. similarly, ϑ x λ xν (a λ 2 ) = a ν 2. therefore, ϑ x λ xν (a λ 1 + a λ 2) = a ν 1 + a ν 2. since, by lemma 2.10, the pairs (aλ1 + a λ 2, aλ) and (ϑ x λ xν (a λ 1 + a λ 2 ), aν) belong to the same element of ā, the pairs (aλ1 +a λ 2 , aλ) and (a ν 1 +a ν 2, aν) also belong to the same element of ā, proving that the sum operation is independent of the element (xλ, xλ) ∈ x. similarly, we prove that the operation of taking the inverse element is independent of (xλ, xλ) ∈ x. the proof of the proposition is completed. � the mappings ̟ā and iā. we put ā ⊠ ā ≡ {(a, b) ∈ ā × ā : π̄(a) = π̄(b)} © agt, upv, 2021 appl. gen. topol. 22, no. 1 162 on sheaves of abelian groups and universality and define the mappings ̟ā : ā ⊠ ā → ā and iā : ā → ā setting ̟ā(a, b) = a + b and iā(a) = −a. proposition 2.12. the mappings ̟ā and iā are continuous. proof. we prove that ̟ā is continuous. let (a1, a2) ∈ ā ⊠ ā and a ≡ a1 + a2. let u be an open neighbourhood of a in ā. we must find open neighbourhoods u1 and u2 of a1 and a2, respectively, in ā such that ̟ā((u1 × u2) ∩ (ā ⊠ ā)) ⊂ u. (2.12.1) without loss of generality, we can suppose that u is an element uāε (h) of the standard base of ā, where ε is a fixed element of τ and h is a fixed element of c(∼t a ) for some fixed t ∈ f. moreover, we can suppose that ε ∈ θa(θ0(τ)). this means that for each aµ ∈ h we have u aµ ε = ran(s µ η ), where η = (θa ◦ θ0) −1(ε) and, therefore, uāε (h) = ∪{i aµ ā (uaµε ) : aµ ∈ h} = ∪{i aµ ā (ran(sµη )) : aµ ∈ h}. let λ be a fixed element of λ such that aλ ∈ h and a ∈ i aλ ā (ran(sλη )). therefore, there exists a point aλ ∈ ran(sλη ) ⊂ aλ such that i aλ ā (aλ) = a, that is (aλ, aλ) ∈ a. let πλ(a λ) = xλ ∈ xλ. then, by relation (2.7.1), ixλ x̄ (xλ) = π̄(a) ≡ x ∈ x̄ and, therefore, (xλ, xλ) ∈ x. by lemma 2.10, there are points aλ1 , a λ 2 ∈ aλ,xλ such that (a λ 1 , aλ) ∈ a1, (a λ 2, aλ) ∈ a2 and (aλ1 + a λ 2, aλ) ∈ a. since (a λ, aλ) ∈ a we have a λ 1 + a λ 2 = a λ ∈ ran(sλη). since the mapping ̟λ is continuous, there exist ε1, ε2 ∈ τ such that a λ 1 ∈ u aλ ε1 , aλ2 ∈ u aλ ε2 and ̟λ((u aλ ε1 × uaλε2 ) ∩ (aλ ⊠ aλ)) ⊂ ran(s λ η). (2.12.2) without loss of generality, we can suppose that ε1, ε2 ∈ θa(θ0(τ)), that is there exist cuts sλη1 and s λ η2 such that ran(sλη1 ) = u aλ ε1 , ran(sλη2) = u aλ ε2 , dom(sλη1) = dom(sλη2) ⊂ dom(s λ η). condition (b) of lemma 2.1 implies that there exists η0 ∈ τ such that s λ η0 = sλη1 + s λ η2 and, therefore, dom(sλη0 ) = dom(s λ ηi ), i = 1, 2. in this case, the left side of the relation (2.12.2) takes the form ̟λ((ran(s λ η1 ) × ran(sλη2 )) ∩ (aλ ⊠ aλ)). (2.12.3) since aλ ⊠ aλ = ∪{aλ,xλ × aλ,xλ : x λ ∈ xλ}, the expression (2.12.3) takes the form ̟λ(∪{(ran(s λ η1 ) ∩ aλ,xλ) × (ran(s λ η2 ) ∩ aλ,xλ) : x λ ∈ xλ}). (2.12.4) since ran(sληi ) ∩ aλ,xλ = ∅ if x λ /∈ dom(sληi), i = 1, 2, (2.12.5) © agt, upv, 2021 appl. gen. topol. 22, no. 1 163 s .d. iliadis and yu. v. sadovnichy and ran(sληi) ∩ aλ,xλ = {s λ ηi (xλ)} if xλ ∈ dom(sληi ), i = 1, 2, (2.12.6) the expression (2.12.4) takes the form ̟λ(∪{{s λ η1 (xλ)} × {sλη2(x λ)} : xλ ∈ dom(sλη0 )}) = ̟λ(∪{{(s λ η1 (xλ), sλη2(x λ))} : xλ ∈ dom(sλη0)}) = ∪{{sλη1(x λ) + sλη2(x λ)} : xλ ∈ dom(sλη0)} = ∪{{(sλη1 + s λ η2 )(xλ)} : xλ ∈ dom(sλη0 )}) = ran(s λ η0 ). thus, relation (2.12.2) implies that ran(sλη0) ⊂ ran(s λ η). let t′ ∈ f and {η1, η2, η, η0} ∪ t ⊂ t ′. denote by h′ the element of c(∼t ′ a ) containing aλ. therefore, h ′ ⊂ h. then, for each aµ ∈ h ′ we have aµ ∼ t ′ a aλ, ran(s µ η1 ) = uaµε1 , ran(s µ η2 ) = uaµε2 . (2.12.7) by condition (4) and (5) of the definitions of the families ra and rx, we have dom(sµη1) = dom(s µ η2 ) = dom(sµη0 ), (2.12.8) ran(sµη1 + s µ η2 ) = ran(sµη0 ) ⊂ ran(s µ η ). (2.12.9) we shall prove that the sets uāεi (h ′) = ∪{i aµ ā (ran(sµηi )) : aµ ∈ h ′}, i = 1, 2, are the required open neighbourhoods ui of ai. obviously, ai ∈ ui. by lemma 2.11 we have ā∆ā = ∪{āx × āx : x ∈ x̄} = {i aν ā (aν,xν ) × i aν ā (aµ,xν ) : aν ∈ a, x ν ∈ xν} using this relation and the relation (2.12.7) we have ̟ā((u ā ε1 (h′) × uāε2(h ′)) ∩ (ā ⊠ ā)) = ̟ā(∪{(i aµ ā (ran(sµη1 )) ∩ i aν ā (aν,xν )) × (i aξ ā (ran(sξη2)) ∩ i aν ā (aν,xν )) : : aξ ∈ h ′, aν ∈ a, x ν ∈ x̄})). (2.12.10) if aν /∈ h ′, then the intersections in the right side of the above equality are empty. therefore, we can suppose that aν ∈ h ′. in this case, relations (2.12.7)(2.12.9) are true if we replace the letter “ µ ” by “ ν ”. let a1 ∈ i aµ ā (ran(sµη1 ))∩ iaν ā (aν,xν ). then relation a1 ∈ i aµ ā (ran(sµη1 )) implies that there exists a µ 1 ∈ ran(sµη1) and (a µ 1 , aµ) ∈ a1 and the relation a1 ∈ i aν ā (aν,xν ) implies that there exists aν1 ∈ aν,xν such that (a ν 1, aν) ∈ a1 and πν(a ν 1) = x ν. from these it follows that aµ ∼a aν and since a µ 1 ∈ u aµ ε1 = ran(s µ η1 ) we have aν1 ∈ uaνε1 = ran(s ν η1 ). therefore, aν1 = s ν η1 (xν). similarly, if a2 ∈ i aξ ā (ran(sξη2 )) ∩ iaν ā (aν,xν ), then there exists a ν 2 ∈ ran(s ν η2 ) and πν(a ν 2) = x ν. thus, aν1, a ν 2 ∈ aν,xν . then, the equality (2.12.10) can be continued as follows: ̟ā(∪{(i aν ā (sνη1(x ν)), iaν ā (sνη2(x ν))) : aν ∈ h ′, xν ∈ dom(sνη0 )}) = © agt, upv, 2021 appl. gen. topol. 22, no. 1 164 on sheaves of abelian groups and universality ∪{iaν ā (sνη1(x ν) + sνη2(x ν)) : aν ∈ h ′, xν ∈ dom(sνη0)} = ∪{iaν ā ((sνη1 + s ν η2 )(xν)) : aν ∈ h ′, xν ∈ dom(sνη0 )} = ∪{iaν ā (ran(sνη0 )) : aν ∈ h ′} ⊂ ∪{iaν ā (ran(sνη)) : aν ∈ h ′} = uāε (h ′) ⊂ uāε (h). we note that for the first equality of the above expression we use the fact that the restriction of the mapping iaν ā on the set aν,xν is an isomorphism. thus, we proved relation (2.12.1), which means that the mapping ̟ā is continuous. similarly, we prove that the mapping iā is continuous. the proof of the proposition is completed. � proposition 2.13. for each point a ∈ ā there exists an open neighbourhood u of a in ā such that π̄ maps u homeomorphically onto an open set of x̄. proof. let a ∈ ā and (aλ, aλ) for some λ ∈ λ. there exists an open neighbourhood of aλ in aλ, which πλ maps homeomorphically onto an open subset of xλ. since b aλ 0 is a base for the open subsets of aλ, without loss of generality, we can suppose that this open neighbourhood is an element v aλη of this base. let t ∈ f such that η ∈ t and h be the element of c(∼t a ) containing aλ. we prove that u ā ε (h), where ε = θa(θ0(η)), is the required open subset u. by lemma 2.8 we have π̄(uāε (h)) = u x̄ δ (l), where δ = θx(η) and l is the element of c(rx) corresponding to h. first, we prove that the restriction of π̄ onto the open subset uāε (h) of ā is oneto-one. indeed, if not, there are two distinct points b1 and b2 of u ā ε (h) such that π̄(b1) = π̄(b2) ≡ x ∈ x̄, that is b1, b2 ∈ āx. let (x ν, xν) ∈ x for some ν ∈ λ. by proposition 2.11 there exist points bν1, b ν 2 ∈ aν,xν such that iaν ā (bν1) = b1 and i aν ā (bν2) = b2, that is (b ν 1, aν) ∈ b1 and (b ν 2, aν) ∈ b2. we have bν1, b ν 2 ∈ u aν ε = v aν η = ran(s ν η), b ν 1 6= b ν 2 and πν(b ν 1) = πν(b ν 2) = x ν, which contradicts the fact that the πν maps homeomorphically the set ran(s ν η) onto an open subset of xν. since the restriction of the mapping π̄ onto the set uāε (h) is one-to-one we can consider the inverse mapping, denoted by s̄ε,h, of the set u x̄ δ (l) onto u ā ε (h). we shall prove that s̄ε,h is continuous. let x1 ∈ u x̄ δ (l) and a1 = s̄ε,h(x1). let also u1 be an arbitrary open neighbourhood of a1 in u ā ε (h). since u ā ε (h) is open in ā, without loss of generality, we can suppose that u1 belongs to the standard base of ā, that is it has the form uāε1(h1). moreover, we can suppose that ε1 = θa(θ0(η1)) for some η1 ∈ τ and h1 ∈ c(∼ t1 a ), where t1 is an element of f such that t ⊂ t1. thus, we have a1 ∈ u ā ε1 (h1) ⊂ u ā ε (h). by lemma 2.8, π̄(uāε1(h1)) = u x̄ δ1 (l1), © agt, upv, 2021 appl. gen. topol. 22, no. 1 165 s .d. iliadis and yu. v. sadovnichy where δ1 = θx(η1) and l1 is the element of c(rx) corresponding to h1. as the above, the restriction of the mapping π̄ onto uāε1(h1) is one-to-one and, therefore, we can consider the inverse mapping, denoted by s̄ε1,h1, of the set ux̄δ1 (l1) onto the set u ā ε1 (h1). the mapping s̄ε1,h1 coincides with the restriction of the mapping s̄ε,h onto the set u x̄ δ1 (l1), that is s̄ε,h(u x̄ δ1 (l1)) = uāε1(h1), which shows that s̄ε,h is continuous. thus, π̄ maps the set u ā ε (h) homeomorphically onto an open subset of x̄. the proof of the proposition is completed. � the final of the proof of theorem 1.3.1 relation (2.7.1) implies that for each (aλ, πλ, xλ) ∈ s the natural embedding i xλ x̄ of xλ into x̄ is the induced mapping of the natural embedding i aλ ā of aλ into ā. proposition 2.11 shows that the embedding iaλ ā is proper. the rest of the proof of theorem 1.3.1 follows immediately from propositions 2.9, 2.11, 2.12 and 2.13. references [1] g. e. bredon, sheaf theory, mcgraw-hill, new york, 1967. 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[14] s. d. iliadis, on embeddings of topological groups, fundamental and applied mathematics 20, no. 2 (2015), 105–112 (russian). journal of mathematical sciences 223, no. 6 (2017), 720–724 (english). © agt, upv, 2021 appl. gen. topol. 22, no. 1 166 on sheaves of abelian groups and universality [15] s. d. iliadis, on isometric embeddings of separable metric spaces, topology and its applications 179 (2015), 91–98. [16] s. d. iliadis, dimension and universality on frames, topology and its applications 201 (2016), 92–109. [17] s. d. iliadis, on spaces continuously containing topological groups, topology and its applications 272 (2020),107072. [18] s. d. iliadis, on actions of spaces continuously containing topological groups, topology and its applications 275 (2020), 107035. © agt, upv, 2021 appl. gen. topol. 22, no. 1 167 () @ appl. gen. topol. 19, no. 2 (2018), 291-305doi:10.4995/agt.2018.10213 c© agt, upv, 2018 on reich type λ−α-nonexpansive mapping in banach spaces with applications to l1([0, 1]) rabah belbaki a, e. karapınar b and amar ould-hammouda a a laboratory of physics mathematics and applications, ens, p.o.box 92, 16050 kouba, algiers, algeria (belbakirab@gmail.com,a.ouldhamouda@yahoo.com) b atilim university, department of mathematics, incek, 068630 ankara, turkey (erdalkarapinar@yahoo.com) communicated by s. romaguera abstract in this manuscript we introduce a new class of monotone generalized nonexpansive mappings and establish some weak and strong convergence theorems for krasnoselskii iteration in the setting of a banach space with partial order. we consider also an application to the space l1([0, 1]). our results generalize and unify the several related results in the literature. 2010 msc: 46t99; 47h10; 54h25. keywords: fixed point; krasnoselskii iteration; monotone mapping; reich type λ − α-nonexpansive mapping; optial property. 1. introduction and preliminaries the study of the existence of fixed point of nonexpansive mappings, initiated in 1965 independently by browder [5], göhde [11] and [16], is one of dynamic research subject in nonlinear functional analysis. in [16], kirk proved that a self-mapping on a nonempty bounded closed and convex subset of a reflexive banach space possesses a fixed point if it is nonexpansive and the corresponding subset has a normal structure. in 1992, veeramani obtained a more general result in this direction by introducing the notion of t−regular set [23]. on the other hand, in 1967, opial introduced in [18] a class of spaces for which the asymptotic center of a weakly convergent sequence coincides with received 24 may 2018 – accepted 09 august 2018 http://dx.doi.org/10.4995/agt.2018.10213 r. belbaki, e. karapınar and a. ould-hammouda the weak limit point of the sequence. a banach space x is said to have the opial property, if for each weakly convergent sequence {xn} in x with limit z, lim inf ‖xn − z‖ < lim inf ‖xn − y‖ for all y ∈ x with y 6= z. in 1972, gossey and lami dozo noticed in [12] that all the spaces of this class have normal structure. it is well known that hilbert spaces, finite dimensional banach spaces and lp-spaces, (1 < p < ∞), have the opial property [8]. in 2008, suzuki introduced in [21] a new class of mappings satisfying the so-called (c)condition which also includes nonexpansive mappings and proved that such mappings on a nonempty weakly compact convex set in a banach space which satisfies opial’s condition have a fixed point. in 2011, falset et al. proposed in [8] mappings satisfying (cλ)-condition, λ ∈ (0, 1), respectively. in [1] aoyama and kohsaka introduced a new class of nonexpansive mappings, and obtained a fixed point result for such mappings. finally, in 2017, in [19] shukla et al proposed a new generalization and introduce the deneralized α−nonexpansive mapping and obtained a fixed theorem for such mappings. all the results cited above were obtained, in the weak case, with opial’s condition. in this report, we propose a generalization of the results of shukla et al. [19] by introducing a class of λ−αgeneralized nonexpansive mapping. in addition, we establish some weak and strong convergence theorems for krasnoselskii iteration in an ordered banach space with partial order ≤. we also consider an application in the context of l1([0, 1]). the presented results in this report, extend, generalize and unify a number of existing results on the the topic in the literature. throughout the paper, n denotes the set of natural numbers and r the set of the real numbers. for a non-empty k of a real banach space x, a mapping t : k → k is said to be nonexpansive if ‖t (x) − t (y)‖ ≤ ‖x − y‖ for all x, y ∈ k. moreover, a selfmapping t is called quasinonexpansive [7] if ‖t (x) − y‖ ≤ ‖x − y‖ for all x ∈ k and y ∈ f(t ), where f(t ) is the set of fixed points of t . definition 1.1 ([12, 22]). the norm of a banach space x is called uniformly convex in every direction, in short, we say that x is uced, if for ε ∈ (0, 2] and z ∈ x with ‖z‖ = 1, there exists δ(ε, z) > 0 such that for all x, y ∈ x with where ‖x‖≤ 1, ‖y‖≤ 1 and x−y ∈{tz : t ∈ [−2,−ε]∪ [ε, 2]} ‖x + y‖≤ 2(1− δ(ε, z)). lemma 1.2 ([21]). for a banach space x, the following are equivalent: (i) x is uced. (ii) if {xn} is a bounded sequence in x , then the function f on x defined by f(x) = lim sup‖xn −x‖ is strictly quasiconvex , that is, f(λx+(1−λ)y) < max{f(x), f(y)} for all λ ∈ (0, 1) and x, y ∈ x with x 6= y. lemma 1.3 ([9]). let (zn) and (wn) be bounded sequences in a banach space x and let λ belongs to (0, 1). suppose that zn+1 = λwn +(1−λ)zn and ‖wn+1− wn‖≤‖zn+1 −zn‖ for all n ∈ n. then lim‖wn −zn‖ = 0. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 292 on reich type λ − α-nonexpansive mapping definition 1.4 ([21]). let k be a nonempty subset of a banach space x. we say that a mapping t : k → k satisfies (c)-condition on k if for x, y ∈ k we have 1 2 ‖x−t (x)‖≤‖x−y‖⇒‖t (x)−t (y)‖≤‖x−y‖. it is clear that each nonexpansive mapping satisfies the condition (c) but the converse is not true. for details and counterexamples see e.g. [10]. definition 1.5 ([8]). let k be a nonempty subset of banach space x and λ ∈ (0, 1). we say that a mapping t : k → k satisfies (cλ) -condition on if for all x, y ∈ k, we have λ‖x−t (x)‖≤‖x−y‖⇒‖t (x)−t (y)‖≤‖x−y‖. note that if λ = 1 2 , then (cλ)-condition implies (c)-condition. for more details and examples, see e.g. falset et al. [8]. throughout the paper, the pair (x,≤) will denote an ordered banach space where x is a banach space endowed with a partial order ” ≤ ”. definition 1.6. a self-mapping t defined on an ordered banach space (x,≤) is said to be monotone if for all x, y ∈ x, x ≤ y ⇒ t (x) ≤ t (y). definition 1.7 ([1]). let k be a nonempty subset of a banach space x. a mapping t : k → k is said to be α-nonexpansive if for all x, y ∈ k and α < 1, ‖t (x) −t (y)‖2 ≤ α‖t (x) −y‖2 + α‖x− t (y)‖2 + (1−2α)‖x−y‖2 definition 1.8 ([19]). let k be a nonempty subset of an ordered banach space (x,≤). a mapping t : k → k will be called a generalized α-nonexpansive mapping if there exists α ∈ (0, 1) such that { 1 2 ‖x−t (x)‖≤‖x−y‖ implies ‖t (x)−t (y)‖≤ α‖t (x)−y‖+ α‖t (y)− x‖+ (1−2α)‖x−y‖ for all x, y ∈ k with x ≤ y. remark 1.9. when α = 0, a generalized-nonexpansive mapping is reduced to a mapping satisfying (c)-condition. the converse is false. for more details and counterexamples see e.g. [19] and [14, 13]. 2. reich type (λ −α)-nonexpansive mappings definition 2.1. let k be a nonempty subset of an ordered banach space (x,≤). a mapping t : k → k will be called reich type (λ−α)-nonexpansive mappings if there exists λ ∈ (0, 1) and α ∈ [0, 1) such that (2.1) λ‖x−t (x)‖≤‖x−y‖⇒‖t (x)−t (y)‖≤ rαt (x, y), where rαt (x, y) := α(‖t (x) −y‖+‖t (y)−x‖) + (1 −2α)‖x−y‖ c© agt, upv, 2018 appl. gen. topol. 19, no. 2 293 r. belbaki, e. karapınar and a. ould-hammouda for all x, y ∈ k with x ≤ y. in addition, if the mapping t is monotone, we say that monotone reich type (λ−α)-nonexpansive mapping. remark 2.2. we point the following special cases: (1) when α = 0, a reich type λ − α-nonexpansive mapping reduced to a mapping satisfying condition (cλ), see e.g. [8]. (2) if λ = 1 2 , it becomes a generalized α-nonexpansive condition. proposition 2.3. let k be a nonempty subset of an ordered banach space (x,≤) and t : k → k be a reich type (λ − α)-nonexpansive mapping with a fixed point z ∈ k with x ≤ z. then t is quasinonexpansive. proof. since z ∈ k fixed point, 0 = λ‖z −t (z)‖≤‖z −x‖ , we have ‖z −t (x)‖≤ α‖z −t (x)‖+ α‖t (z)−x‖+ (1−2α)‖z −x‖≤‖z −x‖. � definition 2.4. let t be a monotone self-mapping on a nonempty convex subset of an ordered banach space (x,≤) . for a fix λ ∈ (0, 1) and for an initial point x1 ∈ k, the krasnoselskii iteration sequence {xn}⊂ k is defined by (2.2) xn+1 = λt (xn) + (1−λ)xn , n ≥ 1. in the sequel we need the following lemmas. lemma 2.5 ([17]). let x, y, z ∈ x and λ ∈ (0, 1). suppose p is the point of segment [x, y] which satisfies ‖x−p‖ = λ‖x−y‖ , then, (2.3) ‖z −p‖≤ λ‖z −y‖+ (1−λ)‖z −x‖ lemma 2.6 ([15]). let k be convex and t : k → k be monotone. assume that x1 ∈ k, x1 ≤ t (x1). then the sequence {xn} defined by (2.2) satisfies: xn ≤ xn+1 ≤ t (xn) ≤ t (xn+1), for n ≥ 1. moreover, if {xn} has two subsequences which converge to y and z, then we must have y = z. it is easy to see that by the mimic of the idea used in lemma 2.6, we get that t (xn+1) ≤ t (xn) ≤ xn+1 ≤ xn, by assuming the initial condition as t (x1) ≤ x1. lemma 2.7. let k be a nonempty convex subset of an ordered banach space (x,≤) and {xn} is the iteration sequence defined by (2.2) in k. let t : k → k be a monotone reich type λ − α-nonexpansive mapping with λ ∈ (1 3 , 1) and α ∈ [0, 1). suppose also that yn = t (xn), n ≥ 1. if, for x1 ∈ k with x1 ≤ y1 = t (x1) we have (2.4) ‖yn −xn+1‖≤ (3λ−1)‖yn −xn‖, for all n ∈ n, then the sequence {‖yn −xn‖} is decreasing. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 294 on reich type λ − α-nonexpansive mapping proof. on account of the definition of krasnoselskii iteration we have (2.5) xn+1 = λyn + (1−λ)xn, with yn = t (xn). it means that xn+1 belongs to the segment ]xn, yn[, and hence we have (2.6) ‖xn −yn‖ = ‖xn −xn+1‖+‖xn+1 −yn‖. furthermore, (2.7) yields that (2.7) ‖xn −xn+1‖ = ‖xn − [λt (xn) + (1−λ)xn]‖ = λ‖xn −t (xn)]‖. on account of the triangle inequality together with the fact that t is monotone reich type (λ−α)-nonexpansive mapping, we derive that (2.8) ‖yn+1 −xn+1‖ ≤‖yn+1 −yn‖+‖yn −xn+1‖ = ‖t (xn+1) −t (xn)‖+‖yn −xn+1‖ ≤ α(‖t (xn) −xn+1‖+‖t (xn+1) −xn‖) +(1−2α)‖xn −xn+1‖+‖yn −xn+1‖ = (1 + α)‖yn −xn+1‖+ α‖yn+1 −xn‖+ (1 −2α)‖xn −xn+1‖ = (1 + α)‖yn −xn+1‖+ α‖yn+1 −xn‖ +(1 + α)‖xn −xn+1‖−3α‖xn −xn+1‖. on account of (2.6) the left hand side of the inequality of (2.8) turns into (2.9) = (1 + α)‖xn −yn‖+ α‖yn+1 −xn‖−3α‖xn −xn+1‖, taking the inequality (2.7) into account, the expression (2.9) turns into (2.10) ≤ (1 + α)‖xn −yn‖+ α‖yn+1 −xn‖−3λα‖xn −yn‖ = (1 + α −3λα)‖xn −yn‖+ α‖yn+1 −xn‖ employing the assumption (2.4) of the lemma, we estimate the expression (2.10) from above as (2.11) = (1 + α−3λα)‖xn −yn‖+ α (3λ−1)‖yn −xn‖ = ‖xn −yn‖. by combining (2.8)(2.11), for each n, we deduce that ‖yn+1 −xn+1‖≤‖xn −yn‖, which complete the proof. � in the following proposition, we extend the goebel-kirk inequality [9] from the class of nonexpansive mappings into the class of monotone generalized(λ− α)-nonexpansive mapping. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 295 r. belbaki, e. karapınar and a. ould-hammouda proposition 2.8. let k be a nonempty convex subset of an ordered banach space (x,≤) . let t : k → k be a monotone reich type (λ−α)-nonexpansive mapping with λ ∈ (1 3 , 1) and α ∈ (0, 1). for x1 ∈ k with x1 ≤ t (x1), we set yn = t (xn) where {xn} is the iteration sequence defined by (2.2) in k satisfies the assumption (2.4). then, we have (2.12) ‖yi+n −xi‖≥ (1−λ) −n[‖yi+n −xi+n‖−‖yi−xi‖] + (1 + nλ)‖yi−xi‖, for all i, n ∈ n. proof. inspired the techniques used in [9], we shall use the method of the induction to prove our assertion. it is evident that (2.12) is trivially true for all i if n = 0. we assume that the inequality (2.12) holds for a given n and for all i. by replacing i by i + 1 in (2.12), we get ‖yi+n+1 −xi+1‖≥ (1−λ) −n[‖yi+n+1 −xi+n+1‖ −‖yi+1 −xi+1‖] + (1 + nλ)‖yi+1 −xi+1‖. (2.13) on the other hand, due to krasnoselskii iteration, we have xn+1 = λyn + (1− λ)xn with yn = t (xn) and also (2.14) ‖xi+1 −xi‖ = ‖λyi + (1−λ)xi −xi‖ = λ‖yi −xi‖. the observation in (2.14) provide to apply lemma 2.5 that yields ‖yi+n+1 −xi+1‖≤ λ‖yi+n+1 −yi‖+ (1 −λ)‖yi+n+1 −xi‖. regarding that t is a monotone reich type (λ−α)-nonexpansive mapping, we have ‖yi+n+1 −xi+1‖ ≤ (1−λ)‖yi+n+1 −xi‖+ λ ∑n k=0 ‖yi+k+1 −yi+k‖ ≤ (1−λ)‖yi+n+1 −xi‖ +λ ∑n k=0(α‖xi+k+1 −yi+k‖+ α‖yi+k+1 −xi+k‖ +(1−2α)‖xi+k+1 −xi+k‖) so, we derive that ‖yi+n+1 −xi‖≥ (1−λ) −1‖yi+n+1 −xi+1‖− (1 −λ) −1λαbin − (1−λ)−1λ(1 −2α)ain, (2.15) where ain = n ∑ k=0 ‖xi+k+1 −xi+k‖ and bin = n ∑ k=0 [‖xi+k+1 −yi+k‖+‖yi+k+1 −xi+k‖]. taking the assumption (2.4) and (2.14) into account, we derive that (2.16) ‖yi+k+1 −xi+k‖+‖yi+k −xi+k‖ ≤ (3λ− 1)‖yi+k −xi+k‖+‖yi+k −xi+k‖ = 3λ‖yi+k −xi+k‖ = 3‖xi+k −xi+k+1‖, c© agt, upv, 2018 appl. gen. topol. 19, no. 2 296 on reich type λ − α-nonexpansive mapping for all k ∈ 0, 1, ..., n. regarding the definition of krasnoselskii iteration we have xi+k+1 = λyi+k + (1 − λ)xi+k, with yi+k = t (xi+k). in other words, xi+k ≤ xi+k+1 ≤ yi+k and we have (2.17) ‖xi+k −yi+k‖ = ‖xi+k+1 −xn+1‖+‖xi+k+1 −yi+k‖. now, by revisiting the inequality (2.16) by keeping the equality (2.17) in mind, we find (2.18) ‖yi+k+1 −xi+k‖+‖xi+k+1 −yi+k‖ = ‖yi+k+1 −xi+k‖+‖yi+k −xi+k‖ −‖xi+k −xi+k+1‖ ≤ 2‖xi+k −xi+k+1‖ which implies bin ≤ 2ain. accordingly, the inequality (2.15) becomes (2.19) ‖yi+n+1 −xi‖≥ (1 −λ) −1‖yi+n+1 −xi+1‖−λ(1 −λ) −1 ain. employing the inequality (2.13) in (2.19), we find that ‖yi+n+1 −xi‖≥(1−λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi+1 −xi+1‖] + (1 −λ)−1(1 + nλ)‖yi+1 −xi+1‖−λ(1 −λ) −1ain. on account of (2.14), the estimation above turns into ‖yi+n+1 −xi‖≥(1−λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi+1 −xi+1‖] + (1 −λ)−1(1 + nλ)‖yi+1 −xi+1‖−λ 2(1−λ)−1cin, (2.20) where cin := n ∑ k=0 ‖yi+k−xi+k‖. by bearing, lemma 2.7, in mind, we find that cin := n ∑ k=0 ‖yi+k −xi+k‖≤ (n + 1)‖yi −xi‖. consequently, (2.20) can be estimated above as ‖yi+n+1 −xi‖≥(1− λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi+1 −xi+1‖] + (1−λ)−1(1 + nλ)‖yi+1 −xi+1‖−λ 2(1−λ)−1(n + 1)‖yi −xi‖ = (1−λ)−(n+1)[‖yi+n+1 −xi+n+1‖−‖yi −xi‖] + [(1−λ)−1(1 + nλ) − (1−λ)−(n+1)]‖yi+1 −xi+1‖ + [(1−λ)−(n+1) −λ2(1−λ)−1(n + 1)]‖yi −xi‖, (2.21) by adding and substraction the same term (1−λ)−(n+1)‖yi+1 −xi+1‖. notice that (1 − λ)−1(1 + nλ) − (1 − λ)−(n+1) ≤ 0. thus, regarding this observation c© agt, upv, 2018 appl. gen. topol. 19, no. 2 297 r. belbaki, e. karapınar and a. ould-hammouda together with lemma 2.7, the inequality (2.21) changed into ‖yi+n+1 −xi‖≥ (1 −λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi −xi‖] + [(1−λ)−1(1 + nλ) − (1−λ)−(n+1)]‖yi −xi‖ + [(1−λ)−(n+1) −λ2(1−λ)−1(n + 1)]‖yi −xi‖ = (1 −λ)−(n+1)[‖yi+n+1 −xi+n+1‖−‖yi −xi‖] + (1 + (n + 1)λ)‖yi −xi‖ which completes the proof of proposition 2.8. � theorem 2.9. let k be a nonempty, convex and compact subset of an ordered banach space (x,≤) . let t : k → k be a monotone reich type λ − αnonexpansive mapping with λ ∈ (1 3 , 1). for x1 ∈ k with x1 ≤ t (x1), we set yn = t (xn) where {xn} is the iteration sequence defined by (2.2) in k satisfies the assumption (2.4). then {xn} converges to some x ∈ k with xn ≤ x and, (2.22) lim n ‖xn −t (xn)‖ = 0 proof. we shall divide the proof in two cases: α = 0 and α ∈ (0, 1). suppose, first, that α = 0 . due to the definition (2.2) of the sequence {xn}, we have λ‖xn −yn‖ = ‖xn −xn+1‖, for all n ≥ 1. on account of lemma 2.6, we have xn ≤ xn+1, for all n ≥ 1. therefore condition (2.1) implies that, ‖t (xn) −t (xn+1)‖ = ‖yn −yn+1‖≤ r α t (xn, xn+1) = ‖xn −xn+1‖, since α = 0. employing lemma 1.3, the inequality above yields that lim n ‖xn −t (xn)‖ = 0. in the following, we shall consider the second case α ∈ (0, 1). the proof of this case mainly adopted from the proof of theorem 3.1 in [15]. since k is compact, there exists a subsequence of {xn} which converges to x ∈ k. on account of lemma 2.6, the sequence {xn} converges to x and xn ≤ x, for n ≥ 1. to show our assertion (2.22), suppose, on the contrary, that lim n ‖xn −t (xn)‖ = r > 0. as x1 ≤ xn ≤ x, we then have (2.23) ‖xn −x1‖≤‖x−x1‖ for all n ≥ 1. due to triangle inequality we have ‖yi+n −xi‖ = ‖t (xi+n) −xi‖≤‖t (xi+n) −xi+n‖+‖xi+n −x1‖+‖x1 −xi‖ ≤‖t (x1) −x1‖+ 2‖x−x1‖ (2.24) c© agt, upv, 2018 appl. gen. topol. 19, no. 2 298 on reich type λ − α-nonexpansive mapping for any i, n ≥ 1, due to (2.23) and lemma 2.7. since all conditions are satisfied in proposition 2.8, we have (2.12). letting i →∞ in the inequality (2.12), we derive that (2.25) lim i→∞ ‖yi+n − xi‖≥ (1 + nλ)r, where we used that lim i→∞ (‖t (xi) −xi‖−‖t (xi+n) −xi+n‖) = r −r = 0, for any n ≥ 1. combining (2.24) and (2.25), we find (1 + nλ)r ≤ lim i→∞ ‖yi+n −xi‖≤‖t (x1)−x1‖+ 2‖x−x1‖ thus, the inequality can be fulfilled only if r = 0 which yields the inequality (2.22). � lemma 2.10. let k be a nonempty subset of an ordered banach space (x,≤) and t : k → k be a monotone reich type (λ−α)-nonexpansive mapping with λ ∈]0, 1 2 ]. then for each x, y ∈ k with x ≤ y : (i) ‖t (x)−t 2(x)‖≤‖x−t (x)‖ (ii) either λ‖x−t (x)‖≤‖x−y‖ or λ‖t (x)−t 2(x)‖≤‖t (x)−y‖ (iii) either ‖t (x)−t (y)‖≤ α‖t (x)−y‖+ α‖x−t (y)‖+ (1−2α)‖x−y‖ or ‖t 2(x) −t (y)‖≤ α‖t (x)−t (y)‖+ α‖t 2(x) −y‖+ (1−2α)‖t (x)−y‖ proof. (i) since we have λ‖x − t (x)‖ ≤ ‖x − t (x)‖ for all λ ∈]0, 1 2 ], by the definition of reich type (λ−α)-nonexpansive mapping we get the desired results. indeed, ‖t (x)−t 2(x)‖≤ α‖x−t 2(x)‖+ (1−2α)‖x−t (x)‖. thus (i) hold for α = 0. (ii) suppose, on the contrary, that λ‖x − t (x)‖ > ‖x − y‖ and ‖t (x) − t 2(x)‖ > ‖t (x) − y‖. then, by triangle inequality together with the assumption (i), we find that ‖x−t (x)‖≤‖x−y‖+‖t (x)−y‖ < λ‖x−t (x)‖+ λ‖t (x) −t 2(x)‖ ≤ 2λ‖x−t (x)‖. since λ ≤ 1 2 we obtain ‖x−t (x)‖ < ‖x−t (x)‖ which is a contradiction. thus, we obtain the desired result. (iii) the proof of (iii) follows from (ii). we skip the details. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 299 r. belbaki, e. karapınar and a. ould-hammouda lemma 2.11. let k be a nonempty subset of an ordered banach space (x,≤) and t : k → k be a monotone reich type (λ−α)-nonexpansive mapping with λ ∈ (0, 1 2 ]. then for each x, y ∈ k with x ≤ y, ‖x−t (y)‖≤ ( 3 + α 1−α )‖x−t (x)‖+‖x−y‖. proof. it is the mimic of the proof of lemma 3.8 of [19]. so, we skip the details. � using the above two lemmas, we can prove the following. theorem 2.12. let k be a nonempty convex and a compact subset of an ordered banach space (x,≤) and be t : k → k a monotone reich type (λ−α)nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. select x1 ∈ k such that x1 ≤ t (x1), and for n ≥ 1, denote yn = t (xn) where {xn} is the iteration sequence defined by (2.2) in k satisfying, for all n ∈ n, the assumption (2.4) . then {xn} converges strongly to a fixed point of t. proof. by theorem 2.9, we have lim n ‖xn −t (xn)‖ = 0. since k is compact, there exist a subsequence {xnk} of {xn} and z ∈ k such that {xnk} converges to z. employing lemma 2.11, we have, ‖xnk −t (z)‖≤ ( 3 + α 1−α )‖xnk −t (xnk)‖+‖xnk −z‖ for all k ∈ n. thus, the sequence {xnk} converges to t (z) and hence t (z) = z. since z is a fixed point of t , by proposition 2.3, we find that ‖xn+1 −z‖≤ λ‖t (xn) −z‖+ (1−λ)‖xn −z‖≤‖xn −z‖ for all n ∈ n. therefore {xn} converges to z . � we say that a banach space x has the opial property [18] if for every weakly convergent sequence {xn} in x with a limit z, fulfils lim inf n→∞ ‖xn −z‖ < lim inf n→∞ ‖xn −y‖, for all y ∈ x with y 6= z. it is a very rich class, for examples, all hilbert spaces, sequence spaces ℓp, (1 < p < ∞), and finite dimensional banach spaces have the opial property. unexpectedly, lp[0, 2π], (p 6= 2) do not have the opial property [9],[10]. proposition 2.13. let k be a nonempty subset of an ordered banach space (x,≤) with the opial property and t : k → k be a monotone reich type (λ − α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. if {xn} converges weakly to z and lim n ‖xn −t (xn)‖ = 0, then t (z) = z. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 300 on reich type λ − α-nonexpansive mapping proof. by lemma 2.11, we have, ‖xn −t (z)‖≤ ( 3 + α 1−α )‖xn −t (xn)‖+‖xn −z‖ for n ∈ n and hence, lim inf n ‖xn −t (z)‖≤ lim inf n ‖xn −z‖ we claim that t (z) = z. indeed, if t (z) 6= z, the opial property implies, lim inf n ‖xn −z‖ < lim inf n ‖xn −t (z)‖ which is a contradiction with inequality (2.22) . � theorem 2.14. let k be a nonempty convex and weakly compact subset of an ordered banach space (x,≤) with the opial property and t : k → k be a monotone reich type (λ − α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. . select x1 ∈ k such that x1 ≤ t (x1), and for n ≥ 1 , denote yn = t (xn) where {xn} is the iteration sequence defined by (2.2) in k satisfying, for all n ∈ n, the assumption (2.4) . then {xn} converges weakly to a fixed point of t . proof. by theorem 2.9, we have lim n ‖xn −t (xn)‖ = 0. since k is weakly compact, there exist a subsequence {xnk} of {xn} and z ∈ k such that {xnk} converges weakly to z . by proposition 2.13, we deduce that z is a fixed point of t . as in the proof of theorem 2.12, we can prove that {‖xn − z‖} is a nonincreasing sequence. we prove our assertion by reductio de absurdum. suppose, on the contrary, that {xn} does not converge to z. then there exist a subsequence {xnj} of {xn} which converges weakly to ω and ω 6= z. we note that t (ω) = ω . from the opial property, lim n ‖xn −z‖ = lim k ‖xnk −z‖ < lim k ‖xnk −ω‖ = lim n ‖xn −ω‖ = lim j ‖xnj −ω)‖ < lim j ‖xnj −z‖ = lim n ‖xn −z‖, a contradiction that complete the proof. � the following theorem directly follows from theorems 2.12 and 2.14. so, to avoid the repetition, we skip the details. theorem 2.15. let k be a convex subset of an ordered banach space (x,≤), and t : k → k be a monotone reich type (λ−α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. assume that either of the following holds: (i) k is compact; (ii) k is weakly compact and x has the opial property. then t has a fixed point. finally, we will give a generalization of a fixed point theorem due to browder [5], göhde [11] and suzuki [21]. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 301 r. belbaki, e. karapınar and a. ould-hammouda theorem 2.16. let k be a convex and weakly compact subset of a uced ordered banach space (x,≤). let t : k → k be a monotone reich type (λ−α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. then t has a fixed point. proof. we construct an iterative sequence {xn} in k by starting x1 ∈ k as xn+1 = 1 2 (t (xn) + xn) with ‖t (xn+1) −xn‖≤‖t (xn) −xn‖, for all n ∈ n. then by theorem 2.9, we have lim n ‖xn −t (xn)‖ = 0 holds. define a continuous convex function f from k to [0, +∞) by f(x) = lim sup n ‖xn −x‖ for all x ∈ k . since k is weakly compact and f is weakly lower semicontinuous, there exists z ∈ k, such that f(z) = min{f(x) : x ∈ k} since, by lemma 2.11: ‖xn −t (z)‖≤ ( 3 + α 1−α )‖xn −t (xn)‖+‖xn −z‖, we then have, f(t (z)) ≤ f(z) . since f(z) is the minimum, f(t (z)) = f(z) holds. if t (z) 6= z, then since f is strictly quasiconvex (lemma 1.2) we have, f(z) ≤ f( z + f(z) 2 ) < max{f(z), f(t (z))} = f(z). which is a contradiction. hence t (z) = z. � 3. application to l1([0, 1]) as an application, we consider l1([0, 1]) the banach space of real valued functions defined on [0, 1] with absolute value lebesgue integrable, i.e., ∫ 1 0 |f(x)|dx < ∞. we recall some definitions which can be found in e.g. [3]. as usual, f = 0 if and only if the set {x ∈ [0, 1] : f(x) = 0} has lebesgue measure 0, then, we say f = 0 almost everywhere. an element of l1([0, 1]) is therefore seen as a class of functions. the norm of any f ∈ l1([0, 1]) is given by ‖f‖ = ∫ 1 0 |f(x)|dx from now on, we will write l1 instead of l1([0, 1]) . recall that f ≤ g if and only if f(x) ≤ g(x) almost everywhere, for any f, g ∈ l1. we adopt the convention f ≤ g if and only if g ≤ f . we remark that order intervals are closed for convergence almost everywhere and convex. recall that an order interval is a subset of the form [f,→) = {g ∈ l1 : f ≤ g} or (←, f] = {g ∈ l1 : g ≤ f}, c© agt, upv, 2018 appl. gen. topol. 19, no. 2 302 on reich type λ − α-nonexpansive mapping for any f ∈ l1. as a direct consequence of this, the subset [f, g] = {h ∈ l1 : f ≤ h ≤ g} = [f,→) ∩ (←, g] is closed and convex, for any f, g ∈ l1. let k be a nonempty subset of l1 which is equipped with a vector order relation ≤ . a map t : k → k is called monotone if for all f ≤ g we have t (f) ≤ t (g). remark 3.1. since l1([0, 1]) fails to be uniformly convex, theorem 2.12 can’t not be used to get a fixed point result for monotone generalized λ − α nonexpansive mappings in l1([0, 1]). as an alternative, we will use an interesting property for the convergence almost everywhere contained in the following lemma. lemma 3.2 ([4]). if (fn) is a sequence of uniformly l p -bounded functions on a measure space, and if fn → f almost everywhere, then lim inf n ‖fn‖ p p = lim inf n ‖fn −f‖ p p +‖f‖ p p for all 0 < p < ∞. in particular, this result holds when p = 1. on account of lemma 2.11 and lemma 3.2, we shall prove the following. theorem 3.3. let k ⊂ l1 be nonempty, convex and compact for the convergence almost everywhere. let t : k → k be a monotone reich type (λ − α)nonexpansive mapping with α ∈]1 3 , 1 2 ]. select f1 ∈ k such that f1 ≤ t (f1),, and for n ≥ 1, denote gn = t (fn) where (fn) is the iteration sequence defined by (2.2) in k satisfying, for all n ∈ n, the assumption (2.4) . then the sequence (fn) converges almost everywhere to some f ∈ k which is a fixed point of t , i.e., t (f) = f. moreover, f1 ≤ f. proof. theorem 2.9 implies that (fn) converges almost everywhere to some f ∈ k where fn → f, for any n ≥ 1. since (fn) is uniformly bounded, lemma 3.2 [4] implies lim inf n ‖fn −t (f)‖ = lim inf n ‖fn −f‖+‖f −t (f)‖ theorem 2.9 implies lim inf n ‖fn −t (fn)‖ = 0. therefore we get lim inf n ‖fn −t (f)‖ = lim inf n ‖fn −f‖+‖f −t (f)‖ on the other hand, we know that each fn ≤ f for each n ≥ 1, so, by assumption (2.1), we have, lim inf n ‖fn−f‖+‖f−t (f)‖≤ lim inf n (α‖fn−t (f)‖+α‖t (fn)−f‖+(1−2α)‖fn−f‖) c© agt, upv, 2018 appl. gen. topol. 19, no. 2 303 r. belbaki, e. karapınar and a. ould-hammouda and, by lemma 2.11, we have, lim inf n ‖fn−f‖+‖f−t (f)‖≤ lim inf n (α 3 + α 1 −α ‖fn−t (f)‖+α‖t (fn)−f‖+(1−2α)‖fn−f‖) again, by application of the theorem 2.9, we obtain, lim inf n ‖fn −f‖+‖f −t (f)‖≤ lim inf n (1−α)‖fn −t (f)‖+ α‖t (fn)−f‖) and like, lim inf n ‖fn −f‖ = ‖t (fn)−f‖ we then have, lim inf n ‖fn −f‖+‖f −t (f)‖≤ lim inf n ‖fn −t (f)‖, that implies ‖f −t (f)‖ = 0 or t (f) = f. � acknowledgements. the authors thanks to anonymous referees for their remarkable comments, suggestion and ideas that helps to improve this paper. references [1] k. aoyama and f. kohsaka, fixed point theorem for α-nonexpansive mappings in banach spaces, nonlinear anal. 74 (2011), 4387–4391. 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[21] t. suzuki, fixed point theorems and convergence theorems for some generalized nonexpansive mappings, j. math. anal. appl. 340, no. 2 (2008), 1088–1095. [22] d. van dulst, equivalent norms and the fixed point property for nonexpansive mappings, j. london math. soc. 25 (1982), 139–144. [23] p. veeramani, on some fixed point theorems on uniformly convex banach spaces, j. math. anal. appl. 167 (1992), 160–166. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 305 @ appl. gen. topol. 23, no. 1 (2022), 135-143 doi:10.4995/agt.2022.15613 © agt, upv, 2022 topological transitivity of the normalized maps induced by linear operators pabitra narayan mandal school of mathematics and statistics, university of hyderabad, hyderabad 500046, india. (pabitranarayanm@gmail.com) communicated by f. balibrea abstract in this article, we provide a simple geometric proof of the following fact: the existence of topologically transitive normalized maps induced by linear operators is possible only when the underlying space’s real dimension is either 1 or 2 or infinity. a similar result holds for projective transformation as well. 2020 msc: 47a16; 37b05. keywords: topological transitivity; supercyclicity; projective transformation; linear transformation; cone transitivity. 1. introduction in general, a topological dynamical system is a pair (x,f), where x is a topological space and f is a self map on x. the study of dynamics is mainly about the eventual behavior of orbits. in our setting, we take x as a metric space and f as a continuous self map on x. for x ∈ x, the forbit of x is denoted by o(f,x) and is defined by {fn(x) : n ∈ n0} where n0 = n ∪{0} and n = {1, 2, 3, ...}. here fn(x) = f ◦ f ◦ ... ◦ f(x) (n-times) and f0(x) = i(x) = x (i denotes the identity map). now f is said to have dense orbit if there exists x ∈ x such that o(f,x) is dense in x and f is said to be topologically transitive if for any two non-empty open sets u, v in x there exists n ∈ n such that fn(u) ∩ v 6= ∅. for several equivalent formulations of topological transitivity, see [3], [9]. having dense orbit and received 13 may 2021 – accepted 28 december 2021 http://dx.doi.org/10.4995/agt.2022.15613 https://orcid.org/0000-0002-8020-3371 p. narayan mandal topological transitivity are not equivalent notions in general. however, in some ‘nice spaces’ both the notions are equivalent. for example, if x is a separable complete metric space without isolated points. then the following assertions are equivalent: (i) f is topologically transitive; (ii) there exists x ∈ x such that o(f,x) is dense in x. this is known as birkhoff transitivity theorem (see [6]). let t : x → x be an invertible continuous linear operator, where x is a real separable hilbert space. hereafter, in this article, x denotes a real separable hilbert space, unless otherwise mentioned. let l(x) be the set of all continuous linear operators on x and gl(x) be the set of all invertible continuous linear operators on x. the map t : x → x induces a map t : sx → sx, where sx := {x ∈ x : ||x|| = 1} and is defined by tx = tx ||tx||, whenever t ∈ gl(x). we call t as the normalized map induced by t . there are some excellent monographs and expository articles which dealt with dynamics of linear operators in great detail (see [2], [4], [6]). here we are interested to study a particular notion namely, the topological transitivity of the map t on sx. more precisely, we ask the following questions: question 1.1. for a given x, does there exist t ∈ gl(x) such that t is topologically transitive on sx? if the answer to the above is positive, we would like to investigate the following also. question 1.2. what are all invertible continuous linear operators whose normalized maps are topologically transitive? let dim(x) be the hilbert dimension of x i.e., the cardinality of an orthonormal basis of x. in this article, as an answer to the question 1.1, we show the following: theorem 1.3. there exists t ∈ gl(x) such that t is topologically transitive on sx if and only if dim(x) ∈{1, 2,∞}. in the context of linear dynamics, topological transitivity and having dense orbit are equivalent (due to birkhoff transitivity theorem) and it is known as hypercyclicity. supercyclicity and positive supercyclicity are two weaker notions than the notion of hypercyclicity. a linear operator t is said to be supercyclic (resp. positive supercyclic) if there exists a vector x ∈ x whose projective orbit (resp. positive projective orbit) r.o(t,x) (resp. r+.o(t,x)) defined by {λtn(x) : n ∈ n0 and λ ∈ r} (resp. {λtn(x) : n ∈ n0 and λ ∈ r+}) is dense in x, where r and r+ denote the set of all real numbers and the set of all positive real numbers respectively. for equivalent formulations on these notions, one can refer [2], [6]. in [7], g. herzog proved that supercyclic vector exists if and only if the dimension of the space is 1, 2, ∞ (over real) and 1, ∞ (over complex). proof of herzog’s theorem is based on the techniques from functional analysis and operator theory. here we essentially reprove the herzog’s theorem in the light of dynamics on unit sphere. although our proof is © agt, upv, 2022 appl. gen. topol. 23, no. 1 136 topological transitivity of the normalized maps induced by linear operators relatively longer, it is still simple. it is intuitive according to our expositions and based on basic and well-known techniques from linear algebra and theory of dynamical systems. in the end, we also list all such invertible continuous linear operators whose normalized maps are topologically transitive as a possible answer to the question 1.2. similarly, the map t ∈ gl(x) induces a map t̃ : sx/∼→ sx/∼, where sx/∼ is the quotient space of sx under the relation ∼. here ‘∼’ identifies the antipodal points. the map t̃ is known as real projective transformation. for a detailed study on dynamics of projective transformation, see [4], [8]. one can ask similar questions as above and expect that similar results also hold for real projective transformation. 2. cone transitivity and basic properties in this section, we introduce a weak notion of topological transitivity, which we call cone transitivity. this notion helps us to study the dynamics of the normalized maps induced by linear operators. first, we start with a definition. definition 2.1 (open cone). an open set v ⊂ x is said to be an open cone if λv ⊂ v for any λ > 0. remark 2.2. for any non-empty open cone v , there exists a non-empty open subset s (namely sx ∩ v ) of sx (in the subspace topology) such that v = ∪x∈slx, where lx := {λx : λ > 0}. definition 2.3 (cone transitivity). a t ∈ l(x) is said to be cone transitive, if for any two non-empty open cones u, v in x, there exists n ∈ n such that tn(u) ∩v 6= ∅. we now provide an alternative definition for the cone transitivity in the following theorem, which is analogous to birkhoff transitivity theorem. theorem 2.4. let t ∈ l(x). then the following are equivalent: (i) there exists x ∈ x such that for any non-empty open cone v , tnx ∈ v for some n ∈ n. (ii) t is cone transitive. proof. let u and v be two non-empty open cones. by (i), there exists x ∈ x such that tn(x) ∈ v and tm(x) ∈ u for some m,n ∈ n. since x does not contain any isolated point, without loss of generality we can choose n > m. therefore tn−m(u) ∩v 6= ∅. hence, (i) implies (ii). since x is separable, it has a countable dense set say {v1,v2,v3, ...}. consider the open balls v ′k of radius � > 0 around vk for each k ≥ 1, form a countable base of the topology of x. let vk be the smallest open cone containing v ′k. if t −n(vk) is the n-th pre-images of vk, then ∪∞n=0t−n(vk) is the set of points which visit vk at least once. we claim that ∪∞n=0t−n(vk) is dense in x. if not, then there exists an non-empty open set u′ in x such that u′ ∩∪∞n=0t−n(vk) = ∅. therefore for each n ∈ n, u′ ∩t−n(vk) = ∅. since © agt, upv, 2022 appl. gen. topol. 23, no. 1 137 p. narayan mandal vk is a cone and t is linear, we have for any λ > 0, λu ′∩t−n(vk) = ∅. therefore if u is the smallest open cone containing u′, then u ∩t−n(vk) = ∅ i.e., tn(u)∩vk = ∅. this is a contradiction to the fact that t is a cone transitive operator. hence ∪∞n=0t−n(vk) is dense in x. therefore ∩∞k=1 ∪ ∞ n=0 t −n(vk) is non-empty, by baire’s category theorem. hence, (ii) implies (i). � remark 2.5. if t is a cone transitive operator, then the x defined in theorem 2.4 is called a cone transitive vector of t . we denote by cv (t) the set of all cone transitive vectors of t . here x does not contain any isolated point. therefore there exists a sequence of natural numbers (nk) such that t nkx ∈ v where nk →∞ and o(x,t) ⊂ cv (t). one can show that the notions of cone transitivity and positive supercyclicity are equivalent. to keep our exposition self-contained and geometrically intuitive, we prefer to use cone transitivity instead of positive supercyclicity. let us see some examples of cone transitivity: example 2.6. any topologically transitive continuous linear operator in infinite dimensional space (known as hypercyclic operator) is cone transitive. to see this: take any two non-empty open sets as any two non-empty open cones. in the next example, we see that there exists a cone transitive linear operator which is not topologically transitive. furthermore, cone transitivity is not an infinite dimensional property like the topological transitivity for linear operators. example 2.7. let t := ( cos θ sin θ −sin θ cos θ ) , where θ π ∈ r\q. here t = t |sr2 . let u and v be two open cones in r 2. then u′ := u∩sr2 and v ′ = v ∩sr2 are two non-empty open sets in sr2 . it is well-known that any irrational circular rotation is topologically transitive on sr2 . therefore there exists n ∈ n such that t n (u′)∩v ′ 6= ∅. hence tn(u)∩v 6= ∅ for some n ∈ n. therefore t is a cone transitive operator on r2. we ask the following question: question 2.8. does there exist any relation among the cone transitivity of t on x and the topological transitivity of t on sx? first, observe that a non-invertible linear operator can not be cone transitive as t(x) is a proper subspace of x, when dim(x) < ∞. however, we find an affirmative answer to question 2.8. in fact, both the notions are equivalent. in this section, we prove the equivalence together with some basic properties of cone transitivity, which are useful throughout our discussions. theorem 2.9. let t ∈ gl(x). then t : sx → sx is topologically transitive if and only if t : x → x is cone transitive. © agt, upv, 2022 appl. gen. topol. 23, no. 1 138 topological transitivity of the normalized maps induced by linear operators proof. (⇒) let u and v be two non-empty open cones in x. in view of remark 2.2, u′ := u ∩sx and v ′ := v ∩sx are two non-empty open sets in sx. since t : sx → sx is topologically transitive, there exists n ∈ n such that t n (u′) ∩v ′ 6= ∅, i.e., there exists x ∈ u′ such that t n x ∈ v ′. since v is an open cone containing v ′, we have tnx ∈ v . this shows that tn(u) ∩v 6= ∅. hence t is cone transitive. (⇐) let u′ and v ′ be two non-empty open sets in sx. take u := ∪λ>0λu′, the smallest open cone containing u′ and v := ∪λ>0λv ′, the smallest open cone containing v ′. since t is cone transitive, there exists n ∈ n such that tn(u)∩v 6= ∅ i.e., there exists x ∈ u such that tnx ∈ v . consider x′ := x||x||. in view of remark 2.2, x′ ∈ u′. since v is an open cone, we have tnx′ ∈ v . therefore t n (x′) = t nx′ ||tnx′|| ∈ v ′. this shows that t n (u′) ∩v ′ 6= ∅. hence t is topologically transitive. � proposition 2.10. let t ∈ l(x). if x is a cone transitive vector of t, then λx is also a cone transitive vector of t for any λ > 0. proof. let v be any non-empty open cone. since x is cone transitive vector of t , there exists n ∈ n such that tnx ∈ v . therefore tn(λx) = λtnx ∈ λv ⊂ v for λ > 0. hence λx is also a cone transitive vector of t for any λ > 0. � remark 2.11. for any t ∈ l(x), tv (t) = { x||x|| : x ∈ cv (t)} and cv (t) := {λx : x ∈ tv (t) and λ > 0}, where tv (t) is the set of all transitive vectors for t . moreover cv (t) is either empty or dense in x. definition 2.12 (linear conjugacy). let t,s ∈ l(x). now t and s are said to be linearly conjugate if there exists p ∈ gl(x) such that s = p−1tp. proposition 2.13. cone transitivity is preserved by linear conjugacy. proof. let t ∈ l(x) be cone transitive. we claim that for any p ∈ gl(x), s = p−1tp is also cone transitive. let u be a non-empty open set and v be a non-empty open cone. then observe that p(u) is a non-empty open set and p(v ) is a non-empty open cone. since t is cone transitive, we have tn(p(u)) ∩p(v ) 6= ∅ for some n ∈ n. on the other hand, psn = tnp . we conclude that psn(u)∩p(v ) 6= ∅, which readily gives sn(u)∩v 6= ∅. hence s is cone transitive. � proposition 2.14. if t,s ∈ gl(x) are linearly conjugate, then t, s are topologically conjugate. proof. let s and t be two invertible linear operators which are linearly conjugate. we claim that s and t are topologically conjugate. since s and t are linearly conjugate, there exists a invertible linear operator p on x such that s = p−1tp. for x ∈ x, p−1 t px = p−1 t( px||px||) = p −1( t(px)/||px|| ||tpx||/||px||) = p−1tpx/||tpx|| ||p−1tpx||/||tpx|| = p−1tpx ||p−1tpx|| = sx. take h = p , then s = h −1th. � theorem 2.15. if t ∈ gl(x) is cone transitive, then t−1 is also cone transitive. © agt, upv, 2022 appl. gen. topol. 23, no. 1 139 p. narayan mandal proof. observe that t −1 = t−1. again the proof goes via topological transitivity of t on sx, by theorem 2.9 and proposition 1.14 in [6]. � remark 2.16. if dim(x) ≥ 2, we may use birkhoff transitivity theorem to show that topological transitivity of t and having dense orbit of t are equivalent. we claim that both the notions are also equivalent for dim(x) = 1. in this case, x ∼= r and sx ∼= {−1, +1}. if t is not topologically transitive but it contains a dense orbit, then we get ||t(1)|| = −||t(−1)||, which is absurd. it is obvious that sx is a baire space with a countable basis of open sets. therefore topological transitivity of t ensures a dense orbit of t. hence in our setting both the notions are equivalent. theorem 2.17 (a necessary condition). let ti ∈ l(xi) for i = 1, 2 and t := t1 ⊕ t2 : x1 ⊕ x2 → x1 ⊕ x2, where each xi is real separable hilbert space. if t is topologically transitive on sx1 ⊕ x2, then ti is also topologically transitive on sxi for i = 1, 2. proof. since t is topologically transitive on sx1 ⊕ x2 , there exists (x1,x2) ∈ sx1 ⊕ x2 such that {t n (x1,x2) : n ∈ n} is dense in sx1 ⊕ x2 . it is enough to prove that ti is cone transitive on xi for i = 1, 2. first, we claim that x1 is cone transitive vector of t1. if not, then there exists an open cone v1 in x1 such that v1 ∩{tn1 x1 : n ∈ n} = ∅. this implies λtn1 x1 /∈ v1 for any λ > 0. let v2 be any open cone in x2. then v1 × v2 is an open cone in x1 ⊕ x2. since λtn1 x1 /∈ v1 for any λ > 0, we have (tn1 x1,t n 2 x2) ||(tn1 x1,t n 2 x2)|| /∈ v1 ×v2. therefore t is not topologically transitive on sx1 ⊕ x2 , which is a contradiction. � the converse of the above theorem is not true in general. we encounter this on several occasions in the proof of the main result. 3. main result theorem 3.1 (main theorem). if dim(x) ∈{1, 2,∞}, then there exists a t ∈ gl(x) such that t is topologically transitive on sx. if dim(x) /∈ {1, 2,∞}, then there exists no t ∈ gl(x) such that t is topologically transitive on sx. proof. the proof depends on the dimension of the space x. thus we prove the result by considering various cases. case a: let dim(x) = 1. then x ∼= r and sx ∼= {+1,−1}. take tx = −x, where x ∈ x. then t is topologically transitive. case b: let dim(x) = 2. then x ∼= r2 and sx ∼= sr2 := {(x1,x2) ∈ r2 : |x1|2 + |x2|2 = 1} = s1. take t := ( cos θ sin θ −sin θ cos θ ) , where θ π is an irrational number. then t : s1 → s1 is an irrational rotation and it is well-known that irrational rotation is topologically transitive on sr2 (for a proof, see example 1.12 in [6]). case c: let dim(x) = ∞. since x is an infinite dimensional separable hilbert space, we have x ∼= l2(z) (see [5]). let t be an invertible hypercyclic operator (see [6]). then t : sx → sx is topologically transitive. © agt, upv, 2022 appl. gen. topol. 23, no. 1 140 topological transitivity of the normalized maps induced by linear operators case d: let 2 < dim(x) < ∞. if s is a linear operator on x, then using real jordan canonical form, s has at least one block-diagonal which is linearly conjugate to one of the following: (1)   r cos θ r sin θ 0 0 −r sin θ r cos θ 0 0 0 0 s cos φ s sin φ 0 0 −s sin φ s cos φ   on r4, where r,s ∈ r\{0}; (2)   λ 1 0 ... 0 0 λ 1 ... 0 .. .. .. .. .. .. .. .. .. .. 0 0 .. λ 1 0 0 0 ... λ   on r n, where λ ∈ r\{0}; (3)   j1 i2 0 ... 0 0 j2 i2 ... 0 .. .. .. .. .. .. .. .. .. .. 0 0 .. jn−1 i2 0 0 0 ... jn   on r 2n, where ji = ( cos θ sin θ −sin θ cos θ ) and i2 = ( 1 0 0 1 ) ; (4)  α 0 00 r cos φ r sin φ 0 −r sin φ r cos φ   on r3, where α,r ∈ r\{0}; (5) ( α 0 0 β ) on r2, where α,β ∈ r\{0}. we now show that each of the above representation can not be cone transitive on their respective invariant subspaces. for (1): let s |r4 = t . by theorem 2.15, we may assume either |s| ≥ |r|, |s| > 1 or |s| = |r| = 1. if possible, let us assume (x,y,u,v) be a cone transitive vector of t such that ||(x,y, 0, 0)|| ≤ ||(0, 0,u,v)||. identify r4 as xyuv-space and tn(x,y,u,v) = (xrn cos nθ + yrn sin nθ,−xrn sin nθ + yrn cos nθ,usn cos nφ + vsn sin nφ,−usn sin nφ + vsn cos nφ). it is clear from the definition that the orbit of (x0,y0, 0, 0) lies on xy-plane and the orbit of (0, 0,u0,v0) lies on uvplane. since (x,y,u,v) is a cone transitive vector of t , we have (x,y) 6= (0, 0) and (u,v) 6= (0, 0). let (x′,y′, 0, 0) be any point in the xy-plane. then ||tn(x,y,u,v) − (x′,y′, 0, 0)|| ≥ ||(0, 0,usn cos nφ + vsn sin nφ,−usn sin nφ + vsn cos nφ)|| = |s|n||(0, 0,u,v)|| and ||tn(x,y,u,v)|| ≤ 2 × max{|r|n, |s|n} ≤ mn, for some m. if |s| > 1, then we take an open 4-ball centered at (x,y, 0, 0) with radius r′, where r′m < ||(x,y,0,0)|| 2 . let v be the smallest cone containing the 4-ball. then tn(x,y,u,v)) /∈ v , for any n ∈ n. if |s| = |r| = 1, then observe that t = t on sr4 . if (x′,y′, 0, 0) is any point in sr4 , similarly, we have ||t n (x,y,u,v) − (x′,y′, 0, 0)|| ≥ ||(0, 0,u,v)||. this means t is not topologically transitive on sr4 . © agt, upv, 2022 appl. gen. topol. 23, no. 1 141 p. narayan mandal for (2): let s |rn = t. by theorem 2.15, we may assume that |λ| ≥ 1. let (x1,x2, ...,xn) be any point in rn. let b be an open ball centered at (0, 0, ...,xn) with radius r < ||(0, 0, ...,xn)|| and v be the smallest open cone b. it is straightforward to verify that there exists c > 0 such that for every (α1,α2, ...,αn) ∈ v , we have |α1| |αn| < c. in particular, if r < ||(0,0,...,xn)|| 2 , then c < 1. if for some n ≥ n0, we have tn (x) ∈ v , then |λnx1 + nλn−1x2 + ... + ( n n− 1 ) λn−n+1xn| < c|λnxn| or, |x1 + n λ x2 + ... + ( n n− 1 ) 1 λn−1 xn| < c|xn|. observe that it is not true, if we choose n0 large enough. therefore there exists a n0 ∈ n, such that tm(x) /∈ v for m ≥ n0. for (3): the proof is in similar lines as that of the previous case (i.e., (2)). thus we omit the details. for (4): let s |r3 = t . here tn(x1,x2,x3) = (αnx1,rnx2 cos nφ+rnx3 sin nφ, −rnx2 sin nφ + rnx3 cos nφ) for any (x1,x2,x3) ∈ r3. if |α| > |r|, then take a sphere centered at (0,x2,x3) with radius r ′, where r′ < ||(0,x2,x3)|| 2 . if v be the smallest cone containing the sphere, then we claim that there exists a n0 ∈ n such that tn(x1,x2,x3) /∈ v for n ≥ n0. if possible, for some n ≥ n0, we have tn (x1,x2,x3) ∈ v . then |α|n||(x1,0,0)|| |r|n||(0,x2,x3)|| < 1, which is not true if we take n0 large enough. if |r| > |α|, then we take the sphere centered at (x1, 0, 0) with radius r′′ < ||(x1,0,0)|| 2 . if |α| = |r|, then ||t n(x1,0,0)|| ||tn(0,x2,x2)|| = ||(x1,0,0)|| ||(0,x2,x3)|| . hence t is not cone transitive. for (5): let s |r2 = t . observe that for any point (x,y), there exists an arbitrarily small � > 0 such that the smallest open cone containing b((x,y),�) contains only finitely many tn(x,y) when |α| 6= |β|. if |α| = |β|, then all tn(x,y) is in the smallest cone containing b((x,y),�), b((−x,y),�), b((x,−y),�) and b((−x,−y),�). hence in both the cases, t is not cone transitive. by theorem 2.15, s is not cone transitive, whenever 2 < dim(x) < ∞. hence the result. � remark 3.2. the proof of theorem 3.1 suggests that a similar argument can also ensure the non-existence of topologically transitive projective transformation on sx/∼, when 2 < dim(x) < ∞. in this case instead of taking a positive open cone, we may need to take a full open cone i.e., instead of λ > 0, we take λ 6= 0. remark 3.3. in contrast with the above result, for n ≥ 2, every n-dimensional compact manifold admits a chaotic homeomorphism. for a proof, see [1]. in particular, for each n ∈ n, there exists a homeomorphism hn on srn such that hn is topologically transitive on srn . © agt, upv, 2022 appl. gen. topol. 23, no. 1 142 topological transitivity of the normalized maps induced by linear operators we conclude by providing a complete list of linear operators whose normalized maps are topologically transitive (similarly, for real projective transformation), which is apparent from the above discussion. since linear conjugacy preserves cone transitivity, we make a list as follows: dim(x) t ∈ gl(x) such that (t,sx) is topologically transitive t ∈ gl(x) such that (t̃,sx/ ∼) is topologically transitive 1 ri, for r < 0. ri, for r 6= 0. 2 ( r cos θ r sin θ −r sin θ r cos θ ) , where θ π ∈ r\q and r 6= 0. ( r cos θ r sin θ −r sin θ r cos θ ) , where θ π ∈ r\q and r 6= 0. ∞ invertible positive supercyclic operators. invertible supercyclic operator. acknowledgements. i profusely thank the anonymous referee for a careful reading of the manuscript and for providing helpful suggestions that significantly improved the original manuscript. i sincerely thank prof. v. kannan and dr. t. suman kumar for their generous support and helpful discussions. i also acknowledge nbhm-dae (government of india) for financial aid (ref. no. 2/39(2)/2016/nbhm/r & d-ii/11397). references [1] j. m. aarts and f. g. m. daalderop, chaotic homeomorphisms on manifolds, topology and its applications 96 (1999), 93–96. [2] f. bayart and e. matheron, dynamics of linear operators, cambridge university press (2009), cambridge. [3] l. s. block and w. a. coppel, dynamics in one dimension, lecture notes in mathematics, 1513, springer-verlag (1992), berlin. [4] s. g. dani, dynamical properties of linear and projective transformation and their applications, indian journal of pure and applied mathematics 35 (2004), 1365–1394. [5] m. fabian, p. habala, p. hájek, v. montesinos and v. zizler, banach space theory: the basis for linear and nonlinear analysis, cms books in mathematics, springer (2011), new york. [6] k.-g. grosse-erdmann and a. peris manguillot, linear chaos, universitext, springer (2011), london. [7] g. herzog, on linear operators having supercyclic vectors, studia mathematica 103 (1992), 295–298. [8] n. h. kuiper, topological conjugacy of real projective transformation, topology 15 (1976), 13–22. [9] r. shah, a. nagar and s. shridharan (ed. by), elements of dynamical systems, hindusthan publishers (2020), delhi. © agt, upv, 2022 appl. gen. topol. 23, no. 1 143 @ appl. gen. topol. 22, no. 1 (2021), 79-89doi:10.4995/agt.2021.13608 © agt, upv, 2021 ideal spaces biswajit mitra and debojyoti chowdhury department of mathematics, university of burdwan, burdwan 713104, west bengal, india. (bmitra@math.buruniv.ac.in, sankha.sxc@gmail.com) communicated by a. tamariz-mascarúa abstract let c∞(x) denote the family of real-valued continuous functions which vanish at infinity in the sense that {x ∈ x : |f(x)| ≥ 1 n } is compact in x for all n ∈ n. it is not in general true that c∞(x) is an ideal of c(x). we define those spaces x to be ideal space where c∞(x) is an ideal of c(x). we have proved that nearly pseudocompact spaces are ideal spaces. for the converse, we introduced a property called “rcc” property and showed that an ideal space x is nearly pseudocompact if and only if x satisfies ”rcc” property. we further discussed some topological properties of ideal spaces. 2010 msc: 54f65; 54g20; 54d45; 54d60; 54d99. keywords: rings of continuous functions; ck(x) and c∞(x); nearly pseudocompact spaces; rcc properties. 1. introduction in this paper, by a space we shall mean completely regular hausdorff space, unless otherwise mentioned. as usual c(x) and c∗(x) are real-valued continuous and bounded continuous functions respectively. they are commutative rings with 1 under usual pointwise addition and multiplication. rigorous and systematic developments of these two rings are made in the classic monograph of l. gillman and m. jerrison entitled “rings of continuous functions”[7]. in fact most of the symbols, definitions and results are hired from the above book. there are two important classes of subrings of c(x), namely, c∞(x) and ck(x), where c∞(x) is the family of all those continuous functions such received 28 april 2020 – accepted 29 october 2020 http://dx.doi.org/10.4995/agt.2021.13608 b. mitra and d. chowdhury that {x : |f(x)| ≥ 1 n } is compact for all n ∈ n and ck(x) is the family of all functions f ∈ c(x) whose support, that is clx(x\z(f)), is compact, where z(f) = {x ∈ x : f(x) = 0}. both the subrings are in fact subrings of c∗(x) also. even more they are ideal in c∗(x). but though ck(x) is an ideal of c(x) , c∞(x) need not be an ideal of c(x). immediate example can be cited if we count x = n. in this paper we have worked on those spaces x for which c∞(x) is an ideal of c(x). for the sake of convenience, we define these spaces as ideal space. in this context it ought to be relevant to mention that azarpanah et. al in [14], [2], already did some works in this area. they have shown that every pseudocompact space is an ideal space and every ideal space is pseudocompact if the space is locally compact. they also introduced ∞-compact space, the space where ck(x) = c∞(x), which is trivially an ideal space. in this paper we have shown that every nearly pseudocompact space, introduced by henriksen and rayburn in [10], is also ideal space. we have further introduced a criteria, so called rcc property, a generalization of locally compact property, to go for converse. in fact we have shown that a nearly pseudocompact space is ideal space and an ideal space is nearly pseudocompact if and only if the space satisfies rcc property. throughout the paper we have given many examples and counter examples. henriksen and rayburn in their paper [10] have shown that every anti-locally realcompact space, i.e., having no point with realcompact neighbourhood, is a nearly pseudocompact space. all the examples of nearly pseudocompact spaces that they cited, are anti-locally realcompact but they did not produce any example of a nearly pseudocompact space which is not anti-locally realcompact. here we have cited such example [example 4.14]. at the end we tried to explore few topological properties of ideal spaces and finally have shown that if xand y are nearly pseudocompact, then x × y is nearly pseudocompact if and only if x × y is ideal space. 2. preliminaries as we already mentioned that most of the basic symbols and terminologies followed the book, the rings of continuous functions, by l. gillman and m. jerrison [7], yet for ready references, we include few basic notations, definitions and related results that will be repeatedly used here. for each f ∈ c(x), z(f) = {x ∈ x : f(x) = 0} is called the zero set of f. the complement of zero set is called cozero set, denoted as coz f. for each space x, βx is the largest compactification of x where every compact-valued continuous function can be continuously extended, referred as stone-čech compactification of x. it is also the largest compactification of x in which x is c∗-embedded. similarly υx is the largest realcompact subspace of βx in which x is c-embedded. a space is realcompact if it can be embedded as a closed subspace in the product of reals. the υx is referred as hewitt-nachbin completion of x or simply hewitt realcompactification of x. compactness and realcompactness can be easily characterized respectively by showing x = βx and x = υx. a space x is © agt, upv, 2021 appl. gen. topol. 22, no. 1 80 ideal spaces pseudocompact (i.e, every realvalued continuous function on x is bounded) if and only if βx = υx or equivalently βx\x = υx\x. both realcompactness and pseudocompactness are significant generalizations of compactness. however the realcompact and pseudocompact jointly enforce compact. a subset s of x is relatively pseudocompact if every continuous function on x is bounded over s. r.l. blair and m.a. swardson proved that a subset s of a space x is relatively pseudocompact if clβxs ⊆ υx [3, proposition 2.6]. henriksen and rayburn in their paper [10] defined a space x to be nearly pseudocompact if υx\x is dense in βx\x, a generalization of pseudocompact space. they have proved the following characterization of nearly pseudocompact spaces. theorem 2.1. a space x is nearly pseudocompact if and only if x = x1 ∪x2, where x1 is a regular closed almost locally compact pseudocompact subset, and x2 is a regular closed anti-locally realcompact subset and int(x1∩x2) = ∅ in the year 1976, rayburn in his paper [13] defined hard set in x. a subset h of x is hard in x if h is closed in x ∪ k where k = clβx(υx\x). it is clear that every hard set in x is closed in x. however he has also provided a characterization of hardness of a closed subset in this paper as follows. theorem 2.2. a closed subset f of x is hard in x if and only if there exists a compact set k such that for any open neighbourhood v of k, there exists a realcompact subset p of x so that f\v and x\p can be completely separated in x. in particular h is hard if it is completely separated from the complement of a realcompact subset of x. henriksen and rayburn in [10] described few more characterizations as follows. theorem 2.3. a space x is nearly pseudocompact if and only if every hard set is compact if and only if every regular hard set (i.e, a regular closed set which is hard in x) is compact. they have further proved the following theorem which is relevant in this paper. theorem 2.4. every regular closed subset of a nearly pseudocompact space is a nearly pseudocompact space. in the year 2005, mitra and acharyya in their paper [12] introduced two subrings of c(x), ch(x) and h∞(x), analogical to the rings ck(x) and c∞(x). as per their definition, ch(x) = {f ∈ c(x) : clx(x\z(f)) is hard in x} and h∞(x) := {f ∈ c(x) : {x ∈ x : |f(x)| ≧ 1 n }is hard in x}. they have shown that ch(x) is an ideal of c(x). but no conclusion was made regarding h∞(x). however they have given the following characterization of nearly pseudocompact spaces using ch(x) and h∞(x). © agt, upv, 2021 appl. gen. topol. 22, no. 1 81 b. mitra and d. chowdhury theorem 2.5. the following statements are equivalent for a space x: (1) x is nearly pseudocompact. (2) ck(x) = ch(x). (3) c∞(x) = h∞(x) (4) ch(x) ⊆ c ∗(x) (5) h∞(x) ⊆ c ∗(x) (6) h∞(x) ⋂ c∗(x) = c∞(x) (7) ch(x) ⋂ c∗(x) = ck(x) 3. ideal spaces-a generalization of nearly pseudocompact space in this section we shall formally study ideal spaces. we begin with the definition of ideal spaces that we already introduced above. definition 3.1. a space x is called an ideal space if c∞(x) is an ideal of c(x). azarpanah and soundarajan in [2], gave a nice characterization of ideal space. theorem 3.2. a space x is an ideal space if and only if every locally compact σ-compact subset is relatively pseudocompact if and only if every open locally compact subset is relatively pseudocompact. they further proved that within the class of local compact spaces, the notions of ideal space and pseudocompact space are identical. however here use of locally compact condition does not require to prove that every pseudocompact space is ideal. on the other hand, it will be shown here that, ideal space generalizes nearly pseudocompact and ∞-compact space. that the ∞-compact space is ideal trivially follows from its definition, hence we shall mainly concentrate on nearly pseudocompact space. before showing that, we shall first show that h∞(x) is indeed an ideal of c(x) which was unanswered in the paper of mitra and acharyya [12]. in fact we shall show that h∞(x) = crc(x), where crc(x) = {f ∈ c(x) : coz f is realcompact} and that crc(x) is an ideal of c(x), follows from the fact that realcompact cozero sets are closed under finite, in fact countable union and realcompactness is a co-zero hereditary property. the notion of crc(x) was introduced by t. isiwata in [11]. further, azarpanah et.al in [[1], theorem 3.8] produced another characterization of this ideal. here we established the following result. theorem 3.3. h∞(x) = crc(x). proof. we know that x\z(f) = ∪n[x ∈ x : |f(x)| 1 n ], for f ∈ c(x). if f ∈ h∞(x), [x ∈ x : |f(x)| ≥ 1 n ] is hard in x and hence realcompact. since [x ∈ x : |f(x)| 1 n ] is a co-zero subset of [x ∈ x : |f(x)| ≥ 1 n ], [x ∈ x : |f(x)| 1 n ] is also realcompact. as countable union of realcompact co-zero set is realcompact, so x\z(f) is also realcompact. so f ∈ crc(x). © agt, upv, 2021 appl. gen. topol. 22, no. 1 82 ideal spaces conversely, if f ∈ crc(x), then the zero set [x ∈ x : |f(x)| ≥ 1 n ] is completely separated from the complement of the realcompact co-zero set x\z(f), for all n. hence the [x ∈ x : |f(x)| ≥ 1 n ] is hard in x, for all n. thus f ∈ h∞(x). � corollary 3.4. every nearly pseudocompact space is an ideal space. proof. since a nearly pseudocompact space, h∞(x) = c∞(x) [theorem 2.5] and h∞(x) is always an ideal of c(x), c∞(x) is also an ideal of c(x). � however the following example shows that the converse is not true. example 3.5. let x1 be a non-compact realcompact space where the closure of the set d of points having compact neighbourhood is compact, e.g. [(−∞, 0) ∩ q] ∪ [0, 1] ∪ [(1, ∞) ∩ q]. then x1 × [0, ω1], where [0, ω1] is the space of all ordinals less than or equal to the first uncountable ordinal ω1, is an ideal space which is not nearly pseudocompact. the reason is quite simple. if we take u, a locally compact, σ-compact space, then u ⊆ d × ω1. now cld × ω1, being pseudocompact space, u is relatively pseudocompact; hence an ideal space. but it is not nearly pseudocompact, as x1 ×{0} being a regular closed subset of x1 × [0, ω1], not nearly pseudocompact as it is non-compact real-compact. in the above example, u is actually contained in a compact space as the right projection of u into [0, ω1] is also locally compact and σ-compact and hence is bounded by a compact subset k. thus u is then contained in cld × k. this type of space is called ∞-compact space. definition 3.6. a space is called ∞-compact if ck(x) = c∞(x). from [2, proposition 2.1], we know the following result. theorem 3.7. a space is ∞-compact if and only if every open locally compact σ-compact subset is bounded by a compact set. this tells us that every ∞-compact space is an ideal space. the following example is an ideal space which is not ∞-compact. example 3.8. as in example 3.5, we take x1 making product with tychonoff plank t . that it is an ideal space but not nearly pseudocompact follows along the same lines of argument as in example 3.5. we show here that it is not even ∞-compact. u = ∪n(1/4, 1/2) × ([0, ω1] × {n}) is open locally compact σ-compact subset. but u is not contained in any compact set as then, it’s projection into tychonoff plank would be covered by a compact set. but the right edge, the copy of n, being a closed subset of u would be compact, which is not true. we therefore have the following parallel strictly forward implications. compact →֒ pseudocompact →֒ nearly pseudocompact →֒ ideal space. compact →֒ ∞ − compact →֒ ideal space. next we give an example of a pseudocompact space which is not ∞-compact. © agt, upv, 2021 appl. gen. topol. 22, no. 1 83 b. mitra and d. chowdhury example 3.9. we take tychonoff plank [0, ω1]×[0, ω0]\{(ω1, ω0)}. the union of the horizontal line [0, ω1] × {n}, n ∈ n is open, locally compact, σ-compact subset but not bounded by any compact set. 4. introduction of rcc property and its significance we start first with the following definition; definition 4.1. a space x is said to satisfy rcc property if the set of points having compact neighbourhood and the set of points having realcompact neighbourhood are same. as for instance, locally compact spaces satisfy rcc property. theorem 4.2. every nearly pseudocompact space satisfies rcc property proof. as clβx(υx\x) = clβx(βx\x), clβx(υx\x)∩x = clβx(βx\x)∩x. but clβx(υx\x) ∩ x and clβx(βx\x) ∩ x are respectively the set of points in x having no realcompact and compact neighbourhoods. hence the set of points having compact neighbourhood is identical with that of having realcompact neighbourhood. hence every nearly pseudocompact space satisfies rcc property. � in 2004, aliabad et. al [14, corollary 1.2] proved that for a locally compact hausdorff space x, c∞(x) is ideal of c(x) if and only if x is pseudocompact space. in the year 1980, henriksen and rayburn, [10, theorem 3.9], proved that regular closed almost locally compact subset of nearly pseudocompact space is pseudocompact which in turn implies that locally compact nearly pseudocompact space is pseudocompact. so we have the following theorem. theorem 4.3. under the assumption of locally compactness, ideal, nearly pseudocompact and pseudocompact spaces are identical. in the next theorem we shall show that under rcc condition, nearly pseudocompact and ideal spaces are same. in [14], aliabad et. al introduced an ideal clσ(x) := {f ∈ c(x) : coz fis locally compact and σ − compact}. clσ(x) is a z − ideal of c(x) and by the result of [14, proposition 3.2], clσ(x) is the smallest z-ideal of c(x) containing c∞(x). further we note that clσ(x) ⊆ crc(x). theorem 4.4. a space x is nearly pseudocompact if and only if x satisfies rcc property and clσ(x) ⊂ c ∗(x) proof. let x be nearly pseudocompact. then x satisfies rcc property by theorem 4.2 . furthermore h∞(x) ⊆ c ∗(x), by theorem 2.5 (5). by theorem 3.3, h∞(x) = crc(x). thus clσ(x) ⊆ crc(x) = h∞(x) ⊆ c ∗(x). conversely, suppose x is not nearly pseudocompact space. since x satisfies rcc property, then the set dx of points with compact neighbourhood is non-empty, otherwise x would be anti-locally realcompact and hence nearly © agt, upv, 2021 appl. gen. topol. 22, no. 1 84 ideal spaces pseudocompact [10, corollary 3.5]. now for all f ∈ h∞(x), each point of x\z(f) has a realcompact neighbourhood [theorem 3.3]. as the space satisfies rcc property, ∀f ∈ h∞(x) we have x\z(f) ⊆ dx. as x is not nearly pseudocompact, by theorem 2.5(5), there exists f ∈ h∞(x) such that f is unbounded on x\z(f). thus there exists a copy n of n in x\z(f), c-embedded in x, [7, corollary 1.20]. we consider a continuous function h on x so that h(n) n3, for all n ∈ n. as n is c-embedded in x, n, hence any of its subset, is closed in x. so clβxn\n is contained in βx\x. so clβx(βx\x) ∪ {m ∈ n : m 6= n} is indeed a closed subsets of βx for each n ∈ n. due to complete regularity, for each n , there exists a continuous function ĝn : βx → r, such that ĝn(x) = { = 0 when x ∈ clβx(βx\x) or x ∈ {m : m 6= n}. = n 3 h(n) when x = n without loss of generality, we assume |ĝn| ≤ 1 as for each n, n 3 h(n) � 1. we take as usual, ĝ = ∑ n gn n2 . then ĝ ∈ c(βx). let g := ĝ| x . clearly ĝ is the stone-extension of g. as ĝ vanishes everywhere on βx\x, closure of {x ∈ x : |g(x)| ≥ 1 n } in βx must not intersect in βx\x. hence {x ∈ x : |g(x)| ≥ 1 n } = clβx{x ∈ x : |g(x)| ≥ 1 n } and is therefore compact. thus g ∈ c∞(x). now as g ∈ c∞(x), g ∈ clσ(x). then hg ∈ clσ(x). moreover hg(n) = n, ∀n and hence is unbounded. � corollary 4.5. a space is nearly pseudocompact if and only if it is an ideal space satisfying rcc property. proof. every nearly pseudocompact space is ideal [theorem 3.4] and satisfies rcc property [theorem 4.2]. conversely suppose x is an ideal space having rcc property, then let f ∈ clσ(x). then x\z(f) is locally compact and σ-compact and hence relatively pseudocompact by theorem 3.2 and hence clσ(x) ⊂ c ∗(x). by the above theorem, x is nearly pseudocompact. � remark 4.6. although it is evident from corollary 4.5 that we can not drop the condition rcc. however in support of the above corollary, we do refer the space given in example 3.5 that does not satisfy rcc property. but it is an ideal space which is not nearly pseudocompact. (0, 1)×[0, ω1] is precisely the set of points which have compact neighbourhood. but each point of x1×[0, ω1] has realcompact neighbourhood x1 is realcompact and [0, ω1] is locally compact. hence the space does not satisfy rcc . in the year 2001, azarpanah and soundararajan [ proposition 2.4, [2]], proved the following result. theorem 4.7 (proposition 2.4, [2]). for any space x, let cψ(x) be the family of all real-valued continuous functions over x with pseudocompact support. © agt, upv, 2021 appl. gen. topol. 22, no. 1 85 b. mitra and d. chowdhury then c∞(x) is subset of cψ(x) if and only if every open locally compact subset of x is relatively pseudocompact. in the year 2005, henriksen and mitra proved the following lemma. theorem 4.8 (lemma 2.10, [9]). a function f ∈ c(x) is in cψ(x) if and only if fg ∈ c∗(x) whenever g ∈ c(x). the following lemma directly follows as a corollary from theorems 3.2, 4.7 and 4.8 above. lemma 4.9. a space x is ideal if and only if ∀f ∈ c(x) and ∀g ∈ c∞(x), fg ∈ c∗(x). proof. suppose x is ideal. by theorem 3.2, every open locally compact subset of x is relatively pseudocompact. by theorem 4.7, c∞(x) ⊆ cψ(x). hence by theorem 4.8, for all f ∈ c(x) and for all g ∈ c∞(x), fg ∈ c ∗(x). conversely suppose ∀g ∈ c∞(x), fg ∈ c ∗(x). by theorem 4.8, we conclude that g ∈ cψ(x). so c∞(x) ⊆ cψ(x). by theorem 4.7, every open locally compact subset of x is relatively pseudocompact and hence by theorem 3.2, x is ideal. � in the year 1990, blair and swardson [proposition 2.6, [3]] proved the following result. theorem 4.10 (proposition 2.6, [3]). a subset a of x is relatively pseudocompact if and only clυx a is compact. w.w. comfort in his paper [4, theorem 4.1] included the following result proved by hager [16]. theorem 4.11. (hager-johnson) let u be open subset of x. if clυxu is compact, then clxu is pseudocompact. the following lemma again trivially follows from theorems 4.10 and 4.11. lemma 4.12. if u is an open relatively pseudocompact subset of x, then clxu is pseudocompact. theorem 4.13. a space is ideal if and only if the closure of its local compactness part is pseudocompact. proof. if x is ideal, then, the set dx of points which have compact neighbourhoods is open and locally compact and hence relatively pseudocompact by [theorem 1.3,[2]]. so clxdx is pseudocompact. conversely, let u be an open locally compact subset of x. then u ⊆ dx ⊆ clxdx. as clxdx is pseudocompact, u is relatively pseudocompact. hence x is ideal as follows from theorem 3.2. � we already mentioned in the introductory section that henriksen and rayburn in [10] did not give any example of nearly pseudocompact, which is not anti-locally realcompact. here we shall produce an example of nearly pseudocompact space which is not anti-locally realcompact. © agt, upv, 2021 appl. gen. topol. 22, no. 1 86 ideal spaces example 4.14. we take any anti-locally realcompact space x. attach (0, 0) of the tychonoff plank with a point y (say) in x. the resulting space is not anti-locally realcompact as its local compact part is t \{(0, 0)} which is also locally realcompact part; that is the space satisfies rcc property. moreover its almost local compact part is t which is pseudocompact and hence an ideal space. this space is not even ∞-compact. but the above theorem 4.5 tells that this space is nearly pseudocompact. 5. few properties of ideal spaces theorem 5.1. regular closed subspace of an ideal space is ideal. definition 5.2 (a.h. stone, [15]). a space x is called feebly compact if every pairwise disjoint locally finite family of open sets of x is finite. i. glicksberg in [8] proved that in a completely regular space, the notion of feebly compact and pseudocompact are identical. in fact he proved that pseudocompact completely regular space is feebly compact and a feebly compact space is pseudocompact. in the same paper he has further shown that in a feebly compact space, closure of any open set is also feebly compact. so through chronological arguments, we conclude that within the class of completely regular spaces, regular closed subspace of a pseudocompact space is pseudocompact and hence,in particular, in a completely regular pseudocompact space, the closure of an open set is also pseudocompact. proof. let x be an ideal space. a be a regular closed subspace of x. so clx(intxa) = a. let da be the set of points in a having compact neighborhood in a. so da is open subset of a. as intxa is dense in a, da∩intxa 6= ∅. but da being open in a, da ∩ intxa open in intxa and hence it is open in x. let x ∈ da ∩ intxa. as x ∈ da, x ∈ ux ⊆ kx,where ux is open in a, kx ⊆ a is compact. again x ∈ intxa. so there exists wx open in x such that x ∈ wx ⊆ a. so x ∈ ux ∩ wx ⊆ ux ⊆ kx. now ux ∩ wx is open in x as wx is open in x. so kx is a compact neighbourhood of x in x. thus x ∈ dx, the set of all points in x having compact neighbourhood in x. so da ∩ intxa ⊆ dx. as intxa is dense in a and da is open in a. so da ⊆ cla(da ∩ intxa). but cla(da ∩ intxa) = clx(da ∩ intxa). so da ⊆ clx(da ∩ intxa) ⊆ clxdx. by theorem 4.13, as x is ideal, clx(dx) is pseudocompact. we denote ω for clxdx. as (da ∩ intxa) is open in x, da ∩ intxa is open in ω. then clω(da∩intxa) is pseudocompact. but clω(da∩intxa) = clx(da∩intxa). so clx(da ∩ intxa) is pseudocompact. again da ⊆ clx(da ∩ intxa) ⊆ a. now we track down the same argument again. let w = clx(da ∩ intxa) ⊆ a. as w is pseudocompact, da being open in a and da ⊆ w , da is open in w also. so clwda is also pseudocompact. but clwda = clada. so clada is pseudocompact. so by theorem 4.13, a is ideal. � theorem 5.3. every open c-embedded subspace of an ideal space is ideal © agt, upv, 2021 appl. gen. topol. 22, no. 1 87 b. mitra and d. chowdhury proof. let u be a open c-embedded subset of x. let p ∈ du ⊆ u. so p has a compact neighbourhood k in u. as u is open in x, k turns out to be compact neighbourhood of p in x. so p ∈ dx ∩ u. conversely as p ∈ dx ∩ u, there exists a compact set k such that p ∈ intxk ⊆ k. as p ∈ u, p ∈ intxk ∩ u, which is also open in x. due to regularity, there exists an open set w in x such that p ∈ w ⊆ clxw ⊆ intxk ∩ u. now w is also open in u and clxw ⊂ k is also compact subset of u and is therefore compact neighbourhood of p in u. so p ∈ du. so du = dx ∩ u. now as x is ideal, clxdx is pseudocompact. as any subset of pseudocompact space is relatively pseudocompact, du being subset of clxdx is relatively pseudocompact subset of x. as u is c-embedded in x, du is relatively pseudocompact subset of u also. again du is open in u. hence cludu is pseudocompact, by above lemma 4.12. so u is ideal by theorem 4.13. � theorem 5.4. product of ideal and compact space is ideal. conversely if x × y is ideal, where y compact, then x is ideal. proof. suppose x is ideal and y is compact. then dx × y = dx×y . so clx×y (dx × y ) = clxdx × y and hence is pseudocompact as clxdx and product of compact and pseudocompact space is pseudocompact. second part follows immediately from the next theorem 5.6 and from the result that the projection on x from x × y , where y is compact, is a perfect map. � the following theorem is immediate. theorem 5.5. finite co-product of ideal spaces is ideal. proof. let x and y be two ideal spaces. then clx ∐ y dx ∐ y = clxdx ∪ cly dy and hence is pseudocompact as clxdx and cly dy are pseudocompact. hence x ∐ y is ideal. � but the result may not be true for arbitrary co-product. for that we take a very simple example, say n, the space of natural numbers with usual topology. then n is indeed countable co-product of singletons. every singleton is compact and hence ideal. but n is popularly known to be non-ideal space. theorem 5.6. let f : x → y be a perfect map. if x is ideal, then y is also ideal. proof. we first note that f−1dy ⊆ dx ⊆ clxdx. hence dy ⊆ f(clxdx). now x is ideal, clxdx is pseudocompact. as f is closed and preserves pseudocompactness, f(clxdx) is also pseudocompact. cly dy being regular closed subset of f(clxdx) is also pseudocompact. hence by theorem 4.13, y is ideal. � corollary 5.7. if a space is not ideal, then so is its absolute. proof. the corollary directly follows from the above theorem 5.6 as there always exist a perfect irreducible map from the absolute of a space onto the space itself. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 88 ideal spaces the next theorem 5.9 follows trivially from the following theorem 5.8 by henriksen and rayburn [10, theorem 3.17]. theorem 5.8. if x and y are nearly pseudocompact, then x × y is nearly pseudocompact if and only if clxdx × cly dy is pseudocompact. theorem 5.9. if x and y are nearly pseudocompact spaces, then x × y is nearly pseudocompact if and only if x × y is ideal. proof. as x × y is nearly pseudocompact, x × y is also ideal. conversely, if x × y is ideal, then clx×y dx×y , is pseudocompact. but clx×y dx×y = clxdx ×cly dy . so clxdx ×cly dy is pseudocompact. hence x ×y is nearly pseudocompact. � theorem 5.10. a realcompact, ideal space is ∞-compact. proof. since x is ideal, clxdx is pseudocompact. since x is realcompact, clxdx is also realcompact. hence it is compact. so clearly every open locally compact subset is bounded by a compact set. hence x is ∞-compact. � references [1] f. azarpanah, m. ghirati and a. taherifar, closed ideals in c(x) with different representations, houst. j. math. 44, no. 1 (2018), 363–383. [2] f. azarpanah and t. soundarajan, when the family of functions vanishing at infinity is an ideal of c(x), rocky mountain j. math. 31, no. 4 (2001), 1–8. [3] r. l. blair and m. a. swardson, spaces with an oz stone-čech compactification, topology appl. 36 (1990), 73–92. [4] w. w. comfort, on the hewitt realcompactification of a product space, trans. amer. math. soc. 131 (1968), 107–118. [5] j. m. domı́nguez, j. gómez and m. a. mulero , intermediate algebras between c∗(x) and c(x) as rings of fractions of c∗(x), topology appl. 77 (1997), 115–130. [6] r. engelking, general topology, heldermann verlag, berlin , 1989 [7] l. gillman and m. jerison, rings of continuous functions, university series in higher math, van nostrand, princeton, new jersey,1960. [8] i. glicksberg, stone-čech compactifications of products, trans. amer. math. soc. 90 (1959), 369–382. [9] m. henriksen, b. mitra, c(x) can sometimes determine x without x being realcompact, comment. math. univ. carolina 46, no. 4 (2005), 711–720. [10] m. henriksen and m. rayburn, on nearly pseudocompact spaces, topology appl. 11 (1980),161–172. [11] t. isiwata, on locally q-complete spaces, ii, proc. japan acad. 35, no. 6 (1956), 263– 267. [12] b. mitra and s. k. acharyya, characterizations of nearly pseudocompact spaces and spaces alike, topology proceedings 29, no. 2 (2005), 577–594. [13] m. c. rayburn, on hard sets, general topology and its applications 6 (1976), 21–26. [14] a. rezaei aliabad, f. azarpanah and m. namdari, rings of continuous functions vanishing at infinity, comm. math. univ. carolinae 45, no. 3 (2004), 519–533. [15] a. h. stone, hereditarily compact spaces, amer. j. math. 82 (1960), 900–914. [16] a. wood hager, on the tensor product of function rings, doctoral dissertation, pennsylvania state univ., university park, 1965. © agt, upv, 2021 appl. gen. topol. 22, no. 1 89 @ appl. gen. topol. 23, no. 2 (2022), 405-424 doi:10.4995/agt.2022.14000 © agt, upv, 2022 best proximity point (pair) results via mnc in busemann convex metric spaces this paper is dedicated in the memory of prof. hans-peter künzi moosa gabeleh a and pradip ramesh patle b a department of mathematics, ayatollah boroujerdi university, boroujerd, iran (gabeleh@abru.ac.ir, gab.moo@gmail.com) b department of mathematics, amity school of engineering and technology, amity university madhya pradesh, gwalior, india. (pradip.patle@gmail.com) communicated by m. abbas abstract in this paper, we present a new class of cyclic (noncyclic) α-ψ and β-ψ condensing operators and survey the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in the setting of reflexive busemann convex spaces. then an application of the main existence result to study the existence of an optimal solution for a system of differential equations is demonstrated. 2020 msc: 53c22; 47h09; 34a12. keywords: coupled best proximity point (pair); cyclic (noncyclic) condensing operator; optimum solution; busemann convex space. 1. introduction in the present paper, we mainly focus on the study of best proximity points for certain classes of mappings t : a ∪ b → a ∪ b for which t(a) ⊆ b and t(b) ⊆ a. such mappings are called cyclic mappings. likewise, if t(a) ⊆ a and t(b) ⊆ b, then t is said to be a noncyclic mapping. for the noncyclic case, the point (p,q) ∈ a×b is said to be a best proximity pair for the mapping t provided that p and q are two fixed points of t which estimates the distance between two sets a and b, that is, p = tp, q = tq and d(p,q) = dist(a,b). received 10 july 2020 – accepted 01 may 2022 http://dx.doi.org/10.4995/agt.2022.14000 https://orcid.org/0000-0001-5439-1631 https://orcid.org/0000-0002-9650-6107 m. gabeleh and p. r. patle the first existence theorem of best proximity points (pairs) was established in [7] for cyclic (noncyclic) relatively nonexpansive mappings. we recall that the mapping t : a∪b → a∪b is called relatively nonexpansive, if d(tx,ty) ≤ d(x,y) for all (x,y) ∈ a×b. in [11] the main result of [7] was extended from banach spaces to reflexive busemann convex spaces. recently gabeleh and markin introduced a class of cyclic (noncyclic) condensing operators by using a notion of measure of noncompactness and then studied the existence of best proximity points (pairs) for such mappings in (strictly convex) banach spaces (see theorem 3.4 and theorem 4.3 of [12]). subsequently gabeleh and künzi generalized the main conclusions of [12] from two points of view. one by considering a class of cyclic (noncyclic) condensing operators of integral type and another one by generalizing (strictly convex) banach spaces to reflexive busemann convex spaces (see theorem 3.9 and theorem 3.11 of [13]). we refer to articles [14, 18] in this direction. for more information about the existence of best proximity points for various classes of non-self mappings, one can see [17, 20, 21, 22]. in the current article, we introduce two new classes of cyclic (noncyclic) condensing operators, called α−ψ and β −ψ condensing operators which are properly contain the class of cyclic (noncyclic) condensing operators introduced by gabeleh and markin in [12]. we establish the existence of best proximity points (pairs) for such mappings in the framework of reflexive busemann convex spaces and then apply our conclusions to present a coupled best proximity point theorem by the notion of measure of noncompactness. finally, the existence of an optimal solution for a system of second order differential equations by using the existence result of best proximity points for the considered extension of cyclic condensing operators is studied. 2. preliminaries in this section, we compile the main notions and notations we will work with along this paper. 2.1. geodesic spaces. let (x,d) be a metric space. by b(x0; r) we denote the closed ball in the space x centered at x0 ∈ x with radius r > 0. consider a and b two nonempty subsets of x. define δx(a) = sup{d(x,y) : y ∈ a} for all x ∈ x, δ(a,b) = sup{δx(b) : x ∈ a}, diam(a) = δ(a,a). throughout this article, we say that the pair (a,b) satisfies a property if both a and b satisfy that property. for example, (a,b) is closed if and only if both a and b are closed. likewise, (a,b) ⊆ (c,d) if and only if a ⊆ c and b ⊆ d. the proximal pair of (a,b) is defined as a0 = {x ∈ a: d(x,y′) = dist(a,b) for some y′ ∈ b}, © agt, upv, 2022 appl. gen. topol. 23, no. 2 406 best proximity point (pair) results via mnc in busemann convex metric spaces b0 = {y ∈ b : d(x′,y) = dist(a,b) for some x′ ∈ a}. proximal pairs may be empty but, in particular, if a and b are nonempty, bonded, closed and convex in a reflexive banach space x, then (a0,b0) is a nonempty, bounded, closed and convex pair in x. we say that (a,b) is proximinal provided that a = a0 and b = b0. in this paper we will mainly work with geodesic spaces. let x,y ∈ x. a geodesic path from x to y is a mapping c : [0, l] ⊆ r → x such that c(0) = x, c(l) = y, and d(c(t),c(t′)) = |t− t′| for all t,t′ ∈ [0, l]. the image of the mapping c forms a geodesic segment which joins x and y and will be denoted by [x,y] whenever it is unique. (x,d) is said to be a (uniquely) geodesic space if every two points in x can be joined by a (unique) geodesic path. a point p ∈ x belongs to a geodesic segment [x,y] if and only if there exists t ∈ [0, 1] such that d(x,p) = td(x,y) and d(y,p) = (1 − t)d(x,y) and for convenience we write p = (1 − t)x⊕ ty. in this situation p = c(tl), where c : [0, l] → x is a geodesic path from x to y. a subset e of a geodesic space x is convex if it contains any geodesic segment joining each two points in e, that is, for any x,y ∈ e we have [x,y] ⊆ e. the (closed) convex hull of a subset e of a geodesic space x is the smallest (closed) convex set containing the set e which is denoted by con(e) and con(e), respectively. lemma 2.1 ([5]). let e be a nonempty subset of a geodesic space x. let g1(e) denote the union of all geodesic segments with endpoints in e. recursively, for n ≥ 2 put gn(e) = g1(gn−1(e)). then con(e) = ∞⋃ n=1 gn(e). it is remarkable to note that in a busemann convex space x the closure of con(e) is convex and so, coincides with con(a) (see [10]). a metric d: x×x → r of a uniquely geodesic space (x,d) is called convex if for any x ∈ x and every geodesic path c : [0, l] → x we have d(x,c(tl)) ≤ (1 − t)d(x,c(0)) + td(x,c(l)), ∀t ∈ [0, 1]. furthermore, (x,d) is said to be busemann convex ([6]) if for any two geodesics c1 : [0, l1] → x and c2 : [0, l2] → x one has d ( c1(tl1),c2(tl2) ) ≤ (1 − t)d(c1(0),c2(0)) + td(c1(l1),c2(l2)) ∀t ∈ [0, 1]. equivalently, a geodesic metric space (x,d) is convex in the sense of busemann if d ( (1 − t)x⊕ ty, (1 − t)z ⊕ tw ) ≤ (1 − t)d(x,z) + td(y,w), for all x,y,z,w ∈ x and t ∈ [0, 1]. it is well-known that busemann convex spaces are uniquely geodesic and with convex metric. a very important class of busemann convex spaces are cat(0) spaces, that is, metric spaces of nonpositive curvature in the sense of gromov (see [5, 6] for a detailed discussion on cat(0) spaces). © agt, upv, 2022 appl. gen. topol. 23, no. 2 407 m. gabeleh and p. r. patle in the sequel we say that a geodesic metric space (x,d) is strictly convex ([3]) if for every r > 0, a,x and y ∈ x with d(x,a) ≤ r, d(y,a) ≤ r and x 6= y, it is the case that d(a,p) < r, where p ∈ [x,y] − {x,y}. we mention that busemann convex spaces are strictly convex with convex metric ([9]). in the next section we will also work with reflexive geodesic spaces which is a generalization of the notion of reflexivity from banach to geodesic spaces. a geodesic space x is said to be reflexive if for every decreasing chain {cα}⊆ x with α ∈ i such that cα is nonempty, bounded, closed and convex for all α ∈ i we have that ⋂ α∈i cα 6= ∅. it was announced in [8] that a reflexive and busemann convex space is complete. the following property of reflexive and busemann convex spaces plays an important role in our coming discussions. proposition 2.2 ([11, proposition 3.1]). if (a,b) is a nonempty, closed and convex pair in a reflexive and busemann convex space x such that b is bounded, then (a0,b0) is nonempty, bounded, closed and convex. we are now ready to state a main existence result of [11]. theorem 2.3 ([11, theorem 3.3, theorem 3.4]). let (a,b) be a nonempty, compact and convex pair in a busemann convex space x. then every cyclic (noncyclic) relatively nonexpansive mapping defined on a∪b has a best proximity point (pair). we mention that the proof of theorem 2.3 is based on the fact that any compact and convex pair in a geodesic space with convex metric has a geometric notion, called proximal normal structure (see proposition 3.10 of [11]). to state an extended version of theorem 2.3 we recall the following concept. definition 2.4 ([12]). let (a,b) be a nonempty and bounded pair in a metric space (x,d) and t : a∪b → a∪b be a cyclic (noncyclic) mapping. we say that t is compact whenever the pair (t(a),t(b)) is compact. theorem 2.5 ([13, theorem 3.2 and theorem 3.4]). let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded and let closed convex hulls of finite sets be compact. assume that t : a∪b → a∪b is a cyclic (noncyclic) relatively nonexpansive mapping. if t is compact, then t has a best proximity point (pair). we finish this section by stating the following auxiliary lemmas. lemma 2.6 ([13, lemma 3.7]). let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d). assume that (e,f) ⊆ (a,b) is a nonempty and proximinal pair with dist(e,f) = dist(a,b). then the pair (con(e), con(f)) is proximinal with dist(con(e), con(f)) = dist(a,b). lemma 2.7 ([13, lemma 3.8]). let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded. © agt, upv, 2022 appl. gen. topol. 23, no. 2 408 best proximity point (pair) results via mnc in busemann convex metric spaces let t : a∪b → a∪b be a cyclic relatively nonexpansive mapping. suppose u0 := a0 and v0 := b0 and for all n ∈ n define un = con(t(un−1)), vn = con(t(vn−1)). then {(u2n,v2n)} is a descending sequence of nonempty, bounded, closed, convex and proximinal pairs which are t -invariant, that is, t is cyclic on u2n∪v2n for all n ∈ n. moreover, un+2 ⊆vn+1 ⊆un ⊆vn−1, dist(u2n,v2n) = dist(a,b), ∀n ∈ n. 2.2. measure of noncompactness on metric spaces. in this section we recall some basic notions of measure of noncompactness which will be used in introducing new classes of condensing operators. let (x,d) be a metric space. by σ we denote the collection of all bounded subsets of x. definition 2.8. a function µ : σ → [0,∞) is called a measure of noncompactness (mnc for brief) if it satisfies the following conditions: (i) µ(a) = 0 iff a is relatively compact, (ii) µ(a) = µ(a) for all a ∈ σ, (iii) µ(a∪b) = max{µ(a),µ(b)} for all a,b ∈ σ. we can survey the following useful properties of an mnc, easily; (1) if a ⊆ b, then µ(a) ≤ µ(b), (2) µ(a∩b) ≤ min{µ(a),µ(b)} for all a,b ∈ σ, (3) if a is a finite set, then µ(a) = 0, (4) if {an} is a decreasing sequence of nonempty, bounded and closed subsets of x such that limn→∞µ(an) = 0, then a∞ := ∩n≥1an is nonempty and compact. in the case that (x,d) is a geodesic metric space, we say µ is invariant w.r.t. the convex hull, whenever µ(con(a)) = µ(a), ∀a ∈ σ. from now on, we assume that the considered mnc is invariant w.r.t. convex hulls. two well-known mnss are due to kuratowski and hausdorff which are denoted by α and χ, respectively and define as below: α(a) = inf{ε > 0 | a ⊆∪nj=1ej : ej ∈ σ, diam(ej) ≤ ε, ∀ 1 ≤ j ≤ n < ∞}, χ(a) = inf{ε > 0 | a ⊆∪nj=1b(xj; rj) : xj ∈ x, rj ≤ ε, ∀ 1 ≤ j ≤ n < ∞}, for all a ∈ σ (see [2] for more details). 3. best proximity points (pairs) we begin our investigations by recalling the following class of functions which will be considered as control functions to introduce a new class of cyclic (noncyclic) condensing operators. let ψ denote the class of functions ψ : [0,∞) → [0,∞) such that limn→∞ψn(t) = 0, for every t > 0, where ψn denotes nth iteration of ψ. it is evident that for every nondecreasing ψ ∈ ψ, for each t > 0, ψ(t) < t. © agt, upv, 2022 appl. gen. topol. 23, no. 2 409 m. gabeleh and p. r. patle let α : x ×x → [0, +∞) be a mapping. t : x → x is called α-admissible ([23]) if for every x,y ∈ x, α(x,y) ≥ 1 =⇒ α(tx,ty) ≥ 1. 3.1. results for α-ψ-condensing operator. let us introduce the following new class of cyclic (noncyclic) mappings. definition 3.1. let (a,b) be a nonempty and convex pair in a banach space x and µ an mnc on x. a mapping t : a∪b → a∪b is said to be a cyclic (noncyclic) α − ψ-condensing operator if for any nonempty, bounded, closed, convex, proximinal and t-invariant pair (k1,k2) ⊆ (a,b) with dist(k1,k2) = dist(a,b) and x ∈ x, we have α(x,tx)µ(t(k1) ∪t(k2)) ≤ ψ(µ(k1 ∪k2)) where ψ ∈ ψ is a nondecreasing function and α : x ×x → [0, +∞). example 3.2. let x = bc(r+) be a banach space and conisder a = {x ∈ bc(r+) : ||x|| ≤ 1} and b = {x ∈ bc(r+) : ||x|| > 1}. let us define t : a∪b → a∪b as tx =   x 2 , if x ∈ a; 2x− 2, if x ∈ b. clearly t is noncyclic mapping. let k1 = a and k2 = b. then for µ(e) = diam(e), we have µ(t(k1) ∪t(k2)) = max{µ(t(k1)),µ(t(k2))} = max { sup { ‖tx(t) −ty(t)‖ : x(t),y(t) ∈ k1 } , sup { ‖tx(t) −ty(t)‖ : x(t),y(t) ∈ k2 }} = max { sup { ‖ x(t) 2 − y(t) 2 ‖ : x(t),y(t) ∈ k1 } , sup { ‖2x(t) − 2 − 2y(t) + 2‖ : x(t),y(t) ∈ k2 }} = max { sup {1 2 ‖x(t) −y(t)‖ : x(t),y(t) ∈ k1 } , sup { 2‖x(t) −y(t)‖ : x(t),y(t) ∈ k2 }} =2µ(t(k2)) = 2µ(k1 ∪k2) which shows that t is not noncyclic condensing mapping. but, t is noncyclic α-ψ-condensing, indeed if we define α : x ×x → x as α(x,tx) = { 1, if x ∈ a; 0, otherwise. then α(x,tx)µ(t(k1) ∪t(k2)) ≤ ψ(µ(k1 ∪k2)), where ψ(t) = t2, t ≥ 0. now we are ready to state and prove our first existence result for a best proximity point. theorem 3.3. let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded and let closed convex hulls of finite sets be compact. assume that µ is an mnc on x and t : a∪b → © agt, upv, 2022 appl. gen. topol. 23, no. 2 410 best proximity point (pair) results via mnc in busemann convex metric spaces a∪b is a cyclic relatively nonexpansive α-admissible α-ψ-condensing operator. then t has a best proximity point if there exists a point x0 in x such that α(x0,tx0) ≥ 1. proof. let us define two sequences: (i) {(u2n,v2n)} consisting of nonempty, bounded, closed, convex, proximinal and t-invariant pairs as defined in lemma 2.7. (ii) let x0 ∈ x and {xn} such that xn = txn−1 for every n ≥ 1. since α(x0,tx0) ≥ 1, α-admissibility of t implies that α(x1,x2) ≥ 1. applying recursion, we get α(xn,xn+1) ≥ 1, for every n ≥ 0. we note that if for some k ∈ n we have max{µ(u2k),µ(v2k)} = 0, then t : u2k∪v2k →u2k∪v2k is a compact and cyclic relatively nonexpansive mapping and so the result follows from theorem 2.5. assume that max{µ(u2n),µ(v2n)} > 0 for all n ∈ n. since t is a α−ψ-condensing operator, µ(u2n+1 ∪v2n+1) ≤ α(xn,xn+1)µ(u2n+1 ∪v2n+1) = α(xn,txn)µ(con(t(u2n)) ∪ con(t(v2n))) ≤ α(xn,txn)µ(t(u2n) ∪t(v2n)) ≤ ψ(µ(u2n) ∪µ(v2n)) < µ(u2n) ∪µ(v2n). repeating this pattern we get the following inequality µ(u2n+1 ∪v2n+1) ≤ ψn(µ(u0 ∪v0)), which yields us µ(u2n+1 ∪v2n+1) → 0, as n →∞. that is, lim n→∞ max{µ(u2n),µ(v2n)} = 0. let u∞ = ∞⋂ n=0 u2n, & v∞ = ∞⋂ n=0 v2n. by the properties of the measure of noncompactness, the pair (u∞,v∞) ⊆ (a,b) is nonempty, compact and convex with dist(u∞,v∞) = dist(a,b). on the other hand, t : u∞ ∪v∞ → u∞ ∪v∞ is a cyclic relatively nonexpansive mapping. now theorem 2.3 ensures that t has a best proximity point. � we now state the noncyclic version of theorem 3.3 in order to study the existence of best proximity pairs. theorem 3.4. let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded and let closed convex hulls of finite sets be compact. assume that µ is an mnc on x and t : a∪b → a∪b is a noncyclic relatively nonexpansive α-admissible and α-ψ-condensing operator. then t admits a best proximity pair if there exists x0 ∈ x such that α(x0,tx0) ≥ 1. © agt, upv, 2022 appl. gen. topol. 23, no. 2 411 m. gabeleh and p. r. patle proof. consider the sequence of pairs (un,vn) ⊆ (a,b) for n ∈ n ∪{0} as in lemma 2.7. it follows from the proof of theorem 3.11 of [13] that {(un,vn)} is a descending sequence of nonempty, bounded, closed, convex and proximinal pairs with dist(un,vn) = dist(a,b) which are t-invariant. if there exists some k ∈ n for which max{µ(uk),µ(vk)} = 0, then from theorem 2.5 the result will be concluded. now let x0 ∈ x. by α-admissibility of t and α(x0,tx0) ≥ 1, we get α(xn,xn+1) ≥ 1. suppose that max{µ(un),µ(vn)} > 0 for all n ∈ n∪{0}. by a discussion as in the proof of theorem 3.3, we can see that limn→∞ max{µ(un), µ(vn)} = 0. now, if we set u∞ = ∞⋂ n=0 un, & v∞ = ∞⋂ n=0 vn, then (u∞,v∞) is a nonempty, compact and convex pair with dist(u∞,v∞) = dist(a,b). hence, the existence of a best proximity pair for the mapping t is deduced from theorem 2.3. � 3.2. results for β-ψ-condensing operator. now we recall the concept of β-admissibility which is introduced by rehman et al. in [19], with a slight modification as follows. definition 3.5. let β : 2x → [0, +∞). a mapping t : x → x is called β-admissible if for every m1,m2 ∈ 2x, we have β(m1 ∪ m2) ≥ 1 =⇒ β(con(t(m1) ∪t(m2))) ≥ 1. we now introduce a new class of cyclic (noncyclic) operators in the following definition. definition 3.6. let (a,b) be a nonempty and convex pair in a banach space x and µ an mnc on x. a mapping t : a∪b → a∪b is said to be a cyclic (noncyclic) β-ψ-condensing operator if for any nonempty, bounded, closed, convex, proximinal and t-invariant pair (k1,k2) ⊆ (a,b) with dist(k1,k2) = dist(a,b), we have β(k1 ∪k2)µ(t(k1) ∪t(k2)) ≤ ψ(µ(k1 ∪k2)) where ψ ∈ ψ is a nondecreasing function and β : 2x → [0, +∞). example 3.7. let x = bc(r+) be a banach space and conisder a = {x ∈ bc(r+) : ||x|| ≤ 1} and b = {x ∈ bc(r+) : ||x|| > 1}. let us define t : a∪b → a∪b as tx =   x 3 , if x ∈ a; 2x− 3 2 , if x ∈ b. © agt, upv, 2022 appl. gen. topol. 23, no. 2 412 best proximity point (pair) results via mnc in busemann convex metric spaces clearly t is noncyclic mapping. let k1 = a and k2 = b. then for µ(e) = diam(e), we have µ(t(k1) ∪t(k2)) = max{µ(t(k1)),µ(t(k2))} = max { sup { ‖tx(t) −ty(t)‖ : x(t),y(t) ∈ k1 } , sup { ‖tx(t) −ty(t)‖ : x(t),y(t) ∈ k2 }} = max { sup { ‖ x(t) 3 − y(t) 3 ‖ : x(t),y(t) ∈ k1 } , sup { ‖2x(t) − 3 2 − 2y(t) + 3 2 ‖ : x(t),y(t) ∈ k2 }} = max { sup {1 3 ‖x(t) −y(t)‖ : x(t),y(t) ∈ k1 } , sup { 2‖x(t) −y(t)‖ : x(t),y(t) ∈ k2 }} =2µ(t(k2)) = 2µ(k1 ∪k2), which shows that t is not noncyclic condensing mapping. but, t is noncyclic β-ψ-condensing. indeed if we define β : 2x → [0, +∞) as β(k1∪k2) = 1, then β(k1 ∪k2)µ(t(k1) ∪t(k2)) ≤ ψ(µ(k1 ∪k2)), where ψ(t) = t2, t ≥ 0. theorem 3.8. let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded and let closed convex hulls of finite sets be compact. assume that µ is an mnc on x and t : a∪b → a∪b is a cyclic relatively nonexpansive β-admissible, β-ψ-condensing operator. then t has a best proximity point if there exist nonempty, bounded, closed and convex x0,y0 ⊆ x such that β(x0 ∪y0) ≥ 1. proof. let us define sequences: (i) {(u2n,v2n)} consisting of nonempty, bounded, closed, convex, proximinal and t-invariant pairs as defined in the lemma 2.7. we note that if for some k ∈ n we have max{µ(u2k),µ(v2k)} = 0, then t : u2k∪v2k →u2k∪v2k is a compact and cyclic relatively nonexpansive mapping and so the result follows from theorem 2.5. assume that max{µ(u2n),µ(v2n)} > 0 for all n ∈ n. since β(u0 ∪ v0) ≥ 1, β-admissibility of t implies that β(u1 ∪v1) ≥ 1. applying recursion, we get β(u2n ∪u2n) ≥ 1, for every n ≥ 0. since t is a β −ψ-condensing operator, µ(u2n+1 ∪v2n+1) ≤ β(u2n ∪u2n)µ(u2n+1 ∪v2n+1) = β(u2n ∪u2n)µ(con(t(u2n)) ∪ con(t(v2n))) ≤ β(u2n ∪u2n)µ(t(u2n) ∪t(v2n)) ≤ ψ(µ(u2n) ∪µ(v2n)). repeating this pattern we get the following inequality µ(u2n+1 ∪v2n+1) ≤ ψn(µ(u0 ∪v0)), which yields us µ(u2n+1 ∪v2n+1) → 0, as n →∞. that is, lim n→∞ max{µ(u2n),µ(v2n)} = 0. © agt, upv, 2022 appl. gen. topol. 23, no. 2 413 m. gabeleh and p. r. patle let u∞ = ∞⋂ n=0 u2n, & v∞ = ∞⋂ n=0 v2n. by the properties of the measure of noncompactness, the pair (u∞,v∞) ⊆ (a,b) is nonempty, compact and convex with dist(u∞,v∞) = dist(a,b). on the other hand, t : u∞ ∪v∞ → u∞ ∪v∞ is a cyclic relatively nonexpansive mapping. now theorem 2.3 ensures that t has a best proximity point. � we now state the noncyclic version of theorem 3.8 in order to study the existence of best proximity pairs. theorem 3.9. let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded and let closed convex hulls of finite sets be compact. assume that µ is an mnc on x and t : a∪b → a∪b is a noncyclic relatively nonexpansive β-admissible and β−ψ-condensing operator. then t admits a best proximity pair if there exist nonempty, bounded, closed and convex x0,y0 ⊆ x such that β(x0 ∪y0) ≥ 1. proof. consider the sequence of pairs (un,vn) ⊆ (a,b) for n ∈ n ∪{0} as in lemma 2.7. it follows from the proof of theorem 3.11 of [13] that {(un,vn)} is a descending sequence of nonempty, bounded, closed, convex and proximinal pairs with dist(un,vn) = dist(a,b) which are t-invariant. if there exists some k ∈ n for which max{µ(uk),µ(vk)} = 0, then from theorem 2.5 the result will be concluded. now by β-admissibility of t and β(u0,v0) ≥ 1, we get β(un,vn) ≥ 1. suppose that max{µ(un),µ(vn)} > 0, ∀n ∈ n∪{0}. by a discussion as in the proof of theorem 3.8, we can see that limn→∞ max{µ(un), µ(vn)} = 0. now, if we set u∞ = ∞⋂ n=0 un, & v∞ = ∞⋂ n=0 vn, then (u∞,v∞) is a nonempty, compact and convex pair with dist(u∞,v∞) = dist(a,b). hence, the existence of a best proximity pair for the mapping t is deduced from theorem 2.3. � 4. coupled best proximity point results in this section, we study the existence of coupled best proximity points in the setting of busemann convex spaces. to this end, we recall the following concepts which first appeared in the ph.d thesis of the first author ([15]). let (a,b) be a nonempty pair in a metric space (x,d) and s : (a×a) ∪ (b×b) → a∪b be a cyclic mapping, that is, s(a×a) ⊆ b and s(b×b) ⊆ a. a point (u,v) ∈ (a×a)∪(b×b) is called a coupled best proximity point for the mapping s provided that d(u,s(u,v)) = d(v,s(v,u)) = dist(a,b). © agt, upv, 2022 appl. gen. topol. 23, no. 2 414 best proximity point (pair) results via mnc in busemann convex metric spaces we also need the following lemma. lemma 4.1 ([1]). suppose that µ1,µ2, · · · ,µn are measures of noncompactness on the metric spaces x1,x2, · · · ,xn, respectively. moreover, assume that the function θ : [0,∞)n → [0,∞) is convex and θ(x1,x2, · · · ,xn) = 0 if and only if xj = 0 for all j = 1, 2, · · · ,n. then µ(e) = θ ( µ1(e1),µ2(e2), · · · ,µn(en) ) , defines a measure of noncompactness on x1×x2×···×xn, where ej denotes the natural projection of e into ej for j = 1, 2, · · · ,n. the next lemma was given in [16]. for the convenience of the reader, we include a proof of this fact. lemma 4.2 (see [16, lemma 4.2]). let (a,b) be a nonempty pair in a metric space (x,d). consider the product space x ×x with the metric d∞ ( (x1,y1), (x2,y2) ) = max{d(x1,x2),d(y1,y2)}, ∀(x1,y1), (x2,y2) ∈ x×x. then the pair (a,b) is proximinal in x if and only if (a × a,b × b) is proximinal in x ×x. proof. note that dist(a × a,b × b) = dist(a,b). in fact, for any (a,a′) ∈ a×a, (b,b′) ∈ b ×b we have dist(a×a,b ×b) = inf( (a,a′),(b,b′) ) ∈(a×a)×(b×b) d∞ ( (a,a′), (b,b′) ) = inf( (a,a′),(b,b′) ) ∈(a×a)×(b×b) max{d(a,b),d(a′,b′)} = dist(a,b). suppose (a,b) is proximinal and (a,a′) ∈ a × a, then we can find b,b′ ∈ b such that d(a,b) = d(a′,b′) = dist(a,b). thus for an element (b,b′) ∈ b × b we have d∞ ( (a,a′), (b,b′) ) = dist(a,b) ( = dist(a × a,b × b) ) , that is, (a×a)0 = a×a. by a similar manner, (b ×b)0 = b ×b and so the pair( (a×a), (b×b) ) is proximinal in x×x. now assume that ( (a×a), (b×b) ) is proximinal and a ∈ a, then (a,a) ∈ a × a and so there exists a point (b,b′) ∈ b × b for which d∞ ( (a,a), (b,b′) ) = dist(a,b) which deduces that d(a,b) = d(a,b′) = dist(a,b), that is, a0 = a. equivalently, b0 = b which implies that (a,b) is proximinal and this completes the proof of lemma. � we now state our first coupled best proximity point result. theorem 4.3. let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded and let closed convex hulls of finite sets be compact. assume that µ is an mnc on x and s : (a×a)∪(b×b) → a∪b is a cyclic mapping satisfying following conditions. (i) let (k1,k2) ⊆ (a,b) and (k′1,k′2) ⊆ (a,b) be nonempty, bounded, closed, convex, proximinal and s-invariant pairs with dist(k1,k2) = © agt, upv, 2022 appl. gen. topol. 23, no. 2 415 m. gabeleh and p. r. patle dist(a,b) = dist(k′1,k ′ 2), γ : x 2 × x2 → [0, +∞) such that for any (x1,y1) ∈ x ×x we have γ((x1,y1), (x2,y2))µ((s(k1×k′1)∪s(k2×k ′ 2))) ≤ 1 2 ψ ( max{µ(k1∪k′1),µ(k2∪k ′ 2)} ) , where ψ ∈ ψ is nondecreasing. (ii) for all (x,y), (u,v) ∈ x ×x and γ ( (x,y), (u,v) ) ≥ 1 we have γ ( (s(x,y),s(y,x)), (s(u,v),s(v,u) ) ≥ 1. (iii) moreover, d(s(x1,x2),s(y1,y2)) ≤ d∞ ( (x1,y1), (x2,y2) ) , ∀(x1,x2) ∈ a×a, ∀(y1,y2) ∈ b ×b. (iv) furthermore, there exists (x0,y0) ∈ x ×x such that γ ( (x0,y0), (s(x0,y0),s(y0,x0)) ) ≥ 1, γ ( (y0,x0), (s(y0,x0),s(x0,y0)) ) ≥ 1. then s admits a coupled best proximity point. proof. set µ̃(e) := max{µ(e1),µ(e2)}, where ej denotes the natural projection of e into ej for j = 1, 2. thus by lemma 4.1 µ̃ is an mnc on x × x. let t : (a×a) ∪ (b ×b) → (a×a) ∪ (b ×b) defined by t(u,v) = (s(u,v),s(v,u)), ∀(u,v) ∈ (a×a) ∪ (b ×b). note that if (u,v) ∈ a×a, then by the fact that s is cyclic, (s(u,v),s(v,u)) ∈ b×b, that is, t(a×a) ⊆ b×b. equivalently, t(b×b) ⊆ a×a. therefore, t is cyclic on (a×a) ∪ (b ×b). besides, for any ((x1,x2), (y1,y2)) ∈ (a×a) × (b ×b) we have d∞ ( t(x1,x2),t(y1,y2) ) = d∞ (( s(x1,x2),s(x2,x1) ) , ( s(y1,y2),s(y2,y1) )) = max{d ( s(x1,x2),s(y1,y2) ) ,d ( s(x2,x1),s(y2,y1) ) } ≤ max{d∞ ( (x1,y1), (x2,y2) ) ,d∞ ( (x2,y2),d(x1,y1) ) } = d∞ ( (x1,x2), (y1,y2) ) . hence, t is relatively nonexpansive. let us now define α : x2 ×x2 → [0, +∞) as follows: α((x1,y1), (x2,y2)) = min{γ((x1,y1), (x2,y2)),γ((y1,x1), (y2,x2))}. by our hypothesis (ii), it is clear that whenever α((x1,y1), (x2,y2)) ≥ 1, we have α(t(x1,y1),t(x2,y2)) ≥ 1, which shows that t is α-admissible. also by hypothesis (iv) it is clear that there exists (x0,y0) ∈ x×x such that α((x0,y0), (y0,x0)) ≥ 1. © agt, upv, 2022 appl. gen. topol. 23, no. 2 416 best proximity point (pair) results via mnc in busemann convex metric spaces moreover, we have α((x1,y1), (x2,y2))µ̃(t(k1 ×k′1) ∪t(k2 ×k ′ 2)) = α((x1,y1), (x2,y2)) max{µ̃ ( t(k1 ×k′1) ) , µ̃ ( t(k2 ×k′2) ) } = α((x1,y1), (x2,y2)) max{µ̃ ( s(k1 ×k′1) ×s(k ′ 1 ×k1) ) , µ̃ ( s(k2 ×k′2) ×s(k ′ 2 ×k2) ) } = α((x1,y1), (x2,y2)) max { max{µ ( s(k1 ×k′1) ) ,µ ( s(k′1 ×k1) ) }, max{µ ( s(k2 ×k′2) ) ,µ ( s(k′2 ×k2) ) } } = α((x1,y1), (x2,y2)) max { max{µ ( s(k1 ×k′1) ) ,µ ( s(k2 ×k′2) ) }, max{µ ( s(k′1 ×k1) ) ,µ ( s(k′2 ×k2) ) } } = α((x1,y1), (x2,y2)) max { µ ( s(k1 ×k′1) ∪s(k2 ×k ′ 2) ) ,µ ( s(k′1 ×k1) ∪s(k ′ 2 ×k2) )} ≤ α((x1,y1), (x2,y2)) [ µ ( s(k1 ×k′1) ∪s(k2 ×k ′ 2) ) + µ ( s(k′1 ×k1) ∪s(k ′ 2 ×k2) )] ≤ 1 2 ψ ( max{µ(k1 ∪k′1),µ(k2 ∪k ′ 2)} ) + 1 2 ψ ( max{µ(k′1 ∪k1),µ(k ′ 2 ∪k2)} ) by hypothesis(i) = ψ ( max{µ(k1 ∪k′1),µ(k2 ∪k ′ 2)} ) = ψ ( max { max{µ(k1),µ(k′1)}, max{µ(k2),µ(k ′ 2)} }) = ψ ( max { µ̃(k1 ×k′1), µ̃(k2 ×k ′ 2) }) = ψ ( µ̃ ( (k1 ×k′1) ∪ (k2 ×k ′ 2) )) . this implies that t is a α-ψ-condensing operator. now, theorem 3.3 ensures that t has a best proximity point, called (p,q) ∈ (a×a) ∪ (b ×b). that is, dist(a,b) = d∞ ( (p,q),t(p,q) ) = d∞ ( (p,q), (s(p,q),s(q,p)) ) = max{d(p,s(p,q)),d(q,s(q,p))}. thus (p,q) is a coupled best proximity point of s. � theorem 4.4. let (a,b) be a nonempty, closed and convex pair in a reflexive and busemann convex space (x,d) such that b is bounded and let closed convex hulls of finite sets be compact. assume that µ is an mnc on x and s : (a×a)∪(b×b) → a∪b is a cyclic mapping satisfying following conditions. (i) for all nonempty, bounded, closed, convex, proximinal and s-invariant pairs (k1,k2) ⊆ (a,b) and (k′1,k′2) ⊆ (a,b) with dist(k1,k2) = dist(a,b) = dist(k′1,k ′ 2) and γ : 2 x×x → [0, +∞) such that γ(k1×k2)µ((s(k1×k′1)∪s(k2×k ′ 2))) ≤ 1 2 ψ ( max{µ(k1∪k′1),µ(k2∪k ′ 2)} ) , where ψ ∈ ψ is nondecreasing. (ii) for any (u,v ) ∈ x ×x and γ(u,v ) ≥ 1 we have γ ( con(s(u ×v ) ×s(v ×u)) ) ≥ 1. (iii) moreover, d(s(x1,x2),s(y1,y2)) ≤ d∞ ( (x1,y1), (x2,y2) ) , ∀(x1,x2) ∈ a×a, ∀(y1,y2) ∈ b ×b, (iv) furthermore, there exists closed and convex x0,y0 ⊆ x such that γ ( (x0 ×y0)) ) ≥ 1 and γ ( y0 ×x0 ) ≥ 1. © agt, upv, 2022 appl. gen. topol. 23, no. 2 417 m. gabeleh and p. r. patle then s admits a coupled best proximity point. proof. let us set µ̃(e) := max{µ(e1),µ(e2)}, where ej denotes the natural projection of e into ej for j = 1, 2. thus by lemma 4.1 µ̃ is an mnc on x ×x. then following proof of theorem 4.3 it is easy to show t is cyclic on (a×a) ∪ (b ×b) and t is relatively nonexpansive. let us now define β : 2x×x → [0, +∞) as follows: β(e1 ×e2) = min{γ(e1 ×e2),γ(e2 ×e1)}. by our hypothesis (ii), it is clear that whenever β(e1 × e2) ≥ 1, we have β(con(t(e1 ×e2))) ≥ 1, which shows that t is β-admissible. also by hypothesis (iv) it is clear that there exist x0,y0 ⊆ x such that β(x0 ×y0) ≥ 1. moreover, we have β(k1 ×k2)µ̃(t(k1 ×k′1) ∪t(k2 ×k ′ 2)) = β(k1 ×k2) max{µ̃ ( t(k1 ×k′1) ) , µ̃ ( t(k2 ×k′2) ) } = β(k1 ×k2) max{µ̃ ( s(k1 ×k′1) ×s(k ′ 1 ×k1) ) , µ̃ ( s(k2 ×k′2) ×s(k ′ 2 ×k2) ) } = β(k1 ×k2) max { max{µ ( s(k1 ×k′1) ) ,µ ( s(k′1 ×k1) ) }, max{µ ( s(k2 ×k′2) ) ,µ ( s(k′2 ×k2) ) } } = β(k1 ×k2) max { max{µ ( s(k1 ×k′1) ) ,µ ( s(k2 ×k′2) ) }, max{µ ( s(k′1 ×k1) ) ,µ ( s(k′2 ×k2) ) } } = β(k1 ×k2) max { µ ( s(k1 ×k′1) ∪s(k2 ×k ′ 2) ) ,µ ( s(k′1 ×k1) ∪s(k ′ 2 ×k2) )} ≤ β(k1 ×k2) [ µ ( s(k1 ×k′1) ∪s(k2 ×k ′ 2) ) + µ ( s(k′1 ×k1) ∪s(k ′ 2 ×k2) )] ≤ 1 2 ψ ( max{µ(k1 ∪k′1),µ(k2 ∪k ′ 2)} ) + 1 2 ψ ( max{µ(k′1 ∪k1),µ(k ′ 2 ∪k2)} ) by hypothesis(i) = ψ ( max{µ(k1 ∪k′1),µ(k2 ∪k ′ 2)} ) = ψ ( max { max{µ(k1),µ(k′1)}, max{µ(k2),µ(k ′ 2)} }) = ψ ( max { µ̃(k1 ×k′1), µ̃(k2 ×k ′ 2) }) = ψ ( µ̃ ( (k1 ×k′1) ∪ (k2 ×k ′ 2) )) . this implies that t is a β-ψ-condensing operator. now, theorem 3.8 ensures that t has a best proximity point, called (p,q) ∈ (a × a) ∪ (b × b). thus (p,q) is a coupled best proximity point of s. � remark 4.5. it is remarkable to note that by a classical theorem due to s. mazur, the convex hull of a compact set in a banach space is again relatively compact. but this fact may not be true in geodesic metric spaces in general (see [4] for more information about the geodesic spaces which satisfies the mazur’s theorem). therefore, we should consider the compactness assumption of closed convex hulls of finite sets in our main existence theorems. © agt, upv, 2022 appl. gen. topol. 23, no. 2 418 best proximity point (pair) results via mnc in busemann convex metric spaces remark 4.6. it is worth noticing that the class of reflexive and busemann convex spaces contains the class of reflexive and strictly convex banach spaces as a subclass. for example consider x = r2 with radial metric defined with d ( (x1,y1), (x2,y2) ) =  ρ ( (x1,y1), (x2,y2) ) ; if (0, 0), (x1,y1), (x2,y2) are colinear, ρ ( (x1,y1), (0, 0) ) + ρ ( (x2,y2), (0, 0) ) ; otherwise, where ρ denotes the usual euclidean metric on r2. then (x,d) is a complete rtree and so is a reflexive and busemann convex space (see [8] for more details). note that the radial metric does not induced with any norm. 5. application this section is dedicated to prove a result which shows the existence of optimum solutions of a system of second order differential equation with two initial conditions. let τ,γ ∈ r+, i = [0,τ] and (e,‖.‖) be a banach space. let b1 = b(α0,γ), b2 = b(β0,γ) where α0,β0 ∈ e. we consider the following system of second order differential equation with two initial conditions (5.1) x ′′ (s) = f(s,x(s)), x(0) = α0, x ′ (0) = α1, y ′′ (s) = g(s,y(s)), y(0) = β0, y ′ (0) = β1, where, f : i × b1 → r, g : i × b2 → r are continuous functions such that ‖f(s,x)‖≤ a1, ‖g(s,y)‖≤ a2, s ∈i and α1,β1 ∈ e. twice integrating (5.1) and usage of initial conditions yields us (5.2) x(s) = α0 + ∫ s 0 (α1 + (s−r)f(r,x(r))dr, y(s) = β0 + ∫ s 0 (β1 + (s−r)g(r,x(r))dr. it is clear that the systems (5.1) and (5.2) are equivalent to each other. let j ⊆ i, s = c(j ,e) be a banach space of continuous mappings from j into e endowed with supremum norm and consider s1 = c(j ,b1) = {x : j → b1 : x ∈s, x(0) = α0, x′(0) = α1}, s2 = c(j ,b2) = {y : j → b2 : y ∈s, y(0) = β0, y′(0) = β1}. so, (s1,s2) is nbcc pair in s. now, for every x ∈ s1 and every y ∈ s2, we have ‖x−y‖ = sup s∈j ‖x(s) −y(s)| ≥ ‖α0 −β0‖. so, dist(s1,s2) = ‖α0−β0‖. let us define operator t : s1∪s2 →s as follows: tx(s) =   β0 + ∫ s 0 ( β1 + (s−r)g(r,x(r)) ) dr, x ∈s1, α0 + ∫ s 0 ( α1 + (s−r)f(r,x(r)) ) dr, x ∈s2. © agt, upv, 2022 appl. gen. topol. 23, no. 2 419 m. gabeleh and p. r. patle it is clear that t is cyclic operator. it is known that w ∈ s1 ∪ s2 is an optimum solution of the system (5.2) if ‖w −tw‖ = dist(s1 ∪s2) is satisfied. equivalently, w is the best proximity point of the operator t . before proving the existence of an optimum solution of system (5.2) we recall an extension of the mean values theorem for integrals, which is presented according to our notations. theorem 5.1 ([12]). for i,j ,b1,b2,f and g as given in the discussion above with s ∈ j we have α0 + ∫ s 0 (α1 + (s−r)f(r,x(r)))dr ∈ α0 + s con ( {α1 + (s−r)f(r,x(r)) : r ∈ [0,s]} ) and β0 + ∫ s 0 (β1 + (s−r)g(r,x(r)))dr ∈ β0 + s con ( {β1 + (s−r)g(r,x(r)) : r ∈ [0,s]} ) . the following theorem shows the existence of optimum solutions for the system (5.1). theorem 5.2. let µ be an arbitrary mnc on s, τ(τa2 +‖β1‖) ≤ γ, τ(τa1 + ‖α1‖) ≤ γ and τ ≤ 1. the system (5.1) has an optimal solution if the following condition holds true: (1) for any bounded pair (k1,k2) ⊆ (b1,b2), there is a nondecreasing function ψ ∈ ψ such that µ(f(j ×k1) ∪g(j ×k2)) < 1 s ψ(µ(n1 ∪n2)). (2) for each x ∈s1 and for all y ∈s2, ‖g(r,x(r)) −f(r,y(r))‖≤ 1 s2 (‖x(r) −y(r)‖−‖β0 −α0‖ + ‖β1 −α1‖s). proof. as the system (5.1) and (5.2) are equivalent to each other, in order to show (5.1) has an optimal solution it is sufficient to show (5.2) has an optimal solution. from the above discussion it is clear that the operator t is cyclic. our first task is to show that t(s1) is a bounded and equicontinuous subset of s2. for each x ∈s1, ‖tx(t)‖ = ‖β0 + ∫ s 0 (β1 + (s−r)g(r,x(r))dr‖ ≤‖β0‖ + ∫ s 0 ‖β1 + (s−r)g(r,x(r)‖dr ≤‖β0‖ + τ(‖β1‖ + τa2) ≤‖β0‖ + γ. © agt, upv, 2022 appl. gen. topol. 23, no. 2 420 best proximity point (pair) results via mnc in busemann convex metric spaces thus t(s1) is bounded. now for s,s′ ∈ j and x ∈s1, ‖tx(s) −tx(s′)‖ = wwww ∫ s 0 (β1 + (s−r)g(r,x(r))dr − ∫ s′ 0 (β1 + (s−r)g(r,x(r))dr wwww ≤ ∣∣∣∣ ∫ s′ s wwβ1 + (s−r)g(r,x(r)wwdr∣∣∣∣ ≤(τa2 + ‖β1‖) ‖s−s′‖ ≤m|s−s′|, where m = τa2 + ‖β1‖, which means that t(s1) is equicontinuous. by a similar argument t(s2) is a bounded and equicontinuous subset of s1. thus an application of the arzela-ascoli theorem concludes that (s1,s2) is relatively compact. now our aim is to show that t is a relatively nonexpansive cyclic β-ψcondensing operator. for each (x,y) ∈ s1 ×s2 with the help of assumption (2), we have ‖tx(s) −ty(s)‖ = wwwwβ0 + ∫ s 0 (β1 + (s−r)g(r,x(r))dr − [α0 + ∫ s 0 (α1 + (s−r)f(r,x(r))dr] wwww ≤‖β0 −α0‖ + wwww ∫ s 0 [ (β1 −α1) + (s−r)(g(r,x(r)) −f(r,x(r)) ] dr wwww ≤‖β0 −α0‖ + ‖β1 −α1‖ s + wwwws ∫ s 0 ( g(r,x(r)) −f(r,x(r) ) dr wwww ≤‖β0 −α0‖ + ‖β1 −α1‖ s + (‖x(s) −y(s)‖−‖β0 −α0‖−‖β1 −α1‖ s) =‖x(s) −y(s)‖. this means that t is relatively nonexpansive. in order to show that t is cyclic α-ψ-condensing, suppose that the pair (k1,k2) ⊆ ( s1,s2 ) is nbcc, proximinal, t-invariant and dist(k1,k2) = dist(s1,s2)(= ‖α0 − β0‖). now © agt, upv, 2022 appl. gen. topol. 23, no. 2 421 m. gabeleh and p. r. patle using theorem 5.1 and assumption (1) we have µ(t(k1) ∪t(k2)) = max{µ(t(k1)),µ(t(k2))} = max { sup s∈j { µ ( {tx(s) : x ∈ k1} )} , sup s∈j { µ ( {ty(s) : y ∈ k2} )}} = max { sup s∈j { µ ({ β0 + ∫ s 0 (β1 + (s−r)g(r,x(r)))dr : x ∈ k1 })} , sup s∈j { µ ({ α0 + ∫ s 0 (α1 + (s−r)f(r,x(r)))dr : x ∈ k1 })}} = max { sup s∈j { µ ({ β0 + s con ( {β1 + (s−r)g(r,x(r)) : r ∈ [0,s]} )} , sup s∈j { µ ({ α0 + s con ( {α1 + (s−r)f(r,x(r)) : r ∈ [0,s]} )}} = max { µ ({ β0 + s ( β1 + con ( {g(r,x(r)) : r ∈ [0,s]} )}) , µ ({ α0 + s ( α1 + con ( {f(r,x(r)) : r ∈ [0,s]} )})} ≤max { sµ ( {g(j ×k1)} ) ,sµ ( {f(j ×k2)} )} =sµ ( {f(j ×k1) ∪g(j ×k2)} ) ≤ s ψ(µ(k1 ∪k2)) s . thus we get µ(t(k1 ∪t(k2))) ≤ ψ(µ(k1 ∪k2). furthermore, we define the function β : 2e → [0, +∞) as follows: β(k1∪k2) = 1. the latter implies that β(k1 ∪k2)µ(t(k1 ∪t(k2))) ≤ ψ(µ(k1 ∪k2)). thus necessary requirements of theorem 3.4 are satisfied. so the operator t has a best proximity point and hence the system (5.1) has an optimal solution. � remark 5.3. we know that, the banach space c(j,x) in theorem 5.2 is not reflexive and the reflexivity condition in theorem 3.4 is essential. in fact, the reflexivity condition in theorem 3.4 was used to prove that the proximal pair (a0,b0) is nonempty. but in theorem 5.2 the proximal pair (s1,s2) is nonempty, automatically because of the fact that (α0,β0) ∈s1 ×s2. as an application of theorem 5.2, we conclude the following existence result of a solution for a system of second order differential equations satisfying the same two initial conditions. corollary 5.4. under the notations defined above in this section and the assumptions of theorem 5.2 if α0 = β0 = x0 and α1 = β1 = x1 , then the system © agt, upv, 2022 appl. gen. topol. 23, no. 2 422 best proximity point (pair) results via mnc in busemann convex metric spaces x ′′ (s) = f(s,x(s)), x(0) = x0, x ′ (0) = x1, y ′′ (s) = g(s,y(s)), y(0) = x0, y ′ (0) = x1, has a solution. 6. conclusions this work elicited some best proximity point (pair) theorems for cyclic (noncyclic) (α−ψ) and (β−ψ) condensing operators in the framework of reflexive busemann convex spaces and by considering an appropriate measure of noncompactness. as an application of the main existence result of best proximity points, we survey the existence of an optimal solution for a system of differential equations. acknowledgements. the authors would like to thank the referees for their useful comments and suggestions. references [1] r. r. akhmerov, m. i. kamenskii, a. s. potapov, a. e. rodkina and b. n. sadovskii, measures of noncompactness and condensing operators, vol. 55, birkhäuser, basel, 1992. 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[23] b. samet, c. vetro and p. vetro, fixed point theorems for α-ψ-contractive type mappings, nonlinear anal. 19 (2012), 2154–2165. © agt, upv, 2022 appl. gen. topol. 23, no. 2 424 @ appl. gen. topol. 21, no. 1 (2020), 111-133 doi:10.4995/agt.2020.12101 c© agt, upv, 2020 fixed point sets in digital topology, 2 laurence boxer department of computer and information sciences, niagara university, niagara university, ny 14109, usa; and department of computer science and engineering, state university of new york at buffalo (boxer@niagara.edu) communicated by v. gregori abstract we continue the work of [10], studying properties of digital images determined by fixed point invariants. we introduce pointed versions of invariants that were introduced in [10]. we introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set. 2010 msc: 54h25. keywords: digital topology; digital image; fixed point; reducible image; retract; wedge; tree. 1. introduction as stated in [10]: digital images are often used as mathematical models of realworld objects. a digital model of the notion of a continuous function, borrowed from the study of topology, is often useful for the study of digital images. however, a digital image is typically a finite, discrete point set. thus, it is often necessary to study digital images using methods not directly derived from topology. in this paper, we examine some properties of digital images concerned with the fixed points of digitally continuous functions; among these properties are discrete measures that are not natural analogues of properties of subsets of rn. received 19 july 2019 – accepted 28 january 2020 http://dx.doi.org/10.4995/agt.2020.12101 l. boxer in [10], we studied rigidity, pull indices, fixed point spectra for digital images and for digitally continuous functions, and related notions. in the current work, we study pointed versions of notions introduced in [10]. we also study such questions as when a set of fixed points fix(f) determines that f is an identity function, or is “approximately” an identity function. some of the results in this paper were presented in [6]. 2. preliminaries much of this section is quoted or paraphrased from [10]. let n denote the set of natural numbers; n∗ = {0}∪n, the set of nonnegative integers; and z, the set of integers. #x will be used for the number of members of a set x. 2.1. adjacencies. a digital image is a pair (x,κ) where x ⊂ zn for some n and κ is an adjacency on x. thus, (x,κ) is a graph for which x is the vertex set and κ determines the edge set. usually, x is finite, although there are papers that consider infinite x. usually, adjacency reflects some type of “closeness” in zn of the adjacent points. when these “usual” conditions are satisfied, one may consider the digital image as a model of a black-and-white “real world” image in which the black points (foreground) are represented by the members of x and the white points (background) by members of zn\{x}. we write x ↔κ y, or x ↔ y when κ is understood or when it is unnecessary to mention κ, to indicate that x and y are κ-adjacent. notations x -κ y, or x y when κ is understood, indicate that x and y are κ-adjacent or are equal. the most commonly used adjacencies are the cu adjacencies, defined as follows. let x ⊂ zn and let u ∈ z, 1 ≤ u ≤ n. then for points x = (x1, . . . ,xn) 6= (y1, . . . ,yn) = y we have x ↔cu y if and only if • for at most u indices i we have |xi −yi| = 1, and • for all indices j, |xj −yj| 6= 1 implies xj = yj. the cu-adjacencies are often denoted by the number of adjacent points a point can have in the adjacency. e.g., • in z, c1-adjacency is 2-adjacency; • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency; • in z3, c1-adjacency is 8-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. the literature also contains several adjacencies to exploit properties of cartesian products of digital images. these include the following. definition 2.1 ([1]). let (x,κ) and (y,λ) be digital images. the normal product adjacency or strong adjacency on x×y , np(κ,λ), is defined as follows. given x0,x1 ∈ x, y0,y1 ∈ y such that p0 = (x0,y0) 6= (x1,y1) = p1, we have p0 ↔np(κ,λ) p1 if and only if one of the following is valid: c© agt, upv, 2020 appl. gen. topol. 21, no. 1 112 fixed point sets in digital topology, 2 • x0 ↔κ x1 and y0 = y1, or • x0 = x1 and y0 ↔λ y1, or • x0 ↔κ x1 and y0 ↔λ y1. building on the normal product adjacency, we have the following. definition 2.2 ([4]). given u,v ∈ n, 1 ≤ u ≤ v, and digital images (xi,κi), 1 ≤ i ≤ v, let x = πvi=1xi. the adjacency npu(κ1, . . . ,κv) for x is defined as follows. given xi,x ′ i ∈ xi, let p = (x1, . . . ,xv) 6= (x′1, . . . ,x ′ v) = q. then p ↔npu(κ1,...,κv) q if for at least 1 and at most u indices i we have xi ↔κi x′i and for all other indices j we have xj = x ′ j. notice np(κ,λ) = np2(κ,λ) [4]. let x ∈ (x,κ). we use the notations n(x) = nκ(x) = {y ∈ x |y ↔κ x} and n∗(x) = n∗κ(x) = nκ(x) ∪{x}. 2.2. digitally continuous functions. we denote by id or idx the identity map id(x) = x for all x ∈ x. definition 2.3 ([16, 3]). let (x,κ) and (y,λ) be digital images. a function f : x → y is (κ,λ)-continuous, or digitally continuous when κ and λ are understood, if for every κ-connected subset x′ of x, f(x′) is a λ-connected subset of y . if (x,κ) = (y,λ), we say a function is κ-continuous to abbreviate “(κ,κ)-continuous.” theorem 2.4 ([3]). a function f : x → y between digital images (x,κ) and (y,λ) is (κ,λ)-continuous if and only if for every x,y ∈ x, if x ↔κ y then f(x) -λ f(y). theorem 2.5 ([3]). let f : (x,κ) → (y,λ) and g : (y,λ) → (z,µ) be continuous functions between digital images. then g ◦ f : (x,κ) → (z,µ) is continuous. it is common to use the term path with the following distinct but related meanings. • a path from x to y in a digital image (x,κ) is a set {xi}mi=0 ⊂ x such that x0 = x, xm = y, and xi -κ xi+1 for i = 0, 1, . . . ,m− 1. if the xi are distinct, then m is the length of this path. • a path from x to y in a digital image (x,κ) is a (2,κ)-continuous function p : [0,m]z → x such that p(0) = x and p(m) = y. notice that in this usage, {p(0), . . . ,p(m)} is a path in the previous sense. definition 2.6 ([3]; see also [14]). let x and y be digital images. let f,g : x → y be (κ,κ′)-continuous functions. suppose there is a positive integer m and a function h : x × [0,m]z → y such that c© agt, upv, 2020 appl. gen. topol. 21, no. 1 113 l. boxer • for all x ∈ x, h(x, 0) = f(x) and h(x,m) = g(x); • for all x ∈ x, the induced function hx : [0,m]z → y defined by hx(t) = h(x,t) for all t ∈ [0,m]z is (2,κ′)−continuous. that is, hx is a path in y . • for all t ∈ [0,m]z, the induced function ht : x → y defined by ht(x) = h(x,t) for all x ∈ x is (κ,κ′)−continuous. then h is a digital (κ,κ′)−homotopy between f and g, and f and g are digitally (κ,κ′)−homotopic in y , denoted f ∼κ,κ′ g or f ∼ g when κ and κ′ are understood. if (x,κ) = (y,κ′), we say f and g are κ-homotopic to abbreviate “(κ,κ)-homotopic” and write f ∼κ g to abbreviate “f ∼κ,κ g”. if further h(x,t) = x for all t ∈ [0,m]z, we say h holds x fixed. if there exists x0 ∈ x such that f(x0) = g(x0) = y0 ∈ y and h(x0, t) = y0 for all t ∈ [0,m]z, then h is a pointed homotopy and f and g are pointed homotopic [3]. if there exist continuous f : (x,κ) → (y,λ) and g : (y,λ) → (x,κ) such that g ◦ f ∼κ,κ idx and f ◦ g ∼λ,λ idy , then (x,κ) and (y,λ) are homotopy equivalent. if there is a κ-homotopy between idx and a constant map, we say x is κ-contractible, or just contractible when κ is understood. theorem 2.7 ([4]). let (xi,κi) and (yi,λi) be digital images, 1 ≤ i ≤ v. let fi : xi → yi. then the product map f : ∏v i=1 xi → ∏v i=1 yi defined by f(x1, . . . ,xv) = (f1(x1), . . . ,fv(xv)) for xi ∈ xi, is (npv(κ1, . . . ,κv),npv(λ1, . . . ,λv))-continuous if and only if each fi is (κi,λi)-continuous. definition 2.8. let a ⊂ x. a κ-continuous function r : x → a is a retraction, and a is a retract of x, if r(a) = a for all a ∈ a. if such a map r satisfies i◦ r ∼κ idx where i : a → x is the inclusion map, then a is a κ-deformation retract of x. a function f : (x,κ) → (y,λ) is an isomorphism (called a homeomorphism in [2]) if f is a continuous bijection such that f−1 is continuous. we use the following notation. for a digital image (x,κ), c(x,κ) = {f : x → x |f is continuous}. given f ∈ c(x,κ), a point x ∈ x is a fixed point of f if f(x) = x. we denote by fix(f) the set {x ∈ x |x is a fixed point of f}. if x ∈ x \ fix(f), we say f moves x. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 114 fixed point sets in digital topology, 2 figure 1. (figure 1 of [9].) the image x discussed in example 3.1. the coordinates are ordered according to the axes in this figure. 3. rigidity and reducibility a function f : (x,κ) → (y,λ) is rigid [10] when no continuous map is homotopic to f except f itself. when the identity map id : x → x is rigid, we say x is rigid [12]. if f : x → y with f(x0) = y0, then f is pointed rigid [12] if no continuous map is pointed homotopic to f other than f itself. when the identity map id : (x,x0) → (x,x0) is pointed rigid, we say (x,x0) is pointed rigid. rigid maps and digital images are discussed in [12, 10]. clearly, a rigid map is pointed rigid, and a rigid digital image is pointed rigid. (note these assertions may seem counterintuitive as, e.g., pointed homotopic functions are homotopic, but the converse is not always true.) we show in the following that the converses of these assertions are not generally true. example 3.1 ([9]). let x = ([0, 2]2z × [0, 1]z)\{(1, 1, 1)}. let x0 = (0, 0, 1) ∈ x. see figure 1. it was shown in [9] that x is 6-contractible (i.e., c1contractible) but (x,x0) is not pointed 6-contractible. the proof of the latter uses an argument that is easily modified to show that any homotopy of idx that moves some point must move x0. it follows that idx is not rigid but is x0-pointed rigid, i.e., that x is not c1-rigid but (x,x0) is c1-pointed rigid. definition 3.2 ([12]). a finite image x is reducible if it is homotopy equivalent to an image of fewer points. otherwise, we say x is irreducible. lemma 3.3 ([12]). a finite image x is reducible if and only if idx is homotopic to a nonsurjective map. let (x,κ) be reducible. by lemma 3.3, there exist x ∈ x and f ∈ c(x,κ) such that idx 'κ f and x 6∈ f(x). we will call such a point a reduction point. in lemma 3.4 below, we have changed the notation of [12], since the latter paper uses the notation “n(x)” for what we call “n∗(x)” or “n∗κ(x)”. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 115 l. boxer lemma 3.4 ([12]). if there exist distinct x,y ∈ x so that n∗(x) ⊂ n∗(y), then x is reducible. in particular, x is a reduction point of x, and x \{x} is a deformation retract of x. remark 3.5 ([12]). a finite rigid image is irreducible. theorem 3.6. let (x,c2) be a digital image in z2. suppose there exists x0 ∈ x such that nc2 (x0) is c2-connected and #nc2 (x0) ∈ {1, 2, 3}. then (x,c2) is reducible. proof. we first show that in all cases, there exists y ∈ nc2 (x0) such that n∗c2 (x0) ⊂ n ∗ c2 (y). (1) suppose #nc2 (x0) = 1. then there exists y ∈ x such that {y} = nc2 (x0). clearly, then, n ∗ c2 (x0) ⊂ n∗c2 (y). (2) suppose #nc2 (x0) = 2. then there exist distinct y,y ′ ∈ x such that {y,y′} = nc2 (x0), which by hypothesis is connected. therefore, {x0,y′}⊂ nc2 (y), so n∗c2 (x0) ⊂ n ∗ c2 (y). (3) suppose #nc2 (x0) = 3. then there exist distinct y,y0,y1 ∈ x such that {y,y0,y1} = nc2 (x0), which by hypothesis is connected. therefore, one of the members of nc2 (x0), say, y, is adjacent to the other two. thus, {x0,y0,y1}⊂ nc2 (y), so n∗c2 (x0) ⊂ n ∗ c2 (y). in all cases we have n∗c2 (x0) ⊂ n ∗ c2 (y). the assertion follows from lemma 3.4. � remark 3.7. if instead we use the c1-adjacency, the analog of the previous theorem is simpler, since if (x,c1) is a digital image in z2 and x0 ∈ x such that nc1 (x0) is nonempty and c1-connected, then #nc1 (x0) = 1. this case is similar to the case #nc2 (x0) = 1 of theorem 3.6 above, so (x,c1) is reducible. 4. pointed homotopy fixed point spectrum in this section, we define pointed versions of the homotopy fixed point spectrum of f ∈ c(x,κ) and the fixed point spectrum of a digital image (x,κ). definition 4.1. let (x,κ) be a digital image. • [10] given f ∈ c(x,κ), the homotopy fixed point spectrum of f is s(f) = {# fix(g) |g ∼κ f}. • given f ∈ c(x,κ) and x0 ∈ fix(f), the pointed homotopy fixed point spectrum of f is s(f,x0) = {# fix(g) |g ∼κ f holding x0 fixed}. definition 4.2. let (x,κ) be a digital image. • [10] the fixed point spectrum of (x,κ) is f(x) = f(x,κ) = {# fix(f) |f ∈ c(x,κ)}. • given x0 ∈ x, the pointed fixed point spectrum of (x,κ,x0) is f(x,x0) = f(x,κ,x0) = {# fix(f) |f ∈ c(x,κ),x0 ∈ fix(f)}. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 116 fixed point sets in digital topology, 2 theorem 4.3 ([10]). let a be a retract of (x,κ). then f(a) ⊆ f(x). the argument used to prove theorem 4.3 is easily modified to yield the following. theorem 4.4. let (a,κ,x0) be a retract of (x,κ,x0). then f(a,κ,x0) ⊆ f(x,κ,x0). theorem 4.5 ([10]). let x = [1,a]z × [1,b]z. let κ ∈{c1,c2}. then s(idx,κ) = f(x,κ) = {i}abi=0. example 4.6. consider the pointed digital image (x,c1,x0) of example 3.1. since f ∈ c(x,c1) and x0 ∈ fix(f) imply f = idx, s(idx,c1,x0) = {#x} = {17}. however, (x,c1) is not rigid. it is easily seen that there is a c1-deformation retraction of x to {(x,y, 0) ∈ x}, which is isomorphic to [1, 3]2z. it follows from theorem 4.3 and theorem 4.5 that {i}9i=0 ⊂ s(idx). since every f ∈ c(x,c1) such that f 'c1 idx and f 6= idx moves every point q of x such that p3(q) = 1, it follows easily that s(idx,c1) = f(x,c1) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17}. 5. freezing sets in this section, we consider subsets of fix(f) that determine that f ∈ c(x,κ) must be the identity function idx. interesting questions include what properties such sets have, and how small they can be. in classical topology, given a connected set x ⊂ rn and a continuous selfmap f on x, knowledge of a finite subset a of the fixed points of f rarely tells us much about the behavior of f on x \a. by contrast, we see in this section that knowledge of a subset of the fixed points of a continuous self-map f on a digital image can completely characterize f as an identity map. 5.1. definition and basic properties. definition 5.1. let (x,κ) be a digital image. we say a ⊂ x is a freezing set for x if given g ∈ c(x,κ), a ⊂ fix(g) implies g = idx. theorem 5.2. let (x,κ) be a digital image. let a ⊂ x. the following are equivalent. (1) a is a freezing set for x. (2) idx is the unique extension of ida to a member of c(x,κ). (3) for every isomorphism f : x → (y,λ), if g : x → y is (κ,λ)continuous and f|a = g|a, then g = f . (4) any continuous g : a → y has at most one extension to an isomorphism ḡ : x → y . c© agt, upv, 2020 appl. gen. topol. 21, no. 1 117 l. boxer proof. 1) ⇔ 2): this follows from definition 5.1. 1) ⇒ 3): suppose a is a freezing set for x. let f : x → y be a (κ,λ)isomorphism. let g : x → y be (κ,λ)-continuous, such that g|a = f |a. then f−1 ◦g|a = f−1 ◦f|a = idx |a = ida . since the composition of digitally continuous functions is continuous, it follows by hypothesis that f−1 ◦g = idx, and therefore that g = f ◦ (f−1 ◦g) = f ◦ idx = f. 3) ⇒ 1): suppose for every isomorphism f : x → (y,λ), if g : x → y is (κ,λ)-continuous and f|a = g|a, then g = f. for g ∈ c(x,κ), a ⊂ fix(g) implies g|a = idx |a, so since idx is an isomorphism, g = idx. 3) ⇒ 4): this is elementary. 4) ⇒ 2): this follows by taking g to be the inclusion of a into x, which extends to idx. � freezing sets are topological invariants in the sense of the following. theorem 5.3. let a be a freezing set for the digital image (x,κ) and let f : (x,κ) → (y,λ) be an isomorphism. then f(a) is a freezing set for (y,λ). proof. let g ∈ c(y,λ) such that g|f(a) = idy |f(a). then g ◦f|a = g|f(a) ◦f|a = idy |f(a) ◦f|a = f|a. by theorem 5.2, g ◦f = f. thus g = (g ◦f) ◦f−1 = f ◦f−1 = idy . by definition 5.1, f(a) is a freezing set for (y,λ). � we will use the following. proposition 5.4 ([10]). let (x,κ) be a digital image and f ∈ c(x,κ). suppose x,x′ ∈ fix(f) are such that there is a unique shortest κ-path p in x from x to x′. then p ⊂ fix(f). let pi : zn → z be the projection to the ith coordinate: pi(z1, . . . ,zn) = zi. the following assertion can be interpreted to say that in a cu-adjacency, a continuous function that moves a point p also moves a point that is “behind” p. e.g., in z2, if q and q′ are c1or c2-adjacent with q left, right, above, or below q′, and a continuous function f moves q to the left, right, higher, or lower, respectively, then f also moves q′ to the left, right, higher, or lower, respectively. lemma 5.5. let (x,cu) ⊂ zn be a digital image, 1 ≤ u ≤ n. let q,q′ ∈ x be such that q ↔cu q′. let f ∈ c(x,cu). (1) if pi(f(q)) > pi(q) > pi(q ′) then pi(f(q ′)) > pi(q ′). (2) if pi(f(q)) < pi(q) < pi(q ′) then pi(f(q ′)) < pi(q ′). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 118 fixed point sets in digital topology, 2 figure 2. illustration for example 5.9 proof. (1) suppose pi(f(q)) > pi(q) > pi(q ′). since q ↔cu q′, if pi(q) = m then pi(q ′) = m− 1. then pi(f(q)) > m. by continuity of f, we must have f(q′) -cu f(q), so pi(f(q ′)) ≥ m > pi(q′). (2) this case is proven similarly. � theorem 5.6. let (x,κ) be a digital image. let x′ be a proper subset of x that is a retract of x. then x′ does not contain a freezing set for (x,κ). proof. let r : x → x′ be a retraction. then f = i ◦ r ∈ c(x,κ), where i : x′ → x is the inclusion map. then f|x′ = idx′ , but f 6= idx. the assertion follows. � corollary 5.7. let (x,κ) be a reducible digital image. let x be a reduction point for x. let a be a freezing set for x. then x ∈ a. proof. since x is a reduction point for x, by lemma 3.4, there is a retraction r : x → x \ {x}. it follows that x \ {x} does not contain a freezing set for (x,κ). � proposition 5.8. let (x,c2) be a connected digital image in z2. suppose x0 ∈ x is such that nc2 (x0) is connected and #nc2 (x0) ∈ {1, 2, 3}. if a is a freezing set for (x,c2), then x0 ∈ a. proof. by the proof of theorem 3.6, we can use lemma 3.4 to conclude that x0 is a reduction point. the assertion follows from corollary 5.7. � proposition 5.8 cannot in general be extended to permit #nc2 (x0) = 4, as shown in the following. example 5.9. let x = {xi}4i=0 ⊂ z 2, where x0 = (0, 0), x1 = (0,−1), x2 = (1, 0), x3 = (0, 1), x4 = (−1, 1). see figure 2. then nc2 (x0) is c2-connected and #nc2 (x0) = 4. it is easily seen that x \{x0} is a freezing set for (x,c2). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 119 l. boxer 5.2. boundaries and freezing sets. for any digital image (x,κ), clearly x is a freezing set. an interesting question is how small a ⊂ x can be for a to be a freezing set for x. we say a freezing set a is minimal if no proper subset of a is a freezing set for x. definition 5.10. let x ⊂ zn. • the boundary of x [15] is bd(x) = {x ∈ x | there exists y ∈ zn \x such that y ↔c1 x}. • the interior of x is int(x) = x \bd(x). proposition 5.11. let a < b, [a,b]z ⊂ [c,d]z, and let f : [a,b]z → [c,d]z be c1-continuous. • if {a,b}⊂ fix(f), then [a,b]z = fix(f). • bd([a,b]z) = {a,b} is a minimal freezing set for [a,b]z. proof. if [a,b]z 6= fix(f), then we have at least one of the following: • for some smallest t0 satisfying a < t0 < b, f(t0) > t0. but then f(t0 − 1) ≤ t0 − 1, so f(t0 − 1) 6-c1 f(t0), contrary to the continuity of f. • for some largest t1 satisfying a < t1 < b, f(t1) < t1. but then f(t1 + 1) ≥ t1 + 1, so f(t1 + 1) 6-c1 f(t1), contrary to the continuity of f. it follows that f|[a,b]z is an inclusion function, as asserted. by taking [c,d]z = [a,b]z and considering all f ∈ c([a,b]z,c1) such that {a,b}⊂ fix(f), we conclude that {a,b} is a freezing set for [a,b]z. to establish minimality, observe that the proper nonempty subsets b of {a,b} allow constant functions c that are c1-continuous non-identities with c|b = idb. � proposition 5.12. let x ⊂ zn be finite. let 1 ≤ u ≤ n. let a ⊂ x. let f ∈ c(x,cu). if bd(a) ⊂ fix(f), then a ⊂ fix(f). proof. by hypothesis, it suffices to show int(a) ⊂ fix(f). let x = (x1, . . . ,xn) ∈ int(a). suppose, in order to obtain a contradiction, x 6∈ fix(f). then for some index j, (5.1) pj(f(x)) 6= xj. since x is finite, there exists a path p = {yi = (x1, . . . ,xj−1,ai,xj+1 . . . ,xn)}mi=1 in x such that a1 < xj < am and ai+1 = ai + 1; y1,ym ∈ bd(a); and {yi}m−1i=2 ⊂ int(a). note x ∈ p . now, (5.1) implies either pj(f(x)) < xj or pj(f(x)) > xj. if the former, then by lemma 5.5, ym 6∈ fix(f); and if the latter, then by lemma 5.5, y1 6∈ fix(f); so in either case, we have a contradiction. we conclude that x ∈ fix(f). the assertion follows. � theorem 5.13. let x ⊂ zn be finite. then for 1 ≤ u ≤ n, bd(x) is a freezing set for (x,cu). proof. the assertion follows from proposition 5.12. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 120 fixed point sets in digital topology, 2 without the finiteness condition used in proposition 5.12 and in theorem 5.13, the assertions would be false, as shown in the following. example 5.14. let x = {(x,y) ∈ z2 |y ≥ 0}. consider the function f : x → x defined by f(x,y) = { (x, 0) if y = 0; (x + 1,y) if y > 0. then f ∈ c(x,c2), bd(x) = z×{0}, and f|bd(x) = idbd(x), but x 6⊂ fix(f), so bd(x) is not a c2-freezing set for x. 5.3. digital cubes and c1. in this section, we consider freezing sets for digital cubes using the c1 adjacency. theorem 5.15. let x = πni=1[0,mi]z. let a = π n i=1{0,mi}. • let y = πni=1[ai,bi]z be such that [0,mi] ⊂ [ai,bi]z for all i. let f : x → y be c1-continuous. if a ⊂ fix(f), then x ⊂ fix(f). • a is a freezing set for (x,c1); minimal for n ∈{1, 2}. proof. the first assertion has been established for n = 1 at proposition 5.11. we can regard this as a base case for an argument based on induction on n, and we now assume the assertion is established for n ≤ k where k ≥ 1. now suppose n = k + 1 and f : x → y is c1-continuous with a ⊂ fix(f). let x0 = π k i=1[0,mi]z ×{0}, x1 = π k i=1[0,mi]z ×{mk+1}. we have that f|x0 and f|x1 are c1-continuous, a ∩ x0 ⊂ fix(f|x0 ), and a∩x1 ⊂ fix(f|x1 ). since x0 and x1 are isomorphic to k-dimensional digital cubes, by theorem 5.3 and the inductive hypothesis, we have( πki=1[0,mi]z ×{0} ) ∪ ( πki=1[0,mi]z ×{mn} ) ⊂ fix(f). then given x = (x1, . . . ,xn) ∈ x, x is a member of the unique shortest c1-path {(x1,x2, . . . ,xk, t)}m1t=0 from (x1,x2, . . . ,xk, 0) ∈ a to (x1,x2, . . . ,xk,mn) ∈ a. by proposition 5.4, x ∈ fix(f). since x was taken arbitrarily, this completes the induction proof that x ⊂ fix(f). by taking y = x and applying the above to all f ∈ c(x,c1) such that a ⊂ fix(f), we conclude that a is a freezing set for (x,c1). minimality of a for n = 1 was established at proposition 5.11. to show minimality of a for n = 2, consider a proper subset a′ of a. without loss of generality, (0, 0) ∈ a\a′, m1 > 0, and m2 > 0. for x ∈ x, let g : x → x be the function g(x) = { x if x 6= (0, 0); (1, 1) if x = (0, 0). suppose y ∈ x is such that y ↔c1 (0, 0). then y = (1, 0) or y = (0, 1), hence g(y) = y ↔c1 (1, 1) = g(0, 0). thus g ∈ c(x,c1), a′ ⊂ fix(g), and g 6= idx. therefore, a′ is not a freezing set for (x,c1), so a is minimal. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 121 l. boxer figure 3. the function g in the proof of example 5.16. members of a\{(0, 0, 0)} are circled. straight line segments indicate c1 adjacencies. curved arrows show the mapping for points in x \ fix(g). the minimality assertion of theorem 5.15 does not extend to n = 3, as shown in the following. example 5.16. let x = [0, 1]3z. let a = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}. see figure 3. then a is a minimal freezing set for (x,c1). proof. note if x ∈ x \ a then for each index i ∈ {1, 2, 3}, x is c1-adjacent to yi ∈ a such that x and yi differ in the ith coordinate. therefore, if f ∈ c(x,c1) such that f(x) 6= x, then c1-continuity requires that for some i we have f(yi) 6= yi. it follows that a is a freezing set for (x,c1). minimality is shown as follows. let a′ be a proper subset of a. without loss of generality, (0, 0, 0) ∈ a\a′. let g : x → x be the function (see figure 3) g(x) =   (1, 1, 0) if x = (0, 0, 0); (1, 1, 1) if x = (0, 0, 1); x otherwise. then g ∈ c(x,c1), g|a′ = ida′ , and g 6= idx. therefore, a′ is not a freezing set for (x,c1). � 5.4. digital cubes and cn. in this section, we consider freezing sets for digital cubes in zn, using the cn adjacency. theorem 5.17. let x = ∏n i=1[0,mi]z ⊂ z n, where mi > 1 for all i. then bd(x) is a minimal freezing set for (x,cn). proof. that bd(x) is a freezing set for (x,cn) follows from theorem 5.13. to show bd(x) is a minimal freezing set, it suffices to show that if a is a proper subset of bd(x) then a is not a freezing set for (x,cn). we must show that there exists (5.2) f ∈ c(x,cn) such that f|a = ida but f 6= idx . c© agt, upv, 2020 appl. gen. topol. 21, no. 1 122 fixed point sets in digital topology, 2 by hypothesis, there exists y = (y1, . . . ,yn) ∈ bd(x) \a. since y ∈ bd(x), for some index j, yj ∈{0,mj}. • if yj = 0 the function f : x → x defined by f(y) = (y1, . . . ,yj−1, 1,yj+1, . . . ,yn), f(x) = x for x 6= y, satisfies (5.2). • if yj = mj the function f : x → x defined by f(y) = (y1, . . . ,yj−1,mj − 1,yj+1, . . . ,yn), f(x) = x for x 6= y, satisfies (5.2). the assertion follows. � 5.5. freezing sets and the normal product adjacency. in the following, pj : ∏v i=1 xi → xj is the map pj(x1, . . . ,xv) = xj where xi ∈ xi. theorem 5.18. let (xi,κi) be a digital image, i ∈ [1,v]z. let x = ∏v i=1 xi. let a ⊂ x. suppose a is a freezing set for (x,npv(κ1, . . . ,κv)). then for each i ∈ [1,v]z, pi(a) is a freezing set for (xi,κi). proof. let fi ∈ c(xi,κi). let f : x → x be defined by f(x1, . . . ,xv) = (f1(x1), . . . ,fv(xv)). then by theorem 2.7, f ∈ c(x,npv(κ1, . . . ,κv)). suppose for all a = (a1, . . . ,av) ∈ a, f(a) = a, hence fi(ai) = ai for all ai ∈ pi(a). since a is a freezing set of x, we have that f = idx, and therefore, fi = idxi . the assertion follows. � 5.6. cycles. a cycle or digital simple closed curve of n distinct points is a digital image (cn,κ) with cn = {xi}n−1i=0 such that xi ↔κ xj if and only if j = i + 1 mod n or j = i− 1 mod n. given indices i < j, there are two distinct paths determined by xi and xj in cn, consisting of the sets pi,j = {xk} j k=i and p ′ i,j = cn \{xk} j−1 k=i+1. if one of these has length less than n/2, it is the shorter path from pi to pj and the other is the longer path; otherwise, both have length n/2, and each is a shorter path and a longer path from pi to pj. in this section, we consider minimal fixed point sets for f ∈ c(cn) that force f to be an identity map. theorem 5.19. let n > 4. let xi,xj,xk be distinct members of cn be such that cn is a union of unique shorter paths determined by these points. let f ∈ c(cn,κ). then f = idcn if and only if {xi,xj,xk}⊂ fix(f); i.e., {xi,xj,xk} is a freezing set for cn. further, this freezing set is minimal. proof. clearly f = idcn implies {xi,xj,xk}⊂ fix(f). suppose {xi,xj,xk} ⊂ fix(f). by hypothesis, there are unique shorter paths p0 from xi to xj, p1 from xj to xk, and p2 from xk to xi, in cn. by c© agt, upv, 2020 appl. gen. topol. 21, no. 1 123 l. boxer proposition 5.4, each of p0, p1, and p2 is contained in fix(f). by hypothesis cn = p0 ∪p1 ∪p2, so f = idcn . hence {xi,xj,xk} is a freezing set. for any distinct pair xi,xj ∈ cn, there is a non-identity continuous self-map on cn that takes a longer path determined by xi and xj to a shorter path determined by xi and xj. thus, {xi,xj} is not a freezing set for cn, so the set {xi,xj,xk} discussed above is minimal. � remark 5.20. in theorem 5.19, we need the assumption that n > 4, as there is a continuous self-map f on c4 with 3 fixed points such that f 6= idc4 [10]. 5.7. wedges. let (x,κ) ⊂ zn be such that x = x0 ∪x1, where x0 ∩x1 = {x0}; and if x ∈ x0, y ∈ x1, and x ↔κ y, then x0 ∈ {x,y}. we say x is the wedge of x0 and x1, denoted x = x0 ∨x1. we say x0 is the wedge point. theorem 5.21. let a be a freezing set for (x,κ), where x = x0 ∨x1 ⊂ zn, #x0 > 1, and #x1 > 1. let x0 ∩x1 = {x0}. then a must include points of x0 \{x0} and x1 \{x0}. proof. otherwise, either a ⊂ x0 or a ⊂ x1. suppose a ⊂ x0. then the function f : x → x given by f(x) = { x if x ∈ x0; x0 if x ∈ x1, belongs to c(x,κ) and satisfies f|a = ida, but f 6= idx. thus a is not a freezing set for (x,κ). the case a ⊂ x1 is argued similarly. � example 5.22. the wedge of two digital intervals is (isomorphic to) a digital interval. it follows from theorem 5.3 and proposition 5.11 that a freezing set for a wedge need not include the wedge point. theorem 5.23. let cm and cn be cycles, with m > 4, n > 4. let x0 be the wedge point of x = cm∨cn. let xi,xj ∈ cm and x′k,x ′ p ∈ cn be such that cm is the union of unique shorter paths determined by xi,xj,x0 and cn is the union of unique shorter paths determined by x′k,x ′ p,x0. then a = {xi,xj,x′k,x ′ p} is a freezing set for x. proof. let f ∈ c(x,κ) be such that a ⊂ fix(f). let p0 be the unique shorter path in cm from xi to xj; let p1 be the unique shorter path in cm from xj to x0; let p2 be the unique shorter path in cm from x0 to xi; let p ′ 0 be the unique shorter path in cn from x ′ k to x ′ p; let p ′ 1 be the unique shorter path in cn from x′p to x0; let p ′ 2 be the unique shorter path in cn from x0 to x ′ k. by proposition 5.4, each of the following paths is contained in fix(f): p0, p1 ∪p ′1 (from xj to x0 to x′p), p2 ∪p ′2 (from xi to x0 to x′k), and p ′ 0. since x = p0 ∪ (p1 ∪p ′1) ∪ (p2 ∪p ′ 2) ∪p ′ 0 ⊂ fix(f), the assertion follows. � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 124 fixed point sets in digital topology, 2 5.8. trees. a tree is an acyclic graph (x,κ) that is connected, i.e., lacking any subgraph isomorphic to cn for n > 2. the degree of a vertex x in x is the number of distinct vertices y ∈ x such that x ↔ y. a vertex of a tree may be designated as the root. we have the following. lemma 5.24 ([10]). let (x,κ) be a digital image that is a tree in which the root vertex has at least 2 child vertices. then f ∈ c(x,κ) implies fix(f) is κ-connected. theorem 5.25. let (x,κ) be a digital image such that the graph g = (x,κ) is a finite tree with #x > 1. let e be the set of vertices of g that have degree 1. then e is a minimal freezing set for g. proof. first consider the case that each vertex has degree 1. since x is a tree, it follows that x = {x0,x1} = e, and e is a freezing set. e must be minimal, since x admits constant functions that are identities on their restrictions to proper subsets of e. otherwise, there exists x0 ∈ x such that x0 has degree of at least 2 in g. this implies #x > 2, and since g is finite and acyclic, #e > 0. since g is acyclic, removal of any member of x \e would disconnect x. if we take x0 to be the root vertex, it follows from lemma 5.24 that e is a freezing set. since #e > 0, for any y ∈ e there exists y′ ∈ x\e such that y′ ↔ y. then the function f : x → x defined by f(x) = { y′ if x = y; x if x 6= y, satisfies f ∈ c(x,κ), f|e\{y} = ide\{y}, and f 6= idx. thus e \{y} is not a freezing set. since y was arbitrarily chosen, e is minimal. � 6. s-cold sets in this section, we generalize our focus from fixed points to approximate fixed points and, more generally, to points constrained in the amount they can be moved by continuous self-maps in the presence of fixed point sets. we obtain some analogues of our previous results for freezing sets. 6.1. definition and basic properties. in the following, we use the pathlength metric d for connected digital images (x,κ), defined [13] as d(x,y) = min{` |` is the length of a κ-path in x from x to y}. if x is finite and κ-connected, the diameter of (x,κ) is diam(x,κ) = max{d(x,y) |x,y ∈ x}. we introduce the following generalization of a freezing set. definition 6.1. given s ∈ n∗, we say a ⊂ x is an s-cold set for the connected digital image (x,κ) if given g ∈ c(x,κ) such that g|a = ida, then for all x ∈ x, d(x,g(x)) ≤ s. a cold set is a 1-cold set. note a 0-cold set is a freezing set. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 125 l. boxer theorem 6.2. let (x,κ) be a connected digital image. let a ⊂ x. then a ⊂ fix(g) is an s-cold set for (x,κ) if and only if for every isomorphism f : (x,κ) → (y,λ), if g : x → y is (κ,λ)-continuous and f |a = g|a, then for all x ∈ x, d(f(x),g(x)) ≤ s. proof. suppose a is an s-cold set for (x,κ). then for all f ∈ c(x,κ) such that f|a = ida and all x ∈ x, we have d(x,f(x)) ≤ s. let f : (x,κ) → (y,λ) be an isomorphism. let g : x → y be (κ,λ)-continuous with f |a = g|a. then ida = f −1 ◦f|a = f−1 ◦g|a. let x ∈ x. then d(x,f−1 ◦g(x)) ≤ s, i.e., there is a κ-path p in x of length at most s from x to f−1 ◦g(x). therefore, f(p) is a λ-path in y of length at most s from f(x) to f ◦f−1 ◦g(x) = g(x), i.e., d(f(x),g(x)) ≤ s. suppose a ⊂ x and for every isomorphism f : (x,κ) → (y,λ), if g : x → y is (κ,λ)-continuous and f|a = g|a, then for all x ∈ x, d(f(x),g(x)) ≤ s. let f ∈ c(x,κ) with f|a = ida. since idx is an isomorphism, for all x ∈ x, d(x,f(x)) ≤ s. thus, a is an s-cold set for (x,κ). � given a digital image (x,κ) and f ∈ c(x,κ), a point x ∈ x is an almost fixed point of f [16] or an approximate fixed point of f [7] if f(x) -κ x. remark 6.3. the following are easily observed. • if a ⊂ a′ ⊂ x and a is an s-cold set for (x,κ), then a′ is an s-cold set for (x,κ). • a is a cold set (i.e., a 1-cold set) for (x,κ) if and only if given f ∈ c(x,κ) such that f|a = ida, every x ∈ x is an approximate fixed point of f. • in a finite connected digital image (x,κ), every nonempty subset of x is a diam(x)-cold set. • if s0 < s1 and a is an s0-cold set for (x,κ), then a is an s1-cold set for (x,κ). note a freezing set is a cold set, but the converse is not generally true, as shown in the following. example 6.4. it follows from definition 6.1 that {0} is a cold set, but not a freezing set, for x = [0, 1]z, since the constant function g with value 0 satisfies g|{0} = id{0}, and g(1) = 0 ↔c1 1. s-cold sets are invariant in the sense of the following. theorem 6.5. let (x,κ) be a connected digital image, let a be an s-cold set for (x,κ), and let f : (x,κ) → (y,λ) be an isomorphism. then f(a) is an s-cold set for (y,λ). proof. let f ∈ c(y,λ) such that f|f(a) = idf(a). then f ◦f|a = f|f(a) ◦f|a = idf(a) ◦f|a = f|a. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 126 fixed point sets in digital topology, 2 by theorem 6.2, for all x ∈ x, d(f ◦f(x),f(x)) ≤ s. substituting y = f(x), we have that y ∈ y implies d(f(y),y) ≤ s. by definition 6.1, f(a) is a cold set for (y,λ). � a is a κ-dominating set (or a dominating set when κ is understood) for (x,κ) if for every x ∈ x there exists a ∈ a such that x -κ a [11]. this notion is somewhat analogous to that of a dense set in a topological space, and the following is somewhat analogous to the fact that in topological spaces, a continuous function is uniquely determined by its values on a dense subset of the domain. theorem 6.6. let (x,κ) be a digital image and let a be κ-dominating in x. then a is 2-cold in (x,κ). proof. let f ∈ c(x,κ) such that f|a = ida. since a is κ-dominating, for every x ∈ x there is an a ∈ a such that x a. then f(x) f(a) = a. thus, we have the path {x,a,f(x)} ⊂ x from x to f(x) of length at most 2. the assertion follows. � theorem 6.7. let (x,κ) be rigid. if a is a cold set for x, then a is a freezing set for x. proof. let f ∈ c(x,κ) be such that f|a = ida. since a is cold, f(x) x for all x ∈ x. therefore, the map h : x × [0, 1]z → x defined by h(x, 0) = x, h(x, 1) = f(x), is a homotopy. since x is rigid, f = idx. the assertion follows. � 6.2. cold sets for cubes. in this section, we consider cold sets for digital cubes in zn. note the hypotheses of proposition 6.8 imply a is c1and c2dominating in bd(x). proposition 6.8. let m,n ∈ n. let x = [0,m]z × [0,n]z. let a ⊂ bd(x) be such that no pair of c1-adjacent members of bd(x) belong to bd(x) \ a. then a is a cold set for (x,c2). further, for all f ∈ c(x,c2), if f|a = ida then f|int(x) = id |int(x). proof. let x = (x0,y0) ∈ x. let f ∈ c(x,c2) such that f|a = ida. consider the following. • if x ∈ a then f(x) = x. • if x ∈ bd(x) \a then both of the c1-neighbors of x in bd(x) belong to a. we will show f(x) -c2 x. let k = {(0, 0), (0,n), (m, 0), (m,n)}⊂ bd(x). – for x ∈ k, consider the case x = (0, 0). then {(0, 1), (1, 0)}⊂ a, so we must have f(x) ∈ n∗c2 ((0, 1)) ∩n ∗ c2 ((1, 0)) ⊂ n∗c2 (x). for other x ∈ k, we similarly find f(x) -c2 x. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 127 l. boxer figure 4. illustration of the proof of proposition 6.8 for the case (x0,y0) ∈ int(x). x = [0, 6]z × [0, 4]z. members of the set a ⊂ bd(x) are marked “a”. corner points such as (0, 4) need not belong to a; also, although we cannot have c1-adjacent members of bd(x) in bd(x)\a, we can have c2adjacent members of bd(x) in bd(x)\a, e.g., (5, 4) and (6, 3). the heavy polygonal line illustrates a c2-path p of length n = 4: p(0) = ql = (2, 0), p(1) = (1, 1), p(2) = (x0,y0) = (1, 2), p(3) = (1, 3), p(4) = qu = (1, 4). – for x ∈ bd(x) \ k, consider the case x = (t, 0). for this case, {(t− 1, 0), (t + 1, 0)}⊂ a, so (t− 1, 0) = f(t− 1, 0) -c2 f(x) -c2 f(t + 1, 0) = (t + 1, 0). therefore, f(x) ∈{x, (t, 1)}, so f(x) -c2 x. for other x ∈ bd(x) \k, we similarly find f(x) -c2 x. • if x ∈ int(x), let l = {(z, 0)}x0+1z=x0−1 and u = {(z,n)} x0+1 z=x0−1. we have l∩a 6= ∅ 6= u ∩a. since no pair of c1-adjacent members of bd(x) belong to bd(x)\a, there exist ql ∈ l∩a, qu ∈ u ∩a such that |p1(ql) −x0| ≤ 1 and |p1(qu ) −x0| ≤ 1. thus, there is an injective c2-path p : [0,n]z → x such that p([0,y0)]z) runs from ql to x and p([y0,n]z) runs from x to qu (note since we use c2-adjacency, there can be steps of the path that change both coordinates see figure 4). therefore, f ◦ p is a path from f(ql) = ql to f(x) to f(qu ) = qu , and p2◦f◦p is a path from p2(ql) = 0 to p2(f(x)) to p2(qu ) = n. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 128 fixed point sets in digital topology, 2 if y′ = p2(f(x)) > y0, then p2 ◦f ◦p |[0,y0]z is a c2-path of length y0 from 0 to y′, which is impossible. similarly, if y′ < y0, then p2 ◦ f ◦ p |[y0,n]z is a c2-path of length n−y0 from y ′ to n, which is impossible. therefore, we must have (6.1) p2 ◦f(x) = y0. similarly, by replacing the neighborhoods of the projections of x on the lower and upper edges of the cube, l and u, by the neighborhoods of the projections of x on the the left and right edges of the cube, l′ = {(0,z)}y0+1z=y0−1 and r = {(m,z)} y0+1 z=y0−1, and using an argument similar to that used to obtain (6.1), we conclude that (6.2) p1 ◦f(x) = x0. it follows from (6.2) and (6.1) that f(x) = x. thus, in all cases, f(x) -c2 x, and f|int(x) = idint(x). � proposition 6.9. let m,n ∈ n. let x = [0,m]z × [0,n]z. let a ⊂ bd(x) be c1-dominating in bd(x). then a is a 2-cold set for (x,c2). further, for all f ∈ c(x,c2), if f|a = ida then f|int(x) = id |int(x). proof. our argument is similar to that of proposition 6.8. let x = (x0,y0) ∈ x. let f ∈ c(x,c2) such that f|a = ida. consider the following. • if x ∈ a then f(x) = x. • if x ∈ bd(x) \a then for some a ∈ a, x -c1 a. therefore, f(x) -c1 f(a) = a. thus, {x,a,f(x)} is a path in x from x to f(x) of length at most 2. • if x ∈ int(x), then as in the proof of proposition 6.8 we have that f(x) = x. thus, in all cases, d(f(x),x) ≤ 2, and f|int(x) = idint(x). � an example of a 2-cold set a that is not a 1-cold set, such that a is as in proposition 6.9, is given in the following. example 6.10. let x = [0, 2]2z. let a = {(0, 2), (1, 0), (2, 2)}⊂ x. then a is c1-dominating in bd(x), so by proposition 6.9, is a 2-cold set for (x,c2). let f : x → x be the function f(0, 0) = (2, 0), f(0, 1) = (1, 1), and f(x) = x for all x ∈ x \ {(0, 0), (0, 1)}. then f ∈ c(x,c2) but d((0, 0),f(0, 0)) = 2, so a is not a 1-cold set. proposition 6.11. let x = ∏n i=1[0,mi]z ⊂ z n, where mi > 1 for all i. let a ⊂ bd(x) be such that a is not cn-dominating in bd(x). then a is not a cold set for (x,cn). proof. by hypothesis, there exists y = (y1, . . . ,yn) ∈ bd(x) \ a such that n(y,cn) ∩a = ∅. since y ∈ bd(x), for some index j we have yj ∈ {0,mj}. let x = (x1, . . . ,xn) ∈ x, for xi ∈ [0,mi]z. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 129 l. boxer • if yj = 0, let f : x → x be defined as follows. f(x) =   (x1, . . . ,xj−1, 2,xj+1, . . . ,xn) if x = y; (x1, . . . ,xj−1, 1,xj+1, . . . ,xn) if x ∈ ncn (y); x otherwise. if u,v ∈ x, u -cn v, then u and v differ by at most 1 in every coordinate. consider the following cases. – if u = y, then v ∈ ncn (y), and clearly f(u) and f(v) differ by at most 1 in every coordinate, hence are cn-adjacent. similarly if v = y. – if u,v ∈ ncn (y), then clearly f(u) and f(v) differ by at most 1 in every coordinate, hence are cn-adjacent. – if u ∈ ncn (y) and v 6∈ ncn (y), then pj(u) ∈ {0, 1}, so pj(f(v)) = pj(v) ∈{0, 1, 2}, and pj(f(u)) = 1. it follows easily that f(u) and f(v) differ by at most 1 in every coordinate, hence are cn-adjacent. similarly if v ∈ ncn (y) and u 6∈ ncn (y) – otherwise, {u,v}∩n∗cn (y) = ∅, so f(u) = u -cn v = f(v). therefore, f ∈ c(x,cn). • if yj = mj, let f : x → x be defined by f(x) =   (x1, . . . ,xj−1,mj − 2,xj+1, . . . ,xn) if x = y; (x1, . . . ,xj−1,mj − 1,xj+1, . . . ,xn) if x ∈ ncn (y); x otherwise. by an argument similar to that of the case yj = 0, we conclude that f ∈ c(x,cn). further, in both cases, f|a = ida, and f(y) 6-cn y. the assertion follows. � 6.3. s-cold sets for rectangles in z2. the following generalizes the case n = 2 of theorem 5.15. proposition 6.12. let x = [−m,m]z × [−n,n]z ⊂ z2, s ∈ n∗, where s ≤ min{m,n}. let a = {(−m + s,−n + s), (−m + s,n−s), (m−s,−n + s), (m−s,n−s)}. then a is a 4s-cold set for (x,c1). proof. let f ∈ c(x,c1) such that f|a = ida. let a′ = [−m + s,m−s]z × [−n + s,n−s]z. by proposition 5.4, bd(a′) ⊂ fix(f). it follows from proposition 5.12 that a′ ⊂ fix(f). thus it remains to show that x ∈ x \ a′ implies d(x,f(x)) ≤ 4s. this is seen as follows. for x ∈ x \a′, there exists a c1-path p of length at most 2s from x to some y ∈ bd(a′). then f(p) is a c1-path from f(x) to f(y) = y of length at most 2s. therefore, p ∪f(p) contains a path from x to y to f(x) of length at most 4s. the assertion follows. � the following generalizes proposition 6.8. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 130 fixed point sets in digital topology, 2 proposition 6.13. let x = [−m,m]z×[−n,n]z ⊂ z2, s ∈ n∗, where m−s ≥ 0, n−s ≥ 0. let a = [−m + s,m−s]z × [−n + s,n−s]z ⊂ x. let a′ ⊂ bd(a) such that no pair of c1-adjacent members of bd(a) belongs to bd(a) \a′. then a′ is a 2s-cold set for (x,c2). further, if f ∈ c(x,c2) and f|a′ = ida′ , then f|a = ida. proof. let f ∈ c(x,c2) be such that f|a′ = ida′ . as in the proof of proposition 6.8, f|a = ida. now consider x ∈ x \a. there is a c2-path p in x from x to some y ∈ a′ of length at most s. then f(p) is a c2-path in x from f(x) to f(y) = y of length at most s. therefore, p ∪f(p) contains a c2-path in x from x to y to f(x) of length at most 2s. the assertion follows. � 6.4. s-cold sets for cartesian products. we modify the proof of theorem 5.18 to obtain the following. theorem 6.14. let (xi,κi) be a digital image, i ∈ [1,v]z. let x = ∏v i=1 xi. let s ∈ n∗. let a ⊂ x. suppose a is an s-cold set for (x,npv(κ1, . . . ,κv)). then for each i ∈ [1,v]z, pi(a) is an s-cold set for (xi,κi). proof. let fi ∈ c(xi,κi). let f : x → x be defined by f(x1, . . . ,xv) = (f1(x1), . . . ,fv(xv)). then by theorem 2.7, f ∈ c(x,npv(κ1, . . . ,κv)). suppose for all i, ai ∈ pi(a), we have fi(ai) = ai. note this implies, for a = (a1, . . . ,av), that f(a) = a. since a is an arbitrary member of the s-cold set a of x, we have that d(f(x),x) ≤ s, for all x = (x1, . . . ,xv) ∈ x, xi ∈ xi, and therefore, d(fi(xi),xi) ≤ s. the assertion follows. � 6.5. s-cold sets for infinite digital images. in this section, we obtain properties of s-cold sets for some infinite digital images. theorem 6.15. let (zn,cu) be a digital image, 1 ≤ u ≤ n. let a ⊂ zn. let s ∈ n∗. if a is an s-cold set for (zn,cu), then for every index i, pi(a) is an infinite set, with sequences of members tending both to ∞ and to −∞. proof. suppose otherwise. then for some i, there exist m or m in z such that m = min{pi(a) |a ∈ a} or m = max{pi(a) |a ∈ a}. if the former, then for z = (z1, . . . ,zn) ∈ zn, define f : zn → zn by f(z) = { (z1, . . . ,zi−1,m,zi+1, . . . ,zn) if zi ≤ m; z otherwise. then f ∈ c(zn,cu) and f|a = ida, but f 6= idzn . thus, a is not an s-cold set. similarly, if m < ∞ as above exists, we conclude a is not an s-cold set. � corollary 6.16. a ⊂ z is a freezing set for (z,c1) if and only if a contains sequences {ai}∞i=1 and {a ′ i} ∞ i=1 such that limi→∞ai = ∞ and limi→∞a ′ i = −∞. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 131 l. boxer proof. this follows from lemma 5.5 and theorem 6.15. � the converse of theorem 6.15 is not generally correct, as shown by the following. example 6.17. let a = {(z,z) |z ∈ z} ⊂ z2. then although p1(a) = p2(a) = z contains sequences tending to ∞ and to −∞, a is not an s-cold set for (z2,c2), for any s. proof. consider f : z2 → z2 defined by f(x,y) = (x,x). we have f ∈ (z2,c2) and f|a = ida, but one sees easily that for all s there exist (x,y) ∈ z2 such that d((x,y),f(x,y)) > s. � 7. further remarks we have continued the work of [10] in studying fixed point invariants and related ideas in digital topology. we have introduced pointed versions of rigidity and fixed point spectra. we have introduced the notions of freezing sets and s-cold sets. these show us that although knowledge of the fixed point set fix(f) of a continuous selfmap f on a connected topological space x generally gives us little information about the nature of f|x\fix(f), if f ∈ c(x,κ) then f|x\fix(f) may be severely limited if a ⊂ fix(f) is a freezing set or, more generally, an s-cold set for (x,κ). acknowledgements. p. christopher staecker and an anonymous reviewer were most helpful. they each suggested several of our assertions, and several corrections. references [1] c. berge, graphs and hypergraphs, 2nd edition, north-holland, amsterdam, 1976. [2] l. boxer, digitally continuous functions, pattern recognition letters 15 (1994), 833– 839. [3] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. [4] l. boxer, generalized normal product adjacency in digital topology, applied general topology 18, no. 2 (2017), 401–427. [5] l. boxer, alternate product adjacencies in digital topology, applied general topology 19, no. 1 (2018), 21–53. [6] l. boxer, fixed points and freezing sets in digital topology, proceedings, interdisciplinary colloquium in topology and its applications in vigo, spain; 55–61. [7] l. boxer, o. ege, i. karaca, j. lopez and j. louwsma, digital fixed points, approximate fixed points, and universal functions, applied general topology 17, no. 2 (2016), 159– 172. [8] l. boxer and i. karaca, fundamental groups for digital products, advances and applications in mathematical sciences 11, no. 4 (2012), 161–180. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 132 fixed point sets in digital topology, 2 [9] l. boxer and p. c. staecker, fundamental groups and euler characteristics of sphere-like digital images, applied general topology 17, no. 2 (2016), 139–158. [10] l. boxer and p. c. staecker, fixed point sets in digital topology, 1, applied general topology, to appear. [11] g. chartrand and l. lesniak, graphs & digraphs, 2nd ed., wadsworth, inc., belmont, ca, 1986. [12] j. haarmann, m. p. murphy, c. s. peters and p. c. staecker, homotopy equivalence in finite digital images, journal of mathematical imaging and vision 53 (2015), 288–302. [13] s.-e. han, non-product property of the digital fundamental group, information sciences 171 (2005), 73–91. [14] e. khalimsky, motion, deformation, and homotopy in finite spaces, in proceedings ieee intl. conf. on systems, man, and cybernetics, 1987, 227–234. [15] a. rosenfeld, digital topology, the american mathematical monthly 86, no. 8 (1979), 621–630. [16] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 133 @ appl. gen. topol. 22, no. 2 (2021), 367-383doi:10.4995/agt.2021.15061 © agt, upv, 2021 lipschitz integral operators represented by vector measures elhadj dahia a and khaled hamidi b a laboratoire de mathématiques et physique appliquées, école normale supérieure de bousaada, 28001 bousaada, algeria (hajdahia@gmail.com) b department of mathematics, university of mohamed el-bachir el-ibrahimi, bordj bou arréridj, 34030 el-anasser, algeria, and laboratoire d’analyse fonctionnelle et géométrie des espaces, university of m’sila, 28000 m’sila, algeria. (khaled.hamidi@univ-bba.dz) communicated by s. romaguera abstract in this paper we introduce the concept of lipschitz pietsch-p-integral mappings, (1 ≤ p ≤ ∞), between a metric space and a banach space. we represent these mappings by an integral with respect to a vector measure defined on a suitable compact hausdorff space, obtaining in this way a rich factorization theory through the classical banach spaces c(k), lp(µ, k) and l∞(µ, k). also we show that this type of operators fits in the theory of composition banach lipschitz operator ideals. for p = ∞, we characterize the lipschitz pietsch-∞-integral mappings by a factorization schema through a weakly compact operator. finally, the relationship between these mappings and some well known lipschitz operators is given. 2010 msc: 47b10; 47l20; 26a16. keywords: lipschitz pietsch-p-integral operators; lipschitz strictly pintegral operators; vector measure representation. introduction the class of p-integral linear operators was introduced in 1969 by persson and pietsch [18] (also known as strictly p-integral or pietsch-p-integral operators) establishing many of its fundamental properties using the theory of received 02 february 2021 – accepted 13 may 2021 http://dx.doi.org/10.4995/agt.2021.15061 e. dahia and k. hamidi vector measures. in 1989, cardassi studied the factorization properties and some results of coincidences for these operators in [6]. the ideal of p-integral polynomials on banach spaces has been defined and characterized by cilia and gutiérrez, in [9] for p = 1 and in [8] for p ≥ 1, as a natural polynomial extension of pietsch-p-integral operators. in this paper we introduce and study the lipschitz version of this concept. we define the lipschitz pietsch-p-integral operator (1 ≤ p ≤∞) as a lipschitz mapping between a pointed metric space and a banach space by an integral representation with respect to a vector measure on the borel σ-algebra of a compact hausdorff space k. special attention is paid to the factorization of these mappings and we compare our class with some well known lipschitz operators defined by a factorization schema or by summability of series. note that the class of lipschitz pietsch-1-integral operators is studied in [5]. in this case, the authors use only factorization schemes to define this concept without using vector measure theory. we describe now the contents of the present paper. after this introduction, in section one we fix notation and basic concepts related to lipschitz mappings and vector measure of interest for our purposes. in section two we extend to lipschitz mappings the concept of pietsch-p-integral operators for p ≥ 1 and we prove a factorization theorem for these mappings through the classical banach spaces c(k) and lp(µ, k). the third section is devoted to study the notion of lipschitz pietsch-∞-integral operators, starting from the representation by a vector measure, we present a characterization given by a factorization through a linear weakly compact operator. finally, in section four we establish the relationship between our class and the class of lipschitz p-summing operators, lipschitz grothendieck-p-integral operators, strongly lipschitz p-nuclear operators and lipschitz weakly compact operators. 1. notation and preliminaries the notation used in the paper is in general standard. e and f are real banach spaces. x, y and z will be pointed metric spaces with a base point denoted by 0 and a metric denoted by d. given a banach space e, e∗ denotes its topological dual, and be its closed unit ball. as usual, l(e, f) denotes the space of all continuous linear operators from e to f with the operator norm. a banach space e will be considered as a pointed metric space with distinguished point 0 and distance d(x, y) = ‖x−y‖. with lip0(x, e) we denote the banach space of all lipschitz mappings from x to e, taking 0 into 0, under the lipschitz norm lip(t ) = inf {c > 0 : ‖t (x) −t (x′)‖≤ cd(x, x′)} . moreover, t is called an isometric embedding if ‖t (x) −t (x′)‖ = d(x, x′) for all x, x′ ∈ x. when e = r, lip0(x, r) is denoted by x # and it is called the lipschitz dual of x. along the paper we consider bx# endowed with the pointwise topology (bx# is a compact hausdorff space in this topology). © agt, upv, 2021 appl. gen. topol. 22, no. 2 368 lipschitz integral operators represented by vector measures it is well know that x# has a predual, namely the space of arens and eells æ(x) of the metric space x [2] (also known as the lipschitz-free banach space f(x) of x [14]). this space is one of the main tools that we will use in the sequel. we summarize some basic properties of æ(x). a molecule on x is a scalar valued function m on x with finite support that satisfies ∑ x∈x m (x) = 0. we denote by m(x) the linear space of all molecules on x. for x, x′ ∈ x the molecule mxx′ is defined by mxx′ = χ{x}−χ{x′}, where χa is the characteristic function of the set a. for m ∈ m(x) we can write m = n∑ i=1 λimxix′i for some suitable scalars λi, and we write ‖m‖m(x) = inf { n∑ i=1 |λi|d(xi, x ′ i), m = n∑ i=1 λimxix′i } , where the infimum is taken over all representations of the molecule m. denote by æ(x) the completion of the normed space (m(x),‖.‖m(x)). the map kx : x →æ(x) defined by kx(x) = mx0 isometrically embeds x in æ(x). for any t ∈ lip0(x, e) there exists a unique linear map tl ∈ l(æ (x) , e) such that t = tl ◦kx, i.e. the following diagram commutes x t // kx �� e æ (x) . tl ;; ① ① ① ① ① ① ① ① ① moreover, ‖tl‖ = lip(t ) (see [19, theorem 2.2.4 (b)]). the operator tl is referred to as the linearization of t . the correspondence t ←→ tl establishes an isometric isomorphism between the banach spaces lip0(x, e) and l(æ (x) , e). in particular, the spaces x# and æ(x)∗ are isometrically isomorphic via the linearization r(f) := fl, where fl(m) = ∑ x∈x f(x)m(x), in particular fl(mxx′) = f(x) −f(x ′), (see [19, theorem 2.2.2]). now we recall some simple notions from vector measure theory. let (ω, σ) be a measurable space and e a banach space. let m : σ −→ e be a countably additive vector measure [11, definition i.1.1]. for every x∗ ∈ e∗, let 〈m, x∗〉 be the scalar signed measure defined by 〈m, x∗〉(a) := 〈m(a), x∗〉 for all a ∈ σ. the semivariation of m is the subadditive real bounded set function ‖m‖ : a ∈ σ −→‖m‖(a) ∈ [0, +∞) defined by ‖m‖(a) = sup{|〈m, x∗〉|(a) : ‖x∗‖≤ 1} , where |〈m, x∗〉| is the variation measure of the signed measure 〈m, x∗〉. according to [17, page 106] and [16, definition 2.1], a measurable (scalar valued) function f is integrable with respect to m if • it is integrable with respect to the scalar measure 〈m, x∗〉 for every x∗ ∈ e∗. © agt, upv, 2021 appl. gen. topol. 22, no. 2 369 e. dahia and k. hamidi • for every a ∈ σ there exists an element mf (a) ∈ e satisfying 〈mf (a), x ∗〉 = ∫ a fd〈m, x∗〉 , x∗ ∈ e∗. we will use the classical notation mf (a) := ∫ a fdm, a ∈ σ. for the general theory of vector measures we refer the reader to the classical monograph [11]. for 1 ≤ p ≤ ∞, linear pietsch-p-integral operators were introduced by persson and pietsch [18] and deeply studied in [6, 10] among others. definition 1.1. the linear operator u : e −→ f , between banach spaces, is pietsch-p-integral (1 ≤ p < ∞) if there are a regular borel countably additive vector measure m of bounded semivariation on b(be∗), (where b(be∗) is the borel σ-algebra of be∗), and a positive regular borel measure µ on be∗ such that u (x) = ∫ be∗ 〈x, x∗〉dm(x∗), x ∈ e, and ∥∥∥∥ ∫ be∗ fdm ∥∥∥∥ ≤ (∫ be∗ |f| p dµ ) 1 p , f ∈ c(be∗). the banach space of these operators is denoted by pip(e, f) under the norm defined by ‖u‖pip = inf µ(be∗) 1 p , where the infimum is taken over all measures µ satisfying the above inequality. for p = ∞, u is called pietsch-∞-integral if there is a regular borel countably additive vector measure m : b(be∗) −→ f of bounded semivariation such that u (x) = ∫ be∗ 〈x, x∗〉dm(x∗), x ∈ e. in this case, ‖t‖pil ∞ = inf ‖m‖(be∗), taking the infimum over all m that satisfy the above equality. in [18, satz 15 and satz 17] we find some canonical linear pietsch-p-integral operators that will be used in the sequel. let k be a compact hausdorff space and ν be a positive regular borel measure on k. let (ω, σ, µ) be a finite measure space and jp, ip be the inclusions of c(k) into lp(k, ν) and of l∞(µ) into lp(µ) respectively for 1 ≤ p < ∞. then jp ∈ pip(c(k), lp(k, ν)) and ip ∈pip(l∞(µ), lp(µ)) with ‖jp‖pip = ‖jp‖ = ν(k) 1 p and ‖ip‖pip = ‖ip‖ = µ(ω) 1 p . remark 1.2. note that by using [8, theorem 2.5], u ∈ pip (e, f) if and only if there are a compact hausdorff space k, an embedding h : e −→ c(k), a regular borel countably additive vector measure m : b(k) −→ f of bounded © agt, upv, 2021 appl. gen. topol. 22, no. 2 370 lipschitz integral operators represented by vector measures semivariation and a positive regular borel measure µ on k such that for all x ∈ e, u(x) = ∫ k h (x) dm, and ∥∥∥∥ ∫ k fdm ∥∥∥∥ ≤ (∫ k |f| p dµ ) 1 p for all f ∈ c (k). in this case ‖u‖pip = inf ‖h‖µ(k) 1 p , where the infimum is taken over all k, m and h as above. 2. lipschitz pietsch-p-integral operators definition 2.1. let x be a pointed metric space, e a banach space and let t ∈ lip0(x, e). for 1 ≤ p < ∞, the mapping t is said to be lipschitz pietsch-p-integral operator if there are a regular borel probability measure space (ω, σ, µ), a linear operator a ∈ l(lp(µ), e) and a lipschitz operator b ∈ lip0(x, l∞(µ)) giving rise to the following commutative diagram (2.1) x b �� t // e l∞(µ) ip // lp(µ). a oo where ip : l∞(µ) −→ lp(µ) is the canonical mapping. the set of all lipschitz pietsch-p-integral mappings from x to e is denoted by pilp (x, e). with each t ∈ pilp (x, e) we associate its lipschitz pietsch-p-integral quantity, ‖t‖pilp = inf ‖a‖lip(b), where the infimum is taken over all µ, a and b as above. remark 2.2. (1) as an easy consequence of the definition, if t ∈ pilp (x, e) we have lip(t ) ≤‖t‖pil p . (2) notice that the definition is the same if we consider a finite regular borel measure space (ω, σ, µ), in this case, for t ∈ pilp (x, e) we have ‖t‖pil p = inf ‖a‖µ(ω) 1 p lip(b), where the infimum is taken over all µ, a and b in (2.1 ). (3) we don’t know if being lipschitz pietsch-p-integrability implies pietschp-integrability whenever the mapping t is linear. the converse is of course clearly true, that is if e and f are banach spaces and t : e −→ f is linear pietsch-p-integral then t is lipschitz pietsch-pintegral and ‖t‖pilp ≤‖t‖pip. we have the following immediate consequence of the definition above. © agt, upv, 2021 appl. gen. topol. 22, no. 2 371 e. dahia and k. hamidi proposition 2.3 (inclusion theorem). let 1 ≤ p ≤ q < ∞. then pilp (x, e) ⊂ pilq (x, e) and ‖t‖pil q ≤‖t‖pil p for all t ∈pilp (x, e). in order to prove the factorization theorem for the class of lipschitz pietschp-integral operators, (1 ≤ p ≤∞) we need the following technical lemma. lemma 2.4. let j : m(x) −→ c(bx#) be the operator defined by j(m)(f) = n∑ i=1 λi(f(xi) −f(x ′ i)), for all m = ∑n i=1 λimxix′i ∈ m(x) and f ∈ bx#. then this operator is an isometric embedding. proof. since æ(x)∗ and x# are isometrically isomorphic via the linearization, for all f ∈ x# there is m∗ ∈ æ (x) ∗ such that fl = m ∗. for all m ∈ m(x) we have ‖j(m)‖ c(b x# ) = sup f∈b x# |j(m)(f)| = sup ‖fl‖≤1 ∣∣∣∣∣ n∑ i=1 λi(fl(mxix′i) ∣∣∣∣∣ = sup ‖m∗‖≤1 |〈m, m∗〉| = ‖m‖æ(x) = ‖m‖m(x) , and the proof follows. � for x ∈ x, we denote by δx the functional δx : x # −→ r defined as δx(f) = f(x), f ∈ x #. let ιx : x −→ c(bx#) the natural lipschitz isometric embedding such that ιx(x) is the restriction of δx to bx#, for all x ∈ x. the following theorem gives a parallel development of the factorization schemes concerning lipschitz pietsch-p-integral operators that highlights the role of the space c (bx#). theorem 2.5. let 1 ≤ p < ∞ and let t ∈ lip0(x, e). then t is lipschitz pietsch-p-integral if and only if there exist a regular borel probability measure ν on bx# and an operator ã ∈ l(lp(ν), e) such that the following diagram commutes (2.2) x ιx �� t // e c (bx#) jp // lp(ν) ã oo , where jp is the canonical map. moreover, ‖t‖pilp = inf {∥∥∥ã ∥∥∥ : t = ã◦ jp ◦ ιx } . proof. we write △ for the proposed infimum. suppose that t admits a factorization (2.2). if j∞ is the canonical inclusion map from c (bx#) to l∞(ν), we have the factorization t = ã◦ ip ◦ j∞ ◦ ιx : x ιx −→ c (bx#) j∞ −→ l∞(ν) ip −→ lp(ν) ã −→ e. © agt, upv, 2021 appl. gen. topol. 22, no. 2 372 lipschitz integral operators represented by vector measures denoting by b = j∞◦ιx, it follows that b ∈ lip0(x, l∞(ν)) and lip(b) ≤ 1, which implies that t is lipschitz pietsch-p-integral and ‖t‖pilp ≤ ∥∥∥ã ∥∥∥ lip(b) ≤ ∥∥∥ã ∥∥∥ . passing to the infimum we get ‖t‖pil p ≤△. conversely, suppose that t ∈ pilp (x, e). fix ε > 0, there are a regular borel probability measure space (ω, σ, µ), an operator a ∈l(lp(µ), e) and a lipschitz mapping b ∈ lip0(x, l∞(µ)) such that t = a◦ ip ◦b : x b −→ l∞(µ) ip −→ lp(µ) a −→ e, and ‖a‖lip(b) ≤‖t‖pil p +ε. let bl ∈l(æ(x), l∞(µ)) be the linearization of the lipschitz mapping b, that is b = bl◦kx and ‖bl‖ = lip(b). consider the natural extension of the isometric embedding j, mentioned in lemma 2.4, to æ(x) which we denote also by j. the injectivity of l∞(µ) assures the existence of an operator b̃l ∈l(c(bx#), l∞(µ)) that extends bl with ∥∥∥b̃l ∥∥∥ = ‖bl‖, that is bl = b̃l ◦j or the following diagram commutes æ (x) bl // j �� l∞(µ) c(bx# ) b̃l 99 s s s s s s s s s . the operator ip : l∞(µ) −→ lp(µ) is p -summing with p-summing norm one then ip ◦ b̃l is too with πp(ip ◦ b̃l) ≤ ∥∥∥b̃l ∥∥∥. by [10, corollary 2.15] there exist a regular borel probability measure ν on bx# and an operator s ∈l(lp(ν), lp(µ)) such that ip ◦ b̃l = s ◦ jp : c (bx#) jp −→ lp(ν) s −→ lp(µ), and πp(ip ◦ b̃l) = ‖s‖ . then t = (a ◦s)◦ jp ◦ (j ◦kx) : x j◦kx −→ c (bx#) jp −→ lp(ν) a◦s −→ e. easy calculations prove that j ◦ kx = ιx, which implies that t admits a factorization of the form (2.2) with ã = a ◦s and we have △ ≤ ∥∥∥ã ∥∥∥ ≤‖a‖πp(ip ◦ b̃l) ≤ ‖a‖ ∥∥∥b̃l ∥∥∥ = ‖a‖lip(b) ≤‖t‖pil p + ε. since this holds for all ε > 0 we arrive at △≤‖t‖pil p . � the next theorem is the main result of this section and provides a characterization of the class of lipschitz pietsch-p-integral operators, that is an integral representation with respect to a vector measure. © agt, upv, 2021 appl. gen. topol. 22, no. 2 373 e. dahia and k. hamidi theorem 2.6. let 1 ≤ p < ∞ and let t ∈ lip0(x, e). then t is lipschitz pietsch-p-integral if and only if there are a compact hausdorff space k, a lipschitz embedding φ : x −→ c(k) with φ(0) = 0, a regular borel countably additive vector measure m : b(k) −→ e of bounded semivariation and a positive regular borel measure µ on k such that (2.3) t (x) = ∫ k φ (x) dm, x ∈ x, and (2.4) ∥∥∥∥ ∫ k fdm ∥∥∥∥ ≤ (∫ k |f| p dµ ) 1 p , for all f ∈ c (k) . in this case ‖t‖pilp = inf { lip(φ)µ (k) 1 p } , where the infimum is taken over all k, φ, m and µ satisfying (2.3) and (2.4). proof. suppose that t ∈pilp (x, e), and fix ε > 0. there are a regular borel probability measure ν on bx# and ã ∈l(lp (ν) , e) such that t = ã ◦ jp ◦ ιx : x ιx −→ c (bx#) jp −→ lp(ν) ã −→ e, and ∥∥∥ã ∥∥∥ ≤‖t‖pil p +ε. the linear operator ã◦jp : c (bx#) −→ e is pietschp-integral with ∥∥∥ã ◦ jp ∥∥∥ pip ≤ ∥∥∥ã ∥∥∥‖jp‖pip ≤ ∥∥∥ã ∥∥∥ ≤‖t‖pilp + ε. by remark 1.2, there are a compact hausdorff space k, an embedding h : c (bx#) −→ c(k), a regular borel countably additive vector measure m : b(k) −→ e of bounded semivariation and a positive regular borel measure µ on k such that for all x ∈ x, t (x) = ã ◦ jp (ιx(x)) = ∫ k h (ιx(x)) dm, ∥∥∥∥ ∫ k fdm ∥∥∥∥ ≤ (∫ k |f| p dµ ) 1 p for all f ∈ c (k) and ‖h‖µ (k) 1 p ≤ ∥∥∥ã◦ jp ∥∥∥ pip + ε. which means that (2.3) and (2.4) are true by taking into account that φ = h ◦ ιx is a lipschitz embedding from x to c(k) vanishing at 0 with lip(φ) ≤‖h‖ . moreover lip(φ)µ (k) 1 p ≤ ∥∥∥ã ◦ jp ∥∥∥ pip + ε ≤‖t‖pilp + 2ε. since this holds for every ε > 0, it follows that lip(φ)µ (k) 1 p ≤‖t‖pilp . conversely, suppose that t satisfies the conditions (2.3) and ( 2.4). by [11, theorem vi.2.1] there exists u ∈ l(c (k) , e) such that u(f) = ∫ k fdm, f ∈ c (k) . consider the canonical mapping jp = ip ◦ j∞ : c (k) j∞ −→ l∞ (k, µ) ip −→ lp (k, µ) , © agt, upv, 2021 appl. gen. topol. 22, no. 2 374 lipschitz integral operators represented by vector measures and define r : jp (c (k)) −→ e by r (jp (f)) := u(f). the linear mapping r is well-defined and continuous with norm ≤ 1 since for all f ∈ c (k) , ‖r (jp (f))‖ = ∥∥∥∥ ∫ k fdm ∥∥∥∥ ≤ (∫ k |f| p dµ ) 1 p = ‖jp (f)‖ . by [12, lemma iv.8.19] we have jp (c (k)) = lp (k, µ) , so r can be extended to a continuous linear operator r̃ : lp (k, µ) −→ e with ∥∥∥r̃ ∥∥∥ ≤ 1. if we put b = j∞ ◦ φ, we obtain b ∈ lip0(x, l∞ (k, µ)) and lip(b) ≤ lip(φ). on the other hand, r̃ ◦ ip ◦b(x) = r̃ ◦ jp ◦φ(x) = u(φ(x)) = ∫ k φ(x)dm = t (x), and therefore t factors as in (2.1), that is t ∈pilp (x, e) with ‖t‖pilp ≤ lip(b) ∥∥∥r̃ ∥∥∥ µ (k) 1 p ≤ lip(φ)µ (k) 1 p . � now we present a relationship between the lipschitz pietsch-p-integral operator and its linearization. theorem 2.7. let t ∈ lip0(x, e) and 1 ≤ p < ∞. then t ∈ pi l p (x, e) if and only if tl ∈pip(æ (x) , e). moreover, we have (2.5) ‖t‖pilp = ‖tl‖pip . proof. suppose that tl ∈pip(æ (x) , e). according to [18, satz 18], for every ε > 0 we can choose a typical factorization of tl tl = a ◦ ip ◦b : æ (x) b −→ l∞ (µ) ip −→ lp (µ) a −→ e, such that a ∈ l(lp (µ) , e) and b ∈ l(æ (x) , l∞ (µ)) with ‖a‖‖b‖ ≤ ‖tl‖pip+ε. it is clear that the mapping r := b◦kx belongs to lip0 (x, l∞ (µ)) and lip(r) ≤ ‖b‖. the factorization t = tl ◦ kx = a ◦ ip ◦ r implies that t ∈pilp (x, e) and ‖t‖pil p ≤‖a‖lip (r) ≤‖tl‖pip + ε. conversely, if t ∈ pilp (x, e), for ε > 0 choose the following factorization of t t = a◦ ip ◦b : x b −→ l∞ (µ) ip −→ lp (µ) a −→ e, such that a ∈ l(lp (µ) , e) and b ∈ lip0(x, l∞ (µ)) with ‖a‖lip(b) ≤ ‖t‖pilp + ε. the uniqueness of the linearization maps gives that tl = (a ◦ ip ◦b)l = a◦ ip ◦bl. then, we have that tl ∈pip(æ (x) , e) with ‖tl‖pip ≤‖a‖‖bl‖ = ‖a‖lip(b) ≤‖t‖pilp + ε. the proof concludes. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 375 e. dahia and k. hamidi the notion of lipschitz operator ideal was introduced by achour, rueda, sánchez-pérez and yahi [1]. this can be seen as an extension of the notion of linear banach operator ideal. a lipschitz operator ideal ilip is a subclass of lip0 such that for every pointed metric space x and every banach space e the components ilip(x, e) := lip0(x, e)∩ilip satisfy (i) ilip(x, e) is a linear subspace of lip0(x, e). (ii) vg ∈ilip(x, e) for v ∈ e and g ∈ x #. (iii) the ideal property: if s ∈ lip0(y, x), t ∈ ilip(x, e) and w ∈ l(e, f), then the composition wt s is in ilip(y, f). a lipschitz operator ideal ilip is a normed (banach) lipschitz operator ideal if there is a function ‖.‖ilip : ilip −→ [0, +∞[ that satisfies (i’) for every pointed metric space x and every banach space e, the pair (ilip(x, e),‖.‖ilip) is a normed (banach) space and lip(t ) ≤‖t‖ilip for all t ∈ilip(x, e). (ii’) ‖idk : k −→ k, idk(λ) = λ‖ilip = 1. (iii’) if s ∈ lip0(y, x), t ∈ilip(x, e) and w ∈l(e, f), then ‖wt s‖ilip ≤ lip(s)‖t‖ilip ‖w‖ . following [1, definition 3.1], there is a way to construct a (banach) lipschitz operator ideal from a (banach) linear operator ideal, called composition method. let a be a (banach) linear operator ideal. a lipschitz mapping t ∈ lip0(x, e) belongs to the composition lipschitz operator ideal a◦ lip0 if there exists a banach space f, a lipschitz operator s ∈ lip0(x, f) and a linear operator u ∈ a(f, e) such that t = u ◦ s. if (a,‖.‖a) is a banach operator ideal we write ‖t‖a◦lip0 = inf ‖u‖a lip(s), where the infimum is taken over all u and s as above. in [1], the authors establish a criterion to decide whenever a lipschitz operator ideal is of composition or not. proposition 2.8 ([1, proposition 3.2]). let x be a pointed metric space, e a banach space and a an operator ideal. a lipschitz operator t ∈ lip0(x, e) belongs to a◦lip0(x, e) if and only if its linearization tl belongs to a(f(x), e). furthermore, if (a,‖·‖a) is a banach operator ideal then (a◦lip0,‖·‖a◦lip0) is banach lipschitz operator ideal with ‖t‖a◦lip0 = ‖tl‖a . by theorem 2.7 and the above criterion, we have the following. proposition 2.9. ( pilp ,‖·‖pil p ) is the banach lipschitz operator ideal generated by the composition method from the banach operator ideal pip. in other words pilp (x, e) = pip ◦lip0(x, e) isometrically © agt, upv, 2021 appl. gen. topol. 22, no. 2 376 lipschitz integral operators represented by vector measures for every pointed metric space x and every banach space e. we say that a pointed metric space w is 1-injective (or an absolute lipschitz retract) if for every metric space x, every subset x0 of x and every lipschitz mapping t ∈ lip0(x0, w) there is a lipschitz mapping t̃ ∈ lip0(x, w) extending t with lip(t ) = lip(t̃). the real banach space l∞(µ) for a finite measure µ is 1-injective (see [4, chapter 1]). by the typical pietsch-p-integral factorization of a lipschitz mapping t , we can find a pietsch-p-integral extension t̃ . proposition 2.10. let x and z be pointed metric spaces with x ⊂ z and let e be a banach space. each lipschitz pietsch-p-integral operator t : x −→ e admits a lipschitz pietsch-p-integral extension t̃ : z −→ e with ‖t‖pil p = ∥∥∥t̃ ∥∥∥ pil p . proof. if t ∈pilp (x, e), then for all ε > 0 there are a regular borel probability measure space (ω, σ, µ), a ∈l(lp(µ), e) and b ∈ lip0(x, l∞(µ)) such that t = a◦ ip ◦b : x b −→ l∞(µ) ip −→ lp(µ) a −→ e, and lip(b)‖a‖ ≤ ‖t‖pil p + ε. since l∞(µ) is 1-injective, b admits an extension b̃ ∈ lip0(z, l∞(µ)) with lip(b̃) = lip(b) i.e., the following diagram commutes x b // i �� l∞(µ) z. b̃ ;; ① ① ① ① ① ① ① ① , where i ∈ lip0(x, z) is the natural isometric embedding. this creates a lipschitz pietsch-p-integral extension t̃ : z −→ e of t having the following factorization t̃ = a ◦ ip ◦ b̃ : z b̃ −→ l∞(µ) ip −→ lp(µ) a −→ e. furthermore, ∥∥∥t̃ ∥∥∥ pilp ≤ lip(b̃)‖a‖ = lip(b)‖a‖≤‖t‖pilp + ε. since this holds for all ε > 0 we get ∥∥∥t̃ ∥∥∥ pil p ≤ ‖t‖pilp . for the reverse inequality, note that ‖t‖pil p = ∥∥∥t̃ ◦ i ∥∥∥ pil p ≤ ∥∥∥t̃ ∥∥∥ pil p . � © agt, upv, 2021 appl. gen. topol. 22, no. 2 377 e. dahia and k. hamidi 3. lipschitz pietsch-∞-integral operators in this section we extend the definition of the class of pietsch-∞-integral linear operators to the case of lipschitz operators and we will show a factorization theorem that characterizes these mappings. definition 3.1. we say that a lipschitz operator t ∈ lip0(x, e) is lipschitz pietsch-∞-integral if there is a regular borel countably additive vector measure m : b(bx#) −→ e of bounded semivariation such that (3.1) t (x) = ∫ b x# f (x) dm(f), x ∈ x. we denote by pil∞(x, e) the set of all these mappings and we put ‖t‖pil ∞ = inf ‖m‖(bx#), taking the infimum over all m such that (3.1) holds. remark 3.2. if t ∈ pil∞(x, e) then lip(t ) ≤ ‖t‖pil ∞ . in order to see this, for ε > 0 choose m such that ‖m‖(bx#) ≤‖t‖pil ∞ + ε and for all x, y ∈ x, ‖t (x)−t (y)‖ ≤ ∫ b x# |f (x) −f (y)|dm(f) ≤ ‖m‖(bx#)d(x, y) ≤ (ε +‖t‖pil ∞ )d(x, y). hence, lip(t ) ≤‖t‖pil ∞ . now we prove the main result of this section. we characterize the pietsch∞-integral lipschitz operators by means of a factorization scheme through a weakly compact linear operator. theorem 3.3. for a lipschitz operator t ∈ lip0(x, e), the following statements are equivalent. (1) t is lipschitz pietsch-∞-integral. (2) there are a compact hausdorff space k, a lipschitz embedding ϕ ∈ lip0(x, c(k)) and a weakly compact linear operator s ∈l(c(k), e) such that the following diagram commutes (3.2) x t // ϕ �� e c(k). s << ② ② ② ② ② ② ② ② (3) there are a regular borel finite measure space (ω, σ, µ), a weakly compact operator r ∈l(l∞(µ), e) and a lipschitz embedding φ ∈ lip0(x, l∞(µ)) © agt, upv, 2021 appl. gen. topol. 22, no. 2 378 lipschitz integral operators represented by vector measures giving rise to the following commutative diagram (3.3) x t // φ �� e l∞(µ). r ;; ① ① ① ① ① ① ① ① ① in addition, ‖t‖pil ∞ = inf ‖s‖lip(ϕ) = inf ‖r‖lip(φ). where the first infimum is taken over all s and ϕ as in (3.2) and the second is taken over all r and φ as in (3.3). proof. (1)=⇒(2). take t ∈ pil∞(x, e). for every ε > 0 choose m satisfying (3.1) and ‖m‖(bx#) ≤ ‖t‖ pil ∞ + ε. consider the linear operator s : c(bx# ) −→ e defined by s(h) = ∫ b x# hdm, for all h ∈ c(bx# ) and the natural lipschitz isometric embedding ιx : x −→ c(bx#). in this case, for all x ∈ x we can write s ◦ ιx(x) = ∫ b x# ιx(x)(f)dm(f) = ∫ b x# f(x)dm(f) = t (x). theorem vi.2.5 in [11] asserts that s is weakly compact with norm ‖s‖ = ‖m‖(bx#) and then ‖s‖lip(ιx) = ‖m‖(bx#) ≤‖t‖pil ∞ + ε. (2)=⇒(3). there is a a regular borel countably additive vector measure m : b(k) −→ e of bounded semivariation such that s(f) = ∫ k fdm for all f ∈ c(k) and ‖m‖(k) = ‖s‖ (see [11, theorem vi.2.1, vi.2.5 and corollary vi.2.14]). it follows that t (x) = s ◦ϕ(x) = ∫ k ϕ(x)dm, x ∈ x. on the other hand, [11, corollary i.2.6 and theorem i.2.1] assures the existence of a regular borel finite measure µ on b(k) such that m(a) = 0 for all a ∈ b(k) which satisfy that µ(a) = 0. define the operator r ∈l(l∞(µ), e) by r(f) = ∫ k fdm, f ∈ l∞(µ) with ‖r‖ = ‖m‖(k) (see [11, theorem i.1.13]). this operator is weakly compact (see [11, definition i.1.14 and theorem vi.1.1]). consequently, r ◦ (j∞ ◦ϕ) = ∫ k j∞ ◦ϕdm = ∫ k φdm = t. © agt, upv, 2021 appl. gen. topol. 22, no. 2 379 e. dahia and k. hamidi (3)=⇒(1). as in the proof of the second implication of theorem 2.5, starting from the diagram (3.3), consider the linearization φl of φ ∈ lip0(x, l∞(µ)) and let φ̃l ∈ l(c(bx#), l∞(µ)) be the extension of φl, i.e., the following diagram commutes æ (x) φl // j �� l∞(µ) c(bx# ). φ̃l 99 s s s s s s s s s s the linear operator r ◦ φ̃l : c(bx#) −→ e is weakly compact. let m be the representing vector measure of r ◦ φ̃l, that is r ◦ φ̃l(f) = ∫ b x# fdm for all f ∈ c(bx#) and ‖m‖(bx#) = ∥∥∥r ◦ φ̃l ∥∥∥. it follows that t (x) = r ◦φ(x) = r ◦ φ̃l ◦j ◦kx(x) = ∫ b x# j ◦kx(x)dm, for all x ∈ x, and then t ∈pil∞(x, e) and ‖t‖pil ∞ ≤‖m‖(bx#) ≤‖r‖ ∥∥∥φ̃l ∥∥∥ = ‖r‖lip(φ). since this is true for every factorization as (3.3), we have‖t‖pil ∞ ≤‖r‖lip(φ). in order to show the reverse inequality, take t ∈ pil∞(x, e) and ε > 0. then there is m : b(bx#) −→ e (as in definition 3.1) such that (3.1) is true and ‖m‖(bx#) ≤ ε + ‖t‖pil ∞ . following the proof of (2)=⇒(3), we can find a regular borel finite measure µ on bx# and a weakly compact operator r ∈l(l∞(µ), e) represented by m such that ‖r‖lip(φ) = ‖r‖ = ‖m‖(bx#) ≤ ε +‖t‖pil ∞ , where φ ∈ lip0(x, l∞(µ)), is the lipschitz embedding defined by φ = j∞◦ιx. the required inequality follows and the second equality follows in a similar way. � 4. some relations of lipschitz pietsch-p-integral operators with other lipschitz operator ideals. 4.1. lipschitz p-summing operators. the definition of the lipschitz psumming operators below was first given by farmer and johnson in [13]. definition 4.1. for a pointed metric space x and a banach space e, the mapping t ∈ lip0(x, e) is called lipschitz p-summing, 1 ≤ p < ∞, if there exists a constant c > 0 such that for all x1, . . . , xn, x ′ 1, . . . , x ′ n in x, (4.1) n∑ i=1 ‖t (xi) −t (x ′ i)‖ p ≤ cp sup f∈b x# n∑ i=1 |f(xi)−f(x ′ i)| p. © agt, upv, 2021 appl. gen. topol. 22, no. 2 380 lipschitz integral operators represented by vector measures in this case we put πlp (t ) = inf {c : satisfying (4.1)}. the set of all lipschitz p-summing operators from x to e is denoted by πlp (x, e). it is well known that (πlp , π l p (·)) is a banach lipschitz operator ideal (see [1, proposition 2.5]). we can establish the following comparison between the classes of lipschitz pietsch-p-integral operators and lipschitz p-summing operators. proposition 4.2. let 1 ≤ p < ∞. every lipschitz pietsch-p-integral operator t : x −→ e is lipschitz p-summing with πlp (t ) ≤‖t‖pil p . proof. if t ∈ pilp (x, e), for ε > 0 we choose a typical lipschitz pietsch-pintegral factorization t = a◦ ip ◦b : x b −→ l∞(µ) ip −→ lp(µ) a −→ e, with ‖a‖lip(b) ≤ ε + ‖t‖pil p . the mapping ip is linear p-summing with πp(ip) = 1 (see [10, page 40]). then it is lipschitz p-summing with π l p (ip) = 1 (see [13, theorem 2]). by the ideal property concerning the lipschitz operator ideal πlp , we have that t ∈ π l p (x, e) and π l p (t ) ≤‖a‖lip(b) ≤ ε+‖t‖pil p . � 4.2. lipschitz grothendieck-p-integral operators. the notion of lipschitz grothendieck-p-integral operators (p ≥ 1) from a pointed metric space x into a banach space e was introduced by jiménez-vargas et al. in [15] (under the name of strongly lipschitz p-integral operators). the mapping t ∈ lip0(x, e) is lipschitz grothendieck-p-integral (in symbols t ∈ gilp (x, e)) if je ◦ t ∈ pi l p (x, e ∗∗), where je : e −→ e ∗∗ is the canonical injection. the class (gilp ,‖·‖gil p ) is a banach lipschitz operator ideal where ‖t‖gilp = ‖je ◦t‖pilp (see [3, remark 4.3 and proposition 4.8]). it is immediate that pilp (x, e) ⊂ gi l p (x, e) and ‖t‖gil p ≤ ‖t‖pil p for all t ∈pilp (x, e). the proof of the next result is an easy adaptation of [5, proposition 3.3]. proposition 4.3. if the banach space e is norm one complemented in e∗∗ (in particular, if e is a dual banach space), then gilp (x, e) ⊂ pi l p (x, e) and ‖t‖gil p = ‖t‖pil p for all t ∈gilp (x, e). 4.3. strongly lipschitz p-nuclear operators. chen and zheng in [7] introduced the concept of strongly lipschitz p-nuclear operators. for a pointed metric space x and a banach space e, a mapping t ∈ lip0(x, e) is strongly lipschitz p-nuclear (1 ≤ p < ∞) if there exist b ∈ lip0(x, ℓ∞) and a ∈l(ℓp, e) and a diagonal operator mλ ∈ l(ℓ∞, ℓp) induced by λ = (λi)i≥1 ∈ ℓp (i.e. mλ((ξi)i≥1) = (λiξi)i≥1) such that t = a◦mλ ◦b : x b −→ ℓ∞ mλ −→ ℓp a −→ e. © agt, upv, 2021 appl. gen. topol. 22, no. 2 381 e. dahia and k. hamidi the banach space of all these mappings is denoted by snlp (x, e) and the norm is defined by ‖t‖sn l p = inf ‖a‖‖mλ‖lip(b), where the infimum is taken over all the above factorizations. proposition 4.4. every strongly lipschitz p-nuclear operator is lipschitz pietschp-integral. moreover, ‖t‖pilp ≤‖t‖sn lp for all t ∈snlp (x, e). proof. given ε > 0, take t ∈ snlp (x, e) with the above factorization such that ‖a‖‖mλ‖lip(b) ≤ ε +‖t‖sn l p . in this case, ℓ∞ and ℓp are the spaces l∞(µ) and lp(µ) with µ the counting measure on n respectively and mλ : l∞(µ) −→ lp(µ) is the multiplication operator induced by λ ∈ lp(µ) (i.e. mλ(f) = λ.f). use [10, page 111] to see that mλ is a pietsch-p-integral linear operator and ‖mλ‖pip = ‖mλ‖ and then it is lipschitz pietsch-p-integral with ‖mλ‖pilp ≤ ‖mλ‖ (by remark 2.2). in view of the ideal property of pilp , we are done. � 4.4. lipschitz weakly compact operators. the definition of lipschitz weakly compact operators is due to jiménez-vargas et al. ([15]). definition 4.5. let x be a pointed metric space and let e be a banach space. the mapping t ∈ lip0(x, e) is called lipschitz weakly compact if the set { t (x)−t (x′) d(x,x′) : x, x′ ∈ x, x 6= x′ } is relatively weakly compact in e. the set of all lipschitz weakly compact operators from x into e is denoted by lip0w(x, e). proposition 2.8 in [15] asserts that every lipschitz grothendieck-p-integral operator is lipschitz weakly compact. so, according to the comments above we have that pilp (x, e) ⊂ lip0w(x, e). acknowledgements. we would like to thank the referee for his/her careful reading and useful suggestions. also, we acknowledge with thanks the support of the general direction of scientific research and technological development (dgrsdt), algeria. references [1] d. achour, p. rueda, e. a. sánchez-pérez and r. yahi, lipschitz operator ideals and the approximation property, j. math. anal. appl. 436 (2016), 217–236. [2] r. f. arens and j. eels jr., on embedding uniform and topological spaces, pacific j. math 6 (1956), 397–403. [3] a. belacel and d. chen, lipschitz (p, r, s)-integral operators and lipschitz (p, r, s)nuclear operators, j. math. anal. appl. 461 (2018) 1115–1137. © agt, upv, 2021 appl. gen. topol. 22, no. 2 382 lipschitz integral operators represented by vector measures [4] y. benyamini and j. lindenstrauss, geometric nonlinear functional analysis, vol. 1, amer. math. soc. colloq. publ., vol. 48, amer. math. soc., providence, ri, 2000. [5] m. g. cabrera-padilla and a. jiménez-vargas, lipschitz grothendieck-integral operators, banach j. math. anal. 9, no. 4 (2015), 34–57. [6] c. s. cardassi, strictly p-integral and p-nuclear operators, in: analyse harmonique: groupe de travail sur les espaces de banach invariants par translation, exp. ii, publ. math. orsay, 1989. [7] d. chen and b. zheng. lipschitz p-integral operators and lipschitz p-nuclear operators, nonlinear anal. 75 (2012), 5270–5282. [8] r. cilia and j. m. gutiérrez, asplund operators and p-integral polynomials, mediterr. j. math. 10 (2013), 1435–1459. [9] r. cilia and j. m. gutiérrez, ideals of integral and r-factorable polynomials, bol. soc. mat. mexicana 14 (2008), 95–124. [10] j. diestel, h. jarchow and a. tonge, absolutely summing operators, cambridge university press, cambridge, 1995. [11] j. diestel and j. j. uhl, jr., vector measures, math. surveys monographs 15, american mathematical society, providence ri, 1977. [12] n. dunford and j. t. schwartz, linear operators, part i:general theory, j. wiley & sons, new york, 1988. [13] j. d. farmer and w. b. johnson, lipschitz p-summing operators, proc. amer. math. soc. 137, no. 9 (2009), 2989–2995. [14] g. godefroy, a survey on lipschitz-free banach spaces, commentationes mathematicae 55, no. 2 (2015), 89–118. [15] a. jiménez-vargas, j. m. sepulcre and m. villegas-vallecillos, lipschitz compact operators, j. math. anal. appl. 415 (2014), 889–901. [16] d. r. lewis, integration with respect to vector measures, pacific j. math. 33 (1970), 157–165. [17] s. okada, w. j. ricker and e. a. sánchez-pérez, optimal domain and integral extension of operators acting in function spaces, operator theory: adv. appl., vol. 180, birkhauser, basel, 2008. [18] a. persson and a. pietsch.p-nuklear und p-integrale abbildungen in banach räumen, studia math. 33 (1969), 19–62. [19] n. weaver, lipschitz algebras, world scientific publishing co., singapore, 1999. © agt, upv, 2021 appl. gen. topol. 22, no. 2 383 () @ appl. gen. topol. 14, no. 2 (2013), 171-178doi:10.4995/agt.2013.1587 c© agt, upv, 2013 the combinatorial derivation igor v. protasov department of cybernetics, kyiv university, volodimirska 64, kyiv 01033, ukraine (i.v.protasov@gmail.com) abstract let g be a group, pg be the family of all subsets of g. for a subset a ⊆ g, we put ∆(a) = {g ∈ g : |ga ∩ a| = ∞}. the mapping ∆ : pg → pg, a 7→ ∆(a), is called a combinatorial derivation and can be considered as an analogue of the topological derivation d : px → px, a 7→ a d , where x is a topological space and a d is the set of all limit points of a. content: elementary properties, thin and almost thin subsets, partitions, inverse construction and ∆-trajectories, ∆ and d. 2010 msc: 20a05, 20f99, 22a15, 06e15, 06e25. keywords: combinatorial derivation; ∆-trajectories; large, small and thin subsets of groups; partitions of groups; stone-čech compactification of a group. 1. introduction let g be a group with the identity e, pg be the family of all subsets of g. for a subset a of g, we denote ∆(a) = {g ∈ g : |ga ∩ a| = ∞}, observe that ∆(a) ⊆ aa−1, and say that the mapping ∆ : pg → pg, a 7→ ∆(a) is the combinatorial derivation. in this paper, on one hand, we analyze from the ∆-point of view a series of results from subset combinatorics of groups (see the survey [9]), and point out some directions for further progress. on the other hand, the ∆-operation is interesting and intriguing by its own sake. in contrast to the trajectory a → received october 2012 – accepted january 2013 http://dx.doi.org/10.4995/agt.2013.1587 i. v. protasov aa−1 → (aa−1)(aa−1)−1 → . . ., the ∆-trajectory a → ∆(a) → ∆2(a) → . . . of a subset a of g could be surprisingly complicated: stabilizing, increasing, decreasing, periodic or chaotic. for a symmetric subset a of g with e ∈ a, there exists a subset x ⊆ g such that ∆(x) = a. we conclude the paper by demonstrating how ∆ and a topological derivation d arise from some unified ultrafilter construction. we note also that ∆(a) may be considered as some infinite version of the symmetry sets well-known in additive combinatorics [11, p. 84]. given a finite subset a of an abelian group g and α > 0, the symmetry set symα(a) is defined by symα(a) = {g ∈ g : |a ∩ (a + g)| > α|a|}. 2. elementary properties claim 2.1. (∆(a))−1 = ∆(a), ∆(a) ⊆ aa−1. claim 2.2. ∆(a) = ∅ ⇔ e /∈ ∆(a) ⇔ a is finite. claim 2.3. for subsets a, b of g, we let ∆(a, b) = {g ∈ g : |ga ∩ b| = ∞} and note that ∆(a ∪ b) = ∆(a) ∪ ∆(b) ∪ ∆(a, b) ∪ ∆(b, a), ∆(a ∩ b) ⊆ ∆(a) ∩ ∆(b) . claim 2.4. if f is a finite subset of g then ∆(fa) = f∆(a)f −1. claim 2.5. if a is an infinite subgroup then a = ∆(a) but the converse statement does not hold, see theorem 6.2. 3. thin and almost thin subsets a subset a of a group g is said to be [8]: • thin if either a is finite or ∆(a) = {e}; • almost thin if ∆(a) is finite; • k-thin (k ∈ n) if |ga ∩ a| 6 k for each g ∈ g \ {e}; • sparse if, for every infinite subset x ⊆ g, there exists a non-empty finite subset f ⊂ x such that ⋂ g∈f ga is infinite; • k-sparse (k ∈ n) if, for every infinite subset x ⊆ g, there exists a subset f ⊂ x such that |f | 6 k and ⋂ g∈f ga is finite. the following statements are from [8]. theorem 3.1. every almost thin subset a of a group g can be partitioned in 3|∆(a)|−1 thin subsets. if g has no elements of odd order, then a can be partitioned in 2|∆a|−1 thin subsets. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 172 the combinatorial derivation theorem 3.2. a subset a of a group g is 2-sparse if and only if x−1x * ∆(a) for every infinite subset x of g. theorem 3.3. for every countable thin subset a of a group g, there is a thin subset b such that a ∪ b is 2-sparse but not almost thin. theorem 3.4. suppose that a group g is either torsion-free or, for every n ∈ n, there exists a finite subgroup hn of g such that |hn| > n. then there exists a 2-sparse subset of g which cannot be partitioned in finitely many thin subsets. by theorem 3.2, every almost thin subset is 2-sparse. by theorems 3.3, 3.4, the class of 2-sparse subsets is wider than the class of almost thin subsets. by theorem 3.3, a union of two thin subsets needs not to be almost thin. by theorem 2.3, a union a1 ∪ . . . ∪ an of almost thin subset is almost thin if and only if ∆(ai, aj) is finite for all i, j ∈ {1, . . . , n}, by claim 2.4, if a is almost thin and k is finite then ka is almost thin. the following statements are from [7]. theorem 3.5. for every infinite group g, there exists a 2-thin subset such that g = xx−1 ∪ x−1x. theorem 3.6. for every infinite group g, there exists a 4-thin subset such that g = xx−1. since ∆(x) = {e} for each infinite thin subset of g, theorem 3.6 gives us x with ∆(x) = {e} and xx−1 = g. 4. large and small subsets a subset a of a group g is called [8]: • large if there exists a finite subset f of g such that g = fa; • ∆-large if ∆(a) is large; • small if (g \ a) ∩ l is large for each large subset l of g; • p-small if there exists an injective sequence (gn)n∈ω in g such that the subsets {gna : n ∈ ω} are pairwise disjoint; • almost p-small if there exists an injective sequence (gn)n∈ω in g such that the family {gna : n ∈ ω} is almost disjoint, i.e. gna ∩ gma is finite for all distinct n, m ∈ ω. • weakly p-small if, for every n ∈ ω, one can find distinct elements g1, . . . , gn of g such that the subsets g1a, . . . , gna are pairwise disjoint. let g be a group, a is a large subset of g. we take a finite subset f of g, f = {g1, . . . , gn} such that g = fa. take an arbitrary g ∈ g. then gia ∩ ga is infinite for some i ∈ {1, . . . , n}, so g−1i g ∈ ∆(a). hence, g = f∆(a) and a is ∆-large. by theorem 3.6, the converse statement is very far from being true. if a is not small then fa is thick (see definition 5.2) for some finite subset f . it follows that ∆(fa) = g. by claim 2.4, ∆(fa) = f∆(a)f −1, so if g is abelian then a is ∆-large. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 173 i. v. protasov j. erde proved that every non-small subset of an arbitrary infinite nonabelian group g ∆-large. it is easy to see that a is p-small (almost p-small) if and only if there exists an infinite subset x of g such that x−1x∩pp −1 = {e} (x−1x∩∆(x) = {e}). a is weakly p-small if and only if, for every n ∈ ω, there exists f ⊂ g such that |f | = n and f −1f ∩ pp −1 = {e}. by [8, lemma 4.2], if aa−1 is not large then a is small and p-small. using the inverse construction from section 6, we can find a such that a is not ∆-large and a is not p-small. every infinite group g has a weakly p-small not p-small subsets [1]. moreover, g has almost p-small not p-small subset and , if g is countable, weakly p-small not almost p-small subset. by [8], every almost p-small subset can be partitioned in two p-small subsets. if a is either almost or weakly p-small then g \ ∆(a) is infinite, but a subset a with infinite g \ ∆(a) could be large: g = z, a = 2z. 5. partitions let g be a group and let g = a1 ∪ . . . an be a finite partition of g. in section 7, we show that at least one cell ai is ∆-large, in particular, aia −1 i is large. if g is infinite amenable group and µ is a left invariant banach measure on g, we can strengthened this statement: there exist a cell ai and a finite subset f such that |f | 6 n and g = f∆(ai). to verify this statement, we take ai such that µ(ai) > 1 n and choose distinct g1, . . . , gm such that µ(gkai ∩ glai) = 0 for all distinct k, l ∈ {1, . . . , m}, and the family {g1ai, . . . , gmai} is maximal with respect to this property. clearly, m 6 n. for each g ∈ g, we have µ(gai ∩ gkai) > 0 for some k ∈ {1, . . . , m} so g−1k g ∈ ∆(ai) and g = {g1, . . . , gm}∆(ai). by [10, theorem 12.7], for every partition a1 ∪. . .∪an of an arbitrary group g, there exist a cell ai and a finite subset f of g such that g = faia −1 i and |f | 6 22n−1−1. s. slobodianiuk strengthened this statement: there are f and ai such that |f | 6 22 n−1−1 and g = f∆(ai). it is an old unsolved problem [5, problem 13.44] whether i and f can be chosen so that g = faia −1 i and |f | 6 n, see also [10, question 12.1]. question 5.1. given any partition g = a1 ∪ . . . ∪ an, do there exist f and ai such that g = f∆(ai) and |f | 6 2n? definition 5.2. a subset a of a group g is called [11]: • thick if g \ a is not large; • k-prethick (k ∈ n) if there exists a subset f of g such that |f | 6 k and fa is thick; • prethick if a is k-prethick for some k ∈ n. by [3, theorem 5.3.2], for a group g, the following two conditions (i) and (ii) are equivalent: c© agt, upv, 2013 appl. gen. topol. 14, no. 2 174 the combinatorial derivation (i) for every partition g = a ∪ b, either g = aa−1 or g = bb−1; (ii) each element of g has odd order. if g is infinite, we can show that these conditions are equivalent to (iii) for every partition g = a ∪ b, either g = ∆(a) or g = ∆(g). 6. inverse construction and ∆-trajectories theorem 6.1. let g be an infinite group, a ⊆ g, a = a−1, e ∈ a. then there exists a subset x of g such that ∆(x) = a. proof. first, assume that g is countable and write the elements of a in the list {an : n < ω}, if a is finite then all but finitely many an are equal to e. we represent g \ a as a union g \ a = ⋃ n∈ω bn of finite subsets such that bn ⊆ bn+1, b−1n = bn. then we choose inductively a sequence (xn)n∈ω of finite subsets of g, xn = {xn0, xn1, . . . , xnn, a0xn0, . . . , anxnn} such that xmx −1 n ∩ bn = {e} for all m 6 n < ∞. after ω steps, we put x = ⋃ n∈ω xn. by the construction, ∆(x) = a. if |a| 6 ℵ0 but g is not countable, we take a countable subgroup h of g such that a ⊆ h, forget about g and find a subset x ⊆ h such that ∆(x) is equal to a in h. since ga ∩ a = ∅ for each g ∈ g \ h, we have ∆(x) = a. at last, let |a| > ℵ0. by above paragraph, we may suppose that |a| = |g|. we enumerate a = {aα : α < |g|} and construct inductively a sequence (xα)α<|g| of finite subsets of g and an increasing sequence (hα)α<|g| of subgroup of g such that if α = 0 or α is a limit ordinal, n ∈ ω, xα+n = {xα+n,0, xα+n,1, . . . , xα+n,n, aαxα+n,0, . . . , aα+nxα+n,n}, xα+n ⊆ hα+n+1 \ hα+n, xα+nx−1α+n ⊆ a ∪ (hα+n+1 \ hα+n). after |g| steps, we put x = ⋃ α<|g| xα. by the construction, ∆(x) = a. � let a1, . . . , am be subsets of an infinite group g such that g = a1∪. . .∪am. by the hindman theorem [4, theorem 5.8], there are exists i ∈ {1, . . . , m} and an injective sequence (gn)n∈ω in g such that fp(gn)n∈ω ⊆ ai, where fp(gn)n∈ω is a set of all element of the form gi1gi2 . . . gil, i1 < . . . , ik < ω, k ∈ ω. we show that there exists x ⊆ fp(gn)n∈ω such that ∆(x) = {e} ∪ fp(gn)n∈ω ∪ (fp(gn)n∈ω)−1. we note that if g is countable, at each step n of the inverse construction, the elements xn0, . . . , xnn can be chosen from any pregiven infinite subset y of g. we enumerate fp(gn)n∈ω in a sequence (an)n∈ω and put y = {gn : n ∈ ω}. using above observation, we get the desired x. if g is countable, we can modify the inverse construction to get x such that ∆(x) = a and |x ∩g1 ∩g2x| < ∞ for all distinct g1, g2 ∈ g\{e}, in particular, x is 3-sparse and, in particular, small. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 175 i. v. protasov another modification, we can choose x such that x ∩ gx 6= ∅ for each g ∈ g. if we take a not large, then we get x which is not p-small and x is not ∆-large, see section 4. theorem 6.2. let g be a countable group such that, for each g ∈ g \ {e}, the set √ g = {x ∈ g : x2 = g} is finite. then the following statements hold: (t r1) given any subset x0 ⊆ g, x0 = x−10 , e ∈ x0, there exists a sequence (xn)n∈ω of subsets of g such that ∆(xn+1) = xn and xm ∩xn = {e}, 0 < m < n < ω. (t r2) there exists a sequence (xn)n∈z of subsets of g such that ∆(xn) = xn+1, xm ∩ xn = {e}, m, n ∈ z, m 6= n. (t r3) there exists a subset a of g such that ∆(a) = a but a is not a subgroup. (t r4) there exists a subset a such that a ⊃ ∆(a) ⊃ ∆2(a) ⊃ . . .. (t r5) there exists a subset a such that a ⊂ ∆(a) ⊂ ∆2(a) ⊂ . . .. (t r6) for each natural natural number n, there exists a periodic ∆-trajectory x0, . . . , xn−1 of length n: x1 = ∆(x0), x2 = ∆(x1), . . . , xn = ∆(xn−1) such that xi ∩ xj = {e}, i < j < n. proof. we use the following simple observation (*) if f is a finite subset of an infinite group g and g /∈ f then the set {x ∈ g : x−1gx /∈ f} is infinite. in constructions of corresponding trajectories, at each inductive step, we use a finiteness of √ g and (*) in the following form: (**) if a ∈ g, f is a finite subset of g, f ∩ {e, a±1} = ∅ then there exists x ∈ g such that {x±1, (ax)±1}{x±1, (ax)±1} ∩ f = ∅. we show how to get a 2-periodic trajectory: x, y , ∆(x) = y , ∆(y ) = x, x ∩ y = {e}. we write g as a union g = ⋃ n∈ω fn of increasing chain {fn : n ∈ ω} of finite symmetric subsets f0 = {e}. we put x0 = y0 = {e} and construct inductively with usage of (**) two chains (xn)n∈ω, (yn)n∈ω of finite subsets of g such that, for each n ∈ ω, xn+1 = {(x(y))±1, (yx(y))±1 : y ∈ y0 ∪ . . . ∪ yn}, yn+1 = {(y(x))±1, (xy(x))±1 : x ∈ x0 ∪ . . . ∪ xn}, (x0 ∪ . . . ∪ xn) ∩ (y0 ∪ . . . ∪ yn) = {e}, xn+1xn+1 ∩ (fn+1 \ (y0 ∪ . . . ∪ yn)) = ∅, yn+1yn+1 ∩ (fn+1 \ (x0 ∪ . . . ∪ xn)) = ∅, (x0 ∪ . . . ∪ xn)xn+1 ∩ (fn+1 \ (y0 ∪ . . . ∪ yn)) = ∅, (y0 ∪ . . . ∪ yn)yn+1 ∩ (fn+1 \ (x0 ∪ . . . ∪ xn)) = ∅. after ω steps, we put x = ⋃ n∈ω xn, y = ⋃ n∈ω yn. � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 176 the combinatorial derivation 7. ∆ and d for a subset a of a topological space x, the subset ad of all limit points of a is called a derived subset, and the mapping d : p(x) → p(x), a → ad, defined on the family of p(x) of all subsets of x, is called the topological derivation, see [6, §9]. let x be a discrete set, βx be the stone-čech compactification of x. we identify βx with the set of all ultrafilters on x, x with the set of all principal ultrafilters, and denote x∗ = βx \ x the set of all free ultrafilters. the topology of βx can be defined by the family {a : a ⊆ x} as a base for open sets, a = {p ∈ βx : a ∈ p}, a∗ = a ∩ g∗. for a filter ϕ on x, we put ϕ = {p ∈ βx : ϕ ⊆ p}, ϕ∗ = ϕ ∩ g∗. let g be a discrete group, p ∈ βg. following [2, chapter 3], we denote cl(a, p) = {g ∈ g : a ∈ gp}, gp = {gp : p ∈ p}, say that cl(a, p) is a closure of a in the direction of p, and note that ∆(a) = ⋂ p∈a∗ cl(a, p). a topology τ on a group g is called left invariant if the mapping lg : g → g, lg(x) = gx is continuous for each g ∈ g. a group g endowed with a left invariant topology τ is called left topological. we note that a left invariant topology τ on g is uniquely determined by the filter ϕ of neighbourhoods of the identity e ∈ g, ϕ and ϕ∗ are the sets of all ultrafilters an all free ultrafilters of g converging to e. for a subset a of g, we have ad = ⋂ p∈(τ∗) cl(a, p), and note that ad ⊆ ∆(a) if a is a neighbourhood of e in (g, τ). now we endow g with the discrete topology and, following [4, chapter 4], extend the multiplication on g to βg. for p, q ∈ βg, we take p ∈ p and, for each g ∈ p , pick some qg ∈ q. then ⋃ g∈p gqp ∈ pq and each member of pq contains a subset of this form. with this multiplication, βg is a compact right topological semigroup. the product pq can also be defined by the rule [2, chapter 3]: a ⊆ g, a ∈ pq ⇔ cl(a, q) ∈ p. if (g, τ) is left topological semigroup then τ is a subsemigroup of βg. if an ultrafilter p ∈ τ is taken from the minimal ideal k(τ) of τ, by [2, theorem 5.0.25]. there exists p ∈ p and finite subset f of g such that fcl(p, p) is neighbourhood of e in τ. in particular, if τ indiscrete (τ = {∅, g}), p ∈ k(βg)) and p ∈ p then cl(p, p) is large. if g is infinite, p ∈ k(βg) is free, so cl(p, p) ⊆ ∆(p) and p is ∆-large. if a group g is finitely partitioned g = a1 ∪ . . . ∪ an, then some cell ai is a member of p, hence ai is ∆-large. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 177 i. v. protasov references [1] t. banakh and n. lyaskovska, weakly p-small not p-small subsets in groups, intern. j. algebra computations 18 (2008), 1–6. [2] m. filali and i. protasov, ultrafilters and topologies on groups, math. stud. monorg. ser., vol. 13, vntl publishers, lviv, 2010. [3] v. gavrylkiv, algebraic-topological structure on superextensions, dissertation, lviv, 2009. [4] n. hindman and d. strauss, algebra in the stone-čech compactification, walter de grueter, berlin, new york, 1998. [5] the kourovka notebook, novosibirsk, institute of math., 1995. [6] k. kuratowski, topology, vol. 1, academic press, new york and london, pwn, warszawa, 1969. [7] ie. lutsenko, thin systems of generators of groups, algebra and discrete math. 9 (2010), 108–114. [8] ie. lutsenko and i. v. protasov, sparse, thin and other subsets of groups, intern. j. algebra computation 19 (2009), 491–510. [9] i. v. protasov, selective survey on subset combinatorics of groups, j. math. sciences 174 (2011), 486–514. [10] i. protasov and t. banakh., ball structure and colorings of groups and graphs, math. stud. monorg. ser., vol. 11, vntl publishers, lviv, 2003. [11] t. tao and v. vu, additive combinatorics, cambridge university press, 2006. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 178 @ appl. gen. topol. 20, no. 1 (2019), 177-191doi:10.4995/agt.2019.10490 c© agt, upv, 2019 cauchy action on filter spaces n. rath school of mathematics and statistics, the university of western australia, crawley, wa 6009, australia. (nanditarath9@gmail.com) communicated by j. galindo abstract a cauchy group (g, d, ·) has a cauchy-action on a filter space (x, c), if it acts in a compatible manner. a new filter-based method is proposed in this paper for the notion of group-action, from which the properties of this action such as transitiveness and its compatibility with various modifications of the g-space (x, c) are determined. there is a close link between the cauchy action and the induced continuous action on the underlying g-space, which is explored here. in addition, a possible extension of a cauchy-action to the completion of the underlying gspace is discussed. these new results confirm and generalize some of the properties of group action in a topological context. 2010 msc: 54a20; 54b05; 54c20; 54c05. keywords: continuous action; cauchy map; g-space; filter space and its modifications; completions. 1. introduction group-action has applications in numerous branches of algebra, especially in finite permutation groups [6] and their primitivity. needless to say that some of these ideas have been very cautiously applied to the topological arena where the action is named as ‘continuous group action’. motivated by the wide range of applications of continuous group-actions to various branches, a new interaction between group-action and set-theoretic topology is demonstrated in this paper. continuous action of a convergence group on a convergence received 06 july 2018 – accepted 18 december 2018 http://dx.doi.org/10.4995/agt.2019.10490 n. rath space already exists in the literature [16, 5]. in particular, there exists an oneto-one correspondence between the homeomorphic representations in h(x) and continuous group actions of convergence groups [16], which is generally unavailable in the case of a topological space x due to the lack of an admissible topology on h(x) [13]. later, this was generalized by boustique et al. [2] and applied to convergence approach spaces by colebunders et al. [5]. as discussed in section 2, there is a close link between a convergence structure (generalization of a topology) [1] and a cauchy space (generalization of a uniform space) [8], so it is reasonable that one would anticipate the corresponding continuous action of a cauchy group on a cauchy space. in this context, a natural question arises–whether the theory of group action can be linked to cauchy continuity? in this paper, this question is partially answered with an introduction of the notion of the cauchy action in section 3, where it is shown that an equivalence relation on the filters is a g-congruence. since an algebraically compatible group structure on a filter space induces a cauchy structure on the space, we need to look no further than a filter semigroup [17] having a continuous action on a filter space. in section 5, attempts have been made to investigate such an action on a filter space and a few of its modifications, while section 4 demonstrates the interaction between a cauchy action and the continuous action on the corresponding g-space. the paper concludes with an extension of a cauchy action to a larger space, namely, the completion of the original filter space in section 6. a word or two must be said about the underlying spaces on which this special type of action is defined. cauchy spaces are generalizations of uniform spaces and metric spaces. if we exclude the last of the three keller’s [8] axioms for a cauchy space, the resulting space is what we call a filter space. a filter semigroup is a filter space with a compatible semigroup operation defined on it. the category fil of filter spaces with the cauchy maps as morphisms forms a topological universe and the category chy of cauchy spaces is a bireflective subcategory of fil [15]. as pointed out by császár [4], there are three different ways of associating a convergence structure with a filter space. out of these three ways, completions corresponding to the k-convergence were discussed in [10]. it is well-known that when a filter space is compatible with a group operation, the three associated convergence structures γ, λ and k [4] coincide. so without loss of generality, császár’s k-convergence [4] is used for defining cauchy actions of filter semigroups in this paper. 2. preliminaries 2.1. filter space. for basic definitions and notations the reader is referred to [10], though some of the frequently used definitions and notations will be mentioned here. let x be a non-empty set and f(x) denote the set of all filters on x. there is a partial order relation ‘≤’, defined on f(x) : f ≤ g if and only if f ⊆ g. if b is a base for the filter f, then f is said to be generated by b, written as f = [b]. in particular, ẋ = [{x}] is the fixed point filter generated c© agt, upv, 2019 appl. gen. topol. 20, no. 1 178 cauchy action on filter spaces by the singleton set {x} and f ∩ g = [{f ∪ g | f ∈ f, g ∈ g}]. if f ∩ g 6= φ for all f ∈ f and g ∈ g, then f ∨ g is the filter [{f ∩ g | f ∈ f, g ∈ g}]. if there exists f ∈ f and g ∈ g such that f ∩ g = φ, then f ∨ g fails to exist. if a ⊆ x and f ∈ f(x) such that f ∩ a 6= φ for all f ∈ f, then fa = [{f ∩ a | f ∈ f}] is called the trace of f on a. definition 2.1. let x be a set and c ⊆f(x). the pair (x, c) is called a filter space, if the following conditions hold : (i) ẋ ∈ c, ∀x ∈ x; (ii) f ∈ c and g ≥ f imply that g ∈ c. any two filters f, g ∈ c are said to be c−linked, written f ∼c g, if there exists a finite number of filters h1, h2, . . . , hn ∈ c such that all of the filters f ∨ h1, h1 ∨ h2,. . . , hn−1 ∨ hn and hn ∨ g exist. note that the relation clinked is an equivalence relation on c. the equivalence class containing f ∈ c is denoted by [f]c (or [f], if c is clear from the context). associated with the filter space (x, c), there is a preconvergence structure pc [10] defined as f pc −→ x if and only if f ∼c ẋ. a filter space (x, c) is a c-filter space(respectively, cauchy space), if it satisfies the following additional condition (iii) (respectively, (iv)), (iii) f, x ∈ x and f ∼c ẋ imply that f ∩ ẋ ∈ c. (iv) f, g ∈ c and f ∼c g imply that f ∩ g ∈ c. associated with the c-filter space (x,c) (respectively, cauchy space), there is a convergence structure qc defined as f qc −→ x if and only if f ∩ ẋ ∈ c. note that even though qc coincides with pc when (x, c) is a c-filter space (respectively, cauchy space), for the astute reader we will use the notation pc and qc for the associated pre-convergence and convergence structures, respectively. example 2.2. let x be the set of real numbers, f = [{(0, 1/n) | n ∈ n}], where (0, 1/n) denotes an open interval in x, and l = {all complements of countable sets}. define c = {ẋ | x ∈ x} ∩ {h | h ≥ f ∩ 0̇} ∩ {k | k ≥ l ∩ 0̇}; then (x, c) is a filter space. note that (x, c) is not a c-filter space and, hence, not a cauchy space. for any a ⊆ x, we define clpc a = {x ∈ x | ∃f ∈ c such that f pc −→ x and a ∈ f} and clpc a = [{clpc a | a ∈ a}]. a filter space is said to be (i) t2 or hausdorff if and only if x = y, whenever ẋ ∼c ẏ, (ii) regular if and only if clpc f ∈ c, whenever f ∈ c, (iii) t3 if and only if it is t2 and regular, (iv) complete if and only if each f ∈ c pc-converges. if (x, c) and (x, d) are two filter spaces such that c ⊆ d, then c is said to be finer than d, written c ≥ d. let a ⊆ x and ca = {g ∈ f(a) | there exists f ∈ c such that fa exists and g ≥ fa}. then (a, ca) is a filter space, called the subspace of (x, c). a mapping f : (x, c) → (y, d) between the filter spaces is called a cauchy map c© agt, upv, 2019 appl. gen. topol. 20, no. 1 179 n. rath if and only if f ∈ c implies that f(f) ∈ d. the map f is a homeomorphism if and only if it is bijective and both f and f−1 are continuous. moreover, f is an embedding of (x, c) into (y, d) if and only if f : (x, c) → (f(x), df(x)) is a homeomorphism. a completion of a filter space (x, c) is a pair ((z, k), φ), where (z, k) is a complete filter space and the map φ : (x, c) → (z, k) is an embedding satisfying the condition clpk φ(x) = z. the t2 wyler completion of a filter space is an example of such a completion [11]; it has been applied to obtain completion and compactification of a family of extension spaces [12]. let fil (respectively, c-fil, chy ) denote the category of all filter spaces (respectively, c-filter spaces, cauchy spaces) as objects with cauchy maps as morphisms. 2.2. filter semigroup. the binary operation in a semigroup (g, ·) can be applied to subsets and filters in an obvious way. for a, b ⊆ g, ab = {xy | x ∈ a, y ∈ b}, and for any two filters f and g in f(g), fg = [{fg | f ∈ f, g ∈ g}]. in particular, for x ∈ x and f ∈f(g), we define x · f = ẋ · f is the filter generated by {x · f | f ∈ f}. the semigroup (g, ·) is abelian if and only if fg = gf, for all f, g ∈f(g). also, for fi, gi ∈f(g), (i = 1, 2), f1 ≥ g1 and f2 ≥ g2 imply that f1f2 ≥ g1g2, which shows that f(g) is a partially ordered semigroup. in particular, when (g, ·) is a group, we define f−1 = [{f −1 | f ∈ f}] for all f ∈ f(g). a triplet (g, d, ·) is called a filter semigroup, if (g, d) ∈ |fil|, (g, ·) is a semigroup with identity e and k, χ ∈ d imply that kχ and χk ∈ d. note that the binary operation b : (g, d) × (g, d) → (g, d), where b(g, h) = g · h is a cauchy map with respect to the cauchy product [9] on g × g. properties of filter semigroups and their completions were investigated in [17]. a filter semigroup (g, d, ·) is a cauchy group, if (g, ·) is a group and d is compatible with the group operations, that is, kχ−1 ∈ d, whenever k and χ are in d. let filsg (respectively, chg) denote the category of all objects as filter semigroups with identity element (respectively, cauchy groups) and morphisms as cauchy maps which are also homomorphisms. the equivalence relation ‘∼d’ preserves the binary operation on the filter semigroup (g, d, ·), that is, ki ∼d χi for i = 1, 2, implies k1k2 ∼d χ1χ2 [17, proposition 3.2]. note that if (g, ·) is a group and (g, d) ∈ |fil|, then (g, d, ·) is a cauchy group ([7], there called pre-cauchy group), since for any two filters k and χ ∈ d, k ∨ χ exists implies that k ∩ χ ≥ kχ−1. example 2.3. if (x, c) is a filter space, then the set of all cauchy maps denoted by c(x, x) is a semigroup with respect to the composition of cauchy maps and identity map. the space (c(x, x), d) is a filter semigroup, where d = {φ ∈ f(c(x, x)) | f ∈ c, pc-convergent ⇒ φ(f) pc-convergent}. a triplet (g, q, ·) is called a preconvergence group, if (g, ·) is a group, (g, q) is a preconvergence space [10], and f q −→ x, g q −→ y imply that fg−1 q −→ xy−1. a preconvergence group is a convergence group, if (g, q) is a convergence space. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 180 cauchy action on filter spaces note that when (g, q) is a convergence group, h(g) the set of all homeomorphisms on g is a complete cauchy group. 2.3. continuous group action. let x be a set and (g, ·) be a group with a binary operation ‘·’ and the identity element e. consider a function µ : x × g −→ x, denoted by µ(x, g) = xg. for any f ⊆ x, g ∈ g, f g = {fg | f ∈ f} and for any a ⊆ g, f a = ∪{f g | g ∈ a}. if f ∈ f(x) and g ∈ g, then fg = [{f g | f ∈ f}], and for any subset a ⊆ g, fa = [{f a | f ∈ f}]. if k ∈ f(g), then fk = [{f a | f ∈ f, a ∈ k}]. the mapping µ is said to be a group action of g on x (or equivalently, x is a g-space), if it satisfies the following conditions: (a1) xe = x, ∀x ∈ x, (a2) (xg)h = xgh, ∀x ∈ x and ∀ g, h ∈ g. this implies that for any k, h ∈ f(g) and f ∈ f(x), f ė = f and (fk)h = fkh. in particular, for any x ∈ x and g ∈ g, ẋġ = xġ = ẋg = ẋg is the fixed ultrafilter with base {xg}. an action of a group g is transitive on x, if for any x, y ∈ x, ∃g ∈ g such that xg = y. an action is said to be quasi-transitive with respect to an equivalence relation ∼ on x, if x, y ∈ x implies that there exists g ∈ g such that xg ∼ y. note that every transitive action is quasi-transitive with respect to ‘=’. an equivalence relation ‘ ∼’ on a g-space a is called a g-congruence [6], if for a, b ∈ a, a ∼ b =⇒ ag ∼ bg for all g ∈ g. remark 2.4. the conditions (a1) and (a2) are equivalent to (a1′) fe = f and (a2′) (fg)h = fgh, respectively, for all f ∈ f(x) and all g, h ∈ g. so every action µ on a set x induces a group action on f(x). in other words, if x is a g-space, then f(x) is a g-space on its own right. hence, in this paper, the group actions are investigated more closely with respect to filters than the individual elements in a set. for a detailed discussion on convergence structures and related notions, the reader is referred to the consolidated work of beattie and butzman [1]. definition 2.5 ([16, definition 5.1]). a group action of a convergence group (g, λ, ·) on a convergence space (x, q) is called a continuous group action, if (ca) ∀f q −→ x and ∀ k λ −→ g, fk q −→ xg, where f ∈ f(x) and k ∈ f(g). the name of this type of action suggests it all: the group action is continuous with respect to the product convergence [3] on x × g. in this case, (x, q) is called a g-space with a continuous action or more precisely, a cg-space. examples of cg-spaces are abundant. for instance, any convergence group is a cg-space with respect to the right-multiplication (a, x) → ax = ax for all a, x ∈ g. the homeomorphism group h(x) acts on the convergence space (x, q) with respect to the action defined as (x, f) → xf = f−1(x) for all x ∈ x and f ∈ h(x). a detailed discussion on continuous group action can be found in [16]. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 181 n. rath 3. cauchy action in theory, a filter-preserving map between two filter spaces (respectively, cauchy spaces) is named as a cauchy map. in alliance with this terminology, a filter-preserving action is reasonable to be named as a cauchy action. throughout this section, (g, d, ·) denotes a filter semigroup with identity e and (x, c) denotes a filter space, unless otherwise specified. definition 3.1. a group action µ : x × g → x, denoted by µ(x, g) = xg ∀x ∈ x and ∀g ∈ g is said to be a cauchy action of g on x, if (cha) ∀f ∈ c and ∀ k ∈ d, fk ∈ c. note that from condition (cha), the map µ : x × g → x on the product space x × g, is a cauchy map. so, in this case, it is natural to call (x, c) a g-space with a cauchy action, or a chg-space in short. later in section 4, it is shown that when (x, c) is a chg-space, (x, pc) is a cg-space. example 3.2. if a filter semigroup (g, d, ·) has a cauchy action on x, then it has a cauchy action on y = x × x, where the action is defined by (x1, x2) g = (x g 1, x g 2), ∀ g ∈ g and x1, x2 ∈ x. here it is assumed that y has the corresponding product structure. example 3.3. consider the right regular representation of the filter semigroup (g, d, ·) on itself, defined by xg = xg for all x, g ∈ g. since the group operation on g is cauchy compatible, f ∈ d, k ∈ d =⇒ fk = fk ∈ d. therefore, this is a cauchy action. if (g, d, ·) is a group, then it also follows that the left regular representation [6] of (g, d, ·) on itself is a cauchy action. theorem 3.4. if the filter semigroup (g, d, ·) has a cauchy action, µ on a filter space (x, c), then there exists the finest structure cf coarser than c such that µ is a cauchy action on (x, cf ). proof. let cf ={t ∈ f(x) | ∃ f ∈ c, k ∈ d such that t ≥ fk}. since ẋė = ẋ, it follows that (x, cf ) is a filterspace. if t ∈ cf , then ∃ f ∈ c and h ∈ d such that t ≥ fh. so for any k ∈ d, t k ≥ (fh)k = fhk , which implies that t k ∈ cf . since (a1) and (a2) hold, µ is, therefore, a cauchy action on (x, cf ). since for any f ∈ c, f = f ė, it follows that cf is coarser than c. let (x, e) be a filter space such that e is coarser than c, and µ is a cauchy action on (x, e). now for any t ∈ cf , t ≥ fh for some f ∈ c and h ∈ d. however, this implies that fh ∈ e, since e is coarser than c and µ is a cauchy action on (x, e). hence cf is finer than e, which proves the theorem. � the next result yields that a cauchy action of a filter semigroup (g, d, ·) on a filter space (x, c) preserves the equivalence relation ‘∼c’ on the filters in c. theorem 3.5. let the filter semigroup (g, d, ·) have a cauchy action on a filter space (x, c). if f, g ∈ c with [f]c = [g]c, and k, l ∈ d with [k]d = [l] d , then [fk]c = [g l]c. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 182 cauchy action on filter spaces proof. let f, g ∈ c such that f ∼c g and k, l ∈ d such that k ∼d l. we show that fk ∼c g l. the filter f ∼ g implies that there exists a finite number of filters h1, h2, . . . , hn in c such that f ∨ h1, h1 ∨ h2, . . . hn ∨ g exist. similarly, there exists a finite number of filters t1, t2, . . . , tr in d such that k ∨ t1, t1 ∨ t2, . . . tr ∨ l exist. assuming r ≥ n, note that fk, gl, htii for i = 1, . . . , n and g tj for j = n + 1, . . . r, are in c. also, since (f × k) ∨ (h1 × t1) exists, it follows that f k∨ ht11 exists. similarly, it can be shown that ht11 ∨ h t2 2 , · · · h tn n ∨ g tn+1 exist. continuing the same argument gtn+1 ∨ gtn+2, . . . , gtr−1 ∨ gtr, gtr ∨ gl exist, which implies that fk ∼c g l. if r < n, then fk∨ ht11 , h t1 1 ∨h t2 2 , . . . h tr r ∨h l r+1, h l r+1 ∨h l r+2, . . . , h l n ∨g l exist which also implies that fk ∼c g l. this completes the proof. � in the following, using remark 2.4 and theorem 3.5, we derive the impact of a cauchy action on the set of filters in c. proposition 3.6. let (x, c) be a chg-space. (i) every cauchy action on (x,c), induces an action on the set of filters in c. (ii) the equivalence relation c-linked on c is a g-congruence. (iii) let c be the set of all pc-convergent filters in c. if µ is a transitive cauchy action on (x, c), then µ is quasi-transitive with respect to the equivalence relation ∼c on c . proof. (i) let µ : x × g −→ x be a cauchy action on (x, c) and g ∈ g. now consider the map µ on c × g. by replacing k with ġ in (cha), fg ∈ c for all f ∈ c, since µ is a cauchy action. also, in view of remark 2.4, (a1) and (a2) hold. therefore, µ : c × g −→ c is an action of g on c, that is, c inherits an action from the cauchy action µ on (x, c). proof of (ii) follows directly from theorem 3.5. to prove (iii), let f, h ∈ c. then, ∃ x, y ∈ x such that f ∼c ẋ and h ∼c ẏ. since µ is transitive, x g = y, for some g ∈ g. by theorem 3.5, f ġ ∼c ẋg =⇒ f ġ ∼c ẏ ∼c h, therefore, µ is quasi-transitive on c with respect to ∼c. � 4. interaction between cauchy action and continuous action continuous action of a convergence group on a convergence space was discussed in [16], and later it was extended to continuous action of preconvergence semigroups on preconvergence spaces by boustique et al. [2] and to approach spaces by colebunders et al. [5]. in this section, the interaction between continuous group action and cauchy action is explored with respect to some admissible convergence (cauchy compatible convergence) structures. note that (g, pd, ·) is a preconvergence semigroup (respectively, convergence group) whenever (g, d, ·) ∈ filsg [17] (respectively, cauchy group [7]). as shown in the following theorem, it turns out that a cauchy action always induces a continuous action on the respective preconvergence (convergence) space. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 183 n. rath theorem 4.1. if the filter semigroup (g, d, ·) has a cauchy action on (i) a filter space (x, c), then the preconvergence group (g, pd, ·) acts continuously on the preconvergence space (x, pc); (ii) a c-filter space (x, c), then the preconvergence group (g, pd, ·) acts continuously on the convergence space (x, qc); (iii) a cauchy space (x, c), then the preconvergence group (g, pd, ·) acts continuously on the convergence space (x, qc). proof. (i) we only need to show that (ca) holds. let f pc −→ x and k pd −→ g. so, f ∈ c, f ∼c ẋ and k ∈ d , k ∼d ġ. since (g, d, ·) has a cauchy action on (x,c), fk ∈ c and by theorem 3.5, fk ∼c (x g)̇, which imply that fk pc −→ xg. (ii) assuming (x, c) is a c-filter space, let f qc −→ x and k pd −→ g. thus, f ∩ ẋ ∈ c, k ∈ d and k ∼d ġ. since (g, d, ·) has a cauchy action on (x, c), fk ∈ c. also, f ∈ c and f ∼c ẋ, which by theorem 3.5, imply that fk ∼c (x g)̇. since (x, c) is a c-filter space, fk ∈ c and fk ∼c (x g)̇ imply that fk ∩(xg)̇ ∈ c. in other words, fk qc −→ xg which proves (ii). proof of (iii) is similar to the proof of (ii). � it is well-known that the category fil is a topological universe. let cfil and chy denote the full subcategories of fil whose objects are c-filter spaces and cauchy spaces, respectively. let pconv (respectively, conv, lim) denote the category of preconvergence spaces (respectively, convergence spaces, limit spaces) with continuous maps as morphisms. if m : fil −→ pconv is defined by m(x, c) = (x, pc) for each object in fil and m(f) = f for each morphism, then m defines a functor on fil. note that m |cf il: cfil −→ conv and m |chy : chy −→ conv with m(x, c) = (x, qc) are also functors on the respective subcategories. let pconv1 (respectively, conv1, lim1) denote the full subcategory of pconv (respectively, conv , lim) whose objects are in the range of the functor m. the objects of pconv1, conv1 and lim1) are, respectively, filter space, c-filter space and cauchy space compatible. a characterization of objects of these subcategories can be found in [10]. a filter semigroup (respectively, cauchy group) (g, d, ·) is said to act continuously on a preconvergence (respectively, convergence) space (x, q), if (g, qd, ·) has a continuous group action [16] on (x, q). remark 4.2. furthermore, if (x, q) ∈ pconv1 (respectively, conv1) and (g, d, ·) is complete, then it has a cauchy action on (x, cq), with cq = {f ∈ f(x) | f q −→ x for some x ∈ x}. a preconvergence (respectively, convergence, limit) group (g, γ·) is said to act continuously on a filter space (respectively, c-filter space, cauchy space) (x, c), if it has a continuous group action on (x, pc) (respectively, (x, qc)). remark 4.3. (a) let (g, γ, ·) ∈ pconv g1 have a continuous action on a complete filter space (x, c). then (g, dγ, ·) has a cauchy action on (x, c). (b) let (g, γ, ·) ∈ conv g1 have a continuous action on a complete c-filter c© agt, upv, 2019 appl. gen. topol. 20, no. 1 184 cauchy action on filter spaces space (respectively, cauchy space) (x, c). then (g, dγ, ·) has a cauchy action on (x, c). proposition 4.4. (a) let (g, γ, ·) ∈ pconv g1 act continuously on (x, q) ∈ pconv1. then (g, cγ, ·) has a cauchy action on (x, cq). (b) let (g, γ, ·) ∈ conv g1 act continuously on (x, q) ∈ conv1. then (g, ·, cγ) has a cauchy action on (x, cq). proof. (a) let k ∈ cγ and f ∈ cq. then k γ −→ g for some g ∈ g and f q −→ x for some x ∈ x. since (g, γ, ·) acts continuously on (x, q), fk q −→ xg, which implies that fk ∈ cq. this completes the proof of (a). proof of (b) is similar to (a). � 5. cauchy action on modifications of a chg-space let ccp and cc denote, respectively, the c-filter modification and cauchy modification [10] of c on x. in fact, ccp (respectively, cc) denotes the finest c-filter structure (respectively, cauchy structure) on x coarser than c. the following proposition shows that these modified spaces also preserve the cauchy action of g on the original filter space (x, c). theorem 5.1. let the filter semigroup (g, d, ·) have a cauchy action on the filter space (x, c). then the following statements are true. (i) if (g, d, ·) is complete, then it has a cauchy action on (x, ccp). (ii) also, (g, d, ·) has a cauchy action on (x, cc). proof. note that (a1) and (a2) hold in both cases (i) and (ii) above, so to complete the proof, we need only show that (cha) holds. (i) note that ccp = c ∪ {f ∩ ẋ1 ∩ ẋ2 ∩ ....ẋn | f ∈ c, f ∼c ẋi, for all i = 1, . . . , n}. let k ∈ d and g ∈ ccp. if g ∈ c, then g k ∈ c ⊆ ccp. if g = f ∩ ẋ1 ∩ ẋ2 ∩ . . . ẋn, where f ∈ c and f ∼c ẋi, for all i = 1, . . . , n, then f k ∈ c. moreover, if (g, d, ·) is complete then there exists t ∈ g such that k ∼d ṫ, so it follows from theorem 3.5 that fk ∼c ẋ t i for each i. this implies g k = fk ∩ ẋt1 ∩ ẋ t 2 · · · ẋ t n ∈ ccp. (ii) recall that cc = c ∪ {g | ∃h1 ∼c h2 ∼c · · · hn ∈ c such that g ≥ ∩ni=1hi}. let k ∈ d and g ∈ cc. then there exist h1, . . . , hn ∈ c such that h1 ∼c h2 ∼c · · · hn and g ≥ ∩ n i=1hi. since (g, d, ·) has a cauchy action on (x, c), hk1 , · · · , h k 1 ∈ c and by theorem 3.5, h k 1 ∼c h k 2 ∼c · · · h k n . also, g ≥ ∩ni=1hi, implies g × k ≥ (∩ n i=1hi) × k. hence, gk = µ(g × k) ≥ µ((∩ni=1hi) × k) = ∩ n i=1µ(hi × k) = ∩ n i=1h k i . therefore, fk ∈ cc, which completes the proof of the theorem. ✷ next, we explore whether a cauchy action can be extended to the regular modification of the filter space (x, c). the regular modification of filter spaces was discussed in [10], and a regularity series for objects in larger categories such as fil and c-fil were introduced. these series are briefly described here for completeness. for a ⊆f(x) and a′ = {ẋ | x ∈ x} ∪ a, define c© agt, upv, 2019 appl. gen. topol. 20, no. 1 185 n. rath (i) pca = {f ∈ f(x) | ∃ g ∈ a ′ with f ≥ g}, (ii) q-ca = {f ∈f(x) | ∃ g ∈ a ′ and a finite number of x1, x2, · · · , xn ∈ x with g ∼a′ xi for each i = 1, · · · n, such that f ≥ g ∩ ẋ1 ∩ · · · ∩ ẋn}, (iii) cca = {f ∈ f(x) | ∃ a finite number of a ′-linked filters h1, · · · , hn such that f ≥ ∩ni=1hi}. proposition 5.2 ([10, proposition 1.11]). for (x, c) ∈ |fil|, pca (respectively, q-ca, cca) is the finest filter structure (respectively, c-filter structure, cauchy structure) on xcontaining a. a regularity series for a filter space (x, c) leads to its regular modification. let r0c = c, r1c = pca1 , where a1 = {cl n pr0c f | f ∈ c, n ∈ n} ∪ {ẋ | x ∈ x} and r2(c) = pca2, where a2 = {cl n pr1c f | f ∈ c, n ∈ n} ∪ {ẋ | x ∈ x}. in general, rβ(c) = pcaβ , where aβ = {cl n prβ−1(c) f | f ∈ c, n ∈ n} ∪ {ẋ | x ∈ x} if β is a non-limit ordinal, and rβ(c) = ∪{rα(c) | α ≤ β} if β is a limit ordinal. from the construction, it can be shown that c = r0c ≥ r1c ≥ · · · ≥ rβ(c) ≥ rβ+1(c), for all ordinal numbers β. the length lr of a regularity series r for a filter space (x, c) is the smallest ordinal number γ for which rγ(c) = rγ+1(c). let rc, called the regular modification of c, be the finest regular filter structure on x, which is coarser than c. lemma 5.3. let the filter semigroup (g, d, ·) have a cauchy action on the filter space (x, rα(c)) where α is any ordinal. then, for any f ∈f(x), we have clprα(c) (f k) ≤ (clprα(c) f) k ≤ fk. proof. the second inequality is straightforward, hence, only proof of the first inequality is given. let t ∈ clpr α(c) (fk), then clpr α(c) (f k) ⊆ t for some f ∈ f and k ∈ k. claim: (clprα(c) f) k ⊆ clprα(c) (f k). let y ∈ (clprα(c) f) k ⇒ y = xg for some g ∈ k and x ∈ clprα(c) f . so, ∃ l pr α(c) −→ x and f ∈ l, but since (g, d, ·) has a cauchy action on (x, rα(c)), by proposition 4.1, l g prα(c) −→ xg = y and f g ∈ lg. however, this implies f k ∈ lg, since f g ⊆ f k. since there exists a filter lg prα(c) −→ y and f k ∈ lg, it follows that y ∈ clpr α(c) (f k). this proves the claim. hence for any t ∈ clpr α(c) (fk), ∃ f ∈ f and k ∈ k such that (clpr α(c) f)k ⊆ t , which completes the proof of the lemma. � remark 5.4. in particular, if the filter semigroup (g, d, ·) has a cauchy action on the filter space (x, c ′ ), then for any l ∈f(x), we have clp c ′ (lk) ≤ (clp c ′ l)k ≤ lk. lemma 5.5. let the filter semigroup (g, d, ·) have a cauchy action on the filter space (x, rα(c)) where α is any ordinal. then for any n ∈ n and f ∈ c, clnprα(c) (fk) ≤ (clnprα(c) f)k ≤ fk. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 186 cauchy action on filter spaces proof. since (g, d, ·) has a cauchy action on the filter space (x, rα(c)), by lemma 5.3, clprα(c) (f k) ≤ (clprα(c) f) k. next, let cliprα(c) (fk) ≤ (cliprα(c) f)k for some n = i. this implies clprα(c) [ cliprα(c) (fk) ] ≤ clprα(c) [ (cliprα(c) f)k ] . now substituting l = cliprα(c) f and c ′ = rα(c), from remark 5.4, it follows that clp c ′ (lk) ≤ (clp c ′ l)k ≤ lk. therefore, cli+1prα(c) (fk) ≤ clprα(c) [ (cliprα(c) f)k ] ≤ [ clprα(c) ( cliprα(c) f )]k = ( cli+1prα(c) f )k . the proof is now complete by applying induction. � theorem 5.6. if (g, d, ·) ∈ |filsg| has a cauchy action on (x, c) ∈ |fil|, then it has a cauchy action on (x, rγ(c)) where γ is any ordinal. proof. it is clear that (a1) and (a2 ) hold for each (x, rα(c)), so we need to prove only (cha), that is, for any ordinal α if f ∈ rα(c) and k ∈ d, then fk ∈ rα(c). this can be proved by transfinite induction. first, let h ∈ r1(c). then ∃ l ∈ c and n ∈ n such that f ≥ clnpc l. using remark 5.4, this yields hk ≥ [clnpc l] k ≥ clnpc [l k]. since (g, d, ·) has a cauchy action on (x, c), lk ∈ c, so that hk ∈ r1(c). this implies that (g, d, ·) has a cauchy action on (x, r1(c)). next assume that for a fixed ordinal α, (g, d, ·) has a cauchy action on (x, rβ(c)) for each ordinal β < α. case 1 let α be a non-limit ordinal. so, if f ∈ rα(c), then f ≥ cl n prα−1(c) l for some l ∈ c and n ∈ n. therefore, for any k ∈ d, fk ≥ [clnprα−1(c) l]k. by assumption, since (g, d, ·) has a cauchy action on (x, rα−1(c)), from lemma 5.5, it follows that [clnprα−1(c) l]k ≥ clnprα−1(c) [lk], which implies fk ≥ clnprα−1(c) [lk]. since lk ∈ c, it follows from definition that fk ∈ rα(c). case 2 let α be a limit ordinal, then f ∈ rα(c) implies that f ∈ rβ(c) for some ordinal β < α. by assumption since (g, d, ·) has a cauchy action on (x, rβ(c)), f k ∈ rβ(c), for any k ∈ d. hence, f k ∈ rα(c). the proof of the proposition is now complete by transfinite induction. � theorem 5.6 leads to the following result. theorem 5.7. (g, d, ·) ∈ |filsg| has a cauchy action on (x, c) ∈ |fil|, then the cauchy action can be extended to the regular modification (x, rc). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 187 n. rath 6. cauchy action on completion of a chg-space in this section, it is assumed that (x, c) is a t2 filter space and (g, d, ·) is a cauchy group, that is, as noted in section 2.2, (g, ·) is a group with a cauchy-compatible group operation. if µ is a cauchy action of (g, d, ·) on (x, c), then in this case, from (cha) it follows that ∀ f ∈ c and ∀ k ∈ d, fk −1 ∈ c. note that for any k ∈ f(g), ė ≥ kk−1 = k−1k. this yields f = f ė ≥ fkk −1 for all f ∈ c. let chg denote the category of all cauchy groups with cauchy maps as morphisms. since every cauchy group is also a filter semigroup, chg is a full subcategory of filsg [17]. consequently, in view of theorem 5.1, the axioms (a1), (a2) and (cha) define a cauchy action of the cauchy group (g, d, ·) on a filter space in general. remark 6.1. in the case when (g, d, ·) is a cauchy group, k ∼d l implies k−1 ∼d l −1, therefore, if it has a cauchy action µ on a filter space (x, c), from theorem 3.5 it follows that [fk −1 ]c = [g l −1 ]c, where f, g ∈ c and k, l ∈ d with k ∼d l and f ∼d g. in particular, [h k]c = [t ]c implies [h]c = [t k −1 ]c, since h = h ė ≥ hkk −1 . lemma 6.2. let the cauchy group (g, d, ·) have a cauchy action on a filter space (x, c), and k ∈ d be pd-convergent. then f k is pc-convergent if and only if f ∈ c is pc-convergent. proof. let k ∼d ġ. if f ∼c ẋ, then f k ∼c ẋg, so that f k is convergent. next, let f ∈ c be nonconvergent. if fk is convergent, then fk ∼c ẏ for some y ∈ x. however, from remark 6.1, since k−1 ∼d ˙g−1, we get f kk −1 ∼c ˙yg −1 . this implies that f = f ė ∼c ˙yg −1 , which is a contradiction. this completes the proof of the lemma. � in particular, for any g ∈ g, f ġ is pc-convergent if and only if f ∈ c is pc-convergent. the wyler completion of a t2 filter space (x, c) was studied in [10] which is briefly summarized here. the completion w∗ = ((x∗, c∗), j), where x∗ = {[f] | f ∈ c}, j : x −→ x∗ is defined as j(x) = [ẋ] ∀ x ∈ x and c∗ ={a ∈ f(x∗) | ∃ f ∈ c such that a ≥ j(f) ∩ ˙[f]}. next, the extension problem is investigated, that is, if the cauchy group (g, d, ·) has a cauchy action on (x, c), then, whether it can be extended to its completion w∗ = ((x∗, c∗), j). define µ∗ : x∗ × g −→ x∗, by µ∗([f], g) = [f ġ], for all f ∈ c and g ∈ g. note that µ∗([ẋ], g) = [ẋg], that is, µ∗(j(x), g) = j(µ(x, g))∀x ∈ x and g ∈ g. lemma 6.3. the map µ∗ is an action of g on (x∗, c∗). proof. since µ is a cauchy action, ẋġ = ẋg ∈ c and pc-convergent, so [ẋg] ∈ x∗. also, if f ∈ c is nonconvergent, then by lemma 6.2, f ġ ∈ c is nonconvergent which implies [f ġ] ∈ x∗. by theorem 3.5, µ∗ is well-defined. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 188 cauchy action on filter spaces next, µ∗([ẋ], e) = [ẋe] = [ẋ] ∀ [ẋ] ∈ x∗, and µ∗([f], e) = [f ė] = [f] for all nonconvergent f ∈ c, therefore, (a1) holds. similarly, it is straightforward to check that (a2) holds. � the action µ∗ is said to be a wyler extension of the cauchy action µ. note that by lemma 6.3, every wyler extension of a cauchy action is an action on the wyler completion of the filter space. further, if µ∗([f], k) = [µ(f, k)] for all pc-nonconvergent filters f ∈ c and k ∈ d, then µ ∗ is called a standard wyler extension of the cauchy action µ. lemma 6.4. define a map h : (x × g) −→ (x∗ × g) by h(x, g) = (j(x), g) for all x ∈ x and g ∈ g, then the following holds: (i) µ∗ ◦ h = j ◦ µ, (ii) for any y ⊆ x and a ⊆ g, µ∗(j(y ), a) = j(µ(y, a)). proof. note that h is well-defined. moreover, by definition, µ∗ ◦ h(x, g) = µ∗(h(x, g)) = µ∗(j(x), g) = µ∗([ẋ], g) = [ẋg] by definition, and this equals [ ˙µ(x, g)] = j(µ(x, g)) = j ◦ µ(x, g), which proves (i). to prove (ii), recall that from (i), [16] µ∗(j(y ), a) = ⋃ a∈a µ∗(j(y ), a) = ⋃ a∈a µ∗(h(y, a)) = ⋃ a∈a j(µ(y, a)), but this equals j( ⋃ a∈a µ(y, a)) = j(µ(y, a)) and this completes the proof. � theorem 6.5. every cauchy action of a complete cauchy group (g, d, ·) on a t2 filter space (x, c) has a standard wyler extension which is also a cauchy action. proof. in view of lemma 6.3, we need only to prove (cha). let a ∈ c∗ and k ∈ d. so a ≥ j(f) ∩ ˙[f] for some pc-nonconvergent f ∈ c or a ≥ j(h) for some pc-convergent h ∈ c. in the latter case, µ ∗(a, k) ≥ µ∗(j(h), k) = j(µ(h, k)) = j(hk) by lemma 6.4. note that µ(h, k) = hk ∈ c, µ being a cauchy action, and also it is pc-convergent by lemma 6.2. hence, µ ∗(a, k) ∈ c∗. on the other hand, if a ≥ j(f) ⋂ ˙[f] for some pc-nonconvergent filter f ∈ c, then µ∗(a, k) ≥ µ∗(j(f) ⋂ ˙[f], k). claim: µ∗(j(f) ⋂ ˙[f] and k) ≥ µ∗(j(f), k) ⋂ µ∗( ˙[f], k). let t ∈ µ∗(j(f), k) ⋂ µ∗( ˙[f], k), then ∃ f ∈ f and k1, k2 ∈ k such that µ∗(j(f), k1) ⋃ µ∗([f], k2) ⊆ t . let l = k1 ⋂ k2, then l ∈ k and µ∗(j(f), l) ⋃ µ∗([f], l) ⊆ µ∗(j(f), k1) ⋃ µ∗([f], k2) ⊆ t. however, since µ∗(j(f), l) ⋃ µ∗([f], l) = µ∗((j(f), l) ⋃ ([f], l)) = µ∗(j(f) ⋃ [f], l), there exist f ∈ f and l ∈ k such that µ∗(j(f) ⋃ [f], l) ⊆ t . this implies that t ∈ µ∗(j(f) ⋂ ˙[f], k), which proves the claim. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 189 n. rath from lemma 6.4 it follows that µ∗(j(f), k) = j(µ(f, k)), and since µ∗ is a standard wyler extension, µ∗([f], k) = [µ(f, k)]. therefore, µ∗(j(f) ⋂ ˙[f], k) ≥ j(µ(f, k)) ⋂ ˙[µ(f, k)], which implies that µ∗(a, k) ≥ j(µ(f, k)) ⋂ ˙[µ(f, k)]. since µ is a cauchy action on (x, c), µ(f, k) ∈ c, therefore, µ∗(a, k) ∈ c∗. this completes the proof of the theorem. � 7. conclusion the notion of cauchy continuity in group-action is a natural blend of settheoretic topology and algebra, so the scope of this work is two-fold. the additional property of a group-action, that is, cauchy continuity leads to a vast area of research in many directions with numerous topological implications. in a nutshell, this is a minuscule attempt by the author to lay the foundation for a new area of 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[10] d. c. kent and n. rath, filter spaces, applied categorical structures 1 (1993), 297– 309. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 190 cauchy action on filter spaces [11] d. c. kent and n. rath, on completions of filter spaces, annals of the new york academy of sciences 767 (1995), 97–107. [12] g. minkler, j. minkler and g. richardson, extensions for filter spaces, acta. math. hungar. 82 (1999), 301–310. [13] v. pestov, topological groups: where to from here, topology proceedings 24 (1999), 421–502. [14] n. c. phillips, equivariant k-theory and freeness of group actions on c∗-algebras, lecture notes in mathematics, springer, new york, 2006. [15] g. preuss, improvement of cauchy spaces, questions answ. general topology 9 (1991), 159–166. [16] n. rath, action of convergence groups, topology proceedings 27 (2003), 601–612. [17] n. rath, completions of filter semigroup, acta. math. hungar. 107 (2005), 45–54. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 191 @ appl. gen. topol. 20, no. 2 (2019), 379-393 doi:10.4995/agt.2019.11417 c© agt, upv, 2019 ideals in b1(x) and residue class rings of b1(x) modulo an ideal a. deb ray a and atanu mondal b a department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata-700019, india (debrayatasi@gmail.com) b department of mathematics and statistics, commerce evening, st. xavier’s college, kolkata700016, india (atanu@sxccal.edu) communicated by f. lin abstract this paper explores the duality between ideals of the ring b1(x) of all real valued baire one functions on a topological space x and typical families of zero sets, called zb-filters, on x. as a natural outcome of this study, it is observed that b1(x) is a gelfand ring but nonnoetherian in general. introducing fixed and free maximal ideals in the context of b1(x), complete descriptions of the fixed maximal ideals of both b1(x) and b ∗ 1(x) are obtained. though free maximal ideals of b1(x) and those of b ∗ 1(x) do not show any relationship in general, their counterparts, i.e., the fixed maximal ideals obey natural relations. it is proved here that for a perfectly normal t1 space x, free maximal ideals of b1(x) are determined by a typical class of baire one functions. in the concluding part of this paper, we study residue class ring of b1(x) modulo an ideal, with special emphasize on real and hyper real maximal ideals of b1(x). 2010 msc: 26a21; 54c30; 13a15; 54c50. keywords: zbfilter; zb-ultrafilter; zb-ideal; fixed ideal; free ideal; residue class ring; real maximal ideal; hyper real maximal ideal. received 20 february 2019 – accepted 01 august 2019 http://dx.doi.org/10.4995/agt.2019.11417 a. deb ray and a. mondal 1. introduction in [1], we have introduced the ring of baire one functions defined on any topological space x and have denoted it by b1(x). it has been observed that b1(x) is a commutative lattice ordered ring with unity containing the ring c(x) of continuous functions as a subring. the collection of bounded baire one functions, denoted by b∗1 (x), is a commutative subring and sublattice of b1(x). certainly, b ∗ 1 (x) ∩c(x) = c∗(x). in this paper, we study the ideals, in particular, the maximal ideals of b1(x) (and also of b∗1 (x)). there is a nice interplay between the ideals of b1(x) and a typical family of zero sets (which we call a zb-filter) of the underlying space x. as a natural consequence of this duality of ideals of b1(x) and zb-filters on x, we obtain that b1(x) is gelfand and in general, b1(x) is non-noetherian. introducing the idea of fixed and free ideals in our context, we have characterized the fixed maximal ideals of b1(x) and also those of b ∗ 1 (x). we have shown that although fixed maximal ideals of the rings b1(x) and b ∗ 1 (x) obey a natural relationship, the free maximal ideals fail to do so. however, for a perfectly normal t1 space x, free maximal ideals of b1(x) are determined by a typical class of baire one functions. in the last section of this paper, we have discussed residue class ring of b1(x) modulo an ideal and introduced real and hyper-real maximal ideals in b1(x). 2. zb-filters on x and ideals in b1(x) definition 2.1. a nonempty subcollection f of z(b1(x)) ([1]) is said to be a zb-filter on x, if it satisfies the following conditions: (1) ∅ /∈ f (2) if z1,z2 ∈ f , then z1 ∩z2 ∈ f (3) if z ∈ f and z′ ∈ z(b1(x)) is such that z ⊆ z′, then z′ ∈ f . clearly, a zb-filter f on x has finite intersection property. conversely, if a subcollection b ⊆ z(b1(x)) possesses finite intersection property, then b can be extended to a zb-filter f (b) on x, given by f (b) = {z ∈ z(b1(x)): there exists a finite subfamily {b1,b2, ...,bn} of b with z ⊇ n⋂ i=1 bi}. indeed this is the smallest zb-filter on x containing b. definition 2.2. a zb-filter u on x is called a zb-ultrafilter on x, if there does not exist any zb-filter f on x, such that u $ f . example 2.3. let a0 = {z ∈ z(b1(r)) : 0 ∈ z}. then a0 is a zb-ultrafilter on r. applying zorn’s lemma one can show that, every zb-filter on x can be extended to a zb-ultrafilter. therefore, a family b of z(b1(x)) with finite intersection property can be extended to a zb-ultrafilter on x. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 380 ideals in b1(x) and residue class rings of b1(x) modulo an ideal remark 2.4. a zb-ultrafilter u on x is a subfamily of z(b1(x)) which is maximal with respect to having finite intersection property. conversely, if a family b of z(b1(x)) has finite intersection property and maximal with respect to having this property, then b is a zb-ultrafilter on x. in what follow, by an ideal i of b1(x) we always mean a proper ideal. theorem 2.5. if i is an ideal in b1(x), then zb[i] = {z(f) : f ∈ i} is a zb-filter on x. proof. since i is a proper ideal in b1(x), we claim ∅ /∈ zb[i]. if possible let ∅ ∈ zb[i]. so, ∅ = z(f), for some f ∈ i. as f ∈ i =⇒ f2 ∈ i and z(f2) = z(f) = ∅, hence 1 f2 ∈ b1(x) [1]. this is a contradiction to the fact that, i is a proper ideal and contains no unit. let z(f),z(g) ∈ zb[i], for some f,g ∈ i. our claim is z(f) ∩z(g) ∈ zb[i]. z(f) ∩z(g) = z(f2 + g2) ∈ zb[i], as i is an ideal and so, f2 + g2 ∈ i. now assume that z(f) ∈ zb[i] and z′ ∈ z(b1(x)) is such that z(f) ⊆ z′. then we can write z′ = z(h), for some h ∈ b1(x). z(f) ⊆ z′ =⇒ z(h) = z(h) ∪ z(f). so, z(h) = z(hf) ∈ zb[i], because hf ∈ i. hence, zb[i] is a zb-filter on x. � theorem 2.6. let f be a zb-filter on x. then z −1 b [f ] = {f ∈ b1(x) : z(f) ∈ f} is an ideal in b1(x). proof. we note that, ∅ /∈ f . so the constant function 1 /∈ z−1b [f ]. hence z−1b [f ] is a proper subset of b1(x). choose f,g ∈ z−1b [f ]. then z(f),z(g) ∈ f and f being a zb-filter z(f) ∩ z(g) ∈ f . now z(f) ∩ z(g) ⊆ z(f − g). hence z(f − g) ∈ f , f being a zb-filter on x. this implies f −g ∈ z−1b [f ]. for f ∈ z−1b [f ] and h ∈ b1(x), z(f.h) = z(f) ∪z(h). as z(f) ∈ f and f is a zb-filter on x, it follows that z(f.h) ∈ f . hence f.h ∈ z−1b [f ]. thus z−1b [f ] is an ideal of b1(x). � we may define a map z : b1(x) → z(b1(x)) given by f 7→ z(f). certainly, z is a surjection. in view of the above results, such z induces a map zb between the collection of all ideals of b1(x), say ib and the collection of all zb-filters on x, say fb(x), i.e., zb : ib → fb(x) given by zb(i) = zb[i], ∀ i ∈ ib. the map zb is also a surjective map because for any f ∈ fb(x), z−1b [f ] is an ideal in b1(x). we also note that zb[z −1 b [f ]] = f . so each zb-filter on x is the image of some ideal in b1(x) under the map zb : ib → fb(x). observation. the map zb : ib → fb(x) is not injective in general. because, for any ideal i in b1(x), z −1 b [zb[i]] is an ideal in b1(x), such that i ⊆ z−1b [zb[i]] and by our previous result zb[z −1 b [zb[i]]] = zb[i]. if one gets an ideal j in b1(x) such that i ⊆ j ⊆ z−1b [zb[i]], then we must have zb[i] = zb[j]. the following example shows that such an ideal is indeed possible to exist. in fact, in the following example, we get countably many ideals in in b1(r) such that the images of all the ideals are same under the map zb. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 381 a. deb ray and a. mondal example 2.7. let f0 : r → r be defined as, f0(x) =   1 q if x = p q , where p ∈ z,q ∈ n and g.c.d. (p,q) = 1 1 if x = 0 0 otherwise it is well known that f0 ∈ b1(r) (see [2]). consider the ideal i in b1(x) generated by f0, i.e., i = 〈f0〉. we claim that f 1 3 0 /∈ i. if possible, let f 1 3 0 ∈ i. then there exists g ∈ b1(r), such that f 1 3 0 = gf0. when x = p q , where p ∈ z,q ∈ n and g.c.d (p,q) = 1, g(x) = q 2 3 . we show that such g does not exist in b1(r). let α be any irrational number in r. we show that g is not continuous at α, no matter how we define g(α). suppose g(α) = β. there exists a sequence of rational numbers {pm qm }, such that {pm qm } converges to α and pm ∈ z,qm ∈ n with g.c.d (pm,qm) = 1, ∀ m ∈ n. if g is continuous at α then {g(pmqm )} converges to g(α), which implies that q 2 3 m converges to β. but qm ∈ n, so {q 2 3 m} must be eventually constant. suppose there exists n0 ∈ n such that ∀ m ≥ n0, qm is either c or −c or qm oscillates between c and −c, for some natural number c, i.e., {pm c } converges to α or −α or oscillates. in any case, {pm qm } cannot converges to α. hence we get a contradiction. so, g is not continuous at any irrational point. it is well known that, if, f ∈ b1(x,y ), where x is a baire space, y is a metric space and b1(x,y ) stands for the collection of all baire one functions from x to y then the set of points where f is continuous is dense in x [4]. therefore, the set of points of r where g is continuous is dense in r and is a subset of q. hence it is a countable dense subset of r (since r is a baire space). but using baire’s category theorem it can be shown that, there exists no function f : r → r, which is continuous precisely on a countable dense subset of r. so, we arrive at a contradiction and no such g exists. hence f 1 3 0 /∈ i. observe that, z(f0) = z(f 1 3 0 ) and i ⊆ z −1 b [zb[i]]. again, f 1 3 0 /∈ i but f 1 3 0 ∈ z −1 b zb[i], which implies i $ z −1 b [zb[i]]. by an earlier result zb[i] = zb[z −1 b [zb[i]]], proving that the map zb : ib → fb(x) is not injective when x = r. observation: 〈f0〉 ( 〈f 1 3 0 〉. analogously, it can be shown that 〈f0〉 ( 〈f 1 3 0 〉 ( 〈f 1 5 0 〉 ( ... ( 〈f 1 2m+1 0 〉 ( ... is a strictly increasing chain of proper ideals in b1(r). hence b1(r) is not a noetherian ring. theorem 2.8. if m is a maximal ideal in b1(x) then zb[m] is a zbultrafilter on x. proof. by theorem 2.5, zb[m] is a zb-filter on x. let f be a zb-filter on x such that, zb[m] ⊆ f . then m ⊆ z−1b [zb[m]] ⊆ z −1 b [f ]. z −1 b [f ] being a proper ideal and m being a maximal ideal, we have z−1b [f ] = m =⇒ c© agt, upv, 2019 appl. gen. topol. 20, no. 2 382 ideals in b1(x) and residue class rings of b1(x) modulo an ideal zb[m] = zb[z −1 b [f ]] = f . hence every zb-filter that contains zb[m] must be equal to zb[m]. this shows zb[m] is a zb-ultrafilter on x. � theorem 2.9. if u is a zb-ultrafilter on x then z −1 b [u ] is a maximal ideal in b1(x). proof. by theorem 2.6, we have z−1b [u ] is a proper ideal in b1(x). let i be a proper ideal in b1(x) such that z −1 b [u ] ⊆ i. it is enough to show that z−1b [u ] = i. now z −1 b [u ] ⊆ i =⇒ zb[z −1 b [u ]] ⊆ zb[i] =⇒ u ⊆ zb[i]. since u is a zb-ultrafilter on x, we have u = zb[i] =⇒ z−1b [u ] = z−1b [zb[i]] ⊇ i. hence z −1 b [u ] = i � remark 2.10. each zb-ultrafilter on x is the image of a maximal ideal in b1(x) under the map zb. let m(b1(x)) be the collection of all maximal ideals in b1(x) and ωb(x) be the collection of all zb-ultrafilters on x. if we restrict the map zb to the class m(b1(x)), then it is clear that the map zb ∣∣∣∣ m(b1(x)) : m(b1(x)) → ωb(x) is a surjective map. further, this restriction map is a bijection, as seen below. theorem 2.11. the map zb ∣∣∣∣ m(b1(x)) : m(b1(x)) → ωb(x) is a bijection. proof. it is enough to check that zb ∣∣∣∣ m(b1(x)) : m(b1(x)) → ωb(x) is injective. let m1 and m2 be two members in m(b1(x)) such that zb[m1] = zb[m2] =⇒ z−1b [zb[m1]] = z −1 b [zb[m1]]. but m1 ⊆ z −1 b [zb[m1]] and m2 ⊆ z−1b [zb[m2]]. by maximality of m1 and m2 we have, m1 = z −1 b [zb[m1]] = z−1b [zb[m2]] = m2. � definition 2.12. an ideal i in b1(x) is called a zb-ideal if z −1 b [zb[i]] = i, i.e., ∀ f ∈ b1(x), f ∈ i ⇐⇒ z(f) ∈ zb[i]. since zb[z −1 b [fb]] = fb, z −1 b [fb] is a zb-ideal for any zb-filter fb on x. if i is any ideal in b1(x), then, z −1 b [zb[i]] is the smallest zb-ideal containing i. it is easy to observe (1) every maximal ideal in b1(x) is a zb ideal. (2) the intersection of arbitrary family of zb-ideals in b1(x) is always a zb-ideal. (3) the map zb ∣∣∣∣ jb : jb → fb(x) is a bijection, where jb denotes the collection of all zb-filters on x. example 2.13. let i = {f ∈ b1(r) : f(1) = f(2) = 0}. then i is a zb ideal in b1(r) which is not maximal, as i ( m̂1 = {f ∈ b1(r) : f(1) = 0}. the ideal i is not a prime ideal, as the functions x− 1 and x− 2 do not belong to i, but their product belongs to i. also no proper ideal of i is prime. more c© agt, upv, 2019 appl. gen. topol. 20, no. 2 383 a. deb ray and a. mondal generally, for any subset s of r,is = {f ∈ b1(r) : f(s) = 0} is a zb-ideal in b1(r). it is well known that in a commutative ring r with unity, the intersection of all prime ideals of r containing an ideal i is called the radical of i and it is denoted by √ i. for any ideal i, the radical of i is given by {a ∈ r : an ∈ i, for some n ∈ n} ([3]) and in general i ⊆ √ i. for if i = √ i, i is called a radical ideal. theorem 2.14. a zb-ideal i in b1(x) is a radical ideal. proof. √ i = {f ∈ b1(x) : ∃ n ∈ n such that fn ∈ i} = {f ∈ b1(x) : such that z(fn) ∈ zb[i] for some n ∈ n} ( as i is a zb-ideal in b1(x) ) = {f ∈ b1(x) : z(f) ∈ zb[i] } = {f ∈ b1(x) : f ∈ i} = i. so i is a radical ideal in b1(x). � corollary 2.15. every zb-ideal i in b1(x) is the intersection of all prime ideals in b1(x) which contains i. next theorem establishes some equivalent conditions on the relationship among zb-ideals and prime ideals of b1(x). theorem 2.16. for a zb-ideal i in b1(x) the following conditions are equivalent: (1) i is a prime ideal of b1(x). (2) i contains a prime ideal of b1(x). (3) if fg = 0, then either f ∈ i or g ∈ i. (4) given f ∈ b1(x) there exists z ∈ zb[i], such that f does not change its sign on z. proof. (1) =⇒ (2) and (2) =⇒ (3) are immediate. (3) =⇒ (4): let (3) be true. choose f ∈ b1(x). then (f ∨0).(f ∧0) = 0. so by (3), f ∨ 0 ∈ i or f ∧ 0 ∈ i. hence z(f ∨ 0) ∈ zb[i] or z(f ∧ 0) ∈ zb[i]. it is clear that f ≤ 0 on z(f ∧ 0) and f ≥ 0 on z(f ∨ 0). (4) =⇒ (1): let (4) be true. to show that i is prime. let g,h ∈ b1(x) be such that gh ∈ i. by (4) there exists z ∈ zb[i], such that |g| − |h| ≥ 0 on z (say). it is clear that, z ∩z(g) ⊆ z(h). consequently z ∩z(gh) ⊆ z(h). since zb[i] is a zb-filter on x, it follows that z(h) ∈ zb[i]. so h ∈ i, since i is a zb-ideal. hence, i is prime. � theorem 2.17. in b1(x), every prime ideal p can be extended to a unique maximal ideal. proof. if possible let p be contained in two distinct maximal ideals m1 and m2. so, p ⊆ m1 ∩ m2. since maximal ideals in b1(x) are zb-ideals and intersection of any number of zb-ideals is zb-ideal, m1 ∩ m2 is a zb-ideal containing the prime ideal p . by theorem 2.16, m1∩m2 is a prime ideal. but in a commutative ring with unity, for two ideals i and j, if, i * j and j * i, c© agt, upv, 2019 appl. gen. topol. 20, no. 2 384 ideals in b1(x) and residue class rings of b1(x) modulo an ideal then i ∩j is not a prime ideal. thus m1 ∩m2 is not prime ideal and we get a contradiction. so, every prime ideal can be extended to a unique maximal ideal. � corollary 2.18. b1(x) is a gelfand ring for any topological space x. definition 2.19. a zb-filter fb on x is called a prime zb-filter on x, if, for any z1,z2 ∈ z(b1(x)) with z1 ∪z2 ∈ fb either z1 ∈ fb or z2 ∈ fb. the next two theorems are analogous to theorem 2.12 in [3] and therefore, we state them without proof. theorem 2.20. if i is a prime ideal in b1(x), then zb[i] = {z(f) : f ∈ i} is a prime zb-filter on x. theorem 2.21. if fb is a prime zb-filter on x then z −1 b [fb] = {f ∈ b1(x) : z(f) ∈ fb} is a prime ideal in b1(x). corollary 2.22. every prime zb-filter can be extended to a unique zb-ultrafilter on x. corollary 2.23. a zb-ultrafilter u on x is a prime zb-filter on x, as u = zb[m], for some maximal ideal m in b1(x). 3. fixed ideals and free ideals in b1(x) in this section, we introduce fixed and free ideals of b1(x) and b ∗ 1 (x) and completely characterize the fixed maximal ideals of b1(x) as well as those of b∗1 (x). it is observed here that a natural relationship exists between fixed maximal ideals of b∗1 (x) and the fixed maximal ideals of b1(x). however, free maximal ideals do not behave the same. in the last part of this section, we find a class of baire one functions defined on a perfectly normal t1 space x which precisely determines the fixed and free maximal ideals of the corresponding ring. definition 3.1. a proper ideal i of b1(x) (respectively, b ∗ 1 (x)) is called fixed if ⋂ z[i] 6= ∅. if i is not fixed then it is called free. for any tychonoff space x, the fixed maximal ideals of the ring b1(x) and those of b∗1 (x) are characterized. theorem 3.2. {m̂p : p ∈ x} is a complete list of fixed maximal ideals in b1(x), where m̂p = {f ∈ b1(x) : f(p) = 0}. moreover, p 6= q =⇒ m̂p 6= m̂q. proof. choose p ∈ x. the map ψp : b1(x) → r, defined by ψp(f) = f(p) is clearly a ring homomorphism. since the constant functions are in b1(x), ψp is surjective and ker ψp = {f ∈ b1(x) : ψp(f) = 0} = {f ∈ b1(x) : f(p) = 0} = m̂p (say). by first isomorphism theorem of rings we get b1(x)/m̂p is isomorphic to the field r. b1(x)/m̂p being a field we conclude that m̂p is a maximal ideal in b1(x). since p ∈ ⋂ zb[m], the ideal m̂p is a fixed ideal. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 385 a. deb ray and a. mondal for any tychonoff space x, we know that p 6= q =⇒ mp 6= mq, where mp = {f ∈ c(x) : f(p) = 0} is the fixed maximal ideal in c(x). since m̂p∩c(x) = mp, it follows that for any tychonoff space x, p 6= q =⇒ m̂p 6= m̂q. let m be any fixed maximal ideal in b1(x). there exists p ∈ x such that for all f ∈ m, f(p) = 0. therefore, m ⊆ m̂p. since m is a maximal ideal and m̂p is a proper ideal, we get m = m̂p. � theorem 3.3. {m̂∗p : p ∈ x} is a complete list of fixed maximal ideals in b∗1 (x), where m̂ ∗ p = {f ∈ b∗1 (x) : f(p) = 0}. moreover, p 6= q =⇒ m̂∗p 6= m̂∗q . proof. similar to the proof of theorem 3.2. � the following two theorems show the interrelations between fixed ideals of b1(x) and b ∗ 1 (x). theorem 3.4. if i is any fixed ideal of b1(x) then i ∩b∗1 (x) is a fixed ideal of b∗1 (x). proof. straightforward. � lemma 3.5. given any f ∈ b1(x), there exists a positive unit u of b1(x) such that uf ∈ b∗1 (x). proof. consider u = 1|f|+1 . clearly u is a positive unit in b1(x) [1] and uf ∈ b∗1 (x) as |uf| ≤ 1. � theorem 3.6. let an ideal i in b1(x) be such that i∩b∗1 (x) is a fixed ideal of b∗1 (x). then i is a fixed ideal of b1(x). proof. for each f ∈ i, there exists a positive unit uf of b1(x) such that uff ∈ i ∩ b∗1 (x). therefore, ⋂ f∈i z(f) = ⋂ f∈i z(uff) ⊇ ⋂ g∈b∗1 (x)∩i z(g) 6= ∅. hence i is fixed in b1(x). � since for any discrete space x, c(x) = b1(x) and c ∗(x) = b∗1 (x), considering the example 4.7 of [3], we can conclude the following: (1) for any maximal ideal m of b1(x), m∩b∗1 (x) need not be a maximal ideal in b∗1 (x). (2) all free maximal ideals in b∗1 (x) need not be of the form m ∩b∗1 (x), where m is a maximal ideal in b1(x). theorem 3.7. if x is a perfectly normal t1 space then for each p ∈ x, χp : x → r given by χp(x) = { 1 if x = p 0 otherwise. is a baire one function. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 386 ideals in b1(x) and residue class rings of b1(x) modulo an ideal proof. for any open set u of r, χ−1p (u) =   x if 0, 1 ∈ u x \{p} if 0 ∈ u but 1 /∈ u {p} if 0 /∈ u but 1 ∈ u ∅ if 0 /∈ u but 1 /∈ u. since x is a perfectly normal space, the open set x\{p} is a fσ set. hence in any case χp pulls back an open set to a fσ set. so χp is a baire one function [5]. � in view of theorem 3.7 we obtain the following facts about any perfectly normal t1 space. observation 3.8. if m is a maximal ideal of b1(x) where x is a perfectly normal t1 space then (1) for each p ∈ x either χp ∈ m or χp − 1 ∈ m. this follows from χp(χp − 1) = 0 ∈ m and m is prime. (2) if χp − 1 ∈ m then χq ∈ m for all q 6= p. for if χq − 1 ∈ m for some q 6= p then z(χp − 1),z(χq − 1) ∈ zb[m]. this implies ∅ = z(χp − 1) ∩ z(χq − 1) ∈ zb[m] which contradicts that zb[m] is a zb-ultrafilter. (3) m is fixed if and only if χp − 1 ∈ m for some p ∈ x. if m is fixed then m = m̂p for some p ∈ x and therefore, χp−1 ∈ m. conversely let χp − 1 ∈ m for some p ∈ x. then {p} = z(χp − 1) ∈ zb[m] shows that m is fixed. (4) m is free if and only if m contains {χp : p ∈ x}. follows from observation (3). the following theorem ensures the existence of free maximal ideals in b1(x) where x is any infinite perfectly normal t1 space. theorem 3.9. for a perfectly normal t1 space x, the following statements are equivalent: (1) x is finite. (2) every maximal ideal in b1(x) is fixed. (3) every ideal in b1(x) is fixed. proof. (1) =⇒ (2): since a finite t1 space is discrete, c(x) = b1(x) = xr. x being finite, it is compact and therefore all the maximal ideals of c(x)( = b1(x) ) are fixed. (2) =⇒ (3): proof obvious. (3) =⇒ (1): suppose x is infinite. we shall show that there exists a free (proper) ideal in b1(x). consider i = {f ∈ b1(x) : x \z(f) is finite} (here finite includes ∅). of course i 6= ∅, as 0 ∈ i. since x is infinite, 1 /∈ i and so, i is proper. we show that, i is an ideal in b1(x). let f,g ∈ i. then x \z(f) and c© agt, upv, 2019 appl. gen. topol. 20, no. 2 387 a. deb ray and a. mondal x \z(g) are both finite. now x \z(f −g) ⊆ x \z(f) ∪ x \z(g) implies that x \z(f −g) is finite. hence f −g ∈ i. similarly, x \z(f.g) ⊆ x \z(f) for any f ∈ i and g ∈ b1(x). so, x \z(f.g) is finite and hence f.g ∈ i. therefore, i is an ideal in b1(x). we claim that i is free. for any p ∈ x, consider χp : x → r given by χp(x) = { 1 if x = p 0 otherwise. using theorem 3.7, χp is a baire one function. also, x \z(χp) = x \ (x \{p}) = {p} = {p} = finite and χp(p) 6= 0. hence, i is free. � 4. residue class ring of b1(x) modulo ideals an ideal i in a partially ordered ring a is called convex if for all a,b,c ∈ a with a ≤ b ≤ c and c,a ∈ i =⇒ b ∈ i. equivalently, for all a,b ∈ a, 0 ≤ a ≤ b and b ∈ i =⇒ a ∈ i. if a is a lattice ordered ring then an ideal i of a is called absolutely convex if for all a,b ∈ a, |a| ≤ |b| and b ∈ i =⇒ a ∈ i. example 4.1. if t : b1(x) → b1(y ) is a ring homomorphism, then ker t is an absolutely convex ideal. proof. let g ∈ ker t and |f| ≤ |g|, where f ∈ b1(x). g ∈ ker t =⇒ t(g) = 0 =⇒ t(|g|) = |t(g)| = 0. since any ring homomorphism t : b1(x) → b1(y ) preserves the order, t(|f|) = 0 =⇒ |t(f)| = 0 =⇒ t(f) = 0 =⇒ f ∈ ker t. � let i be an ideal in b1(x). in what follows we shall denote any member of the quotient ring b1(x)/i by i(f) for f ∈ b1(x). i.e., i(f) = f + i. now we begin with two well known theorems. theorem 4.2 ([3]). let i be an ideal in a partially ordered ring a. the corresponding quotient ring a/i is a partially ordered ring if and only if i is convex, where the partial order is given by i(a) ≥ 0 iff ∃ x ∈ a such that x ≥ 0 and a ≡ x( mod i). theorem 4.3 ([3]). on a convex ideal i in a lattice-ordered ring a the following conditions are equivalent. (1) i is absolutely convex. (2) x ∈ i implies |x| ∈ i. (3) x,y ∈ i implies x∨y ∈ i. (4) i(a∨ b) = i(a) ∨ i(b), whence a/i is a lattice ordered ring. (5) ∀ a ∈ a, i(a) ≥ 0 iff i(a) = i(|a|). remark 4.4. for an absolutely convex ideal i of a, i(|a|) = i(a ∨ −a) = i(a) ∨ i(−a) = |i(a)|, ∀ a ∈ a. theorem 4.5. every zb-ideal in b1(x) is absolutely convex. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 388 ideals in b1(x) and residue class rings of b1(x) modulo an ideal proof. suppose i is any zb-ideal and |f| ≤ |g|, where g ∈ i and f ∈ b1(x). then z(g) ⊆ z(f). since g ∈ i, it follows that z(g) ∈ zb[i], hence z(f) ∈ zb[i]. now i being a zb-ideal, f ∈ i. � corollary 4.6. in particular every maximal ideal in b1(x) is absolutely convex. theorem 4.7. for every maximal ideal m in b1(x), the quotient ring b1(x)/m is a lattice ordered field. proof. proof is immediate. � the following theorem is a characterization of the non-negative elements in the lattice ordered ring b1(x)/i, where i is a zb-ideal. theorem 4.8. let i be a zb-ideal in b1(x). for f ∈ b1(x), i(f) ≥ 0 if and only if there exists z ∈ zb[i] such that f ≥ 0 on z. proof. let i(f) ≥ 0. by condition (5) of theorem 4.3, we write i(f) = i(|f|). so, f −|f| ∈ i =⇒ z(f −|f|) ∈ zb[i] and f ≥ 0 on z(f −|f|). conversely, let f ≥ 0 on some z ∈ zb[i]. then f = |f| on z =⇒ z ⊆ z(f −|f|) =⇒ z(f −|f|) ∈ zb[i]. i being a zb-ideal we get f −|f| ∈ i, which means i(f) = i(|f|). but |f| ≥ 0 on z gives i(|f|) ≥ 0. hence, i(f) ≥ 0. � theorem 4.9. let i be any zb-ideal and f ∈ b1(x). if there exists z ∈ zb[i] such that f(x) > 0, for all x ∈ z, then i(f) > 0. proof. by theorem 4.8, i(f) ≥ 0. but z ∩ z(f) = ∅ and z ∈ zb[i] =⇒ z(f) /∈ zb[i] =⇒ f /∈ i =⇒ i(f) 6= 0 =⇒ i(f) > 0. � the next theorem shows that the converse of the above theorem holds if the ideal is a maximal ideal in b1(x). theorem 4.10. let m be any maximal ideal in b1(x) and m(f) > 0 for some f ∈ b1(x) then there exists z ∈ zb[m] such that f > 0 on z. proof. by theorem 4.8, there exists z1 ∈ zb[m] such that f ≥ 0 on z1. now m(f) > 0 =⇒ f /∈ m which implies that there exists g ∈ m, such that z(f) ∩ z(g) = ∅. choosing z = z1 ∩ z(g), we observe z ∈ zb[m] and f(x) > 0, for all x ∈ z. � corollary 4.11. for a maximal ideal m of b1(x) and for some f ∈ b1(x), m(f) > 0 if and only if there exists z ∈ zb[m] such that f(x) > 0 on z. now we show theorem 4.10 doesn’t hold for every non-maximal ideal i. theorem 4.12. suppose i is any non-maximal zb-ideal in b1(x). there exists f ∈ b1(x) such that i(f) > 0 but f is not strictly positive on any z ∈ zb[i]. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 389 a. deb ray and a. mondal proof. since i is non-maximal, there exists a proper ideal j of b1(x) such that i $ j. choose f ∈ j r i. f2 /∈ i =⇒ i(f2) > 0. choose any z ∈ zb[i]. certainly, z ∈ zb[j] and so, z ∩z(f2) ∈ zb[j] =⇒ z ∩z(f2) 6= ∅. so f is not strictly positive on the whole of z. � in what follows, we characterize the ideals i in b1(x) for which b1(x)/i is a totally ordered ring. theorem 4.13. let i be a zb-ideal in b1(x), then the lattice ordered ring b1(x)/i is totally ordered ring if and only if i is a prime ideal. proof. b1(x)/i is a totally ordered ring if and only if for any f ∈ b1(x), i(f) ≥ 0 or i(−f) ≤ 0 if and only if for all f ∈ b1(x), there exists z ∈ zb[i] such that f does not change its sign on z if and only if i is a prime ideal (by theorem 2.16). � corollary 4.14. for every maximal ideal m in b1(x), b1(x)/m is a totally ordered field. theorem 4.15. let m be a maximal ideal in b1(x). the function φ : r → b1(x)/m (respectively, φ : r → b∗1 (x)/m) defined by φ(r) = m(r), for all r ∈ r, where r denotes the constant function with value r, is an order preserving monomorphism. proof. it is clear from the definitions of addition and multiplication of the residue class ring b1(x)/m that the function is a homomorphism. to show φ is injective. let m(r) = m(s) for some r,s ∈ r with r 6= s. then r − s ∈ m. this contradicts to the fact that m is a proper ideal. hence m(r) 6= m(s), when r 6= s. let r,s ∈ r with r > s. then r−s > 0. the function r − s is strictly positive on x. since x ∈ z(b1(x)), by theorem 4.9, m(r − s) > 0 =⇒ m(r) > m(s) =⇒ φ(r) > φ(s). thus φ is an order preserving monomorphism. � for a maximal ideal m in b1(x), the residue class field b1(x)/m (respectively b∗1 (x)/m) can be considered as an extension of the field r. definition 4.16. the maximal ideal m of b1(x) (respectively, b ∗ 1 (x)) is called real if φ(r) = b1(x)/m (respectively, φ(r) = b∗1 (x)/m) and in such case b1(x)/m is called real residue class field. if m is not real then it is called hyper-real and b1(x)/m is called hyper-real residue class field. definition 4.17 ([3]). a totally ordered field f is called archimedean if given α ∈ f, there exists n ∈ n such that n > α. if f is not archimedean then it is called non-archimedean. if f is a non-archimedean ordered field then there exists some α ∈ f such that α > n, for all n ∈ n. such an α is called an infinitely large element of f and 1 α is called infinitely small element of f which is characterized by the relation 0 < 1 α < 1 n , ∀ n ∈ n. the existence of an infinitely large (equivalently, infinitely small) element in f assures that f is non-archimedean. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 390 ideals in b1(x) and residue class rings of b1(x) modulo an ideal in the context of archimedean field, the following is an important theorem available in the literature. theorem 4.18 ([3]). a totally ordered field is archimedean iff it is isomorphic to a subfield of the ordered field r . we thus get that the real residue class field b1(x)/m is archimedean if m is a real maximal ideal of b1(x). theorem 4.19. every hyper-real residue class field b1(x)/m is non-archimedean. proof. proof follows from the fact that the identity is the only non-zero homomorphism on the ring r into itself. � corollary 4.20. a maximal ideal m of b1(x) is hyper-real if and only if there exists f ∈ b1(x) such that m(f) is an infinitely large member of b1(x)/m. theorem 4.21. each maximal ideal m in b∗1 (x) is real. proof. it is equivalent to show that b∗1 (x)/m is archimedean. choose f ∈ b∗1 (x). then |f(x)| ≤ n, for all x ∈ x and for some n ∈ n. i.e., |m(f)| = m(|f|) ≤ m(n). so there does not exist any infinitely large member in b∗1 (x)/m and hence b ∗ 1 (x)/m is archimedean. � corollary 4.22. if x is a topological space such that b1(x) = b ∗ 1 (x) then each maximal ideal in b1(x) is real. the following theorem shows how an unbounded baire one function f on x is related to an infinitely large member of the residue class field b1(x)/m. theorem 4.23. given a maximal ideal m of b1(x) and f ∈ b1(x), the following statements are equivalent: (1) |m(f)| is infinitely large member in b1(x)/m. (2) f is unbounded on each zero set in zb[m]. (3) for all n ∈ n, zn = {x ∈ x : |f(x)| ≥ n}∈ zb[m]. proof. (1) ⇐⇒ (2): |m(f)| is not infinitely large in b1(x)/m if and only if ∃ n ∈ n such that, |m(f)| = m(|f|) ≤ m(n) if and only if |f| ≤ n on some z ∈ zb[m] if and only if f is bounded on some z ∈ zb[m]. (2) =⇒ (3): choose n ∈ n, we shall show that zn ∈ zb[m]. by (2), zn intersects each member in zb[m]. now zb[m] being a zb-ultrafilter, zn ∈ zb[m]. (3) =⇒ (2): let each zn ∈ zb[m], for all n ∈ n. then for each n ∈ n, |f| ≥ n on some zero set in zb[m]. hence |m(f)| = m(|f|) ≥ m(n), for all n ∈ n. that means |m(f)| is infinitely large member in b1(x)/m. � theorem 4.24. f ∈ b1(x) is unbounded on x if and only if there exists a maximal ideal m in b1(x) such that m(f) is infinitely large in b1(x)/m. proof. let f be unbounded on x. so, each zn in theorem 4.23 is non-empty. we observe that {zn : n ∈ n} is a subcollection of z(b1(x)) having finite c© agt, upv, 2019 appl. gen. topol. 20, no. 2 391 a. deb ray and a. mondal intersection property. so there exists a zb-ultrafilter u on x such that {zn : n ∈ n} ⊆ u . therefore, there is a maximal ideal m in b1(x) for which u = zb[m] and so, zn ∈ zb[m], for all n ∈ n. by theorem 4.23 m(f) is infinitely large. converse part is a consequence of (1) =⇒ (2) of theorem 4.23. � corollary 4.25. if a completely hausdorff space x is not totally disconnected then there exists a hyper-real maximal ideal m in b1(x). proof. it is enough to prove that there exists an unbounded baire one function in b1(x). we know that if a completely hausdorff space is not totally disconnected, then there always exists an unbounded baire one function [1]. � in the next theorem we characterize the real maximal ideals of b1(x). theorem 4.26. for the maximal ideal m of b1(x) the following statements are equivalent: (1) m is a real maximal ideal. (2) zb[m] is closed under countable intersection. (3) zb[m] has countable intersection property. proof. (1) =⇒ (2): assume that (2) is false, i.e., there exists a sequence of functions {fn} in m for which ∞⋂ n=1 z(fn) /∈ zb[m]. set f(x) = ∞∑ n=1 ( |fn(x)|∧ 1 4n ) , ∀x ∈ x. it is clear that, the function f defined on x is actually a baire one function ([1]) and z(f) = ∞⋂ n=1 z(fn). thus, z(f) /∈ zb[m]. hence f /∈ m =⇒ m(f) > 0 in b1(x)/m. fix a natural number m. then z(f1) ⋂ z(f2) ⋂ z(f3)... ⋂ z(fm) = z(say) ∈ zb[m]. now for any point x ∈ z, f(x) = ∞∑ n=m+1 ( |fn(x)|∧ 14n ) ≤ ∞∑ n=m+1 1 4n = 3−14−m. this shows that, 0 < m(f) ≤ m(3−14−m), ∀ m ∈ n. hence m(f) is an infinitely small member in b1(x)/m. so, m becomes a hyper-real maximal ideal and then (1) is false. (2) =⇒ (3): trivial, as ∅ /∈ zb[m]. (3) =⇒ (1): assume that (1) is false, i.e. m is hyper-real. so, there exists f ∈ b1(x) so that |m(f)| is infinitely large in b1(x)/m. therefore for each n ∈ n, zn defined in theorem 4.23, belongs to zb[m]. since r is archimedean, we have ∞⋂ n=1 zn = ∅. thus (3) is false. � so far we have seen that for any topological space x, all fixed maximal ideals of b1(x) are real. though the converse is not assured in general, we show in the next example that in b1(r) a maximal ideal is real if and only if it is fixed. example 4.27. suppose m is any real maximal ideal in b1(r). we claim that m is fixed. the identity i : r → r belongs to b1(r). since m is a real maximal ideal, there exists a real number r such that m(i) = m(r). this c© agt, upv, 2019 appl. gen. topol. 20, no. 2 392 ideals in b1(x) and residue class rings of b1(x) modulo an ideal implies i− r ∈ m. hence z(i− r) ∈ zb[m]. but z(i− r) is a singleton. so, zb[m] is fixed, i.e., m is fixed. in view of observation 3.8(3), we conclude that a maximal ideal m in b1(r) is real if and only if there exists a unique p ∈ r such that χp − 1 ∈ m. if x is a p-space then c(x) possesses real free maximal ideals. in such case however, b1(x) = c(x). consequently, b1(x) possesses real free maximal ideals, when x is a p-space. it is still a natural question, what are the topological spaces x for which b1(x) (⊇ c(x)) contains a free real maximal ideal? references [1] a. deb ray and a. mondal, on rings of baire one functions, applied gen. topol. 20, no. 1 (2019), 237–249. [2] j. p. fenecios and e. a. cabral, on some properties of baire-1 functions, int. journal of math. analysis 7, no. 8 (2013), 393–402. [3] l. gillman and m. jerison, rings of continuous functions, new york: van nostrand reinhold co., 1960. [4] j. r. munkres, topology, second edition, pearson education, delhi, 2003. [5] l. vesely, characterization of baire-one functions between topological spaces, acta universitatis carolinae. mathematica et physica 33, no. 2 (1992), 143–156. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 393 @ appl. gen. topol. 23, no. 1 (2022), 189-199 doi:10.4995/agt.2022.14846 © agt, upv, 2022 selection principles: s-menger and s-rothberger-bounded groups muhammad asad iqbal and moiz ud din khan department of mathematics, comsats university islamabad, pakistan. (m.asadiqbal494@gmail.com, moiz@comsats.edu.pk) communicated by o. valero abstract in this paper, selection principles are defined and studied in the realm of irresolute topological groups. especially, s-menger-bounded and srothberger-bounded type covering properties are introduced and studied. 2020 msc: 22a05; 54d20; 03e75. keywords: irresolute topological group; s-menger-bounded group; srothberger-bounded group; selection principle. 1. introduction many topological properties are defined or characterized in terms of the following two classical selection principles. let p and q be sets consisting of families of subsets of an infinite set x. then: sfin(p,q) denotes the selection hypothesis: for each sequence (pn)n∈n of elements of p there is a sequence (qn)n∈n of finite sets such that for each n, qn ⊂ pn, and ⋃ n∈n qn ∈q. s1(p,q) is the selection hypothesis: for each sequence (pn)n∈n of elements of p there is a sequence (pn)n∈n such that for each n, pn ∈ pn, and {pn : n ∈ n} is an element of q (see [28]). received 25 december 2020 – accepted 17 january 2022 http://dx.doi.org/10.4995/agt.2022.14846 https://orcid.org/0000-0002-9732-1198 m. a. iqbal and m. khan let o denote the family of all open covers of a space x. the property sfin(o,o) (resp. s1(o,o)) is called the menger (resp. rothberger) covering property. for more information about selection principles theory and its relations with other fields of mathematics we refer the reader to see [16, 27, 29, 30]. a topological group is a group with a topology, such that the group operations are continuous. if the group operations are irresolute mappings instead of continuous mappings then we obtain the irresolute topological groups (itg). in the recent years many papers about selection principles and topological groups have appeared in the literature. o-bounded topological groups were introduced by o.okunev. (this notion was also given by kocinac by the same definition but under the name of menger-bounded in an unpublished work of kocinac in [15]. let us now recall [10]. definition 1.1. a topological group (g,∗,τ) is m-bounded (r-bounded) if there is for every sequence (pn)n∈n of neighborhoods (nbd) of 1g, a sequence (qn)n∈n of finite subsets of g (a sequence (pn)n∈n of elements of g) such that g = ⋃ n∈n qn ∗pn (resp. g = ⋃ n∈n pn ∗pn). itgs was first studied by khan, siab and kocinac in [13] where their properties were investigated and their differences from topological groups were established. although many papers on topological groups were published there are very few papers which deal with itgs. our main aim in considering selection principles is to link this with earlier work on irresolute topological groups. hence, section 2 contains several definitions and results which will be needed later on. in section 3 s-menger-bounded, s-rothberger-bounded and s-hurewicz-bounded type covering properties are introduced. 2. preliminaries in this section we recall some basic definitions and results that will enable the casual reader to follow the general ideas presented here. if (g,∗) is a group, and τ a topology on g, then we say that (g,∗,τ) is a topologized group with multiplication mapping µ : g ×g → g, (p,q) 7→ p∗ q and the inverse mapping i : g → g,p 7→ p−1. the identity element of g is denoted by e, or eg when it is necessary, throughout the paper x and y denote topological spaces. for a subset p of x, cl(p) and int(p) will denote the closure and interior of p . we denote f←(q) to define the preimage of a subset q ⊂ y for a mapping f : x 7→ y . the reader is refereed to [7] for undefined topological terminology and notations. a subset p of a topological space x is said to be semi-open [20] if there is an open set r in x such that r ⊂ p ⊂ cl(r). if a semi-open set p contains a point p ∈ x we say that p is a semi-open nbd of p. if x satisfies sfin(so,so) (resp. s1(so,so)), then we say that x has the s-menger (resp. s-rothberger) covering property [18, 26], where so denotes the family of all semi-open covers © agt, upv, 2022 appl. gen. topol. 23, no. 1 190 selection principles: s-menger and s-rothberger-bounded groups of x. throughout so(x) represents the collection of all semi-open sets in x. for terms not defined here we refer the reader to see [26]. definition 2.1. a mapping f : x → y between spaces x and y is called irresolute [6] (resp. pre-semi-open) if for each semi open set q ⊂ y (resp. p ⊆ x), the set f←(q) is semi open in x (resp. f(p) is semi open in y ). definition 2.2. a triplet (g,∗,τ) is called an itg [13] if for each p,q ∈g and each semi-open nbd r of p∗q−1 in g there exist semi-open nbds p of p and q of q such that p ∗q−1 ⊂ r. we note that the union of any family of semi-open sets is semi-open whereas the intersection of two semi-open sets need not be semi-open, thus the family of semi-open sets in a topological space need not be a topology. however in [13] the authors pointed out that if (g,∗,τ) is an itg such that the family so(g) is a topology on g with so(g) 6= τ, then (g,∗,so(g)) is a topological group. (see, observation [13]). lemma 2.3 ([13]). if (g,∗,τ) is an itg, then (1) p ∈ so(g) if and only if p−1 ∈ so(g). (2) if p ∈ so(g) and q ⊂g, then p ∗q and q∗p are both in so(g). lemma 2.4 ([12]). a space x is extremely disconnected if and only if the intersection of any two semi-open subsets of x is semi-open. lemma 2.5 ([23]). let p ⊂ x0 ∈ so(x), then p ∈ so(x) if and only if p ∈ so(x0). lemma 2.6 ([25]). let x0 be a subspace of x and p ∈ so(x0), then p=q∩ x0 for some q ∈ so(x). recall the following notations for collection of covers of a space x. • sω-cover : a semi open cover p of x is semi-ω-cover (sω-cover) [26] if for each finite subset q of x there exists p ∈ p such that q ⊂ p and x is not the member of p. the symbol sω denotes the family of sω-covers of x. • sγ-cover : a semi open cover p of x is a sγ-cover [26] if it is infinite and for every p ∈ x the set {p ∈p = p /∈ p} is finite. the collection of sγ-covers of x will be denoted by sγ. we are particularly interested here in the case where p and q are open covers of topological spaces or topological groups. specifically, let h and g be topological spaces with g a subspace of h. • soh : the collection of semi-open covers of h. • sohg : the collection of covers of g by sets semi-open in h. • sωh : the collection of sω-covers of h. • sωhg : the collection of sω-covers of g by sets semi-open in h. • soh(p) : let (h,∗,τ) be an itg with neutral element eh, if p is a semi-open nbd of eh, then p∗p =: {p∗ q : q ∈ p} is a semi-open nbd © agt, upv, 2022 appl. gen. topol. 23, no. 1 191 m. a. iqbal and m. khan of p. thus, {p∗p : p ∈h} is semi-open cover of h and will be denoted by soh(p). • sonbd(h) : = {p ∈ so : (∃semi-open nbd p of eh) (p = {p∗p : p ∈ h})}. • sωh(p) : for each semi open nbd p of eh, sωh(p) = {q ∗ p : q ⊂ h finite} is an sω-cover of h, where q∗p := {q ∈ q and p ∈ p} when h is not an element of this set. • sωnbd(h) : = {p ∈ sω : (∃ semi-open nbd p of eh)(u = {q∗p : q ⊂h finite})}. • sonh(p) : =for each semi open nbd p of eh, so n h(p) = {q∗p : q ⊂h and 1 ≤| q |≤ n} is an n-cover of group h. lemma 2.7. if (h,∗) is an extremely disconnected itg with the neutral element e, then for each semi-open nighborhood p of e, there exists a symmetric semi-open nbd w of e such that: q = q−1 ⊂ p. 3. s-menger-bounded, s-rothberger-bounded and s-hurewicz-bounded groups babinkostova, kocinac and scheepers in [4] investigated menger-bounded (obounded [9]) and rothberger-bounded groups in the area of selection principles. on analogues to the menger-bounded (o-bounded) and rothberger-bounded groups we examine s-menger-bounded and s-rothberger-bounded groups. we also investigate the internal characterizations of groups having these properties in all finite powers (theorem 3.8, theorem 3.9, and theorem 3.13). to introduce this new concept we use covering properties by semi open sets instead of open sets and the itg properties. semi-menger spaces have been investigated in [26]. we recall that a space x is said to have the semi-menger property (or s-menger property) if it satisfies sfin(so,so). specifically from [26, theorem 3.8] x is s-menger if and only if x satisfies sfin(sω,so). definition 3.1. an itg (g,∗,τ) is: (1) s-menger-bounded if for each sequence (pn)n∈n of semi-open nbds of the neutral element e ∈ g, there exists a sequence (qn)n∈n of finite subsets of g such that g = ⋃ n∈n qn ∗pn. (2) s-rothberger-bounded if for each sequence (pn)n∈n of semi-open nbds of the neutral element e ∈ g, there exists a sequence (pn)n∈n of elements of g such that g = ⋃ n∈n pn ∗pn. (3) s-hurewicz-bounded if there is for each sequence (pn)n∈n of semi open nbdss of neutral element e ∈ g, there exists a sequence (qn)n∈n of finite subsets of g such that each x ∈g belongs to all but finitely many qn ∗pn. let (g,∗) be a subgroup of group (h,∗). then g is s-menger-bounded if the selection principle sfin(onbd(h),ohg) holds, s-rothberger-bounded if the selection principle s1(onbd(h),ohg) holds and s-hurewicz-bounded if the selection principle s1(ωnbd(h),o gp hg) holds. © agt, upv, 2022 appl. gen. topol. 23, no. 1 192 selection principles: s-menger and s-rothberger-bounded groups in subsections 3.1, 3.2 and 3.3 we have verified various properties of each selection principle by taking three types of covering semi open, s-gamma and s-omega and using relation with one another. 3.1. s-menger-bounded groups. in this subsection we have verified some results on s-menger-bounded groups. theorem 3.2. let (h,∗,τ) be an itg and g ≤ h. then sfin(soh,sohg) implies sfin(sωnbd(h),sohg). proof. since sωnbd(h) is a subclass of soh. therefore, the proof follows immediately. � remark 3.3. converse of the theorem 3.2 is not true in general. example 3.4. real line (r, +,τ) with usual topology τ is an itg under the binary operation of addition. it is known [26], that (r, +) is a menger space but not the s-menger space. therefore, there is at least one collection of semi open covers say p1, p2, ..., pn,... for which there exists no collection vn of finite subsets of pn which satisfy ⋃ qn = h. thus sfin(soh,sohg) fails to hold. in order to show that sfin(sωnbd(h),sohg) holds, we follow as under: let (pn)n∈n be a sequence from sωnbd(h). then for each n, pn={q∗pn : q ⊂h finite} and pn ∈ so(h,eh). since q is finite set therefore each pn is finite. then for each n ∈ n we can choose rn of finite subsets of pn such that⋃ rn = g. this proves that, sfin(sωnbd(h),sohg) holds. theorem 3.5. let (h,∗,τ) be an itg and g ≤h. then the following statements are equivalent: (1) s1(sωnbd(h),sohg). (2) sfin(sonbd(h),sohg). (3) sfin(sωnbd(h),sohg). proof. (1) ⇒ (2) is straightforward. (2) ⇒ (3) : since sωnbd(h) is a subclass of sonbd(h). therefore, the proof follows immediately. (3) ⇒ (1) : let (pn)n∈n ∈ sωnbd(h). select a semi-open nbd pn of eh for each n such that pn = sωh(pn). now, apply sfin(sωnbd(h),sohg) to (pn)n∈n : for each n let a finite set rn ⊂ pn such that ⋃ n∈n rn is semi-open cover of g. then each qn is a finite subset of h. put rn = qn ∗pn. then for each n we have rn ∈pn, and, thus {rn}n∈n is a semi-open cover of g. indeed, by writing n = ⋃ n∈n yn here union is disjoint, and applying s1(sωnbd(h),sohg) to each sequence (sω(pk) : k ∈ yn) independently, one finds a sequence (qn : n ∈ n) of subsets of h also finite such that for each p ∈g there are infinitely many with p ∈ qn ∗pn. � theorem 3.6. let (h,∗,τ) be an itg and a semi open set g ≤h. then the following statements are equivalent: (1) s1(sωnbd(h),sohg). © agt, upv, 2022 appl. gen. topol. 23, no. 1 193 m. a. iqbal and m. khan (2) s1(sωnbd(g),sog). proof. (1) ⇒ (2) : let (sω(pn) : n ∈ n) ∈ sωnbd(g), where every pn is a semi-open nbd in g containing the group neutral element. then by lemma 2.6, select qn ∈ so(e,h) for each n such that pn = qn ∩g. now, select rn ∈ so(e,h) for each n such that r−1n ∗rn ⊆ qn. apply s1(sωnbd(g),sog) to (sω(rn) : n ∈ n) ∈ sω(h): we find for each n a set qn ⊂h which is finite such that g ⊆ ⋃ n∈n sn ∗rn. since rn is semi open therefore sn ∗rn is semi open. for each n, and for each m ∈ sn, choose a pm ∈g as follows: pm { ∈g∩m∗rn if nonempty, = e if otherwise. then put a finite set tn = {pm : m ∈ sn}⊂g. for each n we have tn ∗pn ∈ sω(pn) ∈ sωnbd(g). now only remaining to show that g = ⋃ n∈n tn ∗ pn. for let l ∈ tn be given. choose n so that g ∈ sn ∗ rn, and choose m ∈ sn so that l ∈ m ∗ sn. then obviously g ∩ m ∗ rn 6= φ, and so pm ∈ g is belonging to g∩m∗rn. since pm ∈ m∗rn, we have m ∈ pm ∗r−1n , and so l ∈ pm ∗r−1n ∗rn ⊆ pm ∗qn. now p−1m ∗ l ∈g∩qn = pn, and so we have that l ∈ pm ∗pn ⊂ tn ∗pn. (2) ⇒ (1) : by lemma 2.5 the proof is evident. � corollary 3.7. let (h,∗,τ) be an itg and g ≤h. if sfin(soh,sohg) holds then s1(sωnbd(g),sog). proof. by theorem 3.2, theorem 3.5 and theorem 3.6, we have sfin(soh,sohg) ⇒ sfin(sωnbd(h),sohg) ⇒ s1(sωnbd(h),sohg) ⇒ s1(sωnbd(g),sog). � theorem 3.8. let (h,∗,τ) be an extremely disconnected itg and g ≤ h. then the following statements are equivalent: (1) s1(sωnbd(h),so wgp hg ). (2) s1(sωnbd(h),sωhg). proof. (1) ⇒ (2) : let (pn : n ∈ n) ∈ sωnbd(h), and select pn ∈ so(eh,h) for each n with pn = sωh(pn). then define, qn = ⋂ j≤n pj. each qn = sωh(qn) ∈ sωnbd(h). apply s1(sωnbd(h),so wgp hg ) to (qn : n ∈ n), then there is a sequence rn ∈ qn for each n, such that {qn : n ∈ n} is a cover of g and cover is weakly groupable. suppose an increasing sequence p1 < p2 < p3 < ... < pk < ... such that there is for each finite s ⊂ g, a k with s ⊂ ⋃ pk≤j≤pk+1 rj. further, select a finite sn ⊂h with rn = sn ∗qn. since qn is semi open so is rn. for i < p1 set ti = ⋃ j 1 and suppose hn = h×h× ...×h (n copies). let pp = sop p (pp,1 ×pp,2 × ...×pp,n) for each p and define qp = ⋂ j≤nup,j, a semi open nbd of eh. for select a finite set qp ⊂ h such that |qp| ≤ p, and such that {rp ∗ qp : p < ∞} ∈ sωhg. since s1(sonbd(h),so wgp hg ) → sfin(sonbd(h),so wgp hg ) → s1(sωnbd(h),sωhg), as we saw in theorem 3.8. then for each m put gp = rp × rp ×... × rp(n copies), p < ∞. then put sp = gp∗(pp,1×pp,1× ...×pp,n). for each p we have sp ∈ sop p (up,1×up,2× © agt, upv, 2022 appl. gen. topol. 23, no. 1 196 selection principles: s-menger and s-rothberger-bounded groups ...×up,n) and we have {sp : p < ∞}∈ sωhg. by (3) ⇒ (1) of theorem 3.12, s1(sonbd(hn),so wgp hngn ) holds. (2) ⇒ (3) : this is obvious. (3) ⇒ (1) : let pn = son(pn) for each p. write n = ⋃ k<∞bk where for each k, k ≤ min(bk) and bk is infinite, and for p 6= n, bp ∩ bn = ∅. for each k : for p ∈ bk put qm = so(pkm). then (qp : p ∈ bk) is a sequence from sonbd(hk). applying (3) choose for each p ∈ bk an qp ∈ hk such that {qp ∗ pkp : p ∈ bk} is a semi open cover of gk. for each p in bk write qp = (qp(1), ...,qp(k)), and then set φ(qp) = {qp(1), ...,qp(k)}. note that for each p ∈ bk we have |φ(qp)| ≤ k ≤ p, and so φ(qp) ∗ pp is in sop(pp). set qp = φ(qp) ∗ pp for each p, a member of sop(pp) = qp. claim that {qp : p < ∞} is in sωhg. for let r ⊂ g be a finite and put k = |r|. write r = {r1, ...,rk}. suppose q = (r1, ...,rk) ∈ gk. for some p ∈ bk we have q ∈ qp ∗ pkp , and so r ⊂ φ(qp) ∗ pp = qp. now (2) ⇒ (1) of theorem 3.12 implies that s1(sonbd(h),so wgp hg ) holds. � theorem 3.14. let (h,∗,τ) be an extremely disconnected itg and a semi open set g ≤h. then the following statements are equivalent: (1) s1(sonbd(h),so wgp hg ) (2) s1(sonbd(g),so wgp g ) proof. the proof is similar to the proof of theorem 3.6. � 3.3. s-hurewicz-bounded groups. in this subsection we have verified some results on s-hurewicz-bounded groups. theorem 3.15. let (h,∗,τ) be an extremely disconnected itg and g ≤ h. then the following statements are equivalent: (1) s1(sωnbd(h), so gp hg). (2) s1(sωnbd(h),sγhg). proof. (1) ⇒ (2) : for each n ∈ n let pn ∈ sωnbd(h), and select pn ∈ so(eh,h) with pn = sω(pn). put qn = ⋂ j≤n pj. for each n put qn = sω(qn) is in sωnbd(h). then apply s1(sωnbd(h),so gp hg) to (qn : n ∈ n). choose rn ∈ qn such that {rn : n ∈ n} is a groupable semi open cover of g. choose a sequence p1 < p2 < p3 < ... < pk < ... such that x belongs to g, for all but finitely many n, x ∈ ⋃ pn≤j≤pn+1 rj. select finite set sn ⊂ h with rn = sn ∗qn. so rn is also semi open because of itg. now define, for each k, the finite set tk by, tk = { ⋃ i≤m1 si if k ≤ p1⋃ pn≤i≤pn+1 si if pn ≤ k ≤ pn+1 for each n put an = tn ∗pn an element of sω(pn). claim that {an : n ∈ n} is a sγ-cover of g. for consider g is an element of g. select m ∈ n in such a way for all n ≥ m we have g ∈ ⋃ pn pm we have g ∈ ak. it follows that {ak : k ∈ n} is sγ-cover of g. (2) ⇒ (1) : this is evident. � theorem 3.16. let (h,∗,τ) be an itg and a semi open set g ≤ h. then the following statements are equivalent: (1) s1(sωnbd(h),so gp hg). (2) s1(sωnbd(g),so gp g ). proof. the proof is similar to the proof of theorem 3.6. � corollary 3.17. if (h,∗,τ) has property s1(sωnbd(h),sγh), then for each g ≤h, s1(sωnbd(g),sγg) holds. 4. conclusions we have introduced three new types of selection principles in the realm of irresolute topological groups. we have also proved that these new notions are well defined, by means of studying their internal characterizations. kocinac introduced several types of selection principles available in the literature. for future work one can see selection principle in the domain of soft sets. references [1] a. arhangelskii and m. tkachenko, topological groups and related structures, atlantis studies in mathematics, atlantis press, 2008. [2] k. h. azar, bounded topological groups, arxiv:1003.2876. 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[30] b. tsaban, some new directions in infinite-combinatorial topology, in: j. bagaria, s. todorcevic (eds), set theory. trends in mathematics, birkhäuser basel (2006), 225–255. © agt, upv, 2022 appl. gen. topol. 23, no. 1 199 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 97-113 f-door spaces and f-submaximal spaces lobna dridi, sami lazaar and tarek turki ∗ abstract submaximal spaces and door spaces play an enigmatic role in topology. in this paper, reinforcing this role, we are concerned with reaching two main goals: the first one is to characterize topological spaces x such that f(x) is a submaximal space (resp., door space) for some covariant functor f from the category top to itself. t0, ρ and fh functors are completely studied. secondly, our interest is directed towards the characterization of maps f given by a flow (x, f) in the category set, such that (x, p(f)) is submaximal (resp., door) where p(f) is a topology on x whose closed sets are exactly the f-invariant sets. 2010 msc: 54b30, 54d10, 54f65, 46m15. keywords: categories, functors, door spaces, submaximal spaces, primal spaces. 1. introduction among the oldest separation axioms in topology there are some famous ones as t0, t1 and t2. the t0−, t1− and t2−reflections of a topological space have long been of interest to categorical topologists. the first systematic treatment of separation axioms is due to urysohn [34]. more detailed discussion was given by freudenthal and van est [35]. the first separation axiom between t0 and t1 was introduced by j.w. t. youngs [39] who encountered it in the study of locally connected spaces. in 1962, c. e. aull and w. j. thron were interested in separation axioms between t0 and t1-spaces (see [1]). ∗corresponding author: sami lazaar. 98 l. dridi, s. lazaar and t. turki in 2004, karim belaid et al in [4] gave some new separation axioms using the theory of categories and functors in the goal of studying wallman compactification. definition 1.1. let i, j be two integers such that 0 ≤ i < j ≤ 2. let us denote by ti the functor from top to top which takes each topological space x to its ti-reflection (the universal ti-space associated with x). a topological space x is said to be t(i,j)-space if ti(x) is a tj-space (thus we have three new types of separation axioms; namely, t(0,1), t(0,2) and t(1,2)). definition 1.2. let c be a category and f, g two (covariant) functors from c to itself. (1) an object x of c is said to be a t(f,g)-object if g(f(x)) is isomorphic with f(x). (2) let p be a topological property on the objects of c. an object x of c is said to be a t(f,p )-object if f(x) satisfies the property p . one year later, h-p. künzi and t. a. richmond generalized the study of [4] using the ti−ordered reflections (i ∈ {0, 1, 2}) of a partially ordered topological space (x, τ, ≤) and characterized ordered topological spaces whose t0-ordered reflection is t1-ordered (see [28]). on the other hand, recall that a subset a of a topological space x is locally closed if a is open in its closure in x, or equivalently is the intersection of an open subset and a closed subset of x. the study of locally closed sets deals to important results in topology. an investigation is made in certain aspects of the most discrete case, where every subset is locally closed. definition 1.3 ([8, definition 1.1]). a space x is called submaximal if every subset of x is locally closed. one of the reasons to consider submaximal spaces is provided by the theory of maximal spaces. (a topological space x is said to be maximal if and only if for any point x ∈ x, {x} is not open). in [5], the authors give some characterizations of submaximal spaces. theorem 1.4. [5, theorem 3.1] let x be a topological space. then the following statements are equivalent: (i) x is submaximal; (ii) s \ s is closed, for each s ⊆ x; (iii) s \ s is closed and discrete, for each s ⊆ x. furthermore, let a door space be a space in which every subset is either open or closed. clearly, every door space is submaximal. recently, some authors (see [2]) have been interested in topological spaces x that have a compactification noted k(x) which is a door space (resp., submaximal space). in this paper we mainly expose results concerning the following question: how can we characterize topological spaces x such that f(x) is a submaximal f -door spaces and f -submaximal spaces 99 space (resp., door space) where f is a covariant functor from the category top to itself? recall the standard notion of reflective subcategory a of b that is, a full subcategory such that the embedding a −→ b has a left adjoint f : b −→ a (called reflection). further, recall that for all i = 0, 1, 2, 3, 3.1 2 the subcategory topi of ti-spaces is reflective in top, the category of all topological spaces. the tychonoff (resp., functionally hausdorff) reflection of x will be denoted by ρ(x) (resp., fh(x)). specifically, we are interested in t0, ρ and fh functors. in the first section of this paper, we characterize t0-door spaces and t0submaximal spaces. the second section is devoted to the characterization of ρ-door (resp., fhdoor) spaces and ρ-submaximal (resp., fh-submaximal) spaces. in section three, given a flow (x, f) in set we characterize maps f such that (x, p(f)) is submaximal (resp., door). 2. t0-door and t0-submaximal spaces first, let us recall the t0-reflection of a topological space. let x be a topological space. we define the binary relation ∼ on x by x ∼ y if and only if {x} = {y}. then ∼ is an equivalence relation on x and the resulting quotient space t0(x) := x/ ∼ is the t0-reflection of x. the canonical surjection µx : x −→ t0(x) is a quasihomeomorphism. (a continuous map q : x −→ y is said to be a quasihomeomorphism if u 7−→ q−1(u) (resp., c 7−→ q−1(c)) defines a bijection o(y ) −→ o(x) (resp., f(y ) −→ f(x)), where o(x) (resp., f(x)) is the collection of all open subsets (resp., closed subsets) of x [21]). let us give some straightforward remarks about quasihomeomorphisms. remarks 2.1. (1) if f : x −→ y , g : y −→ z are continuous maps such that two of the three maps f, g, g ◦ f are a quasihomeomorphisms, then so is the third one. (2) let q : x −→ y be a quasihomeomorphism. then, according to [4, lemma 2.7], the following properties hold. (a) if x is a t0-space, then q is one-one. (b) if y is a td-space, then q is onto. (c) if y is a td-space and x is a t0-space, then q is a homeomorphism. (d) if x is sober and y is a t0-space, then q is a homeomorphism. now, we introduce some new notations. notations 2.2. let x be a topological space, a ∈ x and a ⊆ x. we denote by: (1) d0(a) := {x ∈ x : {x} = {a}}. (2) d0(a) := ∪[d0(a); a ∈ a]. 100 l. dridi, s. lazaar and t. turki the following remarks follow immediately. remarks 2.3. let x be a topological space and let a be a subset of x. then the following properties hold: (i) d0(a) = µ −1 x (µx(a)). (ii) d0(∅) = ∅, d0(x) = x, d0(∪[ai : i ∈ i]) = ∪[d0(ai) : i ∈ i] and d0(d0(a)) = d0(a). consequently, d0 is a kuratoweski closure. (iii) a ⊆ d0(a) ⊆ a and consequently d0(a) = a. (iv) in particular if a is open (resp., closed ), then d0(a) = a. indeed, µx is an onto quasihomeomorphism and thus by [15, lemma 1.1], it is an open map (resp., a closed map) and µ−1 x (µx(a)) = a for any open (resp., closed) subset a of x. thus, a characterization of t0-spaces and symmetric spaces in term of d0 will be useful. proposition 2.4. let x be a topological space. then the following statements are equivalent: (i) x is a t0-space; (ii) for any subset a of x, d0(a) = a; (iii) for any a ∈ x, d0(a) = {a}. proof. (i) =⇒ (ii) if x is a t0-space, then µx is an homeomorphism and thus d0(a) = µ −1 x (µx(a)) = a. (ii) =⇒ (iii) straightforward. (iii) =⇒ (i) {x} = {y} implies that x ∈ d0(y) = {y} and thus x = y. � recall that a symmetric space is a space in which for any x, y ∈ x, we have x ∈ {y} =⇒ y ∈ {x}. this notion is introduced by n. a. shanin in [30] and rediscovered by a. s. davis in [12]. it is also studied by k. belaid, o. echi and s. lazaar in [4] and called t(0,1)-spaces. proposition 2.5. let x be a topological space. then the following statements are equivalent: (i) x is a t(0,1)-space; (ii) for any a ∈ x, d0(a) = {a}. proof. (i) =⇒ (ii) let x be a t(0,1)-space and a ∈ x, then: d0(a) = {x ∈ x : {x} = {a}} = {x ∈ x : x ∈ {a}} = {a}. (ii) =⇒ (i) x ∈ {y} implies that x ∈ d0(y) and thus {x} = {y}, therefore y ∈ {x}. � proposition 2.6. let x be a topological space. then the following statements are equivalent: (i) x is an alexandroff t(0,1)-space; (ii) for any subset a of x, d0(a) = a. f -door spaces and f -submaximal spaces 101 proof. (i) =⇒ (ii) for any subset a of x, d0(a) = ∪[d0(a) : a ∈ a]. now, since x is t(0,1), we get d0(a) = ∪[{a} : a ∈ a] which is closed because x is an alexandroff space. finally, remarks 2.3 (iii) does the job. (ii) =⇒ (i) clearly, x is a t(0,1)-space. now, let {fi : i ∈ i} be a family of closed subsets of x, then: ∪[fi : i ∈ i] = ∪[fi : i ∈ i] = ∪[d0(fi) : i ∈ i] = d0(∪[fi : i ∈ i]) = ∪[fi : i ∈ i]. � example 2.7. let x be an infinite set equipped with the co-finite topology. clearly x is a t1-space and thus a t(0,1)-space. then for any a ∈ x we have d0(a) = {a}. now, let m ∈ x and a = x \ {m}. it is easily seen that d0(a) = a 6= a = x. therefore a t(0,1)-space need not to be an alexandroff t(0,1)-space. now, we introduce the following definition. definition 2.8. let x be a topological space. x is called a t0-door space if its t0-reflection is a door space. remark 2.9. since every door space is a t0-space, then every door space is a t0-door space. the converse does not hold. indeed, given a set x = {0, 1} such that {0} = {1}, we can easily see that t0(x) is a one point space and thus a door space. however {0} is not open and not closed in x. the following result gives answer about the question mentioned in the introduction concerning door spaces. theorem 2.10. let x be a topological space. then the following statements are equivalent: (i) x is a t0-door space; (ii) for any subset a of x, d0(a) is either open or closed. proof. (i) =⇒ (ii) let a be a subset of x. since x is a t0-door space, then µx(a) is either open or closed and consequently d0(a) = µ −1 x (µx(a)) is either open or closed. (ii) =⇒ (i) let µx(a) be a subset of t0(x), where a ⊆ x. then, d0(a) = µ−1 x (µx(a)) is either open or closed in x and thus µx(a) is either open or closed in t0(x). therefore, x is a t0-door space. � the following result is an immediate consequence of theorem 2.10 and proposition 2.6. corollary 2.11. every t(0,1) alexandroff space is a t0-door space. examples 2.12. (1) a t0-door space need not to be a t(0,1) space. for this let x be a sierpinski pace {0, 1}. then x is a t0-space which is not t1 and thus x is not t(0,1). 102 l. dridi, s. lazaar and t. turki clearly, t0(x) = x is a door space. (2) a t0-door space need not to be an alexandroff space. it is sufficient to choose a door space which is not alexandroff. for this, let x be an infinite set and m ∈ x. equip x with the topology whose closed sets are all subsets of x containing m or all finite subsets of x. hence, if we consider a subset a of x, then two cases arise. if m ∈ a, then a is closed. if not m ∈ x\a and consequently a is open, so x is a door space. however x\{m} = ∪[{x} : x 6= m] is a union of closed subsets of x which is not closed. now, the same study will be devoted to submaximal spaces. that’s why we introduce the following definition. definition 2.13. let x be a topological space. x is called a t0-submaximal space if its t0-reflection is a submaximal space. remark 2.14. since every submaximal space is t0, then every submaximal space is a t0-submaximal space. the converse does not hold. the example in remark 2.9 does the job. theorem 2.15. let x be a topological space. then the following statements are equivalent: (i) x is a t0-submaximal space; (ii) for any subset a of x, we have: a dense =⇒ d0(a) open; (iii) for any subset a of x, d0(a)\d0(a) is a closed set of x. proof. we need a lemma: lemma 2.16. let f : x −→ y be a quasihomeomorphism. then the following statements are equivalent: (i) f is onto; (ii) for any subset a of y, we have f−1(a) = f−1(a). proof of the lemma: (i) =⇒ (ii) clearly, f−1(a) ⊇ f−1(a) for any subset a of y . conversely, let x be in f−1(a) and u be an open subset of x containing x. since f is a quasihomeomorphism, then there exists an open subset v of y such that u = f−1(v ). now, f(x) ∈ v ∩ a and consequently v ∩ a 6= ∅. consider a point y = f(y′) in v ∩ a (since f is onto). clearly, y′ ∈ f−1(v ) ∩ f−1(a) = u ∩ f−1(a) and thus u ∩ f−1(a) is not empty which implies that x ∈ f−1(a). (ii) =⇒ (i) let y be in y , by (ii), f−1({y}) = f−1({y}). since f is a quasihomeomorphism, f−1({y}) is not empty and consequently f−1({y}) is not empty too, which implies that f is onto. proof of the theorem: (i) =⇒ (ii) let a ⊆ x such that a = x. by remarks 2.3 (iii), d0(a) = x and thus µ−1 x (µx(a)) = x. f -door spaces and f -submaximal spaces 103 now, according to lemma 2.16, µ−1 x (µx(a)) = x, which implies that µx(a) = t0(x). since t0(x) is a submaximal space, we get µx(a) open and consequently d0(a) = µ −1 x (µx(a)) is an open set of x. (i) =⇒ (iii) let a be a subset of x, then d0(a)\d0(a) = µ −1 x (µx(a))\µ −1 x (µx(a)) = µ−1x (µx(a))\µ −1 x (µx(a)) = µ−1x (µx(a)\ µx(a)). now, since x is a t0-submaximal space, then µx(a)\ µx(a) is a closed subset of t0(x) and thus µ −1 x (µx(a)\ µx(a)) is a closed subset of x. therefore, d0(a)\d0(a) is closed. (ii) =⇒ (i) let a ⊆ x such that µx(a) is a dense subset of t0(x), that is, µx(a) = t0(x), then µ −1 x (µx(a)) = x. now, according to lemma 2.16, µ−1 x (µx(a)) = x which means that d0(a) = x and thus a = x. by (ii), d0(a) is open and finally µx(a) is open. (iii) =⇒ (i) let a be a subset of x such that d0(a)\d0(a) is closed, then µ−1x (µx(a)\ µx(a)) is a closed subset of x and thus µx(a)\ µx(a) is a closed subset of t0(x). therefore, x is a t0-submaximal space. � 3. ρ-door and ρ-submaximal spaces let x be a topological space, f a subset of x and x ∈ x. x and f are said to be completely separated if there exists a continuous map f : x −→ r such that f(x) = 0 and f(f) = {1}. now, two distinct points x and y in x are called completely separated if x and {y} are completely separated. a space x is said to be completely regular if every closed subset f of x is completely separated from any point x not in f . recall that a topological space x is called a t1-space if each singleton of x is closed. a completely regular t1-space is called a tychonoff space [33]. a functionally hausdorff space is a topological space in which any two distinct points of this space are completely separated. remark here that a tychonoff space is a functionally hausdorff space and consequently a hausdorff space (t2-space). now, for a given topological space x, we define the equivalence relation ∼ on x by x ∼ y if and only if f(x) = f(y) for all f ∈ c(x) (where c(x) designates the family of all continuous maps from x to r). let us denote by x/ ∼ the set of equivalence classes and let ρx : x −→ x/ ∼ be the canonical surjection map assigning to each point of x its equivalence class. since every f in c(x) is constant on each equivalence class, we can define ρ(f) : x/ ∼−→ r by ρ(f)(ρx(x)) = f(x). one may illustrate this situation by the following commutative diagram. 104 l. dridi, s. lazaar and t. turki x ▽ ρ x // x/ ∼ ρ(f) }}z zz z zz z z r �� f ? ? ? ? ? ? ? ? now, equip x/ ∼ with the topology whose closed sets are of the form ∩[ρ(fα) −1(fα) : α ∈ i], where fα : x −→ r (resp., fα) is a continuous map (resp., a closed subset of r). it is well known that, with this topology, x/ ∼ is a tychonoff space (see for instance [36]) and its denoted by ρ(x). the construction of ρ(x) satisfies some categorical properties: for each tychonoff space y and each continuous map f : x −→ y , there exists a unique continuous map f̃ : ρ(x) −→ y such that f̃ ◦ ρx = f. we will say that ρ(x) is the ρ-reflection (or tychonoff-reflection) of x. from the above properties, it is clear that ρ is a covariant functor from the category of topological spaces top into the full subcategory tych of top whose objects are tychonoff spaces. on the other hand, the quotient space x/ ∼ which is denoted by fh(x) is a functionally hausdorff space. the construction fh(x) satisfies some categorical properties: for each functionally hausdorff space y and each continuous map f : x −→ y , there exists a unique continuous map f̃ : fh(x) −→ y such that f̃◦ρx = f. we will say that fh(x) is the functionally hausdorff-reflection of x (or the fh-reflection of x). consequently, it is clear that fh is a covariant functor from the category of topological spaces top into the full subcategory funhaus of top whose objects are functionally hausdorff spaces. notations 3.1. let x be a topological space, a ∈ x and a a subset of x. we denote by: (1) dρ(a) := ∩[f −1(f({a})) : f ∈ c(x)]. (2) dρ(a) := ∪[dρ(a) : a ∈ a]. the following results are immediate. proposition 3.2. let x be a topological space, a ∈ x and a a subset of x. then: (1) dρ(a) = ρ −1 x (ρx(a)). (2) dρ(a) is a closed subset of x. (3) a ⊆ dρ(a) ⊆ ∩[f −1(f(a)) : f ∈ c(x)]. (4) ∀f ∈ c(x), f(a) = f(dρ(a)). now, we give a characterization of functionally hausdorff spaces in term of dρ. f -door spaces and f -submaximal spaces 105 proposition 3.3. let x be a topological space. then the following statements are equivalent: (i) x is a functionally hausdorff space; (ii) for any subset a of x, dρ(a) = a; (iii) for any a ∈ x, dρ(a) = {a}. proof. (i) =⇒ (ii) if x is a functionally hausdorff space, then fh(x) = x and µx is equal to 1x and thus dρ(a) = a. (ii) =⇒ (iii) straightforward. (iii) =⇒ (i) first, remark that dρ(a) = {a} means that for any x ∈ x such that x 6= a, there exists a continuous map f : x −→ r such that f(x) 6= f(a) and thus x is a functionally hausdorff space. � using defintion 1.2 for the functor fh, one may define an other separation axiom: a space x is called t(0,fh) if its t0-reflection is functionally hausdorff. the following result characterize when ρx : x −→ fh(x) is a quasihomeomorphism. proposition 3.4. let x be a topological space. then the following statements are equivalent: (a) x is a t(0,fh)-space; (b) the canonical surjection ρx : x −→ fh(x) is a quasihomeomorphism. proof. (a) =⇒ (b) since x is a t(0,fh)-space, then t0(x) is a functionally hausdorff space and consequently there exists a unique continuous map f : fh(x) −→ t0(x) making commutative the following diagram x ▽ ρ x // fh(x) f yytt tt tt tt t t0(x) "" µx eeeeeeeee that is f ◦ ρx = µx. on the other hand, since fh(x) is a t0-space, there is a unique continuous map g : t0(x) −→ fh(x) such that g ◦ µx = ρx. now, combining the previous equalities we get easily f ◦g = 1t0(x) and g ◦f = 1fh(x) which means that f and g are homeomorphisms and finally ρx is a quasihomeomorphism. (b) =⇒ (a) consider the following commutative diagram x � ρ x // fh(x) 1 t0(x) �� µx t0(ρx ) // t0(fh(x)) = fh(x) 106 l. dridi, s. lazaar and t. turki clearly, t0(ρx) is a quasihomeomorphism between a t0-space and a functionally hausdorff space. now, since fh(x) is a td-space, then according to remarks 2.1 (2.c), t0(ρx) is a homeomorphism which implies that t0(x) is a functionally hausdorff space. � now, we give a characterization of t(0,fh)-spaces in term of dρ. proposition 3.5. let x be a topological space. then the following statements are equivalent: (i) x is a t(0,fh)-space; (ii) for any a ∈ x, dρ(a) = dρ({a}) = {a}. proof. (i) =⇒ (ii) clearly, dρ(a) is a closed subset of x containing a and thus {a} ⊂ dρ(a). conversely, let x ∈ dρ(a), then f(x) = f(a) for any f ∈ c(x). now, suppose that µx(x) 6= µx(a), then since t0(x) is a functionally hausdorff space, there exists a continuous map g from t0(x) to r satisfying g(µx(x)) 6= g(µx(a)) and thus g◦µx is a continuous map of c(x) separating x and y, contradiction. finally, µx(x) = µx(a), that is, {x} = {a} and consequently x ∈ {a}. on the other hand, since x is a t(0,fh)-space, then by proposition 3.4 ρx : x −→ fh(x) is an onto quasihomeomorphism and thus by [15, lemma 1.1] dρ({a}) = ρ −1 x (ρx({a})) = {a}. (ii) =⇒ (i) let µx(x) and µx(a) be two distinct points in t0(x), that is, {a} 6= {x}. then x /∈ {a} or a /∈ {x} which means that x /∈ dρ(a) or a /∈ dρ(x) and consequently there exists a continuous map f from x to r separating a and x. now, by universality of t0, let f̃ be the unique continuous map from t0(x) to r such that f̃ ◦ µx = f. clearly, f̃ is a continuous map separating µx(x) and µx(a). � proposition 3.6. let x be a topological space. then the following statements are equivalent: (i) x is an alexandroff t(0,fh)-space; (ii) for any subset a of x, dρ(a) = a. proof. the same proof as in proposition 2.6. � example 3.7. let r be the real line equipped with usual topology. clearly r is a t(0,f h)-space which is not an alexandroff space. hence, dρ(a) = {a} = {a} for any a ∈ r but dρ(q) = q 6= q = r, where q is the set of rational numbers. let us introduce the following definition. definition 3.8. let x be a topological space. x is called a ρ-door (resp., fh-door) space if its ρ-reflection (resp., fh-reflection) is a door space. by the same way as in theorem 2.10, the following result gives immediately. f -door spaces and f -submaximal spaces 107 theorem 3.9. let x be a topological space. then the following statements are equivalent: (i) x is an fh-door space; (ii) for any subset a of x, dρ(a) is either open or closed. before giving a characterization of ρ-door spaces, let us recall an interesting result which characterizes tychonoff spaces in term of zero-sets (resp., cozerosets). let x be a topological space and a ⊆ x. a is called a zero-set if there exists f ∈ c(x) such that a = f−1({0}). the complement of a zero-set is called a cozero-set. proposition 3.10 ([36, proposition 1.7]). a space is tychonoff if and only if the family of zero-sets of the space is a base for the closed sets (or equivalently, the family of cozero-sets of the space is a base for the open sets). let us state a useful remark. remark 3.11. a closed (resp., open) subset of ρ(x) is of the form ∩[ρ(f)−1({0}) : f ∈ h] (resp., ∪[ρ(f)−1(r⋆) : f ∈ h]) , where h is a collection of continuous maps f : x −→ r. indeed, ρ(x) is a tychonoff space, then the collection {g−1{0}; g : ρ(x) −→ r continuous} (resp., {g−1(r⋆); g : ρ(x) −→ r continuous}) is a basis of closed (resp., open) subsets of ρ(x). according to the universal property of ρ(x), each continuous map g : ρ(x) −→ r may be written as g = ρ(f) with f = g ◦ ρx. theorem 3.12. let x be a topological space. then the following statements are equivalent: (i) x is a ρ-door space; (ii) for any subset a of x, dρ(a) is either an intersection of zero-sets or a union of cozero-sets of x. proof. (i) =⇒ (ii) let a be a subset of x. since ρ(x) is a door space, then ρx(a) ⊆ ρ(x) is either open or closed. • if ρx(a) is closed, then it is equal to ⋂ [ρ(fi) −1({0}) : i ∈ i] (where {fi : i ∈ i} is a family of continuous maps from x to r) and consequently ρ −1 x (ρx(a)) = ⋂ [ρ−1 x (ρ(fi) −1({0})) : i ∈ i] = ⋂ [f−1i ({0}) : i ∈ i]. therefore, dρ(a) is an intersection of zero-sets of x. • if ρx(a) is open, then it is equal to ⋃ [ρ(gi) −1(r⋆) : i ∈ j] (where {gi : i ∈ j} is a family of continuous maps from x to r) and thus ρ−1x (ρx(a)) =⋃ [g−1i (r ⋆) : i ∈ j]. therefore, dρ(a) is a union of cozero-sets of x. (ii) =⇒ (i) conversely, let ρx(a) ⊆ ρ(x), where a is a subset of x. • if dρ(a) = ρ −1 x (ρx(a)) is an intersection of zero-sets of x, then let {fi : i ∈ i} be a family of continuous maps from x to r such that ρ−1x (ρx(a)) =⋂ [ρ−1 x (ρ(fi) −1({0})) : i ∈ i]. now, since ρx is onto, then: 108 l. dridi, s. lazaar and t. turki ρx(a) = ρx( ⋂ [f−1i ({0}) : i ∈ i]) = ρx( ⋂ [ρ−1x (ρ(fi) −1({0}) : i ∈ i]) = ρx(ρ −1 x ( ⋂ [(ρ(fi) −1({0}) : i ∈ i])) = ⋂ [ρ(fi) −1({0}) : i ∈ i] consequently, ρx(a) is a closed subset of ρx. • if dρ(a) = ρ −1 x (ρx(a)) is a union of cozero-sets of x, then let {gi : i ∈ j} be a family of continuous maps from x to r such that ρ−1 x (ρx(a)) =⋃ [ρ−1x (ρ(gi) −1(r⋆)) : i ∈ j]. it is clearly seen, by the same way as in the first case, that ρx(a) =⋃ [ρ(gi) −1(r⋆) : i ∈ j] and thus ρx(a) is an open subset of ρ(x). finally, ρx(a) is either open or closed for every subset a of x which means that ρ(x) is a door space. � definition 3.13. let x be a topological space. x is said to be a ρ-submaximal (resp., fh-submaximal) space if its ρ-reflection (resp., fh-reflection) is submaximal. now, in order to characterize ρ-submaximal spaces and fh-submaximal spaces, we introduce the following definitions: definition 3.14. let x be a topological space. (1) a subset v of x is called a functionally open subset of x (for short f-open) if and only if dρ(v ) is open in x. (2) a subset v of x is called a functionally dense subset of x (for short f-dense) if and only if for any f-open subset w of x, dρ(v ) meets dρ(w). (3) a nonempty subset v of x is said to be a ρ-dense if g(v ) 6= {0} for every nonzero continuous map g from x to r. remarks 3.15. (1) v is an f-open subset of x if and only if ρx(v ) is an open subset of fh(x). (2) clearly, a dense subset is a ρ-dense subset. the converse does not hold. indeed, let x := {0, 1} be the sierpinski space, then it is easily seen that ρ(x) is a one point space, that is, any continuous map f from x to r is constant. hence, any nonempty subset a of x is ρ-dense. now, to conclude choose a = {1}. (3) every f-dense subset of x is ρ-dense. indeed, let u be an open subset of ρ (x) and a an f-dense subset of x, that is ρx(a) dense in fh(x). since u = ∪[ρ (f) −1 (r∗) : f ∈ h], where h is a collection of continuous maps f : x −→ r and ρ−1x (u) = ρ −1 x (∪[ρ (f) −1 (r∗) : f ∈ h]) = ∪[ρ−1x ( ρ (f) −1 (r∗) ) : f ∈ h] = ∪[f−1 (r∗) : f ∈ h], then u is an open subset of fh(x). thus ρx(a) ∩ u 6= ∅. therefore a is ρ-dense. (4) an f-dense subset of x is not necessary dense. indeed, let x = {0, 1} be the sierpinski space. clearly {1} is f-dense but not dense. f -door spaces and f -submaximal spaces 109 proposition 3.16. let x be a topological space and a a subset of x. then the following statements are equivalent: (i) a is a ρ-dense subset of x; (ii) ρx(a) is a dense subset of ρ(x). proof. (i) =⇒ (ii) let a be a ρ-dense subset of x. then for any nonzero continuous map g from x to r, we have a∩g−1(r⋆) 6= ∅. so, let a be in a such that g(a) 6= 0, then ρ(g)(ρx(a)) = g(a) 6= 0 and thus ρx(a)∩ρ(g) −1(r⋆) 6= ∅. now, by remark 3.11, ρx(a) meets every nonempty open subset of ρ(x). (ii) =⇒ (i) let a be a subset of x. since ρx(a) is dense, then for any nonzero continuous map g from x to r, we have ρx(a) ∩ ρ(g) −1(r⋆) 6= ∅ which means that there exists a ∈ a satisfying ρ(g)(ρx(a)) 6= 0 or equivalently g(a) 6= 0. therefore, a is a ρ-dense subset of x. � we are now in a position to give the characterization of ρ-submaximal spaces. theorem 3.17. let x be a topological space. then the following statements are equivalent: (i) x is a ρ-submaximal space; (ii) for any subset a of x, we have: a ρ-dense =⇒ dρ(a) is a union of cozero-sets of x. proof. (i) =⇒ (ii) let a be a ρ-dense subset of x. according to proposition 3.16, ρx(a) is a dense subset of ρ(x). since x is a ρ-submaximal space, then ρx(a) is an open subset of ρ(x) and thus ρx(a) = ⋃ [ρ(f)−1(r⋆) : f ∈ h] (where h is a subfamily of c(x)). so that ρ−1 x (ρx(a)) = ⋃ [ρ−1 x (ρ(f)−1(r⋆)) : f ∈ h]. therefore, dρ(a) = ⋃ [f−1(r⋆) : f ∈ h] is a union of cozero-sets of x. (ii) =⇒ (i) conversely, let a be a subset of x such that ρx(a) = ρ(x). then, by proposition 3.16, a is a ρ-dense subset of x and consequently dρ(a) is a union of cozero-sets of x. hence there exists a subfamily {fi : i ∈ i} of c(x) satisfying ρ−1 x (ρx(a)) = ⋃ [f−1i (r ⋆) : i ∈ i]. then: ρx(a) = ρx( ⋃ [f−1i (r ⋆) : i ∈ i]) = ρx( ⋃ [ρ−1x (ρ(fi) −1(r⋆)) : i ∈ i]) = ρx(ρ −1 x ( ⋃ [ρ(fi) −1(r⋆) : i ∈ i])) = ⋃ [ρ(fi) −1(r⋆) : i ∈ i] finally, ρx(a) is an open subset of ρ(x). � theorem 3.18. let x be a topological space. then the following statements are equivalent: (i) x is fh-submaximal; (ii) for any f-dense subset a of x, dρ(a) is open. proof. (i) =⇒ (ii) let a be an f-dense subset of x. first, let us show that ρx(a) is a dense subset of fh(x). indeed, consider ρx(u) an open subset of fh(x). then 110 l. dridi, s. lazaar and t. turki dρ(u) = ρ −1 x (ρx(u)) is open in x and consequently u is an f-open subset of x. since a is f-dense, dρ(u) ∩ dρ(a) 6= ∅ and thus ρx(u) ∩ ρx(a) 6= ∅. now, since x is fh-submaximal, then ρx(a) is open in fh(x) and dρ(a) = ρ −1 x (ρx(a)) is open in x. (ii) =⇒ (i) let ρx(a) be a dense subset of fh(x), where a is a subset of x, and v an f-open subset of x, that is dρ(v ) is open in x and thus ρx(v ) is open in fh(x). since ρx(a) is dense in fh(x), then ρx(v ) ∩ ρx(a) 6= ∅. thus ρ −1 x (ρx(v )) ∩ ρ −1 x (ρx(a)) 6= ∅. hence dρ(v ) ∩ dρ(a) 6= ∅. therefore, a is an f-dense subset of x. now, by (ii), dρ(a) is open in x and consequently ρx(a) is open in fh(x). therefore, fh(x) is a submaximal space. � 4. alexandroff topology according to kennisson, a flow in a category c is a couple (x, f), where x is an object of c and f : x −→ x is a morphism, called the iterator (see [25] and [26]). now, let (x, f) be a flow in the category set. in [16], the author define the topology p(f) on x with closed sets are exactly those a which are f-invariant (i.e., f(a) ⊆ a). it is clearly seen that for any subset a of x, the topological closure a is exactly ∪[fn(a) : n ∈ n]. in particular for any point x ∈ x, {x} = of (x) = {f n(x) : n ∈ n} called the orbit of x by f. one can see easily that the family {vf (x) : x ∈ x} is a basis of open sets of p(f), where vf (x) := {y ∈ x : f n(y) = x, for some n in n}. clearly, p(f) is an alexandroff topology on x. characterizing maps f such that (x, p(f)) is submaximal, which is one of our main goals, is given by the following result. proposition 4.1. let (x, f) be a flow in set. then the following statements are equivalent: (i) (x, p(f)) is a submaximal space; (ii) f2 = f. proof. (i) =⇒ (ii) let x ∈ x. two cases arise. • if f(x) = x, then f2(x) = f(x). • if f(x) 6= x, then x ∈ {f(x)}c and thus f(x) ∈ {x} ⊆ {f(x)}c, consequently {f(x)}c = x. now, since (x, p(f)) is a submaximal space, then {f(x)}c is open, equivalently {f(x)} is closed and finally f2(x) = f(x). (ii) =⇒ (i) let a be a dense subset of x. since f2 = f, then any point in f(x) is closed and thus, since the topology is principal, every subset of f(x) is closed. in particular every subset of f(a) is closed. on the other hand a = a ∪ f(a) = x, then ac is closed (ac ⊆ f(a)), so a is open. � f -door spaces and f -submaximal spaces 111 example 4.2. consider the map f: n −→ n n 7−→ n + 1 where n is the set of all natural numbers including 0. it is clearly seen that f2 6= f. now, consider the topological space (n, p(f)) and set a = 2n. a is a dense subset of (n, p(f)) which is not open since for each n ∈ n \ {0}, we have 2n − 1 ∈ vf (2n). before giving a characterization of maps f such that (x, p(f)) is door, let us recall that a point x ∈ x is called a fixed point if f(x) = x and we denote by fix(f) the family of all fixed points of x. proposition 4.3. let (x, f) be a flow in set. then the following statements are equivalent: (i) (x, p(f)) is a door space; (ii) |f(fix(f)c)| ≤ 1. proof. (i) =⇒ (ii) suppose that |f(fix(f)c)| ≥ 2. then there exist two distinct points x and y in fix(f)c such that f(x) 6= f(y). set a = {x, f(y)}. clearly a is neither closed (f(x) /∈ a) nor open (f(y) /∈ ac). (ii) =⇒ (i) • if |f(fix(f)c)| = 0, then fix(f) = x and thus (x, p(f)) is the discrete topology which is door. • if |f(fix(f)c)| = 1. let x0 such that f(fix(f) c) = {x0} and let us show that for any point x in x distinct from x0, {x} is open. indeed, set a = f−1({x})\{x}. assume that a 6= ∅ and let y ∈ x such that y 6= x and f(y) = x. then y ∈ fix(f)c and thus f(y) = x0, contradiction. hence a is empty and consequently for any point x ∈ x distinct from x0, vf (x) = {x} is open. now, consider a subset c of x, then it is open if x0 /∈ c and it is closed when it contains x0. finally (x, p(f)) is a door space. � example 4.4. let z be the set of all integers and f: z −→ z n 7−→ |n| where |n| denotes the absolute value of the integer n. then, we have fix(f) = n and thus |f(fix(f)c)| = |n \ {0}| > 1. now, consider the topological space (z, p(f)) and set a = {−1, 2} . clearly a is neither closed (a = {−1, 1, 2}) nor open (vf (2) = {2, −2}). acknowledgements. the authors gratefully acknowledge helpful corrections, comments, and suggestions of the referee which reinforce the presentation of our paper. 112 l. dridi, s. lazaar and t. turki references [1] c. e. aull and w. j. thron, separation axioms between t0 and t1, indag. math. 24 (1962), 26–37. 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(received may 2012 – accepted january 2013) lobna dridi (lobna dridi 2006@yahoo.fr) department of mathematics, tunis preparatory engineering institute. university of tunis. 1089 tunis, tunisia. sami lazaar (salazaar72@yahoo.fr) department of mathematics, faculty of sciences of tunis. university tunis-el manar. “campus universitaire” 2092 tunis, tunisia. tarek turki (tarek turki@hotmail.com) department of mathematics, faculty of sciences of tunis. university tunis-el manar. “campus universitaire” 2092 tunis, tunisia. f-door spaces and f-submaximal spaces. by l. dridi, s. lazaar and t. turki @ appl. gen. topol. 20, no. 1 (2019), 231-236doi:10.4995/agt.2019.10731 c© agt, upv, 2019 when is the super socle of c(x) prime? s. ghasemzadeh and m. namdari department of mathematics, shahid chamran university of ahvaz, ahvaz, iran (s.gh8081@gmail.com, namdari@ipm.ir) dedicated to professor o.a.s. karamzadeh who is not only a “role model” for us, he is also so for many other mathematics teachers and students, alike, in this country communicated by h.-p. a. künzi abstract let scf (x) denote the ideal of c(x) consisting of functions which are zero everywhere except on a countable number of points of x. it is generalization of the socle of c(x) denoted by cf (x). using this concept we extend some of the basic results concerning cf (x) to scf (x). in particular, we characterize the spaces x such that scf (x) is a prime ideal in c(x) (note, cf (x) is never a prime ideal in c(x)). this may be considered as an advantage of scf (x) over cf (x). we are also interested in characterizing topological spaces x such that cc(x) = r + scf (x), where cc(x) denotes the subring of c(x) consisting of functions with countable image. 2010 msc: primary 54c30; 54c40; 54g12; secondary 13c11; 16h20. keywords: super socle of c(x); countably isolated point; countably discrete space; cocountably-disconnected space; one-point lindeĺ’offication. 1. introduction we refer the reader to [7] and [12] for necessary background concerning x, and c(x), the ring of all real-valued continuous functions on a space x. all topological spaces x in this note are infinite completely regular hausdorff, unless otherwise mentioned. cf (x), the socle of c(x), is the sum of all minimal ideals of c(x) which is also the intersection of all essential ideals in c(x). an received 19 september 2018 – accepted 05 november 2018 http://dx.doi.org/10.4995/agt.2019.10731 s. ghasemzadeh and m. namdari ideal in a commutative ring is essential if it intersects every nonzero ideal of the ring nontrivially, see [14], where the socle of c(x) is topologically characterized. it is folklore that one of the main objectives of working in the context of c(x) is to characterize topological properties of a given space x in terms of a suitable algebraic properties of c(x). it turns out, in this regard, the ideal cf (x), plays an appropriate role in the literature, see for examples [2], [3], [5], [6], [8], [9] [14], [15], and [17]. motivated by this role of cf (x), the concept of the super socle, which contains cf (x), is introduced in [11], see also [16]. we are going to extend some of the basic results of the socle of c(x) (i.e., cf (x)) to the super socle of c(x) (i.e., scf (x)). an outline of this article is as follows: in section 2, the concept of the super socle and some preliminary results concerning this ideal, which are frequently used in the subsequent sections, are given. in the next section, we are going to investigate the primeness of the super socle in c(x). this may be considered as an advantage of scf (x) over cf (x) (note, cf (x) is never a prime ideal in c(x)). we also characterize topological spaces x such that cc(x) = r + scf (x), where cc(x) denotes the subring of c(x) consisting of functions with countable image, see [9, proposition 6.6], [10]. 2. preliminaries we begin with the definition of the super socle of c(x) which is motivated by [11, proposition 3.3]. definition 2.1. the set s = {f ∈ c(x) : x\z(f) is countable} is called the super socle of c(x) and it is denoted by scf (x). one can easily see that scf (x) is a z-ideal in c(x). clearly cf (x) ⊆ scf (x), by [11, proposition 3.3]. it is trivial to see that a point in a space x is isolated if and only if it has a finite neighborhood. motivated by this, the next two definitions are natural and are also needed. definition 2.2. an element p ∈ x is called a countably isolated point if p has a countable neighborhood. the set of countably isolated points of x is denoted by ic(x). definition 2.3. if every point of x is countably isolated, then x is called a countably discrete space. thus, x is a countably discrete space in case ic(x) = x. we recite the following results which are in [11]. proposition 2.4 ([11], proposition 2.4). for any space x, ic(x) = ⋃ {coz(f) : f ∈ scf (x)}. corollary 2.5 ([11], corollary 2.5). for any space x, we have the following: (a) scf (x) is not a zero ideal if and only if x has a countably isolated point. (b) scf (x) is a free ideal if and only if x is countably discrete. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 232 when is the super socle of c(x) prime? we remind the reader that an ideal i is said to be regular if for every a ∈ i there exists b ∈ i such that a = aba. theorem 2.6 ([11], theorem 2.10). the following are equivalent for an uncountable space x. (1) x is the one-point lindelöffication of some uncountable discrete space. (2) scf (x) is a regular ideal, and scf (x) = ox for some x ∈ x. 3. the primeness of scf (x) in c(x) we recall that cf (x) is never a prime ideal in c(x), see [1], [8, proposition 1.2]. our aim in this section is to investigate the primeness of the super socle in c(x). we begin with an example to show that scf (x) can be a prime ideal (even a maximal ideal). this may be considered as an advantage of scf (x) over cf (x). example 3.1. let x = y ∪{x0} be the one-point lindelöffication of a countably discrete space y , then we claim that c(x) = r+scf (x) and this shows that scf (x) is a maximal ideal in c(x). to see this, let f ∈ c(x), then we consider two cases. first, let x0 ∈ z(f), then for all n ∈ n, x0 ∈ f −1(−1 n , 1 n ). therefore for all n ∈ n, x \ f−1(−1 n , 1 n ) is countable. so x \ z(f) is countable (note, z(f) = ⋂ n∈n f−1(− 1 n , 1 n )), i.e., f ∈ scf (x) ⊆ r + scf (x). now, let x0 /∈ z(f), then there exists 0 6= r ∈ r, such that f(x0) = r. put g = f − r, hence x0 ∈ z(g), then by what we have already proved g ∈ scf (x) and so f ∈ r + scf (x). in the next theorem, we characterize the spaces x such that scf (x) is a prime ideal in c(x). we need the following well-known lemma, which is in [12, 4i. 4]. let us first emphasize that since every prime ideal p in c(x) is contained in a unique maximal ideal, hence it is either free or it is in the fixed maximal ideal mx for a unique x ∈ x. lemma 3.2. let p be a fixed prime ideal in c(x). then there exists x ∈ x with ox ⊆ p ⊆ mx. theorem 3.3. let scf (x) be a prime ideal in c(x). then x is either a countably discrete space or the one-point lindelöffication of a countably discrete space. proof. if scf (x) is a free ideal in c(x), then by corollary 2.5, x is a countably discrete space. if not, then by lemma 3.2, there exists x ∈ x such that ox ⊆ scf (x). consequently, in view of theorem 2.6, x is either countable or the one-point lindelöffication of an uncountable discrete space (i.e., a countably discrete space), hence we are done. � the following corollary is now immediate. corollary 3.4. let x have at least one non-countably isolated point. then scf (x) is a prime ideal if and only if x is the one-point lindelöffication of a countably discrete spaces. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 233 s. ghasemzadeh and m. namdari finally, motivated by example 3.1 and [9, proposition 6.6], we are interested in characterizing topological spaces x such that cc(x) = r + scf (x), where cc(x) denotes the subring of c(x) consisting of functions with countable image, see [9], [10]. first, we need the following definition. definition 3.5. a space x is called cocountably-disconnected if whenever y is a clopen subset of x, either y or x \ y is countable. clearly, connected spaces, one-point lindelöffication of countably discrete spaces and x = y ∪ ic(x), where ic(x) = x \ y is the countable set of countably isolated points of x and y is connected (e.g., x = (0, 1 2 ) ∪ n as a subspace of r, or more generally a free union of a connected space with a countable space) are some examples of cocountably-disconnected spaces. finally, we conclude this article with our main result which is the counterpart of [9, proposition 6.6]. before stating this main result we should remind the reader that in [10], it is shown that to study cc(x), we may, without loss, consider x to be zero dimentional space. theorem 3.6. let x be a zero-dimentional space. then cc(x) = r+scf (x) if and only if x is cocountably-disconnected and if x = y ∪ ic(x), where ic(x) = x \ y is the set of countably isolated points, every function in cc(x) is constant on y . proof. first, let x be a cocountably-disconnected space with the above properties. we are to show that cc(x) ⊆ r + scf (x). if x is connected, scf (x) = (0), cc(x) = r, and we are done. hence we may assume that x is disconnected, which in turn, implies that the set of countably isolated points of x is nonempty, for x is cocountably-disconnected. now we consider two cases. case i: let ic(x) be finite. since x is cocountably-disconnected, we infer that y = x \ ic(x) must be connected. therefore for each f ∈ cc(x), f is constant, say r, on y (note, since f ∈ cc(x), it is constant on y automatically and no need to make use of our assumption that f is constant on y ). now let f ∈ cc(x) and define g ∈ scf (x) with g(y ) = 0, g(x) = r − f(x) for all x ∈ ic(x). hence f = r+g and we are done. case ii: let us assume that ic(x) is an infinite set. let f ∈ scf (x), hence f(x) = {r1, r2, · · · , rn, · · · } ⊆ r. thus x = ∞⋃ i=1 ai, where ai = f −1(ri) for each i ∈ n. by above hypothesis, f(y ) = rk, for some k ∈ n and since x is a zero-dimensional space, there exists a clopen set uk such that uk ⊆ x \ ak. hence y ⊆ ak ⊆ x \ uk and we infer that uk is countable (note, x \ uk is uncountable). now if we define g ∈ scf (x) with g(x \ uk) = 0, g(x) = f(x) − rk for all x ∈ uk, we have f = g + rk and we are through in this case, too. conversely, let cc(x) = r + scf (x). if x is connected, we are trivially done. therefore we put x = a ∪ b, where a, b are two disjoint clopen subsets of x. we claim that ic(x) 6= ∅, for otherwise scf (x) = (0), hence cc(x) = r. but the function f, with f(a) = {0}, f(b) = {1} which is in cc(x) \ r leads us to c© agt, upv, 2019 appl. gen. topol. 20, no. 1 234 when is the super socle of c(x) prime? a contradiction, hence we must have scf (x) 6= (0). now we claim that x is cocountably-disconnected. to see this, let x = a ∪ b, where a, b are two uncountable disjoint clopen subsets of x and obtain a contradiction. but if f is the function as above, i.e., f(a) = {0}, f(b) = {1}, then f ∈ cc(x) but f /∈ r + scf (x) (note, if f = r + g with r ∈ r, g ∈ scf (x), then either g(a) 6= 0 or g(b) 6= 0 which is impossible, for g must vanish everywhere except on a countable subset of x). finally, we must show that each f ∈ scf (x) is constant on y . to see this, let x1, x2 ∈ y with f(x1) 6= f(x2) and obtain a contradiction. since cc(x) = r + scf (x), we must have f = r + g, for some r ∈ r, g ∈ scf (x). but by proposition 2.4, g is non-zero only on some countably isolated points. whereas either g(x1) 6= 0 or g(x2) 6= 0, which is absurd. � in [11, theorem 2.10], it is observed that, in fact, scf (x) is a maximal ideal in c(x) when x is the one-point lindelöffication of an uncountable discrete space. we conclude this note with the following related remark. remark 3.7. let x be a zero-dimensional cocountably-disconnected space, with x \ ic(x) a singleton and f(ic(x)) a countable set for any f ∈ c(x). then in view of this theorem and corollary 3.4, x is the one-point lindelöffication of a countably discrete space. moreover in this case scf (x) becomes a maximal ideal in c(x), too (note, cc(x) = c(x) = r + scf (x)). in comparison to the latter equality, let us recall that if x is the one-point compactification of a discrete space then cf (x) = r + cf (x), see [9, theorem 6.7]. moreover, in this case cf (x) is a unique proper essential ideal in c f (x) (consequently, a maximal ideal in cf (x)), where cf (x) is the subalgebra of cc(x), a fortiori of c(x), whose elements have finite images, see [13] for general rings with the latter property . acknowledgements. the authors would like to thank professor o. a. s. karamzadeh for introducing the concept of super socle of c(x) and for his helpful suggestions. the authors are also indebted to the well-informed, meticulous referee for reading the article carefully and giving valuable and constructive comments. references [1] f. azarpanah, algebraic properties of some compact spaces, real anal. exchange 25 (2000), 317–328. [2] f. azarpanah, essential ideals in c(x), period. math. hungar. 31 (1995), 105–112. [3] f. azarpanah, intersection of essential ideals in c(x), proc. amer. math. soc. 125 (1997), 2149–2154. [4] f. azarpanah and o. a. s. karamzadeh, algebric characterization of some disconnected spaces, italian. j. pure appl. math. 12 (2002), 155–168. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 235 s. ghasemzadeh and m. namdari [5] f. azarpanah, o. a. s. karamzadeh and s. rahmati, c(x) vs. c(x) modulo its socle, coll. math. 3 (2008), 315–336. [6] t. dube, contracting the socle in ring of continuous functions, rend. semin. mat. univ. padova 123 (2010), 37–53. [7] r. engelking, general topology, heldermann verlag berlin, 1989. [8] a. a. estaji and o. a. s. karamzadeh, on c(x) modulo its socle, comm. algebra 13 (2003),1561–1571. [9] m. ghadermazi, o. a. s. karamzadeh and m. namdari, c(x) versus its functionally countable subalgebra, bull. iranian math. soc. 45 (2019), 173–187. [10] m. ghadermazi, o. a. s. karamzadeh and m. namdari, on the functionally countable subalgebra of c(x), rend. sem. mat. univ. padova 129 (2013), 47–70. [11] s. ghasemzadeh, o. a. s. karamzadeh and m. namdari, the super socle of the ring of continuous functions, math. slovaca 67 (2017), 1001–1010. [12] l. gillman and m. jerison, rings of continuous functions, springer-verlag, 1976. [13] o. a. s. karamzadeh, m. motamedi and s. m. shahrtash, on rings with a unique proper essential right ideal, fund. math. 183 (2004), 229–244. [14] o. a. s. karamzadeh and m. rostami, on the intrinsic topology and some related ideals of c(x), proc. amer. math. soc. 93 (1985), 179–184. [15] o. a. s. karamzadeh, m. namdari and s. soltanpour, on the locally functionally countable subalgebra of c(x), appl. gen. topol. 16, no. 2 (2015), 183–207. [16] s. mehran and m. namdari, the λ-super socle of the ring of continuous functions, categ. general alg. struct. appl. 6 (2017), 37–50. [17] m. namdari and m. a. siavoshi, a note on discrete c-embedded subspaces, mathematica slovaca, to appear. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 236 () @ applied general topology c© universidad politécnica de valencia volume 14, no. 1, 2013 pp. 1-15 upper and lower cl-supercontinuous multifunctions j. k. kohli and chaman prakash arya abstract the notion of cl-supercontinuity (≡ clopen continuity) of functions is extended to the realm of multifunctions. basic properties of upper (lower) cl-supercontinuous multifunctions are studied and their place in the hierarchy of strong variants of continuity of multifunctions is discussed. examples are included to reflect upon the distinctiveness of upper (lower) cl-supercontinuity of multifunctions from that of other strong variants of continuity of multifunctions which already exist in the literature. 2010 msc: 54c05, 54c10, 54c60, 54d20. keywords: upper(lower)cl-supercontinuous multifunction, strongly continuous multifunction, upper(lower) perfectly continuous multifunction, upper(lower) z-supercontinuous multifunction, upper(lower) d-supercontinuous multifunction, upper(lower) supercontinuous multifunction, cl-open set, zero dimensional space, cl-para-lindelöf space, cl-paracompact space, nonmingled multifunction. 1. introduction several weak and strong variants of continuity occur in the lore of mathematical literature which have been studied by host of authors. the strong variants of continuity with which we shall be dealing in this paper include strong continuity due to levine [17], perfect continuity introduced by noiri [19], clopen continuity (cl-supercontinuity) defined by reilly and vamanamurthy [20], and studied by singh[21], and kohli and singh [15], complete continuity initiated by arya and gupta [5] and z-supercontinuity introduced by kohli and kumar 2 j. k. kohli and c. p. arya [14]. multifunctions arise naturally in many areas of mathematics and applications of mathematics and have wide ranging applications in optimization theory, control theory, game theory, mathematical economics, dynamical systems and differential inclusions. recently, there has been considerable interest in trying to extend the notions and results of weak and strong variants of continuity of functions to the realm of multifunctions (see [1], [2], [3], [4], [5], [8], [9], [10], [16], [22], [27]). the present paper is written in continuation of the same theme. in this paper we extend the notion of cl-supercontinuity of functions to the framework of multifunctions and introduce the notions of upper and lower cl-supercontinuous multifunctions and elaborate on their place in the hierarchy of strong variants of continuity of multifunctions. in the process we extend certain result of singh [21] pertaining to cl-supercontinuous functions to the setting of multifunctions. it turns out that class of upper (lower) cl-supercontinuous multifunctions properly includes the class of upper (lower) perfectly continuous multifunctions and so includes all strongly continuous multifunctions [12] and is strictly contained in the class of upper (lower) z-supercontinuous multifunctions [3]. section 2 is devoted to preliminaries and basic definitions, wherein we introduce the notions of upper and lower cl-supercontinuous multifunctions and discuss the interrelations that exist among them and other strong variants of continuity of multifunctions that already exist in the literature. examples are included to reflect upon the distintiveness of the notions so introduced and other strong variants of continuity of multifunctions in the literature. in section 3 we obtain characterizations and study basic properties of upper cl-supercontinuous multifunctions. it turns out that upper cl-supercontinuity of multifunctions is preserved under the shrinking and expansion of range, composition of multifunctions, union of multifunctions, restriction to a subspace, and the passage to the graph multifunction. further, we formulate a sufficient condition for the intersection of two multifunctions to be cl-supercontinuous. moreover, we prove that the graph of an upper cl-supercontinuous multifunction with closed values into a regular space is cl-closed with respect to x. furthermore, an upper cl-supercontinuous multifunction maps mildly compact sets to compact sets. finally it is shown that a closed, open, upper cl-supercontinuous multifunction with paracompact values maps cl-paracompact sets to paracompact sets. section 4 is devoted to the study of lower cl-supercontinuous multifunctions, wherein characterizations of lower cl-supercontinuity are obtained. it is shown that lower cl-supercontinuity is preserved under the shrinking and expansion of range, union of multifunctions, restriction to a subspace and passage to the graph multifunction. further, it is shown that a product of multifunctions is lower cl-supercontinuous if and only if each multifunction is lower cl-supercontinuous. 2. preliminaries and basic definitions throughout the paper we essentially follow the notations and terminology of l. górniewicz. let x and y be nonempty sets. then ϕ : x ⊸ y is called a multifunction from x into y if for each x ∈ x, ϕ(x) is a nonempty subset of upper and lower cl-supercontinuous multifunctions 3 y. let b be a subset of y. then the set ϕ−1+ (b) = {x ∈ x : ϕ(x) ∩ b 6= ∅} is called large inverse image[6]1 of b and the set ϕ−1− (b) = {x ∈ x : ϕ(x) ⊂ b} is called small inverse image of b. the set γϕ = {(x,y) ∈ x × y |y ∈ ϕ(x)} is called the graph of the multifunction. let a be subset of x. then ϕ(a) = ∪{ϕ(x) : x ∈ a} is called image of a. a multifunction ϕ : x ⊸ y is upper semicontinuous (respectively lower semicontinuous) if ϕ−1− (u) (respectively ϕ−1+ (u)) is an open set in x for every open set u in y. a subset u of a topological space x is called a cl-open set if it can be expressed as the union of clopen sets. the complement of a cl-open set will be referred to as a cl-closed set. a subset a of a space x is called regular open if it is the interior of its closure, i.e., a = a ◦ . a collection β of subsets of a space x is called an open complementary system [7] if β consists of open sets such that for each b ∈ β, there exist b1,b2, ...,∈ β with b = ⋃ {x \ bi : i ∈ z +}. a subset u of a space x is called strongly open fσ-set [7] if there exists a countable open complementary system β(u) with u ∈ β(u). a subset h of a space x is called a regular gδ-set [18] if h is the intersection of a sequence of closed sets whose interiors contain h, i.e., if h = ⋂∞ i=1 fi = ⋂∞ i=1 f ◦ i , where each fi is a closed subset of x. the complement of a regular gδ-set is called a regular fσ-set. let x be a topological space and let a ⊂ x. a point x ∈ x is called a θ-adherent point [25] of a if every closed neighbourhood of x intersects a. let clθa denote the set of all θ-adherent point of a. the set a is called θ-closed if a = clθa. the complement of a θ-closed set is referred to as a θ-open set. a point x ∈ x is said to be a cl-adherent point of a if every clopen set containing x intersects a. let [a]cl denote the set of all cl-adherent points of a. then a set a is cl-closed if and only if a=[a]cl. a subset a of a space x is said to be cl-closed if it is the intersection of clopen sets. the complement of a cl-closed set is referred to as a cl-open set. definition 2.1 ([13]). a multifunction ϕ : x ⊸ y from a topological space x into a topological space y is said to be (1) strongly continuous if ϕ−1− (b) is clopen in x for every subset b ⊂ y. (2)upper perfectly continuous if ϕ−1− (v ) is clopen in x for every open set v ⊂ y. (3) lower perfectly continuous if ϕ−1+ (v ) is clopen in x for every open set v ⊂ y. (4) upper completely continuous if ϕ−1− (v ) is regular open in x for every open set v ⊂ y. (5) lower completely continuous if ϕ−1+ (v ) is regular open in x for every open set v ⊂ y. 1however,what we call “large inverse image ϕ −1 + (b)” some authors call it ‘lower inverse image’ and denote it by ϕ−(b); and similarly they call “small inverse image ϕ −1 − (b)” as ‘upper inverse image’ and employ the notation ϕ+(b) for the same. 4 j. k. kohli and c. p. arya definition 2.2. a multifunction ϕ : x ⊸ y from a topological space x into a topological space y is said to be (1) upper z-supercontinuous [3] if for each x ∈ x and each open set v containing ϕ(x), there exists a cozero set u containing x such that ϕ(u) ⊂ v. (2) lower z-supercontinuous[3] if for each x ∈ x and each open set v with ϕ(x) ∩v 6= ∅, there exists a cozero set u containing x such that ϕ(z) ∩v 6= ∅ for each z ∈ u. (3) upper dδ-supercontinuous [4] if for each x ∈ x and each open set v containing ϕ(x), there exists a regular fσ-set u containing x such that ϕ(u) ⊂ v. (4) lower dδ-supercontinuous [4] if for each x ∈ x and each open set v with ϕ(x) ∩ v 6= ∅ , there exists a regular fσ-set u containing x such that ϕ(z) ∩ v 6= ∅ for each z ∈ u. (5) upper d-supercontinuous [1] if for each x ∈ x and each open set v containing ϕ(x), there exists an open fσ-set u containing x such that ϕ(u) ⊂ v. (6) lower d-supercontinuous [1] if for each x ∈ x and each open set v with ϕ(x)∩v 6= ∅, there exists an open fσ-set u containing x such that ϕ(z)∩v 6= ∅ for each z ∈ u. (7) upper d∗-supercontinuous[12] if for each x ∈ x and each open set v containing ϕ(x), there exists a strongly open fσ-set u containing x such that ϕ(u) ⊂ v. (8) lower d∗-supercontinuous [12] if for each x ∈ x and each open set v with ϕ(x) ∩ v 6= ∅, there exists a strongly open fσ-set u containing x such that ϕ(z) ∩ v 6= ∅ for each z ∈ u. (9) upper strongly θ-continuous [16] if for each x ∈ x and each open set v containing ϕ(x), there exists a θ-open set u containing x such that ϕ(u) ⊂ v. (10) lower strongly θ-continuous[16] if for each x ∈ x and each open set v with ϕ(x) ∩ v 6= φ, there exists a θ-open set u containing x such that ϕ(z) ∩ v 6= ∅ for each z ∈ u. definition 2.3 ([21]). the graph γϕ of a multifunction ϕ : x ⊸ y is said to be cl-closed with respect to x if for each (x,y) 6∈ γϕ there exist a clopen set u containing x and an open set v containing y such that (u × v ) ∩ γϕ = ∅. definition 2.4 ([26]). a multifunction ϕ : x ⊸ y is said to have nonmingled point images provided that for x,y ∈ x with x 6= y, the image sets ϕ(x) and ϕ(y) are either disjoint or identical. definition 2.5. a space x is said to be (a) mildly compact[23] if for every clopen cover of x has a finite subcover. in [24] sostak calls mildly compact spaces as clustered spaces. (b) cl-paracompact (cl-para-lindelöf) if every clopen cover of x has locally finite (locally countable) open refinement which covers x. upper and lower cl-supercontinuous multifunctions 5 definition 2.6. we say that a multifunction ϕ : x ⊸ y is (a) upper cl-supercontinuous at x ∈ x if for each open set v with ϕ(x) ⊂ v, there exists a clopen set u containing x such that ϕ(u) ⊂ v. the multifunction is said to be upper cl-supercontinuous if it is upper cl-supercontinuous at each x ∈ x. (b) lower cl-supercontinuous at x ∈ x if for each open set v with ϕ(x)∩v 6= φ, there exists a clopen set u containing x such that ϕ(z) ∩ v 6= ∅ for each z ∈ u. the multifunction is said to be lower cl-supercontinuous if it is lower cl-supercontinuous at each x ∈ x. the following diagram well illustrates the interrelations that exist among various strong variants of continuity of multifunctions defined in definition 2.1, 2.2 and 2.6. figure 1. however, none of the above implications is reversible as is well illustrated by the examples in the sequel and the examples in ([1], [2], [3], [4], [12], [16]). example 2.7. let x = {a,b,c} with the topology ℑx = {∅,x,{a},{b,c}} and let y = {x,y} with the topology ℑy = {∅,y,{y}}. define a multifunction ϕ : (x,ℑx) ⊸ (y,ℑy ) by ϕ(a) = {y}, ϕ(b) = {x,y}, ϕ(c) = {x}. then the multifunction is upper perfectly continuous but not lower perfectly continuous. again, for {x} ⊂ y , ϕ−1− ({x}) = {c} is not clopen which implies that the multifunction ϕ is not strongly continuous. 6 j. k. kohli and c. p. arya example 2.8. let x = {a,b,c} with the topology ℑx = {∅,x,{a},{c},{a,c}, {a,b}} and let y = {x,y} with the topology ℑy = {∅,y,{y}}. define a multifunction ϕ : (x,ℑx) ⊸ (y,ℑy ) by ϕ(a) = {y}, ϕ(b) = {x,y}, ϕ(c) = {y}. then clearly ϕ is lower perfectly continuous. but for {y} ⊂ y ϕ−1− ({y}) = {a} is not clopen, which implies the multifunction ϕ is not strongly continuous. example 2.9. let x = ℜ, set of real numbers with upper limit topology ℑ and let y be same as x with usual topology u. define a multifunction ϕ : (x,ℑ) ⊸ (y,u) by ϕ(x) = {x} for each x ∈ x. then clearly ϕ is upper (lower) cl-supercontinuous. but for ϕ−1− (a,b) = (a,b) = ϕ −1 + (a,b) is not clopen in x, which implies that ϕ is not upper (lower) perfectly continuous. example 2.10. let x be a completely regular space which is not zero dimensional and let y be same as x. then the identity mapping ϕ : x ⊸ y defined by ϕ(x) = {x} for each x ∈ x, is upper (lower) z-supercontinuous but not upper (lower) cl-supercontinuous. 3. properties of upper cl-supercontinuous multifunctions theorem 3.1. for a multifunction ϕ : x ⊸ y from a topological space x into a topological y the following statements are equivalent: (a) ϕ is upper cl-supercontinuous. (b) ϕ−1− (b) is a cl-open set in x for every open set b in y. (c) ϕ−1+ (b) is a cl-closed in x for every closed set b in y. (d) [ϕ−1+ (b)]cl ⊂ ϕ −1 + (b) for every subset b of y. proof. a)⇒ (b). let b be an open subset of y . to show that ϕ−1− (b) is cl-open in x, let x ∈ ϕ−1− (b). then ϕ(x) ⊂ b. since ϕ is upper cl-supercontinuous, therefore, there exists a clopen set h containing x such that ϕ(h) ⊂ b. hence x ∈ h ⊂ ϕ−1− (b) and so is a cl-open set in x. (b)⇒ (c). let b be a closed subset of y . then y \ b is an open subset of y. by (b), ϕ−1− (y \ b) is cl-open set in x. since ϕ −1 − (y \ b) = x \ ϕ −1 + (b), ϕ −1 + (b) is a cl-closed set in x. (c)⇒ (d). since b is closed, ϕ−1+ (b) is a cl-closed set containing ϕ −1 + (b) therefore [ϕ−1+ (b)]cl ⊂ ϕ −1 + (b) . (d)⇒ (a). let x ∈ x and let v be an open set in y such that ϕ(x) ⊂ v. then ϕ(x)∩(y \v ) = ∅ and (y \ v ) = y \v. hence [ϕ−1+ (y \v )]cl ⊂ ϕ −1 + (y \v ) = x\ϕ−1− (v ). since ϕ −1 + (y \v ) is cl-closed, its complement ϕ −1 − (v ) is cl-open set containing x. so there is a clopen set u containing x and contained in ϕ−1− (v ), whence ϕ(u) ⊂ v. thus ϕ is upper cl-supercontinuous. � theorem 3.2. if a multifunction ϕ : x ⊸ y is upper cl-supercontinuous and ϕ(x) is endowed with the subspace topology, then, the multifunction ϕ : x ⊸ ϕ(x) is upper cl-supercontinuous. proof. since ϕ is upper cl-supercontinuous for every open set v of y, ϕ−1− (v ∩ ϕ(x)) = ϕ−1− (v ) ∩ ϕ −1 − (ϕ(x)) = ϕ −1 − (v ) ∩ x = ϕ −1 − (v ) is cl-open and hence ϕ : x ⊸ ϕ(x) is cl-supercontinuous. � upper and lower cl-supercontinuous multifunctions 7 theorem 3.3. if ϕ : x ⊸ y is upper cl-supercontinuous and ψ : y ⊸ z is upper semicontinuous, then ψoϕ is upper cl-supercontinuous. in particular, composition of upper cl-supercontinuous multifunctions is upper cl-supercontinuous. proof. let w be an open set in z. since ψ is upper semicontinuous, ψ−1− (w) is an open set in y . again, since ϕ is upper clsupercontinuous, ϕ−1− (ψ −1 − (w)) = (ψoϕ)−1− (w) is a cl-open set in x. thus ψoϕ : x ⊸ z is upper cl-supercontinuous. � in contrast to theorem 3.2, the following corollary shows that upper clsupercontinuity of a multifunction remains invariant under extension of its range. corollary 3.4. let ϕ : x ⊸ y be upper cl-supercontinuous. if z is a space containing y as a subspace, then ψ : x ⊸ z defined by ψ(x) = ϕ(x) for each x ∈ x is upper cl-supercontinuous. proof. let w be an open set in z. then w ∩ y is an open set in y. since ϕ : x ⊸ y is upper cl-supercontinuous, ϕ−1− (w ∩ y ) is cl-open set in x. now ψ −1 − (w) = {x ∈ x : ψ(x) ⊂ w} = {x ∈ x : ϕ(x) ⊂ w ∩ y }. thus ψ : x ⊸ z is upper cl-supercontinuous. � theorem 3.5. if ϕ : x ⊸ y and ψ : x ⊸ y are upper cl-supercontinuous multifunctions, then ϕ ∪ ψ : x ⊸ y defined by (ϕ ∪ ψ)(x) = ϕ(x) ∪ ψ(x) for each x ∈ x, is upper cl-supercontinuous. proof. let u be an open set in y. since ϕ and ψ are upper cl-supercontinuous, ϕ −1 − (u) and ψ −1 − (u) are cl-open sets in x. since (ϕ ∪ ψ) −1 − (u) = ϕ −1 − (u) ∩ ψ −1 − (u) and since finite intersection of cl-open sets is cl-open, (ϕ ∪ ψ) −1 − (u) is cl-open in x. thus ϕ ∪ ψ is upper cl-supercontinuous. � in general intersection of two upper cl-supercontinuous multifunctions need not be upper cl-supercontinuous. however, in the following theorem we formulate a sufficient condition for the intersection of two multifunctions to be upper cl-supercontinuous. theorem 3.6. let ϕ : x ⊸ y and ψ : x ⊸ y be multifunctions from a space x into a hausdorff space y such that ϕ(x) is compact for each x ∈ x satisfying (1) ϕ is upper cl-supercontinuous, and (2) the graph γψ of ψ is cl-closed with respect to x. then the multifunction ϕ∩ψ defined by (ϕ∩ψ)(x) = ϕ(x)∩ψ(x) for each x ∈ x, is upper cl-supercontinuous. proof. let x0 ∈ x and v be an open set containing ϕ(x0) ∩ ψ(x0). it suffices to find a clopen set u containing x0 such that (ϕ ∩ ψ)(u) ⊂ v. if v ⊃ ϕ(x0), it follows from upper cl-supercontinuity of ϕ. if not, then consider the set k = ϕ(x0) \ v which is compact. now for each y ∈ k, y ∈ y \ ψ(x0). this implies that (x0,y) ∈ x ×y \γψ. since the graph of ψ is cl-closed with respect to x, there exist clopen set uycontaining x0 and an open set vy containing y 8 j. k. kohli and c. p. arya such that γψ ∩ (uy × vy) = ∅. therefore, for each x ∈ uy, ψ(x) ∩ vy = ∅ since k is compact, there exist finitely many in y1,y1, ...,yn in k such that k ⊂ ∪ni=1vyi. let w = ∪ n i=1vyi. then v ∪ w is an open set containing ϕ(x0). since ϕ is upper cl-supercontinuous, there exists a clopen set u0 containing x0 such that ϕ(u0) ⊂ v ∪ w. let u = u0 ∩ (∩ n i=1uyi). then u is a clopen set containing x0. hence for each z ∈ u, ϕ(z) ⊂ v ∪ w and ψ(z) ∩ w = ∅. therefore, (ϕ(z) ∩ ψ(z)) ∩ w = ∅ for each z ∈ u. this proves that ϕ ∩ ψ is upper cl-supercontinuous at x0 � corollary 3.7. let ψ : x ⊸ y be a multifunction from a space x into a compact hausdorff space y such that the graph γψ of ψ is cl-closed with respect to x. then ψ is upper cl-supercontinuous. proof. let the multifunction ϕ : x ⊸ y be defined by ϕ(x) = y for each x ∈ x. now an application of theorem 3.6 yields the desired result. � theorem 3.8. let ϕ : x ⊸ y be any multifunction. then the following statements are true: (a) if ϕ : x ⊸ y is upper cl-supercontinuous and a ⊂ x, then the restriction ϕ|a : a ⊸ y is upper cl-supercontinuous. (b) if {uα : α ∈△} is a cl-open cover of x and if for each α, the restriction ϕα = ϕ|uα : uα ⊸ y is upper cl-supercontinuous, then ϕ : x ⊸ y is upper cl-supercontinuous. proof. (a) let w be an open set in y. since ϕ : x ⊸ y is upper cl-supercontinuous, ϕ −1 − (w) is a cl-open set in x. now ϕ|a(w) = {x ∈ a | ϕ(x) ⊂ w} = {x ∈ a | x ∈ ϕ−1− (w)} = a ∩ ϕ −1 − (w), which is cl-open in x and so ϕ|a is upper cl-supercontinuous. (b) let w be an open set in y. since ϕα = ϕ|uα : uα ⊸ y is upper clsupercontinuous, (ϕα) −1 − (w) is a cl-open set in uα and consequently cl-open in x. since ϕ−1− (w) = ∪α∈ ∧(ϕα) −1 − (w) and since the union of cl-open set is cl-open, ϕ−1− (w) is cl-open set in x. in view of theorem 3.1, ϕ : x ⊸ y is upper cl-supercontinuous. � theorem 3.9. let ϕ : x ⊸ y be a multifunction and let g : x ⊸ x × y defined by g(x) = {(x,y) ∈ x × y |y ∈ ϕ(x)} for each x ∈ x, be the graph multifunction. if g is upper cl-supercontinuous, then ϕ is upper cl-supercontinuous and the space x is zero dimensional. furthermore, if in addition ϕ(x) is compact for each x ∈ x and x is zero dimensional, then g is upper clsupercontinuous whenever ϕ is. proof. suppose that g is upper cl-supercontinuous. by theorem3.3, the multifunction ϕ = pyog is upper cl-supercontinuous, where py : x ×y ⊸ y denotes the projection mapping. to show that x is zero dimensional, let u be an open set in x and let x ∈ u. then u ×y is an open set in x ×y and g(x) ⊂ u ×y. since g is upper cl-supercontinuous, there exists a clopen set w containing x such that g(w) ⊂ u × y and so w ⊂ g−1− (u × y ) = u. hence x ∈ w ⊂ u and upper and lower cl-supercontinuous multifunctions 9 thus x is zero dimensional. conversely, suppose that x is zero dimensional, the multifunction ϕ is upper cl-supercontinuous and ϕ(x) is compact for each x ∈ x. let w be an open set containing g(x) = {x} × ϕ(x). then by wallace theorem [10, p.142] there exist open sets u in x, v in y and g(x) ⊂ u × v ⊂ w. so x ∈ u and ϕ(x) ⊂ v. since x is zero dimensional there exists a clopen set containing x such that x ∈ g1 ⊂ u. again since ϕ is upper cl-supercontinuous, there exists a clopen set g2 containing x such that ϕ(g2) ⊂ v . let g = g1 ∩ g2. then g is a clopen set containing x and it is easily verified that g(g) ⊂ u × v ⊂ w. this proves that g is upper cl-supercontinuous. � the following theorem gives sufficient conditions for the graph of a multifunction to be cl-closed with respect to x. theorem 3.10. if ϕ : x ⊸ y is upper cl-supercontinuous, where y is a regular space and ϕ(x) is closed for each x ∈ x, then the graph γϕ of ϕ is a cl-closed with respect to x. proof. let (x,y) 6∈ γϕ. then y 6∈ ϕ(x). since y is regular, there exist disjoint open sets vy and vϕ(x) containing y and ϕ(x), respectively. since ϕ is upper cl-supercontinuous, there exists a clopen set ux containing x such that ϕ(ux) ⊂ vϕ(x). we assert that (ux × vy) ∩ γϕ = ∅. for, if (h,k) ∈ (ux × vy) ∩ γϕ, then h ∈ ϕ−1− (vϕ(x)), k ∈ vy and k ∈ ϕ(h). hence ϕ(h) ⊂ vϕ(x) and k ∈ ϕ(h) ∩ vy which contradicts the fact that vy and vϕ(x) are disjoint. thus the graph γϕ of ϕ is a cl-closed with respect to x. � the following theorem is a sort of partial converse to theorem3.10 and shows that the multifunctions which have cl-closed graph with respect to x have nice properties. theorem 3.11. if ϕ : x ⊸ y is a multifunction with cl-closed graph with respect to x and k ⊂ y is compact, then ϕ−1+ (k) is cl-closed in x. further, if in addition y is compact, then ϕ is upper cl-supercontinuous. proof. to prove that ϕ−1+ (k) is cl-closed, we shall show that x \ ϕ −1 + (k) is cl-open. to this end, let x ∈ x \ ϕ−1+ (k). then ϕ(x) ∩ k = ∅. since γϕ is cl-closed with respect to x, for each y ∈ k there exist clopen set uy containing x and an open set vy containing y such that (uy × vy) ∩ γϕ = ∅. the collection ω = {vy|y ∈ k} is an open cover of the compact set k. so there exists a finite subset {y1, ...,yn} of k such that k ⊂ ∪ n i=1vyi = v (say). let u = ∩ni=1uyi. then u is a clopen set containing x and since ϕ(u) ∩ k = ∅. thus u ⊂ x \ ϕ−1+ (k) and so x \ ϕ −1 + (k) is cl-open as desired. the last assertion is immediate in view of theorem3.1 and the fact that a closed subset of a compact space is compact. � corollary 3.12. if ϕ : x ⊸ y is a multifunction with ϕ(x) ⊂ k, where k is compact and the graph γϕ of ϕ is cl-closed with respect to x, then ϕ is upper cl-supercontinuous. 10 j. k. kohli and c. p. arya theorem 3.13. let ϕ : x ⊸ y be an upper cl-supercontinuous multifunction such that ϕ(x) is compact for each x ∈ x. if a is a mildly compact set in x, then ϕ(a) is compact. proof. let ω be an open cover of ϕ(a). then ω is also an open cover of ϕ(a) for each a ∈ a. since each ϕ(a) is compact, there exists a finite subset βa ⊂ ω such that ϕ(a) ⊂ ∪b∈βab = va (say). since ϕ is upper clsupercontinuous, there exists a clopen set ua containing a such that ϕ(ua) ⊂ va and so ua ⊂ ϕ −1 − (va). let q = {ua|a ∈ a}. then q is a clopen covering of a. since a is mildly compact, there exists a finite subset {a1, ...,an} of a such that a ⊂ ∪ni=1uai ⊂ ∪ n i=1ϕ −1 − (vai). therefore ϕ(a) ⊂ ϕ(∪ n i=1ϕ −1 − (vai)) = ∪ni=1ϕ(ϕ −1 − (vai)) ⊂ ∪ n i=1vai, where vai = ∪b∈βai b, i = 1, ...,n and each βai is finite. thus ϕ(a) is compact. � we may recall that a space x is called a p-space if every gδ-set in x is open in x. theorem 3.14. let ϕ : x ⊸ y be a closed, open, and upper cl-supercontinuous, nonmingled multifunction from a space x into a p-space y such that ϕ(x) is para-lindelöf for each x ∈ x. if a is a cl-para-lindelöf set in x, then ϕ(a) is para-lindelöf set in y. in particular, if x is cl-para-lindelöf and ϕ is onto, then y is para-lindelöf. proof. let ψ be an open cover of ϕ(a). then ψ is also an open covering of ϕ(x) for each x ∈ a. since ϕ(x) is para-lindelöf, ψ has a locally countable open refinement ψx such that ϕ(x) ⊂ ∪ψx = vx (say). since ϕ is upper clsupercontinuous, there exists a clopen set ux containing x such that ϕ(ux) ⊂ vx. now u = {ux | x ∈ a}is a clopen cover of a. since a is cl-para-lindelöf, u has a locally countable open refinement ω = {wα | α ∈ λ} such that a ⊂ ∪α∈λwα. so for each α ∈ λ there exists a xα ∈ a such that wα ⊂ uxα and hence ϕ(wα) ⊂ ϕ(uα) ⊂ ∪ψxα. let ℜα = {ϕ(wα) ∩ v | v ∈ ψxα} and let ℜ = {r | r ∈ ℜα,α ∈ λ}. we shall show that ℜ is a locally countable open refinement of ψ. since ϕ is open, ϕ(wα) is open and so each r ∈ ℜ is open. let r ∈ ℜ. then r ∈ ℜα for some α ∈ λ, i.e. r = ϕ(wα) ∩ v ⊂ v ⊂ u for some u ∈ ψ. this shows that ℜ is an open refinement of ψ. to show that ℜ is locally countable, let y ∈ ϕ(a). then y ∈ ϕ(x) for some x ∈ a. since ω is locally countable, for each x ∈ a we can choose an open neighborhood gx of x which intersects only countably many members wα1,wα2, ...wαn... of ψ. since ϕ is a nonmingled open multifunction, it follows that h0 = ϕ(gx) is an open neighborhood of y which intersects only countably many members ϕ(wα1),ϕ(wα2 ))...ϕ(wαn )... of the family {ϕ(wα) | α ∈ λ}. furthermore each ℜαk (k = 1, ...,n, ...) is locally countable, hence there exists an open neighborhood hk (k = 1, ...,n, ...) of y which intersects only countably many members of ℜαk (k = 1, ...,n, ...). finally let h = ∩ ∞ k=1hk. since y is p-space, h is an open neighborhood of y which intersects at most countably many member of ℜ, and so ℜ is locally countable. moreover, ϕ(a) ⊂ ϕ(∪α∈λwα) = upper and lower cl-supercontinuous multifunctions 11 ∪α∈λϕ(wα) ⊂ ∪α∈λ(∪ℜα) = ∪{r : r ∈ ℜ}. hence ℜ is a locally countable open refinement of ψ that covers ϕ(a). thus ϕ(a) is para-lindelöf. � theorem 3.15. let ϕ : x ⊸ y be a closed, open, upper cl-supercontinuous nonmingled multifunction from a space x into a space y such that ϕ(x) is paracompact for each x ∈ x. if a is a cl-paracompact, then ϕ(a) is paracompact. in particular, if x is cl-paracompact space and ϕ is onto, then y is paracompact. proof. proof of theorem 3.15 is similar (even simpler) to that of theorem 3.14 and hence omitted. � 4. properties of lower cl-supercontinuous multifunctions theorem 4.1. for a multifunction ϕ : x ⊸ y from a topological space x into a topological space y the following statements are equivalent. (a) ϕ is lower cl-supercontinuous. (b) ϕ−1+ (b) is a cl-open set in x for every open set b in y. (c) ϕ−1− (b) is a cl-closed in x for every closed set b in y. (d) for each x ∈ x and for each open set v with ϕ(x) ∩ v 6= ∅ there exists a cl-open set u containing x such that ϕ(z) ∩ v 6= φ for each z ∈ u. proof. (a)⇒(b). let b be an open subset of y. to show that ϕ−1+ (b) is cl-open in x, let x ∈ ϕ−1+ (b). then ϕ(x) ∩b 6= ∅. since ϕ is lower cl-supercontinuous, there exists a clopen set h containing x such that ϕ(h)∩b 6= ∅ for each h ∈ h. hence x ∈ h ⊂ ϕ−1+ (b) and so ϕ −1 + (b) is a cl-open set in x being a union of clopen sets. (b)⇒(c). let b be a closed subset of y. then y \b is an open subset of y. by (b), ϕ−1+ (y \b) is a cl-open set in x. since ϕ −1 + (y \b) = x \ϕ −1 − (b), ϕ −1 − (b) is a cl-closed set in x. (c)⇒(d). let x ∈ x and let v be an open set in y with ϕ(x) ∩ v 6= ∅. then y \v is a closed set in y with ϕ(x) * (y \v ).therefore, by (c), ϕ−1− (y \v ) = x \ ϕ−1+ (v ) is a cl-closed set in x not containing x and so ϕ −1 + (v ) is a cl-open set in x containing x. let u = ϕ−1+ (v ). then u is a cl-open set containing x such that ϕ(z) ∩ v 6= ∅ for each z ∈ u. the assertion (d)⇒(a) is trivial, since every cl-open set is the union of clopen sets. � theorem 4.2. a multifunction ϕ : x ⊸ y is lower cl-supercontinuous if and only if ϕ([a]cl) ⊂ ϕ(a) for every subset a of x. proof. suppose that ϕ : x ⊸ y is lower cl-supercontinuous. let a be subset of x. then ϕ(a) is a closed subset of y. by theorem 4.1 ϕ−1− (ϕ(a)) is a cl-closed set in x. since a ⊂ ϕ−1− (ϕ(a)) and since [a]cl ⊂ [ϕ −1 − (ϕ(a))]cl = ϕ −1 − (ϕ(a)), ϕ([a]cl) ⊂ ϕ(ϕ −1 − (ϕ(a))) ⊂ ϕ(a). conversely, suppose that ϕ([a]cl) ⊂ ϕ(a) for every subset a of x and let f be a closed set in y . then ϕ−1− (f) is subset of x. by hypothesis, ϕ([ϕ −1 − (f)]cl) ⊂ 12 j. k. kohli and c. p. arya ϕ(ϕ−1− (f)) ⊂ f = f and ϕ −1 − (ϕ([ϕ −1 − (f)]cl)) ⊂ ϕ −1 − (f) so which in its turn implies that [ϕ−1− (f)]cl ⊂ ϕ −1 − (f). hence ϕ −1 − (f) = [ϕ −1 − (f)]cl and so in view of theorem 4.1 ϕ : x ⊸ y is lower cl-supercontinuous. � theorem 4.3. a multifunction ϕ : x ⊸ y is lower cl-supercontinuous if and only if [ϕ−1− (b)]cl ⊂ ϕ −1 − (b) for every subset b of y. proof. suppose that ϕ : x ⊸ y is lower cl-supercontinuous. let b ⊂ y. then b is a closed subset of y. by theorem 4.1, ϕ−1− (b) is a cl-closed subset of x. since, ϕ−1− (b) ⊂ ϕ −1 − (b), [ϕ −1 − (b)]cl ⊂ [ϕ −1 − (b)]cl = ϕ −1 − (b). that is [ϕ−1− (b)]cl ⊂ ϕ −1 − (b). conversely suppose that [ϕ−1− (b)]cl ⊂ ϕ −1 − (b) for every b ⊂ y. let f be any closed subset of y. by hypothesis [ϕ−1− (f)]cl ⊂ ϕ −1 − (f) = ϕ −1 − (f). hence [ϕ−1− (f)]cl = ϕ −1 − (f) and so in view of theorem 4.1 ϕ is lower cl-supercontinuous. � the following theorem shows that lower cl-supercontinuity of a multifunction remains invariant under the shrinking of its range. theorem 4.4. if ϕ : x ⊸ y is lower cl-supercontinuous and ϕ(x) is endowed with subspace topology, then ϕ : x ⊸ ϕ(x) is lower cl-supercontinuous. theorem 4.5. if ϕ : x ⊸ y is lower cl-supercontinuous and ψ : y ⊸ z is lower semicontinuous, then ψoϕ is lower cl-supercontinuous. in particular, composition of two lower cl-supercontinuous multifunctions is upper clsupercontinuous. proof. let w be an open set in z. since ψ is upper semi continuous, ψ−1+ (w) is an open set in y. again since ϕ is lower cl-supercontinuous, ϕ−1+ (ψ −1 + (w))is cl-open in x, and so (ψoϕ)−1+ (w) = ϕ −1 + (ψ −1 + (w)) is a cl-open set in x. thus ψoψ : x ⊸ z is lower cl-supercontinuous. � in contrast to theorem 4.4 the following corollary shows that lower clsupercontinuity of a multifunction is preserved under the expansion of its range. corollary 4.6. let ϕ : x ⊸ y be lower cl-supercontinuous. if z is a space containing y as a subspace, then ψ : x ⊸ z defined by ψ(x) = ϕ(x) for x ∈ x is lower cl-supercontinuous. proof. let w be an open set in z. then w ∩ y is an open set in y. since ϕ : x ⊸ y is lower cl-supercontinuous, ϕ−1+ (w ∩ y ) is cl-open set in x. now, ψ −1 + (w) = {x ∈ x : ψ(x) ∩ w 6= ∅} = {x ∈ x : ϕ(x) ∩ (w ∩ y ) 6= ∅} = ϕ −1 + (w ∩ y ). thus ψ : x ⊸ z is lower cl-supercontinuous. � theorem 4.7. if ϕ : x ⊸ y and ψ : x ⊸ y are lower cl-supercontinuous multifunctions, then the multifunction ϕ ∪ ψ : x ⊸ y defined by (ϕ ∪ ψ)(x) = ϕ(x) ∪ ψ(x) for each x ∈ x, is lower cl-supercontinuous. upper and lower cl-supercontinuous multifunctions 13 proof. let u be an open set in y. then ϕ−1+ (u) and ψ −1 + (u) are cl-open sets in x. since (ϕ ∪ ψ)−1+ (u) = ϕ −1 + (u) ∪ ψ −1 + (u) and since any union of clopen sets is cl-open, (ϕ ∪ ψ)−1+ (u) is cl-open in x. thus ϕ ∪ ψ is lower clsupercontinuous. � theorem 4.8. let ϕ : x ⊸ y be any multifunction. then the following statements are true: (a) if ϕ : x ⊸ y is lower cl-supercontinuous and a ⊂ x, then the restriction ϕ|a : a ⊸ y is lower cl-supercontinuous. (b) if {uα : α ∈ ∆} is a cl-open cover of x and for each α, the restriction ϕα = ϕ|uα : uα ⊸ y is lower cl-supercontinuous, then ϕ : x ⊸ y is lower cl-supercontinuous. proof. (a) let w be an open set in y. since ϕ : x ⊸ y is lower cl-supercontinuous, ϕ −1 + (w) is a cl-open set in x. now, (ϕ|a) −1 + (w) = {x ∈ a | ϕ(x) ∩ w 6= ∅} = {x ∈ a | x ∈ ϕ−1+ (w)} = a ∩ ϕ −1 + (w), which is cl-open in x and so ϕ|a is lower cl-supercontinuous. (b) let w be an open set in y. since ϕα = ϕ|uα : uα ⊸ y is lower clsupercontinuous, (ϕα) −1 + (w) is a cl-open set in uα and consequently cl-open in x. since ϕ−1+ (w) = ∪α∈∆(ϕα) −1 + (w) and since any union of cl-open sets is clopen, ϕ−1+ (w) is cl-open set in x. thus ϕ : x ⊸ y is lower cl-supercontinuous. � theorem 4.9. let {ϕα : x ⊸ xα|α ∈ λ} be a family of multifunctions and let ϕ : x ⊸ ∏ α∈λ xα be defined by ϕ(x) = ∏ α∈λ ϕα(x). then ϕ is lower clsupercontinuous if and only if each ϕα : x ⊸ xα is lower cl-supercontinuous. proof. let ϕ : x ⊸ ∏ α∈λ xα be lower cl-supercontinuous. let pβ : ∏ α∈λ xα −→ xβ be the projection map onto xβ. then pβ being a single valued continuous function is lower semicontinuous. by theorem4.5 ϕβ = pβoϕ is lower cl-supercontinuous for each β ∈ λ. conversely, suppose that ϕβ : x ⊸ xβ is a lower cl-supercontinuous for each β ∈ λ. since the finite intersections and arbitrary union of cl-open sets is clopen, therefore, in view of theorem 4.1 it suffices to prove that ϕ−1+ (b) is a cl-open set for every basic open set b in the product space ∏ α∈λ xα. let b = uα1 ×uα2 ×...×uαn ×( ∏ α6=α1,α2,...,αn xα) be a basic open set in ∏ α∈λ xα now it is easily verified that ϕ−1+ (uα1 × ... × uαn × ( ∏ α6=α1,α2,...,αn xα)) = (ϕα1) −1 + (uα1)∩...∩(ϕαn ) −1 + (uαn ). since each ϕαi is cl-supercontinuous, ϕ −1 + (b) is cl-open in x being the finite intersection of cl-open sets. thus ϕ is lower cl-supercontinuous. � theorem 4.10. for each α ∈ ∆ let ϕα : xα ⊸ yα be a multifunction and let ϕ : ∏ α∈λ xα ⊸ ∏ α∈λ yα be a multifunction defined by ϕ(x) = ∏ ϕα(xα) for each x = (xα) ∈ ∏ α∈λ xα. then ϕ is lower cl-supercontinuous if and only if each ϕα is lower cl-supercontinuous. 14 j. k. kohli and c. p. arya proof. suppose that ϕ : ∏ α∈λ xα ⊸ ∏ α∈λ yα is lower cl-supercontinuous. let uβ be an open set in yβ.then uβ × ∏ α6=β yα is a subbasic open set in ∏ α∈λ yα. so in view of theorem 4.1, ϕ−1+ (uβ × ∏ α6=β yα) is a cl-open set in ∏ α∈λ xα. now it is easily verified that ϕ−1+ (uβ × ∏ α6=β yα) = (ϕβ) −1 + (uβ) × ∏ α6=β xα, and so (ϕβ) −1 + (uβ) is cl-open in xβ. this proves that each ϕβ is lower clsupercontinuous. conversely suppose that ϕα : xα ⊸ yα is lower cl-supercontinuous for each α ∈ λ and let b = vα1 × vα2 × ... × vαn × ( ∏ α6=α1,α2,...,αn yα) be a basic open set in ∏ α∈λ yα. then ϕ −1 + (vα1 × ... × vαn × ( ∏ α6=α1,α2,...,αn yα)) = (ϕα1) −1 + (vα1 ) × ... × (ϕαn ) −1 + (vαn ) × ( ∏ α6=α1,α2,...,αn xα). since each ϕα is lower cl-supercontinuous, ϕ−1+ (b) cl-open in ∏ α∈λ xα and so ϕ is lower clsupercontinuous. � theorem 4.11. let ϕ : x ⊸ y be multifunction and let g : x ⊸ x × y defined by g(x) = {(x,y) ∈ x × y |y ∈ ϕ(x)} for each x ∈ x be the graph multifunction. then g is lower cl-supercontinuous if and only if ϕ is lower cl-supercontinuous and the space x is zero dimensional. proof. suppose that g is lower cl-supercontinuous. by theorem 4.5 the multifunction ϕ = pyog is lower cl-supercontinuous. next we shall show that x is zero dimensional. let u be an open set in x and let x ∈ u. then u × y is an open set in x ×y and g(x)∩(u ×y ) 6= ∅. since g is lower cl-supercontinuous, there exists a clopen set w containing x such that g(z) ∩ (u × y ) 6= ∅ for every z ∈ w and so w ⊂ g−1+ (u × y ) = u. hence x ∈ w ⊂ u and x is zero dimensional. conversely suppose that ϕ is lower cl-supercontinuous. let x ∈ x and let w be an open set with g(x) ∩ w 6= ∅. then there exist open sets u in x and v in y such that g(x) ∩ (u × v ) 6= ∅ and so x ∈ u and ϕ(x) ∩ v 6= ∅. since x is zero dimensional, there exists a clopen set g1 containing x such that x ∈ g1 ⊂ u. again since ϕ is lower clsuper continuous, there exists a clopen set g2 containing x such that ϕ(h)∩v 6= ∅ for each h ∈ g2. let g = g1 ∩g2. then g is a clopen set containing x and it is easily verified that g(h) ∩ w 6= ∅ for each h ∈ g. this proves that g is lower cl-supercontinuous. � references [1] m. akdaǧ, on the upper and lower super d-continuous multifunctions, instanbul univ. sciences faculty, journal of mathehmatics 60 (2001), 101–109. 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(received october 2009 – accepted april 2010) j.k. kohli (jk kohli@yahoo.co.in) dept.of mathematics, hindu college, university of delhi, delhi 110007, india. c. p. arya (carya28@gmail.com) dept.of mathematics, university of delhi, delhi 110007, india. upper and lower cl-supercontinuous[8pt] multifunctions. by j. k. kohli and c. p. arya @ appl. gen. topol. 23, no. 2 (2022), 269-280 doi:10.4995/agt.2022.16143 © agt, upv, 2022 on the group of homeomorphisms on r: a revisit k. ali akbar a and t. mubeena b a department of mathematics, school of physical sciences, central university of kerala, periya 671320, kasaragod, kerala, india (aliakbar.pkd@gmail.com, aliakbar@cukerala.ac.in) b department of mathematics, school of mathematics and computational sciences, university of calicut, thenhipalam 673635, malappuram, kerala, india (mubeenatc@gmail.com, mubeenatc@uoc.ac.in) communicated by f. lin abstract in this article, we prove that the group of all increasing homeomorphisms on r has exactly five normal subgroups, and the group of all homeomorphisms on r has exactly four normal subgroups. there are several results known about the group of homeomorphisms on r and about the group of increasing homeomorphisms on r ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. in this paper, we analyze this structure in some detail. 2020 msc: 20f38; 37b02. keywords: group of homeomorphisms; normal subgroups; dynamical systems; fixed points; conjugacy; bounded functions. 1. introduction there have been several papers discussing about the normal subgroups of the group of homeomorphisms on various metric spaces ([2], [6], [7] and [8]). it is natural to ask: which subsets will arise as normal subgroups of the group of homeomorphisms on a metric space? we provide a proof in the case of the group of (increasing) homeomorphisms on r. this paper contains a detailed received 29 august 2021 – accepted 14 march 2022 http://dx.doi.org/10.4995/agt.2022.16143 k. ali akbar and t. mubeena proof that highlights the differences and similarities between our results and those given in the references. a dynamical system is simply a pair (x,f) where x is a metric space and f : x → x is a continuous function. a point x ∈ x is said to be periodic with period n if fn(x) = x for some n ∈ n, and fm(x) 6= x for 1 ≤ m < n where fn = f ◦f ◦ ...◦f is the composition of f with itself n times. if f(x) = x then we say that x is a fixed point of f. we denote the set of all fixed points of f by fix(f), and the complement of fix(f) by fix(f)c. two dynamical systems (x,f) and (y,g) are said to be conjugate (we simply say f is conjugate to g), if there exists a homeomorphism h from x to y such that h ◦ f = g ◦ h. being conjugate is an equivalence relation in the class of dynamical systems. a homeomorphism h from x to x such that h◦f = f◦h is called a conjugacy of (x,f) or simply a conjugacy. let a,b be two subgroups of a group g. then a is invariant in b if a ⊂ b and if bab−1 ⊂ a for every b ∈ b. a subgroup n of a group g is normal if and only if it is invariant under conjugation, if and only if it is a union of conjugacy classes of g. since our group in this paper is a group of homeomorphisms, the algebraic notion of conjugacy here coincides with topological conjugacy in the sense of dynamical systems theory. for preliminaries from topological dynamics and group theory, the reader may refer [3], [4] and [5]. in this paper, we study the normal subgroups of the group of (increasing) homeomorphisms of r and analyze the topological structure in the aspects of topological dynamics in some detail. the proofs are different from those given in the above references. the classification here is only based on the set of fixed points of members of the normal subgroups. in articles [2], [6] and [8], the set of fixed points or the support of the homeomorphisms are used to classify normal subgroups. first we discuss the ideas of proof involved in the references [2], [6] and [8] to convince the reader that our proof is different from the known ones. for a set x, let π(x) be the group of all permutations (bijections) on x and g be a subgroup of π(x). for a topological space x, let h(x) be the group of homeomorphisms on x. suppose f is a non-empty family of subsets of x. we define s(f,g) = {g ∈ g : fix(g) ⊃ f for some f ∈f}. for s(f,h(x)), we shall write s(f). we say that the family f is ecliptic relative to g whenever it satisfies the following two conditions. (1) if f1,f2 ∈f, then there exists an f3 ∈f such that f3 ⊂ f1 ∩f2, (2) if f1 ∈f and g ∈ g, then there exists an f2 ∈f such that f2 ⊂ g(f1). an ecliptic family which satisfies the following additional condition will be called strictly ecliptic. (3) if f ∈ f and u ⊂ x is open (u 6= ∅), then there exists an h ∈ h(x) such that h(fc) ⊂ u, where fc is the complement of f in x. the objective of the reference [8] is to investigate the normal subgroups for a class of spaces which includes the n-cell bn and the author proved that some of these normal subgroups can be defined in terms of the family of fixed point © agt, upv, 2022 appl. gen. topol. 23, no. 2 270 on the group of homeomorphisms on r: a revisit sets of their elements. for a family f of subsets of x, define s(f,g) = {g ∈ g : fix(g) ⊃ f for some f ∈ f}. it is proved that s(f,g) is a subgroup of g and if f is ecliptic relative to g, then s(f,g) is a normal subgroup of g. if x is a topological space such that for any non-empty, open set u, there is an open set v ⊂ u which is homeomorphic to an open ball in a euclidean space of positive dimension and supposing there is a strictly ecliptic family f on x relative to h(x) and if n is a normal subgroup of h(x), then the author proved that either n ⊃ s(f) or n consists of the identity 1. they also proved that if n is a normal subgroup of h(bn) which contains an element not in h0(bn) = {h ∈ h(bn) : fix(h) ⊃ sn−1(the boundary of bn)}, then n ⊃ h0(bn). the objective of the reference [2] is to analyze the algebraic structure of h(i) in some detail for some interval i. for a subgroup h of π(x), x ∈ x, let hx := isotropy subgroup of h at x. the authors proved that every translation is a product of two involutions and every element of h(i) is a product of at most four involutions. they considered a signature theorem, which provides a useful criteria for the conjugacy in h(i). using this idea, they enumerate completely the normal subgroups of h := h(i) ≤ π(i). let f be the isotropy subgroup h0 of h(r). the idea of proof is as follows. for an interval i, we denote i0 for the interior of i. for a map f : i → i, s(f,x) :=sign(f(x) −x). an element t ∈ f is a translation if it does not have interior points as fixed points and let t denote the set of all translations. also t+ := {t ∈ t : s(t,x) > 0} and t− := {t ∈ t : s(t,x) < 0}. now, let s be a semi-group which is invariant in f, qa = {f ∈ h : f(x) = x for all x in some neighborhood nf (a)} for a ∈ i and q = q0 ∩ q1. the authors proved that if s 6⊂ q0, then s contains an element with at most one interior fixed point. also t+ and t− are complete conjugacy classes in f, t is a complete conjugacy in h and f = tt. using these ideas, the authors also proved that if n is an invariant subgroup of f , then n ⊂ q0 or n ⊂ q1 or n = f. if n is an invariant subgroup of h then either n = h, n = f or n ⊂ q. now the only subgroups of h are h,f,q and {1} since q is simple. if n is normal in q0 (respectively in q1), then either n = {1}, n = q or n = q0 (respectively in q1). hence the only normal subgroups of f are f,q0,q1,q and {1}. the reference [6] is an expository paper, the author provides a relatively complete but concise account of the classification of h := h(i), in terms of a suitable topological signature concept. for φ ∈ h, the author first associated the function s(φ) : r → s = {−1, 0, 1} defined by s(φ)(x) = sign(φ(x) − x). for s ∈ σ := {h : r → s : h is continuous}, let spt(s) = r \ int(s−1(0)), ha = {φ ∈ h : spt(φ) is bounded above}, hb = {φ ∈ h : spt(φ) is bounded below} and hc = {φ ∈ h : spt(φ) is bounded}. the author first observed the following facts: (1) s(φ−1) = −s(φ) (2) for φ1,φ2 ∈ h+ (the set of all increasing homeomorphisms) with s(φ1) ≥ 0 and s(φ2) ≥ 0, it holds s(φ1 ◦φ2) ≥ max{s(φ1),s(φ2)}. © agt, upv, 2022 appl. gen. topol. 23, no. 2 271 k. ali akbar and t. mubeena these facts provide a one-to-one correspondence between the collection of normal subgroups n of h (resp. h+) and σ(n), the family of s-functions closed under the operation (s(φ),s(ψ)) → s(φ−1 ◦ ψ) and closed under topological equivalence. consider the group h(r) = {f : r → r : f is a homeomorphism} under composition of functions, and its subgroups ih(r) = {f ∈ h(r) : f is increasing}, hl = {f ∈ ih(r) : fix(f)c is bounded above} and hr = {f ∈ ih(r) : fix(f)c is bounded below}. our main results prove that: (1) the group ih(r) has exactly five normal subgroups. they are: (a) the whole group ih(r) (b) the trivial group {1} (c) hl (d) hr (e) h = hl ∩hr. (2) for h(r) there are exactly four normal subgroups. they are: (a) the whole group h(r) (b) the trivial group {1} (c) h = {f ∈ ih(r) : fix(f)c is bounded} (d) ih(r). 2. main results let ih([a,b]) denote the group (under composition of functions) of all increasing homeomorphisms on the closed interval [a,b] and let h([a,b]) denote the group (under composition of functions) of all homeomorphisms on the closed interval [a,b]. in fact h(r) and h([a,b]) are topological groups with respect to compact-open topology. this happens since the homeomorphism group on a locally connected and locally compact second countable space is a topological group (see [1]). consider r∗ = r ∪{−∞,∞} with order topology. any closed interval [a,b] in r is homeomorphic to r∗ = r∪{−∞,∞}, and the groups ih([a,b]) and ih(r∗) are isomorphic. we write: (1) ca([a,b]) = {f : [a,b] → [a,b] : f is a homeomorphism such that f(t) > t ∀ t ∈ (a,b)} (2) cb([a,b]) = {f : [a,b] → [a,b] : f is a homeomorphism such that f(t) < t ∀ t ∈ (a,b)}. (3) h = hl ∩hr = {f ∈ ih(r) : fix(f)c is bounded} for f ∈ ca([a,b])∪cb([a,b]), we have f(a) = a and f(b) = b. hence ca([a,b]) and cb([a,b]) are subsets of ih([a,b]). we define ca([a,b]) ◦ cb([a,b]) := {f ◦g : f ∈ ca([a,b]),g ∈ cb([a,b])}. two continuous maps f,g : r → r are said to be order conjugate if there exists an increasing homeomorphism h on r such that h◦f = g◦h. the maps f,g : r → r defined by f(x) = x + 1 and g(x) = x−1 are conjugate to each other through h(x) = −x+ 1 2 ∈ h(r). but f and g are not order conjugate. contrary to assume there is an h ∈ ih(r) such © agt, upv, 2022 appl. gen. topol. 23, no. 2 272 on the group of homeomorphisms on r: a revisit that h◦f = g◦h. then h(x+ 1) = h(x)−1. that is, h(x+ 1)−h(x) = −1 < 0. which is a contradiction to the assumption that h ∈ ih(r) and hence the maps f and g are not order conjugate. lemma 2.1. (1) assume that f,g ∈ ih(r) are such that fix(f) = fix(g) (a) f is conjugate to g; (b) if for every t ∈ r it holds (f(t)−t)(g(t)−t) ≥ 0 then f and g are order conjugate. (2) assume that f,g ∈ ih([a,b]) are such that fix(f) = fix(g) (a) f is conjugate to g; (b) if for every t ∈ [a,b] it holds (f(t) − t)(g(t) − t) ≥ 0 then f and g are order conjugate. proof. (1) assume that f,g ∈ ih(r) are such that fix(f) = fix(g) (a) case 1: fix(f) = fix(g) = ∅ assume that f(0) > 0. for n ∈ n, inductively define f−n = f−n+1 ◦f−1. since f is increasing (fn(0)) increases and thus diverges to ∞, and (f−n(0)) decreases and diverges to −∞. moreover, for t ∈ r there exists unique n ∈ z such that, fn(0) ≤ t < fn+1(0). consider this unique n, and define k : (−∞,f(0)) → (−∞, 1) by k(t) = t f(0) and h : r → r by h(t) = k(f−n(t)) + n for t ∈ r. note that h(f(0)) = 1 and h is a homeomorphism of r. then h◦f(t) = h(t) + 1 ∀ t ∈ r. this h gives a conjugacy from f to x + 1. similarly we can prove that, f is conjugate to x− 1 if f(0) < 0. the maps x + 1 and x− 1 are conjugate to each other. hence the proof. case 2: fix(f) = fix(g) 6= ∅ in this case, define f̃ : r∗ → r∗ by f̃|r (the restriction map f̃ to r)= f, f̃(−∞) = −∞ and f̃(∞) = ∞. similarly define g̃ also. let fix(f̃)c = fix(g̃)c = ∪α(aα,bα) (disjoint union of open intervals). the restriction maps f̃|[aα,bα] : [aα,bα] → [aα,bα] and g̃|[aα,bα] : [aα,bα] → [aα,bα] are increasing with aα = f̃(aα) = g̃(aα) and bα = f̃(bα) = g̃(bα) and fix(f̃|(a,b)) = fix(g̃|(a,b)) = ∅. let hα be a conjugacy from f̃|[aα,bα] to g̃|[aα,bα] for every α. by case 1, this conjugacy hα exists for every α. define h : r∗ → r∗ as h(x) = { hα(x) if x ∈ (aα,bα) x otherwise . then h is a conjugacy from f̃ to g̃. hence h|r is a conjugacy from f to g. (b) suppose (f(t)−t)(g(t)−t) ≥ 0 for all t ∈ r. if fix(f) = fix(g) = ∅ then f(0)g(0) > 0. this implies either both f(0) and g(0) are positive or both f(0) and g(0) are negative. hence both f and g are either order conjugate to x+1 or to x−1. if fix(f) = fix(g) 6= ∅ then consider the maps f̃, g̃ as in case 2 of (1) (a) in the proof of lemma 2.1. if fix(f̃)c = fix(g̃)c = ∪α(aα,bα) (disjoint union © agt, upv, 2022 appl. gen. topol. 23, no. 2 273 k. ali akbar and t. mubeena of open intervals) then the restriction maps f̃|[aα,bα] : [aα,bα] → [aα,bα] and g̃|[aα,bα] : [aα,bα] → [aα,bα] are increasing with aα = f̃(aα) = g̃(aα) and bα = f̃(bα) = g̃(bα), and (f̃(t)−t)(g̃(t)−t) > 0 for all t ∈ (aα,bα). if hα is an order conjugacy from f̃|[aα,bα] to g̃|[aα,bα] for every α then the map h : r ∗ → r∗ defined by h(x) = { hα(x) if x ∈ (aα,bα) x otherwise is an order conjugacy from f to g. hence the proof follows. (2) assume that f,g ∈ ih([a,b]) are such that fix(f) = fix(g). without loss of generality, we can assume that f,g ∈ ih(r∗). then f(−∞) = −∞ and f(∞) = ∞. hence fix(f|r) = fix(g|r). (a) by (1) (a) of lemma 2.1, f|r is conjugate to g|r and hence f is conjugate to g; (b) by (2) (a) of lemma 2.1, if for every t ∈ r it holds (f(t)−t)(g(t)− t) ≥ 0 then f|r and g|r are order conjugate and hence f is order conjugate to g. � for a map f : r → r, if t1, t2 ∈ fix(f) and s /∈ fix(f) for all s ∈ (t1, t2) then we say that t1 and t2 are adjacent. lemma 2.2. let f ∈ ih(r) and let {{aα,bα}}α be the pairs of adjacent fixed points. define g : r → r by g(x) = { x+f(x) 2 if aα < x < bα for some α f(x) otherwise then f is order conjugate to g. proof. the proof follows from lemma 2.1. � proposition 2.3. ca([a,b]) ◦cb([a,b]) = ih([a,b]). proof. let h ∈ ih([a,b]). define ha(x) := { h(x) if h(x) > x x otherwise and hb(x) :={ h(x) if h(x) < x x otherwise . then h = ha ◦ hb. but ha /∈ ca([a,b]) and hb /∈ cb([a,b]). now consider g(x) = x + 1. then g(x) > x for all x ∈ r. let h′a = ha ◦ g and h ′ b = g −1 ◦ hb. then h ◦ g(x) > h(x) for all x. therefore h ◦ g(x) > x whenever h(x) > x. hence h′a(x) > x if h(x) > x. if h(x) ≤ x then h′a(x) = g(x) > x. hence h ′ a ∈ ca([a,b]). similarly we can prove that h′b(x) ∈ cb([a,b]). hence the proof follows since h = h ′ a ◦h ′ b. � corollary 2.4. if n is a normal subgroup of ih([a,b]) that contains an element of either ca([a,b]) or cb([a,b]) then n = ih([a,b]). proof. by lemma 2.1, the sets ca([a,b]) and cb([a,b]) are exactly the conjugacy classes of ih([a,b]), and ca([a,b]) = {f−1 : f ∈ cb([a,b])}. so if subgroup n is normal and intersect either ca([a,b]) or cb([a,b]), then it automatically contains these sets. hence n = ih([a,b]) by proposition 2.3. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 274 on the group of homeomorphisms on r: a revisit we introduce the following notation: for f ∈ ih(r) and t0 ∈ fix(f), we denote ft0 (x) := { x+f(x) 2 if x ≥ t0 f(x) if x < t0 , f∗t0 (x) := { f(x) if x ≥ t0 x if x < t0 , and f∗∗t0 (x) :={ x if x ≥ t0 f(x) if x < t0 . lemma 2.5. let f ∈ ih(r) and let t0 ∈ fix(f). then f∗t0 is order conjugate to f−1t0 ◦f. proof. for f ∈ ih(r), first observe that ft0|[t0,∞) is x+f(x) 2 and f∗t0|[t0,∞) is f(x). hence by lemma 2.1, ft0|[t0,∞) is order conjugate to f ∗ t0 |[t0,∞). for t ∈ [t0,∞), first suppose that f(t) − t ≥ 0. then f(t) ≥ t+f(t) 2 . which implies f−1t0 (f(t)) ≥ f −1 t0 ( t+f(t) 2 ) = t. if f(t) − t ≤ 0 then we can prove that f−1t0 (f(t)) ≤ t. hence (f(t) − t)((f −1 t0 ◦ f)(t) − t) ≥ 0 for all t ∈ [t0,∞). by lemma 2.1, f∗t0 is order conjugate to f −1 t0 ◦f on [t0,∞). also f−1t0 ◦f|(−∞,t0) is the identity function. hence the proof follows. � corollary 2.6. let n be a normal subgroup of ih(r). let f ∈ n and let t0 ∈ fix(f). then f∗t0 ∈ n. proof. by lemma 2.1, ft0 is order conjugate to f and by lemma 2.5, f ∗ t0 is order conjugate to f−1t0 ◦f. hence the proof follows. � proposition 2.7. let n be a normal subgroup of ih(r). if there exists an f ∈ n with fix(f) 6= ∅ is bounded above then hr ⊂ n. proof. let n be a normal subgroup of ih(r) and let f ∈ n such that fix(f) is bounded above and let t0 = sup fix(f), the supremum of fix(f). if g ∈ n with g(t0) = t0 and h ∈ ih(r) with h(t0) = t0 then h|[t0,∞)◦g|[t0,∞)◦h −1|[t0,∞) is the same as h◦g◦h−1|[t0,∞). hence n|[t0,∞)= {g|[t0,∞) : g ∈ n, g(t0) = t0} is a normal subgroup of ih([t0,∞)). since t0 = sup fix(f), either f(t) > t or f(t) < t on (t0,∞). that is, f|[t0,∞] ∈ ca([t0,∞]) or cb([t0,∞]). then by corollary 2.1, it follows that n|[t0,∞) = ih([t0,∞)). now let φ ∈ hr. choose a fixed point s0 of φ such that every number less than s0 is also a fixed point of φ. consider χ(t) = t0 −s0 + φ(t−t0 + s0). if τ(t) = t−t0 + s0 then φ◦τ = τ ◦χ. hence φ is order conjugate to χ. observe that χ = χ∗t0 . hence the order conjugate χ of φ is the identity outside [t0,∞). now χ|[t0,∞) ∈ ih([t0,∞)). hence there exists χ̃ ∈ n such that χ̃ = χ on [t0,∞). thus χ̃∗t0 = χ, and hence χ ∈ n. then φ ∈ n since n is normal. hence hr ⊂ n. � remark 2.8. let n be a normal subgroup of ih(r). if there exists f ∈ n such that fix(f) 6= ∅ is bounded below then by considering analogues arguments involved in the proof of proposition 2.7, we have hl ⊂ n. remark 2.9. let n be a subgroup of ih(r∗). if fix(f) = {−∞,∞} for all f ∈ n then either f ∈ ca([−∞,∞]) or f ∈ cb([−∞,∞]). hence by © agt, upv, 2022 appl. gen. topol. 23, no. 2 275 k. ali akbar and t. mubeena proposition 2.3, n = ih(r∗). from this it follows that, if n is a subgroup of ih(r) with fix(f) = ∅ then n = ih(r). remark 2.10. if n be a normal subgroup of ih(r) with f ∈ n and t0 ∈ fix(f), then analogues to corollary 2.6, we can prove that f∗∗t0 ∈ n. corollary 2.11. let t1 < t2 be adjacent fixed points of some f ∈ n where n is a normal subgroup of ih(r). if g(x) = { f(x) if t1 < x < t2 x otherwise then g ∈ n. proof. observe that g = f∗t1 ◦f ∗ t2 −1. hence g ∈ n by corollary 2.6. � the following two lemmas are important to prove our main theorem. we consider these lemmas before considering our main theorem. we first make a back ground to complete the proof of following lemma. let f : r → r ∈ ih(r) with unique fixed point a. define g : r → r by g(t) = f(a + t) − a for t ∈ r. then g(0) = 0 if and only if f(a) = a, and g is order conjugate to f by the order conjugacy h(t) = a + t for t ∈ r. by lemma 2.1, there are only 3 elements in ih(r) with a unique fixed point upto order conjugacy. let f(x) = { 2x if x ≥ 0 x 2 if x < 0 for x ∈ r and g(x) :=   2x if x ≥ 1 1 2 (3x + 1) if − 1 ≤ x ≤ 1 x−1 2 if x ≤−1 . observe that f has a unique fixed point at 0 and g has unique fixed point at −1. then the map g◦f has no fixed points. by corollary 2.6 and remark 2.10, it follows that, if n is a non-trivial normal subgroup of ih(r) and contains an element with a unique fixed point then it contains an element without fixed points. next, let f,g : r → r be such that f(x) = { x2 + 1 if 0 ≤ x ≤ 1 x + 1 otherwise and g(x) = x − 1. then f and g has no fixed points, and g ◦ f has only two adjacent fixed points 0 and 1. lemma 2.12. the group h is the smallest non-trivial proper normal subgroup of ih(r). proof. let n be a non-trivial normal subgroup of ih(r). suppose there exists f ∈ n with adjacent fixed points t1 < t2. if φ is an element of ih([t1, t2]) such that φ = f|[t1,t2] then by corollary 2.11, the extension φ̃ : r → r defined by φ̃ = { φ(x) if t1 < x < t2 x otherwise is also in n. let h ∈ h = {f ∈ ih(r) : fix(f)c is bounded}. without loss of generality assume that t1 be the infimum of fix(h)c and t2 be the supremum of fix(h) c. then t1, t2 ∈ fix(h). by lemma 2.2, there exists an order conjugate ĥ of h such that φ̃◦ĥ−1 ∈ n. then ĥ ∈ n since ĥ ◦ φ̃ ◦ ĥ−1 ∈ n. which implies h ∈ n. this proves h ⊂ n whenever there exists f ∈ n with adjacent fixed points. by lemma 2.1, any homeomorphisms on r without fixed points is either order conjugate to x + 1 or to x−1. if n contains an element with a unique fixed point then it contains an element without fixed points. therefore it follows that n always contains an element with atleast two adjacent fixed points. hence the proof. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 276 on the group of homeomorphisms on r: a revisit lemma 2.13. let n be a normal subgroup of ih(r). if f ∈ n such that both fix(f) and fix(f)c are not bounded above, then n contains an element such that its set of all fixed points is bounded above. proof. let f ∈ n and t0 ∈ r be a fixed point of f. by corollary 2.6, f∗t0 ∈ n. let {(aα,aα+1)}α be the collection of all intervals of r such that f∗t0 (aα) = aα, f∗t0 (aα+1) = aα+1 and either f ∗ t0 (t) > t or f∗t0 (t) < t for all t ∈ (aα,aα+1). consider a collection of intervals {(bα,bα+1)}α of r such that bα < aα < bα+1 < aα+1 for all α and an increasing homeomorphism g on r which is order conjugate to f∗t0 such that g(bα) = bα, g(bα+1) = bα+1 and either g(t) > t or g(t) < t for all t ∈ (bα,bα+1). this is possible since ih([a,b]) is isomorphic to ih([c,d]) for any intervals [a,b] and [c,d]. without loss of generality we can assume that f∗t0 and g do not have common fixed points which are greater than t0. then f ∗ t0 ◦g ∈ n and fix(f∗t0 ◦g) is bounded above. hence the proof. � remark 2.14. let n be a normal subgroup of ih(r). (1) if there exists f ∈ n such that fix(f) and fix(f)c are not bounded above. then hr ⊂ n by proposition 2.7 and lemma 2.13. (2) if there exists f ∈ n with fix(f) and fix(f)c are not bounded below then hl ⊂ n. this follows by considering analogous arguments involved in the proof of lemma 2.13 and by remark 2.8. now we are ready to prove our main theorems: theorem 2.15. the group ih(r) has exactly five normal subgroups. they are: (1) the whole group ih(r) (2) the trivial group {1} (3) hl (4) hr (5) h = hl ∩hr. proof. let n be a non-trivial normal subgroup of the group ih(r). suppose that the function x+1 is in n. we claim that n = ih(r). consider a function f ∈ ih(r), and let g = (f∨(x−1))∧(x+ 1), where ∨, ∧ denote the maximum, and minimum of functions respectively. then g ∈ ih(r) and it is at a distance ≤ 1 from the diagonal. then fix(f) = fix(g), and f(x)−x and g(x)−x have the same sign between any two adjacent fixed points. hence g is order conjugate to f. now it is enough to prove g ∈ n. as g(x) ≥ x−1, ∀x ∈ r, we have g(x) + 2 > x ∀x ∈ r and therefore the function g(x) + 2 is order conjugate to the function φ(x) = x+ 1, since any two elements of ih(r) whose graphs are above the diagonal are order conjugate. then g + 2 ∈ n and hence φ−1 ◦φ−1 ◦ (g + 2) = g ∈ n, where (g + 2)(x) = g(x) + 2 for x ∈ r. but f is order conjugate to g. therefore f ∈ n. thus n = ih(r). similarly, if x− 1 is in n then also we can prove that n = ih(r). if there is an element f in n without fixed point then n = ih(r). this is because f is order conjugate to either x− 1 or x + 1 by lemma 2.1. © agt, upv, 2022 appl. gen. topol. 23, no. 2 277 k. ali akbar and t. mubeena now, let φ ∈ n be such that it has arbitrarily large fixed points and arbitrarily large non-fixed points (that is, fix(φ) and fix(φ)c are not bounded above). then by lemma 2.13 and by remark 2.14, hr ⊂ n. analogues to the above claim, if there exists φ ∈ n such that fix(φ) and fix(φ)c are not bounded below then hl ⊂ n by remark 2.14. now suppose fix(φ)c is bounded below for every φ ∈ n. then n ⊂ hr. also if fix(φ)c is bounded above for every φ ∈ n then n ⊂ hl. therefore from the following table we conclude that either n ⊂ hl or hr ⊂ n. (1) there exists ψ ∈ n such that then hr ⊂ n fix(ψ) is bounded above (2) there exists φ ∈ n such that then (1) by lemma 2.13 neither fix(φ) nor fix(φ)c is bounded above (3) for every φ ∈ n, fix(φ)c is bounded above then n ⊂ hl similarly by considering the following table analogues to the above table, we can show that n ⊂ hr or hl ⊂ n. (1) there exists ψ ∈ n such that then hl ⊂ n fix(ψ) is bounded below (2) there exists φ ∈ n such that then (1) by remark 2.14 neither fix(φ) nor fix(φ)c is bounded below (3) for every φ ∈ n, fix(φ)c is bounded below then n ⊂ hr now there are only four possibilities for a non-trivial normal subgroup n of ih(r) : case: 1 n ⊂ hl and n ⊂ hr in case: 1, n ⊂ hl ∩hr = h. therefore n = h. case: 2 n = hl case: 3 n = hr case: 4 hr ⊂ n and hl ⊂ n in case 4, hr ∪ hl ⊂ n. let f(x) := { x if x ≥ 0 1 2 x if x < 0 and g(x) :=  2x if x ≥ 1 3x+1 2 if − 1 ≤ x ≤ 1 x if x ≤−1 . then f ∈ hl and g ∈ hr. hence g ◦f ∈ n and it has no fixed points. therefore x + 1 ∈ n. hence n = ih(r). © agt, upv, 2022 appl. gen. topol. 23, no. 2 278 on the group of homeomorphisms on r: a revisit by lemma 2.12, h is the smallest normal subgroup contained in ih(r). hence the other two cases hr ⊂ n ⊂ hl and hl ⊂ n ⊂ hr are not possible. this completes the proof. � remark 2.16. let n be a non-trivial normal subgroup of the group ih(r) and suppose that there is an element in n such that it has only two fixed points. let φ be in n such that it has only two fixed points namely a and b, a < b. consider r∗ = r ∪{−∞,∞} with order topology. if φ ∈ ih(r∗) fixes only −∞ and ∞ then φ|r ∈ ih(r) and it has no other fixed points. let n∗ be a normal subgroup of ih(r∗) containing a map, fixing only −∞ and ∞. consider n∗ = {f|r : f ∈ n∗}. then n∗ is a normal subgroup of ih(r) and φ|r ∈ n∗. then φ|r has no fixed points and hence n∗ = ih(r). therefore n∗ = ih(r∗). then n contains an element without fixed points since ih(r), ih(r∗) and ih([a,b]) are isomorphic. hence n = ih(r). in this case, note that hl ∪hr ⊂ n, and therefore n becomes ih(r) since there is an element in n without fixed points. for a subgroup h of a group g, we denote h ≤ g. theorem 2.17. for h(r) there are exactly four normal subgroups. they are: (1) the whole group h(r). (2) the trivial group {1}. (3) h = {f ∈ ih(r) : fix(f)c is bounded}. (4) ih(r). proof. let g be a group and k ≤ n ≤ g. if k is normal in g then k is normal in n also. the subgroups hl and hr are not normal in h(r). hence by theorem 2.15, if n = ih(r) and g = h(r), then either k = {1} or k = h or k = ih(r). next suppose ih(r) ≤ k ≤ h(r). then the index [h(r) : k] ≤ [h(r) : ih(r)] = 2. therefore either k = h(r) or k = ih(r). finally, suppose there is a normal subgroup n of h(r) such that n = a∪b with a is a proper subgroup of ih(r) and ∅ 6= b ⊂ h(r) \ ih(r). which implies ih(r)∪b is a normal subgroup of h(r). but there is no such normal subgroup for h(r) other than h(r) itself. hence the proof. � acknowledgements. the authors are very thankful to the referee for giving valuable suggestions. the first author acknowledges serb-matrics grant no. mtr/2018/000256 for financial support. the second author acknowledges university of calicut, seed money (u.o. no. 11733/2021/admn; dated: 11.10.2021), india for financial support. © agt, upv, 2022 appl. gen. topol. 23, no. 2 279 k. ali akbar and t. mubeena references [1] r. arens, topologies for homeomorphism groups, american journal of mathematics 68 (1946), 593–610. [2] n. j. fine and g. e. schweigert, on the group of homeomorphisms of an arc, annals of mathematics 62 (1955), 237–253. [3] m. brin and g. stuck, introduction to dynamical systems, cambridge university press, 2002. [4] r. l. devaney, an introduction to chaotic dynamical systems, addison-wesley publishing company advanced book program, redwood city, ca, second edition, 1989. [5] i. n. herstein, topics in algebra, john wiley and sons, 2nd revised edition, 1975. [6] a. g. o’farrell, conjugacy, involutions, and reversibility for real homeomorphisms, irish math. soc. bulletin 54 (2004), 41–52. [7] s. ulam and j. von neumann, on the group of homeomorphisms of the surface of the sphere, (abstract), bull. amer. math. soc. 53 (1947), 506. [8] j. v. whittaker, normal subgroups of some homeomorphism groups, pacific j. math. 10, no. 4 (1960), 1469–1478. © agt, upv, 2022 appl. gen. topol. 23, no. 2 280 @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 11-19 some results and examples concerning whyburn spaces ofelia t. alas, maira madriz-mendoza and richard g. wilson abstract we prove some cardinal inequalities valid in the classes of whyburn and hereditarily weakly whyburn spaces and we construct examples of non-whyburn and non-weakly whyburn spaces to illustrate that some previously known results cannot be generalized. 2010 msc: primary 54d99; secondary 54a25; 54d10; 54g99 keywords: whyburn space, weakly whyburn space, submaximal space, scattered space, semiregular, feebly compact 1. introduction a hausdorff space x is said to be whyburn if whenever a ⊆ x is not closed and x ∈ cl(a) \ a, there is b ⊆ a such that cl(b) \ a = {x}. the space is weakly whyburn if whenever a ⊆ x is not closed, there is b ⊆ a such that |cl(b) \ a| = 1. these classes of spaces have been studied previously in [1], [4] and also earlier in [7] and elsewhere under the names ap-spaces, and wap-spaces. if a ⊆ x, then the whyburn closure of a, denoted by wcl(a) is defined as a∪ ⋃ {cl(c) : c ⊆ a, |cl(c) \a| = 1}. it follows immediately that a space is weakly whyburn if and only if every whyburn closed set is closed. undefined terminology can be found in [2] or [5] and all spaces are assumed to be (at least) hausdorff. 12 o. t. alas, m. madriz-mendoza and r. g. wilson 2. whyburn and weakly whyburn spaces in [4], a pseudocompact whyburn space which is not fréchet was constructed and proposition 2.1 of [7] states that a weakly whyburn compact hausdorff space must have a non-trivial convergent sequence. it is easy to see that this latter result generalizes to countably compact hausdorff spaces and also to feebly compact spaces with an infinite set of isolated points (recall that a space is feebly compact if every locally finite family of non-empty open sets is finite). the question then arises whether this result is true for all pseudocompact or feebly compact spaces. to answer this question we need the following terminology. if y is a non-empty scattered space, then we set y0 = {x : {x} is open} and for each ordinal α, yα = {x : {x} is open in y \ ⋃ {yβ : β < α}}. the dispersion order of y is then the least ordinal for which yα = ∅. for the sequel, we note that for each n ∈ ω, the dispersion order of the countable ordinal ωn + 1 is n + 1. we also need two lemmas, the simple proof of the first of which we omit. lemma 2.1. for n ∈ ω, an infinite scattered subset a of a t1-space x has dispersion order at most n if and only if it is the union of n discrete subspaces. lemma 2.2. if y is a scattered metric space of finite dispersion order n + 1, where n ≥ 1, and x ∈ yn, then for any � > 0, there is an embedding h : ωn + 1 → y such that h(ωn) = x and diam(h[ωn + 1]) ≤ �. proof. the proof is by induction on the dispersion order of y . if n = 1, then each point x ∈ y1 is the limit of a sequence s in y0; s can be taken to have arbitrarily small diameter and s ∪{x} is homeomorphic to ω + 1. suppose now that the result is true for each n < k and let y be a scattered space of dispersion order k + 1. suppose that x ∈ yk and � > 0; pick a sequence 〈xm〉 = s ⊆ yk−1 converging to x such that diam(s) < �/2. since y is hereditarily collectionwise normal, we may find mutually disjoint open sets um such that xm ∈ um; each set um is scattered and has dispersion order k. applying the inductive hypothesis, for each m ∈ ω, we may find an embedding hm : ω k−1 + 1 → um such that hm(ωk−1) = xm and such that diam(tm) < �/4 m, where tm = h[ω k−1 + 1]. let t = ⋃ {tm : m ∈ ω}∪{x}; it is straightforward to check that t is homeomorphic to ωk + 1 since each neighbourhood of x contains all but finitely many of the sets tm; furthermore, diam(t) < �/2 + �/4 + �/16 < �. � example 2.3. there is a whyburn h-closed (hence feebly compact) hausdorff space with no non-trivial convergent sequences. proof. we consider the space x = [0, 1] with the usual metric topology µ. let τ be the topology on x generated by µ∪{x \d : d ⊆ x is µ−discrete}. some results and examples concerning whyburn spaces 13 since {x \d : d ⊆ x is µ−discrete} is a filter of dense subsets of (x,µ) it follows that (x,τ) is h-closed. furthermore, it is clear that (x,τ) is hausdorff and has no convergent non-trivial sequences. even more is true: it follows from lemma 2.1 that every scattered subspace of (x,µ) of finite dispersion order is closed in the topology τ. we will show that (x,τ) is a whyburn space. to this end, suppose that a ⊆ x is not closed and let x ∈ clτ (a) \ a. now in (x,µ), a is the union of a scattered subset c ⊆ a and a dense-in-itself subset b ⊆ a, hence either (i) x ∈ clτ (b) or (ii) x ∈ clτ (c). we consider the cases separately. (i) since b is dense-in-itself, every non-empty open subset of b contains a dense subset homeomorphic to the rationals, q. choose a nested local base at x of µ-closed sets v = {vn : n ∈ ω}; we may assume that vn+1 ⊆ int(vn) and b ∩ (int(vn) \ vn+1) 6= ∅ for each n ∈ ω. since q is universal for countable metric spaces, for each n ∈ ω, in the open subset b ∩ (int(vn) \vn+1) of b we may find a subspace dn homeomorphic to the compact ordinal ω n + 1 which has dispersion order n + 1; let d = ⋃ {dn : n ∈ ω}. it is easy to see that d is scattered and has dispersion order ω and since x ∈ clµ(d) a straightforward argument shows that clτ (d) \d = {x}. (ii) choose a nested local base at x of µ-closed sets v = {vn : n ∈ ω}. if x ∈ clτ (c), since each scattered subspace of (x,µ) of finite dispersion order is τ-closed, it follows that for each n ∈ ω, c ∩ vn has (countably) infinite dispersion order κ and since every countable limit ordinal has cofinality ω, we may assume without loss of generality that κ = ω. then, for each n ∈ ω, using the previous lemma we may find embeddings hn : ω n + 1 → vn ∩ c and it is not hard to see that the maps hn may be chosen so that if m 6= n, then tn ∩tm = ∅, where tk = hk[ωk + 1]. each of the sets tk is µ-compact and τdiscrete but t = ⋃ {tk : k ∈ ω} has infinite dispersion order and so x ∈ clµ(t). furthermore, since for each µ-neighbourhood v of x, the set t \v is τ-closed, it follows that cl(t) \c = {x}. � in the sequel d(x), l(x), t(x) and ψ(x) will denote respectively the density, tightness, lindelöf number and pseudocharacter of a space (x,τ) and ψ(x,x) will denote the pseudocharacter of x in x. if (x,τ) is a hausdorff space and x ∈ x, then let ψc(x,x) = min{|u| : {x} = ⋂ {cl(u) : x ∈ u ∈u ⊆ τ}}. theorem 2.4. a k-space is weakly whyburn if and only if for each nonclosed set a ⊆ x, there is some compact set k ⊆ x and x 6∈ a such that cl(k ∩a) = (k ∩a) ∪{x} = k. proof. the sufficiency is clear, since k∩a is not closed in k. for the necessity, suppose that (x,τ) is a hausdorff weakly whyburn k-space and that a ⊆ x is not closed in x. then there is some compact set c ⊆ x such that c ∩ a is not closed in c. since c is a closed subset of x, c is weakly whyburn and hence there is some x ∈ c \a and a set b ⊆ c ∩a such that cl(b)\a = {x}. clearly cl(b) is the required compact subset of x. � corollary 2.5. a weakly whyburn k-space is pseudoradial. 14 o. t. alas, m. madriz-mendoza and r. g. wilson proof. this is an immediate consequence of the previous lemma and the fact that a compact weakly whyburn space is pseudoradial (see [7]). � the next result extends theorem 3 of [1] to the class of hausdorff spaces. theorem 2.6. if x is a weakly whyburn lindelöf p-space and for each x ∈ x, ψ(x,x) < ℵω, then x is pseudoradial. proof. for any hausdorff space ψc(x,x) ≤ l(x)ψ(x,x) (see 2.8(c) of [3]) and hence ψc(x,x) < ℵω for each x ∈ x. let a ⊆ x be a non-closed set and b ⊆ a such that cl(b)\a = {x} for some x ∈ x. let u = {uα : α < κ} be a family of minimal cardinality κ of open sets in cl(b) such that ⋂ {cl(uα) : α < κ} = {x}. since x is a p-space and x is not isolated, κ is a regular uncountable cardinal. since κ is minimal, for each α ∈ κ we may choose xα ∈ ⋂ {cl(uβ) : β < α}\{x}⊆ a. since cl(b) is lindelöf, the set so constructed {xα : α < κ} must have a complete accumulation point z ∈ cl(b). since ⋂ {cl(uα) : α < κ} = {x} and the well-ordered net s = 〈xα〉α∈κ is finally in each set cl(uβ) it follows that z = x and x is the unique complete accumulation point of s. furthermore, s = 〈xα〉α∈κ must converge to x, for otherwise there would exist a subset of s of size κ with no complete accumulation point. � theorem 2.7. the product of two whyburn spaces, one of which is a k-space and the other is locally compact is weakly whyburn. proof. suppose that x is a whyburn k-space and y is a whyburn locally compact space. it is known (see [7]) that a compact whyburn hausdorff space is fréchet-urysohn and it is easy to see that the same is true of a whyburn hausdorff k-space. it then follows from 3.3.j of [2] that x × y is sequential and hence weakly whyburn. � question 2.8. is the product of two whyburn k-spaces, weakly whyburn? theorem 2.9. if x is weakly whyburn, then |x| ≤ d(x)t(x). proof. if x is finite, the result is trivial; thus we assume that x is infinite. suppose that d(x) = δ, t(x) = κ and d ⊆ x is a dense (proper) subset of cardinality δ. let d = d0 and define recursively an ascending chain of subspaces {dα : α ≤ κ+} as follows: since x is weakly whyburn, there is some x ∈ x\d and bx ⊆ d such that cl(bx) \d = {x}; clearly, we have |cl(bx)| ≤ δ ≤ δκ and we may assume that |bx| ≤ κ. we then define d1 = ⋃ {cl(b) : b ⊆ d0, |b| ≤ κ, |cl(b) \d0| = 1}. clearly d1 ! d0 and since there are at most δκ such sets b it follows that |d1| ≤ δκ. suppose now that for each β < α ≤ κ+ we have defined dense sets dβ such that |dβ| ≤ δκ and dγ ⊆ dλ whenever γ < λ < α. if α is a limit ordinal, then define dα = ⋃ {dβ : β < α} and then |dα| ≤ |α|.δκ ≤ κ+.δκ = δκ. if on the some results and examples concerning whyburn spaces 15 other hand α = β + 1, and dβ x, then since x is weakly whyburn there is some x ∈ x \dβ and bx ⊆ dβ such that cl(bx) \dβ = {x}. again we have that |cl(bx)| ≤ δκ and we may assume that |bx| ≤ κ. now we may define dα = ⋃ {cl(b) : b ⊆ dβ, |b| ≤ κ, |cl(b) \dβ| = 1}. clearly dα ! dβ and since there are at most (δκ)κ such sets b it follows that |dα| ≤ δκ. to complete the proof it suffices to show that for some α ≤ κ+, we have that dα = x. suppose to the contrary that ∆ = ⋃ {dα : α < κ+} 6= x; |∆| ≤ κ+.δκ = δκ. then, since x is weakly whyburn and has tightness κ, there is some z ∈ x \ ∆ and some set b ⊆ ∆ of cardinality at most κ, such that cl(b) \ ∆ = {z}. since the sets {dα : α < κ+} form an ascending chain and cf(κ+) > κ, it follows that for some γ < κ+, b ⊆ ⋃ {dα : α < γ} and hence z ∈ dγ+1, a contradiction. � lemma 2.10. if x is hereditarily weakly whyburn, then |x| ≤ 2d(x). proof. suppose to the contrary that |x| > 2d(x). let ∆ be a dense subset of x of minimal cardinality, a = {a ⊆ ∆ : |clx(a)| ≤ 2d(x)} and y =⋃ {clx(a) : a ∈ a}. since |p(∆)| = 2d(x), it follows that |y | ≤ 2d(x) and hence if we put z = ∆ ∪ (x \ y ), then |z| > 2d(x). now if b ∈ p(∆) \a, then |clx(b) ∩ z| > 2d(x) thus showing that ∆ is whyburn closed in z but not closed. thus z is not weakly whyburn and hence x is not hereditarily weakly whyburn. � 3. the whyburn property in scattered and submaximal spaces we recall our convention that all spaces are hausdorff. a space is said to be submaximal if every dense subset is open. a standard procedure for constructing submaximal topologies is as follows. suppose that (x,τ) is a (hausdorff) space and d is a maximal filter in the family of dense subsets of x. then the topology σ generated by the subbase τ ∪d is submaximal and is called a submaximalization of τ. note that σ is semiregular if and only if τ is semiregular and submaximal (then σ = τ). obviously, a scattered space is submaximal if and only if it has dispersion order 2. as we mentioned earlier, every regular scattered space is weakly whyburn and the katětov extension of ω (see 4.8(n) of [5]) shows that this is not true in the class of urysohn spaces. thus it is natural to ask the following two questions. (1) must a dense-in-itself submaximal whyburn space be regular?, and (2) is every scattered semiregular space whyburn? we give a partial answer to the first question by showing that a submaximalization of a resolvable space is never whyburn and answer the second by 16 o. t. alas, m. madriz-mendoza and r. g. wilson constructing a semiregular scattered space of dispersion order 2 which is not weakly whyburn. recall that a space is resolvable if it possesses two mutually disjoint dense subsets. theorem 3.1. a submaximalization of a resolvable hausdorff space is not weakly whyburn. proof. suppose that (x,τ) is a resolvable t2-space and f is a maximal filter of dense sets in x. we first show that there is f ∈ f such that x \ f is somewhere dense in x. to this end, suppose to the contrary that no such f exists, then for each f ∈ f, uf = x \ f is nowhere dense. now let d and d′ be complementary dense subsets of x; clearly d,d′ 6∈ f. for each f ∈f, since int(f) = x \ cl(x \ f), it follows that int(f) is dense in x and so too are d ∩ int(f) ⊆ d ∩f and d′ ∩ int(f)) ⊆ d′ ∩f. since f is maximal, any dense set which meets each element of f in a dense set is an element of f and so it follows that d ∈f and d′ ∈f contradicting the fact that f is a filter. now let σ be the topology generated by τ ∪f and f ∈ f be such that x \f is somewhere dense; thus intσ(clσ(x \f)) = u 6= ∅. let v = u ∩f , x ∈ clσ(v ) \ f and note that v is infinite. then if b ⊆ v is such that x ∈ clσ(b), it follows that w = intσ(b) 6= ∅. but then, clσ(w) ∩ (x \f) = clτ (w) ∩ (x \f) = clτ (clτ (w) ∩ (x \f)) ∩ (x \f) which is infinite. � an example of a scattered submaximal whyburn (even first countable) space which is not regular (nor even semiregular) is easy to construct. let q denote the rational numbers and x = q×{0, 1} with the following topology: each point of q×{0} is isolated and a basic open neighbourhood of (q, 1) is of the form {(q, 1)}∪ [(uq \{q})×{0}] where uq is a euclidean neighbourhood of q ∈ q. a space x is said to be ω-resolvable if x possesses infinitely many mutually disjoint dense subsets. the construction of the next example depends on the existence of a countable ω-resolvable hausdorff space which is not weakly whyburn. before constructing such a space, the following lemma is needed lemma 3.2. a space x, ω ⊆ x ⊆ βω is hereditarily weakly whyburn if and only if x is scattered. proof. the sufficiency is clear since a subspace of a scattered space is scattered and it was proved in [4] that a regular scattered space is weakly whyburn. furthermore, it is easy to see that if the dispersion order of x is 2, then it is whyburn also. for the inverse implication, suppose that d ⊆ x\ω is dense in itself and let y = ω ∪d; if y were weakly whyburn, then we could find p ∈ d and b ⊆ ω such that cly (b) \ ω = {p}, in other words, cly (b) = b ∪ {p}. however, cly (b) = clβω(b) ∩ y and so cly (b) ∩ d is an open subset of d to which p belongs; since p is not isolated, this set must be infinite. � some results and examples concerning whyburn spaces 17 by way of contrast to the last result we note that under ch the subspace of p-points of βω\ω has character ω1 and it then follows from proposition 2.7 of [4] that this space is whyburn. consider a countable dense-in-itself subset d ⊆ clβq(n)\n ⊆ βq\q (where once again, q denotes the set of rational numbers with the euclidean topology). let x = q∪d; x is a countable tychonoff space which, is clearly ω-resolvable. that n is whyburn closed in x follows from the previous lemma and the fact that clβq(n) is homeomorphic to βω. example 3.3. there is a semiregular scattered space (of dispersion order 2) which is not weakly whyburn. proof. let (z,σ) be an ω-resolvable (dense-in-itself) countable tychonoff space which is not weakly whyburn and let f be an infinite family of mutually disjoint dense subsets of (z,σ) and φ : z →f a bijection. let x = z ×{0, 1} and for each z ∈ z, let vz be an open neighbourhood base at z. we define a topology τ on x = z ×{0, 1} as follows: each point of z ×{0} is isolated and an open neighbourhood of (z, 1) is of the form wv,z = {(z, 1)}∪ (v ×{0}) \ ({(z, 0)}∪φ(z)), where v ∈vz. the space (x,τ) is a scattered space of dispersion order 2 and we proceed to show that it is neither regular nor weakly whyburn. it is easy to see that x is not regular since the open neighbourhood wv,z of (z, 1) contains no closed neighbourhood of that point. to prove that x is semiregular, it suffices to show that each of the sets wv,z is regular open. to see this, suppose that (t, 1) ∈ clx(wv,z) where t 6= z; then since φ(z) is dense in z, each neighbourhood of (t, 1) meets the set φ(z)×{0} showing that (t, 1) 6∈ intx(cl(wv,z)). finally, to show that (x,τ) is not whyburn, it suffices to prove that there is some a ⊆ z such that a×{0} is whyburn closed but not closed in x. however, z is not weakly whyburn and hence there is some a ⊆ z which is whyburn closed but not closed in z and so if b ⊆ a is such that clz(b)\a is nonempty, we must have clz(b)\a has no isolated points (and hence is infinite). we claim that if b ⊆ a is such that clz(b)\a is nonempty then clx(b×{0})\(a×{0}) is infinite. to prove our claim, suppose that s ∈ clz(b) \ a; then either (s, 1) ∈ clx(b ×{0}) \ (a×{0}) or not. if (s, 1) 6∈ clx(b ×{0}) \ (a×{0}) then there is some open neighbourhood u of s in z such that clz(u)∩b ⊆ φ(s) and u contains infinitely many points of clz(b)\a. if s 6= t ∈ u∩(clz(b)\a), then since b ∩u * φ(t), it follows that t ∈ clx(b ×{0}) \ (a×{0}), showing that clx(b ×{0}) \ (a×{0}) is infinite. � 18 o. t. alas, m. madriz-mendoza and r. g. wilson 4. some open questions the space constructed in example 2.3 is not regular, thus we are led to ask: question 4.1. does every (weakly) whyburn pseudocompact tychonoff space have a convergent sequence? a number of dense pseudocompact subspaces of {0, 1}c and ic have been constructed which do not possess a non-trivial convergent sequence (for example see [6]); however, the question of whether such constructions can produce a weakly whyburn space has apparently not been studied. question 4.2. is the bound ψ(x,x) < ℵω necessary in theorem 2.6? question 4.3. suppose that |x| > 2d(x); can x be weakly whyburn? question 4.4. does there exist in zfc a dense whyburn subspace of βω\ω? acknowledgements. this research was supported by the network algebra, topoloǵıa y análisis del promep, project 12611243 (méxico) and fundação de amparo a pesquisa do estado de são paulo (brasil). the third author wishes to thank the departament de matemàtiques de la universitat jaume i for support from pla 2009 de promoció de la investigació, fundació bancaixa, castelló, while working on an early draft of this article. references [1] a. bella, c. costantini and s. spadaro, p-spaces and the whyburn property, houston j. math. 37, no. 3 (2011), 995–1015. [2] r. engelking, general topology, heldermann verlag, berlin, 1989. [3] i. juhász, cardinal functions in topology ten years later, mathematisch centrum, amsterdam, 1980. [4] j. pelant, m. g. tkachenko, v. v. tkachuk and r. g. wilson, pseudocompact whyburn spaces need not be fréchet, proc. amer. math. soc. 131, no. 10 (2003), 3257–3265. [5] j. r. porter and r. g. woods, extensions and absolutes, springer verlag, new york, 1987. [6] e. a. reznichenko, a pseudocompact space in which only sets of complete cardinality are not closed and not discrete, moscow univ. math. bull. 6 (1989), 69–70 (in russian). [7] v. v. tkachuk and i. v. yashchenko, almost closed sets and the topologies they determine, comment. math. univ. carolinae 42, no. 2 (2001), 395–405. (received november 2009 – accepted november 2011) some results and examples concerning whyburn spaces 19 ofelia t. alas (alas@ime.usp.br) instituto de matemática e estat́ıstica, universidade de são paulo, caixa postal 66281, 05311-970 são paulo, brasil maira madriz-mendoza (seber@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, unidad iztapalapa, avenida san rafael atlixco, #186, apartado postal 55-532, 09340, méxico, d.f., méxico richard g. wilson (rgw@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, unidad iztapalapa, avenida san rafael atlixco, #186, apartado postal 55-532, 09340, méxico, d.f., méxico some results and examples concerning[6pt] whyburn spaces. by o. t. alas, m. madriz-mendoza and r. g. wilson @ appl. gen. topol. 22, no. 2 (2021), 417-434doi:10.4995/agt.2021.15231 © agt, upv, 2021 small and large inductive dimension for ideal topological spaces f. sereti university of patras, department of mathematics, 26504, patra, greece (seretifot@gmail.com) communicated by p. das abstract undoubtedly, the small inductive dimension, ind, and the large inductive dimension, ind, for topological spaces have been studied extensively, developing an important field in topology. many of their properties have been studied in details (see for example [1,4,5,9,10,18]). however, researches for dimensions in the field of ideal topological spaces are in an initial stage. the covering dimension, dim, is an exception of this fact, since it is a meaning of dimension, which has been studied for such spaces in [17]. in this paper, based on the notions of the small and large inductive dimension, new types of dimensions for ideal topological spaces are studied. they are called ∗-small and ∗-large inductive dimension, ideal small and ideal large inductive dimension. basic properties of these dimensions are studied and relations between these dimensions are investigated. 2010 msc: 54f45; 54a05; 54a10. keywords: small inductive dimension; large inductive dimension; ideal topological space. 1. introduction and preliminaries the dimension theory is a developing branch of topology, which attracts the interest of many researches (see for example [1, 2, 4–13, 18]). especially, the covering dimension, dim, the small inductive dimension, ind, and the large inductive dimension, ind, are three main topological dimensions, which have received 09 march 2021 – accepted 06 june 2021 http://dx.doi.org/10.4995/agt.2021.15231 f. sereti been studied extensively, and many results in various classes of topological spaces have been proved. simultaneously, the notion of ideal leads to an important chapter in topology (see for example [3,14,15,19]). the main notion of ideal topological space was studied in kuratowski’s monograph [16]. however, the notion of topological dimension has not been investigated under the prism of ideals. the covering dimension is an exception. in the paper [17], the meaning of the ideal covering dimension is inserted and studied in details. thus, in this paper new notions of inductive dimensions for ideal topological spaces are introduced and studied. they are called ∗-small and ∗-large inductive dimension, ideal small and ideal large inductive dimension. especially, in section 2, we insert the meanings of the so-called ∗-small inductive dimension, ind ∗ , and ∗-large inductive dimension, ind ∗ , for an arbitrary ideal topological space and study basic results. in sections 3, we insert and study the meanings of the ideal small inductive dimension, i ind, and the ideal large inductive dimension, i ind, and finally, in section 4, we study additional properties of the ideal topological dimensions. it is considered to be necessary, to recall the main notions and notations that will be used in the rest of this study. especially, the notion of the ideal topological space and the known meanings of the small inductive dimension and the large inductive dimension are presented. the standard notation of dimension theory is referred to [1,5,18]. a nonempty family i of subsets of a set x is called an ideal on x if it satisfies the following properties: (1) if a ∈ i and b ⊆ a, then b ∈ i. (2) if a, b ∈ i, then a ∪ b ∈ i. a topological space (x, τ) with an ideal i is called an ideal topological space and is denoted by (x, τ, i). in [15] the authors defined a new topology τ∗ on x in terms of the kuratowski closure operator cl∗. it is known that the family β∗ = {u \ i : u ∈ τ, i ∈ i} is a basis for τ∗ and the topology τ∗ is finer than τ. especially, if i = {∅}, then τ∗ = τ and if i = p(x), then τ∗ is the discrete topology. in what follows, by an open set (resp. closed set), we mean an open set (resp. closed set) in the topology τ. if a set u is open in the topology τ∗, then we say that u is a ∗-open set. similarly, we define ∗-closed sets. if a ⊆ x, then bdx(a) and bd ∗ x(a) will denote the boundary of a in (x, τ) and (x, τ ∗), respectively. similarly, clx(a) and cl ∗ x(a) will denote the closure of a in (x, τ) and (x, τ∗), respectively. also, if denotes the ideal of all finite subsets of x. definition 1.1. the small inductive dimension of a topological space x, denoted by ind(x), is defined as follows: (i) ind(x) = −1, if x = ∅. © agt, upv, 2021 appl. gen. topol. 22, no. 2 418 small and large inductive dimension for ideal topological spaces (ii) ind(x) 6 k, where k ∈ {0, 1, . . .}, if for every element x ∈ x and for every open subset v of x with x ∈ v , there exists an open subset u of x such that x ∈ u ⊆ v and ind(bdx(u)) 6 k − 1. (iii) ind(x) = k, where k ∈ {0, 1, . . .}, if ind(x) 6 k and ind(x) k − 1. (iv) ind(x) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which ind(x) 6 k is true. definition 1.2. the large inductive dimension of a topological space x, denoted by ind(x), is defined as follows: (i) ind(x) = −1, if x = ∅. (ii) ind(x) 6 k, where k ∈ {0, 1, . . .}, if for every pair (f, v ) of subsets of x, where f is closed, v is open and f ⊆ v , there exists an open set u of x such that f ⊆ u ⊆ v and ind(bdx(u)) 6 k − 1. (iii) ind(x) = k, where k ∈ {0, 1, . . .}, if ind(x) 6 k and ind(x) k − 1. (iv) ind(x) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which ind(x) 6 k is true. 2. the small inductive dimension and the large inductive dimension for ideal topological spaces in this section, based on the notion of the topology τ∗, the ∗-small and ∗large inductive dimension are defined for an ideal topological space (x, τ, i) and basic properties are studied. definition 2.1. the ∗-small inductive dimension of an ideal topological space (x, τ, i), denoted by ind∗(x), is defined as follows: (i) ind∗(x) = −1, if x = ∅. (ii) ind∗(x) 6 k, where k ∈ {0, 1, . . .}, if for every element x ∈ x and for every ∗-open subset v of x with x ∈ v , there exists a ∗-open subset u of x such that x ∈ u ⊆ v and ind∗(bd∗x(u)) 6 k − 1. (iii) ind∗(x) = k, where k ∈ {0, 1, . . .}, if ind∗(x) 6 k and ind∗(x) k−1. (iv) ind ∗ (x) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which ind∗(x) 6 k is true. definition 2.2. the ∗-large inductive dimension of an ideal topological space (x, τ, i), denoted by ind∗(x), is defined as follows: (i) ind∗(x) = −1, if x = ∅. (ii) ind∗(x) 6 k, where k ∈ {0, 1, . . .}, if for every pair (f, v ) of subsets of x, where f is ∗-closed, v is ∗-open and f ⊆ v , there exists a ∗-open set u of x such that f ⊆ u ⊆ v and ind∗(bd∗x(u)) 6 k − 1. © agt, upv, 2021 appl. gen. topol. 22, no. 2 419 f. sereti (iii) ind∗(x) = k, where k ∈ {0, 1, . . .}, if ind∗(x) 6 k and ind∗(x) k−1. (iv) ind∗(x) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which ind∗(x) 6 k is true. we observe that the dimensions ind∗(x) and ind∗(x) are the small and the large inductive dimension, respectively, of the topological space (x, τ∗). the following result will be useful in the rest of this study. theorem 2.3. (1) for any subset a of x, ind∗(a) 6 ind∗(x). (2) for any ∗-closed subset a of x, ind∗(a) 6 ind∗(x). proof. based on definition 2.1 and definition 2.2, (1) and (2) can be proved by induction on the dimension ind∗(x) and ind∗(x), respectively. � however, the following examples prove that these dimensions are different to each other and different from the small inductive dimension, ind, and the large inductive dimension, ind. example 2.4. let x = {a, b, c} with the topology τ = {∅, {b, c}, x}. if we consider the ideal i = {∅, {a}}, then ind∗(x) = 1 and ind∗(x) = 0. example 2.5. (1) we consider the set x = {a, b, c, d} with the topology τ = {∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}, x}. then ind(x) = 1. however, if we consider the ideal i = {∅, {a}, {b}, {a, b}}, then the space (x, τ∗) is the discrete space and thus, ind∗(x) = 0. (2) we consider the set x = {a, b, c} with the topology τ = {∅, x}. then ind(x) = 0. however, if we consider the ideal i = {∅, {b}}, then ind∗(x) = 1. example 2.6. (1) we consider the set x = {a, b, c, d} with the topology τ = {∅, {a}, {a, b}, {a, d}, {a, b, d}, {a, c, d}, x}. then ind(x) = 1. however, if we consider the ideal i = {∅, {a}, {d}, {a, d}}, then the space (x, τ∗) is the discrete space and thus, ind ∗ (x) = 0. (2) we consider the set x = {a, b, c, d} with the topology τ = {∅, {a}, {b}, {a, b}, {a, b, d}, x}. then ind(x) = 0. however, if we consider the ideal i = {∅, {b}, {d}, {b, d}}, then ind∗(x) = 1. remark 2.7. for any ideal topological space (x, τ, i) for which τ = τ∗ we have (1) ind(x) = ind∗(x) and (2) ind(x) = ind ∗ (x). however, the converse of remark 2.7 does not always hold and the following examples prove this claim. © agt, upv, 2021 appl. gen. topol. 22, no. 2 420 small and large inductive dimension for ideal topological spaces example 2.8. (1) let x = {a, b, c} with the topology τ = {∅, {b, c}, x}. then ind(x) = 1. if we consider the ideal i = {∅, {b}}, then ind∗(x) = 1 but τ 6= τ∗. (2) let x = {a, b, c, d} with the topology τ = {∅, {a}, {b}, {a, b}, {a, b, d}, x}. then ind(x) = 0. however, if we consider the ideal i = {∅, {b}}, then ind ∗ (x) = 0 but τ 6= τ∗. moreover, in the following propositions we can prove further relations between these dimensions. proposition 2.9. for every ideal topological space (x, τ, i), where i ⊆ τc, the following are satisfied: (1) ind(x) = ind∗(x) and (2) ind(x) = ind ∗ (x). proof. since i ⊆ τc, we have that β∗ ⊆ τ. therefore, τ = τ∗ and by remark 2.7 we have that ind(x) = ind∗(x) and ind(x) = ind∗(x). � proposition 2.10. for every ideal topological t1-space (x, τ, i), where i ⊆ if , the following are satisfied: (1) ind(x) = ind∗(x) and (2) ind(x) = ind∗(x). proof. since the ideal topological space (x, τ, i) is t1, every i ∈ i is closed in (x, τ). therefore, τ = τ∗ and by proposition 2.9 we have that ind(x) = ind∗(x) and ind(x) = ind∗(x). ✷ let (x, τ, i) be an ideal topological space. if β∗ is a topology on x (and hence τ∗ = β∗), then the ideal i is called τ-simple [14]. also, the ideal i is called τ-codense if i ∩ τ = {∅}, that is each member of i has empty interior with respect to the topology τ [3]. moreover, we state that if a ⊆ x, then the family ia = {a ∩ i : i ∈ i} is an ideal on a. so, we can consider the ideal topological space (a, τa, ia), where τa is the subspace topology on a. the topology (τa) ∗ is equal to the subspace topology (τ∗)a on a [15]. � proposition 2.11. let (x, τ, i) be an ideal topological space. if i is τ-simple and τ-codense and for each closed subset f of x the ideal if is τf -codense, then ind(x) 6 ind∗(x). © agt, upv, 2021 appl. gen. topol. 22, no. 2 421 f. sereti proof. clearly, if ind∗(x) = −1 or ind∗(x) = ∞, then the inequality holds. we suppose that the inequality holds for all integers m < k and we shall prove it for k. let ind∗(x) = k, x ∈ x and v be an open set in x with x ∈ v . then v is a ∗-open set in x. since ind∗(x) = k, there exists a ∗-open set u in x with x ∈ u ⊆ v and ind ∗ (bd ∗ x(u)) 6 k − 1. also, since i is τ-simple, u = w \ i, where w ∈ τ and i ∈ i. we consider the open set v ∩ w in x. then x ∈ u ⊆ v ∩ w ⊆ v . it suffices to prove that ind(bdx(v ∩ w)) 6 k − 1. let y = bdx(v ∩ w) with the subspace topology τy and the ideal iy . we have that iy is τy -simple, as β∗y is a topology on y , and τy -codense, as y is a closed subset of x. also, we should prove that for every closed set k in y , the ideal ik is τk-codense. let k be a closed set in y . since y is closed in x, k is also closed in x and thus, ik is τk-codense. moreover, we observe that bdx(v ∩ w) ⊆ bd ∗ x(u). indeed, we suppose that there exists y ∈ bdx(v ∩w) such that y /∈ bd ∗ x(u). we have that y ∈ x \ (v ∩ w) and thus, y ∈ x \ u. therefore, y /∈ cl∗x(u). that is, there exists a ∗-open set o in x with y ∈ o and o ∩ u = ∅. since i is a τ-simple ideal, we have that o = p \ j, where p ∈ τ and j ∈ i. thus, p is an open set in x with y ∈ p . since y ∈ clx(v ∩ w), we have that p ∩(v ∩w) 6= ∅ or equivalently, (p ∩w)∩v 6= ∅. since (p ∩w)∩v ⊆ p ∩w , we have that p ∩ w 6= ∅. also, the relation o ∩ u = ∅ implies the relation (p \ j) ∩ (w \ i) = ∅ and thus, p ∩ w ⊆ i ∪ j. since i is an ideal and i, j ∈ i, we have that i ∪ j ∈ i. thus, the member i ∪ j of the ideal i has non empty interior with respect to the topology τ, which is a contradiction as i is τ-codense. thus, bdx(v ∩ w) ⊆ bd ∗ x(u). by the subspace theorem for the dimension ind∗ (see theorem 2.3), we have that ind ∗ (bdx(v ∩ w)) 6 ind ∗ (bd ∗ x(u)) 6 k − 1 and by inductive hypothesis, we have that ind(bdx(v ∩ w)) 6 ind ∗(bdx(v ∩ w)). therefore, ind(bdx(v ∩ w)) 6 k − 1. thus, ind(x) 6 k. � proposition 2.12. let (x, τ, i) be an ideal topological space. if i is τ-simple and τ-codense and for each closed subset f of x the ideal if is τf -codense, then ind(x) 6 ind∗(x). proof. it is similar to the proof of proposition 2.11. � proposition 2.13. let (x, τ, i) be an ideal topological space and u be a ∗-open set in x. then ibd∗ x (u) ⊆ (τ ∗ bd∗ x (u)) c. © agt, upv, 2021 appl. gen. topol. 22, no. 2 422 small and large inductive dimension for ideal topological spaces proof. let a ∈ ibd∗ x (u). then there exists i ∈ i such that a = bd ∗ x(u) ∩ i. since i is closed in the space (x, τ∗) [15], that is ∗-closed in x, a is ∗-closed in bd∗x(u), proving the relation of the proposition. � 3. the ideal small inductive dimension i ind and the ideal large inductive dimension i ind for ideal topological spaces in this section, the notions of the ideal small inductive dimension, i ind, and the ideal large inductive dimension, i ind, of an ideal topological space (x, τ, i), are defined, combining the topologies τ and τ∗, and basic properties of these dimensions are investigated. definition 3.1. the ideal small inductive dimension of an ideal topological space (x, τ, i), denoted by i ind(x), is defined as follows: (i) i ind(x) = −1, if x = ∅. (ii) i ind(x) 6 k, where k ∈ {0, 1, . . .}, if for every element x ∈ x and for every open subset v of x with x ∈ v , there exists a ∗-open subset u of x with x ∈ u ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. (iii) i ind(x) = k, where k ∈ {0, 1, . . .}, if i ind(x) 6 k and i ind(x) k − 1. (iv) i ind(x) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which i ind(x) 6 k is true. it is observed that the ideal small inductive dimension i ind is different from the dimensions ind and ind∗ and the following examples prove this assertion. example 3.2. (1) let k > 1. we consider the set xk = {0, 1, 2, . . ., k} and the topology generated by the family {∅, {0}, {0, 1}, . . ., {0, 1, . . . , k}}. then ind(xk) = k. if we consider the ideal i, consisting of all subsets of x, then τ∗ is the discrete topology and therefore, i ind(x) = 0. (2) we consider the indiscrete space (r, τ) and the ideal i = {i ⊆ r : 0 /∈ i}. then τ∗ = {∅}∪{u ⊆ r : 0 ∈ u}, ind∗(r) = 1 and i ind(r) = 0. lemma 3.3. let x be a non-empty set, a, b subsets of x such that a ⊆ b and i an ideal on x. then (ib)a = ia. theorem 3.4. if (x, τ, i) is an ideal topological space and a is a subset of x, then ia-ind(a) 6 i ind(x). proof. obviously, if i ind(x) = −1 or i ind(x) = ∞, then the inequality holds. we suppose that the inequality holds for all integers m < k and we shall prove it for k. let i ind(x) = k, x ∈ a and va be an open set in a with x ∈ va. then there exists an open set v in x such that va = v ∩ a. clearly, x ∈ v . since i ind(x) = k, there exists a ∗-open set u in x with x ∈ u ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. © agt, upv, 2021 appl. gen. topol. 22, no. 2 423 f. sereti we consider the ∗-open set ua = u ∩ a in a. then x ∈ ua ⊆ va. we shall prove that (ia)bd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1. by lemma 3.3, it suffices to prove that ibd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1. since bd∗a(ua) = bd ∗ a(u ∩ a) ⊆ a ∩ bd ∗ x(u) ⊆ bd ∗ x(u), bd∗a(ua) is a subset of bd ∗ x(u) and by inductive hypothesis, we have that (ibd∗ x (u))bd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1. moreover, applying lemma 3.3 we have that (ibd∗ x (u))bd∗ a (ua) = ibd∗a(ua). thus, ibd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1 and so ia-ind(a) 6 k. � proposition 3.5. for any ideal topological space (x, τ, i) we have that i ind(x) 6 min{ind(x), ind∗(x)}. proof. firstly, we shall prove that i ind(x) 6 ind(x). clearly, if ind(x) = −1 or ind(x) = ∞, then the inequality of the proposition holds. we suppose that the inequality holds for all integers m < k and we shall prove it for k. let ind(x) = k, x ∈ x and v be an open set in x with x ∈ v . since ind(x) = k, there exists an open set u in x with x ∈ u ⊆ v and ind(bdx(u)) 6 k − 1. then u is a ∗-open set and by inductive hypothesis, we have that ibdx (u)-ind(bdx(u)) 6 k − 1. since bd∗x(u) ⊆ bdx(u), by theorem 3.4 we have (ibdx (u))bd∗x(u)-ind(bd ∗ x(u)) 6 k − 1. by lemma 3.3 we have that ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. and therefore, i ind(x) 6 k. we shall prove that i ind(x) 6 ind∗(x). clearly, if ind∗(x) = −1 or ind∗(x) = ∞, then the inequality of the proposition holds. we suppose that the inequality holds for all integers m < k and we shall prove it for k. let ind∗(x) = k, x ∈ x and v be an open set in x with x ∈ v . then v is also a ∗-open set in x. since ind∗(x) = k, there exists a ∗-open set u in x with x ∈ u ⊆ v and ind∗(bd∗x(u)) 6 k − 1. then by inductive hypothesis we have that ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1 and thus, i ind(x) 6 k. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 424 small and large inductive dimension for ideal topological spaces proposition 3.6. let (x, τ, i) be an ideal topological space and k ∈ {0, 1, . . .}. if there exists a base b of (x, τ∗) such that ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1, for every u ∈ b, then i ind(x) 6 k. in what follows, we show characterizations for the ideal small inductive dimension, assuming that (x, τ) is a regular space. we state firstly that a subset l of the space (x, τ∗) is called a ∗-partition between two subsets a and b of x if there exist ∗-open sets u, v with a ⊆ u, b ⊆ v , u ∩ v = ∅ and x \ l = u ∪ v . proposition 3.7. let (x, τ, i) be an ideal topological space, where (x, τ) is a regular space. then i ind(x) 6 k, where k ∈ {0, 1, . . .}, if and only if for every point x ∈ x and for every closed subset f of x with x /∈ f, there exists a ∗-partition l between {x} and f with il-ind(l) 6 k − 1. proof. let k ∈ {0, 1, . . .}. firstly, we suppose that i ind(x) 6 k. let x ∈ x and f be a closed subset of x with x /∈ f . then we consider the set v = x\f . then v is an open set in x with x ∈ v . since (x, τ) is a regular space, there exists an open set v1 (and thus a ∗-open set) in x such that x ∈ v1 ⊆ clx(v1) ⊆ v. since i ind(x) 6 k and x ∈ v1, there exists a ∗-open set u in x with x ∈ u ⊆ v1 and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. then x ∈ cl∗x(u) ⊆ cl ∗ x(v1) ⊆ clx(v1) ⊆ v and the set l = bd∗x(u) is a ∗-partition between the sets {x} and f such that il-ind(l) 6 k − 1. conversely, we shall prove that i ind(x) 6 k. let x ∈ x and vx be an open set in x with x ∈ vx. we set f = x \ vx. then f is a closed subset of x with x /∈ f . by assumption there exists a ∗-partition l between {x} and f with il-ind(l) 6 k − 1. that is, there exist ∗-open sets u and v in x with x ∈ u, f ⊆ v , u ∩v = ∅ and x \l = u ∪v . since bd∗x(u) ⊆ l, by theorem 3.4, we have that (il)bd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1 and by lemma 3.3, ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. therefore, i ind(x) 6 k. � corollary 3.8. a non-empty ideal topological space x is ideal zero dimensional (with respect to the ideal small inductive dimension), where (x, τ) is a regular space, if and only if for every point x ∈ x and each closed subset f of x such that x /∈ f the empty set is a ∗-partition between {x} and f. proof. it follows directly by proposition 3.7. � proposition 3.9. if (x, τ, i) is an ideal topological space, where (x, τ) is a regular space, and k ∈ {0, 1, . . .}, then the following statements are equivalent: © agt, upv, 2021 appl. gen. topol. 22, no. 2 425 f. sereti (1) i ind(x) 6 k, (2) for every open neighborhood v of a point x in x, there exists a ∗-open set u and a ∗-closed set e in x such that x ∈ u ⊆ e ⊆ v and ie\u -ind(e \ u) 6 k − 1, (3) for every open neighborhood v of a point x in x, there exists a ∗-open set u in x such that x ∈ u ⊆ cl∗x(u) ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. proof. we prove the implication (1) ⇒ (2). let i ind(x) 6 k and v be an open neighborhood of a point x in x. then by proposition 3.7, there exists a ∗-partition l between {x} and x \ v with il-ind(l) 6 k − 1. we consider disjoint ∗-open sets u and g in x with x ∈ u, x \ v ⊆ g and x \ l = u ∪ g. we set e = x \ g. then e is a ∗-closed set in x with x ∈ u ⊆ e ⊆ v and e \ u = l, completing the proof of this assertion. next, we prove the implication (2) ⇒ (3). let v be an open neighborhood of a point x in x. by (2), there exists a ∗-open set u and a ∗-closed set e in x with x ∈ u ⊆ e ⊆ v and ie\u -ind(e \ u) 6 k − 1. we observe that cl∗x(u) ⊆ e and bd ∗ x(u) ⊆ e \ u. thus, by theorem 3.4, we have that (ie\u )bd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1 and by lemma 3.3 we have that ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. finally, we prove the implication (3) ⇒ (1). let f be a closed set in x and x ∈ x with x /∈ f . we put v = x \ f . then v is an open neighborhood of x. by (3), there exists a ∗-open set u in x with x ∈ u ⊆ cl∗x(u) ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. then u and x \ cl ∗ x(u) are disjoint ∗-open sets in x, containing x and f , respectively, and x \ (u ∪ (x \ cl∗x(u))) = (x \ u) ∩ cl ∗ x(u) = bd ∗ x(u). thus, bd∗x(u) is a ∗-partition between {x} and f and therefore, by proposition 3.7, we have that i ind(x) 6 k. � in the next results we study the ideal small inductive dimension of the topological sum. lemma 3.10. let {xs : s ∈ s} be a set of pairwise disjoint topological spaces and ⊕ s∈s xs be their topological sum. let also for each s ∈ s, is be an ideal of xs. then i = { ⋃ es : es ∈ is for each s ∈ s} is an ideal of ⊕ s∈s xs. theorem 3.11. let {xs : s ∈ s} be a set of pairwise disjoint topological spaces and ⊕ s∈s xs be their topological sum. let also for each s ∈ s, is be an ideal of xs. then i ind( ⊕ s∈s xs) 6 k, where k ∈ {0, 1, . . .} and i is the ideal of lemma 3.10, if and only if is-ind(xs) 6 k for each s ∈ s. © agt, upv, 2021 appl. gen. topol. 22, no. 2 426 small and large inductive dimension for ideal topological spaces proof. let k ∈ {0, 1, . . .}. we suppose that i ind( ⊕ s∈s xs) 6 k and we shall prove that is-ind(xs) 6 k for each s ∈ s. let s ∈ s. in the proof we shall write y = ⊕ s∈s xs. let x ∈ xs and v be an open subset of xs with x ∈ v . then v is an open subset of y . since i ind(y ) 6 k, there exists a ∗-open subset u of y such that x ∈ u ⊆ v and ibd∗ y (u)-ind(bd ∗ y (u)) 6 k − 1. we observe that ixs = is and the set u ∩ xs is a ∗-open subset of xs with x ∈ u ∩ xs ⊆ u ⊆ v . it suffices to prove that (ixs)bd∗ xs (u∩xs)-ind(bd ∗ xs (u ∩ xs)) 6 k − 1 or by lemma 3.3, ibd∗ xs (u∩xs)-ind(bd ∗ xs (u ∩ xs)) 6 k − 1. we have that bd∗xs(u ∩xs) ⊆ bd ∗ y (u). therefore, by theorem 3.4 we have that (ibd∗ y (u))bd∗ xs (u∩xs)-ind(bd ∗ xs (u ∩ xs)) 6 ibd∗ y (u)-ind(bd ∗ y (u)) and by lemma 3.3 we have that ibd∗ xs (u∩xs)-ind(bd ∗ xs (u ∩ xs)) 6 ibd∗ y (u)-ind(bd ∗ y (u)) and therefore, ibd∗ xs (u∩xs)-ind(bd ∗ xs (u ∩ xs)) 6 k − 1. conversely, we suppose that is-ind(xs) 6 k for each s ∈ s and we shall prove that i ind(y ) 6 k. let x ∈ y and v be an open subset of y with x ∈ v . then there exists s ∈ s such that x ∈ xs and the set v ∩xs is an open subset of xs with x ∈ v ∩ xs. since is-ind(xs) 6 k, there exists a ∗-open subset us of xs such that x ∈ us ⊆ v ∩ xs ⊆ v and (is)bd∗ xs (us)-ind(bd ∗ xs (us)) 6 k − 1 or equivalently, (ixs)bd∗ xs (us)-ind(bd ∗ xs (us)) 6 k − 1 and by lemma 3.3, ibd∗ xs (us)-ind(bd ∗ xs (us)) 6 k − 1. the set us is ∗-open subset of y and it suffices to prove that ibd∗ y (us)-ind(bd ∗ y (us)) 6 k − 1. we have that bd∗xs(us) = bd ∗ y (us) and thus, ibd∗ y (us)-ind(bd ∗ y (us)) 6 k − 1. � in what follows, we shall define and study the ideal large inductive dimension i ind of an ideal topological space (x, τ, i). © agt, upv, 2021 appl. gen. topol. 22, no. 2 427 f. sereti definition 3.12. the ideal large inductive dimension of an ideal topological space (x, τ, i), denoted by i ind(x), is defined as follows: (i) i ind(x) = −1, if x = ∅. (ii) i ind(x) 6 k, where k ∈ {0, 1, . . .}, if for every pair (f, v ) of subsets of x, where f is closed, v is open and f ⊆ v , there exists a ∗-open subset u of x such that f ⊆ u ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. (iii) i ind(x) = k, where k ∈ {0, 1, . . .}, if i ind(x) 6 k and i ind(x) k − 1. (iv) i ind(x) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which i ind(x) 6 k is true. we observe that the ideal large inductive dimension is different from the dimensions ind and ind∗ and the following examples prove this assertion. example 3.13. (1) we consider the space x = {a, b, c, d} with the topology τ = {∅, {a}, {a, b}, {a, b, c}, {a, b, d}, x} and the ideal i = {∅, {a}, {b}, {a, b}}. then ind(x) = 1 and i ind(x) = 0. (2) we consider the space x = {a, b, c, d, e} with the topology τ = {∅, {a}, {a, b}, {a, c}, {a, b, c}, {a, b, c, e}, x} and the ideal i = {∅, {c}, {e}, {c, e}}. then ind∗(x) = 2 and i ind(x) = 0. theorem 3.14. if (x, τ, i) is an ideal topological space and a is a closed subset of x, then ia-ind(a) 6 i ind(x). proof. obviously, if i ind(x) = −1 or i ind(x) = ∞, then the inequality holds. we suppose that the inequality holds for all integers m < k and we shall prove it for k. let i ind(x) = k, fa be a closed subset of a and va be an open subset of a such that fa ⊆ va. since a is a closed subset of x, fa is also a closed subset of x. also, there exists an open set v in x such that va = v ∩ a. since i ind(x) = k, there exists a ∗-open set u in x with fa ⊆ u ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. we consider the ∗-open set ua = u ∩ a in a. then fa ⊆ u ∩ a ⊆ v ∩ a or equivalently, fa ⊆ ua ⊆ va. we shall prove that (ia)bd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1. by lemma 3.3 it suffices to prove that ibd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1. since bd∗a(ua) is a closed subset of bd ∗ x(u), by inductive hypothesis we have that © agt, upv, 2021 appl. gen. topol. 22, no. 2 428 small and large inductive dimension for ideal topological spaces (ibd∗ x (u))bd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1. moreover, applying lemma 3.3 we have that (ibd∗ x (u))bd∗ a (ua) = ibd∗a(ua). thus, ibd∗ a (ua)-ind(bd ∗ a(ua)) 6 k − 1 and so ia-ind(a) 6 k. � the assumption that a is a closed subset of x cannot be dropped and the following example justifies this. example 3.15. let x = {a, b, c, d, e, f} with the topology τ = {∅, {f}, {a, f}, {b, f}, {a, b, f}, {a, b, c, f}, {a, b, d, f}, {a, b, c, d, f}, x}. if we consider the ideal i = {∅, {e}}, then i ind(x) = 0. however, if we consider the set a = {a, b, c, d}, which is not closed in x, then ia-ind(a) = 1. proposition 3.16. for any ideal topological space (x, τ, i) we have that i ind(x) 6 min{ind(x), ind∗(x)}. proof. it is similar to proposition 3.5. � the following results characterize the ideal large inductive dimension in various classes of spaces. proposition 3.17. an ideal topological space (x, τ, i), where (x, τ) is a normal space, satisfies the inequality i ind(x) 6 k, where k ∈ {0, 1, . . .}, if and only if for every pair a, b of disjoint closed subsets of x, there exists a ∗partition l between a and b such that il-ind(l) 6 k − 1. proof. it is similar to the proof of proposition 3.7. � corollary 3.18. a non-empty ideal topological space x is ideal zero dimensional (with respect to the ideal large inductive dimension), where (x, τ) is a normal space, if and only if for every disjoint closed subsets a and b of x, the empty set is a ∗-partition between a and b. proof. it follows by proposition 3.17. � proposition 3.19. if (x, τ, i) is an ideal topological space, where (x, τ) is a normal space, and k ∈ {0, 1, . . .}, then the following statements are equivalent: (1) i ind(x) 6 k, (2) for every open neighborhood v of a closed set e of x, there exists a ∗-open set u and a ∗-closed set f in x such that e ⊆ u ⊆ f ⊆ v and if \u -ind(f \ u) 6 k − 1, (3) for every open neighborhood v of a closed set e of x, there exists a ∗-open set u in x with e ⊆ u ⊆ cl∗x(u) ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. proof. it is similar to the proof of proposition 3.9. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 429 f. sereti the following result presents the behavior of the ideal large inductive dimension of the topological sum. theorem 3.20. let {xs : s ∈ s} be a set of pairwise disjoint topological spaces and ⊕ s∈s xs be their topological sum. let also for each s ∈ s, is be an ideal of xs. then i ind( ⊕ s∈s xs) 6 k, where i is the ideal of lemma 3.10, if and only if is-ind(xs) 6 k for each s ∈ s. proof. it is similar to the proof of theorem 3.11. � 4. additional results on dimensions for ideal topological spaces in what follows, properties of the ideal small and ideal large inductive dimension, using different ideals on the underlying sets, are studied. moreover, relations between the dimensions i ind and i ind and the ideal covering dimension i dim are investigated. proposition 4.1. let (x, τ) be a topological space and i1, i2 be two ideals on x. (1) if i1 ⊆ i2, then i2-ind(x) 6 i1-ind(x). (2) if i1 ⊆ i2, then i2-ind(x) 6 i1-ind(x). proof. for the simplicity of the writing, we denote by ∗1 and ∗2 the ∗-open set referred to the ideal topological spaces (x, τ, i1) and (x, τ, i2), respectively. (1) we suppose that the inequality holds for all integers m < k and we shall prove it for k. let i1-ind(x) = k. we shall prove that i2-ind(x) 6 k. let f and v be a closed and an open set in x, respectively, such that f ⊆ v . since i1-ind(x) = k, there exists a ∗1-open set u in x such that f ⊆ u ⊆ v and (i1)bd∗1 x (u)-ind(bd ∗1 x (u)) 6 k − 1. since i1 ⊆ i2, the set u is also a ∗2-open set with bd ∗2 x (u) ⊆ bd∗1 x (u) and thus, applying lemma 3.3 and theorem 3.4, we have that (i1)bd∗2 x (u)-ind(bd ∗2 x (u)) 6 k − 1. also, (i1)bd∗2 x (u) ⊆ (i2)bd∗2 x (u). therefore, by inductive hypothesis, we have that (i2)bd∗2 x (u)-ind(bd ∗2 x (u)) 6 k − 1 and thus, i2-ind(x) 6 k. (2) it is similar to (1). � corollary 4.2. let (x, τ) be a topological space and i1, i2 be two ideals on x. then (1) max{i1-ind(x), i2-ind(x)} 6 i1 ∩ i2-ind(x). (2) max{i1-ind(x), i2-ind(x)} 6 i1 ∩ i2-ind(x). corollary 4.3. let (x, τ) be a topological space, a, b subsets of x and i1 = p(a), i2 = p(b) and i3 = p(a ∪ b) be three ideals on x. then (1) i3-ind(x) 6 min{i1-ind(x), i2-ind(x)}. © agt, upv, 2021 appl. gen. topol. 22, no. 2 430 small and large inductive dimension for ideal topological spaces (2) i3-ind(x) 6 min{i1-ind(x), i2-ind(x)}. proposition 4.4. for any ideal topological space (x, τ, i), where (x, τ) is a t1-space, we have that i ind(x) 6 i ind(x). proof. let (x, τ, i) be an ideal topological space for which (x, τ) is a t1-space. we suppose that the inequality holds for all integers m < k and we shall prove it for k. let i ind(x) = k. we shall prove that i ind(x) 6 k. let x ∈ x and v be an open subset of x with x ∈ v . then the set {x} is a closed subset of x. since i ind(x) 6 k, for the pair ({x}, v ) there exists a ∗-open subset u of x with x ∈ u ⊆ v and ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1. therefore, by inductive hypothesis, we have that ibd∗ x (u)-ind(bd ∗ x(u)) 6 k − 1 and thus, i ind(x) 6 k. � example 4.5. we consider the space x = {a, b, c, d, e} with the topology τ = {∅, {a}, {b}, {a, b}, {a, b, c}, x} and the ideal i = {∅, {d}}. then we observe that the space x is not t1, i ind(x) = 2 and i ind(x) = 0. especially, for the ideal small inductive dimension of x, we have the following statements: – for the element x = a and every open subset v of x with a ∈ v , there exists the ∗-open subset u = {a} of x such that a ∈ u ⊆ v and ibd∗ x ({a})-ind(bd ∗ x({a})) = 1, – for the element x = b and every open subset v of x with b ∈ v , there exists the ∗-open subset u = {b} of x such that a ∈ u ⊆ v and ibd∗ x ({b})-ind(bd ∗ x({b})) = 1, – for the element x = c and every open subset v of x with c ∈ v , there exists the ∗-open subset u = {a, b, c} of x such that c ∈ u ⊆ v and ibd∗ x ({a,b,c})-ind(bd ∗ x({a, b, c})) = 0, – for the element x = d and every open subset v of x with d ∈ v , there exists the ∗-open subset u = x of x such that d ∈ u ⊆ v and ibd∗ x (x)-ind(bd ∗ x(x)) = −1 and – for the element x = e and every open subset v of x with e ∈ v , there exists the ∗-open subset u = x of x such that e ∈ u ⊆ v and ibd∗ x (x)-ind(bd ∗ x(x)) = −1. therefore, i ind(x) = 2. moreover, for the ideal large inductive dimension of x we observe that for every pair (f, v ) of subsets of x, where f is closed, v is open and f ⊆ v , there exists the ∗-open subset u = x of x such that f ⊆ u ⊆ v and ibd∗ x (u)ind(bd∗x(u)) = −1. therefore, i ind(x) = 0. © agt, upv, 2021 appl. gen. topol. 22, no. 2 431 f. sereti for the rest of the paper, we remind that for a topological space (x, τ) a non-empty family c of open subsets of x is called an open cover if the union of all elements of c is x. a family r of subsets of x is said to be a refinement of a family c of subsets of x if each element of r is contained in an element of c. especially, in what follows, for an ideal topological space (x, τ, i) a non empty family c of open sets (respectively, of ∗-open sets) will be said a τ-cover (respectively, a τ∗-cover) of x if the union of all elements of c is x. the order of a family r of subsets of a topological space x is defined as follows: (1) ord(r) = −1, if r consists of the empty set only. (2) ord(r) = k, where k ∈ {0, 1, . . .}, if the intersection of any k+2 distinct elements of r is empty and there exist k+1 distinct elements of r whose intersection is not empty. (3) ord(r) = ∞ if for every k ∈ {1, 2, . . .} there exist k distinct elements of r whose intersection is not empty. definition 4.6 ([17]). the ideal covering dimension, denoted by i dim, is defined as follows: (i) i dim(x) = −1 if and only if x = ∅. (ii) i dim(x) 6 k, where k ∈ {0, 1, . . .}, if for every finite τ-cover c of x there exists a finite τ∗-cover r of x, which is a refinement of c with ord(r) 6 k. (iii) i dim(x) = k, where k ∈ {0, 1, . . .}, if i dim(x) 6 k and i dim(x) k − 1. (iv) i dim(x) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which i dim(x) 6 k is true. proposition 4.7. let (x, τ, i) be an ideal topological space, where (x, τ) is a normal space. if i dim(x) = 0, then i ind(x) = 0. proof. let i dim(x) = 0 and e, f disjoint closed subsets of x. then the family c = {x \ e, x \ f} is a τ-cover of x. since i dim(x) = 0, there exists a finite τ∗-cover r of x, which refines c and has order less than or equal to 0. thus, every point of x is contained in a member of r, a member of r is disjoint from at least one of e and f , and any two members of r are disjoint. hence, u = ⋃ {g ∈ r : g ∩ e 6= ∅} and v = ⋃ {g ∈ r : g ∩ e = ∅} are disjoint ∗-open sets of x with e ⊆ u, f ⊆ v and x = u ∪ v . thus, ∅ is a ∗-partition between e and f with i∅-ind(∅) = −1. therefore, by corollary 3.18, i ind(x) = 0. � corollary 4.8. let (x, τ, i) be an ideal topological space, where (x, τ) is a normal space. if i dim(x) = 0, then i ind(x) = 0. © agt, upv, 2021 appl. gen. topol. 22, no. 2 432 small and large inductive dimension for ideal topological spaces proof. since i dim(x) = 0, by proposition 4.7 we have that i ind(x) = 0 and thus, by proposition 4.4, we have the desired result. � example 4.9. we consider the space x = {a, b, c} with the topology τ = {∅, {a}, {b}, {a, b}, {b, c}, x} and the ideal i = {∅, {b}}. then i ind(x) = i ind(x) = i dim(x) = 0. in general, we observe that the ideal topological dimensions of the types ind, ind and dim are different and the following examples prove this claim. example 4.10. (1) we consider the space x = {a, b, c, d, e} with the topology τ = {∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}, {a, b, c, d}, x} and the ideal i = {∅, {c}}. then i ind(x) = 2 and i dim(x) = 0. (2) we consider the space x = {a, b, c, d} with the topology generated by the family β = {∅, {a}, {a, b}, {a, c}, {a, d}} and the ideal i = {∅, {b}}. then i ind(x) = 1 and i dim(x) = 2. however, under some conditions for topological spaces and ideals we can obtain the following relation between i ind, i ind and i dim. proposition 4.11. let (x, τ, i) be an ideal topological space. if i ⊆ τc and (x, τ) is a separable metric space, then i ind(x) = i ind(x) = i dim(x). proof. by remark 2.7, proposition 2.9 and [17], we have that ind(x) = ind ∗ (x), ind(x) = ind∗(x) and dim(x) = dim∗(x). also, based on the definitions of the ideal topological spaces and the fact that τ = τ∗, we have that i ind(x) = ind(x), i ind(x) = ind(x) and i dim(x) = dim(x). finally, the desired relation follows from the fact that ind(x) = ind(x) = dim(x), whenever (x, τ) is a separable metric space. � acknowledgements. the author would like to thank the reviewer for the careful reading of the paper and the useful comments. © agt, upv, 2021 appl. gen. topol. 22, no. 2 433 f. sereti references [1] m. g. charalambous, dimension theory, a selection of theorems and counterexample, springer nature switzerland ag, cham, switzerland, 2019. 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[19] p. samuels, a topology formed from a given topology and ideal, j. london math. soc. 10, no. 4 (1975), 409–416. © agt, upv, 2021 appl. gen. topol. 22, no. 2 434 () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 193-206 tripled coincidence and fixed point results in partial metric spaces hassen aydi and mujahid abbas abstract in this paper, we introduce the concept of w -compatiblity of mappings f : x × x × x → x and g : x → x and based on this notion, we obtain tripled coincidence and common tripled fixed point results in the setting of partial metric spaces. the presented results generalize and extend several well known comparable results in the existing literature. we also provide an example to support our results. 2010 msc: 54h25, 47h10. keywords: w -compatible mappings, tripled coincidence point, common tripled fixed point, partial metric space. 1. introduction matthews [19, 20] introduced the notion of a partial metric space which is a generalization of usual metric spaces in which the distance of a point to itself is no longer necessarily zero. a partial metric space (see [19, 20]) is a pair (x, p) such that x is a (nonempty) set and p : x × x → r+ (where r+ denotes the set of all non negative real numbers) satisfies (p1) p(x, y) = p(y, x) (symmetry) (p2) if p(x, x) = p(x, y) = p(y, y) then x = y (equality) (p3) p(x, x) ≤ p(x, y) (small self-distances) (p4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z) (triangularity) for all x, y, z ∈ x. in this case we say that p is a partial metric on x. each partial metric p on x generates a t0 topology τp on x with a base of the family open of p-balls {bp(x, ε) : x ∈ x, ε > 0}, where bp(x, ε) = {y ∈ 194 h. aydi and m. abbas x : p(x, y) < p(x, x) + ε} for all x ∈ x and ε > 0. similarly, a closed p-ball is defined as bp[x, ε] = {y ∈ x : p(x, y) ≤ p(x, x) + ε}. definition 1.1 ([19, 20]). (i) a sequence {xn} in a partial metric space (x, p) is called cauchy if lim n,m→∞ p(xn, xm) exists (and finite), (ii) a partial metric space (x, p) is said to be complete if every cauchy sequence {xn} in x converges, with respect to τp, to a point x ∈ x such that p(x, x) = lim n,m→∞ p(xn, xm). notice that for a partial metric p on x, the function ps : x ×x → r+ given by (1.1) ps(x, y) = 2p(x, y) − p(x, x) − p(y, y) is a (usual) metric on x. it is well known and easy to see that (1.2) lim n→∞ ps(x, xn) = 0 ⇔ p(x, x) = lim n→∞ p(x, xn) = lim n,m→∞ p(xn, xm). lemma 1.2 ([19, 20]). (a) a sequence {xn} is cauchy in a partial metric space (x, p) if and only if {xn} is cauchy in the metric space (x, p s). (b) a partial metric space (x, p) is complete if and only if the metric space (x, ps) is complete. for simplicity, we denote from now on x × x · · · x × x︸︷︷︸ k terms by xk where k ∈ n and x a non-empty set. we start by recalling some definitions where x is a non-empty set. definition 1.3 (bhashkar and lakshmikantham [13]). an element (x, y) ∈ x2 is called a coupled fixed point of a mapping f : x2 → x if x = f(x, y) and y = f(y, x). definition 1.4 (lakshmikantham and ćirić [18]). an element (x, y) ∈ x2 is called (i) a coupled coincidence point of mappings f : x2 → x and g : x → x if gx = f(x, y) and gy = f(y, x), and (gx, gy) is called coupled point of coincidence; (ii) a common coupled fixed point of mappings f : x2 → x and g : x → x if x = gx = f(x, y) and y = gy = f(y, x). note that if g is the identity mapping, then definition 1.4 reduces to definition 1.3. in 2011, samet and vetro [21] introduced a fixed point of order n ≥ 3. in particular, for n = 3 we have following definition. definition 1.5 (samet and vetro [21]). an element (x, y, z) ∈ x3 is called a tripled fixed point of a given mapping f : x3 → x if x = f(x, y, z), y = f(y, z, x) and z = f(z, x, y). tripled coincidence and fixed point results 195 note that, berinde and borcut [12] defined differently the notion of a tripled fixed point in the case of ordered sets in order to keep true the mixed monotone property. for more details, see [12]. now, we give the following definitions. definition 1.6. an element (x, y, z) ∈ x3 is called (i) a tripled coincidence point of mappings f : x3 → x and g : x → x if gx = f(x, y, z), gy = f(y, x, z) and gz = f(z, x, y). in this case (gx, gy, gz) is called tripled point of coincidence; (ii) a common tripled fixed point of mappings f : x3 → x and g : x → x if x = gx = f(x, y, z), y = gy = f(y, z, x) and z = gz = f(z, x, y). fixed point theorems on partial metric spaces have received a lot of attention in the last years (see, for instance, [2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 22, 23, 24] and their references). abbas et al. [1] introduced the concept of w-compatible mappings and obtained coupled coincidence point and coupled point of coincidence for mappings satisfying a contractive condition in cone metric spaces. very recently, aydi et al. [11] introduced the concepts of w̃compatible mappings and generalized the results in [1]. the aim of this paper is to introduce the concepts of w-compatible mappings. based on this notion, tripled coincidence point and common tripled fixed point for mappings f : x × x × x → x and g : x → x are obtained in partial metric space. the presented theorems generalize and extend several well known comparable results in the literature. an example is also given in support of our results. definition 1.7 (abbas, khan and radenović [1]). the mappings f : x2 → x and g : x → x are called w-compatible if g(f(x, y)) = f(gx, gy) whenever gx = f(x, y) and gy = f(y, x). definition 1.8. mappings f : x3 → x and g : x → x are called wcompatible if f(gx, gy, gz) = g(f(x, y, z)) whenever f(x, y, z) = gx, f(y, z, x) = gy and f(z, y, x) = gz. 2. main results we present our first result as follows. theorem 2.1. let (x, p) be a partial metric space and f : x3 −→ x and g : x → x be mappings such that f(x3) ⊆ g(x) and g(x) is a complete subspace of x. suppose that for any x, y, z, u, v, w ∈ x, the following condition p(f(x, y, z), f(u, v, w)) ≤ a1p(f(x, y, z), gx) + a2p(f(y, z, x), gy) + a3p(f(z, x, y), gz) + a4p(f(u, v, w), gu) + a5p(f(v, w, u), gv) + a6p(f(w, u, v), gw) + a7p(f(u, v, w), gx) + a8p(f(v, w, u), gy) + a9p(f(w, u, v), gz) + a10p(f(x, y, z), gu) + a11p(f(y, z, x), gv) + a12p(f(z, x, y), gw) + a13p(gx, gu) + a14p(gy, gv) + a15p(gz, gw), 196 h. aydi and m. abbas holds, where ai, i = 1, · · · , 15 are nonnegative real numbers. if 9∑ i=7 ai + 15∑ i=1 ai < 1 or 12∑ i=10 ai + 15∑ i=1 ai < 1, then f and g have a tripled coincidence point. proof. let x0, y0 and z0 be three arbitrary points in x. by given assumptions, there exists (x1, y1, z1) such that f(x0, y0, z0) = gx1, f(y0, z0, x0) = gy1 and f(z0, x0, y0) = gz1. continuing this process, we construct three sequences {xn}, {yn} and {zn} in x such that (2.1) f(xn, yn, zn) = gxn+1, f(yn, zn, xn) = gyn+1 and f(zn, xn, yn) = gzn+1 ∀ n ∈ n. denote δn = p(gxn+1, gxn) + p(gyn+1, gyn) + p(gzn+1, gzn). we claim that (2.2) δn+1 ≤ κδn ∀ n ∈ n, where κ ∈ [0, 1) will be chosen conveniently. first, taking (x, y, z) = (xn, yn, zn) and (u, v, w) = (xn+1, yn+1, zn+1) in the considered contractive condition and using (2.1), we have p(gxn+1, gxn+2) = p(f(xn, yn, zn), f(xn+1, yn+1, zn+1)) ≤ a1p(f(xn, yn, zn), gxn) + a2p(f(yn, zn, xn), gyn) + a3p(f(zn, xn, yn), gzn) + a4p(f(xn+1, yn+1, zn+1), gxn+1) + a5p(f(yn+1, zn+1, xn+1), gyn+1) + a6p(f(zn+1, xn+1, yn+1), gzn+1) + a7p(f(xn+1, yn+1, zn+1), gxn) + a8p(f(yn+1, zn+1, xn+1), gyn) + a9p(f(zn+1, xn+1, yn+1), gzn) + a10p(f(xn, yn, zn), gxn+1) + a11p(f(yn, zn, xn), gyn+1) + a12p(f(zn, xn, yn), gzn+1) + a13p(gxn, gxn+1) + a14p(gyn, gyn+1) + a15p(gzn, gzn+1) = a1p(gxn+1, gxn) + a2p(gyn+1, gyn) + a3p(gzn+1, gzn) + a4p(gxn+2, gxn+1) + a5p(gyn+2, gyn+1) + a6p(gzn+2, gzn+1) + a7p(gxn+2, gxn) + a8p(gyn+2, gyn) + a9p(gzn+2, gzn) + a10p(gxn+1, gxn+1) + a11p(gyn+1, gyn+1) + a12p(gzn+1, gzn+1) + a13p(gxn, gxn+1) + a14p(gyn, gyn+1) + a15p(gzn, gzn+1). then, using (p2) and the triangular inequality (which holds even for partial metrics), one can write for any n ∈ n (1 − a4 − a7 − a10)p(gxn+2, gxn+1) ≤ (a1 + a7 + a13)p(gxn+1, gxn) + (a2 + a8 + a14)p(gyn, gyn+1) +(a3 + a9 + a15)p(gzn, gzn+1) + (a5 + a8 + a11)p(gyn+2, gyn+1) + (a6 + a9 + a12)p(gzn+2, gzn+1). (2.3) tripled coincidence and fixed point results 197 similarly, following similar arguments to those given above, we obtain (1 − a4 − a7 − a10)p(gyn+2, gyn+1) ≤ (a1 + a7 + a13)p(gyn+1, gyn) + (a2 + a8 + a14)p(gzn, gzn+1) +(a3 + a9 + a15)p(gxn, gxn+1) + (a5 + a8 + a11)p(gzn+2, gzn+1) + (a6 + a9 + a12)p(gxn+2, gxn+1), (2.4) and (1 − a4 − a7 − a10)p(gzn+2, gzn+1) � (a1 + a7 + a13)p(gzn+1, gzn) + (a2 + a8 + a14)p(gxn, gxn+1) +(a3 + a9 + a15)p(gyn, gyn+1) + (a5 + a8 + a11)p(gxn+2, gxn+1) + (a6 + a9 + a12)p(gyn+2, gyn+1). (2.5) adding (2.3) to (2.5) we have (2.6) (1−a4−a5−a6−a7−a8−a9−a10−a11−a12)δn+1 ≤ (a1+a2+a3+a7+a8+a9+a13+a14+a15)δn, that is (2.7) δn+1 ≤ κ1 δn ∀ n ∈ n, where κ1 = a1 + a2 + a3 + a7 + a8 + a9 + a13 + a14 + a15 1 − 12∑ i=4 ai . as 9∑ i=7 ai + 15∑ i=1 ai < 1, so 0 ≤ κ1 < 1. hence (2.2) holds for κ = κ1. on the other hand, we have p(gxn+2, gxn+1) = p(f(xn+1, yn+1, zn+1), f(xn, yn, zn)) ≤ a1p(f(xn+1, yn+1, zn+1), gxn+1) + a2p(f(yn+1, zn+1, xn+1), gyn+1) + a3p(f(zn+1, xn+1, yn+1), gzn+1) + a4p(f(xn, yn, zn), gxn) + a5p(f(yn, zn, xn), gyn) + a6p(f(zn, xn, yn), gzn) + a7p(f(xn, yn, zn), gxn+1) + a8p(f(yn, zn, xn), gyn+1) + a9p(f(zn, xn, yn), gzn+1) + a10p(f(xn+1, yn+1, zn+1), gxn) + a11p(f(yn+1, zn+1, xn+1), gyn) + a12p(f(zn+1, xn+1, yn+1), gzn) + a13p(gxn+1, gxn) + a14p(gyn+1, gyn) + a15p(gzn+1, gzn) = a1p(gxn+2, gxn+1) + a2p(gyn+2, gyn+1) + a3p(gzn+2, gzn+1) + a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + a7p(gxn+1, gxn+1) + a8p(gyn+1, gyn+1) + a9p(gzn+1, gzn+1) + a10p(gxn+2, gxn) + a11p(gyn+2, gyn) + a12p(gzn+2, gzn) + a13p(gxn, gxn+1) + a14p(gyn, gyn+1) + a15p(gzn, gzn+1). thus, using again (p2) and the triangular inequality (1 − a1 − a7 − a10)p(gxn+2, gxn+1) ≤ (a4 + a10 + a13)p(gxn+1, gxn) + (a5 + a11 + a14)p(gyn, gyn+1) +(a6 + a12 + a15)p(gzn, gzn+1) + (a2 + a8 + a11)p(gyn+2, gyn+1) + (a3 + a9 + a12)p(gzn+2, gzn+1). (2.8) 198 h. aydi and m. abbas similarly, (1 − a1 − a7 − a10)p(gyn+2, gyn+1) ≤ (a4 + a10 + a13)p(gyn+1, gyn) + (a5 + a11 + a14)p(gzn, gzn+1) +(a6 + a12 + a15)p(gxn, gxn+1) + (a2 + a8 + a11)p(gzn+2, gzn+1) + (a3 + a9 + a12)p(gxn+2, gxn+1) (2.9) and (1 − a1 − a7 − a10)p(gzn+2, gzn+1) ≤ (a4 + a10 + a13)p(gzn+1, gzn) + (a5 + a11 + a14)p(gxn, gxn+1) +(a6 + a12 + a15)p(gyn, gyn+1) + (a2 + a8 + a11)p(gxn+2, gxn+1) + (a3 + a9 + a12)p(gyn+2, gyn+1). (2.10) adding (2.8) to (2.10), we obtain that (2.11) (1−a1−a2−a3−a7−a8−a9−a10−a11−a12)δn+1 ≤ (a4+a5+a6+a10+a11+a12+a13+a14+a15)δn. from (2.11), one can write δn+1 ≤ κ2δn ∀ n ∈ n where κ2 = a4 + a5 + a6 + a10 + a11 + a12 + a13 + a14 + a15 1 − a1 − a2 − a3 − ∑12 i=7 ai . since 12∑ i=10 ai + 15∑ i=1 ai < 1, so 0 ≤ κ2 < 1. thus, (2.2) holds for κ = κ2. by (2.2), we have (2.12) δn ≤ κδn−1 ≤ · · · ≤ κ nδ0. if δ0 = 0, we get p(gx0, gx1)+p(gy0, gy1) = p(gz0, gz1) = 0, that is, gx0 = gx1, gy0 = gy1 and gz0 = gz1. therefore, from (2.1) we have f(x0, y0, z0) = gx1 = gx0, f(y0, z0, x0) = gy1 = gy0 and f(z0, x0, y0) = gz1 = gz0, that is, (x0, y0, z0) is a tripled coincidence point of f and g. now, assume that δ0 6= 0. if m > n, we have p(gxm, gxn) ≤ p(gxm, gxm−1) + p(gxm−1, gxm−2) + · · · + p(gxn+1, gxn), p(gym, gyn) ≤ p(gym, gym−1) + p(gym−1, gym−2) + · · · + p(gyn+1, gyn), and p(gzm, gzn) ≤ p(gzm, gzm−1) + p(gzm−1, gzm−2) + · · · + p(gzn+1, gzn). adding above inequalities and using (2.12), we obtain (for m > n) p(gxm, gxn) + p(gym, gyn) + p(gzm, gzn) ≤ δm−1 + δm−2 + · · · + δn ≤ (κm−1 + κm−1 + · · · + κn)δ0 ≤ κn 1 − κ δ0 → 0 since κ ∈ [0, 1). this implies that (2.13) lim n,m→∞ p(gxm, gxn) = lim n,m→∞ p(gym, gyn) = lim n,m→∞ p(gzm, gzn) = 0. tripled coincidence and fixed point results 199 we deduce that {gxn}, {gyn} and {gzn} are cauchy sequences in (g(x), p) which is complete, then by lemma 1.2, {gxn}, {gyn} and {gzn} are cauchy sequences in the metric subspace (g(x), ps). since is also (g(x), ps) complete, so that there exist x, y, z ∈ x such that (2.14) lim n→∞ ps(gxn, gx) = lim n→∞ ps(gyn, gy) = lim n→∞ ps(gzn, gz) = 0. again, by lemma 1.2 and (2.13), we get that (2.15) p(gx, gx) = lim n→∞ p(gxn, gx) = lim n,m→∞ p(gxm, gxn) = 0, (2.16) p(gy, gy) = lim n→∞ p(gyn, gy) = lim n,m→∞ p(gym, gyn) = 0, and (2.17) p(gz, gz) = lim n→∞ p(gzn, gz) = lim n,m→∞ p(gzm, gzn) = 0. now, we prove that f(x, y, z) = gx, f(y, z, x) = gy and f(z, x, y) = gz. note that p(f(x, y, z), gx) ≤ p(f(x, y, z), f(xn, yn, zn) + p(f(xn, yn, zn), gx) = p(f(x, y, z), f(xn, yn, zn) + p(gxn+1, gx).(2.18) on the other hand, applying the given contractive condition, we obtain p(f(x, y, z), f(xn, yn, zn)) ≤ a1p(f(x, y, z), gx) + a2p(f(y, z, x), gy) + a3p(f(z, x, y), gz) + a4p(f(xn, yn, zn), gxn) + a5p(f(yn, zn, xn), gyn) + a6p(f(zn, xn, yn), gzn) + a7p(f(xn, yn, zn), gx) + a8p(f(yn, zn, xn), gy) + a9p(f(zn, xn, yn), gz) + a10p(f(x, y, z), gxn) + a11p(f(y, z, x), gyn) + a12p(f(z, x, y), gzn) + a13p(gx, gxn) + a14p(gy, gyn) + a15p(gz, gzn) = a1p(f(x, y, z), gx) + a2p(f(y, z, x), gy) + a3p(f(z, x, y), gz) + a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + a7p(gxn+1, gx) + a8p(gyn+1, gy) + a9p(gzn+1, gz) + a10p(f(x, y, z), gxn) + a11p(f(y, z, x), gyn) + a12p(f(z, x, y), gzn) + a13p(gx, gxn) + a14p(gy, gyn) + a15p(gz, gzn). combining above inequality with (2.18) and using a triangular inequality, we have p(f(x, y, z), gx) � a1p(f(x, y, z), gx) + a2p(f(y, z, x), gy) + a3p(f(z, x, y), gz) + a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + a7p(gxn+1, gx) + a8p(gyn+1, gy) + a9p(gzn+1, gz) + a10p(f(x, y, z), gx) + a10p(gx, gxn) + a11p(f(y, z, x), gy) + a11p(gy, gyn) + a12p(f(z, x, y), gz) + a12p(gz, gzn) + a13p(gx, gxn) + a14p(gy, gyn) + a15p(gz, gzn) + p(gxn+1, gx). 200 h. aydi and m. abbas therefore, (1 − a1 − a10)p(f(x, y, z), gx) − (a2 + a11)p(f(y, z, x), gy) − (a3 + a12)p(f(z, x, y), gz) ≤ a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + (1 + a7)p(gxn+1, gx) + a8p(gyn+1, gy) + a9p(gzn+1, gz) + (a10 + a13)p(gx, gxn) + (a11 + a14)p(gy, gyn) + (a12 + a15)p(gz, gzn). (2.19) similarly, we obtain (1 − a1 − a10)p(f(y, z, x), gy) − (a2 + a11)p(f(z, x, y), gz) − (a3 + a12)p(f(x, y, z), gx) ≤ a4p(gyn+1, gyn) + a5p(gzn+1, gzn) + a6p(gxn+1, gxn) + (1 + a7)p(gyn+1, gy) + a8p(gzn+1, gz) + a9p(gxn+1, gx) + (a10 + a13)p(gy, gyn) + (a11 + a14)p(gz, gzn) + (a12 + a15)p(gx, gxn), (2.20) and (1 − a1 − a10)p(f(z, x, y), gz) − (a2 + a11)p(f(x, y, z), gx) − (a3 + a12)p(f(y, z, x), gy) ≤ a4p(gzn+1, gzn) + a5p(gxn+1, gxn) + a6p(gyn+1, gyn) + (1 + a7)p(gzn+1, gz) + a8p(gxn+1, gx) + a9p(gyn+1, gy) + (a10 + a13)p(gz, gzn) + (a11 + a14)p(gx, gxn) + (a12 + a15)p(gy, gyn). (2.21) letting in (2.19)-(2.21) and using (2.15)-(2.17), we get that (1−a1−a2−a3−a10−a11−a12)[p(f(x, y, z), gx)+p(f(y, z, x), gy)+p(f(z, x, y), gz)] = 0. it follows that p(f(x, y, z), gx) = p(f(y, z, x), gy) = p(f(z, x, y), gz) = 0, that is f(x, y, z) = gx, f(y, z, x) = gy and f(z, x, y) = gz. � as consequences of theorem 2.1, we give the following corollaries. corollary 2.2. let (x, p) be a partial metric space. let f : x3 −→ x and g : x → x be mappings such that f(x3) ⊆ g(x) and g(x) is a complete subspace of x. suppose that for any x, y, z, u, v, w ∈ x p(f(x, y, z), f(u, v, w)) ≤ α1[p(f(x, y, z), gx) + p(f(y, z, x), gy) + p(f(z, x, y), gz)] + α2[p(f(u, v, w), gu) + p(f(v, w, u), gv) + p(f(w, u, v), gw)] + α3[p(f(u, v, w), gx) + p(f(v, w, u), gy) + p(f(w, u, v), gz)] + α4[p(f(x, y, z), gu) + p(f(y, z, x), gv) + p(f(z, x, y), gw)] + α5[p(gx, gu) + p(gy, gv) + p(gz, gw)], where αi, i = 1, · · · , 5 are nonnegative real numbers. if α3 + 5∑ i=1 αi < 1/3 or α4 + 5∑ i=1 αi < 1/3, then f and g have a tripled coincidence. tripled coincidence and fixed point results 201 proof. take a1 = a2 = a3 = α1, a4 = a5 = a6 = α2, a7 = a8 = a9 = α3, a10 = a11 = a12 = α4 and a13 = a14 = a15 = α5 in theorem 2.1 with α3 + 5∑ i=1 αi < 1/3 or α4 + 5∑ i=1 αi < 1/3. the result follows. � corollary 2.3. let (x, p) be a partial metric space. let f̃ : x2 −→ x and g : x → x be mappings satisfying f̃(x2) ⊆ g(x), (g(x), p) is a complete subspace of x and for any x, y, u, v ∈ x, p(f̃(x, y), f̃(u, v)) ≤ a1p(f̃(x, y), gx) + a2p(f̃(u, v), gu) + a3p(f̃(u, v), gx) a4p(f̃(x, y), gu) + a5p(gx, gu) + a6p(gy, gv),(2.22) where ai, i = 1, · · · , 6 are nonnegative real numbers such that a3 + 6∑ i=1 ai < 1 or a4 + 6∑ i=1 ai < 1. then f̃ and g have a coupled coincidence point (x, y) ∈ x 2, that is, f̃(x, y) = gx and f̃(y, x) = gy. proof. consider the mappings f : x3 → x defined by f(x, y, z) = f̃(x, y) for all x, y, z ∈ x. then, the contractive condition (2.22) implies that, for all x, y, z, u, v, w ∈ x p(f(x, y, z), f(u, v, w)) ≤ a1p(f(x, y, z), gx) + a2p(f(u, v, w), gu) + a3p(f(x, y, z), gu) + a4p(f(u, v, w), gx) + a5p(gx, gu) + a6p(gy, gv). then f and g satisfy the contractive condition of theorem 2.1. clearly other conditions of theorem 2.1 are also satisfied as f̃(x2) ⊆ g(x) and g(x) is a complete subspace of x. therefore, from theorem 2.1, f and g have a tripled fixed point (x, y, z) ∈ x3 such that f(x, y, z) = gx, f(y, z, x) = gy and f(z, x, y) = gz, that is, f̃(x, y) = gx and f̃(y, x) = gy. this makes end to the proof. � now, we are ready to state and prove a result of common tripled fixed point. theorem 2.4. let f : x3 → x and g : x → x be two mappings which satisfy all the conditions of theorem 2.1. if f and g are w-compatible, then f and g have a unique common tripled fixed point. moreover, common tripled fixed point of f and g is of the form (u, u, u) for some u ∈ x. proof. first, we’ll show that the tripled point of coincidence is unique. suppose that (x, y, z) and (x∗, y∗, z∗) ∈ x3 with    gx = f(x, y, z) gy = f(y, z, x) gz = f(z, x, y), and    gx∗ = f(x∗, y∗, z∗) gy∗ = f(y∗, z∗, x∗) gz∗ = f(z∗, x∗, y∗). 202 h. aydi and m. abbas using contractive condition in theorem 2.1 and (p2), we obtain p(gx, gx∗) = p(f(x, y, z), f(x∗, y∗, z∗)) ≤ a1p(f(x, y, z), gx) + a2p(f(y, z, x), gy) + a3p(f(z, x, y), gz) + a4p(f(x ∗, y∗, z∗), gx∗) + a5p(f(y ∗, z∗, x∗), gy∗) + a6p(f(z ∗, x∗, y∗), gz∗) + a7p(f(x ∗, y∗, z∗), gx) + a8p(f(y ∗, z∗, x∗), gy) + a9p(f(z ∗, x∗, y∗), gz) + a10p(f(x, y, z), gx ∗) + a11p(f(y, z, x), gy ∗) + a12p(f(z, x, y), gz ∗) + a13p(gx, gx ∗) + a14p(gy, gy ∗) + a15p(gz, gz ∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gx ∗, gx) + (a2 + a5 + a8 + a11 + a14)p(gy ∗, gy) + (a3 + a6 + a9 + a12 + a15)p(gz ∗, gz). similarly, we have p(gy, gy∗) = p(f(y, z, x), f(y∗, z∗, x∗)) ≤ (a1 + a4 + a7 + a10 + a13)p(gy ∗, gy) + (a2 + a5 + a8 + a11 + a14)p(gz ∗, gz) + (a3 + a6 + a9 + a12 + a15)p(gx ∗, gx), and p(gz, gz∗) = p(f(z, x, y), f(z∗, x∗, y∗)) ≤ (a1 + a4 + a7 + a10 + a13)p(gz ∗, gz) + (a2 + a5 + a8 + a11 + a14)p(gx ∗, gx) + (a3 + a6 + a9 + a12 + a15)p(gy ∗, gy). adding above three inequalities, we get p(gx, gx∗)+p(gy, gy∗)+p(gz, gz∗) ≤ ( 15∑ i=1 ai)[p(gx, gx ∗)+p(gy, gy∗)+p(gz, gz∗)]. since 15∑ i=1 ai < 1, we obtain p(gx, gx∗) + p(gy, gy∗) + p(gz, gz∗) = 0, which implies that (2.23) gx = gx∗, gy = gy∗ and gz = gz∗, which implies uniqueness of the tripled point of coincidence of f and g, that is, (gx, gy, gz). note that p(gx, gy∗) = p(f(x, y, z), f(y∗, z∗, x∗)) ≤ a1p(f(x, y, z), gx) + a2p(f(y, z, x), gy) + a3p(f(z, x, y), gz) + a4p(f(y ∗, z∗, x∗), gy∗) + a5p(f(z ∗, x∗, y∗), gz∗) + a6p(f(x ∗, y∗, z∗), gx∗) + a7p(f(y ∗, z∗, x∗), gx) + a8p(f(z ∗, x∗, y∗), gy) + a9p(f(x ∗, y∗, z∗), gz) + a10p(f(x, y, z), gy ∗) + a11p(f(y, z, x), gz ∗) + a12p(f(z, x, y), gx ∗) + a13p(gx, gy ∗) + a14p(gy, gz ∗) + a15p(gz, gx ∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gy ∗, gx) + (a2 + a5 + a8 + a11 + a14)p(gz ∗, gy) + (a3 + a6 + a9 + a12 + a15)p(gx ∗, gz). similarly p(gy, gz∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gz ∗, gy) + (a2 + a5 + a8 + a11 + a14)p(gx ∗, gz) + (a3 + a6 + a9 + a12 + a15)p(gy ∗, gx), tripled coincidence and fixed point results 203 and p(gz, gx∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gx ∗, gz) + (a2 + a5 + a8 + a11 + a14)p(gy ∗, gx) + (a3 + a6 + a9 + a12 + a15)p(gz ∗, gy). adding the above inequalities, we obtain p(gx, gy∗)+p(gy, gz∗)+p(gz, gx∗) ≤ ( 15∑ i=1 ai)(p(gx, gy ∗)+p(gy, gz∗)+p(gz, gx∗)), which again yields that (2.24) gx = gy∗, gy = gz∗ and gz = gx∗. in view of (2.23) and (2.24), one can assert that (2.25) gx = gy = gz. that is, the unique tripled point of coincidence of f and g is (gx, gy, gz). now, let u = gx, then we have u = gx = f(x, y, z) = gy = f(y, z, x) = gz = f(z, x, y). since f and g are w-compatible, we have f(gx, gy, gz) = g(f(x, y, z)), which due to (2.25) gives that f(u, u, u) = gu. consequently, (u, u, u) is a tripled coincidence point of f and g, and so (gu, gu, gu) is a tripled point of coincidence of f and g, and by its uniqueness, we get gu = gx. thus, we obtain u = gx = gu = f(u, u, u). hence, (u, u, u) is the unique common tripled fixed point of f and g. this completes the proof. � corollary 2.5. let (x, p) be a cone partial metric space. let f̃ : x2 −→ x and g : x → x be mappings satisfying f̃(x2) ⊆ g(x), (g(x), p) is a complete subspace of x and for any x, y, u, v ∈ x, p(f̃(x, y), f̃(u, v)) ≤ a1p(f̃(x, y), gx) + a2p(f̃(u, v), gu) + a3p(f̃(u, v), gx) + a4p(f̃(x, y), gu) + a5p(gx, gu) + a6p(gy, gv), where ai, i = 1, · · · , 6 are nonnegative real numbers such that a3 + 6∑ i=1 ai < 1 or a4 + 6∑ i=1 ai < 1. if f̃ and g are w-compatible, then f̃ and g have a unique common coupled fixed point. moreover, the common fixed point of f̃ and g is of the form (u, u) for some u ∈ x. 204 h. aydi and m. abbas proof. consider the mappings f : x3 → x defined by f(x, y, z) = f̃(x, y) for all x, y, z ∈ x. from the proof of corollary 2.3 and the result given by theorem 2.4, we have only to show that f and g are w-compatible. let (x, y, z) ∈ x3 such that f(x, y, z) = gx, f(y, z, x) = gy and f(z, x, y) = gz. from the definition of f , we get f̃(x, y) = gx and f̃(y, x) = gy. since f̃ and g are w-compatible, this implies that (2.26) g(f̃(x, y)) = f̃(gx, gy). using (2.26), we have f(gx, gy, gz) = f̃(gx, gy) = g(f̃(x, y)) = g(f(x, y, z)). thus, we proved that f and g are w-compatible mappings, and the desired result follows immediately from theorem 2.4. � remark 2.6. • theorem 2.1 of aydi [5] is a particular case of corollary 2.5 by taking a1 = a1 = a3 = a4 = 0 and g = ix, the identity on x. • theorem 2.4 of aydi [5] is a particular case of corollary 2.5 by taking a3 = a4 = a5 = a6 = 0 and g = ix, the identity on x. • theorem 2.5 of aydi [5] is a particular case of corollary 2.5 by taking a1 = a2 = a5 = a6 = 0 and g = ix, the identity on x. • corollary 2.2 extends theorem 2.9 of samet and vetro [21] to partial metric spaces (corresponding to the case n = 3). • theorem 2.4 extends theorem 2.10 of samet and vetro [21] to partial metric spaces (case n = 3). • theorem 2.4 extends theorem 2.11 of samet and vetro [21] to partial metric spaces ( case n = 3). similar to the corollaries 2.3 and 2.5, by considering f(x, y, z) = fx for all x, y, z ∈ x where f : x → x, we may state the following consequence of theorem 2.4. corollary 2.7. let (x, p) be a partial metric space and f, g : x → x be mappings such that p(fx, fu) ≤ a1p(fx, gx) + a2p(fu, gu) + a3p(fu, gx) + a4p(fx, gu) + a5p(gu, gx)(2.27) for all x, u ∈ x, where ai ∈ [0, 1), i = 1, · · · , 5 and a3 + 5∑ i=1 ai < 1 or a4 + 5∑ i=1 ai < 1. suppose that f and g are weakly compatible, f(x) ⊆ g(x) and g(x) is a complete subspace of x. then the mappings f and g have a unique common fixed point. now, we give an example to illustrate our obtained results. tripled coincidence and fixed point results 205 example 2.8. let x = r+ endowed with the partial metric metric p(x, y) = max(x, y) for all x, y ∈ x. define the mappings f : x3 → x and g : x → x by gx = x 2 and f(x, y, z) = 2x + 3y + 4z 72 . we will check that all the hypotheses of theorem 2.1 are satisfied. note that f(x3) ⊆ g(x) with g(x) is complete in x. now, for all x, y, z, u, v, w ∈ x, we have p(f(x, y, z), f(u, v, w)) = max(f(x, y, z), f(u, v, w)) ≤ max( 2x + 3y + 4z 72 , 2u + 3v + 4w 72 ) ≤ 1 4 [max{ x 2 , u 2 } + max{ y 2 , v 2 } + max{ z 2 , w 2 }] = 1 4 p(gx, gu) + 1 4 p(gy, gv) + 1 4 p(gz, gw). then, the contractive condition is satisfied with ai = 0 for all i = 1, · · · , 12 and a13 = a14 = a15 = 1/4. all conditions of theorem 2.1 are satisfied. consequently, (x, y, z) is a tripled coincidence point of f and g if and only if x = y = z = 0. this implies that f and g are w-compatible. applying our theorem 2.4, we obtain the existence and uniqueness of a common tripled fixed point of f and g. in this example, (0, 0, 0) is the unique common tripled fixed point. acknowledgements. the authors are grateful to the reviewers and the editor for their useful comments. 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(received march 2012 – accepted august 2012) h. aydi (hassen.aydi@isima.rnu.tn) université de sousse, institut supérieur d’informatique et des technologies de communication de hammam sousse, route gp1-4011, h. sousse, tunisie. m. abbas (mujahid@lums.edu.pk) department of mathematics, lahore university of management sciences, 54792lahore, pakistan. tripled coincidence and fixed point results in partial metric spaces. by h. aydi and m. abbas @ appl. gen. topol. 23, no. 1 (2022), 145-156 doi:10.4995/agt.2022.15571 © agt, upv, 2022 common new fixed point results on b-cone banach spaces over banach algebras hojjat afshari a , hadi shojaat b and andreea fulga c a department of mathematics, faculty of sciences, university of bonab, bonab, iran (hojat.afshari@yahoo.com) b department of mathematics, farhangian university, qazvin, iran. (hadishojaat@yahoo.com) c department of mathematics and computer sciences, transilvania university of brasov, brasov, romania (afulga@unitbv.ro) communicated by e. karapinar abstract recently zhu and zhai studied the concepts of cone b-norm and cone bbanach space as generalizations of cone b-metric spaces and they gave a definition of φ-operator and obtained some new fixed point theorems in cone b-banach spaces over banach algebras by using φ-operator. in this paper we propose a notion of quasi-cone over banach algebras, then by utilizing some new conditions and following their work with introducing two mappings t and s we improve the fixed point theorems to the common fixed point theorems. an example is given to illustrate the usability of the obtained results. 2020 msc: 47h10; 54h25; 55m20. keywords: common fixed point; φ-operator; cone b-norm; cone b-banach space. 1. introduction the notion of b-metric was proposed by czerwik [12, 13] to generalize the concept of distance. the analog of the famous banach fixed point theorem was proved by czerwik in the frame of complete b-metric spaces, see also [9, 10, 11]. in [19] e. karapinar generalized some conclusions on the cone banach space, received 06 may 2021 – accepted 27 september 2021 http://dx.doi.org/10.4995/agt.2022.15571 https://orcid.org/0000-0003-1149-4336 https://orcid.org/0000-0002-6838-072x https://orcid.org/0000-0002-6689-0355 h. afshari, h. shojaat and a. fulga in the literature [2] and obtained the existence results of fixed points for selfmappings. also, the cone metric space over banach algebra, proposed by liu and xu (see [23]) and they considered some fixed point results on such new space. in 2001 hussain and shah [17] introduced the notation of cone b-metric space. many researchers continued the work of hussain and shah, and proved some fixed point theorems and common fixed point theorems for multiple operators on these new spaces, and also used them to investigate the existence of the solutions of fractional integral equations (see [3, 5, 4, 6, 8, 14, 15, 16, 18, 24, 25, 27, 28, 26, 20, 21]). recently zhu and zhai [30] studied the concepts of cone b-norm and cone bbanach space as generalizations of cone b-metric spaces. also they introduced the operator φ and obtained some new fixed point theorems in cone b-banach spaces over banach space utilizing the φ-operator. in this paper by introducing a notion of quasi-cone over banach space and also with applying different conditions we examine the existence of some common fixed points of two self-mappings s and t that has led to the development of similar results in the literature. 2. preliminaries let (e,‖ · ‖) be a real banach space, p ⊂ e a cone and θ be the zero of e, also there is a partial ordering ≤ such that ξ ≤ ζ iff ζ − ξ ∈ p. write ξ � ζ for ζ − ξ ∈ intp, where intp is the interior set of p. we say that p is normal if there exists n > 0 such that θ ≤ ξ ≤ ζ implies ‖ ξ ‖≤ n ‖ ζ ‖, for ξ,ζ ∈ e. p is called to be solid if intp 6= ∅. definition 2.1 (see [17]). let x 6= ∅ and s ≥ 1, a mapping % : x × x → e is a cone b-metric if; (i ) θ < %(ξ,ζ) with ξ 6= ζ and %(ξ,ζ) = θ iff ξ = ζ; (ii ) %(ξ,ζ) = %(ζ,ξ); (iii ) %(ξ,ζ) ≤ s[%(ξ,η) + %(η,ζ)], for all ξ,ζ,η ∈ x. the pair (x,%) is said a cone b-metric space, in short, cbms. lemma 2.2 (see [17]). if (x,%) is a cbms. then; (p1) if ξ � ζ and ζ � η, then ξ � η. (p2) if ξ � ζ and ζ � η, then ξ � η. (p3) if θ ≤ ξ � c for c ∈ intp, then ξ = θ. (p4) if c ∈ intp, θ ≤ ξn and ξn → θ, then there exists n0 such that ξn � c for n > n0. (p5) suppose θ � c, if θ ≤ %(ξn,ξ) ≤ ζn and ζn → θ, then eventually %(ξn,ξ) � c, where ξ ∈ x and {ξn}n≥1 is a sequence in x. (p6) if θ ≤ ξn ≤ ζn and ξn → ξ, ζn → ζ, then ξ ≤ ζ, for each cone p. (p7) if ξ ≤ λξ where ξ ∈ p and 0 ≤ λ < 1, then ξ = θ. definition 2.3 (see [19]). let x be a vector space over r. for a cone p ⊂ e and a mapping ‖ · ‖e: x → e if we have; © agt, upv, 2022 appl. gen. topol. 23, no. 1 146 common new fixed point results on b-cone banach spaces over banach algebras (i ) ‖ ξ ‖e≥ θ for ξ ∈ x and ‖ ξ ‖e= θ iff ξ = θ; (ii ) ‖ ξ + ζ ‖e≤‖ ξ ‖e + ‖ ζ ‖e for ξ,ζ ∈ x; (iii )‖ kξ ‖e=| k |‖ ξ ‖e for k ∈ r. then ‖ · ‖e is said a cone norm on x, and (x,‖ · ‖e) is said a cone normed space (cns). if we set %(ξ,ζ) =‖ ξ − ζ ‖e, then every cns is a cms. definition 2.4 (see [19]). let 1 ≤ s ≤ 2, x be a vector space over r, cone p ⊂ e. if ‖ · ‖p: x → e satisfies; (i ) ‖ ξ ‖p≥ θ for ξ ∈ x and ‖ ξ ‖p= θ iff ξ = θ; (ii ) ‖ ξ − ζ ‖p=‖ ζ − ξ ‖p for ξ,ζ ∈ x; (iii ) ‖ ξ + ζ ‖p≤ s[‖ ξ ‖p + ‖ ζ ‖p] for ξ,ζ ∈ x; (i v)‖ kξ ‖p=| k |s‖ ξ ‖p for k ∈ r. then we call ‖ · ‖p a cone-norm on x, and (x,‖ · ‖p), we call it a cone-normed space (cns). obviously, each cns is a cms. in fact, we only need to set %(ξ,ζ) =‖ ξ − ζ ‖p. definition 2.5 (see [30]). suppose that (x,‖ · ‖p) is a cone b-normed space, p ⊂ e is a solid cone, ξ ∈ x and {ξn}n≥1 is a sequence in x. then; (i ) we say that {ξn}n≥1 converges to ξ if for c ∈ e with θ � c, there is a natural number n satisfying ‖ ξn − ξ ‖p� c for n ≥ n. we denote lim n→∞ ξn = ξ or ξn → ξ; (ii ) we say that {ξn}n≥1 is a cauchy if for c ∈ e with θ � c, there exists a natural number n satisfying ‖ ξn − ξm ‖p� c for all n,m ≥ n; (iii ) we say that (x,‖ · ‖p) is complete if every cauchy is convergent. lemma 2.6 (see [30]). suppose (x,‖ · ‖p) is a cone b-normed space, p is a solid cone, ξ ∈ x and {ξn}n≥1 is a sequence in x. then the following conclusions hold: (i ) ‖ ξn − ξ ‖p→ θ(n →∞) iff {ξn} converges to ξ. (ii )‖ ξn − ξm ‖p→ θ(n,m →∞) iff {ξn} is a cauchy. lemma 2.7 ([29]). suppose (e,‖ · ‖) is a real banach space and p is a normal cone in e, then there is an equivalent norm ‖ · ‖1, which satisfies θ ≤ ξ ≤ ζ =⇒‖ ξ ‖1≤‖ ζ ‖1, for ξ,ζ ∈ e, that is, norm ‖ · ‖1 is monotonous. remark 2.8. suppose e is a linear space, ‖ · ‖1 and ‖ · ‖2 are two given norms in e, we say that ‖ · ‖2 is stronger than ‖ · ‖1 if ‖ ξn ‖2→ 0 =⇒‖ ξn ‖1→ 0 (n → ∞). if ‖ · ‖2 is stronger than ‖ · ‖1, and ‖ · ‖1 is stronger than ‖ · ‖2, then ‖ · ‖1 is equivalent ‖ · ‖2. definition 2.9 ([29, 7]). let e be a real banach algebra, that is, for ξ,ζ,η ∈ e, a ∈ r, (i ) ξ(ζη) = (ξζ)η; (ii ) ξ(ζ + η) = ξζ + ξη, (ξ + ζ)η = ξη + ζη; (iii ) a(ξζ) = (aξ)ζ = (aζ)ξ; (i v) ‖ ξζ ‖≤‖ ξ ‖‖ ζ ‖. if banach algebra e with unit element e, such that ξe = eξ = ξ for all ξ ∈ e, then ‖ e ‖= 1. if every non-zero element of e has an inverse element in e, then e is called a divisible banach algebra. © agt, upv, 2022 appl. gen. topol. 23, no. 1 147 h. afshari, h. shojaat and a. fulga definition 2.10 ([7]). let e be a banach algebra with unit element e and p ⊆ e be a cone. p is called algebra cone if e ∈ p and for each ξ,ζ ∈ p, ξζ ∈ p. in our following discussions, ξ = (x,‖ · ‖p) is a cone b-banach space, p is a solid cone and s is a operator defined on d of x. let e := (e,‖ · ‖) be a divisible banach algebra with unit element e. let pe be a normal algebra cone in e with a normal constant n. definition 2.11. let (e,‖ · ‖) be a divisible banach algebra. pe is a normal algebra cone in e. we call the mapping φ : pe → pe is a φ-operator if it satisfies (i ) φ is an increasing operator; (ii ) φ is a continuous bijection and has an inverse mapping φ−1 which is also continuous and increasing; (iii ) φ(ξ + ζ) ≤ φ(ξ) + φ(ζ) for all ξ,ζ ∈ pe; (i v) φ(ξζ) = φ(ξ)φ(ζ) for all ξ,ζ ∈ pe. remark 2.12. by definition 2.11, the part of (iii ), we can get φ−1(ξ)+φ−1(ζ) ≤ φ−1(ξ + ζ) for all ξ,ζ ∈ pe. in fact, note that φ(ξ + ζ) ≤ φ(ξ) + φ(ζ) for all ξ,ζ ∈ pe and φ−1 is also a continuous and increasing operator, then φ−1(φ(ξ + ζ)) ≤ φ−1(φ(ξ) + φ(ζ)), which yields that ξ + ζ ≤ φ−1(φ(ξ) + φ(ζ)). hence, φ−1(φ(ξ)) + φ−1(φ(ζ)) ≤ φ−1(φ(ξ) + φ(ζ)). since φ : pe → pe is a continuous bijection, thus φ−1(ξ)+φ−1(ζ) ≤ φ−1(ξ+ζ), for all ξ,ζ ∈ pe. remark 2.13. by definition 2.11, the part of (i v), we can obtain φ−1(ξζ) = φ−1(ξ)φ−1(ζ), for all ξ,ζ ∈ pe. indeed, from φ(ξζ) = φ(ξ)φ(ζ) for all ξ,ζ ∈ pe and φ−1 : pe → pe is also a continuous, we get φ−1(φ(ξζ)) = φ−1(φ(ξ)φ(ζ)), which yields that ξζ = φ−1(φ(ξ)φ(ζ)). then φ−1(φ(ξ))φ−1(φ(ζ)) = φ−1(φ(ξ)φ(ζ)). thanks to that φ : pe → pe is a continuous bijection, φ−1(ξζ) = φ−1(ξ)φ−1(ζ), for all ξ,ζ ∈ pe. remark 2.14. for example, let e = r, a divisible banach algebra, pe = {ξ ∈ e | ξ ≥ 0} be a normal cone in e, suppose φ : pe → pe, defined by φ(ξ) = ξ 1 5 and then φ−1(ξ) = ξ5, for ξ ∈ pe. we can prove it also satisfies the above conditions. © agt, upv, 2022 appl. gen. topol. 23, no. 1 148 common new fixed point results on b-cone banach spaces over banach algebras 3. main results theorem 3.1. let x be a cone-b-banach space with the coefficient 1 ≤ s ≤ 2, e1 and e2 be divisible banach space with identity elements e1 and e2, also pe1 and pe2 be normal algebra cones in e1 and e2 (respectively). if d and d ′ ⊂ x with d ∩ d′ 6= ∅ are closed and convex, also φ : pe1 ∪ pe2 → pe1 ∪ pe2 is φ-operator and t : d → d′, s : d′ → d satisfying the followings φ(%(η,sξ′)) + φ(%(η′,t ξ)) ≤ k1φ(%(η,ξ)),(3.1) φ(%(ξ′,t η)) + φ(%(ξ,sη′)) ≤ k2φ(%(ξ′,η′)), for all ξ,η ∈ d, ξ′,η′ ∈ d′, where φ(2se1) ≤ k1 < φ(2s+1e1) , φ(2se2) ≤ k2 < φ(2s+1e2) in pe1 and pe2 (respectively). then s and t have a common fixed point in d ∩ d′. proof. let ξ1 ∈ d, η1 ∈ d′ be arbitrary. we introduce two sequences {ξn},{ηn}∈ d ∪ d′, defined by ξ2 = η1+t ξ1 2 ∈ d′, η2 = ξ1+sη1 2 ∈ d, ξ3 = η2+sξ2 2 ∈ d′, η3 = ξ2+tη2 2 ∈ d, ... ξ2n = η2n−1+t ξ2n−1 2 , n = 1, 2, · · · , η2n = ξ2n−1+sη2n−1 2 , n = 1, 2, · · · , ξ2n +1 = η2n +sξ2n 2 , n = 1, 2, · · · , η2n +1 = ξ2n +tη2n 2 , n = 1, 2, · · · . we get η2n −sξ2n = 2(η2n − ( η2n + sξ2n 2 )) = 2(η2n − ξ2n +1), ξ2n −t η2n = 2(ξ2n − ( ξ2n + t η2n 2 )) = 2(ξ2n −η2n +1), η2n +1 −t ξ2n +1 = 2(η2n +1 − ( η2n +1 + t ξ2n +1 2 )) = 2(η2n +1 − ξ2n +2), ξ2n +1 −sη2n +1 = 2(ξ2n +1 − ( ξ2n +1 + sη2n +1 2 )) = 2(ξ2n +1 −η2n +2), which is equivalent to %(η2n,sξ2n ) = ‖ η2n −sξ2n ‖pe1 = ‖ 2(η2n − ξ2n +1) ‖pe1 = 2s ‖ η2n − ξ2n +1 ‖pe1 = 2s%(η2n,ξ2n +1), © agt, upv, 2022 appl. gen. topol. 23, no. 1 149 h. afshari, h. shojaat and a. fulga %(ξ2n,t η2n ) = ‖ ξ2n −t η2n ‖pe2 = ‖ 2(ξ2n −η2n +1) ‖pe2 = 2s ‖ ξ2n −η2n +1 ‖pe2 = 2s%(ξ2n,η2n +1) and %(η2n +1,t ξ2n +1) = ‖ η2n +1 −t ξ2n +1 ‖pe2(3.2) = ‖ 2(η2n +1 − ξ2n +2) ‖pe2 = 2s ‖ η2n +1 − ξ2n +2 ‖pe2 = 2s%(η2n +1,ξ2n +2), %(ξ2n +1,sη2n +1) = ‖ ξ2n +1 −sη2n +1 ‖pe1(3.3) = ‖ 2(ξ2n +1 −η2n +2) ‖pe1 = 2s ‖ ξ2n +1 −η2n +2 ‖pe1 = 2s%(ξ2n +1,η2n +2). substituting ξ′ = ξ2n,ξ = ξ2n +1 and η = η2n,η ′ = η2n +1 in (3.1), we can obtain φ(%(η2n,sξ2n )) + φ(%(η2n +1,t ξ2n +1)) ≤ k1φ(%(η2n,ξ2n +1)). we get φ(2s%(η2n,ξ2n +1)) + φ(2 s%(η2n +1,ξ2n +2)) ≤ k1φ(%(η2n,ξ2n +1)). according to the condition (iii ) of φ-operator, φ(2s(%(η2n,ξ2n +1) + %(η2n +1,ξ2n +2))) ≤ k1φ(%(η2n,ξ2n +1)). remark 2.13 and the property of φ−1 operator, we can get 2s(%(η2n,ξ2n +1) + %(η2n +1,ξ2n +2)) ≤ φ−1(k1)%(η2n,ξ2n +1), by simplifying, we get %(η2n +1,ξ2n +2) ≤ ( φ−1(k1) 2s −e1)%(η2n,ξ2n +1). substituting ξ′ = ξ2n,ξ = ξ2n +1 and η = η2n,η ′ = η2n +1 in (3.1). then one can obtain φ(%(ξ2n,t η2n )) + φ(%(ξ2n +1,sη2n +1)) ≤ k2φ(%(ξ2n,η2n +1)). by (3.2) and (3.3) we get φ(2s%(ξ2n,η2n +1)) + φ(2 s%(ξ2n +1,η2n +2)) ≤ k2φ(%(ξ2n,η2n +1)). according to the condition (iii ) of φ-operator, φ(2s(%(ξ2n,η2n +1) + %(ξ2n +1,η2n +2))) ≤ k2φ(%(ξ2n,η2n +1)). by remark 2.13 and the property of φ−1 operator, we can get 2s(%(ξ2n,η2n +1) + %(ξ2n +1,η2n +2)) ≤ φ−1(k2)%(ξ2n,η2n +1), © agt, upv, 2022 appl. gen. topol. 23, no. 1 150 common new fixed point results on b-cone banach spaces over banach algebras by simplifying, we get %(ξ2n +1,η2n +2) ≤ ( φ−1(k2) 2s −e2)%(ξ2n,η2n +1). thus, %(η2n +1,ξ2n +2) ≤ k′1%(η2n,ξ2n +1) and %(ξ2n +1,η2n +2) ≤ k′2%(ξ2n,η2n +1), where k′1 = φ−1(k1) 2s −e1 and k′2 = φ−1(k2) 2s −e2. repeating this relations, we get %(η2n +1,ξ2n +2) ≤ k′2 n k′1 n %(η1,ξ2),(3.4) %(ξ2n +1,η2n +2) ≤ k′2 n k′1 n %(ξ1,η2). for any m ≥ 1,p ≥ 1, we have one of the following two cases: (i ) m + p = 2r − 1, r ≥ 1,r ∈ n, then we get %(ηm+p,ξm) ≤ s[%(ηm+p,ξm+p−1) + %(ξm+p−1,ξm)] ≤ s%(ηm+p,ξm+p−1) + s2[%(ξm+p−1,ηm+p−2) + %(ηm+p−2,ξm)] ≤ s%(ηm+p,ξm+p−1) + s2%(ξm+p−1,ηm+p−2) + s3%(ηm+p−2,ξm+p−3) + · · · +sp−1%(ξm+2,ηm+1) + s p−1%(ηm+1,ξm) ≤ sk′2 r−2 k′1 r−1 %(η2,ξ1) + s 2k′2 r−3 k′1 r−2 %(ξ2,η1) + s 3k′2 r−4 k′1 r−3 %(η2,ξ1) + · · · +sp−1k′2 2r−p−1 k′1 2r−p %(ξ2,η1) + s p−1k′2 2r−p−2 k′1 2r−p−1 %(η2,ξ1) = (sk′2 r−2 k′1 r−1 + s3k′2 r−4 k′1 r−3 + · · · + sp−1k′2 2r−p−2 k′1 2r−p−1 )%(η2,ξ1) +(s2k′2 r−3 k′1 r−2 + · · · + sp−1k′2 2r−p−1 k′1 2r−p )%(ξ2,η1) = sk′2 r−2 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(η2,ξ1) + s2k′2 r−3 k′1 r−2 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1). (ii ) m + p = 2r, r ≥ 1,r ∈ n, then we get %(ξm+p,ηm) ≤ s[%(ξm+p,ηm+p−1) + %(ηm+p−1,ηm)] ≤ s%(ξm+p,ηm+p−1) + s2[%(ηm+p−1,ξm+p−2) + %(ξm+p−2,ηm)] ≤ s%(ξm+p,ηm+p−1) + s2%(ηm+p−1,ξm+p−2) + s3%(ξm+p−2,ηm+p−3) + · · · +sp−1%(ηm+2,ξm+1) + s p−1%(ξm+1,ηm) ≤ sk′2 r−1 k′1 r−1 %(ξ2,η1) + s 2k′2 r−2 k′1 r−2 %(η2,ξ1) + s 3k′2 r−3 k′1 r−3 %(ξ2,η1) + · · · +sp−1k′2 2r−p k′1 2r−p %(η2,ξ1) + s p−1k′2 2r−p−1 k′1 2r−p−1 %(ξ2,η1) © agt, upv, 2022 appl. gen. topol. 23, no. 1 151 h. afshari, h. shojaat and a. fulga = (sk′2 r−1 k′1 r−1 + s3k′2 r−3 k′1 r−3 + · · · + sp−1k′2 2r−p−1 k′1 2r−p−1 )%(ξ2,η1) +(s2k′2 r−2 k′1 r−2 + · · · + sp−1k′2 2r−p k′1 2r−p )%(η2,ξ1) = sk′2 r−1 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1) + s2k′2 r−2 k′1 r−2 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e1 − s 2 (k′2k ′ 1) 2 %(η2,ξ1). since φ(2se1) ≤ k′2 < φ(2s+1e1) in pe1 and φ(2se2) ≤ k′1 < φ(2s+1e2) in pe2 with 1 ≤ s ≤ 2, we know θ2 ≤ k′2 < e1, θ1 ≤ k′1 < e2, thus θ2 < se2 −k′2 ≤ se2, θ1 < se1 −k′1 ≤ se1. further, ‖ k′2 r−p −θ2 ‖=‖ k′2 r−p ‖≤‖ k′2 ‖ r−p, ‖ k′1 r−p −θ1 ‖=‖ k′1 r−p ‖≤‖ k′1 ‖ r−p,(3.5) since θ2 ≤ k′2 < e2, θ1 ≤ k′1 < e1 and pe is a normal cone in e, by lemma 2.7 we know there is an equivalent norm ‖ · ‖1 and thus 0 ≤‖ k′2 ‖1<‖ e2 ‖1= 1,(3.6) 0 ≤‖ k′1 ‖1<‖ e1 ‖1= 1. by (3.5) and (3.6), we get ‖ k′2 r−p −θ2 ‖1≤‖ k′2 ‖ r−p 1 → 0((r − p) →∞), ‖ k′1 r−p −θ1 ‖1≤‖ k′1 ‖ r−p 1 → 0((r − p) →∞).(3.7) from remark (2.8) and (3.7), ‖ k′2 r−p −θ2 ‖≤‖ k′2 ‖ r−p→ 0((r − p) →∞), ‖ k′1 r−p −θ1 ‖≤‖ k′1 ‖ r−p→ 0((r − p) →∞). thus, lim (r−p)→∞ k′2 (r−p) → θ2, lim (r−p)→∞ k′1 (r−p) → θ1. (3.8) let θ1,θ2 � c be given. by (3.8), sk′2 r−1 k′1 r (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(η2,ξ1)+ s2k′2 r−2 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1) → θ1, sk′2 r k′1 r (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1)+ s2k′2 r−1 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e1 − s 2 (k′2k ′ 1) 2 %(η2,ξ1) → θ2, © agt, upv, 2022 appl. gen. topol. 23, no. 1 152 common new fixed point results on b-cone banach spaces over banach algebras as (r − p) → ∞. making full use of lemma 2.2 (p4), we find m0 ∈ n, such that sk′2 r−1 k′1 r (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(η2,ξ1)+ s2k′2 r−2 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1) � c, sk′2 r k′1 r (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1)+ s2k′2 r−1 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e1 − s 2 (k′2k ′ 1) 2 %(η2,ξ1) � c, for each m > m0. thus sk′2 r−1 k′1 r (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(η2,ξ1)+ s2k′2 r−2 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1) � c, sk′2 r k′1 r (e1e2 − s 2 (k′2k ′ 1) 2 ) p+1 2 e1e2 − s 2 (k′2k ′ 1) 2 %(ξ2,η1)+ s2k′2 r−1 k′1 r−1 (e1e2 − s 2 (k′2k ′ 1) 2 ) p−1 2 e1e1 − s 2 (k′2k ′ 1) 2 %(η2,ξ1) � c, for m > m0 and each p. considering the upper relations we can get; %(ξm+p,ξm) ≤ s(%(ξm+p,ηm) + %(ηm + ξm)), %(ηm+p,ηm) ≤ s(%(ηm+p,ξm) + %(ξm + ηm)). now by lemma ?? part (p1), we can claim that {ξn} and {ηn} are cauchy sequences in d. note that d and d′ are closed and convex and {ξ2n}, {η2n} converges to some ζ,ζ′, that is, ξ2n,η2n → ζ,ζ′ ∈ d ∪ d′. regarding the inequality %(ζ,sη2n +1) ≤ s[%(ζ,ξ2n +1) + %(ξ2n +1,sη2n +1)], %(ζ′,t ξ2n +1) ≤ s[%(ζ′,η2n +1) + %(η2n +1,t ξ2n +1)], and from (3.3), we obtain %(ζ,sη2n +1) ≤ s[%(ζ,ξ2n +1) + 2s%(ξ2n +1,η2n +2)],(3.8) %(ζ′,t ξ2n +1) ≤ s[%(ζ′,η2n +1) + 2s%(η2n +1,ξ2n +2)], let n → ∞, then sη2n +1 → ζ, t ξ2n +1 → ζ. finally, replacing η2n +1 = ζ in (3.8). then one can obtain φ(%(ζ,sζ′)) ≤ s[%(ζ,ξ2n +1) + 2s%(ξ2n +1,η2n +2)], and if ζ = ξ2n +1 φ(%(ζ′,t ζ)) ≤ s[%(ζ′,η2n +1) + 2s%(η2n +1,ξ2n +2)], and by making use of the property iv of φ-operator, we obtain, φ(e1) = e1,φ(e2) = e2. so we get φ(%(ζ,sζ′) = e1, φ(%(ζ′,t ζ) = e2. therefore as n → ∞, we can obtain sζ′ = ζ,t ζ = ζ′. hence considering ζ = ζ′, we conclude ζ = tζ = sζ. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 153 h. afshari, h. shojaat and a. fulga corollary 3.2. let x be a cone-b-banach space with the coefficient 1 ≤ s ≤ 2, e be a divisible banach algebra with identity element e, and also pe be a normal algebra cone in e. if d ⊂ x is closed and convex, φ : pe → pe is an φ-operator and s,t : d → d are mappings satisfying the conditions φ(%(η,sξ)) + φ(%(η,t ξ)) ≤ kφ(%(η,ξ)),(3.9) φ(%(ξ,t η)) + φ(%(ξ,sη)) ≤ kφ(%(ξ,η)), for all ξ,η ∈ d, where φ(2se) ≤ k < φ(2s+1e) in pe. then s and t have a common fixed point in d. proof. if in theorem 3.1 we set, t = s and d = d′, considering the condition of (3.10) and by the proof similar to the proof of theorem 3.1 we deduce the result. � example 3.3. let x = r2 and e = r2 endowed with partial ordered ξ = (ξ1,ξ2) ≤ ζ = (ζ1,ζ2) iff ξ1 ≤ ζ1,ξ2 ≤ ζ2. if p = {(ξ1,ξ2) ∈ e : ξ1 ≥ 0,ξ2 ≥ 0}, we define ‖ (ξ1,ξ2) ‖p= (| ξ1 |2, | ξ2 |2). then (x,‖ · ‖p) is a cone b-banach space with s = 2. for ξ = (ξ1,ξ2) and ζ = (ζ1,ζ2) we define; ξ.ζ = (ξ1ξ2,ζ1ζ2). by the the mentioned definition p is a banach algebra and e := (e,‖ · ‖) is a divisible banach algebra with unit element e = (1, 1), because ξe = eξ = ξ,‖e‖ = 1 and hence e is a multiplicative identity. if we put φ : p → p with φ(ξ = (ξ1,ξ2)) = ( √ ξ1, √ ξ2), then φ satisfies the conditions (i )-(i v) of definition 3.4. also we set; %(ξ,ζ) =‖ ξ − ζ ‖p= (| (ξ1 − ζ1 |2, | ξ2 − ζ2 |2),%(ξ,a) = inf{%(ξ,ζ) : ζ ∈ a} and sξ = ξ 2 ,t ξ = ξ 2 4 . now we define the region d as the following; d = {(ξn,ηn ) : |ηn− ξn 2 |+|ηn− ξn 2 4 | ≤ 2.8|ηn−ξn|, |ξn− ηn 2 |+|ξn− ηn 2 4 | ≤ 2.8|ξn−ηn|,n = 1, 2}. obviously d is closed and convex. φ(%(η,sξ)) + φ(%(η,t ξ)) ≤ φ(%(η, ξ 2 )) + φ(%(η, ξ 2 4 )) ≤ φ(|η1 − ξ12 | 2, |η2 − ξ22 | 2) + φ(|η1 − ξ1 2 4 |2, |η2 − ξ2 2 4 |2) = (|η1 − ξ12 |, |η2 − ξ2 2 |) + (|η1 − ξ1 2 4 |, |η2 − ξ2 2 4 |) = (|η1 − ξ12 | + |η1 − ξ1 2 4 |, |η2 − ξ22 | + |η2 − ξ2 2 4 |) ≤ (|η1 − ξ12 | + |η1 − ξ1 2 4 |, |η2 − ξ22 | + |η2 − ξ2 2 4 |), © agt, upv, 2022 appl. gen. topol. 23, no. 1 154 common new fixed point results on b-cone banach spaces over banach algebras also φ(%(ξ,t η)) + φ(%(ξ,sη)) ≤ φ(%(ξ, η 2 4 )) + φ(%(ξ, η 2 )) ≤ φ(|ξ1 − η1 2 4 |2, |ξ2 − η2 2 4 |2) + φ(|ξ1 − η12 | 2, |ξ2 − η22 | 2) = (|ξ1 − η1 2 4 |, |ξ2 − η2 2 4 |) + (|ξ1 − η12 |, |ξ2 − η2 2 |) = (|ξ1 − η12 | + |ξ1 − η1 2 4 |, |ξ2 − η22 | + |ξ2 − η2 2 4 |) ≤ (|ξ1 − η12 | + |ξ1 − η1 2 4 |, |ξ2 − η22 | + |ξ2 − η2 2 4 |). considering φ(2se) = φ(4, 4) = (2, 2) ≤ k < φ(2s+1e) = φ(8, 8) = (2 √ 2, 2 √ 2), we should have (|η1 − ξ12 | + |η1 − ξ1 2 4 |, |η2 − ξ22 | + |η2 − ξ2 2 4 |) ≤ (2.8, 2.8)(|η1 − ξ1|, |η2 − ξ2|) = (2.8|η1 − ξ1|, 2.8|η2 − ξ2|), (|ξ1 − η12 | + |ξ1 − η1 2 4 |, |ξ2 − η22 | + |ξ2 − η2 2 4 |) ≤ (2.8, 2.8)(|ξ1 −η1|, |ξ2 −η2|) = (2.8|ξ1 −η1|, 2.8|ξ2 −η2|). so according to the definition of region d the conditions of corollary 3.2 are satisfied. hence s and t have a common fixed point. corollary 3.4. let x be a cone-b-banach space with the coefficient 1 ≤ s ≤ 2, e be a divisible banach algebra with identity element e, and also pe be a normal algebra cone in e . if d ⊂ x is closed and convex, φ : pe → pe is an φ-operator and t : d → d is a mapping satisfying the condition φ(%(η,t η)) + φ(%(ξ,t ξ)) ≤ kφ(%(η,ξ)), for all ξ,η ∈ d, where φ(2se) ≤ k < φ(2s+1e) in pe. then t has a fixed point in d. references [1] m. a. alghamdi, s. gulyaz-ozyurt and e. karapinar, a note on extended z-contraction, mathematics 8 (2020), paper no. 195. 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[30] x. zhu and c. zhai, some extension results on cone b-metric spaces over banach space via ϕ-operator, j. anal. 29 (2021), 281–295. © agt, upv, 2022 appl. gen. topol. 23, no. 1 156 @ appl. gen. topol. 21, no. 1 (2020), 35-51 doi:10.4995/agt.2020.11865 c© agt, upv, 2020 on a metric on the space of idempotent probability measures adilbek atakhanovich zaitov tashkent institute of architecture and civil engineering, 13, navoi str, tashkent city, 100011, uzbekistan chirchik state pedagogical institute, 104, amir temur str., chirchik town, 111700, uzbekistan. (adilbek zaitov@mail.ru) communicated by d. werner abstract in this paper we introduce a metric on the space i(x) of idempotent probability measures on a given compact metric space (x, ρ), which extends the metric ρ. it is proven the introduced metric generates the pointwise convergence topology on i(x). 2010 msc: 28c20; 54e35. keywords: compact metrizable space; idempotent measure; metrization. 1. introduction idempotent mathematics is based on replacing the usual arithmetic operations with a new set of basic operations, i. e., on replacing numerical fields by idempotent semirings and semifields. typical example is the so-called max-plus algebra rmax. many authors (s. c. kleene, s. n. n. pandit, n. n. vorobjev, b. a. carré, r. a. cuninghame-green, k. zimmermann, u. zimmermann, m. gondran, f. l. baccelli, g. cohen, s. gaubert, g. j. olsder, j.-p. quadrat, and others) used idempotent semirings and matrices over these semirings for solving some applied problems in computer science and discrete mathematics, starting from the classical paper by s. c. kleene [7]. received 21 may 2019 – accepted 26 november 2019 http://dx.doi.org/10.4995/agt.2020.11865 a. a. zaitov the modern idempotent analysis (or idempotent calculus, or idempotent mathematics) was founded by v. p. maslov and his collaborators [10]. some preliminary results are due to e. hopf and g. choquet, see [2], [5]. idempotent mathematics can be treated as the result of a dequantization of the traditional mathematics over numerical fields as the planck constant h tends to zero taking imaginary values. this point of view was presented by g. l. litvinov and v. p. maslov [11]. in other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the fields of real and complex numbers. the basic paradigm is expressed in terms of an idempotent correspondence principle. this principle is closely related to the well-known correspondence principle of n. bohr in quantum theory. actually, there exists a heuristic correspondence between important, interesting, and useful constructions and results of the traditional mathematics over fields and analogous constructions and results over idempotent semirings and semifields (i. e., semirings and semifields with idempotent addition). a systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. as a result, in parallel with the traditional mathematics over fields, its “shadow,” idempotent mathematics, appears. this “shadow” stands approximately in the same relation to traditional mathematics as classical physics does to quantum theory. recall [10] that a set s equipped with two algebraic operations: addition ⊕ and multiplication �, is said to be a semiring if the following conditions are satisfied: • the addition ⊕ and the multiplication � are associative; • the addition ⊕ is commutative; • the multiplication � is distributive with respect to the addition ⊕: x� (y ⊕z) = x�y ⊕x�z and (x⊕y) �z = x�z ⊕y �z for all x, y, z ∈ s. a unit of a semiring s is an element 1 ∈ s such that 1 � x = x � 1 = x for all x ∈ s. a zero of the semiring s is an element 0 ∈ s such that 0 6= 1 and 0 ⊕ x = x ⊕ 0 = x for all x ∈ s. a semiring s is called an idempotent semiring if x⊕x = x for all x ∈ s. a (an idempotent) semiring s with neutral elements 0 and 1 is called a (an idempotent ) semifield if every nonzero element of s is invertible. note that diöıds, quantales and inclines are examples of idempotent semirings [10]. let r = (−∞, +∞) be the field of real numbers and r+ = [0, +∞) be the semiring of all nonnegative real numbers (with respect to the usual addition and multiplication). consider a map φh : r+ → s = r∪{−∞} defined by the equality φh(x) = h ln x, h > 0. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 36 a metric on the space of idempotent measures let x, y ∈ x and u = φh(x), v = φh(y). put u⊕h v = φh(x + y) and u�v = φh(xy). the imagine φh(0) = −∞ of the usual zero 0 is a zero 0 and the imagine φh(1) = 0 of the usual unit 1 is a unit 1 in s with respect to these new operations. thus s obtains the structure of a semiring r(h) isomorphic to r+. a direct check shows that u⊕h v → max{u, v} as h → 0. the convention −∞�x = −∞ allows us to extend ⊕ and � over s. it can easily be checked that s forms a semiring with respect to the addition u ⊕ v = max{u, v} and the multiplication u � v = u + v with zero 0 = −∞ and unit 1 = 0. denote this semiring by rmax; it is idempotent, i. e., u⊕u = u for all its elements u. the semiring rmax is actually a semifield. the analogy with quantization is obvious; the parameter h plays the role of the planck constant, so r+ can be viewed as a “quantum object” and rmax as the result of its “dequantization”. this passage to rmax is called the maslov dequantization (for details, see [8], [9], [15]). the notion of idempotent (maslov) measure finds important applications in different parts of mathematics, mathematical physics and economics (see the survey article [10] and the bibliography therein). topological and categorical properties of the functor of idempotent measures were studied in [16], [17]. although idempotent measures are not additive and the corresponding functionals are not linear, there are some parallels between topological properties of the functor of probability measures and the functor of idempotent measures (see, for example [15], [14], [16]) which are based on existence of natural equiconnectedness structure on both functors. however, some differences appear when the problem of the metrizability of the space of idempotent probability measures is studied. the problem of the metrizability of the space of the usual probability measures was investigated in [3]. we show that the analog of the metric introduced in [3] (on the space of probability measures) is not a metric on the space of idempotent probability measures. we show the mentioned analog is only a pseudometric. it is well-known that if (x, ρ) is a compact metric space, then the space p(x) of probability measures can be endowed with the kantorovich metric. in [17], m. zarichnyi posed the problem of building a metric on the space of idempotent probability measures. still the problem of existence of a natural metrization of the space i(x) has been open. in this paper we give a positive answer and introduce a metric on the space of idempotent probability measures. 2. idempotent probability measures. preliminaries let x be a compact hausdorff space, c(x) be the algebra of continuous functions on x with the usual algebraic operations. on c(x) the operations ⊕ and � are determined by ϕ⊕ψ = max{ϕ,ψ} and ϕ�ψ = ϕ + ψ where ϕ, ψ ∈ c(x). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 37 a. a. zaitov recall [17] that a functional µ: c(x) → r is said to be an idempotent probability measure on x if it satisfies the following properties: (1) µ(λx) = λ for all λ ∈ r, where λx is a constant function; (2) µ(λ�ϕ) = λ�µ(ϕ) for all λ ∈ r and ϕ ∈ c(x); (3) µ(ϕ⊕ψ) = µ(ϕ) ⊕µ(ψ) for all ϕ, ψ ∈ c(x). for a compact hausdorff space x by i(x) we denote the set of all idempotent probability measures on x. since i(x) ⊂ rc(x), we consider i(x) as a subspace of rc(x). a family of sets of the form 〈µ; ϕ1, . . . , ϕn; ε〉 = {ν ∈ i(x) : |ν(ϕi) −µ(ϕi)| < ε, i = 1, . . . , n} is a base of open neighbourhoods of a given idempotent probability measure µ ∈ i(x) according to the induced topology, where ϕi ∈ c(x), i = 1, . . . , n, and ε > 0. it is obvious that the induced topology and the pointwise convergence topology on i(x) coincide. let x, y be compact hausdorff spaces and f : x → y be a continuous map. it is easy to check that the map i(f) : i(x) → i(y ) determined by the formula i(f)(µ)(ψ) = µ(ψ ◦f) is continuous. the construction i is a normal functor acting in the category of compact hausdorff spaces and their continuous maps. therefore, for each idempotent probability measure µ ∈ i(x) one may determine its support : suppµ = ⋂{ a ⊂ x : a = a, µ ∈ i(a) } . consider functions of the type λ : x → [−∞, 0]. on a given set x we determine a max-plus-characteristic function ⊕χa : x → rmax of a subset a ⊂ x by the rule ⊕χa(x) = { 0 at x ∈ a, −∞ at x ∈ x \a. (2.1) for a singleton {x} we will write ⊕χx instead of ⊕χ{x}. let f1, f2, . . . , fn be a disjoint system of closed sets of a space x, and a1, a2, . . . , an be non-positive real numbers. a function ⊕χ a1, ...,an f1, ...,fn (x) =   a1 at x ∈ f1, . . . , an at x ∈ fn, −∞ at x ∈ x \ n⋃ i=1 fn (2.2) we call the max-plus-step-function defined by the sets f1, f2, . . . , fn and the numbers a1, a2, . . . , an. note that ⊕χaa(x) = a� ⊕χa(x) = { 0 �a at x ∈ a, −∞ at x ∈ x \a = { a at x ∈ a, −∞ at x ∈ x \a c© agt, upv, 2020 appl. gen. topol. 21, no. 1 38 a metric on the space of idempotent measures for a set a in x and a non-positive number a. consequently, for a disjoint system of closed sets f1, f2, . . . , fn in a space x, and non-positive real numbers a1, a2, . . . , an we have ⊕χ a1, ...,an f1, ...,fn (x) = ⊕χa1f1 (x) ⊕ ⊕χa2f2 (x) ⊕ . . . ⊕ ⊕χanfn (x). in the case when f1, f2, . . . , fn are singletons, say fi = {xi}, i = 1, . . . , n, we have (2.3) ⊕χ a1, ...,an {x1}, ...,{xn} = ⊕χa1{x1} ⊕ ⊕χa2{x2} ⊕ . . . ⊕ ⊕χan{xn}. the notion of density for an idempotent measure was introduced in [8], where the main result on the existence on densities for arbitrary measures was proved. a more detailed exposition is given in [9] – the first systematic monograph on the idempotent analysis. later the paper [1] appeared, where further investigations of densities were done. let µ ∈ i(x). then we can define a function dµ : x → [−∞, 0] by the formula (2.4) dµ(x) = inf{µ(ϕ) : ϕ ∈ c(x) such that ϕ ≤ 0 and ϕ(x) = 0}, x ∈ x. the function dµ is upper semicontinuous and is called the density of µ. conversely, each upper semicontinuous function f : x → [−∞, 0] with max{f(x) : x ∈ x} = 0 determines an idempotent measure νf by the formula (2.5) νf (ϕ) = ⊕ x∈x f(x) �ϕ(x), ϕ ∈ c(x). note that a function f : x → r is said to be upper semicontinuous if for each x ∈ x, and for every real number r which satisfies f(x) < r, there exists an open neighbourhood u ⊂ x of x such that f(x′) < r for all x′ ∈ u. it is easy to see that functions defined by (2.1) or by (2.2) are upper semicontinuous. put us(x) = { λ: x → [−∞, 0] ∣∣ λ is upper semicontinuous and there exists a x0 ∈ x such that λ(x0) = 0 } . then we have i(x) = {⊕ x∈x λ(x) �δx : λ ∈ us(x) } . obviously that ⊕ x∈x ⊕χx0 (x) � δx = δx0 , i. e. for a max-plus-characteristic function ⊕χx0 formula (2.5) defines the dirac measure δx0 supported on the singleton {x0}. a set of all dirac measures on a hausdorff compact space x we denote by δ(x). it is easy to notice that the space x and the subspace δ(x) ⊂ i(x) are homeomorphic. this phenomenon gives us the opportunity to consider x as subspace of i(x). let a be a closed subset of a compact hausdorf space x. it is easy to check that ν ∈ i(a) iff {x ∈ x : dν(x) > −∞}⊂ a. hence, supp ν = {x ∈ x : dν(x) > −∞}. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 39 a. a. zaitov it is evident that supp ν = {x1, . . . , xn} if and only if the density dν of ν has the shape (2.3) for singletons {x1}, . . . , {xn} and for some non-negative numbers a1, . . . , an with ai > −∞, i = 1, . . . , n, and max{a1, . . . , an} = 0. in this case ν is said to be an idempotent probability measure with finite support. a subset of i(x) consisting of all idempotent probability measures with finite support we denote by iω(x). consider an idempotent probability measure µ = ⊕ x∈x λ(x)�δx ∈ i(x) and a finite system {u1, . . . , un} of open sets ui ⊂ x such that supp µ∩ui 6= ∅, i = 1, . . . , n, and supp µ ⊂ n⋃ i=1 ui. define a set (2.6) 〈µ; u1 . . . , un; ε〉 = { ν = ⊕ x∈x γ(x) �δx ∈ i(x) : supp ν ∩ui 6= ∅, supp ν ⊂ n⋃ i=1 ui, and |λ(x) −γ(y)| < ε at the points x ∈ supp µ∩ui and y ∈ supp ν ∩ui, i = 1, . . . , n, } . theorem 2.1. the sets of the type (2.6) form a base of the pointwise convergence topology in i(x). proof. let 〈µ; ϕ; ε〉 be a prebase element, where ϕ ∈ c(x), ε > 0 and µ =⊕ x∈x λ(x) � δx ∈ i(x). as ϕ is continuous, for each point x ∈ supp µ there is its open neighbourhood ux in x such that for any point y ∈ ux the inequality |ϕ(x)−ϕ(y)| < ε 2 holds. from the open cover {ux : x ∈ supp µ} in x of supp µ by owing to compactness of supp µ one can choose a finite subcover {ui : i = 1, . . . , n}. further, for every ν = ⊕ x∈x γ(x) � δx ∈ 〈µ; u1, . . . , un; ε2〉 we have |λ(x) −γ(y)| < ε 2 at x ∈ supp µ∩ui and y ∈ supp ν ∩ui. let us estimate the following absolute value |µ(ϕ)−ν(ϕ)| = ∣∣∣∣ ⊕ x∈x λ(x) �ϕ(x) − ⊕ x∈x γ(x) �ϕ(x) ∣∣∣∣ = a. two cases are possible: case 1 : ⊕ x∈x λ(x)�ϕ(x) ≥ ⊕ x∈x γ(x)�ϕ(x). let ⊕ x∈x λ(x)�ϕ(x) = λ(x′)� ϕ(x′). then x′ ∈ ui for some i, and a = ⊕ x∈x λ(x) �ϕ(x) − ⊕ x∈x γ(x) �ϕ(x) = λ(x′) �ϕ(x′) − ⊕ x∈x γ(x) �ϕ(x) ≤ ≤ (for every y ∈ supp ν ∩ui) ≤ ≤ λ(x′) �ϕ(x′) −γ(y) �ϕ(y) = |λ(x′) �ϕ(x′) −γ(y) �ϕ(y)| ≤ ≤ |λ(x′) −γ(y)| + |ϕ(x′) −ϕ(y)| < ε 2 + ε 2 = ε. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 40 a metric on the space of idempotent measures case 2 : ⊕ x∈x λ(x)�ϕ(x) ≤ ⊕ x∈x γ(x)�ϕ(x). let ⊕ x∈x γ(x)�ϕ(x) = γ(x′)� ϕ(x′). then x′ ∈ ui for some i, and a = ⊕ x∈x γ(x) �ϕ(x) − ⊕ x∈x λ(x) �ϕ(x) = γ(x′) �ϕ(x′) − ⊕ x∈x λ(x) �ϕ(x) ≤ ≤ (for every y ∈ supp µ∩ui) ≤ ≤ γ(x′) �ϕ(x′) −λ(y) �ϕ(y) = |γ(x′) �ϕ(x′) −λ(y) �ϕ(y)| ≤ ≤ |λ(x′) −γ(y)| + |ϕ(x′) −ϕ(y)| < ε 2 + ε 2 = ε. so, |µ(ϕ) −ν(ϕ)| < ε. from here ν ∈ 〈µ; ϕ; ε〉, in other words,〈 µ; u1, . . . , un; ε 2 〉 ⊂〈µ; ϕ; ε〉. � theorem 2.1 immediately yields the following statement. corollary 2.2. the subset iω(x) is everywhere dense in i(x) with respect to the pointwise convergence topology. we recall some concepts from [13] and if necessary, modify them for the max-plus case . let x and y be compact hausdorff spaces, f : x → y be a map, f◦ : c(y ) → c(x) be the induced operator defined by equality f◦(ϕ) = ϕ ◦ f, ϕ ∈ c(y ). we say that an operator u: c(x) → c(y ) is a max-plus-linear operator provided u(α � ϕ ⊕ β � ψ) = α � u(ϕ) ⊕ β � u(ψ) for every pair of functions ϕ, ψ ∈ c(x), where −∞ ≤ α, β ≤ 0, α ⊕ β = 0. a max-plus-linear operator u: c(x) → c(y ) is max-plus-regular provided ‖u‖ = sup{‖u(ϕ)‖ : ϕ ∈ c(x), ‖ϕ‖ ≤ 1} = 1 and u(1x) = 1y . a max-pluslinear operator u: c(x) → c(y ) is said to be a max-plus-linear exave for f provided f◦◦u is the identity on f◦(c(y )) or equivalently f◦◦u◦f◦ = f◦. a max-plus-regular exave is a max-plus-linear exave which is a regular operator. if f is a homeomorphic embedding, then a max-plus-linear exave (max-plusregular exave) for f is called max-plus-linear extension operator (max-plusregular extension operator ). if f is a surjective map, then a max-plus-linear exave (max-plus-regular exave) for f is called max-plus-linear averaging operator (max-plus-regular averaging operator ). remind, in category theory a monomorphism (an epimorphism) is a leftcancellative (respectively, right-cancellative) morphism, that is, a morphism f : z → x (respectively, f : x → y ) such that, for each pair of morphisms g1, g2 : y → z the following implication holds f ◦g1 = f ◦g2 ⇒ g1 = g2 (respectively, g1 ◦f = g2 ◦f ⇒ g1 = g2). if u is an exave for f : x → y and y ∈ f(x), then for every function ϕ ∈ c(y ) we have (u◦f◦)(ϕ)(y) = ϕ(y). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 41 a. a. zaitov proposition 2.3. let f : x → y be a map. a max-plus-regular operator u: c(x) → c(y ) is a max-plus-regular extension (respectively, averaging) operator if and only if f◦ ◦u = idc(x) (respectively, u◦f◦ = idc(y )). proof. let u be a max-plus-regular extension (respectively, averaging) operator. then the induced operator f◦ : c(y ) → c(x) is an epimorphism (respectively, monomorphism). that is why equalities f◦ ◦u◦f◦ = f◦ = idc(x) ◦f◦ imply f◦ ◦ u = idc(x) (respectively, f◦ ◦ u ◦ f◦ = f◦ = f◦ ◦ idc(y ) imply u◦f◦ = idc(y )). let u be a max-plus-regular operator and f◦ ◦ u = idc(x). it requires to show f : x → y is an embedding. suppose f(x1) = f(x2), x1, x2 ∈ x. assume there exists a function ϕ ∈ c(x) such that ϕ(x1) 6= ϕ(x2). conversely, we have ϕ(x1) = f ◦ ◦u(ϕ)(x1) = u(ϕ)(f(x1)) = u(ϕ)(f(x2)) = f◦ ◦u(ϕ)(x2) = ϕ(x2). we get a contradiction. so, x1 = x2. let u be a max-plus-regular operator and u◦f◦ = idc(y ). we should show that f : x → y is a surjective map. suppose f is not so. then y \f(x) 6= ∅ and for every y ∈ y \f(x), since the image f(x) is a compact hausdorff space, any ϕ: f(x) → r has different extensions ϕ1, ϕ2 : y → r such ϕ1(y) 6= ϕ2(y). hence, ϕ1 6= ϕ2. on the other hand ϕ1 = u◦f◦(ϕ1) = u◦f◦(ϕ2) = ϕ2. the obtained contradiction finishes the proof. � an epimorphism f : x → y is said to be a max-plus-milutin epimorphism provided it permits a max-plus-regular averaging operator. a compact hausdorff space x is a max-plus-milutin space if there exists a max-plus-milutin epimorphism f : dτ → x [13]. every compact metrizable space is a milutin space ([4], corollary viii.4.6.). analogously, every compact metrizable space is a max-plus-milutin space. 3. an analog of the kantorovich metric it is well-known (see, for example [4]) that every zero-dimensional space of the weight m ≥ ℵ0 embeds into cantor cube dm. consequently, a zerodimensional compact metrizable space is a max-plus-milutin space. let µi = ⊕ x∈x λi(x) � δx ∈ i(x), i = 1, 2. put λ1 2 = λ(µ1, µ2) = {ξ ∈ i(x2) : i(πi)(ξ) = µi, i = 1, 2}, where πi : x × x → x is the projection onto i-th factor, i = 1, 2. we will show the set λ(µ1, µ2) is nonempty. let xi 0 ∈ supp µi be points such that λi(xi 0) = 0, i = 1, 2. then the direct checking shows that i(πi)(ξ) = µi, i = 1, 2, for all ξ ∈ i(x2) of the form ξ = ξ0 ⊕r(µ1,µ2). here ξ0 = 0 � δ(x1 0,x2 0) ⊕ x∈x\{x1 0} λ2(x) � δ(x1 0,x) ⊕ ⊕ x∈x\{x2 0} λ1(x) � δ(x,x2 0) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 42 a metric on the space of idempotent measures is an idempotent probability measure on x2, and r(µ1, µ2) = ⊕ x∈x\{x1 0} y∈x\{x2 0} γ(x, y) �δ(x,y) is some functional on c(x) where −∞≤ γ(x, y) ≤ min{λ1(x), λ2(y)}, x ∈ m, y ∈ n, m ⊂ x \{x1 0}, n ⊂ x \{x2 0}. thus ξ ∈ λ(µ1, µ2), i. e. λ(µ1, µ2) 6= ∅. in fact, here more is proved: it is easy to see if |x| ≥ 2 and |y | ≥ 2 then quantity of the numbers γ(x, y) is uncountable. from here one concludes that the potency of the set λ(µ1, µ2) is no less than continuum potency as soon as each of the supports supp µi, i = 1, 2, contains no less than two points. note that ξ = ξ0 if one takes empty set as k and m. idempotent probability measures ξ ∈ i(x2) with i(πi)(ξ) = µi, i = 1, 2 we will call as a coupling of µ1 and µ2. the following statement is rather evident. proposition 3.1. let µi = ⊕ x∈x λi(x) �δx, i = 1, 2, be idempotent probability measures. then every their coupling ξ = ⊕ (x,y)∈x2 λ1 2(x, y) � δ(x,y) ∈ i(x2) satisfies the following equalities: λ1(x) = ⊕ y∈x λ1 2(x, y), x ∈ x, and λ2(y) = ⊕ x∈x λ1 2(x, y), y ∈ x. consider a compact metrizable space (x, ρ). we define a function ρ0 : i(x)× i(x) → r by the formula ρ0(µ1, µ2) = inf{ξ(ρ) : ξ ∈ λ1 2}. this function was offered by v. v. uspenskii and in [3] it was proved that it is a metric on the space p(x) of probability measures. its analog for idempotent probability measures is not a metric on the space of idempotent probability measures. l. v. kantorovich, g. sh. rubinshtein offer another metric on the space of all measures [6]. for the space of probability measures their metric has the form ρk(µ1, µ2) = inf{ξ(ρ) : ξ ∈ p(x ×x), p(π1)(ξ) −p(π2)(ξ) = µ1 −µ2}. in [12] it was shown that on the space of all probability measures the above metrics ρ0 and ρk coincide. proposition 3.2. for every pair µ1, µ2 ∈ i(x) there exists a coupling ξ ∈ i(x2) of µ1 and µ2 such that ρ0(µ1, µ2) = ξ(ρ). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 43 a. a. zaitov proof. consider a sequence {ξn} of couplings of µ1 and µ2 such that ξn(ρ) −→ ρ0(µ1, µ2). passing in the case of need to a subsequence, owing to the compactness of i(x2), it is possible to assume that {ξn} tends to some ξ ∈ i(x2). since the projections i(πi) are continuous, ξ is a coupling of µ1 and µ2. further, for an arbitrary ε > 0 there exists n0 such that ξn ∈ 〈ξ; ρ; ε〉 for all n ≥ n0, where 〈ξ; ρ; ε〉 is a prebase neighbourhood of ξ in the pointwise convergence topology on i(x2). so, |ξ(ρ) − ξn(ρ)| < ε. consequently, ρ0(µ1, µ2) = ξ(ρ). � proposition 3.3. the function ρ0 is a pseudometric on i(x). proof. since each ξ ∈ i(x2) is order-preserving then the inequality ρ ≥ 0 immediately implies ρ0 ≥ 0. so, ρ0 is nonnegative. obviously, ρ0 is symmetric. let µ1 = µ2 = µ. there exists λ ∈ us(x) such that µ = ⊕ x∈x λ(x) � δx. then ξµ = ⊕ x∈x λ(x) � δ(x,x) is a coupling of µ1 and µ2, and 0 ≤ ρ0(µ1, µ2) = inf{ξ(ρ) : ξ ∈ λ1 2}≤ ξµ(ρ) = ⊕ x∈x λ(x) = 0, i. e. ρ0(µ1, µ2) = 0. let us show that the triangle inequality is true as well. take arbitrary triple µi ∈ i(x), i = 1, 2, 3. let µ1 2, µ2 3 ∈ i(x2) be couplings of µ1 and µ2, and µ2 and µ3, respectively, such that ρ0(µ1, µ2) = µ1 2(ρ) and ρ0(µ2, µ3) = µ2 3(ρ), respectively. for a compact hausdorff space x we put x1 = x2 = x3 = x, x1 2 3 = x 3 = x1 ×x2 ×x3, xij = x 2 = xi ×xj, and let π1 2 3ij : x1 2 3 → xij, π ij k : xij → xk, 1 ≤ i < j ≤ 3, k ∈{i, j}, be corresponding projections. according to corollary 4.3 [17] the functor i is bicommutative. using this fact one can similarly to lemma 4 [3] show that for idempotent probability measures µ2 ∈ i(x2), µ1 2 ∈ i(x1 2), µ2 3 ∈ i(x2 3) such that i(π1 22 )(µ1 2) = µ2 = i(π 2 3 2 )(µ2 3), there exists µ1 2 3 ∈ i(x1 2 3) which satisfies the equalities i(π1 2 31 2 )(µ1 2 3) = µ1 2 and i(π 1 2 3 2 3 )(µ1 2 3) = µ2 3. put (3.1) µ1 3 = i(π 1 2 3 1 3 )(µ1 2 3). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 44 a metric on the space of idempotent measures then according to proposition 3.1 µ1 3 is a coupling of µ1 and µ3. using proposition 3.1, we obtain ρ0(µ1, µ2) + ρ0(µ2, µ3) = µ1 2(ρ) + µ2 3(ρ) = = ⊕ (x1,x2)∈x1 2 dµ1 2 (x1, x2) �ρ(x1, x2) + ⊕ (x2,x3)∈x2 3 dµ2 3 (x2, x3) �ρ(x2, x3) = = ⊕ (x1,x2,x3)∈x1 2 3 dµ1 2 3 (x1, x2, x3) �ρ(x1, x2)+ + ⊕ (x1,x2,x3)∈x1 2 3 dµ1 2 3 (x1, x2, x3) �ρ(x2, x3) ≥ ≥ ⊕ (x1,x2,x3)∈x1 2 3 (dµ1 2 3 (x1, x2, x3) �ρ(x1, x2)+ +dµ1 2 3 (x1, x2, x3) �ρ(x2, x3)) = = ⊕ (x1,x2,x3)∈x1 2 3 dµ1 2 3 (x1, x2, x3) � (ρ(x1, x2) + ρ(x2, x3)) ≥ ≥ ⊕ (x1,x2,x3)∈x1 2 3 dµ1 2 3 (x1, x2, x3) �ρ(x1, x3) = = ⊕ (x1,x3)∈x1 3 dµ1 3 (x1, x3) �ρ(x1, x3) = µ1 3(ρ) ≥ ρ0(µ1, µ3), i. e. ρ0(µ1, µ3) ≤ ρ0(µ1, µ2) + ρ0(µ2, µ3). here dν is the density function of the corresponding measure ν ((2.4), see page 39). � unlike usual probability measures, the function ρ0 is not a metric on i(x). example 3.4. let (x, ρ) be a metric space, x, y ∈ x be points such that ρ(x, y) = 1. consider idempotent probability measures µ1 = 0�δx⊕(−2)�δy and µ2 = 0 � δx ⊕ (−4) � δy. one can directly check that the idempotent probability measure ξ = 0�δ(x,x) ⊕(−2)�δ(y,x) ⊕(−4)�δ(x,y) is a coupling of µ1 and µ2, and ξ(ρ) = 0. that is why ρ0(µ1, µ2) = 0, though µ1 6= µ2. example 3.4 shows that the functors p of probability measures and i of idempotent probability measures are not isomorphic. 4. on a metric on the space of idempotent probability measures let (x, ρ) be a metric compact space. we define distance functions ρ1 : i(x)× i(x) → r and ρ2 : iω(x) × iω(x) → r as follows ρ1(µ1, µ2) = inf{sup{ρ(x, y) : (x, y) ∈ supp ξ} : ξ ∈ λ1 2}, c© agt, upv, 2020 appl. gen. topol. 21, no. 1 45 a. a. zaitov where µ1,µ2 ∈ i(x), and ρ2(µ1, µ2) = inf   ∑ (x,y)∈supp ξ eλ1(x)+λ2(y) ·ρ(x, y)∑ x∈supp µ1 eλ1(x) · ∑ y∈supp µ2 eλ2(y) : ξ ∈ λ1 2   , where µi = ⊕ x∈x λi(x) �δx ∈ iω(x), i = 1, 2. it is easy to notice that ρ2 ≤ ρ1 on iω(x). the following statement has technical character and its proof consists of labour-intensive calculations (similarly calculations were done in [16]). lemma 4.1. for every pair µi = ⊕ x∈x λi(x) �δx ∈ iω(x), i = 1, 2, and for a coupling ξ ∈ iω(x2) of µ1 and µ2 we have ρ2(µ1, µ2) = ∑ (x,y)∈supp ξ eλ1(x)+λ2(y) ·ρ(x, y)∑ x∈supp µ1 eλ1(x) · ∑ y∈supp µ2 eλ2(y) if and only if ρ0(µ1, µ2) = ξ(ρ). theorem 4.2. the function ρ1 is a metric on i(x) which is an extension of the metric ρ. proof. obviously, ρ1 is nonnegative and symmetric. if µ1 = µ2 then similarly to the proof of proposition 3.3 one can show that ρ1(µ1,µ2) = 0. inversely, let ρ1(µ1,µ2) = 0. then there exists a coupling ξ ∈ λ1 2 such that ρ(x, y) = 0 for all (x, y) ∈ supp ξ. consequently supp ξ must lie in the diagonal ∆(x) = {(x, x) : x ∈ x}. applying proposition 3.1, we have dµ1 = dµ2 , which implies µ1 = µ2. now, it remains to check the triangle axiom. but the checking consists only of the repeating of procedure at the proof of proposition 3.3. for every pair of dirac measures δx, δy, x, y ∈ x, the uniqueness of a coupling ξ ∈ i(x2) of δx and δy, ξ = 0 � δ(x,y), implies that ρ1(δx, δy) = ξ(ρ) ⊕ρ(x, y) = 0 � δ(x,y)(ρ) ⊕ρ(x, y) = ρ(x, y). from here we get that ρ1 is an extension of ρ. � lemma 4.3. diam(i(x), ρ1) = diam(x, ρ). proof. indeed, since we may consider x as a subspace of i(x) we get diam(x, ρ) ≤ diam(i(x), ρ1). on the other hand, by construction we have ρ1(µ1, µ2) = inf{sup{ρ(x, y) : (x, y) ∈ supp ξ} : ξ ∈ λ1 2}≤ ≤ sup{ρ(x, y) : (x, y) ∈ supp ξ}≤ sup{ρ(x, y) : (x, y) ∈ x×x} = diam(x, ρ) for an arbitrary pair µ1,µ2 ∈ i(x). consequently, diam(i(x), ρ1) ≤ diam(x, ρ). � c© agt, upv, 2020 appl. gen. topol. 21, no. 1 46 a metric on the space of idempotent measures theorem 4.4. the function ρ2 is a metric on iω(x) which is an extension of the metric ρ. proof. by construction ρ2 is non-negative. it is clear that ρ2 is symmetric. the above noticed inequality ρ2 ≤ ρ1 on iω(x) implies that ρ2 satisfies the identity axiom, i. e. ρ2(µ, ν) = 0 if and only if µ = ν. by definition we have ρ2(δx, δy) = ρ(x, y). for a triple µi ∈ iω(x), i = 1, 2, 3, let µ1 2, µ2 3 ∈ iω(x2) be couplings of µ1 and µ2, and µ2 and µ3, respectively, satisfying proposition 3.2. let µ1 3 ∈ iω(x2) be an idempotent probability measures, defined by (3.1). then proposition 3.1 yields that µ1 3 is a coupling of µ1 and µ3. applying proposition 3.1, lemma 4.1 and theorem 2, we have ρ2(µ1, µ2) + ρ2(µ2, µ3) ≥ ρ2(µ1, µ3). � let µ, ν ∈ i(x).corollary 2.2 implies the existence of sequences {µn}, {νn}⊂ iω(x) converging to µ and ν respectively. we have 0 ≤ ρ2(µn,νn) ≤ρ1(µn,νn) ≤ diam(x, ρ). therefore there exists a limit of the sequence {ρ2(µn,νn)}. put ρi(µ, ν) = lim n→∞ ρ2(µn,νn). now theorem 4.4 gives the following result. corollary 4.5. the function ρi is a metric on i(x) which is an extension of the metric ρ. note that ρi ≤ ρ1. for this reason from lemma 4.3 we obtain the following statement. corollary 4.6. diam(i(x), ρi) = diam(x, ρ). proposition 4.7. let x be a compact metrizable space and a sequence {µn}⊂ i(x) converges to µ0 ∈ i(x) with respect to point-wise convergence topology. then for every open neighbourhood u of the diagonal ∆(x) = {(x, x) : x ∈ x} there exist a positive integer n and a coupling µ0 n ∈ i(x2) of µ0 and µn such that (4.1) ⊕ (x,y)∈x2\u dµ0 n (x, y) �ρ(x, y) = −∞. proof. at first we consider the case of zero-dimensional compact metrizable space x. there exists a disjoint clopen cover {v1, . . . , vn} of x (i. e. a cover, which consists of open-closed sets of x) such that vi × vi ⊂ u for each i = 1, . . . , n. as µn → µ there exists n such that µn ∈ 〈µ; ⊕χv1, ⊕χv2, . . . , ⊕χvn ; ε〉. we will construct a coupling µ0 n ∈ i(x2) of µ0 and µn. there exists a base of the compact metrizable space x consisting of clopen sets v ε1ε2...εk i , 1 ≤ i ≤ s, εk ∈{0, 1}, 1 ≤ k < ∞, such that c© agt, upv, 2020 appl. gen. topol. 21, no. 1 47 a. a. zaitov 1) v 0i ∪v 1 i = vi; 2) v 0i ∩v 1 i = ∅; 3) v ε1ε2...εk0 i ∪v ε1ε2...εk1 i = v ε1ε2...εk i ; 4) v ε1ε2...εk0 i ∩v ε1ε2...εk1 i = ∅. the sets v ε1ε2...εk i ×v ε′1ε ′ 2...ε ′ k i′ form a base of the compact metrizable space x1 2. to determine µ0 n it is enough to construct its density function. let µ0 = ⊕ x∈x λ0(x) � δx, µn = ⊕ x∈x λn(x) �δx. we set λ ε1...εk,ε ′ 1...ε ′ k ii′ = ⊕ (x,y)∈x×x (λ0(x) �λn(y)) �δ(x,y)(⊕χ v ε1...εk i ×v ε′ 1 ...ε′ k i′ ), i. e. λ ε1...εk,ε ′ 1...ε ′ k ii′ = ⊕ (x,y)∈v ε1...εk i ×v ε′ 1 ...ε′ k i′ λ0(x) �λn(y). it is clear that λ ε′1...ε ′ k i′ = s⊕ i=1 λ ε1...εk,ε ′ 1...ε ′ k ii′ and λ ε1...εk i = s⊕ i′=1 λ ε1...εk,ε ′ 1...ε ′ k ii′ , where λ ε1...εk i = ⊕ x∈x λ0(x) � δx(⊕χv ε1...εk i ) = ⊕ x∈v ε1...εk i λ0(x) and λ ε′1...ε ′ k i′ = ⊕ x∈x λn(x) � δx(⊕χ v ε′ 1 ...ε′ k i ) = ⊕ x∈v ε′ 1 ...ε′ k i′ λn(x). put dµ0 n = lim s→∞ s⊕ i, i′=1 ⊕χ λ ε1...εk, ε ′ 1...ε ′ k i i′ v ε1...εk i ×v ε′ 1 ...ε′ k i′ . then dµ0 n is an upper semicontinuous function on x 2 and µ0,n =⊕ (x,y)∈x2 dµ0 n (x, y) � δ(x,y) is a coupling of µ0 and µn with supp µ0,n ⊂ u. consequently, ⊕ (x,y)∈x2\u dµ0 n (x, y) = −∞ and, the equation (4.1) is proved for the zero-dimensional case. now let x be an arbitrary compact metrizable space. there exists a zerodimensional compact metrizable space z, a max-plus-milutin epimorphism f : z → x and a max-plus-regular averaging operator u: c(z) → c(x) corresponding to this epimorphism. the dual max-plus-map u⊕ which we define by the equality u⊕(µ)(ϕ) = µ(u(ϕ)), ϕ ∈ c(z), generates an embedding u⊕ : i(x) → i(z). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 48 a metric on the space of idempotent measures for idempotent probability measures µ′0 = u ⊕(µ0) and µ ′ n = u ⊕(µn) there exists a coupling µ′0,n = ⊕ (x′,y′)∈z2 dµ′0 n (x ′, y′) � δ(x′,y′) ∈ i(z ×z) of µ′0 and µ′n such that ⊕ (x′,y′)∈z2\(f×f)−1(u) dµ′0 n (x ′, y′) �ρ(x′, y′) = −∞. put µ0,n = i(f ×f)(µ′0 n). then for every ϕ ∈ c(x2) we have µ0,n(ϕ) = i(f ×f)(µ′0 n)(ϕ) = µ ′ 0 n(ϕ◦ (f ×f)) = = ⊕ (x′,y′)∈z2 dµ′0 n (x ′, y′) �ϕ◦ (f ×f)(x′, y′) = = ⊕ (x′,y′)∈z2 dµ′0 n (x ′, y′) �ϕ(f(x′), f(y′)) = = ⊕ (x,y)∈x2 dµ′0 n (x, y)) �δ(x,y)(ϕ), i. e. µ0,n = ⊕ (x,y)∈x2 dµ′0 n (x, y)) � δ(x,y). here dµ′0 n (x, y) = ⊕ (x′,y′)∈(f×f)−1(x,y) dµ′0 n (x ′, y′). that is why ⊕ (x,y)∈x2\u dµ′0 n (x, y) �ρ(x, y) = −∞. so, µ0,n = i(f × f)(µ′0 n) satisfies (4.1). it remains to show that µ0,n is a coupling of µ0 and µn. a diagram (4.2) z ×z f×f−−−−→ x ×xyθ121 yπ121 z f−−−−→ x is commutative, where θ121 , π 12 1 are projections onto the first corresponding factors. then i(π121 )(µ0 n) = i(π 12 1 ) ◦ i(f ×f)(µ ′ 0 n) = i(π 12 1 ◦ (f ×f))(µ ′ 0 n) = = (owing to commutativity of the diagram (4.2)) = = i(f ◦θ121 )(µ ′ 0 n) = i(f) ◦ i(θ 12 1 )(µ ′ 0,n) = i(f)(µ ′ 0) = i(f)(u ⊕(µ0)), i. e. for every ϕ ∈ c(x) we have i(π121 )(µ0 n)(ϕ) = i(f)(u ⊕(µ0))(ϕ) = u ⊕(µ0)(ϕ◦f) = u⊕(µ0)(f◦(ϕ)) = = µ0(u◦f◦(ϕ)) = (with respect to proposition 2.3) = µ0(ϕ). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 49 a. a. zaitov thus, i(π121 )(µ0 n) = µ0. similarly, i(π 12 2 )(µ0 n) = µn. the proposition is proved. � theorem 4.8. the metric ρi generates pointwise convergence topology on i(x). proof. let {µn} ⊂ i(x) be a sequence and µ0 ∈ i(x). suppose the sequence converges to µ0 with respect to the pointwise convergence topology but not by ρi. passing in the case of need to a subsequence, it is possible to regard that ρi(µn, µ0) ≥ a > 0 for all positive integer n. consider an open neighbourhood of the diagonal ∆(x): u = { (x, y) ∈ x2 : ρ(x, y) < a 2 } . by virtue of proposition 4.7 there exist a positive integer n and a coupling µ0 n ∈ i(x2) of µ0 and µn such that⊕ (x,y)∈x2\u dµ0 n (x, y) �ρ(x, y) = −∞. therefore, supp µ0 n ⊂ u, and ρi(µn, µ0) ≤ ρ1(µn, µ0) ≤ sup (z,t)∈supp µ0 n {ρ(z, t)} = sup (z,t)∈supp µ0 n {µ0 n(ρ)⊕ρ(z, t)} = = sup (z,t)∈supp µ0 n {( ⊕ (x,y)∈x2 dµ0 n (x, y) �ρ(x, y) ) ⊕ρ(z, t) } = = sup (z,t)∈supp µ0 n {( ⊕ (x,y)∈x2\u dµ0 n (x, y)�ρ(x, y)⊕ sup (x,y)∈u dµ0 n (x, y)�ρ(x, y) ) ⊕ ⊕ρ(z, t) } = sup (z,t)∈supp µ0 n {( sup (x,y)∈u dµ0 n (x, y) �ρ(x, y) ) ⊕ρ(z, t) } ≤ ≤ sup (z,t)∈u {( sup (x,y)∈u dµ0 n (x, y)�ρ(x, y) ) ⊕ρ(z, t) } = sup (z,t)∈u {ρ(z, t)}≤ a 2 . the obtained contradiction finishes the proof. � acknowledgements. the author expresses deep gratitude to the referee for critical comments, suggestions, and useful advice. also the author would like to express gratitude to prof. dirk werner for the revealed shortcomings, the specified remarks. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 50 a metric on the space of idempotent measures references [1] m. akian, densities of idempotent measures 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[17] m. m zarichnyi, spaces and maps of idempotent measures, izv. math. 74, no. 3 (2010), 481–499. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 51 @ appl. gen. topol. 20, no. 2 (2019), 363-377 doi:10.4995/agt.2019.11238 c© agt, upv, 2019 on ideal sequence covering maps sudip kumar pal a , nayan adhikary b and upasana samanta b a department of mathematics, diamond harbour women’s university, wb-743368, india (sudipkmpal@yahoo.co.in) b department of mathematics, jadavpur university, kol-32, wb, india (nayanadhikarysh@gmail.com, samantaupasana@gmail.com) communicated by s. garćıa-ferreira abstract in this paper we introduce the concept of ideal sequence covering map which is a generalization of sequence covering map, and investigate some of its properties. the present article contributes to the problem of characterization to the certain images of metric spaces which was posed by y. tanaka [22], in more general form. the entire investigation is performed in the setting of ideal convergence extending the recent results in [11, 15, 16]. 2010 msc: 54c10; 40a05; 54e35. keywords: sequence covering; sequentially quotient; sn-networks; boundary compact map; ideal convergence. 1. introduction in the past 50 years, papers and surveys on discussion for theory of images of metric spaces were published in large amounts. this issue has become a typical research direction in developments of general topology, which has made outstanding contributions for progress and prosperity of this subject. taking spaces as images of metric spaces, we deal with structures of metric spaces. in 1971, swiec [20, 21] introduced the concept of sequence covering maps which is closely related to the question about compact covering and s-images of metric spaces (see also [4]). in [11] lin discuss about sequence covering maps and its properties also solved many open problem related to this concept. received 15 january 2019 – accepted 11 june 2019 http://dx.doi.org/10.4995/agt.2019.11238 s. k. pal, n. adhikary and u. samanta the notion of statistical convergence, which is an extension of the idea of usual convergence, was introduced by fast [3] and schoenberg [19] and its topological consequences were studied first by fridy [5] and salat [6]. it seems more appropriate to follow the more general approach of [6] where the notion of iconvergence of sequence was introduced by using the ideas of ideal of the set of positive integers. in [15, 16] renukadevi studied some of the results of lin in statistical format. as a natural consequences, in the present paper we continue the investigation proposed in [11] and study similar problems in more general form. in section 3.1 we introduce the concepts of ideal sequence covering map and ideal sequence covering compact map and study some of its properties. also in section 3.2 we propose the concept of ideal sequentially quotient map and examine related properties. the entire investigation is performed in the setting of ideal convergence extending the recent results in [11, 14, 15, 16]. 2. preliminaries we start by recalling the basic definition of ideals and filters. a family i ⊂ 2y of subsets of a nonempty set y is said to be an ideal in y if a,b ∈i implies a∪b ∈i and a ∈i and b ⊂ a imply b ∈i. further an admissible ideal i of y satisfies {x} ∈ y for each x ∈ y. such ideals are called free ideals. if i is a proper non-trivial ideal in y (i.e. y /∈ i , i 6= ∅ ), then the family of sets f(i) = {m ⊂ y : ∃a ∈ i : m = y \ a} is called the filter associated with the ideal i. ifin = {a ⊂ n : a is finite}. it is an ideal and ifin-convergence implies original convergence. now density of a subset a of n is d(a) = limn→∞ 1n|{k : k ∈ a,k ≤ n}|, provided the limit exists. id = {a : a ⊂ n,d(a) = 0} is an ideal and id−convergence implies the notion of statistical convergence (see [7, 12, 17, 26]). throughout all topological spaces are assumed to be hausdorff, all maps are onto and continuous and n stands for the set of all natural number. let x be a topological space and p ⊂ x. a sequence {xn} converging to x in x is eventually in p if {xn : n > k}∪{x}⊂ p for some k ∈ n; it is frequently in p if {xnk} is eventually in p for some subsequence {xnk} of {xn}. throughout by a space we will mean a topological space, unless otherwise mentioned. let us recall some definitions. definition 2.1 ([11]). let x be a space and p ⊂ x. (a) let x ∈ p. then p is called a sequential neighbourhood of x in x if whenever {xn} is a sequence converging to x, then {xn} is eventually in p. (b) p is called a sequentially open subset in x if p is a sequential neighbourhood of x in x for each x ∈ p. definition 2.2 ([9]). let x be a space, and let p be a cover of x. (1) p is a cs-cover of x, if for any convergent sequence s in x, there exists p ∈p such that s is eventually in p. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 364 on ideal sequence covering maps (2) p is an sn-cover of x, if each element of p is a sequential neighbourhood of some point of x and for each x ∈ x, there exists p ∈p such that p is the sequential neighbourhood of x. definition 2.3 ([20]). a space x is strongly fréchet if whenever {an|n ∈ n} is a decreasing sequence of sets in x and x is a point which is in the closure of each an, then for each n ∈ n there exists an element xn ∈ an such that the sequence xn → x. definition 2.4 ([23]). a space x is said to have property ωd if every countably infinite discrete subset has an infinite subset a such that there exists a discrete open family {ux|x ∈ a} with ux ∩a = {x} for each x ∈ a. definition 2.5 ([1]). a class of mappings is said to be hereditary if whenever f : x → y is in the class, then for each subspace h of y , the restriction of f to f−1(h) is in the class. definition 2.6 ([20]). let f : x → y be a mapping. (a) f is a sequence covering map if for every convergent sequence {yn} in y , there is a convergent sequence {xn} in x with each xn ∈ f−1(yn). (b) f is a 1-sequence covering map if for each y ∈ y there exists x ∈ f−1(y) such that whenever {yn} is a sequence converging to y, then there is a sequence {xn} in x converging to x with each xn ∈ f−1(yn). clearly every 1-sequence covering map is a sequence covering map. but the converse is not true, which is shown by the following example. example 2.7. suppose {yn} is a convergent sequence of real numbers with its limit y. let y = {yn : n ∈ n}∪{y} and ∧ = {α : α : n → n} where α be an increasing mapping. define yα = {(yα(k),α) : k ∈ n}∪{(y,α)} for each α ∈ ∧ . the metric dα on yα is defined by dα((p,α), (q,α)) = |p−q|. take x =⊕ α∈ ∧ yα and the metric d on x is defined as d(x1,x2) = min{dα(x1,x2), 1} when x1,x2 ∈ yα for some α ∈ ∧ and d(x1,x2) = 1 when x1 ∈ yα,x2 ∈ yβ for distinct α,β ∈ ∧ . moreover, the topology of y is considered as: each {yn} is open and a basic open set u containing y is of the form {y}∪{yn : n ≥ n0} for some n0. now define a map f : x → y by f(yα(k),α) = yα(k) and f(y,α) = y for each α ∈ ∧ . clearly f is continuous. any convergent sequence {zn} in y is of the form zs(k) = yα(k) for n = s(k),α ∈ ∧ and zn = y otherwise. then choose rn = (yα(k),α) for n = s(k) and rn = (y,α) otherwise. clearly rn ∈ f−1(zn) and rn → (y,α) ∈ f−1(y). so f is a sequence covering map. but for y ∈ y choose any (y,α) ∈ f−1(y). then there is yβ(k) → y, β 6= α but (yβ(k),β) does not converge to (y,α). so f is not 1-sequence covering map. definition 2.8 ([2]). if x is a space that can be mapped onto a metric space by a one-to-one mapping, then x is said to have weaker metric topology. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 365 s. k. pal, n. adhikary and u. samanta definition 2.9 ([6]). a sequence {xn} in a space x is said to be i−convergent to x ∈ x (i.e ilimn→∞xn = x ) if for every neighbourhood u of x, {n ∈ n : xn /∈ u}∈i. lemma 2.10 ([25]). let x be a space with a weaker metric topology. then there is a sequence {pi}i∈n of locally finite open covers of x such that ∩n∈nst(k,pi) = k for each compact subset k of x. definition 2.11 ([10]). let p = ∪{px : x ∈ x} be a cover of a space x such that for each x ∈ x, following conditions (a) and (b) are satisfied: (a) px is a network at x in x, i.e., x ∈ ∩px and for each neighbourhood u of x in x, p ⊂ u for some p ∈px. (b) if u,v ∈px, then w ⊂ u ∩v for some w ∈px. (i) p is called a sn-network of x if each element of px is a sequential neighbourhood of x for each x ∈ x, where px is called a sn-network at x in x. a space y is termed as an snf-countable if y has a sn-network p = ⋃ {py : y ∈ y} such that py is countable and closed under finite intersections for each y ∈ y. (ii) p is called a weak base of x if whenever g ⊂ x, g is open in x if and only if for each x ∈ g, there exists a p ∈px such that p ⊂ g. definition 2.12 ([13]). let a ⊂ x and let o be a family of subsets of x. then o is an external base of a in x if for each x ∈ a and an open set u with x ∈ u there is a v ∈o such that x ∈ v ⊂ u. definition 2.13 ([9]). let f : x → y be a map. (a) f is a boundary compact map if ∂f−1(y) is compact in x for each y ∈ y. (b) f is called sequentially quotient if for each convergent sequence {yn} in y there is a convergent sequence {xk} in x with f(xk) = ynk for each k, where {ynk} is a subsequence of {yn}. see also [14, 16] for more details. definition 2.14 ([9]). let x = {0}∪(n×n). for every n, m ∈ n and f ∈ nn, let w(n,m) = {(n,k) ∈ n × n : k ≥ m}, and l(f) = ∪{w(n,f(n)) : n ∈ n}. then the set x with the following topology is called a sequential fan and denoted briefly as sω : for each x ∈ x, take nx = {{x}}, if x ∈ n×n. nx = {{x}∪l(f) : f ∈ nn}, if x = 0 as a neighbourhood base of x. remark 2.15 ([9]). sequential fan sω is the space obtained by identifying the limits of countably many convergent sequences. definition 2.16 ([18]). let a,b be two non-empty collections of families of subsets of an infinite set x. then s1(a,b) is defined as: for each sequence {an : n ∈ n} of elements of a, there is a sequence {bn : n ∈ n} such that bn ∈ an for each n and {bn : n ∈ n}∈b. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 366 on ideal sequence covering maps 3. main results 3.1. (i,j)-sequence covering map. in this section we introduce a map namely, (i,j)-sequence covering map and investigate some of its properties. definition 3.1. let i and j be two admissible ideals. then (a) f : x → y is said to be a (i,j)-sequence covering map if for any given sequence {yn}, j -converging to y in y , there exists a sequence {xn}, i-convergent to x in x such that xn ∈ f−1(yn) for each n and x ∈ f−1(y). (b) f : x → y is said to be a (i,j)-1-sequence covering map if for each y ∈ y there exists x ∈ f−1(y) such that whenever {yn} is a sequence j -converging to y, there exists a sequence {xn}, which is i-convergent to x in x such that xn ∈ f−1(yn) for each n. these are generalization of sequence covering map and 1-sequence covering map respectively. clearly every (i,j)-1-sequence covering map is a (i,j)-sequence covering map. but the converse is not true which is shown by the next example. example 3.2. y = {yn : n ∈ n}∪{y}. topology of y is defined by each {yn} is open and a basic open set u containing y equals to {yn : n ≥ n0}∪{y} for some n0 ∈ n. suppose {pn} is a convergent sequence of real numbers with its limit p. define xn = pn if n ∈ m where m ∈ f(i) and xn = n, otherwise. clearly {xn}, i-converges to p. let ∧ = {α : α : n −→ n is an increasing mapping }. take xα = {(xk,α) : k ∈ n}∪{(p,α)} and metric dα on xα is defined by dα((m,α), (n,α)) = |m−n|. x = ⊕ α∈ ∧ xα and the metric d on x is defined as d(m1,m2) = min{dα(m1,m2), 1} when m1,m2 ∈ xα for some α ∈ ∧ and d(m1,m2) = 1 when m1 ∈ yα,m2 ∈ yβ for distinct α,β ∈ ∧ . now define a map f : x → y by f(xk,α) = yα(k) and f(p,α) = y for each α ∈ ∧ . clearly f is continuous. convergent sequence {zn} in y is of the form zs(k) = yα(k) α ∈ ∧ and zn = y for n ∈ {s(k) : k ∈ n}c. then choose rs(k) = (xk,α) and rn = (p,α) for n ∈ {s(k) : k ∈ n}c. clearly rn ∈ f−1(zn) and {rn}i-converges to (p,α) ∈ f−1(y). so f is (i,j)-sequence covering map. but for y ∈ y choose any (p,α) ∈ f−1(y). then there is {yβ(k)} → y, β 6= α but {(xk,β)} does not converge to (p,α). so f is not (i,j)-1-sequence covering map. proposition 3.3. let f : x → y and g : y → z be any two maps. also let i,j ,k be admissible ideals. then the following hold: (a) if f is (i,j)-sequence covering map and g is (j ,k)-sequence covering map then g ◦f is (i,k)-sequence covering map. (b) if g ◦f is (i,k)-sequence covering map then g is (i,k)-sequence covering map. proof. (a) let z ∈ z and {zn} be a sequence k-converging to z. since g is (j ,k)-sequence covering map so there exists a sequence {yn}, j -converging to y in y such that yn ∈ g−1(zn) and y ∈ g−1(z). since f is (i,j)-sequence c© agt, upv, 2019 appl. gen. topol. 20, no. 2 367 s. k. pal, n. adhikary and u. samanta covering map so there exists a sequence {xn}, i-converging to x in x such that xn ∈ f−1(yn) and x ∈ f−1(y). so we get xn ∈ (g◦f)−1(zn) and x ∈ (g◦f)−1(z). hence g ◦f is (i,k)-sequence covering map. (b) since f is continuous g is (i,k)-sequence covering map. � proposition 3.4. let i,j be two admissible ideals. then, (a) finite product of (i,j)-sequence covering mappings is (i,j)-sequence covering map. (b) (i,j)-sequence covering mappings are hereditarily (i,j)-sequence covering mappings. proof. (a) let ∏n i=1 fi : ∏n i=1 xi → ∏n i=1 yi be a map where each fi : xi → yi is (i,j)-sequence covering map for i = 1, 2, ...n. let {(yi,n)}n∈n be a sequence j -converging to (yi) in ∏n i=1 yi. then {yi,n} j -converging to yi in yi for each i = 1, 2, ...n. since each fi is (i,j)-sequence covering map there exists {xi,n} i-converges to xi in xi such that fi(xi,n) = yi,n and fi(xi) = yi, i = 1, 2...n. now consider the sequence {(xi,n)}n∈n which is i-convergent to (xi). so ∏n i=1 fi is (i,j)-sequence covering map. (b) let f : x → y be (i,j)-sequence covering map and h be subspace of y. take g = f|f−1(h) such that g : f−1(h) → h is a map. let {yn} be a sequence j -converges to y in h. then {yn}, j -converges to y in y. since f is (i,j)-sequence covering map so there exists a sequence {xn}, i-converging to x in x such that xn ∈ f−1(yn) ⊂ f−1(h) and x ∈ f−1(y) ⊂ f−1(h). hence g is (i,j)-sequence covering map. � lemma 3.5. let ∧ be any index set and let x = ∏ α∈ ∧ xα has the product topology. then {(xα,i)}i∈n is i-converges to (xα) if and only if sequence of each component {xα,i} i-converges to xα. proof. suppose pα : x → xα is a projection. so if {(xα,i)}i∈n i-converges to (xα) then pα({(xα,i)}i∈n)= {xα,i}i∈n i-converges to xα. conversely assume that sequence of each component {xα,i} i-converges to xα. let u be a basic open set containing (xα) then pα(u) = xα for all α ∈ ∧ but finitely many. take the finite set be {α1,α2, ...αk}. since{xαj,i} i-converges to xαj so we get aαj = {i ∈ n : xαj,i /∈ pαj (u)} ∈ i for j = 1, 2, ..k. {i ∈ n : (xα,i) /∈ u} ⊂ ∪kj=1aαj ∈i. hence {(xα,i)}i∈n is i-converges to (xα). � from lemma 3.5 we can say that if f : x = ∏ α∈ ∧ xα → y = ∏α∈∧ yα be the infinite product of (i,j)-sequence covering mappings fα : xα → yα and x,y has the product topology then f is also (i,j)-sequence covering map. the following example shows that the concept of ideal sequence covering map generalizes the concept of statistical sequence covering map and hence sequence covering map. example 3.6. let f be the set of all increasing mappings from n to n. for each f ∈ f consider sf be a i-convergent sequence with its i-limit xf. i.e c© agt, upv, 2019 appl. gen. topol. 20, no. 2 368 on ideal sequence covering maps sf = {xf,n : n ∈ n}∪{xf}. topology of sf is defined as each {xf,n} is open and a basic open set u containing xf is such that au = {n ∈ n : xf,n /∈ u}∈i and there exists at least one u such that d(au) 6= 0. consider the topological sum x = ⊕f∈fsf . let {yn}→ y and y = {yn : n ∈ n}∪{y}. topology of y is defined as each {yn} is open and a basic open set u containing y equals to {yn : n ≥ n0}∪{y} for some n0 ∈ n. let φ : x → y defined as φ(xf,k) = yf(k), for all k ∈ n and φ(xf ) = y. consider any subsequence {yf(k))} of {yn}. let zk = yf(k) then consider corresponding xf,k in sf . then {xf,n}i-converging to xf. now φ(xf,k) = yf(k) = zk and φ(xf ) = y. consider β ∈f and s : n → n is an increasing function. suppose zn = yβ(k),n = s(k) and zn = y, otherwise. then clearly zn → y. consider ps(k) = xβ,k and pn = xβ, otherwise. then φ(ps(k)) = φ(xβ,k) = yβ(k) = zs(k) and φ(pn) = y otherwise. let u be an open set containing xf. hence {n ∈ n : pn /∈ u} = {s(k) : ps(k) /∈ u,k ∈ n}. but {k ∈ n : xβ,k /∈ u} ∈ i. if i has increasing function property i.e b ∈ i ⇒ f(b) ∈i for each increasing function f from n to n then {pn}i-converging to xf. so φ is (i,ifin)−sequence covering map but not sequence covering map. theorem 3.7. let f : x → y be a (i,j)−sequence covering compact map where i,j be two admissible ideals and suppose there exists a sequence {mn} of disjoint infinite subset of n such that mn /∈ i for each n. then for each y ∈ y there exists x ∈ f−1(y) such that whenever u is an open neighbourhood of x, f(u) is a sequential neighbourhood of y. proof. suppose not, that is there exists y ∈ y such that for every x ∈ f−1(y) there exists an open neighbourhood ux of x such that f(ux) is not a sequential neighbourhood of y. since f−1(y) ⊂∪x∈f−1(y) ux and f is a compact map, so there exists x1,x2, . . . ,xn0 ∈ f−1(y) such that f−1(y) ⊂ ∪ n0 i=1 uxi. since each f(uxm ) is not a sequential neighbourhood of y, choose {ym,n}∞n=1 converging to y such that ym,n /∈ f(uxm ) for all m ∈{1, 2, ...n0},n ∈ n. now define yk = ym,k if k ∈ mm,m ∈{1, 2, ...n0} and yk = y, otherwise. then clearly yk → y, which shows that {yk}, j -converging to y. since f is (i,j)-sequence covering map so there exists {xk}, i-converging to x such that xk ∈ f−1(yk) and x ∈ f−1(y). now x ∈ f−1(y) ⊂ ∪n0i=1 uxi. so there exists m0 such that x ∈ uxm0 and {k ∈ n : xk /∈ uxm0}∈i. thus {k ∈ n : f(xk) /∈ f(uxm0 )}∈i which shows that {k ∈ n : yk /∈ f(uxm0 )} ∈ i. but mm0 ⊂{k ∈ n : yk /∈ f(uxm0 )} ∈ i, which contradicts that mm0 6∈ i. thus f(u) is a sequential neighbourhood of y. � theorem 3.8. let i be an admissible ideal and suppose there exists a sequence {mn} of disjoint infinite subset of n such that mn /∈ i for each n. then the following conditions are equivalent for a space y : (a) y is a (i,ifin)-1-sequence covering compact image of a weaker metric topology. (b) y is a (i,ifin)-sequence covering compact image of a weaker metric topology. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 369 s. k. pal, n. adhikary and u. samanta (c) y has a sequence {fi}i∈n of point-finite sn-covers such that ∩i∈nst(y,fi) = {y} for each y ∈ y. (d) y has a sequence {fi}i∈n of point-finite cs-covers such that ∩i∈nst(y,fi) = {y} for each y ∈ y. proof. it is clear that (a) ⇒ (b), (c) ⇐⇒ (d), (c) ⇒ (a).[15] (b) ⇒ (c) suppose f : x → y is a (i,if )−sequence covering compact mapping. as x being a space with a weaker metric topology, there is a sequence {pi}i∈i of locally finite open covers of x such that ∩i∈nst(k,pi) = k for each compact subset k ⊂ x, by lemma 3.5. for each i ∈ n, put fi = f(pi). then fi is a point finite cover of y, since f is compact. by theorem 3.7 for each y ∈ y there exists x ∈ f−1(y) such that for every open neighbourhood ux of x, f(ux) is a sequential neighbourhood of y. since each pi is an open cover of x, there exists p ∈pi such that x ∈ p and so f = f(p) is a sequential neighbourhood of y. choose f′i ⊂ fi which are sequential neighbourhood of y. thus f ′ i is a point finite sn-cover of y. for each y ∈ y, f−1(y) is compact subset of x and ∩i∈nst(f−1(y),pi) = f−1(y). thus ∩i∈nst(y,fi) = {y}. � theorem 3.9. let x be a strongly fréchet space with property ωd. if f : x → y is a closed and (i,j)−sequence covering map, then y is strongly fréchet. proof. clearly y is a fréchet space, since it is a closed image of a strongly fréchet. suppose y is not strongly fréchet. then y contains a homomorphic copy of the sequential fan sω and the copy can be closed in y [9]. consider sω ⊂ y as a closed set where sω = {y} ∪ {ym,n : m,n ∈ ω}. take each sm = {ym,n}n∈ω is a sequence converges to y. take a ∈f(i). for each m ∈ n, choose ymk = y0,k if k ∈ a c and ymk = ym,k, if k ∈ a. then the sequence {ymk} converges to y. since f is (i,j)-sequence covering map there exists xm ∈ f−1(y) and a sequence qm, i-converging to xm such that f(qm) = {ymk}. now for each k ∈ ω take tk = ∪{f−1(sm) : m ≥ k}. suppose that there exists z ∈ x such that for every open neighbourhood u of z, {n ∈ n : xn ∈ u} is infinite. then z ∈∩k∈ntk. since x is strongly fréchet, there exists a sequence {zk} converges to z where zk ∈ tk. but {f(zk)}k∈n does not converges to y, which is a contradiction. so for every z ∈ x there is an open neighbourhood uz of z such that {n ∈ n : xn ∈ uz} is finite. so z cannot be a limit point of {xn}. hence {xn} is closed. suppose the set {xn} is finite. let it be z = xn, n ∈ n′ where n′ is a infinite subset of n. then for every open neighbourhood u of z, {n ∈ n : xn ∈ u} is infinite which is a contradiction. therefore the set {xn}n∈n is infinite, closed and discrete in x. since x has the property ωd there exists an infinite subset {xnj}j∈n and a discrete open family {uj}j∈ω such that uj ∩{xnj}j∈ω = {xnj}. recall that qnj, i-converges to xnj and f(qnj ) = {ynj}. therefore we can take uj ∈ uj∩qnj such that {f(uj)}j∈ω is infinite and contained in {y0,n : n ∈ n}. since {uj}j∈ω is closed in x, {f(uj)}j∈ω is closed in sw, which is a contradiction. thus y is strongly fréchet. � c© agt, upv, 2019 appl. gen. topol. 20, no. 2 370 on ideal sequence covering maps corollary 3.10. every closed and (i,j)-sequence covering image of a metric space is metrizable. proof. suppose f : x → y is closed and a (i,j)-sequence covering map. let x be a metric space. since every metrizable space has ωd property [13], y is strongly fréchet by theorem 3.9. also every strongly fréchet space which is a closed image of a metric space is metrizable [8, 24]. hence y is metrizable. � 3.2. (i,j ,l)sequentially quotient map. in this section we introduce the concept of ideal sequentially quotient map and investigate some of its properties. also we investigate under what condition the mapping will coincide with the ideal sequence covering map. definition 3.11. let i,j ,l be three admissible proper ideals of n. a map f : x → y is said to be (i,j ,l)sequentially quotient map if {yn} is a sequence, j -converging to y there is a sequence {xk}, i-converging to x such that xk ∈ f−1(ynk ), x ∈ f −1(y) and {n ∈ n : xk 6∈ f−1(yn) for all k ∈ n}∈l. remark 3.12. if i = ifin = j and l = id then each (i,j ,l)-sequentially quotient map is a statistically sequentially quotient map. we observe following implications. sequence covering map=⇒ statistically sequentially quotient map and (i,j ,l)sequentially quotient map=⇒ sequentially quotient map. the reverse implications need not be true. we consider the following examples: example 3.13. let f = {a ⊂ n : n\a is infinite and d(a) < 1}. let i be an admissible maximal ideal of n then f(i) is an ultrafilter on n. for each α ∈ f consider a sequence {xα,i : i ∈ n} converging to xα and let {yn : n ∈ n} be a sequence converging to y. define sα = {xα,i : i ∈ α}∪{xα} and y = {yn : n ∈ n}. topologies of sα, α ∈ f and y are defined below. each {xα,i} is open and basic open set containing xα is of the form {xα,i : i ≥ n0, i ∈ α}∪{xα}. each {yn} is open and basic open set containing y is of the form {yn : n ≥ n0}∪{y}. let x = ⊕ α∈f sα. define a map φ : x → y by φ(xα,i) = yi and φ(xα) = y for all α ∈ f. now there is β ∈f(i) with d(β) < 1. suppose β = {nk : k ∈ n} and let xk = xβ,nk. then {xk} converges to xβ and φ(xk) = ynk. also {nk : k ∈ n}∈f(i). consider a subsequence {ymi} of {yn}. let a = {mi : i ∈ n}. now two cases may appear. case 1 : let n\a be infinite. name zi = ymi. consider a set, b which is in f(i) and of density less than 1. let mb be the set of all indices of corresponding elements of {ymi}. clearly d(mb) < 1 and n \ mb is infinite. then there is a sequence converging to xmb. define a sequence {rj} where rj = xmb,mj for j ∈ b and rj = xmb for j 6∈ b then rj → xmb. also b ⊂{nk : φ(rj) = znk}. case 2 : let n\a be finite. consider a set b ∈f(i) and of density less than 1. let mb be the set of all corresponding elements of {ymi}. clearly d(mb) < 1 and n \ mb is infinite. then there is a sequence converging to xmb. define a sequence {rj} where rj = xmb,mj for j ∈ b and rj = xmb for j 6∈ b then c© agt, upv, 2019 appl. gen. topol. 20, no. 2 371 s. k. pal, n. adhikary and u. samanta rj → xmb. also φ(rj) = ymj = zj for j ∈ b and φ(rj) = xmb, j 6∈ b. next consider a sequence {zi} where znk = yk, k ∈ n and zi = y, i 6∈ {nk : k ∈ n}. then zi → y. as f(i) is an ultrafilter on n then either {nk ∈ n} ∈f(i) or its complement belongs to f(i). let f = {nk ∈ n} ∈ f(i) and consider a proper subset of f, f′ that belongs to f(i) and whose density is less than 1. let f′ = {nki : i ∈ n} and let c = {ki : i ∈ n}. also d(c) < 1. define rnki = xc,ki and rj = xc for j 6= nki. then f ′ ⊂{nk : φ(rk) = znk}. consider a sequence {zj} where znk = yα(k) and zj = y for j 6= α(k) where α : n → n is an increasing function. let f = {nk ∈ n}∈f(i). consider an infinite subset b of α(n) where n \ mb is infinite. then there is an infinite subset, f′ of the set of all indices of corresponding elements of znk belonging to f(i). let f′ = {nki : i ∈ n} and c = {α(ki) : i ∈ n}. consider a sequence {rj : j ∈ n} where rj = xc,α(ki), j = nki and rj = xc. therefore rj → xc and f ′ ⊂ {nk : φ(rk) = ynk}. again let n \f = f { ∈ f(i), as f(i) is an ultrafilter on n there is an infinite subset of f{, f{,′, which is a member of f(i) and of density less than 1. let us construct a sequence {rj} where rj = xf{,′ if j ∈ f {,′ and put elements of {xf{,′,i} in rest of the places. but as there is no set in f with density equals to 1 and so φ cannot be a statistically sequentially quotient map but it is an (ifin,ifin,i)-sequentially quotient map. example 3.14. let there be an infinite set a ⊂ n with a /∈i and n\a /∈i. also let y be a convergent sequence with its limit {y}. consider xα = {xi : i ∈ a}∪{xα} and xβ = {zi : i ∈ n \a}∪ (xβ) where xi → xα and zi → xβ. topology of xα is defined as follows: {xi} are open and open set containing xα is of the form {xi : i ≥ i0, i ∈ a}∪{xα}. similarly topologies of xβ and y are defined. define a map f : xα ⊕ xβ → y by f(xi) = yi,f(zi) = yi,f(xα) = f(xβ) = y. then f is a sequentially quotient map but f is not an (ifin,ifin,i)sequentially quotient map. proposition 3.15. (1) let i,j ,k,l be four ideals of n and let f(l) satisfy the property: if {nk : k ∈ n} ∈ f(l) and {ki : i ∈ n} ∈ f(l) then {nki : i ∈ n} ∈ f(l). f : x → y and g : y → z be (i,j ,l)and (j ,k,l)sequentially quotient maps. then g ◦ f : x → z is an (i,k,l)sequentially quotient map. (2) if g ◦f is (i,j ,l)sequentially quotient map then g is so. proof. (1) let {zn} be a sequence, k-converging to z in z. so there is a sequence {yk}, j -converging to y so that for each k ∈ n, g(yk) = znk, g(y) = z and {nk : g(yk) = znk}∈f(l). now as f : x → y is an (i,j ,l)-sequentially quotient map there is a sequence {xi}, i-converging to x such that f(xi) = yki for each i, y = f(x) and {ki ∈ n : yki = f(xi)}∈f(l). now (g◦f)(x) = z for each i. thus (g◦f)(xi) = znki and {nki : i ∈ n}∈f(l)(by our assumptions). therefore g ◦f : x → z is an (i,k,l)sequentially quotient map. (2) let {zn} be a sequence j -converging to z. then there is a sequence {xk}, i-converging to x. now (g ◦ f)(xk) = znk and (g ◦ f)(x) = z. also c© agt, upv, 2019 appl. gen. topol. 20, no. 2 372 on ideal sequence covering maps {nk ∈ n : (g ◦ f)(xk) = znk} ∈ f(l). by continuity of f, it follows that {f(xk)}, i-converges to f(x) and hence g is (i,j ,l)sequentially quotient map. � proposition 3.16. (1) (i,j ,l)sequentially quotient maps is preserved by finite products. (2) (i,j ,l)sequentially quotient maps are hereditarily (i,j ,l)sequentially quotient maps. proof. (1) let fi : xi → yi be (i,j ,l)sequentially quotient map for i = 1, 2, · · · ,n. we define a map f : ∏n i=1 xi → ∏n i=1 yi by f(x1,x2, · · · ,xn ) = (f1(x1),f2(x2), · · · ,fn (xn )). then f is continuous and onto. let {(yi,n) : n ∈ n} be a sequence j -converge to (yi) in ∏n i=1 yi. then {yi,n}, j -converging to yi in yi for i = 1, 2, · · · ,n. now there is a sequence {xi,k}, i-converging to xi such that fi(xi,k) = yi,nk, k ∈ n and {n ∈ n : xi,k 6∈ f −1 i (yi,n) for all k ∈ n} ∈ l for i = 1, 2, · · · ,n, . put x = (xi) ∈ ∏n i=1 xi. then {(xi,k) : k ∈ n} i-converges to (xi). let ni = {n ∈ n : xi,k ∈ f−1i (yi,n) for all k ∈ n} ∈ f(l). put n = n⋂ i=1 ni ∈ f(l). then n ⊂ {n ∈ n : (xi,k) ∈ f−1(yi,n) for all k ∈ n}. therefore f is an (i,j ,l)sequentially quotient map. (2) let f : x → y be an (i,j ,l)sequentially quotient map and let h be a subspace of y. suppose that {yn} is a sequence in h, j -converging to y in h. then there is a sequence {xn}, i-converging to x in x where xk ∈ f−1(ynk ). clearly xk ∈ f−1(h) for each k ∈ n and x ∈ f−1(h). � recall that there is a class of ideals of n, β ‘say’ that satisfies: if a ∈ i then for every strictly increasing function f : n → n, f(a) ∈i. several known ideals are β ideals, for example (1) the (analytic p-ideal) id of natural density zero sets, (2) the (fσ p-ideal) summable ideal i1 n = {a ⊂ n : σ n∈a 1 n < ∞}, (3) the (analytic p-ideal) ilog = {a ⊂ n : lim n→∞ ( σ i∈a∩{1,2,··· ,n} 1 i )/( σ i∈{1,2,··· ,n} 1 i ) = 0} of logarithmic density zero sets. theorem 3.17. let f : x → y be an (i,j ,l)sequentially quotient and boundary compact map where i ∈ β. also let there be an infinite countable partition {mi : i ∈ n} of n such that mi /∈ l for each i ∈ n and i ⊂ l. moreover, let y be snf-countable. then for each non-isolated point y ∈ y, there is a point xy ∈ ∂f−1(y) such that whenever u is an open subset with xy ∈ u, there exists a p ∈py satisfying p ⊂ f(u). proof. let it be false. then there exists a non-isolated point y ∈ y such that for each x ∈ ∂f−1(y) there exists an open neighbourhood ux of x so that c© agt, upv, 2019 appl. gen. topol. 20, no. 2 373 s. k. pal, n. adhikary and u. samanta for all p ∈ py,p 6⊂ f(ux). therefore ∂f−1(y) ⊂ {ux : x ∈ ∂f−1(y)}. since ∂f−1(y) is compact, there exists a finite subfamily u of {ux : x ∈ ∂f−1(y)} that covers ∂f−1(y). name u = {ui : 1 6 i 6 n0}. let py = {pn : n ∈ n} and wy = {fn = n⋂ i=1 pi : n ∈ n}. then wy ⊂ py. also fn+1 ⊂ fn for all n ∈ n. now for each 1 ≤ m ≤ n0, n ∈ n there exists xn,m ∈ fn \ f(um). put yn = xn,1, n ∈ m1, yn = xn,2, n ∈ m2, · · · ,yn = xn,n0, n ∈ mn0 and yn = y, n ∈ n\ ( ⋃ i=1,2,··· ,n0 mi). then {yn} converges to y. as f is (i,j ,l)sequential quotient map there is a sequence {xk}, i-converging to x ∈ ∂f−1(y) in x such that for each k ∈ n, f(xk) = ynk and {n ∈ n : xk 6∈ f −1(yn) for all k} ∈ l. now there is m0 ∈ {1, 2, · · · ,n0} such that x ∈ um0 as x ∈ ∂f−1(y) ⊂ ∪u. hence {k ∈ n : xk 6∈ um0} ∈ i which shows that {nk ∈ n : ynk 6∈ f(um0 )} ∈ i. thus {nk ∈ n : ynk 6∈ f(um0 )} ∈ l as i ⊂ l. therefore {nk ∈ n : f(xk) = ynk} ∈ f(l) and hence {nk ∈ n : ynk ∈ f(um0 )} ∈ f(l) which shows that {n ∈ n : yn ∈ f(um0 )} ∈ f(l). thus {n ∈ n : yn 6∈ f(um0 )} ∈ l. but mm0 /∈ l and for each n ∈ mm0,yn = xn,m0 6∈ f(um0 ), which is a contradiction. hence the theorem. � the following result gives the relation between ideal sequentially quotient map and ideal sequence covering map. theorem 3.18. let f : x → y be an (i,j ,l)sequentially quotient and boundary compact map where x is first countable. suppose j ⊂ i and x satisfies s1(f(i),f(i)). then f is (i,j)-sequence covering map provided y is snf-countable. proof. let y be a non-isolated point in y. as y is snf-countable by theorem 3.17, there exists a point xy ∈ ∂f−1(y) such that whenever u is an open neighbourhood of xy there exists p ∈ py, p ⊂ f(u). let {bn : n ∈ n} be a countable neighbourhood base at xy : bn+1 ⊂ bn, n ∈ n. now for each bn, there exists a pn ∈ py : pn ⊂ f(bn) which shows that f(bn) is a sequential neighbourhood of y ∈ y as each p ∈py is a sequential neighbourhood of y. let {yi} be a sequence, j -converging to y in y, i.e. {i ∈ n : yi 6∈ pn} ∈ j . now for each n, let an = {i ∈ n : yi ∈ f(bn)}. choose in ∈ an, n ∈ n such that {in : n ∈ n}∈f(i). set xj = f−1(yj) if j 6= in, n ∈ n and xj = f−1(yj)∩bn if j = in. let u be an open neighbourhood of xy. then there is bn ⊂ u. therefore {xj}, i-converging to xy and for each j ∈ n, f(xj) = yj, y = f(xy). � the following lemma exhibits the nature of the image of ideal sequentially quotient boundary compact map. lemma 3.19. let ω be the set of all topological spaces such that each compact subset k ⊂ x is metrizable and has a countable neighborhood base in x and c© agt, upv, 2019 appl. gen. topol. 20, no. 2 374 on ideal sequence covering maps f : x → y be an (i,j ,l)sequentially quotient and boundary compact map. if x ∈ ω then y is snf-countable. proof. consider a non-isolated point y ∈ y. ∂f−1(y) 6= ∅, compact and since x ∈ ω there is a countable external base u for ∂f−1(y) in x. let v = {∪u′ : u′ ⊂ u is finite and ∂f−1(y) ⊂ ∪u′}. now f(v) is countable as v is so. now we have to show that f(v) is a sn-network at y. let n be a neighbourhood of y. then clearly ∂f−1(y) ⊂ f−1(n). now for each x ∈ ∂f−1(y) there is ux ∈u : x ∈ ux ⊂ f−1(n). so {ux : x ∈ ∂f−1(y)} covers ∂f−1(y). there is a finite subfamily k of {ux : x ∈ ∂f−1(y)} that covers ∂f−1(y) and ∂f−1(y) ⊂∪k⊂ f−1(n). clearly ∪k∈v and y ∈ f(∪k) ⊂n . let u1 = f(u ′ 1) and u2 = f(u ′ 2) where u ′ 1 and u ′ 2 are elements of v. now ∂f−1(y) ⊂ u′1 ∩ u′2 then as in above we get u′ ∈ v such that ∂f−1(y) ⊂ u′ ⊂ u′1 ∩ u′2. name f(u′) = v ′. then v ′ ⊂ u1 ∩ u2. next we show that each v ∈ f(v) is a sequential neighbourhood of y. let {yn} be a sequence converging to y in y. then there is a sequence {xk}, i-converging to x such that x ∈ ∂f−1(y) and f(xk) = ynk, k ∈ n. now v = f(u) ∈ f(v). so {k ∈ n : xk 6∈ u} ∈ i which shows that {k ∈ n : ynk 6∈ v} ∈ i. suppose that b = {n ∈ n : yn 6∈ v} is an infinite set. then {y′n : n ∈ b} is a subsequence of {yn} converging to y. there is a sequence {x′n}, i-converging to x′ ∈ ∂f−1(y) ⊂ u and its image under f is a subsequence of {y′n : n ∈ b}, hence there is a m ∈ b so that ym ∈ v , which is a contradiction. so b must be a finite set and hence the result. � theorem 3.20. let i, j and l be three ideals of n that satisfy the following conditions. (1) i ∈ β and i satisfies s1(f(i),f(i)). (2) j ⊂i (3) there is a countable infinite partition {mi : i ∈ n} of n such that mi /∈l for each i. then each (i,j ,l)sequentially quotient and boundary compact map f : x → y, is an (i,j)-sequence covering map if x ∈ ω. proof. let f : x → y be an (i,j ,l)sequentially quotient and boundary compact map and let x ∈ ω. by lemma 3.19, it follows that y is snf-countable. as ∂f−1(y) is compact hence applying theorem 3.18, we have f is an (i,j)sequence covering map. � corollary 3.21. let i, j and l be three ideals of n that satisfy the following conditions. (1) i ∈ β and i satisfies s1(f(i),f(i)). (2) j ⊂i (3) there is a countable infinite partition {mi : i ∈ n} of n such that mi /∈l for each i. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 375 s. k. pal, n. adhikary and u. samanta then each (i,j ,l)sequentially quotient and boundary compact map f : x → y, is an (i,j)-sequence covering map if at least one of the following conditions holds. 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[26] a. zygmund, trigonometric series, cambridge univ. press, cambridge, uk (1979). c© agt, upv, 2019 appl. gen. topol. 20, no. 2 377 () @ appl. gen. topol. 14, no. 2 (2013), 147-157doi:10.4995/agt.2013.1575 c© agt, upv, 2013 on the category of profinite spaces as a reflective subcategory abolfazl tarizadeh faculty of basic sciences, university of maragheh, p. o. box 55181-83111, maragheh, iran department of mathematics, institute for advanced studies in basic sciences(iasbs) p. o. box 45195-1159, zanjan, iran. (abtari@iasbs.ac.ir) abstract in this paper by using the ring of real-valued continuous functions c(x), we prove a theorem in profinite spaces which states that for a compact hausdorff space x, the set of its connected components x/∼ endowed with the quotient topology is a profinite space. then we apply this result to give an alternative proof to the fact that the category of profinite spaces is a reflective subcategory in the category of compact hausdorff spaces. finally, under some circumstances on a space x, we compute the connected components of the space t(x) in terms of the ones of the space x. 2010 msc: 03g05, 06e25, 14g32, 18a40. keywords: profinite spaces, connected components, coarser topology, reflective subcategory. 1. introduction a profinite space is a compact hausdorff and totally disconnected topological space. in other words, a space x is profinite if there exists an inverse system of finite discrete spaces for which its inverse limit is homeomorphic to x, consider [2, section 3.4]. recall that a profinite group is a topological group whose underlying space is a profinite space. there are interesting examples of profinite spaces and profinite groups which arise from algebraic geometry, galois theory and topology. for instance, for received april 2012 – accepted march 2013 http://dx.doi.org/10.4995/agt.2013.1575 a. tarizadeh any field k its absolute galois group gal(ks/k) is a profinite group, or more generally the étale fundamental group π1(x,s) of a connected scheme x on a geometric point s : spec(ω) → x is a profinite group [5, theorem 5.4.2]. stone’s duality tells us that any profinite space x is of the form x = spec(b) for some boolean algebra b ([2, theorem 4.1.16]). there is also another characterization of profinite spaces due to craven [3], where he proves that each profinite space is homeomorphic to the space x(f) for some formally real field f and x(f) denotes the set of orderings of the field f endowed with some topology. in partial of this paper, we give an alternative proof to the fact that the category of profinite spaces is a reflective subcategory in category of compact hausdorff spaces. we prove this by using spectra of the boolean algebras and rings of real-valued continuous functions c(x). thanks to the stone-čech compactification functor, one can prove that the category of compact hausdorff spaces is a reflective subcategory in the category of tychonoff spaces. moreover, the category of tychonoff spaces itself is reflective in the category of topological spaces. section 2, contains some preliminaries which will be required in section 3. in section 3, we will use spectra of the boolean algebras to compute the connected components of the spectra of a commutative ring in terms of the its max-regular ideals (theorem 3.5). as an application of this result and also by using some properties of the rings of real-valued continuous functions c(x), we give an alternative proof to a theorem in profinite spaces which states that for a compact hausdorff space x, the set of its connected components x/∼ endowed with the quotient topology is a profinite space (theorem 3.8). this theorem leads us to construct a covariant functor from the category of compact hausdorff spaces k to the category of profinite spaces p, then we will use this categorical construction to show that the category p is a reflective subcategory in the category k (theorem 3.12). finally, under some circumstances on a space x, we compute the connected components of t(x) in terms of the connected components of x(theorem 3.17). consider section 5.7 of the book [2] to see another proof of the theorems 3.8 and 3.12. in that book, these theorems are proven by means of the nearness relation notion. 2. preliminaries in this section, for convenience of the reader and for the sake of completeness we collect some preliminaries which will be required in the next section. for more details on the spectra of the boolean algebras we reference the reader to the book [2, sections 4.1 and 4.2]. definition 2.1. a topological space x is said to be a profinite space if it is compact hausdorff and totally disconnected. totally disconnectedness means c© agt, upv, 2013 appl. gen. topol. 14, no. 2 148 on the category of profinite spaces as a reflective subcategory that there is no connected subset in x other than the single point subsets. definition 2.2. a boolean algebra is a structure (b,∨,∧,c,0,1) with two binary operations ∨,∧ : b × b → b, a unary operation c : b → b and two distinguished elements 0 and 1 in b, such that for all x,y and z in b, (i) the binary operations ∨ and ∧ are commutative and associative, (ii) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), (iii) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), (iv) x ∨ (x ∧ y) = x, x ∧ (x ∨ y) = x, (v) x ∨ xc = 1, x ∧ xc = 0. for the sake of simplicity, the boolean algebra (b,∨,∧,c,0,1) is only denoted by b. the relation x ≤ y ⇐⇒ x ∧ y = x ⇐⇒ x ∨ y = y puts a partial ordering on b. a morphism between two boolean algebras b and b′ is a map ϕ : b → b′ which preserves the binary and unary operations. definition 2.3. let b be a boolean algebra. a filter in b is a subset f ⊆ b such that, (i) 1 ∈ f, (ii) if x,y ∈ f then x ∧ y ∈ f, (iii) if x ∈ f and x ≤ y then y ∈ f. the filter f is called proper if 0 /∈ f. each maximal element of the poset of proper filters ordered by inclusion is called an ultrafilter. definition 2.4. for given boolean algebra b, we denote by spec(b) the set of all its ultrafilters. for every filter h on b consider oh = {f ∈ spec(b) | h * f}. the subsets oh as open subsets constitute a topology on spec(b). this topology is usually called the stone topology and the space spec(b) is called the spectrum of b. also the collection of all ob = {f ∈ spec(b) | b /∈ f} where b ∈ b, as open subsets constitute a basis for the stone topology. lemma 2.5. the spectrum of a boolean algebra is a profinite space. proof. for the proof see [2, proposition 4.1.11]. � all rings that we consider in this paper are commutative with the identity. lemma 2.6. for a ring r, then the set of its idempotents i(r) with the operations e ∨ e′ = e + e′ − ee′, e ∧ e′ = ee′ and ec = 1 − e constitute a boolean algebra. proof. for the proof see [2, proposition 4.2.1]. � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 149 a. tarizadeh remark 2.7. we denote by sp(r) the spectrum of the boolean algebra i(r) which is a profinite space according to the lemma 2.5. definition 2.8. in a ring r, an ideal i of r is called a regular ideal if it is generated by a set of idempotents of r. if moreover i 6= r, then we call it a proper regular ideal. lemma 2.9. for any ideal i of r, then i is a regular ideal if and only if for any a ∈ i there exists an idempotent e ∈ i such that a = ea. proof. see [2, lemma 4.2.7] for its proof. � definition 2.10. for a ring r, any maximal element of the poset of proper regular ideals of r ordered by inclusion is called a max-regular ideal of r. we denote by mr(r) the set of all its max-regular ideals. the subsets oi = {m ∈ mr(r) | i * m} where i is a regular ideal of r, as open subsets constitute a topology on mr(r). lemma 2.11. for a ring r, then there exists a natural map ϕ : sp(r) → mr(r) which is a homeomorphism. proof. consider [2, corollary 4.2.11] for the proof. � remark 2.12. in the above lemma, the explicit description of the map ϕ is as follows. each element f ∈ sp(r) is an ultrafilter of the boolean algebra i(r), define ϕ(f) = m where m = 〈e ∈ i(r) | 1 − e ∈ f〉. in particular, as a consequence of the above lemma, the subsets oe = {m ∈ mr(r) | e /∈ m} where e ∈ r is an idempotent, constitute a basis for the topology of mr(r). definition 2.13. for any ring r, set spec(r) = {p ⊂ r | p is a prime ideal of r}. then the subsets oi = {p ∈ spec(r) | i * p} where i is an ideal of r, constitute a topology for spec(r). this topology is called the zariski topology. we denote by v (i) the complement of oi. also, the subsets d(f) = {p ∈ spec(r) | f /∈ p} where f ∈ r, as open subsets constitute a basis for the topology. for given topological space x, denote by clop(x) the set of all subsets of x which are both open and closed in x. each element of clop(x) is called a clopen of x. throughout this paper, for any topological space x, we denote by x/∼ the set of all its connected components. in the next section, the set x/∼ will be equipped with some topology t which is coarser than the quotient topology and so the canonical projection π : x → (x/∼, t ) will remain continuous for this new topology. recall that for each x ∈ x, π(x) is defined to be the connected component of x which contains x. for any two covariant functors f : c → d and g : d → c , the covariant functor homd(f(−),−) : c op × d → set is called the left hom-set adjunction c© agt, upv, 2013 appl. gen. topol. 14, no. 2 150 on the category of profinite spaces as a reflective subcategory of f . in the similar way, the covariant functor homc (−,g(−)) : c op × d → set is called the right hom-set adjunction of g. definition 2.14. let f : c → d and g : d → c are two covariant functors. the functor f is called a left adjoint to the functor g (or g is called a right adjoint to f) if there exists a natural isomorphism φ : homd(f(−),−) ⇒ homc (−,g(−)) between the left and right hom-set adjunctions of f and g respectively. sometimes, this relationship is indicated by φ : f ⊣ g. definition 2.15. let c be a subcategory of the category d. then c is called a reflective subcategory of d, if there exists a covariant functor f : d → c which is a left adjoint to the inclusion functor i : c → d, i.e., f ⊣ i. in this situation, the functor f is called a reflector. 3. main results although the following result is probably known, we have provided a proof for the sake of completeness. proposition 3.1. for any ring r, consider the space x = spec(r) with the zariski topology. then there exists a one-one correspondence between i(r) and clop(x). proof. define the map η : i(r) → clop(x) by η(e) = d(e) for any e ∈ i(r). since d(e) = v (1 − e), hence d(e) is a clopen and so η is well-defined. we shall show that it is bijective. for the injectivity, let that η(e) = η(e′) for some e,e′ ∈ i(r), then v (1 − e) = v (1 − e′). hence, √ 〈1 − e〉 = √ 〈1 − e′〉, so 1 − e = (1 − e)n ∈ 〈1 − e′〉 for some n ≥ 1. thus 1 − e = r′(1 − e′) for some r′ ∈ r, if we multiply this equation to e′ we get e′ = ee′. by the similar way we also get e = ee′ thus e = e′. now we show that η is surjective. let that u be a clopen of x. since u is closed it is quasi-compact and similarly its complement. write u = n ⋃ i=1 d(fi) as a finite union of standard opens. similarly, write spec(r) \ u = m ⋃ j=1 d(gj) as a finite union of standard opens. but d(figj) = d(fi)∩d(gj) = ∅, and so each of figj is nilpotent, thus if we set i = 〈f1, ...,fn〉 , j = 〈g1, ...,gm〉 then (ij)n = 0 for some sufficiently large integer n ≥ 1. also spec(r) = v (in) ∐ v (jn) because v (in ) = v (i) = m ⋃ j=1 d(gj) and similarly v (j n) = v (j) = n ⋃ i=1 d(fi) so we have r = in + jn. write 1 = x + y with x ∈ in and y ∈ jn. then we have x = x2 because x − x2 = x(1 − x) = xy = 0 for the last equality note that xy ∈ injn = (ij)n = 0. so x is an idempotent and we have η(x) = d(x) = n ⋃ i=1 d(fi) = u. � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 151 a. tarizadeh for a ring r, the elements 0 and 1 are called the trivial idempotents. corollary 3.2. for any ring r, set x = spec(r) with the zariski topology. then the space x is connected if and only if the idempotents of r are only 0 and 1. proof. this is a direct consequence of the above proposition. � proposition 3.3. let m be a regular ideal of r. then m is a max-regular ideal if and only if the idempotents of r/m are trivial. proof. if the set of idempotents of r/m is trivial then it is easy to see that m is a max-regular ideal of r. conversely, suppose that m be a max-regular ideal of r. let x = x + m be an arbitrary idempotent of r/m where x ∈ r, this implies that x − x2 ∈ m. by lemma 2.9, there exists an idempotent e ∈ m so that (x − x2) = e(x − x2) thus we get (1 − e)(x − x2) = 0. this implies that (1 − e)x is an idempotent of r. write x = (1 − e)x + ex which belongs to the regular ideal m + 〈(1 − e)x〉. also (3.1) m ⊆ m + 〈(1 − e)x〉 ⊆ m + 〈x〉. we have m + 〈x〉 = r or m + 〈x〉 6= r. if m + 〈x〉 = r, then write 1 = rx + r1e1 + ... + rnen where r,ri ∈ r and ei ∈ m, if we multiply this equation to 1 − x then we get 1 − x = r(x − x2) + (r1e1 + ... + rnen)(1 − x) which belongs to m and so x = 1, which is a trivial idempotent. if m + 〈x〉 6= r, then since m is a max-regular ideal and m + 〈(1 − e)x〉 is a regular ideal so from (3.1) we get m = m + 〈(1 − e)x〉 but since x ∈ m+〈(1−e)x〉 = m so x = 0, therefore in this case also x is a trivial idempotent. therefore the idempotents of r/m are only trivial. � corollary 3.4. let that m be a max-regular ideal of r. then v (m) is a connected subset of x = spec(r). proof. by the proposition 3.3, the idempotents of r/m are trivial and so by the corollary 3.2, the space spec(r/m) is connected. in other hand, v (m) is naturally homeomorphic to the spec(r/m), so it is also connected. � theorem 3.5. let r be a ring and set x = spec(r) with the zariski topology. then c ⊆ x is a connected component if and only if c is of the form v (m) for some max-regular ideal m of r. proof. in order to prove the assertion, first we define a map f : x → mr(r) by f(p) = 〈e | e ∈ p ∩ i(r)〉. it is easy to check that for any prime p, f(p) is a max-regular ideal of r and so the map is well defined. also f is continuous, because for any basis open oe of mr(r) where e ∈ i(r), we have f−1(oe) = d(e). now let that c is a connected component of x then f(c) is connected subset of c© agt, upv, 2013 appl. gen. topol. 14, no. 2 152 on the category of profinite spaces as a reflective subcategory sp(r), because f is continuous. but by the lemma 2.11, mr(r) is a profinite space and so f(c) = {m} for a max-regular ideal m of r. but we have c ⊆ f−1({m}) = v (m). also by the corollary 3.4, v (m) is a connected subset of x so the inclusion c ⊆ v (m) implies the equality, because c is a connected component. conversely, assume that m be a max-regular ideal of r. again by the corollary 3.4, v (m) is a connected subset of x, thus it is contained in a connected component c of x. by the above paragraph, c = v (n) for some max-regular ideal n of r. so v (m) ⊆ v (n) this implies that n ⊆ √ n ⊆ √ m. but for each element e of a set of idempotent generators of n, we have e = en ∈ m for some n ≥ 1. hence, n ⊆ m. since n is a max-regular ideal, so n = m. � corollary 3.6. for a ring r, set x = spec(r) with the zariski topology. then the set x/∼ with the quotient topology is profinite. proof. in the light of the theorem 3.5, we have x/∼ = {v (m) | m ∈ mr(r)}. hence the map φ : x/∼ → mr(r) given by v (m) m is well-defined and bijective. furthermore, by the remark 2.12, there exists a basis for mr(r) in which any element of this basis is of the form oe = {m ∈ mr(r) | e /∈ m} where e ∈ i(r). so the map φ induces a topology t on x/∼ with the basis {φ−1(oe) | e ∈ i(r)}. therefore with this topology, φ is a homeomorphism. also the topology t is coarser than the quotient topology, because π−1(φ−1(oe)) = d(e) where π : x → x/∼ is the canonical projection. by lemma 2.11, the space mr(r) is profinite, hence the space (x/∼, t ) is also profinite. � for a given ring r, denote by m(r) the set of all maximal ideals of r and consider it as a topological subspace of spec(r). theorem 3.7. let x be a topological space and set r = c(x) the ring of real-valued continuous functions on x. then the set m(r)/∼ with the quotient topology is profinite. proof. first we show that the map ψ : spec(r)/∼ → m(r)/∼ given by v (m) v (m) ∩ m(r) is well-defined and bijective. in order to prove the claim we act as follows, according to [4, theorem 2.11 ], every prime ideal p of r is contained in a unique maximal ideal mp. hence, we obtain a map ψ : spec(r) → m(r) given by ψ(p) = mp. this map is continuous according to [1, corollary 1.6.2.1]. therefore, ψ maps any connected component v (m) of spec(r) to a connected subset ψ(v (m)) = v (m)∩m(r) of m(r). in fact, we show that v (m) ∩ m(r) is a connected component of m(r). for, choose m ∈ v (m) ∩ m(r) and let c is a connected component of m(r) containing m, since the inclusion map m(r) →֒ spec(r) is continuous then c is a connected subset of spec(r), so c is contained in a connected component of spec(r) which is exactly v (m), because m ∈ v (m). thus, we have c ⊆ v (m)∩m(r), but by the connectedness of v (m)∩m(r) we obtain c = v (m) ∩ m(r). therefore, the map ψ : spec(r)/∼ → m(r)/∼ is well-defined (note that the c© agt, upv, 2013 appl. gen. topol. 14, no. 2 153 a. tarizadeh map ψ is in fact induced by ψ). surjectivity of ψ is clear from the preceding argument. for its injectivity, suppose that v (m) ∩ m(r) = v (n) ∩ m(r) for some max-regular ideals m and n of r. if m 6= n then we can choose an idempotent e ∈ m \ n. let m be a maximal ideal of r containing m, then m also contains n and so the regular ideal n + 〈e〉 is contained in m, but since n is a max-regular ideal we get n = n + 〈e〉 which is a contradiction. finally, by using the corollary 3.6, the bijective map ψ : spec(r)/∼ → m(r)/∼ induces a topology {ψ(v ) | v ∈ t } ( we denote it also by t ) on m(r)/∼, one can prove that this topology is nothing but the quotient topology, and with this topology the map ψ becomes a homeomorphism and so the space (m(r)/∼, t ) is profinite. � theorem 3.8. if x is a compact hausdorff space, then the set x/∼ endowed with the quotient topology is a profinite space. proof. for a compact hausdorff space x, according to [4, 4.9.(a)], the map µ : x → m(r) given by µ(x) = mx = {f ∈ r | f(x) = 0} is a homeomorphism where r = c(x). finally, the result implies from the preceding theorem. � remark 3.9. let g : x → y be a continuous map where x is compact hausdorff and y is hausdorff. then g is a closed map, because each closed subset f of x is compact and so g(f) is a compact subset in y , but hausdorffness of y implies that g(f) is closed. remark 3.10. for given topological space x, set r = c(x) the ring of realvalued continuous functions. then one can easily check that the set of idempotents i(r) of r is exactly equal to the set {χ u : u ∈ clop(x)} where χ u is the characteristic function of u. in the sequel we need to this characterization of the idempotents. also this characterization of idempotents implies that x is connected if and only if the space spec(r) is connected. the above theorem 3.8, leads us to a covariant functor f : k → p from the category of compact hausdorff spaces k to the category of profinite spaces p and this categorical construction implies that the category of profinite spaces is a reflective subcategory of the category of compact hausdorff spaces and the reflector is the foregoing functor. hence in what follows, we plan to describe this functor explicitly and then show that this functor actually is a reflector. remark 3.11. for any compact hausdorff space x, set r = c(x) the ring of real-valued continuous functions on x, also set mx = {f ∈ r | f(x) = 0} the maximal ideal of r corresponding to each x ∈ x. then by the theorem 3.8, each connected component of x is of the form cm = {x ∈ x | m ⊆ mx} where m ∈ mr(r). therefore, x/∼ = {cm | m ∈ mr(r)} and the subsets ou = {cm | χ u /∈ m} where χ u is the characteristic function of u ∈ clop(x), as open c© agt, upv, 2013 appl. gen. topol. 14, no. 2 154 on the category of profinite spaces as a reflective subcategory subsets constitute a basis for the topology t as given in the theorem 3.8. finally, define the functor f : k → p, for each compact hausdorff space x, by f(x) = (x/∼, t ) where t is the quotient topology. moreover, for any continuous function f : x → y between the compact hausdorff spaces, then ff : (x/∼, t ) → (y/∼, t ′) is defined for each cm ∈ x/∼, by (ff)(cm ) = cn where cn ∈ y/∼ and f(cm ) ⊆ cn. consider the following commutative diagram x f // πx �� y πy �� (x/∼, t ) f f // (y/∼, t ′) where the vertical arrows are the canonical projections. theorem 3.12. the category of profinite spaces is a reflective subcategory in the category of compact hausdorff spaces. proof. we prove that the functor f : k → p as defined in the remark 3.11, is a left adjoint to the inclusion functor i : p → k . for this purpose, we show that there exists a natural isomorphism µ between the left and right hom-set adjunctions of f and i respectively, µ : hom(f(−),−) ⇒ hom(−, i(−)) . for each object (x,p) ∈ k op × p we define the natural transformation µ x,p : hom((x/∼, t ),p) → hom(x,p) by µx,p (g) = g ◦ πx where g ∈ hom((x/∼, t ),p) and πx : x → x/∼ is the canonical projection. the map µ x,p is injective because πx is surjective. for the surjectivity of µ x,p , suppose that f ∈ hom(x,p). put g = π−1 p ◦ ff : (x/∼, t ) → p (note that if p is already profinite then the natural projection πp : p → (p/∼, t ′) by the remark 3.9, is a homeomorphism). finally, the commutativity of the following diagram x f // πx �� p πp �� (x/∼, t ) f f // (p/∼, t ′) implies that µ x,p (g) = g ◦ πx = f. � on the connected components of t(x). denote by top the category of topological spaces with continuous maps as morphisms. in what follows, first we recall the definition of the classical covariant functor t : top → top and then we obtain our main result concerning to this subsection, i.e. theorem 3.17. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 155 a. tarizadeh definition 3.13. the functor t : top → top is defined for a topological space x, by t(x) = {z ⊆ x | z is a closed and irreducible subset of x}. the subsets t(y ) where y is a closed subset of x, as closed subsets constitute a topology for t(x). moreover, for any continuous map f : x → x′, then the map tf : t(x) → t(x ′ ) is defined for each z ∈ t(x) by (tf)(z) = f(z) which is a continuous map. we will need the following easy fact. lemma 3.14. let x be any topological space and let c be a connected component of it, then t(c) is a connected subset of t(x). proof. suppose that t(c) = (u ∩ t(c)) ∪ (v ∩ t(c)) be a disjoint separation for t(c) where u and v are open subsets of t(x). set u = t(x) \ t(e) and v = t(x) \ t(f) where e and f are closed subsets of x. also set u = x \ e and v = x \f , then c = (u ∩c)∪(v ∩c) is a disjoint separation by the open subsets of c. but by the connectedness of c we get c = u ∩ c or c = v ∩ c. thus we get t(c) = u ∩ t(c) or t(c) = v ∩ t(c). � remark 3.15. note that in the above lemma the assertion is also true for any connected subset of x. more precisely, for any connected subset c of x then the set {z ∈ t(x) | z ⊆ c} is a connected subset of t(x). the proof is similar to the proof of the above lemma. remark 3.16. the structure of the connected components of t(x) in the general case is as follows. each topological space x can be written as a disjoint union of its connected components, i.e., x = ∐ i∈i ci where each ci is a connected component of x. from this fact we easily get t(x) = ∐ i∈i t(ci). hence, each connected component c of t(x), is of the form c = ∐ j∈j t(cj) for some j ⊆ i, because by lemma 3.14, each of the t(ci) is connected. now if j is a finite set then it is just a single point subset. namely, c = t(cj) where j = {j}, because in the finite case each of the t(cj) is a disjoint open subset of c . however, in the general case the set j is not necessarily finite. the following theorem says us that the set j is finite whenever the space x/∼ is totally disconnected with some topology t . theorem 3.17. let x be a topological space so that (x/∼, t ) is totally disconnected with some topology t which is coarser than the quotient topology. then c ⊆ t(x) is a connected component of t(x) if and only if c is of the form t(c) where c is a connected component of x. proof. first we should note that any closed and irreducible subset z of x is also a connected subset of x. we denote by γ(z) the connected component of x which contains z. so this defines a map γ : t(x) → (x/∼, t ). the map c© agt, upv, 2013 appl. gen. topol. 14, no. 2 156 on the category of profinite spaces as a reflective subcategory γ is continuous. in order to prove this, for any closed subset e of x/∼, we will show that γ−1(e ) = t(e) where e = π−1(e ) and π : x → x/∼ is the canonical projection. first let that z ∈ t(e), since z is a nonempty connected subset thus z ⊆ π(z0) = γ(z) for some point z0 ∈ z, but π(z0) ∈ e because z ⊆ e therefore γ(z) ∈ e this shows that t(e) ⊆ γ−1(e ). for the inverse inclusion, suppose that z ∈ γ−1(e ) then γ(z) ∈ e , but z is connected, so for any point z ∈ z we have π(z) = γ(z) ∈ e this shows that z ∈ e = π−1(e ) thus z ⊆ e this means that z ∈ t(e). now for proving the assertion, let that c be any connected component of t(x). since the map γ is continuous and the space (x/∼, t ) is totally disconnected, then we have γ(c ) = {c} for some single point subset {c} of x/∼ where c is a connected component of x. but c ⊆ t(c), because for any point z ∈ c we have γ(z) = c, thus this means that z ∈ t(c). but by the lemma 3.14, t(c) is a connected subset of t(x), also c is a connected component of t(x), thus we get c = t(c). conversely, assume that c is a connected component of x. by the lemma 3.14, t(c) is a connected subset of t(x), thus t(c) is contained in some connected component c of t(x). but we have {c} = γ(t(c)) ⊆ γ(c ) = {c′} for some c ′ ∈ x/∼. thus we get c = c ′ and t(c) = c , so t(c) is a connected component. � acknowledgements. the author would like to give thanks to his ph. d. advisor professor hélène esnault for her guidance and suggestions, professor manuel sanchis for his help and the referee for careful reading of the manuscript. finally, i would like to express my deep gratitude to professor maysam maysami-sadr for his usefull comment which improved the manuscript. references [1] r. bkouche, couples spectraux et faisceaux associés. applications aux anneaux de fonctions, bull. soc. math. france 98 (1970), 253–295. [2] f. borceux and g. janelidze, galois theories, cambridge university press, 2001. [3] t. c. craven, the boolean space of orderings of a field, trans. amer. math. soc. 209 (1975), 225–235. [4] l. gillman and m. jerison, rings of continuous functions, springer, 1976. [5] t. szamuely, galois groups and fundamental groups, cambridge studies in adv. math., vol 117, 2009. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 157 @ appl. gen. topol. 21, no. 2 (2020), 265-284 doi:10.4995/agt.2020.13075 c© agt, upv, 2020 remarks on fixed point assertions in digital topology, 4 laurence boxer department of computer and information sciences, niagara university, usa; and department of computer science and electrical engineering, state university of new york at buffalo (boxer@niagara.edu) communicated by v. gregori abstract we continue the work of [4, 2, 3], in which we discuss published assertions concerning fixed points in digital topology assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon. 2010 msc: 54h25. keywords: digital topology; fixed point; metric space. 1. introduction as stated in [2]: the topic of fixed points in digital topology has drawn much attention in recent papers. the quality of discussion among these papers is uneven; while some assertions have been correct and interesting, others have been incorrect, incorrectly proven, or reducible to triviality. paraphrasing [2] slightly: in [4, 2, 3], we have discussed many shortcomings in earlier papers and have offered corrections and improvements. we continue this work in the current paper. a common theme among many weak papers concerning fixed points in digital topology is the use of a “digital metric space” (see section 2.2 for its definition). this seems to be a bad idea. received 30 january 2020 – accepted 03 august 2020 http://dx.doi.org/10.4995/agt.2020.13075 l. boxer • nearly all correct nontrivial published assertions concerning digital metric spaces use either the adjacency of the digital image or the metric, but not both. where our sources do not use adjacencies, we will state our results using the more general framework of a metric space. • if x is finite (as in a “real world” digital image) or the metric d is a common metric such as any ℓp metric, then (x, d) is uniformly discrete, hence not very interesting either as a topological space or as a metric space. • many of the published assertions concerning digital metric spaces mimic analogues for connected subsets of euclidean rn. often, the authors neglect important differences between the topological space rn and digital images, resulting in assertions that are incorrect, trivial, or trivial when restricted to conditions that many others regard as essential. e.g., in many cases, functions that satisfy fixed point assertions must be constant or fail to be digitally continuous [4, 2, 3]. this paper continues the work of [4, 2, 3] in discussing shortcomings of published assertions concerning fixed points in digital topology. 2. preliminaries we use n to represent the natural numbers, z to represent the integers, and r to represent the reals. a digital image is a pair (x, κ), where x ⊂ zn for some positive integer n, and κ is an adjacency relation on x. thus, a digital image is a graph. in order to model the “real world,” we usually take x to be finite, although there are several papers that consider infinite digital images. the points of x may be thought of as the “black points” or foreground of a binary, monochrome “digital picture,” and the points of zn \x as the “white points” or background of the digital picture. 2.1. adjacencies, connectedness, continuity, fixed point. in a digital image (x, κ), if x, y ∈ x, we use the notation x ↔κ y to mean x and y are κ-adjacent; we may write x ↔ y when κ can be understood. we write x !κ y, or x ! y when κ can be understood, to mean x ↔κ y or x = y. the most commonly used adjacencies in the study of digital images are the cu adjacencies. these are defined as follows. definition 2.1. let x ⊂ zn. let u ∈ z, 1 ≤ u ≤ n. let x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ x. then x ↔cu y if • x ∕= y, • for at most u distinct indices i, |xi − yi| = 1, and • for all indices j such that |xj − yj| ∕= 1 we have xj = yj. definition 2.2 ([14]). a digital image (x, κ) is κ-connected, or just connected when κ is understood, if given x, y ∈ x there is a set {xi}ni=0 ⊂ x such that x = x0, xi ↔κ xi+1 for 0 ≤ i < n, and xn = y. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 266 remarks on fixed point assertions in digital topology, 4 definition 2.3 ([14, 1]). let (x, κ) and (y, λ) be digital images. a function f : x → y is (κ, λ)-continuous, or κ-continuous if (x, κ) = (y, λ), or (digitally) continuous when κ and λ are understood, if for every κ-connected subset x′ of x, f(x′) is a λ-connected subset of y . theorem 2.4 ([1]). a function f : x → y between digital images (x, κ) and (y, λ) is (κ, λ)-continuous if and only if for every x, y ∈ x, if x ↔κ y then f(x) !λ f(y). theorem 2.5 ([1]). let f : (x, κ) → (y, λ) and g : (y, λ) → (z, µ) be continuous functions between digital images. then g ◦ f : (x, κ) → (z, µ) is continuous. we use 1x to denote the identity function on x, and c(x, κ) for the set of functions f : x → x that are κ-continuous. a fixed point of a function f : x → x is a point x ∈ x such that f(x) = x. functions f, g : x → x are commuting if f(g(x)) = g(f(x)) for all x ∈ x. 2.2. digital metric spaces. a digital metric space [8] is a triple (x, d, κ), where (x, κ) is a digital image and d is a metric on x. we are not convinced that this is a notion worth developing; under conditions in which a digital image models a “real world” image, x is finite or d is (usually) an ℓp metric, so that (x, d) is uniformly discrete as a topological space, i.e., there exists ε > 0 such that for x, y ∈ x, d(x, y) < ε implies x = y. typically, assertions in the literature do not make use of both d and κ, so that this notion has an artificial feel. e.g., for a discrete topological space x, all functions f : x → x are continuous, although on digital images, many functions g : x → x are not digitally continuous. we say a sequence {xn}∞n=0 is eventually constant if for some m > 0, n > m implies xn = xm. the notions of convergent sequence and complete digital metric space are often trivial, e.g., if the digital image is uniformly discrete, as noted in the following, a minor generalization of results of [9, 4]. proposition 2.6. let (x, d) be a metric space. if (x, d) is uniformly discrete, then any cauchy sequence in x is eventually constant, and (x, d) is a complete metric space. remarks 2.7. if x is finite or x ⊂ zn and d is an ℓp metric, then (x, d) is uniformly discrete. 2.3. common conditions, limitations, and trivialities. in this section, we state results that limit or trivialize several of the assertions discussed later in this paper. although there are papers that discuss infinite digital images, a “real world” digital image is a finite set. further, most authors writing about a digital metric space choose their metric from the euclidean metric, the manhattan metric, or some other ℓp metric. other frequently used conditions: c© agt, upv, 2020 appl. gen. topol. 21, no. 2 267 l. boxer • the adjacencies most often used in the digital topology literature are the cu adjacencies. • functions that attract the most interest in the digital topology literature are digitally continuous. thus, the use of cu-adjacency and the continuity assumption (as well as the assumption of an ℓp metric) in the following proposition 2.8 should not be viewed as major restrictions. the following is taken from the proof of remark 5.2 of [4]. proposition 2.8. let x be cu-connected. let t ∈ c(x, cu). let d be an ℓp metric on x, and 0 < α < 1 u1/p . let s : x → x such that d(s(x), s(y)) ≤ αd(t(x), t(y)) for all x, y ∈ x. then s must be a constant function. similar reasoning leads to the following. proposition 2.9. let (x, d) be a uniformly discrete metric space. let f ∈ c(x, κ). then if {xn}∞n=1 ⊂ x and limn→∞ xn = x0 ∈ x, then for almost all n, f(xn) = f(x0). other choices of (x, d) need not lead to the conclusion of proposition 2.9, as shown by the following example. example 2.10. let x = n ∪ {0}, d(x, y) = ! ""# ""$ 0 if x = 0 = y; 1/x if x ∕= 0 = y; 1/y if x = 0 ∕= y; |1/x − 1/y| if x ∕= 0 ∕= y. then d is a metric, and limn→∞ d(n, 0) = 0. however, the function f(n) = n+1 satisfies f ∈ c(x, c1) and lim n→∞ d(0, f(n)) = 0 ∕= f(0). proof. example 2.10 of [4] notes that d has the properties of a metric for values of x \ {0}. since clearly d(0, 0) = 0 and d(x, 0) = d(0, x), we must show that the triangle inequality holds when 0 is one of the points considered. we have the following. • if x, y > 0, then d(0, y) = 1/y ≤ 1/x + |1/y − 1/x| = d(0, x) + d(x, y). • similarly, if x, y > 0 then d(x, 0) ≤ d(x, y) + d(y, 0). • if x, y > 0 then d(x, y) = |1/x − 1/y| < 1/x + 1/y = d(x, 0) + d(0, y). thus, the triangle inequality is satisfied. note f ∈ c(x, c1), limn→∞ d(n, 0) = 0, and limn→∞ d(f(n), f(0)) = 1. □ 3. compatible functions and weakly compatible functions the papers [11, 13, 7] discuss common fixed points of compatible and weakly compatible functions (the latter also know as “coincidentally commuting”) and related notions. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 268 remarks on fixed point assertions in digital topology, 4 3.1. compatibility definition and basic properties. definition 3.1 ([6]). suppose s and t are self-functions on a metric space (x, d). consider {xn}∞n=1 ⊂ x such that (3.1) lim n→∞ s(xn) = lim n→∞ t(xn) = t ∈ x. if every sequence satisfying (3.1) also satisfies limn→∞ d(s(t(xn)), t(s(xn))) = 0, then s and t are compatible functions. proposition 3.2. let f, g : x → x be commuting functions on a metric space (x, d). then f and g are compatible. proof. since f and g are commuting, the assertion is immediate. □ definition 3.3 ([11]). let s and t be self-functions on a metric space (x, d). the pair (s, t) satisfies the property e.a. if there exists a sequence {xn}∞n=1 ⊂ x that satisfies (3.1). definition 3.4 ([11]). let s and t be self-functions on a metric space (x, d). the pair (s, t) satisfies the property common limit in the range of t , denoted clr(t), if there exists a sequence {xn}∞n=1 ⊂ x that satisfies lim n→∞ s(xn) = lim n→∞ t(xn) = t(x) for some x ∈ x. proposition 3.5. let s and t be self-functions on a metric space (x, d). (1) suppose the pair (s, t) satisfies the property clr(t). then the pair (s, t) satisfies the property e.a. (2) suppose s and t have a coincidence point, i.e., there exists x ∈ x such that s(x) = t(x). then the pair (s, t) satisfies the property clr(t). (3) if x is finite, then the following are equivalent. (a) (s, t) satisfies the property e.a. (b) (s, t) satisfies the property clr(t). (c) s and t have a coincidence point, i.e., there exists x ∈ x such that s(x) = t(x). proof. (1) suppose the pair (s, t) satisfies the property clr(t). it is trivial that the pair (s, t) satisfies the property e.a. (2) suppose s(x) = t(x). then the sequence xn = x satisfies definition 3.4, so (s, t) has property clr(t). (3) now suppose x is finite. (a) ⇒ (b) and (a) ⇒ (c): suppose (s, t) satisfies the property e.a. let {xn}∞n=1 ⊂ x satisfy (3.1). since x is finite, there is a subsequence {xni} that is eventually constant; say, xni is eventually equal to x ∈ x. hence t(xn) is eventually limni→∞ t(xni) = t(x). thus (s, t) satisfies the property clr(t). similarly, s(xn) is eventually limni→∞ s(xni) = s(x). thus s(x) = lim ni→∞ s(xni) = lim ni→∞ t(xni) = t(x). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 269 l. boxer (b) ⇒ (a): this is shown in part (1). (c) ⇒ (b): this is shown in part (2). □ 3.2. variants on compatibility. in classical topology and real analysis, there are many papers that study variants of compatible (as defined above) functions. several authors have studied analogs of these variants in digital topology. often, the variants turn out to be equivalent. definition 3.6 ([5]). let s, t : x → x. then s and t are weakly compatible or coincidentally commuting if, for every x ∈ x such that s(x) = t(x) we have s(t(x)) = t(s(x)). theorem 3.7. let s, t : x → x. compatibility implies weak compatibility; and if x is finite, weak compatibility implies compatibility. proof. suppose s and t are compatible. we show they are weakly compatible as follows. let s(x) = t(x) for some x ∈ x. let xn = x for all n ∈ n. then lim n→∞ s(xn) = s(x) = t(x) = lim n→∞ t(xn). by compatibility, 0 = lim n→∞ d(s(t(xn)), t(s(xn))) = d(s(t(x)), t(s(x))). thus, s and t are weakly compatible. suppose s and t are weakly compatible and x is finite. we show s and t are compatible as follows. let {xn}∞n=1 ⊂ x such that lim n→∞ s(xn) = lim n→∞ t(xn) = t ∈ x. proposition 2.6 yields that for almost all n, s(xn) = t(xn) = t. since x is finite, there is an infinite subsequence {xni} of {xn}∞n=1 such that xni = y ∈ x, hence s(y) = t(y). therefore, for almost all n and almost all ni, weak compatibility implies s(t(xn)) = s(t(xni)) = s(t(y)) = t(s(y)) = t(s(xni)) = t(s(xn)). it follows that s and t are compatible. □ we have the following, in which we restate (3.1) for convenience. definition 3.8. suppose s and t are self-functions on a metric space (x, d). consider {xn}∞n=1 ⊂ x such that (3.2) lim n→∞ s(xn) = lim n→∞ t(xn) = t ∈ x. • s and t are compatible of type a [6] if every sequence satisfying (3.2) also satisfies lim n→∞ d(s(t(xn)), t(t(xn))) = 0 = lim n→∞ d(t(s(xn)), s(s(xn))). c© agt, upv, 2020 appl. gen. topol. 21, no. 2 270 remarks on fixed point assertions in digital topology, 4 • s and t are compatible of type b [7] if every sequence satisfying (3.2) also satisfies lim n→∞ d(s(t(xn)), t(t(xn))) ≤ (3.3) 1/2 [ lim n→∞ d(s(t(xn)), s(t)) + d(s(t), s(s(xn)))] and lim n→∞ d(t(s(xn)), s(s(xn))) ≤ (3.4) 1/2 [ lim n→∞ d(t(s(xn)), t(t)) + d(t(t), t(t(xn)))]. note this is a correction of the definition as stated in [7], where the inequality here given as (3.4) uses a left side equivalent to lim n→∞ d(t(s(xn)), t(t(xn))) instead of lim n→∞ d(t(s(xn)), s(s(xn))). the version we have stated is the version used in proofs of [7] and corresponds to the version of [12] that inspired the definition of [7]. • s and t are compatible of type c [7] if every sequence satisfying (3.2) also satisfies lim n→∞ d(s(t(xn)), t(t(xn))) ≤ (3.5) 1/2 % limn→∞ d(s(t(xn)), s(t)) + limn→∞ d(s(t), s(s(xn)))+ limn→∞ d(s(t), t(t(xn))) & and lim n→∞ d(t(s(xn)), s(s(xn))) ≤ (3.6) 1/2 % limn→∞ d(t(s(xn)), t(t)) + limn→∞ d(t(t), t(t(xn)))+ limn→∞ d(t(t), s(s(xn))) & . • s and t are compatible of type p [6] if every sequence satisfying (3.2) also satisfies lim n→∞ d(s(s(xn)), t(t(xn))) = 0. we augment theorem 3.7 with the following. theorem 3.9. let (x, d) be a metric space that is uniformly discrete. let s, t : x → x. the following are equivalent. • s and t are compatible. • s and t are compatible of type a. • s and t are compatible of type b. • s and t are compatible of type c. • s and t are compatible of type p. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 271 l. boxer proof. the equivalence of compatible, compatible of type a, and compatible of type p was shown in theorem 3.3 of [2], where the assumption that x is finite or d is an ℓp metric easily generalizes to the assumption that (x, d) is uniformly discrete. compatible of type a implies compatible of type b, by proposition 4.7 of [7]. we show compatible of type b implies compatible, as follows. let s and t be compatible of type b. let {xn}∞n=1 ⊂ x satisfy (3.2). by proposition 2.6, s(xn) = t = t(xn) for almost all n. from (3.3) we have d(s(t), t(t)) ≤ 1/2 [d(s(t), s(t)) + d(s(t), s(t))] = 0, so s(t) = t(t). thus lim n→∞ d(s(t(xn)), t(s(xn))) = lim n→∞ d(s(t), t(t)) = 0. therefore, s and t are compatible. we show compatible implies compatible of type c, as follows. let s and t be compatible. let {xn}∞n=1 ⊂ x satisfy (3.2). by proposition 2.6, s(xn) = t = t(xn) for almost all n, and by compatibility, s(t) = t(t). therefore, lim n→∞ d(s(t(xn)), t(t(xn))) = lim n→∞ d(s(t), t(t)) = 0, so (3.5) is satisfied, and lim n→∞ d(t(s(xn)), s(s(xn))) = d(t(t), s(t)) = 0, so (3.6) is satisfied. thus s and t are compatible of type c. we show compatible of type c implies compatible, as follows. let s and t be compatible of type c. let {xn}∞n=1 ⊂ x satisfy (3.2). by proposition 2.6, s(xn) = t = t(xn) for almost all n. from (3.5) it follows that d(s(t), t(t)) ≤ 1/2 [d(s(t), s(t)) + d(s(t), s(t)) + d(s(t), t(t))], or d(s(t), t(t)) ≤ 1/2 [0 + 0 + d(s(t), t(t))], which implies 0 = d(s(t), t(t)) = lim n→∞ d(s(t(xn)), t(s(xn))). therefore, s and t are compatible. □ 3.3. fixed point assertions of [11]. the following assertion appears as theorem 3.1.1 of [11] and as theorem 4.12 of [7] (there is a minor difference between these: [11] requires µ ∈ (0, 1/2) while [7] requires µ ∈ (0, 1)). assertion 3.10. let (x, d, κ) be a complete digital metric space. let s and t be compatible self-functions on x. suppose (i) s(x) ⊂ t(x); (ii) s or t is continuous; and (iii) for all x, y ∈ x and some µ ∈ (0, 1/2), d(sx, sy) ≤ µ max{d(tx, ty), d(tx, sx), d(tx, sy), d(ty, sx), d(ty, sy)}. then s and t have a unique common fixed point in x. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 272 remarks on fixed point assertions in digital topology, 4 remarks 3.11. the argument given as proof in [11] for this assertion clarifies that the continuity assumed is topological (the classical ε − δ continuity), not digital. further, assertion 3.10 and the argument offered for its proof in [11] are flawed as discussed below (this is the first of several assertions with related flaws; we discuss these assertions together), beginning at remark 3.17. flaws in the treatment of assertion 3.10 in [7] are discussed below, beginning at remark 3.32. the following assertion appears as theorem 3.2.1 of [11]. assertion 3.12. let (x, d, κ) be a complete digital metric space. let s and t be weakly compatible self-functions on x. suppose (i) s(x) ⊂ t(x); (ii) s(x) or t(x) is complete; and (iii) for all x, y ∈ x and some µ ∈ (0, 1/2), d(sx, sy) ≤ µ max{d(tx, ty), d(tx, sx), d(tx, sy), d(ty, sx), d(ty, sy)}. then s and t have a unique common fixed point in x. however, this assertion and the argument offered for its proof are flawed as discussed below, beginning at remark 3.17. the following is theorem 3.3.2 of [11]. theorem 3.13. let (x, d, κ) be a digital metric space. let s, t : x → x be weakly compatible functions satisfying the following. (i) for some µ ∈ (0, 1) and all x, y ∈ x, d(sx, sy) ≤ µd(tx, ty). (ii) s and t satisfy property e.a. (iii) t(x) is a closed subspace of x. then s and t have a unique common fixed point in x. however, this result is limited, as discussed below, beginning at remark 3.17. the following appears as theorem 3.3.3 of [11]. assertion 3.14. let (x, d, κ) be a complete digital metric space. let s and t be weakly compatible self-functions on x. suppose (i) for all x, y ∈ x and some µ ∈ (0, 1/2), d(sx, sy) ≤ µ max{d(tx, ty), d(tx, sx), d(tx, sy), d(ty, sx), d(ty, sy)}; (ii) s and t satisfy the property e.a.; and (iii) t(x) is a closed subspace of x. then s and t have a unique common fixed point in x. however, this assertion and the argument offered for its proof are flawed as discussed below, beginning at remark 3.17. the following is theorem 3.4.3 of [11]. theorem 3.15. let s and t be weakly compatible self-functions on a digital metric space (x, d, κ) satisfying (i) for some µ ∈ (0, 1) and all x, y ∈ x, d(sx, sy) ≤ µd(tx, ty); and c© agt, upv, 2020 appl. gen. topol. 21, no. 2 273 l. boxer (ii) the clr(t) property. then s and t have a unique common fixed point in x. however, this result is quite limited, as discussed below at remark 3.19. the following appears as theorem 3.4.3 of [11]. assertion 3.16. let s and t be weakly compatible self-functions on a digital metric space (x, d, κ) satisfying (i) for all x, y ∈ x and some µ ∈ (0, 1/2), d(sx, sy) ≤ µ max{d(tx, ty), d(tx, sx), d(tx, sy), d(ty, sx), d(ty, sy)}; and (ii) the clr(t) property. then s and t have a unique common fixed point in x. however, this assertion and the argument offered for its proof are flawed as discussed below. remarks 3.17. several times in the arguments offered as proofs for assertions 3.10, 3.12, 3.14, and 3.16, inequalities appear that seem to confuse “min” and “max”. e.g., in the argument for assertion 3.10, it is claimed that the right side of the inequality d(yn, yn+1) ≤ µ max ' d(yn−1, yn), d(yn−1, yn), d(yn−1, yn+1), d(yn, yn), d(yn, yn+1) ( is less than or equal to µd(yn−1, yn+1), which would follow if “max” were replaced by “min”. thus, these assertions as given in [11] must be regarded as unproven. remarks 3.18. further, suppose “min” is substituted for “max” so that (iii) in each of the assertions 3.10 and 3.12 and (i) in each of assertions 3.14 and 3.16 becomes for all x, y ∈ x and some µ ∈ (0, 1/2), d(sx, sy) ≤ µ min{d(tx, ty), d(tx, sx), d(tx, sy), d(ty, sx), d(ty, sy)}. then for all x, y ∈ x, d(sx, sy) ≤ µd(tx, ty). if t ∈ c(x, cu), d is an ℓp metric, and µ < 1/u1/p, then by proposition 2.8, s is constant. it would then follow from compatibility (respectively, from weak compatibility) that s and t have a unique fixed point coinciding with the value of s. remarks 3.19. similarly, in theorems 3.13 and 3.15, if t ∈ c(x, cu), d is an ℓp metric, and µ < 1/u 1/p, then by proposition 2.8, s is constant. it would then follow from compatibility (respectively, from weak compatibility) that s and t have a unique fixed point coinciding with the value of s. 3.4. fixed point assertions of [13]. the following is stated as lemma 3.3.5 of [13]. assertion 3.20. let s, t : (x, d, κ) → (x, d, κ) be compatible. 1) if s(t) = t(t) then s(t(t)) = t(s(t)). 2) suppose limn→∞ s(xn) = limn→∞ t(xn) = t ∈ x. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 274 remarks on fixed point assertions in digital topology, 4 (a) if s is continuous at t, limn→∞ t(s(xn)) = s(t). (b) if s and t are continuous at t, then s(t) = t(t) and s(t(t)) = t(s(t)). but the continuity used in the proof of this assertion is topological continuity, not digital continuity. we observe that if (x, d) is uniformly discrete, then the assumption of continuity need not be stated, as every self-function on x is continuous in the topological sense. the argument given as proof of this assertion in [13] depends on the principle that an → a0 implies s(an) → s(a0) if s is continuous at a0, a valid principle for topological continuity and also for digital continuity if (x, d) is uniformly discrete, but, as shown in example 2.10, not generally true for digital continuity. thus the assertion must be regarded as unproven. we can modify this assertion as follows. notice we do not use a continuity hypothesis, but for part 2) we assume (x, d) is uniformly discrete. lemma 3.21. let s, t : (x, d) → (x, d) be compatible. 1) if s(t) = t(t) then s(t(t)) = t(s(t)). 2) suppose (x, d) is uniformly discrete. if lim n→∞ s(xn) = lim n→∞ t(xn) = t ∈ x, then limn→∞ t(s(xn)) = s(t) = t(t) and s(t(t)) = t(s(t)). proof. we modify the argument of [13]. suppose s(t) = t(t). let xn = t for all n ∈ n. then s(xn) = t(xn) = s(t) = t(t), so d(s(t(t)), t(s(t))) = d(s(t(xn)), t(s(xn))) →n→∞ 0 by compatibility. this establishes 1). suppose limn→∞ s(xn) = limn→∞ t(xn) = t ∈ x. since we assume x is uniformly discrete, we have s(xn) = t(xn) = t for almost all n. therefore, for almost all n, the triangle inequality and compatibility give us d(t(s(xn)), t(t)) ≤ d(t(s(xn)), s(t(xn))) + d(s(t(xn)), t(t)) → 0 + lim n→∞ d(s(t(xn)), t(s(xn))) = 0, so limn→∞ t(s(xn)) = t(t). since x is uniformly discrete, by compatibility we have d(s(t), t(t)) = lim n→∞ d(s(t), t(s(xn))) = lim n→∞ d(s(t(xn)), t(s(xn))) = 0. therefore, s(t) = t(t) and by part 1), s(t(t)) = t(s(t)). □ the following is stated as theorem 3.3.6 of [13]. assertion 3.22. let s and t be continuous compatible functions of a complete digital metric space (x, d, κ) to itself. then s and t have a unique common fixed point in x if for some α ∈ (0, 1), (3.7) s(x) ⊂ t(x) and d(s(x), s(y)) ≤ αd(t(x), t(y)) for all x, y ∈ x. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 275 l. boxer remarks 3.23. the argument given as proof of this assertion in [13] clarifies that the assumed continuity is topological, not digital; the argument is also flawed by its reliance on assertion 3.20, which we have seen is not generally valid. thus, the assertion must be regarded as unproven. as above, we can drop the assumption of continuity from assertion 3.22 if we assume x is uniformly discrete, as shown in the following. theorem 3.24. let s and t be compatible functions of a uniformly discrete metric space (x, d) to itself. if s and t satisfy (3.7) for some α ∈ (0, 1), then they have a unique common fixed point in x. proof. we use ideas from the analogue in [13]. let x0 ∈ x. since s(x) ⊂ t(x), we can let x1 ∈ x such that t(x1) = s(x0), and, inductively, xn ∈ x such that t(xn) = s(xn−1) for all n ∈ n. then for all n > 0, d(t(xn+1), t(xn)) = d(s(xn), s(xn−1)) ≤ αd(t(xn), t(xn−1)). it follows that d(t(xn+1), t(xn)) ≤ αnd(t(x1), t(x0)). by proposition 2.6, there exists t ∈ x such that t(xn) = t for almost all n. our choice of the sequence xn then implies s(xn) = t for almost all n. by lemma 3.21, s(t) = t(t) and s(t(t)) = t(s(t)). then d(s(t), s(s(t))) ≤ αd(t(t), t(s(t))) = αd(s(t), s(t(t))) = αd(s(t), s(s(t))), so (1 − α)d(s(t), s(s(t))) ≤ 0. therefore, d(s(t), s(s(t))) = 0, so s(t) = s(s(t)) = s(t(t)) = t(s(t)). thus s(t) is a common fixed point of s and t . to show the uniqueness of t as a common fixed point, suppose s(x) = t(x) = x and s(y) = t(y) = y. then d(x, y) = d(s(x), s(y)) ≤ αd(t(x), t(y)) = αd(x, y), so (1 − α)d(x, y) ≤ 0, so x = y. □ the following is stated as theorem 3.4.3 of [13]. assertion 3.25. let s and t be weakly compatible functions of a complete digital metric space (x, d, κ) to itself. then s and t have a unique common fixed point in x if either of s(x) or t(x) is complete, and for some α ∈ (0, 1), statement (3.7) is satisfied. remarks 3.26. the argument given in [13] as a proof for assertion 3.25 defines a sequence {xn}∞n=1 ⊂ x such that limn→∞ s(xn) = limn→∞ t(xn) = t ∈ x. from this is claimed that a subsequence of {xn}∞n=1 converges to a limit in x. how this is justified is unclear. therefore, assertion 3.25 as stated is unproven. if we additionally assume that x is finite, then the claim, that a subsequence of {xn}∞n=1 converges to a limit in x, is certainly justified. the following is a version of assertion 3.25 with the additional hypothesis that x is finite. we have not stated an assumption of completeness, since a finite metric space must be complete. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 276 remarks on fixed point assertions in digital topology, 4 theorem 3.27. let s and t be weakly compatible functions of a uniformly discrete metric space (x, d) to itself. then s and t have a unique common fixed point in x if for some α ∈ (0, 1), (3.7) is satisfied. proof. since x is finite, it follows from theorem 3.9 that s and t are compatible. the assertion follows from theorem 3.24. □ note also that assertion 3.22 and theorems 3.24 and 3.27 are limited by proposition 2.8. 3.5. fixed point assertions of [7]. the following appears as proposition 4.10 of [7]. apparently, the authors neglected to state a hypothesis that s and t are compatible; they used this hypothesis in their “proof”, and with this hypothesis, the desired conclusion is correctly reached. assertion 3.28. let s, t ∈ c(x, κ) for a digital metric space (x, d, κ). if s(t) = t(t) for some t ∈ x, then s(t(t)) = t(s(t)) = s(s(t)) = t(t(t)). as stated, this is incorrect, as shown by the following example. example 3.29. let s, t : n → n be the functions s(x) = 2, t(x) = x + 1. then s, t ∈ c(n, c1) and s(1) = t(1) = 2, but s(t(1)) = s(s(1)) = 2, t(s(1)) = t(t(1)) = 3. further, the argument of [7] uses neither the hypothesis of continuity nor the adjacency κ. a corrected version of assertion 3.28 is above at lemma 3.21. the following appears as proposition 4.11 of [7]. assertion 3.30. let (x, d, κ) be a digital metric space and let s, t ∈ c(x, κ). suppose limn→∞ s(xn) = limn→∞ t(xn) = t ∈ x. then (i) limn→∞ t(s(xn)) = s(t); (ii) limn→∞ s(t(xn)) = t(t); and (iii) s(t(t)) = t(s(t)) and s(t) = t(t). the “proof” of this assertion in [7] confuses topological and digital continuity. the following shows that the assertion is not generally true. example 3.31. let s, t : n∪{0} → n∪{0} be the functions s(x) = 0, t(x) = x + 1. let d be the metric of example 2.10. clearly, s, t ∈ c(n∪ {0}, c1), and with respect to d, we have limn→∞ s(n) = 0 = limn→∞ t(n). however, with respect to d we have (i) limn→∞ t(s(n)) = t(0) = 1, s(0) = 0. (ii) limn→∞ s(t(n)) = 0, t(0) = 1; (iii) s(t(0)) = 0, t(s(0)) = 1, s(0) = 0, t(0) = 1. remarks 3.32. we have stated theorem 4.12 of [7] above as assertion 3.10. the argument for this assertion in [7] is flawed as follows. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 277 l. boxer the argument considers the case xn = xn+1 and reaches the statement d(t(xn), t(xn+1)) = d(s(xn−1), s(xn)) ≤ α max{d(t(xn−1), t(xn)), d(t(xn−1), t(xn+1)), d(t(xn), t(xn+1))}. this yields three cases, each of which is handled incorrectly: (1) d(t(xn), t(xn+1)) ≤ αd(t(xn−1), t(xn)). nothing further is stated about this case. (2) d(t(xn), t(xn+1)) ≤ αd(t(xn−1), t(xn+1)). the authors misstate this case as d(t(xn), t(xn+1)) ≤ αd(t(xn), t(xn+1)) and propagate this error forward. (3) d(t(xn), t(xn+1)) ≤ αd(t(xn), t(xn+1)). this implies t(xn) = t(xn+1), since 0 < α < 1, but the authors reach a slightly weaker conclusion differently. they reason that d(t(xn), t(xn+1)) ≤ αnd(t(x0), t(x1)), from an implied induction with the unjustified assumption that this case applies at every level of the induction. later in the argument, the error of confusing topological and digital continuity also appears. therefore, we must consider assertion 3.10 unproven. the following is stated as theorem 4.13 of [7]. assertion 3.33. let s, t : (x, d, κ) → (x, d, κ) be functions that are compatible of type a on a digital metric space, such that (i) s(x) ⊂ t(x); (ii) s or t is (κ, κ)-continuous; and (iii) for all x, y ∈ x and some α ∈ (0, 1), d(sx, sy) ≤ α max{d(tx, ty), d(tx, sy), d(ty, sx), d(tx, sx), d(ty, sy)}. then s and t have a unique common fixed point in x. however, the argument given in [7] to prove this assertion relies on assertion 3.30, which we have shown above is unproven. assertion 3.34. let s, t : (x, d, κ) → (x, d, κ) be functions on a digital metric space satisfying (i), (ii), and (iii) of assertion 3.33. if (a) (stated as theorem 4.14 of [7]) s and t are compatible of type b, or (b) (stated as theorem 4.15 of [7]) s and t are compatible of type c, or (c) (stated as theorem 4.16 of [7]) s and t are compatible of type p, then s and t have a unique common fixed point in x. remarks 3.35. each part of assertion 3.34 must be regarded as unproven, as each has a “proof” in [7] that depends on assertion 3.10, which we have shown above to be unproven. (the arguments in [7] for parts (b) and (c) also depend on the unproven part (a).) note also that, by theorem 3.9, a correct proof of any of (a), (b), or (c) for the case that (x, d) is uniformly discrete would prove the other parts correct for this case. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 278 remarks on fixed point assertions in digital topology, 4 4. commutative and weakly commutative functions the paper [13] discusses common fixed points for commutative and weakly commutative functions on digital metric spaces. definition 4.1. let (x, d) be a metric space. functions f, g : x → x are commutative if f ◦ g(x) = g ◦ f(x) for all x ∈ x. they are weakly commutative if d(f(g(x)), g(f(x))) ≤ d(f(x), g(x)) for all x ∈ x. proposition 4.2 ([13]). let (x, d, κ) be a digital metric space. let t : x → x. then t has a fixed point in x if and only if there is a constant function s : x → x such that s commutes with t. the following is theorem 3.1.4 of [13]. theorem 4.3. let t be a continuous self-function on a complete digital metric space (x, d, κ) into itself. then t has a fixed point in x if and only if there exists α ∈ (0, 1) and a function s : x → x that commutes with t and satisfies (3.7). if (3.7) holds then s and t have a unique common fixed point. we give a modified version of theorem 4.3 as follows. theorem 4.4. let t be a function of a metric space (x, d) into itself. • if t has a fixed point in x, then there exists α ∈ (0, 1) and a function s : x → x that commutes with t and satisfies (3.7). • suppose (x, d) is uniformly discrete. if there exists α ∈ (0, 1) and a function s : x → x that commutes with t and satisfies (3.7), then t has a fixed point in x. proof. it follows from proposition 4.2 that if t has a fixed point, then there is a function s : x → x that commutes with t and satisfies (3.7). suppose x is uniformly discrete. suppose there exists α ∈ (0, 1) and a function s : x → x that commutes with t and satisfies (3.7). then s and t are compatible by proposition 3.2. it follows from theorem 3.24 that t has a fixed point. □ we will use the following. example 4.5 ([4]). let x = {p1, p2, p3} ⊂ z5, where p1 = (0, 0, 0, 0, 0), p2 = (2, 0, 0, 0, 0), p3 = (1, 1, 1, 1, 1). let d be the manhattan metric and let t : (x, c5) → (x, c5) be defined by t(p1) = t(p2) = p1, t(p3) = p2. clearly t(x) ⊂ 1x(x), 1x ∈ c(x, c5), and for all x, y ∈ x we have d(t(x), t(y)) ≤ 2/5 d(1x(x), 1x(y)). however, t ∕∈ c(x, c5) since p2 ↔c5 p3 and t(p2) ∕↔c5 t(p3). in the following, given a function s : x → x and k ∈ n, sk is the k-fold iterate of s, i.e., s1 = s and sj+1 = s ◦ sj. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 279 l. boxer proposition 4.6 ([13]). let t and s be commuting functions of a digital metric space (x, d, κ) into itself. suppose t is continuous and s(x) ⊂ t(x). if there exists α ∈ (0, 1) and k ∈ n such that d(sk(x), sk(y)) ≤ αd(t(x), t(y)) for all x, y ∈ x, then t and s have a common fixed point. remarks 4.7. the continuity hypothesized for proposition 4.6 in [13] is topological continuity, not digital continuity. the assumption is used in the proof to argue that (3.7) implies s is (topologically) continuous. note if x is a discrete topological space, then every self-function on x is topologically continuous. if we were to assume instead that t is digitally continuous, it would not follow from (3.7) that s is digitally continuous, as shown by example 4.5. the following is a modified version of proposition 4.6. in it, there is no continuity assumption. corollary 4.8. let t and s be commuting functions of a complete metric space (x, d) into itself. suppose s(x) ⊂ t(x). if there exists α ∈ (0, 1) and k ∈ n such that d(sk(x), sk(y)) ≤ αd(t(x), t(y)) for all x, y ∈ x, then t and s have a common fixed point. proof. as above, we modify the analogous argument of [13]. we see easily that sk commutes with t and sk(x) ⊂ s(x) ⊂ t(x). by theorem 4.3 whose proof in [13] does not use an adjacency κ, hence is applicable in the more general setting of a metric space there exists a ∈ x such that a is the unique common fixed point of sk and t . then a = sk(a) = t(a). since s and t commute, we can apply s to the above to get s(a) = s(sk(a)) = s(t(a)) = t(s(a)) and, from the first equation in this chain, s(a) = sk(s(a)), so s(a) is a common fixed point of t and sk. since a is unique as a common fixed point of t and sk, we must have a = s(a) = t(a). □ a function t : x → x on a digital metric space (x, d, κ) is a digital expansive function [10] if for some k > 1 and all x, y ∈ x, d(t(x), t(y)) ≥ kd(x, y). however, this definition is quite limited, as shown by the following, which combines theorems 4.8 and 4.9 of [4]. theorem 4.9. let (x, d, κ) be a digital metric space. suppose there are points x0, y0 ∈ x such that d(x0, y0) ∈ {min{d(x, y) | x, y ∈ x, x ∕= y}, max{d(x, y) | x, y ∈ x, x ∕= y}}. then there is no t : x → x that is both onto and a digital expansive function. note the hypothesis of theorem 4.9 is satisfied by every finite digital metric space. the following appears as corollary 3.1.6 of [13]. assertion 4.10. let n ∈ n, k ∈ r, k > 1. let s : x → x be a κcontinuous onto function of a complete digital metric space (x, d, κ) such that d(sn(x), sn(y)) ≥ kd(x, y) for all x, y ∈ x. then s has a unique fixed point. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 280 remarks on fixed point assertions in digital topology, 4 remarks 4.11. theorem 4.9 shows that assertion 4.10 is vacuous for finite digital metric spaces, since s being onto implies sn is onto. similarly, assertion 4.10 is vacuous whenever κ = c1, since x ↔c1 y implies sn(x) !c1 sn(y), hence d(sn(x), sn(y)) ≤ d(x, y). we get corollary 4.12 below by modifying assertion 4.10 to consider contractive rather than expansive functions, making the following changes. • we do not require s to be either continuous or onto, nor do we require completeness. • we use k ∈ (0, 1) rather than k > 1. • we use d(sn(x), sn(y)) ≤ kd(x, y) instead of d(sn(x), sn(y)) ≥ kd(x, y). corollary 4.12. let n ∈ n and let k ∈ (0, 1). let s : x → x for a metric space (x, d) such that d(sn(x), sn(y)) ≤ kd(x, y) for all x, y ∈ x, then s has a unique fixed point. proof. take t = 1x. then this assertion follows from proposition 4.6, whose proof in [13] does not use an adjacency and therefore is applicable to metric spaces. □ we modify assumptions of the second bullet of theorem 4.4 to obtain a similar result with a much shorter proof. theorem 4.13. let (x, d, cu) be a digital metric space, where d is an ℓp metric and x is cu-connected. let t ∈ c(x, cu). suppose we have a function s : x → x such that s commutes with t, s(x) ⊂ t(x), and for some α ∈ (0, 1/u1/p) and all x, y ∈ x, d(s(x), s(y)) ≤ αd(t(x), t(y)). then s is constant, and s and t have a unique common fixed point. proof. it follows from proposition 2.8 that s is a constant function. since s(x) ⊂ t(x), the value x0 taken by s is a member of t(x), and since s commutes with t , t(x0) = t(s(x0)) = s(t(x0)) = x0 = s(x0). since s is constant, x0 is a unique common fixed point. □ the following is theorem 3.2.3 of [13]. theorem 4.14. let t be a function on a complete digital metric space (x, d, κ) into itself. then t has a fixed point in x if and only if there exists α ∈ (0, 1) and a function s : x → x that commutes weakly with t and satisfies (3.7). indeed t and s have a unique common fixed point if (3.7) holds. however, theorem 4.14 is limited by proposition 2.8, which gives conditions implying that the function s must be constant. 5. commuting functions the paper [7] studies common fixed points for commuting functions. the following appears as theorem 3.2 of [7]. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 281 l. boxer assertion 5.1. let ∅ ∕= x ⊂ zn, n ∈ n, and let s and t be commuting functions of a complete digital metric space (x, d, κ) into itself such that (i) t(x) ⊂ s(x); (ii) s ∈ c(x, κ); and (iii) for some α ∈ (0, 1) and all x, y ∈ x, d(t(x), t(y)) ≤ αd(s(x), s(y)). then s and t have a common fixed point in x. remarks 5.2. the argument given as proof for assertion 5.1 in [7] claims that (ii) and (iii) imply t ∈ c(x, κ), but this is incorrect (another instance of confusing topological and digital continuity), as shown in example 4.5. however, if we add to assertion 5.1 the hypothesis that (x, d) is uniformly discrete, then we can delete the continuity assumption and get the following corrected version of assertion 5.1. theorem 5.3. let ∅ ∕= x ⊂ zn, n ∈ n. let s and t be commuting functions of a uniformly discrete metric space (x, d) into itself such that (i) t(x) ⊂ s(x); and (ii) for some α ∈ (0, 1) and all x, y ∈ x, d(t(x), t(y)) ≤ αd(s(x), s(y)). then s and t have a unique common fixed point in x. proof. by proposition 3.2, s and t are compatible. the result follows from theorem 3.24. □ the following is presented as corollary 3.3 of [7]. assertion 5.4. let s and t be commuting functions of a complete digital metric space (x, d, κ) into itself such that (i) t(x) ⊂ s(x); (ii) s ∈ c(x, κ); and (iii) for some α ∈ (0, 1) and k ∈ n we have d(t k(x), t k(y)) ≤ αd(s(x), s(y)) for all x, y ∈ x. then s and t have a unique common fixed point. however, the argument given in [7] for this assertion depends on assertion 5.1, shown above as unproven. assertion 5.4 can be modified as follows. corollary 5.5. let s and t be commuting functions of a metric space (x, d) into itself such that (i) t(x) ⊂ s(x); and (ii) for some α ∈ (0, 1) and k ∈ n we have d(t k(x), t k(y)) ≤ αd(s(x), s(y)) for all x, y ∈ x. if (x, d) is uniformly discrete, then s and t have a unique common fixed point. proof. we use the analogous argument of [7]. clearly, t k commutes with s and t k(x) ⊂ t(x) ⊂ s(x). from theorem 5.3, there is a unique a ∈ x such that a = s(a) = t k(a). by applying the function t and the commuting property, we have t(a) = t(s(a)) = s(t(a)) and t(a) = t(t k(a)) = t k(t(a)), c© agt, upv, 2020 appl. gen. topol. 21, no. 2 282 remarks on fixed point assertions in digital topology, 4 so t(a) is a common fixed point of s and t k. but a is the unique common fixed point of s and t k, so we must have a = t(a), and we have already observed that a = s(a), so a is a common fixed point of s and t . to show the uniqueness of a as a common fixed point, suppose x, y are common fixed points of s and t . from hypothesis (ii), d(x, y) = d(t(x), t(y)) = d(t k(x), t k(y)) ≤ αd(s(x), s(y)) = αd(x, y). since 0 < α < 1, it follows that x = y. □ remarks 5.6. note that theorem 5.3 and corollary 5.5 are limited by proposition 2.8. 6. further remarks we have discussed assertions that appeared in [11, 13, 7]. we have discussed errors or corrections for some, shown some to be limited or trivial, and offered improvements for some. acknowledgements. the suggestions and corrections of an anonymous reviewer are gratefully acknowledged. references [1] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. [2] l. boxer, remarks on fixed point assertions in digital topology, 2, applied general topology 20, no. 1 (2019), 155–175. [3] l. boxer, remarks on fixed point assertions in digital topology, 3, applied general topology 20, no. 2 (2019), 349–361. [4] l. boxer and p. c. staecker, remarks on fixed point assertions in digital topology, applied general topology 20, no. 1 (2019), 135–153. [5] s. dalal, common fixed point results for weakly compatible map in digital metric spaces, scholars journal of physics, mathematics and statistics 4, no. 4 (2017), 196–201. [6] s. dalal, i. a. masmali, and g. y. alhamzi, common fixed point results for compatible map in digital metric space, advances in pure mathematics 8 (2018), 362–371. [7] o. ege, d. jain, s. kumar, c. park and d. y. shin, commuting and compatible mappings in digital metric spaces, journal of fixed point theory and applications 22, no. 5 (2020). [8] o. ege and i. karaca, digital homotopy fixed point theory, comptes rendus mathematique 353, no. 11 (2015), 1029–1033. [9] s.-e. han, banach fixed point theorem from the viewpoint of digital topology, journal of nonlinear science and applications 9 (2016), 895–905. [10] k. jyoti and a. rani, fixed point theorems for β ψ φ-expansive type mappings in digital metric spaces, asian journal of mathematics and computer research 24, no. 2 (2018), 56–66. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 283 l. boxer [11] k. jyoti, a. rani, and a. rani, common fixed point theorems for compatible and weakly compatible maps satisfying e.a. and clr(t) property in digital metric space, ijamaa 13, no. 1 (2017), 117–128. [12] h. k. pathak and m. s. khan, compatible mappings of type (b) and common fixed point theorems of gregus type, czechoslovak math. j. 45 (1995), 685–698. [13] a. rani, k. jyoti, and a. rani, common fixed point theorems in digital metric spaces, international journal of scientific & engineering research 7, no. 12 (2016), 1704–1716. [14] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 284 @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 39-49 aspects of rg-spaces f. abohalfya ∗ and r. raphael † abstract a tychonoff space x which satisfies the property that g(x) = c(xδ) is called an rg-space, where g(x) is the minimal regular ring extension of c(x) inside f(x), the ring of all functions from x to r, and xδ is the topology on x generated by its gδ-sets. we correct an error that we found in the proof of [19, theorem 3.4] and show that rg-spaces must satisfy a finite dimensional condition. we also introduce a new class of topological spaces which we call almost k-baire spaces. the class of almost baire spaces is a particular instance. we show that every rg-space is an almost baire space but not necessarily a baire space. however rg-spaces of countable pseudocharacter must be baire and, furthermore, their dense sets have dense interiors. 2010 msc: primary 54g10; secondary 46e25, 16e50 keywords: almost baire spaces, rg-spaces, blumberg spaces, almost resolvable spaces, spaces of countable pseudocharacter, prime zideal, p-space, almost-p space 1. introduction let x be a tychonoff space, let c(x) be the ring of real-valued continuous functions defined on x, and let f(x) be the ring of all real-valued functions defined on x. it is clear that both of these are commutative semiprime rings sharing the same identity. moreover the ring f(x) is regular (in the sense of von neumann). the (unique) smallest regular ring lying between c(x) and f(x), denoted g(x), was studied intensively in [19]. for any function f ∈ f(x), the quasi-inverse of f is given by: ∗supported by the cultural section of the libyan embassy in canada. †supported by the nserc of canada. 40 f. abohalfya and r. raphael f∗(x) = { 0 if x ∈ z(f) 1 f(x) if x ∈ coz(f) where z(f) = {x : f(x) = 0} and coz(f) = x − z(f). a subset of x is called a zero-set (cozero-set) if it has the form z(f) (coz(f)) for some function f ∈ c(x). the set of all zero-sets in x is denoted z(x). for a topological space x, a point p is called a p-point if p is in the interior of each zero-set containing it. a topological space x is called a p-space if every point in x is a p-point [10, 4l]. a space x is a p-space if and only if c(x) is a regular ring, or equivalently if every gδ-set is open [10, 4j]. a space x is called an almost pspace if every non-empty zero-set has a non-empty interior. details on almost p-spaces appear in [15]. algebraic background, for example on regular rings and quasi-inverses, can be found in [13]. 2. rg-spaces if (x, τ) is a topological space then the family of its gδ-sets forms an open base for a (potentially) stronger topology on x known in the literature as the δ-topology. it is denoted τδ and the new space is written as xδ. in [12] it was shown that g(x) = { ∑ni=1 fig∗i : fi, gi ∈ c(x), n ≥ 1}, so each function in g(x) is continuous in the δ-topology. thus g(x) contains c(x) and is a subring of c(xδ). definition 2.1 ([12]). let x be a topological space. then x is called a regular good space, denoted an rg-space, if g(x) = c(xδ). it is clear from the definition of rg-spaces that every p-space is an rgspace because if x is a p-space, then g(x) = c(x) = c(xδ). there are many examples and non-examples of rg-spaces in the literature, for example in [12] and [19]. interestingly, for any space x, whether rg or not, any function in g(x) is continuous on a dense open subset of x [12]. 3. a theorem revisited and repaired we recall from [10] that a prime ideal p is called a z-ideal if a ∈ p whenever b ∈ p and z(a) = z(b). definition 3.1. by the (krull) z-dimension of a maximal ideal we mean the supremum of the lengths of chains of prime z-ideals lying in it. the krull z-dimension of c(x) is the supremum of the dimensions of the maximal ideals of c(x). our first goal in this note is to revisit [19, theorem 3.4]. for completeness let us recall the result that was claimed. theorem 3.2. if the krull z-dimension of c(x) is infinite then x is not an rg space and rg(x) = ∞. aspects of rg-spaces 41 regrettably, the proof given in [19] is mistaken. the assertion in the last paragraph of the proof that clbk,t contains qk,t and no other prime from the array is not justified because a countably infinite operation is used in defining bk,t. below we give a correct proof of the result. (the method) we use the following idea. a space x can be shown not to be rg if one can apply the following technique to its set of prime z-ideals. by “the method” we mean the selection of a countably infinite array of prime z-ideals belonging to disjoint chains, say dn, of finite but globally unbounded lengths. one also needs a countably infinite family of disjoint clopen sets in xδ such that each clopen set contains precisely one chain from the array in its β(xδ)-closure. then an appropriate function is defined using the method of [5, theorem 3.1 part (3)]. this is done by assigning fixed values (taken from c(x)) on each clopen set, and letting it be zero elsewhere. the constants chosen come from the (proper) containment of a prime in its successor in its chain. by virtue of the topology on xδ the global function thus defined will be in c(xδ) but it will not lie in g(x) because of the unbounded nature of the lengths of the chains dn. lemma 3.3. suppose the following three conditions hold for a space x. 1. x is an rg-space, 2. c(x) has a chain of prime z-ideals of length n, and 3. there is a subpace yδ that is clopen in xδ such that all of the primes in the chain lie in the closure of yδ in β(xδ). then there is a function h ∈ g(x) that vanishes on x − y and has regularity degree at least n. proof. construct a function k as in [19, case 1, page 80]. then get h by multiplying k by the idempotent that is 1 on y and 0 and x − y . the idempotent is in g(x) = c(xδ) so the product is as well. � lemma 3.4. suppose that x is of infinite prime krull z-dimension. then x contains a family of pairwise disjoint chains dn indexed by n, so that the length of the chain dn is equal to n. proof. this is done by an easy induction. the case n = 1 is clear. suppose we have chosen disjoint chains d1, ....dn. since the lengths of the chains is unbounded, there is one of length at least n(n + 1)/2 + (n + 1). even if this chain has some overlap with the chains d1, ..., dn there must be n + 1 primes in it that do not occur in the previous chains. these n + 1 primes give us a new chain dn+1 disjoint from all of the previous chains. � lemma 3.5. assume, if possible, that x is an rg-space of infinite krull zdimension. then x contains a countably infinite family of disjoint subspaces yn, each clopen in the δ-topology such that each yn supports a function hn ∈ c(xδ) that vanishes outside of yn and has regularity degree at least n. proof. since x is of infinite dimension lemma 3.4 applies. let dn have the same meaning as in lemma 3.4. 42 f. abohalfya and r. raphael we construct the disjoint clopen sets inductively as follows. first step. the sets d1 and d2 are finite and disjoint in the boolean space t = spec(xδ) = β(xδ) so they can be separated by complementary clopen sets u and t − u, say d1 ⊂ u, d2 ⊂ t − u. the subspaces a = u ∩ xδ and b = (t − u) ∩ xδ are clopen subsets of xδ so the union of their t -closures is t . they are also rg-spaces in the (relative) x-topology by [12, theorem 2.3(a)]. we know that d1 is in the t -closure of a, d2 is in the t -closure of b, and that the other dn lie wholely or partly in the t -closures of a and b and certainly in the union of the latter. although we know that a and b are rgspaces with union x we do not assert that (at least) one of them is of infinite dimension. this would hold if we knew that x were the free union of a and b by [16, prop 4.7] but we do not know this. fortunately we only want to be able to apply lemma 3.3. observe that one (or both) of a and b is “infinite” with respect to x in the following sense: it is not possible that both a and b have a finite bound on the cardinalities of their intersections with the chains d3, d4, ..... if it is b that has this property then we let y1 = a and continue to the next step in which we will be working inside b. if, on the other hand, there is a global finite bound on the cardinalities of the intersections of the chains d3, d4, ... with b then we let y1 = b in the knowledge that b has a chain of length 2 and therefore one of length 1, and we continue with a. in both cases y1 is clopen in the δ-topology, has a chain of length 1 in its t -closure, and has a complement x − y1 with an “infinite” property. second step. we work with the space x − y1 which has disjoint chains of all finite lengths in its t -closure, in particular, one called s2 of length 2 and one of length 3 called s3. working in the t -closure, again, find a clopen subspace w of t − clt (y1) that separates a2 and a3, say a2 ⊂ w, a3 ⊂ t −clt (y1)−w . if the t -closure of ((t − y1) − w) ∩ x has the “infinite property” we let y2 = w ∩ xδ. if not, we let y2 = ((t − y1) − w) ∩ xδ. in both cases y2 is clopen in xδ, has a chain of length 2 in its t -closure, and (x − y1) − y2 has the ”infinite” property. now one simply continues the process to get the sequence yn. � we can now present our main result of this section. theorem 3.6. if x is an rg-space then x is of finite krull z-dimension. proof. let yn denote the disjoint subsets given by lemma 3.5. each is clopen in xδ. let y = ∪yn. by standard properties of p-spaces y is also clopen. for each n let hn be the function provided by lemma 3.3 on y . let h be defined on x by, h = hn on yn for each n, and h is zero on x − y . then h ∈ c(xδ) and h /∈ g(x) by the argument of [19, theorem 3.1] (see also [19, theorem 3.4)]. remark 3.7. as was shown in [1, p82] the spectrum of the epimorphic hull is the set of prime d-ideals of c(x) under the patch topology. exactly the same arguments show that if dimension is measured using prime d-ideals, then aspects of rg-spaces 43 an infinite dimension is inconsistent with having h(x) be a ring of continuous functions when x is realcompact. note as well that an infinite dimension when measured using prime d-ideals will also prevent x from being an rg space. remark 3.8. notice that the proof of lemma 3.5 contains the following fact of interest. if x is rg, and a is a subset of x which is clopen in x in the δ-topology, then a is also an rg-space. this is simply an application of [12, theorem 2.3(a)]. there is no immediate converse however. the space ψ is known not to be rg. it is discrete in the δ-topology, both n and ψ − n are discrete, and therefore rg, and the idempotent function on ψ which is 1 on n and 0 on ψ − n is certainly in g(ψ). a converse can be achieved with lifting properties, like the c∗-embeddedness of a and its complement. we can get by with a bit less, as follows. proposition 3.9. let x be the union of two disjoint subspaces a and b, suppose a and b are rg-spaces which are g-embedded in x and suppose that the function e that is 1 on a and 0 on b lies in g(x). then x is an rg-space. proof. the proof is straightforward. take f ∈ c(xδ). it suffices to see that both ef and (1 − e)f lie in g(x). let us check the result for ef. the function f|a lies in c(aδ) and therefore in g(a). that means it lifts to a function h ∈ g(x). now eh ∈ g(x) and coincides with ef. similarly (1 − e)f coincides with (1 − e)k for some k ∈ g(x) � corollary 3.10. suppose that a space x is the union of two disjoint lindelof subspaces a and b which are rg one of which is open. suppose further, that the characteristic function of a lies in g(x). then x is an rg-space. proof. suppose that a is the open subspace. since it is lindelof, it is a cozero set of x and is therefore g-embedded in x. the space b is closed in the normal space x and is therefore c-embedded, hence g-embedded in x as well. � 4. nowhere separable spaces the following theorem can be compared with [12, 2.2]. theorem 4.1. let x be an rg-space and (zn) ∞ n=1 be a sequence of nowhere dense zero-sets in x. then ⋃∞ n=1 zn is a nowhere dense subset. proof. let s = ⋃∞ n=1 zn, a1 = z1 and am = zm − ( ⋃m−1 i=1 zi) for each m ≥ 2. then by well-known properties of p-spaces, {an : n ∈ n} is a collection of clopen subsets in xδ, and therefore {an : n ∈ n} ∪ {x − s} is a clopen partition of xδ. let f : xδ −→ r be defined by f(an) = {n + 1} for each n ∈ n and f(x − s) = {1}. then f ∈ g(x), and since x is an rg-space there is a dense open subset d of x on which f is continuous. now suppose that cl( ⋃∞ n=1 zn) has an interior point p. then there is an open subset up containing p such that up ⊆ cl( ⋃∞ n=1 zn), which means that for each y in up and each neighborhood wy of y we have wy ∩ ( ⋃∞ n=1 zn) �= φ. since d is a dense subset, then d ∩ up �= φ. let y ∈ d ∩ up. there are two cases: 44 f. abohalfya and r. raphael (1) if f(y) = 1, then there is an open neighborhood wy of y such that f(wy) ⊆ (0, 3 2 ). so wy ∩ ( ⋃∞ n=1 zn) = φ, which is a contradiction. (2) if f(y) = k + 1, then y ∈ ak, and therefore there is an open neighborhood wy of y such that f(wy) ⊆ (k + 23, k + 43). then wy ⊆ ak ⊆ zk, which is also a contradiction. thus ⋃∞ n=1 zn is a nowhere dense subset of x. � recall that a topological space x is called separable at a point p if there exists an open set o containing p such that o is separable. a topological space x is called nowhere separable if x is not separable at any of its points. details appear in [6]. it is an open question whether an rg-space must have almost p-points. there certainly are separable rg-spaces, even countable ones and these have isolated points. one does have the following implication. theorem 4.2. if x is an rg-space with no almost p-points then x is nowhere separable. proof. let x be an rg-space with no almost p-points that is separable at some point. then there is a countable subset {an : n ∈ n} such that int(cl( ⋃∞ n=1{an})) �= φ. for each an pick a nowhere dense zero-set zn such that an ∈ zn. then int(cl( ⋃∞ n=1 zn)) �= φ which contradicts theorem 4.1. thus x is a nowhere separable. � 5. spaces of countable pseudocharacter and blumberg spaces definition 5.1. a topological space x is said to be of countable pseudocharacter if every point in x is a gδ-set. (this is equivalent to saying that xδ is discrete.) recall that a topological space x is called blumberg if every real-valued function defined on x has a continuous restriction to a dense subset [22]. in [4] j. c. bradford and c. goffman proved that every blumberg space is baire and in [14] r. levy showed that there is consistently a compact hausdorff, and therefore baire, space which is not blumberg. as noted below an rg-space need not be baire and hence need not be blumberg. hoever, we can obtain the blumberg property for one particular class of rg-spaces as follows. theorem 5.2. let x be an rg-space of countable pseudocharacter. then x is blumberg and hence also baire. proof. suppose x is an rg-space of countable pseudocharacter. since xδ is a discrete space, every function on x is in g(x), and thus by [12, prop. 2.1] every real-valued function defined on x can be restricted continuously to a dense open subset. so x satisfies the blumberg property and hence x is a baire space. � definition 5.3 (cf [3]). let x be a topological space. then x is called an almost resolvable space if it is a countable union of sets with void interiors. aspects of rg-spaces 45 theorem 5.4. if x is an rg-space of countable pseudocharacter then x is not an almost resolvable space. proof. let x be an rg-space of countable pseudocharacter and suppose x is almost resolvable. then there is a countable collection {fn : n ∈ n} of sets each with void interior such that x = ⋃∞ n=1 fn. let a1 = f1 and an = fn −( ⋃n−1 m=1 fm) for each n ≥ 2. then {an : n ∈ n} is a a countable collection of disjoint sets with void interior and x = ⋃∞ n=1 an. define f : x −→ r by f(an) = n for each n ∈ n. since xδ is a discrete space, then f ∈ c(xδ) = g(x), which implies that f is continuous on a dense open subset d, which is a contradiction because f is not continuous at any point. thus x is not almost resolvable. � we know from theorem 5.2 that rg-spaces of countable pseudocharacter are baire. in fact, one can do a bit better as follows. lemma 5.5. if x is an rg-space of countable pseudocharacter then every countable union of nowhere dense subsets is nowhere dense. proof. let x be an rg-space and (an) ∞ n=1 be a sequence of nowhere dense subsets of x. let s = ⋃∞ n=1 an, f1 = a1 and fm = am − ( ⋃m−1 i=1 ai) for each m ≥ 2. then {fn : n ∈ n} is a collection of disjoint nowhere dense subsets of x, and therefore {fn : n ∈ n} ∪ {x − s} is a partition of x. now define f : xδ −→ r by f(fn) = {n + 1} for each n ∈ n and f(x − s) = {1}. then f ∈ c(xδ) = g(x), which implies that there is a dense open subset d of x such that f|d is a continuous function. suppose cl(⋃∞n=1 fn) has an interior point p. then there is an open subset up containing p such that up ⊆ cl( ⋃∞ n=1 fn), that is ∀ y ∈ up and for each neighborhood wy of y we have wy ∩ ( ⋃∞ n=1 fn) �= φ. since d ∩ up �= φ, let y be any point in ∈ d ∩ up. again we have two cases: (1) if f(y) = 1, then there is an open neighborhood wy of y such that f(wy) ⊆ (0, 3 2 ). hence wy ∩ ( ⋃∞ n=1 fn) = φ, which is a contradiction. (2) if f(y) = k + 1, then y ∈ fk. so there is an open neighborhood wy of y such that f(wy) ⊆ (k + 23, k + 43), and therefore wy ⊆ fk ⊆ ak, which is a contradiction too. thus ⋃∞ n=1 an is a nowhere dense subset of x. � a topological space x can have a dense subset k such that kc is somewhere dense or even a dense subset. this is will be a relevant point for rg-spaces. lemma 5.6. let x be a topological space. then x is either has the property that every dense subset has a nowhere dense complement or x has a resolvable cozero subspace. proof. suppose d is a dense subset such that dc is somewhere dense. then there is a non-empty cozero subset u such that u ⊆ cl(dc). let a = d ∩ u and b = dc ∩ u. then a and b are disjoint dense subsets of u. hence u is a resolvable cozero subspace. � 46 f. abohalfya and r. raphael theorem 5.7. let x be an rg-space of countable pseudocharacter. then (1) every dense subset of x has a nowhere dense complement. (2) every countable intersection of dense sets has a dense interior. (3) in particular, every dense set has a dense interior. proof. (1) since every cozero subset of x is an rg-space of countable pseudocharacter then it cannot be resolvable. thus the result follows directly by lemma 5.6. (2) let (dn) ∞ n=1 be a sequence of dense subsets of x. let an = x − dn for each n ∈ n. then (an)∞n=1 is a sequence of nowhere subsets of x, which implies that ⋃∞ n=1 an is a nowhere dense subset of x. hence the result follows directly from the fact that cl(t ) c = int(t c) for any subset t of x. (3) this follows directly from (2). � recall that a topological space x is called open hereditarily irresolvable (or simply o.h.i) if each open subspace of x is irresolvable [7, def 1.2]. in [9] ganster proved that a topological space x is open hereditarily irresolvable if and only if every dense set of x has a dense interior. since every rg-space of countable pseudocharacter is open hereditarily irresolvable, one also deduce part (3) of theorem 5.7 from an understanding of ganster’s work. before finishing this section, we use work of tamariz and villegas to prove that under the assumption v = l, every rg-space of countable pseudocharacter has a dense set of isolated points. this matters because the presence of even almost p-points in rg-spaces is an open question. first we recall proposition 4.10 of [21] which says that, assuming v = l, every space without isolated points is almost resolvable. theorem 5.8. assume v = l. then every rg-space of countable pseudocharacter is scattered. proof. since every cozero subspace of x is also an rg-space of countable pseudocharacter, then by proposition 4.10 of [21] and theorem 5.4, every cozero set has an isolated point. it follows immediately that x has a dense set of isolated points. now if y is any subspace of x then y is also of countable pseudocharacter, and since it is c-embedded in x in the δ-topology y is rg itself by [12, theorem 2.3(a)]. thus y has a dense set of isolated points by the first part of the proof, which means that x is scattered. � 6. almost baire spaces recall that a topological space x is called k-baire, where k is a fixed cardinal number, if the intersection of fewer than k dense open sets is dense [20]. thus the usual baire spaces are ℵ1-baire spaces. it is clear that the intersection of all dense open subset of x is a dense subset if and only if x has a dense subset of isolated points, which means that if x has a dense subset of isolated points then x is a k-baire space for any cardinal number k. thus every scattered space is a k-baire space for any cardinal number k. in a general tychonoff space an open subset need not be a cozero-set, and the aspects of rg-spaces 47 collection of all dense cozero-sets can have any cardinality. for these reasons we will introduce the following class of spaces. definition 6.1. let x be a topological space and k be a cardinal number. then we will call x an almost k-baire space if any collection of fewer than k dense cozero-sets has a dense intersection. we will call x almost-baire if x is an almost ℵ1-baire space. if x is a topological space and k is a fixed cardinal number, then x is almost k-baire if and only if the union of fewer than k nowhere dense zero-sets has an empty interior. it is clear that every clopen subspace of almost k-baire space is an almost k-baire space, and a space x is almost k-baire if and only if x has a dense subspace which is almost k-baire. every k-baire space is an almost k-baire space, but the converse is not true in general as we will see next. if x is an rg-space then it is clear from theorem 4.1 that every countable intersection of dense cozero subsets of x has a dense interior. recall that a space x is an almost p-space if and only if every non-empty countable intersection of open sets has a non-empty interior. it is clear that every almost p-space is almost k-baire for each cardinal number k. corollary 6.2. every rg-space is an almost-baire space. proof. this follows directly from theorem 4.1. � rg-spaces need not be baire. in [8] the authors gave two examples. first they gave a regular p-space without isolated points. secondly they gave an example of a tychonoff space x with a dense set of isolated points such that xδ is not a baire space. thus an rg-space does not have to be baire, and consequently an almost k-baire space need not be a k-baire space. corollary 6.3. the cozero-sets and zero-sets of rg-spaces are almost-baire spaces. definition 6.4 (cf [18]). let x be a topological space. then the subset gx is defined to be the intersection of all dense cozero subsets of x. it is clear that gx is the set of almost p-points in x. if x is an rg-space, then it follows from theorem 4.2 that every countable subset of x − gx is a nowhere dense subset of x. proposition 6.5. let x be a topological space. then: (1) x is almost baire implies that every dense open c∗-embedded subset in x is almost baire. (2) x is almost k-baire for each cardinal number k if and only if gx is dense. proof. (1) let x be an almost baire space, let u be a dense open c∗-embedded subset in x and let vn, n = 1, 2, 3, .... be a collection of dense cozero-sets in u. since u is c∗-embedded in x then for each n, there is a dense cozero-set wn in x such that wn ∩ u = vn. but x is almost baire. therefore ⋂∞ n=1 wn 48 f. abohalfya and r. raphael is a dense subset of x which implies that ⋂∞ n=1 wn ∩ u = ⋂∞ n=1 vn is a dense subset of u. thus u is almost baire. (2) (=⇒) let x be almost k-baire for each cardinal number k. suppose there exists a non-empty open subset u such that u ∩ gx = φ. for each x ∈ u choose a nowhere dense zero-set zx such that x ∈ zx and let vx = x − zx. then vx is a dense cozero-set for each x ∈ u and u ∩ ⋂ x∈u vx = φ, which contradicts the fact that x is an almost k-baire space for each cardinal number k. thus gx is a dense subset of x. (⇐=) this is clear from the fact that gx is contained in every dense cozeroset. � 7. open questions question 7.1 (cf [19]). are all rg-spaces of finite regularity degree? question 7.2 (cf corollary 3.10). if x is the union of two disjoint lindelof rg-subspaces, must x be rg? question 7.3. is the intersection of two rg-spaces rg? what about the case where both of the spaces are lindelof? references [1] f. abohalfya, on rg-spaces and the space of prime d-ideals in c(x), phd thesis, concordia university, (2010). [2] r .l. blair and a. w. hager, extensions of zero-sets and real-valued functions, math. zeit. 136 (1974), 41–52. [3] r. bolstein, sets of points of discontinuity, proc. amer. math. soc. 38 (1960), 193– 197. [4] j. c. bradford and c. goffman, metric spaces in which blumberg’s theorem holds, proc. amer. math. soc. 11 (1960), 667–670. [5] w. d. burgess and r. raphael, the regularity degree and epimorphisms in the category of commutative rings, comm. algebra 29, no. 6 (2001), 2489–2500. [6] g. h. butcher, an extension of the sum theorem of dimension theory, duke math. 18 (1951), 859–874. [7] c. bandyopadhyay and c. chattopadhyay, on resolvable and irresolvable spaces, internat. j. math. sci 16, no. 4 (1993), 657–662. [8] r. fox and r. levy, a baire space with first category gδ -topology, top. proc. 9, no. 2 (1984), 293–295. [9] m. ganster, pre-open sets and resolvable spaces, kyungpook math. j. 27 (1987), 135–143. [10] l. gillman and m. jerison, rings of continuous functions, van nostrand, princeton, 1960. [11] m. henriksen, j. martinez and r.g. woods, spaces x in which all prime z-ideals of c(x) are maximal or minimal, comm. math. univ. carol. 44, no. 2 (2003), 261–294. [12] m. henriksen, r. raphael and r. g. woods, a minimal regular ring extension of c(x), fund. math. 172 (2002), 1–17. [13] j. lambek, lectures on rings and modules, blaisdell, toronto, 1966. [14] r. levy, strongly non-blumberg spaces, general topology and appl. 4 (1974), 173– 177. aspects of rg-spaces 49 [15] r. levy, almost p-spaces, can. j. math. 29 (1977), 284–288. [16] j. martinez and e. zenk, dimension in algebraic frames ii: applications of frames of ideals in c(x), comm. math. univ. carol. 46, no. 4 (2005), 607–636. [17] m. henriksen, r. raphael and r. g. woods, a minimal regular ring extension of c(x), fund. math. 172, no. 1 (2002), 1–17. [18] r. raphael and r. g. woods, the epimorphic hull of c(x), top. appl. 105 (2000), 65–88. [19] r. raphael and r. g. woods, on rg-spaces and the regularity degree, appl. gen. topol. 7 (2006), 72–101. [20] f. d. tall, the countable chain condition versus separability-applications of martin’s axiom, general topology appl. 4 (1974), 315–339. [21] a. tamariz-mascara and h. villegas-rodriguez, spaces of continuous functions, box products, and almost-ω resolvable spaces, comment. math. univ. carolinae 43, no. 4 ( 2002), 687–705. [22] w. a. r. weiss, a solution to the blumberg problem, bull. amer. math. soc. 81 (1975), 957–958. (received january 2011 – accepted september 2011) f. abohalfya mathematics and statistics, concordia university, montréal, canada. r. raphael (raphael@alcor.concordia.ca) mathematics and statistics, concordia university, montréal, canada. aspects of rg-spaces. by f. abohalfya and r. raphael @ appl. gen. topol. 20, no. 1 (2019), 281-295doi:10.4995/agt.2019.11057 c© agt, upv, 2019 existence and convergence results for a class of nonexpansive type mappings in hyperbolic spaces rajendra pant a and rameshwar pandey b a department of pure and applied mathematics, university of johannesburg, auckland park 2006, south africa (pant.rajendra@gmail.com; rpant@uj.ac.za) b department of mathematics, visvesvaraya national institute of technology, nagpur 440010, india (rampandey0502@gmail.com) communicated by i. altun abstract we consider a wider class of nonexpansive type mappings and present some fixed point results for this class of mappings in hyperbolic spaces. indeed, first we obtain some existence results for this class of mappings. next, we present some convergence results for an iteration algorithm for the same class of mappings. some illustrative non-trivial examples have also been discussed. 2010 msc: 47h10; 54h25. keywords: reich-suzuki type nonexpansive mapping; hyperbolic metric space; iteration process; nonexpansive mapping. 1. introduction a mapping p from the set of reals r to a metric space (e, ρ) is said to be metric embedding if ρ(p(m), p(n)) = |m − n| for all m, n ∈ r. the image of set r under a metric embedding is called a metric line. the image of a real interval [a, b] = {t ∈ r : a ≤ t ≤ b} under metric embedding is called a metric segment. assume that (e, ρ) has a family f of metric lines such that for each pair u, v ∈ e (u 6= v) there is a unique metric line in f which passes through u and v. this metric line determines a unique metric segment joining u and received 27 november 2018 – accepted 20 february 2019 http://dx.doi.org/10.4995/agt.2019.11057 r. pat and r. pandey v. this segment is denoted by [u, v] and this is an isometric image of the real interval [0, ρ(u, v)]. we denote by γu ⊕ (1 − γ)v, the unique point w of [u, v] which satisfies ρ(u, w) = (1 − γ)ρ(u, v) and ρ(w, v) = γρ(u, v), where γ ∈ [0, 1]. such a metric space with a family of metric segments is called a convex metric space [37]. further, if we have ρ(γu ⊕ (1 − γ)v, γw ⊕ (1 − γ)z) ≤ γρ(u, w) + (1 − γ)ρ(v, z) for all u, v, w, z ∈ e, then e is said to be a hyperbolic metric space [42]. hyperbolic spaces are more general than normed spaces and cat(0) spaces. these spaces are nonlinear. indeed, all normed linear and cat(0) spaces are hyperbolic spaces (cf. [28, 29, 33]). as nonlinear examples, one can consider the hadamard manifolds [6] and the hilbert open unit ball equipped with the hyperbolic metric [14]. a mapping t : e → e is said to be nonexpansive if ρ(t (u), t (v)) ≤ ρ(u, v), for all u, v ∈ e. f(t ) denotes set of fixed points of t . the study of existence of fixed point of nonexpansive type mappings has been of great interest in nonlinear analysis (cf. [15, 46, 3, 4, 7, 8, 10, 11, 27, 35, 39, 41]). fixed point theory of nonexpansive mappings in hyperbolic spaces has been extensively studied (cf. [47, 5, 38, 13, 16, 42, 31, 32]). in the present paper, we consider a wider class of nonexpansive type mapping which properly generalizes some well-known classes of nonexpansive type mappings in hyperbolic spaces. we present some existence and convergence results. since hyperbolic spaces are more general spaces, our theorems extend, generalize and complement many results in the literature. 2. preliminaries let us recall the following definition which is due to kohlenbach [31]: definition 2.1 ([31]). a triplet (e, ρ, h) is said to be a hyperbolic metric space if (e, ρ) is a metric space and h : e × e × [0, 1] → e is a function such that for all u, v, w, z ∈ e and β, γ ∈ [0, 1], the following hold: (k1) ρ(z, h(u, v, β)) ≤ (1 − β)ρ(z, u) + βρ(z, v); (k2) ρ(h(u, v, β), h(u, v, γ)) = |β − γ|ρ(u, v); (k3) h(u, v, β) = h(v, u, 1 − β); (k4) ρ(h(u, z, β), h(v, w, β)) ≤ (1 − β)ρ(u, v) + βρ(z, w). the set seg[u, v] := {h(u, v, β); β ∈ [0, 1]} is called the metric segment with endpoints u and v. now onwards, we write h(u, v, β) = (1−β)u⊕βv. a subset k of e is said to be convex if (1−β)u⊕βv ∈ k, for all u, v ∈ k and β ∈ [0, 1]. when there is no ambiguity, we write (e, ρ) for (e, ρ, h). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 282 existence and convergence results in hyperbolic spaces definition 2.2 ([12, 20]). let (e, ρ) be a hyperbolic metric space. for any a ∈ e, r > 0 and ǫ > 0. set δ(r, ǫ) = inf { 1 − 1 r ρ ( 1 2 u ⊕ 1 2 v, a ) ; ρ(u, a) ≤ r, ρ(v, a) ≤ r, ρ(u, v) ≥ rǫ } . we say that e is uniformly convex if δ(r, ǫ) > 0, for any r > 0 and ǫ > 0. definition 2.3 ([47, 9]). a hyperbolic metric space (e, ρ) is said to be strictly convex if for any u, v, a ∈ e and α ∈ (0, 1); ρ(αu ⊕ (1 − α)v, a) = ρ(u, a) = ρ(v, a), then we must have u = v. every uniformly convex hyperbolic metric space is strictly convex [9]. definition 2.4 ([19]). a hyperbolic metric space (e, ρ) is said to satisfy property (r) if for each decreasing sequence {fn} of nonempty bounded closed convex subsets of e, ∞ ⋂ n=1 fn 6= ∅. uniformly convex hyperbolic spaces satisfy the property (r), see [5]. definition 2.5 ([45]). let k be a subset of a metric space (e, ρ). a mapping t : k → k is said to satisfy condition (i) if there exists a nondecreasing function g : [0, ∞) → [0, ∞) satisfying g(0) = 0 and g(r) > 0 for all r ∈ (0, ∞) such that ρ(u, t (u)) ≥ g(dist(u, f(t ))) for all u ∈ k, here dist(u, f(t )) denotes the distance of u from f(t ), where f(t ) denotes the set of fixed points of t. definition 2.6. a sequence {un} in k is said to be approximate fixed point sequence (a.f.p.s for short) for a mapping t : k → k if lim n→∞ ρ(t (un), un) = 0. let k be a nonempty subset of a hyperbolic metric space (e, ρ) and {un} a bounded sequence in e. for each u ∈ e, define: • asymptotic radius of {un} at u as r({un}, u) := lim sup n→∞ ρ(un, u); • asymptotic radius of {un} relative to k as r({un}, k) := inf{r({un}, u); u ∈ k}; • asymptotic centre of {un} relative to k by a({un}, k) := {u ∈ k; r({un}, u) = r({un}, k)}. lim in [34] introduced the concept of ∆-convergence in a metric space. kirk and panyanak in [30] used lim’s concept in cat(0) spaces and showed that many banach space results involving weak convergence have precise analogs in this setting. definition 2.7 ([30]). a bounded sequence {un} in e is said to ∆-converge to a point u ∈ e, if u is the unique asymptotic centre of every subsequence {unk} of {un}. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 283 r. pat and r. pandey 3. main results the following definition is essentially due to suzuki [46] in a banach space. definition 3.1. let k be a nonempty subset of a hyperbolic metric space e. a mapping t : k → k is said to satisfy condition (c) if for all u, v ∈ k, (3.1) 1 2 ρ(u, t (u)) ≤ ρ(u, v) implies ρ(t (u), t (v)) ≤ ρ(u, v). now, we consider a wider class of nonexpansive type mappings and present some auxiliary and existence results. we also discuss an illustrative example. definition 3.2. let k be a nonempty subset of a hyperbolic metric space e. a mapping t : k → k is said to be reich-suzuki type nonexpansive mappings if there exists a k ∈ [0, 1) such that 1 2 ρ(u, t (u)) ≤ ρ(u, v) implies ρ(t (u), t (v)) ≤ kρ(t (u), u) + kρ(t (v), v) + (1 − 2k)ρ(u, v)(3.2) for all u, v ∈ k. now, we present some basic properties of the above class of mappings. proposition 3.3. a mapping satisfying the condition (c) is a reich-suzuki type nonexpansive mapping but the converse need not be true. proof. when k = 0, then t is a mapping satisfying the condition (c). � the following example shows that the converse need not be true: example 3.4. let k = [−2, 2] be a subset of r endowed with the usual metric, that is, ρ(u, v) = |u − v|. define t : k → k by t (u) =        − u 2 , if u ∈ [−2, 0)\{−1 8 } 0, if u = −1 8 − u 3 , if u ∈ [0, 2]. then for u = −1 8 and v = −1 5 , we have 1 2 ρ(u, t (u)) = 1 16 ≤ 3 40 = ρ(u, v), but ρ(t (u), t (v)) = 1 10 > 3 40 = ρ(u, v). thus t does not satisfy condition (c). now we show that t is a reich-suzuki type nonexpansive mapping with k = 1 2 . we consider different cases as follows: (i) let u, v ∈ [−2, 0)\{−1 8 }; we have ρ(t (u), t (v)) = 1 2 |v − u| ≤ 1 2 |u| + 1 2 |v| ≤ 3 4 |u| + 3 4 |v| = 1 2 ∣ ∣ ∣ u + u 2 ∣ ∣ ∣ + 1 2 ∣ ∣ ∣ v + v 2 ∣ ∣ ∣ = k(ρ(u, t (u)) + ρ(v, t (v))) + (1 − 2k)ρ(u, v). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 284 existence and convergence results in hyperbolic spaces (ii) let u, v ∈ [0, 2]; we have ρ(t (u), t (v)) = 1 3 |u − v| ≤ 1 3 |u| + 1 3 |v| ≤ 2 3 |u| + 2 3 |v| = 1 2 ∣ ∣ ∣ u + u 3 ∣ ∣ ∣ + 1 2 ∣ ∣ ∣ v + v 3 ∣ ∣ ∣ = k(ρ(u, t (u)) + ρ(v, t (v))) + (1 − 2k)ρ(u, v). (iii) let u ∈ [−2, 0)\{−1 8 } and v ∈ [0, 2]; we have ρ(t (u), t (v)) = | v 3 − u 2 | ≤ 1 2 |u| + 1 3 |v| ≤ 3 4 |u| + 2 3 |v| = 1 2 ∣ ∣ ∣ u + u 2 ∣ ∣ ∣ + 1 2 ∣ ∣ ∣ v + v 3 ∣ ∣ ∣ = k(ρ(u, t (u)) + ρ(v, t (v))) +(1 − 2k)ρ(u, v). (iv) let u ∈ [−2, 0)\{−1 8 } and v = −1 8 ; we have ρ(t (u), t (v)) = 1 2 |u| ≤ 3 4 |u| + 1 16 = 1 2 ρ(t (u), u) + 1 2 ρ(t (v), v) = k(ρ(u, t (u)) + ρ(v, t (v))) + (1 − 2k)ρ(u, v). (v) let u ∈ [0, 2] and v = −1 8 ; we have ρ(t (u), t (v)) = 1 3 |u| ≤ 2 3 |u| + 1 16 = 1 2 ρ(t (u), u) + 1 2 ρ(t (v), v) = k(ρ(u, t (u)) + ρ(v, t (v))) + (1 − 2k)ρ(u, v). thus t is a reich-suzuki type nonexpansive mapping with only fixed point 0. notice that the space considered in the above example was a linear space. now we present an example of a hyperbolic space which is not linear. therefore it is a non-trivial example of a hyperbolic space. example 3.5 (see also [17]). let e = {(u1, u2) ∈ r 2; u1, u2 > 0}. define ρ : e × e → [0, ∞) by ρ(u, v) = |u1 − v1| + |u1u2 − v1v2| for all u = (u1, u2) and v = (v1, v2) in e. then it can be easily seen that (e, ρ) is a metric space. now for β ∈ [0, 1], define a function h : e × e × [0, 1] → e by h(u, v, β) = ( (1 − β)u1 + βv1, (1 − β)u1u2 + βv1v2 (1 − β)u1 + βv1 ) . then (e, ρ, h) is a hyperbolic metric space. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 285 r. pat and r. pandey further, suppose k := [1 2 , 3]×[1 2 , 3] ⊂ e and t : k → k be a mapping defined by t (u1, u2) = { (1, 1), if (u1, u2) 6= ( 1 2 , 1 2 ) (3 2 , 3 2 ), if (u1, u2) = ( 1 2 , 1 2 ). then t is a reich-suzuki type nonexpansive mapping which does not satisfy the condition (c). proof. first, we show that (e, ρ, h) is a hyperbolic metric space. for u = (u1, u2), v = (v1, v2), z = (z1, z2) and w = (w1, w2) in e : (k1) ρ(z, h(u, v, β)) = |z1 − (1 − β)u1 − βv1| + |z1z2 − (1 − β)u1u2 − βv1v2| ≤ (1 − β)|z1 − u1| + β|z1 − v1| + (1 − β)|z1z2 − u1u2| + β|z1z2 − v1v2| = (1 − β)ρ(z, u) + βρ(z, v). (k2) ρ(h(u, v, β), h(u, v, γ)) = |(1 − β)u1 + βv1 − (1 − γ)u1 − γv1| + |(1 − β)u1u2 + βv1v2 − (1 − γ)u1u2 − γv1v2| = |β − γ|(|u1 − v1| + |u1u2 − v1v2|) = |β − γ|ρ(u, v). (k3) h(u, v, β) = ( (1 − β)u1 + βv1, (1 − β)u1u2 + βv1v2 (1 − β)u1 + βv1 ) = ( βv1 + (1 − β)u1, βv1v2 + (1 − β)u1u2 βv1 + (1 − β)u1 ) = h(v, u, 1 − β). (k4) ρ(h(u, z, β), h(v, w, β)) = |(1 − β)u1 + βz1 − (1 − β)v1 − βw1| + |(1 − β)u1u2 + βz1z2 − (1 − β)v1v2 − βw1w2| ≤ (1 − β)(|u1 − v1| + |u1u2 − v1v2|) + β(|z1 − w1| + |z1z2 − w1w2|) = (1 − β)ρ(u, v) + βρ(z, w). therefore, (e, ρ, h) is a hyperbolic metric space but not a normed linear space. next, we show that t does not satisfies condition (c) on k. let u = (1 2 , 1 2 ) and v = (11 10 , 11 10 ). then 1 2 ρ(u, t (u)) = 3 2 ≤ 156 100 = ρ(u, v), but ρ(t (u), t (v)) = 7 4 > 156 100 = ρ(u, v). finally, we show that t is reich-suzuki type nonexpansive mapping for k = 1 2 . we consider the following cases: case (i) if u = (u1, u2), v = (v1, v2) 6= ( 1 2 , 1 2 ), then ρ(t (u), t (v)) = 0 ≤ 1 2 (ρ(u, t (u)) + ρ(v, t (v))). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 286 existence and convergence results in hyperbolic spaces case (ii) if u = (u1, u2) 6= ( 1 2 , 1 2 ) and v = (v1, v2) = ( 1 2 , 1 2 ), then 1 2 (ρ(u, t (u)) + ρ(v, t (v))) = 1 2 [|u1 − 1| + |u1u2 − 1|] + 1 2 [ 1 + 10 4 ] = 1 2 [|u1 − 1| + |u1u2 − 1|] + 14 8 ≥ 7 4 = ρ(t (u), t (v)). therefore t is a reich-suzuki type nonexpansive mapping with only fixed point (1, 1). � proof of the following proposition and lemma may be completed on the pattern of [46]. proposition 3.6. let k be a nonempty subset of a hyperbolic metric space e and t : k → k is a reich-suzuki type nonexpansive mapping with a fixed point z ∈ k. then t is quasi-nonexpansive. the following lemma gives the structure of fixed point set for a reich-suzuki type nonexpansive mapping. lemma 3.7. let k be a nonempty subset of a hyperbolic metric space e and t : k → k is a reich-suzuki type nonexpansive mapping. then f(t ) is closed. moreover, if e is strictly convex and k is convex, then f(t ) is convex. the following lemmas will be useful to prove main results of this section, which are modeled on the pattern of [46] and can be proved easily. therefore proof are omitted. lemma 3.8. let k be a nonempty subset of a hyperbolic metric space e and t : k → k is a reich-suzuki type nonexpansive mapping. then for each u, v ∈ k, (i) ρ(t (u), t 2(u)) ≤ ρ(u, t (u)); (ii) either 1 2 ρ(u, t (u)) ≤ ρ(u, v) or 1 2 ρ(t (u), t 2(u)) ≤ ρ(t (u), v); (iii) either ρ(t (u), t (v)) ≤ kρ(t (u), u) + kρ(v, t (v)) + (1 − 2k)ρ(u, v) or ρ(t 2(u), t (v)) ≤ kρ(t 2(u), t (u)) + kρ(t (v), v) + (1 − 2k)ρ(t (u), v). lemma 3.9. let k be a nonempty subset of a hyperbolic metric space e and t : k → k is a reich-suzuki type nonexpansive mapping. then for all u, v ∈ k, we have ρ(u, t (v)) ≤ (3 + k) (1 − k) ρ(u, t (u)) + ρ(u, v). the simplest iteration process is the well-known picard iteration process and is defined as: { u1 ∈ y un+1 = t (un), n ∈ n. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 287 r. pat and r. pandey even though t is nonexpansive and has a fixed point, it is possible that the picard iteration does not converge to the fixed point of t. to overcome from such problems and to get better rate of convergence, a number of iteration processes have been introduced by many authors (cf. [1, 2, 18, 21, 22, 23, 24, 25, 36, 40, 43, 44, 48] and others). we present some convergence results for thakur et al[48] iteration process. one can obtain similar results for other iteration processes using the same line of proofs. for some fixed point u1 ∈ k, the thakur et al. iterative scheme in the framework of hyperbolic metric spaces can be defined as follows [48]: (3.3)          u1 ∈ k un+1 = t (vn) vn = t ((1 − αn)un ⊕ αnwn) wn = (1 − βn)un ⊕ βnt (un), n ∈ n, where {αn} and {βn} are real sequences in (0, 1). lemma 3.10. let k be a nonempty closed convex subset of a hyperbolic metric space e and t : k → k is a reich-suzuki type nonexpansive mapping. let {un} be a sequence with u1 ∈ k defined by (3.3). for any z ∈ f(t ) the following hold: (i) max{ρ(un+1, z), ρ(vn, z), ρ(wn, z)} ≤ ρ(un, z) for all n ∈ n; (ii) lim n→∞ ρ(un, z) = exists; (iii) lim n→∞ d(un, f(t )) exists. proof. let z ∈ f(t ) be arbitrary. by proposition 3.6 and (3.3), we have ρ(wn, z) = ρ((1 − βn)un ⊕ βnt (un), z) ≤ (1 − βn)ρ(un, z) + βnρ(t (un), z) ≤ (1 − βn)ρ(un, z) + βnρ(un, z) = ρ(un, z).(3.4) further, by proposition 3.6, (3.3) and (3.4), we have ρ(vn, z) = ρ(t ((1 − αn)un ⊕ αnwn), z) ≤ ρ((1 − αn)un ⊕ αnwn, z) ≤ (1 − αn)ρ(un, z) + αnρ(wn, z) ≤ (1 − αn)ρ(un, z) + αnρ(un, z) = ρ(un, z).(3.5) further, by proposition 3.6, (3.3) and (3.5), we have ρ(un+1, z) = ρ(t (vn), z) ≤ ρ(vn, z) ≤ ρ(un, z).(3.6) c© agt, upv, 2019 appl. gen. topol. 20, no. 1 288 existence and convergence results in hyperbolic spaces combining (3.4), (3.5) and (3.6) together establishes (i). also by (3.6) the sequence {ρ(un, z)} is bounded and monotone decreasing. therefore lim n→∞ ρ(un, z) exists and this proves (ii). now, since for each z ∈ f(t ), we have ρ(un+1, z) ≤ ρ(un, z) for all n ∈ n. taking infimum over all z ∈ f(t ), we get d(un+1, f(t )) ≤ d(un, f(t )) for all n ∈ n. so, the sequence {d(un, f(t ))} is bounded and decreasing. therefore, lim n→∞ d(un, f(t )) exists. � our next result is prefaced by the following lemmas. lemma 3.11 ([32]). let e be a complete uniformly convex hyperbolic metric space with monotone modulus of uniform convexity δ. then every bounded sequence {un} in k has a unique asymptotic centre with respect to any nonempty closed convex subset k of e. lemma 3.12 ([26]). let (e, ρ) be a uniformly convex hyperbolic metric space with monotone modulus of uniform convexity δ. let z ∈ e and {αn} be a sequence such that 0 < a ≤ αn ≤ b < 1. if {un} and {vn} are sequences in e such that lim sup n→∞ ρ(un, z) ≤ r, lim sup n→∞ ρ(vn, z) ≤ r and lim n→∞ ρ(αnvn ⊕ (1 − αn)un, z) = r for some r ≥ 0, then we have lim n→∞ ρ(un, vn) = 0. theorem 3.13. let k be a nonempty closed convex subset of a complete uniformly convex hyperbolic metric space e and t : k → k is a reich-suzuki type nonexpansive mapping. let {un} be a sequence with u1 ∈ k defined by (3.3) such that {αn} ⊆ [1/2, b) and {βn} ⊆ [a, b] or {αn} ⊆ [a, b] and {βn} ⊆ [a, 1] for some a, b with 0 < a ≤ b < 1. then f(t ) 6= ∅ if and only if {un} is bounded and lim n→∞ ρ(t (un), un) = 0. proof. suppose {un} is a bounded sequence and lim n→∞ ρ(t (un), un) = 0. by lemma 3.11, a({un}, k) 6= ∅, let z ∈ a({un}, k). by definition of asymptotic radius, we have r({un}, t (z)) = lim sup n→∞ ρ(un, t (z)). using lemma 3.9, we have r({un}, t (z)) = lim sup n→∞ ρ(un, t (z)) ≤ (3 + k) (1 − k) lim sup n→∞ ρ(t (un), un) + lim sup n→∞ ρ(un, z) = r({un}, z). by the uniqueness of the asymptotic centre of {un}, we have t (z) = z. conversely, let f(t ) 6= ∅ and z ∈ f(t ). then from lemma 3.10, lim n→∞ ρ(un, z) exists. suppose (3.7) lim n→∞ ρ(un, z) = r. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 289 r. pat and r. pandey by (3.7) and proposition 3.6, we have (3.8) lim sup n→∞ ρ(t (un), z) ≤ r. by (3.7) and (3.4), (3.9) lim sup n→∞ ρ(wn, z) ≤ lim n→∞ ρ(un, z) = r. using (3.9) and proposition 3.6, (3.10) lim sup n→∞ ρ(t (wn), z) ≤ r. by (3.3), lemma 3.10 and proposition 3.6, we have ρ(un+1, z) = ρ(t (vn), z) ≤ ρ(vn, z) = ρ(t ((1 − αn)un ⊕ αnwn), z) ≤ ρ((1 − αn)un ⊕ αnwn, z) ≤ (1 − αn)ρ(un, z) + αnρ(wn, z) ≤ (1 − αn)ρ(un, z) + αnρ(un, z) = ρ(un, z), or (3.11) ρ(un+1, z) ≤ ρ((1 − αn)un ⊕ αnwn, z) ≤ ρ(un, z), it implies that r ≤ lim n→∞ ρ((1 − αn)un ⊕ αnwn, z) ≤ r, then (3.12) lim n→∞ ρ((1 − αn)un ⊕ αnwn, z) = r. from (3.7), (3.9), (3.12) and lemma 3.12, we get (3.13) lim n→∞ ρ(un, wn) = 0. by the triangle inequality, we have ρ(un, z) ≤ ρ(un, wn) + ρ(wn, z), making n → ∞ and using (3.13) (3.14) r ≤ lim inf n→∞ ρ(wn, z). so, by (3.9) and (3.14) we have, (3.15) lim n→∞ ρ(wn, z) = r. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 290 existence and convergence results in hyperbolic spaces now, by (3.3) and proposition 3.6, we have ρ(wn, z) = ρ((1 − βn)un ⊕ βnt (un), z) ≤ (1 − βn)ρ(un, z) + βnρ(t (un), z) ≤ (1 − βn)ρ(un, z) + βnρ(un, z) = ρ(un, z).(3.16) so, making n → ∞ and using equation (3.15) and (3.7), we get (3.17) lim n→∞ ρ((1 − βn)un ⊕ βnt (un), z) = r. by (3.7), (3.8), (3.17) and lemma 3.12, we conclude that lim n→∞ ρ(t (un), un) = 0. � now, we present a result for ∆-convergence. theorem 3.14. let k be a nonempty closed convex subset of a complete uniformly convex hyperbolic metric space e and t : k → k be a reich-suzuki type nonexpansive mapping with f(t ) 6= ∅. let {un} be a sequence with u1 ∈ k defined by (3.3) such that {αn} ⊆ [1/2, b) and {βn} ⊆ [a, b] or {αn} ⊆ [a, b] and {βn} ⊆ [a, 1] for some a, b with 0 < a ≤ b < 1. then the sequence {un} ∆-converges to a fixed point of t. proof. by theorem 3.13, {un} is a bounded sequence. therefore {un} has a ∆-convergent subsequence. we show that every ∆-convergent subsequence of {un} has a unique ∆-limit in f(t ). arguing by contradiction suppose {un} has two subsequences {unj } and {unk} ∆-converging to l and m, respectively. by theorem 3.13, {unj } is bounded and d(t (unj ), unj ) = 0. we claim that l ∈ f(t ). we know that r({unj }, t (l)) = lim sup j→∞ ρ(unj , t (l)). by lemma 3.9, we have r({unj }, t (l)) = lim sup j→∞ ρ(unj , t (l)) ≤ (3 + k) (1 − k) lim sup j→∞ ρ(unj , t (unj )) + lim sup j→∞ ρ(unj , l) ≤ r({unj }, l). since the asymptotic centre of {unj } has a unique element, so t (l) = l. similarly, t (m) = m. by the uniqueness of asymptotic centre of a sequence, we have lim sup n→∞ ρ(un, l) = lim sup j→∞ ρ(unj , l) < lim sup j→∞ ρ(unj , m) = lim sup n→∞ ρ(un, m) = lim sup k→∞ ρ(unk, m) < lim sup k→∞ ρ(unk , l) = lim sup n→∞ ρ(un, l), which is a contradiction, unless l = m. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 291 r. pat and r. pandey theorem 3.15. let k be a nonempty closed and convex subset of a uniformly convex hyperbolic metric space e. let t : k → k be a reich-suzuki type nonexpansive mapping with f(t ) 6= ∅. let {un} be a sequence with u1 ∈ k defined by (3.3) such that {αn} ⊆ [1/2, b) and {βn} ⊆ [a, b] or {αn} ⊆ [a, b] and {βn} ⊆ [a, 1] for some a, b with 0 < a ≤ b < 1. then the sequence {un} converges strongly to a fixed point of t if and only if lim inf n→∞ d(un, f(t )) = 0. proof. suppose that lim inf n→∞ d(un, f(t )) = 0. from lemma 3.10, lim inf n→∞ d(un, f(t )) exists, so (3.18) lim n→∞ ρ(un, f(t )) = 0. in view of (3.18), there exists a subsequence {unk} of the sequence {un} such that ρ(unk , zk) ≤ 1 2k for all k ≥ 1, where {zk} is a sequence in f(t ). by lemma 3.10, we have (3.19) ρ(unk+1, zk) ≤ ρ(unk, zk) ≤ 1 2k . now, by the triangle inequality and (3.19), we have ρ(zk+1, zk) ≤ ρ(zk+1, unk+1) + ρ(unk+1, zk) ≤ 1 2k+1 + 1 2k < 1 2k−1 . a standard argument shows that {zk} is a cauchy sequence. by lemma 3.7, f(t ) is closed, so {zk} converges to some point z ∈ f(t ). now ρ(unk, z) ≤ ρ(unk , zk) + ρ(zk, z). letting k → ∞ implies that {unk} converges strongly to z. by lemma 3.10, lim n→∞ ρ(un, z) exists. hence {un} converges strongly to z. the converse part is obvious. � finally, we give another strong convergence theorem. theorem 3.16. let e be a uniformly convex and complete hyperbolic metric space. let k, t and {un} be same as in theorem 3.13. let t satisfy the condition (i) with f(t ) 6= ∅, then {un} converges strongly to a fixed point of t. proof. from theorem 3.13, it follows that (3.20) lim inf n→∞ ρ(t (un), un) = 0. since t satisfy the condition (i), we have ρ(t (un), un) ≥ g(d(un, f(t ))). from (3.20), we get lim inf n→∞ g(d(un, f(t ))) = 0. since g : [0, ∞) → [0, ∞) is a nondecreasing function with g(0) = 0 and g(r) > 0 for all r ∈ (0, ∞), we have lim inf n→∞ d(un, f(t )) = 0. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 292 existence and convergence results in hyperbolic spaces therefore all the assumptions of theorem 3.15 are satisfied. hence {un} converges strongly to a fixed point of t. � acknowledgements. we are very much thankful to the reviewers for their constructive comments and suggestions which have been useful for the improvement of this paper. references [1] m. abbas and t. nazir, a new faster iteration process applied to constrained minimization and feasibility problems, mat. vesnik 66, no. 2 (2014), 223–234. 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[48] b. s. thakur, d. thakur and m. postolache, a new iterative scheme for numerical reckoning fixed points of suzuki’s generalized nonexpansive mappings, appl. math. comput. 275 (2016), 147–155. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 295 @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 81-89 topological conditions for the representation of preorders by continuous utilities e. minguzzi ∗ abstract we remove the hausdorff condition from levin’s theorem on the representation of preorders by families of continuous utilities. we compare some alternative topological assumptions in a levin’s type theorem, and show that they are equivalent to a polish space assumption. 2010 msc: 54f05, 54e55, 54d50. keywords: preorder normality, utilities, preorder representations, k-spaces. 1. introduction a topological preordered space is a triple (e, t ,≤), where (e, t ) is a topological space endowed with a preorder ≤, that is, ≤ is a reflexive and transitive relation [18]. a function f : e → r is isotone if x ≤ y ⇒ f(x) ≤ f(y), and a utility if it is isotone and additionally “x ≤ y and y � x ⇒ f(x) < f(y)”. in this work we wish to establish sufficient topological conditions on (e, t ) for the representability of the preorder through the family u of continuous utility functions with value in [0,1]. that is, we look for topological conditions that imply the validity of the following property x ≤ y ⇔ ∀f ∈ u,f(x) ≤ f(y). economists have long been interested in the representation of preorders by utility functions [4]. more recently, this mathematical problem has found application in other fields such as spacetime physics [16] and dynamical systems [1]. ∗this work has been partially supported by gnfm of indam and by fqxi. 82 e. minguzzi to start with, it will be convenient to recall some notions from the theory of topological preordered spaces [18]. a semiclosed preordered space e is a topological preordered space such that, for every point x ∈ e, the increasing hull i(x) = {y ∈ e : x ≤ y} and the decreasing hull d(x) = {y : y ≤ x}, are closed. a closed preordered space e is a topological preordered space endowed with a closed preorder, that is, the graph g(≤) = {(x,y) : x ≤ y} is closed in the product topology on e × e. let e be a topological preordered space. a subset s ⊂ e is called increasing if i(s) = s and decreasing if d(s) = s, where i(s) = ⋃ s∈s i(s) and analogously for d(s). a subset s ⊂ e is convex if it is the intersection of an increasing and a decreasing set, in which case we have s = i(s) ∩ d(s). a topological preordered space e is convex if for every x ∈ e, and open set o x, there are an open decreasing set u and an open increasing set v such that x ∈ u ∩ v ⊂ o. notice that according to this terminology the statement “the topological preordered space e is convex” differs from the statement “the subset e is convex” (which is always true). the terminology is not uniform in the literature, for instance lawson [14] calls strongly order convexity what we call convexity. the topological preordered space e is locally convex if for every point x ∈ e, the set of convex neighborhoods of x is a base for the neighborhoods system of x [18]. clearly, convexity implies local convexity. a topological preordered space is a normally preordered space if it is semiclosed preordered and for every closed decreasing set a and closed increasing set b which are disjoint, it is possible to find an open decreasing set u and an open increasing set v which separate them, namely a ⊂ u, b ⊂ v , and u ∩ v = ∅. a regularly preordered space is a semiclosed preordered space such that if x /∈ b, where b is a closed increasing set, then there is an open decreasing set u x and an open increasing set v ⊃ b, such that u∩v = ∅, and analogously, a dual property must hold for y /∈ a where a is a closed decreasing set. we have the implications: normally preordered space ⇒ regularly preordered space ⇒ closed preordered space ⇒ semiclosed preordered space. for normally preordered spaces a natural generalization of urysohn’s lemma holds [18, theor. 1]: if a and b are respectively a closed decreasing set and a closed increasing set such that a ∩ b = ∅, then there is a continuous isotone function f : e → [0,1] such that a ⊂ f−1(0) and b ⊂ f−1(1). a trivial and well known consequence of this fact is (take a = d(y) and b = i(x) with x � y) proposition 1.1. let e be a normally preordered space and let i be the family of continuous isotone functions with value in [0,1], then (1.1) x ≤ y ⇔ ∀f ∈ i,f(x) ≤ f(y). this result almost solves our original problem but for the fact that the family of continuous utility functions is replaced by the larger family of continuous isotone functions. moreover, we have still to identify some topological conditions on (e, t ) in order to guarantee that e is a normally preordered space. it topological conditions for preorder representations 83 is worth noting that eq. (1.1) is one of the two conditions which characterize the completely regularly preordered spaces [18]. let us recall that a kω-space is a topological space characterized through the following property [9]: there is a countable (admissible) sequence ki of compact sets such that ⋃∞ i=1 ki = e and for every subset o ⊂ e, o is open if and only if o ∩ ki is open in ki for every i (here e is not required to be hausdorff). recently, the author proved the following results [17] theorem 1.2. every kω-space equipped with a closed preorder is a normally preordered space. theorem 1.3. every second countable regularly preordered space admits a countable continuous utility representation, that is, there is a countable set {fk} of continuous utility functions fk : e → [0,1] such that x ≤ y ⇔ ∀k,fk(x) ≤ fk(y). using the previous results we obtain the following improvement of levin’s theorem.1,2 [15] [4, lemma 8.3.4] corollary 1.4. let (e, t ,≤) be a second-countable kω-space equipped with a closed preorder, then there is a countable family {uk} of continuous utility functions uk : e → [0,1] such that x ≤ y ⇔ ∀k, uk(x) ≤ uk(y). proof. every closed preordered kω-space is a normally preordered space (theor. 1.2). since e is a second countable regularly preordered space it admits a countable continuous utility representation (theor. 1.3). � with respect to the references we have removed the hausdorff condition.3 another interesting improvement can be found in [5]. in the remainder of the work we wish to compare this result with other reformulations which use different topological assumptions. 1.1. topological preliminaries. since in this work we do not assume hausdorffness of e it is necessary to clarify that in our terminology a topological space is locally compact if every point admits a compact neighborhood. 1these references do not consider the representation problem but rather the existence of just one continuous utility. nevertheless, the argument for the existence of the whole representation is contained at the end of the proof of [4, lemma 8.3.4]. 2the result [8, theor. 1] should not be confused with this one, since their definition of utility differs from our. that theorem can instead be deduced from the stronger theorem 1.2. also note that in their proof they tacitly use a kω-space assumption which can nevertheless be justified. 3in [10] it was first suggested that the hausdorff condition could be removed. this generalization is non trivial and requires some care in the reformulation and generalization of some extendibility results [17]. 84 e. minguzzi definition 1.5. a topological space (e, t ) is hemicompact if there is a countable sequence ki, called admissible, of compact sets such that every compact set is contained in some ki (since points are compact we have ⋃∞ i=1 ki = e, and without loss of generality we can assume ki ⊂ ki+1). the following facts are well known (hausdorffness is not required). every compact set is hemicompact and every hemicompact set is σ-compact. every locally compact lindelöf space is hemicompact, and every first countable hemicompact space is locally compact.4,5 definition 1.6. a topological space e is a k-space if for every subset o ⊂ e, o is open if and only if, for every compact set k ⊂ e, o ∩ k is open in k. we remark that we use the definition given in [21] and so do not include hausdorffness in the definition as done in [7, cor. 3.3.19]. every first countable or locally compact space is a k-space.6 thus under second countability “hemicompact k-space” is equivalent to local compactness. it is easy to prove that an hemicompact k-space is a kω-space and the converse can be proved under t1 separability (see [20, lemma 9.3]). further, in an hemicompact k-space every admissible sequence ki, ki ⊂ ki+1, in the sense of the hemicompact definition is also an admissible sequence in the sense of the kω-space definition. the mentioned results imply the chain of implications compact ⇒ hemicompact k-space ⇒ kω-space ⇒ σ-compact ⇒ lindelöf and the fact that local compactness makes the last four properties coincide. a continuous function f : x → y between topological spaces is said to be a quasi-homeomorphism if the following conditions are satisfied [11, 6]: (i) for any closed set c in x, f−1(f(c)) = c. (ii) for any closed set f in y , f(f−1(f)) = f . every quasi-homeomorphism establishes a bijective correspondence ψf : cl(y ) → cl(x) between the closed sets of y and x through the definition ψf (c) = f−1(c). remark 1.7. if f is surjective (ii) is satisfied. furthermore, a quotient (hence surjective) map which satisfies f−1(f(c)) = c for every closed set c (or equivalently, for every open set) is a quasi-homeomorphism. indeed, if c is closed then f(c) is closed, because of the identity f−1(f(c)) = c and the definition of quotient topology. thus both properties (i)-(ii) hold, and f is a quasihomeomorphism. the given argument also shows that f is closed (and open). furthermore, it can be shown that a quasi-homeomorphism is surjective if and only if it is closed, if and only if it is open [6, prop. 2.4]. 4in order to prove the last claim, modify slightly the proof given in [2, p. 486] replacing “suppose no neighborhood vi has a compact closure” with “suppose x has no compact neighborhood”. 5a first countable hausdorff hemicompact k-space space need not be second countable. indeed, as stressed in [9] not even compactness is sufficient as the unit square with a suitable topology provides a counterexample [19, p. 73]. 6modify slightly the proof in [21, theor. 43.9] topological conditions for preorder representations 85 2. ordered quotient and local convexity on a topological preordered space e the relation ∼, defined by x ∼ y if x ≤ y and y ≤ x, is an equivalence relation. let e/∼ be the quotient space, t / ∼ the quotient topology, and let � be defined by, [x] � [y] if x ≤ y for some representatives (with some abuse of notation we shall denote with [x] both a subset of e and a point on e/∼). the quotient preorder is by construction an order. the triple (e/ ∼, t / ∼,�) is a topological ordered space and π : e → e/∼ is the continuous quotient projection. remark 2.1. taking into account the definition of quotient topology we have that every open (closed) increasing set on e projects to an open (resp. closed) increasing set on e/∼ and all the latter sets can be regarded as such projections. the same holds replacing increasing by decreasing. as a consequence, (e, t ,≤) is a normally preordered space (semiclosed preordered space, regularly preordered space) if and only if (e/∼, t /∼,�) is a normally ordered space (resp. semiclosed ordered space, regularly ordered space). the effect of the quotient π : e → e/∼ on the topological preordered properties has been studied in [13]. remark 2.2. a set s ⊂ e is convex if and only if π(s) is convex. indeed, let u and v be respectively decreasing and increasing sets, we have π(u ∩ v ) = π(u) ∩ π(v ) because: u ∩ v ⊂ π−1(π(u ∩ v )) ⊂ π−1(π(u) ∩ π(v )) = π−1(π(u)) ∩ π−1(π(v )) = u ∩ v . proposition 2.3. let (e, t ,≤) be a topological preordered space. if local convexity holds at x ∈ e then [x] is compact and every open neighborhood of x is also an open neighborhood of [x]. if e is locally convex then every open set is saturated with respect to π (that is π−1(π(o)) = o for every open set o). hence π is a (surjective) quasi-homeomorphism, in particular π is open, closed and proper. proof. let o be an open neighborhood of x and let c be a convex set such that x ∈ c ⊂ o, then [x] = d(x) ∩ i(x) ⊂ d(c) ∩ i(c) = c ⊂ o, thus o is also an open neighborhood for [x]. the compactness of [x] follows easily. let o ⊂ e be an open set and let x ∈ o. we have already proved that [x] ⊂ o. since this is true for every x ∈ o, we have π−1(π(o)) = o. therefore, by remark 1.7, since π is a quotient map it is a quasi-homeomorphism which is open and closed. every such map is easily seen to be proper. � remark 2.4. by the previous result under local convexity the quotient π establishes a bijection between the respective families in e and e/∼ of open sets, closed sets, compact sets, increasing sets, decreasing sets and convex sets. continuous isotone functions on e pass to the quotient on e/∼ and conversely, continuous isotone functions on e/∼ can be lifted to continuous isotone functions on e. as a consequence, many properties are shared between e and e/∼ regarded as topological preordered spaces (one should not apply this observation carelessly, otherwise one would conclude that ≤ is an order and that t is hausdorff). for instance, we have 86 e. minguzzi proposition 2.5. if e is a locally convex closed preordered space then e/∼ is a locally convex closed ordered space. proof. we just prove closure to show how the argument works. if [x] � [y] then x � y. the representatives x and y are separated by open sets [18, prop. 1, chap. 1] ux and uy such that i(ux) ∩ d(uy) = ∅. by local convexity the increasing neighborhood of x, i(ux), projects into an increasing neighborhood π(i(ux)) of [x]. analogously, π(d(uy)) is a decreasing neighborhood of [y] which is disjoint from π(i(ux)). we conclude that � is closed [18, prop. 1, chap. 1]. � the property of closure for the graph of the preorder does not pass to the quotient without additional assumptions [13]. for instance, the previous result holds with “locally convex” replaced by kω-space [17]. remark 2.6. in a topological space (e, t ) the specialization preorder is defined by x � y if x ⊂ y. two points x,y are indistinguishable according to the topology if x � y and y � x, denoted x � y, since in this case they have the same neighborhoods. the quotient under � of the topological space is called kolmogorov quotient or t0-identification and gives a t0-space, sometimes called the t0-reflection of e. the kolmogorov quotient is by construction open, closed and a quasi-homeomorphism. the first statement of proposition 2.3 implies that under local convexity if x ∼ y then x and y have the same neighborhoods, that is, x � y. if the preorder ≤ on e is semiclosed the converse holds because y = x ⊂ i(x) ∩ d(x), which implies, y ∼ x. thus in a locally convex semiclosed preordered space, π is the kolmogorov quotient and e/∼ is the t0-identified space. actually, e/∼ is a t1-space because it is a semiclosed ordered space (remark 2.1) thus [x] ⊂ ie/∼([x]) ∩ de/∼([x]) = {[x]}. if additionally e is a closed preordered space we already know that e/∼ is a closed ordered space. another way to prove that � is closed is to observe that the t0-reflection of a product is the product of the t0-reflections, that is, π × π is the kolmogorov quotient of e × e, and since the kolmogorov quotient is closed it sends the closed graph g(≤) into the graph g(�) which is therefore closed. in summary we have proved proposition 2.7. let e be a locally convex semiclosed preordered space then π : e → e/∼ is the t0-identification of e and e/∼ is t1. furthermore, if e is also a closed preordered space then e/∼ is a closed ordered space and hence t2. the next proposition will be useful (see prop. 3.1) and is an immediate corollary of remark 2.4. proposition 2.8. if (e, t ,≤) is (locally) convex then (e/ ∼, t / ∼,�) is (resp. locally) convex. if (e, t ,≤) is locally convex and locally compact then (e/∼, t /∼,�) is locally compact, and if additionally e is a closed preordered space then every point of e admits a base of closed compact neighborhoods (but e need not be t1). topological conditions for preorder representations 87 3. equivalence of some topological assumptions we wish to clarify the relative strength of some topological conditions that can be used in a levin’s type theorem. let us recall that a polish space is a topological space which is homeomorphic to a separable complete metric space [3, part ii, chap. ix, sect. 6]. a pseudometric is a metric for which the condition d(x,y) = 0 ⇒ x = y, has been dropped [12]. the relation x ≈ y if d(x,y) = 0, is an equivalence relation and the quotient e/≈ is a metric space. a pseudo-metrizable space is a topological space with a topology which comes from some pseudo-metric. in particular, it is hausdorff if and only if it is metrizable because the hausdorff property holds if and only if the equivalence classes are trivial. we say that a space is a pseudo-polish space if it is homeomorphic to a pseudo-metric space and the quotient under ≈ is a polish space. note that every pseudo-polish space is separable. the next result is purely topological (see prop. 2.7) but at some places it makes reference to a preorder. this is done because it is meant to clarify the topological conditions underlying a levin’s type theorem in which the presence of a closed preorder is included in the assumptions. proposition 3.1. let us consider the following properties for a topological space (e, t ) and let ≤ be any preorder on e (e.g. the discrete-order) (i) second-countable kω-space, (ii) second-countable locally compact, (iii) pseudo-metrizable hemicompact k-space, (iv) locally compact pseudo-polish space. then (iv) ⇔ (iii) ⇒ (ii) ⇒ (i). furthermore, if (e, t ,≤) is a locally convex semiclosed preordered space we have (i) ⇒ (ii), and if (e, t ,≤) is a locally convex closed preordered space we have (ii) ⇒ (iii) (note that the discrete-order is locally convex thus the former implication holds also under t1 separability of t and the latter implication holds also under t2 separability of t ). in particular they are all equivalent if (e, t ,≤) is a locally convex closed preordered space (e.g. under hausdorffness). proof. we shall make extensive use of results recalled in the introduction. (ii) ⇒ (i). every second countable locally compact space is an hemicompact k-space and hence a kω-space. (i) ⇒ (ii). assume that (e, t ,≤) is a locally convex semiclosed preordered space. if we prove that e is hemicompact we have finished because first countability and hemicompactness imply local compactness. we have already proved that (e/∼, t /∼) is t1 (prop. 2.7). but (e/∼, t /∼) is a kω-space by a nonhausdorff generalization of morita’s theorem [17] thus e/∼ is hemicompact [20, lemma 9.3]. let k̃i be an admissible sequence on e/∼, since π is proper (prop. 2.3) the sets ki = π −1(k̃i) are compact. they give an admissible sequence for the hemicompact property, indeed if k is any compact on e then 88 e. minguzzi π(k) is compact on e/∼ thus there is some k̃i such that π(k) ⊂ k̃i. finally, k ⊂ π−1(π(k)) ⊂ π−1(k̃i) = ki. (ii) ⇒ (iii). a second countable locally compact space is an hemicompact k-space. since (e, t ,≤) is a locally convex closed preordered space, e/∼ is hausdorff (prop. 2.7). local convexity, local compactness, and second countability pass to the quotient e/∼ (see prop. 2.3,2.8) which is therefore metrizable by urysohn’s theorem. thus e is pseudo-metrizable with the pullback by π of the metric on e/∼. (iii) ⇒ (ii). a pseudo-metrizable space is second countable if and only if it is separable [12, theor. 11 chap. 4] thus it suffices to prove separability. in particular, since e is σ-compact it suffices to prove separability on each compact set kn (of the hemicompact decomposition) with the induced topology (which comes from the induced pseudo-metric). it is known that every compact pseudo-metrizable space is second countable [12, theor. 5, chap. 5] and hence separable, thus we proved that e is second countable. as first countability and the hemicompact property imply local compactness we get the thesis. (iv) ⇒ (iii). e is a separable pseudo-metrizable space thus second countable [12, theor. 11 chap. 4]. second countability and local compactness imply the hemicompact k-space property. (iii) ⇒ (iv). since (iii) ⇒ (ii), e is second countable and locally compact. let d be a compatible pseudo-metric on e and let e/ ≈ be the metric quotient. since π≈ : e → e/≈ is an open continuous map (actually a quasihomeomorphism) and e is second countable and locally compact then e/≈ is second countable and locally compact too. we conclude by [21, 23c] that the one point compactification of e/≈ is metrizable, and by compactness the one point compactification of e/ ≈ is completely metrizable. further, since e/ ≈ is separable its one point compactification is also separable. we conclude that the one point compactification of e/≈ is polish, and since e/≈ is hausdorff and locally compact, e/≈ is an open subset of a polish space hence polish [3, part ii, chap. ix, sect. 6]. � remark 3.2. if e is a locally convex closed preordered space and (iv) holds, as is implied by the assumptions of corollary 1.4, then the local compactness mentioned in (iv) implies, despite the lack of hausdorffness, the stronger versions of local compactness (prop. 2.8). 4. conclusions we have deduced an improved version of levin’s theorem in which the hausdorff condition has been removed. furthermore, some alterative topological assumptions underlying a levin’s type theorem have been compared and we we have shown that the original levin’s theorem included a polish space assumption. topological conditions for preorder representations 89 acknowledgements. the material of this work was initially contained in a first expanded version of [17], and it has benefited from suggestions by an anonymous referee in connection with prop. 2.7. references [1] e. akin and j. auslander, generalized recurrence, compactifications and the lyapunov topology, studia mathematica 201 (2010), 49–63. 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[18] l. nachbin, topology and order, princeton: d. van nostrand company, inc. (1965). [19] l. a. steen and j. a. seebach, jr., counterexamples in topology, new york: holt, rinehart and winston, inc. (1970). [20] n. e. steenrod, a convenient category of topological spaces, michigan math. j. 14 (1967), 133–152. [21] s. willard, general topology, reading: addison-wesley publishing company (1970). (received september 2011 – accepted december 2011) e. minguzzi (ettore.minguzzi@unifi.it) dipartimento di matematica applicata “g. sansone”, università degli studi di firenze, via s. marta 3, i-50139 firenze, italy. topological conditions for the representation of[4pt] preorders by continuous utilities. by e. minguzzi @ appl. gen. topol. 20, no. 2 (2019), 431-447 doi:10.4995/agt.2019.11645 c© agt, upv, 2019 balleans, hyperballeans and ideals dikran dikranjana, igor protasovb, ksenia protasovab and nicolò zavaa a department of mathematical, computer and physical sciences, udine university, 33 100 udine, italy (dikran.dikranjan@uniud.it, nicolo.zava@gmail.com) b department of computer science and cybernetics, kyiv university, volodymyrska 64, 01033, kyiv, ukraine (i.v.protasov@gmail.com, ksuha@freenet.com.ua) communicated by m. sanchis abstract a ballean b (or a coarse structure) on a set x is a family of subsets of x called balls (or entourages of the diagonal in x × x) defined in such a way that b can be considered as the asymptotic counterpart of a uniform topological space. the aim of this paper is to study two concrete balleans defined by the ideals in the boolean algebra of all subsets of x and their hyperballeans, with particular emphasis on their connectedness structure, more specifically the number of their connected components. 2010 msc: 54e15. keywords: balleans; coarse structure; coarse map; asymorphism; balleans defined by ideals; hyperballeans. 1. introduction 1.1. basic definitions. a ballean is a triple b = (x,p,b) where x and p are sets, p 6= ∅, and b : x ×p →p(x) is a map, with the following properties: (i) x ∈ b(x,α) for every x ∈ x and every α ∈ p ; (ii) symmetry, i.e., for any α ∈ p and every pair of points x,y ∈ x, x ∈ b(y,α) if and only if y ∈ b(x,α); received 13 may 2018 – accepted 15 july 2018 http://dx.doi.org/10.4995/agt.2019.11645 d. dikranjan, i. protasov, k. protasova and n. zava (iii) upper multiplicativity, i.e., for any α,β ∈ p , there exists a γ ∈ p such that, for every x ∈ x, b(b(x,α),β) ⊆ b(x,γ), where b(a,δ) = ⋃ {b(y,δ) | y ∈ a}, for every a ⊆ x and δ ∈ p . the set x is called support of the ballean, p – set of radii, and b(x,α) – ball of centre x and radius α. this definition of ballean does not coincide with, but it is equivalent to the usual one (see [11] for details). a ballean b is called connected if, for any x,y ∈ x, there exists α ∈ p such that y ∈ b(x,α). every ballean (x,p,b) can be partitioned in its connected components : the connected component of a point x ∈ x is qx(x) = ⋃ α∈p b(x,α). moreover, we call a subset a of a ballean (x,p,b) bounded if there exists α ∈ p such that, for every y ∈ a, a ⊆ b(y,α). the empty set is always bounded. a ballean is bounded if its support is bounded. in particular, a bounded ballean is connected. denote by [(x) the family of all bounded subsets of a ballean x. if b = (x,p,b) is a ballean and y a subset of x, one can define the subballean b �y = (y,p,by ) on y induced by b, where by (y,α) = b(y,α)∩y , for every y ∈ y and α ∈ p . a subset a of a ballean (x,p,b) is thin (or pseudodiscrete) if, for every α ∈ p, there exists a bounded subset v of x such that ba(x,α) = b(x,α) ∩ a = {x} for each x ∈ a\v . a ballean is thin if its support is thin. bounded balleans are obviously thin. we note that to each ballean on a set x can be associated a coarse structure [12]: a particular family e of subsets of x×x, called entourages of the diagonal ∆x. the pair (x,e) is called a coarse space. this construction highlights the fact that balleans can be considered as asymptotic counterparts of uniform topological spaces. for a categorical look at the balleans and coarse spaces as “two faces of the same coin” see [4]. definition 1.1 ([11, 5]). let b = (x,p,b) be a ballean. a subset a of x is called: (i) large in x if there exists α ∈ p such that b(a,α) = x; (ii) thick in x if, for every α ∈ p , there exists x ∈ a such that b(x,α) ⊆ a; (iii) small in x if, for every α ∈ p , x \b(a,α) is large in x. let bx = (x,px,bx) and by = (y,py ,by ) be two balleans. then a map f : x → y is called (i) coarse if for every radius α ∈ px there exists another radius β ∈ py such that f(bx(x,α)) ⊆ by (f(x),β) for every point x ∈ x; (ii) effectively proper if for every α ∈ py there exists a radius β ∈ px such that f−1(by (f(x),α)) ⊆ bx(x,β) for every x ∈ x; c© agt, upv, 2019 appl. gen. topol. 20, no. 2 432 balleans, hyperballeans and ideals (iii) a coarse embedding if it is both coarse and effectively proper; (iv) an asymorphism if it is bijective and both f and f−1 are coarse or, equivalently, f is bijective and both coarse and effectively proper; (v) an asymorphic embedding if it is an asymorphism onto its image or, equivalently, if it is an injective coarse embedding; (vi) a coarse equivalence if it is a coarse embedding such that f(x) is large in by . we recall that a family i of subsets of a set x is an ideal if a,b ∈i, c ⊆ a imply a∪b ∈i, c ∈i. in this paper, we always impose that x /∈i (so that i is proper) and i contains the ideal fx of all finite subsets of x. because of this setting, a set x that admits an ideal i is infinite, as otherwise x ∈i. we consider the following two balleans with support x determined by i. definition 1.2. (i) the i-ary ballean xi-ary = (x,i,bi-ary), with radii set i and balls defined by bi-ary(x,a) = {x}∪a, for x ∈ x and a ∈i; (ii) the point ideal ballean xi = (x,i,bi), where bi(x,a) = {{x} if x /∈ a, {x}∪a = a otherwise. the balleans xi-ary and xi are connected and unbounded. while xi is thin, xi-ary is never thin (this follows from proposition 1.3 and results from [11] reported in theorem 2.2). for every connected unbounded ballean b with support x one can define the satellite ballean xi, where i = [(x) is the ideal of all bounded subsets of x. proposition 1.3. for every ideal i on a set x, the map idx : xi → xi-ary is coarse, but it is not effectively proper. proof. pick an arbitrary non-empty element f ∈i. since i is a proper ideal, for every k ∈i, there exists xk ∈ x \ (f ∪k). hence, in particular, bi-ary(xk,f) = {xk}∪f 6⊆ {xk} = bi(xk,k). � let b = (x,p,b) be a ballean. then the radii set p can be endowed with a preorder ≤b as follows: for every α,β ∈ p , α ≤b β if and only if b(x,α) ⊆ b(x,β), for every x ∈ x. a subset p ′ ⊆ p is cofinal if it is cofinal in this preorder (i.e., for every α ∈ p , there exists α′ ∈ p ′, such that α ≤b α′). if p ′ is cofinal, then b = (x,p ′,b′), where b′ = b �x×p′. if i is an ideal on a set x, then both the preorders ≤bi and ≤bi-ary on i coincide with the natural preorder ⊆ on i, defined by inclusion. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 433 d. dikranjan, i. protasov, k. protasova and n. zava remark 1.4. let (x,px,bx) and (y,py ,by ) be two balleans and f : x → y be an injective map. we want to give some sufficient conditions that implies the effective properness of f. (i) suppose that there exist two cofinal subsets of radii p ′x and p ′ y of px and py , respectively, and a bijection ψ : p ′ x → p ′ y such that, for every α ∈ p ′ x and every x ∈ x, (1.1) f(bx(x,α)) = by (f(x),ψ(α)) ∩f(x). we claim that, under these hypothesis, f is a coarse embedding and then f : x → f(x) is an asymorphism. first of all, let us check that f is coarse. fix a radius α ∈ px and let α′ ∈ p ′x such that α ≤ α ′. hence f(bx(x,α)) ⊆ f(bx(x,α′)) ⊆ by (f(x),ψ(α′)), for every x ∈ x, where the last inclusion holds because of (1.1). as for the effective properness, since f is bijective, (1.1) is equivalent to bx(x,α) = f −1(by (f(x),ψ(α))), for every x ∈ x, and this yelds to the thesis. in fact, for every β ∈ py , there exists α′ ∈ p ′x such that β ≤ ψ(α ′) and thus, for every x ∈ x, f−1(by (f(x),β)) ⊆ f−1(by (f(x),ψ(α′))) = bx(x,α′). (ii) note that f : x → y is a coarse embedding if and only if f : x → f(x) is a coarse embedding, where f(x) is endowed with the subballean structure inherited by y . suppose that p ′x ⊆ px and p ′′ f(x) ⊆ py are cofinal subsets of radii in x and f(x), respectively, and ψ : p ′x → p ′′ f(x) is a bijection such that (1.1) holds for every x ∈ x. then f is a coarse embedding. (iii) in notations of item (ii), in order to show that p ′′ f(x) is cofinal in f(x), it is enough to provide a cofinal subset of radii p ′y ⊆ py in y and a bijection ϕ: p ′′ f(x) → p ′y such that, for every y ∈ f(x) and every α ∈ p ′′ f(x) , by (y,α)∩ f(x) = by (y,ϕ(α)) ∩f(x). 1.2. hyperballeans. definition 1.5. let b = (x,p,b) be a ballean. define its hyperballean to be exp(b) = (p(x),p, exp b), where, for every a ⊆ x and α ∈ p , (1.2) exp b(a,α) = {c ∈p(x) | a ⊆ b(c,α), c ⊆ b(a,α)}. it is not hard to check that this defines actually a ballean. another easy observation is the following: for every ballean (x,p,b), qexp x(∅) = {∅} and, in particular exp b({∅},α) = {∅} for every α ∈ p , since b(∅,α) = ∅. motivated by this, we shall consider also the subballean exp∗(x) = exp(x) \ {∅}. if b = (x,p,b) is a ballean, the subballean x[ of expb having as support the family of all non-empty bounded subsets of b was already defined and c© agt, upv, 2019 appl. gen. topol. 20, no. 2 434 balleans, hyperballeans and ideals studied in [10]. note that, b is connected (resp., unbounded) if and only if b[ is connected (resp., unbounded). in the sequel we focus our attention on four hyperballeans defined by an ideal i on a set x. in particular, we investigate exp xi and exp xi-ary, as well as their subballeans x[i-ary and x [ i. so exp xi = (p(x),i, exp bi), and, according to (1.2), for a ⊆ x and k ∈i one has (1.3) exp bi(a,k) = {{(a\k) ∪y | ∅ 6= y ⊆ k} if a∩k 6= ∅, {a} otherwise. in fact, fix c ∈ exp bi(a,k). if a∩k = ∅, then c = a (as bi(a,k) = a and, for every a′ ⊆ a, bi(a′,k) = a′). otherwise, c ⊆ bi(a,k) = a∪k. moreover, a ⊆ bi(c,k) if and only if c ∩k 6= ∅ and a ⊆ c ∪k. in other words, exp bi(a,k) = {z ∈p(x) | a\k ( z ⊆ a∪k}, if a∩k 6= ∅. let us now compute the balls in exp xi-ary = (p(x),i, exp bi-ary). as mentioned above, exp bi-ary({∅},k) = {∅} for every k ∈i. fix now a nonempty subset a of x and a radius k ∈ i. then a non-empty subset c ⊆ x belongs to exp bi-ary(a,k) if and only if c ⊆ bi-ay(a,k) = a∪k and a ⊆ bi-ary(c,k) = c ∪k, since both a and c are non-empty. hence exp bi-ary(a,k) = {(a\k) ∪y | y ⊆ a, (a\k) ∪y 6= ∅} = = {z ∈p(x) | a\k ⊆ z ⊆ a∪k, z 6= ∅} for every ∅ 6= a ⊆ x and k ∈i. by putting all together, one obtains that, for every a ⊆ x and every k ∈i, (1.4) exp bi-ary(a,k) = { {z ∈p(x) | a\k ⊆ z ⊆ a∪k, z 6= ∅} if a 6= ∅, {a} otherwise a = ∅. remark 1.6. denote by cx = {0, 1}x the boolean ring of all function x → {0, 1} = z2 and for f ∈ cx let supp f = {x ∈ x | f(x) = 1}. then one has a ring isomorphism  = x : p(x) → cx, sending a ∈p(x) to its characteristic function χa ∈ cx, so (∅) = 0, the zero function. using , one can transfer the ball structure from exp bi-ary to cx: for 0 6= f ∈ {0, 1}x and a ∈ i one has (exp bi-ary( −1(f),a)) = {g | g(x) = f(x), x ∈ x \a} = {g | g �x\a= f �x\a}.(1.5) c© agt, upv, 2019 appl. gen. topol. 20, no. 2 435 d. dikranjan, i. protasov, k. protasova and n. zava while, according to (1.3) and (1.4), the empty set is “isolated” in both balleans exp xi and xi-ary, the set {g | g(x) = 0, x ∈ x \a} = {g | g[x \a] = {0}} (i.e., the functions g with supp g ⊆ a), still makes sense and seems a more natural candidate for a ball of radius a centered at the zero function. taking into account this observation, we modify the ballean structure on cx, denoting by c(x,i) the new ballean, with balls defined by the unique formula suggested by (1.5): (1.6) bc(x,i)(f,a) = {g | g(x) = f(x), x ∈ x \a} = {g | g �x\a= f �x\a}, where a ∈ i; when no confusion is possible, we shall write shortly bc(f,a). in this way (1.7)  �exp∗(xi-ary): exp ∗(xi-ary) → c(x,i) is an asymorphic embedding. the ballean c(x,i), as well as its subballean m(x,i), having as support the ideal {g ∈ cx | supp g ∈ i} of the ring cx, will play a prominent role in the paper (note that m(x,i)\{∅} coincides with (x[i−ary)). 1 if x = n and i = fn, then m(x,i) is the cantor macrocube defined in [10]. motivated by this, the ballean m(x,i), for an ideal i on a set x, will be called the i-macrocube (o, shortly, a macrocube) in the sequel. remark 1.7. one of the main motivations for the above definitions comes from the study of topology of hyperspaces. for an infinite discrete space x, the set p(x) admits two standard non-discrete topologizatons via the vietoris topology and via the tikhonov topology. in the case of the vietoris topology, the local base at the point y ∈ p(x) consists of all subsets of x of the form {z ∈ p(x) | k ⊆ z ⊆ y}, where k runs over the family of all finite subsets of y . the tikhonov topology arises after identification of p(x) with {0, 1}x via the characteristic functions of subsets of x. given an ideal i on x, the point ideal ballean xi can be considered as one of the possible asymptotic versions of the discrete space x, see section 2. with these observations, one can look at exp xi as a counterpart of the vietoris hyperspace of x, and the tikhonov hyperspace of x has two counterparts c(x,i) and exp xi-ary. these parallels are especially evident in the case of the ideal fx of finite subsets of x. 1.3. main results. in this paper we focus on hyperballeans of balleans defined by means of ideals, these are the point ideal balleans and the i-ary balleans. it is known ([11]) that the point ideal balleans are precisely the thin balleans. inspired by this fact, in §2, we give some further equivalent properties (theorem 2.2). 1sometimes we refer to c(x,i) as the i-cartesian ballean. its ballean structure makes both ring operations on c(x,i) coarse maps, while exp(xi-ary) fails to have this property. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 436 balleans, hyperballeans and ideals as already anticipated the main objects of study will be the hyperballeans exp(xi), exp(xi-ary) and c(x,i), where i is an ideal of a set x. by restriction, we will gain also knowledge of their subballeans x[i, x [ i-ary and the i-macrocube m(x,i). since xi, xi-ary and c(x,i) are pairwise different, it is natural to ask whether their hyperballeans are different or not. section §3 is devoted to answering this question, comparing these three balleans from various points of view. in particular, we prove that exp(xi) and exp(xi-ary) are different (corollary 3.2), although they have asymorphic subballeans (theorem 3.3), the same holds for the pair exp xi-ary and c(x,i). moreover, we show that c(x,i) (and in particular, exp∗(xi-ary)) is coarsely equivalent to a subballean of exp(xi); so m(x,i) (and in particular, x[i-ary) is coarsely equivalent to a subballean of x[i the final part of the section is dedicated to a special class of ideals defined as follows. for a cardinal κ and λ ≤ κ consider the ideal kλ = {f ⊆ κ | |f| < λ} of κ. a relevant property of this ideal is homogeneity (i.e., it is invariant under the natural action of the group sym(κ) by permutations of κ 2). for the sake of brevity denote by k the ideal kκ of κ. theorem 3.5 provides a bijective coarse embedding of a subballean of exp(κk) into exp(κk-ary) and, under the hypothesis of regularity of κ, also exp(κk) itself asymorphically embeds into exp(κk-ary). to measure the level of disconnectedness of a ballean b, one can consider the number dsc(b) of connected components of b. although the two hyperballeans exp(xi) and exp(xi-ary) are different, they have the same connected components and in particular, dsc(exp(xi)) = dsc(exp(xi-ary)). moreover, this cardinal coincides with dsc(c(x,i)) + 1 (proposition 4.1). the main goal of section §4 is to compute the cardinal number dsc(c(x,i)). to this end we use a compact subspace i∧ of the stone-čech remained βx\x of the discrete space x. in this terms, dsc(c(x,i)) = w(i∧). 2. characterisation of thin connected balleans let b = (x,p,b) be a bounded ballean. then b is thin. moreover, [(x) = p(x), while every proper subset of x is non-thick and the only small subset is the empty set. hence, we now focus on unbounded balleans. it is known ([11]) that a connected unbounded ballean b is thin if and only if the identity mapping of x defines an asymorphism between b and its satellite ballean. it was also shown that these properties are equivalent to having all functions f : x → {0, 1} being slowly oscillating (such a function is called slowly oscillating if, for every α ∈ p there exists a bounded subset v such that |f(b(x,α))| = 1 for 2consequently, all these permutations become automatically asymorphisms, once we endow κ with the point ideal ballean or the kλ-ary ideal structure. one can easily see that these are the only homogeneous ideals of κ. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 437 d. dikranjan, i. protasov, k. protasova and n. zava each x ∈ x\v ; this is a specialisation (for {0, 1}-valued functions) of the usual more general notion, [11]). theorem 2.2 provides further equivalent properties. for a ballean b = (x,p,b), we define a mapping c : x →p(x) by c(x) = x \{x}. lemma 2.1. let b = (x,p,b) be a connected unbounded ballean. if y is a subset of x, then c(y ) is bounded in exp(b) if and only if there exists α ∈ p such that |b(y,α)| > 1, for every y ∈ y . proof. (→) since c(y ) is bounded in exp(b), there exists α ∈ p such that, for every x,y ∈ y with x 6= y, c(y) ∈ exp b(c(x),α). hence y ∈ x \{x} ⊆ b(x\{y},α) and x ∈ x\{y}⊆ b(x\{x},α), in particular, y ∈ b(y \{y},α) and x ∈ b(y \{x},α), from which the conclusion descends. (←) since, for every y ∈ y , there exists z ∈ y \{y} such that y ∈ b(z,α), c(y) ∈ exp b(x,α). hence c(y ) ⊆ exp b(x,α), and the latter is bounded. � if b is a ballean, denote by bm the subballean of expb whose support is the family of all non-thick non-empty subsets of x. if b is unbounded, then so it is bm. moreover, b[ is a subballean of bm. this motivation for the choice of m comes from the fact that non-thick subsets3 were called meshy in [5] (this term will not be adopted here). theorem 2.2. let b = (x,p,b) be an unbounded connected ballean. then the following properties are equivalent: (i) b is thin; (ii) b = bi, where i = [(x), i.e., b coincides with its satellite ballean; (iii) if a ⊆ x is not thick, then a is bounded; (iv) bm is connected; (v) the map c : x →p(x) is an asymorphism between x and c(x); (vi) every function f : x →{0, 1} is slowly oscillating. proof. the implication (iii)→(iv) is trivial, since item (iii) implies that bm = b[ and the latter is connected. furthermore, (i)↔(ii) and (i)↔(vi) have already been proved in [11]. (iv)→(iii) assume that a ⊆ x is not thick. fix arbitrarily a point x ∈ x. the singleton {x} is bounded, hence non-thick. by our assumption, bm is connected and both a and {x} are non-thick, so there must be a ball centred at x and containing a. therefore, a is bounded. (v)→(i) if b is not thin then there is an unbounded subset y of x satisfying lemma 2.1. since c(y ) is bounded in expb, we see that c is not an asymorphism. (ii)→(v) on the other hand, suppose that b = bi. fix a radius v ∈ i. without loss of generality, suppose that v has at least two elements. now, 3or, equivalently, those subsets whose complement is large c© agt, upv, 2019 appl. gen. topol. 20, no. 2 438 balleans, hyperballeans and ideals pick an arbitrary point x ∈ x. if x ∈ v , then c(bi(x,v )) = {x \{y} | y ∈ v}={a ∈ c(x) | x \ (v ∪{x}) ( a ⊆ x} = expbi(c(x),v ) ∩c(x). if, otherwise, x /∈ v , then c(bi(x,v )) = {x \{x}}={a ∈ c(x) | (x \{x}) \v ( a ⊆ x \{x}} = expbi(c(x),v ) ∩c(x). (i)→(iii) suppose that b is thin and a is an unbounded subset of x. we claim that a is thick. fix a radius α ∈ p and let v ⊆ x be a bounded subset of x such that b(x,α) = {x}, for every x /∈ v . since a is unbounded, there exists a point xα ∈ a\v . hence b(xα,α) = {xα}⊆ a, which shows that a is thick. (iii)→(vi) assume that x does not satisfy (vi), i.e, x has a non-slowlyoscillating function f : x → {0, 1}. take a radius α such that, for every bounded v , there exists x ∈ x \ v such that |f(b(x,α))| = 2. hence a = {x ∈ x | |f(b(x,α))| = 2} is unbounded. decompose a as the disjoint union of a0 = {x ∈ a | f(x) = 0} and a1 = {x ∈ a | f(x) = 1}. since a = a0 ∪a1, either a0 or a1 is unbounded. moreover, for every x ∈ a, both a0 ∩ b(x,α) 6= ∅ and a1 ∩ b(x,α) 6= ∅ and thus a0 and a1 are not thick. � remark 2.3. (i) let us see that one cannot weaken item (iii) in the above theorem by replacing “non-thick” by the stronger property “small”. in other words, a ballean need not be thin provided that all its small subsets are bounded. to this end consider the ω-universal ballean (see [11, example 1.4.6]): an infinite countable set x, endowed with the radii set p = {f : x → [x]<∞ | x ∈ f(x), {y ∈ x | x ∈ f(y)}∈ [x]<∞, ∀x ∈ x}, and b(x,f) = f(x), for every x ∈ x and f ∈ p . since it is maximal (i.e., it is connected, unbounded and every properly finer ballean structure is bounded) by [11, example 10.1.1], then every small subset is finite (by application of [11, theorem 10.2.1]), although it is not thin. (ii) let b be an unbounded connected ballean and x be its support. consider the map cb : b[ → expb such that cb(a) = x \a, for every bounded a. it is trivial that c = cb �x, where x is identified with the family of all its singletons. hence, if cb is an asymorphic embedding, then c is an asymorphic embedding too, and thus b is thin, according to theorem 2.2. however, we claim that cb is not an asymorphic embedding if b is thin and then item (v) in theorem 2.2 cannot be replaced with this stronger property. since b is thin, we can assume that b coincides with its satellite bi (theorem 2.2). fix a radius v ∈i of exp xi and suppose, without loss of generality, that v has at least two elements. for every radius w ∈ i of x[i, pick an element c© agt, upv, 2019 appl. gen. topol. 20, no. 2 439 d. dikranjan, i. protasov, k. protasova and n. zava aw ∈i such that aw ⊆ x \(w ∪v ). hence, cb−1(exp bi(cb(aw ),v )) 6⊆ b[i(aw ,w) = {aw}, which implies that cb is not effectively proper. in fact, since aw ∪v ∈i, exp bi(cb(aw ),v ) = {z ⊆ x | x \ (aw ∪v ) ( z ⊆ x \aw}⊆ cb(x[i), and thus |exp bi(cb(aw ),v ) ∩cb(x[i)| > 1. a characterization of thin (and coarsely thin) balleans in terms of asymptotically isolated balls can be found in [8, theorems 1, 2]. 3. further properties of exp(xi), exp(xi-ary) and c(x,i) let f : x → y be a map between sets. then there is a natural definition for a map exp f : p(x) → p(y ), i.e., exp f(a) = f(a), for every a ⊆ x. if f : x → y is a map between two balleans such that f(a) ∈ [(y ), for every a ∈ [(y ) (e.g., a coarse map), then the restriction f[ = exp f �x[ : x[ → y [ is well-defined. the following proposition can be easily proved. proposition 3.1. let bx = (x,px,bx) and by = (y,py ,by ) be two balleans and let f : x → y be a map between them. then: (i) f : bx → by is coarse if and only if exp f : expbx → expby is coarse if and only if f[ : b[x →b [ y is well-defined and coarse; (ii) f : bx → by is a coarse embedding if and only if exp f : expbx → expby is a coarse embedding if and only if f[ : b[x →b [ y is well-defined and a coarse embedding. for the sake of simplicity, throughout this section, for every ideal i of a set x, the ballean c(x,i) will be identified with −1(c(x,i)) (where  is defined in remark 1.6), whose support is p(x). hence, by this identification, if a ⊆ x and k ∈i, bc(a,k) = {y | a\k ⊆ y ⊆ a∪k}. corollary 3.2. for every ideal i on x, the following statements hold: (i) j = exp idx : exp xi → exp xi-ary is coarse, but it is not an asymorphism; (ii) j : exp xi-ary → c(x,i) is coarse, but it is not an asymorphism; (iii) the same holds for the restriction i = id[x : x [ i → x [ i-ary. proof. since idx : xi → xi-ary is coarse, but it is not effectively proper (proposition 1.3), items (i) and (iii) follow from propositions 3.1. item (ii) descends from the fact that exp(xi-ary) �p(x)\{∅}= c(x,i) �p(x)\{∅}, and qexp xi-ary (∅) = {∅}, while qc(x,i)(∅) = i. � in spite of corollary 3.2, we show now that a cofinal part of exp(xi) asymorphically embeds in exp(xi-ary). c© agt, upv, 2019 appl. gen. topol. 20, no. 2 440 balleans, hyperballeans and ideals for every ideal i on x and x ∈ x consider the families ux = {u ⊆ x | x ∈ u}, the principal ultrafilter of p(x) generated by {x}, and ix = ux ∩i = {f ∈i | x ∈ f}. theorem 3.3. for every ideal i on x and x ∈ x, the following statements hold: (i) if j : exp(xi) → exp(xi-ary) is the map defined in corollary 3.2(i), its restriction j �ux is an asymorphism between the corresponding subballeans; (ii) c(x,i) and, in particular, exp∗(xi-ary) are coarsely equivalent to the subballean of exp(xi) with support ux, witnessed by the map j �ux : exp(xi) �ux→ exp ∗(xi-ary) ⊆ c(x,i). proof. (i) for every c ∈ux and a ∈ix, we have exp bi(c,a) ∩ux = {(c \a) ∪y | x ∈ y ⊆ a} = exp bi-ary(c,a) ∩ux, since c ∩a 6= ∅. hence the conclusion follows by remark 1.4(i), since ix is a cofinal subset of radii of i. (ii) in view of item (i), it remains to see that j(ux) is large in c(x,i). indeed, for every a ∈ux, bc(a,{x}) = {a,a\{x}}, and so bc(j(ux),{x}) = c(x,i), where {x}∈i. � since x[i, x [ i-ary, and m(x,i) are subballeans of exp(xi), exp(xi-ary), and c(x,i) respectively, by taking by restrictions we obtain the following immediate corollary. corollary 3.4. for every ideal i on x and x ∈ x, the following statements hold: (i) j �ix is an asymorphism between the corresponding subballeans of x [ i and x[i-ary; (ii) m(x,i) and, in particular, x[i-ary are coarsely equivalent to the subballean of x[i with support ix, witnessed by the map j �ix : x [ i �ix→ x[i-ary ⊆ m(x,i). 3.1. c(κ,k) and the hyperballeans exp(κk) and exp(κk-ary). now we focus our study on some more specific ideals. for an infinite cardinal κ and for its ideal k = [κ]<κ = {z ⊂ κ | |z| < κ}, consider the two balleans κk and κk-ary. here we investigate some relationships between hyperballeans of those two balleans and the ballean c(κ,k). furthermore, with κ as above, if x < κ, put u≥x = {a ⊆ κ | min a = x} and k≥x = u≥x ∩k = {a ∈k | min a = x}. for every pair of ordinals α ≤ β < κ, let [α,β] = {γ ∈ κ | α ≤ γ ≤ β}. clearly, the cardinal κ is regular if and only if the family pint = {[0,α] | α < κ} is c© agt, upv, 2019 appl. gen. topol. 20, no. 2 441 d. dikranjan, i. protasov, k. protasova and n. zava cofinal in k. for a ballean b = (x,p,b), two subsets a,b of x are called close if a and b are in the same connected component of expb. theorem 3.5. let κ be an infinite cardinal and x < κ. then: (i) the subballean u≥x of exp(κk) is asymorphic to c(κ,k), so exp∗(κk) is the disjoint union of κ pairwise close copies of c(κ,k); (ii) if κ is regular, then exp(κk) asymorphically embeds into exp(κk-ary). proof. (i) fix a bijection g : κ →a, where a is the family of all ordinals α such that x < α < κ. define a map f : c(κ,k) → u≥x ⊆ exp(κk) such that, for every x ⊆ κ, f(x) = g(x)∪{x}. we claim that f is the desired asymorphism. in order to prove it, we want to apply remark 1.4(ii). fix a radius k ∈k (i.e., |k| < κ). then, for every x ⊆ κ, f(bc(x,k)) = f({y ⊆ κ | x \k ⊆ y ⊆ x ∪k}) = = {f(g−1(z)) | x \k ⊆ g−1(z) ⊆ x ∪k} = = {z ∪{x} | g(x) \g(k) ⊆ z ⊆ g(x) ∪g(k)} = = {w ∈u≥x | (g(x) ∪{x}) \ (g(k) ∪{x}) ( w ⊆ ⊆ (g(x) ∪{x}) ∪ (g(k) ∪{x})} = = exp bk(g(x) ∪{x},g(k) ∪{x}) ∩u≥x = exp bk(f(x),g(k) ∪{x}) ∩u≥x. if we show that {g(k) ∪ {x} | k ∈ k} is cofinal in f(κ) = u≥x, then the conclusion follows, since we can apply remark 1.4(ii) by putting ψ(k) = g(k)∪ {x}, for every k ∈k. it is enough to check that, for every x ⊆ κ and k ∈k, exp bk(x,k) ∩u≥x = exp bk(x,ψ(k)) ∩u≥x, which proves the cofinality of ψ(k) in f(κ), in virtue of remark 1.4(iii). the last assertion of item (i) follows from the facts that the family {u≥x | x < κ} is a partition of exp(κk), and, for every x < y < κ, u≥x ∈ exp bk(u≥y, [x,y]), where [x,y] ∈k. (ii) every ordinal α ∈ κ can be written uniquely as α = β + n, where β is a limit ordinal and n is a natural number. we say that α is even (odd) if n is even (odd). we denote by e the set of all odd ordinals from κ and fix a monotonically increasing bijection ϕ: κ → e. for each non-empty f ⊆ κ, let yf ∈ κ such that yf +1 = min ϕ(f) and define f(f) = {yf}∪ϕ(f). moreover, we set f(∅) = ∅. let s = f(exp(κk)). hence the elements of s are the empty set and those subsets a of κ, consisting of odd ordinals and precisely one even ordinal α ∈ a such that α = min a. we claim that f : exp(κk) → s is an asymorphism. since κ is regular, pint ⊆k is a cofinal subset of radii. now fix [0,α] ∈ pint. take an arbitrary subset a of κ. we can assume a to be non-empty, since in that case, there is nothing to be proved. the thesis follows, once we prove that c© agt, upv, 2019 appl. gen. topol. 20, no. 2 442 balleans, hyperballeans and ideals (3.1) f(exp bk(a, [0,α])) = exp bk-ary(f(a), [0,ϕ(α)]) ∩s, since we can apply remark 1.4(i) if we define the bijection ψ([0,β]) = [0,ϕ(β)], for every β < κ, between cofinal subsets of radii. if a ∩ [0,α] = ∅, then also f(a) and [0,ϕ(α)] are disjoint, which implies that exp bk-ary(f(a), [0,ϕ(α)]) ∩s = {f(a)}. otherwise, suppose that a and [0,α] are not disjoint. in particular ya ∈ [0,ϕ(α)]. we divide the proof of (3.1) in this case in some steps. first of all we claim that, for every ∅ 6= z ⊆ κ, (3.2) if a\ [0,α] ⊆ z ⊆ a∪ [0,α], then: z 6= a\ [0,α] if and only if yz ≤ ϕ(α). in fact, if z = a \ [0,α], then min z > α and so min ϕ(z) > ϕ(α). since ϕ(α) ∈ e, ϕ(z) ⊆ e, and yz /∈ e, we have that yz > ϕ(α). conversely, if z 6= a \ [0,α], there exists z ∈ z ∩ [0,α], since z ⊆ a ∪ [0,α]. hence min z ≤ z ≤ α and thus yz < min ϕ(z) ≤ ϕ(α). fix now a subset z ⊆ κ. if f(z) ∈ exp bk-ary(f(a), [0,ϕ(α)]), then, by applying the definitions, ϕ(a)\[0,ϕ(α)] = f(a)\[0,ϕ(α)] ⊆ ϕ(z)∪{yz}⊆ f(a)∪[0,ϕ(α)] = ϕ(a)∪[0,ϕ(a)]. note that ϕ(z) ⊆ e and yz /∈ e. hence ϕ(a\ [0,α]) = ϕ(a) \ϕ([0,α]) ⊆ ϕ(z) ⊆ ϕ(a) ∪ϕ([0,α]) = ϕ(a∪ [0,α]) and yz ≤ ϕ(α). since ϕ is a bijection, we can apply (3.2) and obtain that a \ [0,α] ( z ⊆ a∪ [0,α], which means that z ∈ exp bk(a, [0,α]). hence we have proved the inclusion (⊇) of (3.1). since all the previous implications can be reverted, then (3.1) finally follows. � corollary 3.6. let κ be an infinite cardinal and x < κ. then: (i) the subballean k≥x of κ[k is asymorphic to m(κ,k), so κ [ k is the disjoint union of κ pairwise close k-macrocubes m(κ,k); (ii) if κ is regular then κ[k asymorphically embeds into κ [ k-ary. the proof of item (ii), specified for κ = ω, can be found in [10]. 4. the number of connected components of exp(xi) recall that dsc(b) denotes the number of connected components of a ballean b. clearly, (4.1) dsc(exp(b)) = dsc(exp∗(b)) + 1 ≥ 2 for every non-empty ballean b. we begin with the following crucial observation. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 443 d. dikranjan, i. protasov, k. protasova and n. zava proposition 4.1. for an ideal i on a set x, one has (i) the non-empty subsets y,z of x are close in exp(xi−ary) if and only if y4z ∈i; (ii) two functions f,g ∈ cx are close in c(x,i) if and only if supp f4supp g ∈ i; (iii) for every a ⊆ x, qexp(xi)(a) = qexp(xi−ary)(a), and in particular, (4.2) dsc(exp(xi)) = dsc(exp(xi-ary)) and dsc(exp∗(xi)) = dsc(exp ∗(xi-ary)) = dsc(c(x,i)) (iv) dsc(exp(xi)) = dsc(c(x,i)) + 1. proof. (i) two non-empty subsets y and z of exp(xi-ary) are close if and only if there exists k ∈i such that y ∈ exp bi-ary(z,k), i.e., (4.3) y ⊆ z ∪k and z ⊆ y ∪k. if (4.3) holds, then y4z = (y \z) ∪ (z \y ) ⊆ ((z ∪k) \z) ∪ ((y ∪k) \y ) = k ∈i. conversely, if y4z ∈i, then k = y4z trivially satisfies (4.3). (ii) let y = supp f and z = supp g. if both f,g are non-zero, then y,z are non-empty and the assertion follows from (i) and the asymorphism between exp∗(xi-ary) and c(x,i) \ {0}. if g = 0, then f is close to g if and only if f ∈ (i), i.e., y = −1(f) ∈i. as z = ∅, this proves the assertion in this case as well. (iii) fix a subset a of x. the inclusion qexp(xi)(a) ⊆ qexp(xi-ary)(a) follows from corollary 3.2(i). let us check the inclusion qexp(xi)(a) ⊇ qexp(xi−ary)(a). if a = ∅, the claim is trivial, since qexp y (∅) = {∅}, for every ballean y . otherwise, fix an element x ∈ a. let c ∈ exp bi-ary(a,k), for some k ∈ i. then c 6= ∅, so we can fix also a point y ∈ c and let k′ = k ∪{x,y}∈i. then c ⊆ bi-ary(a,k) = a∪k = bi(a,k′) and a ⊆ bi-ary(c,k) = c ∪k = bi(c,k′), which shows that c ∈ exp bi(a,k′). hence, c ∈qexp(xi)(a). this proves the equality qexp(xi)(a) = qexp(xi−ary)(a). it implies the first as well as the second equality in (4.2). to prove the last equality in (4.2), it suffices to note that qc(x,i)(0) = i, by virtue of (ii). hence, dsc(c(x,i)) = dsc(c(x,i) \{0}). to conclude, use the fact that exp∗(xi-ary) is asymorphic to c(x,i) \{0}. item (iv) follows from (iii) and (4.1) applied to b = exp(xi). � proposition 4.1 allows us to reduce the computations of the number of connected components of all hyperballeans involved to the computation of the cardinal dsc(c(x,i). in the sequel we simply identify cx with the boolean ring c© agt, upv, 2019 appl. gen. topol. 20, no. 2 444 balleans, hyperballeans and ideals p(x). so that functions f ∈ cx are identifies with their support and the ideals i of x are simply the proper ideals of the boolean ring cx = {0, 1}x = zx2 , containing ⊕ x z2. according to proposition 4.1, the connected components of c(x,i) are precisely the cosets f +i of the ideal i, therefore, dsc(c(x,i)) coincides with the cardinality of the quotient ring cx/i: (4.4) dsc(c(x,i)) = |cx/i| = |p(x)/i|. in particular, for every infinite set x and an ideal i of x one has dsc(c(x,i)) = 2 if and only if i is a maximal ideal. this is an obvious consequence of (4.4) as |cx/i| = 2 if and only if the ideal i is maximal. remark 4.2. the cardinality |cx/i| is easy to compute in some cases, or to get at least an easily obtained estimate for |cx/i| from above as we see now. to this end let ι(i) = min{λ | i is an intersection of λ maximal ideals}. then (4.5) |cx/i|≤ min{2ι(i), 2|x|}. indeed, if i = ⋂ {mi | i < ι(i)}, where mi are maximal ideals of b, then b/i embeds in the product ∏ i<ι(i) cx/mi having size ≤ 2 ι(i) as cx/mi ∼= z2 for all i. to conclude the proof of (4.5) it remains to note that obviously |cx/i|≤ |cx| = 2|x|. if ι(i) = n is finite, then dsc(c(x,i)) = 2n. indeed, now i = m1 ∩·· ·∩mn is a finite intersection of maximal ideals and the chinese remainder theorem, applied to the boolean ring cx and the maximal ideals m1, . . . ,mn, provides a ring isomorphism cx/i ∼= n∏ i=1 c/mi ∼= zn2 . in particular, |cx/i| = |zn2 |= 2n. by (4.4), we deduce (4.6) dsc(c(x,i)) = 2n. let us conclude now with another example. for every infinite set x and the ideal i = fx one has dsc(exp(xfx )) = dsc(c(x,fx) = 2 |x|. this follows from (4.4) and |fx| = |x| < 2|x|, which implies |cx/fx| = |cx| = 2|x|. in order to obtain some estimate from below for |cx/i|, we need a deeper insight on the spectrum spec cx of cx. since cx is a boolean ring, spec cx coincides with the space of all maximal ideals of cx, which can be identified with the stone–čech compactification βx when we endow x with the discrete topology. as usual, c© agt, upv, 2019 appl. gen. topol. 20, no. 2 445 d. dikranjan, i. protasov, k. protasova and n. zava • we identify the stone–čech compactification βx with the set of all ultrafilters on x; • the family {a | a ⊆ x}, where a = {p ∈ βx | a ∈ p}, forms the base for the topology of βx; and • the set x is embedded in βx by sending x ∈ x to the principal ultrafilter generated by x. for a filter ϕ on x, define a closed subset ϕ of βx as follows: ϕ = ⋂ {a | a ∈ ϕ}. an ultrafilter p ∈ βx belongs to ϕ if and only if p contains the filter ϕ. in other words, (4.7) ϕ = ⋂ {u | u ∈ ϕ}. for an ideal i on x, we consider the filter ϕi = {x\a | a ∈ i}, and we simply write ϕ when there is no danger of confusion. similarly, for a filter ϕ we define the ideal iϕ = {x\a | a ∈ ϕ} and we simply write i when there is no danger of confusion. finally, let i∧ = ϕi, and note that all ultrafilters in i∧ are non-fixed, i.e., i∧ ⊆ βx \x, as ϕ is contained in the fréchet filter of all co-finite sets on x (since i ⊇ fx). moreover, for a subset a of x one has (4.8) a ∈i if and only if a 6∈ u for all u ∈ ϕi = i∧. as pointed out above, for any x the compact space βx coincides with the spectrum spec cx of the ring cx. for an ideal i on x, ϕi is the set of ultrafilters on x containing ϕ. for u ∈ ϕi the ideal iu is maximal and contains i. more precisely, i = ⋂ u∈i∧ iu. the maximal ideals iu, when u runs over ϕ, bijectively correspond to the maximal ideals of the quotient cx/i; in particular, |i∧| = |spec(cx/i)|. along with remark 4.2, this gives: proposition 4.3. let i be an ideal on set x. if |i∧| = n is finite, then dsc(c(x,i)) = 2n. otherwise, dsc(c(x,i)) = w(i∧). here w(i∧) denotes the weight of the space i∧. the second assertion follows from (4.4) and the equality w(i∧) = |p(x)/i|, its proof can be found in [3, §2]. corollary 4.4. let i be an ideal on a countably infinite set set x such that i∧ is infinite. then dsc(c(x,i)) = 2ω. proof. being an infinite compact subset of βx \x, i∧ contains a copy of βn. therefore, w(i∧) = 2ω. now proposition 4.3 applies. � in this section we have thoroughly investigated the number of connected components of exp(xi), where i is an ideal of a set x. this leaves open the question to estimate dsc(exp(x)), where x is an arbitrary connected ballean. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 446 balleans, hyperballeans and ideals let y be a subballean of x. in particular, dsc(exp(x)) ≥ dsc(exp(y )). if y is thin, we can apply theorem 2.2 so that y = y[(y ). the results from this section give a lower bound of dsc(exp(y )), providing in this way also a lower bound for dsc(exp(x)), since (4.9) dsc(exp(x)) ≥ sup{dsc(exp(z)) | z is a thin subballean of x}. unfortunately, (4.9) doesn’t provide any useful information in the case when every thin subballean of x is bounded. in fact, if z is a non-empty bounded subballean, then exp∗(z) is connected and so dsc(exp(z)) = 2. acknowledgements. the first named author thankfully acknowledges partial financial support via the grant prid at the department of mathematical, computer and physical sciences, udine university. references [1] t. banakh, i. protasov, d. repovs̆ and s. slobodianiuk, classifying homogeneous cellular ordinal balleans up to coarse equivalence, arxiv: 1409.3910v2. 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[9] i. protasov and t. banakh, ball structures and colorings of groups and graphs, mat. stud. monogr. ser 11, vntl, lviv, 2003. [10] i. protasov and k. protasova, on hyperballeans of bounded geometry, arxiv:1702.07941v1. [11] i. protasov and m. zarichnyi, general asymptology, 2007 vntl publishers, lviv, ukraine. [12] j. roe, lectures on coarse geometry, univ. lecture ser., vol. 31, american mathematical society, providence ri, 2003. [13] n. zava, on f-hyperballeans, work in progress. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 447 @ applied general topology © universidad polité ni a de valen iavolume 12, no. 2, 2011pp. 221-225 density of κ-box-produ ts and the existen e ofgeneralized independent familiesstefan ottmar elserabstra tin this paper we will prove a slight generalisation of the hewitt-mar zewski-pondi zery theorem (theorem 2.3 below) on erning thedensity of κ-box-produ ts. with this result we will prove the existen eof generalized independent families of big ardinality ( orollary 2.5 be-low) whi h were introdu ed by wanjun hu.2010 msc: 54a25, 54b10; se ondary: 03e05.keywords: κ-box-produ t, generalized independent family.1. introdu tionlet d(x) denote the density and w(x) the weight of the topologi al spa e x.de�nition 1.1. let µ,κ be two ardinals with ℵ0 ≤ κ ≤ µ and {xi}i∈µ be afamily of topologi al spa es. � κ i∈µxi denotes the κ-box-produ t whi h is indu ed on the full artesian produ t ∏ i∈µ xi by the anoni al base b = { ⋂ i∈i pr −1 i (ui);i ∈ p<κ(µ) and ui is open in xi}where p<κ(µ) := {i ⊆ µ; |i| < κ}.for κ = ℵ0 the κ-box-produ t is the usual ty hono�-produ t [8℄ and for κ+ = µ the κ-box-produ t is the full box-produ t mentioned by kelley [5℄ andbourbaki [1℄. 222 s. o. elserin this paper we will dis uss the density of κ-box-produ ts and the onne -tion with in�nite ombinatori s. the lassi al hewitt-mar zewski-pondi zerytheorem states: d ( � ℵ0 i∈2µxi ) ≤ µ for all spa es xi with d(xi) ≤ µthis has been proven for separable spa es by e. mar zewski [6℄ in 1941. in1944 e. s. pondi zery [7℄ proved a slighty weaker version for hausdor� spa esand in 1947 e. hewitt [3℄ proved the general version as stated above.in theorem 2.4 we will prove: d(�κi∈2µxi) ≤ µ <κ for all spa es xi with d(xi) ≤ µ2. density of κ-box-produ tsin this se tion we will prove a generalisation of theorem 1 in [2℄. to do sowe start with the following de�nition and proposition:de�nition 2.1. let κ,µ be two in�nite ardinals with µ ≥ κ, {xi}i∈i a familyof topologi al spa es and for all i ∈ i let bi be a base of the topology on xi. w ⊆ ∏ i∈i xi is alled a µube if for every i ∈ i there exists wi ⊆ bi with w = ∏ i∈i ( ⋂ wi).proposition 2.2. let x be a set, µ ≥ κ two in�nite ardinals, {xi}i∈i afamily of topologi al spa es, {fi : x → xi}i∈i a family of fun tions and let wbe a subset of ∏i∈i xi whi h is a union of µubes.for every ardinal λ < κ and every tuple 〈{xi}i∈λ ; {ji}i∈λ〉 of families {xi}i∈λ ⊆ x and {ji}i∈λ ⊆ p(i), where all ji are pairwise disjun t and not empty, thereexists a subset q ⊆ w of ardinality less or equal to µ<κ so that for all families {ji;ji ∈ ji}i∈λ the following holds: ( w ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) ⇒ ( q ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) .proof. for every tuple 〈{xi}i∈λ ; {ji}i∈λ〉 with |{i ∈ λ; |ji| > 1}| = 0 the laimis pretty obvious.so we assume that the proposition is valid for ardinals less than ν and let 〈 {xi}i∈λ ; {ji}i∈λ 〉 be a tuple with |{i ∈ λ; |ji| > 1}| = ν.without loss of generality we may assume that |ji| > 1 for all i ∈ ν and |ji| = 1for all other i ≥ ν and that there exists at least one family {ji;ji ∈ ji}i∈λ with w ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅.let p ∈ w be an point so that prji(p) ∈ fji(xi) for all ν ≤ i ∈ λ.then there exists an j ∈ p≤µ(i) with { q ∈ ∏ i∈i xi; ∀j ∈ j : prj(q) = prj(p) } ⊆ w.we hoose for all i ∈ ν and ji ∈ (ji − j) a point qji ∈ fji(xi) and we de�ne apoint q ∈ w as follows: density of κ-box-produ ts 223 pri(q) := { pri(p) , if i ∈ (i − ⋃l∈ν(jl − j)) qjl , if i = jl and jl ∈ (jl − j)by the de�nition of q we have q ∈ (w ∩ ⋂ i∈λ pr −1 ji (fji(xi)) ) for every family {ji;ji ∈ ji}i∈λ su h that for all i ∈ ν: ji ∈ (ji − j).now we have to onsider families {ji;ji ∈ ji}i∈λ with ji ∈ (ji ∩ j) for atleast one i ∈ λ.we de�ne σ := { {j∗i }i∈ν ; |{i ∈ κ;j ∗ i = ji}| < ν ∧ (j ∗ i 6= ji ⇒ j ∗ i ∈ p1(ji ∩ j)) } . ⇒ |σ| ≤ µν ≤ µλ ≤ µ<κfor all σ = {j∗i }i∈ν ∈ σ we de�ne a family {jσi }i∈λ as follows: jσi := { j∗i , if i ∈ ν ji , if i ≥ νfor all these {jσi }i∈λ the proposition already holds, so we an hoose a set qσ ⊆ w with |qσ| ≤ µ<κ and for all families {ji;ji ∈ jσi }i∈λ the followingholds: ( w ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) ⇒ ( qσ ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) .let σ = {ji;ji ∈ ji}i∈ν be a family with w ∩ ⋂i∈λ pr−1ji (fji (xji)) 6= ∅.then σ ∈ σ and qσ ∩ ⋂i∈λ pr−1ji (fji (xji)) 6= ∅.we de�ne q := {q} ∪ ⋃ σ∈σ qσand be ause |q| ≤ µ<κ this is the set we were looking for. �theorem 2.3. let κ and µ be two in�nite ardinals with µ ≥ κ and let �κi∈ixibe a κ-box-produ t with |i| ≤ 2µ and w(xi) ≤ µ for all i ∈ i.then d(w) ≤ µ<κ holds for every subset w ⊆ ∏ i∈i xi whi h is a union of µubes.proof. let |i| = 2µ, so we may assume that i = 2µ.let b∗ be a base of the κ-box-produ t �κi∈µd of the dis rete spa e d = {0; 1}with |b∗| = µ<κ.for all i ∈ 2µ let bi be a base of the topology on xi with |bi| = µ, x be aset with |x| = µ, {fi;fi : x → bi}i∈2µ be a family of surje tive fun tions and ψ : 2µ → ∏ i∈µ d be a bije tion. we de�ne σ := { 〈 {xi}i∈λ ; {ji}i∈λ 〉 ;λ < κ ∧ ∀i,j ∈ λ : xi ∈ x ∧ ∅ 6= ji ⊆ 2 µ ∧ ψ(ji) ∈ b ∗ ∧ (i 6= j ⇒ ji ∩ jj = ∅)} 224 s. o. elserand hoose for every σ ∈ σ a set qσ ⊆ w with all the properties as statedin proposition 2.2. we de�ne q := ⋃ σ∈σ qσ. be ause of |b∗| = µ we have |σ| ≤ µ<κ and therefore |q| ≤ µ<κ. we will now show that q is dense in w .let o be a nonempty open set in w and u an element of the anonial base b of �κi∈2µxi with ∅ 6= u ∩ w ⊆ o. then there exists a set {ji;i ∈ λ} ∈ p<κ(2 µ) and a family {ui;ui ∈ bi}i∈λ with u = ⋂i∈λ pr−1ji (ui).we an hoose for all i ∈ λ pairwise disjun t open sets b∗i ∈ b∗ with ψ(ji) ∈ b∗iand xi ∈ x with fji(xi) = ui.obviously σ := 〈{xi}i∈λ ; {ji}i∈λ〉 is an element of σ and we have the ondi-tion ∅ 6= w ∩ ⋂ i∈λ pr −1 ji (fji(xi)), thus qσ ∩ u 6= ∅ ⇒ q ∩ o ⊇ oσ ∩ w ∩ u = oσ ∩ u 6= ∅therefore q is dense in w and we have d(w) ≤ |q| ≤ µ<κ. �the following is a slight generalisation of the hewitt-mar zewskipondi zerytheorem:theorem 2.4. let κ and λ be two in�nite ardinals with µ ≥ κ and let �κi∈ixia κ-box-produ t with |i| ≤ 2µ and d(xi) ≤ µ for all i ∈ i.then d(�κi∈ixi) ≤ µ<κ.proof. obviously there is a set d whi h is dense in �κi∈ixi and |pri(d)| ≤ µfor all i ∈ i.let �κi∈iwi be the κ-box-produ t of dis rete spa es wi with |wi| = µ andlet f : ∏i∈i wi → d be a ontinuous and surje tive fun tion.be ause ∏ i∈i wi itself is an union of µubes and due to theorem 2.3 there isa dense subset q of w with |q| ≤ µ<κ.let o be a nonempty open set in �κi∈ixi. then d ∩ o 6= ∅ and f−1(d ∩ o)is open in �κi∈iwi.so q ∩ f−1(d ∩ o) 6= ∅ and ∅ 6= f (q ∩ f−1(d ∩ o)) ⊆ f(q) ∩ o.therefore f(q) is dense in �κi∈ixi and d(�κi∈ixi) ≤ µ<κ. �following wanjun hu we de�ne:de�nition 2.5. let s be an in�nite set, κ, λ and θ be three ardinals with κ ≥ ℵ0 and λ ≥ 2. a family i = {iα}α∈τ of partitions iα = {iβα;β ∈ λ} of sis alled a (κ,θ,λ)-generalized independent family, if following holds: ∀j ∈ p<κ(τ)∀f : j → λ : ∣ ∣ ∣ { ⋂ i f(α) α ;α ∈ j } ∣ ∣ ∣ ≥ θwe an now apply 2.4 on this theorem and we re eive the following:corollary 2.6. let κ and λ be two in�nite ardinals with µ ≥ κ.on every set with at least µ<κ elements exists a (κ,1,µ)-generalized indepen-dent family of ardinality 2µ.proof. let s be a set of ardinality µ<κ.for every family {xi}i∈µ of topologi al spa es with d(xi) ≤ λ the following density of κ-box-produ ts 225holds with theorem 2.4: d ( � κ i∈µxi ) ≤ |s|wanjun hu proved in theorem 3.2 in [4℄ that this is equivalent to the existen eof a (κ,1,µ)-generalized independent family of ardinality 2µ on s. �a knowledgements. i am very grateful to prof. dr. ulri h felgner for hissupport and helpful advi e. referen es[1℄ n. bourbaki, livre iii: topologie générale. chapitre 1: stru tures topologiques.chapitre. 2: stru tures uniformes.(2ième edition), (hermann & cie., paris, 1951) p.72.[2℄ r. engelking, cartesian produ ts and dyadi spa es, fund. math. 57 (1965), 287�304.[3℄ e. hewitt, a remark on density hara ters, bull. amer. math. so . 52 (1946), 641�643.[4℄ w. hu, generalized independent families and dense sets of box-produ t spa es, appl.gen. topol. 7, no. 2 (2006), 203�209.[5℄ j. l. kelley, general topology, new york 1955, p. 107.[6℄ e. mar zewski, séparabilité et multipli ation artésienne des espa es topologiques, fund.math. 34 (1937), 127�143.[7℄ e. s. pondi zery, power problems in abstra t spa es, duke math. journ. 11 (1944),835�837.[8℄ a. ty hono�, über die topologis he erweiterung von räumen, math. ann. 102 (1930),544�561. (re eived de ember 2008 � a epted o tober 2009)stefan ottmar elser (stefan.elser�web.de)mathematis hes institut, eberhard karls universität tübingen, auf der mor-genstelle 10, 72076 tübingen, germany density of -box-products and the existence of[5pt] generalized independent families. by s. o. elser () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 207-223 compactification of closed preordered spaces e. minguzzi abstract a topological preordered space admits a hausdorff t2-preorder compactification if and only if it is tychonoff and the preorder is represented by the family of continuous isotone functions. we construct the largest hausdorff t2-preorder compactification for these spaces and clarify its relation with nachbin’s compactification. under local compactness the problem of the existence and identification of the smallest hausdorff t2-preorder compactification is considered. 2010 msc: 54e15 (primary), 54f05, 54e55, 06f30 (secondary). keywords: nachbin compactification, quasi-uniformizable space, completely regularly ordered space. 1. introduction a topological preordered space is a triple (e, t , ≤) where (e, t ) is a topological space and ≤ is a preorder on e, namely a reflexive and transitive relation on e. the preorder is an order if it is antisymmetric. there are many possible compatibility conditions between topology and preorder that can be added to this basic structure. we shall mainly consider the t2-preordered spaces (closed preordered spaces), namely those spaces for which the graph g(≤) = {(x, y) : x ≤ y}, is closed in the product topology t × t of e × e. in this work we shall follow nachbin’s terminology [22] but we remark that in computer science t2-ordered spaces are very much studied and called pospaces. a t2-preordered space e is a t1-preordered space in the sense that for every x ∈ e, i(x) and d(x) are closed where i(x) = {y ∈ e : x ≤ y} is the increasing hull and d(x) = {y ∈ e : y ≤ x} is the decreasing hull. 208 e. minguzzi we recall that an isotone function f : e → r is a function such that x ≤ y ⇒ f(x) ≤ f(y). we shall mostly work with continuous isotone functions with value in [0,1], although we could equivalently work with bounded continuous isotone functions. in this work we shall consider the problem of compactification for t2-preordered spaces. it is understood here that the compactification ce must be endowed with a preorder ≤c which induces ≤ on e, namely if x, y ∈ e, then x ≤ y if and only if x ≤c y. the extended preorder is also demanded to be closed. in the ordered case this problem has been solved by nachbin who proved [4,22,23] that a topological ordered space admits a t2-order compactification if and only if it is a completely regularly ordered space, where a completely regularly preordered space is a topological preordered space for which the following two conditions hold (i) t coincides with the initial topology generated by the set of continuous isotone functions f : e → [0, 1], (ii) x ≤ y if and only if for every continuous isotone function f : e → [0, 1], f(x) ≤ f(y). for future reference let us introduce the equivalence relation x ∼ y on e, given by “x ≤ y and y ≤ x”. let e/∼ be the quotient space, t /∼ the quotient topology, and let . be defined by, [x] . [y] if x ≤ y for some representatives (with some abuse of notation we shall denote with [x] both a subset of e and a point on e/∼). the quotient preorder is by construction an order. the triple (e/∼, t /∼, .) is a topological ordered space and π : e → e/∼ is the continuous quotient projection. nachbin proves [22, prop. 8] that the completely regularly preordered spaces can be characterized as those topological preordered spaces (e, t , ≤) which come from a quasi-uniformity u, in the sense that t = t (u∗) and g(≤) = ⋂ u (see [4,22] for details on quasi-uniformities). note that for these spaces, by (i) above, (e, t ) is completely regular but not necessarily hausdorff (equivalently t1). nevertheless, from (ii) it follows that e is a t2-preordered space, hence t1-preordered thus [x] = d(x)∩i(x) is closed. we conclude that in a completely regularly preordered space, t is t1, and hence (e, t ) is a tychonoff space, if and only if ≤ is an order [22]. in this work we look for topological preordered spaces that admit a hausdorff t2-preordered compactification. since the t2-preorder property is hereditary, and every topological space that admits a hausdorff compactification is tychonoff, the class that we are considering is contained in the family of t2preordered tychonoff spaces. in fact we shall see that all these spaces admit a t2-preorder compactification provided the family of continuous isotone functions determines the preorder. we shall then look for the largest hausdorff t2-preorder compactification and we shall clarify its connection with nachbin’s t2-order compactification. we will end the paper with a discussion of the smallest hausdorff t2-preorder compactification. compactification of closed preordered spaces 209 2. a motivation: the spacetime boundary since the next sections will be particularly abstract, it will be convenient to motivate this study mentioning an application. this author is particularly interested in general relativity, but the reader will easily find other applications in closely related fields, for instance, in dynamical systems theory. this author’s interest for the compactifications of closed preordered spaces comes from the well-known problem of attaching a boundary to a spacetime (physicists term boundary what is known as remainder in topology). we recall that a spacetime is a connected, hausdorff, time oriented lorentzian manifold and is denoted (m, g), where g is the lorentzian metric. in relativity theory the concept of singularity has proved to be quite elusive. one would like to attach a boundary to spacetime so as to distinguish between points at infinity and singularities, where the distinction is made considering the behavior of the riemann tensor near the boundary point (e.g. diverging or not). there have been numerous attempts to construct such a boundary. we mention penrose’s conformal boundary [24], geroch, kronheimer and penrose’s causal boundary [6], scott and szekeres’ abstract boundary [28], and various other proposals by budic and sachs [1], racz [25,26], szabados [30,31], harris [7], flores [5] and others. apart for the case of penrose’s conformal boundary, which cannot be applied in general, one does not demand that spacetime plus the boundary be still a manifold. in general, one wishes just to preserve some notion of continuity and provide a way of extending the causal relation to the boundary. the above constructions are often quite involved. i propose a strategy which takes advantage of the fact that any spacetime is a topological preordered space. let us clarify this point. the causal relation j+ on m is given by the pairs (x, y) of points of m for which there is a c1 curve γ : [0, 1] → m, γ(0) = x, γ(1) = y, which is causal, in the sense that its tangent vector at any point stays in the future causal cone of g. in general j+ might be non-closed, however, there is another relation, intimately connected with j+, which is always closed: the seifert’s relation j+s [18, 29]. the seifert relation turns spacetime into a topological space endowed with a closed relation and, provided some topological conditions are satisfied, it is indeed possible to compactify spacetime along the lines suggested in this work. we do not claim that the compactification constructed in this way, denoted β(e), will be the most physical. indeed, it will add many more points than intuitively required. nevertheless, it will provide an important step since it will dominate any other possible compactification which, therefore, will be obtainable from β(e) through a suitable identification of the boundary points. the possibility of adding a boundary and extending the preorder so as to keep its closure is not known among physicists. it suffices to say that the boundary constructions mentioned above, either apply to very special spacetimes, or do not share this property. 210 e. minguzzi we could also try a different approach by first showing that the spacetime is not only a topological preordered space, but in fact a quasi-pseudo-metric space, and then completing it with a preorder generalization of the cauchy completion. unfortunately, although we could prove, using the results of [20], that most interesting spacetimes are quasi-pseudo-metrizable, the completion would depend on the chosen quasi-pseudo-metric. therefore, this strategy is not entirely viable unless we prove the existence of some natural spacetime quasi-pseudo-metric. let us end this section explaining why we have to generalize nachbin’s compactification to the preordered case, even in those cases in which e is ordered. a key example is provided by misner’s spacetime, a 2-dimensional spacetime which retains several features of the taub-nut spacetime [8]. this spacetime has topology s1 × r and metric g = 2dθdt + tdθ2. the line t = 0 of topology s1 is a closed lightlike geodesic. through any point of the region t ≤ 0 passes a closed causal curve. the topological space e given by the region t ≥ 0 of misner’s spacetime can be endowed with a preorder given by the causal relation. this relation is closed, and the subset t > 0 with the induced topology and preorder is a completely regularly ordered space (indeed it can be shown to be convex and it is normally preordered due to the results of [19]). the set t = 0 represents a natural connected piece which bounds the region t > 0, but nachbin’s compactification cannot dominate a compactification with this piece of boundary since nachbin’s compactification would be ordered while the set t = 0 is a closed null geodesic, and hence any pair of points in this set violates antisymmetry. in summary, although the region t > 0 is ordered, its most natural compactifications are not ordered. evidently, nachbin’s compactification is too restrictive for applications, and the order condition on the compactified space must be relaxed. 3. hausdorff t2-preorder compactifications given two topological preordered spaces (e1, t1, ≤1) and (e2, t2, ≤2) the function h : e1 → e2 is a preorder homeomorphism if h is bijective, continuous and isotone and so is its inverse. we speak of preorder embedding if h is a preorder homeomorphism of e1 on its image h(e1) ⊂ e2, where h(e1) is given the induced topology and induced preorder. we are interested in establishing under which conditions a topological preordered space (e, t , ≤) admits a preorder compactification, namely a preorder embedding c : e → ce into a compact topological preordered space (ce, tc, ≤c) in such a way that c(e) is a dense subset of ce. we shall often identify e with c(e) because c is a preorder homeomorphism between e and c(e). we shall be especially interested in hausdorff t2-preordered compactifications, that is, in those preorder compactifications for which (ce, tc, ≤c) is also a hausdorff t2-preordered space. sometimes we shall write that (ce, tc, ≤c) is a preorder compactification by meaning with this that the map c : e → ce is a preorder compactification. compactification of closed preordered spaces 211 definition 3.1. if c1e, c2e, are two preorder compactifications of e we write c1 ≤ c2 if there is a continuous isotone map c : c2e → c1e such that c◦c2 = c1 (c1 ≤ c2 reads “c2 dominates over c1”). the map c is just an extension to c2e of the preorder homeomorphism c1 ◦ c −1 2 : c2(e) → c1(e). two preorder compactifications are equivalent if c1 ≤ c2 and c2 ≤ c1. we remark that two compactifications may be such that c1e = c2e, c = id, but correspond to different preorders on c1e. in this case c1 ≤ c2 means that, because id must be isotone, g(≤c2) ⊂ g(≤c1) (in our conventions the set inclusion is reflexive). intuitively, to enlarge the compactification means to enlarge the domain ce or to narrow the preorder ≤c or both. from the definition it follows that the relation of domination on the set of all the compactification is a preorder. the next result establishes that it is actually an order provided we pass to the quotient made by the classes of compactifications related by preorder homeomorphisms. proposition 3.2. if two hausdorff preorder compactifications c1, c2, are equivalent, then there is a preorder homeomorphism h : c2e → c1e such that h ◦ c2 = c1. proof. since c1 ≤ c2 there is a continuous isotone map c12 : c2e → c1e such that c12 ◦ c2 = c1 and since c2 ≤ c1 there is a continuous isotone map c21 : c1e → c2e such that c21 ◦ c1 = c2. applying c12 to the latter equation and using the former equation we get c12 ◦ c21 ◦ c1 = c12 ◦ c2 = c1 which implies that c12 ◦c21|c1(e) = idc1e|c1(e). since c1(e) is dense in c1e and c1e is a hausdorff space we have that c12 ◦ c21 = idc1e (e.g. [32, cor. 13.14]). arguing with the roles of 1 and 2 exchanged we get c21 ◦ c12 = idc2e thus c12 and c21 are one the inverse of the other. but they are both isotone thus h := c12 is a preorder homeomorphism. � proposition 3.3. if c1, c2 are two hausdorff preorder compactifications of e and c1 ≤ c2 then the continuous isotone map c : c2e → c1e such that c ◦c2 = c1 satisfies c(c2e) = c1e, c(c2(e)) = c1(e) and c(c2e\c2(e)) = c1e\c1(e). proof. the map c is necessarily onto because c(c2e) is compact and hence closed and the image of c includes c(c2(e)) = c1(e) which is dense in c1e. the preorder compactifications are compactifications so that the last equation follows from [3, theor. 3.5.7]. � let f : e → [0, 1] be a continuous function on a topological space (e, t ), we shall denote by ≤f the total preorder given by “x ≤f y if f(x) ≤ f(y)”. its graph will be denoted with gf . the next proposition establishes some necessary conditions for the existence of a hausdorff t2-preorder compactification. proposition 3.4. if (e, t , ≤) is a subspace of a hausdorff t2-preordered compact space, then e is a t2-preordered tychonoff space and the family of continuous isotone functions f, f : e → [0, 1], is such that x ≤ y if and only if for every f ∈ f, f(x) ≤ f(y) (equivalently g(≤) = ⋂ f∈f gf ). 212 e. minguzzi proof. let e be a subspace of a hausdorff t2-preordered compact space which we denote (e′, t ′, ≤′). since every compact hausdorff space is tychonoff and this property is hereditary, we have that e is tychonoff. the t2-preorder space property is also hereditary thus e is t2-preordered. finally, since every t2preordered compact space is normally preordered [19], for x′, y′ ∈ e′, x′ ≤ y′ iff f(x′) ≤ f(y′) where f : e′ → [0, 1] is any continuous and isotone function on e′ (see e.g. [21, prop. 1.1]). let g be the family of continuous isotone functions, f : e → [0, 1], which come from the restriction of some continuous isotone function f : e′ → [0, 1]. evidently, for x, y ∈ e, x ≤ y iff for every f ∈ g, f(x) ≤ f(y). since f includes g and is made of isotone functions the last claim follows. � 3.1. the largest hausdorff t2-preorder compactification. the next result establishes that the previous necessary conditions are actually sufficient and that there is a hausdorff t2-preordered compactification which dominates over all the other hausdorff t2-preordered compactifications. the locally compact σ-compact hausdorff t2-preordered spaces satisfy these necessary and sufficient conditions [19]. theorem 3.5. let (e, t , ≤) be a t2-preordered tychonoff space, let f be the family of continuous isotone functions f : e → [0, 1], and assume that the preorder is represented by the continuous isotone functions i.e. g(≤) = ⋂ f∈f gf . let β : e → βe be the stone-čech compactification and let f̃ be the set of continuous functions over βe obtained from the (unique) extension1 of the elements of f. there is a largest hausdorff t2-preordered compactification of (e, t , ≤) given by (βe, tβ, ≤β) where g(≤β) = ⋂ f̃∈f̃ gf̃ . every continuous isotone function on e extends to a continuous isotone function on βe. proof. each graph gf̃ is closed because the functions f̃ : βe → [0, 1] are continuous, thus g(≤β) being the intersection of closed sets is closed. further the graphs gf̃ contain the diagonal of βe, thus g(≤β) contains the diagonal. moreover, ≤f̃ is transitive which implies that ≤β is transitive and hence a closed preorder on βe. for every f ∈ f, if x, y ∈ e then f(x) ≤ f(y) iff f̃(x) ≤ f̃(y) thus g(≤) = g(≤β) ∩ (e × e) which proves that (βe, tβ, ≤β) is a preorder compactification. if f : e → [0, 1] is a continuous isotone function on e then its continuous extension to βe, f̃, is such that f̃ ∈ f̃ and by definition of ≤β, g(≤β) ⊂ gf̃ which means that f̃ is isotone. let (ce, tc, ≤c) be another preorder compactification then, since (βe, tβ) is the largest hausdorff compactification [32, theor. 19.9] there is a continuous map h : βe → ce such that h ◦ β = c. the relation on βe, r := (h × h)−1g(≤c) which is clearly reflexive and transitive is also closed in βe × βe because h is continuous. 1note that the extension f̃ is really the extension of f ◦ β−1. compactification of closed preordered spaces 213 the map h extends into a continuous function on βe the preorder homeomorphism c ◦ β−1 : β(e) → c(e) thus r ∩ (β(e) × β(e)) = g(≤β) ∩ (β(e) × β(e)), that is, (β × β)−1r = g(≤). if a function g : βe → [0, 1] is continuous and r-isotone then g ◦ β : e → [0, 1] is continuous and isotone which means that g ∈ f̃ (the extension of a continuous function to a continuous function on βe is unique because β(e) is dense in βe), that is g is also gβ-isotone. since (βe, tβ, r) is a compact t2-preordered space it is normally preordered [19, theor. 2.4] thus r = ⋂ g∈g gg where the intersection is with respect to the family g of all the continuous r-isotone functions on βe. as we have just proved, this family is a subset of f̃ thus g(≤β) ⊂ r. since g(≤β) ⊂ (h × h)−1g(≤c) we conclude that h is isotone and hence that c ≤ β. � theorem 3.6. a hausdorff t2-preorder compactification (ce, tc, ≤c) which shares the properties (a) every continuous function f : e → [0, 1] can be extended to a continuous function on ce, (b) every continuous isotone function f : e → [0, 1] can be extended to a continuous isotone function on ce, is necessarily equivalent to (βe, tβ, ≤β). proof. we already know that c ≤ β because βe is the largest hausdorff t2preorder compactification. since the compactification (ce, tc) shares property (a) it is equivalent with the stone-čech compactification (βe, tβ), in particular there is a continuous map d : ce → βe such that d ◦ c = β. the relation on ce, r := (d × d)−1g(≤β) which is clearly reflexive and transitive is also closed in ce × ce because d is continuous. d extends into a continuous function on ce the preorder homeomorphism β ◦ c−1 : c(e) → β(e) thus r ∩(c(e)× c(e)) = g(≤c)∩(c(e)× c(e)), that is, (c × c)−1r = g(≤). if a function g : ce → [0, 1] is continuous and r-isotone then g ◦ c : e → [0, 1] is continuous and isotone which means by property (b) that g is also gc-isotone (the extension of a continuous function to a continuous function on ce is unique because c(e) is dense in ce). since (ce, tc, r) is a compact t2-preordered space it is normally preordered [19, theor. 2.4] thus r = ⋂ g∈g gg where the intersection is with respect to the family g of all the continuous r-isotone functions on ce. as we have just proved, this family is contained in the family of continuous gc-isotone functions c, ⋂ g∈c gg ⊂ r. finally, note that (ce, tc, ≤c) is also a compact t2-preordered space hence normally preordered and hence with a preorder represented by the continuous gc-isotone functions, g(≤c) = ⋂ g∈c gg, which implies g(≤c) ⊂ r. the inclusion g(≤c) ⊂ (d × d) −1g(≤β) proves that d is isotone and hence that β ≤ c. � adapting the terminology of fletcher and lindgren [4] for ordered compactifications we can say that the next result proves that (βe, tβ, ≤β) is a strict preorder compactification. 214 e. minguzzi theorem 3.7. on (βe, tβ) the closed preorder ≤β is the smallest closed preorder inducing ≤ on e. proof. let ≤r be another closed preorder such that r ∩ (e × e) = g(≤). the map β′ : e → βe, β′ = β, where βe is regarded as the preordered space (βe, tβ, r) is a preorder compactification. since β is the largest β ′ ≤ β, which means that there is a continuous isotone function b : βe → β′e such that b ◦ β = β′. on β(e) the map b coincides with β′ ◦ β−1 = β ◦ β−1 = id, thus b is the identity over βe. the fact that it is isotone means g(≤β) ⊂ r which is the thesis. � theorem 3.8. if (e, t , ≤) is a compact hausdorff t2-preordered space, then its hausdorff t2-preorder compactification β : e → βe constructed in theorem 3.5 is equivalent with the identity id : e → e. proof. the map c : e → e where c = ide and (ce, tc, ≤c) = (e, t , ≤) is a preorder compactification which satisfies both conditions (a) and (b) of theorem 3.6, thus the preorder compactification id is equivalent to β. � the discrete preorder is that preorder for which the increasing hull of a point is made only by the point (thus it is actually an order). the indiscrete preorder is that preorder for which the increasing hull of a point is the whole space. the indiscrete preorder is closed while the discrete preorder requires the hausdorffness of the space, which we assume. corollary 3.9. if ≤ is the discrete (indiscrete) preorder then (βe, tβ, ≤β) is the stone-čech compactification endowed with the discrete (resp. indiscrete) preorder. proof. the discrete preorder ≤d on βe is clearly the smallest closed preorder inducing the discrete preorder ≤, thus ≤d=≤β. for the indiscrete case let x, y ∈ βe and let ox, oy be neighborhoods of x and y respectively. since β(e) is dense there are points x′, y′ ∈ e such that x′ ∈ β(e) ∩ ox, y ′ ∈ β(e) ∩ oy, from β −1(x′) ≤ β−1(y′) since β is isotone we get x′ ≤β y ′ and since ≤β is closed we conclude x ≤ y. � 3.2. the relation with nachbin’s t2-order compactification. in this section we wish to study the relation between the compactification β : e → βe and the nachbin’s compactification n : e → ne in those cases in which e is a completely regularly ordered space so that the latter compactification applies. in this case, although ≤ is an order, ≤β need not be an order. we want to prove that the nachbin’s compactification is obtained by taking the quotient with respect to ∼β. let (e/∼, t /∼, .) be the quotient topological preordered space and let π : e → e/ ∼ be the continuous quotient projection. every open (closed) increasing (decreasing) set on e projects to an open (resp. closed) increasing (resp. decreasing) set on e/∼ and all the latter sets can be regarded as such compactification of closed preordered spaces 215 projections. as a consequence, (e, t , ≤) is a normally preordered space (t1preordered space) if and only if (e/∼, t /∼, .) is a normally ordered space (resp. t1-ordered space). using this fact it is easy to prove (see [19, cor. 4.3]) theorem 3.10. if (e, t , ≤) is a compact t2-preordered space, then (e/∼, t /∼, .) is a compact t2-ordered space. we are ready to establish the connection with the nachbin t2-order compactification. theorem 3.11. let (e, t , ≤) be a t2-preordered tychonoff space such that e/ ∼ is a completely regularly ordered space, then the preorder ≤ is represented by the continuous isotone functions on e. let β : e → βe be the hausdorff t2-preorder compactification constructed in theorem 3.5 and let π : βe → βe/∼β be the quotient projection on the t2-ordered space (βe/∼β , tβ/∼β, .β), then 2 π◦ β ◦ π−1 : e/∼ → βe/∼β is a t2-order compactification equivalent to the nachbin t2-order compactification n : e/∼ → n(e/∼). in particular, up to equivalences, the following diagram commutes e β −−−−→ βe π   y   y π e/∼ n −−−−→ n(e/∼) proof. the order . on e/∼ is represented by the continuous isotone functions because e/∼ is completely regularly ordered. since for x, y ∈ e, x ≤ y iff π(x) . π(y), and the continuous isotone functions on e pass to the quotient, the continuous isotone functions on e represent ≤. the fact that (βe/∼β, tβ/∼β, .β) is t2-ordered follows from theorem 3.10. the expression ϕ := π◦β ◦π−1 gives a well defined function, indeed suppose x, y ∈ e project on the same element [x] ∈ e/∼, then x ∼ y and since β is a preorder embedding β(x) ∼β β(y) which implies π(β(x)) = π(β(y)). the function ϕ is continuous, indeed let o ⊂ βe/∼β be an open subset then β−1(π−1(o)) is open and if x ∈ β−1(π−1(o)) and y ∼ x then as β is a preorder embedding β(y) ∼β β(x), β(x) ∈ π −1(o) which implies β(y) ∈ π−1(o) and hence y ∈ β−1(π−1(o)). the open set β−1(π−1(o)) ⊂ e, being projectable has an open projection by definition of quotient topology which implies that ϕ−1(o) is open. let us prove that ϕ is isotone. let [x] . [y], x, y ∈ e, then x ≤ y and, since β is a preorder embedding, β(x) ≤β β(y), and finally π(β(x)) ≤β π(β(y)) by definition of quotient order. let us prove that ϕ is injective. let [x], [y] ∈ e/∼ be such that ϕ([x]) = ϕ([y]), that is, π(β(x)) = π(β(y)). this equality implies β(x) ∼β β(y), and since β is a preorder embedding x ∼ y, that is, [x] = [y]. 2the inverse π−1 is multivalued but the composition π◦β ◦π−1 is a well defined function. 216 e. minguzzi let us prove that ϕ−1|ϕ(e/∼) : ϕ(e/∼) → e/∼ is isotone. let x, y ∈ e and π(β(x)) .β π(β(y)) then β(x) ≤β β(y) and, since β is a preorder embedding, x ≤ y which implies [x] . [y]. let us prove that ϕ is an embedding. since π is continuous, given an open set n ⊂ e/∼ we have that π−1(n) is open, thus we have only to prove that π ◦ β sends open sets on e of the form π−1(n) to open sets on π ◦ β(e) with the topology induced from βe/∼β. let o ⊂ e be an open set of the form o = π−1(n) with n open set on e/∼ and let x ∈ o (thus [x] ∈ n). since e/∼ is completely regularly ordered space there are [22] a continuous isotone function f̂ : e/∼ → [0, 1] and a continuous anti-isotone function ĝ : e/∼ → [0, 1] such that f̂([x]) = ĝ([x]) = 1 and min(f̂([y]), ĝ([y])) = 0 for [y] ∈ e\n. let us define f = f̂ ◦ π, g = ĝ ◦ π, so that they are respectively continuous isotone and continuous anti-isotone and such that f(x) = g(x) = 1 and min(f(y), g(y)) = 0 for y ∈ e\o. the functions f, g(◦β−1) extend to functions f̃, g̃ : βe → [0, 1] respectively isotone and anti-isotone (extend −g in place of g and take minus the extended function). since they are isotone or anti-isotone there are continuous functions f, g : βe/∼β→ [0, 1], respectively isotone and anti-isotone, such that f̃ = f ◦ π, g̃ = g ◦ π (continuity follows from the universality property of the quotient map [32, theor. 9.4]). the function h = min(f̃, g̃) = min(f, g) ◦ π is continuous and vanishes on β(e\o) and hence min(f, g) vanishes on (π ◦ β)(e\o) = ϕ((e/∼)\n) and equals 1 on [β(x)]β = ϕ(x). since ϕ is injective the open set q = {[w]β ∈ βe/∼β: min(f([w]β), g([w]β)) > 0} contains ϕ(x) and is such that q∩ϕ(e/∼ ) ⊂ ϕ(n) which proves, due to the arbitrariness of [x], that ϕ(n) is open in the topology induced on ϕ(e/∼) by βe/∼β. we infer that ϕ is an embedding and since it is isotone with its inverse it is a preorder embedding. if [z]β ∈ (βe/ ∼β)\ϕ(e/ ∼) and w is an open set containing [z]β then π−1(w) is open and since β is a dense embedding there is some r ∈ e such that β(r) ∈ π−1(w), thus [r] ∈ e/∼ is such that ϕ([r]) ∈ w , that is, ϕ(e/∼) is dense in βe/∼β and hence ϕ is a t2-order compactification. now, let f̂ : e/∼→ [0, 1] be a continuous isotone function, and let f = f̂ ◦π. the function f : e → [0, 1] is a continuous isotone function and we know that there is a continuous isotone function f̃ : βe → [0, 1] which extends f ◦ β−1 : β(e) → [0, 1]. since f̃ is isotone there is some continuous isotone function f : βe/∼β→ [0, 1] (continuity follows from the universality property of the quotient map) such that f̃ = f ◦ π, thus f extends f̂ ◦ ϕ−1 : ϕ(e/∼) → [0, 1]. since the nachbin t2-order compactification is characterized by this extension property [4,22] it follows that ϕ is equivalent to n. finally, ϕ◦π = (π◦β◦π−1)◦π = π◦β which proves that, up to equivalences, the diagram commutes. � compactification of closed preordered spaces 217 corollary 3.12. let e be a completely regularly ordered space, let β : e → βe be the hausdorff t2-preorder compactification constructed in theorem 3.5 and let π : βe → βe/∼β be the quotient projection on the t2-ordered space (βe/∼β, tβ/∼β, .β), then π ◦ β : e → βe/∼β is a t2-order compactification equivalent to the nachbin t2-order compactification n : e → ne. proof. it follows from the previous theorem noting that a completely regularly ordered space is a t2-preordered tychonoff space. � if e is a completely regularly ordered space the preorder compactification β need not be equivalent with the nachbin compactification. consider for instance the interval [0, 1) with the usual topology and order. the nachbin compactification is given by [0, 1] but β([0, 1)) includes many more points. 3.3. the smallest hausdorff t2-preorder compactification. in this section we make some progress in the problem of finding the smallest hausdorff t2-preorder compactification of a topological preordered space in those cases in which it exists. the problem of identifying and characterizing the smallest t2-order compactification was considered in [13,15–17,27]. in this section (e, t , ≤) is a locally compact t2-preordered tychonoff space and f is the family of continuous isotone functions f : e → [0, 1]. accordingly with the necessary conditions singled out in prop. 3.4, we shall assume that the preorder is represented by the continuous isotone functions i.e. g(≤) = ⋂ f∈f gf . let c, c− and c+ be the families of continuous functions in [0, 1] which are constant outside a compact set, which have compact support and which have value 1 outside a compact set, respectively. for every h ⊂ f such that g(≤) = ⋂ h∈h gh we can construct a t2-preorder compactification (ce, tc, ≤c), which we call h-compactification, through the embedding c : e → [0, 1]h∪c identifying ce with the closure of the image. indeed, the family h∪c separates points and has an initial topology coincident with t (thanks to local compactness and the inclusion of c in the family) thus c is an embedding [32, theor. 8.12]. the topology tc is that induced from the product topology in [0, 1]h∪c on ce. we define the t2-preorder � on [0, 1] h∪c as that given by x � y iff xh ≤h yh for every h ∈ h, where ≤h is the usual order on the h-th factor [0,1]. this preorder is closed because the projections πh : [0, 1] h∪c → r are continuous, and hence g(�) = ⋂ h∈h(πh × πh) −1g(≤h) is closed. it is a preorder rather than an order because two points can have the same h-components while being different. the t2-preorder ≤c on ce is that induced by � and is again closed because of the hereditarity of the t2-preorder property. finally, c : e → c(e) is isotone with its inverse because g(≤) = ⋂ h∈h gh. observe that h ◦ c−1 : c(e) → [0, 1] extends to the continuous isotone function πh|ce, that is, the continuous isotone functions belonging to h are extendable to the h-compactification ce keeping the same properties. 218 e. minguzzi remark 3.13. the just defined h-compactification gives back the usual onepoint compactification if the preorder ≤ is indiscrete and h is chosen empty (the additional point is that of coordinates fc, c ∈ c, where fc is the constant value taken by c outside a compact set). if the preorder ≤ is discrete and h is chosen to coincide with c then the compactified space is still the one-point compactification but endowed with the discrete preorder. if h is chosen equal to c−, then the added point is less than any other point. if h is chosen equal to c+, then the added point is greater than any other point. in the next proofs we shall often identify c(e) with e especially when referring to the extension of functions. proposition 3.14. let c : e → ce be a h-compactification. the remainder ce\c(e) endowed with the preorder induced from ≤c is a t2-ordered space. proof. since the t2-preorder property is hereditary the remainder is a t2preordered space. let x, y ∈ ce\c(e) and suppose that x ≤c y ≤c x then x � y � x, that is for the (necessarily unique as c(e) is dense in ce) continuous isotone extension h : ce → [0, 1], h = πh|ce, of h ∈ h we have h(x) ≤ h(y) ≤ h(x), which reads h(x) = h(y). we have only to prove that for every f ∈ c, πf (x) = πf (y) from which it follows x = y. but by local compactness c(e) is open in ce thus ce\c(e) is compact and can be separated by open sets (as ce is hausdorff and compact hence normal) from the compact set outside which f is constant. thus the extension πf |ce of f ∈ c takes a constant value on the whole remainder, which implies πf (x) = πf (y). � we remark that the previous result does not imply that if ≤ is an order then ≤c is an order, but only that if x ≤c y ≤c x, then one point among x and y belongs to c(e) while the other belongs to ce\c(e). proposition 3.15. let (e, t , ≤) be a locally compact t2-preordered tychonoff space then every t2-preordered hausdorff compactification c : e → ce dominates a h-compactification for a family h ⊂ f where h is such that g(≤) = ⋂ h∈h gh. the family h is made by those continuous isotone function with value in [0,1] in e that extend with the same properties to ce. proof. let c1 : e → c1e be a t2-preordered hausdorff compactification. since (c1e, tc1, ≤c1) is a compact t2-preordered space it is normally preordered, thus the family of continuous isotone functions with values in [0, 1], hc1, is such that for x, y ∈ c1e, x ≤c1 y if and only if for every f ∈ hc1 we have f(x) ≤ f(y). let h be made by those functions which are the restriction of the elements of hc1 to e. with this definition g(≤) = ⋂ h∈h gh. let c2 : e → c2e ⊂ [0, 1] h∪c be the h-compactification and let us prove that c1 dominates c2. a continuous isotone map c : c1e → c2e such that c ◦ c1 = c2 can be constructed as follows. by local compactness c1(e) is open and c1e\c1(e) is closed and compact. we consider the family hc1 ∪ cc1 where cc1 is the family of continuous functions with value in [0, 1] on c1e which are constant outside compactification of closed preordered spaces 219 a compact set disjoint from c1e\c1(e). the restriction of the elements of the family cc1 to c1(e) gives back c. by definition, the map c sends x ∈ c1e to the point of [0, 1]hc1∪cc1 whose f coordinate is the value f(x), f ∈ hc1 ∪ cc1. this map is continuous [32, theor. 8.8] and isotone, where we define the preorder on [0, 1]hc1∪cc1 as that determined by the family hc1. let us prove that its image is included in c2e. from the definitions we have that if x ∈ c1(e) then c(x) belongs to c2(e). as c is continuous, and c1(e) is dense in c1e, if x ∈ c1e its image c(x) belongs to the closure of c2(e) namely to c2e. � proposition 3.16. if h2 ⊃ h1 then the h2-compactification dominates over the h1-compactification. proof. indeed, if c2 : e → c2e ⊂ [0, 1] h2∪c is the former and c1 : e → c1e ⊂ [0, 1]h1∪c is the latter preorder compactification, then there is a continuous isotone map c : c2e → c1e such that c ◦ c2 = c1. this map is the restriction to c2e of π : [0, 1] h2∪c → [0, 1]h1∪c where π identifies points with the same coordinates belonging to the set h1 ∪ c. � once a h-compactification is given it is well possible that some f ∈ f\h could be extendable as a continuous isotone function to the whole compactification. let i(h) be the subset of f of so extendable functions. this set being larger than h has again the property that it represents ≤. proposition 3.17. the h-compactification and the i(h)-compactification are equivalent. proof. since h ⊂ i(h) the i(h)-compactification dominates over the h-compactification. for the converse let c2 : e → c2e ⊂ [0, 1] h∪c be the h-compactification and let c1 : e → c1e ⊂ [0, 1] i(h)∪c be the i(h)-compactification. a continuous isotone map c : c2e → c1e such that c ◦c2 = c1 can be constructed as follows. all the functions of i(h) ∪ c extend (uniquely because c2(e) is dense in c2e) from e to c2e thus to every x ∈ c2e we assign the image c(x) given by the point of [0, 1]i(h)∪c having as coordinates the values taken by the functions belonging to i(h) ∪ c. by construction c is continuous [32, theor. 8.8]. let us prove that the image is included in c1e. from the definitions we have that if x ∈ c2(e) then c(x) belongs to c1(e). as c is continuous, and c2(e) is dense in c2e, if x ∈ c2e its image c(x) belongs to the closure of c1(e) namely to c1e. the fact that c is isotone follows immediately from the definition of preorder in [0, 1]i(h)∪c and from the fact that the extension of the function in i(h) to c2e are, by assumption, continuous and isotone. � corollary 3.18. let p(f) denote the family of subsets of f. the map i : p(f) → p(f) is idempotent, namely i(i(h)) = i(h). furthermore, if h1 ⊂ h2 then i(h1) ⊂ i(h2). proof. if a continuous isotone function f : e → [0, 1] can be extended as a continuous isotone function to the i(h)-compactified space, i.e. f ∈ i(i(h)) then, as the h-compactification and the i(h)-compactification are equivalent, 220 e. minguzzi it can be extended as a continuous isotone function to the h-compactified space that is f ∈ i(h). for the last statement, let f ∈ i(h1) that is f : e → [0, 1] can be extended as a continuous isotone function f1 : c1e → [0, 1] to the h1-compactified space. but the h2-compactification dominates over the h1-compactification, that is if c2 : e → c2e is the former and c1 : e → c1e is the latter, there is a continuous isotone function c : c2e → c1e such that c ◦ c2 = c1. the pullback with c of the extension to c1e, namely f2 = f1 ◦ c, is a continuous isotone extension on c2e of f thus f ∈ i(h2). � theorem 3.19. the h-compactification is the smallest hausdorff t2-preordered compactification for which the function belonging to h are extendable as continuous isotone functions to the compactified space. proof. let c : e → ce be a hausdorff t2-preordered compactification for which the functions belonging to h are extendable. by prop. 3.15 the compactification c dominates a g-compactification where g is the set of continuous isotone functions on e with value in [0,1] which are extendable with these properties to ce. thus h ⊂ g and by prop. 3.16 the g-compactification dominates over the h-compactification, thus c dominates the h-compactification. � definition 3.20. the family of invariant sets i is the set of subsets h ⊂ f which satisfy g(≤) = ⋂ h∈h gh and are left invariant by i. the set i is ordered by inclusion. the next theorem serves to define the family of continuous isotone functions s which characterizes the smallest compactification. theorem 3.21. if the smallest hausdorff t2-preorder compactification exists then it is a s-compactification where g(≤) = ⋂ h∈s gh, i(s) = s and s = ⋂ i. proof. suppose that there is a hausdorff t2-preorder compactification which is dominated by all the other hausdorff t2-preorder compactifications, then by prop. 3.15 it is equivalent to a s-compactification where s ⊂ f is such that g(≤) = ⋂ h∈s gh. by prop. 3.17 s can be chosen such that s = i(s), thus belonging to i. clearly, ⋂ i ⊂ s because s ∈ i. suppose that h ∈ i and that f ∈ f, f /∈ h = i(h). this means that f is not extendable as a continuous isotone function to the h-compactified space. if c is the continuous isotone map from the h-compactified space to the s-compactified space (as the scompactification is dominated by all the other compactifications) one has that if f were extendable to the s-compactified space then by pullbacking the extension to the h-compactified space through c one would get an extension in the h-compactified space. the contradiction proves that f /∈ i(s) = s thus s ⊂ h, and finally s ⊂ ⋂ i. � compactification of closed preordered spaces 221 remark 3.22. the smallest compactification does not necessarily exist. for instance, if e is non-compact and endowed with the discrete preorder, the ccompactification dominates over the c−-compactification and the c+-compactification (see remark 3.13), indeed c∓ ⊂ c see prop. 3.16. stated in another way, the one-point compactification endowed with the discrete preorder dominates over that in which the added point is less (resp. greater) than any other point (indeed, the former has a smaller preorder). however, c+ is not contained in i(c−) and conversely, thus the c−− and c+− compactifications differ. actually, it is easy to realize that they are minimal, thus there is no smallest compactification. 4. conclusions we have investigated the compactification of topological preordered spaces, showing the existence of a largest hausdorff t2-preorder compactification for every t2-preordered tychonoff space for which the preorder is represented by the continuous isotone functions. an interesting subclass of this family is that of locally compact σ-compact hausdorff t2-preordered spaces [19]. it turns out that this largest compactification is essentially the stone-čech compactification endowed with a suitable preorder. it can be characterized as the hausdorff t2-preorder compactification for which all the continuous function can be continuously extended and the continuous isotone function do so preserving the isotone property. if the preorder is an order or the quotient space is a completely regularly ordered space it is also possible to show a clean relation with nachbin’s t2-order compactification. we have considered the problem of identifying the smallest hausdorff t2preorder compactification whenever it exists. we have shown that it corresponds necessarily to the compactification obtained demanding the extendibility of a suitable set of continuous isotone functions. generically, this set s is expected to be strictly included in the full set f of continuous isotone functions with value in [0,1]. the approach followed in this work relies on the study of continuous isotone functions and their extension properties. we close noting that filter approaches are also possible. for instance choe and park [2] have constructed a wallman type preorder compactification which has been subsequently extensively investigated in [9–12, 14] together with some variations. for instance, in [10] the authors show that it is possible to obtain the nachbin compactification from the wallman compactification by identifying the points that take the same value on continuous isotone functions. we have followed a similar procedure to show that the nachbin compactification ne can be obtained from the same functional quotient starting from βe. 222 e. minguzzi acknowledgments i thank a referee for pointing out remark 3.22. this work has been partially supported by “gruppo nazionale per la fisica matematica” (gnfm) of “instituto nazionale di alta matematica” (indam). references [1] r. budic and r. k. sachs, causal boundaries for general relativistic spacetimes, j. math. phys. 15 (1974), 1302–1309. [2] t. h. choe and y. s. park, wallman’s type order compactification, pacific j. math. 82 (1979), 339–347. [3] r. engelking, general topology, berlin: helderman verlag (1989). [4] p. fletcher and w. lindgren, quasi-uniform spaces, vol. 77 of lect. notes in pure and appl. math., new york: marcel dekker, inc. (1982). [5] j. l. flores, the causal boundary of spacetimes revisited, commun. math. phys. 276 (2007), 611–643. [6] r. geroch, e. h. kronheimer and r. penrose, ideal points in spacetime, proc. roy. soc. lond. a 237 (1972), 545–567. [7] s. g. harris, universality of the future chronological boundary, j. math. phys. 39 (1998), 5427–5445. [8] s. w. hawking and g. f. r. ellis, the large scale structure of space-time, cambridge: cambridge university press (1973). [9] d. c. kent, on the wallman order compactification, pacific j. math. 118 (1985), 159– 163. [10] d. c. kent, d. liu and t. a. richmond, on the nachbin compactification of products of totally ordered spaces, internat. j. math. & math. sci. 18 (1995), 665–676. [11] d. c. kent and t. a. richmond, separation properties of the wallman ordered compactification, internat. j. math. & math. sci. 13 (1990), 209–222. [12] d. c. kent and t. a. richmond, a new ordered compactification, internat. j. math. & math. sci. 16 (1993), 117–124. [13] h.-p. a. künzi, minimal order compactifications and quasi-uniformities, berlin: akademie-verlag, vol. recent developments of general topology and its applications of mathematical research 67 (1992), pages 181–186. [14] h.-p. a. künzi, a. e. mccluskey and t. a. richmond, ordered separation axioms and the wallman ordered compactification, publ. math. debrecen 73 (2008), 361–377. [15] d. m. liu and d. c. kent, ordered compactifications and families of maps, internat. j. math. & math. sci. 20 (1997), 105–110. [16] t. mccallion, compactifications of ordered topological spaces, proc. camb. phil. soc. 71 (1972), 463–473. [17] s. d. mccartan, separation axioms for topological ordered spaces, proc. camb. phil. soc. 64 (1968), 965–973. [18] e. minguzzi, the causal ladder and the strength of k-causality. ii, class. quantum grav. 25 (2008), 015010. [19] e. minguzzi, normally preordered spaces and utilities, order, to appear (doi: 10.1007/s11083-011-9230-4. arxiv:1106.4457v2). [20] e. minguzzi, quasi-pseudo-metrization of topological preordered spaces, topol. appl. 159 (2012), 2888–2898. [21] e. minguzzi, topological conditions for the representation of preorders by continuous utilities, appl. gen. topol. 13 (2012), 81–89. [22] l. nachbin, topology and order, princeton: d. van nostrand company, inc. (1965). compactification of closed preordered spaces 223 [23] s. nada, studies on topological ordered spaces, ph.d. thesis, southampton (1986). [24] r. penrose, conformal treatment of infinity, new york: gordon and breach, vol. relativity, groups and topology, pages 563–584 (1964). [25] i. rácz, causal boundary of space-times, phys. rev. d 36 (1987), 1673–1675. [26] i. rácz, causal boundary for stably causal spacetimes, gen. relativ. gravit. 20 (1988), 893–904. [27] t. a. richmond, posets of ordered compactifications, bull. austral. math. soc. 47 (1993), 59–72. [28] s. scott and p. szekeres, the abstract boundary: a new approach to singularities of manifolds, j. geom. phys. 13 (1994), 223–253. [29] h. seifert, the causal boundary of space-times, gen. relativ. gravit. 1 (1971), 247–259. [30] l. b. szabados, causal boundary for strongly causal spacetimes, class. quantum grav. 5 (1988), 121–134. [31] l. b. szabados, causal boundary for strongly causal spacetimes ii, class. quantum grav. 6 (1989), 77–91. [32] s. willard, general topology, reading: addison-wesley publishing company (1970). (received march 2012 – accepted july 2012) e. minguzzi (ettore.minguzzi@unifi.it) dipartimento di matematica applicata “g. sansone”, università degli studi di firenze, via s. marta 3, i-50139 firenze, italy. compactification of closed preordered spaces. by e. minguzzi @ appl. gen. topol. 21, no. 2 (2020), 215-233 doi:10.4995/agt.2020.12929 c© agt, upv, 2020 the class of simple dynamical systems k. ali akbar department of mathematics, central university of kerala, kasaragod 671320, kerala, india. (aliakbar.pkd@gmail.com, aliakbar@cukerala.ac.in) communicated by f. balibrea abstract in this paper, we study the class of simple dynamical systems on r induced by continuous maps having finitely many non-ordinary points. we characterize this class using labeled digraphs and dynamically independent sets. in fact, we classify dynamical systems up to their number of non-ordinary points. in particular, we discuss about the class of continuous maps having unique non-ordinary point, and the class of continuous maps having exactly two non-ordinary points. 2010 msc: 54h20; 26a21; 26a48. keywords: special points; non-ordinary points; critical points; order conjugacy; order isomorphism; labeled digraph; dynamically independent set. 1. introduction a dynamical system is a pair (x, f), where x is a metric space and f is a continuous self map on x. two dynamical systems (x, f) and (y, g) are said to be topologically conjugate (or simply conjugate) if there exists a homeomorphism h : x → y (called topological conjugacy) such that h ◦ f = g ◦ h. we simply say that f is conjugate to g, and we write it as f ∼ g. in the case when h happens to be an increasing homeomorphism (for example, when x = r or an interval) we say that f and g are increasingly conjugate or order conjugate. when we are working with a single system, any self conjugacy can utmost shuffle points with same dynamical behavior. therefore a point which is unique up to its behavior must be fixed by every self conjugacy. on the received 02 january 2020 – accepted 17 may 2020 http://dx.doi.org/10.4995/agt.2020.12929 k. ali akbar other hand if a point is fixed by all self conjugacies then it must have a special property (some times it may not be known explicitly). these things motivated to call the set of all points fixed by all self conjugacies as set of special points. for x, y ∈ r, we write x ∼ y if x and y have the same dynamical properties in the dynamical system (r, f). said precisely, x ∼ y if there exists an increasing homeomorphism h : r → r such that h ◦ f = f ◦ h and h(x) = y. it is easy to see that ∼ is an equivalence relation. since the equivalence relation is coming from self conjugacy it is important in the field of topological dynamics. let [x] denote the equivalence class of x ∈ r. let i, j be two subintervals of r. we say that i < j if x < y for all x ∈ i and y ∈ j. the properties of dynamical systems which are preserved by topological conjugacies are called dynamical properties. the points which are unique up to some dynamical property are called dynamically special points. said differently, a special point has a dynamical property which no other point has. we say that a point x is ordinary if, its “like” points near it. that is, an element x ∈ r is ordinary in (r, f) if its equivalence class [x] is a neighbourhood of it. i.e, the equivalence class of x contains an open interval around x. a point which is not ordinary is called non-ordinary. the idea of special points and non-ordinary points are relatively new to the literature (see [1], [2], [8]). recently, we studied the class of simple dynamical systems induced by homeomorphisms. the reader may refer [2] to get an idea of simple systems induced by homeomorphisms having finitely many non-ordinary points. throughout this paper, we will be working with continuous self maps of the real line. since r has order structure, we would like to consider the conjugacies preserving the order. hence the conjugacies which we consider in this paper are order preserving conjugacies (increasing conjugacies). for any continuous map f : r → r, we denote gf↑ for the set of all order conjugacies of f. in a dynamical system (x, f), let n(f) denote the set of all non-ordinary points of f. observe that a point to be special if [x] = {x}. let s(f) denote the set of all special points of f. a point x in a topological space x is said to be rigid if it is fixed by every self homeomorphism of x. for example, the point 1 is rigid in (0, 1]. according to the above definition all rigid points are special, even though there is no role for the map f. by definition, the points of [x] are dynamically the same. we consider the systems for which there are only finitely many equivalence classes. this means there are only finitely many kinds of orbits up to conjugacy. if f : r → r is continuous and per(f) is properly contained in {1, 2, 22, ...}, then f is not li-yorke chaotic (see [7]). also note that, if f : r → r is devaney chaotic then 6 ∈ per(f) (see [5]). therefore, if f : r → r is a continuous map having finitely many non-ordinary points then neither it is li-yorke chaotic nor it is devaney chaotic because of sharkovskii’s theorem. for these reasons we call such systems as simple systems. these are the system in which the phase portrait can be drawn. phase portraits (see [4]) are frequently used to graphically represent the dynamics of a system. a phase portrait consists of a diagram representing possible beginning positions in the system and arrows that indicate the change in these positions under iteration c© agt, upv, 2020 appl. gen. topol. 21, no. 2 216 the class of simple dynamical systems of the function. the drawable systems are interesting to physicist and for this reason the study of the class of simple dynamical systems can be useful. our main results (see section 3) prove that (i) there are exactly 26 continuous maps on r with exactly one nonordinary point and 90 continuous maps on r with exactly two nonordinary points up to order conjugacy. (ii) let c be the class of all continuous self maps of r, having finitely many non-ordinary points. then (a) for every member of c, there exists a maximal dynamically independent set that is a finite union of intervals. (b) two members of c are order conjugate if and only if they have order isomorphic maximal dynamically independent set as in (a), and with isomorphic labeled digraphs. (c) every order isomorphism between such maximal dynamically independent set extends uniquely to an order conjugacy. 2. basic results in this section, we will be discussing some basic results that will be used to prove our main theorems. most of the results are available in [2]. let (x, f) be a dynamical system. by the full orbit of a point x ∈ x we mean the set o(x) = {y ∈ x : fn(x) = fm(y) for some m, n ∈ n}. for any subset a ⊂ r, let o(a) = ! x∈a o(x) = ! x∈a {y ∈ r : fn(y) = fm(x) for some m, n ∈ n}. a point x in a dynamical system (x, f) is said to be a critical point if f fails to be one-to-one in every neighbourhood of x. the set of all critical points of f is denoted by c(f). see [2] and [8], for the following characterization theorem for the set n(f) (and hence for s(f)). theorem 2.1. for continuous self maps of the real line r, the set of all nonordinary points is contained in the closure of the union of full orbits of critical points, periodic points and the limits at infinity (if they exist and are finite). remark 2.2. in theorem 2.1, the inclusion can be strict. proof. consider the map f(x) = x + sinx for all x ∈ r. all integral multiples of π are fixed points for this map but the increasing bijection x &→ x + 2π commutes with f and fixes none of them. □ remark 2.3 ([8]). for polynomials of even degree the equality is true in theorem 2.1. for a dynamical system (x, f), let d = o(c(f) ∪ p(f) ∪ {f(∞), f(−∞)}) where f(∞) and f(−∞) are the limits of f at ∞ and −∞ respectively, provided they are finite. for any subset a of x, we denote ā for the closure of a in x. now we consider the following theorem: c© agt, upv, 2020 appl. gen. topol. 21, no. 2 217 k. ali akbar theorem 2.4 ([8]). for polynomial maps f of r, s(f) has to be either empty or a singleton or the whole d. from the definition, it is clear that the set of special points s(f) is always closed. the following theorem is about the converse and it is proved in [8]. theorem 2.5. given any closed subset f of r, there exists a continuous map f : r → r such that s(f) = f. remark 2.6. for every closed subset f of r there exist continuous maps f : r → r and g : r → r such that s(f) = fix(g) = f . conversely, for every closed subset f of r there exist continuous maps f, g : r → r such that s(f) = fix(g) = f . proof. for every closed subset f of r there exists a strictly increasing continuous bijection f : r → r such that fix(f) = f . this is because, we can define f(x) = x+ 1 2 d(x, f). hence the remark follows from theorem 2.5 since fix(f) is closed. □ the following total order on n is called the sharkovskii’s ordering: 3 ≻ 5 ≻ 7 ≻ 9 ≻ ... ≻ 2 × 3 ≻ 2 × 5 ≻ 2 × 7 ≻ ... ≻ 2n × 3 ≻ 2n × 5 ≻ 2n × 7 ≻ ... ...2n ≻ .... ≻ 22 ≻ 2 ≻ 1 we write m ≻ n if m precedes n (not necessarily immediately) in this order. an n-cycle means a cycle of length n. theorem 2.7 (sharkovskii’s theorem [6]). let m ≻ n in the sharkovskii’s ordering. for every continuous self map of r, if there is an m-cycle, then there is an n-cycle. for any f : r → r, let gf denote the set of all topological conjugacies of f and let gf↑ denote the set of all order conjugacies of f. proposition 2.8 ([2]). if x is an ordinary point of f and if h is a self topological conjugacy of f, then h(x) is ordinary. proposition 2.9 ([2]). if x is a non-ordinary point of f and if h is a self topological conjugacy of f, then h(x) is non-ordinary. now we ask: for a continuous map f : r → r, how the equivalence classes looks like? the following known lemma answer this question. lemma 2.10 ([2]). let f : r → r be continuous. suppose a < b and (a, b) ∩ n(f) = ∅. then x ∼ y for all x, y ∈ (a, b). theorem 2.11 ([2]). let f : r → r be continuous. if |n(f)| = n then |{[x] : x ∈ r}| = 2n + 1. remark 2.12 ([2]). note that, being a point in a particular equivalence class [x] is a dynamical property. there are continuous maps f : r → r having finitely many equivalence classes, but infinitely many non-ordinary points. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 218 the class of simple dynamical systems remark 2.13. (1) if f : r → r has a unique fixed point then it is non-ordinary and vice versa. (2) if f : r → r has finitely many fixed points (critical points) then all fixed (critical) points are special and hence non-ordinary. (3) if there are only finitely many periodic cycles then all periodic points are special. (4) every special point is non-ordinary. but every non-ordinary point may not be special. proof. (1) since the topological conjugacies carry fixed points to fixed points, the unique fixed point must be fixed by every self conjugacy and hence special. suppose x0 ∈ r is the unique non-ordinary point of f. then h(x0) = x0 for all h ∈ gf↑. now, for any h ∈ gf↑ we have h(f(x0)) = f(h(x0)) = f(x0). that is, the point f(x0) is special. since x0 is the only special point, we have f(x0) = x0. (2) it follows from the fact that under a topological conjugacy fixed points will be mapped to fixed points and critical points will be mapped to critical points and the fact that it takes the finite set of fixed points (critical points) to itself bijectively, preserving the order. proof of (3) is easy. (4) it is immediate from the definition that every special point is nonordinary. for the converse, consider the map x &→ x + sin x on r which has countably many fixed points. note that all the fixed points are non-ordinary and they form two distinct equivalence classes, hence they are not special. □ the following proposition says that the class of maps with finitely many non-ordinary points the idea of special points and the idea of non-ordinary point, coincide. proposition 2.14. if f : r → r has only finitely many non-ordinary points then every non-ordinary point is special. proof. since n(f) is finite, it follows from proposition 2.9 that h(n(f)) = n(f) for all h ∈ gf↑. then we must have h(x) = x for all x ∈ n(f), because of the order preserving nature of h. hence all points of n(f) are special. □ proposition 2.15 ([2]). for maps f : r → r with finitely many non-ordinary points, f(x) is non-ordinary whenever x ∈ r is non-ordinary. definition 2.16. for any subset a of r, we write ∂a = a ∩ (x − a) and call it the boundary of a boundary of a set. recall that the properties which are preserved under topological conjugacies are called dynamical properties. hence, if two points x, y in the dynamical system (x, f), differ by a dynamical property, then no conjugacy can map one to the other, from which it follows that, c© agt, upv, 2020 appl. gen. topol. 21, no. 2 219 k. ali akbar proposition 2.17. for any dynamical property p, the points of ∂sp are nonordinary where sp denotes the set of all points in (x, f) having the dynamical property p. corollary 2.18. let f : r → r be constant in a neighbourhood of a point x0. then the end points of the maximal interval around x0 on which f is constant, are non-ordinary. recall that, if f : r → r has a unique non-ordinary point then it is a fixed point. next we consider the following proposition. see [2] for a proof. proposition 2.19 ([2]). let f : r → r be a continuous map. then, (i) if x ∈ r is both critical and ordinary then f is locally constant at x. (ii) if x is ordinary and f is not locally constant at x then f(x) is ordinary. remark 2.20 ([2]). let f : r → r be continuous. then sup f(r), inf f(r), limx→∞ f(x) and limx→−∞ f(x) are special (in particular, non-ordinary) provided they are finite. (note that, for maps with finitely many non-ordinary points both limx→∞ f(x) and limx→−∞ f(x) always exists in r ∪ {−∞, ∞}). proposition 2.21 ([2]). the maps x + 1 and x − 1 on r are topologically conjugate; but not order conjugate. proof. the maps x + 1 and x − 1 are conjugate to each other through −x + 1 2 . if possible, let h be an order conjugacy from f(x) = x + 1 to g(x) = x − 1. then h(x + 1) = h(f(x)) = g(h(x)) = h(x) − 1. i.e, h(x + 1) − h(x) = −1 < 0. which is a contradiction to the assumption that h is increasing. □ remark 2.22. note that for the map x + 1 on r, all points are ordinary. for, if a, b ∈ r, then the map x + b − a is the order conjugacy of x + 1 which maps a to b. recall that for any subset a of a metric space x, (∂a)c = int(a) ∪ int(ac). that is, the complement of ∂a is the union of the interior of a and the interior of ’a complement’. the following proposition gives a characterization for the non-ordinary points of increasing homeomorphisms. proposition 2.23 ([2]). let f : r → r be an increasing bijection and let x ∈ r. then x is non-ordinary if and only if x is in the boundary of fix(f). next we consider: proposition 2.24 ([2]). let f : r → r be a homeomorphism without fixed points. (i) if f(0) > 0 then f is order conjugate with x + 1. (ii) if f(0) < 0 then f is order conjugate with x − 1. we denote graph(f) for the range of the map f. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 220 the class of simple dynamical systems corollary 2.25 ([2]). let f, g : (a, b) → (a, b) be homeomorphisms without fixed points. then f is order conjugate to g if and only if both graph(f) and graph(g) are on the same side of the diagonal. in particular, (i) if f(x) > x for all x ∈ (a, b) then f is order conjugate to x + 1. (ii) if f(x) < x for all x ∈ (a, b) then f is order conjugate to x − 1. remark 2.26. in fact, in the previous corollary, the interval (a, b) can be replaced by any open ray in r. for an increasing bijection f : r → r with finitely many non-ordinary points, all non-ordinary points are fixed points. 3. class of continuous maps we now consider the systems for which there are only finitely many equivalence classes. recall that, if sp denote the set of all points having the dynamical property p then the points of ∂sp (the boundary of sp ) are non-ordinary. as in remark 2.12, being a point in a particular equivalence class is a dynamical property of the point. hence by the very nature of the order conjugacies, it follows that when there are finitely many non-ordinary points (therefore special points) there are only finitely many equivalence classes. hence if f : r → r is an increasing bijections with finitely many non-ordinary points x1 < x2 < ... < xn for some n ∈ n, then these points gives rise to an ordered partition {(−∞, x1), (x1, x2), ..., (xn, ∞)} of r \ {x1, x2, ..., xn}. this partition gives rise to a word w(f) over {a, b, o} of length n + 1 by associating a to f(t) > t ∀ t, b to f(t) < t ∀ t and o to f(t) = t ∀ t. we now study, the class of simple systems induced by continuous maps having finitely many non-ordinary points. we first state the following results without proof. for a proof refer [2]. these results are easy to prove and it will be used to prove our main theorems. proposition 3.1. let f, g be two increasing bijections on r with finitely many (same number of) non-ordinary points. then f and g are order conjugate if and only if w(f) = w(g). proposition 3.2. there is a one to one correspondence between the set of all increasing continuous bijections (up to order conjugacy) on r with exactly n non-ordinary points and the set of all words of length n + 1 on three symbols a,b,o such that oo is forbidden. theorem 3.3. the number of all increasing continuous bijections (up to order conjugacy) on r with exactly n non-ordinary points is equal to an where an = c1(1 + √ 3)n + c2(1 − √ 3)n, c1 = (3 √ 3+5) 2 √ 3 and c2 = (3 √ 3−5) 2 √ 3 . proposition 3.4. two decreasing bijections f and g are order conjugate (respectively topologically conjugate) if and only if f2|[a,∞) and g2|[b,∞) are order conjugate (respectively topologically conjugate) where a and b are the fixed points of f and g respectively. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 221 k. ali akbar proposition 3.5. two decreasing bijections f and g are order conjugate (respectively topologically conjugate) if and only if f2|(−∞,a] and g2|(−∞,b] are order conjugate (respectively topologically conjugate) where a and b are the fixed points of f and g respectively. proposition 3.6. if f is a decreasing bijection from r to r with fixed point a. then f has 2n + 1 non-ordinary points if and only if (f ◦ f)|(a,∞) : (a, ∞) → (a, ∞) has n non-ordinary points. theorem 3.7. if sn denotes the number of decreasing homeomorphisms up to order conjugacy, then sn = " 0 if n is even a n−1 2 if n is odd for all n. in subsections 3.1 and 3.2, we discuss the class of continuous maps with unique (respectively two) non-ordinary points. in the subsection 3.3, we consider the class of continuous maps with finitely many non-ordinary points. 3.1. class of continuous maps having unique non-ordinary point. we consider the following propositions that will help us to prove our main theorem. proposition 3.8. let f, g : r → r be such that (1) f(0) = g(0) = 0. (2) f|(0,∞), g|(0,∞) : (0, ∞) → (0, ∞) are increasing bijections. (3) f|(−∞,0), g|(−∞,0) : (−∞, 0) → (0, ∞) are decreasing bijections. then f is order conjugate to g if and only if f|(0,∞) is order conjugate to g|(0,∞). proof. suppose h : (0, ∞) → (0, ∞) is an order conjugacy from f|(0,∞) to g|(0,∞). for x < 0, define h(x) = (g|(−∞,0))−1hf(x), and h(0) = 0. □ proposition 3.9. let f, g : r → r be such that (1) f(0) = g(0) = 0. (2) f|(−∞,0), g|(−∞,0) : (−∞, 0) → (−∞, 0) are increasing bijections. (3) f|(0,∞), g|(0,∞) : (0, ∞) → (−∞, 0) are decreasing bijections. then f is order conjugate to g if and only if f|(−∞,0) is order conjugate to g|(−∞,0). proof. suppose h : (−∞, 0) → (−∞, 0) is an order conjugacy from f|(−∞,0) to g|(−∞,0). for x > 0, define h(x) = (g|(0,∞))−1hf(x), and h(0) = 0. □ we denote the symbol # for disjoint union. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 222 the class of simple dynamical systems proposition 3.10. (1) let f : (−∞, 0] → (−∞, 0] be an increasing bijection (it follows that f(0) = 0). (a) if f(x) > x for all x ∈ (−∞, 0) then f is order conjugate to x 2 . (b) if f(x) < x for all x ∈ (−∞, 0) then f is order conjugate to 2x. (2) let f : [0, ∞) → [0, ∞) be an increasing bijection (it follows that f(0) = 0). (a) if f(x) > x for all x ∈ (0, ∞) then f is order conjugate to 2x. (b) if f(x) < x for all x ∈ (0, ∞) then f is order conjugate to x 2 . (3) let f : [1, ∞) → [1, ∞) be an increasing bijection (it follows that f(1) = 1). (a) if f(x) > x for all x ∈ (1, ∞) then f is order conjugate to 2x − 1. (b) if f(x) < x for all x ∈ (1, ∞) then f is order conjugate to x+1 2 . proof. (1). proof of (a): let f : (−∞, 0] → (−∞, 0] be an increasing bijection satisfying f(x) > x for all x < 0. it follows that f(0) = 0. note that for any such map# n∈z[f n(x), fn+1(x)) = (−∞, 0) for all points x ∈ (−∞, 0). then f is topologically conjugate to the map x/2. we construct a topological conjugacy h : (−∞, 0] → (−∞, 0]. for this, take any point other than 0, say −1, in the domain. we take an arbitrary increasing homeomorphism h from [−1, f(−1)) to [−1, −1/2). then as noted above, # n∈z[f n(−1), fn+1(−1)) = (−∞, 0). that is, for every x ∈ (−∞, 0), there exists a unique n0 ∈ z such that fn0(x) ∈ [−1, f(−1)). we define h(x) = 2n0h(fn0(x)). this is well defined. it is an increasing homeomorphism from (−∞, 0) to (−∞, 0). this h commutes with f. this h is a conjugacy from f to the map x/2. similarly, we can prove (1).b, (2) and (3). □ theorem 3.11 ([1]). there are exactly 26 maps on r with a unique nonordinary point, up to order conjugacy. proof. by corollary 3.15, propositions 3.8, 3.9, 3.10 (1), and 3.10 (2), the theorem follows. □ definition 3.12. let f : r → r be a continuous map. let i ⊂ r be an interval such that fk(i) = i, and fm(i) ∕= i for all 1 ≤ m < k. then we say that i → f(i) → f2(i) → ... → fk−1(i) → i is a k-cycle through i. cycles through i, j are said to be distinct if fm(i) ∕= fn(j) for all m, n ∈ n. we can represent each member of this class as a graph as follows: let f : r → r be a continuous map with a unique non-ordinary point. now, let i1, i2 be non singleton equivalence classes such that i1 < i2. define a labeled digraph (g, vf ) with vertex set vf = {i1, i2}, an edge from ij to im if f(ij) = im, and a symbol i (respectively d) on this edge whenever f is increasing (respectively decreasing) on ij. label the symbols a, b, o on the least vertex of each k-cycle depends on the graph of fk, k = 1, 2 which is above the diagonal or below the diagonal or on the diagonal respectively. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 223 k. ali akbar note: if f is a decreasing homeomorphism then we can consider the labeled digraph of f2 instead of the labeled digraph f (see proposition 3.4). 3.2. class of continuous maps having exactly two non-ordinary points. next we consider the following proposition. proposition 3.13. let f : r → r, g : r → r be two continuous maps having finitely many non-ordinary points such that n(f) = n(g) and let n(f) c =# ik. for m, n ∈ n, there exist increasing bijections hn : īn → īn and hm : īm → īm such that g ◦ hn(x) = hm ◦ f(x) for all x ∈ īn whenever f(īn) = īm and g(īn) = īm then f ∼ g. proof. define h : r → r by h(x) = hn(x) for all x ∈ īn and n ∈ n. then h is an increasing bijection such that h◦f = g◦h. hence the proposition. □ note that if a continuous bijection has finitely many non-ordinary points then we can assume that these points are arbitrary. because, let a1 < a2 < ... < an be the non-ordinary points of a continuous map f : r → r. let b1 < b2 < ... < bn be arbitrary points in r. let h : r → r be an increasing bijection such that h([ai, ai+1]) = [bi, bi+1] for i = 1, 2, ..., n − 1. then define g = h ◦ f ◦ h−1. we can easily verify b1, b2, ..., bn are the only non-ordinary points of g since h is an order conjugacy from f to g. the following theorem help us to classify the class of continuous maps having finitely many non-ordinary points. theorem 3.14. let f, g : r → r be two continuous maps having finitely many non-ordinary points such that n(f) = n(g). let n(f)c = # n in. (1) if f and g have the same type of monotonicity (i.e., either both f and g are increasing or both are decreasing) in the closure of each equivalence class and contains exactly n distinct cycles of length ki through ijki for i = 1, 2, ..., n and for some jki. then f ∼ g whenever graph (gki|ījki ) and graph (gki|ījki ) are in the same side of the diagonal. (2) if f and g have the same type of monotonicity in the closure of each equivalence class and does not contain any cycle then f ∼ g. proof. for simplicity we consider the case whenever f and g have exactly one cycle through its equivalence classes. let f and g have the same type of monotonicity in the closure of each equivalence class and contain exactly one cycle through its equivalence classes (say of length k and through their equivalence class ij for some j). claim: f ∼ g whenever graph (fk|īj ) and graph (g k|īj ) are same side of the diagonal. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 224 the class of simple dynamical systems let ij = j1 → j2 = f(j1) → ... → jk = fk−1(j1) → j1 be the k-cycle. given that graph (fk|īj) and graph (gk|īj) are same side of the diagonal. then there exists an increasing bijection h : j̄1 → j̄1 such that (3.1) fk ◦ h = h ◦ gk choose h1 = h. find hi : j̄i → j̄i for i = 2, 3, ..., k and h′1 : j̄1 → j̄1 such that f|j̄i ◦ hi = hi+1 ◦ g|j̄i and f ◦ hk = h ′ 1 ◦ g. recursively, we can prove that h′1 = f k ◦ h1 ◦ g−k. this implies h′1 = h by equation (3.1). define sm(x) = x for all x ∈ im whenever f, g are constant on im. in all the other equivalence classes choose h arbitrary and define h′ (or vice versa) such that f ◦ h = h′ ◦ g. this gives a well defined h : r → r such that h(x) = sn(x) where sn : in → in is a continuous increasing bijection obtained as above such that f ◦sn = sm ◦g. this implies f ◦ h = h ◦ g. similarly we can prove the general case of (1) and the case (2). □ corollary 3.15. let f, g be two increasing bijections having finitely many fixed points such that fix(f) = fix(g) and let fix(f) c = # in. if f|īn ∼ g|īn for every n then f ∼ g. remark 3.16. note that the complement of fix(f) is a countable union of open intervals (including rays) whose end points are fixed points. since f is increasing and the end points are fixed, no point in a component interval can be mapped to a point in any other component interval by f. definition 3.17. let a, b ⊂ r, a ∕= b, and f, g : a → b be continuous maps. we say that f is order conjugate to g if there exist increasing bijections hf : a → a and hg : b → b such that f ◦ hf = hg ◦ g. now we consider: proposition 3.18. (1) let f : (−∞, 0] → [0, 1) be a decreasing bijection (it follows that f(0) = 0). then f is order conjugate to −x−x+1. (2) let f : [1, ∞) → [0, 1) be an increasing bijection (it follows that f(1) = 0). then f is order conjugate to x−1 x . (3) let f : [1, ∞) → (0, 1] be a decreasing bijection (it follows that f(1) = 1). then f is order conjugate to 1 x . (4) let f : [1, ∞) → (−∞, 0] be a decreasing bijection (it follows that f(1) = 0). then f is order conjugate to 1 − x. (5) let f : (−∞, 0] → (0, 1] be an increasing bijection (it follows that f(0) = 1). then f is order conjugate to −1 x−1. proof. this proposition easily follows from the following fact. let a, b ⊂ r be intervals such that either both f, g : a → b are decreasing bijections or both are increasing bijections. then there exist increasing bijections hf : a → a and hg : b → b such that f ◦ hf = hg ◦ g. this is because we can always take an arbitrary hf and define hg = f ◦ hf ◦ g−1. □ next we consider: c© agt, upv, 2020 appl. gen. topol. 21, no. 2 225 k. ali akbar proposition 3.19. let f : r → r be a continuous map with exactly two non-ordinary points 0, 1 such that (1) f(0) = 0. (2) either f|[1,∞) : [1, ∞) → [1, ∞) and it is an increasing bijection or f|[1,∞) : [1, ∞) → (0, 1] and it is a decreasing bijection. (3) either f|(−∞,0] : (−∞, 0] → (−∞, 0] and it is an increasing bijection or f|(−∞,0] : (−∞, 0] → [0, 1) and it is a decreasing bijection. then there are only 48 such maps up to order conjugacy. proof. let f : r → r be a continuous map with exactly two non-ordinary points 0, 1 such that it satisfies the three conditions mentioned in the statement of proposition 3.19. then these non-ordinary points provides an ordered partition {(−∞, 0), (0, 1), (1, ∞)} of r \ {0, 1}. now by proposition 2.19, remark 2.20, proposition 3.10, theorem 3.14, and proposition 3.18, the map f : r → r has exactly four choices in the interval (−∞, 0), exactly three choices in the interval (0, 1), and exactly five choices in the interval (1, ∞) up to order conjugacy. but because of the assumed conditions there are only 48 maps up to order conjugacy. see figure 1 (a). □ remark 3.20. there are sixty eight maps (up to order conjugacy) having exactly two non-ordinary points 0, 1 such that both are fixed. proof. by corollary 2.18, if f : r → r be constant in a neighbourhood of a point x0 then the end points of the maximal interval around x0 on which f is constant, are non-ordinary. hence by the arguments involved in the proof of proposition 3.19, this remark follows. see figure 1 (a). □ now we have: proposition 3.21. let f : r → r be a continuous map with exactly two non-ordinary points 0, 1 such that (1) f(1) = 1. (2) either f|[1,∞) : [1, ∞) → [1, ∞) and is an increasing bijection or f|[1,∞) : [1, ∞) → (0, 1] and is a decreasing bijection. (3) either f|(−∞,0] : (−∞, 0] → (−∞, 0] and is a decreasing bijection or f|(−∞,0] : (−∞, 0] → (0, 1] and is a increasing bijection. then there are only 8 such maps up to order conjugacy. proof. proof follows from theorem 3.14, remark 2.20 and propositions 3.10, 3.18 and 2.19 (see figure 1(b)). □ remark 3.22. there are eleven maps on r (up to order conjugacy) having exactly two non-ordinary points 0, 1 such that 1 is a fixed point and the image of 0 is 1. proof. this remark follows from corollary 2.18 together with the results used for the proof of the proposition 3.21 (see figure 1(b)). □ c© agt, upv, 2020 appl. gen. topol. 21, no. 2 226 the class of simple dynamical systems figure 1. maps with exactly two non-ordinary points c© agt, upv, 2020 appl. gen. topol. 21, no. 2 227 k. ali akbar next we have: proposition 3.23. let f : r → r be a continuous map with exactly two non-ordinary points 0, 1 such that (1) f(0) = g(0) = 0. (2) either f|(−∞,0] : (−∞, 0] → (0, 1] and is a decreasing bijection or f|(−∞,0] : (−∞, 0] → (−∞, 0] and is an increasing bijection. (3) either f|[1,∞) : [1, ∞) → (−∞, 0] and is a decreasing bijection or f|[1,∞) : [1, ∞) → [0, 1) and is an increasing bijection. then there are only 8 such maps up to order conjugacy. proof. proof follows from theorem 3.14, remark 2.20 and propositions 3.10, 3.18 and 2.19 (see figure 1 (c)). □ remark 3.24. there are eleven continuous maps on r (up to order conjugacy) having exactly two non-ordinary points 0, 1 such that 0 is a fixed point and the image of 1 is 0. proof. this remark follows from corollary 2.18 together with results used for the proof of the proposition 3.23 (see figure 1 (c)). □ now we are ready to consider our first main theorem: theorem 3.25 (main theorem 1). there are exactly 90 continuous maps on r with exactly two non-ordinary points, up to order conjugacy. proof. let a and b be the two non-ordinary points such that a < b. then {a, b} is invariant under f by proposition 2.15. hence at least one of these two points is a fixed point; the other is either a fixed point or goes to a fixed point. case 1: both a and b are fixed points. without loss of generality we can assume that a = 0 and b = 1. this is because, let f be a continuous map having only two non-ordinary points a and b. let h : r → r be an increasing bijection such that h([a, b]) = [0, 1]. then consider g = hfh−1. then g(0) = 0 and g(1) = 1. from remark 3.20, it follows that that there are only 68 continuous maps of this type up to order conjugacy. case 2: f(a) = a = f(b). without loss of generality we can assume that a = 0, b = 1 (a similar proof as in case 1 will work). from remark 3.24, it follows that there are only 11 maps of this type up to order conjugacy. case 3. f(b) = b = f(a). without loss of generality we can assume that a = 0 and b = 1 (a similar proof as in case 1 will work). from remark 3.22, it follows that there are only 11 maps of this type up to order conjugacy. hence the proof. □ remark 3.26. from corollary 2.18, remark 2.20, theorem 3.14 and propositions 3.10, 3.18 and 2.19, there are 16 somewhere constant continuous maps c© agt, upv, 2020 appl. gen. topol. 21, no. 2 228 the class of simple dynamical systems up to order conjugacy such that interval of constancy is bounded, 31 somewhere constant continuous maps up to order conjugacy such that the interval of constancy is unbounded, 18 nowhere constant continuous maps up to order conjugacy with unique critical point (among them 9 maps having unique critical point as a local maximum, remaining 9 maps having unique critical point as a local minimum), 3 continuous maps up to order conjugacy with exactly two critical points, and 22 continuous maps up to order conjugacy with no critical points. hence there are exactly 90 continuous maps (up to order conjugacy) on r with having exactly two non-ordinary points. it gives another way of counting involved in theorem 3.25. we can represent each member of this class as a digraph as follows. let f : r → r be continuous map having exactly two non-ordinary points. let i1, i2, i3 be non-singleton equivalence classes such that i1 < i2 < i3, and define a graph (g, vf ) with vertex set vf = {i1, i2, i3}, and an edge from ij to im if f(ij) = im, and a symbol i (respectively d) on this edge whenever f is increasing (respectively decreasing) on ij for j = 1, 2, 3. if there is a kcycle, label one of the symbols a, b, o in the least vertex (say j1) of the cycle depends on the graph(fk|j1) is above the diagonal or below the diagonal or on the diagonal respectively. note: if f is a decreasing homeomorphism then we can consider the labeled digraph of f2 instead of the labeled digraph of f because of proposition 3.4. example 3.27. define f : r → r by f(x) = $ % & 2x if x ≤ 0 x2 if 0 < x ≤ 1 2x − 1 if x > 1 the map f : r → r is continuous with exactly two non-ordinary points 0 and 1. the labeled digraph associated to the map f has three vertices i1, i2, i3 with one loop at each vertex with an edge labeling i on each loop. the additional symbol labeled to the vertices i1, i2 and i3 are b, a and a respectively. the associated graph does not have any other edge. example 3.28. f(x) = $ % & −2x + 1 if x ≤ 0 1 if 0 < x ≤ 1 1 x if x > 1 the map f : r → r is continuous with exactly two non-ordinary points 0 and 1. the labeled digraph associated to the map f : r → r has three vertices i1, i2, i3 with a directed edge from i3 to i2 with edge labeling d. the vertex i1 has an additional symbol labeled as a and the remaining vertices does not have any other additional label. the associated graph does not have any other edge. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 229 k. ali akbar 3.3. class of continuous maps having finitely many non-ordinary points. let f : r → r be a continuous map from this class. then the set of all nonordinary points is invariant by proposition 2.15. by corollary 2.18, the end points of the maximal interval around every point on which f is constant are non-ordinary. by proposition 2.19, if x ∈ r is both critical and ordinary then f is locally constant at x, and if x is ordinary then so is f(x) whenever f is not locally constant in a neighbourhood of x. also note that limx→−∞f(x) and limx→+∞f(x) are non-ordinary whenever f has only finitely many nonordinary points. hence the following informations will help us to characterizes the set of all continuous maps from r to r having finitely many non-ordinary points (see theorem 3.32). (1) graph(f) is above the diagonal or below the diagonal or on the diagonal on each equivalence class (a, b) if f|(a,b) is increasing, and f(a) = a, f(b) = b. related to this we will assign the symbols a, b, o on the vertex (a, b) of the labeled digraph of (r, f) depends on the graph(f|(a,b)) is above the diagonal or below the diagonal or on the diagonal respectively. (2) increasing or decreasing or constant on each equivalence class. related to this we will assign symbols i, d on the edge from the equivalence class to itself of the labeled digraph of (r, f) depends on the graph (f) on corresponding equivalence class is increasing or decreasing respectively. (3) if fk(i) = i, fm(i) ∕= i, m < k then consider fk|j and ask (1) for least j ∈ {i, f(i), ..., fk−1(i)} related to this we will assign the symbols a, b, o on the vertex j of the labeled digraph of (r, f) depends on the graph (fk|j) is above the diagonal or below the diagonal or on the diagonal respectively. now we introduce some labeled digraph for each (r, f) as follows: let i1, i2, ..., in be non-singleton equivalence classes such that i1 < i2 < ... < in. define a graph (g, vf ) with vertex set vf = {i1, i2, ..., in}, and define an edge from ij to ik if f(ij) = ik. but this graph would not give full information of the dynamical system (r, f). to achieve this, we give more labels on each edge on the graph of the map, and on the least vertex of each cycle(see (1), (2) and (3) for details). observe that, if f, g are order conjugate then the associated labeled digraph should be isomorphic. note: if f is a decreasing homeomorphism having odd number of nonordinary points then we can consider the graph of f2 instead of f because of proposition 3.4. image of each non-trivial equivalence class let f : r → r be a continuous map having n − 1 non-ordinary points, and i1 < i2 < ... < in be the n non-singleton equivalence classes of f. then f(i1) = im for some m ∈ {1, 2, ..., n} or a constant; and f(i2) = im+1 or a constant if m = 1, and im or im+1 or a constant if m > 1. in general for c© agt, upv, 2020 appl. gen. topol. 21, no. 2 230 the class of simple dynamical systems 3 ≤ k ≤ n − 2, f(ik) = il or il−1 or il+1 or a constant, l depends on m; and f(in) = ij or a constant; and f(in−1) = ij−1 or a constant if j = n, and ij−1 or ij or a constant if j < n. note that j depends on m. note: this information gives the possible choices of edge sets for the assigned labeled digraph. now we consider the following definitions: definition 3.29. a graph isomorphism between two graphs g and h can be defined as a bijection f : vg → vh that such that a pair of vertices u, v is adjacent in vg if and only if the image pair f(u), f(v) is adjacent in vh. in full generality, a graph isomorphism f : g → h is a pair of bijections fv : vg → vh and fe : eg → eh such that for every edge e ∈ eg, the endpoints of e are mapped onto the endpoints of fe(e). a digraph isomorphism is an isomorphism of the underlying graphs such that the edge correspondence preserves all edge directions. a labeled digraph isomorphism is an isomorphism of the underlying digraphs such that the correspondence preserves labeling. two graphs are isomorphic if there is an isomorphism from one to the other, or informally, if their mathematical structures are identical. definition 3.30. let (s, ≤s), (t, ≤t ) be two partially ordered sets. an order isomorphism from (s, ≤s) to (t, ≤t ) is a surjective map h : s → t such that for all u and v in s, h(u) ≤t h(v) if and only if u ≤s v. in this case, the posets s and t are said to be order isomorphic. all surjective order isomorphisms are bijective (see [3]). definition 3.31. let f : r → r be a continuous map. a subset of r is said to be a dynamically independent set if any two points of the set have disjoint orbits. a subset of r is said to be maximal dynamically independent if it is dynamically independent and no other super set is dynamically independent. now we are ready to prove our second main theorem. theorem 3.32 (main theorem 2). let c be the class of all continuous self maps of r, having finitely many non-ordinary points. then (1) for every member of c, there exists a maximal dynamically independent set that is a finite union of intervals. (2) two members of c are order conjugate if and only if they have order isomorphic maximal dynamically independent set as in (1), and with isomorphic labeled digraphs. (3) every order isomorphism between such maximal dynamically independent set as in (1) extends uniquely to an order conjugacy. proof. let c be the class of all continuous self maps of r, having finitely many non-ordinary points. (1) let f ∈ c and let z0 = x1 < x2 < ... < xn be the n non-ordinary points of f. then i1 = (−∞, x1), in+1 = (xn, ∞) and ii = (xi, xi+1), i = 1, ..., n−1 be the n+1 non-singleton equivalence classes by lemma c© agt, upv, 2020 appl. gen. topol. 21, no. 2 231 k. ali akbar 2.11. choose yi ∈ ii whenever f is not a constant on ii for i = 1, 2, ..., n; and let j be the set of all i such that yi has been chosen. let z1 be the least xi not in o(z0). define, inductively, zi+1 as the least xi not in o(zi). let z be the the set of all elements such that zi+1 /∈ o(zi) (it may be empty but always finite). define yi+1 = f ki(yi) if ki is least such that fki(ii) = ii for i ∈ j. then consider m = ' i∈j(yi, yi+1)∪z. then m is a maximal dynamically independent set. note that this m is always non-empty. (2) first part is easy because having maximal dynamically independent set is invariant under order conjugacy. ie., if m1 is a maximal dynamically independent set of f. then h(m1) is a maximal dynamically independent set of g whenever h is an order conjugacy from f to g. conversely, let f, g ∈ c have order isomorphic maximal dynamically independent set as in (1) (say m1 and m2 respectively), and with isomorphic labeled digraphs. consider all non-empty intersection of each equivalence classes of f with m1 and g with m2. observe that there is a one to one correspondence between these intersections. because of maximal dynamical independency, we can extend restriction of the order isomorphism on these sets to a homeomorphism on its corresponding equivalence class. by a similar proof as in theorem 3.14 we can extend it to a unique conjugacy from f to g since f and g have isomorphic labeled digraphs and order isomorphic maximal dynamically independent sets. (3) easily follows from (2). □ 4. summary for n ∈ n, let cn be the class of all continuous self maps of r with finitely many non-ordinary points up to order conjugacy. for each n ∈ n, with the help of theorem 3.32, we can determine the cardinality of cn. note that the cardinality of c1 is 26 and the cardinality of c2 is 90. our main theorem characterizes the class of all continuous self maps of r with finitely many nonordinary points up to order conjugacy. acknowledgements. the author is very thankful to the referee for giving valuable suggestions. the author acknowledges serb-matrics grant no. mtr/2018/000256 for financial support. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 232 the class of simple dynamical systems references [1] k. ali akbar, some results in linear, symbolic, and general topological dynamics, ph. d. thesis, university of hyderabad (2010). [2] k. ali akbar, v. kannan and i. subramania pillai, simple dynamical systems, applied general topology 2, no. 2 (2019), 307–324. [3] a. brown and c. pearcy, an introduction to analysis (graduate texts in mathematics), springer-verlag, new york, 1995. [4] r. a. holmgren, a first course in discrete dynamical systems, springer-verlag, newyork, 1996. [5] s. patinkin, transitivity implies period 6, preprint. [6] a. n. sharkovskii, coexistence of cycles of a continuous map of a line into itself, ukr. math. j. 16 (1964), 61–71. [7] j. smital, a chaotic function with some extremal properties, proc. amer. math. soc. 87 (1983), 54–56. [8] b. sankara rao, i. subramania pillai and v. kannan, the set of dynamically special points, aequationes mathematicae 82, no. 1-2 (2011), 81–90. c© agt, upv, 2020 appl. gen. topol. 21, no. 2 233 @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 21-25 extending maps between pre-uniform spaces adalberto garćıa-máynez and rubén mancio-toledo abstract we give sufficient conditions on a uniformly continuous map f : (x, u) → (y, v ) between completable t1-pre-uniform spaces (x, u), (y, v ) to have a continuous or a uniformly continuous extension ̂f : ̂x → ̂y between the corresponding completions. 2010 msc: 54a20, 54e15 keywords: minimal, round, pre-uniform, completion, extension 1. preliminary results the basic concepts used in this paper: pre-uniformity bases, cauchy or minimal filters, round, weakly round or strongly round filters and completion conditions are given in [1]. the concept of pre-uniform basis appeared in 1970 under the name of structure [3]. however, non hausdorff pre-uniform spaces were very seldom considered in harris monography. t1-pre-uniform spaces have an important property: every cauchy filter contains a unique weakly round filter and every neighborhood filter is weakly round. the set of weakly round filters x̂ of a t1-pre-uniform space has a complete t1-pre-uniform basis û such that the map h: (x,u) → (x̂,û) which assigns to each x ∈ x its neighborhood filter is a uniform embedding. hence, any uniformly continuous map ϕ: (x,u) → (y,v ) between t1-pre-uniform spaces induces a map ϕ̂: (x̂,û) → (ŷ , v̂ ) which sends every weakly round filter ξ ∈ x̂ into the unique weakly round filter n in ŷ which is contained in the cauchy filter ϕ(ξ) = {ϕ(l) | l ∈ ξ}+ . (for every subfamily g of the power set of a set z, we define g+ = {l ⊆ z | for some g ∈ g,g ⊆ l}). if k : (y,v ) → (ŷ , v̂ ) is the canonical uniform 22 a. garćıa-máynez and r. mancio-toledo embedding, i.e. k(y) = neighborhood filter of y, we have the relation ϕ̂ ◦ h = k ◦ ϕ. in this paper, we find conditions on ϕ, u and v which insure that ϕ̂ is continuous or uniformly continuous. 2. main results we start this section with a lemma. lemma 2.1. let (x,u) be a t1-pre-uniform space and suppose (x,τu ) is a t1-space. then every cauchy filter ξ in (x,u) contains a unique minimal filter ξ′. proof. we know ξ′ = {st ∗∗(ξ,α) | α ∈ u}+ is u-minimal and is contained in ξ, where st ∗∗(ξ,α) = ⋃ {l | l ∈ α ∩ ξ} . suppose n ⊆ ξ is another u-minimal filter. therefore, n ′ = n ⊆ ξ′. the minimal property of ξ′ implies that n ′ = n = ξ′. � we give two cases in which ϕ̂ is uniformly continuous. lemma 2.2. suppose for each ξ ∈ x̂ − h(x),ϕ(ξ) = ϕ(ξ)′. then ϕ̂ is uniformly continuous. proof. let β ∈ v and let α ∈ u be such that α ≤ ϕ−1(β). we shall prove that α̂ ≤ ϕ̂−1(β̂ ). let a ∈ α and b ∈ β be such that a ⊆ ϕ−1(b). we claim that â ⊆ ϕ̂−1(b̂). let us take ξ ∈ â. then a ∈ ξ and ϕ(a) ∈ ϕ(ξ). since ϕ(a) ⊆ b, we have also b ∈ ϕ(ξ). therefore, ϕ̂(ξ) = ϕ(ξ) ∈ b̂ and the proof is complete. � lemma 2.3. if (y,v ) is a semi-uniform space, ϕ̂ is uniformly continuous. proof. let β ∈ v . since (y,v ) is a semi-uniform space, there exists a cover γ ∈ v which satisfies the following condition: su) for each c ∈ γ, there exists δc ∈ v and bc ∈ β such that st (c,δc) ⊆ bc. let α ∈ u be such that α ≤ ϕ−1(γ). we shall prove that α̂ ≤ ϕ̂ −1(̂β). if a ∈ α, there exists a set c ∈ γ such that a ⊆ ϕ−1(c). by condition su), there exist δc ∈ v and bc ∈ β such that st (c,δc) ⊆ bc. we claim that â ⊆ ϕ̂−1(b̂c). if ξ ∈ â, we have a ∈ ξ. since ϕ(a) ⊆ c, we have c ∈ ϕ(ξ)+. therefore, st (c,δc) ∈ ϕ(ξ)′ = ϕ̂(ξ). since st (c,δc) ⊆ bc, we conclude that bc ∈ ϕ̂(ξ) and ϕ̂(ξ) ∈ b̂c. � lemma 2.4. let x,y be t2-spaces and let u,v , respectively, be the families of densely finite covers of x,y. let ϕ: x → y be continuous, open and surjective. then ϕ is uniformly continuous as a map from (x,u) onto (y,v ). extending maps between pre-uniform spaces 23 proof. let β be a densely finite cover of y . then we can find a finite subfamily {b1,b2, · · · ,bn} ⊆ β such that b−1 ∪ b − 2 ∪ · · · ∪ bn− = y. if b = b1 ∪ b2 ∪ · · · ∪ bn and α = f−1(β), the hypotheses imply that {a1,a2, . . . ,an} ⊆ α where ai = f −1(bi) for i = 1,2, . . . ,n, satisfies a−1 ∪ a − 2 ∪ · · · ∪ a − n = f −1(b−) = x . hence α is a densely finite cover of x and α ≤ f−1(β) (in fact, α = f−1(β)). then f is uniformly continuous. � lemma 2.5. keep the hypotheses of (2.4). then ϕ̂: x̂ → ŷ is continuous and surjective. proof. let n ∈ x̂ and let t be an open set in y such that ϕ̂(n) = ϕ(n)′ ∈ t̂ . then t ∈ ϕ(n)′. therefore, there exists a cover γ ∈ v such that t ⊇ st ∗∗(ϕ(n),γ). since ϕ: (x,u) → (y,v ) is uniformly continuous (2.4), the filter ϕ(n) is cauchy in (y,v ). select an element n0 ∈ γ ∩ ϕ(n). then ϕ−1(n0) ∈ ϕ−1(γ) ∩ n and n0 ⊆ st ∗∗(ϕ(n),γ) ⊆ t . therefore: ϕ−1(n0) ⊆ st ∗∗(n ,ϕ−1(γ)) = ϕ−1(st ∗∗(ϕ(n),γ)) ⊆ ϕ−1(t). we shall prove that ϕ̂(st (n ,ϕ−1(γ)̂ ) ⊆ t̂ and the continuity of ϕ̂ will follow. let m ∈ st (n ,ϕ−1(γ)̂ ). then there exists an element c ∈ γ such that ϕ−1(c) ∈ m ∩ n . hence, ϕ−1(c) ⊆ st ∗∗(n ,ϕ−1(γ)) ⊆ ϕ−1(t) and c ⊆ st ∗∗(ϕ(n),γ) ⊆ t . we also have c ⊆ st ∗∗(ϕ(m),γ) ∈ ϕ(m) ′ = ϕ̂(m). then t ∈ ϕ̂(m) and ϕ̂(m) ∈ t̂ . � before we prove ϕ̂ is surjective, we need a lemma. lemma 2.6. a non-adherent filter t in (x,u) is u-round if and only if t has as a basis an ultrafilter of open sets. proof. suppose t is a non-adherent round filter in (x,u). let g be the family of open sets in t and take an open set v such that v ∩g = ∅ for every g ∈ g. we have to prove that v ∈ t and that will convert g into an ultrafilter of open sets. since t is non-adherent, the family {x − f−|f ∈ t } is an open cover of x. hence. α = {v,x − v −} ∪ {x − f−|f ∈ t } is a densely finite cover of x. since t is u-cauchy, we have v ∈ t or x − v − ∈ t . if we had x − v − ∈ t , we use the roundness of t and find a cover β ∈ u such that x − v − ⊇ st ∗(t ,β) = ∪{b ∈ β | b ∩ f = ∅ for every f ∈ t }. if g ∈ β ∩ t , we have g ⊆ x − v − and hence v ∩ g = ∅, a contradiction. therefore we must have v ∈ t and g is an ultrafilter of open sets. 24 a. garćıa-máynez and r. mancio-toledo conversely, suppose g is an ultrafilter of open sets. we have to prove that t is u-round. we prove first that t is u-cauchy. let α ∈ u. if t ∩ α = ∅, then a /∈ t ∩ τ for every a ∈ α. let {a1,a2, . . . ,an} ⊆ α be such that x = a1 − ∪a2− ∪· · ·∪an−. since ai /∈ t ∩ τ and t ∩ τ is an ultrafilter of open sets, we can find elements gi ∈ t ∩ τ such that ai ∩ gi = ∅ (i = 1,2, . . . ,n). hence (g1 ∩ g2 ∩ · · · ∩ gn) ∩ (a1 ∪ a2 ∪ · · · ∩an) = ∅. but a1 ∪ a2 ∪ · · · ∪ an is dense in x. hence g1 ∩ g2 ∩ · · · ∩ gn = ∅, a contradiction. we finally prove that t is u-round. pick any element f0 ∈ t and consider the cover α = {f0} ∪ {x − f− | f ∈ t }. clearly st ∗(t ,α) = f0 and hence t is u-round. � in [4] it is proved that every cauchy filter in (x,u), where u is the family of densely finite covers of the hausdorff space (x,τ), contains an u-round filter and by [1],(x,u) has a completion (x̂, û ) where every û-round filter is convergent and the topologyτ ̂u is hausdorff closed. besides the completion (x̂, û ),(x,τ) has the katetov extension kx, which is also hausdorff closed. in this volume we prove that in general, the extensions x̂ and kx are nonequivalent. 3. applications proposition 3.1. let x be a separable, metrizable, dense in itself, 0-dimensional space and let z be a compact, hausdorff, separable space. then there exists a surjective continuous map g : x̂ → z, where x̂ is the completion of the preuniformity basis of x consisting of all densely finite covers of x. proof. the hypothesis imply the existence of mutually disjoint non-empty open sets l1,l2, . . . such that x = ∞ ∪ n=1 ln. the map ϕ: x → n where ln = ϕ−1(n) for each n ∈ n, is continuous, open and surjective. by 2.4, there exists a continuous surjective extension ϕ̂: x̂ → n̂. but n̂ coincides with the stonečech compactification βn of n (because a cover α of n is densely finite if and only if it is finite). on the other hand, by the universal property of βn, there exists a continuous surjective map ψ : βn → z. hence, g = ψ◦ϕ̂ is a continuous surjective map from x̂ onto z. � proposition 3.2. let x be a non-empty completely metrizable separable space. then there exists a continuous surjective map ψ : (n w )̂→ x̂. proof. n w may be identified with the set of irrational numbers and this space satisfies the conditions of (3.1). on the other hand, there exists a continuous open surjective map ϕ: n w → x (see 5.15 in [2]). using 2.4, we complete the proof. � corollary 3.3. if z is a tychonoff separable space which is either compact or completely metrizable, then there exists a continuous surjective map ψ : (nw)̂ → ẑ. extending maps between pre-uniform spaces 25 we finish this paper with a problem: problem 3.4. is every čech-complete separable space a continuous image of (n w )̂ ? references [1] a. garćıa-máynez and r. mancio toledo, completions of pre-uniform spaces, appl. gen. topol. 8, no. 2 (2007), 213–221. [2] a. garćıa-máynez and a. tamariz, topoloǵıa general, ed. porrúa, méxico, 1988. [3] d. harris, structures in topology, memoirs of the american mathematical society, no. 115, providence, rhode island, ams, 1971. [4] r. mancio toledo, los espacios pre-uniformes y sus completaciones, ph. d. thesis, unam, 2006. (received january 2010 – accepted january 2011) a. garćıa-máynez (agmaynez@matem.unam.mx) instituto de matemáticas, universidad nacional autónoma de méxico, área de la investigación cient́ıfica, circuito exterior, ciudad universitaria, 04510 méxico, d.f. méxico rubén mancio-toledo (rmancio@esfm.ipn.mx) escuela superior de f́ısica y matemáticas, instituto politécnico nacional, unidad profesional adolfo lópez mateos, col. lindavista, 07738 méxico, d.f. extending maps between pre-uniform spaces. by a. garcía-máynez and r. mancio-toledo @ appl. gen. topol. 23, no. 2 (2022), 281-286 doi:10.4995/agt.2022.17080 © agt, upv, 2022 the largest topological ring of functions endowed with the m-topology tarun kumar chauhan and varun jindal department of mathematics, malaviya national institute of technology jaipur, jaipur-302017, rajasthan, india (rajputtarun.chauhan@gmail.com, vjindal.maths@mnit.ac.in) communicated by a. tamariz-mascarúa abstract the purpose of this article is to identify the largest subring of the ring of all real valued functions on a tychonoff space x, which forms a topological ring endowed with the m-topology. 2020 msc: 54c30; 54c40; 54h13. keywords: locally bounded functions; real valued functions; rings of functions; m-topology. 1. introduction for a tychonoff (completely regular and hausdorff) space x, rx represents the ring of all real valued functions defined on x. two of its important subrings are c(x), the set of all continuous functions in rx, and c∗(x), the set of all bounded and continuous members of rx. the algebraic properties of the rings c(x) and c∗(x) vis-à-vis topological properties of x have been studied extensively in the literature (see, [3]). besides studying the algebraic properties of c(x), one can also define many interesting topologies on c(x). consequently, one may study the interaction between various algebraic structures of c(x) with the corresponding topology on it. two commonly studied topologies on c(x) and c∗(x) are the u-topology (uniform topology) and the m-topology. both of these topologies can be defined on rx (definitions are given in the next section). received 27 january 2022 – accepted 14 july 2022 http://dx.doi.org/10.4995/agt.2022.17080 t. k. chauhan and v. jindal though the convergence concerning u-topology, popularly known as the uniform convergence, has been known for centuries, the m-topology was introduced by e. hewitt ([5]) in 1948. in many aspects, the u-topology is the most relevant for studying c∗(x), while the m-topology is appropriate to study the ring c(x) (see, theorem 1 and theorem 3 in [5]). the m-topology has been studied in detail in [1, 2, 4, 6, 7, 8, 11, 12]. for more about u-topology and m-topology, we refer readers to the recent research monograph [13]. it is known that c∗(x) equipped with the u-topology forms a topological ring while c(x) need not. however, c(x) equipped with the m-topology is a topological ring. but rx with any of these topologies does not form a topological ring in general. there are several interesting subrings that are intermediate between c∗(x) and rx, such as the ring of all baire one functions and the ring of all locally bounded functions. the main results of the paper (theorems 2.8 and 2.11) help to recognize such subrings, which are topological rings endowed with u-topology and m-topology. it can be shown that c∗(x) is the largest subring of c(x), which is a topological ring under the u-topology. it follows that a subring s(x) of c(x) is a topological ring for the u-topology if and only if s(x) ⊆ c∗(x). this article aims to formulate the largest subrings of rx, which form topological rings when equipped with u-topology and mtopology, respectively. 2. main results throughout this article, x is assumed to be a tychonoff space (though we may specify that it has some additional properties). we first recall the definitions of u-topology and m-topology. definition 2.1. the u-topology or uniform topology (denoted by τu) is determined on rx by taking all sets of the form bu(f,�) = {g ∈ rx : |f(x) −g(x)| < �,∀x ∈ x}; (� > 0 is constant) as a base for the neighborhood system at f ∈ rx. definition 2.2. the m-topology (denoted by τm) is determined on rx by taking all sets of the form bm(f,η) = {g ∈ rx : |f(x) −g(x)| < η(x),∀x ∈ x}; (η ∈ u+(x)) as a base for the neighborhood system at f ∈ rx. here u+(x) represents the set of all positive units in c(x). clearly, τm is finer than τu. these topologies coincide if and only if x is pseudocompact. the u-topology can also be determined on rx by using the positive units of c∗(x) in a way given in the following proposition. proposition 2.3. let τ be the topology on rx determined by taking all sets of the form b(f,γ) = {g ∈ rx : |f(x) −g(x)| < γ(x),∀x ∈ x}; (γ ∈ u∗+(x)) © agt, upv, 2022 appl. gen. topol. 23, no. 2 282 the largest topological ring of functions as a base for the neighborhood system at f for each f ∈ rx, where u∗+(x) denotes the set of all positive units in c∗(x). then τ = τu on rx. there is a considerable difference between the units of the rings c(x) and c∗(x). a function f ∈ c(x) is a positive unit of c(x) if and only if f(x) > 0. but a function f ∈ c∗(x) is a positive unit of c∗(x) if and only if f(x) > 0 and 1 f ∈ c∗(x). equivalently, f ∈ c∗(x) is a positive unit of c∗(x) if and only if inf{f(x) : x ∈ x} > 0. so it is clear that every positive unit of c∗(x) is also a positive unit of c(x), but the converse need not be true. to identify the largest subrings of rx, which are topological rings endowed with τu and τm, we define two families b(x) and d(x) of functions in the following manner. b(x) = {f ∈ rx : ∃ ψ ∈ u∗+(x), |f(x)| < ψ(x) for all x ∈ x}, d(x) = {f ∈ rx : ∃ φ ∈ u+(x), |f(x)| < φ(x) for all x ∈ x}. it is not hard to see that b(x) and d(x) are subrings of rx, and b(x) is the same as the family of all bounded functions in rx. clearly, c(x) ⊆d(x) and b(x) ⊆d(x). also d(x) = b(x) if and only if x is pseudocompact. we now relate d(x) with two important subrings of rx, namely b1(x) and lb(x) which denote respectively, the ring of all baire one functions and the ring of all locally bounded functions. recall that a function f : x → r is called baire one if f is the pointwise limit of a sequence of continuous functions from x to r, and f is called locally bounded if it is bounded on some neighborhood of each point of x. clearly, d(x) ⊆ lb(x). in general, this containment is strict. to see an example of a space x for which d(x) 6= lb(x), we need the following definition and discussion. definition 2.4 (horne, [9]). a space x is called a cb-space if for each h ∈ lb(x), there exists f ∈ c(x) such that |h| ≤ f. the next proposition follows immediately from the definition 2.4. proposition 2.5. d(x) = lb(x) if and only if x is a cb-space. it is known that a cb-space is countably paracompact, and a normal space is a cb-space if and only if it is countably paracompact ([10]). consequently, for a non countably paracompact space x, d(x) 6= lb(x). the following example proves that b1(x) and d(x) may not be comparable. example 2.6. let x = r with the usual topology. define the function f : r → r such that f(x) = 0 for all x ∈ (−∞, 0] and f(x) = 1 x for all x ∈ (0,∞). for each n ∈ n, define a function fn : r → r such that fn(x) =   0 if x ∈ (−∞, 0], n2x if x ∈ (0, 1/n), 1 x if x ∈ [1/n,∞). © agt, upv, 2022 appl. gen. topol. 23, no. 2 283 t. k. chauhan and v. jindal clearly, each fn is continuous and the sequence (fn) converges pointwise to f. therefore, f ∈ b1(x). but f /∈d(x). now define another function h : r → r such that h(x) = 0 for every rational number x and h(x) = 1 for every irrational number x. we can easily see that h ∈d(x) \b1(x). using proposition 2.5 and example 2.6, by theorem 2.8, we can conclude that lb(x) and b1(x) need not form topological rings endowed with the m-topology. theorem 2.7. d(x) endowed with τm is a topological ring. proof. the continuity of the map (f,g) → f + g is easy to check. we only prove that the map (f,g) → fg is continuous. let f,g ∈ d(x). so there exist φf,φg ∈ u+(x) such that |f(x)| < φf (x) and |g(x)| < φg(x) for all x ∈ x. let bm(fg,η) be any basic neighborhood of fg in (d(x),τm) for some η ∈ u+(x). consider the basic neighborhoods bm(f,η1) and bm(g,η2) of f and g respectively for η1 = η 2(1+φg) and η2 = η 2(φf +η1+1) . it is enough to show that for any h1 ∈ bm(f,η1) and h2 ∈ bm(g,η2), we have h1h2 ∈ bm(fg,η). it follows as |(fg)(x) − (h1h2)(x)| ≤ |g(x)||f(x) −h1(x)| + |h1(x)||g(x) −h2(x)| < φg(x)η1(x) + |h1(x)|η2(x) for all x ∈ x < η(x) for all x ∈ x. � our next theorem establish the fact that d(x) is the largest subring of rx which is a topological ring endowed with the m-topology. theorem 2.8. let s(x) be a subring of rx. then the following conditions are equivalent: (a) s(x) endowed with τm is a topological ring; (b) s(x) ⊆d(x). proof. (a) ⇒ (b). suppose s(x) * d(x). let f ∈ s(x) \d(x). we show that pointwise multiplication (f,g) → fg is not continuous at point (0x,f), where 0x is the constant function zero on x. consider the basic neighborhood bm(0x, 1) of the function 0xf = 0x in (s(x),τ m). since f /∈d(x), for every η ∈ u+(x) there exists a point xη ∈ x such that |f(xη)| ≥ 2 η(xη) , that is,∣∣∣∣η(xη)2 f(xη) ∣∣∣∣ ≥ 1. therefore for any η,µ ∈ u+(x), we have η2 ∈ bm(0x,η) and f ∈ bm(f,µ) but η 2 f /∈ bm(0x, 1). (b) ⇒ (a). it follows from theorem 2.7. � corollary 2.9. lb(x) equipped with τm is a topological ring if and only if x is a cb-space. © agt, upv, 2022 appl. gen. topol. 23, no. 2 284 the largest topological ring of functions corollary 2.10. for a first countable space x, the following conditions are equivalent: (a) x is discrete; (b) rx = d(x); (c) rx endowed with τm is a topological ring; (d) rx = lb(x). proof. the implication (a) ⇒ (b) is immediate and the implications (b) ⇒ (c) ⇒ (d) follow from theorem 2.8. (d) ⇒ (a). suppose there is a non-isolated point x0 ∈ x. since x is first countable, there exists a sequence (xn) of distinct points in x \ {x0} which converges to x0. define a function f : x → r such that f(xn) = n for every n ∈ n and f(x) = 0 for all x ∈ x \ {xn : n ∈ n}. it is easy to see that f /∈ lb(x). we arrive at a contradiction. � it should be noted that x being first countable is used only to prove the implication (d) ⇒ (a). it may be interesting to know whether corollary 2.10 is true for any tychonoff space x. theorem 2.11. b(x) is the largest subring of rx, which is a topological ring endowed with τu. proof. it can be proved in a manner similar to theorems 2.7 and 2.8. � corollary 2.12. rx equipped with τu is a topological ring if and only if x is finite. corollary 2.13. every subring of rx which forms a topological ring under τu is also a topological ring under τm. corollary 2.14. for a space x, the following conditions are equivalent: (a) lb(x) endowed with τu is a topological ring; (b) b(x) = lb(x); (c) x is a pseudocompact cb-space; (d) x is countably compact. proof. the equivalence (a) ⇔ (b) follows from theorem 2.11 and (c) ⇔ (d) follows from theorem 9 of [10]. (b) ⇔ (c). it follows from proposition 2.5 and the fact that d(x) = b(x) if and only if x is pseudocompact. � we conclude this article with the following question. question 2.15. for what spaces x, the ring b1(x) endowed with τ m or τu forms a topological ring? © agt, upv, 2022 appl. gen. topol. 23, no. 2 285 t. k. chauhan and v. jindal acknowledgements. we thank the referee for his/her valuable suggestions. the second author acknowledges the support of nbhm research grant 02011/ 6/2020/nbhm(r.p) r&d ii/6277. references [1] f. azarpanah, f. manshoor and r. mohamadian, connectedness and compactness in c(x) with m-topology and generalized m-topology, topol. appl. 159 (2012), 3486–3493. [2] f. azarpanah, m. paimann and a. r. salehi, connectedness of some rings of quotients of c(x) with the m-topology, comment. math. univ. carolin. 56 (2015), 63–76. [3] l. gillman and m. jerison, rings of continuous functions, springer-verlag, new york, 1976. reprint of the 1960 edition, graduate texts in mathematics, no. 43 [4] j. gómez-pérez and w. w. mcgovern, the m-topology on cm(x) revisited, topol. appl. 153, no. 11 (2006), 1838–1848. [5] e. hewitt, rings of real-valued continuous functions. i, trans. amer. math. soc. 64 (1948), 45–99. [6] l. holá and v. jindal, on graph and fine topologies, topol. proc. 49 (2017), 65–73. [7] l. holá and r. a. mccoy, compactness in the fine and related topologies, topol. appl. 109 (2001), 183–190. [8] l. holá and b. novotný, topology of uniform convergence and m-topology on c(x), mediterr. j. math. 14 (2017): 70. [9] j. g. horne, countable paracompactness and cb-spaces, notices. amer. math. soc. 6 (1959), 629–630. [10] j. mack, on a class of countably paracompact spaces, proc. amer. math. soc. 16 (1965), 467–472. [11] g. di maio, l. holá, d. holý and r. a. mccoy, topologies on the space of continuous functions, topol. appl. 86 (1998), 105–122. [12] r. a. mccoy, fine topology on function spaces, int. j. math. math. sci. 9 (1986), 417–424. [13] r. a. mccoy, s. kundu and v. jindal, function spaces with uniform, fine and graph topologies, springer briefs in mathematics, springer, cham, 2018. © agt, upv, 2022 appl. gen. topol. 23, no. 2 286 @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 175-185 morse theory for c*-algebras: a geometric interpretation of some noncommutative manifolds vida milani∗, ali asghar rezaei and seyed m. h. mansourbeigi abstract the approach we present is a modification of the morse theory for unital c*-algebras. we provide tools for the geometric interpretation of noncommutative cw complexes. some examples are given to illustrate these geometric information. the main object of this work is a classification of unital c*-algebras by noncommutative cw complexes and the modified morse functions on them. 2010 msc: 06b30, 46l35, 46l85, 55p15, 55u10. keywords: c*-algebra, critical points, cw complexes, homotopy equivalence, homotopy type, morse function, noncommutative cw complex, poset, pseudo-homotopy type, *-representation, simplicial complex. 1. introduction morse theory is an approach in the study of smooth manifolds by the tools from calculus. the classical morse theory provides a connection between the topological structure of a manifold m and the homotopy type of critical points of a function f : m → r (the morse functions). on a smooth manifold m, a point a ∈ m is a critical point for a smooth function f : m → r, if the induced map f∗ : ta(m) → r is zero. the real number f(a) is then called a critical value. the function f is a morse function if i) all the critical values are distinct and ii) its critical points are non degenerate, i.e. the hessian matrix of second derivatives at the critical points has a ∗corresponding author. 176 v. milani, a. a. rezaei and s. m. h. mansourbeigi non vanishing determinant. the number of negative eigenvalues of this hessian matrix is the index of f at the critical point. the classical morse theory states: theorem ([14]): there exists a morse function on any differentiable manifold and any differentiable manifold has the homotopy type of a cw complex with one λ-cell for each critical point of index λ. so once we have information around the critical points of a morse function on m, we can reconstruct m by a sequence of surgeries. a c*-algebraic approach which links operator theory and algebraic geometry, is obtained via a suitable set of equivalence classes of extensions of commutative c*-algebras. this provides a functor from locally compact spaces into abelian groups([7], [12], [15]). if j and b are two c*-algebras, an extension of b by j is a c*-algebra a together with morphisms j : j → a and η : a → b such that the following sequence is exact: (1.1) 0 −−−−→ j j −−−−→ a η −−−−→ b the aim of the extension problem is the characterization of those c*-algebras a satisfying the above exact sequence. this has something to do with algebraic topology techniques. in the construction of a cw complex, if xk−1 is a suitable subcomplex, ik the unit ball and sk−1 its boundary, then the various solutions for the extension problem of c(xk−1) by c0(i k −sk−1) correspond to different ways of attaching ik to xk−1 along s k−1, via an attaching map ϕk : s k−1 → xk−1 which identify points x ∈ s k−1 with their image ϕk(x) in the disjoint union xk−1 ∪ i k. after the construction of noncommutative geometry [1], there have been attempts to formulate the classical tools of differential geometry and topology in terms of c*-algebras (in some sense the dualization of the notions, [3], [4], [11]). the dual concept of cw complexes , with some regards, is the notion of noncommutative cw complexes ([7] and [15]). the approach of this paper is the geometric study of these structures. so many works have been done on the combinatorial structures of noncommutative simplicial complexes and their decompositions, for example [2], [6], [9], [10]. following these works, together with some topological constructions , we show how a modification of the classical morse theory to the level of c*-algebras will provide an innovative way to explain the geometry of noncommutative cw complexes through the critical ideals of the modified morse function. this leads up to some classification theory. this paper is prepared as follows. after introducing the notion of primitive spectrum of a c*-algebra, we will proceed the topological structure in detail and present some examples. then we will study the noncommutative cw complexes and interpret their geometry by introducing the modified morse function. all these provide tools for the modified morse theory for c*-algebras. the last section is devoted to the proof of the following theorem: morse theory for c*-algebras 177 main theorem: every unital c*-algebra with an acceptable morse function on it is of pseudo-homotopy type as a noncommutative cw complex, having a k-th decomposition cell for each critical chain of order k. 2. the structure of the primitive spectrum the technique we follow to link the geometry, topology and algebra is the primitive spectrum point of view. in fact as we will see in our case it is a promising candidate for the noncommutative analogue of a topological manifold m. we review some preliminaries on the primitive spectrum. details can be found in [5], [11], [13]. let a be a unital c*-algebra. the primitive spectrum of a is the space of kernels of irreducible *-representations of a. it is denoted by prim(a). the topology on this space is given by the closure operation as follows: for any subset u ⊆ prim(a), the closure of u is defined by (2.1) u := {i ∈ prim(a) : ⋂ j∈u j ⊂ i} obviously u ⊆ u. this operation defines a topology on prim(a) (the hullkernel topology), making it into a t0-space ([8]). definition 2.1. a subset u ⊂ prim(a) is called absorbing if it satisfies the following condition: (2.2) i ∈ u,i ⊆ j ⇒ j ∈ u. lemma 2.2. the closed subsets of prim(a) are exactly its absorbing subsets. proof. it is clear from the definition of closed sets. � in the special case, when m is a compact topological space, and a = c(m) is the commutative unital c*-algebra of complex continuous functions on m, a homeomorphism between m and prim(a) is obtained in the following way. for each x ∈ m let ix := {f ∈ a : f(x) = 0}; ix is a closed maximal ideal of a. it is in fact the kernel of the evaluation map (ev)x :a −→ c f 7−→ f(x). now (2.3) i : m → prim(a) defined by i(x) := ix is the desired homeomorphism. let a be an arbitrary unital c*-algebra. to each i ∈ prim(a), there corresponds an absorbing set wi := {j ∈ prim(a) : j ⊇ i}, and an open set oi := {j ∈ prim(a) : j ⊆ i}, 178 v. milani, a. a. rezaei and s. m. h. mansourbeigi containing i. being a t0-space, prim(a) can be made into a partially ordered set (poset) by setting, i < j ⇔ i ⊂ j for i,j ∈ prim(a), remark 2.3. the following statements are equivalent: i) i ⊆ j. ii) oi ⊆ oj. iii) wi ⊇ wj. the topology of prim(a) can be given equivalently by means of this partial order, i < j ⇔ j ∈ {i}, where {i} is the closure of the one point set {i}. since a is unital, prim(a) is compact([8]). let prim(a) = ⋃n i=0 oii be a finite open covering. in general, let us suppose we have a topological space m together with an open covering u = {ui} which is also a topology for m. an equivalence relation on m is set by declaring that any two points x,y ∈ m are equivalent if every open set ui containing either x or y contains the other too. in this way the quotient space of m is made into a finite lattice. in the same way an equivalence relation on prim(a) is given by i ∼ j ⇔ (j ∈ oi ⇔ i ∈ oj). this is of course the trivial relation i ∼ j ⇔ i = j. in each oii choose one ii with respect to the above equivalence relation. since i ⊆ j implies oi ⊆ oj, oiis can be chosen so that ii 6= ij for i 6= j. let i0,i1, ...,in be chosen in this way so that prim(a) is made into a finite lattice for which the points are the equivalence classes of [i0], ..., [in]. for simplicity we show each class [ii] by its only representative ii. let ji0,...,ik := ii0 ∩ ... ∩ iik, where 1 ≤ i0, ..., ik ≤ n,1 ≤ k ≤ n. set wi0,...,ik := {j ∈ prim(a) : j ⊇ ji0,...,ik}. this is a closed subset of prim(a). in what follows we see that the above construction makes it possible to obtain a cell complex decomposition for prim(c(x)) when x has a cw complex structure. in fact the closed sets wi0,...,ik corresponding to ji0,...,ik play the role of chains in the construction. remark 2.4. if ji0,...,ik = 0 for some 1 ≤ i0, ..., ik ≤ n, 1 ≤ k ≤ n, then wi0,...,ik = prim(a). also for each pair of indices (i0, ..., it) ,σ(i0, ..., it+m), wi0,...,it ⊆ wσ(i0,...,it+m) where σ is a permutation on t + m elements and 1 ≤ i0, ..., it+m ≤ n. morse theory for c*-algebras 179 remark 2.5. a sequence x0 ⊂ x1 ⊂ ... ⊂ xn = x is an n-dimensional cw complex structure for a compact topological space x, where x0 is a finite discrete space consisting of 0-cells, and for k = 1, ...,n each k-skeleton xk is obtained by attaching λk number of k-disks to xk−1 via the attaching maps ϕk : ⋃ λk sk−1 → xk−1. in other words (2.4) xk = xk−1 ⋃ (∪λki k) x ∼ ϕk(x) := xk−1 ⋃ ϕk (∪λki k) where ik := [0,1]k and sk−1 := ∂ik. the quotient map is denoted by ρ : xk−1 ⋃ (∪λki k) → xk. for more details see [12]. now let x0 ⊂ x1 ⊂ ... ⊂ xn = x be an n-dimensional cw complex structure for the compact space x. a cell complex structure is induced on prim(c(x)) by the following procedure: let ak = c(xk), k = 0,1, ...,n. set a = c(x) = c(xn) = an. let i : x → prim(c(x)) be the homeomorphism of relation (4). for each k-cell ck in the k-skeleton xk, let ick = ⋂ x∈ck ix = {f ∈ a : f(x) = 0;x ∈ ck}, for 0 ≤ k ≤ n . by considering the restriction of functions on x to xk, ick will be an ideal in ak. definition 2.6. in the above notations, ick is called a k-ideal in a (or ak) and the restriction of its corresponding closed set wi0,...,ik in prim(ak), i.e. wi0,...,ik = {j ∈ prim(ak) : j ⊇ ick } is called a k-chain. in the following two examples we identify the k-ideals and the k-chains for the cw complex structures of the closed interval [0,1] and the 2-torus s1 ×s1. example 2.7. let x0 = {0,1} and x1 = [0,1] be the zero and the one skeleton for a cw complex structure of [0,1]. then we have a0 = c(x0) ≃ c ⊕ c and a = a1 = c(x1). the 0-ideals i0 and i1 and their corresponding 0-chains w0 and w1 are as follow: i0 = {f ∈ a0 : f(0) = 0} ≃ c,i1 = {f ∈ a0 : f(1) = 0} ≃ c, w0 = {j ∈ prim(a0) : j ⊇ i0} ≃ {0},w1 = {j ∈ prim(a0) : j ⊇ i1} ≃ {1}. 180 v. milani, a. a. rezaei and s. m. h. mansourbeigi corresponding to the 1-chain c1 = [0,1], the only 1-ideal is i = ⋂ x∈c1 ix = {f ∈ a : f(x) = 0;x ∈ [0,1]} = 0, with the corresponding 1-chain wi = {j ∈ prim(a) : j ⊇ i} = prim(a) ≃ [0,1]. example 2.8. let x0 = {0},x1 = {α,β},x2 = t 2 = s1 × s1 be the skeletons of a cw complex structure for the 2-torus t 2, where α,β are homeomorphic images of s1 (closed curves with the origin 0). let a0 = c(x0) = c , a1 = c(x1) and a = a2 = c(t 2). the 0-ideal and its corresponding 0-chain are as follow: i0 = {f ∈ a0 : f(0) = 0}, w0 = {j ∈ prim(a0) : j ⊇ i0} ≃ prim(a0) = {0}. also the 1-ideals i1,i2 and 1-chains wi1,wi2 are i1 = {f ∈ a1 : f(α) = 0} = ∩x∈αix ≃ c, i2 = {f ∈ a1 : f(β) = 0} = ∩x∈βix ≃ c, wi1 = {j ∈ prim(a1) : j ⊇ i1} ≃ {α}, wi2 = {j ∈ prim(a1) : j ⊇ i2} ≃ {β}. finally the only 2-ideal and its corresponding 2-chain are i = {f ∈ a : f(t 2) = 0} ≃ 0, wi = {j ∈ prim(a) : j ⊇ i} ≃ t 2. 3. the noncommutative cw complexes in this section we see how the construction of the primitive spectrum of the previous section helps us to study the noncommutative cw complexes. for a continuous map φ : x → y between compact topological spaces x and y , the c*-morphism induced on their associated c*-algebras is denoted by c(φ) : c(y ) → c(x) which is defined by c(φ)(g) := g ◦ φ for g ∈ c(y ). definition 3.1. let a1, a2 and c be c*-algebras. a pull back for c via morphisms α1 : a1 → c and α2 : a2 → c is the c*-subalgebra of a1 ⊕ a2 denoted by pb(c,α1,α2) defined by pb(c,α1,α2) := {a1 ⊕ a2 ∈ a1 ⊕ a2 : α1(a1) = α2(a2)}. for any c*-algebra a, let sna := c(sn → a),ina := c([0,1]n → a),in0 a := c0((0,1) n → a), where sn is the n-dimensional unit sphere. we review the definition of noncommutative cw complexes from [7], [15]. morse theory for c*-algebras 181 definition 3.2. a 0-dimensional noncommutative cw complex is any finite dimensional c*-algebra a0. recursively an n-dimensional noncommutative cw complex is any c*-algebra appearing in the following diagram (3.1) 0 −−−−→ in0 fn −−−−→ an π −−−−→ an−1 −−−−→ 0 ∥ ∥ ∥   y fn   y ϕn 0 −−−−→ in0 fn −−−−→ i nfn δ −−−−→ sn−1fn −−−−→ 0 where the rows are extensions, an−1 an (n − 1)-dimensional noncommutative cw complex, fn some finite (linear) dimensional c*-algebra of dimension λn, δ the boundary restriction map, ϕn an arbitrary morphism (called the connecting morphism), for which (3.2) an = pb(s n−1fn,δ,ϕn) := {(α,β) ∈ i nfn ⊕ an−1 : δ(α) = ϕn(β)}, and fn and π are respectively projections onto the first and second coordinates. with these notations {a0, ...,an} is called the noncommutative cw complex decomposition of dimension n for a = an. for each k = 0,1, ...,n, ak is called the k-th decomposition cell. proposition 3.3. let x be an n-dimensional cw complex containing cells of each dimension k = 0, ...,n. then there exists a noncommutative cw complex decomposition of dimension n for a = c(x). conversely if {a0, ...,an} be a noncommutative cw complex decomposition for the c*-algebra a such that ais (i = 0, ..,n) are unital, then there exists an n-dimensional cw complex structure on prim(a). proof. let x0 ⊂ x1 ⊂ ... ⊂ xn = x be a cw complex structure for x where xk(for each k ≤ n) is the k-skeleton defined in relation (2.4). let ak = c(xk) and i : ⋃ λk sk−1 → ⋃ λk ik , ϕk : ⋃ λk sk−1 → xk−1 be the injection and attaching maps respectively, and c(i) and c(ϕk) be their induced maps. let pb := pb(c( ⋃ λk sk−1),c(ϕk),c(i)), and define θ :c(xk) −→ pb f 7−→ (f ◦ ρ)1 ⊕ (f ◦ ρ)2, where (f ◦ ρ)1 and (f ◦ ρ)2 are the restrictions of (f ◦ ρ) to ⋃ λk ik and xk−1 respectively. θ is well defined since c(ϕk)((f ◦ ρ)1) = c(i)((f ◦ ρ)2). also c(ϕk)(h) = c(i)(g)for (h,g) ∈ pb, and so if f ∈ c(xk) be defined by f(y) = { g(y) y ∈ ⋃ λk ik h(y) y ∈ xk−1 182 v. milani, a. a. rezaei and s. m. h. mansourbeigi then θ(f) = (h,g). now the noncommutative cw complex decomposition of dimension n for a = c(x) is given by {a0, ...,an}. conversely let an be as in (3.2), and ϕ ∗ n : s n−1 → prim(an−1) be the attaching map induced by the connecting morphism ϕn : an−1 → s n−1fn of diagram (3.1). then using the notation in relation (2.4), prim(an) = prim(an−1) ⋃ ϕ∗ n in. we note that ϕ∗n = c(ϕn). furthermore for k ≤ n, ϕ ∗ k(s k−1) is a closed subset of prim(ak−1). it is of the form ϕ∗k(s k−1) = {j ∈ prim(ak−1) : ik−1 ⊂ j} for some ideal ik−1 in ak−1. in fact ik−1 = ⋂ j∈ϕ∗ k (sk−1) j. so prim(a) has an n-dimensional cw-structure with xk = prim(ak) as its k-skeleton for k = 0, ...,n. � example 3.4. following the notations of diagram (3.1), a 1-dimensional noncommutative cw complex decomposition for a = c([0,1]) = c(i) is given by a0 = c ⊕ c,a1 = c([0,1]). let f1 = c, then i10f1 = c0((0,1)),i 1f1 = c([0,1]),s 0f1 = c ⊕ c and ϕ1 = id. also c(i) = pb(s0f1,δ,ϕ1) = {f⊕(λ⊕µ) ∈ c([0,1])⊕(c⊕c) : f(0) = λ,f(1) = µ} together with the maps π :a1 −→ a0 f ⊕ (λ ⊕ µ) 7−→ λ ⊕ µ, f1 :a1 −→ i 1f1 = a1 f ⊕ (λ ⊕ µ) 7−→ f, and δ :i1f1 = a1 −→ s 0 f1 = c ⊕ c f 7−→ f(0) ⊕ f(1). morse theory for c*-algebras 183 4. modified morse theory on c*-algebras in this section, following the study of the morse theory for the cell complexes in [2], [6], [9], [10], with some modification, we define the morse function for the c*-algebras and state and prove the modified morse theory for the noncommutative cw complexes. this is a classification theory in the category of c*-algebras and noncommutative cw complexes. definition 4.1. if a, b are two c*-algebras, two morphisms α,β : a → b are homotopic, written α ∼ β,if there exists a family {ht}t∈[0,1] of morphisms ht : a → b such that for each a ∈ a the map t 7→ ht(a) is a norm continuous path in b with h0 = α and h1 = β.the c*-algebras a and b are said to have the same homotopy type, if there exists morphisms ϕ : a → b and ψ : b → a such that ϕ ◦ ψ ∼ idb and ψ ◦ ϕ ∼ ida. in this case the morphisms ϕ and ψ are called homotopy equivalence. definition 4.2. let a and b be unital c*-algebras. we say a is of pseudohomotopy type as b if c(prim(a)) and b have the same homotopy type. remark 4.3. in the case of unital commutative c*-algebras, by the gns construction [11], c(prim(a)) = a, . so the notions of pseudo-homotopy type and the same homotopy type are equivalent. for a unital c*-algebra a let σ = {wi1,...,ik}1≤i1,...,ik≤n,1≤k≤n be the set of all k-chains (k = 1, ...,n) in prim(a), and γ = {ii1,...,ik}1≤i1,...,ik≤n,1≤k≤n be the set of all k-ideals corresponding to the k-chains of σ for k = 1, ...,n. lemma 4.4. γ is an absorbing set. proof. this follows from the fact that for each ii1,...,ik ∈ γ, j ∈ prim(a) the relation ii1,...,ik ⊂ j is equivalent to j = ii1,...,it for some t ≤ k, meaning j ∈ γ. � definition 4.5. let f : σ → r be a function. the k-chain wk = wi1,...,ik is called a critical chain of order k for f, if for each (k+1)-chain wk+1 containing wk and for each (k − 1)-chain wk−1 contained in wk, we have f(wk−1) ≤ f(wk) ≤ f(wk+1). the corresponding ideal ik to wk is called the critical ideal of order k. definition 4.6. let f has a critical chain of order k. we say f is an acceptable morse function, if it has a critical chain of order i, for all i ≤ k. definition 4.7. a function f : σ → r is called a modified morse function on the c*-algebra a, if for each k-chain wk in σ, there is at most one (k+1)-chain wk+1 containing wk and at most one (k-1)-chain wk−1 contained in wk, such that f(wk+1) ≤ f(wk) ≤ f(wk−1). 184 v. milani, a. a. rezaei and s. m. h. mansourbeigi here we state the discrete morse theory of forman from [9], and state and prove our modification of it. theorem (discrete morse theory): suppose ∆ is a simplicial complex with a discrete morse function. then ∆ is homotopy equivalent to a cw complex with one cell of dimension p for each critical p-simplex. lemma 4.8. if f is an acceptable modified morse function on a, then prim(a) is homotopy equivalent to a cw complex with exactly one cell of dimension p for each critical chain of order p. proof. in the discrete morse theory it suffices to substitute γ for the simplicial complex ∆. since γ is absorbing, it satisfies the properties of the simplicial complex ∆ in the discrete morse theorem. it follows that prim(a) is homotopy equivalent to a cw complex with exactly one cell of dimension p for each critical chain of order p. � now we state our main theorem which provides a condition for a unital c*algebra to admit a noncommutative cw-complex decomposition. this is what we call the geometric condition. theorem 4.9. every unital c*-algebra a with an acceptable modified morse function f on it, is of pseudo-homotopy type as a noncommutative cw complex having a k-th decomposition cell for each critical chain of order k. proof. if a is a unital c*-algebra, then the acceptable modified morse function on a is in fact a function on the simplicial complex of all k-ideals in prim(a) (a function on γ). from lemma 4.8 we conclude that prim(a) is homotopy equivalent to a finite dimensional cw complex ω. from the proposition 3.3 there is a noncommutative cw-complex decomposition for c(ω) making it into a noncommutative cw complex. now c(prim(a))and c(ω) have the same homotopy type, which means a is of psuedo-homotopy type of the noncommutative cw complex c(ω). furthermore since f is acceptable, from the proof of proposition 3.3 it follows that if there exists a critical k-chain for f, then there exists c*-algebras ai for each i ≤ k so that {a0, ...,ak} is a noncommutative cw complex decomposition for c(prim(a)) yielding a noncommutative cw complex decomposition for c(ω) . � morse theory for c*-algebras 185 references [1] a. connes, noncommutative geometry, academic press, 1994. [2] j. cuntz, noncommutative simplicial complexes and the baum-connes conjecture, arxiv: math/0103219. [3] j. cuntz, quantum spaces and their noncommutative topology, notices am. math. soc. 48, no. 8 (2001), 793–799. [4] d. n. diep, the structure of c*-algebras of type i, vestn. mosk. univ., ser. i, no. 2 (1978), 81–87. [5] j. dixmier, les c*-algèbres et leurs représentations, gauthier-villars, (1969). [6] a. duval, a combinatorial decomposition of simplicial complexes, isr. j. math. 87 (1994), 77–87. [7] s. eilers, t. a. loring and g. k. pedersen, stability of anticommutation relations: an application to nccw complexes, j. reine angew math. 499 (1998), 101–143. [8] j. m. g. fell and r. s. doran, representations of *-algebras, locally compact groups and banach *-algebraic bundles, academic press, 1988. [9] r. forman, morse theory for cell complexes, adv. in math. 134 (1998). [10] r. forman, witten-morse theory for cell complexes, topology 37 (1998), 945–979. [11] j. m. gracia-bondia, j. c. varilly and h. figueroa, elements of noncommutative geometry, birkhauser, 2001. [12] a. hatcher, algebraic topology, cambridge univ. press, 2002. [13] g. landi, an introduction to noncommutative spaces and their geometry, arxiv:hepth/9701078. [14] j. milnor, morse theory, annals of math. studies, princeton univ. press, (1963). [15] g. k. pedersen, pullback and pushout constructions in c*-algebras, j. funct. anal. 167 (1999), 243–344. (received february 2011 – accepted july 2011) v. milani (v-milani@cc.sbu.ac.ir, vmilani3@math.gatech.edu) dept. of math., faculty of math. sci., shahid beheshti university, tehran, iran. school of mathematics, georgia institute of technology, atlanta ga, usa. a. a. rezaei (a rezaei@sbu.ac.ir) dept. of math., faculty of math. sci., shahid beheshti university, tehran, iran. s. m. h. mansourbeigi (s.mansourbeigi@ieee.org) dept. of electrical engineering, polytechnic university, ny, usa. morse theory for c*-algebras: a geometric[2pt] interpretation of some noncommutative[2pt] manifolds. by v. milani, a. a. rezaei and s. m. h. mansourbeigi () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 135-150 continuous isomorphisms onto separable groups luis felipe morales lópez abstract a condensation is a one-to-one continuous function onto. we give sufficient conditions for a tychonoff space to admit a condensation onto a separable dense subspace of the tychonoff cube ic and discuss the differences that arise when we deal with topological groups, where condensation is understood as a continuous isomorphism. we also show that every abelian group g with |g| ≤ 2c admits a separable, precompact, hausdorff group topology, where c = 2 ω . 2010 msc: 22a05, 54h11. keywords: condensation, continuous isomorphism, separable groups, subtopology. 1. introduction a condensation is a bijective continuous function. if x and y are spaces and f : x → y is a condensation, we can assume that x and y have the same underlying set and the topology of x is finer than the topology of y . in this case we say that the topology of y is a subtopology of x or that x condenses onto y . the problem of finding conditions under which a space x admits a subtopology with a given property q has been extensively studied by many authors. it is known that every hausdorff space x with nw(x) ≤ κ can be condensed onto a hausdorff space y with w(y ) ≤ κ (see [7, lemma 3.1.18]). similar results remain valid in the classes of regular or tychonoff spaces. in [16], the authors found several necessary and sufficient conditions for a topological space to admit a connected hausdorff or regular subtopology. it is shown in [11] that every non-compact metrizable space has a connected hausdorff subtopology. druzhinina showed in [9] that every metrizable space x with w(x) ≥ 2ω and achievable extent admits a weaker connected metrizable topology. recently, 136 l. morales lópez yengulalp [17] generalized this result by removing the achievable extent condition. in topological groups (and other algebraic structures with topologies), the concept of condensation has a natural counterpart: continuous isomorphism, a homomorphism and a condensation at the same time. at the end of the 70’s, arhangel’skii proved in [2] that every topological group g with nw(g) ≤ κ admits a continuous isomorphism onto a topological group h with w(h) ≤ κ. in [15] shakhmatov gave a construction that implies similar statements for topological rings, modules, and fields. c. hernández modified shakhmatov’s construction and extended that result to many algebraic structures with regular and tychonoff topologies (see [8]). as a corollary to katz’s theorem about isomorphic embeddings into products of metrizable groups (see [1, corollary 3.4.24]) one can easily deduce that if g is an ω-balanced topological group and the neutral element of g is a gδ-set, then there exists a continuous isomorphism of g onto a metrizable topological group. pestov showed that the condition on g being ω-balanced can not be removed (see [14]). in theorem 3.2 of this paper we present conditions that a tychonoff space must satisfy in order to admit a condensation onto a separable dense subspace of the tychonoff cube of weight 2ω. in corollary 4.2 we show that those conditions are not sufficient if we want to have a continuous isomorphism from a topological group to a separable group and in theorem 4.3 we give sufficient and necessary conditions in order for a topological isomorphism from a subgroup of the product of compact metrizable abelian groups onto a separable group to exist. as arhangel’skii showed in [4], every continuous homomorphism of a countably compact group x onto a compact group y of ulam nonmeasurable cardinality is open. in example 4.4 we construct a condensation of a tychonoff countably compact space with cellularity 2ω onto a separable compact space with cardinality 2c thus showing that arhangel’skii result cannot be generalized to arbitrary spaces. finally, we show in theorem 5.11 that every abelian group of cardinality less than or equal to 2c admits a precompact separable hausdorff group topology. 2. notation and terminology we use i for the unit interval [0,1], t for the unit circle, n for the set of positive integers, z for the integers, q for the rational numbers, and r for the set of real numbers. let x be a space. as usual, we denote by w(x), nw(x), χ(x), ψ(x), d(x) the weight, network weight, character, pseudocharacter, and density of x, respectively. we say that z ⊂ x is a zero-set if there exists a real-valued continuous function f : x → r such that z = f−1(0). continuous isomorphisms onto separable groups 137 let {fα : α ∈ a} be a family of functions, where fα : x → yα for each α ∈ a. we denote by △{fα : α ∈ a} the diagonal product of the family {fα : α ∈ a}. suppose that η = {gα : α ∈ a} is a family of topological groups and πη = ∏ α∈a gα is the topological product of the family η. then the σ-product of η, denoted by σπη, is the subgroup of πη consisting of all points g ∈ πη such that |{α ∈ a : πα(g) 6= eα}| ≤ ω and the σ-product of η, denoted by σπη is the subgroup of πη consisting of all points g ∈ πη such that |{α ∈ a : πα(g) 6= eα}| < ω, where πα : πη → gα is the natural projection of πη onto gα and eα ∈ gα is the neutral element of gα, for every α ∈ a. it is easy to see that both σπη and σπη are dense subgroups of πη. a description of properties of these subgroups can be found in [1, section 1.6]. if x is a tychonoff space and g is a topological group, we denote by βx the čech-stone compactification of x (see [7, section 3.6]), and by ρg the rǎıkov completion of g (see [1, section 3.6]). the next definitions are standard in group theory (see [10, section 1.1]). let g be a group, e the neutral element of g, and g ∈ g an element of g distinct from e. we denote by 〈g〉 the cyclic subgroup of g generated by g. the order of g is o(g) = |〈g〉|. if o(g) = ∞ then 〈g〉 is isomorphic to z. the set tor(g) of the elements g ∈ g with o(g) < ∞ is called the torsion part of g. if g is abelian, tor(g) is a subgroup of g. we say that the group g is: • torsion-free if for every element g ∈ g \ {e}, o(g) = ∞; • a torsion group if for every element g ∈ g, o(g) < ∞; • bounded torsion if there exists n ∈ n such that gn = e for every g ∈ g; • unbounded torsion if g is torsion and for each n ∈ n there exists g ∈ g such that o(g) > n; • a divisible group if for every g ∈ g and n ∈ n, there is h ∈ g such that hn = g; • a p-group, for a prime p, if the order of any element of g is a power of p. if g is an abelian torsion group, then g is the direct sum of p-groups gp (see [10, theorem 8.4]). the subgroups gp are called the p-components of g. let p be a prime number. the set of pnth complex roots of the unity, with n ∈ n forms the multiplicative subgroup zp∞ of t. for every prime p, the group zp∞ is divisible. 3. condensations and subtopologies not every space has a separable subtopology. for example, a compact hausdorff space x has a separable hausdorff subtopology only if x is separable. let us extend this fact to a wider class of spaces. we recall that a hausdorff space x is ω-bounded if the closure of any countable subset of x is compact. 138 l. morales lópez proposition 3.1. let x be an ω-bounded non-separable space. then x does not admit a condensation onto a separable hausdorff space. proof. by our assumptions, for every countable subset s of x we have that x \ s̄ 6= ∅. let f : x → y be a condensation onto a hausdorff space y and d a countable subset of y . then s = f−1(d) is a countable subset of x, and s̄ is compact. take an element x ∈ x \ s̄. observe that f(s̄) is compact, d ⊂ f(s̄) and f(x) 6∈ f(s̄), so d cannot be dense in y . � the next theorem gives sufficient conditions on a tychonoff space to admit a condensation onto a separable dense subspace of ic, where c = 2ω. theorem 3.2. let x be a tychonoff space with nw(x) ≤ 2ω. suppose that x contains an infinite, closed, discrete, and c∗-embedded subset a. then x can be condensed onto a separable dense subspace of ic. proof. we can assume that |a| = ω. by the hewitt-marczewski-pondiczery theorem, we know that d(ic) = ℵ0. let d = {dn : n ∈ ω} be a countable dense subset of ic, n a network for x, |n| ≤ 2ω, and a = {xn : n ∈ ω} an enumeration of a. let g : a → d be a bijection, where g(xn) = dn for each n ∈ ω. for every α < c, let fα = pα ◦ g, where pα : i c → i(α) denotes the natural projection of ic to the α-th factor. our goal is to construct a family of continuous functions {gα : x → i}α ω, there is no continuous isomorphism of g onto a separable topological group. theorem 4.1 can be generalized if we replace tκ by the product of any family of compact metrizable abelian groups: theorem 4.3. let η = {gα : α ∈ κ} be a family of compact metrizable groups with κ ≤ 2ω, and g be a subgroup of σ = σπη. then there exists a continuous isomorphism ϕ : g → h of g onto a separable topological group h if and only if ψ(g) ≤ ω. the proof of this fact is almost the same as in the theorem 4.1, and we omitted. arhangel’skii showed in [4, corollary 12] that every continuous homomorphism of a countably compact topological group onto a compact group of ulam nonmeasurable cardinality is open. in particular, if there exists a continuous isomorphism of a countably compact topological group g onto a compact group of ulam nonmeasurable cardinality, then g is compact. the next example shows that one cannot extent this result to topological spaces. example 4.4. there exists a condensation of a countably compact non-separable tychonoff space onto a separable compact space of ulam nonmeasurable cardinality, 2c. let y = βn be the čech-stone compactification of the natural numbers and z = y \ n. by [7, example 3.6.18], z contains a family a of cardinality c consisting of pairwise disjoint non-empty open sets. let π1 : y × z → y and π2 : y × z → z be the natural projections to the first and the second factor respectively. since y is compact, π2 is a closed mapping. by [7, theorem 3.5.8], z is a compact space because it is the remainder of a locally compact space and so, π1 is a closed mapping too. by [7, theorem 3.6.14], every infinite closed subset s of both y and z has cardinality equal to 2c. let m be an infinite subset of y × z. it is clear that 142 l. morales lópez at least one of the set, π1(m) or π2(m), is infinite. suppose that π1(m) is infinite. since the projection π1 is closed, π1(m) is a closed subset of y , so π1(m) and m have cardinality equal to 2 c. our goal is to construct a countably compact non-separable subspace x ⊂ y ×z such that π1(x) = y , π1|x is a one-to-one mapping, and π2(x)∩a 6= ∅ for every a ∈ a. recall that [y ]ω is the family of subsets of y with cardinality ω. let a = {aα : α < c} be a faithful enumeration of a and choose zα ∈ aα for each α < c. let also y = {yβ : β < 2 c} and [y ]ω = {fγ : c ≤ γ < 2 c} be faithful enumerations of y and [y ]ω respectively such that fc ⊂ {yβ : β < c}. we shall define a transfinite sequence {xγ : c ≤ γ < 2 c} of subsets of y × z satisfying the following conditions for each γ with c ≤ γ < 2c: (iγ): xβ ⊂ xγ if c ≤ β < γ; (iiγ): the restriction of π1 to xγ is a one-to-one mapping; (iiiγ): fγ ⊂ π1(xγ); (ivγ): π −1 1 (fγ) ∩ xγ has an accumulation point in xγ; (vγ): |xγ| ≤ |γ|. for every α < c put xα = (yα,zα) and let x ′ c = {xα : α < c}. by our enumeration of [y ]ω, fc ⊂ π1(x ′ c ). put bc = π −1 1 (fc) ∩ x ′ c . since π1 is closed and fc ⊂ π1(bc), the cardinality of π1(bc) is equal to 2 c, so we can choose xc ∈ bc such that π1(xc) 6∈ π1(x ′ c ). put xc = x ′ c ∪ {xc}. conditions (iic), (iiic), (ivc), and (vc) are clearly satisfied, condition (ic) is vacuous. suppose that for some γ with c ≤ γ < 2c, xξ are defined for all ξ, c ≤ ξ < γ. let x̃γ = ⋃ c≤ξ<γ xξ. we have two possibilities. if fγ ⊂ π1(x̃γ), then put x′γ = x̃γ. if fγ \ π1(x̃γ) 6= ∅, then choose an arbitrary point xy ∈ π −1 1 (y) for each y ∈ fγ \ π1(x̃γ) and put x ′ γ = x̃γ ∪ {xy : y ∈ fγ \ π1(x̃γ)}. in both cases, fγ ⊂ π1(x ′ γ). since conditions (iξ) and (vξ) are satisfied for all c ≤ ξ < γ, |x ′ γ| ≤ |γ| < 2 c. let bγ = π −1 1 (fγ) ∩ x ′ γ. since π1 is a closed mapping, |π1(bγ)| = 2 c, so there exists xγ ∈ bγ such that π1(xγ) 6∈ π1(x ′ γ). let xγ = x ′ γ ∪ {xγ}. clearly condition (iγ) is satisfied. since conditions (iξ) and (iiξ) are satisfied for every ξ with c ≤ ξ < γ, π1|x̃γ is a one-to-one mapping. by our definition of x ′ γ, π1|x′γ is a one-to-one mapping too. finally, by our choose of xγ, π1(xγ) 6∈ π1(x ′ γ), so π1|xγ is a one-to-one mapping by (iiγ). as fγ ⊂ π1(x ′ γ) and xγ ∈ xγ, (iiiγ) and (ivγ) are satisfied. since (vξ) and (iξ) are satisfied for every ξ with c ≤ ξ < γ, |x̃γ| ≤ |γ|. as |xγ \ x̃γ| ≤ ω, we conclude that |xγ| ≤ |γ| put x = ⋃ c≤γ<2c xγ and let f : x → y be the restriction of π1 to x. since conditions (iγ) and (iiγ) are satisfied for all γ with c ≤ γ < 2 c, f is a continuous one-to-one function. let y ∈ y be an arbitrary element of y and f ∈ [y ]ω be a subset of y with y ∈ f . then there exists γ, c ≤ γ < 2c such continuous isomorphisms onto separable groups 143 that f = fγ. by (iiiγ), y ∈ f = fγ ⊂ π1(xγ) ⊂ π1(x) = f(x), so f(x) = y . therefore f is a condensation of x onto y . let b be an arbitrary infinite countable subset of x. then f = f(b) is an infinite countable subset of y and there exists γ < 2c such that f = fγ. by (ivγ), b = f −1(f) = π−11 (fγ) ∩ xγ has an accumulation point in xγ and in x. this means that x is countably compact. since a∩π2(x) ⊃ a∩π2(xc) 6= ∅ for every a ∈ a, x cannot be separable. 5. separable group topologies for abelian groups in this section we prove that every abelian group g with |g| ≤ 2c admits a separable precompact hausdorff group topology. to do this, we divide the job in three parts: case 1.: there is x ∈ g with o(x) = ∞. case 2.: g is a bounded torsion group. case 3.: g is an unbounded torsion group. we say that a topological group is monothetic if it has a dense cyclic subgroup. the next result is proved in [12, corollary 25.15]: lemma 5.1. the group tκ is monothetic if and only if κ ≤ c. let us begin with the case when g is a non-torsion group (case 1). theorem 5.2. let g be an abelian group. suppose that |g| ≤ 2c and there is an element x ∈ g of infinite order. then there exists a separable precompact hausdorff group topology on g. proof. the main idea of the proof is to define a monomorphism ϕ : g → tc such that ϕ(g) will be separable. first we do this in the case when g is divisible. let h be a minimal divisible subgroup of g with x ∈ h. since o(x) is infinite, h is isomorphic to q. by lemma 5.1, there exists a ∈ tc such that 〈a〉 = tc. let ϕ : h → tc be a monomorphism such that ϕ(x) = a. for every β < c, put ϕβ = pβ ◦ ϕ, where pβ = t c → t(β) is the projection of t c to the β’s factor. let κ = |g| > ω. since g is divisible, it is isomorphic to the direct sum h ⊕ ⊕ α∈a gα, where each gα is a subgroup of g isomorphic either to q or zp∞ for some prime number p, and a is an index set of cardinality κ (see [10, theorem 23.1]). for each α ∈ a, let ̺α : gα → f be the isomorphism of gα onto f , where f is either q or zp∞ for some prime number p. consider a as a subspace of the space 2c with the product topology. let b be the canonical base of 2c, we know that |b| = c. for each g ∈ g, let hg ∈ h and k ∈ ⊕ α∈a gα be such that g = hg + k. if g ∈ g \ h, then k 6= e and there exists a non-empty finite subset c(g) ⊂ a such that k ∈ ⊕ α∈c(g) gα. for every α ∈ c(g), take kα ∈ gα such that k = 144 l. morales lópez ∑ α∈c(g) kα. choose an arbitrary α(g) ∈ c(g) such that kα(g) is not the identity of the group gα(g). let ug ∈ b be an open set satisfying ug ∩ c(g) = {α(g)}. thus for each g ∈ g \ h we have defined a pair (hg,ug) ∈ h × b. the cardinality of the set p = {(hg,ug) : g ∈ g \ h} is less than or equal to |h × b| = ω · c = c. let p = {pβ : β < c} be an enumeration of p , where pβ is a pair (hβ,uβ) with hβ ∈ h and uβ ∈ b. for each β < c, we define a homomorphism ψβ : ⊕ α∈a gα → t as follows: if ϕβ(hβ) = 1, we can define ψβ such that ψβ|gα = ̺α if α ∈ uβ, and ψβ|gα = 1, otherwise. if ϕβ(hβ) 6= 1, we define ψβ ≡ 1. let ϕβ be the homomorphism defined by ϕβ = ϕβ ⊕ ψβ. it is clear that, for each β < c, ϕβ is an extension of ϕβ, therefore ϕ = △β ω. suppose first that for every g ∈ g, the order of g is a power of a fixed prime number p. since g is a bounded torsion group, we can find k ∈ n with o(g) ≤ pk for each g ∈ g. hence there exists a set {gα : α ∈ a} ⊂ g such that g = ⊕ α∈a〈gα〉 (see [10, theorem 17.2]). for each α ∈ a, let ̺α : 〈gα〉 → t be the monomorphism defined by ̺α(gα) = e 2πi/nα, where nα = o(gα). for every n ≤ k, put an = {α ∈ a : o(gα) = p n} and m = max{n : |an| ≥ ω}. let j0 = {αj : j ∈ ω} be an infinite countable subset of am, g0 = ⊕ α∈j0 〈gα〉, j = ( ⋃ n≤m an) \ j0 and f = ⋃ n>m an. observe that g′ = ⊕ α∈f 〈gα〉 is finite and |g0| = ω. so g = g ′ ⊕ g0 ⊕ ⊕ α∈j〈gα〉. let h = pc, where p is the subgroup of t consisting of all pm-th complex roots of unity. by the hewitt-marczewski-pondiczery theorem, h is separable. let d be a countable dense subgroup of h. since d is a bounded torsion group, it is direct sum of cyclic groups, i.e., d = ⊕ j∈ω〈dj〉. by lemma 5.4 we can assume that o(dj) = p m for every n ∈ ω. let ϕ be a monomorphism ϕ : g0 → h such that ϕ(gαj ) = dj for every n ∈ ω. we will extend this monomorphism to ḡ = g0 ⊕ ⊕ α∈j〈gα〉. for every β < c, let ϕβ = pβ ◦ ϕ, where pβ : h → p(β) is the natural projection of h onto the β-th factor. consider j as a subspace of the space 2c endowed with the product topology. let b be the base of canonical open sets in 2c, |b| = c. for every ḡ ∈ ḡ, there exists ḡ0 ∈ g0 and a finite set c(ḡ) ⊂ j such that ḡ = ḡ0+lḡ, where lḡ ∈ ⊕ α∈c(ḡ)〈gα〉. if ḡ ∈ ḡ\g0, then c(ḡ) 6= ∅. let αḡ ∈ c(ḡ) be an arbitrary element of c(ḡ) and choose uḡ ∈ b such that uḡ ∩ c(ḡ) = {αḡ}. the set s = {(ḡ0,uḡ) : ḡ ∈ ḡ \ g0} has cardinality less than or equal to |g0 × b| = ω · c = c. let s = {sβ : β < c} be an enumeration of s, where sβ is a pair (aβ,uβ) with aβ ∈ g0 and uβ ∈ b. if ϕβ(aβ) = 1, then let ψβ : ⊕ α∈j〈gα〉 → p be a homomorphism such that ψβ|〈gα〉 = ̺α for each α ∈ uβ and ψβ(gα) = 1 if α ∈ j \ uβ. if ϕβ(aβ) 6= 1, put 146 l. morales lópez ψβ ≡ 1. let ϕβ = ϕβ ⊕ ψβ. it is clear that, for each β < c, ϕβ is an extension of ϕβ. therefore ϕ = △β 4π/m, there are two distinct m-th roots of z1 in v . let y1, y2 be two elements of v such that my1 = my2 = z1 and the distance between y1 and y2 is 2π/m. note that y1 and y2 can not be both n-th roots of z2, otherwise the distance between them would be greater than or equal to 2π/n, and it would follow that m ≤ n contradicting the assumptions of the lemma. � continuous isomorphisms onto separable groups 147 lemma 5.7. let k be a countable subgroup of tc and f′ ∈ k, m ≥ 2. suppose that {vα : α < c} is a family of open arcs of t such that 4π/m < l(vα), for every α < c. then there exists f ∈ ∏ {vα : α < c} such that mf = f ′ and nf 6∈ k for each n with 1 ≤ n < m. proof. let k × {1, ...,m − 1} = {(hk,nk) : k ∈ ω} be an enumeration of k ×{1, ...,m−1}. for each k < ω we will define αk < c and xαk ∈ t satisfying the following conditions: (ik): αk 6= αj if j < k; (iik): xαk ∈ vαk ; (iiik): mxαk = f ′(αk); (ivk): nkxαk 6= hk(αk). let α0 < c be an arbitrary ordinal. by lemma 5.6 (with v = vα0, z1 = f′(α0), z2 = h0(α0), n = n0) we can choose an element xα0 ∈ vα0 that satisfies (ii0), (iii0) and (iv0). condition (i0) is vacuous. suppose that for every j < k we have chosen αj and xαj such that conditions (ij) (ivj) are satisfied. we can pick αk < c that satisfies (ik). by lemma 5.6 with v = vαk , z1 = f ′(αk), z2 = hk(αk), and n = nk, we can choose xαk that satisfies (iik) (ivk). finally, for each α ∈ c \ {αk : k ∈ ω} we use lemma 5.6 again with v = vα, z1 = f ′(α), z2 = 1, n = 1 to select xα ∈ vα such that mxα = f ′(α). we define f ∈ tc by f(α) = xα for each α < c. then: • f(α) ∈ vα for each α < c, therefore f ∈ ∏ {vα : α < c}. • mf(α) = mxα = f ′(α) for each α < c, so mf = f′. • given n ∈ {1, ...,m − 1} and h ∈ k, there exists k ∈ ω such that (h,n) = (hk,nk). by conditions (iiik) and (ivk), we have that nf(αk) = nkxαk 6= hk(αk) = h(αk). since h ∈ k is arbitrary, nf 6∈ k for every n < m. � the proof of the following lemma can be found in [1, lemma 1.1.5]: lemma 5.8. let g and g∗ be abelian topological groups, k and k∗ subgroups of g and g∗, respectively. suppose that there exist x ∈ g, x∗ ∈ g∗, m ∈ n, m ≥ 2 and an isomorphism ψ : k → k∗ that satisfy the following conditions: • mx ∈ k and mx∗ ∈ k∗; • nx 6∈ k and nx∗ 6∈ k∗ for every n ∈ n, 1 ≤ n < m; • ψ(mx) = mx∗. then there exists a unique isomorphism ϕ : k + 〈x〉 → k∗ + 〈x∗〉 extending ψ such that ϕ(x) = x∗. now we are going to give some definitions from group theory. a system {a1, ...,ak} of a group g is called independent if n1a1 + ... + nkak = 0 (ni ∈ z) implies 148 l. morales lópez n1a1 = ... = nkak = 0. we say that an infinite system l of the group g is independent if any finite subset of l is independent. by the rank r(g) of an abelian group g is meant the cardinal number of a maximal independent system in g. the torsion-free rank r0(g) is the cardinal of the maximal independent system which contains only elements of infinite order. for each prime number p, the p-rank rp(g) of g is the cardinal of a maximal independent system which contains only elements whose orders are powers of p. the next lemma can be found in [6, lemma 3.17]. lemma 5.9. let g and g∗ be abelian groups such that |g| ≤ r(g∗) and |g| ≤ rp(g ∗) for every prime number p. suppose that h is a subgroup of g satisfying r(h) < r(g∗) and rp(h) < rp(g ∗) for every prime p. if g∗ is a divisible group, then every monomorphism ϕ : h → g∗ can be extended to a monomorphism ψ : g → g∗. now we are in position to prove the following theorem. theorem 5.10. let g be an unbounded torsion abelian group with |g| ≤ 2c. then g admits a separable, precompact, hausdorff group topology. proof. let v be a countable base for the topology of t consisting of open arcs such that t ∈ v. since g is an unbounded torsion group, we can choose a subset s ⊂ g \ {e} such that |ns| = ω for every n ∈ n, where e is the unity of g. consider c as the topological space 2ω and let b be the canonical base for 2ω consisting of non-empty clopen subsets of 2ω. then |b| = ω. let u be the set of all finite open covers of 2ω formed by pairwise disjoint sets. for u ∈ u and α < c let uα,u denote the unique u ∈ u such that α ∈ u. put e = {(u,υ) : u ∈ u and υ : u → v is a function}. for (u,υ) ∈ e, let f(u,υ) = ∏ {υ(uα,u) : α < c}. clearly e is countable. let e = {(uk,υk) : k ∈ ω} be an enumeration of e such that u0 = {2 ω} and υ0(2 ω) = t. for each k < ω, choose nk ∈ n such that 4π/nk < min {l(υ(u)) : u ∈ uk}. by recursion on k ∈ ω we will choose an element xk ∈ s and define a map ϕk : hk = 〈{xj : j ≤ k}〉 → t c satisfying the following conditions: (ik): ϕk(xk) ∈ f(uk,υk); (iik): ϕk is a monomorphism; (iiik): ϕk|hj = ϕj for all j < k. pick any element x0 in s and let ϕ0 : 〈x0〉 → t c be an arbitrary monomorphism. then conditions (i0) and (ii0) are satisfied, while condition (iii0) is vacuous. now let k ∈ n, and suppose that xj ∈ s and a map ϕj satisfying (ij), (iij) and (iiij) have already been constructed for every j < k. put h′k = ⋃ j nk and 4π/m < 4π/nk < l(vα) for every α < c. note that m ≥ 2. by lemma 5.7 we can find f ∈ f(u,υ) = ∏ {υ(uα,u) : α < c} such that mf = f ′ and nf 6∈ k for n < m. put hk = 〈{xj : j ≤ k}〉. by lemma 5.8 we can extend ϕ ′ k to a monomorphism ϕk : hk → t c with ϕk(xk) = f. we are going to verify that ϕk satisfies (ik), (iik) and (iiik). as ϕk(xk) = f ∈ f(u,υ), the condition (ik) is satisfied. by lemma 5.8, ϕk is a monomorphism that extends ϕ′k, so (iik) and (iiik) are satisfied. let h = ⋃ k∈ω hk and ϕ = ⋃ k∈ω ϕk. since (iik) and (iiik) are fulfilled for every k ∈ ω, we have that ϕ : h → tc is a monomorphism. we claim that ϕ(h ∩s) is a dense subset of tc. let w be a non-empty open set of tc. then there exist a finite subset i = {α1, ...,αn} of c and non-empty open arcs vα1, ...,vαn ∈ v such that ∏ α 0}. then f is a filter tied to the point (1, 0) of s1 that has no cluster points in the fiber over (0, 1). it follows that (e, p, t ) is not fibrewise compact. 44 c. m. neira u. 3. the kuratowski-mrówka characterization of fibrewise compactness we begin the main section of this paper with the following observation. remark 3.1. every filter f over a set x determines a topology tf over the set x ⋃ {ω}, where ω /∈ x, as follows: if x 6= ω, the neighborhood filter of x is v(x) = {v ⊂ x ⋃ {ω} : x ∈ v } and the neighborhood filter of ω is v(ω) = {f ⋃ {ω} : f ∈ f}. we denote by xf the topological space (x ⋃ {ω}, tf) (cf. [2]). let (e, p, t ) be a fibrewise topological space and f be a filter over e tied to a point t ∈ t . the function pf : ef −→ t defined by pf (x) = { p(x) if x ∈ e t if x = ω is continuous. that is, (ef , pf , t ) is a fibrewise topological space. to show this, it suffices to verify the continuity of pf at ω. let w be an open neighborhood of t in t . since f is a filter tied to t, one has that w ∈ p(f). then there exists f ∈ f such that p(f ) ⊂ w , hence pf (f ∪ {ω}) ⊂ w . this completes the proof. let (e, pe , t ) and (f, pf , t ) be two fibrewise topological spaces. the fiber product e ∨ f of e with f is the set e ∨ f = {(x, y) ∈ e × f : p(x) = q(y)}. consider e ∨ f with the topology induced by the product topology on e × f . the triplet (e ∨ f, p, t ), where p : e ∨ f −→ t is defined by p(x, y) = pe (x), is a fibrewise topological space. furthermore, (e ∨ f, p, t ) is the product of (e, pe , t ) and (f, pf , t ) in the category of fibrewise topological spaces and fibrewise continuous functions, that is, those continuous functions ϕ : e −→ f satisfying pf ◦ ϕ = pe . theorem 3.2 (kuratowski-mrówka characterization). the fibrewise topological space (e, p, t ) is fibrewise compact, if and only if, for each fibrewise topological space (f, q, t ) the projection π2 : e ∨ f −→ f is a closed map. proof. ⇒ suppose that (e, p, t ) is a fibrewise compact fibrewise topological space and that (f, q, t ) is an arbitrary fibrewise topological space. let b ∈ f , q(b) = t, and o be an open neighborhood of π−12 (b) = et × {b} in e ∨ f . for each x ∈ et there exist a neighborhood ax of x in e and a neighborhood mx of b in f such that ax ∨mx ⊂ o. compactness of et guarantees the existence of x1, ..., xn ∈ et, such that et ⊂ ⋃n i=1 axi . since p is closed, there exists an open neighborhood w of t in t such that p−1(w ) ⊂ ⋃n i=1 axi . let m = ( ⋂n i=1 mxi ) ⋂ q−1(w ). if (y, a) ∈ π−12 (m ), then p(y) = q(a) ∈ w , hence y ∈ axi , for some i ∈ {1, ..., n}. then (y, a) ∈ axi ∨mxi ⊂ π −1 2 (b). this proves that π2 is a closed map. a kuratowski-mrówka type characterization of fibrewise compactness 45 ⇐ suppose that f is a filter over e tied to the point t ∈ t and suppose that f has no cluster points, then for each x ∈ et there exists an open neighborhood ox of x in e and an element fx ∈ f such that ox ⋂ fx = ∅. consider the fibrewise topological space (ef , pf , t ) and the set ∆0 = {(x, x) ∈ e ∨ ef : x ∈ e}. for each x ∈ et, the set ox ∨ (fx ⋃ {ω}) is a neighborhood of (x, ω) in e ∨ ef such that ox ∨ (fx ⋃ {ω}) ⋂ ∆0 = ∅, then (x, ω) /∈ ∆0 for each x ∈ et. this implies that π2(∆0) = e and since e is not a closed subset of ef , because ω ∈ e, it follows that π2 : e ∨ ef −→ ef is not a closed map. � example 3.3. every topological space x can be identified with the fibrewise topological space (x, p, t ), where t consists of a single point and p is the constant map from e to t . the kuratowski-mrówka characterization of the fibrewise compact fibrewise topological spaces asserts that (x, p, t ) is fibrewise compact if and only if π2 : x ∨ y −→ y is closed, for each fibrewise topological space (y, q, t ). unfolding this assertion one finds that every fibrewise topological space (y, q, t ) can be identified with the topological space y : the map q is necessarily the constant map from y to t . furthermore, x ∨ y = {(x, y) ∈ x × y : p(x) = q(y)} = x × y . then “a topological space x is compact if and only if, for each topological space y , each y ∈ y and each open neighborhood o of x × {y} in x × y , there exists an open neighborhood n of y in y , such that x × n ⊂ o.” this result is known in general topology as the tube’s lemma. the second part of the proof of theorem 3.2 implies the following result. corollary 3.4. if (e, p, t ) is a fibrewise topological space such that the projection π2 : e ∨ ef −→ ef is closed for every t-filter f over e, then (e, p, t ) is fibrewise compact. example 3.5. let (e, p, t ) be a covering space. since each fiber has the discrete topology, for (e, p, t ) to be fibrewise compact it is necessary, for the fibers, to be finite. conversely, suppose that (e, p, t ) is a covering space in which every fiber has a finite number of elements. let f be a filter tied to the point t ∈ t and let et = {x1, .., xn}. consider an open neighborhood w of t regularly covered by p and let {oi}i=1,..., n be a partition in slices of ew with xi ∈ oi, for each i = 1, ..., n. to guarantee that the function π2 : e ∨ ef −→ ef is closed, consider ζ ∈ ef and an open neighborhood o of π −1 2 (ζ). if ζ ∈ e, v = {ζ} is an open neighborhood of ζ such that π−12 (v ) ⊂ o. 46 c. m. neira u. suppose that ζ = ω. for each i = 1, ..., n, there exist an open neighborhood ai of xi in e and fi ∈ f in such a way that ai ∨ (f ∪ {ω}) ⊂ o. let v = ⋂n i=1 p(ai). since v ∈ p(f), there exists f ′ ∈ f such that p(f ′) ⊂ v . consider f = f ′ ∩ f1 ∩ ... ∩ fn. if (x, y) ∈ π −1 2 (f ∪ {ω}) and y 6= ω, then p(x) = p(y) ∈ v , therefore x ∈ ai, for some i = 1, ..., n. hence (x, y) ∈ ai ∨ (fi ∪ {ω}) ⊂ o. this shows that que π2 : e ∨ ef −→ ef is a closed map. example 3.6. if (e, p, t ) is a fiber bundle with fiber h and if h is compact, then (e, p, t ) is fibrewise compact. in fact, let f be a filter tied to the point t ∈ t . there exist an open neighborhood w of t and a homeomorphism ϕ : p−1(w ) −→ w × h such that π1ϕ = p. to prove that the function π2 : e ∨ ef −→ ef is closed, consider the following facts. (1) since f is a filter tied to t, then p−1(w ) ∈ f. therefore, the collection g = {f ∈ f : f ⊂ p−1(w )} is a filter over ew tied to the point t. here one is considering p−1(w ) as a fibrewise topological space over w . furthermore, (ew )g is a subspace of ef . (2) since h is compact, the kuratowski-mrówka characterization of compact topological spaces guarantees that w ×h, seen as a fibrewise topological space over w , is fibrewise compact. (3) the commutativity of the diagram p−1(w ) ∨ (ew )g (w × h) ∨ (ew )g (ew )g ϕ×id h h h h h h hhj π2 ? π2 secures that the second projection from p−1(w ) ∨ (ew )g to (ew )g is closed. let o be an open neighborhood of π−12 (ω) in e ∨ ef . then o ∩ (p −1(w ) ∨ (ew )g) is a neighborhood of π −1 2 (ω) in p −1(w ) ∨ (ew )g, hence there exists g ∈ g such that π−12 (g ∪ {ω}) ⊂ o ∩ (p −1(w ) ∨ (ew )g). since g ∈ f, this completes the proof. remark 3.7. if (e, p, t ) is a fibrewise topological space and u is a t-ultrafilter over e that does not converge, then u has no cluster points. again, by the second part of the proof of the previous theorem, it follows that π2 : e ∨eu −→ eu is not a closed map. the last observation implies the following corollary. a kuratowski-mrówka type characterization of fibrewise compactness 47 corollary 3.8. if (e, p, t ) is a fibrewise topological space such that the map π2 : e ∨ eu −→ eu is closed for every t-ultrafilter u over e, then (e, p, t ) is fibrewise compact. references [1] j. adámek, h. herrlich and g. e. strecker, abstract and concrete categories. the joy of cats, wiley & sons, 1990. [2] n. bourbaki, general topology, addison wesley, 1966. [3] d. buhagiar, the category map, mem. fac. sci. eng. shimane univ. series b: mathematical science, 34 (2001), 1–19. [4] e. čech, topological spaces, john wiley and sons, 1966. [5] m. m. clementino, on categorical notions of compact objects, appl. cat. struct. 4 (1996), 15–29. [6] m. m. clementino, e. giuli and w. tholen, topology in a category: compactness, portugal. math. 53 (1996), 397–433. [7] a. dold, fibrewise topology, by i. m. james. book reviews, bulletin (new series) of the american mathematical society, 24 no. 1 (jan., 1991), pp. 246–248. [8] h. herrlich, g. salicrup and g. e. strecker, factorizations, denseness, separation and relatively compact objects, topology appl. 27 (1987), 157–169. [9] i. m. james, fibrewise topology, cambridge university press, 1989. [10] e. g. manes, compact, hausdorff objects, gen. topology appl. 4 (1974), 341–360. [11] g. richter, exponentiability for maps means fibrewise core-compactness, j. pure appl. algebra 187 (2004), 295–303. [12] s. willard, general topology, addison-wesley publishing company, 1970. (received july 2010 – accepted december 2010) clara m. neira u. (cmneirau@unal.edu.co) universidad nacional de colombia, facultad de ciencias, departamento de matemáticas, carrera 30 # 45–03, bogotá, colombia a kuratowski-mrówka type characterization of[6pt] fibrewise compactness. by c. m. neira u. @ appl. gen. topol. 22, no. 1 (2021), 169-181doi:10.4995/agt.2021.14446 © agt, upv, 2021 metric spaces related to abelian groups amir veisi a and ali delbaznasab b a faculty of petroleum and gas, yasouj university, gachsaran, iran (aveisi@yu.ac.ir) b department of mathematics, shahid bahonar university, kerman, iran (delbaznasab@gmail.com) communicated by f. lin abstract when working with a metric space, we are dealing with the additive group (r, +). replacing (r, +) with an abelian group (g, ∗), offers a new structure of a metric space. we call it a g-metric space and the induced topology is called the g-metric topology. in this paper, we are studying g-metric spaces based on l-groups (i.e., partially ordered groups which are lattices). some results in g-metric spaces are obtained. the g-metric topology is defined which is further studied for its topological properties. we prove that if g is a densely ordered group or an infinite cyclic group, then every g-metric space is hausdorff. it is shown that if g is a dedekind-complete densely ordered group, (x, d) a g-metric space, a ⊆ x and d is bounded, then f : x → g with f(x) = d(x, a) := inf{d(x, a) : a ∈ a} is continuous and further x ∈ clxa if and only if f(x) = e (the identity element in g). moreover, we show that if g is a densely ordered group and further a closed subset of r, k(x) is the family of nonempty compact subsets of x, e < g ∈ g and d is bounded, then d ′(a, b) < g if and only if a ⊆ nd(b, g) and b ⊆ nd(a, g), where nd(a, g) = {x ∈ x : d(x, a) < g}, db(a) = sup{d(a, b) : a ∈ a} and d ′(a, b) = sup{da(b), db(a)}. 2010 msc: 54c40; 06f20; 16h20. keywords: g-metric space; l-group; dedekind-complete group; densely ordered group; continuity. received 08 october 2020 – accepted 29 january 2021 http://dx.doi.org/10.4995/agt.2021.14446 a. veisi and a. delbaznasab 1. introduction in this article, a group (g, ∗) (briefly, g) is an abelian group and for readability, we use g1g2 instead of g1 ∗ g2. let x be a set and ≤ relation on x, we recall that the pair (x, ≤) is a partially ordered set (in brief, a poset) if the following conditions hold: x ≤ x, if x ≤ y and y ≤ x, then x = y; if x ≤ y and y ≤ z, then x ≤ z. in a poset, the symbol a∨b denotes sup{a, b}, i.e., the smallest element c, if one exists, such that c ≥ a and c ≥ b. likewise, a∧b stands for inf{a, b}. when both a∨b and a∧b exist, for all a, b ∈ a, then a is called a lattice. a subset s is a sublattice of a provided that, for all x, y ∈ s, the elements x ∨ y and x ∧ y of a belong to s. (thus, it is not enough that x and y have a supremum and infimum in s.) for instance, c(x), the ring of real-valued continuous functions on the topological space x is a lattice. if f, g ∈ c(x), then f ∨ g = f+g+|f−g| 2 ∈ c(x) (note, f ∧ g = −(−f ∨ −g) = f+g−|f−g| 2 ∈ c(x)). in fact, c(x) is a sublattice of rx, the ring of real-valued functions on the set x (note, the partial ordering on rx is: f ≤ g if and only if f(x) ≤ g(x) for all x in x). a poset in which every nonempty subset has both a supremum and an infimum is said to be a lattice-complete. for example, p(x), the set of all subsets of x with inclusion is lattice-complete. union (resp. intersection) of sets is the supremum (resp. the infimum) of them. a totally (or linearly) ordered set is a poset in which every pair of elements is comparable, i.e., x ≤ y or y ≤ x for all x and y in x. we use “ordered sets” instead of “totally ordered sets”. an ordered set is often referred to as a chain. a lattice need not be an ordered set, necessarily, but the converse is always true. we notice that c(x) and cc(x), its subalgebra consisting of elements with countable image, are lattices, while they are not ordered sets, also, they are not lattice-complete necessarily. an ordered set is said to be dedekind-complete provided that every nonempty subset with an upper bound has a supremum, or equivalently, every nonempty subset with a lower bound has an infimum. (for example, r, the set of real numbers is dedekind-complete, but not lattice-complete.) an ordered field f is said to be archimedean if z, the set of integers is cofinal, i.e., for every x ∈ f, there exists n ∈ z such that n ≥ x. for instance, q( √ n) := {a + b √ n : a, b ∈ q, √ n /∈ q} is an archimedean field. theorem 1.1 ([3, theorem 0.21]). an ordered field is archimedean if and only if it is isomorphic to a subfield of the ordered field r. a brief outline of this paper is as follows. in section 2, we introduce the g-metric spaces related to l-groups (i.e., partially ordered groups which are lattices) and study them further. in section 3, the basic topological properties based on the notion of g-disk are studied. we prove that if g is a densely ordered group or an infinite cyclic group, then every g-metric space is hausdorff. it is shown that if g is a dedekind-complete densely ordered group, (x, d) a g-metric space, d is bounded and a ⊆ x, then f : x → g given by f(x) = d(x, a) := inf{d(x, a) : a ∈ a} is continuous and further x ∈ clxa if and only if f(x) = e (the identity element of g). moreover, let © agt, upv, 2021 appl. gen. topol. 22, no. 1 170 metric spaces related to abelian groups f(x) be the family of nonempty closed sets in x, e < g ∈ g, a, b ∈ f(x) and da(b) = sup{d(b, a) : b ∈ b}. then for the g-metric space (f(x), d′) (note, d′(a, b) = sup{da(b), db(a)}), we have d′(a, b) ≤ g if and only if a ⊆ nd(b, ḡ) and b ⊆ nd(a, ḡ), where nd(a, ḡ) = {x ∈ x : d(x, a) ≤ g}. particularly, if g is a densely ordered group and further a closed subset of r, x is a g-metric space and k(x) is the family of nonempty compact sets in x, then d′(a, b) < g if and only if a ⊆ nd(b, g) and b ⊆ nd(a, g), where nd(a, g) = {x ∈ x : d(x, a) < g}. 2. g-metric spaces definition 2.1. a group g with a partial ordering relation ≤ is called a partially ordered group (in brief, a poset group) if the binary operation of g preserves the order, i.e., g1 ≥ g2 implies g1g3 ≥ g2g3 for all g1, g2, g3 ∈ g. (r1) moreover, if a poset group g is a lattice then g is called an l-group. from the above definition, the following facts are evident: g1 ≥ g2 if and only if g1g −1 2 ≥ e; g ≥ e if and only if g−1 ≤ e; if g1 ≥ g3 and g2 ≥ g4, then g1g2 ≥ g3g4. for example, every archimedean field with the addition is an l-group. but zn with the addition of modulo n is not a poset group yet, since this addition does not preserve the order. for an l-group g and g ∈ g, we let |g| = sup{g, g−1} = g ∨ g−1 = |g−1|. example 2.2. consider the group g := z×z×. . .×z (k-times) with the usual addition and the identity element e = (0, 0, . . . , 0). let g1 = (m1, m2, . . . , mk), g2 = (n1, n2, . . . , nk) ∈ g. define g1 ≤ g2 if and only if mi ≤ ni for all i = 1, 2, . . . , k. we see that ≤ is a partial ordering relation on g. also, the condition (r1) in the above definition is satisfied, i.e., g is a poset group. let zi = max{mi, ni} and z′i = min{mi, ni}, where i = 1, 2, . . . , k. let g3 = (z1, z2, . . . , zk) and g4 = (z ′ 1, z ′ 2, . . . , z ′ k). then we obtain g1 ∨ g2 = g3 and g1 ∧ g2 = g4. hence, g is an l-group. by an ordered group, we mean a poset group which is a totally ordered set by its partial ordering relation. it is clear that an ordered group is an l-group. finally, by dedekind-complete group, we mean an ordered group which is a dedekind-complete set with its partial ordering relation, i.e., every nonempty subset with an upper bound has a supremum, or equivalently, every nonempty subset with a lower bound has an infimum. for example, every archimedean field with the addition is a dedekind-complete group. corollary 2.3. if g is an l-group and g1, g2 ∈ g, then |g1g2| ≤ |g1||g2|. proof. we note that g1, g −1 1 ≤ |g1| and g2, g −1 2 ≤ |g2|. definition 2.1 now gives g1g2 ≤ |g1||g2| and also g−11 g −1 2 ≤ |g1||g2|. so we have that |g1g2| = sup{g1g2, (g1g2)−1} ≤ |g1g2|, and the result holds. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 171 a. veisi and a. delbaznasab definition 2.4. let g be a poset group and x a nonempty set. we say the function d : x ×x → g is a g-metric on x, whenever the following conditions hold, for every x, y, z ∈ x. (i) d(x, y) ≥ e (e is the identity element in g), (ii) d(x, y) = e if and only if x = y, (iii) d(x, y) = d(y, x), (iv) d(x, y) ≤ d(x, z)d(z, y) (triangle inequality). the pair (x, d) (briefly, x) is called a g-metric space. evidently, every metric space is a g-metric space, when (g, ∗) = (r, +). if all axioms but the second part of definition 2.4 are satisfied, we call d a g-pseudometric and then x a g-pseudometric space. defining d(x, y) = e for all x and y in x, gives a g-pseudometric on x, called the trivial g-pseudometric, in this case, d is a g-metric if and only if x is the singleton set {x}. although all the material of this section is developed for g-metric spaces, the basic results remain true for g-pseudometric spaces as well. if (x, d) is a g-metric on x and a is a subset of x, then a inherits a g-metric structure from x in an obvious way, making a a g-metric space. in the following example, we will present some examples of g-metric spaces. before it, let x = rn, g1 = (r, +), g2 = ((0, +∞), .), g3 = (r − {−1}, ∗) and g4 = (z2, ⊕), where +, . are usual addition and multiplication, the symbol ⊕ is the addition of modulo 2 and ∗ is defined as follows: x ∗ y = x + y + xy. in g3, the identity element is 0 and the inverse of x is x −1 = −x 1+x . checking of the associative property of ∗ is easy. moreover, let ϕi : gi ×gi → gi, where i = 1, 2, 3, 4, such that ϕ1(x, y) = x−y, ϕ2(x, y) = x y , ϕ3(x, y) = −xy 1+x and ϕ4(x, y) = { 0 if x = y 1 if x 6= y. then, since each ϕi is continuous, each gi is a topological group as subspaces of r with the usual topology. example 2.5. let x = rn, gi, where i = 1, 2, 3, 4, be as defined in the previous discussion. for x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn) ∈ x, ‖x − y‖ is the usual norm, i.e., ‖x − y‖ = ( ∑n i=1(xi − yi)2 ) 1 2 . we claim that each di, the functions below, is a gi-metric and therefore (x, di) is a gi-metric space. we only check that d2 and d3 satisfy (iv) of definition 2.4. other conditions are routine. (1) let d1 : x × x → g1 such that d1(x, y) = ‖x − y‖. (2) let d2 : x × x → g2 such that d2(x, y) = e‖x−y‖. (3) let d3 : x × x → g3 such that d3(x, y) = e‖x−y‖ − 1. (4) let d4 : x × x → g4 such that d(x, y) = 0 if x = y; and 1 if x 6= y. d4 is called a discrete g-metric. we notice that the identity elements in g2 and g3 are 1 and 0 respectively. moreover, d2(x, y) ≥ 1 and d3(x, y) ≥ 0. now, d2(x, y) = e ‖x−y‖ ≤ e(‖x−z‖+‖z−y‖) = e‖x−z‖e‖z−y‖ = d2(x, z)d2(z, y). © agt, upv, 2021 appl. gen. topol. 22, no. 1 172 metric spaces related to abelian groups also, we have d3(x, y) = e ‖x−y‖ − 1 ≤ e‖x−z‖e‖z−y‖ − 1 = (e‖x−z‖ − 1) + (e‖z−y‖ − 1) + (e‖x−z‖ − 1)(e‖z−y‖ − 1) = d3(x, z) + d3(z, y) + d3(x, z)d3(z, y) = d3(x, z)d3(z, y). so d2 and d3 satisfy the triangle inequality of definition 2.4. a g-metric d on a set x is called bounded if d(x, y) ≤ g0, for all x, y ∈ x and some g0 ∈ g. thus, the next result is now immediate. corollary 2.6. let g be an l-group, d a g-metric on x, e < g1 a fixed element in g and d1(x, y) = inf{d(x, y), g1}. then d1 is a bounded g-metric. lemma 2.7. let g be an l-group; a and b are finite subsets of g such that a ≥ e (i.e., a ≥ e, for all a ∈ a) and b ≥ e. then (i) if a ≤ b (i.e., for each a ∈ a there is b ∈ b such that a ≤ b) and e ≤ g ∈ g, then sup(ga) = g sup a ≤ g sup b = sup(gb), where ga = {ga : a ∈ a}. (ii) if a ≥ b (i.e., for each a ∈ a there is b ∈ b such that b ≤ a) and e ≤ g ∈ g, then inf(gb) = g inf b ≤ g inf a = inf(ga). (iii) sup(ab) = sup a sup b, and also inf(ab) = inf a inf b, where ab = {ab : a ∈ a, b ∈ b}. proof. first, we note that by definition of an l-group, each of the finite sets a, b and ab has a supremum and an infimum in g. the proofs of (i) and (ii) are routine. (iii). let sup a = α, sup b = β and sup(ab) = γ. since g is an l-group, it is a poset group. so by definition 2.1, we have ab ≤ αβ, for all a ∈ a and b ∈ b. evidently, γ ≤ αβ. now, we are ready to show that γ = αβ. for the reverse inclusion, let a ∈ a be fixed. then ab ≤ γ implies b ≤ a−1γ. therefore, b is bounded by a−1γ. so β = sup b ≤ a−1γ, in other words, a ≤ β−1γ. since a ∈ a is arbitrary, we deduce that a is bounded by β−1γ. thus, α ≤ β−1γ. this yields αβ ≤ γ, and we are through. the proof of another assertion (infimum) is done similarly. � proposition 2.8. let g be an ordered group. then defining d : g × g → g given by d(g1, g2) = |g1g−12 |, turns g into a g-metric space. proof. we claim that d is a g-metric on g. first, we note that since g is an ordered group, it is an l-group and further g ∈ g gives g ≥ e or g−1 ≥ e. so |g| = |g−1| = sup{g, g−1} = g or g−1. hence, |g| ≥ e and therefore conditions (i)-(iii) of definition 2.4 hold. moreover, if g1, g2, g3 ∈ g, then corollary 2.3 implies d(g1, g2) = |g1g−12 | = |(g1g −1 3 )(g3g −1 2 )| ≤ |g1g −1 3 ||g3g −1 2 | = d(g1, g3)d(g3, g2) © agt, upv, 2021 appl. gen. topol. 22, no. 1 173 a. veisi and a. delbaznasab this gives d satisfies the triangle inequality, i.e., it is a g-metric on g and hence g is a g-metric space. � corollary 2.9. let g be an ordered group, x a nonempty set and f : x → g a function. then d : x × x → g given by d(x, y) = |f(x)f−1(y)| is a gpseudometric on x. moreover, d is a g-metric on x if and only if f is oneone. the next proposition generalizes proposition 2.8. proposition 2.10. let g be an ordered group. then each of the following binary operations on gn (the n product of g), turns it into a g-metric space, where g = (g1, g2, . . . , gn) and g ′ = (g′1, g ′ 2, . . . , g ′ n) are arbitrary elements of gn. (i) d1 : g n × gn → g defined by d1(g, g′) = |g−11 g′1||g −1 2 g ′ 2| . . . |g−1n g′n|. (ii) d2 : g n × gn → g defined by d2(g, g ′) = sup{|g−11 g′1|, |g −1 2 g ′ 2|, . . . , |g−1n g′n|}. proof. we only check that the triangle inequality for d1 and d2. other conditions are routine. (i). let g ′′ = (g ′′ 1 , g ′′ 2 , . . . , g ′′ n) ∈ gn. then d1(g, g ′′ ) = |g−11 g ′′ 1 ||g−12 g ′′ 2 | . . . |g−1n g ′′ n| = |(g−11 g′1)(g′1 −1 g ′′ 1 )||(g−12 g′2)(g′2 −1 g ′′ 2 )| . . . |(g−1n g′n)(g′n −1 g ′′ n)| ≤ |g−11 g′1||g′1 −1 g ′′ 1 ||g−12 g′2||g′2 −1 g ′′ 2 | . . . |g−1n g′n||g′n −1 g ′′ n| = ( |g−11 g ′ 1|g−12 g ′ 2| . . . |g−1n g′n| )( |g′1 −1 g ′′ 1 ||g′2 −1 g ′′ 2 | . . . |g′n −1 g ′′ n| ) = d1(g, g ′)d1(g ′, g ′′ ). notice that the above inequality is obtained by corollary 2.3. (ii). let g ′′ = (g ′′ 1 , g ′′ 2 , . . . , g ′′ n) ∈ gn and let a = {|g−11 g ′′ 1 |, |g−12 g ′′ 2 |, . . . , |g−1n g ′′ n|}, b = {|g−11 g ′ 1||g′1 −1 g ′′ 1 |, |g−12 g ′ 2||g′2 −1 g ′′ 2 |, . . . , |g−1n g′n||g′n −1 g ′′ n|}, b1 = {|g−11 g′1|, |g −1 2 g ′ 2|, . . . , |g−1n g′n|}, and b2 = {|g′1 −1 g ′′ 1 |, |g′2 −1 g ′′ 2 |, . . . , |g′n −1 g ′′ n|}. we notice that g is an l-group. now, according to lemma 2.7, we have a ≤ b ≤ b1b2. therefore, d2(g, g ′′ ) = sup a ≤ sup b ≤ sup(b1b2) = sup b1 sup b2 = d2(g, g′)d2(g′, g ′′ ), which completes the proof. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 174 metric spaces related to abelian groups 3. basic topological concepts in g-metric spaces and some related results we begin with the following definition. definition 3.1. let g be a poset group, (x, d) a g-metric space and x a point of x. given e < g ∈ g, we let nd(x, g) = {y ∈ x : d(x, y) < g}, and call it the g-disk centered at x. also, we put nd(x, g) = {y ∈ x, d(x, y) ≤ g}. a subset u of x is said to be open in x if either u = ∅ or for every x ∈ u there is a g ∈ g such that nd(x, g) ⊆ u. here, x is called an interior point of u. the set of all interior points of u is called the interior of u, denoted by u◦ (or intxu). also, a set f is called closed if and only if its set-theoretic complement is an open set in x. evidently, a set f is closed if and if every g-disk centered at x meets f , then x ∈ f . corollary 3.2. every g-disk nd(x, g) is an open set in x (and hence x r nd(x, g) = {y ∈ x : d(x, y) ≥ g} is a closed set in x). proof. let y ∈ nd(x, g). then g1 = d(x, y) < g. we claim that nd(y, gg−11 ) ⊆ nd(x, g) (note, gg −1 1 > e). to see this, assume that z ∈ nd(y, gg −1 1 ). hence, d(z, x) ≤ d(z, y)d(y, x) < gg−11 g1 = g. this yields z ∈ nd(x, g), i.e., nd(y, gg −1 1 ) ⊆ nd(x, g), and we are done. � definition 3.3. let x be a g-metric space and a ⊆ x. the closure of a in x is denoted by clxa (or briefly cla) and defined by the set cla = ∩{f ⊆ x : f is closed in x and a ⊆ f}. by the above definition, a is closed if and only if a = cla. corollary 3.4. let g be a poset group, (x, d) a g-metric space and ā = {x ∈ x : nd(x, g) ∩ a 6= ∅ for all e < g ∈ g}, where a ⊆ x. then (1) a = cla. (2) if x ∈ x and g ∈ g, then nd(x, g) ⊆ nd(x, g). proof. (1). let x /∈ cla. then x /∈ f , for some closed set f containing a. now, since xrf is open, there exists e < g ∈ g, such that x ∈ nd(x, g) ⊆ xrf . so x /∈ ā. conversely, suppose that x /∈ ā. so for some e < g ∈ g, nd(x, g)∩a = ∅. therefore, the closed set x r nd(x, g) contains a but not x. this gives x /∈ cla, and we are done. (2). suppose that y /∈ nd(x, g). so d(x, y) > g and hence g1 = d(x, y)g −1 > e. we claim that nd(y, g1) ∩ nd(x, g) = ∅. otherwise, for some z ∈ nd(y, g1) ∩ nd(x, g), we have d(x, y) ≤ d(x, z)d(z, y) < gg1 = d(x, y), a contradiction. hence, y /∈ nd(x, g) and we are done. � proposition 3.5. let g be an ordered group and x a g-metric space. then the open sets in x have the following properties: (i) x and ∅ are both open. © agt, upv, 2021 appl. gen. topol. 22, no. 1 175 a. veisi and a. delbaznasab (ii) every union of open sets is open. (iii) every finite intersection of open sets is open. proof. (i) and (ii) are clear. (iii). let x ∈ ⋂n i=1 ui, where ui is an open set in x. take gi ∈ g such that x ∈ nd(x, gi) ⊆ ui. since g is an ordered group, there exists g ∈ g such that g = inf{gi}ni=1 (note, the elements gi form a chain and hence g is one of them). thus, x ∈ nd(x, g) ⊆ ⋂n i=1 nd(x, gi) ⊆ ⋂n i=1 ui, which completes the proof. � by the above proposition, every g-metric d on a set x defines a topology τd on x; members of τd, or, open subsets of x are unions of g-disks. clearly, the family of all g-disks is a base for (x, τd). we call τd the topology induced by the g-metric d (or g-metric topology). remark 3.6. even if g is a dedekind-complete group, a countable intersection of open sets in a g-metric space need not be an open set necessarily. to see this, consider r as a g-metric space, where g is (r, +) or ((0, ∞), .). also, recall the fact that every point a of r is a gδ-set, i.e., {a} = ⋂∞ n=1(a− 1 n , a+ 1 n ). in [2, i.3], an ordered set x is called a densely ordered set, if no cut of x is a jump, or equivalently, for every pair x, y of elements of x satisfying x < y, there exists a z ∈ x, such that x < z < y. definition 3.7. an ordered group g is called a densely ordered group if it is a densely ordered set with its total ordering relation. it is clear that densely ordered groups are infinite. for example, every archimedean field like as r and q( √ n) := {a + b √ n : a, b ∈ q, √ n /∈ q} with the addition is a densely ordered group. z, the group of integers is an ordered group which is not a densely ordered group while zn with the addition of modulo n, is not a poset group yet. from now on, the group g is assumed to be a densely ordered group. proposition 3.8. let g be a densely ordered group, (x, d) a g-metric space, x ∈ x and g ∈ g be fixed, and a = {y ∈ x : d(x, y) > g}. then a is an open set in x (and hence nd(x, g) := {y ∈ g : d(x, y) ≤ g} is closed). proof. let y ∈ a be fixed. then d(x, y) > g. we must show that y is an interior point of a. let g1 = d(x, y)g −1. then g1 > e and d(x, y) = gg1. since g is a densely ordered group, we take e < g2 < g1 and claim that nd(y, g2) is contained in a entirely. to see this, let z ∈ nd(y, g2). then we have d(y, z) < g2 and so g−12 < d −1(y, z). now, the inequality d(x, y) ≤ d(x, z)d(z, y) yields g < gg1g −1 2 < d(x, y)d −1(y, z) ≤ d(x, z). (notice that g < gg1g −1 2 if and only if g2 < g1.) therefore, g < d(x, z), i.e., y is an interior point of a. so a is an open set in x, and we are done. � © agt, upv, 2021 appl. gen. topol. 22, no. 1 176 metric spaces related to abelian groups proposition 3.9. let g be an ordered group and (x, d) a g-metric space. then the following statements hold. (i) if g is a densely ordered group, then x is hausdorff. (ii) if g is an infinite cyclic group, then x is discrete (and so it is first countable). proof. (i). let x, y ∈ x and d(x, y) = g > e. by assumption, since g is a densely ordered set, we can take g1, g2 ∈ g such that e < g1 < g2 < g. now, we claim that two disks nd(x, g1) and nd(y, gg −1 2 ) are disjoint (note, gg −1 2 > e). otherwise, for some x′ ∈ nd(x, g1) ∩ nd(y, gg−12 ), we have d(x, x′) < g1 and d(x′, y) < gg−12 . hence, g = d(x, y) ≤ d(x, x′)d(x′, y) < g1gg −1 2 . therefore, e < g1g −1 2 , or equivalently, g2 < g1, a contradiction. so we are done. (ii). let e < g ∈ g be the generator of g. then g = {gn : n ∈ z}, in fact, we have g ∼= z, the additive group of integers with the generator 1 (or −1). we note that the elements of g form a chain. so we obtain . . . < g−3 < g−2 < g−1 < e < g < g2 < g3 < . . . therefore, for each x ∈ x, nd(x, g) = {y ∈ x : d(x, y) < g} = {y ∈ x : d(x, y) = e} = {x}. this yields x is discrete (note, in this case g is not a densely ordered group). � remark 3.10. by proposition 3.9(ii), if g is an infinite cyclic group then x is first countable. but the converse of that result may be false, since every metric space is first countable, whereas the additive group (r, +) is not even countably generated. in general, the converse of the above proposition does not need to be true. in the next example, we give examples of hausdorff g-metric spaces such that the group g is neither a densely ordered group nor an infinite cyclic group. example 3.11. (i) let d1 : z × z → z defined by d1(m, n) = |m − n|. then, since nd1(m, 1) = {m}, we obtain z is a discrete z-metric space. so it is hausdorff, whereas z is not a densely ordered group. but if z is considered as a q-metric space with the same definition, d1(m, n) = |m − n|, it is a discrete q-metric space while q is a densely ordered group. (ii) let g := z×z with the identity element e = (0, 0). define d2 : z×z → g with d2(m, n) = (|m−n|, |m−n|). by example 2.2, g is an l-group. it is easy to see that d2 is a g-metric on z. now, let g = (1, 1). then nd2 ( m, g ) = {n ∈ z : d2(m, n) < g} = {m}. this yields z is a discrete g-metric space, whereas g = z × z is not a cyclic group (note, it is a finitely generated group which is generated by the set {(0, 1), (1, 0)}). definition 3.12. if (x, dx) (resp. (y, dy )) is a g1(resp. g2-) metric space, a function f : x → y is called continuous at x0 ∈ x if and only if for each e2 < g2 ∈ g2 there is some e1 < g1 ∈ g1 such that dy (f(x0), f(y)) < g2, © agt, upv, 2021 appl. gen. topol. 22, no. 1 177 a. veisi and a. delbaznasab whenever dx(x0, y) < g1. f is called continuous on x, if it is continuous at every x ∈ x. a simple translation of the above definition is: corollary 3.13. a function f : x → y is continuous at x0 ∈ x if and only if for each g2-disk ndy (f(x0), g2) centered at f(x0), there is some g1-disk ndx (x0, g1) centered at x0, such that f(ndx (x0, g1)) ⊆ ndy (f(x0), g2). theorem 3.14. if (x, dx) and (y, dy ) are g1and g2-metric spaces respectively, a function f : x → y is continuous at x0 ∈ x if and only if for each open set v of y containing f(x0), there exists an open set u of x containing x0 such that f(u) ⊆ v. proof. (⇒) : suppose that f is continuous at x0 and v is an open set in y containing f(x0). by definition of open sets, there is g2 ∈ g2 such that f(x0) ∈ ndy (f(x0), g2) ⊆ v . by corollary 3.13, there exists a g1-disk ndx (x0, g1) centered at x0 such that f(ndx (x0, g1)) ⊆ ndy (f(x0), g2) ⊆ v, where g1 ∈ g1. it now suffices to choose u = ndx (x0, g1). (⇐) : consider e2 < g2 ∈ g2 and ndy (f(x0), g2) as an open set in y containing f(x0). by hypothesis, there exists an open set u in x containing x0 such that f(u) ⊆ ndy (f(x0), g2). also, we can take e1 < g1 ∈ g1 such that ndx (x0, g1) ⊆ u. so f(ndx (x0, g1)) ⊆ f(u) ⊆ ndy (f(x0), g2), and we are done. � the following lemma is the counterpart of lemma 2.7 for a dedekindcomplete group g. the only difference is that there a and b were finite subsets of g but here these sets must be bounded. lemma 3.15. let g be a dedekind-complete group and; a and b are bounded subsets of g such that a ≥ e (i.e., a ≥ e, for all a ∈ a) and b ≥ e. then (i) if a ≤ b (i.e., for each a ∈ a there is b ∈ b such that a ≤ b) and e ≤ g ∈ g, then sup(ga) = g sup a ≤ g sup b = sup(gb), where ga = {ga : a ∈ a}. (ii) if a ≥ b (i.e., for each a ∈ a there is b ∈ b such that b ≤ a) and e ≤ g ∈ g, then inf(gb) = g inf b ≤ g inf a = inf(ga). (iii) sup(ab) = sup a sup b, and also inf(ab) = inf a inf b, where ab = {ab : a ∈ a, b ∈ b}. in the remainder of this article, g is assumed to be a dedekind-complete densely ordered group (i.e., a densely ordered group in which every bounded nonempty subset has a supremum and an infimum in g), (x, d) a g-metric space, and d is bounded. the distance of a point x to a set a (⊆ x) is defined by d(x, a) = inf{d(x, a) : a ∈ a}, if a 6= ∅, and d(x, ∅) = e. theorem 3.16. (i) the mapping f : x → g defined by f(x) = d(x, a) is continuous. (ii) x ∈ clxa if and only if f(x) = d(x, a) = e, in fact, clxa = f−1(e). © agt, upv, 2021 appl. gen. topol. 22, no. 1 178 metric spaces related to abelian groups proof. (i). first, by proposition 2.8, we have (g, d′) is a g-metric space, where d′(g1, g2) = |g1g−12 |. let x0 ∈ x, g0 ∈ g and nd′(f(x0), g0) be an open set containing f(x0). then d(x0, a) ≤ d(x0, x)d(x, a), and d(x, a) ≤ d(x, x0)d(x0, a). (r2) now, if we let g1 = {d(x0, a) : a ∈ a} and g2 = {d(x0, x)d(x, a) : a ∈ a}, then g1 and g2 are two subsets of g with the same cardinality and g2 ≥ g1. by lemma 3.15 (ii), we have infa∈a g1 ≤ infa∈a g2. in other words, taking infimum on both sides of each of the inequalities in (r2) with respect to a ∈ a, we obtain inf a∈a d(x0, a) ≤ d(x0, x) inf a∈a d(x, a), and inf a∈a d(x, a) ≤ d(x, x0) inf a∈a d(x0, a). thus, f(x0) ≤ d(x, x0)f(x) and f(x) ≤ d(x, x0)f(x0). hence, f(x0)f−1(x) ≤ d(x, x0) and also f(x)f −1(x0) ≤ d(x, x0), i.e., d(x, x0) is a common upper bound for f(x0)f −1(x) and f−1(x0)f(x). therefore, d′(f(x0), f(x)) = |f(x)f−1(x0)| = sup{f(x0)f−1(x), f−1(x0)f(x)} ≤ d(x, x0). now, for the g0-disk nd(x0, g0) we have f(nd(x0, g0)) ⊆ nd′(f(x0), g0), and we are through. (ii). necessity: first, we note that by corollary 3.4, cla = ā = {x ∈ x : nd(x, g) ∩ a 6= ∅, for all g > e}. if d(x, a) = g > e then d(x, a) ≥ g > e, for all a ∈ a. by assumption, since g is a densely ordered group, we can take g1 ∈ g such that g > g1 > e. now, we observe that nd(x, g1) ∩ a = ∅. hence, x /∈ a. sufficiency: let x /∈ a. then nd(x, g) ∩ a = ∅, for some e < g ∈ g. hence, d(x, a) ≥ g, for all a ∈ a. therefore, d(x, a) ≥ g > e. so d(x, a) 6= e, and we are done. � theorem 3.17. let g be a dedekind-complete densely ordered group, (x, d) a g-metric space, d is bounded, g ∈ g, and let f(x) be the family of all nonempty closed subsets of x. for a, b ∈ f(x) define db(a) = sup{d(a, b) : a ∈ a}, and d′(a, b) = sup{da(b), db(a)}. then the following statements hold. (1) d′ is a g-metric on f(x). we call it the hausdorff g-metric on f(x). (2) d′(a, b) ≤ g if and only if a ⊆ nd(b, ḡ) and b ⊆ nd(a, ḡ), where nd(a, ḡ) = {x ∈ x : d(x, a) ≤ g}. proof. (1). (i) and (iii) of definition 2.4 are evident. let d′(a, b) = e. then db(a) = e = da(b). so d(a, b) = e for all a ∈ a. by theorem 3.16 (ii), a ∈ clb = b, i,e., a ⊆ b. similarly, b ⊆ a. this proves (ii) of definition 2.4. for the proof of triangle inequality, let a, b, c ∈ f(x) and a ∈ a, b ∈ b, c ∈ c. we notice that d(a, b) ≤ d(a, b) and d(b, c) ≤ dc(b). thus, d(a, b) ≤ d(a, b) ≤ d(a, c)d(c, b). taking infimum on both sides of the above inequality with respect to c ∈ c plus lemma 3.15 yield © agt, upv, 2021 appl. gen. topol. 22, no. 1 179 a. veisi and a. delbaznasab d(a, b) ≤ infc∈c{d(a, c)d(c, b)} = infc∈c d(a, c) infc∈c d(c, b). therefore, d(a, b) ≤ d(a, c)d(b, c). since d(b, c) ≤ dc(b), we have d(a, b) ≤ d(a, c)dc(b). taking supremum on both sides of the latter inequality with respect to a ∈ a, we obtain db(a) ≤ dc(a)dc(b). (r3) on the other hand, taking infimum over c ∈ c on both sides of the inequalities d(b, a) ≤ d(a, b) ≤ d(a, c)d(c, b) we obtain d(b, a) ≤ d(a, c)d(b, c) (lemma 3.15). furthermore, d(a, c) ≤ dc(a) gives d(b, a) ≤ dc(a)d(b, c). now, take supremum on both sides of the latter inequality respect to b ∈ b. thus, da(b) ≤ dc(a)dc(b). (r4) combining (r3) and (r4) we get d′(a, b) ≤ dc(a)dc(b) ≤ d′(a, c)d′(c, b). hence, d′ satisfies (iv) of definition 2.4, and we are done. (2). (⇒) : let d′(a, b) ≤ g. then db(a) ≤ g and da(b) ≤ g. hence, d(a, b) ≤ g, for all a in a. so a ⊆ nd(b, ḡ). similarly, b ⊆ nd(a, ḡ). (⇐) : since a ⊆ nd(b, ḡ), it gives d(a, b) ≤ g, for all a in a, and therefore db(a) = supa∈a d(a, b) ≤ g. the assertion da(b) ≤ g is deduced similarly. so d′(a, b) ≤ g, and we are through. � corollary 3.18. let g be a densely ordered group and further a closed subset of r, k(x) the family of nonempty compact subsets of x and a, b ∈ k(x) such that x, d, g, da and d ′ be as defined in theorem 3.17. then d′(a, b) < g if and only if a ⊆ nd(b, g) and b ⊆ nd(a, g), where nd(a, g) = {x ∈ x : d(x, a) < g}. proof. we first recall the fact that a nonempty subset of r has the least-upperbound property (equivalently, the greatest-lower-bound property) if and only if it is closed in r. so g has the least-upper-bound property and hence it is a dedekind-complete densely ordered group. moreover, by proposition 3.9, x is hausdorff and therefore every compact set in x is closed. thus, the conditions of theorem 3.17 are satisfied. the necessary condition is obvious. to prove the sufficiency, let us define f1, f2 : x → g with f1(x) = d(x, a) and f2(x) = d(x, b). now, since a and b are compact subsets of x and further; f1 and f2 are continuous functions on x (theorem 3.16), f1(b) and f2(a) are compact sets in g (note, since g is closed, f1 and f2 are well defined). therefore, sup f1(b) ∈ f1(b) and also sup f2(a) ∈ f2(a). so we have da(b) = sup f1(b) = f1(b1) = d(b1, a), for some b1 ∈ b, and also db(a) = sup f2(a) = f2(a2) = d(a2, b), for some a2 ∈ a. © agt, upv, 2021 appl. gen. topol. 22, no. 1 180 metric spaces related to abelian groups by assumption, we now get da(b) < g and db(a) < g. hence, d ′(a, b) < g, and we are through. � acknowledgements. the authors are grateful to the referee for providing helpful comments and recommendations to improve the quality of the paper. references [1] a. arhangel’skii and m. tkachenko, topological groups and related structures, atlantis press/ world scientific, amsterdam-paris, 2008. 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[10] a. veisi, invariant norms on the subalgebras of c(x) consisting of bounded functions with countable image, jp journal of geometry and topology 21, no. 3 (2018), 167–179. [11] a. veisi, ec-filters and ec-ideals in the functionally countable subalgebra of c ∗(x), appl. gen. topol. 20, no. 2 (2019), 395–405. [12] a. veisi and a. delbaznasab, new structure of norms on rn and their relations with the curvature of the plane curves, ratio mathematica 39 (2020), 55–67. [13] s. willard, general topology, addison-wesley, 1970. © agt, upv, 2021 appl. gen. topol. 22, no. 1 181 buzyakovaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 221-225 on an algebraic version of tamano’s theorem raushan z. buzyakova abstract. let x be a non-paracompact subspace of a linearly ordered topological space. we prove, in particular, that if a hausdorff topological group g contains closed copies of x and a hausdorff compactification bx of x then g is not normal. the theorem also holds in the class of monotonically normal spaces. 2000 ams classification: 54h11,22a05, 54f05 keywords: topological group, hausdorff compactification, normal space, stationary set 1. introduction. this note is devoted to analysis of tamano’s characterization [5] of paracompactness in the context of hausdorff topological groups. the tamano’s argument implies that if a tychonov space x is not paracompact then x × bx is not normal for every hausdorff compactification bx of x. a natural algebraic analysis of this statement leads to the following conjecture: conjecture. let x be a non-paracompact topological space and bx a hausdorff compactification of x. if a topological group g contains closed copies of x and bx then g is not normal. we believe that the conjecture has a good chance for a positive resolution. in this note we give a proof of this conjecture in the class of generalized ordered spaces (=”subspaces of linearly ordered spaces”), or more generally, in the class of monotonically normal spaces. since tamano’s theorem is a criterion it would be natural to ask if given a paracompact space x one can find a hausdorff compactification bx and a normal group g such that g contains closed copies of x and bx. the author does not know if such g and bx exist without additional requirements on x besides paracompactness. it is worth to mention, however, that if x n is lindelöf for every n ∈ ω then the free group f (x ⊕bx) over x ⊕bx is normal (even lindelöf) and contains closed copies of 222 r. z. buzyakova x and bx. this fact is well-known and mentioned, in particular, in the recent survey [4]. also, we would like to mention that in [2] the author proved that if a group g contains closed copies of an uncountable regular cardinal τ and τ + 1, then g contains a closed copy of τ × (τ + 1), which makes g not normal. while some of the ideas of this result could be used to prove the main theorem of this note we use a different approach, which may be helpful in proving the general conjecture. all spaces in this note are assumed to be tychonov. by βx we denote the čech-stone compactification of x. the symbol ⋆ is reserved for group operation. a subspace of a linearly ordered topological space will be called a go-space. a point x ∈ x is a complete accumulation point for an infinite set a ⊂ x if every open neighborhood of x meets a by a subset of cardinality |a|. to prove our main result, we start with four folklore statements, two of which are left without proof. fact 1. let s be a stationary subset of an uncountable regular cardinal τ and bs a hausdorff compactification of s. then there exists a unique point p ∈ bs \ s that is a complete accumulation point for s. moreover, s ∪ {p} is naturally homeomorphic to s ∪ {τ}. the point p in the above fact will be always identified with τ . fact 2. let s be a stationary subset of an uncountable regular cardinal τ , bs a hausdorff compactification of s, and c(s × bs) a hausdorff compactification of s × bs. then there exists a unique point p ∈ c(s × bs) that is the only common and only complete accumulation point for s × {τ} and for {(α, α) : α ∈ s}. the point p in fact 2 will be always identified with (τ, τ ). lemma 1.1. let s be a stationary subset of an uncountable regular cardinal τ . let τ be a limit point for a ⊂ βs \ (s ∪ {τ}) in βs. then clβs (a) ∩ s is closed and unbounded in s. proof. let f be the continuous map from βs to τ + 1 that is the identity on s. since τ is the only complete accumulation point for s in βs, f (a) ⊂ τ . since τ is a limit point for a, f (a) is unbounded in τ . assume the conclusion of lemma is false. then we may also assume that clβs(a) ∩ s = ∅. since f maps the remainder of s in βs to the remainder of s in τ + 1, we have f (clβs(a) \ {τ}) is a closed unbounded subset of τ that does not meet s. this contradicts stationarity of s in τ . � lemma 1.2. let s be a stationary subset of an uncountable regular cardinal τ . if f : s → τ is continuous and unbounded then there exists λ ∈ s such that f (λ) = λ. on an algebraic version of tamano’s theorem 223 proof. since f is unbounded and τ is regular we can select x = {xβ : β < τ} such that (1) xα > max{xβ , f (xβ )} if α > β; (2) f (xα) > max{xβ , f (xβ )} if α > β. observe that property 1 and regularity of τ imply that x is unbounded in τ . since s is stationary there exist λ ∈ s and limit α ∈ τ such that λ is limit for {xβ : β < α} and xβ < λ for all β < α. by 1 and 2 and continuity of f , we have f (λ) = λ. � for our main result we need the following fundamental theorem. theorem (r. engelking and d. lutzer [3]). a go-space x is paracompact iff no closed subspace of x is homeomorphic to a stationary subset of a regular uncountable cardinal. theorem 1.3. let l be a non-paracompact go-space and bl a hausdorff compactification of l. if a topological group g contains closed copies of l and bl, then g is not normal. proof. we may assume that l is a closed subset of g and bl′ ⊂ g is a copy of bl, where l′ is a copy of l with a fixed homeomorphism x ↔ x′. let s be a closed subset of l that is homeomorphic to a stationary subset of an uncountable regular cardinal τ . such an s exists due to theorem’s hypothesis and engelking-lutzer theorem. as agreed earlier, by τ we denote the only complete accumulation point for s in any hausdorff compactification and by τ ′ the only complete accumulation point for s′ in bl′. let h = s × {τ ′} and d = {(α, α′) : α ∈ s}. the sets h and d are closed in s × bs′ and not functionally separated. let hg = ⋆(h) = {α ⋆ τ ′ : α ∈ s} and dg = ⋆(d) = {α ⋆ α ′ : α ∈ s}. claim 1: ⋆̃(τ, τ ′) 6∈ g, where ⋆̃ is the continuous extension of ⋆ over the čechstone compactification. to prove the claim, observe that hg is a closed subset of g homeomorphic to s. this is because multiplication by a constant is a continuous automorphism. by fact 2 and fact 1, (τ, τ ′) is the only complete accumulation point for h in β(g × g)). the set hg does not have a complete accumulation point in g. therefore ⋆̃(τ, τ ′) 6∈ g. the claim is proved. put hα = {(β, τ ′) : β ≥ α, β ∈ s}, hαg = {β ⋆ τ ′ : β ≥ α, β ∈ s}, dα = {(β, β′) : β ≥ α, β ∈ s}, and dαg = {β ⋆ β ′ : β ≥ α, β ∈ s}. 224 r. z. buzyakova claim 2: there exists λ < τ such that hλg ∩ clg(d λ g) = ∅. to prove the claim assume the contrary. then for any α < τ there exists pα ∈ clβ(g×g)(d α) such that ⋆̃(pα) ∈ h α g. by lemma 1.1, clβ(g×g){pα : α < τ} meets d by a closed subset t of cardinality τ . since hg is closed and ⋆ is continuous we have ⋆(t ) ⊂ hg. since |t | = τ we have ⋆̃(τ, τ ′) is a complete accumulation point for ⋆(t ) in βg. by lemma 1.2, there exists (γ, γ′) ∈ t such that ⋆(γ, γ′) = γ ⋆ γ′ = γ ⋆ τ ′. therefore, γ′ = τ ′, contradicting to the fact that (τ, τ ′) 6∈ t . the claim is proved. by claim 2, hλg and clg(d λ g) are closed and disjoint in g. if g were normal, then hλg and clg(d λ g) would have been functionally separated and so would hλ and dλ in g × g. but hλ and dλ are not functionally separated for every λ < τ . � observe that the proof of the theorem uses only one property of l, namely, the fact that l contains a closed copy of a stationary subset of an uncountable regular cardinal τ . since the theorem of engelking and lutzer holds for monotonically normal spaces as well (proved by balogh and rudin [1]) we have the following. theorem 1.4. let x be non-paracompact and monotonically normal and bx a hausdorff compactification of x. if a topological group g contains closed copies of x and bx, then g is not normal. we would like to finish the paper with two questions (which may have been asked before by other authors) related to the discussion in the beginning of this work. question 1. is there a paracompact space that cannot be embedded in a normal group as a closed subspace? question 2. let x n be paracompact for every n ∈ ω. is f (x) normal? references [1] z. balogh and m. e. rudin, monotone normality, topology appl. 47 (1992), no. 2, 115–127. [2] r. z. buzyakova, ordinals in topological groups, fund. math. 196 (2007), no. 2, 127–138. [3] d. j. lutzer, ordered topological spaces, surveys in general topology, pp. 247–295, academic press, new york-london-toronto, ont., 1980. [4] o. v. sipachëva, the topology of free topological group, fundam. prikl. mat. 9 (2003), no. 2, 99–204; english translation in j. math. sci. (n. y.) 131 (2005), no. 4, 5765–5838. [5] h. tamano, on paracompactness, pacific j. math. 10 (1960) 1043–1047. on an algebraic version of tamano’s theorem 225 received april 2008 accepted february 2009 raushan z. buzyakova (raushan buzyakova@yahoo.com) mathematics department, uncg, greensboro, nc 27402 usa @ appl. gen. topol. 22, no. 1 (2021), 109-120doi:10.4995/agt.2021.14045 © agt, upv, 2021 from interpolative contractive mappings to generalized ćirić-quasi contraction mappings kushal roy and sayantan panja department of mathematics, the university of burdwan, purba bardhaman-713104, west bengal, india. (kushal.roy93@gmail.com, spanja1729@gmail.com) communicated by e. karapinar abstract in this article we consider a restricted version of ćirić-quasi contraction mapping for showing that this mapping generalizes several known interpolative type contractive mappings. also here we introduce the concept of interpolative strictly contractive type mapping t and prove a fixed point theorem for such mapping over a t -orbitally compact metric space. some examples are given in support of our established results. finally we give an observation regarding (λ, α, β)-interpolative kannan contractions introduced by gaba et al. 2010 msc: 47h10; 54h25. keywords: fixed point; restricted ćirić-quasi contraction mapping; interpolative strictly contractive type mapping; t -orbitally compact metric space. 1. introduction and preliminaries in the year 1922, s. banach had established a remarkable fixed point theorem, known as ’banach contraction principle’ which is given as follows: theorem 1.1 ([2]). if a mapping t from a complete metric space (x, d) to itself satisfies the following condition (1.1) d(t x, t y) ≤ αd(x, y) for all x, y ∈ x, for some α ∈ [0, 1) then t possesses a unique fixed point in x. received 18 july 2020 – accepted 30 january 2021 http://dx.doi.org/10.4995/agt.2021.14045 k. roy and s. panja several generalizations of this theorem have been made by researchers, working in the area of fixed point theory, by means of different new type contractive mappings. recently e. karapinar [7] proposed a new kannan-type contractive mapping via the notion of interpolation and proved a fixed point theorem over metric space. in his paper, karapinar assumed that the interpolative kannan-type contractive mapping t over a metric space x satisfies the contractive condition for all x, y ∈ x with x 6= t x. but in this situation it is to be noted that if this mapping t has a fixed in x then it will be a constant mapping and therefore t has a unique fixed point trivially. to remove such triviality the authors in [8] assumed that interpolative type mappings satisfy the contractive condition for all x, y ∈ x \ fix(t ), where fix(t ) is the set of all fixed points of t. though in this case an interpolative contractive type mapping may possesses more than one fixed point. definition 1.2 ([7]). in a metric space (x, d), a mapping t : x → x is said to be interpolative kannan-type contractive mapping if it satisfies (1.2) d(t x, t y) ≤ λ[d(x, t x)]α[d(y, t y)]1−α for all x, y ∈ x \ fix(t ), for some λ ∈ [0, 1) and for some α ∈ (0, 1). theorem 1.3. [7] let (x, d) be a complete metric space and t : x → x be an interpolative kannan-type contractive mapping. then t has at least one fixed point in x. as an extension of interpolative kannan-type contractive mappings, karapinar et al. introduced interpolative reich-rus-ćirić type contractions (see [8]). the definition is given below. definition 1.4 ([8]). in a metric space (x, d), a mapping t : x → x is called interpolative reich-rus-ćirić type contraction mapping if it satisfies (1.3) d(t x, t y) ≤ λ[d(x, y)]β[d(x, t x)]α[d(y, t y)]1−α−β for all x, y ∈ x \ fix(t ), for some λ ∈ [0, 1) and for α, β ∈ (0, 1). theorem 1.5 ([8]). let (x, d) be a complete metric space and t : x → x be an interpolative reich-rus-ćirić type contraction mapping. then t has a fixed point in x. further extension of interpolative kannan-type contractive mappings has been given by karapinar et al. [9], which is known as interpolative hardyrogers type contraction. the definition is given as follows. definition 1.6 ([9]). in a metric space (x, d), a mapping t : x → x is said to be interpolative hardy-rogers type contraction mapping if it satisfies (1.4) d(t x, t y) ≤ λ[d(x, y)]β[d(x, t x)]α[d(y, t y)]γ [ 1 2 (d(x, t y) + d(y, t x)) ]1−α−β−γ © agt, upv, 2021 appl. gen. topol. 22, no. 1 110 from interpolative contractive mappings... for all x, y ∈ x \ fix(t ), for some λ ∈ [0, 1) and for α, β, γ ∈ (0, 1) with α + β + γ < 1. theorem 1.7 ([9]). let (x, d) be a complete metric space and t : x → x be an interpolative hardy-rogers type contraction mapping. then t has at least one fixed point in x. recently c. b. ampadu [1] has defined interpolative berinde weak operator in his paper. the definition is given as follows: definition 1.8 ([1]). let (x,d) be a metric space. we say t : x → x is (i) an interpolative berinde weak operator if it satisfies (1.5) d(t x, t y) ≤ λ[d(x, y)]α[d(x, t x)]1−α for all x, y ∈ x \ fix(t ), for some λ ∈ [0, 1) and for some α ∈ (0, 1). (ii) an alternate interpolative berinde weak operator if it satisfies (1.6) d(t x, t y) ≤ λ √ d(x, y)d(x, t x) for all x, y ∈ x \ fix(t ), where λ ∈ [0, 1). any interpolative berinde weak operator is an alternate interpolative berinde weak operator. theorem 1.9 ([1]). in a complete metric space (x, d) an interpolative berinde weak operator t always possesses a fixed point. as a generalization of ’banach contraction principle’, ćirić [3] had introduced a new contractive mapping known as ćirić-quasi contraction mapping and proved a fixed point theorem for such mappings. theorem 1.10 ([3]). let (x, d) be a complete metric space and t : x → x be a self mapping. if t satisfies the contractive condition d(t x, t y) ≤ k max{d(x, y), d(x, t x), d(y, t y), d(x, t y) + d(y, t x) 2 } for all x, y ∈ x, then t has a unique fixed point in x. in the next section we find some new forms of interpolative contractive mappings and show that these interpolative contractive mappings are nothing but ćirić-quasi contraction mappings. 2. main results let (x, d) be a metric space, ∆ik be the set of all interpolative kannan type contractions on x and ∆sk = {t : x → x : d(t x, t y) ≤ λ √ d(x, t x)d(y, t y) for all x, y ∈ x \ fix(t ), where λ ∈ [0, 1)}. theorem 2.1. in a metric space (x, d), ∆ik = ∆sk. proof. clearly ∆sk ⊂ ∆ik. now let t ∈ ∆ik be chosen as arbitrary. then there exists λ ∈ [0, 1) and α ∈ (0, 1) such that d(t x, t y) ≤ λd(x, t x)αd(y, t y)1−α for all x, y ∈ x \ fix(t ). © agt, upv, 2021 appl. gen. topol. 22, no. 1 111 k. roy and s. panja now for any x, y ∈ x \ fix(t ) we have d(t x, t y) ≤ λd(x, t x)αd(y, t y)1−α(2.1) and also due to symmetry d(t x, t y) = d(t y, t x) ≤ λd(y, t y)αd(x, t x)1−α.(2.2) multiplying the inequalities (2.1) and (2.2) it follows that d(t x, t y) ≤ λ √ d(x, t x)d(y, t y), which proves that t ∈ ∆sk and hence ∆ik = ∆sk. � in a metric space (x, d), let ∆ir be the set of all interpolative reichrus-ćirić type contractions on x and ∆sr = {t : x → x : d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)} 1−α 2 for all x, y ∈ x \ fix(t ), where λ ∈ [0, 1), α ∈ (0, 1)}. theorem 2.2. in a metric space (x, d), ∆ir = ∆sr. proof. it is clearly seen that ∆sr ⊂ ∆ir. now let t ∈ ∆ir be chosen arbitrarily. then there exists λ ∈ [0, 1) and α, β ∈ (0, 1) such that d(t x, t y) ≤ λd(x, y)αd(x, t x)βd(y, t y)1−α−β for all x, y ∈ x \ fix(t ). now for any x, y ∈ x \ fix(t ) we have d(t x, t y) ≤ λd(x, y)αd(x, t x)βd(y, t y)1−α−β(2.3) and also due to symmetry we get d(t x, t y) = d(t y, t x) ≤ λd(y, x)αd(y, t y)βd(x, t x)1−α−β.(2.4) multiplying the inequalities (2.3) and (2.4) it follows that d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)} 1−α 2 , which proves that t ∈ ∆sr and hence ∆ir = ∆sr. � remark 2.3. from the theorem 2.2 we observe that, β has no importance to define interpolative reich-rus-ćirić type contraction mappings. let us take ∆ih as the set of all interpolative hardy-rogers type contractions and ∆sh = {t : x → x : d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)}ξ ( 1 2 [d(x, t y) + d(y, t x)] ) 1−α−2ξ for all x, y ∈ x \ fix(t ), where λ ∈ [0, 1), α, ξ ∈ (0, 1) such that α + 2ξ < 1}. theorem 2.4. in a metric space (x, d), ∆ih = ∆sh. proof. ∆sh ⊂ ∆ih trivially. now let t ∈ ∆ih be taken as arbitrary. then there exists λ ∈ [0, 1) and α, β, γ ∈ (0, 1) with α + β + γ < 1 such that d(t x, t y) ≤ λd(x, y)αd(x, t x)βd(y, t y)γ ( 1 2 [d(x, t y) + d(y, t x)] )1−α−β−γ © agt, upv, 2021 appl. gen. topol. 22, no. 1 112 from interpolative contractive mappings... for all x, y ∈ x \ fix(t ). now for any x, y ∈ x \ fix(t ) we have d(t x, t y) ≤ λd(x, y)αd(x, t x)βd(y, t y)γ ( 1 2 [d(x, t y) + d(y, t x)] )1−α−β−γ (2.5) and also due to the symmetry of d we get d(t x, t y) = d(t y, t x) ≤ λd(y, x)αd(y, t y)βd(x, t x)γ ( 1 2 [d(y, t x) + d(x, t y)] )1−α−β−γ . (2.6) multiplying the inequalities (2.5) and (2.6) it follows that d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)} β+γ 2 ( 1 2 [d(y, t x) + d(x, t y)] )1−α−β−γ , which proves that t ∈ ∆sh and hence ∆ih = ∆sh. � remark 2.5. from theorem 2.1, 2.2 and 2.4 it is clear that in each of the definitions, t can be expressed by fewer constants used as powers in the r.h.s. now we consider a version of ćirić-quasi contraction mapping and show that interpolative contractive mappings are special cases of such type of mappings. definition 2.6. let (x, d) be a metric space. a non-identity mapping t : x → x is said to be restricted ćirić-quasi contraction mapping if there exists λ ∈ [0, 1) such that d(t x, t y) ≤ λm(x, y) for all x, y ∈ x \ fix(t ),(2.7) where m(x, y) = max{d(x, y), d(x, t x), d(y, t y), 1 2 [d(y, t x) + d(x, t y)]}. theorem 2.7. in a complete metric space (x, d), a restricted ćirić-quasi contraction mapping possesses at least one fixed point in x. proof. the proof is straight forward so we omit the proof. � clearly any ćirić-quasi contraction mapping is also a restricted ćirić-quasi contraction mapping but the converse is not true in general. the following examples proves our assertion. example 2.8. (i) let x = [0, 1] be the metric space endowed with the usual metric and t : x → x be defined by t (x) =      0 if x = 0 1−x 2 if 0 < x < 1 1 if x = 1. then it can be easily checked that t is a restricted ćirić-quasi contraction mapping but it is not a ćirić-quasi contraction mapping, because t has three fixed points 0, 1 3 and 1. © agt, upv, 2021 appl. gen. topol. 22, no. 1 113 k. roy and s. panja (ii) let x = [1, 2] together with the usual metric and t : x → x be defined by t (x) = { x+1 2 if 1 ≤ x < 2 2 if x = 2. then it can be easily checked that t is a restricted ćirić-quasi contraction mapping but it is not a ćirić-quasi contraction mapping, since t has two fixed points 1 and 2. (iii) let x = [−1, 1] be the metric space endowed with the usual metric and t : x → x be defined by t (x) =      1 2 if x = −1 x if − 1 < x < 1 −1 2 if x = 1. then it can be easily checked that t is a restricted ćirić-quasi contraction mapping but it is not a ćirić-quasi contraction mapping, because t has infinitely many fixed points in x. let ∆iw and ∆ic be the collections of all alternate interpolative berinde weak mappings and restricted ćirić-quasi contraction mappings respectively. now we prove the following theorem. theorem 2.9. in a metric space (x, d) if t ∈ ∆ik ∪ ∆ir ∪ ∆ih ∪ ∆iw then t ∈ ∆ic. proof. let t ∈ ∆ik. then there exists λ ∈ [0, 1) such that d(t x, t y) ≤ λ √ d(x, t x)d(y, t y) for all x, y ∈ x \ fix(t ). thus for any x, y ∈ x \ fix(t ) we have d(t x, t y) ≤ λ √ d(x, t x)d(y, t y) ≤ λ √ m(x, y)2 = λm(x, y).(2.8) if t ∈ ∆ir then there exist λ ∈ [0, 1) and α ∈ (0, 1) such that d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)} 1−α 2 for all x, y ∈ x \ fix(t ). thus for any x, y ∈ x \ fix(t ) we have d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)} 1−α 2 ≤ λm(x, y)α{m(x, y)2} 1−α 2 = λm(x, y).(2.9) choose t ∈ ∆ih. then there exist λ ∈ [0, 1) and α, ξ ∈ (0, 1) with α+2ξ < 1 such that d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)}ξ ( 1 2 [d(x, t y) + d(y, t x)] )1−α−2ξ for all x, y ∈ x \ fix(t ). thus for any x, y ∈ x \ fix(t ) we get d(t x, t y) ≤ λd(x, y)α{d(x, t x)d(y, t y)}ξ ( 1 2 [d(x, t y) + d(y, t x)] )1−α−2ξ ≤ λm(x, y)α{m(x, y)2}ξ (m(x, y)) 1−α−2ξ = λm(x, y).(2.10) © agt, upv, 2021 appl. gen. topol. 22, no. 1 114 from interpolative contractive mappings... consider t ∈ ∆iw . then there exists λ ∈ [0, 1) such that d(t x, t y) ≤ λ √ d(x, y)d(x, t x) for all x, y ∈ x \ fix(t ). thus for any x, y ∈ x \ fix(t ) we have d(t x, t y) ≤ λ √ d(x, y)d(x, t x) ≤ λ √ m(x, y)2 = λm(x, y).(2.11) hence from (2.8), (2.9), (2.10) and (2.11) we have in any case t ∈ ∆ic. � theorem 2.2 [7], corollary 1 [8], theorem 4 [9] and theorem 1.2 [1] follow from our next corollary. corollary 2.10. in a complete metric space (x, d) if t ∈ ∆ik ∪∆ir ∪∆ih ∪ ∆iw then t has a fixed point in x. proof. from theorem 2.9 we see that if t ∈ ∆ik ∪ ∆ir ∪ ∆ih ∪ ∆iw then t ∈ ∆ic. also theorem 2.7 says that a mapping t ∈ ∆ic always possesses fixed point in x. hence the corollary. � any mapping t ∈ ∆ic may not be a member of ∆ik ∪ ∆ir ∪ ∆ih ∪ ∆iw . the next example supports our contention. example 2.11. let us consider x = [0, 1] equipped with the usual metric. also let t : x → x be defined by t (x) =      0 if x = 0 x 2 if 0 < x < 1 1 if x = 1. then clearly t is a restricted ćirić-quasi contraction mapping for 1 2 ≤ λ < 1 but not an usual ćirić-quasi contraction mapping. also by taking x = ǫ and y = 1 − δ with 0 < ǫ, δ < 1 and letting ǫ, δ → 0 we see that t /∈ ∆ik ∪ ∆ir ∪ ∆ih ∪ ∆iw . in metric fixed point theory, our main objective is to check whether a mapping t over a complete metric space x into itself possesses a fixed point in x. in order to satisfy the interpolative kannan type contractive condition (1.2) for a mapping t : x → x, we have to know the set fix(t ) and whenever we know the whole set fix(t ) why we bother about, whether the mapping t satisfies the contractive condition (1.2) ? in one word, to check the existence of fixed points for an interpolative contractive mapping t in x, we have to know the set fix(t ) in advance, which is quite absurd. moreover, theorem 2.7 shows that, if any one of the contractive condition (like banach, kannan, chatterjea) holds ”for all x, y ∈ x \ fix(t )” instead of ”for all x, y ∈ x” then we can easily remove the part uniqueness from the like theorems (banach, kannan, chaterjea), but in each case we have to know first the set fix(t ). © agt, upv, 2021 appl. gen. topol. 22, no. 1 115 k. roy and s. panja from this point of view we can conclude that the theorem 1.3 has no real significance. to avoid such a situation we can redefine the contractive condition (1.2) in the way that is given below. remark 2.12. in a metric space (x, d) if we define an interpolative mapping t : x → x satisfying (2.12) d(t x, t y) ≤ λ √ max{d(x, t x), d(x, y)}. max{d(y, t y), d(x, y)} for all x, y ∈ x and for some λ ∈ [0, 1), then it is seen that t can be a nonconstant function even if t has a fixed point in x. clearly the contractive condition (2.12) can also be taken as d(t x, t y) ≤ λ[max{d(x, t x), d(x, y)}]α[max{d(y, t y), d(x, y)}]1−α for all x, y ∈ x, for α ∈ (0, 1) and for some λ ∈ [0, 1). remark 2.13. it is to be noted that if ba(x), mi(x) and ci(x) are the set of all banach contractions, interpolative contractive mappings satisfying condition (2.12) and ćirić quasi contractions on x respectively then ba(x) ⊂ mi(x) ⊂ ci(x). therefore it is clear that in a complete metric space (x, d) an interpolative contractive mapping t satisfying condition (2.12) has a unique fixed point. 3. interpolative strictly contractive mappings over a compact metric space in this section we prove some fixed point theorems for interpolative strictly contractive type mappings in the framework of a metric space which is weaker than compact metric space. first we recall the definitions of t -orbitally compact metric space with respect to a self mapping t and orbital continuity of a self mapping over a metric space. definition 3.1 ([4]). a metric space (x, d) is said to be t -orbitally compact with respect to a mapping t : x → x if for all x ∈ x, every sequence in the orbit of t at x ∈ x given by o(x, t ) = {x, t x, t 2x, ...} has a convergent subsequence in x. definition 3.2 ([5]). let (x, d) be a metric space. a mapping t : (x, d) → (x, d) is said to be orbitally continuous if u ∈ x and such that u = limi→∞ t nix for some x ∈ x, then t u = limi→∞ t t nix. theorem 3.3. let (x, d) be a metric space and t : x → x be a mapping which satisfies d(t x, t y) < θ(x, y) for all x, y /∈ fix(t ),(3.1) where θ(x, y) = max{ √ d(x, t x)d(y, t y), d(x, y)µ{d(x, t x)d(y, t y)} 1−µ 2 , d(x, y)ν{d(x, t x)d(y, t y)}τ ( 1 2 [d(x, t y) + d(y, t x)] )1−ν−2τ , d(x, y)ξ [ (d(x,t x)+1)d(y,t y) 1+d(x,y) ]1−ξ } with µ, ν, τ, ξ ∈ (0, 1) and ν +2τ < 1. if x is compact (or, t -orbitally compact) © agt, upv, 2021 appl. gen. topol. 22, no. 1 116 from interpolative contractive mappings... then t has atleast one fixed point in x, provided that t is orbitally continuous in x. proof. let x0 ∈ x be chosen as arbitrary. let us construct an iterative sequence {xn}, where xn = t nx0 for all n ≥ 1. if xn = xn+1 for some n ∈ n∪{0} then xn will be a fixed point of t. so without loss of generality we assume that xn 6= xn+1 for all n ≥ 0. now from the contractive condition (3.1) we have d(xn, xn+1) = d(t xn−1, t xn) < θ(xn−1, xn) for all n ≥ 1.(3.2) now we have to consider four cases. case-i: if θ(xn−1, xn) = √ d(xn−1, xn)d(xn, xn+1) then we get d(xn, xn+1) < √ d(xn−1, xn)d(xn, xn+1) ⇒ d(xn, xn+1) < d(xn−1, xn).(3.3) case-ii: if θ(xn−1, xn) = d(xn−1, xn) µ{d(xn−1, xn)d(xn, xn+1)} 1−µ 2 then we have d(xn, xn+1) < d(xn−1, xn) µ{d(xn−1, xn)d(xn, xn+1)} 1−µ 2 ⇒ d(xn, xn+1) 1+µ 2 < d(xn−1, xn) 1+µ 2 ⇒ d(xn, xn+1) < d(xn−1, xn).(3.4) case-iii: if θ(xn−1, xn) = d(xn−1, xn) ν{d(xn−1, xn)d(xn, xn+1)} τ × ( 1 2 [d(xn−1, xn+1) + d(xn, xn)] )1−ν−2τ then we obtain that d(xn, xn+1) < d(xn−1, xn) ν{d(xn−1, xn)d(xn, xn+1)} τ ( 1 2 [d(xn−1, xn+1) + d(xn, xn)] )1−ν−2τ ≤ d(xn−1, xn) ν{d(xn−1, xn)d(xn, xn+1)} τ ( 1 2 [d(xn−1, xn) + d(xn, xn+1)] )1−ν−2τ . (3.5) if d(xn−1, xn) ≤ d(xn, xn+1) then from (3.5) it follows that d(xn, xn+1) < d(xn−1, xn) ν{d(xn−1, xn)d(xn, xn+1)} τ ( 1 2 [d(xn−1, xn) + d(xn, xn+1)] )1−ν−2τ ≤ d(xn, xn+1), a contradiction. (3.6) which implies that d(xn, xn+1) < d(xn−1, xn). © agt, upv, 2021 appl. gen. topol. 22, no. 1 117 k. roy and s. panja case-iv: if θ(xn−1, xn) = d(xn−1, xn) ξ [ (d(xn−1,xn)+1)d(xn,xn+1) 1+d(xn−1,xn) ]1−ξ then we have d(xn, xn+1) < d(xn−1, xn) ξ [ (d(xn−1, xn) + 1)d(xn, xn+1) 1 + d(xn−1, xn) ]1−ξ = d(xn−1, xn) ξd(xn, xn+1) 1−ξ ⇒ d(xn, xn+1) ξ < d(xn−1, xn) ξ ⇒ d(xn, xn+1) < d(xn−1, xn).(3.7) thus from equations (3.3), (3.4), (3.5) and (3.7) we see that d(xn, xn+1) < d(xn−1, xn) for all n ∈ n. so {d(xn−1, xn)} is a monotonically decreasing sequence which is bounded below. therefore there exists some l ≥ 0 such that d(xn−1, xn) → l as n → ∞. now since x is compact (or, t -orbitally compact), {xn} has a convergent subsequence {xnk}, which converges to some u ∈ x. due to the orbital continuity of t it follows that {xnk+1} converges to t u and {xnk+2} converges to t 2u respectively. therefore the continuity of the metric d implies that limk→∞ d(xnk , xnk+1) = d(u, t u) and limk→∞ d(xnk+1, xnk+2) = d(t u, t 2u). so d(u, t u) = l = d(t u, t 2u). if l > 0 then u, t u /∈ fix(t ) and therefore d(t u, t 2u) < θ(u, t u) implies that d(t u, t 2u) < d(u, t u), a contradiction. (3.8) hence l = 0 and t u = u that is u is a fixed point of t. � from the above theorem we get the following immediate corollaries. corollary 3.4. let (x, d) be a metric space and t : x → x be a mapping which satisfies d(t x, t y) < d(x, t x)γd(y, t y)1−γ for all x, y /∈ fix(t ), γ ∈ (0, 1).(3.9) if x is compact (or, t -orbitally compact) then t has a fixed point in x, provided that t is orbitally continuous in x. example 3.5. let x = [0, ∞) with the usual metric, m = {n + ( n + 1 n )2 : n ≥ 2} and t : x → x be defined by t (x) = { n if x = n + ( n + 1 n )2 , n ≥ 2 x if x ∈ x \ m. then t satisfies the contractive condition (3.1) in particular the contractive condition (3.9). also x is t -orbitally compact and t is orbitally continuous on x. here we see that t has infinitely many fixed points in x. corollary 3.6. let (x, d) be a metric space and t : x → x be a mapping which satisfies d(t x, t y) < d(x, y)γd(x, t x)δd(y, t y)1−γ−δ for all x, y /∈ fix(t ), γ, δ ∈ (0, 1). (3.10) © agt, upv, 2021 appl. gen. topol. 22, no. 1 118 from interpolative contractive mappings... if x is compact (or, t -orbitally compact) then t has atleast one fixed point in x, provided that t is orbitally continuous in x. corollary 3.7. let (x, d) be a metric space and t : x → x be a mapping which satisfies d(t x, t y) < d(x, y)γd(x, t x)δd(y, t y)ζ ( 1 2 [d(x, t y) + d(y, t x)] )1−γ−δ−ζ (3.11) for all x, y /∈ fix(t ), where γ, δ, ζ ∈ (0, 1) with γ + δ + ζ < 1. if x is compact (or, t -orbitally compact) then t has a fixed point in x, provided that t is orbitally continuous in x. corollary 3.8. let (x, d) be a metric space and t : x → x be a mapping which satisfies d(t x, t y) < d(x, y)ξ [ (d(x, t x) + 1)d(y, t y) 1 + d(x, y) ]1−ξ for all x, y /∈ fix(t ), ξ ∈ (0, 1). (3.12) if x is compact (or, t -orbitally compact) then t has atleast one fixed point in x, provided that t is orbitally continuous in x. 4. a remark on interpolative kannan contractivity conditions in [6] the authors have defined (λ, α, β)-interpolative kannan contraction and prove a fixed point theorem for such mappings. the definition of the mapping is given as follows: definition 4.1 ([6]). let (x, d) a metric space and t : x → x be a self map. t is called a (λ, α, β)-interpolative kannan contraction, if there exist λ ∈ [0, 1) and α, β ∈ (0, 1) with α + β < 1 such that d(t x, t y) ≤ λd(x, t x)αd(y, t y)β for all x, y ∈ x \ fix(t ).(4.1) theorem 4.2 ([6]). let (x, d) a complete metric space and t : x → x be a (λ, α, β)-interpolative kannan contraction with λ ∈ [0, 1) and α, β ∈ (0, 1), α + β < 1. then t has a fixed point in x. theorem 4.2 is not true in general. the next example proves our assertion. example 4.3. let x = {1 3 , 1 2 } with usual metric and t : x → x be given by t (x) = { 1 2 if x = 1 3 1 3 if x = 1 2 . then t is a (λ, α, β)-interpolative kannan contraction with λ = 3 5 and α = β = 1 3 . here x is complete but t has no fixed point in x. © agt, upv, 2021 appl. gen. topol. 22, no. 1 119 k. roy and s. panja comment/s: (the reason/s why the proof of theorem 2 in [6] fails) in the proof of theorem 2 (see the line number 5 of theorem 2 in page 2 of [6]) the authors used the fact that d(xn, xn+1) 1−β ≤ λd(xn−1, xn) α ≤ λd(xn−1, xn) 1−β whenever α < 1 − β, (4.2) which is actually not true in case 0 < d(xn−1, xn) < 1. therefore the contractive condition d(t x, t y) ≤ λd(x, t x)αd(y, t y)1−α can not be replaced by the contractive condition (4.1). e. karapinar have pointed out a similar idea in example 2 of [10], where he forewarned about the mappings t : {x0, y0} → {x0, y0} defined by t x0 = y0 and t y0 = x0. these particular type of mappings defined on two point sets satisfy the contractive condition (3) (see [10]) but are fixed-points free. acknowledgements. first and second authors acknowledge financial support awarded by the council of scientific and industrial research, new delhi, india, through research fellowship for carrying out research work leading to the preparation of this manuscript. references [1] c. b. ampadu, some fixed point theory results for the interpolative berinde weak operator, earthline journal of mathematical sciences 4 no. 2 (2020), 253–271. [2] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, fund. math. 3 (1922), 133–181. [3] l. b. ćirić, a generalization of banach’s contraction principle, proc. amer. math. soc. 45, no. 2 (1974), 267–273. [4] h. garai, l. k. dey and t. senapati, on kannan-type contractive mappings, numerical functional analysis and optimization 39, no. 13 (2018), 1466–1476. [5] l. b. ćirić, generalized contractions and fixed-point theorems, publ. inst. math. 12 (1971), 19–26. [6] y. u. gaba and e. karapinar, a new approach to the interpolative contractions, axioms 8, no. 4 (2019), 110. [7] e. karapinar, revisiting the kannan type contractions via interpolation. adv. theory nonlinear anal. appl. 2, no. 2 (2018), 85–87. [8] e. karapinar, r. p. agarwal and h. aydi, interpolative reich-rus-ćirić type contractions on partial metric spaces, mathematics 6, no. 11 (2018), 256. [9] e. karapinar, o. alqahtani and h. aydi, on interpolative hardy-rogers type contractions, symmetry 11, no. 1 (2018), 8. [10] a. f. roldán lópez de hierro, e. karapinar and a. fulga, multiparametric contractions and related hardy-roger type fixed point theorems, mathematics 8, no. 6 (2020), 957. © agt, upv, 2021 appl. gen. topol. 22, no. 1 120 () @ applied general topology c© universidad politécnica de valencia volume 12, no. 1, 2011 pp. 17-25 p-compact, p-bounded and p-complete rigoberto vera mendoza ∗ abstract in this paper the nonstandard theory of uniform topological spaces is applied with two main objectives: (1) to give a nonstandard treatment of bernstein’s concept of p-compactness with additional results, (2) to introduce three new concepts (p,q)-compactness, p-totally boundedness and p-completeness. i prove some facts about them and how these three concepts are related with p-compactness. i also give a partial answer to the open question stated in [3] 2010 msc: primary 46a04, 46a20; secondary 46b25. keywords: monad, p-compact, p-totally bounded and p-complete space 1. introduction it was in 1966 that abraham robinson’s book [4] on nonstandard analysis appeared. his methods were based on the theory of models and in particular on the lowenheim-skolem theorem. he introduced extension ∗x (in a superstructure no standard) of any set x (in a standard superstructure) by looking at nonstandard models of their respectives theories. ”infinitely close” or ”infinitely large” were to be found in the enlargement ∗x . in this way he was able to justify proofs using ”infinitesimals” or ”monads” and that was not possible before his discovery, more over, he showed that these methods were able to produce original solutions to unsolved mathematical questions as well. this approach to the nonstandard analysis is based on the axiomatic set theory called zfc (zermelo-fraenkel with axiom choice), all theorems of conventional mathematics remain valid. we start with a superstructure s , the ∗proyecto 9.4 coordinación de la investigación cient́ıfica. 18 r. vera-mendoza sets a, b, x , etc. are set of individuals in s. the extensions or enlargements ∗a, ∗b, ∗x , etc. are sets in the superstructure ŝ. nonstandard analysis is a technique rather than a subject. aside from theorems that tell us that nonstandard notions are equivalent to corresponding standard notions, all the results we obtain can be proved by standard methods. therefore, the subject can only be claimed to be of importance insofar as it leads to simpler, more accesible expositions, or, more important, to mathematical discoveries. in writing formulas we use conventional symbols such that ∈, =, ∀, ∃, ⇒, ∨, ∧, ∼, () etc. 2. some notions of nonstandard theory the nonstandard enlargements satisfies the following properties: a ⊂ ∗a , ∗(a ∩ b) = ∗a ∩ ∗b , ∗(a ∪ b) = ∗a ∪ ∗b , a ⊂ b ⇒ ∗a ⊂ ∗b , ∗(a × b) = ∗a × ∗b , ∗(a\ b) = ∗a\ ∗b , a = ∗a ⇔ a is a finite set. therefore, ∗n \ n, ∗r \ r are non empty sets whose elements are called ”infinite large numbers”. in general, for any infinite set x, ∗x \ x 6=φ each function f : a → b has an ”extension” ∗f : ∗a → ∗b . for this we observe that f ⊂ a × b , so that, ∗f ⊂ ∗a × ∗b given a topological space (x, t) and z ∈ ∗x , the monad of z is µ(z) = ⋂ {∗o | o ∈ t and z ∈ ∗o} the space (x, t) is hausdorff ⇔ ∀ x, y ∈ x , µ(x) ∩ µ(y) = φ we denote by x̂ = ⋃ {µ(x) | x ∈ x} let l denote a language in the standard superstructure and by l̂ we denote de corresponding language in the nonstandard superstructure. a formula of l in which no variable has a free occurrence is called a sentence of l. for each sentence α ∈ l we have correspondent sentence ∗α ∈ l̂ . α = ∗α ⇔ α contains only constants b ∈ s let α be a sentence of l, we will denote |= α if ” α is true in s”; we will denote ∗ |= ∗α if ” ∗α is true in ŝ” transfer principle (tp): the admisible proposition α ∈ (ŝ, £) is true ⇔ ∗α ∈ ( ∗ŝ, ∗£) is true. |= α ⇔ ∗ |= ∗α for instance, a ∩ b 6= φ ⇔ ∗a ∩ ∗b 6= φ , a ⊂ b ⇔ ∗a ⊂ ∗b , etc. the tp provides one of the basic tools of nonstandard analysis. a mathematical theorem that is equivalent to |= α for some sentence α ∈ l can be proved by showing instead that ∗ |= ∗α [1]. a relation r ⊂ a × b is called concurrent if given a1, a2, ..., an ∈ dom(r) there exists b ∈ b such that (ai, b) ∈ r for all i = 1, 2, ..., n p-compact, p-bounded and p-complete 19 concurrence theorem (ct): if r is a concurrent relation then there exists b ∈ ∗b such that ( ∗a, b) ∈ ∗r for all a ∈ a we only will consider hausdorff uniform topological spaces (x, t) , with uniformity u . then, on ∗x we will consider three non-hausdorff topologies: a) τu will be the topology generated by the sets ∗o such that o ∈ t. b) τ̂u will be the uniform topology on ∗x given by the uniformity û = { ∗v | v ∈ u}. c) τi the topology generated by { ∗a | a ⊂ x}. a basis of τu is { ∗v (x) | x ∈ x, v ∈ u}. a basis of τ̂u is { ∗v (z) | z ∈ ∗x, v ∈ u}. for any z ∈ ∗x , µ(z), o(z) and i(z) will denote the monads of z for the topologies a), b) and c) respectively. τu and τ̂u agree on x, thereby µ(x) = o(x) ∀ x ∈ x , in general τu < τ̂u . on the other hand, i(x) = {x} ∀ x ∈ x and τu < τi . so that, i(z) ⊂ µ(z) and o(z) ⊂ µ(z) . the bigger the topology the smaller the monad. if f : (x, t) → (y, t′) is a continuous function then ∗f : ( ∗x, τu ) → ( ∗y, τ ′u ) is a continuous function. any function ∗f : ( ∗x, τi) → ( ∗y, τ ′i ) is continuous. the topology on the set of natural numbers n always be the discrete topology. for any p ∈ ∗n \ n and any topological space x β′px = ⋃ {µ(∗f (p)) | f : n → x}. 3. p-compactness definition 3.1. let p ∈ ∗n \ n be and {zn | n ∈ n} ⊂ ∗x . we will say that z ∈ ∗x is a p-limit of the sequence {zn} ( z = p − lim zn ) if z is an accumulation point of the sequence such that for each o ∈ t , with z ∈ ∗o , p ∈ ∗{n ∈ n | zn ∈ ∗o}. lemma 3.2. if f : x → y is a continuous function and z ∈ ∗x is a p-lim of {zn} ⊂ ∗x then ∗f (z) = p − lim ∗f (zn) in ∗y. proof. let w ⊂ y be an open set such that ∗f (z) ∈ ∗w and let o ⊂ x be open such that z ∈ ∗o and ∗f (∗o) ⊂ ∗w . hence, p ∈ ∗{n ∈ n | zn ∈ ∗o} ⊂ {n ∈ n | ∗f (zn) ∈ ∗w }. � 20 r. vera-mendoza theorem 3.3. z = p − lim xn ⇔ xp ∈ µ(z). proof. ⇒ ) let o ∈ t such that z ∈ ∗o , by definition xp ∈ ∗o. ⇐ ) let o ∈ t be such that z ∈ ∗o . the following sentence is true in the nonstandard structure (∃ n ∈ ∗n )(xn ∈ ∗o) , by the pt (∃ n ∈ n )(xn ∈ o) is true, i.e., a = {n ∈ n | xn ∈ o} 6= ∅ y, p ∈ ∗a. � proposition 3.4. let p, q ∈ ∗n \ n . if there exists f : n → n such that p = q − lim{f (n)} then ∗f (q) ∈ β′pn. proof. p ∈ ∗o ⇒ ∗f (q) ∈ ∗o ⇒ µ(∗f (q)) ⊂ µ(p) . � proposition 3.5. if for {xn} ⊂ x there exists {xnk } → x ∈ x such that p ∈ ∗{nk} , then x = p − lim xn. definition 3.6 ([3]). a topological space (x, t) is p-compact if every sequence in x has a p-limit in ∗x x̂ = ⋃ {µ(x) | x ∈ x}. x ⊂ x̂ ⊂ β′px taking for each x ∈ x f : n → x, f (n) = x theorem 3.7. (x, t) is p-compact ⇔ x̂ = β′px. proof. ⇒ ) the contention left to proof is x̂ ⊃ β′px let {xn} ⊂ x be such that q ∈ µ(xp) . since xp ∈ µ(x) for some x ∈ x , this implies that q ∈ µ(xp) ⊂ µ(x) ⇐ ) nothing left to prove. � corollary 3.8. (x, t) compact ⇒ p-compact for all p ∈ ∗n \ n. proof. x̂ ⊂ β′px ⊂ ∗x = x̂. � definition 3.9. let tp be the topology generated by the set {o ∈ t | ∗f (p)6∈ ∗o for some f : n → x} tp is closed under intersections. lemma 3.10. x tp-compact implies p-compact. proposition 3.11. (x, t) p-compact implies for each numerable tp-cover of x has a finite sub-cover. corollary 3.12. if x is p-compact and tp-lindelof then it is tp-compact. definition 3.13 (comfort order). p ≤c q (in ∗n \ n )if (x, t) q-compact implies p-compact [2]. corollary 3.14. if β′px ⊂ β ′ qx then x q-compact implies p-compact. p-compact, p-bounded and p-complete 21 4. (p, q)-compactness definition 4.1. p ⊗ q = {a ⊂ n × n | (p, q) ∈ ∗a}. definition 4.2. a topological space (x, t) is (p, q)-compact if for each f : n × n → x there exists x ∈ x such that for every x ∈ o ∈ t , ∗ f (p, q) ∈ µ(x) y (p, q) ∈ ∗{(m, n) | f (m, n) ∈ o}. theorem 4.3. (x, t) is (p, q)-compact if and only if it is p and q compact. proof. ⇒ ) let us prove that x is p-compact. let f : n → x and π1 : n × n → n be the projection in the first coordinate, then f ◦ π1 : n × n → x : by hypothesis there exists x ∈ x such that ∗f (p) = (∗f ◦ ∗π1)(p, q) = ∗(f ◦ π1)(p, q) ∈ µ(x) and for each x ∈ o ∈ t (p, q) ∈ ∗{(m, n) | f (m) = (f ◦ π1)(m, n) ∈ o} i.e., {(m, n) | f (m) = (f ◦ π1)(m, n) ∈ o} ∈ p ⊗ q . this implies that p = ∗π1(p, q) ∈ ∗π∗1 ({(m, n) | f (m) = (f ◦ π1)(m, n) ∈ o}) = =∗ {m ∈ n | f (m) ∈ o}. analogous proof for (f ◦ π2) : n × n → x. since x is q-compact, there exists y ∈ x such that ∗f (q) = ∗(f ◦ π2)(p, q) ∈ µ(y) and for each y ∈ u ∈ t , (p, q) ∈ ∗{(m, n) | (f ◦ π2)(m, n) ∈ u} , i.e., {(m, n) | f (n) = (f ◦ π2)(m, n) ∈ u} ∈ p ⊗ q . this implies that q = ∗π2(p, q) ∈ ∗π2( ∗{(m, n) | f (n) = (f ◦ π2)(m, n) ∈ u}) ∈ ∗{n ∈ n | f (n) ∈ u}. ⇐ ) let us consider f : n × n → x . to prove that there exists x ∈ x such that ∗f (p, q) ∈ µ(x) . for each n ∈ n we define fn : n → x as fn(m) = f (m, n) . by hypothesis, there exists xn ∈ x such that ∗f (p, n) = ∗fn(p) ∈ µ(xn) and for each xn ∈ o ∈ t , p ∈ ∗{m ∈ n | fn(m) ∈ o}. let us define gp : n → x as gp(n) = xn ∼ ∗f (p, n) . hence ([1, page 81]) gp ∼ ∗f (p, −) ∈ ∗( ∏ n x) . the set {∗w | w is an open basic set of ∏ n x} is a basis of the topology on ∗( ∏ n x) . by the q-compacity of x , there exists y ∈ x such that ∗gp(q) ∈ µ(y) and for each y ∈ u ∈ t , q ∈ ∗a = ∗{n ∈ n | gp(n) ∈ u}. since gp(n) ∼ ∗f (p, n) , ∗f (p, n) ∈ ∗u ∀ n ∈ a , i.e., the following formula is true (∀ n ∈ n )(n ∈ a ⇒ ∗f (p, n) ∈ ∗u ) . by the pt, the following formula is true (∀ n ∈ ∗n )(n ∈ ∗a ⇒ ∗f (p, n) ∈ ∗u ) . q ∈ ∗a ⇒ ∗f (p, q) ∈ ∗u ⇒ ∗f (p, q) ∈ ⋂ { ∗u | y ∈ u ∈ t} = µ(y). � 22 r. vera-mendoza 5. p-totally bounded and p-complete definition 5.1. all uniform space (x, t) is completely regular, let us denote by {ρj}j the saturated family of pseudometrics that define the topology t given by the uniformity u . we recall that for each z ∈ ∗x o(z) = {y ∈ ∗x | ρj(y, z) ≈ 0 ∀ j ∈ j} [1, 5] µ(z) = ⋂ { ∗v (x) | v ∈ u, x ∈ x, z ∈ ∗v (x)} every v ∈ u is v = {(x, y) ∈ x × x | ρj (x, y) < ǫ} for some ǫ > 0 and some pseudometric ρj , also denoted by v = v ǫ j . for all z ∈ ∗x , µ(z) ⊃ o(z) and for all x ∈ x , µ(x) = o(x) for (x, t) we define the set pns ∗x = {z ∈ ∗x | µ(z) = o(z)}. proposition 5.2. z ∈ pns ∗x ⇒ µ(z) ⊂ pns ∗x. proof. if z ∈ ∗x and w ∈ µ(z) then o(w) ⊂ µ(w) ⊂ µ(z) and o(w) = o(z) if z ∈ pns ∗x then µ(z) = o(z) = o(w) ⊂ µ(w) ⊂ µ(z) , i.e., o(w) = µ(w) thereby w ∈ pns ∗x. � proposition 5.3. z ∈ pns ∗x ⇔ given ǫ > 0 and any pseudometric ρj there exists xj ∈ x such that ∗ρj (z, xj ) < ǫ. proof. ⇒) let z ∈ pns ∗x be given, then, o(z) = µ(z) . for any ǫ > 0 , a pseudometric ρj and v = v ǫ j ∈ u . since o(z) × o(z) ⊂ v , µ(z) × µ(z) ⊂ v , pc tells us that there exists d ∈ ∗fµ(z) such that d ⊂ µ(z) , there exists d ∈ ∗fµ(z) such that d × d ⊂ ∗v , hence, by pt, there exists d ∈ fµ(z) such that d × d ⊂ v . d ∈ fµ(z) ⇒ µ(z) ⊂ ∗d ⇒ {z} × ∗d ⊂ ∗v ⇒ (z, x) ∈ ∗v for all x ∈ d ⊂ x , i.e., ρ(z, x) < ǫ for all x ∈ d ⇐) let w ∈ µ(z) , ǫ > 0 be given and (z, xj ) ∈ ∗v ǫ 2 j = {(x, y) | ρj (x, y) < ǫ 2 } . since w ∈ ∗v ǫ 2 j (xj ) and ∗ρj (w, z) ≤ ∗ρj (w, xj ) + ∗ρj (xj , z) < ǫ 2 + ǫ 2 = ǫ . this implies that w ∈ ∗v ǫj (z) and therefore w ∈ o(z) , i.e., µ(z) ⊂ o(z) ⊂ µ(z). � corollary 5.4. z ∈ pns ∗x ⇔ fµ(z) is a cauchy filter remark 5.5. x̂ ⊂ pns ∗x ⊂ ∗x x̂ = pns ∗x ⇔ x is complete pns ∗x = ∗x ⇔ x is totally bounded definition 5.6. (x, t) is p-totally bounded if for each function f : n → x , ∗f (p) ∈ pns ∗x p-compact, p-bounded and p-complete 23 lemma 5.7. p-compact ⇒ p-totally bounded. proof. x̂ ⊂ pns ∗x ⊂ ∗x. � lemma 5.8. x is p-totally bounded ⇔ β′px ⊂ pns ∗x. proof. ⇒ ) x p-totally bounded ⇒ ∗f (p) ∈ pns ∗x ⇒ µ( ∗f (p)) ⊂ pns ∗x. � corollary 5.9. x is p-totally bounded for all p ∗n \ n ⇔ ∪pβ ′ px ⊂ pns ∗x. lemma 5.10. (x, t) totally bounded ⇒ p-totally bounded for all p ∗n \ n. proof. ⇒) totally bounded implies pns ∗x = ∗x. � lemma 5.11. if (x, t) is a complete space then x p-compact ⇔ p-totally bounded. proof. complete implies x̂ = pns ∗x. � ∗f : ∗n → ∗x always is a τu -continuous function. proposition 5.12. x is p-totally bounded ⇔ every f : n → x is a continuous function in p with the τ̂u topology. proof. ⇒ ) since ∗f (p) ∈ pns ∗x , µ(∗f (p)) = o(∗f (p)) , since ∗f is a continuous function with τu , ∗f (o(p)) = ∗f (µ(p)) ⊂ µ(∗f (p)) = o(∗f (p)) then it also is a continuous function in p with τ̂u (cauchy’s principle, theorem 8.1.4, [5]) ⇐ ) let f : n → x be a continuous function in p ∈ n with τu . to prove that ∗f (p) ∈ pns ∗x , that is, µ(∗f (p)) = o(∗f (p)) , for this only left to prove that µ(∗f (p)) ⊂ o(∗f (p)) we recall that fp = {a ⊂ n | p ∈ ∗a} ⊂ p (n ) is a cauchy filter and so is the filter g generated by the image of f (fp) . claim: µ(g) = µ(∗f (p)). since ∗f (p) ∈ µ(g) , µ(g) ⊂ µ(∗f (p)). on the other hand, ∗f (p) ∈ ∗s ⇒ f −1(s) ∈ fp thereby s ⊃ f (f −1(s)) ∈ g therefore s ∈ g. let v ∈ u and a ∈ fp be given such that ∗f (∗a) ⊂ ∗v (p) (τ̂u -continuity of ∗f in p). this tells us that µ(g) ⊂ o( ∗f (p)) , therefore µ(∗f (p)) ⊂ o(∗f (p)). � corollary 5.13. ∗f (p) ∈ pns ∗x ⇔ ∗f is τ̂u -continuous in p. proposition 5.14. pns ∗x is the biggest subset of ∗x containing x such that τu and τ̂u agree. theorem 5.15. (x, t) is totally bounded ⇔ τu = τ̂(∗)u . 24 r. vera-mendoza corollary 5.16. if ∗x = ∪pβ ′ px and x is p-totally bounded for all p ∗n \ n , then x is totally bounded. definition 5.17. (x, t) is p-complete if any function f : n → x , ∗f (p) ∈ pns ∗x ⇒ ∗f (p) ∈ x̂. proposition 5.18. p-compact ⇔ p-complete and p-totally bounded. definition 5.19. p ≤ta q if q-totally bounded ⇒ p-totally bounded. p ≤cc q if q-complete ⇒ p-complete. questions 2.20 and 2.21 in [3] appeared q 2.20 and q 2.21 (open questions) whose nonstandard versions are: q 2.20: is there p ∈∗ n \ n such that tc (p) ∩ wp = ∅ ? tc (p) = {q ∈ ∗n | q ≤c p and p ≤c q} q 2.21: is β′n ∩ wp 6= for all p ∈∗ n \ n ? or is there p ∈∗ n \ n such that β′n ∩ wp = ∅ ? i will prove that if q 2.21 is true then q 2.20 is true. definition 5.20. for q ∈∗ n \ n , nq = {p ∈ ∗ n | q 6∈ β′pn}. remark 5.21. 1.np ⊂ nq ⇒ p ∈ β ′ qn. 2.for all f : n → n , ∗f (nq) ⊂ nq. definition 5.22. we will say that nq is p-compact if each function f : n → nq →֒ ∗n , its extension f̂ : ∗n → ∗n satisfies f̂ (p) ∈ nq. remark 5.23. nq p-compact ⇒ p ∈ nq. definition 5.24. p ∈∗ n \ n is a weak p-point if p 6∈c ⊂∗ n \ n ⇒ p 6∈ c. we denote the set of weak p-points by wp. theorem 5.25. q ∈ wp ⇒ nq p-compact for all p ∈ nq. theorem 5.26. p ≤c q and p ∈ wp ⇒ np ⊂ nq ⇒ p ∈ β ′ qn. proof. if there is some z ∈ np\ nq then both np is z-compact and q ∈ β ′ z n . hence, there is f : n → n such that q ∈ ∗a ⇔ ∗f (z) ∈ ∗a for all a ⊂ n which implies that ĝ(q) ∈ np since ĝ( ∗f (z)) ∈ np for all g : n → np therefore np is q-compact and, because p ≤c q , np is p-compact then (theorem 5.25) p ∈ np . this contradiction tells us that there is not z ∈ np\ nq. � p-compact, p-bounded and p-complete 25 now we can restate q 2.21: is there p ∈∗ n \ n such that nq is p-compact for all q ∈ wp ? or⋂ q∈wp nq 6= ∅ ? if this last inequality is true, that is, if there is p ∈ ⋂ q∈wp nq then q 6≤c p ∀ q ∈ wp thereby tc (p) ∩ wp = ∅ (q 2.21) acknowledgements. i want to thank to dr. fernando hernández for his valuated suggestions. references [1] m. davis, applied nonstandard analysis, john wiley ny, (1977). [2] s. garcia-ferreira, comfort types of ultrafilters, proc. amer. math. soc. 120, no. 4 (1994), 1251–1260. [3] s. garcia-ferreira, three orderings on β(ω) \ ω, topology appl. 50 (1993), 199–216. [4] a. robinson, non-standard analysis, princeton landmarks in math, (1996). [5] k. d. stroyan and w. a. j. luxemburg, introduction to the theory of infinitesimals, academic press, (1976). (received september 2009 – accepted may 2010) r. vera-mendoza (rigovera@gmail.com) facultad de ciencias f́ısico-matemáticas, universidad michoacana de san nicolás de hidalgo, morelia, michoacán, méxico. p-compact, p-bounded and p-complete. by r. vera-mendoza @ appl. gen. topol. 22, no. 2 (2021), 385-397doi:10.4995/agt.2021.15096 © agt, upv, 2021 fixed point property of amenable planar vortexes james f. peters a and tane vergili b a computational intelligence laboratory, university of manitoba, wpg, mb, r3t 5v6, canada and department of mathematics, faculty of arts and sciences, adiyaman university, 02040 adiyaman, turkey. (james.peters3@umanitoba.ca) b karadeniz technical university, department of mathematics, trabzon, turkey. (tane.vergili@ktu.edu.tr) communicated by i. altun abstract this article, dedicated to mahlon m. day, introduces free group presentations of planar vortexes in a cw space that are a natural outcome of results for amenable groups and fixed points found by m.m. day during the 1960s and a fundamental result for fixed points given by l.e.j. brouwer. 2010 msc: 37c25; 55m20; 54e05; 55u10. keywords: amenable group; cw space; fixed point; planar vortex; presentation. 1. introduction this article introduces consequences of results for amenable groups and fixed points found by m.m. day during the 1960s in terms of free group presentations [17] of planar vortexes in a cw (closure-finite weak) space. results given here spring from a fundamental result for fixed points given by l.e.j. brouwer [3]. theorem 1.1 (brouwer fixed point theorem [18, §4.7, p. 194]). every continuous map from rn to itself has a fixed point. received 11 february 2021 – accepted 20 may 2021 http://dx.doi.org/10.4995/agt.2021.15096 j. f. peters and t. vergili briefly, let σ be a finite group and let m(σ) be a set of bounded, real-valued functions on σ. then σ is amenable, provided there is a mean µ on m(σ) which is both left and right invariant. theorem 1.2 (day abelian group [semigroup] theorem ([6, 7])). every finite group is amenable. the study of amenable groups led to the following extension of the kakutanimarkov theorem by day. theorem 1.3 (day fixed point theorem ([7])). let k be a compact convex subset of a locally convex linear topological space x, and let σ be a semigroup (under functional composition) of continuous affine transformations of k into itself. if σ, when regarded as an abstract semigroup, is amenable, or if it has a left-invariant mean, then there is in k a common fixed point of the family σ. a direct consequence of theorem 1.3 is that each amenable group of a planar vortex has a fixed point in a cw space. definition 1.4. a planar vortex vore is a finite cell complex, which is a collection of path-connected vertices in nested, filled 1-cycles in a cw complex k. a 1-cycle in vore (denoted by cyca) is a sequence of edges with no end vertex and with a nonempty interior. a geometric realization of vore is denoted by |vore| on |k| in the euclidean plane. a nonvoid collection of cell complexes k is a closure finite weak cw space, provided k is hausdorff (every pair of distinct cells is contained in disjoint neighbourhoods [11, §5.1, p. 94]) and the collection of cell complexes in k satisfy the whitehead [19, pp. 315-317], [20, §5, p. 223] cw conditions, namely, the closure of each cell complex is in k and the nonempty intersection of cell complexes is in k. a number of important results concerning fixed points in this paper spring from čech proximities, leading to descriptive proximally continuous maps. a descriptive proximally continuous map is defined over descriptive čech proximity spaces [5, §4.1] in which the description of a nonempty set is in the form of a feature vector derived from probe functions, one for each feature of the set. for the details, see app. c. 2. conjugacy between proximal descriptively continuous maps this section introduces proximal conjugacy between two dynamical systems, which is an easy extension of topological conjugacy [1, §8.1,p. 243]. proximal conjugacy is akin to strongly amenable groups in which each of its proximal topological actions has a fixed point [8]. let ∑ denote either a semigroup or a group. also, let lub, glb denote least upper bound and greatest lower bound, respectively, and let m( ∑ ) be the set of bounded, real-valued functions θ on ∑ for which ‖θ‖ = lubx∈ ∑ |θ(x)| . © agt, upv, 2021 appl. gen. topol. 22, no. 2 386 fixed point property of amenable planar vortexes a mean µ on m( ∑ ) is an element of the m( ∑ )∗ (in the conjugate space b∗ of a banach space b [6, p.510]) such that, for each x ∈ m( ∑ ), we have glbx∈ ∑θ(x) ≤ µ(x) ≤ lubx∈ ∑θ(x). an element of µ of m( ∑ )∗ is left[right] invariant, provided µ(ℓσx) = µ(x) [µ(rσx) = µ(x)], for all x ∈ m( ∑ ), σ ∈ ∑ , where (ℓσx)σ ′ = x(σσ′) and (rσx)σ ′ = x(σ′σ) for all σ′ ∈ σ. definition 2.1 ([6]). a semigroup (also group) ∑ is amenable, provided there is a mean µ on m( ∑ ), which is both left and right invariant. theorem 2.2. a free group presentation of a planar vortex in a cw space is amenable. proof. this result is a direct consequence of theorem 1.2 from day [7] and (i) [6, p.516], since, by construction, every free group g presentation of a planar vortex in a cw space is finite and, by definition, an abelian semigroup. � theorem 2.2 stems from day’s extension of theorem 2.3 (restated by day [7, p. 585]) to cover the case when the family in question is a semigroup. theorem 2.3 (kakutani-markov theorem [9, 10]). let k be a compact convex set in a locally convex linear topological space, and let f be a commuting family of continuous, affine transformations, f, of k into itself. then there is a common fixed point of the functions in f; that is, there is an x in k such that f(x) = x for every f in f. if a planar vortex vore has no hole inside, then we no longer need to require it be convex. it is automatically convex and we have the following. theorem 2.4. for a cw complex k, let (k, δ) be a proximity space that contains a planar vortex vore without a planar hole and let f : vore → vore be proximal continuous. then vore has a fixed point of f. proof. since vore is finite, the topology on the geometric realization |vore| of vore can be regarded as the subspace topology inherited from the euclidean space r2. then if we have the geometric realization of f, denoted |f|, we see that |f| is a continuous affine transformation. this is true since, f maps two near subsets to the two near subsets. also notice that |vore| is convex compact subset of r2 and the collection of maps {|fn| : n = 1, 2 . . .} is amenable, since it is a semigroup under composition. then by theorem 2.3, for the family of continuous affine transformations {|fn| : n = 1, 2, . . .}, there is an x in |vore| such that f(x) = x and so |vore| has a fixed point of |f|. this also allows us to conclude that vore has a fixed point of f without considering the geometric realization. � if a planar vortex vore has a (planar) hole inside, then we could consider a subset vore such that its geometric realization is convex compact. © agt, upv, 2021 appl. gen. topol. 22, no. 2 387 j. f. peters and t. vergili theorem 2.5. for a cw complex k, let (k, δ) be a proximity space that contains a planar vortex vore with a planar hole and let x ⊂ vore such that its geometric realization |x| is convex compact. if f : x → x is proximal continuous, then x has a fixed point under f. proof. by the construction of a planar vortex, there is subset x of vore such that its geometric realization |x| is a convex compact subset of |vore|. then a proximal continuous map f : x → x has a corresponding continuous affine transformation |f| : |x| → |x|. again by theorem 2.3, for the family of continuous affine transformations {|fn| : n = 1, 2, . . . }, there is an x in |x| such that f(x) = x and so |x| has a fixed point of |f|. hence, vore has a fixed point of f without considering the geometric realization. � remark 2.6. from what have observed, notice that any vortex has a locally compact abelian group presentation, since it is a locally compact hausdorff space and its underlying group structure is abelian. in that case, any proximal continuous map from a vortex to itself can be also considered as a group action. hence, by a direct consequence of theorem 2.2 and a result of a generalization of the kakutani-markov theorem 2.3, each amenable vortex has a fixed point. corollary 2.7. if f is a proximal continuous map from a vortex to itself, then f has a fixed point. next, we introduce the (descriptive) proximal conjugate between two proximal (descriptive) continuous maps. note that a (descriptive) c̆ech proximity space x together with a (descriptive) proximal continuous self map on x can be considered as a (descriptive) proximal dynamical system. now we introduce a (descriptive) proximal conjugacy between two (descriptive) dynamical systems, so that the existence of it guarantees the (descriptive) dynamical systems having equivalent flows and related (descriptive) fixed points. definition 2.8. two proximal continuous maps f : (x, δ1) → (x, δ1) and g : (y, δ2) → (y, δ2) are said to be proximal conjugates, provided there exists a proximal isomorphism h : (x, δ1) → (y, δ2) such that g ◦ h = h ◦ f. the function h is called a proximal conjugacy between f and g. the following theorem states that if two proximal continuous maps are proximal conjugate, then their corresponding iterated functions are also proximal conjugate. theorem 2.9. let h be a proximal conjugacy between f : (x, δ1) → (x, δ1) and g : (y, δ2) → (y, δ2). then for each a ⊆ x and n ∈ z+, we have h(f n(a)) = gn(h(a)). proof. the proof follows from the induction on n. � definition 2.10. let (x, δφ) be a descriptive proximity space with a probe function φ : x → rn and a, b ∈ 2x. then a and b are said to be descriptively equal, provided φ(a) = φ(b). in that case, we write a = des b. © agt, upv, 2021 appl. gen. topol. 22, no. 2 388 fixed point property of amenable planar vortexes definition 2.11. let φ(e) ∈ rn be a vector of n real-values that describe a nonempty set e. two proximal descriptive continuous maps f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2) are said to be proximal descriptive conjugates, provided there exists a proximal descriptive isomorphism h : (x, δφ1) → (y, δφ2) such that g ◦ h(a) = des h ◦ f(a) for any a ∈ 2x. the function h is called a proximal descriptive conjugacy between f and g. remark 2.12. we see from the definition of a proximal descriptive conjugacy that g ◦ h(a) and h ◦ f(a) may not be equal but we have φ2(g ◦ h(a)) = φ2(h ◦ f(a)) for a ∈ 2x. moreover g ◦ h(a) = des h◦ f(a) implies g(a) = des h◦ f ◦ h−1(a) and f(a) = des h−1 ◦ g ◦ h(a), so that we have the following commutative diagrams. φ1(f(a)) = φ1(h −1gh(a)) a f(a) h−1gh(a) h(a) gh(a) f h φ1 φ1 g h −1 h−1(c) fh−1(c) c g(c) hfh−1(c) φ2(g(c)) = φ2(hfh −1(c)) f hh −1 g φ2 φ2 remark 2.13. for proximal descriptive conjugates f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2), def. 2.11 tells us that for a ⊆ x and c ⊆ y , we have φ2(g ◦ h(a)) = φ2(h ◦ f(a)), φ1(f ◦ h −1(c)) = φ1(h −1 ◦ g(c)). note that if h is a proximal descriptive conjugacy between f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2), then a = des b implies h(a) = des h(b) for a, b ∈ 2x. theorem 2.14. let h be a proximal descriptive conjugacy between f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2). then for each a ∈ 2 x and n ∈ z+, we have h(fn(a)) = des gn(h(a)). proof. the proof follows from the induction on n. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 389 j. f. peters and t. vergili definition 2.15. let (x, δφ) be a descriptive proximity space with a probe function φ : x → rn, a ∈ 2x, and f : (x, δφ) → (x, δφ) a descriptive proximally continuous map. (i) a is a descriptive fixed subset of f, provided φ(f(a)) = φ(a). (ii) a is an eventual descriptive fixed subset of f, provided a is not a descriptive fixed subset while ft(a) is a descriptive fixed subset for some t 6= 1. (iii) a is an almost descriptive fixed subset of f, provided φ(f(a)) = φ(a) or φ(f(a)) δφ φ(a) so that f(a) ∩ φ a 6= ∅ by lemma c.1. (for the analogy of a point being almost fixed in digital topology, we refer to [2, 16].) corollary 2.16. let h be a proximal descriptive conjugacy between f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2). a) if a is a descriptively fixed subset of f, then h(a) is a descriptively fixed subset of g. b) if a is an eventual descriptively fixed subset of f, then h(a) is an eventual descriptively fixed subset of g. c) if a is an almost descriptively fixed subset of f, then h(a) is an almost descriptively fixed subset of g. proof. a) let a be a descriptively fixed subset of f. that is, φ1(f(a)) = φ1(a). in other words, we have f(a) = des a. since h is a proximal isomorphism, h preserves desciptive proximity h(f(a)) = des h(a). by theorem 2.14, g(h(a)) = des h(a) so that h(a) is a descriptively fixed subset of g. b) let a be an eventual descriptively fixed subset of f. that is, a is not a descriptively fixed subset of f but φ1(f n(a)) = φ1(a) for some positive integer n > 1. in other words, we have fn(a) = des a. since h is a proximal isomorphism, h preserves being equal in a descriptive sense: h(fn(a)) = des h(a). by theorem 2.14, gn(h(a)) = des h(a). note that h(a) is not a descriptively fixed subset of g since a is not a descriptively fixed subset of f and h is an isomorphism. so, h(a) is an eventual descriptively fixed subset of g. c) let a be an almost descriptively fixed subset of f. that is, f(a) = des a or a δφ1 f(a). if f(a) = des a, then we are done. let a δφ1 f(a). since h is a proximal isomorphism, we have h(a) δφ2 h(f(a)). by theorem 2.14, h(a) δφ2 g(h(a)) so that h(a) is a descriptively fixed subset of g. � further, the existence of proximal conjugacy between two dynamical systems of cell complexes such as vortexes also guarantees isomorphic amenable © agt, upv, 2021 appl. gen. topol. 22, no. 2 390 fixed point property of amenable planar vortexes group structures and hence related fixed points, which is another consequence of theorem 2.2. corollary 2.17. if there exists a descriptive proximal conjugacy between two descriptive dynamical systems, then they have isomorphic descriptive fixed subsets. proof. let f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2) be proximal descriptive conjugates and h : (x, δφ1) → (y, δφ2) be the proximal descriptive conjugacy between them. if a ∈ 2x is a descriptive fixed subset of f, then h(a) is a descriptive fixed subset of g by corollary 2.16 so that a and h(a) are descriptively isomorphic. similarly if b ∈ 2y is a descriptive fixed subset of g, then h−1(b) is a descriptive fixed subset of f so that b and h−1(b) are descriptively isomorphic. hence there is a one-to-one correspondence between the set of the descriptive fixed subsets of f and the set of the descriptive fixed subsets of g. � 3. weak conjugacy between descriptive proximally continuous maps this section introduces weak conjugacy between descriptive proximally continuous maps. definition 3.1. two proximally continuous maps f : (x, δ1) → (x, δ1) and g : (y, δ2) → (y, δ2) are said to be weakly proximal conjugates, provided there exists a proximal isomorphism h : (x, δ1) → (y, δ2) such that for any a ∈ 2 x, g ◦ h(a) δ2 h ◦ f(a). note that this also implies that f ◦ h −1(c) δ1 h −1 ◦ g(c) for any c ∈ 2y . the function h is called a weakly proximal conjugacy between f and g. theorem 3.2. let h be a weakly proximal conjugacy between f : (x, δ1) → (x, δ1) and g : (y, δ2) → (y, δ2). then for each a ∈ 2 x and n ∈ z+, we have h(fn(a)) δ2 g n(h(a)). proof. the proof follows from the induction on n. � definition 3.3. two descriptive proximally continuous maps f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2) are said to be weakly proximal descriptive conjugates, provided there exists a proximal descriptive isomorphism h : (x, δφ1) → (y, δφ2) such that g ◦ h(a) δφ2 h ◦ f(a) for any a ∈ 2 x. note that this also implies f ◦ h−1(c) δφ2 h −1 ◦ g(c) for any c ∈ 2y . the function h is called a weakly proximal descriptive conjugacy between f and g. remark 3.4. for weakly proximal descriptive conjugates f : (x, δφ1) → (x, δφ1) and g : (y, δφ2) → (y, δφ2), def. 3.3 and lemma c.1 tell us that for a ∈ 2 x and c ∈ 2y , we have g ◦ h(a) ∩ φ f ◦ h(a) 6= ∅, f ◦ h−1(c) ∩ φ h −1 ◦ g(c) 6= ∅. © agt, upv, 2021 appl. gen. topol. 22, no. 2 391 j. f. peters and t. vergili appendix a. planar vortexes this section briefly looks at planar vortex structures in planar cw spaces. for simplicity, we consider only 2 cycle vortexes containing a pair of nested 1-cycles that intersect or attached to each other via at least one bridge edge. definition a.1. ([15]) let cyca, cycb be a collection of path-connected vertexes on nested filled 1-cycles (with cycb in the interior of cyca) defined on a finite, bounded, planar region in a cw space k. a planar 2 cycle vortex vore is defined by vore = cl(cycb) is contained (nested) in the interior of cl(cyca). ︷︸︸︷ {cl(cyca) : cl(cycb) ⊂ int(cl(cyca))} . a vortex containing adjacent non-intersecting cycles has a bridge edge attached to vertexes on the cycles. definition a.2. a vortex bridge edge is an edge attached to vertexes on a pair of non-intersecting, filled 1-cycles. remark a.3. from def. a.1, the cycles in a 2 cycle vortex can either have nonempty intersection (see, e.g., cyca′ ∩ cycb′ 6= ∅ in |vore′| in fig. 1b) or there is a bridge edge between the cycles (see, e.g., >pq |vore| in fig. 1a). in effect, every pair of vertexes in a 2 cycle vortex is path-connected. remark a.4. the structure of a 2 cycle vortex extends to a vortex with k > 2 nested filled 1-cycles, provided adjacent pairs of cycles cyca, cyca′ in a k-cycle vortex either intersect or there is a bridge edge attached between cyca, cyca′. (a) vortex |vore| with non-intersecting 1-cycles cyca, cycb (b) vortex |vore| with intersecting 1-cycles cyca ′ , cycb ′ figure 1. sample planar 2-cycle vortexes © agt, upv, 2021 appl. gen. topol. 22, no. 2 392 fixed point property of amenable planar vortexes appendix b. free group presentation of a vortex a finite group g is free, provided every element x ∈ g is a linear combination of its basis elements (called generators). we write b to denote a nonempty set of generators { g1, . . . , g|b| } and g(b, +) to denote the free group with binary operation +. example b.1. the basis {g1, g2, g3} generates a group g whose geometric realization is |vore′| in fig. 1b. the + operation on g corresponds to a move from a generator to a neighbouring vertex. for example, b = traversing 3 cyca′ & 3 cycb′ edges to reach b via g1, g2 ︷︸︸︷ 3g1 + 3g2 b = traversing 7 cycb′ edges to reach b via g1, g2 ︷︸︸︷ 4g1 + 3g2 b = traversing 1 cycb′ edge to reach b via g3 ︷︸︸︷ 0g1 + 1g3. the identity element 0 in g is represented by a zero move from a generator g to another vertex (denoted by 0g) and an inverse in g is represented by a reverse move −g. definition b.2. let 2k be the collection of cell complexes in a cw space k, vortex |vore| ∈ 2k, basis b ∈ |vore|, ki the i th integer coefficient in a linear combination ∑ i,j kigj of generating elements gj ∈ b. a free group g presentation of |vore| is a continuous self-map f : 2k → 2k defined by f(|vore|) =    v := ∑ i,j kigj : v ∈ |vore| , gj ∈ b    = |vore| 7→ free group g ︷︸︸︷ g( { g1, . . . , g|b| } , +). appendix c. descriptive proximity spaces this section briefly introduces descriptive čech proximity spaces, paving the way for descriptive proximally continuous maps. the simplest form of proximity relation (denoted by δ) on a nonempty set was introduced by e. čech [4]. a nonempty set x equipped with the relation δ is a čech proximity space (denoted by (x, δ)), provided the following axioms are satisfied. čech axioms (p.0): all nonempty subsets in x are far from the empty set, i.e., a 6 δ ∅ for all a ⊆ x. (p.1): a δ b ⇒ b δ a. © agt, upv, 2021 appl. gen. topol. 22, no. 2 393 j. f. peters and t. vergili (p.2): a ∩ b 6= ∅ ⇒ a δ b. (p.3): a δ (b ∪ c) ⇒ a δ b or a δ c. given that a nonempty set e has k ≥ 1 features such as fermi energy ef e, cardinality ecard, a description φ(e) of e is a feature vector, i.e., φ(e) = (ef e, ecard). nonempty sets a, b with overlapping descriptions are descriptively proximal (denoted by a δφ b). the descriptive intersection of nonempty subsets in a ∪ b (denoted by a ∩ φ b) is defined by a ∩ φ b = i.e., descriptions φ(a) & φ(b) overlap ︷︸︸︷ {x ∈ a ∪ b : φ(x) ∈ φ(a) ∩ φ(b)} . let 2x denote the collection of all subsets in a nonvoid set x. a nonempty set x equipped with the relation δφ with non-void subsets a, b, c ∈ 2 x is a descriptive proximity space, provided the following descriptive forms of the čech axioms are satisfied. descriptive čech axioms (dp.0): all nonempty subsets in 2x are descriptively far from the empty set, i.e., a 6 δφ ∅ for all a ∈ 2 x. (dp.1): a δφ b ⇒ b δφ a. (dp.2): a ∩ φ b 6= ∅ ⇒ a δφ b. (dp.3): a δφ (b ∪ c) ⇒ a δφ b or a δφ c. the converse of axiom (dp.2) also holds. lemma c.1 ([14]). let x be equipped with the relation δφ, a, b ∈ 2 x. then a δφ b implies a ∩ φ b 6= ∅. proof. let a, b ∈ 2x. by definition, a δφ b implies that there is at least one member x ∈ a and y ∈ b so that φ(x) = φ(y), i.e., x and y have the same description. then x, y ∈ a ∩ φ b. hence, a ∩ φ b 6= ∅, which is the converse of (dp.2). � theorem c.2. let k be a cell complex, v or(k) ⊂ k a collection of planar vortexes equipped with the proximity δφ and let vora, vorb ∈ v or(k). then vora δφ vorb implies vora ∩ φ vorb 6= ∅. proof. immediate from lemma c.1. � let (x, δ1) and (y, δ2) be two čech proximity spaces. then a map f : (x, δ1) → (y, δ2) is proximal continuous, provided a δ1 b implies f(a) δ2 f(b), i.e., f(a) δ2 f(b), provided f(a) ∩ f(b) 6= ∅ for a, b ∈ 2 x [12, §1.4]. in general, a proximal continuous function preserves the nearness of pairs of sets [11, © agt, upv, 2021 appl. gen. topol. 22, no. 2 394 fixed point property of amenable planar vortexes §1.7,p. 16]. further, f is a proximal isomorphism, provided f is proximal continuous with a proximal continuous inverse f−1. let (x, δφ1) and (y, δφ2) be descriptive proximity spaces with probe functions φ1 : x → r n, φ2 : y → r n, and a, b ∈ 2x. then a map f : (x, δφ1) → (y, δφ2) is said to be descriptive proximally continuous, provided a δφ1 b implies f(a) δφ2 f(b), i.e., f(a) δφ2 f(b), provided f(a) ∩ φ f(b) 6= ∅. further f is a descriptive proximal isomorphism, provided f and its inverse f−1 are descriptively proximally continuous. definition c.3. let (x, δφ) be a descriptive čech proximity space and f : (x, δφ) → (x, δφ) a descriptive proximally continuous map. a set a ∈ 2 x is said to be descriptively invariant with respect to f, provided φ(f(a)) ⊆ φ(a). notice that if a is a descriptively invariant set with respect to f, then φ(fn(a)) ⊆ φ(a) for all positive integer n. theorem c.4. let (x, δφ) be a descriptive čech proximity space and f : (x, δφ) → (x, δφ) a proximal descriptive continuous map. if {ai}i∈i ⊆ 2 x is a collection of descriptively invariant sets with respect to f, then i) ∪i∈iai is descriptively invariant with respect to f, and ii) ∩i∈iai is descriptively invariant with respect to f. proof. from our assumption, we have φ(f(ai)) ⊆ φ(ai) for all i ∈ i so that i) f(∪i∈iai) = ∪i∈if(ai) φ(f(∪i∈iai)) = φ(∪i∈if(ai)) = ∪i∈iφ(f(ai)) ⊆ ∪i∈iφ(ai) and ii) f(∩i∈iai) ⊆ ∩i∈if(ai) φ(f(∩i∈iai)) ⊆ φ(∩i∈if(ai)) ⊆ ∩i∈iφ(f(ai)) ⊆ ∩i∈iφ(ai) � theorem c.5. let (x, δφ) be a descriptive čech proximity space and f : (x, δφ) → (x, δφ) a descriptive proximally continuous map. if a ∈ 2 x is descriptively invariant with respect to f then clδφa is also descriptively invariant with respect to with respect to f. proof. the descriptive closure of a subset a of x is defined in [13, §1.21.2] as follows: clφa = {x ∈ x | x δφ a}. © agt, upv, 2021 appl. gen. topol. 22, no. 2 395 j. f. peters and t. vergili take an element x in clφa so that x δφ a and φ(x) ∈ φ(a) by lemma c.1. since f is descriptive proximally continuous f(x) δφ f(a) and φ(f(x)) ∈ φ(f(a)) by lemma c.1. we also have φ(f(x)) ∈ φ(a) since a is an invariant set with respect to f. therefore f(x) δφ a and f(x) ∈ clφa. since this holds for all x ∈ clφa, we have f(clφa) ⊆ clφa so that φ(f(clφa)) ⊆ φ(clφa). � acknowledgements. the first author has been supported by the natural sciences & engineering research council of canada (nserc) discovery grant 185986 and instituto nazionale di alta matematica (indam) francesco severi, gruppo nazionale per le strutture algebriche, geometriche e loro applicazioni grant 9 920160 000362, n.prot u 2016/000036 and scientific and technological research council of turkey (tübi̇tak) scientific human resources development (bideb) under grant no: 2221-1059b211301223. references [1] c. adams and r. franzosa, introduction to topology: pure and applied, 1st ed., pearson, london, uk, (2008), 512 pp., isbn-13: 9780131848696. 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[18] e. h. spanier, algebraic topology, mcgraw-hill book co., new york-toronto, ont.,ca, (1966), xiv+528 pp. [19] j. h. c. whitehead, simplicial spaces, nuclei and m-groups, proceedings of the london math. soc. 45 (1939), 243–327. [20] j. h. c. whitehead, combinatorial homotopy. i, bulletin of the american mathematical society 55, no. 3 (1949), 213–245. © agt, upv, 2021 appl. gen. topol. 22, no. 2 397 @ appl. gen. topol. 22, no. 2 (2021), 259-294doi:10.4995/agt.2021.13248 © agt, upv, 2021 on fixed point index theory for the sum of operators and applications to a class of odes and pdes svetlin georgiev georgiev a and karima mebarki b a department of differential equations, faculty of mathematics and informatics, university of sofia, sofia, bulgaria. (svetlingeorgiev1@gmail.com) b laboratory of applied mathematics, faculty of exact sciences,university of bejaia, 06000 bejaia, algeria. (mebarqi karima@hotmail.fr, karima.mebarki@univ-bejaia.dz) communicated by e. a. sánchez-pérez abstract the aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a k-set contraction obtained in [3, 6], to the case of the sum t + f , where t is a mapping such that (i − t ) is lipschitz invertible and f is a k-set contraction. secondly, as illustration of some our theoretical results, we study the existence of non-negative solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (odes for short) posed on the non-negative half-line as well as a class of partial differential equations (pdes for short). 2010 msc: 37c25; 58j20; 47j35. keywords: positive solution; fixed point index; cone; sum of operators; odes; pdes. 1. preliminaries many problems in science lead to nonlinear equations t x + fx = x posed in some closed convex subset of a banach space. in particular, ordinary, fractional, partial differential equations and integral equations can be formulated received 10 march 2020 – accepted 09 april 2021 http://dx.doi.org/10.4995/agt.2021.13248 s. g. georgiev and k. mebarki like these abstract equations. it is the reason for which it becomes desirable to develop fixed point theorems for such equations. when t is compact and f is a contraction there are many classical tools to deal with such problems (see [2], [5], [9], [11] and references therein). the main aim of this paper is to give some recent results for existence of fixed points for some operators that are of the form t + f , where t is an expansive operator and f is a k-set contraction. the positivity of solutions of nonlinear equations, especially ordinary, partial differential equations, and integral equations is a very important issue in applications, where a positive solution may represent a density, a temperature, a velocity, etc. in this paper we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a k-set contraction, obtained in [1, 3, 4, 8, 6, 7], to the case when t is a mapping such that (i − t ) is lipschitz invertible and f is a k-set contraction. we illustrate some of our theoretical results. more precisely, we study the existence of non-negative solutions for the following ivp x′ = f(t, x), t > 0, x(0) = x0, where x0 ∈ r is a given constant, f : [0, ∞) × r → r is a continuous function satisfying a general polynomial growth condition. moreover, we consider an application for an ivp subject to burgers-fisher equation: ut − uxx + α(t)uux = β(t)u(1 − u), t > 0, x ≥ 0, u(0, x) = u0(x), x ≥ 0, where u0 ∈ c2([0, ∞)) and α, β ∈ c([0, ∞)) with α < 0, β ≥ 0 on [0, ∞). the paper is organized as follows. in the next section, we give some auxiliary results. in sections 3 and 4, we will present our contribution in fixed point index theory for the sum of two operators of the form t + f , where t is a mapping such that (i − t ) is lipschitz invertible with constant γ > 0 and f is a k-set contraction when 0 ≤ k < γ−1. we will consider separately two cases: firstly the computation of fixed point index on cones is treated in section 3. then in section 4, we will discuss the computation of fixed point index on translates of cones. applications are given in sections 4 and 5. 2. auxiliary results let x be a linear normed space and i be the identity map of x. the following lemmas give sufficient conditions for i − t to be lipschitz invertible. lemma 2.1 ([12, lemma 2.1]). let (x, ‖.‖) be a normed linear space, d ⊂ x. if a mapping t : d → x is expansive with a constant h > 1, then the mapping © agt, upv, 2021 appl. gen. topol. 22, no. 2 260 fixed point index theory for the sum of operators i − t : d → (i − t )(d) is invertible and ‖(i − t )−1x − (i − t )−1y‖ ≤ (h − 1)−1‖x − y‖ for all x, y ∈ (i − t )(d). lemma 2.2 ([13, lemma 2.3]). let (e, ‖.‖) be a banach space and t : e → e be lipschitzian map with constant β > 0. assume that for each z ∈ e, the map tz : e → e defined by tzx = t x + z satisfies that t pz is expansive and onto for some p ∈ n. then (i − t ) maps e onto e, the inverse of i − t : e → e exists, and ‖(i − t )−1x − (i − t )−1y‖ ≤ γp‖x − y‖ for all x, y ∈ e, where γp = βp − 1 (β − 1)(lip(t p) − 1) · lemma 2.3 ([13, lemma 2.5]). let (x, ‖.‖) be a linear normed space, m ⊂ x. assume that the mapping t : m → x is contractive with a constant k < 1, then the inverse of i − t : m → (i − t )(m) exist, and ‖(i − t )−1x − (i − t )−1y‖ ≤ (1 − k)−1‖x − y‖ for all x, y ∈ (i − t )(m). lemma 2.4 ([13, lemma 2.6]). let (e, ‖.‖) be a banach space and t : e → e be lipschitzian map with constant β ≥ 0. assume that for each z ∈ e, the map tz : e → e defined by tzx = t x + z satisfies that t pz is contractive for some p ∈ n. then (i − t ) maps e onto e, the inverse of i − t : e → e exists, and ‖(i − t )−1x − (i − t )−1y‖ ≤ ρp‖x − y‖ for all x, y ∈ e, where ρp =      p 1−lip(t p) , if β = 1; 1 1−β , if β < 1; β p −1 (β−1)(1−lip(t p)) , if β > 1. 3. fixed point index on cones in all what follows, p will refer to a cone in a banach space (e, ‖.‖), ω is a subset of p, and u is a bounded open subset of p. for r > 0 define the conical shell pr = p ⋂ {x ∈ e : ‖x‖ < r}. assume that t : ω → e is a mapping such that (i−t ) is lipschitz invertible with constant γ > 0 and f : u → e is a k-set contraction. suppose that (3.1) 0 ≤ k < γ−1, (3.2) f(u) ⊂ (i − t )(ω), and (3.3) x 6= t x + fx, for all x ∈ ∂u ⋂ ω. © agt, upv, 2021 appl. gen. topol. 22, no. 2 261 s. g. georgiev and k. mebarki then x 6= (i − t )−1fx, for all x ∈ ∂u and the mapping (i − t )−1f : u → p is a strict γk-set contraction. indeed, (i − t )−1f is continuous and bounded; and for any bounded set b in u, we have α(((i − t )−1f)(b)) ≤ γ α(f(b)) ≤ γkα(b). the fixed point index i ((i − t )−1f, u, p) is so well defined. thus we put (3.4) i∗ (t + f, u ⋂ ω, p) = i ((i − t )−1f, u, p). proposition 3.1. assume that the mapping t : ω ⊂ p → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and tf(u) ⊂ (i − t )(ω) for all t ∈ [0, 1]. if (i − t )−10 ∈ u, and (3.5) x − t x 6= λfx for all x ∈ ∂u ⋂ ω and 0 ≤ λ ≤ 1, then the fixed point index i∗ (t + f, u ⋂ ω, p) = 1. proof. consider the homotopic deformation h : [0, 1] × u → p defined by h(t, x) = (i − t )−1tfx. the operator h is continuous and uniformly continuous in t for each x. moreover, h(t, .) is a strict k-set contraction for each t and the mapping h(t, .) has no fixed point on ∂u. otherwise, there would exist some x0 ∈ ∂u ⋂ ω and t0 ∈ [0, 1] such that x0 − t x0 = t0fx0, which contradicts our assumption. from the invariance under homotopy and the normalization property of the index fixed point, we deduce that i∗ ((i − t )−1f, u, p) = i∗ ((i − t )−10, u, p) = 1. consequently, from (3.4), we deduce that i∗ (t + f, u ⋂ ω, p) = 1, which completes the proof. � as a consequence of proposition 3.1 , we have the two following results. corollary 3.2. assume that the mapping t : ω ⊂ p → e be such that (i −t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and tf(u) ⊂ (i − t )(ω) for all t ∈ [0, 1]. if (i − t )−10 ∈ u, and ‖fx‖ ≤ ‖x − t x‖ and t x + fx 6= x for all x ∈ ∂u ⋂ ω, then the fixed point index i∗ (t + f, u ⋂ ω, p) = 1. proof. it is sufficient to prove that assumption (3.5) is satisfied. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 262 fixed point index theory for the sum of operators corollary 3.3. assume that the mapping t : ω ⊂ p → e be such that (i −t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and tf(u) ⊂ (i − t )(ω) for all t ∈ [0, 1]. if (i − t )−10 ∈ u, fx ∈ p for all x ∈ ∂u ⋂ ω, and fx � x − t x for all x ∈ ∂u ⋂ ω, then the fixed point index i∗ (t + f, ⋂ ω, p) = 1. proof. it is easy to see that assumption (3.5) is satisfied. � proposition 3.4. let u be a bounded open subset of p with 0 ∈ u. assume that the mapping t : ω ⊂ p → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). if fx 6= (i − t )(λx) for all x ∈ ∂u, λ ≥ 1 and λx ∈ ω, then the fixed point index i∗ (t + f, u ⋂ ω, p) = 1. proof. the mapping (i − t )−1f : u → p is a strict γk-set contraction and it is readily seen that the following condition of leray-schauder type is satisfied (i − t )−1fx 6= λx, for all x ∈ ∂u and λ ≥ 1. in fact, if there exist x0 ∈ ∂u and λ0 ≥ 1 such that (i − t )−1fx0 = λ0x0. then fx0 = (i − t )(λ0x0), which contradicts our assumption. the claim then follows from (3.4) and [8, theorem 1.3.7]. � proposition 3.5. let u be a bounded open subset of p with 0 ∈ u ⋂ ω. assume that the mapping t : ω ⊂ p → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). if (3.6) γ‖fx + t 0‖ ≤ ‖x‖ and t x + fx 6= x for all x ∈ ∂u ⋂ ω, then the fixed point index i∗ (t + f, u ⋂ ω, p) = 1. proof. the mapping (i − t )−1f : u → p is a strict γk-set contraction. (i − t ) being lipschitz invertible with constant γ > 0, for each x ∈ u (3.7) ‖(i − t )−1fx‖ = ‖(i − t )−1fx − (i − t )−1(i − t )0‖ ≤ γ‖fx + t 0‖. therefor, from (3.7) and assumption (3.6), we conclude that for all x ∈ ∂u, ‖(i − t )−1fx‖ ≤ γ‖fx + t 0‖ ≤ ‖x‖. our claim then follows from (3.4) and [8, theorem 1.3.7]. � the following result is as straightforward consequence of proposition [8, corollary 1.3.1]. © agt, upv, 2021 appl. gen. topol. 22, no. 2 263 s. g. georgiev and k. mebarki proposition 3.6. assume that the mapping t : ω ⊂ p → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). if further (i − t )−1f(u) ⊂ u, then the fixed point index i∗ (t + f, u ⋂ ω, p) = 1. as a particular case, we obtain corollary 3.7. assume that the mapping t : ω ⊂ p → e be such that (i −t ) is lipschitz invertible with constant γ > 0, f : pr → e is a k-set contraction with 0 ≤ k < γ−1, and f(pr) ⊂ (i − t )(ω). if 0 ∈ ω and (3.8) γ‖fx + t 0‖ < r, for all x ∈ pr, then the fixed point index i∗ (t + f, pr ⋂ ω, p) = 1. proof. from (3.7) and assumption (3.8), for any x ∈ pr, we conclude that ‖(i − t )−1fx‖ ≤ γ‖fx + t 0‖ < r, which implies that (i − t )−1f(pr) ⊂ pr. � taking r > γ 1−γ ‖t 0‖, we get corollary 3.8. assume that the mapping t : ω ⊂ p → e be such that (i − t ) is lipschitz invertible with constant 0 < γ < 1, f : pr → e is a k-set contraction with 0 ≤ k < γ−1, and f(pr) ⊂ (i − t )(ω). if 0 ∈ ω and (3.9) ‖fx‖ ≤ ‖x‖, for all x ∈ pr, then t + f has at least one fixed point in pr ⋂ ω. proposition 3.9. assume that the mapping t : ω ⊂ p → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). if there exists u0 ∈ p∗ such that (3.10) fx 6= (i − t )(x − λu0), for all λ ≥ 0 and x ∈ ∂u ⋂ (ω + λu0), then the fixed point index i∗ (t + f, u ⋂ ω, p) = 0. proof. the mapping (i − t )−1f : u → p is a strict γk-set contraction and for some u0 ∈ p∗ this operator satisfies x − (i − t )−1fx 6= λu0, ∀ x ∈ ∂u, ∀ λ ≥ 0. by (3.4) and [8, theorem 1.3.8], we deduce that i∗ (t + f, u ⋂ ω, p) = i ((i − t )−1f, u, p) = 0. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 264 fixed point index theory for the sum of operators proposition 3.10. assume that the mapping t : ω ⊂ p → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). suppose further that there exists u0 ∈ p∗ such that t (x − λu0) ∈ p, for all λ ≥ 0 and x ∈ ∂u ⋂ (ω + λu0), and one of the following conditions holds: (a) fx x, ∀ x ∈ ∂u. (b) fx ∈ p, ‖fx‖ > n‖x‖, ∀ x ∈ ∂u, and the cone p is normal with constant n. then the fixed point index i∗ (t + f, u ⋂ ω, p) = 0. proof. we show that conditions (a) or (b) imply that fx 6= (i − t )(x − λu0), for all λ ≥ 0 and x ∈ ∂u ⋂ (ω + λu0). on the contrary, assume the existence of λ0 ≥ 0 and x0 ∈ ∂u ⋂ (ω + λ0u0) such that fx0 = (i − t )(x0 − λ0u0). then x0 − fx0 = t (x0 − λ0u0) + λ0u0 ∈ p. if condition (a) holds, then a contradiction is achieved. otherwise, we deduce that fx0 ≤ x0. since p is normal, we deduce that ‖fx0‖ ≤ n‖x0‖, contradicting condition (b) and ending the proof of our proposition. � 4. fixed point index on translates of cones in this section, let e be a banach space, p (p 6= {0}) be a cone in it. given θ ∈ e, we consider the translate of p, namely k = p + θ = {x + θ, x ∈ p}. then k is a closed convex of e, so it is a retract of e. let ω be any subset of k and u be a bounded open of k such that u ⋂ ω 6= ∅. we denote by u and ∂u the closure and the boundary of u relative to k. the fixed point index i∗ (t + f, u ⋂ ω, k) defined by (4.1) i∗ (t + f, u ⋂ ω, k) = i ((i − t )−1f, u, k). is well defined whenever t : ω → e is a mapping such that (i − t ) is lipschitz invertible with constant γ > 0 and f : u → e is a k-set contraction, 0 ≤ k < γ−1 and f(u) ⊂ (i − t )(ω). proposition 4.1. let u be a bounded open subset of k with θ ∈ u. assume that the mapping t : ω ⊂ k → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). if (4.2) fx 6= (i−t )(λx+(1−λ)θ) for all x ∈ ∂u, λ ≥ 1 and λx+(1−λ)θ ∈ ω, then the fixed point index i∗ (t + f, u ⋂ ω, p) = 1. © agt, upv, 2021 appl. gen. topol. 22, no. 2 265 s. g. georgiev and k. mebarki proof. define the homotopic deformation h : [0, 1] × u → k by h(t, x) = t(i − t )−1fx + (1 − t)θ. then, the operator h is continuous and uniformly continuous in t for each x, and the mapping h(t, .) is a strict γk-set contraction for each t. moreover, h(t, .) has no fixed point on ∂u. otherwise, there would exist some x0 ∈ ∂u and t0 ∈ [0, 1] such that 1t0 x0 + (1 − 1 t0 )θ ∈ ω for t0 6= 0, and t0(i − t )−1fx0 + (1 − t0)θ = x0. we may distinguish between two cases: (i) if t0 = 0, then x0 = θ, which is a contradiction. (ii) if t0 ∈ (0, 1], then fx0 = (i − t )( 1t0 x0 + (1 − 1 t0 )θ), which contradicts our assumption. the properties of invariance by homotopy and normalization of the fixed point index guarantee that i ((i − t )−1f, u, k) = i (θ, u, k). consequently, by (4.1), we deduce that i∗ (t + f, u ⋂ ω, k) = 1. � proposition 4.2. let u be a bounded open subset of k with θ ∈ u. assume that the mapping t : ω ⊂ k → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). if (4.3) ‖fx − t θ − θ‖ ≤ ‖x − θ‖ and t x + fx 6= x, for all x ∈ ∂u ⋂ ω, then the fixed point index i∗ (t + f, u ⋂ ω, p) = 1. proof. the mapping (i − t )−1f : u → p is a strict γk-set contraction. since (i − t ) is lipschitz invertible with constant γ > 0, for each x ∈ u (4.4) ‖(i − t )−1fx − θ‖ = ‖(i − t )−1fx − (i − t )−1(i − t )θ‖ ≤ γ‖fx + t θ − θ‖. therefor, from (4.4) and assumption (4.3), we conclude that for all x ∈ ∂u, ‖(i − t )−1fx − θ‖ ≤ γ‖fx + t θ − θ‖ ≤ ‖x − θ‖, which implies the condition (4.5) in proposition 4.1. this completes the proof. � remark 4.3. propositions 4.1,4.2 can be proven directly by appealing to [4, proposition 2.2], and [4, corollary 2.2], respectively. proposition 4.4. let u be a bounded open subset of k. assume that the mapping t : ω ⊂ k → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and ( tf(u) + (1 − t)θ ) ⊂ (i − t )(ω) for all t ∈ [0, 1]. if (i − t )−1θ ∈ u, and (4.5) x − t x 6= λfx + (1 − λ)θ for all x ∈ ∂u ⋂ ω and 0 ≤ λ ≤ 1, then the fixed point index i∗ (t + f, u ⋂ ω, k) = 1. © agt, upv, 2021 appl. gen. topol. 22, no. 2 266 fixed point index theory for the sum of operators proof. define the homotopic deformation h : [0, 1] × u → e by h(t, x) = tfx + (1 − t)θ. then, the operator h is continuous and uniformly continuous in t for each x, and the mapping h(t, .) is a k-set contraction for each t. moreover, t + h(t, .) has no fixed point on ∂u ⋂ ω. otherwise, there would exist some x0 ∈ ∂u ⋂ ω and t0 ∈ [0, 1] such that t x0 + t0fx0 + (1 − t0)θ = x0, then x0−t x0 = t0fx0+(1−t0)θ, leading to a contradiction with the hypothesis. by (4.1), property (c) in [3, theorem 2.3] and the normalization property of the fixed point index, we conclude that i∗ (t + f, u ⋂ ω, k) = i∗ (t + θ, kr ⋂ ω, k) = ((i − t )−1θ, u ⋂ ω, k) = 1. � corollary 4.5. let u be a bounded open subset of k. assume that the mapping t : ω ⊂ k → e be such that (i −t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and ( tf(u) + (1 − t)θ ) ⊂ (i − t )(ω) for all t ∈ [0, 1]. if (i − t )−1θ ∈ u, fx ∈ k for all x ∈ ω ⋂ ∂u, and (4.6) fx � x − t x for all x ∈ ∂u ⋂ ω, then the fixed point index i∗ (t + f, u ⋂ ω, k) = 1. proof. it is easy to see that assumption (4.5) is satisfied. otherwise, there exist some x0 ∈ ∂u ⋂ ω and 0 ≤ λ0 ≤ 1 such that x0 − t x0 = λ0fx0 + (1 − λ0)θ. then fx0 − x0 + t x0 = (1 − λ0)(fx0 − θ) ∈ p, which leads us to a contradiction with (4.6). � proposition 4.6. let u be a bounded open subset of k. assume that the mapping t : ω ⊂ k → e be such that (i − t ) is lipschitz invertible with constant γ > 0, f : u → e is a k-set contraction with 0 ≤ k < γ−1, and f(u) ⊂ (i − t )(ω). if there exists u0 ∈ p∗ such that (4.7) fx 6= (i − t )(x − λu0), for all λ ≥ 0 and x ∈ ∂u ⋂ (ω + λu0), then the fixed point index i∗ (t + f, u ⋂ ω, k) = 0. proof. the mapping (i − t )−1f : u → k is a strict γk-set contraction. suppose that i∗ (t + f, u ⋂ ω, k) 6= 0. then, i ((i − t )−1f, u, p) 6= 0. for each r > 0, define the homotopy: h(t, x) = (i − t )−1fx + tru0, for x ∈ u and t ∈ [0, 1]. © agt, upv, 2021 appl. gen. topol. 22, no. 2 267 s. g. georgiev and k. mebarki the operator h is continuous and uniformly continuous in t for each x. moreover, h(t, .) is a strict k-set contraction for each t and h([0, 1] × u) = (i − t )−1f(u) + tru0 ⊂ k. we check that h(t, x) 6= x, for all (t, x) ∈ [0, 1]×∂u. if h(t0, x0) = x0 for some (t0, x0) ∈ [0, 1] × ∂u, then x0 − t0ru0 = (i − t )−1fx0, and so x0 − t0ru0 ∈ ω. hence (i − t )(x0 − t0ru0) = fx0, for x0 ∈ ∂u ⋂ (ω + t0ru0), contradicting assumption (4.7). by homotopy invariance property of the fixed point index, we deduce that i ((i − t )−1f + ru0, u ⋂ ω, p) = i ((i − t )−1f, u, p) 6= 0. thus the existence property of the fixed point index, for each r > 0, there exists xr ∈ u such that (4.8) xr − (i − t )−1 fxr = ru0. letting r → +∞ in (4.8), the left-hand side of (4.8) is bounded while the right-hand side is not, which is a contradiction. therefore i∗ (t + f, u ⋂ ω, p) = 0, which completes the proof. � 5. applications to ode in this section we investigate the ivp (5.1) x′ = f(t, x), t > 0, x(0) = x0, where x0 ∈ r is a given constant, f : [0, ∞) × r → r is a given function. let l ∈ n and x0, s, r, aj, j ∈ {0, 1, . . . , l}, are positive constants such that (h1): x0 + l ∑ j=0 ( r 2 )j aj < r 2 , (h2): f ∈ c([0, ∞) × r) and 0 ≤ f(y, x) ≤ l ∑ j=0 aj(y)|x|j, y ∈ [0, ∞), x ∈ r, where aj ∈ c([0, ∞)), aj ≥ 0 on [0, ∞) and ∫ ∞ 0 aj(y)dy ≤ aj, j ∈ {0, 1, . . . , l}. © agt, upv, 2021 appl. gen. topol. 22, no. 2 268 fixed point index theory for the sum of operators theorem 5.1. assume that (h1)-(h2) hold. then the ivp (5.1) has a solution x ∈ c1([0, ∞)) such that 0 ≤ x(t) < r 2 , t ∈ [0, ∞). proof. case 1.: let t ∈ [0, 1]. consider the ivp (5.2) x′ = f(t, x), t ∈ (0, 1], x(0) = x0. take ǫ > 0 arbitrarily. let e1 = c([0, 1]) be endowed with the maximum norm and p1 = {x ∈ e1 : x(t) ≥ 0, t ∈ [0, 1]}, ω1 = p1r = {x ∈ p1 : ‖x‖ < r} , u1 = p1 r 2 = { x ∈ p1 : ‖x‖ < r 2 } . for x ∈ e1, define the operators t1x(t) = (1 + ǫ)x(t), f1x(t) = −ǫ ( x0 + ∫ t 0 f(y, x(y))dy ) , t ∈ [0, 1]. note that for any fixed point x ∈ e1 of the operator t1 + f1 we have that x ∈ c1([0, 1]) and it is a solution of the ivp (5.2). (1) for x, y ∈ e1, we have ‖(i − t1)−1x − (i − t1)−1y‖ = 1 ǫ ‖x − y‖, i.e., (i − t1) : e1 → e1 is lipschitz invertible with constant 1ǫ . (2) for x ∈ u1 and t ∈ [0, 1], we have |f1x(t)| = ǫ ( x0 + ∫ t 0 f(y, x(y))dy ) ≤ ǫ  x0 + ∫ t 0 l ∑ j=0 aj(y)(x(y)) jdy   ≤ ǫ  x0 + l ∑ j=0 ( r 2 )j ∫ t 0 aj(y)dy   ≤ ǫ  x0 + l ∑ j=0 ( r 2 )j aj   © agt, upv, 2021 appl. gen. topol. 22, no. 2 269 s. g. georgiev and k. mebarki and |(f1x)′(t)| = ǫ f(t, x(t)) ≤ ǫ l ∑ j=0 aj(y)(x(y)) j ≤ ǫ l ∑ j=0 ( r 2 )j aj(y) ≤ ǫ l ∑ j=0 ( r 2 )j bj thus, ‖f1x‖ ≤ ǫ  x0 + l ∑ j=0 ( r 2 )j aj   and ‖(f1x)′‖ ≤ ǫ l ∑ j=0 ( r 2 )j bj. hence, using the arzela-ascoli theorem, we conclude that f1 : u1 → e is a completely continuous mapping. therefore f1 : u1 → e is a 0-set contraction. (3) let λ ∈ [0, 1] and x ∈ u1 be arbitrarily chosen. then z(t) = λ ( x0 + ∫ t 0 f(y, x(y))ds ) ∈ e1 and z(t) ≤ λ ( x0 + ∫ ∞ 0 f(y, x(y))dy ) ≤ λ  x0 + l ∑ j=0 ∫ ∞ 0 aj(y)(x(y)) j dy   ≤ λ  x0 + l ∑ j=0 ( r 2 )j aj   < λ r 2 ≤ r 2 , t ∈ [0, 1], © agt, upv, 2021 appl. gen. topol. 22, no. 2 270 fixed point index theory for the sum of operators i.e., z ∈ ω1. next, λf1x(t) = −λǫ ( x0 + ∫ t 0 f(y, x(y))dy ) = −ǫz(t) = (i − t1)z(t), t ∈ [0, 1]. thus, λf1(u1) ⊂ (i − t1)(ω1). (4) note that (i − t1)−10 = 0 ∈ u1. (5) assume that there are x ∈ ∂u1 ⋂ ω1 and λ ∈ [0, 1] such that x − t1x = λf1x. if λ = 0, then 0 = x − t1x = −ǫx on [0, 1], whereupon x(t) = 0, t ∈ [0, 1]. this is a contradiction because x ∈ ∂u1. therefore λ ∈ (0, 1]. let t1 ∈ [0, 1] be such that x(t1) = r2. then (i − t1)x(t1) = −ǫx(t1) = −ǫ r 2 = −λǫ ( x0 + ∫ t1 0 f(y, x(y))dy ) , © agt, upv, 2021 appl. gen. topol. 22, no. 2 271 s. g. georgiev and k. mebarki whereupon r 2 = λ ( x0 + ∫ t1 0 f(y, x(y))dy ) ≤ λ ( x0 + ∫ ∞ 0 f(y, x(y))dy ) ≤ λ  x0 + l ∑ j=0 ∫ ∞ 0 aj(y)(x(y)) jdy   ≤ λ  x0 + l ∑ j=0 aj ( r 2 )j   < λ r 2 ≤ r 2 , i.e., r 2 < r 2 , which is a contradiction. by 1, 2, 3, 4, 5 and proposition 3.1, it follows that the operator t1 +f1 has a fixed point in u1. denote it by x1. we have 0 ≤ x1(t) < r 2 , t ∈ [0, 1], and x1 ∈ c1([0, 1]) is a solution of the ivp (5.2). case 2.: let t ∈ [1, 2]. consider the ivp (5.3) x′ = f(t, x), t ∈ (1, 2], x(1) = x1(1). take ǫ > 0 arbitrarily. let e2 = c([1, 2]) be endowed with the maximum norm and p2 = {x ∈ e2 : x(t) ≥ 0, t ∈ [1, 2]}, ω2 = p2r = {x ∈ p2 : ‖x‖ < r} , u2 = p2 r 2 = { x ∈ p2 : ‖x‖ < r 2 } . for x ∈ e2 define the operators t2x(t) = (1 + ǫ)x(t), f2x(t) = −ǫ ( x1(1) + ∫ t 1 f(s, x(s))ds ) , t ∈ [1, 2]. © agt, upv, 2021 appl. gen. topol. 22, no. 2 272 fixed point index theory for the sum of operators note that for x ∈ u2, we have x1(1) + ∫ t 1 f(s, x(s))ds = x0 + ∫ t 0 f(y, x(y))dy ≤ x0 + ∫ ∞ 0 f(y, x(y))dy ≤ x0 + l ∑ j=0 aj(y)(x(y)) j dy ≤ x0 + l ∑ j=0 ajr j < r 2 , t ∈ [1, 2]. as in case 1 we prove that the operator t2 + f2 has a fixed point x2 ∈ u2. we have that 0 ≤ x2(t) < r 2 , t ∈ [1, 2], x2 ∈ c1([1, 2]). note that x1(1) = x2(1), x′1(1) = f(1, x1(1)) = f(1, x2(1)) = x′2(1). thus, x(t) =    x1(t) t ∈ [0, 1] x2(t) t ∈ [1, 2] is a solution to the ivp x′ = f(t, x), t ∈ (0, 2], x(0) = x0. case 3.: consider the ivp x′ = f(t, x), t ∈ (2, 3], x(2) = x2(2). © agt, upv, 2021 appl. gen. topol. 22, no. 2 273 s. g. georgiev and k. mebarki and so on, the function x(t) =                            x1(t) t ∈ [0, 1] x2(t) t ∈ [1, 2] x3(t) t ∈ [2, 3] x4(t) t ∈ [3, 4] . . . is a solution to the ivp (5.1). this completes the proof. � 6. applications to pde in this section we consider the ivp for burgers-fisher equation (6.1) ut − uxx + α(t)uux = β(t)u(1 − u), t > 0, x ≥ 0, (6.2) u(0, x) = u0(x), x ≥ 0, where (a1): u0 ∈ c2([0, ∞)), r1 ≥ u0 ≥ r12 on [0, ∞), where r1 ∈ ( 0, 1 2 ) is a given constant, (a2): α, β ∈ c([0, ∞)), α < 0, β ≥ 0 on [0, ∞) , a ∈ (0, 1) is a constant and g is a positive continuous function on [0, ∞) × [0, ∞) such that 1 − (1 + 2r1)a > 0, ( 4 + 3 2 r1 ) a < 1 2 , and 120 ( 1 + t + t2 + t3 + t4 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) × ∫ t 0 ∫ x 0 g(t1, x1) ( 1 + ∫ t1 0 (β(t2) − α(t2))dt2 ) dx1dt1 ≤ a, t ≥ 0, x ≥ 0. let e = c1([0, ∞), c2([0, ∞))) be endowed with the norm ‖u‖ = { sup (t,x)∈[0,∞)×[0,∞) |u(t, x)|, sup (t,x)∈[0,∞)×[0,∞) ∣ ∣ ∣ ∣ ∂ ∂t u(t, x) ∣ ∣ ∣ ∣ , sup (t,x)∈[0,∞)×[0,∞) ∣ ∣ ∣ ∣ ∂ ∂x u(t, x) ∣ ∣ ∣ ∣ , sup (t,x)∈[0,∞)×[0,∞) ∣ ∣ ∣ ∣ ∂2 ∂x2 u(t, x) ∣ ∣ ∣ ∣ } , provided it exists. © agt, upv, 2021 appl. gen. topol. 22, no. 2 274 fixed point index theory for the sum of operators lemma 6.1. suppose (a1) and (a2). if a function u ∈ e is a solution of the integral equation 0 = ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×u(t2, x2)(1 − u(t2, x2))dx2dt2dx1dt1 − 1 2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 u(t2, x1)dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×(u0(x2) − u(t1, x2))dx2dx1dt1, (t, x) ∈ [0, ∞) × [0, ∞), then it is a solution to the ivp (6.1)-(6.2). proof. we differentiate the considered integral equation five times in t and five times in x and using that g > 0 on [0, ∞) × [0, ∞), we get 0 = g(t, x) ∫ t 0 ∫ x 0 ∫ x1 0 β(t1)u(t1, x2)(1 − u(t1, x2))dx2dx1dt1 − 1 2 g(t, x) ∫ t 0 ∫ x 0 α(t1)(u(t1, x1)) 2 dx1dt1 +g(t, x) ∫ t 0 u(t1, x)dt1 +g(t, x) ∫ x 0 ∫ x1 0 (u0(x2) − u(t1, x2))dx2dx1, (t, x) ∈ [0, ∞) × [0, ∞), © agt, upv, 2021 appl. gen. topol. 22, no. 2 275 s. g. georgiev and k. mebarki whereupon 0 = ∫ t 0 ∫ x 0 ∫ x1 0 β(t1)u(t1, x2)(1 − u(t1, x2))dx2dx1dt1 −1 2 ∫ t 0 ∫ x 0 α(t1)(u(t1, x1)) 2dx1dt1 + ∫ t 0 u(t1, x)dt1 + ∫ x 0 ∫ x1 0 (u0(x2) − u(t1, x2))dx2dx1, (t, x) ∈ [0, ∞) × [0, ∞). the last equation we differentiate twice in x and we get (6.3) 0 = ∫ t 0 β(t1)u(t1, x)(1 − u(t1, x))dt1 − ∫ t 0 α(t1)u(t1, x)ux(t1, x)dt1 + ∫ t 0 uxx(t1, x)dt1 +u0(x) − u(t, x), (t, x) ∈ [0, ∞) × [0, ∞), which we differentiate in t and we obtain 0 = β(t)u(t, x)(1 − u(t, x)) − α(t)u(t, x)ux(t, x) +uxx(t, x) − ut(t, x), (t, x) ∈ [0, ∞) × [0, ∞), i.e., u satisfies (6.1). now we put t = 0 in (6.3) and we get u(0, x) = u0(x), x ∈ [0, ∞). this completes the proof. � © agt, upv, 2021 appl. gen. topol. 22, no. 2 276 fixed point index theory for the sum of operators for u ∈ e, define the operators f1u(t, x) = ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u(t1, x2)dx2dx1dt1, f2u(t, x) = ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×u(t2, x2)dx2dt2dx1dt1 −1 2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 u(t2, x1)dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1, (t, x) ∈ [0, ∞) × [0, ∞). note that if u ∈ e is a fixed point of the operator f2 − f1, then it is a solution of the ivp (6.1)-(6.2). lemma 6.2. suppose (a1), (a2) and r > 0. if u ∈ e and ‖u‖ ≤ r, then ‖f1u‖ ≤ (1 + r)a‖u‖, ‖f2u‖ ≤ ( 3 + r 2 ) ra and f2 : {u ∈ e : ‖u‖ ≤ r} → e is a completely continuous operator. moreover, ‖f1u1 − f1u2‖ ≤ (2r + 1)a‖u1 − u2‖ for any u1, u2 ∈ {u ∈ e : ‖u‖ ≤ r}. © agt, upv, 2021 appl. gen. topol. 22, no. 2 277 s. g. georgiev and k. mebarki proof. take u ∈ {e : ‖u‖ ≤ r} arbitrarily. then |f1u(t, x)| ≤ ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 ≤ r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1)dx1dt1 ≤ r‖u‖t4x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +‖u‖t4x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)a‖u‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂t f1u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 ≤ 4r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1)dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 278 fixed point index theory for the sum of operators ≤ 4r‖u‖t3x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖t3x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)a‖u‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂x f1u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 ≤ 4r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1)dx1dt1 ≤ 4r‖u‖t4x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖t4x5 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)a‖u‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂x2 f1u(t, x) ∣ ∣ ∣ ∣ ≤ 12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 279 s. g. georgiev and k. mebarki ≤ 12r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +12‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1)dx1dt1 ≤ 12r‖u‖t4x4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +12‖u‖t4x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)a‖u‖, t ≥ 0, x ≥ 0, consequently ‖f1u‖ ≤ (1 + r)a‖u‖. next, |f2u(t, x)| ≤ ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −1 2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 280 fixed point index theory for the sum of operators ≤ r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −1 2 r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)4g(t1, x1)dx1dt1 +r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1)dx1dt1 ≤ rt4x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −1 2 r2t4x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +rt5x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +rt4x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) ra, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂t f2u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −2 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 281 s. g. georgiev and k. mebarki ≤ 4r ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2 ∫ t 0 ∫ x 0 x1(t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4r ∫ t 0 ∫ x 0 t1(t − t1)3(x − x1)4g(t1, x1)dx1dt1 +4r ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1)dx1dt1 ≤ 4rt3x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2t3x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4rt4x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +4rt3x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) ra, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂t2 f2u(t, x) ∣ ∣ ∣ ∣ ≤ 12 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −6 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 282 fixed point index theory for the sum of operators ≤ 12r ∫ t 0 ∫ x 0 x21(t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2 ∫ t 0 ∫ x 0 x1(t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12r ∫ t 0 ∫ x 0 t1(t − t1)2(x − x1)4g(t1, x1)dx1dt1 +12r ∫ t 0 ∫ x 0 x21(t − t1)2(x − x1)4g(t1, x1)dx1dt1 ≤ 12rt2x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2t2x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12rt3x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +12rt2x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) ra, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂x f2u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 283 s. g. georgiev and k. mebarki ≤ 4r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)3g(t1, x1)dx1dt1 +4r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1)dx1dt1 ≤ 4rt4x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2t4x4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4rt5x3 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +4rt4x5 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) ra, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂x2 f2u(t, x) ∣ ∣ ∣ ∣ ≤ 12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −6 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 284 fixed point index theory for the sum of operators ≤ 12r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)2g(t1, x1)dx1dt1 +12r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1)dx1dt1 ≤ 12rt4x4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2t4x3 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12rt5x2 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +12rt4x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) ra, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂3 ∂x3 f2u(t, x) ∣ ∣ ∣ ∣ ≤ 24 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +24 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +24 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 285 s. g. georgiev and k. mebarki ≤ 24r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −12r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +24r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)g(t1, x1)dx1dt1 +24r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)g(t1, x1)dx1dt1 ≤ 24rt4x3 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −12r2t4x2 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +24rt5x ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +24rt4x3 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) ra, t ≥ 0, x ≥ 0. consequently ‖f2u‖ ≤ ( 3 + r 2 ) ra, ∥ ∥ ∥ ∥ ∂2 ∂t2 f2u ∥ ∥ ∥ ∥ c0 ≤ ( 3 + r 2 ) ra, ∥ ∥ ∥ ∥ ∂3 ∂x3 f2u ∥ ∥ ∥ ∥ c0 ≤ ( 3 + r 2 ) ra. by the arzela-ascoli theorem, it follows that the operator f2 : {u ∈ e : ‖u‖ ≤ r} → e is a completely continuous operator. let now, u1, u2 ∈ {u ∈ e : ‖u‖ ≤ r}. then |f1u1(t, x) − f1u2(t, x)| ≤ ( ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 286 fixed point index theory for the sum of operators + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2)dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 2rx6t4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +x6t4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)a‖u1 − u2‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂t f1u1(t, x) − ∂ ∂t f1u2(t, x) ∣ ∣ ∣ ∣ ≤ ( 4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 8rx6t3 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +4x6t3 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)a‖u1 − u2‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂x f1u1(t, x) − ∂ ∂x f1u2(t, x) ∣ ∣ ∣ ∣ ≤ ( 4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 287 s. g. georgiev and k. mebarki +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ x1 0 (x1 − x2)dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 8rx5t4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +4x5t4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)a‖u1 − u2‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂x2 f1u1(t, x) − ∂2 ∂x2 f1u2(t, x) ∣ ∣ ∣ ∣ ≤ ( 12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ x1 0 (x1 − x2)dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 24rx4t4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +12x4t4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)a‖u1 − u2‖, t ≥ 0, x ≥ 0. therefore ‖f1u1 − f1u2‖ ≤ (2r + 1)a‖u1 − u2‖. this completes the proof. � theorem 6.3. suppose (a1) and (a2). then the ivp (6.1)-(6.2) has at least one non-negative solution u ∈ c1([0, ∞), c2([0, ∞))). proof. set p = {u ∈ e : u(t, x) ≥ 0, t ≥ 0, x ≥ 0}, ω = {u ∈ p : ‖u‖ ≤ r1, u(t, x) ≤ u0(x), t ≥ 0, x ≥ 0}, u = {u ∈ p : ‖u‖ ≤ r1, 1 2 u0(x) ≤ u(t, x) ≤ u0(x), t ≥ 0, x ≥ 0}. for u ∈ e, define the operators t u(t, x) = −f1u(t, x), su(t, x) = f2u(t, x), t ≥ 0, x ≥ 0. © agt, upv, 2021 appl. gen. topol. 22, no. 2 288 fixed point index theory for the sum of operators (1) let u, v ∈ ω. then (i − t )(u − v) = (i + f1)(u − v) and using lemma 6.2, we get ‖(i − t )(u − v)‖ ≥ ‖u − v‖ − ‖f1(u − v)‖ ≥ (1 − (1 + 2r1)a) ‖u − v‖. thus, i − t : ω → e is lipschitz invertible with γ = 1 1−(1+2r1)a . (2) by lemma 6.2, we have that s : u → e is a completely continuous operator. therefore s : u → e is 0-set contraction. (3) let v ∈ u be arbitrarily chosen. for u ∈ ω, we have −f1u(t, x) + f2v(t, x) ≥ − ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 − ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u(t1, x2)dx2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×v(t2, x2)dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © agt, upv, 2021 appl. gen. topol. 22, no. 2 289 s. g. georgiev and k. mebarki ≥ ( r1 2 − r21 ) ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×(u0(x2) − u(t1, x2))dx2dx1dt1 ≥ 0, t ≥ 0, x ≥ 0, and −f1u(t, x) + f2v(t, x) ≤ ‖f1u‖ + ‖f2v‖ ≤ (1 + r1)r1a + ( 3 + r1 2 ) r1a = ( 4 + 3 2 r1 ) r1a < r1 2 ≤ u0(x), t ≥ 0, x ≥ 0. for u ∈ ω, define the operator lu(t, x) = −f1u(t, x) + f2v(t, x), t ≥ 0, x ≥ 0. then, using lemma 6.2, we get ‖lu‖ ≤ ‖f1u‖ + ‖f2v‖ ≤ r1(1 + r1)a + ( 3 + r1 2 ) r1a = ( 4 + 3 2 r1 ) r1a ≤ r1 2 . consequently l : ω → ω. again, applying lemma 6.2, we obtain ‖lu1 − lu2‖ ≤ (2r1 + 1)a‖u1 − u2‖. therefore l : ω → ω is a contraction operator and there exists a unique u ∈ ω so that u = lu or (i − t )u = sv. then s(u) ⊂ (i − t )(ω). © agt, upv, 2021 appl. gen. topol. 22, no. 2 290 fixed point index theory for the sum of operators (4) assume that there are an u ∈ ∂u and λ ≥ 1 so that su = (i − t )(λu) and λu ∈ ω. then su = (i + f1)(λu) and applying lemma 6.2, we obtain ( 3 + r1 2 ) r1a ≥ ‖su‖ ≥ λ‖u‖ − ‖f1(λu)‖ ≥ λ‖u‖ − (1 + r1)a‖λu‖ = (1 − (1 + r1)a) λ‖u‖ ≥ (1 − (1 + r1)a)‖u‖ = r1(1 − (1 + r1)a), whereupon ( 3 + r1 2 ) a ≥ 1 − (1 + r1)a or ( 4 + 3 2 r1 ) a ≥ 1, which is a contradiction. hence and proposition 3.4, it follows that the operator t + s has at least one fixed point in u ⋂ ω, which is a nontrivial nonnegative solution of the ivp (6.1)-(6.2). this completes the proof. 6.1. example. below, we will illustrate our main result. let h(x) = log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 , l(s) = arctan s11 √ 2 1 − s22 , s ∈ r. then h′(s) = 22 √ 2s10(1 − s22) (1 − s11 √ 2 + s22)(1 + s11 √ 2 + s22) , l′(s) = 11 √ 2s10(1 + s20) 1 + s40 , s ∈ r. therefore −∞ < lim s→±∞ (1 + s + · · · + s9)h(s) < ∞, −∞ < lim s→±∞ (1 + s + · · · + s9)l(s) < ∞. © agt, upv, 2021 appl. gen. topol. 22, no. 2 291 s. g. georgiev and k. mebarki hence, there exists a positive constant c1 so that (1 + s + · · · + s9) ( 1 44 √ 2 log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 + 1 22 √ 2 arctan s11 √ 2 1 − s22 ) ≤ c1, (1 + s + · + s9) ( 1 44 √ 2 log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 + 1 22 √ 2 arctan s11 √ 2 1 − s22 ) ≤ c1, s ∈ [0, ∞). note that by [10](pp. 707, integral 79), we have ∫ dz 1 + z4 = 1 4 √ 2 log 1 + z √ 2 + z2 1 − z √ 2 + z2 + 1 2 √ 2 arctan z √ 2 1 − z2 . let q(s) = s10 (1 + s44) (1 + (1 + s + · · · + s9)2)28 , s ∈ [0, ∞), and g1(t, x) = q(t)q(x), t, x ∈ [0, ∞). then there exists a positive constant a1 such that 720(1 + t + · · · + t6)(1 + x + · · · + x6) ∫ t 0 ∫ x 0 g1(t1, x1)dx1dt1 ≤ a1, t, x ≥ 0. take g(t, x) = g1(t,x) 280a1 , a = 1 50 , r1 = 1 4 . consider the ivp ut − uxx − uux = u(1 − u), t > 0, x ≥ 0, u(0, x) = 1 8 + 1 8(1 + x2) , x ≥ 0. here α = −1, β = 1 on [0, ∞), r1 2 ≤ u0(x) = 1 8 + 1 8(1 + x2) ≤ r1, x ≥ 0, 1 − (1 + r1)a = 39 40 > 0, ( 4 + 3 2 r1 ) a = 7 80 < 1 2 , and 120 ( 1 + t + t2 + t3 + t4 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) © agt, upv, 2021 appl. gen. topol. 22, no. 2 292 fixed point index theory for the sum of operators × ∫ t 0 ∫ x 0 g(t1, x1) ( 1 + ∫ t1 0 (β(t2) − α(t2))dt2 ) dx1dt1 ≤ 240(1 + t) ( 1 + t + t2 + t3 + t4 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) × ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ 720 280a1 ( 1 + t + t2 + t3 + t4 + t5 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) × ∫ t 0 ∫ x 0 g1(t1, x1)dx1dt1 ≤ 1 280 ≤ a. therefore the considered ivp has at least one non-negative solution u ∈ c1([0, ∞), c2([0, ∞))). � acknowledgements. the second author was supported by: direction générale de la recherche scientifique et du développement technologique dgrsdt. mesrs algeria. projet prfu : c00l03un060120180009. references [1] s. benslimane, s. g. georgiev and k. mebarki, expansion-compression fixed point theorem of leggett-williams type for the sum of two operators and application in three-point bvps, studia ubb math, to appear. [2] g. cain and m. nashed, fixed points and stability for a sum of two operators in locally convex spaces, pacific j. math. 39 (1971), 581–592. [3] s. djebali and k. mebarki, fixed point index theory for perturbation of expansive mappings by k-set contraction, topol. meth. in nonlinear anal. 54, no. 2 (2019), 613– 640. [4] s. djebali and k. mebarki, fixed point index on translates of cones and applications, nonlinear studies 21, no. 4 (2014), 579–589. [5] d. edmunds, remarks on nonlinear functional equations, math. ann. 174 (1967), 233– 239. [6] s. g. georgiev and k. mebarki, existence of positive solutions for a class odes, fdes and pdes via fixed point index theory for the sum of operators, commun. on appl. nonlinear anal. 26, no. 4 (2019), 16–40. © agt, upv, 2021 appl. gen. topol. 22, no. 2 293 s. g. georgiev and k. mebarki [7] s. g. georgiev and k. mebarki, existence of solutions for a class of ibvp for nonlinear parabolic equations via the fixed point index theory for the sum of two operators, new trends in nonlinear analysis and applications, to appear. [8] d. guo, y. j. cho and j. zhu, partial ordering methods in nonlinear problems, shangdon science and technology publishing press, shangdon, 1985. [9] m. nashed and j. wong, some variants of a fixed point theorem krasnoselskii and applications to nonlinear integral equations, j. math. mech. 18 (1969), 767–777. [10] a. polyanin and a. manzhirov, handbook of integral equations, crc press, 1998. [11] v. sehgal and s. singh, a fixed point theorem for the sum of two mappings, math. japonica 23 (1978), 71–75. [12] t. xiang and r. yuan, a class of expansive-type krasnosel’skii fixed point theorems, nonlinear anal. 71, no. 7-8 (2009), 3229–3239. [13] t. xiang and s. g. georgiev, noncompact-type krasnoselskii fixed-point theorems and their applications, math. methods appl. sci. 39, no. 4 (2016), 833–863. © agt, upv, 2021 appl. gen. topol. 22, no. 2 294 () @ applied general topology c© universidad politécnica de valencia volume 13, no. 2, 2012 pp. 225-226 correction to: some results and examples concerning whyburn spaces ofelia t. alas, maira madriz-mendoza and richard g. wilson abstract we correct the proof of theorem 2.9 of the paper mentioned in the title (published in applied general topology, 13 no.1 (2012), 11-19). 2010 msc:primary 54d99; secondary 54a25; 54d10; 54g99 keywords: whyburn space, weakly whyburn space, submaximal space, scattered space, semiregular, feebly compact there is an error in the proof of theorem 2.9. a correct proof is as follows. theorem 2.9. if x is weakly whyburn, then |x| ≤ d(x)t(x). proof. if x is finite, the result is trivial; thus we assume that x is infinite. suppose that d(x) = δ, t(x) = κ and d ⊆ x is a dense (proper) subset of cardinality δ. let d = d0 and define recursively an ascending chain of subspaces {dα : α < κ +} as follows: suppose that for some α ∈ κ+ and for each β < α we have defined dense sets dβ such that |dβ| ≤ δ κ and dγ ⊆ dλ whenever γ < λ < α. if α is a limit ordinal, then define dα = ⋃ {dβ : β < α} and then |dα| ≤ |α|.δ κ ≤ κ+.δκ = δκ. if on the other hand α = β + 1, and dβ x, then since x is weakly whyburn there is some x ∈ x \ dβ and bx ⊆ dβ such that |bx| ≤ κ, cl(bx) \ dβ = {x}; thus necessarily, we have that |cl(bx)| ≤ δ κ and we define dα = ⋃ {cl(b) : b ⊆ dβ, |b| ≤ κ, |cl(b)| ≤ δ κ}. clearly dα ! dβ and since there are at most (δ κ)κ such sets b it follows that |dα| ≤ δ κ. if dα = x for some α < κ +, then we are done. if not, then we define ∆ = ⋃ {dα : α < κ +}, and clearly |∆| ≤ κ+.δκ = δκ. 226 o. t. alas, m. madriz-mendoza and r. g. wilson thus to complete the proof it suffices to show that ∆ = x. suppose to the contrary; then, since ∆ is not closed and x is weakly whyburn and has tightness κ, there is some z ∈ x \ ∆ and some set b ⊆ ∆ of cardinality at most κ, such that cl(b) \ ∆ = {z} and hence |cl(b)| ≤ δκ. since the sets {dα : α < κ +} form an ascending chain and cf(κ+) > κ, it follows that for some γ < κ+, b ⊆ ⋃ {dα : α < γ} and hence z ∈ dγ+1 ⊆ ∆, a contradiction. � it should also be noted that theorem 2.6 is not as claimed, an improvement on the cited result of bella, costantini and spadaro, since a lindelöf p-space is regular. (received october 2012 – accepted october 2012) o. t. alas (alas@ime.usp.br) instituto de matemática e estat́ıstica, universidade de são paulo, caixa postal 66281, 05311-970 são paulo, brasil. m. madriz-mendoza, r. g. wilson (seber@xanum.uam.mx, rgw@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, unidad iztapalapa, avenida san rafael atlixco, #186, apartado postal 55-532, 09340, méxico, d.f., méxico. correction to: some results and examples concerning whyburn spaces. by o. t. alas, m. madriz-mendoza and r. g. wilson @ appl. gen. topol. 23, no. 1 (2022), 1-16 doi:10.4995/agt.2022.15668 © agt, upv, 2022 some classes of topological spaces related to zero-sets f. golrizkhatami and a. taherifar department of mathematics, yasouj university, yasouj, iran (f.golrizkhatami@stu.yu.ac.ir, ataherifar@yu.ac.ir) communicated by d. georgiou abstract an almost p -space is a topological space in which every zero-set is regular-closed. we introduce a large class of spaces, c-almost p -space (briefly cap -space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. in this paper we study cap -spaces. it is proved that if x is a dense and z#-embedded subspace of a space t , then t is cap if and only if x is a cap and crz-extended in t (i.e, for each regular-closed zero-set z in x, clt z is a zero-set in t ). in 6p.5 of [8] it was shown that a closed countable union of zero-sets need not be a zero-set. we call x a cz-space whenever the closure of any countable union of zero-sets is a zero-set. this class of spaces contains the class of p -spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and cap spaces. in this paper we study topological properties of cz (resp. cozero complemented)-space and other classes of topological spaces near to them. some algebraic and topological equivalent conditions of cz (resp. cozero complemented)-space are characterized. examples are provided to illustrate and delimit our results. 2020 msc: 54c40. keywords: zero-set; almost p-space; compact space; z-embedded subset. 1. introduction the set of zero-sets in a topological space x, z[x], need not be closed under infinite union. even a countable union of zero-sets need not be a zero-set. for received 21 may 2021 – accepted 20 september 2021 http://dx.doi.org/10.4995/agt.2022.15668 f. golrizkhatami and a. taherifar example, every one-element set in r is a zero-set in r, but q = ⋃ r∈q{r} is not a zero-set. first, we call a countable subfamily of z[x] a cz-family if the union of its elements is a zero-set (cf. definition 3.1). a question for us was: when is any countable subfamily of z[x] a cz-family? we observe that in a space x every countable subfamily is a cz-family if and only if x is a p-space (cf. proposition 3.2). in 5.15 of [8], it is shown that if a countable union of zero-sets belongs to a real z-ultrafilter a, then at least one of them belongs to a. but we need the converse of this fact for our aims. it is shown that for an ideal i of c(x), ⋃∞ i=1 z(fi) ∈ z[i] implies fi ∈ i for some i ∈ n if and only if i is a real maximal ideal (cf. proposition 3.3). we apply this result and prove that for any countable cz-family {z1,z2, ...,zn, ...} of z[x], clβx( ⋃ i∈n zi) = ⋃ i∈n clβxzi if and only if x is a pseudocompact space (cf. theorem 3.6). in a general space, even a closed, countable union of zero-sets need not be a zero-set; see 6p.5 in [8]. this was a motivation for introducing the class of cz-spaces in this paper. in section 4, we introduce cz-spaces as those spaces which in the closure of any countable union of zero-sets is a zero-set (cf. definition 4.1). we observe that a space x is a cz-space if and only if the set of basic z-ideals is closed under countable intersection, i.e., for every countable subset f1,f2, ...,fn, ... of c(x) there exists f ∈ c(x) such that ⋂ i∈n mfi = mf (cf. lemma 4.3). every open z-embedded subset of a cz-space (hence open c∗-embedded and cozero-sets) is a cz-space (cf. proposition 4.4). in section 5, we give some new equivalent conditions algebraic and topological for the class of cozero complemented spaces. it is proved that a space x is cozero complemented if and only if the set of basic zo-ideals is closed under countable intersection, i.e., for every countable subset f1,f2, ...,fn, ... of c(x) there exists f ∈ c(x) such that ⋂ i∈n pfi = pf if and only if for each f ∈ c(x) there exists a g ∈ c(x) such that ann(f) = pg (cf. theorem 5.1). a topological space x is called cap-space if the closure of the interior of every zero-set in x is a zero-set (cf. definition 5.3). this class of spaces contains the class of almost p-spaces and perfectly normal spaces. we conclude that every cz-space is a cozero complemented space and a cap-space (cf. proposition 5.6). examples are given to show that the converse need not be true. we also call a topological space x, crz (resp., cz)-extended in a space t containing it if for each regular-closed zero-set (resp., zero-set) z ∈ z[x], clt z is a zeroset in t (cf. definition 5.7). examples of crz (resp., cz)-extended are given (cf. example 5.8). we prove that for a dense and z#-embedded space x in a space t , x is cap and crz-extended in t if and only if t is cap (cf. theorem 5.13). from this result, we get the following results (cf. corollary 5.13): (1) if x is a weakly lindelöf dense space in a space t , then x is cap and crz-extended in t if and only if t is cap. (2) if x is cz-extended in t , then x is cap if and only if t is a capspace. © agt, upv, 2022 appl. gen. topol. 23, no. 1 2 some class of topological spaces (3) for any completely regular space x, βx is a cap-space if and only if x is a cap and crz-extended in βx. (4) if x is a cz-extended in βx, then βx is a cap-space if and only if x is so. 2. preliminaries in this paper, all spaces are completely regular hausdorff and c(x) (c∗(x)) is the ring of all (bounded) real-valued continuous functions on a space x. for each f ∈ c(x), the zero-set of f denoted by z(f) is the set of zeros of f and cozf is the set x \z(f) which is called the cozero-set of f. the set of all zero-sets in x is denoted by z[x] and for each ideal i in c(x), z[i] is the set of all zero-sets of the form z(f), where f ∈ i. the support of f ∈ c(x), is the set clx(x \z(f)). the space βx is known as the stone-c̆ech compactification of x. it is characterized as that compactification of x in which x is c∗-embedded as a dense subspace. the space υx is the real compactification of x, if x is c-embedded in this space as a dense subspace. for a completely regular hausdorff space x, we have x ⊆ υx ⊆ βx. whenever z = z(f) ∈ z[x], we denote z(fβ) with zβ, where fβ is the extension of f to βx. by a z-ultrafilter on x is meant a maximal z-filter, i.e., one not contained in any other z-filter. when m is a real maximal ideal in c(x), we refer to z[m] as a real-z-ultrafilter. thus, the real z-ultrafilters are those with the countable intersection property. for any p ∈ βx, op (resp., mp) is the set of all f ∈ c(x) for which p ∈ intβx clβx z(f) (resp., p ∈ clβx z(f)). also, for a ⊆ βx, oa (resp., ma) is the intersection of all op(resp., mp) where p ∈ a, and whenever a ⊆ x, we denote it by oa (resp., ma). the intersection of all minimal prime ideals of c(x) (resp., maximal ideals of c(x)) containing f is denoted by pf (resp., mf ). it is proved in [5] that pf = {g ∈ c(x) : intx z(f) ⊆ intx z(g)} and mf = {g ∈ c(x) : z(f) ⊆ z(g)}. an ideal i of c(x) is a z-ideal if for each f ∈ c(x), mf ⊆ i. for f ∈ c(x), ann(f) = {g ∈ c(x) : fg = 0} and it is easy to see that ann(f) = mx\z(f). the reader is referred to [8] for more details on c(x). 3. countable union of zeros-sets definition 3.1. a countable subfamily {z1,z2, ...,zn, ...} of z[x] is called a cz-family if ⋃∞ i=1 zi is a zero-set. if we consider the real numbers r with the usual topology and consider a countable zero-set z ∈ z[r] (e.g., z = z(f), where f(x) = cos x), then {{x} : x ∈ z} is a cz-family and in this space {{x} : x ∈ q}, where q is the set of rational numbers, is not a cz-family of z[r]. recall from [8], a space x is a p-space if c(x) is a regular space, i.e., every zero-set in x is open. the next result shows that whenever x is a non p-space, then there is a countable subfamily of z[x] which is not a cz-family. © agt, upv, 2022 appl. gen. topol. 23, no. 1 3 f. golrizkhatami and a. taherifar proposition 3.2. the following statements are equivalent. (1) every countable subfamily of z[x] is a cz-family. (2) x is a p -space. (3) for every countable subset {z1,z2, ...,zn, ...} of z[x], ⋃ i∈n intx zi is a zero-set. (4) every z-ultrafilter on x is closed under countable union. proof. (1)⇒(2) as every cozero-set is a countable union of zeros-sets, this is evident by [8, 4.j]. (2)⇒(3) by [8, 4.j], every zero-set is open, so this is obvious. (3)⇒(4) let z[mp] be a z-ultrafilter on x. by hypothesis and the fact that every cozero-set is a countable union of the interior of zero-sets, every cozero-set is a zero-set. that is every zero-set is a cozero-set. now assume {z(fi)|i ∈ n} is a countable subset of z[mp]. then for each i ∈ n, there exists a cozero-set x\z(gi) such that z(fi) = x\z(gi). hence, ⋃ i∈n z(fi) = ⋃ i∈n(x\z(gi)) = x\ ⋂ i∈n z(gi) = x\z(g), for some g ∈ c(x). again by hypothesis, x\z(g) is a zero-set, so ⋃ i∈n z(fi) ∈ z[m p]. (4)⇒(1) consider a countable subset s = {z1,z2, ...,zn, ...} of z[x]. for each i ∈ n, define z′i = z1 ∪ z2 ∪ ... ∪ zi. then s ′ = {z′i : i ∈ n} has the finite intersection property and ⋃ s = ⋃ s′. now, consider the collection of all zero-sets in x that contains finite intersections of the members of s′. this is a proper z-filter in x. thus this z-filter is contained in a unique z-ultrafilter, say, z[mp] for some p ∈ βx. for each i ∈ n, we have z′i ∈ z[m p]. thus by hypothesis, ⋃ i∈n zi = ⋃ i∈n z ′ i ∈ z[m p]. so s is a cz-family. � proposition 3.3. let i be an ideal of c(x). then ⋃ i∈n z(fi) ∈ z[i] implies fi ∈ i for some i ∈ n if and only if i is a real maximal ideal. proof. the sufficiency follows from [8, 5.15(a)]. necessity. first, trivially i is a z-ideal. next, for each n ∈ n, put zn = {x ∈ x : |f(x)| ≥ 1/n} and suppose that f /∈ i. we have z(f) ∪ ( ⋃ n∈n zn) = x ∈ z[i]. thus zn ∈ z[i], for some n ∈ n. but zn is of the form z(1 −gf) for some g ∈ c(x), see lemma 2.1 in [2]. thus z(1 − gf) ∈ z[i] for some g ∈ c(x). this shows that 1−gf ∈ i, i.e., i is a maximal ideal. now we want to prove i is a real ideal. by [8, theorem 5.14], it is enough to show that i has the countable intersection property. to see this, let for each n ∈ n, zn ∈ z[i] and ⋂ n∈n zn = ∅. then ⋃ n∈n(x \ zn) = x ∈ z[i]. as every x \ zn is a countable union of zero-sets, we have a zero-set z ∈ z[i] contained in some x \zn. this contradicts with zn ∈ z[i]. � it is well known that υx = {p ∈ βx : mp is real }. now by using the above theorem we obtain the following result. corollary 3.4. let p ∈ βx. then p ∈ υx if and only if for each cz-family {z(fi) : i ∈ n}, p ∈ clβx( ⋃∞ i=1 z(fi)) implies p ∈ clβx z(fi) for some i ∈ n. we again apply proposition 3.3 for proving the next results. © agt, upv, 2022 appl. gen. topol. 23, no. 1 4 some class of topological spaces corollary 3.5. let {z1,z2, ...,zn, ...} be a cz-family in z[x]. (1) clυx( ⋃∞ i=1 zi) = ⋃∞ i=1 clυx zi. (2) the set {zυ1 ,zυ2 , ...,zυn, ...,} is a cz-family in z[υx]. proof. (1) trivially, ⋃∞ i=1 clυx zi ⊆ clυx( ⋃∞ i=1 zi). now for the proof of the other inclusion, let p ∈ clυx( ⋃∞ i=1 zi). then this and hypothesis imply that⋃∞ i=1 zi ∈ z[m p]. hence there exists i ∈ n such that zi ∈ z[mp], by proposition 3.3. this shows that p ∈ clυx zi, and so p ∈ ⋃∞ i=1 clυx zi. so we are done. (2) let ⋃∞ i=1 zi = z, where z ∈ z[x]. by (1), ⋃∞ i=1 z υ i = ⋃∞ i=1 clυx zi = clυx( ⋃∞ i=1 zi) = clυx z = z υ. � the cz-family condition for {z1,z2, ...,zn, ...} in part 1 of the above result is necessary. for, consider q as a subspace of r with the usual topology. as mentioned the set {{x} : x ∈ q} is not a cz-family and for each x ∈ q, {x} is a zero-set. however, clr ⋃ x∈q{x} 6= ⋃ x∈q{x}. theorem 3.6. the following statements hold. (1) for every cz-family {z1,z2, ...,zn, ...} of z[x], we have clβx( ⋃∞ i=1 zi) = ⋃∞ i=1 clβx zi if and only if x is pseudocompact. (2) for every countable subset {z1,z2, ...,zn, ...} of z[x], we have intβx clβx( ⋃∞ i=1 zi) = ⋃∞ i=1 intβx clβx zi, if and only if x is finite. (3) for every countable subset {z1,z2, ...,zn, ...} of z[x], we have intx( ⋃∞ i=1 zi) = ⋃∞ i=1 intx zi if and only if x is a p -space. proof. (1) necessity. let p ∈ βx\υx. then for each n ∈ n, there exists zn ∈ z[mp] such that ⋂∞ n=1 zn = ∅. thus ⋃∞ n=1(x\zn) = x. as each x\zn is a countable union of zero-sets say ⋃∞ m=1 zmn, so we have ⋃∞ n=1( ⋃∞ m=1 zmn) = x. this shows that the set {zmn : m,n ∈ n} is a cz-family in z[x]. so, by hypothesis, ⋃∞ n=1( ⋃∞ m=1 clβx zmn) = βx. thus there exist m,n ∈ n such that zmn ⊆ x \ zn and p ∈ clβx zmn. this shows that zmn ∈ z[mp], a contradiction. sufficiency, we have βx = υx, so this follows from corollary 3.5. (2) necessity. suppose that p ∈ βx and ⋃∞ n=1 zn ∈ z[o p]. then p ∈ intβx clβx( ⋃∞ i=1 zi) = ⋃∞ i=1 intβx clβx zi. thus there exists n ∈ n such that p ∈ intβx clβx zn, i.e., zn ∈ z[op]. by proposition 3.3, op = mp and p ∈ υx (i.e., βx = υx). thus by [8, 7l], x is a p-space and βx = υx implies x is pseudocompact, by [8, 8a.2]. hence by [8, 4k.2], x is finite. sufficiency, x is finite, so this is obvious. (3) necessity. the proof is similar to the (2). sufficiency, x is a p-space. thus by proposition 3.2, ⋃∞ i=1 zi is a zero-set in x and so p ∈ intx( ⋃∞ i=1 zi) implies ⋃∞ i=1 zi ∈ z[op] = z[mp]. thus, by proposition 3.3, zi ∈ z[op], i.e., p ∈ intx zi, for some i ∈ n. the other inclusion always holds, so we are done. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 5 f. golrizkhatami and a. taherifar 4. cz-space definition 4.1. a topological space x is called a cz-space if the closure of any countable union of zero-sets is a zero-set. example 4.2. (1) by proposition 3.2, every p-space is a cz-space. (2) every perfectly normal space (e.g., a metric space) is a cz-space. so the space of real numbers with usual topology is a cz-space which is not a p-space. (3) in [8, 6p.5], n is a closed discrete subset of the space λ = βr\(βn\n), hence is a closed countable union of zero-sets. however it is not a zeroset. thus λ is not a cz-space. (4) in [8, 4.n], s is a non-discrete p-space and hence this is a cz-space. however it contains a closed subset which is not a zero-set. so this is an example of cz which is not a perfectly normal space. let us call a z-ultrafilter f a cz-ultrafilter if for each countable subset {z1,z2, ...,zn, ...} of f, clx( ⋃ i∈n zi) ∈f. lemma 4.3. the following statements are equivalent. (1) the space x is a cz-space. (2) for every countable subset {f1,f2, ...,fn, ...} of c(x) there exists f ∈ c(x) such that ⋂ i∈n mfi = mf . (3) every z-ultrafilter on x is a cz-ultrafilter. proof. (1)⇒(2) consider an arbitrary countable subset {f1,f2, ...,fn, ...} of c(x). by hypothesis, there exists f ∈ c(x) such that clx( ⋃ i∈n z(fi)) = z(f). thus we have: m⋃ i∈n z(fi) = mclx( ⋃ i∈n z(fi)) = mz(f) = mf. trivially m⋃ i∈n z(fi) = ⋂ i∈n mfi . so we are done. (2)⇒(3) let {z(f1),z(f2), ...z(fn), ...} be a countable subset of a z-ultrafilter z[mp]. by hypothesis, ⋂ i∈n mfi = mf , for some f ∈ c(x). this equality shows that m⋃ i∈n z(fi) = mz(f). hence clx( ⋃ i∈n z(fi)) = z(f). thus clx( ⋃ i∈n z(fi)) ∈ z[m p]. (3)⇒(1) consider a countable subset s = {z1,z2, ...,zn, ...} of z[x]. for each i ∈ n, define z′i = z1 ∪ z2 ∪ ... ∪ zi. then s ′ = {z′i : i ∈ n} has the finite intersection property and ⋃ s = ⋃ s′. now, consider the collection of all zero-sets in x that contains finite intersections of members of s′. this is a proper z-filter in x. thus this z-filter is contained in a unique z-ultrafilter, say, z[mp] for some p ∈ βx. for each i ∈ n, we have z′i ∈ z[m p]. thus by hypothesis, clx( ⋃ i∈n zi) = clx( ⋃ i∈n z ′ i) ∈ z[m p]. so x is a cz-space. � proposition 3.2 shows that whenever x is a p-space, then every z-ultrafilter is a cz-ultrafilter. however, if x is a cz-space which is not a p-space (e.g., r with usual topology), then there is a cz-ultrafilter which is not closed under countable union. © agt, upv, 2022 appl. gen. topol. 23, no. 1 6 some class of topological spaces recall from [6], a subset s of the topological space x is z-embedded if each zero-set of s is the restriction to s of a zero-set of x. now we will see that the open z-embedded subsets inherit the cz-property from the space. proposition 4.4. the following statements hold. (1) every open z-embedded subspace of a cz-space is a cz-space. (2) every cozero-set in a cz-space is a cz-space. (3) every open c∗-embedded (resp., c-embedded) subspace of a cz-space is a cz-space. proof. (1) let s be an open z-embedded subspace of x and {zi : i ∈ n} be a countable subset of z[s]. by hypothesis, for each i ∈ n, there exists z′i ∈ z[x] such that z′i ∩ s = zi. x is a cz-space, so there exists z ′ ∈ z[x] with clx( ⋃ i∈n z ′ i) = z ′. it is easy to see that cls( ⋃ i∈n zi) = clx( ⋃ i∈n zi) ∩s = clx( ⋃ i∈n z′i ∩s) ∩s = clx( ⋃ i∈n z′i) ∩s = z ′∩s. this shows that s is a cz-space. (2) by [6, proposition 1.1], every cozero-set in x is an open z-embedded. so this follows from (1). (3) trivially every open c∗-embedded (resp., c-embedded) subspace is a z-embedded set, so this follows from (1). � a space x is an f-space (resp., f ′-space) if disjoint cozero subsets of x are contained in disjoint zero sets (resp., if disjoint cozero subsets have disjoint closures). as every cozero-set is a countable union of zero-sets, whenever x is a cz-space, the closure of every cozero subset is a zero-set. thus we obtain the following result. corollary 4.5. if x is an f ′-space and a cz-space, then it is an f -space. in the sequel we characterize some topological properties of the classes of cz-spaces. recall from [10] that if f : x → y is a continuous surjection map and f(z[x]) ⊆ z[y ], then f is said to be zero-set preserving. the following result is lemma 3.20 of [10]. lemma 4.6. an open perfect surjection is zero-set preserving. theorem 4.7. the following statements hold. (1) if f : x → y is open and zero-set preserving and x is cz, then y is cz. (2) if x is compact and x ×y is cz, then y is cz. proof. (1) let {z(fi) : i ∈ n} be a countable subset of z[y ]. then {f−1(z(fi)) : i ∈ n} ⊆ z[x]. since, for each i ∈ n, f−1(z(fi)) = z(fiof) ∈ z[x]. x is a cz-space, so there exists z(g) ∈ z[x] such that clx( ⋃ i∈n f −1(z(fi))) = z(g). we claim that cly ( ⋃ i∈n z(fi)) = f(z(g)), which is a zero-set in y , by hypothesis. to see this, let y ∈ cly ( ⋃ i∈n z(fi)). we have y = f(x), for some x ∈ x. it is enough to show that x ∈ z(g), i.e., x ∈ clx( ⋃ i∈n f −1(z(fi))). let u be an © agt, upv, 2022 appl. gen. topol. 23, no. 1 7 f. golrizkhatami and a. taherifar open set in x containing x. then f(u) is open in y and containing y and hence f(u)∩( ⋃ i∈n z(fi)) 6= ∅. thus f(u)∩z(fi) 6= ∅, for some i ∈ n. this implies u ∩f−1(z(fi)) 6= ∅, for some i ∈ n. hence u ∩ ( ⋃ i∈n f −1(z(fi))) 6= ∅, i.e., x ∈ clx( ⋃ i∈n f −1(z(fi))). now assume y = f(x) ∈ f(z(g)), where x ∈ z(g) and g be an open set in y containing y. then x ∈ f−1(g), which is open in x. thus f−1(g) ∩ ⋃ i∈n f −1(z(fi))) 6= ∅. hence f−1(g) ∩ f−1(z(fi)) 6= ∅, for some i ∈ n. this implies g∩z(fi) 6= ∅, i.e., g∩ ( ⋃ i∈n z(fi)) 6= ∅. thus y ∈ cly ( ⋃ i∈n z(fi)). (2) the map πy : x ×y → y is an open perfect map (since x is compact) and surjective. thus it is zero-set preserving, by lemma 4.6. so this follows from (1). � as we found algebraic equivalent for a cz-space in lemma 4.3 and the fact that c(x) ' c(υx) we obtain the following result. proposition 4.8. the following statements hold. (1) if c(x) is isomorphic with c(y ) (as two rings) and x is a cz-space, then y is a cz-space. (2) x is a cz-space if and only if υx is a cz-space. (3) if x is pseudocompact and cz, then βx is a cz-space. 5. cz-space and other classes of topological spaces a space x is cozero complemented if, given any cozero set u, there is a disjoint cozero set v such that u∪v is dense in x. in [4], this class of space is called m-space, i.e., every prime zo-ideal of c(x) is minimal. by proposition 1.5 in [4], x is cozero complemented if and only if for every zero-set z ∈ z[x] there exists a zero-set f ∈ z[x] such that z∪f = x and intx z∩intx f = ∅. by corollary 5.5 in [9], this is equivalent to compactness of the space of minimal prime ideals of c(x). in this section we give some another algebraic and topological equivalent conditions for this class of spaces and conclude that every cz-space is a cozero complemented space. some topological properties of cozero complemented spaces are also characterized. we also introduce some other classes of topological spaces which are used in the sequel. in particular, we introduce a large class of topological spaces, which are called cap-spaces (cf. definition 4.3), and observe that these spaces, although they are different from the cozero-complemented spaces, behave in a similar manner as the latter ones. we also provide several examples (cf. examples 5.4 and 5.8). theorem 5.1. the following statements are equivalent. (1) the closure of any countable union of the interior of zero-sets is the closure of the interior of a zero-set. (2) for every countable subset {f1,f2, ...,fn, ...} of c(x) there exists f ∈ c(x) such that ⋂ i∈n pfi = pf . (3) for each countably generated ideal i of c(x), there exists g ∈ c(x) such that ann(i) = pg. (4) for each f ∈ c(x), there exists g ∈ c(x) such that ann(f) = pg. © agt, upv, 2022 appl. gen. topol. 23, no. 1 8 some class of topological spaces (5) x is a cozero complemented space. (6) every support in x is the closure of the interior of a zero-set. proof. (1)⇒(2) let {f1,f2, ...,fn, ...} be a countable subset of c(x). clearly, o⋃ i∈n intx z(fi) = ⋂ i∈n pfi. by hypothesis, clx( ⋃ i∈n intx z(fi)) = clx(intx z(f)), for some f ∈ c(x). since ⋃ i∈n intx z(fi) is open, we have, o⋃ i∈n intx z(fi) = m⋃ i∈n intx z(fi) = mclx( ⋃ i∈n intx z(fi) = mclx(intx z(f)) = mintx z(f) = ointx z(f) = pf. this implies ⋂ i∈n pfi = pf . (2)⇒(3) let i be an ideal of c(x) generated by {f1,f2, ...,fn, ...}. for fi (1 ≤ i ≤ n) , there exists a countable subset {f1i,f2i, ...,fmi, ...} of c(x) such that x \z(fi) = ⋃ m∈n intxz(fmi). trivially we have, ann(i) = m⋃ i∈n(x\z(fi)) = m ⋃ i∈n ⋃ m∈n intxz(fmi) = ⋂ i∈n ⋂ m∈n ointxz(fmi) = ⋂ i∈n ⋂ m∈n pfmi. by hypothesis, ⋂ i∈n ⋂ m∈n pfmi = pg for some g ∈ c(x). so we are done. (3)⇒(4) trivial. (4)⇒(5) let clx(x\z(f)) be a support in x. by hypothesis, clx(intx z(f)) = clx(x \ z(g)), for some g ∈ c(x). get complement of two hands of the equality, we have intx(clx(x \z(f))) = intx(z(g)). hence clx(intx(clx(x \ z(f))) = clx(intx(z(g))). it is easy to see that clx(intx(clx(x \ z(f))) = clx(x \z(f)). thus clx(x \z(f)) = clx(intx(z(g))). (5)⇒(1) let {z(f1),z(f2), ...,z(fn)} be a countable subset of z[x]. by hypothesis, for each i ∈ n, there exists gi ∈ c(x) such that intx z(fi) = intx clx(x \z(gi)). thus we have, clx( ∞⋃ i=1 intx z(fi) = clx( ∞⋃ i=1 intx clx(x \z(gi)) = clx( ∞⋃ i=1 (x \z(gi))) = clx(x \ ∞⋂ i=1 z(gi). there exists g ∈ c(x) such that ⋂∞ i=1 z(gi) = z(g). thus clx( ⋃∞ i=1 intx z(fi)) = clx(x \z(g), which is the closure of the interior of some zero-set, by hypothesis. � proposition 5.2. the following statements are equivalent. (1) every support in x is a zero-set. (2) the closure of any countable union of the interior of zero-sets is a zero-set. © agt, upv, 2022 appl. gen. topol. 23, no. 1 9 f. golrizkhatami and a. taherifar (3) for every countable subset {f1,f2, ...,fn, ...} of c(x) there exists f ∈ c(x) such that ⋂ i∈n pfi = mf . (4) for each countably generated ideal i of c(x), there exists g ∈ c(x) such that ann(i) = mg. (5) for each f ∈ c(x), there exists g ∈ c(x) such that ann(f) = mg. proof. (1)⇒(2) let {z(fi) : i ∈ n} be a countable subset of z[x]. by hypothesis, for each i ∈ n, there exists a cozero-set x \ z(gi) such that intx z(fi) = x \z(gi). thus clx( ⋃∞ i=1 intx z(fi)) = clx( ⋃∞ i=1(x \z(gi))) = clx(x \ ⋂∞ i=1 z(gi)). there exists g ∈ c(x) such that ⋂∞ i=1 z(gi) = z(g). hence clx( ⋃∞ i=1 intx z(fi)) = clx(x \z(g)), which is a zero-set, by hypothesis. (2)⇒(3) let {f1,f2, ....,fn, ...} be a countable subset of c(x). clearly,⋂∞ i=1 pfi = ⋂∞ i=1 ointx z(fi) = o ⋃∞ i=1 intx z(fi). by hypothesis, clx( ⋃∞ i=1 intx z(fi)) = z(f), for some f ∈ c(x). hence o⋃∞ i=1 intx z(fi) = m ⋃∞ i=1 intx z(fi) = mz(f) = mf , i.e., ⋂∞ i=1 pfi = mf . (3)⇒(4) the proof is similar to the proof of (2)⇒(3) of theorem 5.1, step by step. (4)⇒(5) trivial. (5)⇒(1) consider clx(x\z(f)) as a support. by hypothesis, ann(f) = mg for some g ∈ c(x). thus mx\z(f) = mg = mz(g). this implies clx(x \ z(f)) = clx(z(g)) = z(g). � a point p ∈ x is said to be an almost p-point if f ∈ mp, intx z(f) 6= ∅, and x is called an almost p-space if every point of x is an almost p-point. it is easy to see that a space x is an almost p-space if and only if every zero-set in x is regular-closed. the reader is referred to [1], [7], [11] and [13], for more details and properties of almost p -spaces. definition 5.3. a space x is called a c-almost p-space (briefly cap-space) if the closure of the interior of every zero-set in x is a zero-set. example 5.4. (1) clearly every almost p-space is a cap-space. (2) every cz-space is a cap-space. for, let z ∈ z[x]. then clx(x \ z) = x \ intx z = z(f), for some f ∈ c(x). thus clx intx(z) = clx(x\z(f). as x is cz, clx(x\z(f)) is a zero-set, hence clx intx z is a zero-set. this implies every perfectly normal space (hence a metric space) is a cap-space. thus r with usual topology is a cap-space which is not an almost p-space. (3) clearly every oz-space (i.e., a space which in every regular-closed subset is a zero-set) is a cap-space. lemma 5.5. a space x is a cap -space if and only if every basic zo-ideal of c(x) is a basic z-ideal. © agt, upv, 2022 appl. gen. topol. 23, no. 1 10 some class of topological spaces proof. ⇒ let pf be a basic zo-ideal. by hypothesis, there exists g ∈ c(x) such that clx(intx z(f)) = z(g). thus we have, pf = ointx z(f) = mintx z(f) = mclx(intx(z(f))) = mz(g) = mg. ⇐ let z(f) ∈ z[x]. there exists g ∈ c(x) such that pf = mg. since pf = ointx z(f) and mg = mz(g). the equality pf = mg implies clx(intx z(f)) = z(g). so x is a cap-space. � proposition 5.6. the following statements hold. (1) every cz-space is a cozero complemented space. (2) every support in x is a zero-set if and only if x is a cozero complemented space and a cap -space. proof. (1) x is a cz-space. thus every support is a zero-set, so by proposition 5.2, for each f ∈ c(x) there exists g ∈ c(x) such that ann(f) = mg. as ann(f) is a zo-ideal, this implies mg is a z o-ideal and hence equals with pg. now this follows from theorem 5.1. (2) first assume every support in x is a zero-set and z ∈ z[x]. then clx(x \z) = z(f) for some f ∈ c(x). this implies clx(intx(z)) = clx(x \ z(f)), i.e., x is a cozero complemented space, by proposition 1.5 in [4]. again by hypothesis, clx(x\z(f)) is a zero-set. thus x is a cap-space. conversely, let clx(x\z) be a support. by hypothesis, clx(x\z) = clx(intx(z(g))), for some g ∈ c(x). as x is a cap-space, clx(intx(z(g))) is a zero-set. thus clx(x \z) is a zero-set. � part 2 of the above result shows that if x is a cap-space which is not a cozero complemented space or a cozero complemented space which is not a cap-space, then there is a non zero-set support in x. thus x is not a cz-space. to see examples, first, consider the space x as the one point compactification of an uncountable discrete space. then x is an almost pspace, since any non-empty gδ of it contains an isolated point of the space, hence is a cap-space. but this is not a cozero complemented space and hence is not a cz-space, see example 3.3 in [4]. next, consider the space λ = βr \ (βn \ n) in [8, 6p.5]. we have βλ = βr. thus βλ is a cozero complemented and hence λ is a cozero complemented space. however, we know that it is not a cz. on the other hand, by [8, 6p.5], λ is pseudocompact, so if βλ = υλ is cz, then we must have λ is cz, by proposition 4.8, which is not true. thus βλ = βr is not a cz-space while we know that r is a cz-space. henriksen and woods in [10] showed that for an uncountable discrete space d, the stone-c̆hech compactification βd of d is cozero complemented but βd × βd is not, and so by part (1) of proposition 5.6, βd × βd is not a cz-space. definition 5.7. a subspace x of a space t is called crz (resp., cz)extended in t if for each regular-closed zero-set (resp., zero-set) z ∈ z[x], clt z is a zero-set in t . © agt, upv, 2022 appl. gen. topol. 23, no. 1 11 f. golrizkhatami and a. taherifar example 5.8. (1) let x be a c∗-embedded in t . if x is c-embedded in t , then it is cz-extended in t . for, clt z(f) = z(f o), for f ∈ c(x) and fo is the continuous extension of f to t. (2) every pseudocomact space is a cz-extended in βx. for, if x is pseudocompact, then βx = υx and for each z ∈ z[x], clβx z = clυx z = zυ = zβ. (3) consider the infinite p-space x (e.g., the discrete space n). then every zero-set z ∈ z[x] is open and hence clβx z is clopen in βx. thus it is a zero-set in βx. this says that x is a cz-extended in βx. however x need not be c-embedded in βx. (4) trivially every cz-extended in a space containing it, is a crz-extended. but the space σ = n ∪{σ} in [8, 4m] is crz-extended in βς = βn which is not a cz-extended in βn, see [8, 6e]. (5) trivially every cz-extended x in a space containing it, is z-embedded. however a z-embedded need not be a cz-extended. for example, σ is z-embedded in βς, but is not cz-extended in it. recall from [10], let x and t be two completely regular spaces and x be a subspace of t . x is said to be z#-embedded in t if for each f ∈ c(x), there exists a g ∈ c(t) such that clx(intx(z(f))) = clt (intt (z(g)))∩x. the following lemma is proved in [10]. lemma 5.9. if x is a subspace of t that is either open or dense, then the following are equivalent. (1) x is z#-embedded in t . (2) if z ∈ z[x], then there is a zt ∈ z[t ] such that intx z = (intt zt) ∩ x. lemma 5.10. suppose that x is dense or open as well as being z#-embedded in a space t . (1) if t is cap , then so is x. (2) if every support in t is a zero-set, then every support in x is a zero-set. proof. (1) first assume x is dense in t. we show x is a cap-space. let z ∈ z[x]. by hypothesis and lemma 5.9, there is zt ∈ z[t ] such that intx z = intt z t ∩x. thus clx(intx z) = clx(intt zt ∩x) = clt (intt zt ∩ x)∩x = clt (intt zt)∩x. by hypothesis, there exists z(f) ∈ z[t] such that clt (intt z t) = z(f). hence clx(intx z) = z(f) ∩ x which is a zero-set in x. if x is open in t , then for each p ∈ clt (intt zt) ∩ x and u open in x containing p, we have u is open in t and so u∩intx z = u∩intt (zt)∩x 6= ∅. hence clx(intx z) = clt (intt z t) ∩x. so we are done. (2) every support in t is a zero-set, so by proposition 5.6, t is a cozero complemented space and a cap-space. thus x is a cozero complemented space, by lemma 2.5 in [10]. also, by part 1, x is cap. now proposition 5.6 implies every support in x is a zero-set. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 12 some class of topological spaces since every open (dense) z-embedded subset is z#-embedded, and every cozero (resp., c∗-embedded)-subset is z-embedded in any space containing it, the above lemma implies next result. corollary 5.11. the following statements hold. (1) every open (dense) z-embedded subspace of a cap -space is cap . (2) every cozero-set in a cap -space is cap . (3) every open (dense) c∗-embedded in a cap -space is a cap . recall that if every open cover of a space x contains a countable subfamily whose union is dense in x, then x is called weakly lindelöf space. every lindelöf space and every ccc-space is weakly lindelöf space, while an uncountable discrete space is not a weakly lindelöf space. corollary 5.12. if s ∩ w is a weakly lindelöf space where s is a dense subspace and w is an open subspace of a cap -space t , then s ∩w is cap . proof. similar to the proof of theorem 2.6 in [10], we have clw (s ∩ w) = clt (s ∩ w) ∩ w = clt w ∩ w = w . thus s ∩ w is dense in w . so, since s ∩ w is weakly lindelöf, so is w . we have w is open in t, hence this is z#-embedded, by lemma 2.4 of [10]. thus w is cap, by lemma 5.10. but s ∩w is dense in w , so by lemma 2.4 in [10], s ∩w is z#-embedded in w . thus by lemma 5.10, we are done. � theorem 5.13. let x be dense and z#-embedded in a space t . then the following statements hold. (1) t is a cap -space if and only if x is a cap -space and crz-extended in t . (2) every support in t is a zero-set if and only if every support in x is a zero-set and x is crz-extended in t . proof. (1)⇒ part 1 of lemma 5.10 shows x is a cap-space. now, assume z(f) ∈ z[x] be a regular-closed zero-set. then z(f) = clx(intx(z(f))) and by hypothesis, there exists ft ∈ c(t) such that intx z(f) = intt z(ft) ∩ x. t is cap, hence there is a g ∈ c(t) such that clt intt z(ft) = z(g). thus we have, clt z(f) = clt (clx(intx(z(f)))) = clt (clx(intt (z(f t)) ∩x)) = clt (intt (z(f t)) ∩x) = clt (intt (z(ft))) = z(g). this completes the proof. ⇐ let z(ft) ∈ z[t]. then z(ft) ∩ x ∈ z[x]. x is a cap-space, hence clx(intx(z(f t)∩x)) = z(g) for some g ∈ c(x). thus z(g) is a regular-closed set. on the other hand, we have intx(z(f t) ∩x) = intt z(ft) ∩x. to see it, intt z(f t) ∩x is open in x and contained in z(ft) ∩x. thus it is contained in intx(z(f t) ∩ x). now, suppose p ∈ intx(z(ft) ∩ x). then p ∈ x and there is an open subset u of t such that p ∈ u ∩ x ⊆ z(ft). this implies © agt, upv, 2022 appl. gen. topol. 23, no. 1 13 f. golrizkhatami and a. taherifar p ∈ u ⊆ clt u = clt (u ∩x) ⊆ z(ft), i.e., p ∈ intt z(ft)∩x. it is easy to see that; clt (intt (z(f t))) = clt (intt (z(f t)) ∩x) = clt (intx(z(ft) ∩x)) = clt (clx(intx(z(f t) ∩x)) = clt (z(g)). as x is crz-extended in t , clt z(g) is a zero-set in t, so clt (intt (z(f t))) is a zero-set in t , i.e., t is a cap-space. (2) ⇒ part 2 of lemma 5.10 implies every support in x is a zero-set. on the other hand, t is cap, by proposition 5.6. thus by part (1), for any regularclosed zero-set z in x, clt z is a zero-set in t , i.e., x is crz-extended in t . ⇐ every support in x is a zero-set, hence x is a cozero complemented space and a cap-space, by proposition 5.6. this implies t is a cozero complemented space, by theorem 2.8 in [10]. moreover, t is a cap, by part (1). now, again by using proposition 5.6, we have every support in t is a zero-set. � it is easy to see that if x is cz-extended in t, then x is z#-embedded in t . so we conclude the following result from the above theorem. corollary 5.14. the following statements hold. (1) if x is weakly lindelöf and dense in t , then t is cap if and only if x is cap and crz-extended in t . (2) if x is cz-extended in t , then x is cap if and only if t is a cap -space. (3) βx is a cap -space if and only if x is a cap and crz-extended in βx. (4) if x is a cz-extended in βx, then βx is a cap -space if and only if x is so. (5) every support in βx is a zero-set if and only if every support in x is a zero-set and x is crz-extended in βx. (6) if x is a cz-extended in βx, then every support in βx is a zero-set if and only if every support in x is a zero-set. 6. directions and some questions cz − space. as we have shown a space x is cz if and only if the set of basic z-ideals is closed under countable intersection. this shows that this class of spaces is important. so we may have focus on the spaces max (c(x)) and spec(c(x)), with zariski topology, whenever x is a cz-space. on the other hand, since the set l = {mf : f ∈ c(x)} is a lattice with two operations: mf ∨ mg = mf2+g2 and mf ∧ mg = mf ∩ mg = mfg. so, we have x is a cz-space if and only if for every countable subset s of l, ∧s ∈ l. this can help us to investigate more properties of cz-spaces by the lattice properties of l. we have seen that if x is a pseudocompact and cz, then βx is a cz-space. as an example, we have seen that r is a cz-space, but βr is not a cz. however, this is a remainder question as follows: © agt, upv, 2022 appl. gen. topol. 23, no. 1 14 some class of topological spaces question 6.1. when is βx a cz-space? we also have some other questions as follows: question 6.1. is every cz hereditarily cz (i.e., all subspaces are cz)? question 6.2. is the product of two (infinite) cz-spaces a cz? question 6.3. suppose x ×y is a cz-space. must x or y be cz? cap − space. we observed that the class of cap-spaces behaves like cozero-complemented spaces. so we may investigate other properties of capspaces. it also is important to know the relation between cap and cozerocomplemented spaces. we know that x is cozero-complemented if and only if min (c(x)) is compact. so this is a motivation to ask the following questions: question 6.4. what is min (c(x)), when x is a cz? question 6.5. what is max (c(x)), when x is a cz? we also have some questions as follows: question 6.6. characterize the cap spaces which are hereditarily cap. question 6.7. is the product of two (infinite) cap-spaces a cap? acknowledgements. the authors would like to thank the referee for her/his thorough reading of this paper and her/his comments which led to a much improved paper. references [1] f. azarpanah, on almost p-spaces, far east j. math. sci. 1 (2000), 121–132. [2] f. azarpanah, m. ghirati and a. taherifar, closed ideals in c(x) with different reperesentations, houston journal of mathematics 44, no. 1 (2018), 363–383. [3] f. azarpanah, a. a. hesari, a. r. salehi and a. taherifar, a lindelöfication, topology appl. 245 (2018), l46–61. [4] f. azarpanah and m. karavan, on nonregular ideals and zo-ideals in c(x), czechoslovak math. j. 55 (2005), 397–407. [5] f. azarpanah, o. a. s. karamzadeh and a. rezai aliabad, on z-ideals in c(x), fund. math. 160 (1999), 15–25. [6] r. l. blair and a. w. hager, extension of zero-sets and real-valued functions, math. z. 136 (1974), 41–52. [7] f. dashiel, a. hager and m. henriksen, order-cauchy completions and vector lattices of continuous functions, canad. j. math. xxxii (1980), 657-685. [8] l. gillman and m. jerison, rings of continuous functions, springer, 1976. [9] m. henriksen and m. jerison, the space of minimal prime ideals of a commutative ring, trans. amer. math. soc. 115 (1965), 110–130. [10] m. henriksen and g. woods, cozero complemented spaces; when the space of minimal prime ideals of a c(x) is compact, topology appl. 141 (2004), 147–170. © agt, upv, 2022 appl. gen. topol. 23, no. 1 15 f. golrizkhatami and a. taherifar [11] r. levy, almost p-spaces, canad. j. math. 2 (1977), 284–288. [12] a. taherifar, some new classes of topological spaces and annihilator ideals, topology appl. 165 (2014), 84–97. [13] a. i. veksler, p ′-points, p ′-sets, p ′-spaces. a new class of order-continuous measures and functionals, sov. math. dokl. 14 (1973), 1445–1450. © agt, upv, 2022 appl. gen. topol. 23, no. 1 16 @ appl. gen. topol. 23, no. 1 (2022), 213-223 doi:10.4995/agt.2022.11902 © agt, upv, 2022 numerical reckoning fixed points via new faster iteration process kifayat ullah a,∗ , junaid ahmad b and fida muhammad khan c a department of mathematical sciences, university of lakki marwat, lakki marwat 28420, pakistan (kifayatmath@yahoo.com) b department of mathematics, international islamic university, islamabad 44000, pakistan (ahmadjunaid436@gmail.com) c department of mathematics, university of science and technology, bannu 28100, pakistan (fidamuhammad809@gmail.com) communicated by m. abbas abstract in this paper, we propose a new iteration process which is faster than the leading; s [j. nonlinear convex anal. 8, no. 1 (2007), 61–79], thakur et al. [app. math. comp. 275 (2016), 147–155] and m [filomat 32, no. 1 (2018), 187–196] iterations for numerical reckoning fixed points. using this new iteration process, some fixed point convergence results for generalized α-nonexpansive mappings in the setting of uniformly convex banach spaces are proved. at the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes. 2020 msc: 47h09; 47h10. keywords: generalized α-nonexpansive mappings; uniformly convex banach space; iteration process; weak convergence; strong convergence. 1. introduction throughout this paper, we will denote the set of natural numbers by n. let x be a banach space and m be a nonempty subset of x. a mapping received 27 may 2019 – accepted 24 november 2021 http://dx.doi.org/10.4995/agt.2022.11902 https://orcid.org/0000-0002-6991-4287 k. ullah, j. ahmad and f. m. khan t : m → m is said to nonexpansive if ||tx−ty|| ≤ ||x−y||, for all x,y ∈ m. an element p ∈ m is said to be a fixed point of t if p = t(p). from now on, we will denote the set of all fixed points of t by f(t). a mapping t : m → m is said to be a quasi-nonexpansive mapping if f(t) 6= ∅ and ||t(x) −t(p)|| ≤ ||x−p|| for all x ∈ m and p ∈ f(t). it is well-known that f(t) is nonempty in the case when x is uniformly convex, t is nonexpansive and m is closed, bounded and convex; see [6, 7, 10]. a number of generalizations of nonexpansive mappings have been considered by some researchers in recent years. suzuki [17] introduced a new class of mappings known as suzuki generalized nonexpansive mappings which is a condition on mappings called condition (c) and obtained some convergence and existence results for such mappings. note that, a mapping t : m → m is said to satisfy condition (c) if 1 2 ||x−tx|| ≤ ||x−y|| implies ||tx−ty|| ≤ ||x−y||, for each x,y ∈ m. aoyama and kohsaka [4] introduced the class of α-nonexpansive mappings in the framework of banach spaces and obtained some fixed point results for such mappings. a mapping t : m → m is said to be α-nonexpansive if there exists a real number α ∈ [0, 1) such that for all x,y ∈ m, ||tx−ty||2 ≤ α||tx−y||2 + α||x−ty||2 + (1 − 2α)||x−y||2. ariza-puiz et al. [5] proved that the concept of α-nonexpansive is trivial for α < 0. it is obvious that every nonexpansive mapping is 0-nonexpansive and also every α-nonexpansive mapping with f(t) 6= ∅ is a quasi-nonexpansive. note that, in general condition (c) and α-nonexpansive mappings are not continuous (see [17] and [14] ). recently, pant and shukla [14] introduced an interesting class of generalized nonexpansive mappings in banach spaces known as generalized α-nonexpansive mappings which contains the class of suzuki generalized nonexpansive mappings. a mapping t : m → m is said to generalized α-nonexpansive if there exists a real number α ∈ [0, 1) such that for each x,y ∈ m, 1 2 ||x−tx|| ≤ ||x−y|| ⇒ ||tx−ty|| ≤ α||tx−y||+α||ty−x||+(1−2α)||x−y||. once the existence result of a fixed point for a mapping is established, an algorithm to find the value of the fixed point is desirable. the famous banach contraction mapping principle uses picard iteration xn+1 = txn for approximation of fixed point. some other well-known iterations are the mann [11], ishikawa [9], s [3], picard-s [8], noor [12], abbas [1], thakur et al. [19] and so on. speed of convergence plays an important role for an iteration process to be preferred on another iteration process. rhoades [15] mentioned that the mann iteration process for a decreasing function converges faster than the ishikawa iteration process and for an increasing function the ishikawa iteration process is better than the mann iteration process. © agt, upv, 2022 appl. gen. topol. 23, no. 1 214 numerical reckoning fixed points via new faster iteration process the well-known mann [11] and ishikawa [9] iteration schemes are respectively defined as: (1.1) { x1 ∈ m, xn+1 = (1 −αn)xn + αntxn,n ∈ n, where αn ∈ (0, 1). (1.2)   x1 ∈ m, yn = (1 −βn)xn + βntxn, xn+1 = (1 −αn)xn + αntyn,n ∈ n, where αn,βn ∈ (0, 1). in 2007, agarwal et al. [3] introduced the following iteration process known as s iteration: (1.3)   x1 ∈ m, yn = (1 −βn)xn + βntxn, xn+1 = (1 −αn)txn + αntyn,n ∈ n, where αn,βn ∈ (0, 1). they proved that the rate of convergence of iteration process (1.3) is same to the picard iteration xn+1 = txn and faster than the mann [11] iteration process in the class of contraction mappings. in 2016, thakur et al. [19] introduced the following iteration scheme: (1.4)   x1 ∈ m, zn = (1 −βn)xn + βntxn, yn = t ((1 −αn)xn + αnzn) , xn+1 = tyn,n ∈ n, where αn,βn ∈ (0, 1). with the help of a numerical example, they proved that (1.4) is faster than the picard, mann [11], ishikawa [9], s [3], noor [12] and abbas [1] iteration processes in the class of suzuki generalized nonexpansive mappings. recently in 2018, ullah and arshad [20] used a new iteration process known as m iteration: (1.5)   x1 ∈ m, zn = (1 −αn)xn + αntxn, yn = tzn, xn+1 = tyn,n ∈ n, where αn ∈ (0, 1). with the help of a numerical example, they proved that (1.5) is faster than s [3], picard-s [8] and thakur et al. [19] iteration processes for suzuki generalized nonexpansive mappings. problem 1.1. is it possible to develop an iteration process whose rate of convergence is even faster than the iteration process (1.5) ? © agt, upv, 2022 appl. gen. topol. 23, no. 1 215 k. ullah, j. ahmad and f. m. khan as an answer, we introduce the following new iteration called kf iteration scheme: (1.6)   x1 ∈ m, zn = t((1 −βn)xn + βntxn), yn = tzn, xn+1 = t((1 −αn)txn + αntyn),n ∈ n, where αn,βn ∈ (0, 1). with the help of numerical example, we compare the rate of convergence of iteration (1.6) with the leading s (1.3), thakur et al. (1.4) and m (1.5) iteration. 2. preliminaries in this section, we give some preliminaries. let x be a banach space and m be a nonempty closed convex subset of x. let {xn} be a bounded sequence in m. for x ∈ x, set r(x,{xn}) = lim sup n→∞ ||x−xn||. the asymptotic radius of {xn} relative to m is given by r(m,{xn}) = inf{r(x,{xn}) : x ∈ m}. the asymptotic center of {xn} relative to m is the set a(m,{xn}) = {x ∈ m : r(x,{xn}) = r(m,{xn})}. it is well-known that in a uniformly convex banach space setting, a(m,xn) consists of exactly one point. also, a(m,xn) is nonempty and convex when m is weakly compact and convex (see, [18] and [2]). a banach space x is said to uniformly convex if for all ε > 0, there is a λ > 0 such that, for x,y ∈ x with ||x|| ≤ 1, ||y|| ≤ 1 and ||x − y|| ≤ ε, ||x + y|| ≤ 2(1 − λ) holds. note that, a banach space x is said to have opial’s property [13] if for each sequence {xn} in x which weakly converges to x ∈ x and for every y ∈ x, it follows the following lim sup n→∞ ||xn −x|| < lim sup n→∞ ||xn −y||. examples of banach spaces satisfying this condition are hilbert spaces and all lp spaces (1 < p < ∞). we now list some basic facts about generalized α-nonexpansive mappings, which can be found in [14]. proposition 2.1. let x be a banach space, m be a nonempty subset of x and t : m → m be a mapping. (i) if t is a suzuki generalized nonexpansive mapping, then t is a generalized α-nonexpansive mapping. (ii) if t is a generalized α-nonexpansive mapping and has a fixed point, then t is a quasi-nonexpansive mapping. © agt, upv, 2022 appl. gen. topol. 23, no. 1 216 numerical reckoning fixed points via new faster iteration process (iii) if t is a generalized α-nonexpansive mapping. then f(t) is closed. moreover, if x is strictly convex and m is convex, then f(t) is also convex. (iv) if t is a generalized α-nonexpansive mapping. then for each x,y ∈ m, ||x−ty|| ≤ ( 3 + α 1 −α ) ||x−tx|| + ||x−y||. (v) if x has opial property, t is generalized α-nonexpansive, {xn} converges weakly to a point v and limn→∞ ||txn−xn|| = 0, then v ∈ f(t). lemma 2.2 ([16]). let x be a uniformly convex banach space and 0 < p ≤ αn ≤ q < 1 for every n ∈ n. if {xn} and {yn} are two sequences in x such that lim supn→∞ ||xn|| ≤ t, lim supn→∞ ||yn|| ≤ t and limn→∞ ||αnxn + (1 − αn)yn|| = t for some t ≥ 0 then, limn→∞ ||xn −yn|| = 0. 3. main results we open this section with the following important lemma. lemma 3.1. let m be a nonempty closed convex subset of a banach space x and t : m → m be a generalized α-nonexpansive mapping with f(t) 6= ∅. let {xn} be a sequence generated by (1.6), then limn→∞ ||xn − p|| exists for each p ∈ f(t). proof. let p ∈ f(t). by proposition 2.1 part (ii), we have ||zn −p|| = ||t((1 −βn)xn + βntxn) −p|| ≤ ||(1 −βn)xn + βntxn −p|| ≤ (1 −βn)||xn −p|| + βn||txn −p|| ≤ (1 −βn)||xn −p|| + βn||xn −p|| ≤ ||xn −p||, and ||yn −p|| = ||tzn −p|| ≤ ||zn −p||. they imply that, ||xn+1 −p|| = ||t((1 −αn)txn + αntyn) −p|| ≤ ||(1 −αn)txn + αntyn −p|| ≤ (1 −αn)||txn −p|| + αn||tyn −p|| ≤ (1 −αn)||xn −p|| + αn||yn −p|| ≤ (1 −αn)||xn −p|| + αn||zn −p|| ≤ (1 −αn)||xn −p|| + αn||xn −p|| ≤ ||xn −p||. © agt, upv, 2022 appl. gen. topol. 23, no. 1 217 k. ullah, j. ahmad and f. m. khan thus {||xn−p||} is bounded and nonincreasing, which implies that limn→∞ ||xn− p|| exists for all p ∈ f(t). � the following theorem is necessary for the next results. theorem 3.2. let m be a nonempty closed convex subset of a uniformly convex banach space x and t : m → m a generalized α-nonexpansive mapping. let {xn} be a sequence generated by (1.6). then, f(t) 6= ∅ if and only if {xn} is bounded and limn→∞ ||txn −xn|| = 0. proof. suppose that f(t) 6= ∅ and p ∈ f(t). then, by lemma 3.1, limn→∞ ||xn− p|| exists and {xn} is bounded. put (3.1) lim n→∞ ||xn −p|| = t. in view of the proof of lemma 3.1 together with (3.1), we have (3.2) lim sup n→∞ ||zn −p|| ≤ lim sup n→∞ ||xn −p|| = t. by proposition 2.1 part (ii), we have (3.3) lim sup n→∞ ||txn −p|| ≤ lim sup n→∞ ||xn −p|| = t. again by the proof of lemma 3.1, we have ||xn+1 −p|| ≤ (1 −αn)||xn −p|| + αn||zn −p||. it follows that, ||xn+1 −p||− ||xn −p|| ≤ ||xn+1 −p||− ||xn −p|| αn ≤ ||zn −p||− ||xn −p||. so, we can get ||xn+1 −p|| ≤ ||zn −p|| and from (3.1), we have (3.4) t ≤ lim inf n→∞ ||zn −p||. from (3.2) and (3.4), we obtain (3.5) t = lim n→∞ ||zn −p||. from (3.1) and (3.5), we have t = lim n→∞ ||zn −p|| = lim n→∞ ||t((1 −βn)xn + βntxn) −p|| ≤ lim n→∞ ||(1 −βn)xn + βntxn −p|| = lim n→∞ ||(1 −βn)(xn −p) + βn(txn −p)|| ≤ lim n→∞ (1 −βn)||xn −p|| + lim n→∞ βn||txn −p|| ≤ lim n→∞ (1 −βn)||xn −p|| + lim n→∞ βn||xn −p|| ≤ t. © agt, upv, 2022 appl. gen. topol. 23, no. 1 218 numerical reckoning fixed points via new faster iteration process hence, (3.6) t = lim n→∞ ||(1 −βn)(xn −p) + βn(txn −p)||. now from (3.1), (3.3) and (3.6) together with lemma 2.2, we obtain lim n→∞ ||txn −xn|| = 0. conversely, we assume that {xn} is bounded and limn→∞ ||txn−xn|| = 0. let p ∈ a(m,{xn}). by proposition 2.1 part (iv), we have r(tp,{xn}) = lim sup n→∞ ||xn −tp|| ≤ ( 3 + α 1 −α ) lim sup n→∞ ||txn −xn|| + lim sup n→∞ ||xn −p|| = lim sup n→∞ ||xn −p|| = r(p,{xn}). hence, we conclude that tp ∈ a(m,{xn}. since x is uniformly convex, a(m,{xn}) consist of a unique element. thus, we have p = t(p). � first we prove our weak convergence result. theorem 3.3. let x be a uniformly banach space with opial property, m a nonempty closed convex subset of x and t : m → m be generalized αnonexpansive mapping with f(t) 6= ∅. then, {xn} generated by (1.6) converges weakly to an element of f(t). proof. by theorem 3.2, {xn} is bounded and limn→∞ ||txn −xn|| = 0. since x is uniformly convex, x is reflexive. so, a subsequence {xni} of {xn} exists such that {xni} converges weakly to some v1 ∈ m. by proposition 2.1 part (v), we have v1 ∈ f(t). it is sufficient to show that {xn} converges weakly to v1. in fact, if {xn} does not converges weakly to v1. then, there exists a subsequence {xnj} of {xn} and v2 ∈ m such that {xnj} converges weakly to v2 and v2 6= v1. again by proposition 2.1 part (v), v2 ∈ f(t). by lemma 3.1 together with opial property, we have lim n→∞ ||xn −v1|| = lim i→∞ ||xni −v1|| < lim i→∞ ||xni −v2|| = lim n→∞ ||xn −v2|| = lim j→∞ ||xnj −v2|| < lim j→∞ ||xnj −v1|| = lim n→∞ ||xn −v1||. this is a contradiction, so, v1 = v2. thus, {xn} converges weakly to v1 ∈ f(t). � © agt, upv, 2022 appl. gen. topol. 23, no. 1 219 k. ullah, j. ahmad and f. m. khan we now prove our strong convergence result. theorem 3.4. let m be a nonempty closed convex subset of a uniformly convex banach space x and t : m → m be a generalized α-nonexpansive mapping. if f(t) 6= ∅ and lim infn→∞dist(xn,f(t)) = 0 (where dist(x,f(t)) = inf{||x−p|| : p ∈ f(t)}). then, {xn} generated by (1.6) converges strongly to an element of f(t). proof. by lemma 3.1, limn→∞ ||xn − p|| exists, for each p ∈ f(t). so, limn→∞dist(xn,f(t)) exists, thus lim n→∞ dist(xn,f(t)) = 0. therefore, there exists a subsequence {xnk} of {xn} and {vk} in ft such that ||xnk − vk|| ≤ 1 2k for each k ∈ n. by the proof of lemma 3.1, {xn} is nonincreasing, so ||xnk+1 −vk|| ≤ ||xnk −vk|| ≤ 1 2k . therefore, ||vk+1 −vk|| ≤ ||vk+1 −xnk+1|| + ||xnk+1 −vk|| ≤ 1 2k+1 + 1 2k ≤ 1 2k−1 → 0, as k →∞. hence, {vk} is a cauchy sequence in f(t) and so it converges to some p. since, by proposition 2.1 part (iii), f(t) is closed, we have p ∈ f(t). by lemma 3.1, limn→∞ ||xn −p|| exists, hence {xn} converges strongly to p ∈ f(t). � 4. example we compare rate of convergence of our new kf iteration (1.6) with leading s (1.3), m (1.5) thakur et al. (1.4) in slightly general setting using example 4.1, in which t is generalized α-nonexpansive but not suzuki generalized nonexpansive. example 4.1. let m = [0,∞) with absolute valued norm. define a mapping t : m → m by tx = { 0 if x ∈ [ 0, 1 5000 ) x 2 if x ∈ [ 1 5000 ,∞ ) . choose x = 1 8000 and y = 1 5000 . we see that, 1 2 |x − tx| < |x − y| but |tx − ty| > |x − y|. thus, t does not satisfy condition (c) and so t is not suzuki generalized nonexpansive. on the other hand, t is a generalized α-nonexpansive mapping. in fact, for α = 1 3 , we have: case i: when x,y ∈ [ 0, 1 5000 ) , then clearly 1 3 |tx−y| + 1 3 |x−ty| + 1 3 |x−y| ≥ 0 = |tx−ty|. © agt, upv, 2022 appl. gen. topol. 23, no. 1 220 numerical reckoning fixed points via new faster iteration process case ii: when x ∈ [ 1 5000 ,∞ ) and y ∈ [ 0, 1 5000 ) , we have 1 3 |tx−y| + 1 3 |x−ty| + 1 3 |x−y| = 1 3 ∣∣∣x 2 −y ∣∣∣ + 1 3 |x− 0| + 1 3 |x−y| ≥ 1 3 ∣∣∣(x 2 −y ) − (x−y) ∣∣∣ + 1 3 |x| = 1 3 ∣∣∣x 2 ∣∣∣ + 1 3 |x| ≥ 1 3 ∣∣∣x 2 + x ∣∣∣ = 1 2 |x| = |tx−ty|. case iii: when x,y ∈ [ 1 5000 ,∞), we have 1 3 |tx−y| + 1 3 |x−ty| + 1 3 |x−y| = 1 3 ∣∣∣x 2 −y ∣∣∣ + 1 3 ∣∣∣x− y 2 ∣∣∣ + 1 3 |x−y| ≥ 1 3 ∣∣∣(x 2 −y ) + ( x− y 2 )∣∣∣ + 1 3 |x−y| = 1 2 |x−y| + 1 3 |x−y| ≥ 1 2 |x−y| = |tx−ty|. hence, t is a generalized α-nonexpansive mapping with f(t) = {0}. take αn = 0.70 and βn = 0.65. the iterative values for x1 = 10 are given in table 1. figure 1 shows the convergence behaviors of different iterative schemes. clearly the new kf iteration process is moving fast to the fixed point of t as compared to other iteration processes. © agt, upv, 2022 appl. gen. topol. 23, no. 1 221 k. ullah, j. ahmad and f. m. khan table 1. sequences generated by kf (1.6), m (1.5), thakur et al. (1.4) and s (1.3) iteration schemes for mapping t of example 4.1. kf (1.6) m (1.5) thakur et al. (1.4) s (1.3) x1 10 10 10 10 x2 1.0453120000 1.62500000000 1.9312500000 3.8625000000 x3 0.1092678222 0.26406250000 0.3729726562 1.4918906250 x4 0.0114219020 0.04291015625 0.0720303442 0.5762427539 x5 0.0011939456 0.00697290039 0.0139108602 0.2225737636 x6 0.0001248046 0.00113309631 0.0026865348 0.0859691162 x7 0 0.00018412815 0.0005188370 0.0332055711 x8 0 0 0.0001002004 0.0128256518 x9 0 0 0 0.0049539080 x10 0 0 0 0.0019134469 x11 0 0 0 0.0007390688 x12 0 0 0 0.0002854653 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 2 4 6 8 10 12 14 0 1 2 3 4 5 number of iteration v a lu e o f x n kf m thakur et al. s figure 1. convergence behaviors of kf, m, thakur et al. and s iteration processes to the fixed point of the mapping defined in example 4.1 where x1 = 10. © agt, upv, 2022 appl. gen. topol. 23, no. 1 222 numerical reckoning fixed points via new faster iteration process references [1] m. abbas and t. nazir, a new faster iteration process applied to constrained minimization and feasibility problems, mat. vesnik 66, no. 2 (2014) 223–234. 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[21] h. h. wicke and j.m. worrell, jr., open continuous mappings of spaces having bases of countable order, duke math. j. 34 (1967), 255–271. © agt, upv, 2022 appl. gen. topol. 23, no. 1 223 @ appl. gen. topol. 20, no. 1 (2019), 297-305doi:10.4995/agt.2019.11260 c© agt, upv, 2019 extremal balleans igor protasov faculty of computer science and cybernetics, kyiv university, academic glushkov pr. 4d, 03680 kyiv, ukraine (i.v.protasov@gmail.com) communicated by d. georgiou abstract a ballean (or coarse space) is a set endowed with a coarse structure. a ballean x is called normal if any two asymptotically disjoint subsets of x are asymptotically separated. we say that a ballean x is ultranormal (extremely normal) if any two unbounded subsets of x are not asymptotically disjoint (every unbounded subset of x is large). every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold. a normal ballean is ultranormal if and only if the higson′s corona of x is a singleton. a discrete ballean x is ultranormal if and only if x is maximal. we construct a series of concrete balleans with extremal properties. 2010 msc: msc: 54e35. keywords: ballean, coarse structure, bornology, maximal ballean, ultranormal ballean, extremely normal ballean. 1. introduction let x be a set. a family e of subsets of x × x is called a coarse structure if • each e ∈ e contains the diagonal △x, △x = {(x, x) : x ∈ x}; • if e, e′ ∈ e then e ◦ e′ ∈ e and e−1 ∈ e, where e ◦ e′ = {(x, y) : ∃z((x, z) ∈ e, (z, y) ∈ e′)}, e−1 = {(y, x) : (x, y) ∈ e}; • if e ∈ e and △x ⊆ e ′ ⊆ e then e′ ∈ e; • for any x, y ∈ x, there exists e ∈ e such that (x, y) ∈ e. received 20 january 2019 – accepted 21 february 2019 http://dx.doi.org/10.4995/agt.2019.11260 i. protasov a subset e′ ⊆ e is called a base for e if, for every e ∈ e, there exists e′ ∈ e′ such that e ⊆ e′. for x ∈ x, a ⊆ x and e ∈ e, we denote e[x] = {y ∈ x : (x, y) ∈ e}, e[a] = ∪a∈a e[a] and say that e[x] and e[a] are balls of radius e around x and a. the pair (x, e) is called a coarse space [14] or a ballean [10], [12]. each subset y ⊆ x defines the subballean (y, ey ), where ey is the restriction of e to y × y . a subset y is called bounded if y ⊆ e[x] for some x ∈ x and e ∈ e. given a ballean (x, e), a subset y of x is called • large if there exists e ∈ e such that x = e[a]; • small if (x \ a) ∩ l is large for each large subset l; • thick if, for every e ∈ e, there exists a ∈ a such that e[a] ⊆ a. every metric d on a set x defines the metric ballean (x, ed), where ed has the base {{(x, y) : d(x, y) ≤ r} : r ∈ r+}. a ballean (x, e) is called metrizable if there exists a metric d on x such that e = ed. a ballean (x, e) is metrizable if and only if e has a countable base [12, theorem 2.1.1]. let (x, e) be a ballean. a subset u of x is called an asymptotic neighbourhood of a subset y ⊆ x if, for every e ∈ e, e[y ] \ u is bounded. two subsets y, z of x are called • asymptotically disjoint if, for every e ∈ e, e[y ] ∩ e[z] is bounded; • asymptotically separated if y, z have disjoint asymptotic neighbourhoods. a ballean (x, e) is called normal [7] if any two asymptotically disjoint subsets are asymptotically separated. every ballean (x, e) with linearly ordered base of e, in particular, every metrizable ballean is normal [7, proposition 1.1]. a function f : x −→ r is called slowly oscillating if, for any e ∈ e and ε > 0, there exists a bounded subset b of x such that diam f(e[x]) < ε for each x ∈ x b. by [7, theorem 2.2], a ballean (x, e) is normal if and only if, for any two disjoint and asymptotically disjoint subsets y, z of x, there exists a slowly oscillating function f : x −→ [0, 1] such that f|y = 0 and f|z = 1. we say that an unbounded ballean (x, e) is • ultranormal [1] if any two unbounded subsets of x are not asymptotically disjoint; • extremely normal if any unbounded subset of x is large. an unbounded ballean (x, e), is called maximal if x is bounded in any stronger coarse structure. by [13], every unbounded subset of a maximal ballean is large and every small subset is bounded. hence, every maximal ballean is extremely normal. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 298 extremal balleans a family b of subsets of a set x, closed under finite unions and subsets, is called a bornology if ∪b = x. for every ballean x, the family bx of all bounded subsets of x is a bornology. a ballean (x, e) is called discrete (or pseudodiscrete [12], or thin [5]) if, for every e ∈ e, there exists b ∈ bx such that e[x] = {x} for each x ∈ x\b. every bornology b on a set x defines the discrete ballean (x, {eb : b ∈ b}), where eb[x] = b if x ∈ b and eb[x] = {x} if x ∈ x\b. it follows that every discrete ballean x is uniquely determined by its bornology bx, see [12, theorem 3.2.1]. every discrete ballean is normal [12, example 4.2.2]. for a ballean x, the following conditions are equivalent: x is discrete, every function f : x −→ {0, 1} is slowly oscillating, every unbounded subset of x is thick, see [12, theorem 3.3.1] and [3, theorem 2.2]. an unbounded discrete ballean is called ultradiscrete if the family {x\b : b ∈ bx} is an ultrafilter on x. every ultradiscrete ballean is maximal [12, example 10.1.2]. thus, we have got ultradiscrete =⇒ maximal =⇒ extremely normal =⇒ ultranormal and no one arrow can be reversed, see section 3. 2. characterizations let (x, e) be an unbounded ballean. we endow x with the discrete topology, identify the stone-čech compactification βx of x with the set of all ultrafilters on x and denote x♯ = {p ∈ βx : p is unbounded for each p ∈ p}. given any p, q ∈ x♯, we write p ‖ q if there exists e ∈ e such that e[q] ∈ p for each q ∈ q. by [7, lemma 4.1], ‖ is an equivalence relation on x♯. we denote by ∼ the minimal (by inclusion) closed (in x♯ × x♯) equivalence on x♯ such that ‖⊆∼. the compact hausdorff space x♯/ ∼ is called the corona of (x, e), it is denoted by ν(x, e). if every ball {y ∈ x : d(x, y) ≤ r} in the metric space (x, d) is compact then ν(x, ed) is the higson’s corona of (x, d), see [8] and [14]. we say that a function f : x −→ r is constant at infinity if there exists c ∈ r such that, for each ε > 0, the set {x ∈ x : |f(x) − c| > ε} is bounded. we say that f is almost constant if f|x\b =const for some b ∈ bx. if the bornology bx is closed under countable unions then every function, constant at infinity, is almost constant. if f is constant at infinity then f is slowly oscillating. we denote so(x, e) the set of all bounded slowly oscillating functions on x. for a bounded function f : x −→ r, fβ denotes the extension of f to βx. theorem 2.1. for an unbounded normal ballean (x, e), the following conditions are equivalent: (1) every function f ∈ so(x, e) is constant at infinity; (2) ν(x, e) is a singleton; (3) (x, e) is ultranormal. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 299 i. protasov proof. (3) =⇒ (1). we assume that a function f ∈ so (x, e) is not constant at infinity. then there exists distinct a, b ∈ r and p, q ∈ x♯ such that fβ(p) = a, fβ(q) = b. we put ε = |a−b| 4 and choose p ∈ p, q ∈ q such that f(p) ⊂ (a − ε, a + ε), f(q) ⊂ (b − ε, b + ε). given an arbitrary e ∈ e, we take b ∈ bx such that diam f(e[x]) < ε for each x ∈ x \ b. it follows that e[p \ b] ∩ e[q \ b] = ∅ so p and q are asymptotically disjoint and we get a contradiction to (3). (1) =⇒ (2). let p, q ∈ x♯. by proposition 8.1.4 from [12], p ∼ q if and only if fβ(p) = fβ(q) for every f ∈ so (x, e). (2) =⇒ (3). we assume that (x, e) is not ultranormal and choose two disjoint and asymptotically disjoint unbounded subsets a, b of x. we chose p, q ∈ x♯ such that a ∈ p, b ∈ q. since x is normal, there exists f ∈ so (x, e) such that f |a= 0, f |b= 1. then f β(p) 6= fβ(q) and | ν(x, e) |> 1 by proposition 8.1.4 from [12]. � an unbounded ballean x is called irresolvable [11] if x can not be partitioned into two large subsets. theorem 2.2. for an unbounded discrete ballean (x, e), the following conditions are equivalent: (1) (x, e) is ultradiscrete; (2) (x, e) is extremely normal; (3) (x, e) is ultranormal; (4) (x, e) is maximal and irresolvable. proof. (1) =⇒ (2). we denote by p the ultrafilter {x\b : b ∈ bx}. if a is an unbounded subset of x then a ∈ p so x\a is bounded and a is large. (2) =⇒ (3). let a, c be unbounded subsets of x. since a is large, there exists e ∈ e such that e[a] = x so c ⊆ e[a] and a, c are not asymptotically disjoint. (3) =⇒ (1). if (x, e) is not ultradiscrete then there exist two disjoint unbounded subsets a, c of x. since (x, e) is discrete, a and c are asymptotically disjoint. (1) =⇒ (4). this is theorem 10.4.5 from [12]. � let b be a bornology on a set x. following [1], we say that a coarse structure e is compatible with b if b is the bornology of all bounded subsets of (x, e). each bornology b on x defines two coarse structures ⇓ b and ⇑ b, the smallest and the largest coarse structures on x compatible with b. clearly, ⇓ b is the discrete coarse structure defined by b, in particular, (x, ⇓ b) is normal. the coarse structure ⇑ b consists of all entourages e ⊆ x × x such that e = e−1 and e[b] ∈ b for each b ∈ b. but in contrast to ⇓ b, the coarse structure ⇑ b needs not to be normal [2, theorem 12]. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 300 extremal balleans if a ballean (x, e) is maximal then ⇑ bx = e. it follows that every maximal ballean (x, e) is uniquely determined by the bornology of bounded subsets: if (x, e′) is maximal and b(x,e) = b(x,e′) then e = e ′. proposition 2.3. let (x, e) be an extremely normal ballean and let e′ be coarse structure on x such that e ⊆ e′ and (x, e′) is maximal. then b(x,e) = b(x,e′) and e ′ = ⇑ b(x,e). proof. we assume the contrary and pick a ∈ b(x,e′) \ b(x,e). since (x, e) is extremely normal, a is large in (x, e). hence, a is large in (x, e′) so (x, e′) is bounded and we get a contradiction to maximality of (x, e′). � following [2], we say that a ballean (x, e) is relatively maximal if e = ⇑ b(x,e). we recall that two ultrafilters p, q ∈ βx are incomparable if, for every f : x −→ x, we have fβ(p) 6= q, fβ(q) 6= p. proposition 2.4. let (x, e) be an unbounded discrete ballean such that any two distinct utrafilters in x♯ are incomparable. then (x, e) is relatively maximal. proof. we suppose the contrary and choose a coarse structure e′ on x such that e ⊂ e′ and b(x,e) = b(x,e′). then there exists e ′ ∈ e′ such that the set y = {y ∈ x : |e′(y)| > 1} is unbounded in (x, e′). for each y ∈ y , we pick f(y) ∈ e′(y), f(y) 6= y and extend f to x by f(x) = x for each x ∈ x \ y . we take an arbitrary ultrafilter p ∈ (x, e)♯ such that y ∈ p. by the assumption, f(p) is bounded in (x, e) for some p ∈ p, p ⊆ y . then (e′)−1[f(p)] must be bounded in (x, e) and we get a contradiction with the choice of p. � 3. constructions given a family f of subsets of x × x, we denote by 〈f〉 the intersections of all coarse structures, containing each f ∪ △x, f ∈ f and say that 〈f〉 is generated by f. it is easy to see that 〈f〉 has a base of subsets of the form e0 ◦ e1 ◦ . . . ◦ en, where fi ∈ {(f ∪ △x) ∪ (f ∪ △x) −1 : f ∈ f} ∪ {(x, y) ∪ △x : x, y ∈ x}, n < ω. if e1 and e2 are coarse structures, we write e1 ∨ e2 in place of 〈e1 ∪ e2〉 . for lattices of coarse structures, see [11]. proposition 3.1. let (x, e) be an unbounded discrete ballean and let p, q be distinct ultrafilters from x♯. if (x, e) is relatively maximal then fβ(p) 6= q for each f : x −→ x such that, for every b ∈ b(x,e), f(b) ∈ b(x,e) and f−1(b) ∈ b(x,e). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 301 i. protasov proof. we assume the contrary and choose f such that fβ(p) = q. we put f = {(x, f(x)) : x ∈ x} and denote by e′ the coarse structure generated by e and f . since e is discrete and e′ is not discrete, we have e ⊂ e′. we take x ∈ x and e1, . . . , en ∈ e ∪ {f ∪ △x, f −1 ∪ △x}. applying an induction by n and the assumptions f(b) ∈ b(x,e), f −1(b) ∈ b(x,e) for each b ∈ b(x,e), we conclude that e1 ◦. . .◦en[x] ∈ b(x,e). hence, b(x,e) = b(x,e)′ and we get a contradiction to relative maximality of (x, e). � proposition 3.2. every unbounded subballean of maximal (extremely normal, ultranormal) ballean is maximal (extremely normal, ultranormal). proof. we prove only the first statement, the second and third are evident. let (x, e) be a maximal ballean, y be an unbounded subset of x. we assume that (y, ey ) is not maximal and choose a coarse structure e ′ on y such that e |y ⊂ e ′ and (y, e′) is not bounded. we put f = 〈e ∪ e′〉. since (x, e) is maximal and e ⊂ f, (x, f) must be bounded. on the other hand, each bounded subset b of (y, e′) is bounded in (x, e) because otherwise b is large in (x, e) so b is large in (y, ey ). now let x ∈ x, and e1, . . . , en ∈ e ∪{e ′ ∪△x : e ′ ∈ e′}. on induction by n, we see that e1 ◦ . . . ◦ en[x] is bounded in (x, e). hence (x, f) is not bounded and we get a contradiction. � proposition 3.3. let e, e′ be coarse structures on a set x such that e ⊆ e′ . then the following statements hold: (1) if (x, e) is extremely normal then (x, e′ is extremely normal; (2) if (x, e) is ultranormal and b(x,e) = b(x,e′) then (x, e ′) is ultranormal. proof. (1) let a be a subset of x. if a is unbounded in (x, e′) then a is unbounded in (x, e). if a is large in (x, e) then a is large in (x, e′). (2) we assume that some unbounded subsets a, b of (x, e′) are asymptotically disjoint in (x, e). then e′(a)\b ∈ b(x,e′) for each e ′ ∈ e′. since e ⊆ e′ and b(x,e) = b(x,e′), we have e(a)\b ∈ b(x,e′) for each e ∈ e so a, b are asymptotically disjoint in (x, e). � example 3.4. for every infinite regular cardinal κ, we construct a coarse structure mκ on κ such that (κ, mκ) is maximal and b(κ,mκ) = [κ] <κ. we denote by f the family of all coverings of κ defined by the rule: p ∈ f if and only if, for each p ∈ p and x ∈ κ, |p | < κ and | ∪ {p ′ : x ∈ p ′, p ′ ∈ p}| < κ. then mκ is defined by the base {mp : p ∈ f}, where mp = {(x, y) : x ∈ p, y ∈ p for some p ∈ p}. for general construction of coarse structures by means of coverings, see [9] or [12, section 7.5]. clearly, b(κ,mκ) = [κ] <κ and (κ, mκ) is maximal [12, example 10.2.1]. let g be a group and let x be a g-space with the action g × x −→ x, (g, x) 7−→ gx. a bornology i on g is called a group bornology if, for any c© agt, upv, 2019 appl. gen. topol. 20, no. 1 302 extremal balleans a, b ∈ i, we have ab ∈ i, a−1 ∈ i. every group bornology i on g defines a coarse structure ei on x with the base {ea : a ∈ i, e ∈ a}, where e is the identity of g, ea = {(x, y) : y ∈ ax}. moreover, every coarse structure on x can be defined in this way [6]. example 3.5. we define a coarse structure e on ω such that the ballean (ω, e) is extremely normal but (ω, e) is not maximal. let sω denotes the group of all permutations of ω, i = [sω] <ω, e = ei. we show that every infinite subset a of ω is large. we partition a and ω into two infinite subset a = a1 ∪ a2, w = w1 ∪ w2 and choose two permutations f1, f2 of ω so that f1(a1) = w1, f1(w1) = a1, f1(x) = x for each x ∈ ω \ (a1 ∪ w1), f2(a2) = w2, f2(w2) = a2, f2(x) = x for each x ∈ ω \ (a2 ∪ w2) then we put f = {f1, f2, id}, where id is the identity permutation. clearly, ef [a] = ω so e is extremely normal. to see that (ω, e) is not maximal, we note that e ⊆ mω and choose a partition p = {pn : n ∈ ω} of ω such that |pn| = n. then, for each n ∈ n, there exist x ∈ ω such that mp[x] = n. for each h ∈ [sω] <ω and x ∈ ω, we have |eh[x]| ≤ |h|. hence, e ⊂ m<ω. example 3.6. let κ be a cardinal, κ > ω. we construct two coarse structures e, e′ on κ such that e ⊂ e′, (κ, e) is ultranormal but not extremely normal, (κ, e′) is extremely normal but not maximal. let sκ denotes the group of all permutations κ, i = [sκ] <ω, e = ei. clearly, b(κ,e) = [κ] <ω. let a, b be infinite subsets of κ. we take countable subset a′ ⊆ a, b′ ⊆ b and use argument from example 3.5 to choose f = {f1, f2, id} so that b ′ ⊆ ef [a ′]. it follows that a, b are not asymptotically disjoint in (κ, e) so (κ, e) is ultranormal. if a is a countable subset of κ then |eh[a]| = ω for each h ∈ [sκ] <ω so (κ, e) is not extremely normal. we denote by f the discrete coarse structure on κ defined by the bornology [κ]<κ and put e′ = f ∨ e. if a is an unbounded subset of e′ then |a| = κ. applying the arguments from example 3.5, we see that (κ, e′) is extremely normal. the ballean (κ, e′) is not maximal because e′ ⊂ f ∨ ej, where j = [sκ] ≤ω. 4. comments 1. let (x, e) be a ballean, a and b be subsets of x. following [9], we write aδb if and only if there exists e ∈ e such that a ⊆ e[b], b ⊆ e[a]. let e, e′ be coarse structures on a set x such that b(x,e) = b(x,e′). if e and e′ have linearly ordered bases and either so(x, e) = so(x, e′) or δ(x,e) = δ(x,e′) then e = e ′, see [4, theorem 2.1] and [3, theorem 4.2]. we take the coarse structures e and mω on ω from example 3.5. by theorem 2.1, so(ω, e) = so(ω, mω). since b(ω,e) = b(ω,mω)= [ω] <ω and (ω, e), (ω, mω) are extremely normal, we have δ(ω,e) = δ(ω,mω). by the construction, e 6= mω. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 303 i. protasov in this connection we remind question 7.5.1 from [12]: does δ(x,e) = δ(x,e′) imply so(x, e) = so(x, e′) and give the affirmative answer to this question. clearly, δ(x,e) = δ(x,e′) implies b(x,e) = b(x,e′). assume that there exists f ∈ so(x, e) \ so(x, e′). then there exist ε > 0 and e′ ∈ e′ such that, for each b ∈ b(x,e′), one can find yb, zb ∈ x \ b such that (yb, zb) ∈ e ′ but |f(yb) − f(zb)| > ε we put y = {yb : b ∈ b(x,e′)} and choose a function h : x −→ x such that (y, h(y)) ∈ e′, |f(y) − f(h(y))| > ε for each y ∈ y . let p ∈ x♯ and y ∈ p, q be an ultrafilter with the base {h(p) : p ∈ p}. then |fβ(p) − fβ(q)| ≥ ε and there exists p ′ ∈ p such that |f(x) − f(y)| > ε 2 for all x ∈ p ′, y ∈ h(p ′). since f ∈ so(x, e), p ′ and h(p ′) are not close in (x, e) but p ′, h(p ′) are close in (x, e′) . 2. every group g has the natural finitary coarse structure efin with the base {ef : f ∈ [g] <ω, e ∈ f}, where ef = {(x, y) : y ∈ fx}. let g be an uncountable abelian group. by [4, corollary 3.2], (g, efin) is not normal but every function f ∈ so(g, efin) is constant at infinity. this example shows that the assumption of normality in theorem 2.1 can not be omitted. 3. example 3.6 shows that proposition 2.3 does not hold for ultranormal ballean in place of extremely normal. indeed, (κ, e) is ultranormal, b(κ,e) = [κ]<ω, e ⊂ e′, b(κ,e′) = [κ] <κ. we take a maximal coarse structure e′′ such that e′ ⊂ e′′. then e ⊂ e′′. and b(κ,e) 6= b(κ,e′′). 4. following [1], we say that a ballean (x, e) has bounded growth if there is a mapping f : x −→ bx such that • ∪x∈bf(x) ∈ bx for each b ∈ bx; • for each e ∈ e, there exists c ∈ bx such that e[x] ⊆ f(x) for each x ∈ x \ c. clearly, every discrete ballean has bounded growth (f(x) = {x}). let us take the maximal ballean (ω, mω) from example 3.4. since each ball in (ω, mω) is finite, one can use the diagonal process to show that (ω, mω) is not of bounded growth. references [1] t. banakh and i. protasov, the normality and bounded growth of balleans, arxiv: 1810.07979. [2] t. banakh and i. protasov, constructing balleans, arxiv: 1812.03935. [3] d. dikranjan, i. protasov, k. protasova and n. zava, balleans, hyperballeans and ideals, appl. gen. topol, to appear. [4] m. filali and i. protasov, slowly oscillating functions on locally compact groups, appl. gen. topology 6 (2005), 67–77. [5] ie. lutsenko and i. protasov, thin subsets of balleans, appl. gen. topol. 11 (2010), 89–93. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 304 extremal balleans [6] o. v. petrenko and i. v. protasov, balleans and g-spaces, ukr. math. zh. 64 (2012), 344–350. [7] i. v. protasov, normal ball structures, math. stud. 20 (2003), 3–16. [8] i. v. protasov, coronas of balleans, topology appl. 149 (2005), 149–160. [9] i. v. protasov, asymptolic proximities, appl. gen. topol. 9 (2008), 189–195. [10] i. protasov and t. banakh, ball structures and colorings of groups and graphs, math. stud. monogr. ser., vol. 11, vntl, lviv, 2003. [11] i. protasov and k. protasova, lattices of coarse structures, math. stud. 48 (2017), 115–123. [12] i. protasov and m. zarichnyi, general asymptopogy, math. stud. monogr. vol. 12, vntl, lviv, 2007. [13] o. protasova, maximal balleans, appl. gen. topol. 7 (2006), 151–163. [14] j. roe, lectures on coarse geometry, ams university lecture ser. 31, providence, ri, 2003. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 305 cacebuagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 187-195 continuous utility functions on submetrizable hemicompact k-spaces alessandro caterino, rita ceppitelli and francesca maccarino abstract. some theorems concerning the existence of continuous utility functions for closed preorders on submetrizable hemicompact kspaces are proved. these spaces are precisely the inductive limits of increasing sequences of metric compact subspaces and in general are neither metrizable nor locally compact. these results generalize some well known theorems due to levin. 2000 ams classification: 54f05, 91b16. keywords: closed preorder, jointly continuous utility function, hemicompact, submetrizable, k-space. 1. introduction the problem concerning the existence of a continuous utility function (preorder preserving map) on a topological space endowed with a (continuous) preorder is called the utility representation problem. in this paper we study the utility representation problem in connection with a not necessarily linear (total) preorder. several authors were concerned with non-linear preorders. indeed, the use of non-linear preorders (or, more generally, non-linear binary relations) may be viewed as more realistic and adequate in order to explain the behavior of an individual. peleg [27] was the first who presented sufficient conditions for the existence of a continuous utility function for a partial order on a topological space. peleg solved a problem which was posed by aumann [1] in the context of expected utility. in particular, aumann observed that a rational decision-maker may express “indecisiveness” (or equivalently “incomparability”) between two alternatives, so that he is not a priori forced to express “indifference” (see also ok [26]). later, mehta [22, 23] followed the spirit of nachbin [25] in order to show that many general results concerning the continuous utility representation problem can be obtained by combining the classical approach to mathematical 188 a. caterino, r. ceppitelli and f. maccarino utility theory with some of the most important results in elementary topology. herden and pallack [12] generalized the well known debreu theorem concerning the existence of a continuous utility function for a linear preorder on a second countable topological space. in particular, the authors introduced the concept of a weakly continuous preorder on a topological space and in this way provided an appropriate generalization of the notion of a continuous linear preorder. herden and pallack results imply the interesting levin theorem [16] concerning the existence of a continuous utility function for a not necessarily linear closed preorder on a second countable locally compact hausdorff space. in connection with the above results, we prove a new generalization of levin theorem in the case of closed preorders on spaces not necessarily metrizable or second countable. more precisely, we prove the existence of a continuous utility function for a closed preorder on a submetrizable, hemicompact k-space (submetrizable kω-space). an interesting survey on hemicompact k-spaces (kω -spaces) can be found in [11]. submetrizable kω-spaces are also studied in [7, 8]. we recall that submetrizable kω-spaces are exactly the inductive limits of increasing sequences of metric compact subspaces [8]. in [6] the authors give some representation theorems assuming topological conditions that are very close to the ones of our theorems. they prove that if x is the inclusion inductive limit of a countable chain of compact subspaces xn (that is x is a hemicompact k-space) and � is a preorder on x such that every �|xn is closed and order-separable, then � is continuously representable. if we suppose that every �|xn is a linear preorder without jumps, this result is equivalent to our result about submetrizable kω -spaces. in general, the two results are independent. in the last section we are concerned with the existence of the so-called jointly continuous utility functions. this kind of problems are often considered in mathematical economics. let γ be a family of preorders (on a topological space x) endowed with a given topology. the utility representation problem in this context consists in proving the existence of a continuous map u : γ × x −→ r such that u(�, ·) : x −→ r is a utility function for every �∈ γ. in mathematical economics x and γ can be interpreted as a commodity space and respectively as a space of preference relations of economic agents. in the second half of the last century, this problem was extensively treated in the literature. a survey of these results can be found in [5]. we quote [14, 9, 13, 20, 4] where the existence of jointly continuous functions is proved in the case of linear preorders. levin in [16, 17, 18] proved a general theorem for not necessarily linear preorders in second countable locally compact spaces. in the present paper we prove a generalization of levin results in a nonmetrizable setting. more precisely, we prove the existence of a jointly utility continuous function when x is a submetrizable hemicompact k-space and γ is metrizable or both γ and x are submetrizable hemicompact and the product γ × x is a k-space. continuous utility functions on submetrizable hemicompact k-spaces 189 2. preliminaries a reflexive and transitive binary relation on an arbitrary set x is called a preorder �. the preorder � on x is linear if [x � y]∨[y � x] for every x, y ∈ x. an anti-symmetric preorder is said to be an order. if � is a preorder on x the associated asymmetric relation ≺ is defined by [x ≺ y ⇔ (x � y)∧¬(y � x)]. if � is a preorder on a set x, then we will refer to the pair (x, �) as a preordered set. let (x, �) be a preordered set. the family of all the sets of the form ]a, +∞[= {x ∈ x : a ≺ x} and ] − ∞, a[= {x ∈ x : x ≺ a}, where a ∈ x, is a subbasis for a topology on x. this topology, denoted by τ� is the order topology induced by � and (x, τ�) is called a preordered topological space. if (x, �) is a preordered set and τ is a topology on x, then the preorder � is said to be closed with respect to τ if its graph {(x, y) ∈ x × x : x � y} is a closed subset of x × x. the preorder � is said to be continuous with respect to τ (or τ -continuous) if for every a ∈ x the sets [a, +∞[= {x ∈ x : a � x} and ] − ∞, a] = {x ∈ x : x � a} are τ -closed. we recall that if the preorder � is linear, then � is closed iff it is continuous iff τ� ⊂ τ . if (x, �) is a preordered set then a real-valued function f on x is said to be (i) isotone if for every x, y ∈ x, [x � y ⇒ f (x) ≤ f (y)] (ii) preorder-preserving if f is isotone and [x ≺ y ⇒ f (x) < f (y)]. if (x, �) is a linearly preordered set, then (ii) is equivalent to (iii) for every x, y ∈ x, [x � y ⇐⇒ f (x) ≤ f (y)]. if the preorder � is interpreted as a preference relation on a set x of alternatives then a real-valued preorder-preserving function is also called a utility function or a utility representation of the preorder. the notions of network and the corresponding netweight seem useful tools in the theory of representation. spaces with a countable network are a natural generalization of the second countable spaces. a network in a topological space x is a family n of subsets of x such that every open set of x is union of elements of n . as usual, the net weight of x is defined by nw(x) = min{|n| : n is a network for x}. we recall that the netweight is monotone, that is, if x ⊂ y then nw(x) ≤ nw(y ). 3. utility representations for non-linear preorders we consider the problem of the existence of a continuous utility function on a topological preordered space when the preorder is not necessarily linear. we begin with some interesting results due to levin. we remark that the techniques used in this case can be very different from those used when � is a linear preorder. some results on function spaces are sometimes used to solve this kind of problems. if x is a topological space, let c(x, r) be the set of real continuous functions defined on x. we denote by cp(x, r) and ck(x, r) the topological vector 190 a. caterino, r. ceppitelli and f. maccarino spaces (c(x, r), τp) and (c(x, r), τk), where τp is the topology of pointwise convergence and τk is the topology of compact convergence. theorem 3.1 ([16, levin]). let x be a metrizable space such that there exists a sequence {xn} of compact subsets of x with the properties: (1) x = ⋃∞ n=1 xn; (2) a real-valued function f on x is continuous if and only if its restriction to each xn is continuous. then for every closed preorder � there is a continuous utility function on x. note that if x is a second countable locally compact hausdorff space, then x satisfies the hypotheses of theorem 3.1. now, the next result extends the above levin theorem to non-metrizable case. we need some definitions and a lemma that is a well known result on hemicompact k-spaces. a topological space x is said to be hemicompact if there is a sequence {kn} of compact subsets of x which is cofinal in the set of all compact subsets of x, that is, every compact subset of x is contained in kn for a suitable n. a space x is called a k-space if a ⊂ x is open if and only if a ∩ k is open in k for every compact set k of x, that is, x has the weak topology with respect to the family of all compact subsets of x. we recall that the class of the hemicompact k-spaces coincides with the one of the kω-spaces [11]. a kω -space is a space with a kω-decomposition, that is, an increasing sequence {kn} of compact subspaces such that x = ⋃∞ n=1 kn and x has the weak topology with respect to the family {kn}. lemma 3.2 ([10]). let x be a hemicompact k-space and let {kn} be a sequence of compact subsets of x which is cofinal in the set of all compact subsets of x. then a mapping f : x → y is continuous if and only if f|kn is continuous for every n. the following lemma generalizes to hemicompact k-spaces an extension theorem ([16], lemma 2) proved by levin for compact spaces. lemma 3.3. let x be a hemicompact k-space with a closed preorder �, let s be a compact subset of x and let u : s → [0, 1] be a continuous isotone mapping. then there is a continuous isotone function f : x → [0, 1] that extends u. proof. let x = ⋃∞ n=1 xn where xn ⊂ x is compact, xn ⊂ xn+1 for every n ∈ n and {xn} is cofinal in the set of all compact subsets of x. since s is compact, s ⊂ xn̄ for some n̄ ∈ n. by lemma 2 in [16], the function u can be extended to xn̄. using a recursive process and applying lemma 3.2, the function u can be continuously extended to all of x. � continuous utility functions on submetrizable hemicompact k-spaces 191 we recall that a topological space x is said to be submetrizable if there exists a coarser metrizable topology on x. theorem 3.4. let x be a submetrizable space such that there exists a sequence {xn} of compact subsets of x with the properties: (1) x = ⋃∞ n=1 xn; (2) a real-valued function f on x is continuous if and only if its restriction to each xn is continuous. then for every closed preorder � there is a continuous utility function on x. proof. it is not restrictive to suppose that xn ⊂ xn+1, for every n ∈ n. as in the proof of theorem 2 in [5], if h = {u ∈ c(x, r) : x � y =⇒ u(x) ≤ u(y)} is the cone of real continuous isotone mappings defined on x, then x � y if and only if u(x) ≤ u(y) for every u ∈ h. the hypothesis 1) and the submetrizability of x imply that x has a countable network. in fact, every compact submetrizable xn is metrizable and second countable. therefore if bn = {b n i }i∈n is a countable base of the (metrizable) subspace xn, then b = ⋃ bn is a countable network of x. (c(x, r), τp) has a countable network, too (see [21]). since the netweight is monotone, we deduce that also (h, τp) has a countable network and so it is separable. let {un}n∈n be a τp-dense sequence in h. now, it is possible to show that x � y if and only if un(x) ≤ un(y) for every n ∈ n. on the contrary, let x, y ∈ x such that un(x) ≤ un(y) for every n ∈ n but ¬(x � y). then, using lemma 3.3, the function u : {x, y} → r defined by u(y) = 0, u(x) = 1 can be extended to a function ũ ∈ h. let jũ = {g ∈ c(x, r) : |g(x) − ũ(x)| < 1 4 , |g(y) − ũ(y)| < 1 4 }. since g(x) − g(y) > 1 2 for every g ∈ jũ, the density of {un}n∈n in h gets a contradiction. finally, as in levin’s proof, the function u : x → r defined by u(x) = ∞∑ n=1 2−n un(x) 1 + |un(x)| is the desired continuous utility function. � the next result is a direct consequence of theorem 3.4. theorem 3.5. let x be a submetrizable hemicompact k-space and let � be a closed preorder on x. then there is a continuous utility function on x. we recall that every first countable hemicompact space is locally compact and second countable (hence metrizable). but, there are submetrizable hemicompact k-spaces that are not metrizable spaces. 192 a. caterino, r. ceppitelli and f. maccarino example 3.6. let z be the set of integers and r be the set of all real numbers with the usual topology. consider the quotient space y = r/z obtained by identifying the set z to the point y0 ∈ y . then y is a k-space by definition; it cannot be metrizable since it isn’t even first countable (y0 fails to have a countable base of neighbourhoods). moreover, y is submetrizable and hemicompact. example 3.7. the space s′ of tempered distributions ([7], example 3.3) is another example of a submetrizable hemicompact k-space that is not metrizable. in [6] the authors prove some representation theorems assuming topological conditions that are very close to the assumptions of our theorems 3.4 and 3.5. in [6], theorem 2, they prove that if x = lim⊂ xn (inclusion inductive limit) where {xn} is a countable chain of compact subspaces of x and � is a preorder on x such that every �|xn is closed and order-separable, then � is continuously representable. we note that a space x is the inclusion inductive limit of a countable chain of compact subspaces if and only if it is a hemicompact k-space. in fact, every inductive limit of a countable family of compact spaces is a σ-compact k-space since it is a quotient space of the free union s = ∑ xn that is σ-compact and locally compact. the hemicompactness follows by a theorem of steenrod in [28]. conversely, it is easy to prove that every hemicompact k-space is the inclusion inductive limit of a countable chain of compact subspaces. if we suppose that �|xn is a linear preorder without jumps for every n, our theorem 3.5 and theorem 2 in [6] are equivalent. in fact, in this case, since xn is compact then τ�|xn coincides with the topology of the subspace xn. then we get �|xn is order-separable if and only if xn is metrizable. further, an inclusion inductive limit of a countable chain {xn} of compact spaces is submetrizable if and only if xn is metrizable for every n (see [7, 8]). in general, our theorem 3.5 and theorem 2 in [6] are independent. 4. jointly continuous utility functions let x be a topological space and let γ be a family of preorders on x, endowed with a given topology. the problem of the existence of jointly continuous functions consists in proving the existence of a continuous map u : γ × x −→ r such that u(�, ·) : x −→ r is a utility function for every �∈ γ. in [16] levin, using a continuous selection theorem of michael [24], solved the problem when γ is metrizable and x is locally compact and second countable. in the special case in which γ and x are both locally compact and second countable, levin proved the result without the use of michael’s theorem. we begin to extend this result to the case when γ and x are submetrizable and hemicompact and γ×x is a k-space. later (theorem 4.2) we will extend the theorem of levin to the case when x is submetrizable and hemicompact, continuous utility functions on submetrizable hemicompact k-spaces 193 γ is metrizable and γ × x is a k-space. we note that these two results are independent of each other. theorem 4.1. let γ and x be submetrizable hemicompact spaces and suppose γ × x is a k-space. moreover, assume that the set g = {(�, x, y) : x � y} is closed in γ × x × x. then there exists a continuous function u : γ × x → [0, 1] such that, for each � ∈ γ, u(� , ·) is a continuous utility function on x. proof. as in section 5 of [16], the preorder ≤ on γ × x, defined by (�1 , x1) ≤ (�2 , x2) if and only if �1 = �2 and x1 �1 x2, is closed. hence (γ × x, ≤ ) satisfies all the conditions of theorem 3.5 and so there is a continuous utility function u on γ × x with respect to ≤ . the conclusion follows from the definition of the preorder ≤. � theorem 4.2. let γ be a metrizable space, let x be a submetrizable hemicompact space and suppose γ × x is a k-space. moreover, assume that the set g = {(�, x, y) : x � y} is closed in γ × x × x. then there exists a continuous function u : γ × x → [0, 1] such that, for each � ∈ γ, u(� , ·) is a continuous utility function on x. proof. let {kn} be a sequence of compact subsets of x which is cofinal in the set of all compact subsets of x. then the topology τk on ck(x, r) is generated by the countable family of seminorms pn(f ) = supx∈kn | f (x) | . therefore, the locally convex topological vector space ck(x, r) is metrizable. moreover, since x is a k-space then ck(x, r) is also complete ([15], theorem 12) and hence it is a fréchet space. as in the proof of theorem 3.4, the hypotheses of hemicompactness and submetrizability imply that x has a countable network and so cp(x, r) is separable. then ck(x, r) is separable too ([21], cor. 4.2.2). finally, by the hypotheses that γ × x is a k-space, using the same arguments as in theorem 1 in [16], it is possible to construct the desired jointly continuous utility function u. � acknowledgements. the authors would like to thank prof. g. bosi for his appreciated suggestions and comments. 194 a. caterino, r. ceppitelli and f. maccarino references [1] r. aumann, utility theory without the completeness axiom, econometrica 30 (1962), 445–462. 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[18] v. l. levin, functionally closed preorders and strong stochastic dominance, soviet. math. dokl. 32 (1985), 22–26. [19] v. l. levin and a. a. milyutin, the problem of mass transfer with a discontinous cost function and a mass statement of the duality problem for convex extremal problems, russian math. surveys 34 (1978), no. 3, 1–78. [20] a. mas-colell, on the continuous representation of preorders, international economic review, 18 (1977), 509–513. [21] r. a. mccoy and i. ntantu, topological properties of spaces of continuous functions, springer-verlag (1980). [22] g. b. mehta, some general theorems on the existence of order-preserving functions, mathematical social sciences 15 (1988), 135–146. [23] g. b. mehta, preference and utility, in handbook of utility theory, volume 1, eds. s. barberà, p. hammond and c. seidl. dordrecht: kluwer academic publishers, (1998), 1–47. [24] e. michael, continous selections, annals of mathematics 63 (1956), 361–382. [25] l. nachbin,topology and order, van nostrand, princeton (1965). [26] e. a. ok, utility representation of an incomplete preference relation, journal of economic theory 104 (2002), 429–449. [27] b. peleg, utility functions for partially ordered topological spaces, econometrica, 38 (1970), 93–96. continuous utility functions on submetrizable hemicompact k-spaces 195 [28] n. e. steenrod, a convenient category of topological spaces, mich. math. j. 14 (1967), 133–152. received november 2008 accepted november 2009 alessandro caterino (caterino@dmi.unipg.it) dipartimento di matematica e informatica, università degli studi di perugia, via vanvitelli 1, 06123, perugia, italy (corresponding author) rita ceppitelli (matagria@dmi.unipg.it) dipartimento di matematica e informatica, università degli studi di perugia, via vanvitelli 1, 06123, perugia, italy francesca maccarino (maccarinofrancesca@libero.it) via pian del vantaggio 10a, 05018 orvieto (terni), italy @ appl. gen. topol. 23, no. 2 (2022), 391-404 doi:10.4995/agt.2022.16701 © agt, upv, 2022 fixed point results of enriched interpolative kannan type operators with applications mujahid abbas a, rizwan anjum b and shakeela riasat c a department of mathematics, government college university katchery road, lahore 54000, pakistan and department of mathematics and applied mathematics, university of pretoria hatfield 002, pretoria, south africa (mujahid.abbas@up.ac.za) b department of mathematics, riphah institute of computing and applied sciences, riphah international university, lahore, pakistan. (rizwan.anjum@riphah.edu.pk) c abdus salam school of mathematical sciences, government college university, lahore 54600, pakistan. (shakeelariasat@sms.edu.pk) communicated by e. karapinar abstract the purpose of this paper is to introduce the class of enriched interpolative kannan type operators on banach space that contains the classes of enriched kannan operators, interpolative kannan type contraction operators and some other classes of nonlinear operators. some examples are presented to support the concepts introduced herein. a convergence theorem for the krasnoselskii iteration method to approximate fixed point of the enriched interpolative kannan type operators is proved. we study well-posedness, ulam-hyers stability and periodic point property of operators introduced herein. as an application of the main result, variational inequality problem is solved. 2020 msc: 47h10; 47h09; 47j25; 49j40. keywords: fixed point; enriched kannan operators; interpolative kannan type contraction; krasnoselskii iteration; well-posedness; periodic point; ulam-hyers stability; variational inequality problem. received 20 november 2021 – accepted 14 march 2022 http://dx.doi.org/10.4995/agt.2022.16701 mujahid abbas, rizwan anjum and shakeela riasat 1. introduction and preliminaries let (x,d) be a metric space and t : x → x. we denote the set {x ∈ x : tx = x} of fixed points of t by fix(t). solving a fixed point problem of an operator t, denote by fpp(t) is to show that the set fix(t) is nonempty. define t0 = i (the identity map on x) and tn = tn−1 ◦ t, called the nthiterate of t for n ≥ 1. the most simplest iteration procedure to approximate the solution of a fixed point equation tx = x is the method of successive approximations (or picard iteration) given by (1.1) xn = t nx0, n = 1, 2, . . . , where x0 is an initial guess in domain of an operator t. a sequence {xn}∞n=0 in x given by (1.1) is called a t-orbital sequence around x0. the collection of all such sequences is denoted by o(t,x0). if there exists x∗ in x such that fix(t) = {x∗} and the picard iteration associated with t converges to x∗ for any initial guess x0 in x, then t is called a picard operator (see [1, 7, 40]). if there exists k ∈ [0, 1) such that for any x,y ∈ x, we have (1.2) d(tx,ty) ≤ k d(x,y). then t is called a banach contraction operator which, if defined on a complete metric space, is a classical example of a continuous picard operator. thus, it was natural to ask the question whether in the framework of an complete metric space, a discontinuous operator satisfying somewhat similar contractive conditions is a picard operator. this was answered in an affirmative by kannan [26] in 1968. an operator t on x is called kannan contraction operator if there exists a ∈ [0, 0.5) such that for any x,y ∈ x, we have (1.3) d(tx,ty) ≤ a{d(x,tx) + d(y,ty)}. kannan contraction operator defined on a complete metric space is an example of a discontinuous picard operator ([25], [26]). subrahmanyam [41] proved that a metric space x is complete if and only if every kannan contraction operator on x has a fixed point. moreover, connell [18] gave an example of an incomplete metric space x on which every banach contraction operator has a fixed point. this shows that the banach contraction operators do not characterize the completeness of their domain. as long as contractive conditions are concerned, the classes of banach and kannan contraction operators are incomparable ([27]), but the class of kannan contraction operators attracted the attention of several mathematicians because of its connection with a characterization of its metric completeness. kannan’s theorem has been generalized in different ways by many authors to extend the limits of metric fixed point theory in different directions. karapinar introduced a new class of kannan type operators called interpolative © agt, upv, 2022 appl. gen. topol. 23, no. 2 392 enriched interpolative kannan type operators kannan type operators and proved a fixed point result for such operators in the setup of complete metric spaces ([28]). an operator t : x → x is called an interpolative kannan type if there exists a ∈ [0, 1) such that for all x,y ∈ x \fix(t), we have (1.4) d(tx,ty) ≤ a[d(x,tx)]α[d(y,ty)]1−α, where α ∈ (0, 1). the main result in [28] is stated as follows. theorem 1.1 ([28]). let (x,d) be a complete metric space and t an interpolative kannan type operator. then t is a picard operator. for more results in this direction, we refer to [6, 8, 9, 19, 20, 21, 22, 29, 30, 31, 32, 33, 34, 35, 36, 38]) and references mentioned therein. existence, uniqueness, stability, approximation and characterization of fixed points of certain operators are some of the main concerns of a metric fixed point theory. contractive conditions on operators and distance structure of operator’s domain play a vital role to ensure the convergence of iterative methods. if for any x,y ∈ x, we have d(tx,ty) ≤ d(x,y). then t is called a nonexpansive operator. an operator t on x is called asymptotically regular on x if for all x in x, d(tn+1x,tnx) → 0 as n →∞. the concept of asymptotic regularity plays an important role in approximating the fixed points of operators. picard iteration method fails to converge to a fixed point of certain contractive mappings such as nonexpansive mappings on metric spaces. this led to the study of a variety of fixed point iteration procedures in the setup of banach spaces. in this paper, we shall approximate the fixed point of some nonlinear mappings through krasnoselskii iteration method. let λ ∈ [0, 1]. a sequence {xn}∞n=0 given by (1.5) xn+1 = (1 −λ)xn + λtxn, n = 0, 1, 2, . . . is called the krasnoselskii iteration. note that, krasnoselskii iteration {xn}∞n=0 sequence given by (1.5) is exactly the picard iteration corresponding to an averaged operator (1.6) tλ = (1 −λ)i + λt. moreover, for λ = 1 the krasnoselskii iteration method reduces to picard iteration method. also, fix(t) = fix(tλ), for all λ ∈ (0, 1]. on the other hand, browder and petryshyn [16] introduced the concept of asymptotic regularity in connection with the study of fixed points of nonexpansive mappings. as a matter of fact, the same property was used in 1955 by © agt, upv, 2022 appl. gen. topol. 23, no. 2 393 mujahid abbas, rizwan anjum and shakeela riasat krasnoselskii [37] to prove that if k is a compact convex subset of a uniformly convex banach space and t : k → k is a nonexpansive mapping, then for any x0 ∈ k, the sequence (1.7) xn+1 = 1 2 (xn + txn), n ≥ 0, converges to fixed point of t. in proving the above result, krasnoselskii used the fact that if t is nonexpansive which, in general, is not asymptotically regular, then the averaged mapping t1 2 is asymptotically regular. therefore, an averaged operator tλ enriches the class of nonexpansive mappings with respect to the asymptotic regularity. this observation suggested that one could enrich the classes of contractive mappings studied in the framework of metric spaces by imposing certain contractive condition on tλ instead of t itself. employing this approach, the classes such as enriched contractions and enriched φ-contractions [11], enriched kannan contractions [12], enriched chatterjea mappings [14], enriched nonexpansive mappings in hilbert spaces [13], enriched multivalued contractions [3] and enriched ćirić-reich-rus contraction [15], enriched cyclic contraction [5], enriched modified kannan pair [2], enriched quasi contraction [4] were introduced and studied. abbas et al. [3] proved fixed point results by imposing the condition that tλorbital subset is a complete subset of a normed space (see, theorem 3 of [3]). similarly, górnicki and bisht[24] considered the enriched ćirić-reich-rus contraction operators and proved a fixed point theorem by imposing the condition that tλ is asymptotically regular mapping (see, theorem 3.1 of [24]). consistent with [12], let (x,‖·‖) be a normed space. a operator t : x → x is called an enriched kannan contraction or (b,a)-enriched kannan contraction if there exist b ∈ [0,∞) and a ∈ [0, 0.5) such that for all x,y ∈ x, the following holds: (1.8) ‖b(x−y) + tx−ty‖≤ a(‖x−tx‖ + ‖y −ty‖). as shown in [12], several well-known contractive conditions in the existing literature imply the (b,a)-enriched kannan contraction condition. it was proved in [12] that any enriched kannan contraction operator defined on a banach space has a unique fixed point which can be approximated by means of the krasnoselskii iterative scheme. motivated by the work of berinde and păcurar [12], absas et al. [3] and górnicki and bisht [24], we propose a new class of enriched interpolative kannan type operators. the purpose of this paper is to prove the existence of fixed point of such operators. moreover, we study the well-posedness, ulam-hyers stability and periodic point property of the operators introduced herein. finally, an application of our result to solve variational inequality problems is also given. © agt, upv, 2022 appl. gen. topol. 23, no. 2 394 enriched interpolative kannan type operators 2. fixed point approximation of enriched interpolative kannan type operators in the sequel, the notations n and r will denote the set of all natural numbers and the set of all real numbers, respectively. in this section, we present a new class of enriched interpolative kannan type operators, which is first of its kind in the existing literature on metric fixedpoint theory. existence and convergence results of such class of operators are also obtained. first, we introduce the following concept definition 2.1. let (x,‖·‖) be a normed space. a mapping t : x → x is called enriched interpolative kannan type operator if there exist b ∈ [0,∞), a ∈ [0, 1) and α ∈ (0, 1) such that for all x,y ∈ x, we have (2.1) ‖b(x−y) + (tx−ty)‖≤ a ( ‖x−tx‖ )α(‖y −ty‖)1−α. to highlight an involvement of parameters a,b and α in (2.1), we call t a (a,b,α)-enriched interpolative kannan type operator. example 2.2. any interpolative kannan type operator t satisfying (1.4) is a (a, 0,α)-enriched interpolative kannan contraction operator, that is, t satisfies (2.1) with b = 0. we now give an example of an enriched interpolative kannan type operator which is not a interpolative kannan type contraction operator. example 2.3. let (y,µ) be a finite measure space. the classical lebesgue space x = l2(y,µ) is defined as the collection of all borel measurable functions f : y → r such that ∫ y |f(y)|2dµ(y) < ∞. we know that the space x equipped with the norm ‖f‖x = (∫ y |f|2dµ )1 2 is a banach space. define the operator t : l2(y,µ) → l2(y,µ) by tf = g − 3f, where g(y) = 1, ∀ y ∈ y. clearly, g ∈ l2(y,µ) as µ(y ) < ∞. note that t is an (0.5, 3, 0.5)-enriched interpolative kannan type operator but not an interpolative kannan type contraction operator. indeed, if t would be a interpolative kannan type operator then, by (1.4), there would exist a ∈ [0, 1) and α ∈ (0, 1) such that ‖−3f + 3h‖x ≤ a ( ‖4f −g‖x )α(‖4h−g‖x )1−α ∀ f,h ∈ x, which on taking f(y) = 0 and h(y) = 1, for all y ∈ y gives 3α ≤ a < 1, a contradiction. we now present the following result. © agt, upv, 2022 appl. gen. topol. 23, no. 2 395 mujahid abbas, rizwan anjum and shakeela riasat theorem 2.4. let (x,‖·‖) be a normed space, t : x → x a (a,b,α)-enriched interpolative kannan type mapping. then, (1) fix(t) = {x∗}; (2) there exist a tλ-orbital sequence {xn}∞n=0 around x0, given by (2.2) xn+1 = (1 −λ)xn + λtxn; n ≥ 0, converges to x∗ provided that, for x0 ∈ x, tλ-orbital subset o(tλ,x0) is a complete subset of x, where λ = 1 b+1 . proof. we divide the proof into the following two cases. case 1. if b > 0. then λ = 1 b+1 ∈ (0, 1) and the enriched interpolative kannan type operator (2.1) satisfies the following contraction condition:∥∥∥∥ ( 1 λ − 1 ) (x−y) + tx−ty ∥∥∥∥ ≤ a‖x−tx‖α‖y −ty‖1−α and hence ‖(1 −λ)(x−y) + λ(tx−ty)‖≤ aλ‖x−tx‖α‖y −ty‖1−α which can be written in an equivalent form as follows: (2.3) ‖tλx−tλy‖≤ a‖x−tλx‖ α‖y −tλy‖ 1−α , ∀ x,y ∈ x. in view of (1.6), the krasnoselskii iterative sequence defined by (2.2) is exactly the picard’s iteration associated with tλ, that is, (2.4) xn+1 = tλxn, n ≥ 0. take x := xn and y := xn−1 in (2.3) to get ‖xn+1 −xn‖ = ‖tλxn −tλxn−1‖ ≤ a‖xn −tλxn‖ α‖xn−1 −tλxn−1‖ 1−α ≤ a‖xn −xn+1‖ α‖xn−1 −xn‖ 1−α which implies that (2.5) ‖xn+1 −xn‖ 1−α ≤ a‖xn−1 −xn‖ 1−α . as α ∈ (0, 1), (2.6) ‖xn+1 −xn‖≤ a‖xn−1 −xn‖ . inductively, we obtain that (2.7) ‖xn+1 −xn‖≤ an‖x0 −x1‖ . by (2.7) and triangular inequality, we have (2.8) ‖xn −xn+r‖≤ an 1 −a ‖x0 −x1‖ , r ∈ n, n ≥ 1, which, in view of 0 < a < 1 gives that {xn}∞n=0 is a cauchy sequence in the complete subset o(tλ,x0) of x. © agt, upv, 2022 appl. gen. topol. 23, no. 2 396 enriched interpolative kannan type operators next, we assume that there exists an element x∗ in o(tλ,x0) such that limn→∞xn = x∗. note that ‖x∗ −tλx∗‖≤‖x∗ −xn+1‖ + ‖xn+1 −tλx∗‖ ≤‖x∗ −xn+1‖ + ‖tλxn −tλx∗‖ ≤‖x∗ −xn+1‖ + a‖xn −tλxn‖ α‖x∗ −tλx∗‖ 1−α ≤‖x∗ −xn+1‖ + a‖xn −xn+1‖ α‖x∗ −tλx∗‖ 1−α . on taking the limit as n →∞ on both sides of the above inequality, we get that x∗ = tλx ∗. assume that x∗ and y∗ are two fixed points of t. then from (2.3), we have ‖x∗ −y∗‖ = ‖tλx∗ −tλy∗‖≤ a‖x∗ −tλx∗‖ α‖y∗ −tλy∗‖ 1−α ≤ a‖x∗ −x∗‖α‖y∗ −y∗‖1−α , which, gives x∗ = y∗. case 2. b = 0. in this case, the enriched interpolative kannan type operator (2.1) becomes (2.9) ‖tx−ty‖≤ a‖x−tx‖α‖y −ty‖1−α ∀ x,y ∈ x, where a ∈ (0, 1). that is, t is an interpolative kannan type contraction operator and hence by theorem 2.2 of [28], t has a unique fixed point. � the following example illustrate the above theorem. example 2.5. let x = r \ { 1 5 , 4 5 } be endowed with the usual norm and t : x → x be defined by tx = 1 −x, ∀ x ∈ x. clearly t is a (1, 1)-enriched interpolative kannan type operator. now λ = 1 1+b gives that λ = 1 2 . let x0 = 1/2 be the fixed in x. then, we have t1 2 (x0) = 1 2 x0 + 1 2 tx0 = 1 2 ( 1 2 ) + 1 2 ( 1 2 ) = 1 2 . pick x1 = t1 2 x0 = 1/2. continuing this way, we obtained xn+1 = t1 2 xn, where xn+1 = ( 1 2 , 1 2 , 1 2 , . . . ). note that {xn}∞n=0 is a cauchy sequence which converges to 1/2 and 1/2 is the fixed point of t. we now present the following fixed point theorem for (a,b,α)-enriched interpolative kannan type operator on a banach space. corollary 2.6. let (x,‖·‖) be a banach space and t : x → x a (a,b,α)enriched interpolative kannan type operator. then t has a unique fixed point. proof. following arguments similar to those in proof theorem 2.4, the result follows. � if we take b = 0 in the corollary 2.6, we obtain theorem 2.2 of [28] in the setting of banach spaces. © agt, upv, 2022 appl. gen. topol. 23, no. 2 397 mujahid abbas, rizwan anjum and shakeela riasat corollary 2.7 ([28]). let t be an interpolative kannan type operator on a banach space (x,‖·‖). then t is a picard operator. by corollary 2.6, we obtain the following result. corollary 2.8 ([12]). let (x,‖·‖) be a banach space and t : x → x an (b,a)-enriched kannan contraction, that is, for all x,y ∈ x, it satisfies the following inequality; (2.10) ‖b(x−y) + tx−ty‖≤ a { ‖x−tx‖ + ‖y −ty‖ } with b ∈ [0,∞) and a ∈ [0, 0.5). then t has a unique fixed point. proof. take λ = 1 b+1 . obviously, 0 < λ < 1 and the (b,a)-enriched kannan contraction condition (2.10) becomes∥∥∥∥ ( 1 λ − 1 ) (x−y) + tx−ty ∥∥∥∥ ≤ a { ‖x−tx‖ + ‖y −ty‖ } , ∀ x,y ∈ x, which can be written in an equivalent form as follows; (2.11) ‖tλx−tλy‖≤ a { ‖x−tλx‖ + ‖y −tλy‖ } , ∀ x,y ∈ x. by (2.11), tλ is a kannan contraction. it follows from [28] that tλ satisfies condition (2.11) and condition (2.3). since, for λ = 1 b+1 , the inequality (2.3) is same as the condition (2.1). this suggests that t is an enriched interpolative kannan type operator and then the corollary 2.6 leads to the conclusion. � 3. well-posedness, perodic point and ulam-hyers stability results we now present the well-posedness, perodic point and ulam-hyers stability results for (a,b,α)-enriched interpolative kannan type operators. 3.1. well-posedness. let us start with the following definition. definition 3.1 ([39]). let (x,d) be a metric space and t : x → x. the fixed point problem fpp(t) is said to be well-posed if t has unique fixed point x∗(say) and for any sequence {xn} in x satisfying limn→∞d(txn,xn) = 0, we have limn→∞xn = x ∗. since fix(t) = fix(tλ), we conclude that the fixed point problem of t is well-posed if and only if the fixed point problem of tλ is well-posed. well-posedness of certain fixed point problems has been studied by several mathematicians, see for example, [10], [39] and references mentioned therein. we now study the well-posedness of a fixed point problem of mappings in theorem 2.4 and corollary 2.6. theorem 3.2. let (x,‖·‖) be a banach space. suppose that t is an operator on x as in the theorem 2.4. then, fpp(t) is well-posed. © agt, upv, 2022 appl. gen. topol. 23, no. 2 398 enriched interpolative kannan type operators proof. it follows from theorem 2.4 that x∗ is the unique fixed point of t. suppose that limn→∞‖tλxn −xn‖ = 0. using (2.3) we have, ‖xn −x∗‖≤‖xn −tλxn‖ + ‖tλxn −x∗‖ = ‖xn −tλxn‖ + ‖tλxn −tλx∗‖ ≤‖xn −tλxn‖ + a‖xn −tλxn‖ α‖x∗ −tλx∗‖ 1−α that is, (3.1) ‖xn −x∗‖≤‖xn −tλxn‖ . it follows from (3.1) that limn→∞xn = x ∗ provided that limn→∞‖tλxn −xn‖ = 0. this complete the proof. � corollary 3.3. let (x,‖·‖) be banach space. suppose that t is an operator on x as in the corollary 2.6. then the fixed point problem is well-posed. proof. following arguments similar to those in the proof theorem 3.2, the result follows. � 3.2. perodic point result. clearly, a fixed point x∗ of t is also a fixed point of tn for every n ∈ n. however, the converse is false. for example, if we take, x = [0, 1] and define an operator t on x by tx = 1 − x. then t has a unique fixed point 1/2, and for each even integer n, nth-iterate of t is an identity map and hence every point of [0, 1] is a fixed point of tn. also, if x = [0,π], tx = cos x, then every iterate of t has the same fixed point as t. if a map t satisfies fix(t) = fix(tn) for each n ∈ n, then it is said to have a periodic point property p [23]. since fix(t) = fix(tλ), we conclude that the mapping t has property p if and only the mapping tλ has property p. theorem 3.4. let (x,‖.‖) be a banach space. suppose that t is an operator on x as in the theorem 2.4. then t has property p. proof. from theorem 2.4, t has a fixed point. let y∗ ∈ fix(tn). now from (2.3), we have ‖y∗ −tλy∗‖ = ∥∥tnλ y∗ −tλ(tnλ y∗)∥∥ = ∥∥tλ(tn−1λ y∗)−tλ(tnλ y∗)∥∥ ≤ a ∥∥tn−1λ y∗ −tnλ y∗∥∥α ∥∥tnλ y∗ −tn+1λ y∗∥∥1−α , that is, (3.2) ∥∥tnλ y∗ −tn+1λ y∗∥∥α ≤ a∥∥tn−1λ y∗ −tnλ y∗∥∥α . since α ∈ (0, 1), then (3.2) becomes ‖y∗ −tλy∗‖ = ∥∥tnλ y∗ −tn+1λ y∗∥∥ ≤ a∥∥tn−1λ y∗ −tnλ y∗∥∥ ≤ ≤ a2 ∥∥tn−2λ y∗ −tn−1λ y∗∥∥ ≤ . . . ≤ an‖y∗ −tλy∗‖ . now, 0 ≤ a < 1 implies that ‖y∗ −tλy∗‖ = 0 and hence y∗ = ty∗. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 399 mujahid abbas, rizwan anjum and shakeela riasat 3.3. ulam-hyers stability. let (x,d) be a metric space, t : x → x and � > 0 . a point w∗ ∈ x called an �-solution of the fixed point problem fpp(t) if w∗ satisfies the following inequality d(w∗,tw∗) ≤ �. let us recall the notion of ulam-hyers stability. definition 3.5 ([42]). let (x,d) be a metric space, t : x → x and � > 0. the fixed point problem fpp(t) is called generalized ulam-hyers stable if and only if there exists an increasing and continuos function φ : [0,∞) → [0,∞) with φ(0) = 0 such that for each �-solution w∗ ∈ x of the fixed point equation tx = x , there exists a solution x∗ of tx = x in x such that d(x∗,w∗) ≤ φ(�). remark 3.6. if the function φ in the above definition is given by φ(t) = mt for all t ≥ 0, where m > 0, then the fixed point equation tx = x is said to be ulam-hyers stable. theorem 3.7. let (x,‖·‖) be a banach space. suppose that t is an operator on x as in the theorem 2.4. then the fixed point problem is ulam-hyers stable. proof. since fix(t) = fix(tλ), it follows that the fixed point problem tx = x is equivalent to the fixed point problem (3.3) x = tλx. let w∗ be �-solution of the fixed point equation (3.3), that is, (3.4) d(w∗,tλw ∗) ≤ �. using (2.3) and (3.4), we get ‖x∗ −w∗‖ = ‖tλx∗ −w∗‖≤‖tλx∗ −tλw∗‖ + ‖tλw∗ −w∗‖ ≤ a‖x∗ −tλx∗‖ α‖w∗ −tλw∗‖ 1−α + � = �. � 4. application to variational inequality problem variational inequality theory provides some important tools to handle the problems arising in economic, engineering, mechanics, mathematical programming, transportation and others. many numerical methods have been constructed for solving variational inequalities and optimization problems. the aim of this section is to present generic convergence theorems for krasnoselskii type algorithms that solve variational inequality problems. let h be a real hilbert space with inner product 〈 ·, · 〉 , c ⊂ h a closed and convex set and s : h → h. © agt, upv, 2022 appl. gen. topol. 23, no. 2 400 enriched interpolative kannan type operators the variational inequality problem with respect to s and c, denoted by v ip(s,c), is to find x∗ ∈ c such that〈 sx∗,x−x∗ 〉 ≥ 0, ∀ x ∈ h. it is well known [17] that if γ > 0, then x∗ ∈ c is a solution of v ip(s,c) if and only if x∗ is a solution of fixed point problem of pc ◦ (i −γs), where pc is the nearest point projection onto c. amongst many others results in [17], it was proved that if i − γs and pc ◦ (i − γs) are averaged nonexpansive operators, then, under some additional assumptions, the iterative algorithm {xn}∞n=0 defined by xn+1 = pc(i −γs)xn, n ≥ 0, converges weakly to a solution of the v ip(s,c), if such solutions exist. in the case of averaged nonexpansive mappings, the problem of replacing the weak convergence in the above result with strong convergence has received a much attention of researchers. our alternative is to consider v ip(s,c) for enriched interpolative kannan type contraction operators, which are in general discontinuous operators in contrast of nonexpansive operators which are always continuous. in this case, we shall have v ip(s,c) with a unique solution, as shown in the next theorem. moreover, the considered algorithm (4.1) will converge strongly to the solution of the v ip(s,c). theorem 4.1. assume that for γ > 0, pc(i − γs) is enriched interpolative kannan type contraction operator. then there exists λ ∈ (0, 1] such that the iterative algorithm {xn}∞n=0 defined by (4.1) xn+1 = (1 −λ)xn + λpc(i −γg)xn, n ≥ 0, converges strongly to the unique solution x∗ of the v ip(s,c) for any x0 ∈ c. proof. since c is closed, we take x := c and t := pc(i − γs) and apply corollary 2.6. � example 4.2. let x = r2 and for any x = (x1,x2) and y = (y1,y2) in x, the inner product is defined by〈 x,y 〉 = x1y1 + x2y2. then x equipped with the above inner product is a hilbert space. the above inner product gives the norm given by ‖x‖ = (< x,y >)1/2. define s : x → x by s(x) = (1, 0) + x γ ∀ x ∈ x, © agt, upv, 2022 appl. gen. topol. 23, no. 2 401 mujahid abbas, rizwan anjum and shakeela riasat where γ > 0 be fixed real number. for a mapping pc : x → c defined by pc(x) = { x ‖x‖ ; x /∈ c x ; x ∈ c, where c = {x ∈ x : ‖x‖≤ 1}, it is easy to check that pc(i −γs) is a (a, 0,α) enriched interpolative kannan type contraction. by corollary 2.6, pc(i −γs) has a unique solution, which in turns a solution for v ip(s,c). 5. conclusions (1) we introduced a large class of contractive operators, called enriched interpolative kannan type operators, that includes usual interpolative kannan type operators and enriched kannan operators. (2) we presented examples to show that the class of enriched interpolative kannan type operators strictly includes the interpolative kannan type operator in the sense that there exist operators which are not interpolative kannan type and belong to the class of enriched interpolative kannan type operators. (3) we studied the set of fixed point (theorem 2.4) and constructed an algorithm of krasnoselskii type in order to approximate fixed point of enriched interpolative kannan type operators and we proved a strong convergence theorem. (4) we also obtained theorems 3.2, 3.4 and 3.7 for a well-posedness, perodic point and ulam-hyers stability problem of the fixed point problem for enriched interpolative kannan type operators, respectively. (5) as application of our main results (corollary 2.6), we presented two krasnoselskii projection type algorithms for solving variational inequality problems for the class of enriched interpolative kannan type operators, thus improving the existence and weak convergence results for variational inequality problems in [17] to existence and uniqueness as well as to strong convergence theorems. acknowledgements. authors are grateful to the editor in chief and the reviewers for their useful comments and constructive remarks that helped to improve the presentation of the paper. © agt, upv, 2022 appl. gen. topol. 23, no. 2 402 enriched interpolative kannan type operators references [1] r. anjum and m. abbas, fixed point property of a nonempty set relative to the class of friendly mappings, rev. r. acad. cienc. exactas f́ıs. nat. ser. a mat. racsam 116, no. 1 (2022), paper no. 32. 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[42] w. sintunavarat, generalized ulam-hyres stability, well-posedness and limit shadowing of fixed point problems for α-β-contraction mapping in metric spaces, the sci. world j. 2014, article id 569174. © agt, upv, 2022 appl. gen. topol. 23, no. 2 404 @ appl. gen. topol. 21, no. 1 (2020), 1-15 doi:10.4995/agt.2020.11488 c© agt, upv, 2020 topological characterizations of amenability and congeniality of bases sergio r. lópez-permouth and benjamin stanley department of mathematics, ohio university, usa (lopez@ohio.edu,benqstanley@gmail.com) communicated by j. galindo abstract we provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of infinite dimensional algebras. in order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed. a basis b over an infinite dimensional f -algebra a is called amenable if f b, the direct product indexed by b of copies of the field f , can be made into an a-module in a natural way. (mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic a-modules. (not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. we prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions. 2010 msc: 16k40; 16d99; 15b99. keywords: uniform topologies; linear vector spaces; amenable bases; congeniality of bases; schauder bases; infinite-dimensional modules and algebras. 1. introduction and preliminaries the notions of amenability and congeniality for bases of an infinite dimensional algebra were recently introduced in [1]. since then, several papers have received 08 march 2019 – accepted 28 november 2019 http://dx.doi.org/10.4995/agt.2020.11488 s. r. lópez-permouth and b. stanley appeared exploring questions that arise naturally in that context (e.g. [2], [3], [5].) the purpose of our paper is to give topological insights on the naturality of those notions and to show some applications of these new perspectives. in addition, as is the case also in [5], we extend the notions studied in [1] to bases of any infinite dimensional module over an algebra of arbitrary dimension. we start with a brief summary of background definitions and results, which are straightforward adaptations of those in the literature. unless otherwise stated, f denotes a field, a denotes an f-algebra and t denotes a (left) a-module (therefore, itself an f-vector space.) given two sets i and j, an i × j matrix over the field f is a function f : i × j → f . as usual, i indexes rows of f and j indexes its columns. in this sense, the i-th row of f is f|{i}×j and the j-th column of f is f|i×{j}. we say that a matrix f is column finite if |{i ∈ i | f(i,j) 6= 0}| < ∞ for all j ∈ j. similarly, we say that f is row finite if |{j ∈ j | f(i,j) 6= 0}| < ∞ for all i ∈ i. if we have two matrices, f and g, such that f : i ×j → f and g : j ×k → f then, when possible, we define the product fg of f and g, to be the matrix fg(i,k) = ∑ j∈j f(i,j)g(j,k). as infinite sums of non-zero elements are not defined, the product of two matrices is also not necessarily defined. when the product does exist, the result is a matrix with domain i ×k. clearly, if either f is row finite or g is column finite then the product fg exists. we follow the usual definitions of algebras over fields and of unitary modules over rings with unity. this paper contains results from the doctoral dissertation [7]. 1.1. amenability and congeniality. the f-vector space t is isomorphic to f (b), the direct sum of copies of f indexed by b. consequently, f (b) inherits an a-module structure from t . a second vector space, fb, the collection of all infinite linear combinations of elements of b with coefficients in f , clearly contains f (b) ∼= t as a proper subspace. a natural question is for what bases does fb have a (left) module structure over a that naturally extends that of t . for the case t = a, the notion of an amenable basis was introduced in [1] to answer that question. a basis b of t is amenable if for every r ∈a, c ∈b there exist only finitely many d ∈ b such that (rd)c 6= 0. in other words, the set rbc = {d ∈b | (rd)c 6= 0} is finite. it is easy to see that b is amenable if for all r ∈a, the column-finite matrix [`r]b, representing the f-linear map `r : t →t left multiplication by r given by `r(a) = ra, is also row-finite. basically, the row-finiteness requirement allows us to multiply a vector v with infinite support on the left by [`r]b; each entry of the product is the inner product of a vector with finite support and v, a finite sum. in this paper we aim to explain in what sense this module structure is naturally induced by the knowledge of the products of elements from the basis b, as suggested in [1]. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 2 topological characterizations of amenability and congeniality of bases the expression fb = bt serves two purposes: it indicates that b is amenable and gives a name to the a-module structure on fb. given an infinite dimensional module t over an algebra a and an amenable basis b for t , the resulting module bt is said to be a basic module (or the natural module extension of t by b.) the proof of theorem 2.6 of [1] may easily be adapted to get the following result, which guarantees that amenable bases exist when the module considered is countable dimensional over an at most countable dimensional algebra; it is not know in general whether uncountable dimensional algebras must have amenable bases. theorem 1.1. if dima is either finite or countable and dimt is countable then t has an amenable basis. it is easy to see from [1] that some bases, but not necessarily all, are amenable. in fact, the following theorem is a direct consequence of a theorem in [5], which shows that, not only does such a module have a non amenable basis under the hypothesis, but it actually has a contrarian basis (as defined in [5].) theorem 1.2. if a module t has an amenable basis b with an element b ∈b that is not an eigenvector for any left multiplication map lr with r ∈ a\ f , then t also has a non-amenable basis. following [1], given two bases b and c for a module t , we say that b is congenial to c if [i]cb, the matrix representation of the identity map i : t →t with respect to the bases cb and c, is row-finite. in general, this relation is not symmetric; when b is congenial to c and c is congenial to b then we say that b and c are mutually congenial. if b is congenial to c and c is not congenial to b, we say that b is properly congenial to c and the matrix [i]cb is a proper congeniality matrix. in other words, a proper congeniality matrix is a row and column finite matrix whose inverse is column but not row finite. it was proven in [1] that, given two mutually congenial bases b and c, one of them is amenable if and only if the other one is and they induce isomorphic natural module extensions. on the other hand, if b is properly congenial to c, the possible amenability of either basis is largely independent from that of the other one. it is know, however, that if they are both amenable then the map from bt into ct given by left multiplication by the proper congeniality matrix [i]cb is a module homomorphism and, perhaps surprisingly, it is onto. this is one of the main results from [1]. the question about the significance of the potential injectivity of multiplication by [i]cb was left open. the final result in this paper, theorem 3.0.2, shows that a proper congeniality matrix [i]cb never induces a one-to-one left multiplication map and consequently draws a line separating mutual and proper congeniality. our proof of this fact relies on the topological machinery built throughout the paper. proper congeniality has turned out to be one of the most interesting notions in this line of work; it has inspired the notions of simple and projective bases c© agt, upv, 2020 appl. gen. topol. 21, no. 1 3 s. r. lópez-permouth and b. stanley and these types of bases have, in turned, fueled many ongoing projects (see, for example, [1], [2] and [3].) the unit vectors in f (b) are a basis b for f (b) which have the interesting property that they can be, in some sense, a surrogate basis for fb. indeed, any element in fb can be viewed as an infinite linear combination of the elements of b and the only way to represent 0 ∈ fb is using all zero coefficients. so, the elements of b satisfy a stronger sort of linear independence. this is reminiscent of the classical schauder bases where a collection of elements in a topological vector space play the role of a basis because their infinite linear combinations converge to the element that they represent. theorem 1.2.6 basically states that for any basis b there indeed exists a topology on fb that makes b ⊂ fb into a schauder basis. 1.2. nets and convergence. for convenience, we will briefly introduce some topological concepts here. however, most necessary topology notions, such as subspace topologies and discrete topologies (when all subsets are open), may be found in any standard reference (c.g. [8]). the product topology on the cartesian product of a family of topological spaces is the smallest topology making all projection maps continuous. in other words: definition 1.3 (product topologies and projection maps). (1) for a collection of topological spaces {(xi,τi)}i∈i, ∏ i∈i xi = {f : i → ⋃ i∈i xi | f(i) ∈ xi} is a topological space under the topology generated by the base b = {u ⊂ ∏ i∈i xi | u = ∏ i∈i ui where each ui ∈ τi and ui 6= xi only finitely often}. (2) the map πj : ∏ i∈i xi → xj, given by πj(f) = f(j) ∈ xj, is called the j-th projection map. not all topologies may be described in terms of the behavior of sequences, a more powerful notion, called nets, is needed. likewise, although the sequence of images under a continuous map of the elements of a convergent sequence converge to the image of the limit of that sequence, this property alone does not in general characterize continuous functions. in order to obtain a characterization of continuous functions, nets are once again needed. we will start by describing the prerequisite term of directed sets. definition 1.4 (directed sets). a directed set is a non-empty set a with a binary relation (a direction on a) ≤ that satisfies the following: d.1 a ≤ a for all a ∈ a d.2 if a ≤ b and b ≤ c then a ≤ c d.3 if a,b ∈ a then there exists c ∈ a such that a ≤ c and b ≤ c. example 1.5. let x be a topological space and let bx be a local base (also called a neighborhood base) at a point x ∈ x. for u,v ∈ bx define u ≤ v if and only if v ⊂ u, then ≤ is a direction on bx. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 4 topological characterizations of amenability and congeniality of bases definition 1.6 (nets and their convergence). (1) a net is a function f : a → x where a is a directed set and x is a topological space. we write {xα}α∈a to denote the net given by f(α) = xα. (2) for u ⊂ x we say that a net {xα}α∈a is residually (eventually) in u if there exists β ∈ a such that α ≥ β =⇒ xα ∈ u. (3) we say that {xα}α∈a converges to x ∈ x if and only if for all open neighborhoods u of x, {xα}α∈a is eventually in u. the significance of nets is highlighted by the following well-known result. proposition 1.7. for arbitrary topologies, (1) a subset u ⊂ x is open if and only if when a net {xα}α∈a converges to some x ∈ u then {xα}α∈a is residually in u, and (2) a subset f ⊂ x is closed if and only if every convergent net {xα}α∈a ⊂ f converges to some x ∈ f . (3) a function f : x → y is continuous if and only if for every net xα → x we have that f(xα) → f(x). theorem 1.8. when f is equipped with the discrete topology and fb with the corresponding product topology, then f (b) is dense in fb. proof. let u(b1, . . . ,bn|k1, . . . ,kn) = {f ∈ fb | f(bi) = ki, 1 ≤ i ≤ n}, note that sets of this form are open in the product topology as {k} is open in f for all k ∈ f. furthermore, the set {u(b1, . . . ,bn | g(b1), . . . ,g(bn)) | g ∈ f (b), n ∈ n, b1, . . . ,bn ∈b} is a basis for the product topology on fb. clearly u(b1, . . . ,bn | g(b1), . . . ,g(bn)) contains g|{b1,...,bn} ∈ f (b). thus, every open set in fb has non-trivial intersection with f (b). now, let f ∈ fb and let u be open around it. we have seen that u intersects f (b) so that f is a cluster point of f (b) and so, f ∈ cl(f (b)) so that f (b) is dense. � 1.3. uniform topological spaces. the condition of uniform continuity is strictly stronger than continuity. one could say that uniform continuity is the condition of being ‘continuous the same way’ everywhere. the main property of uniformly continuous functions that we will make use of here is that uniformly continuous functions preserve cauchy sequences. that is, if {xn}n∈n is cauchy and f : x → y is uniformly continuous then {f(xn)}n∈n is cauchy in y . we start by recalling the familiar definition, in the context of metric spaces, of a uniformly continuous function. definition 1.9. let x,y be metric spaces, whose metrics are dx,dy respectively. we say a function f : x → y is uniformly continuous if for all � > 0 there exists δ > 0 such that when dx(x,x ′) < δ we have that dy (f(x),f(x ′)) < �. in this paper we deal with potentially uncountable products, which may fail to be metrizable. thus, we introduce uniform spaces, which will allow us to talk about uniformly continuous functions in the absence of a metric. we first develop some notation; the reader may find a more thorough treatment in [8]. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 5 s. r. lópez-permouth and b. stanley definition 1.10. if x is a set, we denote by ∆ the diagonal {(x,x) | x ∈ x} in x × x. if u,v ⊂ x × x, then u ◦ v is the set {(x,y) | for some z ∈ x, (x,z) ∈ v and (z,y) ∈ u}. notice that u and v are just relations on x and ◦ is a natural extension of the notion of composition of functions. the definition we will use comes from the notion that x and y are close together in a metric space if and only if (x,y) is close to the diagonal in x×x. definition 1.11. a diagonal uniformity on a set x is a collection d(x), or just d, of subsets of x ×x, called entourages, which satisfy: (d.a) d ∈d =⇒ ∆ ⊂ d (d.b) d1,d2 ∈d =⇒ d1 ∩d2 ∈d (d.c) d ∈d =⇒ e ◦e ⊂ d for some e ∈d (d.d) d ∈d =⇒ e−1 ⊂ d for some e ∈d where e−1 = {(y,x) | (x,y) ∈ e} (d.e) d ∈d,d ⊂ e =⇒ e ∈d proposition 1.12. if d is a diagonal uniformity and d ∈d then d−1 ∈d. proof. let d ∈d then, by (d.d) we have that there is some e ∈d such that e−1 ⊂ d. therefore, e ⊂ d−1 so by (d.e) we have that d−1 ∈d. � each uniformity gives rise to a topology. to see this, we define neighborhoods of points x ∈ x for a uniform space. definition 1.13. for x ∈ x and d ∈ d let d[x] = {y ∈ x | (x,y) ∈ d}. this is extended to subsets a ⊂ x as follows: d[a] = ⋃ x∈a d[x]. definition 1.14. a collection e ⊂ d is a base for the uniformity on d if for all d ∈d there is some e ∈e such that e ⊂ d. that is, d can be recovered from e by applying (d.e). that these sets, d[x], are neighborhoods is justified by the following theorem which can be found [8]. theorem 1.15. for each x ∈ x, the collection ux = {d[x] | d ∈d} forms a neighborhood base at x, making x a topological space. the way that x is made a topological space is as follows. we say that o is d-open if for all x ∈ o there is some u ∈ d such that u[x] ⊂ o. that is, o is open if and only if is a neighborhood of all its points. this topology will be called the uniform topology on x generated by d. example 1.16. let d� = {(x,y) | d(x,y) < �}⊂ x ×x where x is a metric space with metric d : x ×x → r. the set d = {d� | � > 0} is a uniformity on x. in fact, the uniform topology generated by d coincides with the metric topology on x. example 1.17. let x be a set and let d = {d ⊂ x × x | ∆ ⊂ d}. the uniform topology generated by d is the discrete topology on x. indeed, let a ⊂ x then ∆[x] = {x} ⊂ a for all x ∈ a. this uniformity is called the discrete uniformity on x. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 6 topological characterizations of amenability and congeniality of bases now we generalize the notion of a uniformly continuous function to all uniform spaces. definition 1.18. let x and y be sets with diagonal uniformities d and e respectively. a function f : x → y is uniformly continuous if and only if for each e ∈e, there is some d ∈d such that (x,y) ∈ d =⇒ (f(x),f(y)) ∈ e. as we primarily deal with product spaces in this document it will be useful to have a way to define a uniformity on a product of uniform spaces. in particular, we will do so in such a way that the uniform topology on the product coincides with the product of the uniform topologies. definition 1.19. if xα is a set for each α ∈ a and x = ∏ xα, the αth biprojection is the map pα : x×x → xα×xα defined by pα(x,y) = (πα(x),πα(y)) where πα is the αth projection. that is, πα(x) = x(α) ∈ xα for each x ∈ x. theorem 1.20. if dα is a diagonal uniformity on xα, for each α ∈ a, then the sets p−1α1 (dα1 )∩·· ·∩p −1 αn (dαn ), where dαn ∈dαn for i = 1, . . . ,n, form a base for a uniformity d on x = ∏ xα whose associated topology is the product topology on x. 1.4. complete uniform spaces. let us deal next with completeness for uniform spaces; a notion which, when dealing with metric spaces, relies on cauchy sequences. consequently, we we will lean on the more general concept of cauchy nets. definition 1.21. let x be a uniform space with diagonal uniformity d and let {xλ}λ∈λ ⊂ x be a net. we say that {xλ}λ∈λ is a cauchy net if for all d ∈d there exists λd ∈ λ such that γ,δ > λd =⇒ (xγ,xδ) ∈ d. much like the usual case with uniformly continuous functions and cauchy sequences we have that uniformly continuous functions preserve cauchy nets. that is, if f : x → y where x,y are uniform spaces and {xλ} ⊂ x is a cauchy net then {f(xλ)}⊂ y is a cauchy net. definition 1.22. a uniform space x is complete if every cauchy net converges to some element of x. theorem 1.23. if {xλ}λ∈λ ⊂ x is a cauchy net and f : x → y is uniformly continuous then {f(xλ)}λ∈λ is also a cauchy net. proof. by definition, for each entourage v in x we have that there is some λv ∈ λ such that λ,λ′ > λv implies that (xλ,xλ′) ∈ v . let v be an entourage in y . as f is uniformly continuous there is some entourage u in x such that (x,y) ∈ u implies that (f(x),f(y)) ∈ v . let {xλ}λ∈λ be a cauchy net. let λx,u be such that λ,λ′ ≥ λx,u ⇒ (xλ,xλ′) ∈ u. then, (f(xλ),f(xλ′)) ∈ v so that {f(xλ)}λ∈λ is a cauchy net. � uniformly continuous functions between complete uniform spaces can be continuously extended to the whole space; furthermore, that extension is also uniformly continuous. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 7 s. r. lópez-permouth and b. stanley theorem 1.24. if x,y are complete uniform spaces and a ⊂ x is dense then every f : a → y that is uniformly continuous has a unique uniformly continuous extension f̂ : x → y . the above result, an easy consequence of theorem 8.3.10 in [4], will help us determine when natural module extensions exist. 1.5. topological vector spaces. it turns out that f (b) and fb are topological vector spaces so we write down some useful properties of these spaces. definition 1.25. a topological f−vector space e is an f-vector space e that is endowed with a topology such that the mappings (x,y) → x + y and (λ,x) → λx from e × e to e and f × e to e respectively are continuous. (e ×e and f ×e having the product topologies.) example 1.26. let f be a field and endow f with the discrete topology. then, for non-empty a, x = ∏ α∈a f is a topological vector space with the product topology. proof. we proceed by showing that addition and scalar multiplication are continuous. let + : x ×x → x be such that +(x,y) = x + y. we aim to show that this map is continuous by showing that πj◦+ : x×x → fj is continuous for each j ∈ a. note, of course, that each fj is just a copy of f . note that πj ◦ +(x,y) = πj(x + y) = x(j) + y(j). as singletons make up a base for the topology on f we look at the inverse image of a singleton under this map. (πj ◦ +)−1(z) = ∪x∈fπ−1j (x) × π −1 j (z − x). as π −1 j (x) × π −1 j (z − x) is open in x ×x for each x ∈ f we have that (πj ◦ +)−1(z) is the arbitrary union of open sets and so is open itself. thus πj ◦ + is continuous for each j ∈ a so that + : x ×x → x is continuous. now we show that · : f × x → x defined by ·(λ,f) = λf is continuous. let j ∈ a and note that πj ◦ ·(λ,f) = λf(j). again, we examine the inverse image of singletons, (πj ◦ ·)−1(z) = ∪λ∈f{λ}×π−1j (z λ ) each of which is open in f ×x so that the inverse image is an arbitrary union of open sets. thus, · is continuous as well. � as we saw in definition 1.19 we have that the space defined in example 1.26 is indeed a uniform space. with that in mind, we show that, for linear operators, continuity and uniform continuity are the same. to do this, we rely on some more elementary facts. proposition 1.27. let u be all the basic open sets around 0 as defined by the product topology. let du = {(x,y) | x−y ∈ u} and d = {du | u ∈u}. then, d is a base for the product uniformity, e, on x = ∏ α∈a f . that is d ⊂ e and for every entourage e ∈e there is some entourage d ∈d with d ⊂ e. proof. let e be a base entourage from e, that is, e = p−1α1 (eα1 ) ∩ ·· · ∩ p−1αn (eαn ), where eαn ∈eαn for i = 1, . . . ,n where eα is the discrete uniformity on fα. take u = {x ∈ x | x(αi) = 0 for all i = 1, . . . ,n}. then, du = c© agt, upv, 2020 appl. gen. topol. 21, no. 1 8 topological characterizations of amenability and congeniality of bases {(x,y) | x(αi) = y(αi) for i = 1, . . . ,n} thus du ⊂ e as (x(α),y(α)) ∈ eα for all eα ∈eα. finally, note that du = p−1α1 (∆α1 ) ∩·· ·∩p −1 αn (∆αn ). � proposition 1.28. let x be as in example 1.26 and let f : x → x. then, f is uniformly continuous if and only if for all open sets u around 0 there exists an open set v around 0 such that (x,y) ∈ dv =⇒ (f(x),f(y)) ∈ du where du = {(x,y) | x−y ∈ u}. proof. this follows from the definition of uniformly continuous and the fact that sets of the form du give a base for the uniformity on x. � corollary 1.29. let x be as in example 1.26 and let f : x → x. then, f is uniformly continuous if and only if for all open sets u around 0 there exists an open set v around 0 such that x−y ∈ v =⇒ f(x) −f(y) ∈ u. proposition 1.30. let x be as in example 1.26 and let f : x → x be a linear map. then, f is continuous on all of x if f is continuous at 0. proof. let x ∈ x and let u be open around f(x). let u′ = u − f(x) and note that u′ is open around 0. by continuity at 0 there exists v ′ ⊂ x around 0 such that f(v ′) ⊂ u′. let v = v ′ + x. v is open around x and f(v ) = f(v ′) + f(x) ⊂ u′ + f(x) = u. thus, for linear maps, continuity at 0 is sufficient to show continuity in general. � theorem 1.31. let x be as in example 1.26 and let f : x → x be a linear map. then, f is continuous if and only if f is uniformly continuous. proof. that uniform continuity implies continuity is trivial. let u ⊂ x be an open neighborhood of 0. let v be such that f(v ) ⊂ u which is guaranteed by continuity. now, suppose x − y ∈ v so that f(x − y) ∈ u. however, by linearity of f, f(x−y) = f(x) −f(y) so that x−y ∈ v =⇒ f(x) −f(y) ∈ u so that f is uniformly continuous. � while some of the above results hold for topological vector spaces in general, we opted for leaving our discussion self-contained and focused only on the specific case considered here. 2. topological characterization of amenable bases and the natural module extensions they induce. it would certainly be nice if, in general, we could define rt for r ∈ a and t ∈ fb to be a limit as can be done when the power series are viewed as a module over polynomials with the standard basis. however, if t has a higher dimension than ω this will not work. in particular, elements of fb with uncountable support can not even be found as a limit of elements of t when b is uncountable. that is, the limit of any sequence of finite linear combinations can be at most a countable linear combination, or in the language of our function spaces, a function f : b → f with at most countable support. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 9 s. r. lópez-permouth and b. stanley 2.1. module multiples defined by approximations. for r ∈ a, `r : t → t denotes the map given by `r(t) = rt, γb : f (b) → t denotes the bijection such that γb(f) = ∑ b∈b f(b)b, and ft = [t]b = γ −1 b (t) ∈ f b for t ∈t (the coordinates of t with respect to b.) we think of [t]b as a matrix with one column. also, for r ∈a, the [`r]b represent the map `r with respect to b in the usual way, namely, the rows and columns are labeled by b and, for b ∈ b, the b-th column of [`r]b is given by the equation [`r]b|b×{b} = rfb := [rb]b. with this notation, one gets, as is customary in the theory of matrix representations of linear transformations, the identity granting that, for t ∈ t and r ∈ a, the product [`r]b[t]b is equal to [rt]b = γ −1 b (rt) = rγ −1 b (t) = rft. in other words, multiplication on the left of f ∈ f (b) by [`r]b is a matrix representation of left multiplication by r with respect to the basis b. consequently, for t ∈t the product [`r]b[t]b is equal to (γ−1b ◦ `r ◦γb)(ft). we may then think of left multiplication by [`r]b as a function from f (b) to f (b). as [`r]b already has a function description we avoid using parenthesis when we mean the matrix product and we will write [`r]b· : f (b) → f (b). clearly, this means that [`r]b· = γ−1b ◦ `r ◦γb handling the uncountable dimension situation requires a little finesse beyond the standard approach with sequences. if f ∈ fb has uncountable support, there is no sequence in f (b) that converges to f. we must therefore rely on nets. indeed, recall that a local base at f ∈ fb, bf is given by sets of the form u(b1, . . . ,bn | f(b1), . . . ,f(bn)) and that bf is a directed set under the direction in example 1.5. that is, u ≤ v if and only if v ⊆ u. now we construct a net whose directed set is (bf,≤). for u = u(b1, . . . ,bn | f(b1), . . . ,f(bn)) ∈ bf let fu = f|{b1,...,bn} ∪ 0|b\{b1,...,bn}. definition 2.1. the net constructed above will be called the canonical net for f. claim 2.2. the canonical net for f, {fu | u ∈ bf}, converges to f. proof. let u ∈ bf be a basic open neighborhood of f. let v ∈ bf be such that u ≤ v so that v ⊂ u. therefore, u = u(b1, . . . ,bn | f(b1), . . . ,f(bn)) and v = u(b1, . . . ,bn,bn+1, . . . ,bn+k | f(b1), . . . ,f(bn),f(bn+1), . . . ,f(bn+k)). thus, fv |{b1,...,bn} ≡ fu|{b1,...,bn} so that fv ∈ u. that is, the net is eventually in u for any basic neighborhood of f. � these canonical nets are integral in our description of naturality. note, the canonical net for f consists of better and better approximations of f. therefore, if we can define scalar multiplication naturally, we would like [`r]bfu , u ∈ bf to consist of better and better approximations to [`r]bf. definition 2.3. let r ∈a and f ∈ fb. we say that [lr]b· : fb → fb is the natural extension of [`r]· if {[`r]bfu}→ [lr]bf for all f ∈ fb where {fu}u∈bf is the canonical net described above. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 10 topological characterizations of amenability and congeniality of bases here we must justify the use of the word ‘the’ in our above definition. that is, we must verify that this extension is indeed unique. this is handled by a well-known topological fact. the arbitrary product of hausdorff spaces is again a hausdorff space. as limits are unique in hausdorff spaces we know that {[`r]bfu}u∈bf can converge to at most one element in f b, thus, if one natural extension exists, it is the only one. with it in mind that [`r]bf = rf for r ∈ a and f ∈ f (b) we see that f (b) is a module and {[`r]b · | r ∈a} is a collection of matrices that describe scalar multiplication in f (b). in fact, the definition of γb shows us that this module coincides with t . definition 2.4. we say that af b is the natural extension of the left amodule structure on f (b) if {[lr]b · | r ∈ a} is a set of maps that describe scalar multiplication by r and each [lr]b· is the natural extension of [`r]b·. note, if any [`r]b· fails to have a natural extension then afb can not be a natural extension of f (b) with the aforementioned module structure. note, for any f ∈ fb the scaled canonical net, {rfu}u∈bf is a net in f (b). whether it converges for every f is the determining factor for scaling by r being a valid operation on fb. to help us understand when this happens, we introduce the notion of a uniformity on our topoligical spaces as well as what it means for a net to be cauchy. much like in the case with cauchy sequences in metric spaces, a cauchy net is one such that if we desire two elements of the net to be close to each other, we need only traverse far enough into the net. the notion of closeness will be handled by the uniformity as we lack a metric in general. under this new structure we will utilize the notion of a uniformly continuous function. in particular, uniformly continuous functions map cauchy nets to cauchy nets. therefore, if multiplication by r is found to be uniformly continuous, then we shall be able to define rf as the limu∈bf rfu as this limit will therefore exist. the details are the subject of the following subsection. 2.2. uniform continuity and amenability. we will see that if [`r]b is a row finite matrix, then it is uniformly continuous and, therefore, has a uniformly continuous extension as guaranteed by theorem 1.24. theorem 2.5. if [`r]b· : f (b) → f (b) is row finite then it is uniformly continuous. proof. as [`r]b is row finite we know that for each b ∈ b there are only finitely many b′ ∈ b such that [`r]b(b,b′) 6= 0. as it is f-linear, we show that it is continuous at 0 to establish that it is uniformly continuous. let u(b1, . . . ,bn | 0, . . . , 0) be an open set around 0. that is, u = {f ∈ fb | f(bi) = 0, i = 1, . . . ,n}. let bu = {b ∈b | [`r]b(bi,b) 6= 0 for any i = 1, . . . ,n}. note, bu is finite as each row of [`r]b is finite. let v = u(bu | 0) = {f ∈ fb |f(b) = 0 for all b ∈bu}. now suppose that f1,f2 ∈ f (b) such that f := f1 − f2 ∈ v. then, for i ∈ {1, . . . ,n}, [`r]bf(bi) = ∑ b′∈b[`r]b(bi,b ′)f(b′), yet, [`r]b(bi,b ′) = 0 for all c© agt, upv, 2020 appl. gen. topol. 21, no. 1 11 s. r. lópez-permouth and b. stanley b′ ∈b\bu so that [`r]bf(bi) = ∑ b′∈bu [`r]b(bi,b ′)f(b′) = 0 as f(b′) = 0 for all b′ ∈bu so that [`r]bf ∈u. thus, [`r]b· is uniformly continuous. � theorem 2.6. if [`r]b is row finite then, [lr]b· : fb → fb defined by [lr]bf(b) = ∑ b′∈b [`r]b(b,b ′)f(b′) is the uniformly continuous extension of [`r]b·. proof. as [`r]b is row finite we know that for each b ∈b there are only finitely many b′ ∈ b such that [`r]b(b,b′) 6= 0 so that [lr]bf(b) is defined. therefore, [lr]b is indeed a function from f b to fb. the proof from here is identical to the above proof, except that f1,f2 are taken from f b. � therefore, we have established that if b is amenable, then [`r]b is uniformly continuous, that it has a uniformly continuous extension, and that this extension is the natural extension of [`r]b. therefore, with the collection {[lr]b · | r ∈ a} fb has a left a-module structure. furthermore, this structure is the natural extension of the a-module structure on f (b). lemma 2.7. any injective net {bλ}λ∈λ converges to 0. proof. let u be open around 0 and let b1, . . . ,bn ∈b be such that if f(bi) = 0 for all 1 ≤ i ≤ n then f ∈ u. choose λ ∈ λ such that γ > λ implies that bγ 6∈ {b1, . . . ,bn}. then, γ > λ implies bγ ∈ u. � theorem 2.8. if [`r]b· : f (b) → f (b) is continuous, then [`r]b is row finite. proof. suppose that [`r]b fails to be row finite so that there is some infinite row. that is, there is some b ∈b such that [`r]b(b,b′) 6= 0 for infinitely many b′. let f : n →b be an injective map such that for all b′ ∈ f(n) we have that [`r]b(b,b ′) 6= 0. now, f describes an injective net so that lemma 2.7 applies. by continuity, we see that [`r]bf(n) → 0. therefore, there is some n ∈ n such that for n > n we have that [[`r]bf(n)](b) = 0. this is a contradiction as the b-th coordinate of [`r]bf(n) is non-zero for all n ∈ n. therefore we must conclude that [`r]b is indeed row finite. � corollary 2.9. the basis b is amenable if and only if [`r]b : f (b) → f (b) is uniformly continuous for all r ∈a. theorem 2.10. the left a-module f (b) has a natural extension if and only if b is amenable. 3. characterizations of mutual and proper congeniality it is known that if b is mutually congenial to c then bt and ct are isomorphic. in fact, they are ‘naturally ’isomorphic as the congeniality map is an isomorphism between them. furthermore, for t of countable dimension, it is known that congeniality maps are onto. here we show that if a congeniality map is one-to-one as well as being onto and, as always, continuous, then it is, c© agt, upv, 2020 appl. gen. topol. 21, no. 1 12 topological characterizations of amenability and congeniality of bases in fact, a mutual congeniality map. we do so by establishing continuity of the inverse map. theorem 3.1. let c and b be countable and let g : c ×b → f be a row and column finite matrix such that g· maps fb bijectively onto fc, then g−1 is continuous. the proof of this lemma uses a lot of the same tools that are used in the proof that congeniality maps are onto, which is provided in [1]. proof. enumerate b = {bi}i∈z+ . let v be the row-span of g. for n ∈ z+, define vn = {v ∈ v : v(bj) = 0 if j > n}; one could informally write ‘vn = f{b1,...,bn} ∩ v ’. then v1 ⊆ v2 ⊆ ··· ⊆ ∪∞n=1vi = v . also, for any infinite matrix h and for n ∈ n, h>n or h≥n+1 denote the infinite matrix made up of all rows of h but beginning with the (n + 1)-th. note that vn is a subspace of v and dim vn ≤ n. recursively, choose, for all n ∈ z+, cn, a basis for vn, in such a way that cn ⊂cn+1. then c = ⋃ n∈z+ cn is a basis for v . let f be a matrix such that fg = h is a matrix having, as rows, the elements of c in such a way that the elements of ci appear no later than those of cj when i < j. clearly, f is a row-finite invertible matrix with an inverse f−1 that is also row-finite. let {nk | k ≥ 1} and be strictly increasing sequences of positive integers and let {mk | k ≥ 1} be a sequence of positive integers so that the rows of h may be viewed as a sequence of mk × nk rectangular matrices ck with mk ≤ nk. let `k = ∑k j=1 mj. in the following manner: now we have fg = h =   c1 0 0c2 0 . . .   , and h· : fb → fc → fe where e = {ei}i∈z+ and ei = γc ( f−1c×{ei} ) . we begin by examining the image of sets of the form u p j := u(bj | p) = {s ∈ f b | s(bj) = p} for some bj ∈ b and p ∈ f . let bj ∈ b and p ∈ f be arbitrary. let k = min{k ∈ z+ | j ≤ nk}. let fbnp = {t ∈ f {b1,...bnk} | t(bj) = p} and for t ∈ fbnp let u(t) = {s ∈ fb | s(bj) = t(bj) for all 1 ≤ j ≤ nk} and let t̂ = t ∪ 0|{bj | j>nk} ∈ f b. note that hs(bi) = ht̂(bi) for 1 ≤ i ≤ `k for all s ∈ u(t). thus, hu(t) ⊂ û(t) = {v ∈ fe | v(ej) = ht̂(ej) for all 1 ≤ j ≤ `k}. now, for v ∈ û(t) we show that hx = v has a solution x ∈ u(t). the existence of a solution for the system can be obtained as the limit of a convergent sequence. we build the sequence as follows. let dk = v|{ej | 1 ≤ j ≤ `k} and for every y > k, dy = v|{ej | `y−1+1 ≤ j ≤ `y} and, for all y ≥ k, dy = v|{ej | 1 ≤ j ≤ `y}. clearly, t is such that such that hkt = dk where hy−1 =   c1 0 0 c2 0 . . . cy−1   c© agt, upv, 2020 appl. gen. topol. 21, no. 1 13 s. r. lópez-permouth and b. stanley for y > k. for y > k + 1 suppose a y − 1-th approximation xy−1 has been obtained; in other words, hy−1xy−1 = dy−1. we construct xy, but first we partition cy as ( cy,1 | cy,2 ) where cy,1 has ny−1 columns. we note that the rows of cy,2 are linearly independent. to see this we assume that they aren’t linearly independent and that a1r ′ 1 + . . .amyr ′ my = 0 where r′i is a row of cy,2 and not all ai are zero. now consider the same linear combination, except of the rows of cy. by the linear independence of the rows of cy we know a1r1 + amyrmy 6= 0, however, its non-zero entries must exist in the first ny−1 entries. thus, a1r1 + amyrmy ∈ vy−1 = span{ ⋃y−1 j=1 cj} so that cy is a linearly dependent collection. this is a contradiction, so the rows of cy,2 are indeed linearly independent. now we obtain a solution xy to the equation cy,2xy = dy−1−cy,1xy−1 and setting xy = xy−1 ∪xy. the existence of xy can be assured because my, the number of rows of cy,2, is less than or equal to ny −ny−1, the number of its columns, due to linear independence. we note that xy is a vector of length ny, let sy = xy∪0|{bj | j>ny} ∈ f (b). now, clearly {sy}∞y=1 is a cauchy sequence. that is, if we want sn − sm to have its first non-zero entry after ` then we choose j such that nj > ` and let n,m > nj. let s = limy→∞sk which is guaranteed to exist by completeness. furthermore, (hsy)(ej) = (hyxy)(ej) = dy(ej) for 1 ≤ j ≤ `y and hsy → hs by the continuity of h. let d̂y = dy ∪ 0|{ej | j>`y} and note that d̂y is a cauchy sequence and clearly d̂y → ∪dy = v. additionally, hsy − dy → 0 as (hsy)(ej) = (hyxy)(ej) = dy(ej) for 1 ≤ j ≤ `y. therefore, if we want hsy(ez) − dy(ez) = 0 we need only choose y > y where 1 ≤ z ≤ `y . thus, hsy → v so v = hs. thus, û(t) ⊂ hu(t) so that û(t) = hu(t) which is open in the product topology on fe. then, as u p j = ∪t∈fbnp u(t) we have that huj = h ( ∪ t∈fbnp u(t) ) = ∪ t∈fbnp hu(t) so that huj is open in f e. now, let u = u(bj1, . . . ,bjz | p1, . . . ,pz) be a basic open set in fb. then, u = ⋂z 1=1 u pi ji so that hu = h( ⋂z 1=1 u pi ji ) ⊂ ⋂z 1=1 hu pi ji . however, as g· and f· are bijections we know that fg· = h· is a bijection as well so that hu = h( ⋂z 1=1 u pi ji ) = ⋂z 1=1 hu pi ji so that hu is open in fe. as f is row finite we know that f· : fc → fe is continuous. therefore, f−1hu is open in fc. however, f−1hu = gu so that g· maps open sets to open sets. therefore, g· is an open map. now, by bijectivity we know that (g·)−1 : fc → fb is well-defined. furthermore, [(g·)−1]−1u is open for all open u. thus, (g·)−1 is continuous and therefore is uniformly continuous by virtue of linearity. we also have that (g·)−1|f(c) = g−1 · |f(c) so that g−1· is continuous and g−1· = (g·)−1. � theorem 3.2. let b and c be amenable. then [i]cb· bijectively maps bt onto ct if and only if b is mutually congenial to c. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 14 topological characterizations of amenability and congeniality of bases proof. let b and c be amenable such that [i]cb· bijectively maps bt onto ct . then, [i]cb satisfies the conditions of theorem 3.1 so that ([i] c b·) −1 = [i]bc · is continuous. the other direction is trivial. � references [1] l. m. al-essa, s. r. lópez-permouth and n. m. muthana, modules over infinitedimensional algebras, linear and multilinear algebra 66 (2018), 488–496. [2] p. aydŏgdu, s. r. lópez-permouth and r. muhammad, infinite-dimensional algebras without simple bases, linear and multilinear algebra, to appear. [3] j. dı́az boils, s. r. lópez-permouth and r. muhammad, amenable and simple bases of tensor products of infinite dimensional algebras, preprint. [4] r. engelking, general topology, sigma series in pure mathematics, vol. 6 (1989). [5] s. r. lópez-permouth and b. stanley, on the amenability profile of an infinite dimensional module over an algebra, preprint. [6] p. nielsen, row and column finite matrices, proc. amer. math. soc. 135, no. 9 (2007), 2689–2697. [7] b. stanley, perspectives on amenability and congeniality of bases, ph. dissertation, ohio university, february 2019. [8] s. willard, general topology, dover publications (1970). c© agt, upv, 2020 appl. gen. topol. 21, no. 1 15 @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 91-102 relative dimension r-dim and finite spaces a. c. megaritis abstract in [4] a relative covering dimension is defined and studied which is denoted by r-dim. in this paper we give an algorithm of polynomial order for computing the dimension r-dim of a pair (q, x), where q is a subset of a finite space x, using matrix algebra. 2010 msc: 54f45, 54a05, 65f30. keywords: covering dimension, relative dimension, finite space, incidence matrix. 1. introduction and preliminaries the “ relative dimensions ” or “ positional dimensions ” are functions whose domains are classes of subsets. by a class of subsets we mean a class consisting of pairs (q, x), where q is a subset of a space x. the class of finite topological spaces was first studied by p.a. alexandroff in 1937 in [1]. a topological space x is finite if the set x is finite. in what follows we denote by x = {x1, . . . , xn} a finite space of n elements and by ui the smallest open set of x containing the point xi, i = 1, . . . , n. the cardinality of a set x is denoted by |x| and the first infinite cardinal is denoted by ω. let x = {x1, . . . , xn} be a finite space of n elements. the n × n matrix t = (tij), where tij = { 1, if xi ∈ uj 0, otherwise is called the incidence matrix of x. we observe that uj = {xi : tij = 1}, j = 1, . . . , n. 92 a. c. megaritis we denote by c1, . . . , cn the n columns of the matrix t . let ci = ⎛ ⎜⎜⎜⎝ c1i c2i ... cni ⎞ ⎟⎟⎟⎠ and cj = ⎛ ⎜⎜⎜⎝ c1j c2j ... cnj ⎞ ⎟⎟⎟⎠ be two n × 1 matrices. then, by max ci we denote the maximum max{c1i, c2i, . . . , cni} and by ci + cj the n × 1 matrix ci + cj = ⎛ ⎜⎜⎜⎝ c1i + c1j c2i + c2j ... cni + cnj ⎞ ⎟⎟⎟⎠ . also, we write ci ≤ cj if only if cki ≤ ckj for each k = 1, . . . , n. for the following notions see for example [2]. let x be a space. a cover of x is a non-empty set of subsets of x, whose union is x. a cover c of x is said to be open (closed) if all elements of c is open (closed). a family r of subsets of x is said to be a refinement of a family c of subsets of x if each element of r is contained in an element of c. define the order of a family r of subsets of a space x as follows: (a) ord(r) = −1 if and only if r consists of only the empty set. (b) ord(r) = n, where n ∈ ω, if and only if the intersection of any n + 2 distinct elements of r is empty and there exist n + 1 distinct elements of r, whose intersection is not empty. (c) ord(r) = ∞, if and only if for every n ∈ ω there exist n distinct elements of r, whose intersection is not empty. definition 1.1 (see [4]). we denote by r-dim the (unique) function that has as domain the class of all subsets and as range the set ω ∪ {−1, ∞} satisfying the following condition r-dim(q, x) ≤ n, where n ∈ {−1} ∪ ω if and only if for every finite family c of open subsets of x such that q ⊆ ∪{u : u ∈ c} there exists a finite family r of open subsets of x refinement of c such that q ⊆ ∪{v : v ∈ r} and ord(r) ≤ n. finite topological spaces and the notion of dimension play an important role in digital spaces, computer graphics, and image analysis. in [5] the authors gave an algorithm for computing the covering dimension of a finite topological space using matrix algebra. in this paper we give an algorithm of polynomial order for computing the dimension r-dim of a pair (q, x), where q is a subset of a finite space x, using matrix algebra. relative dimension r-dim and finite spaces 93 2. finite spaces and dimension r-dim in this section we present some propositions concerning the dimension r-dim of a pair (q, x), where q is a subset of a finite space x. proposition 2.1. let x = {x1, . . . , xn} be a finite space and q ⊆ x. then, r-dim(q, x) ≤ k, where k ∈ ω, if and only if there exists a family {uj1, . . . , ujm} such that {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ . . . ∪ ujm and ord({uj1, . . . , ujm) ≤ k. proof. let r-dim(q, x) ≤ k, where k ∈ ω. we prove that there exists a family {uj1, . . . , ujm } such that {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ . . . ∪ ujm and ord({uj1, . . . , ujm) ≤ k. let ν = min{m ∈ ω : there exist j1, . . . , jm ∈ {1, . . . , n} such that {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ · · · ∪ ujm} and c = {uj1, . . . , ujν } be a family such that {xj1, . . . , xjν } ⊆ q ⊆ uj1 ∪ . . . ∪ ujν . since rdim(q, x) ≤ k, there exists a family r = {v1, . . . , vμ} of open subsets of x refinement of c such that q ⊆ v1 ∪ . . . ∪ vμ and ord(r) ≤ k. clearly, it suffices to prove that {uj1, . . . , ujν } ⊆ r. indeed, we suppose that there exists α ∈ {1, . . . , ν} such that ujα /∈ r. since xjα ∈ q, there exists β ∈ {1, . . . , μ} such that xjα ∈ vβ. by the fact that ujα is the smallest open set of x containing the point xjα we have that ujα ⊆ vβ. also, since ujα /∈ r, we have ujα �= vβ. therefore, ujα ⊂ vβ. since r is a refinement of c, there exists γ ∈ {1, . . . , ν} such that vβ ⊆ ujγ . hence, ujα ⊂ ujγ . we observe that q ⊆ (uj1 ∪ . . . ∪ ujν ) \ ujα, which is a contradiction by the choice of ν. thus, c ⊆ r. conversely, we suppose that there exists a family {uj1, . . . , ujm} such that {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ . . . ∪ ujm and ord({uj1 , . . . , ujm) ≤ k. we prove that r-dim(q, x) ≤ k. indeed, let c be a finite family of open subsets of x such that q ⊆ ∪{u : u ∈ c}. it suffices to prove that the family {uj1, . . . , ujm} is a refinement of c. for every i ∈ {1, . . . , m} there exists vi ∈ c such that xji ∈ uji ⊆ vi. this means that the family {uj1, . . . , ujm} is a refinement of c. � proposition 2.2. let x = {x1, . . . , xn} be a finite space, where n > 1, and q ⊆ x. then, r-dim(q, x) ≤ |q| − 1. proof. let q = {xj1, . . . , xjm}. the family {uj1, . . . , ujm} has m elements and, therefore, ord({uj1 , . . . , ujm}) ≤ m − 1. thus, by proposition 2.1, r-dim(q, x) ≤ m − 1 = |q| − 1. � 94 a. c. megaritis note 1. in the following propositions we suppose that x = {x1, . . . , xn} is a finite space with n elements, q ⊆ x, t = (tij), i = 1, . . . , n, j = 1, . . . , n, the incidence matrix of x, and c1, . . . , cn the n columns of the matrix t . we denote by 1q the n × 1 matrix ⎛ ⎜⎜⎜⎝ a1 a2 ... an ⎞ ⎟⎟⎟⎠ , where ai = { 1, if xi ∈ q 0, otherwise. example 2.3. let x = {x1, x2, x3, x4, x5} and q = {x1, x3, x4}. then, 1q = ⎛ ⎜⎜⎜⎜⎝ 1 0 1 1 0 ⎞ ⎟⎟⎟⎟⎠ . proposition 2.4. if cj = 1q and xj ∈ q for some j ∈ {1, . . . , n}, then r-dim(q, x) = 0. proof. since cj = 1q, we have tij = 1 for every xi ∈ q and, therefore, q ⊆ uj. since ord({uj}) = 0, by proposition 2.1, we have r-dim(q, x) = 0. � proposition 2.5. let cji, i = 1, . . . , m, be m columns of the matrix t . then, cj1 + . . . + cjm ≥ 1q if and only if q ⊆ uj1 ∪ . . . ∪ ujm. proof. let cj1 +. . .+cjm ≥ 1q. we prove that q ⊆ uj1 ∪. . .∪ujm. let xi0 ∈ q. by the definition of the matrix t and by the assumption cj1 + . . . + cjm ≥ 1q, there exists κ ∈ {1, . . . , m} such that ti0jκ = 1. since ujκ = {xi : tijκ = 1}, we have xi0 ∈ ujκ . thus, q ⊆ uj1 ∪ . . . ∪ ujm. conversely, we suppose that q ⊆ uj1 ∪ . . . ∪ ujm. then, for every xi ∈ q there exists κ(i) ∈ {1, . . . , m} such that xi ∈ ujκ(i) . therefore, by the definition of the matrix t , tijκ(i) = 1. thus, cj1 + . . . + cjm ≥ 1q. � proposition 2.6 (see proposition 2.6 of [5]). let cji, i = 1, . . . , m, be m columns of the matrix t and k = max(cj1 +. . .+cjm), that is k is the maximum element of the n × 1 matrix cj1 + . . . + cjm. then, ord({uj1, . . . , ujm}) = k − 1. definition 2.7. we define a preorder � on the set of all families {xj1, . . . , xjm} with {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ · · · ∪ ujm by {xj1, . . . , xjm1 } � {xj′1, . . . , xj′m2 } relative dimension r-dim and finite spaces 95 if and only if {uj1, . . . , ujm1 } ⊆ {uj′1, . . . , uj′m2 }. remark 2.8. the space x is t0 if and only if ui = uj implies xi = xj for every i, j (see [1]). therefore, if the space x is t0, then the relation � is an order. we note that if the space x is t0, then there exists exactly one minimal family on the set of all families {xj1, . . . , xjm} with {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪· · ·∪ujm . proposition 2.9. let {xi1, . . . , xiμ} ⊆ q ⊆ {ui1, . . . , uiμ }, ν = min{m ∈ ω : there exist j1, . . . , jm ∈ {1, . . . , n} such that {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ · · · ∪ ujm}, and {xj1, . . . , xjν } ⊆ q ⊆ {uj1, . . . , ujν }. then, {xj1, . . . , xjν } � {xi1, . . . , xiμ}. proof. the proof is similar to that of proposition 2.1. � proposition 2.10. let {xj1, . . . , xjν } be a minimal family on the set of all families {xj1, . . . , xjm} with {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ · · · ∪ ujm. if ord({uj1 , . . . , ujν }) = k ≥ 0, then for every family {xr1, . . . , xrμ} with {xr1, . . . , xrμ} ⊆ q ⊆ ur1 ∪ · · · ∪ urμ we have ord({ur1, . . . , urμ} ≥ k. proof. let {ur1, . . . , urμ} be a family such that {xr1, . . . , xrμ} ⊆ q ⊆ ur1 ∪ · · · ∪ urμ. then, {xj1, . . . , xjν } � {xr1, . . . , xrμ} and, therefore, {uj1, . . . , ujν } ⊆ {ur1, . . . , urμ}. since ord({uj1, . . . , ujν }) = k, we have ord({ur1, . . . , urμ} ≥ k. � proposition 2.11. let {xj1, . . . , xjν } be a minimal family on the set of all families {xj1, . . . , xjm} with {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ · · · ∪ ujm. then, rdim(q, x) = max(cj1 + . . . + cjν ) − 1. proof. let k = max(cj1 + . . . + cjν ). then, by proposition 2.6, we have ord({uj1, . . . , ujν }) = k − 1 and, therefore, by proposition 2.1, rdim(q, x) ≤ k − 1. we prove that rdim(q, x) = k − 1. we suppose that rdim(q, x) < k − 1. then, by proposition 2.1, there exists a family {ur1, . . . , urμ} such that {xr1, . . . , xrμ} ⊆ q ⊆ ur1 ∪ · · · ∪ urμ and ord({ur1, . . . , urμ}) < k − 1. 96 a. c. megaritis since ord({uj1, . . . , ujν }) = k − 1, by proposition 2.10, we have ord({ur1, . . . , urμ}) ≥ k − 1 which is a contradiction. thus, rdim(q, x) = k − 1. � proposition 2.12. let cji, i = 1, . . . , ν, be ν columns of the matrix t such that cj1 + . . . + cjν ≥ 1q and {xj1, . . . , xjν } ⊆ q. if cr1 + . . . + crq � 1q for every {xr1, . . . , xrq } ⊆ q and q < ν, then {xj1, . . . , xjν } is a minimal family on the set of all families {xj1, . . . , xjm} with {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪· · ·∪ujm . proof. since cj1+. . .+cjν ≥ 1q and cr1+. . .+crq � 1q for every {xr1, . . . , xrq } ⊆ q and q < m, by proposition 2.5, we have ν = min{m ∈ ω : there exist j1, . . . , jm ∈ {1, . . . , n} such that {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ · · · ∪ ujm}. thus, by proposition 2.9, {xj1, . . . , xjν } is a minimal family on the set of all families {xj1, . . . , xjm} with {xj1, . . . , xjm} ⊆ q ⊆ uj1 ∪ · · · ∪ ujm. � by propositions 2.11 and 2.12 we have the following corollary. corollary 2.13. let cji, i = 1, . . . , ν, be ν columns of the matrix t such that cj1 + . . . + cjν ≥ 1q and {xj1, . . . , xjν } ⊆ q. if cr1 + . . . + crq � 1q for every {xr1, . . . , xrq } ⊆ q and q < ν, then rdim(q, x) = max(cj1 + . . . + cjν ) − 1. 3. an algorithm for computing the covering dimension in this section we give an algorithm of polynomial order for computing the dimension r-dim(q, x), where q is a subset of a finite space x, using the propositions 2.11 and 2.5. algorithm 3.1. let x = {x1, . . . , xn} be a finite space of n elements, q = {xλ1, . . . , xλl} ⊆ x, and t = (tij) the n × n incidence matrix of x. our intended algorithm contains l − 1 steps: step 1. read the l columns cλ1, . . . , cλl of the matrix t . if some column is equal to 1q, then print r-dim(q, x) = 0. otherwise go to the step 2. step 2. find the sums cλj11 + cλj21 + . . . + cλj(l−1)1 for each {j11, j21, . . . , j(l−1)1} ⊆ {1, . . . , l}. if there exists {j011, j021, . . . , j0(l−1)1} ⊆ {1, . . . , l} such that cλ j0 11 + cλ j0 21 + . . . + cλ j0 (l−1)1 ≥ 1q, then go to the step 3. relative dimension r-dim and finite spaces 97 otherwise print r-dim(q, x) = max(cλ1 + cλ2 + . . . + cλl) − 1. step 3. find the sums cλj12 + cλj22 + . . . + cλj(l−2)2 for each {j12, j22, . . . , j(l−2)2} ⊆ {j011, j021, . . . , j0(l−1)1}. if there exists {j012, j022, . . . , j0(l−2)2} ⊆ {j011, j021, . . . , j0(l−1)1} such that cλ j0 12 + cλ j0 22 + . . . + cλ j0 (l−2)2 ≥ 1q, then go to the step 4. otherwise print r-dim(q, x) = max(cλ j0 11 + cλ j0 21 + . . . + cλ j0 (l−1)1 ) − 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . step l − 2. find the sums cλj1(l−3) + cλj2(l−3) + cλj3(l−3) for each {j1(l−3), j2(l−3), j3(l−3)} ⊆ {j01(l−4), j02(l−4), j03(l−4), j04(l−4)}. if there exists {j01(l−3), j02(l−3), j03(l−3)} ⊆ {j01(l−4), j02(l−4), j03(l−4), j04(l−4)} such that cλ j0 1(l−3) + cλ j0 2(l−3) + cλ j0 3(l−3) ≥ 1q, then go to the step l − 1. otherwise print r-dim(q, x) = max(cλ j0 1(l−4) + cλ j0 2(l−4) + cλ j0 3(l−4) + cλ j0 4(l−4) ) − 1. step l − 1. find the sums cλj1(l−2) + cλj2(l−2) for each {j1(l−2), j2(l−2)} ⊆ {j01(l−3), j02(l−3), j03(l−3)}. if there exists {j01(l−2), j02(l−2)} ⊆ {j01(l−3), j02(l−3), j03(l−3)} such that cλ j0 1(l−2) + cλ j0 2(l−2) ≥ 1, then print r-dim(q, x) = max(cλ j0 1(l−2) + cλ j0 2(l−2) ) − 1. otherwise print r-dim(q, x) = max(cλ j0 1(l−3) + cλ j0 2(l−3) + cλ j0 3(l−3) ) − 1. 98 a. c. megaritis example 3.2. let x = {x1, x2, x3, x4} with the topology τ = {∅, {x2}, {x1, x2}, {x2, x3}, {x1, x2, x3}, x} and q = {x1, x3}. then, 1q = ⎛ ⎜⎜⎝ 1 0 1 0 ⎞ ⎟⎟⎠ . we observe that u1 = {x1, x2}, u2 = {x2}, u3 = {x2, x3}, u4 = x. therefore, t = ⎛ ⎜⎜⎝ 1 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 ⎞ ⎟⎟⎠ , c1 = ⎛ ⎜⎜⎝ 1 1 0 0 ⎞ ⎟⎟⎠ , c3 = ⎛ ⎜⎜⎝ 0 1 1 0 ⎞ ⎟⎟⎠ . moreover, c1 + c3 = ⎛ ⎜⎜⎝ 1 2 1 0 ⎞ ⎟⎟⎠ ≥ 1q and max(c1 + c3) = 2. thus, r-dim(q, x) = max(c1 + c3) − 1 = 1. 4. remarks on the algorithm for computing the covering dimension of finite topological spaces remark 4.1. let a = (αij) be a n × n matrix and b = (βij) a m × m matrix. the kronecker product of a and b (see [3]) is the mn × mn block matrix a ⊗ b = ⎛ ⎜⎝ α11b . . . α1nb ... ... ... αn1b . . . αnnb ⎞ ⎟⎠ . more explicitly, the kronecker product of a and b is the matrix relative dimension r-dim and finite spaces 99 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ α11β11 . . . α11β1m . . . α1nβ11 . . . α1nβ1m ... ... ... ... ... ... α11βm1 . . . α11βmm . . . α1nβm1 . . . α1nβmm ... ... ... ... αn1β11 . . . αn1β1m . . . αnnβ11 . . . αnnβ1m ... ... ... ... ... ... αn1βm1 . . . αn1βmm . . . αnnβm1 . . . αnnβmm ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . let x = {x1, . . . , xn} be a finite space of n elements and y = {y1, . . . , ym} a finite space of m elements. it is known that if tx is the incidence matrix of x and ty is the incidence matrix of y , then the incidence matrix of x × y = {(x1, y1), . . . , (x1, ym), . . . , (xn, y1), . . . , (xn, ym)} is the kronecker product tx ⊗ ty of tx and ty (see [8]). example 4.2. let x = {x1, x2, x3} with the topology τx = {∅, {x2}, {x1, x2}, {x2, x3}, x} and y = {y1, y2, y3, y4} with the topology τy = {∅, {y3}, {y1, y3}, {y2, y3}, {y1, y2, y3}, y }. also, let qx = {x1, x3} and qy = {y1, y2, y3}. then, qx × qy = {(x1, y1), (x1, y2), (x1, y3), (x3, y1), (x3, y2), (x3, y3)} and 1qx = ⎛ ⎝ 10 1 ⎞ ⎠ , 1qy = ⎛ ⎜⎜⎝ 1 1 1 0 ⎞ ⎟⎟⎠ , 1qx×qy = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 1 1 0 0 0 0 0 1 1 1 0 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . the incidence matrix tx of x is tx = ⎛ ⎝ 1 0 01 1 1 0 0 1 ⎞ ⎠ 100 a. c. megaritis and the incidence matrix ty of y is ty = ⎛ ⎜⎜⎝ 1 0 0 1 0 1 0 1 1 1 1 1 0 0 0 1 ⎞ ⎟⎟⎠ . therefore, the incidence matrix tx×y of the product space x × y is tx×y = tx ⊗ ty = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . we observe that c1 + c2 + c9 + c10 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 1 2 0 2 2 4 0 1 1 2 0 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ > 1qx ×qy , cr1 + cr2 + cr3 � 1qx ×qy for every {r1, r2, r3} ⊆ {1, 2, 9, 10}, and max(c1 + c2 + c9 + c10) = 4. thus, r-dim(qx × qy , x × y ) = max(c1 + c2 + c9 + c10) − 1 = 3. also, we observe that r-dim(qx, x) = 1 and r-dim(qy , y ) = 1. remark 4.3. let x = {x1, . . . , xn} be a finite t0-space and q ⊆ x. then, there exists a finite space y homeomorphic to x such that the incidence matrix ty of y is an upper triangular matrix. let h a homeomorphism from x to y such that the incidence matrix ty of y is an upper triangular matrix. in order to calculate the r-dim(q, x) it suffices to calculate r-dim(h(q), y ). relative dimension r-dim and finite spaces 101 example 4.4. let x = {x1, x2, x3} with the topology τx = {∅, {x2}, {x1, x2}, {x2, x3}, x} and q = {x2, x3}. we consider the space y = {y1, y2, y3} with the topology τy = {∅, {y1}, {y1, y2}, {y1, y3}, y }. we observe that the map h : x → y defined by h(x1) = y2, h(x2) = y1, and h(x3) = y3 is a homeomorphism from x to y with h(q) = {y1, y3}. the incidence matrix ty of y is ty = ⎛ ⎝ 1 1 10 1 0 0 0 1 ⎞ ⎠ . since c3 = ⎛ ⎝ 10 1 ⎞ ⎠ = 1h(q), we have r-dim(h(q), y ) = 0. therefore, r-dim(q, x) = 0. proposition 4.5. an upper bound on the number of iterations of the algorithm for computation of the dimension r-dim of a pair (q, x), where q is a subset of a finite space x, is the number 1 2 |q|2 + 3 2 |q| − 3. proof. let |q| = l. we observe that the number of iterations the algorithm performs in steps 1, 2, 3, 4, . . . , l − 2, l − 1 is l, l, l − 1, l − 2, . . . , 4, 3 respectively. thus, the number of iterations the algorithm performs is l + l + (l − 1) + (l − 2) + . . . + 4 + 3 = l + (l − 2)(l + 3) 2 = 1 2 l2 + 3 2 l − 3 = 1 2 |q|2 + 3 2 |q| − 3. � 5. problems in [9] (see also [6] and [7]) two relative covering dimensions are defined and studied which are denoted by dim and dim∗. the given two definitions below are actually the definitions of dimensions dim and dim∗ given in [9] for regular spaces. definition 5.1. we denote by dim the (unique) function with domain the class of all subsets and range the set ω ∪ {−1, ∞}, satisfying the following condition dim(q, x) ≤ n, where n ∈ {−1} ∪ ω if and only if for every finite open cover c of the space x there exists a finite open cover rq of q such that rq is a refinement of c and ord(rq) ≤ n. 102 a. c. megaritis definition 5.2. we denote by dim∗ the (unique) function with domain the class of all subsets and range the set ω ∪ {−1, ∞}, satisfying the following condition dim∗(q, x) ≤ n, where n ∈ {−1} ∪ ω if and only if for every finite open cover c of the space x there exists a finite family r of open subsets of x refinement of c such that q ⊆ ∪{v : v ∈ r} and ord(r) ≤ n. problem 5.3. find an algorithm for computing the dimension dim of a pair (q, x), where q is a subset of a finite space x, using matrix algebra. problem 5.4. find an algorithm for computing the dimension dim∗ of a pair (q, x), where q is a subset of a finite space x, using matrix algebra. acknowledgements. the author would like to thank the referee for very helpful comments and suggestions. references [1] p. alexandroff, diskrete räume, mat. sb. (n.s.) 2 (1937), 501–518. [2] r. engelking, theory of dimensions, finite and infinite, sigma series in pure mathematics, 10. heldermann verlag, lemgo, 1995. viii+401 pp. [3] h. eves, elementary matrix theory, dover publications, inc., new york, 1980. xvi+325 pp. [4] d. n. georgiou and a. c. megaritis, on a new relative invariant covering dimension, extracta mathematicae 25, no. 3 (2010), 263–275. [5] d. n. georgiou and a. c. megaritis, covering dimension and finite spaces, applied mathematics and computation 218 (2011), 3122–3130. [6] d. n. georgiou and a. c. megaritis, on the relative dimensions dim and dim∗ i, questions and answers in general topology 29 (2011), 1–16. [7] d. n. georgiou and a. c. megaritis, on the relative dimensions dim and dim∗ ii, questions and answers in general topology 29 (2011), 17–29. [8] m. shiraki, on finite topological spaces, rep. fac. sci. kagoshima univ. 1 1968 1-8. [9] j. valuyeva, on relative dimension concepts, questions answers gen. topology 15, no. 1 (1997), 21–24. (received november 2011 – accepted march 2012) a. c. megaritis (megariti@master.math.upatras.gr) department of accounting, technological educational institute of messolonghi, 30200 messolonghi, greece relative dimension r-dim and finite spaces. by a. c. megaritis @ appl. gen. topol. 23, no. 2 (2022), 303-314 doi:10.4995/agt.2022.16925 © agt, upv, 2022 cardinal invariants and special maps of quasicontinuous functions with the topology of pointwise convergence mandeep kumar a and brij kishore tyagi b a department of mathematics, university of delhi, delhi 110007, india. (mjakhar5@gmail.com) b department of mathematics, atma ram sanatan dharma college, university of delhi, new delhi 110021, india. (brijkishore.tyagi@gmail.com) communicated by d. n. georgiou abstract for topological spaces x and y , let qp(x, y ) be the space of all quasicontinuous functions from x to y with the topology of pointwise convergence. in this paper, we study the cardinal invariants such as character, weight, density, pseudocharacter, spread and cellularity of the space qp(x, y ). we also discuss the properties of the restriction and induced maps related to the space qp(x, y ). 2020 msc: 54c35; 54c08; 54c30. keywords: quasicontinuous functions; topology of pointwise convergence; character; weight; density; spread; cellularity; induced map; restriction map. 1. introduction kempisty [10] introduced a weaker form of continuity for real-valued functions, named as quasicontinuity. the properties of quasicontinuous functions are discussed in many papers, for example see [2, 13, 15, 16]. the quasicontinuous functions have various applications in different areas of mathematics; for instance topological groups [11], dynamical systems [4] and the study of minimal usco and minimal cusco maps [6]. some examples [3] of received 23 december 2021 – accepted 3 april 2022 http://dx.doi.org/10.4995/agt.2022.16925 https://orcid.org/0000-0001-8773-0927 https://orcid.org/0000-0003-2660-2432 m. kumar and b. k. tyagi quasicontinuous functions are the doubling function d : [0, 1) → [0, 1) defined by d(x) = 2x (mod 1), the extended sin(1/x) function f : r → r defined by f(x) = { sin( 1 x ) if x 6= 0 0 if x = 0, the floor function from r to r defined by bxc = max{n ∈ z : n ≤ x}, and any monotonic left or right continuous function from r to r [15]. the set of all real-valued quasicontinuous maps on a topological space x with the topology of pointwise convergence, denoted by qp(x,r), is studied in [7, 8, 9]. the pointwise convergence of real-valued quasicontinuous maps defined on a baire space is examined in [7]. in [9] metrizability, first countability, closed and compacts subsets of the space qp(x,r) are discussed. the cardinal functions of the space qp(x,r) are studied in [8]. in this paper, we study the results about metrizability, first countability and cardinal functions of the space qp(x,y ) along with the concept of induced and the restriction maps. in a more detail, this paper is organized as follows: in section 3, we define the topology of pointwise convergence on q(x,y ), the set of all quasicontinuous functions from a topological space x to a topological space y . in section 4, when x is hausdorff and y is a nontrivial t1-space, we compare the cardinal functions π-character, character and weight of the space qp(x,y ). moreover, if y is second countable, we characterize these cardinal functions. for a regular space x and a nontrivial t1-space y , we discuss pseudocharacter and spread of the space qp(x,y ). we also show that qp(x,y ) is dense in the space y x. in section 5, we discuss the topological properties of induced maps and the restriction map related to the space qp(x,y ). 2. preliminaries throughout this paper, the symbols x,y,z are topological spaces unless otherwise stated, r is the space of real numbers with the usual topology, n is the set of positive integers, and i is the closed interval [−1, 1]. the topology of a space x is denoted by τ(x). by a nontrivial space we mean a topological space with at least two different points. the symbol ao denotes the interior of a in x and the symbol a denotes the closure of a in x. definition 2.1. a map f : x → y is quasicontinuous [15] at x ∈ x if for every open set u containing x and every open set v containing f(x), there exists a nonempty open set g ⊆ u such that f(g) ⊆ v . if f is quasicontinuous at every point of x, we say that f is quasicontinuous. note that every continuous map is quasicontinuous. conversely, for x = [0, 1) with the usual topology and y = [0, 1) with the sorgenfrey topology, the identity map from x to y is quasicontinuous but nowhere continuous [12]. © agt, upv, 2022 appl. gen. topol. 23, no. 2 304 cardinal invariants and special maps of quasicontinuous functions levine [13] studied quasicontinuous maps under the name of semi-continuity using the terminology of semi-open sets. a subset a of a space x is said to be semi-open (or quasi-open [15]) if a ⊆ ao. a map f : x → y is quasicontinuous if and only if for every open set v in y , f−1(v ) is semi-open in x. 3. quasicontinuous functions and the topology of pointwise convergence let f(x,y ) be the set of all functions and c(x,y ) be the set of all continuous functions from x to y . the function spaces f(x,y ) and c(x,y ) with the topology of pointwise convergence denoted by fp(x,y ) and cp(x,y ), respectively, are widely studied in the literature, for example [1, 5, 14, 17]. for x ∈ x and v ∈ τ(y ), let s(x,v ) = {f ∈ f(x,y ) : f(x) ∈ v}. then fp(x,y ) has a subbase s = {s(x,v ) : x ∈ x,v ∈ τ(y )}. note that f(x,y ) = y x and the topology of pointwise convergence on f(x,y ) is just the product topology on y x. let q(x,y ) be the set of all quasicontinuous functions in f(x,y ). the space q(x,y ) with the topology of pointwise convergence is the subspace q(x,y ) of the space fp(x,y ) and is denoted by qp(x,y ). for x ∈ x and v ∈ τ(y ), denote [x,v ] = {f ∈ q(x,y ) : f(x) ∈ v}. then s′ = {[x,v ] : x ∈ x,v ∈ τ(y )} is a subbase for the space qp(x,y ). observe that for x ∈ x and v1,v2 ∈ τ(y ), we have [x,v1] ∩ [x,v2] = [x,v1 ∩v2]. for x1, . . . ,xn ∈ x and v1, . . . ,vn ∈ τ(y ), denote [x1, . . . ,xn; v1, . . . ,vn] = {f ∈ q(x,y ) : f(xi) ∈ vi, 1 ≤ i ≤ n}. clearly the family b = {[x1, . . . ,xn; v1, . . . ,vn] : xi ∈ x,vi ∈ τ(y ), 1 ≤ i ≤ n,n ∈ n} is a base for the space qp(x,y ). if v is a basis for y then the family b′ = {[x1, . . . ,xn; v1, . . . ,vn] : xi ∈ x,vi ∈v, 1 ≤ i ≤ n,n ∈ n} is also a basis for the space qp(x,y ). if (y,d) is a metric space then for f ∈ q(x,y ); x1, . . . ,xn ∈ x and � > 0, denote o(f,x1, . . . ,xn,�) = {g ∈ q(x,y ) : d(g(xi),f(xi)) < �,1 ≤ i ≤ n}. it is easy to see that the family bf = {o(f,x1, . . . ,xn,�) : x1 . . . ,xn ∈ x,n ∈ n,� > 0} is a local base at f ∈ qp(x,y ). 4. cardinal functions and the space qp(x, y ) in this section, we discuss first countability, metrizability and cardinal functions of the space qp(x,y ). before generalizing some results obtained in [8, 9], first we recall definitions of the cardinal functions for a topological space [8, 14]. a collection v of nonempty open subsets of x is called a local π-base at x ∈ x if for each open set u containing x, there exists v ∈v such that v ⊆ u. the π-character of a point x ∈ x is πχ(x,x) = ℵ0 + min{|v| : v is a local πbase at x}. the π-character of a space x is defined as πχ(x) = sup{πχ(x,x) : © agt, upv, 2022 appl. gen. topol. 23, no. 2 305 m. kumar and b. k. tyagi x ∈ x}. the character of a space x is χ(x) = sup{χ(x,x) : x ∈ x}, where χ(x,x) = ℵ0 + min{|bx| : bx is a base at x}. the weight of a space x is defined by ω(x) = ℵ0 + min{|b| : b is a base for x}. a collection β of nonempty subsets of a space x is called a π-base for x provided that every nonempty open subset of x contains some member of β. the π-weight of a space x is defined by πω(x) = ℵ0 + min{|β| : β is a π-base for x}. the density of a space x is d(x) = ℵ0 + min{|d| : d is a dense subset of x}. the pseudocharacter of a point x ∈ x is ψ(x,x) = ℵ0 + min{|γ| : γ is a family of open sets in x such that ∩γ = {x}}. the pseudocharacter of a space x is defined as ψ(x) = sup{ψ(x,x) : x ∈ x}. the spread of a space x is defined as s(x) = ℵ0 + sup{|d| : d ⊆ x is discrete}. the cellularity or souslin number of a space x is defined by c(x) = ℵ0 + sup{|u| : u is a family of pairwise disjoint nonempty open subsets of x}. a space x is said to have souslin property if c(x) = ℵ0. lemma 4.1 ([8, lemma 4.2]). let x and y be topological spaces and f : x → y be a map such that for any x ∈ x, there exists an open set g in x such that x ∈ g and f(y) = f(x) for all y ∈ g. then f is quasicontinuous. lemma 4.2 ([8, lemma 4.3]). let x and y be topological spaces such that x is hausdorff. for given x1, . . . ,xn ∈ x and (not necessarily distinct) y1, . . . ,yn ∈ y , there exists a quasicontinuous map f : x → y such that f(xi) = yi for each i ∈{1, . . . ,n}. theorem 4.3. let x and y be topological spaces such that x is uncountable hausdorff and y is a nontrivial t1-space. then for any f ∈ qp(x,y ), f does not have a countable local π-base. proof. suppose {un : n ∈ n} be a countable local π-base at some f ∈ qp(x,y ). so for each n, there is a basic open set wn such that wn ⊆ un. then {wn : n ∈ n} is also a countable local π-base at f. let wn = [xn1 , . . . ,x n kn ; v n1 , . . . ,v n kn ] for each n ∈ n. the set a = {xij : 1 ≤ j ≤ ki, i ∈ n} is countable. since x is uncountable, choose x ∈ x \ a. because y is a nontrivial t1space, choose y ∈ y such that y /∈ v for some open set v containing f(x). then w = [x,v ] is an open set containing f. suppose wn ⊆ w for some n ∈ n. by lemma 4.2, let g : x → y be a quasicontinuous function such that g(xni ) ∈ v n i for each i ∈ {1, . . . ,kn} and g(x) = y. then g ∈ wn \w , a contradiction. � corollary 4.4. let x and y be topological spaces such that x is hausdorff and y is a nontrivial t1-space. if the space qp(x,y ) has a countable local π-base at some f ∈ qp(x,y ), then x is countable. corollary 4.5. let x be an uncountable hausdorff space and y be a nontrivial t1-space. then for any f ∈ qp(x,y ), f does not have a countable local base. using corollary 4.4, a more general result than [9, theorem 3.2] is the following. © agt, upv, 2022 appl. gen. topol. 23, no. 2 306 cardinal invariants and special maps of quasicontinuous functions theorem 4.6. let x and y be spaces such that x is hausdorff and y is a nontrivial metrizable space. then the following are equivalent: (a) fp(x,y ) is metrizable. (b) fp(x,y ) is first countable. (c) qp(x,y ) is metrizable. (d) qp(x,y ) is first countable. (e) for any f ∈ qp(x,y ), f has a countable local π-base. (f ) x is countable. proof. clearly (d) implies (e) holds. the assertion (e) implies (f) follows from corollary 4.4. using the facts that qp(x,y ) is a subspace of fp(x,y ) and a countable product of metrizable spaces is metrizable, the rest of the implications can be verified easily. � the result obtained in theorem 4.3 can also be deduced from the following result about the cardinal functions related to the space qp(x,y ). theorem 4.7. let x and y be topological spaces such that x is hausdorff and y is a nontrivial t1-space. then |x| ≤ πχ(qp(x,y )) ≤ χ(qp(x,y )) ≤ ω(qp(x,y )). moreover, if x is infinite and y is second countable, we have |x| = πχ(qp(x,y )) = χ(qp(x,y )) = πω(qp(x,y )) = ω(qp(x,y )). proof. to show |x| ≤ πχ(qp(x,y )), let y1 ∈ y and f ∈ qp(x,y ) be the constant function such that f(x) = y1 for each x ∈ x. let {ut : t ∈ t} be a local π-base at f with |t | ≤ πχ(qp(x,y )). since each ut is a nonempty open subset of qp(x,y ), there exists a basic open set bt = [x t 1, . . . ,x t nt ; v t1 , . . . ,v t nt ] ⊆ ut for each t ∈ t. then the collection bf = {bt : t ∈ t} is also a local π-base at f. for each t ∈ t, let at = {xt1, . . . ,xtnt}. we claim that ⋃ t∈t at = x. let x ∈ x. since y is a nontrivial t1-space, choose y2 ∈ y such that y2 /∈ v1 for some open set v1 in y containing y1. because [x,v1] is an open set containing f and bf is a local π-base at f, there exists t ∈ t such that bt = [x t 1, . . . ,x t nt ; v t1 , . . . ,v t nt ] ⊆ [x,v1]. we claim that x ∈ at = {xt1, . . . ,xtnt}. suppose x /∈ at. since x is hausdorff, there exists an open set u such that x ∈ u and u ∩ at = ∅. because bt is nonempty, let st ∈ bt. then st(x t i) ∈ v t i for each i ∈{1, . . . ,nt}. by lemma 4.2, there is a quasicontinuous map ht : x → y such that ht(xti) = st(x t i) for each i ∈ {1, . . . ,nt}. let us define g : x → y such that g(z) = { y2 if z ∈ u ht(z) if z ∈ x \u by lemma 4.1, g is a quasicontinuous map such that g ∈ bt, but g /∈ [x,v1], which contradicts bt ⊆ [x,v1]. so ⋃ t∈t at = x. hence |x| ≤ πχ(qp(x,y )) ≤ χ(qp(x,y )) ≤ ω(qp(x,y )). if by is a countable base for y then {[x1, . . . ,xn; v1, . . . ,vn] : xi ∈ x,vi ∈ by , 1 ≤ i ≤ n} is a base for the space qp(x,y ). thus ω(qp(x,y )) ≤ |x|. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 307 m. kumar and b. k. tyagi theorem 4.8. let x and y be topological spaces such that x is infinite hausdorff space and y is a nontrivial metrizable space. then |x| = πχ(qp(x,y )) = χ(qp(x,y )). proof. by theorem 4.7, we have |x| ≤ πχ(qp(x,y )) ≤ χ(qp(x,y )). to show χ(qp(x,y )) ≤ |x|, let f ∈ qp(x,y ). if (y,d) is a metric space then the collection bf = {o(f,x1, . . . ,xk, 1n) : x1, . . . ,xk ∈ x,k,n ∈ n} is a local base at f. thus |x| = πχ(qp(x,y )) = χ(qp(x,y )). � lemma 4.9. let x and y be topological spaces such that u1, . . . ,un are nonempty pairwise disjoint open subsets of x and y1, . . . ,yn ∈ y . then there exists a quasicontinuous map g : x → y such that g(ui) = {yi} for each i ∈{1, . . . ,n}. proof. let h = u1 ∪ ·· · ∪ un and y0 ∈ y . for x ∈ h, let k = min{i ∈ {1, . . . ,n} : x ∈ ui}. let us define g : x → y such that g(x) = { yk if x ∈ h y0 if x ∈ x \h by lemma 4.1, the map g is quasicontinuous and g(ui) = {yi} for each i ∈ {1, . . . ,n}. � theorem 4.10. let x and y be topological spaces such that x is hausdorff. then d(qp(x,y )) ≤ ω(x) ·d(y ). proof. let b be a base for x such that |b|≤ ω(x) and u be the family of all finite pairwise disjoint nonempty members of b. let d be a dense set in y such that |d| ≤ d(y ) and v be the family of all nonempty finite subsets of d. for each u = {u1, . . . ,un} ∈ u and y = {y1, . . . ,yn} ∈ v, by lemma 4.9, there exists a quasicontinuous function gu,y : x → y such that gu,y(ui) = yi for each i ∈ {1, . . . ,n}. then g = {gu,y : u ∈ u,y ∈ v} is dense set in qp(x,y ) such that |g| ≤ ω(x) ·d(y ). indeed, for any nonempty basic open set h = [x1, . . . ,xn; v1, . . . ,vn] in qp(x,y ), there exist u = {u1, . . . ,un} ∈ u such that xi ∈ ui and y = {y1, . . . ,yn}∈v such that yi ∈ vi for each i ∈{1, . . . ,n}. thus there is gu,y ∈ g such that gu,y(xi) ∈ vi for each i ∈{1, . . . ,n} and hence gu,y ∈ h ∩g. � corollary 4.11. let x and y be topological spaces such that x is second countable hausdorff and y is separable. then the space qp(x,y ) is separable. lemma 4.12. let x and y be spaces such that x is regular. then for any x ∈ x, any nonempty closed set f ⊆ x such that x /∈ f and y1,y2 ∈ y , there exists a quasicontinuous function f : x → y such that f(x) = y1 and f(f) = {y2}. proof. since x /∈ f and x is regular, there exist open sets u and v such that x ∈ u, f ⊆ v and u ∩ v = ∅. note that x /∈ v . let us define f : x → y © agt, upv, 2022 appl. gen. topol. 23, no. 2 308 cardinal invariants and special maps of quasicontinuous functions such that f(z) = { y1 if z ∈ x \v y2 if z ∈ v by lemma 4.1, f is a quasicontinuous map such that f(x) = y1 and f(f) = {y2}. � theorem 4.13. let x be a regular space and y be any nontrivial space. then d(x) ≤ ψ(qp(x,y )). proof. given a basic open set u = [x1, . . . ,xn; v1, . . . ,vn] in qp(x,y ), let au = {x1, . . . ,xn}. let f0 ∈ qp(x,y ) be the constant function such that f0(x) = y0 for all x ∈ x and γ be a family of open sets with |γ| ≤ ψ(qp(x,y )) such that ∩γ = {f0}. for each g ∈ γ, there exists a basic open set ug = [xg1 , . . . ,x g ng ; v g1 , . . . ,v g ng ] such that f0 ∈ ug ⊆ g. we claim that the set d = ⋃ {aug : g ∈ γ} is dense in x. suppose that x ∈ x \ d and y1 ∈ y such that y1 6= y0. by lemma 4.12, there exists f ∈ qp(x,y ) such that f(x) = y1 and f(d) = {y0}. then f ∈∩γ and f 6= f0, which is a contradiction. thus d is dense in x. � note that if x is a tychonoff space, then the result obtained in theorem 4.13 for y = r can be concluded from the results d(x) = ψ(cp(x,r)) [17, problem 173] and ψ(cp(x,r)) ≤ ψ(qp(x,r)) [17, problem 159]. we cannot expect the equality in between d(x) and ψ(qp(x,r)) even for x = r, because d(r) = ℵ0, while [8, example 5.1] shows that ψ(qp(r,r)) = 2ℵ0 . theorem 4.14. let x be a regular space and y be a nontrivial t1-space. then s(x) ≤ s(qp(x,y )). proof. let d be a discrete subspace of x and {vd : d ∈ d} be a family of open subsets of x such that vd ∩d = {d} for each d ∈ d. choose y1,y2 ∈ y such that y1 6= y2, by lemma 4.12, there exists a quasicontinuous function fd : x → y such that fd(d) = y1 and fd(x \ vd) = {y2}. then the set a = {fd : d ∈ d} is discrete in qp(x,y ). to see this, choose an open set g in y such that y1 ∈ g but y2 /∈ g. then ud = [d,g] is open in qp(x,y ) and ud ∩a = {fd}. thus s(x) ≤ s(qp(x,y )). � if x is a tychonoff space and y = r, then the result obtained in theorem 4.14 can be obtained from the results s(x) ≤ s(cp(x,r)) [17, problem 176] and s(cp(x,r)) ≤ s(qp(x,r)) [17, problem 159]. theorem 4.15. let x and y be topological spaces such that x is hausdorff. then q(x,y ) is dense in fp(x,y ). proof. let w = [x1, . . . ,xn; v1, . . . ,vn] be any nonempty basic open set in fp(x,y ) and f ∈ w . then f(xi) ∈ vi for each i ∈ {1, . . . ,n}. since x is hausdorff and x1, . . . ,xn ∈ x, by lemma 4.2, there exists a quasicontinuous function g : x → y such that g(xi) = f(xi) for each i ∈ {1, . . . ,n}. then g ∈ w ∩q(x,y ). � © agt, upv, 2022 appl. gen. topol. 23, no. 2 309 m. kumar and b. k. tyagi corollary 4.16. let x be a hausdorff space and y be a separable space. then the space qp(x,y ) has the souslin property, that is, c(qp(x,y )) = ℵ0. proof. it is known that if y is separable then the space fp(x,y ) has the souslin property [17, problem 109]. since q(x,y ) is dense in fp(x,y ), c(qp(x,y )) = c(fp(x,y )) [17, problem 110]. thus the space qp(x,y ) has the souslin property. � note that if x is a tychonoff space then c(x,r) is a dense subset of the space qp(x,r) [17, problem 034]. also p(r,r), the set of all polynomials from r to r and u(r,r), the set of all uniformly continuous functions from r to r are dense subsets of the space qp(r,r) [17, problem 041,043]. proposition 4.17. let x and y be topological spaces such that x is hausdorff and y is separable. if f is any locally finite family of nonempty open subsets of qp(x,y ), then f is countable. proof. if possible, suppose f is uncountable. let a be a maximal disjoint family of nonempty open subsets of qp(x,y ) such that each member of a meets at most finitely many members of f. because f is locally finite, the set⋃ a is dense in qp(x,y ). by corollary 4.16, c(qp(x,y )) ≤ℵ0, which implies a is countable. since ⋃ a is dense in qp(x,y ), each u ∈ f intersect some v ∈ a. but every member of a can intersect only finitely many members of f. since a is countable, this implies f is countable, a contradiction. � 5. special maps and the space qp(x, y ) the properties of induced maps related to the space c(x,y ) with the topology of pointwise convergence and others are discussed in [14, chapter ii]. before discussing the properties of induced maps related to the space qp(x,y ), let us first define these maps in view of quasicontinuous maps. note that the composition of two quasicontinuous maps need not be quasicontinuous [16]. however, if f : x → y is quasicontinuous and g : y → z is continuous, then the composition map gof : x → z is quasicontinuous. if g : y → z is continuous, then the induced map g∗ : q(x,y ) → q(x,z) is defined by g∗(f) = g ◦ f for all f ∈ q(x,y ). also if g ∈ q(x,y ), then the induced map g∗ : c(y,z) → q(x,z) is defined as g∗(h) = h ◦ g for all h ∈ c(y,z). theorem 5.1. for a given continuous map g : y → z, the induced map g∗ : qp(x,y ) → qp(x,z) such that g∗(f) = g ◦f is continuous. moreover, if g is an embedding, then g∗ is also an embedding. proof. let f ∈ qp(x,y ) and u be any open set in qp(x,z) containing g∗(f). there is a basic open set v = [x1, . . . ,xn; v1, . . . ,vn] in qp(x,z) such that g∗(f) ∈ v ⊆ u. now w = [x1, . . . ,xn; g−1(v1), . . . ,g−1(vn)] is an open set in qp(x,y ) such that f ∈ w and g∗(w) ⊆ v ⊆ u. thus the map g∗ is continuous. © agt, upv, 2022 appl. gen. topol. 23, no. 2 310 cardinal invariants and special maps of quasicontinuous functions note that if g is injective then g∗ is also injective. now to show g∗ : qp(x,y ) → g∗(qp(x,y )) is an open map, let [x,v ] be any subbasic open set in qp(x,y ). since g is an embedding and v is open in y , there exists an open set w in z such that g(v ) = w ∩ g(y ). we have [x,v ] = [x,g−1(w)] = g−1∗ ([x,w]). then g∗([x,v ]) = [x,w]∩g∗(qp(x,y )) is open in g∗(qp(x,y )). � proposition 5.2. for any space x, there is a continuous map h : qp(x,r) → qp(x,i) such that h(f) = f for each f ∈ qp(x,i). proof. consider the map g : r → i such that g(t) = −1 if t < −1, g(t) = t if t ∈ i = [−1, 1] and g(t) = 1 if t > 1. clearly g is continuous. by theorem 5.1, the map h = g∗ : qp(x,r) → qp(x,i) defined by h(f) = gof is continuous. also h(f) = f for each f ∈ qp(x,i). � theorem 5.3. for a given quasicontinuous map g : x → y , the map g∗ : cp(y,z) → qp(x,z) such that g∗(h) = h ◦ g is continuous. moreover, if g(x) = y , then g∗ is an embedding. proof. let h0 ∈ cp(y,z) and v = [x1, . . . ,xn; v1, . . . ,vn] be any basic open set in qp(x,z) containing g ∗(h0). consider u = [g(x1), . . . ,g(xn); v1, . . . ,vn] open in cp(y,z). then h0 ∈ u and for any h ∈ u, we have g∗(h) ∈ v . thus g∗(u) ⊆ v and hence g∗ is continuous. now suppose that g(x) = y . to see g∗ is an injection, let h,h′ ∈ cp(y,z) such that h 6= h′. then h(y) 6= h′(y) for some y ∈ y . because g(x) = y , let x ∈ g−1(y). then g∗(h)(x) = h(y) 6= h′(y) = g∗(h′)(x). hence g∗(h) 6= g∗(h′). to prove g∗ is an embedding, it suffices to show that (g∗)−1 : g∗(cp(y,z)) → cp(y,z) is continuous. let g∗(f) ∈ g∗(cp(y,z)) and u = [y1, . . . ,yn; v1, . . . ,vn] be any basic open set in cp(y,z) containing f. choose xi ∈ g−1(yi) for each i ∈ {1, . . . ,n}. then v = [x1, . . . ,xn; v1, . . . ,vn] ∩ g∗(cp(y,z)) is open in g ∗(cp(y,z)) containing g ∗(f). to verify (g∗)−1(v ) ⊆ u, let h ∈ v . then h = g∗(h′) for some h′ ∈ cp(y,z). since h = g∗(h′) = h′ ◦ g ∈ v , we have h′ ◦ g(xi) ∈ vi for each i ∈ {1, . . . ,n}. this implies h′(yi) ∈ vi for each i ∈ {1, . . . ,n} so that h′ ∈ u. hence h′ = (g∗)−1(h) ∈ u and we have (g∗)−1(v ) ⊆ u. � for any space x and maps f,g : x → r such that f is continuous and g is quasicontinuous, it is easy to see that the map f + g : x → r defined by (f + g)(x) = f(x) + g(x) is quasicontinuous. proposition 5.4. for any space x, the map s : cp(x,r) × qp(x,r) → qp(x,r) defined by s(f,g) = f + g is continuous. proof. let (f0,g0) ∈ cp(x,r) ×qp(x,r) and u be any open set in qp(x,r) containing h0 = f0 + g0. there exist x1, . . . ,xn ∈ x and � > 0 such that h0 ∈ o(h0,x1, . . . ,xn,�) ⊆ u. then v = o(f0,x1, . . .xn, �2 ) and w = o(g0,x1, . . .xn, � 2 ) are open in cp(x,r) and qp(x,r), respectively. therefore v ×w is open in cp(x,r)×qp(x,r) containing (f0,g0). we claim that © agt, upv, 2022 appl. gen. topol. 23, no. 2 311 m. kumar and b. k. tyagi s(v ×w) ⊆ u. for this, let s(f,g) = f + g ∈ s(v ×w), then |f(xi) + g(xi)− h0(xi)| ≤ |f(xi) − f0(xi)| + |g(xi) − g0(xi)| < � for all i ∈ {1, . . . ,n}. thus f + g ∈ o(h0,x1, . . . ,xn,�) ⊆ u. � lemma 5.5. for any x ∈ x, the evaluation map at x, ex : qp(x,y ) → y defined by ex(f) = f(x) is continuous. proof. let f ∈ qp(x,y ) and v be any open set in y containing f(x). then u = [x,v ] is an open set containing f such that ex(u) ⊆ v . thus ex is continuous. � for any space x and a ⊆ x, a family ba of open subsets of x is called a base at a [5] if each member of ba contains a and for any open set u containing a, there exists b ∈ba such that b ⊆ u. the character of a in x is defined as χ(a,x) = ℵ0 + min{|ba| : ba is a base at a}. note that χ({x},x) = χ(x,x). proposition 5.6. let x be a hausdorff space. if there exists a compact subspace k of the space qp(x,r) such that χ(k,qp(x,r)) ≤ ℵ0, then x is countable. proof. given a basic open set u = [x1, . . . ,xn; v1, . . . ,vn] in qp(x,r), let au = {x1, . . . ,xn}. suppose that {bn : n ∈ n} is a countable base at k in qp(x,r). fix n ∈ n, for each f ∈ k, choose a basic open set unf such that f ∈ unf ⊆ bn. for open cover {u n f : f ∈ k} of k, choose a finite subcover {unf1, . . . ,u n fmn } for some mn ∈ n. let wn = unf1 ∪ ·· · ∪ u n fmn and an = aun f1 ∪·· ·∪aun fmn , then k ⊆ wn ⊆ bn. clearly a = ⋃ {an : n ∈ n} is countable. we claim that a = x. suppose that x ∈ x \ a. by lemma 5.5, the map ex : qp(x,r) → r defined by ex(f) = f(x) is continuous. therefore the set ex(k) is bounded in r. choose m > 0 such that |f(x)| < m for all f ∈ k. since w = [x, (−m,m)] is an open set containing k, there exists k ∈ n such that k ⊆ bk ⊆ w and hence wk = u k f1 ∪ ·· ·∪ukfmk ⊆ w . thus u k f1 = [x1, . . . ,xn; v1, . . . ,vn] ⊆ w such that x /∈ {x1, . . . ,xn}. since x is hausdorff, by lemma 4.2, choose g ∈ qp(x,r) such that g(xi) ∈ vi for each i ∈{1, . . . ,n} and g(x) = m. then g ∈ wk \w , which is a contradiction. � the properties of the restriction map related to the space c(x,r) with the topology of pointwise convergence are discussed in [1]. for y ⊆ x, the restriction map is defined as πy : f(x,z) → f(y,z) such that πy (f) = f|y for all f ∈ f(x,z). note that the restriction of a quasicontinuous map on an open or a dense subset is quasicontinuous [16]. a map f : x → y is called almost onto if f(x) is dense in y . proposition 5.7. let x be a regular space and y be an open subset of x. if the map πy : q(x,r) → q(y,r) such that πy (f) = f|y is injective, then y is dense in x. proof. let h0 ∈ q(x,r) such that h0(x) = 0 for all x ∈ x. suppose that πy is injective but y is not dense in x so that z ∈ x\y . by lemma 4.12, there exists © agt, upv, 2022 appl. gen. topol. 23, no. 2 312 cardinal invariants and special maps of quasicontinuous functions h ∈ q(x,r) such that h(z) = 1 and h(y ) = {0}. we have πy (h) = πy (h0) but h 6= h0, which is a contradiction. hence y is dense in x. � theorem 5.8. let x be a hausdorff space and y ⊆ x be open or dense in x. then the restriction map πy : qp(x,r) → qp(y,r) such that πy (f) = f|y is continuous and almost onto. moreover, πy is a homeomorphism if and only if y = x. proof. consider the natural projection py : rx → ry such that py (x) = x|y . then py is a continuous map [17, problem 107] and πy = py |qp(x,r). therefore πy is continuous. by theorem 4.15, q(x,r) is dense in rx. since py is continuous, ry = py (rx) = py (qp(x,r)) ⊆ py (qp(x,r)). thus πy (qp(x,r)) = py (qp(x,r)) is dense in ry and hence also dense in qp(y,r). now if πy is a homeomorphism and y 6= x. for x ∈ x\y , the set d = {f ∈ qp(x,r) : f(x) = 0} is not dense in qp(x,r), because d∩[x, (0, 1)] = ∅. but πy (d) is dense in qp(y,r). let g = [y1, . . . ,yn; v1, . . . ,vn] be any basic open set in qp(y,r) containing some g. by lemma 4.2, there exists f ∈ qp(x,r) such that f(yi) = g(yi) and f(x) = 0. then f ∈ d such that πy (f) ∈ g. hence πy (d)∩g 6= ∅. because the image of a dense set πy (d) under the map (πy ) −1 is d, which is not dense. this implies that (πy ) −1 is not continuous, which is a contradiction. finally, if y = x then πy is the identity map, and hence a homeomorphism. � acknowledgements. the first author acknowledges the fellowship grant of university grant commission, india with student-id dec18-414765. the authors are thankful to the anonymous referee for his valuable comments and suggestions. references [1] a. v. arhangel’skii, topological function spaces, mathematics and its applications (soviet series), vol. 78, kluwer academic publishers group, dordrecht, 1992. [2] j. borśık, points of continuity, quasicontinuity and cliquishness, rend. istit. mat. univ. trieste 26 (1994), 5–20. [3] r. cazacu and j. d. lawson, quasicontinuous functions, domains, and extended calculus, appl. gen. topol. 8 (2007), 1–33. [4] a. crannell, m. frantz and m. lemasurier, closed relations and equivalence classes of quasicontinuous functions, real anal. exchange 31 (2005/06), 409–424. [5] r. engelking, general topology, sigma series in pure mathematics, vol. 6, heldermann verlag, berlin, 1989. [6] ľ. holá and d. holý, minimal usco maps, densely continuous forms and upper semicontinuous functions, rocky mountain j. math. 39 (2009), 545–562. [7] ľ. holá and d. holý, pointwise convergence of quasicontinuous mappings and baire spaces, rocky mountain j. math. 41 (2011), 1883–1894. © agt, upv, 2022 appl. gen. topol. 23, no. 2 313 m. kumar and b. k. tyagi [8] ľ. holá and d. holý, quasicontinuous functions and the topology of pointwise convergence, topology appl. 282 (2020), article no. 107301. [9] d. holý and l. matej́ıčka, quasicontinuous functions, minimal usco maps and topology of pointwise convergence, math. slovaca 60 (2010), 507–520. [10] s. kempisty, sur les fonctions quasicontinues, fundamenta mathematicae 19 (1932), 184–197. [11] p. s. kenderov, i. s. kortezov and w. b. moors, topological games and topological groups, topology appl. 109 (2001), 157–165. [12] p. s. kenderov, i. s. kortezov and w. b. moors, continuity points of quasi-continuous mappings, topology appl. 109 (2001), 321–346. [13] n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. monthly 70 (1963), 36–41. [14] r. a. mccoy and i. ntantu, topological properties of spaces of continuous functions, lecture notes in mathematics, vol. 1315, springer-verlag, berlin, 1988. [15] t. neubrunn, quasi-continuity, real anal. exchange 14 (1988/89), 259–306. [16] z. piotrowski, a survey of results concerning generalized continuity of topological spaces, acta math. univ. comenian. 52/53 (1987), 91–110. [17] v. v. tkachuk, a cp-theory problem book, topological and function spaces, problem books in mathematics, springer, new york, 2011. © agt, upv, 2022 appl. gen. topol. 23, no. 2 314 @ appl. gen. topol. 23, no. 2 (2022), 235-242 doi:10.4995/agt.2022.17492 © agt, upv, 2022 dynamics of induced mappings on symmetric products, some answers alejandro illanes and verónica mart́ınez-de-la-vega instituto de matemáticas, universidad nacional autónoma de méxico, circuito exterior, cd. universitaria, méxico 04510, ciudad de méxico. (illanes@matem.unam.mx, vmvm@matem.unam.mx) communicated by a. linero abstract let x be a metric continuum and n ∈ n. let fn(x) be the hyperspace of nonempty subsets of x with at most n points. if 1 ≤ m < n, we consider the quotient space f nm(x) = fn(x)/fm(x). given a mapping f : x → x, we consider the induced mappings fn : fn(x) → fn(x) and fnm : f n m(x) → f nm(x). in this paper we study relations among the dynamics of the mappings f, fn and f n m and we answer some questions, by f. barragán, a. santiago-santos and j. tenorio, related to the properties: minimality, irreducibility, strong transitivity and turbulence. 2020 msc: 54f16; 37b02; 54c05; 54f15. keywords: continuum; dynamical system; induced mapping; irreducibility; symmetric product; turbulence. 1. introduction a continuum is a compact connected metric space with more than one point. given a nonempty compact metric space x and integers 1 ≤ m < n we consider the following hyperspaces of x: 2x = {a ⊂ x : a is a nonempty closed subset of x}, fn(x) = {a ∈ 2x : a has at most n points}, received 08 april 2022 – accepted 18 may 2022 http://dx.doi.org/10.4995/agt.2022.17492 https://orcid.org/0000-0002-7109-4038 https://orcid.org/0000-0002-1694-6947 a. illanes and v. mart́ınez-de-la-vega and the quotient space f nm(x) = fn(x)/fm(x). the hyperspace 2x is considered with the hausdorff metric [13, theorem 2.2]. given subsets u1, . . . , uk of x, let 〈u1, . . . , uk〉 = {a ∈ fn(x) : a ⊂ u1 ∪·· ·∪uk and a∩ui 6= ∅ for each i ∈{1, . . . , k}}. then the family of sets of the form 〈u1, . . . , uk〉, where the sets ui are open subsets of x, is a base of the topology in fn(x) [13, theorem 3.1]. the hyperspace fn(x) is called the n th-symmetric product of x. we denote the quotient mapping by qm : fn(x) → f nm(x) (or qnm, if necessary) and we denote by f mx the element in f n m(x) such that qm(fm(x)) = {f mx }. a mapping is a continuous function. given a mapping f : x → x, the induced mapping 2f : 2x → 2x is defined by 2f (a) = f(a) (the image of a under f). the induced mapping fn : fn(x) → fn(x) (also denoted in some papers by fn(f)) is the restriction of 2f to fn(x). the induced mapping f n m : f n m(x) → f nm(x) (also denoted by sf nm(f)) is the mapping that makes commutative the following diagram [8, theorem 4.3, chapter vi]. fn(x) fn // qm �� fn(x) qm �� f nm(x) fnm // f nm(x) a dynamical system is a pair (x, f), where x is a non-degenerate compact metric space and f : x → x is a mapping. given a point p ∈ x, the orbit of p under f is the set orb(p, f) = {fk(p) ∈ x : k ∈ n∪{0}}. a dynamical system (x, f) induces the dynamical systems (2x, 2f ), (fn(x), fn) and (f m n (x), f n m). h. hosokawa [12] was the first author that studied induced mappings to hyperspaces. since then, this topic has been widely studied. the most common problem studied in this area is the following. given a class of mappings m, determine whether one of the following statements implies another: (a) f ∈m, (b) 2f ∈m, (c) fn ∈m, (d) fnm ∈m. of course, this problem has also been considered for other hyperspaces. dynamical properties of induced mappings on symmetric products have been considered in [2], [3], [4], [5], [6], [9], [10], [11] and [14]. in particular, in [4] and [5], the properties of being: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, minimal, irreducible, feebly open and turbulent were studied. © agt, upv, 2022 appl. gen. topol. 23, no. 2 236 dynamics of induced mappings on symmetric products, some answers the aim of this paper is to solve most of the problems posed by f. barragán, a. santiago-santos and j. tenorio in [4] and [5], related to the properties: minimality, irreducibility, strong transitivity and turbulence. throughout this paper the word space means a non-degenerate compact metric space. we are aware that some of our proofs can be copied to obtain results with less restrictions either on the spaces or in the functions, however we consider that, point out the more general setting under each result holds, is worthless and breaks the continuity of the paper. 2. minimality let x be a space. a mapping f : x → x is minimal [1, p.7] if there is no nonempty proper closed subset m of x which is invariant under f (invariance of m means that f(m) ⊂ m); equivalently, if the orbit of every point of x is dense in x. the mapping f is totally minimal if fs is minimal for each s ∈ n. given n ∈ n, in this section we consider the following statements. (1) f is minimal, (2) fn is minimal, and (3) fn1 is minimal. in [4, theorem 4.18], it was proved that (2) implies (3), (3) implies (1), (2) implies (1), (1) does not imply (2) and (1) does not imply (3), for the case that x is a continuum. moreover, in [4, question 4.2] it was asked whether (3) implies (2). the following theorem solves this question and even when it has a very simple proof, it shows that the question and several results on minimal induced mappings are irrelevant. theorem 2.1. let x be a space, f : x → x a mapping and 1 ≤ m < n. then: (a) fn(f1(x)) ⊂ f1(x), (b) fnm(f m x ) = f m x , (c) for each a ∈ fm(x), orb(a, fn) ⊂ fm(x). thus, orb(a, fn) is not dense in fn(x) and fn is not minimal, and (d) orb(f mx , f n m) = {f mx }. thus, orb(f m x , f n m) is not dense in f n m(x) and f n m is not minimal. proof. take a point p ∈ x. then fn({p}) = f({p}) = {f(p)} ∈ f1(x). moreover, fnm(f m x ) = f n m(qm({p})) = qm(fn({p})) = qm({f(p)}) = f mx . this proves (a), (b) and (d). the proof of (c) is similar. � theorem 2.1 (b) implies that the mappings fn and f n m are never minimal or totally minimal. then proved results in which minimality or total minimality of fn or f n m is either assumed or concluded become irrelevant or partially irrelevant, such is the case of the following results by barragán, santiago-santos and tenorio: theorem 4.18, corollary 4.19, corollary 4.20 and corollary 4.21 of [4]; corollary 5.13 and corollary 5.17 of [6]. © agt, upv, 2022 appl. gen. topol. 23, no. 2 237 a. illanes and v. mart́ınez-de-la-vega 3. irreducibility let x be a space. a mapping f : x → x is irreducible [1, p.171] if the only closed subset a of x for which f(a) = x is a = x; given n ∈ n, in this section we consider the following statements. (1) f is irreducible, (2) fn is irreducible, (3) fn1 is irreducible, and (4) fnm is irreducible. using [4, theorem 5.1], in [5, theorem 4.1] it was shown that each one of the statements (2), (3) and (4) implies (1). the authors of [4] and [5] supposed that the spaces are continua, however, it is easy see that the proofs for these results are valid for infinite compact metric spaces without isolated points. the rest of the implications among (1), (2), (3) and (4) are left as questions in [4, questions 5.5] and [5, question 4.2]. the purpose of this section is to complete the proof that, in fact, all the statements (1)-(4) are equivalent. theorem 3.1. let x be a space without isolated points, f : x → x a mapping and 1 ≤ m < n. if f is irreducible, then fn is irreducible. proof. suppose that f is irreducible. claim 1. if u is a nonempty open subset of x, then there exists p ∈ u such that f(p) /∈ f(x \u). in order to prove claim 1, let a = x\u. then a is a proper closed subset of x. since f is irreducible, f is onto. thus there exist q ∈ x such that q /∈ f(a) and p ∈ x such that f(p) = q. observe that p ∈ u. this finishes the proof of claim 1. claim 2. if u is a nonempty open subset of fn(x), then there exists b ∈u such that b ∈ fn(x) \fn−1(x) and f(b) /∈ fn(fn(x) \u). we prove claim 2. since x does not have isolated points, fn(x)\fn−1(x) is dense in fn(x). then there exists d = {p1, . . . , pn}∈ (fn(x)\fn−1(x))∩u. then there exist pairwise disjoint open subsets u1, . . . , un of x such that for each i ∈ {1, . . . , n}, pi ∈ ui and d ∈ 〈u1 . . . , un〉 ⊂ u. by claim 1, for each i ∈{1, . . . , n}, there exists ui ∈ ui such that f(ui) /∈ f(x \ui). define b = {u1, . . . , un}. clearly, b ∈ fn(x) \ fn−1(x). suppose that there exists e ∈ fn(x) \ u such that f(e) = f(b). given i ∈ {1, . . . , n}, let ei ∈ e be such that f(ei) = f(ui). by the choice of ui, ei ∈ ui. thus e ∈ 〈u1 . . . , un〉⊂u, a contradiction. this proves that f(b) /∈ fn(fn(x)\u). this finishes the proof of claim 2. we are ready to prove that fn is irreducible. let a be a proper closed subset of fn(x) and u = fn(x) \a. by claim 2, there exists b ∈ u such that b ∈ fn(x) \fn−1(x) and f(b) /∈ fn(a). therefore fn(a) 6= fn(x) and fn is irreducible. � theorem 3.2. let x be an space without isolated points, f : x → x a mapping and 1 ≤ m < n. if fn is irreducible, then fnm is irreducible. © agt, upv, 2022 appl. gen. topol. 23, no. 2 238 dynamics of induced mappings on symmetric products, some answers proof. suppose that fnm is not irreducible. we will prove that fn is not irreducible. then there exists a proper closed subset a of f nm(x) such that fnm(a) = f nm(x). let b = q−1m (a ∪ {f mx }). then b is a closed subset of fn(x). we check that b 6= fn(x). set u = f nm(x) \a. then u is a nonempty open subset of f nm(x). this implies that q −1 m (u) is a nonempty open subset of fn(x). since x does not have isolated points, fn(x) \ fn−1(x) is dense in fn(x). so, there exists g ∈ (fn(x) \ fn−1(x)) ∩ q−1m (u). thus qm(g) ∈ u \{f mx } = f n m(x) \ (a∪{f mx }). hence g /∈b. therefore b 6= fn(x). now, we prove that fn(b) = fn(x). since fn is surjective, we have that f is surjective [4, theorem 3.2]. take e ∈ fn(x). in the case that e = {q1, . . . , qk}∈ fm(x), with k ≤ m. since f is surjective, for each i ∈{1, . . . , k} there exists pi ∈ x such that f(pi) = qi. thus {p1, . . . , pk} ∈ fm(x) = q−1m (f m x ) ⊂b. therefore e = fn({p1, . . . , pk}) ∈ fn(b). now we suppose that e /∈ fm(x). let a ∈ a be such that fnm(a) = qm(e). let b ∈ fn(x) be such that a = qm(b). then b ∈ b. since qm(e) = fnm(a) = fnm(qm(b)) = qm(fn(b)) and e /∈ fm(x), we have that {e} = q−1m (qm(e)) = fn(b). therefore e ∈ fn(b). we have shown that fn(b) = fn(x). therefore fn is not irreducible. therefore, we have shown that if fnm is not irreducible, then fn is not irreducible. � corollary 3.3. let x be a space without isolated points, 1 ≤ m < n and f : x → x a mapping. then the following statements are equivalent. (1) f is irreducible, (2) fn is irreducible, and (3) fnm is irreducible. 4. strong transitivity let x be a space. a mapping f : x → x is strongly transitive [15, p.369] if for each nonempty open subset u of x, there exists r ∈ n such that⋃r i=0 f i(u) = x. given 1 ≤ m < n, in this section we consider the following statements. (1) f is strongly transitive, (2) fn is strongly transitive, (3) fn1 is strongly transitive, and (4) fnm is strongly transitive. using [4, theorem 4.13], in [5, theorem 3.17] it was shown that (2) implies (1), (2) implies (3), (2) implies (4), (3) implies (1), (4) implies (1), (1) does not imply (2), (1) does not imply (3) and (1) does not imply (4). the authors of [4] and [5] supposed that the spaces are continua, however it is easy to see that the proofs for these results are valid for non-degenerate compact metric spaces without isolated points. © agt, upv, 2022 appl. gen. topol. 23, no. 2 239 a. illanes and v. mart́ınez-de-la-vega the questions whether the rest of the implications hold are contained in [4, question 4.1] and [5, question 3.18]. with the following theorem we show that all these implications hold. theorem 4.1. let x be a space without isolated points, f : x → x a mapping and 1 ≤ m < n. if fnm is strongly transitive, then fn is is strongly transitive. proof. let u be a nonempty open subset of fn(x). fix an element a = {a1, . . . , ak}∈u, where k ≤ n and the cardinality of a is k. let w ′1, . . . w ′ k be pairwise disjoint open subsets of x such that for each i ∈ {1, . . . , k}, ai ∈ w ′i and w ′ = 〈w ′1, . . . , w ′k〉 ⊂ u. for each i ∈ {1, . . . , k}, choose an open subset wi of x such that ai ∈ wi ⊂ clx (wi) ⊂ w ′i . let w = 〈w1, . . . , wk〉. since x does not have isolated points, fn(x) \ fm(x) is dense in fn(x) and the set v = w \ fm(x) is a nonempty open subset of fn(x). observe that qm(v) is a nonempty open subset of f nm(x). by hypothesis there exists r ∈ n such that ⋃r i=0(f n m) i(qm(v)) = f nm(x). we claim that ⋃r i=0(fn) i(w′) = fn(x). take an element b ∈ fn(x). let {bs}∞s=1 be a sequence in fn(x) \ fn−1(x) such that lims→∞bs = b. given s ∈ n, there exist ds ∈ v and is ∈ {0, 1, . . . , r} such that (fnm)is (qm(ds)) = qm(bs). this implies that qm(f is (ds)) = qm(bs). since fn(x) is compact, we may suppose that the sequence {ds}∞s=1 converges to an element d ∈ fn(x) and there exists j ∈ {0, 1, . . . , r} such that for each s ∈ n, is = j. given s ∈ n, qm(fj(ds)) = qm(bs). since bs /∈ fm(x), we obtain that fj(ds) = bs. by the continuity of f j, fj(d) = b. since ds ∈ v ⊂ w ⊂ clfn(x)(w), we conclude that d ∈ clfn(x)(w) ⊂ 〈clx (w1), . . . , clx (wk)〉 ⊂ 〈w ′1, . . . , w ′k〉 = w ′. therefore b ∈ (fn)j(w′). this finishes the proof that⋃r i=0(fn) i(w′) = fn(x), so, ⋃r i=0(fn) i(u) = fn(x) and completes the proof of the theorem. � 5. turbulence let x be a space. a mapping f : x → x is turbulent [7, p.588] if there are compact non-degenerate subsets k and l of x such that k ∩ l has at most one point and k ∪l ⊂ f(k) ∩f(l). given 1 ≤ m < n, in this section we consider the following statements. (1) f is turbulent, (2) fn is turbulent, (3) fn1 is turbulent, and (4) fnm is turbulent. using [4, theorem 5.6] in [5, theorem 4.5] it follows that (1) implies (2), (3) and (4). in [5, questions 4.6], it was asked whether one of the rest of the possible implications holds, when x is a continuum. the following example shows that (2) and (3) does not imply (1), when x is a compact metric space. © agt, upv, 2022 appl. gen. topol. 23, no. 2 240 dynamics of induced mappings on symmetric products, some answers problem 5.1. does one of the statements (2), (3) or (4) implies another for a compact metric space? example 5.2. there exist a non-degenerate compact metric space x and a mapping f : x → x such that f2 and f21 are turbulent but f is not turbulent. define x = {0}∪{1 n : n ∈ n}. for each m ∈ n, let am = 13m−2 , bm = 1 3m−1 and cm = 1 3m . then x = {0}∪{am : m ∈ n}∪{bm : m ∈ n}∪{cm : m ∈ n}. define f : x → x by f(p) =   0, if p = 0, ck, if p = a2k−1, bk, if p = a2k, ak, if p ∈{b2k, c2k, b2k−1, c2k−1}. clearly, f is an onto mapping. suppose to the contrary that f is turbulent. then there are compact nondegenerate subsets k and l of x such that k ∩ l has at most one point and k ∪l ⊂ f(k) ∩f(l). if there exists k ≥ 2 such that ck ∈ k ∪ l, since f−1(ck) = {a2k−1}, we have that a2k−1 ∈ k ∩ l. since f−1(a2k−1) = {b4k−2, c4k−2, b4k−3, c4k−3}, there is p ∈ {b4k−2, c4k−2, b4k−3, c4k−3} ∩ k such that f(p) = a2k−1. since f−1(p) = {ai} for some i > 4k − 3 > 2k − 1, we have that ai ∈ k ∩ l. thus {ai, a2k−1} ⊂ k ∩ l, a contradiction. thus (k ∪ l) ∩ {ck : k ≥ 2} = ∅. similarly, (k ∪ l) ∩{bk : k ≥ 2} = ∅. therefore k ∪ l ⊂ {ak : k ∈ n}∪ {b1, c1}∪{0}. if there exists k ≥ 2 such that ak ∈ k ∪ l, then there exists k′ > 2 such that {bk′, ck′}∪(k ∪l) 6= ∅. this contradicts what we proved in the previous paragraph. thus k∪l ⊂{a1, b1, c1}∪{0}. since ({a1, b1, c1}∪{0})∩f−1(b1) = ∅, we have that b1 /∈ k ∪ l. hence k ∪ l ⊂ {a1, c1}∪{0}. since ({a1, c1}∪ {0}) ∩f−1(a1) = {c1} and ({a1, c1}∪{0}) ∩f−1(c1) = {a1}, we obtain that if {a1, c1}∩ (k ∪ l) 6= ∅, then {a1, c1} ⊂ k ∩ l, a contradiction. this proves that k ∪ l ⊂ {0}, a contradiction. this completes the proof that f is not turbulent. now, we check that f2 is turbulent. define k = {{am, bm}∈ f2(x) : m ∈ n}∪{{0}}, and l = {{am, cm}∈ f2(x) : m ∈ n}∪{{0}}. then k and l are compact non-degenerate subsets of f2(x) and k∩l = {{0}}. given m ∈ n, {am, bm} = {f(c2m), f(a2m)} = f2({c2m, a2m}) ∈ f2(l). moreover, {am, bm} = {f(b2m), f(a2m)} = f2({b2m, a2m}) ∈ f2(k). since {0} = {f(0)} = f({0}}) = f2({0}) ∈ f2(k) ∩ f2(l). we have shown that k⊂ f2(k) ∩f2(l). similarly, l⊂ f2(k) ∩f2(l). therefore, f2 is turbulent. © agt, upv, 2022 appl. gen. topol. 23, no. 2 241 a. illanes and v. mart́ınez-de-la-vega using k0 = q1(k) and l0 = q1(l), it can be proved that f21 is turbulent. acknowledgements. this paper was partially supported by the projects “teoŕıa de continuos, hiperespacios y sistemas dinámicos iii”, (in 106319) of papiit, dgapa, unam; and “teoŕıa de continuos e hiperespacios, dos” (ai-s-15492) of conacyt. the authors wish to thank leonardo espinosa for his technical help during the preparation of this paper. references [1] j. auslander, minimal flows and their extensions, north-holland math. studies, vol. 153. north-holland, amsterdam, 1988. 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[15] w. parry, symbolic dynamics and transformations of the unit interval, trans. amer. math. soc. 122 (1966), 368–378. © agt, upv, 2022 appl. gen. topol. 23, no. 2 242 tkachenkoagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 269-276 compact self t1-complementary spaces without isolated points mikhail tkachenko abstract. we present an example of a compact hausdorff self t1-complementary space without isolated points. this answers question 3.11 from [a compact hausdorff topology that is a t1-complement of itself, fund. math. 175 (2002), 163–173] affirmatively. 2000 ams classification: 54a10, 54a25, 54d30 keywords: alexandroff duplicate; čech–stone compactification; compact; isolated point; t1-complementary; transversal topology 1. introduction we deal with the concept of complementarity in the lattice of t1-topologies on a given infinite set. two elements a, b of an abstract lattice {l, ∨, ∧, 0, 1} with the smallest and greatest elements 0 and 1, respectively, are called complementary if a ∨ b = 1 and a ∧ b = 0. birkhoff noted in [1] that the family l(x) of all topologies on a nonempty set x becomes a lattice when the infimum τ1 ∧ τ2 of τ1, τ2 ∈ l(x) is defined to be the intersection τ1 ∩ τ2 and the supremum τ1 ∨ τ2 is the topology on x with the subbase τ1 ∪ τ2. clearly, the smallest element 0 of l(x) is the coarsest topology {∅, x}, while the greatest element 1 of l(x) is the discrete topology of x. in the case of the lattice l1(x) of all t1-topologies on x, the smallest element 0 of l1(x) is the cofinite topology cf in(x) = {∅} ∪ {x \ f : f ⊆ x, f is finite}. therefore, two topologies τ1, τ2 ∈ l1(x) are complementary in l1(x) if τ1 ∩ τ2 = cf in(x) and τ1 ∪ τ2 is a subbase for the discrete topology on x. it is said that τ1 and τ2 are t1-complementary in this case. the study of complementarity in l1(x) was initiated by a. steiner and e. steiner in [6, 8, 7]. later on, s. watson used an elaborated combinatorics in 270 m. tkachenko [10] to prove that a set x of cardinality c+, where c = 2ω, admits a tychonoff self t1-complementary topology τ . self t1-complementarity of τ means that there exists a bijection f of x onto itself such that the topologies τ and σ = {f −1(u ) : u ∈ τ} are t1-complementary. in [4], d. shakhmatov and the author applied a recursive construction to show that the alexandroff duplicate a(βω \ ω) of βω \ ω is a t1-complement of itself. a(βω \ ω) was the first example of an infinite compact hausdorff space with this property. it is clear that |a(βω \ ω)| = 2c > c, which looks quite similar to the cardinality of watson’s self t1-complementary space in [10]. the necessity of working with topologies on big sets was explained in [4, corollary 3.6]—the existence of a compact hausdorff self t1-complementary space of cardinality less than or equal to c is independent of zf c. the concept of t1-complementarity of topologies is naturally split into transversality and t1-independence. following [5, 9], we say that topologies τ1, τ2 ∈ l1(x) are transversal if τ1 ∨ τ2 is the discrete topology, and t1-independent if τ1 ∧τ2 is the cofinite topology on x. in addition, if the topologies τ1 and τ2 are homeomorphic (i.e., τ2 is obtained from τ1 by means of a bijection of x), we come to the notions of self-transversality and self t1-independence, respectively. a usual way to produce self-transversal topologies is to work with a space that has many isolated points. indeed, suppose that x is a space with topology τ , y ⊆ x, |y | = |x| = |x \ y |, and each point of y is isolated in x. take any bijection f : x → x such that f (x \ y ) = y and put σ = {f −1(u ) : u ∈ τ}. it is easy to see that every point of x is isolated either in τ or in σ, so τ ∨ σ is the discrete topology on x. in other words, the space (x, τ ) is self-transversal. this approach was also adopted in [4, corollary 3.8] to show that the compact space a(βω \ ω) is self-transversal (as a part of the proof that the space is self t1-complementary). this explains question 3.11 from [4]: does there exist a self t1-complementary compact hausdorff space without isolated points? theorem 2.1 answers this question in the affirmative. our space (or, better to say, a series of spaces) is a(βω \ ω) × y , where y is any dense-in-itself compact hausdorff space of cardinality c. it is worth mentioning that the idea of the proof of theorem 2.1 is a natural refinement of the arguments in [4] and [2]. taking y to be the closed unit interval or the cantor set, we obtain in zf c an example of a compact hausdorff space without isolated points which is a t1-complement of itself (see corollary 2.2). further, assuming that 2 ℵ1 = c and taking y = {0, 1}ω1, we get an example of a compact hausdorff space without points of countable character which is again a t1-complement of itself (see corollary 2.3). we finish the article with three open problems about possible cardinalities of compact hausdorff self t1-complementary spaces. 2. the alexandroff duplicate of βω \ ω and products in what follows k denotes βω \ ω, the remainder of the čech–stone compactification of the countable discrete space ω. it is clear that every nonempty compact t1-complementary spaces without isolated points 271 open subset of k has cardinality 2c. we will also use the fact that k contains a pairwise disjoint family λ of open sets such that |λ| = c. the alexandroff duplicate of k is a(k). it is easy to verify that every infinite closed subset of a(k) has cardinality 2c. the reader can find a detailed discussion of the properties of a(x), for an arbitrary space x, in [3]. theorem 2.1. for every compact hausdorff space y with |y | ≤ c, the product space a(k) × y is self t1-complementary. proof. let z = a(k) × y . let also τ be the product topology of z. by recursion of length κ = 2c we will construct a bijection f : z → z such that (1) f ◦ f = idz ; (2) the topology σ = {f (u ) : u ∈ τ} is t1-complementary to τ . let k∗ = a(k) \ k. one of the main ideas of our construction is to use open fibers {x} × y ⊆ z, with x ∈ k∗, to guarantee that each point z ∈ z will be isolated in (z, τ ∨ σ). more precisely, we will construct the bijection f to satisfy the following additional conditions: (3) f (k × y ) = k∗ × y ; (4) for every x ∈ k∗, the image f ({x} × y ) is a discrete subset of k × y . let us show first that every bijection f satisfying conditions (1), (3), and (4) produces the topology σ = f (τ ) transversal to τ . indeed, let π : a(k) × y → a(k) be the projection. take a point z ∈ z such that x = π(z) ∈ k∗. clearly, z ∈ {x} × y and, by (4), f ({x} × y ) is a discrete subset of k × y . hence there exists an open set u in z such that (∗) {f (z)} = u ∩ f ({x} × y ). since the point x is isolated in a(k), the set {x} × y is τ -open in a(k) × y . hence (∗) implies that f (z) is an isolated point of the space (z, τ ∨σ). further, it follows from (1) and (3) that k × y = f (k∗ × y ), and we conclude that every point of k × y is isolated in (z, τ ∨ σ). applying f to both parts of (∗) and taking into account (1), we obtain the equality {z} = f (u ) ∩ ({x} × y ). this means that every point of k∗ × y is isolated in (z, τ ∨ σ). we have thus proved that the topology τ ∨ σ is discrete, i.e., τ and σ are transversal. to guarantee the t1-independence of τ and σ is a more difficult task. we can reformulate the latter relation between τ and σ by saying that f (f ) is not τ -closed in z, for every proper infinite τ -closed set f ⊆ z. let us describe the recursive construction of the bijection f in detail. in what follows the space z always carries the topology τ unless the otherwise is specified. we start with three observations that will be used in our construction of f . the first and the third of them are evident. fact 1. if b is an infinite subset of a(k), then the set b ∩ k has cardinality κ = 2c, where b is the closure of b in a(k). fact 2. if c ⊆ z and the set π(c) is infinite, then the projection π(c∩(k×y )) has cardinality κ, where c is the closure of c in z. 272 m. tkachenko indeed, since the projection π is a closed mapping, we have the equality π(c) = π(c). it follows from |π(c)| ≥ ω and fact 1 that the set π(c) ∩ k has cardinality κ. again, since the mapping π is closed, we see that π−1(x)∩c 6= ∅ for each x ∈ π(c) ∩ k. hence |π(c ∩ (k × y ))| = κ. fact 3. if u is open in z and u ∩ (k × y ) 6= ∅, then |u \ (k × y )| = κ. it is clear that χ(k) ≤ w(k) = c, χ(a(k)) = χ(k) ≤ c, and w(y ) ≤ |y | ≤ c. therefore, χ(z, z) ≤ c for every z ∈ z. since |k × y | = |k| = κ, there exists a base b for k × y in z with |b| ≤ κ. in other words, b is a family of open sets in z with the property that for every z ∈ k × y and every open neighbourhood o of z in z, there exists u ∈ b such that z ∈ u ⊆ o. clearly, we can assume that u ∩ (k × y ) 6= ∅ for each u ∈ b. since κ = κω, we see that |[z]ω × b| = κ, where [z]ω denotes the family of all countably infinite subsets of z. let {(cα, uα) : α < κ} be an enumeration of the set [z] ω × b such that for every pair (c, u ) ∈ [z]ω × b, the set {α < κ : (c, u ) = (cα, uα)} is cofinal in κ. let {zα : α < κ} be a faithful enumeration of z. by recursion on α < κ we will construct sets zα ⊆ z and mappings fα : zα → zα satisfying the following conditions: (iα) |zα| ≤ |α| · c; (iiα) if γ < α, then zγ ⊆ zα; (iiiα) zα ∈ zα+1; (ivα) fα is a bijection of zα onto itself and fα ◦ fα = idzα ; (vα) if γ < α, then fα↾zγ = fγ ; (viα) if z ′, z′′ ∈ zα, π(z ′) = π(z′′), and z′ 6= z′′, then π(fα(z ′)) 6= π(fα(z ′′)); (viiα) fα+1(uα∩zα+1)∩fα+1(cα ∩ zα+1) 6= ∅ provided that the set πfα(cα∩ zα) is infinite; (viiiα) π −1(x) ⊆ zα for each x ∈ π(zα) ∩ k ∗; (ixα) if x ∈ π(zα) ∩ k ∗, then fα({x} × y ) is a discrete subset of k × y ; (xα) fα(zα ∩ (k × y )) ⊆ k ∗ × y . put z0 = ∅ and f0 = ∅. clearly, z0 and f0 satisfy (i0)–(x0). let α < κ, and suppose that a set zβ ⊆ z and a mapping fβ of zβ to itself satisfying conditions (iβ )–(xβ ) have already been defined for all β < α. if α > 0 is limit, we put zα = ⋃ {zβ : β < α} and fα = ⋃ {fβ : β < α}. then the subset zα of z and the mapping fα : zα → zα satisfy (iα)–(xα), except for (iiiα) and (viiα) which are valid for all β < α. suppose now that α = γ+1. let z′γ = zγ ∪{zγ}. since uγ ∩(k×y ) 6= ∅, the cardinality of the set uγ\(k×y ) is κ by fact 3. it follows from |z ′ γ| ≤ |zγ|+1 ≤ |γ+1|·c < κ and |π−1π(z′γ )| ≤ |z ′ γ|·|y | < κ that |(uγ \(k×y ))\π −1π(z′γ )| = κ. therefore, we can pick a point sα ∈ uγ \ π −1(k ∪ π(z′γ )). if πfγ (cγ ∩ zγ ) is infinite, then fγ (cγ ∩ zγ ) ∩ (k × y ) is a closed subset of z whose projection to a(k) has cardinality κ by fact 2. we then use the inequalities |z′γ| < κ and |y | ≤ c to pick a point tα ∈ (k × y ) ∩ fγ (cγ ∩ zγ ) \ compact t1-complementary spaces without isolated points 273 π−1π(z′γ ). otherwise pick an arbitrary point tα ∈ π −1(k \ π(z′γ )); again, such a point exists because |π(z′γ )| ≤ |z ′ γ| < κ = |k|. in either case, tα ∈ k × y . suppose that zγ = (xγ , yγ ), sα = (x ′ α, y ′ α), and tα = (x ′′ α, y ′′ α). notice that x′α ∈ k ∗ \ π(z′γ ) and x ′′ α ∈ k \ π(z ′ γ ). to define zα, we consider the following possible cases. case 1. zγ ∈ zγ . then z ′ γ = zγ and we choose a discrete set dα ⊆ k × {y ′′ α} such that tα ∈ dα, π(dα) ∩ π(zγ ) = ∅, and |dα| = |y |. this is possible since x′′α = π(tα) /∈ π(zγ ) and k contains c pairwise disjoint nonempty open sets, each of cardinality κ. put zα = zγ ∪ dα ∪ ({x ′ α} × y ). it follows from the definition that {zγ, sα, tα} ⊆ zα. since the sets dα, {x ′ α} × y , and zγ are pairwise disjoint, there exists an idempotent bijection fα of zα onto itself such that fα extends fγ , fα({x ′ α} × y ) = dα, and fα(sα) = tα. case 2. zγ /∈ zγ . again, we split this case into two subcases. case 2.1. zγ ∈ k × y , i.e., xγ ∈ k. then we choose a discrete subset dα of k × y such that {zγ , tα} ⊆ dα, dα ∩ zγ = ∅, the restriction of π to dα is one-to-one, and |dα| = |y |. again, this is possible since neither zγ nor tα is in zγ and, by the choice of tα, xγ = π(zγ ) 6= π(tα) = x ′′ α. as in case 1, we put zα = zγ ∪ dα ∪ ({x ′ α} × y ). then {zγ , sα, tα} ⊆ zα. since the sets dα, {x ′ α} × y , and zγ are pairwise disjoint, there exists an idempotent bijection fα : zα → zα such that fα extends fγ , fα(sα) = tα, and fα({x ′ α} × y ) = dα. case 2.2. xγ ∈ k ∗. we choose a discrete set dα ⊆ k × {y ′′ α} such that tα ∈ dα, π(dα) ∩ π(zγ ) = ∅, and |dα| = |y |. then we put zα = zγ ∪ dα ∪ ({xγ , x ′ α} × y ). clearly, {zγ , sα, tα} ⊆ zα. since {xγ , x ′ α} ⊆ k ∗ and {zγ, sα} ∩ zγ = ∅, it follows from (viiiγ ) that ({xγ , x ′ α} × y ) ∩ zγ = ∅. in addition, the set dα is disjoint from both zγ and {xγ , x ′ α} × y , so there exists an idempotent bijection fα of zα onto itself such that fα extends fγ , fα({xγ , x ′ α} × y ) = dα, and fα(sα) = tα. clearly, conditions (iα), (iiα), (iiiγ ), (ivα), (vα), and (viiiα)–(xα) hold true. let us verify conditions (viα) and (viiγ ). we verify (viα) only in case 2.1—the argument in the rest of cases is analogous or even simpler. suppose that z′ and z′′ are distinct elements of zα such that π(z′) = π(z′′). if {z′, z′′} ⊆ zγ , then (vα) and (viγ ) imply that π(fα(z ′)) = π(fγ (z ′)) 6= π(fγ (z ′′)) = π(fα(z ′′)). if {z′, z′′} ⊆ {x′α} × y , then π(fα(z ′)) 6= π(fα(z ′′)) since fα({x ′ α} × y ) = dα and the restriction of π to dα is one-to-one. the case {z′, z′′} ⊆ dα is clearly impossible. finally, suppose that z′ ∈ zγ and z ′′ ∈ zα \zγ (or vice versa). since x ′ α /∈ π(zγ ), if follows from π(z′) = π(z′′) and the definition of zα that z ′′ ∈ dα. our choice of fα implies that fα(dα) = {x ′ α} × y because fα is an idempotent bijection of zα onto itself. hence π(fα(z ′′)) = x′α /∈ π(zγ ) and, therefore, π(fα(z ′′)) 6= π(fα(z ′)). 274 m. tkachenko to check (viiγ ), suppose that πfγ (cγ ∩ zγ ) is infinite. it follows from our construction that sα ∈ uγ ∩ zα and fα(sα) = tα ∈ fγ (cγ ∩ zγ ) which yields tα ∈ fα(uγ ∩ zα) ∩ fα(cγ ∩ zα) 6= ∅. the recursive step is completed. we can now define the bijection f : z → z. from (iiiα) for all α < κ it follows that z = ⋃ {zα : α < κ}. let f = ⋃ {fα : α < κ}. since (iiα), (ivα) and (vα) hold for all α < κ, f is an idempotent bijection of z onto itself. this means that (1) holds. it also follows from (viiiα) and (ixα) for all α < κ that f (k∗ × y ) ⊆ k × y , while (xα) implies that f (k × y ) ⊆ k ∗ × y . since f is a bijection, we conclude that f (k∗ × y ) = k × y and f (k × y ) = k∗ × y , i.e., (3) holds. similarly, conditions (viiiα) and (ixα) for all α < κ together imply the validity of (4). it was shown before the recursive construction that for any bijection f : z → z satisfying (1), (3), and (4), the topologies τ and σ = f (τ ) on z are transversal. it only remains to prove that τ and σ = f (τ ) are t1-independent, for this special bijection f . in other words, we have to verify that for every proper infinite closed subset φ of z, the image f (φ) is not closed in z. let us consider two cases. case a. the projection π(φ) is finite. since φ ⊆ π−1π(φ) and each fiber π−1(x) has cardinality |y | ≤ c, we see that |φ| ≤ c. also, since κc = κ, the cofinality of the cardinal κ is greater than c. applying the equality z = ⋃ {zα : α < κ} and (iiα) for α < κ, we see that φ ⊆ zβ for some β < κ. it is also clear that π−1(x) ∩ φ is infinite for some x ∈ a(k). then (viβ ) yields that the set π(f (φ)) = π(fβ (φ)) is infinite. in its turn, it follows from fact 2 that the closure of f (φ) in z has cardinality κ and, since |φ| ≤ c, the set f (φ) cannot be closed in z. case b. the set π(φ) is infinite. then |φ| = κ, by fact 2. again, we split this case into two subcases. case b.1. (k × y ) \ φ 6= ∅. since cf (κ) > c > ω, the set πfβ (φ ∩ zβ) must be infinite for some β < κ. indeed, otherwise πf (φ) is finite and hence |φ| = |f (φ)| ≤ c, a contradiction. choose a countable set c ⊆ φ ∩ zβ such that πf (c) is infinite. take a point z ∈ (k × y ) \ φ and an element u ∈ b such that z ∈ u ⊆ z \ φ. this is possible because b is a base for k × y in z. note that (c, u ) ∈ [z]ω × b. since the set {α < κ : (c, u ) = (cα, uα)} is cofinal in κ, (c, u ) = (cα, uα) for some α with β ≤ α < κ. from zα ⊇ zβ and cα = c ⊆ zβ we get cα ∩ zα ⊇ cα ∩ zβ = c and, since πf (c) is infinite, so is πf (cα ∩ zα) = πfα(cα ∩ zα). then (viiα) shows that fα+1(uα ∩ zα+1) ∩ fα+1(cα ∩ zα+1) 6= ∅. since f extends fα and φ ⊇ c = cα, it follows that f (uα) ∩ f (φ) ⊇ f (uα) ∩ f (cα) ⊇ fα+1(uα ∩ zα+1) ∩ fα+1(cα ∩ zα+1) 6= ∅. therefore, there exists z∗ ∈ uα such that f (z ∗) ∈ f (φ). it follows from uα = u ⊆ z \ φ that z∗ /∈ φ. since f is a bijection of z, this yields f (z∗) /∈ f (φ). thus f (z∗) ∈ f (φ) \ f (φ), that is, the set f (φ) is not closed in z. compact t1-complementary spaces without isolated points 275 case b.2. k × y ⊆ φ. suppose to the contrary that f (φ) is closed in z. since f (k × y ) = k∗ × y and the latter set is dense in z, we see that k∗ × y ⊆ f (φ) = z. this contradicts our choice of φ as a proper subset of z. we have thus proved that f (φ) fails to be closed in z, i.e., the topologies τ and σ = f (τ ) are t1-independent. since we already know that τ and σ are transversal, this finishes the proof of the theorem. � taking y in theorem 2.1 to be the cantor set or the closed unit interval i = [0, 1], we obtain the following result which answers question 3.11 from [4] in the affirmative: corollary 2.2. there exists an infinite compact hausdorff self t1-complementary space without isolated points. under additional set-theoretic assumptions, one can refine corollary 2.2 as follows: corollary 2.3. let κ be a cardinal with ω ≤ κ < c. it is consistent with zf c that there exists a compact hausdorff self t1-complementary space z such that χ(z, z) ≥ κ for each z ∈ z. proof. one can assume that 2κ = 2ω = c and take y = iκ in theorem 2.1. � the following questions remain open. problem 2.4. let k = βω \ ω. is the product space a(k) × k self t1complementary? problem 2.5. is it true that for every cardinal λ, there exists a compact hausdorff self t1-complementary space z with |z| ≥ λ? here is a stronger version of the above problem: problem 2.6. is it true that for every cardinal λ, there exists a compact hausdorff self t1-complementary space z such that χ(z, z) ≥ λ for all z ∈ z? references [1] g. birkhoff, on the combination of topologies, fund. math. 26 (1936), 156–166. [2] a. b laszczyk and m. g. tkachenko, transversal and t1-independent topologies and the alexandroff duplicate, submitted. [3] r. engelking, on the double circumference of alexandroff, bull. acad. polon. sci. 16 (1968), 629–634. [4] d. shakhmatov and m. g. tkachenko, a compact hausdorff topology that is a t1complement of itself, fund. math. 175 (2002), 163–173. [5] d. shakhmatov, m. g. tkachenko, and r. g. wilson, transversal and t1-independent topologies, houston j. math. 30 (2004), no. 2, 421–433. [6] a. k. steiner, complementation in the lattice of t1-topologies, proc. amer. math. soc. 17 (1966), 884–885. [7] a. k. steiner and e. f. steiner, topologies with t1-complements, fund. math. 61 (1967), 23-38. [8] a. k. steiner and e. f. steiner, a t1-complement of the reals, proc. amer. math. soc. 19 (1968), 177-179. 276 m. tkachenko [9] m. g. tkachenko and iv. yaschenko, independent group topologies on abelian groups, topol. appl. 122 (2002), no. 1-2, 425–451. [10] w. s. watson, a completely regular space which is the t1-complement of itself, proc. amer. math. soc. 124 (1996), no. 4, 1281–1284. received september 2009 accepted november 2009 mikhail tkachenko (mich@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, av. san rafael atlixco 186, col. vicentina, iztapalapa, c.p. 09340, méxico d.f., méxico @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 27-31 hausdorff closed extensions of pre-uniform spaces adalberto garćıa-máynez and rubén mancio-toledo abstract the family of densely finite open covers of a hausdorff space x determines a completable pre-uniformity on x and the canonical completion ̂x is hausdorff closed. we compare ̂x with the katetov extension kx of x and give sufficient conditions for the non-equivalence of kx and ̂x. 2010 msc: 54a20, 54e15 keywords: hausdorff closed space, densely finite-cover, katetov extension, round filter 1. preliminary results for the sake of convenience to the reader, we recall some definitions. they also appear in [2]. a filter t in a pre-uniform space (x, u) is u-cauchy if for every cover α ∈ u, we have t ∩ α �= ∅. a u-cauchy filter t in a pre-uniform space is u-round if for every f0 ∈ t , there exists a cover α ∈ u such that s∗t (t , α) ⊂ f0, where : s∗t (t , α) = ⋃ {a ∈ α|a ∩ f �= ∅ for every f ∈ t } . u-round filters t in a hausdorff pre-uniform space (x, u) satisfy the following conditions: (see [2, theorem 3.8.4 and 3.8.5]) . 1) for every p ∈ x, t adheres to p if and only if t converges to p. 2) every neighborhood filter is u-round 28 a. garćıa-máynez and r. mancio-toledo as a consequence of 1), in hausdorff pre-uniform spaces, a u-round filter t is either non-adherent or converges to a unique point. an ultrafilter of open sets in a topological space (x, τ) is a non-empty subfamily g of τ − {∅} satisfying : 1) if g1, g2 ∈ g, also g1 ∩ g2 ∈ g; 2) if g ∈ g and g ⊆ h, where h ∈ τ, then h ∈ g; 3) if g0 ∈ τ and g0 ∩ g �= ∅ for every g ∈ g, then g0 ∈ g. likewise u-round filters, an ultrafilter of open sets in a hausdorff space x is either non-adherent or converges to a unique point. hausdorff closed spaces are characterized by the property : ([1, p.283]) *) every ultrafilter of open sets is convergent. an open cover α of a topological space x is densely finite if there exists a finite subfamily {a1, a2, . . . , an} ⊆ α such that x = a−1 ∪ a − 2 ∪ · · · ∪ a−n . the family u of densely finite covers of a hausdorff space (x, τ) constitutes a compatible pre-uniform basis which satisfies the condition : **) every u-cauchy filter contains a u-round filter. by [5], (x, u) has a canonical completion (x̂, û) and the topology τ ̂u is hausdorff closed. x̂ consists of all the u-round filters and û consists of all the extension covers α̂ (α ∈ u), where α̂ = { â | a ∈ α } and â = { ξ ∈ x̂ | a ∈ ξ } . the canonical embedding h: x → x̂ assigns to each p ∈ x, its neighborhood filter μp. theorem 2.6 in [3] establishes that a non-adherent filter t in (x, u) is uround if and only if t has as a basis an ultrafilter of open sets. besides the completion (x̂, û), (x, τ) has its katetov extension kx, where: kx = x ∪ {g | g is a non-adherent ultrafilter of open sets} if p ∈ x, a neighborhood basis of p is the filter μp of τ-neighborhoods of p. if g ∈ kx − x, a neighborhood basis of g consists of all the sets {g} ∪ g, where g ∈ g. the resulting topology of kx turns out to be hausdorff closed and kx − x is a closed discrete subspace without interior points, and hence x is open and dense in kx. we wonder what is the relation between kx and x̂. we recall first some definitions : a subset a of a topological space x is c-bounded (or relatively pseudocompact) if for every continuous function ϕ: x → r, ϕ(a) is bounded. a ⊆ x is c-discrete (with respect to x) if for each a ∈ a, there exists an open set ua such that a ∈ ua and the family {ua | a ∈ a} is discrete (with respect to x). the following equivalence is well known (see, for instance [4] : 4.73.3). a subset a of a tychonoff space x is c-bounded if and only if every cdiscrete subset of x contained in a is finite. hausdorff closed extensions of pre-uniform spaces 29 an open set u in a topological space x is wide if there exist two open sets w1, w2 such that : 1) w1 ∪ w2 ⊆ u; 2) w1 ∩ w2 = ∅; 3) w −1 and w − 2 are non-compact. for instance, every non-empty open set u in a nowhere locally compact regular space is wide. we also have : lemma 1.1. every open set u in a tychonoff space which is not c-bounded, is wide. proof. by hypothesis, there exists an infinite discrete family of open sets w1, w2, . . . , where w − i ⊆ u for every i. if s = ⋃∞ i=1 w2i−1 and t = ⋃∞ i=1 w2i, we have s− ∪ t − ⊆ u, s ∩ t = ∅ and none of the sets s−, t − is compact. � 2. main result we give a sufficient condition on a tychonoff space x which insures that the extensions x̂ and kx are non-equivalent. theorem 2.1. let x be a non-compact tychonoff space where every open set with non-compact closure is wide. then x̂ − h(x) is dense in itself, where h: (x, u) → (x̂, û) is the canonical embedding of x into x̂. proof. let us take any element ξ ∈ x̂ − x (we identify each point p ∈ x with its neighborhood filter). let u be an open set in x such that ξ ∈ û. therefore, u ∈ ξ. since the round filter ξ is non-adherent, u− cannot be compact. by hypothesis, u is wide. let s, t be open sets such that s ∪t ⊆ u, s ∩ t = ∅ and s−, t − are both non-compact. by [1, p. 283], s− and t − cannot be hausdorff closed. hence there exist non-adherent ultrafilters of open sets μ1, μ2 in s −, t −, respectively. hence the restrictions μ1 | s and μ2 | t are non-adherent filterbases consisting of open sets in x. take ultrafilters of open sets ξ1, ξ2 in x containing μ1 | s and μ2 | t , respectively. clearly ξ1 and ξ2 are also non-adherent and u belongs to both of them. therefore, at least one of the round filters ξ+1 , ξ + 2 is different from ξ. therefore, û ∩ (x̂ − x) consists of more than one element and x̂ − x is dense in itself. � corollary 2.2. every normal hausdorff metacompact space x satisfies the condition in the theorem. proof. let u ⊆ x be an open set whose closure is non compact. by [4, 4.74.5], the subspace u− cannot be pseudocompact and hence u cannot be c-bounded. � 30 a. garćıa-máynez and r. mancio-toledo corollary 2.3. t if x is paracompact and t2, then x̂ − x is dense in itself, and hence the extensions kx and x̂ are non-equivalent (unless x is compact). lemma 2.4. let u be an open set in a regular hausdorff space x and let ξ ∈ û ∩ (x̂ − x). then u is wide if and only if (û − {ξ}) ∩ (x̂ − x) �= ∅. hence, if u is not wide, we have {ξ} = û ∩ (x̂ − x). proof. reason as in theorem 2.1. � example 2.5. let x be the space of countable ordinals with the order topology. then every uncountable open set in x is wide and hence x̂ − x is dense in itself. proof. let d be the set of non-limit ordinals in x. then d is open, discrete and dense in x. if u ⊆ x is an uncountable open set in x, then u ∩ d is also uncountable (because otherwise u− = (u ∩ d)− would be compact and hence u would be countable). clearly, u ∩d is the union of two uncountable disjoint subsets. hence, u is wide. � example 2.6. the half disk x = { (p, q) ∈ r2 | p2 + q2 ≤ 1, q > 0 } has a noncompact hausdorff closed extension z whose remainder z − x is closed and discrete. however z is not equivalent to x̂ neither to kx. proof. let z = x ∪ {(z, 0) | − 1 ≤ z ≤ 1}. for each (z, 0) ∈ z − x, define μz be the set of unions of {(z, 0)} with upper half open disks in r2 centered at (z, 0) and intersected with x. if z ∈ x, μz consists of all open disks in r2 centered at z and intersected with x. we can now topologize z with the help of these filter bases μz and convert it into a hausdorff, non-regular, extension of x. to see that z is hausdorff closed, we consider a cover of z consisting of elements of the filterbases μz. since { (p, q) ∈ r2 | p2 + q2 ≤ 1, q ≥ 0 } is compact in the usual topology of r2, we could get a finite subcover for this space if we adjoin to the elements of μz (z ∈ z − x) their radii in the x-axis. therefore, the original cover has a finite subfamily which covers x (recall x is dense in z). this argument proves that every open cover of z is densely finite, and hence z is hausdorff closed (see [1]). clearly the remainder z − x is closed and discrete. for each point z ∈ z − x, we can find an infinite family of ultrafilters of open sets in x which have z as a convergence point. this remark proves that z is not equivalent to x̂ neither to kx, because in these extensions, every point of the remainder is the convergence point of a unique ultrafilter of open sets in x. � hausdorff closed extensions of pre-uniform spaces 31 references [1] r. engelking, general topology, polish scientific publishers (warszawa, 1975). [2] a. garćıa-máynez and rubén s. mancio, completions of pre-uniform spaces, appl. gen. topol. 8, no. 2 (2007), 213–221. [3] a. garćıa-máynez and rubén s. mancio, extending maps between pre-uniform spaces, appl. gen. topol. 13, no. 1 (2012), 21–25. [4] a. garćıa-máynez and a. tamariz, topoloǵıa general, porrúa (méxico, 1988). [5] rubén s. mancio, los espacios pre-uniformes y sus completaciones, ph. d. thesis, universidad nacional autónoma de méxico, 2007. (received january 2010 – accepted january 2011) a. garćıa-máynez (agmaynez@matem.unam.mx) instituto de matemáticas, universidad nacional autónoma de méxico, área de la investigación cient́ıfica, circuito exterior, ciudad universitaria, 04510 méxico, d.f. méxico rubén mancio-toledo (rmancio@esfm.ipn.mx) escuela superior de f́ısica y matemáticas, instituto politécnico nacional, unidad profesional adolfo lópez mateos, col. lindavista, 07738 méxico, d.f. hausdorff closed extensions of pre-uniform[6pt] spaces. by a. garcía-máynez and r. mancio-toledo @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 51-60 hereditary separability in hausdorff continua d. daniel and m. tuncali ∗ abstract we consider those hausdorff continua s such that each separable subspace of s is hereditarily separable. due to results of ostaszewski and rudin, respectively, all monotonically normal spaces and therefore all continuous hausdorff images of ordered compacta also have this property. our study focuses on the structure of such spaces that also possess one of various rim properties, with emphasis given to rim-separability. in so doing we obtain analogues of results of m. tuncali and i. lončar, respectively. 2010 msc: primary: 54f15; secondary: 54c05, 54f05, 54f50 keywords: hereditary separability, image of compact ordered space, locally connected continuum, rim-separability 1. introduction it is straightforward that separability is inherited by open subspaces and therefore by regular closed sets [3], and is hereditary in second countable spaces. it has also been shown [14] that separability is inherited by closed sets in luzin spaces. we consider those hausdorff spaces s such that each separable subspace of s is hereditarily separable. such a space will be said to be subhereditarily separable. a motivation for our interest in this property lies partly in its relationship to monotone normality and continuous hausdorff images of ordered compacta. it has been shown that all monotonically normal spaces are sub-hereditarily separable [20]. the closed continuous image of a monotonically normal space is known [10] to be monotonically normal and is therefore subhereditarily separable. in particular, any continuous hausdorff image of an ∗the second author is partially supported by national science and engineering research council of canada grants no:141066-2008. 52 d. daniel and m. tuncali ordered compactum [22] is sub-hereditarily separable. a first countable locally connected sub-hereditarily separable continuum need not be the continuous image of an arc as we demonstrate by example. a central motivation for this study is a sequence of (still un-resolved) questions raised in [8]: question is each locally connected separable suslinian continuum hereditarily separable? question is each (hereditarily) separable locally connected suslinian continuum metrizable? we first note that local connectivity is essential in each of these questions. there is constructed in [7] a compactification of the half-line [0,∞) whose remainder is a homeomorphic copy of a souslin line s (a linearly ordered non-separable continuum having only countably many mutually disjoint subintervals). the resulting continuum z of course fails to be locally connected at each point of s. we also note that in [2], banakh, fedorchuk, nikiel and tuncali proved that under suslin hypothesis, each suslinian continuum is metrizable, and under the negation of suslin hypothesis, they constructed a hereditarily separable, non-metric continuum which is nowhere locally connected. arkhangelskii [1] and shapirovskii [23] have independently shown that if each closed subspace of a separable compact space s is separable then s is hereditarily separable. in addition, it is shown in [8] that each closed zero-dimensional subset of a suslinian continuum is metrizable. it therefore follows that it is sufficient in the initial question above to consider the separability of non-degenerate subcontinua. in [26], it is shown that each separable and connected continuous image of a compact ordered space is metrizable. it is therefore sufficient in the second question above to consider whether the space is the continuous image of a compact ordered space. we also note that in [9] it is shown that if there is an example of a locally connected suslinian continuum which is not the continuous image of an arc then there is such a continuum which is separable. 2. preliminaries all spaces are assumed to be hausdorff. a compactum is a compact space. a continuum is a connected compactum. recall that a continuum in suslinian provided that it possesses only countably many pairwise disjoint non-degenerate subcontinua. an arc is a continuum x which admits a linear ordering such that the order topology coincides with the given topology. it is straightforward that each ordered compactum is contained in an arc. a space x is said to be an iok if it is the continuous image of some compact ordered space k. if k is also connected, then x is said to be ioc. a space s is hereditarily separable provided each subspace of s is separable. a space s is said to be sub-hereditarily separable if and only if each separable hereditary separability in hausdorff continua 53 subspace of s is hereditarily separable. for brevity, we will also say that such a space is shs. a topological space x is said to be monotonically normal (see [10]) provided that there exists a function g which assigns, to each point x ∈ x and each open set u of x containing x, an open set g(x, u) such that (1) x ∈ g(x, u) ⊆ u, (2) if u′ is open and x ∈ u ⊂ u′, then g(x, u) ⊆ g(x, u′), (3) if x and y are distinct points of x, then g(x, x −y)∩g(y, x −x) = ∅. such a function g is called a monotone normality operator on x. it follows from a result of ostaszewski [20] that each monotonically normal compactum is shs (see also [22]). if s ⊆ x, intx(s) will denote the interior of s with respect to x or simply int(s) if the superspace is clear. similarly, bdx(s) (or simply bd(s)) and clx(s) (or simply cl(s)), respectively, will denote the boundary of s and closure of s, respectively, with respect to x. a space s is said to be rim-separable (rim-metrizable) at s ∈ s if and only if s admits a basis of open sets at s with separable (metrizable) boundaries, and s is rim-separable (rim-metrizable) provided that it is rim-separable (rimmetrizable) at each point. it is clear that any rim-metrizable compact space is rim-separable. a relatively comprehensive survey of rim properties may be found in [5]. 3. examples and fundamental properties our interest in spaces that are sub-hereditarily separable was initially spurred by the aforementioned results of ostaszewski [20] and rudin [22], respectively. these results demonstrate that sub-hereditarily separability is a necessary condition in order for a compactum to be an iok. we give a short direct proof of this below. the property is however not sufficient as the example below demonstrates. theorem 3.1. if x is the continuous image of some compact ordered space then x is shs. proof. by the result of arkhangelskii [1] and shapirovskii [23] noted above, it is enough to consider a ⊆ b ⊆ x with a closed in b and b separable and closed in x. from nikiel, purisch, and treybig [18], b is the continuous image of the double arrow space (d = ([0, 1]×{0, 1})−{(0, 0), (1, 1)} with the order topology given by lexicographic order). they have also noted that the double arrow space is hereditarily separable. therefore, b is also hereditarily separable. � let y = [0, 1]2 be equipped with the lexicographic order and let x = y ×[0, 1]. then x is a first countable non-separable locally connected continuum that is both shs and rim-separable. we also note that x is neither an iok [25] nor rim-metrizable [28]. 54 d. daniel and m. tuncali we also note that sub-hereditarily separability is not preserved under arbitrary continuous maps. the sorgenfrey plane r2l (where rl is the real line with the topology generated by the basis b = {[a, b) : a, b ∈ r, a < b}) fails to be hereditarily separable although rl is hereditarily separable. (sub-hereditarily separability therefore fails to be preserved under products.) let r2d denote the plane with the discrete topology; r2d is obviously shs since each separable subset is countable. the identity map id : r2d → r2l is clearly continuous. theorem 3.2. a topological space x is shs if and only if there exists an open continuous onto map f : x → y such that y is shs and f−1(y) is separable for each y ∈ y . proof. suppose a ⊆ b ⊆ x and b is separable. since f(b) is separable and y is shs, there exists a countable dense subset df(a) of f(a). for each y ∈ df(a), let dy denote a countable dense subset of f−1(y). consider d = ∪{dy : y ∈ df(a)}. if d were not dense in a, there exists an a-open set u such that u ∩ d = ∅. assume that u = a ∩ u′ with u′ open in x. then f(u) = f(a) ∩ f(u′), and f(u) ∩ df(a) = ∅, a contradiction. therefore, u ∩ d = ∅ and d is dense in a. � lemma 3.3. suppose f : x → y is a closed continuous map and x is shs. then y is shs. proof. consider a ⊆ b ⊆ y with a closed in b and b separable and closed in y . let d be a countable dense subset in b. for each d ∈ d, select one point from f−1(d) and let dx be the collection of points so selected. then cl(dx) is a hereditarily separable subspace of x. since a ⊆ f(cl(dx)), it follows that a is separable. � the following very simple theorem is an analogue to an important result of treybig [25] if the product x × y of two infinite hausdorff spaces x and y is an iok then each of x and y is metrizable. theorem 3.4. if x and y are compacta and x × y is shs, then each of x and y is shs. proof. if each of x and y are countable, the result is obvious. so, without loss of generality, consider x to be uncountable. suppose a ⊆ b ⊆ x and b is separable. then select a countable (or finite) closed subset y ′ of y . then b × y ′ is separable and thus a × y ′ is separable since x × y is shs. then the continuity and closedness of the projection πx onto x yields that a is separable. � a space z is said to be locally sub-hereditarily separable (or locally shs for brevity) at z ∈ z if and only if there is a shs neighborhood of z in z. theorem 3.5. a compactum x is shs if and only if it is locally shs at each point. hereditary separability in hausdorff continua 55 proof. consider a ⊆ b ⊆ x with a closed in b and b separable and closed in x. for each point a of a, select a b-open separable neighborhood na of a. since x is locally shs, a∩na is separable. by compactness, a is separable. � theorem 3.6. if a first countable compactum x is not shs then the closed subspace s = {p ∈ x : x is not locally shs at p} is not scattered and therefore not countable. proof. assume that s is scattered. s then has an isolated point q and therefore a local basis {un} of open sets at q such that cl(un)∩s = {q} for each n and u1 ⊃cl(u2)⊃ u2 ⊃cl(u3)⊃ ··· . since x is not locally shs at q, we may assume that there is a separable subset b ⊆ u1 and non-separable subset a such that a ⊆ b ⊆ u1. select a countable dense subset qk ⊆ a∩(u1−cl(uk)) for each k = 2, 3, . . . . then q = ∪∞k=2qk is countable dense in a. x is then locally shs at q and therefore q ∈ s, a contradiction. � in [8], it is shown that each suslinian continuum is first countable and that each closed zero-dimensional subset of a suslinian continuum is metrizable. in view of the aforementioned questions (also in [8]), we then note that the following result follows immediately. corollary 3.7. if x is a suslinian continuum and x is not hereditarily separable then s = {p ∈ x : x is not locally hereditarily separable at p} contains a non-degenerate subcontinuum. 4. the class of sub-hereditarily separable continua although several of the results hold in a more general setting (e.g. lemma 4.3), we address in the main in this section spaces that are shs continua. the initial result is a simple analogue to theorem 3.4, p. 1052 of [4]. theorem 4.1. suppose x is a locally connected rim-separable shs continuum. then x admits a basis b such that, for each b ∈b, b is fσ in x, and bd(b) is separable. proof. let b = {o : o is open and o = ∪oi with o1 ⊆cl(o1)⊆ o2 ⊆cl(o2)⊆ ··· , each oi is open in x, bd(oi) is separable for each i, o = x}. then bd(o)⊆cl(∪(bd(oi)) and is therefore separable by hypothesis. � lemma 4.2. for a rim-separable locally connected continuum x and an fσsubset a of x, each compact subset k of cl(a) − a lies in some separable subspace of x. such a k is therefore separable if x is shs. proof. let k be any compact subset of cl(a) − a. we express a = ∪∞n=1an where {an : n = 1, 2, . . .} is an increasing sequence of compact subsets of x. by the rim-separability of x, we may select for each n = 1, 2, . . . an open neighborhood un of an in x with separable boundary bd(un) such that cl(un)∩k = ∅. we now show that k lies in the closure of the separable subspace s = ∪∞n=1bd(un). let x ∈ k and v a connected open neighborhood of x in x. since x ∈ k ⊆cl(a), there exists some n = 1, 2, . . . such that v 56 d. daniel and m. tuncali meets an in some point a. since v is connected and x ∈ cl(un), the set v ∩un is not both closed and open in v . it then follows that v∩bd(un)⊆ v ∩ s is non-empty and x ∈cl(s). � a proof of the following may be found in [6]. lemma 4.3. let x denote a locally connected continuum and suppose that the set t = {p ∈ x : x is not locally an iok at p} is totally disconnected. then x is rim-metric (and therefore rim-separable). corollary 4.4. let x denote a locally connected shs continuum and suppose that the set t = {p ∈ x : x is not locally an iok at p} is totally disconnected. suppose that a is a compact subset of x such that there exists an fσ-subset b of x with a ⊆cl(b)−b. then a is separable. theorem 4.5. suppose x is a rim-separable shs continuum and x fails to be locally connected at x ∈ x. then for each open set o containing x there exists a non-degenerate separable subcontinuum m of x such that m ⊆ o. proof. let x ∈ o with o open in x. since x fails to be locally connected at x ∈ x, x is not connected im kleinen at x. as such [27], there exist open sets u and u′ and mutually disjoint continua c′1, c ′ 2, . . . so that x ∈ u ⊂ cl(u) ⊂ u′ ⊂ cl(u′) ⊂ o and each c′i ⊆ o meets both u and (x − u′). for each i, select a component ci of c ′ i ∩ cl(u′) such that ci meets both u and (x − u′). the limiting continuum of m of {ci}∞i=1 is separable by the reasoning of lemma 4.2 and, by construction, is contained in o. � theorem 4.6. suppose x is a suslinian shs continuum and each pair of points x and y is contained in a separable subcontinuum s(x, y) of x. then x is (hereditarily) separable. proof. let m be a maximal family of non-degenerate pairwise disjoint separable subcontinua of x. since x is suslinian, m = {mi : i = 1, 2, 3, . . .} is at most countable and therefore ∪m had countable dense subset d. d is then dense in x; else there exist open sets u and o such that u ⊂cl(u) ⊂ o and o∩cl(d) = o ∩ (∪m) = ∅. then for p ∈ u and q ∈ (x − o), the closure of the component of s(p, q) ∩ u is a separable continuum, contradicting the maximality of m. � we note that the condition described in the hypothesis of the previous result the existence of a separable continuum containing each pair of distinct points may be generalized and that generalization has been shown to have wide applications in utility theory. in particular, space z is said to be separably connected if for each pair of distinct points x and y in z, there is a separable connected subspace s ⊂ z such that s contains both x and y. the reader is referred to the survey of induráin [11] for results concerning such spaces and their applications. separable continua may arise in locally connected suslinian continua in a somewhat natural way by utilizing related equivalence relations. in particular, hereditary separability in hausdorff continua 57 a locally connected shs suslinian continuum admits an upper semi-continuous decomposition such that the resulting decomposition space is the continuous image of an arc. the following is based on similar constructions employed in work of b. pearson and j. simone (e.g., [24]), respectively. let x be a locally connected suslinian continuum. for each x ∈ x, define mx = {y ∈ x : there exists a separable subcontinuum of x containing x and y}. note that each mx is separable by theorem 4.6. let g = {mx : x ∈ x}. then x/g is hereditarily locally connected and is thus the image of an arc [17], and each element mx ∈ g is a (separable) continuum. corollary 4.7. let x be a suslinian shs locally connected continuum. if x contains no non-degenerate separable subcontinuum then x is an ioc. proof. x is first countable and rim-metrizable (and therefore rim-separable) by [7]. let g denote the decomposition of x as indicated above. if x contains no non-degenerate separable subcontinuum then g is a decomposition of x into singletons. therefore, x is an ioc. � 5. inverse limits of separable continua in this section, we consider rim-separable continua as a sub-class of hereditarily separable continua. in particular, we demonstrate that each rim-separable continuum is the inverse limit of a σ-directed inverse system of separable continua with monotone surjective bonding mappings. the following four results are analogues of various results of m. tuncali [28]. where the proof is not provided, it is an obvious modification of that of tuncali . lemma 5.1. let x be a continuum and let u be an open set in x with separable boundary. let g denote the upper semi-continuous decomposition of cl(u) into components. then cl(u)/g is separable. theorem 5.2. let x be a rim-separable continuum and let y be compact metric. suppose f : x → y is onto and light. then x is separable. proof. let f : x → y be a light mapping of a rim-separable continuum x onto a metrizable continuum y . let b be a countable basis for the topology on y and let {(vn, wn) : n = 1, 2, 3, . . .} be an enumeration of the set {(v, w) : each of v and w is in b and cl(v )⊆ w}. for each n = 1, 2, 3, . . . there is by the rim-separability of x an open neighborhood un ⊆ x of the compact subset f−1(cl(vn)) such that cl(un)⊆ f−1(cl(wn)) and bd(un) is separable. we now show that the separable set s = ∪∞n=1bd(un) is dense in x so that x is separable. let x ∈ x and let u be an open neighborhood of x in x. as the mapping f is light, the pre-image f−1(y) of the point y = f(x) is zero-dimensional. we may assume, by replacing u by a suitable smaller neighborhood if necessary, that u ∩ f−1(y) is both closed and open in f−1(y) and that x − u is non-empty. x−bd(u) is then a neighborhood of f−1(y) and we can find a basic neighborhood w ∈ b of y such that f−1(cl(w))⊆ x−bd(u). select any basic neighborhood v ∈ b of y such that cl(v )⊆ w . 58 d. daniel and m. tuncali then (v, w) = (vn, wn) for some n = 1, 2, 3, . . . by the choice of the open neighborhood un, we have x ∈ f−1(cl(v ))⊆ un ⊆ f−1(w) ⊆ f−1(cl(w))⊆ x−bd(u). since x is a continuum, the open neighborhood u ∩ un of x in x has non-empty boundary in x. this boundary lies in u so that u∩s = ∅. � lemma 5.3. a compactum x is rim-separable if and only if, for each pair p, q of distinct elements of x, there exists a closed separable subset s of x such that s separates x between p and q. theorem 5.4. let x be a rim-separable continuum and let f : x → y be onto and monotone. then y is rim-separable. recall that an inverse system x = {xa, pab, a} is σ-directed provided that for each sequence {ai : i = 1, 2, . . .} there is an a ∈ a such that a ≥ ai for each i = 1, 2, . . . . in theorem 18 of [13] (see also [12]), i. lončar demonstrates that each rim-metrizable continuum x admits a σ-directed inverse system x = {xa, pab, a} such that each xa is a metrizable continuum, each pab is monotone and onto, and x = lim←−x. theorem 5.5. let x be a rim-separable continuum. there exists a σ-directed inverse system x = {xa, pab, a} such that each xa is a separable continuum, each pab is monotone and onto, and x = lim←−x. proof. by mardeśič [15] and by nikiel, tuncali, and tymchatyn (theorem 9.4 of [19]), there is a σ-directed inverse system y = {ya, qab, a} such that each ya is compact metric, each qab is onto, and x = lim←−y. for each a ∈ a, let qa : x → ya denote the natural projection. for each a ∈ a, apply the monotone-light factorization of qa to obtain a space xa, a monotone mapping q′a : x → xa, and a light mapping q′′a : xa → ya such that qa = q′′a ◦q′a. by mardeśič [15], for each pair a, b of elements of a such that a ≤ b in a, there is a monotone mapping pab : xb → xa. we therefore obtain an inverse system x = {xa, pab, a} such that x = lim←−x. applying theorem 5.4 to q′a : x → xa, each xa is rim-separable. applying theorem 5.2 to q′′a : xa → ya, each xa is separable. � from this result, a number of analogues of results of lončar follow. furthermore, the proofs are essentially identical to their proofs. in particular, the two following results (and their proofs) are analogues of theorem 19 and theorem 20, respectively, of [13]. recall that a metric continuum is said to have the property of kelley if and only if, given � > 0, there is a δ > 0 such that if a and b are in x, d(a, b) < � and a ∈ a ∈ c(x) then there exists b ∈ c(x) such that b ∈ b and h(a, b) < �. it is known that each locally connected metric continuum has the property of kelley (see e.g. [16]). the following topological generalization of this property is due to w. j. charatonik. a continuum x is defined to have the property of kelley if, for each a ∈ x, each a in c(x) containing a, and each open set v ∈ c(x) containing a, there exists an open set w containing a and, if b ∈ w , then there exists b ∈ c(x) such that b ∈ b and b ∈ v . lončar (theorem 9 of [12] ) has shown that each locally connected hereditary separability in hausdorff continua 59 continuum has the property of kelley. a continuum x is smooth at the point p ∈ x if for each convergent net {xn} of points of x and for each subcontinuum k of x such that both p and x = lim{xn} are in k, there exists a net {ki} of subcontinua of x such that each ki contains p and some xn and k = lim{ki}. rakowski [21] has shown that a continuum x is smooth at p if and only if for each subcontinuum n of x containing p and for each open set v such that n ⊆ v there exists an open connected set k such that n ⊆ k ⊆ v . recall also that a dendroid is an arcwise connected and hereditarily unicoherent continuum. corollary 5.6. every rim-separable dendroid with the property of kelley is smooth. corollary 5.7. every rim-separable dendroid is the inverse limit of an inverse system of separable dendroids. acknowledgements. the authors acknowledge with gratitude the referee for many helpful comments and suggestions. in particular, the referee’s suggestions led to an improved formulation of lemma 4.2 and a strengthened result in theorem 5.2. references [1] a. v. arkhangelskii, on cardinal invariants, proceed. third prague top. symp. 1971, prague, 1972, 37–46. 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(received march 2011 – accepted september 2011) d. daniel (dale.daniel@lamar.edu) lamar university, department of mathematics, beaumont, texas 77710, usa m. tuncali (muratt@nipissingu.ca) nipissing university, faculty of arts and sciences, north bay, ontario p1b 8l7, canada hereditary separability in hausdorff continua. by d. daniel and m. tuncali 10.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 129 152 on domains witnessing increase ininformationdieter spreenabstract. the paper considers algebraic directed-completepartial orders with a semi-regular scott topology, called regulardomains. as is well known, the category of scott domains andcontinuous maps is cartesian closed. this is no longer true, ifthe domains are required to be regular. two cartesian closedsubcategories of the regular scott domains are exhibited: regulardi-domains with stable maps and strongly regular scott domainswith continuous maps. here a scott domains is strongly regular ifall of its compact open subsets are regular open. if one considersonly embeddings as morphisms, then both categories are closedunder the construction of dependent products and sums. more-over, they are !-cocomplete and their object classes are closedunder several constructions used in programming language seman-tics. it follows that recursive domains equations can be solved andmodels of typed and untyped lambda calculi can be constructed.both kinds of domains can be used in giving meaning to program-ming language constructs.2000 ams classi�cation: 68q55, 06b35, 03b15, 03b40, 18b30keywords: scott domains, di-domains, semi-regular topology, programminglanguage semantics, recursive domain equations, dependent product, dependentsum, lambda calculus 1. introductiondomains and domain-theoretic models of the lambda calculus were discoveredby dana scott [12] in the fall of 1969 when he was working with christopherstrachey at oxford on the semantics of programming languages. in developinghis approach he built on well understood ideas from recursive function theoryusing higher type functionals. but while people in this area were mainly inter-ested in total functions, he started right away to consider also partial objects.similar work was done by yuri l. ershov [6] in russia at nearly the same time.he too was motivated by the theory of functionals in computability theory. 130 dieter spreena domain is a structure modelling the notion of approximation and of com-putation. a computation performed using an algorithm proceeds in discretesteps. after each step there is more information available about the result ofthe computation. in this way the result obtained after each step can be seen asan approximation of the �nal result.to be a bit more formal, a domain is a structure having one binary relationv, a partial order, often called information order, with the intended meaningthat x v y just in case x is an approximation of y or y contains at least asmuch information as x. we also require that a domain should include a leastelement modelling no information. to guarantee the existence of the result ofa computation, that is, of a consistent set of approximations, every directedsubset of a domain must have a least upper bound. directed subsets containconsistent information. the information contained in a �nite set of elements isconsistent, if the set is bounded from above.in any step of a computation a computing device can process at most a �niteamount of information. domain elements containing only �nite information arecalled compact. we shall consider only algebraic domains in this paper. thismeans that every point is the least upper bound of all compact elements belowit.as has already been said, for elements x and y of a domain, x v y meansthat y contains at least the information contained in x. in this paper we areinterested in domains with the property that if the information contained ina compact element can be enlarged, then it can be enlarged in at least twoessentially di�erent ways. this means that for any two distinct compact ele-ments x and y with x v y there is a further compact element z above x whichis inconsistent with y. it follows in this case that x < y not only indicatesthat y properly includes the information contained in x, but that there is alsoa witness for this, namely a point z with x v z such that y and z are notbounded from above. this excludes situations in which x < y, but x uniquelydetermines y.each domain comes with a canonical order-consistent topology, the scotttopology. it turns out that the just mentioned requirement on the informationorder is equivalent to the fact that the scott topology is semi-regular. therefore,we call domains satisfying the above condition regular. it is the aim of thispaper to show that large subclasses of regular domains have su�cient closureproperties so that they can be used for de�ning the meaning of programminglanguage constructs.as we shall see, both the separated and the coalesced sum of families of (atleast two) regular domains are regular again. the same holds for the prod-uct and|in the case of �nite families|for the smash product. moreover, thecategory of regular domains with embedding/projections as morphisms is !-cocomplete: the inverse limit of any !-cochain of regular domains is regularagain. in addition, this category is closed under the construction of dependentsums. on domains witnessing increase in information 131an important closure property needed for the interpretation of programminglanguages that allow procedures taking functions as inputs is closure under thefunction space construction. as is well known, the class of algebraic domainsis not closed under this construction. therefore, one usually restricts oneself tothe subclass of scott domains, when giving meaning to programming languageconstructs. unfortunately, the scott domain of continuous maps between tworegular scott domains need not be regular again. however, we shall present twosubclasses of the regular scott domains that are closed under the correspondingfunction space construction.first, we consider the subclass of regular di-domains. di-domains have beenintroduced by g�erard berry [4] in order to show that certain continuous mapsare not lambda de�nable. with stable maps as morphisms their category iscartesian closed. the same is true for the full subcategory of regular di-domains. we show more generally that the category of di-domains and rigidembedding/projections is closed under the construction of dependent products.since the di-domains are also closed under the other constructions mentionedabove, regular di-domains can be used in giving meaning to functional pro-grams. recursive domain equations have a solution [13] and stable regularmodels for typed and untyped lambda calculi can be constructed [2].the second way we take is to strengthen the regularity requirement. thestronger condition is such that not only all basic scott open sets are regularopen, but all �nite unions of such sets. these are exactly the compact opensubsets of the domain. we call domains satisfying the stronger requirementstrongly regular. as we shall see, this class is closed under nearly all the do-main constructions mentioned earlier. moreover, we show that the category ofstrongly regular scott domains and embedding/projections is closed under theconstruction of dependent products. it follows that the category of stronglyregular scott domains with continuous maps as morphisms is cartesian closed.thus, strongly regular scott domains too are well suited for de�ning the se-mantics of programming languages. again, recursive domain equations have asolution and strongly regular models for typed and untyped lambda calculi canbe constructed.the paper is organized as follows. in section 2 regular domains are intro-duced. an example is given showing that the category of regular scott domainswith continuous maps is not cartesian closed. in the next two sections, twocartesian closed subcategories are exhibited.first, in section 3, regular di-domains are considered and then, in section 4,strongly regular scott domains. in both cases it is proved that the categories ofthese domains with embedding/projections as morphisms are closed under theconstruction of dependent products. further closure properties of the regularand the strongly regular domains are studied in section 5. some �nal remarksappear in section 6. 132 dieter spreen2. regular domainslet (d;v) be a partial order with smallest element ?. we write x < y, ifx v y and x 6= y. for a subset s of d, #s = fx 2 d j (9y 2 s)x v y g and"s = fx 2 d j (9y 2 s)y v xg, respectively, are the lower and the upper setgenerated by s. the subset s is called consistent if it has an upper bound. s isdirected, if it is nonempty and every pair of elements in s has an upper boundin s. d is a directed-complete partial order (cpo) if every directed subset sof d has a least upper bound fs in d, and d is bounded-complete if everyconsistent subset has a least upper bound in d. in particular, if a pair fx;yg isconsistent this is denoted by x " y. in a bounded-complete cpo any consistentpair fx;yg has a least upper bound which is written as x t y.an element x of a cpo d is compact if for any directed subset s of d therelation x v fs always implies the existence of an element u 2 s with x v u.we write k(d) for the set of compact elements of d. if for every y 2 d theset #fyg \ k(d) is directed and y = f#fyg \ k(d), the cpo d is said to bealgebraic and !-algebraic, if, in addition, k(d) is countable. note that insteadof algebraic cpo we also say domain. standard references for domain theoryand its applications are [8, 7, 1, 14, 2].let us �rst consider some examples. for a set a such that ? =2 a seta? = a [ f?g and order it by the at ordering x v y if x = ? or x = y.obviously, all elements of a? are compact. thus, a? is a domain.a further example of a domain is the set p(!) of sets of natural numbers,ordered by set inclusion. here, the �nite sets are exactly the compact elements.the cpo of lazy natural numbers is the set!l = fsn(?) j n 2 ! g [ fsn(0) j n 2 ! g [ fs!(?)gwith s0(?) = ? and s0(0) = 0, ordered byx v y , x = ? _ x = y _ [x = sm(?) ^ (9z)y = sm(z)]:note that the last case includes y = s!(?), setting s!(?) = sm(s!(?)). in thiscase all elements except s!(?) are compact and s!(?) is the least upper boundof all sm(?) with m 2 !.other examples are the cantor domain, that is the set of all �nite andin�nite sequences of 0's and 1's, ordered by the pre�x ordering, and the bairedomain, that is the set of all �nite and in�nite sequences of natural numbers,also ordered by the pre�x ordering. in both cases the �nite sequences are thecompact elements.the product d � e of two cpo's d and e is the cartesian product of theunderlying sets ordered coordinatewise. obviously, k(d �e) = k(d) �k(e).as is well known, on each cpo there is a canonical topology: the scott topol-ogy. a subset x is open, if it is upwards closed with respect to v and intersectseach directed subset of d of which it contains the least upper bound. in casethat d is algebraic, this topology is generated by the sets "fzg with z 2 k(d).let us now introduce the class of domains we want to study in this paper. on domains witnessing increase in information 133de�nition 2.1. a domain d is regular if for all u;v 2 k(d)u 6v v ) (9z 2 k(d))v v z ^ u "� z:note that all domains mentioned in the examples above are regular, exceptin the case of domains a? where a is a singleton. moreover, the product oftwo regular domains is regular again. if d is bounded-complete, the abovecondition can be simpli�ed.lemma 2.2. let d be a bounded-complete domain. then d is regular if andonly if for all u;v 2 k(d)v < u ) (9z 2 k(d))v v z ^ u "� z:proof. the \only if" part is obvious. for the \if" part only the case that u andv are not comparable with respect to the partial order has to be considered.if u "� v take z = v. in the opposite case the least upper bound x of u and vexists, which is compact as well. thus, there is some z 2 k(d) such that v v zand x "� z. it follows that also u "� z. �this shows for two elements u and v of a regular bounded-complete domaincontaining �nite information that if u contains more information than v, thenthere is a witness for this, that is, an element containing at least as muchinformation as v, but the information contained in which is inconsistent withthe information contained in u.as we shall show next, the regularity of a domain, which was de�ned viathe partial order, can also be expressed in topological terms. for a subset xof d let int(x), cl(x) and ext(x), respectively, be the interior, the closureand the exterior of x with respect to the scott topology. then x is regularopen if x = int(cl(x)). moreover, d is semi-regular if all basic open sets"fzg with z 2 k(d) are regular open. note that for z 2 k(d), ext("fzg) =sf"fz0g j z0 2 k(d) ^ z0 "� z g.lemma 2.3. let d be a domain with compact element u. then the followingtwo statements are equivalent:(1) (8v 2 k(d))[u 6v v ) (9z 2 k(d))v v z ^ u "� z].(2) the set "fug is regular open.proof. in order to see that (1) implies (2), assume that there is some v 2int(cl("fug))n"fug. then u 6v v. because of (1) there exists some z 2 k(d)with v v z such that fu;zg is not consistent. it follows that z 2 int(cl("fug)).hence z 2 cl("fug), which implies that "fzg intersects "fug. thus, u and zhave a common upper bound, contradicting what has been said about u and zbefore.for the proof that (1) is a consequence of (2) let v 2 k(d) such that u 6v v.since "fug is regular open, it follows that v 2 ext("fug), which means thatfu;vg cannot be consistent. �corollary 2.4. a domain is regular if and only if its scott topology is semi-regular. 134 dieter spreenmaps between cpo's are not only required to preserve information but alsocomputations: the value under a map of the result of a computation shouldnot contain more information than what is obtained from the approximatingobjects.de�nition 2.5. let d and e be cpo's. a map f : d ! e is said to be scottcontinuous if it is monotone and for any directed subset s of d,f(gs) = gf(s):it is well known that scott continuity coincides with topological continuity.therefore, in what follows we use the shorter term continuous instead of scottcontinuous.denote the collection of all continuous maps from d to e by [d ! e].endowed with the pointwise order, that is, f vp g if f(x) v g(x) for all x 2 d,it is a cpo again. since [d ! e] need not be algebraic, even if d and e are, oneconsiders subclasses of domains which have this property, when using domainsin programming language semantics, e.g. scott domains.de�nition 2.6. a scott domain is a bounded-complete !-algebraic cpo.as is widely known, the category sd of scott domains and continuous maps iscartesian closed: the one-point domain f?g is the terminal object, the domainproduct is the categorical product and the space of continuous maps betweentwo scott domains is the categorical exponent.for two scott domains d and e and elements d 2 k(d) and e 2 k(e) de�nethe step function (d & e): d ! e by(d & e)(x) = (e if d v x,? otherwise.then the compact elements of [d ! e] are exactly the maps of the formff(di & ei) j i � ng, where d0; : : : ;dn 2 k(d) and e0; : : : ;en 2 k(e) sothat whenever i � f0; : : : ;ng is such that fdi j i 2 i g is consistent, then sois fei j i 2 i g.the next example shows that in general the function space [d ! e] of tworegular scott domains d and e need not be regular again.example 2.7. consider the scott domains d and e de�ned by the followingdiagrams: d3 @@ @ @@ @ @ d4 ~ ~~ ~~ ~ ~ @@ @ @@ @ @ d5 ~ ~~ ~~ ~ ~ d1 @@ @@ @ @@ d2 ~~ ~ ~~ ~~ d0 d e3 bb bb bb bb e4 || || || || e1 bb bb bb bb e2 || || || || e0 e on domains witnessing increase in information 135and de�ne f1, f2 2 [d ! e] by f1 = (d2 & e1) and f2 = (d1 & e1) t (d5 &e1). then both maps are compact. obviously, f1 and f2 are not comparablewith respect to the pointwise order, but have a common upper bound, thatis, f2 2 cl("ff1g). we want to show now that "ff2g is included in cl("ff1g),which implies that "ff1g is not regular open. assume to this end that thereis some f 2 [d ! e] with f2 vp f and f1 "� f. then e1 v f(y), if d1 v yor y = d5. moreover, there is some z 2 d such that f1(z) and f(z) have nocommon upper bound. then d2 v z, since for z 2 d with d2 6v z we have thatf1(z) = e0 v f(z). moreover z 62 fd4;d5g. thus z = d2, which implies thatfz;d1g and hence also ff(z);f(d1)g are consistent. it follows that f(z) 6= e2,from which we obtain that f1(z) and f(z) have a common upper bound, incontradiction to our assumption. hence "ff2g is a subset of cl("ff1g).in the following sections we introduce two subclasses of the regular scottdomains which are closed under the construction of function spaces. as weshall see moreover, they are also closed under several other constructions usedin denotational semantics. to this end we need the following de�nitions.de�nition 2.8. an embedding/projection (f;g) from a cpo d to a cpo e isa pair of continuous maps f : d ! e and g : e ! d such that g � f = idd, theidentity map on d, and f � g vp ide. the map f is called embedding and gprojection.normally, we write an embedding/projection as (fl;fr). note that the mapfr is uniquely determined by fl, and vice versa [13]. embeddings are one-to-one and preserve compactness as well as least upper bounds of �nite consistentsets of domain elements [11]. moreover, both the embedding and the projectionare strict, that is, they map least elements onto least elements.suppose (fl;fr) is an embedding/projection from c to d and (gl;gr) isan embedding/projection from d to e. then the composition of (fl;fr) and(gl;gr) is de�ned by(gl;gr) � (fl;fr) = (gl � fl;fr � gr):let sdep denote the category of scott domains and embedding/projections.any partially ordered set may be viewed as a category. for a cpo d, thisis done by letting the objects of the category be the elements of d and lettingthe morphism set between objects x and y be the one-point set f�x;yg preciselywhen x v y and the empty set otherwise.let f : d ! sdep be a functor. recall that this means that f(x) is adomain for each x 2 d, and if x v y then f(�x;y) is an embedding/projectionfrom f(x) to f(y) such that f(�x;x) = idf(x), and if x v y and y v z thenf(�x;z) = f(�y;z) � f(�x;y). when x v y we shall use the notation fx;y forf(�x;y).de�nition 2.9. let d be a cpo. a functor f : d ! sdep is continuous if forany directed subset s of d, f(fs) is a colimit of the diagram(ff(x) j x 2 s g;ffx;y j x;y 2 s ^ x v y g): 136 dieter spreena section of f is a map f : d ! sff(x) j x 2 d g with f(x) 2 f(x), for allx 2 d.de�nition 2.10. let d be cpo and f : d ! sdep be a continuous functor.an section f of f is said to be(1) monotone if flx;y(f(x)) v f(y), for all x;y 2 d with x v y.(2) continuous if f is monotone and for every directed subset s of d,f(gs) = gfflx;fs(f(x)) j x 2 s g:3. regular di-domainsin this section we restrict our attention to the subclass of regular di-domains.de�nition 3.1. a scott domain d is a di-domain if it satis�es the two ax-ioms d and i:� axiom d: for all x;y;z 2 d, if y " z thenx u (y t z) = (x u y) t (x u z):� axiom i: for all u 2 k(d), #fug is �nite.note that in a bounded-complete domain any two elements x and y have agreatest lower bound x u y. moreover, observe that by axiom i all elements ofa di-domain below a compact element are also compact.the elements of a di-domain can be generated by a special kind of compactelements.de�nition 3.2. let d be a bounded-complete domain. an element u 2 d is acomplete prime if for any consistent subset s of d the relation u v fs alwaysimplies the existence of an element x 2 s with u v x.the complete primes in a di-domain d turn out to be those compact elementsthat have a unique element below them in the domain order [15]. as followsfrom a result of winskel, every element in d is the least upper bound of allcomplete primes below it.in the context of di-domains it is usual to work with embedding/projectionsthat satisfy an additional requirement.de�nition 3.3. an embedding/projection (f;g) from a di-domain d to a di-domain e is rigid if for all x 2 d and y 2 e with y v f(x), f � g(y) = y. themap f is called rigid embedding and g rigid projection.note that rigid embeddings preserve greatest lower bounds of �nite sets ofdomain elements. moreover, they map complete primes onto complete primes.as a consequence of this, rigid projections preserve the least upper bounds of�nite consistent sets of domain elements. we denote the category of di-domainsand rigid embedding/projections by direp.for the following de�nition observe that for any bounded-complete domaind and elements x;y;z 2 d such that x;y v z, (xuy;�xuy;x; �xuy;y) is a pullbackof (�x;z; �y;z) in d viewed as a category. on domains witnessing increase in information 137de�nition 3.4. let d be a bounded-complete domain, f : d ! direp a func-tor and f a section of f.(1) the functor f is stable if it is continuous and for all x;y;z 2 d withx;y v z, (f(x u y);fxuy;x;fxuy;y) is a pullback of (fx;z;fy;z).(2) let f be stable. the section f of f is stable if it is continuous and forall x;y 2 d with x " y,f(x u y) = frxuy;x(f(x)) u frxuy;y(f(y)):for the remainder of this section we assume that d is a di-domain andf : d ! direp a stable functor.proposition 3.5. let f be a stable section of f. then for any x 2 d andv 2 k(f(x)) such that v v f(x), there are �u 2 k(d) and �v 2 k(f(�u)) with thefollowing properties:(1) �u v x.(2) v = fl�u;x(�v).(3) �v v f(�u).(4) (8y;z 2 d)[�u;y v z ^ fl�u;z(�v) v fly;z(f(y)) ) �u v y].proof. let x 2 d and v 2 k(f(x)) with v v f(x). sincef(x) = gfflu;x(f(u)) j u 2 k(d) ^ u v xg;there is some û 2 k(d) with v v f l̂u;x(f(û)). because #fûg is �nite, there isa minimal such û, say �u. thus v v fl�u;x(f(�u)). observe that f�u;x is rigid.therefore v = fl�u;x � fr�u;x(v). set �v = fr�u;x(v). then �v 2 f(�u) and �v v f(�u).moreover, �v is compact.now, let y;z 2 d such that �u;y v z and fl�u;z(�v) v fly;z(f(y)). set vy =fry;z � fl�u;z(�v). then vy 2 f(y). moreover, it follows as above that fl�u;z(�v) =fly;z(vy). since f is stable, we have that (f(�uuy);f�uuy;�u;f�uuy;y) is a pullbackof (f�u;z;fy;z). therefore, there is some v̂ 2 f(�u u y) with fl�uuy;�u(v̂) = �v andfl�uuy;y(v̂) = vy. it follows that v̂ v fr�uuy;�u(f(�u)) u fr�uuy;y(f(y)). because f isstable as well, we obtain that v̂ v f(�uuy) and hence that v v fl�uuy;x(f(�uuy)).then �u = �u u y, by the minimality of �u, which implies that �u v y. �this result justi�es the following de�nition.de�nition 3.6. let f be a stable section of f. the settr(f) = f(u;v) j u 2 k(d) ^ v 2 k(f(u)) ^ v v f(u)^ (8y;z 2 d)[u;y v z ^ flu;z(v) v fly;z(f(y)) ) u v y]gis called trace of f.lemma 3.7. let f be a stable section of f. then the trace of f has thefollowing properties:(1) if (u0;v0); : : : ;(un;vn) 2 tr(f) such that fu0; : : : ;ung has least upperbound �u, then fflu0;�u(v0); : : : ;flun;�u(vn)g has a least upper bound, say �v,and (�u; �v) 2 tr(f). 138 dieter spreen(2) if (u;v) 2 tr(f), x;z 2 d and v0 2 f(x) such that u;x v z andv0 v frx;z�flu;z(v), then there exists u0 v u so that flx;z(v0) 2 flu0;z(f(u0))and (u0;fru0;z � flx;z(v0)) 2 tr(f).(3) if (u;v);(u0;v0) 2 tr(f) and for some z 2 d with u;u0 v z, flu;z(v) =flu0;z(v0), then u = u0 and hence v = v0.proof. (1) if fu0; : : : ;ung has least upper bound �u, then we have for i � n thatflui;�u(vi) v f(�u). hence, fflu0;�u(v0); : : : ;flun;�u(vn)g has a least upper bound �v.both, �u and �v are compact. it remains to show that (�u; �v) 2 tr(f).we have already observed that �v v f(�u). now, let y;z 2 d with �u;y v zand fl�u;z(�v) v fly;z(f(y)). then we have for i � n that ui v z and flui;z(vi) vfly;z(f(y)). since (ui;vi) 2 tr(f) it follows that ui v y, for i � n, which impliesthat �u v y.(2) let (u;v) 2 tr(f) and x;z 2 d with u;x v z. moreover, let v0 2 f(x)such that v0 v frx;z � flu;z(v). then flx;z(v0) v flu;z(v). since v is compact, wehave that also flu;z(v) is compact. hence, flx;z(v0) is compact as well. notethat flx;z(v0) v flu;z(v) v flu;z(f(u)) v f(z):therefore, by proposition 3.5, there is some (u0; �v) 2 tr(f) such that u0 v zand flx;z(v0) = flu0;z(�v). because u;u0 v z and flu0;z(�v) = flx;z(v0) v flu;z(f(u)),it follows that u0 v u.(3) if (u;v);(u0;v0) 2 tr(f) and for some z 2 d with u;u0 v z, flu;z(v) =flu0;z(v0), then v0 v f(u0) and thus flu;z(v) = flu0;z(v0) v flu0;z(f(u0)). since(u;v) 2 tr(f), it follows that u v u0. in the same way we obtain that u0 v u.therefore u = u0 and hence flu;z(v) = flu;z(v0), which implies that v = v0, asflu;z is one-to-one. �the next lemma shows how a stable section can be recovered from its trace.lemma 3.8. (1) a stable section f of f can be computed from its trace inthe following way:f(x) = gfflu;x(v) j u v x ^ (u;v) 2 tr(f)g:(2) every set of pairs (u;v) with u 2 k(d) and v 2 k(f(u)) satisfyingproperties (1)-(3) of lemma 3.7 is a trace of the stable section of fde�ned by the formula above.proof. (1) because of property (1) of lemma 3.7 the set of all flu;x(v) with(u;v) 2 tr(f) and u v x is directed and thus has a least upper bound. obvi-ously, the right-hand side of the formula is less than or equal to the left-handside. if v 2 k(f(x)) with v v f(x), then, by proposition 3.5, there is some(u0;v0) 2 tr(f) with u0 v x and v = flu0;x(v). it follows that v is less than orequal to the right-hand side of the formula. since f(x) is the least upper boundof all such v, the same is true for f(x). on domains witnessing increase in information 139(2) we have to show that if x satis�es properties (1)-(3) of lemma 3.7, thenthe map f de�ned byf(x) = gfflu;x(v) j u v x ^ (u;v) 2 x g:is a stable section of f with tr(f) = x.the map f is well de�ned, since the set fflu;x(v) j u v x ^ (u;v) 2 x g isdirected by property 3.7(1) and therefore has a least upper bound.obviously, f is monotone. to see that it is continuous, let s be a directedsubset of d and let v 2 k(f(fs)) with v v f(fs). it follows that there issome (�u; �v) 2 x so that �u v fs and v v fl�u;fs(�v). since �u is compact aswell, there is some x 2 s with �u v x. thus fl�u;x(�v) v f(x), which implies thatv v flx;fs(f(x)). since f(fs) is the least upper bound of all such v, we havethat f(fs) v ffflx;fs(f(x)) j x 2 s g. the converse inequality is obvious,by monotonicity.for the veri�cation of stability it su�ces to show for x;y;z 2 d with x;y v zthat frxuy;x(f(x)) u frxuy;y(f(y)) v f(x u y). let to this end (u;v);(u0;v0) 2 xwith u v x and u0 v y. since the binary operator u is continuous [7], we onlyhave to verify thatfrxuy;x � flu;x(v) u frxuy;y � flu0;y(v0) v f(x u y):set �v = frxuy;x�flu;x(v)ufrxuy;y �flu0;y(v0). then �v v frxuy;x�flu;x(v);frxuy;y �flu0;y(v0). hence, by property 3.7(2), there is some �u v u such that(3.1) flxuy;x(�v) 2 fl�u;x(f(�u))and (�u;fr�u;x � flxuy;x(�v)) 2 x. moreover, there is some �u0 v u0 such that(3.2) flxuy;y(�v) 2 fl�u0;y(f(�u0))and (�u0;fr�u0;y�flxuy;y(�v)) 2 x. note that �u v x v z and �u0 v y v z. in additionfl�u;z � fr�u;x � flxuy;x(�v) = flx;z � fl�u;x � fr�u;x � flxuy;x(�v)= flx;z � flxuy;x(�v) (by (3.1))= flxuy;z(�v)= fl�u0;z � fr�u0;y � flxuy;y(�v) (by (3.2))by property 3.7(3) we therefore have that �u = �u0. thus �u v x u y. it followsthat fr�u;x � flxuy;x(�v) = fr�u;xuy � frxuy;x � flxuy;x(�v) = fr�u;xuy(�v):since (�u;fr�u;xuy(�v)) 2 x and �v 2 fl�u;xuy(f(�u)), by (3.1), we obtain that �v =fl�u;xuy � fr�u;xuy(�v) v f(x u y).next, we show that tr(f) contains x. if (u;v) 2 x then v v f(u). lety;z 2 d such that u;y v z and flu;z(v) v fly;z(f(y)). since flu;z(v) is compact,if follows from the de�nition of f that there is some (û; v̂) 2 x with û v y and 140 dieter spreenflu;z(v) v fly;z � f l̂u;y(v̂). by property 3.7(2) we now obtain that there exists au0 v û so that(3.3) flu;z(v) 2 flu0;z(f(u0))and (u0;fru0;z �flu;z(v)) 2 x. thus, we have that (u;v);(u0;fru0;z �flu;z(v)) 2 x,u;u0 v z and, because of (3.3), flu0;z�fru0;z�flu;z(v) = flu;z(v). therefore u = u0,by property 3.7(3). since u0 v û v y it follows that (u;v) 2 tr(f).finally, we have to verify that also x contains tr(f). let (u;v) 2 tr(f).then v v f(u). as above it follows that there is some u0 v u so that(u0;fru0;u(v)) 2 x and v 2 flu0;u(f(u0)). then v v flu0;u(f(u0)). as a con-sequence of the minimality condition in the de�nition of a trace we obtain thatu v u0. thus u = u0, which means that (u;v) 2 x. �de�ne �df to be the set of all stable sections of f and order it by the stableordering, that is f vs g , tr(f) � tr(g):then �df with the stable ordering is the dependent product of f over d.our next goal is to show that �df is a di-domain. to this end we need thefollowing lemma.lemma 3.9. let f be a stable section of f and i � tr(f). moreover, leti0 = f(x;y) 2 tr(f) j (9(u;v) 2 i)x v u ^ flx;u(y) v v gand x = f(x;y) 2 tr(f) j(9(u0;v0); : : : ;(un;vn) 2 i0)x = fi�n ui ^ y = fi�n flui;x(vi)g:then the following three statements hold:(1) the set x has properties (1)-(3) of lemma 3.7.(2) the stable section g de�ned by x according to lemma 3.8(2) is thesmallest stable section h of f with h vs f and i � tr(h).(3) if the set i is �nite, so is x.proof. (1) property 3.7(1) holds by construction and 3.7(3) is inherited fromtr(f). it remains to show that also 3.7(2) holds. let (u;v) 2 x and forx;z 2 d with u;x v z let v0 2 f(x) such that v0 v frx;z � flu;z(v). since xis contained in tr(f) there is some u0 v u so that flx;z(v0) 2 flu0;z(f(u0)) and(u0;fru0;z � flx;z(v0)) 2 tr(f). we show that (u0;fru0;z � flx;z(v0)) 2 x.because (u;v) 2 x, there are (u0;v0); : : : ;(un;vn) 2 i0 so that u = fi�n uiand v = fi�n flui;u(vi). de�ne u0i = ui u u0, for i � n. then u0 = fi�n u0i. asflx;z(v0) 2 flu0;z(f(u0)) there exists v̂ 2 f(u0) with flx;z(v0) = flu0;z(v̂). thenv̂ = fru0;z � flx;z(v0). for i � n set ~vi = flui;u(vi) u flu0;u(v̂). it follows that~vi v flui;u(vi);flu0;u(v̂). therefore flui;u �frui;u(~vi) = ~vi = flu0;u �fru0;u(~vi). since on domains witnessing increase in information 141(f(u0i);fu0i;u0;fu0i;ui) is a pullback of (fu0;u;fui;u), we obtain that there is somev0i 2 f(u0i) with flu0i;u0(v0i) = fru0;u(~vi) and flu0i;ui(v0i) = frui;u(~vi). thenflu0i;ui(v0i) = frui;u(~vi) v frui;u � flui;u(vi) = vi:thus (u0i;v0i) 2 i0. moreover, flu0i;u0(v0i) = fru0;u(~vi) v fru0;u � flu0;u(v̂) = v̂ andgi�n flu0i;u0(v0i) = gi�n fru0;u(~vi)= fru0;u(gi�n ~vi)= fru0;u(gi�n flui;u(vi) u flu0;u(v̂))= fru0;u(flu0;u(v̂) u (gi�n flui;u(vi)))= fru0;u(flu0;u(v̂) u v)= fru0;u � flu0;u(v̂)= v̂:here, we used thatflu0;u(v̂) = flu0;u � fru0;u � fru;z � flx;z(v0) v fru;z � flx;z(v0) vfru;z � flx;z � frx;z � flu;z(v) v v:it follows that (u0; v̂) 2 x.(2) properties 3.7(1)-(3) imply for any stable section h of f with i � tr(h) �tr(f) that x � tr(h).(3) since embeddings are one-to-one and axiom i holds in d and all f(x)with x 2 d, it follows that if i is �nite, so is i0. obviously kxk � 2ki0k.therefore, also x is �nite in this case. �theorem 3.10. let d be a di-domain and f : d ! direp a stable functor.then �df is a di-domain. the compact elements in �df are exactly thesections with �nite trace.proof. as is easily seen, �df is a cpo. if s is a subset of �df with up-per bound f, then it follows with lemma 3.9 that the section generated bysftr(h) j h 2 s g is the least upper bound of s. thus, �df is bounded-complete. obviously, the sections of f with �nite trace are the compact ele-ments. therefore, �df is a scott domain and axiom i holds. with lemmas 3.7and 3.8 it is readily veri�ed for two section f and g that tr(f) \ tr(g) is thetrace of f ug. by using the two lemmas once more we then obtain that axiom dis satis�ed as well. �in the special case of a constant functor f , say f(x) = e for x 2 d, onehas the well known fact that the space [d !s e] of stable maps from d to eis a di-domain again. indeed, the category of di-domains and stable maps iscartesian closed [7]. 142 dieter spreennow, let rdirep be the category of regular di-domains and rigid embed-ding/projections. we will show that for a stable functor f : d ! rdirep thedependent product domain �df is also regular.theorem 3.11. let d be a di-domain and f : d ! rdirep a stable functor.then �df is a regular di-domain.proof. we only have to verify the condition in lemma 2.2. let f;g 2 k(�df).since tr(f) is �nite, we have that f("fug) is �nite as well. hence there is somex 2"fug so that for all z 2 d with x v z, flx;z(f(x)) = f(z). obviously, we canassume that x is compact. by the formula in lemma 3.8(1) we moreover obtainthat there is some (û; v̂) 2 tr(f) with û v x and f(x) = f l̂u;x(v̂). becausef l̂u;x(v̂) v f l̂u;x(f(û)) v f(x) = f l̂u;x(v̂);we also have that f(û) = v̂. let y = f(x).now, suppose that f < g. then tr(f) & trl(g). thus, there is some(u;v) 2 tr(g) n tr(f).claim 3.12. flu;x(v) 6v y.proof. assume that v v fru;x � f l̂u;x(v̂). by property 3.7(2) there is then someu0 v û so that flu;x(v) 2 flu0;x(f(u0)) and (u0;fru0;x � flu;x(v)) 2 tr(f). itfollows that (u0;fru0;x � flu;x(v)) 2 tr(g). since (u;v) 2 tr(g) as well andflu0;x �fru0;x �flu;x(v) = flu;x(v), we obtain with property 3.7(3) that u = u0 andfru0;x � flu0;x(v) = v. hence (u;v) 2 tr(f), a contradiction.this shows that v 6v fru;x � f l̂u;x(v̂), which implies that flu;x(v) 6v f l̂u;x(v̂) =y. �claim 3.13. (u0;v0) 2 tr(f) ^ u0 " x ) u0 v x ^ flu0;x(v0) v y.proof. with u0 " x we also have that u0 " û. then (u0 t û;flu0;u0tû(v0) tf l̂u;u0tû(v̂)) 2 tr(f), by 3.7(1). hencef l̂u;u0tx(v̂) v flu0tû;u0tx(flu0;u0tû(v0) t f l̂u;u0tû(v̂))v flu0tû;u0tx(f(u0 t û))v f(u0 t x)= f l̂u;u0tx(v̂);(3.4)which implies that flu0;u0tû(v0) t f l̂u;u0tû(v̂) = f l̂u;u0tû(v̂). with 3.7(3) we thushave that û = u0 t û, that is, u0 v û v x. therefore, we obtain from (3.4) thatflu0;x(v0) v f l̂u;x(v̂) = y. �since f(x) is regular and flu;x(v) 6v y there is some �v 2 k(f(x)) so thaty v �v and flu;x(v) "� �v. de�nex = tr(f) [ f(x; ~v)~v 2 k(f(x)) ^ ~v v �v^(@(�u; �v) 2 tr(f))�u v x ^ fl�u;x(�v) = ~vg: on domains witnessing increase in information 143if we have checked that x has properties 3.7(1)-(3), we know that x is thetrace of a section �f 2 k(�df) with f vs �f. suppose that g; �f vs h, forsome h 2 �df . then flu;x(v) v g(x) v h(x) and �v = �f(x) v h(x), which isimpossible as flu;x(v) "� �v. thus g "� �f.it remains to check that x has the properties of lemma 3.7.(1) obviously, it is su�cient to consider the case that n = 1. let (u0;v0),(u1;v1) 2 x so that fu0;u1g is consistent. if (u0;v0);(u1;v1) 2 tr(f) we aredone. so, without restriction, let us assume that (u0;v0) =2 tr(f). then u0 = xand v0 v �v. if (u1;v1) =2 tr(f) too, then u1 = x and v1 v �v. in the oppositecase we obtain with claim 3.13 that u1 v x and flu1;x(v1) v �v. therefore, inboth cases v0 t flu1;x(v1) exists below �v.in order to see that (x;v0 t flu1;x(v1)) 2 x n tr(f) assume that there issome (�u; �v) 2 tr(f) such that �u v x and fl�u;x(�v) = v0 t flu1;x(v1). thenv0 v fl�u;x(�v). with property 3.7(2) it follows that there is some u0 v �u so thatv0 2 flu0;x(f(u0)) and (u0;fru0;x(v0)) 2 tr(f). hence, we have that u0 v x andflu0;x � fru0;x(v0) = v0, contradicting the fact that (u0;v0) 2 x n tr(f).(2) we only have to consider the case that (x; ~v) 2 x n tr(f), z;z0 2 d andv0 2 f(z) such that x;z v z0 and v0 v frz;z0 � flx;z(~v). then frx;z0 � flz;z0(v0) v~v v �v.if there is some (�u; �v) 2 tr(f) with �u v x and fl�u;x(�v) = frx;z0 � flz;z0(v0), itfollows that �u;z v z0. moreover flz;z0(v0) v flx;z0(~v). since the embeddings arerigid, we therefore obtain thatfl�u;z0(�v) = flx;z0 � fl�u;x(�v) = flx;z0 � frx;z0 � flz;z0(v0) = flz;z0(v0):hence flz;z0(v0) 2 fl�u;z0(f(�u)). as �v = fr�u;z0 � flz;z0(v0), we have that (�u;fr�u;z0 �flz;z0(v0)) 2 tr(f). but tr(f) is contained in x.in the opposite case we obtain that (x;frx;z0 � flz;z0(v0)) 2 x n tr(f). notehere that we have already seen that flz;z0(v0) v flx;z0(~v), which implies thatflz;z0(v0) 2 flx;z0(f(x)).(3) let (u0;v0);(u1;v1) 2 x such that u0;u1 v z and flu0;z(v0) = flu1;z(v1),for some z 2 d. the case that both pairs are contained in tr(f) is obvious.without restriction let (u0;v0) 2 x n tr(f). then u0 = x. if (u1;v1) 2 tr(f)we obtain with claim 3.13 that u1 v x, contradicting the fact that (u0;v0) 2x n tr(f). hence (u1;v1) 2 x n tr(f), which means that also u1 = x. �as a special case we obtain that the di-domain of stable maps from a di-domain to a regular di-domain is regular again [9].theorem 3.14. the category of regular di-domains and stable maps is carte-sian closed. 4. strongly regular scott domainsin this section we consider scott domains which satisfy a strengthened reg-ularity condition. 144 dieter spreende�nition 4.1. a domain d is strongly regular, if every �nite union of basicopen sets "fzg with z 2 k(d) is regular open.note that in the scott topology the �nite unions of basic opens are just thecompact open subsets of the domain. by lemma 2.3 every strongly regulardomain is regular. among the examples given in section 2 only the domainsa? generated by �nite nonempty sets a are not strongly regular.the following characterization will turn out to be useful in the remainder ofthis section. we call a subset s of k(d) subbasis, if every compact element ofd is the least upper bound of a �nite subset of s.lemma 4.2. let s be a subbasis of a domain d. then the following threestatements are equivalent:(1) d is strongly regular.(2) for any subset x of s and all v 2 k(d) the next condition holds:(8u 2 x)u 6v v ) (9z 2 k(d))[v v z ^ (8u 2 x)u "� z]:(3) for any �nite subset x of s, sf"fzg j z 2 x g is regular open.proof. in order to see that (1) implies (2), let x be a �nite subset of s andv 2 k(d) such that for all u 2 x, u 6v v. since sf"fug j u 2 x g is regularopen, it follows that "fvg intersects ext(sf"fug j u 2 x g). hence, there issome z 2 k(d) such that v v z and for all u 2 x, u "� z.for the proof that (3) follows from (2), let x be a �nite subset of s and setu = sf"fug j u 2 x g. moreover, assume that there is some v 2 int(cl(u))nu.then we have for all u 2 x that u 6v v. because of (2) there exists some z 2k(d) such that v v z and for all u 2 x, z "� u. it follows that z 2 int(cl(u)).hence z 2 cl(u), which implies that "fzg intersects u. thus, for some u 2 x,u and z have a common upper bound. this contradicts what has been saidabout z before.it remains to show that (1) is a consequence of (3). let y be �nite subsetof k(d). then for any y 2 y there is some �nite subset xy of s such y is theleast upper bound of xy. thus[f"fyg j y 2 y g = [ftf"fug j u 2 xy g j y 2 y g:by distributivity it follows that sf"fyg j y 2 y g is a �nite intersection over�nite unions of basic open sets "fzg with z 2 s. since every such �nite unionis regular open by (3), and the class of regular open sets is closed under �niteintersections, we obtain that sf"fyg j y 2 y g is regular open. �it follows that a domain is strongly regular, exactly if every compact elementthat does not extend �nitely many given compact elements ui can be extendedto a compact element which contains additional information, contradictory tothe information coded into the ui.we will now consider the dependent product of a continuous family of scottdomains. let to this end d be scott domain and f : d ! sdep a continuousfunctor. on domains witnessing increase in information 145de�ne �df to be the set of all continuous sections of f , and order �dfpointwise, that is, letf vp g , (8x 2 d)f(x) vf(x) g(x):then �df with the pointwise order is the dependent product of f over d.as will follow from the next theorem the sections (u & v) with u 2 k(d)and v 2 k(f(u)) form a subbasis of �df . here (u & v) is de�ned by(u & v)(x) = (flu;x(v) if u v x,?f(x) otherwise.the following result has been established in [5, 10].theorem 4.3. let d be a scott domain and f : d ! sdep a continuous func-tor. then �df is a scott domain. the compact elements are exactly the leastupper bounds of �nite consistent sets of sections of the form (u & v).let srsdep be the category of strongly regular scott domains and embed-ding/projections, let d be strongly regular and f : d ! srsdep. we shallshow next that �df is also strongly regular. we do this by using an idea ofu. berger, to which the next lemma is due as well.lemma 4.4. let d be strongly regular and u0; : : : ;un�1;v0; : : : ;vm�1 2 k(d).moreover, let all vj be pairwise distinct. then there exist z0; : : : , zm�1 2 k(d)with the following properties:(1) (8j < m)vj v zj.(2) (8j < m)(8i < n)[ui 6v vj ) ui "� zj].(3) (8j1;j2 < m)[j1 6= j2 ) zj1 "� zj2].proof. the lemma will be proved by induction on m. the case m = 0 is trivial.therefore, let m > 0. moreover, without restriction, let vm�1 be minimalamong v0; : : : ;vm�1. setx = fy 2 fu0; : : : ;un�1;v0; : : : ;vm�2g j y 6v vm�1 g:since d is strongly regular, there is some zm�1 2 k(d) such that vm�1 v zm�1and y "� zm�1, for all y 2 x. thus properties (1) and (2) hold for j = m � 1.because vm�1 is minimal and the vj are pairwise distinct, we moreover havethat vj "� zm�1, for j < m � 1.now, by applying the induction hypothesis to u0; : : : ;un�1 and v0; : : : , vm�2we obtain that there are z0; : : : ;zm�2 2 k(d) such that properties (1)-(3) hold.because of our choice of zm�1 (1) and (2) then also hold for z0; : : : ;zm�1. forproperty (3) we only have to show that for all j < m � 1, zj "� zm�1. assumethat zj " zm�1, for some j < m�1. since vj v zj, it then follows that vj " zm�1as well, which is impossible by our choice of zm�1. �theorem 4.5. let d be a strongly regular scott domain and f : d ! srsdepa continuous functor. then �df is strongly regular. 146 dieter spreenproof. we verify condition (2) in lemma 4.2 for the subbasiss = f(u & y) j u 2 k(d) ^ y 2 k(f(u))g:let to this end gi;a = (ui & yai ) 2 s (i < n, a < bi) such that u0; : : : ;un�1 arepairwise distinct, let f = ff(vj & zj) j j 2 j g 2 k(�df), and assume thatgi;a 6vp f, for i < n and a < bi. we have to construct a section h 2 k(�df)with f vp h and h "� gi;a, for i < n and a < bi.obviously, gi;a 6vp f if and only if yai 6v f(ui). since every domain f(ui)is strongly regular, there is some ~yi 2 k(f(ui)), for each i < n, such thatf(ui) v ~yi and yai "� ~yi, for all a < bi. because d is strongly regular too, itfollows with the preceding lemma that there are �u0; : : : ; �un�1 2 k(d) such thatthe following properties hold:(1) (8i < n)ui v �ui.(2) (8i < n)(8j 2 j)[vj 6v ui ) vj "� �ui].(3) (8i1; i2 < n)[i1 6= i2 ) �ui1 "� �ui2].for i < n let �yi = flui;�ui(~yi). we shall show now that the set f(�ui & �yi) j i 1 there is no canonical “standard adjacency” to use in zn which corresponds naturally to the standard topology of rn. in the case of z2, for example, at least two different adjacency relations seem reasonable: we can view z2 as a rectangular lattice connected by the coordinate grid, so that each point is adjacent to 4 neighbors; or we can additionally allow diagonal adjacencies so that each point is adjacent to 8 neighbors. this is formalized in the following definition from [7] (though these adjacencies had been studied for many years earlier): definition 1.2. let k,n be positive integers with k ≤ n. then define an adjacency relation ck on z n as follows: two points x,y ∈ zn are ck-adjacent if their coordinates differ by at most 1 in at most k positions, and are equal in all other positions. we will also make use of the notion of connectedness. for a,b ∈ z and a < b, let [a,b]z denote the set {a,a + 1, . . . ,b}. this set is called the digital © agt, upv, 2021 appl. gen. topol. 22, no. 2 224 digital homotopy and homology interval from a to b. given two points x,y ∈ (x,κ), a κ-path from x to y is a (c1,κ)-continuous function p : [0,n]z → x with p(0) = x and p(n) = y. when the adjacency relation is understood, a κ-path is called simply a path. a digital image (x,κ) is connected when any two points of x can be joined by a path. a connected component of x is a maximal connected subset of x. the following is the standard definition of digital homotopy. for a,b ∈ z with a ≤ b, let [a,b]z be the digital interval [a,b]z = {a,a + 1, . . . ,b}. definition 1.3 ([2]). let (x,κ) and (y,λ) be digital images. let f,g : x → y be (κ,λ)-continuous functions. suppose there is a positive integer m and a function h : x × [0,k]z → y such that: • for all x ∈ x, h(x,0) = f(x) and h(x,k) = g(x); • for all x ∈ x, the induced function hx : [0,k]z → y defined by hx(t) = h(x,t) for all t ∈ [0,k]z is (c1,λ)−continuous. • for all t ∈ [0,k]z, the induced function ht : x → y defined by ht(x) = h(x,t) for all x ∈ x is (κ,λ)−continuous. then h is a [digital] homotopy between f and g, and f and g are [digitally] homotopic, denoted f ≃ g. if k = 1, then f and g are homotopic in one step. homotopy in one step can easily be expressed in terms of individual adjacencies: proposition 1.4. let f,g : x → y be continuous. then f is homotopic to g in one step if and only if for every x ∈ x, we have f(x) g(x). proof. assume that f is homotopic to g in one step. the homotopy is simply defined by h(x,0) = f(x) and h(x,1) = g(x). then by continuity in the second coordinate of h, we have f(x) = h(x,0) h(x,1) = g(x) as desired. now assume that f(x) g(x) for each x. define h : x × [0,1]z → y by h(x,0) = f(x) and h(x,1) = g(x). clearly h(x,t) is continuous in x for each fixed t, since f and g are continuous. also h(x,t) is continuous in t for fixed x because h(x,0) = f(x) g(x) = h(x,1). thus h is a one step homotopy from f to g as desired. � definition 1.3 is inspired by the concept of homotopy from classical topology, but the classical definition is simpler because it can use the product topology. in classical topology the continuity of the two types of “induced function” is expressed simply by saying that h : x × [0,1] → y is continuous with respect to the product topology on x × [0,1]. given two digital images a and b, we can consider the product a × b as a digital image, but there are several choices for the adjacency to be used. the most natural adjacencies are the normal product adjacencies, which were © agt, upv, 2021 appl. gen. topol. 22, no. 2 225 p. c. staecker defined by boxer in [3]. this generalizes an earlier product construction from [7], which is equivalent to np2. definition 1.5 ([3]). for each i ∈ {1, . . . ,n}, let (xi,κi) be digital images. then for some u ∈ {1, . . . ,n}, the normal product adjacency npu(κ1, . . . ,κn) is the adjacency relation on ∏n i=1 xi defined by: (x1, . . . ,xn) and (x ′ 1, . . . ,x ′ n) are adjacent if and only if their coordinates are adjacent in at most u positions, and equal in all other positions. when the underlying adjacencies are clear, we abbreviate npu(κ1, . . . ,κn) as simply npu. the normal product adjacency is inspired by the various standard adjacencies typically used on zn. on z1, as mentioned above, the standard adjacency is c1. viewing z 2 as the product z2 = z × z, it is easy to see that 4-adjacency is the same as np1(c1,c1), and 8-adjacency is np2(c1,c1). the typical adjacencies used in z3 are 6-, 18-, and 26-adjacency, depending on which types of diagonal adjacencies are allowed. these adjacencies are exactly np1(c1,c1,c1), np2(c1,c1,c1), and np3(c1,c1,c1). boxer showed that the definition of homotopy can be rephrased in terms of an np1 product adjacency: theorem 1.6 ([3, theorem 3.6]). let (x,κ) and (y,λ) be digital images. then h : x × [0,k]z → y is a homotopy if and only if h is (np1(κ,c1),λ)continuous. once we have a notion of homotopy, it is natural to define homotopy equivalence: definition 1.7. digital images x and y are homotopy equivalent when there exist continuous functions f : x → y and g : y → x with g ◦ f ≃ idx and f ◦ g ≃ idy , where idx and idy denote the identity functions on x and y . in this case f and g are called homotopy equivalences. the structure of this paper is as follows: in section 2 we define strong homotopy and give some of its basic properties. in section 3 we show that a homotopy is strong if and only if it can be made “punctuated”, that is, changing by only one point at a time. in section 4 we describe three different digital homology theories existing in the literature. in section 5 we describe a new theory, defined only for images in zn with c1-adjacency, which we call c1-cubical homology, similar to a construction described in [11]. in section 6 we show that the simplicial and singular theories are isomorphic. in section 7 we prove homotopy and strong homotopy invariance properties for the various theories. in section 8 we describe some relationships between the simplicial and cubical theories, and in section 9 we describe some specific examples. the author would like to thank samira jamil and danish ali for helpful conversations concerning early drafts of this paper. © agt, upv, 2021 appl. gen. topol. 22, no. 2 226 digital homotopy and homology 2. strong homotopy theorem 1.6 states that a homotopy is a function h : x ×[0,k]z → y which is continuous when we use the np1 product adjacency in the domain. we will explore the following definition, which simply uses np2 in place of np1. as we will see this imposes extra restrictions on the homotopy h. definition 2.1. let (x,κ) and (y,λ) be digital images. we say h : x × [0,k]z → y is a strong homotopy when h is (np2(κ,c1),λ)-continuous. if there is a strong homotopy h between f and g, we say f and g are strongly homotopic, and we write f ∼= g. if additionally k = 1, we say f and g are strongly homotopic in one step. since we have exchanged np1 for np2 in the definition above, we may say informally that strong homotopy and digital homotopy provide two different but equally natural “digitizations” of the classical topological idea of homotopy. the difference between homotopy and strong homotopy of continuous functions is analogous to the difference between 4-adjacency and 8-adjacency of points in the plane. the strong homotopy relation matches one used in recent work [13] to build a digital homotopy theory in many respect matching the classical one. it is clear from the definition of the normal product adjacency that if two points are np1-adjacent, then they are np2-adjacent. thus any function f : a × b → c which is continuous when using np2 in the domain will automatically be continuous when using np1 in the domain. thus we obtain the following, which justifies the use of the word “strong.” theorem 2.2. let h : x × [0,k]z → y be a strong homotopy. then h is a homotopy. a standard argument shows that strong homotopy is an equivalence relation. theorem 2.3. strong homotopy is an equivalence relation. strong homotopy in one step can be expressed in terms of adjacencies as in proposition 1.4. theorem 2.4. let f,g : x → y be continuous. then f is strongly homotopic to g in one step if and only if for every x,x′ ∈ x with x ↔ x′, we have f(x) g(x′). proof. first assume that f is homotopic to g in one step. the homotopy is simply defined by h(x,0) = f(x) and h(x,1) = g(x). then if x ↔ x′, we will have (x,0) ↔np2 (x ′,1), and thus since h is np2-continuous we have f(x) = h(x,0) h(x′,1) = g(x′) as desired. now assume that f(x) g(x′) for each x. define h : x × [0,1]z → y by h(x,0) = f(x) and h(x,1) = g(x), and we must show that h is np2continuous. take (x,t),(x′, t′) ∈ x × [0,1]z with (x,t) ↔np2 (x ′, t′), and we will show that h(x,t) h(x′, t′). we have a few cases based on the values of t,t′ ∈ {0,1}. © agt, upv, 2021 appl. gen. topol. 22, no. 2 227 p. c. staecker if t = t′, without loss of generality say t = t′ = 0. since (x,t) ↔np2 (x ′, t′), we have x x′, and so we have h(x,t) = h(x,0) = f(x) f(x′) = h(x′,0) = h(x′, t′) since f is continuous. thus h(x,t) h(x′, t′) as desired. finally, if t 6= t′, without loss of generality assume t = 0 and t′ = 1. since (x,t) ↔np2 (x ′, t′), we have x x′, and so we have h(x,t) = h(x,0) = f(x) g(x′) = h(x′,1) = h(x′, t′) and so h(x,t) h(x′, t′) as desired. � by theorem 2.2, if f ∼= g then automatically we have f ≃ g. the converse is not true, however, as the following example shows. one important source of examples in the study of digital images is the digital cycle cn = {c0, . . . ,cn−1}, with adjacency given by ci ↔ ci+1 for each i, where for convenience we always read the subscripts modulo n. thus cn is a digital image of n points which is in many ways topologically analogous to the circle. example 2.5. it is well known that all selfmaps on c4 are homotopic to one another. but we will show that the identity map idc4 : c4 → c4 is not strongly homotopic to any map f with #f(c4) < 4. it will suffice to show that idc4 is not strongly homotopic in 1 step to any such map. without loss of generality assume that f(c4) ⊆ {c0,c1,c2}, and assume for the sake of a contradiction that f is strongly homotopic in one step to idc4. then by theorem 2.4, since c2 ↔ c3 and c0 ↔ c3, we will have c2 f(c3) and also c0 f(c3). thus f(c3) is adjacent to both c0 and c2, but cannot equal c3 since c3 6∈ f(c4). we conclude that f(c3) = c1. by theorem 1.4, this contradicts the fact that f is homotopic to the identity in 1 step. 3. punctuated homotopy given a homotopy h(x,t), we say that h is punctuated if, for each t, there is some xt such that h(x,t) = h(x,t + 1) for all x 6= xt. that is, from each stage of the homotopy to the next, the induced map ht(x) is changing by at most one point at a time. theorem 3.1. any punctuated homotopy is a strong homotopy. proof. let h be a punctuated homotopy, and let (x,t) ↔np2 (x ′, t′). we must show that h(x,t) h(x′, t′). since (x,t) ↔np2 (x ′, t′) we have x x′ and t t′. if t = t′, then we have (x,t) ↔np1 (x ′, t′) and so h(x,t) h(x′, t′) since h is a homotopy. it remains to consider when t ↔ t′ and t 6= t′. in this case, without loss of generality assume t′ = t + 1. since h is a punctuated homotopy, there is some xt such that h(x,t) = h(x,t + 1) for all x 6= xt. when x 6= xt, we have h(x,t) = h(x,t + 1) h(x′, t + 1) = h(x′, t′) since h is a homotopy. similarly we have h(x,t) h(x′, t′) when x′ 6= xt. © agt, upv, 2021 appl. gen. topol. 22, no. 2 228 digital homotopy and homology thus it remains only to consider when x = x′ = xt. that is, we must show that h(xt, t) h(xt, t ′) = h(xt, t + 1), and this is true because h is a homotopy. � we also have a sort of converse to the above. while not every strong homotopy is punctuated, any two strongly homotopic maps can be connected by a punctuated homotopy, provided that the domain is finite. theorem 3.2. let x be a finite digital image, and let f,g : x → y be strongly homotopic. then f and g are homotopic by a punctuated homotopy. proof. by induction, it suffices to show that if f and g are strongly homotopic in one step, then they are homotopic by a punctuated homotopy. enumerate the points of x as x = {x0, . . . ,xn}, and define h : x × [0,n + 1]z → y by: h(xi, t) = { f(xi) if i ≥ t, g(xi) if i < t. then h moves one point at a time, so we need only to show that it has the appropriate continuity properties to be a homotopy. first we show that h(x,t) is continuous in x for fixed t. take x ↔ x′, then h(x,t) ∈ {f(x),g(x)} and h(x′, t) ∈ {f(x′),g(x′)}. since f and g are homotopic in one step we have f(x) f(x′) and g(x) g(x′), and since f and g are strongly homotopic in one step we have f(x) g(x′) and g(x) f(x′). thus in any case we have h(x,t) h(x′, t) as desired. now we show that h(x,t) is continuous in t for fixed x. it suffices to show that h(x,t) h(x,t + 1) for any t. we have h(x,t) ∈ {f(x),g(x)} and also h(x,t + 1) ∈ {f(x),g(x)}. since f and g are homotopic in one step we have f(x) g(x), and thus h(x,t) h(x,t + 1) as desired. � the finiteness assumption above is necessary, as the following example shows. example 3.3. let x = z × {0,1} ⊂ z2, with 8-adjacency, and let f(x,y) = (x,0). then f is strongly homotopic to idx in one step. but f(x,y) and id(x,y) differ for infinitely many values of (x,y) ∈ x. since a punctuated homotopy has finite time interval, and can only change one value at a time, there can be no punctuated homotopy from f to idx. combining the two theorems above gives us a nice characterization of strong homotopy. corollary 3.4. if x is finite, two continuous maps f,g : x → y are strongly homotopic if and only if they are homotopic by a punctuated homotopy. as an application, we show that the identity map on the n-cycle for n ≥ 4 is not strongly homotopic to any other map: example 3.5. let n ≥ 4, and assume for the sake of a contradiction that there is some continuous f : cn → cn with f ∼= idcn and f 6= idcn. without loss of generality, assume that the homotopy from f to idcn is punctuated and © agt, upv, 2021 appl. gen. topol. 22, no. 2 229 p. c. staecker in one step. since the homotopy is punctuated, f moves one point, so without loss of generality we may assume that f(c0) = c1 and f(ci) = ci for all i 6= 0. this contradicts the continuity of f, however, since we will have c0 ↔ cn−1 but c1 = f(c0) 6f(cn−1) = cn−1 since n ≥ 4. 4. digital homology theories in the literature digital topological invariants are typically inspired by classical topology, though most are not literally topological in nature. for example, when x and y are digital images, a digitally continuous function f : x → y is not actually continuous in the classical sense with respect to any topologies on x and y . the digital fundamental group π1(x) is not actually the fundamental group of x with respect to any topology on x, etc. digital homology, however, does fit neatly into the classical theory of homological algebra. each of the homology theories described in [4, 12, 9] is indeed a homology theory of a classical chain complex. though it is not always done in these references, we will make use of results from classical homological algebra whenever possible to avoid the need for specialized definitions and proofs of basic results. we will review the basic homological algebra that will be useful, see e.g. [8]. a chain complex is a sequence of abelian groups c0,c1, . . . and homomorphisms ∂q : cq → cq−1 satisfying ∂q−1 ◦ ∂q = 0 for all q. given some chain complex c = (cq,∂q), for each q we define the cycle and boundary subgroups zq and bq of cq as: zq = ker∂q and bq = im∂p+1. the dimension q homology group of the chain complex is defined as hq = zq/bq. when we are discussing the homology groups of various different chain complexes, we will write hq(c) for the dimension q homology of the chain complex c. given two chain complexes a = (aq,∂ a q ) and b = (bq,∂ b q ), a sequence of homomorphisms fq : aq → bq is a chain map from a to b when fq−1 ◦ ∂ a q = ∂bq ◦ fq for each q. every such chain map induces a well-defined sequence of homomorphisms f∗,q : hq(a) → hq(b). furthermore, this correspondence is functorial in the sense that (f ◦ g)∗,q = f∗,q ◦ g∗,q, and the induced homomorphism of the identity function is the identity homomorphism. when the dimensions are clear, we will omit the subscript q. given three chain complexes a, b, c, and an exact sequence of chain maps: 0 → a f −→ b g −→ c → 0, for each q there is a connecting homomorphism δq : hq(c) → hq−1(a) such that the following sequence is exact: (4.1) · · · → hq+1(c) δq+1 −−−→ hq(a) f∗,q −−→ hq(b) g∗,q −−→ hq(c) δq −→ hq−1(a) → . . . the “long exact sequence” above is the fundamental tool of relative homology theory. © agt, upv, 2021 appl. gen. topol. 22, no. 2 230 digital homotopy and homology 4.1. simplicial homology. digital simplicial homology theory was first defined by arslan, karaca, and öztel in [1], in turkish. this material was extended and published in english in [4]. we review the definitions as presented in [4]. for a digital image x and some q ≥ 0, a q-simplex is defined to be any set of q+1 mutually adjacent points of x. for some ordered list of mutually adjacent points x0, . . . ,xq, the associated ordered q-simplex is denoted 〈x0, . . . ,xq〉. the chain group cq(x) is defined to be the abelian group generated by the set of all ordered q-simplices, where if ρ : {0, . . . ,q} → {0, . . . ,q} is a permutation, then in cq(x) we identify (4.2) 〈x0, . . . ,xq〉 = (−1) ρ〈xρ(0), . . . ,xρ(q)〉, where (−1)ρ = 1 when ρ is an even permutation, and (−1)ρ = −1 when ρ is an odd permutation. the boundary homomorphism ∂q : cq(x) → cq−1(x) is the homomorphism induced by defining: (4.3) ∂q(〈x0, . . . ,xq〉) = q∑ i=0 (−1)i〈x0, . . . , x̂i, . . .xq〉, where x̂i indicates omission of the xi coordinate. it can be verified that ∂q−1 ◦ ∂q = 0, and so (cq(x),∂q) forms a chain complex, and the dimension q homology group hq(x) = zq(x)/bq(x) is defined to be the homology of this chain complex. any continuous function f : x → y induces a homomorphism f#,q : cq(x) → cq(y ) defined by f#,q(〈x0, . . . ,xq〉) = 〈f(x0), . . . ,f(xq)〉, where the right side is interpreted as 0 if the set {f(x0), . . . ,f(xq)} has cardinality less than q. when the value of q is understood, we simply write f#,q = f#. it is easy to check that this f#,q is a chain map, and thus induces a homomorphism f∗,q : hq(x) → hq(x). again, we typically write f∗,q = f∗ when the q is understood. we will remark that the constructions above match exactly the homology of the clique complex of x when viewed as a graph. the clique complex is the simplicial complex built from the complete subgraphs of a given graph, and the homology of this simplicial complex is the same as the digital homology defined above. the free mathematics software sagemath has built-in functions to compute the clique complex of a graph, and further to compute the homology of any finite complex. thus it is easy to implement algorithms to compute the simplicial homology groups of a digital image. source code for computing simplicial homology groups is available at the author’s website for experimentation.1 these definitions and results are all exactly as expected from the classical homology theory of a simplicial complex. as an example, we compute the 1http://faculty.fairfield.edu/cstaecker © agt, upv, 2021 appl. gen. topol. 22, no. 2 231 http://faculty.fairfield.edu/cstaecker p. c. staecker homology groups of the cycle cn for n ≥ 4. the case n = 4 appears as theorem 3.17 of [4]. theorem 4.1. if n ≥ 4, we have: hq(cn) = { z if q ∈ {0,1}, 0 if q > 1. proof. first we prove the case q = 0. the chain group c0(cn) is generated by n different 0-simplices 〈c0〉, . . . ,〈cn−1〉. since ∂0 is a trivial homomorphism, we have z0(cn) = c0(cn). note that for each i, we have 〈ci〉 = (〈ci〉 − 〈ci+1〉) + 〈ci+1〉 = ∂〈ci+1,ci〉 + 〈ci+1〉, and thus 〈ci〉 − 〈ci+1〉 ∈ b0(cn). thus 〈ci〉 and 〈ci+1〉 are equal in h0(cn) for every i. that is, h0(cn) is the group generated by 〈c0〉, and so h0(cn) = z as desired. now for q = 1, first we note that there are no 2-simplices in cn (because n ≥ 4), so c2(cn) is trivial, and thus b1(cn) is trivial. thus h1(cn) will be isomorphic to z1(cn). to determine z1(cn), we must determine which α ∈ c1(cn) satisfy ∂α = 0. any α ∈ c1(cn) can be expressed as: α = w1〈c0,c1〉 + · · · + wn〈cn−1,c0〉 for wi ∈ z, and then ∂α = 0 if and only if: 0 = ∂α = w1(〈c1〉 − 〈c0〉) + · · · + wn(〈c0〉 − 〈cn−1〉) = (wn − w1)〈c0〉 + (w1 − w2)〈c1〉 + · · · + (wn−1 − wn)〈cn−1〉 and thus we have w1 = w2 = · · · = wn since the 〈ci〉 are linearly independent in c1(cn). then we have shown that h1(cn) = z1(cn) is generated by the single element σ = n−1∑ i=0 〈ci,ci+1〉, and thus h1(cn) = z. for q > 1, there are no q-simplices and so cq(cn) is trivial, and thus hq(cn) is trivial. � in the case of cn, we can make a full computation of the induced homomorphisms for any selfmap, using results from [5]. let c : cn → cn be the constant map c(ci) = c0, let l : cn → cn be the “flip map” l(ci) = c−i, and for some integer d, let rd : cn → cn be the rotation rd(ci) = ci+d. theorem 9.3 of [5] shows that there are exactly 3 homotopy classes of selfmaps on cn: any map f : cn → cn is either strongly homotopic to a constant, or is homotopic to the identity and equals rd for some d, or is homotopic to the flip map and equals rd ◦ l for some d. (the strongness of the homotopy to the constant was not mentioned in [5], but the homotopy demonstrated in the proof in that paper is easily made punctuated and therefore strong.) © agt, upv, 2021 appl. gen. topol. 22, no. 2 232 digital homotopy and homology theorem 4.2. let n > 4, and let f : cn → cn be continuous. then for all q > 1, the induced homomorphism f∗,q : hq(cn) → hq(cn) is trivial, for q = 0 the induced homomorphism f∗,0 = id, and for q = 1 we have: f∗,1 =    id if f ≃ idcn, − id if f ≃ l, 0 if f ≃ c. proof. for q > 1 we have already seen that hq(cn) is a trivial group, so we will have f∗ = 0 automatically. when f is homotopic to a constant, as mentioned above, in fact f ∼= c and thus f∗ = c∗ for all q, and so f∗,0 is the identity and f∗,q is trivial for q > 0. now we consider when f is homotopic to the identity or the flip map, for q ∈ {0,1}. first we consider q = 0 and f ≃ idcn. since h0(cn) is generated by 〈c0〉, it suffices to show that f∗(〈c0〉) = 〈c0〉. let d be some integer with f = rd, and we have: f∗(〈c0〉) = 〈f(c0)〉 = 〈cd〉 = 〈c0〉 as desired. exactly the same argument applies for f ≃ l, since we will still have 〈f(c0)〉 = 〈cd〉 for some d. now for q = 1, and f ≃ idcn we must show f∗(σ) = σ, where σ =∑n−1 i=0 〈ci,ci+1〉 as in the proof of theorem 4.1. let f = rd, and we have: f(σ) = n−1∑ i=0 〈f(ci),f(ci+1)〉 = n−1∑ i=0 〈ci+d,ci+1+d〉 = σ as desired. finally we consider q = 1 and f ≃ l, and we must show f∗(σ) = −σ. let f = rd ◦ l, so f(ci) = cd−i, and we have: f(σ) = n−1∑ i=0 〈f(ci),f(ci+1)〉 = n−1∑ i=0 〈cd−i,cd−i−1〉 = n−1∑ i=0 〈c−i+1,c−i〉 = n−1∑ i=0 〈ci+1,ci〉 = n−1∑ i=0 −〈ci,ci+1〉 = −σ as desired. � the three cases of theorem 4.2 suffice to compute f∗ for any selfmap of cn, and we note that in each of the three homotopy classes, the set of induced homomorphisms is different. we obtain a sort of hopf theorem for digital cycles: corollary 4.3. let n > 4, and let f,g : cn → cn be continuous. then f∗,q = g∗,q for each q if and only if f ≃ g. one major difference between the digital theory and classical homology is that the induced homomorphism f∗ is not always a digital homotopy invariant. © agt, upv, 2021 appl. gen. topol. 22, no. 2 233 p. c. staecker example 4.4. by theorem 4.1, the homology group h1(c4) is isomorphic to z. because all maps on c4 are homotopic, the identity map is homotopic to a constant map c. but id∗ : h1(c4) → h1(c4) is the identity homomorphism of z, while c∗ : h1(c4) → h1(c4) is the trivial homomorphism. thus id ≃ c but id∗ 6= c∗. the lack of a homotopy-invariant induced homorphism is a major deficiency in the homology theory of digital images. lacking this homotopy invariance, the homology groups are not well-behaved with respect to typical topological constructions. for example two homotopy equivalent digital images may have different homology groups. as a consequence the digital euler characteristic is not a digital homotopy type invariant. example 7.2 of [6] shows that the hurewicz theorem also fails: that is, that h1(x) may not be isomorphic to the abelianization of the fundamental group of x as defined in [2]. the homology group in dimension zero is easy to predict: theorem 4.5. let x be any digital image with d connected components. then h0(x) ∼= z d. proof. the chain group c0(x) (which equals the group z0(x) of 0-cycles) has basis given by the points of x. it is easy to see that two points x,y ∈ z0(x) are homologous if and only if x and y are in the same connected component of x. � the simplicial homology is also easy to compute when x consists of finitely many isolated points, that is, points which are not adjacent to any other points. theorem 4.6. let x be any digital image consisting of k isolated points. then h0(x) ∼= z k and hq(x) = 0 for q > 0. proof. the statement concerning h0(x) follows immediately from 4.5. since x has no adjacencies, it contains no q-simplices when q > 0. thus hq(x) = 0 when q > 0 as desired. � 4.2. singular homology. singular homology for digital images was defined by d.w. lee in [12]. we will review lee’s definitions, using some different notations to fit more cleanly with the other homology theories. for any natural number q, let ∆q be the standard q-simplex, the digital image consisting of q + 1 mutually adjacent points. viewed as a graph, ∆q is the complete graph of q + 1 vertices. the points of ∆q will be labeled and ordered as ∆q = (e0, . . . ,eq). definition 4.7. let x be a digital image. a singular q-simplex in x is a continuous function φ : ∆q → x. we will write such a singular q-simplex as the ordered list [φ(e0), . . . ,φ(eq)]. for any q ≥ 0, the group of singular q-chains, denoted čq(x), is the free abelian group whose basis is the set of all singular q-simplices of x. © agt, upv, 2021 appl. gen. topol. 22, no. 2 234 digital homotopy and homology the singular boundary operator ∂q : čq(x) → čq−1(x) is defined as follows: (4.4) ∂q[x0, . . . ,xq] = q∑ i=0 (−1)i[x0, . . . , x̂i, . . . ,xq], where as usual x̂i denotes omission of the ith element. when φ is a q-simplex, ∂qφ will be a singular (q−1)-chain. as before, we will often omit the subscript q. note that we are using the same notation to denote the boundary operators in both simplicial and singular homology. in practice this will not cause confusion. the singular q-simplex [x0, . . . ,xq] is very similar to the ordered q-simplex 〈x0, . . . ,xq〉. the main difference algebraically is that, in the singular chain group, we do not identify permutations of the listings. for example we have 〈x0,x1〉 = −〈x1,x0〉 in c1(x), but in č1(x) the basis elements [x0,x1] and [x1,x0] are linearly independent. we also will always have 〈x,x〉 = −〈x,x〉 = 0 in c1(x), while [x,x] will be nontrivial in č1(x). this means that in particular čq(x) has nonzero elements which, when written as lists of points, include repetitions. such elements are always 0 in cq(x) because of (4.2). theorem 3.9 of [12] shows that ∂q−1 ◦ ∂q = 0, and thus (čq(x),∂q) is a chain complex, and the dimension q singular homology group is defined as ȟq(x) = žq(x)/b̌q(x). if f : x → y is continuous, then there is a homomorphism f# : čq(x) → čq(y ) defined on singular chains by f#(φ) = f ◦ φ. this is easily shown to be a chain map, and thus we obtain the induced homomorphism on singular homology f∗ : ȟq(x) → ȟq(y ). we will require an analogue of theorem 4.5 for singular homology. the proof is the same as that of theorem 4.5. theorem 4.8. let x be any digital image with d connected components. then ȟ0(x) ∼= z d. lee’s work provides an analogue of theorem 4.6 for singular homology. the following is a consequence of theorems 3.16 and 3.20 of [12]: theorem 4.9. let x be a digital image which consists of k isolated points. then ȟ0(x) ∼= z k and ȟq(x) = 0 for q > 0. 4.3. cubical homology. cubical homology was introduced by jamil & ali in [9], with definitions mimicking the classical cubical homology as presented in [14]. we will review the definition and basic results. let i = [0,1]z = {0,1}, and we consider i n ⊂ zn as a digital image with c1-adjacency for each n ≥ 1. when n = 0, we define i 0 to be a single point. definition 4.10. let (x,κ) be a digital image. then a q-cube in x is a (c1,κ)-continuous function σ : iq → x. a q-cube is degenerate if there is some i such that the function σ(t1, . . . , tq) does not depend on the coordinate ti. let qq(x) be the free abelian group whose basis is the set of all q-cubes in x, and let dq(x) be the subgroup generated © agt, upv, 2021 appl. gen. topol. 22, no. 2 235 p. c. staecker by the degenerate q-cubes. then the group of cubical q-chains, denoted c̄q(x), is the quotient c̄q(x) = qq(x)/dq(x). the cubical boundary operator is defined in terms of cubical face operators. for some q-cube σ and some i ∈ {1, . . . ,q}, define (q − 1)-cubes aiσ and biσ as: (aiσ)(t1, . . . , tq−1) = σ(t1, . . . , ti−1,0, ti, . . . , tq−1), (biσ)(t1, . . . , tq−1) = σ(t1, . . . , ti−1,1, ti, . . . , tq−1), these ai and bi give the “front face” and “back face” of the cube in each of its q dimensions. the boundary operator ∂q : c̄q(x) → c̄q−1(x) is defined on cubes by the formula: (4.5) ∂q(σ) = q∑ i=1 (−1)i(aiσ − biσ), and extended to chains by linearity. we will sometimes omit the subscript q. a routine calculation shows that ∂q−1 ◦ ∂q = 0, and thus (c̄q(x),∂q) is a chain complex. we obtain cycle and boundary groups z̄q(x) and b̄q(x) and the homology groups h̄q(x) = z̄q(x)/b̄q(x). exactly as in the singular theory, if f : x → y is continuous, then f# : c̄q(x) → c̄q(y ) is defined on cubical chains by f#(σ) = f ◦σ, and this defines the induced homomorphism on cubical homology f∗ : h̄q(x) → h̄q(y ). the most important property of cubical homology which distinguishes it from simplicial homology is the following, which is theorem 3.7 of [9]: theorem 4.11 ([9, theorem 3.7]). let x and y be any digital images and f,g : x → y with induced homomorphisms f∗,g∗ : h̄q(x) → h̄q(x). if f ≃ g, then f∗ = g∗. immediate corollaries include: corollary 4.12 ([9, corollary 3.8]). if x and y are homotopy equivalent, then h̄q(x) ∼= h̄q(y ) for all q. corollary 4.13 ([9, example 3.9]). if x is contractible (i.e. x is homotopy equivalent to a point), then h̄q(x) = { z if q = 0, 0 if q 6= 0. jamil & ali also prove a hurewicz theorem, that h̄1(x) is isomorphic to the abelianization of the fundamental group of x. they also prove results concerning connected components and single points. the following theorems follow from [9, propositions 3.1, 3.2] and corollary 4.13. theorem 4.14. let x be a digital image with d connected components. then h̄0(x) ∼= z d. © agt, upv, 2021 appl. gen. topol. 22, no. 2 236 digital homotopy and homology theorem 4.15. let x be a digital image consisting of k isolated points. then h̄0(x) ∼= z k and h̄q(x) ∼= 0 for q > 0. 5. a new cubical homology theory for images with c1-adjacency ege & karaca in [11] describe another type of cubical homology theory based on classical constructions from [10]. their construction is not generally welldefined for any digital image, but only a digital image which is a “cubical set”, that is, a finite union of “elementary cubes.” we will describe the construction, sometimes using different terminology that is more convenient for making comparisons with the other theories in this paper. ege & karaca’s focus on “cubical sets” requires that the digital image x be a subset of zn, and that we always use c1 as the adjacency relation. this is a significant restriction, but still allows many useful examples and results. all of the following definitions appear in [11]: an elementary interval is a set of the form [a,a + 1]z = {a,a + 1} or [a,a]z = {a}. an elementary interval of 1 point is called degenerate, and one of 2 points is called nondegenerate. an elementary cube is any set: q = j1 × · · · × jq ⊂ z q where each ji is an elementary interval. the dimension of q is the number of nondegenerate factors. an elementary cube of dimension q will be called an elementary q-cube. when x ⊂ zn is a digital image with c1-adjacency, it has a unique maximal expression as a union of elementary cubes. for each q ≥ 0, let c̄c1q (x) be the free abelian group generated by the set of all elementary q-cubes in x. we will give a definition for a boundary operator which differs from the one used in [11], but is rephrased to more closely resemble the boundary operator from the cubical theory. define face operators ai and bi as follows: for an elementary cube q = j1 × · · · × jn, let: aiq = j1 × · · · × ji−1 × {minji} × ji+1 × · · · × jn, biq = j1 × · · · × ji−1 × {maxji} × ji+1 × · · · × jn. note that when ji is a degenerate interval, we have aiq = biq. when ji is nondegenerate, aiq and biq are distinct elementary cubes of dimension one less than the dimension of q. now we define the boundary operator: given an elementary q-cube q = j1 × · · ·×jn, let (j1, . . . ,jq) be the sequence of indices for which jji is nondegenerate. then we define: ∂qq = q∑ i=1 (−1)i(ajiq − bjiq). it can be verified that ∂q−1◦∂q = 0, and thus (c̄ c1 q (x),∂q) is a chain complex. the homology groups are h̄c1q (x) = z̄ c1 q (x)/b̄q(x). we immediately have c1 analogues of theorems 4.14 and 4.15. © agt, upv, 2021 appl. gen. topol. 22, no. 2 237 p. c. staecker theorem 5.1. let x ⊆ zn be a digital image with c1-adjacency having d components. then h̄c10 (x) ∼= zd. proof. the chain group c̄c10 (x) is generated by the points of x, and it is easy to see that two such points are homologous in h̄c10 (x) if and only if there is a path connecting them. (the two points will be the boundary of the chain formed by the path.) thus h̄c10 (x) has a generator for each component of x. � theorem 5.2. let x ⊆ zn be a digital image with c1-adjacency which consists of a set of k isolated points. then h̄c10 (x) ∼= zk and h̄c1q (x) ∼= 0 for q > 0. proof. the first part follows immediately from theorem 5.1. for the second part, observe that if x consists only of isolated points, then the chain group c̄c1q (x) is trivial for all q > 0. � karaca & ege’s presentation in [11] is different from our c1-cubical theory in some important ways. the theory in [11] starts with some digital image (x,κ) endowed with some specified cubical structure. then based on this structure, a homology theory is defined. the work in [11] gives no general system for defining a cubical structure on x, and thus the resulting homology groups are not intrinsic to the digital image (x,κ), but rather depend on the choice of cubical structure. for example the digital image x = [0,1]3 z ⊂ z3 is considered. there is an obvious cubical structure on x as a single elementary 3-cube together with its faces, and we would expect the cubical homology to be z in dimension 0 and trivial in all other dimensions. but instead the calculation in [11, theorem 4.5] gives z in dimensions 0 and 2 and trivial in all other dimensions. this is because the calculation in that theorem (without explicit mention) uses the cubical structure consisting of six 2-cubes together with their faces, but without the “solid” 3-cube. we will see in example 9.5 that h̄c1q (x) is indeed z in dimension 0, and trivial in other dimensions. the author has implemented an algorithm with the free mathematical package sagemath to compute the c1-cubical homology groups of any digital image. source code is available for experimentation at the author’s website.2 the induced homomorphism in this c1-cubical homology theory is nontrivial to develop (there was no effort to define the concept in [11]). given two digital images x and y , both with c1-adjacency, and some continuous f : x → y , we wish to define an induced homomorphism f̄# : c̄ c1 q (x) → c̄ c1 q (y ) which is a chain map. given an elementary q-cube q ⊂ x, we say f is an embedding on q if f(q) is an elementary q-cube in y . in this case let ǫf,q ∈ {−1,1} be the orientation with with f maps q onto f(q). (this orientation could be defined as the determinant of the affine linear map describing f’s restriction to q.) when f is not an embedding on q, we let ǫf,q = 0. 2http://faculty.fairfield.edu/cstaecker © agt, upv, 2021 appl. gen. topol. 22, no. 2 238 http://faculty.fairfield.edu/cstaecker digital homotopy and homology then we define f̄#(q) = ǫf,q(f(q)), where the right side is interpreted as the coefficient ǫf,q times the elementary q-cube f(q) ⊂ y . extending linearly gives the induced homomorphism f̄# : c̄ c1 q (x) → c̄ c1 q (y ). now to obtain an induced homomorphism in homology, we must show that f̄# is a chain map. we have been unable to prove this analytically, although in low dimensions we can show that f̄# is a chain map by computer enumerations. theorem 5.3. let x ⊂ zn and y ⊂ zm be digital images with c1-adjacency and n ≤ 4, and let f : x → y be continuous. then f# : c c1 q (x) → c c1 q (y ) is a chain map. proof. it suffices to show that f#(∂q) = ∂f#(q) for any elementary q-cube q. we may assume that q ≤ n, since there are no q-cubes in x of dimension greater than n. for simplicity, by relabeling points (translating to the origin), we may assume that q = iq = [0,1] q z , and also that f(0, . . . ,0) = (0, . . . ,0). then there are only finitely many possibilities for the behavior of f on q, and we can simply check that f#(∂q) = ∂f#(q) in all cases. we can narrow down the number of cases to check as follows: the set f(q) is contained in a q-dimensional subspace of zm, and so it suffices to consider y ⊂ zq. also note that q is a set of diameter q, and so since f is continuous, f(q) has diameter at most q. since f(q) has diameter q and maps the origin to the origin, we may assume that f(q) ⊂ [−q, . . . ,q] q z . thus the enumeration must only construct all possible continuous functions from iq to [−q, . . . ,q] q z . to further narrow down the search, we assign an ordering to the points of iq and assume without loss of generality that the first nonzero value of f is the point (1,0, . . . ,0). with these filters in place, the computation becomes tractable for q = n ≤ 4. when q = 0 there is only 1 function to check, for q = 1 there are only 2, for q = 2 there are 16, and for q = 3 there are 2128. for q = 4 there are 23,943,296 functions, requiring 4 days to complete the enumeration, which meets the limit of the author’s patience. � obviously it would be preferable to have a human readable proof of the above, using arguments which suffice in any dimension. we state this as a conjecture. conjecture 5.4. let x ⊂ zn and y ⊂ zm be digital images with c1-adjacency, and let f : x → y be continuous. then f# : cq(x) → cq(y ) is a chain map. 6. equivalence of simplicial and singular homologies in this section we show that the simplicial and singular homology theories are equivalent. we will follow the argument used in classical algebraic topology to show that the simplicial and singular homology theories of a simplicial complex are equivalent. we will give a complete proof here, following the general idea used in [8, theorem 2.27]. there is an obvious homomorphism α : čq(x) → cq(x) defined by: α[x0, . . . ,xq] = 〈x0, . . . ,xq〉, © agt, upv, 2021 appl. gen. topol. 22, no. 2 239 p. c. staecker and it is easy to check that this is a chain map. thus we obtain an induced homomorphism α∗ : ȟq(x) → hq(x). in this section we show that α∗ is an isomorphism for each q. for each k ≥ 0, let cq(x k) be the free abelian group generated by the set of q-simplices of dimension k or less. (this group will always either be trivial or equal to cq(x), so it is not very interesting in its own right, but we introduce the notation in order to make an inductive argument below.) for any k, the sequence (cq(x k),∂) is a chain complex, a subcomplex of (cq(x),∂), and we will have cq(x k) ≤ cq(x k+1) ≤ cq(x), where ≤ indicates a subgroup. there are natural inclusion and projection maps which make the following sequence exact: 0 → cq(x k−1) → cq(x k) → cq(x k)/cq(x k−1) → 0, and so we may construct a long exact relative homology sequence as in (4.1): · · · → hq+1(x k,xk−1) → hq(x k−1) → hq(x k) → hq(x k,xk−1) → hq−1(x k−1) → . . . similarly for each k ≥ 0, let čq(x k) be the free abelian group generated by the set of singular q-simplices φ such that φ(∆q) is a simplex of dimension k or less. the group čq(x k) is trivial when q < k but nontrivial for q ≥ k. for any k, the sequence (čq(x k),∂) is a chain complex, a subcomplex of (čq(x),∂), and we will have čq(x k) ≤ čq(x k+1) ≤ čq(x). again from (4.1) we have a long exact relative homology sequence: · · · → ȟq+1(x k,xk−1) → ȟq(x k−1) → ȟq(x k) → ȟq(x k,xk−1) → ȟq−1(x k−1) → . . . the same map α above will restrict to a chain map of čq(x k) → cq(x k) and induce a chain map of čq(x k,xk−1) → cq(x k,xk−1), and thus we obtain the following commutative diagram using the relative homology sequence, where the vertical arrows are induced by α. (6.1) ȟq+1(x k,xk−1) ȟq(x k−1) ȟq(x k) ȟq(x k,xk−1) ȟq−1(x k−1) hq+1(x k,xk−1) hq(x k−1) hq(x k) hq(x k,xk−1) hq−1(x k−1) lemma 6.1. for each q, the function ᾱq : ȟq(x k,xk−1) → hq(x k,xk−1) induced by α is an isomorphism. proof. it is clear from its definition that α is surjective, so ᾱ will be surjective, and it is enough to show that ᾱ is injective. the chain group čq(x k,xk−1) = čq(x k)/čq(x k−1) is the free abelian group generated by all singular q-simplices φ of xk such that φ(∆q) is a simplex of dimension k. when q < k this group is trivial, and so ȟq(x k,xk−1) is trivial, and thus ᾱ is injective for q < k. it remains to consider the case q ≥ k. let ᾱq : čq(x k,xk−1) → cq(x k,xk−1) be the homomorphism induced by α. to show that ker ᾱ∗,q is trivial, it suffices to show that ker ᾱq ⊂ b̌q(x k,xk−1). by the definition of α, we see that kerᾱq © agt, upv, 2021 appl. gen. topol. 22, no. 2 240 digital homotopy and homology is generated by all sums φ + ψ where φ,ψ ∈ čq(x k,xk−1) and ψ is an odd permutation of φ. any odd permutation can be expressed as a composition of an odd number of transpositions, and so to show that kerαq ⊂ b̌q(x k,xk−1), it suffices to show φ + ψ ∈ b̌q(x k,xk−1) whenever ψ is a transposition of φ. that is, given φ = [x0, . . . ,xq] and any i ∈ {0, . . . ,q − 1}, we must show that: (6.2) [x0, . . . ,xi,xi+1, . . . ,xq] + [x0, . . . ,xi+1,xi, . . . ,xq] ∈ b̌k(x q,xq−1). the assumption that φ ∈ čq(x k,xk−1) means that the set {x0, . . . ,xq} consists of exactly k distinct points. let σ = [x0, . . . ,xi+1,xi,xi+1, . . . ,xq] ∈ cq+1(x k,xk−1), and we compute: ∂q+1σ =[x1, . . . ,xi+1,xi,xi+1, . . . ,xq] + · · · + (−1) i[x0, . . . ,xi,xi+1, . . . ,xq] + (−1)i+1[x0, . . . ,xi+1,xi+1, . . . ,xq] + (−1) i+2[x0, . . . ,xi+1,xi, . . . ,xq] + (−1)q+1[x0, . . . ,xi+1,xi,xi+1, . . . ,xq−1] any term above listing fewer than k distinct points will be zero in čq(x k,xk−1), and so most of the terms above are zero. we are left with: ∂q+1σ = (−1) i([x0, . . . ,xi,xi+1, . . . ,xq] + [x0, . . . ,xi+1,xi, . . . ,xq]) which establishes (6.2). � our main theorem requires a very weak finiteness condition on x. any finite digital image will satisfy the condition, as will any image x ⊂ zn with ckadjacency for any k. the main work of the proof has already been accomplished by lemma 6.1, the proof below is simply a homological argument. theorem 6.2. let (x,κ) be a digital image, and assume there is some dimension k for which x contains no simplicies of dimension greater than k. then for each q, we have hq(x) ∼= ȟq(x). proof. we prove the theorem by induction on q and k. for q = 0, we have h0(x) ∼= ȟ0(x), since by theorems 4.5 and 4.8 these homology groups are both zd where d is the number of components of x. when k = 0 then x simply consists of a set of isolated points, and so hq(x) ∼= ȟq(x) by theorems 4.6 and 4.9. now we consider the inductive case. since x contains only simplices of dimension k or less, we will have cq(x) = cq(x k) and čq(x) = čq(x k) for all q. thus hq(x) = hq(x k) and ȟq(x) = ȟq(x k) for all q, and it suffices to show that the vertical arrow in the middle of (6.1) is an isomorphism. by the five lemma (see [8]), it suffices to show that the other 4 vertical arrows are isomorphisms. by induction in k, the second vertical arrow ȟq(x k−1) → hq(x k−1) is an isomorphism, and by induction in k and q the last vertical arrow ȟq−1(x k−1) → hq−1(x k−1) is an isomorphism. by lemma 6.1, the first and fourth arrows are isomorphisms. � the author’s original intention was to additionally prove a cubical version of theorem 6.2, that is, if x ⊂ zn is a digital image with c1-adjacency, then h̄q(x) ∼= h̄ c1 q (x) for each q. © agt, upv, 2021 appl. gen. topol. 22, no. 2 241 p. c. staecker this turned out to be much harder than anticipated: even constructing a chain map β : c̄q(x) → c̄ c1 q (x) is difficult. we believe it should be possible, but we will simply state it as a conjecture: conjecture 6.3. let x ⊂ zn be a digital image with c1-adjacency. then h̄q(x) and h̄c1q (x) are isomorphic. it seems likely that the conjecture could be verified in low dimensions by computer enumerations, but we have not attempted this. 7. homotopy invariance in simplicial and cubical homology in this section we discuss the homotopy invariance of the induced homomorphism on the various homology groups. in [9] it is shown that, when f and g are homotopic, their induced maps on cubical homology groups are the same. we will prove the same property holds for c1-cubical homology, and that a similar result holds for simplicial homology, but in this case the homotopy must be strong. theorem 7.1. if x is finite and f,g : x → y are strongly homotopic, then the induced homomorphisms f∗,g∗ : hq(x) → hq(y ) are equal for each q. proof. by induction and theorem 3.2, it suffices to prove the result when f and g are homotopic by a punctuated homotopy in one step. we mimic the proof for this result in classical homology theory, see for example proposition 2.10 of [8]. for each q, we define the “prism operator” p : cq(x) → cq+1(y ) as follows: for σ ∈ cq(x) with σ = 〈x0, . . . ,xq〉, let: p(σ) = q∑ j=0 (−1)j〈f(x0), . . . ,f(xj),g(xj), . . . ,g(xq)〉, where the term 〈f(x0), . . . ,f(xj),g(xj), . . . ,g(xq)〉 is interpreted as 0 if any of these points are equal. note that the definition of p only makes sense if {f(x0), . . . ,f(xj), g(xj), . . . ,g(xq)} is indeed a set of q + 2 points that are mutually adjacent or equal. this is ensured because f and g are strongly homotopic in 1 step, and thus by theorem 2.4, since {x0, . . . ,xq} are mutually adjacent, the points of {f(x0), . . . ,f(xq),g(x0), . . . ,g(xq)} are mutually adjacent or equal. this is the point at which the proof will fail if the homotopy is not strong. the bulk of the proof consists of proving the following formula: (7.1) ∂(p(σ)) = g#(σ) − f#(σ) − p(∂σ). since f and g are homotopic in one step by punctuated homotopy, there is some x′ ∈ x such that f(x) = g(x) for all x 6= x′. if σ = 〈x0, . . . ,xq〉 with xi 6= x ′ for all i, then f(xi) = g(xi) for each i. thus p(σ) = 0, since 〈f(x0), . . . ,f(xj),g(xj), . . . ,g(xq)〉 will repeat the point f(xj) = g(xj). thus, whenever σ does not use x′, we have p(σ) = 0. formula (7.1) is easy to prove when σ does not use the vertex x′: in that case p(σ) = 0 and thus the left side of (7.1) is 0. for the right side, note that © agt, upv, 2021 appl. gen. topol. 22, no. 2 242 digital homotopy and homology ∂σ also does not use the point x′, and so we have p(∂σ) = 0. also since σ does not use x′, we will have f#(σ) = g#(σ), and thus the right side of (7.1) is also 0, and we have proved (7.1). now we prove (7.1) in the case when σ does use the point x′. without loss of generality assume σ = 〈x′,x1, . . . ,xq〉. in this case we have p(σ) = 〈f(x′),g(x′),g(x1), . . . ,g(xq)〉 + q∑ j=1 (−1)j〈f(x′),f(x1), . . . ,f(xj),g(xj), . . . ,g(xq)〉 = 〈f(x′),g(x′),g(x1), . . . ,g(xq)〉 where most of the terms above are 0 because they repeat the point f(xj) = g(xj). then the left side of (7.1) is ∂(p(σ)) = ∂(〈f(x′),g(x′),g(x1), . . . ,g(xq)〉) = 〈g(x′),g(x1), . . . ,g(xq)〉 − 〈f(x ′),g(x1), . . . ,g(xq)〉 + q∑ i=1 (−1)i+1〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xq)〉 since g(xi) = f(xi), the above simplifies to: ∂(p(σ)) = g#(σ)−f#(σ)+ q∑ i=1 (−1)i+1〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xq)〉. to prove (7.1), it suffices to show that the summation above equals −p(∂(σ)). we have: p(∂(σ)) = p ( 〈x1, . . . ,xq〉 + q∑ i=1 (−1)i〈x′,x1, . . . , x̂i, . . . ,xq〉 ) = p ( q∑ i=1 (−1)i〈x′,x1, . . . , x̂i, . . . ,xq〉 ) where p(〈x1, . . . ,xq〉) = 0 since this simplex does not use x ′. now when we apply p above, the only nonzero terms are those with j = 0 in the definition of p . all others will repeat some point f(xi) = g(xi). thus we have: p(∂(σ)) = q∑ i=1 (−1)i〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xn)〉 = − q∑ i=1 (−1)i+1〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xn)〉 and we have proved (7.1). the formula (7.1) holds when σ is any simplex, and so by linearity it will hold for any chain. now let α ∈ zq(x) be a q-cycle, so ∂(α) = 0 and thus © agt, upv, 2021 appl. gen. topol. 22, no. 2 243 p. c. staecker p(∂(α)) = 0. then by (7.1) we have: g#(α) − f#(α) = ∂p(α) ∈ bq(y ). thus g#(α) and f#(α) differ by a q-boundary, that is, g∗(α) = f∗(α) ∈ hq(y ). � now we prove a homotopy invariance property for c1-cubical homology. since the result concerns the induced homomorphism in c1-cubical homology, we must require that f# : c̄ c1 q (x) → c̄ c1 q (y ) is a chain map. barring a proof of conjecture 5.4, we can only demonstrate the theorem in low dimensions. theorem 7.2. let x ⊆ zm and y ⊆ zn be digital images both with c1adjacency, with m ≤ 3, and let f,g : x → y be homotopic. then the induced homomorphisms f̄∗, ḡ∗ : h̄ c1 q (x) → h̄ c1 q (y ) are equal for each q. if conjecture 5.4 is true, then this holds for any m. proof. as in the proof of theorem 7.1, we may assume that f is homotopic to g in 1 step, that is, that f(x) g(x) for each x. by the same homological argument at the end of the proof of theorem 7.1, it will suffice to define a homomorphism p : c̄c1q (x) → c̄ c1 q+1(y ) which satisfies: (7.2) ∂p(q) = ḡ#(q) − f̄#(q) − p(∂q). for an elementary q-cube q ⊂ x, we define p(q) ∈ c̄c1q+1(y ) as follows: if the set f(q) ∪g(q) is an elementary (q + 1)-cube, then p(q) = f(q) ∪g(q) ∈ c̄c1q+1(y ). otherwise we define p(q) = 0. extending the definition linearly defines a homomorphism p : c̄c1q (x) → c̄ c1 q+1(y ). let t : i × x → y be defined as t(0,x) = f(x) and t(1,x) = g(x). since f(x) g(x) for each x, this function t is continuous (using np1 adjacency in the product i×x). the induced homomorphism t̄# : cq+1(i×x) → cq+1(y ) is closely related to p . in particular p(q) = t̄#(i × q), and when i > 1 we have: t̄#(ai(i × q)) = p(ai−1q), t̄#(bi(i × q)) = p(bi−1q). when i = 1, we instead have t̄#(a1(i × q)) = t̄#({0} × q) = f#(q) and similarly t̄#(b1(i × q) = g#(q). since x ⊂ z3 we may consider i × x ⊂ z4, and so by theorem 5.3 we will have t̄#(∂σ) = ∂t̄#(σ) for any chain σ. thus we have: ∂p(q) = ∂t̄#(i × q) = t̄#(∂(i × q)) = q+1∑ i=1 (−1)i(t#(ai(i × q)) − t#(bi(i × q))) = (g#(q) − f#(q)) + ( q+1∑ i=2 (−1)i(p(ai−1q) − p(bi−1q)) ) = g#(q) − f#(q) − p(∂q) which establishes (7.2). � © agt, upv, 2021 appl. gen. topol. 22, no. 2 244 digital homotopy and homology theorem 7.2 will imply that, if x and y are homotopy equivalent, then they have the same c1-cubical homology groups. we immediately obtain: corollary 7.3. if x ⊂ zn is a contractible digital image with c1-adjacency, and assume n ≤ 3 or that conjecture 5.4 is true. then: h̄c1q (x) = { z if q = 0, 0 if q > 0. 8. relationships between cubical and simplicial homology so far we have exhibited four homology theories: simplicial, singular, cubical, and c1-cubical. the simplicial and singular theories are isomorphic, and we have conjectured that the two cubical theories are isomorphic. thus, subject to the conjecture, there are two different types of homology under discussion: simplicial and cubical. the simplicial and cubical theories are not isomorphic, as we will see in the next section. in this section we consider some relationships that do exist between the simplicial and cubical theories. we have already seen results that imply that the simplicial and cubical theories are the same in dimension 0. theorems 4.5, 4.14, and 5.1 give: theorem 8.1. let x be any digital image. then: h0(x) = h̄0(x) = z d where d > 0 is the number of connected components of x. if x ⊂ zn with c1-adjacency, then also h̄ c1 0 (x) = z d. in dimension 1, there is also a relationship between simplicial and cubical homology. the chain group č1(x) is generated by the set of all maps φ : ∆ 1 → x, while c̄1(x) is generated by the set of all maps σ : i 1 → x. but ∆1 and i1 are the same digital image– so we may identify chain groups č1(x) = c̄1(x), and the singular and cubical boundary maps ∂1 are the same. thus we may also identify the groups of cycles ž1(x) = z̄1(x). this identification induces a natural homomorphism ȟ1(x) → h̄1(x) which simply regards a singular 1-cycle as a cubical 1-cycle. in exactly the same way, when x ⊂ zn with c1-adjacency there is a natural homomorphism h1(x) → h̄ c1 1 (x) induced by the chain map which regards each 1-simplex as an elementary 1-cube. theorem 8.2. for any digital image x, the natural map ȟ1(x) → h̄1(x) is surjective. when x ⊂ zn with c1-adjacency then the map h1(x) → h̄ c1 1 (x) is surjective. proof. we have ȟ1(x) = ž1(x)/b̌1(x) and h̄1(x) = z̄1(x)/b̄1(x). thus it suffices to show that, when we identify ž1(x) = z̄1(x), we have b̌1(x) ≤ b̄1(x). for clarity, write the singular and cubical boundary operators as ∂̌2 : č2(x) → č1(x) and ∂̄1 : c̄2(x) → c̄1(x). when φ = [x0,x1,x2] is a singular 2-simplex, © agt, upv, 2021 appl. gen. topol. 22, no. 2 245 p. c. staecker then define l(φ) = σ to be the following 2-cube: σ(0,0) = x0 σ(0,1) = x1 σ(1,0) = x0 σ(1,1) = x2 and l extends to a map l : č2(x) → c̄2(x). it is routine to check that ∂̌2(φ) = ∂̄2(l(φ)) for any φ ∈ č2(x) when we identify č1(x) = c̄1(x). thus we will have b̌1(x) ≤ b̄1(x) as desired. now when x ⊂ zn with c1-adjacency, similarly we must show that b1(x) ≤ b̄c11 (x) but this is obvious because c2(x) is trivial since a digital image with c1-adjacency contains no 2-simplex. thus b1(x) = ∂(c2(x)) is trivial and so is automatically a subgroup. � since ȟ1(x) ∼= h1(x), the above implies that there is always a surjection h1(x) → h̄1(x). we will see in the next section that h1(x) need not be isomorphic to h̄1(x), and that there need not be any surjection hq(x) → h̄q(x) when q > 1. 9. examples in this section we provide some examples comparing the simplicial and cubical homology groups of various images. most of the examples for simplicial homology appear already in the literature, but no examples for c1-cubical homology have been computed. the simplest examples are the simplicial and cubical homology groups for single points. we have already seen this result as theorems 4.6, 4.15, and 5.2. theorem 9.1. let x be an image consisting of d isolated points. then: hq(x) = h̄q(x) = h̄ c1 q (x) = { zd if q = 0, 0 if q > 0. now we describe some more interesting examples. a fruitful source of example digital images is the digital cycle cm, the digital image consisting of n points x1, . . . ,xm where xi ↔ xj only when j = i ± 1, where we read i and j modulo n. as we will see, the simplicial and cubical homologies for digital cycles (as far as we can compute them) agree in all cases except h1(c4) = z 6= 0 = h̄ c1 1 (c4). we will use a lemma which is straightforward but interesting. for two digital images (x,κ) and (y,λ), we say x embeds in y if x is (κ,λ)-isomorphic to a subset of y . lemma 9.2. the digital cycle cm embeds in (z n,c1) for some n if and only if m is even. proof. when we view (zn,c1) as a graph, it is bipartite, with the two parts given by: s+ = {(x1, . . . ,xn) | x1 + · · · + xn is even}, s− = {(x1, . . . ,xn) | x1 + · · · + xn is odd}. © agt, upv, 2021 appl. gen. topol. 22, no. 2 246 digital homotopy and homology thus cm can only embed into (z n,c1) if it too is bipartite, and this is only possible when m is even. we have shown that if m embeds in (zn,c1), then m is even. for the converse, let m > 0 be even and we will exhibit a subset of (zn,c1) for some n isomorphic to cm. when m = 2, then we may simply use [0,1]z ⊂ z 1, which is isomorphic to c2. when m = 4, then we may use [0,1] 2 ⊂ z2, which is isomorphic to c4. when m = 6, we observe that c6 is isomorphic to the following subset of z3, using c1-adjacency: {(0,0,0),(1,0,0),(1,1,0),(1,1,1),(0,1,1),(0,0,1)} finally when m > 6 is even, the cycle cm is isomorphic to the following subset of z2: ([0,m/2 − 1] × {0,2}) ∪ {(0,1),(m/2 − 1,1)}. � now we can compute the various homology groups of cm. whenever possible (when m is even), we will consider cm as embedded in z n with c1-adjacency. theorem 9.3. for m < 4, we have: hq(cm) = h̄q(cm) = h̄ c1 q (c2) = { z if q = 0, 0 if q > 0. for m = 4, we have: hq(c4) =    z if q = 0, z if q = 1, 0 if q > 1, h̄q(c4) = h̄ c1 q (c4) = { z if q = 0, 0 if q > 0. for m > 4 we have: hq(cm) =    z if q = 0, z if q = 1, 0 if q > 1, h̄q(cm) = { z if q = 0, z if q = 1. when m > 4 is even, we have hq(cm) = h̄ c1 q (cm). proof. the groups hq(cm) were already computed fully in theorem 4.1, so we need only prove the statements concerning h̄q(cm) and h̄ c1 q (cm). for h̄q(cm), we see that h̄0(cm) = z by theorem 8.1, and h̄q(c4) = 0 for q > 0 by corollary 4.13 since c4 is contractible. in fact cm is contractible for all m ≤ 4, and so we have h̄q(cm) = 0 for q > 1 and m ≤ 4. for m > 4 we have h̄1(cm) ∼= z by the hurewicz theorem of [9], since π1(cm) = z for m > 4. for h̄c1q (c2), the required statement is clear because c2 is connected with no elementary cubical q-cycles for any q > 0. for h̄c1q (c4), the statement for q = 0 follows from theorem 8.1. for q = 1 the group of elementary cubical 1-cycles has a single generator, but it is the boundary of an elementary 2-cube, and so h̄c11 (c4) = 0. for q > 1 there are no elementary q-cycles, and so h̄c1q (c4) = 0 when q > 1. © agt, upv, 2021 appl. gen. topol. 22, no. 2 247 p. c. staecker it remains to compute h̄c1q (cm) when m > 4 is even. since cm is connected we have h̄c10 (cm) = z by theorem 8.1. the group of elementary 1-cycles has a single generator which is not the boundary of any elementary cubical 2-chain, and so h̄c11 (cm) = z. for q > 1 there are no elementary cubical q-cycles, and so h̄c1q (cm) = 0 as desired. � it is obviously to be expected that hq(cm) = h̄q(cm) =    z if q = 0, z if q = 1, 0 if q > 1, for all values m > 4 including odd values, though this seems difficult to prove. the definitions of cubical homology make even the group h̄2(c5) very hard to compute by hand. so we will state this as a conjecture: conjecture 9.4. for m > 4, we have: hq(cm) = h̄q(cm) =    z if q = 0, z if q = 1, 0 if q > 1, one example where the simplicial and cubical homologies are quite different is the standard 3-cube i3 = [0,1]3 z taken with c1-adjacency. example 9.5. consider the digital image i3 with c1-adjacency. we have: hq(i 3) =    z if q = 0, z5 if q = 1, z if q = 2, 0 if q > 2, h̄q(i 3) = h̄c1q (i 3) = { z if q = 0, 0 if q > 0. proof. the computation of the simplicial homology was done in theorem 3.20 of [4], where the image in question is called mss′6. the appearance of z 5 as the homology group in dimension 1 is surprising. the generators of h1(i 3) are formed by making 1-cycles around each of the 6 faces of the cube. this produces six 1-cycles which are not boundaries, but they are not linearly independent– each one can be obtained by a combination of the other 5, and thus we have only 5 linearly independent 1-cycles. the details of the computation are given in [4]. in the case of cubical homology, each of these 1-cycles is indeed a boundary of the 2-cube which makes the corresponding face of the cube. indeed i3 is contractible, and so the computations of h̄q(i 3) and h̄c1q (i 3) follow from corollary 4.13 and 7.3. � example 9.6. let x = [0,2]3 z − {(1,1,1)}, taken with c1-adjacency. this digital image is called mss6 in [4], though its homology is not computed, presumably because h1(x) is very large. by theorem 8.1 we have h0(x) = h̄c10 (x) = z. © agt, upv, 2021 appl. gen. topol. 22, no. 2 248 digital homotopy and homology in dimension 1 there are 24 different 1-cycles tracing around unit squares. in h̄c11 (x) these are all trivial since these cycles are boundaries of 2-cubes. in h1(x) these are all nontrivial, although as in i 3 one of these cycles can be written as a sum of the other 23. the computer implementations confirm that h1(x) ∼= z 23 and h̄c11 (x) ∼= 0. in dimension 2 there are no 2-simplices, so h2(x) = 0. the above mentioned 24 unit squares, when taken together, form a 2-cycle in z̄c12 (x) which is not the boundary of any 3-cycle, and thus h̄c12 (x) 6= 0. this example is tractable for our computer implementation, which confirms that h̄c12 (x) = z. for q > 2 there are no q-simplices or elementary q-cubes, so hq(x) = h̄c1q (x) = 0. in summary, we have: hq(x) =    z if q = 0 z23 if q = 1 0 if q > 1 h̄c1q (x) =    z if q = 0 0 if q = 1 z if q = 2 0 if q > 2 note that h2(x) = 0, while h̄ c1 2 (x) ∼= z. thus the example demonstrates that, in contrast with theorem 8.2, there is not always a surjective homomorphism of hq(x) → h̄ c1 q (x) when q > 1. references [1] h. arslan, i. karaca and a. öztel, homology groups of n-dimensional digital images, in: turkish national mathematics symposium xxi (2008), 1–13. 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[15] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. © agt, upv, 2021 appl. gen. topol. 22, no. 2 250 @ appl. gen. topol. 20, no. 1 (2019), 193-210doi:10.4995/agt.2019.10635 c© agt, upv, 2019 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo school of mathematics, statistics and computer science, university of kwazulu-natal, durban, south africa. (izuchukwuc@ukzn.ac.za; 218081063@stu.ukzn.ac.za; 216028272@stu.ukzn.ac.za; mewomoo@ukzn.ac.za) communicated by e. a. sánchez-pérez abstract the main purpose of this paper is to introduce a viscosity-type proximal point algorithm, comprising of a nonexpansive mapping and a finite sum of resolvent operators associated with monotone bifunctions. a strong convergence of the proposed algorithm to a common solution of a finite family of equilibrium problems and fixed point problem for a nonexpansive mapping is established in a hadamard space. we further applied our results to solve some optimization problems in hadamard spaces. 2010 msc: 47h09; 47h10; 49j20; 49j40. keywords: equilibrium problems; monotone bifunctions; variational inequalities; convex feasibility problems; minimization problems; viscosity iterations; cat(0) space. 1. introduction optimization theory is one of the most flourishing areas of research in mathematics that has received a lot of attention in recent time. one of the most important problems in optimization theory is the equilibrium problem (ep) since it includes many other optimization and mathematical problems as special cases; namely, minimization problems, variational inequality problems, complementarity problems, fixed point problems, convex feasibility problems, among received 22 august 2018 – accepted 11 december 2018 http://dx.doi.org/10.4995/agt.2019.10635 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo others (see section 4, for details). thus, eps are of central importance in optimization theory as well as in nonlinear and convex analysis. given a nonempty set c and f : c × c → r, the ep is defined as follows: find x∗ ∈ c such that f(x∗, y) ≥ 0, ∀y ∈ c.(1.1) the point x∗ for which (1.1) is satisfied is called an equilibrium point of f. throughout this paper, we shall denote the solution set of problem (1.1) by ep(f, c). eps have been widely studied in hilbert, banach and topological vector spaces by many authors (see [5, 10, 16, 28]), as well as in hadamard manifolds (see [9, 26]). one of the most popular and effective method used for solving problem (1.1) and other related optimization problems is the proximal point algorithm (ppa) which was introduced in hilbert space by martinet [25] in 1970 and was further extensively studied in the same space by rockafellar [30] in 1976. the ppa and its generalizations have also been studied extensively in banach spaces and hadamard manifolds (see [9, 14, 22, 28] and the references therein). recently, researchers are beginning to extend the study of the ppa and its generalizations to hadamard spaces. for instance, bačák [2] studied the following ppa for finding minimizers of proper convex and lower semicontinuous functionals in hadamard spaces: let x be a hadamard space, then for arbitrary point x1 ∈ x, define the sequence {xn} iteratively by xn+1 = prox f µn (xn),(1.2) where µn > 0 for all n ≥ 1, and prox f µ : x → x is the moreau-yosida resolvent of a proper convex and lower semicontinuous functional f defined by proxfµ(x) = arg min v∈x ( f(v) + 1 2µ d2(v, x) ) .(1.3) bačák [2] proved that (1.2) ∆-convergence to a minimizer of f. in 2016, suparatulatorn et al [33] extended the results of bačák [2] by proposing the following halpern-type ppa for approximating a minimizer of a proper convex and lower semicontinuous functional which is also a fixed point of a nonexpansive mapping in hadamard spaces:      u, x1 ∈ x, yn = prox f µn (xn), xn+1 = αnu ⊕ (1 − αn)t yn n ≥ 1, (1.4) where {αn} ⊂ (0, 1) and µn ≥ λ > 0. they obtained a strong convergence result under some mild conditions. the ppa was also studied by khatibzadeh and ranjbar in [19] for finding zeroes of monotone operators and in [20] for solving variational inequality problems in hadamard spaces. based on the results of suparatulatorn et al [33], khatibzadeh and ranjbar [19], okeke and izuchukwu [27] studied the halpern-type ppa and obtained a strong convergence results for finding a minimizer of a proper convex and lower semicontinuous functional which is also a zero of a monotone operator and a fixed point of a nonexpansive c© agt, upv, 2019 appl. gen. topol. 20, no. 1 194 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space mapping. for more recent important results on ppa in hadamard spaces and other general metric spaces, see [1, 17, 36] and the references therein. very recently, kumam and chaipunya [22] studied ep (1.1) in hadamard spaces. first, they established the existence of an equilibrium point of a bifunction satisfying some convexity, continuity and coercivity assumptions, and they also established some fundamental properties of the resolvent of the bifunction. furthermore, they studied the ppa for finding an equilibrium point of a monotone bifunction in a hadamard space. more precisely, they proved the following theorem. theorem 1.1. let c be a nonempty closed and convex subset of a hadamard space x and f : c × c → r be monotone, ∆-upper semicontinuous in the first variable such that d(j f λ ) ⊃ c for all λ > 0 (where d(j f λ ) means the domain of j f λ ). suppose that ep(c, f) 6= ∅ and for an initial guess x0 ∈ c, the sequence {xn} ⊂ c is generated by xn := j f λn (xn−1), n ∈ n, where {λn} is a sequence of positive real numbers bounded away from 0. then, {xn} ∆-converges to an element of ep(c, f). motivated by the above results of kumam and chaipunya [22], we study some other important properties of the resolvent of monotone bifunctions. we then introduce a viscosity-type ppa comprising of a nonexpansive mapping and a finite sum of resolvent operators associated with these bifunctions. we prove that the sequence generated by our proposed algorithm converges strongly to a common solution of a finite family of equilibrium problems which is also a fixed point of a nonexpansive mapping. furthermore, we applied our results to solve some optimization problems in hadamard spaces. our results serve as a continuation of the work of kumam and chaipunya [22]. they also extend related results from hilbert spaces and hadamard manifolds to hadamard spaces, and they complement some recent important results in hadamard spaces. 2. preliminaries in this section, we recall some basic and useful results that will be needed in establishing our main results. we categorize our study into brief-detailed subsections. 2.1. geometry of hadamard spaces. definition 2.1. let (x, d) be a metric space, x, y ∈ x and i = [0, d(x, y)] be an interval. a curve c (or simply a geodesic path) joining x to y is an isometry c : i → x such that c(0) = x, c(d(x, y)) = y and d(c(t), c(t′) = |t − t′| for all t, t′ ∈ i. the image of a geodesic path is called the geodesic segment, which is denoted by [x, y] whenever it is unique. definition 2.2 ([13]). a metric space (x, d) is called a geodesic space if every two points of x are joined by a geodesic, and x is said to be uniquely geodesic c© agt, upv, 2019 appl. gen. topol. 20, no. 1 195 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo if every two points of x are joined by exactly one geodesic. a subset c of x is said to be convex if c includes every geodesic segments joining two of its points. let x, y ∈ x and t ∈ [0, 1], we write tx ⊕ (1 − t)y for the unique point z in the geodesic segment joining from x to y such that d(x, z) = (1 − t)d(x, y) and d(z, y) = td(x, y).(2.1) a geodesic triangle ∆(x1, x2, x3) in a geodesic metric space (x, d) consists of three vertices (points in x) with unparameterized geodesic segment between each pair of vertices. for any geodesic triangle there is comparison (alexandrov) triangle ∆̄ ⊂ r2 such that d(xi, xj) = dr2(x̄i, x̄j) for i, j ∈ {1, 2, 3}. let ∆ be a geodesic triangle in x and ∆̄ be a comparison triangle for ∆̄, then ∆ is said to satisfy the cat(0) inequality if for all points x, y ∈ ∆ and x̄, ȳ ∈ ∆̄, d(x, y) ≤ dr2(x̄, ȳ).(2.2) let x, y, z be points in x and y0 be the midpoint of the segment [y, z], then the cat(0) inequality implies d2(x, y0) ≤ 1 2 d2(x, y) + 1 2 d2(x, z) − 1 4 d(y, z).(2.3) inequality (2.3) is known as the cn inequality of bruhat and titis [7]. definition 2.3. a geodesic space x is said to be a cat(0) space if all geodesic triangles satisfy the cat(0) inequality. equivalently, x is called a cat(0) space if and only if it satisfies the cn inequality. cat(0) spaces are examples of uniquely geodesic spaces and complete cat(0) spaces are called hadamard spaces. definition 2.4 ([4]). let x be a cat(0) space. denote the pair (a, b) ∈ x×x by −→ ab and call it a vector. then, a mapping 〈., .〉 : (x × x) × (x × x) → r defined by 〈 −→ ab, −→ cd〉 = 1 2 ( d2(a, d) + d2(b, c) − d2(a, c) − d2(b, d) ) ∀a, b, c, d ∈ x is called a quasilinearization mapping. it is easily to check that 〈 −→ ab, −→ ab〉 = d2(a, b), 〈 −→ ba, −→ cd〉 = −〈 −→ ab, −→ cd〉, 〈 −→ ab, −→ cd〉 = 〈−→ae, −→ cd〉 + 〈 −→ eb, −→ cd〉 and 〈 −→ ab, −→ cd〉 = 〈 −→ cd, −→ ab〉 for all a, b, c, d, e ∈ x. a geodesic space x is said to satisfy the cauchy-swartz inequality if 〈 −→ ab, −→ cd〉 ≤ d(a, b)d(c, d) ∀a, b, c, d ∈ x. it has been established in [4] that a geodesically connected metric space is a cat(0) space if and only if it satisfies the cauchyschwartz inequality. examples of cat(0) spaces includes: euclidean spaces r n, hilbert spaces, simply connected riemannian manifolds of nonpositive sectional curvature [29], r-trees, hilbert ball [15], among others. lemma 2.5 (see [23, lemma 7]). let x be a uniformly convex hyperbolic space with modulus of uniform convexity η. for any c > 0, ǫ ∈ (0, 2], λ ∈ [0, 1] and v, x, y ∈ x, we have that d(x, v) ≤ c, d(y, v) ≤ c and d(x, y) ≥ ǫc imply that d((1 − λ)x ⊕ λy, v) ≤ (1 − 2λ(1 − λ)η(c, ǫ))c. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 196 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space if x is a cat(0) space, then x is uniformly convex with modulus of uniform convexity η(c, ǫ) := ǫ 2 8 (see [23, proposition 8]). we end this subsection with the following important lemmas which characterizes cat(0) spaces. lemma 2.6. let x be a cat(0) space, x, y, z ∈ x and t, s ∈ [0, 1]. then (i) d(tx ⊕ (1 − t)y, z) ≤ td(x, z) + (1 − t)d(y, z) (see[13]). (ii) d2(tx ⊕ (1 − t)y, z) ≤ td2(x, z) + (1 − t)d2(y, z) − t(1 − t)d2(x, y) (see [13]). (iii) d2(tx⊕(1−t)y, z) ≤ t2d2(x, z)+(1−t)2d2(y, z)+2t(1−t)〈−→xz, −→yz〉 (see [11]). (iv) d(tw ⊕ (1 − t)x, ty ⊕ (1 − t)z) ≤ td(w, y) + (1 − t)d(x, z) (see [6]). (v) z = tx ⊕ (1 − t)y implies 〈−→zy, −→zw〉 ≤ t〈−→xy, −→zx〉, ∀ w ∈ x (see [11]). (vi) d(tx ⊕ (1 − t)y, sx ⊕ (1 − s)y) ≤ |t − s|d(x, y) (see [8]). lemma 2.7 ([37]). let x be a cat(0) space. for any t ∈ [0, 1] and u, v ∈ x, let ut = tu ⊕ (1 − t)v. then, for all x, y ∈ x, (1) 〈−→utx, −→ uty〉 ≤ t〈 −→ ux, −→ uty〉 + (1 − t)〈 −→ vx, −→ uty〉; (2) 〈−→utx, −→ uy〉 ≤ t〈−→ux, −→uy〉 + (1 − t)〈−→vx, −→ux〉 and (3) 〈−→utx, −→ vy〉 ≤ t〈−→ux, −→vy〉 + (1 − t)〈−→vx, −→vy〉. lemma 2.8 ([35]). let x be a cat(0) space, {xi, i = 1, 2, . . . , n} ⊂ x and αi ∈ [0, 1], i = 1, 2, . . . , n such that ∑n i=1 αi = 1. then, d ( n ⊕ i=1 αixi, z ) ≤ n ∑ i=1 αid(xi, z), ∀z ∈ x. remark 2.9 (see also [35]). for a cat(0) space x, if {xi, i = 1, 2, . . . , n} ⊂ x, and αi ∈ [0, 1], i = 1, 2, . . . , n. then by induction, we can write n ⊕ i=1 αixi := (1 − αn) n−1 ⊕ i=1 αi 1 − αn xi ⊕ αnxn.(2.4) 2.2. the notion of ∆-convergence. definition 2.10. let {xn} be a bounded sequence in a geodesic metric space x. then, the asymptotic center a({xn}) of {xn} is defined by a({xn}) = {v̄ ∈ x : lim sup n→∞ d(v̄, xn) = inf v∈x lim sup n→∞ d(v, xn). it is generally known that in a hadamard space, a({xn}) consists of exactly one point. a sequence {xn} in x is said to be ∆-convergent to a point v̄ ∈ x if a({xnk }) = {v̄} for every subsequence {xnk} of {xn}. in this case, we write ∆lim n→∞ xn = v̄ (see [12]). the concept of ∆-convergence in metric spaces was first introduced and studied by lim [24]. kirk and panyanak [21] later introduced and studied this concept in cat(0) spaces, and proved that it is very similar to the weak convergence in banach space setting. the following lemma is very important as regards to ∆-convergence. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 197 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo lemma 2.11 ([13]). every bounded sequence in a hadamard space always have a △−convergent subsequence. 2.3. existence of solution of equilibrium problems and resolvent operators. theorem 2.12 ([22, theorem 4.1]). let c be a nonempty closed and convex subset of a hadamard space x and f : c × c → r be a bifunction satisfying the following: (a1) f(x, x) ≥ 0 for each x ∈ c, (a2) for every x ∈ c, the set {y ∈ c : f(x, y) < 0} is convex, (a3) for every y ∈ c, the function x 7→ f(x, y) is upper semicontinuous, (a4) there exists a compact subset l ⊂ c containing a point y0 ∈ l such that f(x, y0) < 0 whenever x ∈ c\l. then, problem (1.1) has a solution. in [22], the authors introduce the resolvent of the bifunction f associated with the ep (1.1). they defined a perturbed bifunction f̄x̄ : c × c → r (x̄ ∈ x) of f by f̄x̄(x, y) := f(x, y) − 〈 −→ xx̄, −→ xy〉, ∀x, y ∈ c.(2.5) the perturbed bifunction f̄ has a unique equilibrium called the resolvent operator jf : x → 2c of the bifunction f (see [22]), defined by (2.6) jf (x) := ep(c, f̄x) = {z ∈ c : f(z, y) − 〈 −→ zx, −→ zy〉 ≥ 0, y ∈ c}, x ∈ x. it was established in [22] that jf is well defined. we now recall the following definitions which will be needed in what follows. definition 2.13. let x be a cat(0) space. a point x ∈ x is called a fixed point of a nonlinear mapping t : x → x, if t x = x. we denote the set of fixed points of t by f(t ). the mapping t is said to be (i) firmly nonexpansive (see [19]), if d2(t x, t y) ≤ 〈 −−−→ t xt y, −→ xy〉 ∀x, y ∈ x, (ii) nonexpansive, if d(t x, t y) ≤ d(x, y) ∀x, y ∈ x. from cauchy-schwartz inequality, it is clear that firmly nonexpansive mappings are nonexpasive. definition 2.14. let x be a cat(0) space and c be a nonempty closed and convex subset of x. a function f : c × c → r is called monotone if f(x, y) + f(y, x) ≤ 0 for all x, y ∈ c. definition 2.15. let x be a cat(0) space. a function f : d(f) ⊆ x → (−∞, +∞] is said to be convex if f(tx ⊕ (1 − t)y) ≤ tf(x) + (1 − t)f(y) ∀x, y ∈ x, t ∈ (0, 1). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 198 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space f is proper, if d(f) := {x ∈ x : f(x) < +∞} 6= ∅. the function f : d(f) → (−∞, ∞] is lower semi-continuous at a point x ∈ d(f) if f(x) ≤ lim inf n→∞ f(xn),(2.7) for each sequence {xn} in d(f) such that lim n→∞ xn = x; f is said to be lower semicontinuous on d(f) if it is lower semi-continuous at any point in d(f). lemma 2.16 ([22, proposition 5.4]). suppose that f is monotone and d(jf ) 6= ∅. then, the following properties hold. (i) jf is single-valued. (ii) if d(jf ) ⊃ c, then jf is nonexpansive restricted to c. (iii) if d(jf ) ⊃ c, then f(jf ) = ep(c, f). theorem 2.17 ([22, theorem 5.2]). suppose that f has the following properties (i) f(x, x) = 0 for all x ∈ c, (ii) f is monotone, (iii) for each x ∈ c, y 7→ f(x, y) is convex and lower semicontinuous. (iv) for each x ∈ c, f(x, y) ≥ lim supt↓0 f((1 − t)x ⊕ tz, y) for all x, z ∈ c. then d(jf ) = x and jf single-valued. the following two lemmas will be very useful in establishing our strong convergence theorem. lemma 2.18 ([34]). let {xn} and {yn} be bounded sequences in a metric space of hyperbolic type x and {βn} be a sequence in [0,1] with lim infn→∞ βn < lim supn→∞ βn < 1. suppose that xn+1 = βnxn ⊕ (1 − βn)yn for all n ≥ 0 and lim supn→∞(d(yn+1, yn) − d(xn+1, xn)) ≤ 0. then limn→∞ d(yn, xn) = 0. lemma 2.19 (xu, [38]). let {an} be a sequence of nonnegative real numbers satisfying the following relation: an+1 ≤ (1 − αn)an + αnσn + γn, n ≥ 0, where, (i) {αn} ⊂ [0, 1], ∑ αn = ∞; (ii) lim sup σn ≤ 0; (iii) γn ≥ 0; (n ≥ 0), ∑ γn < ∞. then, an → 0 as n → ∞. 3. main results lemma 3.1. let x be a cat(0) space, {xi, i = 1, 2, . . . , n} ⊂ x, {yi, i = 1, 2, . . . , n} ⊂ x and αi ∈ [0, 1] for each i = 1, 2, . . . , n such that ∑n i=1 αi = 1. then, d ( n ⊕ i=1 αixi, n ⊕ i=1 αiyi ) ≤ n ∑ i=1 αid(xi, yi).(3.1) proof. (by induction). for n = 2, the result follows from lemma 2.6 (iv). now, assume that (3.1) holds for n = k, for some k ≥ 2. then, we prove that c© agt, upv, 2019 appl. gen. topol. 20, no. 1 199 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo (3.1) also holds for n = k + 1. indeed, by remark 2.9, lemma 2.6 (iv) and our assumption, we obtain that d ( k+1 ⊕ i=1 αixi, k+1 ⊕ i=1 αiyi ) ≤ (1 − αk+1)d ( k ⊕ i=1 αi 1 − αk+1 xi, k ⊕ i=1 αi 1 − αk+1 yi ) +αk+1d(xk+1, yk+1) ≤ k+1 ∑ i=1 αid(xi, yi). hence, (3.1) holds for all n ∈ n. � remark 3.2. it follows from (2.6) that the resolvent j f λ of the bifunction f and order λ > 0 is given as (3.2) j f λ (x) := ep(c, f̄x) = {z ∈ c : f(z, y)+ 1 λ 〈−→xz, −→zy〉 ≥ 0, y ∈ c}, x ∈ x, where f̄ is defined in this case as f̄x̄(x, y) := f(x, y) + 1 λ 〈 −→ x̄ x, −→ xy〉, ∀x, y ∈ c, x̄ ∈ x.(3.3) lemma 3.3. let c be a nonempty, closed and convex subset of a hadamard space x and f : c × c → r be a monotone bifunction such that c ⊂ d(j f λ ) for λ > 0. then, the following hold: (i) j f λ is firmly nonexpansive restricted to c. (ii) if f(jλ) 6= ∅, then d 2(jλx, x) ≤ d 2(x, v) − d2(j f λ x, v) ∀x ∈ c, v ∈ f(j f λ ). (iii) if 0 < λ ≤ µ, then d(jfµ x, j f λ x) ≤ √ 1 − λ µ d(x, jfµ x), which implies that d(x, j f λ x) ≤ 2d(x, jfµ x) ∀x ∈ c. proof. (i) let x, y ∈ c, then by lemma (2.16) (i) and the definition of j f λ , we have f(j f λ x, j f λ y) + 1 λ 〈 −−−→ xj f λ x, −−−−−→ j f λ xj f λ y〉 ≥ 0(3.4) and f(j f λ y, j f λ x) + 1 λ 〈 −−−→ yj f λ y, −−−−−→ j f λ yj f λ x〉 ≥ 0.(3.5) adding (3.4) and (3.5), and noting that f is monotone, we obtain 1 λ ( 〈 −−−→ xj f λ x, −−−−−→ j f λ xj f λ y〉 + 〈 −−−→ yj f λ y, −−−−−→ j f λ yj f λ x〉 ) ≥ 0, which implies that 〈−→xy, −−−−−→ j f λ xj f λ y〉 ≥ 〈 −−−−−→ j f λ xj f λ y, −−−−−→ j f λ xj f λ y〉. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 200 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space that is, 〈−→xy, −−−−−→ j f λ xj f λ y〉 ≥ d2(j f λ x, j f λ y).(3.6) (ii) it follows from (3.6) and the definition of quasilinearization that d 2(x, j f λ x) ≤ d2(x, v) − d2(v, j f λ x) ∀x ∈ c, v ∈ f(j f λ ). (iii) let x ∈ c and 0 < λ ≤ µ, then we have that f(j f λ x, jfµ x) + 1 λ 〈 −−−→ xj f λ x, −−−−−→ j f λ xjfµ x〉 ≥ 0(3.7) and f(jfµ x, j f λ x) + 1 µ 〈 −−−→ xj f µ x, −−−−−→ j f µ xj f λ x〉 ≥ 0.(3.8) adding (3.7) and (3.8), and by the monotonicity of f, we obtain that 〈 −−−→ j f λ xx, −−−−−→ jfµ xj f λ x〉 ≥ λ µ 〈 −−−→ jfµ xx, −−−−−→ jfµ xj f λ x〉. by the definition of quasilinearization, we obtain that ( λ µ + 1 ) d2(jfµ x, j f λ x) ≤ ( 1 − λ µ ) d2(x, jµx) + ( λ µ − 1 ) d2(x, j f λ x). since λ µ ≤ 1, we obtain that ( λ µ + 1 ) d 2(jfµ x, j f λ x) ≤ ( 1 − λ µ ) d 2(x, jfµ x), which implies d(jfµ x, j f λ x) ≤ √ 1 − λ µ d(x, jfµ x).(3.9) furthermore, by triangle inequality and (3.9), we obtain d(x, j f λ x) ≤ 2d(x, jfµ x). � remark 3.4. we note here that, if the bifunction f satisfies assumption (i)-(iv) of theorem 2.17, the conclusions of lemma 3.3 hold in the whole space x. lemma 3.5. let c be a nonempty, closed and convex subset of a hadamard space x and t be a nonexpansive mapping on c. let fi : c × c → r, i = 1, 2, . . . , n be a finite family of monotone bifunctions such that c ⊂ d(j fi λ ) for λ > 0. then, for βi ∈ (0, 1) with ∑n i=0 βi = 1, the mapping sλ : c → c defined by sλx := β0x ⊕ β1j f1 λ x ⊕ β2j f2 λ x ⊕ · · · ⊕ βnj fn λ x for all x ∈ c, is nonexpansive and f(t ◦ sλ2) ⊆ ∩ n i=1f(j fi λ1 ) ∩ f(t ) for 0 < λ1 ≤ λ2, where sλ2 : c → c is defined by sλ2x := β0x ⊕ β1j f1 λ2 x ⊕ β2j f2 λ2 x ⊕ · · · ⊕ βnj fn λ2 x for all x ∈ c. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 201 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo proof. since f is monotone, it follows from lemma 3.3 (i) (or lemma 2.16 (ii)) that j fi λ is nonexpansive for λ > 0, i = 1, 2, . . . , n. thus, by lemma 3.1, we obtain d(sλx, sλy) ≤ β0d(x, y) + β1d(j f1 λ x, j f1 λ y) + · · · + βnd(j fn λ x, j fn λ y) ≤ n ∑ i=0 βid(x, y) = d(x, y). hence, sλ is nonexpansive. now, let x ∈ f(t ◦ sλ2) and v ∈ ∩ n i=1f(j fi λ2 ) ∩ f(t ). then, by lemma 3.1, we obtain d(x, v) ≤ d(sλ2x, v) ≤ β0d(x, v) + β1d(j f1 λ2 x, v) + · · · + βnd(j fn λ2 x, v) ≤ n−1 ∑ i=0 βid(x, v) + βnd(j fn λ2 x, v)(3.10) ≤ d(x, v). from (3.10), we obtain that d(x, v) = n−1 ∑ i=0 βid(x, v) + βnd(j fn λ2 x, v) = (1 − βn)d(x, v) + βnd(j fn λ2 x, v), which implies that d(x, v) = d(j fn λ2 x, v) = d(sλ2x, v) = d(β0x ⊕ β1j f1 λ2 x ⊕ β2j f2 λ2 x ⊕ · · · ⊕ βnj fn λ2 x, v). similarly, we obtain d(x, v) = d(j fn−1 λ2 x, v) = · · · = d(j f2 λ2 x, v) = d(j f1 λ2 x, v). thus, (3.11) d(x, v) = d(j fn λ2 x, v) = · · · = d(j f1 λ2 x, v) = d(β0x⊕β1j f1 λ2 x⊕β2j f2 λ2 x⊕· · ·⊕βnj fn λ2 x, v). now, let d(x, v) = c. if c > 0, and there exist ǫ > 0 and i, j ∈ {0, 1, 2, . . ., n}, i 6= j such that d(j fi λ2 x, j fj λ2 x) ≥ ǫc (where j f0 λ2 = i), then by lemma 2.5, we obtain that d(β0x ⊕ β1j f1 λ2 x ⊕ β2j f2 λ2 x ⊕ · · · ⊕ βnj fn λ2 x, v) < c = d(x, v), and this contradicts (3.11). hence, c = 0. this implies that x = v, hence x ∈ ∩ni=1f(j fi λ2 ) ∩ f(t ).(3.12) since 0 < λ1 ≤ λ2, we obtain from lemma 3.3 (iii) and (3.12) that d(x, j fi λ1 x) ≤ 2d(x, j fi λ2 x) = 0, i = 1, 2, . . . , n. hence, x ∈ ∩ni=1f(j fi λ1 ) ∩ f(t ). therefore, we conclude that f(t ◦ sλ2) ⊆ ∩ni=1f(j fi λ1 ) ∩ f(t ). � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 202 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space we now present our strong convergence theorem. theorem 3.6. let c be a nonempty closed and convex subset of a hadamard space x and fi : c × c → r, i = 1, 2, . . . , n be a finite family of monotone and upper semicontinuous bifunctions such that c ⊂ d(j fi λ ) for λ > 0. let t : c → c be a nonexpansive mapping and g : c → c be a contraction mapping with coefficient τ ∈ (0, 1). suppose that γ := ∩ni=1ep(fi, c) ∩ f(t ) 6= ∅ and for arbitrary x1 ∈ c, the sequence {xn} is generated by (3.13) { yn = sλnxn := β0xn ⊕ β1j f1 λn xn ⊕ β2j f2 λn xn ⊕ · · · ⊕ βnj fn λn xn, xn+1 = αng(xn) ⊕ βnxn ⊕ γnt yn, n ≥ 1, where {αn}, {βn} and {γn} are sequences in (0, 1), and {λn} is a sequence of positive real numbers satisfying the following conditions: (i) lim n→∞ αn = 0 and ∑∞ n=1 αn = ∞, (ii) 0 < lim inf n→∞ βn ≤ lim sup n→∞ βn < 1, αn + βn + γn = 1 ∀n ≥ 1, (iii) 0 < λ ≤ λn ∀n ≥ 1 and lim n→∞ λn = λ, (iv) βi ∈ (0, 1) with ∑n i=0 βi = 1. then, {xn} converges strongly to z̄ ∈ γ. proof. step 1: we show that {xn} is bounded. let u ∈ γ, then by lemma 2.8, we obtain that d(xn+1, u) ≤ αnd(g(xn), u) + βnd(xn, u) + γnd(t yn, u) ≤ αnτd(xn, u) + αnd(g(u), u) + βnd(xn, u) + γnd(yn, u) ≤ αnτd(xn, u) + (αn + βn)d(xn, u) + αnd(g(u), u) = (1 − αn(1 − τ))d(xn, u) + αnd(g(u), u) ≤ max { d(xn, u) + d(g(u), u) 1 − τ } ... ≤ max { d(x1, u) + d(g(u), u) 1 − τ } . hence, {xn} is bounded. consequently, {yn}, {g(xn)} and {t (yn)} are all bounded. step 2: we show that lim n→∞ d(xn+1, xn) = 0. observe from remark 2.9, that (3.13) can be rewritten as      yn = sλn xn := β0xn ⊕ β1j f1 λn xn ⊕ · · · ⊕ βnj fn λn xn, wn = αn 1−βn g(xn) ⊕ γn 1−βn t yn, xn+1 = βnxn ⊕ (1 − βn)wn, n ≥ 1. (3.14) c© agt, upv, 2019 appl. gen. topol. 20, no. 1 203 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo now, from (3.14), lemma 2.6 (iv),(vi) and the nonexpansivity of t, we obtain that d(wn+1, wn) = d ( αn+1 1 − βn+1 g(xn+1) ⊕ γn+1 1 − βn+1 t yn+1, αn 1 − βn g(xn) ⊕ γn 1 − βn t yn ) ≤d ( αn+1 1 − βn+1 g(xn+1) ⊕ ( 1 − αn+1 1 − βn+1 ) t yn+1, αn+1 1 − βn+1 g(xn) ⊕ ( 1 − αn+1 1 − βn+1 ) t yn ) +d ( αn+1 1 − βn+1 g(xn) ⊕ ( 1 − αn+1 1 − βn+1 ) t yn, αn 1 − βn g(xn) ⊕ ( 1 − αn 1 − βn ) t yn ) ≤ αn+1 1 − βn+1 τd(xn+1, xn) + ( 1 − αn+1 1 − βn+1 ) d(yn+1, yn) + | αn+1 1 − βn+1 − αn 1 − βn |d(g(xn), t yn) (3.15) without loss of generality, we may assume that 0 < λn+1 ≤ λn ∀n ≥ 1. thus, from (3.14), condition (iv), lemma 3.1 and lemma 3.3 (iii), we obtain d(yn+1, yn) =d ( β0xn+1 ⊕ β1j f1 λn+1 xn+1 ⊕ · · · ⊕ βnj fn λn+1 xn+1, β0xn ⊕ β1j f1 λn xn ⊕ · · · ⊕ βnj fn λn xn ) ≤β0d(xn+1, xn) + n ∑ i=1 βid(j fi λn+1 xn+1, j fi λn xn) ≤β0d(xn+1, xn) + n ∑ i=1 βid(j fi λn+1 xn+1, j fi λn+1 xn) + n ∑ i=1 βid(j fi λn+1 xn, j fi λn xn) ≤d(xn+1, xn) + ( √ 1 − λn+1 λn ) n ∑ i=1 βid(j fi λn+1 xn, xn) ≤d(xn+1, xn) + ( √ 1 − λn+1 λn ) m̄, (3.16) where m̄ := sup n≥1 { ∑n i=1 βid(j fi λn+1 xn, xn) } . substituting (3.16) into (3.15), we obtain that d(wn+1, wn) ≤ αn+1 1 − βn+1 τd(xn+1, xn) + ( 1 − αn+1 1 − βn+1 ) d(xn+1, xn) + ( √ 1 − λn+1 λn )( 1 − αn+1 1 − βn+1 ) m + ∣ ∣ ∣ αn+1 1 − βn+1 − αn 1 − βn ∣ ∣ ∣ d(g(xn), t yn) = [ 1 − αn+1 1 − βn+1 (1 − τ) ] d(xn+1, xn) + ( √ 1 − λn+1 λn )( 1 − αn+1 1 − βn+1 ) m + ∣ ∣ ∣ αn+1 1 − βn+1 − αn 1 − βn ∣ ∣ ∣ d(g(xn), t yn). since lim n→∞ αn = 0, lim n→∞ λn = λ and {g(xn)}, {t yn} are bounded, we obtain that lim sup n→∞ ( d(wn+1, wn) − d(xn+1, xn) ) ≤ 0. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 204 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space thus, by lemma 2.18 and condition (ii), we obtain that lim n→∞ d(wn, xn) = 0.(3.17) hence, by lemma 2.6 we obtain that d(xn+1, xn) ≤ (1 − βn)d(wn, xn) → 0, as n → ∞.(3.18) step 3: we show that lim n→∞ d(xn, t (sλn)xn) = 0 = lim n→∞ d(wn, t (sλn)wn). notice also that (3.13) can be rewritten as xn+1 = αng(xn) ⊕ (1 − αn) ( βnxn ⊕ γnt yn (1 − αn) ) , yn = sλn xn. thus, by lemma 2.6, we obtain that (3.19) d ( xn+1, βnxn ⊕ γnt yn (1 − αn) ) ≤ αnd ( g(xn), βnxn ⊕ γnt yn (1 − αn) ) → 0, as n → ∞. also, from (2.1), we obtain d ( xn, βnxn ⊕ γnt yn (1 − αn) ) = γn 1 − αn d(xn, t yn), which implies from (3.18) and (3.19) that γn 1 − αn d(xn, t yn) ≤ d(xn, xn+1)+d ( xn+1, βnxn ⊕ γnt yn (1 − αn) ) → 0, as n → ∞. hence, lim n→∞ d(xn, t yn) = lim n→∞ d(xn, t (sλn)xn) = 0.(3.20) now, since {xn} is bounded and x is a complete cat(0) space, then from lemma 2.11, there exists a subsequence {xnk } of {xn} such that ∆− lim k→∞ xnk = z̄. again, since t ◦ sλn is nonexpansive (and every nonexpansive mapping is demiclosed), it follows from (3.20), condition (iii), lemma 3.5 and lemma 2.16 (iii) that z̄ ∈ f(t ◦ sλn) ⊆ ∩ n i=1f(j fi λ ) ∩ f(t ) = γ. d(wn, t (sλn)wn) ≤ d(wn, xn) + d(xn, t (sλn)xn) + d(t (sλn)xn, t (sλn)wn) ≤ 2d(wn, xn) + d(xn, t (sλnxn) → 0, as n → ∞.(3.21) step 4: we show that lim sup n→∞ 〈 −−−→ g(z̄)z̄, −−→ xnz̄〉 ≤ 0. now, define tnx = βnx ⊕ (1 − βn)w, where w = αn (1−βn) g(x) ⊕ γn (1−βn) t (sλn)x, then tn is a contractive mapping for each n ≥ 1. thus, there exists a unique fixed point zn of tn ∀n ≥ 1. that is, zm = βmzm ⊕ (1 − βm)wm, where wm = αm (1−βm) g(zm) ⊕ γm (1−βm) t (sλm)zm. moreover, lim m→∞ zm = z ∈ γ (see [31]). thus, we obtain that d(zm, wn) = d(βmzm ⊕ (1 − βm)wm, wn) ≤ βmd(zm, wn) + (1 − βm)d(wm, wn), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 205 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo which implies that d(zm, wn) ≤ d(wm, wn).(3.22) from (3.22) and lemma 2.6(v), we obtain that d2(wm, wn) = 〈 −−−−→ wmwn, −−−−→ wmwn〉 = 〈 −−−−−−−−−−→ wmt (sλm)zm, −−−−→ wmwn〉 + 〈 −−−−−−−−−→ t (sλm)zmwn, −−−−→ wmwn〉 ≤ αm (1 − βm) 〈 −−−−−−−−−−−→ g(zm)t (sλm)zm, −−−−→ wmwn〉 + 〈 −−−−−−−−−→ t (sλmzm)wn, −−−−→ wmwn〉 = αm (1 − βm) 〈 −−−−−−−−−−−→ g(zm)t (sλmzm), −−−−→ wmzm〉 + αm (1 − βm) 〈 −−−−−→ g(zm)wn, −−−→ zmwn〉 + αm (1 − βm) 〈 −−−−−−−−−→ wnt (sλmzm), −−−−→ zmwm〉 + 〈 −−−−−−−−−−−−−−−→ t (sλmzm)t (sλmwn), −−−−→ wmwm〉 + 〈 −−−−−−−→ t (sλmwm), −−−−→ wmwn〉 ≤ αm (1 − βm) d(g(zm), t (sλmzm))d(wm, zm) + αm (1 − βm) 〈 −−−−−→ g(zm)zm, −−−→ zmwn〉 + αm (1 − βm) 〈 −−−−−−−−−→ zmt (sλmzm), −−−→ zmwn〉 + d(t (sλmzm), t (sλmwn))d(wm, wn) + d(t (sλmwn), wn)d(wm, wn) ≤ αm (1 − βm) d(g(zm), t (sλmzm))d(wn, zm) + αm (1 − βm) 〈 −−−−−→ g(zm)zm, −−−→ zmwn〉 + αm (1 − βm) 〈 −−−−−−−−−→ zmt (sλmzm), −−−→ zmwn〉 + d(zm, wm)d(wm, wn) + d(t (sλmwn), wn)d(wn, wm) ≤ αm (1 − βm) d(g(zm), t (sλmzm))d(wn, zm) + αm (1 − βm) 〈 −−−−−→ g(zm)zm, −−−→ zmwn〉 + αm (1 − βm) d(zm, t (sλmzm))d(wm, zm) + d(wm, wn) + d(t (sλmwn), wn)d(wn, wm), which implies that 〈 −−−−−→ g(zm)zm, −−−→ wnzm〉 ≤ d(g(zm), t (sλm)zm)d(wn, zm) + d(zm, t (sλm)zm)d(zm, wm) + (1 − βm) αm d(t (sλn)wn, wn)d(wm, wm). thus, taking lim sup as n → ∞ first, then as m → ∞, it follows from (3.17),(3.20) and (3.21) that lim sup m→∞ lim sup n→∞ 〈 −−−−−→ g(zm)zm, −−−→ wnzm〉 ≤ 0.(3.23) furthermore, 〈 −−−→ g(z)z̄, −−→ xnz̄〉 = 〈 −−−−−−→ g(z̄)g(zm), −−→ xnz〉 + 〈 −−−−−→ g(zm)zm, −−−→ xnwn〉 + 〈 −−−−−→ g(zm)zm, −−−→ wnzm〉 + 〈 −−−−−→ g(zm)zm, −−→ zmz〉 + 〈 −−→ zmz̄, −−→ xnz̄〉 ≤ d(g(z), g(zm))d(xn, z) + d(g(zm), zm)d(xn, wn) + 〈 −−−−−→ g(zm)zm, −−−→ wnzm〉 + d(g(zm), zm)d(zm, z) + d(zm, z̄)d(xn, z̄) ≤ (1 + τ)d(zm, z)d(xn, z̄) + 〈 −−−−−→ g(zm)zm, −−−→ wnzm〉 + [d(xn, wn) + d(zm, z)]d(g(zm), zm), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 206 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space which implies from (3.17), (3.23) and the fact that lim m→∞ zm = z, that lim sup n→∞ 〈 −−−→ g(z)z, −−→ xnz̄〉 = lim sup m→∞ lim sup n→∞ 〈 −−−→ g(z̄)z̄, −−→ xnz̄〉 ≤ lim sup m→∞ lim sup n→∞ 〈 −−−−−→ g(zm)zm, −−−→ wnzm〉 ≤ 0.(3.24) step 5: lastly, we show that {xn} converges strongly to z ∈ γ. from lemma 2.7, we obtain that 〈 −−→ wnz, −−→ xnz̄〉 ≤ αn (1 − βn) 〈 −−−−→ g(xn)z, −−→ xnz〉 + γn (1 − βn) 〈 −−−−−−−→ t (sλn)xnz, −−→ xnz〉 ≤ αn (1 − βn) 〈 −−−−−−→ g(xn)g(z), −−→ xnz〉 + αn (1 − βn) 〈 −−−→ g(z)z, −−→ xnz〉 + γn (1 − βn) d(t (sλn)xn, z)d(xn, z) ≤ αn (1 − βn) τd 2(xn, z) + αn (1 − βn) 〈 −−−→ g(z)z, −−→ xnz〉 + (1 − αn 1 − βn )d2(xn, z) = [ αn (1 − βn) τ + (1 − αn 1 − βn ) ] d2(xn, z) + αn (1 − βn) 〈 −−−→ g(z)z, −−→ xnz〉. thus, from lemma 2.6, we have d2(xn+1, z) ≤ βnd 2(xn, z) + (1 − βn)d 2(wn, z) = βnd 2(xn, z) + (1 − βn)〈 −−→ wnz, −−→ wnz〉 = βnd 2(xn, z) + (1 − βn)[〈 −−→ wnz, −−−→ wnxn〉 + 〈 −−→ wnz, −−→ xnz〉] ≤ [βn + αnτ + γn]d 2(xn, z) + (1 − βn)〈 −−→ wnz, −−−→ wnxn〉 + αn〈g(z)z, xnz〉 ≤ (1 − αn(1 − τ))d 2(xn, z̄) + αn(1 − τ) [ 1 1 − τ 〈 −−−→ g(z̄)z̄, −−→ xnz̄〉 ] + (1 − βn)d(wn, xn)m. (3.25) by (3.17) and applying lemma 2.19 to (3.25), we obtain that {xn} converges strongly to z̄. � corollary 3.7. let c be a nonempty closed and convex subset of a hadamard space x and fi : c × c → r, i = 1, 2, . . . , n be a finite family of monotone and upper semicontinuous bifunctions such that c ⊂ d(j fi λ ) for λ > 0. let g : c → c be a contraction mapping with coefficient τ ∈ (0, 1). suppose that γ := ∩ni=1ep(fi, c) 6= ∅ and for arbitrary x1 ∈ c, the sequence {xn} is generated by (3.26) { yn = sλnxn := β0xn ⊕ β1j f1 λn xn ⊕ β2j f2 λn xn ⊕ · · · ⊕ βnj fn λn xn, xn+1 = αng(xn) ⊕ βnxn ⊕ γnyn, n ≥ 1, where {αn}, {βn} and {γn} are sequences in (0, 1), and {λn} is a sequence of positive real numbers satisfying the following conditions: (i) lim n→∞ αn = 0 and ∑∞ n=1 αn = ∞, (ii) 0 < lim inf n→∞ βn ≤ lim sup n→∞ βn < 1, αn + βn + γn = 1 ∀n ≥ 1, (iii) 0 < λ ≤ λn ∀n ≥ 1 and lim n→∞ λn = λ, c© agt, upv, 2019 appl. gen. topol. 20, no. 1 207 c. izuchukwu, k. o. aremu, a. a. mebawondu and o. t. mewomo (iv) βi ∈ (0, 1) with ∑n i=0 βi = 1. then, {xn} converges strongly to z̄ ∈ γ. corollary 3.8. let c be a nonempty closed and convex subset of a hadamard space x and f : c×c → r be a monotone and upper semicontinuous bifunction such that c ⊂ d(j f λ ) for λ > 0. let t : c → c be a nonexpansive mapping and g : c → c be a contraction mapping with coefficient τ ∈ (0, 1). suppose that γ := ep(f, c) ∩ f(t ) 6= ∅ and for arbitrary x1 ∈ c, the sequence {xn} is generated by { yn = j f λn xn, xn+1 = αng(xn) ⊕ βnxn ⊕ γnt yn, n ≥ 1, (3.27) where {αn}, {βn} and {γn} are sequences in (0, 1), and {λn} is a sequence of positive real numbers satisfying the following conditions: (i) lim n→∞ αn = 0 and ∑∞ n=1 αn = ∞, (ii) 0 < lim inf n→∞ βn ≤ lim sup n→∞ βn < 1, αn + βn + γn = 1 ∀n ≥ 1, (iii) 0 < λ ≤ λn ∀n ≥ 1 and lim n→∞ λn = λ. then, {xn} converges strongly to z̄ ∈ γ. 4. application to optimization problems in this section, we give application of our results to solve some optimization problems. throughout this section, x is a hadamard space and c is a nonempty closed and convex subset of x. 4.1. minimization problem. let h : x → r be a proper convex and lower semicontimnuous function. consider the bifunction fh : c × c → r defined by fh(x, y) = h(y) − h(x), ∀x, y ∈ c. then, fh is monotone and upper semicontinuous (see [22]). moreover, ep(c, fh) = arg minc h, j fh = proxh and d(proxh) = x (see [22]). now, consider the following finite family of minimization problem and fixed point problem: find x ∈ f(t ) such that hi(x) ≤ hi(y), ∀y ∈ c, i = 1, 2 . . . , n,(4.1) where t is a nonexpansive mapping. thus, by setting j fi λn = proxhi λn in algorithm (3.13), we can apply theorem 3.6 to approximate solutions of problem (4.1). 4.2. variational inequality problem. let s : c → c be a nonexpansive mapping. now define the bifunction fs : c × c → r by fs(x, y) = 〈 −−→ sxx, −→ xy〉. then, fs is monotone and j fs = js (see [20, 3]). consider the following finite family of variational inequality and fixed point problems: find x ∈ f(t ) such that 〈 −−→ sixx, −→ xy〉 ≥ 0, ∀y ∈ c, i = 1, 2 . . . , n,(4.2) c© agt, upv, 2019 appl. gen. topol. 20, no. 1 208 a viscosity iterative technique for equilibrium and fixed point problems in a hadamard space where t is a nonexpansive mapping on c. thus, by setting j fi λn = jsi λn in algorithm (3.13), we can apply theorem 3.6 to approximate solutions of problem (4.2). 4.3. convex feasibility problem. let ci, i = 1, 2, . . . , n be a finite family of nonempty closed and convex subsets of c such that ∩ni=1ci 6= ∅. now, consider the following convex feasibility problem: find x ∈ f(t ) such that x ∈ ∩ni=1ci.(4.3) we know that the indicator function δc : x → r defined by δc(x) = { 0, if x ∈ c, +∞, otherwise is a proper convex and lower semicontinuous function. by letting δc = h and following similar argument as in subsection 4.1, we obtain that fδc is monotone and upper semicontinuous, and jfδc = proxδc = pc. therefore, by setting jfi = pci, i = 1, 2, . . . , n in algorithm (3.13), we can apply theorem 3.6 to approximate solutions of (4.3). references [1] k. o. aremu, c. izuchukwu, g. c. ugwunnadi and o. t. mewomo, on the proximal point algorithm and demimetric mappings in cat(0) spaces, demonstr. math. 51 (2018), 277–294. 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[38] h. k. xu, iterative algorithms for nonlinear operators, j. london math. soc. 66, no. 1 (2002), 240–256. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 210 () @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 187-192 some fixed point theorems on the class of comparable partial metric spaces erdal karapinar abstract in this paper we present existence and uniqueness criteria of a fixed point for a self mapping on a non-empty set endowed with two comparable partial metrics. 2010 msc: 46n40,47h10,54h25,46t99. keywords: partial metric space, fixed point theory, comparable metrics. 1. introduction and preliminaries in 1992, matthews [10, 11] introduced the notion of a partial metric space which is a generalization of usual metric spaces in which d(x, x) are no longer necessarily zero. after this remarkable contribution, many authors focused on partial metric spaces and its topological properties (see e.g. [15, 16, 2, 1, 3, 4, 5, 6]). partial metric spaces have extensive potential applications in the research area of computer domains and semantics (see e.g. [7, 12, 8, 13, 14]). consequently, the attention paid to such spaces rapidly increases. a partial metric space (see e.g.[10, 11]) is a pair (x, p) such that x is nonempty set and p : x × x → r+ (where r+ denotes the set of all non negative real numbers) satisfies: (pm1) p(x, y) = p(y, x) (symmetry) (pm2) if p(x, x) = p(x, y) = p(y, y) then x = y (equality) (pm3) p(x, x) ≤ p(x, y) (small self-distances) (pm4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z) (triangle inequality) for all x, y, z ∈ x. we use the abbreviation pms for the partial metric space (x, p). 188 e. karapinar notice that for a partial metric p on x, the function dp : x ×x → r + given by (1.1) dp(x, y) = 2p(x, y) − p(x, x) − p(y, y) is a (usual) metric on x. observe that each partial metric p on x generates a t0 topology τp on x with a base the family open p-balls {bp(x, ε) : x ∈ x, ε > 0}, where bp(x, ε) = {y ∈ x : p(x, y) < p(x, x) + ε} for all x ∈ x and ε > 0. similarly, closed p-ball is defined as bp[x, ε] = {y ∈ x : p(x, y) ≤ p(x, x) + ε} definition 1.1 (see e.g.[10, 11, 1]). (i) a sequence {xn} in a pms (x, p) converges to x ∈ x if p(x, x) = limn→∞ p(x, xn), (ii) a sequence {xn} in a pms (x, p) is called a cauchy sequence if limn,m→∞ p(xn, xm) exists (and finite), (iii) a pms (x, p) is said to be complete if every cauchy sequence {xn} in x converges, with respect to τp, to a point x ∈ x such that p(x, x) = limn,m→∞ p(xn, xm). lemma 1.2 (see e.g.[10, 11, 1] ). (a) a sequence {xn} in a pms (x, p) is cauchy if and only if {xn} is cauchy in a metric space (x, dp), (b) a pms (x, p) is complete if and only if a metric space (x, dp) is complete. moreover, (1.2) lim n→∞ dp(x, xn) = 0 ⇔ p(x, x) = lim n→∞ p(x, xn) = lim n,m→∞ p(xn, xm) in this manuscript, we present some new fixed point theorems on a nonempty set on which there exists two partial metrics with certain conditions. 2. main results the following two lemmas will be used in the proof of the main theorem. lemma 2.1 (see e.g. [3]). let (x, p) be a complete pms. then (a) if p(x, y) = 0 then x = y, (b) if x 6= y, then p(x, y) > 0. lemma 2.2 (see e.g. [1, 3]). assume xn → z as n → ∞ in a pms (x, p) such that p(z, z) = 0. then limn→∞ p(xn, y) = p(z, y) for every y ∈ x. the following theorem is an extension of the result of maia [9]. theorem 2.3. let x be a non-empty set endowed with two partial metrics p1, p2, and let t be a mapping of x into itself. suppose that (i) (x, p1) is complete, (ii) p1(x, y) ≤ p2(x, y) for all x, y ∈ x, (iii) t is continuous with respect to τp1, (iv) t is a contraction with respect to p2, that is, p2(t x, t y) ≤ kp2(x, y) for all x, y ∈ x, where 0 ≤ k < 1. some fixed point theorems on the class of comparable partial metric spaces 189 then t has a unique fixed point in x. proof. fix x ∈ x. we construct a sequence {xn} in the following way: (s1) x0 = x, (s2) xn = t xn−1 = t nx0 for each n ∈ n. then, by assumption (iv) we have p2(xn+1, xn) = p2(t xn, t xn−1) ≤ kp2(xn, xn−1) ≤ · · · ≤ k n p2(t x0, x0). hence, by standard calculations, we get that limn,m→∞ p2(xn, xm) = 0, and by assumption (ii), limn,m→∞ p1(xn, xm) = 0, i.e., {xn} is a cauchy sequence in (x, p1). so, by assumption (i) and lemma 1.2, it converges in (x, dp1) to a point z ∈ x. again by lemma 1.2, (2.1) p1(z, z) = lim n→∞ p1(xn, z) = lim n,m→∞ p1(xn, xm) since limn,m→∞ p1(xn, xm) = 0, then by (2.1) we have p1(z, z) = 0. by the continuity of t and also lemma 2.2, one can get p1(z, z) = limn→∞ p1(z, xn+1) = limn→∞ p(z, t n+1x0) = p1(z, t (limn→∞ t nx0)) = p1(z, t (limn→∞ xn)) = p1(z, t z). hence p(t z, z) = p(z, z) = 0. due to lemma 2.1 the point z is a unique fixed point of t . suppose not, that is, there exist z, y ∈ x such that t z = z and t y = y. then, p2(z, y) = p2(t z, t y) ≤ kp2(z, y). thus, p2(z, y) = 0.regarding lemma 2.1, z = y. � theorem 2.4. let (x, p1) be a pms and t : x → x a mapping. consider the series: (2.2) ∞ ∑ n=0 tnp1(t nx, t ny) suppose that for some t > 1, the series (2.2) converges for every x, y ∈ x. then, for such a point t, the function p2 : x × x → r + defined by p2(x, y) = ∞ ∑ n=0 tnp1(t nx, t ny) is a partial metric on x, moreover, (i) p2 is an upper bound partial metric for p1, (ii) t is a contraction with respect to p2. proof. since t > 1 and p1(t nx, t ny) ≥ 0 for all x, y ∈ x and n ∈ n, then p2(x, y) ≥ 0. it is clear that p2 satisfies (pm1). for the proof of (pm2), assume p2(x, x) = p2(x, y) = p2(x, y) which is equivalent to ∞ ∑ n=0 tnp1(t nx, t nx) = ∞ ∑ n=0 tnp1(t nx, t ny) = ∞ ∑ n=0 tnp1(t ny, t ny) 190 e. karapinar hence ∞ ∑ n=0 tn (p1(t ny, t nx) − p1(t nx, t nx)) = ∞ ∑ n=0 tn (p1(t ny, t nx) − p1(t ny, t ny)) = 0, so p1(t ny, t nx) = p1(t ny, t ny) = p1(t nx, t nx) for all n ∈ n ∪ {0}. in particular, p1(x, y) = p1(x, x) = p1(y, y), and hence, x = y. moreover, (pm3) and (pm4) are obtained by definition. let us prove (i) and (ii). p2(x, y) = ∑ ∞ n=0 tnp1(t nx, t ny) = p1(x, y) + ∑ ∞ n=1 tnp1(t nx, t ny) = p1(x, y) + t ( ∑ ∞ n=0 tnp1(t n+1x, t n+1y) ) = p1(x, y) + tp2(t x, t y) thus, p2(t x, t y) = 1 t (p2(x, y) − p1(x, y)) ≤ 1 t p2(x, y). � theorem 2.5. suppose (x, p1) is a pms and t : x → x is a mapping such that p1(t mx, t my) ≤ kp1(x, y) for some m ∈ n, where 0 ≤ k < 1. then the series p2(x, y) = ∑ ∞ n=0 tnp1(t nx, t ny) converges for t > 1, whatever the points x, y ∈ x. proof. by assumption, p1(t mx, t my) ≤ kp1(x, y) for some m ∈ n, and 0 ≤ k < 1. it yields that p1(t mnx, t mny) ≤ knp1(x, y) for every n integer. then, p2(x, y) = ∑ ∞ n=0 tnp1(t nx, t ny) = ∑ ∞ n=0 tmnp1(t nx, t ny) + ∑ ∞ n=0 tmn+1p1(t mn+1x, t mn+1y) + · · · + ∑ ∞ n=0 tmn+n−1p1(t mn+n−1x, t mn+n−1y) ≤ ∑ ∞ n=0 tmnknp1(x, y) + t ∑ ∞ n=0 tmnknp1(t x, t y) + · · · + tn−1 ∑ ∞ n=0 tmnknp1(t n−1x, t n−1y) just then take t such that: 1 < tn < 1 k , because the series converges regardless of the points x, y ∈ x. � theorem 2.6. let x be a non-empty set endowed with two partial metrics p1, p2, and let t be a mapping of x into itself. suppose that (i) there exists a point x0 ∈ x such that the sequence of iterates {t n(x0)} has a subsequence {t ni(x0)} converging to a point z ∈ x for τp1, (ii) p1(x, y) ≤ p2(x, y) for all x, y ∈ x, (iii) t is continuous at z with respect to p1, (iv) t is contraction with respect to p2, that is, p2(t x, t y) ≤ kp2(x, y) for all x, y ∈ x, where 0 ≤ k < 1. then t has a unique fixed point in x. some fixed point theorems on the class of comparable partial metric spaces 191 proof. fix x0 ∈ x and define xn+1 = t xn for n ∈ n ∪ {0}. as it shown in the proof of theorem 2.3, this sequence {xn} is cauchy with respect to p2. by (ii), the sequence {xn} is also cauchy with respect to p1. by (i), cauchy sequence {xn} has a subsequence {xni} which converges z ∈ x for τp1. thus, {xn} converges to z for τp1. by the continuity of t and also lemma 2.2 one can get p1(z, z) = limn→∞ p1(z, xn+1) = limn→∞ p(z, t n+1x0) = p1(z, t (limn→∞ t nx0)) = p1(z, t (limn→∞ xn)) = p1(z, t z). hence p(t z, z) = p(z, z) = 0. due to lemma 2.1 the point z is a unique fixed point of t . to show uniqueness, assume the contrary. let z and w be two different fixed points. then, by (iv), p2(z, w) = p2(t z, t w) ≤ kp2(z, w) since 0 ≤ k < 1, one can get a contradiction. thus, t has a unique fixed point. � remark 2.7. consider the following condition: (iv)∗ there is a point x0 ∈ x such that the iterated sequence {t n(x0)} is a cauchy sequence with respect to p2. if the condition (iv) is replaced by (iv)∗ in theorem 2.6, the theorem will still guarantee the existence of fixed point. acknowledgements. the author express his gratitude to the referee for constructive and useful remarks and suggestions. references [1] t. abdeljawad, e. karapınar, k. tas, existence and uniqueness of common fixed point on partial metric spaces, appl. math. lett. 24, no. 11 (2011), 1894–1899. 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(received march 2011 – accepted july 2011) erdal karapinar (erdalkarapinar@yahoo.com, ekarapinar@atilim.edu.tr) department of mathematics, atılım university, 06836, i̇ncek, ankara, turkey some fixed point theorems on the class of comparable partial metric spaces. by e. karapinar andriagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 29-37 ∗-half completeness in quasi-uniform spaces athanasios andrikopoulos abstract. romaguera and sánchez-granero (2003) have introduced the notion of t1 ∗-half completion and used it to see when a quasi-uniform space has a ∗-compactification. in this paper, for any quasi-uniform space, we construct a ∗-half completion, called standard ∗-half completion. the constructed ∗-half completion coincides with the usual uniform completion in the uniform spaces and is the unique (up to quasi-isomorphism) t1 ∗-half completion of a symmetrizable quasi-uniform space. moreover, it constitutes a ∗-compactification for ∗-cauchy bounded quasi-uniform spaces. finally, we give an example which shows that the standard ∗-half completion differs from the bicompletion construction. 2000 ams classification: 54e15, 54d35. keywords: quasi-uniform, ∗-half completion, ∗-compactification. 1. introduction and preliminaries the problem of constructing compactifications of quasi-uniform spaces has been investigated by several authors ([4, 3.47], [5], [7]). this notion of quasiuniform compactification is by definition hausdorff. moreover, a point symmetric totally bounded t1 quasi-uniform space may have many totally bounded compactifications (see [5, page 34]) . contrary to this notion, romaguera and sánchez-granero have introduced the notion of ∗-compactification of a t1 quasiuniform space (see [8], [10] and [11]) and prove that: (a) each t1 quasi-uniform space having a t1 ∗-compactification has an (up to quasi-isomorphism) unique t1 ∗-compactification ([11, corollary of theorem 1]); and (b) all the wallmantype compactifications of a t1 topological space can be characterized in terms of the ∗-compactification of its point symmetric totally transitive compatible quasi-uniformities ([9, theorem 1]). the proof of (a) is achieved with the help of the notion of t1 ∗-half completion of a quasi-uniform space, which is introduced in [11]. following ([11, theorem 1]), if a quasi-uniform space (x, u) is t1 30 a. andrikopoulos ∗-half completable (it has a t1 ∗-half completion), then any t1 ∗-half completion of (x, u) is unique up to a quasi-isomorphism. in this paper, we prove that every quasi-uniform space has a ∗-half completion, called standard ∗-half completion, which in the case of a uniform space coincides with the usual one. we also give an example which shows that the standard ∗-half completion and the bicompletion are in general different. while a quasi-uniform space may have many ∗-half completions, here we prove that a symmetrizable quasi-uniform space has an (up to a quasi-isomorphism) unique ∗-half completion. we also prove that the standard ∗-half completion constitutes a ∗-compactification for ∗-cauchy bounded quasi-uniform spaces. let us recall that a quasi-uniformity on a (nonempty) set x is a filter u on x × x such that for each u ∈ u, (i) ∆(x) = {(x, x)|x ∈ x} ⊆ u , and (ii) v ◦ v ⊆ u for some v ∈ u, where v ◦ v = {(x, y) ∈ x × x| there is z ∈ x such that (x, z) ∈ v and (z, y) ∈ v }. the pair (x, u) is called a quasi-uniform space. if u is a quasi-uniformity on a set x, then u−1 = {u −1|u ∈ u} is also a quasi-uniformity on x called the conjugate of u. given a quasiuniformity u on x, u⋆ = u ∨ u−1 will denote the coarsest uniformity on x which is finer than u. if u ∈ u, the entourage u ∩ u −1 of u⋆ will be denoted by u ⋆. the topology τ (u) induced by the quasi-uniformity u on x is {g ⊆ x| for each x ∈ g there is u ∈ u such that u (x) ⊆ g } where u (x) = {y ∈ x|(x, y) ∈ u}. if (x, τ ) is a topological space and u is a quasiuniformity on x such that τ = τ (u) we say that u is compatible with τ . let (x, u) and (y, v) be two quasi-uniform spaces. a mapping f : (x, u) → (y, v) is said to be quasi-uniformly continuous if for each v ∈ v there is u ∈ u such that (f (x), f (y)) ∈ v whenever (x, y) ∈ u . a bijection f : (x, u) → (y, v) is called a quasi-isomorphism if f and f −1 are quasi-uniformly continuous. in this case we say that (x, u) and (y, v) are quasi-isomorphic. a filter b is called u⋆-cauchy if and only if for each u ∈ u there exists b ∈ b such that b×b ⊆ u (see [4, page 48]). a net (xa)a∈a is called u ⋆-cauchy net if for each u ∈ u there exists an a u ∈ a such that (xa, xa′ ) ∈ u whenever a ≥ au , a ′ ≥ a u . we call a u extreme index of (xa)a∈a for u and xa u extreme point of (xa)a∈a for u . a quasi-uniform space (x, u) is half complete if every u⋆-cauchy filter is τ (u)-convergent [2]. following to [11, theorem 1], a ∗-half completion of a t1 quasi-uniform space (x, u) is a half complete t1 quasi-uniform space (y, v) that has a τ (v⋆)-dense subspace quasi-isomorphic to (x, u). in [11, definition 3] also the authors introduce and study the notion of a ∗-compactification a t1 quasi-uniform space. a ∗-compactification of a t1 quasi-uniform space (x, u) is a compact t1 quasi-uniform space (y, v) that has a τ (v ⋆)-dense subspace quasi-isomorphic to (x, u). 2. the ∗-half-completion according to doitchinov [3, definition 1], a net (y β ) β∈b is called a conet of the net (xa)a∈a, if for any u ∈ u there are au ∈ a and βu ∈ b such that ∗ -half completeness in quasi-uniform spaces 31 (yβ, xa) ∈ u whenever a ≥ au and β ≥ βu . in this case, we write (yβ , xa ) −→ 0. we denote (x) the constant net (x a ) a∈a , for which x a = x for each a ∈ a. definition 2.1 (see [1, definitions 1.1(3)]). let (x, u) be a quasi-uniform space. (1) for every u⋆-cauchy net (x a ) a∈a we consider a u⋆-cauchy net (y β ) β∈b which is a conet of (x a ) a∈a , different than (x a ) a∈a . in the following, we consider all the nets a = {(xia)a∈ai |i ∈ i} that have (yβ )β∈b as their conet including (y β ) β∈b itself. in the next, we pick up all the nets b = {(y j β ) β∈bj |j ∈ j} which are conets of all the elements of a. the ordered couple (a, b) have the following properties: (a) for every u ∈ u and every (xa i)a∈ai ∈ a, (yβ j ) β∈bj ∈ b there are indices a u i, β u j such that (y β j , xia) ∈ u whenever a ≥ au i and β ≥ β u j . we call a u i (resp. β u j ) extreme index of (xa i)a∈ai (resp. (yβ j ) β∈bj ) for u and xi a u i (resp. y j β u j ) extreme point of (xa i)a∈ai (resp. (yβ j ) β∈bj ) for u . (b) b contains all the conets of all the elements of a and conversely a contains all the nets whose conets are all the elements of b. we call the ordered pair (a, b) h∗-cut, the nets (xa)a∈a and (yβ )β∈b generator and co-generator of (a, b) respectively. we also say that the pair ((y β ) β∈b , (xa)a∈a) generates the h ∗-cut (a, b). it is clear that different pairs of u⋆-cauchy nets can generate the same h∗cut. the families a and b are called classes (first and second respectively) of the h∗-cut (a, b). in the following, x̃ denotes the set of all h∗-cuts in x. if the above u⋆-cauchy net (x a ) a∈a has not as conet a u⋆-cauchy net different from itself, then we relate to it the h∗-cut which generated by the pair ((xa)a∈a, (xa)a∈a). (2) to every x ∈ x we assign an h∗-cut, denoted φ(x) = (a φ(x) , b φ(x) ), which is generated by the pair ((x), (x)). clearly, x belongs to both of a φ(x) and b φ(x) . thus the class a φ(x) contains all the nets which converge to x in τ u and b φ(x) contains nets which converge to x in τ u−1 . (3) suppose that k = {(x a ) a∈a |(x a ) a∈a is a non τ (u)-convergent u⋆-cauchy net}. let x k = {ξ ∈ x̃| the generator of ξ belongs to k}. then we put x = φ(x) ∪ x k . (4) we often say for a u⋆-cauchy net (xa)a∈a with a conet (yβ )β∈b and u ∈ u that: “finally ((y β ) β , (xa)a) ∈ u ”or in symbols “τ.((yβ )β , (xa)a) ∈ u ”, if there are a u ∈ a and β u ∈ b such that (y β , xa) ∈ u whenever a ≥ a u , β ≥ β u . 32 a. andrikopoulos definition 2.2. let (x, u) be a quasi-uniform space, ξ ∈ x and w ∈ u. (1) we say that a net (t γ ) γ∈γ is w -close to ξ, if for each net (xia)a∈ai ∈ aξ there holds τ.((t γ ) γ , (xia)a) ∈ w . (2) for each u ∈ u denote by u the collection of all pairs (ξ′, ξ′′) for which a co-generator of ξ′ is u -close to ξ′′. the proof of the following result is straightforward, so it is omitted. proposition 2.3. let (x, u) be a quasi-uniform space and let (y β ) β∈b be a co-generator of an h∗-cut ξ in x. then (y β ) β∈b belongs to both of the classes aξ and bξ. as an immediate consequence of definition 2.2 and proposition 2.3 we obtain the following proposition. proposition 2.4. let (x, u) be a quasi-uniform space, ξ′, ξ′′ ∈ x and u ∈ u. if (y β ) β∈b , (y γ ) γ∈γ are co-generators of ξ′ and ξ′′ respectively, then (ξ′, ξ′′) ∈ u if and only if τ.((y β ) β , (y γ ) γ ) ∈ u . corollary 2.5. let (x, u) be a quasi-uniform space and let ξ′, ξ′′ ∈ x. if (y β ) β∈b , (y γ ) γ∈γ are co-generators of ξ′ and ξ′′ respectively, then ξ′ = ξ′′ if and only if (y β , y γ ) −→ 0 in τ (u⋆). the following lemma is obvious. lemma 2.6. let u, v ∈ u. then u ⊆ v if and only if v ⊆ u . theorem 2.7. the family u = {u|u ∈ u} is a base for a quasi-uniformity u on x. proof. by definitions 2.2 and proposition 2.3, it follows that the pair (ξ, ξ) belongs to every element of u and by the previous lemma u is a filter. let now u, w ∈ u be such that w ◦ w ◦ w ⊆ u and x, y ∈ x with (x, y) ∈ w ◦w . then there exists a z in x such that (x, z) ∈ w and (z, y) ∈ w . if (xxa )a∈a , (z z γ ) γ∈γ and (yy β ) β∈b are co-generators of x, z and y respectively, then definition 2.2 and proposition 2.3 imply that τ.((xxa )a, (z z γ ) γ ) ∈ w and τ.((zzγ )γ , (y y β ) β ) ∈ w . we note that, for each (t δ ) δ∈∆ ∈ a y , it holds that τ.(yy β , t δ ) −→ 0. hence, τ.((xxa )a, (tδ )δ ) ∈ w ◦ w ◦ w ⊆ u which implies that (x, y) ∈ u . � proposition 2.8. if ξ ∈ x and (xa)a∈a is a u ⋆-cauchy net which belong to a ξ , then φ(xa) −→ ξ. dually, if (yβ )β∈b is a u ⋆-cauchy net which belong to b ξ , then lim β (φ(y β ), ξ) = 0. proof. let v , u ∈ u such that v ◦ v ⊆ u . if (z γ ) γ∈γ is a co-generator of ξ, then (z γ , x a ) −→ 0. thus there are a v and γ v such that (z γ , x a ) ∈ v for γ ≥ γ v and a ≥ a v . fix an a ≥ a v and pick a net (x δ ) δ∈∆ of a φ(xa ) . then, x δ −→ xa and so (xa, xδ ) ∈ v , whenever δ ≥ δv for some δv ∈ ∆. hence, (z γ , x δ ) ∈ u for γ ≥ γ v and δ ≥ δ v . hence (ξ, φ(xa)) ∈ u , whenever a ≥ av . ∗ -half completeness in quasi-uniform spaces 33 the proof of the dual is similar. � theorem 2.9. the quasi-uniform space (x, u) is a ∗-half completion of (x, u). proof. we firstly prove that (x, u) is half-complete, and secondly that the space (x, u) has a τ (u ⋆ )-dense subspace quasi-isomorphic to (x, u). indeed, let (ξa)a∈a be a u ⋆ -cauchy net of x. in the following, for each a ∈ a, (ya β ) β∈ba denotes a co-generator of ξ a . suppose that w ∈ u. then, there exists a w ∈ a such that (ξγ , ξa) ∈ w whenever γ, a ≥ a w . fix an a ≥ a w and suppose that β(a, w ) is the extreme index of (ya β ) β∈ba for w . we consider the set a⋆ = {(a, w )|a ∈ a, w ∈ u} ordered by (a′, w ′) ≤ (a′′, w ′′) if a′ ≤ a′′ and w ′′ ⊆ w ′. we put y(a, w ) = ya β(a,w ) and we prove that the net {y(a, w )|(a, w ) ∈ a⋆} is a u⋆-cauchy net. indeed, let u ∈ u. pick v ∈ u such that v ◦ v ◦ v ⊆ u . suppose that (a′, w ′), (a′′, w ′′) ≥ (a v , v ). then, (y(a′, w ′), ya ′ β′ ) ∈ (w ′)⋆ ⊆ v ⋆ and (y(a′′, w ′′), ya ′′ β′′ ) ∈ (w ′′)⋆ ⊆ v ⋆ whenever β′ ≥ β′(a′, w ′) and β′′ ≥ β′′(a′′, w ′′). since (ξa)a∈a is a u ⋆ -cauchy net of x, proposition 2.4 implies that τ.((ya ′ β′ ) β′ , (ya ′′ β′′ ) β′′ ) ∈ v ⋆ whenever a′, a′′ ≥ a v . hence, (y(a′, w ′), y(a′′, w ′′)) ∈ v ⋆ ◦ v ⋆ ◦ v ⋆ ⊆ u ⋆. we now prove that (ξa)a∈a is τ (u)-convergent. we have two cases. case 1. (y(a, w )) (a,w )∈a⋆ τ (u)-converges to a point x ∈ x. in this case, we have that (φ(y(a, w ))) (a,w )∈a⋆ τ (u)-converges to φ(x). since (ya β ) β∈ba belongs to b ξa , proposition 2.8 implies that (φ(y(a, w )), ξ a ) −→ 0. hence, from (φ(x), φ(y(a, w ))) −→ 0 we conclude that (ξa)a∈a τ (u)-converges to φ(x). case 2. (y(a, w )) (a,w )∈a⋆ is a non τ (u)-convergent net. let ξ be the h∗-cut in x which is generated by (y(a, w )) (a,w )∈a⋆ . it follows, by proposition 2.8, that (ξ, φ(y(a, w ))) → 0. since (ya β ) β∈ba belongs to b ξa , proposition 2.8 implies that (φ(y(a, w )), ξ a ) −→ 0. the rest is obvious. it remains to prove that (φ(x), u/φ(x) × φ(x)) is a τ (u ⋆ )-dense subspace of (x, u). indeed, let ξ ∈ x and let (y β ) β∈b be a co-generator of it. then, since the co-generator belongs to both of classes of ξ, proposition 2.8 implies that φ(y β ) τ (u ⋆ )-converges to ξ. � in the sequel the ∗-half completion (x, u) constructed above will be called standard ∗-half completion of the space (x, u). the following example shows that the standard ∗-half completion and the bicompletion of a quasi-uniform space are in general different. 34 a. andrikopoulos example 2.10. let x be the set consisting of all nonzero real numbers and let d be the quasi-metric on x given by: d(x, y) = { y − x if x < y 0 otherwise suppose that u is the quasi-uniformity generated by d. let f be the u⋆-cauchy filter generated by {(0, a)|a > 0} and g be the u⋆-cauchy filter generated by {(b, 0)|b < 0}. then a new point is defined by the h∗-cut ξ = (a ξ , b ξ ), where a ξ = {g, f} and b ξ = {f}. hence, x = φ(x) ∪ {ξ}. clearly, ξ defines the point 0 in (x, u). on the other hand, there is exactly one minimal u⋆-cauchy filter coarser than f and g respectively. more precisely, if f 0 and g 0 are any bases for f and g respectively, then {u (f 0 ) |f 0 ∈ f 0 and u is a symmetric member of u⋆} and {u (g 0 ) |g 0 ∈ g 0 and u is a symmetric member of u⋆} are equivalent bases for the minimal u⋆-cauchy filter h̃ coarser than f and g respectively. hence, we have x̃ = i(x) ∪ {h}. the filter h defines the point 0 in (x̃, ũ) as well. we conclude the following: (i) the bicompletion of (x, u) differs from the standard ∗-half completion. indeed, by the definition of ξ and from the propositions 2.3 and 2.8, we conclude that φ(g) and φ(f) converge to 0 with respect to τ (u) and τ (u ⋆ ) respectively. on the other hand, i(g) and i(f) converge to 0 with respect to τ (ũ ⋆ ). (ii) the standard ∗-half completion is not quasi-uniformly isomorphic to its bicompletion. this is true by (i) and the fact that the bicompletion of (x, u) coincides up to a quasi-isomorphism with the bicompletion of (x, u). theorem 2.11. let (x, u) be a uniform space. then, the standard ∗-half completion (x, u) coincides with the usual uniform completion. proof. let (x, u) be a uniform space and let ξ be an h∗-cut in x. suppose that (xa)a∈a ∈ aξ and (yβ )β∈b ∈ bξ . then (yβ , xa) −→ 0 and (xa, yβ ) −→ 0. hence the nets and the conets of ξ coincide. thus, the class of equivalent cauchy nets, of the uniform case, is identified with an h∗-cut and vice versa. hence the “ground set”of the two completions is the x. the rest is obvious. � next, we give an equivalent definition for nets for the definition 5 in [11]. definition 2.12. let (x, u) be a quasi-uniform space. a u⋆-cauchy net (x a ) a∈a on x is said to be symmetrizable if whenever (y β ) β∈b is a u⋆-cauchy net on x such that (y β , x a ) −→ 0, then (x a , y β ) −→ 0. definition 2.13. a quasi-uniform space (x, u) is called symmetrizable if each u⋆-cauchy net on x, including for each x ∈ x the constant net (x), is symmetrizable. it easy to check that a quasi-uniform space is symmetrizable if and only if the bicompletion is t 1 . in this case, the space has only one t 0 ∗-half completion, ∗ -half completeness in quasi-uniform spaces 35 the bicompletion. from theorem 2.9 and [11, theorem 1] we immediate deduce the following result. corollary 2.14. if a t1 quasi-uniform space is symmetrizable, then it has a t1 ∗-half completion which is unique up to a quasi-isomorphism. 3. standard ∗-half completion and ∗-compactification we recall some well known notions from [6]. a net (x a ) a∈a is said to be frequently in s, for some subset s of x, if and only if for all a ∈ a there is some a′ ≥ a such that x a′ ∈ s. a net is said to be eventually in s if and only if there is an a 0 in a such that for all a ≥ a 0 , x a is in s. a point x in x is a cluster point of the net (x a ) a∈a if and only if the net is frequently in all neighborhoods of x. the net (x a ) a∈a converges to x if and only if (x a ) a∈a is eventually in all neighborhoods of x. the tail sets of (x a ) a∈a are the sets ta (a in a) where ta = {xa′ |a ′ ≥ a}. note that the ta have the finite intersection property, by the directedness of the index set a, so they generate a filter, the filter of tails of (x a ) a∈a or the filter associated with the net (x a ) a∈a . then a point x is a cluster point of (x a ) a∈a if and only if x is in cl(ta) for all a (if and only if it is a cluster point of the filter of tails). and x a −→ x if and only if the filter of tails converges to x. this already shows that there is a close relationship between convergence of filters and convergence of nets. definition 3.1 (see [6, page 81]). a universal net in x is one such that for each s ⊂ x, either the net is eventually in s, or it is eventually in x \ s. from the classical theory we have the following statements. (a) a net is a universal net if and only if its associated filter is an ultrafilter. (b) let f be the filter associated with the net (x a ) a∈a and g be a filter with f ⊂ g. then (x a ) a∈a has a subnet whose associated filter is g. (a) and (b) implies that: (c) every net has a universal subnet. (d) a universal net converges to each of its cluster points. (e) a space is compact if and only if every universal net is convergent. definition 3.2 (see [11, definition 6]). a quasi-uniform space (x, u) is called ⋆-cauchy bounded if for each ultrafilter f on x there is a u⋆-cauchy filter g on x such that (g, f) −→ 0. definition 3.2 admits an equivalent definition for nets. definition 3.3. a quasi-uniform space (x, u) is called ⋆-cauchy bounded if for each universal net (x a ) a∈a on x there is a u⋆-cauchy net (y β ) β∈b on x such that (y β , x a ) −→ 0. theorem 3.4. let (x, u) be a ⋆-cauchy bounded quasi-uniform space. then the standard ⋆-half completion (x, u) is a ⋆-compactification of the space (x, u). 36 a. andrikopoulos proof. let (ξ a ) a∈a be a universal net in (x, u). suppose that for any a ∈ a, ξ a = (a ξa , b ξa ). let (ya β ) β∈ba and {y(a, w )|(a, w ) ∈ a⋆} be as in the proof of theorem 2.9. then, {y(a, w )|(a, w ) ∈ a⋆} is a net in x. by the above statement (c), we have that (y(a, w )) (a,w )∈a⋆ has a universal subnet, let {y(a k , w k )|(a k , w k ) ∈ a⋆, k ∈ k}. since (x, u) is ⋆-cauchy bounded, there is a u⋆-cauchy net (x γ ) γ∈γ of x such that (x γ , y(a k , w k )) −→ 0. hence (φ(x γ ), φ(y(a k , w k ))) −→ 0 in (x, u) (1). on the other hand, since the space (x, u) is half-complete, there exists ξ ∈ x such that (φ(x γ )) γ∈γ τ (u)-converges to ξ (2). hence by (1) and (2) we conclude that {φ(y(a k , w k ))|(a k , w k ) ∈ a⋆, k ∈ k} τ (u)-converges to ξ. since {φ(y(a k , w k ))|(a k , w k ) ∈ a⋆, k ∈ k} is a subnet of φ(y(a, w )) (a,w )∈a⋆ we conclude that ξ is a cluster point of the latter. since (ya β ) β∈ba belongs to b ξa , proposition 8 implies that (φ(y(a, w )), ξ a ) −→ 0. hence, ξ is a cluster point of (ξ a ) a∈a . there also holds that (ξ a ) a∈a is a universal net, thus the above statement (d) implies that it τ (u)-converges to ξ. finally, by the above statement (e) we conclude that the space (x, u) is compact. by theorem 9, the space (x, u) has a τ (u ⋆ )-dense subspace quasiisomorphic to (x, u). hence (x, u) is a ⋆-compactification of (x, u). � references [1] a. andrikopoulos, completeness in quasi-uniform spaces, acta math. hungar. 105 (2004), 549-565, mr 2005f:54050. [2] j. deak, on the coincidence of some notions of quasi-uniform completeness defined by filter pairs, stud. sci. math. hungar. 26 (1991), 411-413, mr 94e:94077. [3] d. doitchinov, a concept of completeness of quasi-uniform spaces, topology appl. 38 (1991), 205-217, mr 92b:54061. [4] p. fletcher and w. f. lindgren, quasi-uniform spaces, lectures notes in pure and appl. math. 77 (1978), marc. dekker, new york, mr 84h:54026. [5] p. fletcher, and w. f. lindgren, compactifications of totally bounded quasi-uniform spaces, glasgow math. j. 28 (1986), 31-36, mr 87f:54037. [6] j. kelley, general topology, d.van nostrand company, inc., toronto-new yorklondon, (1955), mr 16, 1136c. [7] h. render, nonstandard methods of completing quasi-uniform spaces, topology appl. 62 (1995), 101-125, mr 96a:54041. [8] s. romaguera and m. a. sánchez-granero, *-compactifications of quasi-uniform paces, stud. sci. math. hung. 44 (2007), 307-316. [9] s. romaguera and m. a. sánchez-granero, a quasi-uniform characterization of wallman type compactifications, stud. sci. math. hung. 40 (2003), 257-267, mr 2004h:54021. [10] s. romaguera and m. a. sánchez-granero, compactifications of quasi-uniform hyperspaces, topology appl. 127 (2003), 409-423, mr 2003j:54011. [11] s. romaguera and m. a. sánchez-granero, completions and compactifications of quasiuniform spaces, topology appl. 123 (2002), 363-382, mr 2003c:54051. ∗ -half completeness in quasi-uniform spaces 37 received january 2008 accepted august 2008 athanasios andrikopoulos (aandriko@cc.uoi.gr) department of economics, university of ioannina, greece () @ applied general topology c© universidad politécnica de valencia volume 11, no. 2, 2010 pp. 117-133 the alexandroff property and the preservation of strong uniform continuity gerald beer abstract. in this paper we extend the theory of strong uniform continuity and strong uniform convergence, developed in the setting of metric spaces in [13, 14], to the uniform space setting, where again the notion of shields plays a key role. further, we display appropriate bornological/variational modifications of classical properties of alexandroff [1] and of bartle [7] for nets of continuous functions, that combined with pointwise convergence, yield continuity of the limit for functions between metric spaces. 2000 ams classification: primary 40a30, 54c35; secondary 54e15, 54c08 keywords: strong uniform continuity, strong uniform convergence, preservation of continuity, variational convergence, bornology, the alexandroff property, the bartle property, shield, quasi-uniform convergence, bornological uniform cover, sticking topology 1. introduction in any introductory analysis course, where one studies functions between metric spaces, one observes that the pointwise limit of a sequence of continuous functions need not be continuous, whereas uniform convergence in fact uniform convergence on compact subsets preserves continuity. on the other hand, it is easy to construct a sequence of piecewise linear continuous realvalued functions on [0, 1] pointwise convergent but not uniformly convergent to the zero function, so uniform convergence on compacta while sufficient is hardly necessary. so one is led to ask, as arzelà first formally did [2, 3], what precisely must be added to pointwise convergence to yield continuity of the limit? for a comprehensive guide to the literature on the preservation of continuity, the reader may consult [18]. perhaps the most satisfying add-on has been given by p. alexandroff [1], one of the founders of general topology. 118 g. beer definition 1.1. let 〈x, d〉 and 〈y, ρ〉 be metric spaces and let f, f1, f2, f3, . . . be a sequence of functions from x to y . then 〈fn〉 is said to have the alexandroff property with respect to f if for each ε > 0 and n0 ∈ n, there exists a strictly increasing sequence 〈nk〉 of integers such that n1 ≥ n0 and a countable open cover {vk : k ∈ n} of x such that ∀k ∈ n, ∀x ∈ vk, we have ρ(f (x), fnk (x)) < ε. alexandroff of course showed that for sequences of continuous functions, in the presence of pointwise convergence to the function f , the alexandroff property is equivalent to continuity of f . in fact, his result is valid without the metrizability assumption on the domain [1, pg. 266]. on the other hand, the alexandroff property alone does not guarantee pointwise convergence: if f = f2 = f4 = f6 = · · · , then 〈fn〉 has the alexandroff property with respect to f no matter how {f1, f3, f5, . . .} are defined. obviously if a sequence has the alexandroff property with respect to f , this property is not in general inherited by its subsequences. so perhaps a more appropriate question to ask in this setting is the following: is there a topology on the set of all functions y x from x to y finer than the topology of pointwise convergence that is somehow intrinsic to the preservation of continuity? that is, is there a topology on y x for which the set of continuous functions c(x, y ) is a closed subset and for which pointwise convergence in c(x, y ) entails convergence in this topology? discovered forty years ago by bouleau [16], the answer to this question also falls out of a general theory of topologies of strong uniform convergence of functions with values in a metric target space 〈y, ρ〉 with respect to a bornology b of nonempty subsets of 〈x, d〉 [13, corollary 6.8]. recall that a bornology b is a cover of x by nonempty subsets that is stable under taking finite unions and under taking nonempty subsets of members of the cover [12, 13, 20]. evidently, the largest bornology is p0(x), the family of all nonempty subsets of x, whereas the smallest is f0(x), the family of nonempty finite subsets of x. strong uniform convergence of a net 〈fλ〉λ∈λ of functions from x to y to f ∈ y x with respect to a particular bornology b is described as follows: for each ε > 0 and each b ∈ b, there exists an index λ0 in the underlying directed set for the net 〈λ, �〉 such that for each λ � λ0 there exists δλ > 0 such that whenever d(x, b) < δλ, then ρ(fλ(x), f (x)) < ε. this notion is fundamentally variational in nature: we insist not only on uniform convergence on members of b but convergence around the edges of elements of b in some almostuniform sense. notice also that since each bornology contains the singletons, we automatically get pointwise convergence, whatever the bornology may be. in the special case of the bornology f0(x), convergence in this sense of a net of continuous functions forces continuity of the limit, and conversely, if the limit is continuous, then this sort of convergence must ensue. for a general bornology b on x, strong uniform convergence is characterized by the preservation of the variational notion of strong uniform continuity of functions on members of b [13, theorem 6.7], that reduces to ordinary pointwise continuity when b is a bornology of relatively compact subsets (see definition 2.2 infra). the alexandroff property and the preservation of strong uniform continuity 119 the purpose of this article is to identify the appropriate bornological alexandroff property that corresponds to strong uniform convergence of functions with respect to an arbitrary bornology on the domain. we choose to do so for nets of functions rather than just for sequences, and we work in the more general context of hausdorff uniform spaces rather than metric spaces, developing in the process the rudiments of the theory of strong uniform continuity and strong uniform convergence in this setting. in particular, our results apply to locally convex spaces, where one might be interested say in bornologies of weakly relatively compact sets. falling out of our analysis is a variational-bornological form of quasi-uniform convergence as defined by bartle [7]. 2. preliminaries let x be a hausdorff topological space. if v is a family of subsets of x we say v is a cover of a ⊆ x provided a ⊆ ∪v . a second family of subsets u is said to refine v if ∀u ∈ u , ∃v ∈ v with u ⊆ v . if 〈x, d〉 is a metric space, we write sd(x, α) for the open ball with center x ∈ x and radius α > 0. if a ⊆ 〈x, d〉 we put sd(a, α) := ∪a∈asd(a, α) = {x : d(x, a) < α}, where d(x, ∅) = ∞ is understood. rephrasing, a bornology on x is a family of nonempty subsets b that contains the singletons, that is stable under finite unions, and whenever b ∈ b and ∅ 6= b0 ⊆ b, then b0 ∈ b. other bornologies of note beyond those mentioned in the introduction are (1) the bornology of relatively compact subsets of x; (2) the subsets b(f ) of x on which some function f with domain x and values in a metric space is bounded; (3) for a metric space 〈x, d〉, the separable subsets of x; (4) for a metric space 〈x, d〉, the d-bounded subsets of x; (5) for a metric space 〈x, d〉, the d-totally bounded subsets of x; (6) for a locally convex topological vector space, the family of subsets that are absorbed by each neighborhood of the origin [19, pg. 51]. bornologies on a metrizable space that are the bounded subsets with respect to some admissible metric have been characterized by hu [21], whereas those that are the totally bounded subsets with respect to some admissible metric have been characterized by beer, costantini and levi [10]. by a base for a bornology, we mean a subfamily of the bornology that is cofinal with respect to inclusion. each of the bornologies listed above have closed bases, and the bornology of metrically bounded sets has a countable closed base. an example of a bornology that does not have a closed base is the family of countable subsets of r. in what follows, letters in bold caps will denote diagonal uniformities. for facts and terminology about diagonal uniformities we will rely totally on the excellent general textbook by willard [24]. if 〈x, d〉 is a hausdorff uniform space and x0 ∈ x and d ∈ d, we write d(x0) for {x ∈ x : (x0, x) ∈ d}. of course, {d(x0) : d ∈ d} forms a local base at x0 for the induced topology. if a ∈ p0(x), we call the uniform neighborhood d(a) := ∪a∈ad(a) an enlargement of a. thus, in the metric context, each set of the form sd(a, α) is an enlargement of a. disjoint subsets a and b of x are called asymptotic if for each d ∈ d, we have d(a) ∩ b 6= ∅. 120 g. beer a bornology b on a hausdorff uniform space is said to be stable under small enlargements [12] if it contains an enlargement of each of its members. the dbounded subsets of a metric space are always stable under small enlargements with respect to the metric uniformity. evidently, the relatively compact sets are stable under small enlargements if and only if x is locally compact. as is well-known, a base for a uniformity consists of its symmetric open entourages [24, pg. 241]. another important fact for our purposes that we regard as a folk-lemma is now stated and proved for completeness. it generalizes the fact that each open cover of a compact metric space has a lebesgue number (a property that actually is characteristic of the larger class of uc metric spaces [6, 8, 23]). it is the provenance of the idea of bornological uniform cover defined in section 4. lemma 2.1. let k be a nonempty compact subset of a hausdorff uniform space 〈x, d〉. then if v is an open cover of k, there exists an entourage d0 such that {d0(x) : x ∈ k} refines v . proof. if this fails, then for each entourage d there exists xd ∈ k such that d(xd) is contained in no element of v . direct d by reverse inclusion, i.e., d1 � d2 provided d1 ⊇ d2. then by compactness of k, the net d 7→ xd has a cluster point p. choose d and v ∈ v such that d(p) ⊆ v . then with a symmetric d1 chosen such that d1 ◦ d1 ⊆ d, ∃d2 ⊆ d1 such that xd2 ∈ d1(p). but then d2(xd2 ) ⊆ v , and we have a contradiction. � the variational notion of strong uniform continuity of a function f on a nonempty subset of the domain, while exhaustively studied in the metric context by beer and levi [13, 14], appears earlier in a paper of beer and diconcilio [11] that characterized those metrics that give rise to the same attouch-wets topologies (see, e.g., [4, 5, 8, 12]) on the closed subsets of x. the definition has a straight-forward extension to the uniform setting. definition 2.2. let 〈x, d〉 and 〈y, t〉 be hausdorff uniform spaces and let f : x → y . we say that f is strongly uniformly continuous on b ∈ p0(x) if for each entourage t ∈ t, ∃d ∈ d such that whenever b ∈ b and x ∈ x satisfy (b, x) ∈ d, then (f (b), f (x)) ∈ t . consistent with the terminology, the condition is stronger than uniform continuity of the restriction of f to b. while we rely exclusively on this formulation, the reader may prefer this more aesthetically pleasing equivalent: for each entourage t ∈ t, there exists a symmetric entourage d ∈ d such that whenever {x, w} ⊆ d(b) and (x, w) ∈ d, then (f (x), f (w)) ∈ t . from this perspective, it is easy to see that if f is strongly uniformly continuous on b, then it is strongly uniformly continuous on cl(b). we now run through some easily verified facts established in [13] in the metric context. strong uniform continuity of f on {x0} is equivalent to the ordinary continuity of f at x0, while strong uniform continuity of f on x is equivalent to global uniform continuity. for each f ∈ c(x, y ), lemma 2.1 easily yields the strong uniform continuity of f on each nonempty compact subset. in general, if the alexandroff property and the preservation of strong uniform continuity 121 f : x → y , then bf := {b ∈ p0(x) : f is strongly uniformly continuous on b} while possibly empty is closed under taking nonempty subsets and finite unions and is thus a bornology if and only if f ∈ c(x, y ). given a bornology b we say f : x → y is strongly uniformly continuous on b provided b ⊆ bf . we write cs b (x, y ) for those functions in c(x, y ) that are strongly uniformly continuous on b and cb(x, y ) for those functions in c(x, y ) whose restriction to each element of b is uniformly continuous. the latter class usually properly contains the former (see example 3.8 and example 4.10 infra). we will be looking at topologies of uniform convergence and strong uniform convergence on y x and their induced relative topologies on c(x, y ) where 〈x, d〉 and 〈y, t〉 are hausdorff uniform spaces. all are determined by uniformities on y x , and all are stronger than the topology of pointwise convergence and thus all are completely regular and hausdorff. let b be a bornology on x. the classical topology of uniform convergence tb on y x [22] has as a base for its entourages [b, t ] := {(f, g) : ∀x ∈ b, (f (x), g(x)) ∈ t } (b ∈ b, t ∈ t). of course, such a topology does not reference the uniformity d at all and makes sense when the domain is not a uniform space. topologies of uniform convergence for spaces of continuous linear transformations of course play a fundamental role in functional analysis, e.g., in the context of normed linear spaces, the norm topology, the weak∗ topology, and the bounded weak∗ topology all fit within this framework (see, e.g., [19]). the topology of strong uniform convergence t s b on y x has as a base for its entourages [b, t ]s := {(f, g) : ∃d ∈ d ∀x ∈ d(b), (f (x), g(x)) ∈ t } (b ∈ b, t ∈ t). these topologies are unchanged by replacing b in their definitions by a base for the bornology or the given uniformities by bases for them. often, we replace t by a symmetric open base in our arguments. for t s b , we may assume without loss of generality that b has a closed base, as strong uniform convergence of a net on b implies strong uniform convergence on the bornology with base {cl(b) : b ∈ b}. when restricting tb to c(x, y ), we may also assume that b has a closed base. 3. coincidence of function spaces topologies evidently, for each hausdorff uniform codomain 〈y, t〉, we have t s b finer than tb on y x (see examples 3.7 and 3.8 infra). the next two results speak to when t s b collapses to tb on y x and on c(x, y ). our first result extends [13, theorem 6.2] to uniform spaces. 122 g. beer theorem 3.1. let b be a bornology on a hausdorff uniform space 〈x, d〉. (1) if b is stable under small enlargements, then for each hausdorff uniform space 〈y, t〉, the standard uniformities for tb and for t s b agree on y x ; (2) if t s b = tb on r x , then b is stable under small enlargements. proof. statement (1) is obvious, for if d(b) ∈ b where d ∈ d and b ∈ b, then [d(b), t ] ⊆ [b, t ]s for any t ∈ t. for statement (2), suppose no enlargement of b0 ∈ b again lies in b. for each superset b of b0 that lies in the bornology and each entourage d, pick xd,b ∈ d(b0)\b, and let χb be the characteristic function of {xd,b : d ∈ d}, defined by χb(x) := { 1 if x = xd,b for some d ∈ d 0 otherwise . direct b0 := {b ∈ b : b0 ⊆ b} by inclusion. then b 7→ χb is easily seen to be uniformly convergent to the zero function on elements of b because whenever b ∈ b and b1 ⊇ b ∪ b0, we have χb1 identically equal to zero on b. but strong uniform convergence fails, as each χb takes on the value 1 on each enlargement of b0. � we now come to coincidence of the function space topologies on c(x, y ). our result is anticipated by recent work in the context of metric spaces [14], but different methods must be employed as sequences do not suffice. the key notion was introduced in [9] (see also [15]). definition 3.2. let b be a bornology on a hausdorff uniform space 〈x, d〉. we say that b is shielded from closed sets if ∀b ∈ b, ∃b1 ∈ b such that b ⊆ b1 and each neighborhood of b1 contains an enlargement of b. the superset b1 of b in the definition is called a shield for b. the terminology we chose can be justified as follows: each nonempty closed set c disjoint from b1 fails to intersect some enlargement of b, so that b1 protects b from the closed set. evidently, if b1 is a shield for b, then b1 contains cl(b) and moreover is a shield for cl(b). the bornology of relatively compact subsets of x is shielded from closed sets, as whenever b is relatively compact, cl(b) serves as a shield for b. evidently each bornology b that is stable under small enlargements is shielded from closed sets. for a bornology shielded from closed sets that is of neither type, in x = [0, ∞), for each n ∈ n with n ≥ 3, put bn := ∞⋃ k=0 [k + 1 n , k + 1 − 1 n ]. our bornology b will consist of all sets of the form e ∪ f where f ∈ f0(x) and for some n ≥ 3, e ⊆ bn. let b = {3} ∪ b3. no enlargement of b lies in the bornology and cl(b) = b fails to shield itself from closed sets, as one can the alexandroff property and the preservation of strong uniform continuity 123 easily construct a sequence in x that is asymptotic to b and does not cluster. nevertheless, the bornology is shielded from closed sets, for if e ∪f ∈ b where e ⊆ bn and f is finite, then bn+1 ∪ f is a shield for e ∪ f . theorem 3.3. let b be a bornology having a closed base on a hausdorff uniform space 〈x, d〉. (1) if b is shielded from closed sets, then for each hausdorff uniform space 〈y, t〉, the standard uniformities for tb and for t s b agree on c(x, y ); (2) if x is normal and t s b = tb on c(x, r), then b is shielded from closed sets. proof. for (1), fix b ∈ b and t an open entourage in t. let b1 be a shield for b in b. it suffices to show that [b1, t ] ⊆ [b, t ] s. to this end, let (f, g) ∈ [b1, t ] be arbitrary and put c = {x ∈ x : (f (x), g(x)) /∈ t }. if c = ∅, then ∀x ∈ x, (f (x), g(x)) ∈ t and in particular (f (x), g(x)) ∈ t for all x lying in any enlargement of b. otherwise, since x 7→ (f (x), g(x)) is continuous and (y ×y )\t is closed, c is nonempty, closed and disjoint from b1. thus for some d ∈ d, c ∩ d[b] = ∅, which means that for each x ∈ d[b], (f (x), g(x)) ∈ t as required. for (2), suppose b0 ∈ b has no shield in b. as any shield for cl(b0) is a shield for b0 as well, we may assume that b0 is closed. direct b0 := {b ∈ b : b0 ⊆ b} by inclusion. for each b ∈ b0, pick cb closed and disjoint from b yet asymptotic to b0. then by normality choose fb ∈ c(x, [0, 1]) with fb(b) = {0} and fb(cb ) = {1}. then if f is the zero function on x, the net 〈fb〉b∈b0 tb-converges to f , whereas t s b -convergence fails: the uniform neighborhood [b0, {(α, β) : |α − β| < 1 2 }]s(f ) ∩ (c(x, y ) × c(x, y )) = {g ∈ c(x, r) : |f (x) − g(x)| < 1 2 ∀x in some enlargement of b0} contains no fb for b ∈ b0. � we now characterize shielded from closed sets for a bornology with closed base in a general hausdorff uniform space in terms of the validity of a dinitype theorem [8, pg. 19]. this result has no antecedent in the metric context. recall that f : x → r is called upper semicontinuous if for each α ∈ r, {x ∈ x : f (x) ≥ α} is a closed subset of x; equivalently, its hypograph {(x, α) : x ∈ x, α ∈ r and α ≤ f (x)} is a closed subset of x × r [8, pg. 20]. theorem 3.4. let b be a bornology with closed base on a hausdorff uniform space 〈x, d〉. the following conditions are equivalent: (1) the bornology b is shielded from closed sets; (2) whenever 〈fλ〉λ∈λ is a net of upper semicontinuous functions tbconvergent from above to f ∈ c(x, r), then the net is t s b -convergent to f . 124 g. beer proof. (1) ⇒ (2). fix b ∈ b and let ε > 0. let b1 be a shield for b in b. it suffices to show that if ∃λ ∈ λ such that ∀x ∈ b1, we have fλ(x) < f (x) + ε, then for some d ∈ d and all x ∈ d(b), we have fλ(x) < f (x) + ε. put c := {x ∈ x : fλ(x) ≥ f (x) + ε}. if c is empty, we are done. otherwise by the upper semicontinuity of fλ − f, c is closed, nonempty and disjoint from b1, so for some d ∈ d we have d(b) ∩ c = ∅ as required. (2) ⇒ (1). suppose b0 ∈ b has no shield in b. without loss of generality we may assume b0 is closed. letting b0 and for each b ∈ b0, cb be exactly as in the proof of theorem 3.3, denote the characteristic function of cb by χb. since cb is closed, χb is upper semicontinuous, and it is verified exactly as in the proof of the last theorem that the net 〈χb〉b0⊆b is tb-convergent (from above) to the zero function but is not t s b -convergent. � as another application of shields, we show that if b is a bornology on a hausdorff uniform space 〈x, d〉 that is shielded from closed sets, then each function on 〈x, d〉 with values in a hausdorff uniform space 〈y, t〉 that is uniformly continuous when restricted to elements of b must lie in cs b (x, y ). our proof bears no resemblance to the one given in the metric context [14]. theorem 3.5. let 〈x, d〉 and 〈y, t〉 be hausdorff uniform spaces and let b be a bornology on x that is shielded from closed sets. suppose f ∈ c(x, y ) is uniformly continuous when restricted to each element of the bornology. then f is strongly uniformly continuous on b. proof. fix b ∈ b and let b1 ∈ b be a shield for b. given t ∈ t, we must produce an entourage d ∈ d so that whenever b ∈ b, x ∈ x and x ∈ d(b), then (f (b), f (x)) ∈ t . choose a symmetric t1 with t1◦t1 ⊆ t , and then by uniform continuity of f on b1 a symmetric entourage d such that {b1, b2} ⊆ b1 and (b1, b2) ∈ d ⇒ (f (b1), f (b2)) ∈ t1. let d̂ satisfy d̂◦d̂ ⊆ d. since f ∈ c(x, y ) for each b1 ∈ b1, ∃ an open neighborhood vb1 of b1 contained in d̂(b1) such that ∀x ∈ vb1 we have ((f (x), f (b1)) ∈ t1. put v = ∪b1∈b1 vb1 . now suppose x ∈ v and b ∈ b satisfy (x, b) ∈ d̂. choosing b1 ∈ b1 with x ∈ vb1 , we have (b, b1) ∈ d so that (f (b), f (b1)) ∈ t1. it follows that (f (b), f (x)) ∈ t . but since b1 is a shield for b there exists a symmetric entourage d ⊆ d̂ for which d(b) ⊆ v . this choice of d does the job. � clearly, in the statement of the last theorem, we must include continuity of f as an assumption (consider the bornology of finite subsets). we also note that the converse of the last result fails in the metric context [14]. there is a uniform topology on y x intermediate in strength between tb and t s b that we wish to introduce, that makes sense when the domain x is only a topological space. with t denoting the topology of x, a base for its entourages consists of these subsets of y x × y x : [b, t ]2 := {(f, g) : ∃u ∈ t such that b ⊆ u and ∀x ∈ u, (f (x), g(x)) ∈ t }, the alexandroff property and the preservation of strong uniform continuity 125 where b runs over the bornology b and t runs over t. it is left to the reader to verify that the standard conditions for a base for a uniformity are satisfied [24]. we denote the induced topology by t 2 b in the sequel. since [b, t ]2 ⊆ [b, t ], clearly, tb is coarser than t 2 b . when the domain is equipped with a diagonal uniformity, we have for each entourage t and each b ∈ b, [b, t ]s ⊆ [b, t ]2, and so t 2 b is coarser than t s b . the intermediate topology on y x for the bornology f0(x) was studied by bouleau under the name sticking topology [16, 17]. the reason that this intermediate topology has attracted no interest whatsoever when restricted to continuous functions is the following. proposition 3.6. let 〈x, t 〉 be a hausdorff space and let 〈y, t〉 be a hausdorff uniform space. suppose b is a bornology on x. then the standard uniformities for tb and t 2 b when restricted to c(x, y ) agree. proof. let b ∈ b and t ∈ t be given. we intend to show that [b, t0] ∩ (c(x, y ) × c(x, y )) ⊆ [b, t ] 2 ∩ (c(x, y ) × c(x, y )), where t0 ∈ t is a symmetric entourage chosen such that t 3 0 ⊆ t . let f and g be continuous functions with (f, g) ∈ [b, t0]. choosing for each b ∈ b, ub ∈ t such that ∀x ∈ ub both (f (b), f (x)) ∈ t0 and (g(b), g(x)) ∈ t0, then with u := ∪b∈b ub, we have ∀x ∈ u, (f (x), g(x)) ∈ t . � example 3.7. we present a pointwise = tf0(x)-convergent sequence of realvalued continuous functions on [0, ∞) that fails to be t 2 f0(x) -convergent. for each n ∈ n define gn : [0, ∞) → r by gn(x) = { 1 − nx if 0 ≤ x ≤ 1/n 0 otherwise . while 〈gn〉 converges pointwise to the characteristic function of the origin χ{0}, the uniform distance between each gn and the limit is one on each neighborhood of {0}. in the next example, we show that t s b can be properly finer than t 2 b even on c(x, r). example 3.8. in the plane r2 equipped with the usual metric, let x be the metric subspace {(n, 0) : n ∈ n} ∪ {(n, 1 n ) : n ∈ n}. since the topology of x is discrete, each real function defined on x is continuous. let us put e := {(n, 0) : n ∈ n}, and consider the bornology b on x consisting of all subsets of the form a ∪ f where a is a (possibly empty) subset of e and f ∈ f0(x). for each n ∈ n, define fn : x → r by fn(x) = { 0 if x = (j, 1 j ) for some j ≤ n 1 otherwise . 126 g. beer evidently, the sequence 〈fn〉 is uniformly convergent to the characteristic function χe of e on elements of the bornology, and since each b ∈ b is open in x, we have t 2 b -convergence. but each fn has uniform distance one from χe on any enlargement of e. by theorem 3.3, the topologies t s b and t 2 b agree on c(x, y ) provided the bornology is shielded from closed sets. actually, they agree on y x under this assumption. proposition 3.9. let b be a bornology on a hausdorff uniform space 〈x, d〉. the following conditions are equivalent: (1) b is shielded from closed sets; (2) for each hausdorff uniform space 〈y, t〉, the natural uniformities on t 2 b and t s b agree on y x ; (3) t 2 b and t s b agree on rx . proof. (1) ⇒ (2). let b ∈ b and t ∈ t be given. choosing a shield b1 ∈ b for b, it is evident that [b1, t ] 2 ⊆ [b, t ]s. (2) ⇒ (3). this is trivial. (3) ⇒ (1). we prove the contrapositive. if (1) fails, there exists b0 ∈ b such that each superset b ∈ b has an open neighborhood vb that contains no enlargement of b0. put b0 := {b ∈ b : b0 ⊆ b}, and ∀b ∈ b0, ∀d ∈ d pick x(b,d) lying in d(b0)\vb. next for each b ∈ b0 write e(b) = {x(b,d) : d ∈ d}. directing b0 by inclusion, we claim that the net 〈χe(b)〉b∈b0 converges in t 2 b to the zero function, which we denote by f . to see this, fix b̂ ∈ b and set b̂1 = b̂ ∪b0 ∈ b0. whenever b̂1 ⊆ b, χe(b) is zero on vb and hence is zero on an open neighborhood of b̂, establishing the claim. on the other hand, the net fails to be t s b -convergent to f , as the uniform distance between each χe(b) and f is one when the characteristic functions are restricted to any enlargement of b0. � 4. the bornological alexandroff property in this section we introduce the bornological alexandroff property, that combined with tb-convergence of nets of functions strongly uniformly continuous on b, is at once necessary and sufficient for t s b convergence, and for strong uniform continuity of the limit. recall that an open cover of a uniform space 〈x, d〉 is called a uniform cover [24] if for some d ∈ d, the cover is refined by {d(x) : x ∈ x}. this naturally leads to the following definition that mixes large and small, the characteristic feature of bornological analysis. definition 4.1. let b be a bornology on a hausdorff uniform space 〈x, d〉. an open cover v of x is called a bornological uniform cover or a b-uniform cover of x if for each b ∈ b there exists an entourage d such that {d(b) : b ∈ b} refines v . if in each case, d can be chosen so that {d(b) : b ∈ b} refines some finite subfamily of v , then the cover is called a b-finitely uniform cover the alexandroff property and the preservation of strong uniform continuity 127 each uniform cover of x is a bornological uniform cover with respect to each bornology on x, and when b = p0(x), each bornological uniform cover is a uniform cover because x ∈ b. given a family a of nonempty subsets of x, we could define the notion of a -uniform cover exactly as in the above definition. but there is no loss of generality in assuming that a is already a bornology in fact a bornology with closed base for if v is a a -uniform cover, then it is a b-uniform cover with respect to the smallest bornology containing {cl(a) : a ∈ a }. similar remarks apply to bornological finitely uniform covers. the next proposition motivates our looking at bornological uniform covers in the present context. proposition 4.2 (cf. [24, theorem 36.8]). let b be a bornology on a hausdorff uniform space 〈x, d〉 and let 〈y, t〉 be a second hausdorff uniform space. then f ∈ c(x, y ) is strongly uniformly continuous on b if and only if for each open uniform cover v of y, {f −1(v ) : v ∈ v } is a b-uniform cover of x. proof. for sufficiency, let b ∈ b and t ∈ t be arbitrary, and choose a symmetric open entourage t0 such that t0 ◦ t0 ⊆ t . as {t0(y) : y ∈ y } is a uniform open cover of y , there exists d ∈ d such that {d(b) : b ∈ b} refines {f −1(t0(y)) : y ∈ y }. now fix b ∈ b and suppose (b, x) ∈ d. choosing y ∈ y with d(b) ⊆ f −1(t0(y)), we have both (f (b), y) ∈ t0 and (y, f (x)) ∈ t0, and so (f (b), f (x)) ∈ t as required. for necessity let v be an open uniform cover of y , and choose an open symmetric entourage t such that {t (y) : y ∈ y } refines v . fix b in the bornology and choose by strong uniform continuity a symmetric entourage d such that whenever (b, x) ∈ (b × x) ∩ d, we have (f (b), f (x)) ∈ t . choosing v ∈ v with t (f (b)) ⊆ v , we have d(b) ⊆ f −1(v ) as required. � the following result is an immediate consequence of lemma 2.1. proposition 4.3. let b be a bornology on a uniform space 〈x, d〉 consisting of a family of relatively compact sets. then each open cover v of x is a b-finitely uniform cover. before we state an appropriate modification of the alexandroff property with respect to strong uniform convergence on bornologies, for the record, we restate the classical alexandroff property for nets of functions defined on a hausdorff space with values in a hausdorff uniform space. definition 4.4. let 〈x, t 〉 be a hausdorff space and let 〈y, t〉 be a hausdorff uniform space. let f : x → y and let 〈fλ〉λ∈λ be a net in y x . then 〈fλ〉λ∈λ is said to have the alexandroff property with respect to f provided for each λ0 ∈ λ and t ∈ t, there exists a cofinal subset λ0 of {λ ∈ λ : λ � λ0} and an open cover {vλ : λ ∈ λ0} of x such that for each λ ∈ λ0, ∀x ∈ vλ, we have (fλ(x), f (x)) ∈ t . definition 4.5. let b be a bornology on a hausdorff uniform space 〈x, d〉 and let 〈y, t〉 be a second hausdorff uniform space. let 〈fλ〉λ∈λ be a net in y x . we say that 〈fλ〉λ∈λ has the bornological alexandroff property with respect 128 g. beer to f : x → y and b provided for each λ0 ∈ λ and t ∈ t, there exists a cofinal subset λ0 of {λ ∈ λ : λ � λ0} and a b-finitely uniform cover {vλ : λ ∈ λ0} such that for each λ ∈ λ0, ∀x ∈ vλ, we have (fλ(x), f (x)) ∈ t . in view of proposition 4.3, we may immediately state proposition 4.6. let b be a bornology of relatively compact subsets on a hausdorff uniform space 〈x, d〉 and let 〈y, t〉 be a second hausdorff uniform space. let 〈fλ〉λ∈λ be a net in y x . then the net has the alexandroff property with respect to f ∈ y x if and only if the net has the bornological alexandroff property with respect to f and b. we now come to our main theorem. theorem 4.7. let 〈x, d〉 and 〈y, t〉 be hausdorff uniform spaces. let b be a bornology on x and let 〈fλ〉λ∈λ be a net in c s b (x, y ) that is tb-convergent to f : x → y . the following conditions are equivalent: (1) f ∈ cs b (x, y ); (2) for each b ∈ b, t0 ∈ t and λ0 ∈ λ, there exists a finite set of indices {λ1, λ2, ..., λn} such that ∀j ≤ n, λj � λ0 and an entourage d̃ ∈ d such that ∀x ∈ d̃(b), ∃j ∈ {1, 2, . . . , n} such that (f (x), fλj (x)) ∈ t ; (3) 〈fλ〉λ∈λ has the bornological alexandroff property with respect to f : x → y and b; (4) 〈fλ〉λ∈λ is t s b -convergent to f . proof. (1) ⇒ (4). fix t ∈ t and b ∈ b, and let t0 be a symmetric entourage with t 30 ⊆ t . fix b ∈ b; by uniform convergence on b, choose λb ∈ λ such that ∀λ � λb, ∀b ∈ b, we have (fλ(b), f (b)) ∈ t0. by strong uniform continuity of each fλ and f on b, there exists a symmetric entourage db,λ ∈ d such that if b ∈ b and x ∈ x and (x, b) ∈ db,λ, we have both (fλ(x), fλ(b)) and (f (x), f (b)) in t0. it follows that ∀λ � λb , ∀x ∈ db,λ[b], (fλ(x), f (x)) ∈ t, which means that (fλ, f ) ∈ [b, t ] s whenever λ � λb . (4) ⇒ (3). fix λ0 ∈ λ and t ∈ t. we will actually produce a residual subset λ0 of {λ : λ � λ0} and an open cover {vλ : λ ∈ λ0} of x such that ∀λ ∈ λ0, fλ restricted to vλ is t -close to f and for each b ∈ b, a single index λ ∈ λ0 and an entourage d such that {d(b) : b ∈ b} refines vλ. to this end, for each b ∈ b, choose by strong uniform convergence an index λb � λ0 and for each λ � λb an open symmetric entourage db,λ such that ∀x ∈ db,λ(b), (fλ(x), f (x)) ∈ t. now fix x0 ∈ x and for each λ � λ{x0}, put aλ := {b ∈ b : λ � λb}. the alexandroff property and the preservation of strong uniform continuity 129 note that ∀λ � λ{x0}, we have {x0} ∈ aλ and the families aλ increase with λ. for each λ � λ{x0}, put vλ := ∪b∈aλ db,λ(b). then • {vλ : λ � λ{x0}} is an open cover of x; • ∀λ � λ{x0}, ∀x ∈ vλ, (fλ(x), f (x)) ∈ t ; • ∀b ∈ b, we can choose an index λ such that λ � λ{x0} and λ � λb, and by construction ∀b ∈ b, db,λ(b) ⊆ vλ. (3) ⇒ (2). fix b ∈ b, t ∈ t, and λ0 ∈ λ. apply the bornological alexandroff property with respect to λ0 and t to obtain a cofinal subset λ0 of {λ ∈ λ : λ � λ0} and a b-finitely uniform cover {vλ : λ ∈ λ0} of x such that ∀λ ∈ λ0, ∀x ∈ vλ, (fλ(x), f (x)) ∈ t. by the definition of bornological finitely uniform cover, there exists an entourage d̃ and a finite set of indices {λ1, λ2, . . . , λn} in λ0 such that {d̃(b) : b ∈ b} refines {vλj : 1 ≤ j ≤ n}. then for each x ∈ d̃(b) we see that x lies in some vλj so that (fλj (x), f (x)) ∈ t . (2) ⇒ (1). fix b ∈ b and t ∈ t. choose a symmetric entourage t0 such that t 30 ⊆ t and then by tb-convergence, an index λ0 ∈ λ such that λ � λ0 ⇒ ∀b ∈ b, (fλ(b), f (b)) ∈ t0. choose relative to b, t0 and λ0 the indices {λ1, λ2, ..., λn} and the entourage d̃ guaranteed by condition (2). by the strong uniform continuity of each fλj on b, we can choose a symmetric entourage d ∈ d such that d ⊆ d̃ and whenever b ∈ b and (x, b) ∈ d we have (fλj (x), fλj (b)) ∈ t0 for j = 1, 2, 3, . . . , n. we intend to show that whenever b ∈ b and x ∈ x and (x, b) ∈ d, we have (f (x), f (b)) ∈ t . for such an x and b, choose j ∈ {1, 2, ..., n} such that (f (x), fλj (x)) ∈ t0. by construction we have these properties: • (f (x), fλj (x)) ∈ t0; • (fλj (x), fλj (b)) ∈ t0 by strong uniform continuity of fλj on b and the choice of d; • (fλj (b), f (b)) ∈ t0 by the choice of λ0 and uniform convergence on b. it follows that (f (x), f (b)) ∈ t as required for the strong uniform continuity of f on b. � 130 g. beer corollary 4.8. let 〈x, d〉 and 〈y, t〉 be hausdorff uniform spaces. then on cs b (x, y ), the topology t s b reduces to tb. corollary 4.9. let 〈x, d〉 and 〈y, ρ〉 be metric spaces and let 〈fn〉 be a sequence in cs b (x, y ) tb-convergent to f : x → y . then f is strongly uniformly continuous on b if and only if each ε > 0 and n0 ∈ n, there exists a strictly increasing sequence 〈nk〉 of integers such that n1 ≥ n0 and a countable open cover {vk : k ∈ n} of x such that ∀k ∈ n, ∀x ∈ vk, we have ρ(f (x), fnk (x)) < ε, and for each b ∈ b, ∃δ > 0 such that {sd(b, δ) : b ∈ b} refines some finite subfamily of {vk : k ∈ n}. in [7, definition 2.2], bartle introduced a property that, combined with pointwise convergence of a net of continuous functions with values in a metric space 〈y, ρ〉 to a function f , is necessary and sufficient for continuity of the limit, provided the domain is a compact hausdorff space: ∀λ0 ∈ λ, ∀ε > 0, there exists a finite set of indices {λ1, λ2, ..., λn} such that ∀j ≤ n, λj � λ0 and for each x ∈ x, ∃ j ≤ n with ρ(f (x), fλj (x)) < ε. bartle called pointwise convergence plus this property quasi-uniform convergence which, unfortunately, is exactly what alexandroff called pointwise convergence plus the alexandroff property [1, pg. 265]. condition (2) of theorem 4.7 may be regarded as a variationalbornological version of bartle’s property, in that it requires not only that the property hold on each b ∈ b but that it almost holds around the edge of each b as well. in the following example, we show that the implication (3) ⇒ (1) in theorem 4.7 fails if in the definition of the alexandroff property, we replace ”b-finitely uniform cover” by ”b-uniform cover”. it suffices to work in the metric context and with sequences of real-valued continuous functions. example 4.10. we revisit the construction in example 3.8. the reader can easily verify that each fn is strongly uniformly continuous on b, while the characteristic function χe of e fails to be strongly uniformly continuous on e as each enlargement of e contains infinitely many points of x\e. for each n ∈ n put vn := e ∪ {(j, 1 j ) : j ≤ n}. by construction, each fn agrees with χe when restricted to vn. by the discreteness of x, for each n0 ∈ n, vn0 := {vn : n ≥ n0} is an open cover of x. notice that each vn0 is actually a uniform cover of x, as it is refined by all open balls of radius 1 2 . in particular, this makes vn0 a b-uniform cover of x. further, each b ∈ b has a finite subcover from vn0 ; in fact b will be contained in vn for all n sufficiently large. but for each ε > 0, {sd(x, ε) : x ∈ e} fails to refine any finite subfamily of vn0 . given a net of functions 〈fλ〉λ∈λ defined on a set x perhaps without further structure with values in a metric or uniform space y that is pointwise convergent to some f : x → y , we can consider the family of nonempty subsets b of x for which 〈fλ〉λ∈λ restricted to b has the bartle property with respect to f . as is easily checked, the family of such sets forms a bornology. the last example shows that uniform convergence on elements of the bornology need not preserve strong uniform continuity on the bornology, and so there is no the alexandroff property and the preservation of strong uniform continuity 131 hope that quasi-uniform convergence in the sense of bartle on elements of the bornology can either. when b is a bornology on x that is shielded from closed sets, we have already seen that cs b (x, y ) = cb(x, y ), and further, t s b reduces to tb on c(x, y ). thus, if we restrict our attention to continuous functions defined on bornologies that are shielded from closed sets, we are reduced to the classical setting. for this reason, the classical function spaces based say on bornologies with a compact base or on the bornology of metrically bounded subsets [22] seem adequate when in fact they more generally are not, as described in some detail in [13]. in the particular case that the bornology is one with a compact base, then as we have noted in section 2, the strongly uniformly continuous functions reduce to c(x, y ). since the alexandroff property for a net of continuous functions coupled with pointwise convergence gives continuity of the limit without assuming any uniform structure on the domain, one would guess that a version of theorem 4.7 can be stated without any uniform structure on the domain. this expectation is of course enhanced by proposition 4.6. the proof of the following result in this direction, obtained by modifying the proof of theorem 4.7, is left to the reader (we suggest the circuit (1) ⇒ (4) ⇒ (3) ⇒ (2) ⇒ (1), noting that (4) ⇒ (3) only requires t 2 f0(x) -convergence). theorem 4.11 (cf. [18, theorem 2.10]). let 〈x, t 〉 be a hausdorff space and 〈y, t〉 be a hausdorff uniform space. suppose b be a bornology on x with compact base, and let 〈fλ〉λ∈λ be a net in c(x, y ) tb-convergent to f : x → y . the following conditions are equivalent: (1) f ∈ c(x, y ); (2) for each nonempty compact subset c of x, t0 ∈ t and λ0 ∈ λ, there exists a finite set of indices {λ1, λ2, ..., λn} such that ∀j ≤ n, λj � λ0 and a neighborhood u of c such that ∀x ∈ u, ∃j ∈ {1, 2, . . . , n} such that (f (x), fλj (x)) ∈ t ; (3) 〈fλ〉λ∈λ has the classical alexandroff property with respect to f ; (4) 〈fλ〉λ∈λ is t 2 b -convergent to f . when b = f0(x) in theorem 4.11, we see that a pointwise limit of a net of continuous functions is continuous if and only if we have t 2 f0(x) -convergence [16, 17]. while we know of no reference for the equivalence of condition (2) with the preservation of continuity under pointwise convergence, it would be remarkable if this has not appeared in the literature. by proposition 3.6, the intermediate topology collapses to the topology of pointwise convergence for continuous functions, so it is in some sense the weakest topology finer than the topology of pointwise convergence preserving continuity. again, when the domain is a uniform space, in view of proposition 3.9, on y x , we get t 2 f0(x) = t s f0(x) because the bornology has a compact base and so is shielded from closed sets. this was the form in which the equivalence 132 g. beer of conditions (1) and (4) of theorem 4.11 was given by beer and levi [13, corollary 6.8] in the setting of metric spaces as a corollary to a general result involving the preservation of strong uniform continuity on bornologies. references 1. p. alexandroff, einführing in die mengenlehre und die theorie der rellen funktionen, deutscher verlag der wissenschaften, berlin, 1964 2. c. arzelà, intorno alla continuitá della somma di infinitá di funzioni continue, rend. r. accad. sci. bologna (1883-84), 79-84. 3. c. arzelà, sulle serie di funzioni, mem. r. accad. sci. ist. bologna, serie 5 (8) (18991900), 131-186 and 701-744. 4. h. attouch, r. lucchetti, and r. wets, the topology of the ρ-hausdorff distance, ann. mat. pura appl. 160 (1991), 303-320. 5. h. attouch and r. wets, quantitative stability of variational systems: i. the epigraphical distance, trans. amer. math. soc. 328 (1991), 695-730. 6. m.atsuji, uniform continuity of continuous functions of metric spaces, pacific j. math. 8 (1958), 11-16. 7. r. bartle, on compactness in functional analysis, trans. amer. math. soc. 79 (1955), 35-57. 8. g. beer, topologies on closed and closed convex sets, kluwer acad. publ., dordrecht, 1993. 9. g. beer, c. costantini, and s. levi, bornological convergence and shields, preprint. 10. g. beer, c. costantini, and s. levi, total boundedness in metrizable spaces, houston j. math., to appear. 11. g. beer and a. di concilio, uniform continuity on bounded sets and the attouch-wets topology, proc. amer. math. soc. 112 (1991), 235-243. 12. g. beer and s. levi, pseudometrizable bornological convergence is attouch-wets convergence, j. convex anal. 15 (2008), 439-453. 13. g. beer and s. levi, strong uniform continuity, j. math. anal. appl. 350 (2009), 568589. 14. g. beer and s. levi, uniform continuity, uniform convergence, and shields, set-valued and variational anal. 18 (2010), 251–275. 15. g. beer and m. segura, well-posedness, bornologies, and the structure of metric spaces, appl. gen. top. 10 (2009), 131-157. 16. n. bouleau, une structure uniforme sur un espace f(e, f), cahiers topologie géom. diff., 11 (1969), 207-214. 17. n. bouleau, on the coarsest topology preserving continuity, preprint, 2006. 18. a. caserta, g. di maio and l. holá, arzelà’s theorem and strong uniform convergence on bornologies, j. math. anal. appl. 371 (2010), 384-392. 19. n. dunford and j. schwartz, linear operators part i, wiley interscience, new york, 1988 20. h. hogbe-nlend, bornologies and functional analysis, north-holland, amsterdam, 1977. 21. s.-t. hu, boundedness in a topological space, j. math pures appl. 228 (1949), 287-320. 22. r. mccoy and i. ntantu, topological properties of spaces of continuous functions, springer verlag, berlin, 1988. 23. j. rainwater, spaces whose finest uniformity is metric, pacific j. math 9 (1959), 567-570. 24. s. willard, general topology, addison-wesley, reading, ma, 1970. the alexandroff property and the preservation of strong uniform continuity 133 received july 2010 accepted october 2010 gerald beer (gbeer@cslanet.calstatela.edu) department of mathematics, california state university los angeles, 5151 state university drive, los angeles, california 90032, usa. the alexandroff property and the preservation of strong uniform continuity. by g. beer @ appl. gen. topol. 21, no. 1 (2020), 135-158 doi:10.4995/agt.2020.12220 c© agt, upv, 2020 fejér monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued banach spaces godwin amechi okeke a and mujahid abbas b a department of mathematics, school of physical sciences, federal university of technology, owerri, p.m.b. 1526 owerri, imo state, nigeria (godwin.okeke@futo.edu.ng) b department of mathematics, government college university, 54000 lahore, pakistan department of mathematics and applied mathematics, university of pretoria ( hatfield campus), lynnwood road, pretoria 0002, south africa (abbas.mujahid@gmail.com) communicated by s. romaguera abstract it is our purpose in this paper to prove some fixed point results and fejér monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued banach spaces. we prove that results in complex valued banach spaces are valid in cone metric spaces with banach algebras. furthermore, we apply our results in solving certain mixed type volterrafredholm functional nonlinear integral equation in complex valued banach spaces. 2010 msc: 47h09; 47h10; 49m05; 54h25. keywords: complex valued banach spaces; fixed point theorems; fejér monotonicity; iterative processes; cone metric spaces with banach algebras; mixed type volterra-fredholm functional nonlinear integral equation. 1. introduction fixed point theory, which is famous in sciences and engineering due to its applications in solving several nonlinear problems in these fields of study became one of the most interesting area of research in the last sixty years. for example, received 16 august 2019 – accepted 18 december 2019 http://dx.doi.org/10.4995/agt.2020.12220 g. a. okeke and m. abbas it has shown the importance of theoretical subjects, which are directly applicable in different applied fields of science. other areas of applications includes optimization problems, control theory, economics and a host of others. in particular, it plays an important role in the investigation of existence of solutions to differential and integral equations, which direct the behaviour of several real life problems for which the existence of solution is critical (see, e.g. [25], [42]). in 1922, banach [12] provided a general iterative method to construct a fixed point result and proved its uniqueness under linear contraction in complete metric spaces. this famous results of banach have been generalized in several directions by many researchers. these generalization were made either by using the contractive condition or by imposing some additional conditions on the ambient space. some of these generalizations of metric spaces includes: rectangular metric spaces, pseudo metric spaces, d-metric spaces, partial metric spaces, g-metric spaces and cone metric spaces (see, e.g. [1], [22], [23]). the notion of complex valued metric spaces was introduced by azam et al. [11] in 2011. they established some fixed point theorems for a pair of mappings satisfying rational inequality. their results is intended to define rational expressions which are meaningless in cone metric spaces. although complex valued metric spaces form a special class of cone metric spaces (see, e.g. [2], [6]), yet the definition of cone metric spaces rely on the underlying banach space which is not a division ring. consequently, rational expressions are not meaningful in cone metric spaces, this means that results involving mappings satisfying rational expressions cannot be generalized to cone metric spaces. inview of this deficiency, azam et al. [11] introduced the concept of complex valued metric spaces. it is known that in cone metric spaces the underlying metric assumes values in linear spaces where the linear space may be even infinite dimensional, whereas in the case of complex valued metric spaces the metric values belong to the set of complex numbers which is one dimensional vector space over the complex field. this instance is the major motivation for the consideration of complex valued metric spaces independently (see, [6]). hence, results in this direction cannot be generalized to cone metric spaces, but to complex valued metric spaces. it is known that complex valued metric space is useful in many branches of mathematics, including number theory, algebraic geometry, applied mathematics as well as in physics including hydrodynamics, mechanical engineering, thermodynamics and electrical engineering (see, e.g. [41]). several authors have obtained interesting and applicable results in complex valued metric spaces (see, e.g. [2], [3], [5], [8], [6], [11], [25], [36], [40], [41], [42]). it is known that there is a close relationship between the problem of solving a nonlinear equation and that of approximating fixed points of a corresponding contractive type operator (see, e.g. [14], [15], [32]). hence, there is a practical and theoretical interests in approximating fixed points of several contractive type operators. since, the introduction of the notion of complex valued metric spaces by azam et al. [11] in 2011, most results obtained in literature by many c© agt, upv, 2020 appl. gen. topol. 21, no. 1 136 fejér monotonicity and fixed point theorems authors are existential in nature (see, e.g. [8], [11], [36], [41], [42]). consequently, there is a gap in literature with respect to the approximation of the fixed point of several nonlinear mappings in this type of space. recently, okeke [32] exploited the idea of complex valued metric spaces to define the concept of complex valued banach spaces and then initiated the idea of approximating the fixed point of nonlinear mappings in complex valued banach spaces. the theory of integral and differential equations is an important aspect of nonlinear analysis and the most applied tool for proving the existence of the solutions of such equations is the fixed point technique (see, e.g. [12], [18], [19], [33]). one of the most frequent and difficult problems faced by scientists in mathematical sciences is nonlinear problems. this is because nature is intrinsically nonlinear (see, e.g. [19]). solving nonlinear equations is cumbersome but important to mathematicians and applied mathematicians such as engineers and physicist. some authors have used the fixed point iterative methods in solving such equations (see, e.g. [18], [19], [33]). in this paper, we apply our results in solving certain mixed type volterra-fredholm functional nonlinear integral equation in complex valued banach spaces. it is our purpose in this paper to prove some fixed point results and fejér monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued banach spaces. we prove that results in complex valued banach spaces are valid in cone metric spaces with banach algebras. our results validates the fact that fixed point theorems in the setting of cone metric spaces with banach algebras are more useful than the standard results in cone metric spaces and that results in cone metric spaces with banach algebras cannot be reduced to corresponding results in cone metric spaces. furthermore, we apply our results in solving certain mixed type volterra-fredholm functional nonlinear integral equation in complex valued banach spaces. our results unify, generalize and extend several known results to complex valued banach spaces, including the results of [4], [9], [10], [18], [19], [28], [33]) among others. 2. preliminaries the following symbols, notations and definitions which can be found in [11] will be useful in this study. let c be the set of complex numbers and z1,z2 ∈ c. define a partial order on c as follows: z1 z2 if and only if re(z1) ≤ re(z2), im(z1) ≤ im(z2). it follows that z1 z2 if one of the following conditions is satisfied: (i) re(z1) = re(z2), im(z1) < im(z2), (ii) re(z1) < re(z2), im(z1) = im(z2), (iii) re(z1) < re(z2), im(z1) < im(z2), (iv) re(z1) = re(z2), im(z1) = im(z2). in particular, we will write z1 � z2 if z1 6= z2 and one of (i), (ii), and (iii) is satisfied and we will write z1 ≺ z2 if only (iii) is satisfied. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 137 g. a. okeke and m. abbas note that (a) a,b ∈ r and a ≤ b =⇒ az bz for all z ∈ c; (b) 0 z1 � z2 =⇒ |z1| < |z2|; (c) z1 z2 and z2 ≺ z3 =⇒ z1 ≺ z3. definition 2.1 ([11]). let x be a nonempty set. suppose that the mapping d : x ×x → c, satisfies: 1. 0 d(x,y), for all x,y ∈ x and d(x,y) = 0 if and only if x = y; 2. d(x,y) = d(y,x) for all x,y ∈ x; 3. d(x,y) d(x,z) + d(z,y), for all x,y,z ∈ x. then d is called a complex valued metric on x, and (x,d) is called a complex valued metric space. recently, okeke [32] defined a complex valued banach space and proved some interesting fixed point theorems in the framework of complex valued banach spaces. definition 2.2 ([32]). let e be a linear space over a field k, where k = r (the set of real numbers) or c (the set of complex numbers). a complex valued norm on e is a complex valued function ‖.‖ : e → c satisfying the following conditions: 1. ‖x‖ = 0 if and only if x = 0, x ∈ e; 2. ‖kx‖ = |k|.‖x‖ for all k ∈ k, x ∈ e; 3. ‖x + y‖‖x‖ + ‖y‖ for all x,y ∈ e. a linear space with a complex valued norm defined on it is called a complex valued normed linear space, denoted by (e,‖.‖). a point x ∈ e is called an interior point of a set a ⊆ e if there exist 0 ≺ r ∈ c such that b(x,r) = {y ∈ e : ‖x−y‖≺ r}⊆ a. a point x ∈ e is called a limit point of the set a whenever for each 0 ≺ r ∈ c, we have b(x,r) ∩ (ane) 6= ∅. the set a is said to be open if each element of a is an interior point of a. a subset b ⊆ e is said to be closed if it contains each of its limit point. the family f = {b(x,r) : x ∈ e, 0 ≺ r} is a sub-basis for a hausdorff topology τ on e. suppose xn is a sequence in e and x ∈ e. if for all c ∈ c, with 0 ≺ c there exists n0 ∈ n such that for all n > n0, ‖xn −xn+m‖≺ c, then {xn} is called a cauchy sequence in (e,‖.‖). if every cauchy sequence is convergent in (e,‖.‖), then (e,‖.‖) is called a complex valued banach space. example 2.3 ([32]). let e = c be the set of complex numbers. define ‖.‖ : c×c → c by ‖z1 −z2‖ = |x1 −x2| + i|y1 −y2| ∀z1,z2 ∈ c, where z1 = x1 +iy1, z2 = x2 +iy2. clearly, (c,‖.‖) is a complex valued normed linear space. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 138 fejér monotonicity and fixed point theorems example 2.4 ([32]). let e = c be the set of complex numbers. define a mapping ‖.‖ : c×c → c by ‖z1−z2‖ = eik|z1−z2|, ∀z1,z2 ∈ c, where k ∈ [0, π 2 ], z1 = x1+iy1, z2 = x2+iy2. then (c,‖.‖) is a complex valued normed linear space. example 2.5 ([32]). let (c[a,b],‖.‖∞) be the space of all continuous complex valued functions on a closed interval [a,b], endowed with the chebyshev norm ‖x−y‖∞ = max t∈[a,b] |x(t) −y(t)|eik, x,y ∈ c[a,b], k ∈ [0, π 2 ]. then (c[a,b],‖.‖∞) is a complex valued banach space, since the elements of c[a,b] are continuous functions, and convergence with respect to the chebyshev norm ‖.‖∞ corresponds to uniform convergence. we can easily show that every cauchy sequence of continuous functions converges to a continuous function, i.e. an element of the space c[a,b]. in 1975, dass and gupta [21] extended the banach contraction mapping principle by proving the following theorem for mappings satisfying contractive condition of the rational type in the framework of complete metric spaces. theorem 2.6 ([21]). let (x,d) be a complete metric space and let t be a mapping on x. assume that there exist α,β ∈ (0, 1) satisfying α + β < 1 and d(tx,ty) ≤ αd(y,ty) 1 + d(x,tx) 1 + d(x,y) + βd(x,y) (2.1) for all x,y ∈ x. then t has a unique fixed point z. moreover {tnx} converges to z for all x ∈ x. the following theorem for a meir-keeler contraction of the rational type was proved in 2013 by samet et al. [39] in the framework of complete metric spaces. theorem 2.7 ([39]). let (x,d) be a complete metric space and t be a mapping from x into itself. we assume that the following hypothesis holds: given ε > 0, there exists δ(ε) > 0 such that 2ε ≤ d(y,ty) 1 + d(x,tx) 1 + d(x,y) + d(x,y) < 2ε + δ(ε) =⇒ d(tx,ty) < ε. (2.2) then t has a unique fixed point ζ ∈ x. moreover, for any x ∈ x, the sequence {tnx} converges to ζ. in 2007, agarwal et al. [7] introduced the s iteration process as follows:  x0 ∈ d, yn = (1 −βn)xn + βntxn, xn+1 = (1 −αn)txn + αntyn, n ∈ n, (2.3) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 139 g. a. okeke and m. abbas in 2014, gürsoy and karakaya [20] introduced the picard-s iterative process as follows:   x0 ∈ d, zn = (1 −βn)xn + βntxn, yn = (1 −αn)txn + αntzn xn+1 = tyn. (2.4) in 2015, thakur et al. [44] introduced the following iterative process:  x0 ∈ d, zn = (1 −βn)xn + βntxn, yn = t((1 −αn)xn + αnzn) xn+1 = tyn. (2.5) the authors in [44] proved that the thakur iterative process (2.5) converges faster than picard, mann [30], ishikawa [26], s [7], noor [31] and abbas [4] iteration processes for suzuki’s generalized nonexpansive mappings. recently, ullah and arshad [45] introduced the m-iteration. they proved that this iterative process converges faster than all of s [7], picard-s [19], picard, mann [30], ishikawa [26], noor [31], sp [35], cr [17], s∗ [27], abbas [4] and normal-s [38] iteration processes. the following is the m-iteration process introduced by ullah and arshad [45] in 2018.  x0 ∈ d, zn = (1 −αn)xn + αntxn, yn = tzn xn+1 = tyn. (2.6) in 2013, khan [28] introduced the picard-mann hybrid iterative process. the iterative process for one mapping case is given by the sequence {mn}∞n=1.  m1 = m ∈ d, mn+1 = tzn, zn = (1 −αn)mn + αntmn, n ∈ n, (2.7) where {αn}∞n=1 is in (0, 1). khan [28] proved that this iterative process converges faster than all of picard, mann and ishikawa iterative processes in the sense of berinde [15] for contractive mappings. recently, okeke and abbas [33] introduced the picard-krasnoselskii hybrid iterative process defined by the sequence {xn}∞n=1 as follows:  x1 = x ∈ d, xn+1 = tyn, yn = (1 −λ)xn + λtxn, n ∈ n, (2.8) where λ ∈ (0, 1). the authors proved that this new hybrid iteration process converges faster than all of picard, mann, krasnoselskii and ishikawa iterative processes in the sense of berinde [15]. they also used this iterative process to find the solution of delay differential equations. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 140 fejér monotonicity and fixed point theorems definition 2.8 ([15]). let {an}∞n=0, {bn}∞n=0 be two sequences of positive numbers that converge to a, respectively b. assume there exists l = lim n→∞ |an −a| |bn − b| . (2.9) 1. if l = 0, then it is said that the sequence {an}∞n=0 converges to a faster than the sequence {bn}∞n=0 to b; 2. if 0 < l < ∞, then we say that the sequences {an}∞n=0 and {bn}∞n=0 have the same rate of convergence. definition 2.9. let d be a nonempty subset of a complex valued banach space (e,‖.‖). the diameter of d is diamd = sup (x,y)∈d×d |‖x−y‖|. (2.10) the distance to d is the function ‖.‖d : e → c : x → inf |‖x−d‖|. (2.11) the following lemma will be useful in this study. lemma 2.10 ([32]). let (e,‖.‖) be a complex valued banach space and let {xn} be a sequence in e. then {xn} converges to x if and only if |‖xn −x‖|→ 0 as n →∞. lemma 2.11 ([32]). let (e,‖.‖) be a complex valued banach space and let {xn} be a sequence in e. then {xn} is a cauchy sequence if and only if |‖xn −xn+m‖|→ 0 as n →∞. lemma 2.12 ([43]). let {βn}∞n=0 be a nonnegative sequence for which one assumes there exists n0 ∈ n, such that for all n ≥ n0 one has satisfied the inequality βn+1 ≤ (1 −µn)βn + µnγn, where µn ∈ (0, 1), for all n ∈ n, ∑∞ n=0 µn = ∞ and γn ≥ 0, ∀n. then the following inequality holds 0 ≤ lim sup n→∞ βn ≤ lim sup n→∞ γn. 3. fejér monotonicity and fixed point theorems in complex valued banach spaces in this section, we prove some fejér monotonicity and fixed point results in the framework of complex valued banach spaces. our results improves and extend some known results in the framework of complex valued banach spaces, including the results of bauschke and combettes [13], cegielski [16] and dass and gupta [21] among others. we begin this section by defining the concept of fejér monotonicity in the framework of complex valued banach spaces and also provide some examples. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 141 g. a. okeke and m. abbas definition 3.1. let d be a nonempty subset of a complex valued banach space (e,‖.‖) and let {xn} be a sequence in e. then {xn}n∈n is fejér monotone with respect to d if for each x ∈ d and each n ∈ n, ‖xn+1 −x‖‖xn −x‖. (3.1) example 3.2. suppose {xn}n∈n is a bounded sequence in c that is increasing (respectively decreasing). then the sequence {xn}n∈n is fejér monotone with respect to [sup{xn}n∈n, +∞) (respectively (−∞, inf{xn}n∈n]). example 3.3. let d be a nonempty subset of a complex valued banach space (e,‖.‖) and t : d → d be a mapping on d with f(t) := {x ∈ d : tx = x} 6= ∅. assume that there exist α,β ∈ (0, 1) satisfying α + β < 1 and ‖tx−ty‖α‖y −ty‖ 1 + ‖x−tx‖ 1 + ‖x−y‖ + β‖x−y‖ (3.2) for all y ∈ f(t). let x0 ∈ d and set xn+1 = txn, ∀n ∈ n. then {xn}n∈n is fejér monotone with respect to f(t). now, using relation (3.2) together with the facts that α,β ∈ (0, 1) and y ∈ f(t), we have ‖xn+1 −y‖ α‖y −ty‖ 1+‖xn−txn‖ 1+‖xn−y‖ + β‖xn −y‖ = α.0 ( 1+‖xn−txn‖ 1+‖xn−y‖ ) + β‖xn −y‖ = β‖xn −y‖ ‖xn −y‖. (3.3) hence, {xn}n∈n if fejér monotone with respect to f(t). proposition 3.4. let d be a nonempty subset of a complex valued banach space (e,‖.‖) and let {xn}n∈n be a sequence in e. suppose that {xn}n∈n is fejér monotone with respect to d. then the following hold: (i) {xn}n∈n is bounded. (ii) for every x ∈ d, (|‖xn −x‖|)n∈n converges. (iii) {‖.‖d(xn)}n∈n is decreasing and converges. (iv) let m ∈ n and let n ∈ n. then |‖xn+m −xn‖|≤ 2‖.‖d(xn). (3.4) proof. (i) suppose x ∈ d. it follows from (3.1) that {xn}n∈n lies in b(x, |‖x0− x‖|). hence, {xn}n∈n is bounded. (ii) by (3.1), we have |‖xn+1 −x‖|≤ |‖xn −x‖|−→ 0 as n →∞. (3.5) hence, by lemma 2.10, we have that {xn}n∈n −→ x as n →∞. (iii) suppose x ∈ d, since {xn}n∈n is fejér monotone, it follows that {‖.‖d(xn+1)}n∈n = inf |‖xn+1 −x‖|≤ inf |‖xn −x‖| = {‖.‖d(xn)}n∈n (3.6) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 142 fejér monotonicity and fixed point theorems hence, by lemma 2.10, we obtain {‖.‖d(xn)}n∈n −→ 0 as n →∞. (iv) since {xn}n∈n is fejér monotone, then by (3.1), we have |‖xn+m −xn‖| ≤ |‖xn+m −x‖| + |‖xn −x‖| ≤ 2|‖xn −x‖|. (3.7) by taking infimum over x ∈ d in (3.7), we have the desired result. the proof of proposition 3.4 is completed. � proposition 3.5. let d be a nonempty closed convex subset of a complex valued banach space (e,‖.‖) and t : d → d be a mapping on d with f(t) := {x ∈ d : tx = x} 6= ∅. assume that there exist λ,β ∈ (0, 1) satisfying λ+β < 1 and ‖tx−ty‖λ‖y −ty‖ 1 + ‖x−tx‖ 1 + ‖x−y‖ + β‖x−y‖ (3.8) for all x,y ∈ d. for arbitrary chosen x0 ∈ d, let the sequence {xn} be generated by the m-iteration process (2.6), where αn ∈ (0, 1) for each n ∈ n, then {xn} is fejér monotone with respect to f(t). proof. suppose p ∈ f(t), then by (2.6), (3.8) and the facts that λ,β ∈ (0, 1) and αn ∈ (0, 1) for all n ∈ n, we obtain ‖xn+1 −p‖ = ‖tyn −p‖ λ‖p−tp‖1+‖yn−tyn‖ 1+‖yn−p‖ + β‖yn −p‖ = λ.0 ( 1+‖yn−tyn‖ 1+‖yn−p‖ ) + β‖yn −p‖ = β‖yn −p‖ = β‖tzn −p‖ β [ λ‖p−tp‖1+‖zn−tzn‖ 1+‖zn−p‖ + β‖zn −p‖ ] λ.0 ( 1+‖zn−tzn‖ 1+‖zn−p‖ ) + β‖zn −p‖ = β‖zn −p‖ = β[‖(1 −αn)xn + αntxn −p‖] (1 −αn)‖xn −p‖ + αn‖txn −p‖ (1 −αn)‖xn −p‖ + αn [ λ‖p−tp‖1+‖xn−txn‖ 1+‖xn−p‖ + β‖xn −p‖ ] = (1 −αn)‖xn −p‖ + αn [ λ.0 ( 1+‖xn−txn‖ 1+‖xn−p‖ ) + β‖xn −p‖ ] = (1 −αn)‖xn −p‖ + αnβ‖xn −p‖ (1 −αn)‖xn −p‖ + αn‖xn −p‖ = ‖xn −p‖. (3.9) this means that ‖xn+1 − p‖ ‖xn − p‖ as desired. therefore, {xn} is fejér monotone. the proof of proposition 3.5 is completed. � theorem 3.6. let d be a nonempty closed convex subset of a complex valued banach space (e,‖.‖) and t : d → d be a mapping on d. assume that there exist λ,β ∈ (0, 1) satisfying λ + β < 1 and ‖tx−ty‖λ‖y −ty‖ 1 + ‖x−tx‖ 1 + ‖x−y‖ + β‖x−y‖ (3.10) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 143 g. a. okeke and m. abbas for all x,y ∈ d. for arbitrary chosen x0 ∈ d, let the sequence {xn} be generated by the m-iteration process (2.6), where αn ∈ (0, 1) for each n ∈ n, and∑∞ n=0 αn = ∞. then {xn} converges strongly to a unique fixed point p of t. proof. we want to show that xn −→ p as n → ∞. now, using relation (2.6) and (3.10), we obtain: ‖xn+1 −p‖ = ‖tyn −tp‖ λ‖p−tp‖1+‖yn−tyn‖ 1+‖yn−p‖ + β‖yn −p‖ = λ.0 ( 1+‖yn−tyn‖ 1+‖yn−p‖ ) + β‖yn −p‖ = β‖yn −p‖. (3.11) next, we obtain the following estimate: ‖yn −p‖ = ‖tzn −tp‖ λ‖p−tp‖1+‖zn−tzn‖ 1+‖zn−p‖ + β‖zn −p‖ = λ.0 ( 1+‖zn−tzn‖ 1+‖zn−p‖ ) + β‖zn −p‖ = β‖zn −p‖ = β‖(1 −αn)xn + αntxn −p‖ β(1 −αn)‖xn −p‖ + βαn‖txn −tp‖ β(1 −αn)‖xn −p‖ + βαn [ λ‖p−tp‖1+‖xn−txn‖ 1+‖xn−p‖ + β‖xn −p‖ ] = β(1 −αn)‖xn −p‖ + β2αn‖xn −p‖ = β(1 −αn(1 −β))‖xn −p‖. (3.12) using (3.12) in (3.11), we have ‖xn+1 −p‖ β‖yn −p‖ β2(1 −αn(1 −β))‖xn −p‖. (3.13) continuing this process gives the following relations  ‖xn+1 −p‖β2(1 −αn(1 −β))‖xn −p‖ ‖xn −p‖β2(1 −αn−1(1 −β))‖xn−1 −p‖ ‖xn−1 −p‖β2(1 −αn−2(1 −β))‖xn−2 −p‖ ... ‖x1 −p‖β2(1 −α0(1 −β))‖x0 −p‖. (3.14) from relation (3.14), we obtain the followings: ‖xn+1 −p‖‖x0 −p‖β2(n+1) n∏ k=0 (1 −αk(1 −β)). (3.15) using the fact that β ∈ (0, 1) and αn ∈ (0, 1) for each n ∈ n, we have (1 −αn(1 −β)) < 1. (3.16) in classical analysis, it is known that 1 −x ≤ e−x for all x ∈ [0, 1]. now using these facts together with relation (3.15), we obtain ‖xn+1 −p‖‖x0 −p‖β2(n+1)e−(1−β) ∑n k=0 αk. (3.17) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 144 fejér monotonicity and fixed point theorems from (3.17), it follows that |‖xn+1 −p‖|≤ |‖x0 −p‖|β2(n+1)e−(1−β) ∑n k=0 αk −→ 0 as n →∞. (3.18) hence, by lemma 2.10, it follows that {xn}−→ p as n →∞. next, we show that the fixed point p of t is unique. now suppose that p∗ is another fixed point of t, then by (3.10), we have ‖p−p∗‖ λ‖p∗ −tp∗‖1+‖p−tp‖ 1+‖p−p∗‖ + β‖p−p ∗‖ = λ.0 ( 1 1+‖p−p∗‖ ) + β‖p−p∗‖ = β‖p−p∗‖. (3.19) relation (3.19) implies that |‖p−p∗‖|≤ β|‖p−p∗‖|. (3.20) which is a contradiction, since β ∈ (0, 1). hence, p = p∗ as desired. the proof of theorem 3.6 is completed. � lemma 3.7. let d be a nonempty closed convex subset of a complex valued banach space (e,‖.‖) and t : d → d be a mapping on d with f(t) 6= ∅. assume that there exist λ,β ∈ (0, 1) satisfying λ + β < 1 and ‖tx−ty‖λ‖y −ty‖ 1 + ‖x−tx‖ 1 + ‖x−y‖ + β‖x−y‖ (3.21) for all x,y ∈ d. for arbitrary chosen x0 ∈ d, let the sequence {xn} be generated by the m-iteration process (2.6), then limn→∞ |‖xn − p‖| exists for any p ∈ f(t). proof. from relation (3.9), it follows that |‖xn+1 −p‖|≤ |‖xn −p‖|−→ 0 as n →∞. (3.22) hence, by lemma 2.10 we see that {‖xn −p‖} is bounded and non-increasing for each p ∈ f(t). therefore, limn→∞ |‖xn −p‖| exists as desired. the proof of lemma 3.7 is completed. � lemma 3.8. let d be a nonempty subset of a complex valued banach space (e,‖.‖). let the sequence {xn}⊆ e be fejér monotone with respect to d. if at least one cluster point x∗ of {xn} belongs to d, then xn → x∗. proof. since every fejér monotone sequence is bounded, it follows that {xn} has a weak cluster point x∗. let a subsequence {xkn} of {xn} converge to x∗ ∈ d. we now prove that {xn} converges to x∗. suppose x′ ∈ e, x′ 6= x∗ is another cluster point of {xn} such that a subsequence {xmn} converges to x′. suppose ε := 1 2 ‖x′ − x∗‖ � 0, let n0 ∈ n be such that ‖xmn − x′‖ ≺ ε and ‖xkn −x∗‖≺ ε, for all n ≥ n0 and let mn > kn0. using the triangle inequality and fejér monotonicity of {xn} with respect to d, we have 2ε = ‖x′ −x∗‖‖x′ −xmn‖ + ‖xmn −x ∗‖≺ 2ε. (3.23) this means that 2ε = |‖x′ −x∗‖|≤ |‖x′ −xmn‖| + |‖xmn −x ∗‖| < 2ε, (3.24) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 145 g. a. okeke and m. abbas which is a contradiction. therefore, by lemma 2.10, it follows that xn → x∗. the proof of lemma 3.8 is completed. � 4. cone metric spaces with banach algebras let a denote a real banch algebra. this means that a is a real banach space in which an operation of multiplication is defined, subject to the following properties (for each x,y,z ∈a, α ∈ r): (i) (xy)z = x(yz); (ii) x(y + z) = xy + xz and (x + y)z = xz + yz; (iii) α(xy) = (αx)y = x(αy); (iv) ‖xy‖≤‖x‖‖y‖. in this paper, we assume that a has a unit; i.e. a multiplicative identity e such that ex = xe = x for each x ∈ a. the inverse of x is denoted by x−1 (see, e.g. rudin [37]). in 2012, öztürk and başarir [34] generalized the concept of cone metric spaces introduced by huang and zhang [23] by replacing a bancach space with a banach algebra a in cone metric spaces. they called this new concept bacone metric spaces. abbas et al. [2] proved that complex valued metric spaces introduced in [11] is a ba-cone metric space, that is a cone metric space over a solid cone in commutative division banach algebra a (see, [2], [34]). perhaps unaware of the work of öztürk and başarir [34], in 2013 liu and xu [29] introduced the concept of cone metric spaces with banach algebras, by replacing banach spaces with banach algebras as the underlying space of cone metric spaces. they proved that fixed point theorems in the setting of cone metric spaces with banach algebras are more useful than the standard results in cone metric spaces and that results in cone metric spaces with banach algebras cannot be reduced to corresponding results in cone metric spaces. . example 4.1 ([29]). let a = mn(r) = {a = (aij)n×n|aij ∈ r for all 1 ≤ i,j ≤ n} be the algebra of all n-square real matrices, and define the norm ‖a‖ = ∑ 1≤i,j≤n |aij|. then a is a real banach algebra with the unit e, the identity matrix. let p = {a ∈a|aij ≥ 0 for all 1 ≤ i,j ≤ n}. then p ⊂a is a normal cone with normal constant m = 1. let x = mn(r), and define the metric d : x ×x →a by d(x,y) = d((xij)n×n, (yij)n×n) = (|xij −yij|)n×n ∈a. then (x,d) is a cone metric space with a banach algebra a. example 4.2 ([29]). let a be the banach space c(k) of all continuous realvalued functions on a compact hausdorff topological space k, with multiplication defined pointwise. then a is a banach algebra, and the constant function f(t) = 1 is the unit of a. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 146 fejér monotonicity and fixed point theorems let p = {f ∈ a|f(t) ≥ 0 for all t ∈ k}. then p ⊂ a is a normal cone with a normal constant m = 1. let x = c(k) with the metric mapping d : x ×x →a defined by d(f,g) = |f(t) −g(t)|, where t ∈k. then (x,d) is a cone metric space with a banach algebra a. example 4.3 ([29]). let a = `1 = {a = (an)n≥0| ∑∞ n=0 |an| < ∞} with convolution as multiplication: ab = (an)n≥0(bn)n≥0 =   ∑ i+j=n aibj   n≥0 . thus a is a banach algebra. the unit e is (1, 0, 0, · · ·). let p = {a = (an)n≥0 ∈ a|an ≥ 0 for all n ≥ 0}, which is a normal cone in a. and let x = `1 with the metric d : x ×x →a defined by d(x,y) = d((xn)n≥0, (yn)n≥0) = (|xn −yn|)n≥0. then (x,d) is a cone metric space with a. motivated by the results above, we now prove that results in complex valued banach spaces (see, e.g. okeke [32]) are true in the context of cone metric spaces with banach algebras. moreover, we show that our results cannot be deduced in cone metric spaces. theorem 4.4. let d be a nonempty closed convex subset of a complete cone metric space with banach algebras (a,‖.‖) and t : d → d be a contraction mapping satisfying the following contractive condition ‖tx−ty‖ϕ(‖x−tx‖) + a‖x−y‖ e + m‖x−tx‖ , ∀x,y ∈ d, a ∈ [0, 1), m ≥ 0, (4.1) where ϕ : c+ → c+ is a monotone increasing function such that ϕ(0) = 0. let {mn} be an iterative sequence generated by the picard-mann hybrid iterative process (2.7) with real sequence {αn}∞n=0 in [0, 1] satisfying ∑∞ n=0 αn = ∞. then {mn} converges strongly to a unique fixed point p of t. proof. we now show that xn → p as n →∞. using (2.7) and (4.1), we obtain: ‖mn+1 −p‖ = ‖tzn −p‖ ϕ(‖p−tp‖)+a‖zn−p‖ e+m‖p−tp‖ = ϕ(‖0‖)+a‖zn−p‖ e+m‖0‖ = a‖(1 −αn)mn + αntmn −p‖ a(1 −αn)‖mn −p‖ + aαn‖tmn −p‖ a(1 −αn)‖mn −p‖ + aαn [ ϕ(‖p−tp‖)+a‖mn−p‖ e+m‖p−tp‖ ] = a(1 −αn)‖mn −p‖ + aαn [ ϕ(‖0‖)+a‖mn−p‖ e+m‖0‖ ] = a(1 −αn)‖mn −p‖ + a2αn‖mn −p‖ = a(1 −αn(1 −a))‖mn −p‖. (4.2) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 147 g. a. okeke and m. abbas using the fact that (1−αn(1−a)) < 1 and a ∈ [0, 1), we obtain the following inequalities from (4.2).  ‖mn+1 −p‖a(1 −αn(1 −a))‖mn −p‖ ‖mn −p‖a(1 −αn−1(1 −a))‖mn−1 −p‖ ‖mn−1 −p‖a(1 −αn−2(1 −a))‖mn−2 −p‖ ... ‖m2 −p‖a(1 −α1(1 −a))‖m1 −p‖. (4.3) from relation (4.3), we derive ‖mn+1 −p‖‖m1 −p‖an+1 n∏ k=1 (1 −αk(1 −a)), (4.4) where (1−αk(1−a)) ∈ (0, 1), since a ∈ [0, 1) and αk ∈ [0, 1] for all k ∈ n. it is well-known in classical analysis that 1 −x ≤ e−x for all x ∈ [0, 1]. using these facts together with relation (4.4), we have ‖mn+1 −p‖‖m1 −p‖an+1 e(1−a) ∑ n k=1 αk . (4.5) therefore, lim n→∞ |‖mn+1 −p‖|≤ { |‖m1 −p‖an+1| |e(1−a) ∑ n k=1 αk| } −→ 0 as n →∞. (4.6) therefore by lemma 2.1 we have that limn→∞‖mn−p‖ = 0. this means that mn → p as n →∞ as desired. next, we show that t has a unique fixed point p ∈ f(t) := {p ∈ d : tp = p}. assume that p∗ is another fixed point of t, then we have ‖p−p∗‖ = ‖tp−tp∗‖ ϕ(‖p−tp‖)+a‖p−p ∗‖ e+m‖p−tp‖ = ϕ(‖0‖)+a‖p−p∗‖ e+m‖0‖ = a‖p−p∗‖. (4.7) this implies that |‖p−p∗‖|≤ |‖p−p∗‖|. (4.8) hence, by lemma 2.10 we have that p = p∗. the proof of theorem 4.4 is completed. � proposition 4.5. let d be a nonempty closed convex subset of a complete cone metric space with banach algebras (a,‖.‖) and let t : d → d be a mapping defined as follows ‖tx−ty‖ϕ(‖x−tx‖) + a‖x−y‖ e + m‖x−tx‖ , ∀x,y ∈ d, a ∈ [0, 1), m ≥ 0, (4.9) where ϕ : c+ → c+ is a monotone increasing function such that ϕ(0) = 0. suppose that each of the iterative processes (2.7) and (2.8) converges to the same fixed point p of t where {αn}∞n=0 and λ are such that 0 < α ≤ λ,αn < 1 c© agt, upv, 2020 appl. gen. topol. 21, no. 1 148 fejér monotonicity and fixed point theorems for all n ∈ n and for some α. then the sequence {xn} generated by the picardkrasnoselskii hybrid iterative process (2.8) have the same rate of convergence as the sequence {mn} generated by the picard-mann hybrid iterative process (2.7). proof. the proof of proposition 4.5 follows similar lines as in the proofs of ([32], proposition 2.2) and theorem 4.4. � remark 4.6. observe that the results in theorem 4.4 and proposition 4.5 was proved for mappings satisfying rational inequality, which is meaningless in cone metric spaces. this means that these results cannot be reduced to some corresponding results in cone metric spaces. 5. applications to a nonlinear integral equation it is our purpose in this section to show that the miterative process (2.6) converges strongly to the solution of a mixed type volterra-fredholm functional nonlinear integral equation in complex valued banach spaces. our results generalize and extend some known results to complex valued banach spaces, including the results of crăciun and şerban [18], gürsoy [19] among others. in 2011, crăciun and şerban [18] considered the following mixed type volterrafredholm functional nonlinear integral equation: x(t) = f ( t,x(t), ∫ t1 a1 · · · ∫ tm am k(t,s,x(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,x(s))ds ) , (5.1) where [a1; b1] × ··· × [am; bm] be an interval in rm, k,h : [a1; b1] × ··· × [am; bm] × [a1; b1] × ··· × [am; bm] × r → r continuous functions and f : [a1; b1] ×···× [am; bm] ×r3 → r. they established the following results. theorem 5.1 ([18]). we assume that: (i) k,h ∈ c([a1,b1] ×···× [am,bm] × [a1,b1] ×···× [am,bm] ×r); (ii) f ∈ c([a1,b1] ×···× [am,bm] ×r3); (iii) there exist α,β,γ nonnegative constants such that: |f(t,u1,v1,w1) −f(t,u2,v2,w2)| ≤ α|u1 −u2| + β|v1 −v2| + γ|w1 −w2|, for all t ∈ [a1,b1] ×···× [am,bm], u1,u2,v1,v2,w1,w2 ∈ r; (iv) there exist lk and lh nonnegative constants such that: |k(t,s,u) −k(t,s,v)| ≤ lk|u−v|, |h(t,s,u) −h(t,s,v)| ≤ lh|u−v|, for all t,s ∈ [a1,b1] ×···× [am,bm], u,v ∈ r; (v) α + (βlk + γlh)(b1 −a1) · · ·(bm −am) < 1. then, the equation (5.1) has a unique solution x∗ ∈ c([a1,b1]×···× [am,bm]). remark 5.2 ([18]). let (b, |.|) be a banach space. then theorem 5.1 remains also true if we consider the mixed type volterra-fredholm functional nonlinear integral equation (5.1) in the banach space b instead of banach space r. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 149 g. a. okeke and m. abbas let d be a nonempty subset of a complex valued banach space (e,‖.‖) and let {mn} be an iterative process defined by the m -iteration associated with f, which is generated as follows:  m0 ∈ d zn = (1 −αn)mn + αntmn gn = tzn mn+1 = tgn, (5.2) where {αn} is a real sequence in (0, 1). consequently, we now obtain the following analogue of theorem 5.1 in complex valued banach spaces. theorem 5.3. we consider the complex valued banach space bc = c([a1,b1]× ···× [am,bm],‖.‖c), where ‖.‖c is the chebyshev’s norm defined by ‖x−y‖c = |x−y|i, ∀x,y ∈bc. we assume that: (i) k,h ∈ c([a1,b1] ×···× [am,bm] × [a1,b1] ×···× [am,bm] ×r); (ii) f ∈ c([a1,b1] ×···× [am,bm] ×r3); (iii) there exist α,β,γ nonnegative constants such that: |f(t,u1,v1,w1) −f(t,u2,v2,w2)| ≤ α|u1 −u2| + β|v1 −v2| + γ|w1 −w2|, for all t ∈ [a1,b1] ×···× [am,bm], u1,u2,v1,v2,w1,w2 ∈ r; (iv) there exist lk and lh nonnegative constants such that: |k(t,s,u) −k(t,s,v)| ≤ lk|u−v|, |h(t,s,u) −h(t,s,v)| ≤ lh|u−v|, for all t,s ∈ [a1,b1] ×···× [am,bm], u,v ∈ r; (v) α + (βlk + γlh)(b1 −a1) · · ·(bm −am) < 1. then, the mixed type volterra-fredholm functional integral equation (5.1) has a unique solution p ∈ c([a1; b1] ×···× [am; bm]). proof. since our analysis is in the complex valued banach space bc = c([a1,b1]× ···×[am,bm],‖.‖c), where ‖.‖c is the chebyshev’s norm defined by ‖x−y‖c = |x−y|i, ∀x,y ∈bc, and the operator a : bc →bc, defined by a(x)(t) = f ( t,x(t), ∫ t1 a1 · · · ∫ tm am k(t,s,x(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,x(s))ds ) (5.3) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 150 fejér monotonicity and fixed point theorems using conditions (iii) and (iv), we have |a(u)(t) −a(v)(t)| α|u(t) −v(t)| + β| ∫ t1 a1 · · · ∫ tm am (k(t,s,u(s))− k(t,s,v(s)))ds|+ γ ∣∣∣∫ b1a1 · · ·∫ bmam (h(t,s,u(s)) −h(t,s,v(s)))ds∣∣∣ α|u(t) −v(t)| + β ∫ t1 a1 · · · ∫ tm am lk|u(s) −v(s)|ds+ γ ∫ b1 a1 · · · ∫ bm am lh|u(s) −v(s)|ds [α + (βlk + γlh)(b1 −a1) · · ·(bm −am)]‖u−v‖c = [α + (βlk + γlh)(b1 −a1) · · ·(bm −am)]|u−v|i. (5.4) it follows from relation (5.4) that |‖a(u)(t) −a(v)(t)‖c| ≤ |[α + (βlk + γlh)(b1 −a1) · · ·(bm −am)]|u−v|i| = [α + (βlk + γlh)(b1 −a1) · · ·(bm −am)]|u−v|. (5.5) using lemma 2.10 in (5.5) together with condition (v), we see that operator a is a contraction, so that by the banach contraction mapping principle, we have that operator a has a unique fixed point f(a) = {p}. this means that our equation (5.1) has a unique solution p ∈ c([a1; b1] ×···× [am; bm]). the proof of theorem 5.3 is completed. � theorem 5.4. suppose that all the conditions (i) (v) in theorem 4.2 are satisfied. let the sequence {mn} be generated by the m-iteration process (5.2), where {αn} ⊂ (0, 1) is a real sequence satisfying ∑∞ n=0 αn = ∞. then the mixed type volterra-fredholm functional integral equation (5.1) has a unique solution, say p ∈ c([a1; b1] ×···× [am; bm]) and the sequence {mn} converges to p. proof. we consider the complex valued banach space bc = c([a1,b1] ×···× [am,bm],‖.‖c), where ‖.‖c is the chebyshev’s norm defined by ‖x − y‖c = |x − y|i, ∀x,y ∈ bc. let {mn} be the sequence generated by the m-iteration process (5.2) for the operator a : bc →bc defined by a(x)(t) = f ( t,x(t), ∫ t1 a1 · · · ∫ tm am k(t,s,x(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,x(s))ds ) . (5.6) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 151 g. a. okeke and m. abbas we want to show that mn −→ p as n →∞. using (5.2), (5.1) and assumptions (i) (v) we obtain ‖mn+1 −p‖c = |a(gn)(t) −a(p)(t)| = |f ( t,gn(t), ∫ t1 a1 · · · ∫ tm am k(t,s,gn(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,gn(s))ds ) − f ( t,p(t), ∫ t1 a1 · · · ∫ bm am k(t,s,p(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,p(s))ds ) | α|gn(t) −p(t)| + β| ∫ t1 a1 · · ·∫ tm am k(t,s,gn(s))ds− ∫ t1 a1 · · · ∫ tm am k(t,s,p(s))ds|+ γ| ∫ b1 a1 · · · ∫ bm am h(t,s,gn(s))ds− ∫ b1 a1 · · · ∫ bm am h(t,s,p(s))ds| α|gn(t) −p(t)| + β ∫ t1 a1 · · · ∫ bm am |k(t,s,gn(s)) −k(t,s,p(s))|ds+ γ ∫ b1 a1 · · · ∫ bm am |h(t,s,gn(s)) −h(t,s,p(s))|ds α|gn(t) −p(t)| + β ∫ t1 a1 · · · ∫ tm am lk|gn(s) −p(s)|ds+ γ ∫ b1 a1 · · · ∫ bm am lh|gn(s) −p(s)|ds [α + (βlk + γlh) ∏m i=1(bi −ai)]‖gn −p‖c. (5.7) next, we have the following estimate ‖gn −p‖c = |a(zn)(t) −a(p)(t)| = |f ( t,zn(t), ∫ t1 a1 · · · ∫ tm am k(t,s,zn(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,zn(s))ds ) − f ( t,p(t), ∫ t1 a1 · · · ∫ bm am k(t,s,p(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,p(s))ds ) | α|zn(t) −p(t)| + β| ∫ t1 a1 · · ·∫ tm am k(t,s,zn(s))ds− ∫ t1 a1 · · · ∫ tm am k(t,s,p(s))ds|+ γ ∣∣∣∫ b1a1 · · ·∫ bmam h(t,s,zn(s))ds−∫ b1a1 · · ·∫ bmam h(t,s,p(s))ds∣∣∣ α|zn(t) −p(t)| + β ∫ t1 a1 · · · ∫ tm am |k(t,s,zn(s)) −k(t,s,p(s))|ds+ γ ∫ b1 a1 · · · ∫ bm am |h(t,s,zn(s)) −h(t,s,p(s))|ds α|zn(t) −p(t)| + β ∫ t1 a1 · · · ∫ tm am lk|zn(s) −p(s)|ds+ γ ∫ b1 a1 · · · ∫ bm am lh|zn(s) −p(s)|ds [α + (βlk + γlh) ∏m i=1(bi −ai)]‖zn −p‖c. (5.8) ‖zn −p‖c (1 −αn)|mn(t) −p(t)| + αn|a(mn)(t) −a(p)(t)| = (1 −αn)|mn(t) −p(t)|+ αn|f ( t,mn(t), ∫ t1 a1 · · · ∫ tm am k(t,s,mn(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,mn(s))ds ) − f ( t,p(t), ∫ t1 a1 · · · ∫ bm am k(t,s,p(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,p(s))ds ) | (1 −αn)|mn(t) −p(t)| + αnα|mn(t) −p(t)| + αnβ ∫ t1 a1 · · ·∫ tm am lk|mn(s) −p(s)|ds +αnγ ∫ b1 a1 · · · ∫ bm am lh|mn(s) −p(s)|ds {1 −αn (1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])}‖mn −p‖c. (5.9) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 152 fejér monotonicity and fixed point theorems using (5.8) and (5.9) in (5.7), together with the fact that [α+(βlk+γlh) ∏m i=1(bi− ai)] < 1 in assumption (v) we obtain ‖mn+1 −p‖c [α + (βlk + γlh) ∏m i=1(bi −ai)] 2× {1 −αn (1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])}‖mn −p‖c {1 −αn (1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])}‖mn −p‖c. (5.10) hence, by induction (5.10) becomes ‖mn+1−p‖c n∏ k=0 { 1 −αk ( 1 − [α + (βlk + γlh) m∏ i=1 (bi −ai)] )} ‖m0−p‖c. (5.11) from the fact that αk ∈ (0, 1) for each k ∈ n, together with assumption (v), we have { 1 −αk ( 1 − [α + (βlk + γlh) m∏ i=1 (bi −ai)] )} < 1. (5.12) it is known in analysis that ex ≥ 1 − x for all x ∈ [0, 1]. therefore (5.11) becomes ‖mn+1 −p‖c ‖m0 −p‖ce−(1−[α+(βlk +γlh ) ∏m i=1 (bi−ai)]) ∑n k=0 αk) = |m0 −p|ie−(1−[α+(βlk +γlh ) ∏m i=1(bi−ai)]) ∑n k=0 αk). (5.13) this means that |‖mn+1 −p‖c| |‖m0 −p‖ce−(1−[α+(βlk +γlh ) ∏m i=1 (bi−ai)]) ∑n k=0 αk)| = |m0 −p|e−(1−[α+(βlk +γlh ) ∏m i=1(bi−ai)]) ∑n k=0 αk) −→ 0 (5.14) as k →∞. therefore, by lemma 2.10, we have xn −→ p as n →∞ as desired. the proof of theorem 5.4 is completed. � we now turn our attention to proving the data dependence of the solution for the integral equation (5.1) via the m-iterative process (5.2). suppose bc is as in theorem 5.3 and the operators t,t̃ : bc → bc are defined by t(x)(t) = f ( t,x(t), ∫ t1 a1 · · · ∫ tm am k(t,s,x(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,x(s))ds ) (5.15) t̃(x)(t) = f ( t,x(t), ∫ t1 a1 · · · ∫ tm am k̃(t,s,x(s))ds, ∫ b1 a1 · · · ∫ bm am h̃(t,s,x(s))ds ) , (5.16) where k,k̃,h,h̃ ∈ c([a1; b1] ×···× [am; bm] × [a1; b1] ×···× [am; bm] ×r). theorem 5.5. let f,k and h be defined as in theorem 5.3 and let {mn} be the iterative sequence generated by the m-iteration process (5.2) associated with t. let {m̃n} be an iterative sequence generated by c© agt, upv, 2020 appl. gen. topol. 21, no. 1 153 g. a. okeke and m. abbas   m̃0 ∈ d, z̃n = (1 −αn)m̃n + αnt̃m̃n, g̃n = t̃ z̃n m̃n+1 = t̃ g̃n, (5.17) where bc is as defined in theorem 5.2 and {αn} is a real sequence in (0, 1) satisfying (a) 1 2 ≤ αn for each n ∈ n, and (b) ∑∞ n=0 αn = ∞. furthermore, suppose (c) there exist nonnegative constants λ1 and λ2 such that |k(t,s,u)−k̃(t,s,u)| ≤ λ1 and |h(t,s,u) − h̃(t,s,u)| ≤ λ2, for all u ∈ r and for all t,s ∈ [a1; b1] × ···× [am; bm]. if p and p̃ are solutions of corresponding nonlinear equations (5.15) and (5.16) respectively, then we have |‖p− p̃‖|≤ 4(βλ1 + γλ2) ∏m i=1(bi −ai) 1 − [α + (βlk + γlh) ∏m i=1(bi −ai) . (5.18) proof. we consider the complex valued banach space bc = c([a1,b1] ×···× [am,bm],‖.‖c), where ‖.‖c is the chebyshev’s norm defined by ‖x − y‖c = |x−y|i, ∀x,y ∈bc. now using (5.1), (5.2), (5.15), (5.16), (5.17) and assumptions (i) (v) together with conditions (a) (c), we have ‖mn+1 − m̃n+1‖c = ‖tgn − t̃ g̃n‖c = |f ( t,gn(t), ∫ t1 a1 · · · ∫ tm am k(t,s,gn(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,gn(s))ds ) − f ( t, g̃n(t), ∫ t1 a1 · · · ∫ tm am k̃(t,s, g̃n(s))ds, ∫ b1 a1 · · · ∫ bm am h̃(t,s, g̃n(s))ds ) | α|gn(t) − g̃n(t)| + β ∫ t1 a1 · · · ∫ tm am |k(t,s,gn(s)) − k̃(t,s, g̃n(s))|ds+ γ ∫ b1 a1 · · · ∫ bm am |h(t,s,gn(s)) − h̃(t,s, g̃n(s))|ds α|gn(t) − g̃n(t)|+ β ∫ t1 a1 · · · ∫ tm am (|k(t,s,gn(s)) −k(t,s, g̃n(s))|+ |k(t,s, g̃n(s)) − k̃(t,s, g̃n(s))|)ds+ γ ∫ b1 a1 · · · ∫ bm am (|h(t,s,gn(s)) −h(t,s, g̃n(s))|+ |h(t,s, g̃n(s)) − h̃(t,s, g̃n(s))|)ds α|gn(t) − g̃n(t)| + β ∫ t1 a1 · · · ∫ tm am (lk|gn(s) − g̃n(s)| + λ1)ds+ γ ∫ b1 a1 · · · ∫ bm am (lh|gn(s) − g̃n(s)| + λ2)ds α‖gn − g̃n‖c + β(lk‖gn − g̃n‖c + λ1) ∏m i=1(bi −ai)+ γ(lh‖gn − g̃n‖c + λ2) ∏m i=1(bi −ai) [α + (βlk + γlh) ∏m i=1(bi −ai)]‖gn − g̃n‖c+ (βλ1 + γλ2) ∏m i=1(bi −ai). (5.19) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 154 fejér monotonicity and fixed point theorems ‖gn − g̃n‖c = ‖tzn − t̃ z̃n‖c = |f ( t,zn(t), ∫ t1 a1 · · · ∫ tm am k(t,s,zn(s))ds, ∫ b1 a1 · · · ∫ bm am h(t,s,zn(s))ds ) − f ( t, z̃n(t), ∫ t1 a1 · · · ∫ tm am k̃(t,s, z̃n(s))ds, ∫ b1 a1 · · · ∫ bm am h̃(t,s, z̃n(s))ds ) | α|zn(t) − z̃n(t)|+ β ∫ t1 a1 · · · ∫ tm am (|k(t,s,zn(s)) −k(t,s, z̃n(s))|+ |k(t,s, z̃n(s)) − k̃(t,s, z̃n(s))|)ds+ γ ∫ b1 a1 · · · ∫ bm am (|h(t,s,zn(s)) −h(t,s, z̃n(s))|+ |h(t,s, z̃n(s)) − h̃(t,s, z̃n(s))|)ds α|zn(t) − z̃n(t)| + β ∫ t1 a1 · · · ∫ tm am (lk|zn(s) − z̃n(s)| + λ1)ds+ γ ∫ b1 a1 · · · ∫ bm am (lh|zn(s) − z̃n(s)| + λ2)ds α‖zn − z̃n‖c + β(lk‖zn − z̃n‖c + λ1) ∏m i=1(bi −ai)+ γ(lh‖zn − z̃n‖c + λ2) ∏m i=1(bi −ai) [α + (βlk + γlh) ∏m i=1(bi −ai)]‖zn − z̃n‖c+ (βλ1 + γλ2) ∏m i=1(bi −ai). (5.20) ‖zn − z̃n‖c (1 −αn)|mn(t) − m̃n(t)| + αn|t(mn)(t) − t̃(m̃n)(t)| (1 −αn)|mn(t) − m̃n(t)| + αn{α|mn(t) − m̃n(t)|+ β ∫ t1 a1 · · · ∫ tm am (lk|mn(s) − m̃n(s)| + λ1)ds+ γ ∫ b1 a1 · · · ∫ bm am (lh|mn(s) − m̃n(s)| + λ2)ds} {1 −αn(1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])}‖mn − m̃n‖c+ αn(βλ1 + γλ2) ∏m i=1(bi −ai). (5.21) using (5.21) in (5.20), together with assumption (v), we have: ‖gn − g̃n‖c {1 −αn(1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])}‖mn − m̃n‖c+ αn(βλ1 + γλ2) ∏m i=1(bi −ai) + (βλ1 + γλ2) ∏m i=1(bi −ai). (5.22) using (5.22) in (5.19) together with assumption (v), we have ‖mn+1 − m̃n+1‖c {1 −αn(1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])}× ‖mn − m̃n‖c+ αn(βλ1 + γλ2) ∏m i=1(bi −ai)+ (βλ1 + γλ2) ∏m i=1(bi −ai)+ (βλ1 + γλ2) ∏m i=1(bi −ai) {1 −αn(1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])}× ‖mn − m̃n‖c+ αn (1 − [α + (βlk + γlh) ∏m i=1(bi −ai)])×( 4(βλ1+γλ2) ∏m i=1 (bi−ai) (1−[α+(βlk +γlh ) ∏ m i=1 (bi−ai)) ) . (5.23) c© agt, upv, 2020 appl. gen. topol. 21, no. 1 155 g. a. okeke and m. abbas from relation (5.23), we choose the sequences βn, µn and γn as follows:   βn = ‖mn − m̃n‖c, µn = αn(1 − [α + (βlk + γlh) ∏m i=1(bi −ai)]) ∈ (0, 1), γn = 4(βλ1+γλ2) ∏m i=1 (bi−ai) (1−[α+(βlk +γlh ) ∏ m i=1(bi−ai)) . (5.24) therefore, from relation (5.23), we see that all the conditions of lemma 2.3 are satisfied. hence, we have ‖p− p̃‖c 4(βλ1 + γλ2) ∏m i=1(bi −ai) (1 − [α + (βlk + γlh) ∏m i=1(bi −ai)) . (5.25) this implies that |‖p− p̃‖c| ≤ 4(βλ1 + γλ2) ∏m i=1(bi −ai) (1 − [α + (βlk + γlh) ∏m i=1(bi −ai)) . 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[45] k. ullah and m. arshad, numerical reckoning fixed points for suzuki’s generalized nonexpansive mappings via new iteration process, filomat 32, no. 1 (2018), 187–196. c© agt, upv, 2020 appl. gen. topol. 21, no. 1 158 kohlisinghagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 33-38 function spaces and strong variants of continuity j. k. kohli and d. singh abstract. it is shown that if domain is a sum connected space and range is a t0-space, then the notions of strong continuity, perfect continuity and cl-supercontinuity coincide. further, it is proved that if x is a sum connected space and y is hausdorff, then the set of all strongly continuous (perfectly continuous, cl-supercontinuous) functions is closed in y x in the topology of pointwise convergence. the results obtained in the process strengthen and extend certain results of levine and naimpally. 2000 ams classification: 54c08, 54c10, 54d10. keywords: strongly continuous function, perfectly continuous function, clsupercontinuous function, sum connected spaces, k-space, topology of pointwise convergence, topology of uniform convergence on compacta, compact open topology, equicontinuity, even continuity. 1. introduction strong variants of continuity occur frequently in many areas of mathematics, more so in many subdisciplines of topology and analysis. in this paper we shall be concerned with three such variants of continuity: strongly continuous functions introduced by levine [5], perfectly continuous functions due to noiri [8], and cl-supercontinuous functions defined by reilly and vamanamurthy [9] under the nomenclature of ‘clopen maps’ and studied by singh [10]. naimpally [7] showed that in contrast to continuous functions, the set of strongly continuous functions is closed in the topology of pointwise convergence if the domain is locally connected and range is hausdorff. in this paper we extend naimpally’s result to a larger framework and further prove that in the proposed framework the set of perfectly continuous functions as well as the set of cl-supercontinuous functions is closed in the topology of pointwise convergence. for our purpose the class of sum connected spaces [4] turns out to be handy 34 j. k. kohli and d. singh and provides an appropriate framework. a space x is sum connected if each x ∈ x has a connected neighbourhood or equivalently each component of x is open in x. the category of sum connected spaces is the coreflective hull of the category of connected spaces and includes all locally connected spaces as well. sum connected spaces are also closely related to natural cover of brown [1] and franklin [2] and michael’s theorem [6] on existence of selections is also closely related to them. for terms used in the paper but not defined, we refer the reader to [3]. 2. definitions, preliminary, observations and examples definitions 2.1. a function f : x → y from a topological space x into a topological space y is said to be (1) strongly continuous [5] if f (ā) ⊂ f (a) for all a ⊂ x. (2) perfectly continuous [8] if for every open set v ⊂ y , f −1(v ) is clopen in x. (3) cl-supercontinuous1 [10] if for each open set v containing f (x) there exists a clopen set u containing x such that f (u ) ⊂ v . definitions 2.2. a space x is said to be (1) an ultra-hausdorff space [11] if for each pair of distinct points in x, there is a clopen set containing one but missing the other; (2) a k-space [3] if a ⊂ x is closed if and only if a ∩ k is closed in k for every compact set k in x. definition 2.3. a family ϑ of functions from a topological space x into a uniform space (y, v) is said to be equicontinuous at a point x ∈ x if for each member v of v there exists a neighbourhood u of x such that f (u ) ⊂ v [f (x)] for each f ∈ ϑ. the family ϑ is said to be equicontinuous if it is equicontinuous at every point. definition 2.4. a family ϑ of functions from a topological space x into a topological space y is said to be evenly continuous if for each x ∈ x, each y ∈ y and each neighbourhood v of y there is a neighbourhood u of x and a neighbourhood w of y such that such that f (u ) ⊂ v whenever f (x) ∈ w . results on strongly continuous functions 2.5. let x and y be topological spaces. (a) if x is discrete, then every function f : x → y is strongly continuous. (b) if x is connected, then a function f : x → y is strongly continuous if and only if f is constant. (c) if f : x → y is strongly continuous, then the collection p = {f −1(y) : y ∈y } is a partition of x into pairwise disjoint clopen subsets of x. so the topology on x is finer than a partition topology (every open set is closed). moreover, the quotient space x/p is discrete. 1‘cl-supercontinuous functions’ are called ‘clopen maps’ in [9]. function spaces and strong variants of continuity 35 (d) if f : x → y is a strongly continuous injection, then x is discrete. further, the hypothesis of injectivity cannot be relaxed. for let x be the set of positive integers equipped with odd-even topology ([12, p. 43]) and y be the set of positive integers with the discrete topology. let f : x → y be defined by f (x) = { k if x = 2k k if x = 2k − 1 then clearly f is a strongly continuous surjection but x is not discrete. (e) if f : x → y is a strongly continuous open surjection, then y is discrete. (f) if f : x → y is a strongly continuous open bijection, then x and y are discrete spaces of same cardinality. results on perfectly continuous functions 2.6. (a) if x is endowed with a partition topology, then every continuous function defined on x is perfectly continuous. (b) a space x is endowed with a partition topology if and only if the identity mapping defined on x is perfectly continuous. if partition topology on x is not discrete, then the identity mapping on x is not strongly continuous. results on cl-supercontinuous functions 2.7. (a) if either of the spaces x and y is a zero dimensional space, then every continuous function f : x → y is cl-supercontinuous. 3. results theorem 3.1 ([5]). for a function f : x → y , the following statements are equivalent. (a) f is strongly continuous. (b) f −1(b) is a clopen subset of x for every subset b of y . (c) f (a′) ⊂ f (a) for every subset a of x, where a’ denotes the set of all limit points of a. theorem 3.2. if f : x → y is cl-supercontinuous function into a t0-space y , then f (c) is a singleton for every nonempty connected subset c of x. proof. assume contrary and let c be a connected subset of x such that f (c) is not a singleton. let f (x), f (y) be any two distinct points of f (c). since y is a t0-space, there is an open set v containing one of the points f (x) and f (y) and missing the other. to be precise, let f (x) ∈ v . then, by f (x) ∈ v , f −1(v )∩c is nonempty proper cl-open subset of c, contradicting the fact that c is connected. � corollary 3.3. let f : x → y be a cl-supercontinuous function into a t0space y . if x is sum connected, then f is constant on each component of x. in particular, if x is connected, then f is constant. 36 j. k. kohli and d. singh theorem 3.4. let f : x → y be a function from a sum connected space x into a t0-space y . then the following statements are equivalent. (a) f is strongly continuous. (b) f is perfectly continuous. (c) f is cl-supercontinuous. proof. the implications (a) ⇒ (b) ⇒ (c) are trivial. to prove that (c) ⇒ (a) we show that f (a′) ⊂ f (a) for every subset a of x. to this end, let x ∈ a′. let c be the component of x containing x. since x is sum connected, c is open in x and so c ∩ a 6= φ. by theorem 3.2 f (c) is a singleton and so f (x) = f (c) = f (c ∩ a) ⊂ f (a). thus f (a′) ⊂ f (a) and so in view of theorem 3.1 f is strongly continuous. � corollary 3.5. if x is a connected space or a locally connected space and y is a t0-space, then the notions of strong continuity, perfect continuity and cl-supercontinuity coincide. let l = l(x, y ), p = (x, y ) and s = s(x, y ) denote the function space of all cl-supercontinuous, respectively perfectly continuous and strongly continuous functions from x into y with the topology of pointwise convergence. theorem 3.6. let y be a hausdorff space and let g ∈ l̄ (where closure is taken in the topology of pointwise convergence). then for each nonempty connected subset c of x, g(c) is a single point. proof. suppose that there exists a nonempty connected subset c of x such that g(c) contains at least two points g(x) and g(y). since x is a hausdorff space, there exist disjoint open sets u and v containing g(x) and g(y), respectively. since g ∈ l̄ , there is a net {f α | α ∈ λ} ⊂ l such that f α (x) → g(x) for each x ∈ x. so there exists a α0 ∈ λ such that for all α ≥ α0, fα(x) ∈ u and f α (y) ∈ v . but f α (x) = f α (y) on c. this contradicts that y is a hausdorff space and proves the result. � theorem 3.7. let x be a sum connected space and y be hausdorff. then l = p = s is closed in the topology of pointwise convergence. proof. this is immediate in view of theorems 3.1,3.4 and 3.6. � proposition 3.8. if x is a sum connected space and y is a compact hausdorff space, then l = p = s is compact in the topology of pointwise convergence. proof. this is immediate in view of theorem 3.7, the fact that l[x] is compact for each x ∈ x and ([3, theorem1, p.218]). � proposition 3.9. if x is a sum connected and (y, v) is a hausdorff uniform space, then l = p = s is equicontinuous and consequently evenly continuous. proof. for each x ∈ x, let c x denote the component of x containing x. since x is sum connected, c x is open. in view of theorem 3.6 for each f ∈ l, f (c x ) = f (x). clearly f (c x ) ⊂ v [f (x)] for all v ∈ v and all f ∈ l. thus l is equicontinuous. by [3, p. 237] l is evenly continuous. � function spaces and strong variants of continuity 37 proposition 3.10. let x be a sum connected space and (y, v) be a hausdorff uniform space. then the topology of pointwise convergence for l = p = s coincides with the topology of uniform convergence on compacta. further, if x is a k-space and l[x] is compact for each x ∈ x, then l = p = s is compact in the topology of pointwise convergence. proof. by proposition 3.9, l is equicontinuous. for an equicontinuous family the topology of pointwise convergence is jointly continuous and so coincides with the topology of uniform convergence on compacta ([3, theorem 15, p. 232]). so by ascoli theorem ([3, theorem 18, p. 234]) it follows that l = p = s is compact in the topology of pointwise convergence. � theorem 3.11. let x be a sum connected space and let y be a hausdorff space. then l = p = s is evenly continuous in the topology of pointwise convergence. further, if in addition x is a regular locally compact space, y a regular space, and l[x] is compact for each x ∈ x. then l = p = s is compact in the topology of pointwise convergence. proof. for each x ∈ x, each y ∈ y and each neighbourhood v of y, let u = c x be the component of x containing x. since x is sum connected, c x = u is open in x. let w = v . now by theorem 3.6 f (u ) = f (c x ) = f (x) ∈ v whenever f (x) ∈ w . so the family l = p = s is evenly continuous. again, by theorem 3.7, l = p = s is closed in the topology of pointwise convergence. hence by ([3, theorem 19, p. 235]) the topology of pointwise convergence on l = p = s is jointly continuous and therefore coincides with the compact open topology. the compactness of l = p = s is immediate in view of ascoli theorem pertaining to evenly continuous families ([3, theorem 21, p. 236]). � references [1] r. f. brown, ten topologies for x×y , quarterly j. math. (oxford) 14 (1963), 303–319. [2] s. p. franklin, natural covers, composito mathematica 21 (1969), 253–261. [3] j. l. kelly, general topology, d. van nostand company, inc., 1955. [4] j. k. kohli, a class of spaces containing all connected and all locally connected spaces, math. nachricten 82 (1978), 121–129. [5] n. levine, strong continuity in topological spaces, amer. math. monthly 67 (1960), 269. [6] e. michael, topologies on spaces of subsets, trans. amer. math. soc. 7 (1951), 152–182. [7] s. a. naimpally, on strongly continuous functions, amer. math. monthly 74 (1967), 166–168. [8] t. noiri, supercontinuity and some strong forms of continuity, indian j. pure. appl. math. 15, no. 3 (1984), 241–250. [9] i. l. reilly and m. k. vamanamurthy, on super-continuous mappings, indian j. pure. appl. math. 14, no. 6 (1983), 767–772. [10] d. singh, cl-supercontinuous functions, applied gen. top. (accepted). [11] r. staum, the algebra of bounded continuous functions into a nonarchimedean field, pac. j. math. 50, no. 1 (1974), 169–185. [12] l. a. steen and j. a. seeback, jr., counter examples in topology, springer verlag, new york, 1978. 38 j. k. kohli and d. singh received august 2006 accepted february 2007 j. k. kohli (jk kohli@yahoo.com) department of mathematics, hindu college, university of delhi, delhi 110 007, india d. singh (dstopology@rediffmail.com) department of mathematics, sri aurobindo college, university of delhi – south campus, delhi 110 017, india filaprotaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 169-175 spread of balleans m. filali and i. v. protasov abstract. a ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. we introduce and study a new cardinal invariant called the spread of a ballean. in particular, we show that, for every ordinal ballean b, spread of b coincides with density of b. 2000 ams classification: 54a25, 54e25, 05a18. keywords: ballean, pseudodiscrete subset, density, cellularity, spread. 1. introduction a ball structure is a triplet b = (x, p, b), where x, p are non-empty sets and, for any x ∈ x and α ∈ p , b(x, α) is a subset of x which is called the ball of radius α around x. it is supposed that x ∈ b(x, α) for all x ∈ x, α ∈ p . the set x is called the support of b, p is called the set of radiuses. given any x ∈ x, a ⊆ x, α ∈ p , we put b⋆(x, α) = {y ∈ x : x ∈ b(y, α)}, b(a, α) = ⋃ a∈a b(a, α). a ball structure b is called a ballean (or a coarse structure) if • for any α, β ∈ p , there exist α′, β′ ∈ p such that, for every x ∈ x, b(x, α) ⊆ b⋆(x, α′), b⋆(x, β) ⊆ b(x, β′); • for any α, β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x, α), β) ⊆ b(x, γ); • for any x, y ∈ x, there exists α ∈ p such that y ∈ b(x, α). let b1 = (x1, p1, b1), b2 = (x2, p2, b2) be balleans. a mapping f : x1 −→ x2 is called a ≺mapping if, for every α ∈ p1, there exists β ∈ p2 such that f (b1(x, α)) ⊆ b2(f (x), β) for every x ∈ x. if f is a bijection such that f 170 m. filali and i. v. protasov and f −1 are ≺mappings, we say that f is an asymorphism. if x1 = x2 and the identity mapping id : x1 −→ x2 is an asymorphism, we identify b1 and b2, and write b1 = b2. for each ballean b = (x, p, b), replacing every ball b(x, α) with b(x, α) ⋂ b∗(x, α), we obtain the same ballean, so every ballean can be determined in such a way that b(x, α) = b∗(x, α) for all x ∈ x, α ∈ p . for motivations for the study of balleans as the asymptotic counterparts of the uniform topological spaces see [1, 5, 6]. 2. spread and density let b = (x, p, b) be a ballean. a subset v ⊆ x is called bounded if there exist x ∈ x and α ∈ p such that v ⊆ b(x, α). a ballean is called bounded if its support x is bounded. given a ballean b = (x, p, b), we say that a subset y ⊆ x is pseudodiscrete, if for every α ∈ p , there exists a bounded subset v of x such that b(y, α) ⋂ y = {y} for every y ∈ y \ v . a ballean b is called pseudodiscrete if its support x is pseudodiscrete. for every subset y ⊆ x, we put | y |b= min{|y \v | : v is a bounded subset of x}, and introduce a new cardinal invariant spread(b) = sup{|y |b : y is a pseudodiscrete subset of x}. we note that |y |b = 0 if and only if y is bounded, so spread(b) = 0 for every bounded ballean. a subset l of x is called large if there exists α ∈ p such that x = b(l, α). the density of b is defined in [4] by den(b) = min{|l| : l is a large subset of x}. clearly, den(b) = 1 for every bounded ballean b, and den(b) is an infinite cardinal for every unbounded ballean b. proposition 2.1. for every ballean b, we have spread(b) ≤ den(b). proof. let l be a large subset of x and y be a pseudodiscrete subset of x. it suffices to show that |l| ≥ |y \v | for some bounded subset v of x. we may suppose that b(x, α) = b∗(x, α) for all x ∈ x, α ∈ p . we take β ∈ p such that x = b(l, β), and choose γ ∈ p such that b(b(x, β), β) ⊆ b(x, γ) for each x ∈ x. since y is pseudodiscrete, there exists a bounded subset v of x such that b(y, γ) ⋂ y = {y} for each y ∈ y \ v. by the choice of γ, the family {b(y, β) : y ∈ y \ v } is disjoint. since b(x, β) ⋂ l 6= ∅ for each x ∈ x, we have l ⋂ b(y, β) 6= ∅ for each y ∈ y \ v. hence |l| ≥ |y \ v |, as required. � in the next example, for every infinite cardinal γ, we construct a ballean b such that den(b) = γ but spread(b) = 0. spread of balleans 171 example 2.2. let x be a set of cardinality γ, κ be an infinite regular cardinal such that κ ≤ γ. we denote by f the family of all subsets of x of cardinality < κ. let p be the set of all mappings f : x −→ f such that, for every x ∈ x, we have x ∈ f (x) and |{y ∈ x : x ∈ f (y)}| < κ. given any x ∈ x and α ∈ p, we put b(x, f ) = f (x) and note that the ball structure b = (x, p, b) is a ballean. we need the regularity of κ to state that |b(b(x, f ), g)| < κ for all x ∈ x and f, g ∈ p. note that a subset v of x is bounded if and only if |v | < κ; and a subset l of x is large if and only if |l| = γ implying that den(b) = γ. now we check that spread(b) = 0. to this end, we take an arbitrary subset y of x such that |y | ≥ κ, write it as y = {yλ : λ ∈ |y |}, and define a mapping f : x −→ f by the rule: f (yλ) = {yλ, yλ+1} for each λ < |y |, and f (x) = {x} for each x ∈ x \ v . then f ∈ p and | b(y, f ) ⋂ y |= 2 for every y ∈ y , so y is not pseudodiscrete and spread(b) = 0. for every ballean b = (x, p, b), we use the preodering ≤ on x defined by the rule: α ≤ β if and only if b(x, α) ⊆ b(x, β) for every x ∈ x. a subset p ′ ⊆ p is called cofinal if, for every α ∈ p , there exists α′ ∈ p ′ such that α′ ≥ α. the cofinality cf (b) is the minimal cardinality of the cofinal subsets of p . a ballean b is called ordinal if p contains a cofinal subset of p ′ which is well-ordered by ≤. replacing p ′ with its minimal cofinal subset, we get the same ballean. hence, we can write b as (x, p, b), where p is a regular cardinal (considered as a set of ordinals). let (x, d) be a metric space. for all x ∈ x and r ∈ r+, we put bd(x, r) = {y ∈ x : d(x, y) ≤ r} and get the metric ballean b(x, d) = (x, r+, bd). clearly, every metric ballean is ordinal. we shall show in theorem 2.3 that spread(b) = den(b) for every unbounded ordinal ballean b. to this end we use another cardinal invariant of a ballean b = (x, p, b), the cellularity of b, defined in [4]. a subset y of x is called thick if, for every α ∈ p, there exists y ∈ y such that b(y, α) ⊆ y . the cellularity of b is the cardinal cell(b) = sup{|f| : f is a disjoint family of thick subsets of x}. by [4, theorem 1], for every ordinal ballean b, we have cell(b) = den(b) and there exists a disjoint family f of cardinality den(b) consisting of thick subsets of x. for every infinite cardinal κ, there exists a metric ballean b with den(b) = κ (see [4, example 1]). theorem 2.3. for every unbounded ordinal ballean b with support x, we have spread(b) = den(b) and there exists a subset y of x such that |y |b = |y | = den(b). 172 m. filali and i. v. protasov proof. let b = (x, ρ, b) where ρ is an infinite regular cardinal, κ = den(b) and cf (κ) be the cofinality of κ. let {fλ : λ ∈ κ} be a disjoint family of thick subsets of x and put f = ⋃ λ∈κ fλ. we fix some element x0 ∈ x and consider four cases. case ρ < cf (κ): we prove the following auxiliary statement. for every α ∈ ρ, there exists a bounded subset z of f such that |z| = κ and the family {b(z, α) : z ∈ z} is disjoint. for every λ ∈ κ, we take y(λ) ∈ fλ such that b(y(λ), λ) ⊆ fλ, and pick f (λ) ∈ ρ such that y(λ) ∈ b(x0, f (λ)). since f maps κ to ρ, by assumption, there exist a subset ∧ ⊆ κ and β ∈ ρ such that | ∧ | = κ and f (λ) = β for every λ ∈ ∧ . put z = {y(λ) : λ ∈ ∧ }. using the auxiliary statement, we can construct inductively a family {yα : α ∈ ρ} of bounded subsets of f such that the family {b(y, α) : y ∈ yα} is disjoint for each α ∈ ρ, b(yα, α) ⋂ b(yα′ , α ′) = ∅ for all distinct α, α′ ∈ ρ, and |yα| = κ for every α ∈ ρ. put y = ⋃ α∈ρ yα. case cf (κ) ≤ ρ < κ: using the assumption and repeating the arguments from the previous case, we get the following auxiliary statement. for every α ∈ ρ and every cardinal κ′ < κ, there exists a bounded subset z of f such that |z| > κ′ and the family {b(z, α) : z ∈ z} is disjoint. using the auxiliary statement, we can construct inductively a family {yα : α ∈ ρ} of bounded subsets of f and an increasing sequence (κα)α∈ρ of cardinals such that κ = sup{κα : α ∈ ρ}, the family {b(y, α) : y ∈ yα} is disjoint for each α ∈ ρ, b(yα, α) ⋂ b(yα′ , α ′) = ∅ for all distinct α, α′ ∈ ρ, and |yα| ≥ κα for every α ∈ ρ. we put y = ⋃ α∈ρ yα. case ρ = κ: for every α ∈ ρ, we take yα ∈ fα such that b(yα, α) ⊆ fα. put y = {yα : α ∈ ρ}. case ρ > κ: this variant is impossible, see case ρ > κ in the proof of theorem 1 from [4]. � by definition, a g−space is a set x endowed with a (left) action g × x −→ x, (g, x) 7−→ g(x) of a group g with identity e such that e(x) = x and g(h(x)) = (gh)(x) for all x ∈ x and g, h ∈ g. now let x be an infinite transitive g−space, i.e., for any x, y ∈ x, there exists g ∈ g such that g(x) = y. let κ be an infinite cardinal such that κ ≤ |x|, and consider fκ = {f ⊆ g : |f | < κ, e ∈ a}. for any x ∈ x and f ∈ fκ, we put b(x, f ) = f (x) = {f (x) : f ∈ f }, and get the ballean b(x, κ) = (x, fκ, b). let l be a large subset of x. we take f ∈ fκ such that b(l, f ) = x. since |f | < |x| and |b(l, f )| ≤ |l||f |, we have |l| = |x|, so den(b(x, κ)) = |x|. spread of balleans 173 theorem 2.4. let x be an infinite transitive g-space, κ be an infinite cardinal such that κ ≤| x |. then the following statements hold. (1) if g = x and g(x) = gx, then spread(b(x, κ)) = |x|. (2) if g is the group of all permutations of x, then spread(b(x, κ)) = 0. (3) let ρ be an infinite cardinal such that ρ ≤ |x|, and let g be the group of all permutations of x with supp(g) < ρ, g ∈ g where supp(g) = {x ∈ x : g(x) 6= x}. if either ρ < κ, or ρ = κ and κ is regular, then spread(b(x, κ)) = |x|. if either ρ > κ, or ρ = κ and κ is singular, then spread(b(x, κ)) = 0. proof. (1) by [2, proposition 4.1], there exists a subset y of g such that |y | = |g| and |gy ⋂ y | ≤ 3 for every g ∈ g, g 6= e. we take an arbitrary f ∈ fκ, and let z = f y ∩ y . since |f | < κ, the subset z of y satisfies |z| ≤ 3|f | < κ, i.e., z is bounded, and b(y, f ) ⋂ y = f y ⋂ y = {y} for every y ∈ y \z. it follows that y is pseudodiscrete. since |y |b = |y |, we have spread(b(x, κ)) = |x|. (2) it suffices to show that every subset y = {yα : α ∈ λ} of x of cardinality λ ≥ κ is not pseudodiscrete. we say that an ordinal α ∈ λ is even if either α is a limit ordinal or α = β + n for some limit ordinal β and some even natural number n. otherwise, we say that α is odd. then we define a permutation f of x by the rule: f (x) =      yα+1 if x = yα and α is even, yα−1 if x = yα and α is odd, x for each x ∈ x\y. we put f = {f, e}. clearly, |b(yα, f ) ⋂ y | = 2 for every α ∈ λ. since y is not bounded, y is not pseudodiscrete. (3) if either ρ < κ, or ρ = κ and κ is regular, we take an arbitrary subset f ∈ fκ and put z = ⋃ {supp(g) : g ∈ f }. then |z| < κ, so it is bounded, and clearly b(x, f ) ⋂ x = {x} for every x ∈ x \ z. it follows that x is pseudodiscrete. since |x|b = |x| we conclude that spread(b(x, κ)) = |x|. if ρ > κ we can use the arguments proving (2) to show that every subset of x of cardinality ≥ κ is not pseudodiscrete. if ρ = κ and κ is singular, we fix an arbitrary subset y of x with |y | = κ and partition y = ⋃ {yβ : β ∈ cf (κ)} so that |yβ| < κ for every β ∈ cf (κ). for every β ∈ cf (κ), we fix a permutation fβ of x such that supp(fβ ) = yβ , f (yβ ) = yβ . then we put f = {e} ⋃ {fβ : β ∈ cf (κ)} 174 m. filali and i. v. protasov and note that f ∈ fκ and |b(y, f ) ⋂ y | ≥ 2 for every y ∈ y . it follows that y is not pseudodiscrete and spread(b(x, κ)) = 0. � theorem 2.5. for every unbounded pseudodiscrete ballean b, we have den(b) = spread(b) and cell(b) = 1. proof. let x be the support of b. by [5, theorem 3.6], there exists a filter ϕ on x such that ⋂ ϕ = ∅ and b = (x, ϕ, b) where b(x, f ) = { {x}, if x ∈ f ; x \ f, if x /∈ f for all x ∈ x and f ∈ ϕ. we put κ = min{|f | : f ∈ ϕ} and note that a subset l of x is large if and only if l ∈ ϕ so den(b) = κ. on the other hand, a subset v of x is bounded if and only if x\v ∈ ϕ. hence, |x|b = κ, and so spread(b) = κ. we note also that every unbounded subset of x is thick. hence, if ϕ is an ultrafilter then cell(b) = 1. � for every pair γ, λ of infinite cardinals with γ < λ, we construct next a ballean b such that den(b) = γ and spread(b) = λ. example 2.6. we take a ballean b1 = (x1, p1, b1) such that spread(b1) = 0, |x1| = den(b1) = γ and each ball b1(x1, α1) is finite (see example 2.2). let b2 = (x2, p2, b2) be a pseudodiscrete ballean such that spread(b2) = |x2| = λ. we consider the ballean b = (x, p, b) with x = x1 × x2, p = p1 × p2 and b((x1, x2), (α1, α2)) = b(x1, α1) × b(x2, α2). since den(b1) = γ and |x| = γ, we see that den(b) = γ. since spread(b2) = λ, we see that spread(b) ≥ λ. let now z be any subset of x with |z| > λ. then there exist an infinite subset y of x1 and a ∈ x2 such that y × {a} ⊆ z. since y is not pseudodiscrete in b1, z is not pseudodiscrete in b. hence, spread(b) = λ. we conclude the exposition with the following open questions. problem 2.7. given a ballean b with support x, does there exist a pseudodiscrete subset y ⊆ x such that |y |b = spread(b) ? problem 2.8. let b = (x, p, b) be a ballean, |x| = κ and let |p | ≤ κ. assume that there exists κ′ < κ such that | b(x, α) |≤ κ′ for all x ∈ x, α ∈ p . is spread(b) = κ? by [4, theorem 2(i)], den(b) = cell(b) = κ. problem 2.9. let b = (x, p, b) be a ballean, |x| = κ and let |p | ≤ κ. assume that κ is regular and |b(x, α)| < κ for all x ∈ x, α ∈ p . is spread(b) = κ? by [4, theorem 2(ii)], den(b) = cell(b) = κ. spread of balleans 175 references [1] a. dranishnikov, asymptotic topology, russian math. surveys 55 (2000), 1085–1129. [2] c. chou, on the size of the set of left invariant means on a semigroup, proc. amer. math. soc. 23 (1969), 199–205. [3] i. v. protasov, normal ball structures, math. stud. 20 (2003), 3–16. [4] i. v. protasov, cellularity and density of balleans, appl. gen. topology 8 (2007), 283– 291. [5] i. v. protasov and m. zarichnyi, general asymptology, math. stud. monogr. ser., vntl, lviv (2006). [6] j. roe, lectures on coarse geometry, ams university lecture ser., 31 (2003). received october 2006 accepted february 2007 mahmoud filali (mahmoud.filali@oulu.fi) dept of math. sciences, university of oulu, fin 90014, oulu, finland igor protasov (islab@unicyb.kiev.ua) department of cybernetics, kyiv university, volodimirska 64, kyiv 01033, ukraine () @ applied general topology c© universidad politécnica de valencia volume 12, no. 1, 2011 pp. 27-33 characterizing meager paratopological groups taras banakh, igor guran and alex ravsky abstract we prove that a hausdorff paratopological group g is meager if and only if there are a nowhere dense subset a ⊂ g and a countable set c ⊂ g such that ca = g = ac. 2010 msc: 22a05, 22a30. keywords: paratopological group, baire space, shift-baire group, shiftmeager group. 1. introduction trying to find a counterpart of the lindeföf property in the category of topological groups i.guran [7] introduced the notion of an ω-bounded group which turned out to be very fruitful in topological algebra, see [9]. we recall that a topological group g is ω-bounded if for each non-empty open subset u ⊂ g there is a countable subset c ⊂ g such that cu = g = u c. a similar approach to the baire category leads us to the notion of an shiftmeager (shift-baire) group. this is a topological group that can(not) be written as the union of countably many translation copies of some fixed nowhere dense subset. the notion of a shift-meager (shift-baire) group can be defined in a more general context of semitopological groups, that is, groups g endowed with a shift-invariant topology τ . the latter is equivalent to saying that the group operation · : g × g → g is separately continuous. if this operation is jointly continuous, then (g, τ ) is called a paratopological group, see [1]. a semitopological group g is defined to be • left meager (resp. right meager) if g = ca (resp. g = ac) for some nowhere dense subset a ⊂ x and some countable subset c ⊂ g; • shift-meager if g is both left and right meager; 28 t. banakh, i. guran and a. ravsky • left baire (resp. right baire) if for every open dense subset u ⊂ x and every countable subset c ⊂ g the intersection ⋂ x∈c xu (resp. ⋂ x∈c u x) is dense in g; • shift-baire if g is both left and right baire. for semitopological groups those notions relate as follows: not shift-baire hh hh hh hh hy �� * ? shift-meager @@i �� * not left baire � left meager � ��3 right meager not right baire q qqk hh hh hh hh hhy meager 6 ? not baire the following theorem implies that for hausdorff paratopological groups all the eight properties from this diagram are equivalent. theorem 1.1. a hausdorff paratopological group g is meager if and only if g is shift-meager. this theorem will be proved in section 4. the proof is based on theorem 2.1 giving conditions under which a meager semitopological group is left (right) meager and theorem 3.2 describing some oscillator properties of 2-saturated hausdorff paratopological groups. 2. shift-meager semitopological groups in this section we search for conditions under which a given meager semitopological group is left (right) meager. following [3], [4] and [5], [6], we define a subset a ⊂ g of a group g to be • left large (resp. right large) if g = f a (resp. g = af ) for some finite subset f ⊂ g; • left p-small (resp. right p-small ) if there is an infinite subset b ⊂ g such that the indexed family {ba}b∈b (resp. {ab}b∈b) is disjoint. theorem 2.1. a meager semitopological group g is left (right) meager provided one of the following conditions holds: (1) g contains a non-empty open left (right) p-small subset; (2) g contains a sequence (un)n∈ω of pairwise disjoint open left (right) large subsets; (3) g contains sequences of non-empty open sets (un)n∈ω and points (gn)n∈ω such that the sets gnunu −1 n (resp. u −1 n ungn), n ∈ n, are pairwise disjoint. characterizing meager paratopological groups 29 proof. (1l) assume that u ⊂ g is a non-empty open left p-small subset. we may assume that u is a neighborhood of the neutral element e of g. it follows that there is a countable subset b = {bn}n∈ω ⊂ g such that bnu ∩ bmu = ∅ for any distinct numbers n 6= m. the countable set b generates a countable subgroup h of g. by an h-cylinder we shall understand an open subset of the form hv g where g ∈ g and v ⊂ u is a neighborhood of e. let u = {hvαgα : α ∈ a} be a maximal disjoint family of h-cylinders in g (such a family exists by the zorn lemma). we claim that ∪u is dense in g. assuming the converse, we could find a point g ∈ g \ ∪u and a neighborhood v ⊂ u of e such that v g ∩ ∪u. taking into account that h · (∪u) = ∪u, we conclude that hv g ∩ ∪u = ∅ and hence u ∪{hv g} is a disjoint family of h-cylinders that enlarges the family u, which contradicts the maximality of u. therefore ∪u is dense in g and hence g \∪u is a closed nowhere dense subset of g. the space g, being meager, can be written as the union g = ⋃ n∈ω mn of a sequence (mn)n∈ω of nowhere dense subsets of g. it is easy to see that the set m = (g \ ∪u) ∪ ⋃ α∈a ⋃ n∈ω bn(mn ∩ vαgα) is nowhere dense in g and g = hm , witnessing that g is left meager. (2l) assume that g contains a sequence (un)n∈ω of pairwise disjoint open left large subsets. for every n ∈ ω find a finite subset fn ⊂ g with g = fn ·un. write g = ⋃ n∈ω mn as countable union of nowhere dense subsets and observe that for every n ∈ ω the subset ⋃ x∈fn x−1(mn ∩ xun) of un is nowhere dense. since the family {un}n∈ω is disjoint, the set m = ⋃ n∈ω ⋃ x∈fn x−1(mn ∩ xun) is nowhere dense in g. since ( ⋃ n∈ω fn) · m = g, the semitopological group g is left meager. (3l) assume that (un)n∈ω is a sequence of non-empty open subsets of g and (gn)n∈ω is a sequence of points of g such that the sets gnunu −1 n , n ∈ ω, are pairwise disjoint. using the zorn lemma, for every n ∈ ω we can choose a maximal subset fn ⊂ g such that the indexed family {xun}x∈fn is disjoint. if for some n ∈ ω the set fn is infinite, then the set un is left p-small and consequently, the group g is left meager by the first item. so, assume that each set fn, n ∈ ω, is finite. the maximality of fn implies that for every x ∈ g there is y ∈ fn such that xun ∩ yun 6= ∅. then x ∈ yunu −1 n and hence g = fnunu −1 n , which means that the open set unu −1 n is left large. since the family {gnunu −1 n }n∈ω is disjoint, it is legal to apply the second item to conclude that the group g is left meager. (1r)−(3r). the right versions of the items (1)–(3) can be proved by analogy. � 30 t. banakh, i. guran and a. ravsky 3. oscillation properties of paratopological groups in this section we establish some oscillation properties of 2-saturated paratopological groups. first, we recall the definition of oscillator topologies on a given paratopological group (g, τ ), see [2] for more details. given a subset u ⊂ g, by induction define subsets (±u )n and (∓u )n, n ∈ ω, of g letting (±u )0 = (∓u )0 = {e} and (±u )n+1 = u (∓u )n, (∓u )n+1 = u −1(±u )n for n ≥ 0. thus (±u )n = u u −1u · · · u (−1) n−1 ︸︷︷︸ n and (∓u )n = u −1u u −1 · · · u (−1) n ︸︷︷︸ n . note that ((±u )n)−1 = (±u )n if n is even and ((±u )n)−1 = (∓u )n if n is odd. by an n-oscillator (resp. a mirror n-oscillator ) on a topological group (g, τ ) we understand a set of the form (±u )n (resp. (∓u )n ) for some neighborhood u of the unit of g. observe that each n-oscillator in a paratopological group (g, τ ) is a mirror n-oscillator in the mirror paratopological group (g, τ −1) and vice versa: each mirror n-oscillator in (g, τ ) is an n-oscillator in (g, τ −1). by the n-oscillator topology on a paratopological group (g, τ ) we understand the topology τn consisting of sets u ⊂ g such that for each x ∈ u there is an n-oscillator (±v )n with x · (±v )n ⊂ u . let us recall [8], [1, p. 342] that a paratopological group (g, τ ) is saturated if each non-empty open set u ⊂ g has non-empty interior in the mirror topology τ −1 = {u −1 : u ∈ τ}. this notion can be generalized as follows. define a paratopological group (g, τ ) to be n-saturated if each non-empty open set u ∈ τn has non-empty interior in the topology (τ −1)n. proposition 3.1. a paratopological group (g, τ ) is 2-saturated if no non-empty open subset u ⊂ g is p-small. proof. to prove that g is 2-saturated, take any non-empty open set u2 ∈ τ2 and find a point x ∈ u2 and a neighborhood u ∈ τ of e such that xu 2u −2 ⊂ u2. by the zorn lemma, there is a maximal subset b ⊂ g such that bu ∩ b′u = ∅ for all distinct points b, b′ ∈ b. by our hypothesis, u is not p-small, which implies that the set b is finite. the maximality of b implies that for each x ∈ g the shift xu meets some shift bu , b ∈ b. consequently, x ∈ bu u −1 and g = ⋃ b∈b bu u −1. it follows that the closure u u −1 of u u −1 in the topology (τ −1)2 has non-empty interior. we claim that u u −1 ⊂ u 2u −2. indeed, given any point z ∈ u u −1 , we conclude that the neighborhood u −1zu of z in the topology (τ −1)2 meets u u −1 and hence z ∈ u 2u −2. now we see that the set u2 ⊃ xu 2u −2 ⊃ xu u −1 has non-empty interior in the topology (τ −1)2, witnessing that the group (g, τ ) is 2-saturated. � characterizing meager paratopological groups 31 by proposition 2 of [2], for each saturated paratopological group (g, τ ) the semitopological group (g, τ2) is a topological group. this results generalizes to n-saturated groups. theorem 3.2. if (g, τ ) is an n-saturated paratopological group for some n ∈ n, then (g, τ2n) is a topological group. proof. according to theorem 1 of [2], (g, τ2n) is a topological group if and only if for every neighborhood u ∈ τ of the neutral element e ∈ g there is a neighborhood v ∈ τ of e such that (∓v )2n ⊂ (±u )2n. since the paratopological group (g, τ ) is n-saturated, the set (±u )n ∈ τ2 contains an interior point x in the mirror topology (τ −1)n. consequently, there is a neighborhood v ∈ τ of e such that (∓v )nx ⊂ (±u )n. now we consider separately the cases of odd and even n. 1. if n is odd, then applying the operation of the inversion to (∓v )nx ⊂ (±u )n, we get x−1(±v )n ⊂ (∓u )n and then (∓v )2n = (∓v )n(±v )n = (∓v )nxx−1(±v )n ⊂ (±u )n(∓u )n = (±u )2n. 2. if n is even, then (∓v )nx ⊂ (±u )n implies x−1(∓v )n ⊂ (±u )n and (∓v )2n = (∓v )n(∓v )n = (∓v )nxx−1(∓v )n ⊂ (±u )n(±u )n = (±u )2n. � according to [2], for each 1-saturated hausdorff paratopological groups (g, τ ) the group (g, τ2) is a hausdorff topological group. for 2-saturated group we have a bit weaker result. theorem 3.3. for any non-discrete hausdorff 2-saturated paratopological group (g, τ ) the maximal antidiscrete subgroup {e} = ⋂ e∈u∈τ (±u )4 of the topological group (g, τ4) is nowhere dense in the topology τ2. proof. to show that {e} is nowhere dense in the topology τ2, fix any nonempty open set u2 ∈ τ2. since g is not discrete, so is the topology τ2 ⊂ τ . consequently, we can find a point x ∈ u2 \ {e}. since g is a hausdorff paratopological group, there is a neighborhood u ∈ τ of e such that e /∈ xu u −1 ⊂ u2. the continuity of the group operation yields a neighborhood v ∈ τ of e such that v 2 ⊂ u and v 2x ⊂ xu . then v 2xv −2 ⊂ xu u −1 6∋ e yields v −1v ∩ v xv −1 = ∅. using the shift-invariantness of the topology τ , find a neighborhood w ∈ τ of e such that w ⊂ v and xw ⊂ v x. since the group g is 2-saturated, the open set xw w −1 ∈ τ2 has non-empty interior in the topology (τ −1)2. consequently, there is a point y ∈ xw w −1 and a neighborhood o ∈ τ of e such that o ⊂ w and o−1yo ⊂ xw w −1. observe that o−1o ∩ o−1yo ⊂ v −1v ∩ xw w −1 ⊂ v −1v ∩ v xv −1 = ∅ and consequently, u2 ∋ y /∈ oo −1oo−1 ⊃ {e}. � problem 3.4. can the topology τ2n be antidiscrete for some hausdorff nsaturated paratopological group? 32 t. banakh, i. guran and a. ravsky problem 3.5. assume that a paratopological group (g, τ ) is 2-saturated. is its mirror paratopological group (g, τ −1) 2-saturated? 4. proof of theorem 1.1 we need to check that each meager hausdorff paratopological group g is left and right meager. if the paratopological group g contains a non-empty open left p-small subset, then g is left meager by theorem 2.1(1). so assume that no non-empty open subset of g is left p-small. in this case proposition 3.1 implies that the paratopological group g is 2-saturated while theorem 3.3 ensures that the topological group (g, τ4) contains a countable disjoint family {wn}n∈ω of nonempty open sets. by the definition of the 4th oscillator topology τ4 each set wn contains a subset of the form xnunu −1unu −1 n where xn ∈ g and un is a neighborhood of the neutral element in the paratopological group g. since the sets xnunu −1 n ⊂ wn, n ∈ ω, are pairwise disjoint, we can apply theorem 2.1(3) to conclude that the paratopological group g is left meager. by analogy we can prove that g is right meager. 5. discussion and open problems the following example shows that without any restrictions, a meager semitopological group needs not be shift-meager. example 5.1. let g be an uncountable group whose cardinality |g| has countable cofinality. endow the group g with the shift-invariant topology generated by the base {g \ a : |a| < |g|}. it is easy to see that a subset a ⊂ g is nowhere dense if and only if it is not dense if and only if |a| < |g|. this observation implies that g is meager (because |g| has countable cofinality). on the other hand, the semi-topological group g is not shift-meager because for every nowhere dense subset a ⊂ g and every countable subset c ⊂ g we get |a| < |g| and hence g 6= ca because |ca| ≤ max{ℵ0, |a|} < |g|. problem 5.2. is each meager paratopological group g shift-meager? problem 5.3. is each meager hausdorff semitopological group shift-meager? problem 5.4. is each left meager semitopological group right meager? also we do not know is the following semigroup version of theorem 1.1 holds. problem 5.5. let s be an open meager subsemigroup of a hausdorff paratopological group g. is s ⊂ ca for some nowhere dense subset a ⊂ s and a countable subset c ⊂ g? characterizing meager paratopological groups 33 references [1] a. arhangel’skii and m. tkachenko, topological groups and related structures, world sci. publ., hackensack, nj, 2008. [2] t. banakh and o. ravsky, oscillator topologies on paratopological groups and related number invariants, algebraical structures and their applications, kyiv: inst. mat. nanu, (2002) 140-152 (arxiv:0810.3028). [3] a. bella and v. malykhin, on certain subsets of a group, questions answers gen. topology 17, no. 2 (1999), 183–197. [4] a. bella, v. malykhin, on certain subsets of a group. ii, questions answers gen. topology 19, no. 1 (2001), 81–94. [5] d. dikranjan, u. marconi and r. moresco, groups with a small set of generators, appl. gen. topol. 4, no. 2 (2003), 327–350. [6] d. dikranjan and i. protasov, every infinite group can be generated by p-small subset, appl. gen. topol. 7, no. 2 (2006), 265–268. [7] i. i. guran, topological groups similar to lindelöf groups, dokl. akad. nauk sssr 256, no. 6 (1981), 1305–1307. [8] i. i. guran. the cardinal invariant of paratopological groups, proc. of ii intern. algebraical conf. in ukraine, dedicated to the memory of l.a. kaluzhnin. – kyiv-vinnitsya, 1999. p.72. (in ukrainian) [9] m. tkachenko, introduction to topological groups, topology appl. 86, no. 3 (1998), 179–231. (received january 2010 – accepted september 2010) taras banakh (t.o.banakh@gmail.com) uniwersytet humanistyczno-przyrodniczy jana kochanowskiego w kielcach, poland, and ivan franko national university of lviv, ukraine igor guran (igor guran@yahoo.com) ivan franko national university of lviv, ukraine alex ravsky (oravsky@mail.ru) institute of applied problems of mechanics and mathematics of national academy of sciences, lviv, ukraine characterizing meager paratopological groups. by t. banakh, i. guran and a. ravsky @ appl. gen. topol. 22, no. 2 (2021), 483-496 doi:10.4995/agt.2021.16562 © agt, upv, 2021 revisiting ćirić type nonunique fixed point theorems via interpolation erdal karapınar * faculty of fundamental science, industrial university of ho chi minh city, ho chi minh, vietnam department of mathematics, cankaya university 06836, incek, ankara-turkey. (erdalkarapinar@yahoo.com) communicated by s. romaguera abstract in this paper, we aim to revisit some non-unique fixed point theorems that were initiated by ćirić, first. we consider also some natural consequences of the obtained results. in addition, we provide a simple example to illustrate the validity of the main result. 2020 msc: 46t99; 47h10; 54h25. keywords: abstract metric space; non-unique fixed point; self-mappings. 1. introduction and preliminaries the notion of ”nonunique fixed point” was suggested and used efficiently by ćirić [16] in 1974. regarding the fact that banach’s fixed point theorem was abstracted from the papers of liouville (1837) and picard (1890), we underline the connection of the fixed point theorem and the solution of the differential equations. as well as the existence, the uniqueness of the solutions of differential equations is desired in most occasions. on the other hand, there are certain types of differential equations that have no unique solution. in connection with this fact, it is necessary to determine that non-unique fixed points are at least as significant as the unique ones. after the initial work of ćirić [16], several authors have published nonunique fixed point results in various conditions in different abstract spaces, see e.g. [16, 37, 1, 21, 35, 36, 23, 24, 25]. *dedicated to professor ljumbor ćirić received 31 october 2021 – accepted 25 november 2021 http://dx.doi.org/10.4995/agt.2021.16562 e. karapınar on the other hand, recently, the notion of interpolative contraction was defined in [27] to revisit the well-known results of kannan [22]. following this pioneering result, several papers on the interpolative contraction have appeared in the literature, see e.g. [28, 19, 9, 4, 26, 8, 34]. one of the most interesting generalization of the metric space is the b-metric space, defined as follow: definition 1.1 ([17, 14]). let x be a nonempty set and let d : x × x −→ [0,∞) satisfy the following conditions for all x,y,u ∈ x, (1.1) (b1) d(x,y) = 0 if and only if x = y(indistancy) (b2) d(x,y) = d(y,x) (symmetry) (b3) d(x,y) ≤ s[d(x,u) + d(u,y)] (modified triangle inequality). then, the map d is called a b-metric and the space (x,d) a b-metric space. it is worthy to note that the notion of ”b-metric” was announced also as ”quasi-metric”, see e.g. [13, 14]. it is also interesting to note that the notion of b-metric has a topology different from that of the standard metric. for example, closed ball is not a closed set. in the same way, the open ball does not form an open set. besides, the b-metric needs not to be continuous. considering the above-mentioned features of the b-metric, we can easily understand why so much research has been done on the b-metric, see e.g. [31, 20, 10, 11, 3, 30, 7, 2, 32, 33, 15, 6, 18, 5]. the following examples are not only standard, but also basic and interesting. example 1.2 ([10, 11]). let x = r. define (1.2) d(x,y) = |x−y|p for p > 1. then d is a b-metric on r. clearly, the first two conditions hold. since |x−y|p ≤ 2p−1[|x−z|p + |z −y|p], the third condition holds with s = 2p−1. thus, (r,d) is a b-metric space with a constant s = 2p−1. example 1.3 ([10, 11]). for p ∈ (0, 1), take x = lp(r) = { x = {xn}⊂ r : ∞∑ n=1 |xn|p < ∞ } . define d(x,y) = ( ∞∑ n=1 |xn −yn|p )1/p . then (x,d) is a b-metric space with s = 21/p. example 1.4 ([10, 11]). let e be a banach space and 0e be the zero vector of e. let p be a cone in e with int(p) 6= ∅ and � be a partial ordering with respect to p . let x be a non-empty set. suppose the mapping d : x×x → e satisfies: © agt, upv, 2021 appl. gen. topol. 22, no. 2 484 revisiting ćirić type nonunique fixed point theorems via interpolation (m1) 0 � d(x,y) for all x,y ∈ x; (m2) d(x,y) = 0 if and only if x = y; (m3) d(x,y) � d(x,z) + d(z,y), for all x,y ∈ x; (m4) d(x,y) = d(y,x) for all x,y ∈ x. then d is called a cone metric on x, and the pair (x,d) is called a cone metric space (cms). recall that a cone p in a banach space (e,‖ ·‖) is called normal if it there exist a real number k ≥ 1 satisfies the following condition: x � y ⇒‖x‖≤ k‖y‖ for all x,y ∈ p. let e be a banach space and p be a normal cone in e with the coefficient of normality denoted by k. let x be a non-empty set and d : x ×x → [0,∞) be defined by d(x,y) = ||d(x,y)||, where d : x × x → e is a cone metric space. then (x,d) is a b-metric space with a constant s := k ≥ 1. in generalization of the contraction condition, several auxiliary functions were considered in the literature. among them, we count the notion of comparison function which was defined by rus [38]. definition 1.5 ([12, 38]). a function φ : [0,∞) → [0,∞) is called a comparison function if it is increasing and φn(t) → 0 as n →∞ for every t ∈ [0,∞), where φn is the n-th iterate of φ. we refer [12, 38] for the basic features and interesting example for comparison functions. among all, we recollect the following lemma that indicates the importance of the comparison functions. lemma 1.6 ([12, 38]). if φ : [0,∞) → [0,∞) is a comparison function, then (1) each iterate φk of φ, k ≥ 1 is also a comparison function; (2) φ is continuous at 0; (3) φ(t) < t for all t > 0. definition 1.7 ([14]). let s ≥ 1 be a real number. a function φ : [0,∞) → [0,∞) is called a (b)-comparison function if (1) φ is increasing; (2) there exist k0 ∈ n, a ∈ [0, 1) and a convergent nonnegative series ∞∑ k=1 vk such that sk+1φk+1(t) ≤ askφk(t) + vk, for k ≥ k0 and any t ≥ 0. the collection of all (b)-comparison functions will be denoted by ψ. berinde [14] also proved the following important property of (b)-comparison functions. lemma 1.8 ([14]). let φ : [0,∞) → [0,∞) be a (b)-comparison function. then © agt, upv, 2021 appl. gen. topol. 22, no. 2 485 e. karapınar (1) the series ∞∑ k=0 skφk(t) converges for any t ∈ [0,∞); (2) the function bs : [0,∞) → [0,∞) defined as bs = ∞∑ k=0 skφk(t) is increasing and is continuous at t = 0. remark 1.9. any (b)-comparison function φ satisfies φ(t) < t and limn→∞φ n(t) = 0 for each t > 0. in this paper, we shall reconsider some of well-known nonunique fixed point theorem via interpolation in the context of b-metric spaces. 2. non-unique fixed points on b-metric space we start this section by considering the analog of the notions, ”orbitally continuous” and ”orbitally complete”, in the framework of b-metric space. definition 2.1 (see [16]). let (x,d) be a b-metric space and t be a self-map on x. (1) t is called orbitally continuous if (2.1) lim i→∞ tnix = z implies (2.2) lim i→∞ ttnix = tz for each x ∈ x. (2) (x,d) is called orbitally complete if every cauchy sequence of type {tnix}i∈n converges with respect to τd. a point z is said to be a periodic point of a function t of period m if tmz = z, where t 0x = x and tmx is defined recursively by tmx = ttm−1x. 2.1. ćirić type non-unique fixed point results. theorem 2.2. for a nonempty set x, we suppose that the function d : x × x → [0,∞) is a b-metric. we presume that a self-mapping t is orbitally continuous and (x,d,s) forms a t -orbitally complete b-metric space with s ≥ 1. if there is ψ ∈ ψ and α ∈ (0, 1) such that (2.3) min{ ( dα(tx,ty)d1−α(x,tx) ) , ( dα(tx,ty)d1−α(y,ty) ) } −min{dα(x,ty),d1−α(tx,y)}≤ ψ(d(x,y)), for all x,y ∈ x, then, for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . proof. starting from an arbitrary x := x0 ∈ x, we shall built a recursive sequence {xn} in the following way: (2.4) x0 := x and xn = txn−1 for all n ∈ n. © agt, upv, 2021 appl. gen. topol. 22, no. 2 486 revisiting ćirić type nonunique fixed point theorems via interpolation we presume that (2.5) xn 6= xn−1 for all n ∈ n. indeed, if for some n ∈ n we observe the inequality xn = txn−1 = xn−1, then, the proof is completed. by replacing x = xn−1 and y = xn in the inequality (2.3), we derive that (2.6) min{ ( dα(txn−1,txn)d 1−α(xn−1,txn−1) ) , ( dα(txn−1,txn)d 1−α(xn,txn) ) } −min{dα(xn−1,txn),d1−α(txn−1,xn)} ≤ ψ(d(xn−1,xn)). it yields that (2.7) min{d1−α(xn,xn+1)dα(xn,xn−1),d(xn,xn+1)}≤ ψ(d(xn−1,xn)). we shall prove that the sequence ( . xn−1,xn)} is non-increasing. suppose, on the contrary, that there is n0 such that d(xn0,xn0+1) > d(xn0−1,xn0 ). since ψ(t) < t for all t > 0, for this case we get d1−α(xn0,xn0+1)d α(xn0,xn0−1) ≤ ψ(d(xn0−1,xn0 )) < d(xn0−1,xn0 ), which implies d(xn0,xn0−1) ≤ d 1−α(xn0,xn0+1)d α(xn0,xn0−1) ≤ ψ(d(xn0−1,xn0 )) < d(xn0−1,xn0 ), that is, a contradiction. thus, we find that for all n ∈ n, (2.8) d(xn,xn+1) ≤ ψ(d(xn−1,xn)) < d(xn−1,xn). recursively, we derive that (2.9) d(xn,xn+1) ≤ ψ(d(xn−1,xn)) ≤ ψ2(d(xn−2,xn−1)) ≤ ···≤ ψn(d(x0,x1)). taking (2.8) into account, we note that the sequence {d(xn,xn+1)} is nonincreasing. in what follows, we shall prove that the sequence {xn} is cauchy. by using the triangle inequality (b3), we get (2.10) d(xn,xn+k) ≤ s[d(xn,xn+1) + d(xn+1,xn+k)] ≤ sd(xn,xn+1) + s{s[d(xn+1,xn+2) + d(xn+2,xn+k)]} = sd(xn,xn+1) + s 2d(xn+1,xn+2) + s 2d(xn+2,xn+k) ... ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . . + sk−1d(xn+k−2,xn+k−1) + s k−1d(xn+k−1,xn+k) ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . . + sk−1d(xn+k−2,xn+k−1) + s kd(xn+k−1,xn+k), © agt, upv, 2021 appl. gen. topol. 22, no. 2 487 e. karapınar since s ≥ 1. combining (2.9) and (2.10), we derive that (2.11) d(xn,xn+k) ≤ sψn(d(x0,x1)) + s2ψn+1d(x0,x1) + . . . + sk−1ψn+k−2(d(x0,x1)) + s kψn+k−1(d(x0,x1)) = 1 sn−1 [snψn(d(x0,x1)) + s n+1ψn+1d(x0,x1) + . . . + sn+k−2ψn+k−2(d(x0,x1)) + s n+k−1ψn+k−1(d(x0,x1))]. inevitably, we derive (2.12) d(xn,xn+k) ≤ 1 sn−1 [pn+k−1 −pn−1] , n ≥ 1,k ≥ 1, where pn = n∑ j=0 sjψj(d(x0,x1)), n ≥ 1. from lemma 1.8, the series ∞∑ j=0 sjψj(d(x0,x1)) is convergent and since s ≥ 1, upon taking limit n →∞ in (2.39), we observe (2.13) lim n→∞ d(xn,xn+k) ≤ lim n→∞ 1 sn−1 [pn+k−1 −pn−1] = 0. we deduce that the sequence {xn} is cauchy in (x,d). taking into account the t-orbitally completeness, we note that there is z ∈ x such that xn → z. owing to the orbital continuity of t, we conclude that xn → tz. consequently, we find z = tz which terminates the proof. � example 2.3. let the set x = {a,b,c,g,e} and d : x × x → [0,∞) be a b-metric (with s = 2) defined as follows d(x,y) a b c g e a 0 1 9 25 16 b 1 0 4 16 9 c 9 4 0 4 1 g 25 16 4 0 1 e 16 9 1 1 0 let also the mapping t : x → x be given as x a b c g e tx b b g e e thus, we have d(tx,ty) ta tb tc tg te ta = b 0 0 16 9 9 tb = b 0 0 16 9 9 tc = g 16 16 0 1 1 tg = e 9 9 1 0 0 te = e 9 9 1 0 0 and d(x,ty) a b c g e ta = b 1 0 4 16 9 tb = b 1 0 4 16 9 tc = g 25 16 4 0 1 tg = e 16 9 1 1 0 te = e 16 9 1 1 0 © agt, upv, 2021 appl. gen. topol. 22, no. 2 488 revisiting ćirić type nonunique fixed point theorems via interpolation we choose α = 1 2 and φ : [0,∞) → [0,∞) as φ(t) = t 2 . we shall denote, m1(x,y) = min{ ( d1/2(tx,ty)d1/2(x,tx) ) , ( d1/2(tx,ty)d1/2(y,ty) ) } m2(x,y) = min{d1/2(x,ty),d1/2(tx,y)}. then we have to consider the following cases: (1) for x = a,y = b and x = g,y = e, we have d(tx,ty) = 0 and obviously, (2.3) holds. (2) for x = a,y = c, we have m1(a,c) = min { d1/2(ta,tc)d1/2(a,ta) ) , ( d1/2(ta,tc)d1/2(c,tc) } = min{4 · 1, 4 · 2} = 4 m2(a,c) = min{5, 2} . thus, m1(a,c) −m2(a,c) = 2 < 92 = φ(d(a,c). (3) for x = a,y = g, we have m1(a,g) = min { d1/2(ta,tg)d1/2(a,ta) ) , ( d1/2(ta,tg)d1/2(g,tg) } = min{3 · 1, 3 · 1} = 1 m2(a,g) = min{4, 4} , and obviously (2.3) holds. (4) for x = a,y = e, we have m1(a,e) = min { d1/2(ta,te)d1/2(a,ta) ) , ( d1/2(ta,te)d1/2(e,te } = min{3 · 1, 3 · 0} = 0 m2(a,e) = min{4, 3} = 3, and (2.3)holds. (5) for x ∈ {b,e} and y ∈ x, since d(x,tx) = 0, we have m1(x,y) = 0 and then (2.3) holds. (6) for x = c and y = g m1(c,g) = min { d1/2(tc,tg)d1/2(c,tc) ) , ( d1/2(tc,tg)d1/2(g,tg } = min{1 · 2, 1 · 1} = 1 m2(c,g) = min{1, 0} = 0. therefore, m1(c,g) −m2(c,g) = 1 < 2 = φ(d(c,g)). (7) for x = c and y = e m1(c,e) = min { d1/2(tc,te)d1/2(c,tc) ) , ( d1/2(tc,te)d1/2(e,te } = min{1 · 2, 1 · 0} = 0 m2(c,e) = min{1, 1} = 1. therefore, m1(c,e) −m2(c,e) = −1 < 1 2 = φ(d(c,e)). then the conditions of theorem 2.2 hold and clearly, t has two fixed points, x = b and x = e. in the next corollaries we give some consequences of the theorem 2.2. corollary 2.4. for a nonempty set x, we suppose that the function d : x × x → [0,∞) is a b-metric. we presume that a self-mapping t on x is orbitally © agt, upv, 2021 appl. gen. topol. 22, no. 2 489 e. karapınar continuous and (x,d,s) forms a t -orbitally complete b-metric space with s ≥ 1. if there is q ∈ [0, 1 s ) and α ∈ (0, 1) such that (2.14) min{ ( dα(tx,ty)d1−α(x,tx) ) , ( dα(tx,ty)d1−α(y,ty) ) } −min{dα(x,ty),d1−α(tx,y)}≤ qd(x,y), for all x,y ∈ x, then for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . proof. it is sufficient to take ψ(t) = qt, where q ∈ [0, 1 s ), in theorem 2.2. � corollary 2.5. let t be an orbitally continuous self-map on the t -orbitally complete metric space (x,d). if there is a comparison function ψ and α ∈ (0, 1) such that (2.15) min{ ( dα(tx,ty)d1−α(x,tx) ) , ( dα(tx,ty)d1−α(y,ty) ) } −min{dα(x,ty),d1−α(tx,y)}≤ ψ(d(x,y)), for all x,y ∈ x, then for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . proof. it is sufficient to take s = 1 in theorem 2.2. � corollary 2.6. let t be an orbitally continuous self-map on the t -orbitally complete metric space (x,d). if there is q ∈ [0, 1) and α ∈ (0, 1) such that (2.16) min{ ( dα(tx,ty)d1−α(x,tx) ) , ( dα(tx,ty)d1−α(y,ty) ) } −min{dα(x,ty),d1−α(tx,y)}≤ qd(x,y), for all x,y ∈ x, then for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . proof. it is sufficient to take ψ(t) = qt, where q ∈ [0, 1), in corollary 2.6. � 2.2. pachpatte type non-unique fixed point results [37]. theorem 2.7. for a nonempty set x, we suppose that the function d : x × x → [0,∞) is a b-metric. we presume that a self-mapping t is orbitally continuous and (x,d,s) forms a t -orbitally complete b-metric space with s ≥ 1. if there exists ψ ∈ ψ and α ∈ (0, 1) such that (2.17) m(x,y) −n(x,y) ≤ ψ(dα(x,tx)d1−α(y,ty)), for all x,y ∈ x, where m(x,y) = min{d(tx,ty),dα(x,y)d1−α(tx,ty),d(y,ty)}, n(x,y) = min{dα(x,tx)d1−α(y,ty),dα(x,ty)d1−α(y,tx)}, then, for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . proof. by verbatim, following the initial lines of the proof of the theorem 2.2, we shall set-up a recursive sequence {xn = txn−1}n∈n, by starting from an arbitrary initial value x0 := x ∈ x. replacing in the inequality (2.17) x = xn−1 and y = xn, we obtain that (2.18) m(xn−1,xn) −n(xn−1,xn) ≤ ψ(dα(xn−1,txn−1)d1−α(xn,txn)), © agt, upv, 2021 appl. gen. topol. 22, no. 2 490 revisiting ćirić type nonunique fixed point theorems via interpolation where m(xn−1,xn) = min{d(txn−1,txn),dα(xn−1,xn)d1−α(txn−1,txn),d(xn,txn)}, n(xn−1,xn) = min{dα(xn−1,txn−1)d1−α(xn,txn),dα(xn−1,txn)d1−α(xn,txn−1)}. by simplifying the above inequality, we get (2.19) m(xn−1,xn) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1)), where m(xn−1,xn) = min{d(xn,xn+1),dα(xn−1,xn)d1−α(xn,xn+1)}. it is clear that the case m(xn−1,xn) = d α(xn−1,xn)d 1−α(xn,xn+1) is not possible for any n ∈ n. if it would be the case, the inequality (2.19) turns into (2.20) dα(xn−1,xn)d 1−α(xn,xn+1) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1)) < dα(xn−1,xn)d 1−α(xn,xn+1), which is a contradiction since ψ(t) < t for all t > 0. consequently, we derive (2.21) d(xn,xn+1) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1)) < dα(xn−1,xn)d 1−α(xn,xn+1), which yields (2.22) d(xn,xn+1) < d(xn−1,xn). on account of the fact that the comparison function ψ is nondecreasing, together with the inequalities (2.21) and (2.22), we find that (2.23) d(xn,xn+1) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1)) < ψ(d(xn−1,xn)), recursively, we obtain that d(xn,xn+1) ≤ ψ(d(xn−1,xn)) ≤ ψ2(d(xn−2,xn−1)) ≤ ···≤ ψn(d(x0,x1)). hence, we conclude that lim n→∞ d(xn+1,xn) = 0. the remaining part of the proof is verbatim repetition of the related lines in the proof of theorem 2.2, so we omit it. � below, we deduce some come consequences of the theorem 2.7 for particular choice of the comparison function ans the constant s. in case of ψ(t) = qt in theorem 2.7 we deduce the following result. corollary 2.8. for a nonempty set x, we suppose that the function d : x × x → [0,∞) is a b-metric. we presume that a self-mapping t is orbitally continuous and (x,d,s) forms a t -orbitally complete b-metric space with s ≥ 1. assume that there exists q ∈ [0, 1 s ) and α ∈ (0, 1), such that (2.24) m(x,y) −n(x,y) ≤ qdα(x,tx)d1−α(y,ty), © agt, upv, 2021 appl. gen. topol. 22, no. 2 491 e. karapınar for all x,y ∈ x, where m(x,y) and n(x,y) are defined as in theorem 2.7. then, for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . if the statements of theorem 2.7 are considered in the context of standard metric space instead of a b-metric space, we shall obtain the following consequence. corollary 2.9. let t be an orbitally continuous self-map on the t -orbitally complete metric space (x,d). suppose that there exist a comparison function ψ and α ∈ (0, 1) such that (2.25) m(x,y) −n(x,y) ≤ ψ(dα(x,tx)d1−α(y,ty)), for all x,y ∈ x, where m(x,y) and n(x,y) are defined as in theorem 2.7. then for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . for ψ(t) = qt in corollary 2.9, the following results is derived. corollary 2.10. let t be an orbitally continuous self-map on the t -orbitally complete standard metric space (x,d). suppose that there exists q ∈ [0, 1) and α ∈ (0, 1) such that (2.26) m(x,y) −n(x,y) ≤ qdα(x,tx)d1−α(y,ty), for all x,y ∈ x, where m(x,y) and n(x,y) are defined as in theorem 2.7. then, for each x0 ∈ x the sequence {tnx0}n∈n converges to a fixed point of t . 2.3. k-type non-unique fixed point results [23]. the following theorem is inspired by the main theorem of [23] theorem 2.11. for a nonempty set x, we suppose that the function d : x ×x → [0,∞) is a b-metric. we presume that a self-mapping t is orbitally continuous and (x,d,s) forms a t -orbitally complete b-metric space with s ≥ 1. assume that there exists real numbers α,β,γ ∈ (0, 1) with α + β + γ < 1 and ψ ∈ ψ. if the following inequality (2.27) dα(tx,ty)dβ(x,tx)dγ(y,ty) [ d(y,tx) + d(x,ty) 2s ]1−α−β−γ ≤ ψ(d(x,y)) holds for all x,y ∈ x, then, t has at least one fixed point. proof. starting from an arbitrary point x = x0 ∈ x, we shall construct a sequence {xn} as follows: (2.28) xn+1 := txn n = 0, 1, 2, ... letting x = xn and y = xn+1, in the inequality (2.27) yields (2.29) dα(txn,txn+1)d β(xn,txn)d γ(xn+1,txn+1)[ d(xn+1,txn)+d(xn,txn+1) 2s ]1−α−β−γ ≤ ψ(d(xn,xn+1)). © agt, upv, 2021 appl. gen. topol. 22, no. 2 492 revisiting ćirić type nonunique fixed point theorems via interpolation on account of (2.28), the statement (2.29) turns into (2.30) dα(xn+1,xn+2)d β(xn,xn+1)d γ(xn+1,xn+2)[ d(xn+1,xn+1)+d(xn,xn+2) 2s ]1−α−β−γ ≤ ψ(d(xn,xn+1)). by elementary calculation and simplification, we derive (2.31) dα(xn+1,xn+2)d β(xn,xn+1)d γ(xn+1,xn+2)[ d(xn,xn+1)+d(xn+1,xn+2) 2 ]1−α−β−γ ≤ ψ(d(xn,xn+1)). suppose that d(xn,xn+1) < d(xn+1,xn+2). then the inequality (2.31) turns into (2.32) dα(xn+1,xn+2)d β(xn,xn+1)d γ(xn+1,xn+2)[d(xn,xn+1)] 1−α−β−γ ≤ dα(xn+1,xn+2)dβ(xn,xn+1)dγ(xn+1,xn+2)[ d(xn,xn+1)+d(xn+1,xn+2) 2s ]1−α−β−γ ≤ ψ(d(xn,xn+1)). then we get (2.33) d1−α−γ(xn,xn+1)d 1−β(xn+1,xn+2) ≤ ψ(d(xn,xn+1)) < d(xn,xn+1). the above inequality can be expressed as (2.34) d1−α−γ(xn,xn+1) < d 1−α−γ(xn,xn+1), which is a contradiction. hence, we conclude that d(xn,xn+1) ≥ d(xn+1,xn+2). so, the inequality above together with (2.32) yields that (2.35) d(xn+1,xn+2) ≤ ψ(d(xn,xn+1)) < d(xn,xn+1) thus, the sequence {d(xn,xn+1)} is non-increasing. recursively, we find that (2.36) d(xn,xn+1) ≤ ψ(d(xn−1,xn)) ≤ ψ2(d(xn−2,xn−1)) ≤ ···≤ ψn(d(x0,x1)). © agt, upv, 2021 appl. gen. topol. 22, no. 2 493 e. karapınar as a next step, we shall show that the sequence {xn} is cauchy. by employing the triangle inequality (b3), we get (2.37) d(xn,xn+k) ≤ s[d(xn,xn+1) + d(xn+1,xn+k)] ≤ sd(xn,xn+1) + s{s[d(xn+1,xn+2) + d(xn+2,xn+k)]} = sd(xn,xn+1) + s 2d(xn+1,xn+2) + s 2d(xn+2,xn+k) ... ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . . +sk−1d(xn+k−2,xn+k−1) + s k−1d(xn+k−1,xn+k) ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . . +sk−1d(xn+k−2,xn+k−1) + s kd(xn+k−1,xn+k), since s ≥ 1. on account of (2.37) and (2.36), we deduce that (2.38) d(xn,xn+k) ≤ sψn(d(x0,x1)) + s2ψn+1d(x0,x1) + . . . + sk−1ψn+k−2(d(x0,x1)) + s kψn+k−1(d(x0,x1)) = 1 sn−1 [snψn(d(x0,x1)) + s n+1ψn+1d(x0,x1) + · · · + sn+k−2ψn+k−2(d(x0,x1)) + sn+k−1ψn+k−1(d(x0,x1))]. consequently, we have (2.39) d(xn,xn+k) ≤ 1 sn−1 [pn+k−1 −pn−1] , n ≥ 1,k ≥ 1, where pn = n∑ j=0 sjψj(d(x0,x1)), n ≥ 1. from lemma 1.8, the series ∞∑ j=0 sjψj(d(x0,x1)) is convergent and since s ≥ 1, upon taking limit n →∞ in (2.39), we obtain (2.40) lim n→∞ d(xn,xn+k) ≤ lim n→∞ 1 sn−1 [pn+k−1 −pn−1] = 0. we deduce that the sequence {xn} is cauchy in (x,d). the remaining part of the proof is verbatim repetition of the related lines in the proof of theorem 2.2. � finally, we state the following consequence of theorem 2.11. corollary 2.12. let t be an orbitally continuous self-map on the t -orbitally complete b-metric space (x,d,s) with s ≥ 1. suppose there exist real numbers © agt, upv, 2021 appl. gen. topol. 22, no. 2 494 revisiting ćirić type nonunique fixed point theorems via interpolation q ∈ [0, 1 s ) and α,β,γ ∈ (0, 1) with α + β + γ < 1. if (2.41) dα(tx,ty)dβ(x,tx)dγ(y,ty) [ d(y,tx) + d(x,ty) 2s ]1−α−β−γ ≤ qd(x,y) holds for all x,y ∈ x, then, t has at least one fixed point. proof. take ψ(t) = qt in the proof of theorem 2.11, where q ∈ [0, 1). � notice also that the above theorem and corollary of this section are valid in the setting of standard metric space. acknowledgements. the author thanks to his institutes. references [1] j. achari, on ćirić’s non-unique fixed points, mat. vesnik 13 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(dpub@pphmj.com, poojasingh 07@rediffmail.com) abstract for a dynamical system (x, f), the passage of various dynamical properties such as transitivity, total transitivity, weakly mixing, mixing, topological exactness, topological conjugacy, to the hyperspace c(x) of x consisting of nonempty closed connected subsets of x, and also to the hyperspace suspension hs(x) of x, have been considered and studied. 2010 msc: 54h20; 37b99. keywords: transitivity; total transitivity; weakly mixing; mixing; topological exactness; topological conjugacy; vietoris topology, hyperspace suspension. 1. introduction throughout the paper, (x,f) denotes a dynamical system, where x is a compact metric space and f is a selfmap on x. by a map, we mean a continuous map. the symbol n denotes the set of positive integers. let (x,f) be a dynamical system. then f is called (i) transitive if for a pair of nonempty open sets u, v of x, there exists an n ∈ n, such that fn(u)∩v 6= φ, (ii) totally transitive if the map fk is transitive, for each k ∈ n, (iii) weakly mixing if for pairs of nonempty open sets u1, u2 and v1, v2 of x, there exists an n ∈ n, such that fn(ui) ∩ vi 6= φ, for i = 1, 2, (iv) mixing if for a pair of nonempty open sets u, v of x, there exists an n ∈ n, such that fn(u) ∩ v 6= φ, n ≥ n, (v) topologically exact if for an open set u of x, there exists an n ∈ n, such that fn(u) = x. it is known that, if f is weakly received 7 november 2013 – accepted 15 march 2014 http://dx.doi.org/10.4995/agt.2014.1841 d. masood and p. singh mixing and u1,u2, . . . ,un, v1,v2, . . . ,vn are nonempty open sets of x, then there exists a k ≥ 1 such that fk(ui) ∩ vi 6= φ, for i = 1,2, . . . ,n [12]. the dynamical system (x,f) is said to possess a dynamical property p , if f has p . two dynamical systems (x,f) and (y,g) are said to be topologically conjugate if there exists a homeomorphism ϕ from x to y such that g ◦ ϕ = ϕ ◦ f. by 2x, we denote the set consisting of all nonempty closed subsets of the space x. on it, (i) the upper vietoris topology τu , has bu ≡ {< u1, . . . ,um >: u1, . . . ,um are open sets of x}, where < u1, . . . ,um > = {a ∈ 2 x : a ⊆ ⋃m i=1 ui}, as a basis, (ii) the lower vietoris topology τl, has bl ≡ {: u ′ 1, . . . ,u ′ m are open sets of x}, where < u′1, . . . ,u ′ m > = {a ∈ 2 x : a ∩ u′i 6= φ, i = 1, . . . ,m}, as a basis, and (iii) the vietoris topology τ, has b ≡ {< v1, . . . ,vn >: v1, . . . ,vn are open sets of x}, where < v1, . . . ,vn >= {a ∈ 2 x : a ⊆ ⋃n i=1 vi, a ∩ vi 6= φ, i = 1, . . . ,n}, as a basis. the hyperspace c(x), denotes the subspace of 2x consisting of nonempty closed connected subsets of x. we shall write c(x) to mean (c(x),τ). the upper and lower vietoris topologies on c(x) are specified by writing (c(x),τu ) and (c(x),τl), respectively. for nonempty open sets u1, . . . ,un of x and u = ui, for some i ≤ n, when m ≥ 1, we may write, as in [1], < u1, . . . ,un > alternatively in the form < u1, . . . ,un,u, . . . ,u ︸︷︷︸ m >. for a nonempty basic open set u = < u1, . . . ,um > of c(x), by ⋃ u we mean ⋃m i=1 ui. observe that x is embedded into c(x) by the embedding x 7→ {x}. also, every continuous selfmap f on x, induces in a natural way, a selfmap f′ on 2x [2]. the hyperspace c(x ) is invariant with respect to f′, and hence we obtain a dynamical system (c(x), f̄), where f̄ = f′|c(x). for k ∈ n, the space fk(x), denotes the collection of those nonempty closed subsets of x that consist of atmost k elements. the set f1(x) consisting of all singletons in x, is a closed subspace of c(x), and the quotient space c(x)/f1(x), denoted by hs(x ), is known as the hyperspace suspension of x [10]. further, for a dynamical system (x,f), the pair (hs(x ), hs(f )) constitutes a dynamical system with the evolution map hs(f ) sending [a] ∈ hs(x) to (qx ◦ f ◦ q −1 x )([a]), where qx denotes the cannonical projection from c(x ) to hs(x ) [5]. that hs(f ) is continuous follows from [4, 4.3, p. 146]. it is pertinent to mention that hs taking x to hs(x ) and f to hs(f ) describes a covariant functor from the category of topological spaces to itself. a brief exposition of our work is as follows. if (x,f) is transitive, then ( (c(x),τu ), f̄ ) is transitive. however, ( (c(x),τl), f̄ ) is transitive if x is pathconnected and f is weakly mixing. the same is the state of affairs so far as the total transitivity and weakly mixing are concerned. it is known that the transitivity and total transitivity do not necessarily lift to c(x) endowed c© agt, upv, 2014 appl. gen. topol. 15, no. 2 176 lifting dynamical properties to hyperspaces and hyperspace suspension with the vietoris topology. for a dynamical system (x,f) with f mixing, it is obtained that f̄ on (c(x),τl) continues to be mixing provided x is pathconnected. all these constitute the contents of section 3. section 4 is devoted to the passage of these dynamical properties to the dynamical system (hs(x),hs(f)) from the dynamical system (x,f). it has been obtained that all these properties get lifted smoothly. finally, it is obtained that topological conjugacy is preserved under the hyperspace suspension. we begin with the preliminaries in section 2, wherein, certain results scattered across various references are stated that we subsequently use. 2. preliminaries a topological graph is a connected compact hausdorff space g for which there exists a finite collection of subspaces ii, i = 1,2, . . . ,s such that g = ⋃s i=1 ii, where each ii is homeomorphic to a compact interval, and each intersection ii ∩ij, for i 6= j is finite. equivalently, a graph is a one-dimensional connected, compact polyhedron. below we state a result related to the lift of transitivity to the hyperspace c(g) of a dynamical system (g,f), where g is a graph and f is transitive. proposition 2.1 ([9, cor. 29]). for a dynamical system (g,f), the map f̄ is never transitive. remark 2.2. note that the map f̄ has been denoted by f̃ in [9]. denoting by f̂ the map induced on k(x) consisting of all nonempty compact subsets of x, we have the following: proposition 2.3 ([7, lemma 5]). for a dynamical system (x,f), the following are equivalent: (1) f is topologically exact. (2) f̂ is topologically exact. remark 2.4. in [7], the map f̂ has been denoted by f̄, which we have used for the map induced on the hyperspace c(x ). proposition 2.5 ([9, lemma 4]). if dynamical systems (x,f) and (y,g) are topologically conjugate, then the same holds for the induced dynamical systems (2x,f′) and (2y ,g′), and also for (c(x), f̄) and (c(y), ḡ). remark 2.6. in [9], the maps f′,g′, f̄ and ḡ have been denoted by f̄, ḡ, f̃ and g̃, respectively. proposition 2.7 ([8]). topological transitivity is preserved under topological conjugacy. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 177 d. masood and p. singh 3. dynamical properties and c(x) theorem 3.1. let (x,f) be a dynamical system with f transitive. then the induced map f̄ on (c(x),τu ) is transitive. proof. consider a pair of nonempty basic open sets u and v of c(x). then ⋃ u and ⋃ v constitute a pair of nonempty open sets in x. since f is transitive, there is a natural number l such that for some y ∈ ⋃ u, fl(y) ∈ ⋃ v. that f̄l(u) intersects v follows by noting that {y} ∈ u and f̄l({y}) ∈ v. � proposition 2.1 shows that there is no dearth of dynamical systems whose evolution maps are transitive possessing induced evolution maps which fail to be transitive. below we provide an example for the sake of illustration. example 3.2. the irrational rotation rα : s 1 → s1 defined by rα(θ) = θ + α, θ ∈ [0,2π), where α is an irrational number, and the multiplicative group s1 is identified with the additive group [0,2π) mod 2π, is a transitive map [3, 11]. let rα be the map induced on c(s1), which is homeomorphic to d2 ≡ {(r,θ) | r ∈ [0,1] and θ ∈ [0,2π)} [2, 6], via the homeomorphism ψ, sending a ∈ c(s1) to (1 − la/2π,mid a) ∈ d 2, where la and mid a, denote the arclength and the middle point of a, respectively. the map g : d2 → d2 defined by g(r,θ) = (r,θ + α), r ∈ [0,1], and θ ∈ [0,2π), is such that g ◦ ψ = ψ ◦ rα, making the systems (c(s 1),rα) and (d 2,g) topologically conjugate. since the orbit of no element of d2 under g is dense in it, the map g is not transitive [8], and hence, by proposition 2.7, rα is also not transitive. theorem 3.3. let (x,f) be a dynamical system with f totally transitive. then the induced map f̄ on (c(x),τu ) is totally transitive. proof. since for a natural number n, fn = f̄n, the result follows from theorem 3.1. � theorem 3.4. let (x,f) be a dynamical system with f weakly mixing. then the induced map f̄ on (c(x),τu ) is weakly mixing. proof. consider pairs of nonempty basic open sets u1,v1 and u2,v2 of c(x). then ⋃ u1, ⋃ v1 and ⋃ u2, ⋃ v2 are pairs of nonempty open sets of x. since f is weakly mixing, there are n ∈ n, y ∈ ⋃ u1 and y ′ ∈ ⋃ u2 such that f n(y) ∈ ⋃ v1, and f n(y′) ∈ ⋃ v2. that f̄ n(u1) ∩ v1 6= φ and f̄ n(u2) ∩ v2 6= φ, follows by observing that {y} ∈ u1, f̄ n({y}) ∈ v1, {y ′} ∈ u2 and f̄ n({y′}) ∈ v2. � theorem 3.5. let (x,f) be a dynamical system with f weakly mixing. if x is pathconnected, then the induced map f̄ on (c(x),τl) is weakly mixing. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 178 lifting dynamical properties to hyperspaces and hyperspace suspension proof. consider pairs < u1, . . . ,um >,< v1, . . . ,vm >, and ,< v ′1, . . . ,v ′ n > of nonempty basic open sets of c(x). then the pairs (ui,vi), i = 1, . . . ,m, and (u′j,v ′ j ), j = 1, . . . ,n, consist of nonempty open sets of x. since f is weakly mixing, there is a k ∈ n, such that fk(ui) ∩ vi 6= φ, i = 1, . . . ,m, and fk(u′j) ∩ v ′ j 6= φ, j = 1, . . . ,n. thus there are elements xi ∈ ui, x ′ j ∈ u ′ j such that fk(xi) ∈ vi, and f k(x′j) ∈ v ′ j , i = 1, . . . ,m, j = 1, . . . ,n. let a and b be paths containing all xi’s, and all x ′ j’s, respectively. then a ∈ < u1, . . . ,um >, b ∈ , and f̄ k(a) ∈ < v1, . . . ,vm >, f̄ k(b) ∈ < v ′1, . . . ,v ′ n >. hence, the result. � because, for a given dynamical system (x,f), the weakly mixing of f implies that f is transitive and also totally transitive, we have the following: corollary 3.6. let (x,f) be a dynamical system with f weakly mixing. if x is pathconnected, then the induced map f̄ on (c(x),τl) is transitive and also totally transitive. theorem 3.7. let (x,f) be a dynamical system with f mixing. then the induced map f̄ on (c(x),τu ) is mixing. proof. for a pair u,v of nonempty basic open sets of c(x), ⋃ u, ⋃ v constitute a pair of nonempty open sets of x. since f is mixing, there is an n ∈ n, such that for n ≥ n, fn( ⋃ u) intersects ⋃ v, which implies that f̄n(u) intersects v, for n ≥ n. hence, the result. � theorem 3.8. let (x,f) be a dynamical system with f mixing. if x is pathconnected, then the induced map f̄ on (c(x),τl) is mixing. proof. consider a pair < u1, . . . ,um >,< v1, . . . ,vm > of nonempty basic open sets of c(x). since f is mixing, for each pair (ui,vi), i = 1, . . . ,m, of open sets of x, there exist ni ∈ n such that f k(ui)∩vi 6= φ, whenever k ≥ ni. set n = max {ni : i = 1, . . . ,m}. then f k(ui) ∩ vi 6= φ, for i = 1, . . . ,m, and k ≥ n. thus for i = 1, . . . ,m, k ≥ n, there are xik ∈ ui such that fk(xik ) ∈ vi. let ak be a path containing all xik ’s. since f̄ k(ak) lies in f̄k(< u1, . . . ,um >) as well as in < v1, . . . ,vm >, the result follows. � 4. dynamical properties and hs(x) we begin with the following: lemma 4.1. for n ∈ n, (hs(f))n ≡ (qx ◦ f n ◦ q−1 x ). proof. the result follows by induction on n. � theorem 4.2. let (x,f) be a dynamical system such that (c(x), f̄) is transitive. then the dynamical system (hs(x),hs(f)) is also transitive. proof. for a pair u,v of nonempty open sets of hs(x), q−1 x (u) and q−1 x (v ) constitute a pair of nonempty open sets of c(x). since f̄ is transitive, there is an n ∈ n such that fn(q−1 x (u)) ∩ q−1 x (v ) 6= φ. applying qx and using lemma 4.1, the result follows. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 179 d. masood and p. singh theorem 4.3. let (x,f) be a dynamical system such that (c(x), f̄) is totally transitive. then the dynamical system (hs(x),hs(f)) is also totally transitive. proof. for n ∈ n, theorem 4.2 and lemma 4.1 provide the transitivity of (hs(f))n. thus, hs(f) is totally transitive. � theorem 4.4. let (x,f) be a dynamical system such that (c(x), f̄) is weakly mixing. then the dynamical system (hs(x),hs(f)) is also weakly mixing. proof. consider pairs u1,u2 and v1,v2 of nonempty open sets of hs(x). then q−1 x (u1),q −1 x (u2) and q −1 x (v1),q −1 x (v2) are pairs of nonempty open sets of c(x). the weakly mixing property of the map f̄ implies the existence of an n ∈ n, such that fn(q−1 x (ui)) ∩q −1 x (vi) 6= φ, for i = 1,2. now arguing as in theorem 4.2, we have the result. � theorem 4.5. let (x,f) be a dynamical system such that (c(x), f̄) is mixing. then the dynamical system (hs(x),hs(f)) is also mixing. proof. for a pair u,v of nonempty open sets of hs(x), q−1 x (u) and q−1 x (v ) constitute a pair of nonempty open sets of c(x). the mixing property of f̄ on c(x) provides an n ∈ n, such that fn(q−1 x (u)) ∩ q−1 x (v ) 6= φ, for n ≥ n. now proceeding as in the proof of theorem 4.2, we have the result. � theorem 4.6. let (x,f) be a dynamical system such that (c(x), f̄) is topologically exact. then the dynamical system (hs(x),hs(f)) is also topologically exact. proof. let u be a nonempty open set of hs(x). then q−1 x (u) is a nonempty open set of c(x). the topological exactness of f̄ on c(x) yields an n ∈ n, such that fn(q−1 x (u)) = c(x). applying qx and using lemma 4.1, the result follows. � theorem 4.7. if the dynamical systems (x,f) and (y,g) are topologically conjugate, then the dynamical systems (hs(x),hs(f)) and (hs(y ),hs(g)) are also topologically conjugate. proof. let ϕ be a conjugacy between the pairs (x,f) and (y,g). then by proposition 2.5, the dynamical systems (c(x), f̄) and (c(y ), ḡ) are topologically conjugate, the induced map ϕ̄ being the topological conjugacy. since ϕ̄ maps f1(x) homeomorphically onto f1(y ), hs(ϕ̄) : hs(x) → hs(y ) is the required topological conjugacy between the dynamical systems (hs(x),hs(f)) and (hs(y ),hs(g)). � acknowledgements. the authors are grateful to their supervisor professor k. k. azad for his valuable guidance and encouragement. the first author thanks the department of science and technology, new delhi, for providing financial support. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 180 lifting dynamical properties to hyperspaces and hyperspace suspension references [1] j. banks, chaos for induced hyperspace maps, chaos solitons fractals 25 (2005), 681– 685. 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[11] h. roman-flores, a note on transitivity in set-valued discrete systems, chaos solitons fractals 17 (2003), 99–104. [12] s. ruette, chaos for continuous interval maps, université paris-sud, 91405 orsay, cedex france (2003). [13] p. sharma and a. nagar, topological dynamics on hyperspaces, appl. gen. topol. 11 (2010), 1–19. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 181 uspagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 197-204 unitary representability of free abelian topological groups vladimir v. uspenskij abstract. for every tikhonov space x the free abelian topological group a(x) and the free locally convex vector space l(x) admit a topologically faithful unitary representation. for compact spaces x this is due to jorge galindo. 2000 ams classification: primary: 22a25. secondary 43a35, 43a65, 46b99, 54c65, 54e35, 54h11 keywords: unitary representation, free topological group, positive-definite function, michael selection theorem 1. introduction with every tikhonov space x one associates the free topological group f (x), the free abelian topological group a(x), and the free locally convex vector space l(x). they are characterized by respective universal properties. for example, l(x) is defined by the following: x is an (algebraic) basis of l(x), and for every continuous mapping f : x → e, where e is a hausdorff locally convex space, the linear extension f̄ : l(x) → e of f is continuous. there are two versions of l(x), real and complex. there also are versions of all these free objects for spaces with a distinguished point. we consider non-pointed spaces. the unitary group u (h) of a hilbert space h will be equipped with the strong operator topology, which is the topology inherited from the tikhonov product hh , or, equivalently, the topology of pointwise convergence. we use the notation us(h) to indicate this topology. a unitary representation of a topological group g is a continuous homomorphism f : g → us(h). such a representation is faithful if f is injective, and topologically faithful if f is a homeomorphic embedding. a topological group is unitarily representable if it is isomorphic to a topological subgroup of us(h) for a hilbert space h (which 198 v. v. uspenskij may be non-separable), or, equivalently, if it admits a topologically faithful unitary representation. all locally compact groups are unitarily representable. for groups beyond the class of locally compact groups this is no longer true: there exist abelian topological groups (even monothetic groups, that is, topologically generated by one element) for which every unitary representation is trivial (sends the whole group to the identity), see [1, theorem 5.1 and remark 5.2]. thus one may wonder what happens in the case of free topological groups: are they unitarily representable? in the non-abelian case, this question is open even if x is compact metric, see [9, questions 35, 36]. the aim of the present note is to answer the question in the positive for l(x) and a(x). theorem 1.1. for every tikhonov space x the free locally convex space l(x) and the free abelian topological group a(x) admit a topologically faithful unitary representation. it suffices to consider the case of l(x), since a(x) is isomorphic to the subgroup of l(x) generated by x [12], [15], see also [5]1. for compact x (or, more generally, for kω-spaces x) theorem 1.1 is due to jorge galindo [4]. however, it was claimed in an early version of [4] that there exists a metrizable space x for which the group a(x) is not isomorphic to a subgroup of a unitary group. this claim was wrong, as theorem 1.1 shows. it is known that (the additive group of) the space l1(µ) is unitarily representable for every measure space (ω, µ). (for the reader’s convenience, we remind the proof in section 3, see fact 3.5.) in particular, if µ is the counting measure on a set a, we see that the banach space l1(a) of summable sequences is unitarily representable. since the product of any family of unitarily representable groups is unitarily representable (consider the hilbert sum of the spaces of corresponding representations), we see that theorem 1.1 is a consequence of the following: theorem 1.2. for every tikhonov space x the free locally convex space l(x) is isomorphic to a subspace of a power of the banach space l1(a) for some a. for a we can take any infinite set such that the cardinality of every discrete family of non-empty open sets in x does not exceed card (a). theorem 1.1 implies the following result from [5]: every polish abelian group is the quotient of a closed abelian subgroup of the unitary group of a separable hilbert space (the non-abelian version of the reduction is explained in section 4, the argument for the abelian case is the same). it is an open question (a. kechris) whether a similar assertion holds for non-abelian polish groups, see [8, section 5.2, question 16], [9, question 34]. 1the result was stated in [12], but the proof was incomplete. i gave a proof in [15] — not knowing that i was rediscovering the wasserstein metric (which is also known under many other names: monge – kantorovich, kantorovich – rubinstein, transportation, earth mover’s) and its basic properties, such as the integer value property. the proof is reproduced in [5], where a historic account is given. unitary representability of free abelian topological groups 199 the fine uniformity µx on a tikhonov space x is the finest uniformity compatible with the topology. it is generated by the family of all continuous pseudometrics on x. it is also the uniformity induced on x by the group uniformity of a(x) or l(x). a fine uniform space is a space of the form (x, µx ). we note the following corollary of theorem 1.2 which may be of some independent interest. corollary 1.3. for every tikhonov space x the fine uniform space (x, µx ) is isomorphic to a uniform subspace of a power of a hilbert space. this need not be true for uniform spaces which are not fine: many separable banach spaces (for example, c0 or lp for p > 2) do not admit a uniform embedding in a hilbert space [3, chapter 8]. since the countable power of an infinite-dimensional hilbert space h uniformly embeds in h (use the fact that h uniformly embeds in the unit sphere of itself [3, corollary 8.11], and uniformly embed the countable power of the unit sphere into the hilbert sum of countably many copies of h), it easily follows that c0 or lp for p > 2 are not uniformly isomorphic to a subspace of a power of a hilbert space. to deduce corollary 1.3 from theorem 1.1, note that the left uniformity on the unitary group us(h) is induced by the product uniformity of h h , hence the same is true for the left uniformity on every unitarily representable group. for any uniform space x one defines the free abelian group a(x) and free locally convex space l(x) in an obvious way. the objects a(x) and l(x) for tikhonov spaces x considered in this paper are special cases of the same objects for uniform spaces, corresponding to fine uniform spaces. question 1.4 (megrelishvili). for what uniform spaces x are the groups a(x) and l(x) unitarily representable? is it sufficient that x be a uniform subspace of a product of hilbert spaces? we prove theorem 1.2 in section 2. the proof depends on two facts: (1) every banach space is a quotient of a banach space of the form l1(a); (2) every onto continuous linear map between banach spaces admits a (possibly non-linear) continuous right inverse. we remind the proof of these facts in section 3. 2. proof of theorem 1.2 let x be a tikhonov space. let t0 be the topology of the free locally convex space l(x). let t1 be the topology on l(x) generated by the linear extensions of all possible continuous maps of x to spaces of the form l1(a). theorem 1.2 means that t1 = t0. in order to prove this, it suffices to verify that (l(x), t1) has the following universal property: for every continuous map f : x → f , where f is a hausdorff lcs, the linear extension of f , say f̄ : l(x) → f , is t1-continuous. since every hausdorff lcs embeds in a product of banach spaces, we may assume that f is a banach space. represent f as a quotient of l1(a) (fact 3.1), let p : l1(a) → f be linear and onto. 200 v. v. uspenskij l1(a) p �� x f // g < 0 such that p(e) = 1 and |1 − p(g)| > a for every g ∈ g \ u . for a measure space (ω, µ) we denote by l1(µ) the complex banach space of (equivalence classes of) complex integrable functions, and by l1 r (µ) the real banach space of (equivalence classes of) real integrable functions. fact 3.4 (schoenberg [10, 11]). if (ω, µ) is a measure space and x = l1 r (µ), the function x 7→ exp(−‖x‖) on x is positive-definite. in other words, for any f1, . . . , fn ∈ l 1 r (µ) the symmetric real matrix (exp(−‖fi − fj‖)) is positive. proof. we invoke bochner’s theorem: positive-definite continuous functions on rn (or any locally compact abelian group) are exactly the fourier transforms of positive measures. for f ∈ l1(rn) we define the fourier transform f̂ by f̂ (y) = ∫ rn f (x) exp (−2πi(x, y)) dx. here (x, y) = ∑n k=1 xkyk for x = (x1, . . . , xn) and y = (y1, . . . , yn). the positive functions p and q on r defined by p(x) = exp(−|x|) and q(y) = 2/(1 + 4π2y2) are the fourier transforms of each other. hence each of them is positive-definite. similarly, the positive functions pn and qn on r n defined by pn(x1, . . . , xn) = exp(− ∑n k=1 |xk|) = ∏n k=1 p(xk) and qn(y1, . . . , yn) =∏n k=1 q(yk) are positive-definite, being the fourier transforms of each other. if m1, . . . , mn are strictly positive masses, the function x 7→ exp(− ∑n k=1 mk|xk|) on rn is positive-definite, since it is the composition of pn and a linear automorphism of rn. this is exactly fact 3.4 for finite measure spaces. the general case easily follows: given finitely many functions f1, . . . , fn ∈ l1 r (µ), we can approximate them by finite-valued functions. in this way we see that the symmetric matrix a = (exp(−‖fi − fj‖)) is in the closure of the set of matrices a′ of the same form arising from finite measure spaces. the result of the preceding paragraph means that each a′ is positive. hence a is positive. � 202 v. v. uspenskij as a topological group, l1(µ) is isomorphic to the square of l1 r (µ). combining facts 3.3 and 3.4, we obtain: fact 3.5. the additive group of the space l1(µ) is unitarily representable for every measure space (ω, µ). see [3, 6] for more on unitarily and reflexively representable banach spaces. 4. open questions let us say that a metric space m is of l1-type if it is isometric to a subspace of the banach space l1 r (µ) for some measure space (ω, µ). a non-abelian version of theorems 1.1 and 1.2 might be the following: conjecture 4.1. for any tikhonov space x the free topological group f (x) is isomorphic to a subgroup of the group of isometries iso (m ) for some metric space m of l1-type. it follows from [16, theorem 3.1] (apply it to the positive-definite function p on r used in the proof of fact 3.4 and defined by p(x) = exp(−|x|)) and fact 3.4 that for every m ⊂ l1 r (µ) the group iso (m ) is unitarily representable. thus, if conjecture 4.1 is true, every f (x) is unitarily representable. this would imply a positive answer to the question of kechris mentioned in section 1: is every polish group a quotient of a closed subgroup of the unitary group of a separable hilbert space? indeed: proposition 4.2. let p be the space of irrationals. if the group f (p ) is unitarily representable, then every polish group is a quotient of a closed subgroup of the unitary group of a separable hilbert space. proof. a topological group is uniformly lindelöf (or, in another terminology, ω-bounded) if for every neighbourhood u of the unity the group can be covered by countably many left (equivalently, right) translates of u . if g is a uniformly lindelöf group of isometries of a metric space m , then for every x ∈ m the orbit gx is separable (see e.g. the section “guran’s theorems” in [14]). if g is a uniformly lindelöf subgroup of the unitary group us(h), where h is a (non-separable) hilbert space, it easily follows that h is covered by separable closed g-invariant linear subspaces and therefore g embeds in a product of unitary groups of separable hilbert spaces. if g is a polish group, there exists a quotient onto map f (p ) → g (because there exists a continuous open onto map p → g, see lemma 4.3 below). the group f (p ), like any separable topological group, is uniformly lindelöf. assume that f (p ) is unitarily representable. then, as we saw in the first paragraph of the proof, f (p ) is isomorphic to a topological subgroup of a power of us(h), where h is a separable hilbert space. an easy factorization argument (see lemma 4.4) shows that there is a group n lying in a countable power of us(h) (and hence isomorphic to a subgroup of us(h)) such that g is a quotient of n . the quotient homomorphism n → g can be extended over the closure of n , so we may assume that n is closed in us(h). � unitary representability of free abelian topological groups 203 the following lemmas were used in the proof above: lemma 4.3. for every non-empty polish space x there exists an open onto map p → x, where p , as above, is the space of irrationals. proof. consider open covers {un} of x such that: • diam u < 2−n for every u ∈ un; • each un is indexed by an = n n; • if t ∈ an, then ut = ⋃ {us : s ∈ an+1 and t = s|n} • if s ∈ an+1 and t = s|n, then us ⊂ ut. for every infinite sequence s ∈ nn let xs be the only point in the intersection⋂ us|n = ⋂ us|n. then the map s 7→ xs from n n (which is homeomorphic to p ) to x is open and onto. � lemma 4.4. let {gα : α ∈ a} be a family of topological groups, k a subgroup of ∏ gα, h a metrizable topological group, f : k → h a continuous homomorphism. then there exists a countable subset b ⊂ a such that f = g◦pb for some continuous homomorphism g : kb → h, where pb : k → ∏ {gα : α ∈ b} is the projection and kb = pb(k). if f is open and onto, then so is g. proof. let b be a countable base at unity of h. for each u ∈ b pick a finite set f = fu ⊂ a such that p −1 f (v ) ⊂ f −1(u ) for some neighbourhood v of unity of ∏ {gα : α ∈ f }. put b = ⋃ {fu : u ∈ b}. � 5. acknowledgement i thank my friends m. megrelishvili and v. pestov for their generous help, which included inspiring discussions and many useful remarks and suggestions on early versions of this paper. references [1] w. banaszczyk, on the existence of exotic banach – lie groups, math. ann. 264 (1983), 485–493. [2] b. bekka, p. de la harpe, and a. valette, kazhdan property (t), available at http://poncelet.sciences.univ-metz.fr/˜bekka/. [3] y. benyamini and j. lindenstrauss, geometric nonlinear functional analysis, vol. 1, ams colloquium publications, vol. 48 (ams, providence, ri, 2000). [4] j. galindo, on unitary representability of topological groups, arxiv:math.gn/0607193. [5] s. gao and v. pestov, on a universality property of some abelian polish groups, fund. math. 179 (2003), 1–15; arxiv:math.gn/0205291. [6] m. megrelishvili, reflexively but not unitarily representable topological groups, topology proceedings 25 (2000), 615–625. [7] e. michael, continuous selections i, ann. math. 63 (1956), 361–382. [8] v. pestov, dynamics of infinite-dimensional groups and ramsey-type phenomena, (publicações dos colóquios de matemática, impa, rio de janeiro, 2005). [9] v. pestov, forty-plus annotated questions about large topological groups, arxiv:math.gn/0512564. [10] i. j. shoenberg, on certain metric spaces arising from euclidean spaces by a change of metric and their imbedding in hilbert space, ann. math. 38 (1937), 787–793. [11] i. j. schoenberg, metric spaces and positive definite functions, trans. amer. math. soc. 44 (1938), 522–536. 204 v. v. uspenskij [12] m. g. tkachenko, on completeness of free abelian topological groups, soviet math. dokl. 27 (1983), 341–345. [13] d. repovš and p. v. semenov, continuous selections of multivalued mappings, kluwer academic publishers, dordrecht–boston–london, 1998. [14] v. v. uspenskii, why compact groups are dyadic, in book: general topology and its relations to modern analysis and algebra vi: proc. of the 6th prague topological symposium 1986, frolik z. (ed.) (berlin: heldermann verlag, 1988), 601–610. [15] v. v. uspenskĭı, free topological groups of metrizable spaces, izvestiya akad. nauk sssr, ser. matem. 54 (1990), 1295–1319; english transl.: math. ussr-izvestiya 37 (1991), 657–680. [16] v. v. uspenskij, on unitary representations of groups of isometries, in book: contribuciones matemáticas. homenaje al profesor enrique outerelo domı́nguez, e. martin-peinador (ed.), universidad complutense de madrid, 2004, 385–389; arxiv:math.rt/0406253. received april 2007 accepted june 2007 vladimir uspenskij (uspensk@math.ohiou.edu) department of mathematics, 321 morton hall, ohio university, athens, ohio 45701, usa lazaarechiagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 227-237 quasihomeomorphisms and lattice equivalent topological spaces othman echi and sami lazaar abstract. this paper deals with lattice-equivalence of topological spaces. we are concerned with two questions: the first one is when two topological spaces are lattice equivalent; the second one is what additional conditions have to be imposed on lattice equivalent spaces in order that they be homeomorphic. we give a contribution to the study of these questions. many results of thron [lattice-equivalence of topological spaces, duke math. j. 29 (1962), 671-679] are recovered, clarified and commented. 2000 ams classification: 54b30, 54d10, 54f65. keywords: quasihomeomorphism; lattice equivalence; sober space; tdspace. 1. introduction let x be a topological space; we denote by γ(x) the lattice of closed sets of x. two topological spaces x and y are said to be lattice equivalent if there is a bijective map from γ(x) to γ(y ) which together with its inverse is order-preserving [8]. the question of characterizing when two topological spaces are lattice equivalent is still open. a lattice equivalence ϕ : γ(x) −→ γ(y ) is said to be induced by a homeomorphism if there is a homeomorphism f : x −→ y such that ϕ(c) = f (c), for each c ∈ γ(x). in [8], thron was concerned in the problem of determining what additional conditions have to be imposed on lattice equivalent spaces in order that they be homeomorphic. a complete answer to this problem is still very far off. 228 o. echi and s. lazaar it is worth noting that, over the years, several researchers dealt with the concept of lattice equivalent topological spaces and representations of an abstract lattice as the family of closed sets on a topological space (see for instance [4], [8], [9]). in 1966, finch [4] proved that a lattice equivalence ϕ : γ(x) −→ γ(y ) is induced by a homeomorphism if and only if the following conditions are satisfied: (i) for each x ∈ x, there exists yx ∈ y such that ϕ({x}) = {yx}. (ii) for each y ∈ y , there exists xy ∈ x such that ϕ −1({y}) = {xy}. (iii) let x ∈ x and y ∈ y . set xx = {t ∈ x : {x} = {t}} and yy = {t ∈ y : {y} = {t}}. then ϕ−1({y}) = {x} implies that |xx| = |yy| (where |xx| denotes the cardinality number of the set xx). in 1972, yip kai wing [9] was interested in quasi-homeomorphisms and lattice-equivalences of topological spaces. this work seems to be very close to our paper; but it is important to announce that none of our results is in yip’s paper [9]. in this connection, the first section will be entirely devoted to quasi-homeomorphisms and comments on yip’s results in his note [9]. the present paper is devoted to shed some light on lattice equivalent topological spaces. let us first recall some notions which were introduced by the grothendieck school (see for example [5] and [6]), such as quasihomeomorphisms and strongly dense subsets. if x is a topological space, we denote by o(x) the set of all open subsets of x. recall that a continuous map q : x −→ y is said to be a quasihomeomorphism if u 7−→ q−1(u ) defines a bijection o(y ) −→ o(x). we say that a subset s of a topological space x is locally closed if it is an intersection of an open set and a closed set of x. a subset s of a topological space x is said to be strongly dense in x, if s meets every nonempty locally closed subset of x. thus a subset s of x is strongly dense if and only if the canonical injection s →֒ x is a quasihomeomorphism. it is well known that a continuous map q : x −→ y is a quasihomeomorphism if and only if the topology on x is the inverse image by q of that on y and the subset q(x) is strongly dense in y [5]. let x be a topological space. if f : x −→ y is continuous, then we define γ(f ) : γ(y ) −→ γ(x), by γ(f )(c) = f −1(c). in particular, if q : x −→ y is a quasihomeomorphism. then the map γ(q) : γ(y ) −→ γ(x) is a lattice equivalence (see for example [1, proposition 1.9]). the following definition is natural. definition 1.1. a lattice equivalence ϕ : γ(x) −→ γ(y ) between two topological spaces is said to be induced by a quasihomeomorphism if there is either a quasihomeomorphism q : y −→ x such that ϕ = γ(q) or a quasihomeomorphism p : x −→ y such that ϕ−1 = γ(p). our main result is a characterization of lattice equivalences induced by a quasihomeomorphism (see theorem 3.4). this result is very close to the one lattice equivalent topological spaces 229 done by finch [4] characterizing lattice equivalences induced by a homeomorphism. as a consequence, many results of thron are recovered, clarified and commented. 2. grothendieck’s quasihomeomorphisms and yip’s quasihomeomorphisms as we have said in the introduction the concept of quasihomeomorphisms was introduced in algebraic geometry by grothendieck ([6], [5]). also, it was shown that this concept arises naturally in the theory of some foliations associated to closed connected manifolds( see the papers [2] and [3]). the following definition is given in [9]. definition 2.1. a continuous map q : x −→ y between topological spaces is said to be a quasihomeomorphism if the following conditions are satisfied: (i) for any closed set c in x, q−1[q(c)] = c. (ii) for any closed set f in y , q[q−1(f )] = f . fortunately, the two notions “grothendieck’s quasihomeomorphism” and “yip’s quasihomeomorphism” coincides, as it is shown in the following. proposition 2.2. let q : x −→ y be a continuous map between topological spaces. then q is a yip’s quasihomeomorphism if and only if it is a grothendieck’s quasihomeomorphism. proof. suppose that q is a grothendieck’s quasihomeomorphism. then γ(q) : γ(y ) −→ γ(x) is a lattice isomorphism, by [1, proposition 1.9]. hence q is a quasihomeomorphism in the sense of yip, by [9, theorem 1]. conversely, if q is a yip’s quasihomeomorphism, then γ(q) : γ(y ) −→ γ(x) is a lattice isomorphism, by [9, theorem 1]; so that γ(q) is bijective; proving that q is a grothendieck’s quasihomeomorphism. � remark 2.3. let q : x −→ y be a quasihomeomorphism between topological spaces. in [9, theorem 2], yip proved that q(x) is dense in y . in fact, it follows from [5, chapter 0, proposition 2.7.1] that q(x) is even strongly dense in y . in [9, theorem 3], the author investigated closed quasihomeomorphisms between t0-spaces. the following result is more precise. proposition 2.4. let q : x −→ y be a quasihomeomorphism. then the following statements are equivalent: (i) q is a surjective mapping; (ii) q is a closed mapping; (iii) q is an open mapping. proof. (ii) =⇒ (i) and (iii) =⇒ (i). let q be closed (resp.open). then, q(x) is a closed (resp.open) subset of y . now, since q−1(q(x)) = q−1(y ), we get q(x) = y , by the definition of a quasihomeomorphism. thus q is onto. 230 o. echi and s. lazaar (i) =⇒ (ii) and (iii). let c be a closed (resp. an open) subset of x. by definition, there is a closed (resp. an open) subset d of y such that c = q−1(d). but, since q is onto, we get q(c) = d, proving that q(c) is closed (resp. open). � example 2.5. let us construct a quasihomeomorphism that is not a surjection. consider an infinite set x and a point α not belonging to x. set y := x ∪ {α}. equip y with the topology whose closed sets are y and the finite subsets of x. hence x is a strongly dense subspace of y ; so that the canonical embedding i : x →֒ y is a quasihomeomorphism. since i is not onto, it is not closed and not open. before stating the next result, we need to recall the notion of sober space (also introduced by the school of grothendieck [5] or [6]). a subspace y of x is called irreducible, if each nonempty open subset of y is dense in y (equivalently, if c1 and c2 are two closed subsets of x such that y ⊆ c1 ∪ c2, then y ⊆ c1 or y ⊆ c2). let c be a closed subset of a space x; we say that c has a generic point if there exists x ∈ c such that c = {x}. recall that a topological space x is said to be sober if any nonempty irreducible closed subset of x has a unique generic point. let x be a topological space and s(x) the set of all nonempty irreducible closed subsets of x [5]. let u be an open subset of x; set ũ = {c ∈ s(x) | u ∩ c 6= ∅}; then the collection {ũ | u is an open subset of x} provides a topology on s(x) and the following properties hold [5]: (i) the map ηx : x −→ s(x), which takes x to {x}, is a quasihomeomorphism. (ii) s(x) is a sober space. (iii) let f : x −→ y be a continuous map. let s(f ) : s(x) −→ s(y ) be the map defined by s(f )(c) = f (c), for each irreducible closed subset c of x. then s(f ) is continuous. (iv) the topological space s(x) is called the soberification of x, and the assignment s, defines a functor from the category top of topological spaces to itself. proposition 2.6. let q : x −→ y be a quasihomeomorphism. then the following properties hold. (1) if x is a t0-space, then q is injective. (2) if x is sober and y is a t0-space, then q is a homeomorphism. proof. (1) let x1, x2 be two distinct points of x with q(x1) = q(x2). then there exists an open subset u of x such that, for example, x1 ∈ u and x2 /∈ u . since there exists an open subset v of y satisfying q−1(v ) = u , we get q(x1) ∈ v and q(x2) /∈ v , which is impossible. it follows that q is injective. (2) firstly, it is obviously seen that if s is a closed subset of y , then s is irreducible if and only if so is q−1(s). lattice equivalent topological spaces 231 now, let us prove that q is surjective. for this end, let y ∈ y . according to the above observation, q−1({y}) is a nonempty irreducible closed subset of x. hence q−1({y}) has a generic point x. thus we have the containments {x} ⊆ q−1({q(x)}) ⊆ q−1({y}) = {x}. so that q−1({q(x)}) = q−1({y}). it follows, from the fact that q is a quasihomeomorphism, that {q(x)} = {y}. since y is a t0-space, we get q(x) = y. this proves that q is a surjective map, and thus q is bijective. but it is easily seen that bijective quasihomeomorphisms are homeomorphisms. � remark 2.7. let q : x −→ y be a quasihomeomorphism. if y is sober and x is a t0-space, then q need not be a homeomorphism. to see this, it suffices to consider a t0-space x which is not sober. then the canonical embedding ηx : x −→ s(x) is a quasihomeomorphism which is not a homeomorphism. 3. lattice equivalence a brouwerian lattice is a complete lattice l for which x ∨ ( ∧ c) = ∧ {x ∨ y | y ∈ c} for all x ∈ l and all c ⊆ l. a morphism of brouwerian lattices is a mapping f : l −→ m that preserves all infima and all finite suprema. let cbl denotes the category of brouwerian lattices and brouwerian lattice maps. then γ : top −→ cbl is a contravariant functor. remark 3.1. let f : x −→ y be a continuous map. then f is rendered invertible by the functor γ (i.e., γ(f ) is a lattice equivalence) if and only if f is a quasihomeomorphism. the following example shows that a lattice equivalence that is induced by a quasihomeomorphism is not necessarily induced by a homeomorphism. example 3.2. let x be a topological space which is not sober. then the canonical quasihomeomorphism ηx : x −→ s(x) induces a lattice equivalence between s(x) and x which is not induced by a homeomorphism. in order to give a complete characterization of lattice equivalence of topological spaces induced by a quasihomeomorphism, we give the following definition. definition 3.3. let x, y be two topological spaces and ϕ : γ(x) −→ γ(y ) a lattice equivalence. we say that ϕ is a point-closure lattice equivalence if one of the following properties is satisfied: (i) for each x ∈ x, there exists yx ∈ y such that ϕ({x}) = {yx}. (ii) for each y ∈ y , there exists xy ∈ x such that ϕ −1({y}) = {xy}. our main result is the following. 232 o. echi and s. lazaar theorem 3.4. let ϕ : γ(x) −→ γ(y ) be a lattice equivalence of topological spaces. then the following statements are equivalent: (i) ϕ is induced by a quasihomeomorphism; (ii) ϕ is a point-closure lattice equivalence. proof. (i) =⇒ (ii) suppose that there is a quasihomeomorphism q : y −→ x such that ϕ = γ(q). then for each d ∈ γ(y ), we have ϕ−1(d) = c with c ∈ γ(x) and q−1(c) = d. hence d ⊆ q−1(q(d)) ⊆ q−1(c) = d. thus ϕ−1(d) = q(d). therefore, for each y ∈ y , we have ϕ−1({y}) = q({y}). but q({y}) = {q(y)}, since q is continuous; so that ϕ−1({y}) = {q(y)}. now, if we suppose that there is a quasihomeomorphism p : x −→ y such that ϕ−1 = γ(p), then we get ϕ({x}) = {p(x)}, for each x ∈ x. it follows that ϕ is a point-closure lattice equivalence. (ii) =⇒ (i) suppose, for instance, that for each x ∈ x, there exists yx ∈ y such that ϕ({x}) = {yx}. for each x ∈ x choose q(x) ∈ y such that ϕ({x}) = {q(x)}. this allows us to define a mapping q : x −→ y . we are aiming to prove that q is a quasihomeomorphism and ϕ−1 = γ(q). it is enough to prove that ϕ−1(g) = q−1(g), for each g ∈ γ(y ). let x ∈ ϕ−1(g). then {x} ⊆ ϕ−1(g). thus {q(x)} ⊆ g; in particular q(x) ∈ g; therefore x ∈ q−1(g). conversely, let x ∈ q−1(g); then q(x) ∈ g. hence {q(x)} ⊆ g; consequently, {x} ⊆ ϕ−1(g). therefore, x ∈ ϕ−1(g). it follows that ϕ−1 = γ(q). if we suppose that for each y ∈ y , there exists a xy ∈ x such that ϕ−1({y}) = {xy}, then by the above argument, there is a quasihomeomorphism p : y −→ x such that ϕ = γ(p). � corollary 3.5. if x is a topological space and y is a t1-space, then each lattice equivalence between them is induced by a quasihomeomorphism. proof. it is easy to check that condition (ii) of definition 3.3 is fulfilled. then we can apply theorem 3.4. � the following result establishes some links between lattice equivalences induced by a homeomorphism and those induced by a quasihomeomorphism. for the proof of the next theorem, we need a lemma. lemma 3.6. let e, f be two topological spaces such that f is a t0-space. if f, g : e −→ f are two continuous maps such that γ(f ) = γ(g), then f = g. proof. let x ∈ e. then f −1({f (x)}) = g−1({f (x)}) and g−1({g(x)}) = f −1({g(x)}). this yields g(x) ∈ {f (x)} and f (x) ∈ {g(x)}. thus {f (x)} = {g(x)}; so that f (x) = g(x), since y is a t0-space. � lattice equivalent topological spaces 233 theorem 3.7. let x, y be two t0-spaces and ϕ : γ(x) −→ γ(y ) a lattice equivalence of topological spaces. then the following statements are equivalent: (i) ϕ is induced by a homeomorphism; (ii) there are two quasihomeomorphisms q : y −→ x and p : x −→ y such that ϕ = γ(q) and ϕ−1 = γ(p). proof. [(i) =⇒ (ii)]. straightforward. [(ii) =⇒ (i)]. let q : y −→ x and p : x −→ y be two quasihomeomorphisms such that ϕ = γ(q) and ϕ−1 = γ(p). we have ϕϕ−1 = 1γ(y ) = γ(1y ) and ϕ −1ϕ = 1γ(x) = γ(1x ). hence γ(pq) = γ(1y ) and γ(qp) = γ(1x ). thus, according to lemma 3.6, we get qp = 1x and pq = 1y . therefore, ϕ is induced by the homeomorphism p. � the following example shows that the t0 axiom cannot be deleted in theorem 3.7. example 3.8. let x, y be two sets with distinct cardinalities. we equip x and y with the indiscrete topology. the unique lattice equivalence between x and y is ϕ : γ(x) −→ γ(y ), defined by ϕ(∅) = ∅ and ϕ(x) = y . it is easily seen that for any two continuous maps q : y −→ x and p : x −→ y , we have ϕ = γ(q) and ϕ−1 = γ(p). moreover, p, q are quasihomeomorphisms. but since the two sets have distinct cardinalities, they cannot be homeomorphic. recall that a topological space x is said to be a td-space if for each x ∈ x, {x} is locally closed. it is easily seen that td is between t0 and t1. the following result will be used in the next corollary. proposition 3.9. every quasihomeomorphism between two td-spaces is a homeomorphism. proof. let q : x −→ y be a quasihomeomorphism between two td-spaces. hence q is injective, by proposition 2.6. on the other hand, q(x) is strongly dense in y and every point-set is locally closed; so that q(x) = y . thus q is a bijective quasihomeomorphism. therefore, q is a homeomorphism. � for the proof of the next corollary, we need a lemma. lemma 3.10. let x, y be two topological spaces and ϕ : γ(x) −→ γ(y ) a lattice equivalence. let g be a closed subset of y . then the following statements are equivalent: (i) g is irreducible in y ; (ii) ϕ−1(g) is irreducible in x. 234 o. echi and s. lazaar proof. since ϕ−1 is also a lattice equivalence, it is enough to show the implication (i) =⇒ (ii). let f1, f2 be two closed subsets of x such that ϕ −1(g) ⊆ f1 ∪ f2. hence g ⊆ ϕ(f1 ∪ f2). but ϕ(f1 ∪ f2) = ϕ(f1) ∪ ϕ(f2). thus g ⊆ ϕ(f1) ∪ ϕ(f2). now, since g is irreducible in y , g ⊆ ϕ(f1) or g ⊆ ϕ(f2). this yields ϕ−1(g) ⊆ f1 or ϕ −1(g) ⊆ f2, proving that ϕ −1(g) is irreducible in x. � corollary 3.11. let x, y be two t0-spaces and ϕ : γ(x) −→ γ(y ) a lattice equivalence. (1) if x or y is a td-space (resp. sober space), then ϕ is induced by a quasihomeomorphism q : x −→ y or p : y −→ x. (2) if x and y are td-spaces (resp. sober spaces), then the mapping q of (1) is a homeomorphism (thus ϕ is induced by a homeomorphism). proof. (1) suppose for example, that x is a td-space and y is a t0-space. thron showed in [8] that for each x ∈ x, there exists a unique yx ∈ y such that ϕ({x}) = {yx}. then ϕ is a point-closure lattice equivalence and according to the proof of theorem 3.4, ϕ is induced by a quasihomeomorphism q : x −→ y ( q takes x to yx). now, suppose that x is sober and y is t0. let y ∈ y ; then, since {y} is an irreducible closed subset of y , ϕ−1({y}) is an irreducible closed subset of x, by lemma 3.10. hence ϕ−1({y}) has a unique generic point. thus ϕ is a point-closure lattice equivalence; so that there is a quasihomeomorphism p : y −→ x which induces ϕ (p takes y ∈ y to the unique generic point of ϕ−1({y})). (2) every quasihomeomorphism between two td-spaces is a homeomorphism, by proposition 3.9. also, every quasihomeomorphism between two sober spaces is a homeomorphism, by proposition 2.6. � corollary 3.12 ([8, theorem 2.1 ]). every lattice equivalence between two td-spaces is induced by a homeomorphism. corollary 3.13 ([8, corollary 2.1 ]). every lattice equivalence between a t0space and a t2-space is induced by a homeomorphism. proof. let x be a t0-space, y a t2-space (thus a td-space) and ϕ : γ(y ) −→ γ(x) a lattice equivalence. then by corollary 3.11, there is a quasihomeomorphism q : y −→ x such that ϕ−1 = γ(q). now, y is a sober space (since it is t2) and x is t0. this forces q to be a homeomorphism, by proposition 2.6. therefore, ϕ is induced by a homeomorphism. � example 3.14. a lattice equivalence between a t1-space (thus a tdspace) and a sober space which is not induced by a homeomorphism. for, let y be an infinite set equipped with the cofinite topology. let α /∈ y , and x = y ∪ {α}. we equip x with the topology whose closed sets are x and lattice equivalent topological spaces 235 the finite subsets of y . hence y is a strongly dense subspace of x; so that the canonical embedding y →֒ x is a quasihomeomorphism; thus it induces a lattice equivalence ϕ. clearly, ϕ is not induced by a homeomorphism, since y is a t1-space and x is not. note that example 3.18 provides nontrivial examples of lattice equivalences between a t1-space and a sober space which are not induced by a homeomorphism. it is worth noting that the td-axiom is the weakest requirement under which [8, theorem 2.1] is true, as shown by thron in the following theorem 3.15 ([8, theorem 2.2]). if x is not a td-space, then there exists a lattice equivalence between x and some other space y , which is not induced by a homeomorphism. looking carefully at the proof of the above theorem, we remark that the lattice equivalence given by the author is not induced by a homeomorphism in both cases, when x is t0 or not; nevertheless, it is induced by a quasihomeomorphism. this rises the natural question whether a lattice equivalence is always induced by a quasihomeomorphism. unfortunately, the answer is negative, as shown by the following nice example. example 3.16. a lattice equivalence of topological spaces that is not induced by a quasihomeomorphism. let x and y be two disjoint infinite sets equipped with the cofinite topology. let α, β /∈ x ∪ y and α 6= β. set x ′ = x ∪ {α} and y ′ = y ∪ {β}. we equip x ′ (resp. y ′ ) with the topology whose closed sets are x ′ (resp. y ′ ) and the finite subset of x (resp. of y ). recall that the free union e + f of disjoint spaces e and f is the set e ∪ f with the topology: u ⊆ e + f is open if and only if u ∩ e is open in e and u ∩ f is open in f . now, consider λ = x ′ + y and ∆ = y ′ + x. it is clear that there exists a unique morphism of lattices ϕ : γ(λ) −→ γ(∆) which satisfies the following properties: – (i) ϕ(x ′ ) = x, ϕ(c) = c, for all finite subset c of x. – (ii) ϕ(y ) = y ′ , ϕ(d) = d, for all finite subset d of y. clearly, ϕ is a lattice equivalence of topological spaces. suppose that ϕ is induced by a quasihomeomorphism. without loss of generality, we may suppose that there is a quasihomeomorphism q : ∆ −→ λ such that ϕ = γ(q). hence q−1(y ) = ϕ(y ) = y ′ ; so that q(β) ∈ y . on the other hand, {β} = y ′ . the continuity of q implies that q(y ′ ) = q({β}) ⊆ q({β}) = {q(β)}. thus y ′ ⊆ q−1({q(β)}) = ϕ({q(β)}) = {q(β)}, a contradiction, since y is infinite. therefore, ϕ is not induced by a quasihomeomorphism. 236 o. echi and s. lazaar in [8, corollary 2.1], thron has written that “if x is a t0-space and y is a t2space, then they are homeomorphic if and only if they are lattice-equivalent”. the following result shows that in [8, corollary 2.1], “t2-space” cannot be replaced by “t1-space”. we need to recall the notion of jacobson space [5]. a topological space x is said to be a jacobson space if the subset of all closed points of x is strongly dense in x. theorem 3.17. (1) if a t0-space x is lattice equivalent to a t1-space y , then x is a jacobson space. (2) there exist a t0-space x and a t1-space y which are lattice equivalent but not homeomorphic (hence any lattice equivalence between them is not induced by a homeomorphism). proof. (1) by corollary 3.5, the lattice equivalence between x and y is induced by a quasihomeomorphism q : y −→ x. according to proposition 2.6, the induced quasihomeomorphism q1 : y −→ q(y ) is bijective. hence y is homeomorphic to the subspace q(y ) of x and q(y ) is strongly dense in x. it suffices to prove that q(y ) is the set x0 of all closed points of x. indeed, x0 ⊆ q(y ), since q(y ) is strongly dense in x. on the other hand, let y ∈ y ; then {q(y)} is closed in q(y ) since y is homeomorphic to q(y ). hence {q(y)} ∩ q(y ) = {q(y)}. let z ∈ {q(y)}; then {z} ∩ q(y ) 6= ∅. thus {z} ∩ q(y ) = {q(y)}. it follows that {q(y)} = {z}. therefore, z = q(y), since x is a t0-space. (2) it suffices to take a jacobson t0-space x which is not t1. let y be the subspace of x whose elements are the closed points of x. hence the canonical embedding of y into x is a quasihomeomorphism. thus x and y are lattice equivalent; however, they are not homeomorphic. � example 3.18. it is easy to give explicit examples of jacobson t0-spaces which are not t1. let r be a hilbert ring which is not a field; i.e., a ring such that the intersection with r of a maximal ideal of the polynomial ring r[t] is maximal (take for example r = k[t1, ..., tn] the polynomial ring on n indeterminates over a field k). let x = spec(r) equipped with the hull-kernel topology. then x is a jacobson space which is not t1. here, if we let y := m ax(r) be the set of all maximal ideals of r, then y is a t1 strongly dense subspace of x. thus the canonical quasihomeomorphism i : y −→ x induces a lattice equivalence between the topological spaces x and y . on the other hand, the space x is sober by [7, proposition 4]. this yields a lattice equivalence between a t1-space and a sober space which is not induced by a homeomorphism. lattice equivalent topological spaces 237 acknowledgements. the authors wish to thank the dgrst 03/u r/15− 03 for its partial support. references [1] k. belaid, o. echi and r. gargouri, a-spectral spaces, topology appl. 138 (2004), 315–322. [2] e. bouacida, o. echi and e. salhi, foliations, spectral topology, and special morphisms, advances in commutative ring theory (fez, 1997), 111–132, lecture notes in pure and appl. math. 205, dekker, new york, 1999. [3] e. bouacida, o. echi and e. salhi, feuilletages et topologie spectrale, j. math. soc. japan 52 (2000), 447–464. [4] p. d. finch, on the lattice-equivalence of topological spaces, j. austral. math. soc. 6 (1966), 495–511. [5] a. grothendieck and j. dieudonné, eléments de géométrie algébrique, die grundlehren der mathematischen wissenschaften, 166, springer-verlag, new york, 1971. [6] a. grothendieck and j. dieudonné, eléments de géométrie algébrique. i. le langage des schémas, inst. hautes etudes sci. publ. math. no. 4, 1960. [7] m. hochster, prime ideal structure in commutative rings, trans. amer. math. soc. 142 (1969), 43–60. [8] w. j. thron, lattice-equivalence of topological spaces, duke math. j. 29 (1962), 671– 679. [9] k. w. yip, quasi-homeomorphisms and lattice-equivalences of topological spaces, j. austral. math. soc. 14 (1972), 41–44. received april 2008 accepted april 2009 othman echi (othechi@yahoo.com,othechi@math.com) king fahd university of petroleum and minerals, department of mathematics & statistics, p.o. box 5046, dhahran 31261, saudi arabia. sami lazaar (salazaar72@yahoo.fr) department of mathematics, university tunis-el manar, faculty of sciences of tunis“campus universitaire”, 2092 tunis, tunisia bouassidaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 253-262 the jordan curve theorem in the khalimsky plane ezzeddine bouassida abstract. the connectivity in alexandroff topological spaces is equivalent to the path connectivity. this fact gets some specific properties to z2, equipped with the khalimsky topology. this allows a sufficiently precise description of the curves in z 2 and permit to prove a digital jordan curve theorem in z 2 . 2000 ams classification: 54d05, 54d10, 68u05, 68u10, 68r10. keywords: topological space, alexandroff topology, khalimsky topology, simple closed curve, jordan curve theorem. 1. introduction the computer sciences, the medical imagery, the robotic sciences and other applied sciences, make more and more useful the study of discrete sets. there is an approach where the graph theory takes place, it permits to check some results as the jordan curve theorem: if γ is an n-connected closed curve in z 2, then z2 \ γ has two and only two n-connectivity components (n + n = 12, n = 4, 8) (see [8]). this result is a kind of generalization of the classical jordan curve theorem in r2 stating that: if γ is a simple closed curve in r2, then r2 \ γ has two and only two connectivity components. a generalization of this theorem to the discrete sets needs to define topologies on this kind of sets. on z2, more than one is developed (khalimsky, marcus, wyse), see [5, 9]. nowadays, the khalimsky’s one is one of the most important concepts of the digital topology. it is well known that a criterion of the convenience of a topology on z2 is the validity of an analogue of the jordan curve theorem. in [5] kopperman, khalimsky and meyer stated a generalization in z2 equipped with the khalimsky topology. 254 e. bouassida our purpose in this note is to present a new proof of the khalimsky’s jordan curve theorem using the specificity of the khalimsky’s plane as an alexandroff topological space and the specific properties of connectivity on these spaces. first of all, we present the result about the connectivity in alexandroff spaces useful for our purpose: it is proved that in an alexandroff topological space that the connectivity is equivalent to the c.o.t.s-arc-connectivity which is equivalent to the c.o.t.s-path-connectivity [1, 3, 9]. the c.o.t.s (connected ordered topological space) is described in [5, 8]. here our c.o.t.s is an interval of z equipped with the khalimsky topology. following this, we present the khalimsky plane and we describe the specialization order, our aim is to understand the behaviour of the line-complex of an arc. this leads to a geometric description of arcs and simple closed curves. the specificity of the khalimsky topology gives us some properties of the points in (z2, κ) and there adjacency sets. this allows a sufficiently precise description of the arcs in (z2, κ) and permit to prove the digital jordan curve theorem. 2. alexandroff topological spaces definition 2.1. let (x, τ ) be a topological space, (x, τ ) is said to be an alexandroff topological space , or shortly an a-space if any intersection of elements of τ is an element of τ . it is well known that a topological space (x, τ ) is an a-space if and only if every point x ∈ x has a smallest open neighborhood denoted by n (x). the set b = {n (x); x ∈ x} is a base for the topology τ . recall that, given an alexandroff topology τ on a set x, the specialization preorder (�) on x, associated to τ is defined by: ∀(x, y) ∈ x × x, (x � y ⇔ y ∈ n (x) ⇔ x ∈ {y}) where {y} is the closure of the point {y}. it is proved in [2] (see [1, 5]) that: theorem 2.2. on a given set x the specialization preorder determines a one to one correspondence between the t0-alexandroff topologies and the partial orders. in what follows, we present some terminologies and definitions necessary for understanding the results of the next section. recall that a topological space (x, τ ) is a t0-space if for each pair of distinct points of x, there exist a neighborhood of one of them not containing the other. a topological space, (x, τ ), is said to be connected if it is a non empty set and the only subsets which are both open and closed are the empty set ∅ and x. a subset a of a topological space (x, τ ) is called connected if it is connected as a topological space with the induced topology, equivalently: a is non empty and for all non empty open subsets u and v of x, we have: (u ∩ a 6= ∅, v ∩ a 6= ∅) ⇒ u ∩ v ∩ a 6= ∅. the jordan curve theorem in the khalimsky plane 255 a connectivity component (or shortly a component) of a topological space is a connected subset which is maximal with respect to the inclusion. a component is always closed. in an a-space (x, τ ), a subset of distinct points {x, y}, is connected if and only if either x ∈ n (y) or y ∈ n (x). we say that x and y are adjacent or y is adjacent to x. let x and y be two points in a topological space (x, τ ), a path (resp. an arc) linking x and y is a couple (i, γ) where i = [a, b] is a compact interval of r equipped with the usual topology and γ a continuous map (resp. an homeomorphism) of i onto x such that γ(a) = x and γ(b) = y. a topological space (x, τ ) is said to be path (resp. arc) connected if for any two points x and y in (x, τ ), there exists a path (resp. an arc) linking x and y. when (x, τ ) is a point set, the path (resp. arc) connectivity is generalized, by the c.o.t.s connectivity, see [2, 9]. a connected ordered topological space (c.o.t.s) is a connected topological space l with the property: for any x1, x2, x3 distinct points in l, there is an i such that xj and xk lie in different components of l \ {xi} where {i, j, k} = {1, 2, 3}. a c-path (resp. c-arc) in the topological space (x, τ ) is a couple (i, γ) where i is a c.o.t.s and γ is a continuous map (resp. an homeomorphism) from i onto x. the space (x, τ ) (resp. a nonempty subset a of x ) is said to be c.o.t.s path connected (c.p.c) (resp. c.o.t.s arc connected (c.a.c)) if for any two points x and y in (x, τ ) (resp. in a) there exists a c-path (resp. c-arc) in x (resp. in a) joining x and y. we recall from [1, 3, 5, 9], the next result. theorem 2.3. let (x, τ ) be an alexandroff topological space and a ⊆ x a subset. then the following conditions are equivalent. (i) a is path-wise connected (p.c). (ii) a is c.o.t.s-path-wise connected (c.p.c). (iii) a is c.o.t.s-arc-wise connected (c.a.c). (iv ) a is connected. the following proposition, (see [7, 8]), establishes the relationship between continuity in a-spaces and preorders. as a consequence, the rich theory of preordered sets can be put to work here. proposition 2.4. let x and y be two t0-alexandroff topological spaces and f : x → y be an application, the following conditions are equivalent. (i) f is a continuous map. (ii) f is an increasing map for the specialization pre-orders of x and y . 3. the khalimsky plane let b = {{2n + 1}, {2n − 1, 2n, 2n + 1}, n ∈ z} be a subset of p(z); b is a basis of a topology κ on z. the topological space (z, κ) is a t0-alexandroff topological space called the khalimsky or the digital line. in (z, κ), the point 256 e. bouassida set {x} is open (resp. closed) if and only if x is odd (resp. even), n (2n + 1) = {2n + 1}, n (2n) = {2n − 1, 2n, 2n + 1}. a half line is open if and only if its end point is open. the specialization order is as follows: ... − 3 � −2 � −1 � 0 � 1 � 2 � 3... a pair {x, y}, x 6= y, is a connected subset of (z, κ) if and only if y = x + 1. in this note the khalimsky plane (z2, κ) is the cartesian product (z, κ) × (z, κ) equipped with the product topology. let (x, y) be a point in (z2, κ), (x, y) is an open (resp. closed) point if both x and y are odd (resp. even), (x, y) is said to be a pure point. otherwise (x, y) is said to be a mixed point, open-closed (resp. closed-open) if x is odd (resp. even) and y is even (resp. odd). the smallest neighborhood n (x, y) of the point (x, y) of (z2, κ) is: – n (x, y) = {(x, y)} when (x, y) is an open point. – n (x, y) = {(x, y − 1), (x, y), (x, y + 1)} when (x, y) is an open-closed point. – n (x, y) = {(x − 1, y), (x, y), (x + 1, y)} when (x, y) is a closed-open point. – n (x, y) = {(x − 1, y − 1), (x − 1, y), (x − 1, y + 1), (x, y − 1), (x, y), (x, y + 1), (x + 1, y − 1), (x + 1, y), (x + 1, y + 1)} = {(a, b) ∈ z2, ‖(a, b) − (x, y)‖∞ ≤ 1} when (x, y) is a closed point. we denote by a(x, y) the adjacency set of (x, y) and we have: – if (x, y) is a pure point a(x, y) = {(x − 1, y − 1), (x − 1, y), (x − 1, y + 1), (x, y − 1), (x, y + 1), (x + 1, y − 1), (x + 1, y), (x + 1, y + 1)} – if (x, y) is a mixed point a(x, y) = {(x − 1, y), (x, y + 1), (x + 1, y), (x, y − 1)} the specialization order � on z2 is as follow: ∀(x, y) ∈ z2, (x, y) � (x, y) if (x, y) is a closed point, we have (x, y) ≺ (a, b) for all (a, b) such that ‖(a, b) − (x, y)‖∞ = 1. if (x, y) is a open point, we have (a, b) ≺ (x, y) for all (a, b) such that ‖(a, b) − (x, y)‖∞ = 1. if (x, y) is a open-closed point, (x, y) ≺ (x, y + 1), (x, y) ≺ (x, y − 1), (x, y) ≻ (x − 1, y) and (x, y) ≻ (x + 1, y). if (x, y) is a closed-open point, (x, y) ≺ (x − 1, y), (x, y) ≺ (x + 1, y), (x, y) ≻ (x, y + 1) and (x, y) ≻ (x, y − 1). the connectivity graph of (z2, κ) has all the points of z2 as vertices. two vertices (x, y) and (a, b) are connected if and only if ‖(a, b) − (x, y)‖∞ = 1 and (x, y) and (a, b) are not simultaneously mixed points. we note that two edges can not cross. the following remark is useful to imagine the geometric behaviour of various lines in z2. the jordan curve theorem in the khalimsky plane 257 remark 3.1. if (p + q) is even, the connectivity graph is invariant under the action of the translation t(p,q) . if p and q are even, then, the hass diagram (graph of the order) of (z2, �) is invariant under the action t(p,q). if p and q are odd and (p + q) is even, then t(p,q) transforms (z 2, �) in (z2, �). the connectivity graph and the hass diagram on z2 ar invariant under a rotation by π 2 . 4. paths, arcs and curves in the khalimsky plane the spaces (z, κ) and (z2, κ) are equipped with the khalimsky topology. an interval i = [a, b] = {a, a + 1, a + 2, ..., a + n = b} is a c.o.t.s. thus give the following: definition 4.1. a path (resp. an arc) in (z2, κ) is a couple (i, γ), where i = [a, b] = {a, a + 1, a + 2, ..., a + n = b} is an interval in (z, κ) and γ is a continuous map (resp. an homeomorphism) from i into (z2, κ). it is obvious that the image γ(i) = γ is connected. denoting by γ̂ the linecomplex of γ, γ̂ is the broken line in r2 having the points of γ as vertices and completed with the edges linking the consecutive vertices of γ. among the several interesting consequences of proposition (2.4), one can check the following: proposition 4.2. let γ̂ be the line-complex of an arc (i = [a, b] = {a, a + 1, a + 2, ..., a + n = b}, γ) in (z2, κ) and denote by γa, γa+1, ...γa+n the vertices of γ̂. then the following hold: (i) for all i ∈ {a, a+1, a+2, ..., a+n = b}, we have, either γi−1 � γi � γi+1 or γi−1 � γi � γi+1. (ii) if the vertex γi is a mixed point of (z 2, κ), then γ̂, remains a straight line at this point. (iii) if the vertex γi is a pure point of (z 2, κ), then γ̂, can rotate by π 4 , π 2 , 3π 2 , 7π 4 or 2π (that means γ̂ can’t have an acute angle π 4 ). proof. the map γ is one-to-one, thus γ don’t have double points. (i) without loss of generality, we can suppose that a ∈ 2z, so a � a + 1 � a + 2 � a + 3 � a + 4.... since the map γ is continuous, it is strictly increasing. hence: γa � γa+1 � γa+2 � γa+3 � .... (ii) let γi be a mixed point (closed-open for example) in γ. according to the parity of i, we have, i − 1 � i � i + 1 or i − 1 � i � i + 1. so we must have γi−1 � γi � γi+1 or γi−1 � γi � γi+1. in the first case, the three points must have the same second coordinate and in the second case, they must have the same first coordinate. this implies that γ̂ still straight. the same argument can be used when γi is open-closed (iii) let γi be a pure point in γ, then γi is comparable (smaller or greater) to p for all p in {p ∈ z2, ‖p −γi‖∞ = 1}. thus γi+1 ∈ {p ∈ z 2, ‖p −γi‖∞ = 1}. 258 e. bouassida this ables γ̂ to turn at p by the angle α = 0 or π 4 or π 2 or 3π 4 or π or 5π 4 or 3π 2 or 7π 4 or 2π. assume α = π, then we have γi−1 = γi+1 which is forbidden. now if α = 3π 4 or 5π 4 , then the points γi−1 and γi+1 are adjacent and we get γi+1 � γi−1 or γi+1 � γi−1. the map γ −1 is continuous, so (γi+1 � γi−1 ⇒ i + 1 � i − 1) and (γi+1 � γi−1 ⇒ i + 1 � i − 1) which is impossible. � it follows readily from the previous proposition, that we have the following remarks. remark 4.3. let (i = [a, b], γ) be an arc, γ(i) = γ and γ̂ its line-complex. (i) when we follow γ̂, we don’t meet an acute angle. we can turn only on the pure vertices. these restrictions disappear if (i = [a, b], γ) is only a path. to see this, consider the map γ : {0, 1, 2, 3} → z2, γ(0) = (0, 0), γ(1) = (0, 1) = γ(2), γ(3) = (1, 1). one can check easily that γ is continuous and γ̂ rotate at (0, 1) which is a mixed point. (ii) if pi (a + 1 ≤ i ≤ a + n − 1) is a vertex in γ̂, then pi has two adjacent points in γ. (iii) it is forbidden to have two mixed points in consecutive positions in γ̂. 5. the jordan curve theorem this section is mainly concerned with the jordan curve theorem. the purpose of this section is to prove the jordan curve theorem. we start with the following definitions. definition 5.1. a simple closed curve (s.c.c) in (z2, κ) is the image of an interval i = [a, a + n] = {a, a + 1, ..., a + n} by a continuous map γ in (z2, κ) such that γ(a) = γ(a + n) and any connected subset of γ is the image of an arc. remark 5.2. our definition of the closed simple curve meets the kiselmann’s one for the khalimsky jordan curve which is a homeomorphic image of a khalimsky circle z/mz, where m is an even integer ≥ 4. as an initial step toward understanding the structure of (s.c.c), we check the following. proposition 5.3. if γ is a simple closed curve (s.c.c) in (z2, κ), then card(γ) is an even integer ≥ 4. proof. assume that γ = {p1, p2, p3}, γ̂ is a triangle in the connectivity graph of (z2, κ). thus, two of the angles of γ̂ are π 4 which is in contradiction with the definition of an arc. thus card(γ) ≥ 4. assume now that γ = {p1, p2, p3, p4}. if one of the pi, (1 ≤ i ≤ 4) is a mixed point, then, γ̂ can not rotate at this pi. to be closed, γ needs to be a triangle with acute angles. thus {p1, p2, p3, p4} are pure points. the linecomplex γ̂ of γ is a square , two of its vertices are closed points and the others are open points and we have the following order: the jordan curve theorem in the khalimsky plane 259 ...p1 � p2 � p3 � p4 � p1 � p2 � p3 � p4... or ...p1 � p2 � p3 � p4 � p1 � p2 � p3 � p4... assume that γ = {p1, p2, p3, p4, p5}. the fact that any subset of γ is the image of an arc yields: ...p1 � p2 � p3 � p4 � p5 � p1 � p2 � p3... or ...p1 � p2 � p3 � p4 � p5 � p1 � p2 � p3... which is incoherent in the both cases. this incoherence disappears when card(γ) is even. � let γ be a (s.c.c) in (z2, κ), we denote by a the subset of z2 interior to γ̂ and b the subset of z2 exterior to γ̂. our aim is to prove that a and b are connectivity components of z2 \ γ. we start our search with the case where card(γ) ≤ 8. proposition 5.4. let γ be a (s.c.c) in (z2, κ) and card(γ) ≤ 8. denote by a the subset of z2 interior to γ̂ and b the subset of z2 exterior to γ̂.then a and b are two connectivity components of z2 \ γ. proof. if card(γ) = 4, the unique curve (modulo the geometric translations and the rotation mentioned in remark(0.3.1)) is the square where the four vertices are pure points and a is a mixed point p . the smaller neighborhood of p is n (p ) = {p1, p, p3} where p1 and p3 are the two open vertices in γ. the set {p2, p, p4} is closed in (z 2, κ), thus a is a component of z2 \ γ. an example of such γ is γ = {p1 = (−1, 1), p2 = (0, 2), p3 = (1, 1), p4 = (0, 0)}, here a = {p } = {(0, 1)}. if m and n are two pure points in b, we can avoid (γ ∪ a) by the linecomplex of a path j nm where all its vertices are pure points. if m is a mixed point in b, m has an adjacent pure point m1 in b otherwise m = p . let n be a pure point in b (n 6= m ), we can avoid (γ ∪ a) by the line-complex of a path j nm1 where all the vertices are pure points. we add to j nm1 the point m and we have j n m ⊂ b a path joining m and n , so b is (c.p.c). hence b is a connectivity component of z2 \ γ. assume card(γ) = 6, γ̂ can not turn at a mixed point, can not have an acute angle and each point of γ has two and only two adjacent points in γ. these constraints get off the possibility of closed simple curve γ with card(γ) = 6. modulo the geometric translation and the rotation mentioned in remark(0.3.1)), there is four type of closed simple curves γ with card(γ) = 8. it is easy to verify directly that in the four cases a and b are connectivity components of z 2 \ γ. � 260 e. bouassida to prove the general case we need the following lemmas: lemma 5.5. let γ be a (s.c.c) in (z2, κ). let γ̂, a, and b defined as before, we have: (i) if p ∈ a, then a(p ) ⊂ a ∪ γ. (ii) if p ∈ b, then a(p ) ⊂ b ∪ γ. (iii) if p is a pure point in a and a 6= {p }, then p has an adjacent mixed point in a. (iv ) if p is a mixed point in a and a 6= {p }, then p has an adjacent pure point in a. proof. (i) let p be a mixed point in a, a(p ) = {p1, p2, p3, p4}, where pi, 1 ≤ i ≤ 4 are pure points. p1 and p3 have the same second coordinate with p , and p2 and p4 have the same first coordinate with p . if pi ∈ b then γ̂ must run between pi and p , which is impossible, so a(p ) ⊂ a ∪ γ. let p be a pure point in a, a(p ) = {p1, p2, p3, p4, m1, m2, m3, m4} where mi is a mixed point and pi is a pure point (i = 1, 2, 3, 4). m1 and m3 have the same second coordinate with p , m2 and m4 have the same first coordinate with p , p1 and p2 have the same second coordinate with m2, p3 and p4 have the same second coordinate with m4. if mi ∈ b, then γ̂ must run between mi and p , which is impossible. thus mi ∈ a ∪ γ. if pi ∈ b, then γ̂ links mi and mi+1, which is impossible too, thus a(p ) ⊂ a ∪ γ. (ii) the same proof as in (i). (iii) if p is a pure point in a, then a(p ) ⊂ a ∪ γ, and card(a(p )) = 8. the assumption a(p ) ⊂ γ leads to γ = a(p ) and a = {p }, a contradiction. thus we get a(p ) ∩ a 6= ∅. if one of the mi’s belong to a(p ) ∩ a, we obtain the desired conclusion. if one of the pi’s belong to a(p ) ∩ a, the two mixed points mi and mi+1 are in a too, otherwise γ̂ turns in a mixed point, thus p has an adjacent mixed point in a. (iv ) let p be a mixed point in a, then a(p ) = {p1, p2, p3, p4}. if a(p ) ⊂ γ, then γ = a(p ) and a = {p } which contradict the hypothesis a 6= {p }, so a(p ) ∩ a 6= ∅. � lemma 5.6. let γ be a (s.c.c) in (z2, κ) such that card(γ) > 8, and let γ1 and γ2 be two successive points in γ, (‖γ1−γ2‖∞ = 1), then a(γ1)∩a(γ2)∩a 6= ∅ and a(γ1) ∩ a(γ2) ∩ b 6= ∅. proof. modulo the geometric translation and the rotation mentioned in remark(0.3.1), four cases may occur: first case: γ1 and γ2 are successive points in a straight line parallel to one of the coordinates axes on γ̂, then one of the two points is pure (suppose it γ1), and the other is mixed. we denote a(γ1) = {p1, p2, p3, p4, m1, m2, m3, m4} the jordan curve theorem in the khalimsky plane 261 and a(γ2) = {p ′ 1, p ′ 2, p ′ 3, p ′ 4}, a(γ1) ∩ a(γ2) ∩ a = {p ′ 3 = p2} or {p ′ 1 = p1}, a(γ1) ∩ a(γ2) ∩ b = {p ′ 1 = p1} or {p ′ 3 = p2}. second case:γ1 and γ2 are successive points in a straight diagonal line on γ̂, then the two points are pure. we denote a(γ1) = {p1, p2, p3, p4, m1, m2, m3, m4} and a(γ2) = {p ′ 1, p ′ 2, p ′ 3, p ′ 4, m ′ 1, m ′ 2, m ′ 3, m ′ 4}, a(γ1)∩a(γ2)∩a = {m ′ 4 = m3} or {m ′2 = m1}, a(γ1) ∩ a(γ2) ∩ b = {m ′ 2 = m1} or {m ′ 4 = m3}. third case: the line-complex γ̂ rotate at γ1 by π 4 , in this case γ2 is a pure point and we have: a(γ1) ∩ a(γ2) ∩ a = {m ′ 4 = m3} or {m ′ 2 = m1}, a(γ1) ∩ a(γ2) ∩ b = {m ′ 2 = m1} or {m ′ 4 = m3}. fourth case: the line-complex γ̂ rotate at γ1 by π 2 , in this case γ2 is a mixed point and we have: a(γ1) ∩ a(γ2) ∩ a = {p ′ 4 = p4} or {p ′ 2 = p1}, a(γ1) ∩ a(γ2) ∩ b = {p ′ 2 = p1} or {p ′ 4 = p4}. thus a(γ1) ∩ a(γ2) ∩ a 6= ∅ and a(γ1) ∩ a(γ2) ∩ b 6= ∅. � as a consequence of the previous lemma, we get: corollary 5.7. let γ be as in the previous lemma and let p and q two points in γ, then, there exists a path in a linking a(p ) to a(q). proposition 5.8. let γ be a (s.c.c) in (z2, κ), and let γ̂, a, and b defined as before. if u and v are two points of a, then, there exists a path j vu linking u and v . proof. let u ∈ a, if a(u ) ∩ γ 6= ∅, we choose γ(u ) ∈ a(u ) ∩ γ. otherwise, consider a point u1 ∈ a(u ). if a(u1) ∩ γ 6= ∅, we choose γ(u ) ∈ a(u1) ∩ γ, otherwise we consider a(u2) where u2 ∈ a(u1), etc...after finite steps, we obtain γ(u ) ∈ γ and a path in a, j γ(u) u linking u and γ(u ). let v ∈ a, (u 6= v ), and consider j γ(v ) v . now, using the previous corollary, we obtain a path in a, j γ(u) γ(v ) , linking a(γ(u )) and a(γ(v )). the needed path is: j vu = {j γ(u) u \ γ(u )} ∪ j γ(v ) γ(u) ∪ {j vγ(v ) \ γ(v )}. � corollary 5.9. let γ be a (s.c.c) in (z2, κ), and let γ̂, a, and b defined as in the previous proposition. if u and v are two points of b, there exists a path j vu linking u and v . we close this section by the following results. theorem 5.10. let γ be a (s.c.c) in (z2, κ), then γ share z2 in two components, both of them is (c.p.c). corollary 5.11 (khalimsky jordan curve theorem). let γ be a (s.c.c) in ((z2, κ), then z2 \ γ has exactly two and only two connectivity components. 262 e. bouassida references [1] f. g. arenas, alexandroff spaces, acta math. univ. comenianea, vol. lxviii, 1 (1999), 17–25. [2] e. bouacida, o. echi and e. salhi, topologies associées à une relation binaire et relation binaire spectrale, boll. mat. ital., vii. ser., b 10 (1996), 417–439. [3] e. bouacida and n. jarboui, connectivity in a-spaces, jp journal of geometry and topology. 7 (2007), 309–320. [4] n. bourbaki, topologie générale. elément de mathématique, première partie, livre iii, chapitr 1-2, tird edition. paris: hermann. [5] e. d. khalimsky, r. kopperman and p. r. meyer, computer graphics and connected topologies on finite closed sets, topology appl. 36 (1967), 1–17. [6] c. o. kisselman, digital jordan curve theorems, lecture notes in computer science, springer, berlin, vol. 1953 (2000). [7] c. o. kisselman, digital geometry and mathematical morphology, lecture notes, uppsala university, departement of mathematics, (2002). [8] t. y. kong, r. kopperman and p. r. meyer, a topological approach to digital topology, american. math. monthly. 98 (1991), 901–917. [9] j. slapal, digital jordan curves, topology appl. 153 (2006), 3255–3264. received june 2007 accepted december 2007 ezzeddine bouassida (ezzeddine−bouassida@yahoo.fr) faculty of scienses of sfax, departement of mathematics. 3018 sfax, p.d.box:802, tunisia. @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 1-10 on cofree s-spaces and cofree s-flows behnam khosravi abstract let s-tych be the category of tychonoff s-spaces for a topological monoid s. we study the cofree s-spaces and cofree s-flows over topological spaces and we prove that for any topological space x and a topological monoid s, the function space c(s, x) with the compactopen topology and the action s · f = (t �→ f(st)) is the cofree s-space over x if and only if the compact-open topology is admissible and tychonoff. finally we study injective s-spaces and we characterize injective cofree s-spaces, when the compact-open topology is admissible and tychonoff. as a consequence of this result, we characterize the cofree s-spaces and cofree s-flows, when s is a locally compact topological monoid. 2010 msc: 54c35, 22a30, 20m30, 54d45, 54h10. keywords: s-space, s-flow, cofree s-space, cofree s-flow compact-open topology, injective s-space. 1. introduction and preliminaries there are many works about s-spaces or more specially g-spaces and their applications, and some authors study the free and projective s-spaces (gspaces) and their applications [1, 7, 10, 11, 13, 14, 15, 16, 18]. also, there are some results about injective and cofree boolean s-spaces (see [1]). recall that, for a monoid s, a set a is a left s-set (or s-act) if there is, so called, an action μ : s × a → a such that, denoting μ(s,a) := sa, (st)a = s(ta) and 1a = a. the definitions of an s-subset a of b and an shomomorphism (also called s-map) between s-sets are clear. in fact s-maps are action-preserving maps: f : a → b with f(sa) = sf(a), for s ∈ s, a ∈ a. each monoid s can be considered as an s-set with the action given by its multiplication. let s be a monoid and a be an s-set. recall that for s ∈ s, the 2 b. khosravi s-homomorphism λs : a → a is defined by y �→ sy for any y ∈ a. similarly, for a ∈ a, the s-map ρa : s → a is defined by t �→ ta for any t ∈ s. let c be a concrete category over d and u : c → d be the forgetful functor. an object k in c with a morphism ψ : k → d in d, where d ∈ d, is the cofree object over d, if for every morphism f : c → d in d there exists a unique morphism f̃ : c → k in c such that ψ ◦ f̃ = f in d. for any two topological spaces x and y , we denote the set of all continuous maps from x to y by c(x,y ). if τ is a topology on the set c(x,y ), then the corresponding space is denoted by cτ (x,y ). the category of all tychonoff spaces is denoted by tych. note that all of the spaces in this note are tychonoff (completely regular and hausdorff). a monoid s with a hausdorff topology τs such that the multiplication · : s × s → s is (jointly) continuous, is called a topological monoid. for a topological monoid s, an s-space is an s-set a with a topology τa such that the action s × a → a is (jointly) continuous. the category of all tychonoff s-spaces with continuous s-maps is denoted by s-tych (see [14, 15, 16, 18]). a compact hausdorff s-space is called an s-flow (see [2, 13]). let y and z be two topological spaces. a topology on the set c(y,z) is called splitting if for every space x, the continuity of a map g : x × y → z implies that of the map g̃ : x → cτ (y,z) defined by g̃(x)(y) = g(x,y), for every x ∈ x and y ∈ y (this topology is also called proper [3, 8] or weak [6]). a topology τ on c(y,z) is admissible if the mapping ω(y,f) := f(y) from y × c(y,z) into z is continuous in y and f. equivalently, a topology τ on c(y,z) is admissible if for every topological space x, the continuity of an f : x → cτ (y,z) implies the continuity of ̂f : x × y → z, where ̂f(x,y) := f(x)(y) for every (x,y) ∈ x × y (see [8]) (the latter definition is usually used as the definition of admissible topology, but we use the former). a topological space y is said to be exponential if for every space x there is an admissible and splitting topology on c(y,x) (see [6]). for any topological spaces x and y , we denote the set c(x,y ) with the compact-open topology by cco(x,y ). for any compact subset k of x and an open set u in y , by (k,u) we mean the set {f ∈ c(x,y )|f(k) ⊆ u}. for topological topics and facts about stone-cech compactification, we refer to [3, 9, 17]. in this note, we study the cofree and injective s-spaces and s-flows. recall that for a set e and a monoid s, es, the set of all functions from s to e with the action sf := (t �→ f(st)), for any function f : s → e and s ∈ s, is the cofree s-set over e (see [12]). in section 2, we study the cofree s-spaces and s-flows over topological spaces. as a consequence of these results, we characterize the cofree s-spaces and the cofree s-flows over topological spaces, when the compact-open topology is admissible and tychonoff (more specially, when s is locally compact). finally, in section 3, we study injective cofree s-spaces and s-flows over topological spaces, when the compact-open topology is admissible and tychonoff. note that we state and prove our results for on cofree s-spaces and cofree s-flows 3 topological monoids and plainly all of our results hold for topological groups and g-spaces. 2. the cofree s-spaces and s-flows over a topological space one of the main steps in the study of injective objects in a category is the study of cofree objects. these objects can be used for presenting injective cover for objects in a category1. in this section, first we study the cofree tychonoff sspaces over a tychonoff space, then we study the cofree s-flow over a compact space2. finally in this section, as a consequence of these results, we will show that if s is a locally compact topological monoid, then the cofree s-space and the cofree s-flow exist and we present them explicitly. remark 2.1. it is a known fact that the cofree s-set over a set e, is the set es of all functions from s to e with the action defined by s · f := (t �→ f(st)) for all f ∈ es, s ∈ s and t ∈ s. let e and d be two sets. recall that for an s-set a and any function h : a → e, the s-homomorphism h : a → es defined by h(a) := h ◦ ρa is the unique s-map such that ψ ◦ h = h, where ψ : es → e is defined by ψ(f) := f(1), for any f ∈ es (see [12]). remark 2.2. (i) let s be a topological monoid and x be a topological space. the s-space s×x with the product topology and the action λ1 : s×(s×x) → s × x, t(s,x) = (ts,x), is denoted by l(x). for a topological space x, the s-space x with the trivial action, sx := x, is denoted by t(x). (ii) note that for any topological space x and a non-empty topological space y , if we define cx(y) := x for every y ∈ y , and c := {cx|x ∈ x}, then c as a subspace of cco(y,x) is homeomorphic to x. so the function j : x → cco(y,x) defined by j(x) := cx is an embedding from x to cco(y,x). from now on, we denote this embedding by jx for any topological space x. by theorem 2.9 in [6], we have remark 2.3. let x be a topological space. then the following are equivalent (a) x is exponential; (a) for every space y , there exists a splitting and admissible topology on c(x,y ); (c) x is core compact. note that for hausdorff spaces (and more generally for sober spaces) core compactness is the same as local compactness (see [5]). furthermore, it is a known fact that if x is locally compact, then the compact-open topology on c(x,y ) is admissible and splitting. 1note that the cofree objects in an arbitrary category are not injective in general. 2one can easily see that if the cofree s-flow exists over a space x, then x is compact. so this assumption is not an extra assumption. 4 b. khosravi theorem 2.4. let s be a topological monoid. then the following are equivalent (a) for every tychonoff space x, the compact-open topology on c(s,x) is admissible and tychonoff; (b) for every space x, cco(s,x) is tychonoff and cco(s,x) with the action defined by sf = (s′ �→ f(ss′)) is the cofree s-space over x. proof. (a) ⇒ (b) let x be a topological space. first we show that cco(s,x) with its introduced action is an s-space, then we show that it has the cofree universal property. first, note that since s is a topological monoid, the action is well-defined. let f ∈ c(s,x), s ∈ s and (k,u) be subbasis element for cco(s,x) containing sf. therefore for any k ∈ k, f(sk) = (sf)(k) ∈ u. since by the assumption, the compact-open topology on c(s,x) is admissible and since for any k ∈ k, we have ω(sk,f) ∈ u, there exist open neighborhoods of and wsk of f and sk, in cco(s,x) and s, respectively such that ω(wsk,of ) ⊆ u. on the other hand, since s is a topological monoid, for sk ∈ wsk, there exist open sets wks and wk in s which contain {s} and {k}, respectively and wks · wk ⊆ wsk. since k is compact and obviously {wk}k∈k forms an open cover for k, there exist k1, · · · ,kn in k such that k ⊆ ∪ni=1wki . define ws := ∩ni=1wkis . clearly for ws ∈ τs we have ω(ws · k,of ) ⊆ ω(ws · (∪ni=1wki),of ) ⊆ u ⇒ sf ∈ wsof ⊆ (k,u). hence cco(s,x) with its introduced action is an s-space. now we prove the universal property. first note that the function ψ : cco(s,x) → x is continuous. let (a,τa) be an s-space and h : (a,τa) → x be continuous. we show that for any s-space (a,τa) and a continuous function h : (a,τa) → x, the function h : (a,τa) → cτ (s,x) defined by h(a) = (s �→ h(sa)) is continuous and ψ ◦ h = h. first, note that for every a ∈ a and s ∈ s, h(a)(s) = h(sa) = h ◦ ρa(s), and since (a,τa) is an s-space, for every a ∈ a, h(a) ∈ c(s,x), so h is a well defined function. consider the continuous function h ◦ λ : s × a → a → x (s,a) �→ sa �→ h(sa) since τ is splitting, the function ˜(h ◦ λ) : a → cτ (s,x), where ˜(h ◦ λ)(a) := (s �→ (h ◦ λ)(s,a)), is continuous. therefore, h is continuous. hence cco(s,x) with its introduced action is the cofree s-space over x. (b) ⇒ (a) let λ denote the action of the cofree s-space over x. it is a known fact that the compact-open topology is splitting. on the other hand, since cco(s,x) with λ is an s-space and since ψ : cco(s,x) → x is continuous, ω = ψ ◦ λ is continuous. therefore the compact-open topology is admissible. therefore, the compact-open topology is admissible and tychonoff. � as a quick consequence of the above theorem, we have on cofree s-spaces and cofree s-flows 5 corollary 2.5. let s be a locally compact topological monoid. then for any tychonoff space x, cco(s,x) with the action defined by sf = f ◦ λs is the cofree s-space over x. proof. since s is locally compact, by remark 2.3, the compact-open topology on c(s,x) is admissible. on the other hand, by [5, corollary 3.8], cco(s,x) is tychonoff. so by the above theorem we have the result. � theorem 2.6. let s be a completely regular topological monoid3. then the following are equivalent: (a) for every compact space x, the compact-open topology on c(s,x) is admissible and tychonoff; (b) for every compact space x, cco(s,x) is completely regular and there exists an action ˜λ : s×β(cco(s,x)) → β(cco(s,x)) such that ˜λ|cco(s,x) coincides with the action of cco(s,x) and β(cco(s,x)) is the cofree s-flow over the space x. proof. (a)⇒ (b) let s be a topological monoid such that the compact-open topology on c(s,x) is admissible, for every compact space x. let x be a compact space and let λ denote the action of the cofree s-space cco(s,x). first, we introduce ˜λ and we show that it is continuous, then we prove the universal property. since s × cco(s,x) is tychonoff, β(s × cco(s,x)) exists. by the characteristic of the stone-cech compactification, for the continuous action λ : s × cco(s,x) → cco(s,x), there exists a continuous function λ : β(s × cco(s,x)) → β(cco(s,x)) such that λ|s×cco(s,x) = λ. fix an arbitrary t ∈ s and define k : cco(s,x) → s × cco(s,x) as follows k(f) := (t,f), for every f ∈ cco(s,x). consider the closure of k(cco(s,x)) in β(s × cco(s,x)). it is obvious that there exists a compact space b such that the closure of k(cco(s,x)) in β(s × cco(s,x)) is equal to {t} × b. again by the characteristic of the stone-cech compactification, there exists a continuous function h : β(cco(s,x)) → b such that h ◦ i = k, where i is the natural inclusion map from cco(s,x) to β(cco(s,x)). define λ ′ := λ|s×b. now we define ˜λ := λ′ ◦ (ids × h) : s × β(cco(s,x)) → s × b → β(cco(s,x)) and we show that ˜λ is an action. let b ∈ β(cco(s,x)) and s,s′ ∈ s. then since g ∈ β(cco(s,x)), there exists a net (fα) ⊆ cco(s,x) such that fα → g. ˜λ(ss′,g) = ˜λ(ss′, limαfα) = limαλ ′(ss′,k(fα)) = limαλ(ss ′,fα) = limαλ(s,λ(s ′,fα)) = ˜λ(s, ˜λ(s ′,g)). therefore ˜λ is a continuous action and β(cco(s,x)) with action ˜λ is an s-flow. now we prove the universal property. first, let ˜ψ : β(cco(s,x)) → x be the continuous extension of ψ : cco(s,x) → x which exists by the characteristic of the stone-cech compactification. to prove the universal property, we show that for any s-flow (f,τf ) and a continuous function l : (f,τf ) → x, there 3clearly for a topological group, this assumption is not necessary. 6 b. khosravi exists a continuous s-map l : (f,τf ) → β(cco(s,x)) such that ˜ψ ◦ l = l. let (f,τf ) be an s-flow and let l : (f,τf ) → x be a continuous function. since by theorem 2.4, cco(s,x) with action λ is the cofree s-space over x, there exists a continuous s-map l : (f,τf ) → cco(s,x) such that ψ ◦ l = l. since ˜ψ|cco(s,x) = ψ, we have clearly ˜ψ ◦ l = l. therefore β(cco(s,x)) with action ˜λ is the cofree s-flow over the space x. (b) ⇐ (a) suppose that there exists a continuous action ˜λ on β(cco(s,x)) such that β(cco(s,x)) with this action is an s-flow and ˜λ|cco(s,x) = λ, where λ is the action of cco(s,x). therefore cco(s,x) is an s-space. on the other hand, since ψ : cco(s,x) → x is continuous, ω = ψ ◦ λ : s × cco(s,x) → cco(s,x) → x is continuous. therefore the compact-open topology on c(s,x) is admissible and tychonoff. � recall that for a topological group g, a g-space (a.τa) is called g-compactificable or g-tychonoff, if (a,τa) is a g-subspace of a g-flow (compact g-space) (see [14, 15]). since s is locally compact, it is a tychonoff space. therefore by the above theorem, we have corollary 2.7. (a) let s be a locally compact topological monoid. then for any compact space x, there exists an action ˜λ on β(cco(s,x)) such that β(cco(s,x)) with this action is the cofree s-flow over the space x. (b) let g be a locally compact topological group and xbe a topological space. then the cofree g-space over x is g-compactificable. 3. injective cofree s-spaces and s-flows recall that by an embedding of topological spaces (s-spaces) we mean a homeomorphism (homeomorphism s-map) onto a subspace (an s-subset). a topological space (s-space) z is called injective over an embedding of topological spaces (of s-spaces) j : x ↪→ y if any continuous map (continuous s-homomorphism) f : x → z extends to a continuous map (continuous shomomorphism) f : y ↪→ z along j. a space (an s-space) is injective if it is injective over embeddings (see [5]). proposition 3.1. let s be a completely regular topological monoid and (e,τ) be an s-space. if (e,τ) is an injective s-space, then (|e|,τ) is injective in tych. proof. suppose that we are given the following diagram in tych i x ↪→ y f ↓ (e,τ) on cofree s-spaces and cofree s-flows 7 where x and y are topological spaces and f is a continuous function. now consider the following diagram id × i l(x) ↪→ l(y ) id × f ↓ s × e ↙ λ ↓ h e since (e,τ) is an injective s-space, there exists a continuous s-map h from l(y ) to (e,τ) such that h(id × i) = λ(id × f). (note that λ(id × f) is a continuous s-homomorphism.) note that for any topological space z, the spaces z and z × {1} with the product topology are homeomorphic. furthermore, we have (id × i)|{1}×x : {1} × x ↪→ {1} × y , and λ ◦ (id × i)(1,x) = h ◦ (id × i)(1,x) = f(x). now, define h′ := h|{1}×y : {1} × y → e and consider the following diagram (id × i)|{1}×x {1} × x −→ {1} × y ⊆ s × y g1 ↓ ↓ g2 x −→ y f ↓ i e where g1 and g2 are the following homeomorphisms g1 : {1} × x → x, and g2 : {1} × y → y (1,x) �→ x (1,y) �→ y. since g1 and g2 are homeomorphisms and, h ′((id×i)|{1}×x) = λ((id×f)|{1}×x) = fg1, we have h′ ◦ (id × i)|{1}×x ◦ g−11 = f. (i) on the other hand, since the rectangular in the above diagram is commutative, we have (id × i)|{1}×x ◦ g−11 = g−12 ◦ i. (ii) now, define f◦ := h′ ◦ g−12 . clearly, by the relations (i) and (ii), f◦ is a continuous function from y to e such that f◦i = h′g−12 i = h ′ ◦ (id × i)|{1}×x ◦ g−11 = f. � similarly, we can prove that proposition 3.2. let s be a compact topological monoid and (f,τf ) be an sflow. if (f,τf ) is an injective s-flow, then (|f |,τf ) is injective in the category of compact hausdorff spaces. 8 b. khosravi it is known that the cofree s-spaces are not injective in general. in the next proposition, we characterize the injective cofree s-spaces when s is a locally compact topological monoid. proposition 3.3. for a locally compact monoid s and a topological space x, the cofree s-space over x, cco(s,x) is injective in s-tych if and only if x is injective in tych. proof. (⇒) suppose that we are given the following diagram in tych z f ↓ x ↪→ y consider the following diagram in s-tych. t(z) jx ◦f ↓ cco(s,x) ↪→ t(y ) since cco(s,x) is injective, there exists a continuous s-homomorphism h : t(y ) → cco(s,x) such that h ◦ i = jx ◦ f. therefore, f = ψ ◦ h ◦ i. take k := ψ ◦ h. hence f = k ◦ i and x is injective. (⇐) suppose that we are given i : (a,τa) ↪→ (b,τb) and f : (a,τa) → cco(s,x) for two s-spaces (a,τa) and (b,τb). consider the following diagram i (a,τa) ↪→ (b,τb) f ↓ cco(s,x) ψ ↓ x since x is injective in tych, there exists g : (b,τb) → x such that g ◦ i = ψ ◦ f. since cco(s,x) is the cofree s-space over x, there exists h : (b,τb) → cco(s,x) such that ψ ◦ h = g. we claim that h ◦ i = f. clearly we have ψ ◦ h ◦ i = ψ ◦ f. so for every a ∈ a and s ∈ s, we have h ◦ i(a)(s) = h ◦ i(a) ◦ λs(1) = ψ(h ◦ i(a) ◦ λs) = ψ(f(a) ◦ λs) = f(a) ◦ λs(1) = f(a)(s). hence, h ◦ i = f, as we wanted. so cco(s,x) is an injective s-space. hence cco(s,x) is injective in tych. � similarly we have proposition 3.4. for a completely regular monoid s and a compact hausdorff space x, the cofree s-flow over x, β(cco(s,x)) is injective in the category of s-flows if and only if x is injective in the category of compact hausdorff spaces. as an immediate result of propositions 3.1 and 3.3, we have proposition 3.5. let s be a locally compact monoid. cco(s,x) is injective in tych if and only if cco(s,x) is injective in s-tych. on cofree s-spaces and cofree s-flows 9 proof. (⇐) since for any space z, t(z) is an s-space, the result is obvious. (⇒) suppose that cco(s,x) is injective in tych and we are given i : (a,τa) ↪→ (b,τb) and f : (a,τa) → cco(s,x) for two s-spaces (a,τa) and (b,τb). since cco(s,x) is injective in tych, there exists a continuous function g : (b,τb) → cco(s,x) such that g ◦ i = f. i (a,τa) ↪→ (b,τb) f ↓ ↙g cco(s,x) ψ ↓ x since cco(s,x) is the cofree s-space over x and ψ ◦ g : (b,τb) → x is continuous, there exists a continuous s-homomorphism h : (b,τb) → cco(s,x) such that ψ ◦ h = ψ ◦ g. clearly we have ψ ◦ h ◦ i = ψ ◦ g ◦ i. so, by the same argument as the proof of proposition 3.3, we have h ◦ i = f. hence cco(s,x) is injective in s-tych. � acknowledgements. the author would like to imply his gratitude to the referee and professor sanchis for their kindness and helps. references [1] r. n. ball, s. geschke and j. n. hagler, injective and projective t-boolean algebras, j. pure appl. algebra 209, no. 1 (2007), 1–36. 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(received november 2009 – accepted december 2011) behnam khosravi (behnam kho@yahoo.com) department of mathematics, institute for advanced studies in basic sciences, zanjan 45137-66731, iran on cofree s-spaces and cofree s-flows. by b. khosravi arbeagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 207-212 the diagonal of a first countable paratopological group, submetrizability, and related results a. v. arhangel’skii and a. bella abstract. we discuss some properties stronger than gδ -diagonal. among other things, we prove that any first countable paratopological group has a gδ-diagonal of infinite rank and hence also a regular gδdiagonal. this answer a question recently asked by arhangel’skii and burke. 2000 ams classification: 54b20, 54h13 keywords: gδ-diagonal of rank n, semitopological and paratopological groups, souslin number, submetrizability. a semitopological group is a group with a topology such that the multiplication is separately continuous. a paratopological group is a group with a topology such that the multiplication is jointly continuous. in the sequel, all spaces are assumed to be hausdorff. for notations and undefined notions we refer to [5]. the starting point of the present note was to answer problem 25 in [1]. in that paper the authors proved that a first-countable abelian paratopological group has a regulargδ-diagonal, raising the question of whether the abelianity of the group could be dropped. a topological space x has a regular gδ-diagonal if there exists a countable family {un : n < ω} of open subsets of x × x such that ∆(x) = ⋂ {un : n < ω}. here ∆(x) denotes the diagonal {(x, x) : x ∈ x} of x. the star of a collection u with respect to a set a is the set st(u, a) = ⋃ {u : u ∈ u and u ∩ a 6= ∅}. when a = {x}, we simply write st(u, x). we put st1(u, x) = st(u, x) and recursively define stn+1(u, x) = st(u, stn(u, x)). we say that a space x has a gδ-diagonal of rank n if there exists a countable collection {uk : k < ω} of open covers ofx such that ⋂ {stn(uk, x) : k < ω} = {x} for each x ∈ x. if a space has a gδ-diagonal of any possible rank, then we say that it has a gδ-diagonal of infinite rank. 208 a. v. arhangel’skii and a. bella zenor has pointed out in [9] that a gδ-diagonal of rank 3 is always regular. for the reader’s benefit, we provide here the simple proof. proposition 1. a topological space x with a gδ-diagonal of rank 3 has also a regular gδ-diagonal. proof. let {un : n < ω} be a sequence of open covers of x witnessing the rank 3 of the diagonal and put vn = ⋃ {u × u : u ∈ un}. we will check that ∆(x) = ⋂ {vn : n < ω}. let (x, y) ∈ x × x \ ∆(x) and choose n in such a way that y /∈ st3(un, x). then, select ux, uy ∈ un such that x ∈ ux and y ∈ uy. we must have ux × uy ∩ vn = ∅, otherwise there would exist some u ∈ un such that ux ∩ u 6= ∅ 6= uy ∩ u and this in turn would imply that y ∈ st3(un, x). � we would like to mention that the requirement for the diagonal to be an intersection of closed neighbourhoods implies a stronger separation axiom. proposition 2. if the diagonal of the space x is the intersection of a collection of closed neighbourhoods, then xsatisfies the urysohn separation axiom. proof. let x, y ∈ x and x 6= y. fix an open set u ⊆ x × x such that ∆(x) ⊆ u and (x, y) /∈ u . clearly, we may find open sets v, w ⊆ x such that (x, y) ∈ v × w ⊆ x × x \ u . let us assume that there exists a point z ∈ v ∩ w and fix an open set o ⊆ x such that (z, z) ∈ o × o ⊆ u . as we have v × w ∩ o × o 6= ∅, we reach a contradiction and so v ∩ w = ∅. � corollary 1. every space with a regular gδ-diagonal is a urysohn space. the usual ψ-space over the integers is a pseudocompact space with a gδdiagonal of rank 2 which does not have a regular gδ-diagonal. the latter fact maybe derived from a result of mc arthur [7], stating that any pseudocompact space with a regular gδ-diagonal is compact.at the moment, we do not know the answer to the following: problem 1. is every regular gδ diagonal always of rank 2? we are interested in the above problem also because a positive answer would allow us to deduce a recent remarkable result of buzyakova [4], stating that a ccc-space with a regular gδ-diagonal has cardinality at most c, from an old result of bella [3], saying the same for a gδ-diagonal of rank 2. if we assume a diagonal of higher rank, we can weaken the hypothesis on the cellularity in the last mentioned result. theorem 1. if x is a space with a gδ-diagonal of rank 4 and cellularity at most c, then |x| ≤ c. proof. let {un : n < ω} be a collection of open covers of x satisfying the formula ⋂ {st4(un, x) : n < ω} = {x} for each x ∈ x. let an ⊆ x be maximal with respect to the property thatst2(un, x) ∩ an = {x}. as the family {st(un, x) : x ∈ an} consists of pairwise disjoint sets, we have|an| ≤ c. moreover, {st2(un, x) : x ∈ an} is a cover of x. for any x ∈ x and any n < ω, the diagonal of a first countable paratopological group 209 choosexn ∈ an such that x ∈ st 2(un, xn) and let φ(x) = {st 2(un, xn) : n < ω}. since we are assuming that ⋂ {st4(un, x) : n < ω} = {x}, we have ⋂ φ(x) = {x} and so the map x 7→ φ(x) is one-to-one. now, an easy counting argument shows that |x| ≤ c. � as it is well-known [8], there are ccc-spaces withgδ -diagonal and arbitrarily large cardinality. nevertheless, the following question remains open. problem 2. is theorem 1 still true if we assume the diagonal to be of rank 2 or 3? now we move on to the case of a space with a group structure. let g be a semitopological group. a local π-base p at the neutral element e is called t-linked (here t stands for translation) if p x∩xp 6= ∅ for any p ∈ p and x ∈ g. this notion is instrumental to the following: lemma 1. let g be a semitopological group and p be a t-linked local π-base at the neutral element. then, for any p ∈ p, the collection u(p ) = {p x∩xp : x ∈ g} is a cover of g. proof. fix p ∈ p and x ∈ g. since p x ∩ xp 6= ∅, we may find p1, p2 ∈ p such thatp1x = xp2. observe that p p −1 1 x ∋ p1p −1 1 x = x and p −1 1 xp ∋ p −1 1 xp2 = p−11 p1x = x and so x ∈ p p −1 1 x ∩ p −1 1 xp . � lemma 2. if g is a paratopological group, then for any pair of distinct points y, z ∈ g and any integer n there exists a neighbourhood p of the neutral element e such thatp ny ∩ zp n = ∅. theorem 2. if g is a paratopological group with a countable t-linked local π-base at the neutral element, then g has a gδ-diagonal of infinite rank. proof. for any p ⊆ g and any x ∈ g we putp [x] = xp ∩ p x. let p be a countable t-linked local π-base at the neutral element e.for any p ∈ p, let u(p ) = {xp ∩ p x : x ∈ g} = {p [x] : x ∈ g}. by lemma 1, each u(p ) is a cover of g. we will show that the collection γ = {u(p ) : p ∈ p} witnesses the infinite rank of the diagonal. towards this end, fix an integer n and a pair of distinct points y, z ∈ g. we have to check that there is some p ∈ p such that ∗ z /∈ stn(u(p ), y) by lemma 2, we may fix p ∈ p in such a way that p ny∩zp n = ∅. the failure of (*) for this p means that there exist x1, . . . , xn−1 ∈ g such thaty ∈ p [x1], z ∈ p [xn−1] and p [xi]∩p [xi+1] 6= ∅ for 1 ≤ i ≤ n−1. the previous conditions are equivalent to the existence of point s p1, . . . , pn, q0, . . . , qn−1 ∈ p satisfying y = x1q0, z = pnxn and pixi = xi+1qi. from the equalityxi = p −1 i xi+1qi, we may easily arrive at the formula y = p−11 p −1 2 · · · p −1 n zqn−1qn−2 · · · q1q0. it follows that ypnpn−1 · · · p2p1y = zqn−1qn−2 · · · q1q0. hence, p ny ∩ zp n is nonempty, a contradiction. � 210 a. v. arhangel’skii and a. bella as a local base is always t-linked, we have: corollary 2. every first countable paratopological group has a gδ-diagonal of infinite rank. as a local π-base in an abelian group is clearly t-linked, we have: corollary 3. every abelian paratopological group of countable π-character has a gδ-diagonal of infinite rank. obviously, corollaries 2 and 3 suggest the following: problem 3. does a (hausdorff, regular, tychonoff ) paratopological group of countable π-character have a gδ-diagonal of rank n for each integer n? from corollary 2 and proposition 1 we immediately obtain the following recent result of c. liu [6], answering problem 25 in [1]: corollary 4. any first countable paratopological group has a regular gδ-diagonal. similarly, from corollary 3 and proposition 1 we obtain: corollary 5. every abelian paratopological group of countable π-character has a regular gδ-diagonal. corollary 6. any first countable paratopological group with cellularity at most c has cardinality at most c. observe that in any first countable paratopological group there is one countable family of open covers witnessing that g has a gδ-diagonal of rank n for each n < ω. we finish this note with a sufficient condition for the submetrizability of a semitopological group, which slightly generalizes theorem 28 in [1]. a semitopological group g is said to be ω -narrow if for every open neighbourhood u of the neutral element e of g there is a countable subset a ⊆ g such that au = g. theorem 3. every separable semitopological group is ω-narrow. proof. fix a countable subset a of g such that a−1 is dense in g, and let u be any open neighbourhood of the neutral element ein g.let us show that au = g. take any b ∈ g. there is an open setw such thatb−1 ∈ w and w b ⊆ u . since a−1 is dense in g, there is a ∈ a such that a−1 ∈ w . then a−1b ∈ u and hence, b ∈ au . thus, au = g. � lemma 3. suppose that g is a semitopological group, and that y, z are any two distinct elements of g. then there is an open neighbourhood u of the neutral element e in g such that for the family γu = {xu : x ∈ g} we have y /∈ st(z, γu ). proof. clearly, we may assume that z = e. since g is hausdorff, there is an open neighbourhood u of e such that u ∩ u y = ∅. then y /∈ u−1u .we now show that u is the neighbourhood we are looking for. indeed, take any x ∈ g such that e ∈ xu . then x ∈ u−1 and hence, xu ⊆ u−1u . it follows that y is not in xu . thus, y /∈ st(z, γu ). � the diagonal of a first countable paratopological group 211 theorem 4. suppose that g is a tychonoff ω-narrow semitopological group of countable π-character. then the space g is submetrizable. proof. let p be a countable local π-base at the neutral element e. since the space g is tychonoff, we may assume that every u ∈ p is a cozero-set. for each u ∈ p, put γu = {xu : x ∈ g}. since g is ω-narrow, there is a countable subcover ηu ⊆ γu of g. put b = ∪{ηu : u ∈ p}. then b is a countable family of cozero-sets in g. the family b is t1-separating. indeed, fix any distinct y and z in g. by lemma 2, there is u ∈ p such that for γu = {xu : x ∈ g} we have y /∈ st(z, γu ). by the choice of ηu , there is v ∈ ηu such that z ∈ v . then y /∈ v ∈ b. thus, the family b is t1-separating.it remains to make a standard step: for every v ∈ b fix a continuous real-valued function fv on g such that v = {x ∈ g : fv (x) 6= 0}, and take the diagonal product of functions fv where v runs over the countable set b. the resulting function is the desired one-to-one continuous mapping of the space g onto a separable metrizable space. � problem 4. is every (hausdorff, regular) semitopological (paratopological) group with countable souslin number ω-narrow? problem 5. let g be a paratopological (semitopological) (hausdorff, regular) group of countable extent. must g be ω-narrow? problem 6. let g be a paratopological (semitopological) (hausdorff, regular) group of countable extent. must g be submetrizable in connection with the last open question, we have the following partial result: theorem 5. if g is a first countable normal (weakly m -normal) paratopological group of the countable extent, then g can be condensed onto a separable metrizable space (hence, g is submetrizable). proof. indeed, by theorem 2, g has a rank 5-diagonal. besides, g is starlindelöf. it remains to apply a result from [2]. � clearly, “countable extent” can be replaced by “star-lindelöf” in theorem 5. another open problem, already formulated in [1], is whether every first countable regular paratopological group is submetrizable. in connection with problem 4, observe that if the answer is “yes”, then every tychonoff first countable paratopological group with countable souslin number is submetrizable. 212 a. v. arhangel’skii and a. bella references [1] a. v. arhangel’skii and d. burke, spaces with regular gδ-diagonal, topology appl. 153, no. 11 (2006), 1917–1929. [2] a. v. arhangel’skii and r. buzyakova, the rank of the diagonal and submetrizability, comment. math. univ. carolinae 47, no. 4 (2006), 585–597. [3] a. bella, more on cellular extent and related cardinal functions, boll. un. mat. ital. 7, no. 3a (1989), 61–68. [4] r. z. buzyakova, cardinalities of ccc-spaces with regular gδ-diagonal, topology appl. 153, no. 11 (2006), 1696–1698. [5] r. engelking, general topology, 1977. [6] c. liu, a note on paratopological groups, comment. math. univ. carolinae 47, no. 4 (2006), 633–640. [7] v. mc arthur, gδ-diagonal and metrization theorems, pacific j. math. 44 (1973), 213–217. [8] v. v. uspenskii, large fσ-discrete spaces having the souslin property, comment. math. univ. carolinae 25, no. 2 (1984), 257–260. [9] p. zenor, on spaces with regular gδ-diagonal, pacific j. math. 40 (1972), 959–963. received march 2006 accepted may 2006 a. v. arhangelskii (arhangel@math.ohiou.edu) department of mathematics, 321 morton hall, ohio university, athens, ohio 45701, usa. a. bella (bella@dmi.unict.it) dipartimento di matematica, cittá universitaria, viale a.doria 6, 95125, catania, italy. @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 33-38 on density and π-weight of lp(βn, r, μ) giuseppe iurato abstract the new topological concept of selective separability is able to give some estimates for density and π-weight of the lebesgue space lp(βn, r, μ) with 1 ≤ p < +∞. in particular, we deduce a purely topological proof of the non-separability of such a space. 2010 msc: 28a33, 28b20, 54c65, 54d65 keywords: separability, selective separability, lebesgue space in integration theory, it is important to establish the separability or not of lebesgue spaces of the type lp, with 1 ≤ p < +∞. in general, the usual proof of this type of results for certain lebesgue spaces, is conducted through methods of real analysis. in this work, we use some concepts and methods of pure general topology in proving the non-separability of a particular lebesgue space. further, we provide some estimates for density and π-weight of such a space. 1. introduction in the context of infinite-combinatorial topology, m. scheepers (see [14]) has introduced and studied a particular selection principle sfin(d,d) connected with some problems of topological diagonalization of covers of a given completely regular topological space x, whose dense set family is d. subsequently, a. bella and coworkers (see [3]) have extended and generalized some formal properties introduced by m. scheepers in [14], into a pure topological context, proposing the notion of selective separability, defined as follows. a completely regular topological space (or a t3 1 2 -space), say (x,τ), is said to be selectively separable (or m-separable) if, for any sequence {dn;n ∈ n} of dense sets of x, there exists a sequence {fn;n ∈ n} of finite subsets of x such that fn ⊆ dn for each n ∈ n and ⋃ n∈n fn is dense in x. 34 g. iurato a selectively separable space is a separable space too. in fact, if we put dn = x for any n ∈ n, then there exists a finite subset fn ⊆ x for each n ∈ n such that ⋃ n∈n fn = x, with ⋃ n∈n fn countable. there exist however separable spaces that are not selectively separable, as, for example, the tychonov cube ic of weight c = 2ℵ0, when i = [0,1] is equipped with the usual euclidean topology. one of the purposes of this note is that of discussing some elementary properties of selectively separable spaces in view of their applications. 2. preliminary concepts given an arbitrary topological space (x,τ), we put τ∗ = τ \ {∅}. a base of x is a family (∅ �=)b ⊆ τ∗ such that every element of τ∗ is the union of elements of b. a π-base of x is a family (∅ �=)b ⊆ τ∗ such that, for every a ∈ τ∗, there is a b ∈ b such that b ⊆ a. any base of x is also a π-base, but the inverse, in general, is not true. the minimal cardinality of a base [π-base] of x, is called the weight [π-weight] of x, say w(x) [πw(x)]. the property πw(x) ≤ w(x) holds true; moreover, if βn is the čech-stone compactification of the discrete space n, then it is possible to prove that πw(βn) = ℵ0 < c = w(βn). if d(x) is the density of x, it is immediate to prove that d(x) ≤ πw(x). in fact, if b is a π-base of x such that card b = πw(x), and a is a dense subset of x, then we have a ∩ b �= ∅ for any b ∈ b. in addition, if we choose a xb ∈ a ∩ b for every b ∈ b, then the set {xb;b ∈ b} is dense in x, so that d(x) ≤ card {xb;b ∈ b} ≤ card b = πw(x). therefore, we have the following relation between cardinal functions (1) d(x) ≤ πw(x) ≤ w(x). if (x,τ) is a hausdorff topological space, then it is known that it has an infinite disjoint base, say b. it follows that ℵ0 ≤ π(x), because, if b̃ is any π-base of x, then, by the definition of π-base, there exists a ab ∈ b̃, for each b ∈ b, such that ab ⊆ b, for which card b̃ ≥ ℵ0 being b an infinite disjoint family. since b̃ is an arbitrary π-base of x, it follows that πw(x) ≥ ℵ0. in particular, if (x,τ) is a metrizable space, then it follows that d(x) = πw(x) = w(x) ≥ ℵ0 (see [11], section 6, remark 6.2, and [7]). the space (x,τ) has countable fan tightness if, for any x ∈ x and any sequence {an;n ∈ n} of subsets of x such that x ∈ ⋂ n∈n an, there exists a finite set bn ⊆ an, for each n ∈ n, such that x ∈ ⋃ n∈n bn. in this case, we write concisely vet (x) ≤ ℵ0. the space (x,τ) has countable tightness if, for any x ∈ x and a ⊆ x with x ∈ a, there exists a countable set b ⊆ a such that x ∈ b. in this case, we write concisely t(x) ≤ ℵ0. we have (vet(x) ≤ ℵ0) ⇒ (t(x) ≤ ℵ0), but the inverse is, in general, not true. on density and π-weight of lp(βn, r, μ) 35 3. selective separability: basic properties we here recall some properties of selectively separable spaces. theorem 3.1. every dense subspace of a selectively separable space is also selectively separable. proof. if s is a dense subspace of a selectively separable space (x,τ) and {dn;n ∈ n} is a sequence of dense subsets of (s,τs), it follows that each dn is also dense in x (as a consequence of the density of s in x), so that, for every n ∈ n, there exists a finite set fn ⊆ dn such that ⋃ n∈n fn x = x. hence, we have ⋃ n∈n fn s = ⋃ n∈n fn x ∩ s = x ∩ s = s. � theorem 3.2. a topological space (x,τ) has the property that (1) if πw(x) = ℵ0, then x is selectively separable, (2) if x is separable and vet (x) ≤ ℵ0, then x is selectively separable. proof. if πw(x) = ℵ0, let b = {bn;n ∈ n} be a π-base of x. if {dn;n ∈ n} is a sequence of dense subsets of x, then we have dn ∩ bm �= ∅ for all n,m ∈ n, so that we can choose a point xn ∈ dn ∩bn for each n ∈ n. hence, {xn;n ∈ n} is dense in x, and thus x is selectively separable. this proves (1). if vet (x) ≤ ℵ0, and x is separable, let {an;n ∈ n} be a countable dense subset of x, and let {dn;n ∈ n} be a sequence of dense subsets of x. since dk = x = {an;n ∈ n0} for each k ∈ n, we have, for each n ∈ n, an ∈ dk for infinitely many k ∈ n, so that an ∈ dk ∀k ∈ ln, with ln infinite subset of n. therefore, we may always choose a disjoint family {ln;n ∈ n} of infinite subsets of n in such a way that1 n = ⋃ n∈n ln. hence an ∈ ⋂ k∈ln dk for every n ∈ n, and since x has countable fan tightness, it follows that there exists a finite set fk ⊆ dk, for each k ∈ ln, such that an ∈ ⋃ k∈ln fk, whence we have {an;n ∈ n} ⊆ ⋃ n∈n ( ⋃ k∈ln fk ) ⊆ ⋃ n∈n ⋃ k∈ln fk, so that( x = {an;n ∈ n} ⊆ ⋃ n∈n ⋃ k∈ln fk ⊆ x ) ⇒ ( x = ⋃ n∈n ⋃ k∈ln fk ) . the countability of ⋃ n∈n ⋃ k∈ln fk proves the selective separability of x. this proves (2). � remark 3.3. in general, the proposition (1) of theorem 3.2, is not valid if we suppose πw(x) < ℵ0. moreover, since w(x) = ℵ0 for every separable 1for instance, if {pm; m ∈ n} is the infinite sequence of the prime numbers of n, then ln = {pnm; m ∈ n} ∀n ∈ n, verifies as required. 36 g. iurato metrizable spaces x, from what has been said in section 2, it follows that w(x) = πw(x) = d(x) = ℵ0 for such spaces2. if x is a completely regular topological space, and cp(x) is the space of continuous functions f : x → r, equipped with the pointwise topology (see [6], section 2.6.), then it is possible to prove the following theorem 3.4. cp(x) is selectively separable if and only if x is separable and vet (x) = ℵ0. the proof is given in [3]. nevertheless, the study of c(x) from this viewpoint has been only done with respect to the pointwise topology (see [2]). 4. the lebesgue space lp(βn,r,μ) following [15], counterexamples 110 and 111, if we endow n with the discrete topology, said βn the čech-stone compactification of the discrete space n, then βn is a compact hausdorff space. let us recall that ℵ0 = πw(βn) < w(βn) = c. the definition of lp(βn,r,μ) can be given in three ways3 as follows. 1. following [10], chap. 7, and4 [4], chapter ix, section 5, let μ be a prescribed radon measure on βn, so that we can consider the lebesgue space lp(βn,r,μ) of the lp-sommable functions f : βn → r, with 1 ≤ p < +∞. it is known that the space of functions with compact support c0(βn,r) is dense in lp(βn,r,μ), and since βn is compact, we have that c0(βn,r) = c(βn,r) is dense too in lp(βn,r,μ). 2. taking into account that βn is a hausdorff compact space, let μ be a positive borel measure on βn (see [13], chap. 3, and [9], chap. 10), defined over the borel σ-algebra generated by the topology of βn, and having a countable base (see [9], section 37.1, and [8], chap. iii, section 3, problems 418 420). hence, reasoning as in 1., it follows that c(βn,r) is dense in lp(βn,r,μ). 3. let x be a compact space with a countable base (w(x) = ℵ0), and let μ be a baire measure on x. then lp(x,r,μ) is a separable space for each 1 ≤ p < +∞. indeed (see [12], chap. iv, section 4, and problem 43 (a)), if {bn;n ∈ n} is a countable base of x, for all n,m ∈ n, n �= m, 2from here, it follows that the concepts of separability and selective separability are the same, in the case of metric spaces. 3there exists a fourth way, based on integration theory over separable measure spaces (see [5], and [16], chap. 7, section 5), leading to the same results. 4section 5 of n. bourbaki’s chapter ix, deals with measures on a completely regular space. these results can be applied to βn since this is a hausdorff compact space, and hence a completely regular space (see [6], section 3.3.). on density and π-weight of lp(βn, r, μ) 37 with bn ∩ bm = ∅, it is possible to define fn,m ∈ c(x,r) in such a way that fn,m = 0 on bn and fn,m = 1 on bm. hence, the fn,m can be used to define a countable dense set in c(x,r), whence a countable dense set in lp(x,r,μ), being c(x,r) dense in lp(x,r,μ). nevertheless, this proof is no longer valid when πw(x) = ℵ0, hence for x = βn since ℵ0 = πw(βn) < w(βn) = c, so that we cannot say that such a lebesgue space is separable, but only that it has a dense subset (namely c(x,r)). on the other hand, it is well-known too as the banach space (respect to the supremum norm) of all real bounded sequences, namely l∞(n,r), is not separable as metric space. since each f ∈ l∞(n,r) is continuous and bounded, by the universal properties of βn (see [6], section 3.6.), it is possible, in a unique manner5, to extend it to a continuous function ψ(f) : βn → r. it is possible to prove (see [1], sections 2.17 and 2.18) that the correspondence f → ψ(f) is an isometric isomorphism from l∞(n,r) to c(βn,r), hence a homeomorphism. it follows that c(βn,r) is not separable, but dense in lp(βn,r,μ), with 1 ≤ p < +∞. finally, it also follows that the space lp(βn,r,μ) cannot be separable: indeed, as metric space, taking into account what has been said in section 2 and in remark 3.3, we have the following density and π-weight estimates for such a space (2) ℵ0 < d(lp(βn,r,μ)) = πw(lp(βn,r,μ)) = w(lp(βn,r,μ)). hence, if lp(βn,r,μ) were separable, then we would have w(lp(βn,r,μ)) = ℵ0 (as separable metric space), hence πw(lp(βn,r,μ)) = ℵ0, so that, by (1) of theorem 3.2, lp(βn,r,μ) would be selectively separable, whence it would follow that every dense subspace of it would be selectively separable too (by theorem 3.1), and this is impossible since c(βn,r) is a no separable dense subspace of lp(βn,r,μ), with 1 ≤ p < +∞. following the lines of this paper, it would be of a certain interest to analyze the possible role played by the various notions of tightness, fan tightness (see [2] for the topological function spaces case) and others properties of selectively separable spaces, in the case of a generic lebesgue space lp(x,r,μ), with 1 ≤ p < +∞. 5recalling, on the other hand, that n is dense in βn. 38 g. iurato references [1] c. d. aliprantis and k. c. border, infinite dimensional analysis, springer-verlag, berlin, 2006. [2] a.v. arkhangel’skii, topological function spaces, kluwer academic publishers, dordrecht, 1992. [3] a. bella, m. bonanzinga, m. v. matveev and v. v. tkachuck, selective separability: general facts and behaviour in countable spaces, topology proceedings 32 (2008), 15–30. [4] n. bourbaki, intégration, chapitres i-ix, hermann, paris, 1952–1969. [5] n. dunford and j. t. schwartz, linear operators, part i, john wiley and sons, new york, 1976. [6] r. engelking, general topology, polish scientific publishers, warszawa, 1977. [7] i. juhász, cardinal functions in topology ten years later, mathematisch centrum, amsterdam, 1983. [8] a. a. kirillov and a. d. gvishiani, theorems and problems in functional analysis, springer-verlag, new york-berlin, 1982 (italian edition: a. a. kirillov, a. d. gviv̌iani, teoremi e problemi dell’analisi funzionale, edizioni mir, mosca, 1983). [9] a. n. kolmogorov and s. v. fomin, introductory real analysis, dover publications, inc., new york, 1970. [10] g. letta, teoria elementare dell’integrazione, boringhieri, torino, 1976. [11] v. i. ponomarev, on spaces co-absolute with metric spaces, russian math. surveys 21, no. 4 (1966), 87–114. [12] m. reed and b. simon, methods of modern mathematical physics, vol. i, revised and enlarged edition, academic press, new york, 1980. [13] w. rudin, real and complex analysis, macgraw-hill, ljubljana, 1970. [14] m. scheepers, combinatorics of open covers vi: selectors for sequences of dense sets, quaest. math. 22 (1999), 109–130. [15] l. a. steen and j. a. seebach, counterexamples in topology, holt, rinehart and winston, inc., new york, 1970. [16] a. tesei, istituzioni di analisi superiore, bollati boringhieri, torino, 1997. (received july 2010 – accepted august 2011) giuseppe iurato (giuseppe.iurato@unipa.it) department of physics and related technologies, university of palermo, avenue of science, building 18, i-90128, palermo, italy on density and -weight of lp(n,r,). by g. iurato gansteagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 243-247 on bτ -closed sets maximilian ganster and markus steiner abstract. this paper is closely related to the work of cao, greenwood and reilly in [10] as it expands and completes their fundamental diagram by considering b-closed sets. in addition, we correct a wrong assertion in [10] about βgs-spaces. 2000 ams classification: 54a05 keywords: b-closed, qr-closed 1. introduction and preliminaries in recent years quite a number of generalizations of closed sets has been considered in the literature. we recall the following definitions: definition 1.1. let (x, τ ) be a topological space. a subset a ⊆ x is called (1) α-closed [18] if cl(int(cl(a))) ⊆ a, (2) semi-closed [15] if int(cl(a)) ⊆ a, (3) preclosed [17] if cl(int(a)) ⊆ a, (4) b-closed [3] if int(cl(a)) ∩ cl(int(a)) ⊆ a, (5) semi-preclosed [2] or β-closed [1] if int(cl(int(a))) ⊆ a. the complement of an α-closed (resp. semi-closed, preclosed, b-closed, βclosed) set is called α-open (resp. semi-open, preopen, b-open, β-open). the smallest α-closed (resp. semi-closed, preclosed, b-closed, β-closed) set containing a ⊆ x is called the α-closure (resp. semi-closure, preclosure, b-closure, β-closure) of a and shall be denoted by clα(a) (resp. cls(a), clp(a), clb(a), clβ(a)). in 2001, cao, greenwood and reilly [10] introduced the concept of qr-closed sets to deal with various notions of generalized closed sets that had been considered in the literature so far. if p = {τ, α, s, p, β} and q, r ∈ p then a subset a ⊆ x is called qr-closed if clq(a) ⊆ u whenever a ⊆ u and u is r-open. (for convenience we denote cl(a) by clτ (a) and open (resp. semi-open, preopen) by τ -open (resp. s-open, p-open).) 244 m. ganster and m. steiner in the following we shall consider the expanded family p ∪{b}. as in corollary 2.6 of [10], it is easily established that the concept of a bτ -closed set yields the only new type of sets that can be gained by utilizing the b-closure (resp. the b-interior) in the context of qr-closed sets. thus we give definition 1.2. let (x, τ ) be a topological space. a subset a ⊆ x is called bτ -closed if clb(a) ⊆ u whenever a ⊆ u and u is open. the complement of a bτ -closed set is called bτ -open. remark 1.3. the concepts of ss-closed (resp. sτ -closed, pτ -closed, βτ -closed) sets have been first introduced in the literature under the name of sg-closed [5] (resp. gs-closed [4], gp-closed [16], gsp-closed [11]) sets. we also consider the following classes of topological spaces: definition 1.4. a topological space (x, τ ) is called (1) sg-submaximal if every codense subset of (x, τ ) is ss-closed, (2) tgs if every sτ -closed subset of (x, τ ) is ss-closed, (3) extremally disconnected if the closure of each open subset of (x, τ ) is open, (4) resolvable if (x, τ ) is the union of two disjoint dense subsets. for undefined concepts we refer the reader to [10] and [9] and the references given there. 2. bτ-closed sets and their relationships in [10] the relationships between various types of generalized closed sets have been summarized in a diagram. we shall expand this diagram by adding b-closed sets and bτ -closed sets. proposition 2.1. every ss-closed set in a topological space (x, τ ) is b-closed. proof. let a ⊆ x be ss-closed and let x ∈ clb(a). since singletons are either preopen or nowhere dense (see [14]), we distinguish two cases. if {x} is preopen, it is also b-open and hence {x} ∩ a 6= ∅, i.e. x ∈ a. if {x} is nowhere dense, then x \ {x} is semi-open. suppose that x /∈ a. then a ⊆ x \ {x} and, since a is ss-closed, we have clb(a) ⊆ cls(a) ⊆ x \ {x}. hence x /∈ clb(a), a contradiction. therefore clb(a) ⊆ a, and so a is b-closed. � the remaining relationships in the following diagram can easily be established. on bτ -closed sets 245 ss-closed �� &&m m mm m mm mm mm preclosed xxqq qq qq qq qq q &&m mm mm mm mm mm b-closed �� &&n nn nn nn nn nn sτ -closed �� pτ -closed xxpp pp pp pp pp p bτ -closed �� β-closed // βτ -closed we now address the question of when the above implications can be reversed. proposition 2.2. let (x, τ ) be a topological space. then: (1) each b-closed set is ss-closed iff (x, τ ) is sg-submaximal. (2) each b-closed set is sτ -closed iff (x, τ ) is sg-submaximal. (3) each sτ -closed set is b-closed iff (x, τ ) is tgs. (4) each bτ -closed set is b-closed iff (x, τ ) is tgs. (5) each bτ -closed set is sτ -closed iff (x, τ ) is sg-submaximal. (6) each bτ -closed set is pτ -closed iff (x, τ ) is extremally disconnected. (7) each bτ -closed set is β-closed iff (x, τ ) is tgs. (8) each β-closed set is b-closed iff cl(w ) is open for every open resolvable subspace w of (x, τ ). proof. we will only show (1). the other assertions can be proved in a similar manner using the standard methods that can be found in [10], and (8) has been shown in [13]. first recall that a space is sg-submaximal iff every preclosed set is ss-closed (see [7]). if every b-closed set is ss-closed then every preclosed set ss-closed, i.e. (x, τ ) is sg-submaximal. conversely, suppose that (x, τ ) is sg-submaximal and let a be b-closed. then a is the intersection of a semi-closed and a preclosed set (see [3]). since every semi-closed set is ss-closed, by hypothesis, a is the intersection of two ssclosed sets. since the arbitrary intersection of ss-closed sets is always ss-closed (see [12]), we conclude that a is ss-closed. � proposition 2.3. let (x, τ ) be a topological space. then the following statements are equivalent: (1) each βτ -closed set is bτ -closed. (2) each β-closed set is bτ -closed. proof. the necessity is clear, so we only have to show the sufficiency. let a be a βτ -closed set and u be an open subset of x such that a ⊆ u . since 246 m. ganster and m. steiner a is βτ -closed we have clβ (a) ⊆ u . now, clβ (a) is β-closed and hence bτ closed by hypothesis. therfore clb(a) ⊆ clb(clβ (a)) ⊆ u and thus our claim is proved. � remark 2.4. if a ⊆ x, then the largest b-open subset of a is called the b-interior of a and is denoted by bint(a). it is well known that bint(a) = (cl(int(a)) ∪ int(cl(a))) ∩ a (see [3]). consequently, a subset a is bτ -open iff for every closed subset f satisfying f ⊆ a we have f ⊆ cl(int(a)) ∪ int(cl(a)). we shall now present one of our major results. theorem 2.5. let (x, τ ) be a topological space. then the following are equivalent: (1) each β-closed set is b-closed. (2) each β-closed set is bτ -closed. (3) cl(w ) is open for every open resolvable subspace w of (x, τ ). proof. it is obvious that (1) ⇒ (2). furthermore, it has been shown in [13] that (3) ⇔ (1), so we only have to prove that (2) ⇒ (3). if w = ∅, we are done, so let w be a nonempty open resolvable subspace and let e1 and e2 be disjoint dense subsets of (w, τ|w ). suppose that there exists a point x ∈ cl(w ) \ int(cl(w )). let s = e1 ∪ cl({x}). it is easily checked that int(s) = ∅, cl(s) = cl(w ) and that s is β-open. by hypothesis, s is bτ open and so, since cl({x}) ⊆ s, we conclude that {x} ⊆ cl({x}) ⊆ int(cl(s)) = int(cl(w )). this is, however, a contradiction to our assumption and so cl(w ) has to be open. � 3. a remark on βgs-spaces in [10] a space has been called βgs if every βτ -closed subset of (x, τ ) is sτ closed. we observe that this is equivalent to the property that every β-closed subset is sτ -closed, see [6]. using some of our previous results, we are now able to give the following characterization. theorem 3.1. let (x, τ ) be a topological space. then the following are equivalent: (1) (x, τ ) is a βgs-space. (2) every β-closed set is b-closed and (x, τ ) is sg-submaximal. (3) every β-closed set is ss-closed. note that the last property in the above theorem has been fully characterized in [8]. it was shown there that every β-closed set is ss-closed iff (x, τ α) is gsubmaximal, where τ α denotes the α-topology of (x, τ ) (see [18]), and a space is called g-submaximal if each codense set is τ τ -closed. so the claim in [10] that there exists a βgs-space whose α-topology is not g-submaximal turns out to be wrong now. in fact, one can easily check that example 3.5 of [10] is false. so βgs is not a new topological property as we have just seen. on bτ -closed sets 247 references [1] m. e. abd el-monsef, s.n. el-deeb, and r.a. mahmoud, β-open sets and β-continuous mappings, bull. fac. sci. assiut univ. 12 (1983), 77–90. [2] d. andrijević, semi-preopen sets, mat. vesnik 38 (1986), 24–32. [3] d. andrijević, on b-open sets, mat. vesnik 48 (1996), 69–64. 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[18] o. nj̊astad, on some classes of nearly open sets, pacific j. math. 15 (1965), 961–970. received march 2006 accepted july 2006 m. ganster (ganster@weyl.math.tu-graz.ac.at) department of mathematics, graz university of technology, steyrergasse 30, a-8010 graz, austria m. steiner (msteiner@sbox.tugraz.at) department of mathematics, graz university of technology, steyrergasse 30, a-8010 graz, austria kareagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 301-310 on some variations of multifunction continuity a. kanibir and i. l. reilly abstract. this paper considers six classes of multifunctions between topological spaces, namely almost ℓ−continuous multifunctions, k-almost c−continuous multifunctions, nearly continuous multifunctions, almost nearly continuous multifunctions, super continuous multifunctions, and δ−continuous multifunctions. we relate these classes of multifunctions to others, and provide characterizations of related concepts especially in terms of appropriate changes of topology. 2000 ams classification: 54a10, 54c05, 54c60, 54c10 keywords: almost ℓ−continuity, k-almost c−continuity, semi continuiy, super continuity, δ−continuity, nearly continuity, change of topology, multifunction. 1. introduction and preliminaries there are several notions of generalized continuity for multifunctions in the recent literature, for example almost continuity and ℓ−continuity [17], ccontinuity [10] and [17], almost c-continuity [14], nearly continuity [5], almost nearly continuity [6], super continuity [1], δ−continuity [2] and [15]. in this paper we consider each of the following continuity properties of multifunctionsalmost ℓ−continuity, k-almost c−continuity, nearly continuity, almost nearly continuity, δ−continuity, and super continuity. throughout this paper, the closure (resp. interior) of a subset b in a topological space (y, τ ) is denoted by clb (resp. intb). then b is called regular open if b = int(clb). the family of all regular open sets in (y, τ ) which is denoted by ro(y, τ ), forms a base for a topology τs on y , known as the semiregularization of τ . in general τs ⊆ τ , and if τs = τ then (y, τ ) is called a semiregular space. a set a is said to be δ−closed [18] if for each point x /∈ a there exists a regular open set containing x which has empty intersection with a. a set b is δ−open [18] if and only if its complement is δ−closed. for any topological space (y, τ ) the collection of all δ−open sets forms a topology on x which coincides with τs . for a topological space (y, τ ), the cocompact topology 302 a. kanibir and i. l. reilly of τ on y is denoted by c(τ ) and defined by c(τ ) = {∅} ∪ {u ∈ τ : y − u is τ−compact}. the almost cocompact topology of τ on y is denoted by e(τ ) and it has as a base e′(τ ) = {u ∈ ro(y, τ ) : y − u is τ−compact}. these topologies are considered by gauld [7] and [8]. if (y, τ ) is a topological space then the colindelöf topology of τ on y is denoted by ℓ(τ ) and defined by ℓ(τ ) = {∅} ∪ {u ∈ τ : y − u is τ−lindelöf}, considered by gauld, mrsevic, reilly and vamanamurthy [9]. recall that a subset b of a topological space (y, τ ) is called n-closed in x if every regular open cover of b in (y, τ ) has a finite subcover. the concept of n-closed subsets was first considered by carnahan [3]. if (y, τ ) is a topological space then the almost con-closed topology of τ on y is denoted by p(τ ) and it has as a base p′(τ ) = {u ∈ ro(y, τ ) : y − u is n-closed relative to τ}. the almost colindelöf topology of τ on y is denoted by q(τ ) and it has as a base q′(τ ) = {u ∈ ro(y, τ ) : y − u is τ−lindelöf}. these topologies are considered by konstadilaki-savvopoulou and reilly [12] and [13]. by a multifunction f : (x, σ) → (y, τ ), we mean a point-to-set correspondence from (x, σ) into (y, τ ), and we always assume that f (x) 6= ∅ for all x ∈ x. for each b ⊆ y , f +(b) = {x ∈ x : f (x) ⊆ b} and f −(b) = {x ∈ x : f (x) ∩ b 6= ∅}. for each a ⊆ x, f (a) = ∪x∈af (x). as usual, f is said to be a surjection if f (x) = y . moreover f : (x, σ) → (y, τ ) is called upper semicontinuous, abbreviated as u.s.c. (resp. lower semicontinuous, abbreviated as l.s.c.) at a point x ∈ x if for each open set v in y with f (x) ⊆ v ( resp. f (x) ∩ v 6= ∅ ), there exists an open set u containing x such that f (u ) ⊆ v (resp. f (z)∩v 6= ∅ for every z ∈ u ), or equivalently, if f +(v ) (resp. f −(v )) is open in (x, σ) for every open set v of (y, τ ). a subset k ⊆ f (x0) is said to be a kernel [4] for f at x0, if the multifunction fk : x → y defined by: fk (x) = k , if x = x0 fk (x) = f (x) ∩ (y − f (x0)) , otherwise is u.s.c. at x0. for a multifunction f : x → y , the graph multifunction gf : x → x × y is defined as gf (x) = {x} × f (x) for every x ∈ x. this paper discusses how several versions of multifunction continuity behave with respect to a change of topology approach. the results presented here show that, in general, ”lower” multifunction continuities behave well from this point of view. however, ”upper” multifunction continuities do not behave at all well from this perspective. this behaviour of multifunctions can not be predicted from a consideration of corresponding behaviour of (single-valued) functions. it is unexpected, and a somewhat surprising situation. 2. some properties the following basic properties of almost ℓ−continuity and k-almost c−continuity are useful in the sequel: on some variations of multifunction continuity 303 definition 2.1 ([11]). a multifunction f : x → y is called (a) upper almost ℓ−continuous (resp. upper k-almost c−continuous), or u.a.ℓ−continuous (resp. u.k-a.c−continuous), at x ∈ x if for each regular open subset v of y with f (x) ⊆ v and having lindelöf (resp. compact) complement, there is an open neighbourhood u of x such that f (u ) ⊆ v. (b) lower almost ℓ−continuous (resp. lower k-almost c−continuous), or l.a.ℓ−continuous (resp. l.k-a.c−continuous), at x ∈ x if for each regular open subset v of y with f (x) ∩ v 6= ∅ and having lindelöf (resp. compact) complement, there is an open neighbourhood u of x such that f (z) ∩ v 6= ∅ for every point z ∈ u . (c) almost ℓ−continuous (resp. k-almost c−continuous) at x ∈ x if it is both u.a.ℓ−continuous (resp.u.k-a.c−continuous) and l.a.ℓ−continuous (resp. l.k-a.c−continuous) at x ∈ x. (d) almost ℓ−continuous (resp. u.a.ℓ−continuous ; l.a.ℓ−continuous, ka.c−continuous, u.k-a.c−continuous, l.k-a.c−continuous) if it is almost ℓ−continuous (resp. u.a.ℓ−continuous; l.a.ℓ−continuous, k-a.c− continuous, u.k-a.c−continuous, l.k-a.c−continuous) at each point of x. theorem 2.2 ([11]). let f : (x, σ) → (y, τ ) be a multifunction. then (a) f is upper (resp. lower) almost ℓ−continuous if and only if f +(v ) (resp. f −(v )) is open for each v ∈ q′(τ ). (b) f is upper (resp. lower) k-almost c−continuous if and only if f +(v ) (resp. f −(v )) is open for each v ∈ e′(τ ). theorem 2.3. let f : (x, σ) → (y, τ ) be a multifunction and x0 ∈ x. then f is u.s.c. at x0 if and only if f (x0) is a kernel for f at x0. proof. (⇒) let f : (x, σ) → (y, τ ) be u.s.c. at x0 and k = f (x0). we will show that fk : (x, σ) → (y, τ ) is u.s.c. at x0. let fk (x0) ⊆ v where v ∈ τ. since fk (x0) = f (x0) and f is u.s.c. at x0, there exists an open set u containing x0 such that f (u ) ⊆ v. therefore fk (u ) ⊆ v and so fk is u.s.c. at x0 and hence f (x0) is a kernel for f at x0. (⇐) let k = f (x0) be a kernel for f at x0 and f (x0) ⊆ v where v ∈ τ. then fk (x0) = f (x0) ⊆ v. since fk is u.s.c. at x0, there exists an open set u containing x0 such that fk (u ) ⊆ v. therefore f (u ) ⊆ v since f (x0) ⊆ v and hence f is u.s.c. at x0. � theorem 2.4. let f : (x, σ) → (y, τ ) be a multifunction and f (x0) ∈ q ′(τ ) (resp. f (x0) ∈ e ′(τ )). then f is upper almost ℓ−continuous (resp. upper k-almost c−continuous ) at x0 ∈ x if and only if f (x0) is a kernel for f at x0. proof. (⇒) let k = f (x0) ∈ q ′(τ ) and fk (x0) = f (x0). we will show that fk : (x, σ) → (y, τ ) is u.s.c. at x0. let fk (x0) ⊆ v where v ∈ τ. 304 a. kanibir and i. l. reilly since fk (x0) ∈ q ′(τ ) and f is u.a.ℓ−continuous multifunction, there exists an open set u containing x0 such that f (u ) ⊆ fk (x0) = f (x0) ⊆ v. therefore fk (u ) ⊆ v and hence fk is u.s.c at x0 ∈ x. this shows that f (x0) is a kernel for f at x0. (⇐) it follows from theorem 2.3 and the fact that every upper semi continuous multifunction is upper almost ℓ−continuous [11]. the proof for upper k-almost c−continuity is similar. � theorem 2.5 ([16]). a subset a of (y, τ ) is n-closed with respect to τ if and only if a is compact in (y, τs ). definition 2.6. a multifunction f : (x, σ) → (y, τ ) is called (a) upper (resp. lower) δ−continuous [2], or u.δ.c (resp. l.δ.c), if for each x ∈ x and for each open subset v of y with f (x) ⊆ v (resp. f (x) ∩ v 6= ∅), there is an open neighbourhood u of x such that f (int(clu )) ⊆ int(clv ) (resp. f (z) ∩ int(clv ) 6= ∅ for every point z ∈ int(clu )). (b) upper (resp. lower) super continuous [1], or u. sup .c (resp. l. sup .c), if for each x ∈ x and for each open subset v of y with f (x) ⊆ v (resp. f (x) ∩ v 6= ∅), there is an open set u containing x such that f (int(clu )) ⊆ v (resp. f (z) ∩ v 6= ∅ for every point z ∈ int(clu )). (c) upper (resp. lower) c−continuous [10], or u.c.c (resp. l.c.c), if for each x ∈ x and for each open subset v of y with f (x) ⊆ v (resp. f (x) ∩ v 6= ∅) and having compact complement, there is an open set u containing x such that f (u ) ⊆ v (resp. f (z) ∩ v 6= ∅ for every point z ∈ u ). (d) upper (resp. lower) ℓ−continuous [17], or u.ℓ.c (resp. l.ℓ.c), if f +(v ) (resp. f −(v )) is open for each open subset v ⊆ y having lindelöf complement. (e) upper (resp. lower) nearly continuous [5] if for each x ∈ x and for each open subset v of y with f (x) ⊆ v (resp. f (x) ∩ v 6= ∅) and having n-closed complement, there is an open set u containing x such that f (u ) ⊆ v (resp. f (z) ∩ v 6= ∅ for every point z ∈ u ). (f) upper (resp. lower) almost nearly continuous [6] if for each x ∈ x and for each open subset v of y with f (x) ⊆ v (resp. f (x) ∩ v 6= ∅) and having n-closed complement, there is an open set u containing x such that f (u ) ⊆ int(clv ) (resp. f (z) ∩ int(clv ) 6= ∅ for every point z ∈ u ). the following characterizations are important for our discussion. theorem 2.7. a multifunction f : (x, σ) → (y, τ ) is (a) upper (resp. lower) δ−continuous [2] if and only if f +(v ) (resp. f −(v )) is δ−open for each regular open subset v ⊆ y. (b) upper (resp. lower) super continuous [1] if and only if f +(v ) (resp. f −(v )) is δ−open for each open subset v ⊆ y. on some variations of multifunction continuity 305 (c) upper (resp. lower) c−continuous [10] and [17] if and only if f +(v ) (resp. f −(v )) is open for each open subset v ⊆ y having compact complement. (d) upper (resp. lower) nearly continuous [5] if and only if f +(v ) (resp. f −(v )) is an open set for any open subset v ⊆ y having n-closed complement. (e) upper (resp. lower) almost nearly continuous [6] if and only if f +(v ) (resp. f −(v )) is an open set for any regular open subset v ⊆ y having n-closed complement. the next six results indicate how change of topology is related to these concepts of near continuity of multifunctions. theorem 2.8. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σ) → (y, τs ) is upper (resp. lower) c−continuous if and only if f : (x, σ) → (y, τ ) is upper (resp. lower) nearly continuous. proof. (⇒) let v be an open subset having n-closed complement. then y − v is compact in (y, τs ) by theorem 2.5. since f : (x, σ) → (y, τs ) is upper c−continuous, f +(v ) ∈ σ. hence f : (x, σ) → (y, τ ) is upper nearly continuous. the proof for the case f lower nearly continuous is analogous. (⇐) let v ∈ τs and having compact complement. then y − v is n-closed with respect to τ by theorem 2.5. since f : (x, σ) → (y, τ ) is upper nearly continuous, f +(v ) ∈ σ. hence f : (x, σ) → (y, τ ) is upper c−continuous. the proof for the case f lower c−continuous is analogous. � corollary 2.9. let f : (x, σ) → (y, τ ) be a multifunction and y be semi regular. then f : (x, σ) → (y, τ ) is upper (resp. lower) c−continuous if and only if f : (x, σ) → (y, τ ) is upper (resp. lower) nearly continuous. theorem 2.10. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σ) → (y, τs ) is upper (resp. lower) k-almost c−continuous if and only if f : (x, σ) → (y, τ ) is upper (resp. lower) almost nearly continuous. proof. (⇒) let v be a regular open subset having n-closed complement. then y − v is compact in (y, τs ) by theorem 2.5. since f : (x, σ) → (y, τs ) is upper k-almost c−continuous, f +(v ) ∈ σ. hence f : (x, σ) → (y, τ ) is upper almost nearly continuous. the proof for the case f lower almost nearly continuous is analogous. (⇐) let v ∈ q′(τs ). then y − v is n-closed with respect to τ by theorem 2.5. since f : (x, σ) → (y, τ ) is upper almost nearly continuous, f +(v ) ∈ σ. hence f : (x, σ) → (y, τ ) is upper k-almost c−continuous. the proof for the case f lower k-almost c−continuous is analogous. � corollary 2.11. let f : (x, σ) → (y, τ ) be a multifunction and y be semi regular. then f : (x, σ) → (y, τ ) is upper (resp. lower) k-almost c−continuous if and only if f : (x, σ) → (y, τ ) is upper (resp. lower) almost nearly continuous. 306 a. kanibir and i. l. reilly we have the following corollary since upper (resp. lower) almost ℓ−continuous multifunctions are upper (resp. lower) k-almost c−continuous [11]. corollary 2.12. if f : (x, σ) → (y, τs ) is upper (resp. lower) almost ℓ−continuous, then f : (x, σ) → (y, τ ) is upper (resp. lower) almost nearly continuous. theorem 2.13. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σ) → (y, τ ) is lower almost nearly continuous if and only if f : (x, σ) → (y, p(τ )) is lower semi continuous. proof. (⇒) let v ∈ p(τ ). we can write v = ∪α∈λvα where vα is a regular open set having n-closed complement for each α ∈ λ. we have f −(∪α∈λvα) = ∪α∈λf −(vα). from theorem 2.7(e) we have that f −(vα) is an open set for each α ∈ λ. so f −(v ) is an open set. hence f : (x, σ) → (y, p(τ )) is l.s.c. (⇐) obvious. � a result analogous to theorem 2.13 for upper almost nearly continuous does not hold as the following example shows. example 2.14 ([11]). let x = y = {1, 2, 3, 4} and σ = {{1}, {1, 3, 4}, x, ∅} be the topology on x and τ = {{1}, {2}, {1, 2}, y, ∅} be the topology on y. let f be defined as f (1) = {4}, f (2) = {1, 2}, f (3) = {3}, f (4) = {4}. the family p′(τ ) = {{1}, {2}, y, ∅} is a base consisting of regular open sets having n-closed complement in y for p(τ ) = τ. then for any regular open set v having n-closed complement we have f +(v ) ∈ σ. therefore f : (x, σ) → (y, τ ) is upper almost nearly continuous. the topology p(τ ) contains the set {1, 2} but f +({1, 2}) = {2} /∈ σ. hence f : (x, σ) → (y, p(τ )) is not u.s.c. we have the following corollary by theorem 2.13 and since lower semi continuous multifunctions are lower almost ℓ−continuous multifunctions [11]. corollary 2.15. if the multifunction f : (x, σ) → (y, τ ) is lower almost nearly continuous, then f : (x, σ) → (y, p(τ )) is l.a.ℓ−continuous. the proof of each of the next three results is straightforward from theorem 2.7 and definition 2.6(d). note that propositions 2.16 and 2.23 provide change of topology results for upper super continuity, and that this is an unusual circumstance. these are exceptional results, as the counter-examples 2.14 and 2.20 show. proposition 2.16. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σ) → (y, ℓ(τ )) is u. sup .c.(resp. l. sup .c.) if and only if f : (x, σs ) → (y, τ ) is u.ℓ.c.(resp. l.ℓ.c.) proposition 2.17. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σ) → (y, τ ) is l.δ.c. if and only if f : (x, σ) → (y, τs ) is l. sup .c. proposition 2.18. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σs ) → (y, τ ) is l.a.ℓ−continuous if and only if f : (x, σ) → (y, q(τ )) is l.δ.c. on some variations of multifunction continuity 307 theorem 2.19. let f : (x, σ) → (y, τ ) be a multifunction. then the following statements are equivalent: (a) f : (x, σs ) → (y, τ ) is l.a.ℓ−continuous. (b) f : (x, σs ) → (y, q(τ )) is l. sup .c. (c) f : (x, σs ) → (y, q(τ )) is l.δ.c. (d) f : (x, σs ) → (y, q(τ )) is l.s.c. (e) f : (x, σs ) → (y, q(τ )) is l.ℓ.c. proof. the proofs of (b)⇒ (c), (c)⇒ (d), (d)⇒ (e), (e)⇒ (a) are immediate. we prove only (a)⇒ (b). let v ∈ q(τ ). then we can write v = ∪α∈λvα where vα is a regular open set having lindelöf complement, for each α ∈ λ. we have f −(∪α∈λvα) = ∪α∈λf −(vα). therefore f −(vα) ∈ σs by hypothesis. so f −(v ) ∈ σs and therefore is δ−open. hence f : (x, σs ) → (y, q(τ )) is l. sup .c. � a result analogous to theorem 2.19 for u.a.ℓ−continuous, u. sup .c., u.δ.c., u.s.c., u.ℓ.c. does not hold as the following example shows. example 2.20. let us redefine the topology σ in example 2.14 as follows σ = {{1, 2}, {3, 4}, ∅, x}. no other changes are made to example 2.14. it is obvious that σs = σ and q(τ ) = τ. then f is u.a.ℓ−continuous and u.δ.c but not u. sup .c., u.s.c. and u.ℓ.c even if x is regular and lindelöf and y is semi regular and lindelöf. proposition 2.21 ([12]). if ( y, τ ) is a lindelöf and semi regular space, then q(τ ) = τ. we have the following corollary by theorem 2.19 and proposition 2.21. corollary 2.22. let x be a semi regular topological space and let y be a semi regular and lindelöf topological space and f : (x, σ) → (y, τ ) be a multifunction. then the following statements are equivalent: (a) f : (x, σ) → (y, τ ) is l.a.ℓ−continuous. (b) f : (x, σ) → (y, τ ) is l. sup .c. (c) f : (x, σ) → (y, τ ) is l.δ.c. (d) f : (x, σ) → (y, τ ) is l.s.c. (e) f : (x, σ) → (y, τ ) is l.ℓ.c. a result analogous to corollary 2.22 for u.a.ℓ−continuous, u. sup .c., u.δ.c., u.s.c., u.ℓ.c. does not hold as example 2.20 shows. the next two results are analogues of propositions 2.16 and 2.18 respectively. their proofs are straightforward from theorem 2.7. proposition 2.23. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σ) → (y, c(τ )) is u. sup .c.(resp. l. sup .c.) if and only if f : (x, σs ) → (y, τ ) is u.c.c.(resp. l.c.c.). 308 a. kanibir and i. l. reilly proposition 2.24. let f : (x, σ) → (y, τ ) be a multifunction. then f : (x, σs ) → (y, τ ) is l.k-a.c−continuous if and only if f : (x, σ) → (y, e(τ )) is l.δ.c. similarly to theorem 2.19, we can obtain the following characterizations. theorem 2.25. let f : (x, σ) → (y, τ ) be a multifunction. then the following statements are equivalent: (a) f : (x, σs ) → (y, τ ) is l.k-a.c−continuous. (b) f : (x, σs ) → (y, e(τ )) is l. sup .c. (c) f : (x, σs ) → (y, e(τ )) is l.δ.c. (d) f : (x, σs ) → (y, e(τ )) is l.s.c. (e) f : (x, σs ) → (y, e(τ )) is l.c.c. a result analogous to theorem 2.25 for u.k-a.c−continuous, u. sup .c., u.δ.c., u.s.c., u.c.c. does not hold as example 2.20 shows. corresponding to proposition 2.21 we have the following result, which seems to be new. proposition 2.26. if ( y, τ ) is a compact and semi regular space, then e(τ ) = τ. corollary 2.27. let x be a semi regular topological space and let y be a semi regular and compact topological space and f : (x, σ) → (y, τ ) be a multifunction. then the following statements are equivalent: (a) f : (x, σ) → (y, τ ) is l.k-a.c−continuous. (b) f : (x, σ) → (y, τ ) is l. sup .c. (c) f : (x, σ) → (y, τ )is l.δ.c. (d) f : (x, σ) → (y, τ )is l.s.c. (e) f : (x, σ) → (y, τ )is l.c.c. (f) f : (x, σ) → (y, τ ) is lower nearly continuous. (g) f : (x, σ) → (y, τ )is lower almost nearly continuous. proof. this is an immediate consequence of theorem 2.25, proposition 2.26, and corollaries 2.9 and 2.11. � a result analogous to corollary 2.27 for u.k-a.c−continuous, u. sup .c., u.δ.c., u.s.c., u.c.c., upper almost nearly continuous, upper nearly continuous does not hold as example 2.20 shows. several properties of upper and lower almost ℓ−continuous multifunctions have been given in [11]. we now provide some more. we can also obtain similar sets of results [which are parallel to the following results] for u.ka.c−continuous ( resp. l.k-a.c−continuous) multifunctions. we shall leave them unstated. the following result is immediate. the proof is left to the reader. on some variations of multifunction continuity 309 theorem 2.28. let (x, σ), (y, τ ), (z, ϑ) be topological spaces and let f : x → y and g : y → z be multifunctions. if f : x → y is upper (resp. lower) semi continuous and g : y → z is upper (resp. lower) almost ℓ−continuous, then f ◦ g : x → z is upper(resp. lower) almost ℓ−continuous multifunction. theorem 2.29. let f : (x, σ) → (y, τ ) be a multifunction and let a ⊆ x be a nonempty subset. if f is u.a.ℓ−continuous (resp. l.a.ℓ−continuous) then the restriction multifunction f |a : a → y is u.a.ℓ−continuous (resp. l.a.ℓ−continuous). proof. we prove only the assertion for f |a u.a.ℓ−continuous, the proof for f |a l.a.ℓ−continuous being analogous. let x ∈ a and v be a regular open subset of (y, τ ) having lindelöf complement such that (f |a)(x) ⊆ v. since f is u.a.ℓ−continuous and (f |a)(x) = f (x), there exists u ∈ σ containing x such that f (u ) ⊆ v. then x ∈ a ∩ u and a ∩ u is open in a. moreover (f |a)(a ∩ u ) ⊆ v. this shows that f |a is u.a.ℓ−continuous. � theorem 2.30. let f : (x, σ) → (y, τ ) be a multifunction. let {vα : α ∈ λ} be an open cover of x. if the restriction multifunction f α = f |vα is u.a.ℓ−continuous (resp. l.a.ℓ−continuous) multifunction for each α ∈ λ, then f is u.a.ℓ−continuous (resp. l.a.ℓ−continuous). proof. let v ∈ q′(τ ). since fα is u.a.ℓ−continuous f + α (v ) ⊆ intvα (f +(v )) and since vα is open, we have f +(v ) ∩ vα ⊆ intvα (f +(v ) ∩ vα) and f +(v ) ∩ vα ⊆ int(f +(v )) ∩ vα. since {vα : α ∈ λ} is an open cover of x, f +(v ) = int(f +(v )). hence f is u.a.ℓ−continuous. the proof for l.a.ℓ−continuous is similar. � corollary 2.31. let {vα : α ∈ λ} be an open cover of x. a multifuntion f : x → y is u.a.ℓ−continuous (resp. l.a.ℓ−continuous) if and only if the restriction f |vα : vα → y is u.a.ℓ−continuous (resp. l.a.ℓ−continuous) for each α ∈ λ. proof. this is an immediate consequence of theorems 2.29 and 2.30. � theorem 2.32. let f : (x, σ) → (y, τ ) be multifunction and let x × y be a lindelöf space. if the graph multifunction of f is lower (resp. upper) almost ℓ−continuous multifunction, then f is lower (resp. upper) almost ℓ−continuous multifuction. proof. let x ∈ x and let v ∈ q′(τ ) with f (x)∩v 6= ∅. then gf (x)∩(x×v ) 6= ∅ and x × v is a regular open set having lindelöf complement. since the graph multifunction gf is lower almost ℓ−continuous, there exists an open neighbourhood u of x such that gf (z) ∩ (x × v ) 6= ∅ for every point z ∈ u. thus f (z) ∩ v 6= ∅. hence f is lower almost ℓ−continuous. the proof of the upper almost ℓ−continuity of f is similar. � 310 a. kanibir and i. l. reilly references [1] m. akdag, on super continuous multifunctions, acta math.hungar. 99(1-2)(2003),143153. 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[10] l. hola, v. balaz and t. neubrunn, remarks on c-continuous multifunctions, acta math. univ. comenianae l-li (1987), 51–58. [11] a. kanibir and i. l. reilly, on almost ℓ−continuous multifunctions, hacettepe j. math. stat. 35, no. 2 (2006), 181–188. [12] ch. konstadilaki-savvopoulou and i. l. reilly, almost ℓ−continuous functions, glasnik matematicki 25, no. 45 (1990), 363–370. [13] ch. konstadilaki-savvopoulou and i. l. reilly, on almost n-continuous functions, j. austral. math. soc. (seriesa) 59 (1995), 118–130. [14] y. kucuk, almost c-continuous multifunctions, pure and applied mathematical science xxxix, no.1-2(1994), 1–9. [15] y. kucuk, on some characterizations of δ−continuous multifunctions, demonstratio mathematica xxviii, no. 3 (1995), 587–595. [16] t. noiri, remarks on locally nearly compact spaces, boll. un. mat. ital. 10, no. 4 (1974), 36–43. [17] k. sakalova, continuity properties of multifunctions, acta math. univ. comenianae lvi-lvii (1989), 159–165. [18] n. v. velicko, h-closed topological spaces, mat. sb. 70, no. 112 (1966), 98-112. received april 2007 accepted june 2008 a. kanibir (kanibir@hacettepe.edu.tr) department of mathematics, hacettepe university, 06532 beytepe, ankara, turkey i. l. reilly (i.reilly@auckland.ac.nz) department of mathematics, university of auckland, p.b. 92019, auckland, new zealand dasagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 197-206 ∆-normal spaces and decompositions of normality a. k. das abstract. generalizations of normality, called (weakly) (functionally) ∆-normal spaces are introduced and their interrelation with some existing notions of normality is studied. ∆-regular spaces are introduced which is a generalization of seminormal, semiregular and θ-regular space. this leads to decompositions of normality in terms of ∆-regularity, seminormality and variants of ∆-normality. 2000 ams classification: 54d10, 54d15, 54d20. keywords: δ-open sets, δ-closed sets (weakly) (functionally) θ-normal spaces, (weakly) (functionally) ∆-normal spaces, θ-regular spaces, seminormal spaces. 1. introduction and preliminaries to investigate properly the existing notions of general topology, topologists adopted various techniques. decomposition of a given topological property in terms of two weaker properties is one of them. none of the existing classical notions of general topology remain untouched of decomposition process. since normality is an important property, its decomposition is desirable. first step in this direction was initiated by vigilino [18] and singal and arya [13], where a decomposition of normality was given in terms of almost normal spaces and seminormal spaces. another decomposition of normality was given in [6] in terms of θ-normality and its variants. mack [10] introduced δ-normal spaces and the same has been utilised in [8] to give a factorization of normality. in an attempt to get another decomposition of normality in terms of seminormal spaces, in this paper we introduce the notion of ∆-normal spaces. let x be a topological space and let a ⊂ x. throughout the present paper the closure of a set a will be denoted by a or cla and the interior by inta. a set u ⊂ x is said to be regularly open [9] if u = intu . the complement of 198 a. k. das a regularly open set is called regularly closed. a point x ∈ x is called a θlimit point (respectively δ-limit point) [17] of a if every closed (respectively regularly open) neighbourhood of x intersects a. let clθa (respectively clδa) denotes the set of all θ-limit point (respectively δ-limit point) of a. the set a is called θ-closed (respectively δ-closed) if a = clθa (respectively a = clδa). the complement of a θ-closed (respectively δ-closed) set will be referred to as a θ-open (respectively δ-open) set. the family of θ-open sets as well as the family of δ open sets form topologies on x. the topology formed by the set of δ-open sets is the semiregularization topology whose basis is the family of regularly open sets. a space x is said to be almost regular [12] if every regularly closed set and a point out side it are contained in disjoint open sets. a space is called semi-normal [18] if for every closed set f and each open set u containing f, there exists a regular open set v such that f ⊂ v ⊂ u . a space is called almost normal [13] if every pair of disjoint closed sets one of which is regularly closed are contained in disjoint open sets and a space x is said to be mildly normal [15] ( or κ-normal [16]) if every pair of disjoint regularly closed sets are contained in disjoint open sets. a space is almost completely regular [13] if for every regularly closed set a and a point x /∈ a, there exists a continuous function f : x → [0, 1] such that f (x) = 0 and f (a) = 1. a space x is said to be nearly compact[14] if every open covering of x admits a finite subcollection the interiors of the closures of whose members cover x. a subset g of a space x is called a regular gδ-set if it is the intersection of a sequence of closed sets whose interiors contain g, i.e., g = ∞ ⋂ n=1 fn = ∞ ⋂ n=1 f on , where each fn is a closed subset of x. the complement of a regular gδ-set is called a regular fσ-set[10]. in a topological space, every zero set is a regular gδ-set and every regular gδ-set is θ-closed. in general the θ-closure operator is a čech closure operator (see [11]) but not a kuratowski closure operator, since θ-closure of a set may not be θ-closed (see [4]). however, the following modification yields a kuratowski closure operator. definition 1.1 ([5]). let x be a topological space and let a ⊂ x. a point x ∈ xis called a uθ-limit point of a if every θ-open set u containing x intersects a. let auθ denote the set of all uθ-limit points of a. lemma 1.2 ([7]). the correspondence a → auθ is a kuratowski closure operator. it turns out that the set auθ is the smallest θ-closed set containing a. definition 1.3. a topological space x is said to be (i) θ-normal [6] if every pair of disjoint closed sets one of which is θclosed are contained in disjoint open sets; (ii) weakly θ-normal[6] if every pair of disjoint θ-closed sets are contained in disjoint open sets; 199 (iii) functionally θ-normal [6] if for every pair of disjoint closed sets a and b one of which is θ-closed there exists a continuous function f : x →[0,1] such that f (a) = 0 and f (b)=1; (iv) weakly functionally θ-normal (wf θ-normal)[6] if for every pair of disjoint θ-closed sets a and b there exists a continuous function f : x → [0,1] such that f (a) = 0 and f (b)= 1; and (v) θ-regular[6] if for each closed set f and each open set u containing f , there exists a θ-open set v such that f ⊂ v ⊂ u . (vi) σ-normal[8] if for each closed set f and each open set u containing f , there exists a regular fσ set v such that f ⊂ v ⊂ u . 2. ∆-normal spaces definition 2.1. a topological space x is said to be (i) ∆-normal if every pair of disjoint closed sets one of which is δ-closed are contained in disjoint open sets; (ii) weakly ∆-normal if every pair of disjoint δ-closed sets are contained in disjoint open sets; (iii) weakly functionally ∆-normal (wf ∆-normal) if for every pair of disjoint δ-closed sets a and b there exists a continuous function f : x → [0,1] such that f (a) = 0 and f (b)= 1. theorem 2.2. for a topological space x, the following statements are equivalent. (a) x is ∆-normal. (b) for every closed set a and every δ-open set u containing a there exists an open set v such that a ⊂ v ⊂ v ⊂ u . (c) for every δ-closed set a and every open set u containing a there exists an open set v such that a ⊂ v ⊂ v ⊂ u . (d) for every pair of disjoint closed sets a and b one of which is δ-closed there exists a continuous function f : x → y such that f (a) = 0 and f (b) = 1. (e) for every pair of disjoint closed sets one of which is δ-closed are contained in disjoint θ-open sets. (f) for every δ-closed set a and every open set u containing a there exists a θ-open set v such that a ⊂ v ⊂ vuθ ⊂ u . (g) for every closed set a and every δ-open set u containing a there exists a θ-open set v such that a ⊂ v ⊂ vuθ ⊂ u . (h) for every pair of disjoint closed sets a and b, one of which is δ-closed there exist θ-open sets u and v such that a ⊂ u , b ⊂ v and uuθ ∩ vuθ = φ. proof. to prove the assertion (a) ⇒ (b), let x be a ∆-normal space and let u be an δ-open set containing a closed set a. now a is closed set which is disjoint from the δ-closed set x − u . by ∆-normality of x there are disjoint open sets 200 a. k. das v and w containing a and x − u , respectively. then a ⊂ v ⊂ x − w ⊂ u . since x − w is closed, a ⊂ v ⊂ v ⊂ u . to prove the implication (b) ⇒ (c), let u be a open set containing a δ-closed set a. then x − a is an δ-open set containing the closed set x − u . so by hypothesis there exists an open set w such that x − u ⊂ w ⊂ w ⊂ x − a. let v = x − w . then a ⊂ v ⊂ x − w ⊂ u . since x − w is closed, a ⊂ v ⊂ v ⊂ u . to prove the implication (c) ⇒ (d), let a be a δ-closed set disjoint from a closed set b. then a ⊂ x − b = u1 (say). since u1 is open, there exists a open set u1/2 such that a ⊂ u1/2 ⊂ u 1/2) ⊂ u1. again, since closure of an open set is δ-closed, u 1/2 is a δ-closed set, so there exist open sets u1/4 and u3/4 such that a ⊂ u1/4 ⊂ u 1/4 ⊂ u1/2 and u 1/2 ⊂ u3/4 ⊂ u 3/4 ⊂ u1. continuing the above process, we obtain for each dyadic rational r, a δ-open set ur satisfying r < s implies u r ⊂ us. let us define a mapping f : x →[0,1] by f (x) = { inf { x : x ∈ ur } if x belongs to some ur, 1 if x does not belongs to any ur. clearly f is well defined and f (a) = 0, f (b) = 1. now it remains to prove that f is continuous. to this end we first observe that if x ∈ ur, then f (x) ≤ r. similarly f (x) ≥ r if x /∈ u r. to prove continuity, let x ∈ x and (a, b) be an open interval containing f (x). now choose two dyadic rationals p and q such that a < p < f (x) < q < b. let u = uq −u p. then u is an open set containing x. now for y ∈ u , y ∈ uq. so f (y) ≤ q. also as y ∈ u , y /∈ u p. thus f (y) ≥ q. and so f (y) ∈ [p, q]. therefore f (u ) ⊂ [p, q] ⊂ (a, b). hence f is continuous. to prove the assertion (d) ⇒ (e), let a, b be disjoint closed sets in x, where b is δ-closed. by the hypothesis there exists a continuous function f : x →[0,1] such that f (a) = 0 and f (b) = 1. since every continuous function lifts back every θ-open set to θ-open set, the set f −1[0, 1/2) and f −1 (1/2, 1] are disjoint θ-open sets containing a and b respectively. to prove (e) ⇒ (f), let a be a δ-closed set in x and let u be an open set containing a. since a and x −u are disjoint, by hypothesis there exist disjoint θ-open sets v and w such that a ⊂ v and x −u ⊂ w . so a ⊂ v ⊂ x −w ⊂ u . since x − w is θ-closed and vuθ is the smallest θ-closed set containing v , a ⊂ v ⊂ vuθ ⊂ u . to prove (f) ⇒ (g), let a be a closed set contained in a δ-open set u . then x − u is a δ-closed set contained in the open set x − a. by hypothesis, there exists a θ-open set w such that x−u ⊂ w ⊂ wuθ ⊂ x−a. let v = x−wuθ. then a ⊂ v ⊂ x − w ⊂ u . since x − w is θ-closed and vuθ is the smallest θ-closed set containing v , a ⊂ v ⊂ vuθ ⊂ u . to prove (g) ⇒ (h), let a be a closed set disjoint from a δ-closed set b. then x − b is a δ-open set containing a. so there exists a θ-open set w such that a ⊂ w ⊂ wuθ ⊂ x − b. again by hypothesis there exists a θ-open set u such that a ⊂ u ⊂ uuθ ⊂ w ⊂ wuθ ⊂ x − b. let v = x − wuθ, then u and v are θ-open sets containing a and b respectively and uuθ ∩ vuθ = ∅. 201 the assertion (h) ⇒ (a) is obvious. � theorem 2.3. a topological space x is weakly ∆-normal if and only if for every δ-closed set a and a δ-open set u containing a there is an open set v such that a ⊂ v ⊂ v ⊂ u . proof. let x be a weakly ∆-normal space and u be a δ-open set containing a δ-closed set a. then a and x − u are disjoint δ-closed sets in x. thus by weak ∆-normality of x there are disjoint open sets v and w containing a and x − u , respectively. then a ⊂ v ⊂ x − w ⊂ u . since x − w is closed, a ⊂ v ⊂ v ⊂ u . conversely, let a and b be two disjoint δ-closed sets in x. then u = x −b is a δ-open set containing the δ-closed set a. thus by the hypothesis there exists an open set v such that a ⊂ v ⊂ v ⊂ u . then v and x − v are disjoint open sets containing a and b, respectively. hence x is weakly ∆-normal. � the following diagram is immediate from the definitions. normal �� almost normal ((p pp pp pp pp pp pp pp pp pp pp pp pp pp pp p ∆-normal // 66mmmmmmmmmmmm �� wf∆-normal �� ((qq qq qq qq qq qq q fθ-normal // �� wfθ-normal �� w∆-normal vvmmm mm mm mm mm mm // κ-normal θ-normal // wθ-normal none of the above implication is reversible (see examples 2.4 2.8 below, [6, example 3.6 3.8] and [5, example 3.4]). example 2.4. a functionally θ-normal and weakly functionally ∆-normal space which is not ∆-normal. let x = {a, b, c, d} and τ = {{a, b}, {b}, {b, c}, {c}, {b, c, d}, {a, b, c}, x, φ}. here the δ-closed set { c, d } and closed set { a } can not be separated by disjoint open sets. thus the space is not ∆-normal but the space is functionally θ-normal and weakly functionally ∆-normal. example 2.5. a functionally θ-normal space which is not weakly ∆-normal. let x be the set of positive integers. define a topology on x by taking every odd integer to be open and a set u ⊂ x is open if for every even integer p ∈ u , the predecessor and successor of p are also in u . the space is not weakly ∆normal as disjoint δ-closed sets { 2, 3, 4 } and { 6 } cannot be separated by disjoint open sets. but the space is functionally θ-normal. example 2.6. a weakly ∆-normal space which is not weakly functionally ∆-normal 202 a. k. das let x denote the interior of the unit square s in the plane together with the points (0, 0) and (1, 0), i.e. x = ints∪ {(0, 0), (1, 0) }. every point in ints has the usual euclidean neighourhoods. the points (0, 0) and (1, 0) have neighbourhoods of the form un and vn respectively, where un = {(0, 0)} ∪ {(x, y) : 0 < x < 1/2, 0 < y < 1/n} and vn = {(1, 0)} ∪ {(x, y) : 1/2 < x < 1, 0 < y < 1/n}. the space x is weakly ∆-normal, since every pair of disjoint δ-closed sets are separated by disjoint open sets. however, the δ-closed sets {(0, 0)} and {(1, 0)} do not have disjoint closed neighbourhoods and hence cannot be functionally separated. example 2.7. a ∆-normal space which is not normal. let x = {a, b, c, d} and τ = {{a, b}, {b, c}, {b} ,{d}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}, x, φ } . example 2.8. an almost normal space which is not weakly ∆-normal. let x = {a, b, c} and τ = {{b, c}, {a, c}, {c}, x, φ }. theorem 2.9. for a hausdorff space x, the following statements are equivalent. (a) x is normal. (b) x is ∆-normal. (c) x is functionally θ-normal. (d) x is θ-normal. proof. the implications (a) ⇒ (b) ⇒ (c) ⇒ (d) are immediate from definitions and theorem 2.2. the implication (d) ⇒ (a) is shown in [6, theorem 3.5]. � lemma 2.10. in an almost regular space every δ-closed set is θ-closed. theorem 2.11. in an almost regular space the following statements are equivalent. (a) x is ∆-normal. (b) x is functionally θ-normal. (c) x is θ-normal. proof. by lemma 2.10, every δ-closed set is θ-closed. thus in an almost regular space every θ-normal space is ∆-normal. � theorem 2.12. in an almost regular space the following statements hold. (a) every weakly functionally θ-normal space is weakly functionally ∆-normal. (b) every weakly θ-normal space is weakly ∆-normal. recall that a space x is an ro-space [2] if for every open set u in x, x ∈ u implies {x} ⊂ u . ro-spaces are called s1-spaces in [1]. theorem 2.13. a ∆-normal ro-space is almost completely regular. proof. let x be a ∆-normal ro-space. let a be a regular closed set and x /∈ a. then x ∈ x − a. since x is ro, {x} ∈ x − a. so {x} is a closed set disjoint from the δ-closed set a. thus by theorem 2.2, there exists a continuous 203 function f : x → [0, 1] such that f ({x}) = 0 and f (a) = 1. hence x is almost completely regular. � theorem 2.14. a hausdorff weakly functionally ∆-normal space is almost completely regular. proof. let x be a hausdorff weakly functionally ∆-normal space. let x ∈ x and a be a regularly closed set in x such that x /∈ a. as x is hausdorff, { x } and a are disjoint δ-closed sets. thus by weak functional ∆-normality of x there exists a continuous function f : x → [0, 1] such that f (x) = 0 and f (a) = 1. so f is almost completely regular. � theorem 2.15. a hausdorff weakly ∆-normal space is almost regular. proof. let x be a hausdorff weakly ∆-normal space. let a be a regularly closed set not containing x. by [3, 2.3], every singleton in x is θ-closed. so { x } and a are disjoint δ-closed sets which can be separated by disjoint open sets by weak ∆-normality. � corollary 2.16. a hausdorff weakly ∆-normal space is weakly functionally θ-normal. proof. it is immediate in view of theorem 2.15 and the fact that an almost regular weakly θ-normal space is weakly functionally θ-normal (see [5, theorem 5.18]). � 3. decompositions of normality definition 3.1. a topological space x is said to be ∆-regular if for every closed set f and each open set u containing f , there exists a δ-open set v such that f ⊂ v ⊂ u . clearly, every θ-regular space as well as every semi-normal space is ∆regular. theorem 3.2. every semiregular space is ∆-regular. proof. let f be a closed set contained in an open set u . for every x ∈ f there exists a regular open set ux such that x ∈ ux ⊂ u . let ⋃ x∈f ux = v . thus f ⊂ v ⊂ u , where v need not be regular open but δ-open. hence x is ∆-regular. � the following diagram well illustrates the interrelations that exist among variants of regularity and normality. 204 a. k. das normal // �� z-normal // σ-normal wwoo oo oo oo o oo seminormal �� θ-regular wwooo oo oo oo oo o ∆-regular semiregularoo regularoo ggooooooooooo however, none of the above implications is reversible as is well exhibited by the following examples and example 3.8 in [8]. example 3.3. a ∆-regular space which is not θ-regular. every open set in example 2.5 is regular open and the only θ-open set in x is x itself. so the space is ∆-regular but not θ-regular. example 3.4. a ∆-regular space which is not semiregular. let x = {a, b, c} and let τ = {{a}, {a, b},x, φ}. this space is ∆-regular but not semiregular. theorem 3.5. an ro ∆-regular space is semiregular. proof. let x be an ro ∆-regular space. let x ∈ x and let u be an open set containing x. since x is an ro-space, {x} ⊂ u . so by θ-regularity of x, there exists a δ-open set v such that {x} ⊂ v ⊂ u . since v is the union of regular open sets, there exists a regular open set w such that x ∈ w ⊂ v ⊂ u . so x is semiregular. � theorem 3.6. every nearly compact θ-regular space is ∆-normal. proof. let a and b be two disjoint closed sets where a is δ-closed. since every δ-closed subset of a nearly compact space is n-closed relative to x, a is n-closed relative to x. now a ⊂ x − b. thus by θ-regularity of x, there exists a θ-open set v such that a ⊂ v ⊂ x − b. now for every x ∈ a, there exists an open set ux such that x ∈ ux ⊂ ux ⊂ v . thus u = {intux : x ∈ a} is a regular open cover of a. since a is n-closed relative to x, there exist a finite subcollection {intuxi : x ∈ a} which covers a. then p = n ⋃ i=1 intuxi and q = n ⋂ i=1 (x − uxi ) are disjoint open sets containing a and b respectively. hence x is ∆-normal. � remark 3.7. even a compact ∆-regular space need not be weakly ∆-normal. e.g.; let x = {a, b, c} and τ = {{b, c}, {a, c}, {c}, φ, x}. here the space is compact and ∆-regular but not θ-regular. the following theorem and the corollary provides factorizations of normality in terms of ∆-regularity, seminormality and variants of ∆-normality. 205 theorem 3.8. in a ∆-regular space the following statements are equivalent. (a) x is normal. (b) x is ∆-normal. (c) x is weakly functionally ∆-normal. (d) x is weakly ∆-normal. proof. the implications (a) ⇒ (b) ⇒ (c) ⇒ (d) are immediate. to prove (d) ⇒ (b), let x be ∆-regular, w∆-normal space. let a and b be two disjoint closed subsets of x, where one of them is δ-closed say a. then a ⊂ x − b. thus by ∆-regularity of x, there exists a δ-open set u such that a ⊂ u ⊂ (x − b). so a and x − u are disjoint δ-closed sets which can be separated by disjoint open sets by weak ∆-normality. hence x is δ-normal. to show that (b) ⇒ (a), let a and b be two disjoint closed subsets of x. since x is ∆-regular, there is a δ-open set w such that a ⊂ w ⊂ x − b. then x − w is a δ-closed set containing b. by ∆-normality of x, there exist disjoint open sets u and v containing a and x − w , respectively and so a and b respectively. � corollary 3.9. in a seminormal space the following statements are equivalent. (a) x is normal. (b) x is ∆-normal. (c) x is weakly functionally ∆-normal. (d) x is weakly ∆-normal. corollary 3.10. in a semiregular space the following statements are equivalent. (a) x is normal. (b) x is ∆-normal. (c) x is weakly functionally ∆-normal. (d) x is weakly ∆-normal. theorem 3.11. in a θ-regular space the following statements are equivalent. (a) x is normal. (b) x is ∆-normal. (c) x is functionally θ-normal. (d) x is weakly functionally ∆-normal. (e) x is θ-normal. (f) x is weakly functionally θ-normal. (g) x is weakly ∆-normal. (h) x is weakly θ-normal. proof. the implications (a) ⇒ (b) ⇒ (c) ⇒ (e) ⇒ (h), (a) ⇒ (b) ⇒ (d) ⇒ (f) ⇒ (h) and (a) ⇒ (b) ⇒ (d) ⇒ (g) ⇒ (h) are immediate. to prove (h) ⇒ (a), let x be a θ-regular weakly θ-normal space by [6, theorem 3.11], x is normal. � 206 a. k. das references [1] á. császár, general topology, adam higler ltd, bristol, 1978. [2] a.s. davis, indexed systems of neighbourhoods for general topological spaces, amer. math. monthly 68 (1961), 886–893. [3] r.f. dickman, jr. and j.r. porter, θ-perfect and θ-absolutely closed functions, illinois j. math. 21 (1977), 42–60. [4] j.e. joseph, θ-closure and θ-subclosed graphs, math. chron. 8 (1979), 99–117. [5] j.k. kohli and a.k. das, on functionally θ-normal spaces, applied gen. topol. 6(2005), no. 1, 1–14. [6] j.k. kohli and a.k. das, new normality axioms and decompositions of normality, glasnik mat. 37 (2002), no. 57, 163–173. [7] j.k. kohli and a.k. das, a class of spaces containing all generalized absolutely closed (almost compact) spaces, applied gen. topol. 7(2006), no. 2, 233–244. [8] j.k. kohli and d. singh, weak normality properties and factorizations of normality, acta. math. hungar. 110 (2006), no. 1-2, 67–80. [9] c. kuratowski, topologie i, hafner, new york, 1958. [10] j. mack, countable paracompactness and weak normality properties, trans. amer. math. soc. 148 (1970), 265–272. [11] m. mrsevic and d. andrijevic on θ-connectedness and θ closure spaces, topology appl. 123 (2002), 157–166. [12] m.k. singal and s.p. arya, on almost regular spaces, glasnik mat. 4 (1969), no. 24, 89–99. [13] m.k. singal and s.p. arya, on almost normal and almost completely regular spaces, glasnik mat. 5 (1970), no. 25, 141–152. [14] m.k. singal and a. mathur, on nearly compact spaces, boll. u.m.i. 4 (1969), 702–710. [15] m.k. singal and a.r. singal, mildly normal spaces, kyungpook math j. 13 (1973), 27–31. [16] e.v. stchepin, real valued functions and spaces close to normal, sib. j. math. 13 (1972), no. 5, 1182–1196. [17] n.v. veličko h-closed topological spaces, amer. math. soc, transl. 78 (1968), no. 2, 103–118. [18] g. vigilino, seminormal and c-compact spaces, duke j. math. 38 (1971), 57–61. received december 2008 accepted october 2009 a. k. das (ak.das@smvdu.ac.in, akdasdu@yahoo.co.in) school of applied physics and mathematics, shri mata vaishno devi university, kakriyal, katra-182320, jammu and kashmir, india. niyaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 151-159 products of straight spaces with compact spaces kusuo nishijima and kohzo yamada dedicated to the memory of jan pelant abstract. a metric space x is called straight if any continuous real-valued function which is uniformly continuous on each set of a finite cover of x by closed sets, is itself uniformly continuous. let c be the convergent sequence {1/n : n ∈ n} with its limit 0 in the real line with the usual metric. in this paper, we show that for a straight space x, x × c is straight if and only if x × k is straight for any compact metric space k. furthermore, we show that for a straight space x, if x × c is straight, then x is precompact. note that the notion of straightness depends on the metric on x. indeed, since the real line r with the usual metric is not precompact, r × c is not straight. on the other hand, we show that the product space of an open interval and c is straight. 2000 ams classification: primary 54c30, 54d05; secondary 54c20, 54e35 keywords: straight spaces, convergent sequence, precompact, uniformly continuous. 1. introduction all spaces are metric spaces and one fixed metric on a space x will be denoted by dx , and c(x) denotes the set of all continuous real-valued functions of a space x. let (x, dx ) and (y, dy ) be metric spaces. for a subspace m of x, we consider the restriction dx|m×m to m × m as a metric on m , which is denoted by dm . a metric dx×y on the product space x × y will be defined by dx×y ((x1, y1), (x2, y2)) = √ (dx (x1, x2))2 + (dy (y1, y2))2. in this paper we study notions that use metrics in their definitions. however, the symbols of metrics will simply be denoted by d or be often omitted except when it is necessary to be clear which metric we consider. let x be a metric space and {fi : i = 1, 2, . . . , n} be a finite closed cover of x. then it is well-known that every function f on x is continuous if the 152 k. nishijima and k. yamada restriction f|fi of f is continuous on fi for each i = 1, 2, . . . , n. however, it is not valid for uniform continuity. indeed, consider the subspace x = {eiθ : 0 < θ < 2π} of the complex plane with the euclidean metric and the function f (eiθ) = θ defined on x. then the function f is not uniformly continuous on x, but its restrictions on {eiθ : 0 < θ ≤ π} and {eiθ : π ≤ θ < 2π} are uniformly continuous. the following facts are useful to determine whether a given continuous function on a metric space is uniformly continuous or not. lemma 1.1. let f ∈ c(x). then the following are equivalent: (1) f is uniformly continuous; (2) for every pair of sequences {xn} and {yn} in x if lim n→∞ d(xn, yn) = 0, then lim n→∞ |f (xn) − f (yn)| = 0. (3) for every pair of sequences {xn} and {yn} in x if lim n→∞ d(xn, yn) = 0, then there are subsequences {xkn } of {xn} and {ykn } of {yn} such that lim n→∞ |f (xkn ) − f (ykn )| = 0. applying lemma 1.1, it is easy to see that the above function f (eiθ) = θ is not u.c., because let αn = π n and βn = 2π − π n for each n ∈ n, and if we consider the sequences {xn = eiαn } and {yn = eiβn }, then lim n→∞ d(xn, yn) = 0, but lim n→∞ |f (xn) − f (yn)| = 2π. recently, berarducci, dikranjan and pelant [3] defined the following notion. definition 1.2 ([3]). a metric space x is straight if whenever x is the union of finitely many closed sets, then f ∈ c(x) is uniformly continuous (briefly, u.c.) iff its restriction to each of the closed sets is u.c. recall that a metric space x is called u c [1, 2] provided every continuous function on x is u.c. and a metric space is called uniformly locally connected if for every ε > 0 there is δ > 0 such that any two points at distance < δ lie in a connected set of diameter < ε. clearly, all compact spaces are u c and all u c spaces are straight. berarducci, dikranjan and pelant [3] prove that all uniformly locally connected spaces are straight. hence, since the real line r and an open interval in r with the usual metric are clearly uniformly locally connected, they are straight, and of course, they are not u c. the product space of two compact spaces is compact, and hence u c. however, in general, the product space x ×y of a u c space x and a compact space y need not be u c. indeed, atsuji’s result [2, theorem 6] yields that if the product space x × y of a non-compact and non-uniformly discrete u c space x and a space y is u c, then y must be uniformly discrete or finite (recall that a space is uniformly discrete if there is δ > 0 such that any two distinct points are at distance at least δ). on the other hand, there are non-compact and non-uniformly discrete straight spaces whose products with compact spaces are straight, for example, r × i is uniformly locally connected, and hence straight, where i means that the unit closed interval. products of straight spaces with compact spaces 153 in this paper, we consider properties of a straight space whose product with any compact space is straight. let x be a straight space and c be the convergent sequence {1/n : n ∈ n} with its limit 0 in the real line with the usual metric. recall that a metric space x is precompact if for every ε > 0 there are finite points x1, x2, . . . , xn in x such that x = n ⋃ k=1 bε(xk), where bε(x) = {z ∈ x : d(x, z) < ε}. then we will show the following: (1) x × c is straight if and only if x × k is straight for any compact space k; (2) if x × c is straight, then x is precompact. we can know, from the result (2), that r × c is not straight. on the other hand, we prove that the product space of an open interval and c is straight. however, we cannot decide whether the inverse implication of the result (2) is valid or not (cf. acknowledgement). 2. results we first introduce the terminology that is defined in [3]. let x be a metric space. a pair e and f of closed sets of x is u-placed if d(eε, fε) > 0 for every ε > 0, where eε = {x ∈ e : d(x, e ∩ f ) ≥ ε} and fε = {x ∈ f : d(x, e ∩ f ) ≥ ε}. note that if e ∩ f = ∅, then eε = e and fε = f . hence, a partition x = e ∪ f of x into clopen sets is u-placed iff d(e, f ) > 0. berarducci and dikranjan and pelant give the following characterizations of straight spaces in the same paper. theorem 2.1 ([3]). for a metric space x the following are equivalent: (1) x is straight; (2) whenever x is the union of two closed sets, then f ∈ c(x) is u.c. iff its restriction to each of the closed sets is u.c.; (3) every pair of closed subsets, which form a cover of x, is u-placed. according to theorem 2.1, we can conclude that the space q of rational numbers and the space r \ q of irrational numbers with the usual metric are not straight. applying lemma 1.1 and theorem 2.1, we will show the following, which says that for given straight space x, it suffices to check whether x × c is straight in order to know whether x × k is straight for any compact space k. theorem 2.2. for a straight space x x × c is straight if and only if x × k is straight for any compact space k. proof. assume that x × c is straight and let k be a compact space. from the definition of the straightness we assume that k is an infinite compact space. to show that x × k is straight, take a closed cover {e, f } of x × k and f ∈ c(x × k) on x × k such that the restrictions f|e and f|f are u.c. if we can show that f is u.c., then, from theorem 2.1, our proof is complete. 154 k. nishijima and k. yamada consider sequences {xn} and {yn} in x ×k such that lim n→∞ dx×k (xn, yn) = 0. we shall find subsequences {xkn } of {xn} and {ykn } of {yn} such that lim n→∞ |f (xkn ) − f (ykn )| = 0. we denote the projection of x × k onto k by πk . we consider the following cases. case 1: πk ({xn : n ∈ ω}) is a finite set. take a subsequence {xkn } of {xn} and z ∈ k such that πk (xkn ) = z for each n ∈ ω. case 1.1: πk ({ykn : n ∈ ω}) is a finite set. in this case, since d(xkn , ykn ) converges to 0, there is an infinite subset y ⊆ {ykn : n ∈ ω} for which πk (y ) = {z}. so we may assume that {ykn : n ∈ ω} is the infinite set. put ez = e∩(x×{z}) and fz = f ∩(x×{z}). then we can know that (i) {ez, fz} is a closed cover of x × {z} and (ii) the restrictions f|ez and f|fz are u.c. since x is straight and isometric to x ×{z}, x ×{z} is straight. it follows that the restriction f|x×{z} is u.c. observe that the sequences {xkn } and {ykn } lie in x × {z} and lim n→∞ dx×{z}(xkn , ykn ) = 0. hence, we have that lim n→∞ |f (xkn ) − f (ykn )| = lim n→∞ |f|x×{z}(xkn ) − f|x×{z}(ykn )| = 0. case 1.2: πk ({ykn : n ∈ ω}) is an infinite set. since k is compact, πk ({ykn : n ∈ ω}) contains a non-trivial convergent sequence. we may assume that πk ({ykn : n ∈ ω}) is the non-trivial convergent sequence and also πk (ykm ) 6= πk (ykn ) if m 6= n. note that d(xkn , ykn ) converges to 0. hence, it follows that z is the convergent point of the sequence {πk (ykn )}. put zn = πk (ykn ) for each n ∈ ω and z = {zn : n ∈ ω} ∪ {z}. define a mapping g : x × c → x × z by g(x, 1/n) = (x, zn) for each x ∈ x and n ∈ ω, and g(x, 0) = (x, z) for each x ∈ x. clearly, g is a uniformly homeomorphism. put h = g−1(e ∩ (x × z)), i = g−1(f ∩ (x × z)), akn = g −1(xkn ), bkn = g −1(ykn ) for each n ∈ ω, and h = f ◦ g : x × c → r. then we can show the following: (i) {h, i} is a closed cover of x × c, (ii) lim n→∞ dx×c (akn , bkn ) = 0, and (iii) h|h = f|e∩(x×z) ◦ g|h , and h|i = f|f ∩(x×z) ◦ g|i , and hence h|h and h|i are u.c. products of straight spaces with compact spaces 155 since x × c is straight, h is u.c. hence lim n→∞ |h(akn ) − h(bkn )| = 0. it follows that lim n→∞ |f (xkn ) − f (ykn )| = lim n→∞ |f (g(akn )) − f (g(bkn ))| = lim n→∞ |h(akn ) − h(bkn )| = 0. case 2: πk ({xn : n ∈ ω}) is an infinite set. since x is compact, we can pick a subsequence {xkn } of {xn} and z ∈ k such that πk (xkm ) 6= πk (xkn ) if m 6= n and {πk (xkn )} converges to z. note that lim n→∞ d(xkn , ykn ) = 0. hence, this yields that the sequence {πk (ykn )} also converges to z. let {zn : n ∈ ω} be an enumeration of πk ({xkn : n ∈ ω}∪{ykn : n ∈ ω}) such that zm 6= zn if m 6= n. then, the sequence {zn} converges to z. consider the same mapping g : x × c → x × ({zn : n ∈ ω} ∪ {z}) as in case 1.2. then, with the same argument in case 1.2, if we put akn = g −1(xkn ) and bkn = g −1(ykn ) for each n ∈ ω and h = g ◦ f , then we can show that h is u.c., and hence lim n→∞ |h(akn ) − h(bkn )| = 0. consequently, lim n→∞ |f (xkn ) − f (ykn )| = lim n→∞ |f (g(akn )) − f (g(bkn ))| = lim n→∞ |h(akn ) − h(bkn )| = 0. therefore, in any case, we can find subsequences {xkn } of {xn} and {ykn } of {yn} such that lim n→∞ |f (xkn ) − f (ykn )| = 0. it follows, from lemma 1.1, that f is u.c. consequently, x × k is straight. � the following result gives a necessary condition of x for which x × c is straight. theorem 2.3. for a straight space x if x × c is straight, then x is precompact. proof. put y = x × c. suppose that x is not precompact and pick ε > 0 and an infinite set {xn : n ∈ n} such that bε(xm) ∩ bε(xn) = ∅ if m 6= n. for each n ∈ n let an = (xn, 1 n ) ∈ y and bn = (xn, 1 n + 1 ) ∈ y . clearly, lim n→∞ dy (an, bn) = 0. hence, we can find n ∈ n such that bn ∈ bε/2(an) for every n ≥ n . put m = y \ ⋃ n≥n bε(an). then m is a closed subset of y . for each n ≥ n put an = (x × { 1 i : i ≤ n}) ∩ bε(an), bn = (x × ({ 1 i : i ≥ n + 1} ∪ {0})) ∩ bε(an). 156 k. nishijima and k. yamada note that the collection {bε(an) : n ∈ n} is closed discrete in y , and hence so are {an : n ∈ n} and {bn : n ∈ n}. if we put e = m ∪ ⋃ n≥n an and f = m ∪ ⋃ n≥n bn, then we have that (a) {e, f } is a closed cover of y , (b) e ∩ f = m , (c) for each n ≥ n d(an, e ∩ f ) = d(an, m ) ≥ d(an, y \ bε(an)) = ε, and (d) for each n ≥ n d(bn, e ∩ f ) = d(bn, m ) ≥ d(bn, y \ bε(an)) = ε 2 , because bn ∈ bε/2(an). the conditions (c) and (d) imply that {an : n ≥ n} ⊆ eε/2 and {bn : n ≥ n} ⊆ fε/2. since lim n→∞ d(an, bn) = 0, it follows that d(eε/2, fε/2) = 0. that is, the pair e and f is not u-placed. consequently, by theorem 2.1 we can prove that x × c is not straight. � remark 2.4. since r is a straight space that is not precompact, theorem 2.3 says that r × c is not straight. indeed, we can construct a pair of closed sets e and f which is not u-placed. for example, let e = ⋃ n∈n ([2n, ∞) ×{ 1 n }) and f = ⋃ n∈n ((−∞, 2n]×{ 1 n })∪r×{0}. then {e, f } is a closed cover of r×c and e ∩ f = {(2n, 1 n ) : n ∈ n}. put xn = (2n + 1, 1 n ) and yn = (2n + 1, 1 n + 1 ) for each n ∈ n. then we can see that lim n→∞ d(xn, yn) = 0, {xn : n ∈ n} ⊆ e1/2 and {yn : n ∈ n} ⊆ f1/2. hence d(e1/2, f1/2) = 0. this means that the pair e and f is not u-placed. corollary 2.5. for a complete straight space x the following are equivalent: (1) x is precompact; (2) x is compact; (3) x × c is straight; (4) x × k is straight for any compact space k. we don’t know whether the inverse implication of theorem 2.3 is true or not, however, we can show that the product space of an open interval and c is straight (cf. acknowledgment). we need the following lemmas. lemma 2.6. a metric space x which is represented as a topological sum of a family {xα : α ∈ a} of a spaces is straight if inf{d(xα, xβ ) : α 6= β} > 0. the following lemma is introduced in [3, theorem 5.3] and proved in [4, proposition 2.4]. lemma 2.7 ([3, 4]). let x be a metric space and x = k ∪ y , where k is a compact subspace of x and y is a closed subset of x. then x is straight iff y is straight. theorem 2.8. the product space of a half open interval and c is straight. products of straight spaces with compact spaces 157 proof. let x = (a, b] × c, where a < b. to show that x is straight, let e and f be closed sets in x with e ∪ f = x and take an arbitrary (small) positive number ε > 0. to avoid confusion we use notations such as exε and (e ∩ y )yε , and which mean that exε = {x ∈ e : dx (x, e ∩ f ) ≥ ε} and (e ∩ y )yε = {x ∈ e ∩ y : dy (x, e ∩ f ∩ y ) ≥ ε}, where e, f and y are subsets of a space x. according to theorem 2.1, we shall show that the pair e and f is u-placed. assuming that b = a + 1 and we can pick n ∈ n for which 1 n + 1 < ε√ 2 ≤ 1 n , put u = (a, a + ε√ 2 ) × ({ 1 n : n ≥ n + 1} ∪ {0}) and y = x \ u. case 1. u ∩ (e ∩ f ) 6= ∅. in this case, since the diameter of u is less than ε, u ⊆ bxε (e ∩ f ∩ u ) ⊆ bxε (e ∩ f ). thus (2.1) exε ∪ f xε ⊆ x \ u = y. it follows that dx (e x ε , f x ε ) = dy (e x ε , f x ε ). to show that e x ε ⊆ (e ∩ y )yε , let x ∈ exε . then x ∈ e and dx (x, e ∩ f ) ≥ ε. since x ∈ e ∩ y by (2.1) and dy (x, (e ∩ y ) ∩ (f ∩ y )) ≥ dx (x, e ∩ f ) ≥ ε, we can see that x ∈ (e ∩ y )yε . therefore exε ⊆ (e ∩ y )yε . in the same way, we can show that f xε ⊆ (f ∩ y )yε . on the other hand, lemma 2.6 and lemma 2.7 yield that y is straight, and hence dy ((e ∩ y )yε , (f ∩ y )yε ) > 0. so, we can get that dx (e x ε , f x ε ) ≥ dx ((e ∩ y )yε , (f ∩ y )yε ) = dy ((e ∩ y )yε , (f ∩ y )yε ) > 0. case 2. u ∩ (e ∩ f ) = ∅. in this case, for every p ∈ { 1 n : n ≥ n + 1} ∪ {0} (a, a + ε√ 2 ) × {p} ⊆ e ∪ f(2.2) (e ∩ f ) ∩ ((a, a + ε√ 2 ) × {p}) = ∅.(2.3) since every (a, a + ε√ 2 ) × {p} is connected, (a, a + ε√ 2 ) × {p} ⊆ e or (a, a + ε√ 2 ) × {p} ⊆ f. 158 k. nishijima and k. yamada now, we assume that (a, a + ε√ 2 ) × {0} ⊆ e. then, from the conditions (2.2) and (2.3), we can find m ≥ n + 1 such that (a, a + ε√ 2 ) × ({ 1 n : n ≥ m} ∪ {0}) ⊆ e. put v = (a, a + ε 2 √ 2 ) × ({ 1 n : n ≥ m + 1} ∪ {0}) and z = x \ v . then e ∩ f ⊆ f ⊆ z. lemma 2.6 and lemma 2.7 claim that z is straight. so, we can say that (2.4) dz ((e ∩ z)zε , (f ∩ z)zε ) > 0. here, we shall show that (2.5) exε ∩ z ⊆ (e ∩ z)zε and f xε ⊆ (f ∩ z)zε . let x ∈ exε ∩ z. then x ∈ e ∩ z and dx (x, e ∩ f ) ≥ ε. since e ∩ f ⊆ z, dz (x, (e ∩ z) ∩ (f ∩ z)) = dz (x, e ∩ f ) = dx (x, e ∩ f ) ≥ ε. it follows that x ∈ (e ∩ f )zε , and hence exε ∩ z ⊆ (e ∩ z)zε . next, let x ∈ f xε . then x ∈ f and dx (x, e ∩ f ) ≥ ε. since f ⊆ z, x ∈ f ∩ z and dz (x, (e ∩ z) ∩ (f ∩ z)) = dz (x, e ∩ f ) = dx (x, e ∩ f ) ≥ ε. it follows that x ∈ (f ∩ z)zε , and hence f xε ⊆ (f ∩ z)zε . the conditions (2.4) and (2.5) yield that (2.6) dx (e x ε ∩ z, f xε ) ≥ dx ((e ∩ z)zε , (f ∩ z)zε ) = dz ((e ∩ z)zε , (f ∩ z)zε ) > 0. furthermore, since v = (a, a + ε 2 √ 2 ) × ({ 1 n : n ≥ m + 1} ∪ {0}) and ( (a, a + ε√ 2 ) × ( 1 n : n ≥ m} ∪ {0}) ) ∩ f = ∅, we can see that dx (v, f ) > 0, and hence (2.7) dx (e x ε ∩ v, f xε ) > 0. the fact exε = (e x ε ∩ v ) ∪ (exε ∩ z) and the conditions (2.6) and (2.7) yield that dx (e x ε , f x ε ) > 0. in any case, we can get dx (e x ε , f x ε ) > 0. consequently, we can conclude that x = (a, b] × c is straight. with the same argument we can prove that [a, b) × c is also straight. � corollary 2.9. the product space of an open interval and c is straight. proof. let x = (a, b) × c, where a < b. take real numbers c and d for which a < c < d < b and put y = (a, c] × c, z = [d, b) × c and k = [c, d] × c. then theorem 2.7 and lemma 2.6 yield that y ∪z is straight. therefore, since y ∪ z is a straight closed subspace of x, k is compact and x = (y ∪ z) ∪ k, applying lemma 2.8, we can show that x is straight. � products of straight spaces with compact spaces 159 finally, we obtain the following from corollary 2.5 and corollary 2.9. corollary 2.10. the product of an open interval and a compact metric space is straight. acknowledgements. when the second author gave a talk about the results in this paper at the iii japan-mexico joint meeting in topology and its applications, he has learned from professor dikranjan that a. berarducci, d. dikranjan and j. pelant in [4] proved independently that r × c is not straight and the product space of an open interval and c is straight. after that, professor dikranjan sent us their paper [5] quite recently, in which they obtain the interesting result such as the inverse implication of theorem 2.3 is valid. therefore, we can know now that for a straight space x x × k is straight for every compact metric space k if and only if x is precompact. references [1] m. atsuji, uniform continuity of continuous functions on metric spaces, pacific j. math. 8 (1958), 11–16. [2] m. atsuji, uniform continuity of continuous functions on metric spaces, canad. j. math. 13 (1961), 657–663. [3] a. berarducci, d. dikranjan and j. pelant, an additivity theorem for uniformly continuous functions, topology appl. 146-147 (2005), 339–352. [4] a. berarducci, d. dikranjan and j. pelant, local connectedness and extension of uniformly continuous functions, preprint. [5] a. berarducci, d. dikranjan and j. pelant, products of straight spaces, preprint. received august 2005 accepted august 2006 kusuo nishijima (lkusuo@ezweb.ne.jp) faculty of education, shizuoka university, shizuoka, 422-8529 japan. kohzo yamada (eckyama@ipc.shizuoka.ac.jp) faculty of education, shizuoka university, shizuoka, 422-8529 japan. @ appl. gen. topol. 23, no. 2 (2022), 425-436 doi:10.4995/agt.2022.16940 © agt, upv, 2022 fredholm theory for demicompact linear relations aymen ammar, slim fakhfakh and aref jeribi department of mathematics, university of sfax, faculty of sciences of sfax, soukra road km 3.5, b.p 1171, 3000, sfax, tunisia (ammar aymen84@yahoo.fr, sfakhfakh@yahoo.fr, aref.jeribi@fss.rnu.tn) communicated by d. werner abstract we first attempt to determine conditions on a linear relation t such that µt becomes a demicompact linear relation for each µ ∈ [0,1) (see theorems 2.4 and 2.5). second, we display some results on fredholm and upper semi-fredholm linear relations involving a demicompact one (see theorems 3.1 and 3.2). finally, we provide some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see theorems 4.2 and 4.3). 2020 msc: 47a06. keywords: demicompact linear relations; fredholm theory; block matrix. 1. introduction throughout this work, x, y and z are vector spaces over the field k = r or c. a mapping t, whose domain is a linear subspace d(t) := {x ∈ x : tx 6= ∅} of x, is called a linear relation (or a multivalued linear operator) if for all x,z ∈d(t) and non-zero scalars α; we have tx + tz = t(x + z) αtx = t(αx). evidently, the domain of linear relation is a linear subspace. received 25 december 2021 – accepted 10 may 2022 http://dx.doi.org/10.4995/agt.2022.16940 a. ammar, s. fakhfakh and a. jeribi in this notation, lr(x,y ) denotes the class of all linear relations on x into y , if x = y simply denotes lr(x,x) := lr(x). if t maps the points of its domain to singletons, then it is said to be a single valued linear operator (or simply an operator). the simplest naturally occurring example of a multivalued linear operator is the inverse t−1 of a linear map t from x to y defined by the set of solutions t−1y := {x ∈ x : tx = y} for equation tx = y. each linear relation is identified only by its graph, g(t), which is defined by g(t) := {(x,y) ∈ x ×y : x ∈d(t) and y ∈ tx}. the inverse of t is the linear relation, t−1 expressed by g(t−1) := {(y,x) ∈ y ×x : (x,y) ∈ g(t)}. the subspace n(t) := {x ∈d(t) such that (x, 0) ∈ g(t)} is called the null space of t , and t is called injective if n(t) = {0}, that is, if t−1 is a single valued linear operator. t−1(0) := n(t). the range of t is the subspace r(t) := {y ∈ y,∃x ∈d(t) : (x,y) ∈ g(t)} and t is called surjective if r(t) = y . if t is injective and surjective, then we state that t is bijective. the quantities α(t) := dim (n(t)) and β(t) := codim(r(t)) = dim(y/r(t)) are called the nullity (or the kernel index) and the deficiency of t, respectively. we also write β(t) := codim(r(t)). the index of t is defined by i(t) := α(t) − β(t). if α(t) and β(t) are infinite, then t is said to have no index. let m be a subspace of x such that m ∩d(t) 6= ∅ and let t ∈ lr(x,y ); then, the restriction t|m , is the linear relation indicated by g(t|m ) := {(m,y) ∈ g(t) : m ∈ m} = g(t) ∩ (m ×y ). for s, t ∈lr(x,y ) and r ∈lr(y,z), the sum s + t and the product rs are the linear relations determined by g(t + s) := {(x,y + z) ∈ x ×y : (x,y) ∈ g(t) and (x,z) ∈ g(s)}, and g(rs) := {(x,z) ∈ x ×z : (x,y) ∈ g(s), (y,z) ∈ g(r) for some y ∈ y}, respectively and if λ ∈ k, the λt is computed by g(λt) := {(x,λy) : (x,y) ∈ g(t)}. if t ∈lr(x) and λ ∈ k, then the linear relation λ−t is identified by g(λ−t) := {(x,y −λx) : (x,y) ∈ g(t)}. © agt, upv, 2022 appl. gen. topol. 23, no. 2 426 fredholm theory for demicompact linear relations let t ∈lr(x,y ). we write qt for the quotient map from y into y/t(0). clearly, qtt is an operator. for all x ∈ d(t), we define ‖tx‖ := ‖qttx‖, and the norm of t is defined by ‖t‖ := ‖qtt‖. we note that ‖tx‖ and ‖t‖ are not real norms. in fact, a non-zero relation can have a zero norm. t is said to be closed if its graph g(t) is a closed subspace of x ×y . the closure of t denoted by t is defined in terms of its graph g(t) := g(t). we denote by cr(x,y ) the class of all the closed linear relations on x into y , if x = y which simply denotes cr(x,x) := cr(x). if t is an extension to t, we say that t is closable. let t ∈lr(x,y ). we say that t is continuous if for each neighbourhood v in r(t), the inverse image t−1(v ) is a neighbourhood in d(t) equivalently if ‖t‖ < ∞; open if t−1 is continuous, bounded if d(t) = x and t is continuous, bounded below if it is injective and open and compact if qtt(bd(t)) is compact in y (bd(t) := {x ∈d(t) : ‖x‖≤ 1}). we denote by kr(x,y ) the class of all the compact linear relations on x into y , if x = y simply denotes kr(x,x) := kr(x). if x is a normed linear space, then x ′ will denote the dual space of x, i.e., the space of all the continuous linear functionals x′ which are defined on x, with the norm ‖x′‖ = inf{λ : |x′x| ≤ λ‖x‖ for all x ∈ x}. if k ⊂ x and l ⊂ x ′ , we shall adopt the following notations: k⊥ := {x′ ∈ x ′ : x′ = 0 for all x ∈ k}, l> := {x ∈ x : x′ = 0 for all x′ ∈ l}. clearly, k⊥ and l> are closed linear subspaces of x ′ and x, respectively. let t ∈lr(x,y ). the adjoint of t, which is t ′, is defined by g(t ′) = g(−t−1)⊥ ⊂ y ′ ×x ′ where 〈 (y,x), (y′,x′) 〉 := 〈x,x′〉 + 〈y,y′〉. this means that (y′,x′) ∈ g(t ′) if, and only if, y′y −x′x = 0 for all (x,y) ∈ g(t). similarly, we have y′y = x′x for all y ∈ tx, x ∈d(t). hence, x′ ∈ t ′y if, and only if, y′tx = x′x for all x ∈d(t). definition 1.1 ([7, definition, v.1.1]). (i) a linear relation t ∈lr(x,y ) is said to be upper semi-fredholm and denoted by t ∈f+(x,y ), if there exists a finite codimensional subspace m of x for which t|m is injective and open. (ii) a linear relation t is said to be lower semi-fredholm and denoted by t ∈f−(x,y ), if its conjugate t ′ is upper semi-fredholm. if x = y , this simply denotes f+(x,y ) and f−(x,y ) by respectively f+(x) and f−(x). for the case, when x and y are banach spaces, we extend the class of closed single valued fredholm type operators provided earlier to include closed multivalued operators. note that the definitions of f+(x,y ) and f−(x,y ) are, © agt, upv, 2022 appl. gen. topol. 23, no. 2 427 a. ammar, s. fakhfakh and a. jeribi respectively, consistent with φ+(x,y ) := {t ∈cr(x,y ) : r(t) is closed, and α(t) < ∞}, φ−(x,y ) := {t ∈cr(x,y ) : r(t) is closed, and β(t) < ∞}. if x = y , this simply denotes φ+(x,y ) and φ−(x,y ) by respectively φ+(x) and φ−(x). lemma 1.2 ([1, lemma 2.1]). let t : d(t) ⊆ x −→ y be a closed linear relation. then, (i) t ∈ φ+(x,y ) if, and only if, qtt ∈ φ+(x,y/t(0)). (ii) t ∈ φ−(x,y ) if, and only if, qtt ∈ φ−(x,y/t(0)). definition 1.3 ([9]). let x be a banach space. let d be a bounded subset of x. we define γ(d), the kuratowski measure of noncompactness of d, to be inf{d > 0 such that d can be covered by a finite number of sets of a diameter less than or equal to d }. the following proposition displays some properties of the kuratowski measure of noncompactness which are frequently used. proposition 1.4 ([9]). let d and d′ be two bounded subsets of x. then, we have the following properties: (i) γ(d) = 0 if, and only if, d is relatively compact. (ii) if d ⊆ d′, then γ(d) ≤ γ(d′). (iii) γ(d + d′) ≤ γ(d) + γ(d′). (iv) for every α ∈ c, γ(αd) = |α|γ(d). the linear relations, which were introduced into a functional analysis by j. von neumann, were motivated by the need to consider adjoints of nondensely defined linear differential operators. these linear relations were widely investigated in a large number of papers (see, for example, [2], [3] and [5]). the notion of demicompactess for linear operators (that is, single valued operators) was introduced into the functional analysis by w.v petryshyn [10], to discuss fixed points. since this notion has be come a hot area of research triggering significant scientific concern, several research papers such as [8, 10] invested it in their investigation. in 2012, w. chaker, a. jeribi and b. krichen achieved some results on fredholm and upper semi-fredholm operators involving demicompact operators [6]. in what follows, we shall present two definitions set forward by a. ammar, h. daoud and a. jeribi in 2017 [4], who extended the concept of demicompact and k-set-contraction of linear operators on multivalued linear operators and developed some pertinent properties. definition 1.5 ([4, definition 3.1]). a linear relation t : d(t) ⊆ x −→ x is said to be demicompact if for every bounded sequence {xn} in d(t), such © agt, upv, 2022 appl. gen. topol. 23, no. 2 428 fredholm theory for demicompact linear relations that qi−t (i − t)xn = qt (i − t)xn → x ∈ x/t(0), there is a convergent subsequence of qtxn. definition 1.6 ([4, definition 4.1]). t : d(t) ⊆ x → y is a linear relation, while δ1 and δ2 are respectively kuratowski measures of noncompactness in x/d and y , where d is a closed subspace of n(t). let k ≥ 0, t is said to be k −d-set-contraction if, for any bounded subset b of d(t), qtt(b) is a bounded subset of y/t(0) and δ2(qttb) ≤ kδ1(qdb). if d = {0}, then t is said to be k−{0}-set-contractive linear relation or simply k-set-contractive. according to these definitions and referring to certain notations and some basic concepts of demicompact linear relations, we elaborate the following propositions. proposition 1.7. let t : d(t) ⊆ x −→ x be a closed single-valued linear operator. (i) [6, theorem 4] if t is demicompact, then i − t is an upper single-valued linear operator semi-fredholm. (ii) [6, theorem 5] if µt is demicompact for each µ ∈ [0, 1], then i − t is a single-valued linear operator fredholm and i(i −t) = 0. the basic objective of this paper is to attempt to answer the following question ”under which conditions does the linear relation µt for each µ ∈ [0, 1) become a demicompact linear relation ?” subsequently, we shall exhibit some results on fredholm linear relations and upper semi-fredholm demicompact linear relations. thereafter, we shall display some results about a demicompact block matrix of linear relations. the rest of the current paper is organized as follows. in section 2 which is entitled ”auxiliary results on demicompact linear relation”, we provide conditions so that any linear relation becomes a demicompact linear relation and we present the results deriving from these demicompact relations (see theorems 2.4 and 2.5). in section 3 which is entitled ”fredholm and upper semi-fredholm linear relations”, we investigate fredholm linear relations as well as upper semifredholm demicompact linear relations (see theorems 3.1 and 3.2). finally we exhibit some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see theorems 4.2 and 4.3). 2. auxiliary results on demicompact linear relations in this section, we try to answer the following question ”under which conditions does the linear relation µt for each µ ∈ [0, 1) become a demicompact linear relation ?” we then present some fundamental results about demicompact linear relations. lemma 2.1. let t : d(t) ⊆ x −→ x be a linear relation. if i − qt is compact, then t is demicompact if, and only if, qtt is demicompact. © agt, upv, 2022 appl. gen. topol. 23, no. 2 429 a. ammar, s. fakhfakh and a. jeribi proof. we suppose that t is demicompact. let {xn} be a bounded sequence of d(t) such that xn −qttxn → y. we have xn −qttxn = (i −qt )xn + qtxn −qttxn.(2.1) based upon eq. (2.1) and considering that i−qt is compact and {xn−qttxn} is a convergent sequence, {qtxn − qttxn} has a convergent subsequence. when the latter is added to demicompact t , we get {qtxn}, as a convergent subsequence. on the other side, we have xn = xn −qtxn + qtxn = (i −qt )xn + qtxn. since i −qt is compact and {qtxn} has a convergent subsequence, {xn} has a convergent subsequence. conversely, we suppose that qtt is demicompact. let {xn} be a bounded sequence of d(t) such that qtxn −qttxn → y. we have qtxn −qttxn = −(i −qt )xn + xn −qttxn.(2.2) according to eq. (2.2), and considering the fact that i −qt is compact and {qtxn −qttxn} is a convergent sequence, we infer that {xn −qttxn} has a convergent subsequence. bearing in mind the fact that qtt is demicompact and {xn − qttxn} has a convergent subsequence, we obtain {xn} as a convergent subsequence. on the other side, we have qtxn = qtxn −xn + xn = −(i −qt )xn + xn. besides, we have i − qt which is compact and {xn} which has a convergent subsequence. thus, {qtxn} has a convergent subsequence. � proposition 2.2. let t : d(t) ⊆ x −→ x be a continuous linear relation. if t is a k −t(0)-set-contraction, then µt is demicompact for each µk < 1. proof. let {xn} be a bounded sequence of d(t) such that qµtxn−qµtµtxn → y. we have qµtxn = qµt (xn −µtxn) + qµtµtxn.(2.3) suppose that γ({qµtxn}) 6= 0. therefore, using eq. (2.3) and proposition 1.4, we obtain γ({qµtxn}) ≤ γ({qµt (xn −µtxn)}) + γ({qµtµtxn}) ≤ µkγ({qµtxn}) < γ({qµtxn}). however, the result is not accurate. it follows that γ({qµtxn}) = 0. hence, {qµtxn} is relatively compact. � an immediate consequence of proposition 2.2 is the following corollary: corollary 2.3. let k ≥ 0 and t : d(t) ⊆ x −→ x be a continuous linear relation. if t is a k −t(0)-set-contraction, then 1 1+k t is demicompact. © agt, upv, 2022 appl. gen. topol. 23, no. 2 430 fredholm theory for demicompact linear relations theorem 2.4. let t : d(t) ⊆ x −→ x be a linear relation. if m ∈ n∗, (qtµt) m is compact for each µ ∈ [0, 1) and i − qµt is compact, then µt is demicompact for each µ ∈ [0, 1). proof. let {xn} be a bounded sequence of d(t) such that yn = qµtxn −qµtµtxn → y. let’s consider the various cases for m: case 1: for m = 1. using lemma 2.1, we notice that, µt is demicompact for each µ ∈ [0, 1). case 2: for m ∈ n∗ \{1}, we have m−1∑ k=0 (qµtµt) kqµtxn = m−1∑ k=0 (qµtµt) kyn + m−1∑ k=0 (qµtµt) k+1xn qµtxn + m−1∑ k=1 (qµtµt) kqµtxn = m−1∑ k=0 (qµtµt) kyn + m−2∑ k=0 (qµtµt) k+1xn +(qµtµt) mxn qµtxn + m−2∑ k=0 (qµtµt) k+1qµtxn = m−1∑ k=0 (qµtµt) kyn + m−2∑ k=0 (qµtµt) k+1xn +(qµtµt) mxn. since qttqt and (qtt) nqt are single-valued linear operators for all n ≥ 1, we get m−2∑ k=0 (qµtµt) k+1qµtxn which is single-valued. therefore, qµtxn = m−1∑ k=0 (qµtµt) kyn + m−2∑ k=0 (qµtµt) k+1(i −qµt )xn +(qµtµt) mxn. as a matter of fact, γ({qµtxn}) ≤ m−1∑ k=0 γ((qµtµt) k)γ({yn}) + γ((qµtµt)m)γ({xn}) + m−2∑ k=0 γ((qµtµt) k+1)γ(i −qµt )γ({xn}) = 0. we conclude that γ({qµtxn}) = 0. hence, {qµtxn} is relatively compact. thus, there is a convergent subsequence of {qµtxn}. � theorem 2.5. let t : d(t) ⊆ x −→ x be a linear relation and k ≥ 0. (i) if m ∈ n∗, (qtt)m and i −qµt are compact, then 11+kt is demicompact. © agt, upv, 2022 appl. gen. topol. 23, no. 2 431 a. ammar, s. fakhfakh and a. jeribi (ii) if (qtµt) m is compact for each µ ∈ [0, 1) and m > 0, then µt is demicompact for each µ ∈ [0, 1). (iii) if m ∈ n∗, γ((qtµt)m) ≤ k and i −qµt is compact, then µt is demicompact for each 0 ≤ µmk < 1. (iv) if m ∈ n∗, γ(tm) ≤ k and i−q 1 1+k t is compact, then 1 1+k t is demicompact. proof. (i) an immediate consequence of theorem 2.4 for µ = 1 1+k . (ii) likewise, based on the preceding proof of theorem 2.4, we obtain qµtxn = m−1∑ k=0 (qµtµt) kyn + m−2∑ k=0 (qµtµt) k+1(i −qµt )xn +(qµtµt) mxn. as a matter of fact, γ({qµtxn}) ≤ m−1∑ k=0 γ((qµtµt) k)γ({yn}) + γ((qµtµt)m)γ({xn}) + m−2∑ k=0 γ((qµtµt) k+1)γ(i −qµt )γ({xn}) = 0. we conclude that γ({qµtxn}) = 0. hence, {qµtxn} is relatively compact. thus, there is a convergent subsequence of {qµtxn}. (iii) let {xn} be a bounded sequence of d(t) such that yn = qµtxn − qµtµtxn → y. suppose that γ({qµtxn}) 6= 0. we have γ({qµtxn}) ≤ m−1∑ k=0 γ((qµtµt) k)γ({yn}) + γ((qµtµt)m)γ({xn}) + m−1∑ k=1 γ((qµtµt) k)γ(i −qµt )γ({xn}) ≤ µmγ((qµtt)m)γ({xn}) ≤ µmγ((qtt)m)γ({xn}) ≤ µmkγ({xn}) < γ({xn}). however, this result is not accurate. it follows that γ({qµtxn}) = 0. hence, {qµtxn} is relatively compact. (iv) an immediate consequence of (iii) for µ = 1 1+k resides in the fact that, we have k (1+k)m < 1 for each k ≥ 0. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 432 fredholm theory for demicompact linear relations 3. fredholm and upper semi-fredholm demicompact linear relations in this section, we set forward some results on fredholm and upper semifredholm linear relations involving demicompact linear relations. in particular, in both theorems stated below, we extend proposition 1.7, to linear relations: theorem 3.1. let t : d(t) ⊆ x −→ x be a closed linear relation. if t is demicompact and i − qt is compact, then i − t is an upper semi-fredholm relation. proof. let t be a demicompact and i−qt be a compact linear relation. using lemma 2.1, we infer that qtt is demicompact. based on the latter and using proposition 1.7 (i), we obtain i−qtt which is an upper semi-fredholm single valued linear operator. on the other side, qi−t (i −t) = qt (i −t) = qti −qtt + i − i = −(i −qt )i + i −qtt. since i−qt is compact and i−qtt is an upper single valued linear operator semi-fredholm, we notice that qi−t (i − t) is an upper single valued linear operator semi-fredholm. using lemma 1.2, we obtain i−t which is an upper semi-fredholm relation. � theorem 3.2. let t : d(t) ⊆ x −→ x be a closed linear relation. if µt is demicompact and i −qt is compact, then i −t is a fredholm relation and i(i −t) = 0. proof. let t be a demicompact linear relation and i−qt be a compact operator. applying lemma 2.1, we get qtt which is a demicompact linear relation. using proposition 1.7 (ii) and demicompact qtt , we obtain i −qtt , which is a single valued linear operator fredholm and i(i −qtt) = 0. on the other side, qi−t (i −t) = qt (i −t) = qti −qtt + i − i = −(i −qt )i + i −qtt. moreover, we have i − qt which is compact and i − qtt which is a single valued linear operator fredholm and i(i −qtt) = 0. therefore, qi−t (i −t) is a fredholm operator of index zero. using lemma 1.2, we notice that i −t is a fredholm relation and i(i −t) = 0. � proposition 3.3. let t : d(t) ⊆ x −→ x be a continuous linear relation. if t is demicompact, k −t(0) is a set-contraction and i −qt is compact, then i −t is a fredholm relation and i(i −t) = 0. © agt, upv, 2022 appl. gen. topol. 23, no. 2 433 a. ammar, s. fakhfakh and a. jeribi proof. since t is demicompact, k − t(0) is a set-contraction and i − qt is compact, grounded on corollary 2.3, we deduce that 1 1+k t is demicompact. using theorem 3.2, we obtain i−t which is fredholm relation and i(i−t) = 0. � 4. demicompact block matrix of linear relations in this section, a block matrix of linear relations l is identified. afterwards, some results, where this block matrix of linear relations l becomes a demicompact block matrix of linear relations, are displayed. in the banach space x ⊕y , we consider the linear relation l provided by the block matrix of linear relations (4.1) l = ( a b c d ) , where a : d(a) ⊆ x −→ x, b : d(b) ⊆ y −→ x, c : d(c) ⊆ x −→ y and d : d(d) ⊆ y −→ y are linear relations with their natural domain d(l) := ( d(a) ∩d(c) ) ⊕ ( d(b) ∩d(d) ) . the graph of l is defined by g(l) := {( (x1,x2),(y1,y2) ) : (x1,x2) ∈ d(l), y1 ∈ ax1 + bx2 and y2 ∈ cx1 + dx2 } . lemma 4.1 ([5, remark 2.3]). let l = ( a b c d ) be a block matrix of linear relations where a : d(a) ⊆ x −→ x, b : d(b) ⊆ y −→ x, c : d(c) ⊆ x −→ y and d : d(d) ⊆ y −→ y . if b(0) ⊂ a(0) and c(0) ⊂ d(0), then qll = ( qaa qab qdc qdd ) . theorem 4.2. let a : d(a) ⊆ x −→ x and d : d(d) ⊆ y −→ y be two demicompact linear relations. then, m = ( a 0 0 d ) is a demicompact linear relation. proof. let {tn} = ( xn yn ) be a bounded sequence of d(m) such that qmtn − qmmtn is convergent. we have qmtn −qmmtn = ( qa 0 0 qd ) ( xn yn ) − ( qaa 0 0 qdd ) ( xn yn ) = ( qaxn −qaaxn qdyn −qddyn ) . since {xn} is a bounded sequence of d(a), qaxn−qaaxn are convergent and a is a demicompact linear relation; then {qaxn} has a convergent subsequence. similarly, we get {qdyn} which has a convergent subsequence. hence, {qmtn} has a convergent subsequence. © agt, upv, 2022 appl. gen. topol. 23, no. 2 434 fredholm theory for demicompact linear relations � theorem 4.3. let a : d(a) ⊆ x −→ x and d : d(d) ⊆ y −→ y be two demicompact linear relations and let b : d(b) ⊆ y −→ x and c : d(c) ⊆ x −→ y be two linear relations. if qa(i −b) and qd(i −c) are compact and b(0) ⊆ a(0) and c(0) ⊆ d(0), then l = ( a b c d ) is a demicompact linear relation. proof. let {tn} = ( xn yn ) be a bounded sequence of d(l) such that qltn −qlltn is convergent. using lemma 4.1, we obtain qltn −qlltn = ( qa qa qd qd ) ( xn yn ) − ( qaa qab qdc qdd ) ( xn yn ) = ( qaxn + qayn −qaaxn −qabyn qdxn + qdyn −qdcxn −qddyn ) = ( qaxn −qaaxn + qa(i −b)yn qdyn −qddyn + qd(i −c)xn ) . we have {xn} which is a bounded sequence and qa(i −b) which is compact. then, {qa(i − b)yn} is bounded. since {qaxn − qaaxn + qa(i − b)yn} is convergent and {qa(i−b)yn} is bounded, {qaxn−qaaxn} is convergent. subsequently, using the fact that a is a demicompact linear relation, {qaxn}, therefore, has a convergent subsequence. similarly, we get {qdyn} which has a convergent subsequence. as a matter of fact, {qltn} has a convergent subsequence. � an immediate consequence of theorem 4.3 is the following corollary: corollary 4.4. let a : d(a) ⊆ x −→ x and d : d(d) ⊆ y −→ y be two demicompact linear relations and let b : d(b) ⊆ y −→ x and c : d(c) ⊆ x −→ y be two linear relations. thus, the block matrix l = ( a b c d ) is a demicompact linear relation, if one of the following conditions holds: a.: qa(i−b) and qc(i−d) are compact and b(0) ⊆ a(0) and d(0) ⊆ c(0). b.: qb(i−a) and qd(i−c) are compact and a(0) ⊆ b(0) and c(0) ⊆ d(0). c.: qb(i−a) and qc(i−d) are compact and a(0) ⊆ b(0) and d(0) ⊆ c(0). acknowledgements. the authors wish to express their gratitude to the referee for his valuable comments which helped to improve the quality of this paper. © agt, upv, 2022 appl. gen. topol. 23, no. 2 435 a. ammar, s. fakhfakh and a. jeribi references [1] f. abdmouleh, t. álvarez, a. ammar and a. jeribi, spectral mapping theorem for rakocević and schmoeger essential spectra of a multivalued linear operator, mediterr. j. math. 12, no. 3 (2015), 1019–1031. 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[8] a. jeribi, spectral theory and applications of linear operator and block operator matrices, springer-verlag, new york, 2015. [9] k. kuratowski, sur les espaces complets, fund. math. 15 (1930), 301–309. [10] w. v. petryshyn, remarks on condensing and k-set-contractive mappings, j. math. appl. 39 (1972),3 717–741. © agt, upv, 2022 appl. gen. topol. 23, no. 2 436 () @ appl. gen. topol. 14, no. 2 (2013), 179-193doi:10.4995/agt.2013.1671 c© agt, upv, 2013 discrete dynamics on noncommutative cw complexes vida milani a and seyed m. h. mansourbeigi b a dept. of math., faculty of math. sci., shahid beheshti university, tehran 1983963113, iran. school of mathematics, georgia institute of technology, atlanta ga 30332, usa (v-milani@sbu.ac.ir; vmilani3@math.gatech.edu) b dept. of electrical engineering, polytechnic university, ny 11747, usa department of electrical engineering, sccc, brentwood, ny 11717, usa (mansous@sunysuffolk.edu) abstract the concept of discrete multivalued dynamical systems for noncommutative cw complexes is developed. stable and unstable manifolds are introduced and their role in geometric and topological configurations of noncommutative cw complexes is studied. our technique is illustrated by an example on the noncommutative cw complex decomposition of the algebra of continuous functions on two dimensional torus. 2010 msc: 46l85, 55u10, 54h20, 34d35. keywords: closed hemi-continuous, c*-algebra, cw complexes, discrete dynamical system, modified morse function, noncommutative cw complex, open hemi-continuous, stable manifold, unstable manifold. 1. introduction the theory of cw complexes was invented by whitehead in 1949 [14]. the concept of cw complex structures on topological manifolds has been a great development in the category of topological spaces [8]. it is a well known fact that the topology of a manifold can be reconstructed from the commutative c*-algebra of continuous functions on it [7, 10]. in other words commutative c*-algebras play as the dual concept for topological manifolds. away from received january 2013 – accepted august 2013 http://dx.doi.org/10.4995/agt.2013.1671 v. milani and s. m. h. mansourbeigi commutativity, c*-algebras are still substitutes for noncommutative topological manifolds and provide building blocks of noncommutative topology theory [2, 4, 10]. in the category of noncommutative manifolds, noncommutative cw complexes were introduced in [6, 13]. a great development in the theory of noncommutative topology would be the study of noncommutative manifolds (c*-algebras) which are endowed with a noncommutative cw complex structure. in the study of cw complexes there exist two classical approaches. one approach comes from differential topology and morse theory [11]. the second one is the dynamics point of view and the relation between dynamical properties of a flow and the homological configuration of the cw complex. our aim is to develop the two approaches in the framework of noncommutative topology in order to study noncommutative cw complexes: dynamics primitive spectrum −−−−−−−−−−−−→ nccw complexes c*-algebra ←−−−−−−− diff. topology in this regard our first attempt was the development of the morse theory approach in [12]. in the present paper we are developing the second approach. in both approaches we apply techniques from combinatorial topology [3, 5] and the primitive spectrum of c*-algebras [10] as basic tools. the paper is organized as follows. in section 2 we review fundamental notions in the theory of noncommutative cw complexes. we explain the role of the primitive spectrum as a bridge between cw complexes and noncommutative cw complexes. section 3 is devoted to a review from [12] on the basics of modified morse theory on noncommutative cw complexes. discrete multivalued dynamical systems have been introduced in [1, 9]. in section 4 we develop discrete multivalued dynamical systems on noncommutative cw complexes and provide tools to relate a dynamical picture to the topology and geometry of noncommutative cw complexes. we will see how the dynamical properties of the trajectories are related to the configuration of noncommutative cw complexes. in this section, stable and unstable manifolds are introduced and some of their properties are studied. an example will serve to illustrate our dynamical construction. section 5 is devoted to the explanation of this example. in this section we study the noncommutative cw complex structure of c(t 2): the algebra of continuous functions on the 2-dimensional torus. we associate a discrete multivalued dynamical system with it. we shall see how the configuration of this noncommutative cw complex is explained by the stable and unstable manifolds. 2. noncommutative cw complexes in this section we review basic definitions and results on the theory of noncommutative cw complexes from [6, 13]. we explain the technique of the primitive spectrum and its role as a link between cw complexes and noncommutative cw complexes. details on the structure of primitive spectrum can be found in [7, 10, 12]. first we review the concept of cw complex structure for a topological space from [8]. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 180 discrete dynamics on noncommutative cw complexes a sequence x0 ⊂ x1 ⊂ ... ⊂ xn = x is an n-dimensional cw complex structure for a compact topological space x, where x0 is a finite discrete space consisting of 0-cells, and for k = 1, ...,n each k-skeleton xk is obtained by attaching λk number of k-disks to xk−1 via the attaching maps ϕk : ⋃ λk sk−1 → xk−1. in other words (2.1) xk = xk−1 ⋃ (∪λki k) x ∼ ϕk(x) := xk−1 ⋃ ϕk (∪λki k) where ik := [0,1]k and sk−1 := ∂ik. the quotient map is denoted by ρ : xk−1 ⋃ (∪λki k) → xk. for a continuous map φ : x → y between compact topological spaces x and y , the c*-morphism induced on their associated c*-algebra of functions is denoted by c(φ) : c(y ) → c(x) which is defined by c(φ)(g) := g ◦φ for g ∈ c(y ). definition 2.1. let a1, a2 and c be c*-algebras. a pull back for c via morphisms α1 : a1 → c and α2 : a2 → c is the c*-subalgebra of a1 ⊕ a2 denoted by pb(c,α1,α2) defined by pb(c,α1,α2) := {a1 ⊕a2 ∈ a1 ⊕a2 : α1(a1) = α2(a2)}. for any c*-algebra a, let sna := c(sn → a),ina := c([0,1]n → a),in0 a := c0((0,1) n → a), where sn is the n-dimensional unit sphere. definition 2.2. a 0-dimensional noncommutative cw complex is any finite dimensional c*-algebra a0. recursively an n-dimensional noncommutative cw complex is any c*-algebra appearing in the following diagram 0 −−−−→ in0 fn −−−−→ an π −−−−→ an−1 −−−−→ 0 ∥ ∥ ∥   y fn   y ϕn 0 −−−−→ in0 fn −−−−→ i nfn δ −−−−→ sn−1fn −−−−→ 0 where the rows are extensions, an−1 an (n−1)-dimensional noncommutative cw complex, fn some finite (linear) dimensional c*-algebra of dimension λn, δ the boundary restriction map, ϕn an arbitrary morphism (called the connecting morphism), for which an = pb(s n−1fn,δ,ϕn) := {(α,β) ∈ i nfn ⊕an−1 : δ(α) = ϕn(β)}, and fn and π are respectively projections onto the first and second coordinates. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 181 v. milani and s. m. h. mansourbeigi with these notations {a0, ...,an} is called the noncommutative cw complex decomposition of dimension n for a = an. for each k = 0,1, ...,n, ak is called the k-th decomposition cell. let a be a unital c*-algebra. the primitive spectrum of a is the space of kernels of irreducible *-representations of a. it is denoted by prim(a). the topology on this space is given by the closure operation as follows: for any subset u ⊆ prim(a), the closure of u is defined by u := {i ∈ prim(a) : ⋂ j∈u j ⊂ i} obviously u ⊆ u. this operation defines a topology on prim(a) (the hullkernel topology), making it into a t0-space [10]. definition 2.3. a subset u ⊆ prim(a) is called absorbing if it satisfies the following condition: i ∈ u,i ⊆ j ⇒ j ∈ u. remark 2.4. the closed subsets of prim(a) are exactly its absorbing subsets. in the special case, when m is a compact topological space, and a = c(m) is the commutative unital c*-algebra of complex continuous functions on m, a homeomorphism between m and prim(a) is obtained in the following way. for each x ∈ m let ix := {f ∈ a : f(x) = 0}; ix is a closed maximal ideal of a. it is in fact the kernel of the evaluation map (ev)x :a −→ c f 7−→ f(x). now i : m → prim(a) defined by i(x) := ix is the desired homeomorphism. let x0 ⊂ x1 ⊂ ... ⊂ xn = x be an n-dimensional cw complex structure for the compact space x. a cell complex structure is induced on prim(c(x)) by the following procedure: let ak = c(xk), k = 0,1, ...,n. set a = c(x) = c(xn) = an. consider the homeomorphism i : x → prim(c(x)). for each k-cell ck in the k-skeleton xk, let ick = ⋂ x∈ck ix = {f ∈ a : f(x) = 0;x ∈ ck}, for 0 ≤ k ≤ n . by considering the restriction of functions on x to xk, ick will be an ideal in ak. in the above notations, the closed sets wi0,...,ik := {j ∈ prim(ak) : j ⊇ ick} are corresponded to the ideals ick. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 182 discrete dynamics on noncommutative cw complexes in general we can have proposition 2.5 ([12]). let x be an n-dimensional cw complex containing cells of each dimension k = 0, ...,n. then there exists a noncommutative cw complex decomposition of dimension n for a = c(x). conversely if {a0, ...,an} be a noncommutative cw complex decomposition for the c*-algebra a such that ais (i = 0, ..,n) are unital, then there exists an n-dimensional cw complex structure on prim(a). example 2.6. let x0 = {0,1}and x1 = [0,1] be the zero and the one skeleton for a cw complex structure of [0,1]. then we have a0 = c(x0) ≃ c⊕ c and a = a1 = c(x1). the 0-ideals i0 and i1 and their corresponding 0-chains w0 and w1 are as follow: i0 = {f ∈ a0 : f(0) = 0}≃ c,i1 = {f ∈ a0 : f(1) = 0}≃ c, w0 = {j ∈ prim(a0) : j ⊇ i0} = {i0},w1 = {j ∈ prim(a0) : j ⊇ i1} = {i1}. corresponding to the 1-chain c1 = [0,1], the only 1-ideal is i = ⋂ x∈c1 ix = {f ∈ a : f(x) = 0;x ∈ [0,1]} = {0}, with the corresponding 1-chain wi = {j ∈ prim(a) : j ⊇ i} = prim(a) ≃ [0,1]. proposition (2.5) can be extended to an arbitrary unital c*-algebra. let a be an arbitrary unital c*-algebra. to each i ∈ prim(a), there corresponds an absorbing set wi := {j ∈ prim(a) : j ⊇ i}, and an open set oi := {j ∈ prim(a) : j ⊆ i}, containing i. we have the following equivalent statements: i ⊆ j ⇔ oi ⊆ oj ⇔ wi ⊇ wj in [12] we have seen how prim(a) is made into a finite lattice with vertices i0, ...,in. let ji0,...,ik := ii0 ∩ ...∩iik, where 1 ≤ i0, ..., ik ≤ n,1 ≤ k ≤ n.set wi0,...,ik := {j ∈ prim(a) : j ⊇ ji0,...,ik}. as we have seen in [12], these are the k-chain closed subsets of prim(a) having the following property if ji0,...,ik = 0 for some 1 ≤ i0, ..., ik ≤ n, 1 ≤ k ≤ n, then wi0,...,ik = prim(a). also for each pair of indices (i0, ..., it) , σ(i0, ..., it+m), wi0,...,it ⊆ wσ(i0,...,it+m) c© agt, upv, 2013 appl. gen. topol. 14, no. 2 183 v. milani and s. m. h. mansourbeigi where σ is a permutation on t + m + 1 elements and 1 ≤ i0, ..., it+m ≤ n. in the case of prim(c(x)) when x has a cw complex structure, the kchains are the closed sets wi0,...,ik = {j ∈ prim(ak) : j ⊇ ick} corresponding to the k-ideals ick [12]. 3. basics of modified morse theory on c*-algebras the first step towards understanding the geometry of noncommutative cw complexes was the idea of modified morse theory on c*-algebras that we have done in [12]. in this section we review some of the results. for a unital c*-algebra a let σ = {wi1,...,ik}1≤i1,...,ik≤n,1≤k≤n be the set of all k-chains (k = 1, ...,n) in prim(a), and γ = {ii1,...,ik}1≤i1,...,ik≤n,1≤k≤n be the absorbing set of all k-ideals corresponding to the k-chains of σ for k = 1, ...,n. we recall the following definitions from [12]. definition 3.1. let f : σ → r be a function. the k-chain wk = wi1,...,ik is called a critical chain of order k for f, if for each (k+1)-chain wk+1 containing wk and for each (k −1)-chain wk−1 contained in wk, we have f(wk−1) ≤ f(wk) ≤ f(wk+1). the corresponding ideal ik to wk is called the critical ideal of order k. definition 3.2. let f has a critical chain of order k. we say f is an acceptable morse function, if it has a critical chain of order i, for all i ≤ k. definition 3.3. a function f : σ → r is called a modified morse function on the c*-algebra a, if for each k-chain wk in σ, there is at most one (k+1)-chain wk+1 containing wk and at most one (k-1)-chain wk−1 contained in wk, such that f(wk+1) ≤ f(wk) ≤ f(wk−1). definition 3.4. if a, b are two c*-algebras, two morphisms α,β : a → b are homotopic, written α ∼ β,if there exists a family {ht}t∈[0,1] of morphisms ht : a → b such that for each a ∈ a the map t 7→ ht(a) is a norm continuous path in b with h0 = α and h1 = β.the c*-algebras a and b are said to have the same homotopy type, if there exists morphisms ϕ : a → b and ψ : b → a such that ϕ◦ψ ∼ idb and ψ ◦ϕ ∼ ida. in this case the morphisms ϕ and ψ are called homotopy equivalence. definition 3.5. let a and b be unital c*-algebras. we say a is of pseudohomotopy type as b if c(prim(a)) and b have the same homotopy type. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 184 discrete dynamics on noncommutative cw complexes remark 3.6. in the case of unital commutative c*-algebras, by the gns construction [7], c(prim(a)) = a, . so the notions of pseudo-homotopy type and the same homotopy type are equivalent. theorem 3.7. if f is an acceptable modified morse function on a, then prim(a) is homotopy equivalent to a cw complex with exactly one cell of dimension p for each critical chain of order p. consequently every unital c*algebra a with an acceptable modified morse function f on it, is of pseudohomotopy type as a noncommutative cw complex having a k-th decomposition cell for each critical chain of order k. 4. dynamical systems on noncommutative cw complexes in this section we develop tools to relate a dynamical picture to the topology and geometry of noncommutative cw complexes. we will see how the dynamical properties of the trajectories are related to the homological configuration of noncommutative cw complexes. definition 4.1. let x,y be topological spaces, p(y ) be the power set of y (the set of all subsets of y ) and f : x →p(y ) be a mapping. • the mapping f is called open hemi-continuous at x ∈ x, if for each open subset b ⊆ y such that f(x) ⊆ b, there exists an open set u ⊆ x containing x such that f(u) := ⋃ {f(x) : x ∈ u}⊆ b. • the mapping f is called closed hemi-continuous at x ∈ x, if for each closed subset b ⊆ y such that f(x) ⊆ b, there exists a closed set k ⊆ x containing x such that f(k) := ⋃ {f(x) : x ∈ k}⊆ b. • the mapping f is with compact value if for all x ∈ x, f(x) ⊆ y is a compact subset. definition 4.2. let x be a topological space and p(x) be the power set of x. a mapping ϕ : x ×z →p(x) is a discrete multivalued dynamical system on x if the following conditions satisfy: • for each n ∈ z the mapping fn : x → p(x) defined by fn(x) := ϕ(x,n), for all x ∈ x, is closed hemi-continuous for n ∈ z+ and is open hemi-continuous for n ∈ z−. • the mapping f1 is with compact value. • for all x ∈ x, ϕ(x,0) = {x}. • for all n,m ∈ z with nm ≥ 0 and for all x ∈ x, ϕ(ϕ(x,n),m) = ϕ(x,n + m). • for all x,y ∈ x, x ∈ ϕ(y,−1) ⇔ y ∈ ϕ(x,1). c© agt, upv, 2013 appl. gen. topol. 14, no. 2 185 v. milani and s. m. h. mansourbeigi remark 4.3. with the above notations if we let (ϕ(x,1)) := f(x), then it follows that for all x ∈ x and n ≥ 1, ϕ(x,n) = fn(x), where fn(x) = f(fn−1(x)) := ⋃ {f(z) : z ∈ fn−1(x)} is defined inductively. so f : x → x is called the generator of the discrete multivalued dynamical system. let a be a unital c*-algebra and let prim(a) be the topological space associated with it as in the construction of the previous sections. define two mappings f,g : prim(a) →p(prim(a)) f(i) = wi = {j ∈ prim(a) : j ⊇ i} g(i) = oi = {j ∈ prim(a) : j ⊆ i} . lemma 4.4. for all i,j ∈ prim(a) we have j ∈ oi ⇔ i ∈ wj. proof. we have j ∈ oi ⇔ j ⊆ i ⇔ i ∈ wj. � for the mappings f,g defined above we have proposition 4.5. the mapping f is closed hemi-continuous and the mapping g is open hemi-continuous. proof. let i ∈ prim(a), w ⊆ prim(a) be closed and f(i) = wi ⊆ w. we show that there exists a closed subset k ⊆ prim(a) with i ∈ k and f(k) ⊆ w. set k := wi. then we have f(wi) = ⋃ {f(j) : j ∈ wi} = ⋃ {f(j) : j ⊇ i}⊆ wi ⊆ w. since for j ⊇ i, we have wj ⊆ wi. now let i ∈ prim(a), o ⊆ prim(a) be open and g(i) = oi ⊆ o. we show that there exists a open subset u ⊆ prim(a) with i ∈ u and g(u) ⊆ o. set u := oi. then we have g(oi) = ⋃ {g(j) : j ∈ oi} = ⋃ {g(j) : j ⊆ i}⊆ oi ⊆ o. since for j ⊆ i, we have oj ⊆ oi. � in the following we will see how the above mappings f,g generate a discrete multivalued dynamical system on prim(a). let ϕ : prim(a) ×z+ →p(prim(a)) be defined in the following way: for all i ∈ prim(a), set ϕ(i,1) := f(i) = wi. for n ∈ z +, define ϕ(i,n) inductively by ϕ(i,n) := fn(i) = f(fn−1(i)) = ⋃ {f(in−1) : in−1 ∈ f n−1(i)} c© agt, upv, 2013 appl. gen. topol. 14, no. 2 186 discrete dynamics on noncommutative cw complexes = ⋃ in−1⊇in−2 ... ⋃ i1⊇i f(in−1) = ⋃ in−1⊇...⊇i1⊇i f(in−1). the mapping ϕ has the following property: lemma 4.6. for all i ∈ prim(a) and all n,m ∈ z+, we have ϕ(i,n + m) = ϕ(ϕ(i,n),m). proof. we have ϕ(i,n + m) = fn+m(i). on the other hand ϕ(ϕ(i,n),m) = ⋃ {ϕ(j,m) : j ∈ ϕ(i,n)} = ⋃ {fm(j) : j ∈ fn(i)} = fm(fn(i)) = fm+n(i). � in the same way let ψ : prim(a) × z− → p(prim(a)) be defined in the following way: for all i ∈ prim(a), set ψ(i,−1) := g(i) = oi. for n ∈ z +, define ψ(i,−n) inductively by ψ(i,−n) := gn(i) = g(gn−1(i)) = ⋃ {g(in−1) : in−1 ∈ g n−1(i)} = ⋃ in−1⊆in−2 ... ⋃ i1⊆i g(in−1) = ⋃ in−1⊆...⊆i1⊆i g(in−1). the mapping ψ has the following property: lemma 4.7. for all i ∈ prim(a) and all n,m ∈ z+, we have ψ(i,−n−m) = ψ(ψ(i,−n),−m). proof. we have ψ(i,−n−m) = gn+m(i). on the other hand ψ(ψ(i,−n),−m) = ⋃ {ψ(j,−m) : j ∈ ψ(i,−n)} = ⋃ {gm(j) : j ∈ gn(i)} = gm(gn(i)) = gm+n(i) . � proposition 4.8. let f,g,ϕ,ψ be as before. let θ : prim(a) ×z →p(prim(a)) be defined by θ(i,n) = ϕ(i,n) = fn(i); θ(i,−n) = ψ(i,−n) = gn(i); θ(i,0) = {i} for all i ∈ prim(a),n ∈ z+. then θ defines a discrete multivalued dynamical system on prim(a) with generators f,g. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 187 v. milani and s. m. h. mansourbeigi proof. we have to check the properties of definition (4.2) for θ. first of all for each i ∈ prim(a), θ(i,1) is compact. moreover from proposition (4.5), the hemi-continuity property satisfies for θ. also • for all i ∈ prim(a), we have θ(i,0) = {i}. • for all n,m ∈ z with nm ≥ 0 it follows from lemmas (4.6) and (4.7), θ(i,n + m) = θ(θ(i,n),m). and eventually from the lemma (4.4), for all i,j ∈ prim(a) we have j ∈ θ(i,1) = f(i) = wi ⇔ i ∈ θ(j,−1) = g(i) = oi. � remark 4.9. with the notations of the previous proposition, if for each w ⊆ prim(a) we define f−1(w) := {j ∈ prim(a) : f(j) ⊆ w}, then the proof of the above proposition shows that g = f−1. for this reason sometimes we refer to f as the only generator of the system. definition 4.10. let a be a unital c*-algebra, θ : prim(a)×z →p(prim(a)) be a discrete multivalued dynamical system with generator f , k,m ∈ z+ and [−k,m] be an interval in z containing 0 ∈ z. let {ii}−k≤i≤m be a sequence in prim(a) such that ∀−k ≤ i ≤ m ; ii+1 ∈ f(ii). define a map α : [−k,m] → prim(a) by α(i) = ii, for all −k ≤ i ≤ m. obviously α(i + 1) ∈ f(α(i)). with these notations α is called a solution for f and the sequence{ii}−k≤i≤m is called a trajectory for f passing through α(0) = i0. with these notations: proposition 4.11. if α : [−k,m] → prim(a) is a solution for f, then for each i ∈ [−k,m], α(i) ∈ fi(α(0)). proof. we prove the statement by induction on k,m. the induction is in two parts: positive and negative parts of the interval. for k = 0,m = 1, we have α(1) ∈ f(α(0)). now suppose α(i) ∈ fi(α(0)), for 0 ≤ i ≤ m. we show that α(m + 1) ∈ fm+1(α(0)). we have fm+1(α(0)) = f(fm(α(0))) = ⋃ {f(j) : j ∈ fm(α(0))}. set j = α(m). then f(α(m)) ⊆ fm+1(α(0)). on the other hand we have α(m + 1) ∈ f(α(m)). so α(m + 1) ∈ fm+1(α(0)). so the induction on the positive part is completed. now we go through the second part of the induction. the proof of this part is the same as the first part with a minor difference. we just have to note that for k = −1,m = 0, we have α(0) ∈ f(α(−1)) = wα(−1). consequently α(−1) ∈ oα(0), which means α(−1) ∈ g(α(0)) = f −1(α(0)). now if α(i) ∈ fi(α(0)), for −k ≤ i ≤−1. we can easily see that α(−k −1) ∈ f−k−1(α(0)). � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 188 discrete dynamics on noncommutative cw complexes in what follows θ : prim(a) × z → p(prim(a)) is a discrete multivalued dynamical system on prim(a) with generator f . definition 4.12. let α be a solution for f and {ii}−k≤i≤m be a trajectory for f passing through α(0) = i0. the ideal i0 is called a fixed point for f if there exist w ⊆ prim(a) such that for all n, fn(i0) = w . consequently for all n, α(n) ∈ w . definition 4.13. the unstable manifolds of f at point i ∈ prim(a) is defined by wu(i,f) = ⋃ n≥1 fn(i). in the same way the stable manifold of f at i is defined by w s(i,f) = ⋃ n≥1 f −n(i) = ⋃ n≥1 g n(i). proposition 4.14. let i,j ∈ prim(a) and wu(i,f) ⋂ ws(j,f) 6= ∅. then there exists a trajectory {li}0≤i≤m for f from i to j, i.e. l0 = i,lm = j. proof. let l ∈ wu(i,f) ⋂ ws(j,f). then l ∈ wu(i,f) = ⋃ n≥1 f n(i). so there exists n0 ≥ 1 such that l ∈ fn0(i) = fn0−1(f(i)) = ⋃ {fn0−1(d) : d ∈ f(i)} so there exists l1 ∈ f(i) such that l ∈ fn0−1(l1) = f n0−2(f(l1)) = ⋃ {fn0−2(d) : d ∈ f(l1)} so there exists l2 ∈ f(l1) with l ∈ f n0−2(l2). continuing in this process we obtain a sequence {l1, ...,ln0} with the property that li+1 ∈ f(li) and l ∈ fn0−i(li) for all 1 ≤ i ≤ n0 −1. now the sequence {l0,l1, ...,ln0} with l0 = i,ln0 = l is a trajectory for f from i to l. on the other hand we have l ∈ ws(j,f) = ⋃ n≥1 f −n(j) = ⋃ n≥1 g n(j). so there exists m ≥ 1 such that l ∈ gm(j) = gm−1(g(j)) = ⋃ {gm−1(d) : d ∈ g(j)} so there exists dm−1 ∈ g(j) such that l ∈ gm−1(dm−1) = g m−2(g(dm−1)) = ⋃ {gm−2(d) : d ∈ g(dm−1)} so there exists dm−2 ∈ g(dm−1) with l ∈ g m−2(dm−2). continuing in this process we obtain a sequence {d1, ...,dm−1} with the property that dm−i ∈ g(dm−i+1), for all 2 ≤ i ≤ m− 1 and l ∈ g m−i(dm−i), for 1 ≤ i ≤ m − 1. this means that dm−i+1 ∈ f(dm−i), for all 2 ≤ i ≤ m − 1. set dm = j. from dm−1 ∈ g(j), it follows that j ∈ f(dm−1). now the sequence {d0,d1, ...,dm−1,dm} with d0 = l,dm = j is a trajectory for f from l to j. if we rename di = ln0+i, for 0 ≤ i ≤ m, then the sequence {l0,l1, ...,ln0+m} is a trajectory for f from i to j. � c© agt, upv, 2013 appl. gen. topol. 14, no. 2 189 v. milani and s. m. h. mansourbeigi in the next section we go through an example to have a better understanding of the constructions of this section. 5. dynamical system on c(t 2) in [12] we explained how the cw complex structure for a compact topological space x induces a noncommutative cw complex structure on the algebra c(x) of continuous functions on x. in this part we apply the techniques of the previous section to introduce a discrete multivalued dynamical system on the noncommutative cw complex structure of c(t 2): the algebra of continuous functions on the 2-dimensional torus. we compute the stable and nonstable manifolds and explain the geometry of the noncommutative cw complex by its stable and unstable manifolds. consider the following cw complex structure for the torus t 2. x0 = {0} is the one point set, x1 = {α,β}, where α,β are closed curves homeomorphic images of the circle s1, starting at point 0 and x2 = t 2. the noncommutative cw complex decomposition on c(t 2) is induced as: a0 = c(x0),a1 = c(x1),a = a2 = c(t 2). to each x ∈ x there corresponds an ideal ix ∈ prim(c(x)) defined by ix := {f ∈ c(x) : f(x) = 0}. we can partition prim(a) into three classes of ideals: there is only one 0-ideal defined by i0 := {f ∈ a : f(0) = 0}. there are two 1-ideals defined by iα := {f ∈ a : f(x) = 0;x ∈ α} = ⋂ x∈α ix, iβ := {f ∈ a : f(x) = 0;x ∈ β} = ⋂ x∈β ix. there is one 2-ideal defined by i := {f ∈ a : f(x) = 0;x ∈ t 2} = {0}. obviously i ⊆ iα,iβ ⊆ i0. we have w0 = {j ∈ prim(a) : j ⊇ i0} = {i0} wi = {j ∈ prim(a) : j ⊇ i} = {i0,iα,iβ,i} wα = {j ∈ prim(a) : j ⊇ iα} = {i0,iα} wβ = {j ∈ prim(a) : j ⊇ iβ} = {i0,iβ} and w0 ⊆ wα,wβ ⊆ wi. on the other hand we have o0 = {j ∈ prim(a) : j ⊆ i0} = {i0,iα,iβ,i} oi = {j ∈ prim(a) : j ⊆ i} = {i} oα = {j ∈ prim(a) : j ⊆ iα} = {i,iα} oβ = {j ∈ prim(a) : j ⊆ iβ} = {i,iβ} and oi ⊆ oα,oβ ⊆ o0. now we start our computations. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 190 discrete dynamics on noncommutative cw complexes • the ideal i0: we have θ(i0,1) = f(i0) = w0 = {i0}. also θ(i0,2) = f 2(i0) = f(f(i0)) = f(w0) = ⋃ {f(j) : j ∈ w0} = f(i0) = {i0}. continuing with this process, we see that for each n ≥ 1 we have θ(i0,n) = f n(i0) = {i0} = w0. we have θ(i0,−1) = g(i0) = o0 = {i0,iα,iβ,i} θ(i0,−2) = g 2(i0) = g(g(i0)) = ⋃ {g(j) : j ∈ g(i0)} = ⋃ {g(j) : j = i0,iα,iβ,i} = g(io) ⋃ g(iα) ⋃ g(iβ) ⋃ g(i) = o0 = {i0,iα,iβ,i} in the same way we see that for all n ≥ 1, θ(i0,−n) = f −n(i0) = g n(i0) = {i0,iα,iβ,i} = o0 • the ideal iα: we have θ(iα,1) = f(iα) = wα = {i0,iα}. also θ(iα,2) = f 2(iα) = f(f(iα)) = f(wα) = ⋃ {f(j) : j ∈ wα} = ⋃ {f(j) : j = i0,iα} = f(i0) ⋃ f(iα) = wα. continuing with this process, we see that for each n ≥ 1 we have θ(iα,n) = f n(iα) = wα. for the negative part we have θ(iα,−1) = g(iα) = oα = {iα,i} θ(iα,−2) = g 2(iα) = g(g(iα)) = ⋃ {g(j) : j ∈ g(iα)} = ⋃ {g(j) : j = iα,i} = g(iα) ⋃ g(i) = oα = {iα,i} in the same way we see that for all n ≥ 1, θ(iα,−n) = f −n(iα) = g n(iα) = oα = {iα,i} • the ideal iβ: for this ideal as in the case of iα we can see that for all n ∈ z +, θ(iβ,n) = f n(iβ) = {iβ,i0} θ(iβ,−n) = f −n(iβ) = {iβ,i} • the ideal i: we have θ(i,1) = f(i) = wi = {i0,iα,iβ,i}. also θ(i,2) = f2(i) = f(f(i)) = f(wi) = ⋃ {f(j) : j ∈ wi} = f(i0) ⋃ f(iα) ⋃ f(iβ) ⋃ f(i) = {i0,iα,iβ,i}. continuing with this process, we see that for each n ≥ 1 we have θ(i,n) = fn(i) = wi = {i0,iα,iβ,i} we have θ(i,−1) = g(i) = oi = {i} c© agt, upv, 2013 appl. gen. topol. 14, no. 2 191 v. milani and s. m. h. mansourbeigi θ(i,−2) = g2(i) = g(g(i)) = ⋃ {g(j) : j ∈ g(i)} = g(i) = {i} in the same way we see that for all n ≥ 1, θ(i,−n) = f−n(i) = gn(i) = {i} = oi now we explain the trajectories of the system and find the fixed points and observe the behavior of the stable and unstable manifolds at the fixed points. first we consider the sequence {i,iα,i0}. for this sequence we have iα ∈ f(i), i0 ∈ f(iα) therefore the sequence defines a trajectory for the system. now if we define a curve σ : [0,2] ⊆ z → prim(a) by σ(0) = i, σ(1) = iα, σ(2) = i0 then we have σ(n) ∈ f(σ(n − 1)) and σ(n) ∈ fn(σ(0)) for n = 1,2. on the other hand fn(σ(0)) = fn(i) = wi. so i is a fixed point for this trajectory. the unstable and stable manifolds would be wu(i,f) = ⋃ n=1,2 fn(i) = wi = {i0,iα,iβ,i}. ws(i,f) = ⋃ n≥1 f−n(i) = ⋃ n=1,2 gn(i) = oi = {i}. from the above calculations we can conclude that the whole prim(a) is unstable and the ideal i corresponded to the critical chain wi of the modified discrete function on prim(a) is stable, we refer to [12] for details on critical chains. this critical chain corresponds to the maximum point of the morse height function on t 2. since the compact torus t 2 is homeomorphic to the space prim(a) [7], this is a natural conclusion comparing to the unstability of torus. we have another beautiful interpretation: wu(i0,f) = ⋃ n fn(i0) = w0 = {i0}. ws(i0,f) = ⋃ n≥1 f−n(i0) = ⋃ n≥1 gn(i0) = o0 = {i0,iα,iβ,i}. which means that the stable and unstable manifolds are interchanged along suitable trajectories. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 192 discrete dynamics on noncommutative cw complexes references [1] m. allili, d. corriveau, s. deriviere, t. kaczynski and a. trahan, discrete dynamical system framework for construction of connections between critical regions in lattice height data, j. math. imaging and vision 28 (2007), 99–111. 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[10] g. landi, an introduction to noncommutative spaces and their geometry, lecture notes in physics monographs 51 (springer , 1998). [11] j. milnor, morse theory, annals of math. studies, (princeton univ. press, 1963). [12] v. milani, a. a. rezaei and s. m. h. mansourbeigi, morse theory for c*-algebras: a geometric interpretation of some noncommutative manifolds, applied general topology 12 (2011), 175–185. [13] g. k. pedersen, pull back and pushout constructions in c*-algebras, j. funct. analysis 167 (1999), 243–344. [14] j. h. c. whitehead, combinatorial homotopy, i. bulletin of the american society 55 (1949), 1133–1145. c© agt, upv, 2013 appl. gen. topol. 14, no. 2 193 vroegrijkagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 277-287 uniformizable and realcompact bornological universes tom vroegrijk abstract. bornological universes were introduced some time ago by hu and obtained renewed interest in recent articles on convergence in hyperspaces and function spaces and optimization theory. one of hu’s results gives us a necessary and sufficient condition for which a bornological universe is metrizable. in this article we will extend this result and give a characterization of uniformizable bornological universes. furthermore, a construction on bornological universes that the author used to find the bornological dual of function spaces endowed with the bounded-open topology will be used to define realcompactness for bornological universes. we will also give various characterizations of this new concept. 2000 ams classification: 54e15, 54e35, 54c25. keywords: bornology; uniform space; totally bounded; realcompactness. 1. introduction a bornological universe (x, b) consists of a topological space x and a bornology b on the underlying set of x. bornological universes play a key role in recent publications on convergence structures on hyperspaces [2, 3, 4, 14], topologies on function spaces [7] and optimization theory [5, 6]. in [11] hu defines a bornological universe (x, b) to be metrizable if there is a metric d on x such that the topology defined by d is the original topology on x and the elements of b are exactly the sets that are bounded for this metric. in this article the following characterization of metrizable bornological universes is given: 278 t. vroegrijk theorem 1.1. a bornological universe (x, b) is metrizable iff the following conditions are satisfied: (1) x is metrizable, (2) b has a countable base, (3) for each b1 ∈ b there is a b2 ∈ b such that b1 ⊆ ◦ b2 keeping this result of hu in mind we can ask ourselves when a bornological universe is uniformizable. the first problem that we encounter is how to define uniformizability for bornological universes. each uniform space has an underlying topological space, but there are (at least) two natural bornologies that we can associate with a uniformity: the bornology of sets that are bounded in the sense of bourbaki and the bornology of totally bounded sets. this gives us two possible ways to define uniformizable bornological universes. we will see, however, that both are equivalent. one of the various characterizations of uniformizability for bornological universes that we will give is being isomorphic to a subspace of a product of real lines. this property was also studied by beer in [1], but his definition of a product bornology is different from the one that we will use here. in [16] the author uses generalized bornologies (see [15]) to extend the duality between locally convex topological vector spaces and bornological vector spaces to vector spaces with a topology defined by extended quasinorms. one of the objects that is encountered in this article is the space c(x) of continuous, real valued maps on a tychonoff space x endowed with the bounded-open topology for some bornology b on x. to describe the bornological dual of this object a realcompact extension υb(x) of x is introduced. the way this space υb(x) is defined for a bornological universe (x, b) is similar to the definition of the hewitt realcompactification for topological spaces. we will use this to define realcompactness for bornological universes. 2. uniformizable bornological universes definition 2.1. a bornology on a set x is a set b ⊆ 2x that satisfies the following conditions: b1 {x} ∈ b for each x ∈ x, b2 if b ∈ b and a ⊆ b, then a ∈ b, b3 a ∪ b ∈ b whenever a, b ∈ b. the elements of a bornology are called bounded sets and a map that preserves boundedness is called a bounded map. a topological space x endowed with a bornology b will be called a bornological universe. it is easily verified that the category of bornological universes and bounded, continuous maps is a topological construct. a source (fi : (x, b) → (xi, bi))i∈i in this category is initial if x is endowed with the topology that is initial for the source (fi : x → xi)i∈i and a set b ⊆ x is bounded iff fi(b) is bounded for each i ∈ i. this means that if (x, b) is a bornological universe and y ⊆ x, uniformizable and realcompact bornological universes 279 the subspace structure on y consists of the subspace topology on y and the bornology of all subsets of y that are bounded in x. the product structure on a product of a family of bornological universes (xi, bi)i∈i on the other hand is defined as the product topology and the bornology of sets b ⊆ ∏ i∈i xi for which each projection πi(b) is bounded. definition 2.2. a subset b of a uniform space (x, u) is called bounded in the sense of bourbaki (see [8]) if for each entourage u ∈ u we can find an n ∈ n and a finite set k ⊆ x such that b ⊆ u n(k). for a locally convex topological vector space the classical notion of boundedness coincides with boundedness in the sense of bourbaki with respect to the canonical uniformity. in a metric space (x, d) we have that each set that is bounded for the metric d is bounded in the sense of bourbaki for the underlying uniformity, but the converse is in general not true. if (x, u) is a uniform space, then the sets that are bounded in the sense of bourbaki form a bornology on x. a second bornology that we can associate with the uniformity u is the bornology of totally bounded subsets. a subset b of a uniform space is totally bounded if for each entourage u there is a finite partition (bi) n i=1 of b such that bi × bi ⊆ u for each i ∈ i. this is equivalent with saying that for each entourage u ∈ u there is a finite set k ⊆ x such that b ⊆ u (k), so obviously each totally bounded set is bounded in the sense of bourbaki. the proof of the following theorem can be found in [10]. theorem 2.3. let (x, u) be a uniform space. the following statements are equivalent: (1) a is bounded in the sense of bourbaki, (2) a is bounded for each uniformly continuous pseudometric d on x, (3) each real valued, uniformly continuous map is bounded on a. the bornology of sets that are bounded in the sense of bourbaki and the bornology of totally bounded sets are two natural bornologies associated with a uniform space. at first sight it therefore looks like we can define two notions of uniformizability for topological bornological spaces. we will see that they are actually equal. definition 2.4. we will call a bornological universe (x, b) bb-uniformizable if there is a uniform space (x, u) for which x is the underlying topological space and b is the set of subsets of x that are bounded in the sense of bourbaki for u. in the case that b is equal to the set of totally bounded subsets of the uniform space (x, u) we will call (x, b) tb-uniformizable. let (x, u) be a uniform space and take v ⊆ u. if each entourage in u contains a finite intersection of elements of v, then v is called a base for u. for a set c of real valued maps on a set x we will denote the uniform space that is initial for the source c as (x, c). define u ǫf as the set that contains all (x, y) ∈ x × x for which |f (x) − f (y)| < ǫ. the entourages u ǫf , form a base for the uniformity of (x, c). 280 t. vroegrijk lemma 2.5. let v be a base for a uniformity u. if for each v ∈ v we can find a finite partition (bi) n i=0 of b such that bi × bi ⊆ v for all 0 ≤ i ≤ n, then b is totally bounded. proof. each entourage u ∈ u contains a set v0 ∩. . .∩vm where each vk is in v. for all 0 ≤ k ≤ m we can find a finite partition pk of b such that bk ×bk ⊆ vk for each bk ∈ pk. if we define p as {b0 ∩ . . . ∩ bm|∀0 ≤ k ≤ m : bk ∈ pk}, then p is a finite partition of b and each of each of its elements p satisfies p × p ⊆ u . � lemma 2.6. for a set c of real valued maps on a set x, the following statements are equivalent: (1) f (b) is bounded for all f ∈ c, (2) b is totally bounded in (x, c), (3) b is bounded in the sense of bourbaki in (x, c). proof. if we assume that the first statement is true and we take ǫ > 0 and f ∈ c, then we can find a finite partition (ci) n i=0 of f (b) such that each element of this partition has diameter smaller than ǫ. the sets bi, defined as b ∩ f −1(ci), form a finite partition of b and for each 0 ≤ i ≤ n holds bi × bi ⊆ u ǫ f . applying lemma 2.5 gives us that b is totally bounded in (x, c). we already saw that the second statement implies the third. now suppose that b is bounded in the sense of bourbaki. this yields that it is bounded for each uniformly continuous metric on (x, c). since df , where df (x, y) is defined as |f (x) − f (y)|, is a uniformly continuous metric for each f ∈ c, we have that f (b) is bounded. � definition 2.7. we will call a bornology b on a topological space saturated if it contains each b that satisfies the following condition: if (gn)n is a sequence of open sets such that each bounded set is contained in some gn and gn ⊆ gn+1 for all n ∈ n, then there is an m ∈ n for which b ⊆ gm. this definition is equivalent with saying that b is equal to the intersection of all bornologies that contain b and are generated by a sequence (gn)n of open sets that satisfies gn ⊆ gn+1 for all n ∈ n. from here on the set of all continuous, real valued maps on x that are bounded on all elements of b will be denoted as cb(x). proposition 2.8. the following statements are equivalent: (1) cb(x) is an initial source, (2) (x, b) is bb-uniformizable, (3) x is completely regular and b is saturated. uniformizable and realcompact bornological universes 281 proof. if the first statement is true, then we have by definition that the underlying topological space of (x, cb(x)) is the original topology in x. we know from lemma 2.6 that the set of subsets of x that are bounded in the sense of bourbaki in (x, cb(x)) is exactly b. now suppose that (x, b) is bb-uniformizable. we automatically obtain that x is completely regular. if b is an unbounded subset of x, then we can find a uniformly continuous metric d on x for which b is unbounded. take an arbitrary x0 ∈ x and define gn as the open ball with center x0 and radius n + 1. this is an increasing sequence of open sets that satisfies the conditions stated in definition 2.7. furthermore, we know that b is not contained in any of the sets gn. hence we have that b is saturated. for a completely regular space x, the initial topology for the source that consists of all continuous maps into [0, 1] is the original topology. clearly, these maps are all in cb(x). if we now assume that the third statement is true and that b is an unbounded set, then we can find a sequence (gn)n such that each bounded set is contained in some gn, gn ⊆ gn+1 for all n ∈ n and no gn contains b. choose a sequence (xn)n in b such that xn ∈ b \ gn. let φn be a map into [0, n] that vanishes on gn and attains the value n in xn. define the map φ as ∑ n φn. since φ is equal to ∑n k=0 φk on each gn we obtain that φ is a continuous map. by definition, φ is bounded on all elements of b and unbounded on b. � a subset of a topological space is called relatively pseudocompact if it is mapped to a bounded subset of r by all real valued, continuous maps on x. the previous proposition yields that each relatively pseudocompact subset of a uniformizable bornological universe is bounded. proposition 2.9. an object (x, b) is bb-uniformizable iff it is tb-uniformizable. proof. if (x, b) is bb-uniformizable, then cb(x) is initial. from lemma 2.6 we obtain that (x, cb(x)) is a uniform space with underlying topological space x for which the totally bounded subsets are exactly the sets in b. now let (x, b) be a tb-uniformizable object and b an unbounded subset of x. we can find a uniformly continuous metric d on x and a sequence (xn)n in b such that for distinct numbers n and m it holds that d(xn, xm) ≥ 1. define kn as the set {xm|m ≥ n} and gn as the set of all x ∈ x that lie at a d-distance strictly greater than 1/(n + 3) from kn. this is an increasing sequence of open sets and gn ⊆ gn+1 for all n ∈ n. this sequence also satisfies that each bounded set is contained in some gn. if this were not the case, then there would be a bounded set that contains a countable subset with the property that all of its elements, for the metric d, lie at a distance greater than 1/3 of each other. this would imply that this bounded set is not totally bounded. since b is not contained in any of the sets gn we obtain that b is saturated. � 282 t. vroegrijk from here on a bb-uniformizable bornological universe will be simply called uniformizable. we will say that a bornological universe (x, b) is hausdorff if x is hausdorff. proposition 2.10. the following statements are equivalent: (1) (x, b) is hausdorff and uniformizable, (2) (x, b) is isomorphic to a subspace of a product of real lines. proof. if (x, b) is hausdorff uniformizable, then the source cb(x) is initial and separating. this yields that the map from x to rcb(x) that sends an element x to (f (x))f∈cb(x), is in fact an embedding. to prove the converse, we need to show that each subspace of a product of real lines is hausdorff uniformizable. we can endow each subset x of a product rα of real lines with the uniformity that it inherits from the product uniformity on rα. this uniformity is the initial one for the source that consists of all projection maps to r. we know that the underlying topology of this uniformity is the original topology on x. lemma 2.6 grants us that a subset of x is bounded in the sense of bourbaki for this uniformity iff each of its projections is bounded. � an article by beer (see [1]) also contains a necessary and sufficient condition for a bornological universe to be embeddable into a product of real lines. the product bornology, however, is in this article defined as the bornology generated by all sets for which at least one projection is bounded, while we use the bornology of sets for which all projections are bounded. 3. realcompact bornological universes a subset b of a locally convex topological vector space e is called bounded if each continuous seminorm is bounded on b. if e satisfies the condition that each seminorm that is bounded on all bounded sets is automatically continuous, then e is called bornological. the bornological objects form a concretely coreflective subcategory of the category of locally convex topological vector spaces. let b be a bornology on a tychonoff space x that consists only of relatively pseudocompact sets. the set of all real valued, continuous maps on x endowed with the bounded-open topology is a locally convex topological vector space. in [13] schmets gives a characterization of the bornological coreflection of this topological vector space using the hewitt realcompactification υ(x) of x. if the elements of b are no longer supposed to be relatively pseudocompact, then the bounded-open topology is no longer a vector topology. in [16], however, an extension of the classical duality between topological and bornological vector spaces is given that allows us to define the bornological coreflection of this space. in that same article a characterization of this bornological coreflection is given using a realcompact space υb(x) that contains x as a dense uniformizable and realcompact bornological universes 283 subspace. we will use these spaces to define realcompactness for bornological universes and give various characterizations of this new concept. let (x, b) be a bornological universe where x is a tychonoff space. the space υb(x) is defined as {x ∈ β(x)|∀f ∈ cb(x) : f β(x) 6= ∞}. here f β is the unique map from β(x) to the one-point compactification of r that satisfies the condition that its restriction to x is equal to f . proposition 3.1. υb(x) is a realcompact tychonoff space that contains each b ∈ b as a relatively compact subspace, i.e. the closure in υb(x) of an element in b is compact. proof. the proof of the first statement can be found in [9]. furthermore, it is a well-known fact that the relatively pseudocompact subsets of x are relatively compact in υ(x). the proof of the second statement is completely analogous. � definition 3.2. a bornological universe (x, b) will be called realcompact if υb(x) is equal to x. proposition 3.3. (x, b) is realcompact iff x is realcompact as a topological space and b contains only relatively compact subsets. proof. that this condition is necessary follows from the previous proposition. if this condition is satisfied, then each real valued, continuous map is automatically bounded on all elements of b. this means that cb(x) is equal to c(x) and that υb(x) is by definition equal to the hewitt realcompactification of x. since we assumed x to be realcompact we obtain that υb(x) equals x. � proposition 3.4. (υb(x), c(υb(x)) is the completion of (x, cb(x)). proof. because υb(x) is a realcompact tychonoff space, we obtain that the uniform space (υb(x), c(υb(x)) is complete (see [9]). to prove that it is the completion of (x, cb(x)) we need to show that (x, cb(x)) →֒ (υb(x), c(υb(x)) is a dense embedding. because all b ∈ b are relatively compact in υb(x) we have that the restriction of a map f ∈ c(υb(x)) to x is bounded and because the source c(υb(x)) is initial this yields that (x, cb(x)) →֒ (υb(x), c(υb(x)) is uniformly continuous. moreover, because each f ∈ cb(x) can be extended to a real valued, continuous map on υb(x) — the extension being the restriction of f β to υb(x) — we obtain that this map is actually a dense embedding. � corollary 3.5. (x, b) is realcompact iff (x, cb(x)) is complete. by a character on an algebra we mean a non-zero, real valued algebra morphism. one of the possible characterizations of realcompactness of a tychonoff space x is the following: for each character τ on c(x) there is an x ∈ x such that τ (f ) = f (x) for each f ∈ c(x). this means that for a realcompact topological space each character on the algebra of real valued, continuous maps is equal to a point-evaluation. in the setting for bornological universes we have the following result: 284 t. vroegrijk proposition 3.6. (x, b) is realcompact iff each character on cb(x) is equal to a point-evaluation. proof. suppose that this condition is satisfied and that y is an element of υb(x). this element y defines a character on cb(x) by sending an f to f β(y) and thus we can find an x ∈ x such that f (x) = f β (y) for all f ∈ cb(x). because the continuous maps from β(x) into [0, 1] separate points we obtain that x = y. if x is equal to υb(x), then all elements of b are relatively compact and therefore cb(x) is equal to c(x). if we combine this result with the fact that υb(x), and therefore x, is realcompact as a topological space, then we get that each character on the algebra of bounded continuous maps on (x, b) is equal to a point-evaluation. � for a tychonoff space x both compactness and realcompactness can be characterized using z-ultrafilters. in particular, x is compact iff each z-ultrafilter is fixed and x is realcompact iff each z-ultrafilter with the countable intersection property is fixed. we will see that for bornological universes (x, b) with a normal underlying topological space x the notion of realcompactness can be described with z-ultrafilters as well. definition 3.7. let (x, b) be a bornological universe. a decreasing sequence (zn)n of zero-sets is called unbounded if for each b ∈ b there is an n ∈ n for which b ∩ zn = φ. lemma 3.8. if b is a subset of a normal space and z is a zero-set that intersects with all zero-sets that contain b, then z intersects with b. proof. if this were not the case, then we could find a continuous map from x into [0, 1] such that f (b) ⊆ {0} and f (z) ⊆ {1}. this would imply that there is a zero-set that contains b and does not intersect with z. � lemma 3.9. a subset b of a topological space x is relatively compact iff each open cover of x contains a finite subcover of b. proposition 3.10. let (x, b) be a bornological universe with a normal underlying topological space x. (x, b) is realcompact iff each z-ultrafilter that does not contain a decreasing, unbounded sequence of zero-sets, is fixed. proof. let (x, b) be realcompact and f a z-ultrafilter on x that does not contain a decreasing, unbounded sequence of zero-sets. we want to prove that f has the countable intersection property. take a decreasing sequence (zn)n of zero-sets in f . by definition we can find a b ∈ b such that for each n ∈ n we have b ∩ zn 6= φ. because b is compact, the z-filter generated by the sets b ∩ zn is fixed. hence we obtain that ⋂ n zn is not empty. this means that f has the countable intersection property and that, because x is realcompact, f is fixed. to prove that this condition is sufficient we first take a z-ultrafilter f on x with the countable intersection property. since each singleton in x is bounded uniformizable and realcompact bornological universes 285 we obtain that f does not contain any decreasing, unbounded sequence of zerosets and thus, that f is fixed. this yields that x is realcompact. now we take a b ∈ b and an open cover g of x. let z be the set of all zero-sets that contain a set x \ g with g ∈ g and all zero-sets that contain b. if we assume that g does not contain a finite subcover of b, then z has the finite intersection property and we can find a z-ultrafilter f that contains z. from lemma 3.8 we obtain that each element of f intersects with b and this of course implies that f does not contain any decreasing, unbounded sequences of zero-sets. because f is not fixed we have to conclude that our original assumption was false and that g does contain a finite subcover of b. � when we look at the proof it is clear that if we do not assume the space x to be normal, the condition stated in the previous proposition is still necessary. the last results that we will encounter all concern hausdorff uniformizable bornological universes (x, b), i.e. x is tychonoff and b is saturated. we already saw that in a hausdorff uniformizable bornological universe (x, b) all relatively pseudocompact sets, and therefore all relatively compact sets, are contained in b. we also know that in a realcompact bornological universe all bounded sets are relatively compact. this means that if a bornological universe (x, b) is uniformizable and realcompact, the space x is realcompact and b is the bornology of relatively compact sets (which is equal to the bornology of relatively pseudocompact sets for realcompact spaces). since in this case all continuous, real valued maps on x are bounded we automatically obtain that the converse implication is true as well. proposition 3.11. (x, b) is hausdorff uniformizable and realcompact iff it is isomorphic to a closed subspace of a product of real lines. proof. each realcompact topological space x is isomorphic to a closed subset c of a product rα of real lines. in such a space a subset b is bounded iff each of its projections is bounded or, equivalently, it is contained in a compact subset of rα. since c is closed this is equivalent to saying that the closure of b in c is compact. this means that this condition is necessary. we know that a topological space that is isomorphic to a closed subset of a product of real lines is realcompact and that the closure of a bounded set in such a space is compact, so this condition is also sufficient. � definition 3.12. for a hausdorff uniformizable bornological universe (x, b) we define υ(x, b) as the object (υb(x), υ(b)), where υ(b) denotes the set of all relatively compact subsets of υb(x). this is again a hausdorff uniformizable bornological universe and by definition it is also realcompact. proposition 3.13. let f : (x, b) → (y, b′) be a morphism. if we define υ(f ) as the restriction of β(f ) to υb(x), then υ(f ) : υ(x, b) → υ(y, b ′) is a morphism. 286 t. vroegrijk proof. what we need to prove is that β(f ) is a morphism from υb(x) to υb′ (y ). let x be an element of υb(x). for each g ∈ cb′ (y ) we have that g ◦ f is an element of cb(x) and thus that (g ◦ f ) β (x) is real. because the map (g ◦ f )β is equal to gβ ◦ β(f ) we obtain that β(f )(x) is in υb′ (y ). that υ(f ) is bounded is trivially true, since relatively compact sets are mapped to relatively compact sets. � corollary 3.14. υ(x, b) is the realcompact reflection of (x, b) in the category of hausdorff uniformizable bornological universes and bounded, continuous maps. proof. we know that (x, b) →֒ υ(x, b) is a morphism. furthermore, if (y, b′) is realcompact, then υ(y, b) is equal to (y, b) and thus we obtain for each map f : (x, b) → (y, b′) that is bounded and continuous that υ(f ) is a morphism from υ(x, b) to (y, b). � the following proposition extends the well-known theorem that states that the hewitt realcompactification of a dense subspace y of x equals the hewitt realcompactification of x iff y is c-embedded in x. proposition 3.15. let y be a dense subspace of x and let b′ be the subspace bornology on y derived from b. the following statements are equivalent: (1) if (z, b′′) is realcompact, then each map f : (y, b′) → (z, b′′) that is continuous and bounded has an extension to (x, b), (2) each realvalued map f on (y, b′) that is continuous and bounded has an extension to (x, b), (3) υ(y, b′) = υ(x, b). proof. the first statement implies the second because r endowed with the bornology of relatively compact subsets is a realcompact bornological universe. from the second statement we obtain that each realvalued, continuous map f that is bounded on y has an extension to x . this implies that β(y ) = β(x). by definition we have that υb′ (y ) ⊆ υb(x). now take an x ∈ υb(x), an f ′ ∈ cb′ (y ) and let f be the extension of f ′ to x. since they coincide on a dense subset we have that (f ′)β equals f β. hence we get that (f ′)β (x) is equal to f β (x) and therefore an element of the reals. because this is true for all maps in cb′ (y ) we obtain that υb(x) is a subset of υb′ (y ). from the foregoing we obtain that υb(x) is equal to υb′ (y ) and thus that υ(x, b) equals υ(y, b ′). suppose the third statement is true and take a map f : (y, b′) → (z, b′′), where f is bounded and continuous and (z, b′′) is realcompact. we know that the map υ(f ) : υ(y, b) → υ(z, b′′) is bounded and continuous and that υ(z, b′′) is equal to (z, b′′). this implies that the restriction of υ(f ) to x is a map to (z, b′′) that extends f . � uniformizable and realcompact bornological universes 287 references [1] g. beer, embeddings of bornological universes, set-valued analysis 16 (2008), 477–488. 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[10] j. hejcman, boundedness in uniform spaces and topological groups, czechoslovak mathematical journal 84 (1959), 544–563. [11] s. hu, boundedness in a topological space, journal de mathématiques pures et appliquées 28 (1949). [12] s. hu, introduction to general topology, (holden-day, san francisco, 1966). [13] j. schmets, espaces de fonctions continues, lecture notes in mathematics 519 (1976). [14] a. lechicki, s. levi, a. spakowski, bornological convergences, journal of mathematical analysis and applications 297 (2004), 751–770. [15] t. vroegrijk, pointwise bornological spaces, topology and its applications 156 (2009), 2019–2027. [16] t. vroegrijk, pointwise bornological vector spaces, topology and its applications, to appear. received october 2009 accepted december 2009 t. vroegrijk (tom.vroegrijk@ua.ac.be) university of antwerp, middelheimlaan 1, 2000 antwerpen, belgium. () @ applied general topology c© universidad politécnica de valencia volume 12, no. 1, 2011 pp. 35-39 hypercyclic abelian semigroup of matrices on cn and rn and k-transitivity (k ≥ 2) adlene ayadi∗ abstract we prove that the minimal number of matrices on c n required to form a hypercyclic abelian semigroup on c n is n + 1. we also prove that the action of any abelian semigroup finitely generated by matrices on cn or r n is never k-transitive for k ≥ 2. these answer questions raised by feldman and javaheri. 2010 msc: 37c85, 47a16 keywords: hypercyclic, tuple of matrices, semigroup, subgroup, dense orbit, transitive, semigroup action. 1. introduction let k = r or c. following feldman from [6], by an p-tuple of matrices, we mean a finite sequence of length p (p ≥ 1) of commuting matrices a1, a2, . . . , ap on kn. we will let g = {ak11 a k2 2 . . . a kp p : k1, k2, . . . , kp ∈ n} be the semigroup generated by a1, a2, . . . , ap. for a vector x ∈ k n, the orbit of x under the action of g on kn is og(x) = {ax : a ∈ g}. for a subset e ⊂ k n, denote by e (resp. ◦ e ) the closure (resp. interior) of e. a subset e ⊂ kn is called g-invariant if a(e) ⊂ e for any a ∈ g. the orbit og(x) ⊂ k n is dense (resp. locally dense) in kn if og(x) = k n (resp. ˚ og(x) 6= ∅). the semigroup g is called hypercyclic (or also topologically transitive) (resp. locally hypercyclic) if there exists a vector x ∈ kn such that og(x) is dense (resp. locally dense) in ∗this work is supported by the research unit: systèmes dynamiques et combinatoire: 99ur15-15 36 a. ayadi k n. for an account of results and bibliography on hypercyclicity, we refer to the book [3] by bayart and matheron. on the other part, let k ≥ 1 be an integer. denote by (kn) k the k-fold cartesian product of kn. for every u = (x1, . . . , xk) ∈ (k n)k, the orbit of u under the action of g on (kn)k is denoted o k g(u) = {(ax1, . . . , axk) : a ∈ g}. when k = 1, okg(u) = og(u). we say that the action of g on k n is ktransitive if, the induced action of g on (kn) k is hypercyclic, this is equivalent to that for some u ∈ (kn) k , ok g (u) = (kn)k. a 2-transitive action is also called weak topological mixing and 1-transitive means hypercyclic. in [6], feldman showed that in cn there exist a hypercyclic semigroup generated by (n + 1)-tuple of diagonal matrices on cn and that no semigroup generated by n-tuple of diagonalizable matrices on kn can be hypercyclic. if one remove the diagonalizability condition, costakis et al. proved in [4] that there exists a hypercyclic semigroup generated by n-tuple of non diagonalizable matrices on rn. however, they show in [5] that there exist a hypercyclic semigroup generated by (n + 1)-tuple of diagonalizable matrices a1, . . . , an+1 on rn. the main purpose of this paper is twofold: firstly, we give a general result (with respect to the results above) by showing that the minimal number of matrices on cn required to form a hypercyclic tuple in cn is n + 1. this answer a question raised by feldman in ([6], section 6). secondly, we prove that the action of any abelian semigroup finitely generated by matrices on kn is never k-transitive for k ≥ 2. this answer a question of javaheri in ([7], problem 3). our principal results are the following: theorem 1.1. for every n ≥ 1, any abelian semigroup generated by n matrices on cn is not locally hypercyclic. theorem 1.2. let g be an abelian semigroup generated by p matrices (p ≥ 1) on kn (k = r or c). then the action of g on kn is never k-transitive for k ≥ 2. 2. on hyercyclic semigroups let mn(k) be the set of all square matrices of order n ≥ 1 with entries in k and gl(n, k) be the group of invertible matrices of mn(k). let g be an abelian semigroup generated by p matrices (p ≥ 1) on kn and we let g′ = g∩gl(n, k). hypercyclic abelian semigroup of matrices on cn and rn and k-transitivity (k ≥ 2) 37 lemma 2.1. under the notation above, let k ≥ 1 be an integer and u ∈ (kn)k. then (i) ok g (u) = (kn)k if and only if ok g′ (u) = (kn)k. (ii) ˚ ok g (u) = ∅ if and only if ˚ ok g′ (u) = ∅. in particular, if the action of g on kn is k-transitive so is the action of g′ on k n. proof. (i) suppose that ok g′ (u) = (kn) k for some u ∈ (kn) k . then since ok g′ (u) ⊂ ok g (u), we see that ok g (u) = (kn) k . conversely, suppose there exists u ∈ (kn) k such that ok g (u) = (kn) k . denote by (a1, . . . , ap) an p-tuple of matrices on k n which generate the semigroup g. one can suppose that for some 0 ≤ r ≤ p, a1, . . . , ar ∈ gl(n, k) and ar+1, . . . , ap ∈ mn(k)\gl(n, k). then g ′ = g ∩ gl(n, k) is the semigroup generated by a1, . . . , ar. for k = 1, . . . , r, write im(ak) = ak(k n) the range of ak. then im(ak) is a vector subspace of k n of dimension < n, hence ◦ im(ak) = ∅. if r = p then g = g′ and so (i) is obvious. if r = 0 then for every u ∈ (kn)k, okg(u) ⊂ p ⋃ k=1 (im(ak)) k ∪ {u}. since ˚p ⋃ k=1 (im(ak))k = ∅, ◦ ok g (u) = ∅. if 0 < r < p then o k g(u) ⊂   r ⋃ j=1 (im(aj )) k   ∪ okg′ (u). it follows that (kn) k ⊂   r ⋃ j=1 (im(aj )) k   ∪ ok g′ (u) and therefore ok g′ (u) = (kn) k . the proof of (ii) is the same as for (i). � lemma 2.2 ([2], corollary 1.5). let g be an abelian subgroup of gl(n, c). if g is generated by n matrices (n ≥ 1), it has no dense orbit. lemma 2.3 ([6], corollary 5.7). let g be an abelian semigroup generated by p matrices (p ≥ 1) on cn. then every locally dense orbit of g is dense in cn. from lemmas 2.2 and 2.3, we obtain the following: corollary 2.4. any abelian semigroup generated by n matrices (n ≥ 1) of gl(n, c) is not locally hypercyclic. 38 a. ayadi proof of theorem 1.1. let g be an abelian semigroup generated by n matrices on cn and we let g′ = g ∩ gl(n, c). by corollary 2.4, ˚ ok g′ (u) = ∅ for every u ∈ (cn)k and hence by lemma 2.1, ˚ ok g (u) = ∅. the proof is complete. � 3. on k-transitivity (k ≥ 2) let recall first the following result: proposition 3.1 ([1], theorem 4.1). let g be an abelian subgroup of gl(n, k) (k = r or c). then there exists a g-invariant dense open subset u in kn such that if, u, v ∈ u and (bm)m∈n is a sequence of g such that lim m→+∞ bmu = v then lim m→+∞ b−1m v = u. corollary 3.2. let g be an abelian subgroup of gl(n, k) (k = r or c) and let u be a g-invariant dense open subset of kn as in proposition 3.1. then for every k ≥ 2, if v ∈ u k and w ∈ ok g (v) ∩ u k then ok g (v) ∩ u k = ok g (w) ∩ u k. proof. write v = (v1, . . . , vk), w = (w1, . . . , wk) ∈ u k. suppose that w ∈ ok g (v) ∩ u k. then there exists a sequence (bm)m∈n in g such that lim m→+∞ (bmv1, . . . , bmvk) = (w1, . . . , wk). then lim m→+∞ bmvj = wj , for every 1 ≤ j ≤ k. since vj , wj ∈ u , so by proposition 3.1, lim m→+∞ b−1m wj = vj and hence lim m→+∞ (b−1m w1, . . . , b −1 m wk) = v ∈ o k g (w). it follows that ok g (v) ∩ u k = ok g (w) ∩ u k. � proof of theorem 1.2. suppose the action of g is k-transitive (k ≥ 2), then there exists v = (v1, . . . , vk) ∈ (k n)k so that ok g (v) = (kn)k. we let g′ = g ∩ gl(n, k). by lemma 2.1, ok g′ (v) = (kn)k. denote by g′′ the group generated by g′ and by u a g′′-invariant dense open subset in kn as in proposition 3.1. then ok g′′ (v) = (kn)k and hence v ∈ u k. write w := (v1, . . . , v1). then w ∈ u k and by corollary 3.2, ok g′′ (w) = (kn)k (since u k is dense in (kn)k). it follows that og′′ (v1) = k n. let ϕ : kn −→ (kn) k be the homomorphism defined by ϕ(x) = (x, . . . , x), x ∈ kn. then okg′′ (w) = ϕ(og′′ (v1)) ⊂ ϕ(k n). as ϕ(kn) is a vector subspace of (kn) k of dimension n < nk, og′′ (w) cannot be dense in (k n)k (since k ≥ 2), this is a contradiction and the theorem is proved. � hypercyclic abelian semigroup of matrices on cn and rn and k-transitivity (k ≥ 2) 39 references 1. a. ayadi and h. marzougui, dynamic of abelian subgroups of gl(n, c): a structure theorem, geom. dedicata 116 (2005), 111–127. 2. a. ayadi and h. marzougui, dense orbits for abelian subgroups of gl(n, c), foliations 2005: world scientific, hackensack, nj (2006), 47–69. 3. f. bayart and e. matheron, dynamics of linear operators, cambridge tracts in math., 179, cambridge university press, 2009. 4. g. costakis, d. hadjiloucas and a. manoussos, dynamics of tuples of matrices, proc. amer. math. soc. 137, no. 3 (2009), 1025–1034. 5. g. costakis, d. hadjiloucas and a. manoussos, on the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple, j. math. anal. appl. 365 (2010), 229– 237. 6. n. s. feldman, hypercyclic tuples of operators and somewhere dense orbits, j. math. anal. appl. 346 (2008), 82–98. 7. m. javaheri, topologically transitive semigroup actions of real linear fractional transformations, j. math. anal. appl. 368 (2010), 587–603. (received june 2010 – accepted january 2011) adlene ayadi (adleneso@yahoo.fr) university of gafsa, department of mathematics, faculty of science of gafsa, 2112, gafsa, tunisia. hypercyclic abelian semigroup of matrices[3pt] on cn and rn and k-transitivity (k2). by a. ayadi chendengagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 301-307 lower homomorphisms on additive generalized algebraic lattices xueyou chen and zike deng ∗ abstract. in this paper, with the additivity property ([8]), the generalized way-below relation ([15]) and the maximal system of subsets ([6]) as tools, we prove that all lower homomorphisms between two additive generalized algebraic lattices form an additive generalized algebraic lattice, which yields the classical theorem: the function space between t0-topological spaces is also a t0-topological space with respect to the pointwise convergence topology. 2000 ams classification: 06b30, 06b35, 54d35, 54h10 keywords: additivity, generalized way below relation, lower homomorphism, upper adjoint. 1. introduction the notions of a directed set, a way-below relation, a continuous lattice and an algebraic lattice were introduced in [12], and applied in the study of domain theory, topological theory, lattice theory, etc. as a generalization, d. novak introduced the notions of a system of subsets, a generalized way-below relation, and defined a generalized continuous lattice (m-continuous lattice) and a generalized algebraic lattice in [15]. in the study of topological theory and lattice theory, many researchers are interested in the topological representation of a complete lattice. for example: suppose (x, t ) is a topological space. all open sets t of a topological space may be viewed as a frame and a frame may also be viewed as an open set lattice. about frame (locale) theory, see ([13]). on the other hand, suppose (x, c) is a co-topological space and c the set of all closed subsets of a topological space on x. d. drake, w. j. thron, s. papert ∗this work was partially supported by the national natural science foundation of china (grant no. 10471035/a010104) and natural science foundation of shandong province (grant no. 2003zx13). 302 x. chen and z. deng considered c as a complete lattice (c, ∪, ∩, ∅, x)([11, 16]). but unfortunately the correspondence between complete lattices and t0-topological spaces is not one-to-one. to solve the problem, on the basis of [1, 11, 15, 16], deng also investigated generalized continuous lattices. he introduced the notions of the maximal system of subsets, additivity property, and homomorphisms in [5, 6, 7, 10]. finally, the question was settled in [8, 9], he obtained the equivalence between the category of additive generalized algebraic lattices with lower homomorphisms and the category of t0-topological spaces with continuous mappings. this paper is a sequel of [2, 3, 4, 8, 9]. in section 2, we begin an overview of generalized continuous lattices, deng’s work, and some separation axioms, which surveys as preliminaries. in section 3, we prove that all lower homomorphisms between additive generalized algebraic lattices form a additive generalized algebraic lattice, and investigate some results about separation axioms. 2. preliminaries we introduce some notions for each area, i.e., generalized continuous lattices and additive generalized algebraic lattices. 2.1. generalized continuous lattices. in [15], d. novak introduced the notions of a generalized way-below relation and a system of subsets. let (p, ≤) be a complete lattice, ≺ is said to be a generalized way-below relation if (i) a ≺ b ⇒ a ≤ b, (ii) a ≤ b ≺ c ≤ d ⇒ a ≺ d. obviously, it is a natural generalization of a way-below relation ([12]). m ⊆ 2p is said to be a system of subsets of p , if for a ∈ p , there exists s ∈ m , such that ↓ a =↓ s, where ↓ a = {b | b ≤ a}, ↓ s = ∪{↓ c | c ∈ s}. there are three kinds of common used system of subsets: (i) the system of all finite subsets, (ii) the system of all directed sets and (iii) the system of all subsets. by means of the notion of a system of subsets, he defined a generalized waybelow relation. suppose m is a system of subsets. for a, b ∈ p , a is said to be way-below b with respect to m , in symbols a ≺m b, if for every s ∈ m with b ≤ ∨s, then a ∈↓ s. clearly ≺m is a generalized way-below relation induced by m ([15]). we will denote ≺m as ≺. (p, ≺) is called a generalized continuous lattice, if for every a ∈ p , we have a = ∨ ⇓ a, where ⇓ a = {b | b ≺ a}. a ∈ p is called a compact element, if a ≺ a. let k(≺) = {a ∈ p | a ≺ a}. (p, ≺) is called a generalized algebraic lattice, if for every a ∈ p , we have a = ∨{↓ a ∩ k(≺)}. for further study, see [1, 17]. lower homomorphisms on additive generalized algebraic lattices 303 2.2. additive generalized algebraic lattices. suppose (p, ≺) is a generalized continuous lattice. deng introduced the notion of a maximal system of subsets generated by ≺, that is, m (≺) = {s ⊆ p | ∀a ∈ p with a ≺ ∨s, then a ∈↓ s}. suppose (p, ≺) is a generalized algebraic lattice. deng defined a new property: (p, ≺) is additivity, if for a, b, c ∈ p with a ≺ b ∨ c implies a ≺ b or a ≺ c ([8, 9]). he investigated the connection between additive generalized algebraic lattices and t0-topological spaces as follows. on one hand, suppose (p, ≺) is a generalized algebraic lattice, let x = k(≺), and t : p → 2x , t (a) =↓ a ∩ k(≺). if (p, ≺) is additive, then t satisfies: (1) t (0) = ∅, (2) t (1) = x, (3) for s ∈ m (≺) = m (k(≺)), t (∨s) = ∪t (s), (4) for s ⊆ p, t (∧s) = ∩t (s), (5) t (a ∨ b) = t (a) ∪ t (b). if c = t (p ), then (x, c) is a t0 co-topological space, and (p, ≺) is isomorphic to (x, c) (see [8, 9]). on the other hand, assume (x, c) is a co-topological space and let q = {{x}− | x ∈ x} be the collection of closure of all singletons. clearly q is a ∨−base for c, i.e., a ∈ c, a is a closed subset, and we have a = ∨ ↓ a. m (q) = {s | s ⊆ x, for a ∈ q, a ≤ ∨s we have a ∈↓ s} is a system of subsets induced by q, then (c, ≺m(q)) is a additive generalized algebraic lattice with k(≺m(q)) = q. in this case, a ≺m(q) b, for a, b ∈ c if and only if a ⊆ {x}− for some x ∈ b. it is clear that ≺m(q) is the specialization order ([12]) which is essentially in topological theory and domain theory. furthermore, (c, ≺m(q)) is an example of additive generalized algebraic lattice. for another example in commutative ring, see [9]. suppose (p1, ≺1), (p2, ≺2) are two generalized continuous lattices. h : p1 → p2 is said to be a lower homomorphism if it preserves arbitrary joins and the generalized way-below relations. thus a lower homomorphism h is residuated. if g be its upper adjoint, we have (g, h) is a galois connection ([7]). the lower homomorphism also corresponds to the closed mapping. so he obtained the equivalence between the category of additive generalized algebraic lattices with lower homomorphisms and the category of t0-topological spaces with continuous mappings in [8, 9]. from the point of view of deng’s work ([8, 9]), an additive generalized algebraic lattice is an algebraic abstraction of a topological space. thus topological theory may be directly constructed on it. the work will benefit the study of the theory of topological algebra and the possible application on additive generalized algebraic lattices. in [2, 3, 4], we constructed stone compactification, tietze extension theorem, separation axioms. in this paper, we will prove that all lower homomorphisms between additive generalized algebraic lattices form an additive generalized algebraic lattice. in [2, 3, 4], we defined some separation axioms. 304 x. chen and z. deng definition 2.1. (p, ≺) is said to be regular, if for x ∈ k(≺), b ∈ p , x 6≺ b, then x ∧ b = 0. definition 2.2. a family of elements 〈cα | α ∈ [0, 1] & α is a rational number 〉 is called a scale of (p, ≺), if it satisfies: for α < β, we have cα ≺ cβ. for a, b ∈ p , if there exists a scale 〈cα〉, such that a ≤ c0, c1 ≤ b. we use the symbol a � b to indicate the relation. (p, ≺) is said to be completely regular, if ∀a ∈ l, a = ∧{b | a � b}. definition 2.3. (p, ≺) is said to be normal, if for a, b ∈ p , a ∧ b = 0, then there exist c, d ∈ p , such that a ∧ c = 0, b ∧ d = 0 and c ∨ d = 1. for other notions and results cited in this paper, see [2, 3, 4, 8, 9, 15]. 3. lower homomorphisms definition 3.1. suppose p1 and p2 are two additive generalized algebraic lattices. ∀p ∈ k(≺1), q ∈ k(≺2), we define 〈p, q〉(a) = { q if p ≺1 a 0 if p 6≺1 a ∀a ∈ p1. lemma 3.2. 〈p, q〉 is a lower homomorphism. proof. first, we have to show that 〈p, q〉 preserves arbitrary join. suppose {aα} ⊆ p1. if p ≺1 ∨ aα, we obtain 〈p, q〉( ∨ aα) = q. since p1 is additive, by p ≺1 ∨ aα, there exists aα0 , such that p ≺1 aα0 . so 〈p, q〉(aα0 ) = q, thus 〈p, q〉( ∨ aα) = q = ∨ 〈p, q〉(aα). if p 6≺1 ∨ aα, then 〈p, q〉 ∨ aα = 0. by this, we have p 6≺1 aα for every α. so 〈p, q〉(aα) = 0, which implies that 〈p, q〉( ∨ aα) = 0 = ∨ 〈p, q〉(aα). second, we have to prove that 〈p, q〉 preserves the generalized way-below relation. given a, c ∈ p1, and a ≺1 c, if p ≺1 a, then p ≺1 c, we have 〈p, q〉(a) = q, 〈p, q〉(c) = q, thus 〈p, q〉(a) ≺2 〈p, q〉(c); if p 6≺1 a, p ≺1 c, then 〈p, q〉(a) = 0, 〈p, q〉(c) = q, thus 〈p, q〉(a) ≺2 〈p, q〉(c); if p 6≺1 a, p 6≺1 c, then 〈p, q〉(a) = 0, 〈p, q〉(c) = 0, thus 〈p, q〉(a) ≺2 〈p, q〉(c). by the above proof, we obtain that 〈p, q〉 also preserves the generalized way below relation. � by lemma 3.2, 〈p, q〉 is a lower homomorphism. let gpq be its upper adjoint. then (〈p, q〉, gpq) is a galois connection. let [p1 → p2] be the set of all lower homomorphisms from p1 to p2 and suppose h1, h2 ∈ [p1 → p2]. then we may define h1 ∨ h2 : p1 → p2, for every p ∈ k(≺1), (h1 ∨h2)(p) = h1(p)∨h2(p). similarly, (h1 ∧h2)(p) = h1(p)∧h2(p). so [p1 → p2] is a complete lattice with the minimal element 0 and the maximal element 1, where 0(p) = 0, 1(p) = 1 for every p ∈ k(≺1). we also define h1 ≤ h2, if for every p ∈ k(≺1), we have h1(p) ≤2 h2(p), where ≤2 is the partial order on p2. similarly, h1 ≺ ∗ h2, if for every p ∈ k(≺1), we have h1(p) ≺2 h2(p). lower homomorphisms on additive generalized algebraic lattices 305 lemma 3.3. ≺∗ is a generalized way below relation on [p1 → p2]. proof. we have to show (1) and (2), (1) h1 ≺ ∗ h2 ⇒ h1 ≤ h2, (2) h1 ≤ h2 ≺ ∗ h3 ≤ h4 ⇒ h1 ≺ ∗ h4. the proof is trivial. � lemma 3.4. 〈p, q〉 is a compact element of [p1 → p2]. proof. by the definition of ≺∗, the proof is trivial. � clearly, k(≺∗) = {〈p, q〉 | p ∈ k(≺1), q ∈ k(≺2)}. lemma 3.5. if h is a lower homomorphism, then ∀p ∈ k(≺1), h(p) ∈ k(≺2). proof. see [8, 9]. � lemma 3.6. if h ∈ [p1 → p2], q ≤2 h(p), we have 〈p, q〉 ≺ ∗ h. proof. ∀a ∈ p1, 〈p, q, 〉(a) = { q if p ≺1 a 0 if p 6≺1 a if p ≺1 a, 〈p, q〉(a) = q ≤2 h(p) ≤2 h(a); if p 6≺1 a, 〈p, q〉(a) = 0 ≤2 h(a). thus we have 〈p, q〉(a) ≤2 h(a) for all a ∈ p1, thus 〈p, q〉 ≤ h. by lemma 3.4, since 〈p, q〉 is a compact element, we obtain 〈p, q〉 ≺∗ h. � lemma 3.7. if h ∈ [p1 → p2], if p ∈ k(≺1), then h(p) = 〈p, h(p)〉(p). proof. for p ∈ k(≺1), since h is a lower homomorphism, h(p) ∈ k(≺2) (see [8]). thus we have h(p) = 〈p, h(p)〉(p). � note 1. ∨ 〈pα, qα〉 does not preserve the way below relation in general. example 3.8. without the assumption of additive property, lemma 3.7 does not hold. suppose p1, p2 are two classical algebraic lattices [12]. if k(≺2) 6= p2, there exists e ∈ p2, e 6∈ k(≺2). since p2 is algebraic, there exists a directed set {qα} ⊆ k(≺2), such that e = ∨qα. we define 〈0, qα〉 : p1 → p2, ∀x ∈ p1, 〈0, qα〉(x) = qα. ce : p1 → p2, ∀x ∈ p1, ce(x) = e. it is easy to show that {〈0, qα〉} is also a directed set in [p1 → p2], which preserves the way-below relation, but ce = ∨〈0, qα〉 does not hold. proposition 3.9. ∀h ∈ [p1 → p2], h = ∨ p∈k(≺1) ∨ q≤2h(p) 〈p, q〉. proof. for every a ∈ p1 and since p1 is generalized algebraic, we have a = ∨{p | p ∈ k(≺1)}, and h preserves arbitrary joins. thus it suffices to prove that for every p ∈ k(≺1), h(p) = ∨ p∈k(≺1) ∨ q≤2h(p) 〈p, q〉(p) = ∨ q≤2h(p) 〈p, q〉(p). since q ≤2 h(p), we have 〈p, q〉(p) = q ≤2 h(p). by lemma 3.7, 〈p, h(p)〉(p) = h(p), we obtain h(p) = ∨ q≤2h(p) 〈p, q〉(p). � 306 x. chen and z. deng proposition 3.10. suppose p1 and p2 are two generalized algebraic lattices. then [p1 → p2] is a generalized algebraic lattice. proof. by proposition 3.9, we have h = ∨(↓ h ∪ k(≺∗)) for h ∈ [p1 → p2]. so [p1 → p2] is a generalized algebraic lattice. � proposition 3.11. [p1 → p2] is additive. proof. suppose 〈p, q〉 ∈ k(≺∗), h1, h2 ∈ [p1 → p2], and 〈p, q〉 ≺ ∗ h1 ∨ h2. we have 〈p, q〉(p) = q ≺2 (h1 ∨ h2)(p) = h1(p) ∨ h2(p). since p2 is additive, we have q ≺∗ h1(p), or q ≺ ∗ h2(p). by this, we obtain 〈p, q〉 ≺ ∗ h1, or 〈p, q〉 ≺ ∗ h2. thus [p1 → p2] is additive. � by propositions 3.10 and 3.11, we obtain [p1 → p2] is an additive generalized algebraic lattice. from the point of of view of topological theory, the result corresponds to the classical theorem: the function space between two t0-topological spaces is also t0-topological space with respect to the pointwise convergence topology. proposition 3.12. if (p2, ≺2) is regular, then [p1 → p2] is also regular. proof. for 〈p, q〉 ∈ k(≺∗), h ∈ [p1 → p2], if 〈p, q〉 6≺ ∗ h, by the definition of 〈p, q〉, we have 〈p, q〉(p) 6≺2 h(p), so q 6≺2 h(p). since (p2, ≺2) is regular, we obtain q ∧ h(p) = 0, which implies that 〈p, q〉 ∧ h = 0, thus [p1 → p2] is regular. � proposition 3.13. if (p2, ≺2) is completely regular, then [p1 → p2] is also completely regular. proof. for h1 � h2, by the definition of ≺ ∗, it is equivalent to: for every p ∈ k(≺1), h1(p)�h2(p). since (p2, ≺2) is completely regular, so h1(p) = ∧{h2(p) | h1(p)�h2(p)}, which implies that h1 = ∧{h2 | h1 �h2}. proposition 3.13 holds. � proposition 3.14. if (p2, ≺2) is normal, then [p1 → p2] is also normal. proof. if h1, h2 ∈ [p1 → p2], and h1 ∧ h2 = 0, then for any p ∈ k(≺1), h1(p) ∧ h2(p) = 0. for h1(p) ∧ h2(p) = 0, since (p2, ≺2) is normal, there exist cp, dp ∈ p2, such that h1(p) ≺2 cp, h2(p) ≺2 dp, and cp ∨ dp = 1. let hcp = ∨ q≺2cp 〈p, q〉, hdp = ∨ q≺2dp 〈p, q〉, so h1 ≺ ∗ hcp and h2 ≺ ∗ hdp . let hc = ∧p∈k(≺1)hcp , hd = ∨ p ∈ k(≺1)hdp . it is easy to prove h1 ≺ ∗ hc, h2 ≺ ∗ hd, and hc ∨ hd = 1. thus [p1 → p2] is also normal. � based on the above work, we constructed tietze extension theorem in [3]. proposition 3.15 (tietze extension theorem). (p, ≺) is normal iff for every closed lower sublattice (q, ≺q) of (p, ≺), and a lower homomorphism h : (q, ≺q) → (cj , ≺j ), there exists a lower homomorphism h : (p, ≺) → (cj , ≺j ), such that h|q = h lower homomorphisms on additive generalized algebraic lattices 307 the proof can be seen in [3]. acknowledgements the authors would like to thank the editor juanjo font for his english revision of the paper, which has helped to improve the paper significantly. references [1] h. j. bandelt, m-distributive lattices, arch math 39 (1982), 436–444. [2] x. chen, q. li and z. deng, stone compactification on additive generalized algebraic lattice, applied general topology, to appear. [3] x. chen, q. li, f. long and z. deng, tietze extension theorem on additive generalized algebraic lattice, acta mathematica scientia (a)(in chinese), to appear. [4] x. chen, z. deng and q. li, separation axioms on additive generalized algebraic lattice, j. of shandong univ. technology (in chinese) 20 (2006), 5–8. [5] z. deng, generalized-continuous lattices i, j. hunan univ. 23, no. 3 (1996), 1–3. [6] z. deng, generalized-continuous lattices ii, j. hunan univ. 23, no. 5 (1996), 1–3. [7] z. deng, homomorphisms of generalized-continuous lattices, j. hunan univ. 26, no. 3 (1999), 1–4. [8] z. deng, topological representation for generalized-algebraic lattices, (in w. charles. holland, edited: ordered algebraic structures, algebra, logic and applications vol 16, 49-55 gordon and breach science publishers, 2001.) [9] z. deng, additivity of generalized algebraic lattices and t0-topology, j. hunan univ. 29, no. 5 (2002), 1–3. [10] z. deng, representation of strongly generalized-continuous lattices in terms of complete chains, j. hunan univ. 29, no. 3 (2002), 8–10. [11] d. drake and w. j. thron, on representation of an abstract lattice as the family of closed sets of a topological space, tran. amer. math. soc. 120 (1965), 57–71. [12] g. gierz et al., a compendium of continuous lattices, berlin, speringerverlag, 1980. [13] p. t. johnstone, stone spaces, cambridge univ press, cambridge, 1983. [14] j. l. kelly, general topology, van nostrand princeton, nj, 1995. [15] d. novak, generalization of continuous posets, tran. amer. math. soc 272 (1982), 645–667. [16] s. papert, which distributive lattices are lattices of closed sets?, proc. cambridge. phil. soc. 55 (1959), 172–176. [17] q. x. xu, construction of homomorphisms of m-continuous lattices, tran. amer. math. soc. 347 (1995),3167–3175. received june 2006 accepted june 2007 xueyou chen (chenxueyou0@yahoo.com.cn) school of mathematics and information science, shandong university of technology, zibo, shandong 255049, p. r. china. zike deng school of mathematics and economics, hunan university, changsha, hunan 410012, p.r. china. alaswilsonagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 273-281 on complete accumulation points of discrete subsets ofelia t. alas and richard g. wilson∗ abstract. we introduce a class of spaces in which every discrete subset has a complete accumulation point. properties of this class are obtained and consistent examples are given to show that this class differs from the class of countably compact and the class of compact spaces. a number of questions are posed. 2000 ams classification: primary 54a25, 54a35, 54d99 keywords: discrete subset, complete accumulation point, compact space, countably compact space, discretely complete space, us-spacewrite here some important concepts used in your paper. 1. introduction a classical theorem of general topology states that a hausdorff space is compact if and only if each infinite subset has a complete accumulation point (for details, we refer the reader to [3], 3.12.1). additionally, it was shown in [14] that a hausdorff space is compact if and only if the closure of every discrete subspace is compact. on comparing these results, it is natural to ask whether one can characterize compactness in terms of complete accumulation points of discrete sets: question 1.1. is it true that if every discrete subspace of a hausdorff space x has a complete accumulation point in x, then x is compact? shortly we shall see that the answer to this question is consistently, no. we make the following definition. definition 1.2. a t1-space x is said to be discretely complete if every infinite discrete subspace has a complete accumulation point in x. ∗research supported by consejo nacional de ciencia y tecnoloǵıa (méxico), grant 38164e and fundação de amparo a pesquisa do estado de são paulo (brasil) 274 o. t. alas and r. g. wilson clearly every compact topological space is discretely complete and each discretely complete space is countably compact. furthermore, since an accumulation point of a countable subset of a t1-space is a complete accumulation point of that set, it follows that: remark 1.3. a countably compact t1-space with countable spread is discretely complete; in particular, each hereditarily separable countably compact t1-space is discretely complete. we do not know if there exists a model of zf c in which every discretely complete hausdorff space is compact. since a discretely complete space is countably compact, and it is well-known that a countably compact, linearly lindelöf space is compact, it is immediate that a completely discrete, linearly lindelöf space is compact. furthermore, a space is linearly lindelöf if and only if every uncountable subset of regular cardinality has a complete accumulation point. so a non-compact t1-space which is discretely complete must contain some (non-discrete) subset of uncountable regular cardinality which has no complete accumulation point. in [10], ostaszewski constructed a non-compact, perfectly normal, hereditarily separable, countably compact tychonoff space assuming ch + ♣(≡ ♦); it follows immediately from remark 1.3 that this is an example of a discretely complete non-compact space. in [4], fedorčuk constructed using ♦, a hereditarily separable compact hausdorff space in which every infinite closed set has cardinality 2c. in the sequel, this space is used to show that a product of two discretely complete spaces need not be countably compact. both of the examples we have just mentioned are s-spaces (regular, hereditarily separable but not hereditarily lindelöf) and it is well-known that the existence of an s-space is independent of zf c. we also note that the existence of a discretely complete, non-compact regular space of countable spread is independent of zf c since it was shown in [2] that under the proper forcing axiom each regular countably compact space of countable spread is compact. furthermore, peter nyikos has informed us that the same result holds in the case of countably compact hausdorff spaces. however, it seems not to be known whether or not there exists in zf c a compact hausdorff space x such that |x| > 2s(x) and as we shall show later, the existence of such a space gives rise to a non-compact discretely complete hausdorff space. all spaces in the sequel are assumed to be t1. all notation and terminology not specifically defined below can be found in [3], [7] and [11]. 2. the results the examples of fedorčuk [4] and ostaszewski [10] cited above in section 1, both have countable spread, but a simple construction allows us to produce discretely complete, non-compact tychonoff spaces of arbitrary spread. on complete accumulation points of discrete subsets 275 lemma 2.1. if in a model of zf c there is a discretely complete, non-compact tychonoff space of countable spread, then for each cardinal κ there is a discretely complete, non-compact tychonoff space of spread κ. proof. let x be a discretely complete non-compact tychonoff space of countable spread and let z = (κ × x) ∪ {∞} where κ has the discrete topology, κ × x has the product topology and basic neighbourhoods of ∞ are of the form {∞} ∪ [(κ \ f ) × x], where f ⊆ κ is finite. we denote the projection from κ × x → κ by π. clearly s(z) = κ and if d ⊆ z is an infinite discrete set, there are two possibilities: 1) if π(d) is infinite, then ∞ is a complete accumulation point of d, or 2) if π(d) = {α0, . . . , αn} is finite, then d is countable and for some j ∈ {0, . . . , n}, d ∩ ({αj} × x) is countably infinite and hence d has a complete accumulation point in {αj} × x. � the class of discretely complete spaces, like that of compact space is closed under taking continuous images and closed subsets. lemma 2.2. the continuous image of a discretely complete space is discretely complete. proof. suppose x is discretely complete and f : x → y is continuous and surjective. let d = {dα : α ∈ κ} be a discrete subset of y ; then if for each α ∈ κ we pick xα ∈ f −1[{dα}] it is clear that {xα : α ∈ κ} is a discrete subset of x and hence must have a complete accumulation point, p say. then if q = f (p) and v is an open neighbourhood of q, it follows that |{α : xα ∈ f −1[v ]}| = κ and hence |{α : dα ∈ v }| = κ, showing that q is a complete accumulation point of d. � the proof of the following trivial result is left to the reader. lemma 2.3. a closed subspace of a discretely complete space is discretely complete however, unlike the class of compact spaces, the class of discretely complete spaces is not closed under the taking of products. example 2.4. it is consistent with zfc that there exist two discretely complete tychonoff spaces whose product is not countably compact. proof. let x denote the (above mentioned) compact space constructed in [4] under ♦. let s be a fixed countably infinite subset of x; note that since x has no non-trivial convergent sequences, s is not compact and hence not countably compact. we enumerate the infinite discrete subsets of s as d = {dα : α < c}. since |cl(d0)| = 2 c, we may choose distinct points a00, b 0 0 ∈ cl(d0) \ s. if for some α < c, we have chosen sets a0,α = {a 0 β : β < α} and b0,α = {b 0 β : β < α}, such that a0,α ∩ b0,α = ∅ and a 0 β , b 0 β ∈ cl(dβ ) \ s for each β < α, then again, since |cl(dα)| = 2 c and |s ∪ a0,α ∪ b0,α| ≤ c, we can choose distinct points a0α, b 0 α ∈ cl(dα) \ (s ∪ a0,α ∪ b0,α). let k1 = s ∪ {a 0 α : α < c} and l1 = s ∪ {b 0 α : α < c}. 276 o. t. alas and r. g. wilson clearly k1 and l1 have cardinality c and hence |[k1] ω| = |[l1] ω| = c. enumerate the countably infinite discrete subsets of k1 and l1 as {d1,α : α < c} and {e1,α : α < c} respectively. since, for each α < c, the sets cl(d1,α) and cl(e1,α) have cardinality 2 c we can repeat the process described in the previous paragraph so as to obtain, for each α < c, sets a1,α = {a 1 β : β < c} and b1,α = {b 1 β : β < c}, such that a1,α∩b1,α = ∅ and a 1 β ∈ cl(d1,β )\(k1∪l1) and b1β ∈ cl(e1,β ) \ (k1 ∪ l1); finally. let k2 = k1 ∪ a1,α and l2 = l1 ∪ b1,α. having defined sets kβ and lβ for each β < α < ω1 = c, if α is a limit ordinal, then define kα = ⋃ {kβ : β < α}, and lα = ⋃ {lβ : β < α}; if α is a successor ordinal then repeat the process of the previous paragraph. thus we define kα and lα of x for each α < ω1 = c. clearly s ⊆ kα ⊆ kγ and s ⊆ lα ⊆ lγ whenever α < γ < c and kα ∩ lα = s. let k = ⋃ {kα : α < c} and l = ⋃ {lα : α < c}. if t is a countable subset of k (respectively, l), then there is some α < c such that t ⊆ kα (respectively, t ⊆ lα) and hence t has an accumulation point in kα+1 (respectively, lα+1). thus both k and l are countably compact. since x is hereditarily separable, it follows that both k and l have countable spread and are not compact since they are proper dense subsets of cl(s). it follows from remark 1.3 that both k and l are discretely complete. however, {(s, s) : s ∈ s} ∼= s is a closed subspace of k × l which is not countably compact. hence k × l is not countably compact. � as our next result shows, even the product of a compact hausdorff space and a discretely complete space need not be discretely complete (although it will certainly be countably compact). theorem 2.5. if x is a discretely complete, but non-compact t1-space, then there is a compact hausdorff space y such that x×y is not discretely complete. proof. since x is not compact, there is a subset a = {aα : α ∈ κ} ⊆ x which has no complete accumulation point. since x is discretely complete, it is countably compact, and hence κ must be uncountable. let y be the alexandroff compactification of the discrete space d(κ) of cardinality κ and let {dα : α ∈ κ} be an enumeration of d(κ). we consider the set c = {(aα, dα) : α ∈ κ} ⊆ x × y . since {dα} is open for each α ∈ κ, it follows that c is discrete and if x × y were discretely complete, then c would have a complete accumulation point, say p = (x0, y0). thus if u is a neighbourhood of x0 and v is a neighbourhood of y0, then |{α : (aα, dα) ∈ u × v }| = κ. thus for each neighbourhood u of x0, |{α : aα ∈ u}| = κ, showing that x0 is a complete accumulation point of a, a contradiction. � corollary 2.6. the property of being discretely complete is not an inverse invariant of perfect mappings. in contrast to theorem 2.5, we have the following results. recall that a space is initially κ-compact if every open cover of size at most κ has a finite subcover. the following two lemmas are immediate consequences of theorems 2.2 and 5.2 of [12]. on complete accumulation points of discrete subsets 277 lemma 2.7. a space is initially κ-compact if and only if every infinite subset of cardinality at most κ has a complete accumulation point. lemma 2.8. if x is compact and y is initially κ-compact, then x × y is initially κ-compact. theorem 2.9. if x is a compact space of weight κ and y is a discretely complete space which is initially κ-compact, then x × y is discretely complete. proof. note that lemma 2.8 implies that x×y is initially κ-compact. suppose that d = {(xα, yα) : α ∈ λ} is an infinite discrete subset of x × y . there are three cases to be considered. 1) if λ ≤ κ, then since x × y is initially κ-compact, it follows from lemma 2.7 that d has a complete accumulation point in x × y . 2) if cof (λ) > κ, then fix a base b of x of size κ and for each b ∈ b, define ib = {α ∈ λ : there is an open neighbourhood wα of yα such that (b × wα) ∩ d = {(xα, yα)}}. since cof (λ) > κ, there is some b ∈ b such that |ib| = λ. the set yb = {yα : α ∈ ib} is discrete in y and hence has a complete accumulation point q ∈ y . but then, if for each x ∈ x, (x, q) is not a complete accumulation point of d, then for each x ∈ x we can find open neighbourhoods ux of x and vx of q such that |(ux × vx) ∩ d| < |d|. the open cover {ux : x ∈ x} of x has a finite subcover {ux1, . . . , uxn} and if we let v = ⋂ {vxk : 1 ≤ k ≤ n}, it follows that |(x ×v )∩d| < |d| which contradicts the fact that q is a complete accumulation point of yb. 3) if λ > κ ≥ cof (λ), then we can find regular cardinals {λα : α ∈ cof (λ)} such that κ < λα < λ and sup{λα : α ∈ cof (λ)} = λ. now write d = ⋃ {dα : α ∈ cof (λ)} where |dα| = λα. by 2), each of the discrete sets dα has a complete accumulation point (pα, qα) ∈ x × y , and since this latter space is initially κcompact, it again follows from lemma 2.7 that the set {(pα, qα) : α ∈ cof (λ)} has a complete accumulation point (p, q) ∈ x × y . now any neighbourhood v of (p, q) contains cof (λ)-many points (pα, qα) and hence λα-many points of dα for cof (λ)-many α. it follows that (p, q) is a complete accumulation point of d. � corollary 2.10. if x is a compact metrizable space and y is discretely complete, then x × y is discretely complete. proof. since x is metrizable, w(x) = ω. the space y is discretely complete, hence countably compact, that is to say, initially ω-compact; the result now follows from the theorem. � we now show that a construction very similar to that used in example 2.4 can in fact be carried out on any compact hausdorff space in which |x| > 2s(x) in order to construct a non-compact discretely complete space. the construction is reminiscent of the classical construction of a countably compact, non-compact dense subspace of βω of size c (see [5], 9.15). 278 o. t. alas and r. g. wilson theorem 2.11. in any model of zfc in which there exists a compact hausdorff space x with |x| > 2s(x), there exists a non-compact discretely complete tychonoff space. proof. suppose κ = s(x); by theorem 2.17 of [7], there is a dense subspace e0 ⊆ x of cardinality at most 2 κ. we enumerate the discrete subsets of e0 as {d0α : α ∈ λ0}, where λ0 ≤ |e0| s(x) ≤ (2κ)κ = 2κ < |x|. since x is compact, each subset d0α has a complete accumulation point x 0 α ∈ x; let e1 = e0 ∪{x 0 α : α ∈ λ0}. clearly |e1| ≤ max{λ0, 2 κ} < |x|. having constructed subsets eα for each α < β < κ+, with the property that |eα| ≤ 2 κ < |x| and such that eµ ⊆ eν whenever µ < ν, we construct eβ as follows: if β is a limit ordinal then eβ = ⋃ {eα : α ∈ β}. if β = γ + 1, then since |eγ| ≤ 2 κ, we can enumerate the discrete subsets of eγ as {d γ α : α ∈ λγ} where λγ ≤ 2 κ < |x|. again, since x is compact, each of the sets dγα has a complete accumulation point in x and for each α ∈ λγ we choose one such, xγα. let eβ = eγ ∪ {x γ α : α ∈ λγ}. it is immediate that |eβ| ≤ 2 κ < |x|. now let e = ⋃ {eα : α < κ +}; clearly |e| ≤ κ+.2κ = 2κ and so e x. furthermore, if s ⊆ e is discrete, then |s| ≤ κ and hence there is some α ∈ κ+ such that s ⊆ eα and so s has a complete accumulation point in eα+1 and hence in e as well. � we note in passing that by applying the technique of gryzlov [6], the space e in the previous theorem can even be made to be normal. before stating a generalization of this result we need some definitions. recall that a space x is a kc-space if every compact subspace of x is closed, x is an sc-space (see for example, [1]) if every convergent sequence together with its limit forms a closed subset of x and x is a us-space (see, [8]) if every convergent sequence in x has a unique limit. it is easy to see that kc ⇒ sc ⇒ u s ⇒ t1. a similar technique to that used in theorem 2.11 can be used to prove the following result. we leave the details to the reader. theorem 2.12. in any model of zfc in which there exists a compact kcspace x with |x| > d(x)s(x), there exists a non-compact discretely complete kc-space. as mentioned in the introduction, we do not know if there is a zf c example of a discretely complete, non-compact hausdorff space; however, discretely complete, non-compact u s-spaces exist in zf c. example 2.13. there is in zfc a non-compact u s-space which is discretely complete. on complete accumulation points of discrete subsets 279 proof. our aim is to define a topology σ on the set ω1 such that (ω1, σ) is a non-compact, discretely complete u s-space; we begin by defining a topology τ (used in [13]) generated by the following sub-base: {{β : β < α} : α ∈ ω1} ∪ {c : ω1 \ c is finite}. clearly τ is a t1-topology which is weaker than the order topology on ω1 and hence (ω1, τ ) is countably compact but not lindelöf, since the open cover {{β : β < α} : α ∈ ω1} has no countable subcover. furthermore, if a ⊂ ω1 has order type (induced by the order on ω1) greater than or equal to ω + 1, then a is not discrete, and hence every discrete subset of (ω1, τ ) is countable. that this space is discretely complete but not lindelöf and hence not compact is now a consequence of the remarks following definition 1.2. however, in (ω1, τ ), every injective sequence converges to an uncountable number of points. to obtain a u s-space we use a construction used for compact spaces by künzi and van der zypen, [8]. let a = {aα : α ∈ i} be a maximal almost disjoint (mad) family of injective sequences in ω1, where aα = {x n α : n ∈ ω} and for each α ∈ i, choose a limit ℓα 6∈ aα. denote the set {x n α : n ≥ m} ∪ {ℓα} by a m α and let σ be the topology generated by the subbase τ ∪ {x \ amα : m ∈ ω, α ∈ i}. we claim that (ω1, σ) is a u s-space which is discretely complete and since it is not lindelöf, it is not compact. in order to show that the space is discretely complete, we first show that its spread is countable. to this end, suppose that b is an uncountable subset of ω1 and we write b = ⋃ {bα : α ∈ ω1} where the sets bα are mutually disjoint and countably infinite. consider the countably infinite set b0 ≡ bβ0 . since a is a mad family, there must exist aα0 ∈ a such that b0 ∩ aα0 is infinite. let b0 = sup(aα0 ∪ b0 ∪ {ℓα0}) < ω1, clearly b ∩ (b0, ω1) is uncountable. let β1 = min{β ∈ ω1 : |bβ ∩ (b0, ω1)| = ω}. again, since a is a mad family, there is some aα1 ∈ a such that aα1 ∩ bβ1 ∩ (b0, ω1) is infinite; let b1 = sup(aα1 ∪ bβ1 ∪ {ℓα1}). continuing this process, we obtain a family {bβn : n ∈ ω} of countably infinite, mutually disjoint subsets of b, each of which intersects an element of a in an infinite set. let s = ⋃ {aαn ∩ bβn ∩ (bn−1, ω1) : n ∈ ω} and let b ∈ b ∩ (sup(s), ω1). we claim that b ∈ clσ(s), thus showing that b is not discrete. suppose to the contrary that b 6∈ clσ(s); then there is some basic closed set containing s but not b. since all τ -closed sets contain a cofinal interval of ω1, it follows that there must be some finite subset of a, say {aγ1 , aγ1 , . . . , aγn} such that b 6∈ ⋃ {aγm : 1 ≤ m ≤ n} ∪ {ℓγm : 1 ≤ m ≤ n} ⊇ s. clearly then there is some j ∈ {1, . . . , n} and some k ∈ ω such that aγj ∩ aαk ∩ bβk is infinite, which since there are only finitely many possible such j but infinitely many such k, contradicts the fact that a is an almost disjoint family. thus b ∈ cl(s) and to show that (ω1, σ) is discretely complete, it now suffices to show that it is countably compact. however, if t = {tn : n ∈ ω} is a countably infinite subset of ω1, then there is some aλ ∈ a such that aλ ∩ t is infinite. it is immediate that ℓλ ∈ cl(t ). finally, the proof that (ω1, σ) is a u s-space follows exactly as in [8]. � 280 o. t. alas and r. g. wilson recall that a space is weakly lindelöf if every open cover has a countable dense subsystem. the spaces βω and ω∗ = βω \ ω are the source of many examples and counterexamples, thus the following is of interest. theorem 2.14. [ch] for each p ∈ ω∗, ω∗ \ {p} is not discretely complete. proof. in the proof of corollary 1.5.4 of [9] it is shown (in zf c) that ω∗ \ {p} is not weakly lindelöf. hence there is an open cover u of ω∗ \ {p} with no countable dense subsystem. now, using ch, we may enumerate u as {uα : α < ω1}. choose x0 ∈ u0 and let α0 = 0. suppose now that for some λ < ω1, we have chosen points xγ , indices αγ ∈ ω1 and elements uαγ ∈ u for each γ < λ. note that since αγ is a countable ordinal for all γ < λ, it follows that cl( ⋃ {uξ : ξ < αγ , γ < λ}) 6= ω ∗ \ {p} and hence we can define αλ = min{β < ω1 : uβ \ cl( ⋃ {uξ : ξ < αγ , γ < λ}) 6= ∅} and choose xλ ∈ uαλ \ cl( ⋃ {uξ : ξ < αγ , γ < λ}). note that this construction ensures that xλ 6∈ cl{xγ : γ < λ} and xγ 6∈ cl(uαλ ) for each γ > λ and so {xα : α ∈ ω1} is discrete. furthermore, the discrete subset {xα : α ∈ ω1} has no complete accumulation point in ω ∗ \ {p}, since each of the open sets uα contains only countably many points of the set {xα : α ∈ ω1}. � 3. open questions below, we repeat the principal open questions regarding discretely complete spaces. question 3.1. is it consistent with zfc that every tychonoff discretely complete space is compact? question 3.2. is there in zfc, an sc (or even a kc or hausdorff ) example of a discretely complete space which is not compact? references [1] o. t. alas and r. g. wilson, minimal properties between t1 and t2, houston j. math. 32, no. 2 (2006), 493–504. [2] z. balogh, a. dow, d. h. fremlin and p. j. nyikos, countable tightness and proper forcing, bulletin a.m.s. (n. s.) 19, no. 1 (1988), 295–298. [3] r. engelking, general topology, heldermann verlag, berlin 1989. [4] v. fedorčuk, on the cardinality of hereditarily separable compact hausdorff spaces, soviet math. doklady 16 (1975), 651–655. [5] l. gillman and m. jerison, rings of continuous functions, van nostrand, princeton, 1960. on complete accumulation points of discrete subsets 281 [6] a. gryzlov, cardinal invariants and compactifications, comm. math. univ. carolinae 35, no. 1 (1994), 403–408. [7] i. juhasz, cardinal funcions in topology ten years later, mathematical centre tracts 123, mathematisch centrum, amsterdam, 1980. [8] h.-p. künzi and d. van der zypen, maximal (sequentially) compact topologies, preprint, http : //arxiv.org/ps cache/math/pdf/0306/0306082.pdf [9] j. van mill, an introduction to βω, in handbook of set-theoretic topology, north holland, amsterdam, 1984. [10] a. j. ostaszewski, on countably compact perfectly normal spaces, j. london math. soc. 14 (1976), 501–516. [11] m. e. rudin, lectures on set theoretic topology, cbms, regional conference series in mathematics, no. 23, a. m. s., providence, 1975. [12] r. m. stephenson, initially κ-compact and related spaces, in handbook of set-theoretic topology, north holland, amsterdam, 1984. [13] a. tamariz and r. g. wilson, example of a t1 topological space without a noetherian base, proceedings a.m.s. 104, no. 1 (1988), 310–312. [14] v. v. tkachuk, spaces that are projective with respect to classes of mappings, trans. moscow math. soc. 50 (1988), 139–156. received may 2006 accepted march 2007 o. t. alas (alas@ime.usp.br) instituto de matemática e estat́ıstica, universidade de são paulo, caixa postal 66281, 05311-970 são paulo, brasil r. g. wilson (rgw@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, unidad iztapalapa, avenida san rafael atlixco, #186, apartado postal 55-532, 09340, méxico, d.f., méxico kosiagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 1-12 almost cl-supercontinuous functions j. k. kohli and d. singh abstract. reilly and vamanamurthy introduced the class of ‘clopen maps’ (≡ ‘cl-supercontinuous functions’). subsequently generalizing clopen maps, ekici defined and studied almost clopen maps (≡ almost cl-supercontinuous functions). continuing in the spirit of ekici, here basic properties of almost clopen maps are studied. behavior of separation axioms under almost clopen maps is elaborated. the interrelations between direct and inverse transfer of topological properties under almost clopen maps are investigated. the results obtained in the process generalize, improve and strengthen several known results in literature including those of ekici, singh, and others. 2000 ams classification: primary: 54c05, 54c10; secondary: 54d10, 54d15, and 54d 20. keywords: almost clopen map, almost cl-supercontinuous function, (almost) z-supercontinuous function, clopen almost closed graphs, almost zero dimensional space, hyperconnected space. 1. introduction variants of continuity occur in almost all branches of mathematics and applications of mathematics. the strong variants of continuity with which we shall be dealing in this paper include strongly continuous functions introduced by levine [13], perfectly continuous functions considered by noiri ([18], [19]), clopen maps (≡ cl-supercontinuous functions) defined by reilly and vamanamurthy [21], and studied by singh [26], z-supercontinuous functions initiated by kohli and kumar [12], and supercontinuous functions introduced by munshi and bassan [16]. the variants of continuity which are independent of continuity and will be dealt with in this paper include regular set connected functions (≡ almost perfectly continuous functions) defined by dontchev, ganster and reilly [3], almost clopen maps (≡ almost cl-supercontinuous functions) studied by ekici [4], almost z-supercontinuous functions [11] and δ-continuous functions defined by noiri [17]. moreover, the weak forms of continuity which will 2 j. k. kohli and d. singh crop up in our discussion include almost continuous functions due to singal and singal [24], θ-continuous functions [5], quasi θ-continuous functions [20], weakly continuous functions [14], faintly continuous functions [15], dδ-continuous functions [9], z-continuous functions [23], and others. the purpose of this paper is to study properties of almost cl-supercontinuous functions (≡ almost clopen maps). in the process we generalize, improve and refine several known results in the literature including those of ekici [4], singh [26], and others. section 2 is devoted to basic definitions, preliminaries and nomenclature. in section 3 of this paper we study basic properties of almost cl-supercontinuous functions. it is shown that (i) almost cl-supercontinuity is preserved under the expansion of range as well as under the shrinking of range if f (x) is δ-embedded in y ; (ii) a mapping into a product space is almost cl-supercontinuous if and only if its composition with each projection map onto the co-ordinate space is almost cl-supercontinuous; (iii) if x is almost zero-dimensional, then f is almost cl-supercontinuous if and only if the graph function is almost clsupercontinuous. section 4 is devoted to the behavior of separation axioms under almost clsupercontinuous functions wherein interrelations between direct and inverse transfer of separation properties are investigated. in the process we generalize and considerably improve upon certain results of ekici [4], and singh [26]. in section 5, we interrelate (almost) cl-supercontinuity and connectedness. in the process we prove the existence and nonexistence of certain (almost) cl-supercontinuous functions. in section 6, we consider clopen almost closed graphs and obtain refinements of certain results of ekici [4]. 2. preliminaries and basic definitions 2.1. nomenclature. reilly and vamanamurthy [21] call a function clopen continuous if for each open set v containing f (x) there is a clopen (closed and open) set u containing x such that f (u ) ⊂ v . similarly, ekici [4] calls a function almost clopen if for each x ∈ x and each regular open set v containing f (x) there is a clopen set u containing x such that f (u ) ⊂ v . moreover, dontchev, ganster and reilly [3] call a function regular set connected if f −1(v ) is clopen in x for every regular open set v in y . however, as was also pointed out in [26] that in the topological folklore the phrase “clopen map” is used for the functions which map clopen sets to open sets and hence therein the “clopen continuous maps” of reilly and vamanamurthy are renamed as “cl-supercontinuous functions”, a better nomenclature since it represents a strong form of supercontinuity introduced by munshi and bassan [16]. in the same sprit in this paper we rename “almost clopen maps” studied by ekici [4] as “almost cl-supercontinuous functions” and ”regular set connected functions” defined by dontchev, ganster and reilly [3] as “almost perfectly continuous functions”, respectively. for the convenience of the reader and for the clarity of presentation we give here the precise definitions of all these variants of continuity. almost cl-supercontinuous functions 3 definition 2.2. a function f : x → y from a topological space x into a topological space y is said to be (i) strongly continuous [13] if f (ā) ⊂ f (a) for each subset a of x. (ii) perfectly continuous [18] if f −1(v ) is clopen in x for every open set v ⊂ y . (iii) almost perfectly continuous (≡ regular set connected [3]) if f −1(v ) is clopen for every regular open set v in y . (iv) cl-supercontinuous [26] (≡ clopen map [21]) if for each open set v containing f (x) there is a clopen set u containing x such that f (u ) ⊂ v . (v) almost cl-supercontinuous (≡ almost clopen map [4]) if for each x ∈ x and each regular open set v containing f (x) there is a clopen set u containing x such that f (u ) ⊂ v . (vi) z-supercontinuous [12] if for each x ∈ x and each open set v containing f (x) there is a cozero set u containing x such that f (u ) ⊂ v . (vii) almost z-supercontinuous [11] if for each x ∈ x and each regular open set v containing f (x) there is a cozero set u containing x such that f (u ) ⊂ v . (viii) supercontinuous [16] if for each x ∈ x and each open set v containing f (x) there is a regular open set u containing x such that f (u ) ⊂ v . (ix) δ-continuous [17] if for each x ∈ x and each regular open set v containing f (x) there is a regular open set u containing x such that f (u ) ⊂ v . (x) almost continuous [24] if for each x ∈ x and each regular open set v containing f (x) there is an open set u containing x such that f (u ) ⊂ v . remark 2.3. the original definitions of the concepts (v), (vii), (viii), (ix) and (x) in definitions 2.2 are slightly different from the ones which first appeared in the literature but are equivalent to ones given here, and are the simplest and most convenient to work with. the following implications are immediate from the definitions and well known (or easily verified). strongly continuous ⇓ perfectly continuous ⇒ almost perfectly continuous ⇓ ⇓ cl-supercontinuous ⇒ almost cl-supercontinuous ⇓ ⇓ z-supercontinuous ⇒ almost z-supercontinuous ⇓ ⇓ supercontinuous ⇒ δ-continuous ⇓ ⇓ continuous ⇒ almost continuous however, it is well known that none of the above implications is reversible. 4 j. k. kohli and d. singh 3. basic properties of almost cl-supercontinuous functions definition 3.1. a set g in a topological space x is said to be cl-open [26] ( δ-open [29]) if for each x ∈ g, there exist a clopen (regular open) set h such that x ∈ h ⊆ g, equivalently g is the union of clopen (regular open) sets. the complement of a cl-open (δ-open) set is referred to as cl-closed ( δ-closed) set. theorem 3.2. let f : x → y and g : y → z be functions. then the following statements are true. (a) if f is cl-supercontinuous and g is continuous, then g◦f is cl-supercontinuous. (b) if f is cl-supercontinuous and g is almost continuous, then g ◦ f is almost cl-supercontinuous. (c) if f is almost cl-supercontinuous and g is δ-continuous, then g ◦f is almost cl-supercontinuous. (d) if f is almost cl-supercontinuous and g is supercontinuous, then g ◦ f is cl-supercontinuous. proof. the assertion (a) is due to singh (see [26, theorem 2.10]) and (b) is due to ekici [4, theorem 13(2)]. to prove (c); let w ⊂ z be a regular open set. since g is δ-continuous, g−1(w ) is a δ-open set in y , i.e. g−1(w ) = ⋃ α vα, where each vα is a regular open set in y (see [17]). since f is almost cl-supercontinuous, each f −1(vα) is cl-open in x. thus (g ◦ f )−1(w ) = f −1(g−1(w )) = f −1( ⋃ α vα) = ⋃ α f −1(vα) being the union of cl-open sets is cl-open in x and so g ◦ f is almost clsupercontinuous. to prove (d); let w be an open set in z. since g is supercontinuous, g−1(w ) is δ-open set in y , i.e. g−1(w ) = ⋃ α vα, where each vα is a regular open set in y (see[16]). since f is almost cl-supercontinuous, f −1(vα) is a cl-open set in x for each α. thus (g ◦ f )−1(w ) = f −1(g−1(w )) = f −1( ⋃ α vα) = ⋃ α f −1(vα) being the union of cl-open sets is cl-open in x. hence g ◦ f is cl-supercontinuous. � remark 3.3. the assertion (c) of theorem 3.2 represents a simultaneous generalization of parts (1), (4), (5) and (6) of theorem 13 of ekici [4]. theorem 3.4. let {xα : α ∈ λ} be a cl-open cover of x. if for each α fα = f|xα is almost cl-supercontinuous, then f is almost cl-supercontinuous. proof. let v be a regular open subset of y . then f −1(v ) = ∪{f −1α (v ) : α ∈ λ}. since each fα is almost cl-supercontinuous, each f −1 α (v ) is cl-open in xα and hence in x. thus f −1(v ) being the union of cl-open sets is cl-open and so f is almost cl-supercontinuous. � remark 3.5. since every clopen set is cl-open, theorem 3.4 is an improvement of theorem 11 of ekici [4]. our next result gives a sufficient condition for the preservation of almost clsupercontinuity under the shrinking of range. first we formulate the concept of a δ-embedded set which seems to be of considerable significance in itself. almost cl-supercontinuous functions 5 definition 3.6. a subset s of a space x is said to be δ-embedded in x if every regular open set in s is the intersection of a regular open set in x with s or equivalently every regular closed set in s is the intersection of a regular closed set in x with s. theorem 3.7. let f : x → y be an almost cl-supercontinuous function. if f (x) is δ-embedded in y , then f : x → f (x) is almost cl-supercontinuous. proof. let v1 be a regular open set in f (x). since f (x) is δ-embedded in y , there exists a regular open set v in y such that v1 = v ∩ f (x). again, since f is almost cl-supercontinuous, f −1(v ) is cl-open in x. now f −1(v1) = f −1(v ∩ f (x)) = f −1(v ) ∩ f −1(f (x)) = f −1(v ) and so f : x → f (x) is almost cl-supercontinuous. � remark 3.8. in contrast to theorem 3.7, it is easily verified that almost clsupercontinuity is preserved under the expansion of range. the following lemma due to singal and singal [24] will be used in the sequel. lemma 3.9 ([24]). let {xα : α ∈ λ} be a family of spaces and let x = ∏ xα be the product space. if x = (xα) ∈ x and v is a regular open set containing x, then there exists a basic regular open set ∏ vα such that x ∈ ∏ vα ⊂ v , where vα is a regular open set in xα for each α ∈ λ and vα = xα for all except finitely many α1, α2 . . . αn ∈ λ. our next result shows that a mapping into a product space is almost clsupercontinuous if and only if its composition with each projection map onto a co-ordinate space is almost cl-supercontinuous. theorem 3.10. let {fα : x → xα : α ∈ λ} be a family of functions and let f : x → ∏ αελ xα be defined by f (x) = (fα(x)) for each x ∈ x. then f is almost cl-supercontinuous if and only if each fα : x → xα is almost cl-supercontinuous. proof. let f : x → ∏ αελ xα be almost cl-supercontinuous. since projection maps are δ-continuous, then in view of theorem 3.2 (c) the composition fα = pα ◦ f , where pα denotes the projection of ∏ αελ xα onto α th-coordinate space xα, is almost cl-supercontinuous for each α. conversely, suppose that each fα : x → xα is almost cl-supercontinuous. to show that the function f is almost cl-supercontinuous, it is sufficient to show that f −1(v ) is cl-open for each regular open set v in the product space∏ αελ xα. in view of lemma 3.9, it is clear that each regular open set v in the product space ∏ xα is the union of basic regular open sets of the form ∏ vα where each vα is regular open in xα and vα = xα for each α except finitely many indices α1, α2 . . . αn. thus each basic regular open set in ∏ xα is the finite intersection of sub-basic regular open sets of the form vβ × ∏ α6=β xα, where vβ is a regular open set in xβ. since arbitrary unions and finite intersections of cl-open sets is cl-open, it suffices to prove that f −1(s) is cl-open for every subbasic regular open set s in the product space∏ αελ xα. let vβ × ∏ α6=β xα be a subbasic regular open set in ∏ αελ xα. then 6 j. k. kohli and d. singh f −1(vβ × ∏ α6=β xα) = f −1(pβ −1(vβ )) = f −1 β (vβ ) is cl-open in x. hence f is almost cl-supercontinuous. � definition 3.11 ([7]). a space x is said to be almost zero dimensional at x ∈ x if for every regular open set v containing x there exists a clopen set u containing x such that u ⊂ v . the space x is said to be almost zero dimensional if it is almost zero dimensional at each x ∈ x. theorem 3.12 ([7]). a space x is almost zero dimensional if and only if each regular open set in x is cl-open. theorem 3.13. let f : x → y be a function and g : x → x × y , defined by g(x) = (x, f (x)) for each x ∈ x, be the graph function. then g is almost clsupercontinuous if and only if f is almost cl-supercontinuous and x is almost zero dimensional. proof. let g : x → x × y be almost cl-supercontinuous. then in view of theorem 3.2 (c) it is immediate that the composition f = py ◦ g is almost cl-supercontinuous, where py is the projection from x × y onto y (see also [4,theorem12]). to prove that x is almost zero dimensional, let u be a reguar open set in x and let x ∈ u . then u × y is a regular open set containing g(x). since g is almost cl-supercontinuous, there exists a clopen set w containing x such that g(w ) ⊂ u × y . thus x ∈ w ⊂ u , which shows that u is a cl-open and so the space x is almost zero dimensional. to prove sufficiency, let x ∈ x and let w be a regular open set containing g(x). by lemma 3.9 there exist regular open sets u ⊂ x and v ⊂ y such that (x, f (x)) ∈ u × v ⊂ w . since x is almost zero dimensional, there exists a clopen set g1 in x containing x such that x ∈ g1 ⊂ u . since f is almost cl-supercontinuous, there exists a clopen set g2 in x containing x such that f (g2) ⊂ v . let g = g1 ∩ g2. then g is a clopen set containing x and g(g) ⊂ u × v ⊂ w . this proves that g is almost cl-supercontinuous. � 4. separation axioms definitions 4.1. a space x is said to be (i) ultra hausdorff [27] if for each pair of distinct points x and y in x there exist disjoint clopen sets u and v containing x and y, respectively. (ii) ultra t1 (≡clopen t1 [4]) if for each pair of distinct points x and y in x there exist clopen sets u and v containing x and y, respectively such that y /∈ u and x /∈ v . (iii) ultra t0-space if for each pair of distinct points x and y in x there exists a clopen set u containing one of the points x and y but not the other. proposition 4.2. for a topological space x the following statements are equivalent. (a) x is an ultra hausdorff space. (b) x is an ultra t1-space. (c) x is an ultra t0-space. almost cl-supercontinuous functions 7 proof. clearly (a)⇒(b)⇒(c). to prove (c)⇒(a), let x be an ultra t0-space and let x, y be any two distinct points in x. then there exists a clopen set u containing one of the points x and y but not the other. to be precise assume that x ∈ u then u and x \ u are disjoint clopen sets containing x and y, respectively and so x is an ultra hausdroff space. � definitions 4.3. a topological space x is said to be (i) δt1-space (≡ r-t1 space [4]) if for each pair of distinct points x and y in x there exist regular open sets u and v containing x and y, respectively such that y /∈ u and x /∈ v . (ii) δt0-space if for each pair of distinct points x and y in x there exists a regular open set containing one of the points x and y but not the other. hausdorff space ⇒ δt1-space ⇒ δt0-space ⇓ ⇓ t1-space ⇒ t0-space example 4.4. the real line with co-finite topology is a t1-space which is not δt0 and so not a δt1-space. it is shown in [26] that if f : x → y is a cl-supercontinuous injection into a t0-space y , then x is an ultra-hausdorff space. in contrast, for an almost cl-supercontinuous injection we have the following. theorem 4.5. let f : x → y be an almost cl-supercontinuous injection. if y is a δt0-space, then x is an ultra-hausdorff space. proof. let x1 and x2 be two distinct points in x. then f (x1) 6= f (x2). since y is a δt0-space, there exists a regular open set v containing one of the points f (x1) or f (x2) but not the other. to be precise, assume that f (x1) ∈ v . since f is an almost cl-supercontinuous function, there exists a clopen set u containing x1 such that f (u ) ⊂ v . then u and x \ u are disjoint clopen sets containing x1 and x2 respectively and so x is ultra-hausdorff. � remark 4.6. the above theorem generalizes theorems 20 and 22 of ekici [4]. further, ekici ([4,theorem23]) proved that the equalizer of two almost clsupercontinuous functions into a hausdorff space is closed. here we obtain the following stronger version. theorem 4.7. let f, g : x → y be almost cl-supercontinuous functions into a hausdorff space y . then the equalizer e = {x ∈ x : f (x) = g(x)} of the functions f and g is a cl-closed subset of x. proof. to prove that e is cl-closed, we shall show that x \ e is cl-open. to this end, let x ∈ x \ e. then f (x) 6= g(x). since y is hausdorff, there exist disjoint open sets u1 and v1 containing f (x) and g(x), respectively. then u = (ū1) 0 and v = (v̄1) 0 are disjoint regular open sets containing f (x) and g(x), respectively. since f and g are almost cl-supercontinuous functions, there exist clopen sets g1 and g2 containing x such that f (g1) ⊂ u and g(g2) ⊂ v . 8 j. k. kohli and d. singh then g = g1 ∩ g2 is a clopen set containing x. since u and v are disjoint, clearly g ⊂ x \ e and so x \ e is cl-open. � the following theorem represents an strengthening of theorem 24 of ekici [4]. theorem 4.8. let f : x → y be an almost cl-supercontinuous function into a hausdorff space y . then the set a = {(x1, x2) ∈ x × x : f (x1) = f (x2)} is a cl-closed subset of x × x. proof. let (x, y) /∈ a. then f (x) 6= f (y). since y is hausdorff, there exist disjoint open sets u1 and v1 containing f (x) and f (y), respectively. then u = (ū1) 0 and v = (v̄1) 0 are disjoint regular open sets containing f (x) and f (y), respectively. since f is almost cl-supercontinuous, there exist clopen sets g1 and g2 containing x and y, respectively such that f (g1) ⊂ u and f (g2) ⊂ v . then g1 × g2 is a clopen subset of x × x containing (x, y) and (g1 × g2)∩a = φ. hence g1 × g2 ⊂ (x × x)\ a and so (x × x)\ a is cl-open being the union of clopen sets. thus a is a cl-closed subset of x × x. � definitions 4.9. a space x is said to be (i) almost regular [22] if for each regularly closed set f and each x /∈ f there exist disjoint open sets u and v containing x and f , respectively. (ii) mildly normal [25] if for every pair of disjoint regular closed sets a and b there exist disjoint open sets u and v containing a and b, respectively. the following theorem shows that the hypothesis that “x is regular” in theorem 27 of ekici [4] is superfluous and hence can be omitted. theorem 4.10. let f : x → y be an almost cl-supercontinuous open bijection. then y is an almost regular space. proof. let f be a regular closed subset of y and let y be a point outside f . then f −1(y) ∩ f −1(f ) = φ and f −1(y) is a singleton. since f is almost cl-supercontinuous, f −1(f ) is a cl-closed subset of x. hence x \f −1(f ) is a clopen subset of x containing f −1(y). so there exists a clopen set g containing f −1(y) such that g ⊂ x \ f −1(f ). then g and x \ g are disjoint clopen sets containing f −1(y) and f −1(f ), respectively. since f is an open bijection, f (g) and f (x \ g) are disjoint open sets containing y and f , respectively. so y is an almost regular space. � definitions 4.11. a space x is said to be weakly ∆-normal [2] (weakly θnormal [8], [10]) if each pair of disjoint δ-closed (θ-closed) are contained in disjoint open sets. the following theorem represents a significant improvement of theorem 28 of ekici [4]. theorem 4.12. let f : x → y be an almost cl-supercontinuous open bijection defined on a weakly θ-normal space x. then y is mildly normal. almost cl-supercontinuous functions 9 proof. let a and b be disjoint regular closed subsets of y . since f is almost cl-supercontinuous, f −1(a) and f −1(b) are disjoint cl-closed subsets of x. since every cl-closed set is θ-closed and since x is weakly θ-normal, there exist disjoint open sets u and v containing f −1(a) and f −1(b), respectively. since f is an open bijection, f (u ) and f (v ) are disjoint open sets containing a and b, respectively and hence y is mildly normal. � corollary 4.13. let f : x → y be an almost cl-supercontinuous open bijection defined on a weakly ∆-normal space x. then y is mildly normal. corollary 4.14 (ekici [4, theorem 28]). if f is an almost cl-supercontinuous open bijection from a normal space x onto a space y , then y is mildly normal. proof. every normal space is a weakly θ-normal space. � 5. connectedness ekici [4] calls a space x almost connected if x can not be written as a disjoint union of two nonempty regular open sets. we observe that a space is connected if and only if it can not be expressed as a disjoint union of two nonempty clopen sets and hence it can not be written as the disjoint union of two nonempty regular open sets. thus the notion of almost connectedness introduced by ekici is precisely connectedness. moreover, the hypothesis of theorem 30 of ekici [4] is too strong and can be considerably weakened, since connectedness is preserved under functions satisfying fairly mild continuity conditions. the known such weakest variant of continuity is slight continuity [6]. a function f : x → y is said to be slightly continuous if f −1(v ) is open in x for every clopen subset v of y . thus connectedness is preserved under each of the following variants of continuity listed in the following diagram, each of which is weaker than continuity except δ-continuity (which is independent of continuity). almost cl-supercontinuous ⇓ continuous δ-continuous [17] ⇓ ⇓ almost continuous [24] ⇓ θ-continuous [5] ⇓ ⇓ quasi-θ-continuous [20] weakly continuous [14] ⇓ ⇓ faintly continuous [15] ⇓ dδ-continuous [9] ⇓ z-continuous [23] ⇓ slightly-continuous [6] 10 j. k. kohli and d. singh definition 5.1 ([27], [1]). a space x is said to be hyperconnected if every nonempty open set in x is dense in x. ekici [4] showed that an almost cl-supercontinuous image of a connected space is hyperconnected. in contrast, our next result shows that cl-supercontinuous image of a connected space is indiscrete. theorem 5.2. let f : x → y be a cl-supercontinuous function from a connected space x onto a space y . then y is an indiscrete space. proof. suppose that y is not indiscrete and let v 6= y be an open set in y . since f is cl-supercontinuous, by [26, theorem 2.2] f −1(v ) is a nonempty proper cl-open subset of x. so there exists a nonempty proper clopen subset of x, contradicting the fact that x is connected. � thus there exists no cl-supercontinuous function from a connected space onto a non indiscrete space. in contrast it is shown in [26, theorem 4.9] that there exist no non constant cl-supercontinuous function from a connected space into a t0-space. 6. clopen almost closed graphs definition 6.1 ([4]). the graph g(f ) of a function f : x → y is said to be clopen almost closed if for each (x, y) /∈ g(f ) there exists a clopen set u of x and a regular open set v containing y such that (u × v ) ∩ g(f ) = φ. the following theorem represents an improved version of theorem 35 of ekici [4] which was essentially proved by him. however, for the convenience of the reader we include its proof. theorem 6.2. let f : x → y be an injection such that its graph g(f ) is clopen almost closed. then x is ultra hausdorff. proof. let x, y ∈ x, x 6= y. since f is an injection, (x, f (y)) /∈ g(f ). in view of almost closedness of the graph g(f ), there exist a clopen set u of x and a regular open set v containing f (y) such that (u × v ) ∩ g(f ) = φ. then f (u ) ∩ v = φ and hence u ∩ f −1(v ) = φ. therefore y /∈ u . then u and x \ u are disjoint clopen sets containing x and y, respectively. hence x is ultra hausdorff. � finally, we point out that in the hypothesis of [4, theorem 41, part 3], it is sufficient to assume x to be countably compact instead of compact. the next result is an strengthening of theorem 39 of ekici [4] which was essentially proved by him. however, for the sake of completeness and continuity of presentation, we include its proof. theorem 6.3. let f : x → y be a function such that the graph g(f ) of f is clopen almost closed in x × y . then f −1(k) is cl-closed in x for every n -closed subset k of y . almost cl-supercontinuous functions 11 proof. let k be an n -closed subset of y . to prove that f −1(k) is cl-closed, we shall show that x \ f −1(k) is cl-open. to this end, let x ∈ x \ f −1(k). then for each y ∈ k, (x, y) /∈ g(f ). so there exists a clopen set uy containing x and a regular open set vy containing y such that (uy × vy ) ∩ g(f ) = φ and hence f (uy) ∩ vy = φ. the collection {vy : y ∈ k} is a cover of k by regular open sets in y . so there exist finitely many y1, . . . , yn ∈ k such that k ⊂ ⋃n i=1 yyi . let u = ⋂n i=1 uyi . then u is a clopen set containing x such that f (u ) ∩ k = φ. hence u ⊂ x \ f −1(k) and so u is cl-open being the union of clopen sets. � references [1] n. ajmal and j. k. kohli, properties of hyperconnected spaces, their mapping into hausdorff space and embedding into hyperconnected spaces, acta math. hungar. 60, no. 1-2 (1992), 41–49. 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[10] j. k. kohli and d. singh, weak normality properties and factorizations of normality, acta math. hungar. 110, no. 1–2 (2006), 67–80. [11] j. k. kohli, d. singh and r. kumar, generalizations of z-supercontinuous functions and dδ-supercontinuous functions, applied general topology, to appear. [12] j. k. kohli and r. kumar, z-supercontinuous functions, indian j. pure appl. math. 33, no. 7 (2002), 1097–1108. [13] n. levine, strong continuity in topological spaces, amer. math. monthly 67 (1960), 269. [14] n. levine, a decomposition of continuity in topological spaces, amer. math. monthly 68 (1961), 44–46. [15] p. e. long and l. l. herrington, tθ-topology and faintly continuous functions, kyungpook math. j. 22 (1982), 7–14. [16] b. m. munshi and d. s. bassan, super-continuous mappings, indian j. pure appl. math. 13 (1982), 229–236. [17] t. noiri, on δ-continuous functions, j. korean math. soc. 16 (1980), 161–166. 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[29] n. v. velicko, h-closed topological spaces, amer. math. soc. transl. 78, no. 2 (1968), 103–118. received september 2007 accepted april 2008 j. k. kohli (jk kohli@yahoo.com) department of mathematics, hindu college, university of delhi, delhi 110 007, india d. singh (dstopology@rediffmail.com) department of mathematics, sri aurobindo college, university of delhi-south campus, delhi 110 017, india () @ applied general topology c© universidad politécnica de valencia volume 12, no. 1, 2011 pp. 49-66 introduction to generalized topological spaces irina zvina abstract we introduce the notion of generalized topological space (gt-space). generalized topology of gt-space has the structure of frame and is closed under arbitrary unions and finite intersections modulo small subsets. the family of small subsets of a gt-space forms an ideal that is compatible with the generalized topology. to support the definition of gt-space we prove the frame embedding modulo compatible ideal theorem. we provide some examples of gt-spaces and study key topological notions (continuity, separation axioms, cardinal invariants) in terms of generalized spaces. 2010 msc: 54a05, 06d22. keywords: generalized topology, generalized topological space, gt-space, compatible ideal, modulo ideal, frame, order generated by ideal. 1. introduction the notion of i-topological space was presented in [9]. that was our first try to develop the concept of topological space modulo small sets. generalized topological space presented in this paper also acts modulo small sets that are encapsulated in an ideal, but has completely other form that makes things easier. the new form let us easily profit from compatibility of ideal with generalized topology, since now it is the basic property of generalized space (we had to assume compatibility in the preceding work). we start with developing the frame-theoretical framework for generalized spaces and prove frame embedding modulo compatible ideal theorem that let us to make the definitions of generalized topological notions more transparent. 50 i. zvina 2. frame embedding modulo compatible ideal in what follows, we assume that a frame (t,≤,∨,∧) and a complete boolean lattice (f,≤,∪,∩, c) such that t ⊆ f are fixed. we also assume that the inclusion from t into f is an order embedding preserving zero. for all a,b ∈ f we define the operation \ as follows: a \ b = a ∩ bc. with respect to the fixed element z ∈ f , we say that a ∈ f is z-empty iff a ≤ z, and z-nonempty in other case [8]. if the converse is not stated, we always assume that some z ∈ f is fixed. we use the notion of z-emptiness in order to use the following equivalent form of infinite distributivity. proposition 2.1 ([8]). the following are equivalent: (id1) t is infinitely distributive, i.e. for every b in t and every subset a ⊆ t, b ∧ ∨ a = ∨ a∈a (b ∧ a), (id2) for every d,z ∈ t and every a ⊆ t, if d is z-nonempty and d ≤ ∨ a then there exists a0 ∈ a such that d ∧ a0 is z-nonempty. the notion of compatible ideal in topological spaces was studied in numerous papers of t.r. hamlett and d. janković [3, 4, 5, 6, 7]. we generalize this notion in the natural way. definition 2.2. we say that an element a ∈ f is small with respect to a subfamily h ⊆ f iff there exists a family u ⊆ t such that a ≤ ∨ u and a ∩ u ∈ h for all u ∈ u. denote the family of all small elements by sm(h). we do not specify t in the notation sm(h) since t is always fixed. the family h is said to be compatible with t , denote by h ∼ t , iff it contains all small elements from sm(h). an element a ∈ f is said to be join-generated by the family u ⊆ t iff a ≤ ( ∨ u ) \ ( ⋃ u ). an element b ∈ f is said to be meet-generated by the family v ⊆ t iff v is finite and b ≤ ( ⋂ v ) \ ( ∧ v ). in what follows, we are going to consider the following families: (1) g – the family of all joinand meet-generated elements, (2) i(g) – the family of finite joins of elements of g, (3) c(g) – the family of finite joins of elements of sm(g). lemma 2.3. the families i(g) and c(g) are ideals. proof. both i(g) and c(g) are closed under finite joins. thus, we have to show that both families are lower sets. consider b ∈ f and a ∈ i(g) with b ≤ a. then there exists finite a ⊆ g such that a = ⋃ a. then b = b ∩ a = b ∩ ⋃ a = ⋃ a∈a (b ∩ a). clearly, b ∩ a ≤ a and, hence, b ∩ a ∈ g holds for every a ∈ a, since a ∈ g and g is a lower set. hence, b lies in i(g) as a finite join of elements of g. introduction to generalized topological spaces 51 assume that b ∈ f and a ∈ sm(g) with b ≤ a. then there exists a family u ⊆ t such that a ≤ ∨ u and a ∩ u ∈ g for all u ∈ u. clearly, it holds that b ≤ ∨ u and b ∩ u ∈ g for all u ∈ u, since g is a lower set. we conclude that sm(g) is a lower set. consider b ∈ f and a ∈ c(g) with b ≤ a. then there exists finite a ⊆ sm(g) such that a = ⋃ a. then b = b ∩ a = b ∩ ⋃ a = ⋃ a∈a (b ∩ a). clearly, b ∩ a ≤ a and, hence, b ∩ a ∈ sm(g) holds for every a ∈ a, since a ∈ sm(g) and sm(g) is a lower set. hence, b lies in c(g) as a finite join of elements of sm(g). � lemma 2.4. the following inclusions hold: g ⊆ sm(g), i(g) ⊆ c(g), i(g) ⊆ sm(i(g)) and c(g) ⊆ sm(c(g)). proof. assume that h ⊆ f is a lower set, a ∈ h and there exists a family u ⊆ t such that a ≤ ∨ u. then a ∈ sm(h). since for every element a ∈ g there is u ∈ t such that a ≤ ∨ u, we conclude that g ⊆ sm(g). the other inclusions are the corollary from the later one. � lemma 2.5. given u = { u1,u2, ...,uk} ⊆ t and a z-nonempty v ∈ t such that v ≤ u1, then there exists a z-nonempty w ∈ t that satisfies w ≤ v and one of the following: (i) w ≤ ∧ u; (ii) w ∧ uj is z-empty for some uj ∈ u. proof. define the decreasing chain v = { v1,v2, ...,vk } as follows: v1 = v and vi = vi−1 ∧ ui for i ∈ { 2, 3, ...,k }. at least one element of this chain is z-nonempty since v1 = v. put w = vj where vj is the last z-nonempty element of the chain. clearly, it holds that w ≤ v. there are two possibilities: either j < k and then w ∧ uj+1 is z-empty, or j = k and then w ≤ ∧ u. the lemma is proved. � lemma 2.6. given z ∈ t, a z-nonempty y ∈ t satisfying y\z ≤ ⋃ a for some a = { a1,a2, ...,an } ⊆ g, then there exist a z-nonempty w ∈ t and aj ∈ a such that w ≤ y and w ∩ aj = 0. proof. for every ai ∈ a we denote by ui the corresponding family of elements from t by that ai is joinor meet-generated. we define the covering family as follows: c1 = { z } ∪ u1 ∪ u2 ∪ ... ∪ un. it holds that y ≤ ∨ c1. by proposition 2.1, we conclude that v1 = y ∧ u is z-nonempty for some u ∈ c1. without loss of generality, we assume that v1 ≤ u ∈ u1. there are two possibilities for a1: either a1 is join-generated and then v1 ∩ a1 = 0, we put w = v1 and the proof is complete, or a1 is meet-generated and then, applying lemma 2.5 for u1 and v1, we obtain a znonempty element w1 ≤ v1. if w1 ≤ ∧ u1 then w1 ∩ a1 = 0; we put w = w1 and the proof is complete. if the other case, we possess the element u1 ∈ u1 52 i. zvina such that w1 ∧u1 is z-empty. we repeat the whole process from the beginning. define the covering family c2 as follows: c2 = { z } ∪ { u1 } ∪ u2 ∪ u3 ∪ ... ∪ un. it holds that w1 ≤ ∨ c2. again, by proposition 2.1, v2 = w1 ∧u is z-nonempty for some u ∈ c2. without loss of generality, we assume that v2 ≤ u ∈ u2. again, there are two possibilities for a2: either a2 is join-generated and then w = v2 satisfies the conditions of the theorem, or a2 is meet-generated and we apply lemma 2.5 for u2 and v2 and obtain a z-nonempty element w2 ≤ v2. if it holds that w2 ≤ ∧ u2 then w = w2 is the one we need. in the other case, we possess the element u2 ∈ u2 such that w2 ∧ u2 is z-empty. we continue in the same way as above defining the covering family c3. through the process, we obtain the decreasing chain of z-nonempty elements y ≥ v1 ≥ w1 ≥ v2 ≥ . . . . if the process stops at some w = vj or w = wj , where j ∈ { 1, . . . ,n}, it means that the proof is complete. let us consider the other case. that is, we assume that the process did not stop and we possess the chain y ≥ w1 ≥ w2 ≥ · · · ≥ wn. as above, we define the covering family cn+1: cn+1 = { z } ∪ { u1 } ∪ { u2 } ∪ · · · ∪ { un−1 } ∪ { un }. it holds that wn is z-nonempty and wn ≤ ∨ cn+1. hence, by proposition 2.1, there exists u ∈ cn+1 such that wn∧u is z-nonempty. but such u does not exist. means the process stopped at some previous step. the proof is complete. � lemma 2.7. for every y,z ∈ t, it holds that y \ z ∈ i(g) iff y ≤ z. proof. if y ≤ z then y \ z = 0 ∈ i(g). let us proof the other implication. assume that y\z 6= 0. then there exists the family a1 = { a1,a2, . . . ,an } ⊆ g such that y \ z ≤ ⋃ a1. applying lemma 2.6, we obtain a z-nonempty element w1 ≤ y such that w1 ∩ a = 0 for some a ∈ a1. without loss of generality, assume that w1 ∩ a1 = 0. then from w1 \ z ≤ ⋃ a1 and infinite distributivity of f we imply that w1\z = (w1\z)∩ ⋃ a1 = ⋃ a∈a1 ((w1\z)∩a) = ⋃ a∈a2 ((w1\z)∩a) = (w1\z)∩ ⋃ a2, that is w1 \ z ≤ ⋃ a2 where a2 = { a2, . . . ,an }. we continue this process and obtain the decreasing chain of y ≥ w1 ≥ · · · ≥ wn of z-nonempty elements. for the last element wn it holds that wn \ z ≤ 0. means wn is z-empty. but an element cannot be both z-nonempty and z-empty. hence, our assumption that y \ z 6= 0 was false, and we conclude that y \ z ∈ i(g) implies y ≤ z. � lemma 2.8. for every y,z ∈ t, it holds that y \ z ∈ c(g) iff y ≤ z. proof. if y ≤ z then y \ z = 0 ∈ c(g). let us proof the other implication. assume that y \z 6= 0. then there exists the family a = { a1,a2, . . . ,an } ⊆ sm(g) such that y \ z ≤ ⋃ a. for every ai ∈ a we denote by ui the corresponding family of elements from t (def. 2.2). we define the covering family as follows: c1 = { z } ∪ u1 ∪ u2 ∪ ... ∪ un. introduction to generalized topological spaces 53 it holds that y ≤ ∨ c1. by proposition 2.1, we conclude that v1 = y ∧ u1 is z-nonempty for some u1 ∈ c1.without loss of generality, we assume that u1 ∈ u1. consider the following auxiliary families: d1 = u2 ∪ ... ∪ un and a1 = { a2, . . . ,an }. assume that v1 ∧ u is z-empty for all u ∈ d1. then it follows from infinite distributivity that v1 ∩ ⋃ a1 ≤ (v1 ∩ ∨ d1) \ z ∈ g. on the other hand, v1∩a1 ≤ u1∩a1 ∈ g since a1 is a small element. hence, v1\z = v1∩ ⋃ a ∈ i(g). applying lemma 2.7, we conclude that v1 ≤ z – a contradiction! means, v2 = v1 ∧ u2 = y ∧ u1 ∧ u2 is z-nonempty for some u2 ∈ d1. without loss of generality assume that u2 ∈ u2. as above, the assumption that v2 ∧ u is z-empty for all u ∈ u3 ∪ · · · ∪ un will bring us to the contradiction. hence, v3 = v2 ∧ u3 = y ∧ u1 ∧ u2 ∧ u3 is z-nonempty for, without loss of generality, some u3 ∈ u3. we continue this process till we obtain a z-nonempty vn = u1 ∧ · · · ∧ un. since all ai ∈ a are small elements, we imply that ui ∩ ai ∈ g, for all ui that form vn. hence, it holds that vn \z ∈ i(g). this is a contradiction with lemma 2.7, and we conclude that the assumption that y \ z 6= 0 is false. the proof is complete. � lemma 2.9. consider a ∈ f and u ⊆ t such that a ≤ ∨ u and a ∩ u ∈ c(g) for all u ∈ u. then there exist b,c ∈ f such that a = b ∪ c, b ∈ sm(g) and c ∩ u ∈ sm(g) for all u ∈ u. proof. fix u ∈ u. since a ∩ u ∈ c(g), there exist a = {a1, . . . ,an } ⊆ sm(g) such that a ∩ u = ⋃ a. for every ai ∈ a there is vi ⊆ t satisfying ai ≤ ∨ vi and ai ∩v ∈ g for all v ∈ vi. write v = n ⋃ i=1 vi. then a∩u ⊆ u and a∩u ⊆ ∨ v . hence, there exists bu ∈ g such that (a ∩ u) \ bu ≤ u ∧ ∨ v = ∨ v∈v ( u ∧ v ) . write ci = ai \ b u and wi = {u ∧ v | v ∈ vi }. consider c1. it holds that c1 ∩ w ∈ g for all w ∈ w1. on the other hand, n ∨ i=1 wi ≤ u. hence, w \ n ∨ i=2 wi ∈ g holds for all w ∈ w1. applying lemma 2.7, we conclude that w ≤ n ∨ i=2 wi for all w ∈ wi. thus, c1 = 0. we continue this process for all ci where i = 2, . . . ,n − 1. at the end, we will imply that c1 = · · · = cn−1 = 0 and cn = (a ∩ u) \ b u. it is clear that cn ∈ sm(g). write c u = cn. we do the same steps for each u ∈ u. write b = ⋃ {bu | u ∈ u } and c = ⋃ { cu | u ∈ u }. then b and c satisfy the necessary properties. the proof is complete. � lemma 2.10. it holds that c(g) is an ideal, sm(c(g)) = c(g) and, hence, c(g) ∼ t. 54 i. zvina proof. we already proved that c(g) is an ideal and that c(g) ⊆ sm(c(g)) (lemmas 2.3 and 2.4). let us prove that sm(c(g)) ⊆ c(g). consider a ∈ f and u ⊆ t such that a ≤ ∨ u and a ∩ u ∈ c(g) for all u ∈ u. applying lemma 2.9 and since c(g) contains finite joins of small elements, without loss of generality, we can assume that a ∩ u ∈ sm(g) for all u ∈ u. fix u ∈ u. then there exists a family v u ⊆ t such that a ∩ u ≤ ∨ v u and a ∩ u ∩ v ∈ g for all v ∈ v u. consider the family w u = {v ∧ u | v ∈ v u}. clearly, ∨ w u ≤ u and there exists cu ∈ g such that (a ∩ u) \ cu ≤ ∨ w u and cu ≤ u. denote b = ⋃ {(a ∩ u) \ cu | u ∈ u} and c = {cu | u ∈ u}. then a = b ∪ c and both b and c are small elements. we conclude that a ∈ c(g). � proposition 2.11. given an ideal i ⊆ f, then the relation � defined as follows is a preorder on f: a � b iff a \ b ∈ i. the relation ≈ defined as follows is an equivalence on f: a ≈ b iff a \ b ∈ i and b \ a ∈ i. proof. the reflexivity holds for � and ≈ since 0 ∈ i. the transitivity for � and ≈ holds since i is closed under finite joins. the symmetry follows from the definition of ≈. � the relations � and ≈ considered in the previous proposition are called the preorder generated by the ideal i and the equivalence generated by the ideal i, respectively. now we are ready to prove the frame embedding modulo compatible ideal theorem. briefly speaking, this theorem says that an order embedding of a frame into a complete boolean lattice preserving zero is always a frame embedding modulo compatible ideal. the assumption t ⊆ f simplifies the notations but is not essential. we could speak as well about an order embedding ϕ: t → f preserving zero and consider the image ϕt ⊆ f instead of t . theorem 2.12. let (t,∨,∧) be a frame and (f,∪,∩, ∗) be a complete boolean lattice such that t ⊆ f and the inclusion from t into f is an order embedding preserving zero. then i = c(g) is the least ideal satisfying the following: (i) for every u ⊆ t there is a ∈ i such that ∨ u = ( ⋃ u) ∪ a; (ii) for every v,w ∈ t there is b ∈ i such that v ∧ w = (v ∩ w) \ b; (iii) i ∩ t = {0}; (iv) for every u,v ∈ t, it holds that u � v iff u ≤ v; (v) i ∼ t. proof. it follows from the construction of i that (i) and (ii) hold. the statement (iv) is proved in lemma 2.8 and applying lemma 2.8 for all y ∈ t and z = 0 we imply (iii). the statement (v) is proved in lemma 2.10. introduction to generalized topological spaces 55 the least ideal satisfying (i),(ii) and (v) should contain all elements of sm(g). on the other hand, the least ideal that contains all elements of sm(g) is c(g). hence, c(g) is the least ideal satisfying (i), (ii) and (v). � remark 2.13. in theorem 2.12, if we assume that t is not only a frame but is a complete completely distributive lattice is it then possible to construct the ideal in such a way that (i), (iii)-(v) hold and (ii) holds for arbitrary subfamilies of t ? the counterexample for it was introduced in [8]. 3. definition of gt-space. examples we start with the immediate corollary from theorem 2.12 that is the motivation for the following definition of generalized topological space. corollary 3.1. let x be a nonempty set. assume that t ⊆ 2x forms a frame with respect to ⊆ and ∅,x ∈ t. then there exists the least ideal i ⊆ 2x such that: (i) for every u ⊆ t holds ∨ u \ ⋃ u ∈ i; (ii) for every v,w ∈ t holds (v ∩ w) \ (v ∧ w) ∈ i; (iii) t ∩ i = {∅}; (iv) u � v (prop. 2.11), i.e. u \v ∈ i, implies u ⊆ v for every u,v ∈ t; (v) u ≈ v (prop. 2.11), i.e. u∆v ∈ i, implies u = v for every u,v ∈ t; (vi) the ideal i is compatible with t, write i ∼ t (def. 2.2), i.e. a ⊆ x and u ⊆ t with a ⊆ ∨ u and a ∩ u ∈ i, for all u ∈ u, imply that a ∈ i. definition 3.2. let x be a nonempty set. a family t ⊆ 2x is called a generalized topology (or topology modulo ideal ) and the pair (x,t ) is called a generalized topological space (gt-space for short, or topological space modulo ideal ) provided that: (gt1) ∅,x ∈ t ; (gt2) (t,⊆) is a frame. the elements of x are called points, and the elements of t are called open sets. we say that y ⊆ x is a neighborhood of a point x iff there is u ∈ t such that x ∈ u ⊆ y . we use the notation t (x) for the family of all open neighborhoods of a point x. an ideal j ⊆ 2x satisfying (i)-(v) of corollary 3.1 is called suitable. in there is no chance for confusion, we keep the notation i, sometimes with the appropriate index, to denote the least suitable ideal (the existence of it is proved in corollary 3.1). if the ideal is not specified in a definition or construction then it is always the least suitable ideal. if there is no specification or index, we use the symbols � and ≈ to denote the preoder and equivalence, respectively, generated by the least suitable ideal (prop. 2.11). we keep the notations ∨ and ∧ for the frame operations of generalized topology. 56 i. zvina a topological space is a trivial example of gt-space where the least suitable ideal consists only of the empty set. in order to distinguish between topological spaces and gt-spaces that are not topological spaces, we provide the following classification. definition 3.3. a gt-space is called (1) crisp gt-space (crisp space for short) iff its least suitable ideal is {∅}; (2) proper gt-space iff it is not crisp. example 3.4 (right arrow gt-space). consider the real line r and the family of “right arrows” a = { [a,b) | a,b ∈ r ∪ {−∞, +∞} and a < b} . we say that a subfamily a′ ⊆ a is well separated iff for all [a,b), [c,d) ∈ a′ it holds that b < c or d < a. construct the family tra as follows: tra = { ∅, r } ∪ { ⋃ a ′ | a′ ⊆ a and a′ is well separated } . clearly tra is a complete lattice and it is also easy to see that tra is infinitely distributive (e.g. by proposition 2.1). then the pair (r,tra) forms a gt-space. the respective least ideal for this gt-space is the family d of nowhere dense subsets of the real line. let us prove that. consider a nowhere dense subset a ⊆ r and the open subset u = ∨ {v ∈ tra | v ∩ a = ∅ }. fix a point a ∈ a and assume that a /∈ u. assume that we could find the closest “right arrow” on the right from the point a, that is there exists b = min { r ∈ u | r > a}. then there is an interval (c,d) ⊆ (a,b) such that (c,d) ∩ a = ∅, since a is nowhere dense, and, hence, there is a non-empty open v ⊆ (c,d) with v ∩ a = ∅. the latter means that v ⊆ u but this is a contradiction and we conclude that such minimal point b does not exist. then the part of u lying on the right from the point a is a union of a well separated subfamily of “right arrows”. this and the fact that a is nowhere dense imply that there exists an interval (c,d) such that a < c, (c,d) ∩ u = ∅ and (c,d) ∩ a = ∅. hence, there is a non-empty open v ⊆ (c,d) with v ∩ u = ∅ and v ∩ a = ∅. the latter is a contradiction and we conclude that our assumption that a /∈ u is false. thus, we proved that a ⊆ u. this means that u = r and, hence, a belongs to the least suitable ideal as a join-generated subset of r. example 3.5. consider the family t qra = {u ∩ q | u ∈ tra } where tra is a generalized topology from the previous example. then (q,t qra) is also a gtspace. example 3.6. let (x,t ) be a topological space. the family of all regular open subsets of x is denoted by r(t ). it is known [2] that r(t ) forms a frame with respect to ⊆. clearly, ∅,x ∈ r(t ). hence, (x,r(t )) is a gt-space. introduction to generalized topological spaces 57 example 3.7. let t be a usual topology on r2. consider the following family τ = { r 2 \ u | u ∈ r(t ) } . then (x,τ) is a gt-space. briefly speaking, this is a generalized topology consisting of “2-dimensional figures without 1-dimensional protuberances and cracks”. example 3.8. let x be a nonempty set x and s ⊆ 2x . assume that s separates the elements of x, is a complete, completely distributive lattice with respect to ⊆, contains ∅ and x, arbitrary meets coincide with intersections, and finite joins with unions. then (x,s) is called a texture [1]. a ditopology [1] on a texture (x,s) is a pair (τ,κ) of subsets of s, where the set of open sets τ and the set of closed sets κ satisfy the following: (1) s, ∅ ∈ τ, (4) s, ∅ ∈ κ, (2) g1,g2 ∈ τ implies g1 ∩ g2 ∈ τ, (5) k1,k2 ∈ κ implies k1 ∪ k2 ∈ κ, (3) g ⊆ τ implies ∨ g ∈ τ, (6) k ⊆ κ implies ⋂ k ∈ κ. let (τ,κ) be a ditopology on a texture (x,s). then (x,τ) forms a gt-space. 4. closed sets. interior and closure operators the concept of gt-space makes it possible to preserve the classical definition for closed sets. nevertheless, it would be interesting to consider the notion of “closed subset modulo ideal” in the further research. definition 4.1. let (x,t ) be a gt-space. a subset y ⊆ x is called closed iff y = x \ u for some u ∈ t . proposition 4.2. let (x,t ) be a gt-space. the family of all closed subsets of x is a complete lattice where the join and meet operations are the following: ∨ a = x \ ∧ {x \ a | a ∈ a} and ∧ a = x \ ∨ {x \ a | a ∈ a} where a is a family of closed subsets of x. proof. the proof is an easy exercise since t is a complete lattice. � proposition 4.3. let (x,t ) be a gt-space. for all closed subsets a,b ⊆ x, it holds that a � b iff a ⊆ b. proof. consider closed subsets a,b ⊆ x such that a � b. then x \ a and x \ b are open, and it holds that x \ b � x \ a. the latter implies that x \ b ⊆ x \ a, and, hence, a ⊆ b. � we use the set operator ψ [3] and the local function ∗ [5] as interior and closure operators in gt-spaces. note that these operators would not make so much topological sense in our framework if the ideal would not be compatible with a gt-topology (def. 3.1). indeed, to prove that ψ(a) � a � a∗ holds for all a ⊆ x in a gt-space (x,t ) we need the ideal i to be compatible with t . 58 i. zvina definition 4.4. let (x,t ) be a gt-space. the operators ∗ : 2x → 2x and ψ : 2x → 2x are defined as follows, for all a ⊆ x: ψ(a) = {x ∈ x | exists u ∈ t (x) such that u � a}, a∗ = {x ∈ x | for all u ∈ t (x) it holds that a ∩ u /∈ i}. the operator ψ is called the interior operator (interior operator modulo ideal ) and ∗ is called the closure operator (closure operator modulo ideal). theorem 4.5. in a gt-space (x,t ), the following hold for every a,b ⊆ x: (i) ψ(a) = ∨ {u ⊆ x | u � a, u is open}, and a∗ = ∧ {b ⊆ x | a � b, b is closed} (def. 4.1); (ii) ψ(a) = x \ (x \ a)∗; (iii) a is open iff a = ψ(a), and a is closed iff a = a∗; (iv) ψ(x) = x and ∅∗ = ∅; (v) ψ(a) � a � a∗; (vi) ψ(ψ(a)) = ψ(a) and (b∗)∗ = b∗; (vii) ψ(a ∩ b) = ψ(a) ∧ ψ(b) and (a ∪ b)∗ = a∗ ∨ b∗. proof. the statement (i) is a straight corollary from the definition of the interior and closure operators. let us prove (ii). consider a subset a ⊆ x. a point x ∈ x belongs to ψ(a) if there exists u ∈ t (x) such that u � a, that is u ∩ (x \ a) ∈ i. the latter means that x /∈ (x \ a)∗ and, hence, x ∈ x \ (x \ a)∗. then it follows that ψ(a) ⊆ x \ (x \ a)∗. assume that x ∈ x \ (x \ a)∗, that is x /∈ (x \ a)∗. then there exists v ∈ t (x) such that v ∩ (x \ a) ∈ i. hence, v � a and we conclude that x ∈ ψ(a). the proof is complete. for (iii)-(vii), we provide the proofs only for the interior operator. the proofs for the closure operator could be done in the similar way. let us prove (iii). if a is open then a � a, and u � a implies u ⊆ a for all open u. hence, a = ψ(a). on the other hand, if a = ψ(a) then a is the join of all such open u that u � a, that is a is open. the statement (iv) is obvious. to prove (v), it is enough to remember that i ∼ t . the property (vi) is a straight corollary from (iii). to prove (vii), let us consider the following chain of implications: u ⊆ ψ(a ∩ b) iff u � a ∩ b iff u � a and a � b iff u ⊆ ψ(a) ∩ ψ(b) iff u ⊆ ψ(a) ∧ ψ(b). the proof is complete. � 5. generalized continuous (g-continuous) mappings we generalize the notion of continuous mapping in a natural way. like in the previous section, note that the proof of the essential theorem 5.2 is not possible without the ideal being compatible with a gt-topology (def. 3.1). definition 5.1. let (x,tx ) and (y,ty ) be gt-spaces. a mapping f : x → y is called a generalized continuous mapping (or g-continuous mapping for short) provided that there exists a frame homomorphism h: ty → tx such that h(u) ≈ f−1(u) holds for every u ∈ ty . introduction to generalized topological spaces 59 the g-continuous mapping f is called a generalized homeomorphism (or g-homeomorphism for short) iff f is a bijection and f−1 is g-continuous. theorem 5.2. given gt-spaces (x,tx ) and (y,ty ), and a g-continuous mapping f : x → y , then the following hold: (i) the corresponding frame homomorphism h: ty → tx is unique; (ii) f−1(b) ∈ ix holds for all b ∈ iy . proof. assume that there exists a frame homomorphism g : ty → tx such that g(u) ≈ f−1(u) for every u ∈ ty . then h(u) ≈ g(u) for every u ∈ ty . since h(u) ∈ tx and g(u) ∈ tx , it follows from corollary 3.1 that h(u) = g(u) for all u ∈ ty . hence, h = g, and we proved (i). assume that b ⊆ y is join-generated. then there exists a family v ⊆ ty such that b ⊆ ( ∨ v ) \ ( ⋃ v ). denote by a the preimage of b: a = f−1(b) ⊆ f−1 ( ∨ v ) \ ⋃ v ∈v f−1(v ). divide a in three subsets and prove that they belong to the ideal ix : (1) a \ h ( ∨ v ) ∈ ix holds since h ( ∨ v ) ≈ f−1 ( ∨ v ); (2) ( a ∩ h ( ∨ v ) ) \ ⋃ v ∈v h(v ) ∈ ix holds since h ( ∨ v ) = ∨ v ∈v h(v ) and h ( ∨ v ) \ ⋃ v ∈v h(v ) ∈ ix ; (3) a ∩ ⋃ v ∈v h(v ) ∈ ix holds since a ∩ h(v ) ⊆ h(v ) \ f−1(v ) ∈ ix , for all v ∈ v, and ix ∼ tx . in a similar way, we prove that the preimage of every meet-generated subset of y lies in the ideal ix . now, consider a subset b ⊆ y and a family v ⊆ ty satisfying the following: for all v ∈ v, it holds that b ∩ v is a joinor meet-generated subset of y . denote by a = f−1(b). as above, divide a in three subsets and prove that they belong to the ideal ix : (4) a \ h ( ∨ v ) ∈ ix and ( a ∩ h ( ∨ v ) ) \ ⋃ v ∈v h(v ) ∈ ix as above; (5) a ∩ ⋃ v ∈v h(v ) ∈ ix holds since a ∩ h(v ) ≈ a ∩ f−1(v ) = f−1(b ∩ v ) ∈ ix , for all v ∈ v, and ix ∼ tx . the rest of the proof is obvious. � proposition 5.3. given gt-spaces (x,tx ), (y,ty ) and (z,tz ) and g-continuous mappings f : x → y and g : y → z, then the composition g ◦ f : x → z is also a g-continuous mapping. proof. denote by hf and hg the corresponding frame homomorphisms for f and g, respectively. it is known that the composition of frame homomorphisms is also a frame homomorphism. hence, hf ◦ hg is a frame homomorphism. we 60 i. zvina have to show that f−1 ( g−1(v ) ) ≈ hf ( hg(v ) ) holds for all v ∈ tz . the latter holds, since f−1(b) ∈ ix for all b ∈ iy . � the next proposition makes it easier to check if a mapping is g-continuous in some particular cases. we will use it when proving theorem 6.5 (urysohn lemma for gt-spaces). definition 5.4. let (x,t ) be a gt-space. a family b ⊆ t is called a base for t provided that for every open set u there exists a subfamily b0 ⊆ b such that u = ∨ b0. proposition 5.5. let (x,tx ) and (y,ty ) be gt-spaces, b ⊆ ty be a base, and f : x → y be a mapping. assume that (1) f−1(b) ∈ ix holds for all b ∈ iy , (2) for every v ∈ b there is v ′ ∈ tx such that v ′ ≈ f−1(v ) ⊆ v ′. then f is a g-continuous mapping. proof. define a mapping h0 : b → tx such that h0(v ) ≈ f −1(v ) ⊆ h0(v ), for all v ∈ b. such a mapping exists under our assumption, and is unique since (x,tx ) is a gt-space. take an open subset u ⊆ y , and consider a family v ⊆ b such that u = ∨ v. then f−1(u) = f−1 ( ∨ v ) ≈ f−1 ( ⋃ v ) = ⋃ v ∈v f−1(v ) ⊆ ⋃ v ∈v h0(v ) ≈ ∨ v ∈v h0(v ). denote a = ⋃ v ∈v h0(v ) \ ⋃ v ∈v f−1(v ). then a ⊆ ∨ v ∈v h0(v ) and a ∩ h0(v ) ∈ ix , for all v ∈ v. since ix ∼ tx , we conclude that a ∈ ix . hence, f−1(u) ≈ ∨ v ∈v h0(v ). consider v1, v2 ⊆ b such that u = ∨ v1 = ∨ v2. then, according to the latter observation, it holds that f−1(u) ≈ ∨ v ∈v h0(v1) ≈ ∨ v ∈v h0(v2). since both joins ∨ v ∈v h0(v1) and ∨ v ∈v h0(v2) are open in tx , we conclude that they are equal. define the mapping h: ty → tx as follows, u ∈ ty : h(u) = h0(u), if u ∈ b; h(u) = ∨ v ∈v h0(v ), if ∨ v = u /∈ b and v ⊆ b. then h(u) ≈ f−1(u) holds for all u ∈ ty . note that h preserves arbitrary joins of the elements of v. let us prove that h preserves arbitrary joins of arbitrary open subsets of y . consider u ⊆ ty and a family v ⊆ b satisfying ∨ u = ∨ v. we have to show that ∨ u∈u h(u) = h ( ∨ u ) . introduction to generalized topological spaces 61 it holds that ∨ u∈u h(u) ≈ ⋃ u∈u h(u), and it follows from the assumption (1) and the construction of h that ⋃ u∈u f−1(u) = f−1 ( ⋃ u ) ≈ f−1 ( ∨ u ) ≈ h ( ∨ u ) = = h ( ∨ v ) = f−1 ( ∨ v ) ≈ f−1 ( ⋃ v ) ≈ ⋃ v ∈v f−1(v ). hence, to complete the proof, it is enough to show that the subsets a,b ⊆ x defined as follows are elements of the ideal ix : a = ⋃ u∈u h(u) \ ⋃ u∈u f−1(u) and b = ⋃ v ∈v f−1(v ) \ ⋃ u∈u h(u). the subset a lies in the ideal, since ix ∼ tx . under our assumption, for every v ∈ v there is u ∈ u such that v ⊆ u. then, for every such v and u, it holds that f−1(v ) ⊆ h(v ) ⊆ h(u). thus, ⋃ v ∈v f−1(v ) ⊆ ⋃ u∈u h(u), that is b = ∅ ∈ ix . the proof is complete. � in the following example we show that for given gt-spaces (x,tx ) and (y,ty ) and a frame homomorphism h: ty → tx it is possible that there does not exist a g-continuous mapping f : x → y such that h is its corresponding frame homomorphism. example 5.6. consider the gt-spaces (r,tra) and (q,tra) (examples 3.4 and 3.5). assume that there exists a g-continuous mapping of the given gt-spaces f : r → q such that the corresponding frame homomorphism is the identity mapping id : tra → tra. we mentioned already that the least suitable ideal of the gt-space (r,tra) is the family d of nowhere dense subsets of r. since f is a g-continuous mapping, it follows that u∆f−1(u) ∈ d for every u ∈ tra. hence, |u∆f −1(u)| ≤ ℵ0 holds for every u ∈ tra. assume that |f−1(q)| ≤ ℵ0 for every q ∈ q. then ℵ0 ≥ |f −1q| 6= |r|. the later inequality is a contradiction, and we conclude that there exists q ∈ q such that |f−1(q)| > ℵ0. denote uq = [q, +∞). since |uq∆f −1(u)q| ≤ ℵ0, we imply that |uq ∩ f −1(u)q| > ℵ0. then there exists p ∈ q such that |[p, +∞) ∩ f−1(q)| > ℵ0. then it follows that |[p, +∞)∆f−1[p, +∞)| > ℵ0, and we conclude that f is not a g-continuous mapping. 6. separation axioms in this section, we consider t1 separation axiom and normal spaces. we use the notation i for the real number interval [0, 1] with the usual topology of open intervals. definition 6.1. the gt-space (x,t ) is said to be t1 iff for every x,y ∈ x there is u ∈ t such that x ∈ u and y /∈ u. 62 i. zvina proposition 6.2. given a t1 gt-space (x,t ) and x ∈ t, then one and only one of the following holds: { x} ∈ i or {x} is closed. proof. consider the set a = { ⋃ u ∈ t | x /∈ u } = x \ {x}. there are two possibilities: a is open or a is not open. assume that a is open. then { x} is closed and it cannot be that { x} ∈ i, since it is impossible that x and a are both open, x 6= a and x ≈ a. assume that {x} ∈ i. then it is impossible that a is open, since it cannot be that x and a are both open, x 6= a and x ≈ a. hence, {x} is not closed. � in the following example we show that the property of t1 gt-space from the previous proposition is necessary but not sufficient for t1. example 6.3. let x = { 0 } ∪ [1, 2). consider a family a = { { 0 } ∪ [1,b) | b ∈ (1, 2) } ∪ { (a,b) | a,b ∈ (1, 2) and a ≤ b } . define t as a family consisting of empty set, the elements of a and such disjoint units of elements of a that, for every u ∈ t and b ∈ (1, 2), it holds that (1,b) ⊆ u implies { 0, 1 } ⊆ u. then (x,t ) is a gt-space and the least suitable ideal is i = { { ∅ } ,{ 0 } ,{ 1 } ,{ 0, 1 } }. for every point x ∈ (1, 2), it holds that {x} is a closed set, since { 0 } ∪ [1,x) ∪ (x, 2) ∈ t. on the other hand, { x} is not an element of the ideal. the sets { 0 } and { 1 } are elements of the ideal and are not closed sets. this gt-space is not t1 since we cannot separate 0 and 1. definition 6.4. a gt-space (x,t ) is called normal iff for every disjoint nonempty closed a,b ⊆ x there exist u,v ∈ t such that a ⊆ u, b ⊆ v and u ∧ v = ∅. theorem 6.5 (urysohn’s lemma for gt-spaces). let (x,t ) be a normal gt-space and a,b ⊆ x be disjoint nonempty closed subsets. assume that finite meets of open subsets coincide with intersections. then there exists a g-continuous mapping f : x → i such that f(a) = 0 and f(b) = 1. proof. let us organize the rational numbers from i into a sequence r0,r1,r2, . . . where r0 = 0 and r1 = 1. for every rational number r, we are going to define an open subset wr such that the following is satisfied for every index k: (6.1) w∗ri ⊆ wrj for ri < rj and i,j ≤ k. consider disjoint nonempty closed subsets a,b ⊆ x. since x is normal, there exist u,v ∈ t such that a ⊆ u, b ⊆ v and u ∧ v = ∅. denote w0 = u and w1 = x \ b. then w0 � x \ v = (x \ v ) ∗ ⊆ w1. by theorem 4.5, we conclude that w∗0 ⊆ w1, and, hence, ( 6.1) is satisfied for k = 1. assume that the open subsets wri satisfying ( 6.1) are already constructed for all i ≤ n ≥ 1. denote by rl = max{ri | i ≤ n and ri < rn+1} andrm= min{ri | i ≤ n and rn+1 < ri}. introduction to generalized topological spaces 63 it holds that rl ≤ rm, and, hence, w ∗ rl ⊆ wrm . since x is normal, there exist open u,v such that w∗rl ⊆ u, x \ wrm ⊆ v , and u ∧ v = ∅. then u � x \ v = (x \ v )∗ ⊆ wrm , and u ∗ ⊆ wrm . denote wn+1 = u. by the finite induction, we obtained the sequence wr0,wr1,wr2, . . . satisfying (6.1) for all indexes k, and (6.2) a ⊆ w0 and b ⊆ x \ w1. we define the function f : x → i as follows: f(x) = { inf{r | x ∈ wr}, if x ∈ w1, 1, if x ∈ x \ w1. consider x ∈ x and a,b ∈ i. it holds that f(x) a iff there are rational numbers r and r′ such that a < r < r′ and x /∈ wr′ . then it follows from (6.1) that x /∈ w ∗ r ⊆ wr′ and f −1 ( (a, 1] ) = ⋃ {x \ w∗r | r > a}. for every a < b, define the corresponding open subset of x as follows: ua,b = ( ∨ {x \ w∗r | r > a} ) ∧ ( ∨ {wr | r < b} ) . then, under the assumption of the current theorem, the following holds for all intervals (a,b), which form a base for the topology of i: f−1 ( (a,b) ) = ( ⋃ {x \ w∗r | r > a} ) ∩ ( ⋃ {wr | r < b} ) ⊆ ( ∨ {x \ w∗r | r > a} ) ∩ ( ∨ {wr | r < b} ) = ua,b ∈ t and ua,b \ f −1 ( (a,b) ) ∈ ix . since i is a crisp space (def. 3.3), it holds that f−1ii = f −1{∅} ⊆ ix . then, by proposition 5.5, the mapping f is g-continuous. finally, it follows from (6.2) that f(a) = 0 and f(b) = 1. � 7. normalized spaces. cardinal invariants we introduce the notion of normalized gt-space. we exploit it as an auxiliary tool to prove some results in this section. definition 7.1. let (x,t ) be a gt-space. the operator n : t → 2x is called the normalization operator provided that, for every u ∈ t : un = {x ∈ u | u ∧ v 6= ∅ for all v ∈ t (x)}. the family t n = {un | u ∈ t} is called the normalization of t . proposition 7.2. given a gt-space (x,t ), then the following hold: (i) t n is a frame, isomorphic to t; (ii) (x,t n ) is a gt-space. and the following conditions are equivalent: 64 i. zvina (iii) t = t n ; (iv) u ∧ v = ∅ iff u ∩ v = ∅ for all u,v ∈ t. definition 7.3. a gt-space (x,t ) is called normalized iff t = t n . cardinal invariant is a function associating a cardinal number to each space and taking the same value on homeomorphic spaces. definition 7.4. let (x,t ) be a gt-space and b ⊆ t be a base (def. 5.4). the smallest cardinal number of the form |b|, where b is a base, is called the weight of the given space and is denoted by w(x,t ), w(x), or w(t ). definition 7.5. let (x,t ) be a gt-space. a family n ⊆ 2x is called a network for t provided that for every open set u there exists a subfamily n0 ⊆ n and a ∈ i such that u = a ∪ ( ⋃ n0 ). the smallest cardinal number of the form |n|, where n is a network, is called the network weight of the given space and is denoted by nw(x,t ), nw(x), or nw(t ). the statement of the following theorem is the immediate corollary from the fact that every base is a network. theorem 7.6. in a gt-space (x,t ), it holds that nw(t ) ≤ w(t ). theorem 7.7. let (x,t ) be a gt-space. assume that nw(t ) ≤ m. then for every family u of of open sets there exists a subfamily u0 ⊆ u such that |u0| ≤ m and ∨ u0 = ∨ u. proof. fix a network n with |n| ≤ m. consider a subfamily n0 ⊆ n with the following property: m ∈ n0 iff there exists u ∈ u satisfying m ⊆ u. clearly, it holds that |n0| ≤ m and for every u ∈ u holds u \ ( ⋃ n0 ) ∈ i. consider a mapping ϕ: n0 → u such that m ⊆ ϕ(m), for every m ∈ n0. denote u0 = ϕn0. then it holds that |u0| ≤ |n0| ≤ m and ⋃ n0 ⊆ ∨ u0. take u ∈ u. since u \ ( ⋃ n0 ) lies in the ideal, it holds that u \ ( ∨ u0 ) also lies in the ideal, and, hence, u ⊆ ∨ u0. thus, we proved the inclusion ∨ u ⊆ ∨ u0. the converse inclusion is obvious. the proof is complete. � definition 7.8. let (t,x) be a gt-space. a family c ⊆ t is called a cover of x iff ∨ c = x. a subfamily c0 ⊆ c is called a subcover iff it is a cover. the smallest cardinal number m such that, for every cover c there exists a subcover c0 satisfying |c0| ≤ m is called the lindelöf number of the given space and is denoted as l(x,t ), l(x), or l(t ). the following result is the immediate corollary from theorem 7.7. theorem 7.9. in a gt-spaced (x,t ), it holds that l(t ) ≤ nw(t ). lemma 7.10. in a normalized gt-space (x,t ), for every u ⊆ t and every v ∈ t that satisfies v ∩ ( ∨ u ) 6= ∅ there exists a nonempty open w ⊆ v such that w ⊆ ⋃ u. proof. under the assumption, v ∩ ( ∨ u ) 6= ∅ implies that w0 = v ∧ ( ∨ u ) is nonempty. then there exists u ∈ u such that w0 ∧ u 6= ∅, since w0 ⊆ ∨ u. put w = w0 ∧ u. it holds that w ⊆ u ⊆ ⋃ u. the proof is complete. � introduction to generalized topological spaces 65 lemma 7.11. in a normalized gt-space (x,t ), for every finite family u ⊆ t and every v ∈ t that satisfies v ∩ u1 6= ∅ for some u1 from u there exists a nonempty open w ⊆ v such that the intersection of w and ( ⋂ u ) \ ( ∧ u ) is empty. proof. put u = { u1,u2, . . . ,uk }, and a = ( ⋂ u ) \ ( ∧ u ). (1) v ∩ u1 6= ∅ implies w1 = v ∧ u1 is nonempty. if w1 ∩ a = ∅ then put w = w1. (2) if w1 ∩ a 6= ∅ then w2 = w1 ∧ u2 is nonempty. if w2 ∩ a = ∅ then put w = w2. . . . (k) if wk−1 ∩ a 6= ∅ then wk = wk−1 ∧ uk is nonempty. if the process stops at some w = wi, where 1 ≤ i ≤ k − 1, then the proof is complete. otherwise, we obtain a nonempty w = wk and then w ⊆ ∧ u. hence, w ∩ a = ∅. the proof is complete. � proposition 7.12. in a normalized gt-space (x,t ), for every nonempty v ∈ t and every a ∈ i the inclusion a ⊆ v implies that there exists a nonempty open w ⊆ v such that w ∩ a = ∅. proof. apply lemma 7.10 and lemma 7.11 to verify that for every v ∈ t and a = a1 ∪ a2 ∪ · · · ∪ an ∈ i, where every ai is joinor meet-generated, the inclusion a ⊆ v implies that there exists a nonempty open w ⊆ v such that w ∩ a1 = ∅. the remainder of the proof is obvious. � definition 7.13. let (x,t ) a gt-spaces. a subset a ⊆ x is called dense if a∗ = x (def. 4.4). the smallest cardinal number of the form |a|, where a is a dense subset of x, is called the density of the given gt-space and is denoted by d(x,t ), d(x), or d(t ). proposition 7.14. in a normalized gt-space (x,t ), it holds d(t ) ≤ nw(t ). proof. let n be a network for x satisfying nw(t ) = |n|. fixing an arbitrary point in every set from n, we construct the set a. then it holds that |a| ≤ |n| = nw(t ). on the other hand, for every x ∈ x and every v ∈ t (x), if v ∩ a ∈ i then, by proposition 7.12, there exists an open subset w such that w ∩ a = ∅. but the latter equality is impossible, since a has a nonempty intersection with every member of the network. therefore, we conclude that a is dense, and hence d(x) ≤ |a| ≤ |n| = nw(t ). the proof is complete. � definition 7.15. in a gt-space (x,t ), the least cardinal number m such that |s| ≤ m holds for every family of nonempty open subsets s with the property that u∧v = ∅ holds for all u,v ∈ s is called the suslin number and is denoted by c(x,t ), c(x), or c(t ). proposition 7.16. in a normalized gt-pace (x,t ), it holds that c(t ) ≤ d(t ). proof. consider a ⊆ x such that |a| = d(t ), and a family of nonempty open subsets s with the property that u ∧ v = ∅ holds for all u,v ∈ s. assume 66 i. zvina that |a| < |s|. then there exists a nonempty u ∈ s such that a ∩ u = ∅. the latter statement is a contradiction, since a is dense. hence, we conclude that |s| ≤ |a|, and this holds for all s ⊆ t with the mentioned property. then c(t ) ≤ d(t ). � proposition 7.17. let (x,t ) be a gt-space and (x,t̃) be its normalization (def. 7.1). then the following hold: (i) c(t ) = c(t̃); (ii) d(t ) = d(t̃); (iii) nw(t ) = nw(t̃ ); (iv) w(t ) = w(t̃ ); (v) l(t ) = l(t̃). proof. we omit the proof, since it is a long but rather easy exercise. � the following result is the natural corollary from propositions 7.14, 7.16, and 7.17. theorem 7.18. in a gt-space (x,t ), it holds c(t ) ≤ d(t ) ≤ nw(t ). references [1] l. m. brown and m. diker, ditopological texture spaces and intuitionistic sets, fuzzy sets syst. 98, no. 2 (1998), 217–224 [2] s. givant and p. halmos, introduction to boolean algebras, springer science+business media, llc (2009) [3] t. r. hamlett and d. janković, ideals in topological spaces and the set operator ψ, boll. unione mat. ital., vii. ser., b 4, no. 4 (1990), 863–874. [4] t. r. hamlett and d. janković, ideals in general topology, general topology and applications, proc. northeast conf., middletown, ct (usa), lect. notes pure appl. math. 123 (1990), 115–125. [5] d. janković and t. r. hamlett, new topologies from old via ideals, am. math. mon. 97, no.4 (1990), 295–310. [6] d. janković and t. r. hamlett, compatible extensions of ideals, boll. unione mat. ital., vii. ser., b 6, no.3 (1992), 453–465. [7] d. janković, t. r. hamlett and ch. konstadilaki, local-to-global topological properties, math. jap. 52, no.1 (2000), 79–81. [8] i. zvina, complete infinitely distributive lattices as topologies modulo an ideal, acta univ. latviensis, ser. mathematics, 688 (2005), 121–128 [9] i. zvina, on i-topological spaces: generalization of the concept of a topological space via ideals, appl. gen. topol. 7, no. 1 (2006), 51–66 (received november 2010 – accepted february 2011) irina zvina (irinazvina@gmail.com) department of mathematics, university of latvina, zellu str. 8, lv-1002, riga, latvia. introduction to generalized topological spaces. by i. zvina mirandaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 177-184 on pseudo-k-spaces annamaria miranda abstract. in this note a new class of topological spaces generalizing k-spaces, the pseudo-k-spaces, is introduced and investigated. particular attention is given to the study of products of such spaces, in analogy to what is already known about k-spaces and quasi-k-spaces. 2000 ams classification: 54d50, 54d99, 54b10, 54b15 keywords: quotient map, product space, locally compact space, (locally) pseudocompact space, pseudo-k-space. 1. introduction the first example of two k-spaces whose cartesian product is not a k-space was given by dowker (see [2]). so a natural question is when a k-space satisfies that its product with every k-spaces is also a k-space. in 1948 j.h.c. whitehead proved that if x is a locally compact hausdorff space then the cartesian product ix × g, where ix stands for the identity map on x, is a quotient map for every quotient map g. using this result d.e. cohen proved that if x is locally compact hausdorff then x × y is a k-space for every k-space y (see theorem 3.2 in [1]). later the question was solved by michael who showed that a k-space has this property iff it is a locally compact space (see [5]). a similar question, related to quasi-k-spaces, was answered by sanchis (see [8]). quasi-k-spaces were investigated by nagata (see [7]) who showed that “a space x is a quasi-k-space (resp. a k-space) if and only if x is a quotient space of a regular (resp. paracompact) m -space (see [6]). the study of quasi-k-spaces suggests to define a larger class of spaces simply replacing countable compactness with pseudocompactness in the definition. this note begins with the study of general properties about pseudo-k-spaces which leads on results about products of pseudo-k-spaces, in analogy with those known about k-spaces and more generally about quasi-k-spaces. for terminology and notations not explicitly given we refer to [3]. 178 a. miranda 2. pseudo-k-spaces we consider pseudocompact spaces which are not necessarily tychonoff. recall that definition 2.1. a topological space x is called pseudocompact if every continuous real-valued function defined on x is bounded. definition 2.2. a topological space x is called locally compact (resp. locally countably compact) if each point of x has a compact (resp. countably compact) neighborhood. in analogy with the definitions of locally compact (resp. locally countably compact) space we have the following definition 2.3. a topological space x is called locally pseudocompact if each point of x has a pseudocompact neighborhood. clearly a locally compact space is locally pseudocompact and we have proposition 2.4. the cartesian product of a locally pseudocompact space x and a locally compact space y is locally pseudocompact. proof. it suffices to observe that corollary 3.10.27 in [3] holds even if the pseudocompact factor is not necessarily tychonoff. � proposition 2.5. if all spaces xs are pseudocompact then the sum ⊕s∈s xs, where xs 6= ∅ for s ∈ s, is locally pseudocompact. now we are going to define a new class of spaces which is larger than the class of k-spaces. definition 2.6. a topological space x is called a pseudo-k-space if x is a hausdorff space and x is the image of a locally pseudocompact hausdorff space under a quotient mapping. in other words, pseudo-k-spaces are hausdorff spaces that can be represented as quotient spaces of locally pseudocompact hausdorff spaces. clearly every locally pseudocompact hausdorff space is a pseudo-k-space. we can compare this kind of spaces with the one of quasi-k-spaces. to this aim recall that definition 2.7. a hausdorff space x is a quasi-k-space if, and only if, a subset a ⊂ x is closed in x whenever the intersection of a with any countably compact subset z of x is closed in z. on pseudo-k-spaces 179 condition (2) in theorem 2.11 yields proposition 2.8. every quasi-k-space is a pseudo-k-space. the following example will show that the class of quasi-k-spaces is strictly contained in the class of pseudo-k-spaces. definition 2.9. a hausdorff space x is called h-closed if x is a closed subspace of every hausdorff space in which it is contained. for a hausdorff space x, this definition is equivalent to say that every open cover {us}s∈s of x contains a finite subfamily {us1 , us2 , ..., usk } such that u s1 ∪ u s1 ∪ ... ∪ u s1 = x. example 2.10. a h-closed space which is not a quasi-k-space. let ℑ be the family of all free ultrafilters on n, let kn = n ∪ ℑ be the katětov extension of n. we have that (1) kn is a h-closed space; (2) kn is not a quasi-k-space. it is enough to show that all countably compact subsets of kn have finite cardinality. let y ⊂ x = kn be countably compact. ℑ is closed and discrete in x so y ∩ ℑ is closed and discrete in y , therefore y ∩ ℑ = {p1, . . . , pn}. hence y = s ∪ {p1, . . . , pn}, where s ⊂ n. assume that s is infinite. since p1, . . . , pn are distinct ultrafilters, there exists s1 ⊂ s such that |s1| = ω, s1 ∈ p1 and s1 /∈ pi for every i 6= 1. in fact let hi ∈ p1 such that hi /∈ pi for every i 6= 1, then s1 = n⋂ i=1 hi ∈ p1 and s1 /∈ pi for every i 6= 1, otherwise s1 ∈ pi and s1 ⊂ hi implies hi ∈ pi. moreover s1 is infinite. indeed, if p is an ultrafilter, a = {x1, . . . , xn} and a ∈ p, then {xi} /∈ p implies that n\{xi} ∈ p, for every i, so n⋂ i=1 n\{xi} = n\a ∈ p, a contradiction. now, let g ⊂ s1 such that |g| = ω and |s1\g| = ω. then g ∈ p1 or n\g ∈ p1. since s1 ∈ p1 it follows that g ∈ p1 or s1\g ∈ p1. let us suppose that s1\g ∈ p1. then g /∈ p1. therefore g /∈ pi for every i. since g /∈ pi ∀ i ∈ {1, . . . , n}, it follows that for every i there exists ai ∈ pi such that g ∩ ai = ∅, so vi = ai ∪ {pi} is an open neighborhood of pi such that vi ∩ g = ∅, therefore pi /∈ g for every i, hence g is closed in y and, since g ⊂ n, g is also discrete. so g is an infinite closed discrete subspace of the countably compact space y , a contradiction. hence s is finite. in conclusion, since any h-closed space is a pseudocompact space, kn is a pseudo-k-space which is not a quasi-k-space. now we give two useful characterizations of pseudo-k-spaces. 180 a. miranda theorem 2.11. let x be a hausdorff space. the following conditions are equivalent: (1) x is a pseudo-k-space. (2) for each a ⊂ x, the set a is closed provided that the intersection of a with any pseudocompact subspace z of x is closed in z. (3) x is a quotient space of a topological sum of pseudocompact spaces. proof. (1)⇒(2) let x be a pseudo-k-space and let f : y → x be a quotient mapping of a locally pseudocompact hausdorff space y onto x. suppose that the intersection of a set a with any pseudocompact subspace p of x is closed in p . take a point y ∈ f −1(a) and a neighborhood u ⊂ y of the point y such that u is pseudocompact. since the space f (u ) is pseudocompact (see theorem 3.10.24 [3] which holds even if the range space y is not tychonoff), the set a ∩ f (u ) is closed in f (u ). now, if y 6∈ f −1(a) then f (y) 6∈ a ∩ f (u ) so there exists an open set t in x containing f (y) such that t ∩ (a ∩ f (u )) = ∅. it follows that f −1(t ) ∩ f −1(a) ∩ u = ∅ where the set f −1(t ) ∩ u represents a neighborhood of y disjoint from f −1(a). this is a contradiction. then y ∈ f −1(a). (2)⇒(3) now consider a hausdorff space x and denote by p(x) the family of non-empty pseudocompact subspaces of x. let x̃ = ⊕{p : p ∈ p(x)}. the surjective mapping f : ∇p ∈p(x), ip : x̃ → x, where ip is the embedding of the subspace p in the space x, is continuous (see proposition 2.1.11 [3]). suppose now that a is closed in x̃, this means a ∩ p closed in p , for every pseudocompact subset p of x. then, by (2), a is closed in x. it follows that f is a quotient map. (3)⇒(1) if x is a quotient space of a topological sum of pseudocompact spaces then x is a pseudo-k-space, by proposition 2.5. � corollary 2.12. a hausdorff space x is a pseudo-k-space if, and only if, a subset a ⊂ x is open in x whenever the intersection of a with any pseudocompact subset p of x is open in p . regarding the continuity of a mapping whose domain is a pseudo-k-space we have the following theorem 2.13. a mapping f of a pseudo-k-space x to a topological space y is continuous if and only if for every pseudocompact subspace p ⊂ x the restriction f|p : p → y is continuous. from the definition of a pseudo-k-space we obtain theorem 2.14. if there exists a quotient mapping f : x → y of a pseudo-kspace x onto a hausdorff space y , then y is a pseudo-k-space. on pseudo-k-spaces 181 theorem 2.11 yields theorem 2.15. the sum ⊕s∈s xs is a pseudo-k-space if and only if all spaces are pseudo-k-spaces. 3. on products of pseudo-k-spaces the cartesian product of two pseudo-k-spaces need not be a pseudo-k-space. so, when a pseudo-k-space satisfies that its product with every pseudo-k-space is also a pseudo-k-space? proposition 2.4 states that the cartesian product of a locally compact space and a locally pseudocompact hausdorff space is a locally pseudocompact space. this result, together with definition 2.6, yields theorem 3.1. the cartesian product x × y of a locally compact hausdorff space x and a pseudo-k-space y is a pseudo-k-space. proof. let g : z → y be a quotient mapping of a locally pseudocompact hausdorff space z onto a pseudo-k-space y . the cartesian product f : idx ×g : x × z → x × y is a quotient mapping, by virtue of the whitehead theorem (see lemma 4 in [9], or theorem 3.3.17 in [3]). now, since, by proposition 2.4, x × z is a locally pseudocompact hausdorff space, it follows that x × y is a pseudo-k-space. � the previous theorem gives a sufficient condition to obtain that the cartesian product of two pseudo-k-spaces is a pseudo-k-space. this condition, for regular spaces, is also necessary, as we will see in theorem 3.4. now, starting from a regular space x which is not locally compact, we define, following a construction introduced by michael in [5], a normal pseudo-k-space y (x) such that the product x × y (x) is not a pseudo-k-space. this enable us not only to give examples of two pseudo-k-spaces whose product is not a pseudo-k-space, but also to show theorem 3.4. suppose that x is a regular space which is not locally compact at some x0 ∈ x. let {uα}α∈a be a local base of non-compact closed sets at x0. for every α ∈ a let λ(α) be a limit ordinal and {fλ}λ<λ(α) be a well-ordered family of nonempty closed subsets of uα whose intersection is empty. each λ(α)+ 1, equipped with the order topology, is a compact hausdorff space. therefore λ(α) + 1 is a normal pseudo-k-space. then, by theorem 2.15 jointly with theorem 2.27 in [3], the topological sum λ = ⊕{λ(α) + 1 : α ∈ a} is a normal pseudo-k-space. now, let us denote by y (x) the quotient space obtained by identifying all the final points λ(α) ∈ λ(α) + 1 to a single points y0. 182 a. miranda we have the following theorem 3.2. the space y (x) is a normal pseudo-k-space. moreover, if p is a pseudocompact subset of y (x), then |{α ∈ a : p ∩ λ(α) 6= ∅}| < ω. proof. let us denote by g : λ −→ y (x) the canonical projection defining y (x). it is easy to verify that g is a closed mapping. so, since the normality preserves under closed mappings, it follows that y (x) is normal. moreover, since g is a continuous surjective closed map, then g is a quotient mapping. then, by theorem 2.14, the space y (x) is a pseudo-k-space. now, suppose that there exists b ⊂ a, |b| ≥ ω, such that a pseudocompact subset p of y (x) meets each element of the family {λ(α) : α ∈ b}. observe that for every α ∈ a, since λ(α) is open in y (x), the set λ(α) ∩ p is open in p . then the set {λ(α) ∩ p : α ∈ b} is a locally finite family of non-empty open subsets of p . since p is a tychonoff space, this is equivalent to say that p is not pseudocompact (see theorem 3.10.22 in [3]), a contradiction. � theorem 3.3. let x be a regular space which is not locally compact at a point x0. the cartesian product x × y (x) is not a pseudo-k-space. proof. let x be a regular space which is not locally compact at a point x0. let us show that the cartesian product x × y (x) is not a pseudo-k-space. it suffices to find a subset h of x × y (x), which is not closed even if the intersection of h with any pseudocompact subspace p of the space x × y (x) is closed in p . recall that, in the definition of y (x), the set a denotes an index set and to each α ∈ a is associated a limit ordinal λ(α) such that ⋂ λ<λ(α) fλ is empty. now fix α ∈ a and λ ∈ λ(α) + 1 and define eλ = ⋂ µ<λ fµ. then eλ(α) = ∅. moreover the set sα = ∪{eλ × {λ} λ ∈ λ(α) + 1} is closed in x × (λ(α) + 1), which implies that it is closed in x × λ. denote by g the canonical projection g : λ −→ y (x) and by h the function idx × g, and define the set h = ⋃ α∈a h(sα) ⊂ x × y (x). we shall show that h is the set we are searching for. first let us prove that the intersection of h with any pseudocompact subset p of x × y (x) is closed in p . the projection py(p ) is a pseudocompact subset in y (x) so, by virtue of theorem 3.2, we have |{α ∈ a : py(p ) ∩ λ(α) 6= ∅}| < ω then p meets finitely many x × g(λ(α) + 1) = x × (λ(α) ∪ {y0}) ⊃ h(sα). now, since h(sα) is closed in x × y (x) for each α ∈ a, it follows that the set h ∩ p = ⋃ α∈a (h(sα) ∩ p ) is closed in p . on pseudo-k-spaces 183 now let us show that h is not closed in x × y (x). the point (x0, y0) ∈ x ×y (x) belongs to h but does not belong to h. take a neighborhood u ×v of (x0, y0), u open in x, v open in y (x), and let uβ a closed non-compact neighborhood uβ ⊂ u , for some β ∈ a. now, consider the canonical projection g : λ → y (x) , and fix λ ∈ g−1(v ) ∩ λ(β). the set h(eλ × {λ}) 6= ∅ is contained in (u × v ) ∩ h. therefore (x0, y0) ∈ h. suppose that (x0, y0) ∈ h, then (x0, y0) ∈ h(sα) for some α ∈ a. this is a contradiction. � theorems 3.1 and 3.3 provide the following characterization for locally compact spaces. theorem 3.4. let x be a regular space. the following conditions are equivalent: (1) x is locally compact. (2) x × y is a pseudo-k-space, for each pseudo-k-space y . proof. (1)⇒(2) it follows from theorem 3.1. (2)⇒(1) let x be a regular space which is not locally compact at a point x0. then, by virtue of theorems 3.2 and 3.3, the space y (x) is a pseudo-k-space such that x × y (x) is not a pseudo-k-space. � in terms of products of mappings we have theorem 3.5. let x be a regular space. the following conditions are equivalent: (1) x is locally compact. (2) idx × g is a quotient map with domain a locally pseudocompact hausdorff space, for every quotient map g with domain a locally pseudocompact hausdorff space y . proof. (1)⇒(2) it comes directly from whitehead theorem (see theorem 3.3.17 in [3]) and proposition 2.4. (2)⇒(1) if x is not locally compact then we can consider y (x), defined as before, and the projection map g : λ → y (x), which is a quotient map with domain the locally pseudocompact hausdorff space λ. it is easy to show that h = idx × g is not a quotient map with domain a locally pseudocompact hausdorff space. indeed if h was a quotient map with domain a locally pseudocompact hausdorff space then x × y (x) should be a pseudo-k-space, but x × y (x) is not a pseudo-k-space by virtue of theorem 3.3. � 184 a. miranda references [1] d. e. cohen, spaces with weak topology, quart. j. math., oxford 5 (1954), 77–80. [2] c. h. dowker, topology of metric complexes, amer. j. math. 74 (1952), 555–577. [3] r. engelking, general topology, sigma ser. pure math. 6 (heldermann, berlin, 1989). [4] t. jech, set theory, academic press, 1978. [5] e. michael, local compactness and cartesian product of quotient maps and k-spaces, ann. inst. fourier 18 (1968), 281–286. [6] k. morita, product of normal spaces with metric spaces, math. ann. 154 (1964), 365– 382. [7] j. nagata, quotient and bi-quotient spaces of m -spaces, proc. japan acad. 45 (1969), 25–29. [8] m. sanchis, a note on quasi-k-spaces, rend. ist. mat. univ. trieste suppl. xxx (1999), 173–179. [9] j. h. c. whitehead, a note on a theorem due to borsuk, bull. amer. math. soc. 54 (1948), 1125–1132. received february 2007 accepted october 2007 a. miranda (amiranda@unisa.it) dip. di matematica e informatica, università di salerno, via ponte don melillo, 84084 fisciano (salerno), italy () @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 101-134 dual attachment pairs in categorically-algebraic topology anna frascella, cosimo guido and sergey a. solovyov∗ abstract the paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. the new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation “∈” called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. following the recent interest of the fuzzy community in topological systems of s. vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inherent topology, but are capable of providing a natural transformation between two topological theories. we also outline a more general setting for developing the attachment theory, motivated by the concept of (l, m)-fuzzy topological space of t. kubiak and a. šostak. 2010 msc: 03e72, 54a40, 18b30, 18c99. keywords: dual attachment pair, (lattice-valued) categorically-algebraic topology, (l, m)-fuzzy topology, (localic) algebra, (pre)image operator, quasi-coincidence relation, quasi-frame, spatialization, topological system, variety. ∗this research was partially supported by the esf project of the university of latvia no. 2009/0223/1dp/1.1.1.2.0/09/apia/viaa/008. 102 a. frascella, c. guido and s. a. solovyov 1. introduction motivated by the abundance of lattice-valued topological theories available in the literature and the lack of interaction means between them, this paper makes another step towards developing a new approach to (lattice-valued) topological structures deemed to incorporate in itself the majority of the existing settings. based in category theory and universal algebra, the new framework is called categorically-algebraic (catalg) topology [64] to underline its generating theories. it originates from point-set lattice-theoretic (poslat) topology of s. e. rodabaugh [61, 62], developed in the framework of lattice-valued powerset theories (motivated by algebraic theories (in clone form) of e. g. manes [46], the basic example given by the theory of the powerset functor on the category set of sets and maps, appended with its induced contravariant powerset functor), where the underlying algebraic structures for topology are semi-quantales. we replace semi-quantales with algebras (possibly having a class of non-finitary operations) from an arbitrary variety and consider an abstract category as the ground for topology. the framework obtained in this manner includes the most important approaches to (lattice-valued) topology, providing convenient means of intercommunication between them, and (that is more essential) ultimately erasing the border between lattice-valued and crisp developments. moreover, the amount of building blocks of the proposed theory is reduced to minimum, postulating the so-called “plug and play approach”, when additional requirements on the underlying setting are motivated by the need of additional properties. in particular, we never employ the framework of monadic topology, developed by several authors in the literature [18, 29, 31], as being too restrictive for our current purposes. briefly speaking, we propagate the slogan: achieve more with less. on the other hand, it should be underlined immediately, that all essential properties of modern (lattice-valued) topology (e.g., compactness, separation axioms, connectedness, etc.) can be incorporated in the catalg setting. it is the theory of catalg spaces [67], which is currently undertaking the job. moreover, the new framework is rapidly progressing in several other directions [65, 71, 72, 73, 74], influencing each other dramatically. it is the main purpose of this paper to show one of the important applications of the new theory, i.e., the development of a fruitful topological setting, based in a catalg generalization of the set-theoretic membership relation “∈”. the starting point for the proposed research topic lies in the concept of quasicoincidence between a fuzzy point and a fuzzy set, introduced by p.-m. pu and y.-m. liu [49, definition 2.3′] with the aim to extend the standard approach to topology through neighborhood structures by fuzzifying the above relation “∈”. given a set x, a fuzzy point ax (a map from x to the unit interval i = [0,1], taking value a at x and 0 elsewhere) is said to be quasi-coincident with a fuzzy set α (a map x α −→ i) provided that 1 − α(x) < a. later on, y.-m. liu and m.-k. luo [43, definition 2.3.1] used a completely distributive lattice l, equipped with an order-reversing involution (−)′, to generalize the definition to a 66 (α(x))′. moreover, y.-m. liu [42] showed that the quasi-coincidence dual attachment pairs in categorically-algebraic topology 103 relation is the unique membership relation, which satisfies the four principles of a “reasonable” (generating a fruitful topological theory) membership relation. the next step was done by c. guido (and v. scarciglia) [25, 26, 27], who removed the requirement on the existence of an involution and introduced a lattice-valued analogue of the relation in question under the name of attachment. given a complete lattice l, an attachment a in l is a family {fa |a ∈ l\{⊥}} of completely prime ( ∨ s ∈ fa implies s ⋂ fa 6= ∅) filters of l, indexed by its elements, with an additional requirement f⊥ = ∅. an l-point ax is said to be attached to an l-set α (denoted ax aα) provided that α(x) ∈ fa. the new notion not only generalizes quasi-coincidence relation (take l = i and let fa = {b | 1 − a < b} for every a ∈ l), but also induces a functor from the category l-top of l-topological spaces [33] to the category top of topological spaces, which takes an l-topological space (x,τ) to the crisp space (sx,τ ⋆), where sx is the set of l-points of x and τ ⋆ = {α⋆ |α ∈ τ} with α⋆ = {ax |ax aα}. the new functor is closely related to the well-known hypergraph functors [20, 32, 36, 53, 55, 58] used in lattice-valued mathematics, bringing them under the common roof of attachment, and thereby removing the difference in their definition by various authors, that prevents them from gaining in popularity in applications. the employed machinery is based in the concept of topological system of s. vickers [75], introduced to merge point-set (topological spaces) and pointless (their underlying algebraic structures locales [35]) topology, which recently has raised an interest among the fuzzy researchers [12, 13, 14, 74] as a possible framework to incorporate both lattice-valued topology and its underlying algebraic structures (in most cases, particular kinds of the already mentioned semi-quantales). in [73], s. solovyov generalized the approach even further, taking into consideration the fact that there exists a one-to-one correspondence between completely prime filters of a complete lattice l and points of l (frame homomorphisms from l to the two-element frame 2 = {⊥,⊤} [35]), which opens a possibility to define an attachment as a map l a −→ frm(l,2) (omitting the requirement f⊥ = ∅ as never influencing the essential properties of the theory). the above-mentioned catalg approach to topology in hand, he introduced the notion of variable-basis attachment for an arbitrary variety of algebras. definition 1.1. let a be a variety of algebras and let a (−)∗ −−−→ setop be a functor, which takes an a-algebra a to its underlying set. an (a-)attachment is a triple f = (ωf,σ f, ), where ωf and σf are a-algebras, and ωf −→ a(ωf,σf) is a map. an attachment morphism f1 f −→ f2 is a pair of ahomomorphisms (ωf1,σf1) (ωf,σ f) −−−−−−→ (ωf2,σf2) such that for every a1 ∈ ωf1 and every a2 ∈ ωf2, ( 2(a2))(ω f(a1)) = (σf ◦ 1((ωf) ∗op(a2)))(a1). atta is the category of attachments and their homomorphisms, concrete over the product category a × a. the main achievement of [73] is a common framework (based in catalg attachment) for the majority of instances of hypergraph functor, providing a 104 a. frascella, c. guido and s. a. solovyov convenient tool for exploring their features, and the explicit study of categorical properties of attachment and its generated functors (in the sense of [25, 26]), which appear to have a right adjoint for a particular attachment type called spatial, generalizing the respective property of the hypergraph functor of u. höhle [32]. the advantage of the last result is the extension of the achievement of u. höhle to all of the above instances of hypergraph functor, taking the appropriate underlying variety in each case. moreover, this fact illustrates the main contribution of the catalg setting itself, whose essence is: prove once for many. it is the goal of this paper to continue the once started line of research by presenting a dual version of attachment. the meta-mathematical inducement for the new approach was given by the observation that both partially ordered sets and categories provide the means for dualization of their results. the real push, however, was taken up by the authors in their wish to change the setting of lattice-valued attachment of [26] from filters to ideals. after a brief discussion on the topic, the following crucial observations came to light. observation 1. the case of a complete chain l provides a possibility to define an attachment a on l as a family f⊥ = ∅ and fa = {b ∈ l |a < b} for every a ∈ l\{⊥}. more particularly, given an l-point ax and an l-set α, ax aα iff α(x) ∈ fa iff a < α(x). if α(x) 6= ⊤, then gα(x) = {b ∈ l |b < α(x)} is a completely prime ideal of l, which is the notion dual to that of completely prime filter. this suggests the family g⊤ = ∅ and ga = {b ∈ l |b < a} for every a ∈ l\{⊤} as a possible substitute for a, thereby turning the attachment condition ax a α from α(x) ∈ fa into a ∈ gα(x). observation 2. in the wake of [73], the concluding section of [26] introduced the notion of generalized attachment, based in the category qfrm of quasi-frames (definition 2.2 of this paper), which are complete lattices, with the respective morphisms preserving arbitrary ∨ and binary ∧ (the empty meet is excluded). the new category gives rise to the notion of completely prime quasi-filter of a complete lattice l as the preimage of {⊤} under a quasi-frame map (quasi-point) l p −→ 2, which allows, apart from the standard filters, also the empty one. a generalized attachment in a quasi-frame l is then a map l −→ qfrm(l,2) (cf. definition 1.1). the particular case of a complete chain l suggests the following definition of the map : ( (a))(b) = { ⊤, a < b, ⊥, otherwise, providing a generalized attachment f = (l,2, ). brief consideration brings a new map l � −→ 2l defined by (�(a))(b) = ( (b))(a), which induces the triple g = (l,2,�). an important property of the map is its preservation of ∨ and binary ∧, i.e., (�( ∨ s))(b) = ⊤ iff b < ∨ s iff b < s for some s ∈ s iff (�(s))(b) = ⊤ for some s ∈ s iff ( ∨ s∈s �(s))(b) = ⊤, whereas (�(s∧t))(b) = ⊤ iff b < s ∧ t iff b < s, b < t iff (�(s))(b) ∧ (�(t))(b) = ⊤. on the other hand, the map �(a) is not ∨ -preserving for a 6= ⊥, since (�(a))(⊥) = ⊤ 6= ⊥. in dual attachment pairs in categorically-algebraic topology 105 such a manner, the so-called dual attachment pair (f,g) arises, where duality means ax f α iff ( (a))(α(x)) = ⊤ iff �(α(x))(a) = ⊤ iff ax gα. the above remarks provide an opening for a new definition of attachment, called in this paper dual attachment. apart from concrete applications to the already developed theory, the concept represents a catalg extension of the notion of “duality” in mathematics (should not be mixed with the theory of catalg dualities [71, 72] dealing with topological representations of algebraic structures). the attentive reader will see that catalg “duality” is neither categorical duality (as, e.g., the dual of a category), nor algebraic duality (as, e.g., the dual of a partially ordered set), but truly categorically-algebraic “duality”. it will be the topic of our forthcoming papers to find the proper place for such kind of dualities in mathematics, whereas this manuscript is bound to consider categorical properties of dual attachment and the functors arising from it. it appears that the concept still retains a close relation to topological systems of s. vickers. on the other hand, the results of this paper clearly show that the nature of the two notions is essentially different, the latter being equipped with an internal topology extracted by the procedure of spatialization of systems introduced by s. vickers [75], whereas the former providing a way of interaction (natural transformation) between two topological theories, resulting in a functor between the categories of the respective topological structures. the achievement finally resolves the question (posed in the fuzzy community) on relationships between the two concepts and a possible common framework for both of them (non-existent due to the principally different categorical perspectives of the notions). moreover, the just mentioned crucial property of attachment gave rise to the study of one of the authors on general relationships between catalg topological theories and their induced catalg topological structures (see section 6 of this manuscript for the respective definitions), partly announced during the presentation of [65] and currently being developed as the subject of a forthcoming paper, similar by the approach (but not the results) to the widely used in categorical algebra algebraic theories of f. w. lawvere [41]. this paper uses both category theory and universal algebra, relying more on the former. the necessary categorical background can be found in [1, 28, 45, 46]. for the notions of universal algebra we recommend [7, 9, 23, 46]. although the authors tried to make the paper as much self-contained as possible, some details are still omitted and left to the reader. 2. dual attachment with its induced categories and functors in this section, we introduce the notion of dual attachment and consider its related categories and functors. the cornerstone of the approach is the concept of algebra. the structure is to be thought of as a set with a family of operations defined on it, satisfying certain identities, e.g., semigroup, monoid, group and also (that is different from the standard theory of universal algebra) complete lattice, frame, quantale. the classes of finitary algebras (those induced by a set of finitary operations) are usually described in universal algebra as either varieties or equational classes [7, 9, 23], which coincide due to the well-known 106 a. frascella, c. guido and s. a. solovyov hsp-theorem of g. birkhoff [6]. to incorporate the algebraic structures used in lattice-valued topology (where set-theoretic unions are usually replaced by arbitrary joins), this paper extends the approach of varieties to cover its needs. definition 2.1. (1) let ωωω = (nλ)λ∈λ be a (possibly proper) class of cardinal numbers. an ωωω-algebra is a pair (a,(ωaλ )λ∈λ), comprising a set a and a family of maps anλ ω a λ −−→ a (nλ-ary primitive operations on a). an ωωωhomomorphism (a,(ωaλ )λ∈λ) ϕ −→ (b,(ωbλ )λ∈λ) is a map a ϕ −→ b such that ϕ ◦ ωaλ = ω b λ ◦ ϕ nλ for every λ ∈ λ. alg(ωωω) is the construct of ωωω-algebras and ωωω-homomorphisms. (2) let m (resp. e) be the class of ωωω-homomorphisms with injective (resp. surjective) underlying maps. a variety of ωωω-algebras is a full subcategory of alg(ωωω) closed under the formation of products, m-subobjects and e-quotients. the objects (resp. morphisms) of a variety are called algebras (resp. homomorphisms). (3) given a variety a, a reduct of a is a pair (‖ − ‖,b), where b is a variety such that ωωωb ⊆ ωωωa and a ‖−‖ −−→ b is a concrete functor. from now on, every concrete category is supposed to be equipped with the underlying functor | − | to its respective ground category (cf. definition 1.1). for the sake of shortness, the fact will be never mentioned explicitly again. an experienced reader will probably be able to find numerous examples to back the new notion. below, we extend the list with several more items, all of which (except the last one) come from the realm of lattice-valued topology [61, 62] and will be used throughout the paper. definition 2.2. (1) given ξ ∈ { ∨ , ∧ }, a ξ-semilattice is a partially ordered set having arbitrary ξ. cslat(ξ) is the variety of ξ-semilattices. (2) a semi-quantale (s-quantale) is a ∨ -semilattice equipped with a binary operation ⊗ (multiplication). squant is the variety of s-quantales. (3) an s-quantale is called demorgan provided that it is equipped with an order-reversing involution (−)′. dmsquant is the variety of demorgan s-quantales. (4) an s-quantale is called unital (us-quantale) provided that its multiplication has the unit . usquant is the variety of us-quantales. (5) a quantale is an s-quantale whose multiplication is associative and distributes across ∨ from both sides. quant is the variety of quantales. (6) a quasi-frame (q-frame) is an s-quantale whose multiplication is ∧. qfrm is the variety of q-frames. (7) a semi-frame (s-frame) is a unital q-frame. sfrm is the variety of s-frames. (8) a frame is an s-frame which is a quantale. frm is the variety of frames. dual attachment pairs in categorically-algebraic topology 107 (9) a closure semilattice (c-semilattice) is a ∧ -semilattice, with the singled out bottom element ⊥. csl is the variety of c-semilattices. the reader should bear in mind that all varieties of definition 2.2 have complete lattices as objects. moreover, all of them except dmsquant, quant and frm are reducts of the variety clat of complete lattices. to continue the topic, we remark that cslat( ∨ ) is a reduct of squant; squant is a reduct of usquant and qfrm; usquant is a reduct of sfrm; set is a reduct of any variety. also notice that the categories sfrm and qfrm, having essentially the same objects (complete lattices), differ significantly on morphisms. the last item of definition 2.2 was motivated by the concept of strong (⊤-preserving) quantale homomorphism [37, 38], and would provide an additional example for the concept of catalg topology introduced later on in the paper. for the sake of convenience, from now on we use the following notations, which differ from the respective category-theoretic ones (see, e.g., [14, 57, 61] for the motivation). an arbitrary variety is denoted a, b, c, etc. the categorical dual of a variety a is denoted loa (the “lo” comes from “localic”), whose objects (resp. morphisms) are called localic algebras (resp. homomorphisms). several other categories introduced in the paper (but always related to varieties) employ similar notation for their duals. following the already accepted designation of [35], the dual of frm is denoted loc , whose objects are called locales. to distinguish maps (or, more generally, morphisms) and homomorphisms, the former are denoted f,g,h, reserving ϕ,ψ,φ for the latter. given a homomorphism ϕ, the respective localic one is denoted ϕop and vice versa. given an algebra a of a variety a (or an object of a related category), sa stands for the subcategory of loa comprising the identity 1a on a as the only morphism. we will occasionally use the notation saa, to underline the originating variety of the algebra a. given a set x, an algebra a and an element a ∈ a, x ax −−→ a denotes the constant map with value a. a few words are due to the many-valued framework employed in the paper. following [73], we extend the concept of lattice-valued set to that of algebraic one, which is defined as follows (recall the underlying functor of alg(ωωω)). definition 2.3. let x be a set and let a be an algebra of a variety a. an (a-)algebraic set in x is a map x α −→ |a|. the underlying idea of the new setting is based in a direct algebraization of the classical frameworks of l. a. zadeh [77] and j. a. goguen [21], which can be easily restored by choosing an appropriate variety a. despite the fact that the theory of algebraic sets provides a nice challenge for research, the current paper will not develop the topic off the bounds of its interests. to distinguish algebraic sets from other maps, from now on, they will be denoted α, β, γ. all preliminaries in their places, the new notion of attachment is ready to introduce (recall that set stands for the category of sets and maps). definition 2.4. let b be a variety and let b (−)∗ −−−→ setop be a functor such that b∗ = |b|. a dual (b-)attachment is a triple g = (ωg,σg,�), where 108 a. frascella, c. guido and s. a. solovyov ωg, σg are b-algebras, and ωg � −→ σg|ω g| is a b-homomorphism. a dual attachment morphism g1 f −→ g2 is then a pair of b-homomorphisms (ωg1,σg1) (ωf,σ f) −−−−−−→ (ωg2,σg2) such that for every b1 ∈ ωg1 and every b2 ∈ ωg2, (�2(ωf(b1)))(b2) = (σf ◦ �1(b1))((ω f) ∗op(b2)). attb is the category of dual attachments and their homomorphisms, concrete over the product category b × b. to convince the reader that definition 2.4 gives a category, we check the closure under composition. given two attb-morphisms g1 f −→ g2, g2 g −→ g3 and b1 ∈ ωg1, b3 ∈ ωg3, one easily gets that (�3(ω(g ◦ f)(b1)))(b3) = σg ◦ (�2(ωf(b1)))((ω g) ∗op(b3)) = σg ◦ σf ◦ (�1(b1))((ω f) ∗op ◦ (ωg)∗ op (b3)) = σ(g◦f)◦(�1(b1))((ω g ◦ ωf) ∗op(b3))=(σ(g◦f)◦(�1(b1)))(((ω(g ◦ f)) ∗op)(b3)). an attentive reader will notice striking similarities between the categories atta (definition 1.1) and attb (to distinguish the new type of attachment, capital letters are used in the notation of the respective category). a somewhat deeper insight into their nature reveals not less striking differences in their behavior, one of which being ready for display on the spot. in [73], a full embedding a � � ea // atta was provided, showing that atta gave a proper extension of its underlying variety a. in the new framework, a similar procedure results in an (in general, non-full) embedding under certain requirements only. proposition 2.5. suppose there exists a nullary operation ωλ0 of b, satisfying the identity ωλ(〈ωλ0 〉nλ) = ωλ0 for every b-operation ωλ (implying that ωλ0 is the unique nullary operation of b). then there exists an (in general, non-full) embedding b � � eb // attb, eb(b1 ϕ −→ b2) = (b1,b1,�1) (ϕ,ϕ) −−−→ (b2,b2 �2), with bi �i −−→ b |bi| i given by �i(b) = ω bi λ0 . proof. to show that the functor is correct on objects, notice that given λ ∈ λb and bi ∈ b for i ∈ nλ, (�(ω b λ (〈bi〉nλ)))(b) = ω b λ0 (b) = ωbλ0 = ω b λ (〈ω b λ0 〉nλ) = ωbλ (〈(�(bi))(b)〉nλ) = (ω b |b| λ (〈�(bi)〉nλ))(b) for every b ∈ b. to check the correctness on morphisms, notice that given b1 ∈ b1 and b2 ∈ b2, one gets, (�2(ϕ(b1)))(b2) = ω b2 λ0 (b2) = ω b2 λ0 = ϕ(ωb1 λ0 ) = (ϕ ◦ ωb1 λ0 )(ϕ∗op(b2)) = (ϕ ◦ (�1(b1)))(ϕ ∗op(b2)). the embedding properties of eb follow directly from the definition of the functor. the claim on non-fullness requires an additional assumption that there exist two different b-homomorphisms b1 ϕ // ψ // b2. then eb(b1) (ϕ,ψ) −−−→ eb(b2) is an attb-morphism, since given b1 ∈ b1 and b2 ∈ b2, (�2(ϕ(b1)))(b2) = ω b2 λ0 (b2) = ω b2 λ0 = ψ(ωb1 λ0 ) = (ψ ◦ ωb1 λ0 )(ϕ∗op(b2)) = (ψ ◦ (�1(b1)))(ϕ ∗op(b2)). by the fact that ϕ 6= ψ, we obtain that (ϕ,ψ) is not in the image of eb, thereby concluding the proof of the proposition. � an example for proposition 2.5 is the variety cslat( ∨ ) of ∨ -semilattices, which gives rise to the respective non-full embedding. the variety frm of dual attachment pairs in categorically-algebraic topology 109 frames, however, does not fit into the proposed framework, having more than one nullary operation, but its reduct qfrm suits well. in one word, in some cases the category attb provides a proper extension of its underlying variety. to continue, we need additional notions from the framework of categoricallyalgebraic (catalg) topology, introduced recently [64] as an extension of the pointset lattice-theoretic (poslat) topology of s. e. rodabaugh [57, 61]. the full development of the theory will be given in section 6 of this paper, whereas here, we just borrow some of its building blocks. by analogy with its predecessor, the new setting is based in a generalization of the backward powerset theory employed by the classical topological setting. the intuition for the new concept comes from the so-called (pre)image operators [61], well-known for every working mathematician. recall that given a set map x f −→ y , there exist the maps p(x) f → −−→ p(y ) (resp. p(y ) f ← −−→ p(x)) such that f→(s) = {f(x) |x ∈ s} (resp. f←(t) = {x |f(x) ∈ t}). the operators have already been extended to powersets of lattice-valued sets (see [10, 21, 56, 77]) and the latter one can be lifted to a more general setting. proposition 2.6. given a variety a, every subcategory c of loa induces a functor set × c (−)← −−−→ loa defined by ((x1,a1) (f,ϕ) −−−→ (x2,a2)) ← = a x1 1 ((f,ϕ)←)op −−−−−−−→ ax22 with (f,ϕ) ←(α) = ϕop ◦ α ◦ f. proof. the proof consists of easy calculations and can be found in [69, 70]. � for the sake of convenience, the functor set × sa (−)← −−−→ loa (the socalled fixed-basis approach, whereas the full framework is referred to as the variable-basis approach) is denoted by (−)←a , omitting the notation for 1a in its definition. the functor of proposition 2.6 has the merit of incorporating in itself the majority of the approaches to powersets of many-valued mathematics. the most crucial of its properties is the fact that it gives rise to a category of catalg (strictly speaking, its variety-based reduction [67]) topological spaces, providing a common framework for many approaches to (lattice-valued) topology. definition 2.7. let a be a variety and let c be a subcategory of loa. a c-topological space (c-space) is a triple (x,a,τ), where (x,a) is a set × cobject, and τ (c-topology on (x,a)) is a subalgebra of ax. a c-continuous map (x1,a1,τ1) (f,ϕ) −−−→ (x2,a2,τ2) is a set × c-morphism (x1,a1) (f,ϕ) −−−→ (x2,a2) such that ((f,ϕ) ←)→(τ2) ⊆ τ1. c-top is the category of c-spaces and c-continuous maps, which is concrete over the product category set × c. the category sa-top is denoted a-top, whose objects (resp. morphisms) are shortened to (x,τ) (resp. f). it should be underlined that the category c-top is a particular instance of a more general approach to catalg topology, developed in section 6 of this paper. the main advantages of the new framework have already been described in an abstract way in introduction and would be illustrated by concrete examples in section 6. at the moment, the reader should notice that apart from serving 110 a. frascella, c. guido and s. a. solovyov as a convenient tool for developing the attachment theory, the new setting provides the (much needed) means of interaction between hugely diversified (lattice-valued) topological theories available in the modern literature. it appears that the framework of attachment provides a more general category for topology than c-top. definition 2.8. given a variety b and a subcategory d of loattb, a d-topological space (d-space) is a triple (x,g,τ), where (x,g) is a set × d-object, and τ (d-topology on (x,g)) is a subalgebra of (ωg)x. a dcontinuous map (x1,g1,τ1) (f,g) −−−→ (x2,g2,τ2) is then a set × d-morphism (x1,g1) (f,g) −−−→ (x2,g2) with ((f,(ωg) op )←)→(τ2) ⊆ τ1. d-top is the category of d-spaces and d-continuous maps, concrete over the category set× d. for the sake of brevity, the category sg-top is denoted g-top, employing the shortened notations of the category a-top. under the assumption used at the beginning of proposition 2.5, there exists the embedding functor lob-top � � etop // loattb-top, etop((x1,b1,τ1) (f,ϕ) −−−→ (x2,b2,τ2)) = (x1,eb(b1),τ1) (f,eb(ϕ)) −−−−−−→ (x2,eb(b2),τ2), which in general is not full (using the machinery of the proof of proposition 2.5, etop(∅,b1,b ∅ 1 ) (!,(ϕ,ψ)) −−−−−→ etop(∅,b2,b ∅ 2 ) is continuous, but never belongs to the image of etop). the new functor makes the diagram lob-top |−| �� etop // loattb-top |−| �� set × lob � � 1set×e op b // set × loattb commute, showing that (in some cases) the category loattb-top provides a proper extension of the category lob-top. moreover, it appears that the former category induces another functor, which has more importance in the current developments. the new definition requires an additional (and very significant) notion related to catalg topology. this time, it is the concept of topological system introduced by s. vickers [75] as a common framework for incorporating both topological spaces and their underlying algebraic structures – locales [35], thereby trying to merge point-set and pointless topology. recently, the notion was successfully extended to include the case of lattice-valued topologies, the most significant results in the field achieved by j. t. denniston, a. melton, s. e. rodabaugh [11, 12, 13, 14], c. guido [25, 26] and s. solovyov [66, 74]. dual attachment pairs in categorically-algebraic topology 111 definition 2.9. let a be a variety and let c, d be subcategories of loa. a (c,d)-topological system ((c,d)-system) is a tuple d = (ptd,σd,ωd, |=), where (ptd,σd,ωd) is a set × c × d-object and ptd × ωd |= −→ σd is a map (σd-satisfaction relation on (pt d,ωd)) such that ωd |=(x,−) −−−−−→ σd is an a-homomorphism for every x ∈ ptd. a (c,d)-continuous map d1 f −→ d2 is a set×c×d-morphism (ptd1,σd1,ωd1) (pt f,(σf)op,(ωf)op) −−−−−−−−−−−−−→(ptd2,σd2,ωd2) such that for every x ∈ ptd1 and every b ∈ ωd2, it follows that |=1(x,ωf(b)) = σf(|=2(pt f(x),b)). (c,d)-topsys is the category of (c,d)-systems and (c,d)-continuous maps, concrete over the product category set × c × d. for the sake of shortness, the category (loa,loa)-topsys is denoted loa-topsys, whereas the category (sa,loa)-topsys is denoted a-topsys. to provide the intuition for the concept, we list two important examples. example 2.10. loc-topsys is precisely the category of lattice-valued topological systems introduced by j. t. denniston, a. melton and s. e. rodabaugh in [12]. its subcategory 2-topsys ( 2 is the two-element frame {⊥,⊤}) is isomorphic to the category topsys of s. vickers [75]. example 2.11. given a set k, the subcategory k-topsys of loset-topsys is isomorphic to the category chu(set,k) (or just chuk) comprising chu spaces over a given set k [5, 48]. in particular, chu2 is the category cont of contexts of formal concept analysis [19, 76], and also the category intsys of interchange systems introduced recently by j. t. denniston, a. melton and s. e. rodabaugh [13] in connection with certain aspects of program semantics (the so-called predicate transformers) initiated by e. w. dijkstra [16]. sharing the same definition, the categories chu2, cont and intsys have quite different motivating theories. the framework of definition 2.9 is closely related to the category loa-top, allowing the extension of the system spatialization procedure, introduced by s. vickers [75] to extract their inherent topology. theorem 2.12. (1) there exists a full embedding loa-top � � e // loa-topsys defined by e((x1,a1,τ1) (f,ϕ) −−−→ (x2,a2,τ2)) = (x1,a1,τ1, |=1) (f,ϕ,((f,ϕ)←)op) −−−−−−−−−−−→ (x2,a2,τ2, |=2), where |=i(x,α) = α(x). (2) there exists a functor loa-topsys spat −−−→ loa-top, which is defined by the formula spat(d1 f −→ d2) = (pt d1,σd1,τ1) (pt f,(σf)op) −−−−−−−−→ (pt d2,σd2,τ2), where τj = {|=j(−,b) |b ∈ ωdj}. (3) spat is a right-adjoint-left-inverse to e. (4) the category loa-top is isomorphic to a full (regular mono)-coreflective subcategory of the category loa-topsys. 112 a. frascella, c. guido and s. a. solovyov the attentive reader has probably already guessed that the name “spat” in the second item of theorem 2.12 comes from “spatialization”. the new category of definition 2.8 in hand, we can proceed to the definition of a new functor. proposition 2.13. there is a functor loattb-top eatt −−−−→ lob-topsys, which is given through the formula eatt((x1,g1,τ1) (f,g) −−−→ (x2,g2,τ2)) = (x1×| ωg1|,σg1,τ1, |=1) (f×(ω g)∗op,(σg)op,((f,(ωg)op)←)op) −−−−−−−−−−−−−−−−−−−−−−−−→(x2×| ωg2|,σg2, τ2, |=2), where |=i((x,b),α) = (�i(α(x)))(b). proof. to show that the functor in question is correct on objects, notice that given λ ∈ λb and αi ∈ τ for i ∈ nλ, it follows that |=((x,b),ωτλ(〈αi〉nλ)) = (�((ω τ λ(〈αi〉nλ))(x)))(b) = (�(ω ωg λ (〈αi(x)〉nλ)))(b) = (ω (σ g)|ω g| λ (〈�(αi(x))〉nλ))(b) = ω σg λ (〈(�(αi(x)))(b)〉nλ) = ωσgλ (〈|=((x,b),αi)〉nλ). to check the preservation of continuity, use the fact that for (x,b) ∈ x1×| ωg1| and α ∈ τ2, |=1((x,b),(f,(ω g) op )←(α)) = (�1(((f,(ω g) op )←(α))(x)))(b) = (�1(ωg ◦ α ◦ f(x)))(b) = (σg ◦ �2(α ◦ f(x)))((ω g) ∗op(b)) = σg ◦ |=2((f(x),(ω g) ∗op(b)),α) = σg ◦ |=2(f × (ωg) ∗op(x,b),α). � it should be noticed at once that despite the notation, the functor of proposition 2.13 never needs to be an embedding. in fact, the merits of the functor in question are highly dependant on the properties of the employed functor b (−)∗ −−−→ setop. on the other hand, it is possible to restrict the domain of eatt and obtain an embedding. below we suggest two possible approaches, the first of which being rather straightforward. proposition 2.14. given a dual b-attachment g such that ωg is non-empty, the restriction g-top egatt=eatt | σg-topsys g-top −−−−−−−−−−−−−−−−→ σg-topsys is an embedding. proof. given a g-continuous map (x1,τ1) f −→ (x2,τ2), e g att((x1,τ1) f −→ (x2,τ2)) = (x1 × | ωg|,τ1, |=1) (f×1|ω g|,(f ← ω g) op) −−−−−−−−−−−−−→ (x2 × | ωg|,τ2, |=2) (recall our shortened notation for fixed-basis topological spaces) that implies the desired property, the condition on ωg excluding the case of the constant functor mapping everything to the empty system. � the second approach is more sophisticated. the restriction in question is provided by the concept of stratified topological space (the idea of stratification is due to r. lowen [44], the term itself coined by p.-m. pu and y.-m. liu [50]). dual attachment pairs in categorically-algebraic topology 113 definition 2.15. loattb-top ∅k is the full subcategory of loattb-top of non-empty stratified spaces, i.e., spaces (x,g,τ) such that both x and ωg are non-empty, and for every a ∈ ωg, the constant map ax is in τ. the notation “(−)k” for stratified spaces comes from [51, 52] and is already widely accepted among the researchers, motivating us to follow their steps. proposition 2.16. the restriction loattb-top ∅k e ∅k att =eatt|loattb-top ∅k −−−−−−−−−−−−−−−−−−→ lob-topsys provides an embedding. proof. let (x1,g1,τ1) (f1,g1)// (f2,g2) // (x2,g2,τ2) be a pair of loattb-top∅k-morphisms. to show that e∅k att embeds objects, notice that e∅k att (x1,g1,τ1) = e∅k att (x2,g2,τ2) implies x1 × | ωg1| = x2 × | ωg2|, σg1 = σg = σg2, τ1 = τ = τ2 and (x1 × | ωg1|) × τ |= 1 −−→ σg = (x2 × | ωg2|) × τ |= 2 −−→ σg. the assumption on non-emptiness (which can not be avoided) provides x1 = x = x2 and | ωg1| = y = | ωg2|. to show that ωg1 = ωg2, take some x0 ∈ x and then, given λ ∈ λb and bi ∈ y for i ∈ nλ, ω ωg1 λ (〈bi〉nλ) = (ω τ1 λ (〈bi〉nλ))(x0) = (ω τ2 λ (〈bi〉nλ))(x0) = ω ω g2 λ (〈bi〉nλ), implying ωg1 = ωg = ωg2. to show that �1 = �2, employ the existing x0 to get that for every b1,b2 ∈ ωg, (�1(b1))(b2) = (�1(b1(x0)))(b2) = |=1((x0,b2),b1) = |=2((x0,b2),b1) = (�2(b1))(b2). to show faithfulness of e∅k att , use the fact that e∅k att (f1,g1) = e ∅k att (f2,g2) implies f1×(ωg1) ∗op = f2 ×(ωg2) ∗op, σg1 = σg = σg2 and (f1,(ωg1) op )← = (f2,(ωg2) op )←. the non-emptiness requirement provides f1 = f = f2 and also (ωg1) ∗op = (ωg2) ∗op. to verify that ωg1 = ωg2, use the fact that given b ∈ ωg2, (ωg1)(b) = (ωg1 ◦ b ◦ f)(x0) = ((f,(ωg1) op )←(b))(x0) = ((f,(ωg2) op )←(b))(x0) = (ω g2)(b). � at the end of this section, we finally define the main object of our interest, namely, a particular functor. it provides an analogue of the functor h, introduced in [73] as a generalization of the functor l-top (−)⋆ −−−→ top of [26] (already mentioned in introduction), with the aim to produce a convenient framework for studying categorical properties of the hypergraph functors. it is one of the main goals of this paper to explore the nature of the new functor and its relationships to its predecessors. definition 2.17. there exists a functor loattb-top hatt −−−−→ lob-top = loattb-top eatt −−−−→ lob-topsys spat −−−→ lob-top, hatt((x1,g1,τ1) (f,g) −−−→ (x2,g2,τ2))=(x1×| ωg1|,σg1, τ̃1) (f×(ω g)∗op,(σg)op) −−−−−−−−−−−−−→(x2×| ωg2|,σg2, τ̃2), where τ̃i = {α̃ = (�i(α(−)))(−) |α ∈ τi}. given a loattb-object g, the respective fixed-basis functor g-top hgatt −−−−→ σg-top is defined by the formula hgatt((x1,τ1) f −→ (x2,τ2)) = (x1 × | ωg|, τ̃1) f×1|ω g| −−−−−−→ (x2 × | ωg|, τ̃2). 114 a. frascella, c. guido and s. a. solovyov unlike [73], we are not going to touch the topic of hypergraph functors in this paper, restricting our attention to the functor hatt itself. we begin with the remark that certain properties of the dual attachment g can help to provide an embedding property for the resulting functor hgatt. definition 2.18. a dual attachment g is called (1) ω-spatial provided that for every b1,b2 ∈ ωg such that b1 6= b2, �(b1) 6= �(b2) (� is injective). (2) σ-spatial provided that for every b1,b2 ∈ ωg such that b1 6= b2, there exists some b ∈ ωg such that (�(b))(b1) 6= (�(b))(b2). after brief consideration, the reader will easily see that ω-spatiality and σspatiality are quite different notions. a nice example on the topic is provided by the category attset. the map i � −→ ii, taking every a ∈ i to the identity 1i, gives a dual attachment g = (i,i,�), which is σ-spatial but not ω-spatial. on the other hand, changing the definition to take every a ∈ i to the constant map ai, provides an attachment which is ω-spatial but not σ-spatial. examples for more complicated varieties can be found in [73]. proposition 2.19. given an ω-spatial attachment g with the property that ωg is non-empty, the functor g-top hgatt −−−−→ σg-top is an embedding. proof. it will be enough to show the injectivity on objects. given two spaces (x1,τ1) and (x2,τ2) such that h g att(x1,τ1) = h g att(x2,τ2), the non-emptiness of ωg implies x1 = x = x2. to show that τ1 = τ2, notice that given α1 ∈ τ1, α̃1 ∈ τ̃1 = τ̃2 and, therefore, α̃1 = α̃2 for some α2 ∈ τ2. given x ∈ x, (�1(α1(x)))(b) = α̃1(x,b) = α̃2(x,b) = (�2(α2(x)))(b) for every b ∈ ωg and that implies α1(x) = α2(x) by ω-spatiality of g. as a result, α1 = α2, providing τ1 ⊆ τ2. the converse inclusion is similar. � 3. dual attachment pairs this section clarifies the word “dual” in the term “dual attachment” used in this paper. the motivation for the choice comes from a particular property of attachment found out in [26], namely, the existence of a functor l-top (−)⋆ −−−→ top (already mentioned in introduction), which takes an l-topological space (x,τ) to the crisp space (sx,τ ⋆), where sx is the set of l-points of x, and τ⋆ consists of the sets α⋆ for every α ∈ τ, comprising precisely those l-points, which are attached to the particular α in question. a catalg analogue of the above-mentioned functor has been already considered in [73], whose counterpart for the current setting is given in definition 2.17. it is the main purpose of this section to show that both functors coincide in case the respective attachments form a dual attachment pair. we begin with the definition of a generalized version of the above-mentioned functors, which require some additional preliminaries contained in the following definition and proposition. dual attachment pairs in categorically-algebraic topology 115 definition 3.1. let a be a variety and let (‖−‖,b) be a reduct of a. a dual b-attachment g is called a-derived provided that there exist a-algebras aω, aς such that ωg = ‖aω‖, σg = ‖aς‖. proposition 3.2. let a be a variety, let (‖ − ‖,b) be a reduct of a and let a be an a-algebra. there exist two functors: (1) ‖a‖-top xt‖a‖a −−−−−→ a-top defined by xt‖a‖a((x1,τ1) f −→ (x2,τ2)) = (x1,〈τ1〉) f −→(x2,〈τ2〉) with 〈τi〉 the a-subalgebra of a generated by τi; (2) a-top rda‖a‖ −−−−−→ ‖a‖-top defined by rda‖a‖((x1,τ1) f −→ (x2,τ2)) = (x1,‖τ1‖) f −→ (x2,‖τ2‖). if g is an a-derived dual b-attachment, then there is a functor aω-top h g att −−−−→ aς-top defined by commutativity of the following diagram: aω-top rdaω ω g �� h g att // aς-top g-top hgatt // σg-top. xtς gaς oo proof. it is enough to verify that the functor xt‖a‖a preserves continuity. we use the simple fact that given a homomorphism a1 ϕ −→ a2 and a subset s ⊆ a1, ϕ→(〈s〉) = 〈ϕ→(s)〉 [70]. as a result, (f←a ) →(〈τ2〉) = 〈(f ← a ) →(τ2)〉 ⊆ 〈τ1〉. � the reader should notice that “xt” (resp. “rd”) is the abbreviation for “extension” (resp. “reduction”), used to underline the action of the functor in question, i.e., to extend (resp. reduce) the algebraic structure. both functors will play an important role in the subsequent developments. to compare the new functor with the already existing setting of [73], one should recall some results from its approach to the concept of attachment. proposition 3.3. there exists a functor loatta-top eatt −−−→ loa-topsys, which is given by the formula eatt((x1,f1,τ1) (f,g) −−−→ (x2,f2,τ2)) = (x1 × | ωf1|,σf1,τ1, |=1) (f×(ωg)∗op,(σ g)op,((f, (ωg)op)←)op) −−−−−−−−−−−−−−−−−−−−−−−−→(x2×| ωf2|,σf2,τ2, |=2), |= i ((x,a),α) = ( i(a))(α(x)). the reader is advised to pay attention to the important fact that the only difference in the definition of the functors of propositions 2.13, 3.3 concerns the respective satisfaction relation. definition 3.4. there exists a functor loatta-top hatt −−−→ loa-top = loatta-top eatt −−−→ loa-topsys spat −−−→ loa-top, hatt((x1,f1,τ1) (f,g) −−−→ (x2,f2,τ2)) = (x1×| ωf1|,σf1, τ̂1) (f×(ωg)∗op,(σ g)op) −−−−−−−−−−−−−→ (x2×| ωf2|,σf2, τ̂2), 116 a. frascella, c. guido and s. a. solovyov where τ̂i = {α̂ = ( i(−))(α(−)) |α ∈ τi}. given a loatta-object f , the respective fixed-basis functor f-top hfatt −−−→ σf-top is given by hfatt((x1,τ1) f −→ (x2,τ2)) = (x1 × | ωf |, τ̂1) f×1|ω f| −−−−−−→ (x2 × | ωf |, τ̂2). by analogy with proposition 3.2(3), one obtains the functor aω-top h f att −−−→ aς-top (notice the difference in the notations). the crucial question arises on when the two functors hfatt and h g att coincide, and that is precisely the point for dual attachment pairs to come in play. definition 3.5. let a be a variety, let (‖ − ‖,b) and (‖ − ‖,c) be reducts of a, and let f and g be a b-attachment and a dual c-attachment respectively. (1) both f and g are called reduct attachments. (2) the pair (f,g) is an attachment pair w.r.t. (a,b,c). (3) (f,g) is a related attachment pair provided that both f and g are a-derived, and afω = aω = a g ω, a f σ = aς = a g σ. (4) (f,g) is a dual attachment pair provided that (f,g) is a related attachment pair, and for every a1,a2 ∈ aω, ( (a1))(a2) = (�(a2))(a1) w.r.t. the maps |aω| // � // set(|aω|, |aς|). every related attachment pair gives two functors aω-top h f att // h g att // aς-top. with a dual attachment pair in hand, these two functors coincide. proposition 3.6. given a dual attachment pair (f,g), hfatt = h g att. proof. since the case of morphisms is clear, it will be enough to show equality of the functors on objects. given an aω-space (x,τ), it is sufficient to verify that 〈τ̂〉 = 〈τ̃〉. given α ∈ τ, α̂(x,a) = (( (−))(α(−)))(x,a) = ( (a))(α(x)) = (�(α(x)))(a) = ((�(α(−)))(−))(x,a) = α̃(x,a) for every (x,a) ∈ x × |aω|. thus τ̂ = τ̃, implying 〈τ̂〉 = 〈τ̃〉. � a good illustration of proposition 3.6 is provided by observation 2 from introduction, which gives a dual attachment pair (f,g) w.r.t. the varieties (clat,qfrm,qfrm), where f = (l,2, ) is based on a complete chain l and the map ( (a1))(a2) = { ⊤, a1 < a2, ⊥, otherwise. to get more intuition for the attachment pair (f,g), one can represent every map (a) as a particular subset of l (the preimage of {⊤} under the map in question), i.e., (a) = {b ∈ l |a < b} =↾ a (notice that (⊤) = ∅). the dual attachment � is then the collection of sets �(a) = {b ∈ l |b < a} =⇂ a (�(⊥) = ∅ now) for every a ∈ l. it is easy to see that in this particular case, the attachment duality reduces to the usual, well-known in mathematics, order-theoretic duality. the main point of proposition 3.6 is that the functors dual attachment pairs in categorically-algebraic topology 117 l-top h f att // h g att // (2-top ∼= top) generated by f and g coincide. on the other hand, straightforward computations backed by [73] show that hfatt is precisely the functor l-top (−)⋆ −−−→ top of [26]. it follows that a two-fold representation of the already well-known notion is obtained. the reader should pay attention to the fact that the case l = 2 does not result in the identity functor on top, since a topological space (x,τ) is taken to the space (x × |2|,〈 ˆ‖τ‖〉 = 〈 ˜‖τ‖〉), which has a different carrier set. the reader will easily find other examples on the topic. it is important to underline, however, that the case of an unrelated attachment pair can provide completely different (incomparable) functors hfatt, h g att. 4. existence of dual attachment pairs the previous section showed that the case of a dual attachment pair has the crucial property of equality of the derived functors. on the other hand, the functors of a just related attachment pair need not coincide which gives dual attachment pairs even more importance. a natural question on the existence of dual attachment pairs arises. this short section clarifies the situation. proposition 4.1. let f be a b-attachment which is a reduct attachment w.r.t. a. there exists a dual c-attachment g such that (f,g) is a dual attachment pair iff the following conditions are fulfilled: (1) f is a-derived (ωf = ‖aω‖ and σf = ‖aς‖); (2) there exists a reduct (‖−‖,c) of a such that ‖aω‖ −→ c(‖aω‖,‖aς‖). proof. for the necessity, notice that, firstly, afω = aω = a g ω, a f σ = aς = a g σ, and, secondly, given λ ∈ λc and ai ∈ ‖aω‖ for i ∈ nλ, ( (a))(ω ‖aω‖ λ (〈ai〉nλ)) = (�(ω ‖aω‖ λ (〈ai〉nλ)))(a) = (ω ‖aς‖ |aω| λ (〈�(ai)〉nλ))(a) = ω ‖aς‖ λ (〈(�(ai))(a)〉nλ) = ω ‖aς‖ λ (〈( (a))(ai)〉nλ) for every a ∈ ‖aω‖. the sufficiency is slightly more sophisticated. define the required dual attachment g by ωg = ‖aω‖, σg = ‖aς‖ (the respective reducts are taken in the variety c) together with ωg � −→ (σg)|ω g| given by (�(a1))(a2) = ( (a2))(a1). the only challenge now is to show that � is a c-homomorphism. given λ ∈ λc and ai ∈ ωg for i ∈ nλ, (�(ω ‖aω‖ λ (〈ai〉nλ)))(a) = ( (a))(ω ‖aω‖ λ (〈ai〉nλ)) = ω ‖aς‖ λ (〈( (a))(ai)〉nλ) = ω ‖aς‖ λ (〈(�(ai))(a)〉nλ) = (ω ‖aς‖ |aω| λ (〈ai〉nλ))(a) for every a ∈ ωg. � an example for the proposition is provided by the dual attachment pair (f,g) with f = (l,2, ), mentioned at the end of the previous section, the second of the requirements (the first being obvious) verified in introduction as follows: ( (a))( ∨ s) = ⊤ iff a < ∨ s iff a < s for some s ∈ s iff ( (a))(s) = ⊤ for some s ∈ s iff ( ∨ s∈s (a))(s) = ⊤, whereas ( (a))(s ∧ t) = ⊤ iff a < s ∧ t 118 a. frascella, c. guido and s. a. solovyov iff a < s and a < t iff ( (a))(s) ∧ ( (a))(t) = ⊤. the converse way (from dual attachment to attachment) is equally easy and can be run through as follows. proposition 4.2. let g be a dual c-attachment which is a reduct attachment w.r.t. a. there exists a b-attachment f such that (f,g) is a dual attachment pair iff the following conditions are fulfilled: (1) g is a-derived (ωg = ‖aω‖ and σg = ‖aς‖); (2) there exists a reduct (‖ − ‖,b) of a such that ‖aω‖ � −→ ‖aς‖ |aω| is a b-homomorphism. proof. for the necessity notice that, firstly, agω = aω = a f ω, a g σ = aς = a f σ, and, secondly, given λ ∈ λb and ai ∈ ‖aω‖ for i ∈ nλ, ((�(ω ‖aω‖ λ (〈ai〉nλ)))(a) = ( (a))(ω ‖aω‖ λ (〈ai〉nλ)) = ω ‖aς‖ λ (〈( (a))(ai)〉nλ) = ω ‖aς‖ λ (〈(�(ai))(a)〉nλ) = (ω ‖aς‖ |aω| λ (〈�(ai)〉nλ))(a) for every a ∈ ‖aω‖. to show the sufficiency, define an attachment f by ωf = ‖aω‖, σf = ‖aς‖ together with ωf −→ b(ω f,σf) given by ( (a1))(a2) = (�(a2))(a1). it should be verified that (a) is a b-homomorphism for every a ∈ ωf . given λ ∈ λb and ai ∈ ωf for i ∈ nλ, ( (a))(ω ‖aω‖ λ (〈ai〉nλ)) = (�(ω ‖aω‖ λ (〈ai〉nλ)))(a) = (ω ‖aς‖ |aω| λ (〈�(ai)〉nλ))(a) = ω ‖aς‖ λ (〈(�(ai))(a)〉nλ) = ω ‖aς‖ λ (〈( (a))(ai)〉nλ). � we close this section with the remark (already mentioned in introduction) that the concept of duality for attachment used in this paper is developed in the framework of arbitrary algebras (possibly) void of any kind of order relation and, therefore, our current setting is not the duality induced by a partial order. on the other hand, the approach is neither a duality of category theory, since the underlying category of the respective attachment is not dualized. based on the employed framework of catalg topology, the type of duality presented in the paper could be called categorically-algebraic. it will be the topic of our forthcoming papers to find the proper place of the new notion in mathematics. the current manuscript will continue exploring another aspects of attachment. 5. natural transformations induced by attachment it was shown in [26] that every attachment a in a complete lattice l provides a frame homomorphism lx (−)⋆ −−−→ p(sx) for every set x (simply taking α ∈ l x to the set α⋆ of all l-points attached to α), which gives rise to the already mentioned functor l-top (−)⋆ −−−→ top. this section extends the map to our catalg setting, resulting in consequences important for the whole development. proposition 5.1. let f be a b-attachment and let g be a dual c-attachment. for every set x, there exist a b-homomorphism (ωf)x (−) −−−→ (σf)x×|ωf| dual attachment pairs in categorically-algebraic topology 119 defined by α (x,b) = ( (b))(α(x)), and a c-homomorphism (ωg)x (−)� −−−→ (σg)x×|ω g| defined by α�(x,b) = (�(α(x)))(b). if (f,g) is a related attachment pair, then the maps have the same (co)domain |aω| x (−)� // (−) // |aς| x×|aω| . if (f,g) is a dual attachment pair, then the maps coincide. proof. to prove the first claim, we show that the map (−)� provides a chomomorphism. given λ ∈ λc and 〈αi〉nλ ∈ (ωg) x, (ω (ω g)x λ (〈αi〉nλ)) �(x,b) = (�((ω (ω g)x λ (〈αi〉nλ))(x)))(b) = (�(ωωgλ (〈αi(x)〉nλ)))(b) = (ω (σ g)|ω g| λ (〈�(αi(x))〉nλ))(b) = ωσgλ (〈(�(αi(x)))(b)〉nλ) = ω σg λ (〈α � i (x,b)〉nλ) = (ω (σg)x×|ω g| λ (〈α�i 〉nλ))(x,b) for every (x,b) ∈ x × | ωg|. the second claim is obvious. for the last statement, notice that given α ∈ |aω| x, α�(x,a) = (�(α(x)))(a) = ( (a))(α(x)) = α (x,a) for every (x,a) ∈ x × |aω|. � after a closer scrutiny, it appears that the homomorphisms of proposition 5.1 are actually components of natural transformations (the fact, never mentioned in [26]). to prove the claim, start with the preliminary remark that given a dual c-attachment g, there exists a functor set (−×|ωg|)←σ g −−−−−−−−−→ loc defined by commutativity of the following triangle: set (−×|ω g|)←σ g '' −×|ω g| // set (−)←σ g �� loc, where − × | ωg| is the standard product functor [1] defined by the formula (− × | ωg|)(x1 f −→ x2) = x1 × | ωg| f×1|ω g| −−−−−−→ x2 × | ωg|. proposition 5.2. every dual c-attachment g provides a natural transformation (− × | ωg|)←σ g ((−)�) op −−−−−−→ (−)←ω g. proof. given a map x1 f −→ x2, one has to verify commutativity of the diagram (ωg)x2 f ← ω g �� (−)�x2 // (σg)x2×|ω g| (f×1|ω g|) ← σ g �� (ωg)x1 (−)�x1 // (σg)x1×|ω g|, and that follows from the fact that (((−)�x1 ◦f ← ωg)(α))(x,b) = (α◦f) �(x,b) = (�(α ◦f(x)))(b) = α�(f(x),b) = (α� ◦ (f × 1|ω g|))(x,b) = (((f × 1|ω g|) ← σ g ◦ (−)�x2)(α))(x,b) for every α ∈ (ωg) x2 and every (x,b) ∈ x1 × | ωg|. � 120 a. frascella, c. guido and s. a. solovyov similarly, one gets a natural transformation (−×| ωf |)←σ f ((−) ) op −−−−−−→ (−)←ω f for a given b-attachment f , which reduces to the setting of [26] for b being the variety frm of frames. in case of a dual attachment pair, (as one might expect) both natural transformations coincide. to generalize the passage of [26] from a natural transformation to a functor, we need some additional notions from the realm of catalg topology, presented briefly in the subsequent section. 6. categorically-algebraic topology and attachment in this section we recall from [71] basic concepts of categorically-algebraic (catalg) topology (see also [64, 66, 72]), which bring to light the crucial property of attachment, i.e., generation of a functor between two topological settings. the approach is motivated by the currently dominating in the fuzzy community point-set lattice-theoretic (poslat) topology introduced by s. e. rodabaugh [55] and developed by p. eklund, c. guido, u. höhle, t. kubiak, a. šostak and the initiator himself [17, 24, 33, 39, 40, 57]. the main advantage of the new setting is the fact that apart from incorporating as special subcases the most important approaches to (lattice-valued) topology and providing convenient means of interaction between them, the catalg framework ultimately erases the border between crisp and many-valued developments, producing a theory which underlines the algebraic essence of the whole (not only lattice-valued) mathematics, thereby propagating algebra as the main driving force of modern exact sciences. it should be noticed immediately that some parts of the theory have already been used throughout the paper (definition 2.7). the current section provides a more rigid foundation for the approach and backs it by several motivating examples, to give the flavor of fruitfulness of the new theory. the setting is based in a mixture of powerset theories of [61, definition 3.5] (see also [60, 62]) and topological theories of [1, exercise 22b]. definition 6.1. a variety-based backward powerset theory (vbp-theory) in a category x (the ground category of the theory) is a functor x p −→ loa to the dual of a variety a. to get the intuition for the concept, the reader is advised to recall the functor of proposition 2.6, providing the main example for the notion and incorporating many approaches to powerset operators popular in lattice-valued mathematics. example 6.2. (1) set×s2 p=(−)← −−−−−−→ locbool, where cbool is the variety of complete boolean algebras (complete, complemented, distributive lattices) and 2 = {⊥,⊤}, provides the standard preimage operator, mentioned before proposition 2.6. (2) set × si z=(−)← i −−−−−−→ dmloc (cf. definition 2.2), where i = [0,1] is the unit interval, gives the fixed-basis fuzzy approach of l. a. zadeh [77]. dual attachment pairs in categorically-algebraic topology 121 (3) set × sl g1=(−) ← l −−−−−−→ loc provides the fixed-basis l-fuzzy approach of j. a. goguen [21]. the setting was changed to set × sl g2=(−) ← l −−−−−−→ louquant in [22]. the machinery can be generalized to an arbitrary variety a and the theory set × sa saa=(−) ← a −−−−−−−→ loa, which unites the previous items in one common fixed-basis framework. (4) set × c rc1 =(−) ← −−−−−−−→ dmloc, where c is a subcategory of dmloc, gives the variable-basis poslat approach of s. e. rodabaugh [54]. the setting has been extended to set × c rc2 =(−) ← −−−−−−−→ lousquant in [61] and then reduced to set × loc r3=(−) ← −−−−−−→ loc in [11, 14]. (5) set×fuzlat e=(−)← −−−−−→ fuzlat provides the variable-basis approach of p. eklund [17], motivated by those of s. e. rodabaugh [54] and b. hutton [34]. notice that fuzlat is the dual of the variety hut of completely distributive demorgan frames called hutton algebras [57]. the machinery can be generalized to an arbitrary variety a and the theory set×c sca=(−) ← −−−−−−−→ loa, which unites the previous items in one common variable-basis framework. on the next step, we provide another level of abstraction, which has never been used in the above-mentioned theories of s. e. rodabaugh. definition 6.3. let x be a category and let t = (p,(‖ − ‖,b)) contain a vbp-theory x p −→ loa in the category x and a reduct (‖ − ‖,b) of a. the variety-based topological theory (vt-theory) in x induced by t is the functor x t=‖−‖op◦p −−−−−−−−→ lob. since a vt-theory t is completely determined by the respective pair t , we use occasionally the notation (p,b) instead of t . it is important to underline that the aim of an additional level of abstraction is to remove the unused topological structure provided by powerset theories, the move, motivated by the observation that the standard backward powerset theory is based in boolean algebras, whereas the respective topological theory is reduced to frames (the case of closure spaces mentioned below provides another good example). on the other hand, the case of coincidence between powerset and topological theories is not excluded in our framework. the reader will see that the subsequent developments will often provide a topological theory only, without any explicit reference to its generating powerset theory. definition 6.4. let t be a vt-theory in a category x. top(t) is the category, concrete over x, whose objects (t-topological spaces) are pairs (x,τ), comprising an x-object x and a subalgebra τ of t(x) (t-topology on x), and whose morphisms (x,τ) f −→ (y,σ) are those x-morphisms x f −→ y , which satisfy ((t(f)) op )→(σ) ⊆ τ (t-continuity). 122 a. frascella, c. guido and s. a. solovyov the significance of the category top(t) is the fact that it unites many of the existing topological frameworks in mathematics. to give the reader the flavor of their abundance and the fruitfulness of the new unifying framework, we provide a short list of examples illustrating the notion of catalg topology. example 6.5. (1) top((p,frm)) is isomorphic to the category top of topological spaces and continuous maps. (2) top((p,csl)) is isomorphic to the category cls of closure spaces and continuous maps, studied by d. aerts et al. [2, 3]. (3) top((z,frm)) is isomorphic to the category i-top of fixed-basis fuzzy topological spaces, introduced by c. l. chang [8]. (4) top((g2,uquant)) is isomorphic to the category l-top of fixed-basis l-fuzzy topological spaces of j. a. goguen [22]. (5) top((rci ,usquant)) is isomorphic to the category c-topi, i ∈ {1,2} for variable-basis poslat topology of s. e. rodabaugh [54, 61]. (6) top((e,frm)) is isomorphic to the category fuzz for variable-basis poslat topology of p. eklund [17], motivated by those of s. e. rodabaugh [54] and b. hutton [34]. (7) top((saa,a)) (resp. top((s loa a ,a))) is isomorphic to the fixed(resp. variable-) basis category a-top (resp. loa-top) used in the former approach to catalg topologies of [70] (resp. [69]) as well as in the previous sections of this paper (definition 2.7). the reader should notice the fact that the second item of example 6.5 is never included in the setting of topological theories of s. e. rodabaugh [61], which are based explicitly on s-quantales (and, therefore, on ∨ -semilattices), whereas closure spaces rely on c-semilattices (and, therefore, on ∧ -semilattices). it appears (as an experienced reader might guess) that in order to deal successfully with the categories of the form top(t), it is enough to consider their generating topological theories t . proposition 6.6. let x t1 −→ loa, y t2 −→ loa be vt-theories, let x f −→ y be a functor, and let t2 ◦ f η −→ t1 be a natural transformation. there exists a functor top(t1) hη −−→ top(t2) given by hη((x1,τ1) f −→ (x2,τ2)) = (f(x1),(η op x1 )→(τ1)) f(f) −−−→ (f(x2),(η op x2 )→(τ2)). proof. it will be enough to show that the functor hη preserves continuity and that follows immediately from commutativity of the diagram t1(x2) (t1(f)) op �� η op x2 // t2 ◦ f(x2) (t2◦f(f)) op �� t1(x1) η op x1 // t2 ◦ f(x1), dual attachment pairs in categorically-algebraic topology 123 since ((t2 ◦ f(f)) op )→((η op x2 )→(τ2)) = ((t2 ◦ f(f)) op ◦ η op x2 )→(τ2) = (η op x1 ◦ (t1(f)) op )→(τ2) ⊆ (η op x1 )→(τ1). � as an example of the obtained result, one can look at proposition 5.2 and the remark just afterward, providing two functors top((−)←ω g) h�op −−−→ top((−)←σ g) and top((−) ← ω f ) h op −−−→ top((−)←σ f ), which essentially are the fixed-basis functors g-top hgatt −−−−→ σg-top and f-top hfatt −−−→ σf-top of definitions 2.17 and 3.4 respectively. moreover, it appears that a more general framework is available. start by defining the required topological theories, together with an additional functor and a natural transformation. proposition 6.7. there exist topological theories (1) set×loattc t att ω −−−−→ loc, where t attω ((x1,g1) (f,g) −−−→ (x2,g2)) = (ωg1) x1 ((f,(ω g)op)←)op −−−−−−−−−−−→ (ωg2) x2 ; (2) set × loc t att σ −−−−→ loc = set × loc (−)← −−−→ loc; together with a functor set×loattc katt×ω −−−−→set×loc, katt×ω ((x1,g1) (f,g) −−−→ (x2,g2)) = (x1 × | ωg1|,σg1) (f×(ωg)∗op,(σ g)op) −−−−−−−−−−−−−→ (x2 × | ωg2|,σg2) and a natural transformation t attς ◦ k att ×ω ((−)�) op −−−−−−→ t attω given by the maps of proposition 5.1. proof. since the definitions of the functors are straightforward, the only thing to verify is correctness of the definition of the natural transformation. consider a set×loattc-morphism (x1,g1) (f,g) −−−→ (x2,g2) and check commutativity of the diagram (ωg2) x2 (f,(ωg)op)← �� (−)�(x2,g2) // (σg2) x2×|ω g2| (f×(ω g)∗op,(σg)op)← �� (ωg1) x1 (−)�(x1,g1) // (σg1) x1×|σ g1|. given α ∈ (ωg2) x2 and (x,b) ∈ x1 × | ωg1|, ((−)�(x1,g1) ◦ (f,(ω g) op )←(α))(x,b) = (ωg ◦ α ◦ f)�(x,b) = (�1(ωg ◦ α ◦ f(x)))(b) = (σg ◦ �2(α ◦ f(x))((ω g) ∗op))(b) = σg ◦ α�(f(x),(ω g)∗ op (b)) = (σg ◦ α� ◦ (f × (ωg)∗ op ))(x,b) = ((f × (ωg)∗ op ,(σg) op )← ◦ (−)�(x2,g2)(α))(x,b). � 124 a. frascella, c. guido and s. a. solovyov propositions 6.6, 6.7 give a functor top(t attω ) h�op −−−→ top(t attς ), which is essentially (up to the change of the notation for the underlying variety) the functor loattc-top hatt −−−−→ loc-top of definition 2.17. in a similar way, one obtains the functor top(t attω ) h op −−−→ top(t attς ), which essentially provides the functor loattb-top hatt −−−→ lob-top of definition 3.4. the reader should notice the significant difference between the ways of obtaining the functors hatt, hatt by propositions 6.6, 6.7 and in definitions 2.17, 3.4, the latter relying on the framework of topological systems, whereas the former being based explicitly on catalg topology. it is up to the reader to decide, which way is more applicable in his/her framework. we would just like to notice that the final results of this section clearly show one of the main advantages of the notion of attachment, i.e., the fact that it provides a way of moving (natural transformation) between two topological theories, resulting in a functor between the categories of the respective topological structures. 7. categorically-algebraic attachment the attentive reader will easily notice that although the definition of objects of the category attb of definition 2.4 provides a straightforward generalization of the attachment notion of [26], the definition of morphism comes essentially “out of the blue”, the only its justification being the existence of the functor loattb-top eatt −−−−→ lob-topsys of proposition 2.13. the simple reason for the occurrence is the fact that the case of attachment morphisms has never been treated in [25, 26] and, therefore, there is actually nothing to compare with. it is the main goal of this section to provide a more trustworthy justification for the definition of morphisms of the category attb. we start with some new functors, which will be used in the subsequent procedures. for the sake of convenience, in what follows, we change the notation (kept until now) for the underlying variety of attb from b to a. proposition 7.1. given a variety a, there exists a functor setop×a (−)↼ −−−→ a defined by the formula ((x1,a1) (f,ϕ) −−−→ (x2,a2)) ↼ = ax11 (f,ϕ)↼ −−−−→ ax22 with (f,ϕ)↼(α) = ϕ◦α◦fop. the new functor satisfies the equality setop×a (−)↼ −−−→ a = (set × loa (−)← −−−→ loa)op. proof. correctness of the definition of the functor follows from the last claim (backed by proposition 2.6), which is a consequence of the fact that given α ∈ ax11 , (f,ϕ) ↼(α) = ϕ ◦ α ◦ fop = (fop,ϕop)←(α). � to underline the motivating setting of the functor (−)←, the new one uses a similar notation (−)↼. the other functors are collected in the next definition. dual attachment pairs in categorically-algebraic topology 125 definition 7.2. every variety a equipped with a functor a (−)∗ −−−→ setop such that a∗ = |a|, gives rise to the following three functors: (1) a × a π1 −−→ a, π1((a1,a ′ 1) (ϕ,ψ) −−−→ (a2,a ′ 2)) = a1 ϕ −→ a2 (the first projection functor); (2) a × a k −→ setop × a = a × a (−)∗×1a −−−−−−→ setop × a; (3) setop × a p −→ a = setop × a (−)↼ −−−→ a. the next definition shows a more general approach to attachment, with the notion of morphism coming from the existing framework of comma categories. definition 7.3. att ∗ a is the full subcategory of the comma category (π1 ↓ p ◦ k), whose objects are precisely those (π1 ↓ p ◦ k)-objects π1(a1,a2) ϕ −→ p ◦ k(a′1,a ′ 2) for which (a1,a2) = (a ′ 1,a ′ 2). a natural question on the equivalence of definition 7.3 and definition 2.4 arises. it is the purpose of the next result to answer it positively. proposition 7.4. the categories atta and att∗a are isomorphic. proof. an att∗a-object g is a triple (ωg,σg,�), where (ωg,σg) is the object of the product category a × a and π1(ωg,σg) � −→ p ◦ k(ωg,σg) = ωg � −→ (σg)|ω g| is an a-homomorphism. an att∗a-morphism g1 f −→ g2 is a pair of a-homomorphisms (ωg1,σg2) (ω f,σ f) −−−−−−→ (ωg2,σg2) making the following diagram commute: π1(ωg1,σg1) π1(f) �� 1 // p ◦ k(ωg1,σg1) p◦k(f) �� π1(ωg2,σg2) �2 // p ◦ k(ωg2,σg2) that in its turn provides commutativity of the next diagram: ωg1 ωf �� 1 // (σg1) |ω g1| ((ω f)∗,σ f)↼ �� ωg2 �2 // (σg2) |ω g2|, which is equivalent to the fact that given b1 ∈ ωg1 and b2 ∈ ωg2, it follows that (�2 ◦ ωf(b1))(b2) = ((((ω f) ∗,σf)↼ ◦ �1)(b1))(b2) = σf ◦ �1(b1) ◦ (ωf)∗ op (b2) = (σf ◦ �1(b1))((ω f) ∗op(b2)). taking together, the remarks provide an isomorphism atta k −→ att∗a, k(g1 f −→ g2) = g1 f −→ g2. � having introduced an equivalent definition of attachment, we are going to generalize the results from the end of the last section to the new framework, 126 a. frascella, c. guido and s. a. solovyov namely, derive a respective natural transformation between topological theories. now, however, we would like to provide a more explicit description of the machinery employed. to make the things easier, we begin with several additional properties of powerset operators. proposition 7.5. given a variety a and a set x, there exists a functor a (−)x→ −−−→ a, (a1 ϕ −→ a2) x → = a x 1 ϕ x → −−→ ax2 with ϕ x →(α) = ϕ ◦ α. proof. to show that the functor is correct on morphisms (provides an ahomomorphism), notice that given λ ∈ λa and αi ∈ a x 1 for i ∈ nλ, it follows that (ϕx→(ω a x 1 λ (〈αi〉nλ)))(x) = ϕ◦ω a1 λ (〈αi(x)〉nλ) = ω a2 λ (〈ϕ ◦ αi(x)〉nλ) = (ω a x 2 λ (〈ϕ ◦ αi〉nλ))(x) = (ω a x 2 λ (〈ϕx→(αi)〉nλ))(x) for every x ∈ x. � notice that the morphism action of the functor of proposition 7.5 has already been considered in [69, 70], where it has been observed (following, e.g., [56, 59]) that the variable-basis functor of proposition 2.6 splits up as follows: a x2 2 f ← a2 �� (ϕop)x2→ // (f,ϕ)← '' a x2 1 f ← a1 �� a x1 2 (ϕop)x1→ // ax11 . in view of proposition 7.5 and the above-mentioned remark, every commutative diagram in a variety a a1 ψ1 �� ϕ1 // a′1 ψ2 �� a2 ϕ2 // a′2 and every map x1 f −→ x2, provide the following commutative diagram: (7.1) ax21 (f,ψ op 1 ) ← '' (ψ1) x2 → �� (ϕ1) x2 → // a′1 x2 (ψ2) x2 → �� (f,ψ op 2 ) ← xx a x2 2 f ← a2 �� (ϕ2) x2 → // (f,ϕ op 2 ) ← n n n n n ''nn n n n a′2 x2 f ← a′ 2 �� a x1 2 (ϕ2) x1 → // a′2 x1. the last needed property of powerset operators is in the next proposition. dual attachment pairs in categorically-algebraic topology 127 proposition 7.6. for a set×loa×loa-morphism (x1,a1,b1) (f,ϕop,ψop) −−−−−−−→ (x2,a2,b2), there exist a-homomorphisms (b |ai| i ) xi θi −−→ b xi×|ai| i defined by (θi(α))(x,a) = (α(x))(a), making the following diagram commute: (b |a2| 2 ) x2 (f,((ϕ∗,ψ)↼)op)← �� θ2 // b x2×|a2| 2 (f×ϕ∗op,ψop)← �� (b |a1| 1 ) x1 θ1 // b x1×|a1| 1 . proof. there are two simple challenges to deal with. to show that θi is an a-homomorphism, notice that given λ ∈ λa and αj ∈ (b |ai| i ) xi for j ∈ nλ, it follows that (θi(ω (b |ai| i )xi λ (〈αj〉nλ)))(x,a) = ((ω (b |ai| i )xi λ (〈αj〉nλ))(x))(a) = (ω b |ai| i λ (〈αj(x)〉nλ))(a) = ω bi λ (〈(αj(x))(a)〉nλ) = ω bi λ (〈(θi(αj))(x,a)〉nλ) = (ω b xi×|ai| i λ (〈θi(αj)〉nλ))(x,a) for every (x,a) ∈ xi×|ai|. commutativity of the above-mentioned diagram follows from the fact that given α ∈ (b |a2| 2 ) x2 and (x1,a1) ∈ x1 × |a1|, ((θ1 ◦ (f,((ϕ ∗,ψ)↼) op )←)(α))(x1,a1) = (θ1((ϕ ∗,ψ)↼ ◦ α ◦ f))(x1,a1) = ((ϕ ∗,ψ)↼ ◦ α ◦ f(x1))(a1) = ψ ◦ (α ◦ f(x1)) ◦ ϕ ∗op(a1) = ψ ◦ (θ2(α))(f(x1),ϕ ∗op(a1)) = ψ◦θ2(α)◦(f ×ϕ ∗op)(x1,a1) = ((f ×ϕ ∗op,ψop)←◦ θ2)(x1,a1). � everything in its place, we consider the analogues for our current setting of the functors of proposition 6.7, which are set × loatt∗a t att ω −−−−→ loa, set × loatt∗a katt×ω −−−−→ set × loa and set × loa t att σ −−−−→ loa. our aim is to provide an equivalent description of the natural transformation between them given in proposition 6.7. the new approach is bound to clarify its nature, vaguely touched in proposition 5.1. proposition 7.7. there is a natural transformation t attς ◦ k att ×ω ((−)�) op −−−−−−→ t attω defined by t att ω (x,g) (−)�(x,g) −−−−−−→ t attς ◦ k att ×ω (x,g) = (ωg) x � x −−→ (σg|ω g|)x θ −→ σgx×|ωg|, where the map θ comes from proposition 7.6. proof. the claim is a consequence of the fact that every set × loatt∗amorphism (x1,g1) (f,g) −−−→ (x2,g2) makes the following diagram commute: (ωg2) x2 (f,(ωg)op)← �� x2 2 // ((σg2) |ω g2|)x2 (f,(((ωg)∗,σg)↼)op)← �� θ2 // (σg2) x2×|ωg2| (f×(ωg)∗op,(σ g)op)← �� (ωg1) x1 � x1 1 // ((σg1) |ω g1|)x1 θ1 // (σ g1) x1×|ω g1|, 128 a. frascella, c. guido and s. a. solovyov where the left rectangle uses commutativity of the second diagram of proposition 7.4 in the form of resulting diagram (7.1), and the right rectangle is a direct consequence of proposition 7.6. � the reader should pay attention to the fact that the new description clarifies the categorical background of the natural transformation in question, which was defined in an algebraic way in propositions 5.1, 6.7. categorical approach brings more universality in play, reducing dramatically the algebraic dependence on points, thereby opening a possibility to define attachment in a more general category than set. it will be the subject of our forthcoming papers to provide such an extended definition of attachment. 8. conclusion: lattice-valued categorically-algebraic topology the notion of dual attachment introduced in this paper clarified completely the categorical nature of the concept of attachment considered in [25, 26, 73]. as the main achievement, we showed that it provides a way of interaction (natural transformation) between two topological theories (propositions 6.7, 7.7), which results in a functor between the respective categories of topological structures (see remarks after proposition 6.7). it was the framework of categoricallyalgebraic (catalg) topology, which helped to discover this important property. moreover, we have already remarked (see, e.g., example 6.5) that the catalg approach incorporates the majority of the existing (lattice-valued) topological settings, erasing the border between crisp and many-valued frameworks. a significant drawback of the concept, however, is its inability to include the theory of (l,m)-fuzzy topological spaces of t. kubiak and a. šostak [40]. the striking difference of their approach from those used in the paper is the fact that a topological space is defined as a pair (x,t), where t is a lattice-valued subalgebra of t(x) for an appropriate topological theory t of definition 6.3. this observation in hand, the concept of lattice-valued catalg topology has been introduced in [65], based in a suitably defined notion of lattice-valued algebra. the latter notion has already appeared in [68], motivated by the concept of fuzzy group of a. rosenfeld [63] and its generalization of j. m. anthony and h. sherwood [4]. the employed machinery goes in line with the general procedure of, e.g., j. n. mordeson and d. s. malik [47] as follows. definition 8.1. let a, l be varieties, the latter having the variety cslat( ∨ ) as a reduct, and let c be a subcategory of l. an (a,c)-algebra is a triple (a,µ,l), where a is an a-algebra, l is a c-algebra and |a| µ −→ |l| is a map such that for every λ ∈ λ and every 〈ai〉nλ ∈ a nλ, ∧ i∈nλ µ(ai) ≤ µ(ω a λ (〈ai〉nλ)). an (a,c)-homomorphism (a1,µ1,l1) (ϕ,ψ) −−−→ (a2,µ2,l2) is an a × c-morphism (a1,l1) (ϕ,ψ) −−−→ (a2,l2) such that ψ ◦ µ1(a) ≤ µ2 ◦ ϕ(a) for every a ∈ a1. c-a is the category of (a,c)-algebras and (a,c)-homomorphisms, which is concrete over the product category a × c. dual attachment pairs in categorically-algebraic topology 129 an important moment arising from the definition should be underlined at once. in [15], a. di nola and g. gerla introduced the category c(τ) as a “general approach to the theory of fuzzy algebras”. it is easy to see that c(τ) is isomorphic to the subcategory clat-alg(ωωω) = of clat-alg(ωωω), with the same objects and with morphisms (a1,µ1,l1) (ϕ,ψ) −−−→ (a2,µ2,l2) satisfying the identity ψ ◦ µ1 = µ2 ◦ ϕ (reflected in the notation “(−) =”). moreover, [15] started to develop the theory of fuzzy universal algebra, some results of which can be easily extended to our approach. bound to the topological nature of this paper, we will only notice that the category c-a provides a more appropriate fuzzification of universal algebra, which fuzzifies not only algebras, but also (and that is more important) their respective homomorphisms. the related topological stuff is an easy modification of definitions 6.3, 6.4. definition 8.2. let t be a vt-theory in a category x, let l be a variety having cslat( ∨ ) as a reduct, and let l be a subcategory of lol. an l-valued vttheory in x induced by t and l is the pair (t,l). definition 8.3. let (t,l) be an l-valued vt-theory in a category x. ltop(t) is the category, concrete over x × l, whose objects (l-valued t-topological spaces) are triples (x,t,l), comprising an x-object x, an l-object l and a (b,lol)-algebra (t(x),t,l) (l-valued t-topology on x), and whose morphisms (x,t,l) (f,ψ) −−−→ (y,s,m) are those x × l-morphisms (x,l) (f,ψ) −−−→ (y,m), which satisfy the property of (t(x),t,l) (t(f),ψ) −−−−−→ (t(y ),s,m) being a lo(lol-b)-morphism (l-valued t-continuity). it appears that lattice-valued catalg topology is truly a universal one, incorporating all (up to the knowledge of the authors) existing topological settings (including the catalg one). the following examples justify the fruitfulness of the new notion (the reader should notice that an l-valued vt-theory (t,l) is occasionally denoted by (p,b,lol), to underline its building blocks). example 8.4. (1) ltop((ssl clat ,sfrm,s cdclat m )), where clat is the variety of complete lattices and cdclat is its subcategory of completely distributive lattices, provides a categorical accommodation of the theory of (l,m)fuzzy topological spaces of t. kubiak and a. šostak [40]. (2) ltop((r3,frm,frm)) is isomorphic to the category loc-f 2 top of (l,m)-fuzzy topological spaces of j. t. denniston, a. melton and s. e. rodabaugh [11], which was introduced as a variable-basis counterpart of the above-mentioned approach of t. kubiak and a. šostak. (3) ltop((p,frm,sdmlocl )) provides the approach of u. höhle [30]. (4) ltop(t) for l = s cslat( ∨ ) 2 is isomorphic to the category top(t) introduced in definition 6.4. in view of the above-mentioned remarks, it seems natural to consider the notion of attachment in the more general lattice-valued framework, and, therefore, one can postulate the following open problem. 130 a. frascella, c. guido and s. a. solovyov problem 8.5. what will be the concept of lattice-valued catalg attachment? it will be the topic of our further research to extend the already developed framework to the new setting. table of categories a, b, c: varieties of algebras. 107 alg(ωωω): ωωω-algebras. 106 atta: variety-based attachments. 103 attb: dual variety-based attachments. 108 att ∗ a: (π1 ↓ p ◦ k)-objects π1(a1,a2) ϕ −→ p ◦ k(a′1,a ′ 2) for which (a1,a2) = (a ′ 1,a ′ 2). 125 c-a: lattice-valued algebras. 128 cbool: complete boolean algebras. 120 (c,d)-topsys: variable-basis variety-based topological systems. 111 chu(set,k): chu spaces. 111 clat: complete lattices. 107 cont: contexts. 111 csl: closure semilattices. 107 cslat(ξ): ξ-semilattices. 106 c-top: variable-basis variety-based topological spaces. 109 dmsquant: demorgan semi-quantales. 106 frm: frames. 106 fuzlat: dual of hut. 121 hut: hutton algebras. 121 intsys: interchange systems. 111 loa: the dual category of a variety a. 107 loattb-top ∅k: non-empty stratified variable-basis variety-based topological spaces. 113 loc: locales. 107 l-top: fixed-basis lattice-valued topological spaces. 103 ltop(t): lattice-valued topological spaces induced by a topological theory t . 129 qfrm: quasi-frames. 106 quant: quantales. 106 sa: subcategory of loa whose only morphisms is the identity 1a. 107 set: sets. 102 sfrm: semi-frames. 106 squant: semi-quantales. 106 top: topological spaces. 103 top(t): topological spaces induced by a topological theory t . 121 usquant: unital semi-quantales. 106 131 references [1] j. adámek, h. herrlich and g. e. strecker, abstract and concrete categories: the joy of cats, dover publications, 2009. 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(received december 2010 – accepted march 2011) 134 anna frascella (frascella anna@libero.it) department of mathematics “e. de giorgi”, university of salento, p. o. box 193, 73100 lecce, italy. cosimo guido (cosimo.guido@unisalento.it) department of mathematics “e. de giorgi”, university of salento, p. o. box 193, 73100 lecce, italy. sergey a. solovyov (sergejs.solovjovs@lu.lv, sergejs.solovjovs@lumii.lv) department of mathematics, university of latvia, zellu iela 8, lv-1002 riga, latvia. institute of mathematics and computer science, university of latvia, raina bulvaris 29, lv-1459 riga, latvia. dual attachment pairs in categorically-algebraic[2pt] topology. by a. frascella, c. guido and s. a. solovyov reviewagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 311-313 review of the book of vladimir kovalevsky “geometry of locally finite spaces” sergei matveev the title of the book may create an impression that it is for geometers and topologists. that is right, but i think that the main purpose of the book is to construct a broad road for specialists in digital geometry, image recognition, and other branches of computer and applied mathematics. where to? into the world of computational and algorithmic topology. topological notions such as surface, manifold, connectedness, boundary, orientation, dimension unavoidably appear when you start to think about basic theoretical problems in the above-mentioned branches of science. however, existing topological books are devoted to the purely theoretical interior problems of topology and are very far from practical applications. the theory of locally finite spaces presented in the 330-page monograph of professor vladimir kovalevsky fills this gap. the theoretic part of the book contains numerous new definitions and theorems which express in a form understandable for non-specialists applications of those notions of set topology which are needed for computer imagery. the part devoted to applications presents some algorithms for investigating topological and geometrical properties of digitized twoand three-dimensional images. the book assumes a certain level of preparation, but some elementary knowledge in topology and in image processing would be sufficient. one of the important features of this book is that it provides a means which gives the possibility to overcome the existing discrepancy between theory and applications: the traditional way of research consists in making theory in euclidean space with real coordinates while applications deal only with finite discrete sets and rational numbers. the reason is that even the smallest part of the euclidean space cannot be explicitly represented in a computer and computations with irrational numbers are impossible since there exists no arithmetic of irrational numbers. the author demonstrates that locally finite spaces are explicitly representable in a computer, only rational coordinates are used and the theory of those spaces is in 312 s. matveev accordance with classical topology. the book consists of 14 chapters: an introductory section followed by thirteen main sections. the introductory section presents a short retrospect to the origin of the book followed by an overview of the contents and of the aims of the monograph. section 2 presents a new set of axioms and a proof that classical axioms of a topological space follow from the new axioms as theorems. this means that a locally finite space satisfying the new axioms (alf spaces) is a particular case of a classical space: it is a t0 alexandroff space which is not a t1 space. i would like to stress that the axioms are actually designed to applications. in particular, they are very convenient while working with computer presentations of geometric objects. section 3 is devoted to the theory of the spaces under consideration. it contains numerous theorems about the properties of alf spaces and definitions of balls, spheres and of the dimension of the space elements. these definitions are important for describing combinatorial homeomorphisms between spaces. this section also contains a new generalization of the orientation of simplicial complexes to the general case of abstract complexes and a generalization of the classical notion of a boundary. section 4 considers mappings among locally finite spaces. the author has demonstrated that classical homeomorphisms based on continuous maps being applied to locally finite spaces degrade to isomorphisms. he suggests to replace them by a much more convenient notion of connectedness preserving correspondences (cpms), which can map one space element to many. he also has demonstrated that a combinatorial homeomorphism based on elementary subdivisions of space elements uniquely defines a continuous cpm whose inverse is also continuous. sections 6 to 9 are devoted to a new concept of digital geometry which reflects euclidean geometry numerically. there is among others a new and complete theory of digital straight segments being considered as one-dimensional complexes rather than sequences of pixels. a digital straight segment is considered here as a subset of the boundary of a digital half-plane rather than as digitization of a euclidean straight line. this theory leads to interesting efficient applications to image analysis (section 11.3). sections 10 to 13 are devoted to applications. they contain descriptions of numerous algorithms based on the theory of locally finite spaces. there are among them algorithms for tracing and encoding boundaries in twoand three-dimensional digital images, exactly reconstructing images from their boundary codes, labeling connected components, computing skeletons, constructing convex hulls and others. some sections of the book contain problems to be solved which will stimulate further research. particularly interesting is section 14 ”topics for discussion”. the author discusses here the possibility to avoid irrational numbers and to use finite differences instead of derivatives. he demonstrates the possibilities of his approach while presenting an inference of the taylor formula based on finite differences. the book is unique: it is the first one in its own way presenting a self-contained theory of locally finite spaces that is independent of the hausdorff topology. it also contains a concept of digital geometry that is independent of euclidean geometry. i do not know review 313 any similar books. the book is carefully and clearly written and contains numerous well-made excellent illustrations. it is understandable for those whose interests are close to digital geometry / topology as well as for specialists in topology craving for applications. this book would be very useful and guiding for students and researchers, or for anyone interested in digital topology, digital geometry and computer imagery. it also may be of interest for physicists working on quantum gravity since it presents a well founded theory of spaces that could serve as the base of this branch of physics. my main conclusion is that the book should certainly find its way into university libraries and onto many private book shelves. sergei matveev (matveev@csu.ru) corresponding member of ras department of mathematics, chelyabinsk state university, kashirin brothers street 129, chelyabinsk 454021, russia. bufemiagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 205-212 on the topology of generalized quotients józef burzyk, cezary ferens and piotr mikusiński abstract. generalized quotients are defined as equivalence classes of pairs (x, f ), where x is an element of a nonempty set x and f is an element of a commutative semigroup g acting on x. topologies on x and g induce a natural topology on b(x, g), the space of generalized quotients. separation properties of this topology are investigated. 2000 ams classification: primary 54b15, secondary 20m30, 54d55. keywords: generalized quotients, semigroup acting on a set, quotient topology, hausdorff topology. 1. preliminaries let x be a nonempty set and let s be a commutative semigroup acting on x injectively. for (x,ϕ), (y,ψ) ∈ x × s we write (x,ϕ) ∼ (y,ψ) if ψx = ϕy. this is an equivalence relation in x × s. finally, we define b(x,s) = (x × s)/∼, the set of generalized quotients. the equivalence class of (x,ϕ) will be denoted by x ϕ . elements of x can be identified with elements of b(x,s) via the embedding ι : x → b(x,s) defined by ι(x) = ϕx ϕ , where ϕ is an arbitrary element of s. the action of g can be extended to b(x,s) via ϕx ψ = ϕx ψ . if ϕx ψ = ι(y), for some y ∈ x, we will write ϕx ψ ∈ x and ϕ x ψ = y. for instance, we have ϕx ϕ = x. other properties of generalized quotients and several examples can be found in [2] and [4]. if x is a topological space and g is a commutative semigroup of continuous maps acting on x, equipped with its own topology, then we can define the product topology on x × g and then the quotient topology on b(x,s) = (x × g)/∼. 206 j. burzyk, c. ferens and p. mikusiński it is easy to show that the embedding ι : x → b(x,s) is continuous. moreover, the map x ψ 7→ ϕx ψ is continuous for every ϕ ∈ g. these and other topological properties of generalized quotients can be found in [1]. in this note we will always assume that the topology on g is discrete. in most examples, it is a natural assumption. let y be a topological space and let ∼ be an equivalence relation. if y ∈ y , then by [y] we denote the equivalence class of y, that is, [y] = {w ∈ y : w ∼ y}. the map q : y → y/∼, defined by q(y) = [y], is called the quotient map. a subset u ⊂ y is called saturated if y ∈ u implies [y] ⊂ u. in other words, u is saturated if u = q−1(q(u)). let z = y/∼. a set v ⊂ z is open (in the quotient topology) if and only if v = q(u) for some open saturated u ⊂ y . whenever convenient, we use convergence arguments. the sequential convergence defined by the topology of b(x,g) is not easily characterized. the following theorem is often useful. theorem 1.1. let xn ϕn ∈ b(x,g), n ∈ n. if there exist a ψ ∈ g and a y ∈ x such that xn ϕn = yn ψ , for all n ∈ n, and yn → y in the topology of x, then xn ϕn → y ψ in the topology of b(x,g). proof. if u is an open neighborhood of y ψ in b(x,g), then (y,ψ) ∈ q−1(u). since q−1(u) is open in x × g, there exists an open v ⊂ x such that (y,ψ) ∈ v ×{ψ} ⊂ q−1(u). but then yn ∈ v for almost all n ∈ n, because yn → y in the topology of x. hence, (yn,ψ) ∈ q −1(u) for almost all n ∈ n or, equivalently yn ψ = xn ϕn ∈ u for almost all n ∈ n. � in this note we investigate some separation properties of the topology of b(x,g). 2. general separation properties we are interested in the general question whether a separation property of x is inherited by b(x,g). first we consider t1. theorem 2.1. if x is t1 and the topology of g is discrete, then b(x,g) is t1. proof. if x ϕ ∈ b(x,g), then (x × g) \ q−1 ( x ϕ ) is an open saturated subset of x × g. � now we give an example of a banach space x and a semigroup g of continuous injections on x for which b(x,g) is not hausdorff. if f,g : r → r and the set {t ∈ r : f(t) 6= g(t)} is meager in the usual topology of r, then we will write f ≃ g. let b(r) be the space of all bounded real-valued functions on r and let x = b(r)/≃. with respect to the norm ‖[f]‖ = inf{‖g‖∞ : g ≃ f} x is a banach space. let g = {[f] ∈ x : {t ∈ r : f(t) = 0} is a meager set in r} . on the topology of generalized quotients 207 then g is a semigroup of injections acting on x by pointwise multiplication. note that b(x,g) can be identified with rr/≃. to show that the topology of b(x,g) is not hausdorff we need two simple lemmas. in what follows, we will not distinguish between functions and equivalence classes of functions. the indicator function of a set a will be denoted by ia. lemma 2.2. if (an) is a sequence of subsets of r such that an ⊂ an+1, for each n ∈ n, and r \ ⋃∞ n=1 an is meager, then for each f ∈ c(r) the sequence fn = fian is convergent to f in b(x,g). proof. define a function g : r → r as follows g(t) = { 1 if t ∈ a1, 1 n if t ∈ an \ an−1. it is easy to see that fng → fg in x. consequently fn → f in b(x,g). � corollary 2.3. if a set u ⊂ b(x,g) is sequentially open and f g ∈ u, then for each r ∈ r there exists a open neighborhood v ⊂ r of r such that fir\v g ∈ u. lemma 2.4. if (an) is a sequence of subsets of r such that an+1 ⊂ an, for each n ∈ n, and the set ⋂∞ n=1 an is meager, then for each f ∈ x the sequence (fn), where fn = fian , is convergent to 0 in b(x,g). proof. use g(t) = { 1 if t /∈ a1, 1 n if t ∈ an \ an+1. � theorem 2.5. if u is a nonempty sequentially open subset of b(x,g), then u is sequentially dense in b(x,g). proof. it is enough to prove that there exists a sequence fn ∈ u such that fn → 0 in b(x,g). consider an arbitrary element f/g ∈ u and assume that (rn) is a sequence of all rational numbers. then, by corollary 2.3, there exits a neighborhood v1 of r1 such, that f1 = fir\v1 g ∈ u. next we find a neighborhood v2 of r2 such, that f2 = fir\(v1∪v2) g ∈ u. by induction, we construct a sequence vn ⊂ r such that vn is a neighborhood of rn and fn = fir\(v1∪...∪vn) g ∈ u. the set ⋃∞ n=1 vn is open and dense in r. hence, the complement of ⋃∞ n=1 vn is a meager set. by lemma 2.4, fir\(v1∪...∪vn) → 0 in b(x,g), and consequently fn → 0 in b(x,g). � 208 j. burzyk, c. ferens and p. mikusiński since x in this example is a banach space, no separation property of x above t1 will be inherited by the topology of b(x,g) without additional assumptions. in the remaining part of this note we give examples of theorems that discribe special situations in which the topology of b(x,g) is hausdorff. 3. hausdorff property in special cases first we introduce some notation and make some useful observations. if u ⊂ x × g, then u = ⋃ ϕ∈g uϕ × {ϕ}, where uϕ ⊂ x. for every ψ ∈ g let πψ : x × g → x be the projection defined by πψ   ⋃ ϕ∈g uϕ × {ϕ}   = uψ. if a ⊂ x × g, then the smallest saturated set containing a will be denoted by σa. we have the following straightforward characterization on σa. proposition 3.1. if a ⊂ x × g, then σa = ⋃ ϕ,ψ∈g ϕ−1ψπϕa × {ψ}. in other words, for every ψ ∈ g, we have πψσa = ⋃ ϕ∈g ϕ−1ψπϕa. corollary 3.2. a set a ⊂ x × g is saturated if and only if ϕ−1ψπϕa ⊂ πψa for every ϕ,ψ ∈ g. theorem 3.3. if x is hausdorff and every ϕ ∈ g is an open map, then b(x,g) is hausdorff. proof. let x ϕ and y ψ be two distinct elements of b(x,g). it suffices to find open and saturated subsets of x × g that separate (x,ϕ) and (y,ψ). since ψx 6= ϕy and x is hausdorff, there exist open and disjoint sets u,v ⊂ x such that ψx ∈ u and ϕy ∈ v . define a = ψ−1u × {ϕ} and b = ϕ−1v × {ψ}. consider the sets σa and σb. by proposition 3.1, σa and σb are open sets. if (z,γ) ∈ πγσa, then z ∈ ϕ −1γψ−1u, again by proposition 3.1. this means that ϕz = γψ−1u for some u ∈ u. hence, (z,γ) ∼ (ψ−1u,ϕ). similarly, if (z,γ) ∈ πγσb, there exists a v ∈ v such that (z,γ) ∼ (ϕ −1v,ψ). therefore, (ψ−1u,ϕ) ∼ (ϕ−1v,ψ), which implies u = v, contradicting u ∩ v = ∅. � for topological spaces x and y , by c(x,y ) we denote the space of continuous maps from x to y . for a continuous ϕ : x → x, by ϕ∗ : c(x,y ) → c(x,y ) we denote the adjoint map, that is, (ϕ∗f)x = f(ϕx) where f ∈ c(x,y ). on the topology of generalized quotients 209 theorem 3.4. let x be a topological space, g a commutative semigroup of continuous injections from x into x, equipped with the discrete topology, such that ϕ(x) is dense in x for all ϕ ∈ g. let y be a hausdorff space and let f ⊂ c(x,y ) be such that f separates points in x and for every ϕ ∈ g we have f ⊂ ϕ∗(f). then the topology of b(x,g) is hausdorff. proof. first note that, since ϕ(x) is dense in x, ϕ∗ is a injection. for f ∈ f and ϕ ∈ g define fϕ to be the unique function in f such that ϕ ∗fϕ = f. then, for any ϕ,ψ ∈ g, we have ψ∗fψ = f = (ϕψ) ∗fϕψ and hence ψ ∗fψ = ψ ∗ϕ∗fϕψ. since ψ∗ is injective, we have fψ = ϕ ∗fϕψ. thus, fψ(x) = ϕ ∗fϕψ(x) = fϕψ(ϕx) for any x ∈ x. consider two distinct elements f1 and f2 of b. without loss of generality, we can assume that f1 = x1 ϕ and f2 = x2 ϕ , for some x1 6= x2. there exists an f ∈ f such that f(x1) 6= f(x2). let ω1, ω2 ⊂ y be open disjoint neighborhoods of f(x1) and f(x2), respectively. for every ψ ∈ g let uψ = ϕ −1 ( f−1 ψ (ω1) ) and vψ = ϕ −1 ( f−1 ψ (ω2) ) . we will show that u = ⋃ ψ∈g uψ × {ψ} and v = ⋃ ψ∈g vψ × {ψ} are disjoint saturated open sets that separate (x1,ϕ) and (x2,ϕ). it suffices to prove that the sets are saturated. since the sets are defined the same way, we will only prove it for u. suppose x ∈ uψ and (x,ψ) ∼ (y,γ). then γx = ψy and fγ(ϕy) = fψγ(ϕψy) = fψγ(ϕγx) = fψ(ϕx) ∈ ω1. thus y ∈ uγ. � example 3.5. let x = {x ∈ c(r) : x(0) = 0}, with the topology of uniform convergence on compact sets, and let g = {λn : n ∈ n0}, where λx(t) = ∫ t 0 x(s) ds and n0 denotes the set of all nonnegative integers. to show that the topology of b(x,g) is hausdorff we use theorem 3.4 with y = r and f = {f ∈ d(r) : f 6= 0}, where d(r) is the space of smooth functions with compact support. if f ∈ f and x ∈ x, then we define f(x) = ∫ ∞ −∞ f(t)x(t) dt. clearly, λn is injective and λn(x) is dense in x for every n ∈ n. moreover, f separates points in x. if f ∈ f and n ∈ n, then there exists a g ∈ f such that f(x) = g(λnx) for every x ∈ x, namely g = (−1)nf(n). thus all the assumptions of the theorem are met. the assumption that x(0) = 0, in the definition of x, may seem artificial. it is made for convenience and it does not affect the final result. note that for any x ∈ c(r) we have x λn = λx λn+1 and λx ∈ x. one can prove that, in general, b(x,g) = b(gx,g) for any g ∈ g (see [1]). in the next theorem we assume that g is generated by a single function, that is, g = {ϕn : n ∈ n0}. 210 j. burzyk, c. ferens and p. mikusiński proposition 3.6. let g = {ϕn : n ∈ n0} and a ⊂ x × g. a is saturated if and only if, for all i,j ∈ n0, (3.1) z ∈ πia if and only if ϕ jz ∈ πi+ja, where πk = πϕk . proof. assume that (3.1) holds for some a ⊂ x×g, x ∈ πna, and ϕ ny = ϕmx for some y ∈ x and m ∈ n0. if n ≤ m, then y = ϕ m−nx. hence, if we take j = m − n, i = n, and z = x, we obtain y = ϕm−nx ∈ πma, by (3.1). if n > m, then x = ϕn−my, and thus, ϕn−mx ∈ πna. hence y ∈ πm, by (3.1). therefore a is saturated. assume now that a ⊂ x × g is saturated. then, by corollary 3.2, we have ϕjπia ⊂ πi+ja. hence, if z ∈ πi, then ϕ jz ∈ πi+ja. now, conversely, if ϕjz ∈ πi+ja, then z ∈ πi since (z,ϕ i) ∼ (ϕjz,ϕi+j) and a is saturated. � corollary 3.7. if g = {ϕn : n ∈ n0}, then a ⊂ x × g is saturated if and only if πj−1a = ϕ −1πja for every j ∈ n. theorem 3.8. if x is a normal space, ϕ : x → x is a closed and continuous injection, and g = {ϕn : n ∈ n0}, then b(x,g) is a hausdorff space. proof. consider two distinct points in b(x,g). without loss of generality, we can assume that they are represented by x ϕn and y ϕn for some x,y ∈ x and n ∈ n. then x 6= y and there exist open sets un,vn ⊂ x such that x ∈ un, y ∈ vn, and un ∩ vn = ∅. since ϕ is a closed injective map, ϕ(un) and ϕ(vn) are disjoint closed sets. whereas x is normal, there exist open sets un+1,vn+1 ⊂ x such that ϕ(un) ⊂ un+1, ϕ(vn) ⊂ vn+1, and un+1 ∩ vn+1 = ∅. similarly, by induction, we can construct open sets un+k,vn+k ⊂ x such that ϕ(un+k) ⊂ un+k+1, ϕ(vn+k) ⊂ vn+k+1, and un+k+1 ∩ vn+k+1 = ∅, for all k = 1, 2, . . . . now, for m = n,n + 1,n + 2, . . . , we define open subsets of x × g: u′m = m ⋃ j=0 ( ϕj−mum ) × {ϕj} and v ′m = m ⋃ j=0 ( ϕj−mvm ) × {ϕj}. note that u′n ⊂ u ′ n+1 ⊂ . . . , v ′ n ⊂ v ′ n+1 ⊂ . . . , and u ′ m∩v ′ m = ∅ for all m ≥ n. finally, let u = ∞ ⋃ m=n u′m and v = ∞ ⋃ m=n v ′m. clearly, u and v are disjoint open subsets of x ×g such that (x,ϕn) ∈ u and (y,ϕn) ∈ v . since u and v are defined the same way, it suffices to show that on the topology of generalized quotients 211 u is saturated. note that πju = ∞ ⋃ m=n ϕj−mum if j = 0, . . . ,n, and πju = ∞ ⋃ m=j ϕj−mum if j > n. since πj−1u = ϕ −1πju for every j ∈ n, it follows that u is saturated by corollary 3.7. � corollary 3.9. if x is a compact hausdorff space and g is generated by a continuous injection, then b(x,g) is a hausdorff space. now we consider the case when x has an algebraic structure, namely x is a topological semigroup. a nonempty set x with an associative operation (x,y) → xy from x × x into x is called a semigroup. if the topology of x is hausdorff and the semigroup operation is continuous (with respect to the product topology on x × x), then x is called a topological semigroup. our main result follows from a theorem of lawson and madison (see theorem 1.56 in [3]). theorem 3.10 (lawson and madison). let s be a locally compact σ-compact semigroup and let r be a closed congruence on s. then s/r is a topological semigroup. an equivalence ∼ in a semigroup a is called a congruence if a ∼ b implies ca ∼ cb for all c ∈ a. if (x, ·) is a semigroup and g is a commutative semigroup of injective homomorphisms on x, then x × g is a semigroup with respect to the binary operation ∗ defined by (x,ϕ) ∗ (y,ψ) = ((ψx) · (ϕy),ϕψ), where x,y ∈ x and ϕ,ψ ∈ g. lemma 3.11. the equivalence ∼ in x × g defined by (x,ϕ) ∼ (y,ψ) if ψx = ϕy is a congruence with respect to ∗. proof. let (x,ϕ), (y,ψ), (z,γ) ∈ x × g and (x,ϕ) ∼ (y,ψ). then (x,ϕ) ∗ (z,γ) = ((γx) · (ϕz),ϕγ) and (y,ψ) ∗ (z,γ) = ((γy) · (ψz),ψγ). since ψx = ϕy and g is commutative, we have ψγ((γx) · (ϕz)) = (ψγγx) · (ψγϕz) = (ϕγγy) · (ϕγψz) = ϕγ((γy) · (ψz)), which means (x,ϕ) ∗ (z,γ) ∼ (y,ψ) ∗ (z,γ). � 212 j. burzyk, c. ferens and p. mikusiński a relation ∼ in a topological space y is called closed if {(a,b) ∈ y ×y : a ∼ b} is a closed subset of y × y with respect to the product topology. lemma 3.12. if x is hausdorff, then ∼ is a closed relation in x × g. proof. we have to show that the set r = {((x,ϕ), (y,ψ)) : (x,ϕ), (y,ψ) ∈ x × g and (x,ϕ) ∼ (y,ψ)} is closed in (x × g) × (x × g). consider ((x,ϕ), (y,ψ)) /∈ r. then (x,ϕ) 6∼ (y,ψ) and hence ψx 6= ϕy. since x is hausdorff, there are open and disjoint u,v ⊂ x such that ψx ∈ u and ϕy ∈ v . then (x,ϕ) × (y,ψ) ∈ (ψ−1(u) × {ϕ}) × (ϕ−1(v ) × {ψ}). clearly, (ψ−1(u) × {ϕ}) × (ϕ−1(v ) × {ψ}) is open and disjoint with r. � in view of the above lemmas, the theorem of lawson and madison gives us the following result. theorem 3.13. if x is a hausdorff semigroup and (x ×g) is locally compact σ-compact, then b(x,g) is hausdorff. references [1] d. bradshaw, m. khosravi, h. m. martin and p. mikusiński, on categorical and topological properties of generalized quotients, preprint. [2] j. burzyk and p. mikusiński, a generalization of the construction of a field of quotients with applications in analysis, int. j. math. sci. 2 (2003), 229–236. [3] j. h. carruth, j. a. hildebrant, and r. j. koch, the theory of topological semigroups (marcel dekker, new york, 1983). [4] p. mikusiński, generalized quotients with applications in analysis, methods appl. anal. 10 (2004), 377–386. received may 2007 accepted december 2007 józef burzyk institute of mathematics, technical university of silesia, kaszubska 23, 44-100, gliwice, poland cezary ferens (c.ferens@wp.pl) ul. batorego 77/1, 43-100 tychy, poland piotr mikusiński (piotrm@mail.ucf.edu) department of mathematics, university of central florida, orlando, fl 328161364, usa fujitakatoagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 239-244 almost periodic points and minimal sets in topological spaces chikara fujita and hisao kato abstract. in the paper [almost periodic points and minimal sets in ω-regular spaces, topology appl. 154 (2007), 2873–2879], mai and sun showed that several known results concerning almost periodic points and minimal sets of maps can be generalized from regular spaces to ωregular spaces. also, they have three unsolved problems. in this paper, we answer to all problems which remain unsolved in the paper of mai and sun. in fact we prove some general theorems which give counter examples of the problems1. 2000 ams classification: primary 54h20; secondary 54d10, 37b20, 37b35. keywords: almost periodic points, minimal sets, ω-regular space. 1. introduction in this section, we need the following terminology and concepts. let n = {0, 1, 2, ..., } be the set of natural numbers and let z = {0, ±1, ±2, ..., } be the set of integers. for a set a, |a| denotes the cardinality of the set a. if f : x → x is a map (=continuous function) of a topological space x, then f 0 = id and f n (n ≥ 1) denotes the composition with itself n times. the orbit of a point x ∈ x under f , denoted by o+(x, f ), is the set {f n(x)| n ∈ n}. also if f : x → x is a homeomorphism, then we put f −n = (f −1)n (n ≥ 1), where f −1 is the inverse of f . the two-sided orbit of a point x ∈ x under f , denoted by o±(x, f ), is the set {f n(x)| n ∈ z}. a point x ∈ x is called a periodic point of f if there exists a positive number n ∈ n such that f n (x) = x. a point x ∈ x is called an almost periodic point of f provided that for any neighborhood u of x in x, there exists n ∈ n such that {f n+i(x)| i = 0, 1, 2, ..., n} ⋂ u 6= φ for all n ∈ n. we denote the set of all almost periodic points of f by ap (f ). a subset w of x is invariant of f if w 6= φ and f (w ) ⊆ w . a subset w of x is a minimal set of f if w is a closed invariant set of f and w does not contain 1in [6], fedeli and le donne gave counter examples of the problems by different methods. 240 ch. fujita and h. kato any proper closed invariant set of f . a map f : x → x is minimal if x is a minimal set of f . it is well known that if f : s1 → s1 is an irrational rotation of the unit circle s1, then f is minimal. a topological space x is a t1-space if for any distinct points x and y in x, there exist open sets u and v of x such that x ∈ u , y ∈ v , y /∈ u and x /∈ v . a topological space x is a hausdorff space if for any distinct points x and y in x, there exist disjoint open sets u and v of x such that x ∈ u and y ∈ v . a topological space x is a regular space if for any closed subset w of x, any point x ∈ x − w , there exist disjoint open sets u and v such that x ∈ u and w ⊆ v . a topological space x is an ω-regular space if for any closed subset w of x, any point x ∈ x − w and any countable subset a of w , there exist disjoint open sets u and v such that x ∈ u and a ⊆ v . the following theorem is well known (see [1], [2] and [3]). theorem 1.1. let x be a compact hausdorff space and let f : x → x be a map. then the followings hold. (1) if x ∈ x is any almost periodic point of f , then cl(o+(x, f )) is a minimal set of f . (2) all points in any minimal set of f are almost periodic points. in [5], mai and sun showed that several results related to theorem 1.1 can be generalized from regular spaces to ω-regular spaces. in this paper, we answer to all problems which remain unsolved in the paper of mai and sun. consequently, we know that currently “ω-regular space” is the best concept in generalization. 2. closure of almost periodic orbits in hausdorff spaces in [5], mai and sun proved that if x is an ω-regular space and f : x → x is a map, then the closure of every almost periodic orbit of f is a minimal set of f . related to this result, they have the following problem (problem 2.5 of [5]). problem 2.1. is the closure of an almost periodic orbit in a hausdorff space a minimal set ? by use of zorn’s lemma, we see that if x is any compact hausdorff space and f : x → x is a map, then there is a minimal set of f . also, there are many kinds of minimal homeomorphisms f : x → x of some uncountable compact metric spaces x. we have the following theorem which gives a counter example of this problem. theorem 2.2. suppose that (x, t ) is an uncountable compact hausdorff space and f : (x, t ) → (x, t ) is a minimal homeomorphism. let λi (i = 1, 2) be any disjoint nonempty sets with |λ1| + |λ2| = |x|. then there exists a topology tb of x and points aλ ∈ x (λ ∈ λ1), bλ ∈ x (λ ∈ λ2) such that (1) (x, tb) is a hausdorff space with t ⊆ tb, (2) f : (x, tb) → (x, tb) is a homeomorphism with x = ap (f ), almost periodic points and minimal sets in topological spaces 241 (3) the family {o±(aλ, f )| λ ∈ λ1} ⋃ {o±(bλ, f )| λ ∈ λ2} is a decomposition of x, (4) cl(o+(aλ, f )) = x (λ ∈ λ1), o ±(bλ, f ) is a closed set of (x, tb) and hence cl(o±(bλ, f )) 6= x (λ ∈ λ2), which implies that f : (x, tb) → (x, tb) is not minimal, and (5) if λ ∈ λ2 and z ∈ x \ o ±(bλ, f ), then z and the closed set o ±(bλ, f ) can not be separated by any two open disjoint sets, in particular (x, tb) is not an ω-regular space. proof. note that for any x1, x2 ∈ x, o ±(x1, f ) ⋂ o±(x2, f ) 6= φ if and only if o±(x1, f ) = o ±(x2, f ). since each o ±(x, f ) is a countable set, we can choose aλ ∈ x (λ ∈ λ1), bλ ∈ x (λ ∈ λ2) such that the family {o±(aλ, f )| λ ∈ λ1} ⋃ {o±(bλ, f )| λ ∈ λ2} is a decomposition of x. we put kλ = o ±(bλ, f ) (λ ∈ λ2). we consider the topology tb on x as follows: for x 6∈ ⋃ λ∈λ2 kλ, we consider that x has the (open) neighborhood base b(x) = {u \ km | x ∈ u ∈ t , km = ⋃ λ∈m kλ, m ⊆ λ2}. for x ∈ kλ(x) (λ(x) ∈ λ2), we consider that x has the neighborhood base b(x) = {u \ km | x ∈ u ∈ t , km = ⋃ λ∈m kλ, m ⊆ λ2 \ {λ(x)}}. in fact, the family {b(x)| x ∈ x} satisfies the following properties. (1) if b ∈ b(x), then x ∈ b. (2) if b1, b2 ∈ b(x), then there is b ∈ b(x) such that b ⊆ b1 ⋂ b2. (3) if b ∈ b(x) and x′ ∈ b, there is b′ ∈ b(x′) such that b′ ⊆ b. hence we obtain the topology tb on x from the neighborhood bases b(x) (x ∈ x). in the definition of b(x), if we consider the case m = φ, we see that t ⊆ tb. hence (x, tb) is a hausdorff space such that kλ = o ±(bλ, f ) is a closed set for λ ∈ λ2. let z ∈ x \ kλ. suppose, on the contrary, that there exist open disjoint sets w1 and w2 of (x, tb) such that z ∈ w1, kλ ⊆ w2. we may assume that w1 = u \ km , where z ∈ u ∈ t , km = ⋃ λ∈m kλ for some m ⊆ λ2. since f : (x, t ) → (x, t ) is minimal, there is a point b ∈ o+(bλ, f ) ⊆ kλ such that b ∈ u . take an open neighborhood (u ′ \ km′ ) of b in (x, tb ) such that u ′ ⊆ u and (u ′ \ km′ ) ⊆ w2. choose a point aλ1 ∈ x (λ1 ∈ λ1). then we have a point a ∈ o +(aλ1 , f ) such that a ∈ u ′. we see that a ∈ (u \ km ) ⋂ (u ′ \ km′ ) ⊆ w1 ⋂ w2. this is a contradiction. since f (kλ) = kλ (λ ∈ λ2), we can easily prove that f : (x, tb) → (x, tb) is continuous, and hence f : (x, tb) → (x, tb) is homeomorphism. to avoid confusing, we express the homeomorphism f : (x, tb) → (x, tb) by fb. since f : (x, t ) → (x, t ) is minimal, by theorem 1.1 we see x = ap (f ). by the definition of the neighborhood bases b(x) (x ∈ x), we can easily prove that 242 ch. fujita and h. kato x = ap (fb). similarly, by use of the fact cl(o +(aλ, f )) = x, we can prove that for each λ ∈ λ1, cl(o +(aλ, fb)) = x. � 3. minimal sets in locally compact t1-spaces in [5], mai and sun proved that if x is a locally compact hausdorff space and f : x → x is a map, then each minimal set of f is compact. related to this result, they have the following problem (problem 3.10 of [5]). problem 3.1. let f : x → x be a map of a locally compact t1-space x. is each minimal set of f compact? we have the following theorem which gives a counter example of this problem. theorem 3.2. there exists a locally compact t1-space z and a map f : z → z such that (1) z is a minimal set of f , (2) z = ap (f ), and (3) z is not compact. proof. let x and y be arbitrary infinite sets. consider the cofinite topology tx on x, i.e., tx = {u| u ⊆ x and |x \ u| < ∞} ⋃ {φ}. it is well known that (x, tx ) is a compact t1-space. let z = x ⋃ y , where x ⋂ y = φ. we consider the following topology tz on z. for z ∈ x ⊆ z, z has the neighborhood base b(z) = {u| z ∈ u ∈ tx}. for z ∈ y , we consider that z has the neighborhood base b(z) = {{z} ⋃ u| u 6= φ and u ∈ tx }. in fact, the family {b(z)| z ∈ z} satisfies the following properties. (1) if v ∈ b(z), then z ∈ v . (2) if v1, v2 ∈ b(z), then there is v ∈ b(z) such that v ⊆ v1 ⋂ v2. (3) if v ∈ b(z) and z′ ∈ v , there is w ∈ b(z′) such that w ⊆ v . hence we obtain the topology tz on z from the neighborhood bases b(z) (z ∈ z). note that (z, tz ) is a locally compact t1-space and it is not compact, because that y is an infinite set. we can easily see that if z ∈ y and u is a neighborhood of z in (z, tz ), then u ⋂ x 6= φ. take an arbitrary function g : x → x such that g has no periodic point and |g−1(x)| is finite for each x ∈ x. since g is a finite-to-one function, we see that g : (x, tx ) → (x, tx ) is continuous. define a map f : z → z by f (z) = g(z) for each z ∈ x, and f (z) = h(z) for each z ∈ y , where h : y → x is an arbitrary function. then we can easily see that f : (z, tz ) → (z, tz ) is continuous. since g has no periodic point, we see that cl(o+(z, f )) = z for each z ∈ z and z = ap (f ). note that z = (z, tz ) is a (unique) minimal set of f and z is a locally compact t1-space, but z is not compact. � almost periodic points and minimal sets in topological spaces 243 4. points in closures of almost periodic points in hausdorff spaces in [5], mai and sun proved that if x is an ω-regular space and f : x → x is a map, then all points in the closure of any almost periodic orbit of f are almost periodic. related to this result, they have the following problem (problem 4.4 of [5]). problem 4.1. does the closure of any almost periodic orbit in a hausdorff space contain only almost periodic points? we have the following theorem which gives a counter example of this problem. theorem 4.2. suppose that (x, t ) is an uncountable compact hausdorff space and f : (x, t ) → (x, t ) is a minimal homeomorphism. let λi (i = 1, 2) be any disjoint nonempty sets with |λ1| + |λ2| = |x|. then there exists a topology tb of x and points aλ ∈ x (λ ∈ λ1), bλ ∈ x (λ ∈ λ2) such that (1) (x, tb) is a hausdorff space with t ⊆ tb, (2) f = ftb : (x, tb) → (x, tb) is a homeomorphism, (3) the family {o±(aλ, f )| λ ∈ λ1} ⋃ {o±(bλ, f )| λ ∈ λ2} is a decomposition of x, and (4) o±(aλ, f ) ⊆ ap (f ) and cl(o +(aλ, f )) = x for λ ∈ λ1, (5) o±(bλ, f )) is a discrete closed set of (x, tb) and o ±(bλ, f ) ⋂ ap (f ) = φ for λ ∈ λ2, (6) if λ ∈ λ2 and z ∈ x \ o ±(bλ, f ), then z and the closed set o ±(bλ, f ) can not be separated by any two open disjoint sets, in particular (x, tb) is not an ω-regular space. proof. as in the proof of theorem 2.2, we obtain points aλ ∈ x (λ ∈ λ1), bλ ∈ x (λ ∈ λ2) such that the family {o±(aλ, f )| λ ∈ λ1} ⋃ {o±(bλ, f )| λ ∈ λ2} is a decomposition of x. we put kλ = o ±(bλ, f ) (λ ∈ λ2). we consider the topology tb on x as follows: for x 6∈ ⋃ λ∈λ2 kλ, we consider that x has the (open) neighborhood base b(x) = {u \ km | x ∈ u ∈ t , km = ⋃ λ∈m kλ, m ⊆ λ2}. for x ∈ kλ(x) (λ(x) ∈ λ2), we consider that x has the neighborhood base b(x) = {(u \ km ) ⋃ {x}| x ∈ u ∈ t , km = ⋃ λ∈m kλ, λ(x) ∈ m ⊆ λ2}. in fact, the family {b(x)| x ∈ x} satisfies the following properties. (1) if b ∈ b(x), then x ∈ b. (2) if b1, b2 ∈ b(x), then there is b ∈ b(x) such that b ⊆ b1 ⋂ b2. (3) if b ∈ b(x) and x′ ∈ b, there is b′ ∈ b(x′) such that b′ ⊆ b. 244 ch. fujita and h. kato hence we obtain the topology tb on x from the neighborhood bases b(x) (x ∈ x). by the definition of b(x), we see that t ⊆ tb. hence (x, tb) is a hausdorff space. also, by the definition of b(x), kλ = o ±(bλ, f ) is a discrete closed set for λ ∈ λ2. since f has no periodic point, we see that o ±(bλ, f ) ⋂ ap (f ) = φ for λ ∈ λ2. as in the proof of theorem 2.2, we see that o ±(aλ, f ) ⊆ ap (f ) and cl(o+(aλ, f )) = x for λ ∈ λ1. also we see that each point z ∈ x \ o ±(bλ, f ) and the closed set o±(bλ, f ) can not be separated by any two open disjoint sets, in particular (x, tb) is not an ω-regular space. this completes the proof. � references [1] g. d. birkhoff, dynamical systems, revised editions, amer. math. soc. colloq. publ., vol. ix, amer. math. soc., providence, ri, 1966. [2] w. h. gottschalk, orbit-closure decompositions and almost periodic properties, bull. amer. math. soc. 50 (1944), 915–919. [3] w. h. gottschalk, almost periodic points with respect to transformation semi-groups, ann. math. 47 (1946), 762–766. [4] j. l. kelley, general topology, springer-verlag, new york, 1975. [5] j.-h. mai and w.-h. sun, almost periodic points and minimal sets in ω-regular spaces, topology appl. 154 (2007), 2873–2879. [6] a. fedeli and a. le donne, on almost periodic orbits and minimal sets, topology appl. 156 (2008), no. 2, 473–475. received september 2008 accepted may 2009 chikara fujita (chikara@math.tsukuba.ac.jp) institute of mathematics, university of tsukuba, ibaraki, 305-8571 japan. hisao kato (hisakato@sakura.cc.tsukuba.ac.jp) institute of mathematics, university of tsukuba, ibaraki, 305-8571 japan. () @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 81-94 extensions defined using bornologies alessandro caterino and m. cristina vipera abstract many extensions of a space x such that the remainder y is closed can be constructed as b-extensions, that is, by defining a topology on the disjoint union x ∪ y , provided there exists a map, satisfying some conditions, from a basis of y into the family of the subsets of x which are “unbounded” with respect to a given bornology in x. we give a first example of a (nonregular) extension with closed remainder which cannot be obtained as b-extension. extensions with closed discrete remainders and extensions whose remainders are retract are mostly considered. we answer some open questions about separation properties and metrizability of b-extensions. 2010 msc: 54d35, 54d10, 54d20. keywords: boundedness, bornology, topological extension, b-extension. 1. introduction. the construction of the one-point compactification of a locally compact space can be generalized by taking as open neighborhoods of the new point the complements of the closed members of any boundedness. a nonempty family fx of subsets of a space x is said to be a boundedness if it is closed with respect to subsets and finite unions. fx is said to be closed (open) if every bounded set, that is, every member of fx, is contained in a closed (resp. open) bounded set ([10]). the one-point extension naturally associated to a boundedness fx in x is denoted by o(fx). in order to obtain t1 one-point extensions of a t1 space x we need that all singletons of x are bounded. in this case the boundedness is also called bornology. it turns out that all possible t1 one-point extensions of x can be defined in this way. in fact, every extension ax of x determines a closed bornology hx(ax) 82 a. caterino and m. c. vipera in x, namely the family of sets whose closure in x is also closed in ax. if ax is a one-point extension, then ax is equivalent to o(hx(ax)). if x is hausdorff, then o(fx) will be hausdorff provided every point has a bounded closed neighborhood. it is known that every hausdorff n-point compactification x ∪{x1. . . . , xn} can be obtained associating to every xi an open non-relatively compact subset ui of x, where ui∩uj = ∅ and x \( ⋃ ui) is compact. using a suitable bornology, an analogous result was proved for hausdorff n-point extensions ([5]). it is natural to try a generalization of this kind of construction to other hausdorff extensions. the question is whether it is possible to obtain every extension ax of x, in which x is open, using a bornology in x and some kind of correspondence between a basis of the remainder y = ax \ x and the family of unbounded open subsets of x. this idea inspired the construction of the so-called esh-compactifications, first defined in [4]. the authors used an essential semilattice homomorphism π from a basis of a compact space k into the family of nonrelatively compact open subsets of a locally compact space x to obtain a compactification of x whose remainder is homeomorphic to k. large families of compactifications (in most cases even the stone-čech compactification) can be obtained in this way. the word “essential” stands for “up to relatively compact sets”. this construction can also be generalized using different (closed) bornologies. an extension which can be obtained in this way is said to be a b-extension. in ([5]) b-extensions were first introduced and mostly used in order to construct lindelöf extensions of locally lindelöf spaces. two questions naturally arise: which extensions are b-extensions which conditions a bornology fx must satisfy to produce a b-extension ax = x ∪y preserving some specific topological property of x and y . in this paper we give a first example of a (non-regular) extension which is not a b-extension. we do not know if every regular extension can be obtained as b-extension. the problem is still open even for compactifications. in section 5 we will show that every regular extension such that the remainder is a retract is a b-extension. some results are known about the second question. if fx is a closed bornology in a regular (normal) space x, then the extension o(fx) is regular (resp. normal) if and only if fx is open ([12]). it was proved in ([6]) that o(fx) is tychonoff if and only if x is tychonoff and fx is functionally open. this means that for every (closed) f ∈ fx there is an open w ∈ fx such that f and x \ w are completely separated. this concept was used in [3] to obtain a topological and bornological immersion of a tychonoff space into a cube. it is also known that o(fx) is metrizable if and only x is metrizable and fx is induced by a metric ([2], [6]). in [6] the results about regularity, normality and metrizability of one-point extensions were extended to b-extensions with compact remainder. it was also observed that the moore-niemytzki plane and the mrówka space can be extensions defined using bornologies 83 naturally obtained as b-extensions. these examples was used to prove that the results about normality and metrizability do not hold if we remove the compactness hypothesis. the case of a regular extension with regular noncompact remainder was not solved there. in section 3 of this paper we mostly study b-extensions with closed discrete remainders. we prove that all regular extensions of this kind are b-extensions. we also show that the condition that the given bornology is open is not in general either necessary or sufficient to obtain a regular b-extension. we give conditions on the bornology fx which are equivalent to the b-extension being regular (tychonoff) when the remainder is discrete and x is regular (resp. tychonoff). in section 4 we discuss the weight of a b-extension. we prove by an example (the so-called butterfly space) that the weight of a b-extension ax can be greater than max{w(x), w(y ), χ(ax)}. in section 5 we study a particular case of b-extensions, when the map from the open subsets of the remainder y and the unbounded open subsets of x is induced by a map from x to y . in this case almost all results about separation and metrizability properties of one-point extensions can be generalized. 2. basic definitions. we recall that a boundedness (or bornology) fx is said to be local if every x ∈ x has a neighborhood in fx. in this case one can also say that x is locally bounded with respect to fx. an open bornology is obviously local. if ax is a regular extension of x then hx(ax) is local. a boundedness fx is nontrivial if x is not bounded. a basis for a boundedness is a cofinal subfamily. an open and closed bornology with a countable basis is said to be of metric type (or m-boundedness), since the usual boundedness induced by a metric has these properties. it was proved in [11] that every m-boundedness in a metrizable space is induced by a compatible metric. let x, y be hausdorff spaces, fx a nontrivial closed local bornology on x and b an open basis of y , closed with respect to finite unions. let us denote by tx the topology of x. a map π = π(b, fx) : b → (tx \ fx) ∪ {∅}, is said to be a b-map provided that b0) u 6= ∅ implies π(u) 6= ∅, for every u ∈ b; b1) if {ui}i∈a ⊂ b is a cover of y , then x \ [ ⋃ i∈aπ(ui) ] ∈ fx; b2) if u, v ∈ b then π(u ∪ v ) ∆ [π(u) ∪ π(v )] ∈ fx; b3) if u, v ∈ b and cly (u) ∩ cly (v ) = ∅ then π(u) ∩ π(v ) ∈ fx. putting on the disjoint union x ∪ y the topology generated by tx ∪ {u ∪ (π(u) \ f) | u ∈ b, f = clx(f) ∈ fx}, 84 a. caterino and m. c. vipera we obtain a dense extension of x, denoted by x ∪π y . if y is t3 the extension is hausdorff. if y is t2 not t3 we can obtain a hausdorff extension if we replace b3) by the stronger condition • if u, v ∈ b and u ∩ v = ∅ then π(u) ∩ π(v ) ∈ fx. a b-extension of x is any extension of x which can be constructed in this way. as we have already mentioned, the family of b-extensions includes esh-compactifications and hausdorff extensions ax with finite remainder. moreover normal extensions with 0-dimensional remainder are b-extensions ([5]). if ax = x ∪π y is a b-extension, where π = π(b, fx), then π is also a b-map with respect to hx(ax) and x ∪π(b,hx (ax)) y is equivalent to ax (see [5]). 3. extensions with discrete remainders. all spaces will be hausdorff and the word “extension” will always mean “dense extension”. theorem 3.1. let ax be a regular extension of x such that y = ax \ x is closed in ax. suppose there is a basis b of y consisting of open and closed sets and for every b ∈ b there are disjoint open subsets u and v of ax such that b ⊂ u and y \ b ⊂ v . then ax is a b-extension (with respect to hx(ax)). the proof is essentially the same as the one of theorem 1.5 in [5]. in fact, the hypothesis that ax is normal was used only to find disjoint open neighborhood of b and y \ b for every b in the clopen basis. for a discrete space y we denote by b0 the basis consisting of all finite subsets of y . by the above theorem we easily obtain the following result. theorem 3.2. every regular extension ax such that y = ax \ x is closed and discrete is a b-extension. proof. it suffices to take as b, in the above theorem, the family b0. � if ax = x ∪π y is a b-extension, where π = π(b, fx) and y is discrete, then b contains b0, hence b can be replaced by b0 and π by its restriction (see [6], lemma 4.7). in the previous theorem, the hypothesis that ax is regular cannot be removed. in fact we can give an example of a nonregular hausdorff extension ax, such that y = ax \ x is closed and discrete but ax cannot be obtained as a b-extension. example 3.3. let y = m∪{0} ⊂ r, where m = { 1 n ∣ ∣ n ∈ n } . put x = r\y and let ax = (r, t ), where t is generated by the union of the usual topology and the family {(−a, a) \ m | a ∈ r+} ((r, t ) is often used as example of hausdorff nonregular space). t induces the euclidean topology on x and the discrete topology on y . suppose ax is a b-extension, that is ax = x ∪π y . we can suppose π = π(b0, hx(ax)) (see above). by definition, {0} ∪ π({0}) extensions defined using bornologies 85 is open in ax so it must contain a set of the form (−a, a) \ m with a ∈ r+. let m ∈ n such that 1 m < a. the open set { 1 m } ∪ π ({ 1 m }) must contain an interval (b, c), with 0 < b < 1 m < c. the intersection (b, c) ∩ (−a, a) \ m ⊂ x cannot be in hx(ax) because 1 m belongs to its closure. this means that π does not satisfy b3), a contradiction. for any subset a of a space x we denote by frx(a) the boundary of a in x. lemma 3.4. let ax = x ∪π y be a b-extension, where y is t3 and π = π(b, fx). then, for every b ∈ b we have (i) clax(π(b)) = cly (b) ∪ clx(π(b)); (ii) frax(b ∪ π(b)) = fry (b) ∪ frx(π(b)). proof. (i) we only have to prove that if for y ∈ y , y /∈ cly (b) then y /∈ clax(π(b)). let b ′ ∈ b such that y ∈ b′ ⊂ cly (b ′) ⊂ y \ cly (b). then, by b3, π(b) ∩ π(b′) = k ∈ fx. therefore y ∈ b ′ ∪ (π(b′) \ k) which is disjoint from π(b). then the conclusion follows. (ii) let b ∈ b. clearly we have frax(b ∪π(b)) = (clax(b ∪π(b)))\(b ∪ π(b)). we note that clax(b)\(b ∪π(b)) ⊂ clax(π(b))\(b ∪π(b)). in fact if y ∈ clax(b)\(b∪π(b)) then y ∈ y and, for every b ′ ∈ b and f ∈ fx such that y ∈ b′, we have (b′ ∪(π(b′)\f))∩π(b) 6= ∅ otherwise, for some b′ ∈ b, (b′ ∪ π(b′) \ f) ∩ (b ∪ π(b)) = b ∩ b′ would be a non-empty open subset of ax = x ∪π y contained in y . it follows that (clax(b ∪π(b)))\ (b ∪π(b)) = (clax(π(b)))\(b∪π(b)). by (i) we get (clax(π(b)))\(b∪π(b)) = (cly (b)∪ clx(π(b))) \ (b ∪ π(b)) and finally the conclusion follows by the obvious equalities (cly (b)∪clx (π(b)))\ (b ∪π(b)) = (cly (b)\ b)∪((clx (π(b))\ π(b)) = fry (b) ∪ frx(π(b)). � remark 3.5. if b consists of open and closed sets then (ii) becomes (ii′) frax(b ∪ π(b)) = frx(π(b)), hence frx(π(b)) is closed in ax. from now on, for a b-extension x ∪π y , where π = π(b, fx) and y is discrete, we will always put b = b0 and uy = π({y}) for every y ∈ y . corollary 3.6. let ax = x∪π y be a b-extension, where y is discrete. then, for every y ∈ y we have (i) clax(uy) = {y} ∪ clx(uy); (ii) frx(uy) = frax({y} ∪ uy), hence frx(uy) is closed in ax. proposition 3.7. let ax = x ∪π y be a b-extension of a regular space x, where π = π(b, fx) and b is a basis of y consisting of open and closed subsets of y . then (i) if ax is regular and b is compact then frx(π(b)) belongs to fx; (ii) if fx is open and, for each b ∈ b, frx(π(b)) belongs to fx, then ax is regular. proof. (i) let ax be regular and let b ∈ b, b compact. by the remark 3.5 a = frx(π(b)) is closed in ax. then for every y ∈ b there exists 86 a. caterino and m. c. vipera an open subset wy of ax that contains a and is disjoint from a basic open neighborhood by ∪ (π(by) \ ky) of y. we can suppose wy ⊂ x. from the compactness of b it follows that b ⊂ b′ = ⋃n i=1 byi for some y1, . . . , yn ∈ b. hence π(b) \ π(b′) = π(b ∪ b′)∆[π(b) ∪ π(b′)] ∈ fx by b2) and clearly π(b) \ ( ⋃n i=1 π(byi)) also belongs to fx. if we put w = ⋂n i=1 wyi then w ∩ ( ⋃n i=1 π(byi)) ⊂ ⋃n i=1 kyi and w ∩ π(b) is bounded. now let x ∈ a and let v be any neighborhood of x in x. then v ∩w ∩π(b) 6= ∅ and this means that x ∈ clx(w ∩ π(b)). therefore a ⊂ clx(w ∩ π(b)), which is bounded. (ii) let x ∈ x. the hypotheses imply that x has a local basis consisting of bounded closed subset of x, which are also closed neighborhoods in ax. now let y ∈ y and let f be a closed subset of ax such that y /∈ f . then there are b ∈ b and a closed member k of fx such that y ∈ b ∪ (π(b) \ k), which is disjoint from f . put v = ax\[clax(b∪π(b))] = ax\[b∪clx(π(b))] = ax\[b∪π(b)∪frx(π(b))]. then we have f \ v = (f ∩ (b ∪ π(b))) ∪ (f ∩ frx(π(b))) ⊂ k ∪ frx(π(b)). since k ∪frx(π(b)) is bounded, it is contained in an open member w of fx. then v ∪ w is an open subset of ax which contains f and is disjoint from b ∪ [π(b) \ clx(w)], which is a basic neighborhood of y. � proposition 3.8. let ax = x∪π y be a b-extension of x, where y is discrete and π = π(b0, fx). suppose x is regular. then (i) if ax is regular, then for each y ∈ y , frx(uy) belongs to fx; (ii) if fx is open and, for each y ∈ y , frx(uy) belongs to fx, then ax is regular. proof. it easily follows by proposition 3.7, since frx(uy) ∈ fx for every y ∈ y implies that frx(π(b)) ∈ fx for every b ∈ b0. � the following example shows that, for a b-extension ax = x ∪π y , the conditions that x, y are regular and fx is open do not ensure that ax is regular, even if y is discrete. example 3.9. let x be the upper half plane, defined by {(x, y) ∈ r2 | y > 0}, with the usual topology, and y be the x-axis with the discrete topology. let fx = {a ⊂ x | d(a, y ) > 0}, where d is the euclidean metric. clearly fx is an open boundedness and x is locally bounded. for every z = (a, 0) ∈ y , put uz = {(x, y) ∈ x | |x − a| < y < 1}. uz is clearly unbounded. let π : b0 → tx \ fx be defined by b 7→ ⋃ z∈b uz. it is easy to see that π is a b-map. then we can define the hausdorff bextension ax = x ∪π y . let z be any point of y . clearly e = frx(uz) is unbounded, so by proposition 3.7, ax is not regular. in fact, e is closed extensions defined using bornologies 87 in ax and z /∈ e, but no open set containing e can be disjoint from a basic neighborhood {z} ∪ (uz \ k), with k ∈ fx. remark 3.10. in the above example, although fx is open, hx(ax) is not open. by lemma 3.4(ii) and proposition 3.8(ii), if ax = x ∪π(b,fx) y , where x is regular, y is discrete and hx(ax) is open, then ax is regular. however, as we will see below, the condition that hx(ax) is open is not necessary to obtain a regular b-extension. we notice that, for any extension ax with closed remainder y , hx(ax) is open if and only if y is separated, by disjoint open subsets of ax, from every closed subset of ax which is contained in x. there exists a tychonoff b-extension ax of a space x, with discrete remainder, which cannot be obtained as b-extension with respect to any open boundedness fx. in particular hx(ax) is not open. example 3.11. the tychonoff plank t can be seen as an extension ax of x = ω1 × (ω + 1) with closed discrete remainder y = {ω1} × ω. by theorem 3.2, ax is a b-extension. put f = ω1 × {ω}. clearly hx(ax) is not open, otherwise, f and y would be contained in disjoint open subsets of t . now suppose that fx is a closed boundedness on x, and π = π(b, fx) a b-map such that t = x ∪π y . put yn = (ω1, n) for every n, so that y = {yn}n∈n. for sake of simplicity, for every y ∈ y , we will write π(y) instead of π({y}). since {yn} ∪ π(yn) is open in t , π(yn) must contain a set of the form (βn, ω1) × {n}, where βn < ω1. property b1) implies that g = f \ ( ⋃ n∈n π(yn) ) belongs to fx. now suppose fx is open. then g and y are contained in disjoint open subsets of t . this implies g ⊂ [0, α] × {ω}, where α < ω1, hence (α, ω1) × {ω} ⊂ ⋃ n∈n π(yn). let m ∈ n such that m = {γ > α | (γ, ω) ∈ π(ym)} is uncountable. for every γ ∈ m let vγ = (η1(γ), η2(γ)) × (nγ, ω] be a basic open neighborhood of (γ, ω) contained in π(ym). let h ∈ n such that h = {γ ∈ m | nγ = h} is uncountable and let k ∈ n, k > max(h, m). then (βk, ω1) × {k}, which is contained in π(yk), contains all the points (γ, k) with γ ∈ h. but every (γ, k), with γ ∈ h belongs to vγ ⊂ π(ym). then the uncountable set {(γ, k) | γ ∈ h} is contained in both π(yk) and π(ym). this means that yk ∈ clt (π(yk) ∩ π(ym)). since k 6= m, by b3) π(yk) ∩ π(ym) belongs to fx, so no point of y can be in its closure, contradiction. the following proposition provides a condition which is equivalent to the regularity of x ∪π y where y is discrete. proposition 3.12. let ax = x ∪π y be a b-extension of x, where y is discrete and π = π(b0, fx). suppose x is regular. then ax is regular if and only if, for every y ∈ y and for every closed subset f of x such that f ∩ uy is bounded, there is an open subset w of x containing f ∩ clx(uy) such that w ∩ uy is bounded. proof. suppose ax is regular. let y ∈ y and f ∩ uy ∈ fx, where f is closed in x. this is clearly equivalent to y /∈ clax(f). then there exists an 88 a. caterino and m. c. vipera open subset w1 of ax that contains clax(f) and is disjoint from a basic open neighborhood {y} ∪ (uy \ k) of y. put w = w1 ∩ x. then f ∩ clx(uy) ⊂ w and w ∩ uy ⊂ k, so that w ∩ uy is bounded. conversely, let y ∈ ax and let g be a closed subset of ax such that y /∈ g. we can suppose y ∈ y (see the proof of proposition 3.7(ii)). put f = g ∩ x. then y /∈ clax(f), hence f ∩ uy is bounded. by hypothesis, there exists an open subset w of x such that f ∩ clx(uy) ⊂ w and w ∩ uy is bounded. we know that v = ax \ clax(uy) contains all points of y except y (see lemma 3.4). since y /∈ g, we have g \ v = f \ v ⊂ w . therefore we have g ⊂ w ∪v. clearly [{y}∪(uy \w)]∩(v ∪w) = ∅. we have {y}∪(uy \w) = {y} ∪ (uy \ (w ∩ uy)) ⊃ {y} ∪ (uy \ clx(w ∩ uy)), which is a basic open neighborhood of y disjoint from v ∪ w . � we recall that every one-point extension ax of x can be obtained as o(fx) for a suitable closed bornology fx; moreover we have hx(ax) = fx ([6]). then, by theorem 3.3 in [6], a one-point extension ax is tychonoff if and only if hx(ax) is functionally open. the following proposition provides a sufficient condition for the complete regularity of a b-extension with discrete remainder. proposition 3.13. let ax = x ∪π y be a b-extension of x, where y is discrete and π = π(b0, fx). suppose x is tychonoff. if fx is functionally open and, for each y ∈ y , frx(uy) belongs to fx, then ax is tychonoff proof. let z ∈ ax and let f be a closed subset of ax such that z /∈ f . first suppose z ∈ x. let h be a bounded open neighborhood of z in x which is disjoint from f . since x is tychonoff, there exists a continuous function f : x → i such that f(z) = 1 and f(x \ h) = 0. the map f̂ : ax → i defined by f̂ |x= f, f̂(y ) = 0, is continuous because no y ∈ y belongs to clax(h). clearly f̂ separates z from f . let now z ∈ y . put a = frx(uz), t = clx(uz) = uz ∪ a and t ∗ = clax(t ) = {z} ∪ t . t ∗, with the topology induced by ax, is a one-point extension of t . we want to prove that ht (t ∗) is functionally open. let g be a subset of t which is closed in t ∗. there exists a neighborhood {z}∪(uz \k) which is disjoint from g, where k is a closed member of fx. since g is closed in t and is contained in k ∪ a, g is a closed member of fx. then there is an open w ∈ fx such that g and x \ w are completely separated. put v = w ∩ t . since t is unbounded, t \ v = t \ w is nonempty and completely separated from g. moreover, clt ∗(v ) ⊂ clax(w) which does not meet y . then v ∈ ht (t ∗). we have proved that ht (t ∗) is functionally open, that is, t ∗ is tychonoff. let f : t ∗ → i be a continuous function such that f(z) = 1, f((f ∩ t ) ∪ a) = 0. we define an extension f̂ : ax → i of f putting f̂(ax \ t ∗) = 0. since f is equal to 0 on a, which is the boundary of t ∗ in ax, ĥ is continuous. moreover f̂ separates z from f . � in the above proposition the hypothesis frx(uy) ∈ fx for every y is clearly necessary (see proposition 3.7(i)). the examples 3.9 and 3.11 show that the extensions defined using bornologies 89 condition that fx is functionally open is a neither sufficient nor necessary condition. in view of corollary 3.6(ii), the condition that hx(ax) is functionally open is sufficient, but it is not necessary (see example 3.11 again). it is easy to see the following proposition 3.14. for every normal extension ax, hx(ax) is functionally open. corollary 3.15. every normal extension ax such that y = ax \ x is closed and discrete is a b-extension with respect to a functionally open boundedness. proof. by the proof of theorem 3.1, ax is a b-extension with respect to hx(ax). � however, there exist nonnormal b-extensions ax with discrete remainder where x is normal and hx(ax) is functionally open. example 3.16. let ψ be the mrówka space. it is known that ψ is a nonnormal tychonoff space (see for instance [9]), and is a b-extension of n with respect to the boundedness fn of the finite subsets of n (see [6], example 4.10). clearly hn(m) = fx is functionally open, since its members are clopen. 4. weight of b-extensions. for a boundedness fx on x we put µ(fx) = min{|c| : c is a basis of fx}. proposition 4.1. let ax = x ∪π y be any b-extension of a space x,where π = π(b, fx). then we have w(ax) ≤ max{w(x), w(y ), µ(fx)}. proof. by [6] lemma 4.7, we can suppose |b| = w(y ). let b1 be a basis of x with |b1| = w(x) and c a basis of fx with |c| = µ(fx). it is easy to see that b1 ∪ {u ∪ (π(u) \ f) | u ∈ b, f ∈ c} is a basis for the topology of ax whose cardinality is max{w(x), w(y ), µ(fx)}. � the weight of a b-extension ax = x ∪π y can be greater than max{w(x), w(y ), χ(ax)}. example 4.2. let us consider the so called butterfly space, that is the space z = (r × r, t ) where t can be described as follows. let us denote the xaxis by y and put x = (r × r) \ y . the points of x have the same open neighborhoods as in the ordinary topology. let pa = (a, 0) ∈ y and let r ∈ r +. we put rr(a) = (a− r, a+ r)× (− 1 r , 1 r ) and we denote by c1r (a) and c 2 r (a) the 90 a. caterino and m. c. vipera (closed) circles of radious 1 r and center (a, 1 r ), (a, −1 r ), respectively (so that the circles are tangent to the x-axis in pa). we also put br(a) = [rr(a) \ (c 1 r (a) ∪ c 2 r (a))] ∪ {pa}. a local basis for pa will be the family {br(a)}r∈r+. if we consider the subfamily {br(a)}r∈ q, we obtain a countable local basis, hence χ(z) = ω. it is known that w(z) = c. the topology induced on both x and y are the usual ones. we also observe that the sets of the form x ∩ cir(a) = c i r(a) \ {pa} are closed in z. the butterfly space can be considered a dense extension ax of x such that y = ax \ x is naturally homeomorphic to the real line r. we want to prove that ax can be obtained as a b-extension x ∪π r with respect to the boundedness hx(ax). let b be the family of finite unions of bounded open intervals in r. given an open interval (a − r, a + r), with a ∈ r, r ∈ r+, we put π(a − r, a + r) = br(a)∩x, which is clearly an open unbounded subset of x. for a finite disjoint union u = ⋃ i (ai − ri, ai + ri) we put π(u) = ⋃ i π(ai − ri, ai + ri). we want to prove that π is a b-map. the property b3) is obviously satisfied, since for u1, u2 ∈ b such that u1∩u2 = ∅, we have π(u1) ∩ π(u2) = ∅. let {uj}j∈j ⊂ b be a cover of r. for uj = ⋃n i=1(ai − ri, ai + ri), where the union is disjoint, we put wj = ⋃n i=1 bri(ai), so that one has wj ∩ x = π(uj). therefore x \ ( ⋃ j π(uj) ) = x \ ( ⋃ j wj ) = ax \ ( ⋃ j wj ) ∈ hx(ax). we have proved that b1) is satisfied. now we prove b2) in case u, v are intervals, u = (a−r, a+r), v = (b−s, b+s). the general case will easily follow. if u and v are disjoint the proof is trivial, hence we can suppose that u ∪ v is an interval (c − t, c + t). we want to show that e = π(c − t, c + t)△[π(a − r, a + r) ∪ π(b − s, b + s)] is bounded. we have e ⊂ (c− t, c+ t)× (r\ {0}). moreover every x ∈ e must be in x \ π(c − t, c + t) = x \ bt(c) or in x \ [π(a − r, a + r) ∪ π(b − s, b + s)] = (x \ br(a)) ∩ (x \ bs(b)). suppose x ∈ e, and x /∈ bt(c). then either x ∈ (c−t, c+t)×[r\(−1 t , 1 t )], which is clearly bounded, or x ∈ (c1t (c)∪c 2 t (c))∩x which is also bounded. similarly we can prove that, if x ∈ e and x /∈ br(a), x /∈ bs(b) then x belongs to a bounded set. therefore e is contained in a finite union of bounded sets and so it is bounded. we have proved that π is a b-map. if we identify r with the x-axis y in z = (r × r, t ), then the topology of x ∪π r is equal to t . in fact, for a ∈ r, r ∈ r +, we have br(a) = u ∪ π(u), where u = (a − r, a + r), identified with (a − r, a + r) × {0}. conversely, every set of the form u ∪ (π(u) \ f), where u ∈ b and f is a closed member of extensions defined using bornologies 91 hx(ax), is open in z, because u ∪ π(u) is a union of sets of the form br(a) and f is closed in z by the definition of hx(ax). 5. extensions whose remainders are retracts. given a nontrivial local closed bornology on x, a continuous mapping f from x to any space y is said to be b-singular with respect to fx if f −1(u) /∈ fx for every nonempty open subset u of y . if f is b-singular, then the map π : ty → (tx \ fx) ∪ {∅}, π(u) = f −1(u), is clearly a b-map. the b-extension induced by π, denoted by x ∪f y , is said to be b-singular ([5], [7]). regular extensions are b-singular if and only if the remainder is a retract. in fact we have theorem 5.1. let (ax, tax) be a regular extension of x such that there exists a retraction g : ax → y = ax \ x. then f = g|x is b-singular with respect to hx(ax) and (ax, tax) = x ∪f y . conversely, for every b-singular extension ax = x ∪f y , the map f̂ : ax → y defined by f̂|x = f and f̂|y = 1y is a continuous extension of f, hence a retraction. proof. let g : ax → y be a retraction and let u be a nonempty open subset of y . it is easy to see that every point of u belongs to clax(f −1(u)), hence f−1(u) /∈ hx(ax). since ax is t3, x is locally bounded with respect to hx(ax). now we will prove that the topology of x ∪f y coincides with tax. first we observe that tx is contained in both topologies. let u1 = u ∪ [f −1(u) \ f ] where u is open in y and f is a closed member of hx(ax). then u1 = g−1(u) \ f ∈ tax. this proves the first inclusion. let us choose any w ∈ tax. we only need to prove that, for every y ∈ w ∩ y , there is a basic open set u1 in x ∪f y such that y ∈ u1 ⊂ w . let u be an open neighborhood of y in y such that y ∈ u ⊂ cly (u) ⊂ w ∩ y . put a = g−1(u) \ w which is a subset of x. we want to prove that no point of y belongs to clax(a). this is obvious for z ∈ w . if z ∈ y \ w then z /∈ cly (u). let v be an open neighborhood of z in y such that u ∩ v = ∅. then g−1(v )∩g−1(u) = ∅, that is, g−1(v ) is a neighborhood of z in (ax, tax) which is disjoint from a. we have proved that clx(a) is a (closed) member of hx(ax). then u1 = g −1(u) \ clx(a) = u ∪ [f −1(u) \ clx(a)] is a basic neighborhood of y in x ∪f y and, by the definition of a, we have u1 ⊂ w . we have proved (ax, tax) = x ∪f y . let now ax = x ∪f y a b-singular extension and let u be open in y . then f̂−1(u) = u ∪ f−1(u) is a basic open set of x ∪f y . � in [6], theorem 4.11, it was proved that a b-singular extension x ∪f y of x, with respect to an open (and closed) bornology fx, is regular provided x and 92 a. caterino and m. c. vipera y are both regular. if we replace “open” by “functionally open”, we obtain an analogous result for the tychonoff property. theorem 5.2. let x, y be tychonoff spaces and let f : x → y be a bsingular map with respect to a functionally open bornology fx. then ax = x ∪f y is tychonoff. proof. first suppose that y is compact. let q : x ∪f y → o(fx) = x ∪ {p} be the natural mapping which takes every point of y to the point p. by [5], proposition 1.1, q is a quotient map. let now z ∈ ax and let a be a closed subset of ax with z /∈ a. first suppose z ∈ x and put b = a ∪ y . then q(b) is a closed subset of o(fx) which does not contain z. since o(fx) is tychonoff ([6], theorem 3.3), z and q(b) are separated by a continuous function g from o(fx) to i, where i is the unit interval. clearly g ◦ q separates z and a. now, let z ∈ y and a ⊂ x. then a = q(a) is closed in o(fx). let g : o(fx) → i be a function such that g(p) = 1 and g(a) = 0. then g ◦ q separates z from a. note that g ◦ q maps all of y onto 1. finally let c = a ∩ y 6= ∅ and z ∈ y . take a map v : y → i such that v(c) = 0 and v(z) = 1. put h = v ◦ f̂ : ax → i and u = h−1([0, 1/2)). then a \ u is a closed subset of ax contained in x. we can take, as before, a function u : ax → i such that u(y ) = 1 and u(a \ u) = 0. then the map h ∧ u is less than 1/2 in u ∪ a and maps z to 1. we have proved that ax is tychonoff in case y is compact. let now y be any tychonoff space and let (k, k) be any compactification of y , where k : y → k is the embedding. then f1 = k ◦ f : x → k is b-singular and x ∪f1 k is tychonoff. it is easy to see that ax = x ∪f y is a subspace of x ∪f1 k, hence ax is also tychonoff. this completes the proof. � lemma 5.3. let ax = x ∪f y be a b-singular extension of x and suppose v be a locally finite family in y . then the family v1 = {v ∪ f −1(v ) | v ∈ v} is locally finite in ax. proof. let y ∈ y . there is an open neighborhood ny of y in y such that ny ∩ v 6= ∅ for only finitely many v ∈ v. this implies that ny ∪ f −1(ny) meets only finitely many members of v1. let x ∈ x and y = f(x). then f−1(ny) is a neighborhood of x which meets only finitely many members of v1. � theorem 5.4. let ax = x ∪f y be a b-singular extension of x with respect to an m-boundedness fx. suppose x and y are metrizable. then ax is metrizable. proof. since x ∪f y is t3 by [6], theorem 4.11, we need only to prove that it admits a σ-locally finite basis. by hypothesis, fx has a countable basis {mk}k∈n, where mk is open and clx(mk) is bounded. we have ⋃ k∈n mk = extensions defined using bornologies 93 x. let c = ⋃ n∈n cn be a basis for tx, where every cn is a locally finite family. put ckn = {c ∩ mk | c ∈ cn}, n, k ∈ n. similarly, let u = ⋃ n∈n un be a basis of ty , where every un is locally finite. for every n, k ∈ n, let ukn = {u ∪ [f −1(u) \ clx(mk)] | u ∈ un}. we claim that s = ( ⋃ n,k∈n ckn ) ∪ ( ⋃ n,k∈n ukn ) is a σ-locally finite basis for ax. let w be an open subset of ax and let x be in w . if x ∈ x, then x ∈ mk for some k and there is c ∈ cn, for some n, such that x ∈ c ⊂ w . then x ∈ c ∩ mk ⊂ w , where c ∩ mk ∈ c k n. if x ∈ y , then there is u ∈ un for some n and f = clx(f) ∈ fx such that x ∈ u ∪ (f−1(u) \ f) ⊂ w . we have f ⊂ mk for some k, hence x ∈ u ∪ [f−1(u) \ clx(mk)] ⊂ (u ∪ (f −1(u) \ f)) ⊂ w, where u ∪ [f−1(u) \ clx(mk)] ∈ u k n. every ckn is locally finite. in fact, for every x ∈ x, there is a neighborhood that meets only finitely many members of cn, hence of c k n. if x ∈ y , any basic neighborhood of x of the form v ∪ [f−1(v ) \ clx(mk)] meets no member of ckn. by lemma 5.3, ukn is locally finite for every n, k. this completes the proof. � references [1] g. beer, on metric boundedness structures, set-valued anal. 7 (1999), 195–208. [2] g. beer, on convergence to infinity, monathsh. math. 129 (2000), 267–280. [3] g. beer, embeddings of bornological universes, set-valued anal. 16, no. 4 (2008), 477– 488. [4] a. caterino, g. d. faulkner, m. c. vipera, construction of compactifications using essential semilattice homomorphisms, proc. amer. math. soc. 116 (1992), 851–860. [5] a. caterino and s. guazzone, extensions of unbounded topological spaces, rend. sem. mat. univ. padova 100 (1998), 123–135. [6] a. caterino, f. panduri and m. c. vipera, boundedness, one-point extensions and b-extensions, math. slovaca. 58, no. 1 (2008), 101–114. [7] r. e. chandler and g. d. faulkner, singular compactifications: the order structure, proc. amer. math. soc. 100 (1987), 377–382. [8] r. engelking, general topology, heldermann, berlin, 1989. [9] l. gillman and m. jerison, rings of continuous functions, van nostrand, princeton, 1960. [10] s. t. hu, boundedness in a topological space, j. math. pures appl. 28 (1949), 287–320. [11] s. t. hu, introduction to general topology, holden-day inc., san francisco, 1969. [12] m. c. vipera, some results on sequentially compact extensions, comment. math. univ. carolinae 39 (1998), 819–831. (received november 2008 – accepted august 2009) 94 a. caterino and m. c. vipera alessandro caterino (caterino@dipmat.unipg.it) dipartimento di matematica e informatica, università degli studi di perugia, via vanvitelli 1, 06123, perugia, italy. (corresponding author) m. cristina vipera (vipera@dipmat.unipg.it) dipartimento di matematica e informatica, università degli studi di perugia, via vanvitelli 1, 06123, perugia, italy. extensions defined using bornologies. by a. caterino and m. c. vipera tkachukagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 39-48 condensations of cp(x) onto σ-compact spaces v. v. tkachuk ∗ abstract. we show, in particular, that if nw(nt) ≤ κ for any t ∈ t and c is a dense subspace of the product ∏ {nt : t ∈ t } then, for any continuous (not necessarily surjective) map ϕ : c → k of c into a compact space k with t(k) ≤ κ, we have ψ(ϕ(c)) ≤ κ. this result has several applications in cp-theory. we prove, among other things, that if k is a non-metrizable corson compact space then cp(k) cannot be condensed onto a σ-compact space. this answers two questions published by arhangel’skii and pavlov. 2000 ams classification: primary 54h11, 54c10, 22a05, 54d06; secondary 54d25, 54c25 keywords: condensation, continuous image, lindelöf σ-space, σ-compact space, topology of pointwise convergence, network weight, tightness, lindelöf space. 1. introduction. a weaker topology on a space x can be considered an approximation of the topology of x. if this approximation has some nice properties then we can obtain a lot of useful information about the space x. thus it is natural to find out when a space has a weaker compact topology. this is an old topic and an extensive research has been done here both in general topology and descriptive set theory. the quest for nice condensations of function spaces had its origin in functional analysis after banach asked whether every separable banach space has a weaker compact metrizable topology. this problem was solved positively by pytkeev [9]. answering a question of arhangel’skii, casarrubias–segura showed in [5] that function spaces of cantor cubes have a weaker lindelöf topology but it is consistent that some of them do not have a weaker compact topology. ∗research supported by consejo nacional de ciencia y tecnoloǵıa (conacyt) de méxico, grant 400200-5-38164-e 40 v. v. tkachuk arhangel’skii and pavlov [4] studied systematically when cp(x) has a weaker compact topology and formulated some open questions on weaker σ-compact topologies on cp(x). it is also worth mentioning that marciszewski [7] gave a consistent example of a space x ⊂ r such that cp(x) does not have a weaker σ-compact topology. in this paper we consider product spaces n = ∏ t∈t nt such that nw(nt) ≤ κ for all t ∈ t . we prove that if c is a dense subspace of n and ϕ : c → k is a continuous (not necessarily surjective) map of c into a compact space k with t(k) ≤ κ then ψ(ϕ(c)) ≤ κ, i.e., every closed subset of ϕ(c) is the intersection of at most κ-many open subsets of ϕ(c). this result has several important applications in cp-theory. we establish, in particular, that if x is an ω-monolithic space such that l(cp(x)) = t(cp(x)) = ω and cp(x) condenses onto a σ-compact space then x is cosmic. as a consequence, if x is a non-metrizable corson compact space then cp(x) does not condense onto a σ-compact space. this answers questions 29 and 30 of the paper of arhangel’skii and pavlov [4]. any compact space of countable tightness has countable π-character (see [1, theorem 2.2.20]). this easily implies that if cp(x) embeds in such a compact space then x is countable. therefore it is natural to conjecture that every continuous image of cp(x) has a countable network whenever it embeds in a compact space of countable tightness. another reason to believe that this conjecture might be true is a theorem of tkachenko [12] which states that if a compact space k of countable tightness is a continuous image of a lindelöf σ-group then k is metrizable. at the present moment nothing contradicts the hypothesis that if g is a lindelöf σ-group and ϕ : g → k is a continuous map, where k is compact and t(k) ≤ ω then ϕ(g) has a countable network. we prove this conjecture for the spaces cp(x) with the lindelöf σ-property. 2. notation and terminology. all spaces under consideration are assumed to be tychonoff. if x is a space then τ(x) is its topology and τ∗(x) = τ(x) \ {∅}; given an arbitrary set a ⊂ x let τ(a,x) = {u ∈ τ(x) : a ⊂ u}. if x ∈ x then we write τ(x,x) instead of τ({x},x). the space r is the real line with its natural topology and n = ω \ {0}. if x and y are spaces then cp(x,y ) is the space of real-valued continuous functions from x to y endowed with the topology of pointwise convergence. we write cp(x) instead of cp(x, r). the expression x ≃ y says that the spaces x and y are homeomorphic. a family n of subsets of a space z is called a network if for any u ∈ τ(z) there is n ′ ⊂ n such that ⋃ n ′ = u. the network weight nw(z) of a space z is the minimal cardinality of a network in z. a space x is called cosmic if the network weight of x is countable. if x ∈ x then a family a is a network of x at the point x if x ∈ ⋂ a and for any u ∈ τ(x,x) there is a ∈ a such that a ⊂ u. a family b ⊂ τ∗(x) is a π-base of x at a point x ∈ x if for any u ∈ τ(x,x) there is b ∈ b with b ⊂ u. the minimal cardinality of a π-base of x at x is condensations of cp(x) onto σ-compact spaces 41 denoted by πχ(x,x) and πχ(x) = sup{πχ(x,x) : x ∈ x}. if ϕ is a cardinal invariant then hϕ(x) = sup{ϕ(y ) : y ⊂ x} is the hereditary version of ϕ. if x is a space and f is a closed subset of x then pseudocharacter ψ(f,x) of the set f in the space x is the minimal cardinality of a family u ⊂ τ(f,x) such that⋂ u = f ; let ψ(x) = sup{ψ({x},x) : x ∈ x} and ψ(x) = sup{ψ(f,x) : f is a closed subset of x}. the tightness t(x) of a space x is the minimal cardinal κ such that, for any a ⊂ x, if x ∈ a then there is b ⊂ a with |b| ≤ κ such that x ∈ b. we use the russian term condensation to denote a continuous bijection. a space z is called κ-monolithic if for any a ⊂ z with |a| ≤ κ, we have nw(a) ≤ κ. if we have a product z = ∏ t∈t zt and a ⊂ t then za = ∏ t∈a zt is the a-face of z and πa : z → za is the natural projection. a set f ⊂ z depends on a ⊂ t if π−1 a πa(f) = f ; if f depends on a set of cardinality ≤ κ then we say that f depends on at most κ-many coordinates. a set e ⊂ z covers a face za if πa(e) = za. suppose that, for every t ∈ t we have a family nt of subsets of zt and let n = {nt : t ∈ t}. if we have a faithfully indexed set a = {t1, . . . , tn} ⊂ t and ni ∈ nti for each t ≤ n then let [t1, . . . , tn,n1, . . . ,nn] = {x ∈ z : x(ti) ∈ ni for all i = 1, . . . ,n}. a set h ⊂ z is called n -standard (or standard if n is clear) if h = [t1, . . . , tn,n1, . . . ,nn] for some t1, . . . , tn ∈ t and ni ∈ nti for all i ≤ n. in this case we let supp(h) = a and r(h) = n. we also consider that h = z is the unique standard subset of z such that r(h) = 0. given any point x ∈ z and a ⊂ t the set 〈x,a〉 = {y ∈ z : y(t) = x(t) for any t ∈ a} is closed in z. if a ⊂ t then the face za is called κ-residual if |t \a| ≤ κ. say that a non-empty closed set f ⊂ k is κ-large if, for any x ∈ f and any finite a ⊂ t , the set 〈x,a〉 ∩ f covers a κ-residual face of k. all other notions are standard and can be found in [6] and [3]. 3. nice continuous images of function spaces. our results will be obtained by strengthening a result of shirokov [11]. although our modifications of shirokov’s method are minimal, we give complete proofs because the paper [11] has never been translated and, even in russian, it is completely out of access for a western reader. in particular, we present the proof of the following lemma established in [11]. lemma 3.1. given an infinite cardinal κ suppose that nw(nt) ≤ κ for any t ∈ t and n = ∏ t∈t nt. assume that c ⊂ n is dense in n, and we have a compact extension kt of the space nt for any t ∈ t. if a set f ⊂ k = ∏ t∈t kt is κ-large then there exists a gκ-set g in the space k such that f ⊂ g and f ∩ c = g ∩ c. in particular, f ∩ c is a gκ-subset of c. proof. we can assume, without loss of generality, that k \ f 6= ∅. for every t ∈ t fix a network nt in the space nt such that |nt| ≤ κ; we will need the family mt = {clkt (n) : n ∈ nt}. if m = {mt : t ∈ t} then the m-standard subsets of k will be called standard. it is easy to see that (1) the family h of all standard subsets of k is a network in k at every x ∈ c. 42 v. v. tkachuk given standard sets p and p ′ say that p ′ � p if p = [t1, . . . , tn,m1, . . . ,mn] and there exists a natural k ≤ n such that p ′ = [ti1, . . . , tik,mi1, . . . ,mik ] for some distinct i1, . . . , ik ∈ {1, . . . ,n}; if k < n then we write p ′ ≺ p . we also include here the case when k = 0 so p ′ = k � p for any standard set p . say that a standard set p is minimal if p ∩ f = ∅ but p ′ ∩ f 6= ∅ whenever p ′ ≺ p . it follows from (1) that (2) for any x ∈ c \ f there exists a minimal standard set p such that x ∈ p . it will be easy to finish our proof if we establish that (3) the family s of minimal standard sets has cardinality not exceeding κ. assume, toward a contradiction that |s| > κ. then we can choose s0 ⊂ s such that |s0| = κ + and there exists n ∈ ω with r(p) = n for all p ∈ s0. observe first that (4) if a ⊂ t , a set d ⊂ k covers the face kt \a and a standard set p is disjoint from d then supp(p) ∩ a 6= ∅. indeed, if supp(p) = {t1, . . . , tk} ⊂ t \ a and p = [t1, . . . , tk,m1, . . . ,mk] then it follows from πt \a(d) = kt \a that there exists a point x ∈ d such that x(ti) ∈ mi for all i ≤ k. therefore x ∈ d ∩ p which is a contradiction. the set f being κ-large, there exists a1 ⊂ t with |a1| ≤ κ such that f covers the face kt \a1 . the property (4) shows that supp(p) ∩a1 6= ∅ for any p ∈ s0. there exists a point t1 ∈ a1 such that the family s ′ 0 = {p ∈ s0 : t1 ∈ p} has cardinality κ+. since |mt1| ≤ κ, we can find a family s1 ⊂ s ′ 0 and m1 ∈ mt1 such that |s1| = κ + and [t1,m1] � p for any p ∈ s1. proceeding by induction assume that k < n and we have a set ak ⊂ t with |ak| ≤ κ and a family sk such that |sk| = κ + and, for some t1, . . . , tk ∈ ak and mi ∈ mti (i = 1, . . . ,k), we have [t1, . . . , tk,m1, . . . ,mk] � p for every p ∈ sk. therefore p = [t1, . . . , tk,s1, . . . ,sn−k,m1, . . . ,mk,e1, . . . ,en−k] for every p ∈ sk; let q(p) be the set in which s1 and e1 are omitted from the definition of p , i.e., q(p) = [t1, . . . , tk,s2, . . . ,sn−k,m1, . . . ,mk,e2, . . . ,en−k]. it is clear that q(p) ≺ p ; since p is minimal, the set q(p) intersects f for each p ∈ sk. fix a set r ∈ sk and let f ′ = f ∩ q(r). the set f being κ-large, we can find a ⊂ t with |a| ≤ κ such that f ′ covers the face kt \a and hence the face kt \(a∪ak) as well. let ak+1 = a∪ak and observe that every set p ∈ sk is disjoint from f ′; this, together with (4) shows that supp(p)∩ak+1 6= ∅. suppose for a moment that p = [t1, . . . , tk,s1, . . . ,sn−k,m1, . . . ,mk,e1, . . . ,en−k] ∈ sk and {s1, . . . ,sn−k} ∩ ak+1 = ∅. since f ′ covers the face kt \ak+1 , we can find a point x ∈ f ′ such that x(si) ∈ ei for all i ≤ n − k; since also x(ti) ∈ mi for all i ≤ k because x ∈ q(r), we conclude that x ∈ f ′∩p . this contradiction implies that {s1, . . . ,sn−k}∩ak+1 6= ∅ and hence the set supp(p)\{t1, . . . , tk} intersects the set ak+1 for any p ∈ sk \ {r}. therefore we can choose a family sk+1 ⊂ sk of cardinality κ + together with a point tk+1 ∈ ak+1 \ {t1, . . . , tk} and a set mk+1 ∈ mtk+1 such that we have [t1, . . . , tk+1,m1, . . . ,mk+1] � p for any p ∈ sk+1. as a consequence, our inductive procedure can be continued to construct a family sn ⊂ s such that condensations of cp(x) onto σ-compact spaces 43 |sn| = κ + while [t1, . . . , tn,m1, . . . ,mn] � p for any p ∈ sn. recalling that r(p) = n, we conclude that we have the equality p = [t1, . . . , tn,m1, . . . ,mn] for each p ∈ sn; this contradiction shows that |s| ≤ κ, i.e., (3) is proved. it is straightforward that g = k \ ( ⋃ s) is a gκ-subset of k such that f ⊂ g and f ∩ c = g ∩ c. � the following result generalizes theorem 1 of [11]. theorem 3.2. given an infinite cardinal κ suppose that nw(nt) ≤ κ for any t ∈ t and c ⊂ n = ∏ t∈t nt is a dense subspace of n. assume additionally that we have a continuous (not necessarily surjective) map ϕ : c → l of c into a compact space l. if y ∈ c′ = ϕ(c) and hπχ(y,l) ≤ κ then ψ(y,c′) ≤ κ. proof. there is no loss of generality to assume that c′ is dense in l. choose a compact extension kt of the space nt for any t ∈ t ; then k = ∏ t∈t kt is a compact extension of both n and c. there exist continuous maps φ : βc → l and ξ : βc → k such that φ|c = ϕ and ξ(x) = x for any x ∈ c. it is clear that both φ and ξ are surjective. for every t ∈ t fix a network nt in the space nt such that |nt| ≤ κ and let mt = {clkt (n) : n ∈ nt}. if m = {mt : t ∈ t} then the m-standard subsets of k will be called standard. our first step is to prove that (5) the set fy = ξ(φ −1(y)) is κ-large. fix a point x ∈ fy , a finite a ⊂ t and consider the set p = 〈x,a〉 = {x′ ∈ k : x′(t) = x(t) for all t ∈ a}. it follows from p ∩ fy 6= ∅ that ξ−1(p) ∩ φ−1(y) 6= ∅ and hence y ∈ q = φ(ξ−1(p)). the set q is compact and it follows from hπχ(y,l) ≤ κ that we can choose a π-base b of the space q at the point y such that |b| ≤ κ. for every b ∈ b pick a set ob ∈ τ(l) such that ∅ 6= ob ∩ q ⊂ ob ∩ q ⊂ b. it follows in a standard way from c(k) ≤ κ that (6) for any u ∈ τ∗(l), the set clk (ϕ −1(u)) depends on at most κ-many coordinates and coincides with the set ξ(clβc (φ −1(u))). apply (6) to find a set s ⊂ t of cardinality at most κ for which a ⊂ s and the set db = ξ(clβc (φ −1(ob ))) depends on s for any b ∈ b. the face kt \s is residual; to show that p ∩ fy covers kt \s fix any point w ∈ kt \s and consider the set e = {z ∈ k : πt \s (z) = w and πs (z) ∈ πs (p)}. clearly, e is a non-empty compact subset of p . fix any b ∈ b; it follows from ob ∩q 6= ∅ that there is a point u ∈ ξ −1(p) such that φ(u) ∈ ob; thus u ∈ φ −1(ob ) which shows that ξ(u) ∈ db ∩ p . define a point u′ ∈ k by the equalities πt \s (u ′) = w and πs (u ′) = πs (ξ(u)). since the sets db and p depend on s, we conclude that u ′ ∈ db ∩ p . on the other hand, πs (u ′) ∈ πs (p) so u ′ ∈ e, and therefore e ∩ db 6= ∅. as a consequence, φ(ξ−1(e)) ∩ ob 6= ∅ and hence φ(ξ −1(e)) ∩ b 6= ∅ for any b ∈ b; since φ(ξ−1(e)) is a closed subset of q and b is a π-base of q at y, we must have y ∈ φ(ξ−1(e)) which implies that ξ−1(e) ∩ φ−1(y) 6= ∅ and hence e ∩ fy 6= ∅. if v ∈ e ∩ fy then w = πt \s (v) ∈ πt \s (p ∩ fy ); the 44 v. v. tkachuk point w ∈ kt \s was chosen arbitrarily so p ∩fy covers kt \s and hence (5) is proved. by lemma 3.1 there exists a gκ-set g in the space k such that fy ⊂ g and g ∩ c = fy ∩ c = ϕ −1(y). therefore we can choose a family f of compact subsets of k such that |f| ≤ κ and c \ fy ⊂ ⋃ f ⊂ k \ fy . for any f ∈ f the set wf = l \ φ(ξ −1(f)) is an open neighbourhood of y in l and it is straightforward that h = ⋂ {wf : f ∈ f} is a gκ-subset of l such that h ∩ c′ = {y}. � corollary 3.3. suppose that c is a dense subspace of a product n = ∏ t∈t nt such that nw(nt) ≤ κ for each t ∈ t. assume that k is a compact space with t(k) ≤ κ and ϕ : c → k is a continuous (not necessarily surjective) map; let c′ = ϕ(c). then every closed subspace of c′ is a gκ-set, i.e., ψ(c ′) ≤ κ; in particular, ψ(c′) ≤ κ. proof. fix a non-empty closed set f ′ in the space c′ and let f = clk (f ′). consider the quotient map p : k → kf obtained by contracting the set f to a point and let q = p|c′. it is easy to see that we have the inequalities t(kf ) ≤ t(k) ≤ κ; denote by y the point of the space kf represented by f and let c′′ = p(c′). it follows from [1, theorem 2.2.20] that hπχ(y,kf ) ≤ κ so theorem 3.2, applied to the map p ◦ ϕ, implies that ψ(y,c′′) ≤ κ. since f ′ = q−1(y), we conclude that f ′ is a gκ-subset of c ′. � corollary 3.4. suppose that c is a dense subspace of a product n = ∏ t∈t nt such that nw(nt) ≤ κ for each t ∈ t. assume additionally that l(c) ≤ κ and k is a compact space with t(k) ≤ κ such that there exists a continuous (not necessarily surjective) map ϕ : c → k. if c′ = ϕ(c) then hl(c′) ≤ κ. proof. we have l(c′) ≤ κ while every closed subspace of the space c′ is a gκ-set by corollary 3.3. now, a standard proof shows that hl(c ′) ≤ κ. � corollary 3.5. if c is a dense subspace of a product of cosmic spaces and k is a compact space then, for any continuous map ϕ : c → k, we have ψ(ϕ(c)) ≤ t(k). the last corollary has several applications in cp-theory. let us start with the following observation. proposition 3.6. (folklore). if the space of a topological group g embeds in a compact space of countable tightness then g is metrizable. in particular, if cp(x) embeds in a compact space of countable tightness then cp(x) is second countable and hence x is countable. proof. assume that g is a dense subspace of a compact space k with t(k) ≤ ω. then πχ(g,g) = πχ(g,k) ≤ ω (see [1, theorem 2.2.20]) and hence we have the equality χ(g,g) = πχ(g,g) = ω for any g ∈ g (see [2, proposition 1.1]) so g is metrizable. � the following result is a curious generalization of proposition 3.6 for the case of condensations. condensations of cp(x) onto σ-compact spaces 45 corollary 3.7. for any x, the space cp(x) condenses onto a space embeddable in a compact space of countable tightness if and only if cp(x) condenses onto a second countable space. proof. apply corollary 3.5 and the equality ψ(cp(x)) = iw(cp(x)). � however, it would be interesting to find out whether any continuous image of cp(x) embeddable in a compact space of countable tightness has to be cosmic or even metrizable. it follows from corollary 3.3 that such an image is a perfect space. the following theorem shows that this conjecture is true when cp(x) is a lindelöf σ-space. theorem 3.8. suppose that ϕ : cp(x) → k is a continuous (not necessarily surjective) map and k is a compact space with t(k) ≤ ω; let y = ϕ(cp(x)). then (i) y is a perfect space of countable π-weight; (ii) if cp(x) is a lindelöf σ-space then y is cosmic. proof. that y is perfect is an immediate consequence of corollary 3.3. since ω1 is a precaliber of cp(x), it has to be also a precaliber of y and hence of y . the space y being compact, the cardinal ω1 is a caliber y ; it follows from t(y ) ≤ ω that y has a point-countable π-base [10]. this implies that πw(y ) = ω and hence πw(y ) = ω as well, i.e., we settled (i). if cp(x) is a lindelöf σ-space then cp(x) × cp(x) is lindelöf. the space y × y is a continuous image of cp(x) × cp(x) ≃ cp(x ⊕ x) so we can apply corollary 3.4 to convince ourselves that y ×y is hereditarily lindelöf and hence y condenses onto a second countable space. this, together with the lindelöf σ-property of y implies that nw(y ) ≤ ω and hence (ii) is proved. � in the sequel we will need the following lemma from [13]. lemma 3.9. if cp(x) = ⋃ n∈ω fn and every fn is closed in cp(x) then there exists n ∈ ω such that cp(x) embeds in fn. theorem 3.10. suppose that l(xn) = ω for all n ∈ n and cp(x) is lindelöf. if cp(x) condenses onto a σ-compact space y then the space x is separable and ψ(y ) = ω. proof. fix a condensation ϕ : cp(x) → y and a family {kn : n ∈ ω} of compact subsets of y such that y = ⋃ n∈ω kn. the set fn = ϕ −1(kn) is closed in cp(x) for every n ∈ ω. if n ∈ ω and s is an uncountable free sequence in kn then s ′ = ϕ−1(s) is an uncountable free sequence in fn which is impossible because l(fn) ≤ l(cp(x)) = ω and t(fn) ≤ t(cp(x)) = ω. this contradiction shows that kn has no uncountable free sequences and therefore t(fn) ≤ ω for any n ∈ ω. apply lemma 3.9 to see that there exists n ∈ ω such that c ≃ cp(x) for some c ⊂ fn. since ϕ|c maps c into kn, corollary 3.5 shows that ψ(ϕ(c)) ≤ ω. since ϕ|c is a condensation, we have ψ(cp(x)) = ψ(c) ≤ ψ(ϕ(c)) = ω and hence d(x) = ψ(cp(x)) = ω, i.e., x is separable as promised. 46 v. v. tkachuk it follows from ψ(cp(x)) = ω that cp(x) \ {f} is an fσ-set for any f ∈ cp(x). the space cp(x) being lindelöf, cp(x) \ {f} is lindelöf as well. therefore y \ {y} is lindelöf for any y ∈ y ; this implies that ψ(y ) ≤ ω. � corollary 3.11. suppose that x is an ω-monolithic space such that cp(x) is lindelöf and xn is lindelöf for any n ∈ n. if cp(x) condenses onto a σ-compact space y then nw(x) = nw(y ) = ω. proof. theorem 3.10 shows that the space x must be separable so nw(x) = ω by ω-monolithity of x. therefore nw(y ) ≤ nw(cp(x)) = nw(x) = ω. � the following result gives a complete answer (in a much stronger form) to problems 29 and 30 from the paper [4]. corollary 3.12. if x is an ω-monolithic compact space such that cp(x) is lindelöf and can be condensed onto a σ-compact space then x is metrizable. in particular, if x is a non-metrizable corson compact space then cp(x) does not condense onto a σ-compact space. corollary 3.13. under ma+¬ch if k is a compact space such that cp(k) is lindelöf and can be condensed onto a σ-compact space then x is metrizable. proof. it is a result of reznichenko (see [3, theorem iv.8.7]) that ma+¬ch together with the lindelöf property cp(k) implies that k is ω-monolithic so we can apply corollary 3.12 to see that k is metrizable. � corollary 3.14. assume that cp(x) is a lindelöf σ-space and there exists a condensation of cp(x) onto a σ-compact space y . then nw(x) = nw(y ) = ω. proof. denote by υx the hewitt realcompactification of x. it is evident that cp(x) is a continuous image of the space cp(υx). besides, z = υx is a lindelöf σ-space by [8, corollary 3.6] ; since cp(z) is also a lindelöf σ-space (see [14, theorem 2.3]), corollary 3.11 is applicable to z and we can conclude that nw(z) = nw(y ) = ω. since x ⊂ z, we have nw(x) ≤ nw(z) = ω. � 4. open problems. there are still many opportunities for discovering interesting facts about condensations of function spaces. the list below shows some possible lines of research in this direction. problem 4.1. suppose that k is a compact space of countable tightness and ϕ : cp(x) → k is a continuous map. is it true that ϕ(cp(x)) is cosmic or even metrizable? problem 4.2. suppose that cp(x) is lindelöf, k is a compact space of countable tightness and ϕ : cp(x) → k is a continuous map. is it true that ϕ(cp(x)) is cosmic or even metrizable? condensations of cp(x) onto σ-compact spaces 47 problem 4.3. suppose that cp(x) is hereditarily lindelöf, k is a compact space of countable tightness and ϕ : cp(x) → k is a continuous map. is it true that ϕ(cp(x)) is cosmic or even metrizable? problem 4.4. suppose that x is compact, k is a compact space of countable tightness and ϕ : cp(x) → k is a continuous map. is it true that ϕ(cp(x)) is cosmic or even metrizable? problem 4.5. is it true that, for any cardinal κ and any compact space k with t(k) ≤ ω, if ϕ : rκ → k is a continuous map then ϕ(rκ) is cosmic or even metrizable? problem 4.6. suppose that k is a compact space of countable tightness, g is a topological group with the lindelöf σ-property and ϕ : g → k is a continuous map. is it true that ϕ(g) is cosmic? problem 4.7. suppose that cp(x) is lindelöf and there exists a condensation of cp(x) onto a σ-compact space y . must y be cosmic? problem 4.8. suppose that cp(x) is lindelöf and there exists a condensation of cp(x) onto a σ-compact space y . must x be separable? problem 4.9. suppose that cp(x) condenses onto a space of countable πweight. must x be separable? problem 4.10. suppose that cp(x) is lindelöf and ϕ : cp(x) → y is a continuous onto map. is it true that every compact subspace of y has countable tightness? problem 4.11. suppose that k is eberlein compact and ϕ : cp(k) → y is a continuous surjective map of cp(x) onto a σ-compact space y . must y be cosmic? problem 4.12. suppose that k is corson compact and ϕ : cp(k) → y is a continuous surjective map of cp(x) onto a σ-compact space y . must y be cosmic? problem 4.13. suppose that x is a space such that cp(x) has the lindelöf σ-property and ϕ : cp(k) → y is a continuous surjective map of cp(x) onto a σ-compact space y . must y be cosmic? problem 4.14. suppose that cp(x) is lindelöf and x n is also lindelöf for any n ∈ n. assume additionally that there exists a condensation of cp(x) onto a σ-compact space y . must y be cosmic? problem 4.15. suppose that k is a perfectly normal compact space. is it true that every σ-compact continuous image of cp(x) has a countable network? 48 v. v. tkachuk references [1] a. v. arhangel’skii, the structure and classification of topological spaces and cardinal invariants (in russian), uspehi mat. nauk 33, no. 6 (1978), 29–84. 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[10] b. e. shapirovsky, cardinal invariants in compact spaces (in russian), seminar gen. topol., ed. by p.s. alexandroff, moscow univ. p.h., moscow, 1981, 162–187. [11] l. v. shirokov, on bicompacta which are continuous images of dense subspaces of topological products (in russian), soviet institute of science and technology information (viniti), registration number 2946–81, moscow, 1981. [12] m. g. tkachenko, factorization theorems for topological groups and their applications, topology appl. 38 (1991), 21–37. [13] v. v. tkachuk, decomposition of cp(x) into a countable union of subspaces with ”good” properties implies ”good” properties of cp(x), trans. moscow math. soc. 55 (1994), 239–248. [14] v. v. tkachuk, behaviour of the lindelöf σ-property in iterated function spaces, topology appl. 107 (2000), 297–305. received january 2008 accepted january 2009 vladimir v. tkachuk (vova@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, av. san rafael atlixco, 186, col. vicentina, iztapalapa, c.p. 09340, méxico d.f. kohlisinghkumaraggarwalagt.dvi @ applied general topology c© universidad politécnica de valencia volume 11, no. 1, 2010 pp. 43-55 between continuity and set connectedness j. k. kohli, d. singh, rajesh kumar and jeetendra aggarwal abstract. two new weak variants of continuity called ‘r-continuity’ and ‘f-continuity’ are introduced. their basic properties are studied and their place in the hierarchy of weak variants of continuity, that already exist in the literature, is elaborated. the class of r-continuous functions properly contains the class of continuous functions and is strictly contained in each of the three classes of (1) faintly continuous functions studied by long and herrignton (kyungpook math. j. 22(1982), 7-14); (2) d-continuous functions introduced by kohli (bull. cal. math. soc. 84 (1992), 39-46), and (3) f-continuous functions which in turn are strictly contained in the class of z-continuous functions studied by singal and niemse (math. student 66 (1997), 193-210). so the class of r-continuous functions is also properly contained in each of the classes of d∗-continuous functions, dδ-continuous function and set connected functions. 2000 ams classification: primary 54c05, 54c08, 54c10 secondary 54d10, 54d15. keywords: almost continuous function, d-continuous function, z-continuous function, quasi θ-continuous function, faintly continuous function, functionally hausdorff space, zero set. 1. introduction functions occur everywhere in mathematics and applications of mathematics. continuous functions play a prominent role in topology, analysis and many other branches of mathematics. in many situations/applications in geometry, topology, functional analysis and complex analysis, continuity is not sufficient and a condition stronger than continuity is required. on the other hand a condition strictly weaker than continuity is sufficient to meet the demand of a particular situation. several strong variants of continuity occur in the lore of mathematical literature and applications of mathematics (see for example [6, 12, 13, 14, 19, 20, 24, 27, 30, 32, 34, 38]), while others are weaker than 44 j. k. kohli, d. singh, r. kumar and j. aggarwal continuity (see for example [7, 11, 15, 23, 25, 28, 35, 36, 37]), and yet others are independent of continuity (see for example [17, 18, 21, 22, 26, 31]). in this paper we restrict ourselves to the study of weak variants of continuity and introduce two new weak variants of continuity called ‘r-continuity’ and ‘f-continuity’ and study their basic properties. we discuss their interrelations and interconnections with other weak variants of continuity that already exist in the literature. sufficient conditions on range are given for an f-continuous (r-continuous) function to be continuous. the other weak variants of continuity with which we shall be dealing in this paper include among others, almost continuous functions [36], θ-continuous function [7], d-continuous functions [11], faintly continuous functions [28] and z-continuous functions [35]. it turns out that the class of r-continuous functions properly includes the class of continuous functions and is strictly contained in each of the three classes of (i) d-continuous functions [11], (ii) faintly continuous functions [28], and (iii) f-continuous functions; while the class of f-continuous functions properly contains the class of r-continuous functions and is strictly contained in the class of z-continuous functions [35]. this in turn implies that the class of r-continuous functions is also properly contained in each of the classes of d∗-continuous functions [37], dδ-continuous functions [15], z-continuous functions [35] and set connected functions [23]. the organization of the paper is as follows. section 2 is devoted to preliminaries and basic definitions. in section 3, we introduce the notions of ‘r-continuous function’ and ‘f-continuous function’ and elaborate on their place in the hierarchy of weak variants of continuity that already exist in the literature. examples are included to reflect upon the distinctiveness of the old and new variants of continuity discussed in the paper. section 4 is devoted to the study of basic properties of r-continuous functions and f-continuous functions. in section 5, we discuss the properties of graphs of r-continuous (f-continuous) functions. in section 6, we consider retopologization of the range of an r-continuous (f-continuous) function and conclude with alternative proofs of certain results of preceding sections. in section 7, we discuss conditions on the range of a function under which certain weak variants of continuity are identical among themselves and/or coincide with continuity. 2. preliminaries and basic definitions a collection β of subsets of a space x is called an open complementary system [8] if β consists of open sets such that for every b ∈ β, there exist b1, b2, . . . , ∈ β. with b = ∪{x \bi : i ∈ n}. a subset a of a space x is called a strongly open fσ -set [8] if there exists a countable open complementary system β(a) with a ∈ β(a). the complement of a strongly open fσ-set is called strongly closed gδ-set. a subset a of a space x is called regular gδ-set [29] if a is the intersection of a sequence of closed sets whose interiors contain a, i.e., if a = ∞ ⋂ n=1 fn = ∞ ⋂ n=1 f 0n , where each fn is a closed subset of x (here f 0n denotes the interior of fn). the complement of a regular gδ-set is between continuity and set connectedness 45 called a regular fσ-set. a point x ∈ x is called a θ-adherent point [40] of a ⊂ x if every closed neigbourhood of x intersects a. let clθa denote the set of all θ-adherent points of a. the set a is called θ-closed it a = clθa. the complement of a θ-closed set is referred to as a θ-open set. definition 2.1. a function f : x → y is said to be (i) dδ-continuous [15] if for each point x ∈ x and each regular fσ-set v containing f (x) there is an open set u containing x such that f (u ) ⊂ v . (ii) d∗-continuous [37] if for each point x ∈ x and each strongly open fσset v containing f (x) there is an open set u containing x such that f (u ) ⊂ v . (iii) almost continuous [36] if for each x ∈ x and each open set v containing f (x) there is an open set u containing x such that f (u ) ⊂ (v )0. (iv) d-continuous [11] if for each x ∈ x and each open fσset v containing f (x) there is an open set u containing x such that f (u ) ⊂ v . (v) z-continuous [35] if for each x ∈ x and each cozero set v containing f (x) there is an open set u containing x such that f (u ) ⊂ v . (vi) θ-continuous [7] if for each x ∈ x and each open set v containing f (x) there is an open set u containing x such that f (u ) ⊂ v . (vii) weakly continuous [25] if for each x ∈ x and each open set v containing f (x) there exists an open set u containing x such that f (u ) ⊂ v . (viii) faintly continuous [28] if for each x ∈ x and each θ-open set v containing f (x) there exists an open set u containing x such that f (u ) ⊂ v . (ix) quasi θ-continuous function [33] if for each x ∈ x and each θ-open set v containing f (x) there exists a θ-open set u containing x such that f (u ) ⊂ v . (x) set connected mapping [23] if whenever x is connected between a and b; f (x) is connected between f (a) and f (b). (xi) cl-continuous∗ if f −1(v ) is open in x for every clopen set v ⊂ y . definition 2.2. a topological space x is said to be (i) functionally regular [2] if for each closed set a in x and a point x 6∈ a there exists a continuous real-valued function f defined on x such that f (x) 6∈ f (a); or equivalently for each x ∈ x and each open set u containing x there exists a zero set z in x such that x ∈ z ⊂ u . (ii) r0-space [4] if for each open set u in x and each x ∈ u , {x} ⊂ u . (iii) d-regular [8] if it has a base of open fσ-sets. (iv) d-completely regular [8] if it has a base of strongly open fσ-sets (v) dδ-completely regular [16] if it has a base of regular fσ-sets. definition 2.3. a collection k of subsets of a space x is said to be (i) closure preserving if cl(∪l) = ∪{cl(l)|l ∈ l} for any subcollection l of k; and ∗cl-continuous functions have been referred to as slightly continuous in ([10], [15]). 46 j. k. kohli, d. singh, r. kumar and j. aggarwal (ii) hereditarily closure preserving if whenever a subset h(k) ⊂ k is chosen for each k ∈ k, then the resulting collection h = {h(k)|k ∈ k} is closure preserving. 3. r-continuous functions and f -continuous functions let x be a topological space. an open subset u of x is said to be r-open [20] if for each x ∈ u there exists a closed set b such that x ∈ b ⊂ u or equivalently, if u is expressible as a union of closed sets. an open subset w of x is said to be f-open [19] if for each x ∈ w there exists a zero set z such that x ∈ z ⊂ w or equivalently, if w is expressible as a union of zero sets. definition 3.1. a function f : x → y from a topological space x into a topological space y is said to be f-continuous (r-continuous) at x ∈ x, if for each f -open (r-open) set v in y containing f (x) there exists an open set u containing x such that f (u ) ⊂ v . the function f is said to be f-continuous (r-continuous) if it is f-continuous (r-continuous) at each x ∈ x. the following diagram (figure 1) well illustrates the interrelations and interconnections that exist between r-continuity, f-continuity, and certain other weak variants of continuity that already exist in the literature (see definitions 2.1). figure 1 examples in [1, 11, 15, 35, 36, 37] and the following examples show that none of the above implications is reversible. between continuity and set connectedness 47 example 3.2. let x be the real line endowed with indiscrete topology and y be the real line equipped with the right order topology [39, example 50]. let f : x → y be the identity function from x onto y . then f is an r-continuous function which is not continuous. example 3.3. let x be the real line endowed with cofinite topology and let y be the real line equipped with cocountable topology. let f be the identity mapping of x onto y . then f is d-continuous but not r-continuous. example 3.4. let x denote the real line endowed with the usual euclidean topology u. let y denote the real line endowed with the topology generated by u together with all sets of the form (a, b) − k, where k = {1/n : n ∈ z+}. let f denote the identity mapping of x onto y . then f is faintly continuous but not r-continuous. example 3.5. let x = y = z+, where z+ is the set of positive integers. let x be endowed with indiscrete topology τ and y be endowed with relative prime integer topology τ ∗ [39, example 60]. then every real-valued continuous function defined on (y, τ ∗) is constant. let f denote the identity function from x onto y . then f is f-continuous but not r-continuous. example 3.6. let x be the real line endowed with usual topology u and let y be the real line endowed with the topology u∗ generated by u and the addition of all sets of the form q∩u , where u ∈ u and q is the set of rationals. the space (y, u∗) is a functionally hausdorff lindelöf space and so by [2, theorem 3] it is a functionally regular space. let g : (x, u) → (y, u∗) denote the identity mapping. then g is not an f-continuous function since q is f -open in y but its inverse image is not open in x. however, we claim that g is a z-continuous function. to this end, let f : (r, u∗) → (r, u) be any continuous function. to show that f : (r, u) → (r, u) is continuous, let x ∈ r. the following cases arise: case i: x is irrational. clearly f is u − u continuous at x. case ii: x is rational. let ǫ > 0 be given. then there exist an open interval i containing x such that f (q ∩ i) ⊂ (f (x) − ǫ/2, f (x) + ǫ/2). now, since q is dense in (r, u∗), clu∗ (q∩i) = clu∗ (i). therefore, f (i) ⊂ f (clu∗ i) = f (clu∗ (q ∩ i)) ⊂ f (q ∩ i) ⊂ [f (x) − ǫ/2, f (x) + ǫ/2] ⊂ (f (x) − ǫ, f (x) + ǫ). that is f (i) ⊂ (f (x) − ǫ, f (x) + ǫ). thus it turns out that the spaces (r, u∗) and (r, u) have same classes of zero sets. consequently, g is z-continuous. proposition 3.7. a function f : x → y from a topological space into a functionally regular space (r0-space) y is continuous if and only if it is f-continuous (r-continuous). 48 j. k. kohli, d. singh, r. kumar and j. aggarwal 4. basic properties of r-continuous functions and f-continuous functions theorem 4.1. let f : x → y be a function from a topological space x into a topological space y. then the following statements are equivalent. (a) f is f-continuous (r-continuous) (b) the set f −1(v ) is open for every f -open (r-open) subset v ⊂ y . (c) the set f −1(b) is closed for every f -closed (r-closed) subset b ⊂ y . (d) for each x ∈ x and each net {xα}α∈λ which converges to x, the net {f (xα)}α∈λ is eventually in each f -open (r-open) set containing f (x). proof. easy. � theorem 4.2. let f : x → y be any function. then the following statements are true. (a) if f : x →y is f-continuous (r-continuous) and a ⊂ x, then f|a : a→y is f-continuous (r-continuous). (b) if {uα : α ∈ λ} is an open cover of x and for each α ∈ λ, fα = f|uα is f-continuous (r-continuous), then f is f-continuous (r-continuous). (c) if {fβ : β ∈ λ} is a hereditarily closure preserving closed cover of x and if for each β ∈ λ, fβ = f|fβ is f-continuous (r-continuous), then f is f-continuous (r-continuous). proof. (a) let v be an f -open (r-open) subset of y . then f −1(v ) is an open set in view of f-continuity (r-continuity) of f . so (f|a)−1(v ) = f −1(v ) ∩ a is an open subset of a. (b) let v be an f -open (r-open) subset of y . then f −1(v ) =∪{f −1α (v ) : α∈λ}. since each fα is f-continuous (r-continuous), each f −1 α (v ) is open in uα and hence in x. thus f −1(v ) being the union of open sets is open. (c) let b be an f -closed (r-closed) subset of y . then f −1(b) =∪{f −1 β (b)|β ∈λ}. since each fβ is f-continuous (r-continuous), each f −1 β (b) is closed in fβ and hence in x. again, since {fβ : β ∈ λ} is a hereditarily closure preserving closed cover of x, the collection {f −1 β (b) : β ∈ λ} is a closure preserving collection of closed sets. consequently, their union f −1(b) is closed. � theorem 4.3. if f : x → y is continuous and g : y → z is f-continuous (r-continuous), then g ◦ f is f-continuous (r-continuous). proof. let w be an f -open (r-open) subset of z. then g−1(w ) is open in y . again, since f is continuous f −1(g−1(w )) = (g ◦ f )−1(w ) is open in x. so g ◦ f is f-continuous (r-continuous). � example 4.4. in general f-continuity of g ◦ f need not imply continuity of f . for example, let x be the real line with smirnov’s deleted sequence topology [39, example 64], y be the real line with countable complement extension topology [39, example 63] and z be the real line equipped with usual topology. let f : x → y and g : y → z be the identity mappings. then g ◦ f and g are f-continuous. however, f is not a continuous function. between continuity and set connectedness 49 theorem 4.5. let f : x →y be a quotient map. then a function g : y →z is f-continuous (r-continuous) if and only if g◦f is f-continuous (r-continuous). proof. to prove necessity, let w be an f -closed (r-closed) subset of z. then g−1(w ) is a closed subset of y and since f is a quotient map, (g ◦ f )−1(w ) = f −1(g−1(w )) is closed. hence g ◦ f is f-continuous. to prove sufficiency, let v be an f -open (r-open) subset of z. then (g ◦ f )−1(v ) = f −1(g−1(v )) is open in x. since f is a quotient map, g−1(v ) is open in y and so g is f-continuous (r-continuous). � theorem 4.6. let f : x → y be either an open or closed surjection and let g : y → z be any function such that g ◦ f is f-continuous (r-continuous). then g is f-continuous (r-continuous). proof. assume that f is open (respectively, closed) and let v be an f -open (ropen) (respectively, f -closed (r-closed)) subset of z. since g◦f is f-continuous (r-continuous), (g ◦ f )−1(v ) = f −1(g−1(v )) is open (respectively, closed). since f is an open (respectively, closed) surjection, f (f −1(g−1(v ))) = g−1(v ) is open (respectively, closed) and consequently, g is f-continuous (r-continuous). � theorem 4.7. let f : x → ∏ α∈λ xα be a function into a product space. if f is f-continuous (r-continuous), then each pα ◦f is f-continuous (r-continuous), where pα denotes the projection map pα : ∏ α∈λ xα → xα. proof. let fβ be an f -closed (r-closed) subset of xβ. now (pβ ◦ f ) −1(fβ ) = f −1(p−1 β (fβ )) = f −1 ( fβ ×( ∏ α6=β xα) ) . since f is f-continuous (r-continuous) and since fβ × ( ∏ α6=β xα ) is an f -closed (r-closed) set in the product space ∏ α∈λ xα, f −1 ( fβ × ( ∏ α6=β xα ) ) is a closed set in x and consequently pα ◦ f is f-continuous (r-continuous). � theorem 4.8. let fα : xα → yα be a mapping for each α ∈ λ and let f : πxα → πyα be the mapping defined by f ((xα)) = (fα(xα)) for each point (xα) in πxα. if f is f-continuous (r-continuous), then fα is f-continuous (r-continuous) for each α ∈ λ. proof. let fβ be an f -closed (r-closed) subset of yβ . then fβ× ( ∏ α6=β yα ) is an f -closed (r-closed) subset of the product space ∏ α∈λ yα. since f is f-continuous (r-continuous), f −1 ( fβ × ∏ α6=β yα ) = f −1 β (fβ )× ( ∏ α6=β xα ) is closed in ∏ α∈λ xα. consequently, f −1 β (fβ ) is closed in xβ and so fβ is f-continuous (r-continuous). � 50 j. k. kohli, d. singh, r. kumar and j. aggarwal definition 4.9. let f : x → y be any function. then the function g : x → x × y defined by g(x) = (x, f (x)) is called the graph function with respect to f . theorem 4.10. let f : x → y be a function such that the graph function g : x → x × y is f-continuous (r-continuous). then f is f-continuous (r-continuous). proof. let x ∈ x and let v be an f -open set containing f (x). since y −v is an f -closed (r-closed) set, so is x × (y − v ) = x × y − p−1y (v ), where py denotes the projection of x × y onto y . therefore, p−1y (v ) is an f -open (r-open) subset of x × y containing g(x). since g is f-continuous (r-continuous), there is an open set u containing x, such that g(u ) ⊂ p−1y (v ). it follows that py(g(u )) = f (u ) ⊂ v and so f is f-continuous (r-continuous). � 5. properties of graphs of f-continuous (r-conttnuous) functions in this section we study how the graph of an f-continuous (r-continuous) function f : x → y is situated in the product space x × y . first we prove the following proposition which will be used in the sequel. proposition 5.1. for a topological space x, the following statements are equivalent. (a) x is functionally hausdorff. (b) every pair of distinct points in x are contained in disjoint cozero sets. (c) every pair of distinct points in x are contained in disjoint f -open sets. (d) for every pair of distinct points in x there exists an f -open set containing one of the points but not the other. proof. the equivalence of (a)-(c) is discussed in [19, proposition 5.2]. the implication (c) ⇒ (d) is trivial. to show that (d) ⇒ (a), let x, y ∈ x, x 6= y and let v be an f -open set containing one of the points x and y but not the other. to be precise, suppose that x ∈ v . since v is an f -open set, there exists a zero set z such that x ∈ z ⊂ v . let f : x → [0, 1] be a continuous function with z as its zero set. then f (x) = 0 and f (y) 6= 0 and so x is functionally hausdorff. � definition 5.2. the graph g(f ) of a function f : x → y is said to be (i) cozero closed with respect to x × y if for each (x, y) 6∈ g(f ) there exist cozero sets u and v containing x and y, respectively such that (u × v ) ∩ g(f ) = φ; (ii) cozero closed with respect to x if for each (x, y) 6∈ g(f ) there exists a cozero set u containing x and an open set v containing y such that (u × v ) ∩ g(f ) = φ; and (iii) f-closed with respect to x × y if for each (x, y) /∈ g(f ) there exist f -open sets u (in x) and v (in y ) containing x and y respectively such that (u × v ) ∩ g(f ) = φ. between continuity and set connectedness 51 theorem 5.3. let f : x → y be an injection such that its graph g(f ) is cozero closed with respect to x. then x is functionally hausdorff. proof. let x1, x2 ∈ x, x1 6= x2. since f is an injection (x1, f (x2)) 6∈ g(f ). in view of cozero closedness of the graph g(f ), there exists a cozero set u containing x1 and an open set v containing f (x2) such that (u ×v )∩g(f ) = φ and hence u ∩f −1(v ) = φ. therefore x2 6∈ u . since every cozero set is f -open, in view of proposition 5.1, x is functionally hausdorff. � theorem 5.4. let f : x → y be a z-continuous function into a functionally hausdorff space y . then the graph g(f ) of f is cozero closed with respect to x × y . proof. suppose (x, y) 6∈ g(f ). then f (x) 6= y. since y is functionally hausdorff, in view of proposition 5.1, there exist disjoint cozero sets v and w containing f (x) and y, respectively. since f is z-continuous, by [35, theorem 2.3] u = f −1(v ) is a cozero set containing x and f (u ) ⊂ f ⊂ y − w . consequently, u × v contains no point of g(f ) and so g(f ) is cozero closed with respect to x × y . � corollary 5.5. let f : x → y be an f-continuous function. if y is functionally hausdorff, then f has an f -closed graph with respect to x × y . definition 5.6 ([26]). a function f : x → y is said to have strongly closed graph, if for each (x, y) 6∈ g(f ), there exist open sets u and v containing x and y, respectively such that u × v contains no point of g(f ). theorem 5.7. let f : x → y be an r-continuous function into a hausdorff space y . then f has strongly closed graph. proof. suppose (x, y) 6∈ g(f ). then f (x) 6= y. in view of hausdorffhess of y , there are disjoint open sets v and w such that f (x) ∈ v and y ∈ w and v is an r-open set. since f is r-continuous, f −1(v ) is open. then u = f −1(v ) is an open set containing x and f (u ) ⊂ v ⊂ y − w consequently, u × w contains no point of g(f ) and so g(f ) is strongly closed x × y . � singal and niemse [35] showed that the equalizer of two z-continuous functions into a functionally hausdorff space is z-closed ([35, theorem 4.3]). consequently, the equalizer of two f-continuous into a functionally hausdorff space is z-closed and hence f -closed. in contrast the following result shows that the equalizer of two r-continuous functions into a hausdorff space is closed. theorem 5.8. let f : x → y be f-continuous (r-continuous) injection into a functionally hausdorff (hausdorff ) space y . then x is hausdorff. proof. let x and y be any two distinct points in x. then f (x) 6= f (y). since y is functionally hausdorff (hausdorff) there exist disjoint f -open (r-open) sets u and v containing f (x) and f (y) respectively. since f is f-continuous (r-continuous), f −1(u ) and f −1(v ) are disjoint open sets containing x and y, respectively. hence x is hausdorff. � 52 j. k. kohli, d. singh, r. kumar and j. aggarwal 6. change of topology in this section we show that if the range of an f-continuous (r-continuous) function is retopologized in an appropriate way then f is simply a continuous function. this fact in conjunction with the properties of continuous functions leads to alternative proofs of certain results of section 4. for let (x, τ ) be a topological space. let τf (respectively τr) denote the collection of all f -open (respectively r-open) subsets of x. it is easily verified that the collection τf (respectively τr) is a topology on x. moreover, since every zero set is a closed set, τf ⊂ τr ⊂ τ . theorem 6.1. a space (x, τ ) is functionally regular (respectively r0-space) if and only if τ = τf (respectively τ = τr). theorem 6.2. for a topological space (x, τ ) the following statements are equivalent. (a) (x, τ ) is functionally regular. (b) every f-continuous function f : y → (x, τ ) from a topological space y into (x, τ ) is continuous. (c) the identity mapping 1x : (x, τf ) → (x, τ ) is continuous. proof. (a) ⇒ (b). let u ∈ τf . since (x, τ ) is functionally regular space, u is f -open. in view of f-continuity of f, f −1(u ) is open in y and so f is continuous. (b) ⇒ (c). by the definition of τf , the identity mapping 1x : (x, τf ) → (x, τ ) is f-continuous and hence in view of (b), it is continuous. (c) ⇒ (a). by (c) every open set in (x, τ ) is f -open and so in view of theorem 6.1 it is a functionally regular space. � theorem 6.3. for a topological space (x, τ ), the following statements are equivalent. (a) (x, τ ) is an r0-space. (b) every r-continuous function f : y → (x, τ ) from a topological space y into (x, τ ) is continuous. (c) the identity mapping 1x : (x, τr) → (x, τ ) is continuous. proof of theorem 6.3 is similar to that of theorem 6.2 and hence omitted. many of the results studied in section 4 now follow from theorem 6.2 or 6.3 and the corresponding standard properties of continuous functions. furthermore, as an application of theorems 6.2 and 6.3 we provide with an alternative proofs of following well known results in the literature. theorem 6.4 (aull [3]). any product of functionally regular spaces is functionally regular. proof. let {xα : α ∈ λ} be any collection of functionally regular spaces. let x = ∏ xα. to show that x is functionally regular, in view of theorem 6.2 it is sufficient to prove that every f-continuous function f : y → x is continuous. between continuity and set connectedness 53 thus it is sufficient to prove that πα ◦ f is continuous for each α ∈ λ, where πα : x → xα denote the projection onto the α-th co-ordinate space xα. to this end, let uα be an f -open subset of xα. let uα = ⋃ γ∈λuα zαγ , where each zαγ is a zero set in xα. it is easily verified that uα × ( ∏ β 6=α xβ ) is an f -open subset of x. since f is f-continuous, f −1 ( uα × ( ∏ β 6=α xβ ) ) is open in y . again, since (πα ◦ f ) −1(uα) = f −1(π−1α (uα)) = f −1 ( uα × ( ∏ β 6=α xβ ) ) , πα ◦ f is f-continuous. since each xα is functionally regular, in view of theorem 6.1 each πα ◦ f is continuous. this completes the proof of theorem 6.4. � theorem 6.5. any product of r0-spaces is an r0-space. proof of theorem 6.5 ([5]) is similar to that of theorem 6.4 except for obvious modifications and hence omitted. 7. when do weak forms of continuity imply continuity? variants of continuity defined in sections 2 and 3 (see definitions 2.1 and 3.1) are distinct from each other and are strictly weaker than continuity in general. however, if the range space y is suitably augmented, then many of them coincide among themselves and/or identical with continuity. for the convenience of reader, we summarize the following observations which are either easily verified or well elaborated in the literature (see the corresponding papers cited in the references). 7.1. y is an r0-space if and only if continuity and r-continuity are identical notions. 7.2. y is a semiregular space if and only if continuity and almost continuity coincide [36]. 7.3. y is a regular space if and only if continuous ≡ almost continuous ≡ θcontinuous ≡ quasi θ-continuous ≡ weakly continuous ≡ faintly continuous ≡ r-continuous ([7], [25], [28]). 7.4. y is a d-regular space if and only if continuous ≡ d-continuous ≡ rcontinuous [11]. 7.5. y is a d-completely regular space if and only if continuous ≡ d-continuous ≡ d∗-continuous ≡ r-continuous [37]. 7.6. y is a functionally regular space if and only if continuous ≡ f-continuous ≡ r-continuous 7.7. y is a dδ-completely regular space if and only if continuous ≡ almost continuous ≡ θ-continuous ≡ quasi θ-continuous ≡ weakly continuous ≡ faintly continuous ≡ dδ-continuous ≡ d-continuous ≡ r-continuous [16]. 54 j. k. kohli, d. singh, r. kumar and j. aggarwal 7.8. y is a completely regular space if and only if all the classes of functions from (1) to (12) in the diagram (figure 1) are identical. 7.9. y is a zero dimensional space if and only if all the classes of functions from (1) to (14) in the diagram (figure 1) coincide. acknowledgements. the research of second author was partially supported by university grants commission, india. the fourth author gratefully acknowledges jrf fellowship awarded by the council of scientific and industrial research, india. references [1] s. p. arya and m. deb, on θ-continuous mappings, math. student 42 (1974), 81–89. 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[40] n. v. velic̆ko, h-closed topological spaces, amer. math. soc. transl. 78, no. 2 (1968), 103–118. received july 2009 accepted april 2010 j. k. kohli (jk kohli@yahoo.com) department of mathematics, hindu college, university of delhi, delhi 110 007, india d. singh (dstopology@rediffmail.com) department of mathematics, sri aurobindo college, university of delhi-south campus, delhi 110 017, india rajesh kumar (rkumar2704@yahoo.co.in) department of mathematics, acharya narendra dev college, university of delhi, govindpuri, kalkaji, delhi 110 019, india jeetendra aggarwal (jitenaggarwal@gmail.com) department of mathematics, university of delhi, delhi 110 007, india @ appl. gen. topol. 23, no. 2 (2022), 453-462 doi:10.4995/agt.2022.16641 © agt, upv, 2022 the ε-approximated complete invariance property g. garćıa departamento de matemáticas, uned, 03202 elche, alicante, spain (gonzalogarciamacias@gmail.com) dedicated to my teacher and friend prof. dr. gaspar mora communicated by o. okunev abstract in the present paper we introduce a generalization of the complete invariance property (cip) for metric spaces, which we will call the εapproximated complete invariance property (ε-acip). for our goals, we will use the so called degree of nondensifiability (dnd) which, roughly speaking, measures (in the specified sense) the distance from a bounded metric space to its class of peano continua. our main result relates the ε-acip with the dnd and, in particular, proves that a densifiable metric space has the ε-acip for each ε > 0. also, some essentials differences between the cip and the ε-acip are shown. 2020 msc: 54h25; 54c50; 54c10 ; 54e40. keywords: complete invariance property (cip); set of fixed points; peano continua; α-dense curves; degree of nondensifiability. 1. introduction in 1973 ward [20] introduced the following concept: definition 1.1. a topological space x has the complete invariance property (cip) if for every non-empty and closed c ⊂ x there is a continuous mapping f : x −→ x such that fix(f) = c, where fix(f) stands for the set of fixed points of f. received 11 november 2021 – accepted 16 august 2022 http://dx.doi.org/10.4995/agt.2022.16641 https://orcid.org/0000-0001-6840-3226 g. garćıa as is mentioned in [8], some spaces known to have the cip include n-cells, dendrites, convex subsets of banach spaces, compact manifolds without boundary, and all compact triangulable manifolds with or without boundary. it is convenient to recall that a peano continuum is a compact, connected and locally connected metric space (x,d), or equivalently, by the hahn-mazurkiewicz theorem (see, for instance, [19, 21]), x is the continuous image of the unit interval i = [0, 1]. in [20] was asked the following: has every peano continuum the cip? the answer is negative: in [8, 9] are given some examples of n-dimensional peano continua, with n > 1, that fail to have the cip. however, for n = 1 the situation is very different: theorem 1.2 (martin and tymchatyn [10], 1980). every 1-dimensional peano continuum has the cip. since the publication of the ward’s paper, many others works have been devoted to the study and analysis of the cip and other issues related with it, see [2, 5, 6, 7, 11, 12, 13, 22] and references therein. so, it seems that the study of the cip problem, and its variants, is an interesting and actual topic. on the other hand, the so called degree of nondensifiability (dnd), explained in detail in section 2, has been used to prove, under suitable conditions, the existence of fixed points of continuous self mappings defined into a non-empty, bounded, closed and convex subset of a banach space (see [3] and references therein). in the present paper, for a given metric space (x,d), we introduce the concept of ε-approximated complete invariance property (ε-acip), which generalizes the cip one and, by using the dnd, we relate in our main result (see theorem 3.2) this novel concept with the dnd of a bounded metric space. in particular, our main result proves that densifiable metric spaces (and therefore every peano continuum) have the ε-acip for each ε > 0. also, and as consequence of our main result, we derive some properties for the ε-acip which are not satisfied by the cip, namely, that the ε-acip is preserved (in the specified sense) by the countable or finite products of bounded metric spaces or by the continuous image of a bounded metric space. 2. the degree of nondensifiability in this section, and for a better comprehension of the manuscript, we recall the concepts of α-dense curves and densifiable sets and also that of degree of nondensifiability. as in section 1, (x,d) will be a metric space and we denote by b(x) the class of non-empty and bounded subsets of x. in 1997 cherruault and mora introduced in [15] the following concepts: definition 2.1. let α ≥ 0 and b ∈ b(x). a continuous mapping γ : i −→ (x,d) is said to be an α-dense curve in b if it satisfies: (i) γ(i) ⊂ b. (ii) for any x ∈ b there is y ∈ γ(i) such that d(x,y) ≤ α. © agt, upv, 2022 appl. gen. topol. 23, no. 2 454 the ε-approximated complete invariance property the class of α-dense curves in b is denoted by γα,b. the set b is said to be densifiable if γα,b 6= ∅ for each α > 0. for a detailed exposition of the α-dense curves and densifiable sets, see [1, 14, 17]. some comments are necessary before to continue: (i) let us note that, given b ∈ b(x), γα,b 6= ∅ for each α ≥ diam(b), the diameter of b. indeed, fixed x0 ∈ b, the mapping γ(t) = x0 is an α-dense curve in b for each α ≥ diam(b). (ii) if b = in for some integer n > 1 then a 0-dense curve is, precisely, a space-filling curve (see [19]), i.e. a continuous mapping from i onto in. so, we can say that the α-dense curves are a generalization of the space-filling curves. (iii) by recalling that the hausdorff distance between b1,b2 ∈b(x) is given by dh(b1,b2) = max { sup b1∈b1 inf b2∈b2 d(b1,b2), sup b2∈b2 inf b1∈b1 d(b1,b2) } , is clear that if γ is an α-dense curve in b ∈b(x), then dh(b,γ(i)) ≤ α. we also recall that dh is pseudometric, and is a metric if x is complete, and a metric in the class of non-empty, bounded and closed subsets of x. next, we show some examples. example 2.2 (a compact and connected but not densifiable set). let, in the euclidean plane, the set b = { (x, sin(1/x)) : x ∈ [−1, 0) ∪ (0, 1] }⋃{ (0,y) : y ∈ [−1, 1] } . then, given any continuous γ : i −→ r2 with γ(i) ⊂ b, γ(i) has to be contained is some of the three connected components of b. so, if 0 < α < 1, there is not an α-dense curve in b, and consequently b is not densifiable. example 2.3 (a densifiable set without the cip). consider, in the euclidean plane, the sets b1 = { (x, sin(1/x)) : x ∈ (0, 1] } , b2 = { (0,y) : y ∈ [−1, 1] } , and let b = b1 ∪b2, often called the topologist’s sine. then, is easy to prove that b is densifiable. in the following lines we will show that b has not the cip. define the set c = b ∩ (i × [−1, 0]), and assume that there is a continuous f : b −→ b such that f(c) = c. as c ∩b1 = f(c ∩b1) ⊂ f(b1) and f(b1) is path-wise connected, f(b1) = b1. hence, as b is compact, we have b = b1 = f(b1) ⊂ f(b) = f(b), where the bar stands for the closure. this means that f is surjective and therefore f(b2) = b2. so, there is a continuous surjection ϕ : [−1, 1] −→ [−1, 1] such that f(0,y) = (0,ϕ(y)) for all y ∈ [−1, 1]. hence there exists a ∈ [−1, 1] such that ϕ(a) = 1. © agt, upv, 2022 appl. gen. topol. 23, no. 2 455 g. garćıa set b = ϕ(1). as [−1, 0] is the set of fixed points of ϕ, we conclude that a ∈ (0, 1) and b ∈ [−1, 1). next, define ψ : [a, 1] −→ [−1, 1]2 as ψ(x) = (x,ϕ(x)) and denote by ∆ the diagonal of [−1, 1]2. let us note that ψ([a, 1]) ∩ ∆ = ∅ because ϕ does not have any fixed point in [a, 1]. hence, ψ is a path (see the below definition) in [−1, 1] \ ∆. but, the set [−1, 1]\∆ has two components ω1 and ω2 which are above and below of ∆, respectively. then, ψ(a) = (a, 1) ∈ ω1 and ψ(1) = (1,b) ∈ ω2, which is contradictory. so, b does not have the cip as claimed. following willard [21], we recall that a topological space y is said to be path-wise connected (resp. arc-wise connected ) if for any x,y ∈ b there is a continuous (resp. a one-to-one continuous) f : i −→ b, often called path (resp. arc) such that f(0) = x and f(1) = y. however, if y is a hausdorff space (and, in particular, a metric space), both concepts are equivalents (see [21, corollary 31.6]. here, as we work with metric spaces, for our goals is more convenient to use the term arc-wise connected. at this point, we can state the following result (see [17]): proposition 2.4. let b ∈b(x) be arc-wise connected. then b is densifiable if, and only if, it is precompact. although, by the hahn-mazurkiewicz theorem, in is a peano continuum and in particular densifiable, the above result also demonstrates us that in is densifiable. moreover, we can give an explicit expression of an α-dense curve in in, γ, for an arbitrarily small α > 0, such that γ(i) is also a 1-dimensional peano continuum: example 2.5 (1-dimensional peano continua densifying in). fixed n > 1, for a given integer k ≥ 1 define γk : i −→ rn as γk(t) = ( t, 1 2 (1 − cos(πmt)), . . . , 1 2 (1 − cos(πmk−1t)) ) , for all t ∈ i. then, γk is a √ n−1 k -dense curve in in (see [1, proposition 9.5.4]) remark 2.6. other examples of α-dense curves in more general subsets of rn than in can be found in [18]. from the concepts of α-dense curves, we can define the so called degree of nondensifiability, which was introduced by mora and mira in [16] and analyzed in [4]: definition 2.7. given b ∈ b(x), we define the degree of nondensifiability, dnd, of b as φd(b) = inf { α ≥ 0 : γα,b 6= ∅ } . as we have pointed out above, γα,b 6= ∅ for each α ≥ diam(b) and therefore the dnd is well defined. also, let us note that, for a given b ∈ b(x), φd(b) © agt, upv, 2022 appl. gen. topol. 23, no. 2 456 the ε-approximated complete invariance property measures (in the specified sense) the distance from b to the class of its peano continua. example 2.8 (see [16]). let b be the closed unit ball of a banach space v , and d the distance in v induced by its norm. then, φd(b) =   0, if v is finite dimensional 1, if v is infinite dimensional . some properties of the dnd are given in the next result. (see [4, 16]). proposition 2.9. the dnd satisfies the following: (1) if φd(b) = 0, then b is precompact. moreover, if b is precompact and arc-wise connected then φd(b) = 0. (2) φd(b) = φd(b), for each b ∈b(x) where, as usual, the bar stands for the closure. on the other hand, for our main result we will use theorem 1.2 and the dnd. so, we will need that the α-dense curves used in the definition of the dnd be 1-dimensional peano continua. note that, a priori, an α-dense curve is not necessarily a 1-dimensional peano continua: for instance, a n-dimensional peano continua or, in particular, the space-filling curves in in given in [19]. however, in the next result, we prove that the dnd can be defined by means of α-curves such that the image of i under these curves be a 1-dimensional peano continua. theorem 2.10. given b ∈ b(x) and α > 0, let γ(1)α,b ⊂ γα,b be the class of α-dense curves in b such that γ(1)(i) is a 1-dimensional peano continuum for all γ(1) ∈ γ(1)α,b. by putting φ (1) d (b) = inf { α ≥ 0 : γ(1)α,b 6= ∅ } , we have φd(b) = φ (1) d (b). proof. let α be such that α > φd(b) and γ : i −→ (x,d) an α-dense curve in b. so, by the compactness of γ(i), given any ε > 0 there exists a finite set {y1, . . . ,yn}⊂ γ(i) (without loss of generality we assume n > 1) such that (2.1) b ⊂ n⋃ i=1 b̄d(yi,α + ε), b̄d(yi,α + ε) being the closed ball centered at yi of radius α + ε. as γ(i) is a peano continuum it is arc-wise connected (see, for instance, [21, theorem 31.2]), for each i = 1, . . . ,n− 1 there exists a one-to-one continuous hi : i −→ γ(i) with hi(0) = yi and hi(1) = yi+1. in particular, each hi(i) is a 1dimensional peano continuum, for i = 1, . . . ,n. define, for each i = 1, . . . ,n−1, © agt, upv, 2022 appl. gen. topol. 23, no. 2 457 g. garćıa the one-to-one continuous τi : i −→ [ i−1n−1, i n ] as τi(t) = i−1+t n−1 for all t ∈ i. then, the mapping γ(1) : i −→ (x,d) given by γ(1)(t) = hi ( τi(t) ) , for t ∈ [ i− 1 n− 1 , i n ], i = 1, . . . ,n− 1, is continuous, γ(1)(i) ⊂ γ(i) ⊂ b and γ(1)(i) is a 1-dimensional peano continuum because it is the finite union of 1-dimensional peano continua. also, from (2.1) we have γ(1) ∈ γ(1)α+ε,b . by the arbitrariness of ε > 0, we conclude that φ (1) d (b) ≤ α and by the arbitrariness of α > φd(b), the inequality φ (1) d (b) ≤ φd(b) holds. on the other hand, if γ ∈ γ(1)α,b, from the inclusion γ (1) α,b ⊂ γα,b, we have γ ∈ γα,b. thus, φd(b) ≤ φ (1) d (b) and the proof is now complete. � to conclude this section, we give a result for the dnd of the product of bounded metric spaces. proposition 2.11. let λ be a finite set or λ = n, and (xλ,dλ)λ∈λ a family of metric spaces such that diam(xλ) ≤ m for certain m > 0 and all λ ∈ λ. put φ∗ = sup{φdλ(xλ) : λ ∈ λ}, x ∗ = ∏ λ∈λ xλ and d ∗(x,y) = max{dλ(x,y) : λ ∈ λ} if λ is finite or d∗(x,y) = ∑ k≥1 2 −kdk(x,y) if λ = n, for all x,y ∈ x∗. then, φd∗(x ∗) ≤ φ∗. moreover if λ is finite, then the equality holds. proof. firstly, note that (x∗,d∗) is, effectively, a bounded metric space and therefore φd∗(x ∗) is well defined (in fact, φd∗(x ∗) ≤ m). assume, λ = n and let α > φ∗. let, for each k ≥ 1, γk : i −→ xk an α-dense curve in xk. so, for each k ≥ 1, given xk ∈ xk there is tk ∈ i such that (2.2) dk(xk,γk(tk)) ≤ α. let ω = (ωk)k≥1 : i −→ in be a space-filling curve (see [19, section 7.5]). that is, ω (and hence each coordinate function ωk) is continuous and ω(i) = i n. define γ : i −→ x∗ as γ(t) = ( γk(ωk(t)) ) k≥1, for all t ∈ i. it is clear that γ is continuous and γ(i) ⊂ x∗. also, given (xk)k≥1 ∈ x∗ take (tk)k≥1 ⊂ i satisfying (2.2) and t ∈ i such that ω(t) = (tk)k≥1. so, we have d∗ ( (xk)k≥1,γ(t) ) = ∑ k≥1 dk ( xk,γk(ωk(t)) ) 2k = ∑ k≥1 dk ( xk,γk(tk) ) 2k ≤ α. and consequently γ is an α-dense curve in x∗. then, φd∗(x ∗) ≤ α and letting α → φ∗, we conclude φd∗(x∗) ≤ φ∗. © agt, upv, 2022 appl. gen. topol. 23, no. 2 458 the ε-approximated complete invariance property if λ is finite, without loss of generality we assume λ = {1, . . . ,n} for some n > 1, we take ω = (ω1, . . . ,ωn) : i −→ in a space-filling curve (again, [19]) and the proof follows in a totally analogous way that above. assume φd∗(x ∗) < φ∗ and take φd∗(x ∗) < α < φ∗ and an α-dense curve in x∗, put γ = (γ1, . . . ,γn) : i −→ (x∗,d∗). then, fixed 1 ≤ k ≤ n, the mapping γk : i −→ (xk,dk) is continuous and one can check straightforwardly that it is an α-dense curve in xk. but, this is not possible as α < φ ∗ ≤ φdk(xk). � 3. the main result we start this section with the following definition: definition 3.1. given ε ≥ 0, we will say that a metric space (x,d) has the εapproximated complete invariance property (ε-acip) if for each non-empty and closed c ⊂ x there is a continuous fε : x −→ x such that dh(c, fix(fε)) ≤ ε. the following facts are clear from the definitions: (i) if (x,d) is bounded, then (x,d) has ε-acip for every ε ≥ diam(x). (ii) the 0-acip is, precisely, the cip. also, the cip implies the ε-acip for each ε > 0, but as we will see below, the inverse implication does not hold in general. that is to say, there are metric spaces with the ε-acip for all ε > 0, but such metric spaces do not have the cip. now, we are ready to state and prove our main result: theorem 3.2. let (x,d) a bounded metric space. then, (x,d) has the εacip for each ε > φd(x). in particular, if x is densifiable then it has the ε-acip for each ε > 0. proof. let ε be such that ε > φ(x). let any c ⊂ x non-empty and closed, and γε : i −→ (x,d) and ε-dense curve such that γε(i) is a 1-dimensional peano continuum. such ε-dense curve exists by virtue of theorem 2.10. define the set gc = { x ∈ γε(i) : d(x,c) ≤ ε, for some c ∈ c } ⊂ x. it is clear that the set gc is non-empty and closed. thus, by theorem 1.2, there is fε : x −→ x with fix(fε) = gc. now, let c ∈ c. as γε is an ε-dense curve in x, there is x ∈ γε(i) with d(x,c) ≤ ε. then, x ∈ gc and therefore x = fε(x). so, we have infx∈fix(fε) d(c,x) ≤ ε and from the arbitrariness of c ∈ c, we infer (3.1) sup c∈c inf x∈fix(fε) d(c,x) ≤ ε. likewise for a given x ∈ fix(fε), as x ∈ gc, d(c,x) ≤ ε for some c ∈ c. consequently, infc∈c d(c,x) ≤ ε and noticing the arbitrariness of x ∈ fix(fε) (3.2) sup x∈fix(fε) inf c∈c d(c,x) ≤ ε. © agt, upv, 2022 appl. gen. topol. 23, no. 2 459 g. garćıa so, from (3.1) and (3.2), we have dh(c, fix(fε)) ≤ ε. if x is densifiable then, by the definition of the dnd, φd(x) = 0 and therefore has the ε-acip for each ε > 0. � an immediate consequence of the above result is the following: corollary 3.3. every peano continuum has the ε-acip for each ε > 0. as we have said above, in general, the ε-acip for each ε > 0 does not imply the cip. we illustrate this fact in the following examples. example 3.4. let x be the topologist’s sine of example 2.3. then, x is densifiable but does not have the cip. however, by theorem 3.2 x has the ε-acip for each ε > 0. example 3.5. let x be a n-dimensional peano continuum without the cip (see [8, 9]). then, by corollary 3.3, x has the ε-acip for each ε > 0. so, in general, we cannot replace the condition ε > φd(x) by ε ≥ φd(x) in theorem 3.2. this fact is explained by the following ones: (i) there is not necessarily a φd(x)-dense curve in x. indeed, for instance, the topologist’s sine x of example 2.3 satisfies φd(x) = 0 but there is not a 0-dense curve in x: otherwise, x would be a peano continuum, which is not possible because it is not locally connected. (ii) even if x = γ(i), for certain continuous γ : i −→ x, if γ(i) is not a 1-dimensional peano continuum, we cannot apply theorem 1.2 in the proof of theorem 3.2 to derive that x has the cip (see also example 3.5). we have remarked above that if (x,d) is bounded, then (x,d) has ε-acip for every ε ≥ diam(x). this bound can be improved by theorem 3.2: example 3.6. let x be the set given in example 2.2. then, diam(x) = 2 and φd(x) = 1. so, by theorem 3.2, x has ε-acip for every ε > 1. as was proved in [6], the cip need not be preserved by self-products. however, bearing in mind proposition 2.11 and theorem 3.2, we have the following result for the product of bounded metric spaces: corollary 3.7. with the notation of proposition 2.11, (x∗,d∗) has the εacip for each ε > φ∗. in particular, the finite or countable product of peano continua has the ε-acip for each ε > 0. example 3.8. let (x,d) be the 1-dimensional peano continuum given in [6, theorem 2.2]. then, x×x does not have the cip. however, by corollary 3.7, x ×x has the ε-acip for each ε > 0. also, the cip need not to be preserved by continuous mappings. indeed, take any metric space (x,d) that does not have the cip and τ the discrete topology on x. then, (x,τ) has the cip and the identity mapping g : (x,τ) −→ (x,d) is continuous. however, for the ε-acip we have the following: © agt, upv, 2022 appl. gen. topol. 23, no. 2 460 the ε-approximated complete invariance property corollary 3.9. let (x,d) and (y,d′) be bounded metric spaces and g : (x,d) −→ (y,d′) continuous. then (y,d′) has the ε-acip for each ε > ωφd(x)(g), where ωr(g) = sup { d′(f(x),f(y)) : x,y ∈ x,d(x,y) ≤ r } , is the modulus of continuity of g of order r, for r ≥ 0. proof. it is immediate to check that if γ : i −→ (x,d) is an α-dense curve in (x,d), then g◦γ : i −→ (y,d′) is a ωα(g)-dense curve in (y,d′). therefore, we infer that φd′(y ) ≤ ωφd(x)(g) and by theorem 3.2, (y,d ′) has the ε-acip for each ε > ωφd(x)(g). � on the other hand, it is important to stress that the reciprocal of theorem 3.2 is not true in general: there are metric spaces with the ε-acip for all ε > 0 (in fact, with the cip) that are not densifiable: example 3.10. let x be the closed unit ball of an infinite dimensional banach space. from the comments of section 1, x has the cip and therefore the εacip for all ε > 0. however, from example 2.8, φd(x) = 1 and noticing proposition 2.9 x is not densifiable. so, we conclude our exposition with the following question: if (x,d) is a bounded metric space having the ε-acip, for some ε > 0, under what conditions can we relate, in some way, ε and φd(x)? acknowledgements. the author is very grateful to the anonymous referee for his/her comments and suggestions, which have substantially improved the presentation of the paper. references [1] y. cherruault and g. mora, optimisation globale. théorie des courbes α-denses, económica, paris, 2005. 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[22] d. x. zhou, complete invariance property with respect to homeomorphism over frame multiwavelet and super-wavelet spaces, journal of mathematics 2014 (2014), article id 528342, 6 pages. © agt, upv, 2022 appl. gen. topol. 23, no. 2 462 couzviagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 39-50 low separation axioms via the diagonal maŕıa luisa colasante, carlos uzcátegui and jorge vielma∗ abstract. in the context of a generalized topology g on a set x, we give in this article characterizations of some separation axioms between t0 and t2 in terms of properties of the diagonal in x × x. 2000 ams classification: 54a05, 54d10. keywords: generalized topologies, intersection structures, envelope operations, kerneled and saturated sets. 1. introduction a well known elementary fact says that a topological space x is hausdorff iff the diagonal ∆ is closed in x × x. in this paper we show that behind this observation there is a general pattern which includes several separation axioms below t2 (namely t0, t1/4, t1/2, t1, r0 and r1). these low separation axioms have been studied in a more general setting where, instead of open sets, other kind of subsets are used: semi-open sets, α-open sets, λ-open sets, etc. ([1], [6], [8]). these families (called generalized topologies in [5]) always contain ∅ and x and are closed under arbitrary unions (but not necessarily under finite intersections). on the other hand, in the study of low separation axioms, set operations similar to the closure operator are frequently used. these operations are naturally extended to the context of a generalized topology g (and are then called envelope operations [5]). for instance, kg(a) corresponds to the topological closure of a, χg(a) corresponds to the kernel of a [7] (i.e. the intersection of all open sets containing a) and satg(a) corresponds to the union of the closure of points in a. our characterizations of the separation axioms are in terms of the behavior of ∆ under kg, χg and satg. an example of our results is that g satisfies t1 iff χg(∆) = ∆. we will also give a characterization of low separation axioms in terms of saturated sets. a set in a topological space is said to be saturated when it ∗the authors would like to thank the partial support provided by the universidad de los andes cdcht grant a-1335-05-05. 40 m. l. colasante, c. uzcátegui and j. vielma contains the closure of each of its points. it is known that a topology satisfies the axiom r0 iff every open set is saturated [5]. another notion of saturation was studied in [4]. we extend these notions and study its connection with low separation axioms. the paper is organized as follows. in section 2 we recall the basic separation axioms and state some facts about generalized topologies and envelope operations. the results about the properties of the diagonal and the separation axioms are shown in section 3. in section 4 we study the family of saturated sets and its connection with the separation axioms. finally, in section 5 we analyze the axioms t1/2 and t1/4. 2. preliminaries we follow the notations and definitions used in [5]. a subset g of the power set p(x) of a set x is a generalized topology (briefly gt ) on x if {∅, x} ⊆ g and g is closed under arbitrary unions. if g is a generalized topology, then the family of complements of sets in g is usually called an intersection structure. in this article g will always denote a generalized topology. definition 2.1 ([5]). an envelope operation on x is a mapping ρ : p(x) → p(x) such that (i) a ⊆ ρa for a ⊆ x. (ii) if a ⊆ b, then ρa ⊆ ρb for all a ⊆ b ⊆ x. (iii) ρa = ρρa for a ⊆ x. more generally, ρ : p(x) → p(x) is called a weak envelope if (i) and (ii) are satisfied. examples of envelope operations are given below. definition 2.2. let a ⊆ x. (i) χg(a) = ⋂ {h ∈ g : a ⊆ h}. (ii) kg(a) = {x ∈ x : k ∩ a 6= ∅ for each k ∈ g with x ∈ k}. (iii) satg(a) = ⋃ x∈a kg({x}). the operator χg and kg were defined in [5] and shown to be envelope operations. it is straightforward to show that satg is an envelope. it is also easy to see that χg(a) = a for all a ∈ g. moreover, x ∈ χg(y) if and only if y ∈ kg(x) for any x, y ∈ x (where we write ρ(x) instead of ρ({x}) for any set operator ρ). when g is a topology, kg is the closure operator cl and χg(a) is the kernel of a, frequently denoted by â or ker(a). notice that, in general, if τ is the topology generated by a gt g, then kg 6= clτ and χg 6= kerτ (for instance, in r take g to be the gt generated by the collection of intervals of the form (−∞, a) and (a, +∞)). now we formulate the fundamental separation axioms in terms of an arbitrary gt ([5]). (t0) for all x, y ∈ x, with x 6= y there is k ∈ g containing precisely one of x and y. low separation axioms via the diagonal 41 (t1) for all x, y ∈ x with x 6= y there is k ∈ g such that x ∈ k, y /∈ k. (t2) for all x, y ∈ x, x 6= y, there are k, k ′ ∈ g such that x ∈ k, y ∈ k′ and k ∩ k′ = ∅. (r0) for all x, y ∈ x, if kg(x) 6= kg(y), then kg(x) ∩ kg(y) = ∅. (r1) for all x, y ∈ x, if kg(x) 6= kg(y), then there are k, k ′ ∈ g disjoint such that kg(x) ⊆ k and kg(y) ⊆ k ′. proposition 2.3 ([5]). g satisfies (r0) iff for all x, y ∈ x, if there is k ∈ g such that x ∈ k and y 6∈ k, then there is k′ ∈ g such that x 6∈ k′ and y ∈ k′. two of the recently widely studied separation axioms below t1 can be stated for a generalized topology g as follows: (t1/2) for all x ∈ x, {x} ∈ g or {x} = kg(x). (t1/4) for all x ∈ x, {x} = χg(x) or {x} = kg(x). in the rest of this section we introduce some notions and present some basic facts about the envelope operations χg, kg and satg that will be used in the sequel. in order to simplify the notation, we will write χ(a), k(a) and sat(a) avoiding the use of g. we say that a set a is closed (resp. kerneled) iff k(a) = a (resp. χ(a) = a). when g is a topology, kerneled sets are usually called λ-sets [7]. a subset a ⊆ x is said g-saturated (or just saturated) if sat(a) = a, equivalently if k(x) ⊆ a for all x ∈ a. note that sat(a) is the smallest saturated set containing a, and that a is saturated if and only if x\a is kerneled, where x\a denotes the complement of a. in particular sat(x) = k(x), for any x ∈ x. the collection of all saturated subsets of x is denoted by s(x). it is easy to see that s(x) is closed under arbitrary unions and arbitrary intersections. proposition 2.4 ([5]). a is closed iff x\a ∈ g. proof. let a closed. if y ∈ x\a then y /∈ k(a), and hence there exists by ∈ g such that y ∈ by and by ∩a = ∅. thus x\a = ⋃ y∈x\a by ∈ g. the converse is obvious. � since our analysis of the separation axioms will be in terms of the behavior of the diagonal ∆, we need to introduce the gt on the product x × x. let g be a gt on x, then the family g2 below is a gt in x2: g 2 = { d ⊆ x × x : d = ⋃ α aα × bα, with aα, bα ∈ g } . in this article the operators k, χ and sat on x × x refer to the generalized topology g2. proposition 2.5. for any (x, y) ∈ x × x the following holds: (i) χ(x, y) = χ(x) × χ(y). (ii) k(x, y) = k(x) × k(y). proof. (i) let (p, q) ∈ χ(x, y) and a, b ∈ g with x ∈ a and y ∈ b. then (x, y) ∈ d = a × b ∈ g2 and so (p, q) ∈ a × b. thus p ∈ χ(x) and q ∈ χ(y). 42 m. l. colasante, c. uzcátegui and j. vielma conversely, if (p, q) ∈ χ(x) × χ(y) and d ∈ g2, then (x, y) ∈ d = ⋃ α aα × bα, with aα, bα ∈ g. there is α such that x ∈ aα and y ∈ bα and hence (p, q) ∈ aα × bα ⊆ d. this implies that (p, q) ∈ χ(x, y). � proposition 2.6. (i) if a = ⋃ i∈i ai, then χ(a) = ⋃ i∈i χ(ai). (ii) if a = ⋃ i∈i ai, then sat(a) = ⋃ i∈i sat(ai). proof. (i) was proved in [5] and (ii) is obvious. � proposition 2.7. let a be a subset of x × x. then (i) χ(a) = ⋃ (x,y)∈a χ(x) × χ(y). (ii) sat(a) = ⋃ (x,y)∈a k(x) × k(y). proof. the result follows directly from propositions 2.5 and 2.6. � to end this section, we introduce three more operations. let a ⊆ x, then define kθ(a) = {x ∈ x : a ∩ k(d) 6= ∅ for each d ∈ g such that x ∈ d} kλ(a) = k(a) ∩ χ(a) kµ(a) = sat(a) ∩ χ(a) it is easy to see that kθ is a weak envelope on x such that k(a) ⊆ kθ(a) for all a ⊆ x. also, it is straightforward to show that kλ and kµ are envelope operations on x (more generally, the finite intersection of envelope operations is again an envelope). when g is a topology, kθ is the well known clθ operator [9, 4] and kλ is the clλ operator [2]. the clλ-closed sets (i.e. sets such that clλ(a) = a) are usually called λ-closed sets and their complements λ-open sets [1]. if g is the gt consisting of the λ-open sets, then k = clλ. 3. separation axioms as properties of the diagonal we will denote by ∆ the diagonal in x × x. in this section we show that the separation axioms can be characterized in terms of χ(∆), sat(∆) and k(∆). besides ∆ there are two others binary relations which play an important role in what follows. (x, y) ∈ lg iff ∀a ∈ g [x ∈ a → y ∈ a] (x, y) ∈ eg iff ∀a ∈ g [x ∈ a ↔ y ∈ a] . notice that ∆ ⊆ eg ⊆ lg. moreover, lg is transitive relation and eg is an equivalence relation on x. the main result of this section is summarized in the following table. t2 ⇔ k(∆) = ∆ t1 ⇔ χ(∆) = ∆ ⇔ sat(eg) = ∆ t0 ⇔ ∆ = eg r0 ⇔ χ(∆) = eg ⇔ eg = sat(∆) r1 ⇔ k(∆) = eg in order to show these results we need several auxiliary lemmas. low separation axioms via the diagonal 43 lemma 3.1. (i) (x, y) ∈ lg iff y ∈ χ(x) iff x ∈ k(y). (ii) (x, y) ∈ eg iff k(x) = k(y) iff χ(x) = χ(y). proof. since χ(x) = ∩{a ∈ g : x ∈ a}, (i) follows directly from the definition of lg. part (ii) follows from the symmetry of the relation eg. � the following result characterizes χ, k, and sat for the diagonal ∆ on x × x. in particular, it shows that k(∆), χ(∆) and sat(∆) are symmetric (and obviously reflexive) relations. lemma 3.2. (i) (x, y) ∈ k(∆) iff a ∩ b 6= ∅ for all a, b ∈ g such that x ∈ a and y ∈ b. (ii) (x, y) ∈ χ(∆) iff k(x) ∩ k(y) 6= ∅. (iii) (x, y) ∈ sat(∆) iff χ(x) ∩ χ(y) 6= ∅. proof. (i) let (x, y) ∈ k(∆) and let a, b ∈ g with x ∈ a and y ∈ b. then (x, y) ∈ a × b ∈ g2 and thus a × b ∩ ∆ 6= ∅, which implies a ∩ b 6= ∅. reciprocally, let d ∈ g2 containing (x, y). then d = ⋃ α aα × bα, with aα, bα ∈ g. it follows that (x, y) ∈ aα × bα for some α. by assumption aα ∩ bα 6= ∅. if z ∈ aα ∩ bα, then (z, z) ∈ d ∩ ∆. therefore (x, y) ∈ k(∆). (ii) (x, y) ∈ χ(∆) = ⋃ x∈x χ(x) × χ(x) if and only if there is z ∈ x such that (x, y) ∈ χ(z) × χ(z) for some z ∈ x if and only if (z, z) ∈ k(x) × k(y). (iii) follows by a similar argument as (ii). � from lemma 3.2(i), it follows that (x, y) ∈ k(∆) iff ∀a ∈ g [x ∈ a → y ∈ k(a)] iff ∀b ∈ g [y ∈ b → x ∈ k(b)]. therefore we have the following fact about the operator kθ (defined in section 2). lemma 3.3. (x, y) ∈ k(∆) iff y ∈ kθ(x) iff x ∈ kθ(y). we prove that the envelope operations k, χ and sat coincide on the relations ∆, lg and eg. lemma 3.4. (i) k(∆) = k(lg) = k(eg). (ii) χ(∆) = χ(lg) = χ(eg). (iii) sat(∆) = sat(lg) = sat(eg). proof. (i). since ∆ ⊆ eg ⊆ lg, it suffices to show that lg ⊆ k(∆). let (x, y) ∈ lg and let a, b ∈ g with (x, y) ∈ a × b. by lemma 3.1, x ∈ k(y) and thus y ∈ χ(x). since χ(x) ⊆ a then y ∈ a. it follows that (y, y) ∈ a × b. therefore (x, y) ∈ k(∆), by definition of k(∆). (ii). as in (i) we only show that lg ⊆ χ(∆). let (x, y) ∈ lg. by proposition 2.7, χ(∆) = ⋃ z∈x χ(z) × χ(z). thus, if (x, y) /∈ χ(∆), then in particular y /∈ χ(x) and this implies that there is a ∈ g containing x such that y /∈ a, a contradiction. (iii). if (x, y) ∈ lg then, by lemma 3.1(i) and proposition 2.7, (x, y) ∈ k(y) × k(y) ⊆ sat(∆). hence lg ⊆ sat(∆). � 44 m. l. colasante, c. uzcátegui and j. vielma proposition 3.5. (i) g satisfies (t2) iff kθ(x) = {x} for each x ∈ x. (ii) g satisfies (t1) iff k(x) = {x} for each x ∈ x iff χ(x) = {x} for each x ∈ x. (iii) g satisfies (t0) iff kλ(x) = {x} for each x ∈ x. that is to say, k(x) ∩ χ(x) = {x} for each x ∈ x. proof. (i) first note that, if a, b ∈ g, then a ∩ b = ∅ iff a ∩ k(b) = ∅. suppose g satisfies (t2). given x ∈ x and y 6= x, there exist a, b ∈ g such that x ∈ a, y ∈ b and a∩b = ∅, then y /∈ kθ(a) and, in particular, y /∈ kθ(x). therefore kθ(x) = {x} for each x ∈ x. conversely, suppose kθ(x) = {x} for each x ∈ x. given x, y ∈ x, if x 6= y then y /∈ kθ(x). thus (x, y) /∈ k(∆) and hence, by lemma 3.2, there exist a, b ∈ g such that x ∈ a, y ∈ b and a ∩ b = ∅, which shows that g satisfies (t2). (ii) g satisfies (t1) iff given x ∈ x and y 6= x, there exist a ∈ g such that x ∈ a and y /∈ a, iff given x ∈ x and y 6= x, y /∈ k(x), iff k(x) = {x} for each x ∈ x. for the second part, note that if k(x) = {x} for each x ∈ x, then the set x\{x} = ⋃ y 6=x k(y) is saturated for each x ∈ x, and thus {x} is kerneled for each x ∈ x. a similar argument shows the reverse implication. (iii) g satisfies (t0) iff given x ∈ x and y 6= x, y /∈ k(x) or x /∈ k(y), iff y /∈ k(x) or y /∈ χ(x), iff kλ(x) = k(x) ∩ χ(x) = {x} for each x ∈ x. � now we start showing the main results of this section. theorem 3.6. (i) g satisfies (t2) iff k(∆) = ∆ iff ∆ is closed. (ii) g satisfies (t1) iff χ(∆) = ∆ iff lg = ∆ iff ∆ is saturated iff ∆ is kerneled. (iii) g satisfies (t0) iff eg = ∆. proof. (i) by proposition 3.5(i), g satisfies (t2) iff kθ(x) = {x} for each x ∈ x, iff y 6= x implies y /∈ kθ(x) iff (x, y) /∈ k(∆). the second part is obvious. (ii) suppose g satisfies (t1). from proposition 3.5(ii), k(x) = {x} for each x ∈ x. if (x, y) ∈ χ(∆), then k(x) ∩ k(y) 6= ∅ and thus x = y. on the other hand, if χ(∆) = ∆ and x 6= y, then (x, y) /∈ χ(∆) and thus k(x) ∩ k(y) = ∅. in particular x /∈ k(y), so there exists a ∈ g such that x ∈ a and y /∈ a. therefore g satisfies (t1). the second and third parts follow from lemma 3.1(i) and proposition 3.5(ii). the last part is obvious. (iii) g satisfies (t0) iff for all x 6= y, y /∈ χ(x) or x /∈ χ(y) iff χ(x) 6= χ(y) iff (x, y) /∈ eg iff eg = ∆. � theorem 3.7. g satisfies (r0) iff χ(∆) = eg iff χ(∆) = lg iff eg is kerneled iff eg is saturated. proof. since eg ⊆ χ(eg) = χ(∆), then χ(∆) = eg iff χ(∆) ⊆ eg. from proposition 2.3, g satisfies (r0) iff x, y ∈ x implies k(x) = k(y) or k(x)∩k(y) = ∅. therefore g satisfies (r0) iff χ(∆) = eg. the second part of the equivalence follows from the fact that eg ⊆ lg ⊆ χ(lg) = χ(∆). the third part is obvious. low separation axioms via the diagonal 45 on the other hand, since y ∈ k(x) iff x ∈ χ(y), then g satisfies (r0) iff the sets χ(x), x ∈ x, form a partition of x, iff sat(∆) = eg. � theorem 3.8. g satisfies (r1) iff k(∆) = eg iff k(∆) = lg iff eg is closed. proof. g satisfies (r1) iff x, y ∈ x, and k(x) 6= k(y), implies the existence of a, b ∈ g such that x ∈ a, y ∈ b and a ∩ b = ∅ iff (x, y) /∈ eg implies (x, y) /∈ k(∆) iff k(∆) ⊆ eg. since eg ⊆ k(eg) = k(∆), it follows that g satisfies (r1) iff k(∆) = eg. the other two equivalences are obvious. � corollary 3.9. (i) g satisfies (r0) iff k(x) = χ(x) for each x ∈ x. (ii) g satisfies (r1) iff kθ(x) = χ(x) = k(x) for each x ∈ x. proof. (i) and (ii) follow from theorems 3.7 and 3.8 respectively. � remark 3.10. if x is a topological space, and g is the family of the λ-open sets, then kg(x) and χg(x) are usually denoted clλ(x) and λker(x) respectively [3]. these envelope operations satisfy that clλ(x) = λker(x) for all x ∈ x. in fact, since every open set and every closed set is λ-open, then λker(x) ⊆ cl(x)∩ker(x) = clλ(x). on the other hand, since every λ-open set is the union of an open set and a saturated set, then clλ(x) ⊂ a for every λ-open set a containing x. from this and corollary 3.9, every topological space x is λ-r0, a fact that was unnoticed by the authors of [3]. 4. relations, saturated sets and separation axioms in this section we will introduce the notion of a saturated set with respect to a binary relation (like k(∆), lg and eg). we will show that the results of the previous section can be stated in terms of algebraic properties of the collection of saturated sets. let e be a binary relation on a set x (i.e. e ⊆ x × x). we will always assume that e contains the diagonal ∆. we say that a subset a ⊆ x is e-saturated if whenever x ∈ a and (y, x) ∈ e, then y ∈ a. the family of e-saturated sets will be denoted by s[e]. the following result shows that the notion of an e-saturated set is a natural generalization of a g-saturated set. proposition 4.1. (i) a ∈ s[lg] iff for each x ∈ a, k(x) ⊆ a, i.e. s(x) = s[lg]. (ii) a ∈ s[eg] iff kλ(x) ⊆ a, for each x ∈ a. (iii) a ∈ s[k(∆)] iff kθ(x) ⊆ a, for each x ∈ a. proof. the proof follows from the fact that (y, x) ∈ lg iff y ∈ k(x), (y, x) ∈ eg iff k(x) = k(y) iff y ∈ kλ(x) = k(x) ∩ χ(x), and (y, x) ∈ k(∆) iff y ∈ kθ(x). � we show below a general fact about saturated sets which will be used several times in the sequel. 46 m. l. colasante, c. uzcátegui and j. vielma lemma 4.2. let e be a binary relation over x. (i) s[e] is closed under arbitrary unions and intersections. (ii) if e is a symmetric relation, then s[e] is a complete atomic boolean algebra. moreover, s[e] = s[f ] where f is the smallest equivalence relation containing e and the f -equivalence classes are the atoms of s[e]. (iii) s[e] = p(x) iff e = ∆. proof. (i) is obvious. (ii) to get the result it is enough to prove that s[e] is closed under complements. let a ∈ s[e]. if x\a /∈ s[e], there exists x, y ∈ x such that y ∈ a and (y, x) ∈ e but x /∈ a. from the symmetry of e, it follows that (x, y) ∈ e which implies that x ∈ a, a contradiction. let f be the transitive closure of e, that is to say, (x, y) ∈ f if there are xi ∈ x, i = 0, · · · , n such that x = x0, y = xn and (xi, xi+1) ∈ e. it is easy to check that f is the smallest equivalence relation containing e. therefore s[f ] ⊆ s[e]. on the other hand, it is routine to verify that each f -equivalence class [x]f is e-saturated. moreover, if z ∈ a ⊆ [x]f and a is e-saturated, then [z]f = [x]f and thus a = [x]f . hence the f -equivalence classes are the atoms of s[e] and s[e] = s[f ]. (iii) one direction is obvious. for the other, suppose e is not equal to ∆ and let (x, y) ∈ e with x 6= y. then {y} is not e-saturated. � remark 4.3. (i) since k(∆), χ(∆) and eg are symmetric relations (lemma 3.2), then s[k(∆)], s[χ(∆)] and s[eg] are complete atomic boolean algebras. now from theorem 3.6 and lemma 4.2, it follows immediately that g satisfies t2 iff s[k(∆)] = p(x) iff every cofinite set belongs to s[k(∆)]. clearly the axioms t1 and t0 are characterized in an analogous way. (ii) if g is a topology, s[k(∆)] is denoted by bθ(x) in [4]. it was proved there that bθ(x) is complete boolean algebra. note that this result is an immediate consequence of lemma 4.2(ii). our next results deal with the axioms (r0) and (r1). theorem 4.4. the following are equivalent. (i) g satisfies (r0). (ii) s[lg] is a complete atomic boolean algebra. (iii) g ⊆ s[lg]. proof. the equivalence (i) ↔ (iii) was proved in [5] lemma 3.2. it is clear that g satisfies (r0) iff lg is a symmetric relation. therefore (i) → (ii) follows from lemma 4.2(ii). for the reverse implication, note that if x ∈ x and z ∈ k(x), then k(z) ⊆ k(k(x)) = k(x), thus k(x) ∈ s(x). suppose s[lg] is a complete boolean algebra, and let y ∈ x and x ∈ k(y). if y /∈ k(x), then y ∈ x\k(x) ∈ s(x) and we will have that x ∈ k(y) ⊆ x\k(x), a contradiction. thus y ∈ k(x) which shows that (ii) → (i). � low separation axioms via the diagonal 47 theorem 4.5. the following are equivalent. (i) g satisfies (r1). (ii) s[k(∆)] is a complete atomic boolean algebra and the sets k(x) (x ∈ x) are its atoms. (iii) g ⊆ s[k(∆)]. proof. (i) → (ii). suppose g satisfies (r1). since k(∆) is symmetric, then by lemma 4.2 s[k(∆)] is a complete atomic boolean algebra. since k(∆) = lg (theorem 3.8), then each k(x) is k(∆)-saturated. to show that the sets k(x) are the atoms, let z ∈ a ⊆ k(x) with a a k(∆)-saturated set. then z ∈ k(x) and thus (z, x) ∈ k(∆). by symmetry (x, z) ∈ k(∆) and as a is k(∆)-saturated, then x ∈ a. hence a = k(x). (ii) → (iii). suppose (ii) holds. we will show that s[lg] = s[k(∆)] and the result will follow from theorem 4.4. since lg ⊆ k(lg) = k(∆) (lemma 3.4), then s[k(∆)] ⊆ s[lg]. conversely, if a is lg-saturated, then a is equal to the union of the sets k(x) with x ∈ a. but by hypothesis each k(x) belongs to the complete algebra s[k(∆)], thus a ∈ s[k(∆)]. (iii) → (i). suppose g ⊆ s[k(∆)]. we will show that k(∆) = lg, and from this and theorem 3.8 the result follows. let (x, y) ∈ k(∆). given a ∈ g with y ∈ a, then a ∈ s[k(∆)] and thus x ∈ a. then x ∈ k(y) and therefore (x, y) ∈ lg. since lg ⊆ k(lg) = k(∆), we conclude that k(∆) = lg. � 5. t1/2 and t1/4 in this section we characterize the axioms t1/2 and t1/4 in terms of properties of the diagonal and also in terms of properties of the family of saturated sets. we start with a general result about envelope operations. lemma 5.1. let g be a generalized topology on x and let ρ be an envelope such that ρ(x) = k(x) for all x ∈ x. for each a ⊆ x, the following are equivalent: (i) a = ρ(a) ∩ χ(a). (ii) ρ(x) ⊆ ρ(a) \ a, for all x ∈ ρ(a) \ a. proof. (i) → (ii). suppose a = ρ(a) ∩ χ(a) and let x ∈ ρ(a) \ a. then x /∈ χ(a) and thus there exists h ∈ g such that a ⊆ h and x /∈ h. let y ∈ ρ(x) ⊂ ρ(a). if y ∈ a, then y ∈ h and it must be that x ∈ h since y ∈ k(x) = ρ(x), a contradiction. thus y /∈ a and therefore ρ(x) ⊆ ρ(a) \ a. (ii) → (i). conversely, suppose ρ(x) ⊆ ρ(a) \ a, for all x ∈ ρ(a) \ a. let z ∈ ρ(a)∩χ(a). if z /∈ a, then ρ(z) ⊆ ρ(a)\a and it is clear that a ⊂ x \ρ(z). since ρ(z) = k(z), z ∈ χ(a) and x \ k(z) ∈ g (proposition 2.4), then it must be that z ∈ x \ k(z), a contradiction. therefore a = ρ(a) ∩ χ(a). � recall from section 2 the definition of the envelope operations kλ(a) = k(a) ∩ χ(a) and kµ(a) = sat(a) ∩ χ(a), a ⊂ x. we denote a ′ = k(a) \ a and a∗ = sat(a) \ a. the following result is an immediate consequence of lemma 5.1. 48 m. l. colasante, c. uzcátegui and j. vielma corollary 5.2. let a ⊆ x. then (i) a = kλ(a) iff a ′ ∈ s(x). (ii) a = kµ(a) iff a ∗ ∈ s(x). it is known that a topological space x is t1/2 iff every subset of x is λ-closed [1]. this result inspired part of the theorems 5.3 and 5.4 that follows. theorem 5.3. the following are equivalent. (i) g satisfies (t1/4). (ii) a = kµ(a) for all a ⊂ x. (iii) a∗ ∈ s(x) for all a ⊂ x. (iv) if sat(a) ⊂ χ(a), then sat(a) = a. proof. the equivalence (ii) ↔ (iii), follows from corollary 5.2(ii). (i) ↔ (ii). suppose g satisfies (t1/4) and let a ⊂ x. let a1 = {x ∈ x \ a : χ(x) = {x}} and a2 = x \ (a ∪ a1). notice that a1 is kerneled (by proposition 2.6) and a2 is saturated (since k(x) = {x} for every x ∈ a2). since a = ac1 ∩ a c 2, then sat(a) ⊆ a c 1 and χ(a) ⊆ a c 2. therefore a = sat(a) ∩ χ(a). conversely, suppose that a = sat(a)∩χ(a) for all a ⊂ x and let x ∈ x. if {x} is not kerneled, then x \ {x} is not saturated. since x is the only saturated set containing x \ {x}, this set must be kerneled, and thus {x} is saturated. (iv) ↔ (i). suppose (iv) holds and let x ∈ x. if {x} is not kerneled, then x \ {x} is not saturated. hence there exists y ∈ sat(x \ {x}) such that y /∈ χ(x \ {x}), which implies that x \ {x} is kerneled and therefore {x} is closed. conversely, suppose (i) holds and let a ⊂ x such that sat(a) ⊂ χ(a). if a is not saturated, there is x ∈ sat(a) \ a. by hypothesis {x} is kerneled or closed. if {x} is kerneled, then x \ {x} is a saturated set containing a, thus x \ {x} contains sat(a), a contradiction. if {x} is closed, then x \ {x} is kerneled and hence x \{x} ⊃ χ(a) ⊃ sat(a), again a contradiction. therefore sat(a) = a. � by replacing the envelope sat by the envelope k in the proof of theorem 5.3, we obtain the following result. theorem 5.4. the following are equivalent. (i) g satisfies (t1/2). (ii) a = kλ(a) for all a ⊂ x. (iii) a′ ∈ s(x) for all a ⊂ x. (iv) for all a ⊂ x, if k(a) ⊂ χ(a), then k(a) = a. the following two results show that the axioms (t1/2) and (t1/4) can also be characterized in terms of properties of ∆. theorem 5.5. g satisfies (t1/2) iff ∆ = ∆1 ∪ ∆2, where ∆1 ∈ g 2 and ∆2 is saturated. proof. (⇒) suppose g satisfies (t1/2). let a1 = {x ∈ x : {x} ∈ g} and a2 = {x ∈ x : k(x) = {x}}, and let ∆i = ⋃ x∈ai {x} × {x}, i = 1, 2. by definition of low separation axioms via the diagonal 49 (t1/2), it is obvious that ∆ = ∆1 ∪ ∆2. also, ∆1 = ⋃ x∈a1 {x} × {x} ∈ g2 and sat(∆2) = ⋃ x∈a2 k(x) × k(x) = ⋃ x∈a1 {x} × {x} = ∆2. (⇐) suppose ∆ = ∆1 ∪ ∆2, where ∆1 ∈ g 2 and ∆2 is saturated. since ∆1 ⊂ ∆, there exists a set b1 ⊂ x such that ∆1 = ⋃ x∈b1 {x} × {x}. also, ∆1 ∈ g 2 implies that ∆1 = ⋃ α aα × bα, with aα, bα ∈ g. thus, for each x ∈ b1, {x} × {x} = aα × bα for some α, and visceversa, and it follows that {x} ∈ g for each x ∈ b1. on the other hand, ∆2 = sat(∆2) = ⋃ x∈b2 k(x)×k(x) ⊂ ∆, for some set b2 ⊂ x, and it follows that k(x) = {x}, for each x ∈ b2. it is clear that x = b1 ∪ b2. therefore g satisfies (t1/2). � theorem 5.6. g satisfies (t1/4) iff ∆ = ∆1 ∪ ∆2, where ∆1 is kerneled and ∆2 is saturated. proof. (⇒) suppose g satisfies (t1/4). let a1 = {x ∈ x : χ(x) = {x}} and a2 = {x ∈ x : k(x) = {x}} and let ∆i = ⋃ x∈ai {x} × {x}. it is clear that ∆ = ∆1 ∪ ∆2, χ(∆1) = ∆1, and sat(∆2) = ∆2. (⇐) suppose ∆ = ∆1 ∪∆2, where ∆1 is kerneled and ∆2 is saturated. there exists b1 ⊂ x such that ∆1 = ⋃ x∈b2 χ(x)×χ(x). since ∆1 ⊂ ∆, it follows that χ(x) = {x} for each x ∈ b1. also, there exist b2 ⊂ x such that k(x) = {x}, for each x ∈ b2, and since x = b1 ∪ b2, we conclude that g satisfies (t1/4). � some results found in [2] and [3] can be obtained directly from those proved here, by considering the generalized topologies given by the α-open sets and the λ-open sets respectively. acknowledgements. we would like to thank professor salvador garćıaferreira for helpful discussions related to this research. he visited universidad de los andes during the month of april 2006, with a partial support given by the program intercambio cient́ıfico. references [1] f. arenas, j. dontchev, and m. ganster, on λ-sets and the dual of generalized continuity, questions and answers in gen. top.15, no. 1 (1997), 3–13. [2] m. caldas, d. n. georgiou, and s. jafari, characterizations of low separation axioms via α-open sets and α-closure operator, bol. soc. paran. mat., 21, no. 1-2(2003), 1–14. [3] m. caldas and s. jafari, on some low separation axioms via λ-open sets and λ-closure operator, rend. circ. mat. palermo, serie ii(tomo liv) (2005), 195–208. [4] m. caldas, s. jafari and m.m. kovár, some properties of θ-open sets, divulgaciones matemáticas 12, no. 2(2004), 161–169. [5] a. csaszar, separation axioms for generalized topologies, acta math. hungar. 104 (2004), 63–69. [6] n. levine, semi-open sets and semicontinuity in topological spaces, amer. math. monthly 70 (1963), 36–41. 50 m. l. colasante, c. uzcátegui and j. vielma [7] h. maki, generalized λ-sets and the associated closure operator, the special issue in conmmemoration of proffesor kazusada ikeda’s retirement, (1986), 139–146. [8] o. nj̊astad, on some classes of nearly open sets, pacific. j. math. 15 (1965), 961–970. [9] n. v. veličko, h-closed topological spaces, mat sb. 70 (1966), 98–112. english transl., amer. math. soc. transl., 78 (1968), 102–118. received august 2006 accepted september 2007 m. l. colasante (marucola@ula.ve) departamento de matemáticas, facultad de ciencias, universidad de los andes, mérida 5101, venezuela. c. uzcátegui (uzca@ula.ve) departamento de matemáticas, facultad de ciencias, universidad de los andes, mérida 5101, venezuela. j. vielma (vielma@ula.ve) departamento de matemáticas, facultad de ciencias, universidad de los andes, mérida 5101, venezuela. @ applied general topology c© universidad politécnica de valencia volume 13, no. 1, 2012 pp. 61-79 random selection of borel sets ii bernd günther abstract the theory of random borel sets as presented in part i of this paper is developed further. special attention is payed to the reconstruction of the topology of the underlying space from our presentation of the measure algebra, to an analysis of capacities in context of random borel sets, to inspection processes on the unit segment and to the markov process of random allocation. 2010 msc: primary 60d05; secondary 54h05 keywords: random borel sets 1. introduction we present four topics from the theory of random borel sets as laid out in the first part of this paper: 1. it is well known [5, thm.13.1] that the topology of a polish space can be modified such that an arbitrarily prescribed borel subset is turned into a closed and open subset without modifying the borel algebra as a whole, and in consequence it is not possible to characterize open subsets or closed subsets in the standard borel algebra by invariant properties. in spite of this deficiency, we are going to demonstrate in section 2 that our “dyadic” algebra y [2, §5] provides us with a special representation of the standard borel algebra that easily distinguishes open sets from general borel sets. this discrepancy should not come as a surprise, because y has built in g∞-invariance [2, §3] but fails the invariance under the larger group consisting of all measure preserving homeomorphisms [2, §9], characteristic for the standard measure algebra. evidently, y has a more rigid structure, and the isomorphism [2, §6] forgets certain properties. 62 b. günther 2. in section 3 we investigate the possibility to apply choquet capacities in context of random borel sets in a manner similar to the closed case. this turns out to be infeasible. however, a simple description of random borel sets in terms of stochastic processes can be given, leading to an elegant and abstract characterization. for practical and numerical purposes the “dyadic algebra” is superior, as will be seen for instance in section 5. 3. in [8] straka and štěpán used inspection processes to deal with random borel sets on the unit segment, but they seem to have overlooked the fact that not every inspection process corresponds to a random set. the problem will be dealt with in section 4. 4. the combinatorial concept of random allocation describes the probabilistic allotment of objects to certain places and the observation of the time evolution of their distribution [6]. in each round one may allocate a single object, or a fixed finite number of objects, or an infinite set. an example for the second case consists of the numbered lots drawn in each round in a lottery drawing, where one may observe which numbers appeared at least once after r rounds. thus in the i-th run we select a finite random set bi consisting of the numbers drawn, and observe the growth of the accumulated set cr := ⋃r i=1 bi over the markov time r. if the lottery is fair, the random variable bi assumes each set containing the correct number of elements with equal probability. in more general examples, one may relax the assumption that the sets bi have a predetermined size but still require all sets of the same size to have the same probability. we may play the same game using infinite sets for bi, in our case borel sets. having equal probability for all sets of the same measure would require location independence of the distribution of bi, but we know from [2, §9] that the best we can achieve is g∞-invariance. in section 5 we compute expected value and second moment of the size of the accumulated sets cr in a numerically feasible way. this paper uses the same terms and notations as part i; in particular x denotes a compact metric space carrying a non atomic measure μ with supp(μ) = x; y (μ) denotes the borel algebra of μ-measurable sets, identifying two sets if they differ only in a 0-set. y denotes the “coordinate representation” of y (μ) introduced in [2, §5]. 2. reconstruction of the topology the isomorphism between the algebras y and y (μ) was proved using resolutions of either of two species, labelled type i and type ii [2, def.4.4], and it was already observed that the type i resolutions establish a close link between measure and topology. this is illustrated by the following proposition: proposition 2.1. if a type i resolution is used for the isomorphism theorem [2, thm.6.1], then: i) the borel set corresponding to a sequence (xnm) can be chosen as open if and only if xnm = supn≥n 2 n−n# { k : 2n−nm ≤ k < 2n−n+1,xnk = 1 } for random selection of borel sets ii 63 all n,m. ii) the borel set corresponding to a sequence (xnm) can be chosen as closed if and only if xnm = infn≥n 2n−n# { k : 2n−nm ≤ k < 2n−n+1,xnk > 0 } for all n,m. iii) a borel set is jordan measurable (i.e. its boundary is a 0-set [3, ch.3]) if and only if it satisfies both (i) and (ii). proof. to prove (i) let us assume xnm = supn≥n 2 n−n# { k : 2n−nm ≤ k < 2n−n+1,xnk = 1 } for all n,m and set bn := (⋃ m:xnm=1 anm )◦ . we observe bn+1 ⊇ bn because xnm = 1 ⇒ xn+1,2m = xn+1,2m+1 = 1 and consider the open set b := ⋃ n bn. therefore μ(anm ∩ b) = limn→∞ μ(anm ∩ bn) = limn→∞ ∑ k:xnk=1,2n−nm≤k<2n−n+1m μ(ank) = limn→∞ 2 −n# {k : xnk = 1, 2n−nm ≤ k < 2n−n+1m } = 2nxnm hence the sequence (xnm) corresponds to the open set b under our isomorphism theorem (if necessary we replace μ by 1 ϕ μ). conversely, if b is open we can use inner regularity to find a compact subset k ⊆ anm ∩ b whose measure is as close to that of anm ∩ b as we please and then choose n ≥ n such that maxk diamank is a lebesgue number for the open covering consisting of b and �k. the limit condition follows. (ii) is dual to (i) and does not require separate proof. sufficiency of (iii) follows from (i) and (ii). it remains to establish necessity, so let us consider a sequence (xnm) satisfying both (i) and (ii). we construct the sets cn :=(⋃ m:xnm=0 anm )◦ and c := ⋃ n cn, and just like above we see that c is an open set representing the complement of b, i.e. μ ( b��c ) = 0. but by construction b ∩ c = ∅ and therefore ∂b ⊆ � (b ∪ c) = b��c must be a 0-set. � remark 2.2. this means that (xnm) represents an open set if and only if limn→∞ 2n−n ∑ 2n−nm≤k<2n−n+1m 0 0 for all m,n, and remark 2.2 implies 0 = limn→∞ 2n−n ∑ 2n−nm≤k<2n−n+1m (1 − x′nk) = 1 − x′nm for all 64 b. günther m,n, in particular μ(a′) = 1 since x′00 = 1. hence �a′ is an open subset of x with measure 0, and since supp(μ) = x we must have a′ = x, i.e. a is dense. similarly, if the element (x′′nm)0≤m<2n,n∈n0 ∈ y represents an open subset a′′ contained in a, then x′′nm ≤ xnm < 1 and remark 2.2 implies 0 = limn→∞ 2n−n ∑ 2n−nm≤k<2n−n+1m x ′′ nk = x ′′ nm, in particular μ(a ′′) = 0 since x′′00 = 0. since a ′′ is open it must be empty and a has empty interior. � 3. the capacity problem our construction of probability measures for random borel sets [2, thm.7.1] requires resolutions and thus utilizes a particular “coordinate representation” of the standard borel algebra. in contrast, random closed sets can be defined by capacities in a coordinate independent way. a coordinate free presentation can be given in the borel setting too, but rather different from the closed case. we start by preparing some tools. 3.1. finite partitions. definition 3.1. a finite partition b = {b1, . . .bn} of x is a finite set of borel subsets bi ⊆ x such that (1) ∀i : μ(bi) > 0 (2) ∀i �= j : μ(bi ∩ bi) = 0 (3) ∑ i μ(bi) = 1. we set mesh b := maxi diambi. we observe that the mesh of a partition is defined by metric properties, not by set theoretic ones. proposition 3.2. for any two borel subsets a,b ⊆ x and each ε > 0 there exists δ > 0 such that for each finite partition c = {c1, . . .cn} of x with mesh c < δ (3.1) ∣∣∣∣∣μ(a ∩ b) − ∑ i μ(a ∩ ci)μ(b ∩ ci) μ(ci) ∣∣∣∣∣ < ε proof. for every finite partition c of x we define an orthogonal projection operator tc in the hilbert space l2(μ) by (3.2) tcf := ∑ i ∫ ci fdμ μ(ci) χci a short calculation shows that tc is the orthogonal projection onto the finite dimensional subspace spanned by the indicator functions χci. now suppose ε > 0 and a continuous function f are given; we choose δ > 0 such that d(x,y) < δ ⇒ d(f(x),f(y)) < ε, observing that f must be uniformly continuous by compactness of x. then for mesh c < δ we must have |tcfx − fx| < ε uniformly and therefore ‖tcf − f‖2 ≤ ε. now, finally, suppose a and b are given and choose continuous function f,g such that ‖χa − f‖2 < ε and ‖χb − g‖2 < ε; then using the case above pick random selection of borel sets ii 65 δ > 0 such that ‖tcf − f‖2 ≤ ε and ‖tcg − g‖2 ≤ ε for mesh c < δ. observing ‖tc‖ ≤ 1 we obtain ‖tcχa − χa‖2 ≤ ‖tc (χa − f)‖2 + ‖tcf − f‖2 + ‖f − χa‖2 ≤ ‖χa − f‖2 + ‖tcf − f‖2 + ‖f − χa‖2 ≤ 3ε and similarly ‖tcχb − χb‖2 ≤ 3ε. now |〈χa,χb〉 − 〈tcχa,tcχb〉| ≤ |〈χa − tcχa,χb〉| + |〈tcχa,χb − tcχb〉| ≤ ‖χa − tcχa‖2 ‖χb‖2 + ‖tcχa‖2 ‖χb − tcχb‖2 ≤ 6ε. since 〈χa,χb〉 = μ(a ∩ b) and 〈tcχa,tcχb〉 = ∑ i μ(a∩ci)μ(b∩ci) μ(ci) condition (3.1) follows. � corollary 3.3. for every borel subset a ⊆ x and each ε > 0 there exists δ > 0 such that for each finite partition b = {b1, . . .bn} with mesh b < δ (3.3) 0 ≤ μ(a) − ∑ i μ(a ∩ bi)2 μ(bi) < ε proof. ∑ i μ(a∩bi)2 μ(bi) = ∑ i μ(bi) ( μ(a∩bi) μ(bi) )2 ≤ ∑i μ(bi) μ(a∩bi)μ(bi) = ∑i μ(a ∩ bi) = μ(a). the rest follows from an application of proposition 3.2 to the case a = b. � observe that (3.4) mεδ := ⋂ b:mesh b<δ { a ∣∣∣∣∣ ∑ i μ(a ∩ bi)2 μ(bi) ≥ μ(a) − ε } ⊆ y (μ) is an event because the uncountable intersection may be replaced by an intersection over a dense countable family of finite partitions without affecting the result. our proposition states ∀ε > 0 : ⋃n mε 1n = y (μ). 3.2. generating systems of the borel algebra. the following proposition clearly distinguishes random borel sets from their closed counterpart, because in the closed setting the subsets of the form yb = { a ⊆ x ∣∣∣a ∩ b �= ∅} generate the borel algebra and thus pave the way for capacities: proposition 3.4. the σ-algebra generated by the subsets (3.5) yb := { a ⊆ x ∣∣∣μ(a ∩ b) > 0} ⊆ y (μ) with b ranging over all borel subsets of x is strictly smaller than the borel algebra on y (μ); for instance it does not contain the event m := { a ⊆ x ∣∣∣μ(a) > 12 } . proof. suppose the contrary. then by [4, ch.i,§5,thm.d] there exists a sequence of borel sets bn such that m is contained in the σ-algebra f generated by the countably family of events ybn. consider a ′ ∈ m, i.e. a′ ⊆ x with μ(a′) > 1 2 . clearly f cannot distinguish a′ from any other subset a′′ such that for all n: μ(a′ ∩ bn) > 0 ⇔ μ(a′′ ∩ bn) > 0; hence any such subset must be contained in m and therefore should satisfy μ(a′′) > 1 2 . set γ := { n ∣∣∣μ(a′ ∩ bn) > 0} ⊆ n. using the intermediate value theorem for non atomic measures we can find a subset cn ⊆ a′ ∩bn for each n ∈ γ, such 66 b. günther that 0 < μ(cn) < 2 −n−2, and then in particular μ (⋃ n∈γ cn ) < 1 4 and thus a′′ := ⋃ n∈γ cn ⊆ a′ with μ(a′′) < 14. for n ∈ γ we have a′′ ∩ bn ⊇ cn and therefore μ(a′′ ∩ bn) > 0, while for n ∈ n\γ we obtain a′′ ∩bn ⊆ a′ ∩bn and μ(a′ ∩ bn) = 0. from the above we conclude a′′ ∈ m, a contradiction. � proposition 3.5. the borel algebra on y (μ) is generated by the events (3.6) yt,b := { a ⊆ x ∣∣∣μ(a ∩ b) < t} ⊆ y (μ) with b ranging over all borel subsets of x and 0 < t < 1. actually it is sufficient to restrict b to a dense subset of y (μ) (such as all open or all compact subsets of x) and t to a dense subset of ]0,1[. proof. define metric d on y (μ) by d(a,b) := max (μ(a \ b) ,μ(b \ a)). since 1 2 μ(a�b) ≤ d(a,b) ≤ μ(a�b) this generates the customary topology. since μ(b \ a) = μ(b) − μ(b ∩ a) we obtain μ(b \ a) < ε ⇔ ∀δ > μ(b)−ε : μ(a ∩ b) < δ. hence the ε-ball around b with respect to the metric d is given by { a ⊆ x ∣∣∣μ (a ∩ �b) < ε} ∩ ⋂ n { a ⊆ x ∣∣∣μ(a ∩ b) < μ(b) − ε + 1 n } = yε,�b ∩ ⋂ n yμ(b)−ε+ 1 n ,b (3.7) therefore these sets generate the topology and hence all borel sets in y (μ). � unfortunately, the events yt,b do not constitute a ring, i.e. they are are incompatible with unions or intersections. therefore the construction of measures will require us to consider finite intersections: yt1,...tr,b1,...br := { a ⊆ x ∣∣∣μ(a ∩ b1) < t1 and . . .μ(a ∩ br) < tr} = r⋂ i=1 yti,bi. 3.3. the probability space. for every finite partition b = {b1, . . .bn} we consider the map qb : y (μ) → in(3.8) a �→ ( μ(a ∩ b1) μ(b1) , . . . μ(a ∩ bn) μ(bn) ) (3.9) and denote by νb the image of the probability measure on y (μ) on in under the map qb. in will be equipped with the metric db ( tj, t ′ j ) := √∑ j μ(bj) ( tj − t′j )2 . random selection of borel sets ii 67 consider a finite partition c = {c1, . . .cm}. then for any other finite partition b = {b1, . . .bn}, not necessarily a refinement of c, we define a map qbc : i n → im(3.10) (t1, . . . tn) �→ (u1, . . .um)(3.11) ui := ∑ j μ(bj ∩ ci) μ(ci) tj(3.12) notice that the map qbc is a contraction: by convexity for each i:⎛ ⎝∑ j μ(bj ∩ ci) μ(ci) ( tj − t′j )⎞⎠ 2 ≤ ∑ j μ(bj ∩ ci) μ(ci) ( tj − t′j )2 , hence dc ( qbc (tj) ,q b c ( t′j )) = √∑ i μ(ci) (∑ j μ(bj ∩ci) μ(ci) ( tj − t′j ))2 ≤√∑ i,j μ(ci) μ(bj∩ci) μ(ci) ( tj − t′j )2 = √∑ j μ(bj) ( tj − t′j )2 = db ( tj, t ′ j ) . b = {b1, . . .bn} is called refinement of c = {c1, . . .cm} if for any two indices i,j either μ(ci ∩ bj) = 0 or μ(ci \ bj) = 0. under this condition equation (3.12) assumes the form ui = ∑ j:μ(ci∩bj) =0 μ(bj) μ(ci) tj. if b is a refinement of c and a an arbitrary finite partition, then qbc qab = qac and qbc qb = qc theorem 3.6. the system of measures νb satisfies the following two conditions: (1) qbc νb = νc for every refinement b of c. (2) for each ε > 0 there exists δ > 0 such that (3.13) νb ( (tk) ∈ in : ∑ k μ(bk) ( tk − 1 2 )2 > 1 4 − ε ) > 1 − ε for each finite partition b = {b1, . . .bn} with mesh b < δ. conversely: every system of probability measures satisfying these properties derives from a probability measure p on y (μ) as system of images νb = qbp. proof. we extend definition (3.8) to a map qb : z(μ) → in, qb(f) = (t1, . . . tn), tk := 1 μ(bk) ∫ bk fdμ; by reasons of compactness this is easily checked to be an inverse limit representation. it now suffices to show that the subset y (μ) ⊆ z(μ) corresponds precisely to the set m of all those f ∈ z(μ), whose representations (t1, . . . tn) = qb(f) satisfy the condition ∀ε > 0∃δ > 0∀ mesh b < δ : ∑ k μ(bk) ( tk − 12 )2 > 1 4 − ε. corollary 3.3 readily implies y (μ) ⊆ m. but for f ∈ z(μ) \ y (μ) we can find ε > 0 with μ(ε < f < 1 − ε) > ε; let c = {c1,c2,c3} be the partition c1 := {f ≤ ε}, c2 := {ε < f < 1 − ε}, c3 := {f ≥ 1 − ε}. if b is any refinement of c, then (t1, . . .tn) = qb(f) satisfies ∑ k μ(bk) ( tk − 12 )2 ≤ 1 4 − ε2 and therefore f �∈ m. � 68 b. günther example 3.7. suppose we are given a random closed set a with capacity functional τ(k) = p (a ∩ k �= 0). considering this as random borel set we get for any finite partition b = {b1, . . .bn} the corresponding capacity as νb ( ∏ i [0, ti[) = p (∀i : μ(a ∩ bi) < tiμ(bi)) = 1 − sup { τ ( ⋃ i ki) ∣∣∣∀i : ki compact,ki ⊆ bi,μ(ki) > (1 − ti) μ(bi)}. 3.4. the process aspect. we may now describe random borel sets as stochastic processes as follows: theorem 3.8. for any random borel set a consider the process assigning to each compact set (each open set, each borel set) k the random variable yk := μ(a∩k) μ(k) with values in i. (for μ(k) = 0 we may define yk arbitrarily or disregard it entirely.) then: (1) yk1∪k2 = μ(k1) μ(k1∪k2)yk1 + μ(k2) μ(k1∪k2)yk2 provided k1 ∩ k2 = ∅. (2) for each finite partition k = {k1, . . .kn} consider the sum ∑ i μ(ki)y 2 ki . then ∑ i μ(ki)y 2 ki → yx a.s. if mesh k → 0. (notice yx = μ(a).) conversely, any process satisfying these two properties derives from a random borel set, whose distribution is uniquely characterized by these properties. the proof is a direct application of theorem 3.6 making a few observations: consider two sets k,l. then |yk − yl| = ∣∣∣ μ(k\l)μ(k) yk\l + μ(k∩l)μ(k) yk∩l− μ(l\k) μ(l) yl\k − μ(k∩l)μ(l) yk∩l ∣∣∣ ≤ μ(k\l)μ(k) + μ(l\k)μ(l) + μ(k ∩ l) ∣∣∣ 1μ(k) − 1μ(l) ∣∣∣. this immediately implies yk = yl if μ(k�l) = 0; moreover ykn → yk uniformly if kn → k in the topology of y (μ). since the compact sets and the open sets are dense in y (μ) they suffice to define yb for any borel set b as uniform limit. furthermore, just like in the proof of [2, lem.4.1] it can be shown that the compact sets, whose boundary is of measure 0, (or the open sets, whose boundary is of measure 0) are dense in y (μ). this readily implies that any finite partition of the compact space x into borel sets can be arbitrarily closely approximated by a finite partition into compact sets or into open sets. condition (2) means ∑ i μ(ki) ( yki − 12 )2 = ∑ i μ(ki)y 2 ki − ∑i μ(ki)yki + 14 =∑ i μ(ki)y 2 ki − yx + 14 and is preserved by the extension to borel sets. now observe corollary 3.3. then define the measure νb as the joint distribution of the random variables yb1, . . .ybn. lemma 3.9. for any borel set b, the random variable yb is the a.s.-limit∑ i μ(bi) μ(b) y 2bi → yb with b = {b1, . . .bn} ranging over the partitions of b such that mesh b → 0. proof. since 0 ≤ ybi we have ∑ i μ(bi) μ(b) y 2bi ≤ ∑ i μ(bi) μ(b) ybi = yb for each partition of b. now set c := �b and supplement every partition of b with one of c to obtain a partition of x without increasing the mesh. then also∑ i μ(ci) μ(c) y 2ci ≤ yc. furthermore μ(b) ∑ i μ(bi) μ(b) y 2bi + μ(c) ∑ i μ(ci) μ(c) y 2ci → yx by theorem 3.8, which can happen only if ∑ i μ(bi) μ(b) y 2bi → yb and ∑ i μ(ci) μ(c) y 2ci → yc. � random selection of borel sets ii 69 proposition 3.10. for any two random borel sets represented by processes yk resp. y ′ k we can define a new process (3.14) yk ∧ y ′k := lim ∑ i μ(ki) μ(k) ykiy ′ ki as a.s.-limit taken over all finite partitions k = {k1, . . .kn} of k such that mesh k → 0. this new process represents the intersection of the two given random borel sets. proof. apply proposition 3.2. � lemma 3.11. two random borel sets represented by processes yk resp. y ′ k are independent as random variables if and only if for any two borel sets (compact sets, open sets) k and l the random variables yk and y ′ l are independent. proof. apply proposition 3.5. � definition 3.12. a random borel set represented by a process yk is called isotropic, if for every ε > 0 there exists a finite partition b = {b1, . . .bn} of x with mesh b < ε, such that for any two indices 1 ≤ i �= j ≤ n there exists a permutation π ∈ sn with π(i) = j, such that the joint distribution of ykπ(1), . . .ykπ(n) coincides with the joint distribution of yk1, . . .ykn. notice that under the condition above in particular e (yki) = e ( ykj ) for any two indices i,j, and since yx = ∑ i μ(ki)yki we obtain e (yx) =∑ i μ(ki)e (yki) = e ( ykj ) ∑ i μ(ki) = e ( ykj ) for each j. proposition 3.13. consider two independent random borel sets a and b, at least one of which is assumed isotropic. then e (μ(a ∩ b)) = e (μ(a)) e (μ(b)). proof. assume y ′k is isotropic and apply proposition 3.10: e (μ(a ∩ b)) = e (yx ∧ y ′x) = lim ∑ i μ(ki)e (yki)e ( y ′ki ) = (lim ∑ i μ(ki)e (yki))e (y ′ x) = (lime ( ∑ i μ(ki)yki))e (y ′ x) = e (yx)e (y ′ x) = e (μ(a)) e (μ(b)). � 4. random borel sets on the unit segment and inspection processes in [8] random borel sets on the unit segment were uniquely characterized by their inspection processes, neglecting the fact that not every inspection process corresponds to a random borel set. every random variable ϕ satisfying conditions (1–2) below can be interpreted as an inspection process of a random variable with values in a certain function space and hence is more general than a random set; the special case or random sets is determined by condition (3) below. for any random borel subset x of the unit segment i we consider the inspection process xt := λ(x ∩ [0, t]) = ∫ t 0 χxdλ, where λ is lebesgue measure on i. observe that xt as a function of t is continuous, increasing and almost everywhere differentiable with derivative almost surely equal to χx [7, 70 b. günther thm8.8,p.168]. that means xt is the primitive (syn. antiderivative) of the indicator function χx. observe that this implies that the paths xt are much better behaved than the paths of brownian motion, which are almost surely nowhere differentiable [1, thm.12.25,p.261]. we consider the set of functions ϕ : i → r subject to the conditions (1) ϕ(0) = 0 (2) ∀0 ≤ x ≤ y ≤ 1 : 0 ≤ ϕ(y) − ϕ(x) ≤ y − x (3) sup ∑ k [ϕ(tk+1)−ϕ(tk)]2 tk+1−tk = ϕ(1), where the supremum is taken over all finite decompositions of the unit segment 0 = t0 < t1 < · · · < tn = 1. notice that the functions satisfying conditions (1) and (2) constitute a compact convex set z of the space of all continuous functions on i equipped with the topology of uniform convergence by the theorem of arzela-ascoli. by y ⊂ z we denote the subspace of functions satisfying all three conditions (1–3). lemma 4.1. for all x,y ∈ r and α,β > 0 with α + β = 1 the inequality (x + y)2 ≤ x2 α + y2 β holds. proof. 0 ≤ (√ β α x − √ α β y )2 = β α x2 − 2xy + α β y2 = ( 1 α − 1 ) x2 − 2xy +( 1 β − 1 ) y2 = 1 α x2 + 1 β y2 − (x + y)2. � lemma 4.2. all functions ϕ ∈ z satisfy sup ∑k [ϕ(tk+1)−ϕ(tk)]2tk+1−tk ≤ ϕ(1). moreover, if the decomposition 0 = u0 < u1 < · · · < un = 1 is a refinement of 0 = t0 < t1 < · · · < tn = 1, then ∑n−1 k=0 [ϕ(uk+1)−ϕ(uk)]2 uk+1−uk ≥ ∑n−1 k=0 [ϕ(tk+1)−ϕ(tk)]2 tk+1−tk . proof. ∑n−1 k=0 [ϕ(tk+1)−ϕ(tk)]2 tk+1−tk = ∑n−1 k=0 (tk+1 − tk) [ ϕ(tk+1)−ϕ(tk) tk+1−tk ]2 ≤ ∑n−1k=0 (tk+1− tk) [ ϕ(tk+1)−ϕ(tk) tk+1−tk ] = ∑n−1 k=0 [ϕ(tk+1) − ϕ(tk)] = ϕ(1). now consider three points 0 ≤ v1 < v2 < v3 ≤ 1; then an application of lemma 4.1 to x := ϕ(v2) − ϕ(v1), y := ϕ(v3) − ϕ(v3), α := v2−v1v3−v1 and β := v3−v2 v3−v1 yields [ϕ(v3)−ϕ(v1)]2 v3−v1 ≤ [ϕ(v2)−ϕ(v1)]2 v2−v1 + [ϕ(v3)−ϕ(v2)]2 v3−v2 , and this concludes the proof. � remark 4.3. for every function ϕ ∈ z there is a lebesgue measurable function ψ : i → i with ϕ(t) = ∫ t 0 ψ(v)dv for all t ∈ i (consider positive measure defined on intervals [x,y] by ϕ(y) − ϕ(x) and apply the radon-nikodym theorem). lemma 4.4. suppose ϕ is the antiderivative ϕ(t) = ∫ t 0 ψ(v)dv of a measurable function ψ : i → i, and suppose a sequence of decompositions ζn ={ 0 = t (n) 0 < t (n) 1 < · · · < t (n) nn = 1 } of the unit segment is given with limn→∞ meshζn = 0. then limn→∞ ∑nn−1 k=0 [ ϕ ( t (n) k+1 ) −ϕ ( t (n) k )]2 t (n) k+1 −t(n) k = ∫ 1 0 ψ(v)2dv. random selection of borel sets ii 71 proof. we define a sequence of functions fn : i → i by fn(t) := ϕ ( t (n) k+1 ) −ϕ ( t (n) k ) t (n) k+1 −t(n) k if t (n) k ≤ t < t (n) k+1. by [7, thm8.8,p.168] fn → ψ almost everywhere, and consequently f2n → ψ2 almost everywhere. hence by lebesgue’s dominated convergence theorem limn→∞ ∫ 1 0 fn(v) 2dv = ∫ 1 0 ψ(v)2dv. � proposition 4.5. a function ϕ ∈ z is the antiderivative of an indicator function if and only if it satisfies condition (3). proof. in view of lemma 4.4 condition 3 translates to ∫ 1 0 ψ(1 − ψ)dλ = 0. this means that ψ can assume values different from 0 or 1 only on a 0-set, i.e. ψ almost surely equals an indicator function. � by proposition 4.5 we may identify inspection processes satisfying condition (3) with random borel sets. evidently, this condition is a relation among the increments of the inspection process, and one suspects that these increments cannot be independent. proposition 4.6. suppose the inspection process xt := λ(x ∩ [0, t]) = ∫ t 0 χxdλ corresponding to a random borel set x has independent increments. then there exists a fixed (deterministic) borel set b such that x ≡ b a.s. proof. we consider two numbers 0 ≤ x < y ≤ 1 and an integer n ∈ n. setting h := y−x n , tk := x + kh, 0 ≤ k ≤ n, we telescope xy − xx =∑n k=0 ( xtk+1 − xtk ) . furthermore set bt := e (xt). independence of increments then implies var (xy − xx) = var n∑ k=0 ( xtk+1 − xtk ) (4.1) = n∑ k=0 var ( xtk+1 − xtk ) (by independence)(4.2) = n∑ k=0 ( e (( xtk+1 − xtk )2) − (e (xtk+1 − xtk ))2 ) (4.3) = he n∑ k=0 ( xtk+1 − xtk )2 tk+1 − tk − h n∑ k=0 ( btk+1 − btk )2 tk+1 − tk (4.4) → he ( xtk+1 − xtk ) − h n∑ k=0 ( btk+1 − btk )2 tk+1 − tk (by condition 3)(4.5) = h ( btk+1 − btk ) − h n∑ k=0 ( btk+1 − btk )2 tk+1 − tk (4.6) now var (xy − xx) = 0 is obtained by letting n → ∞ and hence h → 0; this means xy − xx ≡ by − bx a.s. hence the function t �→ bt must also satisfy 72 b. günther condition 3 and defines the borel set b required by our theorem. (on a sideline we may observe that the function t �→ bt satisfies conditions 1 and 2, therefore∑n k=0 (btk+1 −btk ) 2 tk+1−tk ≤ y − x.) � 5. an application to random allocation our random allocation experiment will be developed from a random borel set that is a stronger version of example [2, ex.7.5]; that means we construct a probability measure on the space of borel sets b by prescribing the distribution of the composite random variable b �→ μ(b) (condition (1) below) and then inductively lifting the measure over the fibers of the projection maps pn+1n : i 2n+1 → i2n familiar from [2, §7] (condition (2) below). the intended computation of first and second moments requires some detailed preparations. so we suppose we are given (1) a random variable x00 distributed on i and (2) another random variable u symmetrically distributed on [−1,1] such that (5.1) ∀ε > 0 : p (|u| ≥ 1 − ε) > 0 then we define random variables xnm for n > 0, 0 ≤ m < 2n inductively by picking identically distributed copies unm of u, independent from one another and from all xrk, r ≤ n, and setting xn+1,2m = xnm + unm min (xnm,1 − xnm)(5.2) xn+1,2m+1 = xnm − unm min (xnm,1 − xnm)(5.3) observe that the scaling factor min (xnm,1 − xnm) is chosen such that it ensures 0 ≤ xn+1,2m ≤ 1 and 0 ≤ xn+1,2m+1 ≤ 1. also notice that this is almost the same as [2, ex.7.5], with the difference that our random variables need not possess a density, and with [2, eq.7.8] replaced by the much weaker assumption (5.1). the system of random variables is invariant under g∞. for fixed n, the joint distribution of the random variables xn0, . . .xn,2n−1 defines a probability measure νn on i 2n, and we claim that [2, eq.7.1] is satisfied and hence a probability measure on the space of borel sets y (μ) is obtained. by gn-invariance e (( xnm − 12 )2) is independent of m and we can define (5.4) an := e (( xnm − 1 2 )2) = e ( 2−n 2n−1∑ m=0 ( xnm − 1 2 )2) by jensen’s inequality the sum sn := 2 −n ∑2n−1 m=0 ( xnm − 12 )2 increases with n; hence, if we can show an → 14, we can conclude that sn converges to 14 almost surely and therefore in probability. thus [2, eq.7.1] will be proved. random selection of borel sets ii 73 we visualize our random variables as tree with root x00, such that each xnm has two descendents xn+1,2m and xn+1,2m+1. fix 0 < ε < 1 2 and set q := 1 2 p (|u| ≥ 1 − 2ε) > 0. we claim (5.5) p (∣∣∣∣xnm − 12 ∣∣∣∣ ≥ 12 − ε ) ≥ q for all n > 0. for, assume m = 2k is even and xn−1,k ≥ 12, then xnm = xn−1,k +un−1,k (1 − xn−1,k) = (1 − un−1,k)xn−1,k +un−1,k ≥ 12 (1 − un−1,k) + un−1,k = 12 (1 + un−1,k). but since un−1,k ≥ 1 − 2ε with probability at least q we obtain xnm ≥ 1 − ε with probability at least q provided xn−1,k ≥ 12. the cases xn−1,k ≤ 12 or m odd are handled similarly. now observing the independence of un−1,k from xn−1,k we conclude that the event anm that xnm itself or at least one of its ancestors xrk in our tree satisfies ∣∣xrk − 12∣∣ ≥ 12 − ε has probability p (anm) ≥ 1 − (1 − q)n. we claim an ≥ ( 1 2 − ε )2 [1 − (1 − q)n]. lim infn an ≥ ( 1 2 − ε )2 will follow, and therefore, since ε was arbitrary, limn an = 1 4 . to this end consider the event a (r) nm ⊆ anm such that the first ancestor of xnm satisfying ∣∣xrk − 12∣∣ ≥ 12 − ε occurs at level r and observe that a (r) nm is invariant under the subgroup of gn leaving the range 2n−rk ≤ � < 2n−r(k + 1) invariant; therefore (5.6) ∫ a (r) nm ( xnm − 1 2 )2 dp = ∫ a (r) nm 2r−n 2n−r(k+1)−1∑ �=2n−rk ( xn� − 1 2 )2 dp however, by jensen’s inequality we have 2r−n ∑2n−r(k+1)−1 �=2n−rk ( xn� − 12 )2 ≥( 2r−n ∑2n−r(k+1)−1 �=2n−rk xn� − 12 )2 = ( xrk − 12 )2 ≥ (1 2 − ε )2 on a (r) nm and therefore ∫ a (r) nm ( xnm − 12 )2 dp ≥ ( 1 2 − ε )2 p ( a (r) nm ) . hence ∫ anm ( xnm − 12 )2 dp = ∑ r ∫ a (r) nm ( xnm − 12 )2 dp ≥ ( 1 2 − ε )2 ∑ r p ( a (r) nm ) = ( 1 2 − ε )2 p (anm) ≥ ( 1 2 − ε )2 [1 − (1 − q)n]. this concludes the proof. assumption 1. for the remainder of this subsection we assume that the random variable u is uniformly distributed on [−1,1]. 74 b. günther observe that with this assumption e ( h(xnm) ∣∣∣x10 = t) = e (h(xn−1,m) ∣∣∣x00 = t) for any function h! in consequence e ( h(xnm) ∣∣∣x00 = t)(5.7) = 1 2 +1∫ −1 e ( h(xnm) ∣∣∣x10 = t + u min (t,1 − t))du(5.8) = 1 2 +1∫ −1 e ( h(xn−1,m) ∣∣∣x00 = t + u min (t,1 − t)) du(5.9) = 1 2 min(t,1 − t) min(1,2t)∫ max(0,2t−1) e ( h(xn−1,m) ∣∣∣x00 = v) dv(5.10) applying the foregoing to h(t) = t2 leads to the functions fn(t) = e ( x2nm ∣∣∣x00 = t), whose special role has already been investigated in [2, §8]: f0(t) := t 2(5.11) fn+1(t) := 1 2 min(t,1 − t) min(1,2t)∫ max(0,2t−1) fn(u)du(5.12) f1(t) = t 2 + 1 3 [min(t,1 − t)]2(5.13) by induction one can easily prove fn(t) ≤ t for all n. since f1 ≥ f0 we also infer by induction fn+1 ≥ fn for all n. consequently there must be a limit function f∞(t) = limn→∞ fn(t), at least lower semicontinuous, and satisfying t2 ≤ f∞(t) ≤ t with (5.14) f∞(t) = 1 2 min(t,1 − t) min(1,2t)∫ max(0,2t−1) f∞(u)du equation (5.14) immediately implies that f∞ is continuous on the whole interval i, even at 0 and 1. by induction one can prove (5.15) fn(t) − fn(1 − t) = 2t − 1 for all n, thus gn(t) := t − fn(t) is symmetric around 12, gn(1 − t) = gn(t), g0(t) = t(1 − t), gn+1(t) = 12t 2t∫ 0 gn(u)du for 0 ≤ t ≤ 12 and gn ↓ 0 for n → ∞. random selection of borel sets ii 75 0.1 0.2 0.3 0.4 0.5 0.05 0.10 0.15 0.20 0.25 figure 1. first few iterates of gn. lemma 5.1. under assumption 1, f∞(t) = t for all t ∈ i. proof. suppose the contrary and consider the function h(t) := t − f∞(t); then h is continuous, h ≥ 0, h(0) = h(1) = 0, h(t) = 1 2 min(t,1−t) min(1,2t)∫ max(0,2t−1) h(u)du and there exists t0 ∈]0,1[ such that h(t0) = max u∈i h(u) > 0. if t0 ≤ 12, then h(t0) = 1 2t0 2t0∫ 0 h(u)du < 1 2t0 2t0∫ 0 h(t0)du = h(t0), a contradiction. similarly, if t0 ≥ 12, then h(t0) = 12(1−t0) 1∫ 1−2t0 h(u)du < 1 2(1−t0) 1∫ 1−2t0 h(t0)du = h(t0) and we arrive again at a contradiction. � observe that lemma 5.1 provides an independent proof that our construction leads to a well defined probability on y . the first few iterates fn of the fredholm equation (5.12) (or equivalently, of gn, cf. figure 1) can be evaluated in closed form, later ones can be obtained numerically. now we repeat our random borel set drawing experiment independently and with identical distribution, at the i-th run obtaining a borel set bi. then cr := ⋃r i=1 bi models the set of elements allocated up to run r; the set valued random variables cr constitute a markov chain over the time variable r. theorem 5.2. let p be the probability measure on i obtained as the distribution of x00, i.e. of the measure of the borel set obtained in a single run of our random experiment. then at markov time r the accumulated borel set has the expected measure 1 − [∫ (1 − x)dp(x) ]r and variance (5.16) ∞∑ m=0 2−m−1 [∫ (2fm(1 − x) − fm+1(1 − x)) dp(x) ]r − [∫ (1 − x)dp(x) ]2r (the case of a dirac measure p located at the “fixed weight” x is displayed in figure 2.) 76 b. günther 0.2 0.4 0.6 0.8 1.0 0.02 0.04 0.06 0.08 figure 2. square root of variance for markov times r = 2,5,10. the proof requires a few preparations, starting with a stronger continuity property of the ∧-product than given in [2, lem.5.2]: proposition 5.3. the ∧-product is a jointly continuous map ∧ : zh×zh → zh in the hilbert space topology and ‖a ∧ x − a ∧ y‖2 ≤ √ ‖a‖2 · ‖x − y‖2. proof. we recall that for all z = (znm)0≤m<2n,n≥0 ∈ y the sequence 2−n ∑2n−1 m=0 z 2 nm is increasing with supn∈n0 2 −n ∑2n−1 m=0 z 2 nm = ‖z‖22 (cf. [2, eq.5.4]). for any two vectors z′,z′′ ∈ z we have ‖z′ − z′′‖22 ≤ 1 because all |z′nm − z′′nm| ≤ 1. this applies in particular to z = a ∧ x − a ∧ y. hence for each 0 ≤ α < ‖a ∧ x − a ∧ y‖22 ≤ 1 we can find n ∈ n0 such that 2−n ∑2n−1 m=0 z 2 nm > α. now pick numbers 0 ≤ αm < 1 such that 2−n ∑2n−1 m=0 αm ≥ α and z2nm > αm for each 0 ≤ m < 2n; by definition of the ∧-product there exists n0 ≥ n such that for all n ≥ n0 and each 0 ≤ m < 2n the inequality 2n−n ∑(m+1)2n−n−1 k=m2n−n ank |xnk − ynk| > √ αm ≥ αm holds (remember αm ≤ 1, hence √ αm ≥ αm). by summing over all m we obtain 2−n ∑2n −1 k=0 ank |xnk− ynk| > 2−n ∑2n−1 m=0 αm ≥ α and hence, using the cauchy-schwarz inequality: ‖a‖2 · ‖x − y‖2 ≥ √ 2−n ∑2n −1 k=0 a 2 nk · √ 2−n ∑2n −1 k=0 (xnk − ynk) 2 > α. since α < ‖a ∧ x − a ∧ y‖22 was arbitrary this implies ‖a‖2·‖x − y‖2 ≥ ‖a ∧ x − a ∧ y‖ 2 2. this implies continuity of the ∧-product with respect to the hilbert space topology because ‖x ∧ y − x0 ∧ y0‖2 ≤ √ ‖x0‖2 · ‖y − y0‖2+ √ ‖y‖2 · ‖x − x0‖2 for x,x0,y,y0 ∈ z. � lemma 5.4. for each x = (xnm) ∈ z consider the sequence x(r) = ( x (r) nm ) ∈ z defined by (5.17) x(r)nm := { xr,�m2r−n r ≤ n xnm r ≥ n random selection of borel sets ii 77 then limr→∞ x(r) = x in the hilbert space topology of z (cf. [2, §5]). observe that x ∈ z ⇒ x(r) ∈ z, but in general x(r) �∈ y even if x ∈ y . proof. x (r) n,2m = x (r) n,2m+1 for n > r, therefore ∥∥x − x(r)∥∥2 = ∑∞n=r+1 ∑2n−1−1m=0 2−n−1 (xn,2m − xn,2m+1)2. for r → ∞ this is the remainder of a convergent series x200 + ∑∞ n=1 ∑2n−1−1 m=0 2 −n−1 (xn,2m − xn,2m+1)2 = ‖x‖2 and hence converges to 0 for r → ∞. � corollary 5.5. for each finite sequence of vectors xi = (xi;nm)0≤m<2n,n∈n0 ∈ z, 1 ≤ i ≤ s: (5.18) ( s∧ i=1 xi ) nm = lim n→∞ 2n−n (m+1)2n−n−1∑ k=m2n−n s∏ i=1 xi;nk proof. by induction on s one can easily show (5.19) ( s∧ i=1 x (r) i ) nm = ⎧⎪⎪⎨ ⎪⎪⎩ 2n−r (m+1)2r−n−1∑ k=m2r−n s∏ i=1 xi;rk r ≥ n s∏ i=1 xi;r,�m2r−n r ≤ n the corollary now follows from continuity of the ∧-product taking the limit r → ∞. � proof of theorem 5.2. we switch to complements, setting b′i := �bi, c′r := �cr = ⋂r i=1 b ′ i, thus translating unions into intersections, which is equivalent but in better agreement with our preparations. now the proof resembles that of [2, thm.8.2]. in coordinate representation, let the random borel set b′i correspond to the process ξ (i) nm, then by corollary 5.5 c ′ r corresponds to ξnm = limn→∞ 2 n−n∑(m+1)2n−n−1 k=m2n−n ∏r i=1 ξ (i) nk. therefore e ( ξ00|ξ(1)00 , . . .ξ (r) 00 ) = lim n→∞ 2−n 2n −1∑ k=0 e ( r∏ i=1 ξ (i) nk ∣∣∣ξ(1)00 , . . .ξ(r)00 ) (5.20) = lim n→∞ 2−n 2n −1∑ k=0 r∏ i=1 e ( ξ (i) nk|ξ (i) 00 ) (5.21) = r∏ i=1 ξ (i) 00(5.22) 78 b. günther and with ξ00 = μ(c ′ r) = 1 − μ(cr), ξ(i)00 = μ(b′i) = 1 − μ(bi) this leads to 1 − e (μ(cr)) = ∫ e ( ξ00 ∣∣∣ξ(1)00 = 1 − t1, . . .ξ(r)00 = 1 − tr ) (p ⊗ · · · ⊗ p) (dt1 · · ·dtr) (5.23) = ∫ r∏ i=1 (1 − ti) (p ⊗ · · · ⊗ p) (dt1 · · ·dtr)(5.24) = [ 1 − ∫ tp(dt) ]r (5.25) now the second moments are obtained as follows: e ( ξ200|ξ(1)00 , . . .ξ (r) 00 ) = lim n→∞ 2−2n ⎡ ⎣2n −1∑ k=0 r∏ i=1 e ( ξ (i)2 nk |ξ (i) 00 ) + 2n −1∑ a =b=0 r∏ i=1 e ( ξ (i) naξ (i) nb|ξ (i) 00 )⎤⎦ (5.26) = lim n→∞ 2−2n [ 2n −1∑ k=0 r∏ i=1 fn ( ξ (i) 00 ) + 2n −1∑ a =b=0 r∏ i=1 ( 2fv(a,b)−1 ( ξ (i) 00 ) − fv(a,b) ( ξ (i) 00 ))](5.27) integrating the summand 2−2n ∑2n −1 k=0 ∏r i=1 fn ( ξ (i) 00 ) over the product measure p ⊗ · · · ⊗ p leads to 2−n [∫ fn(1 − t)p(dt) ]r and drops out in the limit n → ∞. this leaves us with (5.28) e ( μ(c′r) 2 ) = lim n→∞ 2−2n 2n −1∑ a =b=0 [∫ ( 2fv(a,b)−1 (1 − t) − fv(a,b) (1 − t) ) p(dt) ]r observing # {(a,b)|v(a,b) = m} = 22n−m we arrive at e ( μ(c′r) 2 ) = ∞∑ m=1 2−m [∫ (2fm−1 (1 − t) − fm (1 − t))p(dt) ]r (5.29) var ( μ(cr) 2 ) = var ( μ(c′r) 2 ) = ∞∑ m=1 2−m [∫ (2fm−1 (1 − t) − fm (1 − t))p(dt) ]r − [∫ (1 − t)p(dt) ]2r (5.30) � random selection of borel sets ii 79 references [1] l. breiman, probability, volume 7 of classics in applied mathematics. siam, 1992. [2] b. günther, random selection of borel sets, appl. gen. topol. 11, no. 2 (2010), 135–158. [3] h. hadwiger, vorlesungen über inhalt, oberfläche und isoperimetrie, volume 93 of grundlehren. springer, 1957. [4] p. r. halmos, measure theory, volume 18 of gtm. springer, 1974. [5] a. s. kechris, classical descriptive set theory, volume 156 of gtm. springer, 1995. [6] v. f. kolchin, b. a. sevast’yanov and v. p. chistyakov, random allocations, scripta series in mathematics. john wiley & sons, 1978. [7] w. rudin, real and complex analysis, series in higher mathematics, macgraw-hill, 2nd edition, 1974. [8] f. straka and j. štěpán, random sets in [0,1], in j. visek and s. kubik, editors, information theory, statistical decision functions, random processes, prague 1986, volume b, pages 349–356. reidel, 1989. (received july 2011 – accepted march 2012) b. günther (dr.bernd.guenther@t-online.de) db systel gmbh, development center databases t.svd32, weilburger straße 28, 60326 frankfurt am main, germany random selection of borel sets ii. by b. günther gmaynezmancioagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 213-221 completions of pre–uniform spaces adalberto garćıa–máynez and rubén s. mancio t. abstract. in this paper we prove the existence of a completion of a t0–pre-uniform space (x, u), with the property that each cauchy filter in (x, u) contains a weakly round filter. 2000 ams classification: 54a05, 54a20, 54e15. keywords: pre–uniform space, complete pre–uniform space, uniform continuity, pre–uniformity basis, semi–uniformity basis, unimorphism, unimorphism embedding, equivalent, admisible and compatible pre–uniformities, balanced filter, cauchy, minimal cauchy, round, weakly round, and strongly round filters, proper pre–uniform space. 1. introduction by a pre–uniform space we mean a pair (x, u) where x is a set and u is a non–empty family of covers of x satisfying certain properties. every preuniform space (x, u) determines a topology tu in x and the convergence properties of filters in (x, tu ) constitute an important area of study. pre–uniform spaces generalize the semi–uniform spaces introduced by morita in [3]. the most important filters to be considered are cauchy filters in (x, u), i.e., filters in x which intersect every cover of u. in many important examples, for a cauchy filter η in (x, u) to converge it is necessary and sufficient that η has an adherence point, i.e., a point which belongs to the closure of every member of η. we consider an increasing chain of four important subclasses of the class of cauchy filters: strongly round filters, round filters, weakly round filters and minimal filters. every cauchy filter in a semi–uniform space contains an strongly round filter and hence, a minimal filter. however, this is not true in more general pre–uniform spaces. we give all the necessary definitions in next section. as in the case of uniform or semi–uniform spaces, we say that a pre–uniform space (x, u) is complete if every cauchy filter in (x, u) converges. we have 214 a. garćıa–máynez and r. mancio four less restrictive types of completeness if we require only that the filters in one of the four classes defined above are convergent. if (x, u) is a pre–uniform space and a ⊆ x, the family of cover restrictions ua = { α|a : α ∈ u } determines a pre–uniform space (a, ua) and we say then that (a, ua) is a subspace of (x, u) or that (x, u) is an extension of (a, ua). it is easy to see that tu |a = tua and this justifies the use of the words “subspace” and “extension”. in this paper we prove that every t0–pre–uniform space (x, u) has a canonical t1 extension (x̂, û ), where every weakly round filter converges. we also prove that a necessary and sufficient condition for (x̂, û ) to be complete is that every cauchy filter in (x, u) contains a weakly round filter. as a corollary of this, we deduce the known result that every semi–uniform space is completable. these are the main results of this paper. 2. pre–uniform spaces and uniform continuity all the definitions and notation agreements of this paper appear in the doctoral thesis of the second author. for convenience to the reader, we shall include the most important ones. note 1. if x is a set, f ⊆ x, p ∈ x and α is a cover of x, then: s t (p, α) = s t ({p} , α) = ∪ {l ∈ α : p ∈ l} s t (f, α) = ∪ {l ∈ α : l ∩ f 6= ∅} . definition 2.1. let u be a non–empty family of covers of a set x. the topology tu in x induced by u is defined as follows: l ⊆ x belongs to tu iff for every x ∈ l, we may find a finite collection {α1, α2, . . . , αn} ⊆ u such that n⋂ i=1 s t (x, αi) ⊆ l. note 2. suppose α1, α2, . . . , αn, α, β are covers of a set x. we denote: α1 ∧ α ∧ · · · ∧ αn = {l1 ∩ l2 ∩ · · · ∩ ln : l1 ∈ α1, l2 ∈ α2, · · · , ln ∈ αn} . α ≤ β means that α refines β i.e., there exists a map λ : α → β such that a ⊆ λ (a) for every a ∈ α. clearly α1 ∧ α2 ∧ · · · ∧ αn ≤ αi for every i ∈ {1, 2, . . . , n}. if u1, u2 are collections of covers of a set x, we write u1 ≤ u2 if for every α ∈ u1, there exists a finite collection {β1, β2, . . . , βn} ⊆ u2 such that β1 ∧ β2 ∧ · · · ∧ βn ≤ α. u1 and u2 are equivalent if u1 ≤ u2 and u2 ≤ u1. clearly u1 ⊆ u2 ⇒ u1 ≤ u2 ⇒ tu1 ⊆ tu2 . if u is a collection of covers a set x, u+ denotes the family of covers γ of x such that for some α ∈ u, we have α ≤ γ. clearly, the families u and u+ are equivalent and hence they generate the same topology on x. completions of pre–uniform spaces 215 definition 2.2. a non-empty collection u of covers of a set x is a pre– uniformity basis on x if u satisfies the following two conditions: 1) whenever α1, α2, . . . , αn ∈ u there exists a cover β ∈ u such that β ≤ α1 ∧ α2 ∧ · · · ∧ αn. 2) for each α ∈ u, there exists a cover β ∈ u such that β ≤ ◦ α, where ◦ α = {int tu l : l ∈ α}. hence, every pre–uniformity basis u is equivalent to a pre–uniformity basis u′ where each cover α ∈ u′ is open with respect to the topology tu . definition 2.3. a pre–uniform space is a pair (x, u), where x is a set and u is a pre–uniformity basis on x. remark 2.4. if (x, u) is a pre–uniform space and a ⊆ x, then u |a ={ α|a : α ∈ u } is a pre–uniformity basis on a and the topologies tu |a and tu |a coincide. we say then that ( a, u |a ) is a pre–uniform subspace of (x, u). remark 2.5. let (x, u) be a pre–uniform space. then, for every a ⊆ x, we have: (2.1) inttu a = {x ∈ a : there exists αx ∈ u such that s t (x, αx) ⊆ a} . definition 2.6. 1) a map f : (x, u) → (y, v) between pre–uniform spaces is uniformly continuous if for every β ∈ v, we may find a cover α ∈ u such that α ≤ { f −1 (b) : b ∈ β } . 2) a bijection ϕ : (x, u) → (y, v) between pre–uniform spaces is said to be a unimorphism if ϕ and ϕ−1 are both uniformly continuous maps and we say, in this case, that (x, u) and (y, v) are unimorphic spaces. 3) a map g : (x, u) → (y, v) from the pre–uniform space (x, u) into the pre–uniform space (y, v) is a unimorphic embedding if g (x) is dense in y and g is a unimorphism from (x, u) onto ( g (x) , v |g (x) ) . remark 2.7. 1) if f : (x, u) → (y, v) is uniformly continuous, then f : (x, tu ) → (y, tv ) is continuous. 2) si u, v are pre–uniformity bases in the same set x, then the identity map id : (x, u) → (x, v) is uniformly continuous iff v ≤ u. therefore, id is a unimorphism iff u and v are equivalent. 216 a. garćıa–máynez and r. mancio definition 2.8 (see [2], definition 1.2.2, page 10). 1) a non–empty family u of covers of a set x is a semi–uniformity basis on x if u satisfies 2.2.1 and: su) for every α ∈ u, there exists a cover β ∈ u such that for every b ∈ β, we may find a cover γ b ∈ u and a set a b ∈ α such that s t (b, γ b) ⊆ a b. 2) a semi–uniform space is a pair (x, u), where x is a set and u is a semi–uniformity basis on x. 3) u is a uniformity basis on x if u satisfies condition 2.2.1 and the stronger condition: u) for every α ∈ u, there exists a cover β ∈ u such that: {st (b, β) : b ∈ β} ≤ α. the following facts are well known: theorem 2.9. 1) let u be a semi–uniformity basis on a set x. then tu is a a regular topology on x. conversely, if (x, t ) is a regular topological space, there exists a semi–uniformity basis u on x such that t = tu . 2) let u be a uniformity basis on a set x. then tu is a a completely regular topology on x. conversely, if (x, t ) is a completely regular topological space, there exists a uniformity basis u on x such that t = tu . 3. filters and completeness we recall now some definitions about filters. let x be a set. a filter in x is a non–empty subfamily f of the power set p (x) which satisfies the following properties: i) ∅ /∈ f. ii) f1, f2 ∈ f ⇒ f1 ∩ f2 ∈ f. iii) f ∈ f, f ⊆ l ⊆ x ⇒ l ∈ f. a filter base in x is a non–empty subfamily η of p (x) satisfying the properties: i) ∅ /∈ η. ii) n1, n2 ∈ η ⇒ ∃ n3 ∈ η such that n3 ⊆ n1 ∩ n2. for any subfamily g ⊆ p (x), we denote: g+ = {a ∈ p (x) : g ⊆ a for some g ∈ g} . if g1, g2 ⊆ p (x), g1 ≤ g2 means that g + 2 ⊆ g + 1 , i.e., g1 ≤ g2 iff every element of g2 contains an element of g1. g1 ∼ g2 (g1 is equivalent to g2) means that g1 ≤ g2 and g2 ≤ g1, i.e., g1 ∼ g2 iff g + 1 = g + 2 . this is clearly an equivalence relation in p (p (x)). if we restrict ourselves to filter basis in x, ∼ is still an equivalence relation. it is easy to prove that every equivalence class contains completions of pre–uniform spaces 217 exactly one filter, namely, if η is a filter base in x, η+ is the only filter in x which satisfies the relation η ∼ η+. let (x, t ) be a topological space and let η be a filter in x. a point x ∈ x is an adherence point of η (in symbols, η 7→ x) if every neighborhood of x intersects every element of η. equivalently, η 7→ x iff x ∈ cℓ (n) (= the t – closure of n) for every n ∈ η. x is a convergence point of η (in symbols, η → x) if every neighborhood of x belongs to η. a filter η in a set x is an ultrafilter in x if η is not properly contained in any other filter in x. two filters η1, η2 in x mingle (in symbols, η1 ↔ η2) if every element of η1 intersects every element of η2. remark 3.1. two filters η1, η2 in x mingle iff there exists a filter η in x such that η1 ∪ η2 ⊆ η. a filter η in a topological space (x, t ) is balanced if every adherence point of η is also a convergence point of η. for any x ∈ x, ηx denotes the filter of t –neighborhoods of x. remark 3.2. let η1, η2 be filters in a topological space (x, t ) and let x ∈ x. then: 1) η1 → x implies η1 7→ x. 2) η1 → x and η1 ↔ η2 imply that η2 7→ x. 3) η1 7→ x iff η1 ↔ ηx. 4) η1 → x iff η1 ⊇ ηx. 5) η1 ≤ η2 and η1 7→ x imply that η2 7→ x. 6) η1 ≤ η2 and η2 → x imply that η1 → x. 7) every ultrafilter in x is balanced. 8) every filter in x without adherence points is balanced. 9) every convergent filter in a hausdorff space is balanced. definition 3.3. a filter η in a pre–uniform space (x, u) is u–cauchy (or cauchy in (x, u)) if for every α ∈ u, we have η ∩ α 6= ∅. remark 3.4. if ϕ : (x, u) → (y, v) is a uniformly continuous map and if f is a u–cauchy filter, then ϕ (f) + is a v–cauchy filter. hence, if u and v are equivalent pre–uniform bases on x, then (x, u) and (x, v) have the same cauchy filters. definition 3.5. for every cauchy filter f in a pre–uniform space (x, u), define: f ′ = {st (f, α) : f ∈ f, α ∈ u} + f r = { s∗t (f, α) : α ∈ u }+ f rr = { s∗∗t (f, α) : α ∈ u }+ , where s∗t (f, α) = ∪ {a ∈ α : a ∩ f 6= ∅ for every f ∈ f} 218 a. garćıa–máynez and r. mancio and s∗∗t (f, α) = ∪ {a : a ∈ f ∩ α} . remark 3.6. if f is a cauchy filter in (x, f), then f ′, f r, f rr are also filters in x and we have: f ′ ⊆ f r ⊆ f rr ⊆ f. definition 3.7. let f be a cauchy filter in a pre–uniform space (x, u). 1) f es minimal if f does not properly contain any other cauchy filter in (x, u). 2) f is weakly round if f = f rr. 3) f is round if f = f r. 4) f is strongly round if f = f ′. we summarize in a theorem the most important relations among these different kinds of filters. the proofs can be found in [2]. theorem 3.8. 1) every strongly round filter is round. 2) every round filter is weakly round. 3) every weakly round filter is minimal. 4) two round filters f1, f2 in a pre–uniform space mingle iff f1 = f2. 5) every neighborhood filter is weakly round. 6) if (x, u) is a semi–uniform space and if f is a cauchy filter in (x, u), then f ′ is also a cauchy filter in (x, u) and f ′ is contained in any cauchy filter g contained in f. in fact, f ′ is a strongly round filter in (x, u) and f ′′ = f ′. definition 3.9. a pre–uniform space (x, u) is complete if every cauchy filter in (x, u) converges. lemma 3.10. let ϕ : (x, u) → (y, v) be a unimorphism between pre–uniform spaces and let η be a weakly round filter in x. then ϕ (η) = {ϕ (n) : n ∈ η} is a weakly round filter in (y, v). proof. fix an element n ∈ η and let α ∈ u be such that: h (α) = ∪ {l : l ∈ η ∩ α} ⊆ n. let β ∈ v be such that β ≤ ϕ (α) = {ϕ (a) : a ∈ α}. we shall prove that k (β) = ∪ {b : b ∈ ϕ (η) ∩ β} ⊆ ϕ (n). for each b ∈ β select an element ab ∈ α such that b ⊆ ϕ (ab). therefore, if also b ∈ ϕ (η), we have ϕ (ab) ∈ ϕ (η) and ab ∈ η (recall ϕ is a bijective map). therefore, k (β) ⊆ ∪ {ϕ (a) : a ∈ η ∩ α} = ϕ (h (α)) ⊆ ϕ (n) . � completions of pre–uniform spaces 219 lemma 3.11. let η be a weakly round filter in a pre–uniform space (x, u) and let a ⊆ x. then η|a is a weakly round filter in (a, ua). proof. obvious. � definition 3.12. a complete pre–uniform space (y, v) is a completion of a pre–uniform space (x, u) if there exists a unimorphic embedding ϕ : (x, u) → (y, v). definition 3.13. a pre–uniform space (x, u) is proper if every cauchy filter in (x, u) contains a weakly round filter. note 3. let (x, u1) be a t0–pre–uniform space and choose an open pre– uniformity basis u equivalent to u1. let x̂ = {ξ : ξ is a weakly round filter in (x, u)}. for every a ⊆ x and every α ∈ u, define: â = { ξ ∈ x̂ : a ∈ ξ } α̂ = { â : a ∈ α } and let û = {α̂ : α ∈ u}. for every x ∈ x, let ϕ (x) = ηx = filter of tu – neighborhoods of x. theorem 3.14. keep the notation of 3 and suppose the topology tu is t0. then û is an open pre–uniformity basis in x̂ and ϕ is a unimorphic embedding of (x, u) into ( x̂, û ) . besides, every weakly round filter in ( x̂, û ) is convergent. proof. everything is proved in [2], except the last part. let f be a weakly round filter in ( x̂, û ) . define: η = { a ∈ ∪ {α : α ∈ u} : â ∈ f }+ . it is easy to prove that η is a cauchy filter in (x, u). we shall prove that η is weakly round. choose an element l ∈ η. by the definition of η, there exists a cover α ∈ u and an element a ∈ α such that a ⊆ l and â ∈ f. since f is weakly round in ( x̂, û ) , there exists a cover β ∈ u such that ∪ { b̂ : b̂ ∈ f ∩ β̂ } ⊆ â. take an element b ∈ η ∩ β. therefore, b̂ ∈ f ∩ β̂ and b̂ ⊆ â. conversely, b = ϕ−1 ( b̂ ) ⊆ ϕ−1 ( â ) = a ⊆ l. therefore, η is weakly round in (x, u) and η ∈ x̂. it remains to prove that f → η. take a set w ∈ t û containing η. since ∪ {α̂ : α ∈ u} is a basis for the topology t û , we may assume that w coincides with â, where a ∈ α for some α ∈ u. this implies that a ∈ η and, therefore, â ∈ f. hence, the filter of t û –neighborhoods of η is contained in f and f → η. � 220 a. garćıa–máynez and r. mancio theorem 3.15. let (x, u) be a t0–pre–uniform space. then (x, u) admits at least one completion iff (x, u) is proper. proof. (⇒) let (y, v) be a completion of (x, u) and let ϕ : (x, u) → (y, v) be a unimorphic embedding. let η be a cauchy filter in (x, u). then f = ϕ (η) + is a cauchy filter in (y, v) and hence, there exists an element y ∈ y such that f → y. let fy be the filter of tv –neighborhoods of y. by 3.8.5, fy is a weakly round filter in (y, v). besides, fy ⊆ f because f → y. using lemmas 3.10 and 3.11, we deduce that η0 = ϕ−1 ( fy| ϕ (x) ) is a weakly round filter in (x, u) contained in η. (⇐) we shall prove that ( x̂, û ) is a completion of (x, u). let f be a cauchy filter in ( x̂, û ) and let: η = { a ∈ ⋃ α∈u α : â ∈ f }+ . it is easy to prove that η is a cauchy filter in (x, u). since (x, u) is proper, there exists a weakly round filter η0 in (x, u) contained in η. let us prove that f → η0. let α ∈ u and a ∈ α be such that η0 ∈ â. therefore, a ∈ η0 ⊆ η and â ∈ f. hence, every tû –neighborhood of η0 belongs to f, i.e., f → η0. � corollary 3.16. let (x, u) be a t0–pre–uniform space where every cauchy filter in (x, u) contains a round filter. then ( x̂, û ) is a hausdorff completion of (x, u) and the only round filters in ( x̂, û ) are the neighborhood filters. proof. see [2]. � example 3.17. let (x, u) be a hausdorff topological space and let u be the family of open covers α of x such that some finite subfamily λ ⊆ α covers a dense subset of x. then u is an open pre–uniformity basis on x satisfying the condition of 3.16. in this case, t = tu and the hausdorff completion ( x̂, û ) is a hausdorff closed extension of (x, u). proof. see [2] and [5]. � completions of pre–uniform spaces 221 references [1] a. garćıa-máynez and a. támariz mascarúa , topoloǵıa general, méxico., porrúa, 1988. [2] r. mancio-toledo, los espacios pre–uniformes y sus completaciones, ph. d. thesis, universidad nacional autónoma de méxico, 2006. [3] k. morita, on the simple extension of a space with respect to a uniformity, i, ii, iii and iv, proc. japan acad. 27 (1951), 65–72, 130–137, 166–171 and 632–636. 14(1953), 68–69, 571. [4] k. morita and j. nagata, topics in general topology, amsterdam, elsevier science publishers b. v., 1989. [5] r. j. porter and r. g. woods, extensions and absolutes of hausdorff spaces, new york., springer-verlag new york inc., 1988. received march 2006 accepted january 2007 adalberto garćıa–máynez cervantes (agmaynez@matem.unam.mx) instituto de matemáticas, universidad nacional autónoma de méxico, area de la investigación cient́ıfica, circuito exterior, ciudad universitaria, distrito federal, c.p. 04510, méxico rubén s. mancio toledo (rmancio@esfm.ipn.mx) escuela superior de f́ısica y matemáticas del instituto politécnico nacional, edificio no. 9, unidad profesional adolfo lópez mateos, colonia lindavista, méxico d.f., c.p. 07738, méxico pajooheshagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 1-14 topological and categorical properties of binary trees h. pajoohesh to my friend, tehmineh fadaei for her 40’th birthday abstract. binary trees are very useful tools in computer science for estimating the running time of so-called comparison based algorithms, algorithms in which every action is ultimately based on a prior comparison between two elements. for two given algorithms a and b where the decision tree of a is more balanced than that of b, it is known that the average and worst case times of a will be better than those of b, i.e., t a(n) ≤ t b(n) and t w a (n) ≤ t w b (n). thus the most balanced and the most imbalanced binary trees play a main role. here we consider them as semilattices and characterize the most balanced and the most imbalanced binary trees by topological and categorical properties. also we define the composition of binary trees as a commutative binary operation, *, such that for binary trees a and b, a ∗ b is the binary tree obtained by attaching a copy of b to any leaf of a. we show that (t, ∗) is a commutative po-monoid and investigate its properties. 2000 ams classification: 06a12, 06f05, 16b50. keywords: algorithm, decision tree, lower topology, semilattice. 1. introduction here we bring some definitions and statements from [11]: definition 1.1. binary trees are reversed trees with a root node, in which every internal node has exactly two children. the order of the leaves is insignificant, so a tree is determined (up to permutation on the leaves) by the lengths of the paths from the root node to each leaf ( the distance of the leaf from the root). thus we can represent equivalence classes of the rooted binary trees with n leaves by sequences of n non-negative integers, which give the path-length of each leaf. 2 h. pajoohesh for example, the path-length sequence < 1 3 3 4 4 4 4 > represents a binary tree with n = 7 leaves, of which one has path-length 1, two have path-length 3, and four have path-length 4. this is shown in the following: remark 1.2. notice that both of the following binary trees are the same for us. we bring the following theorem from [6]: theorem 1.3. if we denote the set of binary trees with n leaves by tn, < x1 ... xn >∈ tn if and only if ∑n i=1 1/2 xi = 1. our interest in decision trees stems from the fact that in order to carry out the running time analysis of so-called “comparison-based algorithms”, i.e. algorithms in which every action is ultimately based on a prior comparison between two elements, the notion of a decision tree is a fundamental tool [2]. most sorting and searching algorithms are comparison-based. decision trees are binary trees representing the comparisons carried out during the computation of a comparison-based algorithm. the path-length from the root (“input”) to a leaf (“output”) gives the comparison time for the algorithm (i.e. the total number of comparisons) to compute the output corresponding to the given input. the concept of the path-length of a tree is of great importance in the analysis of algorithms, since this quantity is often directly related to the execution time [6]. if we consider decision trees, then the path-length represents the number of comparisons made while producing the result for a given input list. therefore, the sum of the path-length sequence represents the total number of comparisons made by the sorting algorithm over all possible permutations of the input list. topological and categorical properties of binary trees 3 for example, if we consider some sorting algorithm a which sorts a list of size 3 then this will produce a decision tree with six leafs, each of which corresponds to the computation involved in producing the sorted list for one of the six possible input lists which can exist. assume that the path-length sequence for the decision tree of a is < 2 3 3 2 3 3 >,(see the following diagram). as mentioned previously, the path-length sequences form an equivalence class so we can represent this tree with the sequence < 2 2 3 3 3 3 >. from this sequence we can see that two of the input lists took two comparisons to be sorted and the other four input lists took three comparisons each. cb abcacb cab ca cb cba bca bac c>acb a,b,c definition 1.4. consider x = < a1 ... aj p aj+2 ... q + 1 q + 1 ... an >, y = < a1 ... aj p + 1 p + 1aj+3 ... q ... an >∈ tn. then we say y can be obtained from x via a ternary exchange. if q = p + 1 then we call this ternary exchange, a minimal ternary exchange. for every x =< x1 ... xn >∈ tn, the level of x is defined by l(x) = (n − 1)(n + 2)/2 − ∑n i=1 xi. this type of balancing exchange has the effect of moving a subtree consisting of two leaves at level q in the tree to a leaf (which then becomes an internal node) at level p in the tree. an example of this is shown in the following: <1 3 3 3 3> <2 2 2 3 3> 4 h. pajoohesh it is obvious that these minimal balancing exchanges have the effect of increasing the level of balance of a tree by a value of one. consequently, they also have the effect of decreasing the sum of the path-lengths of a sequence by a value of one. therefore, every time we increase the level of balance of a tree’s path-length sequence (through minimal balancing exchanges), we simultaneously decrease the sum of its path-lengths. definition 1.5. if x, y ∈ tn then we define x ≤ y if there are l1, ..., lm ∈ tn such that y = l1, x = lm and for each i, 1 ≤ i < m, li+1 is obtained from li via a ternary exchange. so if x ≤ y then we say that x is more balanced than y. example 1.6. in the following diagram, x =< 1 2 3 3 > and y =< 2 2 2 2 >, p = 1 and q + 1 = 3. so y is obtained from x via a ternary exchange which is actually a minimal ternary exchange because q = p + 1. thus y is more balanced than x, what is seen in the diagrams as well. x y the path-length from the root (“input”) to a leaf (“output”) gives the comparison time for the algorithm (i.e. the total number of comparisons) to compute the output corresponding to the given input. so the total time is the summation of all path-lengths from the root to the leaves. for instance in example 1.6 we can see that for x, the total summation of path-lengths is 1 + 2 + 3 + 3 = 9 while for y, 2 + 2 + 2 + 2 = 8. so it is seen in this example that when y is more balanced than x then the summation of the path length sequence for y is less than the summation of the path length sequence for x. it is true in general that if the binary tree corresponding to the c.b. algorithm (comparison based algorithm)a is less than or equal to the binary tree corresponding to the c.b. algorithm b then the running(total) time of a (which is actually the sum of path length sequence of the binary tree corresponding to it) is less than the running time of b. in other words a is faster than b. in other words when x =< x1 ... xn > is more balanced than y =< y1 ... yn > then∑n i=1 xi ≤ ∑n i=1 yi the average time analysis and the worst and the best case analysis time for a c.b. algorithm a which is defined as the usual, is denoted respectively by t (a),t w (a) and t b(a). it was shown in [9, 10] that given two algorithms a and b where the decision tree of a is more balanced than that of b, we have that the average and worst case times of a will be better than those of b, i.e., t a(n) ≤ t b(n) and t w a (n) ≤ t w b (n). topological and categorical properties of binary trees 5 proposition 1.7 ([11]). tn with the above order is a lattice. definition 1.8. the smallest element of tn is called the most balanced binary tree and the biggest one is called the most imbalanced binary tree. the notion of balance is very important in the context of running times of algorithms. a binary tree is the most balanced if the height of the left subtree of every node never differs by more than ±1 from the height of its right subtree [7]. if a decision tree is balanced then its sum of path-lengths will be minimised [6]. this in turn means that the computation time for the algorithm which produced this decision tree will be minimised. 2. topologies on ordered trees in this section we look at binary trees as a semi-lattice. they are dcpos with the induced order. we study the scott and lower topology on them and will show that every join preserving map from a binary tree to another is scott continuous. then we characterize the most imbalanced binary trees by using the lower topology. definition 2.1. consider the poset x and x, y ∈ x such that x < y. we say x is a lower cover of y and will denote x ≺ y if there is no element between them. in other words x ≺ y if and only if {t|x < t < y} = ∅. definition 2.2. an ordered tree is a conditionally complete ∨-semilattice t so that for every x ∈ t , i) |{y ∈ t |y ≺ x}| = 2 or 0 ii) |{y ∈ t |x ≺ y}| = 1 or 0 by a conditionally complete join semilattice we mean a ∨-semilattice such that every non-empty subset has a join. definition 2.3. for every ordered tree, t by ∨ t we mean the biggest element of t and by m(t ) we mean the set of minimal elements of t . every partial order on a dcpo (a poset such that every directed set has a join) induces two topologies on this set, scott topology and lower topology and their join which is called lawson topology. we recall that if (x, ≤) is a dcpo, then a ⊆ x is called scott open if it is an uppersetwhich means that if a ∈ a and a ≤ b then b ∈ aand whenever for directed set d ⊆ x, ∨ d ∈ a then d ∩ a 6= ∅. the collection of the scott open sets is a topology and is called scott topology. the topology generated by x− ↑ x, x ∈ x where ↑ x = {y ∈ x|y ≥ x} is called lower topology and is denoted by α. we study these topologies in this section. theorem 2.4. let t1, t2 be ordered trees. then every join preserving map f from t1 to t2 preserves arbitrary join and hence is scott continuous. 6 h. pajoohesh proof. consider {xi|i ∈ i} ⊆ t1. we show that f ( ∨ i∈i xi) = ∨ i∈i f (xi). since f is join preserving and hence order preserving which gives ∨ i∈i f (xi) ≤ f ( ∨ i∈i xi). (*) now if m, n ≺ ∨ xi, m 6= n, then it is obvious that there are xl, xh, where h, l ∈ i, xl 6= xh, such that xl ≤ m and xh ≤ n. thus xl ∨ xh = ∨ xi, so f (xl) ∨ f (xh) = f ( ∨ xi). this shows f ( ∨ xi) ≤ ∨ f (xi). (**) from (*) and (**) we get f ( ∨ xi) = ∨ f (xi) and the proof is complete. � definition 2.5. let t be an ordered tree, then for every x ∈ t , we define the length of x be the distance of x to ∨ t and we denote it by l(x). more precisely for x let x = x0 ≺ x1 ≺ ... ≺ ∨ t = xm then l(x) = m. also we define the length of t , l(t ) = max{l(x)|x ∈ m(t )}. remark 2.6. by the construction of ordered trees for every element of an ordered tree t , there is a unique path which connects it to ∨ t . definition 2.7. for every ordered tree, we define, s(t ), to be the increasing sequence corresponding to the distance of the minimal elements to ∨ t . definition 2.8. in the lower topology, α, a ∈ α is called principal if there exists x ∈ a such that a =↓ x. obviously every minimal element is an open set. we call them trivial open sets. definition 2.9. the principal α-open set v is called atomic if c(v ) = {w ∈ α|w ( v , w is nontrivial and principal } with inclusion is a chain. it is called strongly atomic if c(v ) has more than one element. theorem 2.10. let t be an ordered tree with n minimal elements. then (1) every principal open set in the lower topology is atomic if and only if for some n ∈ {2, 3, ...}, s(t ) = (1, ..., n − 2, n − 1, n − 1). (2) every strongly atomic open set consists at least 3 minimal elements with 3 different lengths. proof. (1) since every principal open set is atomic, t itself is atomic. let x1, y1 ≺∨ t . we show that one of x1 and y1 is a minimal element. equivalently we show that ↓ x1 ∩ t = {x} or ↓ y1 ∩ t = {y}. first of all since t is a lower set ↓ x1 =↓ x1 ∩ t and ↓ y1 =↓ y1 ∩ t . clearly these two sets are non-empty. now if ↓ x1 ∩ t and ↓ y1 ∩ t both have more than one element, then both ↓ x1 =↓ x1 ∩ t and ↓ y1 =↓ y1 ∩ t are nontrivial principal open sets which are contained in t and neither of them contains the other one which contradicts the atomic property of t . thus either x1 or y1 is a minimal element. let x1 be minimal element. notice that if both are minimal elements then s(t ) = (1, 1). now by assumption ↓ y1 is atomic. now if x2, y2 ≺ y1 then again with the same reason x2 is a minimal element or y2 is a minimal element. topological and categorical properties of binary trees 7 let x2 be a minimal element. then again if we consider y2 and assume that x3, y3 ≺ y2 then we get another minimal element, x3 and so on. thus we get a sequence of minimal elements x1, x2, ..., xn−2 where by their structure l(xi) = i for i = 1, 2, ..., n − 2. now when we consider yn−2 since t has n minimal elements this time xn−1 and yn−1 both are minimal elements and their lengths are both n − 1. thus s(t ) = (1, ..., n − 2, n − 1, n − 1). now assume that s(t ) = (1, ..., n − 2, n − 1, n − 1). we have to show that every principal open set is atomic. let k be principal and∨ k = e. if l(e) = s and m, n ≺ e we show that m is a minimal element or n is a minimal element. if neither of them is a minimal element then t (m) = {x ∈ t |x ≤ m} is an ordered tree so there are at least two minimal elements with the same length. we have the same result for n, too. thus again t (n) = {x ∈ t |x ≤ m} has two minimal elements with the same length. thus in the ordered tree there are 4 minimal elements with at most 2 distinct lengths where the construction of s(t ) doesn’t allow this. thus either m or n is a minimal element. if both are minimal elements then c(k) = ∅ and hence k is atomic. now if m is a minimal element but n is not, then ↓ n ∈ c(k) and is the maximum element of c(k). inductively we can get a finite sequence which shows that c(k) is a chain. (2) let m be a strongly atomic open set and l( ∨ m ) = s. then by what we have proved already if x, y ≺ ∨ m , one of them is a minimal element. let x be a minimal element then l(x) = s + 1. now if z, t ≺ y and both are minimal elements then c(m ) = {{y, z, t}}, has one element which contradicts m being strongly atomic. thus one of z and t is a minimal element and the other is not, which means that we will have another two lengths aside from the length of x and the proof is complete. � remark 2.11. we can improve the above theorem to an infinite ordered tree and notice that in this case we will have: (1) every principal open set in the lower topology is atomic if and only if s(t ) = (1, 2, 3, ...). (2) every strongly atomic open set consists at least 3 minimal elements with 3 different lengths. 3. category of ordered trees in this section we introduce the category of binary trees whose objects are binary trees and morphisms are strictly length preserving maps. then we characterize the most balanced binary trees with 2k leaves, k ∈ {0, 1, 2, ...} as weak initial objects of this category. 8 h. pajoohesh definition 3.1. let t1, t2 be ordered trees. a map f : m(t1) → m(t2) is length preserving if for minimal elements m, n ∈ t1, l(m) ≤ l(n) implies l(f (m)) ≤ l(f (n)). it is obvious that l(x) = l(y) implies l(f (x)) = l(f (y)). theorem 3.2. let t1, t2 be ordered trees. then every length preserving map f : t1 → t2 can be extended to an order preserving map, g : t1 → t2. proof. define g : t1 → t2 such that for a ∈ m(t1), g(a) = f (a) and for the other points g(a) = ∨ t2. then clearly g is order preserving. � theorem 3.3. the class of (finite) ordered trees with length preserving maps is a category. we denote it by (flt ) lt . proof. this is because the composition of two length preserving maps is a length preserving map. � theorem 3.4. t1, t2 ∈ flt are isomorphic if and only if s(t1) = s(t2). proof. first assume that t1 is isomorphic to t2. thus there are length preserving maps, f : m(t1) → m(t2) and g : m(t2) → m(t1) such that f ◦g = idm(t2) and g ◦ f = idm(t1). this implies |m(t1)| = |m(t2)|. also notice that l(x) < l(y), x, y ∈ t1, implies l(f (x)) < l(f (y)), because if l(f (x)) = l(f (y)) then l(x) = l(g(f (x)) = l(g(f (y)) = l(y). also if l(x) = l(y) then l(f (x)) = l(f (y)), because if for example l(f (x)) < l(f (y)), then by the same proof l(x) = l(g(f (x))) < l(g(f (y))) = l(y). thus we can say l(x) = l(y) if and only if l(f (x)) = l(f (y)). so the proof will be complete if we show that l(x) = l(f (x)). but |{l(x) : x ∈ t1}| = |{l(x) : x ∈ t2}|, also for every a ∈ t1, |{x ∈ t1 : l(x) = l(a)}| = |{x ∈ t2 : l(x) = l(f (a))}|. now the only thing that has remained is, {l(x)|x ∈ t1} = {l(x)|x ∈ t2}. we prove this by induction on the cardinality of |{l(x)|x ∈ t1}|. if |{l(x)|x ∈ t1}| = 1, then {l(x)|x ∈ ti} = {ki}, for i = 1, 2. then ti has 2 ki , where i = 1, 2, minimal elements and since m(t1) = m(t2) , 2 k1 = 2k2 and hence k1 = k2. now assume that |{l(x)|x ∈ t1}| = |{l(x)|x ∈ t2}| = p and let l(ti) = mi, for i = 1, 2. we show that m1 = m2. for this we derive isomorphic ordered trees t ′i from ti for i = 1, 2 such that l(t ′ i ) = l(ti) − 1. for this take t ′ i = ti − {x ∈ m(ti)|l(x) = mi}, for i = 1, 2. now m(t ′ i ) = ai∪̇bi, for i = 1, 2 where ai = m(ti) ∩ m(t ′ i ) and bi = m(t ′ i ) − ai. notice that f (a1) = a2 and g(a2) = a1. by the definition of ordered trees, 2|b1| = |{x ∈ m(t1) : l(x) = m1}| = |{x ∈ m(t2) : l(x) = m2}| = 2|b2|. since |b1| = |b2|, there is a one one and onto function, h : b1 → b2. now define f ′ : m(t ′1) → m(t ′ 2) such that f ′(x) = f (x), if x ∈ a1 and f ′(x) = h(x) if x ∈ b1 and define g′ : m(t ′2) → m(t ′ 1) such that g ′(x) = g(x), if x ∈ a2 and g ′(x) = h−1(x) if x ∈ b2. then f ′ and g′ are length preserving maps such that g′ ◦ f ′ = idm(t ′ 1) and f ′ ◦ g′ = idm(t ′ 2). thus t ′ 1 and t ′ 2 are isomorphic. topological and categorical properties of binary trees 9 now if n1 = max{l(x)|x ∈ t1, l(x) < m1} and n2 = max{l(x)|x ∈ t2, l(x) < m2}. then m1 − n1 = m2 − n2. assume the contrary for example m1 − n1 < m2 − n2 then if we reduce the length of t1 and t2 like above m1 − n1 times then the new ordered trees obtained from t1 and t2 are isomorphic, because in every stage of reducing the length we get isomorphic ordered trees. but the new ordered tree obtained from t1 has p − 1 different lengths and the new ordered tree obtained by t2 has again p different lengths. but it is impossible to have two isomorphic ordered trees with different number of different lengths. so m1 −n1 = m2 −n2. now we reduce the length m1 −n1 times in both ordered trees, then we will have two isomorphic ordered trees with p−1 different lengths and by using induction and by the proof, ∀x ∈ t1, l(x) < m1, l(x) = l(f (x)). thus in particular n1 = n2 and since m1 −n1 = m2 −n2 we get m1 = m2. this proves that if for x ∈ t1, l(x) = m1 then l(x) = l(f (x)) and also we showed that ∀x ∈ t1, l(x) < m1, l(x) = l(f (x)). thus for every x ∈ t1, l(x) = l(f (x)) and the proof of this part is complete. the converse of the theorem is trivial because of the definition of length, length preserving maps and s(t ). � now we introduce a useful subcategory of lt (flt ) whose objects are the object of lt (flt ) but whose morphisms are strictly length preserving maps-that is if t1, t2 are trees and f : m(t1) → m(t2), whenever l(x) < l(y) for x, y ∈ t1, then l(f (x)) < l(f (y)). we denote this subcategory by slt (fslt ). lemma 3.5. if f ∈ m or(lt ) is a morphism of slt then l(f (x)) = l(f (y)) if and only if l(x) = l(y). proof. we know from length preserving maps, l(x) = l(y) implies l(f (x)) = l(f (y)). also if l(f (x)) = l(f (y)) then l(x) = l(y) because if for example l(x) < l(y) then l(f (x)) < l(f (y)) which contradicts l(f (x)) = l(f (y)). � theorem 3.6. t1, t2 ∈ fslt are isomorphic if and only if s(t1) = s(t2). remark 3.7. theorems 3.4 and 3.6 are not true for infinite ordered trees. for example consider t1 and t2 such that s(t1) = (1, 2, 3, ...) and s(t2) = (2, 3, 4, ...). define f : m(t1) → m(t2) such that l(f (x)) = l(x) + 1 and g : m(t2) → m(t1) such that l(g(x)) = l(x) − 1. then g ◦ f = idm(t1) and f ◦ g = idm(t2). so t1 is isomorphic to t2 but s(t1) 6= s(t2). now we recall from (cf [1]) that the skeleton subcategory of a category is a full subcategory such that for any object in the original category, there exists a unique isomorphic object in the skeleton subcategory theorem 3.8. the skeletons of fslt and flt are the same. proof. it is true because we can characterize the skeletons of both slt and lt by the path-length sequences which is exactly the characterization of binary trees. � 10 h. pajoohesh corollary 3.9. consider the binary tree t : (1) t is the most imbalanced iff when we consider it as a semilattice every open set in t is atomic (2) if t is the most balanced then it has no atomic open set and so has no strongly atomic open set. proof. by the above theorem and theorem 2.10. � now why have we introduced this special subcategory of lt ? because by this subcategory we can state some of the important issues in computer science which are useful in practice as categorical properties. one of the most important types of binary trees in computer science are the most balanced binary trees. in the following theorem we characterize these binary trees by categorical structures. first we recall the following definition. definition 3.10. an object, w in the category c is called weak initial if for every object of c there is at least one morphism from w to it. theorem 3.11. every object of lt (flt ) is weak initial. proof. consider a an object of lt . now for every object c of lt , if z ∈ m(c), then define f : m(a) → m(c), such that for every x ∈ m(a), f (x) = z. � theorem 3.12. t ∈ obj(slt ) is weak initial if and only if it is the most balanced binary tree with 2k leaves, k ∈ {0, 1, 2, ...}. proof. let t be the most balanced binary tree with 2k leaves, k ∈ {0, 1, 2, ...}. then for every binary tree w , define a constant map from m(t ) to m(w ) which maps every leaf of t to a fixed leaf of w . then this map is strictly length preserving. conversely let t be a binary tree such that for every binary tree w there is a strictly length preserving map from m(t ) to m(w ). if t is not the most balanced binary tree with 2k leaves, k ∈ {0, 1, 2, ...} then t has at least two leaves with different lengths. but in this case there is not any strictly length preserving map from t to the binary tree with 2 leaves. � 4. composition of comparison based algorithms sometimes a c.b. algorithm (comparison based algorithm) b acts on some outputs of the c.b. algorithm a. here we study this situation, and give a formula for it. we try to show it as an algebraic operation. let the binary tree corresponding to a be < a1 ... am > and the binary tree corresponding to b be < b1 ... bn >. let c be the algorithm obtained when b acts on the output of a at the leaf xk of a. we can obtain the binary tree corresponding to c by attaching a copy of b to the leaf xk. so c = [< c1 ... ck−1 ck,1 ... ck,n ck+1 ... cm >] such that ci = ai for i = 1, ..., k − 1, k + 1, ..., m and ck,i = ak + bi, for i = 1, ..., n and has (m − 1) + n leaves. by [< c1 ... ck−1 ck,1 ... ck,n ck+1 ... cm >] we mean the increasing sequence obtaining from the sequence < c1 ... ck−1 ck,1 ... ck,n ck+1 ... cm >. if b acts on more than one output of the algorithm a the final binary tree is obtained similarly. topological and categorical properties of binary trees 11 now consider c.b. algorithm c obtained when b acts on all outputs of the c.b. algorithm a. so if we consider their correspondence binary trees, a copy of the binary tree corresponding to a is attached to every leaf of the binary tree corresponding to a. thus in this case the decision binary tree corresponding to c will be [< cij >] such that cij = ai + bj, where i = 1, ..., m and j = 1, ..., n. we denote c by a ∗ b. now we try to formalize this definition. we bring the following from [3]. definition 4.1. a po-groupoid (or m-poset), (m, ., ≤) is a poset m with a binary multiplication, ., which satisfies the isotonicity condition a ≤ b implies x.a ≤ x.b and a.x ≤ b.x (*) for all a, b, x ∈ m . when multiplication is (commutative) associative, m is called a (commutative) po-semigroup. a po-semigroup with identity 1, such that x.1 = 1.x = x for all x ∈ m is called a partly ordered monoid, or pomonoid. a po-group is (g, +, ≤) such that (g, +) is a group which satisfies (∗). an l-group is a po-group such that (g, ≤) is a lattice. if tn is the set of binary trees with n leaves, n ∈ n, then as was mentioned when a ∈ tn and b ∈ tm then a ∗ b ∈ tmn. take t = ⋃ n∈n tn. then ∗ : t × t → t defined by ∗(a, b) = a ∗ b is a binary operation on t . we call ∗ product. now we define ≤t on t as follows: a ≤t b if and only if there is n ∈ n such that a, b ∈ tn and a ≤ b. now we have the following theorem: theorem 4.2. (t, ∗, ≤t ) is a commutative po-monoid. proof. one can easily see that if a ∈ tn and b ∈ tm then a ∗ b = b ∗ a. also the binary tree with one leaf (< 0 >), is the identity because for every a ∈ tn where n ∈ n, a∗ < 0 >=< 0 > ∗a = a. also notice that if a ≤t b then for every c ∈ t , a ∗ c ≤t b ∗ c. because if for example the binary tree a is obtained from the binary tree b via a ternary exchange then a ∗ c is obtained from b ∗ c via m ternary exchanges, where c ∈ tm. � remark 4.3. notice that (t, ∗, ≤t ) is not group-because, for example there is no binary tree l such that l∗ < 11 >=< 11 > ∗l =< 0 >. corollary 4.4. if the c.b. algorithm a is faster than the c.b. algorithm b, then for every c.b. algorithm c, a ∗ c is faster than b ∗ c. corollary 4.5. if the algorithm c is the product of the c.b. algorithm b with a and the algorithm d is the product of a with b. then c and d are the same. for proving the next theorem we need the definition of a multi-set. definition 4.6. a multi-set is a finite set-like object in which order is ignored but multiplicity is explicity significant. thus contrary to sets, multi-sets allow for the repetition of elements. therefore multi-sets {1, 2, 3} and {3, 2, 1} are considered to be equivalent, but {1, 2, 2, 3} and {1, 2, 3} differ. also for the 12 h. pajoohesh multi-sets a and b by a − b we mean the set a with the elements in common with the set b removed according to their multiplicity. for example if {1, 2, 2, 2, 3} and {1, 2, 2, 3, 4} then a − b = {2}. theorem 4.7. consider a, b and c in t then: (1) am = bm, where m ∈ n implies a = b. (2) a ∗ c = b ∗ c implies a = b proof. (1) it is enough to show that a2 = b2 implies a = b. since a2 = b2, there exists some n ∈ n such that a ∈ tn and b ∈ tn. let a =< a1 ... an−1 an−1 > and b =< b1 ... bn−1 bn−1 >. assume that a 6= b and let i be the first position that ai and bi differ. assume ai < bi. since aj = bj, for j = 1, 2, ..., i − 1, the multi-sets {aj + ak|j, k = 1, ..., i−1} and {bj +bk|j, k = 1, ..., i−1} are equal. thus the multi-sets a2 −{aj + ak|j, k = 1, ..., i − 1} and b 2 −{bj + bk|j, k = 1, ..., i − 1} are equal. notice that since these sequences are increasing and a2 = b2, the smallest number that appears in a2 is 2a1 and in b 2 is 2b1, so a1 = b1. thus i > 1. now the smallest element that appears in a2 −{aj + ak|j, k = 1, ..., i − 1} is a1 + ai and the smallest element that appears in b2 − {bj + bk|j, k = 1, ..., i − 1} is b1 + bi = a1 + bi > a1 + ai which contradicts a2 −{aj + ak|j, k = 1, ..., i−1} = b 2 −{bj + bk|j, k = 1, ..., i − 1}. thus a2 = b2. similar method yields that am = bm implies a = b. (2) by the same technique we get a = b. � theorem 4.8. for the c.b. algorithms a1, a2, where a1 ∈ tn and a2 ∈ tm: (1) t a1∗a2 (nm) = t a1 (n) + t a2 (m). (2) t wa1∗a2 (nm) = t w a1 (n) + t wa2 (m). (3) t ba1∗a2 (nm) = t b a1 (n) + t ba2 (m). proof. (1) let a1 =< x1 ... xn > and a2 =< y1 ... ym > then a1 ∗ a2 = [< zij >], zij = xi + yj, where i = 1, ..., n and j = 1, ..., m. thus nmt a1∗a2 (nm) = ∑ i,j zij = ∑ i,j (xi + yj) = ∑ i ( ∑ j (xi + yj )) = ∑ i (mxi + ∑ j yj ) = m ∑ i xi + n ∑ j yj = mnt a1 (n) + nmt a2 (m). so t a1∗a2 (nm) = t a1 (n) + t a2 (m) (2) a1 ∗ a2 is obtained when we attach a copy of a2 to every leaf of a1. so the longest path-length in a1 ∗ a2 is the leaf which is the result of the longest length in a2 attaching to the longest length in a1. so the longest length in a1 ∗ a2 is the summation of longest lengths in a1 and a2. so t w a1∗a2 (nm) = t wa1 (n) + t w a2 (m) (3) it is proved similar to the previous part. � topological and categorical properties of binary trees 13 definition 4.9. a nontrivial binary tree is called prime if it can not been written as the product of two nontrivial binary trees (binary trees with at least two leaves) otherwise it is decomposable. remark 4.10. notice that the binary tree with one leaf is not neither prime nor decomposable, we denote it by 1. theorem 4.11. if the binary tree t is decomposable, then it is not atomic. proof. let t be the product of 2 nontrivial binary trees a and b. then the length of every leaf of t is at least 2. now consider ∨ t , then there are x, y such that x, y ≺ ∨ t . since t doesn’t have any leaf with length less than 2, x, y are not leaves. now {t|t ≤ x} and {t|t ≤ y} neither a and b contains the other one. so c(t ) is not a chain and so t is not atomic. � theorem 4.12. every element of tn, when n is prime natural number is a prime binary tree. proof. if c ∈ tn is the product of a and b, where a ∈ tj and b ∈ tk, j, k ∈ {2, 3, ...} then a ∗ b ∈ tjk. thus n = jk for j, k ∈ {2, 3, ...}, which is a contradiction. � remark 4.13. the converse of the previous theorem is not true. for example < 1 2 3 3 >∈ t4 is prime while 4 is not a prime number. i tried to prove a theorem like the fundamental theorem of arithmetic for binary trees in the sense that ”every binary tree is 1 or prime or can be decomposed uniquely apart from rearrangement to the prime binary trees.” i worked on it for a while and i could not prove it. then i decided to leave it in the article as a conjecture. but while mr diarmuid early was helping me with the english he found the following counter example. example 4.14. consider a =< 1 2 2 >, b =< 1 3 3 3 5 5 5 5 >, c =< 1 2 3 3 > and d =< 1 2 4 4 4 4 >. four of these binary trees are prime and a ∗ b = c ∗ d =< 2 3 3 4 4 4 5 5 5 5 5 5 6 6 6 6 7 7 7 7 7 7 7 7 >. so it is seen that you can write a binary tree as the product of prime binary trees in several ways. acknowledgements. i would like to thank dr michel schellekens and mr diarmuid early for their useful comments. 14 h. pajoohesh references [1] j. adamek, h. herrlich and g. strecker, abstract and concrete categories, john wiley and sons, inc., 1990. [2] a. aho, j. hopcroft and j. ullman, data structures and algorithms, addison-wesley series in computer science and information processing, addison-wesley, 1983. [3] g. birkhoff. lattice theory, american mathematical society colloquium publications, volume xxv, 1967. [4] b. c. pierce, basic category theory for computer scientists, foundations of computing series 1991. [5] g. k. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. w. mislove and d. s. scott, a compendium of continuous lattices, springer-verlag, berlin, 1980. [6] d. knuth, the art of computer programming vol. 1, addison-wesley, 1973. [7] d. knuth, the art of computer programming vol. 3, addison-wesley, 1973. [8] r. kopperman and r.wilson, on the role of finite, hereditarily normal spaces and maps in the genesis of compact hausdorff spaces, topology appl. 135 (2004), 265– 275. [9] m. o’keeffe, h. pajoohesh and m. schellekens, on the relation between balance and speed of algorithms, hadronic journal, to appear. [10] m. o’keeffe, h. pajoohesh and m. schellekens, decision trees of algorithms and a semivaluation to measure their distance, electron. notes theor. comput. sci. (proceeding of mfcsit 2004), to appear. [11] d. stott parker and p. ram, the construction of huffman codes is a submodular (”convex”) optimization problem over a lattice of binary trees, sociely for industrial and applied mathematics 28, no. 5, 1875–1905. received june 2005 accepted november 2005 h. pajoohesh (hpajoohesh@mec.cuny.edu) medgar evers college, department of mathematics, 1150 carroll street, brooklyn, ny 11225-2298, usa. @ applied general topology c© universidad politécnica de valencia volume 11, no. 2, 2010 pp. 67-88 convergence semigroup categories h. boustique, p. mikusiński and g. richardson abstract. properties of the category consisting of all objects of the form (x, s, λ) are investigated, where x is a convergence space, s is a commutative semigroup, and λ : x × s → x is a continuous action. a ”generalized quotient” of each object is defined without making the usual assumption that for each fixed g ∈ s, λ(., g) : x → x is an injection. 2000 ams classification: 54a20, 54b15, 54b30 keywords: convergence space, convergence semigroup, continuous action, categorical properties. 1. introduction and preliminaries . the notion of a topological group acting continuously on a topological space has been the subject of numerous research articles. park [13, 14] and rath [16] studied these concepts in the larger category of convergence spaces. this is a more natural category to work in since the homeomorphism group on a space can be equipped with a coarsest convergence structure making the group operations continuous. moreover, unlike in the topological context, quotient maps are productive in the category of all convergence spaces with continuous maps as morphisms. this property played a key role in the proof of several results contained in [3]; for example, see theorem 4.5 [3]. given a topological semigroup acting on a topological space, burzyk et al. [5] introduced a ”generalized quotient space.” elements of this space are equivalence classes determined by an abstraction of the method used to construct the rationals from the integers. generalized quotient spaces are used in the study of generalized functions [10, 11, 12]. moreover, generalized quotients in the category of convergence spaces are defined in [3] for the case whenever λ(., g) : x → x is injective, where λ is a continuous action of a convergence semigroup s on a convergence space x. generalized quotients are defined and studied here without the requirement that λ(., g) is injective. furthermore, 68 h. boustique, p. mikusiński and g. richardson the category consisting of objects of the form (x, s, λ) is investigated. the terminology used here involving categories can be found in adamek et al. [1]. basic definitions and concepts needed in the area of convergence spaces are given in this section. let x be a set, 2x the power set of x, and let f(x) denote the set of all filters on x. recall that b ⊆ 2x is a base for a filter on x provided b 6= ∅, ∅ /∈ b, and b1, b2 ∈ b implies that there exists b3 ∈ b such that b3 ⊆ b1 ∩ b2. moreover, [b] denotes the filter on x whose base is b; that is, [b] = {a ⊆ x : b ⊆ a for some b ∈ b}. fix x ∈ x, define ẋ to be the filter whose base is b = {{x}}. if f : x → y and f ∈ f(x), then f→f denotes the image filter on y whose base is {f(f) : f ∈ f}. a convergence structure on x is a function q : f(x) → 2x obeying: (cs1) x ∈ q(ẋ) for each x ∈ x (cs2) x ∈ q(f) implies that x ∈ q(g) whenever f ⊆ g. the pair (x, q) is called a convergence space. the more intuitive notation f q −→ x is used for x ∈ q(f). a map f : (x, q) → (y, p) between two convergence spaces is called continuous whenever f q −→ x implies that f→f p −→ f(x). let conv denote the category whose objects consist of all the convergence spaces, and whose morphisms are all the continuous maps between objects. the collection of all objects in conv is denoted by |conv|. if p and q are two convergence structures on x, then p ≤ q means that f p −→ x whenever f q −→ x. in this case, p(q) is said to be coarser(finer) than q(p), respectively. also, for f, g ∈ f(x), f ≤ g means that f ⊆ g, and f(g) is called coarser(finer) than g(f), respectively. it is well-known that conv possesses initial and final convergence structures. in particular, if (xj, qj) ∈ |conv| for each j ∈ j, then the product convergence structure r on x = × j∈j xj is given by h r −→ x = (xj) iff π→j h qj −→ xj for each j ∈ j, where πj denotes the j th projection map. also, if f : (x, q) → y is a surjection, then the quotient convergence structure σ on y is defined by: h σ −→ y iff there exists x ∈ f−1(y) and f q −→ x such that f→f = h. in this case, σ is the finest convergence structure on y making f : (x, q) → (y, σ) continuous. convergence quotient maps are defined and studied by kent [7]. moreover, beattie and butzmann [2] and preuss [15] are good references for convergence space results. unlike the category of all topological spaces, conv is cartesian closed and thus has suitable function spaces. in particular, let (x, q), (y, p) ∈ |conv| and let c(x, y ) denote the set of all continuous functions from x to y . define ω : (x, q)×c(x, y ) → (y, p) to be the evaluation map given by ω(x, f) = f(x). there exists a coarsest convergence structure c on c(x, y ) such that w is jointly continuous. more precisely, c is defined by : φ c −→ f iff w→(f × φ) p −→ f(x) whenever f q −→ x. this compatibility between (x, q) and (c(x, y ), c) is an example of a continuous action in conv. let sg denote the category whose objects consist of all the semigroups (with an identity element), and whose morphisms are all the homomorphisms convergence semigroup categories 69 between objects. further, (s, ., p) is said to be a convergence semigroup provided : (s, .) ∈ |sg|, (s, p) ∈ |conv|, and γ : (s, p) × (s, p) → (s, p) is continuous, where γ(x, y) = x.y. let csg be the category whose objects consist of all the convergence semigroups, and whose morphisms are all the continuous homomorphisms between objects. an action of a semigroup on a topological space is used to define ”generalized quotients” in [5]. below is rath’s [16] definition of an action in the convergence space context. let (x, q) ∈ |conv|, (s, ., p) ∈ |csg|, λ : x × s → x, and consider the following conditions: (a1) λ(x, e) = x for each x ∈ x (e is the identity element) (a2) λ(λ(x, g), h) = λ(x, g.h) for each x ∈ x, g, h ∈ s (a3) λ : (x, q) × (s, ., p) → (x, q) is continuous. then (s, .)((s, ., p)) is said to act(act continuously) on (x, q) whenever a1a2 (a1-a3) are satisfied and, in this case, λ is called the action (continuous action), respectively. for sake of brevity, (x, s) ∈ a(ac) denotes the fact that (s, .) is commutative and (s, ., p) ∈ |csg| acts (acts continuously) on (x, q) ∈ |conv|, respectively. moreover, (x, s, λ) ∈ a indicates that the action is λ. remark 1.1. fix a set x; then the set of all convergence structures on x with the ordering p ≤ q defined above is a complete lattice. indeed, if (x, qj) ∈ |conv|, j ∈ j, then sup j∈j qj = q 1 is given by f q1 −→ x iff f qj −→ x, for each j ∈ j. dually, inf j∈j qj = q 0 is defined by f q0 −→ x iff f qj −→ x, for some j ∈ j. it is easily verified that if ((x, qj), (s, ., p), λ) ∈ ac for each j ∈ j, then both ((x, q1), (s, ., p), λ) and ((x, q0), (s, ., p), λ) belong to ac. the notion of ”generalized quotients” determined by a commutative semigroup acting on a topological space is investigated in [5]. elements of the semigroup in [5] are assumed to be injections on the given topological space. lemma 1.2 ([5]). suppose that (s, x, λ) ∈ a, and λ(., g) : x → x is an injection, for each g ∈ s. define (x, g) ≈ (y, h) on x × s iff λ(x, h) = λ(y, g). then ≈ is an equivalence relation on x × s. in the context of lemma 1.2, let 〈(x, g)〉 be the equivalence class containing (x, g), b(x, s) denote the quotient set (x×s)/ ≈, and define ϕ : (x×s, r) → b(x, s) to be the canonical map, where r = q × p is the product convergence structure. equip b(x, s) with the convergence quotient structure σ. then k σ −→ 〈(y, h)〉 iff there exist (x, g) ≈ (y, h) and h r −→ (x, g) such that ϕ→h = k. properties of (b(x, s), σ) are investigated in [3]. whenever the hypothesis that λ(., g) is an injection for each g ∈ s fails, one can still define a generalized quotient by extending ≈ to an equivalent relation as defined in (r2) below. 70 h. boustique, p. mikusiński and g. richardson assume that ((x, q), (s, ., p), λ) ∈ a, and consider the following relations: (r1) (x, f) ≈ (y, g) in x × s iff λ(x, g) = λ(y, f) (r2) (x, f) ∼ (y, g) in x × s iff there exists (zi, hi) ∈ x × s, satisfying (x, f) ≈ (z1, h1) ≈ (z2, h2) ≈ · · · ≈ (zn, hn) ≈ (y, g), for some n ≥ 1 (r3) x ≃ y in x iff there exists g ∈ s such that λ(x, g) = λ(y, g). according to lemma 1.1 [5], (r1) is an equivalence relation provided that λ(., g) is an injection for each g ∈ s. however, (r2) is an equivalence relation without assuming that λ(., g) is an injection for each g ∈ s. moreover, it easily follows that (r3) is an equivalence relation. given ((x, q), (s, ., p), λ) ∈ a; denote x∗ = x/ ≃, ξ : x → x∗ the canonical map ξ(x) = [x], and let q∗ be the quotient structure on x∗ in conv determined by ξ : (x, q) → x∗. define ϕ : x × s → b(x, s) = (x × s)/ ∼ to be the canonical map ϕ(x, g) = 〈(x, g)〉, and let σ denote the quotient structure on b(x, s) in conv determined by ϕ : (x × s, r) → b(x, s), where r = q × p. likewise, define ϕ∗ : x∗ × s → b(x∗, s) = (x∗ × s)/ ≈ by ϕ∗([x], g) = 〈([x], g)〉. let r∗ denote the product structure on x∗ × s, and let σ∗ be the quotient structure on b(x∗, s) determined by ϕ∗ : x∗ ×s → b(x∗, s). define λ∗ : x∗ ×s → x∗ by λ∗([x], g) = [λ(x, g)], and denote η : b(x, s) → b(x∗, s) by η(〈(x, g)〉) = 〈([x], g)〉. it is shown below that these definitions are welldefined, and their properties are investigated. lemma 1.3. assume that ((x, q), (s, ., p), λ) ∈ a. then (a) (x, f) ∼ (y, g) iff there exists h ∈ s such that (x, hf) ≈ (y, hg) (b) x ≃ y iff for each g ∈ s, λ(x, g) ≃ λ(y, g) (c) λ∗(., g) : x∗ → x∗ is an injection, for each fixed g ∈ s (d) η : b(x, s, λ) → b(x∗, s, λ∗) is a bijection. proof. (a): suppose that (x, f) ∼ (y, g). then there exists (zi, hi) ∈ x × s, 1 ≤ i ≤ n, such that (x, f) ≈ (z1, h1), (z1, h1) ≈ (z2, h2),...,(zn, hn) ≈ (y, g). verification is illustrated whenever n = 2; that is, (x, f) ≈ (z1, h1), (z1, h1) ≈ (z2, h2), and (z2, h2) ≈ (y, g). then λ(x, h1) = λ(z1, f), λ(z1, h2) = λ(z2, h1), and λ(z2, g) = λ(y, h2). it is shown that (x, h2h1f) ≈ (y, h2h1g). indeed, since (s, .) is commutative, λ(x, h2h1g) = λ(λ(x, h1), h2g) = λ(λ(z1, f), h2g) = λ(λ(z1, h2), fg) = λ(λ(z2, h1), fg) = λ(λ(z2, g), h1f) = λ(λ(y, h2), h1f) = λ(y, h2h1f). hence (x, h2h1f) ≈ (y, h2h1g) and, in general h = h1h2 · · · hn. conversely, assume that (x, hf) ≈ (y, hg) for some h ∈ s. it is shown that (x, f) ≈ (λ(y, h), hg) and (λ(y, h), hg) ≈ (y, g). indeed, by hypothesis, λ(x, hg) = λ(y, hf) = λ(λ(y, h), f) and thus (x, f) ≈ (λ(y, h), hg). also, λ(λ(y, h), g) = λ(y, hg) and hence (λ(y, h), hg) ≈ (y, g). it follows that (x, f) ∼ (y, g). (b): suppose that x ≃ y; then there exists h ∈ s such that λ(x, h) = λ(y, h). assume that g ∈ s. then (λ(x, g), h) ≈ (λ(y, g), h)). indeed, since (s, .) is commutative, λ(λ(x, g), h) = λ(λ(x, h), g) = λ(λ(y, h), g) = λ(λ(y, g), h), and thus λ(x, g) ≃ λ(y, g). conversely, x = λ(x, e) ≃ λ(y, e) = y. convergence semigroup categories 71 (c): recall that x∗ = x/ ≃= {[x] : x ∈ x} and λ∗([x], g) = [λ(x, g)]. first, note that λ∗ is well-defined. indeed, if x ≃ y and g ∈ s, then by (b), λ(x, g) ≃ λ(y, g) implies that λ∗([x], g) = λ∗([y], g), and thus λ∗ is well-defined. next, it is shown that for g ∈ s fixed, λ∗(., g) is an injection. suppose that λ∗([x], g) = λ∗([y], g); then λ(x, g) ≃ λ(y, g) and it follows by (r3) that there exists h ∈ s such that λ(λ(x, g), h) = λ(λ(y, g), h). equivalently, λ(x, gh) = λ(y, gh), and again by (r3), x ≃ y. hence λ∗(., g) is an injection. (d): recall that η(〈(x, g)〉) = 〈([x], g)〉, x ∈ x, g ∈ s. first, observe that η is well-defined. indeed, assume that (x, g) ∼ (y, h); then by (a), there exists k ∈ s such that (x, kg) ≈ (y, kh). hence λ(x, kh) = λ(y, kg), and thus λ∗([x], kh) = [λ(x, kh)] = [λ(y, kg)] = λ∗([y], kg). therefore ([x], kg) ≈ ([y], kh) on x∗ × s, and thus by (a) and (c), ([x], g) ≈ ([y], h). it follows that 〈([x], g)〉 = 〈([y], h)〉, and hence η is well-defined. next, it is shown that η is an injection. suppose that 〈([x], g)〉 = η(〈(x, g)〉) = η(〈(y, h)〉) = 〈([y], h)〉; then ([x], g) ≈ ([y], h) in x∗ × s. hence [λ(x, h)] = λ∗([x], h) = λ∗([y], g) = [λ(y, g)], and thus λ(x, h) ≃ λ(y, g). according to (r3), there exists k ∈ s such that λ(x, hk) = λ(λ(x, h), k) = λ(λ(y, g), k) = λ(y, gk). hence (x, gk) ≈ (y, hk), and by part (a), (x, g) ∼ (y, h). therefore 〈(x, g)〉 = 〈(y, h)〉, and thus η is an injection. clearly η is a surjection, and consequently η is a bijection. � 2. action categories . consider the triple (x, (s, .), λ), where x ∈ |set|, (s, .) ∈ |sg|, (s, .) is commutative, and λ : x × s → x is an action. define x to be the category whose objects consist of all triples (x, (s, .), λ), and whose morphisms are all pairs (f, k) : (x, (s, .), λ) → (y, (t, .), µ) obeying: (b1) f : x → y is a map, k : (s, .) → (t, .) is a homomorphism (b2) f ◦ λ = µ ◦ (f × k). furthermore, let c denote the category consisting of all objects ((x, q), (s, ., p), λ) ∈ ac, and whose morphisms (f, k) : ((x, q), (s, .), λ) → ((y, qy ), (t, ., pt ), µ) satisfy: (c1) f : (x, q) → (y, qy ) is continuous, k : (s, ., p) → (t, ., pt ) is a continuous homomorphism (c2) f ◦ λ = µ ◦ (f × k). clearly idx ×ids : ((x, q), (s, ., p), λ) → ((x, q), (s, ., p), λ) is the identity morphism in c. also, observe that the composition of two c-morphisms is again a cmorphism. indeed, suppose that (f, k) : ((x, q), (s, ., p), λ) → ((y, qy ), (t, ., pt ), µ) and (h, l) : ((y, qy ), (t, ., pt ), µ) → ((z, qz), (r, ., pr), δ) are two c-morphisms. clearly (c1) is satisfied. it remains to verify (c2). since (f, k) and (h, l) each obeys (c2), f ◦ λ = µ ◦ (f × k) and h ◦ µ = δ ◦ (h × l). if (x, g) ∈ x × s, then δ ◦ (h × l) ◦ (f × k)(x, g) = δ ◦ (h × l)(f(x), k(g)) = (h ◦ µ)(f(x), k(g)) = h(µ ◦ (f × k))(x, g) = h(f ◦ λ)(x, g) = ((h ◦ f) ◦ λ)(x, g), and it follows that (h ◦ f) ◦ λ = δ ◦ (h × l) ◦ (f × k). hence (c2) is valid and c is a category; likewise, x , is also a category. 72 h. boustique, p. mikusiński and g. richardson if u : c → x denotes the faithful functor defined by u((x, q), (s, ., p), λ) = (x, (s, .), λ), then (c, u) is a concrete category over the base category x . let d denote the full subcategory of c consisting of all objects ((x, q), (s, ., p), λ) ∈ |c| such that λ(., g) : x → x is an injection, for each fixed g ∈ s. then (d, u ◦ e) is also a concrete category over x , where e : d → c denotes the inclusion functor. theorem 2.1. the category d is reflective in c. proof. given ((x, q), (s, .p), λ) ∈ |c|, consider ((x∗, q∗), (s, ., p), λ∗), where (x∗, q∗) ∈ |conv| and λ∗ are defined in section 1. then λ∗ : x∗ × s → x∗ is an action, and by lemma 1.2(c), λ∗(., g) : x∗ → x∗ is an injection. recall that q∗ is the quotient structure in conv determined by the canonical map ξ : (x, q) → (x∗, q∗), ξ(x) = [x]. since quotient maps are productive in conv, ξ × ids is a quotient map. moreover, observe that λ∗ ◦(ξ ×ids) = ξ ◦λ, ξ ◦λ is continuous, and hence λ∗ : (x∗, q∗)×(s, ., p) → (x∗, q∗) is a continuous map. if follows that ((x∗, q∗), (s, .p), λ∗) ∈ |d|. moreover, (ξ, ids) : ((x, q), (s, ., p), λ) → ((x∗, q∗), (s, ., p), λ∗) is a c morphism. clearly (c1) is satisfied, and as mentioned above, ξ ◦ λ = λ∗ ◦ (ξ × ids); hence (ξ, ids) is a c-morphism. assume that (f, k) : ((x, q), (s, ., p), λ) → ((y, qy ), (t, ., pt ), µ) is a cmorphism, where ((y, qy ), (t, ., pt ), µ) ∈ |d|. then f ◦ λ = µ ◦ (f × k). define f∗ : x∗ → y by f∗([x]) = f(x). observe that f∗ is well-defined. indeed, if x1 ∈ [x], then by (r3), λ(x, g) = λ(x1, g) for some g ∈ s. hence µ(f(x), k(g)) = (µ ◦ (f × k))(x, g) = (f ◦ λ)(x, g) = (f ◦ λ)(x1, g) = (µ ◦ (f × k))(x1, g) = µ(f(x1), k(g)), and thus by (r3), f(x) ≃ f(x1). since y∗ = y , f∗ : x∗ → y is well-defined. moreover, f∗ ◦ ξ = f is continuous, ξ : (x, q) → (x∗, q∗) is a quotient map in conv, and thus it follows that f∗ is continuous. finally, (µ ◦ (f∗ × k))([x], g) = µ(f(x), k(g)) = (µ ◦ (f × k))(x, g) = (f ◦ λ)(x, g) = (f∗ ◦ λ∗)([x], g), for each ([x], g) ∈ x∗ × s, and thus f∗ ◦ λ∗ = µ ◦ (f∗ × k). hence (f∗, k) : ((x∗, q∗), (s, ., p), λ∗) → ((y, q y ), (t, ., pt ), µ) is a d-morphism. therefore d is a reflective subcategory of c. � theorem 2.2. the concrete category (c, u) over x is topological. proof. assume that (fj, kj) : ((x, (s, .), λ) → u((xj, qj), (sj, ., pj), λj), j ∈ j, is a source in x . define f q −→ x (g p −→ g) iff for each j ∈ j, f→j f qj −→ fj(x)(k → j g pj −→ kj(g)), respectively. then q(p) is the initial structure in conv (csg) determined from fj : x → (xj, qj) (kj : (s, .) → (sj, ., pj)), j ∈ j, respectively. next, it is shown that λ : (x, q) × (s, p) → (x, q) is continuous. assume that f q −→ x and g p −→ g; then f→j f qj −→ fj(x) and k → j g pj −→ kj(g), for each j ∈ j. employing the hypothesis, fj ◦ λ = λj ◦ (fj × kj), it follows that f→j (λ →(f × g)) = λ→j (fj × kj) →(f × g) = λ→j (f → j f × k → j g) qj −→ λj(fj(x), kj(g)) = (λj ◦ (fj × kj))(x, g) = fj(λ(x, g)), for each j ∈ j. hence λ→(f×g) q −→ λ(x, g), and thus λ is a continuous action. then ((x, q), (s, ., p), λ) convergence semigroup categories 73 ∈ |c|, and thus (fj, kj) : ((x, q), (s, ., p), λ) → ((xj, qj), (sj, ., pj), λj) is a cmorphism. finally, suppose that (f, k) : u((z, qz), (t, ., pt ), µ) → u((x, q), (s, ., p), λ) is a x-morphism such that (fj × kj) ◦ (f × k) : ((z, q z), (t, ., pt ), µ) → ((xj, qj), (sj, ., pj), λj) is a c-morphism. since q(p) is the initial structure in conv (csg) determined by fj : x → (xj, qj) (kj : (s, .) → (sj, ., pj)), j ∈ j, it follows that f : (z, qz) → (x, q) and k : (t, ., pt ) → (s, ., p) are continuous, respectively. moreover, p and q are the unique structures possessing these properties. since (f, k) is an x-morphism, f ◦ µ = λ ◦ (f × k), and thus (f, k) is a c-morphism. therefore (c, u) is topological. � remark 2.3. theorem 2.2 shows that products exist in c. in particular, suppose that ((xj, qj), (sj, ., pj), λj) ∈ |c|, for each j belonging to set j. denote the product set by x = × j∈j xj, the product semigroup by (s, .) = × j∈j (sj, .), the jth projection map by πj1 × πj2 : x × s → xj × sj, and define λ : x × s → x by λ((xj)j∈j , (gj)j∈j ) = (λj(xj, gj))j∈j . it follows that λ is an action, and (x, (s, .), λ) ∈ |x|. observe that for each j ∈ j, πj1 ◦ λ = λj ◦ (πj1 × πj2), and thus (πj1, πj2) : (x, (s, .), λ) → u((xj, qj), (sj, ., pj), λj) is an x-morphism. then by theorem 2.2, ((x, q), (s, ., p), λ) is the unique u-initial lift, where q(p) are product structures in conv (csg), respectively. it was shown in theorem 2.2 that every u-structured source has a unique u-initial lift. this also implies that every u-structured sink has a unique u-final lift; for example, see theorem 21.9 [1]. quotient morphisms in c are considered in the next theorem. recall that if ((x, q), (s, ., p), λ) ∈ |c|, then b(x, s) = (x×s)/ ∼ denotes the generalized quotient, where ∼ is the equivalence relation defined in (r2). let r = q×p; then ϕ : (x×s, r) → (b(x, s), σ) is the quotient map ϕ(x, g) = 〈(x, g)〉, and σ is the corresponding quotient structure in conv. theorem 2.4. assume that ((x, q), (s, ., p), λ) ∈ |c|. then (a) a surjective x-morphism (f, k) : u((x, q), (s, .p), λ) → (y, (t, .), µ) has a unique u-final lift to a quotient map (f, k) : ((x, q), (s, ., p), λ) → ((y, qy ), (t, ., pt ), µ) in c. (b) (ϕ, ids) : ((x ×s, r), (s, ., p), λ) → ((b(x, s), σ), (s, ., p), λb) is a quotient map in c, where λ((x, g), h) := (λ(x, h), g) and λb(〈(x, g)〉, h) = 〈(λ(x, h), g)〉. (c) (η, ids) : ((b(x, s, λ), σ), (s, ., p), λb) → ((b(x∗, s, λ∗), (s, ., p), λ ∗ b) is a c-isomorphism, where λ∗b(〈([x], g)〉, h) = 〈(λ∗([x], h), g)〉 = 〈([λ(x, h)], g)〉. proof. (a): fix (y, t) ∈ y × t , and define h q y −−→ y (k p t −−→ t) iff there exists f q −→ x ∈ f−1(y) (g p −→ g ∈ k−1(t)) such that f→f = h (k→g = k), respectively. then qy (pt ) is the quotient structure in conv (csg). it is shown that the action µ : (y, qy )×(t, ., pt ) → (y, qy ) is continuous. indeed, suppose that h q y −−→ y and k p t −−→ t; then there exist f q −→ x ∈ f−1(y) and g p −→ g ∈ k−1(t) 74 h. boustique, p. mikusiński and g. richardson such that f→f = h and k→g = k. since (f, k) is an x-morphism, µ◦(f ×k) = f ◦ λ, and thus µ→(h × k) = µ→(f→f × k→g) = (µ ◦ (f × k))→(f × g) = (f ◦ λ)→(f × g) q y −−→ (f ◦ λ)(x, g) = (µ ◦ (f × k))(x, g) = µ(y, t). therefore µ is continuous, and (f, k) is a c-morphism. next, suppose that (f, k) : u((y, qy ), (t, ., pt ), µ) → u((z, qz), (r, ., pr), δ) is an x-morphism such that (f, k) ◦ (f, k) : ((x, q), (s, ., p), λ) → (z, qz), (r, ., pr), δ) is a c-morphism. since qy and pt are quotient structures in conv and csg, f and k are continuous maps, and thus (f, k) is a c-morphism. moreover qy and pt are unique and hence (f, k) is a quotient map in c. (b): observe that λ is an action. indeed, λ((x, g), e) = (λ(x, e), g) = (x, g). moreover, λ(λ((x, g), h), k) = λ((λ(x, h), g), k) = (λ(λ(x, h), k), g) = (λ(x, hk), g) = λ((x, g), hk), and thus λ is an action. note that λ is the composition of the continuous maps :((x, g), h) 7→ ((x, h), g) 7→ (λ(x, h), g). therefore λ is a continuous action, and thus ((x × s, r), (s, ., p), λ) ∈ |c|. first, it is shown that λb is well-defined. it must be shown that if (x, g) ∼ (x1, g1), then λb(〈(x, g)〉, h) = λb(〈(x1, g1)〉, h). according to lemma 1.3(a), there exists k ∈ s such that (x, kg) ≈ (x1, kg1), or λ(x, kg1) = λ(x1, kg). note that λ(λ(x, h), kg1) = λ(λ(x, kg1), h) = λ(λ(x1, kg), h) = λ(λ(x1, h), kg), and thus (λ(x, h), kg) ≈ (λ(x1, h), kg1). again, by lemma 1.3(a), (λ(x, h), g) ∼ (λ(x1, h), g1), and thus λb is well-defined. next, λb is an action. indeed, λb(〈(x, g)〉, e) = 〈(λ(x, e), g)〉 = 〈(x, g)〉, and λb(λb(〈(x, g)〉, h), k) = λb(〈(λ(x, h), g)〉, k) = 〈(λ(λ(x, h), k), g)〉 = 〈(λ(x, hk), g)〉 = λb(〈(x, g)〉, hk). hence λb is an action. since ϕ (ids) are quotient maps in conv (csg), respectively, it remains to show that ϕ ◦ λ = λb ◦ (ϕ × ids). let ((x, g), h) ∈ (x × s) × s; then (λb ◦ (ϕ × ids))((x, g), h) = λb(〈(x, g)〉, h) = 〈(λ(x, h), g)〉 = 〈λ((x, g), h)〉 = (ϕ ◦ λ)((x, g), h). therefore ϕ ◦ λ = λb ◦ (ϕ × ids), and thus by part (a), λb is continuous, and (ϕ, ids) : ((x × s, r), (s, ., p), λ) → ((b(x, s), σ), (s, ., p), λb) is a quotient map in c. (c): recall from lemma 1.3(d) that η is a bijection, where η : (b(x, s, λ), σ) → ((b(x∗, s, λ∗), σ∗) is defined by η(〈(x, g)〉 = 〈([x], g)〉. it is shown that η is a homeomorphism in conv. observe that the diagram below commutes: (x × s, r) ϕ (b(x, s), σ) (x∗ × s, r∗) ξ × ids ? ϕ∗ (b(x∗, s), σ∗) η ? since ϕ is a quotient map in conv, η is continuous iff η ◦ ϕ is continuous. however, η ◦ ϕ = ϕ∗ ◦ (ξ × ids) is continuous by construction, and thus η is continuous. also, ϕ∗ is a quotient map in conv, and hence η −1 is continuous iff η−1 ◦ ϕ∗ is continuous. since ξ × ids is a quotient map, η −1 ◦ ϕ∗ is continuous iff (η−1 ◦ ϕ∗) ◦ (ξ × ids) is continuous. however, the latter map is simply convergence semigroup categories 75 ϕ, and hence η−1 ◦ ϕ∗ is continuous. therefore η −1 is continuous, and thus η : (b(x, s), σ) → (b(x∗, s), σ∗) is a homeomorphism in conv. since η is a homeomorphism, ((b(x, s), σ), (s, ., p), λb) ∈ |c|, ((b(x∗, s), σ∗), (s, ., p), λ∗b) ∈ |c|, and it remains to show that η ◦ λb = λ ∗ b ◦ (η × ids) and η−1 ◦ λ∗b = λb ◦ (η −1 × ids). let (〈(x, g)〉, h) ∈ b(x, s) × s; then λ ∗ b ◦ (η × ids)(〈(x, g)〉, h) = λ ∗ b(〈([x], g)〉, h) = 〈(λ∗([x], h), g)〉 = 〈([λ(x, h)], g)〉 = η(〈(λ(x, h), g)〉) = (η◦λb)(〈(x, g)〉, h). hence η◦λb = λ ∗ b ◦(η×ids). moreover, since η is a bijection, λb = η −1 ◦λ∗b ◦(η ×ids) and λb ◦(η −1 ×ids) = η −1 ◦λ∗b. therefore (η, ids) is a c-isomorphism. � remark 2.5. (i): assume that ((x, q), (s, ., p), λ) ∈ |c|. an object in conv is called hausdorff whenever each filter converges to at most one element. it is shown in theorem 4.1 [3] that (b(x, s, λ), σ) is hausdorff iff (x, q) is hausdorff, provided that λ(., g) is an injection for each g ∈ s. since λ∗(., g) is an injection for each g ∈ s, and (b(x, s, λ), σ) and (b(x∗, s, λ∗), σ∗) are homeomorphic, it follows that (b(x, s, λ), σ) is hausdorff iff (x∗, q∗) is hausdorff. (ii): suppose that (f, k) : ((x, q), (s, ., p), λ) → ((y, qy ), (t, ., pt ), µ) is an isomorphism in c. in particular, f−1 (k−1) is an isomorphism in conv (csg), respectively. moreover, since f ◦ λ = µ ◦ (f × k), then µ = f ◦ λ ◦ (f−1 × k−1) and λ = f−1 ◦ µ◦ (f × k). hence each action can be found from the other. it is not enough just to assume that f and k are isomorphisms in conv and csg, respectively. 3. extensions consideration is given to embedding objects in c into objects which have nicer properties such as compactness. recall that a convergence space (x, q) is said to be hausdorff whenever each filter converges to at most one element. a (hausdorff) convergence space (x, q) is regular (t3) provided clqf q −→ x whenever f q −→ x, respectively. moreover, (x, q) is compact if each ultrafilter on x q-converges. given (x, q) ∈ |conv|; (x, πq) denotes the pretopological modification of (x, q), where f πq −→ x iff f ≥ ∩{g : g q −→ x} (neighborhood filter at x). it is shown that in theorem 1 [18] that (x, q) ∈ |conv| has a t3 compactification iff πq is a hausdorff completely regular topology, and q and πq agree on ultrafilter convergence. in this case, there exists a t3 compactification (x∗, q∗, δ) such that δ : (x, q) → (x∗, q∗) is a dense embedding, and if f : (x, q) → (y, ρ) is a continuous function, where (y, ρ) is compact t3, then there exists a continuous map f∗ : (x∗, q∗) → (y, ρ) such that f∗ ◦ δ = f lemma 3.1. suppose that (f1, k1) and (f, k) are c-morphisms, f(k) is a morphism in conv (csg), respectively, f1(x) and k(s) are dense, and the diagram below commutes. then (f, k) is also a c-morphism. 76 h. boustique, p. mikusiński and g. richardson ((x, q), (s, ., p), λ) (f1, k1) ((x1, q1), (s1, ., p1), λ1) ((y, qy ), (t, ., pt ), µ) (f, k) ? � (f, k) proof. it remains to show that f ◦ λ1 = µ ◦ (f × k). fix (z, g) ∈ x1 × s1. since f1(x)(k1(s)) is dense, there exists f(g) ∈ f(x)(f(s)) such that f→1 f q1 −→ z(k→1 g p1 −→ g), respectively. employing the assumptions, it follows that (µ ◦ (f × k))(z, g) = µ(f(z), k(g)) = µ(f(lim f→1 f), k(lim k → 1 g) = µ(lim(f ◦ f1) →f, lim(k ◦ k1) →g) = lim µ→(f→f × k→g) = lim(µ ◦ (f × k))→(f × g) = lim(f ◦ λ)→(f × g) = lim((f ◦ f1) ◦ λ) →(f × g) = lim f →(f1 ◦ λ)→(f ×g) = lim f →(λ1 ◦(f1 ×k1)) →(f ×g) = lim(f ◦λ1) →(f→1 f ×k → 1 g) = (f ◦ λ1)(lim(f → 1 f × k → 1 g)) = (f ◦ λ1)(z, g). hence f ◦ λ1 = µ ◦ (f × k), and thus (f, k) is a c-morphism. � theorem 3.2. assume that (x, q) has a t3-compactification, and (f, k) : ((x, q), (s, ., p), λ) → ((y, qy ), (t, ., pt ), µ) is a c-morphism, where (y, qy ) is compact t3, and p is the discrete structure. then, using the notations above, (δ, ids) is a dense c-embedding and, moreover, (f ∗, k) is a c-morphism such that the diagram below commutes, for some λ∗: ((x, q), (s, ., p), λ) (δ, ids) ((x∗, q∗), (s, ., p), λ∗) ((y, qy ), (t, ., pt ), µ) (f, k) ? � (f ∗ , k) proof. fix g ∈ s. then the subspace x∗ × {g} of x∗ × s is a t3 compactification of x × {g} which possesses the continuous extension property. let λg : x × {g} → x be the function λg(x, g) = λ(x, g), x ∈ x. since δ ◦ λg is continuous, there exists a continuous extension λ∗g such that the diagram below commutes: x × {g} δ ◦ λg x∗ x∗ × {g} δ × idg ? λ ∗ g convergence semigroup categories 77 define λ∗ : x∗ × s → x∗ by λ∗(z, g) = λ∗g(z, g), for each z ∈ x ∗, g ∈ s. since p is the discrete structure on s, it follows that λ∗ is continuous and, moreover, since the diagram above commutes, (λ∗ ◦(δ ×ids))(x, g) = (λ ∗ g ◦(δ × idg))(x, g) = (δ ◦ λg)(x, g) = (δ ◦ λ)(x, g), for each (x, g) ∈ x × s. hence λ ∗ ◦ (δ ×ids) = δ ◦λ, and thus (δ, ids) : ((x, q), (s, ., p), λ) → ((x ∗, q∗), (s, ., p), λ∗) is a c-morphism. next, it is shown that λ∗ : x∗ × s → x∗ is a action. as shown above, λ∗◦(δ×ids) = δ◦λ, and thus if x ∈ x, then λ ∗(δ(x), e)) = (λ∗◦(δ×ids))(x, e) = (δ ◦ λ)(x, e) = δ(λ(x, e)) = δ(x). moreover, if z ∈ x∗ − δ(x), then since δ(x) is dense in x, there exists a filter f on x such that δ→f q ∗ −→ z . however, λ∗ is continuous whenever p is the discrete structure, and thus λ∗→(δ→f × ė) q ∗ −→ λ∗(z, e). moreover, λ∗→(δ→f × ė) = (λ∗→ ◦(δ ×ids)) →(f × ė) = (δ ◦λ)→(f × ė) = δ→(λ→(f × ė)) = δ→f q ∗ −→ z. since (x∗, q∗) is hausdorff, λ∗(z, e) = z. let z ∈ x and g, h ∈ s; it is shown that λ∗(λ∗(z, g), h) = λ∗(z, gh). first, suppose that z = δ(x). then λ∗(λ∗(δ(x), g), h) = λ∗(λ∗ ◦ (δ × ids)(x, g), h) = λ∗((δ ◦ λ)(x, g), h) = λ∗(δ(λ(x, g)), h) = (λ∗ ◦ (δ × ids))(λ(x, g), h) = (δ ◦ λ)(λ(x, g), h) = δ(λ(λ(x, g), h)) = δ(λ(x, gh)) = (δ ◦ λ)(x, gh) = (λ∗ ◦ (δ × ids))(x, gh) = λ ∗(δ(x), gh). hence λ∗(λ∗(δ(x), g), h) = λ∗(δ(x), gh). further, assume that z ∈ x∗ − δ(x); it is shown that λ∗(λ∗(z, g), h) = λ∗(z, gh). there exists a filter f on x such that δ→f q ∗ −→ z. since λ∗ is continuous whenever p is the discrete structure, λ∗→(δ→f × ġ) q ∗ −→ λ∗(z, g). employing λ∗◦(δ×ids) = δ◦λ, λ ∗→(δ→f ×ġ) = [λ∗◦(δ×ids)] →(f ×ġ) = (δ◦λ)→(f ×ġ), and thus λ∗→[(δ◦λ)→(f×ġ)×ḣ] q ∗ −→ λ∗(λ∗(z, g), h). however, λ∗→[(δ◦λ)→(f× ġ) × ḣ] = [λ∗ ◦ (δ × ids)] →(λ→(f × ġ) × ḣ) = (δ ◦ λ)→(λ→(f × ġ) × ḣ) = δ→[λ→(λ→(f × ġ) × ḣ)] = δ→(λ→(f × ˙gh)) = [λ∗ ◦ (δ × ids)] →(f × ˙gh) = λ∗→(δ→f × ġh) q ∗ −→ λ∗(z, gh). since (x∗, q∗) is hausdorff, it follows that λ∗(λ∗(z, g), h) = λ∗(z, gh), and thus λ∗ is a continuous action. it remains to show that (δ−1 δ(x) , ids) : ((δ(x), q ∗|δ(x)), (s, ., p), λ ∗|δ(x)×s) → ((x, q), (s, ., p), λ) is a c-morphism. since δ is a embedding, only (δ−1 ◦ λ∗)|δ(x)×s = λ ◦ (δ −1 δ(x) × ids) must be verified. using the fact that λ ∗ ◦ (δ × ids) = δ ◦ λ and δ is an embedding, λ ∗|δ(x)×s = δ ◦ λ ◦ (δ −1 δ(x) × ids) and δ−1 δ(x) ◦ λ∗|δ(x)×s = λ ◦ (δ −1 δ(x) × ids). hence (δ −1 δ(x) , ids) is also a c-morphism, and thus (δ, ids) : ((x, q), (s, ., p), λ) → ((x ∗, q∗), (s, ., p), λ∗) is an embedding in c. finally, since (δ, ids), (f, k) are c-morphisms and δ(x) is dense in x ∗, it follows from lemma 3.1 that (f∗, k) is a c-morphism. � suppose that (x, q) has the t3 compactification (x∗, q∗, δ) mentioned above, and assume that ((x, q), (s, ., p), λ) ∈ |c|. define g p ∗ −→ g iff for each h q ∗ −→ z, λ∗→(h × g) q ∗ −→ λ∗(z, g). 78 h. boustique, p. mikusiński and g. richardson corollary 3.3. assume that ((x, q), (s, ., p), λ) ∈ |c|, and (x∗, q∗, δ) is the t3 compactification of (x, q) described above. then p∗ is the coarsest structure on s such that ((x∗, q∗), (s, ., p∗), λ∗) ∈ |c|. proof. first, it is shown that (s, p∗) ∈ |conv|. fix g ∈ s, suppose that h q ∗ −→ z; then according to theorem 3.1, λ∗ : (x∗, q∗) × (s, p) → (x∗, q∗) is continuous whenever p has the discrete structure. hence λ∗→(h × ġ) q ∗ −→ λ∗(z, g), and thus ġ p ∗ −→ g. moreover, it easily follows that if k p ∗ −→ g and l ≥ k, then l p ∗ −→ g. hence (s, p∗) ∈ |conv|. next, assume that gi p ∗ −→ gi, i = 1, 2, and h q ∗ −→ z; then λ∗→(h × g1g2) = λ ∗→(λ∗→(h × g1) × g2) q ∗ −→ λ∗(λ∗(z, g1), g2) = λ ∗(z, g1g2). hence g1g2 p ∗ −→ g1g2, and thus (s, ., p ∗) ∈ |csg|. it follows from the definition that p∗ is the coarsest structure such that ((x∗, q∗), (s, ., p∗), λ∗) ∈ |c|. � let (x, q) ∈ |conv|. it is shown in [17] that if (x, q) is hausdorff, then there exists a hausdorff compactification (x̂, q̂), where j : (x, q) → (x̂, q̂) denotes a dense embedding, and x̂ = j(x) ∪ {α : α = g is an ultrafilter on x which fails to q-converge}. let f ∈ f(x); then f̂ denotes the filter on x̂ whose base is {f̂ : f ∈ f}, and f̂ = j(f) ∪ {α ∈ x̂ : f ∈ α}. define h q̂ −→ j(x) iff h ≥ f̂ for some f q −→ x, and h q̂ −→ α iff h ≥ ĝ for some α = g. according to [17], if f : (x, q) → (y, qy ) is continuous, (y, qy ) is compact t3, then f̂ : (x̂, q̂) → (y, qy ) is a continuous extension of f, where f̂(j(x) = f(x) and f̂(α) = lim f→f in (y, qy ). moreover, it is easily verified that the above results are valid whenever (x, q) fails to be hausdorff. given any (x, q) ∈ |conv|, it is shown in proposition 2.1 [8] that there exists a finest regular convergence rq which is coarser than q. define x ∼ y iff ẋ rq −→ y; let sx be the set of all equivalence classes, and denote the corresponding quotient map by φ : x → sx. let sq be the quotient structure in conv determined by: φ : (x, rq) → (sx, sq). according to proposition 1.3 [9], (sx, sq) is t3. moreover, it is shown in [8] and [9] that if f is continuous, then the maps below are continuous and the diagram commutes: (x, q) idx (x, rq) φx (sx, sq) (y, p) f ? idy (y, rp) f ? φy (sy, sp) fs ? , where fs([x]) = f(x). in particular, if f : (x, q) → (y, qy ) is continuous and (y, qy ) is compact t3, then the maps below are continuous and the diagram commutes: convergence semigroup categories 79 (x, q) j (x̂, q̂) id x̂(x̂, rq̂) φ (sx̂, sq̂) (y, qy ) f ? idy (y, qy ) f̂ ? idy (y, qy ) f̂ ? idy (y, qy ) f̂s ? theorem 3.4. assume that (f, k) : ((x, q), (s, ., p), λ) → ((y, qy ), (t, ., pt ), µ) is a c-morphism, where (x, q) is hausdorff, p is the discrete structure on s, and (y, qy ) is compact t3. using the notations defined above, (f̂, k) and (f̂s, k) are c-morphisms such that the diagram below commutes: ((x, q), (s, ., p), λ) (j, ids) ((x̂, rq̂), (s, ., p), λ̂) (φ, ids) ((sx̂, sq̂), (s, ., p), λ̂s) ((y, qy ), (t, ., pt ), µ) (f̂, k)) ? � (f̂s , k) (f, k) proof. it follows from (3.1) that f̂, f̂s are continuous, and the diagram commutes. using the notations given above, define λ̂ : x̂ × s → x̂ by λ̂(j(x), g) = (j ◦ λ)(x, g), and if α = f, λ̂(α, g) = lim(j ◦ λ)→(f × ġ) in (x̂, q̂). observe that λ̂(â × b) ⊆ clrq̂(j ◦ λ)(a × b) whenever a ⊆ x and b ⊆ s, and thus, since p is discrete, λ̂ : (x̂, q̂) × (s, p) → (x̂, rq̂) is continuous. according to theorem 6.3 [8], r(q̂ × p) = rq̂ × p, and thus λ̂ : (x̂, rq̂) × (s, p) → (x̂, rq̂) is continuous. moreover, define λ̂s : sx̂ × s → sx̂ by λ̂s([z], g) = [λ̂(z, g)]. note that λ̂s is well-defined. indeed, if z1 ∼ z2, then ż1 rq̂ −→ z2, and since λ̂ : (x̂, rq̂) × (s, p) → (x̂, rq̂) is continuous, λ̂ ˙(z1, g) = λ̂ →(ż1 × ġ) rq̂ −→ λ̂(z2, g). hence λ̂(z1, g) ∼ λ̂(z2, g), and thus λ̂s is well-defined. observe that the diagram below commutes and φ × ids is a quotient map: (x̂, rq̂) × (s, p) λ̂ (x̂, rq̂) (sx̂, sq̂) × (s, p) φ × ids ? λ̂s (sx̂, sq̂) φ ? since φ × ids is a quotient map and φ × λ̂ is continuous, it follows that λ̂s is continuous. then ((x̂, rq̂), (s, ., p), λ̂), ((sx̂, sq̂), (s, ., p), λ̂s) ∈ |c|, and it is 80 h. boustique, p. mikusiński and g. richardson straightforward to verify that (j, ids) and (φ, ids) are c-morphisms. moreover, employing lemma 3.1, (f̂, k) and (f̂s, k) are also c-morphisms. � a commutative semigroup can be embedded in a group iff it is cancellative. one way of constructing the group is by means of equivalence classes of ordered pairs just as one forms the rationals from the integers. let cg denote the category whose objects consist of all the commutative convergence groups, and having all the continuous group homomorphisms as its morphisms. let (s, .) ∈ |sg| be commutative and cancellative; then a special case of theorem 1.24 [6] shows that (s, .) is embedded in the group (s̄, .), where elements in s̄ can be expressed in the form gh−1, for some g, h ∈ s. the natural injection is denoted by j : (s, .) → (s̄, .), where j(g) = g for all g ∈ s. this notation is used below. lemma 3.5. assume that (s, ., p) ∈ |csg| is commutative and cancellative. then there is a finest structure p̄ on s̄ such that (s̄, ., p̄) ∈ |cg| and j : (s, p) → (s̄, p̄) is continuous. proof. let h ∈ f(s̄); then h−1 denotes the filter on s̄ whose base is {h−1 : h ∈ h}. define p̄ on s̄ as follows: k p̄ −→ gh−1 iff there exist g p −→ g1 and h p −→ h1 such that k ≥ (j→g)(j→h)−1 and g1h −1 1 = gh −1. clearly ˙gh−1 p̄ −→ gh−1 since ġ p −→ g and ḣ p −→ h. also, if l ≥ k and k p̄ −→ gh−1, then l p̄ −→ gh−1. hence (s̄, p̄) ∈ |conv|. next, it is shown that multiplication in (s̄, ., p̄) is a continuous operation. suppose that ki p̄ −→ gih −1 i , gi p −→ gi, hi p −→ hi and ki ≥ (j →gi)(j →hi) −1, i = 1, 2. since (s, .) is commutative and j is a homomorphism, k1.k2 ≥ (j→g1.j →g2) (j →h1) −1(j→h2) −1 = j→(g1.g2)(j →(h1.h2)) −1. however, g1.g2 p −→ g1g2, h1.h2 p −→ h1h2, and thus k1.k2 p̄ −→ (g1g2)(h1h2) −1 = (g1h −1 1 ) (g2h −1 2 ). hence multiplication in (s̄, ., p̄) is continuous. finally, inversion is a continuous operation. indeed, suppose that k p̄ −→ gh−1, g p −→ g, h p −→ h, and k ≥ (j→g)(j→h)−1. then k−1 ≥ (j→h)(j→g)−1 p̄ −→ hg−1 = (gh−1)−1, and thus (s̄, .p̄) ∈ |cg| assume that (s̄, ., r) ∈ |cg| such that j : (s, p) → (s̄, r) is continuous and k p̄ −→ gh−1. then there exist g p −→ g1, h p −→ h1 such that k ≥ (j →g)(j→h)−1 and g1h −1 1 = gh −1. it follows that j→h r −→ g1, j →h r −→ h1, and since (s̄, ., r) ∈ |cg|, (j→g).(j→h)−1 r −→ g1h −1 1 . therefore k r −→ gh−1, and thus p̄ ≥ r. � suppose that ((x, q), (s, ., p), λ) ∈ |c|, and recall that (b(x, s), σ) denotes the generalized quotient space determined by ϕ : (x × s, r) → (x × s)/ ∼= b(x, s). it is shown in theorem 2.4(b) that ((b(x, s), σ), (s, ., p), λb) ∈ |c|, where λb(〈(x, g)〉, h) = 〈(λ(x, h), g)〉. the next result shows that (s̄, ., p̄) also acts continuously on (b(x, s), σ). convergence semigroup categories 81 lemma 3.6. assume that ((x, q), (s, ., p), λ) ∈ |c|, where (s, .) is cancellative, and j : (s, ., p) → (s̄, p̄) denote the natural injection. then there exists a continuous action λ̄b such that ((b(x, s), σ), (s̄, ., p̄), λ̄b) ∈ |c| and (idb, j) : ((b(x, s), σ), (s, ., p), λb) → ((b(x, s), σ), (s̄, ., p̄), λ̄b) is a c-morphism. proof. define λ̄b : b(x, s)×s̄ → b(x, s) by λ̄b(〈(x, g)〉, kl −1) = 〈(λ(x, k), gl)〉), where x ∈ x and g, k, l ∈ s. first, it is shown that λ̄b is well-defined. assume that (x, g) ∼ (x1, g1) and kl −1 = k1l −1 1 . employing lemma 1.3(a), there exists h ∈ s such that (x, hg) ≈ (x1, hg1), and thus λ(x, hg1) = λ(x1, hg). then (λ(x, k), hgl) ≈ (λ(x1, k1), hg1l1); indeed, λ(λ(x, k), hg1l1) = λ(x, khg1l1) = λ(λ(x, hg1), kl1) = λ(λ(x1, hg), kl1) = λ(λ(x1, hg), k1l) = λ(x1, hgk1l) = λ(λ(x1, k1), hgl). hence (λ(x, k), hgl) ≈ (λ(x1, k1), hg1l1), and again by lemma 1.3(a), (λ(x, k), gl) ∼ (λ(x1, k1), g1l1). therefore, λ̄b is well-defined. observe that λ̄b is an action. indeed, λ̄b(〈(x, g)〉, e) = 〈(λ(x, e), g)〉 = 〈(x, g)〉, and λ̄b(λ̄b(〈(x, g)〉, k1l −1 1 ), k2l −1 2 ) = λ̄b(〈(λ(x, k1), gl1)〉, k2l −1 2 ) = 〈(λ(λ(x, k1), k2), gl1l2)〉 = 〈(λ(x, k1k2), gl1l2)〉 = λ̄b(〈(x, g)〉, k1l −1 1 k2l −1 2 ). hence λ̄b is an action. furthermore, λ̄b : (b(x, s), σ) × (s̄, p̄) → (b(x, s), σ) is continuous. suppose that h σ −→ 〈(x, g)〉 and m p̄ −→ kl−1; then there exist (x1, g1) ∼ (x, g), k1l −1 1 = kl −1, f q −→ x1, g p −→ g1, k p −→ k1, and l p −→ l1 such that h ≥ ϕ→(f×g) and m ≥ k.l−1. it follows that λ̄b → (h×m) ≥ λ̄b → (ϕ→(f×g)× k.l−1) = ϕ→(λ→(f × k) × g.l) σ −→ ϕ(λ(x1, k1), g1l1) = 〈(λ(x1, k1), g1l1)〉 = λ̄b(〈(x1, g1)〉, k1l −1 1 ) = λ̄b(〈(x, g)〉, kl −1) since λ̄b is well-defined. therefore λ̄b is continuous, and thus ((b(x, s), σ), (s̄, ., p̄), λ̄b) ∈ |c| finally, (idb, j) : ((b(x, s), σ), (s, ., p), λb) → ((b(x, s), σ), (s̄, ., p̄), λ̄b) is a c-morphism since idb ◦ λb = λ̄b ◦ (idb × j). � lemma 3.7. suppose that (f, k) : ((x, q), (s, ., p), λ) → ((y, qy ), (t, ., pt ), µ) is c-morphism, and assume that (t, ., pt ) ∈ |cg|. if (x, g), (y, h) ∈ x × s and (x, g) ∼ (y, h), then µ(f(x), (k(g))−1)) = µ(f(y), (k(h))−1). proof. since (x, g) ∼ (y, h), it follows from lemma 1.3(a) that there exists l ∈ s such that (x, lg) ≈ (y, lh); hence λ(x, lh) = λ(y, lg) and thus f(λ(x, lh)) = f(λ(y, lg)). observe that µ(f(x), (k(g))−1) = µ(f(x), (k(g))−1k(lh)(k(lh))−1) = µ(µ(f(x), k(lh)), (k(g))−1(k(lh))−1) = µ((µ◦(f×k))(x, lh), (k(g))−1(k(lh))−1). since (f, k) is a c-morphism, f ◦ λ = µ ◦ (f × k), and thus (µ ◦ (f × k))(x, lh) = f(λ(x, lh)) = f(λ(y, lg)) = ((µ ◦ (f × k))(y, lg). therefore, µ(f(x), (k(g))−1) = µ((µ ◦ (f × k))(y, lg), (k(g))−1(k(lh))−1) = µ(µ(f(y), k(l)k(g)), (k(g))−1(k(h))−1(k(l))−1) = µ(f(y), (k(h))−1). � define β : (x, q) → (b(x, s), σ) by β(x) = 〈(x, e)〉, and observe that β is continuous. indeed, if f q −→ x, then β→f = ϕ→(f × ė) σ −→ ϕ(x, e) = 〈(x, e)〉. hence β is continuous. therefore (β, j) : ((x, q), (s, ., p), λ) → ((b(x, s), σ), (s̄, ., p̄), λ̄b) is a c-morphism since (λ̄b ◦ (β × j))(x, g) = λ̄b(〈(x, e)〉, g) = 〈(λ(x, g), e)〉 = β(λ(x, g)) = (β ◦ λ)(x, g). 82 h. boustique, p. mikusiński and g. richardson the following result was proved in the topological context under the assumptions that λ(., g) is injective for each fixed g ∈ s, and s is equipped with the discrete topology [4]. here j : (s, .) → (s̄, .) is an embedding, and consequently (s, .) must be cancellative. a canonical map is used in [4] which is not necessarily an embedding but does not require that (s, .) be cancellative. theorem 3.8. assume ((x, q), (s, ., p), λ) ∈ |c|, (s, .) is cancellative, and let j : (s, ., p) → (s̄, ., p̄) denote the injection j(g) = g given in lemma 3.5. if (f, k) is a c-morphism and (t, ., pt ) ∈ |cg|, then there exists a c-morphism (f, k) such that the diagram below is commutative: ((x, q), (s, ., p), λ) (β, j) ((b(x, s), σ), (s̄, ., p̄), λ̄b) ((y, qy ), (t, ., pt ), µ) (f, k) ? � (f, k) proof. define f : b(x, s) → y by f(〈(x, g)〉) = µ(f(x), (k(g))−1), and k : s̄ → t as k(mn−1) = k(m)(k(n))−1. if follows from lemma 3.7 that f is well-defined, and it is easily shown that k is a well-defined group homomorphism. observe that f is continuous. indeed, assume that h σ −→ 〈((x, g))〉; then there exist (x1, g1) ∼ (x, g), f q −→ x1, g p −→ g1 such that h ≥ ϕ →(f ×g). hence f →h ≥ f →(ϕ→(f × g)) = µ→(f→f × (k→g)−1) q y −−→ µ(f(x1), (k(g1)) −1) = f(〈(x1, g1)〉) = f(〈(x, g)〉), and thus f is continuous. it easily follows that k is continuous and the diagram commutes. it remains to show that f ◦ λ̄b = µ◦(f ×k). note that (µ◦(f ×k))(〈(x, g)〉, mn−1) = µ(µ(f(x), (k(g))−1), k(m) (k(n))−1) = µ(f(x), k(m)(k(gn))−1). since (f, k) is a c-morphism, f ◦ λ = µ ◦ (f × k), and therefore f(λ(x, m)) = µ(f(x), k(m)). hence (f◦λ̄b)(〈(x, g)〉, mn −1) = f(〈(λ(x, m), gn)〉) = µ(f(λ(x, m)), (k(gn))−1) = µ(µ(f(x), k(m)), (k(gn))−1) = µ(f(x), k(m)(k(gn))−1), and thus f ◦ λ̄b = µ ◦ (f × k). therefore, (f, k) is a c-morphism. � corollary 3.9. suppose that the hypotheses of theorem 3.8 are satisfied except that (t, .) is cancellative, commutative, and (t, ., pt ) ∈ |csg|. then there exists a c-morphism (f, k) such that the diagram below is commutative: convergence semigroup categories 83 ((x, q), (s, ., p), λ) (βx, js) ((b(x, s), σx), (s̄, ., p̄), λ̄b) ((y, qy ), (t, ., pt ), µ) (f, k) ? (βy , jt ) ((b(y, t ), σy ), (t̄ , ., p̄t ), µ̄b) (f, k) ? 4. examples and special cases first, some examples of generalized quotient spaces are presented, and then the work is concluded with some results pertaining to the case whenever the semigroup is generated by a single element. in the following examples, n denotes a natural number, and let n designate the set of all natural numbers. example 4.1. let x = c∞(rn) be the space of all continuous complexvalued functions defined on rn, and equipped with the topology of uniform convergence on compact sets. denote s = {f ∈ c∞(rn) : f has compact support, f ≥ 0, and ∫ f = 1}. then s is a semigroup with respect to the convolution (f ∗ g)(u) = ∫ rn f(v)g(u − v)dv. define the action λ of s on the space x by convolution, that is, λ(x, f) = x ∗ f. note that λ is not injective. then, for x, y ∈ x, x ≃ y iff x ∗ f = y ∗ f for some f ∈ s. under this equivalence relation some functions will be identified with 0 and, moreover, b(x∗, s) is a space of generalized functions. clearly, not all continuous functions can be identified with elements of b(x∗, s), and thus b(x∗, s) is a proper extension of x∗. every element of b(x∗, s) has derivatives of all orders defined by dα〈([x], g)〉 = 〈([dαx], g)〉. it is not difficult to check that this operation is well-defined and continuous. consider the situation when s is generated by a single map. let x be a nonempty set, and let g : x → x be a non-injective map. then s = {gn : n = 0, 1, 2, ...} is a commutative semigroup with respect to composition, and x ≃ y iff there exists an n such that λ(x, gn) = λ(y, gn). if g is a surjection, then λ∗(., g n) : x∗ → x∗ is a surjection for every n ∈ n. consequently, b(x∗, s) = x∗ (identifiable); hence, in order to produce a 84 h. boustique, p. mikusiński and g. richardson proper extension of x∗, it is necessary to start with a g that is not surjective. example 4.2. let x be a normed space, t : x → x a norm preserving linear operator, and assume that s is the semigroup generated by t . since t is injective, x∗ = x. for sake of brevity, denote 〈(x, t n)〉 by x t n ∈ b(x, s), and define ∥ ∥ ∥ x t n ∥ ∥ ∥ = ‖x‖ . if x t n = y t m , then t mx = t ny, and hence ∥ ∥ ∥ x t n ∥ ∥ ∥ = ‖x‖ = ‖t mx‖ = ‖t ny‖ = ‖y‖ = ∥ ∥ ∥ y t m ∥ ∥ ∥ . therefore, ‖·‖ is well-defined in b(x, s). one can also verify that ‖·‖ is a norm in b(x, s); in particular, ∥ ∥ ∥ x t n + y t m ∥ ∥ ∥ = ∥ ∥ ∥ ∥ t mx + t ny t m+n ∥ ∥ ∥ ∥ = ‖t mx + t ny‖ ≤ ‖t mx‖ + ‖t ny‖ = ‖x‖ + ‖y‖ = ∥ ∥ ∥ x t n ∥ ∥ ∥ + ∥ ∥ ∥ y t m ∥ ∥ ∥ . consequently, this extension of a normed space (x, ‖·‖) produces a normed space (b(x, s), ‖·‖). moreover, the map defined by t x t n = t x t n is a norm preserving bijection on b(x, s). in particular, given a vector space x, let {e1, e2, ...} be a hamel basis, and define a norm by ∥ ∥ ∥ ∥ ∥ ∥ k ∑ j=1 αjepj ∥ ∥ ∥ ∥ ∥ ∥ = k ∑ j=1 |αj|. then t ( k ∑ j=1 αjepj ) = k ∑ j=1 αjepj+1 and u( k ∑ j=1 αjepj ) = k ∑ j=1 αje2pj are norm preserving operators on x. in both cases, b(x, s) is a proper extension of x. convergence semigroup categories 85 example 4.3. (i) consider an arbitrary nonempty set z and denote x = zn. define g : x → x by g((xn)) = (xn+1), and let s = {g n : n = 0, 1, 2, ...}. then (xn) ≃ (yn) iff there exists an n0 such that xn = yn for all n ≥ n0. that is, two sequences are equivalent whenever they are eventually equal. note that g is a surjection, and thus b(x∗, s) = x∗ . (ii) let e be a vector space, x = en, and denote s = {(αn) ∈ {0, 1} n : card{n ∈ n : αn 6= 0} < n0}. then s is a commutative semigroup with respect to termwise multiplication, and define λ((xn), (αn)) = (αnxn). observe that λ(., (αn)) is neither injective nor surjective, and b(x∗, s) = x∗. recall that a continuous surjection f : (x, q) → (y, p) in conv is called a quotient map if whenever f p −→ y, then there exists g q −→ x ∈ f−1(y) such that f→g = f. moreover, a continuous surjection f is said to be a proper map provided that for each ultrafilter f on x, f→f p −→ y implies that f q −→ x, for some x ∈ f−1(y). given that s = {gn : n = 0, 1, 2, ...}, two special cases are considered below by requiring that g be either a quotient or proper map. theorem 4.4. assume that (x, q) ∈ |conv|, g : (x, q) → (x, q) is a quotient map in conv, s = {gn : n = 0, 1, 2, ...}, and define the action by λ(x, gn) = gn(x), n = 0, 1, 2, .... then (a) λ∗(., g n) : (x∗, q∗) → (x∗, q∗) is a homeomorphism, for each fixed n = 0, 1, 2, ... (b) β∗ : (x∗, q∗) → (b(x∗, s), σ∗) is a homeomorphism whenever s is equipped with the discrete topology, where β∗([x]) = 〈([x], e)〉. proof. (a): denote λ∗([x], g) = [λ(x, g)] by g∗([x]) = [g(x)]. consider the commutative diagram below: (x, q) g (x, q) (x∗, q∗) ξ ? g∗ (x∗, q∗) ξ ? since ξ is a quotient map in conv, g∗ is continuous iff g∗ ◦ ξ is continuous. however, g∗ ◦ ξ = ξ ◦ g is continuous, and thus g∗ is continuous. moreover, g∗ is injective. indeed, if [g(x)] = g∗([x]) = g∗([z]) = [g(z)], then g(x) ≃ g(z) and thus gn+1(x) = gn(g(x)) = gn(g(z)) = gn+1(z) for some n ≥ 0, and thus 86 h. boustique, p. mikusiński and g. richardson [x] = [z]. hence g∗ is injective. since g is onto, it follows that g∗ is a continuous bijection. next, it is shown that g−1 ∗ is continuous. again, since ξ is a quotient map, g−1 ∗ is continuous iff g−1 ∗ ◦ ξ is continuous. since the diagram commutes, g∗ ◦ ξ = ξ ◦ g, and thus ξ = g −1 ∗ ◦ ξ ◦ g. assume that f q −→ x; it must be shown that (g−1 ∗ ◦ ξ)→f q∗ −→ g−1 ∗ ([x]). since g is a quotient map in conv, there exists h q −→ z such that g→h = f and g(z) = x. then ξ→h q∗ −→ [z], and thus (g−1 ∗ ◦ ξ)→f = (g−1 ∗ ◦ ξ ◦ g)→h = ξ→h q∗ −→ [z]. since g(z) = x, g∗([z]) = [x], and thus (g −1 ∗ ◦ ξ)(x) = g−1 ∗ ([x]) = [z]. hence (g∗ ◦ ξ) →f → (g−1 ∗ ◦ ξ)(x)), and thus g−1 ∗ ◦ ξ is continous. therefore g−1 ∗ is continuous, and hence g∗ is a homeomorphism. further, since the finite composition of quotient maps is again a quotient in conv, gn is also a quotient map, and it follows that (gn)∗ is also a homeomorphism, n = 0, 1, 2, .... (b): note that if 〈([x], e)〉 = β∗([x]) = β∗([z]) = 〈([z], e)〉, then [x] = λ∗([x], e) = λ∗([z], e) = [z], and thus β∗ is injective. given 〈([z], g n)〉 ∈ b(x∗, s), since g n is onto, gn(x) = z for some x ∈ x. then β∗([x]) = 〈([x], e)〉 = 〈([z], g n)〉, and thus β∗ is a bijection. observe that β∗ = ϕ∗ ◦ σ, where σ([x]) = ([x], e), and hence β∗ is a continuous bijection. it remains to show that β−1 ∗ is continuous. assume that f ∈ f(x∗) such that β→ ∗ f σ∗ −→ β∗([x]); it must be shown that f q∗ −→ [x]. since ϕ∗ : (x∗ × s, r∗) → (b(x∗, s), σ∗) is a quotient map in conv and β → ∗ f σ∗ −→ β([x]) = 〈([x], e)〉, there exists k r∗ −→ ([z], gn) such that ϕ→ ∗ k = β→ ∗ f, where r∗ = q∗ × p and 〈([z], gn)〉 = 〈([x], e)〉. in particular, λ∗([z], e) = λ∗([x], g n), or [z] = [gn(x)]. since s has the discrete topology, there exists h q∗ −→ [z] such that h × ġn ≤ k, and thus ϕ→ ∗ (h × ġn) ≤ ϕ→ ∗ k = β→ ∗ f. fix h ∈ h; then there exists f ∈ f such that β∗(f) ⊆ ϕ∗(h×{g n}). if [s] ∈ f , then β∗([s]) = 〈([s], e)〉 = 〈([t], g n)〉 for some [t] ∈ h. in particular, [gn(s)] = λ∗([s], g n) = λ∗([t], e) = [t], and thus (gn)∗(f) ⊆ h. therefore (g n)→ ∗ f ≥ h q∗ −→ [z], and since (gn)∗ is a homeomorphism, f q∗ −→ (gn)−1 ∗ ([z]). however, (gn)∗([x]) = [g n(x)] = [z], and thus (gn)−1 ∗ ([z]) = [x]. hence f q∗ −→ [x], and it follows that β∗ is a homeomorphism. � the final result is the analogue of theorem 4.4 whenever g is a proper map. object (x, q) ∈ |conv| is said to be a choquet space provided that f q −→ x whenever each ultrafilter g ≥ f, g q −→ x; that is, q−convergence is determined by q−convergence of the ultrafilters on x. given any (x, q) ∈ |conv|, define q̂ by : f q̂ −→ x iff for each ultrafilter g ≥ f, g q −→ x. note that q̂ and q agree on ultrafilter convergence. moreover, it follows that q̂ is the finest choquet structure on x which is coarser than q. suppose that f : (x, q) → (y, p) is a map between two convergence spaces such that f→f p −→ f(x) whenever f is an ultrafilter on x for which f q −→ x. it is easily shown that f : (x, q̂) → (y, p̂) is continuous. verification of the next result makes use of the preceding fact along with an argument similar to that given in theorem 4.4. the proof is omitted. convergence semigroup categories 87 theorem 4.5. suppose that (x, q) ∈ |conv|, g : (x, q) → (x, q) is a proper map, s = {gn : n = 0, 1, 2, ...}, and action λ(x, gn) = gn(x), n ≥ 0. then (a) λ∗(., g n) : (x∗, q̂∗) → (x∗, q̂∗) is a homeomorphism, for each fixed n ≥ 0 (b) β∗ : (x∗, q̂∗) → (b(x∗, s), γ∗) is a homeomorphism whenever s has the discrete topology p, and γ∗ denotes the quotient structure in conv determined from ϕ∗ : (x∗, q̂∗) × (s, p) → (x∗ × s)/ ≈. employing theorem 4.5, along with the fact that a continuous surjection g : (x, q) → (x, q) is a proper map whenever (x, q) ∈ |conv| is compact and hausdorff, gives the concluding result. corollary 4.6. assume that (x, q) ∈ |conv| is compact, hausdorff , and g : (x, q) → (x, q) is a continous surjection. suppose that s = {gn : n = 0, 1, 2, ...} has the discrete topology. then β∗ : (x∗, q̂∗) → (b(x∗, s), γ∗) is a homeomorphism. references [1] j. adamek, h. herrlich and g. e. strecker, abstract and concrete categories, wiley pub., 1990. [2] r. beattie and h.-p. butzmann, convergence structures and applications to functional analysis, kluwer acad. pub., dordrecht, 2002. [3] h. boustique, p. mikusiński and g. richardson, convergence semigroup actions: generalized quotients, appl. gen. topol. 10 (2009), 173–186. [4] d. bradshaw, m. khosravi, h. m. martin and p. mikusiński, on categorical and topological properties of generalized quotients (seminar notes), 2005. [5] j. burzyk, c. ferens and p. mikusiński, on the topology of generalized quotients, appl. gen. topol. 9 (2008), 205–212. [6] a. h. clifford and g. b. preston, the algebraic theory of semigroups, volume 1, amer. math. soc., providence, r.i, 1961. [7] d. kent, convergence quotient maps, fund. math. 65 (1969), 197–205. [8] d. kent and g. richardson, the regularity series of a convergence space, bull. aust. math. soc. 13 (1975), 21–44. [9] d. c. kent and g. d. richardson, the regularity series of a convergence space ii, bull. aust. math. soc. 15 (1976), 223–243. [10] m. khosravi, pseudoquotients: construction, applications, and their fourier transform, ph.d. dissertation, univ. of central florida, orlando, fl, 2008. [11] p. mikusiński, boehmians and generalized functions, acta math. hungar. 51 (1988), 271–281. [12] p. mikusiński, generalized quotients with applications in analysis, methods and applications of anal. 10(2003), 377–386. [13] w. park, convergence structures on homeomorphism groups, math. ann. 199(1972), 45–54. [14] w. park, a note on the homeomorphism group of the rational numbers, proc. amer. math. soc. 42(1974), 625–626. [15] g. preuss, foundations of topology: an approach to convenient topology, kluwer acad. pub., dordrecht, 2002. [16] n. rath, action of convergence groups, topology proc. 27 (2003), 601–612. 88 h. boustique, p. mikusiński and g. richardson [17] g. d. richardson, a stone-cěch compactification for limit spaces, proc. amer. math. soc. 25 (1970), 403–404. [18] g. d. richardson and d. c. kent, regular compactifications of convergence spaces, proc. amer. math. soc. 31 (1972), 571–573. received november 2009 accepted september 2010 gary richardson (garyr@mail.ucf.edu) department of mathematics, university of central florida,orlando, fl 32816, usa, fax: (407) 823-6253, tel: (407) 823-2753 convergence semigroup categories. by h. boustique, p. mikusinski and g. richardson songagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 249-258 on countable star-covering properties yan-kui song ∗ abstract. we introduce two new notions of topological spaces called a countably starcompact space and a countably absolutely countably compact (= countably acc) space. we clarify the relations between these spaces and other related spaces and investigate topological properties of countably starcompact spaces and countably acc spaces. some examples showing the limitations of our results are also given. 2000 ams classification: 54d20, 54b10, 54d55 keywords: countably compact, acc, starcompact, countably starcompact, countably acc 1. introduction by a space, we mean a topological space. let us recall that a space x is countably compact if every countable open cover of x has a finite subcover. fleischman defined in [4] a space x to be starcompat if for every open cover u of x, there exists a finite subset b of x such that st(b, u) = x, where st(b, u) = ⋃ {u ∈ u : u ∩ b 6= ∅}. he proved that every countably compact space x is starcompact. conversely, van douwen-reed-roscoe-tree [2] proved that every starcompact t2-space is countably compact, but this does not hold for t1-spaces (see example 2.5 below). strengthening the definition of starcompactness, matveev defined in [5] a space x to be absolutely countably compact (= acc) if for every open cover u of x and every dense subspace d of x, there exists a finite subset f of d such that st(f, u) = x. every acc t2-space is countably compact (see [5]), but an acc t1-space need not be countably compact (see example 2.4 below). restricting the definitions of starcompactness and absolutely countably compactness to countable open covers, we define the following classes of spaces: ∗the author is supported by nsfc project 10571081 and by the natural science foundation of the jiangsu higher education institutions of china (grant no 07kjb110055). 250 y.-k. song definition 1.1. a space x is countably starcompact if for every countable open cover u of x, there exists a finite subset b of x such that st(b, u) = x. definition 1.2. a space x is countably absolutely countably compact (= countably acc) if for every countable open cover u of x and every dense subspace d of x, there exists a finite subset f of d such that st(f, u) = x. the purpose of this paper is to clarify the relationship among these spaces and to consider topological properties of a countably starcompact space and a countably acc space, respectively. the systematic study of countable starcovering properties is very important to supply the research of star-covering properties. from the definitions and above remarks, we have the following diagram, where a → b means that every a-space is a b-space: countably compact   y acc −−−−→ starcompact −−−−→ countably starcompact   y   y t2 countably acc countably compact diagram 1 the cardinality of a set a is denoted by |a|. for a cardinal κ, κ+ denotes the smallest cardinal greater than κ. let c denote the cardinality of the continuum, ω the first infinite cardinal and ω1 = ω +. as usual, a cardinal is the initial ordinal and an ordinal is the set of smaller ordinals. for each ordinals α, β with α < β, we write (α, β) = {γ : α < γ < β}, [α, β) = {γ : α ≤ γ < β} and [α, β] = {γ : α ≤ γ ≤ β}, other terms and symbols will be used as in [3]. 2. relations among spaces in this section, we consider the relations among countably acc spaces, countably starcompact spaces and other related spaces. proposition 2.1. every countably compact space is countably acc and every countably acc space is countably starcompact. proof. let x be a countably compact space. let u be a countable open cover of x and let d be a dense subspace of x. then, there exists a finite subcover {u1, u2, . . . un} of u, since x is countably compact. pick a point xi ∈ ui ∩ d for i = 1, 2, . . . n. then, st({x1, x2, · · · , xn}, u) = x, which shows that x is countably acc. hence, every countably compact space is countably acc. it follows immediately from the definitions that every countably acc space is countably starcompact. � on countable star-covering properties 251 proposition 2.2. for a t2-space x, the following conditions are equivalent: (1) x is countably compact; (2) x is countably acc; (3) x is countably starcompact. proof. the implications (1) ⇒ (2) and (2) ⇒ (3) are true by proposition 2.1. it remains to show that (3) ⇒ (1). suppose that x is not countably compact. then, there exists an infinite closed discrete subset d = {xn : n ∈ ω} of x. for each m ∈ ω, let dm = {xn : 2 m 6 n < 2m+1}; then |dm| = 2 m. since x is a t2-space, there exists a collection um = {un : 2 m 6 n < 2m+1} of pairwise disjoint open sets in x such that un ∩ d = {xn} for each n ∈ ω. take such a collection um for each m ∈ ω and let u = {x \ d} ∪ ⋃ m∈ω um. then, u is a countable open cover of x. let b be any finite subset of x with |b| = k. since |b| < 2k = |uk| and uk is disjoint, some un ∈ uk does not intersect b. then, xn /∈ st(b, u), because un is only member of u containing xn. hence, x is not countably starcompact. this proves that (3) ⇒ (1). � proposition 2.3. every countably starcompact space x is pseudocompact. proof. let f be a continuous real-valued function on x, and let un = {x ∈ x : n − 1 < f (x) < n + 1} for each n ∈ z. then, u = {un : n ∈ z} is a countable open cover of x. since x is countably starcompact, there exists a finite subset b of x such that st(b, u) = x. since u is point-finite, the set {u ∈ u : u ∩ b 6= ∅} is finite, say {un1 , un2 , . . . unk }. if we put m = max{|ni| + 1 : i = 1, 2, . . . k}, then |f (x)| ≤ m for each x ∈ x. hence, every continuous real-valued function on x is bounded, which means that x is pseudocompact. � summing up the above results, we have the following diagram, where the implications (1)–(6) hold for arbitrary spaces and the inverses of implications (2)–(5) also hold for t2-spaces: acc   y (1) countably compact (2) −−−−→ countably acc   y (3)   y (4) acc (6) −−−−→ starcompact (5) −−−−→ countably starcompact   y t2 countably compact diagram 2 252 y.-k. song in the rest of this section, we give examples which show the implications (1)–(6) in diagram 2 cannot be reversed in the realm of t1-spaces. the first one shows that the inverses of the implications (2) and (3) do not hold for t1-spaces. example 2.4. there exists an acc t1-space which is not countably compact. proof. let κ be an infinite cardinal and a a set of cardinality κ. define x = κ+ ∪ a. we topologize x as follows: κ+ has the usual order topology and is an open subspace of x, and a basic neighborhood of a ∈ a is of the form gβ (a) = (β, κ +) ∪ {a}, where β < κ+. then, x is a t1-space which is not countably compact, because a is infinite discrete closed in x. to show that x is absolutely countably compact, let u be an open cover of x. let d be the set of all isolated points of κ+. then, d is dense in x and every dense subspace of x includes d. thus, it is suffices to show that there exists a finite subset f ⊆ d such that st(f, u) = x. since κ+ is absolutely countably compact, there is a finite subset f ′ ⊆ d such that κ+ ⊆ st(f ′, u). for each a ∈ a, there is β(a) < κ+ such that gβ(a)(a) is included in some member of u. if we choose β ∈ d with β > sup{β(a) : a ∈ a}, then a ⊆ st(β, u). let f = f ′ ∪ {β}. then, st(f, u) = x. hence, x is absolutely countably compact, which completes the proof. � � the second example shows that the converses of the implications (3), (4) and (6) in diagram 2 do not hold for t1-spaces. example 2.5. there exists a starcompact t1-space which is not countably acc. proof. let y = (ω + 1) × ω1, where both ω + 1 and ω1 have the usual order topologies and y has the tychonoff product topology. let x = ω ∪ y . we topologize x as follows: y is an open subspace of x; a basic neighborhood of a point n ∈ ω takes the form oα(n) = {n} ∪ ((n, ω] × (α, ω1)) where α < ω1. then, x is a t1-space. to show that x is starcompact, let u be an open cover of x. then, there exists a finite subset f1 of y such that y ⊆ st(f1, u), since y is countably compact. for each n ∈ ω, there is αn < ω1 such that oαn (n) is included in some member of u. if we choose α0 < ω1 with α0 > sup{αn : n ∈ ω}, then ω ⊆ st(〈ω, α0〉, u). let f0 = f1 ∪ {〈ω, α0〉}. then, x = st(f0, u), which shows that x is starcompact. next, we show that x is not countably acc. let d = ω × ω1. then, d is dense in x. therefore, it is suffices to show that there exists a countable open cover v of x such that st(a, v) 6= x for any finite subset a of d. let us consider the countable open cover v = {y } ∪ {o0(n) : n ∈ ω}. on countable star-covering properties 253 let a be any finite subset of d. then, there exists n ∈ ω such that ([n, ω] × ω1) ∩ a = ∅. hence, n /∈ st(a, v), since o0(n) is only element of v such that n ∈ o0(n). this shows that x is not countably acc. � the third example shows that the converses of the implications (1) and (5) in diagram 2 do not hold for t1-spaces. example 2.6. there exists a countably acc t1-space which is not a starcompact space. proof. let x = ω1 ∪ a, where a = {aα α ∈ ω1} is a set of cardinality ω1. we topologize x as follows: ω1 has the usual order topology and is an open subspace of x; a basic neighborhood of a point aα ∈ a takes the form oβ (aα) = {aα} ∪ (β, ω1) where β < ω1. then, x is a t1-space. to show that x is countably acc, let u be a countable open cover of x. let d be the set of all isolated points of ω1. then, d is dense in x and every dense subspace of x includes d. thus, it is suffices to show that there exists a finite subset f ⊆ d such that st(f, u) = x. since ω1 is absolutely countably compact, there is a finite subset f ′ ⊆ d such that ω1 ⊆ st(f ′, u). let v = {u ∈ u : u ∩ a 6= ∅}. for each u ∈ v, there exists a βu < ω1 such that (βu , ω1) ⊆ u . since v is countable, we can choose β ∈ d with β > sup{βu : u ∈ v}. thus, a ⊆ st(β, v) ⊆ st(β, u), since β ∈ u for each u ∈ v. let f = f ′ ∪ {β}. then, x = st(f, u), which shows that x is countably acc. next, we show that x is not starcompact. let us consider the open cover v = {ω1} ∪ {oα(aα) : α < ω1}. let a be any finite subset of x. then, there exists α < ω1 such that a ∩ ((α, ω1) ∪ {aβ : β > α}) = ∅. choose β > α. then aβ /∈ st(a, v), since oα(aα) is only element of v containing aα for each α ∈ ω1. this shows that x is not starcompact. � remark 2.7. pavlov [8] proved that a countably compact space need not be acc even if it is a normal t2-space. 3. discrete sum and subspaces we begin with a proposition which follows immediately from the definitions of a countably starcompact space and a countably acc space: proposition 3.1. the discrete sum of a finite collection of countably starcompact (resp. countably acc) spaces is countably starcompact (resp. countably acc). it is well known that a closed subspace of a countably compact space is countably compact. however, a similar result does not hold for starcompactness, countable starcompactness and countable acc properties. in fact, the following example shows that these properties are not preserved by taking regular closed subspaces. 254 y.-k. song example 3.2. there exists an acc t1-space having a regular-closed subspace which is not countably starcompact. proof. let s1 = ω ∪ r be the isbell-mrówka space [7], where r is a maximal almost disjoint family of infinite subsets of ω so that |r| = c. sinces1 is not countably compact, s1 is not countably starcompact by proposition 2.2. let s2 = c + ∪ a, where a is a set of cardinality c. we topologize s2 as follows: c+ has the usual order topology and is an open subspace of s2, and a basic neighborhood of a ∈ a takes the form gβ (a) = (β, c +) ∪ {a}, where β < c+. we assume that s1 ∩ s2 = ∅. let ϕ : r → a be a bijection. let x be the quotient space obtained from the discrete sum s1 ⊕ s2 by identifying r with ϕ(r) for each r ∈ r. let π : s1 ⊕ s2 → x be the quotient map. it is easy to check that π(s1) is a regular-closed subset of x, however, it is not countably starcompact, since it is homeomorphic to s1. next, we show that x is acc. for this end, let u be an open cover of x. let s be the set of all isolated points of c+ and let d = π(s ∪ ω). then, d is dense in x and every dense subspace of x includes d. thus, it is suffices to show that there exists a finite subset f of d such that x = st(f, u). by the proof of example 2.4, s2 is acc. since π(s2) is homeomorphic to the space s2, π(s2) is acc, hence, there exists a finite subset f0 of π(s) such that π(s2) ⊆ st(f0, u). on the other hand, since π(s1) is homeomorphic to s1, every infinite subset of π(ω) has an accumulation point in π(s1). hence, there exists a finite subset f1 of π(ω) such that π(ω) ⊆ st(f1, u). for, if π(ω) 6⊆ st(b, u) for any finite subset b ⊆ π(ω), then, by induction, we can define a sequence {xn : n ∈ ω} in π(ω) such that xn 6∈ st({xi : i < n}, u) for each n ∈ ω. by the property of π(s1) mentioned above, the sequence {xn : n ∈ ω} has a limit point x0 in π(s1). pick u ∈ u such that x0 ∈ u . choose n < m < ω such that xn ∈ u and xm ∈ u . then, xm ∈ st({xi : i < m}, u), which contradicts the definition of the sequence {xn : n ∈ ω}. let f = f0 ∪ f1. then, x = st(f, u). hence, x is acc, which completes the proof. � 4. mappings it is well known that a continuous image of a countably compact space is countably compact (see [3]) and a continuous image of a starcompact space is starcompact (see [2]). similarly, we have the following proposition. proposition 4.1. a continuous image of a countably starcompact space is countably starcompact. proof. suppose that x is a countably starcompact space and f : x → y a continuous onto map. let u be a countable open cover of y . then, v = {f −1(u ) : u ∈ u} is a countable open cover of x. since x is countable starcompact, there exists a finite set b ⊆ x such that st(b, v) = x. let f = f (b). then, f is a finite set of y and st(f, u) = y . hence, y is countably starcompact. � on countable star-covering properties 255 matveev showed in [5, example 3.1] that a continuous image of an acc space need not be acc. now, we give an example showing that a continuous image of an acc t1-space need not be countably acc. example 4.2. there exist an acc t1-space x and a continuous map f : x → y onto a space y which is not countably acc. proof. let x1 = (ω + 1)× ω1 with the tychonoff product topology, where both ω + 1 and ω1 have the usual order topologies. then, x1 is acc by [5, theorem 2.3], since ω +1 is a first countable, compact space and ω1 is acc by [5, theorem 1.8]. let x2 = ω1 ∪ ω. we topologize x2 as follows: ω1 has the usual order topology and is an open subspace of x2, and a basic neighborhood of n ∈ ω takes the form gβ (n) = (β, ω1) ∪ {n}, where β < ω1. by the proof of example 2.4, x2 is acc. let x = x1 ⊕ x2 be the discrete sum of x1 and x2. then, x is acc by proposition 1.3 [5]. let y = x1 ∪ x2. we topologize y as follows: x1 is an open subspace of y ; a basic neighborhood of a point β < ω1 ⊆ x2 takes the form oγ,m(β) = ([m, ω] × ω1) ∪ (γ, β], where γ < β and m ∈ ω. a basic neighborhood of a point n ∈ ω takes the form oα(n) = ([n, ω] × ω1) ∪ (α, ω1) ∪ {n}, where α < ω1; to show that y is not countably acc. let d = ω × ω1. then, d is dense in y . therefore, it is sufficient to show that there exists a countable open cover v of y such that st(a, v) 6= y for any finite subset a of d. let us consider the countable open cover v = {x1 ∪ ω1} ∪ {o0(n) : n ∈ ω}. let a be any finite subset of d. then, there exists a n ∈ ω such that ([n, ω] × ω1) ∩ a = ∅. hence, n /∈ st(a, v), since o0(n) is the only element of v such that n ∈ o0(n) for each n ∈ ω, which shows that y is not countably acc. let f : x → y be the identity map. then, f is continuous. this completes the proof. � recall from [5] or [6] that a continuous mapping f : x → y is varpseudoopen provided inty f (u ) 6= ∅ for every nonempty open set u of x. in [5], it was proved that a continuous varpseudoopen image of an acc space is acc. similarly, we prove the following proposition. proposition 4.3. a continuous varpseudoopen image of a countably acc space is countably acc. proof. suppose that x is a countably acc space and f : x → y is a continuous varpseudoopen onto map. let u be a countable open cover of y and d a dense subspace of y . then, v = {f −1(u ) : u ∈ u} is a countable open cover 256 y.-k. song of x, and f −1(d) is a dense subspace of x since f is a varpseudoopen map. hence, there exists a finite set b ⊆ f −1(d) such that st(b, v) = x. let f = f (b). then, f is a finite set of d and st(f, u) = y , which shows that y is a countably acc space. � now, we consider preimages. it is well known that a perfect preimage of a countably compact space is countably compact (see [3, theorem 3.10.10]) but a perfect preimage of an acc space need not be acc (see [1, example 3.2]). now, we give an example showing that (1) a perfect preimage of a starcompact space need not be starcompact, (2) a perfect preimage of a countably starcompact space need not be countably starcompact, and (3) a perfect preimage of a countably acc space need not be countably acc. our example uses the alexandorff duplicate a(x) of a space x: the underlying set of a(x) is x × {0, 1}; each point of x × {1} is isolated and a basic open neighborhood of 〈x, 0〉 ∈ x × {0} is a set of the from (u × {0}) ∪ ((u × {1}) \ {〈x, 1〉}), where u is an open neighborhood of x in x. example 4.4. there exists a perfect onto map f : x → y such that y is an acc t1-space, but x is not countably starcompact. proof. let y = ω1 ∪ ω. we topologize y as follows: ω1 has the usual order topology and is an open subspace of y , and a basic neighborhood of n ∈ ω takes the form gβ (n) = (β, ω1) ∪ {n}, where β < ω1. by the proof of example 2.4, y is an acc space. let x = a(y ) be the alexandorff duplicate of y . then, x is not countably starcompact, since ω × {1} is countable discrete, open and closed in x and countable starcompactness is preserved by open and closed set. let f : x → y be the projection. then, f is a perfect onto map. this completes the proof. � 5. products it is well known that the product of a countably compact space and a compact space is countably compact. however, the product of an acc tychonoff space with a compact t2-space need not be acc (see [5, example 2.2]). also, in [4, example 3], an example was given showing that the product of a starcompact t1-space with a compact metric space need not be starcompact. now, we show that the same example also shows that the product of a countably starcompact (resp. countably acc) t1-space with a compact metric space need not be countably starcompact (resp. countably acc). example 5.1 (fleischman). there exist an acc t1-space x and a compact metric space y such that x × y is not countably starcompact. on countable star-covering properties 257 proof. let x = ω1 ∪ a and ω1 ∩ a = ∅, where a = {an : n ∈ ω} is a countable set. we topologize x as follows: ω1 has the usual order topology and is an open subspace of x, and a basic neighborhood of each an ∈ a takes the form gβ (an) = (β, ω1) ∪ {an}, where β < ω1. then, x is an acc t1-space by the proof of example 2.4. let y = ω + 1 with the usual order topology. then, y is a compact metric space. next, we prove that x ×y is not countably starcompact. let un = [n, ω1)∪ {an} and vn = (n, ω] for each n ∈ ω. let u = {un × vn : n ∈ ω} ∪ {x × {n} : n ∈ ω}. then, u is a countable open cover of x × y. let f be a finite subset of x × y . then, there exists a n ∈ ω such that (x × {n}) ∩ f = ∅. hence, 〈an, n〉 /∈ st(f, u), since x × {n} is the only element of u such that 〈an, n〉 ∈ x × {n} for each n ∈ ω. this completes the proof. � remark 5.2. by example 5.1, we can see that (1) an open perfect preimage of a starcompact space need not be starcompact, (2) an open perfect preimage of a countably starcompact space need not be countably starcompact, and (3) an open perfect preimage of a countably acc space need not be countably acc. acknowledgements. the author would like to thank prof. h. ohta for his kind help and valuable suggestions. references [1] m. bonanzinga, preservation and reflection of acc and hacc spaces, comment. math. univ. carolinae 37, no. 1 (1996), 147–153. [2] e. k. van douwen, g. m. reed, a. w. roscoe and i. j. tree, star covering properties, topology appl. 39 (1991), 71–103. [3] r. engelking, general topology, revised and completed edition, heldermann verlag, berlin, 1989. [4] w. m. fleischman, a new extension of countable compactness, fund. math. 67 (1970), 1–7. [5] m. v. matveev, absolutely countably compact spaces, topology appl. 58 (1994), 81–92. [6] m. v. matveev, a survey on star-covering properties, topology atlas, preprint no 330, 1998. [7] s. mrówka, on complete regular spaces, fund. math. 41 (1954), 105–106. [8] o. i. pavlov, a normal countably compact not absolutely countably compact space, proc. amer. math. soc. 129 (2001), 2771–2775. [9] y. song,on some questions on star covering properties, q and a in general and topology 18 (2000), 87–92. 258 y.-k. song received april 2006 accepted november 2006 yan-kui song (songyankui@njnu.edu.cn) institute of mathematics, school of mathematics and computer sciences, nanjing normal university, nanjing 210097, p. r. of china. aroelagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 185-188 a note on a fixed point theorem for ray oriented maps a. arockiasamy and a. anthony eldred abstract. in this paper, we will prove a fixed point theorem for a ray-oriented map defined on a nonempty closed bounded convex subset of a banach space. 2000 ams classification: 47h10 , 54h25. keywords: fixed point, ray oriented map. notations let x be a banach space and k be a non-empty subset of x. let t : k → k be a mapping. let r x be the ray passing through the segment < x, t x > and so r x := {(1 − λ)x + λt x : λ ∈ r}. let < x, y > be defined to be as {(1 − λ)x + λy : λ ∈ [0, 1]} and (x, y) := {(1 − λ)x + λy : λ ∈ (0, 1)}. for any x1, x2 ∈ rx, we say that x1 ≤ x2 whenever λ1 ≤ λ2 where x1 = (1 − λ1)x + λ1t x and x2 = (1 − λ2)x + λ2t x for some λ1 , λ2 ∈ r. 1. introduction let x be a normed linear space and let k be a nonempty closed bounded convex subset of x. suppose t : k → k is a mapping satisfying the following conditions: (i). for some element x0 of k, rx0 ⋂ k is invariant under t and (ii). for each element x ∈ r x0 ⋂ k, t | < x, t x > ⋂ k is continuous. then, we will prove that there exists y0 ∈ rx0 ⋂ k such that < y0, t y0 >⊆ rx0 ⋂ k is invariant under t . moreover,the above theorem will be followed by a corollary as in the following: suppose t : [a, b] → [a, b] is a mapping where a, b ∈ ℜ. if for each x ∈ [a, b], the map t restricted to the segment joining x and t x is continuous. then we will prove that there exists an invariant interval under t and so it will have a fixed point in [a, b]. this result extends one dimensional brouwer’s result for a larger class of mappings which need not be continuous. also one can find some similar treatment for the convergence of fixed point in the real line by beardon [1]. for further important fixed point results one can refer to [2]. 186 a. arockiasamy and a. anthony eldred 2. main results theorem 2.1. let x be a normed linear space and let k be a nonempty closed bounded convex subset of x. suppose t : k → k is a mapping satisfying the following conditions: (1) for some element x0 of k, rx0 ⋂ k is invariant under t and (2) for each element x ∈ r x0 ⋂ k, t | < x, t x > ⋂ k is continuous then, t has a fixed point in r x0 ⋂ k. note: when we say t | < x, t x > is continuous, we mean that t is right continuous at x and left continuous at t x if x < t x. proof. assume that the conclusion of the theorem is false. that is, t does not have a fixed point in r x0 ⋂ k. therefore, for every b ∈ r x0 ⋂ k, < b, t b > is not invariant under t. fix y0 ∈ rx0 ⋂ k and let x0 ∈< y0, t y0 > such that t x0 /∈< y0, t y0 > . let g x0 = r x0 ⋂ k. now we can easily prove that a = {λ ∈ r : (1 − λ)x0 + λt x0 ∈ k} is bounded. let α = inf a and β = sup a. let a = (1 − α)x o + αt x o and b = (1 − β)x o + βt x o . therefore, there exists a sequence {α n } ∈ a such that {α n } converges to α. hence (1 − α n )x o + α n t x o converges to (1 − α)x o + αt x o . therefore it is easy to see that a ∈ g xo and b ∈ g xo . hence g xo = {(1−λ)a + λb : 0 ≤ λ ≤ 1}. now, define a map g : g x0 −→ g x0 by g(z) =    x0 if z ≤ x0, z if z ∈ (x0, t x0), t x0 if z ≥ t x0, . since g and t are continuous, got :< x0, t x0 >−→< x0, t x0 > is also continuous. hence the map got has a fixed point z0 ∈< x0, t x0 > . case 1: z0 = x0 then x0 = z0 = got (z0) = got (x0) = g(t x0) = t x0. hence x0 = t x0, contradicting our assumption. case 2: z0 ∈ (x0, t x0). if t z0 ≤ x0, then z0 = (got )(z0) = g(t z0) = x0, contradicting z0 ∈ (x0, t x0]. if t z0 ∈< x0, t x0 >, then z0 = (got )(z0) = g(t z0) = t z0, again contradicting our assumption, < z0, t z0 > is not invariant under t . therefore, (2.1) t z0 ≥ t x0 a note on a fixed point theorem for ray oriented maps 187 that is , (2.2) z0 = got (z0) = g(t z0) = t x0 substituting (2.2) in (2.1) we get (2.3) t 2x0 ≥ t x0 now let us construct b = {x ∈ r x0 ⋂ k : x < t x < t 2x}. moreover it is bounded above and so it must have a least upper bound. therefore let u be the least upper bound of b. then there exists x n ∈ b such that x n → u. suppose t u < u, then there exists a positive integer n such that for all n ≥ n , x n ∈< u, t u > . then since t | < u, t u > is continuous, t x n → t u. since x n < t x n , u ≤ t u, a contradiction. therefore, u ≤ t u. since t | < u, t u > is not invariant, by 2.3 we have t 2u ≥ t u. therefore, u < t u < t 2u . hence u ∈ b. but again, t 3u ≥ t 2u. therefore, u < t u < t 2u < t 3u. hence t u ∈ b, which is a contradiction. therefore there exists a y0 ∈ rx0 ⋂ k such that t | < y0, t y0 > is invariant. hence t has a fixed point in < y0, t y0 > . � corollary 2.2. suppose t : [a, b] → [a, b] is a mapping where a, b ∈ ℜ. for each element x ∈ [a, b], t | < x, t x > is continuous. then t has a fixed point in [a, b]. remark 2.3. there exists a discontinuous mapping t satisfying the conditions of corollary 2.2. ( t : [0, 1] → [0, 1] by t (0) = 0 and t (x) = 1 for 0 < x ≤ 1). acknowledgement the authors are indebted to dr. s. romaguera for his valuable comments on this paper. moreover, the authors are grateful to dr. p. veeramani, department of mathematics, indian institute of technology madras, chennai 600 036 (india) and dr. s. somasundaram, department of mathematics, manonmaniam sundaranar university, tirunelveli 627 012 (india) for the knowledge they have acquired from the meaningful discussions with them. 188 a. arockiasamy and a. anthony eldred references [1] a. f. beardon, contractions of the real line, the mathematical association of america, monthly113(june-july 2006), 557-558. [2] m. a. khamsi and w. a. kirk, an introduction to metric spaces and fixed point theory, a wiley-interscience publication, 2001. received february 2007 accepted october 2007 a. arockiasamy (arock − sj@rediffmail.com) department of mathematics, st.xavier’s college, palayamkottai, tirunelveli627 002, india. anthony eldred department of mathematics,indian institute of technology madras, chennai600 036, india. @ appl. gen. topol. 23, no. 2 (2022), 325-331 doi:10.4995/agt.2022.17006 © agt, upv, 2022 results about s2-paracompactness ohud alghamdi a and lutfi kalantan b a department of mathematics, faculty of science and arts in almandaq, al-baha univesity, p.o.box 1988, al-baha 65581, saudi arabia (ofalghamdi@bu.edu.sa) b department of mathematics, king abdulaziz university, saudi arabia (lkalantan@kau.edu.sa and lnkalantan@hotmail.com) communicated by m. a. sánchez-granero abstract we present new results regarding s2-paracompactness, that we established in [1], and its relation with other properties such as s-normality, epinormality and l-paracompactness. 2020 msc: 54c10; 54d20. keywords: separable; paracompact; s-paracompact; s2-paracompact; lparacompact; l2-paracompact; s-normal; l-normal. 1. introduction in this paper, we present some new results about s-paracompactness and s2-paracompactness. first, we introduce the significant notations. an order pair will be denoted by 〈x,y〉. the sets of positive integers, rational numbers, irrational numbers and real numbers will be denoted by n,q,p and r, respectively. the closure and the interior of the subset a of a topological space x will be denoted respectively by a and int(a). throughout this paper, a t1 normal space is called t4 and a t1 completely regular space is called tychonoff space (t3 1 2 ). in the definitions of compactness, countable compactness, paracompactness, and local compactness we do not assume t2. moreover, in the definition of lindelöfness we do not assume regularity. also, the ordinal γ is the set of all ordinal α such that α < γ. we denote the first infinite ordinal by ω and the first uncountable ordinal by ω1. received 13 january 2022 – accepted 30 august 2022 http://dx.doi.org/10.4995/agt.2022.17006 o. alghamdi and l. kalantan definition 1.1. a topological space x is called s-paracompact if there exist a paracompact space y and a bijective function f : x −→ y such that for every separable subspace a ⊆ x we have that f |a: a −→ f(a) is a homeomorphism. moreover, if y is t2 paracompact, then x is s2-paracompact [1]. 2. s2-paracompactness and other topological properties 2.1. s2-paracompactness and l2-paracompactness. recall from [5] that a topological space x is called l-paracompact if there exist a paracompact space y and a bijective function f : x −→ y such that f |b: b −→ f(b) is a homeomorphism for all lindelöf subspace b of x. in addition, if y is t2 paracompact, then x is l2-paracompact. recall from [6] that a topological space x is called p-space if it is t1 and every gδ set is open. the countable complement topology defined on r, (r,cc) (see [9, example 20]), is an example of a space that is s2-paracompact but not l2-paracompact. it is s2-paracompact because it is p-space, (see [1]), but not l2-paracompact because it is lindelöf and not paracompact space. in fact, it is not even l-paracompact. we still do not have an answer for the following question: does there exist an l-paracompact space which is not s-paracompact? theorem 2.1. if x is l-paracompact (resp. l2-paracompact) such that for any separable subspace a ⊆ x there exists a lindelöf subspace b ⊆ x such that a ⊆ b, then x is s-paracompact (resp. s2-paracompact). proof. let x be l-paracompact such that for any separable subspace a ⊆ x there exists a lindelöf subspace b ⊆ x such that a ⊆ b. then, there exist a paracompact space y and a bijective function f : x −→ y such that f |b: b −→ f(b) is a homeomorphism for every lindelöf subspace b ⊆ x. let a be any separable subspace of x. then, there exists a lindelöf subspace b of x such that a ⊆ b. then, f |a: a −→ f(a) is a homeomorphism. � a similar proof as in theorem 2.1 yields the following corollaries. corollary 2.2. if x is s-paracompact (resp. s2-paracompact) such that for any lindelöf subspace b ⊆ x, there exists a separable subspace a with b ⊆ a. then, x is l-paracompact (resp. l2-paracompact). recall from [7] that a space x is called c-paracompact if there exist a paracompact space y and a bijection f : x −→ y such that f |k: k −→ f(k) is a homeomorphism for every compact subspace k ⊆ x. moreover, if y is t2 paracompact, we say that x is c2-paracompact. corollary 2.3. if x is s-paracompact (resp. s2-paracompact) such that for any compact subspace b ⊆ x, there exists a separable subspace a of x with a ⊆ b. then, x is c-paracompact (resp. c2-paracompact). © agt, upv, 2022 appl. gen. topol. 23, no. 2 326 results about s2-paracompactness application of corollary 2.3: take (r,cc), the countable complement topology on r. since a ⊂ r is compact if and only if a is finite, we can say that every compact subspace is contained in a separable subspace of r. thus, (r,cc) is c2-paracompact. recall from [3, 4.4.f] that a space x is locally separable if each element of x has a separable open neighborhood. theorem 2.4. every s-paracompact (resp. s2-paracompact) hereditarly locally separable is l-paracompact (resp. l2-paracompact). proof. let x be s-paracompact (resp. s2-paracompact) and hereditarly locally separable and let b be any lindelöf subspace of x. then, b is a locally separable lindelöf subspace of x. pick ux to be a separable open neighborhood of each x ∈ b. then, {ux}x∈b is an open cover of b. let u be a countable open subcover of {ux}x∈b and let dx be a countable dense subset of each ux ∈ u. then, d = ⋃ dx is a countable dense subset of b, implying that b is separable. therefore, since every lindelöf subspace of x is separable and x is s-paracompact (resp. s2-paracompact), then x is l-paracompact (resp. l2-paracompact). � problem 2.5. does there exist a topological space that is l-paracompact (resp. l2-paracompact) but not locally separable or not s-paracompact (resp. s2paracompact)? note that local separability is essential in theorem 2.4. for example, (r,cc) is s-paracompact not locally separable. observe that (r,cc) is not l-paracompact. theorem 2.6. let x be a topological space such that the only separable or lindelöf subspaces are the countable ones. then, x is s-paracompact (resp. s2-paracompact) if and only if x is l-paracompact (resp. l2-paracompact). proof. let x be any topological space such that the only separable or lindelöf subspaces are the countable ones. suppose that x is s-paracompact (resp. s2paracompact). if b is any lindelöf subspace of x, then b is countable, implying that b is separable. hence, x is l-paracompact (resp. l2-paracompact). conversely, suppose that x is l-paracompact (resp. l2-paracompact) and a is any separable subspace of x. then, a is countable, implying that a is lindelöf. hence, x is s-paracompact (resp. s2-paracompact). � application of theorem 2.6: consider ω1 with its usual ordered topology. let a be any uncountable subset of ω1. then a is not bounded. hence, {[0,α] : α < ω1} is an open cover of a that has no countable subcover, which implies that a is not lindelöf. since ω1 satisfies the condition in theorem 2.6, then ω1 is l2-paracompact because it is s2-paracompact, (see [1]). © agt, upv, 2022 appl. gen. topol. 23, no. 2 327 o. alghamdi and l. kalantan a family {as}s∈s of subsets of a space x is called point-finite if for each x ∈ x, the set {s ∈ s : x ∈ as} is finite, (see [3]). recall from [9] that a space x is metacompact if every open cover of x has a point-finite open refinement. theorem 2.7. any hereditarly metacompact l-paracompact (resp. l2-paracompact) is s-paracompact (resp. s2-paracompact). proof. let x be l-paracompact (resp. l2-paracompact) hereditarly metacompact and let a be any separable subspace of x. then, a is a separable metacompact subspace of x. suppose that a is not lindelöf. then, there exists an open cover of a, say w = {wα : α ∈ λ}, which has no countable subcover. let u be a point-finite open refinement of w. then, u is uncountable by our assumption. let d be the countable dense subset of a. hence, d∩u 6= ∅ for all u ∈u implying that there exists d ∈ d contained in uncountable members of u which contradicts the fact that u is a point-finite family. hence, a is lindelöf, implying that x is s-paracompact (resp. s2-paracompact). � problem 2.8. does there exist a topological space which is s-paracompact (resp. s2-paracompact) but not hereditarly metacompact or l-paracompact (resp. l2-paracompact)? 2.2. s2-paracompactness and epinormality. definition 2.9. a topological space (x,τ) is epinormal if there exists a coarser topology, say v, such that (x,v) is t4, (see [2]). since every epinormal space is hausdorff as it is proved in [2], then the countable complement topology on r, (r,cc), is an example of s2-paracompact that is not epinormal. on the other hand, the following example shows that there exists an epinormal space which is not s2-paracompact. example 2.10. let a = {〈x, 0〉 : 0 < x ≤ 1} and b = {〈x, 1〉 : 0 ≤ x < 1}. figure 1. this figure illustrates the neighborhood system of strong parallel line topology (x,σ). © agt, upv, 2022 appl. gen. topol. 23, no. 2 328 results about s2-paracompactness the strong parallel line topology σ on x = a ∪ b is the unique topology generated by the following neighborhood system: for each 〈t, 0〉 ∈ a, let b(〈t, 0〉) = {u : u = {〈x, 0〉 : 0 ≤ a < x ≤ t}∪{〈x, 1〉 : a < x < t}}, and for each 〈t, 1〉 ∈ b, let b(〈t, 1〉) = {v : v = {〈x, 1〉 : t ≤ x < b ≤ 1}}, (see [9, example 96]). since (x,σ) is separable and not paracompact space because it is a hausdorff and not regular topological space, then (x,σ) cannot be s2-paracompact. define τ on x to be the unique topology that is generated by the following neighborhood system: every element in a has the same local base as σ and for each element 〈t, 1〉 ∈ b, let b(〈t, 1〉) = {v : v = {〈x, 0〉 : t < x < b ≤ 1}∪{〈x, 1〉 : t ≤ x < b}}. the topology (x,τ) is named weak parallel line, (see [9, example 96]). figure 2. this figure illustrates the neighborhood system of weak parallel line topology (x,τ). define a relation � on x as follows: for 〈x,y〉 and 〈k,l〉 ∈ x, we write 〈x,y〉 � 〈k,l〉 if and only if either x < k or x = k and y = 0 < l = 1, or x = k and y = l. then, (x,τ) is a linearly ordered topological space (lots). since any lots is t4, we have (x,τ) is t4 and τ is coarser than σ, hence, we get that (x,σ) is epinormal. theorem 2.11. any s2-paracompact fréchet space is epinormal. proof. let (x,τ) be s2-paracompact fréchet space. without loss of generality, assume that (x,τ) is not normal. let (y, τ ′) be a t2 paracompact space and let f : x −→ y be a bijective function such that f |a: a −→ f(a) is a homeomorphism for every separable subspace a ⊆ x. then, f is continuous since x is fréchet. consider v = {f−1(u) : u ∈ τ ′ }. then, v is a topology on x and since any open set in v is open in τ by continuity of f, we get that v is coarser than τ. observe that f : (x,v) −→ (y, τ ′) is a homeomorphism. therefore, (x,v) is t2 paracompact and, hence, t4. � for the converse of theorem 2.11, we have the left ray topological space defined on r, (r,l), as an example of epinormal fréchet space that is not s2-paracompat since it is separable and not paracompact space. © agt, upv, 2022 appl. gen. topol. 23, no. 2 329 o. alghamdi and l. kalantan 2.3. s2-paracompactness and s-normality. recall that a topological space x is s-normal if there exist a normal space y and a bijective function f : x −→ y such that f |a: a −→ f(a) is a homeomorphism for each separable subspace a of x, (see [4]). from the definition of s-normality, it is clear that any s2-paracompact is s-normal. however, we show in the following example that this relation is not reversible. example 2.12. an example of a t4 topological space that is s-normal but not s2-paracompact is the sigma product σ(0) as a subspace of 2 ω1 , where 2 = {0, 1} considered with the discrete topology. it is not s2-paracompact since it cannot be condensed onto a t2 paracompact space, (see [8]). theorem 2.13. let x be fréchet and lindelöf space such that any finite subspace of x is discrete. x is s-normal if and only if x is s2-paracompact. proof. let y be a normal space and let f : x −→ y be a bijective function such that f |a: a −→ f(a) is a homeomorphism for each separable subspace a of x. without loss of generality, let x have more than one element. thus, y is t1 since any finite subspace of x is separable and discrete. by continuity of f and since x is lindelöf, then y is lindelöf. since y is t3 lindelöf, then y is t2 paracompact. thus, x is s2-paracompact. conversely, assume that x is s2 paracompact. let y be a t2 paracompact space and let f : x −→ y be a bijective function such that f |a: a −→ f(a) is a homeomorphism for each separable subspace a of x. hence, since y is t2 paracompact, then y is t4. therefore, x is s-normal. � recall that from [3] that a topological space x is locally metrizable if there exists a metrizable open nighborhood for each x ∈ x.. theorem 2.14. if x is hereditary lindelöf, s2-paracompact and locally metrizable, then x is t2 paracompact and, hence, t4. proof. set ux to be a metrizable open nieghborhood of each x ∈ x. since x is lindelöf, then there exists a countable set e such that x ⊆ ⋂ x∈e ux. now, since x is hereditary lindelöf, then ux is lindelöf as a subspace of x for every x ∈ x. hence, ux is separable being lindelöf and metrizable for every x ∈ x. since x is s2-paracompact and separable, then x is t2 paracompact and, hence, t4. � let (x,τ) be a topological space and let m be a proper nonempty subset of x. the discrete extension of the topological space (x,τ) is defined by the following neighborhood system: for each x ∈ x \m, let b(x) = {{x}} and for each x ∈ m, let b(x) = {u ∈τ: x ∈ u}. we denote the discrete extension of x by xm , (see [9]). © agt, upv, 2022 appl. gen. topol. 23, no. 2 330 results about s2-paracompactness the following example shows that the discrete extension of s2-paracompact need not to be s2-paracompact. example 2.15. consider (r,rs), the rational sequence topology on r, (see [9, example 65]). since rs is separable and not paracompact, then it is not s2-paracompact. also, because it is a tychonoff locally compact space, we can set x = r ∪{p} to be the one point compactification of it. x is t2 compact, which implies that x is s2-paracompact. consider xr, the discrete extension of x. since {p} is closed and open subset in xr, then r is a closed subspace of xr. however, since (r,rs) is not normal, we conclude that xr cannot be normal. since xr is t2 and not normal space, then xr is not paracompact. since xr is separable as q∪{p} is a countable dense subset of xr, then xr is not s2-paracompact. references [1] o. alghamdi, l. kalantan and w. alagal, s-paracompactness and s2-paracompactness, filomat 33, no. 17 (2019), 5645–5650. [2] s. alzharani and l. kalantan, epinormality, j. nonlinear sci. appl. 9 (2016), 5398–5402. [3] r. engelking, general topology, pwn, warszawa, 1977. [4] l. kalantan and m. alhomieyed, s-normality, j. math. anal. 9, no. 5 (2018), 48–54. [5] l. kalantan, l-paracompactness and l2-paracompactness, hacet. j. math. stat. 48, no. 3 (2019), 779–784. [6] a. k. misra, a topological view of p -spaces, gen. topol. appl. 2 (1972), 349–362. [7] m. m. saeed, l. kalantan and h. alzumi, c-paracompactness and c2-paracompactness, turk. j. math. 43 (2019), 9–20. [8] m. m. saeed, l. kalantan and h. alzumi, result about c2-paracompactness, eur. j. pure appl. math. 14, no. 2 (2021), 351–357. [9] l. steen and j. a. seebach, counterexamples in topology, dover publications, 1995. [10] v. zaitsev, on certain classes of topological spaces and their bicompactification, dokl. akad. nauk sssr 178 (1968), 778–779. © agt, upv, 2022 appl. gen. topol. 23, no. 2 331 @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 193-211 universal elements for some classes of spaces d. n. georgiou, s. d. iliadis and a. c. megaritis∗ abstract in the paper [4] two dimensions, denoted by dm and dm, are defined in the class of all hausdorff spaces. the dimension dm does not have the universality property in the class of separable metrizable spaces because the family of all such spaces x with dm(x) ≤ 0 coincides with the family of all totally disconnected spaces in which there are no universal elements (see [5]). in [3] we gave the dimension-like functions dm ik,ib ie and dm ik,ib ie , where ik is a class of subsets, ie a class of spaces and ib a class of bases and we proved that in the families ip(dm ik,ib ie ≤ κ) and ip(dm ik,ib ie ≤ κ) of all spaces x for which dm ik,ib ie (x) ≤ κ and dm ik,ib ie (x) ≤ κ, respectively there exist universal elements. in this paper, we give some new dimension-like functions and define using these definitions classes of spaces in which there are universal elements. 2010 msc: 54b99, 54c25. keywords: dimension-like function, saturated class, universal space. 1. introduction and preliminaries agreement. all spaces are assumed to be t0-spaces of weight ≤ τ, where τ is a fixed infinite cardinal. the set of all finite subsets of τ is denoted by f and the first infinite cardinal is denoted by ω. the cardinality of a set x is denoted by |x|. the class of all ordinals is denoted by o. we also consider two symbols: −1 and ∞. it is assumed that −1 < α < ∞ for every α ∈ o. in the proof of the main results of this paper widely we use notions and notations from [2]. for this reason we start given some of them. ∗work supported by the caratheodory programme of the university of patras. 194 d. n. georgiou, s. d. iliadis and a. c. megaritis we shall use the symbol “ ≡ ” in order to introduce new notations without mention this fact. if “ ∼ ” is an equivalence relation on a non-empty set x, then the set of all equivalence classes of ∼ is denoted by c(∼). let s be an indexed collection of spaces. an indexed collection m ≡ {{uxδ : δ ∈ τ} : x ∈ s} (1) where {uxδ : δ ∈ τ} is an indexed base for x, is called a co-mark of s. the co-mark m of s is said to be a co-extension of a co-mark m + ≡ {{v xδ : δ ∈ τ} : x ∈ s} of s if there exists a one-to-one mapping θ of τ into itself such that for every x ∈ s and for every δ ∈ τ, v xδ = u x θ(δ). the corresponding mapping θ is called an indicial mapping from m+ to m. let r1 ≡ {∼ s 1: s ∈ f} and r0 ≡ {∼ s 0: s ∈ f} be two indexed families of equivalence relations on s. it is said that r1 is a final refinement of r0 if for every s ∈ f there exists t ∈ f such that ∼ t 1⊆∼ s 0. an indexed family r ≡ {∼s: s ∈ f} of equivalence relations on s is said to be admissible if the following conditions are satisfied: (a) ∼∅= s × s, (b) for every s ∈ f the number of ∼s-equivalence classes is finite, and (c) ∼s ⊆ ∼t, if t ⊆ s. we denote by c(r) the set ∪{c(∼s) : s ∈ f}. the minimal ring of subsets of s containing c(r) is denoted by c♦(r). consider the co-mark (1) of s. we denote by rm ≡ {∼ s m: s ∈ f} the indexed family of equivalence relations ∼sm on s defined as follows: for every x, y ∈ s we set x ∼sm y if and only if there exists an isomorphism i of the algebra of subsets of x generated by the set {uxδ : δ ∈ s} onto the algebra of subsets of y generated by the set {uyδ : δ ∈ s} such that i(u x δ ) = u y δ , for every δ ∈ s. also, we set ∼∅ m = s × s. an admissible family r of equivalence relations on s is said to be m-admissible if r is a final refinement of rm. let r ≡ {∼s: s ∈ f} be an m-admissible family of equivalence relations on s. on the set of all pairs (x, x), where x ∈ s and x ∈ x, we consider an equivalence relation, denoted by ∼mr , as follows: (x, x) ∼ m r (y, y ) if and only if x ∼s y for every s ∈ f, and either x ∈ uxδ and y ∈ u y δ or x /∈ u x δ and y /∈ uyδ for every δ ∈ τ. the set of all equivalence classes of the relation ∼ m r is denoted by t(m, r) or simply by t. for every h ∈ c♦(r) the set of all a ∈ t(m, r) for which there exists an element (x, x) ∈ a such that x ∈ h is denoted by t(h). for every δ ∈ τ and h ∈ c♦(r) we denote by utδ (h) the set of all a ∈ t(m, r) for which there exists an element (x, x) ∈ a such that x ∈ h and x ∈ uxδ . universal elements for some classes of spaces 195 for every subset κ of τ and l ∈ c♦(r) we set (1) bt ♦ ≡ {utδ (h) : δ ∈ τ and h ∈ c ♦(r)}. (2) bt ♦,κ ≡ {u t δ (h) : δ ∈ κ and h ∈ c ♦(r)}. (3) bl ♦,κ ≡ {u t δ (h) ∈ b t ♦,κ : h ⊆ l}. under some simple (set-theoretical) conditions on r the set bt♦ is a base for a topology on the set t(m, r) such that the corresponding space is a t0-space of weight ≤ τ. moreover, if for every x ∈ s the set {uxδ : δ ∈ κ} is a base for x, then the set bt♦,κ is a base for the same topology on t(m, r). therefore, the family bl♦,κ is a base for t(l). (see corollary 1.2.8 and proposition 1.2.9 in [2]). for every element x of s there exists a natural embedding ixt of x into the space t(m, r) defined as follows: for every x ∈ x, ixt (x) = a, where a is the element of t(m, r) containing the pair (x, x). thus, we have constructed containing space t(m, r) for s of weight ≤ τ. suppose that for every x ∈ s a subset qx of x is given. the set q ≡ {qx : x ∈ s} (2) is called a restriction of s. let if be a class of subsets. a restriction q of an indexed collection s of spaces is said to be a if-restriction if (qx, x) ∈ if for every x ∈ s. consider the restriction (2) of s. the trace on q of the co-mark m of s is the co-mark m|q ≡ {{u x δ ∩ q x : δ ∈ τ} : qx ∈ q} of q. the trace on q of an equivalence relation ∼ on s is the equivalence relation on q denoted by ∼|q and defined as follows: q x∼|qq y if and only if x ∼ y . let r ≡ {∼s: s ∈ f} be an indexed family of equivalence relations on s. the trace on q of the family r is the family r|q ≡ {∼ s|q : s ∈ f} of equivalence relations on q. the trace on q of an element h of c♦(r) is the element h|q ≡ {q x ∈ q : x ∈ h} of c♦(r|q). the m-admissible family r of equivalence relations on s is said to be (m, q)admissible if r|q is an m|q-admissible family of equivalence relations on q. if r is an (m, q)-admissible family of equivalence relations on s, then we can consider the containing space t(m|q, r|q) for the indexed collection q corresponding to the co-mark m|q and the m|q-admissible family r|q. the containing space t(m|q, r|q) is denoted briefly by t|q. there exists a natural embedding of t(m|q, r|q) into t(m, r). so we can consider the containing space t(m|q, r|q) as a subspace of the space t(m, r). the subsets of this form will be called specific subsets of t(m, r). a class ip of spaces is said to be saturated if for every indexed collection s of spaces belonging to ip there exists a co-mark m+ of s satisfying the following 196 d. n. georgiou, s. d. iliadis and a. c. megaritis condition: for every co-extension m of m+ there exists an m-admissible family r+ of equivalence relations on s such that for every admissible family r of equivalence relations on s, which is a final refinement of r+, and for every l ∈ c♦(r) the space t(l) belongs to ip. the co-mark m+ is said to be an initial co-mark of s corresponding to the class ip and the family r is said to be an initial family of s corresponding to the co-mark m and the class ip. agreement. in what follows we denote by ν a fixed cardinal greater than ω and less than or equal to τ. notation. for every dimension-like function dfν, with as domain the class of all spaces and as range the class o ∪ {−1, ∞}; and for every α ∈ {−1} ∪ o, we denote by ip(dfν ≤ α) the class of all spaces x with dfν(x) ≤ α. 2. the dimension-like functions: dm ik,ib ie,ν and dm ik,ib ie,ν in this section we give some new dimension-like functions and define using these definitions classes of spaces in which there are universal elements. the proofs of these results are similar to the proofs of the results in [3], for this reason are omitted. definition 2.1 (see [1]). let a and b be two disjoint subsets of a space x. we say that a subset l of x separates a and b if there exist two open subsets u and w of x such that: (a) a ⊆ u, b ⊆ w , (b) u ∩ w = ∅, and (c) x \ l = u ∪ w . definition 2.2 (see [3]). a class ie of spaces is said to be ib-hereditaryseparated, where ib is a class of bases, if for every element x of ie there exists a ib-base bx ≡ {uδ : δ ∈ τ} for x such that for every two elements uδ1 and uδ2 of b x with cl(uδ1) ∩ cl(uδ2) = ∅ there exists a subset l of x separating the sets cl(uδ1) and cl(uδ2) and belonging to ie. we note that if ie is ib-hereditary-separated, then ∅ ∈ ie. this follows by the fact that the empty set is the unique subset of x separating the elements ∅ and x of bx. definition 2.3. let ib be a class of bases, ie a ib-hereditary-separated class of spaces, and ik a class of subsets with (x, x) ∈ ik for every space x. we denote by dm ik,ib ie,ν and dm ik,ib ie,ν the dimension-like functions with as domain the class of all spaces and as range the class o ∪ {−1, ∞} satisfying the following conditions: (1) dm ik,ib ie,ν (x) = dm ik,ib ie,ν (x) = −1 if and only if x ∈ ie. (2) dm ik,ib ie,ν (x) ≤ α, α ∈ o, if and only if there exists a ib-base b x ≡ {uδ : δ ∈ τ} for x such that for every two elements uδ1, uδ2 of b x with cl(uδ1) ∩ cl(uδ2) = ∅ there exists a subset l of x separating cl(uδ1) and cl(uδ2) with dm ik,ib ie,ν (l) < α. universal elements for some classes of spaces 197 (3) dm ik,ib ie,ν (x) ≤ α, α ∈ o, if and only if x = ∪{si : i ∈ ν} such that: (a) the subset si of x is closed, (b) (si, x) ∈ ik, and (c) dm ik,ib ie,ν (si) ≤ α, i ∈ ν. therefore, dm ik,ib ie,ν (x) = ∞ (respectively, dm ik,ib ie,ν (x) = ∞) if and only if the inequality dm ik,ib ie,ν (x) ≤ α (respectively, dm ik,ib ie,ν (x) ≤ α) is not true for every α ∈ o. remark 2.4. (1) in order that the above definition to be well defined we need to show that if for a space x we have dm ik,ib ie,ν (x) = dm ik,ib ie,ν (x) = −1, then dm ik,ib ie,ν (x) ≤ 0 and dm ik,ib ie,ν (x) ≤ 0. for dimension-like function dm ik,ib ie,ν this follows immediately by the fact that x ∈ ie and the class ie is ib-hereditary-separated. for dimension-like function dm ik,ib ie,ν , we have (a) x = {si : i ∈ ν}, where si = x, (b) (x, x) ∈ ik, and (c) dm ik,ib ie,ν (x) = −1 ≤ 0, which means that dm ik,ib ie,ν (x) ≤ 0. (2) for ν = ω the dimension-like functions dm ik,ib ie,ν and dm ik,ib ie,ν coincide with the dimension-like functions dm ik,ib ie and dm ik,ib ie , respectively which are defined in [3]. proposition 2.5. for every space x we have dm ik,ib ie,ν (x) ≤ dm ik,ib ie,ν (x). proposition 2.6. for every space x, dm ik,ib ie,ν (x) ∈ {−1, ∞}∪τ + and, therefore, dm ik,ib ie,ν (x) ∈ {−1, ∞} ∪ τ +. theorem 2.7. let ib be a saturated class of bases, ie a saturated ib-hereditaryseparated class of spaces, and ik a saturated class of subsets with (x, x) ∈ ik for every space x. then, for every κ ∈ {−1} ∪ ω the classes ip(dm ik,ib ie,ν ≤ κ) and ip(dm ik,ib ie,ν ≤ κ) are saturated. corollary 2.8. for every κ ∈ ω in the classes ip(dm ik,ib ie,ν ≤ κ) and ip(dm ik,ib ie,ν ≤ κ) there exist universal elements. corollary 2.9. let ip be one of the following classes (a) the class of all (completely) regular spaces of weight ≤ τ, (b) the class of all (completely) regular countable-dimensi-onal spaces of weight ≤ τ, (c) the class of all (completely) regular strongly countable-dimensional spaces of weight ≤ τ, (d) the class of all (completely) regular locally finite-dimensional spaces of weight ≤ τ, and 198 d. n. georgiou, s. d. iliadis and a. c. megaritis (e) the class of all (completely) regular spaces x of weight ≤ τ such that ind(x) ≤ α ∈ τ+. then, for every κ ∈ ω in the classes ip(dm ik,ib ie,ν ≤ κ) ∩ ip and ip(dm ik,ib ie,ν ≤ κ) ∩ ip there exist universal elements. 3. the dimension-like functions: w-dm ik,ib ie,ν and w-dm ik,ib ie,ν definition 3.1. a class ie of spaces is said to be ib-weakly-hereditary-separated, where ib is a class of bases, if for every element x of ie there exists a ib-base bx ≡ {uδ : δ ∈ τ} for x such that for every two elements uδ1 and uδ2 of b x with cl(uδ1)∩uδ2 = ∅ there exists a subset l of x separating the sets cl(uδ1) and uδ2 and belonging to ie. we note that if ie is ib-weakly-hereditary-separated, then ∅ ∈ ie. this follows by the fact that the empty set is the unique subset of x separating the elements ∅ and x of bx. definition 3.2. let ib be a class of bases, ie a ib-weakly-hereditary-separated class of spaces, and ik a class of subsets with (x, x) ∈ ik for every space x. we denote by w-dm ik,ib ie,ν and w-dm ik,ib ie,ν the dimension-like functions with as domain the class of all spaces and as range the class o ∪ {−1, ∞} satisfying the following conditions: (1) w-dm ik,ib ie,ν (x)=w-dm ik,ib ie,ν (x) = −1 if and only if x ∈ ie. (2) w-dm ik,ib ie,ν (x) ≤ α, where α ∈ o, if and only if there exists a ib-base bx ≡ {uδ : δ ∈ τ} for x such that for every two elements uδ1, uδ2 of bx with cl(uδ1) ∩ uδ2 = ∅ there exists a subset l of x separating cl(uδ1) and uδ2 with w-dm ik,ib ie,ν (l) < α. (3) w-dm ik,ib ie,ν (x) ≤ α, α ∈ o, if and only if x = ∪{si : i ∈ ν} such that: (a) the subset si of x is closed, (b) (si, x) ∈ ik, and (c) wdm ik,ib ie,ν (si) ≤ α, i ∈ ν. therefore, w-dm ik,ib ie,ν (x) = ∞ (respectively, w-dm ik,ib ie,ν (x) = ∞) if and only if the inequality w-dm ik,ib ie,ν (x) ≤ α (respectively, w-dm ik,ib ie,ν (x) ≤ α) is not true for every α ∈ o. remark 3.3. in order that the above definition to be well defined we need to show that if for a space x we have w-dm ik,ib ie,ν (x)=w-dm ik,ib ie,ν (x) = −1, then w-dm ik,ib ie,ν (x) ≤ 0 and w-dm ik,ib ie,ν (x) ≤ 0. for dimension-like function w-dm ik,ib ie,ν this follows immediately by the fact that x ∈ ie and the class ie is ib-weakly-hereditary-separated. for dimension-like function w-dm ik,ib ie,ν , we have (a) x = {si : i ∈ ν}, where si = x, (b) (x, x) ∈ ik, and (c) w-dm ik,ib ie,ν (x) = −1 ≤ 0, which means that w-dm ik,ib ie,ν (x) ≤ 0. universal elements for some classes of spaces 199 proposition 3.4. for every space x we have w-dm ik,ib ie,ν (x) ≤ w-dm ik,ib ie,ν (x). (3) proof. let w-dm ik,ib ie,ν (x) = α ∈ {−1, ∞} ∪ o. the inequality (3) is clear if α = −1 or α = ∞. suppose that α ∈ o. we have x = ∪{si : i ∈ ν}, where si = x. since (si, x) = (x, x) ∈ ik and w-dm ik,ib ie,ν (si) = w-dm ik,ib ie,ν (x) ≤ α, the condition (3) of definition 3.2 implies that w-dm ik,ib ie,ν (x) ≤ α. � proposition 3.5. for every space x, w-dm ik,ib ie,ν (x) ∈ {−1, ∞} ∪ τ +, and, therefore w-dm ik,ib ie,ν (x) ∈ {−1, ∞} ∪ τ +. proof. suppose that the proposition is not true. let α be the minimal element of o \ τ+ such that there exists a space x with w-dm ik,ib ie,ν (x) = α. let bx = {uδ : δ ∈ τ} be the ib-base for x mentioned in condition (2) of definition 3.2. denote by p the set of all pairs (δ1, δ2) ∈ τ × τ with cl(uδ1) ∩ uδ2 = ∅. for every (δ1, δ2) ∈ p let l(δ1, δ2) be a subset of x separating the sets cl(uδ1) and uδ2 with w-dm ik,ib ie,ν (l(δ1, δ2)) = β(δ1, δ2) < α. first we suppose that β(δ1, δ2) < τ + for every (δ1, δ2) ∈ p . since |p | ≤ τ there exists an ordinal β ∈ τ+ such that β(δ1, δ2) < β for every (δ1, δ2) ∈ p . then, w-dm ik,ib ie,ν (l(δ1, δ2)) < β and, by condition (2) of definition 3.2, w-dm ik,ib ie,ν (x) ≤ β, which is a contradiction. now, we suppose that there exists (δ1, δ2) ∈ p such that τ + ≤ β(δ1, δ2). since w-dm ik,ib ie,ν (l(δ1, δ2)) = β(δ1, δ2), there exist closed subsets s l(δ1,δ2) i of l(δ1, δ2), i ∈ ν, such that: (a) l(δ1, δ2) = ∪{s l(δ1,δ2) i : i ∈ ν}, (b) (s l(δ1,δ2) i , l(δ1, δ2)) ∈ ik, and (c) w-dm ik,ib ie,ν (s l(δ1,δ2) i ) = βi ≤ β(δ1, δ2) < α. if βi < τ + for all i ∈ ν, then there exists an ordinal β ∈ τ+ such that βi ≤ β, which means that w-dm ik,ib ie,ν (s l(δ1,δ2) i ) ≤ β. therefore, w-dm ik,ib ie,ν (l(δ1, δ2)) ≤ β < τ + ≤ β(δ1, δ2), which is a contradiction. thus, there exists i ∈ ν such that τ+ ≤ w-dm ik,ib ie,ν (s l(δ1,δ2) i ) < α. the last relation contradicts the choice of the ordinal α completing the proof of the proposition. � 200 d. n. georgiou, s. d. iliadis and a. c. megaritis theorem 3.6. let ib be a saturated class of bases, ie a saturated ib-weaklyhereditary-separated class of spaces, and ik a saturated class of subsets such that (x, x) ∈ ik for every space x. then, for every κ ∈ {−1} ∪ ω the classes ip(w-dm ik,ib ie,ν ≤ κ) and ip(w-dm ik,ib ie,ν ≤ κ) are saturated. proof. we prove the theorem by induction on κ. let κ = −1. then, a space x belongs to ip(w-dm ik,ib ie,ν ≤ −1) if and only if x belongs to ie, that is ip(w-dm ik,ib ie,ν ≤ −1) = ie. therefore, ip(w-dm ik,ib ie,ν ≤ −1) is a saturated class of spaces. similarly, the class ip(w-dm ik,ib ie,ν ≤ −1) is saturated. let κ ∈ ω. suppose that the classes ip(w-dm ik,ib ie,ν ≤ m) and ip(w-dm ik,ib ie,ν ≤ m) are saturated, m ∈ {−1}∪κ. we prove that the classes ip(w-dm ik,ib ie,ν ≤ κ) and ip(w-dm ik,ib ie,ν ≤ κ) are also saturated. first we prove that ip(w-dm ik,ib ie,ν ≤ κ) is a saturated class. let s be an indexed collection of elements of ip(w-dm ik,ib ie,ν ≤ κ). for every x ∈ s let bx ≡ {v xε : ε ∈ τ} be an indexed ib-base for x satisfying condition (2) of definition 3.2. then, there exist (a) an indexed set {lxη : η ∈ τ} of subsets of x, (b) two indexed sets {w xη : η ∈ τ} and {o x η : η ∈ τ} of open subsets of x, and (c) a one-to-one mapping ϕ of τ × τ onto τ such that (1) for every ε1, ε2 ∈ τ and η = ϕ(ε1, ε2) we have (d) cl(v xε1 ) ⊆ w x η , v x ε2 ⊆ oxη , (e) w xη ∩ o x η = ∅, and (f) x \ lxη = w x η ∪ o x η , in the case, where cl(v xε1 )∩v x ε2 = ∅, and lxη = ∅ in the case, where cl(v x ε1 )∩ v xε2 6= ∅. (2) for every η ∈ τ, w-dm ik,ib ie,ν (l x η ) ≤ κ − 1. for every η ∈ τ we set lη = {l x η : x ∈ s}, wη = {w x η : x ∈ s}, and oη = {o x η : x ∈ s}. by the above property (2), lη is an indexed collection of elements of the class ipκ−1 ≡ ip(w-dm ik,ib ie,ν ≤ κ − 1). by inductive assumption the class ipκ−1 is saturated. therefore, there exists an initial co-mark m+ lη of lη corresponding to the class ipκ−1. denote by mη a co-mark of s such that its trace on lη is universal elements for some classes of spaces 201 a co-extension of the co-mark m+ lη . the existence of such a co-mark is easily proved. consider the co-indication n ≡ {{v xε : ε ∈ τ} : x ∈ s} of the ib-co-base b ≡ {bx : x ∈ s} of s. since ib is a saturated class of bases there exists an initial co-mark m+ib of s corresponding to the co-indication n of b and the class ib. in particular, m+ib is a co-extension of n. by lemma 2.1.2 of [2], there exists a co-mark m+ of s, which a co-extension of the co-marks m+ib and mη for every η ∈ τ. in particular, m + is a co-extension of n. we show that m+ is an initial co-mark of s corresponding to the class ip(w-dm ik,ib ie,ν ≤ κ). indeed, let m ≡ {{uxδ : δ ∈ τ} : x ∈ s} be an arbitrary co-extension of m+. then, m is a co-extension of the co-marks m + ib, n, and mη for every η ∈ τ. denote by ϑ an indicial mapping from n to m. then, for every x ∈ ie, v xε = u x ϑ(ε), ε ∈ τ. obviously, the co-mark m|lη is a co-extension of the co-mark m+ lη of lη. let r+ib be an initial family of equivalence relations on s corresponding to the co-mark m, the co-indication n of b, and the class ib. let also r+ lη be an initial family of equivalence relations on lη corresponding to the co-mark m|lη and the class ipκ−1. denote by rη the family of equivalence relations on s such that the trace on lη of rη is the family r + lη . by lemma 2.1.1 of [2], there exists an admissible family r+ of equivalence relations on s, which is a final refinement of the families r+ib and rη for every η ∈ τ. in particular, r+ is m-admissible. without loss of generality, we can suppose that r+ is (m, wη)-admissible, (m, oη)-admissible, (m, co(wη))admissible, and (m, co(oη))-admissible. we prove that r + is an initial family of s corresponding to the co-mark m of s and the class ip(w-dm ik,ib ie,ν ≤ κ). for this purpose we consider an arbitrary admissible family r of equivalence relations on s, which is a final refinement of r+, and prove that for every l ∈ c♦(r) the space t(l) belongs to ip(w-dm ik,ib ie,ν ≤ κ). let l ∈ c ♦(r). since ib is a saturated class, we have (bl ♦,ϑ(τ) , t(l)) ∈ ib. we show that the base bl ♦,ϑ(τ) of t(l) satisfies condition (2) of definition 3.2, that is for every utδ1(h1) and u t δ2 (h2) of b l ♦,ϑ(τ) (where h1, h2 ⊆ l), with clt(l)(u t δ1 (h1)) ∩ u t δ2 (h2) = ∅ (4) there exists a subset l of t(l) separating clt(l)(u t δ1 (h1)) and u t δ2 (h2) such that w-dm ik,ib ie,ν (l) ≤ κ − 1. consider two elements utδ1(h1), u t δ2 (h2) of b l ♦,ϑ(τ) satisfying relation (4). first we suppose that h1 ∩ h2 = ∅. then, 202 d. n. georgiou, s. d. iliadis and a. c. megaritis (g) clt(l)(u t δ1 (h1)) ⊆ t(h1), u t δ2 (h2) ⊆ t(l \ h1), (h) t(h1) ∩ t(l \ h1) = ∅, and (i) t(l) = t(h1) ∪ t(l \ h1). therefore, the empty set separates the sets clt(l)(u t δ1 (h1)) and u t δ2 (h2). since w-dm ik,ib ie,ν (∅) = −1 < κ, we have w-dm ik,ib ie,ν (t(l)) ≤ κ. now, we suppose that h1 ∩ h2 6= ∅. let h = h1 ∩ h2, ϑ −1(δ1) = ε1, ϑ−1(δ2) = ε2, and η = ϕ(ε1, ε2). we prove that t(h|lη) separates the sets clt(l)(u t δ1 (h1)) and u t δ2 (h2), and w-dm ik,ib ie,ν (t(h|lη )) ≤ κ − 1 < κ. since ipκ−1 is a saturated class of spaces, the subspace t(h|lη ) of t(m|lη , r|lη) belongs to ipκ−1. hence, w-dm ik,ib ie,ν (t(h|lη )) ≤ κ − 1 < κ. we prove that the subset t(h|lη ) of t(l) separates clt(l)(u t δ1 (h1)) and utδ2(h2). suppose that x ∈ h. since the subsets cl(v x ε1 ) and v xε2 of x are disjoint, by condition (1) we have (k) cl(v xε1 ) ⊆ w x η , v x ε2 ⊆ oxη , (l) w xη ∩ o x η = ∅, and (m) x \ lxη = w x η ∪ o x η . the above relations imply that (n) clt(l)(u t δ1 (h)) ⊆ t(h|wη ) = t|wη ∩ t(h), utδ2(h) ⊆ t(h|oη ) = t|oη ∩ t(h), (o) t(h|wη ) ∩ t(h|oη ) = ∅, and (p) t(h) \ t(h|lη) = t(h|wη ) ∪ t(h|oη ). since the restriction wη of s is open and the family r is (m, co(wη))admissible, by lemma 1.4.7 of [2], the subset t|wη of t is open. similarly, the subset t|oη of t is open. also, since the subset t(h) of t is open and t(h) ⊆ t(l), the sets t(h|wη) and t(h|oη ) are open in t(l). setting w = t(h1 \ h) ∪ t(h|wn) and o = t(l \ h1) ∪ t(h|on) we have (q) clt(l)(u t δ1 (h1)) ⊆ w , u t δ2 (h2) ⊆ o, (r) w ∩ o = ∅, and (s) t(l) \ t(h|lη ) = w ∪ o. therefore, the subset t(h|lη ) of t(l) separates the sets clt(l)(u t δ1 (h1)) and utδ1(h2). thus, the class ip(w − dm ik,ib ie,ν ≤ κ) is saturated. now, we prove that the class ip(w-dm ik,ib ie,ν ≤ κ) is saturated. let s be a indexed collection of elements of ip(w-dm ik,ib ie,ν ≤ κ). for every x ∈ s there exists an indexed set {qxi : i ∈ ν} of subsets of x such that universal elements for some classes of spaces 203 (3) x = ∪{qxi : i ∈ ν}. (4) for every i ∈ ν, the subset qxi of x is closed and (q x i , x) ∈ ik. (5) for every i ∈ ν, w-dm ik,ib ie,ν (q x i ) ≤ κ. we set qi = {q x i : x ∈ s}, i ∈ ν. by the preceding, the class ip ≡ ip(w-dm ik,ib ie,ν ≤ κ) is saturated. by property (5), qi is an indexed collection of elements of the class ip. therefore, there exists an initial co-mark m+ qi of qi corresponding to the class ip. denote by mi a co-mark of s such that its trace on qi is a co-extension of the co-mark m + qi . by property (4), the restriction qi of s is a ik-restriction. since ik is a saturated class of subsets, for every i ∈ ν there exists an initial co-mark m+ik,i of s corresponding to the ik-restriction qi. by lemma 2.1.2 of [2], there exists a co-mark m+ of s, which a co-extension of the co-marks mi and m + ik,i for every i ∈ ν. we show that m + is an initial co-mark of s corresponding to the class ip(w-dm ik,ib ie,ν ≤ κ). indeed, let m ≡ {{uxδ : δ ∈ τ} : x ∈ s} be an arbitrary co-extension of m+. then, m is a co-extension of the co-marks mi and m + ik,i and the co-mark m|qi is a co-extension of the co-mark m + qi of qi, i ∈ ν. let r+ qi be an initial family of equivalence relations on qi corresponding to the co-mark m|qi and the class ip. denote by ri the family of equivalence relations on s such that the trace on qi of ri is the family r + qi . let also r+ik,i be an initial family of equivalence relations on s corresponding to the co-mark m and the ik-restriction qi. by lemma 2.1.1 of [2], there exists an admissible family r+ of equivalence relations on s, which is a final refinement of the families ri and r + ik,i, i ∈ ν. therefore, r+ is an m-admissible family. we prove that r+ is an initial family of s corresponding to the co-mark m of s and the class ip(w-dm ik,ib ie,ν ≤ κ). for this purpose, we consider an arbitrary admissible family r of equivalence relations on s, which is a final refinement of r+. then, r is a final refinement of the families ri and r + ik,i for every i ∈ ν. we need to prove that for every l ∈ c♦(r), t(l) ∈ ip(w-dm ik,ib ie,ν ≤ κ). let l ∈ c♦(r). it suffices to show that t(l) = ∪{ti(l) : i ∈ ν} such that (t) the subset ti(l) of t(l) is closed, (u) (ti(l), t(l)) ∈ ik, and (v) w-dm ik,ib ie,ν (ti(l)) ≤ κ, i ∈ ν. we set ti(l) = t(l|qi ), i ∈ ν. it is easy to verify that the subset t(l|qi ) of t(l) is closed and t(l) = ∪{t(l|qi) : i ∈ ν}. since ik is a saturated class 204 d. n. georgiou, s. d. iliadis and a. c. megaritis of subsets, (t(l|qi ), t(l)) ∈ ik. since ip is a saturated class, the subspace t(l|qi) of t(m|qi, r|qi) belongs to ip. hence, w-dm ik,ib ie,ν (t(l|qi )) ≤ κ. thus, by condition (3) of definition 3.2, w-dm ik,ib ie,ν (t(l)) ≤ κ proving that the class ip(w-dm ik,ib ie,ν ≤ κ) is saturated. � corollary 3.7. for every κ ∈ ω in the classes ip(w-dm ik,ib ie,ν ≤ κ) and ip(w-dm ik,ib ie,ν ≤ κ) there exist universal elements. corollary 3.8. let ip be one of the following classes (a) the class of all (completely) regular spaces of weight ≤ τ, (b) the class of all (completely) regular countable-dimensi-onal spaces of weight ≤ τ, (c) the class of all (completely) regular strongly countable-dimensional spaces of weight ≤ τ, (d) the class of all (completely) regular locally finite-dimensional spaces of weight ≤ τ, and (e) the class of all (completely) regular spaces x of weight ≤ τ such that ind(x) ≤ α ∈ τ+. then, for every κ ∈ ω in the classes ip(w-dm ik,ib ie,ν ≤ κ) ∩ ip and ip(w-dm ik,ib ie,ν ≤ κ) ∩ ip there exist universal elements. 4. the dimension-like functions: s-dm ik,ib ie,ν and s-dm ik,ib ie,ν definition 4.1. a class ie of spaces is said to be ib-strong-hereditary-separated, where ib is a class of bases, if for every element x of ie there exists a ib-base bx ≡ {uδ : δ ∈ τ} for x such that for every two elements uδ1 and uδ2 of b x with uδ1 ∩ uδ2 = ∅ there exists a subset l of x separating the sets uδ1 and uδ2 and belonging to ie. we note that if ie is ib-strong-hereditary-separated, then ∅ ∈ ie. this follows by the fact that the empty set is the unique subset of x separating the elements ∅ and x of bx. definition 4.2. let ib be a class of bases, ie a ib-strong-hereditary-separated class of spaces, and ik a class of subsets with (x, x) ∈ ik for every space x. we denote by s-dm ik,ib ie,ν and s-dm ik,ib ie,ν the dimension-like functions with as domain the class of all spaces and as range the class o ∪ {−1, ∞} satisfying the following conditions: (1) s-dm ik,ib ie,ν (x)=s-dm ik,ib ie,ν (x) = −1 if and only if x ∈ ie. (2) s-dm ik,ib ie,ν (x) ≤ α, where α ∈ o, if and only if there exists a ib-base bx ≡ {uδ : δ ∈ τ} for x such that for every two elements uδ1, uδ2 of universal elements for some classes of spaces 205 bx with uδ1 ∩ uδ2 = ∅ there exists a subset l of x separating uδ1 and uδ2 with s-dm ik,ib ie,ν (l) < α. (3) s-dm ik,ib ie,ν (x) ≤ α, α ∈ o, if and only if x = ∪{si : i ∈ ν} such that: (a) the subset si of x is closed, (b) (si, x) ∈ ik, and (c) sdm ik,ib ie,ν (si) ≤ α, i ∈ ν. therefore, s-dm ik,ib ie,ν (x) = ∞ (respectively, s-dm ik,ib ie,ν (x) = ∞) if and only if the inequality s-dm ik,ib ie,ν (x) ≤ α (respectively, s-dm ik,ib ie,ν (x) ≤ α) is not true for every α ∈ o. remark 4.3. in order that the above definition to be well defined we need to show that if for a space x we have s-dm ik,ib ie,ν (x)=s-dm ik,ib ie,ν (x) = −1, then s-dm ik,ib ie,ν (x) ≤ 0 and s-dm ik,ib ie,ν (x) ≤ 0. for dimension-like function s-dm ik,ib ie,ν this follows immediately by the fact that x ∈ ie and the class ie is ib-strong-hereditary-separated. for dimension-like function s-dm ik,ib ie,ν , we have (a) x = {si : i ∈ ν}, where si = x, (b) (x, x) ∈ ik, and (c) s-dm ik,ib ie,ν (x) = −1 ≤ 0, which means that s-dm ik,ib ie,ν (x) ≤ 0. proposition 4.4. for every space x we have s-dm ik,ib ie,ν (x) ≤ s-dm ik,ib ie,ν (x). (5) proof. let s-dm ik,ib ie,ν (x) = α ∈ {−1, ∞} ∪ o. the inequality (5) is clear if α = −1 or α = ∞. suppose that α ∈ o. we have x = ∪{si : i ∈ ν}, where si = x. since (si, x) = (x, x) ∈ ik and s-dm ik,ib ie,ν (si) = s-dm ik,ib ie,ν (x) ≤ α, the condition (3) of definition 4.2 implies that s-dm ik,ib ie,ν (x) ≤ α. � proposition 4.5. for every space x, s-dm ik,ib ie,ν (x) ∈ {−1, ∞} ∪ τ +, and, therefore s-dm ik,ib ie,ν (x) ∈ {−1, ∞} ∪ τ +. proof. suppose that the proposition is not true. let α be the minimal element of o \ τ+ such that there exists a space x with s-dm ik,ib ie,ν (x) = α. let bx = {uδ : δ ∈ τ} be the ib-base for x mentioned in condition (2) of definition 4.2. denote by p the set of all pairs (δ1, δ2) ∈ τ × τ with uδ1 ∩ uδ2 = ∅. for every (δ1, δ2) ∈ p let l(δ1, δ2) be a subset of x separating the sets uδ1 and uδ2 with s-dm ik,ib ie,ν (l(δ1, δ2)) = β(δ1, δ2) < α. first we suppose that β(δ1, δ2) < τ + for every (δ1, δ2) ∈ p . since |p | ≤ τ there exists an ordinal β ∈ τ+ such that β(δ1, δ2) < β for every (δ1, δ2) ∈ p . then, 206 d. n. georgiou, s. d. iliadis and a. c. megaritis s-dm ik,ib ie,ν (l(δ1, δ2)) < β and, by condition (2) of definition 4.2, s-dm ik,ib ie,ν ≤ β, which is a contradiction. now, we suppose that there exists (δ1, δ2) ∈ p such that τ + ≤ β(δ1, δ2). since s-dm ik,ib ie,ν (l(δ1, δ2)) = β(δ1, δ2), there exist closed subsets s l(δ1,δ2) i of l(δ1, δ2), i ∈ ν, such that: (a) l(δ1, δ2) = ∪{s l(δ1,δ2) i : i ∈ ν}, (b) (s l(δ1,δ2) i , l(δ1, δ2)) ∈ ik, and (c) s-dm ik,ib ie,ν (s l(δ1,δ2) i ) = βi ≤ β(δ1, δ2) < α. if βi < τ + for all i ∈ ν, then there exists an ordinal β ∈ τ+ such that βi ≤ β, which means that s-dm ik,ib ie,ν (s l(δ1,δ2) i ) ≤ β. therefore, s-dm ik,ib ie,ν (l(δ1, δ2)) ≤ β < τ + ≤ β(δ1, δ2), which is a contradiction. thus, there exists i ∈ ν such that τ+ ≤ s-dm ik,ib ie,ν (s l(δ1,δ2) i ) < α. the last relation contradicts the choice of the ordinal α completing the proof of the proposition. � theorem 4.6. let ib be a saturated class of bases, ie a saturated ib-stronghereditary-separated class of spaces, and ik a saturated class of subsets such that (x, x) ∈ ik for every space x. then, for every κ ∈ {−1} ∪ ω the classes ip(s-dm ik,ib ie,ν ≤ κ) and ip(s-dm ik,ib ie,ν ≤ κ) are saturated. proof. we prove the theorem by induction on κ. let κ = −1. then, a space x belongs to ip(s-dm ik,ib ie,ν ≤ −1) if and only if x belongs to ie, that is ip(s-dm ik,ib ie,ν ≤ −1) = ie. therefore, ip(s-dm ik,ib ie,ν ≤ −1) is a saturated class of spaces. similarly, the class ip(s-dm ik,ib ie,ν ≤ −1) is saturated. let κ ∈ ω. suppose that the classes ip(s-dm ik,ib ie,ν ≤ m) and ip(s-dm ik,ib ie,ν ≤ m) are saturated, m ∈ {−1} ∪ κ. we prove that the classes ip(s-dm ik,ib ie,ν ≤ κ) and ip(s-dm ik,ib ie,ν ≤ κ) are also saturated. first we prove that ip(s-dm ik,ib ie,ν ≤ κ) is a saturated class. let s be an indexed collection of elements of ip(s-dm ik,ib ie,ν ≤ κ). for every x ∈ s let bx ≡ {v xε : ε ∈ τ} be an indexed ib-base for x satisfying condition (2) of definition 4.2. then, there exist (a) an indexed set {lxη : η ∈ τ} of subsets of x, (b) two indexed sets {w xη : η ∈ τ} and {o x η : η ∈ τ} of open subsets of x, and (c) a one-to-one mapping ϕ of τ × τ onto τ such that universal elements for some classes of spaces 207 (1) for every ε1, ε2 ∈ τ and η = ϕ(ε1, ε2) we have (d) v xε1 ⊆ w x η , v x ε2 ⊆ oxη , (e) w xη ∩ o x η = ∅, and (f) x \ lxη = w x η ∪ o x η , in the case, where v xε1 ∩v x ε2 = ∅, and lxη = ∅ in the case, where v x ε1 ∩v xε2 6= ∅. (2) for every η ∈ τ, s-dm ik,ib ie,ν (l x η ) ≤ κ − 1. for every η ∈ τ we set lη = {l x η : x ∈ s}, wη = {w x η : x ∈ s}, and oη = {o x η : x ∈ s}. by the above property (2), lη is an indexed collection of elements of the class ipκ−1 ≡ ip(s-dm ik,ib ie,ν ≤ κ − 1). by inductive assumption the class ipκ−1 is saturated. therefore, there exists an initial co-mark m+ lη of lη corresponding to the class ipκ−1. denote by mη a co-mark of s such that its trace on lη is a co-extension of the co-mark m+ lη . the existence of such a co-mark is easily proved. consider the co-indication n ≡ {{v xε : ε ∈ τ} : x ∈ s} of the ib-co-base b ≡ {bx : x ∈ s} of s. since ib is a saturated class of bases there exists an initial co-mark m+ib of s corresponding to the co-indication n of b and the class ib. in particular, m+ib is a co-extension of n. by lemma 2.1.2 of [2], there exists a co-mark m+ of s, which a co-extension of the co-marks m+ib and mη for every η ∈ τ. in particular, m + is a co-extension of n. we show that m+ is an initial co-mark of s corresponding to the class ip(s-dm ik,ib ie,ν ≤ κ). indeed, let m ≡ {{uxδ : δ ∈ τ} : x ∈ s} be an arbitrary co-extension of m+. then, m is a co-extension of the co-marks m + ib, n, and mη for every η ∈ τ. denote by ϑ an indicial mapping from n to m. then, for every x ∈ ie, v xε = u x ϑ(ε), ε ∈ τ. obviously, the co-mark m|lη is a co-extension of the co-mark m+ lη of lη. let r+ib be an initial family of equivalence relations on s corresponding to the co-mark m, the co-indication n of b, and the class ib. let also r+ lη be an initial family of equivalence relations on lη corresponding to the co-mark m|lη and the class ipκ−1. denote by rη the family of equivalence relations on s such that the trace on lη of rη is the family r + lη . by lemma 2.1.1 of [2], there exists an admissible family r+ of equivalence relations on s, which is a final refinement of the families r+ib and rη for every 208 d. n. georgiou, s. d. iliadis and a. c. megaritis η ∈ τ. in particular, r+ is m-admissible. without loss of generality, we can suppose that r+ is (m, wη)-admissible, (m, oη)-admissible, (m, co(wη))admissible, and (m, co(oη))-admissible. we prove that r + is an initial family of s corresponding to the co-mark m of s and the class ip(s-dm ik,ib ie,ν ≤ κ). for this purpose we consider an arbitrary admissible family r of equivalence relations on s, which is a final refinement of r+, and prove that for every l ∈ c♦(r) the space t(l) belongs to ip(s-dm ik,ib ie,ν ≤ κ). let l ∈ c ♦(r). since ib is a saturated class, we have (bl ♦,ϑ(τ) , t(l)) ∈ ib. we show that the base bl ♦,ϑ(τ) of t(l) satisfies condition (2) of definition 4.2, that is for every utδ1(h1) and u t δ2 (h2) of b l ♦,ϑ(τ) (where h1, h2 ⊆ l), with utδ1(h1) ∩ u t δ2 (h2) = ∅ (5) there exists a subset l of t(l) separating utδ1(h1) and u t δ2 (h2) such that s-dm ik,ib ie,ν (l) ≤ κ − 1. consider two elements utδ1(h1), u t δ2 (h2) of b l ♦,ϑ(τ) satisfying relation (5). first we suppose that h1 ∩ h2 = ∅. then, (g) utδ1(h1) ⊆ t(h1), u t δ2 (h2) ⊆ t(l \ h1), (h) t(h1) ∩ t(l \ h1) = ∅, and (i) t(l) = t(h1) ∪ t(l \ h1). therefore, the empty set separates the sets utδ1(h1) and u t δ2 (h2). since s-dm ik,ib ie,ν (∅) = −1 < κ, we have s-dm ik,ib ie,ν (t(l)) ≤ κ. now, we suppose that h1 ∩ h2 6= ∅. let h = h1 ∩ h2, ϑ −1(δ1) = ε1, ϑ−1(δ2) = ε2, and η = ϕ(ε1, ε2). we prove that t(h|lη) separates the sets utδ1(h1) and u t δ2 (h2), and s-dm ik,ib ie,ν (t(h|lη )) ≤ κ − 1 < κ. since ipκ−1 is a saturated class of spaces, the subspace t(h|lη ) of t(m|lη , r|lη) belongs to ipκ−1. hence, s-dm ik,ib ie,ν (t(h|lη )) ≤ κ − 1 < κ. we prove that the subset t(h|lη ) of t(l) separates u t δ1 (h1) and u t δ2 (h2). suppose that x ∈ h. since the subsets v xε1 and v x ε2 of x are disjoint, by condition (1) we have (k) v xε1 ⊆ w x η , v x ε2 ⊆ oxη , (l) w xη ∩ o x η = ∅, and (m) x \ lxη = w x η ∪ o x η . the above relations imply that (n) utδ1(h) ⊆ t(h|wη ) = t|wη ∩ t(h), u t δ2 (h) ⊆ t(h|oη ) = t|oη ∩ t(h), (o) t(h|wη ) ∩ t(h|oη ) = ∅, and (p) t(h) \ t(h|lη) = t(h|wη ) ∪ t(h|oη ). since the restriction wη of s is open and the family r is (m, co(wη))admissible, by lemma 1.4.7 of [2], the subset t|wη of t is open. similarly, universal elements for some classes of spaces 209 the subset t|oη of t is open. also, since the subset t(h) of t is open and t(h) ⊆ t(l), the sets t(h|wη) and t(h|oη ) are open in t(l). setting w = t(h1 \ h) ∪ t(h|wn) and o = t(l \ h1) ∪ t(h|on) we have (q) utδ1(h1) ⊆ w , u t δ2 (h2) ⊆ o, (r) w ∩ o = ∅, and (s) t(l) \ t(h|lη ) = w ∪ o. therefore, the subset t(h|lη) of t(l) separates the sets u t δ1 (h1) and u t δ1 (h2). thus, the class ip(s-dm ik,ib ie,ν ≤ κ) is saturated. now, we prove that the class ip(s-dm ik,ib ie,ν ≤ κ) is saturated. let s be a indexed collection of elements of ip(s-dm ik,ib ie,ν ≤ κ). for every x ∈ s there exists an indexed set {qxi : i ∈ ν} of subsets of x such that (3) x = ∪{qxi : i ∈ ν}. (4) for every i ∈ ν, the subset qxi of x is closed and (q x i , x) ∈ ik. (5) for every i ∈ ν, s-dm ik,ib ie,ν (q x i ) ≤ κ. we set qi = {q x i : x ∈ s}, i ∈ ν. by the preceding, the class ip ≡ ip(s-dm ik,ib ie,ν ≤ κ) is saturated. by property (5), qi is an indexed collection of elements of the class ip. therefore, there exists an initial co-mark m+ qi of qi corresponding to the class ip. denote by mi a co-mark of s such that its trace on qi is a co-extension of the co-mark m + qi . by property (4), the restriction qi of s is a ik-restriction. since ik is a saturated class of subsets, for every i ∈ ν there exists an initial co-mark m+ik,i of s corresponding to the ik-restriction qi. by lemma 2.1.2 of [2], there exists a co-mark m+ of s, which a co-extension of the co-marks mi and m + ik,i for every i ∈ ν. we show that m + is an initial co-mark of s corresponding to the class ip(s-dm ik,ib ie,ν ≤ κ). indeed, let m ≡ {{uxδ : δ ∈ τ} : x ∈ s} be an arbitrary co-extension of m+. then, m is a co-extension of the co-marks mi and m + ik,i and the co-mark m|qi is a co-extension of the co-mark m + qi of qi, i ∈ ν. let r+ qi be an initial family of equivalence relations on qi corresponding to the co-mark m|qi and the class ip. denote by ri the family of equivalence relations on s such that the trace on qi of ri is the family r + qi . let also r+ik,i be an initial family of equivalence relations on s corresponding to the co-mark m and the ik-restriction qi. 210 d. n. georgiou, s. d. iliadis and a. c. megaritis by lemma 2.1.1 of [2], there exists an admissible family r+ of equivalence relations on s, which is a final refinement of the families ri and r + ik,i, i ∈ ν. therefore, r+ is an m-admissible family. we prove that r+ is an initial family of s corresponding to the co-mark m of s and the class ip(s-dm ik,ib ie,ν ≤ κ). for this purpose, we consider an arbitrary admissible family r of equivalence relations on s, which is a final refinement of r+. then, r is a final refinement of the families ri and r + ik,i for every i ∈ ν. we need to prove that for every l ∈ c♦(r), t(l) ∈ ip(s-dm ik,ib ie,ν ≤ κ). let l ∈ c♦(r). it suffices to show that t(l) = ∪{ti(l) : i ∈ ν} such that (t) the subset ti(l) of t(l) is closed, (u) (ti(l), t(l)) ∈ ik, and (v) s-dm ik,ib ie,ν (ti(l)) ≤ κ, i ∈ ν. we set ti(l) = t(l|qi ), i ∈ ν. it is easy to verify that the subset t(l|qi ) of t(l) is closed and t(l) = ∪{t(l|qi) : i ∈ ν}. since ik is a saturated class of subsets, (t(l|qi ), t(l)) ∈ ik. since ip is a saturated class, the subspace t(l|qi) of t(m|qi, r|qi) belongs to ip. hence, s-dm ik,ib ie,ν (t(l|qi)) ≤ κ. thus, by condition (3) of definition 4.2, s-dm ik,ib ie,ν (t(l)) ≤ κ proving that the class ip(s-dm ik,ib ie,ν ≤ κ) is saturated. � corollary 4.7. for every κ ∈ ω in the classes ip(s-dm ik,ib ie,ν ≤ κ) and ip(s-dm ik,ib ie,ν ≤ κ) there exist universal elements. corollary 4.8. let ip be one of the following classes (a) the class of all (completely) regular spaces of weight ≤ τ, (b) the class of all (completely) regular countable-dimensi-onal spaces of weight ≤ τ, (c) the class of all (completely) regular strongly countable-dimensional spaces of weight ≤ τ, (d) the class of all (completely) regular locally finite-dimensional spaces of weight ≤ τ, and (e) the class of all (completely) regular spaces x of weight ≤ τ such that ind(x) ≤ α ∈ τ+. then, for every κ ∈ ω in the classes ip(s-dm ik,ib ie,ν ≤ κ) ∩ ip and ip(s-dm ik,ib ie,ν ≤ κ) ∩ ip there exist universal elements. 5. questions question 5.1. does there exists a universal element in the class of all spaces x with dm ik,ib ie,ν (x) ≤ α or in the class of all spaces x with dm ik,ib ie,ν (x) ≤ α, where α is an ordinal. universal elements for some classes of spaces 211 question 5.2. does there exists a universal element in the class of all spaces x with w-dm ik,ib ie,ν (x) ≤ α or in the class of all spaces x with w-dm ik,ib ie,ν (x) ≤ α, where α is an ordinal. question 5.3. does there exists a universal element in the class of all spaces x with s-dm ik,ib ie,ν (x) ≤ α or in the class of all spaces x with s-dm ik,ib ie,ν (x) ≤ α, where α is an ordinal. references [1] r. engelking, theory of dimensions, finite and infinite, sigma series in pure mathematics, 10. heldermann verlag, lemgo, 1995. viii+401 pp. [2] s. d. iliadis, universal spaces and mappings, north-holland mathematics studies, 198. elsevier science b.v., amsterdam, 2005. xvi+559 pp. [3] d. n. georgiou, s. d. iliadis and a.c. megaritis, dimension-like functions and universality, topology appl. 155 (2008), 2196–2201. [4] a. k. o’ connor, a new approach to dimension, acta math. hung. 55, no. 1-2 (1990), 83–95. [5] r. pol, there is no universal totally disconnected space, fund. math. 79 (1973), 265– 267. (received april 2011 – accepted july 2011) d. n. georgiou (georgiou@math.upatras.gr) department of mathematics, university of patras, 265 04 patras, greece. s. d. iliadis (iliadis@math.upatras.gr) department of mathematics, university of patras, 265 04 patras, greece. a. c. megaritis (megariti@master.math.upatras.gr) department of mathematics, university of patras, 265 04 patras, greece. universal elements for some classes of spaces. by d. n. georgiou, s. d. iliadis and a. c. megaritis jlmagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 161-170 finite products of filters that are compact relative to a class of filters francis jordan, iwo labuda and frédéric mynard abstract. filters whose product with every countable based countably compact filter is countably compact are characterized. 2000 ams classification: 54a20, 54b10, 54d30. keywords: compact, countably compact, filters, product space, product filters. 1. introduction two families a and b of subsets of a topological space x mesh (denoted a#b), if a ∩ b 6= ∅ whenever a ∈ a and b ∈ b. given a class d of filters on x, we say that a filter f on x is d-compact at a ⊂ x if d ∈ d, d#f =⇒ adh d ∩ a 6= ∅. if f = {x} and a = x, we recover the notion of a d-compact space. instances include compact spaces (when d is the class f of all filters), countably compact spaces (when d is the class fω of countably based filters), lindelöf spaces (when d is the class f∧ω of countably deep filters) among others. we recall that a filter f is countably deep if ⋂ a ∈ f whenever a is a countable subfamily of f. in [11], [12], j. vaughan investigated stability under product of d-compact spaces under fairly general conditions on the class d. however, even for simple cases like that of d being the class of countably based filters, no internal characterization of spaces whose product with every countably compact space is countably compact is known (known characterization involve the stone-čech compactification of the space). we investigate the problem in the framework of d-compact filters. in section 3 below, we characterize filters whose product with every d-compact filter is d-compact. the result turns out to be interesting for at least two reasons. it leads to a discussion of exotic filters in section 4 which provides a deeper perspective on the productivity problem for variants of compactness like countable compactness. next, the theorem allows other applications. indeed, many classes of maps can be characterized as those preserving 162 f. jordan, i. labuda and f. mynard d-compact filters [10]. several local topological properties can be characterized in terms involving d-compact filters. consequently, our product theorem for d-compact filters can be applied to the investigations of stability under product of global properties (d-compact spaces), local properties (fréchetness and variants) and maps [9]. 2. terminology and basic facts our terminology is standard and compatible with [7]. the word ‘space’ refers to ‘topological space’. a filter on a set x is a non-empty family of subsets stable by supersets and finite intersections. the only filter containing the empty set is the degenerate filter 2x . the set of filters on a set x is preordered by inclusion. denoting a↑ = {b ⊂ x : ∃a ∈ a, a ⊂ b}, the infimum filter f ∧ g is {f ∪ g : f ∈ f, g ∈ g}↑. the supremum f ∨ g of two filters f and g exists whenever f#g and is {f ∩ g : f ∈ f, g ∈ g}↑. we use blackboard fonts to denote classes of filters. for instance f denotes the class of all filters (with unspecified set) and f(x) denote the family of filters on x. analogously, fω denotes the class of countably based filters (set unspecified) and fω(x) the family of countably based filters on x. the class of principal filters is denoted f1. a filter is free if its intersection is empty. if d is a class of filters, we denote by d∅ the class of free d-filters. we denote by f∅ the free part f ∨ ( ⋂ f) c of a filter f, and by f• its principal part ⋂ f. one or the other may be the degenerate filter {∅}↑ = 2x . we always have f = f∅ ∧ f•, with the convention that g ∧ {∅}↑ = g. let d be a class of filters and let a and b be two families of subsets of a space x. we say that a is d-compact at b if d ∈ d, d#a=⇒adhx d#b. if b = {x}, we drop ‘at b’ and if b = a, we say that a is d-selfcompact. notice that a subset a of x is compact (resp. countably compact, lindelöf) if and only if {a} is d-selfcompact for the class d = f of all (resp. the class d = fω of countably based, the class d = f∧ω of countably deep) filters. compactness relative to the class f1 of principal filters is trivial only for principal filters. it becomes an important concept for general filters. for instance, it is instrumental in characterizing convergence in terms of compactness. lemma 2.1. let x be a space and let f ∈ f(x). the following are equivalent: (1) x ∈ limx f; (2) f is compact at {x}; (3) f is f1-compact at {x}. recall that x is fréchet if, for each a ⊂ x and each x in the closure of a, there exists a sequence (xn)n∈n on a converging to x. a space x is fréchet if and only if its neighborhood filters are fréchet filters in the following sense: a filter f is fréchet if a#f =⇒ ∃h ∈ fω, h#a and h ≥ f. finite products of compact filters 163 similarly, a space x is strongly fréchet if, for each x ∈ ⋂ n∈n cl an with a decreasing sequence (an)n∈n of subsets of x, there exists xn ∈ an such that x ∈ lim(xn)n∈n. a space x is strongly fréchet if and only if its neighborhood filters are strongly fréchet filters in the following sense: a filter f is strongly fréchet if d#f, d ∈ fω =⇒ ∃h ∈ fω, h#d and h ≥ f. fréchet and strongly fréchet filters are instances of the following general concept. let d and j be classes of filters. a filter f is called d to j meshable-refinable, or a (d/j)#≥-filter, if (2.1) d ∈ d, d#f =⇒ ∃j ∈j, j #d and j ≥ f. similarly, a filter f is called d to j sup-meshable, or a (d/j)#∨-filter, if (2.2) d ∈ d, d#f =⇒ ∃j ∈j, j #d ∨ f. a filter f is called d to j meshable, or a (d/j)#-filter, if (2.3) d ∈ d, d#f =⇒ ∃j ∈j, j #d. let j be a set of filters on a set x. by x ⊕ j we mean the set x ∪ j endowed with the topology in which all points of x are isolated and the neighborhood filters of the points j of j are of the form nx⊕j({j }) = j ∧{j }. by the very definition of x ⊕ j, we have proposition 2.2. a free filter on x is d to j meshable if and only if the filter it generates is d-compact in x ⊕ j. a space x is d-based if its neighborhood filters are d-filters; it is (d/j)accessible if it is (d/j)#≥-based. with j being the class of countably based filters, (d/j)#≥-filters are fréchet (resp. strongly fréchet, productively fréchet [5], weakly bisequential and bisequential filters), whenever d stands for the class of principal (resp. countably based, strongly fréchet, countably deep and all) filters [1], [6]. the notion of total countable compactness was first introduced by z. froĺık [2] for a study of products of countably compact and pseudocompact spaces. the property has been rediscovered under various names by several authors (see [11, p. 212]). a topological space is totally countably compact if every countably based filter has a finer (equivalently, meshes a) compact countably based filter. the name comes from total nets of pettis. as one of the possible generalizations of total countable compactness of a set, we say that a filter f is compactly countably meshable (at a) if for every countably based filter h#f, there exists a countably based filter c#h which is compact (at a). more generally, in order to maximize the number of cases handled by the result, we introduce the following key notion: let d and j be classes of filters. a filter f is called compactly d to j meshable (at a), or f is a compactly (d/j)#-filter, if for every d-filter d#f there exists a j-filter j #d which is compact (at a). it turns out to be the notion permitting characterizations of filters whose product with every d-compact filter (of a certain class) is d-compact . with 164 f. jordan, i. labuda and f. mynard various instances of d and j, the concept is also instrumental in the characterization of a wide variety of notions, from total countable compactness, total lindelöfness and total pseudocompactness [12] to fréchetness, strong fréchetness, productive fréchetness and bisequentiality [4], and properties of maps and their range (see [10] for details). in particular, we have proposition 2.3. let d and j be classes of filters on a space x. the following are equivalent. (1) x is (d/j)-accessible; (2) for every d ∈ d, adhd ⊂ ⋃ {lim j : j #d, j ∈ j} ; (3) x ∈ lim f =⇒ f is compactly d to j meshable at {x}. we also note that f compactly (d/j)# =⇒ f d-compact =⇒ x is (d/j)-accessible f compactly (d/j)# f compact =⇒ f∈(d/j)#≥ f compactly (d/j)# . in particular, a countably compact filter on a strongly fréchet space is compactly countably meshable and a bisequential filter is compactly f to fω meshable if and only if it is compact. if d and j are two classes of filters, we say that j is d-composable if for any x and y, the (possibly degenerate) filter hf generated by {hf : h ∈ h, f ∈ f} belongs to j(y ) whenever f ∈j(x) and h ∈ d(x × y ), with the convention that every class of filters contains the degenerate filter. if a class d is dcomposable, we simply say that d is composable. notice that (2.4) h# (f × g) ⇐⇒ hf#g ⇐⇒ h−g#f, where h−g = {h−g = {x ∈ x : (x, y) ∈ h and y ∈ g} : h ∈ h, g ∈ g}↑. observe also [6] that a composable class of filters that contains principal filters is stable under finite products. 3. a product theorem theorem 3.1. let d be a composable class of filters containing all principal filters and let j be a d-composable class of filters. let a ⊂ x. the following are equivalent. (1) f is a compactly (j/d)#filter at a ⊂ x; (2) for every space y , every b ⊂ y and every j-filter g which is d-compact at b, the filter f × g is d-compact at a × b; (3) for every space y , every b ⊂ y and every j-filter g which is d-compact at b, the filter f × g is f1-compact at a × b. moreover, if f ∈ (j∅/d∅)#∨, then the above are also equivalent to: (4) for every space y and for every d-compact j-filter g, the filter f × g is d-compact at a × y. finite products of compact filters 165 proof. (1) =⇒ (2). let d be a d-filter such that d#f ×g. as j is d-composable, d−g is a j-filter and d−g#f. since f is a compactly (j/d)#-filter at a, there exists a d-filter c#d−g which is compact at a. now dc#g and dc is a d-filter, so that there exists a filter m#dc which is convergent to some y in b and meshes with dc. the filter d−m meshes with c, which is compact at a. therefore, there exists u#d−m which is convergent to some point x in a. the filter u × m meshes with d and converges to (x, y) ∈ a × b. (2) =⇒ (3) is clear, as f1 ⊂ d and (3) =⇒ (4) is clear. (3) =⇒ (1). if f is not compactly (j/d)# at a, then there exists a j-filter j #f such that for every d-filter d#j , there exists an ultrafilter ud ≥ d such that limud ∩ a = ∅. consider the topological space y =x ⊕ {ud : d#j , d ∈ d}. then the filter ĵ generated by j on y is d-compact at b = {ud : d#j , d ∈ d} but f × ĵ is not f1-compact at a × b. indeed, ∆ = {(x, x) : x ∈ x} ⊂ x × y is in f1 and ∆#f × ĵ because f#j . but adh∆ ∩ a × b = ∅. indeed, if h is a filter on ∆, then there exists a filter h0 on x such that h is generated by {{(x, x) : x ∈ h} : h ∈ h0}. if (x, ud) ∈ limx×y h ∩ (a × b) then ud ∈ limy h0 (and x ∈ limx h0). hence as a filter on x, h0= ud, so that limx h0 ∩ a = ∅. thus x /∈ a. (4) =⇒ (1) if f ∈ (j∅/d∅)#∨. indeed, if f is not compactly (j/d)# at a, then the filter generated by j on y = x ⊕ {ud : d#j , d ∈ d} (constructed as in (3) =⇒ (1)) is d-compact . since f ∈ (j∅/d ∅ )#∨, there exists a free d-filter d0#j ∨ f. the filter ̂d0 × d0 generated on x × y by d0 × d0 is a d-filter by composability of d [6]. moreover ( ̂d0 × d0 ) # (f × j ) . but adhx×y ( ̂d0 × d0 ) ∩ (a × y ) = ∅. indeed, consider an ultrafilter finer than ̂d0 × d0, which can be written u × u. as d0 is free, u converges in y only if u = ud for some d ∈ d such that d#j . but in this case, limx u ∩a = ∅. � observe that (3) =⇒ (1) only uses f1⊂ d and no other composability assumption. notice that in the special case where j = d, the condition f ∈ (j∅/d∅)#∨ is always verified. therefore, in the case j = d = fκ, and a = x, we get corollary 3.2. the product filter f ×g is κ-compact for every κ-compact and κ-based filter g if and only if f is a compactly (fκ/fκ)#-filter. in particular, if κ = ω, we obtain the announced result on the productivity of countable compactness at the level of filters. 166 f. jordan, i. labuda and f. mynard corollary 3.3. (1) the product filter f ×g is countably compact for every countably based and countably compact filter g if and only if f is compactly countably meshable. (2) let f be a countably based filter. the product filter f × g is countably compact for every strongly fréchet and countably compact filter g if and only if f is compactly countably meshable. proof. (2) follows from the simple observation that if f is a countably based and compactly (fω/fω)#-filter, then it is also a compactly ((fω/fω)#≥ /fω)#filter. � in particular, if f = {x}, we obtain that the product of a totally countably compact space with not only countably compact spaces but also strongly fréchet countably compact filters is countably compact, generalizing [11, theorem 2]. more generally, the part (1 =⇒ 3) of theorem 3.1 applied to principal filters f = {x} and g = {y }, for various instances of d = j allows to recover results of j. vaughan [11], and also to provide new variants. for instance: corollary 3.4. the product of a lindelöf space and a compactly (f∧ω/f∧ω)meshable space is lindelöf. denote by o(a) the family {o open: ∃a ∈ a, a ⊂ o} whenever a is a family of subsets of a space x. further, o(d) will denote the class of d-filters d such that d = o(d) ↑ . a topological space x is feebly compact if and only if {x} is o(fω )-compact . completely regular feebly compact spaces are called pseudocompact. corollary 3.5. the product of a compactly (o(fω )/o(fω))-meshable space and a feebly compact space is feebly compact. other applications of theorem 3.1 to results of stability under product of local topological properties (like fréchetness and its variants) and of maps (variants of perfect maps and of quotient maps) are presented in [9]. 4. d-exotic filters this section is devoted to a discussion of the additional assumption that f ∈ (j∅/d∅)#∨ in point (4) of theorem 3.1. the very fact that such an additional condition is needed is surprising. in our experience with d-compact filters, this is the only instance we know of, where it makes a significant difference to consider d-compact filters at a proper subset of x or simply d-compact filters (that is, at x). as noticed before, this condition is always fulfilled if d = j. let d be a class of filters. a filter f is d-exotic if every ultrafilter u finer than f is d-deep, that is, d∈ d and d ≤ u implies that d• ∈ u. notice that every fixed ultrafilter is d-exotic, whatever the class d is. a d-exotic filter is non-trivial if it has a finer free d-deep ultrafilter. recall [6] that a class j of filters is d-steady if j ∨ d ∈ j whenever j ∈ j and d ∈ d. finite products of compact filters 167 theorem 4.1. let d be an f1-steady class of filters and let f be a filter on x. the following are equivalent. (1) f is d-exotic; (2) d• 6= ∅ whenever d is a d-filter d meshing with f; (3) f is d-compact on x endowed with the discrete topology. if moreover d is composable and contains fixed ultrafilters, then the above conditions are equivalent to (4) for every space y , and for every d-compact d-filter g , the filter f ×g is d-compact in x × y with discrete x. proof. (1) =⇒ (2). let d#f be a d-filter. then there exists an ultrafilter u finer than both f and d. since f is d-exotic, u is d-deep so that d• ∈ u and consequently d• 6= ∅. (2) =⇒ (3) because adhd = d• in the discrete topology of x. (3) =⇒ (1). let u be an ultrafilter finer than f and let d∈ d with d ≤ u. then d#f so that adhd = d• 6= ∅. observe that d• ∈ u. otherwise, (d•) c ∈ u and we would have (d•) c #d. hence d∅ would be a free d-filter meshing with f and so having empty adherence in the discrete topology of x. hence u is d-deep. (2) =⇒ (4). let x carry the discrete topology, let g be a d-compact d-filter on y and let d ∈d such that d# (f × g) . if d is composable, then d−g ∈ d. moreover d−g#f. by (2), (d−g) • 6= ∅. let x ∈ (d−g) • . then d{x}#g and d{x} ∈ d, so that there exists a convergent filter l meshing with d{x}. then( {x}↑ × l ) #d and {x}↑ × l is x × y convergent. hence adhx×y d 6= ∅. (4) =⇒ (2). if f is not d-exotic, then there exists a free d-filter d#f. the filter d is convergent, hence compact in y = x ⊕{d}. however, f ×d is not d-compact. indeed, the filter generated by {(x, x) : x ∈ d}d∈d on x × y is a d-filter meshing with f × d, but its adherence in x × y is empty if x carries the discrete topology, since any filter finer than d is free, hence does not converge in x. � corollary 4.2. let d be an f1-steady class of filters. a filter f /∈ (j ∅/d ∅ )#∨ if and only if there exists a free j-filter j meshing with f such that f ∨ j is d-exotic. so the assumption that f ∈ (j∅/d ∅ )#∨ in theorem 3.1 is empty, if nontrivial d-exotic filters do not exist. this is often the case, as shown by the following observation. proposition 4.3. let x be an infinite set. if the class d(x) of filters contains the cofinite filter c on x, then there is no free d-deep ultrafilter on x, and therefore, no non trivial d-exotic filter on x. proof. assume that u is a free d-deep ultrafilter. then u ≥ c because u is free. since c ∈ d, c• ∈ u and therefore c• 6= ∅; a contradiction. � 168 f. jordan, i. labuda and f. mynard notice that if d ⊂ j are two classes of filters, then every j-exotic filter is also d-exotic. in other words, if there is no non-trivial d-exotic filter, then there is no non-trivial j-exotic filter. as the cofinite filter on an infinite set is almost principal, hence productively fréchet [6], the class of d-exotic filters is trivial when d is the class of almost principal, of productively fréchet, of strongly fréchet, of fréchet, of countably tight, or of countably fan-tight filters, among others (see [6] for details). recall that a filter f is almost principal [6] if there exists f0 ∈ f such that |f \f0| < ω for every f ∈ f. this, however, does not take care of the case d = fω. in fact, the existence of fω-exotic filters depends on the cardinality of the underlying set. recall that the cardinality of x is measurable if there exists a countably additive {0, 1}-measure on 2x . the free ultrafilter formed by the sets of measure 1 is then fω-deep. theorem 4.4. the following are equivalent: (1) card (x) is measurable; (2) the cofinite filter on x is not a bisequential filter; (3) there exists a non trivial fω-exotic (ultra)filter on x; (4) there exists an fω-compact (ultra)filter on x with the discrete topology; (5) x endowed with the discrete topology is not realcompact. proof. (1) ⇐⇒ (2) is proven in [8, example 10.15] though in the following different language: the one point compactification of a discrete space x is bisequential if and only if card(x) is not measurable. as the one point compactification of x is x ⊕ c, its bisequentiality amounts to that of the filter c. (1) ⇐⇒ (3) is clear and (3) ⇐⇒ (4) follows from theorem 4.1. finally (1) ⇐⇒ (5) is [3, theorem 12.2]. � consequently, if the cardinality of the set is non measurable, then the class of (f/fω)#≥-exotic filters, and therefore that of fω-exotic filters is trivial. corollary 4.5. let x be a set of non measurable cardinality and let f be a bisequential filter on x. then f × g is countably compact for every countably compact filter g if and only if f is compact. proof. in view of theorem 3.1 and theorem 4.4, f × g is countably compact for every countably compact filter g if and only if f is a compactly (f/fω)#filter. as observed before, this condition is equivalent to compactness for a bisequential filter. � corollary 4.6. let x be a set of non measurable cardinality and let f be a filter on x. let f̂ denote the filter generated by f on the space x ⊕{f}. then f̂ × g is countably compact for every countably compact filter g if and only if f is bisequential. proof. if f is bisequential, then f̂ is both compact and bisequential on x ⊕ {f}. in view of corollary 4.5, f̂ × g is countably compact for every countably compact filter g. finite products of compact filters 169 conversely, if f is not bisequential, then there exists an ultrafilter u0 finer than f such that every countably based filter d coarser than u is not finer than f. in other words, for every countably based d ≤ u, there exists fd ∈ f such that f cd#d. let ud denote an ultrafilter of d∨f c d. the filter û0 generated by u0 on y = x ⊕ {ud : d ∈ fω, d ≤ u0} is countably compact. but f̂× û0 is not countably compact. indeed, there exists a free countably based filter d coarser than u0 on x, because the cardinality of x is non measurable. the filter d̃ generated on (x ⊕ {f}) × y by {(x, x) : x ∈ d, d∈ d} is countably based and meshes with f̂× û0. but adh(x⊕{f})×y d̃ = ∅. indeed, if a filter finer than d̃ has a convergent y -projection then its y -projection is one of the ultrafilters ud. then its (x ⊕ {f})-projection is also ud and therefore does not mesh with f. consequently, the (x ⊕ {f})-projection does not converge in x ⊕ {f}. � on the other hand, if the cardinality of x is measurable, an fω-deep ultrafilter on x defines a p -point in the čech-stone compactification βx of the discrete topological space x. therefore, the set of ultrafilters finer than an fω-exotic filter is a closed subset of p -points of the compact set βx, hence a compact set of p -points. but every countably compact space of p -points is finite [3, 4.k.1]. therefore: proposition 4.7. if the cardinality of x is measurable, the fω-exotic filters on x are the infima of finitely many fω-deep ultrafilters on x. neither theorem 4.4 nor proposition 4.3 apply to the question of existence of non-trivial f∧ω-exotic filters. however, we observe that f∧ω-exotic ultrafilters can exist only on a countable set. indeed, proposition 4.8. let x be an uncountable set. if the class d(x) contains the cocountable filter cω then there is no non-trivial d-exotic filter that does not contain any countable set. proof. let f be a d-exotic filter that does not contain any countable set. there is a free ultrafilter u finer than f which does not contain any countable set. therefore, u ≥ cω. moreover, as an ultrafilter of f, it is d-deep, and cω ∈ d, so that c•ω ∈ u. a contradiction, because c • ω = ∅. � in particular, the cocountable filter on an uncountable set is countably deep. therefore f∧ω-exotic filters must contain a countable set. moreover, no f∧ωfilter on a countable set is free, so that every filter containing a countable set is f∧ω-exotic. corollary 4.9. a free filter is f∧ω-exotic if and only if it contains a countable set. as a final remark on d-exotic filters, notice that a class d of filters generating many d-exotic filters cannot contain the cofinite filter or the cocountable filter. roughly speaking, it seems that a naturally defined class of filters that does not 170 f. jordan, i. labuda and f. mynard contain the cofinite filter would have to be defined either in terms of cardinality of a base — in which case the existence of exotic filters implies the existence of measurable cardinals — or in terms of depth of the filter, in which case the class would contain the cocountable filter. so, in the latter case, the question of existence of exotic filters is, in a sense, reduced to sets of small cardinality. references 1. s. dolecki, active boundaries of upper semicontinuous and compactoid relations; closed and inductively perfect maps, rostock. math. coll. 54 (2000), 51–68. 2. z. froĺık, the topological product of two pseudocompact spaces, czech. math. j. 85 (1960), no. 10, 339–349. 3. l. gillman and m. jerison, rings of continuous functions, van nostrand, 1960. 4. f. jordan and f. mynard, espaces productivement de fréchet, c. r. acad. sci. paris, ser i 335 (2002), 259–262. 5. , productively fréchet spaces, czech. math. j. 54 (2004), no. 129, 981–990. 6. , compatible relations on filters and stability of local topological properties under supremum and product, top. appl. 153 (2006), 2386–2412. 7. j. kelley, general topology, van nostrand, 1955. 8. e. michael, a quintuple quotient quest, gen. topology appl. 2 (1972), 91–138. 9. f. mynard, products of compact filters and applications to classical product theorems, top. appl. 154 (2007), no. 4, 953–958. 10. , relations that preserve compact filters, applied gen. top. 8 (2007),171–185. 11. j. vaughan, products of topological spaces, gen. topology appl. 8 (1978), 207–217. 12. j. e. vaughan, countably compact and sequentially compact spaces, vol. handbook of set-theoretic topology, pp. 569–602, elsevier science, 1984. received september 2005 accepted january 2006 francis jordan (fjordan@georgiasouthern.edu) dept. mathematical sciences, georgia southern university, po box 8093, statesboro, ga 30460-8093, u.s.a. iwo labuda (mmlabuda@olemiss.edu) department of mathematics, university of mississippi,university, ms 38677, u.s.a. frédéric mynard (fmynard@georgiasouthern.edu) dept. mathematical sciences, georgia southern university, po box 8093, statesboro, ga 30460-8093, u.s.a. druzhininaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 139-150 connected metrizable subtopologies and partitions into copies of the cantor set irina druzhinina abstract. we prove under martin’s axiom that every separable metrizable space represented as the union of less than 2 ω zerodimensional compact subsets is zero-dimensional. on the other hand, we show in zf c that every separable completely metrizable space without isolated points is the union of 2 ω pairwise disjoint copies of the cantor set. 2000 ams classification: 54b15, 54c05, 54c10, 54d05, 54d30, 54e35, 54e50 keywords: metrizable space, completely metrizable space, condensation, connected metrizable subtopology, cantor set, zero-dimensional space, martin’s axiom 1. introduction we say that a space x condenses onto a space y if there exists a continuous bijection f : x → y . a space x has a weaker connected topology (also called connected subtopology) if x condenses onto a connected space. in the article [6] on connected subtopologies, the following two results were proved: (a) let cα be the cantor set, for each α ∈ a, and x = ⊕ {cα : α ∈ a} the disjoint topological union of the spaces cα. the space x has a weaker t3 connected topology iff it has a weaker tychonoff connected topology iff |a| ≥ ω1. (b) if iα is the unit segment [0, 1], for each α ∈ a, and x = ⊕ {iα : α ∈ a} is the disjoint topological union of the spaces iα, then space x has a weaker t3 connected topology iff |a| ≥ ω. in section 3 of this article, we study the problem of when the spaces of the form ⊕ {cα : α ∈ a} or ⊕ {iα : α ∈ a} admit connected metrizable subtopologies. here we mention three more results on connected subtopologies presented in [4] that will be used in the sequel. 140 i. druzhinina theorem 1.1. let x be a metrizable space of weight κ ≤ 2ω. if there exists a closed set p ⊆ x which admits a condensation onto a connected non-compact metrizable space, then x condenses onto a connected separable metrizable space. theorem 1.2. every metrizable space of weight 2ω admits a condensation onto a connected separable metrizable space. theorem 1.3. every metrizable space of weight κ ≥ 2ω with achievable extent admits a weaker connected metrizable topology. recall that a metrizable space x has an achievable extent if x contains a closed discrete subset of the size equal to the weight of x. the authors of [6] rose the following problem: problem 1.4. is it true in zf c that the topological sum of ω1 copies of the cantor set condenses onto a connected compact space? delay and just in [3] gave the negative answer to the above problem under the assumption that the real line cannot be covered by ω1 nowhere dense sets. in section 2 of the article we show under m a that if separable metrizable space x is a union of less than 2ω zero-dimensional compact subspaces, then x is also zero-dimensional. in particular, under m a + ¬ch, all metrizable subtopologies on the space ⊕ {cα : α ∈ ω1} are zero-dimensional. this result and theorem 1.2 together imply that the existence of a connected metrizable subtopology on the topological sum of ω1 copies of the cantor set does not depend on zf c. we also prove in section 2 that the topological sum ⊕ {iα : α ∈ a}, where each iα is a copy of the unit segment [0, 1], admits a connected metrizable subtopology iff |a| ≥ ω. in connection with problem 1.4 we show in section 3 that the topological sum of 2ω copies of the cantor set condenses onto the closed unit interval and, even more, it condenses onto every compact metrizable space without isolated points. finally, we generalize the latter fact and prove that every separable complete metrizable space without isolated points can be represented as a disjoint union of 2ω copies of the cantor set. the reader can consult kunen’s book [5] for details about martin’s axiom (for short, m a). 2. connected metrizable subtopologies let cα be the cantor set for each α < ω1 and let x = ⊕ {cα : α < ω1}. we consider the question whether x admits a connected metrizable subtopology. notice that under ch, the answer is affirmative, since one can apply theorem 1.2. the next theorem, combined with the well-known fact that every metrizable space of weight ≤ 2ω admits a separable metrizable subtopology (see lemma 2.5 in [4]), implies that the answer to the question is “no” under m a + ¬ch. in what follows the family of all finite subsets of a set a is denoted by [a]<ω. connected metrizable subtopologies 141 theorem 2.1. suppose that m a holds. let x be a separable metrizable space represented as x = ⋃ α<κ kα, where κ < 2 ω and every kα is a zerodimensional compact subspace of x. then x is also zero-dimensional. proof. let b be a countable base for x. pick a point x0 ∈ x and take any element w ∈ b such that x0 ∈ w . we have to show that there exists a clopen set u in x such that x0 ∈ u ⊆ w . notice that for every α < κ, the open subspace kα ∩ w of kα is σ-compact since kα is compact and metrizable. hence, we can represent each kα in the form kα = ⋃ n∈ω kα,n ∪(kα\w ), where the sets kα,n are compact and satisfy kα,n ⊆ w . therefore, we can assume from the very beginning that kα ⊆ w or kα ∩ w = ∅ for each α < κ. let us consider the following family: p = { (f, γ, λ) : f ∈ [κ]<ω, γ, λ ∈ [b]<ω, x0 ∈ ⋃ γ, ⋃ γ ⊆ w, ⋃ γ ∩ ⋃ λ = ∅, ⋃ α∈f kα ⊆ ( ⋃ γ) ∪ ( ⋃ λ) } . it is easy to see that the family p is not empty. now we introduce a partial order ≤ in p by the following rule: (f1, γ1, λ1) ≤ (f2, γ2, λ2) ⇐⇒ f2 ⊆ f1 & γ2 ⊆ γ1 & λ2 ⊆ λ1. we claim that the poset (p, ≤) satisfies the countable chain condition. indeed, let q ⊆ p be a subset of cardinality ℵ1. since |b| = ℵ0 there exist q∗ ⊆ q, γ∗ ∈ [b]<ω, and λ∗ ∈ [b]<ω such that |q∗| = ℵ1 and all elements of q ∗ have the form (f, γ∗, λ∗) with f ∈ [κ]<ω. take two elements p1 = (f1, γ ∗, λ∗) and p2 = (f2, γ ∗, λ∗) of q∗. it is easy to see that p = (f1 ∪ f2, γ ∗, λ∗) is in p, p ≤ p1 and p ≤ p2. this proves our claim. in fact, almost the same argument shows that (p, ≤) is σ-centered. for every α ∈ κ, put dα = {(f, γ, λ) ∈ p : α ∈ f }. let us verify that dα is dense in (p, ≤). take an element p = (f, γ, λ) ∈ p. it suffices to find an element p ∗ ∈ dα such that p ∗ ≤ p . if kα ⊆ ⋃ γ ∪ ⋃ λ, then p ∗ = (f ∪{α}, γ, λ) ∈ dα and, clearly, p ∗ ≤ p . otherwise, we put t1 = kα∩ ⋃ γ and t2 = kα ⋂ ⋃ λ. then t1 and t2 are disjoint, and we may assume that t1 6= ∅. then, clearly, kα ⊆ w . since kα is zero-dimensional, there exist clopen sets a and b in kα such that a ∪ b = kα, a ∩ b = ∅, and t1 ⊆ a, t2 ⊆ b. evidently, the sets a and b are closed in x. therefore, the sets a ∪ ⋃ γ and b ∪ ⋃ λ are closed in x and disjoint. we choose disjoint open sets o1 and o2 in x such that a ∪ ⋃ γ ⊆ o1 and b ∪ ⋃ λ ⊆ o2. in addition, we can take o1 with o1 ⊆ w , since a ∪ ⋃ γ ⊆ w . let γ′ be an open cover of a by elements of the base b such that ⋃ γ′ ⊆ o1, and λ ′ an open cover of b by elements of b such that ⋃ λ′ ⊆ o2. it is easy to see that p ∗ = (f ∪ {α}, γ ∪ γ′, λ ∪ λ′) is an element of p. in addition, p ∗ ∈ dα and p ∗ ≤ p . in the case when t1 = ∅ we take p ∗ = (f ∪ {α}, γ, λ ∪ λ′). then again p ∗ ∈ dα and p ∗ ≤ p . 142 i. druzhinina we have thus shown that dα is dense in (p, ≤) for each α < κ. it is also clear that the cardinality of the family d = {dα : α ∈ κ} is not greater than κ < 2ω. hence, m a implies that there exists a d-generic filter g in (p, ≤), that is, a filter g ⊆ p such that g ∩ dα 6= ∅ for each α < ω1. now we define two open subsets u and v of x by u = ⋃ { ⋃ γ : ∃ f ∈ [ κ]<ω ∃ λ ∈ [b]<ω such that (f, γ, λ) ∈ g }, v = ⋃ { ⋃ λ : ∃ f ∈ [ κ]<ω ∃ γ ∈ [b]<ω such that (f, γ, λ) ∈ g }. let us check that the sets u and v satisfy the following conditions: (1) x0 ∈ u ⊆ w ; (2) u ∩ v = ∅; (3) u ∪ v = x. condition (1) follows from the definition of p. to show that (2) holds we note that if p1 = (f1, γ1, λ1) ∈ g and p2 = (f2, γ2, λ2) ∈ g, then there exists p = (f, γ, λ) ∈ g such that p ≤ p1 and p ≤ p2. therefore, γ1 ∪ γ2 ⊆ γ and λ1 ∪ λ2 ⊆ λ. since ⋃ γ ∩ ⋃ λ = ∅, we have that ⋃ γ1 ∩ ⋃ λ2 = ∅ and⋃ γ2 ∩ ⋃ λ1 = ∅. hence, ⋃ γ ∩ ⋃ λ = ∅ for arbitrary γ and λ that are used to form u and v . to check (3) we note that for each α < κ, there exists p = (f ∗, γ∗, λ∗) ∈ g ∩ dα. hence, α ∈ f ∗ and kα ⊆ ( ⋃ γ∗) ∪ ( ⋃ λ∗). since⋃ γ∗ ⊆ u and ⋃ λ∗ ⊆ v , we have kα ⊆ u ∪ v . it follows from (1) and (3) that v 6= ∅ and, hence, u and v are clopen sets in x. thus, the clopen set u ⊆ x satisfies x0 ∈ u ⊆ w . � corollary 2.2. the existence of a connected metrizable subtopology on the space x = ⊕ {cα : α ∈ ω1}, where each cα is a copy of the cantor set, does not depend on zf c. in the second part of this section we study the problem whether there exists a connected metrizable subtopology on the space x = ⊕ {iα : α ∈ a}, where each iα is the unit segment. example 2.3. let in be the unit segment [0, 1] for each n ∈ ω and x =⊕ {in : n ∈ ω}. then the space x condenses onto a connected non-compact subspace of the plane r2. proof. let {rn : n ∈ n} be the set of rational numbers in [0, 1]. for each n ∈ n, put jn = {(x, rn) ∈ r 2 : 1/n ≤ x ≤ 1} and j0 = {0} × [0, 1]. let us consider y = ⋃ {jn : n ∈ ω}. we claim that y is a connected subspace of r2. indeed, let u be a clopen set of y such that j0 ⊆ u . since j0 is compact, there exists ǫ > 0 such that if ũ = {(x, y) ∈ r 2 : 0 ≤ x < ǫ, 0 ≤ y ≤ 1}, then j0 ⊆ ũ ∩ y ⊆ u . take n0 ∈ n such that 1/n0 < ǫ. then for every n ≥ n0, cn = (1/n, rn) ∈ ũ ∩ y ⊆ u . since cn ∈ jn, jn is connected, and u is a clopen set in y , we conclude that jn ⊆ u for every n ≥ n0. connected metrizable subtopologies 143 take m < n0 and let v be an open neighborhood of the point cm = (1/m, rm) in y . evidently, |v ∩ {(1/m, rn) : n ∈ n}| = ℵ0. therefore, there exists k > n0 such that (1/m, rk) ∈ jk ∩ v ⊆ u ∩ v 6= ∅. it follows that cm = (1/m, rm) ∈ cly (u ). since cm ∈ jm, we conclude that jm ⊆ u for every m < n0. thus, u = y and this proves that y is connected. for each n ∈ ω, take a continuous bijection fn : in → jn. then the sum of the functions fn, say, f = ∇n∈ωfn : x → y is a continuous bijection (see proposition 2.1.11 in [2]). hence, f is a condensation of x onto the connected separable metrizable space y ⊆ r2. finally, y is not closed in r2 and is not compact. � corollary 2.4. let x = ⊕ {iα : α ∈ a}, where each iα is the unit segment [0, 1]. then the space x admits a connected metrizable subtopology if and only if |a | ≤ 1 or |a | ≥ ω. proof. if 1 < |a | < ω, then x is a disconnected compact space and, hence, it does not admit a connected hausdorff subtopology. if |a | = ω, it follows from example 2.3 that x has a connected separable metrizable subtopology. if ω < |a | ≤ 2ω, then x ⊇ p = ⊕ {in : n ∈ ω}. the set p is closed in x and condenses onto a connected non-compact metrizable space. apply theorem 1.1 to conclude that x admits a connected separable metrizable subtopology. let |a | > 2ω. in each iα, take a point xα. then p = {xα : α ∈ a} is a closed discrete set in x whose size equal to the weight of x, that is, x has achievable extent. it follows from theorem 1.3 that x admits a connected metrizable subtopology. � 3. condensations of the disjoint topological union of 2ω copies of the cantor set in theorem 3.3 we prove that the topological sum of 2ω copies of the cantor set condenses onto the unit interval i = [0, 1]. then we extend this fact to compact metrizable spaces without isolated points (theorem 3.8). we finish with theorem 3.10 that shows that the conclusion is valid for every separable complete metrizable space without isolated points. we will use the following notation. the cantor set is c = {0, 1}ω. if a ⊆ ω, then πa : c → {0, 1} a is the projection and, for n ∈ ω, πn is the projection of c onto the nth factor. if a ⊆ b ⊆ ω, then πba is the projection of {0, 1} b onto {0, 1}a. for an open canonical set u ⊆ {0, 1}a, we put coord(u ) = {i ∈ a : |πi(u )| = 1}. if a ⊆ b ⊆ ω and f ∈ {0, 1}a, then the set o(f ) = (πba ) −1(f ) is called the cylinder over f in {0, 1}b. 144 i. druzhinina let x be a space. we say that s ⊆ x is a first category set in x if s = ⋃∞ n=1 sn, where each sn is a nowhere dense subset of x. we start with a lemma. lemma 3.1. let s be a subset of the cantor set c. if there exists a family {an : n ∈ ω} of pairwise disjoint infinite subsets of ω such that ω = ⋃ {an : n ∈ ω} and |{0, 1} an \ πan (s)| = c for each n ∈ ω, then c is the union of c = 2ω pairwise disjoint copies of the cantor set such that s intersects each of these copies in at most one point. proof. let cn = {0, 1} an be a copy of the cantor set for each n ∈ ω. clearly, c ∼= ∏∞ n=0 cn. for every n ∈ ω, put sn = πan (s). since |cn \ sn| = c, it is possible to represent each cn as the union cn = ⋃ {pn,α : α ∈ γn}, where each pn,α is a doubleton, pn,α ∩ pn,β = ∅ if α 6= β, |sn ∩ pn,α| ≤ 1, and γn is an index set of cardinality c. for an element γ = (γn)n∈ω of γ = ∏∞ n=0 γn, put cγ = ∏∞ n=0 pn,γn . it is routine to verify that: (i) cγ ⊆ c is a copy of the cantor set for every γ ∈ γ; (ii) c = ⋃ {cγ : γ ∈ γ} and |γ| = 2 ω; (iii) cγ ∩ cλ = ∅ if γ 6= λ; (iv) |s ∩ cγ| ≤ 1 for every γ ∈ γ. our lemma is proved. � corollary 3.2. let s be a countable subset of the cantor set c. then c is the union of 2 ω pairwise disjoint copies of the cantor set such that s intersects each of these copies in at most one point. proof. clearly, there exists a family {an : n ∈ ω} of disjoint infinite sets such that ω = ⋃∞ n=0 an. we have that |πan (s)| = ℵ0 and, therefore, the complement {0, 1}an \ πan (s) has cardinality c, for each n ∈ ω. now apply lemma 3.1. � theorem 3.3. the unit segment i = [0, 1] is the union of 2 ω pairwise disjoint copies of the cantor set. proof. we take the cantor set c = {0, 1} ω and consider the mapping f : c → i defined by the formula f (x) = ∞∑ i=0 xi 2i+1 , where xi is ith coordinate of the point x ∈ c. it is known that f is a continuous mapping onto i (see 3.2.b of [2]). although f is not one-to-one, it is easy to see that there is a countable infinite subset s ⊂ c such that |f −1(f (x))| = 2 for every x ∈ s and |f −1(f (x))| = 1 for x ∈ c \ s. it follows from corollary 3.2 that there exists a representation c = ⋃ {cγ : γ ∈ γ} such that the family {cγ : γ ∈ γ} and the set s satisfy conditions (i)–(iv) in the proof of lemma 3.1. put kγ = f (cγ ). since |cγ ∩ s| ≤ 1, it is evident that fγ = f ↾cγ : cγ → kγ is a homeomorphism; therefore, connected metrizable subtopologies 145 kγ is a copy of the cantor set for every γ ∈ γ. notice that i = ⋃ {kγ : γ ∈ γ}. it is easy to see that if cγ ∩ s = ∅, then kγ does not meet any other kλ, and if cγ ∩ s = {x}, then there is a unique λ ∈ γ \ {γ} such that kγ ∩ kλ 6= ∅. in this case, kγ ∩ kλ = {y}, where y = f (x). hence, for every y ∈ f (s) there exists a unique pair (αy , βy) ∈ [γ] 2 such that kαy ∩ kβy = {y}. put ly = kαy ∪ kβy . clearly, each ly is a copy of the cantor set and ly ∩ lz = ∅ if y 6= z. we obtain that i = ⋃ {kγ : γ ∈ γ, cγ ∩ s = ∅} ∪ ⋃ {ly : y ∈ f (s)}, that is, i is the union of 2ω disjoint copies of the cantor set. � in the sequel we will show that theorem 3.3 remains valid if the unit interval i is replaced by a compact metrizable space without isolated points. first we need some lemmas. the following result is trivial. lemma 3.4. if f ⊆ {0, 1}ω is a nowhere dense set and a ⊆ ω is finite, then πω\a(f ) is a nowhere dense subset of {0, 1} ω\a, where πω\a is the projection of {0, 1}ω onto {0, 1}ω\a. lemma 3.5. let s ⊆ c = {0, 1}ω be a first category set. then there are infinite disjoint sets a and b such that ω = a ∪ b and the sets πa(s) and πb(s) are of the first category in {0, 1} a and in {0, 1}b, respectively. proof. let s = ⋃∞ m=0 fm, where every fm is a nowhere dense subset of c. we will construct by induction two families {an : n ∈ ω} and {bn : n ∈ ω} which satisfy following conditions for each n ∈ ω: (i) an, bn ∈ [ω] <ω; (ii) an ∩ bn = ∅; (iii) an ⊆ an+1, an+1 \ an 6= ∅; (iv) bn ⊆ bn+1, bn+1 \ bn 6= ∅; (v) for every f ∈ {0, 1} an , there exists a family {u0f , u1f , . . . , unf } of open canonical subsets of {0, 1} ω\bn such that π ω\bn an (uif ) = {f}, uif ∩πω\bn (fi) = ∅ and an ⊆ coord(uif ) ⊆ an+1, for each i ≤ n; (vi) for every g ∈ {0, 1} bn , there exists a family {u0g, u1g, . . . , ung} of open canonical subsets of {0, 1} ω\an+1 such that π ω\an+1 bn (uig) = {g}, uig ∩ πω\an+1 (fi) = ∅ and bn ⊆ coord(uig ) ⊆ bn+1, for each i ≤ n; (vii) n ∈ an ∪ bn. we start with a0 = {0} and b0 = {1}. now suppose that for some n ∈ ω we have constructed {ak : k ≤ n} and {bk : k ≤ n} which satisfy (i)–(vii). let us construct an+1. take f ∈ {0, 1} an and let o(f ) be the cylinder over f in {0, 1} ω\bn. for each i = 0, 1, . . . , n, the set πω\bn (fi) is nowhere dense in {0, 1} ω\bn (see lemma 3.4). hence, for each i ≤ n there exists an open canonical set uif ⊆ {0, 1} ω\bn such that uif ⊆ o(f ) and uif ∩πω\bn (fi) = ∅. 146 i. druzhinina clearly, π ω\bn an (uif ) = {f}, an ⊆ coord(uif ) and coord(uif ) ∩ bn = ∅. put ãn+1 = ⋃ { n⋃ i=0 coord(uif ) : f ∈ {0, 1} an} and an+1 = min(ω \ (an ∪ bn)). we set an+1 = ãn+1 ∪{an+1}. it is easy to see that an+1 is finite, an ⊆ an+1 , an+1 \ an 6= ∅, and an+1 ∩ bn = ∅. the choice of an+1 and (vii) together imply that n + 1 ∈ an+1 ∪ bn. in addition, for every f ∈ {0, 1} an and each i ≤ n we have that coord(uif ) ⊆ an+1. now we construct bn+1. take g ∈ {0, 1} bn and let o(g) be the cylinder over g in {0, 1}ω\an+1. for each i = 0, 1, . . . , n, the set πω\an+1 (fi) is nowhere dense in {0, 1}ω\an+1 (see lemma 3.4). hence, for each i ≤ n there exists an open canonical set uig ⊆ {0, 1} ω\an+1 such that uig ⊆ o(g) and uig ∩πω\an+1 (fi) = ∅. clearly, π ω\an+1 bn (uig) = {g}, bn ⊆ coord(uig ) and coord(uig ) ∩ an+1 = ∅. let b̃n+1 = ⋃ { n⋃ i=1 coord(uig ) : g ∈ {0, 1} bn} and bn+1 ∈ (ω \ (an+1 ∪ bn)). we set bn+1 = b̃n+1 ∪ {bn+1}. it follows from the construction that bn+1 is finite, an+1 ∩ bn+1 = ∅, bn ⊆ bn+1, and bn+1 \ bn 6= ∅. in addition, coord(uig ) ⊆ bn+1, for every g ∈ {0, 1} bn and each i ≤ n. therefore, an+1 and bn+1 satisfy conditions (i)–(vii). thus we have constructed two families {an : n ∈ ω} and {bn : n ∈ ω}. now we define a = ⋃ {an : n ∈ ω} and b = ⋃ {bn : n ∈ ω}. it follows from (i)–(iv) that a and b are disjoint infinite subset of ω. condition (vii) implies that a ∪ b = ω. it remains to verify that πa(s) and πb (s) are of the first category in {0, 1}a and in {0, 1}b, respectively. to prove that πa(s) = ⋃∞ m=0 πa(fm) is of the first category in {0, 1} a, it suffices to verify that πa(fm) is nowhere dense in {0, 1} a, for each m ∈ ω. indeed, fix an open canonical non-empty set u ⊆ {0, 1}a and m ∈ ω. as coord(u ) ⊆ a is finite, (iii) implies that there is k ∈ ω such that coord(u ) ⊆ ak. let n = max{k, m}, then coord(u ) ⊆ an and m ≤ n. take an arbitrary f ∈ πaan (u ) ⊆ {0, 1} an . by (v), there exists an open canonical set umf in {0, 1}ω\bn such that π ω\bn an (umf ) = {f}, umf ∩ πω\bn (fm) = ∅ and an ⊆ coord(umf ) ⊆ an+1 ⊆ a. put v = π ω\bn a (umf ) and let k = coord(u ). we have: k ⊆ an ⊆ coord(umf ) = coord(v ) (1) and πak (u ) = {π an k (f )} = π ω\bn k (umf ) = π a k (v ). (2) it follows from (1) and (2) that v ⊆ u . to verify that v ∩ πa(fm) = ∅, we assume the contrary and choose an element g ∈ v ∩ πa(fm). since coord(v ) ⊆ an+1 ⊆ a, we have that π a an+1 (g) ∈ πaan+1 (v ) ∩ πan+1 (fm). it follows from the definition of v that πaan+1 (v ) = π ω\bn an+1 (umf ). hence, connected metrizable subtopologies 147 πaan+1 (g) ∈ π ω\bn an+1 (umf ) ∩ πan+1 (fm). take an element g̃ ∈ πω\bn (fm) such that π ω\bn an+1 (g̃) = πaan+1 (g). then g̃ ∈ umf , since coord(umf ) ⊆ an+1. we obtain a contradiction, as umf ∩ πω\bn (fm) = ∅. we have thus proved that πa(fm) is nowhere dense in {0, 1} a for each m ∈ ω and, hence, πa(s) =⋃∞ m=1 πa(fm) is of the first category in {0, 1} a. the verification that πb(s) =⋃∞ m=1 πb(fm) is a first category set in {0, 1} b is similar. � lemma 3.6. let s be a set of the first category in the cantor set c. there is a decomposition ω = ⋃∞ n=0 an such that each an is infinite, an ∩ am = ∅ if n 6= m, and πan (s) is a first category set in {0, 1} an for every n ∈ ω. proof. the proof consists of applying ω times the previous lemma. � if f : x → y is a continuous mapping and u ⊆ x, then we put f #(u ) = y \ f (x \ u ). in the following lemma we collect some well-known facts that will be frequently used in the sequel. lemma 3.7. (a) let f : x → y be a continuous closed and irreducible mapping. if u ⊆ x is open, then v = f −1(f #(u )) is an open dense subset of u (see [1, problem 109, chapter vi]). (b) the inverse image of a closed nowhere dense set under a continuous closed irreducible mapping is a closed nowhere dense set. (c) if x is a gδ-set in a compact space and x does not have isolated points, then |x| ≥ c (see [2, 3.12.11 (b)]). (d) a non-empty open subset of the cantor set can be represented as the union of countably many pairwise disjoint copies of the cantor set. theorem 3.8. every compact metrizable space without isolated points can be represented as the union of 2 ω pairwise disjoint copies of the cantor set. proof. we consider the cantor set c and a compact metrizable space without isolated points y . we can assume that y ⊆ iω, where i is the closed unit interval. let f : c → i be the continuous mapping onto i considered in the proof of theorem 3.3. now define g : cω → iω by g = ∏ {gi : i ∈ ω} where gi = f for each i ∈ ω. since g is a perfect mapping, there exists a closed subset c′ ⊆ cω such that g(c′) = y and g↾c′: c ′ → y is irreducible (see [2, 3.1.c (a)]). clearly, c′ does not have isolated points and hence is homeomorphic to c. therefore, we can assume from the very beginning that there exists a continuous irreducible mapping of c onto y which is denoted by the same letter f . let s = {x ∈ c : |f −1(f (x))| > 1}. our first step is to verify that s is of the first category in c. for each m ∈ n, we choose a finite covering {um1, . . . , umkm} of c by clopen sets of diameter less than or equal to 1/m with respect to a given metric on c. for each m ∈ n and each i = 1, 2, . . . , km, the set vmi = f −1(f #(umi)) is open and dense in umi by (a) of lemma 3.7 and, hence, fmi = umi \ vmi is closed 148 i. druzhinina and nowhere dense in c. therefore, for each m ∈ n the set fm = ⋃km i=1 fmi is closed and nowhere dense in c. put f = ⋃∞ m=1 fm. we claim that s = f . indeed, for x ∈ c \ s we have that |f −1(f (x))| = 1. from this, if x ∈ umi for some m ∈ n and i ∈ {1, 2, . . . , km}, then x ∈ f −1(f #(uni)) = vmi, that is, x /∈ fmi = umi \ vmi. now, it is evident that x /∈ f . thus, f ⊆ s. for x ∈ s, let z = f (x). there exists y ∈ c \ {x} such that z = f (y). we choose m ∈ n and i ∈ {1, 2, . . . , km} such that x ∈ umi and y /∈ umi. then z = f (y) ∈ f (y \umi), that is, z /∈ f #(umi). we have that x ∈ f −1(z)∩umi ⊆ umi \ f −1(f #(umi)) = fmi ⊆ fm ⊆ f . therefore, s ⊆ f . this implies that s = f = ⋃∞ m=1 fm , where every fm is a closed nowhere dense subset of c. apply lemma 3.6 to find a family {an : n ∈ ω} of infinite 1 disjoint subsets of ω whose union is equal to ω and such that πan (s) is a first category set in cn = {0, 1} an for each n ∈ ω. it follows that the complement cn \ πan (s) is a dense subset of the cantor set cn. in addition, cn \ πan (s) is a gδ-set in cn, since every set πan (fm) is closed in cn. then by lemma 3.7(c), we obtain that |cn \ πan (s)| = c for each n ∈ ω. thus the family {an : n ∈ ω} and the set s ⊆ c satisfy the conditions of lemma 3.1 and, hence, we can represent c in the form c = ⋃ {cγ : γ ∈ γ}, where |γ| = 2 ω and every cγ is a copy of c. in addition, cγ ∩ cλ = ∅ if γ 6= λ and |s ∩ cγ| ≤ 1 for each γ ∈ γ. let kγ = f (cγ ) for each γ ∈ γ. since f ↾cγ is a homeomorphism, kγ is a copy of the cantor set for each γ ∈ γ and we have that y = ⋃ {kγ : γ ∈ γ}. note that the family {kγ : γ ∈ γ} has the following properties: (i) if cγ ∩ s = ∅, then kγ ∩ kλ = ∅ for each λ ∈ γ \ {γ}; (ii) if cγ ∩ s 6= ∅, then cγ ∩ s = {s} for some s ∈ s; in this case for each λ ∈ γ \ {γ} we have: (ii1) kγ ∩ kλ = ∅ if cλ ∩ f −1(f (s)) = ∅; (ii2) kγ ∩ kλ = {f (s)} if cλ ∩ f −1(f (s)) 6= ∅. now for every y ∈ f (s), we define ky = {kγ : γ ∈ γ, cγ ∩ f −1(y) 6= ∅}. it is easy to see that ⋃ ky ∩ ⋃ kz = ∅ if y 6= z and y = ⋃ {kγ : γ ∈ γ, cγ ∩ s = ∅} ∪ ⋃ { ⋃ ky : y ∈ f (s)}. (∗) consider the family ky for some y ∈ f (s). choose an element γy ∈ γ such that kγy ∈ ky. for every λ ∈ γ\{γy} such that kλ ∈ ky, put lλ = kλ\{y}. every lλ is an open subset of the cantor set, so it can be represented as the union of countably many disjoint copies of the cantor set (item (d) of lemma 3.7). since ⋃ ky = ⋃ {lλ : kλ ∈ ky, λ 6= γy} ∪ kγy , for every y ∈ f (s), the set ⋃ ky can be represented as the union of some number κy of disjoint copies of the cantor set, where |ky| ≤ κy ≤ |ky| · ω. taking into account the equality (∗), we obtain the conclusion of the theorem. � remark 3.9. let y be a compact metrizable space without isolated points, c the cantor set and f : c → y a continuous irreducible mapping. if s is a set connected metrizable subtopologies 149 of the first category in c and {x ∈ c : |f −1(f (x))| > 1} ⊆ s, then repeating the argument in the proof of theorem 3.8 one can represent y as the union of 2ω pairwise disjoint copies of the cantor set such that every copy meets the set f (s) in at most one point. due to the above observation it is possible to extend theorem 3.8 as follows: theorem 3.10. every separable completely metrizable space without isolated points can be represented as the union of 2ω pairwise disjoint copies of the cantor set. proof. let x be a separable completely metrizable space without isolated points. we can assume that x ⊆ iω. then y = cliω (x) is a metrizable compactification of x. since x has no isolated points and is dense in y , it follows that y is a compact metrizable space without isolated points. the space x is čech-complete, therefore, the remainder r = y \ x is a fσ-set in y . hence, r = ⋃∞ n=1 fn, where every fn is a closed nowhere dense subset of y . consider a continuous irreducible mapping f : c → y , where c is the cantor set (see the proof of theorem 3.8). then each f −1(fn) is a closed nowhere dense subset of c, by (b) of lemma 3.7, and s′ = f −1(r) = ⋃∞ n=1 f −1(fn). in the proof of theorem 3.8 we have established that s′′ = {x ∈ c : |f −1(f (x))| > 1} is the union of countably many closed nowhere dense subsets of c. put s = s′ ∪ s′′ and apply remark 3.9 to conclude that y = ⋃ {kγ : γ ∈ γ}, where |γ| = 2ω, every kγ is a copy of the cantor set, kγ ∩ kλ = ∅ if γ 6= λ and |kγ ∩ f (s)| ≤ 1 for each γ ∈ γ. since r ⊆ f (s), the set r ∩ kγ is empty or consists of one point. for each γ ∈ γ, we define k̃γ = kγ \ (kγ ∩ r). it is evident that x = ⋃ {k̃γ : γ ∈ γ}. every k̃γ is a copy the cantor set or an open subset of this copy. in the last case k̃γ can be represented as a union of ω pairwise disjoint copies of the cantor set, by (d) of lemma 3.7. this finishes the proof. � references [1] a. v. arhangel’skii and v. i. ponomarev, fundamentals of general topology: problems and exercises (translated from the russian), mathematics and its applications, d. reidel publishing co., dordrecht, 1984. xvi+415 pp. [2] r. engelking, general topology, helderman verlag, berlin 1989. [3] p. delaney and w. just, two remarks on weaker connected topologies, comment. math. univ. carolin. 40 no. 2 (1999), 327–329. [4] i. druzhinina, condensations onto connected metrizable spaces, houston j. math. 30 no. 3, (2004), 751–766. [5] k. kunen, set theory, north holland, 1980. [6] m. g. tkachenko, v. v. tkachuk, v. uspenskij, and r. g. wilson, in quest of weaker connected topologies, comment. math. univ. carolin. 37 no. 4 (1996), 825–841. 150 i. druzhinina received may 2004 accepted april 2005 irina druzhinina (mich@xanum.uam.mx) departamento de matemáticas, uam, iztapalapa, av. san rafael atlixco 186, col. vicentina, del. iztapalapa, c.p. 09340, méxico d.f. songagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 13-20 embedding into discretely absolutely star-lindelöf spaces ii yan-kui song ∗ abstract. a space x is discretely absolutely star-lindelöf if for every open cover u of x and every dense subset d of x, there exists a countable subset f of d such that f is discrete closed in x and st(f, u) = x, where st(f, u) = ⋃ {u ∈ u : u ∩f 6= ∅}. we show that every hausdorff star-lindelöf space can be represented in a hausdorff discretely absolutely star-lindelöf space as a closed gδ-subspace. keywords: star-lindelöf, absolutely star-lindelöf, centered-lindelöf 2000 ams classification: 54d20, 54g20 1. introduction by a space, we mean a topological space. a space x is absolutely starlindelöf (see [1]) (discretely absolutely star-lindelöf)(see [12, 13]) if for every open cover u of x and every dense subset d of x, there exists a countable subset f of d such that st(f, u) = x (f is discrete and closed in x and st(f, u) = x, respectively), where st(f, u) = ⋃ {u ∈ u : u ∩ f 6= ∅}. a space x is star-lindelöf (see [4, 7] under different names) (discretely starlindelöf)(see [9, 16]) if for every open cover u of x, there exists a countable subset (a countable discrete closed subset, respectively) f of x such that st(f, u) = x. it is clear that every separable space and every discretely starlindelöf space are star-lindelöf as well as every space of countable extent(in particular, every countably compact space or every lindelöf space). a family of subsets is centered (linked) provided every finite subfamily (every two elements, respectively) has nonempty intersection and a family is called ∗the author acknowledges support from the nsf of china grant 10571081 and project supported by the national science foundation of jiangsu higher education institutions of china (grant no 07kjb110055) 14 y.-k. song σ-centered (σ-linked) if it is the union of countably many centered subfamilies(linked subfamilies, respectively). a space x is centered-lindelöf (linkedlindelöf) (see [2, 3]) if for every open cover u of x has σ-centered (σ-linked) subcover. from the above definitions, it is not difficult to see that every discretely absolutely star-lindelöf space is absolutely star-lindelöf, every discretely absolutely star-lindelöf space is discretely star-lindelöf, every absolutely starlindelöf space is star-lindelöf, every star-lindelöf space is centered-lindelöf, every centered-lindelöf space is linked-lindelöf. bonanzinga and matveev [2] proved that every hausdorff (regular, tychonoff) linked-lindelöf space can be represented as a closed subspace in a hausdorff (regular, tychonoff, respectively)star-lindelöf space. they asked if every hausdorff (regular, tychonoff) linked-lindelöf space can be represented as a closed gδ-subspace in a hausdorff (regular, tychonoff, respectively) starlindelöf space. the author [10] gave a positive answer to their question. the author [10] showed that every hausdorff (regular, tychonoff) linked-lindelöf space can be represented as a closed gδ-subspace in a hausdorff (regular, tychonoff, respectively) absolutely star-lindelöf space. the author [13] showed that every separable hausdorff (regular, tychonoff, normal) star-lindelöf space can be represented in a hausdorff (regular, tychonoff, normal, respectively) discretely absolutely star-lindelöf space as a closed gδ-subspace. the author [14] showed that every hausdorff linked-lindelöf space can be represented in a hausdorff discretely absolutely star-lindelöf space as a closed subspace and asked the following question: question 1.1. is it true that every hausdorff (regular, tychonoff) linkedlindelöf-space can be represented a closed gδ-subspace in a hausdorff (regular, tychonoff, respectively) discretely absolutely star-lindelöf space? the purpose of this note is to give a construction showing every hausdorff linked-lindelöf space can be represented in a hausdorff discretely absolutely star-lindelöf space as a closed gδ-subspace, which give a positive answer to the above question in the class of hausdorff spaces. throughout this paper, the cardinality of a set a is denoted by |a|. let ω denote the first infinite cardinal. for a cardinal κ, let κ+ be the smallest cardinal greater than κ. as usual, a cardinal is the initial ordinal and an ordinal is the set of smaller ordinals. when viewed as a space, every cardinal has the usual order topology. for each pair of ordinals α, β with α < β, we write [α, β] = {γ : α ≤ γ ≤ β} and (α, β) = {γ : α < γ < β}. other terms and symbols that we do not define will be used as in [5]. 2. embedding into discretely absolutely star-lindelöf spaces as a closed gδ-subspaces first, we show that every hausdorff star-lindelöf space can be represented in a hausdorff discretely absolutely star-lindelöf space as a closed gδ-subspace. embedding into discretely absolutely star-lindelöf spaces ii 15 recall the alexandorff duplicate a(x) of a space x. the underlying set of a(x) is x × {0, 1}; each point of x × {1} is isolated and a basic neighborhood of a point 〈x, 0〉 ∈ x ×{0} is of the from (u ×{0})∪((u ×{1})\{〈x, 1〉}), where u is a neighborhood of x in x. it is well-known that a(x) is hausdorff(regular, tychonoff, normal) iff x is, a(x) is compact iff x is and a(x) is lindelöf iff x is. recall from [6] that a space x is absolutely countably compact (=acc) if for every open cover u of x and every dense subset d of x, there exists a finite subset f of d such that st(f, u) = x. it is not difficult to show that every hausdorff acc space is countably compact (see [6]). in our construction, we use the following lemma. lemma 2.1 ([8, 15]). if x is countably compact, then a(x) is acc. moreover, for any open cover u of a(x), there exists a finite subset f of x × {1} such that a(x) \ st(f, u) ⊆ x × {0} is a finite subset consisting of isolated points of x × {0}. theorem 2.2. every hausdorff star-lindelöf space can be represented in a hausdorff discretely absolutely star-lindelöf space as a closed gδ-subspace. proof. if |x| ≤ ω, then x is separable. the author [13] showed that every separable hausdorff (regular, tychonoff, normal) space can be represented in hausdorff (regular, tychonoff, normal, respectively) discretely absolutely starlindelöf space as a closed gδ-subspace. let x be a star-lindelöf space with |x| > ω and let t be x with the discrete topology and let y = t ∪ {∞}, where ∞ /∈ t be the one-point lindelöfication of t . pick a cardinal κ with κ ≥ |x|. define s(x, κ) = x ∪ (y × κ+). we topologize s(x, κ) as follows: y × κ+ has the usual product topology and is an open subspace of s(x, κ), and a basic neighborhood of a point x of x takes the form g(u, α) = u ∪ (u × (α, κ+)), where u is a neighborhood of x in x and α < κ+. then, it is easy to see that x is a closed subset of s(x, κ) and s(x, κ) is hausdorff if x is hausdorff. let r(x) = a(s(x, κ)) \ (x × {1}). then, r(x) is hausdorff if x is hausdorff. let p(r(x)) = ((x × {0}) × {ω}) ∪ (r(x) × ω) be the subspace of the product of r(x) × (ω + 1). for each n ∈ ω, let xω = (x × {0}) × {ω} and xn = r(x) × {n} for each n ∈ ω. then, p(r(x)) = xω ∪ ∪n∈ω xn. 16 y.-k. song from the construction of the topology of p(r(x)), it is not difficult to see that x can be represented in p(r(x)) as a closed gδ-subspace, since x is homeomorphic to xω, and p(r(x)) is hausdorff if x is hausdorff. we show that p(r(x)) is discretely absolutely star-lindelöf. to this end, let u be an open cover of p(r(x)). without loss of generality, we assume that u consists of basic open sets of p(r(x)). let s be the set of all isolated points of κ+ and let dn1 = (((t × s) × {0}) × {n}) ∪ (((t × κ +) × {1}) × {n}), dn2 = (({∞} × κ +) × {1}) × {n} and dn = dn1 ∪ dn2 for each n ∈ ω. if we put d = ∪n∈ωdn. then, every element of d is isolated in p(r(x)), and every dense subset of p(r(x)) contains d. thus, it is sufficient to show that there exists a countable subset f of d such that f is discrete closed in p(r(x)) and st(f, u) = p(r(x)). for each x ∈ x, there exists a ux ∈ u such that 〈〈x, 0〉, ω〉 ∈ ux, hence there exist αx < κ +, nx ∈ ω and an open neighborhood vx of x in x such that ((vx × {0}) × [nx, ω]) ∪ (a(vx × (αx, κ +)) × [nx, ω)) ⊆ ux. if we put v = {vx : x ∈ x}, then v is an open cover of x. for each n ∈ ω, let x′ n = ∪{x : nx = n}, then x = ∪n∈ωx ′ n . for each x′ ∈ x \ x′ n , there exists a ux′ ∈ u such that 〈〈x′, 0〉, n〉 ∈ ux′ . hence, there exist αx′ < κ + and an open neighborhood vx′ of x ′ in x such that ((vx′ × {0}) × {n}) ∪ (a(vx′ × (αx′ , κ +)) × {n}) ⊆ ux′ . if we put vn = {vx : x ∈ x ′ n } ∪ {vx′ : x ′ ∈ x \ x′ n }. then, vn is an open cover of x. hence, there exists a countable subset f ′ n of x such that x = st(f ′ n , u), since x is star-lindelöf. if we pick αn0 > max{sup{αx : x ∈ x ′ n }, sup{αx′ : x ′ ∈ x \ x′ n }}. then, αn0 < κ +, since |x| ≤ κ. let xn1 = ((x × {0}) × {n}) ∪ (a(t × [αn0, κ +)) × {n}); xn2 = a(t × [0, αn0]) × {n} and xn3 = a({∞} × κ +) × {n}). then, xn = xn1 ∪ xn2 ∪ xn3. let fn1 = ((f ′ n × {αn0}) × {1}) × {n}. then, fn1 is a countable subset of dn1 and ((x′ n × {0}) × {ω}) ∪ xn1 ⊆ st(fn1, u), since ux ∩ fn1 6= ∅ for each x ∈ x ′ n and ux′ ∩ fn1 6= ∅ for each x ′ ∈ x \ x′ n . since fn1 ⊆ dn1 and fn1 is countable. then, fn1 is closed in xn by the embedding into discretely absolutely star-lindelöf spaces ii 17 construction of the topology of xn. hence, fn1 is closed in p(r(x)), since xn is open and closed in p(r(x)). on the other hand, since y is lindelöf and [0, αn0] is compact, then y × [0, αn0] is lindelöf, hence xn2 = a(y × [0, αn0]) × {n} is lindelöf. for each α ≤ αn0, there exists a uα ∈ u such that 〈〈〈∞, α〉, 0〉, n〉 ∈ uα. hence, there exists an open neighborhood vα of α in κ + and an open neighborhood v ′ α of ∞ in y such that (a(v ′ α × vα) × {n}) \ (〈〈〈∞, α〉, 1〉, n〉) ⊆ uα. let v′ n = {vα : α ≤ αn0}. then, v ′ n is an open cover of [0, αn0]. hence, there exists a finite subcover vα1 , vα2 , ...vαm , since [0, αn0] is compact. let tn = ∪{t \ v ′ αi : i ≤ m}. then, tn is a countable subset of t . for each i ≤ m, we pick xi ∈ dn ∩ uαi . let f ′ n2 = {xi : i ≤ m}. then, f ′ n2 is a finite subset of dn and ((({∞} × [0, αn0]) × {0}) × {n}) ∪ (a((t \ tn) × [0, αn0]) × {n}) ⊆ st(f ′ n2, u). for each t ∈ tn, since {t} × [0, αn0] is compact, then a({t} × [0, αn0]) × {n} is compact, hence there exists a finite subset ft of dn such that a({t} × [0, α0]) × {n} ⊆ st(ft, u). let f ′′ n2 = ∪{ft : t ∈ tn}. then, f ′′ n2 is countable, since tn is countable. since f ′′ n2 ∩ (a(y × {α}) × {n}) is countable for each α < κ + and f ′′ n2 ∩ (a({t} × κ+) × {n} is finite for each t ∈ t , then f ′′ n2 is closed in xn by the construction of the topology of xn, hence fn2 is closed in p(r(x)), since xn is open closed in p(r(x)). by the definition of f ′′ n2, we have a(tn × [0, αn0]) × {n} ⊆ st(f ′′ n2, u). if we put fn2 = f ′ n2 ∪ f ′′ n2. then, fn2 is a countable subset of dn and f ′′ n2 is closed in p(r(x)), since f ′ n1 is finite and and f ′′ n2 is closed in p(r(x)). by the definition of fn2, we have xn2 ∪ ((({∞} × [0, αn0]) × {0}) × {n}) ⊆ st(fn2, u). finally, we show that there exists a finite subset fn of dn such that xn3 ⊆ st(fn3, u). since {∞}×κ + is countably compact, then, by lemma 2.1, a({∞}× κ+) × {n} is acc and there exists a finite subset f ′ n3 ⊆ dn2 such that en = xn3 \ st(f ′ n3, u) ⊆ (({∞} × κ +) × {0}) × {n} is a finite subset and each point of en is an isolated point of (({∞}× κ +)×{0})×{n}. for each point x ∈ en, there exists a ux ∈ u such that x ∈ ux. for each point x ∈ en, pick dx ∈ dn ∩ ux. let f ′′ n3 = {dx : x ∈ e}, then f ′′ n3 is a finite subset of dn and e ⊆ st(f ′′ n3, u). if we put fn3 = f ′ n3 ∪ f ′′ n3, then fn3 is a finite subset of dn and xn3 ⊆ st(fn3, u). 18 y.-k. song if we put fn = fn1 ∪ fn2 ∪ fn3, then fn is a countable subset of dn such that ((x′ n × {0}) × {ω}) ∪ xn ⊆ st(fn, u). since fn1 and fn2 are closed in p(r(x)), fn3 is finite and each point of fn is isolated, then fn is discrete closed in p(r(x)). let f = ∪n∈ωfn. then, f is a countable subset of d and st(f, u) = ∪n∈ωst(fn, u) ⊇ ∪n∈ω(((x ′ n × {0}) × {ω}) ∪ xn) = p(r(x)). since each point of f is isolated, then f is discrete in p(r(x)). since fn is discrete closed in xn and xn is open closed in p(r(x)) for each n ∈ ω, then f has not accumulation points in r(x) × ω. on the other hand, since f is countable and κ ≥ |x| > ω, then every point of xω is not accumulation point of f by the construction of the topology of p(r(x)). this shows that f is closed in p(r(x)), which completes the proof. � since every discretely absolutely star-lindelöf space is discretely star-lindelöf, the next corollary follows from theorem 2.2. corollary 2.3. every hausdorff star-lindelöf space can be represented in a hausdorff discretely star-lindelöf space as a closed gδ-subspace. since every discretely absolutely star-lindelöf space is absolutely star-lindelöf, the next corollary follows from theorem 2.2. corollary 2.4. every hausdorff star-lindelöf space can be represented in a hausdorff absolutely star-lindelöf space as a closed gδ-subspace. the author [10] proved that every hausdorff (regular, tychonoff) linkedlindelöf space can be represented a closed gδ-subspace in hausdorff (regular, tychonoff, respectively) star-lindelöf space. thus, we have the next corollary. corollary 2.5. every hausdorff linked-lindelöf space can be represented in a hausdorff discretely absolutely star-lindelöf space as a closed gδ-subspace. on the separation of theorem 2.2, song [14] showed that r(x) is tychonoff if x is a locally-countable (ie., each point of x has a neighborhood u with |u| ≤ ω) tychonoff space. thus, we have the following proposition by the construction of the topology of p(r(x)). proposition 2.6. if x is a locally countable tychonoff space, then p(r(x)) is tychonoff. by theorem 2.2 and proposition 2.6, we have the next corollary. corollary 2.7. every locally-countable, star-lindelöf tychonoff space can be represented in a discretely absolutely star-lindelöf tychonoff space as a closed gδ-subspace. the author [10] proved that every hausdorff (regular, tychonoff) linkedlindelöf space can be represented a closed gδ-subspace in hausdorff (regular, tychonoff, respectively) star-lindelöf space. thus, we have the following corollary by corollary 2.7. embedding into discretely absolutely star-lindelöf spaces ii 19 corollary 2.8. every locally-countable, linked-lindelöf tychonoff space can be represented in a discretely absolutely star-lindelöf tychonoff space as a closed gδ-subspace. remark 2.9. in theorem 2.2, even if x is locally-countable normal, r(x) need not be normal (hence, p(r(x)) need not be normal). indeed, x×{0} and a({∞} × κ+) are disjoint closed subsets of r(x) that can not be separated by disjoint open subsets of r(x). thus, the author does not know if every locally countable, normal star-lindelöf space can be represented in a normal discretely absolutely star-lindelöf space as a closed gδ-subspace. remark 2.10. the author does not know if every regular (tychonoff, normal) star-lindelöf space can be represented in a regular (tychonoff, normal, respectively) discretely absolutely star-lindelöf space as a closed subspace or as a closed gδ-subspace. references [1] m. bonanzinga, star-lindelöf and absolutely star-lindelöf spaces, quest. answers gen. topology 16 (1998), 79–104. [2] m. bonanzinga and m. v. matveev, closed subspaces of star-lindelöf and related spaces, east-west j. math. 2 (2000), no. 2, 171–179. [3] m. bonanzinga and m. v. matveev, products of star-lindelöf and related spaces, houston j. math. 27 (2001), 45–57. [4] e. k. van douwn, g. m. reed, a. w. roscoe and i. j. tree, star covering properties, topology appl. 39 (1991), 71–103. [5] r. engelking, general topology, revised and completed edition, heldermann verlag, berlin (1989). [6] m. v. matveev, absolutely countably compact spaces, topology appl. 58 (1994), 81–92. [7] m. v. matveev, a survey on star-covering properties, topological atlas 330 (1998). [8] w.-x. shi, y.-k. song and y.-z. gao, spaces embeddable as regular closed subsets into acc spaces and (a)-spaces, topology appl. 150 (2005), 19–31. [9] y.-k. song, discretely star-lindelöf spaces, tsukuba j. math. 25 (2001), no. 2, 371–382. [10] y.-k. song, remarks star-lindelöf spaces, quest. answers gen. topology 20 (2002), 49–51. [11] y.-k. song, closed subsets of absolutely star-lindelöf spaces ii , comment. math. univ. carolinae 44 (2003), no. 2, 329–334. [12] y.-k. song, regular closed subsets of absolutely star-lindelöf spaces, questions answers gen. topology 22 (2004), 131–135. [13] y.-k. song, some notes on star-lindelöf spaces, questions answers gen. topology 24 (2006), 11–15. [14] y.-k. song, embedding into discretely absolutely star-lindelöf spaces, comment. math. univ. carolinae 12 (2007), no. 2, 303–309. [15] j. e. vaughan, absolutely countably compactness and property (a), talk at 1996 praha symposium on general topology. [16] y. yasui and z.-m. gao, spaces in countable web, houston j. math. 25 (1999), 327–335. 20 y.-k. song received november 2007 accepted march 2009 yan-kui song (songyankuinjnu.edu.cn) institute of mathematics, school of mathematics and computer sciences, nanjing normal university, nanjing, 210097, p. r. china () @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 135-141 mappings on weakly lindelöf and weakly regular-lindelöf spaces anwar jabor fawakhreh and adem kılıçman abstract in this paper we study the effect of mappings and some decompositions of continuity on weakly lindelöf spaces and weakly regular-lindelöf spaces. we show that some mappings preserve these topological properties. we also show that the image of a weakly lindelöf space (resp. weakly regular-lindelöf space) under an almost continuous mapping is weakly lindelöf (resp. weakly regular-lindelöf). moreover, the image of a weakly regular-lindelöf space under a precontinuous and contracontinuous mapping is lindelöf. 2010 msc: primary: 54a05, 54c08, 54c10; secondary: 54c05, 54d20. keywords: lindelöf, weakly lindelöf and weakly regular-lindelöf spaces. almost continuous and almost precontinuous functions. 1. introduction among the various covering properties of topological spaces a lot of attention has been made to those covers which involve open and regular open sets. in 1959 frolik [9] introduced the notion of a weakly lindelöf space that afterward was studied by several authors. in 1982 balasubramanian [2] introduced and studied the notion of nearly lindelöf spaces. in 1984 willard and dissanayake [18] gave the notion of almost lindelöf spaces and in 1996 cammaroto and santoro [4] introduced the notion of weakly regular-lindelöf spaces on using regular covers. some generalizations of lindelöf spaces have been recently studied by the authors (see [7]) and by song and zhang [17]. moreover, decompositions of continuity have been recently of major interest among general topologist. they have been studied by many authors, including singal and singal [16], mashhour et al. [11], abd el-monsef et al. [1], nasef 136 a. j. fawakhreh and a. kılıçman and noiri [12], noiri and popa [13], dontchev [5], dontchev and przemski [6], kohli and singh [10] and many other topologists. throughout the present paper, spaces mean topological spaces on which no separation axioms are assumed unless explicitly stated otherwise. the interior and the closure of any subset a of a space x will be denoted by int(a) and cl(a) respectively. by regular open cover of x we mean a cover of x by regular open sets in (x, τ). the purpose of this paper is to study effect of mappings and decompositions of continuity on weakly lindelöf and weakly regular-lindelöf spaces. we also show that some mappings preserve these topological properties. we conclude that the image of a weakly lindelöf space (resp. weakly regular-lindelöf space) under an almost continuous mapping is weakly lindelöf (resp. weakly regularlindelöf). moreover, the image of a weakly regular-lindelöf space under a precontinuous and contra-continuous mapping is lindelöf. 2. preliminaries recall that a subset a ⊆ x is called regular open (regular closed) if a = int(cl(a)) (a = cl(int(a))). a space (x, τ) is said to be semiregular if the regular open sets form a base for the topology. it is called almost regular if for any regular closed set c and any singleton {x} disjoint from c, there exist two disjoint open sets u and v such that c ⊆ u and x ∈ v . note that a space x is regular if and only if it is semiregular and almost regular [14]. moreover, a space x is said to be submaximal if every dense subset of x is open in x and it is called extremally disconnected if the closure of each open set of x is open in x. a space x is said to be nearly paracompact [15] if every regular open cover of x admits an open locally finite refinement. definition 2.1. let (x, τ) and (y, σ) be topological spaces. a function f : x −→ y is said to be (1) almost continuous [16] if f−1(v ) is open in x for every regular open set v in y . (2) precontinuous [11] ( resp. β-continuous [1] ) if f−1(v ) ⊆ int(cl(f−1(v ))) ( resp. f−1(v ) ⊆ cl(int(cl(f−1(v )))) ) for every open set v in y . (3) almost precontinuous ( resp. almost β-continuous ) [12] if for each x ∈ x and each regular open set v in y containing f(x), there exists a set u in x containing x with u ⊆ int(cl(u)) ( resp. u ⊆ cl(int(cl(u))) ) such that f(u) ⊆ v . (4) contra-continuous [5] if f−1(v ) is closed in x for every open set v in y . note that almost continuity as well as precontinuity implies almost precontinuity and almost precontinuity as well as β-continuity implies almost βcontinuity but the converses, in general, are not true (see [6], [11] and [13]). mapping on weakly lindelöf spaces 137 definition 2.2. a topological space x is said to be nearly lindelöf [2] ( resp. almost lindelöf [18] ) if, for every open cover {u α : α ∈ ∆} of x, there exists a countable subset {α n : n ∈ n} ⊆ ∆ such that x = ⋃ n∈n int(cl(u αn )) ( resp. x = ⋃ n∈n cl(u αn ) ) . 3. mappings on weakly lindelöf spaces definition 3.1 ([9]). a topological space x is said to be weakly lindelöf if for every open cover {u α : α ∈ ∆} of x there exists a countable subset {α n : n ∈ n} ⊆ ∆ such that x = cl( ⋃ n∈n u αn ). it is obvious that every nearly lindelöf space is almost lindelöf and every almost lindelöf space is weakly lindelöf, but the converses are not true (see [4]). moreover, it is well known that the continuous image of a lindelöf space is lindelöf and in [8] it was shown that a δ-continuous image of a nearly lindelöf space is nearly lindelöf. for weakly lindelöf spaces we give the following theorem. theorem 3.2. let (x, τ) and (y, σ) be topological spaces. let f : x −→ y be an almost continuous surjection from x into y . if x is weakly lindelöf then y is weakly lindelöf. proof. let {u α : α ∈ ∆} be an open cover of y . then {int(cl(u α )) : α ∈ ∆} is a regular open cover of y . since f is almost continuous, f−1(int(cl(u α ))) is an open set in x. thus {f−1(int(cl(u α ))) : α ∈ ∆} is an open cover of the weakly lindelöf space x. so there exists a countable subset {α n : n ∈ n} ⊆ ∆ such that x = cl( ⋃ n∈n f −1(int(cl(u αn )))) ⊆ cl( ⋃ n∈n f −1(cl(u αn ))) = cl(f−1( ⋃ n∈n cl(u αn ))) ⊆ cl(f−1(cl( ⋃ n∈n u αn ))). since cl( ⋃ n∈n u αn ) is regular closed in y and f is almost continuous, we have f−1(cl( ⋃ n∈n u αn )) is closed in x. so x = cl(f−1(cl( ⋃ n∈n u αn ))) = f−1(cl( ⋃ n∈n u αn )). since f is surjective, y = f(x) = f(f−1(cl( ⋃ n∈n u αn ))) = cl( ⋃ n∈n u αn ). which implies that y is weakly lindelöf and completes the proof. � corollary 3.3. the almost continuous image of a weakly lindelöf space is weakly lindelöf. 138 a. j. fawakhreh and a. kılıçman since every continuous function is almost continuous, (proposition 3.5, [17]) becomes a direct corollary of theorem 3.2 above. corollary 3.4. weakly lindelöf property is a topological property. note that a weakly lindelöf, semiregular and nearly paracompact space x is almost lindelöf (see [4], theorem 3.8). so depending on theorem 3.2 above we conclude the following two corollaries. corollary 3.5. let f : x −→ y be an almost continuous surjection from x into y . if x is weakly lindelöf and y is semiregular and nearly paracompact then y is almost lindelöf. corollary 3.6. let f : x −→ y be an almost continuous surjection from x into y . if x is weakly lindelöf and y is regular and nearly paracompact then y is lindelöf. proposition 3.7. let f : x −→ y be an almost β-continuous surjection. if x is submaximal, extremally disconnected and weakly lindelöf then y is weakly lindelöf. proof. this follows immediately from theorem 3.2 above and ([12], theorem 4.3). � proposition 3.8. let f : x −→ y be an almost precontinuous surjection. if x is submaximal and weakly lindelöf then y is weakly lindelöf. proof. this follows immediately from theorem 3.2 above and ([12], theorem 4.4). � 4. mappings on weakly regular-lindelöf spaces definition 4.1 ([3]). an open cover {u α : α ∈ ∆} of a topological space x is called regular cover if, for every α ∈ ∆, there exists a nonempty regular closed subset c α of x such that c α ⊆ u α and x = ⋃ α∈∆ int(c α ). definition 4.2 ([4]). a topological space x is said to be weakly regularlindelöf if every regular cover {u α : α ∈ ∆} of x admits a countable subset {α n : n ∈ n} ⊆ ∆ such that x = cl( ⋃ n∈n u αn ). now we prove the following theorem. theorem 4.3. let (x, τ) and (y, σ) be topological spaces. let f : x −→ y be an almost continuous surjection from x into y . if x is weakly regular-lindelöf then y is weakly regular-lindelöf. proof. let f : x −→ y be an almost continuous function from the weakly regular-lindelöf space x onto y . let {u α : α ∈ ∆} be a regular cover of y . (i. e. for every α ∈ ∆ there exists a regular closed set c α ⊆ u α with y = ⋃ α∈∆ int(c α ).) but ⋃ α∈∆ int(c α ) ⊆ ⋃ α∈∆ c α ⊆ ⋃ α∈∆ u α ⊆ ⋃ α∈∆ int(cl(u α )). mapping on weakly lindelöf spaces 139 so ⋃ α∈∆ f −1(int(c α )) ⊆ ⋃ α∈∆ f −1(c α ) ⊆ ⋃ α∈∆ f −1(u α ) ⊆ ⋃ α∈∆ f −1(int(cl(u α ))). thus x = f−1(y ) = f−1( ⋃ α∈∆ int(c α )) = ⋃ α∈∆ f−1(int(c α )). since c α is regular closed and f is almost continuous, f−1(c α ) is closed. thus cl(int(f−1(c α ))) ⊆ f−1(c α ) ⊆ f−1(int(cl(u α ))). since int(c α ) is regular open in y and f is almost continuous, f−1(int(c α )) is open in x. thus ⋃ α∈∆ int(cl(int(f−1(c α )))) = ⋃ α∈∆ int(f−1(c α )) ⊇ ⋃ α∈∆ int(f−1(int(c α ))) = ⋃ α∈∆ f −1(int(c α )) = x. it means that, for every α ∈ ∆, there exists a regular closed set cl(int(f−1(c α ))) ⊆ f−1(int(cl(u α ))) such that x = ⋃ α∈∆ int(cl(int(f −1(c α )))). thus {f−1(int(cl(u α ))) : α ∈ ∆} is a regular cover of the weakly regular-lindelöf space x. so there exists a countable subset {α n : n ∈ n} ⊆ ∆ such that x = cl( ⋃ n∈n f−1(int(cl(u αn )))) ⊆ cl( ⋃ n∈n f−1(cl(u αn ))) = cl(f−1( ⋃ n∈n cl(u αn ))) ⊆ cl(f−1(cl( ⋃ n∈n u αn ))). since f is almost continuous and cl( ⋃ n∈n u αn ) is regular closed, f−1(cl( ⋃ n∈n u αn )) is closed in x. so x = cl(f−1(cl( ⋃ n∈n u αn ))) = f−1(cl( ⋃ n∈n u αn )). thus y = f(x) = f(f−1(cl( ⋃ n∈n u αn ))) ⊆ cl( ⋃ n∈n u αn ). this implies that y is weakly regular-lindelöf and completes the proof. � corollary 4.4. the almost continuous image of a weakly regular-lindelöf space is weakly regular-lindelöf. corollary 4.5. weakly regular-lindelöf property is a topological property. note that every regular and weakly regular-lindelöf space x is weakly lindelöf and if x, moreover, is nearly paracompact then it is lindelöf (see [7]). so depending on theorem 4.3 above we conclude the following two corollaries. corollary 4.6. let f : x −→ y be an almost continuous mapping from x onto y . if x is weakly regular-lindelöf and y is regular then y is weakly lindelöf. corollary 4.7. let f : x −→ y be an almost continuous mapping from x onto y . if x is weakly regular-lindelöf and y is regular and nearly paracompact then y is lindelöf. 140 a. j. fawakhreh and a. kılıçman next we prove the following proposition. proposition 4.8. the image of a weakly regular-lindelöf space under a precontinuous and contra-continuous mapping is lindelöf. proof. let f : x −→ y be a contra-continuous and precontinuous mapping from the weakly regular-lindelöf space x into y . let u = {u α : α ∈ ∆} be an open cover of f(x). for each x ∈ x, let u αx ∈ u such that f(x) ∈ u αx . since f is contra-continuous, f−1(u αx ) is closed in x. since f is precontinuous, f−1(u αx ) ⊆ int(cl(f−1(u αx ))) = int(f−1(u αx )). so f−1(u αx ) = int(f−1(u αx )). it follows that f−1(u αx ) is closed and open in x and hence {f−1(u αx ) : x ∈ x} is a regular cover of the weakly regular-lindelöf space x. thus there exists a countable subfamily {x n : n ∈ n} such that x = cl( ⋃ n∈n f−1(u αxn )) = cl(f−1( ⋃ n∈n u αxn )). since ⋃ n∈n u αxn is open in y and f is contra-continuous, f−1( ⋃ n∈n u αxn ) is closed in x. thus cl(f−1( ⋃ n∈n u αxn )) = f−1( ⋃ n∈n u αxn ). so x = f−1( ⋃ n∈n u αxn ). since f is surjective, f(x) = f(f−1( ⋃ n∈n u αxn )) ⊆ ⋃ n∈n u αxn . this implies that f(x) is lindelöf and completes the proof. � as in weakly lindelöf spaces we give the following propositions. proposition 4.9. let f : x −→ y be an almost β-continuous surjection. if x is submaximal, extremally disconnected and weakly regular-lindelöf then y is weakly regular-lindelöf. proof. the proof follows immediately from ([12], theorem 4.3) and theorem 4.3 above. � proposition 4.10. let f : x −→ y be an almost precontinuous surjection. if x is submaximal and weakly regular-lindelöf then y is weakly regular-lindelöf. proof. the proof follows immediately from ([12], theorem 4.4) and theorem 4.3 above. � references [1] m. e. abd el-monsef, s. n. el-deep and r. a. mahmoud, β-open sets and β-continuous mappings, bull. fac. sci. assiut univ. 12 (1983), 77–90. [2] g. balasubramanian, on some generalizations of compact spaces, glasnik mat. 17, no. 37 (1982), 367–380. [3] f. cammaroto and g. lo faro, spazi weakly compact, riv. mat. university parma 4, no. 7(1981), 383–395. [4] f. cammaroto and g. santoro, some counterexamples and properties on generalizations of lindelöf spaces, internat. j. math. math. sci. 19, no. 4(1996), 737–746. mapping on weakly lindelöf spaces 141 [5] j. dontchev, contra-continuous functions and strongly s-closed spaces, internat. j. math. math. sci. 19 (1996), 303–310. [6] j. dontchev and m. przemski, on the various decompositions of continuous and some weakly continuous functions, acta math. hungar. 71, no. 1-2 (1996), 109–120. [7] a. j. fawakhreh and a. kılıçman, on generalizations of regular-lindelöf spaces, internat. j. math. math. sci. 27, no. 9 (2001), 535–539. [8] a. j. fawakhreh and a. kılıçman, mappings and some decompositions of continuity on nearly lindelöf spaces, acta math. hungar. 97, no. 3 (2002), 199–206. [9] z. frolik, generalization of compact and lindelöf spaces, czechoslovak math. j. 9, no. 84 (1959), 172–217. [10] j. k. kohli and d. singh, between strong continuity and almost continuity, appl. gen. topol. 11, no. 1 (2010), 29–42. [11] a. s. mashhour, m. e. abd el-monsef and s. n. el-deep, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. egypt 53 (1982), 47–53. [12] a. a. nasef and t. noiri, some weak forms of almost continuity, acta math. hungar. 74, no. 3 (1997), 211–219. [13] t. noiri and v. popa, on almost β-continuous functions, acta math. hungar. 79, no. 4 (1998), 329–339. [14] m. k. singal and s. p. arya, on almost regular spaces, glasnik math. ser. iii 4, no. 24(1969), 89–99. [15] m. k. singal and s. p. arya, on nearly paracompact spaces, mat. vesnik 6, no. 21 (1969), 3–16. [16] m. k. singal and a. r. singal, almost-continuous mappings, yokohama math. j. 16 (1968), 63–73. [17] y. song and y. zhang, some remarks on almost lindelöf spaces and weakly lindelöf spaces, mat. vesnik 62, no. 1 (2010), 77–83. [18] s. willard and u. n. b. dissanayake, the almost lindelöf degree, canad. math. bull. 27, no. 4 (1984), 452–455. (received december 2010 – accepted march 2011) a. j. fawakhreh (abuanas7@hotmail.com) department of mathematics, collage of science, qassim university, p.o. box 6644, buraydah 51402, saudi arabia. a. kılıçman (akilicman@putra.upm.edu.my) department of mathematics and institute for mathematical research, university putra malaysia, 43400 upm, serdang, selangor, malaysia. mappings on weakly lindelöf and weakly [3pt] regular-lindelöf spaces. by a. j. fawakhreh and a. kılıçman dikranjangiordanoagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 85-119 arnautov’s problems on semitopological isomorphisms dikran dikranjan and anna giordano bruno abstract. semitopological isomorphisms of topological groups were introduced by arnautov [2], who posed several questions related to compositions of semitopological isomorphisms and about the groups g (we call them arnautov groups) such that for every group topology τ on g every semitopological isomorphism with domain (g, τ ) is necessarily open (i.e., a topological isomorphism). we propose a different approach to these problems by introducing appropriate new notions, necessary for a deeper understanding of arnautov groups. this allows us to find some partial answers and many examples. in particular, we discuss the relation with minimal groups and non-topologizable groups. 2000 ams classification: primary 22a05, 54h11; secondary: 18a20, 20f38, 20k45. keywords: a-complete topology, heisenberg group, markov group, minimal group, open mapping theorem, permutations group, semitopological isomorphism, tăımanov topology, topologizable group. 1. introduction it is easy to prove that for every continuous isomorphism f : (g, τ ) → (h, σ) of topological groups, there exist a topological group (g̃, τ̃ ) containing g as a topological subgroup and an open continuous homomorphism f̃ : (g̃, τ̃ ) → (h, σ) extending f [2, theorem 1] (see also [14, theorem 1.1] for continuous surjective homomorphisms). the following notion is motivated by the fact that it is not always possible to prove the existence of such g̃ and f̃ , asking g to be also a normal subgroup of g̃ (see also [1] for topological rings). 86 d. dikranjan and a. giordano bruno definition 1.1 ([2, definition 2]). a continuous isomorphism f : (g, τ ) → (h, σ) of topological groups is semitopological if there exist a topological group (g̃, τ̃ ) containing g as a topological normal subgroup and an open continuous homomorphism f̃ : (g̃, τ̃ ) → (h, σ) extending f . in other words semitopological isomorphisms are restrictions of open continuous surjective homomorphisms to normal subgroups. obviously the class of semitopological isomorphisms contains the class of topological isomorphisms. arnautov characterized semitopological isomorphisms [2, theorem 4]. we give his characterization in terms of commutators and of thin subsets, as done in [14]. for a neighborhood u of the neutral element eg of a topological group g call a subset m of g u -thin if ⋂ {x−1u x : x ∈ m} is still a neighborhood of eg (i.e., there exists a neighborhood u1 of eg in g such that xu1x −1 ⊆ u for every x ∈ m ). the subsets m of g that are u -thin for every u are precisely the thin sets in the sense of tkachenko [29, 30]. for example compact sets are thin. theorem 1.2 ([2, theorem 4]). let (g, τ ) and (h, σ) be topological groups. let f : (g, τ ) → (h, σ) be a continuous isomorphism. then f is semitopological if and only if for every u ∈ v(g,τ )(eg): (a) there exists v ∈ v(h,σ)(eh ) such that f −1(v ) is u -thin; (b) for every g ∈ g there exists vg ∈ v(h,σ)(eh ) such that [g, f −1(vg)] ⊆ u . in [14] we extended the notion of semitopological isomorphism introducing semitopological homomorphisms. we defined new properties and considered particular cases in order to give internal conditions, similar to those of theorem 1.2, which are sufficient or necessary for a continuous surjective homomorphism to be semitopological. finally we established various stability properties of the class of all semitopological homomorphisms. many particular cases are considered and they turn out to be useful also in this paper as well as other particular results; for those we will give references. in section 2 we give general properties of semitopological isomorphisms and see some stability properties of the class si of all semitopological isomorphisms. in fact it has been proved in [2] that the class si is stable under taking subgroups, quotients and products, but not under taking compositions. the aim of this paper is to discuss and answer the following problems raised by arnautov [2]: problem a ([2, problem 14]). find groups g such that for every group topology τ on g every semitopological isomorphism f : (g, τ ) → (h, σ), where (h, σ) is a topological group, is open. arnautov’s problems on semitopological isomorphisms 87 problem b ([2, problem 13]) let k be a class of topological groups. find (g, τ ) ∈ k such that every semitopological isomorphism f : (g, τ ) → (h, σ) in k is open. the third problem concerns compositions: problem c ([2, problem 15]) (a) which are the continuous isomorphisms of topological groups that are compositions of semitopological isomorphisms? (b) is every continuous isomorphism of topological groups composition of semitopological isomorphisms? 1.1. the open mapping theorem and its weaker versions. according to the banach’s open mapping theorem every surjective continuous linear map between banach spaces is open [3]. as a generalization, pták [22] introduced the notion of b-completeness for the class of linear topological spaces. it was based on the property weaker than openness, that can be formulated also in the larger class of topological groups as follows: a homomorphism f : g → h of topological groups is called almost open, if for every neighborhood u of eg in g the image f (u ) is dense in some neighborhood of eh in h. a topological group g is b-complete (respectively, br-complete) if every continuous almost open surjective homorphism (respectively, isomorphism) from g to any hausdorff group is open. these groups were intensively studied in the sixties and the seventies ([4], [15], [16], [27]). it was shown by husain [16] that locally compact groups as well as complete metrizable groups are b-complete. brown [4] found a common generalization of this fact by proving that čech-complete groups are b-complete. the following notion introduced by choquet (see döıtchinov [11]) and stephenson [26] in 1970 takes us closer to the spirit of banach’s open mapping theorem: definition 1.3. a hausdorff group topology τ on a group g is minimal if for every continuous isomorphism f : (g, τ ) → h, where h is a hausdorff topological group, f is a topological isomorphism. call g totally minimal if for every continuous homomorphism f : (g, τ ) → h, where h is hausdorff, f is open. clearly, the totally minimal groups are precisely the topological groups that satisfy the banach’s open mapping theorem. since all surjective homomorphisms between precompact groups are almost open, a precompact group is br-complete (respectively, b-complete) if and only if it is minimal (respectively, totally minimal). in particular, the br-complete precompact abelian groups coincide with the minimal abelian groups as every minimal abelian group is precompact according to the celebrated prodanov-stoyanov’s theorem. according to this theorem, an infinite minimal abelian group is never discrete. this radically changes in the non-abelian case. in the forties markov 88 d. dikranjan and a. giordano bruno asked whether every infinite group g is topologizable (i.e., admits a non-discrete hausdorff group topology). definition 1.4. a group g is: • markov if the discrete topology δg is the unique hausdorff group topology on g (i.e., δg is minimal); • totally markov if g/n is markov for every n ⊳ g. obviously totally markov implies markov and finite groups are totally markov, while every simple markov group is totally markov. denote by m and mt the classes of all markov and totally markov groups respectively. markov’s question (on whether m contains infinite groups), was answered only thirty-five years later by shelah [24] (who needed ch for his example, resolving simultaneously also kurosh’ problem) and ol′shanskii [21] (who made use of the properties of remarkable adian’s groups). a smaller class of groups arose in the solution of a specific problem related to categorical compactness in [10]: namely the subclass of mt consisting of those groups g ∈ mt such that every subgroup of g belongs to mt as well (these groups were named hereditarily non-topologizable by lukács [18]). it is still an open question whether an infinite hereditarily non-topologizable group exists ([9, 10, 18]). a possibility to relax the strong requirement in the open mapping theorem in the definition of minimal groups is to restrict the class of topological groups: definition 1.5. let k be a class of topological groups. a topological group (g, τ ) ∈ k is k-minimal if (g, σ) ∈ k and σ ≤ τ imply τ = σ. when k is the class of all metrizable abelian groups, k-minimal groups are precisely the minimal abelian groups that are metrizable [8], but in general a kminimal group need not be minimal. anyway, if h is the class of all hausdorff topological groups, then h-minimality is precisely the usual minimality. recently new generalizations of minimality for topological groups were considered (relative minimality and co-minimality, cf. [7, 25]). 1.2. main results. the next definition reminds the br-completeness (since we impose openness only on certain continuous isomorphisms, namely, the semitopological ones): definition 1.6. a group topology τ on g is a-complete if for every group topology σ on g, σ ≤ τ and idg : (g, τ ) → (g, σ) semitopological imply τ = σ. finally, we can formulate the notion that captures the core of problem a: definition 1.7. a group g is an arnautov group if every group topology on g is a-complete (i.e., if for every pair of group topologies τ, σ on g with σ < τ , idg : (g, τ ) → (g, σ) is not semitopological). hence problem a can be formulate also as follows: characterize the groups g such that every group topology on g is a-complete, that is, characterize the arnautov groups. arnautov’s problems on semitopological isomorphisms 89 we denote by a the class of all arnautov groups. tăımanov [28] introduced the group topology tg on a group g, which has the family of the centralizers of the elements of g as a prebase of the filter of the neighborhoods of eg. this topology was introduced with the aim of the topologization of abstract groups with hausdorff group topologies. since idg : (g, δg) → (g, σ) is semitopological if and only if σ ≥ tg (see [14, corollary 5.3] or remark 5.12) and we are studying arnautov groups, we need to impose that tg is discrete and we introduce the following notion. definition 1.8. a group g is: • tăımanov if tg = δg; • totally tăımanov if g/n is tăımanov for every n ⊳ g. obviously every simple tăımanov group is totally tăımanov. we denote by t and tt the classes of tăımanov and totally tăımanov groups respectively. since problem a in its full generality seems to be hard to handle (because of two universal quantifiers), we start considering a particular case, that is when the discrete topology on a group g is a-complete and we prove that for a group g the discrete topology is a-complete if and only if g ∈ t (see theorem 5.13). moreover we extend this result for almost trivial topologies (which are obtained from the trivial ones by extension, as their name suggests — see section 3), characterizing in theorem 5.15 when an almost trivial topology is a-complete in terms of t. moreover tt contains a, but we do not know if they coincide (see theorem 5.16 and question 5.17). example 5.18 considers properties of the permutations group s(z) related to problem a. first of all it shows that a-completeness has a behavior different from that of the usual minimality. indeed we see that s(z) admits at least two different but comparable a-complete group topologies. moreover s(z) is not tăımanov and consequently not arnautov. nevertheless s(z)/sω(z) is totally tăımanov but we do not know if it is also arnautov (see question 5.20). this question can be seen as a first step in answering the following one, which could give an infinite example of a simple infinite markov group without assuming ch (see question 5.27): does s(z)/sω(z) ∈ m? but the situation can be reversed: if s(z)/sω(z) ∈ m then s(z)/sω(z) ∈ a, in view of corollary 5.26(b), which says that every simple markov group is necessarily arnautov. thanks to this property we have the unique infinite arnautov group that we know at the moment, that is shelah group, which is an infinite simple markov group constructed under ch [24] (see example 5.29). the next definition, combining definition 1.6 (a-completeness) and definition 1.5 (k-minimality) will allow us to handle easier problem b. 90 d. dikranjan and a. giordano bruno definition 1.9. for a class k of topological groups, a topological group (g, τ ) from k is ak-complete if (g, σ) ∈ k, σ ≤ τ and idg : (g, τ ) → (g, σ) semitopological imply τ = σ. let g be the class of all topological groups. remark 1.10. (a) obviously k-minimality implies ak-completeness and k-minimality coincides with ak-completeness whenever all groups in k ⊆ g are abelian. (b) moreover a-completeness coincides with ag-completeness. so problem a can be seen as a particular case of problem b, namely with k = g. (c) if k ⊆ k′ are classes of topological groups, then for every g ∈ k ak′ complete implies ak-complete. in particular, if k ⊆ g and g ∈ k, then g a-complete implies g ak-complete. clearly ah-completeness is a generalization of minimality, since h-minimality is precisely the usual minimality, which is intensively studied, as noted in section 1.1. this is a strict generalization as shown by example 6.1. a topological group g has small invariant neighborhoods (i.e., g is sin ) if g is thin (i.e., it has a local base at eg of neighborhoods invariant under conjugation). we prove that a topological group, which is sin and ah-complete, is a-complete if and only it has trivial center (see remark 6.8). in particular, if g is a group with trivial center, its discrete topology is ah-complete if and only if g ∈ t (see corollary 6.6). so also in this case tăımanov groups play a central role. moreover we give an example of a small class k in which each element is ak-complete (see example 6.14). this class is built on the heisenberg group hr :=   1 r r 0 1 r 0 0 1   , that is the group of upper unitriangular 3 × 3 matrices over r, endowed with different group topologies. the group hr is nilpotent of class 2. in a forthcoming paper [6] we extend this example for generalized heiseberg groups, that is, the group of upper unitriangular 3 × 3 matrices over a unitary ring a. in example 7.5 we resolve negatively item (b) of problem c. moreover theorem 7.2 answers partially (a), in the case when the topologies on the domain and on the codomain are the discrete and the indiscrete one respectively. since we consider the trivial topologies, the condition that we find is exclusively algebraic. indeed we prove that idg : (g, δg) → (g, ιg) is composition of n semitopological isomorphisms if and only if g is nilpotent of class ≤ n, where n ∈ n+. arnautov’s problems on semitopological isomorphisms 91 notation and terminology. we denote by r, q, z, p, n and n+ respectively the field of real numbers, the field of rational numbers, the ring of integers, the set of primes, the set of natural numbers and the set of positive integers. let g be a group and x, y ∈ g. we denote by [x, y] the commutator of x and y in g, that is [x, y] = xyx−1y−1 and for x ∈ g and a subset y of g let [x, y ] = {[x, y] : y ∈ y }. more in general, if h and k are subgroups of g, let [h, k] = 〈[h, k] : h ∈ h, k ∈ k〉, and in particular the derived g′ of g is g′ = [g, g], that is, the subgroup of g generated by all commutators of elements of g. the center of g is z(g) = {x ∈ g : xg = gx, ∀g ∈ g} and for g ∈ g the centralizer of g in g is cg(g) = {x ∈ g : xg = gx}. the diagonal map ∆ : g → g×g is defined by ∆(g) = (g, g) for every g ∈ g. if h is another group, we denote by p1 : g × h → g and p2 : g × h → h the canonical projections on the first and the second component respectively. if f : g → h is a homomorphism, denote by γf the graph of f , that is the subgroup γf = {(g, f (g)) : g ∈ g} of g × h. if τ is a group topology on g then denote by v(g,τ )(eg) the filter of all neighborhoods of eg in (g, τ ) and by bτ a base of v(g,τ )(eg). if x is a subset of g, x τ stands for the closure of x in (g, τ ). if n is a normal subgroup of g and π : g → g/n is the canonical projection, then τq is the quotient topology of τ in g/n . moreover nτ denotes the subgroup {eg} τ . the discrete topology on g is δg and the indiscrete topology on g is ιg. for undefined terms see [12, 13]. 2. properties of semitopological isomorphisms in the next remark we discuss the possibility to consider only the case of one group g endowed with two different topologies τ ≥ σ taking idg : (g, τ ) → (g, σ) as the continuous isomorphism: remark 2.1. let (g, τ ), (h, η) be topological groups and f : (g, τ ) → (h, η) a continuous isomorphism. consider the topology σ = f −1(η) on g. this topology σ is coarser than τ and so idg : (g, τ ) → (g, σ) is a continuous isomorphism and (g, σ) is topologically isomorphic to (h, η). in particular idg : (g, τ ) → (g, σ) is semitopological if and only if f : (g, τ ) → (h, η) is semitopological. moreover the next proposition shows that semitopological is a “local” property, like the stronger property open. the proof is a simple application of theorem 1.2. proposition 2.2. let g be a group and τ, σ group topologies on g such that σ ≤ τ . then idg : (g, τ ) → (g, σ) is semitopological if there exists a τ -open subgroup n of g such that idg ↾n : (n, τ ↾n ) → (n, σ ↾n ) is semitopological. 92 d. dikranjan and a. giordano bruno the following theorems show the stability of the class of semitopological isomorphisms under taking subgroups, quotients and products. theorem 2.3 ([2, theorems 7 and 8]). let g be a group, σ ≤ τ group topologies on g and suppose that idg : (g, τ ) → (g, σ) is semitopological. (a) if a is a subgroup of g, then ida : (a, τ ↾a) → (a, σ ↾a) is semitopological. (b) if a is a normal subgroup of g, then idg/a : (g/a, τq) → (g/a, σq ) is semitopological. theorem 2.4 ([2, theorem 9], [14, theorem 6.15]). let {gi : i ∈ i} be a family of groups and {τi : i ∈ i}, {σi : i ∈ i} families of group topologies such that σi ≤ τi are group topologies on gi for every i ∈ i. then idgi : (gi, τi) → (gi, σi) is semitopological for every i ∈ i if and only if ∏ i∈i idgi :∏ i∈i (gi, τi) → ∏ i∈i (gi, σi) is semitopological. the next lemma shows a cancellability property of compositions of semitopological isomorphisms. lemma 2.5 ([14, theorem 6.11]). let σ ≤ τ be group topologies on a group g. if idg : (g, τ ) → (g, σ) is semitopological, then for a group topology ρ on g such that σ ≤ ρ ≤ τ , idg : (g, τ ) → (g, ρ) is semitopological. in a particular case, that is for initial topologies, the converse implication of theorem 2.3(b) holds true: lemma 2.6. let g be a group and n a normal subgroup of g. let σ ≤ τ be group topologies on g/n and σi ≤ τi the respective initial topologies on g. then idg : (g, τi) → (g, σi) is semitopological if and only if idg/n : (g/n, τ ) → (g/n, σ) is semitopological. in the next theorem we consider the particular cases when one of the two topologies on g is trivial: theorem 2.7 ([2, corollary 5], [14, corollary 5.11]). let g be a group and τ a group topology on g. then: (a) idg : (g, δg) → (g, τ ) is semitopological if and only if cg(g) is τ -open for every g ∈ g; (b) idg : (g, τ ) → (g, ιg) is semitopological if and only if g ′ ≤ nτ . since z(g) ⊆ cg(g) for every g ∈ g, by (a) idg : (g, δg) → (g, τ ) semitopological implies z(g) τ -open. the condition g′ ≤ nτ in (b) is equivalent to say that g ′ is indiscrete endowed with the topology inherited from (g, τ ). moreover, as noted in [14], it implies that (g, τ ) is sin. for sin groups condition (a) of theorem 1.2 is always verified, since sin groups are thin, so only condition (b) remains: arnautov’s problems on semitopological isomorphisms 93 proposition 2.8. let g be a group and σ ≤ τ group topologies on g. suppose that (g, τ ) is sin. then idg : (g, τ ) → (g, σ) is semitopological if and only if for every u ∈ v(g,τ )(eg) and for every g ∈ g there exists vg ∈ v(g,σ)(eg) such that [g, vg] ⊆ u . the next lemma gives a simple necessary condition of algebraic nature for a continuous isomorphism to be semitopological. lemma 2.9. let g be a group and σ ≤ τ group topologies on g, such that idg : (g, τ ) → (g, σ) is semitopological. then [g, nσ] ≤ nτ . proof. by theorem 1.2, for every u ∈ v(g,τ )(eg) and every g ∈ g, there exists vg ∈ v(g,τ )(eg) such that [g, vg] ⊆ u . consequently [g, nσ] ⊆ u for every g ∈ g, so [g, nσ] ⊆ nτ for every g ∈ g and hence [g, nσ] ≤ nτ . � corollary 2.10. let g be a group and τ a group topology on g. if τ is hausdorff, then idg : (g, τ ) → (g, ιg) is semitopological if and only if g is abelian. proof. if idg : (g, τ ) → (g, ιg) is semitopological, by lemma 2.9 g ′ ≤ nτ = {eg} and hence g is abelian. if g is abelian every continuous isomorphism is semitopological. � in particular idg : (g, δg) → (g, ιg) is semitopological if and only if the group g is abelian. proposition 2.11. let g be a group and σ ≤ τ group topologies on g, such that idg : (g, τ ) → (h, σ) is semitopological. if z(g) = {eg} and τ is hausdorff, then σ is hausdorff as well. proof. since nτ = {eg} and [g, nσ] ≤ nτ by lemma 2.9, using the hypothesis z(g) = {eg} we conclude that nσ = {eg}. � 3. almost trivial topologies in this section we introduce a class of group topologies containing the trivial ones and with nice stability properties; moreover we extend theorem 2.7 to this class. definition 3.1. [14, definition 5.13] a topological group (g, τ ) is almost trivial if nτ is open in (g, τ ). since in this case τ is completely determined by the normal subgroup n := nτ of g, we denote an almost trivial topology on g by ζn , underling the role of the normal subgroup. every group topology on a finite group is almost trivial and every almost trivial group is sin. for example, for a group g, the discrete and the indiscrete topologies (i.e., the so-called trivial topologies) are almost trivial, with δg = ζ{eg} and ιg = ζg. this justifies the term used in definition 3.1. 94 d. dikranjan and a. giordano bruno lemma 3.2. let g be a simple non-abelian group and let τ be a group topology on g. then either nτ = g or nτ = {eg}, that is, either τ = ιg or τ is hausdorff, respectively. if τ is almost trivial, then τ is either discrete or indiscrete. the almost trivial topologies help also to express in simple terms topological properties: remark 3.3. given a topological group (g, τ ) and a normal subgroup n of g, it is possible to consider the group topology obtained “adding” to the open neighborhoods also n (since it is normal, it suffices to add n to the prebase of the neighborhoods and all the intersections u ∩ n , with u ∈ v(g,τ )(eg), give the neighborhoods of eg in the new topology). this new topology is sup{τ, ζn }. for example, if g is a group and τ its profinite topology, with bτ = {nα}α, where the nα are all the normal subgroups of g of finite index, then τ = supα ζnα . more in general, if τ is a linear topology on g, that is bτ = {nα}α, where nα are normal subgroups of g, then τ = supα ζnα . if (g, τ ) is a topological group, let τ̄ denote the quotient topology of (g, τ ) with respect to the normal subgroup nτ , which is indiscrete. then τ̄ is hausdorff. moreover (g, τ ) is almost trivial if and only if (g/nτ , τ̄ ) is discrete. analogously it is possible to consider the case when a topological group (g, τ ) has a discrete normal subgroup d such that (g/d, τq) is indiscrete. for groups with this property we have a strong consequence: lemma 3.4. let (g, τ ) be a topological group such that d is a discrete normal subgroup of (g, τ ) and (g/d, τq) is indiscrete. then (g, τ ) ∼= d × nτ , where d is discrete and nτ is indiscrete. in particular τ is almost trivial. proof. pick a symmetric neighborhood w of eg in g such that w 3∩d = {eg}. since (g/d, τq) is indiscrete, d is dense in g, so g = dw . let w1, w2 ∈ w . then there exists d ∈ d such that w1w2 ∈ dw . let w1w2 = dw for some w ∈ w . then d = w1w2w −1 ∈ w 3 ∩ d = {eg}. so w1w2 = w ∈ w . since w is symmetric, this proves that w is an open subgroup of m with w ∩ d = {eg}. hence the restriction of the canonical projection g → g/d to w gives a topological isomorphism w ∼= (g/d, τq). this shows that w is an indiscrete group. since nτ ≤ w is closed, we deduce that w = nτ . this proves that nτ is open in τ and that (g, τ ) ∼= d × nτ . � 3.1. permanence properties of the almost trivial topologies. the assignment n 7→ ζn defines an order reversing bijection between the complete lattice n (g) of all normal subgroups of a group g and the complete lattice at (g) of all almost trivial group topologies on g. let us note that the complete lattice at (g) is not a sublattice of the complete lattice t (g) of all group topologies on g. indeed, the meet of a family {ζni : i ∈ i} in at (g) is simply ζ⋂ i∈i ni , whereas the meet of a family {ζni : i ∈ i} in t (g) is the group topology having as prebase of the neighborhoods at eg the family {ζni : i ∈ i} (in arnautov’s problems on semitopological isomorphisms 95 other words, the latter topology may be strictly weaker than the former one in case i is infinite). the next lemma shows, among others, that the class of almost trivial groups is closed under taking subgroups and quotients. lemma 3.5. let (g, ζn ) be an almost trivial group, where n is a normal subgroup of g. (a) for every subgroup h of g: (a1) the topology induced on h by ζn is almost trivial and coincides with ζh∩n ; (a2) the following conditions are equivalent: (i) h is ζn -open; (ii) h is ζn -closed; (iii) h ≥ n . (b) for every normal subgroup n0 of g the quotient topology of ζn on g/n0 is almost trivial and coincides with ζn0n/n0 . remark 3.6. in connection to item (a1) of the previous lemma notice that if h is an open subgroup of a topological group g and h is almost trivial, then also g is almost trivial. now we show that the class of almost trivial groups is stable also with respect to taking finite products. lemma 3.7. let g1, g2 be groups and n1, n2 normal subgroups of g1, g2 respectively. then ζn1 × ζn2 = ζn1×n2 on g1 × g2. the next lemma follows directly from the definitions. lemma 3.8. let g be a topological group and n an indiscrete normal subgroup of g such that g/n is almost trivial. then g is almost trivial. we want to generalize this lemma and we need the following concept. definition 3.9. for a class of topological groups p one says that p has the three space property, if a topological group g belongs to p whenever n ∈ p and g/n ∈ p for some normal subgroup n of g. for example the class of all discrete groups and the class of all indiscrete groups have the three space property. so the next result shows that the class of all almost trivial groups is the smaller class with the three space property containing all discrete and all indiscrete groups. proposition 3.10. the class of almost trivial groups has the three space property. proof. we have to prove that, in case g is a group and n a normal subgroup of g, if τ is a group topology on g such that (n, τ ↾n ) and (g/n, τq ) are almost trivial, then (g, τ ) is almost trivial. let m be the normal subgroup of g containing n such that m/n = nτq . then m/n is indiscrete and open in g/n . consequently, m is open in (g, τ ). to end the proof we need to verify that m is almost trivial (see remark 3.6). 96 d. dikranjan and a. giordano bruno if τ ↾n is hausdorff, equivalently it is discrete, since it is almost trivial, and by lemma 3.4 m is almost trivial. so we consider now the general case. the subgroup n1 := nτ ∩ n is the closure of {eg} in m . then n1 is a normal subgroup of n . now the normal subgroup n/n1 of the hausdorff quotient group m/n1 is almost trivial and consequently discrete. moreover, the quotient (m/n1)/(n/n1) ∼= m/n is indiscrete. so by the previous case the group m/n1 is almost trivial. since the group n1 is indiscrete, we can conclude with lemma 3.8. � 3.2. semitopological isomorphisms between almost trivial topologies. since every almost trivial group is sin, it is possible to apply proposition 2.8 instead of theorem 1.2 to verify if a continuous isomorphism is semitopological. in case the topology on the domain or that on the codomain is almost trivial, the conditions of theorem 1.2 become simpler: proposition 3.11. let (g, σ) be a topological group and let σ ≤ τ be group topologies on g. (a) if τ is almost trivial, then idg : (g, τ ) → (g, σ) is semitopological if and only if for every g ∈ g there exists vg ∈ v(g,σ)(eg) such that [g, vg] ⊆ nτ . (b) if σ is almost trivial, then idg : (g, τ ) → (g, σ) is semitopological if and only if nσ is u -thin for every u ∈ v(g,τ )(eg) and [g, nσ] ≤ nτ . proof. (a) follows from proposition 2.8. (b) the necessity of the condition that nσ is u -thin for every u ∈ v(g,τ )(eg) follows from theorem 1.2, while the necessity of [g, nσ] ≤ nτ follows from lemma 2.9. the sufficiency of the two conditions is a consequence of theorem 1.2. � if τ is hausdorff in this proposition, then (b) becomes n ≤ z(g). so we have the following corollary, which can be also seen as a consequence of proposition 2.11. corollary 3.12. let g be a group. if τ is a hausdorff group topology on g, then for every non-central τ -open subgroup n of g idg : (g, τ ) → (g, ζn ) is not semitopological. combining together the two items of proposition 3.11 we have precisely the following corollary, which is the “almost trivial version” of theorem 1.2. furthermore it shows that the necessary condition of lemma 2.9 becomes also sufficient in the case of almost trivial topologies. corollary 3.13. [14, lemma 5.15] let g be a group and ζn ≥ ζl almost trivial group topologies on g. then idg : (g, ζn ) → (g, ζl) is semitopological if and only if [g, l] ≤ n . the next example is a consequence of this corollary. arnautov’s problems on semitopological isomorphisms 97 example 3.14. let g be a group and ζn an almost trivial group topology on g. consider (g, δg) idg −−→ (g, ζn ) idg −−→ (g, ιg). then: (a) idg : (g, δg) → (g, ζn ) is semitopological if and only if n ≤ z(g); (b) idg : (g, ζn ) → (g, ιg) is semitopological if and only if g ′ ≤ n . on a group g it is possible to consider the almost trivial topology generated by g′, that is ζg′ . a group g is perfect if g = g ′, and g is perfect if and only if ζg′ = ιg. remark 3.15. with this topology generated by the derived group, we can write again theorem 2.7(b) as: let g be a group and τ a group topology on g. then idg : (g, τ ) → (g, ιg) is semitopological if and only if τ ≤ ζg′ . remark 3.16. let n be a normal subgroup of a group g and let ζn be the respective almost trivial topology on g. let τ be a group topology on g. then idg : (g, ζnτ ) → (g, τ ) is continuous. moreover, if ζl is another almost trivial topology on g such that idg : (g, ζl) → (g, τ ) is continuous, then idg : (g, ζl) → (g, ζnτ ) is continuous. (g, ζnτ ) // (g, τ ) (g, ζl) :: u u u u u u u u u eek k k k k consequently idg : (g, ζl) → (g, τ ) semitopological implies idg : (g, ζl) → (g, ζnτ ) semitopological by lemma 2.5, that is, [g, nτ ] ≤ l by corollary 3.13. 4. tăımanov groups let f ∈ [g]<ω be a finite subset of g and cg(f ) = ⋂ x∈f cg(x) the centralizer of f in g. then c = {cg(f ) : f ∈ [g] <ω} is a family of subgroups of g closed under finite intersections. then the tăımanov topology tg has c as local base at eg, that is btg = c. we collect in the next lemma the first properties of this topology. lemma 4.1. let g be a group. then: (a) ntg = z(g); (b) tg is hausdorff if and only if z(g) = {eg}; (c) g is abelian if and only if tg = ιg; (d) in case g is finitely generated, tg is almost trivial; in particular tg = δg if and only if z(g) is trivial. 98 d. dikranjan and a. giordano bruno 4.1. permanence properties of the class t. the following results show that the tăımanov topology has nice properties. the next proposition proves that it is a functorial topology with respect to continuous surjective homomorphisms. proposition 4.2. let g be a group. then every surjective homomorphism f : (g, tg) → (h, th ) is continuous. proof. let h ∈ h and consider g ∈ g such that f (g) = h. then f (cg(g)) ⊆ ch (h). this proves the continuity of f : (g, tg) → (h, th ). � on the other hand, the next example shows that the tăımanov topology is not functorial with respect to open surjective homomorphisms. example 4.3. for hz :=   1 z z 0 1 z 0 0 1   the group of upper unitriangular 3×3 matrices over z, the canonical projection π : (hz, thz ) → (hz/z(hz), thz/z(hz)) is not open. indeed, since hz/z(hz) =: g is abelian, tg = ιg by lemma 4.1(c). moreover note that g ∼= z × z. let h =   1 0 0 0 1 1 0 0 1   ∈ hz. then chz (h) =   1 0 z 0 1 z 0 0 1   and π(chz (h)) ∼= {0} × z, which is not open in (g, ιg). lemma 4.4. let g = ∏ i∈i gi. then ∏ i∈i tgi ≤ tg. if i = {1, . . . , n} is finite, then tg1 × . . . × tgn = tg. proof. since all the canonical projections πi : (g, tg) → (gi, tgi ) are continuous by proposition 4.2, ∏ i∈i tgi ≤ tg. suppose now that i = {1, . . . , n} is finite. if f is a finite subset of g1 × . . . × gn, then it is contained in a finite subset of the form f1 × . . . × fn, where each fi is a finite subset of gi for i = 1, . . . , n. moreover cg(f1 × . . . × fn) = cg1 (f1) × . . . × cgn (fn). this proves that tg = tg1 × . . . × tgn . � proposition 4.5. (a) if g ∈ t, then z(g) = {eg}. (b) if g ∈ tt, then g is perfect. proof. (a) follows from lemma 4.1(b). (b) since g/g′ is abelian and in t, g/g′ is trivial in view of lemma 4.1(c), that is g = g′. � it follows from (a) that every non-trivial abelian group g 6∈ t. the next result about products is a consequence of lemma 4.4. proposition 4.6. the class t is closed under taking finite products. arnautov’s problems on semitopological isomorphisms 99 proof. let g1, g2 ∈ t and g := g1 × g2. by lemma 4.4 tg = tg1 × tg2 and so tg = δg, that is g ∈ t. this can be extended to all finite products. � the next example in particular shows that t is not closed under taking quotients and subgroups since the groups in (b) and (c) have abelian quotients (so they are not in tt) and non-trivial abelian subgroups. example 4.7. (a) a finite group g ∈ t if and only if z(g) = {eg}. this follows from lemma 4.1, but can be also simply directly proved. (b) let g = ( r∗ r 0 1 ) . then g ∈ t. indeed, for f = {( 2 0 0 1 ) , ( 1 1 0 1 )} cg(f ) = {eg}. (c) every non-abelian free group f (x) of rank > 1 is in t. indeed, for f = {a, b}, where a, b ∈ x are generators of f (x), cf (x)(f ) = {ef (x)}. this example shows also that the condition “surjective” in proposition 4.2 cannot be removed: if g is one of the groups in (b) or (c), then g has some non-trivial abelian subgroup h. since h is abelian, th = ιh , while tg = δg. consequently the injective homomorphism (h, th ) → (g, tg) is far from being continuous. remark 4.8. since non-abelian free groups of rank > 1 are tăımanov (as shown in example 4.7(c)), • there exist arbitrarily large tăımanov groups; moreover, every nonabelian subgroup of a non-abelian free group is tăımanov, being free [23]; • every group is quotient of a tăımanov group, since every group is quotient of a non-abelian free group of rank > 1 [23]. it is not clear if this holds also for subgroups: question 4.9. is every group subgroup of a tăımanov group? theorem 4.11 answers positively the question in the abelian case. for an abelian group g and p ∈ p in what follows we denote by rp(g) the p-rank of g. lemma 4.10. let g be an abelian group with r2(g) = 0. then there exists h ∈ t such that g ≤ h and [h : g] = 2. proof. let f : g → g be defined by f (x) = −x for every x ∈ g. moreover let h := g ⋊ 〈f〉 (⋊ denotes the semidirect product). then ch (0, f ) = 〈(0, f )〉 and (0, f ) 6∈ ch (g, idg) for every g ∈ g \ {0}. consequently for f = {(g, idg), (0, f )}, with g ∈ g \ {0}, ch (f ) = {eh}, that is h ∈ t. since f has order 2, g has index 2 in g. � theorem 4.11. for every abelian group g there exists a group h ∈ t containing g as a subgroup and such that |h| = ω · |g|. 100 d. dikranjan and a. giordano bruno proof. let g be an abelian group. then g ⊆ d(g) = g1⊕g2, where r2(g1) = 0 and r2(g2) = r2(d(g)). then there exists h1 ∈ t such that g1 ≤ h1 and |h1| = 2 · |g1| by lemma 4.10. now consider g2 = ⊕ r2(g) z(2∞). if r2(g) ≤ ω, then g2 is contained in ⊕ ω z(2 ∞). let then σ be the shift ⊕ ω z(2 ∞) → ⊕ ω z(2 ∞) defined by (xn)n 7→ (xn−1)n for every (xn)n ∈ ⊕ ω z(2 ∞). then σn has no non-zero fixed point for every n ∈ z, n 6= 0. claim. let g be an abelian group and let f be an automorphism of g such that f n has no non-zero fixed point for every n ∈ z, n 6= 0. then there exists h ∈ t such that g ≤ h and |h| = ω · |g|. proof of the claim. let h := g ⋊ 〈f〉. then ch (0, f ) = 〈(0, f )〉 and (0, f ) 6∈ ch (g, idg) for every g ∈ g \ {0}. consequently for f = {(g, idg), (0, f )}, with g ∈ g \ {0}, ch (f ) = {eh}, that is h ∈ t. since f has infinite order |h| = ω · |g|. � by the claim there exists a group h2 ∈ t such that g2 ≤ ⊕ ω z(2 ∞) ≤ h2 and |h2| = ω. suppose that r2(g) ≥ ω. then g2 ∼= ⊕ r2(g) ( ⊕ ω z(2 ∞)). let σ̃ : ⊕ r2(g) ( ⊕ ω z(2 ∞)) → ⊕ r2(g) ( ⊕ ω z(2 ∞)) be defined by σ̃ ↾⊕ ω z(2∞)= σ. then σ̃n has no non-zero fixed point for every n ∈ z, n 6= 0, and again the claim gives a group h2 ∈ t that contains g2 as a subgroup and such that |h2| = |g2|. let h := h1 ⊕ h2. by proposition 4.6 h ∈ t. moreover |h| = ω · |h1| · |h2| = ω · |g1| · |g2| = ω · |g|. � lemma 4.13 shows that to prove that a group is tăımanov it suffices to consider a convenient quotient with a finite normal subgroup and check whether it is tăımanov. claim 4.12. let g be a group with z(g) = {eg}. if there exists a finite subset f of g such that cg(f ) is finite, then there exists another finite subset f1 ⊇ f of g such that cg(f1) = {eg}. in particular g ∈ t. proof. let cg(f ) = {eg, g1, . . . , gn}. since z(g) = {eg}, for every i ∈ {1, . . . , n} there exists hi ∈ g such that [gi, hi] 6= eg; in particular gi 6∈ cg(hi). let f1 = f ∪ {h1, . . . , hn}. then gi 6∈ cg(f1) for every i ∈ {1, . . . , n}. since cg(f1) ⊆ cg(f ) = {eg, g1, . . . , gn}, this proves that cg(f1) = {eg}. � lemma 4.13. let g be a group with z(g) = {eg} and let n be a normal finite subgroup of g such that g/n ∈ t. then g ∈ t. proof. let f1 be a finite subset of g/n such that cg/n (f1) = {eg/n }. let π : g → g/n be the canonical projection and let f be a finite subset of g such that π(f ) = f1. since π(cg(f )) ⊆ cg/n (f1) = {eg/n }, cg(f ) ⊆ n . since n is finite, claim 4.12 applies to conclude that g ∈ t. � arnautov’s problems on semitopological isomorphisms 101 the next is an example of a totally tăımanov group. example 4.14. we denote by g := so3(r) the group of all orthogonal matrices 3 × 3 with determinant 1 and coefficients in r. then g ∈ t. since g is simple, g ∈ tt. indeed, g = ⋃ α tα, where tα ∼= t and tα is generated by an element α of g, that is, 〈α〉 = tα. moreover cg(α) contains tα as a finite index subgroup and for α, β ∈ g with α 6= β and tα 6= tβ, tα ∩tβ is finite. then cg(α)∩cg(β) is finite. by claim 4.12 g ∈ t. 4.2. the permutations groups. for a set x, x ∈ x and a subgroup h of s(x) let oh (x) := {h(x) : h ∈ h}, stab x := {ρ ∈ s(x) : ρ(x) = x}, and sx := stab x ∩ h. moreover h induces a partition of x, that is x = ⋃ x∈rh oh (x), where rh ⊆ x is a set of representing elements. if τ ∈ s(x), then stab x = (stab τ (x))τ . remark 4.15. let x be a set and h a subgroup of s(x). if τ ∈ ns(x)(h), then: (a) τ (oh (x)) = oh (τ (x)); (b) sx = (sτ (x)) τ ; indeed, sx = stab x ∩ h = (stab τ (x)) τ ∩ hτ = (stab τ (x) ∩ h)τ = (sτ (x)) τ ; (c) τ induces a permutation τ̃ of rh . indeed, τ (oh (x)) = oh (τ (x)) by (a); so we can define τ̃ (x) = y, where y ∈ rh is the representing element of oh (τ (x)). then cs(x)(h) =    τ ∈ ⋂ x∈rh \supp τ̃ nh (sx) · stab x : [h, τ ] ⊆ ⋂ x∈supp τ̃ sx    . we describe the subgroups of s(x) with trivial centralizer: lemma 4.16. let x be a set and h a subgroup of s(x). then cs(x)(h) = {idx} if and only if the following conditions hold: (a) sx = nh (sx) for every x ∈ rh , and (b) sx and sy are not conjugated in h for every x, y ∈ rh with x 6= y. proof. let τ ∈ cs(x)(h)\{idx}. there exists x ∈ rh such that y := τ (x) 6= x. indeed, if τ (x) = x for every x ∈ rh , then for every z ∈ x, there exist h ∈ h and x ∈ rh such that z = h(x), and so τ (z) = τ (h(x)) = h(τ (x)) = h(x) = z. by remark 4.15(a,b) τ (oh (x)) = oh (τ (x)) and sx = (sy) τ = sy. 102 d. dikranjan and a. giordano bruno if y ∈ oh (x), then τ ↾oh (x): oh (x) → oh (x) is a bijection and y = h0(x) for some h0 ∈ h; then τ (h(x)) = h(τ (x)) = hh0(x) for every h ∈ h. let h ∈ sx. since τ is well-defined, h(x) = x implies hh0(x) = h0(x), that is (h0) −1hh0(x) = x. this is equivalent to h h0 ∈ sx, that is h0 ∈ nh (sx). but h0 6∈ sx and this contradicts (a). suppose now that y 6∈ oh (x) and so oh (x) ∩ oh (y) = ∅. let z ∈ rh ∩ oh (y). then y = h0(z) for some h0 ∈ h. by remark 4.15(b) sz = (sy) h0 = (sx) h0 and this contradicts (b). assume that there exists h0 ∈ nh (sx)\sx for some x ∈ rh . let τ : x → x be defined by τ (x) = h0(x), τ (h(x)) = hh0(x) for every h ∈ h and τ (y) = y for every y ∈ x \ oh (x). this τ is well-defined. indeed, if h1(x) = h2(x) for some h1, h2 ∈ h, that is, h −1 2 h1 ∈ sx; then h1h0(x) = h2h0(x), equivalently h−10 (h −1 2 h1)h0(x) = x, that is, h −1 0 (h −1 2 h1)h0 ∈ sx, which holds true by the hypothesis that h0 ∈ nh (sx). moreover, it is possible to check that τ ∈ s(x). by the definition τ h = hτ for every h ∈ h and so idx 6= τ ∈ cs(x)(h). suppose that sx = (sz ) h0 for some x, z ∈ rh and h0 ∈ h. then for y = h−10 (z) ∈ oh (z) we have sy = (sz) h0 = sx by remark 4.15(b). define τ : x → x as τ (x) = y, τ (h(x)) = h(y) for every h ∈ h and τ (w) = w for every w ∈ x \ oh (x). then τ is well-defined; indeed, if h1(x) = h2(x) for some h1, h2 ∈ h, that is, h −1 2 h1 ∈ sx, then h1(y) = h2(y), equivalently, h −1 2 h1 ∈ sy, which holds true since sx = sy. moreover, it is possible to check that τ ∈ s(x). by the definition τ h = hτ for every h ∈ h and so idx 6= τ ∈ cs(x)(h). � proposition 4.17. for a cardinal κ the following conditions are equivalent: (a) there exists a set x with |x| = κ and s(x) ∈ t; (b) there exists a set x with |x| = κ such that there exists a finitely generated subgroup h of s(x) such that sx = nh (sx) for every x ∈ rh and sx, sy are not conjugated for every x, y ∈ rh with x 6= y. if κ > ω, then the following condition is equivalent to the previous: (c) there exists a finitely generated group h admitting a family s = {sα : α < κ} of subgroups of h such that sα = nh (sα) for every α < κ. proof. (a)⇔(b) the condition s(x) ∈ t is equivalent to the existence of a finite subset f of s(x) such that cs(x)(f ) = {idx}. let h = 〈f 〉. then cs(x)(h) = cs(x)(f ) and so equivalently cs(x)(h) = {idx}. by lemma 4.16 we have the conclusion. (b)⇒(c) since κ > ω, and each oh (x) is countable, |rh| = κ. so {sx : x ∈ rh} is the family requested in (c). (c)⇒(b) since κ > ω and h is countable, we can suppose that s has the property that sα and sβ are not conjugated in h for every α, β < κ with α 6= β. indeed, every subgroup sα of h has at most countably many conjugated subgroups in h, so we can restrict the family s taking only one element for every class of conjugation, finding a subfamily of the same cardinality κ as s. arnautov’s problems on semitopological isomorphisms 103 define xα := {hsα : h ∈ h} for every α < κ and x := ⋃ α<κ xα. moreover let xα := idh sα ∈ xα for every α > κ. in particular |x| = κ. moreover h acts on x by multiplication on the left and oh (xα) = xα for every α < κ. there exists a group homomorphism ϕ : h → s(x); let h̃ := ϕ(h) ≤ s(x). then h̃ is finitely generated and the action of h̃ on x is the same as the action of h on x. then o h̃ (xα) = xα for every α < κ and rh̃ = {xα : α < κ}. moreover ϕ(sα) = stab xα ∩ h̃ =: sxα . since sα = nh (sα) for every α < κ and sα and sβ are not conjugated for every α < β < κ, it is possible to prove that sxα = nh (sxα ) for every xα ∈ rh̃ and sxα and sxβ are not conjugated for every xα, xβ ∈ rh̃ with xα 6= xβ . so the properties in (b) are satisfied. � theorem 4.18. let x be a set with |x| > 2. (a) if |x| ≤ ω, then s(x) ∈ t. (b) if |x| > c, then s(x) 6∈ t. proof. (a) assume that 2 < |x| < ω. since z(s(x)) is trivial, s(x) ∈ t by example 4.7(a). assume that |x| = ω. we can suppose x = z. let h = 〈σ, τ〉, where τ = (−1, 1) and σ is the shift, that is σ(n) = n + 1 for every n ∈ z. then oh (0) = z and so rh = {0}. moreover s0 = 〈τ〉 and hence nh (s0) = s0. by proposition 4.17 s(x) ∈ t. (b) let h be a finitely generated subgroup of s(x). since oh (x) is countable for every x ∈ rh , |rh| = |x| > c. since h is countable, it has at most c subgroups and so there exists a subset s of rh such that |s| > c and sx = sy for every x, y ∈ s. by proposition 4.17 s(x) 6∈ t. � question 4.19. let x be a set. (a) is s(x) ∈ t if |x| = ω1? (b) is s(x) ∈ t if |x| = c? (c) is s(x) ∈ t if ω < |x| ≤ c? remark 4.20. question 4.19 can be formulated in equivalent terms thanks to proposition 4.17. indeed, if x is a set of cardinality κ with ω < κ ≤ c, then s(x) ∈ t if and only if there exists a finitely generated group h admitting a family s = {sα : α < κ} of subgroups of h such that sα = nh (sα) for every α < κ. so question 4.19 becomes: does there exist a finitely generated group h with a “large” (i.e., of cardinality κ with ω < κ ≤ c) family of self-normalizing subgroups? 5. problem a we start considering a stability property of the the class a of arnautov groups. theorem 5.1. the class a is closed under taking quotients. 104 d. dikranjan and a. giordano bruno proof. let g ∈ a and let n be a normal subgroup of g. let σ ≤ τ be group topologies on g/n such that idg/n : (g/n, τ ) → (g/n, σ) is semitopological. then idg : (g, τi) → (g, σi) is semitopological by lemma 2.6. since g ∈ a, τi = σi and hence τ = σ. � in section 5.2 we will comment the stability of a under taking subgroups and products. example 5.2. (a) obviously every indiscrete group g is a-complete. (b) let g be a group. let gab = g/g ′ be the abelianization of g and endow gab with the discrete topology and with the indiscrete topology: (g, ζg′ ) −−−−→ (gab, δgab ) idg y yidgab (g, ιg) −−−−→ (gab, ιgab ) if g 6= g′ then idgab is a semitopological non-open isomorphism, because gab is abelian, and idg is a semitopological non-open isomorphism too, in view of remark 3.15. so (g, ζg′ ) is not a-complete. (c) an abelian topological group g is a-complete if and only if g is indiscrete. in particular the only abelian arnautov group is g = {eg} (as (g, δg) must be indiscrete). the next proposition generalizes the example in (b). proposition 5.3. a topological group g with indiscrete derived group g′ is a-complete precisely when g is indiscrete. proof. the conclusion follows from remark 3.15. � example 5.4. let g be a group and τ a group topology on g. (a) if (g, τ ) is sin, then it is a-complete if and only if for every group topology σ < τ on g there exist u ∈ v(g,τ )(eg) and g ∈ g such that [g, vg] 6⊆ u for every vg ∈ v(g,σ)(eg) (this follows from proposition 2.8). (b) if (g, τ ) is hausdorff and τ ≤ ζg′ (as already noted after theorem 2.7, this condition yields τ sin), then g is abelian and consequently τ > ιg implies that (g, τ ) is not a-complete (supposing that g is not a singleton). proposition 5.5. let g be a group and n a normal subgroup of g. let τ be a group topology on g/n and τi the initial topology of τ on g. then τ is a-complete if and only if τi is a-complete. proof. let idg/n : (g/n, τ ) → (g/n, σ) be semitopological, where σ ≤ τ is another group topology on g. by lemma 2.6 also idg : (g, τi) → (g, σi) is semitopological and the hypothesis implies that τi = σi. consequently τ = σ. arnautov’s problems on semitopological isomorphisms 105 suppose that τ is a-complete. let σ < τi be another group topology on g and consider the quotient topology σq of σ on g/n . so we have the following situation: (g, τi) idg −−−−→ (g, σ) π y yπ (g/n, τ ) idg/n −−−−→ (g/n, σq ). since σ < τi, it follows that nσ ≥ nτi = n . consequently σ is the initial topology of σq and so σq < τ , otherwise σ = τi. by hypothesis idg/n : (g/n, τ ) → (g/n, σq ) is not semitopological. to conclude that also idg : (g, τi) → (g, σ) is not semitopological apply theorem 2.3. � corollary 5.6. let g be a group and τ a group topology on g. consider the quotient g/nτ and the quotient topology τq of τ on g/nτ . then τ is a-complete if and only if τq is a-complete. proof. since τ is the initial topology of τq, it suffices to apply proposition 5.5. � now we give a necessary condition for a group to be arnautov. proposition 5.7. for a group g the following conditions are equivalent: (a) idg : (g, τ ) → (g, ιg) is semitopological for no group topology τ > ιg on g; (b) g is perfect. proof. (a)⇒(b) since idg : (g, ζg′ ) → (g, ιg) is a semitopological isomorphism by theorem 2.7(b), our hypothesis (a) implies ζg′ = ιg and hence g = g ′. (b)⇒(a) suppose g = g′; then ζg′ = ιg. if idg : (g, τ ) → (g, ιg) is a semitopological isomorphism, then τ ≤ ζg′ = ιg by theorem 2.7(b), so τ = ιg. this means that idg is open. � therefore, if a group g is arnautov, then for every non-indiscrete group topology τ on g idg : (g, τ ) → (g, ιg) is not semitopological. in particular proposition 5.7 implies that every arnautov group is perfect. corollary 5.8. let g be a simple non-abelian group and τ a group topology on g. if τ > ιg, then idg : (g, τ ) → (g, ιg) is not semitopological. a consequence of these results is that every minimal simple non-abelian group (g, τ ) is a-complete. indeed, if σ ≤ τ is another group topology on g and idg : (g, τ ) → (g, σ) is semitopological, then by lemma 3.2 either σ is hausdorff or σ = ιg. since g is simple and non-abelian, g is perfect. then proposition 5.7 implies that σ is not indiscrete and so σ has to be hausdorff. the minimality of τ yields that σ = τ . this consequence is improved by the next result. proposition 5.9. if (g, τ ) is a minimal group and z(g) = {eg}, then (g, τ ) is a-complete. 106 d. dikranjan and a. giordano bruno proof. let σ ≤ τ be a group topology on g and suppose that idg : (g, τ ) → (g, σ) is semitopological. by proposition 2.11 σ is hausdorff and so σ = τ by the minimality of τ . � example 5.10. every simple finite non-abelian group g is an arnautov group. indeed, the only group topologies on g are the trivial ones and idg : (g, δg) → (g, ιg) is not semitopological by corollary 5.8. the following remark could be used as a test to verify if a group is arnautov. remark 5.11. if g ∈ a, then for every group topology τ on g and for every normal subgroup n of g, • idg : (g, sup{τ, ζn }) → (g, τ ) is not semitopological if sup{τ, ζn } > τ ; • idg : (g, sup{τ, ζn }) → (g, ζn ) is not semitopological if sup{τ, ζn } > ζn . 5.1. when the discrete topology is a-complete. remark 5.12. [14, corollary 5.3] we can formulate theorem 2.7(a) in terms of the tăımanov topology: let g be a group and σ a group topology on g. then idg : (g, δg) → (g, σ) is semitopological if and only if σ ≥ tg, that is, nσ ≤ ntg = z(g). consequently the tăımanov topology is the coarsest topology σ on a group g such that idg : (g, δg) → (g, σ) is semitopological. so, since in this section we consider the case when the discrete topology is a-complete, we have to impose that the tăımanov topology is discrete, that is, the group is tăımanov. this also motivates definition 1.8. the next theorem solves a particular case of problem a, that is, it characterizes the groups for which the discrete topology is a-complete. theorem 5.13. let g be a group. then δg is a-complete if and only if g ∈ t. proof. suppose that δg > tg. then idg : (g, δg) → (g, tg) is semitopological by remark 5.12. this proves that δg is not a-complete. suppose that δg = tg. let τ < δg be a group topology on g. then idg : (g, δg) → (g, τ ) is not semitopological by remark 5.12. this proves that δg is a-complete. � by proposition 4.5(a) the equivalent conditions of this theorem imply that the group has trivial center. the next example shows that they can be strictly stronger than having trivial center. moreover this is an example of a tăımanov group which has an infinite non-abelian subgroup that is not tăımanov. example 5.14. consider s(n) and let g := sω(n) be the subgroup of s(n) of the permutations with finite support, that is sω = ⋃∞ n=1 sn. then z(g) = {eg}. if f is a finite subset of g, then there exists n ∈ n+ such that f ⊆ sn and c(sn) = s(n \ {1, . . . , n}) is infinite. therefore tg < δg and so g 6∈ t. arnautov’s problems on semitopological isomorphisms 107 anyway in the finite case the three conditions are equivalent, as stated by example 4.7(a). the next theorem characterizes the almost trivial topologies that are acomplete. it covers theorem 5.13. theorem 5.15. let g be a group and n ⊳ g. then (g, ζn ) is a-complete if and only if g/n ∈ t. proof. suppose that ζn is a-complete. since ζn is the initial topology of δg/n , it follows that δg/n is a-complete by proposition 5.5. by theorem 5.13 this is equivalent to g/n ∈ t. suppose now that g/n ∈ t. by theorem 5.13 this is equivalent to say that δg/n is a-complete and so ζn is a-complete by corollary 5.6. � the next theorem offers a relevant necessary condition for a group to be arnautov: theorem 5.16. if g ∈ a, then g ∈ tt. proof. the conclusion follows from theorems 5.1 and 5.13. � so the next question naturally arises. question 5.17. does g ∈ tt imply g ∈ a? we shall give a positive answer to this question in a particular case in proposition 5.25. the next examples show that a group can admit two a-complete topologies that are one strictly finer than the other. example 5.18. let g := s(z) and s := sω(z) > a := aω (z), which are the only proper normal subgroups of g. (a) the point-wise convergence topology t on g is a-complete: t is minimal and z(g) is trivial, so proposition 5.9 applies. (b) the discrete topology δg is a-complete by theorems 4.18 and 5.13. (c) we show that z(g/a) = s/a and |s/a| = 2. the group s/a has only one non-trivial element, that is, s/a = 〈π(τ )〉, where π : g → g/a is the canonical projection and τ = (12) ∈ g. indeed, if σ ∈ s and σ 6∈ a, then τ σ ∈ a and so π(σ) ∈ 〈π(τ )〉. moreover τ 6∈ a. since s/a is a non-trivial normal subgroup of g/a and it has size 2, it is central; since s/a is the unique non-trivial normal subgroup of g/a, s/a = z(g/a). (d) it follows from (c) that g 6∈ t by proposition 4.5(a). (e) by (d) ζa is not a-complete in view of theorem 5.15, hence g 6∈ a. (f) moreover it is possible to prove that g/s ∈ t. consequently g/s ∈ tt, being simple. this is an example of a group g which is not arnautov but with δg acomplete. moreover, since the subgroup of g generated by the shift σ is abelian 108 d. dikranjan and a. giordano bruno and so not a-complete, while δg is a-complete, this example shows also that a subgroup of an a-complete group need not be a-complete. example 5.19. consider the group g := so3(r). as shown by example 4.14, g ∈ t. consequently δg is a-complete by theorem 5.13. moreover the usual compact topology τ of g is a-complete, because τ compact implies minimal, z(g) is trivial and so proposition 5.9 applies. a first step to find an answer to question 5.17 is to consider the following. question 5.20. (a) does s(z)/sω(z) ∈ a? (b) does so3(r) ∈ a? 5.2. totally markov groups. our aim is to provide examples of groups in a. the next results shows that for totally markov groups the topologies are all almost trivial and so to verify if a continuous isomorphism of a totally markov group is semitopological is simple, thanks to corollary 3.13. proposition 5.21. a group g ∈ mt if and only if every group topology on g is almost trivial. proof. suppose that g ∈ mt and let τ be a group topology on g. then the quotient topology of τ on g/nτ is hausdorff and hence discrete, being g ∈ mt. so nτ is open in (g, τ ) and therefore τ is almost trivial. suppose that the group g 6∈ mt. then there exists a normal subgroup n of g such that there exists a hausdorff non-discrete group topology σ on g/n . let π : g → g/n be the canonical projection and τ = π−1(σ). therefore nτ = n (because n = ⋂ {v : v ∈ v(g/n,σ)(eg/n )} in g/n ). since σ is non-discrete n is not open and so τ is not almost trivial. � proposition 3.10, together with proposition 5.21, immediately implies that mt is closed under extensions: definition 5.22. for a class of abstract groups p one says that p is closed under extensions, if a group g belongs to p whenever n ∈ p and g/n ∈ p for some normal subgroup n of g. moreover we have the same result for m: theorem 5.23. the classes m and mt are closed under extensions. in particular, m and mt are closed under finite direct products. proof. that mt is closed under extensions is a direct consequence of propositions 3.10 and 5.21. suppose that the group g has a normal subgroup n such that n ∈ m and g/n ∈ m. we show that g ∈ m. to this end let τ be a hausdorff group topology on g. then τ ↾n = δn . consequently: (i) n is closed in (g, τ ), and (ii) π : (g, τ ) → (g/n, τq ) is a local homeomorphism. arnautov’s problems on semitopological isomorphisms 109 by (i) (g/n, τq ) is hausdorff and so discrete. in view of (ii) τ = δg. � in view of theorem 5.13, a necessary condition for a-completeness of δg for a group g is z(g) = {eg}. for markov groups also the converse implication holds: corollary 5.24. let g ∈ m. then g ∈ t if and only if z(g) = {eg}. proof. if g ∈ t, apply theorem 5.13. suppose z(g) = {eg}. then tg is hausdorff by lemma 4.1(b) and so tg = δg. � in the following proposition we characterize totally markov groups which are a-complete or arnautov. in particular it shows that for a totally markov group it is equivalent to be arnautov and to be totally tăımanov, which is precisely the answer to question 5.17 in the particular case of totally markov groups. proposition 5.25. let g ∈ mt. (a) if τ is a group topology on g, the following conditions are equivalent: (i) (g, τ ) is a-complete; (ii) g/nτ ∈ t; (iii) for every n ⊳ g, if [g, n ] ≤ nτ ≤ n , then n = nτ . (b) the following conditions are equivalent: (i) g ∈ a; (ii) g ∈ tt; (iii) z(g/n ) = {eg/n } for every n ⊳ g; (iv) [g, n ] = n for every n ⊳ g. proof. (a) the equivalence (i)⇔(ii) follows from lemma 5.21 and theorem 5.15. the equivalence (i)⇔(iii) follows from lemma 5.21 and corollary 3.13. (b) the equivalence (i)⇔(ii) follows from (a) and the equivalence (ii)⇔(iii) follows from corollary 5.24. (iii)⇒(iv) let n be a normal subgroup of g. then [g, n ] is a normal subgroup of g and z(g/[g, n ]) is trivial by hypothesis. since n/[g, n ] ≤ z(g/[g, n ]) also n/[g, n ] is trivial, that is n = [g, n ]. (iv)⇒(i) by lemma 5.21 every group topology on g is almost trivial. so let l be a normal subgroup of g. for every normal subgroup n of g such that [g, n ] ≤ l ≤ n , n = l because [g, n ] = n by hypothesis. this proves that ζl is a-complete by (a). consequently g ∈ a. � this proposition covers example 5.10. corollary 5.26. (a) a finite group g ∈ a if and only if g ∈ tt. (b) for every g ∈ m simple, g ∈ a. in example 5.18 we have seen that s(z) 6∈ a, but s(z)/sω(z) ∈ tt. in relation to question 5.20 we consider the following, which has also its own interest. in example 4.14 we have seen that so3(r) ∈ tt, but clearly so3(r) 6∈ m. 110 d. dikranjan and a. giordano bruno question 5.27. does s(z)/sω(z) ∈ m? a positive answer to this question would imply that s(z)/sω(z) ∈ a, that is a positive answer to question 5.20, in view of corollary 5.26(b), since s(z)/sω(z) is simple. from another point of view, in order to answer question 5.27, it is possible to consider first question 5.20 which involves a weaker condition. example 5.28. let v = (fpm ) n, where m, n ∈ n+, p ∈ p and (n, p m −1) = 1. define g to be the semidirect product of sl(v ) and v . then [g, v ] = v . moreover every normal subgroup of g contains v and so, since sl(v ) is simple, v is the unique non trivial normal subgroup of g. then g ∈ a by corollary 5.26(a). example 5.29. (a) corollary 5.26(b) provides an example of an infinite arnautov group. indeed shelah [24] constructed a simple markov (hence totally markov) group m under ch. (b) the group m contains a subgroup isomorphic to z, which is abelian and so not in a. (c) in general a totally markov group need not be an arnautov group, that is, mt 6⊆ a; for example g := m × z(2) ∈ mt but g 6∈ a. item (b) of this example shows that a is not stable under taking subgroups. question 5.30. is a stable under taking (finite) direct products? and under taking (finite) powers? in the next example we give examples of arnautov groups which are not simple. moreover we see a particular case (that of markov simple groups) in which finite powers of arnautov groups are arnautov. example 5.31. let m ∈ m be simple; by corollary 5.26(b) m ∈ a. we show that m n ∈ mt and also m n ∈ a, for every n ∈ n+. since m ∈ m is simple, m ∈ mt. by theorem 5.23 m n ∈ mt for every n ∈ n+. so m n ∈ a by proposition 5.25(b): for every normal subgroup n of m n, n = m k for some k ≤ n up to topological isomorphisms, and consequently [m n, n ] = [m n, m k] = m k = n . the next are corollaries of propositions 3.10 and 5.21. corollary 5.32. let g be a group and n1 ≤ n2 be normal subgroups of g with n2/n1 ∈ mt. then every group topology τ on g with ζn2 ≤ τ ≤ ζn1 is almost trivial. in particular, (a) if n2 ∈ mt, then every group topology τ on g with τ ≥ ζn2 is almost trivial; and (b) if g/n1 ∈ mt, then every group topology τ on g with τ ≤ ζn1 is almost trivial. proof. (a) since n2 ∈ mt, by proposition 5.21 τ ↾n2 is almost trivial. moreover τq ≥ (ζn2 )q = δg/n2 on g/n2, and so τq = δg/n2 and in particular it is almost trivial. by proposition 3.10 τ is almost trivial. arnautov’s problems on semitopological isomorphisms 111 obviously, n1 ≤ nτ ≤ n2. therefore, the quotient topology τq of (g, τ ) with respect to n1 satisfies δg/n1 ≥ τq ≥ ζn2/n1 . to the normal subgroup n2/n1 ∈ mt of the group g/n1 and τq ≥ ζn2/n1 we apply (a) to claim that τq is almost trivial. since τq was obtained from τ via a quotient with respect to the τ -indiscrete normal subgroup n1, by lemma 3.8 τ is almost trivial. (b) follows from the previous part. � corollary 5.33. let g be a group and n1 ≤ n2 be normal subgroups of g with [n2 : n1] finite. then g admits only finitely many group topologies τ with ζn2 ≤ τ ≤ ζn1 and they are all almost trivial. proof. apply corollary 5.32 to conclude that every group topology τ on g such that ζn2 ≤ τ ≤ ζn1 is almost trivial. moreover these τ are finitely many because [n2 : n1] is finite. � remark 5.34. a group g is hereditarily non-topologizable in case every subgroup of g is totally markov [18]. thus hereditarily non-topologizable ⇒ totally markov ⇒ markov. consequently every group topology on a hereditarily non-topologizable group is almost trivial. if a hereditarily non-topologizable group g is arnautov, then every quotient of g is arnautov. while infinite arnautov groups exist (see example 5.29(a)), it is not known if there exists any infinite non-topologizable group. the existence of such a group would solve an open problem from [10]. 6. problem b we start by underlying an important aspect of ak-completeness compared to k-minimality, where k is a class of topological groups. indeed, let us recall first that ag-completeness coincides with a-completeness and implies akcompleteness (see remark 1.10). the k-minimal groups are precisely the indiscrete groups, whenever k contains all indiscrete groups. this fails to be true for ak-completeness. in fact, the group g = s(z), equipped with either the discrete or the pointwise convergence topology, is a-complete (so ak-complete, for every k ⊆ g) as shown by example 5.18(a,b). more generally for every nontrivial g ∈ t, the (obviously) non-indiscrete group (g, δg) is a-complete (so ak-complete, for every k ⊆ g) by theorem 5.13. as we have seen in section 5 a-complete (i.e., ag-complete) groups are not easy to come by. in order to have a richer choice of groups, we consider akcomplete groups for appropriate subclasses k of g. in case the subclass k is completely determined by an algebraic property (i.e., for every group topology τ on g, (g, τ ) ∈ k if and only if (g, δg) ∈ k), then obviously a topological group (g, τ ) ∈ k is ak-complete if and only if it is a-complete. a typical example to this effect is the class of all topological abelian groups, or more generally the class of all topological groups such that the underlying group belongs to a 112 d. dikranjan and a. giordano bruno fixed variety v (in the sense of [20]) of abstract groups. we formulate an open question for a specific v in question 6.13. in the sequel we consider subclasses k ⊆ g of a different form, most often k ⊆ h. since h-minimality coincides with minimality, ah-completeness is a generalization of minimality. it is a strict generalization in view of (a) of the next example. example 6.1. (a) the group (s(z), δs(z)) is a-complete, as shown by example 5.18(b), and consequently ah-complete, but it is not minimal: δs(z) and the point-wise convergence topology t are both hausdorff. (b) let g ∈ t be non-torsion. then (g, δg) is a-complete by theorem 5.13, and in particular it is ah-complete. on the other hand, by our hypothesis there exists x ∈ g of infinite order, that is 〈x〉 is abelian and so not ah-complete. this shows that in general a subgroup of an ah-complete group need not be ah-complete. (this is noted after example 5.18 for the particular case of (s(z), δs(z)).) anyway ah-completeness coincides with minimality in the abelian case: proposition 6.2. if g is an abelian group and (g, τ ) ∈ h, then (g, τ ) is ah-complete if and only if it is minimal. this proposition gives a partial answer to problem b for the subclass of h of abelian topological groups. the problem remains open for the larger class h: question 6.3. when is a topological group (g, τ ) ∈ h ah-complete? and in which cases is (g, δg) ah-complete? the next example, that extends example 5.18(a), motivates lemma 6.5. example 6.4. for an infinite topologically simple (i.e., there exists no nontrivial closed normal subgroup) hausdorff non-abelian group (g, τ ), minimal implies a-complete. in fact z(g) = {eg} and proposition 5.9 applies. the next lemma and corollary provide partial answers to question 6.3. lemma 6.5 in particular covers the previous example, since it implies that every minimal group with trivial center is a-complete (in view of the fact that minimal implies ah-minimal). lemma 6.5. let g be a group with z(g) = {eg} and let τ be a hausdorff group topology on g. then (g, τ ) is ah-complete if and only if (g, τ ) is acomplete. proof. if (g, τ ) is a-complete, then it is ah-complete. suppose that (g, τ ) is ah-complete. let σ ≤ τ be a group topology on g such that idg : (g, τ ) → (g, σ) is semitopological. by proposition 2.11 σ is hausdorff. then σ = τ . this proves that (g, τ ) is a-complete. � arnautov’s problems on semitopological isomorphisms 113 this lemma implies proposition 5.9, since minimal groups are ah-complete. corollary 6.6. let g be a group. then z(g) = {eg} and δg is ah-complete if and only if g ∈ t. proof. if z(g) = {eg} and δg is ah-complete, then δg is a-complete by lemma 6.5 and so g ∈ t by theorem 5.13. assume that g ∈ t. by theorem 5.13 δg is a-complete and so ahcomplete. moreover z(g) = {eg} by proposition 4.5(a). � lemma 6.5 suggests the following question: is z(g) = {eg} a necessary condition for the validity of the implication (g, τ ) ah-complete ⇒ (g, τ ) acomplete? according to corollary 6.6 the answer is “yes” in case τ is the discrete topology. proposition 6.7. let (g, τ ) be a sin hausdorff group. if (g, τ ) is a-complete, then z(g) = {eg}. proof. suppose that z(g) 6= {eg}. we want to see that (g, τ ) fails to be a-complete. consider the topology t := τ ∧ ζz(g), which has as a local base at eg the family bt = {u · z(g) : u ∈ v(g,τ )(eg)}. since τ is hausdorff and t is not hausdorff (because z(g) 6= {eg}), τ > t . so it remains to prove that idg : (g, τ ) → (g, t ) is semitopological. since (g, τ ) is sin, it suffices to prove that for every u ∈ v(g,τ )(eg) and for a fixed g ∈ g there exists vg ∈ bt such that [g, vg] ⊆ u and then apply proposition 2.8. so let u ∈ v(g,τ )(eg) and g ∈ g. since (g, τ ) is sin, there exists u ′ ∈ v(g,τ )(eg) such that u ′u ′ ⊆ u and gu ′g−1 ⊆ u ′. let vg = u ′·z(g) ∈ bt . then [g, vg] = [g, u ′] ⊆ u ′u ′ ⊆ u . since we have proved that idg : (g, τ ) → (g, t ) is semitopological and τ > t , then (g, τ ) fails to be a-complete. � remark 6.8. as a consequence of lemma 6.5 and proposition 6.7 we have the following equivalence between a-completeness and the purely algebraic property of having trivial center. indeed, if (g, τ ) ∈ h is ah-complete, then z(g) = {eh} implies (g, τ ) a-complete by lemma 6.5. moreover, if (g, τ ) is sin, in view of proposition 6.7 also the converse implication holds, that is, (g, τ ) is a-complete if and only if z(g) = {eg}. corollary 6.9. let (g, τ ) be a hausdorff group with z(g) 6= {eg}. (a) if (g, τ ) is sin and ah-complete, then it is not a-complete. (b) if (g, τ ) is sin and minimal, then it is not a-complete. (c) if (g, τ ) is compact, then it is not a-complete. this corollary produces in particular examples of ah-complete groups which are not a-complete (e.g., compact groups with non-trivial center), showing that the implication (g, τ ) ah-complete ⇒ (g, τ ) a-complete may fail to be true, also for non-discrete groups. in particular in example 6.12 shows a group, with non-trivial center, which does not admit any compact topology, but admits minimal linear (so sin) topologies, that are not a-complete by corollary 6.9. 114 d. dikranjan and a. giordano bruno proposition 6.10. let g be a group such that g ∈ t and let f be a finite group. then δg×f is ah-complete. proof. let τ be a hausdorff group topology on g×f and suppose that idg×f : (g × f, δg×f ) → (g × f, τ ) is semitopological. by remark 5.12 τ ≥ tg×f . but tg×f = tg × tf = δg × tf by lemma 4.4. so τ ≥ δg × tf . since τ is hausdorff, τ = δg×f , and this proves that δg×f is ah-complete. � using this proposition we can give examples of ah-complete groups which are not a-complete, as the following. another example of an ah-complete group which is not a-complete is in example 6.12. example 6.11. let g = s(z) × z(2). by theorem 4.18(a) s(z) ∈ t. then (g, δg) is ah-complete by proposition 6.10. since z(g) = {idz} × z(2) is not trivial, g 6∈ t by proposition 4.5(a). consequently g is not a-complete by theorem 5.13. example 6.12. let p ∈ p and let g be the group hz (see example 4.3) equipped with the product topology t = p (τp, τp, τp) where τp is the p-adic topology of z. a base of t is given by the family of the (normal) subgroups formed by the matrices of the form   1 pnz pnz 0 1 pnz 0 0 1   . clearly g is sin. then (g, t ) is minimal [5, 7], so ah-complete. moreover (g, t ) is a-complete by corollary 6.9. considering sin groups in example 5.4, proposition 6.7 and corollary 6.9 we have weakened the commutativity from a topological point of view. a different way to weaken commutativity, but algebraically, is to consider nilpotent topological groups: question 6.13. if (g, τ ) is a nilpotent topological group, when is (g, τ ) acomplete? the following example is dedicated to a very particular case of this question. example 6.14. consider the class k0 r := {(hr, p (τ, τ, τ )) : τ is a ring topology on r}, where p (τ, τ, τ ) denotes the product topology on g. then every g ∈ k0 r is ak0 r -complete. indeed, let τ ≥ σ be ring topologies on r such that (hr, p (τ, τ, τ )), (hr, p (σ, σ, σ)) ∈ k 0 r . suppose that idr : (hr, p (τ, τ, τ )) → (hr, p (σ, σ, σ)) is semitopological. by theorem 1.2, for every u ′ =   1 u u 0 1 u 0 0 1   ∈ v(hr,p (τ,τ,τ ))(ehr ) and h = arnautov’s problems on semitopological isomorphisms 115   1 1 0 0 1 0 0 0 1   there exists vh =   1 v v 0 1 v 0 0 1   ∈ v(hr ,p (σ,σ,σ))(ehr ) such that [h, vh] ⊆ u ′. in particular this implies v ⊆ u and hence σ ≥ τ , that is σ = τ . in a forthcoming paper [6] we extend this result to the more general case of generalized heisenberg groups on an arbitrary unitary ring a. 7. problem c problem c is about compositions of semitopological isomorphisms. in order to measure more precisely the level of being semitopological, we introduce the next notion. definition 7.1. let g be a group, σ ≤ τ group topologies on g and n ∈ n+. then idg : (g, τ ) → (g, σ) is n-step semitopological if there exist n − 1 group topologies σ ≤ λn−1 ≤ . . . ≤ λ1 ≤ τ on g such that idg : (g, τ ) → (g, λ1), idg : (g, λ1) → (g, λ2), . . . , idg : (g, λn−1) → (g, σ) are semitopological. obviously idg : (g, τ ) → (g, σ) is 1-step semitopological if and only if it is semitopological. moreover a continuous isomorphism of topological groups is composition of semitopological isomorphisms if and only if it is n-step semitopological for some n ∈ n+. let g be a non-trivial group. the lower central series of g is defined by γ1(g) = g and γn(g) = [g, γn−1(g)] for every n ∈ n, n ≥ 2. the upper central series of g is defined by z0(g) = {eg}, z1(g) = z(g) and zn(g) is such that zn(g)/zn−1(g) = z(g/zn−1(g)) for every n ∈ n, n ≥ 2. a group g is nilpotent if and only if γn(g) = {eg} for some n ∈ n+, if and only if zm(g) = g for some m ∈ n+. the minimum n ∈ n+ such that γn+1(g) = {eg}, that is, the minimum n ∈ n+ such that zn(g) = g, is the class of nilpotency of g. our main theorem about n-step semitopological isomorphisms is the following. it is an answer to problem c(a) in the particular case when the topologies on the domain and on the codomain are the discrete and the indiscrete one respectively. theorem 7.2. let g be a group and n ∈ n+. then idg : (g, δg) → (g, ιg) is n-step semitopological if and only if g is nilpotent of class ≤ n. proof. if idg : (g, δg) → (g, ιg) is n-step semitopological, then there exist n − 1 group topologies λn−1 ≤ . . . ≤ λ1 on g such that idg : (g, δg) → (g, λ1), idg : (g, λ1) → (g, λ2), . . . . . . , idg : (g, λn−2) → (g, λn−1), idg : (g, λn−1) → (g, ιg) are semitopological. by theorem 2.7(b) g′ ⊆ v for every v ∈ v(g,λn−1)(eg). since idg : (g, λn−2) → (g, λn−1) is semitopological, theorem 1.2 implies that for every u ∈ v(g,λn−2)(eg) and for every g ∈ g there exists vg ∈ v(g,λn−1)(eg) such that [g, vg] ⊆ u . consequently [g, g ′] ⊆ u for every u ∈ v(g,λn−2)(eg). 116 d. dikranjan and a. giordano bruno hence γ3(g) = [g, g ′] ⊆ u for every u ∈ v(g,λn−2)(eg). proceeding by induction we have that γn(g) ⊆ u for every u ∈ v(g,λ1)(eg). by theorem 2.7(a) cg(g) is λ1-open for every g ∈ g. thus γn(g) ⊆ z(g) and this implies that g is nilpotent of class ≤ n (γn+1(g) = {eg}). conversely, if g is nilpotent of class ≤ n, consider on g the group topologies ζz(g), ζz2(g), . . . , ζzn−1(g). then idg : (g, δg) → (g, ζz(g)) is semitopological by theorem 2.7(a) and idg : (g, ζzn−1(g)) → (g, ιg) is semitopological because g′ ≤ zn−1(g) since g/zn−1(g) is abelian and applying theorem 2.7(b). for every i = 1, . . . , n − 1, by corollary 3.13 idg : (g, ζzi (g)) → (g, ζzi+1(g)) is semitopological if and only if [g, zi+1(g)] ≤ zi(g) and this holds true since zi+1(g)/zi(g) = z(g/zi(g)). � as a particular case of n = 2 in this theorem, we find [2, example 12], which witnesses that the composition of semitopological isomorphisms is not semitopological in general. indeed idg : (g, δg) → (g, ιg) is not semitopological, whenever g is not abelian. for n ∈ n+, let n-s := {fn ◦ . . . ◦ f1 : fi ∈ s}. observe that s = 1-s ⊂ 2-s ⊂ . . . ⊂ n-s ⊂ (n + 1)-s ⊂ . . . , where all inclusions are proper by the previous theorem. define also ∞-s := ⋃∞ n=1 n-s and observe that it is closed under compositions. moreover ∞-s is closed also under taking subgroups, quotients and finite products, in the following sense: lemma 7.3. let n ∈ n+, let g be a group and τ ≥ σ group topologies on g such that idg : (g, τ ) → (g, σ) is n-step semitopological. (a) if a is a subgroup of g, then idg ↾a= ida : a → a is n-step semitopological. (b) if a is a normal subgroup of g, then idg/a : (g/a, τq) → (g/a, σq ) is n-step semitopological. proof. (a) by hypothesis there exist n − 1 group topologies σ ≤ λn−1 ≤ . . . ≤ λ1 ≤ τ on g such that idg : (g, τ ) → (g, λ1), idg : (g, λ1) → (g, λ2), . . . , idg : (g, λn−1) → (g, σ) are semitopological. theorem 2.3(a) implies that ida : (a, τ ↾a) → (a, λ1 ↾a), ida : (a, λ1 ↾a) → (a, λ2 ↾a), . . . . . . , ida : (a, λn−1 ↾a) → (a, σ ↾a) are semitopological and so ida : (a, τ ↾a) → (a, σ ↾a) is n-step semitopological. (b) follows from theorem 2.3(b). � arnautov’s problems on semitopological isomorphisms 117 the following lemma shows that for each n ∈ n+ the class n-s is closed under taking products. in particular it implies that ∞-s is closed under taking finite products. lemma 7.4. let n ∈ n+, let {gi : i ∈ i} be a family of groups and {τi : i ∈ i}, {σi : i ∈ i} two families of group topologies such that σi ≤ τi are group topologies on gi and idgi : (gi, τi) → (gi, σi) is n-step semitopological for every i ∈ i. then ∏ i∈i idgi : ∏ i∈i (gi, τi) → ∏ i∈i (gi, σi) is n-step semitopological. proof. it follows from theorem 2.4. � the following example shows that ∞-s is not closed under taking infinite direct products and answers negatively (b) of problem c. in fact we construct a continuous isomorphism which is not composition of semitopological isomorphisms. example 7.5. for every n ∈ n+ let gn be a nilpotent group of class n. then ∏∞ n=1 idgn : ∏∞ n=1(gn, δgn ) → ∏∞ n=1(gn, ιgn ) is n-step semitopological for no n ∈ n+. indeed idgn+1 : (gn+1, δgn+1 ) → (gn+1, ιgn+1 ) is not n-step semitopological whenever n ∈ n+, in view of theorem 7.2, because gn+1 is not nilpotent of class ≤ n. the next example is another particular case in which we answer problem c(a). example 7.6. let n ∈ n+, let g be a totally markov group and τ, σ group topologies on g. every group topology on g is almost trivial by proposition 5.21. then idg : (g, τ ) → (g, σ) is n-step semitopological if and only if [g, [g, [...[g ︸︷︷︸ n , nσ]]]] ≤ nτ . in fact, suppose that idg : (g, τ ) → (g, σ) is n-step semitopological. then there exist group topologies σ ≤ λn−1, ≤ . . . , ≤ λ1 ≤ τ on g such that idg : (g, τ ) → (g, λ1), idg : (g, λ1) → (g, λ2), . . . . . . , idg : (g, λn−1) → (g, σ) are semitopological. by corollary 3.13 [g, nσ] ⊆ nλ1 , [g, nλ1 ] ⊆ nλ2 , . . . , [g, nλn−1 ] ⊆ nτ and hence [g, [g, [...[g ︸︷︷︸ n , nσ]]]] ≤ nτ . assume that [g, [g, [...[g ︸︷︷︸ n , nσ]]]] ≤ nτ . let nλ1 = [g, nσ], nλ2 = [g, nλ1 ], . . . , nλn−1 = [g, nλn−2 ]. by corollary 3.13 and our assumption idg : (g, τ ) → (g, λ1), idg : (g, λ1) → (g, λ2), . . . , idg : (g, λn−1) → (g, σ) are semitopological. 118 d. dikranjan and a. giordano bruno references [1] v. i. arnautov, semitopological isomorphisms of topological rings (russian), mathematical investigations (1969) 4:2 (12), 3–16. 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[29] m. g. tkachenko, completeness of topological groups (russian), sibirsk. mat. zh. 25 (1984), no. 1, 146–158. arnautov’s problems on semitopological isomorphisms 119 [30] m. g. tkachenko, some properties of free topological groups (russian), mat. zametki 37 (1985), no. 1, 110–118, 139. received july 2008 accepted october 2008 dikran dikranjan (dikran.dikranjan@dimi.uniud.it) dipartimento di matematica e informatica, università di udine, via delle scienze, 206 33100 udine, italy anna giordano bruno (anna.giordanobruno@dimi.uniud.it) dipartimento di matematica e informatica, università di udine, via delle scienze, 206 33100 udine, italy chenlidengagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 309-317 stone compactification of additive generalized-algebraic lattices xueyou chen, qingguo li and zike deng∗ abstract. in this paper, the notions of regular, completely regular, compact additive generalized algebraic lattices ([7]) are introduced, and stone compactification is constructed. the following theorem is also obtained. theorem: an additive generalized algebraic lattice has a stone compactification if and only if it is regular and completely regular. 2000 ams classification: 06b30, 06b35, 54d35, 54h10 keywords: additivity, generalized way-below relation, lower homomorphism, upper adjoint. 1. introduction the notions of directed sets, way-below relations, continuous lattices, algebraic lattices were introduced in [12], and applied in the study of domain theory, topological theory, lattice theory, etc.. as a generalization, d. novak introduced the notions of generalized continuous lattices (m-continuous lattices) and generalized algebraic lattices in [15]. in the study of topological theory and lattice theory, many researchers are interested in the topological representation of a complete lattice. for example: suppose (x, t ) is a topological space, all open sets t of a topological space may be viewed as a frame, and a frame may also be viewed as an open sets lattice. for the theory of frame (locale), please refer to [13]. on the other hand, suppose (x, c) is a co-topological space. c is the set of all closed subsets. d. drake, w. j. thron and s. papert considered c as a complete lattice (c, ∪, ∩, ∅, x)([11, 16]). but unfortunately the correspondence between complete lattices and t0-topological spaces is not one-to-one. ∗this work was partially supported by the national natural science foundation of china (grant no. 10471035/a010104) and natural science foundation of shandong province (grant no. 2003zx13). 310 x. chen, q. li and z. deng to solve the problem, deng also investigated generalized continuous lattices on the basis of [1, 11, 15, 16]. he introduced the notions of maximal systems of subsets, additivity property, homomorphisms, direct sums, lower sublattices in [5, 6, 9, 10]. finally, the question was settled in [7, 8]. he obtained the equivalence between the category of additive generalized algebraic lattices with lower homomorphisms and the category of t0-topological spaces with continuous mappings. this paper is a sequel of [2, 7, 8]. in section 2, we begin with an overview of generalized continuous lattices, deng’s work, which surveys preliminaries. in section 3, we introduce the notions of regular, completely regular and compactness on an additive generalized algebraic lattice, and obtain a stone compactification. 2. preliminaries we introduce some notions for each area, i.e., generalized continuous lattices and additive generalized algebraic lattices. 2.1. generalized continuous lattices. in [15], d. novak introduced the notions of generalized way-below relations and systems of subsets. let (p, ≤) be a complete lattice, ≺ is said to be a generalized way-below relation if (i) a ≺ b ⇒ a ≤ b, (ii) a ≤ b ≺ c ≤ d ⇒ a ≺ d. obviously, it is a natural generalization of the notion of a way-below relation ([12]). m ⊆ 2p is said to be a system of subsets of p , if for a ∈ p , there exists s ∈ m , such that ↓ a =↓ s, where ↓ a = {b | b ≤ a}, ↓ s = ∪{↓ a | a ∈ s}. there are three kinds of common used system of subsets: (i) the system of all finite subsets, (ii) the system of all directed sets and (iii) the system of all subsets. by means of the notion of systems of subsets, he defined a generalized waybelow relation. suppose m is a system of subsets. for a, b ∈ p , a is said to be way-below b with respect to m , in symbols a ≺m b, if for every s ∈ m with b ≤ ∨s, then a ∈↓ s. clearly ≺m is a generalized way-below relation induced by m ([15]). we will denote ≺m as ≺. (p, ≺) is called a generalized continuous lattice, if for every a ∈ p , we have a = ∨ ⇓ a, where ⇓ a = {b | b ≺ a}. a ∈ p is called a compact element, if a ≺ a. let k(≺) = {a ∈ p | a ≺ a}. (p, ≺) is called a generalized algebraic lattice, if for every a ∈ p , we have a = ∨{↓ a ∩ k(≺)}. further study, see [1, 17]. 2.2. additive generalized algebraic lattices. suppose (p, ≺) is a generalized continuous lattice, deng introduced the notion of a maximal system of subsets generated by ≺, that is, stone compactification of additive generalized-algebraic lattices 311 m (≺) = {s ⊆ p | ∀a ∈ p with a ≺ ∨s, then a ∈↓ s}. suppose (p, ≺) is an generalized algebraic lattice, deng defined a new property: (p, ≺) is additivity, if for a, b, c ∈ p with a ≺ b ∨ c implies a ≺ b or a ≺ c ([7]). he investigated the connection between additive generalized algebraic lattices and t0-topological spaces as follows. from the one direction, suppose (p, ≺) is an generalized algebraic lattice, let x = k(≺), and t : p → 2x , t (a) =↓ a ∩ k(≺). if (p, ≺) is additivity, then t satisfies: (1) t (0) = ∅, (2) t (1) = x, (3) for s ∈ m (≺) = m (k(≺ )), t (∨s) = ∪t (s), (4) for s ⊆ p, t (∧s) = ∩t (s), (5) t (a ∨ b) = t (a) ∪ t (b). let c = t (p ), then (x, c) is a t0 co-topological space, and (p, ≺) is isomorphic to (x, c), see [7]. from the other direction, we assume (x, c) is a co-topological space, let q = {{x}− | x ∈ x} be the collection of closure of all singletons. clearly q is a ∨−base for c, i.e., a ∈ c, a is a closed subset, we have a = ∨ ↓ a. m (q) = {s | s ⊆ x, for a ∈ q, a ≤ ∨s we have a ∈↓ s} is a system of subsets induced by q, then (c, ≺m(q)) is an additive generalized algebraic lattice with k(≺m(q)) = q. in this case, a ≺m(q) b for a, b ∈ c if and only if a ⊆ {x}− for some x ∈ b. it is clearly that ≺m(q) is the specialization order ([12]) which is essentially in topological and domain theory. suppose (p1, ≺1), (p2, ≺2) are two generalized continuous lattices, and h : p1 → p2 is said to be a lower homomorphism if it preserves arbitrary joins and the generalized way-below relations. thus a lower homomorphism h is residuated. let g be its upper adjoint, we have (g, h) is a galois connection ([7]). the lower homomorphism also corresponds to the closed mapping. so he obtained the equivalence between the category of additive generalized algebraic lattices with lower homomorphisms and the category of t0-topological spaces with continuous mappings in [7]. from the point of view of deng’s work ([7, 8]), an additive generalized algebraic lattice is algebraic abstraction of a topological space. thus topological theory may be directly constructed on it. the work will benefit the study of the theory of topological algebra and the possible application about additive generalized algebraic lattices. in [2], we constructed tietze extension theorem. furthermore, (c, ≺m(q)) is an example of additive generalized algebraic lattice. for another example in commutative ring, see [7]. for other notions and results cited in this paper, please refer to [7, 15]. 3. stone compactification in the section, (p, ≺) denotes an additive generalized algebraic lattice. it is t0, but not t1 ([7]). k(≺) is the set of all compact elements of (p, ≺). definition 3.1. for a ∈ p , a∗ = ∧{x|a ∨ x = 1} 312 x. chen, q. li and z. deng note 1. since (p, ≺) is a complete lattice, we have a ∨ 1 = 1 for every a ∈ p , so the existence of a∗ is obvious. proposition 3.2. (1) ∀a ∈ p , a ∨ a∗ = 1 (2) a ≺ b ⇒ b∗ ≤ a∗ (3) a ∧ b = 0 ⇒ a ≤ b∗ proof. (1) ∀y ∈ k(≺), if y ≺ a, then y ≺ a ∨ a∗. if y 6≺ a, then ∀x ∈ {x | a ∨ x = 1}, y ≺ y ≤ 1 = a ∨ x. since (p, ≺) is additive, we have y ≺ x, which implies y ≺ a∗. hence y ≺ a ∨ a∗. furthermore (p, ≺) is algebraic, 1 = ∨(↓ 1 ∩ k(≺)), we obtain a ∨ a∗=1. (2) it is clear. (3) ∀y ≺ a, if y = 0, certainly y ≺ b∗. if y 6= 0, a ∧ b = 0, so b ∧ y = 0, which implies y 6≺ b. by (1), we have y ≺ 1 = b ∨ b∗. since (p, ≺) is additive, so y ≺ b∗. thus a = ∨ ⇓ a = ∨{y | y ≺ a} ≤ b∗ � note 2. on (p, ≺), ∀a ∈ p , a ∧ a∗ = 0 is false in general. we introduce the notion of regular on (p, ≺). definition 3.3. (p, ≺) is said to be regular, if for x ∈ k(≺), b ∈ p , x 6≺ b, then x ∧ b = 0. note 3. let (x, t ) be a point-set topological space, if ∀z ∈ x, a closed set a ⊆ x, z 6∈ a if and only if {z}− 6⊆ a, which equivalent to {z}− 6≺ a according to the definition of ≺. if (x, t ) is regular, then there exist u, v two open sets, such that z ∈ u, a ⊆ v and u ∩ v = ∅. we obtain {z}− ∩ a = ∅. otherwise if {z}− ∩ a 6= ∅, there exists y ∈ {z}− ∩ a, so y ∈ a ⊆ v . by y ∈ {z}−, we have {z} ∩ v 6= ∅, thus z ∈ v , a contradiction. definition 3.3 coincides with the above definition when (x, c) is a co-topological space. the notion of compactness is defined as follows. definition 3.4. (p, ≺) is said to be compact if for every d ⊆ p , ∧d = 0 implies that there exists a finite subset d0 ⊆ d satisfying ∧d0 = 0. that is to say, if d has the finite intersection property, then ∧d 6= 0. we introduce the notions of a scale, completely regular on (p, ≺). definition 3.5. a family of elements 〈cα ∈ p | α ∈ [0, 1] & α is a rational number 〉 is called a scale of (p, ≺), if it satisfies: for α < β, we have cα ≺ cβ. for a, b ∈ p , if there exists a scale 〈cα〉, such that a ≤ c0, c1 ≤ b. we denote the relation by a � b. (p, ≺) is said to be completely regular, if ∀a ∈ p, a = ∧{b | a � b}. stone compactification of additive generalized-algebraic lattices 313 suppose (pα, ≺α) is a family of additive generalized algebraic lattices, α ∈ λ (a index set), then (πpα, ≺π) is the direct product, and prα : πpα → pα, ∀a = (aα) ∈ πpα, prα(a) = aα, prα is onto upper adjoint. qα : (pα, ≺α) → (πpα, ≺π) is the lower homomorphism of prα ([9]). by the definitions of prα and qα, we know that qα preserves the generalized way-below relation, and obtain the following proposition. proposition 3.6. suppose (pα, ≺α) is regular, completely regular for every α ∈ λ, then (πpα, ≺π) is also regular, completely regular. proof. it is trivial. � since every inclusion mapping is a lower homomorphism, it is obvious that every lower sublattice of regular, completely regular (p, ≺) is also regular, completely regular. proposition 3.7 (tychonoff product theorem). suppose for every α ∈ λ, (pα, ≺α) is compact, then (πpα, ≺π) is also compact. proof. it is similar to bourbaki’s proof ([14]). (1) let b ⊆ πpα be the maximal with respect to the finite intersection property ([14]) (2) prα : πpα → pα is the onto upper adjoint, then for some α ∈ λ, {prα(b) | b ∈ b} also has the finite intersection property. since (pα, ≺α) is compact, by definitions 3.4, ∧{prα(b) | b ∈ b} 6= 0, so there exists c ∈ k(≺α), c 6= 0, c ≺ ∧{prα(b) | b ∈ b}. (3) qα is the lower homomorphism of prα, so by c ≺α prα(b), we obtain qα(c) ≺ b for every b ∈ b, and qα(c) 6= 0, qα(c) ∈ πpα. thus ∧b 6= 0, which shows that (πpα, ≺π) is compact. � suppose i = [0, 1], the topology on i induced by ρ(x, y) = |x − y|. ci denotes the family of all closed subsets, thus (i, ci ) is a co-topology on i. according to proposition 4.2 ([7]), let q = {{r}− | r ∈ [0, 1]}, m (q) generated by q. ci ordered by inclusion relation, forms a complete lattice. the generalized way-below relation ≺i induced by m (q), and m (≺i ) = m (q). then (ci , ≺i ) is an additive generalized algebraic lattice. by definitions 3.3, 3.4 and 3.5, (ci , ≺i ) is regular, completely regular and compact. furthermore, by propositions 3.6 and 3.7, (πci , ≺π) is also regular, completely regular and compact. by [7] lemma 4.5, the system of subsets m (≺i ) is the collection of classes of closed subsets such that the union of any class is still closed. i.e., ∨s = ∪s for every s ∈ m (≺i ), and ∪s ∈ ci . by the property of closed sets, for d ⊆ ci , we have ∧d = ∩d ∈ ci . lemma 3.8. for a, b ∈ p , suppose a � b, then there exists a lower homomorphism h : (p, ≺) → (ci , ≺i ), such that a ≤ g(0) and g(i) ≤ b. 314 x. chen, q. li and z. deng proof. the upper adjoint g : (ci , ≺i ) → (p, ≺) is first defined. since a � b, then there exists a scale 〈cα〉, such that a ≤ c0, c1 ≤ b and cα ≺ cβ for α < β. this implies {cα} is an increasing function of α. for [α, β] ∈ ci , g([α, β]) = eα ∧ dβ , where eα = ∨r≥αcr, dα = ∨r≤αcr. by [5] theorem 3, we obtain eα, dα ∈ m (≺). (1) for (ci , ≺i ), the closed interval is [α, β], and the elementary closed set fλ = n⋃ i=1 [αi, βi], the closed set f = ∩fλ. since for every s ∈ m (≺i ), by [7] lemma 4.5, ∨s = ∪s. so for every s ∈ m (≺i ), we have g(s) ∈ m (≺). (2) by ∨s = ∪s, we obtain g(∨s) = g(∪s) = ∨g(s) for every s ∈ m (≺i ), (3) since for s ⊆ ci , ∧s = ∩s, we know that g also preserves arbitrary meets, i.e., g(∧s) = ∧g(s). by the above proof, g is an upper adjoint. thus h : (p, ≺) → (ci , ≺i ) is a lower homomorphism. g(i) = g([0, 1]) = e0 ∧ d1 ≤ b g(0) = g({0}) = e0 ∧ d0 ≥ a. � proposition 3.9 (tychonoff embedding theorem). suppose (p, ≺) is an additive generalized algebraic lattice, then (p, ≺) is regular, completely regular iff (p, ≺) is isomorphic to a lower sublattice of (πci , ≺π). proof. by proposition 3.6, (πci , ≺π) is regular, completely regular, and every lower sublattice of (πci , ≺π) is also regular, completely regular, so the proof is trivial on the other hand, suppose (p, ≺) is an additive generalized algebraic lattice, let s = {(gs, hs) | gs : (ci , ≺i ) → (p, ≺) is an upper adjoint, hs : (p, ≺) → (ci , ≺i ) is a lower homomorphism of gs}, s 6= ∅ taking: h: (p, ≺) → (πci , ≺π) the direct product of (ci , ≺i ) by index set of s, ∀a ∈ p , h(a) = πhs(a). by the property of {hs}, h is also a lower homomorphism, so g : (πci , ≺π) → (p, ≺) is the upper adjoint of h. we show (p, ≺) is isomorphic to a lower sublattice of (πci , ≺π), it suffices to prove h is one-to-one on k(≺). ∀x, y ∈ k(≺), x 6= y, then we may assume x 6≺ y. since (p, ≺) is regular, so x ∧ y = 0, which follows that h(x) 6= h(y). thus (p, ≺) is isomorphic a lower sublattice of (πci , ≺π), which generated by h(k(≺)), and h(k(≺)) ⊆ k(≺π). � proposition 3.10 (stone compactification). suppose (p, ≺) is regular, completely regular, then there exists a regular, completely regular compact additive generalized algebraic lattice (βp, ≺β ), such that (p, ≺) is isomorphic to a dense lower sublattice of (βp, ≺β ). stone compactification of additive generalized-algebraic lattices 315 proof. by proposition 3.9, (p, ≺) is isomorphic to a lower sublattice of (πci , ≺π). let (βp, ≺β ) be the closure of the lower sublattice, and the compactness of (βp, ≺β ) follows from proposition 3.7. � in general, (βp, ≺β ) is said to be a stone compactification of (p, ≺). note 4. clearly, if the generalized way-below relation ≺ satisfies the interpolation property, then (p, ≺) is completely regular by the choice axiom. as the end of this paper, we embark on an alternative description of (βp, ≺β ) by means of ideals of (p, ≺). definition 3.11. i ⊆ p is said to be an ideal if (1) for any finite e ⊆ i, ∨e ∈ i, (2) z ∈ i, x ≤ z implies x ∈ i. idl(p ) denotes all ideals of p , and certainly idl(p ) is a complete lattice, the order is the inclusion order. ∀i ∈ idl(p ), ↓ i = {j | j ≤ i}, where j ≤ i iff j ⊆ i definition 3.12. for i, j ∈ idl(p ), a binary relation on idl(p ) is defined as: i ≺∗ j if and only if ∨i ≺ ∨j holds on (p, ≺). lemma 3.13 ([15]). ≺∗ is a generalized way-below relation on idl(p ). proof. (1) i ≺∗ j if and only if ∨i ≺ ∨j holds on (p, ≺). then ∀a ∈ i, a ≤ ∨i ≺ ∨j, so a ∈ j. that is, i ⊆ j, thus i ≤ j. (2) i1 ≤ i2 ≺ ∗ i3 ≤ i4, which implies that ∨i1 ≤ ∨i2 ≺ ∨i3 ≤ ∨i4 holds on (p, ≺). so we have ∨i1 ≺ ∨i4, thus i1 ≺ ∗ i4. � lemma 3.14. idl(p ) is algebraic. proof. for i, j ∈ idl(p ), i ≺∗ j implies ∨i ≺ ∨j on (p, ≺). since (p, ≺) is algebraic, there exists c ∈ k(≺), such that ∨i ≤ c ≤ ∨j. furthermore ↓ c ∈ idl(p ). by c ∈ k(≺), so c ≺ c on (p, ≺), hence ↓ c ≺∗↓ c on idl(p ). i.e, ↓ c ∈ k(≺∗) by definition 3.11. considering i ≤↓ c ≤ j and ↓ c ∈ k(≺∗), thus (idl(p ), ≺∗) is algebraic. � lemma 3.15. idl(p ) is continuous. proof. it is trivial ([4]). � lemma 3.16. idl(p ) is additive. proof. for i ≺∗ j1 ∨ j2, where i, j1, j2 ∈ idl(p ), then on (p, ≺), ∨i ≺ ∨(j1 ∨ j2) = (∨j1) ∨ (∨j2) holds. since (p, ≺) is additive, it follows that ∨i ≺ ∨j1 or ∨i ≺ ∨j2, thus i ≺ ∗ j1 or i ≺ ∗ j2, which proves lemma 3.16. � proposition 3.17. (idl(p ), ≺∗) is an additive generalized algebraic lattice. proof. by lemmas 3.14, 3.15, 3.16. � lemma 3.18. for any regular (p, ≺), idl(p ) is also regular. 316 x. chen, q. li and z. deng proof. it is obvious that on (idl(p ), ≺∗), k(≺∗) = {↓ x | x ∈ k(≺)}. then ∀ ↓ x ∈ k(≺∗), ∀j ∈ idl(p ), if ↓ x 6≺∗ j, which implies x 6≺ ∨j by definition 3.11. since (p, ≺) is regular, x ∈ k(≺), ∨j ∈ p , x 6≺ ∨j, then x ∧ (∨j) = 0. so we obtain that ↓ x ∧ j = 0. it follows that idl(p ) is regular. � for i ∈ idl(p ), i is called completely regular, if ∀a ∈ i, there exists b ∈ i, such that a � b. let r(p ) = {i is completely regular in idl(p )}, then we have lemma 3.19. suppose (p, ≺) is completely regular, then (r(p ), ≺∗) is also completely regular. proof. it is trivial. � lemma 3.20. suppose (p, ≺) is compact, then (r(p ), ≺∗) is also compact. proof. for a family {iα|α ∈ λ} satisfying ∧iα = 0. since (p, ≺) is a complete lattice, iα = ∨{↓ x | x ∈ iα}, we may assume iα =↓ aα. then ∧iα = ∧(↓ aα) =↓ (∧aα), thus ↓ (∧aα) = 0, it follows that ∧aα = 0. furthermore (p, ≺) is compact, by definition 3.4, there exist a1, a2, · · · , am satisfying m∧ i=1 ai = 0. by this, it is easy to prove m∧ i=1 (↓ ai) = 0. that is, m∧ i=1 ii = 0. thus (r(p ), ≺ ∗) is compact. � proposition 3.21. suppose (p, ≺) is compact, regular and completely regular, then (p, ≺) and r(p ) are isomorphic. proof. by lemmas 3.18, 3.19, 3.20, and h : p → r(p ), ∀a ∈ p , h(a) =⇓ a = {b | b ≺ a}, certainly h(a) ∈ r(p ). a ≺ b holds on (p, ≺) if and only if h(a) ≺∗ h(b) holds on (r(p ), ≺∗). since (p, ≺) is continuous, ∀a ∈ p , a = ∨ ⇓ a, so (p, ≺) is embedded into r(p ), and h preserves the generalized way-below relation. it is trivial to prove h is one-to-one. � by proposition 3.10, suppose (p, ≺) is compact, regular, completely regular, then (βp, ≺β ) and (r(p ), ≺ ∗) are also isomorphic by propositions 3.10 and 3.21, the following theorem is also obtained. theorem 3.22. an additive generalized algebraic lattice (p, ≺) has a stone compactification iff it is regular, completely regular. note 5. (1) according to [3], the class of generalized continuous lattices includes completely distributive lattices and traditional continuous lattices ([15]) as its special cases. (2) according to [4], the traditional algebraic lattice is generalized algebraic lattice ([4]), and completely distributive lattice is also generalized algebraic lattice ([4]). stone compactification of additive generalized-algebraic lattices 317 acknowledgements we are grateful to the editor for his valuable comments and suggestions. references [1] h. j. bandelt, m-distributive lattices, arch math 39 (1982), 436–444. [2] x. chen, q. li, f. long and z. deng, tietze extension theorem on additive generalized algebraic lattice, acta. mathematica scientia (a)(in chinese), accepted. [3] z. deng, generalized-continuous lattices i, j. hunan univ. 23, no. 3 (1996), 1–3. [4] z. deng, generalized-continuous lattices ii, j. hunan univ. 23, no. 5 (1996), 1–3. [5] z. deng, homomorphisms of generalized-continuous lattices, j. hunan univ. 26, no. 3 (1999), 1–4. [6] z. deng, direct sums and sublattices of generalized-continuous lattices, j. hunan univ. 28, no. 1 (2001), 1–4. [7] z. deng, topological representation for generalized-algebraic lattices, (in w.charles. holland, edited: ordered algebraic structures, algebra, logic and applications vol 16, 49-55 gordon and breach science publishers, 2001.) [8] z. deng, additivity of generalized algebraic lattices and t0-topology, j. hunan univ. 29, no. 5 (2002), 1–3. [9] z. deng, structures of generalized-continuous lattices, j. hunan univ. 28, 6 (2001), 1–4. [10] z. deng, representation of strongly generalized-continuous lattices in terms of complete chains, j. hunan univ. 29, no. 3 (2002), 8–10. [11] d. drake and w. j. thron, on representation of an abstract lattice as the family of closed sets of a topological space, trans. amer. math. soc. 120 (1965), 57–71. [12] g. gierz et, al, a compendium of continuous lattices, berlin, speringerverlag, 1980. [13] p. t. johnstone, stone spaces, cambridge univ press, cambridge, 1983. [14] j. l. kelly, general topology, van nostrand princeton, nj, 1995. [15] d. novak, generalization of continuous posets, trans. amer. math. soc 272 (1982), 645–667. [16] s. papert, which distributive lattices are lattices of closed sets?, proc. cambridge. phil. soc. 55 (1959), 172–176. [17] q. x. xu, construction of homomorphisms of m-continuous lattices, trans. amer. math. soc. 347 (1995), 3167–3175. received december 2005 accepted may 2006 xueyou chen (chenxueyou0@yahoo.com.cn) college of mathematics and information science, shandong university of technology, zibo,shandong 255012, p. r. china. qingguo li college of mathematics and economics, hunan university, changsha, hunan 410012, p. r.china. zike deng college of mathematics and economics, hunan university, changsha, hunan 410012, p. r.china. @ appl. gen. topol. 15, no. 1 (2014), 85-92doi:10.4995/agt.2014.1897 c© agt, upv, 2014 baire spaces and hyperspace topologies revisited steven bourquin and lászló zsilinszky department of mathematics and computer science, the university of north carolina at pembroke, pembroke, nc 28372, usa (steven.bourquin@uncp.edu, laszlo@uncp.edu) abstract it is the purpose of this paper to show how to use approach spaces to get a unified method of proving baireness of various hyperspace topologies. this generalizes results spread in the literature including the general (proximal) hit-and-miss topologies, as well as various topologies generated by gap and excess functionals. it is also shown that the vietoris hyperspace can be non-baire even if the base space is a 2nd countable hausdorff baire space. 2010 msc: primary 54b20; secondary 54e52, 54a05 keywords: baire spaces, approach spaces, (proximal) hit-and-miss topologies, weak hypertopologies, banach-mazur game 1. introduction a topological space is a baire space [7] provided countable collections of dense open subsets have a dense intersection. baireness is one of the weakest completeness properties, and yet, it has fundamental applications throughout mathematics. this is why there has been interest in investigating baireness, along with other completeness properties, in the theory of hyperspace topologies which in turn have applications in various branches of mathematics on their own ([1], [12]). in the present paper we will continue in this research by exhibiting some common features of the plethora of studied hyperspaces as far as their baireness is concerned. indeed, following the unified exposition of hyperspace topologies introduced in [20], we prove baireness of these general hyperspaces in received 16 november 2013 – accepted 13 february 2014 http://dx.doi.org/10.4995/agt.2014.1897 s. bourquin and l. zsilinszky one theorem, thereby generalizing and extending several results about specific hyperspaces spread in the literature ([1],[14],[19],[23],[21],[3],[4],[9]). in hyperspace theory, it is customary to assume that the base space is hausdorff (or at least t1), since then singletons are closed, and the base space embeds into the hyperspace. it was observed that imposing separation axioms on the base space is frequently not necessary to obtain results on hypertopologies (see [6], [19], [24], [10]), which is the case throughout this paper as well. at the end, an example is provided that shows the limitations of baireness results for the vietoris topology. 2. preliminaries throughout the paper ω stands for the non-negative integers, p(x) for the power set, cl(x) for the nonempty closed subsets of a topological space x, and ec for the complement of e ⊂ x in x. the general description of the hyperspaces we will use was given in [20], here we just provide the definitions so the paper is self-contained, more detail can be found in [20], and the references therein. suppose that (x, δ) is an approach space [13], i.e. x is a nonempty set and δ : x × p(x) → [0, ∞] is a so-called distance (on x) having the following properties: ∀x ∈ x : δ(x, {x}) = 0(d1) ∀x ∈ x : δ(x, ∅) = ∞(d2) ∀x ∈ x ∀a, b ⊂ x : δ(x, a ∪ b) = min{δ(x, a), δ(x, b)}(d3) ∀x ∈ x ∀a ⊂ x ∀ε > 0 : δ(x, a) ≤ δ(x, bε(a)) + ε,(d4) where bε(a) = {x ∈ x : δ(x, a) ≤ ε}; we will also use the notation sε(a) = {x ∈ x : δ(x, a) < ε}. every approach space (x, δ) generates a topology τδ on x defined by the closure operator: ā = {x ∈ x : δ(x, a) = 0}, a ⊂ x. the functional d : p(x) × p(x) → [0, ∞] will be called a gap provided: ∀a, b ⊂ x : d(a, b) ≤ inf a∈a δ(a, b)(g1) ∀x ∈ x ∀a ⊂ x : d({x}, a) = inf y∈{x} δ(y, a)(g2) ∀a, b, c ⊂ x : d(a ∪ b, c) = min{d(a, c), d(b, c)}.(g3) in the sequel (unless otherwise stated) x will stand for an approach space (x, δ) with a gap d (denoted also as (x, δ, d)). remark 2.1. (1) examples of distances: • [13] let x be a topological space. for x ∈ x and a ⊂ x define δt(x, a) = { 0, if x ∈ ā, ∞, if x /∈ ā. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 86 baire spaces and hyperspace topologies revisited then δt is a distance on x, and τδt coincides with the topology of x. • let (x, u) be a uniform space. then u is generated by the family d of uniform pseudo-metrics on x such that d ≤ 1 for all d ∈ d, and d1, d2 ∈ d implies max{d1, d2} ∈ d. then δu(x, a) = sup d∈d d(x, a), x ∈ x, a ⊂ x defines a (bounded) distance on x and τδu coincides with the topology induced by u on x. • let (x, d) be a metric space. then δm(x, a) = d(x, a) is a distance on x and τδm is the topology generated by d on x. (2) examples of gaps: • for an arbitrary approach space (x, δ) d(a, b) = inf a∈a δ(a, b) is clearly a gap on x. this is how we are going to define the gap dt (resp. dm) in topological (metric) spaces using the relevant distance δt (resp. δm). • let (x, u) be a uniform space generated by the family d of uniform pseudo-metrics on x bounded above by 1. for a, b ⊂ x define du(a, b) = sup d∈d inf a∈a d(a, b). then du is a gap on x. (3) excess functional: for all a, b ⊂ x define the excess of a over b by e(a, b) = sup a∈a δ(a, b). the symbols et, eu, em will stand for the excess in topological, uniform and metric spaces, respectively defined via δt, δu, δm. 3. hyperspace topologies for e ⊂ x write e− = {a ∈ cl(x) : a ∩ e 6= ∅}, e+ = {a ∈ cl(x) : a ⊂ e} and e++ = {a ∈ cl(x) : d(a, ec) > 0}. in what follows ∆1 ⊂ cl(x) is arbitrary and ∆2 ⊂ cl(x) is such that ∀ε > 0 ∀a ∈ ∆2 ⇒ sε(a) is open in x. denote d = d(∆1, ∆2) = ⋃ k∈ω(∆1 ∪ {∅}) k+1 × (∆2 ∪ {x}) k+1 × (0, ∞)2k+2. whenever referring to some s, t ∈ d, we will assume that for some k, l ∈ ω s = (s0, . . . , sk; s̃0, . . . , s̃k; ε0, . . . , εk; ε̃0, . . . , ε̃k) t = (t0, . . . , tl; t̃0, . . . , t̃l; η0, . . . , ηl; η̃0, . . . , η̃l). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 87 s. bourquin and l. zsilinszky for s ∈ d denote m(s) = ⋂ i≤k (bεi(si)) c ∩ sε̃i(s̃i) and s∗ = ⋂ i≤k {a ∈ cl(x) : d(a, si) > εi and e(a, s̃i) < ε̃i}. for u0, . . . , un ∈ τδ \ {∅} and s ∈ d denote (u0, . . . , un)s = ⋂ i≤n u−i ∩ s ∗, [u0, . . . , un]s = ∏ i≤n (ui ∩ m(s)) × ∏ i>n m(s). it is easy to see that the collections b∗ = {(u0, . . . , un)s : u0, . . . , un ∈ τδ \ {∅}, s ∈ d, n ∈ ω}, b = {[u0, . . . , un]s : u0, . . . , un ∈ τδ \ {∅}, s ∈ d, n ∈ ω} form a base for topologies on cl(x) and xω, respectively; denote them by τ∗, and τ, respectively. remark 3.1. note that τ is a “pinched-cube” topology as defined in [3], [23]; indeed, we just need to take ∆ = {m(s)c : s ∈ d}. remark 3.2. • let (x, τ) be a topological space. let ∆1 = ∆ and ∆2 = {x}. then for b ∈ ∆ and ε, η > 0, {a ∈ cl(x) : dt(a, b) > ε} = (b c)+ and {a ∈ cl(x) : et(a, x) < η} = cl(x). thus τ ∗ = τ+ is the general hit-and-miss topology on cl(x) (see [17], [1], [8], [19], [3]). choosing ∆ = cl(x) we get the most studied hit-and-miss topology, the socalled vietoris topology τv (cf. [15], [5], [1]); a typical base element for τv is (u0, . . . , un) = {a ∈ cl(x) : a ⊆ ⋃ i≤n ui and a ∩ ui 6= ∅ for all i ≤ n}. • let (x, u) be a uniform space. let ∆1 = ∆ and ∆2 = {x}. then for b ∈ ∆ and ε, η > 0, {a ∈ cl(x) : du(a, b) > ε} = (b c)++ and {a ∈ cl(x) : eu(a, x) < η} = cl(x). thus τ ∗ = τ++ is the proximal hit-and-miss topology on cl(x) (see [1], [19], [3]). • let (x, d) be a metric space. let ∆1, ∆2 ⊂ cl(x) be such that ∆1 contains the singletons. then τ∗ coincides with the weak hypertopology τweak generated by gap and excess functionals (see [2], [9], [20]). as we indicated in the introduction, we do not assume any separation axiom on x, however, the following property seems to be necessary: we will say that (x, δ, d) has property (p) provided ∀x ∈ x ∀k ∈ ω ∀a0, . . . , ak ∈ cl(x) ∀ε0, . . . , εk > 0 ∃y ∈ {x} : δ(x, ai) > εi =⇒ d({y}, ai) > εi for all i ≤ k. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 88 baire spaces and hyperspace topologies revisited this property is satisfied in uniform and metric spaces, and a topological space x has property (p) iff x is weakly-r0 [24], i.e. for all open u ⊆ x and x ∈ u there is a y ∈ {x} with {y} ⊂ u iff each nonempty difference of open sets contains a nonempty closed set. we will say that the family d is weakly quasi-urysohn [20] provided for all ∅ 6= (u0, . . . , un)s ∈ b ∗ there is a t ∈ d such that ∅ 6= (u0, . . . , un)t ⊂ (u0, . . . , un)s and (*) ∀e countable : e ⊂ m(t ) =⇒ e ∈ s∗. the translation of this property for the (proximal) hit-and-miss topologies is as follows: given a topological (uniform) space x (resp. (x, u)), the family ∆ ⊆ cl(x) is said to be (uniformly) weakly quasi-urysohn provided whenever s ∈ ∆ is disjoint to some nonempty open ui ⊆ x (i ≤ n), there exists t ∈ ∆ such that ui ∩ t c 6= ∅ for all i ≤ n, s ⊂ t , and ∀e countable : (e ⊂ t c =⇒ e ⊂ sc) (resp. e ⊂ u[s]c for some u ∈ u). we will say that x is (weakly) quasi-regular provided every nonempty open u ⊂ x has a nonempty open subset v such that v ⊆ u (resp. e ⊂ u for all countable e ⊂ v ). it is not hard to see that if x is weakly quasi-regular, then cl(x) is weakly quasi-urysohn. proposition 3.3 ([20, lemma 3.1]). labelzs2 suppose that (x, δ, d) has property (p), and (u0, . . . , un)s, (v0, . . . , vm)t ∈ b ∗. then (u0, . . . , un)s ⊂ (v0, . . . , vm)t implies m(s) ⊂ m(t ), and for all j ≤ m there exists i ≤ n with m(s) ∩ ui ⊂ m(t ) ∩ vj. although it would be possible to establish baireness of (cl(x), τ∗) using the so-called banach-mazur game, as done in [14], [19], or [3], we have chosen a different method, which makes the proofs more transparent and actually improves on results of the above mentioned papers. we will need some auxiliary material: if y, z are topological spaces, the mapping f : y → z is said to be feebly continuous [7] provided intf−1(u) 6= ∅ for each open u ⊂ z with f−1(u) 6= ∅; further f is δ-open [7] provided f(a) is somewhere dense in z for every somewhere dense a of y . proposition 3.4 ([7, theorem 4.7]). if f is a feebly continuous δ-open function from a baire space onto a space y , then y is a baire space. the following theorem generalizes [14, theorem 3.8], [19, theorem 4.1], and [3, theorem 2.5]: theorem 3.5. suppose that x has property (p) and d is a weakly quasiurysohn family. if (xω, τ) is a baire space, then so is (cl(x), τ∗). proof. denote s(x) = {a ∈ cl(x) : a is separable}. then s(x) is dense in (cl(x), τ∗), since even the set of closures of finite subsets of x is. thus, it suffices to prove that (s(x), τ∗ ↾s(x)) is a baire space. define the mapping f : xω → s(x) via f((xk)k) = {xk : k ∈ ω}, where (xk)k ∈ x ω. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 89 s. bourquin and l. zsilinszky we will show that f is a feebly continuous δ-open function, so proposition 3.4 will apply. to see feeble continuity, take a u = (u0, . . . , un)s∩s(x) such that f −1(u) 6= ∅. now if we take the t ∈ d from weak quasi-urysohnness of d corresponding to (u0, . . . , un)s, then (*) virtually claims that ∅ 6= [u0, . . . , un]t ⊂ f −1(u). to justify δ-openness of f, take an a ⊂ xω which is dense in some ∅ 6= v = [v0, . . . , vm]p ∈ b. then f(a) is dense in v ∗ = (v0, . . . , vm)p ∩ s(x). indeed, if u∗ = (u0, . . . , un)s ∩ s(x) is a nonempty open subset of v ∗, take the t ∈ d from weak quasi-urysohnness of d corresponding to (u0, . . . , un)s. then in view of proposition ??, [u0, . . . , un]t is a nonempty open subset of v. consequently, we can find an (xk)k ∈ a ∩ [u0, . . . , un]t . then e = {xk : k ∈ ω} ⊂ m(t ), so by (*), f((xk)k) = e ∈ s ∗, and hence f((xk)k) ∈ f(a) ∩ u∗. � a collection p of nonempty open subsets in a space x is a π-base, if every nonempty open subset of x contains at least one member of p; moreover, p is a countable-in-itself π-base [22], provided each member of p contains countably many members of p. corollary 3.6. suppose that x is a baire space with a countable-in-itself πbase and it has property (p). suppose that d is a weakly quasi-urysohn family. then (cl(x), τ∗) is a baire space. proof. since τ is a pinched-cube topology (see remark 3.1), it follows by [23, theorem 2.1], that (xω, τ) is a baire space, so theorem 3.5 applies. � 4. applications the following theorem improves [19, corollary 4.2], and [3, theorem 4.1,theorem 5.1]: theorem 4.1. suppose that x is a weakly-r0 (resp. uniform) space having a countable-in-itself π-base, which is also a baire space. if ∆ ⊂ cl(x) is a (uniformly) weakly quasi-urysohn subfamily, then (cl(x), τ+) (resp. (cl(x), τ++)) is a baire space. the previous theorem yields the following corollary, which slightly generalizes [14, corollary 3.9] (see also [4, corollary 2.2]): theorem 4.2. let (x, τ) be a weakly-r0, weakly quasi-regular baire space having a countable-in-itself π-base. then (cl(x), τv ) is a baire space. in the next example we show that weak quasi-regularity of x is an essential condition in the baireness results concerning the vietoris topology. to do this we need to recall some basic facts about the banach-mazur game bm(x) (see [7] or [11]) played by players α and β who take turns choosing sets from a fixed π-base p in the topological space x: β starts by picking v0, then α responds by choosing some u0 ⊆ v0. continuing like this, while each player picks a c© agt, upv, 2014 appl. gen. topol. 15, no. 1 90 baire spaces and hyperspace topologies revisited nonempty open set contained in the previous choice of the opponent, one gets a run v0, u0, . . . , vn, un, . . . of bm(x), which is won by α, if ⋂ i<ω ui 6= ∅, otherwise, β wins. the key result about the banach-mazur game is that x is not a baire space iff β has a winning tactic in bm(x), i.e. there is a function t : p ∪ {∅} → p such that if t(∅) = v0, and t(un) = vn+1 for each n < ω, then β wins. example 4.3. there exists a hausdorff, baire, 2nd countable space such that (cl(x), τv ) is not baire. proof. let e be the euclidean topology on r, and b ⊂ r be a bernstein set [7], i.e. such that both b and r \ b meet every uncountable gδ subset of r. define a finer than e topology eb on r as follows: eb = {u ∪ v ∩ b : u, v ∈ e}. then x = (r, eb) is clearly hausdorff and 2nd countable. in order to prove that (cl(x), τv ) is β-favorable, we will play the banachmazur game using the following π-base for the hyperspace: pv = {(i0 ∩ b, . . . , in ∩ b) : i0, . . . , in are pairwise disjoint open intervals}. define a tactic tv for β in bm(cl(x)) as follows: put tv (∅) = b +; moreover, if u = (i0 ∩ b, . . . , in ∩ b) ∈ pv , for each i ≤ n, choose disjoint bounded open intervals j0i , j 1 i so that j 0 i ∪ j 1 i e ⊆ ii, and put tv (u) = (j 0 0 ∩ b, j 1 0 ∩ b, . . . , j 0 n ∩ b, j 1 n ∩ b). let b+, u0, tv (u0), . . . , uk, tv (uk), . . . be a run of bm(cl(x)) compatible with tv . suppose that there is some a ∈ ⋂ k uk. then a is a eb-closed subset of b, and since the e-, and eb-closure of subsets of b coincide, a would be a eclosed, and in view of the definition of tv , also dense-in-itself subset of b, which contradicts the definition of the bernstein set. consequently, ⋂ k uk = ∅, and β wins the run. � remark 4.4. it follows by [18, example 2.7], that the previous example is not quasi-regular, but it is not even weakly quasi-regular by theorem 4.2. it is also an example showing that quasi-regularity cannot be removed in the key result of [4, theorem 2.1], stating that if x is a quasi-regular space such that xω is a baire space, then (cl(x), τv ) is a baire space, since by a result of oxtoby [16] products of 2nd countable baire spaces are baire. our last result generalizes [9, theorem 4.2] and [20, corollary 6.1]: theorem 4.5. let (x, d) be a separable metric baire space, and ∆1, ∆2 ⊂ cl(x) be such that ∆1 contains the singletons. then (cl(x), τweak) is a baire space. proof. it follows from corollary 3.6, since d(∆1, ∆2) is weakly quasi-urysohn by [20, remark 3.1(iii)]. � c© agt, upv, 2014 appl. gen. topol. 15, no. 1 91 s. bourquin and l. zsilinszky references [1] g. beer, topologies on closed and closed convex sets, kluwer, dordrecht, 1993. 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[20] l. zsilinszky, topological games and hyperspace topologies, set-valued anal. 6 (1998), 187–207. [21] l. zsilinszky, baire spaces and weak topologies generated by gap and excess functionals, math. slovaca 49 (1999), 357–366. [22] l. zsilinszky, products of baire spaces revisited, fundam. math. 183 (2004), 115–121. [23] l. zsilinszky, on baireness of the wijsman hyperspace, boll. unione mat. ital. sez. a mat. soc. cult. (8) 10 (2007) 1071–1079. [24] l. zsilinszky, on separation axioms in hyperspaces, rendiconti del circolo matematico di palermo (2) 45 (1996), 75–83. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 92 @ appl. gen. topol. 23, no. 2 (2022), 243-253 doi:10.4995/agt.2022.16720 © agt, upv, 2022 a urysohn lemma for regular spaces ankit gupta a and ratna dev sarma b a department of mathematics, bharati college, (university of delhi), delhi 110058, india (ankitsince1988@yahoo.co.in) b department of mathematics, rajdhani college (university of delhi), delhi 110015, india (ratna sarma@yahoo.com) communicated by s. garćıa-ferreira abstract using the concept of m-open sets, m-regularity and m-normality are introduced and investigated. both these notions are closed under arbitrary product. m-normal spaces are found to satisfy a result similar to urysohn lemma. it is shown that closed sets can be separated by m-continuous functions in a regular space. 2020 msc: 54a05; 54d10; 54d15; 54c08. keywords: regularity; normality; m-normality; m-regularity; urysohn lemma. 1. introduction nowadays topological approaches are being investigated in various diverse field of science and technology such as computer graphics, evolutionary theory, robotics[4, 9, 10] etc. to name a few. in a finite topological space, the intersection of all open neighbourhoods of a point p is again an open neighbourhood of p, which is the smallest one. it is called the minimal neighbourhood of p. however, in general framework, we define the minimal open sets or m-open sets as the ones obtained by taking the arbitrary intersections of the open sets. they have been studied in the recent past by several researchers [2, 6]. in this paper, we further use them for construction of new notions in topology, namely m-regularity and m-normality. these two notions are distinct from the already existing notions of regularity and normality, and are found to have received 23 november 2021 – accepted 05 may 2022 http://dx.doi.org/10.4995/agt.2022.16720 https://orcid.org/0000-0003-0760-4467 https://orcid.org/0000-0002-4181-3732 a. gupta and r. d. sarma several interesting properties. both of them are closed under arbitrary product. m-normal spaces are found to exhibit urysohn lemma type property. finally it is shown that even in regular spaces, disjoint pair of closed sets can be separated by mappings, the so called m-continuous mappings. in that sense, the last result of the paper may be treated as urysohn lemma for regular spaces. here it may be mentioned that the classical urysohn lemma is unprovable in zf[1, 5]. the usual proof of urysohn lemma in kelley [8] uses the axioms of dependent choice to successfully select open sets separating previously chosen sets [1]. similar choices have been made in our proof also, making it valid only in zfc. 2. preliminaries definition 2.1 ([6]). let (x,τ) be a topological space. a set a ⊆ x is called m-open if a can be expressed as intersection of a subfamily of open sets. the complement of an m-open set is called an m-closed set. the collection of m-open sets of a topological space (x,τ) is denoted by m. clearly, every open set is m-open. in a finite space, open sets are the only m-open sets. the following example gives an idea about the abundance of m-open sets. example 2.2. let x = n be the set of natural numbers, equipped with the co-finite topology. then every subset of x is m-open. proposition 2.3 ([6]). for a topological space (x,τ), we have the following results: (i) ∅,x ∈m; (ii) m is closed under arbitrary union; (iii) m is closed under arbitrary intersection. definition 2.4. [6] let (x,τ) and (y,µ) be two topological spaces. then a function f : x → y is said to be m-continuous at a point x ∈ x if for every open neighbourhood v of f(x) there exists an m-open set u containing x in x such that f(u) ⊆ v . a function f : x → y is said to be m-continuous [6] if it is m-continuous at each point x of x. since every open set is m-open. therefore every continuous function is mcontinuous. but the converse need not be true. example 2.5. let x = n be the set of natural numbers equipped with the co-finite topology and y = {a,b,c} with the topology µ = {∅,{a},{b,c},y}. then consider a function f : x → y defined as: f(x) =   a if x < 10, b if 10 ≤ x < 100, c otherwise. © agt, upv, 2022 appl. gen. topol. 23, no. 2 244 a urysohn lemma for regular spaces since every subset a ⊆ n is m-open under the co-finite topology, function f is m-continuous. but f is not a continuous function as f−1({a}) = {x ∈ n |x < 10} is not an open set in n under the co-finite topology. theorem 2.6. let (x,τ) and (y,µ) be two topological spaces. then for a function f : (x,τ) → (y,µ), the following are equivalent: (i) f is m-continuous; (ii) inverse image of every open subset of y is m-open; (iii) inverse image of every closed subset of y is m-closed. proof. (i) ⇒ (ii): let u be any open subset of y and let x ∈ f−1(u). then f(x) ∈ u. therefore there exists an m-open subset v in x such that x ∈ v and f(v ) ⊆ u. thus x ∈ v ⊆ f−1(u), therefore f−1(u) is an m-open neighbourhood of x. hence f−1(u) is m-open. (ii) ⇒ (iii): let a be any closed subset of y . then y �a is open and therefore f−1(y \a) is m-open, that is, x\f−1(a) is m-open. hence f−1(a) is m-closed. (iii) ⇒ (i): let m be an open neighbourhood of f(x), therefore y \m is closed, and consequently f−1(y \m) is m-closed. thus f−1(m) is m-open and hence x ∈ f−1(m) = n (say). then, we have n is an m-open neighbourhood of x such that f(n) ⊆ m. � in next result, we prove that arbitrary product of m-open sets is again mopen under the product topology. here we use the fact that “an open set v of a product topology can be realized in the form v = ∏ i∈λ vi, where vi ∈ τi and vi = xi except for finitely many i ′s.” this can be verified using the concept of basic open sets and the fact that(⋃ i∈i ai ) ×  ⋃ j∈j bj  × . . . = ⋃ (i,j,...)∈i×j×... (ai ×bj × . . .) . we also provide below the following results, which will be used in our paper. lemma 2.7 ([3, p. 28]). let {aα,β} be an arbitrary family of non-empty sets. then we have ⋂ β (∏ α aα,β ) = ∏ α  ⋂ β aα,β   lemma 2.8 ([7, p. 34]). let {(xα,τα)|α ∈a} be an arbitrary family of topological spaces and let aα ⊆ xα for each α ∈a. then we have cl (∏ α vα ) = ∏ α (cl(vα)). theorem 2.9. let (xα,τα) be topological spaces and uα be an m-open set in (xα,τα). then the product of u ′ αs is an m-open set in the product topology∏ α τα of x = ∏ α xα. © agt, upv, 2022 appl. gen. topol. 23, no. 2 245 a. gupta and r. d. sarma proof. let {(xα,τα)}α be a family of topological spaces and a be an m-open set in the product topology x = ∏ α xα. then there exists open sets ui in x such that a = ⋂ i ui. since ui is an open set in the product topology of x, therefore there exists open set uα,i ∈ τα with uα,i = xα for all but finitely many α’s such that ui = ∏ α uα,i. hence a = ⋂ i (∏ α uα,i ) . using the fact that ⋂ β (∏ α aα,β ) = ∏ α  ⋂ β aα,β  , in view of lemma 2.7, we have a = ⋂ i (∏ α uα,i ) = ∏ α (⋂ i uα,i ) . hence the proof. � now, we will show that every subset of a t1-topological space (x,τ) is mopen. theorem 2.10. every subset of a t1-space is m-open. proof. let (x,τ) be a topological space, which is t1. let a be a non-empty subset of x. then every singleton {x}⊆ x is a closed set. therefore, consider a = ⋂ x∈x\a x \{x}. since every singleton is closed therefore x\{x} is an open set in x. hence the arbitrary intersection of open sets is m-open, thus a is m-open. � 3. m-regular spaces definition 3.1. a topological space (x,τ) is said to be m-regular if for each pair consisting of a point x and an m-closed set b not containing x, there exists a disjoint pair of an m-open set u and an open set v containing x and b respectively. in other words, for every pair, x and b with x /∈ b, where b is an m-closed set, there exist an m-open set u and an open set v such that x ∈ u, b ⊆ v and u ∩v = ∅. now, we provide a characterization for m-regularity. theorem 3.2. let (x,τ) be a topological space. then x is m-regular if and only if for a given point x ∈ x and an m-open neighbourhood u of x, there exists an m-open neighbourhood v of x such that x ∈ v ⊆ cl(v ) ⊆ u. proof. suppose that x is m-regular. let x ∈ x and u ⊆ x, an m-open neighbourhood of x, be given. then b = x \ u is an m-closed set disjoint from x. by the given hypothesis, there exists a disjoint pair of an m-open set v and an open set w containing x and b respectively. thus we have, x ∈ v and x \ u = b ⊆ w , that is, x ∈ v ⊆ x \ w ⊆ u. hence we have, x ∈ v ⊆ cl(v ) ⊆ cl(x \ w) = x \ w ⊆ u. therefore we have, x ∈ v ⊆ cl(v ) ⊆ u. © agt, upv, 2022 appl. gen. topol. 23, no. 2 246 a urysohn lemma for regular spaces conversely, suppose that a point x ∈ x and an m-closed set b not containing x are given. then u = x \ b is an m-open set containing x. then, by the hypothesis, there exists an m-open neighbourhood v of x such that x ∈ v ⊆ cl(v ) ⊆ u. thus we have, a disjoint pair of m-open set v and an open set x \ cl(v ) which contains x and b respectively. therefore (x,τ) is mregular. � with the help of following example, we show that an m-regular space need not be regular. example 3.3. let x = n be the set of natural numbers, equipped with the co-finite topology. then every subset a of x is m-open. therefore x is a m-regular space but not a regular space. but, every regular topological space is m-regular. for this, we have the following result: theorem 3.4. every regular space is m-regular. proof. let (x,τ) be a topological space which is regular. we have to show that x is m-regular. for this, let v be any m-open subset of x and let x ∈ v . then, we have v = ⋂ j∈j vj, where v ′ j s are open sets in x. therefore, as x ∈ v = ⋂ j∈j vj, we have x ∈ vj for all j ∈ j. since the space x is given to be regular, thus there exists open set wj such that x ∈ wj ⊆ cl(wj) ⊆ vj, for all j ∈ j. now, consider w = ⋂ j∈j wj an m-open set in x, we have x ∈ w ⊆ ⋂ j∈j cl(wj) ⊆ ⋂ j∈j vj = v . thus we have, x ∈ w ⊆ cl(w) ⊆ v , where w is an m-open subset of x. hence x is m-regular. � from the theorem 2.10, one can conclude that every t1-space is m-regular. but the converse of the above statement is not true. that is, an m-regular space need not be t1. for this, we have the following example: example 3.5. let x = {a,b,c,d} be a non-empty set equipped with a topology τ = {∅,{a,b},{c,d},x}. then (x,τ) is an m-regular space but it is not a t1-space. next we will provide the decomposition of t1-space with the help of mregularity. theorem 3.6. every m-regular t0-space is t1. proof. let (x,τ) be a topological space. let x,y ∈ x be a pair of distinct points of an m-regular t0-space x. let there exists an open set v in x such that x ∈ v but y /∈ v . since every open set is m-open and the space x is given to be m-regular, therefore there exists an m-open set u in x such that © agt, upv, 2022 appl. gen. topol. 23, no. 2 247 a. gupta and r. d. sarma x ∈ u ⊆ cl(u) ⊆ v . then consider, w = x \ cl(u) is open in x such that x /∈ w and y ∈ w . thus x is t1. � from the theorem 3.6, one can state the following: theorem 3.7. every t0-space is m-regular if and only if it is t1. however a t0-space need not be m-regular. example 3.8. let x be a sierpiński space, that is, x = {a,b} with topology τ = {∅,{a},x}. then (x,τ) is a t0-space but not m-regular. in our next result, we show that hausdorffness is a sufficient condition for m-regularity. theorem 3.9. every hausdorff space is m-regular. proof. let (x,τ) be a hausdorff space. let x and b be a pair of a point and an m-closed set such that x /∈ b. then for every y ∈ b, we have x 6= y. therefore by the given hypothesis, there exists a disjoint pair of open sets uy and vy such that x ∈ uy and y ∈ vy. then consider v = ⋃ y {vy |y ∈ b}, an open cover of b and u = ⋂ y {uy |y ∈ b} is an m-open set containing x. thus we have a disjoint pair consisting an open set v and an m-open set u containing b and x respectively. therefore x is m-regular. � the converse, however need not be true. example 3.10. let x = {a,b,c,d} be a non-empty set equipped with a topology τ = {∅,{a,b},{c,d},x}. then (x,τ) is an m-regular space but it is not a hausdorff space. our next result is on the product of m-regular spaces. theorem 3.11. any arbitrary product of m-regular spaces is again m-regular. proof. let {(xα,τα)}α be a family of m-regular spaces and x = ∏ α xα. let x = (xα) ∈ x be a point and u be an m-open neighbourhood of x ∈ x. since u is an m-open set in x, therefore u = ⋂ i ui, where ui is an open set in x under the product topology. therefore, we have ui = ∏ α uα,i, where uα,i ∈ τα. hence, we have u = ⋂ i (∏ α uα,i ) . we use the fact that ⋂ β (∏ α aα,β ) = ∏ α  ⋂ β aα,β   in view of lemma 2.7. therefore, we have u = ⋂ i (∏ α uα,i ) = ∏ α (⋂ i uα,i ) and x ∈ u. hence we have xα ∈ ⋂ i uα,i, © agt, upv, 2022 appl. gen. topol. 23, no. 2 248 a urysohn lemma for regular spaces where ⋂ i uα,i = wα (say) is an m-open set in (xα,τα). since (xα,τα) is m-regular, therefore there exists an m-open set vα of (xα,τα) such that xα ∈ vα ⊆ clα(vα) ⊆ wα. now, v = ∏ α vα is an m-open set in x in view of theorem 2.9. again cl(v ) = cl (∏ α vα ) = ∏ α clα (vα), in view of lemma 2.8. thus, we have an m-open set v in x such that x ∈ v ⊆ cl(v ) ⊆ ∏ α wα ⊆ u. hence x is m-regular. � 4. m-normal spaces definition 4.1. a topological space (x,τ) is said to be m-normal if for each disjoint pair consisting of a closed set a and an m-closed set b, there exists a disjoint pair consisting of an m-open set u and an open set v in x containing a and b respectively. remark 4.2. an m-normal space need not be m-regular. for this, consider the sierpiński space mentioned in the example 3.8. the space (x,τ) is m-normal but not m-regular. in our next result, we provide a characterization for m-normality. theorem 4.3. let (x,τ) be a topological space. then (x,τ) is m-normal if and only if for a given closed set c and an m-open set d such that c ⊆ d, there is an m-open set g such that c ⊆ g ⊆ cl(g) ⊆ d. proof. let c and d be the closed and m-open sets respectively such that c ⊆ d. then x \d is an m-closed set such that c ∩(x \d) = ∅. then, from the m-normality, there exist an m-open set g and an open set v such that c ⊆ g, x\d ⊆ v and g∩v = ∅. therefore x\v ⊆ d and hence c ⊆ g ⊆ x\v ⊆ d, where x \v is a closed set. hence c ⊆ g ⊆ cl(g) ⊆ cl(x \v ) = x \v ⊆ d. conversely, consider d and c as closed and m-closed sets respectively such that c ∩d = ∅. then x \c is m-open set containing d. then by the given hypothesis, there exist an m-open set g such that d ⊆ g ⊆ cl(g) ⊆ x \ c. thus, we have d ⊆ g, c ⊆ v and g∩v = ∅, where v = x \ cl(g), an open set. hence x is m-normal. � from the example 3.3, one can easily verify that m-normality doesn’t imply normality. here the space x is m-normal but it is not normal. in the following result, we show that every normal space is m-normal. theorem 4.4. every normal space is m-normal. proof. let (x,τ) be a topological space which is normal. we have to show that x is m-normal. for this, let a be any closed subset of x and let v be an m-open subset of x such that a ⊆ v . then, we have v = ⋂ j∈j vj, where v ′ j s © agt, upv, 2022 appl. gen. topol. 23, no. 2 249 a. gupta and r. d. sarma are open sets in x. therefore, we have a ⊆ v = ⋂ j∈j vj, that is, a ⊆ vj for all j ∈ j. since the space x is given to be normal, thus there exists open set wj such that a ⊆ wj ⊆ cl(wj) ⊆ vj, for all j ∈ j. now, consider w = ⋂ j∈j wj, an m-open set in x, we have a ⊆ w ⊆ ⋂ j∈j cl(wj) ⊆ ⋂ j∈j vj = v . thus we have, a ⊆ w ⊆ cl(w) ⊆ v , where w is an m-open subset of x. hence x is m-normal. � one specialty of m-normality is that it is preserved under arbitrary product. theorem 4.5. any arbitrary product of m-normal spaces is again m-normal. proof. let {(xα,τα)}α be a family of m-normal spaces and x = ∏ α xα. let a ⊆ x be a closed set and u be an m-open set such that a ⊆ u. since u is an m-open set in x, therefore u = ⋂ i ui, where ui is an open set in x under the product topology. thus, we have ui = ∏ α uα,i, where uα,i ∈ τα and uα,i = xα for all but finitely many α’s, as explained before theorem 2.9 . hence, we have u = ⋂ i (∏ α uα,i ) . we use the fact that ⋂ β (∏ α aα,β ) = ∏ α  ⋂ β aα,β   following lemma 2.7. thus we have u = ⋂ i (∏ α uα,i ) = ∏ α (⋂ i uα,i ) . similarly, we have a = ∏ α aα, where aα is a closed set in xα. since a ⊆ u, thus we have aα ⊆ ⋂ i uα,i. let ⋂ i uα,i = wα, an m-open set in xα. we have aα ⊆ wα and since xα is an m-normal space, therefore, there exists an m-open set vα such that aα ⊆ vα ⊆ cl(vα) ⊆ wα. thus we have, ∏ α aα ⊆ ∏ α vα ⊆∏ α clα(vα) ⊆ ∏ α wα. now, ∏ α vα is m-open in view of theorem 2.9. also, cl (∏ α vα ) = ∏ α (cl(vα)), in view of lemma 2.8. hence we have v = ∏ α vα, an m-open set in x such that a ⊆ v ⊆ cl(v ) = ∏ α (cl(vα)) = cl (∏ α vα ) . it follows that a ⊆ v ⊆ cl(v ) ⊆ ∏ α wα ⊆ u. hence x is m-normal. � following result for m-normal spaces is analogous to the well known urysohn lemma for normal spaces. © agt, upv, 2022 appl. gen. topol. 23, no. 2 250 a urysohn lemma for regular spaces theorem 4.6. let (x,τ) be a topological space. then for each pair of disjoint subsets a and b of x, one of which is closed and the other is m-closed, there exists an m-continuous function f on x to [0, 1] (resp. [a,b] for any real number a,b, a < b ), such that f(a) = {0} and f(b) = {1} (resp. f(a) = {a}, and f(b) = {b} ) provided x is m-normal. proof. let (x,τ) be an m-normal space. let c0 and c1 be two disjoint sets, where c0 is closed and c1 is m-closed in x. since c0 ∩ c1 = ∅, therefore c0 ⊆ x\c1. let p be the set of all dyadic rational numbers in [0, 1]. we shall define for each p in p , an m-open set up of x, in such a way that whenever p < q, we have cl(up) ⊂ uq. first we define u1 = x \ c1, an m-open set such that c0 ⊆ u1. since x is m-normal space, by theorem 4.3, there exists an m-open set u1/2 such that c0 ⊆ u1/2 ⊆ cl(u1/2) ⊆ u1. similarly, there also exist another m-open sets u1/4 and u3/4 such that c0 ⊆ u1/4 ⊆ cl(u1/4) ⊆ u1/2 ⊆ cl(u1/2) ⊆ u3/4 ⊆ cl(u3/4) ⊆ u1, because cl(u1/4) is again a closed set. continuing the process, we define ur, for each r ∈ p such that c0 ⊆ ur ⊆ cl(ur) ⊆ u1 and cl(ur) ⊆ us whenever r < s, for r,s ∈ p . let us define q(x) to be the set of those dyadic rational numbers p such that the corresponding m-open sets up contains x: q(x) = {p | x ∈ up} now we define a function f : x → [0, 1] as f(x) = inf q(x) = inf{p | x ∈ up} clearly, f(c0) = {0} and f(c1) = {1}. then we show that f is the desired m-continuous function. for a given point x0 ∈ x and an open interval (c,d) in [0, 1] containing the point f(x0). we wish to find an m-open neighbourhood u of x0 such that f(u) ⊆ (c,d). let us choose rational numbers p and q such that c p, we have x0 ∈ cl(up) ⊆ us. thus x0 ∈ us for all s > p. this implies that f(x0) ≤ p, as f(x0) = inf{s |x0 ∈ us}. this contradicts the fact that f(x0) > p. similarly, as f(x0) < q, therefore x0 ∈ uq and hence u = uq \ cl(up) is the desired neighbourhood of x0. hence f is an m-continuous function on x to [0, 1] with f(c0) = {0} and f(c1) = {1}. this completes the proof. � since every closed set is m-closed, we get the following result: corollary 4.7. let (x,τ) be an m-normal topological space. then for each pair of disjoint closed subsets a and b of x, there exists an m-continuous function f on x to [0, 1] (resp. [a,b] for any real number a,b, a < b ), such that f(a) = {0} and f(b) = {1} (resp. f(a) = {a}, and f(b) = {b} ). © agt, upv, 2022 appl. gen. topol. 23, no. 2 251 a. gupta and r. d. sarma remark 4.8. from the proof of theorem 4.6, it is clear that the proof is in line of the usual proof of urysohn lemma in kelley[8], wherein the choice function plays its role. hence the proof is valid for zfc. again, it has been pointed out in [1] that the axiom of multiple choice also implies urysohn lemma, since one can use the intersection of the finitely many separating open sets provided by mc. essentially the same argument shows that dmc implies urysohn lemma. since similar working is followed in our theorem 4.6, hence the variant of urysohn lemma in our paper is also valid for zf with dmc. our next theorem provides the relation between regularity and m-normality. theorem 4.9. every regular space is m-normal. proof. let a and b be two disjoint sets such that a is closed and b is m-closed. since a∩b = ∅, therefore b ⊆ x \a, where x \a is an open set containing the m-closed set b. then, by the given hypothesis, for each b ∈ b ⊆ x \ a, there exists an open set ub such that b ∈ ub ⊆ cl(ub) ⊆ x \a. thus, we have a collection d = {ub |b ∈ b} which covers b. further, if d ∈d, then cl(d) is disjoint from a because cl(d) ⊆ x \a. consider v = ⋃ {d |d ∈d}. then v is an open set in x which contains b, an m-closed set. since d lies in some ub whose closure is disjoint from a , therefore w = ⋃ {cl(d) |d ∈d} is disjoint from a. therefore, v = ⋃ {d |d ∈d} and x \ w are two disjoint subsets of x. now w , being union of closed sets, is m-closed. thus we have one open set and one m-open set containing the sets b and a respectively. hence x is m-normal. � in view of theorem 4.6 and 4.9, one can observe the following: theorem 4.10. let (x,τ) be a regular space. then for every disjoint pair of sets consisting of a closed set a and an m-closed set b, there always exists an m-continuous mapping f from x to [a,b] such that f(a) = {a} and f(b) = {b}. the last and the final result of this section is a simple corollary of theorem 4.10. however its importance lies in revealing the fact that even in regular spaces, closed sets can be separated by mappings, the so-called m-continuous mappings. in that sense, this result may be treated as urysohn lemma for regular spaces. theorem 4.11. let (x,τ) be a regular space. then for every disjoint pair of closed sets a and b, there always exists an m-continuous mapping f from x to [a,b] such that f(a) = {a} and f(b) = {b}. © agt, upv, 2022 appl. gen. topol. 23, no. 2 252 a urysohn lemma for regular spaces acknowledgements. the authors sincerely acknowledge the reviewers of the paper for their valuable suggestions. references [1] a. blass, injectivity, projectivity and the axiom of choice, trans. am. math. soc. 255 (1970), 31–59. [2] c. boonpok, ξµ-sets in generalized topological spaces, acta math. hungar. 96 (2012), 269–285. [3] j. dugundji, topology, allyn and bacon (1966). [4] e. fabrizi and a. saffiotti, behavioral navigation on topology-based maps, in: proc. of the 8th symp. on robotics with applications, maui, hawaii, 2000. [5] c. good and i. j. tree, continuing horrors of topology without choice, topology appl. 63, no. 1 (1995), 79–90. [6] a. gupta and r. d. sarma, on m-open sets in topology, in: conference proceedings “3rd international conference on innovative approach in applied physical, mathematical/statistical, chemical sciences and energy technology for sustainable development”, 7–11. [7] i. m. james, topologies and uniformities, springer-verlag (1987). [8] j. l. kelley, general topology, d. van nostrand, princeton, n. j., (1955). [9] v. kovalesky and r. kopperman, some topology-based image processing algorithms, ann. ny. acad. sci. 728 (1994), 174–182. [10] b. m. r. stadler and p. f. stadler, generalized topological spaces in evolutionary theory and combinatorial chemistry, j. chem. inf. comp. sci. 42 (2002), 577–585. © agt, upv, 2022 appl. gen. topol. 23, no. 2 253 gonzalezhrusakagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 207-219 more on ultrafilters and topological games r. a. gonzález-silva and m. hrušák∗ abstract. two different open-point games are studied here, the g-game (of bouziad [4]) and the gp-game (introduced in [11]), defined for each p ∈ ω∗. we prove that for each p ∈ ω∗, there exists a space in which none of the players of the gp-game has a winning strategy. nevertheless a result of p. nyikos, essentially shows that it is consistent, that there exists a countable space in which all these games are undetermined. we construct a countably compact space in which player ii of the gp-game is the winner, for every p ∈ ω ∗. with the same technique of construction we built a countably compact space x, such that in x ×x player ii of the g-game is the winner. our last result is to construct ω1-many countably compact spaces, with player i of the g-game as a winner in any countable product of them, but player ii is the winner in the product of all of them in the g-game. 2000 ams classification: primary 54a20, 91a05: secondary 54d80, 54g20 keywords: open-point game, ultrafilter, g-space, gp-space, countably compact 1. introduction and preliminaries in [15] g. gruenhage introduced a local game on topological spaces, so called open-point game (here denoted as the w -game). given a topological space x and a point x ∈ x, the rules of the open-point game are as follows: two players i and ii play infinitely many innings, in the n-th inning player i choosing a neighborhood un of x and player ii responding with a point xn ∈ un. after ω-many innings we declare a winner, using the sequence (xn)n<ω of the moves ∗the first listed author gratefully acknowledges support received from promep grant no. 103.5/07/2636. the second author was supported partially by dgapa grant no. in108802 and partially by gačr grant 201/00/1466. 208 r. a. gonzález-silva and m. hrušák of the second player. we say that player i wins the w (x, x)-game if the sequence (xn)n<ω converges to x, otherwise player ii is declared a winner. this game and its variations (see [4], [11] and [17]) have proved useful in studying local and convergence properties of topological spaces. these variants have the same rules and only differ from the w -game in the way a winner is declared. following a. bouziad [4], we say that player i wins the g(x, x)-game if {xn : n < ω} has an accumulation point in x, otherwise, player ii is the winner. here we are mainly concerned with an ultrafilter version of the open-point game as introduced and studied in [11] and [12]. recall the definition of the p-limit of a sequence (r. a. bernstein [2]). let p be a free filter on ω. a point x of a space x is said to be the p-limit of a sequence (xn)n<ω in x (x = p-lim xn) if for every neighborhood u of x, {n < ω : xn ∈ v } ∈ p. now, we are ready to define the gp-game, a parametrized version of the above mentioned g-game. let p be a free ultrafilter on ω. we say that player i wins the gp(x, x)-game if p-lim xn exists (in x). otherwise, player ii wins. in what follows we are mostly concerned with the question as to whether either player has a winning strategy in one of the above mentioned games. a strategy for one of the players is an algorithm that specifies each move of the player in every possible situation. more precisely, a strategy for player i in the open-point game is any sequence of functions σ = {σn : n (x) n × x n → n (x) : n < ω}. a sequence (xn)n<ω in x is called a σ-sequence if xn+1 ∈ σn+1(〈x0, ..., xn〉; 〈v0, ..., vn〉) = vn+1, for each n < ω. a strategy σ for player i is a winning strategy in the g(x, x)-game (respect. w (x, x)-game, gp(x, x)-game), if each σ-sequence has an accumulation point in x (respect. xn → x, or there exist y ∈ x such that p-lim xn = y). a space x is called a g-space (respect. w -space, gp-space) if player i has a winning strategy in the g(x, x)-game (resp. w (x, x)-game, gp(x, x)-game), for every x ∈ x. similarly, one defines a strategy for player ii. it is a sequence of functions ρ = {ρn : x n×n (x) n+1 → x : n < ω}, such that ρn(〈x0, ..., xn−1〉; 〈v0, ..., vn〉) ∈ vn, for each n < ω. a sequence 〈(vn, xn) : n < ω〉 where vn ∈ n (x) and xn ∈ vn is called a ρ-sequence, if xn = ρn(〈x0, ..., xn−1〉; 〈v0, ..., vn〉) ∈ vn, for each n < ω. a strategy ρ for player ii is a winning strategy in the g(x, x)-game (respect. w (x, x)-game, gp(x, x)-game), if for each ρ-sequence, 〈(vn, xn) : n < ω〉, the set {xn : n < ω} does not have cluster point in x (resp. xn 6→ x, or the p-limit of the sequence {xn} does not exist). we denote the fact that player i has a winning strategy in the g(x, x)-game, by i ↑ g(x, x). if he does not have a winning strategy we write i ↓ g(x, x). when i ↑ g(x, x) for every x ∈ x, this is denoted by i ↑ g(x). the meaning more on ultrafilters and topological games 209 of ii ↑ g(x, x), ii ↓ g(x, x) is defined analogously with the same notation used for the w -game or gp-game. the following implications are easy consequences from definitions, i ↑ w (x) =⇒ i ↑ gp(x) =⇒ i ↑ g(x). they can not be reversed in general, as shown for the spaces ω∗, β(ω) \ {q ∈ ω∗ : q ≤rf p}), but they are equivalent to first countability if x is a countable space (see proposition 2.5). dually, ii ↑ w (x) ⇐= ii ↑ gp(x) ⇐= ii ↑ g(x). these implications are also strict, the same examples work. in the next diagram, one can see relationships of these games with other concepts of general topology (for more details, see [13]). x is compact   y x is first countable x is p-compact −→ x is countably compact   y   y   y i ↑ w (x) −→ i ↑ gp(x) −→ i ↑ g(x) sharma proved in [23] that x is strongly frechet ⇐⇒ ii ↓ w (x), where a space x is called strongly fréchet iff for every point x ∈ x, and every sequence (an)n<ω of subsets of x with x ∈ an for each n < ω, there exists a sequence {xn} such that xn ∈ an for every n ∈ ω and xn → x. the notation used here is mostly standard. the stone-čech compactification βω of the countable discrete space ω is identified with the set of all ultrafilters on ω and its remainder ω∗ = βω \ ω denotes the set of all free ultrafilters on ω. if f : ω → x is a function into a compact space x, f̂ denotes its (unique) extension to βω. two ultrafilters are said to be of the same type (in βω) if there is a permutation f of ω such that f̂ takes one to the other. the set of ultrafilters of the same type as a fixed ultrafilter p, is denoted by t (p). for p, q ∈ ω∗, p ≤rk q denotes that p is rudin-keisler bellow q and means that there is f : ω → ω such that f̂ (q) = p. the relation p ≤rf q is the rudin-froĺık order and it means that there is an embedding f : ω → βω such that f̂ (p) = q. 2. indeterminacy of the games gp, w and g we say that a game is determined on a space x if for every point of x one of the players (not the same for all points) has a winning strategy, otherwise, the game is undetermined. for nice definable spaces the games are typically determined. however, they are not determined in general. in this section we are going to work with the indeterminacy of the games gp, g and w . for this, let us introduce the following notation. let y be a set. a subset t of y <ω is a tree if whenever t ∈ t and s ∈ y <ω with s ⊆ t, then s ∈ t. let t be an element of the tree t, the set of successors of t, {y ∈ y : t⌢y ∈ t} is denoted by succt(t). a function f : ω → y , is said 210 r. a. gonzález-silva and m. hrušák to be a branch of t, if f ↾n∈ t for every n < ω. the set of branches of t is denoted by [t]. next we will show that for every p ∈ ω∗, there is a countably compact space such that no player of the gp-game has a winning strategy. to that end the following lemmas will be useful. the following fact is a standard reformulation of the existence of a winning strategy for player ii (see e.g [17]). lemma 2.1. suppose that x is a topological space, x ∈ x and p ∈ ω∗. then the following are equivalent: (1) ii ↑ gp(x, x). (2) ii has a wining strategy ρ′ in the gp(x, x)-game such that x 6∈ rng(ρ ′) (3) there exists a tree t such that i. for every t ∈ t, x ∈ succt(t) \ {x}. ii. for every f ∈ [t], p-lim f (n) does not exist in x. proof. 1=⇒2. let ρ = {ρn : n < ω} be a winning strategy for player ii in the gp(x, x)-game. we say that a sequence 〈v0, y0, v1, y1, ..., vn, yn〉 is ρ-legal, if the v0, ..., vn are neighborhoods of x, and for each i ∈ {0, ..., n}, we have ρi(〈y0, ..., yi−1〉, 〈v0, ..., vi〉) = yi ∈ vi. we will recursively define a winning strategy ρ′ such that: (a) x 6∈ rng(ρ′) and (b) for every ρ′-legal sequence 〈v0, x0, v1, x1, ..., vn, xn〉, there is a unique ρ-legal sequence 〈v0, y0, v1, y1, ..., vn, yn〉 such that yi = xi whenever yi 6= x. if n = 0, let ρ′0(v0) be equal to ρ0(v0) if ρ0(v0) 6= x otherwise ρ ′ 0(v0) is any element of v0 \ {x}. for the inductive step, let 〈v0, x0, v1, x1, ..., xn−1, vn〉 be sequence of moves where the xi are played according to the strategy ρ ′. consider 〈v0, x0, v1, x1, ..., vn−1, xn−1〉. by the inductive hypothesis there is a unique ρ-legal sequence 〈v0, y0, v1, y1, ..., vn−1, yn−1〉 such that yi = xi whenever yi 6= x. define ρ′n(〈x0, ..., xn−1〉; 〈v0, ..., vn〉) as follows: it is equal to ρn(〈y0, ..., yn−1〉; 〈v0, ..., vn〉) if ρn(〈y0, ..., yn−1〉; 〈v0, ..., vn〉) 6= x, otherwise is any point of vn \ {x}. it is clear that (a) holds and that ρ′ is a strategy. now lets see that (b) holds. let 〈v0, x0〉 be a ρ ′-legal, then we have two cases, x0 is equal to ρ0(v0) or not, in any case (b) holds. now suppose that the statement (b) is true for any ρ′-legal sequence of length n and let 〈v0, x0, v1, x1, ..., vn, xn〉 be ρ ′-legal sequence, so the subsequence 〈v0, x0, v1, x1, ..., vn−1, xn−1〉 holds (b), hence there is a unique ρ-legal sequence 〈v0, y0, v1, y1 , ..., vn−1, yn−1〉 fulling (b), and ρn(〈y0, ..., yn−1〉; 〈v0, ..., vn〉) = yn, so 〈v0, y0, v1, y1, ..., vn, yn〉 is the unique ρ-legal sequence. finally lets see that ρ′ is a winning strategy for player ii, for this, let (xn)n<ω be a sequence of moves of player ii according to strategy ρ ′. then there is exists a unique sequence (yn)n<ω which is constructed by segments of (xn)n<ω; the difference between (xn)n<ω and (yn)n<ω are the points yn which more on ultrafilters and topological games 211 are x. since the p-lim yn is not in x then the p-lim xn is not in x, so ρ ′ is a winning strategy. 2=⇒3. let t′′ = {l ∈ (n (x) × x)<ω : l is a ρ′−legal sequence} and define t ′ = {g↾n : g ∈ [t ′′] and g is inf inite}. note that each f ∈ [t′] is a gp-play a cording to strategy ρ′, hence t′ 6= ∅ and if sf = (xfn)n<ω is the subsequence generated by the points of f , then this sequence does not have a p-limit in x. set t = {sf↾n: f ∈ [t ′]}, with sf↾n⊆ s g↾m if and only if f↾n⊆ g↾m. to see that i holds, pick t ∈ t and a neighborhood u of x. from the construction of t, choose a branch f ∈ [t′] such that t = sf↾n. let (v f n )n<ω be the subsequence generated by the neighborhoods of f . then ρ′|t|(〈t(0), t(1), ..., t(|t| − 1)〉; 〈v f 0 , v f 1 , ..., v f |t|−1 , u〉) ∈ u , hence u ∩ (succt(t) \ {x}) 6= ∅. finally, if g ∈ [t], then g = sf for some f ∈ [t′], so p-lim g(n) does not exist in x, this fulfilling condition ii. 3=⇒1. take a tree t fulfilling clauses i and ii. for each n ∈ ω, define ρn : x n × n (x) n+1 → x, such that ρn(〈x0, ..., xn−1〉; 〈v0, ..., vn〉) ∈ vn ∩ succt(〈x0, ..., xn−1〉). let ρ = {ρn : n < ω}. it is straightforward to see that ρ is a winning strategy for player ii in the gp(x, x)-game, as in any play the resulting sequence is a branch of the tree t, and by ii, it does not have a p-limit in x. � the next result, due to z. froĺık, is used in the proof of lemma 2.3 and also later on in the text. lemma 2.2 (froĺık). if f, g : ω → ω∗ are embeddings and p ∈ ω∗. then, f̂ (p) = ĝ(p) if and only if {n < ω : f (n) = g(n)} ∈ p. lemma 2.3. let p ∈ ω∗ and t ⊆ (ω∗)<ω be a countable tree, such that (1) for each t ∈ t, |succt(t)| ≥ 2. (2) for each f ∈ [t], f is an embedding in ω∗. (3) if f, g ∈ [t], f 6= g, then |f ∩ g| < ℵ0. then, f̂ (p) 6= ĝ(p) for any two elements f, g ∈ [t], and in particular the set p[t] = {p-lim f (n) : f ∈ [t]} has cardinality c. proof. follows from clauses 2 and 3, and lemma 2.2. � the idea to construct a space x in which the gp-game is undetermined (for p ∈ ω∗ fixed), is to construct recursively a space x ⊂ ω∗, diagonalizing across all the possible strategies for players i and ii. there are two obvious obstacles to doing this. if we don’t know x, then we can’t say too much about the strategies. another obstacle, is that there are going to be at least 2|x| possible strategies. fortunately lemma 2.3 can be used to overcome both obstacles. the space x is going to be constructed in c-many steps, so the cardinality of {t ⊆ x <ω : t satisfies the conditions of lemma 2.3 } is at most c. 212 r. a. gonzález-silva and m. hrušák theorem 2.4. for each p ∈ ω∗, there exists a countably compact space x such that for every x ∈ x, i ↓ gp(x, x) and ii ↓ gp(x, x). proof. fix a bijection φ : c → c × c such that, for φ(α) = (φ0(α), φ1(α)), we have φ0(α), φ1(α) ≤ α, for each α < c. by recursion we are going to construct for each ν < c, spaces xν , yν and a sequence of trees {t ν α : α < c}, such that (1) x0 ⊂ ω ∗ is countable and dense in itself, and y0 = ∅. (2) xη ⊂ xµ y yη ⊂ yµ, for all η < µ < ν. (3) |xµ| ≤ |µ + ω| y |yµ| ≤ |µ|, for all µ < ν. (4) xµ ∩ yη = ∅, for all η < µ < ν. (5) {tνα : α < c} is an enumeration of all trees in x <ω ν satisfying the conditions of lemma 2.3. (6) if µ + 1 < ν, then xµ+1 ∩ p[t φ0(µ) φ1(µ) ] 6= ∅ and yµ+1 ∩ p[t φ0(µ) φ1(µ) ] 6= ∅. the construction of the space x0 can be done using theorem 1.4.7 of [18]. for a limit ordinal ν, define xν = ⋃ µ<ν xµ and yν = ⋃ µ<ν yµ. when ν = µ + 1, define xν = xµ ∪ {pµ} and yν = yµ ∪ {qµ}, where pµ, qµ ∈ ω ∗ have the property that pµ 6= qµ and pµ, qµ ∈ p[t φ0(µ) φ1(µ) ] \ (xµ ∪ yµ). let x = ⋃ ν 0, pick inductively points xs⌢0 6= xs⌢1 in x and a clopen neighborhood vn of x with the followings properties: xs⌢ 0, xs⌢1 ∈ succt (t ⌢xs|1 ⌢xs|2 ⌢...⌢xs), xs⌢ 0, xs⌢1 6∈ {xr : r ∈ 2 ≤|s|+1 \ {s⌢0, s⌢1}}, xs⌢ 0, xs⌢1 ∈ vn, for each s ∈ 2 n−1 and n − 1 ≥ 0, xs⌢ 0, xs⌢1 6∈ vn, for each s ∈ 2 0. let ts = t ⌢〈xs|1 , xs|2 , ..., xs〉, and define t = {ts : s ∈ 2 <ω}. so t is a subtree of t like lemma 2.3. hence there exists γ < c with t φ0(γ) φ1(γ) = t. however from this fact, there is a branch f ∈ [t] with p-lim f (n) ∈ x. � the proof of the following fact is analogous to the proof given in [15, theorem 3.3] for the w-game. we have already mentioned that, for a countable space x, the existence of a winning strategy for player i in the g-game on x is equivalent to x being first countable. proposition 2.5. in a tychonoff countable space x, the following statements are equivalent for a fixed element x in x: (1) χ(x, x) = ℵ0. (2) i ↑ g(x, x). proof. 1 =⇒ 2. it is straightforward to define a winning strategy for player i using a countable local base. 2 =⇒ 1. suppose that χ(x, x) > ℵ0. let σ be any strategy for player i. enumerate the range of σ as {vn : n < ω}. as x is zero-dimensional, we can get for each n < ω, a clopen subset un such that (1) un+1 ⊂ un, for every n < ω. (2) ⋂ n<ω un = {x}. (3) un ⊂ vn, for every n < ω. since χ(x, x) > ℵ0, there exists a neighborhood v of x such that |un \ v | = ℵ0 for each n < ω. take xn ∈ un \ v for each n < ω. then x 6∈ {xn : n < ω}. now, if y ∈ x \{x}, then there exist n < ω with y 6∈ un, hence x \ un ∈ n (y), so |(x \ un) ∩ {xn : n < ω}| < ℵ0, i.e. y 6∈ {xn : n < ω}. hence the sequence {xn : n < ω} does not have cluster points. it is easy to see that it contains a subsequence which is σ-sequence without cluster points. therefore the strategy σ is not winning. � theorem 1.12 of [21] essentially says that it is consistent that there exist countable dense-in-themselves spaces on which our three games are undetermined. we will need the following version of this result. 214 r. a. gonzález-silva and m. hrušák theorem 2.6 (p. nyikos). assume p > ω1. if d is a countable dense subset of 2ω1 , then i ↓ g(d) and ii ↓ w (d). from this theorem and the implications between the games w , gp and g, we have the next corollary. corollary 2.7 (p > ω1). there exists a topological countable group g such that the games w , g and gp are undetermined in g. 3. player ii and countable compactness if x is countably compact, player i has a (trivial) winning strategy in the g-game. this is no longer true for the gp-game. in fact, it is easy to construct (for a fixed p ∈ ω∗) a countably compact space x such that ii ↑ gp(x). now, we will construct a countably compact space x such that ii ↑ gp(x) for every p ∈ x and then show that there is a countably compact space x such that ii ↑ g(x × x), which is a strengthening of results of novak and terasaka’s examples (see [24, lemma 3.1]). recall the definition of the relative type, introduced by z. froĺık. let y ∈ [ω∗]ω be discrete and p ∈ y ∗ = y βω \ y . the relative type of p with respect to y is t (ĥ(p)), where h : y → ω is an embedding. it is going to be denote by t (p, y ). now, for a subset s of βω and p ∈ ω∗, define t [p, s] = {t (p, y ) : y ∈ [s]ω and y is homeomorphic to ω}. froĺık proved that t [p, ω∗] has cardinality c. theorem 3.1. there exists a countably compact space x such that ii ↑ gp(x, x) for every p ∈ ω ∗ and x ∈ x. proof. the space x is going to be the union of {xν : ν < ω1}, where each xν is constructed recursively. suppose that for each µ < ν < ω1 we have xµ such that (1) x0 ⊆ ω ∗ is countable and dense in it self. (2) x0 is a dense subset of xµ, for each µ < ν. (3) |xµ| ≤ c, for each µ < ν. (4) xη ⊂ xµ, if η < µ < ν. (5) if µ + 1 < ν, then every countable discrete subset of xµ has a cluster point in xµ+1. (6) for each x ∈ xµ \ x0, {y ∈ xµ : t [x, x0] ∩ t [y, x0] 6= ∅} = {x}. we can assume the existence of the space x0, using theorem 1.4.7 of [18]. now, we show how to construct xν . when ν is a limit ordinal, define xν = ⋃ µ<ν xµ. if ν is a successor ordinal, say ν = µ + 1 then we have from clause 3, that the set of all embeddings from ω to xµ has size c, let {fα : α < c} be an enumeration of this set. for each α < c, pick a point pα ∈ fα[ω] β(ω) such that more on ultrafilters and topological games 215 for all y ∈ xµ, t [pα, x0] ∩ t [y, x0] = ∅, and also t [pα, x0] ∩ t [pβ, x0] = ∅, for all β < α. define xµ+1 = xµ ∪ {pα : α < c}. notice that our space x = ⋃ ν<ω1 xν is countably compact and also for each p ∈ ω∗, |{y ∈ x \ x0 : t (p) ∈ t [y, x0]}| ≤ 1. therefore, |{y ∈ x : t (p) ∈ t [y, x0]}| ≤ ω. fix p ∈ ω∗ and x ∈ x. let’s see that ii ↑ gp(x, x). it follows from the previous observation that the set a = {q ∈ x : t (p) ∈ t [q, x0]} is countable. enumerate it as {qi : i < ω}. for each i < ω fix an embedding fi : ω → x0 such that f̂i(p) = qi. the strategy of player ii is to choose in the n-th step g(n) ∈ x0 \ {f0(n), f1(n), ..., fn(n)} such that the function g : ω → x0 defined in this way is an embedding. from lemma 2.2, we have ĝ(p) 6∈ a. and hence t (p) ∈ t [ĝ(p), x0], then ĝ(p) 6∈ x. so this is a winning strategy for player ii in the gp(x, x)-game. � in the construction of the next example, we use a space which is countable, dense in itself and extremally disconnected. this space is defined for a fixed ultrafilter p ∈ ω∗ and it is denoted by seq(p), its underlying set is ω<ω, the set of all finite sequences in ω. a set u ⊂ ω<ω is open if and only if for every t ∈ u , {n < ω : t⌢n ∈ u} ∈ p (see [7], [20], [5], [25]). lemma 3.2. there exists a countable dense-in-itself space x ⊂ ω∗ such that, for any x ∈ x there exists a sequence {vn : n < ω} ⊂ n (x) with the following property: if {xn : n < ω} ⊂ x and xn ∈ ⋂ m≤n vm for each n < ω then x 6∈ {xn : n < ω}. proof. let p ∈ ω∗ be not a p-point and consider the space seq(p). using theorem 1.4.7 of [18], we can take an homeomorphic copy of seq(p) inside of ω∗. so, now it is sufficient to prove that seq(p) is the desired space. since p is not a p-point, there exists a sequence {un : n < ω} ⊂ p without pseudointersection in p. take x ∈ seq(p) and define vn = {t ∈ seq(p) : x ⊆ t and t(|x|) ∈ un}. if (xn)n<ω is a sequence such that xn ∈ ⋂ m≤n vm, then xn(|x|) ∈ um, for every n > m. hence w = {xn(|x|) : n < ω} 6∈ p. so u = ω \ w ∈ p, this implies that v = {t ∈ x : x ⊆ t and t(|x|) ∈ u} is a neighborhood of x, disjoint from {xn : n < ω}. � it is easy to see that the product of at most ω-many w -spaces (gp-spaces), is also a w -space (gp-space). however, this is not true for g-spaces, as we will see in the next example. an application of the following example, is the existence of a countably compact space whose product is not countably compact. example 3.3. there exists a countably compact space x such that ii ↑ g(x × x). proof. let x be the space constructed in theorem 3.1, with the condition that, the space x0 is homeomorphic to seq(p) where the free ultrafilter p is not a 216 r. a. gonzález-silva and m. hrušák p-point. let’s see that ii ↑ g((x, y), x × x), for a fixed point (x, y) ∈ x × x. by ∆ we denote the set {(x, x) : x ∈ x0}. case (i): (x, y) ∈ x × x \ (x0 × x0). let {(xn, yn) : n < ω} be an enumeration of all the points in x0×x0. for each n < ω, let wn ∈ n ((xn, yn))\ n ((x, y)) clopen such that x0 × x0 \ ⋃ m≤n wm is infinite for every n < ω and also (xm, ym) 6∈ wn for every m < n (this is possible because the space x0 is a subspace of ω∗ dense in it self). let v0 be the first move of player i, player ii responds with a point (g(0), h(0)) ∈ v0 ∩ (x0 × x0), and at the same time he chooses clopen sets a0 ∈ n (g(0)) \ n (x) and b0 ∈ n (h(0)) \ n (y), such that (x0 × x0) \ [(a0 × x0) ∪ (x0 × b0) ∪ ∆)] is infinite. inductively players i and ii produce a sequence of points in x0 × x0, {(g(n), h(n)) : n < ω}, and sequences of clopen sets {an : n < ω} and {bn : n < ω}, such that, if the moves of player i are denoted by vn ′ s then: (1) (g(0), h(0)) ∈ v0, (2) (g(n), h(n)) ∈ vn ∩ (x0 × x0 \ [ ⋃ m≤n wm ∪ ⋃ m ν. then by theorem 2.6 of [12], we obtain that i ↑ gpµ ( ∏ ν<µ xν ), this shows the first part of the theorem. notice that from the linearity of the rf-order and the properties of the ultrafilters pν ’s, it follows that ⋂ ν<ω1 xν = ω. now, fix x ∈ ∏ ν<ω1 xν , we will show that ii ↑ g(x, ∏ ν<ω1 xν ). indeed, assume that player i has chosen at the n-th step vn = ⋂ α∈fn [α, vα], where fn ∈ [ω1] <ω, vα ∈ n (x(α)) for each α ∈ fn and [α, vα] = {f ∈ ∏ ν<ω1 xν : f (α) ∈ vα}. the strategy of player ii is to choose at the n-th step xn ∈ ∏ ν<ω1 xν such that xn(α) = { x(α) if α ∈ fn, n if α ∈ ω1 \ ( ⋃ m≤n fm). from the fact that ⋂ ν<ω1 xν = ω, we have that for β = sup( ⋃ n<ω fn) the set {xn|[β,ω1) = n : n < ω} does not have a cluster point. so {xn : n < ω} is close and discrete and hence player ii wins. � 218 r. a. gonzález-silva and m. hrušák references [1] a. v. arkhangel’skii, classes of topological groups, russian math. surveys 36 (1981), no. 3, 151–174. [2] a. r. bernstein, a new kind of compactness for topological spaces, fund. math. 66 (1970), 185–193. [3] d. booth, ultrafilters on a countable set, ann. math. logic 2 (1970), 1–24. [4] a. bouziad, the ellis theorem and continuity in groups, topology appl. 50 (1993),73– 80. [5] a. blaszczyk and a. szymański, cohen algebras and nowhere dense ultrafilters, bulletin of the polish acad. of sciences math. 49 (2001), 15–25. [6] w. comfort and s. negrepontis, the theory of ultrafilters, springer-verlag, berlin, 1974. [7] a. dow, a. v. gubbi and a. szymański, rigid stone spaces within zfc, proc. amer. math. soc. 102 (1988), 745–748. [8] r. engelking, general topology, sigma series in pure mathematics vol. 6, heldermann verlag berlin, 1989. [9] z. froĺık, sums of ultrafilters, bull. amer. math. soc. 73 (1967), 87–91. [10] s. garćıa-ferreira, three orderings on ω∗, topology appl. 50 (1993), 199–216. [11] s. garćıa-ferreira and r. a. gonzález-silva, topological games defined by ultrafilters, topology appl. 137 (2004), 159–166. [12] s. garćıa-ferreira and r. a. gonzález-silva, topological games and product spaces, coment. math. univ. carolinae 43 (2002), no. 4, 675–685. [13] j. gerlits and zs. nagy, some properties of c(x), i, topology appl. 14 (1982), 151–161. [14] j. ginsburg and v. saks, some applications of ultrafilters in topology, pacific j. math. 57 (1975), 403–418. [15] g. gruenhage, infinite games and generalizations of first countable spaces, gen. topol. appl. 6 (1976), 339–352 [16] m. hrušák, fun with ultrafilters and special functions, manuscript. [17] m. hrušák, selectivity of almost disjoint families, acta univ. caroline-math. et physica, 41 (2000), no. 2, 13–21. [18] jan van mill, an introduction to βω en: handbook of set-theoretic topology, editors k. kunen y j. e. vaughan, north-holland, (1984), 505-567. [19] k. kunen, weak p -points in n∗, colloq. math. soc. jános bolyai 23, budapest (hungary), 741–749. [20] w. f. lindgren and a. szymański, a non-pseudocompact product of countably compact spaces via seq, proc. amer. math. soc. 125 (1997), no. 12, 3741–3746. [21] p. nyikos, subsets of ωω and the fréchet-urysohn and αi-properties, topology appl. 48 (1992) 91–116. [22] p. simon, applications of independent linked families, colloq. math. soc. jános bolyai 41 (1983), 561–580. [23] p. l. sharma, some characterizations of the w -spaces and w-spaces, gen. topol. appl. 9 (1978), 289–293. [24] j. e. vaughan, countably compact sequentially compact spaces, in: handbook of settheoretic topology, editors j. van mill and j. vaughan, north-holland, 571-600. [25] j. e. vaughan, two spaces homeomorphic to seq(p), coment. math. univ. carolinae 42 (2001), no. 1, 209–218. received december 2008 accepted september 2009 more on ultrafilters and topological games 219 r. a. gonzález-silva (rgonzalez@culagos.udg.mx) departamento de ciencias exactas y tecnológicas (udg) enrique dı́az de león 1144, col. paseos de la montaña, 47460, lagos de moreno jalisco, méxico m. hrušák (mhrusak@matmor.unam.mx) instituto de matemáticas (unam) a.p. 61-3 xangari, 58089 morelia, michoacán, méxico @ appl. gen. topol. 23, no. 2 (2022), 315-323 doi:10.4995/agt.2022.17021 © agt, upv, 2022 on set star-lindelöf spaces sumit singh department of mathematics, dyal singh college, university of delhi, lodhi road, new delhi110003, india. (sumitkumar405@gmail.com) communicated by f. mynard abstract a space x is said to be set star-lindelöf if for each nonempty subset a of x and each collection u of open sets in x such that a ⊆ ⋃ u, there is a countable subset v of u such that a ⊆ st( ⋃ v,u). the class of set star-lindelöf spaces lie between the class of lindelöf spaces and the class of star-lindelöf spaces. in this paper, we investigate the relationship between set star-lindelöf spaces and other related spaces by providing some suitable examples and study the topological properties of set starlindelöf spaces. 2020 msc: 54d20; 54e35. keywords: menger; star-lindelöf; strongly star-lindelöf; set star-lindelöf; covering; star-covering; topological space. 1. introduction and preliminaries arhangel’skii [1] defined a cardinal number sl(x) of x: the minimal infinite cardinality τ such that for every subset a ⊂ x and every open cover u of a, there is a subfamily v ⊂u such that |v|≤ τ and a ⊆ ⋃ v. if sl(x) = ω, then the space x is called slindelöf space. following this idea, kočinac and konca [7] introduced and studied the new types of selective covering properties called set-covering properties (for a similar studies, see [4, 14, 15, 16, 17]). a space x is said to have the set-menger [7] property if for each nonempty subset a of x and each sequence (un : n ∈ n) of collections of open sets in x such that for each n ∈ n, a ⊆ ⋃ un, there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un and a ⊆ ⋃ n∈n ⋃ vn. the author [13] noticed received 16 january 2022 – accepted 18 april 2022 http://dx.doi.org/10.4995/agt.2022.17021 https://orcid.org/0000-0001-9701-3091 s. singh that the set-menger property is nothing but another view of menger covering property. recently, the author [12] defined and studied set starcompact and set strongly starcompact spaces (also see [8]). in this paper, we consider the classes of set star-lindelöf spaces and set strongly star-lindelöf spaces already introduced in [9] and recently studied in [4]. note that in fact in the class of t1 spaces, set strongly star-lindelöfness is equivalent to the property having countable extent [[4], proposition 3.1]. if a is a subset of a space x and u is a collection of subsets of x, then st(a,u)= ⋃ {u ∈u : u ∩a 6= ∅}. we usually write st(x,u) = st({x},u). throughout the paper, by “a space” we mean “a topological space”, n, r and q denotes the set of natural numbers, set of real numbers, and set of rational numbers, respectively, the cardinality of a set is denoted by |a|. let ω denote the first infinite cardinal, ω1 the first uncountable cardinal, c the cardinality of the set of all real numbers. an open cover u of a subset a ⊂ x means elements of u open in x such that a ⊆ ⋃ u = ⋃ {u : u ∈u}. we first recall the classical notions of spaces that are used in this paper. definition 1.1 ([5]). a space x is said to be (1) starcompact if for each open cover u of x, there is a finite subset v of u such that x = st( ⋃ v,u). (2) strongly starcompact if for each open cover u of x, there is a finite subset f of x such that x = st(f,u). definition 1.2 ([12, 8]). a space x is said to be (1) set starcompact if for each nonempty subset a of x and each collection u of open sets in x such that a ⊆ ⋃ u, there is a finite subset v of u such that a ⊆ st( ⋃ v,u). (2) set strongly starcompact if for each nonempty subset a of x and each collection u of open sets in x such that a ⊆ ⋃ u, there is a finite subset f of a such that a ⊆ st(f,u). definition 1.3. a space x is said to be (1) star-lindelöf [5] if for each open cover u of x, there is a countable subset v of u such that x = st( ⋃ v,u). (2) strongly star-lindelöf [5] if for each open cover u of x, there is a countable subset f of x such that x = st(f,u). note that the star-lindelöf spaces have a different name such as 1-starlindelöf and 1 1 2 -star-lindelöf in different papers (see [5, 10]) and the strongly star-lindelöf space is also called star countable in [10, 21]. it is clear that, every strongly star-lindelöf space is star-lindelöf. recall that a collection a ⊆ p(ω) is said to be almost disjoint if each set a ∈a is infinite and the sets a ⋂ b are finite for all distinct elements a,b ∈a. for an almost disjoint family a, put ψ(a) = a ⋃ ω and topologize ψ(a) as follows: for each element a ∈ a and each finite set f ⊂ ω, {a} ⋃ (a \ f) is a basic open neighborhood of a and the natural numbers are isolated. the © agt, upv, 2022 appl. gen. topol. 23, no. 2 316 on set star-lindelöf spaces spaces of this type are called isbell-mrówka ψ-spaces [2, 11] or ψ(a) space. for other terms and symbols, we follow [6]. the following result was proved in [8]. theorem 1.4 ([8]). every countably compact space is set strongly starcompact. note that in the class of hausdorff spaces strongly starcompactness, set strongly starcompactness and countable compactness are equivalent [4, proposition 2.2]. 2. set star-lindelöf and related spaces in this section, we give some examples showing the relationship among set star-lindelöf spaces, set strongly star-lindelöf spaces, and other related spaces. first we define our main definition. definition 2.1. a space x is said to be (1) set star-lindelöf if for each nonempty subset a of x and each collection u of open sets in x such that a ⊆ ⋃ u, there is a countable subset v of u such that a ⊆ st( ⋃ v,u). (2) set strongly star-lindelöf if for each nonempty subset a of x and each collection u of open sets in x such that a ⊆ ⋃ u, there is a countable subset f of a such that a ⊆ st(f,u). note that in the class of t1 spaces the set strongly star-lindelöfness is equivalent to the property to have a countable extent [4, proposition 3.1]. note that there is a misprint in the statement of the definition of relatively∗ set star strongly-compact in [4]: the authors write that set f is a finite subset of a but the original definition asks that f is contained in a and bonanzinga and maesano use exactly this last fact during all the paper. we have the following diagram from the definitions and [4, proposition 3.1]. however, the following examples show that the converse of these implications are not true. set strongly starcompact → set starcompact ↓ ↓ lindelof → countable extent ↔t1 set strongly star −lindelof → set star −lindelof ↓ ↓ strongly star −lindelof → star −lindelof example 2.2. (i) the discrete space ω has countable extent but it is not set starcompact space. (ii) the space [0,ω1) has countable extent but it is not lindelöf. (iii) let y be a discrete space with cardinality c. let x = y ∪{y∗}, where y∗ /∈ y topologized as follows: each y ∈ y is an isolated point and a set u © agt, upv, 2022 appl. gen. topol. 23, no. 2 317 s. singh containing y∗ is open if and only if x \u is countable. then x has countable extent but it is not countably compact. bonanzinga [3] proved that every isbell-mrówka space is a tychonoff strongly star-lindelöf space with uncountable extent (hence, it is not set strongly starlindelöf). note that in [3] strongly star-lindelöf is called star-lindelöf. the following lemma was proved by song [18]. lemma 2.3 ([18, lemma 2.2]). a space x having a dense lindelöf subspace is star-lindelöf. the following example shows that the lemma 2.3 does not hold if we replace star-lindelöf space by a set star-lindelöf space. example 2.4. there exists a tychonoff space x having a dense lindelöf subspace such that x is not set star-lindelöf. proof. let d(c) = {dα : α < c} be a discrete space of cardinality c and let y = d(c) ∪{d∗} be one-point compactification of d(c). let x = (y × [0,ω)) ∪ (d(c) ×{ω}) be the subspace of the product space y × [0,ω]. then y × [0,ω) is a dense lindelöf subspace of x and by lemma 2.3, x is star-lindelöf. in [4, proposition 3.4] shows that if x is a space such that there exists a closed and discrete subspace d of x having uncountable cardinality and a disjoint family u = {oa : a ∈ d} of open neighborhoods of points a ∈ d, then x is not set star-lindelöf. so, we conclude that x is not set star-lindelöf. � bonanzinga and maesano [4, example 3.5] constructed an example of a tychonoff separable (hence set star-lindelöf) non set strongly star-lindelöf space. remark 2.5. (1) in [12], singh gave an example of a tychonoff set starcompact space x that is not set strongly starcompact. (2) it is known that there are star-lindelöf spaces that are not strongly star-lindelöf (see [5, example 3.2.3.2] and [5, example 3.3.1]). now we give some conditions under which star-lindelöfness coincides with set star-lindelöfness and strongly star-lindelöfness coincide with set strongly star-lindelöfness. recall that a space x is paralindelöf if every open cover u of x has a locally countable open refinement. song and xuan [19] proved the following result. theorem 2.6 ([19, theorem 2.24]). every regular paralindelöf star-lindelöf spaces are lindelöf. we have the following theorem from theorem 2.6 and the diagram. © agt, upv, 2022 appl. gen. topol. 23, no. 2 318 on set star-lindelöf spaces theorem 2.7. if x is a regular paralindelöf space, then the following statements are equivalent: (1) x is lindelöf; (2) x is set strongly star-lindelöf; (3) e(x) = ω; (4) x is set star-lindelöf; (5) x is strongly star-lindelöf; (6) x is star-lindelöf. a space is said to be metalindelöf if every open cover of it has a pointcountable open refinement. bonanzinga [3] proved the following result. theorem 2.8 ([3]). every strongly star-lindelöf metalindelöf spaces are lindelöf. we have the following theorem from theorem 2.8 and the diagram. theorem 2.9. if x is a metalindelöf space, then the following statements are equivalent: (1) x is lindelöf; (2) x is set strongly star-lindelöf; (3) e(x) = ω; (4) x is strongly star-lindelöf. 3. properties of set star-lindelöf spaces in this section, we study the topological properties of set star-lindelöf spaces. theorem 3.1. if x is a set star-lindelöf space, then every open and closed subset of x is set star-lindelöf. proof. let x be a set star-lindelöf space and a ⊆ x be an open and closed set. let b be any subset of a and u be a collection of open sets in (a,τa) such that cla(b) ⊆ ⋃ u. since a is open, then u is a collection of open sets in x. since a is closed, cla(b) = clx(b). applying the set star-lindelöfness property of x, there exists a countable subset v of u such that b ⊆ st( ⋃ v,u). hence a is a set star-lindelöf. � consider the alexandorff duplicate a(x) = x ×{0, 1} of a space x. the basic neighborhood of a point 〈x, 0〉 ∈ x ×{0} is of the form (u ×{0}) ⋃ (u × {1} \ {〈x, 1〉}), where u is a neighborhood of x in x and each point 〈x, 1〉 ∈ x ×{1} is an isolated point. theorem 3.2. if x is a t1-space and a(x) is a set star-lindelöf space. then e(x) < ω1. proof. suppose that e(x) ≥ ω1. then there exists a discrete closed subset b of x such that |b| ≥ ω1. hence b ×{1} is an open and closed subset of © agt, upv, 2022 appl. gen. topol. 23, no. 2 319 s. singh a(x) and every point of b ×{1} is an isolated point. thus a(x) is not set star-lindelöf by theorem 3.1. � theorem 3.3. let x be a space such that the alexandorff duplicate a(x) of x is set star-lindelöf. then x is a set star-lindelöf space. proof. let b be any nonempty subset of x and u be an open cover of b. let c = b ×{0} and a(u) = {u ×{0, 1} : u ∈u}. then a(u) is an open cover of c. since a(x) is set star-lindelöf, there is a countable subset a(v) of a(u) such that c ⊆ st( ⋃ a(v),a(u)). let v = {u ∈u : u ×{0, 1}∈ a(v)}. then v is a countable subset of u. now we have to show that b ⊆ st( ⋃ v,u). let x ∈ b. then 〈x, 0〉 ∈ st( ⋃ a(v),a(u)). choose u × {0, 1} ∈ a(u) such that 〈x, 0〉 ∈ u ×{0, 1} and u ×{0, 1}∩ ( ⋃ a(v)) 6= ∅, which implies u ∩ ( ⋃ v) 6= ∅ and x ∈ u. therefore x ∈ st( ⋃ v,u), which shows that x is set star-lindelöf space. � on the images of set star-lindelöf spaces, we have the following result. theorem 3.4. a continuous image of set star-lindelöf space is set starlindelöf. proof. let x be a set star-lindelöf space and f : x → y is a continuous mapping from x onto y . let b be any subset of y and v be an open cover of b. let a = f−1(b). since f is continuous, u = {f−1(v ) : v ∈ v} is the collection of open sets in x with a = f−1(b) ⊆ f−1(b) ⊆ f−1( ⋃ v) = ⋃ u. as x is set star-lindelöf, there exists a countable subset u′ of u such that a ⊆ st( ⋃ u′,u). let v′ = {v : f−1(v ) ∈ u′}. then v′ is a countable subset of v and b = f(a) ⊆ f(st( ⋃ u′,u)) ⊆ st( ⋃ f({f−1(v ) : v ∈ v′}),v) = st( ⋃ v′,v). thus y is set star-lindelöf space. � next, we turn to consider preimages of set strongly star-lindelöf and set star-lindelöf spaces. we need a new concept called nearly set star-lindelöf spaces. a space x is said to be nearly set star-lindelöf in x if for each subset y of x and each open cover u of x, there is a countable subset v of u such that y ⊆ st( ⋃ v,u). for the strong version of this property (see [4]). theorem 3.5. if f : x → y is an open and perfect continuous mapping and y is a set star-lindelöf space, then x is nearly set star-lindelöf. © agt, upv, 2022 appl. gen. topol. 23, no. 2 320 on set star-lindelöf spaces proof. let a ⊆ x be any nonempty set and u be an open cover of x. then b = f(a) is a subset of y . let y ∈ b. then f−1{y} is a compact subset of x, thus there is a finite subset uy of u such that f−1{y}⊆ ⋃ uy. let uy = ⋃ uy. then vy = y \ f(x \ uy) is a neighborhood of y, since f is closed. then v = {vy : y ∈ b} is an open cover of b. since y is set star-lindelöf, there exists a countable subset v′ of v such that b ⊆ st( ⋃ v′,v). without loss of generality, we may assume that v′ = {vyi : i ∈ n′ ⊆ n}. let w = ⋃ i∈n′ uyi . since f −1(vyi ) ⊆ ⋃ {u : u ∈uyi} for each i ∈ n′. then w is a countable subset of u and f−1( ⋃ v′) = ⋃ w. next, we show that a ⊆ st( ⋃ w,u). let x ∈ a. then there exists a y ∈ b such that f(x) ∈ vy and vy ⋂ ( ⋃ v′) 6= ∅. since x ∈ f−1(vy) ⊆ ⋃ {u : u ∈uy}, we can choose u ∈uy with x ∈ u. then vy ⊆ f(u). thus u ⋂ f−1( ⋃ v′) 6= ∅. hence x ∈ st(f−1( ⋃ v′),u). therefore x ∈ st( ⋃ w,u), which shows that a ⊆ st( ⋃ w,u). thus x is nearly set star-lindelöf. � it is known that the product of star-lindelöf space and compact space is a star-lindelöf (see [5]). problem 3.6. does the product of set star-lindelöf space and a compact space is set star-lindelöf ? the following example shows that the product of two countably compact (hence, set star-lindelöf) spaces need not be set star-lindelöf. example 3.7. there exist two countably compact spaces x and y such that x ×y is not set star-lindelöf. proof. let d(c) be a discrete space of the cardinality c. we can define x =⋃ α<ω1 eα and y = ⋃ α<ω1 fα, where eα and fα are the subsets of β(d(c)) which are defined inductively to satisfy the following three conditions: (1) eα ⋂ fβ = d(c) if α 6= β; (2) |eα| ≤ c and |fα| ≤ c; (3) every infinite subset of eα (resp., fα) has an accumulation point in eα+1 (resp, fα+1). © agt, upv, 2022 appl. gen. topol. 23, no. 2 321 s. singh those sets eα and fα are well-defined since every infinite closed set in β(d(c)) has the cardinality 2c (see [20]). then x×y is not set star-lindelöf, since the diagonal {〈d,d〉 : d ∈ d(c)} is a discrete open and closed subset of x ×y with the cardinality c. � van douwen-reed-roscoe-tree [5, example 3.3.3] gave an example of a countably compact x (hence, set star-lindelöf) and a lindelöf space y such that x × y is not strongly star-lindelöf. now we use this example to show that x ×y is not set star-lindelöf. example 3.8. there exists a countably compact space x and a lindelöf space y such that x ×y is not set star-lindelöf. proof. let x = [0,ω1) with the usual order topology. let y = [0,ω1] with the following topology. each point α < ω1 is isolated and a set u containing ω1 is open if and only if y \u is countable. then, x is countably compact and y is lindelöf. it is enough to show that x ×y is not star-lindelöf. for each α < ω1, uα = x × {α} is open in x × y . for each β < ω1, vβ = [0,β] × (0,ω1] is open in x ×y . let u = {uα : α < ω1}∪{vβ : β < ω1}. then u is an open cover of x ×y . let v be any countable subset of u. since v is countable, there exists α′ < ω1 such that uα /∈ v for each α > α′. also, there exists α′′ < ω1 such that vβ /∈ v for each β > α′′. let β = sup{α′,α′′}. then uβ ⋂ ( ⋃ v) = ∅ and uβ is the only element containing 〈β,β〉. thus 〈β,β〉 /∈ st( ⋃ v,u), which shows that x is not star-lindelöf. � van douwen-reed-roscoe-tree [5, example 3.3.6] gave an example of hausdorff regular lindelöf spaces x and y such that x ×y is star-lindelöf. now we use this example and show that the product of two lindelöf spaces is not set star-lindelöf. example 3.9. there exists a hausdorff regular lindelöf spaces x and y such that x ×y is not set star-lindelöf. proof. let x = r\q have the induced metric topology. let y = r with each point of r\q is isolated and points of q having metric neighborhoods. hence both spaces x and y are hausdorff regular lindelöf spaces and first countable too, so x×y hausdorff regular and first countable. now we show that x×y is not set star-lindelöf. let a = {(x,x) ∈ x × y : x ∈ x}. then a is an uncountable closed and discrete set (see [[5], example 3.3.6]). for (x,x) ∈ a, ux = x ×{x} is the open subset of x × y . then u = {ux : (x,x) ∈ a} is an open cover of a. let v be any countable subset of u. then there exists (a,a) ∈ a such that (a,a) /∈ ⋃ v and thus ( ⋃ v) ⋂ ua = ∅. but ua is the only element of u containing (a,a). thus (a,a) /∈ st( ⋃ v,u), which completes the proof. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 322 on set star-lindelöf spaces acknowledgements. the author would like to thank the referee for several suggestions that led to an improvement of both the content and exposition of the paper. references [1] a. v. arhangel’skii, an external disconnected bicompactum of weight c is inhomogeneous, dokl. akad. nauk sssr. 175 (1967), 751–754. [2] m. bonanzinga and m. v. matveev, some covering properties for ψ-spaces, mat. vesnik. 61 (2009), 3–11. [3] m. bonanzinga, star lindelöf and absolutely star-lindelöf spaces, quest. answ. gen. topol. 16 (1998), 79–104. [4] m. bonanzinga and f. maesano, some properties defined by relative versions of starcovering properties, topology appl. 306 (2022), article no. 107923. [5] e. k. van douwen, g. k. reed, a. w. roscoe and i. j. tree, star covering properties, topology appl. 39 (1991), 71–103. [6] r. engelking, general topology, pwn, warszawa, 1977. [7] lj. d. r. koc̆inac and s. konca, set-menger and related properties, topology appl. 275 (2020), article no. 106996. [8] lj. d. r. koc̆inac, s. konca and s. singh, set star-menger and set strongly star-menger spaces, math. slovaca 72 (2022), 185–196. [9] lj. d. r. koc̆inac and s. singh, on the set version of selectively star-ccc spaces, j. math. (2020) article id 9274503. [10] m. v. matveev, a survey on star-covering properties, topology atlas, preprint no 330 1998. [11] s. mrówka, on completely regular spaces, fund. math. 41 (1954), 105–106. [12] s. singh, set starcompact and related spaces, afr. mat. 32 (2021), 1389–1397. [13] s. singh, remarks on set-menger and related properties, topology appl. 280 (2020), article no. 107278. [14] s. singh, on set-star-k-hurewicz spaces, bull. belg. math. soc. simon stevin 28, no. 3 (2021), 361–372. [15] s. singh, on set-star-k-menger spaces, publ. math. debrecen 100 (2022), 87–100. [16] s. singh, on set weak strongly star-menger spaces, submitted. [17] s. singh and lj. d. r. kočinac, star versions of hurewicz spaces, hacet. j. math. stat. 50, no. 5 (2021), 1325–1333. [18] y. k. song, remarks on neighborhood star-lindelöf spaces, filomat 27, no. 1 (2013), 149–155. [19] y. k. song and w. f. xuan, remarks on new star-selection principles in topology, topology appl. 268 (2019), paper no. 106921. [20] r. c. walker, the stone-čech compactification, berlin, 1974. [21] w. f. xuan and w. x. shi, notes on star lindelöf spaces, topology appl. 204 (2016), 63–69. © agt, upv, 2022 appl. gen. topol. 23, no. 2 323 () @ applied general topology c© universidad politécnica de valencia volume 12, no. 1, 2011 pp. 1-13 the structure of the poset of regular topologies on a set ofelia t. alas and richard g. wilson abstract we study the subposet σ3(x) of the lattice l1(x) of all t1-topologies on a set x, being the collections of all t3 topologies on x, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. we show that each regular topology which is not r-closed does have such a predecessor and as a corollary we obtain a result of costantini that each non-compact tychonoff space has an immediate predecessor in σ3. we also consider the problem of when an r-closed topology is maximal r-closed. 2010 msc: primary 54a10; secondary 06a06, 54d10 keywords: lattice of t1-topologies, poset of t3-topologies, upper topology, lower topology, r-closed space, r-minimal space, submaximal space, maximal r-closed space, dispersed space 1. introduction in a previous paper [3], we studied the problem of when a jump can occur in the order of the lattice l1(x); that is to say, when there exist t1-topologies τ and τ+ on a set x such that whenever µ is a topology on x such that τ ⊆ µ ⊆ τ+ then µ = τ or µ = τ+. the existence of jumps in l1(x) and in the subposet of hausdorff topologies, has been studied in [5], [2], [10] and [16]; in the last two articles an immediate successor τ+ was said to be a cover of (or simply to cover) τ. in the above cited paper [3], when a topology τ has a cover τ+ we have called τ a lower topology and τ+ an upper topology and we continue to use this terminology here. in the present work we study the structure of the subposet σ3(x) of all t3-topologies of the lattice l1(x), on a set x with a view to deciding which elements of this partially ordered sets have and which do not have covers. 2 o. t. alas and r. g. wilson in [1] it was shown that a t3-topology on x which is not feebly compact is an upper topology in σ3(x) and in [6], costantini showed that every noncompact tychonoff topology on x is upper in σ3(x). in section 2 of this paper we generalize both these results by showing that every t3-topology which is not r-closed is upper in σ3(x). (a t3-space is r-closed if it is closed in every embedding in a t3-space.) in section 3 we consider the problem of the existence of spaces which are maximal with respect to being r-closed and in section 4 we study lower topologies in σ3. in the final section we pose a number of open problems. a set x with a topology ξ will be denoted by (x,ξ) and if p ∈ x, then ξ(p,x) denotes the collection of all open sets in x which contain p. the closure (respectively, interior) of a set a in a topological space (x,τ) will be denoted by clτ (a) (respectively, intτ (a)) or simply by cl(a) (respectively int(a)) when no confusion is possible. all undefined terms can be found in [7] or [13] and all spaces in this article are (at least) t3. a comprehensive survey of results on r-closed spaces and many open questions can be found in [8]. we make the following formal definitions. definition 1.1. say that two (distinct) t3-topologies τ1 and τ2 on a set x are adjacent in σ3(x) if whenever σ ∈ σ3(x) and τ1 ⊆ σ ⊆ τ2, then either σ = τ1 or σ = τ2. we say that τ1 is a lower topology in σ3(x), τ2 is an upper topology in σ3 and τ2 is an immediate successor of τ1. for a topology τ, τ + will always denote an immediate successor of τ. a t3-topology on x is r-minimal if there is no weaker t3-topology on x; it is well known that an r-minimal topology is r-closed. clearly an r-minimal topology is not upper in σ3(x). in the sequel, whenever the space x is understood, we will write σ3 instead of σ3(x). in [11], it was shown that the structure of basic intervals in σ3 is essentially different from those of the poset σt of tychonoff spaces in that not every finite interval is isomorphic to the power set of a finite ordinal. the following result is lemma 22 of [11]. lemma 1.2. if σ is an immediate successor of τ in σ3, then τ and σ differ at precisely one point. an open filter (that is, a filter with a base of open sets) f is a regular filter if for each u ∈ f there is v ∈ f such that cl(v ) ⊆ u. a simple application of zorn’s lemma shows that every regular filter can be embedded in a maximal regular filter and furthermore, in a regular space, if a maximal regular filter has an accumulation point, then it must converge to that point. by theorem 4.14 of [4], a t3-space is r-closed if and only if every regular filter has an accumulation point or equivalently, if and only if every maximal regular filter converges. 2. upper topologies in σ3 the next result generalizes theorem 2.14 of [1]. regular topologies on a set 3 theorem 2.1. each t3-topology which is not r-closed is upper in σ3. proof. suppose that (x,σ) is a t3-space which is not r-closed. then there is some maximal regular filter f in (x,σ) which is not fixed. pick p ∈ x and define a new topology τ on x as follows: τ = {u ∈ σ : p 6∈ u} ∪ {u ∈ σ : p ∈ u ∈ f}. the topologies τ and σ differ only at the point p and hence for each a ⊆ x, clτ (a) ⊆ clσ(a) ∪ {p}. we first show that (x,τ) is a t3-space; suppose that c ⊆ x is τ-closed and q 6∈ c. there are three cases to consider. 1) if p 6∈ c ∪{q}, then there are σ-open sets u,v separating c and q in x \{p} and u,v are τ-open. 2) if p ∈ c, then c is σ-closed and hence there are disjoint σ-open sets u and v such that c ⊆ u and q ∈ v . furthermore, since f is a free regular filter, there is w ∈ f such that q 6∈ clσ(w) and hence q 6∈ clτ (w) = clσ(w) ∪ {p}. it is now clear that u ∪ w and u \ clτ (w) are disjoint τ-open sets containing c and q respectively. 3) if p = q, then since c is τ-closed and p 6∈ c, it follows that there is some element w ∈ f such that w ∩ c = ∅. furthermore, since c is σ-closed, there are disjoint sets u,v ∈ σ such that c ⊆ u and p ∈ v . since f is a regular filter, there is some t ∈ f such that clσ(t ) ⊆ w . since clτ (t ) = clσ(t ) ∪ {p}, it is now clear that u \ clτ (t ) and v ∪ t are disjoint τ-open sets containing c and p respectively. we claim that τ is the immediate predecessor of σ in σ3. to see this, suppose that µ is a t3-topology on x such that τ µ σ; note that µ differs from σ and τ only at the point p. if there is some µ-neighbourhood u of p which misses some element f ∈ f, then if w is a σ-open neighbourhood of p, it follows that w ∪ f is a τ-open, hence µ-open neighbourhood of p. but then (w ∪ f) ∩ u = w ∩ u ⊆ w is a µ-open neighbourhood of p, implying that µ = σ. hence every µ-neighbourhood of p must meet every element of f; we claim that this implies that µ = τ. to prove our claim, let vp be the filter of µ-open neighbourhoods of p and let g be the open filter generated by {f ∩ v : f ∈ f and v ∈ vp}. we will show that g is a regular filter in (x,σ), thus contradicting the maximality of f. however, if f ∈ f and v ∈ vp, then there is w ∈ vp and h ∈ f such that v ⊇ clµ(w) ⊇ clσ(w) and clσ(h) ⊆ f . hence w ∩ h ∈ g and clσ(w ∩ h) ⊆ f ∩ v . � in [6], the concept of a strongly upper topology was defined. (a topology τ is strongly upper if whenever µ τ, there is an immediate predecessor τ− of τ such that µ ⊆ τ− τ.) a simple modification of the above proof shows that every regular topology which is not r-closed is in fact strongly upper. clearly every r-closed tychonoff space is compact and hence the following result of [6] is an immediate corollary. 4 o. t. alas and r. g. wilson corollary 2.2. every tychonoff topology which is not compact is (strongly) upper in σ3. every completely hausdorff topology possesses a weaker tychonoff topology (the weak topology induced by the continuous real-valued functions). thus every completely hausdorff r-minimal topology is compact. the following question then arises: question 2.3. is every completely hausdorff t3-topology which is not compact an upper topology in σ3? question 2.4. is every regular topology which has a compact hausdorff subtopology an upper topology in σ3? 3. maximal r-closed topologies recall that a space is submaximal if every dense set is open (we do not assume that a submaximal space must be dense-in-itself). it follows from 7m of [13] that each h-closed topology is contained in a maximal h-closed topology and that a space is maximal h-closed if and only if it is h-closed and submaximal. however, as we show below, the class of submaximal r-closed spaces is much more restricted. recall that a space is feebly compact if every locally finite family of open sets is finite. theorem 3.1. each submaximal regular, feebly compact topology has an isolated point. proof. suppose that (x,τ) is feebly compact, submaximal and has no isolated points. fix p ∈ x and let c be a maximal cellular family of open sets in x so that for each c ∈ c, we have p 6∈ clτ (c). the subset ⋃ c is dense in x and hence f = x \ ⋃ c is closed and discrete. since p ∈ f , there are disjoint open sets u and v so that p ∈ u and f \ {p} ⊆ v . let s = {u ∩c : c ∈ c and u ∩ c 6= ∅}; since p is not isolated, it follows that s is an infinite cellular family of open sets. since x is feebly compact, this family must have an accumulation point in f , and hence its only accumulation point is p. for each u ∩ c ∈ s, pick xc ∈ u ∩c; since x has no isolated points, the set {xc : u ∩c ∈ s}∪{p} is closed and discrete and hence there are disjoint opens sets u′ and v ′ such that p ∈ u′ and {xc : u ∩c ∈ s} ⊆ v ′. it follows immediately that the infinite family of non-empty open sets {c ∩ u ∩ v ′ : u ∩ c ∈ s} has no accumulation point in x, contradicting the fact that x is feebly compact. � the following theorem is a result of scarborough and stone [14]. for completeness we include the simple proof. regular topologies on a set 5 theorem 3.2. an r-closed topology is feebly compact. proof. suppose to the contrary that u = {un : n ∈ ω} is an infinite discrete family of open subsets of (x,τ). for each n ∈ ω, pick xn ∈ un. it is then straightforward to check that the family b = {u ∈ τ : u ⊇ {xn : n ≥ k} for some k ∈ ω} is a regular filter base on x with no accumulation point, contradicting the fact that (x,τ) is r-closed. � corollary 3.3. each submaximal r-closed space has an isolated point. lemma 3.4. an r-closed space which is scattered and of dispersion order 2 is compact. proof. suppose x = x0 ∪ x1 where x0 is the set of isolated points and x1 is the set of accumulation points of x. for each p ∈ x1, there is a closed neighbourhood u of p such that u ⊆ x0 ∪ {p}. it is clear that u is clopen and so x is 0-dimensional and hence tychonoff. thus x is compact. � stephenson’s examples (see [15] and [9]) show that the previous result is false for r-closed scattered spaces of dispersion order 3. since a subspace of a submaximal space is submaximal, the closure of the set of isolated points of an r-closed submaximal space is scattered of dispersion order 2. thus: corollary 3.5. each submaximal r-closed space is a compact scattered space of dispersion order 2. proof. suppose that (x,τ) is an r-closed submaximal space and let x0 denote the set of isolated points of x; by corollary 3.3, x0 6= ∅. let c = cl(x \ cl(x0)); if c = ∅ then we are done, so suppose to the contrary. then c is a submaximal space without isolated points and so again by corollary 3.3, (c,τ|c) is not feebly compact. thus there is an infinite locally finite family f of open sets in (c,τ|c). but then, {f ∩ (x \ cl(x0)) : f ∈ f} is an infinite locally finite family of open sets in x, implying that x is not feebly compact, which is a contradiction. � theorem 3.6. a submaximal r-closed space is maximal r-closed. proof. suppose that (x,τ) is a submaximal r-closed space. by the previous corollary, x is compact scattered of dispersion order 2; let x0 denote the set of isolated points of x and x1 = x \ x0. suppose that σ ! τ is a regular topology on x which differs from τ at a point p. then there is some σ-open neighbourhood u of p which is not τ-open and hence does not contain any τ-neighbourhood of p; there is also a compact τ-neighbourhood v of p such that v ⊆ x0 ∪ {p}. it is then clear that v \ u is an infinite σ-closed subset of x0, implying that (x,σ) is not feebly compact. � 6 o. t. alas and r. g. wilson lemma 3.7. a feebly compact regular space of countable pseudocharacter is first countable. proof. suppose that (x,τ) is feebly compact regular space and ψ(x,p) = ω. there is a family b = {bn : n ∈ ω} of open sets such that ⋂ {bn : n ∈ ω} = {p} and for each n ∈ ω, cl(bn+1) ⊆ bn. if b is not a local base at p, then there is some open neighbourhood u of p such that for each n ∈ ω, bn * u. it is straightforward to check that the family of open sets {bn \ (clτ (bn+1 ∪ u)) : n ∈ ω} is an infinite locally finite family of open sets, contradicting the fact that x is feebly compact. � the next theorem should be compared with theorem 2.20 of [12]. theorem 3.8. a regular feebly compact first countable topology is maximal among regular feebly compact topologies. proof. suppose that (x,τ) is a regular feebly compact first countable space and σ ! τ is a regular topology on x; we will show that (x,σ) is not feebly compact. to this end, suppose that u ∈ σ \ τ; then x \ u is σ-closed but not τ-closed and so since (x,τ) is first countable, there is some sequence {pn} in x \ u convergent (in (x,τ)) to p ∈ u. by lemma 4.1 of [2], there is a family of disjoint τ-open sets {un : n ∈ ω} whose only accumulation point (in (x,τ)) is p and such that pn ∈ un for each n ∈ ω. now by regularity of (x,σ) there is w ∈ σ such that p ∈ w ⊆ clσ(w) ⊆ u; then, the collection of sets u = {un \ clσ(w) : n ∈ ω} is a locally finite collection of open subsets of (x,σ) and so if an infinite number of elements of u are non-empty, then (x,σ) is not feebly compact. however, if for some n0 ∈ ω, un \ clσ(w) = ∅ for all n ≥ n0, then pn ∈ un ⊆ clσ(w) for all n ≥ n0 contradicting the fact that pn ∈ x \ u ⊆ x \ clσ(w). � the following result is now an immediate consequence of theorems 3.2 and 3.8 and lemma 3.7. corollary 3.9. an r-closed space of countable pseudocharacter is maximal r-closed. remark 3.10. note that we have proved something a little stronger: if (x,τ) is r-closed and σ ⊇ τ differs from τ at a point of countable pseudocharacter, then (x,σ) is not r-closed. corollary 3.11. a regular space with a strictly weaker r-closed first countable topology is upper in σ3. corollary 3.12. a first countable compact hausdorff space is maximal rclosed. question 3.13. is a fréchet compact hausdorff space maximal r-closed? regular topologies on a set 7 4. lower topologies a point p is a maximal regular point of a regular space (x,τ) if the trace of the regular filter vτp generated by τ(p,x) on x \ {p} is a maximal regular filter. lemma 4.1. a point p in a regular topological space (x,τ) is a maximal regular point of x if and only if whenever τ σ is a regular topology on x such that σ|(x \ {p}) = τ|(x \ {p}) then p is an isolated point of (x,σ). proof. for the sufficiency suppose that the regular filter vτp generated by τ(p,x) when restricted to x \ {p} is not maximal. then there is some regular filter f ! vτp |(x \ {p}). define σ to be that topology on x generated by the subbase τ ∪ {v ∪ {p} : v ∈ f}; it is straightforward to show that σ is a regular topology on x strictly finer that τ in which p is not an isolated point. to show the necessity, suppose that p is a maximal regular point of (x,τ). then if σ ! τ and σ|(x \ {p}) = τ|(x \ {p}), it follows that the trace of the neighbourhood filter vσp at p on x \ {p} is strictly larger than the trace of the neighbourhood filter vτp at p on x \ {p} and since σ|(x \ {p}) = τ|(x \ {p}), vσp |(x \ {p}) is a τ-open collection strictly larger than the maximal regular filter vτp |(x \ {p}). it follows that p is an isolated point of (x,σ). � it was essentially shown in theorem 2.13 of [1] that a point of first countability in a space is not a maximal regular point. corollary 4.2. if (x,τ) has a maximal regular point then τ is a lower topology in σ3. in [3] we characterized lower topologies in the poset of hausdorff spaces as those having a closed subspace with a maximal point. example 4.10 below shows that having a closed subspace with a maximal regular point does not guarantee that a topology is lower in σ3. however, we have the following result: lemma 4.3. if σ ∈ σ3(x) is a simple extension of τ ∈ σ3(x) which differs from τ at precisely one point p ∈ x, then σ is upper and each lower topology µ corresponding to σ has a closed subspace with a maximal regular point. proof. it was shown in [6] that if a t3-topology σ is a simple extension of a t3-topology τ that differs from τ at precisely one point p, then σ is upper in σ3(x) and is generated by the subbase τ ∪ {u ∪ {p}} for some u ∈ τ. clearly µ∪{u ∪{p}} is also a subbase for σ and hence p is an isolated point of a = (x \u) ∪ {p} in the topology σ but not in µ. thus p is a maximal regular point of (a,µ|a). � remark 4.4. if τ is a lower topology in σ3 and τ and τ + differ at p ∈ x then there is some u0 ∈ τ such that u0 ∪ {p} ∈ τ + \ τ. then since τ+ is 8 o. t. alas and r. g. wilson regular, for each n ≥ 1 there is un ∈ τ such that un ∪ {p} ∈ τ + \ τ and un ∪ {p} ⊆ clτ + (un) ∪ {p} ⊆ un−1 ∪ {p}. it is clear that τ + is generated by the subbase τ ∪ {un ∪ {p} : n ∈ ω} and hence the character of p in (x,τ +) is no greater than its character in (x,τ). a family s = {sn : n ∈ ω} is said to be strongly decreasing at p if for each n ∈ ω, cl(sn+1) ∪ {p} ⊆ sn ∪ {p}. we now formulate the above remark as a lemma: lemma 4.5. let (x,τ) be a t3-space; if τ has an immediate successor τ + ∈ σ3, then there is p ∈ x and a family u = {un : n ∈ ω} ⊆ τ which is strongly decreasing at p, such that for each n ∈ ω, un ∪ {p} 6∈ τ and τ + is generated by the subbase τ ∪ {un ∪ {p} : n ∈ ω}. this result allows us to characterize (rather abstractly it must be said) lower topologies in σ3 in the next theorem. in order to simplify the notation somewhat, when w = {wn : n ∈ ω} ⊆ τ and v = {vn : n ∈ ω} ∈ τ are strongly decreasing families at (a fixed) p ∈ x, τw will denote the topology generated by τ ∪ {wn ∪ {p} : n ∈ ω} and w ∩ v will denote the family {wn ∩vn : n ∈ ω} which is also strongly decreasing at p. theorem 4.6. a topology τ on x is lower in σ3 if and only if there is p ∈ x and a strongly decreasing family u = {un : n ∈ ω} ⊆ τ at p such that whenever v = {vn : n ∈ ω} ⊆ τ is strongly decreasing at p and τu = τu∩v, then either τv = τu or τv = τ. proof. suppose that τ is not lower and fix p ∈ x; if u = {un : n ∈ ω} ⊆ τ is strongly decreasing at p, then there is σ ∈ σ3 such that τ σ τu . we may then choose a strongly decreasing family (at p) v = {vn : n ∈ ω} ⊆ σ, such that for each n ∈ ω, vn ∪ {p} ∈ σ \ τ and so τ τv τu . however, since for each n ∈ ω, vn ∪ {p} ∈ τu , we have that (un ∩ vn) ∪ {p} ∈ τu which implies that τu = τu∩v , giving a contradiction. conversely, suppose that τ is lower in σ3; by lemma 4.5, there is p ∈ x and a strongly decreasing family u at p such that τ+ = τu . then, if v = {vn : n ∈ ω} ⊆ τ is a strongly decreasing family at p such that τu = τu∩v it follows that for each n ∈ ω, vn ∪ {p} ∈ τu and so τv ⊆ τu∩v = τu . � theorem 4.7. a compact lots is maximal r-closed. proof. suppose that (x,τ,<) is a compact lots and σ ! τ. then there is some u ∈ σ \ τ and p ∈ u such that u is not a τ-neighbourhood of p, and hence lp \ u is cofinal in lp \ {p} or rp \ u is cofinal in rp \ {p}, where lp = {x ∈ x : x ≤ p} and rp = {x ∈ x : x ≥ p}. it is easy to see that (x,τ) is maximal r-closed if and only if both of the compact subspaces (lp,τ) and (rp,τ) are maximal r-closed. thus, if p is a point of first countability of (x,τ), then it is also of first countability in both (lp,τ) and (rp,τ) and so the result is an immediate consequence of remark 3.10. suppose then that χ(p,x) > ω, say χ(p,lp) = κ > ω (where κ is a regular uncountable cardinal); in the sequel we consider only the subspace lp. let regular topologies on a set 9 v ∈ σ be such that p ∈ v ⊆ clσ(v ) ⊆ u, then clearly, either, v = {p} or v \ {p} is a cofinal σ-closed subset of lp \ {p}. if the former occurs, then clearly lp \ {p} is open and closed in (lp,σ) which then cannot be r-closed. if v \ {p} is cofinal in lp \ {p} then, inductively we may construct interpolating sequences {vn : n ∈ ω} ⊆ v \ {p} and {wn : n ∈ ω} ⊆ lp \ u such that wn < vn < wn+1 for all n ∈ ω. since (x,<) is complete, q = sup{vn : n ∈ ω} = sup{wn : n ∈ ω} exists. now for each n ∈ ω, let on = v ∩ (wn,wn+1). the sets {on : n ∈ ω} are σ-open and their only possible accumulation point in (x,σ) is q. there are now two possibilities: 1) if q ∈ clσ({wn : n ∈ ω}), then q ∈ lp \ u and so q is not an accumulation point in (x,σ) of the family {on : n ∈ ω}, showing that (x,σ) is not feebly compact and hence not r-closed. 2) if on the other hand, q 6∈ clσ({wn : n ∈ ω}), then {wn : n ∈ ω}) is closed and discrete in (x,σ). since σ is regular, we may construct a discrete family of σ-open sets {wn : n ∈ ω} such that wn ∈ wn, again showing that (x,σ) is not feebly compact. � the same proof essentially shows that: theorem 4.8. if (x,τ,<) is a lots and χ(p,lp) > ω, then p is a maximal regular point of lp. corollary 4.9. a compact lots is lower in σ3 if and only if it is not first countable. proof. the sufficiency follows from theorem 4.8 and corollary 4.2. the necessity was proved in theorem 2.13 of [1]. � compactness is essential in the previous theorem. it is straightforward to show that the one-point lindelofication of a discrete space of cardinality ω1 is a lots but is neither first countable nor lower in σ3. from theorem 4.8 we see that if κ is an uncountable regular cardinal, then κ is a maximal regular point of κ + 1 (with the order topology). example 4.10. let κ denote the first ordinal of cardinality c+ and let x denote the set (κ + 1) × [0, 1], τ the product topology on x and σ the topology generated by τ ∪ {(κ, 1)}. we will show that σ = τ+. to this end, suppose that µ is a regular topology such that τ µ ⊆ σ; clearly µ differs from τ and σ only at the point (κ, 1) and hence there is some open µ-neighbourhood v which is not a τ-neighbourhood of (κ, 1) and some µ-neighbourhood u of (κ, 1) such that clµ(u) ⊆ v . since κ > c, there are a number of possibilities: 1) there is an infinite set j = {rn : n ∈ ω} ⊆ [0, 1) with 1 ∈ cl(j) and for each n ∈ ω a set sn ⊆ κ such that either, a) sn is cofinal in κ or b) κ ∈ sn and ⋃ {sn × {rn} : n ∈ j} ∩ v = ∅. or, 2) there is a cofinal set sω ⊂ κ such that (sω × {1}) ∩ v = ∅; furthermore, since v \ {(κ, 1)} is τ-open, we may assume that sω is τ-closed in κ. 10 o. t. alas and r. g. wilson if 1a) occurs, then {κ} × j ⊆ x \ v ⊆ x \ clµ(u); and if 1b) occurs, then since v \ {(κ, 1)} is τ-open, it follows that {κ} × j ⊆ x \ v ⊆ x \ clµ(u). thus in either case 1a) or 1b), there is an infinite subset j ⊆ [0, 1) with 1 ∈ cl(j) such that {κ} × j ⊆ x \ v ⊆ x \ clµ(u). it then follows that for each rn ∈ j there is αn ∈ κ such that ⋃ {(αn,κ] × {rn} : n ∈ j} ⊆ x \ clµ(u). letting α = sup{αn : n ∈ j} ∈ κ we have that (α,κ] × j ⊆ x \ clµ(u) and so (α,κ) × {1} ⊆ x \ u. again using regularity of (x,µ), there is some µ-open neighbourhood w of (κ, 1) such that clµ(w) ⊆ u and hence clµ(w) ∩ ((α,κ) × {1}) = ∅. if on the other hand, 2) occurs, then since clµ(u) is also τ-closed and it follows that clµ(u) ∩ (κ×{1}) is a τ-closed subset of κ×{1}). thus, since κ is a regular cardinal with uncountable cofinality and clµ(u) ∩ sω = ∅, it follows that there is some α ∈ κ such that clµ(u) ∩ ((α,κ) × {1}) = ∅. thus in both cases 1) and 2) we have shown that there is a µ-open neighbourhood o of (κ, 1) and α ∈ κ such that clµ(o) ∩ ((α,κ) × {1}) = ∅. now, since 1 ∈ cl(j), it follows that {(rn, 1] : n ∈ j} is a local base at 1 and so for each α < γ ∈ κ, there is rnγ ∈ j and oγ open in κ such that oγ × (rnγ , 1] ⊆ x \ clµ(o). now denoting by ln the set {γ : nγ = n ∈ j} and by mn the set ⋃ {oγ : γ ∈ ln} we have that for each n ∈ j, mn × (rn, 1] ⊆ x\clµ(o). however, ⋃ {mn : n ∈ j} ⊇ (α,κ) and hence there is a finite subset {mn1, . . . ,mnk } which covers (α,κ). letting r = max{rn1, . . . ,rnk }, we have that (α,κ) × (r, 1] ⊆ s \ clµ(o) and hence o ∩ ((α,κ + 1] × (r, 1]) ⊆ {κ} × [0, 1]. since o∩(x\{(κ, 1)}) is τ-open this shows that o∩((α,κ+1]×(r, 1]) = {(κ, 1)}, that is to say, (κ, 1) is an isolated point of (x,µ). of course, for each r ∈ [0, 1], the same argument applies to the point (κ,r) ∈ x. thus each point of x is either a maximal regular point or a point of first countability; it follows that (κ + 1) × [0, 1] is maximal r-closed and is lower in σ3. now let l denote the ordered set (κ+ 1) ⊕ ω−1 (that is to say, κ+ 1 with its usual ordering followed by ω with its reverse ordering, with the order topology) and y = l×[0, 1] with the product topology τ. the space y is the product of two lots, is not first countable and contains x as a closed subspace. nonetheless, we claim that y is not lower in σ3. to see this suppose that τ σ and that τ and σ differ at precisely one point p ∈ y . by theorem 2.13 of [1], p is not a point of first countability, hence p = (κ,r) ∈ {κ} × [0, 1]. clearly the neighbourhood filter vσp of p in (y,σ) must differ from that in (y,τ), v τ p , either on the subset (κ + 1) × [0, 1] or on y \ (κ × [0, 1]). suppose then that the traces of vσp and v τ p on (κ + 1) × [0, 1] are the same; then v σ p and v τ p differ on z = y \ (κ × [0, 1]), however, (z,τ) is first countable and hence again by theorem 2.13 of [1] there are t3-topologies on it lying strictly between τ and σ. thus τ and σ differ on (κ + 1) × [0, 1] and so by what we showed above, p must be an isolated point of ((κ + 1) × [0, 1],σ) and hence also of ({κ} × [0, 1],σ). however, the topology on y \ (κ × [0, 1]) obtained by declaring {κ} × ([0, 1] \ {r}) to be closed is not regular, and an argument similar to that regular topologies on a set 11 employed in theorem 2.13 of [1] shows that there is no topology, minimal in the class of regular topologies larger than it. with a little more work, using the fact that [0, 1] is second countable, it is possible to substitute ω1 instead of κ in the previous example. however the following questions remain open. question 4.11. if a regular topology is lower does some closed subspace have a maximal regular point? question 4.12. is there an internal concrete characterization of lower topologies in σ3? 5. first countable regular topologies denote by σ′3(x) the partially ordered set of first countable t3-topologies on a set x. theorem 5.1. there are no jumps in σ′3(x); between any two first countable t3-topologies on x there are at least c incomparable first countable t3topologies. proof. suppose that ξ and τ are two first countable t3-topologies on x which differ precisely at the point x ∈ x, let {vn : n ∈ ω} and {wn : n ∈ ω} be nested local bases at x in the topologies ξ and τ respectively. we may now choose a sequence {xm}m∈ω which converges to x in (x,ξ) but not in (x,τ) and by passing to a subsequence if necessary, we may assume that xm ∈ vm and {xm : m ∈ ω} is a closed, discrete subset of (x,τ). for each m ∈ ω, let {unm : n ∈ ω} be a local base of τ-open sets at xm such that x 6∈ clτ (u n+1 m ) ⊆ unm ⊆ vm for each m,n ∈ ω; since (x,τ) is regular, we may assume that {u1m : m ∈ ω} is a discrete family of τ-open sets. note that each set u n m is ξ-open and for each n ∈ ω, the family {unm : m ∈ ω} has x as its unique accumulation point in (x,ξ). now let a be an almost disjoint family of subsets of ω of size c and for each a ∈ a we define fa = {u ∈ τ : if x ∈ u then there is n ∈ ω and some finite f ⊆ ω such that u ⊇ ⋃ {unm : m ∈ a \ f}}. it is clear that this is a sub-base for a first countable topology µa ⊆ τ on x and since {xm}m∈a converges to x in (x,µa) it follows that µa 6= τ. furthermore, since unm ⊆ vm for each m,n ∈ ω, it follows that ξ ⊆ µa and since {xm}m∈ω\a does not converge to x in (x,µa) it follows that µa 6= ξ. finally, note that if a,b ∈ a are distinct, then µa and µb are incomparable topologies. finally, we need to show that each topology µa is regular. to this end, suppose that x ∈ u ∈ µa; then there is some finite set f ⊆ ω such that u ⊇ ⋃ {unm : m ∈ a\f}}. it follows that ⋃ {clτ (u n+1 m ) : m ∈ a\f} is a µa-closed neighbourhood of x which is contained in u. if x 6= z ∈ u ∈ τ, then there is some τ-closed neighbourhood w ⊆ u of z and some n ∈ ω such that w ∩ ⋃ {unm : m ∈ ω} = ∅ and hence w is a µa-closed neighbourhood of z contained in u. thus (x,µa) is regular. � 12 o. t. alas and r. g. wilson in theorem 2.13 of [1] it was shown that a sequential t3-topology of countable pseudocharacter is not a lower topology in σ3. however, we do not know the answer to the following question: question 5.2. is every first countable t3-topology which is not r-minimal, upper in σ3? 6. some more open problems the supremum of a chain of regular topologies is regular. thus a positive answer to the first question would imply a positive answer to the second. question 6.1. is the supremum of a chain of r-closed topologies r-closed? question 6.2. is every r-closed topology contained in a maximal r-closed topology ? note: there are maximal r-closed topologies which are not compact. in [15], stephenson gave an example under ch of a first countable non-compact rclosed topology by corollary 3.9, this must be maximal r-closed. in [9] it was shown that the same construction can be done in zfc. this space is scattered and has dispersion order 3. the topology contains a weaker compact hausdorff topology of dispersion order 3 (which is clearly not maximal r-closed). question 6.3. is a maximal r-closed topology which is not r-minimal, upper in σ3? stephenson’s examples show that maximal r-closed topologies need not be lower. finally, the most general question of all: question 6.4. is every regular topology which is not r-minimal an upper topology in σ3? acknowledgements. research supported by programa integral de fortalecimiento institucional (pifi), grant no. 34536-55 (méxico) and fundação de amparo a pesquisa do estado de são paulo (brasil). the second author wishes to thank the departament de matemàtiques de la universitat jaume i for support from pla 2009 de promoció de la investigació, fundació bancaixa, castelló, during the preparation of the final version of this paper. regular topologies on a set 13 references [1] o. t. alas, s. hernández, m. sanchis, m. g. tkachenko and r. g. wilson, adjacency in the partial orders of tychonoff, regular and locally compact topologies, acta math. hungar. 112, no. 3 (2006), 2005–2025. 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(received october 2008 – accepted september 2009) o. t. alas (alas@ime.usp.br) instituto de matemática e estat́ıstica, universidade de são paulo, caixa postal 66281, 05311-970 são paulo, brasil. r. g. wilson (rgw@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, unidad iztapalapa, avenida san rafael atlixco, #186, apartado postal 55-532, 09340, méxico, d.f., méxico. the structure of the poset of regular topologies on a set. by o. t. alas and r. g. wilson @ appl. gen. topol. 23, no. 2 (2022), 363-376 doi:10.4995/agt.2022.17418 © agt, upv, 2022 fixed point theorems for f-contraction mapping in complete rectangular m-metric space mohammad asim a , samad mujahid b and izhar uddin b a department of mathematics, faculty of science, shree guru gobind singh tricentenary university, gurugram, haryana, india. (mailtoasim27@gmail.com) b department of mathematics, jamia millia islamia, new delhi-110025, india. (mujahidsamad721@gmail.com, izharuddin1@jmi.ac.in) communicated by i. altun abstract in this paper, we prove a fixed point result for f -contraction principle in the framework of rectangular m-metric space. an example is also adopted to exhibit the utility of our result. finally, we apply our fixed point result to show the existence of solution of fredholm integral equation. 2020 msc: 47h10; 54e50. keywords: fixed point; f -contraction; rectangular m-metric space; integral equation. 1. introduction fixed point theory is an important and very active branch of functional analysis. it provides essential tools for solving problems arising in various branches of mathematical analysis. it guarantees the uniqueness and existence of the solution of integral and differential equations. in 1922, a polish mathematician stefan banach gives a contraction principle [10], which is one of the most wellknown and important discovery in mathematics. received 31 march 2022 – accepted 22 august 2022 http://dx.doi.org/10.4995/agt.2022.17418 https://orcid.org/0000-0002-2209-4488 https://orcid.org/0000-0002-5650-0296 https://orcid.org/0000-0002-2424-2798 m. asim, s. mujahid and i. uddin in the literature, there are two ways to generalized the banach contraction principle either change the contraction condition or alter the metric space. in fixed point theory several contractions defined in metric space such that boyd and wong’s nonlinear contraction principle [11], meir-keeler contraction [20, 1, 6], suzuki contraction [33], kannan contraction [17], ćirić generalized contraction [14], ćirić’s quasi contraction [15], weak-contraction [29], chatterjea contraction [13], zamfirescu contraction [35] and f-suzuki contraction [27] and many more [9, 25]. in 2012, wardowski [34] introduced a new type of contraction for real-valued mapping f defined on positive real numbers and satisfying some conditions and obtained a fixed point theorem for it. after that several authors have worked on f-contraction mapping in different metric space. in 2014, kumam and piri [27] applied weaker condition on self map and extended the result of wardowski [34]. in 2014, minak et al. [21] obtained result for generalized f-contractions including ćirić type generalized f-contraction and almost fcontraction on complete metric space. in 2017, kumam et al. [28] introduced the f-contraction in the setting of complete asymmetric metric spaces and extend several results. in 2018, kadelburg and radenović [16] obtained the result on concerning f-contraction in b-metric space. in 2019, luambano et al. [18] introduced the fixed point theorem for f-contraction in partial metric space and obtained certain results for it with suitable examples and many more [31, 30, 23, 22]. in 2014, asadi et al. [4] introduced m-metric space, which extends the p-metric space given by matthews [19] and proved the banach contraction principle for it. several authors have worked in this metric space [23, 3, 24, 2]. in 2000, branciari [12] introduced rectangular metric space which is the another generalization of metric space. in 2018, özg̈ur et al. [26] introduced rectangular m-metric space. they were inspired by the work of branciari [12] and shukla [32], who defined partial rectangular metric spaces which is the generalization of rectangular metric space. in 2019, asim et al. [5, 7, 8] generalized the rectangular m-metric space as rectangular mb-metric space, extended rectangular mrξ-metric space and mνmetric space. in this article, we establish the fixed point theorem for f-contraction in rectangular mmetric space. throughout the article r is the set of all real numbers, r+ is the set of all positive real numbers and n is the set of all natural numbers. 2. preliminaries in this section, we collect some basic notions, definitions, examples and auxiliary results. in 2000, branciari [12] introduced rectangular metric space. the definition is as follows: © agt, upv, 2022 appl. gen. topol. 23, no. 2 364 fixed point theorems for f-contraction mapping in complete rectangular m-metric space definition 2.1 ([12]). let x be a non-empty set. a function r : x×x → r+ is said to be a rectangular metric on x, if it satisfies the following (for all x,y ∈ x and for all distinct point u,v ∈ x \{x,y}): (1) r(x,y) = 0, if and only if x = y, (2) r(x,y) = r(y,x) and (3) r(x,y) ≤ r(x,u) + r(u,v) + r(v,y). then, the pair (x,r) is called a rectangular metric space. after that, shukla [32] introduced partial rectangular metric space. the definition is as follows: definition 2.2 ([32]). let x be a non-empty set. a function ρ : x×x → r+ is said to be a partial rectangular metric on x, if it satisfies the following conditions (for any x,y ∈ x and for all distinct point u,v ∈ x \{x,y}): (1) x = y if and only if ρ(x,y) = ρ(x,x) = ρ(y,y), (2) ρ(x,x) ≤ ρ(x,y), (3) ρ(x,y) = ρ(y,x) and (4) ρ(x,y) ≤ ρ(x,u) + ρ(u,v) + ρ(v,y) −ρ(u,u) −ρ(v,v). then, the pair (x,ρ) is called a partial rectangular metric space. in 2014, asadi et al. [4] generalized the partial metric space to m-metric space and obtained certain theorems related to m-metric space. notation: the following notations are useful in the sequel: (i) mxy := m(x,x) ∨m(y,y) = min{m(x,x),m(y,y)} and (ii) mxy := m(x,x) ∧m(y,y) = max{m(x,x),m(y,y)}. definition 2.3 ([4]). let x be a non-empty set. a function m : x×x → r+ is called a m-metric, if it satisfying the following conditions: (1) m(x,x) = m(y,y) = m(x,y) ⇐⇒ x = y, (2) mxy ≤ m(x,y), (3) m(x,y) = m(y,x) and (4) (m(x,y) −mxy) ≤ (m(x,z) −mxz) + (m(z,y) −mzy). then, the pair (x,m) is called an m-metric space. in 2018, özg̈ur et al. [26] introduced rectangular m-metric space and definition are as follows: notation: the following notations are useful in the sequel: (i) mrxy := mr(x,x) ∨mr(y,y) = min{mr(x,x),mr(y,y)} and (ii) mrxy := mr(x,x) ∧mr(y,y) = max{mr(x,x),mr(y,y)}. definition 2.4 ([26]). let x be a non-empty set. a function mr : x×x → r+ is called mr-metric, if it satisfying the following conditions: (rm1) mr(x,x) = mr(y,y) = mr(x,y) ⇐⇒ x = y, (rm2) mrxy ≤ mr(x,y), (rm3) mr(x,y) = mr(y,x) and (rm4) (mr(x,y)−mrxy ) ≤ (mr(x,u)−mrxu) + (mr(u,v)−mruv ) + (mr(v,y)− mrvy ) for all u,v ∈ x/{x,y}. then, the pair (x,mr) is called an rectangular m-metric space. © agt, upv, 2022 appl. gen. topol. 23, no. 2 365 m. asim, s. mujahid and i. uddin example 2.5 ([26]). let mr be an mr-metric. put (i) mwr (x,y) = mr(x,y) − 2mrxy + mrxy (ii) msr(x,y) = mr(x,y) −mrxy when x 6= y and msr(x,y) = 0 if x = y. then, mwr and m s r are ordinary metrics. definition 2.6 ([26]). let (x,mr) be an rectangular mmetric space. then, (1) a sequence {xn} in x converges to a point x, if and only if (2.1) lim n→∞ (mr(xn,x) −mrxn,x) = 0. (2) a sequence {xn} in x is said to be mr-cauchy sequence, if and only if (2.2) lim n,m→∞ (mr(xn,xm) −mrxn,xm ) and limn,m→∞(mr(xn,xm) −mrxn,xm ) exist and finite. (3) an rectangular m-metric space is said to be mr-complete, if every mrcauchy sequence {xn} converges to a point x such that (2.3) lim n→∞ (mr(xn,x) −mrxn,x) = 0 and limn→∞(mr(xn,x) −mrxn,x) = 0. lemma 2.7 ([26]). let (x,mr) be a rectangular m-metric space. then, (1) {xn} is an mr-cauchy sequence in (x,mr) if and only if it is a cauchy sequence in the metric space (x,mwr ). (2) (x,mr) is mr-complete if and only if the metric space (x,m w r ) is complete. furthermore, lim n→∞ (mwr (xn,x) = 0 ⇔ lim n→∞ (mr(xn,x)−mrxn,x) = 0, limn→∞(mr(xn,x)−mrxn,x) = 0. likewise the above definition holds also for msr. lemma 2.8 ([26]). assume that xn → x as n → ∞ in an rectangular mmetric space (x,mr). then, lim n→∞ (mr(xn,y) −mrxn,y ) = mr(x,y) −mrx,y,∀ y ∈ x. lemma 2.9 ([26]). assume that xn → x and yn → y as n → ∞ in an rectangular m-metric space (x,mr). then, lim n→∞ (mr(xn,yn) −mrxn,yn ) = mr(x,y) −mrx,y. lemma 2.10. [26] assume that xn → x and yn → y as n → ∞ in an rectangular m-metric space (x,mr). then, mr(x,y) = mrx,y . further if mr(x,x) = mr(y,y), then x = y. lemma 2.11 ([26]). let {xn} be a sequence in an rectangular m-metric space (x,mr), such that there exists k ∈ [0, 1) such that (2.4) mr(xn+1,xn) ≤ kmr(xn,xn−1) for all n ∈ n. © agt, upv, 2022 appl. gen. topol. 23, no. 2 366 fixed point theorems for f-contraction mapping in complete rectangular m-metric space then, (a) lim n→∞ mr(xn,xn−1) = 0, (b) lim n→∞ mr(xn,xn) = 0, (c) lim n,m→∞ mrxn,xm = 0 and (d) {xn} is an mr-cauchy sequence. proof. [26] using the definition of convergence and inequality (2.4), the proof of the condition (a) follows easily. from the condition (rm2) and the condition (a), we get lim n→∞ min{mr(xn,xn),mr(xn−1,xn−1)} = lim n→∞ mrxnxn−1 ≤ lim n→∞ mr(xn,xn−1) = 0. therefore, the condition (b) holds. since lim n→∞ mr(xn,xn) = 0, the condition (c) holds. using the previous conditions and the definition (2.6), we see that the condition (d) holds. � in 2012, wardowski [34] introduced f-contraction and the definition are as follows: definition 2.12 ([34]). let f : r+ → r be a mapping satisfying: (f1) f is strictly increasing, i.e. for all α,β ∈ r+ such that α < β,f(α) < f(β), (f2) for each sequence {αn}n∈n of positive numbers lim n→∞ αn = 0 if and only if lim n→∞ f(αn) = −∞, (f3) there exists k ∈ (0, 1) such that lim α→0+ αkf(α) = 0. denote 4f by the collection of all those functions which satisfy the conditions (f1-f3). a mapping t : x → x is said to be an f-contraction if there exists τ > 0 such that (2.5) d(tx,ty) > 0 ⇒ τ + f(d(tx,ty)) ≤ d(x,y) ∀ x,y ∈ x. some examples related to f-contraction [34] are: example 2.13. let f : r+ → r be given by the formula f(α) = ln(α), it is clear that f satisfies (f1)-(f3) ((f3) for any k ∈ (0, 1)). example 2.14. let f : r+ → r be given by the formula f(α) = ln(α)+α,α > 0. then, f satisfies (f1)-(f3). example 2.15. let f : r+ → r be given by the formula f(α) = − 1√α, α > 0. then, f satisfies (f1)-(f3) ((f3) for any k ∈ (1/2, 1)). © agt, upv, 2022 appl. gen. topol. 23, no. 2 367 m. asim, s. mujahid and i. uddin 3. main results the following definition is new version of the f-contraction for a rectangular m-metric space. definition 3.1. let (x,mr) be an rectangular m-metric space. the mapping t : x → x is said to be an f-contraction on x, if there exists τ > 0 and f ∈4f such that ∀ x,y ∈ x (3.1) mr(tx,ty) > 0 ⇒ τ + f(mr(tx,ty)) ≤ f(mr(x,y)). theorem 3.2. let (x,mr) be a complete rectangular m-metric space and let t : x → x be a continuous f -contraction. then, t has a unique fixed point x∗ ∈ x and for every x0 ∈ x a sequence {tn(x0)}n∈n is convergent to x∗. proof. let x0 ∈ x be arbitrary and fixed. we define a sequence {xn}n∈n ⊂ x, xn+1 = txn, n = 0, 1, · · · . denote ξn = mr(xn,xn+1) − mrxn,xn+1 , n = 0, 1, · · · , if there exist n0 ∈ n for which xn0+1 = xn0. then, txn0 = xn0 and the proof is finished. suppose that xn+1 6= xn for every n ∈ n. then ξn > 0, for all n ∈ n. using(3.1), then following holds for every n ∈ n. (3.2) f(ξn) ≤ f(ξn−1) − τ ≤ f(ξn−2) − 2τ ≤ ... ≤ f(ξ0) −nτ from (3.2) we obtain lim n→∞ f(ξn) = −∞ that together with condition (f2) gives (3.3) lim n→∞ mr(xn,xn+1) −mrxn+1,xn = 0. we shall prove that lim n→∞ mr(xn,xn+2) = 0. we assume that xn 6= xm for every n,m ∈ n,n 6= m. indeed, suppose that xn = xm for some n = m + k with k > 0. f(mr(xm,xm+1) −mrxm,xm+1 ) = f(mr(xn,xn+1) −mrxn,xn+1 ) = f(mr(xm+k,xm+k+1) −mrxm+k,xm+k+1 ) ≤ f(mr(xm,xm+1) −mrxm,xm+1 )) −kτ < f(mr(xm,xm+1) −mrxm,xm+1 )) a contradiction. therefore, mr(xn,xm)−mrxn,xm > 0 for every n,m ∈ n with n 6= m. (3.4) τ + f(mr(xn,xn+2)) ≤ f(mr(xn−1,xn+1)), ∀ n ∈ n. hence (3.5) f(mr(xn,xn+2)) ≤ f(mr(xn−1,xn+1)) − τ © agt, upv, 2022 appl. gen. topol. 23, no. 2 368 fixed point theorems for f-contraction mapping in complete rectangular m-metric space f(mr(xn,xn+2)) ≤ f(mr(xn−2,xn)) − 2τ ≤ ···≤ f(mr(x0,x2)) −nτ. taking limit as n →∞ in above inequality, we get lim n→∞ f(mr(xn,xn+2)) = −∞. then, from the condition (f2) of definition (2.12), we conclude that (3.6) lim n→∞ (mr(xn,xn+2)) = 0. next, we shall show that {xn}n∈n is a mr-cauchy sequence, that is lim n,m→∞ (mr(xn,xm)) −mrxn,xm = 0, n,m ∈ n. now, from definition (2.12) there exist k ∈ (0, 1) such that lim n→∞ mr(xn,xn+1) k f(mr(xn,xn+1)) = 0. f(mr(xn,xn+1)) ≤ f(mr(x0,x1)) −nτ. we have mr(xn,xn+1) k (f(mr(xn,xn+1))−f(mr(x0,x1))) ≤ mr(xn,xn+1) k (f(mr(x0,x1))−nτ) (3.7) mr(xn,xn+1) k (f(mr(xn,xn+1)) −f(mr(x0,x1))) ≤−mr(xn,xn+1) k nτ ≤ 0. taking limit as n →∞ in above inequality, we conclude that lim n→∞ mr(xn,xn+1) k nτ = 0. then, there exist n1 ∈ n such that nmr(xn,xn+1) k ≤ 1, ∀ n ≥ n1 mr(xn,xn+1) ≤ 1 n1/k , ∀ n ≥ n1. now, from definition (2.12) there exists k ∈ (0, 1) such that (3.8) lim n→∞ (mr(xn,xn+2)) kf(mr(xn,xn+2)) = 0. since f(mr(xn,xn+2)) ≤ f(mr(x0,x2)) −nτ. we have (mr(xn,xn+2)) kf(mr(xn,xn+2)) ≤ (mr(xn,xn+2))k(f(mr(x0,x2)) −nτ) (mr(xn,xn+2)) k(f(mr(xn,xn+2)) −f(mr(x0,x2))) ≤−nτ(mr(xn,xn+2))k (mr(xn,xn+2)) ≤ 1n1/k . next, we show that {xn} is mr-cauchy sequence, that is lim n→∞ (mr(xn,xn+p) −mrxn,xn+p ) = 0, ∀ p ∈ n. © agt, upv, 2022 appl. gen. topol. 23, no. 2 369 m. asim, s. mujahid and i. uddin the cases p = 1 and p = 2 are proved respectively by (3.3) and (3.4). now, we take p ≥ 3. it is sufficient to examine two cases. case 1. firstly, let p is odd that is p = 2m + 1 for any m ≥ 1,n ∈ n. from the condition (rm4) of definition of the mr-metric, we get mr(xn,xn+p) = mr(xn,xn+2m+1) (mr(xn,xn+p) −mrxn,xn+p ) ≤ (mr(xn,xn+1) −mrxn,xn+1 ) +(mr(xn+1,xn+2) −mrxn+1,xn+2 ) +(mr(xn+2,xn+p) −mrxn+2,xn+p ) (mr(xn,xn+2m+1) −mrxn,xn+2m+1 ) ≤ (mr(xn,xn+1) −mrxn,xn+1 ) +(mr(xn+1,xn+2) −mrxn+1,xn+2 ) +(mr(xn+2,xn+2m+1) −mrxn+2,xn+2m+1 ) ≤ (mr(xn,xn+1) −mrxn,xn+1 ) +(mr(xn+1,xn+2) −mrxn+1,xn+2 ) +(mr(xn+2,xn+3) −mrxn+2,xn+3 ) + ... +(mr(xn+2m,xn+2m+1) −mrxn+2m,xn+2m+1 ) ≤ [(mr(xn,xn+1) −mrxn,xn+1 ) +(mr(xn+2,xn+3) −mrxn+2,xn+3 ) + ... +(mr(xn+2m−2,xn+2m−1) −mrxn+2m−2,xn+2m−1 )] +[(mr(xn+1,xn+2) −mrxn+1,xn+2 ) + ... +(mr(xn+2m−1,xn+2m) −mrxn+2m−1,xn+2m ) +(mr(xn+2m,xn+2m+1) −mrxn+2m,xn+2m+1 )] (mr(xn,xn+p) −mrxn,xn+p ) ≤ n+p−1∑ i=n (mr(xi,xi+1) −mrxi,xi+1 ) ≤ ∞∑ i=n (mr(xi,xi+1) −mrxi,xi+1 ) ≤ ∞∑ i=n 1 i1/k . © agt, upv, 2022 appl. gen. topol. 23, no. 2 370 fixed point theorems for f-contraction mapping in complete rectangular m-metric space from the above from the convergence of the series n+p−1∑ i=n 1 i1/k ⇒ lim n→∞ (mr(xn,xn+p) −mrxn,xn+p ) = 0. case 2. secondly, assume p is even that is p = 2m for any m ≥ 1,n ∈ n. from the condition (rm4) of definition of the mr-metric, we get (mr(xn,xn+p) −mrxn,xn+p ) ≤ (mr(xn,xn+2) −mrxn,xn+2 ) +(mr(xn+2,xn+3) −mrxn+2,xn+3 ) +(mr(xn+3,xn+2m) −mrxn+3,xn+2m ) (mr(xn,xn+2m) −mrxn,xn+2m ) ≤ (mr(xn,xn+2) −mrxn,xn+2 ) +(mr(xn+2,xn+3) −mrxn+2,xn+3 ) +(mr(xn+3,xn+4) −mrxn+3,xn+4 ) + ... +(mr(xn+2m−3,xn+2m−2) −mrxn+2m−3,xn+2m−2 ) +(mr(xn+2m−2,xn+2m−1) −mrxn+2m−2,xn+2m−1 ) +(mr(xn+2m−1,xn+2m) −mrxn+2m−1,xn+2m ) ≤ (mr(xn,xn+2) −mrxn,xn+2 ) +(mr(xn+2,xn+3) −mrxn+2,xn+3 ) +(mr(xn+3,xn+4) −mrxn+3,xn+4 ) +(mr(xn+4,xn+5) −mrxn+4,xn+5 ) + ... +(mr(xn+2m−2,xn+2m−1) −mrxn+2m−2,xn+2m−1 ) +(mr(xn+2m−1,xn+2m) −mrxn+2m−1,xn+2m ) ≤ (mr(xn,xn+2) −mrxn,xn+2 ) + n+2m−1∑ i=n+2 (mr(xi,xi+1) −mrxi,xi+1 ) ≤ (mr(xn,xn+2) −mrxn,xn+2 ) + n+p−1∑ i=n+2 (mr(xi,xi+1) −mrxi,xi+1 ) ≤ (mr(xn,xn+2) −mrxn,xn+2 ) + ∞∑ i=n+2 (mr(xi,xi+1) −mrxi,xi+1 ) ≤ (mr(xn,xn+2) −mrxn,xn+2 ) + ∞∑ i=n+2 1 i1/k ≤ 1 n1/k + ∞∑ i=n+2 1 i1/k . © agt, upv, 2022 appl. gen. topol. 23, no. 2 371 m. asim, s. mujahid and i. uddin from the above from the convergence of the series n+2m−1∑ i=n+2 1 i1/k ⇒ lim n→∞ (mr(xn,xn+p) −mrxn,xn+p ) = 0. by lemma (2.9), we obtain that for any n,m ∈ n, msr(xn,xm) = mr(xn,xm) −mrxn,xm → 0 as n →∞. this implies that {xn}n∈n is a mr-cauchy sequence with respect to msr and converges by lemma (2.10). thus, lim n,m→∞ msr(xn,xn+2m+1) = 0 and lim n,m→∞ msr(xn,xn+2m) = 0. we received by lemma (2.7) is that {xn} is an mr-cauchy sequence. from the completeness of x, there exist x∗ ∈ x such that lim n→∞ xn = x ∗. finally, the continuity of t yields (mr(tx ∗,x∗) −mrtx∗,x∗ ) = limn→∞(mr(txn,xn) −mrtxn,xn ) = lim n→∞ (mr(xn+1,xn) −mrxn+1,xn ) = 0. now, we show that the uniqueness of a fixed point of t . assume that t has two distinct fixed points x,y ∈ x, such that x = tx,y = ty. from the condition (3.1), we have f(mr(x,y)) = f(mr(tx,ty)) < τ + f(mr(tx,ty)) ≤ f(mr(x,y)), which is contradiction. hence, t has unique fixed point. � example: let x = [0, 1] and mr(x,y) = |x|+|y| 2 , for all x,y ∈ x. then, (x,mr) is complete rectangular m-metric space. define a mapping t : x → x such that t(x) = x 2 , for all x ∈ x. define the function f : r+ → r by f(r) = ln(r), for all x,y ∈ x such that mr(tx,ty) > 0 this implies that τ + f(mr(tx,ty)) = τ + ln( |x|+|y| 4 ). let τ ≤ ln 2. then τ + ln( |x| + |y| 4 ) ≤ ln 2 + ln( |x| + |y| 4 ) = f(mr(x,y)). thus, the contractive condition is satisfied for all x,y ∈ x. hence, all hypotheses of the theorem (3.2) are satisfied and t has a unique fixed point x = 0. © agt, upv, 2022 appl. gen. topol. 23, no. 2 372 fixed point theorems for f-contraction mapping in complete rectangular m-metric space 4. applications in this section, we apply theorem (3.2) to investigate the existence and uniqueness of solution of the fredholm integral equation [7]. let x = c([a,b],r) be the set of continuous real valued functions defined on [a,b]. now, we consider the following fredholm type integral equation: (4.1) x(p) = ∫ b a g(p,q,x(p))dq + h(p), for p,q ∈ [a,b] where g,h ∈ c([a,b],r). define mr : x ×x → r+ by (4.2) mr(x(p),y(p)) = sup p∈[a,b] (|x(p)| + |y(p)|) 2 , ∀ x,y ∈ x. then, (x,mr) is an mr-complete in rectangular m-metric space. theorem 4.1. suppose that there exist τ > 0 and for all x,y ∈ c([a,b],r) |g(p,q,x(p)) + g(p,q,y(p)) + 2h(p)| ≤ e−τ (b−a) |x(p) + y(p)|, ∀ p,q ∈ [a,b]. then, the integral equation (4.1) has a unique solution. proof. define t : x → x by, t(x(p)) = ∫ b a g(p,q,x(p))dq + h(p), ∀ p,q ∈ [a,b]. observe that existence of a fixed point of the operator t is equivalent to the existence of a solution of the integral equation (4.1). now, for all x,y ∈ x. we © agt, upv, 2022 appl. gen. topol. 23, no. 2 373 m. asim, s. mujahid and i. uddin have mr(tx,ty) = ∣∣∣∣t(x(p)) + t(y(p))2 ∣∣∣∣ = ∣∣∣∣∣ ∫ b a ( g(p,q,x(p)) + g(p,q,y(p)) + 2h(p) 2 )dq ∣∣∣∣∣ ≤ ∫ b a ∣∣∣∣(g(p,q,x(p)) + g(p,q,y(p)) + 2h(p)2 )dq ∣∣∣∣ ≤ e−τ (b−a) ∫ b a |x(p) + y(p)| 2 dq ≤ e−τ (b−a) ∫ b a |x(p)| + |y(p)| 2 dq ≤ e−τ (b−a) sup p∈[a,b] (|x(p)| + |y(p)|) 2 ( ∫ b a dq) ≤ e−τ (b−a) mr(x,y)(b−a) ≤ e−τmr(x,y). thus, the condition (3.1) is satisfied 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[35] t. zamfirescu, fixed point theorems in metric spaces, arch. math. 23 (1972), 292–298. © agt, upv, 2022 appl. gen. topol. 23, no. 2 376 () @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 95-100 some remarks on chaos in topological dynamics huoyun wang ∗ and heman fu abstract bau-sen du introduced a notion of chaos which is stronger than liyorke sensitivity. a tds (x, f) is called chaotic if there is a positive ε such that for any x and any nonempty open set v of x there is a point y in v such that the pair (x, y) is proximal but not ε-asymptotic. in this article, we show that a tds (t, f) is transitive but not mixing if and only if (t, f) is li-yorke sensitive but not chaotic, where t is a tree. moreover, we compare such chaos with other notions of chaos. 2010 msc: 37b05, 54h20, 37b20, 58k15. keywords: sensitivity, chaos, tree maps. 1. introduction throughout this paper a topological dynamical system is a pair (x, f) (tds for short), where x is a compact metric space with a metric d and f : x → x is a continuous surjective map. a tds (x, f) is nontrivial if x contains at least two points. chaotic behavior is a manifestation of the complexity of the dynamical system. now we recall some concepts of complexity. a tds (x, f) is sensitive [3], if there exists a positive ε such that for any x in x and any open neighborhood u of x, there exist y ∈ u and a positive integer n with d(fn(x), fn(y)) > ε. let ε be a positive number. a subset c in x is a li-yorke ε-scrambled set of a tds (x, f), if any pair (x, y) of distinct points x and y in c is proximal but not ε-asymptotic, that is, lim inf n→∞ d(fn(x), fn(y)) = 0 and lim sup n→∞ d(fn(x), fn(y)) > ε. ∗supported by national nature science funds of china (11071084), and guangzhou education bureau (08c016). 96 h. wang and h. fu a tds (x, f) is li-yorke ε-chaotic [5], if it has an uncountable li-yorke ε-scrambled set. in 2003, akin and kolyada [1] introduced the notion of li-yorke sensitivity which links the li-yorke version of chaos with sensitivity. a tds (x, f) is called li-yorke sensitive [1] if there is a positive ε such that every x in x is a limit of points y in x such that the pair (x, y) is proximal but not ε-asymptotic. what is the nature of chaos? various people have various understandings. in [4], bau-sen du believed that chaos should involve not only nearby points could diverge apart but also faraway points could get close to each other. therefore in 2006, he [4] proposed a new definition of chaos as follows, which is stronger than li-yorke sensitivity. a tds (x, f) is called chaotic [4] if there is a positive ε such that for any x and any nonempty open set v of x there is a point y in v such that the pair (x, y) is proximal but not ε-asymptotic. there is a tds (x, f) which is li-yorke sensitive but not chaotic (see [4, theorem 4]). the present article goes on studying the nature of chaos, and is written on basis of the preprint [4]. this article is organized as follows. in section 2, we investigate the chaos of transitive maps on trees. we show that a tds (t, f) is transitive but not mixing if and only if (t, f) is li-yorke sensitive but not chaotic, where t is a tree. finally, we compare the chaos with other notion of the same. 2. the chaos of transitive maps on trees in this section, the chaos of transitive maps on trees are investigated. by a tree we mean a connected compact one-dimensional polyhedron, which does not contain any subset homeomorphic to a circle and which contains a subset homeomorphic to an interval. let t be a tree. given point x ∈ t , we define the valence of x, val(x), as the number of connected components of t − {x}. each point of valence 1 is an endpoint of t . a subtree of a tree t is a subset of t , which is a tree itself. a tds (x, f) is (topologically) transitive if for any two nonempty open sets u and v there exists a positive integer n such that fn(u)∩v 6= ∅. a tds (x, f) is (topologically) weakly mixing if (x × x, f × f) is (topologically) transitive. a tds (x, f) is (topologically) totally transitive if (x, fn) is transitive for any positive integer n. a tds (x, f) is (topologically) mixing if for any two nonempty open sets u and v there exists a positive integer n such that fn(u) ∩ v 6= ∅ for any n ≥ n. a tds (x, f) is minimal if the set orb(x, f) = {fn(x) : n = 0, 1, 2, · · ·} is dense in x for any x of x. the main aim of this section is to prove the following result. some remarks on chaos in topological dynamics 97 theorem 2.1. let t be a tree and let (t, f) be transitive. then the following results hold. (1) (t, f) is mixing if and only if it is chaotic; (2) (t, f) is not mixing if and only if it is li-yorke sensitive but not chaotic. remark 2.2. theorem 2.1 may not be true for a general tds. let g be an irrational rotation of the unit circle s1, then (s1, g) is minimal and totally transitive but not li-yorke sensitive. we need the following lemmas which come from [8] and [2] respectively. lemma 2.3. let t be a tree and let (t, f) be transitive. then p(f) = t , where p(f) denotes the closure of the set of all periodic points of f. lemma 2.4. let t be a tree and let (t, f) be transitive. then exactly one of the following alternatives holds. (1) (t, f) is totally transitive. (2) there is a positive integer n0 such that there are an interior fixed point y and subtrees t1, t2, · · · , tn0 of t with ∪ n0 i=1ti = t , ti ∩ tj = {y} for i 6= j and f(ti) = ti+1(mod n0) for 1 ≤ i ≤ n0. moreover, (ti, f n0|ti) is transitive, 1 ≤ i ≤ n0. proposition 2.5. let f : t → t be a tree map. (t, f) is mixing if and only if it is totally transitive. proof. let us denote by e(t ) the set of endpoints of the tree t and suppose (t, f) is totally transitive. suppose that u and v are nonempty open connected subsets of t . we may assume that u is contained in t − e(t ). since (t, f) is transitive, then p(f) = t by lemma 2.3. let y be any periodic point in u which orbit orb(y, f) is contained in t − e(t ). let x be any periodic point in v . let m be a common multiple of the periods of x and y, and set g = fm. then every point of orb(y, f) ∪ {x} is a fixed point of g. let k = ∪∞n=0g n(v ). then k is a connected subset of t , since x is a fixed point of g. since (t, g) is transitive, then k is a dense connected subset of t . this implies k contains t − e(t ). for any u ∈ orb(y, f), there is an integer ku ≥ 0 such that u ∈ g ku(v ). let k = max{ku : u ∈ orb(y, f)}. since every point u of orb(y, f) is a fixed point of g, then fkm(v ) = gk(v ) which contains orb(y, f). thus fn(v ) contains point y for any n ≥ km. this implies fn(v ) ∩ u 6= ∅, hence (t, f) is mixing. conversely, it is obvious. � proposition 2.6. let t be a tree and let (t, f) be transitive. then exactly one of the following alternatives holds. (1) (t, f) is mixing. (2) there is a positive integer n0 such that there are an interior fixed point y and subtrees t1, t2, · · · , tn0 of t with ∪ n0 i=1ti = t , ti ∩ tj = {y} for 98 h. wang and h. fu i 6= j and f(ti) = ti+1(mod n0) for 1 ≤ i ≤ n0. moreover, (ti, f n0|ti) is mixing, 1 ≤ i ≤ n0. proof. suppose that (t, f) is transitive. if (t, f) is totally transitive, then (t, f) is mixing by proposition 2.5. otherwise, there is a positive integer n0 such that there are an interior fixed point y and subtrees t1, t2, · · · , tn0 of t with ∪n0i=1ti = t , ti ∩ tj = {y} for i 6= j and f(ti) = ti+1(mod n0) for 1 ≤ i ≤ n0. moreover, (ti, f n0|ti) is transitive, 1 ≤ i ≤ n0. then f n0|ti satisfies the condition (1) of lemma 2.4, since y is a fixed point of fn0|ti, and y is the endpoint of ti. hence (ti, f n0|ti) is mixing by lemma 2.4 and proposition 2.5. � by [1, theorem 3.4 and lemma 3.8], the following lemma holds. lemma 2.7. if a nontrivial tds (x, f) is weakly mixing, then it is chaotic. proof of theorem 2.1. if (t, f) is mixing, then (t, f) is chaotic by lemma 2.7. otherwise, by proposition 2.6 there are a positive integer n0 such that there is an interior fixed point y and subtrees t1, t2, · · · , tn0 of t with ∪ n0 i=1ti = t , ti ∩ tj = {y} for i 6= j and f(ti) = ti+1(mod n0) for 1 ≤ i ≤ n0. moreover, (ti, f n0|ti) is mixing, 1 ≤ i ≤ n0. since every (ti, f n0|ti) is chaotic, hence (t, f) is li-yorke sensitive. next we show that (t, f) is not chaotic. let x be a periodic point of f in the interior of t1. let v be an open subset of t which is contained in t2. since x and v are jumping alternatively and never get close to each other, then (t, f) is not chaotic. hence, theorem 2.1 holds. 2 3. comparison of various notions of chaos in this section x will denote a general compact metric space. below, we discuss the interrelations between the notions of chaos. a tds (x, f) is devaney’s chaotic [3] if it is transitive and the set of periodic points of f is dense in x. recall that a mycielski set is a countable union of cantor sets, while a cantor set is a set homeomorphic to the standard middle-third cantor set on the real line. the following lemmas come from [1] and [6] respectively. lemma 3.1. for a tds (x, f) the following conditions are equivalent. 1. (x, f) is sensitive. 2. there exists a positive ε such that {(x, y) ∈ x×x : lim supn→∞ d(f n(x), fn(y)) > ε} is a dense gδ set of x × x. lemma 3.2 (mycielski). let x be a separable complete metric space without isolated point. if for each natural number n, rn is a residual set of x n, then there is a mycielski set k of x such that (1) for any nonempty open set u of x, k ∩ u contains a nonempty perfect set; (2) for each natural number n, for all x1, x2, · · · , xn mutually distinct points in k, (x1, x2, · · · , xn) ∈ rn. some remarks on chaos in topological dynamics 99 theorem 3.3. if a tds (x, f) is chaotic, then there is a dense mycielski set k in x such that k is a li-yorke ε-scrambled set for some positive ε. hence, (x, f) is li-yorke ε-chaotic. proof. since (x, f) is chaotic, then (x, f) is sensitive, this implies x without isolated point. there is a positive ε such that c1(ε) := {(x, y) ∈ x × x : lim supn→∞ d(f n(x), fn(y)) > ε} is a dense gδ set of x × x, by lemma 3.1. moreover, c2 := {(x, y) ∈ x × x : lim infn→∞ d(f n(x), fn(y)) = 0} is a dense gδ set of x × x because (x, f) is chaotic. put r2 = c1(ε) ∩ c2. so theorem 3.3 holds by lemma 3.2. � remark 3.4. there is a tds with positive topological entropy, which is liyorke ε-chaotic and devaney’s chaotic but not chaotic. let f(x) = 1 2 + 2x if 0 ≤ x ≤ 1 4 ; f(x) = 3 2 − 2x if 1 4 ≤ x ≤ 1 2 ; f(x) = 1 − x if 1 2 ≤ x ≤ 1. then f : [0, 1] → [0, 1] is transitive but not mixing. since f is transitive, then the set of periodic points of f is dense in x. this implies that ([0, 1], f) is devaney’s chaotic. because f has positive topological entropy, then ([0, 1], f) is li-yorke ε-chaotic for some positive ε but not chaotic. remark 3.5. there is a tds which is chaotic with zero topological entropy. a minimal system (x, f) which is weakly mixing and has zero topological entropy has been built, such as in [7]. then the minimal system (x, f) is chaotic, and has zero topological entropy. remark 3.6. if a tds (x, f) is li-yorke sensitive, the product system (x × y, f × g) is li-yorke sensitive for any tds (y, g) (see [1, theorem 3.11]). however, for any tds (x, f) which is chaotic, there is a tds (y, g) such that the product system (x × y, f × g) is not chaotic. actually, let g be an irrational rotation of the unit circle s1. the product system (x × s1, f × g) is li-yorke sensitive but not chaotic for any chaotic tds (x, f). remark 3.7. there is a tds (x, σ) which is chaotic but not devaney’s chaotic. let ( ∑ 2, σ) be the full shift over two letters {0, 1}. let a1 = 1, a2 = 101, · · · , an+1 = an0 nan. then x = limn→∞ anan · · · ∈ ∑ 2. put x = orb(x, σ). then (x, σ) is mixing and has a fixed point 00 · · · which is the unique minimal subset of x (see [9]). (x, σ) is chaotic but not devaney’s chaotic since the set of periodic points of σ is not dense in x. there exists a tds which is chaotic but not transitive, a fortiori not weakly mixing(see [4, theorem 5]). however we have remark 3.8. if a tds (x, f) is minimal, then (x, f) is chaotic if and only if it is weakly mixing. if (x, f) is chaotic, then the proximal cell p(f)(x) is dense in x for any x of x, where p(f)(x) = {y ∈ x : lim infn→∞ d(f n(x), fn(y)) = 0}. hence (x, f) is weakly mixing by [1, theorem 3.7]. conversely, it is clear. 100 h. wang and h. fu below, we summarize the interrelations between the notions of chaos, see figure 1. positive entropy ; : chaos ; : devaney’s chaos weakly mixing ⇓ ⇑+minimal w 6t u 6v li-yorke sensitivity li-yorke ε-chaos figure 1. relations of various notions of chaos acknowledgements. the authors would like to thank the referee for the careful reading and many valuable comments. references [1] e. akin and s. kolyada, li-yorke sensitivity, nonlinearity 16 (2003), 1421–1433. [2] l. alseda, s. kolyada, j. llibre and l. snoha, entropy and periodic points for tree maps, trans. amer. math. soc. 351 (1997), 1551–1573. [3] r. devaney, chaotic dynamical systems, addison-wesley, redwood city, 1980. [4] b. du, on the nature of chaos, arxiv: math.ds/0602585 v1 26 feb 2006. [5] t. li and j. yorke, period 3 implies chaos, amer. math. monthly 82 (1975), 985–992. [6] j. mycielski, independent sets in topological algebras, fund. math. 55 (1964), 139–147. [7] l. wang, z. chen and g. liao, the complexity of a minimal sub-shift on symbolic spaces, j. math. anal. appl. 37 (2006), 136–145. [8] x. ye, the center and the depth of the center of a tree map, bull. austral. math. soc. 48 (1993), 347–350. [9] x. ye, w. huang and s. shao, an introduction to topolgical dynamics, science press, bejing, 2008. [chinese] (received september 2010 – accepted july 2011) huoyun wang (wanghuoyun@126.com) department of mathematics, guangzhou university, guangzhou, 510006, people’s republic of china heman fu (dbfhm@163.com) college of mathematics and information sciences, zhaoqing university, zhaoqing, 526061, people’s republic of china some remarks on chaos in topological dynamics. by h. wang and h. fu @ appl. gen. topol. 23, no. 2 (2022), 345-361 doi:10.4995/agt.2022.16332 © agt, upv, 2022 the zariski topology on the graded primary spectrum of a graded module over a graded commutative ring saif salam and khaldoun al-zoubi department of mathematics and statistics, jordan university of science and technology, p.o.box 3030, irbid 22110, jordan (smsalam19@sci.just.edu.jo, kfzoubi@just.edu.jo) communicated by p. das abstract let r be a g-graded ring and m be a g-graded r-module. we define the graded primary spectrum of m, denoted by psg(m), to be the set of all graded primary submodules q of m such that (grm (q) :r m) = gr((q :r m)). in this paper, we define a topology on psg(m) having the zariski topology on the graded prime spectrum specg(m) as a subspace topology, and investigate several topological properties of this topological space. 2020 msc: 13a02; 16w50. keywords: graded primary submodules; graded primary spectrum; zariski topology. 1. introduction and preliminaries let g be a multiplicative group with identity e and r be a commutative ring with identity. then r is called a g-graded ring if there exist additive subgroups rg of r indexed by the elements g ∈ g such that r = ⊕rg g∈g and rgrh ⊆ rgh for all g,h ∈ g. the elements of rg are called homogeneous of degree g. if r ∈ r, then r can be written uniquely as ∑ rg g∈g , where rg is the component of r in rg. the set of all homogeneous elements of r is denoted received 20 september 2021 – accepted 29 may 2022 http://dx.doi.org/10.4995/agt.2022.16332 https://orcid.org/0000-0002-1330-2556 https://orcid.org/0000-0001-6082-4480 s. salam and k. al-zoubi by h(r), i.e. h(r) = ⋃ rg g∈g . let r be a g-graded ring and i be an ideal of r. then i is called g-graded ideal of r if i = ⊕ g∈g (i ⋂ rg). by i �g r, we mean that i is a g-graded ideal of r, (see [13]). the graded radical of i is the set of all a = ∑ g∈g ag ∈ r such that for each g ∈ g there exists ng > 0 with a ng g ∈ i. by gr(i) (resp. √ i) we mean the graded radical (resp. the radical) of i, (see [18]). the graded prime spectrum specg(r) of a graded ring r consists of all graded prime ideals of r. it is known that specg(r) is a topological space whose closed sets are v rg (i) = {p ∈ specg(r) | i ⊆ p} for each graded ideal i of r (see, for example, [14, 16, 18]). let r be a g-graded ring and m a left r-module. then m is said to be a g-graded r-module if m = ⊕mg g∈g with rgmh ⊆ mgh for all g,h ∈ g, where mg is an additive subgroup of m for all g ∈ g. the elements of mg are called homogeneous of degree g. if x ∈ m, then x can be written uniquely as ∑ xg g∈g , where xg is the component of x in mg. the set of all homogeneous elements of m is denoted by h(m), i.e. h(m) = ⋃ mg g∈g . let m = ⊕mg g∈g be a g-graded r-module. a submodule n of m is called a g-graded r-submodule of m if n = ⊕ g∈g (n ⋂ mg). by n ≤g m (resp. n 0 the sets u1 := { b | |〈χa, χb − χa〉| < ε 2 } = { b | µ (a \ b) < ε 2 } and u2 := { b | |〈χ∁a, χb − χa〉| < ε 2 } = { b | µ (b \ a) < ε 2 } are weak neighborhoods of a such that u1 ∩ u2 ⊆ {b | µ (a△b) < ε}, hence the weak topology is finer than the group topology on y (µ). the last statement is obvious. � we observe that the set operations ∩ and ∁ as well as the measure function µ are continuous on y (µ). as an auxiliary object we denote by z(µ) ⊂ l2(µ) the set of all functions 0 ≤ f ≤ 1. this set inherits the norm topology and the weak topology from l2(µ); the latter is compact by the banach-alaoglu theorem. 138 b. günther lemma 2.2. y (µ) is weakly dense in z(µ). proof. consider a function g ∈ z(µ) and a weak neighborhood of g defined as the set of all h ∈ z(µ) such that |〈fi, h − g〉| < 1 with suitable functions f1, . . . fn ∈ l 2(µ). without loss of generality (observe ‖h − g‖2 ≤ 1) we may assume that each fi is a step function fi = ∑mi j=1 αij χaij and that g is a step function g = ∑mn+1 j=1 αn+1,j χan+1,j with 0 ≤ αn+1,j ≤ 1 and aij ∩ aik = ∅ for j 6= k. if b1, . . . bn is the collection of all intersections aij ∩ aℓk with non zero measure, then we may write fi = ∑n j=1 βij χbj and g = ∑n j=1 γj χbj with 0 ≤ γj ≤ 1 and bj ∩ bk = ∅ for j 6= k. choose points xj ∈ bj . then by our standing assumption 0 = µ ({xj}) < µ (bj ) and therefore sierpiński’s mean value theorem ensures the existence of sets cj ⊆ bj with µ (cj ) = γj µ (bj ). ⋃n j=1 cj ∈ y (µ) is contained in the given weak neighborhood of g. � there are two intrinsic characterizations of y (µ) as a subset of z(µ). first: y (µ) is the extreme set of the convex set z(µ), because any f ∈ z(µ) can be written as f = 1 2 f 2 + 1 2 [ 1 − (1 − f )2 ] . second: pointwise multiplication provides us with a product on z(µ) (let’s not worry about its continuity here), and y (µ) is the set of idempotents. furthermore, on y (µ) the product equals the intersection of sets. these observations will be utilized in section 5. y (µ) can easily be identified as a weak gδ in z(µ); in particular it is polish (cf. [8, exc.17.43,p.117] and [2, ch.ix,§6.1,thm.1]) it should be observed that lemma 2.2 states a much stronger property than would be obtained by an application of the krein-milman theorem [3, ch.ii,§7.1,thm1], which would merely assure us that the convex hull of y (µ) is weakly dense in z(µ). however, the property is familiar from lindenstrauss’ proof of liapounoff’s theorem [10], applied to the measures fiµ. 3. isometries of cantor’s discontinuum for us, cantor’s discontinuum is the compact abelian groupc = zn2 , equipped with the dyadic ultrametric |t − t′|2 = 2 − min{n:tn−t′n 6=0}, t = (tn), t ′ = (t′n). observe that the ordering of the coordinates enters essentially. furthermore, c is a probability space equipped with the haar measure. by g∞ we denote the group of isometries of c; by the arzela-ascoli theorem this group is compact. to obtain a simple description we consider the projection maps pn : z n 2 → z n 2 onto the first n coordinates. it follows immediately from the definition that every isometry must factor over pn and provide us with a ladder of permutations πn ∈ s2n : random selection of borel sets 139 definition 3.1. a permutation π : zn2 ≈ z n 2 is called filtered, if there exists a commutative ladder of permutations (not necessarily automorphisms) as in (3.1) with π = πn. the group of all filtered permutations is denoted gn. (3.1) zn2 πn≈ �� p n n−1 // // z n−1 2 πn−1≈ �� // // · · · // // z22 π2≈ �� p 2 1 // // z12 π1≈ �� zn2 p n n−1 // // z n−1 2 // // · · · // // z22 p 2 1 // // z12 we obtain g∞ = lim←−n gn; as inverse limit of finite discrete (hence compact) groups this is a compact group. hence any isometry is measure preserving. furthermore, the action of g∞ on c is transitive but not 2-transitive: indeed, for two pairs of points x, y ∈ c and x′, y′ ∈ c, an isometry γ ∈ g∞ with γx = x′ and γy = y′ exists if and only if |x − y|2 = |x ′ − y′|2. 4. review of measurable dyadic spaces the classical result obtained by hausdorff and alexandroff states that every compactum can be represented as a dyadic space, i.e. as continuous image of cantor’s discontinuum. we have to squeeze measure theoretic properties out of this theorem. the dyadic resolutions we are about to construct should be compared to the “rastering” of an image and will be the fundamental tool in our analysis of the space of borel sets in section 6. lemma 4.1. every point of x ∈ x has a fundamental sequence of open neighborhoods u with µ (∂u ) = 0. proof. for a given point x ∈ x choose a continuous function ϕ : x → i with ϕ−1(0) = x. let c ⊂ i be the at most countable subset of all points t ∈ i with µ ( ϕ−1(t) ) > 0; for any t ∈ i \ c the open neighborhood u := ϕ−1 ([0, t[) of x satisfies ∂u ⊆ ϕ−1(t) and therefore µ (∂u ) = 0. � lemma 4.2. suppose a is a locally closed subset of x with µ(a) > 0 and µ (∂a) = 0. then for any 0 < β < µ(a) there exists a locally closed subset b ⊂ a with µ(b) = β and µ (∂b) = µ (∂ (a \ b)) = 0. we recall that a set is locally closed if it is the intersection of a closed and an open set [2, ch.i,§3.4]. this adds a condition about the boundary to sierpiński’s theorem [16]. proof. we construct two sequences of open subsets un, vn ⊂ a ◦ such that un ∩ vn = ∅, un ⊆ un+1, vn ⊆ vn+1, β − 1 n ≤ µ (un) < β, µ(a) − β − 1 n ≤ µ (vn) < µ(a) − β and µ (∂un) = µ (∂vn) = 0. clearly we can start with u1 = v1 = ∅. at inductive stage n, if µ (un) = β or µ (vn) = µ(a) − β we are finished, so let us assume 0 < β − µ (un) < µ(a) − µ (un) − µ (vn). let k ⊆ a◦ \ ( u n ∪ v n ) be a compact subset with µ(k) > β − µ (un). using atomicity of µ and lemma 4.1 we can cover k by finitely many open subsets wi of a◦ \ ( u n ∪ v n ) with µ (wi) < 1 n+1 and µ (∂wi) = 0. let j be the maximal 140 b. günther number such that µ (w1 ∪ . . . ∪ wj−1) < β−µ (un). then µ (w1 ∪ . . . ∪ wj ) ≥ β−µ (un) and, since µ (wj ) < 1 n+1 , µ (w1 ∪ . . . ∪ wj−1) >= β−µ (un)− 1 n+1 . set un+1 := un ∪ w1 ∪ . . . ∪ wj−1. again, if µ (un+1) = β we are finished. otherwise we have µ(a) − µ (un+1) − µ (vn) > µ(a) − µ (vn) − β > 0. let k′ ⊆ a◦ \ ( u n+1 ∪ v n ) be a compact subset with µ (k′) > µ(a) − µ (vn) − β and cover k′ by finitely many open subsets w ′i of a ◦ \ ( u n+1 ∪ v n ) with µ (w ′i ) < 1 n+1 and µ (∂w ′i ) = 0. let j be the maximal number such that µ ( w ′1 ∪ . . . ∪ w ′ j−1 ) < µ(a)−µ (vn)−β. then µ ( w ′1 ∪ . . . ∪ w ′ j−1 ) ≥ µ(a)− µ (vn) − β − 1 n+1 , set vn+1 := vn ∪ w ′ 1 ∪ . . . ∪ w ′ j−1. now u := ⋃ n un and v := ⋃ n vn are disjoint open subsets of a ◦ with µ(u ) = β and µ(v ) = µ(a) − β. since ∂u ⊆ a \ (u ∪ v ) we must have µ (∂u ) = 0. � lemma 4.3. suppose we are given real numbers βi > 0 with β := ∑n−1 i=0 βi ≤ 1. then for any 0 < ε < 1 and n ≥ 4 ε maxi β βi we can find numbers ki ∈ n (in particular ki > 0) such that ∑n−1 i=0 ki = n and ∣ ∣ ∣ βi ki − β n ∣ ∣ ∣ ≤ ε n . proof. set ϑ := maxi β βi . first choose integer k′i ∈ z with ∣ ∣ ∣ n βi β − k′i ∣ ∣ ∣ ≤ 1 2 . then |n − ∑ i k′i| = ∣ ∣ ∣ ∑n−1 i=0 ( n βi β − k′i )∣ ∣ ∣ ≤ n 2 and hence, by adjusting at most n 2 cases ki := k ′ i ± 1 we can assure ∑n−1 i=0 ki = n and ∣ ∣ ∣ n βi β − ki ∣ ∣ ∣ ≤ 3 2 . since n ≥ 4ϑ ≥ 4β βi we have n βi β ≥ 4 and in particular ki > 0. this gives us ∣ ∣ ∣ kiβ n βi − 1 ∣ ∣ ∣ ≤ 3β 2n βi ≤ 3ϑ 2n ≤ 1 2 and therefore ∣ ∣ ∣ n βi kiβ − 1 ∣ ∣ ∣ ≤ 3ϑ n , hence ∣ ∣ ∣ βi ki − β n ∣ ∣ ∣ ≤ 3βϑ n 2 ≤ 3ϑ n 2 ≤ 3ε 4n . � definition 4.4. a type i resolution of x consists of a double sequence of locally closed subsets anm ⊆ x, n ∈ n0, 0 ≤ m < 2 n, subject to the following conditions: (1) a00 = x (2) anm ∩ ank = ∅ for m 6= k (3) an+1,2m ∪ an+1,2m+1 = anm (4) lim n→∞ max2 n−1 m=0 diam anm = 0 (5) µ (∂anm) = 0 (6) there exists a sequence of numbers εn > 0 with ∑ n εn < ∞ such that 1−εn 2 µ (anm) ≤ µ (an+1,2m) , µ (an+1,2m+1) ≤ 1+εn 2 µ (anm) proposition 4.5. every compactum satisfying our standing assumption has a type i resolution. proof. we construct the sets anm by induction on n, pushing ahead from step n to step n + n for a suitable number n and then assembling the intermediate sets as pairwise disjoint unions. observing lemma 4.1 we can chop up anm into a disjoint union of locally closed sets anm = ∐r j=0 bj with random selection of borel sets 141 µ (∂bj) = 0 and diam bj ≤ 1 2 diam anm; moreover, since supp µ = x we have µ (bj ) > 0. for any 0 < ε < 1 lemma 4.3 ensures the existence of numbers kj ∈ n such that ∑ j kj = 2 n and ∣ ∣ ∣ 1 kj µ (bj) − 2 −n µ (anm) ∣ ∣ ∣ ≤ ε n 2n , provided 2n n ≥ 4(anm) εµ(bj ) . using lemma 4.2 we can partition each bj into a disjoint union of kj locally closed sets bj = ∐ ℓ∈ij an ℓ, #ij = kj , with µ (∂an ℓ) = 0 and µ (an ℓ) = 1 kj µ (bj ), so that in particular ∣ ∣µ (an ℓ) − 2 −n µ (anm) ∣ ∣ ≤ ε n 2n . each of the intermediate sets an+v,w, 0 ≤ v ≤ n , 2 vm ≤ w < 2v(m + 1), is the disjoint union an+v,w = ∐ ℓ∈j an ℓ, #j = 2 v. thus 2−vµ (an+v,w) = ∑ ℓ∈j 2 −vµ (an ℓ), and by convexity ∣ ∣2−vµ (an+v,w) − 2 −n µ (anm) ∣ ∣ ≤ ε n 2n . equivalently, ∣ ∣ ∣ 2n−v µ(an+v,w) µ(anm) − 1 ∣ ∣ ∣ ≤ ε n µ(anm) and therefore2 ∣ ∣ ∣ 2µ(an+v+1,2w) µ(an+v,w) − 1 ∣ ∣ ∣ ≤ 4ε n µ(anm) and ∣ ∣ ∣ 2µ(an+v+1,2w+1) µ(an+v,w) − 1 ∣ ∣ ∣ ≤ 4ε n µ(anm) if ε is chosen small enough. this takes care of condition 6 in definition 4.4, where the steps from n + 1 to n + n contribute a total of at most ∑n v=1 4ε n µ(anm) = 4ε µ(anm) to the sum of all error terms εn. � remark: the proof shows that we can arrange for the total error ∑ n εn to be arbitrarily small. example 4.6. on cantor’s discontinuum, construct a type i resolution as follows. for 0 ≤ m < 2n consider the dual expansion m = ∑n−1 k=0 εn−k2 k and set cnm := p −1 n (ε1, . . . εn). notice that cnm is closed and open, diam (cnm) = 2−n−1 and ν (cnm) = 2 −n. lemma 4.7. for any type i resolution the finite unions of the sets anm constitute a dense subset of the space of all borel sets y (µ). proof. inner and outer regularity of µ [8, thm.17.10] and small diameter of the sets anm. � theorem 4.8. each compactum x satisfying our standing assumption can be represented as continuous image of cantors discontinuum f : c ։ x, such that there exists a measurable inverse function g : x → c whose points of discontinuity constitute a 0-set, with f g = idx strictly and gf = idc a.s. moreover, there exists a continuous, strictly positive density function ϕ : c → r with g∗µ = ϕν and f∗ν = 1 ϕg µ, where ν is haar measure on c. ϕ may be chosen as close to 1 as we please. notice for instance that the fibers of f must be non void 0-sets. x can be changed into cantor’s discontinuum by altering it at a 0-set. the probability spaces (x, µ) and (c, ϕν) are measure isomorphic. proof. observe that for any sequence of numbers mn with mn+1 = 2mn or mn+1 = 2mn + 1 for each n we have 2 nµ (anm) = ∏n−1 k=0 2µ(ak+1,mk+1 ) µ(ak,mk ) and 2observe that |x − 1| ≤ ε ≤ 1 2 and |y − 1| ≤ ε ≤ 1 2 imply ∣ ∣ ∣ x y − 1 ∣ ∣ ∣ ≤ 4ε 142 b. günther that the product converges uniformly for all such sequences mn. hence, if we define a continuous function ϕn : c → r to assume the value 2 nµ (anm) on cnm, then this function will converge uniformly to a continuous function ϕ : c → r, ϕ > 0. for n ≥ n we have ∫ cnm ϕn dν = ∑2n −n(m+1)−1 k=2n −nm ∫ cn k ϕn dν = ∑2n −n(m+1)−1 k=2n −nm µ (an k) = µ (anm) and therefore ∫ cnm ϕdν = µ (anm). define fn : c → x to be the continuous map that assumes on cnm a constant value contained in anm. this sequence of functions converges uniformly to a map f : c → x with f (cnm) ⊆ anm. for a point x ∈ x consider the unique sequence mn with x ∈ anm. there is a unique point y ∈c with y ∈ cnmn for all n, therefore f (y) ∈ ⋂ anmn . hence f (y) = x. define gn : x → c to be the function that assumes on anm a constant value contained in cnm; notice that gn is measurable and is continuous except possibly at ⋃ m ∂anm. this sequence converges uniformly to a function g : x → c with g (anm) ⊆ cnm that is measurable and is continuous except possibly at x0 := ⋃ n,m ∂anm; notice µ (x0) = 0. since f g (anm) ⊆ f (cnm) ⊆ anm we must have f g = idx strictly, by the same argument as above. g (anm) ⊆ cnm implies anm ⊆ g −1 (cnm), but since for fixed n these sets constitute a partition of x we must have anm = g −1 (cnm). hence ∫ cnm ϕdν = µ (anm) = µ ( g−1cnm ) = (g∗µ) cnm. since the finite unions of the sets cnm generate all borel sets we conclude ϕν = g∗µ. now let y0 := f −1 (x0) ⊆c be the inverse image of the singularity set of g. then g−1 (y0) = g −1f −1 (x0) = (f g) −1 (x0) = x0 and in particular ∫ y0 ϕdν = µ ( g−1y0 ) = µ (x0) = 0. since the continuous density ϕ is everywhere positive we conclude ν (y0) = 0. let’s consider a point y ∈ c \ y0; for n pick m such that y ∈ cnm. then f (y) ∈ anm \ x0 ⊆ anm and therefore gf (y) ∈ cnm. this can happen for arbitrary n only if gf (y) = y. we claim f −1a◦nm ⊆ cnm. for a point y ∈ f −1a◦nm we pick k such that y ∈ cnk, then f (y) ∈ ank. if we had k 6= m we could conclude ank ∩anm = ∅ and hence ank ∩ a ◦ nm = ∅ and hence f (y) 6∈ a ◦ nm, thus arriving at a contradiction. therefore we have cnm ⊆ f −1anm ⊆ f −1 (a◦nm ∪ x0) ⊆ cnm ∪ y0. this implies f −1 (anm \ x0) = cnm \ y0, in particular ν ( f −1 (anm) \ y0 ) = ν ( f −1 (anm \ x0) ) = 2−n and hence ν ( f −1anm ) = 2−n. this implies ∫ anm 1 ϕg dµ = ∫ anm df∗ν and therefore 1 ϕg µ = f∗ν. finally, ϕ can be made arbitrarily close to 1 because the error sum ∑ n εn in condition 6 of definition 4.4 is completely at our disposal. � notice that theorem 4.8 allows to transport the group action of g∞ on c onto x by πx := f πg(x). the map x 7→ πx is a measure isomorphism of 1 ϕg µ and is continuous except at a 0-set, and the equation (πσ)x = π (σx) holds for almost all x, the exception set depending on σ. the action is transitive in the strict sense, i.e. for each x ∈ x the orbit equals the entire space g∞x = x. random selection of borel sets 143 definition 4.9. a type ii resolution of x consists of a double sequence of borel subsets anm ⊆ x, n ∈ n0, 0 ≤ m < 2 n, subject to the following conditions: (1) a00 = x (2) anm ∩ ank = ∅ for m 6= k (3) an+1,2m ∪ an+1,2m+1 = anm (4) µ (an+1,2m) = µ (an+1,2m+1) = 1 2 µ (anm) (5) the finite unions of the sets anm are dense in y (µ). type ii resolutions have the advantage of reproducing the measure on x exactly, but otherwise they are considerably weaker. easy examples such as taking x as disjoint union of two closed segments of length 1 3 and 2 3 show that the properties of type i and type ii resolutions are mutually exclusive in general. proposition 4.10. every compactum satisfying our standing assumption has a type ii resolution. evidently, this holds for the unit segment; the general case then follows from the isomorphism theorem for measures (cf. [8, thm.17.41], [5, §41]). for comparison: using type ii resolutions instead of type ii in the proof of theorem 4.8 just reproduces the ordinary isomorphism theorem. however, here it is not necessary to adjust our measure by a density function. proposition 4.11. for any compactum x satisfying our standing assumption there exists a measurable function g : x → c such that g∗µ = ν, where ν is haar measure on c. moreover, for any borel subset a ⊆ x there exists a borel subset b ⊆c such that µ ( a△g−1b ) = 0. thus y (µ) ≈ y (ν). 5. the coordinate space let z denote the set of all sequences of real numbers xnm, n ∈ n0, 0 ≤ m < 2n, subject to the conditions 0 ≤ xnm ≤ 1(5.1) xnm = 1 2 (xn+1,2m + xn+1,2m+1)(5.2) z is a closed subset of the hilbert cube and thus inherits a compact topology, that will be called the weak topology. notice that z is convex. lemma 5.1. for all (xnm) ∈ z (5.3) x2nm + ∞ ∑ r=n+1 (m+1)2r−n−1−1 ∑ k=m2r−n−1 2n−r−1 (xr,2k − xr,2k+1) 2 ≤ xnm proof. we show by induction on n ≥ n that (5.4) x2nm + n ∑ r=n+1 (m+1)2r−n−1−1 ∑ k=m2r−n−1 2n−r−1 (xr,2k − xr,2k+1) 2 = 2n−n (m+1)2n −n−1 ∑ k=m2n −n x2n k 144 b. günther the inductive step is as follows: 2n−n (m+1)2n −n−1 ∑ k=m2n −n x2n k + (m+1)2n −n−1 ∑ k=m2n −n 2n−n−2 (xn +1,2k − xn +1,2k+1) 2 (5.5) = 2n−n−2 (m+1)2n −n−1 ∑ k=m2n −n [ (xn +1,2k + xn +1,2k+1) 2 + (xn +1,2k − xn +1,2k+1) 2 ] (5.6) = 2n−n−1 (m+1)2n −n−1 ∑ k=m2n −n ( x2n +1,2k + x 2 n +1,2k+1 ) (5.7) = 2n−n−1 (m+1)2n +1−n−1 ∑ k=m2n +1−n x 2 n +1,k(5.8) similarly one shows (5.9) 2n−n (m+1)2n −n−1 ∑ k=m2n −n xn k = xnm and the asserted lemma follows from x2n k ≤ xn k. � this implies in particular that z is contained in the hilbert space of all sequences satisfying (5.2), equipped with the scalar product (5.10) 〈xnm, x ′ nm〉 := x00x ′ 00 + ∞ ∑ n=1 2n−1−1 ∑ m=0 2−n−1 (xn,2m − xn,2m+1) ( x′n,2m − x ′ n,2m+1 ) thence z inherits another topology, finer than the one above. lemma 5.2. on z there is a product (xnm) = (x ′ nm) ∧ (x ′′ nm) defined by (5.11) xnm = lim n→∞ 2n−n (m+1)2n −n−1 ∑ k=m2n −n x′n kx ′′ n k it satisfies xnm ≤ √ x′nmx ′′ nm and is continuous as a function zh × zh → zw, the suffixes indicating hilbert space topology and weak topology, respectively. the bilinear map zw ×zw → zw is separately continuous [3, ch.iii,§5.1]. the ∧-product is commutative and associative. random selection of borel sets 145 proof. for all n ≥ n we obtain 2n−n (m+1)2n −n−1 ∑ k=m2n −n x′n kx ′′ n k + (m+1)2n −n−1 ∑ k=m2n −n 2n−n−2 ( x ′ n +1,2k − x ′ n +1,2k+1 )( x ′′ n +1,2k − x ′′ n +1,2k+1 ) (5.12) = 2n−n−2 (m+1)2n −n−1 ∑ k=m2n −n [ ( x′n +1,2k + x ′ n +1,2k+1 )( x′′n +1,2k + x ′′ n +1,2k+1 ) + ( x′n +1,2k − x ′ n +1,2k+1 )( x′′n +1,2k − x ′′ n +1,2k+1 ) ] (5.13) = 2n−n−1 (m+1)2n −n−1 ∑ k=m2n −n ( x′n +1,2kx ′′ n +1,2k + x ′ n +1,2k+1x ′′ n +1,2k+1 ) (5.14) = 2n−n−1 (m+1)2n +1−n−1 ∑ k=m2n +1−n x′n +1,kx ′′ n +1,k(5.15) lemma 5.1 and the cauchy-schwarz inequality imply that the perturbation term in (5.12) converges to 0. we obtain (5.16) xnm := lim n→∞ 2n−n (m+1)2n −n−1 ∑ k=m2n −n x′n kx ′′ n k = x′nmx ′′ nm + ∞ ∑ r=n+1 (m+1)2r−n−1−1 ∑ k=m2r−n−1 2n−r−1 ( x′r,2k − x ′ r,2k+1 )( x′′r,2k − x ′′ r,2k+1 ) this demonstrates the asserted joint continuity condition right away, as well as the relation xnm ≤ √ x′nmx ′′ nm via another application of lemma 5.1 and the cauchy-schwarz inequality. obviously the so defined sequence xnm satisfies (5.1) and (5.2). commutativity and associativity are easily checked. for fixed (x′mk) the convergence of the series (5.16) is uniform, this implies separate continuity on zw × zw. � proposition 5.3. for any x = (xnm) ∈ z the following conditions are equivalent: (1) x is an extreme point of z. (2) x ∧ x = x (3) x200 + ∑∞ n=1 ∑2n−1−1 k=0 2 −n−1 (xn,2k − xn,2k+1) 2 = x00 (4) limn→∞ 2 −n ∑2n−1 m=0 ( xnm − 1 2 )2 = 1 4 146 b. günther proof. with 1− x := (1 − xnm) we get x = 1 2 x∧ x + 1 2 [1 − (1 − x) ∧ (1 − x)], hence (1)⇒(2). (3) is just the 00-component of (2). in terms of our scalar product (5.10) condition (3) means ‖x‖ 2 = x00. now assume x = 1 2 x′ + 1 2 x′′. then, observing lemma 5.1 we can conclude x00 = ∥ ∥ 1 2 x′ + 1 2 x′′ ∥ ∥ 2 = 1 4 ‖x′‖ 2 + 1 2 〈x′, x′′〉 + 1 4 ‖x′′‖ 2 ≤ ( 1 2 ‖x′‖ + 1 2 ‖x′′‖ )2 ≤ ( 1 2 √ x′00 + 1 2 √ x′′00 )2 ≤ 1 2 x′00 + 1 2 x′′00 = x00. therefore we must have 〈x ′, x′′〉 = ‖x′‖‖x′′‖, i.e. x′ and x′′ must be collinear, and 1 2 √ x′00 + 1 2 √ x′′00 = √ 1 2 x′00 + 1 2 x′′00, i.e. x ′ 00 = x ′′ 00. this implies x′ = x′′ = x and establishes the equivalence of the first three conditions, and the equation 2−n 2n−1 ∑ k=0 ( xnm − 1 2 )2 = 2−n 2n−1 ∑ k=0 x2nm − x00 + 1 4 = ( x00 − 1 2 )2 + n ∑ r=1 2r−1−1 ∑ k=0 2−r−1 (xr,2k − xr,2k+1) 2 (5.17) takes care of (4). � remark 5.4. equation (5.17) shows that the sequence 2−n ∑2n−1 m=0 ( xnm − 1 2 )2 increases with n and has limit ≤ 1 4 , for all (xnm) ∈ z. definition 5.5. we denote by y ⊆ z the subspace of all points satisfying the equivalent conditions of proposition 5.3. now observe that on y we have ‖(xnm)‖ 2 = x00. hence for fixed (x ′ nm) the norm distance ‖(x′nm − x ′′ nm)‖ = √ x′00 + x ′′ 00 − 2〈(x ′ nm) , (x ′′ nm)〉 is continuous as a function of (x′′nm) on yw. therefore the weak topology and the hilbert space topology coincide on y . this subspace is closed with respect to the ∧-product, because for x, y ∈ y we have (x ∧ y) ∧ (x ∧ y) = (x ∧ x) ∧ (y ∧ y) = x ∧ y, hence x ∧ y ∈ y . thus ∧ induces a continuous product on y . y is a weak gδ in z, in particular it is polish. we can easily establish the density of y in z. for given r, n and (xnm) ∈ z we will construct (ynm) ∈ y with |xn m − yn m| ≤ 2 −r; then |xnm − ynm| ≤ 2−r for n ≤ n follows from (5.2). pick numbers km ∈ n0 such that ∣ ∣xn m − km 2r ∣ ∣ ≤ 2−r. we now define yn +r,ℓ such that it assumes the value 1 exactly km times in the range m2r ≤ ℓ < (m+1)2r and is 0 otherwise. observing (5.2) this defines (ynm) uniquely, and yn m = km 2r . moreover, (ynm) ∈ y because ynm − 1 2 = ±1 2 for n ≥ n + r. the group g∞ we encountered in section 3 acts continuously on y . suppose we are given a ladder of filtered permutations πn like in diagram (3.1), and consider the dual expansion m = ∑n−1 i=0 εn−i2 i of a number 0 ≤ m < 2n. set (ε′1, . . . ε ′ n) := πn (ε1, . . . εn) and πn(m) := ∑n−1 i=0 ε ′ n−i2 i. since xn,πn(m) = 1 2 ( xn+1,πn+1(2m) + xn+1,πn+1(2m+1) ) this induces an operation of g∞ on z, random selection of borel sets 147 continuous in the topology of your choice. condition 4 of proposition 5.3 is obviously invariant under g∞, therefore g∞y ⊆ y . 6. the isomorphism theorem theorem 6.1. let (anm) be a resolution of either type, and define a map h : y (µ) → y , h(b) = (xnm) by xnm := 2 n ∫ b∩anm 1 ϕg dµ (where ϕ and g are as in theorem 4.8) in case of type i, and xnm := 2 nµ (b ∩ anm) in case of type ii. h is a homeomorphism. the intersection corresponds to the ∧-product, and the complement of a set represented by (xnm) corresponds to (1 − xnm). notice that in particular, x00 = ∫ b 1 ϕd dµ in case of type i and x00 = µ(b) in case of type ii. proof. we consider h as map h : z(µ) → z and observe that it is continuous if the spaces are equipped with either weak or hilbert space topology, using the same choice for both spaces. we will show h (y (µ)) ⊆ y below. let b = ⋃ i anmi be a finite union of elements of the resolution (n is some fixed number); then xnm = 1 if m appears among the mi and xnm = 0 otherwise. the collection of all sequences of this form has been recognized as dense at the end of section 5, and by compactness (weak topology) we have h (z(µ)) = z. the equation h (f1f2) = h (f1) ∧ h (f2) is immediate for finite unions ⋃ i anmi and hence for step functions ∑ i αiχanmi , but since the map (f1, f2) 7→ h (f1) ∧ h (f2) is continuous as a map z(µ)h × z(µ)h → zw and since the described step functions are norm dense, it must hold generally. in particular we obtain h(f ) = h ( f 2 ) = h(f ) ∧ h(f ) and therefore h(f ) ∈ y if f ∈ y (µ). now assume h (f1) = h (f2). for abbreviation, let’s write µ̃ = 1 ϕg µ in case i and µ̃ = µ in case ii, then by definition the 00-component of h equals h00(f ) = ∫ f dµ̃. therefore ∫ f1f2dµ̃ = h00 (f1f2) = (h (f1) ∧ h (f2))00 = (h (f1) ∧ h (f1))00 = h00 ( f 21 ) = ∫ f 21 dµ̃ and similarly ∫ f1f2dµ̃ = ∫ f 22 dµ̃. therefore ∫ (f1 − f2) 2 dµ̃ = 0. hence h : z(µ) ≈ z is a homeomorphism (in either kind of topology). � the reader may notice that the homeomorphism h transports the “a.s.action” of g∞ on x defined after theorem 4.8 to the action on y from section 5. 7. probability measures on the space of borel sets for us, a probability measure on y (µ) ≈ y is a borel probability measure ν on the compact space z (weak topology) such that ν (z \ y ) = 0, this being computable by condition 4 of proposition 5.3. the definition of the compact space z may be rephrased as follows: denote by pn+1n : i 2n+1 → i2 n the map pn+1n (x0, . . . x2n+1−1) = ( x′0, . . . x ′ 2n−1 ) , x′m := 1 2 (x2m + x2m+1), then z = lim←−n i2 n taken along the maps pn+1n . let pn : z → i2 n be the natural projection, and consider the measures νn := pnν on 148 b. günther i2 n . they determine ν uniquely [4, ch.iii,no.4,§5]. we arrive at the following characterization: theorem 7.1. a probability measure ν on y corresponds bijectively to a sequence of probability measures νn = pnν on i 2n such that pn+1n νn+1 = νn and for all ε > 0 (7.1) lim n→∞ νn { 2−n 2n−1 ∑ m=0 ( xnm − 1 2 )2 ≤ 1 4 − ε } = 0 ν is invariant under g∞ if and only if each νn is invariant under gn. the reader will have noticed that the sequence of numbers in (7.1) is decreasing, and we just have to exclude a strictly positive limit. also, the event in (7.1) is invariant under gn because gn simply permutes the coordinates xnm. we can now construct the measures νn inductively subject to the conditions above, starting with an arbitrary measure ν0 on the unit segment. νn+1 can be chosen gn+1-invariant if νn is gn-invariant. the inductive step requires the distribution of mass along the fibers of pn+1n , to which end we must surmount a difficulty displayed in figure 1. assume ε > 0 and n > n are fixed. we say that a point (xk) ∈ i 2n with 2−n ∑2n−1 k=0 ( xk − 1 2 )2 ≤ 1 4 − ε is critical if the entire fiber ( pnn )−1 (xk) is contained in the ball 2 −n ∑2n −1 k=0 ( x′k − 1 2 )2 ≤ 1 4 −ε. figure 1. critical and non critical fibers of q : i2 → i lemma 7.2. let 0 < x < 1 and consider the “projection” q : im → i, q (x1, . . . xm) = 1 m ∑m k=1 xk. then max { ∑m k=1 ( xk − 1 2 )2 : 1 m ∑m k=1 xk = x } = m−1 4 + ( mx −⌊mx⌋− 1 2 )2 . random selection of borel sets 149 proposition 7.3. (xk) ∈ i 2n is critical if and only if 2n−1 ∑ k=0 ( 2n−nxk − ⌊ 2n−nxk ⌋ − 1 2 )2 < 2n−2 − 2n ε. hence it suffices to choose n large enough such that 2n−nε > 1 4 to exclude any critical points. example 7.4. we define νn inductively using a function ϕ : i 2n ×i2 n → r+ such that ϕ (x, x′) 6= 0 ⇒ pnn (x ′) = x(7.2) ∀x ∈ i2 n : ∫ x′∈(pn n )−1(x) ϕ (x, x′) λ (dx′) = 1(7.3) where λ denotes lebesgue measure on r2 n −2n . then for any function f : i2 n → r we define (7.4) ∫ f dνn := ∫ ∫ ϕ (x, x′) f (x′) λ (dx′) νn (dx) one could for example take the following choice: a (x) :=    x ′ ∈ ( pnn )−1 (x) : 2−n 2n −1 ∑ k=0 ( x′k − 1 2 )2 > 1 4 − ε    (7.5) ϕ (x, x′) := { λ (a (x)) −1 x′ ∈ a (x) 0 x′ 6∈ a (x) (7.6) notice that λ (a (x)) > 0 if n is chosen according to proposition 7.3. then (7.7) νn  (x′k) ∈ i 2n : 2−n 2n −1 ∑ k=0 ( x′k − 1 2 )2 > 1 4 − ε   = 1 example 7.5. let’s consider our random borel sets as stochastic process as follows: at time n+1 we split up the random variable xnm into two random variables xn+1,2m, xn+1,2m+1 subject to the condition xnm = 1 2 (xn+1,2m + xn+1,2m+1), thus picking a point in the fiber displayed in figure 1. this forces the difference of the new values into the interval xn+1,2m−xn+1,2m+1 ∈ [−2 min (xnm, 1 − xnm) , 2 min (xnm, 1 − xnm)]. except for the necessary scaling this is done independently and with identical distribution defined by a density function ϕn : [−1, +1] → r + subject to the conditions ϕn(−t) = ϕn(t), ∫ +1 −1 ϕn(t)dt = 1 and (7.8) lim n→∞    ∫ |t|≥1−ε ϕn(t)dt    2n = 1 for each ε > 0. for instance we could use ϕn(t) := cn exp ( (nt)2 ) , with suitable normalization factors cn. 150 b. günther this leads to measures νn on i 2n with density functions φn : i 2n → r + defined inductively as follows: φn+1 (x0, . . . x2n+1−1) := φn (x̃0, . . . x̃2n−1) 2n−1 ∏ i=1 ϕn ( x2m−x2m+1 2 min(x̃m,1−x̃m) ) 2 min (x̃m, 1 − x̃m) (7.9) x̃m := 1 2 (x2m + x2m+1)(7.10) where φ0 : i → r + must satisfy ∫ 1 0 φ(t)dt = 1, otherwise arbitrary. gninvariance is immediate, and the following lemma ensures the assumptions of theorem 7.1: lemma 7.6. suppose ε > 0 and δ > 0 are given. choose (1) r ∈ n such that 2−r < 3ε 1−ε , (2) ϑ > 0 such that (1 − ϑ)r > 1 − δ, (3) n ∈ n such that for all n ≥ n the inequality [ ∫ |t|≥1−2−r−2ε ϕn(t)dt ]2n ≥ 1 − ϑ holds. then for all n ≥ n +r we obtain νn ( (xk) ∈ i 2n : 2−n ∑2n−1 k=0 ( xk − 1 2 )2 ≥ 1 4 − ε ) > 1 − δ. proof. let us define points (xsm)0≤m<2s ∈ i 2s for n−r ≤ s ≤ n by downward induction xnm := xm and xs−1,m := 1 2 (xs,2m + xs,2m+1); we consider the coordinate xs−1,m as “parent” of the “children” xs,2m and xs,2m+1. this provides us with a set of trees with nodes labeled xsm, with root nodes xn−r,m and leave nodes xm. a leave node xm = xnm will be called “good” if at least one element of its chain of ancestors xs,ms for n − r ≤ s ≤ n satisfies ∣ ∣xs,ms − 1 2 ∣ ∣ ≥ 1 2 − 2−r−2ε. since we must necessarily have ms2 n−s ≤ m < (ms + 1) 2 n−s and 2−(n−s) ∑(ms+1)2 n−s i=ms2n−s xi = xs,ms we conclude 1 2 − 2−r−2ε ≤ 2−(n−s) ∣ ∣ ∣ ∑(ms+1)2 n−s i=ms2n−s ( xi − 1 2 ) ∣ ∣ ∣ ≤ 2−(n−s) ∑(ms+1)2 n−s i=ms2n−s ∣ ∣xi − 1 2 ∣ ∣ ≤ 2−(n−s)−1 (2n−s − 1) + 2−(n−s) ∣ ∣xm − 1 2 ∣ ∣ and therefore ∣ ∣xm − 1 2 ∣ ∣ ≥ 1 2 − 2n−s−r−2ε ≥ 1 2 − ε 4 for every good leave node xm. we claim that with probability ≥ (1 − ϑ)r at most one leave node in each of the 2n−r trees is bad, more generally: at most one level ℓ node in each tree is bad with probability ≥ (1 − ϑ)ℓ for 1 ≤ ℓ ≤ r. we start by considering level 1, i.e. the 2n−r pairs of children xn−r+1,2m, xn−r+1,2m+1 of the root nodes. at least one child of a root node is good with probability ∫ |t|≥1−2−r−2ε ϕn(t)dt; hence each of the trees contains at most one bad node of level 1 with probability [ ∫ |t|≥1−2−r−2ε ϕn(t)dt ]2n−r ≥ (1 − ϑ)2 −r ≥ 1 − ϑ. by definition both children of a good level ℓ node are good level ℓ + 1 nodes, and we have at most 2n−r random selection of borel sets 151 bad ones at level ℓ with probability (1 − ϑ)ℓ. each of these has at least one good child with probability ≥ 1 − ϑ by the same estimate as above, leading to a probability ≥ (1 − ϑ)ℓ+1 of our event at level ℓ + 1. we conclude that with probability ≥ 1 − δ at least 2n − 2n−r leave nodes xm satisfy ∣ ∣xm − 1 2 ∣ ∣ ≥ 1 2 − ε 4 and therefore ( xm − 1 2 )2 ≥ 1 4 − ε 4 . hence 2−n ∑2n−1 m=0 ( xm − 1 2 )2 ≥ (1 − 2−r) ( 1 4 − ε 4 ) ≥ 1 4 − ε. � example 7.7. we could opt to push the entire mass of i2 n onto its 1-skeleton. this choice will be discussed in depth in section 10. 8. conditional expectation and variance the coordinates xnm in y may be considered as random variables ξnm : y → i with ξnm = 1 2 (ξn+1,2m + ξn+1,2m+1) and limn→∞ 2 −n ∑2n−1 m=0 ( ξnm − 1 2 )2 = 1 4 a.s. the intersection of random sets is represented by the ∧-product. we will be using a g∞-invariant probability measure on y as constructed in section 7; furthermore we will assume that ξ00 equals the measure of our random set, so either type ii resolutions have to be used or adjustment by a density function must be allowed. for two numbers 0 ≤ k 6= m < 2n we consider their dual expansions k = ∑n−1 i=0 δn−i2 i m = ∑n−1 i=0 εn−i2 i and define v(m, k) := min{i : εi 6= δi} = − lb |m − k|2. we prepare to answer question 1. lemma 8.1. there exists a sequence of functions fn : i → i such that e (ξnm|ξ00) = ξ00(8.1) e ( ξ 2 nm|ξ00 ) = fn (ξ00)(8.2) e (ξnmξnk|ξ00) = 2fv(m,k)−1 (ξ00) − fv(m,k) (ξ00) if m 6= k(8.3) lim n→∞ fn = idi ν0-a.s.(8.4) proof. trivially, ξ00 is g∞-invariant. invariance of ν therefore implies that e (ξnm|ξ00) is independent of m, and (8.1) follows from ξ00 = 2 −n ∑2n−1 m=0 ξnm. the same argument shows that e ( ξ2nm|ξ00 ) is independent of m, and (8.2) may be taken as definition of the function fn. (8.4) follows from limn→∞ 2 −n ∑2n−1 m=0 ξ 2 nm = ξ00 a.s. again by g∞-invariance, e (ξnmξnk|ξ00) =: f (n, v(m, k), ξ00) for m 6= k depends only on n, v(m, k) and ξ00. observing ξnmξnk = 1 4 (ξn+1,2mξn+1,2k + ξn+1,2m+1ξn+1,2k + ξn+1,2mξn+1,2k+1 +ξn+1,2m+1ξn+1,2k+1) and v(2m, 2k) = v(2m + 1, 2k) = v(2m, 2k + 1) = v(2m + 1, 2k + 1) = v(m, k) we can drop the first argument of f and write e (ξnmξnk|ξ00) = f (v(m, k), ξ00). 152 b. günther in the equation ξ2nm = 2 2(n−n ) ( ∑(m+1)2n −n−1 ℓ=m2n −n ξn ℓ )2 = 22(n−n ) ( ∑(m+1)2n −n−1 ℓ=m2n −n ξ2n ℓ + ∑(m+1)2n −n−1 a6=b=m2n −n ξn aξn b ) for n ≥ n we count the number of pairs a, b with specific dyadic distance and obtain fn (ξ00) = 2 n−n fn (ξ00)+ ∑n r=n+1 2 n−rf (r, ξ00) = 2 n−n−1fn +1 (ξ00)+ ∑n +1 r=n+1 2 n−rf (r, ξ00) and hence 2fn (ξ00) = fn +1 (ξ00) + f (n + 1, ξ00). � theorem 8.2. for two any two independent random variables a and b assuming borel subsets of x as values we obtain (8.5) e (µ(a ∩ b)|µ(a), µ(b)) = µ(a)µ(b) (8.6) var (µ(a ∩ b)|µ(a), µ(b)) = ∞ ∑ n=0 2−n−1 [ 2fn (µ(a)) − fn+1 (µ(a)) − µ(a) 2 ] [2fn (µ(b)) −fn+1 (µ(b)) − µ(b) 2 ] here var (η|f) = e ( η2|f ) − e (η|f) 2 . the functions fn are those from lemma 8.1. in general context, this is about all that can be said concerning intersections of independent random sets. more specific results will be obtained in section 10. proof. in coordinate representation, let the random borel set a correspond to the process ξ′nm and b to the independent process ξ ′′ nm, then a∩b corresponds to ξnm = limn→∞ 2 n−n ∑(m+1)2n −n−1 k=m2n −n ξ′n kξ ′′ n k. therefore e (ξ00|ξ ′ 00, ξ ′′ 00) = lim n→∞ 2−n 2n −1 ∑ k=0 e (ξ′n kξ ′′ n k|ξ ′ 00, ξ ′′ 00)(8.7) = lim n→∞ 2−n 2n −1 ∑ k=0 e (ξ′n k|ξ ′ 00) e (ξ ′′ n k|ξ ′′ 00)(8.8) = ξ′00ξ ′′ 00(8.9) and that proves (8.5). similarly, (8.10) e ( ξ200|ξ ′ 00ξ ′′ 00 ) = lim n→∞ 2−2n   2n −1 ∑ k=0 e ( ξ′2n k|ξ ′ 00 ) e ( ξ′′2n k|ξ ′′ 00 ) + 2n −1 ∑ a6=b=0 e (ξ′n aξ ′ n b|ξ ′ 00) e (ξ ′′ n aξ ′′ n b|ξ ′′ 00)   counting the number of pairs with specific dyadic distance and applying lemma 8.1 now proves (8.2). � random selection of borel sets 153 9. impossibility of complete location invariance through the action of g∞ our compactum x is “measure homogeneous”. for any two raster blocks anm and ank of the same level n there is a transformation π ∈ g∞ taking the one to the other modulo a 0-set. because of the failure of 2-transitivity this does not extend to more general subsets, for instance, anm = an+1,2m∪an+1,2m+1 cannot be transformed into an+1,2m∪an+1,2m+2. consequently, question 2 in the introduction does not have a general answer derivable from knowledge of µ(a) alone. one could try to improve this state of affairs by picking a larger transformation group than g∞. this, however, turns out to be impossible except in trivial cases. if we had such a group whose operation was at least 2-transitive, then (8.3) would imply that 2fn−1 − fn is ν0-almost surely independent of n and therefore, observing (8.4), fn = idi ν0-a.s. for all n. but then e (ξnm (1 − ξnk) |ξ00) = ξ00 − fn (ξ00) = 0 ν0-a.s. for all n, m, k, which is only possible if all mass of ν0 is located at the two points 0 and 1. 10. the sierpiński example let e∞ ⊆ z be the set of all points (xnm), such that at each level n, all xnm ∈{0, 1} with at most one permissible exception. since any such sequence satisfies 2−n ∑2n−1 k=0 ( xnk − 1 2 )2 ≥ 1 4 − 2−n we actually have e∞ ⊆ y . it can also be described as the set of all points of z which are carried to the 1-skeleton of i2 n by the natural projection map pn : z → i 2n , hence e∞ is a compact subspace of y that could be called its 1-skeleton. figure 2. sierpiński’s universal curve not the entire 1-skeleton of the cubes will be used by this construction. the projection p20 : i 2 → i maps the four vertices (0, 1, 0, 1), (1, 0, 0, 1), (0, 1, 1, 0) and (1, 0, 1, 0) to the interior point ( 1 2 , 1 2 ) ; these vertices must be avoided. hence we define e1 as 1-skeleton of i 2 and, for n > 1, en ⊆ i 2n as the 154 b. günther 1-skeleton of ( pnn−1 )−1 en−1. this inverse image consists of a collection of 2dimensional faces, one for each edge of en−1, whose interior is disregarded. hence en is obtained from en−1 by replacing each edge by the boundary of a square. e4 is displayed in figure 2. evidently, e∞ = lim←−n en is homeomorphic to sierpiński’s universal curve [7, ex.i.1.11,p.9]. en consists of 4 n edges, labeled σab for a = (a1, . . . an) ∈ i̇ n and b = (b1, . . . bn) ∈ z n 2 as follows: for any number 0 ≤ m < 2 n we construct the dual expansion m = ∑n−1 i=0 mn−i2 i; if now k is such that mk 6= bk but ∀ℓ < k : mℓ = bℓ, then we stipulate that the points (x0, . . . x2n−1) ∈ σab must satisfy the equation xm = ak. identifying m with the sequence m = (m1, . . . mn) ∈ z n 2 , the condition means in terms of the dyadic ultrametric distance in z n 2 : xm = a− lb|m−b| 2 in particular, for any filtered permutation g ∈ gn we have gσab = σa,gb. observe that we obtain one equation for all coordinates except for xr with r = ∑n−1 i=0 bn−i2 i. furthermore (10.1) pn0 (σab) = [ n ∑ k=1 ak2 −k, 2−n + n ∑ k=1 ak2 −k ] in particular, any such interval is covered 2n-fold. we want to construct a g∞-invariant probability measure on e∞, starting from a probability measure ν0 on the unit segment with 0 point masses and supp ν0 = i. equation (10.1) tells us how to distribute mass along the edge σab, where all such edges bearing the same first index a will be served evenly. a compatible sequence of measures on en is obtained, leading to a measure on e∞ ⊂ y . for this measure we can give a much stronger version of theorem 8.2 and can determine the conditional distribution of µ(a ∩ b) given µ(a) and µ(b) completely: theorem 10.1. suppose two borel sets a and a′ are randomly and independently chosen. if µ(a) and µ (a′) are given, then µ (a ∩ a′) can assume only countably many values. these occur with the following probabilities: (10.2) p ( µ (a ∩ a′) = a′n ∞ ∑ k=n+1 2−kak + an ∞ ∑ k=n+1 2−ka′k + n−1 ∑ k=1 aka ′ k2 −k ∣ ∣ ∣ µ(a) = t, µ (a′) = t′ ) = 2−n for all n ∈ n, where t = ∑∞ k=1 ak2 −k and t′ = ∑∞ k=1 a ′ k2 −k are dual expansions and the “sierpiński” measure constructed above is used on y (µ). proof. since ν0 is assumed not to have any point masses it is sufficient to give the proof for irrational numbers t, t′, where the dyadic expansion is unique. we define for b, b′ ∈ zn2 : (10.3) nn (b, b ′) := 2−n ∑ m6=b,b′ a− lb|m−b| 2 a′− lb|m−b′| 2 random selection of borel sets 155 then nn converges almost surely to µ (a ∩ a ′). for the evaluation of the product on the right hand side of (10.3) we have to distinguish three cases (observe the special properties of the dyadic ultrametric): (1) |m − b|2 < |m − b ′|2 = |b − b ′|2 (2) |b − b′|2 < |m − b|2 = |m − b ′|2 (3) |m − b′|2 < |m − b|2 = |b − b ′|2 observing # { m ∈ zn2 | |m|2 = 2 −k } = 2n−k we obtain (10.4) 2nnn (b, b ′) = a′ − lb|b−b′| 2 ∑ 0<|m−b|2<|b−b ′| 2 a− lb|m−b|2 + ∑ |m−b|2=|m−b ′| 2 >|b−b′| 2 a− lb|m−b|2 a ′ − lb|m−b|2 +a− lb|b−b′| 2 ∑ 0<|m−b′| 2 <|b−b′| 2 a ′ − lb|m−b′| 2 (10.5) = a′ − lb|b−b′| 2 ∑ 0<|m|2<|b−b ′| 2 a− lb|m|2 + a− lb|b−b′|2 ∑ 0<|m|2<|b−b ′| 2 a′ − lb|m|2 + ∑ |m|2>|b−b ′| 2 a− lb|m|2 a ′ − lb|m|2 (10.6) = a′ − lb|b−b′| 2 ∑ k>− lb|b−b′| 2 2n−kak + a− lb|b−b′| 2 ∑ k>− lb|b−b′| 2 2n−ka′k + ∑ k<− lb|b−b′| 2 2n−kaka ′ k the theorem follows. � 11. on the relation of random closed and random borel sets in the introduction it has been emphasized that our approach to random borel sets is not an extension of the theory of random closed sets. in this section we are going to investigate the relation. let t = 2x denote the hyperspace of x, i.e. the space of non void closed subsets carrying the vietoris topology. since any closed subset is borel we obtain a natural, non continuous function q : t → y (µ). proposition 11.1. there exists a finer topology on t generating the same borel sets, turning t into a polish space and q : t → y (µ) into a continuous map. in particular, q : t → y (µ) is measurable with respect to the vietoris topology. proof. i) for any fixed b ∈ y (µ) the graph γ (fb) ⊆ t × y (µ) of the upper semicontinuous function fb : t → r, fb(a) := µ (a ∩ b) is a gδ, hence polish. for fb is the infimum of a decreasing sequence of continuous functions ϕn ↓ fb [2, ch.ix,§1.6,prop.5] and γ (fb) = ⋂ n { (x, y) ∈ t × y (µ) ∣ ∣fb(x) − 1 n < y < ϕn(x) + 1 n } . 156 b. günther ii) for any dense sequence (bn)n∈n in y (µ) the graph γ(f ) of the function f : t → rn with coordinates fbn is polish because it is homeomorphic to ∏ n γ (fbn ). iii) the graph γ ( f̃ ) of the function f̃ : t → ry (µ) with coordinates fb is polish; we show that it is homeomorphic to γ(f ). the restriction from all borel sets to the sequence (bn)n∈n provides us with a natural (hence continuous) projection map π : γ ( f̃ ) → γ(f ), π ( a, f̃ (a) ) = (a, f (a)) which is bijective. to show that the inverse π−1 is bijective too it suffices to prove that for each borel set b the function gb : γ(f ) → r, gb (a, f (a)) = fb(a) is continuous. this is evidently true if b is an element of our dense sequence, because then fb(a) is simply the b-coordinate of f (a). in the general situation we select a subsequence (bnk )k∈n of borel sets converging to b. then ∣ ∣ ∣ gb (a, f (a)) − gbnk (a, f (a)) ∣ ∣ ∣ = ∣ ∣ ∣ fb(a) − fbnk (a) ∣ ∣ ∣ = |µ (a ∩ b) −µ (a ∩ bnk )| ≤ µ (b△bnk ) and therefore limk→∞ gbnk = gb uniformly. iv) the graph γ(q) ⊆ t × y (µ) of q : t → y (µ) is polish and therefore a gδ in t × y (µ). we claim γ(q) ≈ γ ( f̃ ) and have to show that the map (a, q(a)) ↔ ( a, f̃ (a) ) is continuous in both directions. for any borel set b the b-coordinate of ( a, f̃ (a) ) equals fb(a) = µ (a ∩ b) which is a continuous function of q(a) because intersection is continuous on y (µ). for the reverse it is sufficient to observe d (q(a), q (a0)) = µ (a△a0) = f∁a0 (a)−fa0 (a) + µ (a0). v) we now identify t with the graph γ(q) by means of the bijection i : t ≈ γ(q) defined by i(a) = (a, q(a)) and consider the topology on t obtained by transporting back the topology of γ(q) over i. since i−1 (γ ∩ (u × y (µ))) = u the new topology is finer than the vietories topology. under this identification the map q corresponds to the projection map γ →֒ t × y (µ) → y (µ) and is therefore continuous. it remains to show that for any borel subset b ⊆ γ(q) the inverse i−1(b) ⊆ t is borel with respect to the vietoris topology. we observe that the natural projection π : t × y (µ) → t provides us with a continuous bijection π : γ(q) → t and our inverse image i−1(b) = q(b) is the continuous bijective image of a borel set. we now observe that the spaces t and γ(q), being polish, are in particular lusin [2, ch.ix,§6.4,prop.12]. then b as borel subset of a lusin space is a lusin space itself [2, ch.ix,§6.7,thm.3], hence its continuous bijective image π(b) is again a lusin space and therefore borel. � proposition 11.1 allows to consider any random closed set, i.e. any random variable with values in t , as random borel set by composition with the measurable map q : t → y (µ). however, this may involve a loss of information by generating a coarser event algebra. it can be shown that no information is lost if and only of there exists a subset b ⊆ t of probability 1 such that q is one-to-one on b. since this applies for instance to random closed sets which are almost certainly regular closed [11, def.4.29,p.63] this covers quite a random selection of borel sets 157 few examples. the obvious counterexamples are random closed sets that have almost certainly measure 0, such as random finite sets or buffon’s needle. on the other hand, random borel sets are better adapted to image processing than random closed sets, for instance because of their relation to wavelets (see below). it is no accident that random borel sets cannot distinguish sets that differ only by a 0-set, since such a small difference would not be visible in an image. lemma 11.2. the sequence of vectors e(n k) = ( e (n k) nm ) n≥0,0≤m<2n with n ≥ 0 and 0 ≤ k < 2n−1, defined by (11.1) e(n k)nm =          1 n = 0, k = 0 2 n −1 2 n ≥ n > 0, k2n+1−n ≤ m < ( k + 1 2 ) 2n+1−n −2 n −1 2 n ≥ n > 0, ( k + 1 2 ) 2n+1−n ≤ m < (k + 1)2n+1−n 0 else constitutes a complete on-system in the hilbert space considered in section 5. for any vector x = (xnm) we have 〈 x, e(00) 〉 = x00 and 〈 x, e(n k) 〉 = 2− n +1 2 (xn,2k − xn,2k+1) for n > 0. proof. observing that our system of numbers can satisfy e (n k) n,2m − e (n k) n,2m+1 6= 0 only if n = n > 0 and m = k all computations are rather straightforward. (5.1) checks easily, and so does ∥ ∥e(n k) ∥ ∥ 2 = 1. the relations 〈 x, e(00) 〉 = x00 and 〈 x, e(n k) 〉 = 2− n +1 2 (xn,2k − xn,2k+1) for n > 0 are obvious. this immediately implies orthonormality; furthermore any vector x perpendicular to all e(n k) must satisfy x00 = 0, xn,2k = xn,2k+1 for all n > 0 and all k as well as (5.1) and hence x = 0. � from the lemma above it should be clear that our approach to random borel sets is essentially an expansion in terms of the on-base e(n k). on the unit segment this corresponds to the l2-functions 2− n +1 2 ( χ[ 2k 2n , 2k+1 2n [ − χ[ 2k+1 2n , 2k+2 2n [ ) , i.e. to the haar wavelet [18, def.1.1] or rather to those constituents of the haar wavelet that live on the unit segment. references [1] p. alexandroff and h. hopf, topologie, chelsea, 1972. [2] n. bourbaki, general topology, elements of mathematics, hermann and addisonwesley, 1966. [3] n. bourbaki, topological vector spaces i-v, elements of mathematics, springer, 1987. [4] n. bourbaki, integration i (chapters 1-6), elements of mathematics, springer, 2004. [5] p. r. halmos,measure theory, volume 18 of gtm, springer, 1974. [6] f. hausdorff, mengenlehre, göschens lehrbücherei. walter de gruyter & co., 2nd edition, 1927. [7] s. b. nadler jr, continuum theory, volume 158 of pure and applied mathematics, marcel dekker, inc., 1992. 158 b. günther [8] a. s. kechris, classical descriptive set theory, volume 156 of gtm, springer, 1995. [9] s. li, y. ogura and v. kreinovich, limit theorems and applications of set-valued and fuzzy set-valued random variables, theory and decisions library, kluwer, 2002. [10] j. lindenstrauss, a short proof of liapounoff’s convexity theorem, j. math. mech. 15 (1966), no. 6, 971–972. [11] i. molchanov, theory of random sets, probability and its applications. springer, 2005. [12] h. e. robbins, on the measure of random set, ann. math. statist. 15 (1944), 70–74. [13] h. e. robbins, on the measure of random set ii, ann. math. statist. 16 (1945), 342–347. [14] c. rosendal, the generic isometry and measure preserving homeomorphism are conjugate to their powers, fund. math. 205 (2009), 1–27. [15] r. schneider and w. weil, stochastic and integral geometry, probability and its applications, springer, 2008. [16] w. sierpiński, sur les fonctions d’ensemble additives et continues, fund. math. 3 (1922), 240–246. [17] f. straka and j. štěpán, random sets in [0,1], in j. visek and s. kubik, editors, information theory, statistical decision functions, random processes, prague 1986, volume b, pages 349–356, reidel, 1989. [18] p. wojtaszczyk, a mathematical introduction to wavelets, volume 37 of student text, london mathematical society, 1997. received april 2010 accepted september 2010 b. günther (dr.bernd.guenther@t-online.de) db systel gmbh, development center databases t.sid32, hahnstraße 40, 60528 frankfurt am main, germany random selection of borel sets. by b. günther kanibirreillyagt.dvi @ applied general topology c© universidad politécnica de valencia volume 11, no. 1, 2010 pp. 57-65 on almost cl-supercontinuous functions a. kanibir and i. l. reilly ∗ abstract. recently the class of almost cl-supercontinuous functions between topological spaces has been studied in some detail. we consider this class of functions from the point of view of change(s) of topology. in particular, we conclude that this class of functions coincides with the usual class of continuous functions when the domain and codomain have been retopologized appropriately. some of the consequences of this fact are considered in this paper. 2000 ams classification: primary 54c60, 54c05, 54c10; secondary 54d10, 54d20. keywords: change of topology, cl-supercontinuity, almost cl-supercontinuity, continuity, almost continuity, perfect continuity, almost perfect continuity, slightly continuous. 1. introduction there can be no argument that the notion of continuity is one of the most important concepts in the whole of mathematics. over the years many generalizations and variants of continuous functions between topological spaces have been introduced. very recently, kohli and singh [7], have considered the class of ”almost cl-supercontinuous ” functions. this is a new name for the class of ”almost clopen ” functions introduced by ekici [4] as a generalization of the class of ”clopen continuous ” mappings defined by reilly and vamanamurthy [11], and studied in some detail by singh [13], under the name of cl-supercontinuous functions. in particular, kohli and singh [7] noted that almost cl-supercontinuity is a variant of continuity which is ” independent of continuity ”. a primary purpose of this paper is to argue exactly the opposite of this view. we advocate that the distinction made by kohli and singh [7] between the concepts of ∗the second author gratefully acknowledges the award of a fellowship for visiting scientists by the scientific and technological research council of turkey (tubitak) 58 a. kanibir and i. l. reilly almost cl-supercontinuity and continuity must be interpreted very strictly. it is our contention that almost cl-supercontinuity is just continuity in disguise. indeed, if the domain and codomain spaces of an almost cl-supercontinuous function f are retopologized in a suitable fashion (see theorem 4.1 ), then f is simply a continuous function. this observation puts the notion of almost cl-supercontinuity in a more natural context, and it allows alternative proofs of some of the results of kohli and singh [7]. what we are claiming, in the language of category theory, is that an almost cl-supercontinuous function f arises because the wrong source and target have been chosen for the morphism f in the category of topological spaces and continuous functions. in section 2 we consider the basic properties of semi-regular topologies and of quasi-topologies, which we shall need in this paper. the relevant definitions of the classes of functions we consider in this paper are provided in section 3. section 4 examines the class of almost cl-supercontinuous functions studied by kohli and singh [7], especially from the perspective of change(s) of topology. kohli and singh [7, 2.1] and singh [13, introduction] make a case for a change in nomenclature, and rename the clopen continuous functions defined by reilly and vamanamurthy [11] as cl-supercontinuous functions. names are largely a matter of personal taste, although the original term at least suggests what the definition might be. however, the use of the clprefix notation leads to the possibility of serious confusion. from [13, definition 2.1] we see that the subsets called cl-open by singh [13] are precisely the quasi-open sets of the quasi-topology, see our section 2, and the cl-closed sets are the quasiclosed sets. the difference between clopen sets and cl-open sets is significant, but the notations are too similar. similarly [13, definitions 2.4 and 2.6] show that cl-adherence and cl-convergence are precisely adherence and convergence in the quasi-topology, so why not use the terms quasi-adherence and quasiconvergence. this usage of the cl-prefix leads to [13, definition 2.9] where a function is defined to be cl-open if it takes clopen sets to open sets. but as pointed out by singh [ 13, introduction p293] ”in the topological folklore the phrase ” clopen map” is used for the functions which map clopen sets to open sets”. this desire for purity of nomenclature has created the absurd situation where ”clopen maps” should now be termed ”cl-open maps”. despite our preference for the original term ”clopen continuous”, we shall use the term ”cl-supercontinuous” throughout this paper. our notation and terminology are standard, see for example dugundji [3]. no separation properties are assumed for topological spaces unless explicitly stated. we denote the interior of the subset a of the topological space (x,τ) by τinta, and the closure of a by τcla. 2. two topologies in a topological space (x,τ) a set a is called τ regular open if a = τint(τcla) and τ regular closed if a = τcl(τinta). we let ro(x,τ) denote the collection of all regular open sets in (x,τ). since the intersection of two regular open sets is regular open, the collection of all τ regular open sets forms a base for a on almost cl-supercontinuous functions 59 smaller topology τs on x, called the semi-regularization of τ. the space (x,τ) is said to be semi-regular if τs = τ. any regular space is semi-regular, but the converse is false. in 1968, velicko [16] introduced the notion of δ-open and δ-closed sets in a space (x,τ). a point x ∈ x is said to be a δ-cluster point of the subset a of (x,τ) if a ∩ u 6= ∅ for every τ regular open set u containing x. the set of all δ-cluster points of a is called the δ-closure of a and denoted by [a]δ. if a =[a]δ then a is called δ-closed, and the complement of a δ-closed set is called δ-open. for any space (x,τ) the collection of all δ-open sets forms a topology τδ on x, and the definitions of δ-adherent point, δ-closure, δ-convergence of filterbases and so on are the usual definitions applied to the topology τδ. it turns out that using δ-open sets is another way to describe the semi-regularization topology, that is, τδ = τs for any space (x,τ). semiregularization topologies are considered in some detail by mrsevic, reilly and vamanamurthy [8], especially from the change of topology perspective. let (x,τ) be a topological space. the quasi-component of a point x ∈ x is the intersection of all clopen subsets of x which contain the point x. the quasi-topology on x is the topology having as base the collection of all clopen subsets of (x,τ). since any clopen subset of x is regular open (and regular closed), the quasi-topology is smaller than the semi-regularization topology in general. the closure of each point in the quasi-topology is precisely the quasicomponent of that point. we shall call the open ( resp. closed) subsets of the quasi-topology quasi-open (resp. quasi-closed) and denote the quasi-topology by τq. for a given topological space (x,τ), the space (x,τq) is called by staum [15] the ultraregular kernel of x. recall that a space (x,τ) is zero-dimensional if it has a base of clopen sets for the topology τ. we observe that (x,τ) is zero-dimensional if and only if τq = τ. we have noted that for any topological space (x,τ) we have τq ⊂ τs ⊂ τ. quasi-topologies are considered by dontchev, ganster and reilly [2, section 3], and by singh [13, section 5] who used the notation τ∗ for τq. an unusual feature of quasi-topologies is their behaviour with respect to the lower separation properties, t0, t1 and hausdorff. in fact, any quasi-topology has either all or none of these three separation properties. to be precise, if (x,τ) is such that (x,τq) is t0, then (x,τq) is hausdorff. let a and b be distinct points of x, so that there is a quasi-open subset m of x containing one of a and b but not the other. assume that a ∈ m. then there is a clopen subset d of x such that a ∈ d ⊂ m. now b ∈ x −d and x −d is clopen and disjoint from d, so that {d, x −d} is a hausdorff separation of a and b in (x,τq). in particular, (x,τq) is hausdorff if and only if (x,τq) is t0. 3. definitions here we provide a list of definitions of the variations of continuity that we consider in this paper. 60 a. kanibir and i. l. reilly a function f : (x,τ) → (y,σ) between topological spaces is defined to be (1) almost continuous [12] if for each x ∈ x and for each regular open set v containing f(x) there is an open set u containing x such that f(u) ⊂ v. (2) δ-continuous [10] if for each x ∈ x and for each regular open set v containing f(x) there is a regular open set u containing x such that f(u) ⊂ v. (3) supercontinuous [9] if for each x ∈ x and for each open set v containing f(x) there is a regular open set u containing x such that f(u) ⊂ v. (4) cl-supercontinuous [13] ( or clopen continuous [11] ) if for each x ∈ x and for each open set v containing f(x) there is a clopen set u containing x such that f(u) ⊂ v. (5) almost cl-supercontinuous [7] ( or almost clopen [4] ) if for each x ∈ x and for each regular open set v containing f(x) there is a clopen set u containing x such that f(u) ⊂ v. (6) slightly continuous [6] if f−1(m) is open in x for each clopen set m in y. 4. change of topology the fundamental defining characteristic of the class of almost cl-supercontinuous functions is given by the following result, especially the equivalence of (1) and (4), which is proved immediately from the definitions. we observe that ekici [4 theorem 6 (1), (10), (11), (12) ] obtained this theorem, but did not subsequently use it in his paper. on the other hand, it is the cornerstone of our approach to this topic. theorem 4.1. let f be a function from a topological space (x,τ) to a topological space (y,σ). then the following are equivalent: (1) f : (x,τ) → (y,σ) is almost cl-supercontinuous, (2) f : (x,τq) → (y,σ) is almost continuous, (3) f : (x,τ) → (y,σs) is cl-supercontinuous, (4) f : (x,τq) → (y,σs) is continuous. the equivalence of theorem 4.1 (1) and (4) shows that almost cl-supercontinuity is a µ-continuity property in the sense of gauld, mrsevic, reilly and vamanamurthy [5]. the containment σs ⊂ σ and the equivalence of (1) and (3) in theorem 4.1 show that, in general, cl-supercontinuity is stronger than almost cl-supercontinuity, but that for semi-regular codomains they are equivalent. it seems that the diagram of implications given by kohli and singh [7, page 3] can create confusion. we reproduce their diagram here. on almost cl-supercontinuous functions 61 strongly continuous ⇓ perfectly continuous ⇒ almost perfectly continuous ⇓ ⇓ cl-supercontinuous ⇒ almost cl-supercontinuous ⇓ ⇓ z-supercontinuous ⇒ almost z-supercontinuous ⇓ ⇓ supercontinuous ⇒ δ-continuous ⇓ ⇓ continuous ⇒ almost continuous we take this diagram to mean that one can find a function between topological spaces (x,τ) and (y,σ) having one of these properties but not one of the stronger properties. one must regard the topologies on x and y as fixed for this interpretation. on the other hand, theorem 4.1 above shows that four of these continuity type concepts coincide if one is willing to change the topology on x or on y or on both x and y. in particular, it shows that the concept of almost cl-supercontinuity coincides with the classical notion of continuity under appropriate changes of topology on domain and co-domain. this diagram of implications [7, page 3] strongly suggests that such a situation cannot occur. theorem 5.1 of singh [13] is a corollary of the method of proof of theorem 4.1. theorem 4.2. the function f : (x,τ) → (y,σ) is cl-supercontinuous if and only if f : (x,τq) → (y,σ) is continuous. this result reveals that whenever τq = τ then any continuous function f : (x,τ) → (y,σ) is cl-supercontinuous. thus if (x,τ) is zero-dimensional we have that any continuous function with domain (x,τ) is cl-supercontinuous. the topological properties of the codomain space are not germaine to this discussion. our theorem 4.1 can be used to provide elegant alternative proofs of some of the results of kohli and singh [7]. for example, consider the matter of composition of functions [7, theorem 3.2]. first we need to observe the following equivalences. (1) f : (x,τ) → (y,σ) is δ-continuous if and only if f : (x,τs) → (y,σs) is continuous [ 10, theorem 2.5]. (2) f : (x,τ) → (y,σ) is cl-supercontinuous if and only if f : (x,τq) → (y,σ) is continuous [ 13, theorem 5.1]. (3) f : (x,τ) → (y,σ) is almost continuous if and only if f : (x,τ) → (y,σs) is continuous [ 8, proposition 12]. 62 a. kanibir and i. l. reilly (4) f : (x,τ) → (y,σ) is supercontinuous if and only if f : (x,τs) → (y,σ) is continuous [ 9, theorem 2.1]. (5) f : (x,τ) → (y,σ) is slightly continuous if and only if f : (x,τ) → (y,σq) is continuous [ theorem 5.3 (a) of singh [13] ]. change of topology allows us to prove the next theorem simply by observing that the composition of two continuous functions is continuous. there is no need to give proofs going back to first principles like those presented by kohli and singh [7] and ekici [4, theorem 13]. note that part of theorem 2.10 of singh [13] is a special case of (1) of theorem 4.3, while theorem 2.17 of singh [13] is (5) of our theorem 4.3. theorem 4.3. let f : (x,τ) → (y,σ) and g : (y,σ) → (z,ψ) be functions. (1) if f is cl-supercontinuous and g is continuous then g ◦ f is clsupercontinuous. (2) if f is cl-supercontinuous and g is almost continuous then, g ◦ f is almost cl-supercontinuous. (3) if f is almost cl-supercontinuous and g is δ-continuous, then g ◦f is almost cl-supercontinuous. (4) if f is almost cl-supercontinuous and g is supercontinuous, then g ◦ f is cl-supercontinuous. (5) if f is slightly continuous and g is cl-supercontinuous, then g ◦ f is continuous. (6) if f is slightly continuous and g is almost cl-supercontinuous, then g ◦ f is almost continuous. proof. (1) f : (x,τq) → (y,σ) and g : (y,σ) → (z,ψ) are continuous, so that g ◦ f : (x,τq) → (z,ψ) is continuous. (2) f : (x,τq) → (y,σ) and g : (y,σ) → (z,ψs) are continuous, so that g ◦ f : (x,τq) → (z,ψs) is continuous. (3) f : (x,τq) → (y,σs) and g : (y,σs) → (z,ψs) are continuous, so that g ◦ f : (x,τq) → (z,ψs) is continuous. (4) f : (x,τq) → (y,σs) and g : (y,σs) → (z,ψ) are continuous, so that g ◦ f : (x,τq) → (z,ψ) is continuous. (5) f : (x,τ) → (y,σq) and g : (y,σq) → (z,ψ) are continuous, so that g ◦ f : (x,τ) → (z,ψ) is continuous. (6) f : (x,τ) → (y,σq) and g : (y,σq) → (z,ψs) are continuous, so that g ◦ f : (x,τ) → (z,ψs) is continuous. � on almost cl-supercontinuous functions 63 in section 4 of their paper, kohli and singh [7] consider separation properties. their definition 4.1 defines three classes of spaces, namely ultra-hausdorff, ultra-t1 and ultra-t0 spaces. we observe that 1) (x,τ) is ultra-hausdorff if and only if (x,τq) is hausdorff, 2) (x,τ) is ultra-t1 if and only if (x,τq) is t1, and 3) (x,τ) is ultra-t0 if and only if (x,τq) is t0. in section 2 above we noted that (x,τq) is hausdorff if and only if (x,τq) is t0. this fact provides a change of topology proof of proposition 4.2 of [7]. definition 4.3 of [7] defines two classes of spaces. we note that 1) (x,τ) is δt1 if and only if (x,τs) is t1, and 2) (x,τ) is δt0 if and only if (x,τs) is t0. mrsevic, reilly and vamanamurthy [8, proposition 1] have observed that (x,τ) is hausdorff if and only if (x,τs) is hausdorff. this fact explains why there is no analagous definition of δt2 spaces. we now provide an alternative proof of theorem 4.5 of kohli and singh [7]. proposition 4.4. let f : (x,τ) → (y,σ) be an almost cl-supercontinuous injection. if (y,σ) is δt0, then (x,τ) is ultra-hausdorff. proof. note that f : (x,τq) → (y,σs) is a continuous injection, and that (y,σs) is t0. thus (x,τq) is t0, so that (x,τq) is hausdorff. that is, (x,τ) is ultra-hausdorff. � theorem 4.7 of kohli and singh [7] is a standard result restated in this new setting. proposition 4.5. let f,g : (x,τ) → (y,σ) be almost cl-supercontinuous functions and (y,σ) be hausdorff. then e = {x ∈ x : f(x) = g(x)} is quasi-closed in (x,τ). proof. we have that f,g : (x,τq) → (y,σs) are continuous, by theorem 4.1 (1) and (4), and (y,σs) is hausdorff. so by dugundji [3, page 140 1.5 (1)] we have that e is closed in (x,τq), and therefore e is quasi-closed in (x,τ). � another standard theorem applied in this context yields the following result. proposition 4.6. if f : (x,τ) → (y,σ) is almost cl-supercontinuous and (y,σ) is hausdorff, then g(f), the graph of f, is closed in (x × y,τq × σs). 64 a. kanibir and i. l. reilly proof. we have that f : (x,τq) → (y,σs) is continuous and (y,σs) is hausdorff. thus, by dugundji [3, page 140 1.5 (3)] the graph g(f) of f is closed in (x × y,τq × σs). � this result is a simpler version of much of the work in section 6 of kohli and singh [7]. recall that a topological space (x,τ) is called mildly compact [15], or a clustered space [14], if every clopen cover of x has a finite subcover. furthermore, (x,τ) is called nearly compact [1] if every regular open cover of x has a finite subcover. in fact, (x,τ) is mildly compact if and only if (x,τq) is compact, while (x,τ) is nearly compact if and only if (x,τs) is compact, carnahan [ 1,theorem 4.1]. we can now use the preservation of compactness by continuous functions together with our theorem 4.1 to provide an alternative proof of theorem 41 (1) of ekici [4]. theorem 4.7. let f : (x,τ) → (y,σ) be an almost cl-supercontinuous surjection. if (x,τ) is mildly compact, then (y,σ) is nearly compact. proof. since (x,τ) is mildly compact, (x,τq) is compact. as f : (x,τ) → (y,σ) is almost cl-supercontinuous, we have by theorem 4.1 that f : (x,τq) → (y,σs) is continuous. furthermore, f is surjective, so that (y,σs) is compact. hence (y,σ) is nearly compact. � by now we have provided sufficient evidence to be able to claim unequivocally that change of topology approaches are significant. they provide new insights into some of the developments taking place in general topology. they permit elegant alternative proofs of existing results, and allow the creation of new results. references [1] d. carnahan, locally nearly compact spaces, boll. u.m.i. 4 (1972), 146–153. [2] j. dontchev, m. ganster and i. reilly, more on almost s-continuity, indian j. math. 41 (1999), 139–146. [3] j. dugundji, topology, allyn and bacon, boston, mass. 1966. [4] e. ekici, generalization of perfectly continuous, regular set connected and clopen functions, acta math. hungar. 107, no. 3 (2005), 193–205. [5] d. gauld, m. mrsevic, i. l. reilly and m. k. vamanamurthy, continuity properties of functions, coll. math. soc. janos bolyai 41 (1983), 311–322. [6] r. c. jain, the role of regularly open sets in general topology, ph.d. thesis, meerut univ., institute of advanced studies, meerut, india (1980). [7] j. k. kohli and d. singh, almost cl-supercontinuous functions, appl. gen. topol. 10, no. 1, (2009), 1–12. [8] m. mrsevic, i. l. reilly and m. k. vamanamurthy, on semi-regularization topologies, j. austral. math. soc. a 38 (1985), 40–54. [9] b. m. munshi and d. s. bassan, super-continuous mappings, indian j. pure appl. math. 13 (1982), 229–236. [10] t. noiri, on δ-continuous functions, j. korean math. soc. 16 (1980), 161–166. [11] i. l. reilly and m. k. vamanamurthy, on super-continuous mappings, indian j. pure appl. math. 14, no. 6 (1983), 767–772. on almost cl-supercontinuous functions 65 [12] m. k. singal and a. r. singal, almost continuous mappings, yokohama math. 3 (1968), 63–73. [13] d. singh, cl-supercontinuous functions, appl. gen. topol. 8, no. 2 (2007), 293–300. [14] a. sostak, on a class of topological spaces containing all bicompact and connected spaces, general topology and its relation to modern analysis and algebra iv: proceedings of the 4th prague topological symposium, (1976) part b, 445–451. [15] r. staum, the algebra of bounded continuous functions into a nonarchimedean field, pacific j. math. 50 (1974), 169–185. [16] n. velicko, h-closed topological spaces, amer. math. soc. transl. 78, no. 2 (1968), 103– 118. received october 2009 accepted march 2010 a. kanibir (kanibir@hacettepe.edu.tr) department of mathematics, hacettepe university, 06532, beytepe, ankara, turkey i. l. reilly (i.reilly@auckland.ac.nz) department of mathematics, university of auckland, p.b. 92019, auckland, new zealand cogoreagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 263-280 on the continuity of factorizations w. w. comfort, ivan s. gotchev and luis recoder-núñez∗ abstract. let {xi : i ∈ i} be a set of sets, xj := ∏ i∈j xi when ∅ 6= j ⊆ i; y be a subset of xi , z be a set, and f : y → z. then f is said to depend on j if p, q ∈ y , pj = qj ⇒ f (p) = f (q); in this case, fj : πj [y ] → z is well-defined by the rule f = fj ◦ πj |y . when the xi and z are spaces and f : y → z is continuous with y dense in xi , several natural questions arise: (a) does f depend on some small j ⊆ i? (b) if it does, when is fj continuous? (c) if fj is continuous, when does it extend to continuous fj : xj → z? (d) if fj so extends, when does f extend to continuous f : xi → z? (e) if f depends on some j ⊆ i and f extends to continuous f : xi → z, when does f also depend on j? the authors offer answers (some complete, some partial) to some of these questions, together with relevant counterexamples. theorem 1. f has a continuous extension f : xi → z that depends on j if and only if fj is continuous and has a continuous extension fj : xj → z. example 1. for ω ≤ κ ≤ c there are a dense subset y of [0, 1]κ and f ∈ c(y, [0, 1]) such that f depends on every nonempty j ⊆ κ, there is no j ∈ [κ]<ω such that fj is continuous, and f extends continuously over [0, 1]κ. example 2. there are a tychonoff space xi , dense y ⊆ xi , f ∈ c(y ), and j ∈ [i]<ω such that f depends on j, πj [y ] is c-embedded in xj , and f does not extend continuously over xi . 2000 ams classification: primary 54b10, 54c20; secondary 54c45. keywords: product space, dense subspace, continuous factorization, continuous extensions of maps, c(x). ∗the authors gratefully thank gary gruenhage for helpful e-mail correspondence. 264 w. w. comfort, i. s. gotchev and l. recoder-núñez 1. introduction in addition to the notation given in the abstract, we adopt these conventions. ω is (the cardinality of) the set of non-negative integers and i is an index set (usually infinite). α, κ, and λ are cardinals and [i]<κ := {j ⊆ i : |j| < κ}. (xi )κ denotes xi := ∏ i∈i xi with the κ-box topology (so (xi )ω = xi ) and σλ(p) := {x ∈ xi : |{i ∈ i : xi 6= pi}| < λ} whenever p ∈ xi . by a (canonical) basic open set in (xi )κ we mean a set of the form u = ui = πi∈i ui with ui open in xi and with r(u) := {i ∈ i : ui 6= xi} ∈ [i] <κ. (in the terminology of [4], r(u) is the restriction set of the (basic) open set u.) the symbol r denotes the real line with its usual topology, the cardinality of r is denoted by c, the cardinality of the set x is denoted by |x|, and the closure of x by x. the weight of a space x is w(x) := min{|b| : b is a base for x} + ω, the density of x is d(x) := min{|d| : d is dense in x} + ω, χ(x,x) denotes the character (i.e., the local weight) of the point x in the space x, and χ(x) := sup{χ(x,x) : x ∈ x}. finally, a pairwise disjoint collection of nonempty open sets in x is called a cellular family and the cellularity of x is c(x) := sup{|u| : u is a cellular family in x} + ω. for spaces y and z we denote by c(y,z) the set of continuous functions from y into z. we write c(x) := c(x, r) and (in contrast with the convention used in [8] and elsewhere) we write c∗(x) := c(x, [0, 1]). a subspace y of a space x is c(z)-embedded if every f ∈ c(y,z) extends to f ∈ c(x,z); then as usual [8], a c(z)-embedded space y ⊆ x with z = r is said to be c-embedded; if z = [0, 1] then y is said to be c∗-embedded. our spaces are not subjected to any standing separation hypothesis. when a specific property is wanted, as in remark 3.4, section 4, and theorems 5.5 and 5.6, we state it explicitly. definition 1.1. when xi , y , z and f are as in the abstract and f depends on j ⊆ i, the function fj : πj [y ] → z (defined by the relation f = fj ◦ πj |y ) is a factorization of f. for additional topological definitions not given above, see [8], [4], [16], [11], or [7]. the point of departure of our investigation is a lemma given in the book “chain conditions in topology” by w. w. comfort and s. negrepontis [4], together with a question those authors posed. we give these now, paraphrasing slightly to facilitate the present exposition. lemma 1.2. [4, 10.3] let ω ≤ κ ≤ α, {xi : i ∈ i} be a set of nonempty topological spaces, and y be a subspace of (xi )κ such that πj [y ] = xj for every nonempty j ⊂ i with |j| < α. let also z be a topological space and f be a continuous function from y to z such that f depends on < α coordinates. then there is continuous f : (xi )κ → z such that f ⊂ f. question 1.3. [4, p. 235] let ω ≤ κ ≤ α, {xi : i ∈ i} be a set of nonempty topological spaces, and y be a dense subspace of (xi )κ. let z be a space such that πj [y ] is c(z)-embedded in xj for every nonempty j ∈ [i] <α. if on the continuity of factorizations 265 f ∈ c(y,z) and there is j ∈ [i]<α such that f depends on j, must f extend continuously over (xi )κ? what about the case κ = ω? in fact, the present authors do not know the answer to question 1.3 even if all the spaces xi are assumed metrizable. let us specialize to the case κ = ω. it is clear from theorem 3.5(c) below that if the function f in lemma 1.2 depends on some j ∈ [i]<α then the factorization fj : xj → z is continuous. then since πj [y ] = xj , the function f := fj ◦ πj is a continuous extension of f that depends on j. thus lemma 1.2 has this consequence. theorem 1.4. let α ≥ ω, xi be a product space, and y be a subspace of xi such that πj [y ] = xj for every nonempty j ∈ [i] <α. let also z be a topological space and f ∈ c(y,z) depend on some j ∈ [i]<α. then fj is continuous and f := fj ◦ πj : xi → z is a continuous extension of f that depends on j. questions (a) through (e) of the abstract are subsidiary to a more compelling very general question: “when is a continuous function defined on a dense subset of a product space continuously extendable over the full product?” this question and question (a) of our abstract have generated a huge literature. among the works in this vein, representing a variety of approaches, we mention these familiar papers: h. corson [5], r. engelking [6], i. glicksberg [9], m. hušek [12], [13], [14], a. mishchenko [18], n. noble [19], n. noble and m. ulmer [20], m. ulmer [22], [23]. the textbooks [7] and [4] strive for comprehensive bibliographies. as is indicated in [4], many of the published results responding positively to these questions generalize to product spaces with the κ-box topology. in contrast, questions (b) through (e) of the abstract have been almost totally ignored in the literature. in this paper, always with f ∈ c(y,z) and usually with y dense in xi , we study some of these questions and their relation to question 1.3. in section 2 we give some examples of discontinuous factorizations fj ; in section 3 we give some sufficient conditions for the existence of continuous factorizations; in section 4 we study when the existence of a factorization of a function defined on a dense subspace of a product space implies the existence of a factorization of its continuous extension to the full product; in section 5 we give some sufficient conditions for a positive answer to question 1.3; and in section 6 we pose a question related to question 1.3. 2. discontinuous factorizations: some examples in this section, responding to question (b) of the abstract, we give examples showing that a factorization of a continuous function defined on a dense subset of a product space need not be continuous (2.3, 2.8, 2.9); and when it is continuous the initial function may (2.8) or may not (2.13) extend continuously over the full product. concerning question (e) of the abstract, 2.3 and 2.9 give examples of a function f ∈ c(y,z) with an extension f ∈ c(xi,z) such that, 266 w. w. comfort, i. s. gotchev and l. recoder-núñez for certain j ⊆ i, f does and f does not depend on j. as to question (d), the answer is “always” (4.6). we use in what follows the familiar fact (see for example [7, 2.3.15] or [3, 3.18]) that for κ ≤ c the product of κ-many separable spaces is separable. for convenience we give statements with each xi = [0, 1], but routine generalizations (for example, with each xi = r) are clearly valid. lemma 2.1. let i be an index set with 0 < |i| ≤ c and xi = [0, 1] i . let d = {x(n) : n < ω} be a (countable) dense subset of xi , and for n < ω let y(n) ∈ xi satisfy |y(n)i − x(n)i| < 1 n+1 for each i ∈ i. then e := {y(n) : n < ω} is dense in xi . proof. each nonempty open u ⊆ xi contains a (basic) set of the form n(p,f,ǫ) := ∏ i∈f (pi − ǫ,pi + ǫ) × [0, 1] i\f with p ∈ xi , f ∈ [i] <ω and ǫ > 0. the open set v := n(p,f, ǫ 2 ) satisfies |v ∩ d| = ω, and with n chosen so that 1 n+1 < ǫ 2 and x(n) ∈ v we have y(n) ∈ n(p,f,ǫ) ∩ e ⊆ u ∩ e. � lemma 2.2. let i be an index set with 0 < |i| ≤ c and xi = [0, 1] i . there is a countable dense subset e of xi such that for each i ∈ i the restriction πi|e : e → [0, 1]i = [0, 1] is an injection. proof. let d = {x(n) : n < ω} be dense in xi . define y(0)i = x(0)i for each i, and if y(n)i has been defined for all n < m choose y(m)i ∈ [0, 1] so that y(m)i /∈ {y(0)i, . . . ,y(m − 1)i} and |y(m)i − x(m)i| < 1 m+1 . then by lemma 2.1 the set e := {y(n) : n < ω} is as required. � example 2.3. there is a product space xi , a (countable) dense subspace y = {y(n) : n < ω} ⊆ xi , f ∈ c(y ), and an index j ∈ i, such that (i) the function fj : πj [y ] → r given by fj (y(n)j ) := f(y(n)) is welldefined, and (ii) fj is not continuous on πj [y ]. one may arrange in addition that f : y → r extends continuously over xi . proof. let each xi = r or xi = [0, 1] with |i| = c, and set xi := ∏ i∈i xi and y := e = {y(n) : n < ω} as in lemma 2.2. choose and fix two different coordinates i,j ∈ i and define f : y → r by f := πi|y . note that f extends continuously over xi . note also that the function fj : πj [y ] → r given by fj(y(n)j ) := f(y(n)) = y(n)i is well-defined, since if y(n),y(m) ∈ y with y(n)j = y(m)j then m = n. now choose distinct x,y ∈ y and let p = xi and q = yj. thus fj (q) 6= p. let also nk be a sequence such that y(nk)i → p, y(nk)j → q. then fj (y(nk)j ) = f(y(nk)) = y(nk)i → p 6= fj(q), so fj is not continuous. � theorem 2.4. let i be an index set with 0 < |i| ≤ c and xi = [0, 1] i . there is a dense subspace y of xi such that for each i ∈ i the restriction πi|y : y ։ [0, 1]i = [0, 1] is a bijection onto [0, 1]. on the continuity of factorizations 267 proof. begin with e = {y(n) : n < ω} as in lemma 2.2 and for i ∈ i let {y(η)i : ω ≤ η < c} be a faithful enumeration of the set [0, 1]\πi[e]. then y(η) ∈ xi and the set y := {y(η) : η < c} = e ∪ {y(η) : ω ≤ η < c} is as required. � remark 2.5. in example 2.3 the set πj [y ] is a countable, dense subspace of rj = r or of [0, 1]j = [0, 1], hence is not c ∗-embedded. if we take i, xi , and y as in theorem 2.4 with i = {i,j}, |y | = c, xi = r 2 and with both πj and πi surjections from y onto xj = r and xi = r, respectively, then we can arrange the essential features of that argument. note that the function fj : πj [y ] → r given by fj(y(η)j ) := f(y(η)) = y(η)i is still well-defined and discontinuous since its restriction to the countable subspace e is discontinuous, though fj ◦ πj = f with f(y(η)) = y(η)i is continuous on y and as before, f extends continuously over all of r2. in example 2.3 the function f depends on the set {i} and also depends on the set {j} but the function fi is continuous while the function fj is not. the following proposition shows, more generally, that if (continuous) f : y → z depends on nonempty disjoint sets j1,j2 ⊂ j with fj1 continuous, then either f is a constant function or fj2 is nowhere continuous. proposition 2.6. let xi be a product space, z be a hausdorff space, and y be a dense subspace of xi . let also j1 and j2 be nonempty disjoint subsets of i, f : y → z be a non-constant (continuous) function that depends on j1 and j2, and fj1 is continuous. then fj2 is discontinuous at every point of πj2 [y ]. proof. let y ∈ y . we shall show that fj2 is discontinuous at yj2 . let fj2 (yj2 ) = z1, hence f(y) = z1 and fj1 (yj1 ) = z1. since the function f is not constant there is x ∈ y such that f(x) 6= z1. let f(x) = z2, hence fj1 (xj1 ) = z2. since z is a hausdorff space we can find two disjoint open sets u1 and u2 in z such that z1 ∈ u1 and z2 ∈ u2. the function fj1 is continuous at xj1 . therefore there is a basic open neighborhood vj1 of xj1 in xj1 such that fj1 [vj1 ∩πj1 [y ]] ⊂ u2. now, assume that there exists a basic open neighborhood vj2 of yj2 in xj2 such that fj2[vj2 ∩ πj2 [y ]] ⊂ u1. since y is dense in xi and j1 ∩ j2 = ∅ there is t ∈ y such that t ∈ π −1 j1 [vj1 ] ∩ π −1 j2 [vj2 ], hence tj1 ∈ vj1 ∩ πj1 [y ] and tj2 ∈ vj2 ∩ πj2 [y ]. then f(t) = fj1 (tj1 ) = fj2 (tj2 ), so f(t) ∈ u1 ∩ u2 = ∅, a contradiction. � corollary 2.7. let xi be a product space, z be a hausdorff space, and y be a dense subspace of xi . let also j1 and j2 be nonempty disjoint subsets of i, f : y → z be a (continuous) function that depends on j1 and j2, and fj1 and fj2 are continuous. then f is constant. example 2.3 suggests the speculation that if some f ∈ c(y,z) depends on a set j with fj discontinuous, and if f extends continuously over xi , then one can find another set j1 ⊂ i with |j1| = |j| such that f depends on j1 and fj1 is continuous. we show in corollary 2.9 below that this can fail: there exist a cardinal number α ≥ ω, a product space xi , a hausdorff space z, a dense 268 w. w. comfort, i. s. gotchev and l. recoder-núñez subspace y of xi , and a continuous function f : y → z such that for every j ∈ [i]<α, f depends on j but fj is not continuous, and the function f can be extended to a continuous function f on xi . (there is, however, a fragment of question 1.3—logically, an equivalent formulation—which survives in the face of corollary 2.9. we give the statement in 6.1 below.) in what follows we take i, xi , and y as in theorem 2.4 with |i| ≥ ω, |y | = c, and with each πi|y : y ։ [0, 1] a bijection onto [0, 1]; the indexing y = {y(η) : η < c} plays no further role. we note that for every set z, every function f : y → z depends (vacuously) on each set {i} with i ∈ i. thus, for every f : y → z and ∅ 6= j ⊆ i the function fj : πj [y ] → z is well-defined by the rule fj (pj ) := f(p) (p ∈ y ). theorem 2.8. given y ⊆ xi = [0, 1] i as above, let c = {in : n < ω} ∈ [i]ω (faithfully indexed) and define g : xi → [0, 1] by g(x) := ∑ n<ω xin 2n = ∑ n<ω πin (x) 2n . then for ∅ 6= j ⊂ i the function gj : πj [y ] → [0, 1] is continuous if and only if c ⊆ j. proof. (we note that the series defining g converges uniformly on xi , so g : xi → [0, 1] is continuous.) if. let pj ∈ πj [y ] ⊆ xj with p ∈ y , and let p(λ)j be a net in πj [y ] (with p(λ) ∈ y ) such that p(λ)j → pj . then p(λ)i → pi for each i = in ∈ c, so g(p(λ)) → g(p), i.e., gj (p(λ)j ) → gj (pj ). only if. we show (when c ⊆ j fails) that gj is continuous at no point pj ∈ πj [y ]. fix i = in ∈ c\j and p ∈ y and define q ∈ xi by qi := pi if i 6= i ∈ i, |qin − pin | = 1 4 . since qi = pi for all i 6= in, in particular for all i ∈ c\{in}, we have |g(p) − g(q)| = 1 4 · 1 2n . there is a net p(λ) in y such that p(λ) → q. since pj = qj we have p(λ)j → qj = pj ∈ πj [y ], but from the continuity of g on xi we have gj (p(λ)j ) = g(p(λ)) → g(q) 6= g(p) = gj (pj )). thus gj is not continuous on πj [y ]. � corollary 2.9. for ω ≤ κ ≤ c there are a dense subset y of [0, 1]κ and continuous f : y → [0, 1] such that (a) f depends on every nonempty j ⊆ κ; (b) each restricted projection πη|y : y ։ [0, 1]η = [0, 1] (η < κ) is a bijection onto [0, 1]; (c) there is no j ∈ [κ]<ω such that fj : πj [y ] → [0, 1] is continuous; and (d) f extends to a continuous function f : [0, 1]κ → [0, 1]. proof. from theorem 2.8, taking f := g|y . � the examples just given show that a function g ∈ c(xi,z) may fail to depend on a set j, even when f := g|y : y → z does depend on j. we discuss this in greater detail in section 4. on the continuity of factorizations 269 discussion 2.10. we draw the reader’s attention to two hypotheses in question 1.3. (a) there is j ∈ [i]<α such that f depends on j; and (b) πj [y ] is c(z)-embedded in xj for every nonempty j ∈ [i] <α. we end this section with examples showing that if either (a) or (b) is not satisfied then the resulting weaker questions can be answered in the negative. again we specify to the case κ = ω. example 2.11. there are α ≥ ω, a tychonoff product space xi , a dense subspace y ⊂ xi , and f ∈ c(y ) such that πj [y ] is c-embedded in xj for all nonempty j ∈ [i]<α, but f has no continuous extension from xi to r and f does not depend on any proper nonempty subset j ∈ [i]<α. proof. we are aware of two relevant constructions from the literature. (1) [the case α = ω] let n denote the countably infinite discrete space. it is shown in [1] that there is a sequence {yk : k < ω} of spaces, with in each case n ⊆ yk ⊆ β(n) := xk, such that y := πk<ω yk is not pseudocompact but yj = πk∈j yk is pseudocompact whenever j ∈ [ω] <ω. it follows from glicksberg’s theorem [9] that β(yj ) = πk∈j xk for each j ∈ [ω] <ω (so πj [y ] = yj is c ∗-embedded in xj ), but the relation β(y ) = πk<ω xk fails (so some f ∈ c∗(y ) has no continuous extension from xi = πk<ω xk to r). if f depends on some nonempty j ∈ [ω]<ω then (since y is a product space) theorem 3.5(a) shows fj ∈ c ∗(yj ). then fj extends to fj ∈ c ∗(xj ) and we have f ⊆ fj ◦ πj ∈ c ∗(xi ), a contradiction. (2) [arbitrary α ≥ ω] ulmer [22], [23] has given many examples, enhanced and extended in our work [2], of a tychonoff product space xi and a σαproduct subspace y ⊂ xi which is not c-embedded in xi (so some f ∈ c(y ) has no continuous extension from xi to r). if f depends on some nonempty j ∈ [i]<α then fj ∈ c(πj [y ]) since y is a σα-space (see theorem 3.5(b)). πj [y ] is trivially c-embedded in xj since πj [y ] = xj , and we have the contradiction f ⊆ fj ◦ πj ∈ c(xi ). � example 2.12. there are a tychonoff product space xi = πi∈i xi, a dense subspace y ⊂ xi , a function f ∈ c(y ), and j ∈ [i] <ω such that (a) f depends on j, (b) πj [y ] is c-embedded in xj , and (c) f does not extend continuously over xi . proof. in [2, 3.2] we have shown, extending arguments introduced by ulmer [22], [23], that there are a tychonoff space xi′ = πi∈i′ xi with |xi| = |i ′| = ω for each i ∈ i′, q′ ∈ xi′ , and a continuous function f ′ : xi′\{q ′} → [0, 1] which does not extend continuously over xi′ . we arrange the notation so that no symbol in i′ is named 0, and we set i := i′ ∪ {0} and x0 := [0, 1]. since {q′} × x0 is closed and nowhere dense in xi , there is a countable dense subset {x(n) : n < ω} of xi which misses {q ′} × x0, and a routine modification of the argument in lemmas 2.1 and 2.2 gives a dense set e = {y(n) : n < ω}, also missing {q′} × x0, such that the restricted projection π0|e : e → x0 is 270 w. w. comfort, i. s. gotchev and l. recoder-núñez an injection. (arguing recursively one lets y(m)i′ = x(m)i′ and one chooses (distinct) points y(m)0 ∈ [0, 1] such that |y(m)0 − x(m)0| < 1 m+1 .) since |(xi′\{q ′})\πi′ [e]| = |[0, 1]\π0[e]| = c, there is a set y such that e ⊆ y ⊆ xi and πi′ [y ] = xi′\{q ′} and π0|y is a bijection onto x0. we define f : y → [0, 1] by f(r,x′) := f′(x′) for (r,x′) ∈ y ⊆ x0 × (xi′\{q ′}) ⊆ x0 × xi′ = xi . to see that f is continuous on y it is enough to note that if y(λ) is a net in y such that y(λ) → y = (r,x′) ∈ y then y(λ)i′ → x ′ with y(λ)i′ ∈ xi′\{q ′} and hence f(y(λ)) = f′(y(λ)i′ ) → f ′(x′) = f(y). clearly f depends on the coordinate 0 ∈ i, i.e., f depends on j := {0} ∈ [i]<ω, and π0[y ] is trivially c-embedded in x0 since π0[y ] = x0. also f depends on i ′, so there can be no continuous extension f of f over xi : according to the implication (i) ⇒ (iv) of theorem 4.6 below (with our f′, i′, and f playing the role there of f, j, and g, respectively), the existence of such continuous f would yield f′ ⊆ fi′ ∈ c(xi′, [0, 1]), a contradiction. � example 2.13. there is a tychonoff product space xi = πi∈i xi such that for every nonempty j ∈ [i]≤ω there are a dense subspace y ⊂ xi and a function f ∈ c(y ) such that (a) f depends on j, (b) fj is continuous, (c) fj does not extend continuously over xj , and (d) f does not extend continuously over xi . proof. let {xi : i ∈ i} be a set of metrizable spaces without isolated points, let j ⊆ i satisfy 0 < |j| ≤ ω, fix p ∈ xj and set d := xj \{p}. some g ∈ c(d, [0, 1]) admits no continuous extension over xj , and then y := π −1 j (d) and f := g ◦ πj |y are as required (with fj = g and πj [y ] = d). � 3. existence of continuous factorizations in this section we give some conditions that imply the continuity of functions of the form fj . we begin with the following observation. lemma 3.1. let x, y , and z be spaces and f : x → y , g : x → z, and h : z → y be functions such that f = h ◦ g. if f is continuous and g is open, then h|g[x] is continuous. proof. take u an open set in y . then (h|g[x]) −1[u] = g[f−1[u]]. � theorem 3.2. let xi be a product space, j be a nonempty subset of i, and y be a nonempty subspace of xi such that πj [u ∩ y ] = πj [u] ∩ πj [y ] for every basic open set u of xi . let also z be a space and f ∈ c(y,z) depend on j. then fj : πj [y ] → z is continuous. proof. since f = fj ◦ πj |y with f continuous and πj |y open, the continuity of fj follows from lemma 3.1. � on the continuity of factorizations 271 lemma 3.3. let xi be a product space, j be a nonempty proper subset of i, and y be a subset of xi . the set πi\j [π −1 j (πj (y)) ∩ y ] is dense in xi\j for every y ∈ y if and only if πj [u] ∩ πj [y ] = πj [u ∩ y ] for every basic open set u in xi . proof. let the set πi\j [π −1 j (πj (y)) ∩ y ] be dense in xi\j for every y ∈ y and u be a basic open set in xi . we shall prove that πj [u] ∩ πj [y ] = πj [u ∩ y ]. let t ∈ πj [u] ∩ πj [y ] and let x ∈ y be such that xj = t and xi\j ∈ πi\j [u] ∩ πi\j [y ]. (such a point x exists since πi\j [π −1 j (t) ∩ y ] is dense in xi\j .) then x ∈ u ∩ y , hence t ∈ πj [u ∩ y ]. therefore πj [u] ∩ πj [y ] = πj [u ∩ y ]. now, let πj [u]∩πj [y ] = πj [u∩y ] for every basic open set u in xi . we shall prove that πi\j [π −1 j (πj (y)) ∩ y ] is dense in xi\j for every y ∈ y . let y ∈ y and v be a basic open set in xi\j . then w = π −1 i\j [v ] is a basic open set in xi and πj (y) ∈ πj [w ]∩πj [y ] = πj [w ∩y ]. therefore there exists x ∈ w ∩y such that xj = yj and xi\j ∈ πi\j [w ] = v , hence xi\j ∈ πi\j [π −1 j (πj (y))∩y ]∩v . thus πi\j [π −1 j (πj (y)) ∩ y ] is dense in xi\j . � remark 3.4. (a) it is clear that when xi , y and j satisfy the (equivalent) conditions of lemma 3.3, the function πj |y is an open map. it is useful to note that the converse can fail. for an example, let x0 and x1 be nonempty t1-spaces with each |xi| > 1, fix (x0,x1) ∈ x0 ×x1 and set y := ((x0\{x0})× x1)∪{(x0,x1)}. then π0|y is an open map, but π1[π −1 0 (π0(x0,x1))∩y ], which is the singleton set {π1(x0,x1)} = {x1}, is not dense in x1. (alternatively: with u := x0 × (x1\{x1}) we have x0 ∈ (π0[u] ∩ π0[y ])\π0[u ∩ y ].) (b) in a trivial way, using lemma 3.1, fj will be continuous provided y is a nonempty open subspace of a product space xi and f ∈ c(xi,z) depends on j ⊂ i. theorem 3.5. let xi be a product space, j be a nonempty proper subset of i, α = |j|+, and y be a nonempty subspace of xi . let also z be a space and f ∈ c(y,z) depend on j. then fj : πj [y ] → z is a continuous function if (a) y = xi ; or (b) [y ∈ y ] ⇒ σα(y) ⊆ y ; or (c) [j′ ⊆ i, |j′| ≤ |j|] ⇒ πj′ [y ] = xj′ ; or (d) [j′ ⊆ i,j′ = j ∪ f with |f | < ω] ⇒ πj′ [y ] = xj′ ; or (e) π−1 j [πj [y ]] = y ; or (f) πi\j [π −1 j (πj (y)) ∩ y ] is dense in xi\j for every y ∈ y ; or (g) πj [u] ∩ πj [y ] = πj [u ∩ y ] for every basic open set u in xi ; or (h) πj [y ] × {yi\j } ⊆ y for some y ∈ y . proof. clearly (a) ⇒ (b), (b) ⇒ (c), and (c) ⇒ (d). to see that (d) ⇒ (g), let u be a basic open set in xi and xj ∈ πj [u] ∩ πj [y ] with x ∈ u, and define j′ := j ∪ r(u); then since πj′ [y ] = xj′ there is y ∈ y such that yj′ = xj′ , so xj = yj ∈ πj [u ∩ y ]. we conclude that πj [u] ∩ πj [y ] = πj [u ∩ y ]. if (e) holds then πi\j [y ] = xi\j , so (f) holds, and if (f) holds then (g) holds by lemma 3.3. thus by theorem 3.2 the function fj is continuous under any 272 w. w. comfort, i. s. gotchev and l. recoder-núñez of the conditions (a), (b), (c), (d), (e), (f), and (g). if (h) holds and h is the natural homeomorphism from πj [y ] onto y ′ := πj [y ] × {yi\j }, then f|y ′ is defined and fj = f|y ′ ◦ h. � remark 3.6. theorem 3.5 serves to show that any dense subspace y of a product space xi for which some f ∈ c(y,z) depends on j ⊆ i with fj discontinuous must have properties in common with the spaces from our examples in section 2. thus if the answer to question 1.3 is “no” then the witnessing example must have some of the properties of the dense spaces y in our examples in section 2. 4. factorizations and continuous extensions let g ∈ c(xi,z), y be a dense subset of xi , f := g|y , and j ⊂ i. we know from example 2.3 and corollary 2.9 that g may fail to depend on j even if f does depend on j. in this section we give additional conditions sufficient to ensure that this counterintuitive phenomenon cannot occur. theorem 4.1. let xi be a product space, j be a nonempty proper subset of i, and y be a nonempty subspace of xi such that πj [y ] × {yi\j } ⊆ y for some y ∈ y . let also z be a space and g ∈ c(xi,z) be such that the function f := g|y depends on j. then fj is continuous and has a continuous extension fj : xj → z. therefore f has a continuous extension f : xi → z that depends on j. proof. the function fj is continuous according to theorem 3.5(h). let t = xj × {yi\j }. if h := g|t then h : t → z is continuous and depends on j. therefore, according to theorem 3.5(h) again, the function hj : xj → z is continuous. since hj |πj [y ] = fj the function fj := hj is a continuous extension of fj . then the function f := fj ◦ πj is a continuous extension of f that depends on j, as required. � corollary 4.2. let xi be a product space, j be a nonempty proper subset of i, and y be a dense subspace of xi such that πj [y ] × {yi\j } ⊆ y for some y ∈ y . let also z be a hausdorff space and g ∈ c(xi,z) be such that the function f := g|y depends on j. then g depends on j. proof. the domain of agreement of two continuous functions from a fixed space to a hausdorff space is closed [7, 2.1.9], so with z hausdorff we have f = g in theorem 4.1. � the following two corollaries are immediate from corollary 4.2. corollary 4.3. let α ≥ ω, xi be a product space, z be a hausdorff space, g ∈ c(xi,z), j be a nonempty proper subset of i, and y ⊆ xi be a σα-space. then g depends on j if and only if f := g|y depends on j. on the continuity of factorizations 273 corollary 4.4. let xi be a product space, z be a hausdorff space, g ∈ c(xi,z), j be a nonempty proper subset of i, and y be a dense subspace of xi such that y = πj [y ] × πi\j [y ]. then g depends on j if and only if f := g|y depends on j. the following theorem is a special case of corollary 4.4. the countable case (|j| ≤ ω) is mentioned without proof by n. noble and m. ulmer in the proof of proposition 2.1 in [20]. theorem 4.5. let xi be a product space, z be a hausdorff space, g ∈ c(xi,z), yi be a dense subspace of xi for every i ∈ i, y = ∏ i∈i yi, and j be a nonempty proper subset of i. then g depends on j if and only if f := g|y depends on j. the following theorem contains conditions equivalent to the continuity of a given factorization. theorem 4.6. let xi be a product space, z be a hausdorff space, g ∈ c(xi,z), y be a dense subspace of xi , f := g|y , and j be a nonempty subset of i. then the following are equivalent. (i) f depends on j and fj : πj [y ] → z is continuous. (ii) h := g| π −1 j [πj [y ]] depends on j. (iii) g depends on j. (iv) f depends on j, fj is continuous, and fj has a continuous extension fj : xj → z. proof. (i) ⇒ (ii). let x,y ∈ π−1 j [πj [y ]] and z ∈ y be such that xj = yj = zj . then there exists a net (xα) ⊂ y with limit x. since g is continuous we have (g(xα)) → g(x). therefore (fj ((xα)j )) → g(x) for g(xα) = fj ((xα)j ) for each α, and since fj is continuous (fj ((xα)j )) → fj (xj ). thus g(x) = fj (xj ) = fj (zj ) = g(z). similarly we have g(y) = g(z). therefore g(x) = g(y). (ii) ⇒ (iii). let x,y ∈ xi be such that xj = yj . if xj ∈ πj [y ], then x,y ∈ π−1 j [πj [y ]] and therefore g(x) = g(y) for g|π−1 j [πj [y ]] depends on j. now, let xj /∈ πj [y ]; then x,y /∈ π −1 j [πj [y ]]. since πj −1[πj [y ]] is dense in xi there exists a net (xα) ⊂ π −1 j [πj [y ]] such that (xα) → x. then (g(xα)) → g(x) for g is continuous. for each α, we define wα(i) = xα(i) for all i ∈ j and wα(i) = y(i) for all i ∈ i \j. it is clear that the net (wα) → y. then (g(wα)) → g(y) for g is continuous. also, wα,xα ∈ π −1 j [πj [y ]] and (wα)j = (xα)j for every α and since h = g| π −1 j [πj [y ]] depends on j we have g(wα) = g(xα) for every α. therefore g(x) = g(y). thus g depends on j. (iii) ⇒ (iv). if g depends on j then gj is a continuous function which extends fj . (iv) ⇒ (i). obvious. � the following theorem is immediate from theorem 4.6. 274 w. w. comfort, i. s. gotchev and l. recoder-núñez theorem 4.7. let xi be a product space, z be a hausdorff space, g ∈ c(xi,z), and j be a nonempty subset of i. the function g depends on j if and only if there exists a dense subspace y of xi such that f := g|y depends on j and fj is continuous. as an illustrative application of some of our results we provide now in theorem 4.10 a proof of a generalization of a classical theorem of a. m. gleason (see [21, p. 401], [15], or [6]). more general versions of that theorem, with different proofs, can be found in [18], [12], or [14]. lemma 4.8. let α ≥ ω, xi be a product space, y ⊆ xi be a dense subset with |y | ≤ α, z be a t1-space such that χ(z) ≤ α, and f ∈ c(y,z). then there exists j ∈ [i]≤α such that f depends on j and fj is continuous. proof. for y ∈ y let {u(y)a : a ∈ a} be a local base at f(y) in z with |a| ≤ α. since the function f is continuous at y, for every u(y)a, a ∈ a, we can find a basic open neighborhood v (y)a of y in xi such that f[v (y)a ∩ y ] ⊂ u(y)a. let j := ∪y∈y (∪a∈a r(v (y)a)). if two points x,y ∈ y are such that xj = yj then f(x) = f(y), hence f depends on j. now, let w ∈ πj [y ] and y ∈ y be such that w = yj . let also u be an open neighborhood of fj (w) = f(y) in z. then there exists b ∈ a such that f[v (y)b ∩ y ] ⊂ u and since r(v (y)b) ⊂ j we have πj [v (y)b] ∩ πj [y ] = πj [v (y)b ∩ y ]. therefore fj [πj [v (y)b] ∩ πj [y ]] ⊂ u. we conclude that fj is continuous at w. � lemma 4.8 can be “localized” as follows. corollary 4.9. let xi be a product space, z be a t1-space, y ⊆ xi , j be an infinite subset of i, f ∈ c(y,z) depend on j, y ∈ y , and χ(f(y),z) ≤ |j|. then there exists jy ⊆ i such that |jy| = |j|, f depends on jy, and fjy is continuous at yjy . proof. let {ua : a ∈ a} be a local base at f(y) in z with |a| ≤ |j|. since the function f is continuous at y, for every ua, a ∈ a, we can find a basic open neighborhood va of y in xi such that f[va ∩ y ] ⊂ ua. let jy = j ∪ (∪a∈a r(va)). then |jy| = |j|, f depends on jy, and fjy is continuous at yjy . � theorem 4.10. let α ≥ ω, {xi : i ∈ i} be a set of spaces with d(xi) ≤ α for each i ∈ i, z be a hausdorff space with χ(z) ≤ α, and g ∈ c(xi,z). then g depends on ≤ α coordinates. proof. we assume without loss of generality, replacing z by g[xi ] if necessary, that g : xi → z is a surjection. since d(xi) ≤ α for each i ∈ i the cellularity c(xi ) of xi is ≤ α (see [7, 2.3.17], [4, 3.28], [16, 5.6]). thus, c(z) ≤ α and since χ(z) ≤ α and z is hausdorff we have |z| ≤ 2c(z)χ(z) = 2α (see [16, 2.15(b)]), [11]). for z ∈ z let {u(z)a : a ∈ a} be a local base at z in z with |a| ≤ α. the function g is continuous, hence the set g−1[u(z)a] is open for every z ∈ z and on the continuity of factorizations 275 every a ∈ a. since xi is a product of spaces with d(xi) ≤ α for each i ∈ i, the closure of every open set depends on ≤ α many coordinates (see [4, 10.13]). for z ∈ z and a ∈ a, let j(z,a) ∈ [i]≤α be a nonempty set such that the set g−1[u(z)a] depends on it and let k := ∪z∈z (∪a∈aj(z,a)). if x,y ∈ xi are such that xk = yk and g(x) = z then y ∈ g−1[u(z)a] ⊆ g −1[u(z)a] for every a ∈ a and since z is hausdorff {z} = ∩a∈au(z)a, thus y ∈ g −1(z). therefore g(x) = g(y), hence g depends on k. thus, gk : xk → z is continuous and since |k| ≤ 2α it follows from hewitt-marczewski-pondiczery theorem (see [7, 2.3.15]) that xk contains a dense set y ∈ [xk ] ≤α. then by lemma 4.8 (with xk now in place of xi ) there exists j ∈ [k] ≤α such that gk depends on j. therefore g depends on j. � remark 4.11. as it is clear from the proof of theorem 4.10, the hypothesis χ(z) ≤ α there may be relaxed to the condition 2χ(z) ≤ 2α. since χ(z) ≤ w(z) for every space z, theorem 4.10 gives a proof of theorem 10.14 in [4]: corollary 4.12. let α ≥ ω, {xi : i ∈ i} be a set of spaces with d(xi) ≤ α for i ∈ i, z be a hausdorff space with w(z) ≤ α, and g ∈ c(xi,z). then g depends on ≤ α coordinates. the foregoing results afford several conditions which ensure that (under suitable hypotheses) a function g : xi → z depends on a set j ⊆ i iff the restricted function g|y does so. we record two instances of particular interest. theorem 4.13. let xi be a product space, z be a hausdorff space, g ∈ c(xi,z), j be a nonempty proper subset of i, and y be a dense subset of xi . if either (a) [j′ ⊆ i, |j′| ≤ |j|] ⇒ πj′ [y ] = xj′ , or (b) πi\j [π −1 j (πj (y)) ∩ y ] is dense in xi\j for every y ∈ y , then g depends on j iff f := g|y depends on j. proof. surely if g depends on j then f = g|y depends on j. for the reverse implications, it is enough to note from theorem 3.5 that fj is continuous, so the implication (i) ⇒ (iii) of theorem 4.6 applies. � 5. on the comfort-negrepontis question (question 1.3) in this section we give some sufficient conditions for a positive answer to the comfort–negrepontis question. we begin with a workable equivalent condition which in particular cases is readily verified (or, as in section 2, refuted). theorem 5.1. let α ≥ ω, xi be a product space, z be a space, y ⊆ xi be a dense subspace such that πj [y ] is c(z)-embedded in xj for every nonempty j ∈ [i]<α, and f ∈ c(y,z). the function f has a continuous extension f : xi → z that depends on < α coordinates if and only if there exists a nonempty j ∈ [i]<α such that f depends on j and fj is continuous. 276 w. w. comfort, i. s. gotchev and l. recoder-núñez proof. if f ∈ c(xi,z) depends on some nonempty j ∈ [i] <α then fj is continuous (see theorem 3.2). therefore fj is continuous. if f depends on j and fj is continuous, then since πj [y ] is c(z)-embedded in xj there is fj such that fj ⊆ fj ∈ c(xj,z). then f := fj ◦ πj is a continuous extension of f and f depends on j. � it is easily seen from the proof of the above theorem that if we need to extend a particular function f that depends on a given nonempty set j ⊆ i then we do not require the strong hypothesis that πj [y ] is c(z)-embedded in xj . we need only that fj is continuous and that it extends to continuous fj : xj → z. therefore theorem 5.1 could be restated as follows. theorem 5.2. let xi be a product space, z be a space, j be a nonempty subset of i, y ⊆ xi be a dense subspace, and f ∈ c(y,z) depend on j. the function f has a continuous extension f : xi → z that depends on j if and only if fj is continuous and has a continuous extension fj : xj → z. the next theorem generalizes theorem 1.4 and gives conditions sufficient that the answer to question 1.3 is in the affirmative. theorem 5.3. let xi be a product space, j be a nonempty proper subset of i, α = |j|+, and y be a nonempty subspace of xi . let also z be a space and f ∈ c(y,z) depend on j. if (a) y = xi ; or (b) [y ∈ y ] ⇒ σα(y) ⊆ y ; or (c) [j′ ⊆ i, |j′| ≤ |j|] ⇒ πj′ [y ] = xj′ ; or (d) [j′ ⊆ i,j′ = j ∪ f with |f | < ω] ⇒ πj′ [y ] = xj′ ; or (e) π−1j [πj [y ]] = y and πj [y ] is c(z)-embedded in xj ; or (f) πi\j [π −1 j (πj (y)) ∩ y ] is dense in xi\j for every y ∈ y and πj [y ] is c(z)-embedded in xj ; or (g) πj [u] ∩ πj [y ] = πj [u ∩ y ] for every basic open set u in xi and πj [y ] is c(z)-embedded in xj ; or (h) πj [y ] × {yi\j } ⊆ y for some y ∈ y and πj [y ] is c(z)-embedded in xj then f has a continuous extension f : xi → z that depends on j. proof. as in theorem 3.5, clearly (a) ⇒ (b), (b) ⇒ (c), (c) ⇒ (d), and (d) ⇒ (g). also (e) ⇒ (f) and (f) ⇒ (g). if (g) holds then fj in continuous by theorem 3.5; thus f := fj ◦ πj |y is a continuous extension of f that depends on j, so under any of the conditions (a), (b), (c), (d), (e), (f), and (g) f has a continuous extension that depends on j. similarly, if (h) holds then fj is continuous by theorem 3.5; thus f has a continuous extension that depends on j. � a partial answer, in the positive, to question 1.3 is given in theorem 5.6. on the continuity of factorizations 277 lemma 5.4. let x be a space, p ∈ x, and (wn) be a sequence of open subsets of x such that wn+1 ⊆ w x n+1 ⊆ wn and ∩n wn = {p}. then (wn) is locally finite at each point of x\{p}. proof. we are to show for p 6= x ∈ x that some neighborhood u of x meets wn for only finitely many n. given x, it is enough to choose k such that x /∈ w x k and to set u := x\w x k . � we say as usual that a subset y of a space x is sequentially closed if [yn ∈ y and yn → p ∈ x] ⇒ p ∈ y , and x is a sequential space if every sequentially closed subset of x is closed. theorem 5.5. let x be a tychonoff space with countable pseudocharacter and let y be a c∗-embedded subset of x. then y is sequentially closed in x. proof. suppose there is a sequence yn ∈ y such that yn → p ∈ x\y , and let (un) be a (countable) local pseudobase at p in x. let vn0 := u0 and choose yn0 ∈ vn0 . suppose that vnk and ynk have been defined. let nk+1 > nk be such that ynk+1 ∈ vnk+1 , where vnk+1 is a neighborhood in x of p such that vnk+1 ⊆ v x nk+1 ⊆ vnk ∩unk and ynk /∈ v x nk+1 . with the sequences vnk and ynk so defined, choose a function fk ∈ c(x, [0, 1]) such that fk ≡ 0 for k even, and for odd k, using the tychonoff property of x, such that 0 ≤ fk ≤ 1, fk(ynk ) = 1, and fk ≡ 0 on (x\vnk ) ∪ v x nk+1 . since the sequence vnk \v x nk+1 is pairwise disjoint, the function f := σk fk : x → [0, 1] is well-defined. according to lemma 5.4, the sequence (vnk ) is locally finite at each point of x\{p}, so the sequence ((vnk \v x nk+1 ) ∩ y ) is locally finite in y ; thus g := f|y ∈ c ∗(y ). for each neighborhood u of p in x there is k such that yn2k,yn2k+1 ∈ u. we have yn2k+1 ∈ u ∩ y with g(yn2k+1 ) = 1 and yn2k ∈ vn2k ∩ y with g(yn2k ) = 0. we conclude that g does not extend continuously to p, a contradiction. � theorem 5.6. let α ≤ ω1 and s be a class of tychonoff spaces such that if sn ∈ s (n < ω, repetitions permitted) then πn<ω sn ∈ s and πn<ω sn is a sequential space. let also {xi : i ∈ i} ⊆ s with ψ(xi) ≤ ω for each i ∈ i, and y be a dense subspace of xi such that πj [y ] is c ∗-embedded in xj for every nonempty j ∈ [i]<α. if f ∈ c∗(y ) and there is j ∈ [i]<α such that f depends on j, then f extends continuously over xi . proof. by virtue of lemma 1.2 (taking κ = ω and z = r there), it suffices to prove that πj [y ] = xj for all j ∈ [i] <α. take j ∈ [i]<α. from ψ(xi) ≤ ω (all i ∈ j) and |j| ≤ ω it follows easily that ψ(xj ) ≤ ω, so πj [y ] is sequentially closed in xj by theorem 5.5 and hence closed in xj , so πj [y ] = xj . � remark 5.7. the class of all tychonoff, first countable spaces of course satisfies the hypotheses required of s in the statement of theorem 5.6, but other less familiar classes of spaces do so as well. for example, one may take for s the class of all tychonoff, bi-sequential spaces (see [17, definition 3.d.1]) with 278 w. w. comfort, i. s. gotchev and l. recoder-núñez countable pseudocharacter but not first countable. (it is known that every bisequential space is sequential ([17]); the class of bi-sequential spaces is closed under countable products ([17]); there exists a bi-sequential tychonoff space without countable pseudocharacter ([17, example 10.4]); and there exists a bisequential tychonoff space with countable pseudocharacter which is not first countable (see example 5.8 below).) we are indebted to gary gruenhage for the following example. example 5.8. let k := [0, 1] be the interval [0, 1] with the usual topology. for every rational q ∈ k, choose the constant sequence sq := {q} and for every irrational r ∈ k, choose a sequence sr of rational numbers in k converging to r. define s := ∪x∈ksx and x := s ∪ {∞}. let t be the smallest topology on x such that: (1) for every q ∈ s, {q} is t -open in x; and (2) for every finite subset f of s and every finite subset g of [0, 1] the set n(∞,f,g) := {∞} ∪ [s\(f ∪ ∪x∈gsx)] is a t -open neighborhood of ∞. it follows from [10, proposition 3.2] that x is bi-sequential. the family {n(∞, ∅,{q}) : q ∈ s} is countable and {∞} = ∩q∈sn(∞, ∅,{q}). hence ψ(x) = ω. now take a countable family {n(∞,fn,gn) : n < ω} of neighborhoods of ∞. choose an irrational r ∈ k not in ∪n<ωgn. then n(∞, ∅,{r}) contains no n(∞,fn,gn), n < ω. therefore x is not first countable. since each of the t -basic open sets specified in (1) and (2) is also t -closed, the hausdorff space (x,t ) is a tychonoff space. 6. concluding remarks we remark that most of the results of the foregoing sections admit natural generalizations to the κ-box topology. since these are routine and their formulation adds little to our broad understanding, we leave the details to the interested reader. we note explicitly, however, that question 1.3 remains open even in the case κ = ω. we anticipate that question 1.3 has a negative answer. when seeking a positive response, however, we have been drawn to the following question, which is closely related to theorem 2.8. clearly, a positive answer to 6.1 will respond positively also to 1.3 (in the case that z is a hausdorff space). problem 6.1. let α ≥ ω, xi be a product space, z be a space, and y be a dense subspace of xi such that πj [y ] is c(z)-embedded in xj for every nonempty j ∈ [i]<α. if f ∈ c(y,z) depends on some nonempty j′ ∈ [i]<α, must there be a nonempty j ∈ [i]<α such that f depends on j and fj is continuous? returning finally to question 1.3, we remark that if a counterexample exists then there will be a product space xi , a cardinal number α ≥ ω, a dense subspace y of xi , a space z, and a function f ∈ c(y,z) such that πj [y ] is c(z)-embedded in xj for every j ∈ [i] <α, f depends on some nonempty j ∈ [i]<α, and f does not extend continuously over xi . in particular, then, fj must be discontinuous for each j ∈ [i] <α on which f depends. on the continuity of factorizations 279 references [1] w. w. comfort, a nonpseudocompact product space whose finite subproducts are pseudocompact, math. annalen 170 (1967), 41–44. 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[23] m. ulmer, c-embedded σ-spaces, pacific j. math. 46 (1973), 591–602. received october 2007 accepted may 2008 280 w. w. comfort, i. s. gotchev and l. recoder-núñez w. w. comfort (wcomfort@wesleyan.edu) department of mathematics and computer science, wesleyan university, middletown, ct 06459, usa ivan s. gotchev (gotchevi@ccsu.edu) department of mathematical sciences, central connecticut state university, 1615 stanley street, new britain, ct 06050, usa luis recoder-núñez (recoderl@ccsu.edu) department of mathematical sciences, central connecticut state university, 1615 stanley street, new britain, ct 06050, usa rodabaughagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 77-108 functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics s. e. rodabaugh ∗ abstract. this paper studies various functors between (latticevalued) topology and (lattice-valued) bitopology, including the expected “doubling” functor ed : l-top → l-bitop and the “cross” functor e× : l-bitop → l 2-top introduced in this paper, both of which are extremely well-behaved strict, concrete, full embeddings. given the greater simplicity of lattice-valued topology vis-a-vis latticevalued bitopology and the fact that the class of l2-top’s is strictly smaller than the class of l-top’s encompassing fixed-basis topology, the class of e×’s makes the case that lattice-valued bitopology is categorically redundant. as a special application, traditional bitopology as represented by bitop is (isomorphic in an extremely well-behaved way to) a strict subcategory of 4-top, where 4 is the four element boolean algebra; this makes the case that traditional bitopology is a special case of a much simpler fixed-basis topology. 2000 ams classification: primary: 03e72, 54a10, 54a40, 54e55. secondary: 06f07, 18a22, 18a30, 18a35, 18b30. keywords: unital-semi-quantale, unital quantale, (fixed-basis) topology, (fixed-basis) bitopology, order-isomorphism, categorical (functorial) embedding, redundancy. 1. introduction and preliminaries 1.1. motivation. bitopology has a long and distinguished history spanning five decades and a literature of some 700 papers [29] with traditional bitopology playing a wide range of roles in baire spaces, homotopy and algebraic topology, generalizations of metric spaces, biframes, programming semantics, etc. ∗support of youngstown state university via a sabbatical for the 2005–2006 academic year is gratefully acknowledged. 78 s. e. rodabaugh first defined and used in [31, 32, 3, 4], a bitopological space was originally defined as a triple ((x, t) , (x, s) , e) with (x, t) , (x, s) topological spaces and e : (x, t) → (x, s) a continuous bijection—cf. [3]. but if we set t ′ = {e→ (u ) : u ∈ t} , then t′ is a topology on x and the continuity of e insures that idx : (x, t ′) → (x, s) is continuous, i.e., that t′ ⊃ s. it is therefore not surprising that almost immediately [4] the original definition was replaced by the simpler, equivalent definition that a bitopological space is a triple (x, t, s) with t, s topologies on x with t ⊃ s, t being called the strong topology and s the weak topology. even in the broader lattice-valued topology setting, this definition plays a categorical role (proposition 3.5 below). since a quasi-pseudo-metric p on a set x determines its conjugate quasipseudo-metric q, namely by q (x, y) = p (y, x) , quasi-pseudo-metrics necessarily occur in conjugate pairs which generate pairs of topologies that need not be related. thus the definition of a traditional bitopological space was generalized in [22] to its modern form to be an ordered triple (x, t, s) with t, s topologies on x (and no relationship assumed between t and s). further, a bicontinuous mapping f : (x, t1, t2) → (y, s1, s2) is a mapping f : x → y satisfying t1 ⊃ (f ←) → (s1) , t2 ⊃ (f ←) → (s2) , i.e., f : (x, t1) → (y, s1) and f : (x, t2) → (y, s2) are both continuous. with the composition and identities of set, one has the category bitop, which is a topological construct and hence strongly complete and strongly cocomplete along with many other properties. there is a voluminous literature for bitop concerning separation, compactness, connectedness, completion, connections to uniform and quasi-uniform spaces, homotopy groups and algebraic topology, relationships to bilocales [2], a recently emerging role in programming semantics [25], etc. a significant part of the recent literature on bitopology is in lattice-valued mathematics [30, 27, 50, 51]. letting l be a us-quantale (subsection 1.2 below) and x a set, the triple (x, τ, σ) is an l-bitopological space if τ, σ are l-topologies on x (subsection 1.5); and such spaces with l-bicontinuous mappings comprise the category l-bitop. this category is a topological construct, strongly complete, strongly cocomplete, and so on. the schemum {l-bitop : l ∈ |usquant|} essentially includes bitop via its functorial isomorph 2-bitop. this paper studies functorial relationships between (lattice-valued) bitopology and (lattice-valued) topology in sections 2–3. the expected functor ed strictly embeds l-top into l-bitop, a functor we dub the “doubling” functor; and to fully study ed, it is necessary to construct several functors from l-bitop to l-top whose relationships with ed lead us to conclude that ed is extremely well-behaved. but on the other hand, for each l ∈ |usquant| , the direct product l2 ∈ |usquant| and there is an embedding e× of lbitop into l2-top (3.4.1) which is extremely well-behaved (subsubsections 3.4.2, 3.4.3) if l is a u-quantale (subsection 1.2) and a strict embedding if l bitopology as topology 79 is consistent (subsubsection 3.4.1). given that this embedding is strict (for consistent l) and that the l2’s form a proper subclass of usquant—which means (lattice-valued) bitopology is properly “contained” in the proper subclass { l2-bitop : l ∈ |usquant| } of {l-top : l ∈ |usquant|} , it follows (lattice-valued) topology (twice) strictly generalizes bitopology. in section 4 we summarize some metamathematical facts: given that lattice-valued topology is fundamentally simpler than lattice-valued bitopology—a membership lattice and one topology vis-a-vis a membership lattice and two topologies, it follows that topology and the class of embeddings e×’s make lattice-valued bitopology categorically redundant; and as a special application, traditional bitopology bitop strictly embeds in an extremely well behaved way into 4top, the latter being lattice-valued topology based on the four-element boolean algebra 4, so that traditional bitopology both is a strictly special case of the simpler lattice-valued topology and demonstrates the necessity of lattice-valued topology. on the other hand, this last fact points the way for bringing over into lattice-valued topology successful ideas from the extensive literature of traditional bitopology; in particular, traditional bicompactness mandates, via the embedding of bitop into 4-top, the compactness of [5] for lattice-valued topology (corollary 4.7). 1.2. lattice theoretics. a semi-quantale (l,≤,⊗) (s-quantale) is a complete lattice (l,≤) equipped with a binary operation ⊗ : l × l → l, with no additional assumptions, called a tensor product; an ordered semi-quantale (os-quantale) is an s-quantale in which ⊗ is isotone in both variables; a complete quasi-monoidal lattice (cqml) [20, 41] is an os-quantale for which ⊤ is an idempotent element for ⊗; a unital semi-quantale (us-quantale) is an s-quantale in which ⊗ has an identity element e ∈ l called the unit [33]—units are unique; a quantale is an s-quantale with ⊗ associative and distributing across arbitrary ∨ from both sides (implying ⊥ is a two-sided zero) [20, 33, 49]; and a unital quantale (u-quantale) is a us-quantale which is a quantale; and a strictly two-sided quantale (st-quantale) is a u-quantale for which e = ⊤ [20]. all quantales are os-quantales. the notions of s-quantales, os-quantales, and us-quantales are from [45, 46]. squant comprises all semi-quantales together with mappings preserving ⊗ and arbitrary ∨ ; osquant is the full subcategory of squant of all osquantales; usquant is a subcategory of squant comprising all us-quantales together with all mappings preserving arbitrary ∨ , ⊗, and e; quant is the full subcategory of osquant of all quantales; and uquant is the full subcategory of uosquant of all unital quantales. note uos-quantales for which ⊗ = ∧ (binary) are semiframes and sfrm is the full subcategory of uosquant of all semiframes; and u-quantales for which ⊗ = ∧ (binary) are frames— in which case e = ⊤—and frm is the full subcategory of uquant of all frames. semiframes equipped with an order-reversing involution are complete demorgan algebras; and s-quantales equipped with a semi-polarity 80 s. e. rodabaugh [16] (∀α, β ∈ l, α ≤ β ⇒ β′ ≤ α′ and α ≤ (α′) ′ ) are complete semidemorgan s-quantales. throughout this paper, the requirement of us-quantale [u-quantale] can be relaxed to s-quantale [quantale, resp.] if one wishes to consider the relationships between q-topology and q-bitopology ([46] and subsection 1.5 below). justifying the above lattice-theoretic notions is a wealth of examples (see [17, 20, 21, 23, 33, 35, 39, 40, 41, 44, 46] and their references). the lattice 2 = {⊥,⊤} with ⊥ 6= ⊤; and a lattice is consistent if it contains 2 and inconsistent if it is singleton (with ⊥ = ⊤). 1.3. powerset operators. let x ∈ |set| and l ∈ |squant|. then lx is the l-powerset of x of all l-subsets of x. the constant l-subset member of lx having value α is denoted α. all order-theoretic operations (e.g., ∨ , ∧ ) and algebraic operations (e.g., ⊗) on l lift point-wise to lx and are denoted by the same symbols. in the case l ∈ |usquant| , the unit e lifts to the constant map e, which is the unit of ⊗ as lifted to lx . the operator ℘∅ : |set| → |set| is useful in this paper, where ℘∅ (x) denotes the poset of all the nonempty subsets of x. let l ∈ |squant| , x, y ∈ |set| , and f : x → y be in set. then the standard (traditional) image and preimage operators f→ : ℘ (x) → ℘ (y ) , f← : ℘ (x) ← ℘ (y ) are f→ (a) = {f (x) ∈ y : x ∈ a} , f← (b) = {x ∈ x : f (x) ∈ b} , and the zadeh image and preimage operators f→l : l x → ly , f←l : l x ← ly [53] are f→l (a) (y) = ∨ {a (x) : x ∈ f← ({y})} , f←l (b) = b ◦ f. if l is understood, it may be dropped providing the context distinguishes these operators from the traditional operators. it is observed that f→ and f← are naturally isomorphic to f→2 and f ← 2 , resp. it is well-known [36, 37, 39, 40, 46] that each f←l preserves arbitrary ∨ , arbitrary ∧ , ⊗, and all constant maps, as well as the unit e if l ∈ |usquant|; each f→l preserves arbitrary ∨ ; f→ ⊣ f←, f→l ⊣ f ← l ; f→ and f→l are left-inverses [right-inverses] of f ← and f←l , resp., if f is surjective [injective, resp.]; and f→, f←, f→l , f ← l are all order-isomorphisms if and only if f is a bijection. powerset operators and the powerset theories underlying lattice-valued mathematics are studied extensively in [6, 14, 7, 8, 15, 10, 11, 36, 37, 39, 40, 46]. 1.4. category theoretics. the main reference for categorical notions is [1], to which we refer the reader for various properties of functors as well as various versions of the adjoint functor theorem and related notions. the proving of functorial adjunctions is done via lifting (or major) and naturality (or minor) diagrams in the manner of [28, 36, 37, 41]. bitopology as topology 81 1.5. topology and bitopology. given l ∈ |usquant|, the category l-top comprises objects of the form (x, τ ), where τ ⊂ lx is closed under arbitrary ∨ and binary ⊗ and contains e—so that τ is a sub-us-quantale of lx , together with morphisms f : (x, τ ) → (y, σ), where f : x → y is a function and τ ⊃ (f←l ) → (σ) , namely f←l (v) ∈ τ for each v ∈ σ. the objects (x, τ ) are called l-topological spaces and τ is an l-topology on x comprising l-subsets of x; and the morphisms f are called l-continuous. cf. [20, 41, 46]. similarly, the category l-bitop comprises objects of the form (x, τ, σ) , where τ, σ are l-topologies on x, together with morphisms f : (x, τ1, τ2) → (y, σ1, σ2), where f : x → y is a function and τ1 ⊃ (f ← l ) → (σ1) , τ2 ⊃ (f ← l ) → (σ2) . the objects (x, τ, σ) are called l-bitopological spaces and (τ, σ) is an lbitopology on x; and the morphisms f are called l-bicontinuous. if the l is clear in context, it may be dropped from the labels. as noted in subsection 1.1, the traditional category bitop is isomorphic to 2-bitop (cf. 3.25 below) and embeds into each l-bitop, and similarly top is isomorphic to 2-top and embeds into each l-top. each of l-top and l-bitop has the base l of the category fixed and so is part of fixed-basis (lattice-valued) topology and fixed-basis (lattice-valued) bitopology, resp. the disciplines of fixed-basis topology and fixed-basis bitopology are encompassed by the respective classes {l-top : l ∈ |usquant|} , {l-bitop : l ∈ |usquant|} . both l-top and l-bitop are topological over set and have small fibres, hence are [co]complete and [co]well-powered, and hence are strongly [co]complete with many other nice properties (see 3.36 and its proof below). the categorical product for l-top is given in, or adapted from, [12, 52] (cf. [20, 41]) and for {(xγ , τγ ) : γ ∈ γ} denoted by   ∏ γ ∈γ (xγ , τγ ) ,{πγ}γ ∈γ   , ∏ γ ∈γ (xγ , τγ ) ≡ (×γ ∈γxγ , πγ ∈γτγ ) , where {πγ : γ ∈ γ} are the projections. the binary l-topological product for two spaces (x, τ ) , (y, σ) is denoted (x, τ ) π (y, σ) or (x × y, τ π σ) with projections {π1, π2}. the categorical product for l-bitop for {(xγ , τγ ) : γ ∈ γ} is   ∏ γ ∈γ (xγ , τγ , σγ ) ,{πγ}γ ∈γ   , ∏ γ ∈γ (xγ , τγ , σγ ) ≡ (×γ ∈γxγ , πγ ∈γτγ , πγ ∈γσγ ) , where πγ ∈γτγ , πγ ∈γσγ are the l-topological product topologies in each slot and the projections are as above. 82 s. e. rodabaugh an l-topology τ is weakly stratified [20] if {α : α ∈ l} ⊂ τ , nonstratified if it is not weakly stratified, and anti-stratified [9, 35] if {α : α ∈ l, α ∈ τ} = {⊥, e} ; so a weakly stratified topology contains all constant l-subsets, while an antistratified topology contains precisely the constant l-subsets ⊥ and e (which are the same if l is inconsistent with ⊥ = ⊤). an l-topological space is weakly stratified [anti-stratified] if its topology is weakly stratified [anti-stratified], and an l-bitopological space is weakly stratified [anti-stratified] if both topologies are weakly stratified [anti-stratified]. the inclusionist position that the axioms of a fixed-basis topology must allow for all types of stratification has recently received additional, emphatic confirmations from both lattice-valued frames [35] and topological systems in domain theory [9]. the following definition and proposition are needed in this paper. definition 1.1. let x be a set and let l be a us-quantale. then the ltopological fibre, respectively, l-bitopological fibre on x is l-t (x) ≡ { τ ⊂ lx : (x, τ ) ∈ |l-top| } , l-bt (x) ≡ {(τ, σ) : (x, τ, σ) ∈ |l-bitop|} . proposition 1.2. let x be a set, let l be a us-quantale, and recall ℘∅ from subsection 1.3. (1) l-t (x) is a complete meet subsemilattice of ℘ ( lx ) ; and since each l-topology is nonempty, l-t (x) ⊂ ℘∅ ( lx ) . (2) l-bt (x), ordered coordinate-wise by inclusion, is a complete meet subsemilattice of ℘ ( lx ) ×℘ ( lx ) ; and further, l-bt (x) ⊂ ℘∅ ( lx ) × ℘∅ ( lx ) . proof. the first part of (1) is well-known, and the second part of (1) is trivial. now (2) follows from (1) since l-bt (x) = l-t (x) × l-t (x) ⊂ ℘∅ ( lx ) × ℘∅ ( lx ) . � finally, we need the notion of a subbase of an l-topology τ on x [41]. we say σ ⊂ lx is a subbase of τ , written τ = 〈〈σ〉〉 , if τ = ⋂ {τ′ ∈ l-t (x) : σ ⊂ τ′} , the right-hand side always existing by proposition 1.2(1), and we say β ⊂ lx is a base of τ , written τ = 〈β〉 , if ∀u ∈ τ, ∃bu ⊂ β, u = ∨ bu. one can always pass from a subbase σ to a topology τ through a base β in the traditional way, written τ = 〈β〉 = 〈〈σ〉〉 , if and only if ⊗ is associative and distributes across arbitrary ∨ , i.e., if and only if l is a u-quantale. bitopology as topology 83 2. functorial interpretations of topology as bitopology for each us-quantale l, this section records a simple (and expected) “doubling” embedding ed : l-top → l-bitop. the behavior of ed w.r.t. limits and colimits—it preserves, reflects, detects both—is examined completely in subsections 3.1–3.2 below. it emerges that ed is an extremely well-behaved embedding. proposition 2.1. let l be a us-quantale. define ed : l-top → l-bitop by the following correspondences: ed (x, τ ) = (x, τ, τ ) , ed (f ) = f. then ed is a concrete, full, strict embedding; and so l-top is isomorphic to a full subcategory of l-bitop. proof. all details are straightforward. � 3. functorial interpretations of bitopology as topology this section records several interpretations of bitopology as topology, the most important of which would seem to be the extremely well-behaved embedding e× of subsection 3.4. 3.1. fl, fr, f∧ : l-bitop → l-top and behavior of ed : l-top → lbitop w.r.t. limits. this subsection constructs the concrete, faithful, full forgetful functors—the “left-forgetful” functor fl : l-bitop → l-top and the “right-forgetful” fr : l-bitop → l-top—as well as the concrete, faithful “meet” functor f∧ : l-bitop → l-top and shows f∧ is the left-adjoint of ed of the previous section and that each of fl, fr is a left-adjoint of ed under certain restrictions. proposition 3.1. let l be a us-quantale and define fl, fr : l-bitop → ltop as follows: fl (x, τ, σ) = (x, τ ) , fl (f ) = f, fr (x, τ, σ) = (x, σ) , fr (f ) = f. then each of fl, fr is a concrete, faithful, full, object-surjective functor, but need not be an embedding. proof. we comment only on fl. trivially, fl is a concrete, faithful, objectsurjective functor. as for fullness, let f : (x, τ ) → (y, σ) in l-top; then f : (x, τ, τ ) → (y, σ, σ) is l-bicontinuous, so is in l-bitop, and maps to f : (x, τ ) → (y, σ). now suppose that either [|x| ≥ 1 and |l| ≥ 3] or [|x| ≥ 2 and |l| ≥ 2]; then ∃τ, σ ∈ l-t (x) with τ 6= σ, so that fl (x, τ, σ) = (x, τ ) = fl (x, τ, τ ), and hence fl does not inject objects and is not an embedding. � proposition 3.2. let l be a us-quantale and define f∧ : l-bitop → l-top as follows: f∧ (x, τ, σ) = (x, τ ∩ σ) , f∧ (f ) = f. 84 s. e. rodabaugh then f∧ is a concrete, faithful, object-surjective functor, but need not be full nor an embedding. proof. since l-top has complete fibres, f∧ is well-defined on objects. now let f : (x, τ1, τ2) → (y, σ1, σ2) be l-bicontinuous. since the image operator of the zadeh preimage operator preserves ⊂, it follows τ1 ⊃ (f ← l ) → (σ1) , τ2 ⊃ (f ← l ) → (σ2) ⇒ τ1 ∩ τ2 ⊃ (f ← l ) → (σ1) ∩ (f ← l ) → (σ2) ⊃ (f ← l ) → (σ1 ∩ σ2) , so that f : (x, τ1 ∩ τ2) → (y, σ1∩σ2) is l-continuous. immediately, f∧ is a concrete, faithful functor which surjects objects. now let l be a complete demorgan algebra (with ⊗ = ∧ (binary)) and consider each of the l-bitopological spaces (r (l) , τl (l) , τl (l)) and (r (l) , τl (l) , τ (l)) , where r (l) is the l-fuzzy real line, τl (l) is the left-hand l-topology on r (l) determined by the lt operators and τ (l) is the standard l-topology on r (l) [43]. then (r (l) , τl (l) , τl (l)) 6= (r (l) , τl (l) , τ (l)) and f∧ (r (l) , τl (l) , τl (l)) = (r (l) , τl (l)) = (r (l) , τl (l) ∩ τ (l)) = f∧ (r (l) , τl (l) , τ (l)) , showing that f∧ does not inject objects, so is not an embedding. now letting f : r (l) → r (l) be idr(l), we have f : f∧ (r (l) , τl (l) , τ (l)) → f∧ (r (l) , τ (l) , τl (l)) is l-continuous, but f : (r (l) , τl (l) , τ (l)) → (r (l) , τ (l) , τl (l)) cannot be l-bicontinuous (because of the first slot). the concreteness of f∧ implies there exists no g ∈ l-bitop with f∧ (g) = f , so f∧ is not full. � theorem 3.3. let l be a us-quantale. then f∧ ⊣ ed, this adjunction is a monocoreflection, and f∧ takes l-bitop to a monocoreflective subcategory of l-top. on the other hand, ed ⊣/ f∧. proof. let (x, τ1, τ2) ∈ |l-bitop| , choose η = id : (x, τ1, τ2) → edf∧ (x, τ1, τ2) = (x, τ1 ∩ τ2, τ1 ∩ τ2) , and note η is an l-continuous injection. now let (y, σ) ∈ |l-top| , suppose f : (x, τ1, τ2) → ed (y, σ) = (y, σ, σ) is l-bicontinuous, and note τ1 ⊃ (f ← l ) → (σ) , τ2 ⊃ (f ← l ) → (σ) ⇒ τ1 ∩ τ2 ⊃ (f ← l ) → (σ) , making f : f∧ (x, τ1, τ2) = (x, τ1 ∩ τ2) → (y, σ) l-continuous. then f = f is the unique choice making f = f ◦ η. the naturality diagram now follows by concreteness as do the other claims concerning f∧ ⊣ ed. finally, given f∨ of subsection 3.2 and ed ⊣ f∨ of 3.9 below, ed ⊣/ f∧ since f∧ ≇ f∨ and right-adjoints are essentially unique. � bitopology as topology 85 definition 3.4. l-bitop (⊂) [l-bitop (⊃)] is the full subcategory of lbitop of all spaces (x, τ, σ) in which τ ⊂ σ [τ ⊃ σ]. note bitop (⊂) and bitop (⊃) (essentially setting l = 2) express the original sense of traditional bitopology [3, 4]. proposition 3.5. let l be a us-quantale. then fl |l-bitop(⊂) = f∧|l-bitop(⊂), fr |l-bitop(⊃) = f∧|l-bitop(⊃). hence fl |l-bitop(⊂) ⊣ ed and fr |l-bitop(⊃) ⊣ ed, but ed ⊣/ fl |l-bitop(⊂) and ed ⊣/ fr |l-bitop(⊃). proof. the restricted forgetful functors obviously coincide with the meet functor. observing that ed maps into each of l-bitop (⊂) and l-bitop (⊃), the claimed adjunctions are then immediate from 3.3. the claimed non-adjunctions follow from 3.10 below. � corollary 3.6. let l be a us-quantale. the following hold: (1) ed preserves all strong limits and f∧ preserves all strong colimits. (2) fl preserves the strong colimits of l-bitop (⊂), fr preserves the strong colimits of l-bitop (⊃), and ed preserves strong limits into each of l-bitop (⊂) and l-bitop (⊃). proposition 3.7. for each us-quantale l, ed : l-top → l-bitop reflects and detects all limits and hence lifts all limits and is transportable. proof. the details are straightforward using 3.6 and proposition 13.34 [1]. � 3.2. f∨ : l-bitop → l-top and behavior of ed : l-top → l-bitop w.r.t. colimits. this subsection constructs the concrete, faithful “join” functor f∨ : l-bitop → l-top and shows it is the right-adjoint of ed of the previous section. proposition 3.8. let l be a us-quantale and define f∨ : l-bitop → l-top as follows: f∨ (x, τ, σ) = (x, τ ∨ σ) , f∨ (f ) = f, where τ ∨ σ = 〈〈τ ∪ σ〉〉 then f∨ is a concrete, faithful, object-surjective functor, but need not be full nor an embedding. proof. since l-top has complete fibres, f∨ is well-defined on objects. now let f : (x, τ1, τ2) → (y, σ1, σ2) be l-bicontinuous. since the image operator of the zadeh preimage operator preserves unions, then τ1 ⊃ (f ← l ) → (σ1) , τ2 ⊃ (f ← l ) → (σ2) ⇒ τ1 ∨ τ2 ⊃ τ1 ∪ τ2 ⊃ (f ← l ) → (σ1) ∪ (f ← l ) → (σ2) = (f ← l ) → (σ1 ∪ σ2) , 86 s. e. rodabaugh so that f : (x, τ1 ∨ τ2) → (y, σ1 ∨ σ2) is l-subbasic continuous. by theorem 3.2.6 of [41] as restricted to the fixed-basis case and then adapted to the usquantalic case, f : (x, τ1 ∨ τ2) → (y, σ1 ∨ σ2) is l-continuous. immediately, f∨ is a concrete, faithful functor which surjects objects. now let l be a complete demorgan algebra (with ⊗ = ∧ (binary)) and consider each of the l-bitopological spaces (r (l) , τl (l) , τr (l)) and (r (l) , τ (l) , τ (l)) , where r (l) is the l-fuzzy real line, τl (l) is the left-hand l-topology on r (l) determined by the lt operators, τr (l) is the left-hand ltopology on r (l) determined by the rt operators, and τ (l) is the standard l-topology on r (l) [43]. then (r (l) , τl (l) , τr (l)) 6= (r (l) , τ (l) , τ (l)) and f∨ (r (l) , τl (l) , τr (l)) = (r (l) , τl (l) ∨ τr (l)) = (r (l) , τ (l)) = f∨ (r (l) , τ (l) , τ (l)) , showing that f∨ does not inject objects, so is not an embedding. now letting f : r (l) → r (l) be idr(l), we have f : f∨ (r (l) , τl (l) , τr (l)) → f∨ (r (l) , τr (l) , τl (l)) is l-continuous, but f : (r (l) , τl (l) , τr (l)) → (r (l) , τr (l) , τl (l)) cannot be l-bicontinuous. the concreteness of f∨ implies there exists no g ∈ l-bitop with f∨ (g) = f , so f∨ is not full. � theorem 3.9. let l be a us-quantale. then ed ⊣ f∨, this adjunction is an isoreflection, and f∨ takes l-bitop to an isoreflective subcategory of l-top. on the other hand, f∨ ⊣/ ed. proof. let (x, τ ) ∈ |l-top| , choose η = id : (x, τ ) → f∨ed (x, τ ) = (x, τ ∨ τ ) = (x, τ ) , and note η is an l-homeomorphism. now let (y, σ1, σ2) ∈ |l-bitop| , suppose f : (x, τ ) → f∨ (y, σ1, σ2) = (y, σ1 ∨ σ2) is l-continuous, and note τ ⊃ (f←l ) → (σ1 ∨ σ2) ⊃ (f ← l ) → (σ1 ∪ σ2) ⊃ (f ← l ) → (σ1) , (f ← l ) → (σ2) , making f : ed (x, τ ) = (x, τ, τ ) → (y, σ1, σ2) l-bicontinuous. then f = f is the unique choice making f = f ◦ η. the naturality diagram now follows by concreteness, as do the other claims concerning ed ⊣ f∨. finally, given f∧ of subsection 3.1 and f∧ ⊣ ed of 3.3 above, f∨ ⊣/ ed since f∧ ≇ f∨ and left-adjoints are essentially unique. � corollary 3.10. let l be a us-quantale. the following hold: (1) ed preserves all strong colimits and f∨ preserves all strong limits. (2) ed ⊣/ fl |l-bitop(⊂) and ed ⊣/ fr |l-bitop(⊃), and hence ed ⊣/ fl and ed ⊣/ fr. bitopology as topology 87 proof. (1) is immediate. as for (2), it is clear that fl |l-bitop(⊂), fr |l-bitop(⊃) ≇ f∨ |l-bitop(⊂), f∨ |l-bitop(⊃), resp., implying ed ⊣/ fl |l-bitop(⊂) and ed ⊣/ fr |l-bitop(⊃) by the essential uniqueness of the right-adjoint in 3.9; and hence ed ⊣/ fl and ed ⊣/ fr. � proposition 3.11. for each us-quantale l, ed : l-top → l-bitop reflects and detects all colimits. proof. the details are straightforward. � 3.3. fπ : l-bitop → l-top. this subsection constructs the non-concrete, faithful “product” functor fπ : l-bitop → l-top which, when appropriately restricted, is an embedding. it need not preserve finite products and hence lacks a left-adjoint. proposition 3.12. let l be a us-quantale and define fπ : l-bitop → l-top as follows: fπ (x, τ, σ) = (x × x, τ π σ) , fπ (f ) = f × f, where τ π σ is the l-product topology on x × x (subsection 1.5). then fπ is a non-concrete, faithful functor which need not be full nor object-surjective nor an embedding. proof. immediately fπ is well-defined on objects. let f : (x, τ1, τ2) → (y, σ1, σ2) be l-bicontinuous and let v ∈ σ1 π σ2 be a subbasic open set of the form (π1) ← l (s1) with s1 ∈ σ1. then given (x1, x2) ∈ x × x, (f × f ) ← l (v) (x1, x2) = (π1) ← l (s1) (f (x1) , f (x2)) = s1 (π1 (f (x1) , f (x2))) = s1 (f (x1)) = f←l (s1) (x1) = f←l (s1) (π1 (x1, x2)) = (π1) ← l (f←l (s1)) (x1, x2) , so that (f × f ) ← l (v) = (π1) ← l (f←l (s1)) ∈ τ1 π τ2; and similarly, if v is a subbasic open set of the form (π2) ← l (s2) with s2 ∈ σ2, (f × f ) ← l (v) ∈ τ1 π τ2. so fπ (f ) : fπ (x, τ1, τ2) → fπ (y, σ1, σ2) is l-subbasic continuous and hence l-continuous (cf. theorem 3.2.6 of [41]). it is easy to show fπ preserves composition and identities—and so is a functor—and is faithful and need not be full nor object-surjective. to see that fπ need not inject objects, let l = {⊥, α, β,⊤} be a chain with ⊗ = ∧ (binary), x = {x} , τ1 = {⊥, α,⊤} , and τ2 = { ⊥, β,⊤ } . then (x, τ1, τ2) 6= (x, τ2, τ1) , yet fπ (x, τ1, τ2) = fπ (x, τ2, τ1). � proposition 3.13. fπ does not preserve binary products and hence has no left-adjoint. proof. let (x, τ1, τ2) , (y, σ1, σ2) be given with x 6= y . then the carrier set of fπ [(x, τ1, τ2) π (y, σ1, σ2)] is (x × y ) × (x × y ) and the carrier set of fπ (x, τ1, τ2) π fπ (y, σ1, σ2) is (x × x)×(y × y ), clearly not the same. � 88 s. e. rodabaugh definition 3.14. letting l be a us-quantale, l-nbitop is the full subcategory of all spaces (x, τ, σ) satisfying the condition that each open l-subset u 6= ⊥ in each of τ, σ is l-normalized, i.e., has the property that ∨ x∈x u (x) = e. if l is an st-quantale, then the notion of l-normalized subsets coincides with the usual notion, namely ∨ x∈x u (x) = ⊤. theorem 3.15. let l be a u-quantale. then fπ |l-nbitop : l-nbitop → ltop is an embedding. this embedding does not preserve binary products and hence has no left-adjoint. proof. because of 3.12, it suffices to show fπ as restricted injects objects. for two distinct objects, let us consider (x, τ1, σ) 6= (x, τ2, σ) with τ1 6= τ2; all other cases are similar and left to the reader. suppose w.l.o.g. there is u ∈ τ1 − τ2 and assume τ1 π σ = τ2 π σ on x × x. then setting ⊠ ≡⊗◦×, ∃ {uγ ⊠ vγ}γ ∈γ ⊂ τ2 π σ such that (π1) ← l (u) = ∨ γ ∈γ (uγ ⊠ vγ ) . applying the surjectivity of π1 and properties of zadeh image operators (subsection 1.3), we obtain the contradiction u = (π1) → l ((π1) ← l (u)) = (π1) → l   ∨ γ ∈γ (uγ ⊠ vγ )   = ∨ γ ∈γ ((π1) → l (uγ ⊠ vγ )) = ∨ γ ∈γ uγ ∈ τ2, where we have used the fact, for each γ ∈ γ and each x ∈ x, that (π1) → l (uγ ⊠ vγ ) (x) = ∨ y ∈x (uγ ⊠ vγ ) (x, y) = ∨ y ∈x (uγ (x) ⊗ vγ (y)) = uγ (x) ⊗ ∨ y ∈x vγ (y) = uγ (x) ⊗ e = uγ (x) . bitopology as topology 89 the non-preservation of products follow for the restricted functor as in the proof of 3.13. � corollary 3.16. fπ ◦ gχ : bitop → top is an embedding. this embedding does not preserve binary products and hence has no left-adjoint. proof. the first statement is a corollary of 3.15 as follows: given any non-empty subset a of set x, χa : x → 2 is normalized; 2-nbitop = 2-bitop; and gχ : bitop → 2-bitop is a categorical isomorphism. the non-preservation of products follows for the composite functor as in the proof of 3.13. � remark 3.17. corollary 3.16 furnishes an embedding of bitop into top; but this is not enough to say that top may be categorically regarded as a generalization of bitop since fπ ◦ gχ is not sufficiently well-behaved. this motivates the search for a better behaved embedding of bitopology into topology conducted in the next subsection. 3.4. e× : l-bitop → l 2-top. this subsection constructs the concrete, full, strict “cross” embedding e× : l-bitop → l 2-top, establishes its behavior w.r.t. limits and colimits—for appropriate l, e× preserves both and detects and reflects the former, and shows that e× is essentially neutral w.r.t. stratification issues. it follows that e× is an extremely well-behaved embedding. 3.4.1. construction of e× : l-bitop → l 2-top. proposition 3.18 (cf. [16]). let x be a set. (1) for each set l the mapping ϕx : l x × lx → ( l2 )x given by ϕx (a1, a2) = a1 × a2, i.e., ϕx (a1, a2) (x) = (a1 (x) , a2 (x)) is a bijection with inverse mapping ϕ−1 x : lx × lx ← ( l2 )x given by ϕ−1 x (a) = (π1 ◦ a, π2 ◦ a) , where π1, π2 are the projections from l 2 to l. (2) if l is a poset, then ϕx is an order-isomorphism. (3) if l is a semi-demorgan s-quantale, then ϕx preserves semi-complements. (4) if l is an [u]s-quantale, then ϕx is an [u]s-quantalic isomorphism (i.e., ϕx also preserves tensor products [and the unit]). proof. the details of (1)−(3) are the same as, or analogous to, those of lemma 4.4.1 of [16]. the details of (4) are straightforward. � corollary 3.19. ϕ→x : ℘ ( lx × lx ) → ℘ ( ( l2 )x ) is an order-isomorphism. proof. this is immediate from 3.18(1) using subsection 1.3. � proposition 3.20. let a, b be nonempty sets. then ζ : ℘∅ (a) × ℘∅ (b) → ℘∅ (a × b) given by ζ (c, d) = c × d is an order-isomorphism onto its image, i.e., an order-embedding. 90 s. e. rodabaugh proof. clearly ζ is well-defined. as for injectivity, let (c1, d1) 6= (c2, d2). then there are several cases, and a typical case is c1 6= c2, d1 = d2. then w.l.o.g. there is x ∈ c1 − c2. since d1 = d2 6= ∅, there is y ∈ d1 = d2. so (x, y) ∈ (c1 × d1) − (c2 × d2) ; hence ζ (c1, d1) 6= ζ (c2, d2). since all orderings in question are coordinate-wise, it follows that both ζ and ζ−1 (on im (ζ)) are isotone. � proposition 3.21. let x be a set, l be a us-quantale, and ζ denote any restriction of the ζ of 3.20. (1) ζ : ℘∅ ( lx ) ×℘∅ ( lx ) → ℘∅ ( lx × lx ) is an order-isomorphism onto its image. (2) ζ : l-bt (x) → ℘∅ ( lx × lx ) is an order-isomorphism onto its image. proof. conjoin proposition 1.2 and 3.20. � lemma 3.22. let x be a set and l be a us-quantale, and put e× : l-bt (x) → l 2-t (x) by e× = ϕ → x ◦ ζ. then e× is an order-isomorphism onto its image. proof. it must be first verified that e× actually maps into l 2-t (x). let (τ1, τ2) ∈ l-bt (x). then τ1, τ2 are l-topologies on x and hence sub-usquantales of lx . it is straightforward to check that as direct products, ζ (τ1, τ2) = τ1 × τ2 ⊂ l x × lx and τ1 × τ2 is a sub-us-quantale of l x × lx . it follows e× (τ1, τ2) = ϕ → x (ζ (τ1, τ2)) = ϕ → x (τ1 × τ2) ⊂ ℘ ( ( l2 )x ) and that e× (τ1, τ2) is a sub-us-quantale of ( l2 )x , namely an l2-topology on x. hence e× (τ1, τ2) ∈ l 2-t (x). the remaining claims concerning e× follow from 3.19 and 3.21. � theorem 3.23. let l be a us-quantale, let f ∈ l-bitop ((x, τ1, τ2) , (y, σ1, σ2)) , and put e× (x, τ1, τ2) = (x, e× (τ1, τ2)) , e× (f ) = f. then e× : l-bitop → l 2-top is a concrete, full embedding; and hence l-bitop is concretely isomorphic to a full subcategory of l2-top. further, if l is consistent, e× is a strict embedding (not a functorial isomorphism). proof. it is immediate from 3.22 that e× is well-defined at the object-level into l2-top. it must be now checked that e× is well-defined at the morphism-level, i.e., that f : (x, τ1, τ2) → (y, σ1, σ2) is l-bicontinuous implies f : (x, e× (τ1, τ2)) → (y, e× (σ1, σ2)) is l 2-continuous. to that end, let v ∈ e× (σ1, σ2) = ϕ → y (σ1 × σ2) . bitopology as topology 91 then ∃ (v1, v2) ∈ σ1 × σ2 with v = ϕy (v1, v2) . now let x ∈ x. then f←l (v) (x) = v (f (x)) = ϕy (v1, v2) (f (x)) = (v1 (f (x)) , v2 (f (x))) = (f←l (v1) (x) , f ← l (v2) (x)) . since f is l-bicontinuous, u1 ≡ f ← l (v1) ∈ τ1, u2 ≡ f ← l (v2) ∈ τ2; and so choosing u = ϕx (u1, u2) ∈ e× (τ1, τ2) , we have f←l (v) = u, finishing the proof that f is l2-continuous. since e× is concrete (with respect to the usual forgetful functors), it is immediate that e× is a functor and that e× injects hom-sets. to verify that e× is full, we show that f : (x, e× (τ1, τ2)) → (y, e× (σ1, σ2)) is l 2-continuous implies f : (x, τ1, τ2) → (y, σ1, σ2) is l-bicontinuous. let v ∈ σ1 and note ⊥∈ σ2. then (v,⊥) ∈ σ1×σ2, so that ϕy (v,⊥) ∈ e× (σ1, σ2). hence f ← l (ϕy (v,⊥)) ∈ e× (τ1, τ2) by the l 2-continuity of f . it follows ∃u ∈ e× (τ1, τ2) , and hence ∃ (u1, u2) ∈ τ1 × τ2, such that f←l (ϕy (v,⊥)) = u = ϕx (u1, u2) . now let x ∈ x. then (u1 (x) , u2 (x)) = ϕx (u1, u2) (x) = f←l (ϕy (v,⊥)) (x) = (v (f (x)) ,⊥) = (f←l (v) (x) ,⊥) , so that u1 (x) = f ← l (v) (x) . it follows f ← l (v) = u1 ∈ τ1. similarly, it can be shown that if v ∈ σ2, then f ← l (v) ∈ τ2. hence f is l-bicontinuous. for e× to be an embedding, it remains to show that e× injects objects. to that end let (x, τ1, τ2) 6= (y, σ1, σ2) . if x 6= y, we are done. so suppose that x = y and that (τ1, τ2) 6= (σ1, σ2). then immediately by 3.22, e× (τ1, τ2) 6= e× (σ1, σ2) . it follows that e× (x, τ1, τ2) 6= e× (y, σ1, σ2) . finally, the strictness of e×, when l is consistent, follows from 3.24 below. � many more properties of e× are developed in the next three subsections which show that it is an extremely well-behaved embedding. 92 s. e. rodabaugh counterexample 3.24. if e× were to surject objects, then e× would be a functorial isomorphism. this however is usually not the case. let l be any consistent us-quantale, note l ⊃ 2 = {⊥, e}, and consider the l2-topological space (x, τ ) with x nonempty and τ the indiscrete l2-topology τ = { (⊥,⊥), (e, e) } . suppose a space (x, e× (τ1, τ2)) from the image of e× is (x, τ ). this forces ϕ→x (τ1 × τ2) = e× (τ1, τ2) = τ. noting {⊥, e}⊂ τ1, {⊥, e}⊂ τ2, it follows ϕx (⊥, e) ∈ ϕ → x (τ1 × τ2) , ϕx (⊥, e) = (⊥, e) /∈ τ, a contradiction. hence the space (x, τ ) is not in the image of e×. hence, for consistent l it is the case that e× is not a functorial isomorphism, but a strict embedding. this justifies examining e×’s behavior w.r.t. limits and colimits in the next subsubsections 3.4.2–3.4.3 as well as characterizing ∣ ∣e→× (l-bitop) ∣ ∣ in 3.28. corollary 3.25. let 4 be the 4-element boolean algebra {⊥, α, β,⊤}with ⊗ = ∧ (binary). then the traditional category bitop of bitopological spaces and bicontinuous maps concretely, fully, strictly embeds into 4-top as a full monocoreflective subcategory that is closed under all limits and colimits. proof. consider the bitopological version gχ : bitop → 2-bitop of the characteristic functor given by gχ (t) = {χu : u ∈ t} , gχ (s) = {χv : v ∈ s} , gχ (x, t, s) = (x, gχ (t) , gχ (s)) , gχ (f ) = f. then this bitopological gχ is a concrete functorial isomorphism. now clearly by the direct product of us-quantales, 22 ∼= 4, so by 3.23 and 3.24, 2-bitop concretely, fully, strictly embeds into 4-top. hence via the composition e× ◦ gχ : bitop →֒ 4-top, bitop concretely, fully, strictly embeds into 4-top. for the monoreflectivity claim, see 3.26 below; and the claim regarding limits and colimits follows from subsubsections 3.4.2–3.4.3 below, the limit claim needing the observation that 4 is a u-quantale with ⊗ = ∧. � bitopology as topology 93 3.4.2. behavior of e× : l-bitop → l 2-top w.r.t. colimits. since for all consistent l the full concrete embedding e× is not a functorial isomorphism, but only a strict embedding, it is worthwhile to investigate its behavior w.r.t. limits and colimits. this subsection shows for any us-quantale l that the embedding e× has a right-adjoint—and hence preserves colimits. the next subsection then shows step by step for l a u-quantale that the special adjoint functor theorem constructs for e× a left-adjoint—and hence e× preserves limits; and further the next subsection shows for any us-quantale l that the embedding e× reflects and detects all limits and is transportable. therefore, this subsection—in concert with the preceding and subsequent subsections— shows that e× is an extremely well-behaved embedding. theorem 3.26 (e× ⊣ fπ). let l be a us-quantale and put the “projection” functor fπ : l-bitop ← l 2-top as follows: fπ (x, τ ) = (x, fπ (τ )) , fπ (f ) = f, where the fibre level of fπ fπ (τ ) = (π1 ◦ τ ≡{π1 ◦ u : u ∈ τ} , π2 ◦ τ ≡{π2 ◦ u : u ∈ τ}) uses the projections π1, π2 : l × l → l for the us-quantalic (direct) product. then the following hold: (1) fπ is a concrete embedding which is not full and does not lift limits. (2) e× ⊣ fπ , so e× preserves all strong colimits and fπ preserves all strong limits. (3) fπ need not detect limits nor be transportable. (4) fπ ◦ e× = idl-bitop. (5) l-bitop is isomorphic (via e×) to a full monocoreflective subcategory of l2-top. (6) l2-top is isomorphic (via fπ) to an isoreflective subcategory of l-bitop. proof. ad(1). since us-quantalic projections preserve arbitrary joins, the tensor, and the unit, it follows that fπ (τ ) ∈ l-bt (x); and hence (x, fπ (τ )) ∈ |l-bitop| and fπ is well-defined at the object level. as for morphisms, let f : (x, τ ) → (y, σ) be l2-continuous in l2-top. then, given v ∈ σ, the identities f←l (π1 ◦ v) = π1 ◦ f ← l (v) , f ← l (π2 ◦ v) = π2 ◦ f ← l (v) are easily checked and immediately imply that f : (x, fπ (τ )) → (y, fπ (σ)) is l-bicontinuous in l-bitop. now by the concreteness of fπ, it is immediately a concrete and faithful functor. to show that fπ is an embedding, it remains to check that fπ injects objects: but if u, v : x → l 2 are distinct, there exists x ∈ x such that w.l.o.g. π1 (u (x)) 6= π1 (v (x)) ; which implies that if τ 6= σ as l2-topologies on x, then fπ (τ ) 6= fπ (σ) as l-bitopologies on x, showing that fπ injects objects. 94 s. e. rodabaugh to see that fπ need not be full, let l = 2, write the boolean algebra l 2 = 4 as {(⊥,⊥) , (⊥,⊤) , (⊤,⊥) , (⊤,⊤)}, let x = {x} , and choose τ = { (⊥,⊥), (⊤,⊤) } , σ = { (⊥,⊥), (⊥,⊤), (⊤,⊥), (⊤,⊤) } . then it follows idx : (x, τ ) → (x, σ) is not l 2-continuous (since σ is not a subset of τ ). now π1 ◦ (⊥,⊥) = π1 ◦ (⊥,⊤) = ⊥, π1 ◦ (⊤,⊥) = π1 ◦ (⊤,⊤) = ⊤, π2 ◦ (⊥,⊥) = π2 ◦ (⊤,⊥) = ⊥, π2 ◦ (⊥,⊤) = π2 ◦ (⊤,⊤) = ⊤, so that fπ (τ ) = (π1 ◦ τ, π2 ◦ τ ) = ({⊥,⊤} , {⊥,⊤}) = (π1 ◦ σ, π2 ◦ σ) = fπ (σ) , implying idx : (x, fπ (τ )) → (x, fπ (σ)) is l-bicontinuous. the concreteness of fπ implies there exists no g ∈ l-top with fπ (g) = idx , so fπ is not full. to see that fπ need not lift limits, let the diagram in l 2-top be the space (x, σ) of the preceding paragraph. then the image of this diagram is the space (x, fπ (σ)) in l-bitop. now the space (x, fπ (τ )), together with the arrow idx : (x, fπ (τ )) → (x, fπ (σ)), is a limit of the diagram (x, fπ (σ)): any l-bicontinuous f : (z, υ1, υ2) → (x, fπ (σ)) trivially factors uniquely through idx . but as seen in the preceding paragraph, there is no g ∈ l-top with fπ (g) = idx , which means there is no limiting cone of (x, σ) in l 2-top which fπ carries over to the limit idx : (x, fπ (τ )) → (x, fπ (σ)) in l-bitop. hence fπ need not lift limits. ad(2). let (x, τ1, τ2) ∈ |l-bitop| be given. then fπe× (x, τ1, τ2) = fπ (x, ϕ → x (τ1 × τ2)) ≡ (x, τ̂1, τ̂2) , where it follows that τ̂1 = {π1 ◦ ϕx (u, v) : u ∈ τ1, v ∈ τ2} , τ̂2 = {π1 ◦ ϕx (u, v) : u ∈ τ1, v ∈ τ2} . we choose the right unit η to be the identity mapping id : x → x. then for each x ∈ x, (π1 ◦ ϕx (u, v)) (x) = π1 (u (x) , v (x)) = u (x) , (π2 ◦ ϕx (u, v)) (x) = π2 (u (x) , v (x)) = v (x) , which immediately gives the l-bicontinuity of η. for universality of the lifting, let (x, τ ) ∈ ∣ ∣l2-top ∣ ∣ be given, along with an l-bicontinuous map f : (x, τ1, τ2) → (x, fπ (τ )). choosing f̄ = f , we now check f̄ : e× (x, τ1, τ2) → (x, τ ) is an l-continuous map from e× (x, τ1, τ2) to (x, τ ) by letting v ∈ τ and x ∈ x. then the l-bicontinuity of f implies f←l (π1 ◦ u) ∈ τ1, f ← l (π2 ◦ u) ∈ τ2, bitopology as topology 95 from which it follows ϕx (f ← l (π1 ◦ u) ∈ τ1, f ← l (π2 ◦ u) ∈ τ2) ∈ ϕ → x (τ1 × τ2) . further, we note f←l (u) (x) = u (f (x)) = (π1 (u (x)) , π2 (u (x))) = (f←l (π1 ◦ u) (x) , f ← l (π2 ◦ u) (x)) = ϕx (f ← l (π1 ◦ u) ∈ τ1, f ← l (π2 ◦ u) ∈ τ2) (x) . finally, it is immediate that f̄ is the unique l-continuous map from e× (x, τ1, τ2) to (x, τ ) such that f = f̄ ◦ η, completing the universality of the lifting. the naturality diagram now follows by concreteness. ad(3). this is an immediate consequence of (1), (2), and proposition 13.34 [1]. ad(4). since τ̂1 = τ1, τ̂2 = τ2 in the proof of (2), it is immediate that fπ ◦ e× = idl-bitop. ad(5). using fπ ◦e× = idl-bitop, the components of the left unit (counit) of e× ⊣ fπ furnish the needed monocoreflection arrows to l 2-topological spaces from the e× image of l-bitop. ad(6). using fπ ◦ e× = idl-bitop, the components of the right unit of e× ⊣ fπ furnish the needed isocoreflection arrows to l-bitopological spaces from the fπ image of l 2-top. � remark 3.27. we collect some facts concerning e×, fπ , and their fibredependent constructions, where l is a us-quantale: (1) fπ ⊣/ e× if l is consistent. this is a consequence of 3.24. (2) e× ⊣ fπ need not be a categorical equivalence. this follows from (1). (3) for each (x, τ1, τ2) ∈ |l-bitop| , fπe× (τ1, τ2) = fπ (ϕ → x (τ1 × τ2)) = (π1 ◦ ϕ → x (τ1 × τ2) , π2 ◦ ϕ → x (τ1 × τ2)) = (τ1, τ2) . (4) for each (x, τ ) ∈ ∣ ∣l2-top ∣ ∣ , “e× (fπ (τ )) = ϕ → x ((π1 ◦ τ ) × (π2 ◦ τ )) ⊃ τ ” always holds; but for l consistent, “e× (fπ (τ )) = ϕ → x ((π1 ◦ τ ) × (π2 ◦ τ )) ⊂ τ ” need not hold. the latter statement is another version of (1). theorem 3.28 (characterization of ∣ ∣e→× (l-bitop) ∣ ∣). let l be a us-quantale and (x, τ ) ∈ ∣ ∣l2-top ∣ ∣. then (x, τ ) ∈ ∣ ∣e→× (l-bitop) ∣ ∣ if and only if e× (fπ (τ )) = τ , i.e., both inequalities of 3.27(4) hold. 96 s. e. rodabaugh 3.4.3. behavior of e× : l-bitop → l 2-top w.r.t. limits. the question of a left-adjoint for e× is open for general us-quantales l; and it is our conjecture is that for general us-quantales l, e× would not preserve products or intersections and hence would not have a left-adjoint. but on the other hand, this section shows e× has a left-adjoint (and therefore preserves all limits) for l any uquantale. we point out that our proof of this left-adjoint is existential (via the special adjoint functor theorem) and not constructive; and it is an additional open question whether there is a direct construction of this left adjoint not essentially factoring through our proof. it is further proved that e× reflects and detects limits and is transportable. lemma 3.29 (preservation of products). for each u-quantale l, e× : lbitop → l2-top preserves arbitrary (small) products. sublemma 3.30. let l be a u-quantale and suppose x is a set and τ1, τ2 are l-topologies on x with respective subbases σ1, σ2, namely τ1 = 〈〈σ1〉〉 , τ2 = 〈〈σ2〉〉 , such that {⊥, e}⊂ σ1 ∩ σ2. then (*) ϕ→x (τ1 × τ2) = 〈〈ϕ → x (σ1 × σ2)〉〉 . proof. to see that “⊃” holds in (*), note that σ1 × σ2 ⊂ τ1 × τ2, ϕ→x (σ1 × σ2) ⊂ ϕ → x (τ1 × τ2) , 〈〈ϕ→x (σ1 × σ2)〉〉⊂ ϕ → x (τ1 × τ2) . for “⊂” in (*), we first invoke the associativity of ⊗ and its infinite distributivity over ∨ to write members of τ1, τ2 as joins of tensor products of members of σ1, σ2, respectively. more precisely, consider these typical members ∨ α∈a1   ⊗ β ∈b1 uαβ   , ∨ α∈a2   ⊗ β ∈b2 vαβ   of τ1, τ2, respectively, where a1, a2 are arbitrary indexing sets, b1, b2 are arbitrary finite indexing sets, each uαβ ∈ σ1, each vαβ ∈ σ2, and where w.l.o.g. we assume a1 ∩ a2 = ∅ = b1 ∩ b2. next, we augment the uαβ ’s and vαβ ’s as follows, using the assumption that {⊥, e}⊂ σ1 ∩ σ2: α ∈ a1, β ∈ b2, uαβ ≡ e, α ∈ a2, β ∈ b1 ∪ b2, uαβ ≡⊥, α ∈ a2, β ∈ b1, vαβ ≡ e, α ∈ a1, β ∈ b1 ∪ b2, vαβ ≡⊥. bitopology as topology 97 it follows that as maps from x to l that ∨ α∈a1 ∪a2   ⊗ β ∈b1 ∪b2 uαβ   = ∨ α∈a1   ⊗ β ∈b1 uαβ   , ∨ α∈a1 ∪a2   ⊗ β ∈b1 ∪b2 vαβ   = ∨ α∈a2   ⊗ β ∈b2 vαβ   . we thus have that a typical member   ∨ α∈a1   ⊗ β ∈b1 uαβ   , ∨ α∈a2   ⊗ β ∈b2 vαβ     of τ1 × τ2 may be rewritten as   ∨ α∈a1 ∪a2   ⊗ β ∈b1 ∪b2 uαβ   , ∨ α∈a1 ∪a2   ⊗ β ∈b1 ∪b2 vαβ     = ∨ α∈a1 ∪a2   ⊗ β ∈b1 ∪b2 (uαβ , vαβ )   , the latter being the form of a typical member of 〈〈σ1 × σ2〉〉 . to complete the proof of “⊂”, we invoke the fact that ϕ→x is an order-isomorphism preserving all tensor products (3.18(4)) to conclude that ϕ→x (τ1 × τ2) ⊂ ϕ → x 〈〈σ1 × σ2〉〉 = 〈〈ϕ → x (σ1 × σ2)〉〉 . � proof of 3.29. recall the categorical products in l-bitop use the categorical product of l-top in each slot as well as the usual projections for the morphisms of the product (subsection 1.5), and let {(xγ , (τ γ 1 , τ γ 2 ))}γ ∈γ ⊂ |l-bitop|. because of the concreteness of e×, the validity of e×   ∏ γ ∈γ (xγ , (τ γ 1 , τ γ 2 )) ,{πγ}γ ∈γ   =   ∏ γ ∈γ e× (xγ , (τ γ 1 , τ γ 2 )) ,{πγ}γ ∈γ   holds if and only if we have the equality of topologies (**) ϕ→×γ ∈ γxγ (πγ ∈γ τ γ 1 × πγ ∈γ τ γ 2 ) = πγ ∈γ ϕ → xγ (τ γ 1 × τ γ 2 ) , where “×” denotes as usual the direct product of us-quantales. for convenience, “lhs” and “rhs” respectively denote the left-hand side and right-hand side of (**). let a subbasic open subset w be given from rhs. then w may be written as follows: w = (πβ ) ← l ( ϕxβ ( t β 1 , t β 2 )) , 98 s. e. rodabaugh where ( t β 1 , t β 2 ) ∈ τ β 1 × τ β 2 for a fixed index β ∈ γ. given {xγ}γ ∈γ ∈×γ ∈γxγ , then w ( {xγ}γ ∈γ ) = ϕxβ ( t β 1 , t β 2 )( πβ ( {xγ}γ ∈γ )) = ϕxβ ( t β 1 , t β 2 ) (xβ ) = ( t β 1 (xβ ) , t β 2 (xβ ) ) = ([ (πβ ) ← l ( t β 1 )]( {xγ}γ ∈γ ) , [ (πβ ) ← l ( t β 2 )]( {xγ}γ ∈γ )) . this shows w is in lhs, lhs contains a subbasis of rhs, and so lhs contains rhs. for the reverse direction, let z be in lhs. then ∃ (u1, u2) ∈ πγ ∈γ τ γ 1 × πγ ∈γ τ γ 2 with z = ϕ×γ ∈ γxγ (u1, u2) . since the τ γ 1 ’s and τ γ 2 ’s contain {⊥, e} and since these l-subsets are preserved by the zadeh preimage operators of all the projection maps, the usual subbasis for each of πγ ∈γ τ γ 1 and πγ ∈γ τ γ 2 contains {⊥, e}. thus 3.30 applies to say it suffices to let u1, u2 be subbasic in their respective l-product topologies ∏ γ ∈γ τ γ 1 , ∏ γ ∈γ τ γ 2 ; so we may write u1 = (πα) ← l (tα1 ) , u2 = (πβ ) ← l ( t β 2 ) , where tα1 ∈ τ α 1 , t β 2 ∈ τ β 2 for fixed indices α, β ∈ γ. let {xγ}γ ∈γ ∈ ×γ ∈γxγ . then recalling that l has a unit e for ⊗ and that e is the corresponding unit for ⊗ lifted to lx , we have z ( {xγ}γ ∈γ ) = ϕ×γ ∈ γxγ (u1, u2) ( {xγ}γ ∈γ ) = ( u1 ( {xγ}γ ∈γ ) , u2 ( {xγ}γ ∈γ )) = ( (πα) ← l (tα1 ) ( {xγ}γ ∈γ ) , (πβ ) ← l ( t β 2 )( {xγ}γ ∈γ )) = ( tα1 ( πα ( {xγ}γ ∈γ )) , t β 2 ( πβ ( {xγ}γ ∈γ ))) = ( tα1 (xα) , t β 2 (xβ ) ) = ( tα1 (xα) ⊗ e, e ⊗ t β 2 (xβ ) ) = (tα1 (xα) , e (xα)) ⊗ ( e (xβ ) , t β 2 (xβ ) ) = ( [(πα) ← l (ϕxα (t α 1 , e))]⊗ [ (πβ ) ← l ( ϕxβ ( e, t β 2 ))] ) ( {xγ}γ ∈γ ) , the last line being the evaluation at {xγ}γ ∈γ by a tensor of open subsets of rhs and hence of an open subset of rhs. thus z is in rhs, so lhs is contained in rhs, completing the proof of the theorem. 2 lemma 3.31. for each us-quantale l, e× : l-bitop → l 2-top preserves equalizers. bitopology as topology 99 sublemma 3.32. let l be a us-quantale, (x, τ, σ) ∈ |l-bitop| , z ⊂ x, and τ (z) , σ (z) , e× (τ, σ) (z)be the l-subspace topologies on z given by τ (z) = { u |z : u ∈ τ } , σ (z) = { v |z : v ∈ σ } , e× (τ, σ) (z) = ϕ → x (τ × σ) (z) (cf. [41]). then e× (τ, σ) (z) = e× (τ (z) , σ (z)) . restated, e× respects subspace topologies. proof. let u ∈ τ, v ∈ σ, z ∈ z. then ϕx (u, v) |z (z) = (u (z) , v (z)) = ( u |z (z) , v |z (z) ) = ϕx ( u |z , v |z ) (z) . this implies e× (τ, σ) (z) = ϕ → x (τ × σ) (z) = ϕ → x (τ (z) × σ (z)) = e× (τ (z) , σ (z)) . � proof of 3.31. a categorical proof based upon the concreteness of e×, f and fπe× = idl-bitop (3.26) does not work since it would generally require that e×fπ (τ ) ⊂ τ , which need not be true by (3.27(4)). it is necessary to look at the actual construction of equalizers in each of l-bitop and l2-top and show that e× carries the former into the latter. it can be checked that the equalizer of f, g : (x, τ1, τ2) ⇉ (y, σ1, σ2) in l-bitop is given by ((z, τ1 (z) , τ2 (z)) , →֒) , where z = {x ∈ x : f (x) = g (x)} , and that the equalizer of f, g : e× (x, τ1, τ2) ⇉ e× (y, σ1, σ2) in l 2-top is given by ((z, e× (τ1, τ2) (z)) , →֒) using the same z. because of the concreteness of e×, the issue is whether e× (τ1, τ2) (z) is the same as e× (τ1 (z) , τ2 (z)), and this is settled in 3.32. 2 corollary 3.33. for each u-quantale l, e× : l-bitop → l 2-top preserves all small limits. in particular, for each frame l, e× preserves all small limits. proof. it is not difficult to show that l-bitop is topological over set w.r.t. the usual forgetful functor; and since set is complete, it follows that l-bitop is complete (theorem 21.16 [1]). conjoin 3.29 and 3.31 to get that e× preserves equalizers and (all) products; and then apply proposition 13.4 [1] to finish the proof. � lemma 3.34. for each u-quantale l, e× : l-bitop → l 2-top preserves all intersections. proof. as in the proof of 3.31, it is necessary to look at the actual construction of intersections in each of l-bitop and l2-top and show that e× carries the former into the latter. since this is trivially the case if the indexing class of the intersection is empty, we assume sequens that the indexing class is nonempty. 100 s. e. rodabaugh to describe intersections in l-bitop, let {((xγ , τ γ 1 , τ γ 2 ) , mγ )}γ ∈γ be a class of subobjects of (y, σ1, σ2)—by the well-poweredness of l-bitop (subsection 1.5), this class is not proper, i.e., we may take γ as a set; form the product ( (×γ ∈γxγ , πγ ∈γ τ γ 1 , πγ ∈γ τ γ 2 ) ,{πγ}γ ∈γ ) of these subobjects in l-bitop; let x ≡ { {xγ}γ ∈γ : ∀β, δ ∈ γ, mβ (xβ ) = mδ (xδ) } ⊂×γ ∈γxγ ; fix ζ ∈ γ; and put m ≡ mζ ◦ πζ ◦ →֒ : x → y . then equipping x with the l-subspace topologies [πγ ∈γ τ γ 1 ] (x) , [πγ ∈γ τ γ 2 ] (x) , respectively, it can be shown that ((x, [πγ ∈γ τ γ 1 ] (x) , [πγ ∈γ τ γ 2 ] (x)) , m) is the required intersection in l-bitop. we now consider in l2-top the image {(e× (xγ , τ γ 1 , τ γ 2 ) , mγ )}γ ∈γ = {((xγ , e× (τ γ 1 , τ γ 2 )) , mγ )}γ ∈γ , under e× of the family {((xγ , τ γ 1 , τ γ 2 ) , mγ )}γ ∈γ , which image by the functoriality of e× is a sink of subobjects for e× (y, σ1, σ2). using the x and m of the preceding paragraph, it can be shown that ((x, [πγ ∈γ e× (τ γ 1 , τ γ 2 )] (x)) , m) is the required intersection in l2-top. to show that e× takes the l-bitop intersection to the l 2-top intersection, we note e× ([πγ ∈γ τ γ 1 ] (x) , [πγ ∈γ τ γ 2 ] (x)) = e× (πγ ∈γ τ γ 1 , πγ ∈γ τ γ 2 ) (x) (by 3.32) = [πγ ∈γ e× (τ γ 1 , τ γ 2 )] (x) (by proof of 3.29 (**)), which shows e× (x, [πγ ∈γ τ γ 1 ] (x) , [πγ ∈γ τ γ 2 ] (x)) = (x, [πγ ∈γ e× (τ γ 1 , τ γ 2 )] (x)) . � theorem 3.35. for each u-quantale l, e× : l-bitop → l 2-top preserves all strong limits. proof. this follows from 3.33, 3.34, and definition 13.1(3) [1]. � theorem 3.36. for each u-quantale l, e× : l-bitop → l 2-top has a left adjoint. bitopology as topology 101 proof. first, l-bitop has small fibres and is a topological construct (proof of 3.33); hence, l-bitop is complete and well-powered with coseparators by corollary 21.17 [1]. second, proposition 12.5 [1] now gives l-bitop is strongly complete. third, since e× preserves all strong limits (3.35), the special adjoint functor theorem 18.17 [1] now implies e× is a right-adjoint. finally, apply proposition 18.9 [1]. � proposition 3.37. for each us-quantale l, e× : l-bitop → l 2-top reflects and detects all limits and hence lifts all limits and is transportable. proof. the details are straightfoward using the preservation of limits by fπ, fπ ◦ e× = idl-bitop, 3.36, and proposition 13.34 [1]. � 3.4.4. behavior of e× : l-bitop → l 2-top w.r.t. stratification issues. this subsubsection shows e× is essentially neutral w.r.t. stratification issues. lemma 3.38. let l be a us-quantale, (x, τ, σ) ∈ |l-bitop|, and (γ, δ) ∈ l2. then (γ, δ) ∈ e× (τ, σ) if and only if γ ∈ τ and δ ∈ σ. proof. it is straightforward to check that (γ, δ) ∈ ϕ→x (τ × σ) ⇔ ∃u ∈ τ, ∃v ∈ σ, ∀x ∈ x, (u (x) , v (x)) = (γ, δ) ⇔ γ ∈ τ, δ ∈ σ. � theorem 3.39. let l be a us-quantale,(x, t) ∈ |top| , (x, τ ) ∈ |l-top| , and (x, τ, σ) ∈ |l-bitop|. the following hold: (1) e× (x, τ, σ) always has (⊥,⊥), (⊥, e), (e,⊥), (e, e) as open subsets. (2) e× (x, τ, σ) is anti-stratified if and only if l is inconsistent. (3) for l = 2, e× (x, τ, σ)is weakly stratified. (4) e×gχ (x, t) is weakly stratified for |l| = 2 and non-stratified for |l| > 2. (5) e× (x, τ, σ) is weakly stratified if and only if (x, τ, σ) is weakly stratified. (6) e× (x, τ, σ) is non-stratified if and only if (x, τ, σ) is non-stratified. (7) statements (1–3, 5–6) with e× (x, τ, σ) replaced with e×fd (x, τ ) and (x, τ, σ) replaced with (x, τ ). proof. (1) follows from 3.38 given that {⊥, e}⊂ τ ∩ σ; (2, 3, 4) follow from (1) and the fact that 4 may be taken as precisely {(⊥,⊥) , (⊥, e) , (e,⊥) , (e, e)}; (5) follows from 3.38; (6) contraposes (5); and (7) is immediate from the other statements. � 4. summary this paper surveys the relationship between (lattice-valued) bitopology and (lattice-valued) topology by examing a variety of functorial relationships —ed, fl, fr, f∧, f∨, fπ, e×, fπ —when l is a us-quantale. from this overview 102 s. e. rodabaugh of these functors and their properties, the following metamathematical conclusions emerge: (1) if it were assumed that the underlying lattice l of membership values is not allowed to change, then this survey would support the following viewpoint: (a) (lattice-valued) bitopology is strictly more general than (latticevalued) topology in an extremely well-behaved way—justified by ed; and (b) (lattice-valued) topology is not more general than (lattice-valued) bitopology—justified by fl, fr, f∧, f∨, fπ in comparison with ed, though the variety of ways in which bitopological spaces may be interpreted as topological spaces is rather striking. (2) if it were assumed that the underlying lattice l of membership values is allowed to change (e.g., to the direct s-quantalic product l2), then this survey would support the following viewpoint: (a) (lattice-valued) topology is strictly more general than (latticevalued) bitopology in an extremely well-behaved way—justified by e×; and (b) (lattice-valued) bitopology is not more general than (lattice-valued) topology—justified by fπ in comparison with e×, though fπ is a rather interesting interpretation of topological spaces as bitopological spaces. (3) this paper supports viewpoint (2) against viewpoint (1) for the following reasons: (a) we are in fact allowed to choose whatever underlying lattice of membership values we wish, so in fact the underlying assumption of (1) is false and the underlying assumption of (2) is true. the class of embeddings e× stands and must be reckoned with. (b) topology (lattice-valued) is fundamentally simpler than bitopology (lattice-valued): (i) an l-bitopological space (x, τ, σ) adds to the ground object x three parameters—l, τ, σ; while an m -topological space (x, τ ) adds to the ground object x two parameters—m, τ . (ii) when passing (via e×) from the l-bitopological space (x, τ, σ) to the l2-topological space, the complexity of two topologies is isolated in the underlying lattice of membership values, leaving behind one topology. (c) topology (lattice-valued) is strictly more general than bitopology (lattice-valued) in each of two ways: (i) for each l ∈ |usquant| , the direct product l2 ∈ |usquant| and l-bitop embeds as a strict subcategory of l2-top (via e×), which is extremely well-behaved if l ∈ |uquant| . (ii) the class { l2-top : l ∈ |usquant| } bitopology as topology 103 representing the field of fixed-basis bitopology using us-quantales is a strictly proper subclass of the class {l-top : l ∈ |usquant|} representing the field of fixed-basis topology using us-quantales (and not every us-quantale is a direct square of another usquantale), and this strictness holds if the class is indexed by |uquant| . (iii) thus when one proves a theorem in fixed-basis topology, it is strictly more general w.r.t. coverage of categories and coverage of objects in each category in which bitopological spaces are embedded. (d) the upshot of (a, b, c) is that (lattice-valued) bitopology is categorically redundant, particularly for underlying unital quantales: (lattice-valued) topology is fundamentally simpler and strictly more general. fixed-basis bitopology is a complicated version of restricted subcategories of categories from a restricted class of categories of fixed-basis topological spaces. for lattice-theoretic bases larger than 2, workers in lattice-valued bitopology should now be working in lattice-valued topology. (4) the above arguments apply to traditional bitopology in a more subtle way. on the one hand, traditional bitopology is isomorphic—in an extremely well-behaved way—to a strictly proper, extremely wellbehaved subcategory of the much simpler 4-topology (bitop embeds into 4-top: 3.25 above); restated, traditional bitopology is a restricted subcase of a particular kind of fuzzy topology (namely 4-topology) and therefore traditional bitopology is categorically redundant vis-a-vis fixed-basis lattice-valued topology. on the other hand, the crisp lattice 2 underlying bitop is so extremely simple that it is really a question of two topologies in bitop vis-a-vis the lattice 4 and one topology in 4-top; restated, moving from (x, t, s) to (x, e× (t, s)) means moving from the parameters (2, t, s) to the parameters (4, e× (t, s)) , with the increased complexity in going from 2 to 4 offset by going from the two topologies t, s to the one 4-topology e× (t, s) , noting that each of t, s is more complex than 4. at the very least, workers in traditional bitopology should consider working in 4-topology. (5) the above arguments for redundancy in some sense are even stronger than those used in [16] to show that various versions of “intuitionistic” topologies or topologies comprising double subsets are redundant and a categorically special case of fixed-basis topology since the e×’s of this paper are strict embeddings and not functorial isomorphisms (when l is consistent) as in [16]. (6) the rich history and literature of traditional bitopology, including interesting separation and compactness axioms which “mix” together the two topologies, are now immediately part of the literature of 4-top 104 s. e. rodabaugh since the functorial embedding e×◦gχ is an embedding at the powerset and fibre levels in which these axioms are formulated. the precise shape of these axioms as packaged by e×◦gχ in 4-top is, however, an open question. answering this question may teach us how to use successful axioms of traditional bitopology to formulate successful axioms for fixed-basis topology. we illustrate (6) by showing that from traditional bicompactness e× induces the compactness of [5] for lattice-valued topology and by discussing the relationship between the respective tihonov theorems for the two categories bitop and 4-top. as repeatedly shown in [36, 37, 38, 42, 34], chang’s original axiom of compactness [5] for lattice-valued topology, dubbed localic compactness in [38] and simply compactness in [19, 42], has been extraordinarily successful and justified with regard to classes of representations of l-spatial locales, l-coherent locales, distributive lattices, boolean algebras, traditional compact hausdorff spaces, classes of stone-čech compactifications, classes of stone-weierstraß theorems [42], etc; indeed, for l a frame, only this compactness axiom (and the very closely related axiom of [20]) has an unrestricted compactification reflector for all of l-top. further, its tihonov theorem, namely the goguen-tihonov theorem [12], is one of the few tihonov theorems in the fuzzy literature which does not need the classical theorem in its proof; and hence it generalizes and explains both the statement and the proof of the classical theorem. we need the statement of this theorem. let l be any complete lattice and let κ be a cardinal. we say ⊤ is κ-isolated [12] in l if for each a ⊂ l −{⊤} with |a| ≤ κ, ∨ a < ⊤. theorem 4.1 (goguen-tihonov [12]). let l be a complete lattice and γ be an indexing set. then ⊤ is |γ|-isolated in l if and only if each collection {(xγ , τγ ) : γ ∈ γ} ⊂ l-top of compact spaces (in the sense of [5]) yields a compact product ∏ γ ∈γ (xγ , τγ ). corollary 4.2. the traditional tihonov theorem holds: for any indexing set γ, ∏ γ ∈γ (xγ ,tγ ) is compact if and only if each (xγ ,tγ ) is compact. proof. the forward direction—the easier direction—can be given the usual proof. as for the backward direction—the harder direction, we proceed as follows. first, the backward direction transfers directly, via the functorial isomorphism gχ : top → 2-top, to the claim that each collection {(xγ , τγ ) : γ ∈ γ}⊂ 2-top of compact spaces (in the sense of [5]) yields a compact product ∏ γ ∈γ (xγ , τγ ); and this claim holds immediately from 4.1 since in the lattice 2, ⊤ is κ-isolated in 2 for each cardinal κ, and so the claim holds for each indexing set γ. � a traditional bicompact bitopological space (x, t, s) is defined by saying that x is compact w.r.t. each of the topologies t, s. given the construction of products in bitop (subsection 1.5), we immediately have the usual tihonov theorem for traditional bitopology. bitopology as topology 105 corollary 4.3. for any indexing set γ, ∏ γ ∈γ (xγ , tγ , sγ ) is bicompact if and only if each (xγ , tγ , sγ ) is bicompact. corollary 4.4. let γ be an indexing set. then each collection {(xγ , τγ ) : γ ∈ γ} ⊂ 4-top of compact spaces (in the sense of [5]) yields a compact product ∏ γ ∈γ (xγ , τγ ) if and only if |γ| = 0 or 1. proof. letting 4 be written as {⊥, a, b,⊤} with a, b unrelated, this is immediate from 4.1 since ⊤ is κ-isolated in 4 if and only if κ ≤ 1. � the plot thickens with the next definition, theorem, and corollary. definition 4.5. let l ∈ |usquant| . an l-bitopological space (x, τ1, τ2) is (l-)bicompact if x is compact (in the sense of [5]) w.r.t. each of τ1 and τ2. theorem 4.6. for each l ∈ |usquant|, e× : l-bitop → l 2-top preserves bicompactness to compactness in the sense of [5]. proof. let a bicompact l-topological space (x, τ1, τ2) be given and let {uγ × vγ : γ ∈ γ} be a cover of x from the l2-topology e× (τ1, τ2). if γ is finite, then this cover is its own finite subcover; so we assume γ is not finite. now (⊤,⊤) = (⊤,⊤) = ∨ γ∈γ (uγ × vγ ) = ∨ γ∈γ uγ × ∨ γ∈γ vγ , forcing each of {uγ : γ ∈ γ} and {vγ : γ ∈ γ} to be covers of x from τ1 and τ2, respectively. the bicompactness yields two finite subcovers which we may respectively write as follows: {ui : i = 1, ..., m} , {vi : i = m + 1, ..., m + n} . then |γ| ≥ m + n and m+n ∨ i=1 (ui × vi) = m+n ∨ i=1 ui × m+n ∨ i=1 vi ≥ m ∨ i=1 ui × m+n ∨ i=m+1 vi = ⊤×⊤ = (⊤,⊤), showing that {ui × vi : i = 1, ..., m + n} is the needed subcover of x. � corollary 4.7. the functorial embedding e×◦gχ : bitop → 4-top preserves bicompactness to compactness in the sense of [5]. proof. since gχ : bitop → 4-bitop preserves traditional bicompactness to the bicompactness of 4.5, the corollary follows from 4.6. � we close this discussion of (6) above with a few comments. first, traditional bicompactness mandates the compactness of [5] for lattice-valued topology (4.7). second, we note (e×gχ) → (bitop) is isomorphic to bitop and closed under all products (in 4-top) (3.23, 3.29): this means that the cardinality unrestricted tihonov theorem for bitop (4.3) transfers to a cardinality unrestricted tihonov theorem for the subcategory (e×gχ) → (bitop) of 4-top w.r.t. the compactness of [5]. third, it now follows (4.4, 4.7) that e× 106 s. e. rodabaugh is not object-onto (already known) and that the special cardinality restriction of the goguen-tihonov theorem for 4-top resides outside the subcategory (e×gχ) → (bitop). acknowledgements. the referees are thanked for their thorough reading and helpful criticisms which significantly improved this paper. the author is especially grateful to the referee who found an important error in the original submission following corollary 4.4, the correction of which sheds light on the topological properties of the e× functor. references [1] j. adámek, h. herrlich, g. e. strecker, abstract and concrete categories: the joy of cats, wiley interscience pure and applied mathematics (john wiley & sons, brisbane/chicester/new york/singapore/toronto, 1990). 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[53] l. a. zadeh, fuzzy sets, information and control 8 (1965), 338–353. received october 2006 accepted april 2007 s. e. rodabaugh (rodabaug@math.ysu.edu) department of mathematics and statistics, youngstown state university, youngstown, ohio 44555-3609, usa carcraagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 15-19 cancellation of 3-point topological spaces s. carter and f. j. craveiro de carvalho ∗ abstract. the cancellation problem, which goes back to s. ulam [2], is formulated as follows: given topological spaces x, y, z, under what circumstances does x × z ≈ y × z (≈ meaning homeomorphic to) imply x ≈ y ? in [1] it is proved that, for t0 topological spaces and denoting by s the sierpinski space, if x × s ≈ y × s then x ≈ y . this note concerns all nine (up to homeomorphism) 3-point spaces, which are given in [4]. 2000 ams classification: 54b10 keywords: homeomorphism, cancellation problem, 3-point spaces. 1. two cancellation results below x and y denote t1 topological spaces. proposition 1.1. let s be a topological space with a unique closed singleton {p}. if there is a homeomorphism φ : x×s → y ×s then φ(x×{p}) = y ×{p}. proof. we shall show that φ(x×{p}) ⊂ y ×{p} which, using similar arguments, will be enough to prove that φ(x × {p}) = y × {p} and, consequently, that x ≈ y . let us suppose that for some x ∈ x,y ∈ y and q ∈ s \ {p} we have φ(x,p) = (y,q). then {(y,q)} is closed and, therefore, (y × s) \ {(y,q)} is open. let r belong to the topological closure of {q},r 6= q. then (y,r) ∈ (y ×s) \ {(y,q)} and we must have open sets uy,ur, containing y and r, respectively, such that uy × ur ⊂ (y × s) \ {(y,q)}. we reach a contradiction since (y,q) belongs to uy × ur. � ∗the second named author gratefully acknowledges financial support from fundação para a ciência e tecnologia, lisboa, portugal. 16 s. carter and f. j. craveiro de carvalho an example of such an s is obtained as follows. let s be a set with 4 elements at least. let a,b ∈ s and denote by s1 the complement of the subset they form. take then as basis for a topology on s the set {{a},{a,b},s1}. if s happens to have just 4 points then it is the only minimal, universal space with such a number of elements [3]. proposition 1.2. let s be a topological space with a dense, open singleton {p} and such that, for every q ∈ s \ {p}, the topological closure of {q} is finite. if there is a homeomorphism φ : x × s → y × s then φ(x × {p}) = y × {p}. proof. let {p} be an open, dense singleton in s. we will show that φ(x × {p}) = y × {p} which, as observed before, is enough to conclude that x ≈ y . assume that for some x ∈ x,y ∈ y and q 6= p we have φ(x,p) = (y,q). consider the closed set {y}×{q}, the bar denoting closure, its image φ−1({y}× {q}), which is also closed, and suppose that {q} has s elements. also, observe that p /∈ {q}. since (x,p) belongs to φ−1({y}×{q}) and this set has s elements, there is an r in {q} such that (x,r) does not belong to this set. there are then open sets ux,ur, containing x and r, respectively, with ux×ur ⊂ (x×s)\φ −1({y}×{q}). we have a contradiction since (x,p) ∈ ux × ur. � an example for s can be the following door space. let s be a set and fix p ∈ s. define u ⊂ s to be open if it is empty or contains p. 2. 3-point spaces we go on assuming that x,y are t1 topological spaces though such assumption is not used in propositions 2.1 and 2.2 below. if we now consider s = {a,b,c} to be one of the 3-point spaces [4], we see that propositions 1.1 and 1.2 of §1 allow us to deduce immediately that s can be cancelled except in the following cases s is discrete, s has {{a},{b},{a,c}} as a topological basis, s is trivial. if s is discrete the situation is not as simple as one might be led to think. let us take the following example. let s = z, here z stands for the integers with the discrete topology, and consider the discrete spaces x = {0, 1, . . . ,n − 1},n ≥ 2,y = {0}. now define φ : {0, 1, . . . ,n − 1} × z → {0} × z by φ(x,r) = (0,nr + x). this map is a homeomorphism and however z cannot be cancelled. we can say something when the spaces x,y have a finite number of connected components. proposition 2.1. let s be a finite discrete space and assume that x has a finite number of connected components. if x × s ≈ y × s then x ≈ y . cancellation of 3-point topological spaces 17 proof. the connected components of x × s or y × s are of the type x′ × {x},y ′ ×{y}, where x′,y ′ are components of x and y , respectively. it follows that y has the same number of components as x. let us consider in the sets of connected components of x and connected components of y the homeomorphism equivalence relation and take an equivalence class of components of x, say {x1, . . . ,xk}. the subspace k⋃ i=1 xi × s has kn components, where n is the cardinal of s. the same happens with φ( k⋃ i=1 xi × s), where φ is a homeomorphism between x × s and y × s. let p ∈ s. for every i = 1, . . . ,k, φ(xi × {p}) = yi × {qi}, where the qi’s belong to s and the yi’s are components of y homeomorphic to the xi’s. assume that the equivalence class to which the yi’s belong is {y1, . . . ,yl}. then φ( k⋃ i=1 xi × {p}) ⊂ l⋃ j=1 yj × s. consequently, also φ( k⋃ i=1 xi × s) ⊂ l⋃ j=1 yj × s. using the inverse homeomorphism φ−1, we are led to conclude that the reverse inclusion holds and, therefore, φ( k⋃ i=1 xi × s) = l⋃ j=1 yj × s. so k⋃ i=1 xi × s and l⋃ j=1 yj × s have the same number of components and it follows that k = l. from each component class in x choose a representative and use φ to establish a homeomorphism between that representative and a component in y . these homeomorphisms can then be used to conclude that every component of x is homeomorphic to a component of y . since components are closed and finite in number, x is homeomorphic to y . � proposition 2.2. let x and y be topological spaces with the same finite number of connected components and s be a discrete space. assume, moreover, that neither space has two homeomorphic components. if x × s ≈ y × s then x ≈ y . proof. let xi, i = 1, . . . ,n, be the components of x and fix p ∈ s. if φ is a homeomorphism between x × s and y × s then there are qi ∈ s,i = 1, . . . ,n, such that φ(xi × {p}) = yi × {qi}, i = 1, . . . ,n, where, due to our assumption on the non-existence of homeomorphic components, the yi’s are the components of y . hence φ induces a homeomorphism φi : xi → yi, i = 1, . . . ,n. again, since the number of components is finite and they are closed, the φi’s can be used to obtain a homeomorphism between x and y . � proposition 2.3. let s have {{a},{b},{a,c}} as basis. if φ : x ×s → y ×s is a homeomorphism then φ(x × {b}) = y × {b}. 18 s. carter and f. j. craveiro de carvalho proof. let πs : y × s → s denote the standard projection. the image πs (φ(x × {b})) is open and, therefore, it is either {b} or contains a. assume that for some x ∈ x,y ∈ y we have φ(x,b) = (y,a). the subset {(x,b)} is closed and, consequently, the same happens with {(y,a)}. hence (y × s) \ {(y,a)} is open and contains (y,c). we must then have an open neighbourhood uy of y such that uy × {a,c} ⊂ (y × s) \ {(y,a)}. again we have a contradiction and φ(x × {b}) = y × {b}. � to conclude the proof that a non-discrete 3-point space can be cancelled it only remains to deal with the case where s is trivial. above we have an example of a homeomorphism φ : x × s → y × s which does take a slice x ×{x} onto a slice y ×{y}. more examples can be obtained. take x = y , with at least 2 elements, a trivial space s with also, at least, 2 elements and let ψ : s → s be a fixed point free bijection. fix x0 ∈ x and define φ : x × s → x × s by φ(x,s) = (x,s), for x 6= x0, and φ(x0,s) = (x0,ψ(s)). then φ is a bijection and φ({x} × s) = {x} × s, for x ∈ x. since open sets in x × s are of the form u × s, u open in x, and φ(u × s) = u × s, φ is a homeomorphism. obviously no slice x × {x} is mapped onto a similar slice. proposition 2.4. let s be a finite trivial space. if x × s ≈ y × s then x ≈ y . proof. open (closed) sets in x × s and y × s are of the form u × s, where u is open (closed). we are going to define f : x → y as follows. let x ∈ x. then {x} is closed and so are {x} × s and φ({x} × s), where φ : x × s → y × s is a homeomorphism. hence φ({x}×s) = c ×s, for some closed set c in y . since s is finite, c is a singleton and we make {f(x)} = c. this way we obtain an f which is a bijection since we began with a bijective φ. if c is closed in x, φ(c × s) = f(c) × s is closed in y × s. consequently f(c) is closed in y . therefore f is closed and f−1 is continuous. taking φ−1, we would conclude that f is continuous the same way. � we can now state. theorem 2.5. for x and y t1 topological spaces and s a non-discrete 3-point topological space, if x × s ≈ y × s then x ≈ y . 3. a particular case we will no longer assume x,y to be t1 and will suppose that s has a unique isolated point a. moreover, the singleton {a} will be assumed to be closed. that is, for instance, the case where s = {a,b,c} and {{a},{b,c}} is an open basis. cancellation of 3-point topological spaces 19 proposition 3.1. let s have a unique isolated point a. assume that {a} is closed. for x,y connected with, at least, an isolated point each, if φ : x×s → y × s is a homeomorphism then φ(x × {a}) = y × {a}. proof. let πs : y × s → s denote the standard projection, as before. the image πs (φ(x ×{a})) is open and connected. therefore it is either {a} or some open, connected subset of s, which naturally does not contain a. let the latter be the case. if x ∈ x is an isolated point then {(x,a)} is open and the same happens to its image under πs ◦ φ. this is impossible because {a} is the unique open singleton of s. � examples of spaces satisfying the conditions of proposition 3.1 are, again, some door spaces. let z be a set. fix p ∈ z and define u ⊂ z to be open if u = z or p /∈ u. references [1] b. banaschewski and r. lowen, a cancellation law for partially ordered sets and t0 spaces, proc. amer. math. soc. 132 (2004). [2] r. h. fox, on a problem of s. ulam concerning cartesian products, fund. math. 27 (1947). [3] k. d. magill jr, universal topological spaces, amer. math. monthly 95 (1988). [4] j. r. munkres, topology, a first course, prentice-hall, inc., 1975. received july 2006 accepted november 2006 s. carter (s.carter@leeds.ac.uk) school of mathematics, university of leeds, leeds ls2 9jt, u.k. f. j. craveiro de carvalho (fjcc@mat.uc.pt) departamento de matemática, universidade de coimbra, 3001-454 coimbra, portugal @ applied general topology c© universidad politécnica de valencia volume 11, no. 2, 2010 pp. 89-93 thin subsets of balleans ievgen lutsenko and igor protasov abstract. a ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. we characterize the ideal generated by the family of all thin subsets in an ordinal ballean, and apply this characterization to metric spaces and groups. 2000 ams classification: 54a25, 54e25, 05a18. keywords: ballean, thin subsets, ideal. let g be a group with the identity e. a subset a ⊆ g is called thin if |ga ∩ a| < ℵ0 for every g ∈ g, g 6= e. for thin subsets, its modifications, applications and references see [4]. we denote by tg the family of all thin subsets of g. then the smallest ideal t ∗g (in the boolean algebra of all subsets of g) containing tg is the family of all finite unions of thin subsets. thus, to characterize t ∗g, we need some test which, for given a ⊆ g and m ∈ n, detect whether a can be represented as a union of 6 m thin subsets. let (x, d) be a metric space. we say that a subset a ⊆ x is thin if, for every r ∈ r+, there exists a bounded subset y ⊆ x such that a ∩ b(x, r) = {x} for every x ∈ a \ y , where bd(x, r) = {x ∈ y : d(x, y) 6 r}. as in the group case, to characterize the ideal t ∗(x, d) generated by the family t (x, d) of all thin subsets of (x, d), we ask for a test recognizing if a subset a ⊆ x is a union of 6 m thin subsets. it is easy to see that a subset a ⊆ g is thin if and only if, for every finite subset f of g containing e, there exists a finite subset y of g such that a ∩ fg = {g} for every x ∈ a \ y . following [1], we say that fg is a ball of radius f around g. from this point of view, the definitions of the thin subsets in groups and metric spaces are very similar syntactically. to formalize this similarity we use the ballean approach from [5]. a ballean is a set endowed with some family of 90 ie. lutsenko and i. protasov its subsets which are called the balls. the property of the family of ball are postulated in such a way that the balleans can be considered as the counterparts of the uniform topological spaces (see section 1 for precise definition). in section 1 we define the thin subsets of a ballean and, for every ordinal ballean, characterize the ideal generated by the thin subsets. the group and metric spaces have the natural ballean structures. in section 2 we apply the result from section 1 to justify the following two tests. a subset a of a metric space x can be partitioned in 6 m thin subsets if and only if, for every r ∈ r+, there exists a bounded subset y ⊆ x such that |a ∩ b(x, r)| 6 m for every x ∈ a \ y . a subset a of a countable group g can be partitioned in 6 m thin subsets if and only if, for every finite subset f of g, there exists a finite subset y of g such that |a ∩ fx| 6 m for every x ∈ a \ y . we do not know whether this test is effective for an uncountable group. 1. ballean context a ball structure is a triple b = (x, p, b), where x, p are not-empty sets and, for every x ∈ x and α ∈ p , b(x, α) is a subset of x which is called a ball of radius α around x. it is supposed that x ∈ b(x, α) for all x ∈ x and α ∈ p . the set x is called the support of b, p is called the set of radii. given any x ∈ x, a ⊆ x, we put b∗(x, α) = {y ∈ x : x ∈ b(y, α)}, b(a, α) = ⋃ a∈a b(a, α). a ball structures b is called a ballean if • for any α, β ∈ p , there exist α′, β′ ∈ p such that, for every x ∈ x, b(x, α) ⊆ b∗(x, α′), b∗(x, β) ⊆ b(x, β′); • for any α, β ∈ p , there exist γ ∈ p such that, for every x ∈ x, b(b(x, α), β) ⊆ b(x, γ). we note that a ballean can also be defined in terms of entourages of diagonal in x × x. in this case it is called a coarse structures [7]. a ballean b is called connected if, for any x, y ∈ x, there exists α ∈ p such that y ∈ b(x, y). all balleans under considerations are suppose to be connected. replacing each ball b(x, α) to b(x, α) ∩ b∗(x, α), we may suppose that b∗(x, α) = b(x, α) for all x ∈ x, α ∈ p . a subset y ⊆ x is called bounded if there exist x ∈ x and α ∈ p such that y ⊆ b(x, α). we use a preordering 6 on the support x of b defined by the rule: α 6 β if and only if b(x, α) 6 b(x, β) for every x ∈ x. a subset p ′ ⊆ p is called cofinal if, for every α ∈ p , there exists α′ ∈ p ′ such that α 6 α′. a ballean b is called ordinal if there exists a cofinal subset p ′ ⊆ p well ordered by 6. let b = (x, p, b) be a ballean, m ∈ n. we say that a subset a ⊆ x is m-thin if, for every α ∈ p , there exists a bounded subset yα ⊆ x such that |b(x, α) ∩ a| 6 m for every x ∈ a \ yα. a 1-thin subset is called thin. thus, thin subsets of balleans 91 a is thin if, for every α ∈ p , there exists a bounded subset yα of x such that b(x, α) ∩ a = {x} for every x ∈ a \ yα. in the terminology of [5], the thin subsets are called pseudodiscrete. for pseudodiscreteness see also [2], [6]. we use the following notation: t (b) is the family of all thin subsets of x; tm(b) is the family of all m-thin subsets of x;⋃ m t (b) is the family of all unions of 6 m thin subsets of x; t ∗(b) is the ideal generated by t (b). clearly, t ∗(b) = ⋃ m∈n( ⋃ m t (b)). lemma 1.1. for every ballean b, we have ⋃ m t (b) ⊆ tm(b). proof. let a1, . . . , an be thin subsets of x. for every α ∈ p , we pick γ(α) ∈ p such that b(b(x, α), α) = b(x, γ(α)). for all α ∈ p and i ∈ {1, . . . , m}, we choose a bounded subset yα(i) such that b(x, α) ∩ ai = {x} for every x ∈ ai \yα(i), and put yα = yα(1)∪. . .∪yα(m). we take an arbitrary element a ∈ (a1 ∪. . .∪am)\yα and suppose that |b(a, α)∩(a1 ∪. . .∪am)| > m. then there exists j ∈ {1, . . . , m} such that |aj ∩b(a, α)| > 2. let b, c ∈ ai ∩b(a, α), b 6= c. then c ∈ b(b, γ(α)) contradicting the choice of yα(j). � the following theorem gives a characterization of t ∗(b) in the case of an ordinal ballean b. theorem 1.2. for every ordinal ballean b and m ∈ n, we have tm(b) =⋃ m t (b). proof. in view of lemma 1.1, it suffices to show that tm(b) ⊆ ⋃ m t (b). let a ∈ tm(b). we may suppose that p is will ordered by 6. we construct inductively a family {yα : α ∈ p} of bounded subsets of x such that |b(x, α)∩ a| for every x ∈ a\yα and yα ⊆ yβ for all α 6 β. then we consider a graph γ with the set of vertices a and the set of edges e defined as follows: {x, y} ∈ e if and only if x 6= y and there exists α ∈ p such that x, y ∈ a\yα and y ∈ b(x, α). we show that deg(x) 6 m − 1 for every x ∈ a, where deg(x) = |{y ∈ a : {x, y} ∈ e}|. we suppose the contrary and choose x ∈ a and distinct vertices y1, . . . , ym such that {x, yi} ∈ e for every i ∈ {1, . . . , m}. by the definition of e, for every i ∈ {1, . . . , m}, there exists αi ∈ p and a bounded subset yαi of x such that yi ∈ b(x, αi) and x, yi ∈ a \ yαi. let α = max{α1, . . . , αm} and α = αj. then y1, . . . , ym ∈ b(x, αj) and y1, . . . , ym ∈ a \ yαj because yαi ⊆ yαj for all i ∈ {1, . . . , m}, so we get a contradiction with the choice of yαj because |b(x, αj) ∩ a| 6 m. by [3, corollary 12.2], the chromatic number of γ does not exceed m. hence a can be partitioned a = a1 ∪ . . . ∪ak, κ 6 m so that, for every i ∈ {1, . . . , k} and x, y ∈ ai, we have {x, y} /∈ e. we show that each subset ai is thin. for every α ∈ p , we put zα = b(yα, α). suppose that there exists x ∈ ai \ zα such that |b(x, α) ∩ ai| > 1. let y ∈ b(x, α)∩ai and y 6= x. since x /∈ zα then y /∈ yα. thus x, y ∈ a\ yαi and y ∈ b(x, α), so {x, y} ∈ e contradicting the choice of ai. � 92 ie. lutsenko and i. protasov 2. applications theorem 2.1. let (x, d) be a metric space, m ∈ n. a subset a ⊆ x can be partitioned in 6 m thin subsets if and only if , for every r ∈ r+, there exists a bounded subset y of x such that |b(x, r) ∩ a| 6 m for every x ∈ a \ y . proof. we consider (x, d) as the ballean b(x, d) = (x, r+, bd). clearly, b(x, d) is ordinal so we can apply theorem 1.2. � let g be a group, κ be an infinite cardinal, fκ(g) = {f ⊆ g : |f | < κ, e ∈ f}. we consider the ballean bκ(g) = (g, fκ(g), b), where b(g, f) = fg for all g ∈ g, f ∈ fκ(g). if κ > |g|, bκ(g) is bounded. for κ = |g|, bκ(g) is ordinal. indeed, let g0 = e, {gα : α < κ} be a numeration of g, fα = {gβ : β 6 α}. then the well ordered by ⊆ family f = {fα : α < κ} is cofinal in f. we say that a subset a ⊂ g is κ-thin if |ga ∩ a| < κ for every g ∈ g, g 6= e. in the case κ = ℵ0, we get the thin subsets defined in the very beginning of the paper. lemma 2.2. let a be a subset of a group g. if a is thin in the ballean bκ(g) then a is κ-thin. if a is κ-thin and κ is regular then a is thin in the ballean bκ(g). proof. let a be thin in bκ(g). for every g ∈ g, g 6= e, we put fg = {e, g} and choose a bounded subset yg in bκ(g) such that b(x, f) ∩ a = {x} for every x ∈ a \ yg. then gx /∈ a for every x ∈ a \ yg so ga ∩ a ⊆ yg. since yg is bounded in bκ(g) then |yg| < κ and a is κ-thin. let a be κ-thin, f ∈ fκ(g). we put y = ⋃ {ga ∩ a : g ∈ f \ {e}}. since |ga ∩ a| < κ, |f | < κ and κ is regular, |y | < κ so y is bounded in bκ(g). for every x ∈ a \ y , we have fx ∩ a = {x} hence a is thin in bκ(g). � theorem 2.3. let g be a group of regular cardinality κ, m ∈ n. a subset a ⊆ g can be partitioned in 6 m κ-thin subsets if and only if, for every f ⊂ g, |f | < κ, there exists a subset y ⊆ g such that |y | < κ and |fx ∩ a| 6 m for every x ∈ a \ y . proof. since the ballean bκ(g) is ordinal, in view of lemma 2.2, we can apply theorem 1.2. � remark 2.4. a subset a of a group g is called almost thin if the set ∆(a) = {g ∈ g : ga ∩ a is infinite} is finite. by [4, theorem 3.1], every almost thin subset of a group g can be partitioned in 3|∆(a)|−1 thin subsets, but the union of two thin subsets needs not to be almost thin [4, theorem 3.2]. thin subsets of balleans 93 references [1] a. bella and v. malykhin, on certain subsets of groups, questions answers gen. topology 17 (1999), 183–187. [2] m. filali and i. v. protasov, spread of balleans, appl. gen. topol. 9 (2008), 169–175. [3] f. harary, graph theory, addison-wesley publ. comp., 1969. [4] ie. lutsenko and i. v. protasov, sparse, thin and other subsets of groups, internat. j. of algebra comput. 19 (2009), 491–510. [5] i. protasov and m. zarichnyi, general asymptology, math. stud. monogr. ser., vol. 11, vntl publishers, lviv, 2007. [6] o. protasova, pseudodiscrete balleans, algebra discrete math. 4 (2006), 81–92. [7] j. roe, lecture on coarse geometry, ams university lecture ser. 31 (2003). received december 2009 accepted september 2010 ie. lutsenko (ie.lutsenko@gmail.com) departament of cybernetics, kyiv university, volodimirska 64, kyiv, 01033, ukraine i. protasov (i.v.protasov@gmail.com) departament of cybernetics, kyiv university, volodimirska 64, kyiv, 01033, ukraine thin subsets of balleans. by ie. lutsenko and i. protasov @ appl. gen. topol. 23, no. 2 (2022), 437-451 doi:10.4995/agt.2022.16655 © agt, upv, 2022 remarks on fixed point assertions in digital topology, 5 laurence boxer department of computer and information sciences, niagara university, niagara university, ny 14109, usa; and department of computer science and engineering, state university of new york at buffalo, usa (boxer@niagara.edu) communicated by s. romaguera abstract as in [6, 3, 4, 5], we discuss published assertions concerning fixed points in “digital metric spaces” assertions that are incorrect or incorrectly proven, or reduce to triviality. 2020 msc: 54h25 keywords: digital topology, fixed point, metric space 1. introduction as stated in [3]: the topic of fixed points in digital topology has drawn much attention in recent papers. the quality of discussion among these papers is uneven; while some assertions have been correct and interesting, others have been incorrect, incorrectly proven, or reducible to triviality. paraphrasing [3] slightly: in [6, 3, 4, 5], we have discussed many shortcomings in earlier papers and have offered corrections and improvements. we continue this work in the current paper. authors of many weak papers concerning fixed points in digital topology seek to obtain results in a “digital metric space” (see section 2.1 for its definition). this seems to be a bad idea. we quote [5]: received 14 november 2021 – accepted 19 march 2022 http://dx.doi.org/10.4995/agt.2022.16655 https://orcid.org/0000-0001-7905-9643 l. boxer • nearly all correct nontrivial published assertions concerning digital metric spaces use either the adjacency of the digital image or the metric, but not both. • if x is finite (as in a “real world” digital image) or the metric d is a common metric such as any `p metric, then (x,d) is uniformly discrete as a topological space, hence not very interesting. • many of the published assertions concerning digital metric spaces mimic analogues for subsets of euclidean rn. often, the authors neglect important differences between the topological space rn and digital images, resulting in assertions that are incorrect, trivial, or trivial when restricted to conditions that many regard as essential. e.g., in many cases, functions that satisfy fixed point assertions must be constant or fail to be digitally continuous [6, 3, 4]. since the publication of [5], additional papers concerning fixed points in digital metric spaces have come to our attention. this paper continues the work of [6, 3, 4, 5] in discussing shortcomings of published assertions concerning fixed points in digital metric spaces. many of the definitions and assertions we discuss were written with typographical and grammatical errors, and mathematical flaws. we have quoted these by using images of the originals so that the reader can see these errors as they appear in their sources (we have removed or replaced with a different style labels in equations and inequalities in the images to remove confusion with labels in our text). 2. preliminaries much of the material in this section is quoted or paraphrased from [5]. we use n to represent the natural numbers, z to represent the integers, and r to represent the reals. a digital image is a pair (x,κ), where x ⊂ zn for some positive integer n, and κ is an adjacency relation on x. thus, a digital image is a graph. in order to model the “real world,” we usually take x to be finite, although there are several papers that consider infinite digital images. the points of x may be thought of as the “black points” or foreground of a binary, monochrome “digital picture,” and the points of zn\x as the “white points” or background of the digital picture. for this paper, we need not specify the details of adjacencies or of digitally continuous functions. a fixed point of a function f : x → x is a point x ∈ x such that f(x) = x. 2.1. digital metric spaces. a digital metric space [9] is a triple (x,d,κ), where (x,κ) is a digital image and d is a metric on x. the metric is usually taken to be the euclidean metric or some other `p metric. we are not convinced that the digital metric space is a notion worth developing. typically, assertions © agt, upv, 2022 appl. gen. topol. 23, no. 2 438 remarks on fixed point assertions in digital topology, 5 figure 1. definition of kannan digital contraction in [11] figure 2. fixed point assertion for kannan digital contractions in [11] in the literature do not make use of both d and κ, so that “digital metric space” seems an artificial notion. e.g., for a discrete topological space x, all functions f : x → x are continuous, although on digital images, many functions g : x → x are not digitally continuous (digital continuity is defined in [2], generalizing an earlier definition [15]). 3. assertions for contractions in [11] the paper [11] claims fixed point theorems for several types of digital contraction functions. serious errors in the paper are discussed below. 3.1. fixed point for kannan contraction. figure 1 shows the definition appearing in [11] of a kannan digital contraction. figure 2 shows a fixed point assertion for kannan digital contractions in [11]. the “proof” of this assertion has errors discussed below. • in the fourth and fifth lines of the “proof” is the claim that λ[d(xn,xn−1) + d(xn−1,xn−2)] ≤ 2λd(xn,xn−1). since λ > 0, this claim implies d(xn−1,xn−2) ≤ d(xn,xn−1), so if any d(xn,xn−1) is positive, the sequence {xn} is not a cauchy sequence, contrary to a claim that appears later in the argument. © agt, upv, 2022 appl. gen. topol. 23, no. 2 439 l. boxer figure 3. the example of [11], pp. 10769 10770 figure 4. a “theorem” of [11], p. 10770 • towards the end of the existence argument, the authors claim that a kannan digital contraction is digitally continuous. this assumption is contradicted by example 4.1 of [6]. in light of these errors, we must conclude that the assertion of figure 2 is unproven. 3.2. example of pp. 10769 10770. this example is shown in figure 3. one sees easily the following. • d(0, 1) = d(0, 2) = 0, so d, contrary to the claim, is not a metric. • t(1) = 1/2 6∈ x. 3.3. fixed point for generalization of kannan contraction. figure 4 shows an assertion of a fixed point result on p. 10770 of [11]. note “and let f satisfies” should be “and let t satisfy”. © agt, upv, 2022 appl. gen. topol. 23, no. 2 440 remarks on fixed point assertions in digital topology, 5 figure 5. definition 3.2 of [13], used in [11, 14]. more importantly: in the argument offered as proof of this assertion, the authors let x0 ∈ x, and, inductively, xn+1 = t(xn), an+1 = d(xn,xn+1). they claim that by using the statements marked (i) and (ii) in figure 4, it follows that an+1 = d(xn,xn+1) ≤ υ(d(xn,xn−1) + d(xn−1,xn−2)) (3.1) < 2d(xn,xn−1) = 2an, which, despite the authors’ claim, does not show that the sequence {an} is decreasing. however, what correctly follows from the statements marked (i) and (ii) in figure 4 is an+1 = d(xn,xn+1) = d(t(xn−1),t(xn)) ≤ υ(d(xn−1,t(xn−1)) + d(xn,t(xn))) = υ(d(xn−1,xn) + d(xn,xn+1)) (3.2) = υ(an + an+1) < an + an+1 2 . from this we see that an+1 < an, so the sequence {an} is decreasing and bounded below by 0, hence tends to a limit a ≥ 0. the authors then claim that if a > 0 then an+1 ≤ y (an). however, what we showed in (3.2) does not support this conclusion, which is not justified in any obvious way. since the authors wish to contradict the hypothesis that a > 0 in order to derive that the sequence {xn} is a cauchy sequence, we must regard the assertion shown in figure 4 as unproven. 3.4. fixed point for zamfirescu contraction. a zamfirescu digital contraction is defined in figure 5. this notion is used in [11, 14], and will be discussed in the current section and in section 6. figure 6 shows an assertion found on p. 10770 of [11]. the argument offered as “proof” of this assertion considers cases. for an arbitrary x0 ∈ x, a sequence is inductively defined via xn+1 = t(xn). for © agt, upv, 2022 appl. gen. topol. 23, no. 2 441 l. boxer figure 6. another “theorem” of [11], p. 10770 convenience, let us define m(x,y) = max { d(x,y), d(x,tx) + d(y,ty) 2 , d(x,ty) + d(y,tx) 2 } . the argument considers several cases. • case 1 says if d(xn+1,xn) = m(xn+1,xn) then, by implied induction, d(xn+1,xn) ≤ λnd(x1,x0). but this argument is based on the unproven assumption that this is also the case for all indices i < n; i.e., that d(xi+1,xi) = m(xi+1,xi). • case 2 says if d(xn+1,txn+1) + d(xn,txn) 2 = max   d(xn+1,xn), d(xn+1,t xn+1)+d(xn,t xn) 2 , d(xn+1,t xn)+d(xn,t xn1 ) 2   , then d(xn+1,xn) = d(txn,txn−1) ≤ λ d(xn,xn−1) + d(xn−1,xn−2) 2 ≤ λd(xn,xn−1). but in order for the second inequality in this chain to be true, we would need d(xn−1,xn−2) ≤ d(xn,xn−1), and no reason is given to believe the latter. • case 3 says that if d(xn+1,t xn)+d(xn,t xn+1) 2 = m(xn+1,xn) then d(xn+1,xn) = d(txn,txn−1) ≤ λ d(xn,xn−2) + d(xn−1,xn−1) 2 . the correct upper bound, according to the definition shown in figure 5, is λ d(xn+1,txn) + d(xn,txn+1) 2 = λ d(xn+1,xn+1) + d(xn,xn+2) 2 = λ d(xn,xn+2) 2 . © agt, upv, 2022 appl. gen. topol. 23, no. 2 442 remarks on fixed point assertions in digital topology, 5 figure 7. definition of rhoades digital contraction, [11], p. 10769 figure 8. fixed point assertion for rhoades digital contraction, [11], p. 10771 further, the conclusion reached by the authors for this case, that the distances d(xn+1,xn) are bounded above by an expression that tends to 0 as n →∞, depends on the unproven hypothesis that an analog of this case holds for all indices i < n. thus all three cases considered by the authors are handled incorrectly. we must conclude that the assertion of figure 6 is unproven. 3.5. fixed point for rhoades contraction. figure 7 shows the definition appearing in [11] of a rhoades digital contraction. the paper [11] claims the fixed point result shown in figure 8 for such functions. the argument offered in “proof” of this assertion has errors that are discussed below. for convenience, let m(x,y) = max { d(x,y), d(x,tx) + d(y,ty) 2 , d(x,ty), d(y,tx) } . the authors’ argument considers cases corresponding to which of the embraced expressions above gives the value of m(xn+1,xn). in each case, the authors assume without proof that the same case is valid for m(xi+1,xi), for all indices i < n. additional errors: • in case 2, the inequality d(txn,txn−1) ≤ λ d(xn,xn−1) + d(xn−1,xn−2) 2 © agt, upv, 2022 appl. gen. topol. 23, no. 2 443 l. boxer should be, according to figure 7, d(xn,xn+1) = d(txn,txn−1) ≤ λ d(xn,txn) + d(xn+1,txn−1) 2 = λ d(xn,xn+1) + d(xn+1,xn) 2 = λd(xn,xn+1). note this implies (3.3) xn = xn+1, which would imply the existence of a fixed point. also, the authors claim that λ d(xn,xn−1) + d(xn−1,xn−2) 2 ≤ λd(xn,xn−1), which is equivalent to d(xn−1,xn−2) ≤ d(xn,xn−1). no reason is given in support of the latter; further, it undermines the later claim that {xn} is a cauchy sequence, since the authors did not deduce (3.3). • in case 3, it is claimed that λd(xn,txn−1) ≤ λd(xn,xn−1). this should be corrected to λd(xn,txn−1) = λd(xn,xn) = 0, which would guarantee a fixed point. • in case 4, we see the claim d(xn−1,txn) = d(xn−1,xn−1) = 0. this should be corrected to d(xn−1,txn) = d(xn−1,xn+1). in view of these errors, we must regard the assertion shown in figure 8 as unproven. 4. assertions for weakly compatible maps in [1] in this section, we show that the assertions of [1] are trivial or incorrect. 4.1. theorem 3.1 of [1]. definition 4.1. [8] let s,t : x → x. then s and t are weakly compatible or coincidentally commuting if, for every x ∈ x such that s(x) = t(x) we have s(t(x)) = t(s(x)). © agt, upv, 2022 appl. gen. topol. 23, no. 2 444 remarks on fixed point assertions in digital topology, 5 figure 9. the assertion stated as theorem 3.1 of [1] the assertion stated as theorem 3.1 in [1] is shown in figure 9. in this section, we show the assertion is false except in a trivial case. note if d is any `p metric (including the usual euclidean metric) then the requirements of closed subsets of x are automatically satisfied, since (x,d) is a discrete space. theorem 4.1. if functions g,h,p,q satisfy the hypotheses of theorem 3.1 of [1] then g = h = p = q. therefore, each of the pairs (p,g) and (q,h) has a unique common point of coincidence if and only if n consists of a single point. proof. we observe the following. • the inequality in the assertion simplifies as ψ(d(px,qy)) ≤− 1 2 ψ(dg,h (x,y)). since the function ψ is non-negative, we have (4.1) ψ(d(px,qy)) = ψ(dg,h (x,y)) = 0. therefore, we have (4.2) p(x) = q(y) for all x,y ∈ n. • in the equation for dg,h in figure 9, the pairs of points listed on the right side should be understood as having d applied, i.e., (4.3) dg,h (x,y) = max { d(g(x),h(y)),d(g(x),p(x)),d(h(y),q(y)), 1 2 [d(g(x),q(y)) + d(h(y),p(x))] } . since ψ(x) = 0 if and only if x = 0, we have from (4.1) that dg,h (x,y) = 0, so (4.2) and (4.3) imply g(x) = p(x) = q(x) = h(x) for all x ∈ n. © agt, upv, 2022 appl. gen. topol. 23, no. 2 445 l. boxer figure 10. the assertion stated as example 3.1 of [1] we conclude that (p,g) and (q,h) are pairs of functions whose respective common points of coincidence are unique if and only if n consists of a single point. � 4.2. example 3.1 of [1]. in figure 10, we see the assertion presented as example 3.1 of [1]. note the following. • if “x = [4, 40]” is meant to mean the real interval from 4 to 40, then x is not a digital image, as all coordinates of members of a digital image must be integers. perhaps x is meant to be the digital interval [4, 40]z = [4, 40] ∩z. • the function g is not defined on all of [4, 40]z, appears to be doubly defined for some values of x, (notice the incomplete inequality at the end of the 3rd line) and is not restricted to integer values. • the function h is not defined on all of [4, 40]z and is doubly defined for x ∈{13, 14}. • the function q is not defined for x ∈ {9, 10, 11, 12}, and is doubly defined for x ∈{14, 15}. • the “above theorem” referenced in the last sentence of figure 10 is the assertion discredited by our theorem 4.1, which shows that p = q = g = h. clearly, the assertion shown in figure 10 fails to satisfy the latter. thus, example 3.1 of [1] is not useful. 4.3. corollary 3.2 of [1]. the assertion presented as corollary 3.2 of [1] is presented in figure 11. notice that “weakly mappings” is an undefined phrase. no proof is given in [1] for this assertion, and it is not clear how the assertion might follow from previous assertions of the paper (which, as we have seen above, are also flawed). © agt, upv, 2022 appl. gen. topol. 23, no. 2 446 remarks on fixed point assertions in digital topology, 5 figure 11. the assertion stated as corollary 3.2 of [1] figure 12. the assertion presented as theorem 4 of [12] perhaps “weakly mappings” is intended to be “weakly compatible mappings”. at any rate, by labeling this assertion as a corollary, the authors suggest that it follows from the paper’s flawed “theorem” 3.1. the assertion presented as “corollary” 3.2 of [1] must be regarded as undefined and unproven. 5. assertion for coincidence and fixed points in [12] figure 12 shows the assertion presented as theorem 4 of [12]. the assertion as stated is false. flaws in this assertion include: • “d” apparently should be “l”, and “xx” apparently should be “x”. • no restriction is stated for the value of h. therefore, we can take h = 0, leaving the inequality in i) as b(fx,fy) ≥ 0; since b is a metric, this inequality is not a restriction. thus f and g are arbitrary; they need not have a coincidence point or fixed points. 6. assertion for zamfirescu contractions in [14] let f : x → x, where (x,d,κ) is a digital metric space. recall that a zamfirescu digital contraction [13] is defined in figure 5. we show the assertion presented as theorem 4.1 of [14] in figure 13. observe: • the symbol θ has distinct uses in this assertion. in the first line, θ is introduced as both the metric and the adjacency of x. since our discussion below does not use an adjacency, we will assume θ is the metric d. © agt, upv, 2022 appl. gen. topol. 23, no. 2 447 l. boxer figure 13. the assertion presented as theorem 4.1 of [14] • the symbol ϕ seems intended to be a real number satisfying some restriction, but no restriction is stated. alternately, it may be that ϕ is intended to be a function to be applied to the max value in the statement, but no description of such a function appears. perhaps most important, we have the following. theorem 6.1. if y has more than one point and d is any `p metric, then no function t satisfies the hypotheses shown in figure 13. proof. suppose there is such a function t. by choice of d, there exist u0,v0 ∈ x such that d(u0,v0) = min{d(x,y) | x,y ∈ x,x 6= y}. by the inequality stated in item 1 of figure 13, d(tu0,tv0) = 0. this contradicts the hypothesis that t is injective. � 7. assertion for expansive map in [7] the paper [7] claims to have a fixed point theorem for digital metric spaces. however, it is not clear what the authors intend to assert, as the paper has many undefined and unreferenced terms and many obscuring typographical errors. the assertion stated as “preposition” 2.7 of [7] (and also as an unlabeled proposition on p. 10769 of [11]) was shown in example 4.1 of [6] to be false. the definition of what this paper calls a generalized (α − φ) ψ expansive mapping for random variable is shown in figure 14. we observe the following. • this is not the same as a β − ψ − φ expansive mapping defined in [10]. • notice the ρ that appears intended to be the adjacency of the digital metric space. this is significant in our discussion later. • the set ψ is not defined anywhere in the paper. perhaps it is meant to be the set ψ of [10]. • the functions d, α, and m all have a third parameter a that appears to be extraneous, since each of these functions is used later with only two parameters. © agt, upv, 2022 appl. gen. topol. 23, no. 2 448 remarks on fixed point assertions in digital topology, 5 figure 14. the statement presented as definition 3.1 of [7] figure 15. the assertion presented as theorem 3.3 of [7] • one supposes µ and ω must be non-negative, but this is not stated. the assertion presented as theorem 3.3 of [7] is shown in figure 15. in the statement of this assertion: • “dgms” appears, although it is not defined anywhere in the paper. perhaps it represents “digital metric space”. • the term “exclusive rfp” is not defined anywhere in the paper. one supposes the “fp” is for “fixed point”. in the argument offered as “verification” of this assumption, we note the following. • the second line of the verification contains an undefined operator, + n which perhaps is meant to be +. • the same line contains part of the phrase “un is a unique point of s.” what the authors intend by this is unclear. • at the start of the long statement (3e), it is claimed that m(ξun,ξun+1) is the maximum of three expressions. the second term of the expression © agt, upv, 2022 appl. gen. topol. 23, no. 2 449 l. boxer for m(ξun,ξun+1) applies ρ to a numeric expression. this makes no sense, since ρ is the adjacency of y (see figure 14). notice also that figure 14 shows no such term in its expression for the function m. the use of ρ, as a numeric value that has neither been defined nor restricted to some range of values, propagates through both of the cases considered. • in the expression for m(ξun,ξun+1), the third term, ω(ξun,ξun−1), should be ωd(ξun,ξun−1) according to figure 14. this error repeats several times in statement (3e). other errors are present, but we have established enough to conclude that whatever the authors were trying to prove is unproven. 8. further remarks we have shown that nearly every assertion introduced in the papers [11, 1, 12, 14, 7] is incorrect, unproven due to errors in the “proofs,” or trivial. these papers are part of a larger body of highly flawed publications devoted to fixed point assertions in digital metric spaces, and emphasize our contention that the digital metric space is not a worthy subject of study. references [1] s. k. barve, q. a. kabir and r. d. daheriya, unique common fixed point theorem for weakly compatible mappings in digital metric space, international journal of scientific research and reviews 8, no. 1 (2019), 2114–2121. [2] l. boxer, a classical construction for the digital fundamental group, journal of mathematical imaging and vision 10 (1999), 51–62. [3] l. boxer, remarks on fixed point assertions in digital topology, 2, applied general topology 20, no. 1 (2019), 155–175. [4] l. boxer, remarks on fixed point assertions in digital topology, 3, applied general topology 20, no. 2 (2019), 349–361. [5] l. boxer, remarks on fixed point assertions in digital topology, 4, applied general topology 21, no. 2 (2020), 265–284. [6] l. boxer and p. c. staecker, remarks on fixed point assertions in digital topology, applied general topology 20, no. 1 (2019), 135–153. [7] c. chauhan, j. singhal, s. shrivastava, q. a. kabir and p. k. jha, digital topology with fixed point, materials today: proceedings 47 (2021), 7167–7169. [8] s. dalal, common fixed point results for weakly compatible map in digital metric spaces, scholars journal of physics, mathematics and statistics 4, no. 4 (2017), 196– 201. [9] o. ege and i. karaca, digital homotopy fixed point theory, comptes rendus mathematique 353, no. 11 (2015), 1029–1033. [10] k. jyoti and a. rani, fixed point theorems for β ψ φ-expansive type mappings in digital metric spaces, asian journal of mathematics and computer research 24, no. 2 (2018), 56–66. [11] k. jyoti and a. rani, fixed point theorems with digital contractions, international journal of current advanced research 7, no. 3(e) (2018), 10768–10772. © agt, upv, 2022 appl. gen. topol. 23, no. 2 450 remarks on fixed point assertions in digital topology, 5 [12] a. mishra, p. k. tripathi, a. k. agrawal and d. r. joshi, a contraction mapping method in digital image processing, international journal of recent technology and engineering 8, no. 4s5 (2019), 193–196. [13] l. n. mishra, k. jyoti, a. rani and vandana, fixed point theorems with digital contractions image processing, nonlinear science letters a 9, no. 2 (2018), 104–115. [14] k. rana and a. garg, various contraction conditions in digital metric spaces, advances in mathematics: scientific journal 9, no. 8 (2020), 5433–5441. [15] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. © agt, upv, 2022 appl. gen. topol. 23, no. 2 451 estyagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 259-265 cl(r) is simply connected under the vietoris topology n. c. esty abstract. in this paper we present a proof by construction that the hyperspace cl(r) of closed, nonemtpy subsets of r is simply connected under the vietoris topology. this is useful in considering the convergence of time scales. we also present a construction of the (known) fact that this hyperspace is also path connected, as part of the proof. 2000 ams classification: 54b20, 54d05 keywords: hyperspace, vietoris topology, simply connected, path connected, time scales 1. introduction spaces of all non-empty and closed subsets of a topological space (or hyperspaces) are a critical part of the study of time scales. the theory of time scales attempts to organize the solution methods for differential and difference equations which, when considered under the same equation, sometimes have very similar solutions and sometimes have wildly different solutions. the approach is to consider a dynamic equation over an unknown domain, which is a non-empty and closed subset of r, or in other words, a point in cl(r). such points of cl(r) are called time scales. for a good introduction to the theory of time scales, see [2]. when studying time scales in this context, there are immediate and interesting questions involving convergence: if a sequence of time scales converges, and we consider solutions of the same dynamic equation over each member of the sequence, will the solutions converge? of course, a formalized concept for convergence of functions over different domains is needed (in addition to a formalized concept of “sameness”). results for some of these questions have been given in [12], when the solutions are unique and the dynamic equation is sufficiently continuous. 260 n. c. esty central to this discussion is the topology on the space of time scales. there are several well known topologies on hyperspaces, including the hausdorff metric topology and the vietoris topology. the hausdorff metric topology of cl(r) is, happily, metrizable, but it has the unfortunate property that under it, [−n, n] does not converge to r. since this convergence would be useful in the context of time scales, we turn instead to the vietoris topology. this topology is not metrizable on cl(r); however, on hyperspaces associated to compact metrizable spaces it coincides with the hausdorff metric topology. in 1951, it was shown by michael that cl(r) was completely regular, seperable, and first countable; see [14]. the statement that it is locally compact turned out to contain an error – a correction to the problematic proposition can be found in [5] – however it is now known that cl(r) is not locally compact. a result of ivanova, keesling and velichko says that if the vietoris topology on cl(x) is normal, then x is compact: see [10], [11], and [16]. it follows that cl(r) is not a normal space. in 2003, hola, pelant and zsilinszky showed that cl(r) is not developable and that it is submetrizable; see [8]. it is also known to be strong alpha-favorable; this follows from statements in [19]. more attention has been paid to hyperspaces in the case that x is compact. it was shown as far back as 1931 by borsuk and mazurkiewicz that for a metrizable continuum x, both the hyperspace k(x) of compact subsets of x and c(x), the hyperspace of subcontinua of x, are path connected [4]. the non-metrizable case was investigated by mcwaters [13] and ward [17]. local path connectedness of k(x) and c(x) was shown to be equivalent to local connectedness of x if x is compact in [18] in 1939. for topological properties on compact vietoris hyperspaces, see the 1978 book of nadler [15], or the more recent book by illanes and nadler, [9]. in 2002, in [6], costantini and kubis showed that under the vietoris topology, cl(r) is pathwise connected but not locally connected. they actually showed a stronger statement, applying to a wider class of topologies, and giving conditions for path-wise connectedness. they also give several results for the hyperspace of closed, bounded sets under the hausdorff metric topolgoy, including that it is an absolute retract. for the reader who is not familiar with it, in section 2 we will briefly discuss the vietoris topology on a general topological space x. then in section 4 we shall prove: theorem 1.1. under the vietoris topology, cl(r) is simply connected. for the purposes of the proof, we will also present an alternate proof that cl(r) is path connected, by constructing an explicit path from any point in cl(r) to r; this will be done in section 3. this will assist in the construction of a nullhomotopy of an arbitrary loop in section 4. cl(r) is simply connected under the vietoris topology 261 2. the vietoris topology suppose that you have a topological space (x, τ ). the vietoris topology is one of a group of topologies called “hit and miss” topologies. the name is indicative of the fact that open sets in the space cl(x) are given by those subsets of x which “hit” certain specific open sets of x and “miss” their complements. for a full discussion of hit and miss topologies, see [1]. let u1, . . . , un be a finite collection of open sets in x, i.e. members of τ . we define an open set in cl(x), denoted b =< u1, . . . , un >, to be all those non-empty and closed subsets a of x satisfying the following two properties: (1) a ∩ ui 6= ∅, for i = 1, . . . , n. (“hit”) (2) a ⊂ ⋃n i=1 ui (“miss”) the collection of all such sets, for any finite collection of ui, forms a basis for the vietoris topology on cl(x). when x = r, it is not hard to see that under this topology, the sequence of time scales tn = [−n, n] does in fact converge to r. an alternative way of looking at the vietoris topology is to use the fact that it is the supremum of the upper and lower vietoris topologies, the first of which is generated by all sets of the form u + = {a ∈ cl(x) : a ⊂ u}, and the second of which is generated by sets of the form u − = {a ∈ cl(x) : a ∩ u 6= ∅}, where u is a τ -open set. subbase elements of the vietoris topology are of the form u + with u ∈ τ and ⋂ u∈u u −, with u ⊂ τ finite. 3. path connected in the following, we will consider cl(r) endowed with the vietoris topology. theorem 3.1. cl(r) is path connected. proof. let t ∈ cl(r) be an arbitrary point of the hyperspace. in the future, to distinguish between points of cl(r) and points of r, we will refer to the former as time scales. we construct a path from t to r. as t 6= ∅, choose t0 ∈ t. define γ : [0, 1] → cl(r) by γ(1) = r, and for s ∈ [0, 1), γ(s) = t ∪ [t0 − s 1 − s , t0 + s 1 − s ] we denote by a(s) the closed interval [t0 − s 1−s , t0 + s 1−s ]. note that γ(s) is clearly nonempty and closed, as it is the finite union of closed sets, the first of which is always nonempty. note also that lims→1 s 1−s diverges to infinity. 262 n. c. esty first we must show that γ is continuous. it is enough to show γ is continuous with respect to the upper and lower vietoris topologies. first let γ(s0) ∈ u +, where u + is a basic open set in the upper vietoris topology. then γ(s0) ⊂ u . if s0 = 1, then γ(s0) = r ⊂ u = r, so clearly for all s ∈ [0, 1], γ(s) ⊂ u and γ(s) ∈ u +. assume s0 6= 1. as u is open and γ(s0) is compact, there exists some ǫ > 0 such that b(γ(s0), ǫ) ⊂ u . by continuity of f (x) = x 1−x , there exists some δ > 0 such that if |s − s0| < δ, then |f (s) − f (s0)| < ǫ, and therefore a(s) ⊂ u , so γ(s) ∈ u +. next suppose γ(s0) ∈ u − 1 ∩ · · · ∩ u − n , a basic open set in the lower vietoris topology. if s0 = 1, then there is some ǫ > 0 such that if s ∈ (1 − ǫ, 1], a(s)∩ui 6= ∅ for all i. in addition, if t ∈ u − i , then γ(s) ∈ u − i for all s ∈ [0, 1]. so assume that t /∈ u − i and s0 6= 1. therefore a(s0) ∈ u − 1 ∩ · · · ∩ u − n . choose ti ∈ a(s0) ∩ ui, and let di > 0 be such that b(ti, di) ⊂ ui ∩ a(s0) for all i ∈ {1, . . . , n}. as f is continuous, we can find δ > 0 such that if |s − s0| < δ, then |f (s) − f (s0)| < min{d1, . . . , dn}. then ti ∈ γ(s) for all i and therefore γ(s) ∈ u −1 ∩ · · · ∩ u − n . � 4. simply connected it is enough to show that all loops with a particular base point t are nullhomotopic. we choose the base point to be r, and assume that we are given an arbitrary loop based at r, i.e. a continuous map f : [0, 1] → cl(r) with f (0) = f (1) = r. lemma 4.1. there exists a continuous map x : [0, 1] → r such that x(s) ∈ f (s). proof. we define x : [0, 1] → r by letting x(s) be the point of f (s) which is closest to the origin, choosing the positive point in the case of a tie. this map is well-defined because each f (s) ∈ cl(r), meaning it is a nonempty and closed subset of the real line, so such a point exists. we claim that this map is in fact a loop in r. it is easy to see that x(0) = x(1) = 0, so we need only check continuity. notationally we will sometimes write xs for x(s). fix s0 ∈ [0, 1] and fix ǫ > 0. consider f (s0). we know that x(s0) ∈ f (s0) by definition of x. consider the ball around f (s0) in cl(r) b1 =< r, (xs0 − ǫ/2, xs0 + ǫ/2) > continuity of f implies there exists a δ1 > 0 such that if s ∈ (s0 −δ1, s0 +δ1), then f (s) ∈ b1. in particular, f (s) ∩ (xs0 − ǫ/2, xs0 + ǫ/2) 6= ∅. therefore the closest point of f (s) to the origin can have distance from the origin no greater than |xs0 | + ǫ/2. cl(r) is simply connected under the vietoris topology 263 should xs0 be within ǫ/2 of the origin, we can let δ = δ1 at this point. if not, consider b2 =< (−∞, −|xs0| + ǫ/2), (|xs0| − ǫ/2, ∞) > because f (s0) contains no points closer to the origin than xs0 , f (s0) ∈ b2. by continuity of f , there exists some δ2 > 0 such that s ∈ (s0 − δ2, s0 + δ2) implies f (s) ∈ b2. but this means that the closest point of f (s) to the origin can have distance from the origin no less than |xs0 | − ǫ/2. choose δ = min{δ1, δ2}. then for all s ∈ (s0 − δ, s0 + δ), the closest point of f (s) to the origin lies within ǫ/2 of either xs0 or −xs0 . by choosing the positive one in all tie cases, we ensure that in fact xs is within ǫ/2 of xs0 . therefore x is a continuous function. � given an arbitrary point p0 ∈ t, let γt : [0, 1] → cl(r) be the map defined by γt(s) = t ∪ [p0 − s 1−s , p0 + s 1−s ] for s ∈ [0, 1) and γt(1) = r. we know from the proof of theorem 3.1 in section 3 that this map is a continuous path from t to r. since γ depends on the point p0, we will sometimes write γt(p0, s). we wish to find a homotopy from f to the constant loop c(s) = r, s ∈ [0, 1]. in other words, we require a continuous map f : [0, 1] × [0, 1] → cl(r) with the following properties: (1) for all s ∈ [0, 1], f (s, 0) = f (s), i.e., at time zero we have the original loop f . (2) for all s ∈ [0, 1], f (s, 1) = r, i.e., at time one we have the constant loop c. (3) for all t ∈ [0, 1], f (0, t) = f (1, t) = r, i.e., at all other times we do, in fact, have loops based at r. theorem 4.2. f (s, t) = γf (s)(x(s), t) is a homotopy from f to the constant loop. proof. it is easy to see that f has the three properties listed. continuity is all that remains to check. fix a particular point (s0, t0). it is clear that f is continuous in t because the path γf (s0) is continuous. let us check continuity in s. note that if t = 1, then f (s, t) = r for all s. therefore we need only consider the case t0 6= 1. again, we check continuity with respect to the upper and lower vietoris topologies. we know that f (s0, t0) = fs0 ∪ [xs0 − t0 1−t0 , xs0 + t0 1−t0 ] for brevity, we will refer to that closed interval as is0 . 264 n. c. esty let f (s0, t0) ∈ u +, a basic open set in the upper vietoris topology. as f is continuous, there exists some δ1 > 0 such that if |s − s0| < δ1, then fs ∈ u +. because is0 is compact, there is some ǫ > 0 such that b(is0 , ǫ) ⊂ u . by continuity of x(s), there exists some δ2 such that if |s − s0| < δ2, then |xs − xs0| < ǫ. then it is clear that is ⊂ b(is0 , ǫ), and so is ∈ u +. let |s − s0| < min{δ1, δ2}, and we have that f (s, t0) ∈ u +. next let f (s0, t0) ∈ u − 1 ∩ · · · ∩ u − n , a basic open set in the lower vietoris topology. if fs0 ∈ u − i , then by continuity of f , there exists some δ > 0 such that if |s − s0| < δ, fs ∈ u − i , and so f (s, t0) ∈ u − i . so we can suppose without loss of generality that fs0 /∈ u − i for all i. therefore is0 ∈ u − 1 ∩ · · · ∩ u − n . we use a reasoning similar to that in the proof of theorem 3.1. take ti ∈ is0 ∩ ui, and let di such that b(ti, di) ⊂ is0 ∩ ui. there exists some δ > 0 such that when |s − s0| < δ, |xs − xs0| < min{d1, . . . , dn} and therefore ti ∈ is for all i. thus we have that f (s, t0) ∈ u − 1 ∩ · · · ∩ u − n . � acknowledgements. i would like to thank drs. bonita lawrence and ralph oberste-vorth for suggesting this problem. references [1] g. beer and r. k. tamaki, on hit-and-miss hyperspace topologies, comment. math. univ. carolinae 34 (1993), 717–728. [2] m. bohner and a. peterson, dynamic equations on time scales, birkhauser; 2001. [3] m. bohner and a. peterson, advances in dynamic equations on time scales, birkhauser; 2003. [4] k. borsuk and s. marzurkiewicz, sur l’hyperspace d’un continu, c. r. soc. sc. varsovie 24 (1931), 142–152. [5] c. constantini, s. levi and j. pelant, compactness and local compactness in hyperspaces, topology applications 123 (2002), 573–608. [6] c. constantini and w. kubis, paths in hypserspaces , app. gen. top. 4 (2003), 377–390. [7] d. w. curtis, hyperspaces of noncompact metric spaces, comp. math. 40 (1980), 139– 152. [8] l. hola, j. pelant and l. zslinszky, developable hyperspaces are metrizable, app. gen. top. 4 (2003), 351–360. [9] a. illanes and s. nadler, hyperspaces, marcel-dekker (1999). [10] v. m. ivanova, on the theory of the space of subsets, dokl. akad. nauk. sssr 101 (1955), 601–603. [11] j. keesling, on the equivalence of normality and compactness in hyperspaces, pacific j. math. 33 (1970), 657–667. [12] b. lawrence and r. oberste-vorth, solutions of dynamic equations with varying time scales, proc. int. con. of difference equations, special functions and applications (2006). cl(r) is simply connected under the vietoris topology 265 [13] m. m. mcwaters, arcs, semigroups and hyperspaces, can. j. math. 20 (1968), 1207– 1210. [14] e. michael, topologies on spaces of subsets, trans. am. math. soc. 71 (1951), 152–182. [15] s. nadler, hyperspaces of sets, marcel-dekker (1978). [16] n. v. velichko, on spaces of closed subsets, sibirskii matem. z. 16 (1975), 627–629. [17] l. e. ward, arcs in hyperspaces which are not compact, proc. amer. math. soc. 28 (1971), 254–258. [18] m. wojdyslawski, retractes absolus et hyperespaces des continus, fund. math. 32 (1939), 184–192. [19] l. zsilinszky, topological games and hyperspace topologies, set-valued anal. 6 (1998), 187–207. received april 2006 accepted september 2006 n. c. esty (esty@marshall.edu) department of mathematics, marshall university, one john marshall drive, huntington, wv, 45669, usa protasovagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 189-195 asymptotic proximities i. v. protasov abstract. a ballean is a set endowed with some family of subsets which are called the balls. the properties of the family of balls are postulated in such a way that the balleans can be considered as a natural asymptotic counterparts of the uniform topological spaces. we introduce and study an asymptotic proximity as a counterpart of proximity relation for uniform topological space. 2000 ams classification: 54e05, 54e15. keywords: ballean, determining covering, proximity. 1. introduction and preliminaries a ball structure is a triple b = (x, p, b) where x, p are non-empty sets and, for any x ∈ x and α ∈ p , b(x, α) is a subset of x which is called a ball of radius α around x. it is supposed that x ∈ b(x, α) for all x ∈ x, α ∈ p . the set x is called the support of b, p is called the set of radii. given any x ∈ x, a ⊆ x, α ∈ p , we put b ∗(x, α) = {y ∈ x : x ∈ b(y, α)}, b(a, α) = ⋃ a∈a b(a, α). a ball structure is called • lower symmetric if, for any α, β ∈ p , there exist α′, β′ ∈ p such that, for every x ∈ x, b ∗(x, α′) ⊆ b(x, α), b(x, β′) ⊆ b∗(x, β); • upper symmetric if, for any α, β ∈ p , there exist α′, β′ ∈ p such that, for every x ∈ x, b(x, α) ⊆ b∗(x, α′), b∗(x, β) ⊆ b(x, β′); 190 i. v. protasov • lower multiplicative if, for any α, β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x, γ), γ) ⊆ b(x, α) ∩ b(x, β); • upper multiplicative if, for any α, β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x, α), β) ⊆ b(x, γ). let b = (x, p, b) be a lower symmetric and lower multiplicative ball structure. then the family { ⋃ x∈x b(x, α) × b(x, α) : α ∈ p } is a base of entourages for some (uniquely determined) uniformity on x. on the other hand, if u ⊆ x × x is a uniformity on x, then the ball structure (x, u, b) is lower symmetric and lower multiplicative, where b(x, u ) = {y ∈ x : (x, y) ∈ u}. thus, the lower symmetric and lower multiplicative ball structures can be identified with the uniform topological spaces. we say that a ball structure is a ballean if b is upper symmetric and upper multiplicative. a structure on x, equivalent to a ballean, can also be defined in terminology of entourages. in this case it is called a coarse structure [5]. for motivations to study balleans see [1],[4],[5]. let b1 = (x1, p1, b1) and b2 = (x2, p2, b2) be balleans. a mapping f : x1 → x2 is called a ≺-mapping if, for every α ∈ p1, there exists β ∈ p2 such that, for every x ∈ x1, f (b1(x, α)) ⊆ b2(f (x), β). a bijection f : x1 −→ x2 is called an asymorphism between b1 and b2 if f and f −1 are ≺-mappings. let b1, b2 be balleans with common support x. we say that b1 ≺ b2 if the identity mapping id:x → x is a ≺-mapping of b1 to b2. if b1 ≺ b2 and b2 ≺ b1, we say that b1 and b2 coincide and write b1 = b2. let b = (x, p, b) be a ballean. a subset y ⊆ x is called bounded if there exist α ∈ p such that y ⊆ b(x, α) for some x ∈ y . a family f of subsets of x is called uniformly bounded if there exists α ∈ p such that f ⊆ b(x, α) for all f ∈ f, x ∈ f . we use the following observation: the ballean b1 and b2 with common support coincide if and only if every family of subsets of x uniformly bounded in b1 is uniformly bounded in b2 and vise versa. for an arbitrary ballean b = (x, p, b) we define preordering 6 on the set p by the rule: α 6 β if and only if b(x, α) ⊆ b(x, β) for every x ∈ x. a subset p ′ ⊆ p is called cofinal if, for every α ∈ p , there exists α′ ∈ p ′ such that α 6 α′. a ballean b is called connected if, for any x, y ∈ x, there exists α ∈ p such that y ∈ b(x, α). a connected ballean b is called ordinal if there exists a well-ordered by 6 subset p ′ of p . asymptotic proximities 191 every metric space (x, d) determines the metric ballean (x, r+, bd) where bd(x, r) = {y ∈ x : d(x, y) ≤ r}. a ballean is called metrizable if it is asymorphic to some metric ballean. by [4, theorem 9.1], a ballean b = (x, p, b) is metrizable if and only if b is connected and p has a countable cofinal subset. clearly, every metrizable ballean is ordinal. we begin the proper exposition with characterization (section 2) of families of coverings of a set x which determine a ballean on x. then we introduce and study (section 3) an asymptotic proximity as an equivalence relation σ on the family p(x) of all subsets of a set x such that y ⊆ z ⊆ y ′ and y σy ′ imply y σz. every proximity σ determines some ballean b(σ) on x. given a ballean b = (x, p, b), we say that the subsets y, z of x are close if there exists α ∈ p such that y ⊆ b(z, α), z ⊆ b(y, α). the closeness relation is a prototype for the asymptotic proximity. we show (theorem 3.1) that, given an asymptotic proximity σ on p(x), the closeness σ′ defined by b(σ) is finner then σ. on the other hand (theorem 3.4), if b = (x, p, b) is a ballean and σ is a closeness on p(x) determined by b, then σ = σ′ where σ′ is closeness determined by b(σ). in section 4 we examine the question whether the closeness on p(x) arising from a ballean b = (x, p, b) determines b. in general case this is not so, but our main result (theorem 4.2) gives a positive answer in the case of ordinal (in particular, metrizable) balleans. 2. determining coverings let x be a set, f be a family of subsets of x, y ⊆ x. we put st(y, f) = ⋃ {f ∈ f : y ⋂ f 6= ∅}. given any x ∈ x, we write st(x, f) instead of st({x}, f). for two families f, f′ of subsets of x, we put st(f, f′) = {st(f, f′) : f ∈ f}. a family f of subsets of xis called hereditary if, for any subsets f, f ′ of x such that f ∈ f and f ′ ⊆ f , we have f ′ ⊆ f. a family f of subsets of x is called a covering if ⋃ f = x. we say that a family {fα, α ∈ p } of hereditary coverings of x is star stable if, for any α, β ∈ p , there exist γ ∈ p such that st(fα, fβ ) ⊆ fγ . let {fα : α ∈ p } be a family of star stable coverings of x. we consider a ball structure b = (x, p, b), where b(x, α) = st(x, fα), and show that b is a ballean. given any x ∈ x and α ∈ p , we have b(x, α) = {y ∈ x : y ∈ st(x, fα)}, b ∗(x, α) = {y ∈ x : x ∈ st(y, fα)}. since y ∈ st(x, fα) if and only if x ∈ st(y, fα), then b ∗(x, α) = b(x, α), so b is upper symmetric. 192 i. v. protasov given any x ∈ x and α, β ∈ p , we choose α′ ∈ p and γ ∈ p such that st(fα, fα) ⊆ fα′ and st(fα′ , fβ ) ⊆ fγ . then we have b(b(x, α), β) = st(st(x, fα), fβ) ⊆ st(x, fγ ) = b(x, γ), so b is upper multiplicative. we note that a subset y of x is bounded in b if and only if y ∈ fα for some α ∈ p . a family f of subsets of x is bounded in b if and only if there exists α ∈ p such that f ⊆ fα. thus we have shown that every star stable family of coverings of x determines some ballean on x. on the other hand, let b = (x, p, b) be an arbitrary ballean on x. for every α ∈ p , we put fα = {f ⊆ x : f ⊆ b(x, α) f or some x ∈ x}. then the ballean on x determined by the star stable family {fα : α ∈ p } of coverings of x coincides with b. 3. proximities and closeness let x be a set, p(x) be a family of all subsets of x. let σ be an equivalence on p(x) such that, for all y, y ′, z ∈ p(x), y ⊆ z ⊆ y ′, y σy ′ =⇒ y σz. we say that σ is (an asymptotic) proximity and describe a way in which σ defines some ballean b(σ) on x. we call a family f of subsets of x to be non-expanding with respect to σ if, for every subset y of x, we have y σ(y ⋃ st(y, f)). we note that every subfamily of non-expanding family is non-expanding. let f1, f2 be non-expanding with respect to σ families of subsets of x. we show that the family st(f1, f2) is also non-expanding with respect to σ. we fix an arbitrary subset y of x and put f′2 = {f ′ ∈ f2 : y ⋂ f ′ 6= ∅}. since f′2 is non-expanding, we have y σ(y ⋃ ⋃ f′2). we put z = y ⋃ ⋃ f′2 and f′1 = {f ∈ f1 : f ⋂ f ′ 6= ∅ f or some f ′ ∈ f′2}. since f′1 is non-expanding, we have zσ(z ⋃ ⋃ f′1). asymptotic proximities 193 we put t = z ⋃ ⋃ f′1. since f2 is non-expanding, we have t σ(t ⋃ ( ⋃ {f ∈ f2 : f ⋂ t 6= ∅})). we put h = t ⋃ ( ⋃ {f ∈ f2 : f ⋂ t 6= ∅}). then y σh and y ⊆ h. by the construction of h, we have y ⊆ y ⋃ ( ⋃ {s ∈ st(f1, f2) : s ⋂ y 6= ∅}) ⊆ h. since σ is a proximity, we conclude y σ(y ⋃ ( ⋃ {s ∈ st(f1, f2) : s ⋂ y 6= ∅})). in particular, we proved that the family of all non-expanding (with respect to σ) hereditary covering of x is star stable. following section 2, we define b(σ) by means this family of coverings. we note that a subset y of x is bounded in b(σ) if and only if the family {y } is non-expanding, equivalently, {y}σy for every y ∈ y . a family f of subsets of x is uniformly bounded in b(σ) if and only if f is non-expanding. let b = (x, p, b) be a ballean. we consider a relation σ on p(x) defined by the rule: y σz if and only if there exists α ∈ p such that y ⊆ b(z, α), z ⊆ b(y, α). it is easy to see that σ is a proximity; we call it a closeness defined by b. we note that y, z are close if and only if there exists a uniformly bounded covering f of x such that ⋃ {f ∈ f : f ⋂ y 6= ∅} = ⋃ {f ∈ f : f ⋂ z 6= ∅}. theorem 3.1. let x be a set, σ be a proximity on p(x), σ′ be a closeness defined by b(σ). then σ′ ⊆ σ. proof. we remind that a family f of subsets of x is uniformly bounded in b(σ) if and only if f is non-expanding with respect to σ. let y, z ∈ p(x) and y σ′z. then there exists a non-expanding (with respect to σ) family f of subsets of x such that y ⊆ ⋃ f, z ⊆ ⋃ f and y ⋂ f 6= ∅, z ⋂ f 6= ∅ for every f ∈ f. it follows that y σ( ⋃ f) and zσ( ⋃ f), so y σz. 2 � the following two examples show that the proximity σ from theorem 3.1 could be much more coarse than σ′. example 3.2. let x be an infinite set. we define an equivalence σ on p(x) by the rule: y σz if and only if either y, z are finite, or y, z are infinite. then a subset y of x is bounded in b(σ) if and only if y is finite; a family f of subsets of x is uniformly bounded in b(σ) if and only if each subset f ∈ f is finite and, for every x ∈ x, the set {f ∈ f : x ∈ f } is finite. we show that y σ′z if and only if either y, z are finite, or y, z are infinite and |y | = |z|. we should only check that if y, z are infinite and |y | = |z| then y σ′z. to this end we fix some bijection f : y −→ z, and put f = {{y, f (y)} : y ∈ y }. then f is uniformly bounded in b(σ), y σ′( ⋃ f) and zσ′( ⋃ f), so y σ′z. now if x is uncountable than σ is coarser than σ′. 2 194 i. v. protasov example 3.3. let x be a well-ordered set. we define an equivalence σ on p(x) by the rule: y σz if and only if miny = minz. then a subset y is bounded in b(σ) if and only if y is a singleton. it follows that y σ′z if and only if y = z. 2 theorem 3.4. let b = (x, p, b) be a ballean, σ be a closeness defined by b, σ′ be a closeness defined by b(σ). then σ = σ′. proof. by theorem 3.1, σ ⊆ σ′. to see that σ ⊆ σ′ it suffices to note that every uniformly bounded in b family of subsets of x is non-expanding with respect to σ. � 4. does closeness determine a ballean? let b1 and b2 be balleans with common support x, σ1 and σ2 be closeness on p(x) defined by b1 and b2. is b1 = b2 provided that σ1 = σ2? we give a negative answer to this general question, but prove one partial statement (theorem 4.2) in positive direction. example 4.1. let x be a countable set. we consider two families ϕ1, ϕ2 of coverings of x. a family ϕ1 is defined by the rule: f ∈ ϕ1 if and only if every subset f ∈ f is finite, and the set {f ∈ f : x ∈ f } is finite for every x ∈ x. a family ϕ2 is defined by the rule: f ∈ ϕ2 if and only if there exists a natural number n such that |f | ≤ n for every f ∈ f, and there exists a natural number m such that |{f ∈ f : x ∈ f }| ≤ m for every x ∈ x. clearly, the families ϕ1 and ϕ2 are star-stable. let b1 and b2 be balleans on x determined by ϕ1 and ϕ2. using arguments from example 3.2, it is easy to see that b1 and b2 define the same closeness σ: y σz if and only if either y, z are finite, or y, z are infinite. then we take a partition {fn : n ∈ ω} of x such that |fn| = n for every n ∈ ω. clearly, f is uniformly bounded in b1, but f is not uniformly bounded in b2. it follows that b1 is stronger than b2. it is worth to mark that example 4.1 gives a ballean b with the closeness σ such that b 6= b(σ). to see this, we put b = b2 and note that b(σ) = b1. theorem 4.2. let b1 = (x1, p1, b1) and b2 = (x2, p2, b2) be ordinal balleans with common support and the same closeness. then b1 = b2. proof. we assume on the contrary that, say, b2 ≺ b1 does not hold, and choose β ∈ p2 such that, for every α ∈ p1, there exists x(α) ∈ x such that b2(x(α), β) * b1(x(α), α). we may suppose that p1 is well-ordered. in the proof of theorem 2.1 from [3] we constructed inductively a subset y = {y(α) : α ∈ p1} of x such that the family {b1(y(α), α) : α ∈ p1} is disjoint and, for every α′ ∈ p , b2(y(α ′), β) * ⋃ {b1(y(α), α) : α ∈ p1}. we put z = b2(y, β). then y, z are close in b2, but y, z are not close in b1, whence a contradiction. � asymptotic proximities 195 references [1] a. dranishnikov, asymptotic topology, russian math. surveys 55 (2000), 71–116. [2] r. engelking, general topology, pwn, warszava, 1985. [3] m. filali and i. v. protasov, s lowly oscillating function on locally compact groups, applied general topology 6, no. 1 (2005) 67-77. [4] i. v. protasov and t. banakh, ball structures and colorings of groups and graphs, math. stud. monogr. ser. v.11, 2003. [5] j. roe, lectures on coarse geometry, ams university lecture series, 31 (2003). received march 2007 accepted december 2007 i. v. protasov (protasov@unicyb.kiev.ua) department of cybernetics, kyiv national university, volodimirska 64, kyiv 01033, ukraine. jujeleleagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 213-228 applications of pre-open sets young bae jun, seong woo jeong, hyeon jeong lee and joon woo lee abstract. using the concept of pre-open set, we introduce and study topological properties of pre-limit points, pre-derived sets, preinterior and pre-closure of a set, pre-interior points, pre-border, prefrontier and pre-exterior. the relations between pre-derived set (resp. pre-limit point, pre-interior (point), pre-border, pre-frontier, and preexterior) and α-derived set (resp. α-limit point, α-interior (point), α-border, α-frontier, and α-exterior) are investigated. 2000 ams classification: 54a05, 54c08. keywords: pre-limit point, pre-derived set, pre-interior, pre-closure, preinterior points, pre-border, pre-frontier, pre-exterior. 1. introduction the notion of α-open set was introduced by nj̊astad [14]. since then it has been widely investigated in several literatures (see [1, 3, 4, 5, 6, 7, 9, 10, 12, 15]). in [2], caldas introduced and studied topological properties of α-derived, αborder, α-frontier, and α-exterior of a set by using the concept of α-open sets. the notion of pre-open set was introduced by mashhour et al. [8]. in this paper, we introduce the notions of pre-limit points, pre-derived sets, pre-interior and pre-closure of a set, pre-interior points, pre-border, pre-frontier and pre-exterior by using the concept of pre-open sets, and study their topological properties. we provide relations between pre-derived set (resp. pre-limit point, pre-interior (point), pre-border, pre-frontier, and pre-exterior) and α-derived set (resp. αlimit point, α-interior (point), α-border, α-frontier, and α-exterior). 214 y. b. jun, s. w. jeong, h. j. lee and j. w. lee 2. preliminaries through this paper, (x, t ) and (y, k ) (simply x and y ) always mean topological spaces. a subset a of x is said to be pre-open [11] (respectively, α-open [14] and semi-open [13]) if a ⊂ int(cl(a)) (respectively, a ⊂ int(cl(int(a))) and a ⊂ cl(int(a))). the complement of a pre-open set (respectively, an α-open set and a semi-open set) is called a pre-closed set (respectively, an α-closed set and a semi-closed set). the intersection of all pre-closed sets (respectively, α-closed sets and semi-closed sets) containing a is called the pre-closure (respectively, α-closure and semi-closure) of a, denoted by clp(a) (respectively, clα(a) and cls(a)). a subset a is also pre-closed (respectively, α-closed and semi-closed) if and only if a = clp(a) (respectively, a = clα(a) and a = cls(a)). we denote the family of pre-open sets (respectively, αopen sets and semi-open sets) of (x, t ) by t p (respectively, t α and t s). obviously, we have the following relations. open set (closed set) α-open set (α-closed set) pre-open set (pre-closed set) semi-open set (semi-closed set) ? � �� h hhj none of these implications is reversible in general. 3. pre-open sets and α-open sets definition 3.1 ([11, 14]). a subset a of x is said to be pre-open (respectively, α-open) if a ⊆ int(cla) (respectively, a ⊆ int(cl(inta))). the complement of a pre-open set (respectively, an α-open set) is called a pre-closed set (respectively, an α-closed set). the intersection of all pre-closed sets (respectively, α-closed sets) containing a is called the pre-closure (respectively, α-closure) of a, denoted by clp(a) (respectively, clα(a)). a subset a is also pre-closed (respectively, α-closed) if and only if a = clp(a) (respectively, a = clα(a)). we denote the family of pre-open sets (respectively, α-open sets) of (x, t ) by t p (respectively, t α). example 3.2. let t = {∅, x, {a}, {c, d}, {a, c, d}} be a topology on x = {a, b, c, d, e}. then we have t α = t ∪ {{a, b, c, d}, {a, c, d, e}}, t p = t ∪ {{c}, {d}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, e}, {a, d, e}, {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}}. applications of pre-open sets 215 4. applications of pre-open sets definition 4.1. let a be a subset of a topological space (x, t ). a point x ∈ x is said to be pre-limit point (resp. α-limit point) of a if it satisfies the following assertion: (∀g ∈ t p( resp. t α)) (x ∈ g ⇒ g ∩ (a \ {x}) 6= ∅). the set of all pre-limit points (resp. α-limit points) of a is called the prederived set (resp. α-derived set) of a and is denoted by dp(a) (resp. dα(a)). denote by d(a) the derived set of a. note that for a subset a of x, a point x ∈ x is not a pre-limit point of a if and only if there exists a pre-open set g in x such that x ∈ g and g ∩ (a \ {x}) = ∅ or, equivalently, x ∈ g and g ∩ a = ∅ or g ∩ a = {x} or, equivalently, x ∈ g and g ∩ a ⊆ {x}. example 4.2. let x = {a, b, c} with topology t = {x, ∅, {a}}. then we have the followings: (i) t p = {x, ∅, {a}, {a, b}, {a, c}} = t α. (ii) if a = {c}, then d(a) = {b} and dα(a) = dp(a) = ∅. (iii) if b = {a} and c = {b, c}, then dp(b) = {b, c}, dp(c) = ∅ and dp(b ∪ c) = {b, c}. theorem 4.3. if a topology t on a set x contains only ∅, x, and {a} for a fixed a ∈ x, then t p = t α. proof. let a ∈ x and let a be an element of t p. then a ∈ a. in fact, if not then a 6⊆ int(cl(a)) = int({a}c) = ∅. hence a /∈ t p, a contradiction. now since int(a) = {a}, we have int(cl(int(a))) = int(cl({a})) = int(x) = x which contains a, that is, a ∈ t α. note that t α ⊆ t p. thus t α = t p. � example 4.4. let x = {a, b, c, d, e} with topology t = {x, ∅, {a}, {c, d}, {a, c, d}, {b, c, d, e}}. then t p = {x, ∅, {a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {c, e}, {d, e}, {a, b, c}, {a, b, d}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}, {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}, {b, c, d, e}} 216 y. b. jun, s. w. jeong, h. j. lee and j. w. lee and t α = {x, ∅, {a}, {c, d}, {a, c, d}, {b, c, d}, {c, d, e}, {a, b, c, d}, {a, c, d, e}, {b, c, d, e}}. consider subsets a = {a, b, c} and b = {b, d} of x. then d(a) = {b, d, e}, dp(a) = ∅, int(a) = {a}, intp(a) = a, intα(a) = {a}, clp(a) = a, clα(a) = x, clp(b) = b, clα(b) = {b, c, d, e}, int(b) = ∅, intp(b) = b, intα(b) = ∅. example 4.5. consider a topology t = {x, ∅, {a}, {a, b}, {a, c, d}, {a, b, c, d}, {a, b, e}} on x = {a, b, c, d, e}. then t p = {x, ∅, {a}, {a, b}, {a, c}, {a, d}, {a, e}, {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e} {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}} = t α. for subsets a = {c, d, e} and b = {b} of x, we have d(a) = {c, d} d(b) = {e}. dp(a) = ∅ dp(b) = ∅. dα(a) = ∅ dα(b) = ∅. int(a) = ∅ int(b) = ∅, intp(a) = ∅, intp(b) = ∅, intα(a) = ∅, intα(b) = ∅, clp(a) = {c, d, e}, clp(b) = {b}, clα(a) = {c, d, e}, clα(b) = {b}, clp({b, d}) = {b, d}, clα({b, d}) = {b, d}, int({b, d}) = ∅, intp({b, d}) = ∅, intα({b, d}) = ∅. lemma 4.6. if there exists a ∈ x such that {a} is the smallest element of (t \ {∅}, ⊆), then every non-empty pre-open set contains ⋂ {gi | gi ∈ t \ {∅}; i = 1, 2, 3, · · ·}. proof. if {a} is the smallest element of (t \ {∅}, ⊆), then ⋂ {gi | gi ∈ t \ {∅}; i = 1, 2, 3, · · ·} = {a}. let a be a non-empty pre-open set in x. if a /∈ a, then cl(a) ⊆ {a} and so a * int(cl(a)) ⊆ int({a}c) = ∅ which is a contradiction. hence a ∈ a, and so the desired result is valid. � applications of pre-open sets 217 theorem 4.7. let t be a topology on a set x. if there exists a ∈ x such that {a} is the smallest element of (t \ {∅}, ⊆), then t α = t p. proof. it is sufficient to show that t p ⊆ t α. let a ∈ t p. if a = ∅, then clearly a ∈ t α. assume that a 6= ∅. then a ∈ a by lemma 4.6. since {a} ⊆ int(a), it follows that x = cl({a}) ⊆ cl(int(a)) so that a ⊆ x = int(x) ⊆ int(cl(int(a))). hence a is an α-open set. � theorem 4.8. let t1 and t2 be topologies on x such that t p 1 ⊆ t p 2 . for any subset a of x, every pre-limit point of a with respect to t2 is a pre-limit point of a with respect to t1. proof. let x be a pre-limit point of a with respect to t2. then (g∩a)\{x} 6= ∅ for every g ∈ t p2 such that x ∈ g. but t p 1 ⊆ t p 2 , so, in particular, (g ∩ a) \ {x} 6= ∅ for every g ∈ t p1 such that x ∈ g. hence x is a pre-limit point of a with respect to t1. � the converse of theorem 4.8 is not true in general as seen in the following example. example 4.9. consider topologies t1 = {x, ∅, {a}} and t2 = {x, ∅, {a}, {b, c}, {a, b, c}} on a set x = {a, b, c, d}. then t p 1 = t1 ∪ {{a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}} and t p 2 = t2 ∪ {{b}, {c}, {a, b}, {a, c}, {a, d}, {a, b, d}, {a, c, d}}. note that t p 1 ⊆ t p 2 and c is a pre-limit point of a = {a, b} with respect to t1, but it is not a pre-limit point of a with respect to t2. lemma 4.10. if {ai | i ∈ λ} is a family of pre-open sets in x, then ⋃ i∈λ ai is a pre-open set in x where λ is any index set. proof. straightforward. � in example 3.2, we see that {a, b, c, e} ∩ {a, b, d, e} = {a, b, e} /∈ t p, which shows that the intersection of two pre-open sets is not pre-open in general. thus we know that for any topology t on a set x, t p may not be a topology on x. proposition 4.11. if i (resp. d) is the indiscrete (resp. discrete) topology on a set x, then i p (resp. d p) is a topology on x. proof. straightforward. � 218 y. b. jun, s. w. jeong, h. j. lee and j. w. lee theorem 4.12. for any subsets a and b of (x, t ), the following assertions are valid: (1) dp(a) ⊆ dα(a). (2) if a ⊆ b, then dp(a) ⊆ dp(b). (3) dp(a) ∪ dp(b) ⊆ dp(a ∪ b) and dp(a ∩ b) ⊆ dp(a) ∩ dp(b). (4) dp(dp(a)) \ a ⊆ dp(a). (5) dp(a ∪ dp(a)) ⊆ a ∪ dp(a). proof. (1) it suffices to observe that every α-open set is pre-open. (2) let x ∈ dp(a) and let g ∈ t p with x ∈ g. then (g ∩ a) \ {x} 6= ∅. since a ⊆ b, it follows that (g ∩ b) \ {x} 6= ∅ so that x ∈ dp(b). (3) straightforward by (2). (4) let x ∈ dp(dp(a)) \ a and let g ∈ t p with x ∈ g. then g ∩ (dp(a) \ {x}) 6= ∅. let y ∈ g ∩ (dp(a) \ {x}). then y ∈ g and y ∈ dp(a), and so g ∩ (a \ {y}) 6= ∅. if we take z ∈ g ∩ (a \ {y}), then x 6= z because x /∈ a. hence (g ∩ a) \ {x} 6= ∅. therefore x ∈ dp(a). (5) let x ∈ dp(a ∪ dp(a)). if x ∈ a, the result is obvious. assume that x /∈ a. then g ∩ ((a ∪ dp(a)) \ {x}) 6= ∅ for all g ∈ t p with x ∈ g. hence (g ∩ a) \ {x} 6= ∅ or g ∩ (dp(a) \ {x}) 6= ∅. the first case implies x ∈ dp(a). if g∩(dp(a)\{x}) 6= ∅, then x ∈ dp(dp(a)). since x /∈ a, it follows similarly from (4) that x ∈ dp(dp(a)) \ a ⊆ dp(a). therefore (5) is valid. � in general, in theorem 4.12, the reverse inclusion of (1), (4) and (5), and the converse of (2) may not be true, and the equality in (3) does not hold as seen in the following example. example 4.13. (1) consider the topology t on x = {a, b, c, d, e} described in example 3.2. for a subset a = {b, c, d} of x, we have dα(a) = {b, c, d, e} and dp(a) = {b, e}. this shows that the reverse inclusion of theorem 4.12(1) is not true. now let x = {a, b, c, d} with a topology t = {x, ∅, {a}, {d}, {a, b}, {a, d}, {c, d}, {a, b, d}, {a, c, d}}. then t p = t . for two subsets a = {a, c} and b = {a, b, d} of x, we get dp(a) = {b} ⊆ {b, c} = dp(b), but a * b. this shows that the converse of theorem 4.12(2) is not valid. now consider two subsets a = {a, b} and b = {b, c, d} of x in example 3.2. then dp(a) = {b, e} = dp(b), and so dp(a ∩ b) = ∅ ⊆ dp(a) ∩ dp(b). thus the equality in theorem 4.12(3) is not valid. (2) consider a topology t = {x, ∅, {b, c}, {b, c, d}, {a, b, c}} on x = {a, b, c, d}. then t p = {x, ∅, {b}, {c}, {a, b}, {a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. let a = {a, b} and b = {a, c} be subsets of x. then dp(a) = ∅ = dp(b), and so dp(a) ∪ dp(b) = ∅ ⊂ {a, d} = dp(a ∪ b). for a subset a = {a, b, c} of x, we have dp(dp(a)) = dp({a, d}) = ∅, dp(dp(a)) \ a = ∅ ⊆ dp(a) = {a, d}, applications of pre-open sets 219 and so the equality in theorem 4.12(4) is not valid. now for a subset b = {b, c} of x, we get dp(b) = {a, d}, and so b ∪dp(b) = x and dp(x) = {a, d} ⊆ x. this shows that dp(b ∪ dp(b)) 6= b ∪ dp(b) = x. hence the equality in theorem 4.12(5) is not valid. theorem 4.14. let a be a subset of x and x ∈ x. then the following are equivalent: (i) (∀g ∈ t p) (x ∈ g ⇒ a ∩ g 6= ∅). (ii) x ∈ clp(a). proof. (i) ⇒ (ii) if x /∈ clp(a), then there exists a pre-closed set f such that a ⊆ f and x /∈ f. hence x \ f is a pre-open set containing x and a ∩ (x \ f ) ⊆ a ∩ (x \ a) = ∅. this is a contradiction, and hence (ii) is valid. (ii) ⇒ (i) straightforward. � corollary 4.15. for any subset a of x, we have dp(a) ⊆ clp(a). proof. straightforward. � theorem 4.16. for any subset a of x, clp(a) = a ∪ dp(a). proof. let x ∈ clp(a). assume that x /∈ a and let g ∈ t p with x ∈ g. then (g ∩ a) \ {x} 6= ∅, and so x ∈ dp(a). hence clp(a) ⊆ a ∪ dp(a). the reverse inclusion is by a ⊆ clp(a) and corollary 4.15. � theorem 4.17. let a and b be subsets of x. if a ∈ t p and t p is a topology on x, then a ∩ clp(b) ⊆ clp(a ∩ b). proof. let x ∈ a∩clp(b). then x ∈ a and x ∈ clp(b) = b ∪dp(b). if x ∈ b, then x ∈ a ∩ b ⊆ clp(a ∩ b). if x /∈ b, then x ∈ dp(b) and so g ∩ b 6= ∅ for all pre-open set g containing x. since a ∈ t p, g ∩ a is also a pre-open set containing x. hence g ∩ (a ∩ b) = (g ∩ a) ∩ b 6= ∅, and consequently x ∈ dp(a ∩ b) ⊆ clp(a ∩ b). therefore a ∩ clp(b) ⊆ clp(a ∩ b). � example 4.18. let t = {x, ∅, {b}, {b, c}, {b, c, d}} be a topology on a set x = {a, b, c, d}. then t p = {x, ∅, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}} which is a topology on x. let a = {a, b} and b = {b, c} be subsets of x. then a ∩ clp(b) = {a, b} 6= x = clp(a ∩ b). this shows that the equality in theorem 4.17 is not true in general. example 4.19. consider t and t p which are given in example 4.13(2). note that t p is not a topology on x. for subsets a = {a, b} and b = {b, c} of x, we have a ∩ clp(b) = {a, b} * {b} = clp(a ∩ b). this shows that if t p is not a topology on x then the result in theorem 4.17 is not true in general. theorem 4.20. let a and b subsets of x. if a is pre-closed, then clp(a ∩ b) ⊆ a ∩ clp(b). 220 y. b. jun, s. w. jeong, h. j. lee and j. w. lee proof. if a is pre-closed, then clp(a) = a and so clp(a ∩ b) ⊆ clp(a) ∩ clp(b) = a ∩ clp(b) which is the desired result. � lemma 4.21. a subset a of x is pre-open if and only if there exists an open set h in x such that a ⊆ h ⊆ cl(a). proof. straightforward. � lemma 4.22. the intersection of an open set and a pre-open set is a pre-open set. proof. let a be an open set in x and b a pre-open set in x. then there exists an open set g in x such that b ⊆ g ⊆ cl(b). it follows that a ∩ b ⊆ a ∩ g ⊆ a ∩ cl(b) ⊆ cl(a ∩ b). now since a∩g is open, it follows from lemma 4.21 that a∩b is pre-open. � theorem 4.23. let a and b be subsets of x. if a is open, then a ∩ clp(b) ⊆ clp(a ∩ b). proof. it is by theorem 4.17 and lemma 4.22. � theorem 4.24. if a is a subset of a discrete topological space x, then dp(a) = ∅. proof. let x be any element of x. recall that every subset of x is open, and so pre-open. in particular, the singleton set g := {x} is pre-open. but x ∈ g and g ∩ a = {x} ∩ a ⊆ {x}. hence x is not a pre-limit point of a, and so dp(a) = ∅. � theorem 4.25. for every subset a of x, we have a is pre-closed if and only if dp(a) ⊆ a. proof. assume that a is pre-closed. let x /∈ a, i.e., x ∈ x \ a. since x \ a is pre-open, x is not a pre-limit point of a, i.e., x /∈ dp(a), because (x \a)∩(a\ {x}) = ∅. hence dp(a) ⊆ a. the reverse implication is by theorem 4.16. � theorem 4.26. let a be a subset of x. if f is a pre-closed superset of a, then dp(a) ⊆ f. proof. by theorem 4.12(2) and theorem 4.25, a ⊆ f implies dp(a) ⊆ dp(f ) ⊆ f. � theorem 4.27. let a be a subset of x. if a point x ∈ x is a pre-limit point of a, then x is also a pre-limit point of a \ {x}. proof. straightforward. � applications of pre-open sets 221 definition 4.28 ([2]). let a be a subset of a topological space x. a point x ∈ x is called an α-interior point of a if there exists an α-open set g containing x such that g ⊆ a. the set of all α-interior points of a is called the α-interior of a and is denoted by intα(a). based on the above definition, we give the notion of a pre-interior point. definition 4.29. let a be a subset of a topological space x. a point x ∈ x is called a pre-interior point of a if there exists a pre-open set g such that x ∈ g ⊆ a. the set of all pre-interior points of a is called the pre-interior of a and is denoted by intp(a). example 4.30. let (x, t ) be a topological space which is given in example 4.4. we know that a is the only pre-interior point of a = {a, b, e}, i.e., intp(a) = {a}. theorem 4.31. let a be a subset of x. then every α-interior point of a is a pre-interior point of a, i.e., intα(a) ⊆ intp(a). proof. if x is an α-interior point of a, then there exists an α-open set g containing x such that g ⊆ a. since every α-open set is pre-open, it follows that x is a pre-interior point of a. � the following example shows that there exists a pre-interior point of a which is not an α-interior point of a. example 4.32. in example 4.4, intα(a) = {a} and intp(a) = {a, b, c}. hence b and c are pre-interior points of a. but they are not α-interior points of a. proposition 4.33. for subsets a and b of x, the following assertions are valid. (1) intp(a) is the union of all pre-open subsets of a; (2) a is pre-open if and only if a = intp(a); (3) intp(intp(a)) = intp(a); (4) intp(a) = a \ dp(x \ a). (5) x \ intp(a) = clp(x \ a). (6) x \ clp(a) = intp(x \ a). (7) a ⊆ b ⇒ intp(a) ⊆ intp(b). (8) intp(a) ∪ intp(b) ⊆ intp(a ∪ b). (9) intp(a ∩ b) ⊆ intp(a) ∩ intp(b). proof. (1) let {gi | i ∈ λ} be a collection of all pre-open subsets of a. if x ∈ intp(a), then there exists j ∈ λ such that x ∈ gj ⊆ a. hence x ∈ ⋃ i∈λ gi, and so intp(a) ⊆ ⋃ i∈λ gi. on the other hand, if y ∈ ⋃ i∈λ gi, then y ∈ gk ⊆ a for some k ∈ λ. thus y ∈ intp(a), and ⋃ i∈λ gi ⊆ intp(a). accordingly, intp(a) = ⋃ i∈λ gi. (2) straightforward. 222 y. b. jun, s. w. jeong, h. j. lee and j. w. lee (3) it follows from (1) and (2). (4) if x ∈ a \ dp(x \ a), then x /∈ dp(x \ a) and so there exists a pre-open set g containing x such that g ∩ (x \ a) = ∅. thus x ∈ g ⊆ a and hence x ∈ intp(a). this shows that a \ dp(x \ a) ⊆ intp(a). now let x ∈ intp(a). since intp(a) ∈ t p and intp(a) ∩ (x \ a) = ∅, we have x /∈ dp(x \ a). therefore intp(a) = a \ dp(x \ a). (5) using (4) and theorem 4.16, we have x \ intp(a) = x \ (a \ dp(x \ a)) = (x \ a) ∪ dp(x \ a) = clp(x \ a). (6) using (4) and theorem 4.16, we get intp(x \ a) = (x \ a) \ dp(a) = x \ (a ∪ dp(a)) = x \ clp(a). (7) straightforward. (8) and (9) they are by (7). � the converse of (7) in proposition 4.33 is not true in general as seen in the following example. example 4.34. consider a topological space (x, t ) which is described in example 4.4. let a = {a, b} and b = {a, c, d} be subsets of x. then intp(a) = {a} ⊆ intp(b) = {a, c, d}. definition 4.35 ([2]). for any subset a of x, the set bα(a) := a \ intα(a) is called the α-border of a, and the set frα(a) := clα(a) \ intα(a) is called the α-frontier of a. definition 4.36. for any subset a of x, the set bp(a) := a \ intp(a) is called the pre-border of a, and the set frp(a) := clp(a) \ intp(a) is called the pre-frontier of a. note that if a is a pre-closed subset of x, then bp(a) = frp(a). example 4.37. (1) let (x, t ) be the topological space which is described in example 4.4. let a = {a, b, e} be a subset of x. then intp(a) = {a}, and so bp(a) = {b, e}. since a = {a, b, e} is pre-closed, clp(a) = {a, b, e} and thus frp(a) = {b, e}. (2) consider the topological space (x, t ) which is given in example 3.2. for a subset a = {b, c, d} of x, we have intp(a) = {c, d} and clp(a) = {b, c, d, e}. hence bp(a) = {b} and frp(a) = {b, e}. applications of pre-open sets 223 proposition 4.38. for a subset a of x, the following statements hold: (1) bp(a) ⊆ bα(a). (2) a = intp(a) ∪ bp(a). (3) intp(a) ∩ bp(a) = ∅. (4) a is a pre-open set if and only if bp(a) = ∅. (5) bp(intp(a)) = ∅. (6) intp(bp(a)) = ∅. (7) bp(bp(a)) = bp(a). (8) bp(a) = a ∩ clp(x \ a). (9) bp(a) = a ∩ dp(x \ a). proof. (1) since intα(a) ⊆ intp(a), we have bp(a) = a \ intp(a) ⊆ a \ intα(a) = bα(a). (2) and (3). straightforward. (4) since intp(a) ⊆ a, it follows from proposition 4.33(2) that a is pre-open ⇔ a = intp(a) ⇔ bp(a) = a \ intp(a) = ∅. (5) since intp(a) is pre-open, it follows from (4) that bp(intp(a)) = ∅. (6) if x ∈ intp(bp(a)), then x ∈ bp(a) ⊆ a and x ∈ intp(a) since intp(bp(a)) ⊆ intp(a). thus x ∈ bp(a) ∩ intp(a) = ∅, which is a contradiction. hence intp(bp(a)) = ∅. (7) using (6), we get bp(bp(a)) = bp(a) \ intp(bp(a)) = bp(a). (8) using proposition 4.33(6), we have bp(a) = a \ intp(a) = a \ (x \ clp(x \ a)) = a ∩ clp(x \ a). (9) applying (8) and theorem 4.16, we have bp(a) = a ∩ clp(x \ a) = a ∩ ((x \ a) ∪ dp(x \ a)) = a ∩ dp(x \ a). this completes the proof. � lemma 4.39. for a subset a of x, a is pre-closed if and only if frp(a) ⊆ a. proof. assume that a is pre-closed. then frp(a) = clp(a) \ intp(a) = a \ intp(a) ⊆ a. conversely suppose that frp(a) ⊆ a. then clp(a) \ intp(a) ⊆ a, and so clp(a) ⊆ a since intp(a) ⊆ a. noticing that a ⊆ clp(a), we have a = clp(a). therefore a is pre-closed. � theorem 4.40. for a subset a of x, the following assertions are valid: (1) frp(a) ⊆ frα(a). (2) clp(a) = intp(a) ∪ frp(a). (3) intp(a) ∩ frp(a) = ∅. (4) bp(a) ⊆ frp(a). 224 y. b. jun, s. w. jeong, h. j. lee and j. w. lee (5) frp(a) = bp(a) ∪ (dp(a) \ intp(a)). (6) a is a pre-open set if and only if frp(a) = bp(x \ a). (7) frp(a) = clp(a) ∩ clp(x \ a). (8) frp(a) = frp(x \ a). (9) frp(a) is pre-closed. (10) frp(frp(a)) ⊆ frp(a). (11) frp(intp(a)) ⊆ frp(a). (12) frp(clp(a)) ⊆ frp(a). (13) intp(a) = a \ frp(a). proof. (1) since clp(a) ⊆ clα(a) and intα(a) ⊆ intp(a), it follows that frp(a) = clp(a) \ intp(a) ⊆ clα(a) \ intp(a) ⊆ clα(a) \ intα(a) = frα(a). (2) straightforward. (3) intp(a) ∩ frp(a) = intp(a) ∩ (clp(a) \ intp(a)) = ∅. (4) since a ⊆ clp(a), we have bp(a) = a \ intp(a) ⊆ clp(a) \ intp(a) = frp(a). (5) using theorem 4.16, we obtain frp(a) = clp(a) \ intp(a) = (a ∪ dp(a)) ∩ (x \ intp(a)) = (a \ intp(a)) ∪ (dp(a) \ intp(a)) = bp(a) ∪ (dp(a) \ intp(a)). (6) assume that a is pre-open. then frp(a) = bp(a) ∪ (dp(a) \ intp(a)) = ∅ ∪ (dp(a) \ a) = dp(a) \ a = bp(x \ a) by using (5), proposition 4.38(4), proposition 4.33(2) and proposition 4.38(9). conversely suppose that frp(a) = bp(x \ a). then ∅ = frp(a) \ bp(x \ a) = (clp(a) \ intp(a)) \ ((x \ a) \ intp(x \ a)) = a \ intp(a) by (4) and (5) of proposition 4.33, and so a ⊆ intp(a). since intp(a) ⊆ a in general, it follows that intp(a) = a so from proposition 4.33(2) that a is pre-open. (7) using proposition 4.33(5), we have clp(a) ∩ clp(x \ a) = clp(a) ∩ (x \ intp(a)) = clp(a) \ intp(a) = frp(a). (8) it follows from (7). applications of pre-open sets 225 (9) we have clp(frp(a)) = clp(clp(a) ∩ clp(x \ a)) ⊆ clp(clp(a)) ∩ clp(clp(x \ a)) = clp(a) ∩ clp(x \ a) = frp(a). obviously frp(a) ⊆ clp(frp(a)), and so frp(a) = clp(frp(a)). hence frp(a) is pre-closed. (10) this is by (9) and lemma 4.39. (11) using proposition 4.33(3), we get frp(intp(a)) = clp(intp(a)) \ intp(intp(a)) ⊆ clp(a) \ intp(a) = frp(a). (12) we obtain frp(clp(a)) = clp(clp(a)) \ intp(clp(a)) ⊆ clp(a) \ intp(a) = frp(a). (13) we get a \ frp(a) = a \ (clp(a) \ intp(a)) = a ∩ ((x \ clp(a)) ∪ intp(a)) = ∅ ∪ (a ∪ intp(a)) = intp(a). this completes the proof. � the converses of (1) and (4) of theorem 4.40 are not true in general as seen in the following example. example 4.41. in example 3.2, let a = {a, b, c}. then frp(a) = {e} ( {b, c, d, e} = frα(a), which shows that the reverse inclusion of theorem 4.40(1) is not valid. also, example 4.37(2) shows that the reverse inclusion of theorem 4.40(4) is not valid in general. definition 4.42 ([2]). for a subset a of x, extα(a) = intα(x \ a) is said to be an α-exterior of a. definition 4.43. for a subset a of x, the semi-interior of x \ a is called the pre-exterior of a, and is denoted by extp(a), that is, extp(a) = intp(x \ a). example 4.44. let (x, t ) be a topological space in example 4.4. for subsets a = {a, b, c} and b = {b, d} of x, we have extp(a) = {d, e} and extp(b) = {a, c, e}. 226 y. b. jun, s. w. jeong, h. j. lee and j. w. lee theorem 4.45. for subsets a and b of x, the following assertions are valid. (1) extα(a) ⊆ extp(a). (2) extp(a) is pre-open. (3) extp(a) = x \ clp(a). (4) extp(extp(a)) = intp(clp(a)) ⊇ intp(a). (5) a ⊆ b ⇒ extp(b) ⊆ extp(a). (6) extp(a ∪ b) ⊆ extp(a) ∩ extp(b). (7) extp(a ∩ b) ⊇ extp(a) ∪ extp(b). (8) extp(x) = ∅, extp(∅) = x. (9) extp(a) = extp(x \ extp(a)). (10) x = intp(a) ∪ extp(a) ∪ frp(a). proof. (1) using theorem 4.31, we have extα(a) = intα(x \ a) ⊂ intp(x \ a) = extp(a). (2) it follows from lemma 4.10 and proposition 4.33(1). (3) it is straightforward by proposition 4.33(6). (4) applying (5) and (7) of proposition 4.33, we get extp(extp(a)) = extp(intp(x \ a)) = intp(x \ intp(x \ a)) = intp(clp(a)) ⊃ intp(a). (5) assume that a ⊂ b. then extp(b) = intp(x \ b) ⊆ intp(x \ a) = extp(a) by using proposition 4.33(7). (6) applying proposition 4.33(9), we get extp(a ∪ b) = intp(x \ (a ∪ b)) = intp((x \ a) ∩ (x \ b)) ⊆ intp(x \ a) ∩ intp(x \ b) = extp(a) ∩ extp(b). (7) using proposition 4.33(8), we obtain extp(a ∩ b) = intp(x \ (a ∩ b)) = intp((x \ a) ∪ (x \ b)) ⊇ intp(x \ a) ∪ intp(x \ b) = extp(a) ∪ extp(b). (8) straightforward. (9) using proposition 4.33(3), we have extp(x \ extp(a)) = extp(x \ intp(x \ a)) = intp(x \ a) = extp(a). (10) straightforward. � applications of pre-open sets 227 let (x, t ) be a topological space which is given in example 4.4. take a = {d, e}. then extα(a) = {a} and extp(a) = {a, b, c}. thus the reverse inclusion of theorem 4.45(1) is not valid. let a = {b, e} and b = {c, d, e}. then extp(b) = {a} ⊆ {a, c, d} = extp(a). this shows that the converse of (5) in theorem 4.45 is not valid. now let a = {d, e} and b = {c}. then extp(a ∪ b) = {a} 6= {a, b} = {a, b, c} ∩{a, b, d, e} = extp(a) ∩ extp(b) which shows that the equality in theorem 4.45(6) is not valid. finally let a = {a, b} and b = {c, d, e}. then extp(a ∩ b) = {a, b, c, d, e} and extp(a) ∪ extp(b) = {a, c, d, e}. this shows that the equality in theorem 4.45(7) is not valid. references [1] d. andrijevic, some properties of the topology of α-sets, mat. vesnik 36 (1984), 1–10. [2] m. caldas, a note on some applications of α-sets, int. j. math. math. sci. 2003, no. 2 (2003), 125–130. [3] m. caldas and j. dontchev, on spaces with hereditarily compact α-topologies, acta math. hung. 82 (1999), 121–129. [4] s. jafari and t. noiri, contra-α-continuous functions between topological spaces, iranian int. j. sci. 2, no. 2 (2001), 153–167. [5] s. jafari and t. noiri, some remarks on weak α-continuity, far east j. math. sci. 6, no. 4 (1998), 619–625. [6] s. n. maheshwari and s. s. thakur, on α-irresolute mappings, tamkang j. math. 11 (1980), 209–214. [7] s. n. maheshwari and s. s. thakur, on α-compact spaces, bull. inst. math. acad. sinica 13 (1985), 341–347. [8] a. s. mashhour, m. e. abd el-monsef and s. n. el-deeb, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. egypt, 53 (1982), 47–53. [9] h. maki, r. devi and k. balachandran, generalized α-closed sets in topology, bull. fukuoka univ. ed. part iii 42 (1993), 13–21. [10] h. maki and t. noiri, the pasting lemma for α-continuous maps, glas. mat. 23(43) (1988), 357–363. [11] a. s. mashhour, i. a. hasanein and s. n. el-deeb, a note on semi-continuity and precontinuity, indian j. pure appl. math. 13, no. 10 (1982), 1119–1123. [12] a. s. mashhour, i. a. hasanein and s. n. el-deeb, α-continuous and α-open mappings, acta math. hungar. 41, no. 3-4 (1983), 213–218. [13] n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. monthly 70 (1963), 36–41. [14] o. nj̊astad, on some classes of nearly open sets, pacific j. math. 15 (1965), 961–970. [15] i. l. reilly and m. k. vamanamurthy, on α-sets in topological spaces, tamkang j. math. 16 (1985), 7–11. [16] j. tong, on decomposition of continuity in topological spaces, acta math. hungar. 54, no. 1-2 (1989), 51–55. received may 2007 accepted february 2008 228 y. b. jun, s. w. jeong, h. j. lee and j. w. lee young bae jun (skywine@gmail.com) department of mathematics education (and rins), gyeongsang national university, chinju 660-701, korea seong woo jeong (liveinworld@hanmail.net) department of mathematics education (and rins), gyeongsang national university, chinju 660-701, korea hyeon jeong lee (jfield@hanmail.net) department of mathematics education (and rins), gyeongsang national university, chinju 660-701, korea joon woo lee (jwlee−angel@hanmail.net) department of mathematics education (and rins), gyeongsang national university, chinju 660-701, korea () @ applied general topology c© universidad politécnica de valencia volume 12, no. 1, 2011 pp. 15-16 the equality of the patch topology and the ultrafilter topology: a shortcut luz m. ruza and jorge vielma abstract in this work r denotes a commutative ring with non-zero identity and we prove that the patch topology and the ultrafilter topology defined on the prime spectrum of r are equal, in a different way as the given by marco fontana and k. alan loper in ([2]). 2010 msc: 54e18, 54f65, 13c05. keywords: patch topology, ultrafilter topology, prime spectrum of a ring. 1. terminology and basic definitions let r be a commutative ring with non-zero identity. spec(r) denotes the set of all prime ideals of r. for every proper subset i of r, we denote by v (i) the set of all prime ideals of r containing i, and d0(i) = spec(r)−v (i). v (a) will denote the set v (ar) and d0(a) the set d0(ar). the zariski topology tz on spec(r) is the one that has as its closed sets those of the form v (i) ([1]). the patch topology on spec(r) is defined as the smallest topology having the collections v (i) and d0(a) as closed sets. let c be a subset of spec(r), and let ω be an ultrafilter on c. it was shown in ([2]) that the set pω = {a ∈ r : v (a) ∩ c ∈ ω} is a prime ideal of r. the set c is said to be ultrafilter-closed if for every ultrafilter ω on c, pω ∈ c. the ultrafilter-closed sets define a topology on spec(r) called the ultrafilter topology ([2]), and is denoted by τu . in this work we prove that the patch topology and the ultrafilter topology are equal, in a different way as the given by fontana and loper in ([2]). 16 l. m. ruza and j. vielma 2. the shortcut theorem 2.1. the ultrafilter topology τu is compact. proof. let u be a non principal ultrafilter in spec(r). we want to prove that u τu -converge to pu . let θ be a τu -open set containing pu . suppose that a = θc belongs to u and let ua = {u ∩ a : u ∈ u} be the ultrafilter on a induced by u. since a is τu -closed, then pua ∈ a. if a ∈ pu , v (a) ∈ u, then v (a) ∩ a ∈ ua and therefore a ∈ pua . also, if b ∈ pua it follows that v (b) ∩ a ∈ ua and there exists u ∈ u such that v (b) ∩ a = u ∩ a. since u ∩ a ∈ u, then v (b) ∩ a ∈ u which implies that v (b) ∈ u and so b ∈ pu . therefore pu = pua ∈ a which is a contradiction. � corollary 2.2. the ultrafilter topology and the patch topology are equal. proof. since the patch topology is hausdorff ([3]), weaker than the ultrafilter topology ([2]) and the well known fact that any compact topology does not admit a weaker hausdorff topology unless they are equal, the result follows. � references [1] m. atiyah and i. g. macdonald, introduction to commutative algebra, addison-wesley, 1969. [2] m. fontana and k. a. loper, the patch topology and the ultrafilter topology on the prime spectrum of a commutative ring, comm. algebra 36 (2008), 2917–2922. [3] m. hochster, prime ideal structure in commutative rings, trans. amer. math. soc. 142 (1969), 43–60. (received november 2008 – accepted september 2009) luz m. ruza (ruza@ula.ve) universidad de los andes, departamento de matemática, merida, venezuela jorge vielma (vielma@ula.ve) universidad de los andes, departamento de matemática, merida, venezuela the equality of the patch topology and the[0.2cm] ultrafilter topology: a shortcut. by l. m. ruza and j. vielma @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 213-220 matkowski’s type theorems for generalized contractions on (ordered) partial metric spaces salvador romaguera ∗ abstract we obtain extensions of matkowski’s fixed point theorem for generalized contractions of ćirić’s type on 0-complete partial metric spaces and on ordered 0-complete partial metric spaces, respectively. 2010 msc: 54h25, 47h10, 54e50 keywords: matkowski’s fixed point theorem; generalized contraction; 0complete partial metric space; ordered partial metric space 1. introduction and preliminaries partial metric spaces, and their equivalent weightable quasi-metric spaces, were introduced by matthews [14] in the context of his studies on denotational semantics for dataflow networks. in fact, this class of spaces provides a suitable framework to construct computational models for metric spaces and related structures (see e.g. [10, 14, 21, 22, 24, 25, 26]). it also provides an appropriate setting to discuss, with the help of techniques of fixed points of denotational semantics, the complexity analysis of several algorithms which can be defined by recurrence equations (see e.g. [9, 20, 23]). these facts explain, in part, the recent extensive research on fixed points for self maps in partial metric spaces ([1, 2, 3, 5, 8, 11, 13, 19], etc). in this note we try to reach a new advance in this direction by obtaining extensions of matkowski’s fixed point theorem [15, theorem 1.2] for generalized contractions on 0-complete partial metric space and on ordered 0-complete partial metric spaces, respectively. these results extend, generalize and unify some theorems from the current literature. ∗the author thanks the support of the ministry of science and innovation of spain, under grant mtm2009-12872-c02-01 214 s. romaguera throughout this paper the letter ω will denote the set of all nonnegative integer numbers. let us recall [14] that a partial metric on a set x is a function p : x × x → [0,∞) such that for all x,y,z ∈ x : (i) x = y ⇔ p(x,x) = p(x,y) = p(y,y); (ii) p(x,x) ≤ p(x,y); (iii) p(x,y) = p(y,x); (iv) p(x,z) ≤ p(x,y) + p(y,z) − p(y,y). a partial metric space is a pair (x,p) where p is a partial metric on x. each partial metric p on x induces a t0 topology τp on x which has as a base the family of open balls {bp(x,ε) : x ∈ x,ε > 0}, where bp(x,ε) = {y ∈ x : p(x,y) < p(x,x) + ε} for all x ∈ x and ε > 0. matthews observed in [14, p. 187] that a sequence (xn)n∈ω in a partial metric space (x,p) converges to some x ∈ x with respect to τp if and only if limn→∞ p(x,xn) = p(x,x). next we recall some useful concepts and facts on completeness of partial metric spaces. if p is a partial metric on x, then the function ps : x × x → [0,∞) given by ps(x,y) = 2p(x,y) − p(x,x) − p(y,y), is a metric on x. furthermore, a sequence (xn)n∈ω in x converges to some x ∈ x with respect to τps if and only if limn,m→∞ p(xn,xm) = limn→∞ p(x,xn) = p(x,x). a sequence (xn)n∈ω in a partial metric space (x,p) is called a cauchy sequence if there exists (and is finite) limn,m p(xn,xm) [14, definition 5.2]. a partial metric space (x,p) is said to be complete if every cauchy sequence (xn)n∈ω in x converges, with respect to τp, to a point x ∈ x such that p(x,x) = limn,m p(xn,xm) [14, definition 5.3]. it is well known (see, for instance, [14, p. 194]) that a sequence in a partial metric space (x,p) is a cauchy sequence in (x,p) if and only if it is a cauchy sequence in the metric space (x,ps), and that a partial metric space (x,p) is complete if and only the metric space (x,ps) is complete. romaguera introduced in [18] the notions of a 0-cauchy sequence in a partial metric space and of a 0-complete partial metric space. a sequence (xn)n∈ω in a partial metric space (x,p) is called 0-cauchy if limn,m p(xn,xm) = 0. we say that a partial metric space (x,p) is 0-complete if every 0-cauchy sequence in x converges, with respect to τp, to a point x ∈ x such that p(x,x) = 0. in this case, p is said to be a 0-complete partial metric on x. note that every 0-cauchy sequence in (x,p) is a cauchy sequence in (x,p), and that every complete partial metric space is 0-complete. on the other hand, the partial metric space (q ∩ [0,∞),p), where q denotes the set of rational numbers and the partial metric p is given by p(x,y) = max{x,y}, provides an easy example of a 0-complete partial metric space which is not complete. generalized contractions on (ordered) partial metric spaces 215 2. fixed points for 0-complete partial metric spaces given a partial metric space (x,p) and f : x → x a map, we define m(x,y) := max { p(x,y),p(x,fx),p(y,fy), 1 2 [p(x,fy) + p(y,fx)] } , for all x,y ∈ x (compare e.g. [5, 19]). the proof of [19, lemma 2] shows the following. lemma 2.1. let (x,p) be a partial metric space, f : x → x a map and x0 ∈ x such that f nx0 6= f n+1x0 and p(fn+1x0,f n+2x0) ≤ ϕ(m(f nx0,f n+1x0)), for all n ∈ ω, where ϕ : [0,∞) → [0,∞) satisfies ϕ(t) < t for all t > 0. then the following hold: (a) m(fnx0,f n+1x0) = p(f nx0,f n+1x0 ) for all n ∈ ω. (b) p(fn+1x0,f n+2x0) ≤ ϕ(p(f nx0,f n+1x0)) < p(f nx0,f n+1x0) for all n ∈ ω. remark 2.2. recall [15, 16] that if ϕ : [0,∞) → [0,∞) is a nondecreasing function such that limn→∞ ϕ n(t) = 0 for all t > 0, then ϕ(t) < t for all t > 0 and thus ϕ(0) = 0. now we prove the main result of this section. theorem 2.3. let (x,p) be a 0-complete partial metric space and f : x → x a map such that p(fx,fy) ≤ ϕ(m(x,y)), for all x,y ∈ x, where ϕ : [0,∞) → [0,∞) is a nondecreasing function such that limn→∞ ϕ n(t) = 0 for all t > 0. then f has a unique fixed point z ∈ x. moreover p(z,z) = 0. proof. let x0 ∈ x. for each n ∈ ω put xn = f nx0. thus xn+1 = fxn for all n ∈ ω. if there is k ∈ ω such that xk = xk+1, then xk is a fixed point of f. moreover if fz = z for some z ∈ x, it follows that p(z,xk) = p(fz,fxk) ≤ ϕ(m(z,xk)) = ϕ(p(z,xk)), so, p(xk,z) = 0, i.e., z = xk. so xk is the unique fixed point of f, and, clearly, p(xk,xk) = 0 by the contraction condition. hence, we shall assume that fnx0 6= f n+1x0 for all n ∈ ω. thus p(xn,xn+1) > 0 for all n ∈ ω. by lemma 2.1 (b), p(xn,xn+1) ≤ ϕ(p(xn−1,xn)) for all n ∈ n, and then p(xn,xn+1) ≤ ϕ n(p(x0,x1)), for all n ∈ ω. so lim n→∞ p(xn,xn+1) = 0. 216 s. romaguera now choose an arbitrary ε > 0. since, by remark 2.2, ϕ(ε) < ε, then there is nε ∈ n such that p(xn,xn+1) < ε − ϕ(ε), for all n ≥ nε. therefore p(xn,xn+2) ≤ p(xn,xn+1) + p(xn+1,xn+2) < ε − ϕ(ε) + ϕ(p(xn,xn+1)) ≤ ε − ϕ(ε) + ϕ(ε) = ε, for all n ≥ nε. so p(xn,xn+3) ≤ p(xn,xn+1) + p(xn+1,xn+3) < ε − ϕ(ε) + ϕ(m(xn,xn+2)), for all n ≥ nε. now suppose that there is n ≥ nε such that m(xn,xn+2) > ε. then, from m(xn,xn+2) = max{p(xn,xn+2),p(xn,xn+1),p(xn+1,xn+2), 1 2 [p(xn,xn+3) + p(xn+1,xn+2)} ≤ max{ε, 1 2 [p(xn,xn+3) + ϕ(ε)]}. it follows that m(xn,xn+2) ≤ 1 2 [p(xn,xn+3) + ϕ(ε)], so p(xn,xn+3) < ε − ϕ(ε) + ϕ(m(xn,xn+2)) < ε − ϕ(ε) + m(xn,xn+2) ≤ ε − ϕ(ε) + 1 2 [p(xn,xn+3) + ϕ(ε)]. we deduce that m(xn,xn+2) < ε, a contradiction. therefore p(xn,xn+3) < ε, and following this process, p(xn,xn+k) < ε, for all n ≥ nε and k ∈ n. consequently lim n,m→∞ p(xn,xm) = 0, and thus (xn)n∈ω is a 0-cauchy sequence in the 0-complete partial metric space (x,p). hence there is z ∈ x such that lim n,m→∞ p(xn,xm) = lim n→∞ p(z,xn) = p(z,z) = 0. finally, the fact that z is the unique fixed point of f follows similarly to the last part of the proof of [19, theorem 4]. � generalized contractions on (ordered) partial metric spaces 217 in a recent paper [12], jachymski showed the equivalence between several generalized contractions on (ordered) metric spaces. since the key of his study is lemma 1 of the cited paper, then jachymski’s approach also holds in the partial metric framework. as an instance, we shall combine this lemma with theorem 2.3 above to deduce the following (compare [1, corollary 2.1]). corollary 2.4. let (x,p) be a 0-complete partial metric space and f : x → x a map such that ψ(p(fx,fy)) ≤ ψ(m(x,y)) − φ(m(x,y)), for all x,y ∈ x, where ψ,φ : [0,∞) → [0,∞) are nondecreasing functions such that ψ is continuous on [0,∞) and ψ−1(0) = φ−1(0) = {0}. then f has a unique fixed point z ∈ x. moreover p(z,z) = 0. proof. by [12, lemma 1 (ii)⇒(viii)], there exists a continuous and nondecreasing function ϕ : [0,∞) → [0,∞) such that ϕ(t) < t for all t > 0, and p(fx,fy) ≤ ϕ(m(x,y)) for all x,y ∈ x. from continuity of ϕ it follows that limn→∞ ϕ n(t) = 0 for all t > 0. theorem 2.3 concludes the proof. � remark 2.5. theorem 2.3 and corollary 2.4 extend several fixed point theorems for complete metric spaces due to dutta and choudhury, khan, swaleh and sessa, and rhoades, among others (see [12, theorems 1 and 3, and the bibliography]). theorem 2.3 also improves [11, theorem 3.2], [5, theorem 1] and [19, theorem 4]. 3. fixed points for ordered 0-complete partial metric spaces our main purpose in this section is to prove an ordered counterpart of theorem 1. in this way, we shall extend the main result of agarwal, el-gebeily and o’regan in [6] (see also [17, theorem 3.11]). by an ordered (0-complete) partial metric space we mean a triple (x,�,p) such that � is a partial order on x and p is a (0-complete) partial metric on x. an ordered partial metric space (x,�,p) is called regular if for any nondecreasing sequence (xn)n∈ω for �, which converges to some z ∈ x with respect to τp, it follows xn � z for all n ∈ ω. we say that a self map f of a partial metric space (x,p) is continuous if it is continuous from (x,τp) into itself. theorem 3.1. let (x,�,p) be an ordered 0-complete partial metric space and f : x → x a nondecreasing map for �, such that p(fx,fy) ≤ ϕ(m(x,y)), for all x,y ∈ x with x � y, where ϕ : [0,∞) → [0,∞) is a nondecreasing function such that limn→∞ ϕ n(t) = 0 for all t > 0. if there is x0 ∈ x such that x0 � fx0, and f is continuous or (x,�,p) is regular, then f has a fixed point z ∈ x such that p(z,z) = 0. moreover, the set of fixed points of f is a singleton if and only it is well-ordered. 218 s. romaguera proof. for each n ∈ ω put xn = f nx0. thus xn+1 = fxn for all n ∈ ω. since x0 � fx0 and f is nondecreasing for �, it follows that xn � xn+1 for all n ∈ ω, so (xn)n∈ω is a nondecreasing sequence in (x,�). if there is k ∈ ω such that xk = xk+1, then xk is a fixed point of f. hence, we shall assume that fnx0 6= f n+1x0 for all n ∈ ω. thus p(xn,xn+1) > 0 for all n ∈ ω. then, the proof of theorem 2.3 shows (note, in particular, that xn � xn+k for all n,k ∈ ω) that (xn)n∈ω is a 0-cauchy sequence in (x,p). hence there is z ∈ x such that lim n,m→∞ p(xn,xm) = lim n→∞ p(z,xn) = p(z,z) = 0. we show that z is a fixed point of f. indeed, if f is continuous, we deduce that limn→∞ p(fz,xn) = p(fz,fz). since p(z,fz) ≤ p(z,xn) + p(xn,fz), for all n ∈ ω, it follows, taking limits as n → ∞, that p(z,fz) ≤ p(fz,fz), so p(z,fz) = p(fz,fz). hence, since z � z, we have p(z,fz) ≤ ϕ(m(z,z)) = ϕ(0) = 0, and thus z = fz, and p(z,z) = 0. if f is not continuous, it follows from regularity of (x,�,p) that xn � z for all n ∈ ω. assume p(z,fz) > 0. then, there is n0 ∈ n such that m(z,xn−1) = p(z,fz) for all n ≥ n0. thus p(z,fz) ≤ p(z,xn) + p(xn,fz) ≤ p(z,xn) + ϕ(m(z,xn−1)) = p(z,xn) + ϕ(p(z,xn−1)) ≤ p(z,xn) + p(z,xn−1), for all n ≥ n0. taking limits as n → ∞, we deduce that p(z,fz) = 0, a contradiction. we conclude that z = fz. finally, if the set of fixed point is well-ordered and u ∈ x is a fixed point of f,we deduce, assuming u � z, that p(u,z) = p(fu,fz) ≤ ϕ(m(u,z)) = ϕ(p(u,z)), so p(u,z) = 0, i.e., u = z. this concludes the proof. � from [12, lemma 1] and theorem 3.1 we deduce the following ordered counterpart of corollary 2.4. corollary 3.2. let (x,�,p) be an ordered 0-complete partial metric space and f : x → x a nondecreasing map for �, map such that ψ(p(fx,fy)) ≤ ψ(m(x,y)) − φ(m(x,y)), for all x,y ∈ x with x � y, where ψ,φ : [0,∞) → [0,∞) are nondecreasing functions such that ψ is continuous on [0,∞) and ψ−1(0) = φ−1(0) = {0}. if there is x0 ∈ x such that x0 � fx0, and f is continuous or (x,�,p) is regular, then f has a fixed 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(received september 2011 – accepted november 2011) s. romaguera (sromague@mat.upv.es) instituto universitario de matemática pura y aplicada, universitat politècnica de valència, camı́ de vera s/n, 46022 valencia, spain matkowski's type theorems for generalized contractions on (ordered) partial metric spaces. by s. romaguera rajveeraagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 21-28 best proximity pair theorems for relatively nonexpansive mappings v. sankar raj and p. veeramani ∗ abstract. let a, b be nonempty closed bounded convex subsets of a uniformly convex banach space and t : a∪b → a∪b be a map such that t (a) ⊆ b, t (b) ⊆ a and ‖t x − t y‖ ≤ ‖x − y‖, for x in a and y in b. the fixed point equation t x = x does not possess a solution when a ∩ b = ∅. in such a situation it is natural to explore to find an element x0 in a satisfying ‖x0 − t x0‖ = inf{‖a − b‖ : a ∈ a, b ∈ b} = dist(a, b). using zorn’s lemma, eldred et.al proved that such a point x0 exists in a uniformly convex banach space settings under the conditions stated above. in this paper, by using a convergence theorem we attempt to prove the existence of such a point x0 (called best proximity point) without invoking zorn’s lemma. 2000 ams classification: 47h10 keywords: best proximity pair, relatively nonexpansive map, cyclic contraction map, strictly convex space, uniformly convex banach space, fixed point, metric projection. 1. introduction let a, b be nonempty subsets of a normed linear space (x, ‖ · ‖) and a map t : a ∪ b → a ∪ b is said to be a relatively nonexpansive map if it satisfies (i) t (a) ⊆ b, t (b) ⊆ a and (ii) ‖t x − t y‖ ≤ ‖x − y‖, for all x ∈ a, y ∈ b. note that a relatively nonexpansive map need not be continuous in general. but if a ∩ b is nonempty, then the map t restricted to a ∩ b is a nonexpansive self map. if the fixed point equation t x = x does not possess a solution it is natural to explore to find an x0 ∈ a satisfying ‖x0 − t x0‖ = dist(a, b) = inf{‖a − b‖ : a ∈ a, b ∈ b}. a point x0 ∈ a is said to be a best proximity point for t if it satisfies ‖x0 − t x0‖ = dist(a, b). ∗corresponding author. 22 v. sankar raj and p. veeramani in [2], eldred et.al introduced a geometric concept called proximal normal structure which generalizes the concept of normal structure introduced by milman and brodskii [7]. definition 1.1 (proximal normal structure [2]). a convex pair (k1, k2) in a banach space is said to have proximal normal structure if for any closed, bounded, convex proximal pair (h1, h2) ⊆ (k1, k2) for which dist(h1, h2) = dist(k1, k2) and δ(h1, h2) > dist(h1, h2), there exists (x1, x2) ∈ h1 × h2 such that δ(x1, h2) < δ(h1, h2), δ(x2, h1) < δ(h1, h2) where δ(h1, h2) = sup{ ‖h1 − h2‖ : h1 ∈ h1, h2 ∈ h2}. using the concept of proximal normal structure, eldred et.al [2] proved the existence of best proximity points for relatively nonexpansive mappings. theorem 1.2 ([2]). let (a, b) be a nonempty, weakly compact convex pair in a banach space (x, ‖ · ‖), and suppose (a, b) has proximal normal structure. let t : a ∪ b → a ∪ b be a relatively nonexpansive map. then there exists (x, y) ∈ a × b such that ‖x − t x‖ = ‖t y − y‖ = dist(a, b). the proof of the above theorem invokes zorn’s lemma and the proximal normal structure idea. also it has been proved that every closed bounded convex pair (a, b) of a uniformly convex banach space has proximal normal structure and every compact convex pair has proximal normal structure. in this paper, by using a convergence theorem we attempt to prove the existence of a best proximity point without invoking zorn’s lemma. 2. preliminaries in this section we give some basic definitions and concepts which are useful and related to the context of our results. we shall say that a pair (a, b) of sets in a banach space satisfies a property if each of the sets a and b has that property. thus (a, b) is said to be convex if both a and b are convex. (c, d) ⊆ (a, b) ⇔ c ⊆ a, d ⊆ b etc. dist(a, b) = inf{‖x − y‖ : x ∈ a, y ∈ b} a0 = {x ∈ a : ‖x − y‖ = dist(a, b) f or some y ∈ b} b0 = {y ∈ b : ‖x − y‖ = dist(a, b) f or some x ∈ a} let x be a normed linear space and c be a nonempty subset of x. then the metric projection operator pc : x → 2 c is defined as pc (x) = {y ∈ c : ‖x − y‖ = dist(x, c)}, f or each x ∈ x. it is well known that the metric projection operator pc on a strictly convex banach space x is a single valued map from x to c, where c is a nonempty weakly compact convex subset of x. in [6], kirk et.al proved the following lemma which guarantees the nonemptiness of a0 and b0. best proximity pair theorems for relatively nonexpansive mappings 23 lemma 2.1 ([6]). let x be a reflexive banach space and a be a nonempty closed bounded convex subset of x, and b be a nonempty closed convex subset of x. then a0 and b0 are nonempty and satisfy pb (a0) ⊆ b0, pa(b0) ⊆ a0. in [8], sadiq basha and veeramani proved the following result. lemma 2.2 ([8]). if a and b are nonempty subsets of a normed linear space x such that dist(a, b) > 0, then a0 ⊆ ∂(a) and b0 ⊆ ∂(b) where ∂(c) denotes the boundary of c in x for any c ⊆ x. suppose (a, b) is a nonempty weakly compact convex pair of subsets in a banach space x. consider the map p : a ∪ b → a ∪ b defined as p (x) = { pb (x), if x ∈ a pa(x), if x ∈ b (2.1) if x is a strictly convex banach space, then the map p is a single valued map and satisfies p (a) ⊆ b, p (b) ⊆ a. proposition 2.3. let a, b be nonempty weakly compact convex subsets of a strictly convex banach space x. let t : a ∪ b → a ∪ b be a relatively nonexpansive map and p : a ∪ b → a ∪ b be a map defined as in (2.1). then t p (x) = p (t x), for all x ∈ a0 ∪ b0. proof. let x ∈ a0. then there exists y ∈ b such that ‖x − y‖ = dist(a, b). by the uniqueness of the metric projection on a strictly convex banach space, we have pb(x) = y, pa(y) = x. since t is relatively nonexpansive, we have ‖t x − t y‖ ≤ ‖x − y‖ = dist(a, b). ie pa(t x) = t y. this implies that pa(t x) = t pb(x) � this observation will play an important role in this article. in [3], eldred and veeramani introduced a notion of cyclic contraction and studied the existence of best proximity point for such maps. we make use of the main results proved in [3] to obtain best proximity pair theorems for relatively nonexpansive mappings. definition 2.4 ([3]). let a and b be nonempty subsets of a metric space x. a map t : a ∪ b → a ∪ b is said to be a cyclic contraction map if it satisfies : (1) t (a) ⊆ b, t (b) ⊆ a (2) there exists k ∈ (0, 1) such that d(t x, t y) ≤ kd(x, y)+(1−k)dist(a, b) for each x ∈ a, y ∈ b we can easily see that every cyclic contraction map satisfies d(t x, t y) ≤ d(x, y), for all x ∈ a, y ∈ b. in [3], a simple existence result for a best proximity point of a cyclic contraction map has been given. it states as follows: theorem 2.5 ([3]). let a and b be nonempty closed subsets of a complete metric space x. let t : a ∪ b → a ∪ b be a cyclic contraction map, let x0 ∈ a and define xn+1 = t xn, n = 0, 1, 2, · · · . suppose {x2n} has a convergent subsequence in a. then there exists x ∈ a such that d(x, t x) = dist(a, b). 24 v. sankar raj and p. veeramani in uniformly convex banach space settings, the following result proved in [3] ensures the existence, uniqueness and convergence of a best proximity point for a cyclic contraction map. we use this result to prove our main results. theorem 2.6 ([3]). let a and b be nonempty closed and convex subsets of a uniformly convex banach space. suppose t : a ∪ b → a ∪ b is a cyclic contraction map, then there exists a unique best proximity point x ∈ a( that is with ‖x − t x‖ = dist(a, b) ). further, if x0 ∈ a and xn+1 = t xn, then {x2n} converges to the best proximity point. we need the notion of ”approximatively compact set” to prove a convergence result in the next section. definition 2.7 ([9]). let x be a metric space. a subset c of x is said to be approximatively compact if for any y ∈ x, and for any sequence {xn} in c such that d(xn, y) → dist(y, c) as n → ∞, then {xn} has a subsequence which converges to a point in c. in a metric space, every approximatively compact set is closed and every compact set is approximatively compact. also a closed convex subset of a uniformly convex banach space is approximatively compact. 3. main results the following convergence theorem will play an important role in our main results. theorem 3.1. let x be a strictly convex banach space and a be a nonempty approximatively compact convex subset of x and b be a nonempty closed subset of x. let {xn} be a sequence in a and y ∈ b. suppose ‖xn − y‖ → dist(a, b), then xn → pa(y). proof. suppose that {xn} does not converges to pa(y), then there exists ε > 0 and a subsequence {xnk } of {xn} such that ‖xnk − pa(y)‖ ≥ ε(3.1) since {xnk } is a sequence in a such that ‖xnk − y‖ → dist(a, b), and a is approximatively compact, {xnk } has a convergent subsequence {xn′ k } such that xn′ k → x for some x ∈ a. then ‖xn′ k − y‖ → ‖x − y‖ also, ‖xn′ k − y‖ → dist(a, b) implies ‖x − y‖ = dist(a, b). by the uniqueness of pa we have x = pa(y). but from (3.1) we have ε ≤ ‖xn′ k − pa(y)‖ =⇒ 0 < ε ≤ ‖x − pa(y)‖ =⇒ x 6= pa(y) which is a contradiction. hence xn → pa(y). � the above theorem generalizes the following convergence result proved in [3] ([3],corollary 3.9) for a strictly convex banach space. best proximity pair theorems for relatively nonexpansive mappings 25 corollary 3.2 ([3]). let a be a nonempty closed convex subset and b be nonempty closed subset of a uniformly convex banach space. let {xn} be a sequence in a and y0 ∈ b such that ‖xn−y0‖ → dist(a, b). then xn converges to pa(y0). remark 3.3. let x be a normed linear space, let a be a nonempty closed convex subset of x, and b be a nonempty approximatively compact convex subset of x. if a0 is compact, then b0 is also compact. proof. if b0 is empty, then nothing to prove. assume b0 is nonempty. let {yn} be a sequence in b0. then for each n ∈ n, there exists xn ∈ a0 such that ‖xn − yn‖ = dist(a, b). since a0 is compact, there exists a convergent subsequence {xnk } which converges to some x ∈ a0. consider the inequality, ‖ynk − x‖ ≤ ‖ynk − xnk ‖ + ‖xnk − x‖ → dist(a, b). since b is approximatively compact, {ynk} has a convergent subsequence {yn′k } converges to some y ∈ b. since b0 is closed, it implies that y ∈ b0. hence b0 is compact. � now we prove our main results. theorem 3.4. let x be a uniformly convex banach space. let a be a nonempty closed bounded convex subset of x and b be a nonempty closed convex subset of x. let t : a ∪ b → a ∪ b be a relatively nonexpansive map. then there exist a sequence {xn} in a0 and x ∗ ∈ a0 such that (1) xn w −→ x∗ (2) ‖x∗ − t x∗‖ ≤ dist(a, b) + lim inf n ‖t xn − t x ∗‖. proof. by lemma 2.1, a0 is nonempty, hence there exist x0 ∈ a0 and y0 ∈ b0 such that ‖x0 − y0‖ = dist(a, b). for each n ∈ n, define a map tn : a ∪ b → a ∪ b by tn(x) =          1 n y0 + ( 1 − 1 n ) t x, if x ∈ a 1 n x0 + ( 1 − 1 n ) t x, if x ∈ b (3.2) since a and b are convex and t is a relatively nonexpansive map, for each n ∈ n, tn(a) ⊆ b, tn(b) ⊆ a. also for each x ∈ a, y ∈ b, ‖tn(x) − tn(y)‖ ≤ 1 n ‖x0 − y0‖ + ( 1 − 1 n ) ‖t x − t y‖ ≤ ( 1 − 1 n ) ‖x − y‖ + 1 n dist(a, b).(3.3) this implies that for each n ∈ n, tn is a cyclic contraction on a ∪ b. hence by theorem 2.6, for each n ∈ n there exists xn ∈ a such that ‖xn − tnxn‖ = dist(a, b).(3.4) 26 v. sankar raj and p. veeramani hence xn ∈ a0, for each n ∈ n. since a0 is bounded, b0 is also bounded, and t (a0) ⊆ b0, tn(a0) ⊆ b0. also observe that for any x ∈ a0, ‖tnx − t x‖ ≤ 1 n ‖y0 − t x‖ ≤ 1 n δ(b0) → 0 as n → ∞.(3.5) since a0 is a closed bounded convex set, {xn} has a weakly convergent subsequence. without loss of generality, let us assume that {xn} itself weakly converges to x∗, for some x∗ ∈ a0. then xn − t x ∗ w −→ x∗ − t x∗. since ‖ · ‖ is weakly lower semi continuous, and by (3.4), (3.5) we have ‖x∗ − t x∗‖ ≤ lim inf n ‖xn − t x ∗‖ ≤ lim inf n {‖xn − tnxn‖ + ‖tnxn − t xn‖ + ‖t xn − t x ∗‖} ≤ lim inf n { dist(a, b) + 1 n δ(b0) + ‖t xn − t x ∗‖ } ≤ dist(a, b) + lim inf n ‖t xn − t x ∗‖ hence the theorem. � we use the above theorem to prove : theorem 3.5. let x be a uniformly convex banach space. let a be a nonempty closed bounded convex subset of x such that a0 is compact, and b be a nonempty closed convex subset of x. let t : a ∪ b → a ∪ b be a relatively nonexpansive map. then there exist x∗ ∈ a such that ‖x∗ − t x∗‖ = dist(a, b). proof. by theorem 3.4, there exist a sequence {xn} in a0 and x ∗ ∈ a0 such that xn w −→ x∗ and satisfies the inequality ‖x∗ − t x∗‖ ≤ dist(a, b) + lim inf n ‖t xn − t x ∗‖. since a0 is compact, xn converges to x ∗ strongly. the proof will be complete if we show that ‖t xn − t x ∗‖ → 0. claim : ‖t xn − t x ∗‖ → 0 as n → ∞. it is enough to show that ‖t xn − pa(t x ∗)‖ → dist(a, b) as n → ∞. then by theorem 3.1, we have t xn → pb(pa(t x ∗)) = t x∗. consider ‖xn − pbx ∗‖ ≤ ‖xn − x ∗‖ + ‖x∗ − pbx ∗‖ → dist(a, b). since t is relatively nonexpansive we have, ‖t xn − pat x ∗‖ = ‖t xn − t (pbx ∗)‖ ≤ ‖xn − pbx ∗‖ → dist(a, b). this ends the claim and hence the theorem. � theorem 3.6. let x be a strictly convex banach space, let a be a nonempty closed convex subset of x such that a0 is a nonempty compact set and b be a nonempty closed convex subset of x. let t : a ∪ b → a ∪ b be a relatively nonexpansive map. then there exists x∗ ∈ a such that ‖x∗−t x∗‖ = dist(a, b). best proximity pair theorems for relatively nonexpansive mappings 27 proof. since a0 is nonempty and compact, we can construct a sequence of cyclic contraction maps tn : a ∪ b → a ∪ b as in theorem 3.4. we use theorem 2.5 for an existence of best proximity point xn ∈ a0 such that ‖xn − tnxn‖ = dist(a, b). since a0 is compact, {xn} has a convergent subsequence {xnk } such that xnk → x ∗ for some x∗ ∈ a0. as in the proof of theorem 3.5, we can show t xnk → t x ∗. the proof ends by considering the following inequality, ‖x∗ − t x∗‖ ≤ ‖x∗ − xnk ‖ + ‖xnk − tnk xnk ‖ + ‖tnk xnk − t xnk ‖ + ‖t xnk − t x ∗‖ and by observing ‖tnk xnk − t xnk ‖ ≤ 1 nk δ(b0) → 0. � we give below some situations where a0 is a compact subset of a. example 3.7. let a be a unit ball in a strictly convex banach space x and b be a closed convex subset of x with dist(a, b) > 0. then a0 contains atmost one point. proof. clearly a0 is a bounded convex subset of a, moreover by lemma 2.2, a0 is contained in the boundary of a. ie a0 ⊆ ∂a. suppose x1, x2 ∈ a0 with x1 6= x2, then by strict convexity ‖ x1+x2 2 ‖ < 1 which implies that x1+x2 2 /∈ ∂a, a contradiction to the convexity of a0. hence a0 contains atmost one point. � example 3.8. let a be a nonempty closed bounded convex subset of a uniformly convex banach space x and b be a nonempty closed convex subset of x such that span(b) is finite dimensional with dist(a, b) > 0. then a0 and b0 are nonempty compact subsets of a, b respectively. proof. let {yn} be a sequence in b0 then there exists a sequence {xn} in a0 such that ‖xn − yn‖ = dist(a, b). since {xn} is bounded, {yn} is also a bounded sequence in b0. since b is finite dimensional, {yn} has a convergent subsequence. hence b0 is compact. then by remark 3.3, a0 is also a compact set. � corollary 3.9. let a be a nonempty closed bounded convex subset of a uniformly convex banach space x and b be a nonempty closed convex subset of x such that span(b) is finite dimensional with dist(a, b) > 0. let t : a ∪ b → a ∪ b be a relatively nonexpansive map. then there exists x ∈ a such that ‖x − t x‖ = dist(a, b). acknowledgements. the authors would like to thank the referee for useful comments and suggestions for the improvement of the paper. the first author acknowledges the council of scientific and industrial research(india) for the financial support provided in the form of a junior research fellowship to carry out this research work. 28 v. sankar raj and p. veeramani references [1] handbook of metric fixed point theory, edited by w. a. kirk and brailey sims, kluwer acad. publ., dordrecht, 2001. mr1904271 (2003b:47002) [2] a. anthony eldred, w. a. kirk and p. veeramani, proximal normal structure and relatively nonexpansive mappings, studia math. 171, no. 3 (2005), 283–293. [3] a. anthony eldred and p. veeramani, existence and convergence of best proximity points, j. math. anal. appl. 323 (2006), 1001–1006. [4] m. a. khamsi and w. a. kirk, an introduction to metric spaces and fixed point theory, wiley-interscience, new york, 2001. mr1818603 (2002b:46002) [5] w. a. kirk, p. s. srinivasan and p. veeramani, fixed points for mappings satisfying cyclic contractive conditions, fixed point theory 4, no. 1 (2003), 79–89. [6] w. a. kirk, s. reich and p. veeramani, proximal retracts and best proximity pair theorems, numer. funct. anal. optim. 24 (2003), 851–862. [7] d. p. milman and m. s. brodskii, on the center of a convex set, dokl. akad. nauk. sssr (n.s) 59 (1948), 837–840. [8] s. sadiq basha and p. veeramani, best proximity pair theorems for multifunctions with open fiber, j. approx. theory. 103 (2000), 119–129. (2000) [9] s. singh, b. watson and p. srivastava, fixed point theory and best approximation: the kkm-map principle, kluwer acad. publ., dordrecht, 1997. mr1483076 (99a:47087) received december 2007 accepted january 2008 v. sankar raj (sankar rajv@iitm.ac.in) department of mathematics, indian institute of technology madras, chennai 600 036, india. p. veeramani (pvmani@iitm.ac.in) department of mathematics, indian institute of technology madras, chennai 600 036, india. jartkaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 191-201 when is an ultracomplete space almost locally compact? d. jardón and v. v. tkachuk ∗ abstract. we study spaces x which have a countable outer base in βx; they are called ultracomplete in the most recent terminology. ultracompleteness implies čech-completeness and is implied by almost local compactness (≡having all points of non-local compactness inside a compact subset of countable outer character). it turns out that ultracompleteness coincides with almost local compactness in most important classes of isocompact spaces (i.e., in spaces in which every countably compact subspace is compact). we prove that if an isocompact space x is ω-monolithic then any ultracomplete subspace of x is almost locally compact. in particular, any ultracomplete subspace of a compact ω-monolithic space of countable tightness is almost locally compact. another consequence of this result is that, for any space x such that υx is a lindelöf σ-space, a subspace of cp(x) is ultracomplete if and only if it is almost locally compact. we show that it is consistent with zfc that not all ultracomplete subspaces of hereditarily separable compact spaces are almost locally compact. 2000 ams classification: primary: 54h11, 54c10, 22a05, 54d06. secondary: 54d25, 54c25. keywords: ultracompleteness, čech-completeness, countable type, pointwise countable type, lindelöf σ-spaces, splittable spaces, eberlein compact spaces, almost locally compact spaces, isocompact spaces ∗research supported by consejo nacional de ciencia y tecnoloǵıa (conacyt) of mexico grants 94897 and 400200-5-38164-e. 192 d. jardón and v. tkachuk 1. introduction. in 1987 ponomarev and tkachuk introduced in [12] strongly complete spaces as those which have countable outer character in βx. in 1998 romaguera studied the same class calling its spaces cofinally čech-complete; he proved in [13] that a metrizable space has a cofinally complete metric if and only if it is cofinally čech-complete. buhagiar and yoshioka gave in [5] an internal characterization of cofinal čech-completeness and renamed it ultracompleteness; in this paper we will use their term for this class. it is easy to see that ultracompleteness lies between čech-completeness and local compactness so, to check whether or not a space x is ultracomplete, it is natural to deal with the set x0 of points of non-local compactness of x to find out whether x0 is small in some sense. the first results in this direction were obtained in [12]: on the one hand, if x is ultracomplete then x0 has to be a bounded subset of x; on the other hand, if x0 is contained in a compact subset of countable outer character in x (in this paper we will follow [10] calling such spaces x almost locally compact) then the space x is ultracomplete. this places ultracomplete spaces between čech-complete and almost locally compact ones so the natural question is when an ultracomplete space has to be almost locally compact. it was proved in [12] that ultracompleteness coincides with almost local compactness in the class of paracompact spaces. in [10] the same result was established for the class of dieudonné complete spaces as well as for eberlein–grothendieck ones. we develop the methods from [10] to find more classes in which ultracompleteness coincides with almost local compactness. the principal object of our considerations is the class of isocompact spaces, i.e., the spaces in which every countably compact subset is compact. this class is quite a wide one: it contains all sequential dieudonné spaces, all spaces with a gδ-diagonal as well as some spaces dealt with in cp-theory, such as the splittable spaces and the spaces cp(x) for which υx is a lindelöf σ-space. we prove that if x is an isocompact ω-monolithic space then a subspace of x is ultracomplete if and only if it is almost locally compact. consequently, ultracompleteness coincides with almost local compactness in subspaces of the spaces cp(x) such that υx is lindelöf σ. this result gives a positive answer (in a much stronger form) to problem 3.9 from [10]. another consequence is coincidence of ultracompleteness and almost local compactness in subspaces of compact ω-monolithic spaces of countable tightness. we give examples of compact spaces (some of them in zfc and some consistent) which show that neither ω-monolithity nor countable tightness can be omitted here. it is worth mentioning that an easy consequence of results of [10] is that, in any subspace of a first countable compact space, ultracompleteness coincides with almost local compactness (in fact, this coincidence even holds in realcompact spaces of countable pseudocharacter). when is an ultracomplete space almost locally compact? 193 we also prove that the coincidence of ultracompleteness and almost local compactness takes place in splittable spaces and give some easy observations which help to solve problems 3.7 and 3.10 from [10]. 2. notation and terminology. all spaces under consideration are assumed to be tychonoff. the space r is the set of real numbers with its natural topology. for any space x we denote by cp(x) the space of continuous real-valued functions on x endowed with the topology of pointwise convergence. the stone-čech compactification of a space x is denoted by βx. the outer character of a subspace a ⊂ x, denoted by χ(a, x), is the minimal of the cardinalities of all outer bases of a in x. a space x is čech-complete if it is a gδ-set in βx. a topological space x is called ultracomplete if χ(x, βx) ≤ ω. it is clear that any ultracomplete space is also čech-complete. the space x is of (pointwise) countable type if for any compact f ⊂ x (x ∈ x) there exists a compact k ⊂ x such that f ⊂ k (x ∈ k) and χ(k, x) ≤ ω. a space x is called hemicompact if there is a countable family {fn : n < ω} of compact subsets of x such that for any compact k ⊂ x there exists n ∈ ω for which k ⊂ fn. a space x is called scattered if any y ⊂ x has an isolated point. a space x is ω-monolithic if for any y ⊂ x with |y | ≤ ω we have nw(y ) ≤ ω, where nw(y ) is the network weight of the space y . given a space x the family τ (x) is its topology and τ ∗(x) = τ (x) \ {∅}; if x ∈ x then τ (x, x) = {u ∈ τ (x) : x ∈ u}. the tightness t(x) of a space x is the smallest cardinal κ such that for any a ⊂ x and x ∈ a there exists b ⊂ a with |b| ≤ κ such that x ∈ b. a subset a of a space x is bounded in x if every f ∈ cp(x) is bounded on the set a. a space x is called lindelöf σ if it has a compact cover c and a sequence of closed sets f = {fn : n ∈ ω} such that for each set c ∈ c and any u ∈ τ (x) with c ⊂ u there is an f ∈ f such that c ⊂ f ⊂ u . a space x is called splittable if, for each f ∈ rx , there exists a countable n ⊂ cp(x) such that f ∈ n (the closure is taken in rx ). the rest of our terminology is standard and follows [8]. 3. ultracomplete subspaces of isocompact spaces. the following fact, (see [12]), is useful for working with ultracomplete spaces. theorem 3.1. for any space x, the following conditions are equivalent: (i) χ(x, cx) ≤ ω for some compactification cx of the space x; (ii) χ(x, kx) ≤ ω for every compactification kx of the space x; (iii) χ(x, βx) ≤ ω for the stone–čech compactification βx of the space x; (iv) cx \ x is hemicompact for some compactification cx of the space x; (v) kx \ x is hemicompact for every compactificaction kx of the space x; (vi) βx \ x is hemicompact. 194 d. jardón and v. tkachuk a space x is called ultracomplete if it satisfies one of the conditions of theorem 3.1. definition 3.2. call a space x almost locally compact if there is a compact k ⊂ x such that χ(k, x) ≤ ω and x0 = {x ∈ x : x is not locally compact at x} ⊂ k. given a space x let n (x) denote the family of all countably infinite closed and discrete subspaces of x. definition 3.3. a countable family u ⊂ n (x) marks a point x ∈ x, if for any w ∈ τ (x, x) there exists d ∈ u such that the set d∩w is infinite. a point x ∈ x is called marked in x if it is marked by some countable u ⊂ n (x). it is easy to see that if x is marked by a family u, then y = {x} ∪ ( ⋃ u) is a countable set which reflects the non-local countable compactness of the space x at the point x. the proof of the following statement is an easy exercise. proposition 3.4. if every point of a countable subspace a of a space x is marked then all points of the set a are marked as well. in particular, if t(x) = ω then the set of all marked points of x is closed in x. it is clear that, in any space, only the points of non-local countable compactness can be marked. if x is a space and a point x ∈ x has a countable local base {bn : n ∈ ω} such that, for every n ∈ ω the set bn is not countably compact then, choosing a countably infinite closed discrete dn ⊂ bn for each n ∈ ω we obtain a family u = {dn : n ∈ ω} which marks the point x. we push further this idea in the following theorem to characterize the points of non-local countable compactness in a reasonably general class of spaces. theorem 3.5. suppose that x is a space of pointwise countable type such that t(k) ≤ ω for any compact k ⊂ x. then a point x ∈ x is marked in x if and only if x is not a point of local countable compactness of x. proof. we already saw that only sufficiency must be proved so assume that x ∈ x is not a point of local countable compactness in x. by proposition 3.4, the set m of all marked points of x is ω-closed in x, i.e., a ⊂ m for any countable a ⊂ m ; suppose that x ∈ x \ m . the space x being of pointwise countable type, there is a compact k ⊂ x such that x ∈ k and χ(k, x) ≤ ω. the set f = m ∩ k is ω-closed in k; since t(k) ≤ ω, the set f is closed in k and hence in x. therefore k \f is an open neighbourhood of the point x in the space k; this makes it possible to find a closed k′ ⊂ k such that k′ is a gδ-subset of k and x ∈ k′ ⊂ k \ f . it is straightforward that χ(k′, x) ≤ χ(k′, k) · χ(k, x) ≤ ω so we can find an outer base {bn : n ∈ ω} of the set k ′ in x such that bn+1 ⊂ bn for all n ∈ ω. since x ∈ bn, it is possible to choose a countably infinite closed discrete dn ⊂ bn for any n ∈ ω. we claim that the family u = {dn : n ∈ ω} ⊂ n (x) marks some point of k′. indeed, if this is not so, then every y ∈ k′ has a neighbourhood uy when is an ultracomplete space almost locally compact? 195 such that uy ∩ dn is finite for any n ∈ ω. since k ′ is compact, there exist y1, . . . , yk ∈ k ′ such that k′ ⊂ u = uy1 ∪ . . . ∪ uyk . there is n ∈ ω such that bn ⊂ u and hence bn+1 ⊂ u . therefore dn+1 ⊂ u while uyi ∩ dn+1 is finite for every i ≤ k; this contradiction shows that some y ∈ k′ is marked by u. however, all marked points are in m which does not meet k′; this final contradiction proves that x is marked in x. � example 3.6. let ξ be a free ultrafilter on ω; the space x = ω ∪{ξ} (with the topology induced from βω) is countable and non-locally countably compact at ξ. however, if ξ is a p -point of βω\ω then ξ is not marked in x. therefore, under ch, there is a countable space whose set of points of non-local (countable) compactness does not coincides with the set of marked points of x. proof. since p -points in βω \ ω exist under ch, it suffices to show that ξ is not marked in x if ξ is a p -point. suppose that d = {dn : n ∈ ω} ⊂ n (x) marks the point ξ. then dn /∈ ξ for any n ∈ ω and hence un = (βω \ ω) \ dn is an open neighbourhood of ξ in βω \ ω (the bar denotes the closure in the space βω). since ξ is a p -point, there is a clopen w ⊂ βω\ω with ξ ∈ w ⊂ ⋂ n∈ω un. choose a set v ∈ ξ such that v ∩ (βω \ ω) ⊂ w ; it is straightforward that v ∪ {ξ} is a clopen neighbourhood of ξ in x such that dn ∩ v is finite for any n ∈ ω. this contradiction shows that the point ξ is not marked in x. � recall that a space x is called isocompact if every countably compact subspace of x is compact. corollary 3.7. if x is an isocompact space of pointwise countable type then a point x ∈ x is marked if and only if x is not a point of local compactness of x. proof. by isocompactness of x, if x is not a point of local compactness of x then x is not a point of local countable compactness of x. besides, any compact k ⊂ x has countable tightness—it is an easy exercise that any isocompact compact space has countable tightness. thus we can apply theorem 3.5 to conclude that x is marked. � theorem 3.8. if x is an isocompact ω-monolithic space then a subspace y ⊂ x is ultracomplete if and only if it is almost locally compact. proof. we only must prove necessity so assume that y ⊂ x is ultracomplete. the space y is of pointwise countable type being čech-complete so the set m of all points at which y is not locally compact coincides with the set of marked points of y by corollary 3.7. if m is not countably compact then it has a countably infinite closed discrete subspace s. any point s ∈ s is marked by a countable family of ds ⊂ n (y ). the family d = ⋃ {ds : s ∈ s} is countable; since any closed subspace of an ultracomplete space is ultracomplete, the space e = cly ( ⋃ d) is ultracomplete; besides, nw(e) ≤ ω because y is ω-monolithic. now, even in čech-complete spaces the weight and the network weight coincide so w(e) = nw(e) = ω. 196 d. jardón and v. tkachuk therefore e is a metrizable space and hence the subspace e0 of points at which e is not locally compact, is a compact space by [12, corollary 5]. observe that any s ∈ s is marked in y by the family ds and it is clear that the family ds marks the point s in the space e as well. therefore s ∈ e0 for all s ∈ s which shows that s ⊂ e0 is an infinite closed discrete subset of e0, a contradiction with compactness of e0. thus m is countably compact and hence compact by isocompactness of y . finally observe that y is of countable type so there is a compact k ⊂ y for which m ⊂ k and χ(k, y ) ≤ ω, i.e., y is almost locally compact. � to apply theorem 3.8, let us look at some well-known classes of isocompact spaces. the following fact gives a positive answer to problem 3.9 of [10]. corollary 3.9. suppose that x is a space such that υx is lindelöf σ. then a subspace y ⊂ cp(x) is ultracomplete if and only if it is almost locally compact. in particular, this is true if cp(x) is a lindelöf σ-space or x is pseudocompact. proof. it is known that, for such x, the space cp(x) is ω-monolithic and isocompact (see [2, proposition iv.9.10] and [2, theorem ii.6.34]. � corollary 3.10. if x is a compact ω-monolithic space of countable tightness then any ultracomplete y ⊂ x is almost locally compact. in particular, a subspace of a corson compact space is ultracomplete if and only if it is almost locally compact. proof. the space x is fréchet–urysohn and hence isocompact so theorem 3.8 does the rest. � corollary 3.10 shows that it is natural to ask whether the same result can be proved if we omit ω-monolithity of the space x. we will show later that it is impossible, at least, consistently. however, the conclusion of corollary 3.10 remains valid if we strengthen countable tightness of x to first countability. in fact, the following much more general statement is true. proposition 3.11. if x is a hereditarily realcompact space (in particular, if x is a realcompact space of countable pseudocharacter) then a subspace y ⊂ x is ultracomplete if and only if it is almost locally compact. proof. it suffices to observe that any ultracomplete realcompact space is almost locally compact by [10, theorem 2.4]. � another important class of isocompact spaces is given by splittable spaces. recall that a space x is splittable if, for any a ⊂ x there is a continuous map f : x → rω such that a = f −1f (a). the class of splittable spaces is isocompact because every pseudocompact splittable space is compact and metrizable (see [4, theorem 3.2]). however, a splittable space need not be ω-monolithic so we cannot apply theorem 3.8 directly. theorem 3.4 of [3] shows that if x is a splittable space of non-measurable cardinality then x is hereditarily realcompact so proposition 3.11 works to establish that any when is an ultracomplete space almost locally compact? 197 ultracomplete subspace of x is almost locally compact. however, the same result can be proved without assuming anything about the cardinality of x. proposition 3.12. for any splittable space x, a subspace y ⊂ x is ultracomplete if and only if y is almost locally compact. proof. since splittability is hereditary, it suffices to prove that any splittable ultracomplete space y is almost locally compact. it follows from [4, corollary 2.15] that y is first countable. let m be the set of points of non-local compactness of y . if m is not countably compact then fix a countably infinite closed discrete d ⊂ m and apply corollary 3.7 to find a family ds ⊂ n (y ) which marks the point s for any s ∈ d; consider the family d = ⋃ s∈d ds. the set z = ⋃ d is countable; since y is first countable, the set f = z has cardinality at most c. splittable spaces of cardinality at most c have a weaker second countable topology [4, corollary 2.19]. therefore f is an ultracomplete realcompact space which implies that f is almost locally compact by [10, theorem 2.4]. in particular, the set f0 of the points of non-local compactness of f is compact. however, d marks all points of d in f ; this shows that d ⊂ f0 is a closed and discrete subspace of f0 which is a contradiction. therefore m is countably compact and hence compact (and metrizable). the space y being of countable type, there is a compact k ⊂ y such that m ⊂ k and χ(k, y ) ≤ ω, i.e., y is almost locally compact. � example 3.13. it was proved in [6] that the space y = ωω1 is ultracomplete and has no points of local compactness. this example disproves many hypothesis showing, in particular, that (1) a first countable ultracomplete space need not be almost locally compact (compare with proposition 3.11); (2) the restriction on tightness cannot be omitted in corollary 3.10; (3) an ultracomplete subspace of a σ-product of real lines need not be almost locally compact (this draws a limit for possible generalizations of the statement on corson compact spaces in corollary 3.10); (4) there is a space x such that x ω is lindelöf while some ultracomplete subspace of cp(x) is not almost locally compact (therefore the σ-property cannot be omitted in corollary 3.9); (5) there exists a homogeneous non-locally compact ultracomplete space. this gives a negative answer to problem 3.10 of [10] being of interest also because any ultracomplete topological group has to be locally compact—this was proved in [11, corollary 2.13]. proof. it is evident that the space y is first countable and not almost locally compact; besides, y is a subspace of (ω1+1) ω which is an ω-monolithic compact space. since ω1 embeds in a σ-product of real lines, so does ω ω 1 . this proves (1)–(3). it is known that any σ-product of real lines is homeomorphic to a space cp(x) where x is the lindelöfication of an uncountable discrete space. since x ω is lindelöf (see [2, proposition iv.2.21]), we also have (4). finally, a famous theorem of dow and pearl [7, theorem 2] says that the countably 198 d. jardón and v. tkachuk infinite power of any first countable zero-dimensional space is homogeneous. since the space ω1 is zero-dimensional and first countable, we conclude that ωω1 is homogeneous; this settles (5). � example 3.14. it is consistent with zfc that there is a hereditarily separable compact space x such that some ultracomplete y ⊂ x is not almost locally compact. this shows, in a much stronger form, that ω-monolithity cannot be omitted in corollary 3.10. proof. fedorchuk proved in [9] that it is consistent with zfc that there is a hereditarily separable compact space x of cardinality 2c without non-trivial convergent sequences. it is easy to see that any infinite scattered compact space has a convergent sequence so x is not scattered; fix a non-empty closed dense-in-itself set p ⊂ x. if d is a countable dense subset of p then y = p \d is dense in p so it does not have points of local compactness. another easy fact is that any countable space without convergent sequences is hemicompact; thus d is hemicompact so we can apply theorem 3.1 to see that y is an ultracomplete space without points of local compactness. therefore y is not almost locally compact. � it follows from [12, lemma 4] that any ultracomplete space without points of local compactness is pseudocompact. so far, all examples of such ultracomplete spaces were countably compact. we will see that the following statement (which seems to be of interest in itself) implies that there are zfc examples of ultracomplete non-countably compact spaces without points of local compactness. theorem 3.15. the space {0, 1}c has a dense countable subspace without nontrivial convergent sequences. proof. it was proved in [1, theorem 2.3] that {0, 1}c has a dense countable irresolvable subspace d. consider the set e = {x ∈ d: there is a non-trivial sequence in d which converges to x}. if e is dense in d then, enumerating the relevant countable family of convergent sequences, it is standard to construct disjoint a, b ⊂ d such that the sets a ∩ s and b ∩ s are infinite for any sequence s from this family. an immediate consequence is that both a and b are dense in d which is a contradiction. thus there is a non-empty open set u in the space d with u ∩ e = ∅; therefore u has no non-trivial convergent sequences. if v is open in {0, 1}c and v ∩ d = u then u is dense in v . it is easy to find an open set w in the space {0, 1}c such that w ⊂ v and w is homeomorphic to {0, 1}c. it is immediate that u ′ = u ∩ w is dense in w ; identifying w with {0, 1}c, we conclude that u ′ is the promised countable dense subspace of {0, 1}c. � example 3.16. there exists a dense subspace x of the space {0, 1}c which is ultracomplete, non-countably compact and has no points of local compactness. proof. apply theorem 3.15 to fix a countable dense set d ⊂ {0, 1}c which has no convergent sequences; we can assume, without loss of generality, that the when is an ultracomplete space almost locally compact? 199 point u ∈ {0, 1}c whose all coordinates are equal to zero, belongs to d. it is easy to see that d is hemicompact so the space x = {0, 1}c\d is ultracomplete by theorem 3.1. the density of x in {0, 1}c is evident; it follows from density of d in {0, 1}c that x has no points of local compactness. to see that x is not countably compact, consider, for any α < c, the point uα ∈ {0, 1} c defined by uα(α) = 1 and uα(β) = 0 for any β ∈ c \ {α}. it is easy to see that the set a = {uα : α < c} ∪ {u} is homeomorphic to the one-point compactification of a discrete space of cardinality c. in particular, any countably infinite subset of a \ {u} is a sequence which converges to u. since the set d has no convergent sequences, the set d′ = d ∩ a is finite; it is straightforward that the set a \ d′ is an infinite (even uncountable) closed discrete subspace of x so x is not countably compact. � we will finish this paper with a couple of observations on the set x1 of points of first countability of a compact space x. it was proved in [10, example 2.17] that, in a countable eberlein–grothendieck space x, the set of points of non-local compactness of x can be compact without x being ultracomplete. since little is known yet on ultracompleteness of x1 even in scattered eberlein compact spaces, it is worth mentioning that such a situation is not possible in the space x1 whenever x is a scattered compact space. proposition 3.17. if x is a scattered compact space and the set m of all points of non-local compactness of x1 is compact, then x1 is ultracomplete. proof. since x \ x1 is lindelöf by [10, proposition 2.19], the set x1 is of countable type, i.e., every compact subset of x1 is contained in a compact subspace of countable outer character in x1. in particular, this is true for m so x1 is almost locally compact and hence ultracomplete by [12, lemma 9]. � the following result gives a positive answer to problem 3.7 from [10]. proposition 3.18. if x is a compact space then l(x \ y ) ≤ c for every y ⊂ x1. proof. if u ⊂ τ (x) is a cover of z = x \ y then k = x \ ( ⋃ u) is a compact set contained in x1. we have χ(k) ≤ ω so |k| ≤ c; an easy consequence is that k is the intersection of ≤ c-many open subsets of the space x. hence the set u = ⋃ u is a union of ≤ c-many compact subsets of x. therefore we can find u′ ⊂ u such that |u′| ≤ c and ⋃ u′ = u ⊃ z. thus the family u′ is a subcover of z of cardinality at most c. � 4. open problems. the are quite a few interesting open questions on coincidence of ultracompleteness and almost local compactness. the list below shows that the topic of this paper still has a strong potential for development. problem 4.1. does there exist in zfc a compact space x of countable tightness such that some ultracomplete y ⊂ x is not almost locally compact? 200 d. jardón and v. tkachuk problem 4.2. does there exist an isocompact space x such that some ultracomplete subspace of x is not almost locally compact? problem 4.3. must any non-empty isocompact ultracomplete space have points of local compactness? problem 4.4. let x be a sequential compact space. is it true that every ultracomplete y ⊂ x is almost locally compact? problem 4.5. let x be a fréchet–urysohn compact space. is it true that every ultracomplete y ⊂ x is almost locally compact? problem 4.6. let x be a space with a gδ-diagonal. is it true that any ultracomplete subspace of x is almost locally compact? problem 4.7. is there a zfc example of a countable space x with some point of non-local compactness which is not marked in x? problem 4.8. let x be a homogeneous ultracomplete space without points of local compactness. must x be countably compact? problem 4.9. let x be a scattered compact space. is it true that every ultracomplete y ⊂ x is almost locally compact? problem 4.10. let x be a scattered compact space of countable tightness. is it true that every ultracomplete y ⊂ x is almost locally compact? what happens if x is fréchet–urysohn or sequential? references [1] o. t. alas, m. sanchis, m. g. tkachenko, v. v. tkachuk and r. g. wilson, irresolvable and submaximal spaces: homogeneity versus σ-discreteness and new zfc examples, topology appl. 107 (2000), 259–273. [2] a. v. arkhangel’skii, topological function spaces, kluwer academic publishers, 1992. [3] a. v. arkhangel’skii, a survey of cleavability, topology appl. 54(1993), 141–163. [4] a. v. arhangel’skii and d. b. shakhmatov, approximation by countable families of continuous functions (in russian), proc. of i.g. petrovsky’s seminar 13 (1988), 206– 227. [5] d. buhagiar and i. yoshioka, ultracomplete topological spaces, acta math. hungar. 92 (2001), 19–26. [6] d. buhagiar and i. yoshioka, sums and products of ultracomplete topological spaces, topology appl. 122 (2002), 77–86. [7] a. dow and e. pearl, homogeneity in powers of zero-dimensional first countable spaces, proc. amer. math. soc. 125:8 (1997), 2503–2510. [8] r. engelking, general topology, heldermann verlag, 1989. [9] v. v. fedorchuk, on the cardinality of hereditarily separable compact haudorff spaces, soviet math. dokl. 16:3 (1975), 651–655. [10] d. jardón and v. v. tkachuk, ultracompleteness in eberlein-grothendieck spaces, bol. soc. mat. mex. (3)10 (2004), 209–218. [11] m. lópez de luna and v. v. tkachuk, čech-completeness and ultracompleteness in ”nice” spaces, comment. math. univ. carolinae 43:3 (2002), 515–524. [12] v. i. ponomarev and v. v. tkachuk, the countable character of x in βx compared with the countable character of the diagonal in x×x (in russian), vestnik mosk. univ. 42:5 (1987), 16–19. when is an ultracomplete space almost locally compact? 201 [13] s. romaguera, on cofinally complete metric spaces, q&a in gen. topology 16 (1998), 165–169. received may 2005 accepted september 2005 daniel jardón arcos (jardon60@hotmail.com) departamento de matemáticas, universidad autónoma metropolitana, san rafael atlixco, 186, col. vicentina, iztapalapa, c.p. 09340, méxico d.f. vladimir tkachuk (vova@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, san rafael atlixco, 186, col. vicentina, iztapalapa, c.p. 09340, méxico d.f. gmaynezagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 223-237 a survey on wallman bases adalberto garćıa-máynez c. abstract. wallman bases are frequently used in compactification processes of topological spaces. however, they are also related with quasi–uniform structures and they are useful to characterize some topological properties. we present a brief survey on the subject which supports these statements. 2000 ams classification: 54c25, 54c45, 54d35, 54d80. keywords: wallman basis, annular basis, ultrafilter, perfect extension, wallman type, regular wallman, equivalent compactifications, cover uniformity basis, quasi–uniformity, transitive, totally bounded, symmetric, point symmetric, locally symmetric. 1. historical background among compactification methods in topology, we bring out two of them, which are perhaps the most useful: • wallman’s method of ultrafilters. • completion of totally bounded uniform spaces. the main advantadge of the method of ultrafilters is its generality: it may be applied to any t1-space. wallman applied this method for the first time in 1938 ([17]) and he proved that any t1–space x has a t1–compactification ωx which coincides with the stone–čech compactification if x is normal. the concept of uniform space was introduced by a. weil, in 1937 ([18]) and he proved, among other things, that every uniform space (x, u) has unique uniform completion (unique up to uniform equivalences). the notion of totally bounded uniform spaces was formally introduced by n. bourbaki in 1940, although it was used implicitly by weil in the well known characterization of compactness in uniform spaces: a uniform space (x, u) is compact (i.e., the topology tu induced by u is compact) iff (x, u) is complete and satisfies an extra condition which is precisely total boundedness. 224 a. garćıa-máynez since total boundedness is not lost under uniform extensions, we conclude that the uniform completion of a totally bounded uniform t1–space x is a compactification of x. the method of wallman can be applied in smaller families of closed sets and produce a compactification in exactly the same way. one of the requirements is that the family of complements of the given closed sets is a basis for the topology of the space. 2. annular bases and quasi–uniformities every topological space (x, t ) has many bases, i.e., subfamilies b of t such that every u ∈ t can be expressed as a union of some members of b. annular bases are not so numerous: a basis b of t is annular if it satisfies two conditions: i) ∅ ∈ b and x ∈ b. ii) b, b′ ∈ b implies that b ∩ b′ ∈ b and b ∪ b′ ∈ b. it is possible that t is the only annular basis of itself. a less strong condition is that t has a minimal annular basis, i.e., an annular basis which is contained in every other annular basis. this happens in any locally compact 0–dimensional space. a successful generalization of uniform spaces was found in 1960 ([1]). only completely regular spaces admit uniform structures which produce their topology. however, any topological space admits compatible quasi–uniform structures. we briefly state the main definitions. a quasi–uniformity u on a set x is a filter in x × x satisfying the following properties: i) the diagonal △ (x) = {(x, x) | x ∈ x} is contained in each member of u, i.e., each u ∈ u is a reflexive relation in x, and ii) whenever u ∈ u, there exists an element v ∈ u such that v ◦ v ⊆ u. a connector on x is, by definition, a reflexive relation on x. therefore, every element of a quasi–uniformity is a connector and the family of all connectors on x is the largest quasi–uniformity on x. each quasi–uniformity u on a set x determines a topology tu on x according to the following definition: ∗) a subset a of x belongs to tu iff for every x ∈ a, we may find an element ux ∈ u such that: ux (x) = {y ∈ x | (x, y) ∈ ux} ⊆ a. a quasi–uniformity u on a topological space (x, t ) is compatible (with t ) if t = tu . a connector u on a topological space (x, t ) is a neighbornet if for each x ∈ x, u (x) = {y ∈ x | (x, y) ∈ u} is a t –neighborhood of x. a sequence u1, u2, . . . of neighbornets of a topological space (x, t ) is normal if for every n ∈ n, we have u2n+1 = un+1 ◦ un+1 ⊆ un. a neighbornet u is normal if wallman bases 225 it belongs to a normal sequence of neighbornets. clearly, if u is a compatible quasi–uniformity on a topological space (x, t ), each member u ∈ u is a normal neighbornet of (x, t ). in an arbitrary topological space (x, t ) there exists at least one compatible quasi–uniformity, the pervin quasi–uniformity (see [9]). among all compatible quasi–uniformities on a topological space (x, t ), there exists one, the fine quasi–uniformity, which contains every other compatible quasi–uniformity: take all normal neighbornets on (x, t ). spaces where the fine quasi–uniformity is the only compatible quasi-uniformity are characterized in [8]. such spaces have only one annular basis, the topology itself. a basis η for a quasi–uniformity u is simply a filterbase in x × x which generates u. a quasi–uniformity basis η is transitive if each element n ∈ η is transitive (i.e., n ◦ n ⊆ n) and symmetric if each n ∈ η is symmetric (i.e., n = n−1). a connector a on a set x is totally bounded if there exists a finite cover {e1, e2, . . . , es} of x such that ei × ei ⊆ a for every i = 1, 2, . . . , s. a quasi–uniformity u on x is totally bounded if each member of u is totally bounded. a quasi–uniformity u on x is transitive (resp., a uniformity) if u has a transitive (resp., a symmetric) basis. the pervin quasi–uniformity on a topological space is both transitive and totally bounded. an important fact observed by romaguera and sánchez granero is the following: there exists a bijection between the family of all transitive and totally bounded compatible quasi–uniformities on a topological space (x, t ) and the family of all annular bases of t . this bijection, which respects inclusions, is described in [10]. of course, the quasi-uniformity corresponding to t is the pervin quasi–uniformity. 3. wallman bases a wallman basis b on (x, t ) is an annular basis of t satisfying the extra condition: wb) if x ∈ b ∈ b, there exists a closed set h such that x ∈ h ⊆ b and such that x − h ∈ b. whenever we restrict ourselves to a definite basis b of a topological space (x, t ), by a cobasic set we simply mean a closed set h ⊆ x such that x − h ∈ b. c (b) denotes the family of all cobasic sets with respect to b. so, condition wb) is equivalent to say that every basic set b ∈ b is a union of cobasic sets h ∈ c (b). a quasi–uniformity u on a set x is said to be point symmetric if for each u ∈ u and each x ∈ x, there exists a symmetric connector vx ∈ u such that vx (x) ⊆ u (x) (i.e., (x, y) ∈ vx implies (x, y) ∈ u). in the correspondence between transitive totally bounded compatible quasi– uniformities of a topological space (x, t ) and the family of annular bases of t , if we restrict ourselves to point–symmetric quasi–uniformities, we have a bijection with the family of all wallman bases of (x, t ) (see [10]). 226 a. garćıa-máynez not every topological space (x, t ) admits wallman bases. however, a very mild restriction on t guarantees their existence: theorem 3.1. a topological space (x, t ) admits wallman bases iff t itself is a wallman basis, i.e., iff every open set u ∈ t is a union of closed sets. the condition in the theorem above is equivalent to the following property: r0) if x, y ∈ x and y ∈ cℓ ({x}), then cℓ ({y}) = cℓ ({x}). spaces satisfying condition r0) are simply called r0–spaces. it is obvious that every t1–space (i.e., spaces where every finite set is closed) is r0. in fact, the condition t1 is equivalent to the conditions r0 and: t0) if x, y ∈ x and y ∈ cℓ ({x}) and x ∈ cℓ ({y}), then x = y. a wallman basis b on (x, t ) is regular if whenever x ∈ b ∈ b, there exist elements d ∈ b and h ∈ c (b) such that x ∈ d ⊆ h ⊆ b. a quasi–uniformity u on x is locally symmetric if for each u ∈ u and x ∈ x, there exists a symmetric element vx ∈ u such that v 2 x (x) ⊆ u (x). going back to the correspondence theorem, we obtain: theorem 3.2. the quasi–uniformity ub which corresponds to a wallman basis b of (x, t ) is locally symmetric iff b is regular. normality is perhaps the most widely used property in general topology; it may understood in at least four different ways: a) topological spaces, b) open covers, c) connectors and d) wallman bases. of course, the first is the most important: a topological space is normal if for every pair h k of disjoint closed sets, there exist disjoint open sets uh, uk such that h ⊆ uh and k ⊆ uk. the last one is closely related to this: a wallman basis b of a topological space (x, t ) is normal if for every pair h, k of disjoint cobasic sets, there exist disjoint basic sets bh, bk ∈ b such that h ⊆ bh and k ⊆ bk. combining these two definitions, we have: an r0–space (x, t ) is normal iff the wallman basis t is normal. however a space with a normal wallman basis may not be normal. in fact, every locally compact hausdorff space (x, t ) has a normal wallman basis: define b as the family of open sets b such that cℓ (b) or x − b is compact. more generally, if (x, t ) is a hausdorff space and the family b of open sets v ∈ t with compact boundary is a basis of t , then b is a normal wallman basis of (x, t ). these spaces were called peripherally compact by gordon t. whyburn and it is quite clear that every locally compact hausdorff space is peripherally compact but the converse may not be true. there are many examples in the literature of locally compact hausdorff spaces which are not normal (see [12], ex. 106 and [2], ex. 2, p.239). on other side, every completely regular space (x, t ) has a normal wallman basis, the collection of cozero sets: v ⊆ x is a cozero set if there exists a continuous function: f : (x, t ) → (r, td) (td usual topology of r) wallman bases 227 such that v = f −1 (r − {0}) = {x ∈ x | f (x) 6= 0}. a remarkable property of normal wallman basis is the existence of scales between any pair of disjoint cobasic sets: definition 3.3. let a, b be disjoint subsets of a topological space (x, t ) and let d = { m 2n | m, n ∈ n } be the family of dyadic rationals in the open unit interval (0, 1). a scale between a and b is a map ϕ : d → p (x) such that whenever d1, d2 ∈ d, d1 < d2, we have cℓx (ϕ (d1)) ⊆ intx (ϕ (d2)). every scale ϕ : d → p (x) determines a continuous map f : x → [0, 1 ] from x to the closed unit interval such that: f −1 (0) = ∩ {ϕ (r) | r ∈ d} and f −1 (1) = x − ∪ {ϕ (r) | r ∈ d} . the map f is defined exactly as in the proof of urysohn’s lemma: f (x) = { 0 if x ∈ ∩ {ϕ (r) | r ∈ d} sup {r ∈ d | x /∈ ϕ (r)} otherwise . if b is a normal wallman basis of a topological space and if h, k are disjoint cobasic sets, apply the normality of b and obtain two disjoint basic sets bh, bk such that h ⊆ bh and k ⊆ bk. proceed to define the scale ϕ by induction on the power of two in the denominator of a dyadic rational m 2k ∈ d, starting with ϕ ( 1 2 ) = bh. inductively, if ϕ has already been defined in: dn = { m 2k ∈ d | k ≤ n } and in such a way that for each d ∈ dn we have ϕ (d) ∈ b and h ⊆ ϕ (d) ⊆ fd for some cobasic set fd contained in x−k and such that whenever d1, d2 ∈ dn, d1 < d2, there exists a cobasic set f such that: (3.1) ϕ (d1) ⊆ f ⊆ ϕ (d2) , we proceed to define ϕ on dn+1: let m 2n+1 ∈ dn+1, where m is odd and 1 < m < 2n+1 − 1. then d1 = m − 1 2n+1 and d2 = m + 1 2n+1 both belong to dn. by the induction hypothesis, there exists a cobasic set f such that (3.1) holds. then f and x − ϕ (d2) are disjoint cobasic sets. since b is normal, there exist disjoint basic sets b, b′ such that f ⊆ b and x − ϕ (d2) ⊆ b ′. define ϕ ( m 2n+1 ) = b. if m = 1, we find disjoint basic sets b, b′ containing h and x − ϕ ( 1 2n ) and define ϕ ( 1 2n ) = b. if m = 2n+1 − 1, we find disjoint 228 a. garćıa-máynez basic sets b, b′ containing f 2n−1 2n and k and define ϕ ( 2n+1 − 1 2n+1 ) = b. this completes the inductive construction. a very important consequence of this result, is that every topological space which admits a normal wallman basis is completely regular. hence we have the following characterization: theorem 3.4. a topological space (x, t ) is completely regular iff t admits a normal wallman basis. in fact, every completely regular infinite t2–space of weight α has a normal wallman basis of cardinality α. completely regular hausdorff spaces, and only them, admit hausdorff compactifications (see [3], 3.30.3). every wallman basis of a t1–space (x, t ) yields a t1–compactification of (x, t ). to construct such a compactification, we define the concept of wallman ultrafilter: sea b be a wallman basis of x and let c (b) be the family of cobasic sets. a non-empty subfamily ξ of c (b) is a wallman ultrafilter if ξ satisfies the following conditions: 1) each element h ∈ ξ is non-empty. 2) for every pair of elements h, k ∈ ξ, h ∩ k also belongs to ξ. 3) if h ∈ ξ and h ⊆ k ∈ c (b), then k ∈ ξ. 4) an element k ∈ c (b) belongs to ξ iff k ∩ h 6= ∅ for every h ∈ ξ. observe every point p ∈ x determines a wallman ultrafilter, namely ξp = {h ∈ c (b) | p ∈ h}. the collection of wallman ultrafilters (respect to b) is denoted as x (b). there is a natural map v : x → x (b) which assigns to every p ∈ x its fixed ultrafilter ξp. for every a ⊆ x we define a subset a ∗ of x (b) by means of the formula: a∗ = {ξ ∈ x (b) | for some f ∈ ξ, f ⊆ a} . this operator a 7→ a∗ respects inclusions and for every pair of subsets c, d of x, we have: (c ∩ d) ∗ = c∗ ∩ d∗. the formula (c ∪ d) ∗ = c∗ ∪ d∗ is also valid provided that c and d both belong to b ∪ c (b) (see [3]). the family b∗ = { b∗ | b ∈ b } is an annular basis for a compact t1–topology t ∗ of x (b). the natural map v : (x, t ) → ( x (b) , t ∗ ) is then injective, continuous, open onto its range v (x) and with v (x) dense in x (b). therefore, the pair (v, x (b)) is a t1–compactification of x, called the wallman compactification of x with respect to the basis b. wallman bases 229 4. topological and embedding properties a general problem we may set ourselves is the following: given a topological property p and an embedding property j, find conditions on a wallman basis b of (x, t ) which insure that ( x (b) , t ∗ ) has property p or that x is j–embedded in x (b). we give first some definitions: definition 4.1. two wallman bases of (x, t ) are equivalent if every pair of disjoint cobasic sets with respect to any one of the bases, are contained in disjoint cobasic sets with respect to the other. it is not difficult to prove that if b is a normal wallman basis of a tychonoff space x and if there exists a pair of non–compact disjoint cobasic sets, then there exists a normal wallman basis b ′ for the same topology which is not equivalent to b. definition 4.2. an extension z of a space x is perfect if whenever we have a separation in x: x − k = u ∪ v, k closed in x, u, v, open and disjoint we also have a separation in z: x − cℓz (k) = u1 ∪ v1, u1 ⊇ u, v1 ⊇ v, u1, v1, open and disjoint in z. a simple characterization of perfect extensions can be given if we use the operator e : tx → tz between the topologies of x and z, where e (u) = z − cℓz (x − u) . then z is a perfect extension of x iff for every pair u, v of disjoint open sets in x, we have e (u ∪ v) = e (u) ∪ e (v). definition 4.3. if z is an extension of x, we say x is locally connected in z if z has a basis b such that for every b ∈ b, b ∩ x is a connected subset of x. it is easy to see that if x is locally connected in z, then z is a perfect extension of x. it is well known that if x is a tychonoff space, then the stone–čech compactification β x is a perfect extension of x and x is c∗–embedded in β x. another important example of a perfect compactification is the wallman compactification of a peripherally compact hausdorff space x, where b is the family of open subsets of x with compact boundary. we state a few classical results in this context: theorem 4.4. let b a wallman basis of a t1–space (x, t ). then ( x (b) , t ∗ ) is a hausdorff space iff b is normal. 230 a. garćıa-máynez theorem 4.5. if b is a normal wallman basis of a tychonoff space (x, t ), then x is c∗–embedded in x (b) iff b is equivalent to the cozero wallman basis of (x, t ). the following results were proved by myself: ([5]) theorem 4.6. if b is a wallman basis of a t1–space (x, t ), then x (b) is a perfect compactification of x iff b satisfies the following condition: ∗) if k ⊆ b, where k ∈ c (b) and b ∈ b and if l is an open set in x such that b ∩ f r (l) = ∅, then there exists a basic set bl ∈ b such that k ∩ l ⊆ bl ⊆ b ∩ l. condition ∗) is clearly satisfied if every clopen subset of a basic set is also a basic set. theorem 4.7. if b is a wallman basis of a t1–space (x, t ) satisfying the property: (4.1) b ∈ b, e component of b ⇒ e ∈ b, then x is locally connected in x (b) iff b satisfies the following condition: ∗∗) if k ⊆ b, where k ∈ c (b) and b ∈ b, then there exists a finite collection c1, c2, . . . , cn of connected subsets of x such that: k ⊆ n⋃ i=1 ci ⊆ b. we say b is locally connected if it satisfies the property 4.1. condition ∗∗) is satisfied if x is locally connected, hausdorff and peripherally compact and x has only a finite number of components. before stating two more results, we need a definition: definition 4.8. a hausdorff compactification z of a space x is of wallman type if x has a wallman basis b such that z and x (b) are equivalent compactifications of x. we have then: theorem 4.9 ([14]). if z is a compact metrizable space and if x is dense in z, then z is a wallman type compactification of x. theorem 4.10 ([5]). if z is a compact and hausdorff and if x is gδ–dense in z (i.e., every non-empty gδ set in z intersects x) then z is a wallman type compactification of x. before going on, we need some more definitions: definition 4.11. a topological space (x, t ) is s–metrizable if there exists a metric d on x inducing t and satisfying the following property: s) for every ε > 0, there exists a finite cover c1, c2, . . . , cn of x consisting of connected sets of diameter < ε. wallman bases 231 every s–metrizable space is locally connected, separable and has only a finite number of connected components. we characterize now s–metrizability: theorem 4.12. the following three properties of a topological space (x, t ) are equivalent: a) x is s–metrizable. b) x has a perfect locally connected metrizable compactification z ([4]). c) t has a countable normal locally connected wallman basis consisting of open domains. (this equivalence is easily obtainable from results of [14]). another topological concept which leads to many open problems is weak pseudocompactness: definition 4.13. a tychonoff space (x, t ) is weakly pseudocompact if (x, t ) has a hausdorff compactification in which x is gδ–dense. a non-compact locally compact t2–lindelöf space x cannot be weakly pseudocompact. obviously, every tychonoff pseudocompact space is weakly pseudocompact. the hedgehog j (α), where α ≥ ℵ1 is an example of a weakly pseudocompact space which is not pseudocompact. as far as i know, it is an open problem if rα (α ≥ ℵ1) is weakly pseudocompact. we have however, the following characterization: theorem 4.14 ([6]). a tychonoff space (x, t ) is weakly pseudocompact iff t has a normal wallman basis b such that every countable cover of x with elements of b has finite subcover. 5. cover uniformities and wallman bases there is a strong relation between normal wallman bases and cover uniformities. the treatment of uniform spaces thru covers instead of connectors must be seen as an alternative, but the two treatments are equivalent. definition 5.1. a cover uniformity basis on a set x is a non–empty family g of covers of x such that for every pair of covers α, β ∈ g, there exists a cover γ ∈ g which refines baricentrically each of the covers α, β, i.e., for every p ∈ x we may find elements ap ∈ α, bp ∈ β such that st ( p, γ) = ∪ {c | p ∈ c ∈ γ} ⊆ ap ∩ bp. a cover uniformity on x is simply a cover uniformity basis g on x with the additional property: ∗) if α ∈ g and if β is a cover of x refined by α, then β ∈ g. each cover uniformity basis g on x is contained in a unique smallest cover uniformity on x, namely g ⊆ g+ where g+ = {β | β is a cover of x and some cover α ∈ g refines β} . 232 a. garćıa-máynez two cover uniformity bases g1, g2 on x are equivalent if g + 1 = g + 2 . each cover uniformity basis g on x determines a topology tg on x defined by: l ∈ tg ⇔ ∀x ∈ l, ∃ αx ∈ g ⋔ st (x, αx) ⊆ l. it is easy to see that two equivalent cover uniformity bases determine the same topology on x but the converse is not true in general. a standard result (but not so obvious), states that for every cover uniformity basis g on a set x, tg is a completely regular topology on x. besides, tg is t1 (and hence, tg is a tychonoff topology on x) iff for every x ∈ x, we have ∩ {st (x, α) | α ∈ g} = {x}. in our terminology, a cover uniform space is a pair (x, g), where g is a cover uniformity basis on x. it is easy to see that for every subset a ⊆ x, g|a ={ α|a | α ∈ g } is a cover uniformity basis on a and ( a, tg|a ) is a subspace of (x, tg). so, by a cover uniform subspace of (x, g), we simply mean a pair( a, g|a ) , where a ⊆ x. the correspondence between the concepts of cover uniform space and uniform space is very simple: given a cover uniform space (x, b) we construct a filter basis fb of symmetric connectors of x, namely, each α ∈ b determines the symmetric connector: e (α) = ∪ {l × l | l ∈ α} . the filter f +b = {t ⊆ x × x | e (α) ⊆ t for some α ∈ b} is a uniformity on x and the topologies tb, tfb coincide. conversely, given a symmetric quasi– uniformity basis f on x, each f ∈ f determines an indexed cover: αf = {f (x) | x ∈ x} . the family b = {αf | f ∈ f} is a cover uniformity basis: let αf1 , αf2 ∈ b and let g ∈ f be such that g2 ⊆ f1 ∩ f2. then α △ g = {st (x, αg) | x ∈ x} refines both covers αf1 , αf2 : in fact, if z ∈ st (x, αg) and g (y) contains both points x, z, then (y, z) , (y, x) ∈ g and so (x, y) ∈ g−1 = g and (x, z) ∈ g2 ⊆ f1 ∩ f2 and z ∈ f1 (x) ∩ f2 (x). in this case we have also the same topologies tb = tf . � a map ϕ : (x, g) → (y, h) between cover uniform spaces is uniformly continuous if for every ε ∈ h we may find a cover δ ∈ g such that δ refines ϕ−1 (ε) = { ϕ−1 (e) | e ∈ ε } . ϕ is a unimorphism if ϕ es bijective and both maps ϕ : (x, g) → (y, h), ϕ−1 : (y, h) → (x, g) are uniformly continuous. ϕ : (x, g) → (y, h) is a unimorphic embedding if ϕ is a unimorphism from ϕ : (x, g) → ( ϕ (x) , h|ϕ (x) ) and if ϕ (x) is dense in y. it is obvious that if g1, g2 are cover uniformity bases on x, then the identity map j : (x, g1) → (x, g2) is a unimorphism iff g1 and g2 are equivalent. we have several important concepts in uniform space theory: wallman bases 233 definition 5.2. a) a filter f on a cover uniform space (x, g) is cauchy if f ∩ α 6= ∅ for every α ∈ g. b) a cover uniform space (x, g) is complete if every cauchy filter on (x, g) is convergent (with respect to the topology tg). c) a cover uniform space (x, g) is totally bounded if every cover α ∈ g has a finite subcover for x. d) a cover uniform space (y, h) is a cover completion of a cover uniform space (x, u) if (y, h) is complete and if there exists a unimorphic embedding ϕ : (x, u) → (y, h). the easiest example of a completion is the metric completion: let (x, d) be a metric space and let ( x̃, d̃ ) be a metric completion of (x, d). for each ε > 0 define αε = { v dε (x) | x ∈ x } ; α̃ε = { v d̃ε (x) | x ∈ x } . then gd = {αε | ε > 0} is a cover uniformity basis on x, gd̃ = {α̃ε | ε > 0} is a cover uniformity basis on x̃ and ( x, g d̃ ) is a cover completion of (x, gd ). we state without proof a few important theorems on cover uniform space theory. (see [3], chap. 7). theorem 5.3. let ( a, g|a ) be a dense subspace of a cover uniform space (x, g) and let ϕ : ( a, g|a ) → (y, h) be a uniformly continuous map into a complete t2 cover uniform space (y, h). then ϕ has unique continuous extension ϕ̃ : (x, tg) → (y, th) and this unique extension is uniformly continuous as a map from (x, g) to (y, h). theorem 5.4. every hausdorff cover uniform space (x, g) has a cover completion ( x̃, g̃ ) and every other cover completion (y, h) is unimorphic to ( x̃, g̃ ) . theorem 5.5. a cover uniform space (x, g) is totally bounded iff every ultrafilter f on x is cauchy. theorem 5.6. let ( a, g|a ) be a dense subspace of a cover uniform space (x, g). then ( a, g|a ) is totally bounded iff (x, g) is totally bounded. also, every subspace of a totally bounded cover uniform space is totally bounded. since every adherence point of an ultrafilter in a topological space is a convergence point and since a topological space is compact iff every ultrafilter converges, we obtain as a corollary the result of a a. weil mentioned in the introduction: 234 a. garćıa-máynez theorem 5.7. the topology tg of a cover uniform space (x, g) is compact iff (x, g) is complete and totally bounded. normal wallman bases of tychonoff spaces determine totally bounded cover uniform spaces: theorem 5.8 ([3]). let b be a normal wallman basis of a tychonoff space (x, t ). then the family u (b) of finite covers of x consisting of elements of b is a totally bounded cover uniformity basis on x and t = tu(b). the cover completion ( x̃, ũ (b) ) is a hausdorff compactification of (x, t ) which is equivalent to the wallman compactification ( x (b) , t ∗ ) . theorem 5.9 ([3]). let x, y be a tychonoff spaces with respective normal wallman bases bx, by. a map ϕ : (x, u (bx)) → (y, u (by)) is uniformly continuous iff ϕ satisfies the following condition: ∗) whenever h, k ∈ c (by) are disjoint, there exist disjoint cobasic sets h1, k1 ∈ c (bx) such that ϕ −1 (h) ⊆ h1 and ϕ −1 (k) ⊆ k1. as a corollary of this last theorem, we obtain the well known universal property of the stone–čech compactification of a tychonoff space x: theorem 5.10. let ϕ : x → y be a continuous map from a tychonoff space x into a compact hausdorff space y. then ϕ has a continuous extension ϕ1 : β x → y. proof. apply last theorem taking as bx, by the cozero bases of x, y, respectively. apply then the extension theorem of uniformly continuous maps into complete spaces. � take a look of [3] for further applications of these theorems. 6. the wallman type compactification problem every annular basis b of a compact hausdorff space z is a normal wallman basis of z. if x is a dense subspace of z, then b|x = {b ∩ x | b ∈ b} is a wallman basis of x but b|x may not be normal. assuming b|x is normal, we wonder under what conditions z and x (b|x) are equivalent compactifications of x. the answer is quite simple: z and x (b|x) are equivalent compactifications of x iff z is the only member of b containing x (see [13]). if this happens, we say then that b has the trace property respect to x. if x is gδ dense in z, then the cozero basis of z has the trace property respect to x. if b consists of open domains (i.e., sets which coincide with the interior of their closures) then we have a better result: z has the trace property respect to any dense subspace of z. this suggests the following definition: wallman bases 235 definition 6.1. a compact t2–space z is regular wallman if z is a wallman type compactification of each of its dense subspaces. as we saw before, if the compact hausdorff space z has an annular basis consisting of open domains, then z is regular wallman. this happens in compact metric spaces and, more generally, in arbitrary products of compact metrizable spaces (see [14]). even better, the stone–čech compactification of any metrizable space has an annular basis consisting of open domains and hence every metrizable space has a normal wallman basis consisting of open domains. (see [7]). consider the following two questions: q1 . is every compact t2–space regular wallman? q2 . is every hausdorff compactification of a tychonoff space x of wallman type? r. c. solomon ([11]) answered in the negative the first question proving that some closed subspace of the cube ik, where k = (2c) + , is not regular wallman. as far as the second question is concerned, a. k. steiner and e. f. steiner reduced q2 to an equivalent problem ([15], 1977): q′2 . is every t2–compactification of a discrete space x of wallman type? in the same year (1977), ([16]) ul’janov answered q2 in the negative exhibiting a compactification of the discrete space with 2c elements which is not of wallman type. he also proved that the continuum hypothesis is equivalent to the fact that every hausdorff compactifications of the set of natural numbers is of wallman type. observe that if a tychonoff space x has a normal wallman basis b consisting of open domains, then b∗ is an annular basis of x (b) consisting of open domains and hence x (b) is regular wallman. we finish this section with two questions which, as far as i know, still remain open: q3 . is every hausdorff wallman type compactification of a metrizable space regular wallman? q4 . is every normal wallman basis of a tychonoff space x equivalent to a subfamily of the cozero basis of x? 7. a topological dream i conclude this brief survey on wallman bases with a topological dream: “if p is any topological property, there exists a list of conditions on annular basis of a space x which are equivalent to property p”. we have some examples where this dream becomes true: 1) a topological space x is compact and pseudo–metrizable iff x has a countable normal wallman basis such that every cobasic set is pseudocompact. 236 a. garćıa-máynez 2) a hausdorff space x is compact iff x has an annular basis such that every cobasic set is compact. 3) a hausdorff space x is locally compact iff x has a normal wallman basis such that every basic set has compact closure or compact complement. 4) a tychonoff space x is almost compact (i.e., it admits only one compatible uniformity) iff x has a normal wallman basis b such that in every pair of disjoint cobasic sets, at least one them is compact. by a previous remark, a tychonoff space x is almost compact iff it admits only one (except for equivalence) normal wallman basis. we have previously characterized s–metrizable and weakly pseudo–compact spaces specifying the existence of a normal wallman basis with some properties. consider the following long list of topological properties of tychonoff spaces. (i do not include any definition): compact metrizable eberlein compact compact locally compact lindelöf ? locally compact paracompact realcompact � locally compact realcompact � locally compact topologically complete � locally compact ? almost locally compact ultra– complete čech complete p countable type ? point countable type �k can you characterize each of these properties in a similar way? wallman bases 237 references [1] á. császár, fondements de la topologie générale, budapest-paris, 1960. [2] j. dujundji, topology, allyn and bacon, inc., boston, 1966. [3] a. garćıa–máynez and a. tamariz, topoloǵıa general, porrúa, méxico, 1988. [4] a. garćıa–máynez, property c, wallman bases and s–metrizability, topology and its applications 12 (1981), 237–246. [5] a. garćıa–máynez, on wallman bases and compactifications, bolet́ın de la sociedad matemática mexicana 3a serie vol 11, 2 (2005), 283–292. [6] s. garćıa, ferreira s. and a. garćıa–máynez, on weakly-pseudocompact spaces, houston journal of mathematics. 20 (1994), 145–159. [7] t. kimura, the stone–čech compactifications, the stone–čech remainder and the regular wallman property, proc. amer. math. soc. 99 (1987), 193–198. [8] h. p. a. künzi, topological spaces with a unique compatible quasi–uniformity, canad. math. bull. 29 (1986), 40–43. [9] w. j. pervin, quasi–uniformization of topological spaces, math. ann. 147 (1962), 316– 317. [10] s. romaguera and m. a. sánchez-granero, a quasi–uniform characterization of wallman type compactifications, studia sci. math. hungar. 40 (2003), 257–267. [11] c. r. solomon, a hausdorff compactification that is not regular wallman, general topology and its applications 7 (1977), 59–63. [12] l. a. steen and j. a. jr. seebach, counterexamples in topology, springer-verlag, new york, 2nd. edition, 1970. [13] e. f. steiner, wallman spaces and compactifications, fundamenta mathematicae 61 (1968), 295–304. [14] a. k. steiner and e. f. steiner, products of compact metric spaces are regular wallman, indag. math. 30 (1968), 428–430. [15] a. k. steiner and e. f. steiner, on the reduction of the wallman compactification problem to discrete spaces, general topology and its applications 7 (1977), 35–37. [16] v. m. ul’janov, solution of the fundamental problem of bicompact extensions of wallman type, (russian) dokl. akad. nauk sssr. 233(1977), 1056–1059. (english translation) soviet math. dokl. 18 (1977), 567–571. [17] h. wallman, lattices and topological spaces, ann. of math. 39 (1938), 112–126. [18] a. weil,, sur les espaces à structure uniforme et sur la topologie générale, paris 1938. received march 2006 accepted february 2007 adalberto garćıa–máynez cervantes (agmaynez@matem.unam.mx) instituto de matemáticas, universidad nacional autónoma de méxico, area de la investigación cient́ıfica, circuito exterior, ciudad universitaria, distrito federal, c.p. 04510, méxico singhhematulinpantagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 121-130 new coincidence and common fixed point theorems s. l. singh, apichai hematulin and rajendra pant abstract. in this paper, we obtain some extensions and a generalization of a remarkable fixed point theorem of proinov. indeed, we obtain some coincidence and fixed point theorems for asymptotically regular non-self and self-maps without requiring continuity and relaxing the completeness of the space. some useful examples and discussions are also given. 2000 ams classification: 54h25; 47h10. keywords: coincidence point; fixed point; banach contraction; quasicontraction; asymptotic regularity. 1. introduction the well-known banach fixed point theorem has been generalized and extended by many authors in various ways. recently, proinov [15] has obtained two types of generalizations of banach’s fixed point theorem. the first type involves meirkeeler type conditions (see, for instance, cho et al. [3], jachymski [6], lim [10], matkowski [11], park and rhoades [14]) and the second type involves contractive gauge functions (see, for instance, boyd and wong [1] and kim et al. [9]). proinov [15] obtained equivalence between these two types of contractive conditions and also obtained a new fixed point theorem. inspired by jungck [7], naimpally et al. [13], proinov [15] and romaguera [19], we obtain coincidence theorems on a very general setting and derive various fixed point theorems. some special cases are also discussed. in all that follows y is an arbitrary non-empty set, (x, d) a metric space and n := {1, 2, 3, ..., }. for t, f : y → x, let c(t, f ) denote the set of coincidence points of t and f , that is c(t, f ) := {z ∈ y : t z = f z}. the following definition comes from sastry et al. [20] and s. l. singh et al. [21]. 122 s. l. singh, a. hematulin and r. pant definition 1.1. let s, t and f be maps on y with values in a metric space (x, d). the pair (s, t ) is asymptotically regular with respect to f at x0 ∈ y if there exists a sequence {xn} in y such that f x2n+1 = sx2n, f x2n+2 = t x2n+1, n = 0, 1, 2, ..., and lim n→∞ d(f xn, f xn+1) = 0. if y = x and s = t then we get the definition of asymptotic regularity of t with respect to f due to rhoades et al. [18]. further if y = x, s = t and f is the identity map on x, then we get the usual definition of asymptotic regularity for a map t due to browder and peteryshyn [2]. definition 1.2 ([16]). let (x, d) be a metric space and t, f : x → x. then the self-maps t and f are r-weakly commuting if there exists a positive real number r such that d(t f x, f t x) ≤ rd(t x, f x) for all x ∈ x. following itoh and takahashi [5] and singh and mishra [22], we have the following definition for a pair of self-maps on a metric space x. definition 1.3. let t, f : x → x. then the pair (t, f ) is (it)-commuting at z ∈ x if t f z = f t z. they are (it)-commuting on x (also called weakly compatible, by jungck and rhoades [8]) if t f z = f t z for all z ∈ x such that t z = f z. definition 1.4 ([15] definition 2.1 (i)). let φ denote the class of all functions ϕ : r+ → r+ satisfying: for any ε > 0 there exists δ > ε such that ε < t < δ implies ϕ(t) ≤ ε. 2. main results proinov [15] obtained the following result generalizing some fixed point theorems of jachymski [6] and matkowski [11]. theorem 2.1 ([15, th. 4.1]). let t be a continuous and asymptotically regular self-map on a complete metric space (x, d) satisfying the following conditions: (p1): d(t x, t y) ≤ ϕ(d(x, y)), for all x, y ∈ x; (p2): d(t x, t y) < d(x, y), for all distinct x, y ∈ x, where d(x, y) = d(x, y) + γ[d(x, t x) + d(y, t y)], γ ≥ 0 and ϕ ∈ φ. then t has a unique fixed point. moreover if d(x, y) = d(x, y) + d(x, t x) + d(y, t y) and ϕ is continuous and satisfies ϕ(t) < t for all t > 0, then continuity of t can be dropped. for a self-map t : x → x the quasi-contraction due to ćirić [4] is as follows (c) d(t x, t y) ≤ qm (x, y), where m (x, y) = max{d(x, y), d(x, t x), d(y, t y), d(x, t y), d(y, t x)}, 0≤ q < 1. we remark that following the listing of conditions due to rhoades [17] the condition (c) is the condition (24). according to rhoades [17] the condition (25): new coincidence and common fixed point theorems 123 d(t x, t y) < m (x, y), is the most general condition among the contractive conditions. the following example shows that (p1) is more general than condition (c). example 2.2. let x = {1, 2, 3} with the usual metric d and t : x → x such that t 1 = 1, t 2 = 3, t 3 = 1. then t satisfies (c) with q > 1. clearly, the condition (p1) is satisfied with ϕ(t) = t 2 for all t > 0 and ϕ(0) = 0 and γ ≥ 1. evidently t can not satisfy the conditions (24) and (25) listed by rhoades [17]. first we extend the scope of theorem 2.1 by introducing a dummy map f in theorem 2.1. this idea comes essentially from jungck [7]. we remark that the requirement “ϕ(t) < t for all t > 0” in theorem 2.1 is redundant as this is the consequence of definition 1.4. we shall use this fact in the proof of the following theorem. theorem 2.3. let t and f be self-maps on a complete metric space (x, d) such that (a1): t (x) ⊆ f (x); (a2): d(t x, t y) ≤ ϕ(g(x, y)) for all x, y ∈ x, where g(x, y) = d(f x, f y) + γ[d(f x, t x) + d(f y, t y)], γ ≥ 0 and ϕ ∈ φ is continuous; (a3): d(t x, t y) < g(x, y) for all distinct x, y ∈ y ; (a4): (t, f ) is asymptotically regular at x0 ∈ x. if t is continuous then t has a fixed point provided that t and f are r-weakly commuting. further if f is continuous and γ = 1 then t and f have a unique common fixed point provided that t and f are r-weakly commuting. proof. pick x0 ∈ x. define a sequence {yn} by yn+1 = t xn = f xn+1, n = 0, 1, 2, ... this can be done since the range of f contains the range of t . let us fix ε > 0. since ϕ ∈ φ, there exists δ > ε such that for any t ∈ (0, ∞), (2.1) ε < t < δ ⇒ ϕ(t) ≤ ε. without loss of generality we may assume that δ ≤ 2ε. since the pair (t, f ) is asymptotically regular, lim n→∞ d(yn, yn+1) = 0. hence, there exists an integer n ≥ 1 such that (2.2) d(yn, yn+1) < δ − ε 1 + 2γ for all n ≥ n. by induction we shall show that (2.3) d(yn, ym) < δ + 2γε 1 + 2γ for all m, n ∈ n with m ≥ n ≥ n . 124 s. l. singh, a. hematulin and r. pant let n ≥ n be fixed. obviously, (2.3) holds for m = n. assuming (2.3) to hold for an integer m ≥ n, we shall prove it for m + 1. by the triangle inequality, we get d(yn, ym+1) ≤ d(yn, yn+1) + d(yn+1, ym+1) or (2.4) d(yn, ym+1) ≤ d(yn, yn+1) + d(t xn, t xm). we claim that (2.5) d(t xn, t xm) ≤ ε. to prove (2.5), we consider two cases. case 1.: let g(xn, xm) ≤ ε. by (a2) and (a3), d(t xn, t xm) ≤ g(xn, xm) ≤ ε, and (2.5) holds. case 2.: let g(xn, xm) > ε. by (a2), (2.6) d(t xn, t xm) ≤ ϕ(g(xn, xm)). by the definition of g(x, y), g(xn, xm) = d(yn, ym) + γ[d(yn, yn+1) + d(ym, ym+1)]. from (2.2) and (2.3), g(xn, xm) < δ + 2γε 1 + 2γ + 2γ δ − ε 1 + 2γ = δ. now by (2.1), ε < g(xn, xm) < δ ⇒ ϕ(g(xn, xm)) ≤ ε. so (2.6) implies (2.5). from (2.5), (2.4) and (2.2), it follows that d(yn, ym+1) ≤ δ − ε 1 + 2γ + ε = δ + 2γε 1 + 2γ . this proves(2.3). since δ ≤ 2ε, (2.3) implies that d(yn, ym) < 2ε for all integers m and n with m ≥ n ≥ n . so {yn} is a cauchy sequence. since the space x is complete the sequence {yn} has a limit. call it z. suppose t is continuous. then t t xn → t z and t f xn → t z. since t and f are r-weakly commuting, d(t f xn, f t xn) ≤ rd(t xn, f xn). making n → ∞, f t xn → t z. if z 6= t z, then by (a2), d(t xn, t t xn) ≤ ϕ(g(xn, t xn) = ϕ(d(f xn, f t xn) + γ[d(f xn, t xn) + d(f t xn, t t xn)]). making n → ∞, d(z, t z) ≤ ϕ(d(z, t z) < d(z, t z), a contradiction. it follows that z = t z. new coincidence and common fixed point theorems 125 if f continuous and γ = 1. then f f xn → f z and f t xn → f z. since t and f are r-weakly commuting, d(t f xn, f t xn) ≤ rd(t xn, f xn). making n → ∞, t f xn → f z. if z 6= f z, then by (a2), d(t xn, t f xn) ≤ ϕ(g(xn, f xn) = ϕ(d(f xn, f f xn) + γ[d(f xn, t xn) + d(f f xn, t f xn)]). making n → ∞, d(z, f z) ≤ ϕ(d(z, f z) < d(z, f z), a contradiction. it follows that z = f z. now if z 6= t z, then by (a2), d(t z, t f xn) ≤ ϕ(g(z, f xn) = ϕ(d(f z, f f xn) + [d(f z, t z) + d(f f xn, t f xn)]). making n → ∞, d(t z, f z) ≤ ϕ(d(t z, f z) < d(t z, f z), a contradiction. it follows that t z = f z = z, and z is a common fixed point of f and t . uniqueness follows easily. � we remark that theorem 2.1 is obtained from theorem 2.3 as a corollary. notice that conditions (p1) and (p2) come respectively from (a2) and (a3) when f is the identity map on x. further, the continuity of only one map is needed. the following example shows the superiority of theorem 2.3 over theorem 2.1. example 2.4. let x = [0, ∞) with usual metric d. let t : x → x such that t x = { x if x is rational, 0 if x is irrational. theorem 2.1 is not applicable to this map t as it is not continuous. however, if we take a (dummy) map f : x → x such that f x = 2x for all x ∈ x then t and f satisfy all the hypotheses of theorem 2.3. notice that f is continuous and t 0 = f 0 = 0. now we modify certain requirements of theorem 2.3 a slightly to obtain a new result. theorem 2.5. let t and f be maps on an arbitrary non-empty set y with values in a metric space (x, d) such that (b1): t (y ) ⊆ f (y ); (b2): d(t x, t y) ≤ ϕ(g(x, y)) for all x, y ∈ y , where g(x, y) = d(f x, f y) + γ[d(f x, t x) + d(f y, t y)], 0 ≤ γ ≤ 1, and ϕ : r+ → r+ continuous; 126 s. l. singh, a. hematulin and r. pant (b3): (t, f ) is asymptotically regular at x0 ∈ y . if t (y ) or f (y ) is a complete subspace of x then (i): c(t, f ) is non-empty. further, if y = x, then (ii): t and f have a unique common fixed point provided that t and f are (it)-commuting at a point u ∈ c(t, f ). proof. pick x0 ∈ y . define a sequence {yn} by yn+1 = t xn = f xn+1, n = 0, 1, 2..., this can be done since the range of f contains the range of t . since the pair (f, t ) is asymptotically regular, lim n→∞ d(yn, yn+1) = 0. first we shall show that {yn} is a cauchy sequence. suppose {yn} is not cauchy. then there exists µ > 0 and increasing sequences {mk} and {nk} of positive integers such that for all n ≤ mk < nk, d(ymk , ynk ) ≥ µ and d(ymk , ynk−1) < µ. by the triangle inequality, d(ymk , ynk ) ≤ d(ymk , ynk−1) + d(ynk−1, ynk ). making k → ∞, d(ymk , ynk ) < µ. thus, d(ymk , ynk ) → µ as k → ∞. now by (b2), d(ymk+1, ynk+1) = d(t xmk , t xnk ) ≤ ϕ(g(xmk , xnk )) = ϕ(d(f xmk , f xnk ) + γ[d(f xmk , t xmk ) + d(f xnk , t xnk )]). making k → ∞, µ ≤ ϕ(µ) < µ, a contradiction. therefore {yn} is cauchy. suppose f (y ) is complete. then {yn} being contained in f (y ) has a limit in f (y ). call it z. let u ∈ f −1z. then f u = z. using (b2), d(t u, t xn) ≤ ϕ(d(f u, f xn) + γ[d(t u, f u) + d(t xn, f xn)]). making n → ∞, d(t u, z) ≤ ϕ(γd(t u, z)) < d(t u, z), a contradiction. therefore t u = z = f u. this proves (i). now if y = x and the pair(t, f ) is (it)-commuting at u then t f u = f t u and t t u = t f u = f t u = f f u. in view of (b2), it follows that d(t u, t t u) < ϕ(g(u, t u)) = ϕ(d(f u, f t u) + γ[d(t u, f u) + d(t t u, f t u)]) < d(t u, t t u), a contradiction. therefore t t u = t u and f t u = t t u = t u = z. this proves (ii). new coincidence and common fixed point theorems 127 in the case t (y ) is a complete subspace of x, the condition (b1) implies that sequence {yn} converges in f (y ), and the previous proof works. the uniqueness of common fixed point follows easily. � the following result generalizes an important result of proinov [15, cor. 4.3] corollary 2.6. let t and f be maps on an arbitrary non-empty set y with values in metric space (x, d) such that (c1): t (y ) ⊆ f (y ); (c2): d(t x, t y) ≤ ϕ(m (x, y)), for all x, y ∈ y , where m (x, y) = max{d(f x, f y), d(f x, t x), d(f y, t y), 1 2 [d(f x, t y)+ d(f y, t x)]} and ϕ : r+ → r+ continuous. if t (y ) or f (y ) is a complete subspace of x then conditions (i) and (ii) of above theorem 2.5 hold. now we obtain a new common fixed point theorem for three non self-maps. theorem 2.7. let s, t and f be maps on an arbitrary non-empty set y with values in a metric space (x, d). let (s, t ) be asymptotically regular with respect to f at x0 ∈ y and the following conditions are satisfied: (d1): s(y ) ∪ t (y ) ⊆ f (y ); (d2): d(sx, t y) ≤ ϕ(h(x, y)), for all x, y ∈ x, where h(x, y) = d(f x, f y) + γ[d(sx, f x) + d(t y, f y)], 0 ≤ γ ≤ 1, and ϕ : r+ → r+ continuous. if s(y ) or t (y ) or f (y ) is a complete subspace of x then (i): c(s, f ) is non-empty; (ii): c(t, f ) is non-empty. further, if y=x then (iii): s and f have a common fixed point provided that s and f are (it)-commuting at a point u ∈ c(s, f ). (iv): t and f have a common fixed point provided that t and f are (it)-commuting at a point v ∈ c(t, f ). (v): s, t and f have a unique common fixed point provided that (iii) and (iv) both are true. proof. let x0 be an arbitrary point in y . since (s, t ) is asymptotically regular with respect to f , then there exists a sequence {xn} in y such that f x2n+1 = sx2n, f x2n+2 = t x2n+1, n = 0, 1, 2, ..., and lim n→∞ d(f xn, f xn+1) = 0. now we shall show that {f xn} is cauchy sequence. suppose {f xn} is not cauchy. then there exists µ > 0 and increasing sequences {mk} and {nk} of positive integers, such that for all n ≤ mk < nk, d(f xmk , f xnk ) ≥ µ and d(f xmk , f xnk−1) < µ. by the triangle inequality, d(f xmk , f xnk ) ≤ d(f xmk , f xnk−1) + d(f xnk−1, f xnk ). 128 s. l. singh, a. hematulin and r. pant making k → ∞, we get d(f xmk , f xnk ) < µ. thus d(f xmk , f xnk ) → µ as k → ∞. by (d2) we have d(f xmk+1, f xnk+1) = d(sxmk , t xnk ) ≤ ϕ(h(xmk , xnk )) = ϕ(d(f xmk , f xnk ) + γ[d(sxmk , f xmk ) + d(t xnk , f xnk )]). making k → ∞ µ ≤ ϕ(µ) < µ, a contradiction. thus {f xn} is cauchy sequence. suppose f (y ) is a complete subspace of x. then {yn} being contained in f (y ) has a limit in f (y ). call it z. let u = f −1z. thus f u = z for some u ∈ y . note that the subsequences {f x2n+1} and {f x2n+2} also converge to z. now by (d2), d(su, t2n+1) ≤ ϕ(d(f u, f2n+1) + γ[d(su, f u) + d(t2n+1, f2n+1)]). making n → ∞, d(su, f u) ≤ ϕ(γd(su, f u)) < d(su, f u) a contradiction. therefore su = f u = z. this proves (i). since s(y )∪t (y ) ⊆ f (y ). therefore there exists v ∈ y such that su = f v. we claim that f v = t v. using (d2), d(f v, t v) = d(su, t v) ≤ ϕ(d(f u, f v) + γ[d(su, f u) + d(t v, f v)]) = ϕ(γd(f v, t v)) < d(f v, t v), which is a contradiction. therefore t v = f v = su = f u. this proves (ii). now if y = x, (s, f ) and (t, f ) are (it)-commuting then sf u = f su and ssu = sf u = f su = f f u, t f v = f t v and t t v = t f v = f t v = f f v. in view of (d2), it follows that d(ssu, su) = d(ssu, t v) ≤ ϕ(d(f su, f v) + γ[d(ssu, f su) + d(t v, f v)]) = ϕ(γd(ssu, su)) < d(ssu, su). therefore ssu = su = f su, su is a common fixed point of s and f . similarly, t v is a common fixed point of t and f . since su = t v, we conclude that su is a common fixed point of s, t and f . the proof is similar when s(y ) or t (y ) are complete subspaces of x since, s(y ) ∪ t (y ) ⊆ f (y ). uniqueness of the common fixed point follows easily. � acknowledgements. the authors are indebted to the referee and prof. salvador romaguera for their perspicacious comments and suggestions. new coincidence and common fixed point theorems 129 references [1] d. w. boyd and j. s. w. wong, on nonlinear contractions, proc. amer. math. soc. 20 (1969), 458–464. 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[22] s. l. singh and s. n. mishra, coincidence and fixed points of nonself hybrid contractions, j. math. anal. appl. 256 (2001), 486–497. 130 s. l. singh, a. hematulin and r. pant received august 2008 accepted january 2009 s. l. singh (vedicmri@gmail.com) 21, govind nagar rishikesh 249201, india apichai hematulin department of mathematics, nakhonratchasima rajabhat university, nakhoratchasima, thailand rajendra pant (pant.rajendra@gmail.com) srm university modinagar, ghaziabad (u.p.) 201204, india @ appl. gen. topol. 23, no. 1 (2022), 17-30 doi:10.4995/agt.2022.16405 © agt, upv, 2022 representations of bornologies homeira pajoohesh department of mathematics, medgar evers college, brooklyn, ny, u.s.a. (hpajoohesh@mec.cuny.edu) communicated by j. rodŕıguez-lópez abstract bornologies abstract the properties of bounded sets of a metric space. but there are unbounded bornologies on a metric space like p(ir) with the euclidean metric. we show that by replacing [0, ∞) with a partially ordered monoid every bornology is the set of bounded subsets of a generalized metric mapped into a partially ordered monoid. we also prove that the set of bornologies on a set is the join completion of the equivalence classes of a relation on the power set of the set. 2020 msc: 54c99; 06d22; 06b10. keywords: bornology; metrizablity; frame. 1. introduction let x be a topological space. a bornology on x is a family of subsets a of x which satisfies the following axioms: • a is closed under finite union and ⋃ a = x • a ∈a and b ⊆ a implies b ∈a a typical example of a bornology is the set of bounded subsets of a metric space. another example is the set of finite subsets of a given set; it is denoted by f. one can easily verify that every bornology on a set contains f. the bornology generated by a given subset h of a set x is the smallest bornology containing h and in [4] is called a principal bornology and it can be easily seen that it is ↓ h ∪f, where ↓ h = {y ⊆ x : y ⊆ h}. received 29 september 2021 – accepted 24 january 2022 http://dx.doi.org/10.4995/agt.2022.16405 https://orcid.org/0000-0003-4035-0032 h. pajoohesh given bornologies a and b on the sets x and y respectively, a function f : x → y is called bounded map provided that f(m) ∈b if m ∈a. the theory of bornologies play an important role in functional analysis, see [5] and [7]. bornologies also have been considered in the theory of locally convex spaces, see [11]. bornologies on topological spaces are the generalization of the set of bounded sets of a metric space, the sets whose elements are within a fixed distance from each other. metrics are useful when we talk about distance between the points and sets, closeness of sets and points, and more importantly one can talk about cauchy sequences and convergence. but, some of these notions fail when we deal with topological spaces. these concepts are all related to boundedness. in this article we overcome this issue by relating bornologies with boundedness. in [3], it is proved that a bornology a on a non-empty set x is coincident with the set of bounded sets of a metric d : x ×x → [0,∞) if and only if a has a countable base. therefore, when one wants to relate a bornology with uncountable base with boundedness, the boundedness or the set of radii must come from a more general setting than the real numbers. here we consider generalized metrics into partially ordered monoids, po-monoids. po-monoids are monoids with an order which is cooperative with the binary relation. this will enable us to talk about distance between the points and sets and more importantly about bounded sets and bornologies. in [12] hu considered bornologies on topological spaces and he proved the following theorem: hu’s theorem: let b 6= p(x) be a bornology on a normal topological space x. then there is a continuous function f : x → [0,∞) such that b =↓ {{x ∈ x : f(x) < t} : t ≥ 0} if and only if b has a countable base and every x ∈ x is in the interior of some element b ∈b. in the next section we prove a similar theorem for every bornology on a set x by allowing the range of the function be a lattice rather than [0,∞). in section three we show that every bornology is the set of bounded sets of a generalized metric space. in the last section we show that the set of bornologies on a set x is a frame and we investigate the properties of this frame. 2. induced bornologies by compatible functions let l be a poset, d be a directed subset of l, and α : x → l be a function, for every t ∈ d define nt,α = {x ∈ x : α(x) < t} and nt,α = {x ∈ x : α(x) ≤ t}. we call nt,α and nt,α respectively the strict sublevel and the sublevel of x with respect to α and d. define bα,d = {a ⊆ x : ∃t ∈ d such that a ⊆ nt,α} and bα,d = {a ⊆ x : ∃t ∈ d such that a ⊆ nt,α}. one can easily see that nt,α ⊆ nt,α and therefore, bα,d ⊆bα,d. we say α and d are compatible if for every x ∈ x there is a t ∈ d such that α(x) < t. © agt, upv, 2022 appl. gen. topol. 23, no. 1 18 representations of bornologies theorem 2.1. let (l,≤) be a poset and d be a directed subset of l. if α : x → l is a function compatible with d, then bα,d and bα,d are bornologies on x. proof. we prove bα,d is a bornology and the proof that bα,d is a bornology is similar and we will leave it to the reader. one can easily see that if b ⊆ a ∈ bα,d then b ∈bα,d. next if a ⊆ nt,α and b ⊆ ns, then for every x ∈ a∪b either α(x) < t or α(x) < s. since d is directed, there is an r ∈ d such that s,t ≤ r. since t,s ≤ r, we have α(x) < r. thus, a ∪ b ⊆ nr and therefore, bα,d is closed under finite union. since α is a function compatible with d, one can see that ⋃ bα,d = x � proposition 2.2. let l and j be posets and d be a directed subset of l. if α : x → l is a function compatible with d and f : l → j is an order preserving map, then bf◦α,f(d) is a bornology on x such that bα,d ⊆ b̄f◦α,f(d). proof. first of all note that f(d) is a directed subset of j because for every f(a),f(b) ∈ f(d), we have f(a),f(b) ≤ f(r), where r ∈ d with a,b ≤ r. next note that if m ⊆ e ∈ bf◦α,f(d) then obviously, m ∈ bf◦α,f(d). also if e,t ∈ bf◦α,f(d) then there are f(r),f(s) ∈ f(d) such that e ⊆ {x ∈ x : (f ◦ α)(x) < f(r)} and t ⊆ {x ∈ x : (f ◦ α)(x) < f(s)}. consequently, e ∪ t ⊆ {x ∈ x : (f ◦ α)(x) < f(p)}, where p ∈ d and r,s ≤ p and so, e ∪t ∈bf◦α,f(d). also for every x ∈ x, since α is compatible with d, there is a t ∈ d such that α(x) < t and therefore, x ∈{x ∈ x : (f ◦α)(x) < f(t)}. for the second part, note that if a ⊆ {x : α(x) < t}. then, a ⊆ {x : f(α(x)) ≤ f(t)}. � proposition 2.3. let (l,≤) be a poset and d be a directed subset of l. if α : x → l is a function compatible with d and f : r → x is a function, then bα◦f,d is a bornology on r and f : r → x is a bounded map between (r,bα◦f,d) and (x,bα,d). proof. note that if m ⊆ e ∈bα◦f,d then m ∈bα◦f,d. also if e,j ∈bα◦f,d then there are r,s ∈ d such that e ⊆{x ∈ x : (α◦f)(x) < r} and j ⊆{x ∈ x : (α ◦ f)(x) < s}. consequently, e ∪ j ⊆ {x ∈ x : (α ◦ f)(x) < p} where p ∈ d and r,s ≤ p and so, e ∪ j ∈ bα◦f,d. also for every x ∈ r we have f(x) ∈ x. since α is compatible with d, there is a t ∈ d such that α(f(x)) < t and therefore, x ∈{x ∈ x : (α◦f)(x) < t}. for the second part, suppose c ∈bα◦f,d. we show that f(c) ∈bα,d. since c ∈ bα◦f,d, there is an r ∈ d such that c ⊆ {x ∈ r : (α◦f)(x) < r}. thus, for every c ∈ c we have α(f(c)) < r and so, f(c) ∈{x ∈ x : α(x) < r}. thus, f(c) ⊆{x ∈ x : α(x) < r}∈bα,d. � in order to show the converse of theorem 2.1, we need to employ a second relation on the partially ordered set that is compatible with the original order © agt, upv, 2022 appl. gen. topol. 23, no. 1 19 h. pajoohesh of the partial ordered set. in [16] and [20] a second relation was called a good relation , so we adopt the same name. definition 2.4. a good relation ≺ on a poset (g,≤) is a subset of ≤ such that for each a,b,c,d ∈ g: (trans) a ≤ b ≺ c ≤ d ⇒ a ≺ d. example 2.5. • on each poset , ∅ is a good relation. • if (l,≤) is a partially ordered set, then < is a good order on l • let g = r × h with the lexicographic order, h any poset. define (r,h) ≺ (s,k) if r < s; then ≺ is a good relation on g. theorem 2.6. let a be a bornology on a set x. there is a lattice (l,≤) equipped with a good relation ≺, a directed subset d of l, and a function α : x → l compatible with d such that a = b∗α,d, where b ∗ α,d = {a ⊆ x : ∃t ∈ d such that a ⊆ {x ∈ x : α(x) ≺ t}}. moreover, a = {{x ∈ x : α(x) ≺ t} : t ∈ d}, the strict sublevel of x with respect to α and d. proof. suppose that a is a bornology on a set x. for each k ∈ a let lk = [0, 1] × [0, 1] equipped with the lexicographic order: (q,r) ≤ (s,t) if q < s or q = s and r ≤ t. then lk is a lattice. let l = ∏ k∈alk and define a good relation on l as follows: if (gk)k∈a, (hk)k∈a ∈ l then (gk)k∈a ≺ (hk)k∈a if and only if gk < hk for every k ∈a. one can easily see that ≺ obeys the good order condition. for each k ∈a let dk = [0, 1] × (0, 1] and d = {r = ((r1k,r 2 k))k∈a ∈ l : ∃m ∈a such that r 1 k = 0 for every k ⊇ m} we prove that d is a directed subset of l. let r = ((r1k,r 2 k))k∈a ∈ d and s = ((s1k,s 2 k))k∈a ∈ d. by definition, there is an m ∈ a such that r 1 k = 0 for every k ⊇ m and there is an e ∈ a such that s1k = 0 for every k ⊇ e. now define t1k = { 0, if k ⊇ m ∪e; r1k ∨s 1 k, otherwise . one can see that t = ((t1k, t 2 k))k∈a where t 2 k = r 2 k∨s 2 k is in d and r,s ≤ t. therefore, d is a directed set. for k ∈a define ϕk : x → [0, 1] such that ϕk(z) = { 0, if z ∈ k 1, if z 6∈ k . now define the function α : x → l by (α(x))k = (0,ϕk(x)). we prove that b∗α,d = a. first we show that b ∗ α,d ⊆ a. suppose that w ∈ b∗α,d. there is a t ∈ d such that w ⊆ {x ∈ x : α(x) ≺ t}. let t = ((t1k, t 2 k))k ∈ d. for every x ∈ w we have, α(x) ≺ t and therefore for every k ∈ a we have (0,ϕk(x)) < (t1k, t 2 k). since t ∈ d, there is a c ∈ a © agt, upv, 2022 appl. gen. topol. 23, no. 1 20 representations of bornologies such that t1k = 0 for every k ⊇ c and in particular, t 1 c = 0. now we show that w ⊆ c and therefore w ∈a by way of contradiction, suppose that w 6⊆ c. let z ∈ w \ c. then (α(z))c = (0,ϕc(z)) = (0, 1) 6< (0, t2c) = (t 1 c, t 2 c) and therefore, z 6∈ {x ∈ x : α(x) ≺ t} which is a contradiction. consequently, w ⊆ c. hence, b∗α,d ⊆a. next , we prove that a ⊆ b∗α,d. suppose h ∈ a. we will prove that h = {x ∈ x : α(x) ≺ th}, where th = (t1k, t 2 k) = { (0, 0.5), if k ⊇ h (1, 1), if k 6⊇ h . note that for every x ∈ h we have α(x)k = (0,ϕk(x)) = { (0, 0), if x ∈ k (0, 1), if x 6∈ k . there are two cases either x ∈ k or x 6∈ k. if x ∈ k, then α(x)k = (0,ϕk(x)) = (0, 0) < (t 1 k, t 2 k). suppose now that x 6∈ k. in this case, α(x)k = (0,ϕk(x)) = (0, 1). from x 6∈ k we have k 6⊇ h, which implies that (t1k, t 2 k) = (1, 1). hence, α(x)k = (0,ϕk(x)) = (0, 1) < (1, 1) = (t 1 k, t 2 k). by cases, α(x) ≺ th. therefore, h ⊆{x ∈ x : α(x) ≺ th}. therefore, a = b∗α,d. since h was an arbitrary element of a in the previous paragraph, showing {x ∈ x : α(x) ≺ th} ⊆ h completes proving a = {{x ∈ x : α(x) ≺ t} : t ∈ d}. note that if z 6∈ h, then ϕh(z) = 1. thus, α(z)h = (0,ϕh(z)) = (0, 1) 6< (th)h = (0, 0.5). therefore, z 6∈ {x ∈ x : α(x) ≺ th} and so, {x ∈ x : α(x) ≺ t} ⊆ h. consequently, a ⊆ {{x ∈ x : α(x) ≺ t} : t ∈ d} ⊆ b∗α,d = a. therefore, a = {{x ∈ x : α(x) ≺ t} : t ∈ d} and the proof is complete. � in the next theorem we will simplify the lattice in theorem 2.6 and will obtain a similar result. here we work with sublevels of x rather than strict sublevels. theorem 2.7. let a be a bornology on a set x. then there is a poset (l,≤), a directed subset d of l, and a function α : x → l compatible with d such that a = bα,d = {{x ∈ x : α(x) ≤ t} : t ∈ d}. proof. suppose that a is a bornology on a set x. let ∏ k∈a[0, 1] be ordered coordinatewise and define α : x → ∏ k∈a[0, 1] by α(x) = (ϕk(x))k∈a where ϕk(x) is defined as theorem 2.6. for every h ∈ a define (sh)k ={ 0, if k ⊇ h; 1, if k 6⊇ h. . define d = {sh : h ∈ a} ⊆ l. one can see that if sh and sp are in d then, sh,sp ≤ sh∪p and therefore d is directed. we show that h = nsh for every h ∈ a which implies a = bα,d. first, we show that h ⊆ nsh by proving that if a ∈ h, then α(a)k ≤ (sh)k for every k ∈ a. it is enough to show that α(a)k = 1 implies (sh)k = 1. note that α(a)k = 1 implies a 6∈ k. so, a ∈ h \k. thus, k 6⊇ h, which implies that (sh)k = 1. therefore, h ⊆ nsh . © agt, upv, 2022 appl. gen. topol. 23, no. 1 21 h. pajoohesh next suppose a ∈ nsh . so, α(a)k ≤ (sh)k for every k ∈a. we prove that a ∈ h. by way of contradiction, suppose that a 6∈ h. in this case, α(a)h = 1. on the other hand, (sh)h = 0 which is a contradiction. thus, nsh ⊆ h. therefore, h = nsh . consequently, a = {{x ∈ x : α(x) ≤ t} : t ∈ d}. thus, bα,d, the bornology generated by {{x ∈ x : α(x) ≤ t} : t ∈ d}, equals a as a is a bornology. therefore, {{x ∈ x : α(x) ≤ t} : t ∈ d} = a = bα,d. � we conclude this section by representing bornologies on a set as special subsets of the power set of the set. let a be a bornology on a set x. define: ν : a ↪→ ∏ k∈p(x){0, 1} by ν(h)k =   0, if k ∈a and k ⊇ h; 1, if k ∈a and k 6⊇ h 0, if k 6∈a . lemma 2.8. let a be a bornology on a set x. • ν(∅) = (0)k∈a, • ν(h ∪t) = ν(h) ∨ν(t) and therefore, ν is order preserving. • ν is one-to-one. proof. we leave the proof of the first claim to the reader. for the second claim note that if (ν(h ∪ t))k = 0 then k ⊇ h ∪ t and so, k ⊇ h and k ⊇ t . therefore, (ν(h))k = (ν(t))k = 0. on the other hand if (ν(h ∪ t))k = 1 then k 6⊇ h ∪t and therefore, either k 6⊇ h or k 6⊇ t . thus, (ν(h))k = 1 or (ν(t))k = 1. consequently, ν(h ∪t) = ν(h) ∨ν(t) = 1 for the third claim note that if a,b ∈ a and a 6= b, either a\b 6= ∅ or b \a 6= ∅. without lost of generality suppose that a\b 6= ∅ and z ∈ a\b. then, (ν(a))(a∪b)\{z} = 1 but (ν(b))(a∪b)\{z} = 0 and therefore, ν(a) 6= ν(b). consequently, ν is one-to-one. � theorem 2.9. suppose l ⊆ ∏ k∈p(x){0, 1}. consider the following four conditions. (a) for every r ∈ l and k ∈ p(x) if rk = 0, then rm = 0 for every m ⊇ k. (b) for every r ∈ l and k ∈ p(x) if rk = 1, then rm = 1 for every m ⊆ k. (c) for every r,s ∈ l and m,n ∈ p(x) if rm = 1 and sn\m = 1, then there is an t ∈ l such that tm∪n = 1. (d) if x has more than one element, then for every x ∈ x there is an t ∈ l such that t{x} = 1. if l satisfies conditions (a)-(d), then al = {m ⊆ x : ∃r ∈ l such that rm = 1} is a bornology on x. conversely, if a 6= p(x) is a bornology on the set x, then ν(a) ⊆ ∏ k∈p(x){0, 1} satisfies conditions (a)-(d). moreover, aν(a) = a. © agt, upv, 2022 appl. gen. topol. 23, no. 1 22 representations of bornologies proof. suppose that l ⊆ ∏ k∈p(x){0, 1} satisfies conditions (a)-(d). we prove that al = {m ⊆ x : ∃r ∈ l such that rm = 1} is a bornology on x. suppose t ⊆ m ∈al. there exists an r ∈ l such that rm = 1. so, rt = 1 by property (b). thus, t ∈al. next we show that m,n ∈al implies m∪n ∈al. since m,n ∈ al, there are s,t ∈ l such that sm = 1 and tn = 1. since tn = 1, by property (b) we have tn\m = 1. so, by property (c) there is an r ∈ l such that rm∪n = 1. therefore, m ∪n ∈al. finally, since for every x ∈ x there is an r ∈ l such that r{x} = 1, we have {x}∈al. therefore, ⋃ al = x. for the converse suppose a is a bornology on the set x. we first show that conditions (a) and (b) are satisfied. to this end let r = ν(h) ∈ aν(a), where h ∈ a. for condition (a) assume that rk = 0 and k ∈ p(x) and m ⊇ k. if m 6∈ a then rm = 0 by definition of ν. if m ∈ a then k ∈a and hence k ⊇ h. consequently, m ⊇ h and therefore, rm = 0. so, ν(a) satisfies condition (a). for condition (b), assume rk = 1. we show that rm = 0 for m ⊆ k. since rk = 1, we have k ∈ a and k 6⊇ h. thus, m ⊆ k implies m 6⊇ h. therefore, rm = 1. so, ν(a) satisfies condition (b). for condition (c) assume that m,n ∈ p(x) and r,s ∈ ν(a) are such that rm = 1, and sn\m = 1. let h,j ∈ a be such that r = ν(h) and s = ν(j). notice that m ∈a, m 6⊇ h, n \m ∈a, and n \m 6⊇ j. thus, m ∪n ∈a. now, let t = ν(m ∪ n ∪{z}) where z 6∈ m ∪ n. therefore, rm∪n = 1. so, ν(a) satisfies condition (c). for condition (d) note that for every y 6= x we have, ν({y}){x} = 1. at last we prove aν(a) = a. let b ∈ a. since a 6= p(x), there is a z ∈ x \b. now, ν(b ∪{z})b = 1 and so, b ∈aν(a). therefore, a⊆aν(a). on the other hand, if b 6∈ a, then by definition of ν we have ν(h)b = 0 for every h ∈a and therefore b 6∈aν(a). consequently, aν(a) = a. � definition 2.10. let l ⊆ ∏ k∈p(x){0, 1}. by the kernel of l we mean ker(l) = {k ⊆ x : ∀r ∈ l,rk = 0} and kerc(l), the co-kernel of l, will be defined by p(x) \ker(l) definition 2.11. we say l ⊆ ∏ k∈p(x){0, 1} is called a ν-subset of ∏ k∈p(x){0, 1} if for every r = (rk)k∈p(x) ∈ l there is an h ⊆ x such that for every k ∈ kerc(l) we have rk = 1 if and only if k 6⊇ h. remark 2.12. by what we learned for every ν-subset of ∏ k∈p(x){0, 1} we can define a bornology on x and conversely we can assign a ν-set to every bornology. 3. generalized metrics the goal of this section is to show that for every bornology a on a set x we can find an algebraic structure m and a generalized metric d : x ×x → m so that a is the set of bounded subsets of x. since d is a metric, at minimum m must be equipped with a binary operation + and have a 0. also, there should be an order on m that is compatible with +. © agt, upv, 2022 appl. gen. topol. 23, no. 1 23 h. pajoohesh there is a long tradition of allowing metrics to take values in structures more general than the non-negative reals or to satisfy weaker axioms. kelley, in [13], lists several references going back as far as [18] in 1941. for more recent related work see [8], [9], [10], [15], [14], [17], [19], and [20]. here we study generalized metrics that satisfy all of the axioms of metrics except that their values are in a po-monoid equipped with a partial order rather than [0,∞). the motivation of this paper comes from our some previous work in generalized metrics. in [17], topologies induced by positive filters on abelian lattice ordered groups were defined and used to show that some of the topologies on c(x) such as the m-topology or the uniform topology are obtained that way. later in [20], k-metrics, another generalization of metrics, were defined and topologies induced by positive filters on general `-groups were studied and the methods developed there were used later on in [16] to show that even though non-t0 completely regular spaces cannot be subspaces of powers of [0, 1] similar results can be obtained by replacing r with a non-archimedean partially ordered group, which can be given a natural euclidean-like bitopological structure. to pursue our goal here it seems that partially ordered monoids, po-monoids for short, are the natural candidates for the values of the generalized metric we are after. we bring the following definitions from [6]. definition 3.1. the structure (m, +, 0,≤) is an (abelian) po-monoid if m 6= {0} and (m, +, 0) is an (abelian) monoid which is equipped with a partial order ≤ such that for every a,x,y ∈ m, x ≤ y implies a+x ≤ a+y and x+a ≤ y+a. we call the po-monoid m abelian, if x + y = y + x for every x,y ∈ m. a po-monoid is a good candidate for the values of a metric and can be used as a set of radii for inducing topology as it is equipped with an order and a binary operation. example 3.2. both (r, +,≤) and ([0, 1],⊕,≤), where ⊕ is the truncated sum, are abelian po-monoids. another example is g = r×m with the lexicographic order and coordinatewise addition, where m is any po-monoid. let m to be the set of order preserving maps on a poset p with composition of functions. for f,g ∈ m define f ≤ g provided f(x) ≤ g(x) for every x ∈ p. then m with the composition of functions and ≤ is a non-abelian po-monoid. definition 3.3. let (m, +, 0,≤) be po-monoid such that 0 ≤ m for every m ∈ m. we define an m-metric on a set x to be a map d : x × x → m satisfying the axioms for a metric, except that it maps into m rather than [0,∞). metrics on a set induce a topology on the set. here we define a topology induced by a generalized metric into a monoid m. by m+ we mean {r ∈ m : r > 0}. © agt, upv, 2022 appl. gen. topol. 23, no. 1 24 representations of bornologies definition 3.4. let (m, +, 0,≤) be a po-monoid with 0 the smallest element, e be a subset of m+, and d : x ×x → m be an m-metric. for every r ∈ e let nr(x) = {y ∈ x : d(x,y) < r} and nr(x) = {y ∈ x : d(x,y) ≤ r}. we call sets of the form nr(x) open balls and sets of the form nr(x) closed balls. define τe = {t ⊆ x : ∀x ∈ t∃r ∈ e such that nr(x) ⊆ t}∪{x}. we say a set a is e-bounded if there is an r ∈ e such that a×a ⊆{(x,y) : d(x,y) < r}. we call a bounded if a is m+-bounded. theorem 3.5. let (m, +, 0,≤) be a po-monoid with 0 the smallest element, e ⊆ m+, and d : x ×x → m be an m-metric on the set x. then • if e is a down directed set, τe is a topology on x. • if m = ↓↓e = {x : ∃e ∈ e such that x < e} then the set of e-bounded subsets of x form a bornology. proof. for the first part suppose e is a down directed set. we show that τe is a topology on x. it is straightforward to show that τe is closed under union and x,∅ ∈ τe. we show that τe is closed under intersection. let a,b ∈ τe and x ∈ a∩b, then there are r,s ∈ e such that nr(x) ⊆ a and ns(x) ⊆ b. since e is a down directed set, there is a t ∈ e such that t ≤ r,s. now one can easily see that nt(x) ⊆ a∩b. hence, τe is a topology. we now show the second part. assume m = ↓↓e = {x : ∃e ∈ e such that x < e}. we show the collection of e-bounded sets is a bornology. suppose that a and b are bounded. we prove that a ∪ b is bounded. since a and b are bounded, there are r,s ∈ e such that d(x,y) < r for every x,y ∈ a and d(x,y) < s for every x,y ∈ b. let a be a fixed element of a and b be a fixed element of b. thus, for every x ∈ a and y ∈ b, we have 0 ≤ d(x,y) ≤ d(x,a) + d(a,b) + d(b,y) ≤ r + d(a,b) + d(b,y) ≤ r + d(a,b) + s. since m = ↓↓e, there is an e ∈ e such that r + d(a,b) + s < e. thus, for every x,y ∈ a∪b we have d(x,y) < e and therefore, a∪b is e-bounded. it is obvious that if a is e-bounded and h ⊆ a then h is e-bounded. finally note that the union of all e-bounded sets is x as every singleton set is e-bounded. � corollary 3.6. let (m, +, 0,≤) be a po-monoid with 0 the smallest element, e ⊆ m+, and d : x×x → m be an m-metric on the set x. if m+ = ↓↓m+ = {x : ∃e > 0 such that x < e} then the set of bounded subsets of x form a bornology. theorem 3.7. let a be a bornology on a set x. then there is an abelian po-monoid (m, +, 0,≤) with 0 the smallest element and an m-metric d on x such that a is the set of bounded subsets of d. proof. suppose a is a bornology on a set x. for each k ∈ a let sk = [0,∞)× [0,∞). define +k : sk ×sk → sk coordinatewise and let ≤k be the lexicographic order. let s = ∏ k∈ask. then s with the coordinatewise addition and coordinatewise order forms a po-monoid. define m ⊆ s by, © agt, upv, 2022 appl. gen. topol. 23, no. 1 25 h. pajoohesh m = {((x1k,x 2 k))k∈a ∈ s : ∃c ∈a such that x 2 k = 0 for every k ⊇ c}. we show that m with inherited addition and order from s is a po-monoid. it is enough to show that m is a monoid. note that (0, 0)k∈a ∈ m because by letting c = ∅ we have x2k = 0 for every k and obviously it is the smallest element of m. suppose ((x1k,x 2 k))k∈a and ((y 1 k,y 2 k))k∈a are elements of m. then, by definition of m there are c,d ∈ a such that x2k = 0 for every k ⊇ c and y2k = 0 for every k ⊇ d. it is straightforward to verify that x2k + y 2 k = 0 for every k ⊇ c ∪ d ∈ a and we leave it to the reader. therefore, ((x1k,x 2 k))k∈a + ((y 1 k,y 2 k))k∈a ∈ m. thus, m is a monoid. define d : x × x → m by (d(x,y))k = (0, |ϕk(x) − ϕk(y)|), where ϕ is defined as in theorem 2.6. first we show that d is well defined. note that whenever k ⊇ {x,y} we have ϕk(x) = ϕk(y) = 0 and therefore, |ϕk(x) − ϕk(y)| = 0. consequently d(x,y) ∈ m. next we prove that d is an m-metric. one can easily see that for every x,y ∈ x, d(x,x) = 0 and d(x,y) = d(y,x). we show that d(x,z) ≤ d(x,y) + d(y,z) for every x,y,z ∈ x. due to the definition of d, it is enough to show that for every k ∈ a the case of d(x,z)2k = 1 and d(x,y)2k = d(y,z) 2 k = 0 is impossible. note that d(x,z) 2 k = 1 implies that one of x and z belongs to k and the other one does not belong. without loss of generality, suppose that x ∈ k and z 6∈ k. then d(x,y)2k = 0 and x ∈ k implies y ∈ k. on the other hand, d(y,z)2k = 0 and z 6∈ k implies y 6∈ k, which is in contradiction with y ∈ k. thus, the case d(x,z)2k = 1 and d(x,y)2k = d(y,z) 2 k = 0 is not possible and therefore the triangularity condition holds. next, we show that d(x,y) = ((0, 0))k∈a implies x = y. note that if x 6= y then (d(x,y)){x} = (0, |ϕ{x}(x) −ϕ{x}(y)|) = (0, |0 − 1|) 6= (0, 0) and therefore d(x,y) 6= ((0, 0))k∈a. consequently, d is an m-metric. next we show that the set of bounded subsets of x equals a. suppose h ∈ a. we show that h × h ⊆ {(x,y) : d(x,y) < th}, where (th)k ={ (0, 1), if k ⊇ h; (1, 1), if k 6⊇ h. . note that for every x,y ∈ h, dk(x,y) = (0, |ϕk(x) − ϕk(y)|) < (1, 1) = (th)k when k 6⊇ h. on the other hand x,y ∈ h implies x,y ∈ k when k ⊇ h. therefore, in this case dk(x,y) = (0, |ϕk(x) −ϕk(y)|) = (0, 0) < (th)k. thus, d(x,y) < th for every x,y ∈ h and therefore h is bounded. for the reverse inclusion suppose that a is a bounded subset of x. thus, there is a t = ((t1k, t 2 k))k∈a ∈ m such that d(x,y) < (t 1 k, t 2 k) for every x,y ∈ a. by definition there is a q ∈ a such that t2k = 0 for every k ⊇ q. we prove that a ⊆ q. assuming the contrary, if a 6⊆ q then there is a z ∈ a \ q. if a = {z} obviously a ∈ a and we are done. if a 6= {z}, consider y ∈ a \ {z}. then, r = q ∪{y} ∈ a. then since r ⊇ q, we have (t1r, t 2 r) = (0, 0). now, dr(y,z) = (0, |ϕr(y) − ϕr(z)|) = (0, |0 − 1|) 6< (0, 0), which is a contradiction. thus, a equals the set of bounded sets of the metric d. � © agt, upv, 2022 appl. gen. topol. 23, no. 1 26 representations of bornologies in the previous theorem for every a in the bornology a define bta = {p ⊆ x : ∀x,y ∈ p,d(x,y) < ta}, where (ta)k = { (0, 1), if k ⊇ a; (1, 1), if k 6⊇ a., then proposition 3.8. let a be a bornology on the set x then bta =↓ a = {p : p ⊆ a} for every a ∈a. remark 3.9. for a bornology a on a set x let s = ∏ k∈a[0,∞). then s with the coordinatewise addition and coordinatewise order forms a po-monoid. define m ⊆ s by, m = {(xk)k∈a ∈ s : ∃c ∈a such that xk = 0 for every k ⊇ c}. similarly, it can be shown that m with inherited addition and order from s is a po-monoid and (0)k∈a is in m. next define d : x × x → m by (d(x,y))k = |ϕk(x) −ϕk(y)|. similarly, d is an m-metric on x. then by an argument similar to the one in the previous theorem one can verify that for every a ∈ a we have, a × a ⊆ {(x,y) : d(x,y) ≤ ta}, where tak = { 0, if k ⊇ a; 1, if k 6⊇ a. 4. the lattice of bornologies on a set in this section we consider, bx, the set of bornologies on a set x. we prove some of the properties of it as a lattice. the ultimate goal of this section is to prove that bx is the join-completion of p(x) modulo finite sets. the set of bornologies on a set x, bx, forms a complete lattice, where f is the smallest element and p(x) is the largest element; for b1,b2 ∈bx we have b1 ∧b2 = b1 ∩b2 and b1 ∨b2 = {b1 ∪b2 : b1 ∈ b1,b2 ∈ b2}. so, one can see that if s ⊆bx, then ∨ s = {b1 ∪·· ·∪bn : bi ∈bi ∈s}; see [4]. a complete lattice l is a frame if a∧ ∨ s = ∨ s∈s(a∧s) for every a ∈ l and s ⊆ l. a frame x is normal if x ∨ y = 1 implies the existence of u and v in x satisfying u∧v = 0 and x∨u = y ∨v = 1. theorem 4.1. the set of bornologies on a set x is a normal frame. proof. it is enough to show that a∩ ∨ s ⊆ ∨ {a∩b : b ∈s}. let h ∈a∩ ∨ s. there are bi ∈ bi ∈ s, i = 1, · · · ,n such that h = b1 ∪ ·· · ∪ bn. so, h = (h ∩ b1) ∪ ·· · ∪ (h ∩ bn). since h ∩ bi ∈ a∩bi, we conclude that h ∈ ∨ {a∩b : b ∈s}. for normality, suppose that a∨b = p(x). if a = f or b = f then we are done. so, assume that a 6= f and b 6= f. since a∨b = p(x), there are r ∈ a and s ∈ b such that r ∪ s = x. now, let u = f ∪{s} and v = f ∪{r\s}. obviously, u ∩v = f, a∨u = p(x), and b∨v = p(x). � in [1] they define more general structures which they call l-bornologies and they show that the set of l-bornologies is a frame. however, they do not state © agt, upv, 2022 appl. gen. topol. 23, no. 1 27 h. pajoohesh that it is a frame. they just prove the distributivity of meet over arbitrary join. consider p(x) and define the relation / on p(x) by a/b if (a\b)∪(b\a) is finite. let [p(x)] be the set of equivalence classes of the relation /. define v on [p(x)] by [h] v [k] if and only if h \k is finite. lemma 4.2. let x be a set. then [p(x)] with v forms a lattice. proof. it is straightforward to show that v is well defined and we will leave it to the reader. obviously, for every [h] we have [h] = [h]. we leave to the reader to prove that v is antisymmetric. we show that v is transitive. assume [g] v [h] and [h] v [k] then [g] v [k]. note that since g \ k ⊆ (g\h)∪(h\k) and both g\h and h\k are finite, we have g\k is finite. thus, [g] v [k]. therefore, v is transitive. next we show that [h ∪k] = [h] ∨ [k] and [h ∩k] = [h] ∧ [k] for every [h], [k] ∈ [p(x)]. note that h ⊆ k implies [h] v [k]. thus, [h ∩ k] v [h]∧[k] v [h]∨[k] v [h∪k] for every [h], [k] ∈ [p(x)]. we now show that [h ∪k] v [h] ∨ [k]. let [m] ∈ [p(x)] be such that let [h], [k] v [m]. since h \m is finite and k \m is finite, (h ∪k) \m is finite. so, [h ∪k] v [m]. similarly, if [p ] ∈ p(x) is such that [p ] v [h], [k] then p \ h is finite and p \k is finite. so, p \ (h ∩k) is finite. thus, [p ] v [h ∩k]. � let bx be the set of bornologies on the set x. define θ : [p(x)] → bx by θ([h]) =↓ h ∪ f, where f is the bornology of finite subsets of x and ↓ h = {y : y ⊆ h}. recall that bornologies of ↓ h ∪f form are called principal bornologies. lemma 4.3. the map θ : [p(x)] →bx is an embedding. proof. it is straightforward to show that θ is well defined and order preserving and we will leave it to the reader. note that if θ([h]) ⊆ θ([k]) then h ∈ θ([k]) and therefore, h = z ∪e where z ⊆ k and e is a finite subsets of x. thus, h \k is finite and therefore, [h] v [k]. � corollary 4.4. if i(bx) is the set of principal bornologies of a set x, then θ : [p(x)] → i(bx) is an isomorphism. definition 4.5. let l be a lattice and c be a complete lattice. we say c is a join-completion of l if l is a sublattice of c and every element of c is join of elements of l. the concept of a meet-completion is defined dually. the join-completions and meet-completions were introduced by banaschewski in [2] and were extensively studied and extended by schimidt in [21] and [22]. theorem 4.6. for every set x, the frame bx is a join-completion of i(bx); it is a meet-completion if and only if x is finite. further, if l is another complete lattice which is join-completion of i(bx), there is a unique isomorphism from bx to l fixing i(bx) and therefore, l must be a frame. © agt, upv, 2022 appl. gen. topol. 23, no. 1 28 representations of bornologies proof. let a be a bornology on a set x. then, a = ∨ h∈a(↓ h ∪f) and therefore, i(bx) is join-dense in l. next we show that if x is infinite then i(bx) is not meet-dense in bx. it is enough to show that i(b(z)) is not meet-dense in bz. let a = ∨ p∈p (↓ pz∪f), where p is the set of prime numbers. we show that a cannot be the infimum of principal bornologies. suppose to the contrary that a = ∧ i∈i(↓ hi ∪f) =⋂ i∈i(↓ hi∪f). then, pz ∈ (↓ hi∪f) for every p ∈ p and every i ∈ i. thus,⋃ p∈p pz ∈ (↓ hi ∪f) for every i ∈ i. therefore, ⋃ p∈p pz ∈ ⋂ i∈i(↓ hi ∪f). but ⋃ p∈p pz 6∈a which is a contradiction. if i : i(bx) → l is an embedding, define f : bx → l by f(a) = ∨ h∈a i(↓ h ∪f). then f is an isomorphism. � corollary 4.7. the set of bornologies on a set x is the join-completion of [p(x)]. acknowledgements. i would like to thank jerry beer for answering my questions as well as reading and commenting on this manuscript references [1] m. abel and a. šostak, towards the theory of l-bornological spaces, iran. j. fuzzy syst. 8, no. 1 (2011), 19–28. 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[10] r. c. flagg and r. kopperman, continuity spaces: reconciling domains and metric spaces, chic. j. theoret. comput. sci. 77 (1977), 111–138. [11] h. hogbe-nlend, bornologies and functional analysis, north-holland, amsterdam-new york-oxford, 1977. [12] s. t. hu, boundedness in a topological space, j. math. pures appl. 228 (1949), 287–320. [13] j. l. kelley, general topology, d. van nostrand, 1955. [14] r. kopperman, all topologies come from generalized metrics, am. math. monthly 95 (1988), 89–97. © agt, upv, 2022 appl. gen. topol. 23, no. 1 29 h. pajoohesh [15] r. kopperman, s. matthews and h. pajoohesh, partial metrizability in value quantales, applied general topology 5, no. 1 (2004), 115–127. [16] r. kopperman and h. pajoohesh, representing topologies using partially ordered semigroups, topology and its applications 249 (2018), 135–149. [17] r. kopperman, h. pajoohesh and t. richmond, topologies arising from metrics valued in abelian `-groups, algebra universalis 65 (2011), 315–330. [18] j. p. lasalle, topology based upon the concept of pseudo-norm, proc. nat. acad. sci. 27 (1941), 448–451. [19] s. g. matthews, partial metric topology, in: proc. 8th summer conference on topology and its applications, ed s. andima et al., new york academy of sciences annals 728 (1994), 183–197. [20] h. pajoohesh, k-metric spaces, algebra universalis 69, no. 1 (2013), 27–43. [21] j. schmidt, universal and internal properties of some completions of k-join-semilattices and k-join-distributive partially ordered sets, j. reine angew. math. 255 (1972), 8–22. [22] j. schmidt, each join-completion of a partially ordered set is the solution of a universal problem, j. austral. math. soc. 17 (1974), 406–413. © agt, upv, 2022 appl. gen. topol. 23, no. 1 30 protaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 283-291 cellularity and density of balleans i. v. protasov abstract. a ballean is a set x endowed with some family f of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. then we define the asymptotic counterparts for dense and open subsets, introduce two cardinal invariants (density and cellularity) of balleans and prove some results concerning relationship between these invariants. we conclude the paper with applications of obtained partitions of left topological group in dense subsets. 2000 ams classification: 54a25, 54e25, 05a18. keywords: ballean, large and thick subsets, density, cellularity. 1. introduction every infinite group g can be partitioned in |g|-many subsets dense in every totally bounded group topology on g. in [5] this statement was extracted from the following combinatorial claim. for every infinite group g there exists a disjoint family f of cardinality |g| such that, for every f ∈ f and every finite subset k of g, there exists g ∈ f such that kg ⊆ f . each subset f ∈ f looks like a set with non-empty interior in some structure dual to uniform topological space. to explain this duality we need some definitions and notations. a ball structure is a triple b = (x,p,b) where x, p are non-empty sets and, for any x ∈ x and α ∈ p , b(x,α) is a subset of x which is called a ball of radius α around x. it is supposed that x ∈ b(x,α) for all x ∈ x, α ∈ p . the set x is called the support of b, p is called the set of radii. given any x ∈ x, a ⊆ x, α ∈ p , we put b∗(x,α) = {y ∈ x : x ∈ b(y,α)},b(a,α) = ⋃ a∈a b(a,α). 284 i. v. protasov a ball structure is called • lower symmetric if, for any α,β ∈ p , there exist α′,β′ ∈ p such that, for every x ∈ x, b∗(x,α′) ⊆ b(x,α),b(x,β′) ⊆ b∗(x,β); • upper symmetric if, for any α,β ∈ p , there exist α′,β′ ∈ p such that, for every x ∈ x, b(x,α) ⊆ b∗(x,α′),b∗(x,β) ⊆ b(x,β′); • lower multiplicative if, for any α,β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x,γ),γ) ⊆ b(x,α) ∩ b(x,β); • upper multiplicative if, for any α,β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x,α),β) ⊆ b(x,γ). let b = (x,p,b) be a lower symmetric and lower multiplicative ball structure. then the family { ⋃ x∈x b(x,α) × b(x,α) : α ∈ p } is a base of entourages for some (uniquely determined) uniformity on x. on the other hand, if u ⊆ x × x is a uniformity on x, then the ball structure (x,u,b) is lower symmetric and lower multiplicative, where b(x,u) = {y ∈ x : (x,y) ∈ u}. thus, the lower symmetric and lower multiplicative ball structures can be identified with the uniform topological spaces. a ball structure is said to be a ballean if it is upper symmetric and upper multiplicative. in entourage form the balleans arouse in coarse geometry [10] under name coarse structures and independently in combinatorics [6] under name uniform ball structures. now we define the mappings which play the parts of uniformly continuous and uniformly open mappings on the ballean stage. let b1 = (x1,p1,b1) and b2 = (x2,p2,b2) be balleans. a mapping f : x1 → x2 is called a ≺ −mapping if, for every α ∈ p1, there exists β ∈ p2 such that, for every x ∈ x1, f(b1(x,α)) ⊆ b2(f(x),β). a mapping f : x1 → x2 is called ≻-mapping if, for every β ∈ p2, there exists α ∈ p1 such that, for every x ∈ x1 b2(f(x),β) ⊆ f(b1(x,α)). if f : x1 → x2 is a bijection such that f is a ≺-mapping and f is a ≻mapping, we say that f is an asymorphism and b1, b2 are asymorphic. given an arbitrary ballean b = b(x,p,b), we can replace every ball b(x,α) by b∗(x,α) ∩ b(x,α) and get an asymorphic ballean in which b∗(x,α) = cellularity and density of balleans 285 b(x,α). in what follows we shall assume that b∗(x,α) = b(x,a) for all x ∈ x, α ∈ p . we need also some classification of subsets of x for a ballean b = (x,p,b). given a subset a ⊆ x, we say that a is • large if there exists α ∈ p such that x = b(a,α); • small if x \ b(a,α) is large for every α ∈ p ; • thick if, for every α ∈ p there exists a ∈ a such that b(a,α) ⊆ a. for some special balleans these types of subsets were introduced in [1] and [2]. we note also that large, small and thick subsets of a ballean may be considered as asymptotic duplicates of dense, nowhere dense and subsets with non-empty interior of uniform spaces. following this (non-formal) duality between uniform spaces and balleans, we define the density d(b) and cellularity c(b) as d(b) = min{|l| : l ⊆ x,l is large}, c(b) = sup{|f| : f is a disjoint family of thick subsets of x}. as in the case of uniform spaces, density of a ballean is much more easy to calculate or evaluate than its cellularity, so our main goal is to find some relationships between d(b) and c(b). 2. observations (1) let b = (x,p,b) be a ballean, t be a thick subset of x and l be a large subset of x. then there exists α ∈ p such that x = b(l,α) and b(x,α) ⊆ t for some x ∈ t , so l ∩ t 6= ∅. since every large subset meets every thick subset, we have c(b) 6 d(b). (2) given α ∈ p and y ⊆ x, we say that y is α-discrete if the family {b(y,α) : y ∈ y } is pairwise disjoint. by zorn lemma, every αdiscrete subset y of x is contained in some maximal (by inclusion) α-discrete subset z of x. if y ∈ x then b(y,α) ∩ b(z,α) 6= ∅. we choose β ∈ p such that b(b(x,α),α) ⊆ b(x,β) for every x ∈ x. then y ∈ b(z,β) and z is large. on the other hand, let l be a large subset of x, x = b(l,α) and let z be a maximal α-disjoint subset of y . then |z| 6 |l| and z is large. hence, d(b) can be defined as the minimal cardinality of maximal α-disjoint subsets of x where α runs over p . (3) let (x,d) be a metric space. given any x ∈ x, n ∈ ω, we put bd(x,n) = {y ∈ x : d(x,y) 6 n} and say that b(x,d) = (x,ω,bd) is a metric ballean. a ballean b is called metrizable if b is asymorphic to some metric ballean. to characterize metrizable balleans we need two definitions. a ballean b = (x,p,b) is called connected if, for any x,y ∈ x, there exists α ∈ p such that y ∈ b(x,α). 286 i. v. protasov we define the preordering 6 on p by the rule: α 6 β if and only if b(x,α) ⊆ b(x,β) for every x ∈ x. a subset p ′ ⊆ p is called cofinal if, for every α ∈ p there exists β ∈ p ′ such that α 6 β. the cofinality cf(b) is the minimal cardinality of the cofinal subsets of p . by [6, theorem 9.1], a ballean b is metrizable if and only if b is connected and cf(b) 6 ℵ0. for approximation of arbitrary balleans via metrizable balleans see [7]. (4) a connected ballean b = (x,p,b) is called ordinal if there exists a well-ordered by ≤ cofinal subset of p . replacing p to its minimal cofinal subset, we get the asymorphic ballean. hence, we can write b as (x,β,b), where β is a regular cardinal (considered as a set of ordinals). we note that every metrizable ballean is ordinal, and metric balleans are the main subject of asymptotic topology [3]. (5) a subset y ⊆ x is called bounded if there exist y ∈ y and α ∈ p such that y ⊆ b(y,α). a ballean is called bounded if its support is bounded. clearly, d(b) = c(b) = 1 for every bounded ballean b. 3. results theorem 3.1. for every ordinal ballean b, c(b) = d(b) and there exists a disjoint family f of cardinality d(b) consisting of thick subsets of x. proof. let b = (x,ρ,b), κ = d(b) and cf(κ) be the cofinality of κ. if b is bounded, we use observation 5, so we assume that b is unbounded. we fix some element x0 ∈ x and consider four cases. case ρ < cfκ. we prove the following auxiliary statement. for every α < ρ, there exist β, α < β < ρ and an α-discrete subset yα of x such that x0 ∈ yα, b(yα,α) ⊆ b(x0,β) and |yα| = κ. let z be a maximal α-discrete subset of x such that x0 ∈ z. by observation 2, |z| > κ. for every λ < ρ, we put zλ = z ∩ b(x0,λ). since z = ⋃ λ<ρ zλ, |z| > κ and ρ < cf(κ), there exists µ < ρ such that |zµ| > κ. we choose β < ρ such that α < β and b(b(x0,µ),α) ⊆ b(x0,β). then b(zµ,α) ⊆ b(x0,β) and we can choose a subset yα ⊆ zµ such that x0 ∈ y and |yα| = κ. using the auxiliary statement and regularity of ρ, we can define inductively a mapping f : ρ → ρ and a family {yf (α) : α < ρ} of subsets of x such that, for every α < ρ, f(α) > α, yf (α) is f(α)-discrete, |yf (α)| = κ and b(yf (α),f(α)) ⊆ b(x0,f(α + 1)) \ b(x0,f(α)). for every α < ρ, we enumerate yf (α) = {y(f(α),λ) : λ < κ} and, for every λ < κ, put tλ = ⋃ α<ρ b(y(f(α),λ),f(α)). clearly, every subset tλ is thick and the family {tλ : λ < κ} is disjoint. cellularity and density of balleans 287 case cf(κ) 6 ρ < κ. we prove the following auxiliary statement. for any α < ρ and k′ < κ, there exist β, α < β < ρ and an α-discrete subset yα of x such that x0 ∈ yα, b(yα,α) ⊆ b(x0,β) and |yα| > κ ′. let z be a maximal α-discrete subset of x such that x0 ∈ z. by observation 2, |z| > κ. let i be a cofinal subset of κ such that |i| = ρ. for every λ ∈ i, we put zλ = z ∩ b(x0,λ). clearly, z = ⋃ λ∈i zλ. if |zλ| 6 κ ′ for every λ ∈ i, then |z| 6 κ′|i| = κ′ρ < κ. hence, there exists µ ∈ i such that |zµ| > κ ′. we choose β < ρ such that α < β and b(b(x0,µ),α) ⊆ b(x0,β). then b(zµ,α) ⊆ b(x0,β) and we put yα = zµ. let ϕ : ρ → κ be an injective mapping such that ϕ(ρ) is cofinal in κ and α < β < ρ implies ϕ(α) < ϕ(β) < κ. using the auxiliary statement and regularity of κ, we can define inductively a mapping f : ρ → ρ and a family {yf (α) : α < ρ} of subsets of x such that • (i) f(ρ) is cofinal in ρ and α < β < ρ implies f(α) < f(β) < ρ; • (ii) yf (α) is f(α)-discrete; • (iii) b(yf (α),f(α)) ⊆ b(x0,f(α + 1)) \ b(x0,f(α)); • (iv) |yf (α)| = ϕ(α). for every α < ρ, we enumerate yf (α) = {y(f(α),λ) : λ < ϕ(α)}. then for any α and γ such that ϕ(α) 6 γ < ϕ(α + 1), we put tγ = ⋃ { b(f(β),γ) : α + 1 < β < ρ } . by (i), tγ is thick. by (ii) and (iii), the family f = { tγ : ϕ(α) 6 γ < ϕ(α + 1),α < ρ } is disjoint. since ϕ(ρ) is cofinal in κ, by (iv), we have |f| = κ. case ρ = κ. using the assumption, we can construct inductively the subset {yα : α < κ} of x such that the family {b(yα,α) : α < κ} is disjoint. then we partition κ = ⋃ λ<κ iλ into κ cofinal subsets and, for every λ < κ, put tλ = ⋃ { b(yα,α) : α ∈ iλ } . clearly, every subset tλ is thick and the family {tλ : λ < κ} is disjoint. case ρ > κ. we show that this variant is impossible. suppose the contrary. let z be a large subset of x such that |z| = κ and x = b(z,α). for every z ∈ z, we pick α(z) < ρ such that b(z,α) ⊆ b(x0,α(z)). since ρ is regular and κ < ρ, there exists β < ρ such that β > α(z) for every z ∈ z. then b(z,α) ⊆ b(x0,β) for every z ∈ z, so x = b(x0,β) and b is bounded. � corollary 3.2. for every metrizable ballean b, c(b) = d(b) and there exists a disjoint family f of cardinality d(b) consisting of thick subsets of x. theorem 3.3. let b = (x,p,b) be a ballean, |x| = κ and let |p | 6 κ. then c(b) = d(b) = κ and there exists a disjoint family f of cardinality κ consisting of thick subsets of x provided that one of the following conditions is satisfied: • (i) there exists κ′ < κ such that b(x,α) 6 κ′ for all x ∈ x and α ∈ p; • (ii) |b(x,α)| < κ for all x ∈ x, α ∈ p and κ is regular. 288 i. v. protasov proof. let l be a large subset of x, α ∈ p and x = b(l,α). then each of the assumptions (i) and (ii) gives |l| = κ so d(b) = κ. (i) let z be a subset of x such that |z| < κ, α ∈ p . let y be a maximal (by inclusion) α-discrete subset of x such that b(z,α) ∩ b(y,α) = ∅. if x ∈ x then b(x,α) ∩ b(z ∪ y,α) 6= ∅. hence, z ∪ y is large and |y | = κ. we fix a bijection f : p × κ → κ and note that, for every α ∈ p , the set f(α,κ) is cofinal in κ (as a set of ordinals). we define also two mappings ϕ : κ → p and ψ : κ → κ be the rule: if f(α,λ) = γ then ϕ(γ) = α, ψ(γ) = λ. we take an arbitrary ϕ(0)-discrete subset y0 of x such that |y0| = ψ(0). assume that, for some γ < κ, we have defined the family {yλ : λ < γ} of subsets of x such that each subset yλ is ϕ(λ)-discrete, |yλ| = ψ(λ) and the family {b(yλ,ϕ(λ)) : λ < γ} is disjoint. put z = ⋃ λ<γ b(yλ,ψ(λ)). in view of above paragraph there exists a ϕ(γ)-discrete subset yγ such that |yγ| = ψ(γ) and z ∩ b(yγ,ϕ(γ)) = ∅. after κ steps we get the family {yγ : γ < κ}. for every γ < κ, we enumerate yγ = {y(λ,γ) : λ < |yγ|} and put t0 = ⋃ γ<κ b(y(0,γ),ϕ(γ)). since ϕ is surjective, t0 is thick. assume that, for some δ < κ, we have defined disjoint family {tµ : µ < δ} of thick subsets of x. put t = ⋃ µ<δ tµ. to define tδ we denote i = {γ : γ < κ,yγ \t 6= ∅} and put tδ = ⋃ γ∈i b(y(δ,γ),ϕ(γ)). after κ steps we put f = {tδ : δ < γ}. since f(α,κ) is cofinal in κ for every α ∈ p , every subset tδ is thick. (ii) let γ < κ and {yλ : λ < γ} be a family of subsets of x such that |yλ| < κ for every λ < γ. let {pλ : λ < γ} be a subset of p . we put z = ⋃ λ<γ b(yλ,λ). by (ii), |z| < κ. hence, for any α ∈ p and κ′ < κ, we can take an α-disjoint subset y(γ,α) of x such that b(y (γ,α),α) ∩z = ∅ and |y (γ,α)| > κ ′. using this remark, we can construct the family f as in (i). � 4. examples we show that, for every infinite cardinal κ, there exists a metric space x such that d(b(x)) = κ. example 4.1. let i be a (non-directed) graph with the set of vertices ω and the set of edges {(i, i+ 1) : i ∈ ω}. we consider the set {iγ : γ < κ} of copies of i, identify the terminal vertices of these copies and denote by γ the resulting graph. let x = v (γ) be the set of vertices of γ. we endow x with path metric: the distance between two vertices u,v ∈ x is the length of the shortest path between u and v. if l is a large subset of x, then l∩v (iγ ) is infinite for every γ < κ, so |l| = κ and d(b(x)) = κ. the next two examples show that cellularity of a ballean could be much more smaller than its density. cellularity and density of balleans 289 example 4.2. let x be a set and ϕ be a filter on x. for any x ∈ x and f ∈ ϕ, we put bϕ(x,f) = { {x}, if x ∈ f ; x \ f , if x /∈ f and consider the ballean b(x,ϕ) = (x,ϕ,bϕ). a ballean b = (x,p,b) is called pseudodiscrete if, for every α ∈ p , there exists a bounded subset v of x such that b(x,α) = {x} for every x ∈ x \ v . by [8], a ballean b is pseudodiscrete if and only if there exists a filter ϕ on x such that b is asymorphic to b(x,ϕ). now let x be infinite and ∩ϕ = ∅. then b(x,ϕ) is an unbounded connected ballean. a subset l ⊆ x is large if and only if l ∈ ϕ, so d(b) = min{|f | : f ∈ ϕ}. on the other hand, let t be a thick subset of x, f ∈ ϕ. we take x ∈ x such that bϕ(x,f) ⊆ x. then either x ∈ f or x \ f ⊆ t . it follows that t is cofinal with respect to ϕ, i.e.: f ∩ t 6= ∅ for every f ∈ ϕ. hence, if ϕ is an ultrafilter then any two thick subsets of x have non-empty intersection and c(b(x,ϕ)) = 1. example 4.3. let x be an infinite set of regular cardinality κ. denote by f the family of all subsets of x of cardinality < κ. let p be a set of all mappings f : x → f such that, for every x ∈ x, we have x ∈ f(x) and ∣ ∣ { y ∈ x : x ∈ f(y) } ∣ ∣ < κ. given any x ∈ x and f ∈ p , we put b(x,f) = f(x) and consider the ball structure b = (x,p,b). since b∗(x,f) = {y ∈ x : x ∈ f(y)}, b is upper symmetric. since κ is regular, b is upper multiplicative. hence, b is a ballean. clearly, b is connected and unbounded. if l is a large subset of x, by regularity of κ, we have |l| = κ. if a is a subset of x and |a| = κ, we fix an arbitrary bijection h : a → x and put f(x) = { {x}, if x /∈ a; {x,h(x)}, if x ∈ a then f ∈ p and b(a,f) = x. hence, a subset l of x is large if and only if |l| = κ, so d(b) = κ. if a ⊆ x and |x \ a| = κ, by observation 1 and above paragraph, a is not thick. it means that any two thick subsets of x are not disjoint and c(b) = 1. now we compare cellularity and density with another cardinal invariant of balleans, namely resolvability, defined in [9]. given a ballean b = (x,p,b) and a cardinal κ, we say that b is κresolvable if x can be partitioned in κ-many large subsets. the resolvability of b is the cardinal r(b) = sup{κ : b is κ-resolvable}. if b is a ballean from example 4.1, by [9, theorem 2.3], r(b) = ℵ0. by corollary 3.2, d(b) = c(b) = κ. if b is a ballean from example 4.2, defined by free ultrafilter, then c(b) = r(b) = 1, but d(b) = min{|f | : f ∈ ϕ}. 290 i. v. protasov if b is a ballean from example 4.3, then r(b) = d(b) = κ, c(b) = 1. the above remarks show that there are no direct correlations between resolvability on one hand and density or cellularity on the other hand. 5. applications let g be a group with the identity e endowed with some topology. then g is called left topological if all the left shifts x 7→ gx, g ∈ g are continuous. let g be an infinite left topological group, |g| = κ, γ be an infinite cardinal such that γ 6 κ. we say that g is • totally bounded if, for every nieghbourhood u of e, there exists a finite subset f of g such that g = fu; • γ-bounded if, for every neighbourhood u of e, there exists a subset f of g such that |f | < γ and g = fu; • weakly bounded if, for every neighbourhood u of e, there exists a subset f of g such that |f | < κ and g = fu. in this terminology, totally bounded groups are ℵ0-bounded and weakly bounded groups are κ-bounded. we denote by fγ the family of all subsets f of x such that e ∈ f , f = f −1 and |f | < γ. given any g ∈ g and f ∈ fγ , we put b(g,f) = fg and denote by b(g,γ) the ballean (g,fγ,b). if γ = ℵ0 then a subset l is large (with respect to b(g,ℵ0)) if and only if g = fl for some finite subset f of g. by theorem 3.2 (case (ii) for κ = ℵ0 and case (i) for κ > ℵ0), there exists a disjoint family f of cardinality κ consisting of thick subsets. by observation 1, every subset f ∈ f meets every large subset l. it follows that f ∩ gu 6= ∅ for every g ∈ g and every neighbourhood u of e in every totally bounded topology τ on g, so f is dense in τ. hence, g can be partitioned to κ subsets dense in each totally bounded topology. if γ < κ, the same arguments applying to b(g,γ) and theorem 3.2 (i) prove that g can be partitioned to κ subsets dense in every γ-bounded topology on g. if γ = κ and κ is regular, we apply either theorem 3.1 or theorem 3.2 (ii) and conclude that g can be partitioned in κ-many subsets dense in every weakly bounded topology on g. what about γ = κ and κ is singular? this is old (and unsolved) problem posed by the author [4, problem 13.45] in the following weak form. problem 5.1. every infinite group g of regular cardinality κ can be partitioned g = a1 ∪a2 so that fa1 6= g and fa2 6= g for every subset f of g such that |f | < κ. is the same true for groups of singular cardinalities? cellularity and density of balleans 291 references [1] a. bella and v. i. malyhkin, small and other subsets of a group, q and a in general topology, 11 (1999), 183–187. [2] t. j. carlson, n. hindman, j. mcleod and d. strauss, almost disjoint large subsets of semigroups, topology appl., to appear. [3] a. dranishnikov, asymptotic topology, russian math. survey, 52 (2000), 71–116. [4] the kourovka notebook, amer. math. soc. transl. (2), 121, providence, 1983. [5] v. i. malykhin and i. v. protasov, maximal resolvability of bounded groups, topology appl. 73 (1996), 227–232. [6] i. protasov and t. banakh, ball structures and colorings of groups and graphs, math. stud. monogr. ser. v. 11, 2003. [7] i. v. protasov, uniform ball structures, algebra and discrete math. 2002, n1, 129–141. [8] i. v. protasov, normal ball structures, math. stud. 20 (2003), 3–16. [9] i. v. protasov, resolvability of ball structures, appl. gen. topology 5 (2004), 191–198. [10] j. roe, lectures on coarse geometry, ams university lecture series, 31, 2003. received may 2006 accepted february 2007 igor protasov (islab@unicyb.kiev.ua) department of cybernetics, kyiv university, volodimirska 64, kyiv 01033, ukraine yangagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 49-68 pointwise convergence and ascoli theorems for nearness spaces zhanbo yang ∗ abstract. we first study subspaces and product spaces in the context of nearness spaces and prove that u-n spaces, c-n spaces, pn spaces and totally bounded nearness spaces are nearness hereditary; t-n spaces and compact nearness spaces are n -closed hereditary. we prove that n2 plus compact implies n -closed subsets. we prove that totally bounded, compact and n2 are productive. we generalize the concepts of neighborhood systems into the nearness spaces and prove that the nearness neighborhood systems are consistent with existing concepts of neighborhood systems in topological spaces, uniform spaces and proximity spaces respectively when considered in the respective sub-categories. we prove that a net of functions is convergent under the pointwise convergent nearness structure if and only if its cross-section at each point is convergent. we have also proved two ascoli-arzelà type of theorems. 2000 ams classification: 54e17, 54e05, 54e15, 68u10 keywords: nearness spaces, subspace, product space, neighborhood system, pointwise convergent, ascoli’s theorem 1. introduction as a natural extension of geometry, the concept of “near/apart” has been a center for topology and related studies. topology characteries the “nearness” between a point and a set. proximity [14] is an axiomatization of “nearness” between two sets. contiguity [10] describes the concept of nearness among the elements of a finite family of sets. the concept of “nearness space” introduced by herrlich [8] in 1974 attempts to characterize the nearness of an arbitrary collection of sets. the category of nearness spaces, the most general among the aforementioned structures, can be used as a unifying framework. the ∗this work was in part supported by a grant from the 2008 faculty research fund of the university of the incarnate word. 50 z. yang categories of several aforementioned structures can all be “nicely embedded” into the category of the nearness spaces as (either bireflective or bicoreflective) sub-categories ([8]). in recent years, the notion of “nearness” in a number of different variations has found new applications in digital topology, image processing and pattern recognition areas, perhaps due to the fact that those structures are “richer” than classical topology. in 1995, latecki and prokop [12] used a weaker version of proximity spaces called semi-proximity spaces (sp-spaces). they talked about the possibility of describing all digital pictures used in computer vision and computer graphs as non-trivial semi-proximity spaces, which is not possible in classical topology. they also discussed the application of “semi-proximity continuous functions” in well-behaved operations such as thinning on digital images. in 1996, chaudhuri [4] introduced a new definition for the neighbors of an arbitrary point p . this new definition used a “centroid criterion” to capture the idea that the neighbors of p should be as near to p and as symmetrically paced around p as possible. this new definition could be used for pattern classification, clustering and low-level description of dot patterns. in 2000, ptak and kropatsch [16] discussed the application of proximity spaces in studying of digital images. they showed by examples that the “proximate complexity” of a finite covering in a digital picture might be too high to be adequately depicted in a finite topological space, which might indicate another conceptual advantage of proximities over topologies. most recently, in 2007, wolski and peters [18, 15] investigated approximation spaces in the context of topological structures which axiomatized certain notion of nearness. peters [15] pointed out particularly that the concept of “nearness” was not confined to spatial nearness, or geometrical likeness. it was possible to introduce a nearness relation that could be used to determine the “nearness” among sets of objects that were spatially far away and, yet, “qualitatively” near to each other. the main objectives of this paper are to establish a ”pointwise convergent” nearness structure on a function space made of a family of functions from x to y and to establish two versions of the ascoli-arzelà theorems for nearness spaces that relate the compactness of the underlining space y with that of the function space. since the function space is really a subspace of the product space y x , we begin with the nearness structures on a subspace and discuss the hereditary properties for a number of important concepts in nearness spaces. we then define the nearness structure on a product spaces and discuss its various properties. the nearness structure on a function space is then introduced as a subspace of the product space y x . we will end the discussion with two ascoli-arzelà type of theorems. some work in the past, such as [2] (1979), [7] (1979) and [1] (2006) have discussed a number of results with respect to subspaces and product spaces. most of the results in that paper were dealing with topological nearness (t-n ) spaces and do not duplicate what are to be presented in this paper. for the purpose of clarity and being self-contained, we will still give the definition of ”subspace” and ”product” space here and prove relevant results. pointwise convergence for nearness spaces 51 the classical ascoli-arzelà theorem was proved in the 19th century first by ascoli and then independently by arzelà. it characterizes compactness of sets of continuous real-valued functions on the interval [0,1] with respect to the topology of uniform convergence. it is commonly known that the issue came from the fact that a convergent sequence of continuous functions may not converge to a continuous function. so the natural question is: under what conditions the limit of a convergent sequence of continuous functions is still continuous. it turned out that the concept of equicontinuous was used to characterize the condition needed (see [11]) in topological spaces. in 1970, [13] discussed ascoli’s theorem for the spaces of multifunctions. in 1981, [6] discussed ascoli’s theorem for topological categories. in 1984, [3] discussed ascoli’s theorem for a class of merotopic spaces. in 1993, [5] studied a version of the ascoli’s theorem for set valued proximally continuous functions. in 2001, [17] proved a version of ascoli’s theorem for sequential spaces. as far as we know, no nearness space version of the ascoli’s theorem has been established yet at this time. we have practical reason to be interested in this topic. in many cases, a digital image processing algorithm is essentially the application of a sequence of deformation functions to a digital plane. for example, [12] proved that a deletion of a simple point (a point that does not affect the connectness of the digital picture) can be regarded as a sp-continuous function. hence a thinning algorithm that preserves connectness can be arranged as a sequence of spcontinuous functions. we may be able to use the tools of function spaces, and the results on convergence of function sequences to study the image processing algorithms, which opens a new set of doors. the rest of this paper is organized as follows: section 2 is a collection of the major definitions involving nearness spaces that are relevant to this paper. section 3 studies the nearness structures on a subspaces. section 4 is about the product spaces and the function spaces. the summary at the end concludes this paper. 2. notation and definitions in this section, we define the basic concepts used throughout this paper. we will use the language of categories in some of our discussions. for readers who are not familiar with category theory, a category is basically a family of objects with a particular type of structures. for example, we can talk about the category of all topological spaces, the category of all groups, etc. the so called “morphism” from one object to another is a function that preserves the structure on the objects. for example, a “morphism” in the category of topological spaces would be a continuous function. a ”morphism” in the category of all groups would be a homomorphism. an embedding from one category into another category is a way to assign each object from one category to an object of the other category in some injective manner that also preserves the morphisms. for readers who are interested at further information about category theory, please see [20]. 52 z. yang the readers can see kelley [11] or any common general topology text book for terms in general topology. 2.1. basic notations. let x be a set, p(x) represents the power set of x. p0(x) = x, p1(x) = p(x), . . . , pn(x) = p(pn−1(x)) a, b, . . . represent elements in p(x), i.e. subsets in x a , b, . . . represent elements in p2(x), i.e. subsets in p(x) ξ, η, . . . represent elements in p3(x), i.e. subsets in p2(x) a c = {x − a : a ∈ a } for each b ⊆ x, a (b) = ⋃ {a : a ∩ b 6= φ, a ∈ a } ξa denotes a ∈ ξ, ξa denotes a /∈ ξ aξ b denotes {a, b} ∈ ξ. aξ b denotes {a, b} /∈ ξ clξa = {x : {x} ξ a}, intξa = x − clξ(x − a) clξa = {clξa : a ∈ a }, intξa = {intξa : a ∈ a } a ∨ b = {a ∪ b : a ∈ a , b ∈ b} a ∧ b = {a ∩ b : a ∈ a , b ∈ b} a ≺ b if and only if for any b ∈ b , there is a ∈ a such that a ⊆ b. this is referred to as ”b co-refines a ”. 2.2. definitions related to nearness structure. we will restate some major definitions about nearness spaces here (due to [8]): (i) let x be a non-empty set. the ordered pair (x, ξ) is said to be a nearness space, or n -space, if the following are satisfied: (n1) if ⋂ a 6= φ, then ξa . (n2) if ξb, and for each a ∈ a , there exists a b ∈ b such that b ⊆ clξa, then ξa , i.e. b ≺ clξa . (n3) if ξa and ξb, then ξ(a ∨ b). (n4) if φ ∈ a, then ξa . (ii) let x be a set. let (x, ξ) and (y, η) be two n-spaces. a function f : x → y is said to be a (ξ, η) n-preserving map, or an n-preserving map, or simply an n-mapping, if one of the following two equivalent conditions is satisfied: (m1) if ξa , then η f (a ), where f (a ) = {f (a) : a ∈ a } (m2) if ηb, then ξ f −1(b), where f −1(b) = {f −1(b) : b ∈ b}. we will use the notation near to represent the category of all nearness spaces with n-mappings. (iii) an n-space is called an n1-space, if the following is satisfied: (n0) if {x}ξ{y}, then x = y. (iv) an n-space is called an n2 space, if for any x, y ∈ x, x 6= y implies the existence of a ⊆ x and b ⊆ x such that a ∩ b = φ, ξ{{x}, x − a} and ξ{{y}, x − b}. (v) an n-space is called a t-n space, if the following is satisfied: (t) if ξa , then ⋂ clξa 6= φ. a t − n2 space is an n2 space that also satisfies the condition (t). pointwise convergence for nearness spaces 53 we will use the symbol t − near to represent the subcategory of near, consists of all t-n spaces with n-mappings. (vi) an n-space is called a u-n space, if the following is satisfied: (u) if ξa , then there exists a b such that ξb and for each b ∈ b, there is an a ∈ a such that bc(x − b) ⊆ x − a. we will use the notation u − near to represent the subcategory of near, consists of all u-n spaces with n-mappings. (vii) an n-space is called a c-n space, if the following is satisfied: (c) if ξa , then there is a finite subcollection b ⊆ a such that ξb. we will use the notation c − near to represent the subcategory of near, consists of all c-n spaces with n-mappings. (viii) an n-space is called a p-n space, if it satisfies both of the conditions (u) and (c). we will use the notation p − near to represent the subcategory of near, consists of all p-n spaces with n-mappings. (ix) an n-space is called a totally bounded space, if one of the following equivalent conditions is satisfied: (b1) if ξa , then there is a finite subcollection b ⊆ a such that ⋂ b = φ. (b2) if f is a filter on x, then ξf . (x) an n-space is called a compact space, if it satisfies both condition (t) and (c). (xi) let x be a set. {ξα : α ∈ λ} is a family of n-structures on x. the least upper bound, denoted by ξ = sup{ξα : α ∈ λ}, is defined as follows: ξa if and only if there are finitely many ai’s such that for each i = 1, 2, ...n, ξiai, and a ≺ n ∨ i=1 ai. (xii) let (x, ξ) be a nearness space. then the topology induced by the closure operator a 7→ clηa is denoted by tξ. 3. nearness structure on subspaces we will begin by giving the definition of a ”nearness subspace”, then proceed to show that subspaces as defined here are well-defined (theorem 3.2), act ”natural” (lemma 3.3) and produces a topology that is consistent with the subspace topology (theorem 3.8). definition 3.1. let (x, ξ) be a nearness space and x0 ⊆ x. define ξ0 on x0 as follows: ξ0 = {a ⊆ p(x0) : ξa }. we will denote such ξ0 as ξ0 = ξ|x0 and refer to it as the “nearness structure on the subspace induced by the nearness structure ξ”. theorem 3.2. let (x, ξ) be a nearness space, and x0 ⊆ x. ξ0, as defined in definition 3.1, is a nearness structure on x0. 54 z. yang the proof is an easy deduction from the fact that ξ0 is consists of the type of a that are in ξ already. we will skip the details. the following lemma is also easy to prove. lemma 3.3. let (x, ξ) be a nearness space and x0 ⊆ x. let i : x0 → x be the inclusion map, then ξ0 = i −1(ξ). hence i : x0 → x is n-preserving. lemma 3.4. let x be a set, (y, η) be a nearness space. let f : x → y be an injective map. then tf −1(η) = f −1(tη). proof. clf −1(η)a = {x : {x}f −1(η)a} = {x : f (x)η f (a)} = {x : f (x) ∈ clη(f (a))} = {x : x ∈ f −1(clη(f (a))} = f −1(clη(f (a)). � we will next exam whether some of the common properties are hereditary. i.e. whether a particular condition or property can be “inherited” by its subspaces from their “mother” spaces. it turns out that being a t-n space is not hereditary (example 3.5). neither was being a compact n-space. those two properties can be inhered by n-closed subspaces. many of the other properties are hereditary. example 3.5. a subspace of a t-n space may not be a t-n space. let x = r, the real line with an ordinary open interval topology t . then (x, t ) is a r0space, hence corresponding to a t-n space (x, ξ) ([8] theorem 2.2). now we let x0 = x − {0}, a = (−∞, 0) and b = (0, ∞). then clξa ∩ clξb 6= φ, but clξ0 a ∩ clξ0 b = φ. this means that the nearness subspace (x0, ξ0) does not satisfy condition (t), hence not an t-n space. definition 3.6. let (x, ξ) be a nearness space and (x0, ξ0) be a subspace. (x0, ξ0) is said to be a n-closed subspace, if for any a ⊆ x0, we have clξ0 a = clξa. theorem 3.7. let (x, ξ) be a nearness space and (x0, ξ0) be a subspace. then (x0, ξ0) is a n-closed subspace if and only if clξ0 x0 = clξx0. proof. the necessity is obvious. we now will prove the sufficiency. take any a ⊆ x0, then clξ0 a = clξa ∩ x0 = clξa ∩ clξx0 = clξ(a ∩ x0) = clξa. pointwise convergence for nearness spaces 55 � theorem 3.8. let (x, ξ) be a nearness space and (x0, ξ0) be a subspace. then (a) tξ0 = tξ|x0 . (b) if (x, ξ) is a t-n space and (x0, ξ0) is an n-closed subspace, then (x0, ξ0) is also a t-n space. proof. (a) tξ0 = ti−1(ξ) = i −1(tξ) = tξ ∣ ∣ x0 . (b) let ξ0a0, then ξa0. it follows from the assumption of (x0, ξ0) being an n-closed subspace that clξa0 = clξ0 a0. therefore ⋂ {clξ0 a0 : a0 ∈ a0} = ⋂ {clξa0 : a0 ∈ a0} 6= φ. so (x, ξ0) satisfies the condition (t). � theorem 3.9. let (x, ξ) be a nearness space and (x0, ξ0) be a subspace. then if (x, ξ) is a u-n space, so is (x0, ξ0). moreover, uξ0 = uξ|x0 , where uξ0 and uξ denotes the uniformity induced by the nearness structure ξ0 and ξ respectively. uξ|x0 is the uniformity uξ restricted to x0. proof. if ξ0a0, then ξa0. since (x, ξ) is an u-n space, there exists a ξb that satisfies the condition (u) with respect to a0. let b0 = {b ∩ x0 : b ∈ b}. from (n2) we can see that ξ0b0. for each b0 ∈ b0 = {b ∩ x0 : b ∈ b}, there is a b ∈ b such that b = b ∩ x0. so by condition (u), there should be an a0 ∈ a0 such that a0 ⊆ ⋂ {c : b ∪ c 6= x, c ∈ b}. also because a0 ⊆ x0, we have a0 = a0 ∩ x0 ⊆ ⋂ {c : b ∪ c 6= x, c ∈ b} ∩ x0 = ⋂ {c ∩ x0 : b ∪ c 6= x, c ∈ b} ⊆ ⋂ {c0 : b0 ∪ c0 6= x0, c0 ∈ b0} therefore, ξ0 satisfies the condition (u). furthermore, for any a0 ∈ p 2(x), we have the following equivalent deductions: a0 ∈ u |i−1(ξ) ⇔ a c0 /∈ i −1(ξ) ⇔ a c0 /∈ ξ0 ⇔ a0 ∈ u |ξ0 ⇔ a0 ∈ i −1(uξ). therefore, uξ0 = ui−1(ξ) = i −1(uξ) = uξ|x0 . � 56 z. yang theorem 3.10. let (x, ξ) be a nearness space and (x0, ξ0) be a subspace. if (x, ξ) is a c-n space, so is (x0, ξ0). moreover, cξ0 = cξ|x0 , where cξ0 and cξ denote the contiguity induced by the nearness structure ξ0 and ξ respectively. cξ|x0 is the contiguity cξ restricted to x0. proof. for any a0 ∈ p 2(x) and ξ0a0, then ξa0. by condition (c), a0 has a finite subcollection b0 such that ξb0. this implies that ξ0b0. and furthermore, a0 ∈ cξ0 ⇔ a0 ∈ ξ0 and a0 is finite. ⇔ a0 ∈ ξ, a0 is finite ⇔ a0 ∈ cξ ⇔ a0 ∈ cξ|x0 . � since a p-n space is one that satisfies condition (u) and (c), the following theorem is obvious from the theorems 3.9 and 3.10: theorem 3.11. let (x, ξ) be a nearness space and (x0, ξ0) be a subspace. then if (x, ξ) is a p-n space, so is (x0, ξ0). since a compact nearness space is one that satisfies condition (t) and (c), the following theorem is obvious from theorems 3.8 and 3.10: theorem 3.12. let (x, ξ) be a nearness space and (x0, ξ0) be a n-closed subspace. then if (x, ξ) is a compact nearness space, so is (x0, ξ0). lemma 3.13. let (x, ξ) be a t-n space. then (x, clξ) is topologically compact if and only if (x, ξ) is a compact nearness space. proof. recall that (x, ξ) is a compact nearness space if and only if condition (t) and (c) are satisfied. necessity: if (x, clξ) is topologically compact. take an a such that ξa . we want to show that condition (c) is met by showing that a has a finite subcollection b such that ξb. we will first claim that ⋂ clξa = φ. if not, then by (n1), ξclξa would be true. for each a ∈ a , there would be a clξa ∈ clξa such that clξa ⊆ clξa. by (n2), ξa would be true, and that would contradict to the assumption of ξa . so ⋂ clξa = φ must be true. (clξa ) c is an open cover of x. since (x, clξ) is topologically compact, we will let b be the finite subcollection of a and (clξb) c is an open cover of x. this implies that ⋂ clξb = φ. by condition (t), ξb is true. hence condition (c) has been met. sufficiency: if (x, ξ) is a compact nearness space, which means that it meets condition (t) and (c). any open cover of (x, clξ) can be expressed as the complement collection of a collection clξa and ⋂ clξa = φ. from condition (t), ξclξa is true. from condition (c), there must be a finite pointwise convergence for nearness spaces 57 subcollection of clξa , say clξb, such that ξclξb is true. it follows that ⋂ clξb = φ. then (clξb) c is the finite subcover of the original open cover. hence (x, clξ) is topologically compact. � from lemma 3.13, one can easily see the following is true: lemma 3.14. let (x, ξ) be a compact nearness space, (y, η) be a t-n space and f : (x, ξ) → (y, η) be n-preserving, then (y, η) is a compact nearness space. definition 3.15. let x be a set. define a partial order among all possible nearness structures on x as follows: if ξ1 and ξ2 are two nearness structures on a set x, then ξ1 ≤ ξ2 if and only if ξ1 ⊇ ξ2. lemma 3.16. (i) let x be a set and ξ1 and ξ2 be two nearness structures on a set x. ξ2 ≤ ξ1 if and only if i : (x, ξ1) → (x, ξ2) is n-preserving. (ii) let (x, ξ) and (y, η) be two nearness spaces. then f : (x, ξ) → (y, η) is n-preserving if and only if ξ ≥ f −1(η). proof. (i) ξ2 ≤ ξ1 ⇔ ξ2 ⊇ ξ1 ⇔ i(ξ1) ⊆ ξ2 ⇔ i : (x, ξ1) → (x, ξ2) is n − preserving. (ii) f : (x, ξ) → (y, η) is n − preserving ⇔ ξ ⊆ f −1(η) ⇔ ξ ≥ f −1(η). � theorem 3.17. let (x, ξ) be a t − n2 space and (x0, ξ0) be a n-compact subspace. then (x0, ξ0) is n-closed. theorem 3.18. if (x, ξ) is totally bounded, and x0 ⊆ x, then (x0, ξ0) is totally bounded. proof. ξ0a0 implies that ξa0. hence, by condition (b1) for (x, ξ), there is a b0, a finite subcollection of a0, such that ⋂ b0 = φ. so (x0, ξ0) satisfies (b1). � the following lemma proved by hunsaker and sharma as corollary (2.5) in their 1974 paper [9] is used to prove the next theorem. lemma 3.19. let f : (x, ξα) → (y, ηα) be an n-preserving map for each α ∈ λ. then f : (x, sup{ξα}) → (y, sup{ηα}) is an n-preserving map. 58 z. yang the following theorem is needed to ensure that the concept of nearness structure on the function space, which will be introduced in next section, is well defined. theorem 3.20. if {ξα : α ∈ λ} is a family of nearness structures on x and x0 ⊆ x. then sup α∈λ {ξα} ∣ ∣ ∣ ∣ x0 = sup α∈λ { ξα|x0}. proof. let i : x0 → x be the inclusion map. for each α ∈ λ, by lemma 3.3, i : (x0, ξα|x0 ) → (x, ξα) is n-preserving. from lemma 3.19, i : (x0, sup α∈λ {ξα ∣ ∣ ∣ ∣ x0 }) → (x, sup α∈λ {ξα}) is still n-preserving. by lemma 3.16, sup α∈λ {ξα ∣ ∣ ∣ ∣ x0 } ≥ i−1(sup α∈λ {ξα}) = sup α∈λ {ξα} ∣ ∣ ∣ ∣ x0 . moreover, for each α ∈ λ, sup α∈λ {ξα} ⊆ ξα, hence sup α∈λ {ξα} ∣ ∣ ∣ ∣ x0 ⊆ ξα|x0 . it follows that sup α∈λ {ξα} ∣ ∣ ∣ ∣ x0 ⊆ sup α∈λ {ξα ∣ ∣ ∣ ∣ x0 }. i.e. sup α∈λ {ξα} ∣ ∣ ∣ ∣ x0 ≥ sup α∈λ { ξα|x0}. therefore, sup α∈λ {ξα} ∣ ∣ ∣ ∣ x0 = sup α∈λ { ξα|x0}. � 4. the pointwise convergent nearness structure on function space the following theorem shows that the least upperbound nearness structure is a generalization of the respective least upperbound structure when considered in each of the subcategory of t − near, u − near and p − near respectively. pointwise convergence for nearness spaces 59 figure 1. commutative diagram of the natural projections theorem 4.1. if {ξα : α ∈ λ} is a family of nearness structures on x and let ξ = sup α∈λ {ξα}. then, when considered in t − near, u − near, or p − near, ξ will induce the least upper bound of the respective structures induced by {ξα : α ∈ λ} in the respective type of spaces. proof. let f be the isomorphic functor from t − near to r0 − top. we just need to prove that f is order preserving. assume ξ1 ≤ ξ2. by lemma 3.16, ξ1 ≤ ξ2 ⇔ i : (x, ξ2) → (x, ξ1) is n-preserving ⇔ f (i) is a morphism from f [(x, ξ2)] to f [(x, ξ1)]1) ⇔ tξ2 ⊇ tξ1 . this shows that f does preserve the order. the proofs for other two types are parallel and therefore omitted. � the following theorem ensures that the product nearness structure in the categorical sense is the largest nearness structure on the product space that makes all natural projections n-preserving. theorem 4.2. if {(xα, ξα) : α ∈ λ} is a family of nearness spaces and let pα : ∏ α∈λ xα → xα be the natural projection map. then the nearness structure ξ∗ = sup α∈λ {p −1α (ξα)} is exactly the categorical product of {(xα, ξα) : α ∈ λ}. proof. it would suffice to show that for any n-space (x, ξ), and any family of n-preserving maps {fα : x → xα : α ∈ λ}, there is an unique n-preserving map f : x → ∏ α∈λ xα such that ∀α ∈ λ, pα ◦ f = fα. i.e. the diagram in figure 1 is commutative. we will first make a claim that for any map f : x → ∏ xα, f is n-preserving if and only if for each α ∈ λ, pα ◦ f is n-preserving. in fact, the necessity is obvious. let us assume that for each α ∈ λ, pα ◦ f is n-preserving. from lemma 3.19, f is (ξ, sup α∈λ {p −1α }) n-preserving. i.e. it is (ξ, ξ ∗) n-preserving. now we consider the family of n-preserving maps {fα : x → xα : α ∈ λ}. define f in the natural way (usually known as the “evaluation map”): ∀x ∈ x, f (x) = (fα(x))α∈λ. then pα ◦ f = fα. since each fα is n-perserving, each pα ◦ f is n-preserving. by earlier proof, f is n-preserving. we also know that such an f is unique from its definition. � 60 z. yang the following purely categorical lemma should be obvious: lemma 4.3. if c is a category, {aα : α ∈ λ} is a family of objects. ∏ α∈λ aα is the categorical product in c. d is another category isomorphic to c with f as the isomorphic functor from c to d. then f ( ∏ α∈λ aα) = ∏ α∈λ f (aα). we now officially define the nearness structure on the function space: definition 4.4. if (y, ξ) is a n-space, xis a non-empty set. f ⊆ y x . if we consider y x as a product and let ξ∗ be the product nearness structure as defined in theorem 4.2. let ξρ = ξ ∗| f . then ξρ is said to be the pointwise convergent nearness structure on f ⊆ y x . notice that if {ex : y x → y : x ∈ x} is the family of natural projections, then ξ∗ = sup x∈x {e−1x (ξ)}. so by theorem 3.20, ξρ = ξ ∗| f = sup x∈x {e−1x (ξ)} ∣ ∣ ∣ ∣ f = sup x∈x { e−1x (ξ) ∣ ∣ f }. the readers may refer to [11] for the concept of product topology, product uniformity, pointwise convergent topology and pointwise convergent uniformity. refer to [14] for the concepts of product proximity and pointwise convergent proximity. we would like to make sure that the product nearness structure and the pointwise convergent nearness structure is a generalization of the respective structures in topological spaces, uniform spaces and proximity spaces respectively. theorem 4.5. when considered in each of the subcategory t − near, u − near, or p − near, (i) ξ∗ will induce the tychonoff product topology, product uniformity or the product proximity respectively. (ii) ξρ will induce the pointwise convergent topology, the pointwise convergent uniformity or the pointwise convergent proximity respectively proof. the first conclusion can be obtained from theorems 4.1 and 4.2. the second conclusion can be obtained from theorems 3.8, 3.9 and 3.10. � next we will try to generalize the concept of ”neighborhood”, which is essential when characterizing ”convergence”. definition 4.6. if (x, ξ) is a n-space, and a ⊆ x. a subset u is called a nearness neighborhood of a if there is a ξa such that a ⊆ a c(a) ⊆ u . the notation n earn (a) represents the collection of all nearness neighborhood of a subset a. if the set a contains only one point x, then we simplify the notation from n earn ({x}) to n earn (x). if there are two types of neighborhood system on a space that characterize the same convergence, i.e. being convergent under one neighborhood system is equivalent to being convergent under the other neighborhood system, then we consider them as ”equivalent” neighborhood systems. this is typically pointwise convergence for nearness spaces 61 characterized by the condition that any neighborhood under one system always contains a neighborhood in the other system. for example, consider the two dimensional cartesian plane. the neighborhood of circular disks centered at a point is equivalent to the neighborhood of squares centered at the same point. we will try to show that the nearness neighborhood generalizes the topological neighborhood([11]), uniform neighborhood ([11]), and proximal neighborhood ([14]) by showing that the nearness neighborhood system is really equivalent to the respective neighborhood system in the respective subcategories. theorem 4.7. let (x, ξ) be a n-space, x ∈ x and a ⊆ x. (i) in t − near, n earn (x) is equivalent to a topological neighborhood system at x. (ii) in u − near, n earn (x) is equivalent to a uniform neighborhood system at x. (iii) in p − near, n earn (a) is equivalent to a proximal neighborhood system at a. proof. (i) take an u ∈ n earn (x), there will be an a /∈ ξ such that x ∈ a c(x) ⊆ u . for this a , by conditions (n1) and (n2), ⋂ clξa = ∅. so (clξa ) c is a cover of x. there will be an a ∈ a such that x0 ∈ x − clξa ⊆ x − a ⊆ a c(x0). the set x − clξa is an open neighborhood of x. on the other hand, if we take an open neighborhood v of x under the topology tξ, then x /∈ x − v and x − v is closed, i.e. clξ(x − v ) = x − v . let a = {{x}, x − v }, then a /∈ ξ. and a c(x) = x − (x − v ) = v ⊆ v . this shows that v is a nearness neighborhood also. (ii) take an u ∈ n earn (x), there will be an a /∈ ξ such that x ∈ a c(x) ⊆ u . for this a , since a /∈ ξ, a c is a cover of x, which means that there has to be an a ∈ a such that x ∈ x − a. let ua = { ⋃ a∈a ((x − a) × (x − a)) : a /∈ ξ}, then ua [x] = {y : (x, y) ∈ ua } = {y : ∃a ∈ a ∋ (x, y) ∈ (x − a) × (x − a)} = {y : ∃a ∈ a ∋ x ∈ (x − a), y ∈ (x − a)} = ⋃ {x − a : x ∈ x − a} = a c(x) this “equal” relation shows the equivalency between the uniform neighborhood system and the nearness neighborhood system. (iii) [14] stated that a set b is a proximal ( δ−) neighborhood of a set a if a ≪ b. in [19], the proximity ≪ξ induced by a nearness structure is 62 z. yang defined as a ≪ξ b if and only if there is a a /∈ ξ such that a c(a) ⊆ b. so the equivalency between the nearness neighborhood system and the proximal neighborhood systems is obvious. � the following theorem demonstrates the consistency of the n -converges with those previously established concepts of convergent nets. with the establishment of the previous theorem, the proof should be obvious. theorem 4.8. the n-convergence, as it is defined in definition 4.9, when considered in t − near, u − near and p − near, is equivalent to the convergence with respect to the corresponding types of structures, respectively. a set d is said to be a directed set, if it is endowed with a reflexive and transitive binary relation ≥ such that ∀m, n ∈ d, ∃p ∈ d s.t. p ≥ m and p ≥ n. i.e. for any two elements of d, there is always another element that precedes them. (see [11]) as a generalization of sequences, a net in a set x is a function x : d → x, where d is a directed set. we typically write a net as {xd : d ∈ d}. we would like to exam the relation of the convergency of a net of functions and that of the nets obtained by fixing the net of functions at any arbitrary point x of x. of course, we expect the two convergences are to be equivalent. theorem 4.7 shows exactly that. definition 4.9. if (x, ξ) is a n-space, {xd : d ∈ d} is a net in x. we say {xn : n ∈ d} n-converges to a point x0 ∈ x, if for any ξa , there is an n ∈ d such that for each n ≥ n, n ∈ d, we have xn ∈ a c(x0). theorem 4.10. if (y, ξ) is a n-space, xis a non-empty set. {fn : n ∈ d} is a net in f ⊆ y x . then {fn : n ∈ d} n-converges to a function f in (f , ξρ) if and only if for any x ∈ x, {fn(x) : n ∈ d}, as a net in (y, ξ), n-converges to the point f (x). proof. necessity: take an arbitrary point x ∈ x, take an ξa , since the natural projection map ex : y x → x is n-preserving, we have ξρe −1 x (a ), so ξρ( e −1 x (a ) ∣ ∣ f ). since {fn : n ∈ d} is n-convergent to f in f . there is an n ∈ d, such that for each m > n, m ∈ d, we have fm ∈ ( e −1 x (a ) ∣ ∣ f )c (f ). i.e. there is an a ∈ a , such that {fm, f} ⊆ f − e −1 x (a). so fm(x) = ex(fm) /∈ a and f (x) = ex(f ) /∈ a. pointwise convergence for nearness spaces 63 i.e. fm(x) ∈ x − a, and f (x) ∈ x − a therefore, fm(x) ∈ a c (f (x)). sufficiency: assume that {fn : n ∈ d} is a net in f ⊆ y x . furthermore, for any x ∈ x, assume that {fn(x) : n ∈ d}, as a net in x, n-converges to the point f (x). we now arbitrarily take a b such that ξρb. by the definition of ξ∗ as the least upper bound, there should be finitely many bi ⊆ p(x), i = 1, 2, ..., n, such that for each i, we have bi /∈ e −1 x (ξ) ∣ ∣ f , and b ≺ ∨bi. ξexi (bi), so there is an mi such that for any n ≥ mi, there should be an ai ∈ exi (bi), {fn(xi), f (xi)} ⊆ x − ai. since ai ∈ exi (bi), there exists a bi ∈ bi and ai = exi (bi). so fn(xi) /∈ exi (bi), and f (xi) /∈ exi (bi). i.e. exi (fn) /∈ exi (bi), and exi (f ) /∈ exi (bi). therefore, {fn, f} ⊆ f − bi, or we can say that fn ∈ b c i (f ). since there are only finitely many mi’s. we will let n = max{m1, ..., mn}. then for any n ≥ n , fn ∈ ∧(b c i )(f ) = (∨bi) c (f ). hence fn ∈ b c (f ). � the following corollary is associated with the concept of ”accumulation points” in classical topology. corollary 4.11. if (x, ξ) is a n-space, b is a subset of x, {xd : d ∈ d} is a net in b. then (i) if {xd : d ∈ d} is n-convergent to x0 ∈ x, then x0 ∈ clξb. (ii) in t − near, for any x0 ∈ clξb, there is a {xd : d ∈ d} in b and {xd : d ∈ d} is n-convergent to x0. proof. (i) if, to the contrary, x0 /∈ clξb. then a = {{x0}, b} /∈ ξ. since {xd : d ∈ d}, and a c = {x − {x0}, x − b}, there is an n ∈ d such that for any n ≥ n , xn ∈ a c(x0) = x − b. but this contradicts to the assumption that {xd : d ∈ d} ⊆ b. hence x0 ∈ clξb must be true. (ii) in t − near, from theorem 4.8(i), n-convergence is equivalent to topological convergence and clξb is the topological closure of the set b. the conclusion must be true due to classical topology. � the following corollary is a natural consequence of the corollary 4.11: corollary 4.12. let (x, ξ) be a n-space and b ⊆ x. (i) if b is n-closed, then any convergent net in b must converge to a point in b. (ii) in t − near, if the limit of any convergent net in b always remains in b, then b is n-closed. 64 z. yang the next several theorems show that the properties of being totally bounded, compact or n2 are productive respectively. theorem 4.13. let x be a set, (y, η) be a n-space, and f : x → y be a n-preserving map. then f −1(η) is totally bounded if and only if η is totally bounded. proof. assume that f −1(η) is totally bounded. we would like to show that η is totally bounded by showing that it meets condition (b1). arbitrarily take a b such that ηb. then f −1(b) /∈ f −1(η). there should be a finite subcollection of b, say b0 ⊆ b, such that ⋂ f −1(b0) = φ. hence ⋂ b0 = φ. now we assume that η is totally bounded. arbitrarily take a /∈ f −1(η), then ηf (a ). so there should be a finite subcollection of a , say a0 ⊆ a , such that ⋂ f (a0) = φ. now we can easily see that ⋂ a0 = φ. � theorem 4.14. let {ξα : α ∈ λ} be a family of nearness structures on x. let ξ = sup{ξα : α ∈ λ}. then ξ is totally bounded if and only if for each α ∈ λ, ξα is totally bounded. proof. first we assume that for each α ∈ λ, ξα is totally bounded. then for any ξa , there should be finitely many ξαi , i = 1, 2, ...n as well as aαi , i = 1, 2, ...n such that aαi /∈ ξαi and a ≺ n ∨ i=1 aαi . since each ξαi is totally bounded, each aαi contains a finite subcollection bαi and ⋂ bαi = φ. n ∨ i=1 bαi is a finite subcollection of n ∨ i=1 aαi . we will claim that ⋂ n ∨ i=1 bαi = ∅. take an arbitrary point x ∈ x, then for each i = 1, 2, ..., n, there is a bxi ∈ bαi such that x /∈ b x αi . so x /∈ n ∪ i=1 bxαi . hence x /∈ ⋂ n ∨ i=1 bαi . this shows that ⋂ n ∨ i=1 bαi = φ. so ξ is totally bounded. now we assume that ξ is totally bounded. arbitrarily take a ξα. then for any ξαaα, we have ξaα. since ξ is assumed to be totally bounded, aα must have finite subcollection with empty intersection. � theorem 4.15. if {ξα : α ∈ λ} is a family of nearness structures on x . its product ( ∏ α∈λ xα, ∏ α∈λ ξα) is totally bounded if and only if each (xα, ξα) is totally bounded. particularly, if (y, η) is totally bounded and f ⊆ y x where x is a non-empty set, then (f , ξρ) is totally bounded. proof. the first conclusion can be deduced from theorem 4.13 and theorem 4.14. the second conclusion can be deduced from the theorem 3.18. � the next lemma, due to herrlich ([8], 4.5 proposition, part (2)), will be used in the proof of the following theorem. lemma 4.16. for a t-n space, the following conditions are equivalent: (1) (x, ξ) is contigual; (2) (x, ξ) is totally bounded; (3) (x, ξ) is compact. pointwise convergence for nearness spaces 65 theorem 4.17. if {ξα : α ∈ λ} is a family of nearness structures on x. if each (xα, ξα) is compact, then its product ( ∏ α∈λ xα, ∏ α∈λ ξα) is also compact. proof. recall that a compact nearness space satisfies condition (t) and (c). it is easy to verify that the product is a t-n space, if each (xα, ξα) is a t-n space. according to lemma 4.16, a t-n space is c-n if and only it it is totally bounded. so the conclusion of this theorem follows from theorem 4.15. � theorem 4.18. if {ξα : α ∈ λ} is a family of nearness structures on x . if each (xα, ξα) is n2, then the product (x, ξ) = ( ∏ α∈λ xα, ∏ α∈λ ξα) is also n2. proof. by the definition of product of nearness structures, ξ = sup{p −1α (ξα) : α ∈ λ}, where pα : ∏ α∈λ xα → xα are natural projection maps. let x, y ∈ x and x 6= y, there should be at least one α ∈ λ such that pα(x) 6= pα(y). since ξα is n2, there are aα ⊆ x and bα ⊆ x such that aα ∩ bα = ∅ and pα(x) ∈ xα − clξα (xα − aα), pα(y) ∈ xα − clξα (xα − bα). it is easy to see that p −1α (aα) ∩ p −1 α (αb) = ∅. we will try to show that x ∈ x − clξ(x − p −1 α (aα)), and y ∈ x − clξ(x − p −1 α (bα)). take an arbitrary point z ∈ p −1α (xα − clξα (xα − aα)), then pα(z) ∈ xα − clξα (xα − aα). and {{pα(z)}, xα − aα} /∈ ξα. {{p −1α (pα(z))}, p −1 α (xα − aα)} /∈ p −1 α (ξα). {{z}, x − p −1α (aα)} /∈ p −1 α (ξα). the last statement is true since z ∈ p −1α (pα(z)) and p −1α (xα − aα) ⊆ x − p −1 α (aα). therefore, z ∈ x − cl p −1 α (ξα) (x − p −1α (aα)). this shows that p −1α (xα − clξα (xα − aα)) ⊆ x − clp −1α (ξα)(x − p −1 α (aα)). hence x ∈ p −1α (xα − clξα (xα − aα)) ⊆ x − cl p −1 α (ξα) (x − p −1α (aα)) ⊆ x − clξ(x − p −1 α (aα)). 66 z. yang by similar argument, y ∈ p −1α (xα − clξα (xα − bα)) ⊆ x − cl p −1 α (ξα) (x − p −1α (bα)) ⊆ x − clξ(x − p −1 α (bα)). therefore, (x, ξ) is a n2space. � now we will try to establish the relation between the compactness of the underderlining set y and a function space f ⊆ y x . theorem 4.19. let x be a set, and (y, η) be a compact n-space. f ⊆ y x . then (i) the condition (a) is sufficient for (f , ξρ) to be compact. (ii) if (y, η) is also n2, then the condition (a) is also necessary for (f , ξρ) to be compact. (a) f is n-closed in (y x , ξ∗). proof. (i) since (y, η) is a compact n-space. by theorem 4.17, (y x , ξ∗) is compact. by theorem 3.12, f ⊆ y x , as an n-closed subset of a compact space, is also compact under the subspace nearness structure. (ii) if (y, η) is a n2-space. by theorem 4.18, (y x , ξ∗) is an n2space. then by theorem 3.17, (f , ξρ), as a compact subspace of an n2space, is n-closed. � theorem 4.20. if x is a set, and (y, η) is an n-space. f ⊆ y x . then (i) the conditions (a) and (b) are sufficient for (f , ξρ) to be compact. (ii) if (y, η) is also n2, then the conditions (a) and (b) are also necessary for (f , ξρ) to be compact. (a) f is n-closed in (y x , ξ∗) (b) for any x ∈ x, f [x] = {f (x) : f ∈ f } is contained in a compact subspace of (y, η). proof. (i) assume that (yx, ηx) is a compact nearness subspace of (y, η) with ηx = η|yx and f [x] ⊆ yx ⊆ y . then f ⊆ ( ∏ x∈x yx, ∏ x∈x ηx) and the later space is compact, according to theorem 4.17. it is easy to see that ξρ = ∏ x∈x ηx|f . since f is n-closed also, it is n-compact due to theorem 3.12. (ii) if (y, η)is a n2-space, and (f , ξρ) is compact. it follows from theorem 3.17 that (a) is true. and since the evaluation map ex : (f , ξρ) → (y, η) is n-preserving. by lemma 3.14, f [x] = {f (x) : f ∈ f } = {ex(f ) : f ∈ f } is also compact. � pointwise convergence for nearness spaces 67 summary. this paper essentially lays the foundation for some possible applications of the theory of nearness function spaces in digital topology. the main results is the introduction of the pointwise convergent nearness spaces in the function spaces in such a way that is consistent with the existing and established structures. two ascoli’s type of theorems on nearness spaces are established also. any deformation of a digital image (such as thinning) can be considered as a function from the digital plane to itself. of course, we would prefer those functions to preserve some properties of the digital image, such as ”nearness”. when a sequence of deformations are applied to a digital image, we would like to be able to make some type of projection or prediction about the final images based on the type of deformations involved in the sequence. we believe that the line of work presented in this paper will be helpful to address those issues. references [1] h. l. bentley, j. w. carlson and h, herrlich, on the epireflective hull of top in near, topology appl. 153 (2007), 3071-3084. [2] h. l. bentley and h. herrlich, compleness is productiver, categorical topology (dold a. and eckmann b., eds.), 719 proceedings, berlin , springer-verlag, berlin, heidelberg, new york (1978), 13–17. [3] h. l. bentley and h. herrlich, ascoli’s theorem for a class of merotopic spaces, convergence structures (dold a. and eckmann b. eds.), proc. bechyne conf. , springerverlag, berlin, heidelberg, new york (1984), 47–54. [4] b. b. chaudhuri, a new definition of neighborhood of a point in multi-dimensional space, pattern recognition letters 17 (1996), 11–17. [5] p. r. fu, proximity on function space of set-valued functions, journal of mathematics 13, no. 3 (1993), 347–350. [6] j. w. gray, ascoli’s theorem for topological categories, categorical aspects of topology and analysis (dold a. and eckmann b. eds.), 915 proceedings, berlin , springer-verlag, berlin, heidelberg, new york (1978), 86–104. [7] n. c. heldermann, concentrated nearness spaces, categorical topology (dold a. and eckmann b., eds.), 719 proceedings, berlin , springer-verlag, berlin, heidelberg, new york (1978), 122–136. [8] h. herrlicht, a concept of nearness, general topology and application 5 (1974), 191– 212. [9] w. n. hunsaker and p. l. sharma, nearness structure compatible with a topological space, arch. math. xxv (1974), 172–178. [10] v. m. ivanova and a. a. ivanov, contiguity spaces and bicompact extensions, izv. akad. nauk. sssr 23 (1959), 613–634. [11] j. l. kelley, general topology, (d. van norstrand, princeton toronto london new work, 1974). [12] l. latecki and f. prokop, semi-proximity continuous functions in digital images, pattern recognition letters 16 (1995), 1175–1187. [13] y. f. lin and d. rose, ascoli’s theorem for spaces of multifunctions, pacific journal of mathematics 34, no. 3 (1970), 741–747. [14] s. a. naimpally and b. d. warrack, proximity spaces, (cambridge university press, london 1970). [15] j. f. peters, a. skowron and j. stepaniuk, nearness of objects: extension of approximation space model, fundamenta informaticae 79 (2007), 497–512. 68 z. yang [16] p. ptak and w. g. kropatsch, nearness in digital images and proximity spaces, dgci 2000, lncs 1953 (2000), 69–77. [17] g. sonck, an ascoli theorem for sequential spaces, int. j. math. math. sci. 26, no. 5 (2001), 303–315. [18] m. wolski, approximation spaces and nearness structures, fundamenta informaticae 79 (2007), 567–577. [19] z. yang, a new proof on embedding the category of proximity spaces into the category of nearness spaces, fundamenta informaticae 88, no. 1-2 (2008), 207–223. [20] j. adàmek, h. herrlich and g. e. strecker, abstract and concrete categories, the joy of cats, (john wiley and sons, new work 1970). received may 2008 accepted october 2008 zhanbo yang (yang@uiwtx.edu) department of mathematical sciences, university of the incarnate word, 4301 broadway, san antonio, tx 78209, usa melinagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 51-66 continuous extension in topological digital spaces erik melin abstract. we give necessary and sufficient conditions for the existence of a continuous extension from a smallest-neighborhood space (alexandrov space) x to the khalimsky line. using this result, we classify the subsets a ⊂ x such that every continuous function a → z can be extended to all of x. we also consider the more general case of mappings x → y between smallest-neighborhood spaces, and prove a digital no-retraction theorem for the khalimsky plane. 2000 ams classification: primary 54c20; secondary 54c05, 54f07 keywords: khalimsky topology, digital geometry, alexandrov space, continuous extension 1. introduction the classical tietze extension theorem states that if x is a normal topological space and a is a closed subset of x, then any continuous map from a into the closed interval [a,b] can be extended to a continuous function on all of x into [a,b]. in digital geometry it is more natural to study functions from a digital space (defined below) to the integers. having equipped the spaces with suitable topologies, we may consider continuous functions and the corresponding extension problem. we solve this problem for a fairly general class of topological digital spaces and for continuous functions with the khalimsky line as codomain. the khalimsky topology is a topology that appears naturally in the study of digital geometry, cf. kong [12]. the notion of straight lines in khalimsky spaces has been studied in [18] and more general curves and surfaces in [17]. the results in this paper are generalizations of a previous work [19], where the digital spaces considered were restricted to zn. digital topology can be considered from a graph-theoretical point of view; one has a set of points and an adjacency relation, i.e., a symmetric binary 52 e. melin relation defining which points are adjacent. for example, in zn one possibility is to say that points p and q are adjacent if ‖p − q‖∞ = 1. in such, graph based, spaces no topology, in the usual meaning, is directly involved. nevertheless, a theory for continuous functions has been developed in this setting, for example by boxer [2]. herman and webster [5] and kovalevsky [16] have independently argued that cellular complexes are appropriate topological spaces for digital geometry. the results of this paper applies to such spaces, since they are topologically equivalent to spaces of the type introduced in this article. see, for example, klette [10]. we shall only consider digital spaces that are also topological spaces and where the topological notion of connectedness agrees with the notion of connectedness in digital spaces. more details on the theory of such spaces and applications to image analysis can be found in for example the survey by kong and rosenfeld [14], in [13] by kong et al. and in koppermans’s [15] article. 2. background this section is devoted to some basic definitions and results that we will need. those unfamiliar with the subject might find kiselman’s [9] lecture notes helpful. 2.1. smallest-neighborhood spaces. in any topological space, a finite intersection of open sets is open, whereas the stronger requirement that an arbitrary intersection of open sets is open, is not satisfied in general. in a classical article, alexandrov [1] considers topological spaces that fulfill the stronger requirement, where arbitrary intersections of open sets are open. in his paper, alexandrov called such spaces diskrete räume (discrete spaces). unfortunately, this terminology is not possible today since the term discrete topology is occupied by the topology where every set is open. instead, following kiselman [9], we will call such spaces smallest-neighborhood spaces. another name that has been used is alexandrov spaces. let nx (x) denote the intersection of all neighborhoods of a point x in a topological space x. if there is no danger of ambiguity, we will just write n (x). in a smallest-neighborhood space, n (x) is always open and thus a neighborhood of x; clearly n (x) is the smallest neighborhood of x. we may note that x ∈ n (y) if and only if y ∈ c (x), where c (x) = {x} is the topological closure of the singleton set containing x. note that the familiar construction of the closure, as an intersection of closed sets, is dual to the construction of the smallest neighborhood. conversely, the existence of a smallest neighborhood around every point implies that an arbitrary intersection of open sets is open; hence this existence could have been used as an alternative definition of a smallest-neighborhood space. a topological space x is called connected if the only sets which are both closed and open are the empty set and x itself. note that n (x) considered as a subspace of x is always connected and that a two-point set {x,y} is connected continuous extension in digital spaces 53 if and only if x ∈ n (y) or y ∈ n (x). a point x is called open if the set {x} is open, and is called closed if {x} is closed. if a point x is either open or closed it is called pure, otherwise it is called mixed. kolmogorov’s separation axiom, also called the t0 axiom, states that given two distinct points x and y, there is an open set containing one of them but not the other. an equivalent formulation is that n (x) = n (y) implies x = y for every x and y. this axiom is quite natural to impose; if x and y have the same neighborhood, then they are indistinguishable from a topological point of view and should perhaps be identified. the separation axiom t1, on the other hand, is too strong. it states that points are closed and in a smallest-neighborhood space this means that every set is closed. hence only spaces with the discrete topology are smallest-neighborhood spaces satisfying the t1 axiom. remark 2.1. there is a correspondence between smallest-neighborhood spaces and partially ordered sets; namely if we define x 4 y if and only if y ∈ n (x). this relation is always reflexive and transitive and is anti-symmetric if and only the space is t0. then the order is a partial order and is called the specialization order. it was introduced by alexandrov [1]. it is not hard to see that a function is continuous if and only if it is increasing for the specialization order. a consequence is that our results can be formulated in the language of partially ordered sets instead of topologies, if one prefers. 2.2. topological digital spaces. in the graph-theoretical setting, a digital space is called connected if it is path connected under the adjacency relation— see [14] for details. given a topological space x we may try to identify it with a digital space in this sense. it is natural to define the binary adjacency relation such that distinct points x and y are adjacent if and only if {x,y} is connected (as a subspace of x). if x is not connected in the topological sense, it is obvious that it cannot be path connected under the adjacency relation above. on the other hand, the following lemma shows that a connected smallest-neighborhood space is indeed also a connected digital space in the graph-theoretical sense. the result is also stated in for example [4, lemma 4.2.1], however, the present proof is shorter. lemma 2.2. let x be a connected smallest-neighborhood space. then for any pair of points x and y of x there is a finite sequence (x0, . . . ,xn) such that x = x0 and y = xn and {xj,xj+1} is connected for i = 0, 1, . . . ,n − 1. proof. let x be a point in x, and denote by y the set of points which can be connected to x by such a finite sequence. obviously x ∈ y . suppose that y ∈ y . it follows that n (y) ⊂ y and c (y) ⊂ y . thus y is open, closed and nonempty. since x is connected this reads y = x. � with this lemma as a motivation, we give the following definition: a topological digital space is a connected smallest-neighborhood space. 54 e. melin figure 1. construction of the khalimsky line. 2.3. examples of topological digital spaces. we shall present some examples of digital spaces and examine some properties of these examples. eckhardt and latecki [3] and kong [11] give more examples of topologies on z2 and z3. example 2.3. every finite topological space is obviously a smallest-neighborhood space; therefore every connected finite space is a digital space. properties of finite spaces are studied by stong [20]. many results there are also true for infinite smallest-neighborhood spaces. example 2.4 (the khalimsky line [6, 7]). we shall construct a topology on the digital line, z, originally introduced by efim khalimsky. let us identify with each even integer m the closed, real, interval [m − 1/2,m + 1/2] and with each odd integer the open interval ]m − 1/2,m + 1/2[. these intervals form a partition of the euclidean line r and we may therefore consider the quotient space. identifying each interval with the corresponding integer gives us the khalimsky topology on z (see figure 1). since r is connected, the khalimsky line is connected. it follows easily that an even point is closed and that an odd point is open. in terms of smallest neighborhoods, we have n (m) = {m} if m is odd and n (n) = {n ± 1,n} if n is even. as will be seen in the next section, the khalimsky topology is of particular importance in the study of topological digital spaces. therefore we will here make some further definitions. let a and b be integers. a khalimsky interval is an interval [a,b] ∩ z of integers with the topology induced from the khalimsky line. we will denote such an interval by [a,b]z. a khalimsky arc is a homeomorphic image of a khalimsky interval into any topological space. unless otherwise stated, we will assume that z is equipped with the khalimsky topology from now on. this makes it meaningful to consider continuous functions f : z → z. we will discuss some properties of such functions; more details can be found in [9]. suppose that f is continuous. since m = {m,m + 1} is connected it follows that f(m) is connected, but this is the case only if |f(m) − f(m + 1)| 6 1. hence f is lipschitz with lipschitz constant 1; we say f is lip-1. lip-1, however, is not sufficient for continuity. if y = f(x) is odd, then u = f−1({y}) must be open, so if x is even then x ± 1 ∈ u. this means that f(x ± 1) = f(x). it is straightforward to prove the following, see [19, lemma 2]. proposition 2.5. f : z → z is continuous if and only if for every p,q ∈ z where p 6= q, one of the following conditions holds (1) |f(q) − f(p)| < |q − p| (2) |f(q) − f(p)| = |q − p| and p ≡ f(p) (mod 2). continuous extension in digital spaces 55 corollary 2.6. a function f : a → z, a ⊂ z can be extended to a continuous function f : z → z if and only if the condition in the above proposition holds for every p,q ∈ a where p 6= q. the following two examples give spaces with topologies that are derived from the khalimsky topology. example 2.7 (khalimsky n-space). the khalimsky plane is the cartesian product of two khalimsky lines and in general, khalimsky n-space is zn with the product topology. points with all coordinates even are closed and points with all coordinates odd are open. points with both even and odd coordinates are mixed. example 2.8. we may consider a quotient space zm = z/mz for some even integer m > 2 (if m is odd, we will end up identifying open and closed points, which results in a space with the indiscrete topology, sometimes called the chaotic topology). such a space is called a khalimsky circle. if m > 4, zm is a compact space which is locally homeomorphic to the khalimsky line. in a similar manner, we may construct a khalimsky torus, by identifying the edges of a khalimsky rectangle, i.e., a product of two khalimsky intervals (of even length). our final example shows that there are topological digital spaces of arbitrarily high cardinality. example 2.9. let j be any index set and let k be the disjoint union of j copies of the khalimsky line. this space is of course not connected, but we may identify the point 0 in every copy and then consider the quotient space k0. it is easy to see that k0 is connected. note also that n (0) may be a very large set in this space. 2.4. arc-connected spaces. let x be a topological digital space satisfying the t0 axiom and let x,y ∈ x. the following theorem shows that we can actually find a khalimsky arc in x with endpoints x and y, i.e., there is an interval i = [a,b]z and a function ψ : i → x such that ψ(a) = x, ψ(b) = y and ψ is a homeomorphism of i and ψ(i). we call a topological space having this property khalimsky arc-connected or just arc-connected if it is clear from the context what is intended. note that an arc-connected space is always connected; this can be proved with the same argument as is used to prove that a path-connected space (in the usual topological sense) is connected. remark 2.10. in [7] the result below was proved for finite spaces only. another difference in their version is that it was not required there that the space in question satisfied the t0-axiom. this is because they allowed an arc with two points {x,y} to have the indiscrete topology {∅,{x,y}}, which is clearly necessary if n (x) = n (y) in x. with our definition, this type of two-point arc is not allowed; hence we must require the space to be t0. theorem 2.11. a t0 topological digital space is khalimsky arc-connected. 56 e. melin proof. let x be a topological digital space and let a,b ∈ x. first, use lemma 2.2 to get a finite, connected sequence of points (x0, . . . ,xn) such that x0 = a and xn = b. then use the finiteness of the sequence to choose a subsequence y = (y0, . . . ,ym) that is minimal with respect to connectedness, and such that y0 = a and ym = b. with this, we mean that y (considered as a subset of x) has no connected proper subset containing a and b. suppose that i > j and that {yi,yj} is connected. then i = j + 1; for if not, the sequence (y0, . . . ,yj,yi, . . . ,ym) would be a connected proper subset of the minimal sequence, which is a contradiction. by the result of the above paragraph, there are two possibilities for ny (y0): either ny (y0) = {y0} or ny (y0) = {y0,y1}. (if x ∈ ny (y), then x and y are connected.) let s = 1 in the first case and s = 0 in the second case. for a clearer notation, let us agree to re-index the sequence {yi}, so that ys is its first element and ym+s its last. we will show that the function ψ : i = [s,s + m]z → y,i 7→ yi is a homeomorphism. note that y is t0 since x is t0. clearly ψ is surjective, and by minimality also injective. to show that ψ is a homeomorphism, it suffices to show that (2.1) ny (yi) = ψ(ni (i)) for every i ∈ i. for i = s this holds by the choice of s. we use finite induction. suppose that s < k 6 s + m, and that (2.1) holds for i = k − 1. we consider two cases: case 1: k is odd. then ni (k) = {k}. we must show that yk−1 and yk+1 are not in ny (yk). but since eq. (2.1) holds for i = k − 1, and k ∈ ni (k − 1), we know that yk ∈ ny (yk−1). it follows that yk−1 6∈ ny (yk) (y is t0) and also that yk+1 6∈ ny (yk), since ny (yk−1) ∩ ny (yk) is a neighborhood of yk which does not include yk+1. case 2: k is even. we must show that yk−1 ∈ ny (yk) and yk+1 ∈ ny (yk) provided k < s + m. now, ny (yk−1) = {yk−1} by assumption, so clearly yk−1 belongs to ny (yk). if k < s+m and yk ∈ ny (yk+1), then ny (yk)∩ny (yk+1) would be a neighborhood of yk not containing yk−1. this is a contradiction, and hence yk+1 ∈ ny (yk). � this result shows the fundamental importance of khalimsky’s topology in the theory of t0 digital spaces; in any such space, a minimal connected subset containing two points x and y has the topological structure of a finite khalimsky interval. we state this important result formally: corollary 2.12. a subspace of a t0 digital space is a minimal connected subspace containing points x and y if and only if it is a khalimsky arc with endpoints x and y. note that this result implies that the khalimsky topology is the only topology on z (up to translation) such that z is connected in the intuitive sense (i.e., {m,m + 1} is connected for every m) and such that removing one point separates the digital line into two components. continuous extension in digital spaces 57 3. the arc metric let x be a t0 topological digital space and let a be a khalimsky arc in x. we define the length of a, denoted l(a), to be the number of points in a minus one, l(a) = |a| − 1. this means that an arc consisting of just one element has length zero. the arc metric is defined to be the minimal length of an arc between two points x and y. definition 3.1. suppose that x is a t0 digital space. then we define the arc metric ρ on x to be ρx (x,y) = min{l(a); a ⊂ x is a khalimsky arc between x and y}. if there is no danger of ambiguity, we may write just ρ(x,y). the set of khalimsky arcs between two points is not empty by theorem 2.11 and their lengths form a discrete set, so the minimum does exist. therefore ρ is well defined and it is very easy to check that it satisfies the axioms for a metric. remark 3.2. if y is a subspace of x, the arc metric on y need not equal the restriction of the arc metric on x. we only have the inequality: ρx (x,y) 6 ρy (x,y) for every x,y ∈ y . for example, consider a khalimsky circle k with at least 6 points. take any b ∈ k and consider its two neighbors x and y. then ρk (x,y) = 2 but ρkr{b}(x,y) = card(k) − 2 > 2. the following proposition will give some idea of the relation between this metric and continuous functions on x. proposition 3.3. suppose that x is a smallest-neighborhood space and that f : x → z is continuous. then f is lip-1 for the arc metric. proof. suppose that x,y ∈ x and let a be an arc of minimal length connecting x and y. by definition, a is a homeomorphic image of a khalimsky interval; there exist a homeomorphism η : i → a ⊂ x. clearly the composition f ◦ η : i → z is continuous and hence lip-1. but the length of i equals l(a) and by combining these facts it follows that |f(x) − f(y)| 6 l(a) = ρ(x,y), i.e., f is lip-1 for the arc metric. � remark 3.4. on the khalimsky line we have ρ(m,n) = |m − n|. as we know already from the study of continuous functions z → z (proposition 2.5) the lip-1 condition is not sufficient for continuity. let a be a khalimsky arc with l(a) > 1 in a topological space x. as a subspace, we may think of a as a khalimsky interval, or rather it is—up to homeomorphism. therefore it makes sense to speak of open and closed points of a and every point in a is either open or closed. note that a point, open in a, may very well be closed in another khalimsky arc b. if, however, the point is pure in x, then it is either always open or always closed. the following generalized concept turns out to be important in what follows. 58 e. melin figure 2. the point (1, 0) is closed in one minimal arc connecting it to the point (2, 1) and open in another. definition 3.5. let x be a t0 topological digital space and let x,y ∈ x be distinct points in x. we say that x is open with respect to y if there is a khalimsky arc a between x and y of length ρ(x,y) such that x is open in a, and we say that x is closed with respect to y if there is a khalimsky arc b between x and y of length ρ(x,y) such that x is closed in b. clearly a point x is either open or closed w.r.t. y, but can it be both? the answer is in the affirmative; a simple example can be found in the khalimsky plane z2. let x = (1, 0) and y = (2, 1). then ρ(x,y) = 2, and there are two arcs of length 2 between x and y. in the one passing (1, 1), x is closed, in the one passing through (2, 0), x is open. this is illustrated in figure 2. 4. extensions of continuous functions the following definition gives a condition on a function which is stronger than lip-1. it is a generalization of definition 7 in [19]. definition 4.1. let x be a t0 topological digital space, let a ⊂ x and consider a function f : a → z. suppose x and y are two distinct points in a. if one of the the following conditions hold (1) |f(x) − f(y)| < ρ(x,y) or (2) |f(x) −f(y)| = ρ(x,y) and x is open (closed) w.r.t. y implies that f(x) is odd (even), then we say that the function is strongly lip-1 (in x) with respect to (the points) x and y. if the function is strongly lip-1 (in x) with respect to every pair of distinct points in a then we simply say that f is strongly lip-1 (w.r.t. x). we emphazise that (w.r.t x) means with respect to the (global) arc metric on x and not to the arc metric of a. confer remark 3.2. the relation f is strongly lip-1 w.r.t. x and y is symmetric in x and y, and the statement can intuitively be thought of as a guarantee that there is enough distance between x in y along the shortest arcs (and thus any arc) connecting these points for the function to change continuously from f(x) and f(y) along these arcs, cf. proposition 2.5. proposition 4.2. let x be a t0 topological digital space and f : x → z a function. then f is continuous if and only if f is strongly lip-1 w.r.t. x. continuous extension in digital spaces 59 proof. assume first that f is continuous. the proof that f then is strongly lip1 is practically the same as for proposition 3.3. suppose that x is open (closed) w.r.t. y and let a be a minimal arc connecting these points such that x is open (closed) in a. let η and i be as in the proof of proposition 3.3. as before, f ◦η is continuous. hence either |f(x) − f(y)| < ρ(x,y) or |f(x) − f(y)| = ρ(x,y) and η−1(x) agrees in parity with f(x), which by assumption means that f(x) is odd (even). but then f is strongly lip-1 w.r.t. x and y and therefore f is strongly lip-1. to prove the converse, assume that f is strongly lip-1. it is sufficient to show that nx (x) ⊂ f −1(nz(f(x))) for every x ∈ x. suppose therefore that y ∈ nx (x). note that {x,y} is an arc connecting x and y and that x is closed w.r.t. y. we have to consider two cases: case 1: f(x) is odd. since x is closed w.r.t. y, |f(x) − f(y)| < ρ(x,y) = 1. it follows that f(y) = f(x) ∈ nz(f(x)). case 2: f(x) is even. then |f(x) − f(y)| 6 ρ(x,y) = 1. but f(x) even also implies nz(x) = {f(x),f(x) ± 1}, so that f(y) ∈ nz(f(x)). � now we will show that a strongly lip-1 function defined on a subset of x can be extended to a strongly lip-1 function defined on all of x. we start with the following lemma. lemma 4.3. suppose that x and y are two distinct points in a t0 topological digital space x, f : {x,y} → z a function that is strongly lip-1 w.r.t. x. then it is possible, for any point p ∈ x, to extend the function to f : {x,y,p} → z so that f is strongly lip-1 w.r.t. x. proof. suppose for definiteness that f(x) 6 f(y). let a = (a1, . . . ,am) be a minimal arc connecting x and p and let b = (b1, . . . ,bn) be a minimal arc connecting y and p. if f(x) is even and x is open w.r.t. p, we can take a such that x is open in a (and if f(x) is odd, ensure that x is closed in a if x is closed w.r.t. p) and similarly for b and y. consider now the set a∪b. it is connected and hence it includes a (minimal) arc c connecting x and y. if p is included in this arc we may argument as follows: by minimality, c = (a1, . . . ,am−1,p,bn−1, . . .b1) and by the strongly lip-1 assumption, f can be continuously extended along the arc; in particular a value is assigned to p and this value will be our f(p). since we have chosen a to be the arc yielding the strongest restriction possible for f to be strongly lip-1 w.r.t. x and p in x, and it succeeds in a by proposition 4.2, f must be strongly lip-1 w.r.t. x and p in x. similarly f is strongly lip-1 w.r.t. y and p in x, so f is indeed the required extension. by the triangle inequality, we always have ρ(x,y) 6 ρ(x,p) + ρ(y,p). suppose that we have equality. this implies, with c defined as above that c = a ∪ b and we are done. next consider the possibility ρ(x,y) = ρ(x,p) + ρ(y,p) − 1. then it may happen that p is not included in c; we may have c = (a1, . . . ,am−1,bn−1, . . .b1). we can still do a continuous extension along c, say that it takes the value α at am−1 and β at bn−1. if α = β it is clear that we can set f(p) = α, but what if β = α + 1? assume for definiteness that α is 60 e. melin odd and β is even. if we set f(p) = α, we will have no problem along a, but we may have along b, namely if p is closed in b. we now show that if it is so, then we can set f(p) = β, or in other words that if p is closed in b, then p is also closed in a. assume therefore, on the contrary, that p is open in a. our assumptions can be summarized to the following: am−1 ∈ nc (bn−1) ⊂ n (bn−1) (since α is odd and the extension along c is not constant at am−1 it follows that am−1 is open), p ∈ na(am−1) ⊂ n (am−1) (since p is assumed to be open in a) and bn−1 ∈ nb(p) ⊂ n (p) (since p is closed in b) and the latter implies p 6∈ n (bn−1) as we assume the space to be t0. the contradiction follows from the following equation p ∈ n (am−1) = n (am−1) ∩ n (bn−1) ⊂ n (bn−1) 6∋ p thus the bad configuration cannot occur, and it only remains to consider the case ρ(x,y) 6 ρ(x,p) + ρ(y,p) − 2. now, there is a maximal value m that f(p) can take, if f is to be strongly lip-1 w.r.t. x and p, and every integer between f(x) and m will also make f strongly lip-1. the following lower estimation of m is immediate; m > f(x) + ρ(x,p) − 1. similarly, there is a minimal value n for f(p), so that f is strongly lip-1 w.r.t. y and p; n 6 f(y) − ρ(y,p) + 1. it is therefore sufficient to show that m > n. assume on the contrary that m < n, in other words that f(x) + ρ(y,p) + 1 < f(y) − ρ(y,p) + 1. but then ρ(x,y) 6 ρ(x,p) + ρ(y,p) − 2 < f(y) − f(x) this contradicts the fact that f is strongly lip-1. � proposition 4.4. suppose that x is a t0 topological digital space, that a is a subspace of x and that f : a → z is strongly lip-1 w.r.t. x. then f can be extended to all of x so that the extended function is still strongly lip-1. proof. if a is the empty set or a is all of x the lemma is trivially true, so we need not consider these cases further. first we show that for any point where f is not defined we can define it so that the new function still is strongly lip-1. to this end, let p be any point in x r a. for every x ∈ a it is possible to extend f to fx defined on a ∪ {p} so that the new function is strongly lip-1 w.r.t x and p—for example by letting fx(p) = f(x). it is also clear that there is a minimal (say nx) and a maximal (say mx) value that fx(p) can attain if it still is to be strongly lip-1 w.r.t. x, and that fx(p) may also attain every integer value in between nx and mx. thus the set of possible values is in fact an interval [nx,mx]z. now define r = ⋂ x∈a [nx,mx]z if r is empty, then there is an x and a y such that mx < ny. this means that it is impossible to extend f at p so that it is strongly lip-1 with respect to both x and y. but this cannot happen according to lemma 4.3 and therefore r cannot be empty. define f̃(p) to be, say, the smallest integer in r and f̃(x) = f(x) if x ∈ a. then f̃ : a ∪ {p} → z is still strongly lip-1. continuous extension in digital spaces 61 we will now use zorn’s lemma to show existence of a strongly lip-1 extension. let e be the set of all (graphs of) strongly lip-1 extensions of f. an element in e is thus a subset of x × z. a partial order on e is defined by set inclusion. let c be a chain in e. an upper bound for c is given by ⋃ c. thus e is inductive. by zorn’s lemma, e has a maximal element and it corresponds to a function f : y → z. it remains to be shown that y = x. suppose that z ∈ x ry . then, by the first part of the proof, there is a strongly lip-1 extension of f , y ∪ {z} → z, which contradicts the maximality of f . we conclude that f must in fact be the required extension of f. � we now turn to the main theorem of this section. theorem 4.5 (continuous extensions). let a be a subspace of a t0 topological digital space x and let f : a → z be any function. then f can be extended to a continuous function on all of x if and only if f is strongly lip-1 w.r.t. x. proof. that the function must be strongly lip-1 follows from proposition 4.2. for the converse, first use proposition 4.4 to find a strongly lip-1 extension to all of x and then proposition 4.2 again to conclude that this extension is in fact continuous. � remark 4.6. the case with a non-t0 digital space x can be handled. it is not hard to see that if f : x → z is continuous and n (x) = n (y), then f(x) = f(y). suppose that a ⊂ x and that f is a function on a. if f does not satisfy the implication: if n (x) = n (y) then f(x) = f(y), then f cannot be continuously extended. otherwise we can identify the points with the same neighborhood and consider the quotient space (which is t0) with the induced function f̃. it follows that f can be continuously extended if and only if f̃ can be extended, i.e., if and only if f̃ is strongly lip-1. remark 4.7. also the case where x is not connected can be handled; we may consider extension of the function on each connected component separately. we may include both these remarks in a modified version of theorem 4.5. for this we need to consider the extended arc metric ρ̃x defined on any smallestneighborhood space x as follows. ρ̃x (x,y) = { 1/2 if x 6= y and n (x) = n (y) ρx (x,y) otherwise, where ρx is defined by the same equation as before. ρx (x,y) = min{l(a); a ⊂ x is a khalimsky arc between x and y}. note that if x is not connected and x and y belong to different connected components, we get ρ̃x (x,y) = min ∅ = +∞. hence we allow here infinite distance between points. if we replace the arc metric ρ with the extended arc metric ρ̃ in definition 4.1, we may speak about strongly lip-1 functions defined on a smallest-neighborhood space x. using this extended version of the strongly lip-1 condition, we have the following version of theorem 4.5. 62 e. melin figure 3. a) a completely arc-connected set. b) the set is not completely arc-connected. note that both sets are connected, however. theorem 4.8. let a be a subspace of a smallest-neighborhood space x and let f : a → z be any function. then f can be extended to a continuous function on all of x if and only if f is strongly lip-1 w.r.t. x. 5. completely arc-connected sets the goal of this section is to use theorem 4.5 to classify the subsets of a topological digital space such that any continuous function defined there can be continuously extended to all of x. this is a digital analogue of the classical tietze extension theorem. definition 5.1. let x be a t0 topological digital space and a a subset of x. then a is called completely arc-connected in x if a is connected, ρa = ρx ∣ ∣ a and for every pair x, y of distinct points in a, it is true that if x is open (closed) w.r.t. y in x then x is also open (closed) w.r.t. y in a. it is easy to see that a is completely arc-connected in x if and only if the following conditions are satisfied for every pair of distinct points x and y: if x is open (closed) w.r.t. y in x, then a contains an arc m connecting x and y, such that x is open (closed) in m and l(m) = ρx (x,y). example 5.2. the set {(0, 0), (1, 1), (0, 2), (−1, 1)} in the khalimsky plane is completely arc-connected. the set m = {(1, 0), (1, 1), (2, 1)} is not, since (1, 0) is both closed and open w.r.t. (2, 1) in z2, but only closed w.r.t. (2, 1) in m. adding the point (2, 0) gives the set m ∪ {(2, 0)}, which is completely arc-connected in z2. the sets are shown in figure 3. the following theorem is now very easy to prove. theorem 5.3. suppose that x is a t0 topological digital space and that a ⊂ x is completely arc-connected in x. if f : a → z is continuous in a, then f can be extended to a continuous function f defined on all of x. proof. by proposition 4.2, f is strongly lip-1 w.r.t. a. we have to check that f is also strongly lip-1 w.r.t. x. let x and y be distinct points in a. since ρx (x,y) = ρa(x,y), the conclusion follows easily if |f(x) − f(y)| < ρ(x,y). suppose that |f(x) − f(y)| = ρ(x,y). if x is open (closed) w.r.t. y in x, then x is also open (closed) w.r.t. y in a. therefore f(x) is odd (even) and thus strongly lip-1 w.r.t. x. � continuous extension in digital spaces 63 theorem 5.4. let x be a t0 topological digital space. if a ⊂ x is not completely arc connected then there is a continuous function f : a → z that cannot be extended to all of x. proof. if a is not connected, the conclusion is immediate. suppose therefore that a is connected but not completely arc-connected and let x and y be points in a such that some condition for arc-connectedness fails for the pair x and y. we shall define a function g : {x,y} → z and then we will show that g can be extended to a continuous function on a but not to a continuous function on x. suppose first that ρa(x,y) > ρx (x,y). let g(x) = 0 if x is open w.r.t. y in x, and g(x) = 1 otherwise. then define g(y) = g(x) + ρa(x,y) − 1. we get |g(x) − g(y)| = ρa(x,y) − 1 < ρa(x,y), so g is strongly lip-1 w.r.t. a and can therefore be continuously extended to g: a → z by theorem 4.5. on the other hand, we have that |g(x) − g(y)| = |g(x) − g(y)| = ρa(x,y) − 1 > ρx (x,y), and by construction g(x) and x do not match in parity. therefore, again by theorem 4.5, g cannot be extended to all of x. the remaining case is ρa(x,y) = ρx (x,y). here x must be both open and closed w.r.t. y in x, but only open or only closed w.r.t. y in a. define the function g as g(x) = 0 if x is closed w.r.t. y in a and g(x) = 1 otherwise. let g(y) = g(x) + ρ(x,y). clearly |g(x) − g(y)| = ρa(x,y) but since x and g(x) match in parity, g is strongly lip-1 w.r.t. a and can therefore be continuously extended to g: a → z. we also have |g(x) − g(y)| = ρx (x,y), however, x is both closed and open w.r.t. y in x, so g is not strongly lip-1 in x. this completes the proof. � example 5.5. the set m of example 5.2 is not completely arc-connected. define a function on m by f(1, 0) = 0, f(1, 1) = 1 and f(2, 1) = 2. then f is continuous on m but f cannot be continuously extended to m ∪ {(2, 0)}: f(1, 0) = 0 implies that f(2, 0) = 0 and f(2, 1) = 2 implies that f(2, 0) = 2, which is contradictory. 6. continuous functions between digital spaces suppose that x and y are t0 topological digital spaces and that a ⊂ x. let f : a → y be a continuous function. what can we say about the possibilities to extend f to all of x? consider any continuous function g : x → y and let x,y be distinct points in x. if b is a minimal arc in x connecting x and y, then g(b) is a connected subset of y —hence g(b) contains an arc connecting g(x) and g(y). from these observations we get the following chain of inequalities (6.1) ρy (g(x),g(y)) 6 card(g(b)) − 1 6 card(b) − 1 = ρx (x,y) if f is to be extended, it must therefore satisfy this inequality for every pair of points where f is defined. we state this result as a proposition: 64 e. melin proposition 6.1. suppose that x and y are t0 topological digital spaces and that g : x → y is continuous. then g is lip-1 for the arc metric. this result is a generalization of proposition 3.3. guided by the sharpened version of that result, proposition 4.2, one might expect that a similar sharpening is possible here too. indeed, suppose we have equality in (6.1). then g is a continuous bijection between the khalimsky arcs b and g(b). but then g−1 is automatically continuous, as is easy to check, so b and g(b) are homeomorphic. we conclude that if x is open (closed) w.r.t. y in x, then f(x) is open (closed) w.r.t. f(y) in y . definition 6.2. let us call a function f : x → y strongly lip-1 w.r.t. the arc metric if for every x,y ∈ x, x 6= y either (1) ρy (f(x),f(y)) < ρx (x,y) or (2) ρy (f(x),f(y)) = ρx (x,y) and x is open (closed) w.r.t. y in x implies that f(x) is open (closed) w.r.t. f(y) in y . this definition is a generalization of definition 4.1 since ρz(m,n) = |m − n| for points m and n on the khalimsky line. we also get the following theorem, corresponding to proposition 4.2. theorem 6.3. let x and y be t0 topological digital spaces. then f : x → y is continuous if and only if f is strongly lip-1 w.r.t. the arc metric. proof. one direction is already clear; it remains to be proved that if f is strongly lip-1 w.r.t. the arc metric, then f is continuous. assume that f is strongly lip-1. it is sufficient to show that nx (x) ⊂ f −1(ny (f(x))) for every x ∈ x. let y ∈ nx (x). then ρx (x,y) = 1. if ρy (f(x),f(y)) = 0 then f(x) = f(y) and the conclusion follows. the other possibility is that ρy (f(x),f(y)) = 1. since y ∈ nx (x), y is open w.r.t. x. hence there is an arc b of length 1 in y such that f(y) is open in b. but clearly b = {f(x),f(y)} and this means that f(y) ∈ ny (f(x)). � however, strongly lip-1 is not sufficient for an extension to exist; there is no analogue of proposition 4.4. proposition 6.5 below demonstrates this. we need to introduce some new notation and a result to formulate it. let x be any topological space. a khalimsky jordan curve in x is a homeomorphic image of a khalimsky circle, c.f. example 2.8. we will let x be the khalimsky plane, z2. the following theorem is proved by khalimsky et al. in [7]. kiselman [8] has given a simplier proof. theorem 6.4 (khalimsky jordan curve theorem). let j be any khalimsky jordan curve in z2. then the complement z2 r j has exactly two connectivity components. it is easy to see that one of these components must be finite. this component together with its boundary j will be denoted dj and called a khalimsky disk. the component itself, i.e., dj r j, will be called the interior of dj and is denoted by int(dj ). continuous extension in digital spaces 65 proposition 6.5. let j be any khalimsky jordan curve in z2 and suppose that f = (f1,f2) : dj → dj is a continuous mapping such that f ∣ ∣ j is the identity mapping of j. then f is the identity mapping of dj . proof. let x = (x1,x2) be a point in int(dj ). let a = min{m ∈ n; x + (m, 0) ∈ j} and b = min{m ∈ n; x − (m, 0) ∈ j}. clearly 0 < a,b < ∞. let p = x + (a, 0) and q = x − (b, 0). by assumption we have f(p) = p and f(q) = q, so that |f1(x) − p1| 6 ρ(f(x),p) = ρ(f(x),f(p)) 6 ρ(x,p) = |x1 − p1| = a here f1(x) is the first coordinate of f(x), and the first inequality follows since |xi − yi| 6 ρ(x,y) for the i:th coordinate of any two points x,y in a khalimsky subspace. the second inequality follows from theorem 6.3. similarly we have |f1(x) − q1| 6 b. but since |p1 − q1| = a + b, the only possibility is f1(x) = x1. using the same argument in the other coordinate direction we get f2(x) = x2 and hence f(x) = x as required. � if x is any topological space and a a subspace of x, a retraction of x onto a is a continuous map ϕ: x → a such that ϕ ∣ ∣ a is the identity map of a. if such a map exist we say that a is a retract of x. the well-known classical no-retraction theorem states that there is no retraction of the unit disk b2 of r 2 onto the circle s1, or more generally of bn onto sn−1. in view of this, proposition 6.5 in particular states the there is no retraction from dj onto j. note also that f ∣ ∣ j , being the identity of j, is strongly lip-1 w.r.t. the arc-metric on j but cannot be extended to dj . 7. conclusion we have shown that a condition somewhat stronger than a lipschitz condition with lipschitz constant 1 for the arc metric is equivalent to continuity for a function f : x → y , where x and y are t0 digital spaces. we call such a function strongly lip-1 w.r.t. x. moreover, if y = z with the khalimsky topology, then any function defined on a subset of x such that f is strongly lip-1 w.r.t. x can be extended to a continuous defined on all of x. using this result, we have proved a digital analogue of the classical tietze extension theorem; the subsets of a such that any continuous function a → z can be extended to all of x are precisely the subsets a where the notion of being strongly lip-1 w.r.t. a agrees with the notion of being strongly lip-1 w.r.t. x. we have also proved a digital version of the no-retraction theorem for a disk in the plane. this result shows that if the codomain y is a more general space than z, then the strongly lip-1 condition need not be sufficient for a continuous extension to exist. 66 e. melin references [1] p. alexandrov, diskrete räume. mat. sb. 2 (1937), 501–519. [2] l. boxer, digitally continuous functions, pattern recognition lett. 15 (1994), 833–839. [3] u. eckhardt and l. j. latecki, topologies for the digital spaces z2 and z3 computer vision and image understanding 90 (2003), 295–312. [4] g. t. herman, geometry of digital spaces, applied and numerical harmonic analysis. birkhäuser boston inc., boston, ma, 1998. [5] g. t. herman and d. webster, a topological proof of a surface tracking algorithm, computer vision, graphics, and image processing 23 (1983), 162–177. [6] e. khalimsky, topological structures in computer science, j. appl. math. simulation 1 (1987), 25–40. [7] e. khalimsky, r. kopperman and p. r. meyer, computer graphics and connected topologies on finite ordered sets, topology appl. 36 (1990), 1–17. [8] c. o. kiselman, digital jordan curve theorems, in g. borgefors, i. nyström, and g. sanniti di baja, editors, discrete geometry for computer imagery, volume 1953 of lecture notes in computer science, pages 46–56, 2000. [9] c. o. kiselman, digital geometry and mathematical morphology. lecture notes, uppsala university, 2004. available at www.math.uu.se/~kiselman. [10] reinhard klette, topologies on the planar orthogonal grid, in 6th international conference on pattern recognition (icpr’02), volume ii, pages 354–357, 2002. [11] t. y. kong, topological adjacency relations on zn,theoret. comput. sci. 283 (2002), 3-28. [12] t. y. kong, the khalimsky topologies are precisely those simply connected topologies on zn whose connected sets include all 2n-connected sets but no (3n − 1)-disconnected sets, theoret. comput. sci. 305 (2003), 221–235. [13] t. y. kong, r. kopperman and p. r. meyer, a topological approach to digital topology, amer. math. monthly 98 (1991), 901–917. [14] t. y. kong and a. rosenfeld, digital topology: introduction and survey, comput. vision graph. image process. 48 (1989), 357–393. [15] r. kopperman, topological digital topology. in ingela nyström, gabriella sanniti di baja, and stina svensson, editors, dgci, volume 2886 of lecture notes in computer science, pages 1–15. springer, 2003. [16] v. a. kovalevsky, finite topology as applied to image analysis, comput. vision graph. image process. 46 (1989), 141–161. [17] e. melin, how to find a khalimsky-continuous approximation of a real-valued function, in reinhard klette and jovisa žunić, editors, combinatorial image analysis, volume 3322 of lecture notes in computer science, pages 351–365, 2004. [18] e. melin, digital straight lines in the khalimsky plane, math. scand. 96 (2005) 49–62. [19] e. melin, extension of continuous functions in digital spaces with the khalimsky topology, topology appl. 153 (2005) 52–65. [20] r. e. stong, finite topological spaces, trans. amer. math. soc. 123 (1966) 325–340. received august 2006 accepted december 2007 erik melin (melin@math.uu.se) uppsala university, department of mathematics, box 480, se-751 06 uppsala, sweden @ appl. gen. topol. 15, no. 1 (2014), 65-84doi:10.4995/agt.2014.1823 c© agt, upv, 2014 unified common fixed point theorems under weak reciprocal continuity or without continuity zoran kadelburg a, mohammad imdad b and sunny chauhan c a university of belgrade, faculty of mathematics, studentski trg 16, 11000 beograd, serbia. (kadelbur@matf.bg.ac.rs) b department of mathematics, aligarh muslim university, aligarh 202 002, india. (mhimdad@yahoo.co.in) c near nehru training centre, h. no. 274, nai basti b-14, bijnor 246 701, uttar pradesh, india. (sun.gkv@gmail.com) abstract the purpose of this paper is two fold. firstly, using the notion of weak reciprocal continuity due to pant et al. [weak reciprocal continuity and fixed point theorems, ann. univ. ferrara sez. vii sci. mat. 57(1), 181–190 (2011)], we prove unified common fixed point theorems for various variants of compatible and r-weakly commuting mappings in complete metric spaces employing an implicit relation which covers a multitude of contraction conditions yielding thereby known as well as unknown results as corollaries. secondly, we point out that more natural results can be proved under relatively tighter conditions if we replace the completeness of the space by completeness of suitable subspaces. the realized improvements in our results are also substantiated using appropriate examples. 2010 msc: primary 47h10; secondary 54h25. keywords: metric space; compatible mappings; r-weakly commuting mappings; weak reciprocal continuity; coincidentally commuting; implicit relation. 1. introduction and preliminaries indeed, banach contraction principle is a fundamental and fruitful result of metric fixed point theory. after this classical result, several authors have received 1 november 2013 – accepted 22 january 2014 http://dx.doi.org/10.4995/agt.2014.1823 z. kadelburg, m. imdad and s. chauhan contributed to the vigorous development of metric fixed point theory in different ways (e.g. [4, 7, 10, 11, 12, 17, 30]). as patterned in jungck [13], a result on the existence of common fixed points generally involves conditions on commutativity, continuity and contraction along with a suitable condition on the containment of range of one mapping into the range of other. hence, one is always required to improve one or more of these conditions to prove a new fixed point theorem. jungck [13] obtained a common fixed point theorem for a pair of commuting mappings. in 1982, sessa [29] formulated the notion of weak commutativity and established a common fixed point theorem for such pairs. jungck [14] generalized the notion of weakly commuting mappings by introducing the concept of compatible mappings. thereafter, pant [20] defined r-weak commutativity and proved common fixed point results for r-weakly commuting mappings in metric spaces. the study of common fixed points of non-compatible mappings is equally natural which was indeed noted in pant [22]. in 1997, pathak et al. [24] introduced the notions of r-weakly commuting mappings of types (ag) and (af ) and utilize the same to prove common fixed point theorems in metric spaces. in 1998, pant [21] introduced the notion of reciprocal continuity and utilize the same to prove results on common fixed points which also remains a point of discontinuity of the involved mappings. recently, pant et al. [23] improved the notion of reciprocal continuity by introducing weak reciprocal continuity and observed that weak reciprocal continuity is applicable to compatible as well as to non-compatible mappings. in fact, they proved the following result. theorem 1.1. let f and g be weakly reciprocally continuous self-mappings of a complete metric space (x,d) such that (i) f(x) ⊆ g(x); (ii) d(fx,fy) ≤ ad(gx,gy) + bd(fx,gx) + cd(fy,gy), for some a,b,c ≥ 0 with a + b + c < 1 and for all x,y ∈ x. if f and g are either compatible or r-weakly commuting of type (ag) or rweakly commuting of type (af ) then f and g have a unique common fixed point. the notions and definitions utilized thus far and also to be utilized in the sequel are presented in the following multitude of definitions: definition 1.2. let f,g : x → x be two self-mappings of a metric space (x,d). then the pair (f,g) is said to be (1) commuting if fgx = gfx, for all x ∈ x, (2) weakly commuting [29] if d(fgx,gfx) ≤ d(fx,gx), for all x ∈ x, (3) r-weakly commuting [20] if there exists some real number r > 0 such that d(fgx,gfx) ≤ rd(fx,gx) for all x ∈ x, (4) pointwise r-weakly commuting [20] if given x ∈ x there exists some real number r > 0 such that d(fgx,gfx) ≤ rd(fx,gx), c© agt, upv, 2014 appl. gen. topol. 15, no. 1 66 unified common fixed point theorems... (5) r-weakly commuting of type (ag) [24] if there exists some real number r > 0 such that d(ffx,gfx) ≤ rd(fx,gx) for all x ∈ x, (6) r-weakly commuting of type (af ) [24] if there exists some real number r > 0 such that d(fgx,ggx) ≤ rd(fx,gx) for all x ∈ x, (7) compatible [14] if lim n→∞ d(fgxn,gfxn) = 0 for each sequence {xn} in x such that lim n→∞ fxn = lim n→∞ gxn, (8) g-compatible [25] if lim n→∞ d(ffxn,gfxn) = 0 for each sequence {xn} in x such that lim n→∞ fxn = lim n→∞ gxn, (9) f-compatible [25] if lim n→∞ d(fgxn,ggxn) = 0 for each sequence {xn} in x such that lim n→∞ fxn = lim n→∞ gxn, (10) non-compatible [22] if there exists a sequence {xn} in x such that lim n→∞ fxn = lim n→∞ gxn but lim n→∞ d(fgxn,gfxn) is either nonzero or nonexistent, (11) reciprocally continuous [21] if lim n→∞ fgxn = ft and lim n→∞ gfxn = gt whenever {xn} is a sequence such that lim n→∞ fxn = lim n→∞ gxn = t for some t ∈ x, (12) weakly reciprocally continuous [23] if lim n→∞ fgxn = ft or lim n→∞ gfxn = gt whenever {xn} is a sequence such that lim n→∞ fxn = lim n→∞ gxn = t for some t ∈ x. (13) coincidentally commuting [5] (or weakly compatible [15]) if they merely commute at their coincidence points. in the foregoing definitions, it is assumed that there is at least one sequence in the underlying space meeting the prescribed requirements. definition 1.3 ([8]). two families of self mappings {fi}mi=1 and {gk}nk=1 are said to be pairwise commuting if (1) fifj = fjfi for all i,j ∈ {1,2, . . . ,m}, (2) gkgl = glgk for all k,l ∈ {1,2, . . . ,n}, (3) figk = gkfi for all i ∈ {1,2, . . . ,m} and k ∈ {1,2, . . . ,n}. for more details on systematic comparisons and illustrations of earlier described notions, we refer to murthy [19], singh and tomar [31], pant et al. [23] and kadelburg et al. [16]. in this paper, utilizing the notion of implicit relation due to popa et al. [27], we prove common fixed point results for variants of compatible (g-compatible and f-compatible) and r-weakly commuting (r-weakly commuting of type (ag) and r-weakly commuting of type (af )) mappings, employing an implicit relation which is a slightly refined form of an implicit function due to popa [26]. further, we prove a more general result for coincidentally commuting mappings without any requirement of weak reciprocal continuity. some related results are also derived besides furnishing illustrative examples which exhibit the superiority of our results over some of the known ones. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 67 z. kadelburg, m. imdad and s. chauhan 2. implicit relations implicit functions are often used very effectively to cover various contraction conditions in one go rather than proving a separate theorem for each contraction condition. the first ever attempt to coin an implicit relation can be traced back to popa [26]. in 2008, ali and imdad [1] introduced a new class of implicit functions which covers several classes of contractions conditions. since then, many mathematicians utilized implicit relations with various properties to prove a number of fixed point theorems (e.g. [3, 27, 28]). in order to describe the implicit function we intend to use here, let φ be the family of lower semi-continuous functions φ : r6+ → r satisfying the following conditions: (φ1): φ is non-increasing in variables t5 and t6, (φ2): there exists h ∈ (0,1) such that for u,v ≥ 0, (φ2a): φ(u,v,v,u,u + v,0) ≤ 0 or (φ2b): φ(u,v,u,v,0,u + v) ≤ 0 implies u ≤ hv, (φ3): φ(u,u,0,0,u,u) > 0, for all u > 0. the following examples of functions φ ∈ φ appeared in popa [26] and imdad and ali [7]. example 2.1. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − k max { t2, t3, t4, 1 2 (t5 + t6) } , where k ∈ (0,1). example 2.2. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − t1(αt2 + βt3 + γt4) − ηt5t6, where α > 0, β,γ,η ≥ 0, α + β + γ < 1 and α + η < 1. example 2.3. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t 3 1 − αt21t2 − βt1t3t4 − γt25t6 − ηt5t26, where α > 0, β,γ,η ≥ 0, α + β < 1 and α + γ + η < 1. example 2.4. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t 3 1 − k t23t 2 4 + t 2 5t 2 6 1 + t2 + t3 + t4 , where k ∈ (0,1). example 2.5. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − αt22 − β t5t6 1 + t23 + t 2 4 , where α > 0, β ≥ 0 and α + β < 1. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 68 unified common fixed point theorems... example 2.6. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − α max{t22, t23, t24} − β max{t3t5, t4t6} −γt5t6 where α > 0, β,γ ≥ 0, α + 2β < 1 and α + γ < 1. example 2.7. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − k max { t2, t3, t4, t5 2 , t6 2 } , where k ∈ (0,1). example 2.8. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − k max { t2, t3 + t4 2 , t5 + t6 2 } , where k ∈ (0,1). example 2.9. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − (αt2 + βt3 + γt4 + ηt5 + λt6), where α + β + γ + η + λ < 1. example 2.10. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − k 2 max {t2, t3, t4, t5, t6} , where k ∈ (0,1). example 2.11. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − [αt2 + βt3 + γt4 + η(t5 + t6)], where α + β + γ + 2η < 1. since verifications of requirements (φ1), (φ2) and (φ3) for examples 2.1–2.11 are straightforward, the details are not included. here, one may notice that some other well known contraction conditions can also be deduced as particular cases of implicit relation of popa [26]. in order to strengthen this viewpoint, we include some further examples and utilize them to demonstrate how this implicit relation can cover several other known contraction conditions and is also good enough to yield further natural contraction conditions as well. example 2.12. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) =    t1 − a1 t23 + t 2 4 t3 + t4 − a2t2 − a3(t5 + t6), if t3 + t4 6= 0; t1, if t3 + t4 = 0, where ai ≥ 0 with at least one ai non-zero and a1 + a2 + 2a3 < 1. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 69 z. kadelburg, m. imdad and s. chauhan example 2.13. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) =    t1 − a1t2 − a2t3t4 + a3t5t6 t3 + t4 , if t3 + t4 6= 0; t1, if t3 + t4 = 0, where a1,a2,a3 ≥ 0 such that 1 < 2a1 + a2 < 2. example 2.14. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − a1 [ a2 max { t2, t3, t4, 1 2 (t5 + t6) } +(1 − a2) [ max { t22, t3t4, t5t6, t3t6 2 , t4t5 2 }] 1 2 ] , where a1 ∈ (0,1) and 0 ≤ a2 ≤ 1. example 2.15. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − a1 max { t22, t 2 3, t 2 4 } − a2 max { t3t5 2 , t4t6 2 } −a3t5t6, where a1,a2,a3 ≥ 0 and a1 + a2 + a3 < 1. very recently, popa et al. [27] proved several fixed point theorems satisfying suitable implicit relations from which husain and sehgal [6] type contraction conditions can be deduced. a slight modification in condition (φ1) is used as follows: (φ′1): φ is decreasing in variables t2, . . . , t6. hereafter, let φ : r6+ → r be a continuous function which satisfies the conditions φ′1, φ2 and φ3 and let φ ′ be the family of such functions φ. in this paper, we employ such implicit relations to prove our results. but before we proceed further, let us furnish some examples to highlight the utility of the modifications instrumental herein. example 2.16. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − ψ ( max{t2, t3, t4, 1 2 (t5 + t6)} ) , where ψ : r+ → r+ is an increasing upper semi-continuous function with ψ(0) = 0 and ψ(t) < t for each t > 0. example 2.17. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t1 − ψ (t2, t3, t4, t5, t6) , where ψ : r5+ → r+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 70 unified common fixed point theorems... example 2.18. define φ : r6+ → r as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − ψ ( t22, t3t4, t5t6, t3t6, t4t5 ) , where ψ : r5+ → r+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. clearly, apart from these examples, there are many other functions which meet the requirements (φ′1), (φ2) and (φ3). 3. results under weak reciprocal continuity in the following proposition, we notice that under the prescribed setting, the set of common fixed point of the involved mappings is always singleton provided such points exist. proposition 3.1. let (x,d) be a metric space and let f,g : x → x be two mappings satisfying (3.1) φ(d(fx,fy),d(gx,gy),d(fx,gx),d(fy,gy),d(gx,fy),d(gy,fx)) ≤ 0, for all x,y ∈ x where φ enjoys the property (φ3). then f and g have at most one common fixed point in x. proof. let, on the contrary, the mappings f and g have two common fixed points w and w′ such that w 6= w′. on using (3.1), we have φ(d(fw,fw′),d(gw,gw′),d(fw,gw),d(fw′,gw′),d(gw,fw′),d(gw′,fw)) ≤ 0, or φ(d(w,w′),d(w,w′),d(w,w),d(w′,w′),d(w,w′),d(w′,w)) ≤ 0, so that φ(d(w,w′),d(w,w′),0,0,d(w,w′),d(w′,w)) ≤ 0, which contradicts (φ3), yielding thereby w = w ′. � let f and g be two mappings from a metric space (x,d) into itself satisfying the following condition: (3.2) f(x) ⊆ g(x). for an arbitrary point x0 ∈ x there exists a point x1 ∈ x such that fx0 = gx1. continuing in this way, one can inductively define a sequence {yn} in x such that (3.3) yn = fxn = gxn+1, for all n = 0,1,2, . . . . lemma 3.2. let f and g be two mappings from a metric space (x,d) into itself which satisfy conditions (3.1) (for some φ ∈ φ′) and (3.2), and let {yn} be defined by (3.3). then {yn} is a cauchy sequence. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 71 z. kadelburg, m. imdad and s. chauhan proof. on using (3.1) with x = xn and y = xn+1, we have φ ( d(fxn,fxn+1),d(gxn,gxn+1),d(fxn,gxn), d(fxn+1,gxn+1),d(gxn,fxn+1),d(gxn+1,fxn) ) ≤ 0, or φ ( d(yn,yn+1),d(yn−1,yn),d(yn,yn−1), d(yn+1,yn),d(yn−1,yn+1),d(yn,yn) ) ≤ 0, or φ ( d(yn,yn+1),d(yn−1,yn),d(yn−1,yn), d(yn,yn+1),d(yn−1,yn) + d(yn,yn+1),0 ) ≤ 0, implying thereby d(yn,yn+1) ≤ hd(yn−1,yn) (due to (φ2a)) where h ∈ (0,1). in general, for all n = 0,1,2, . . ., we have d(yn,yn+1) ≤ hd(yn−1,yn). again using (3.1) with x = xn+2 and y = xn+1, we get φ ( d(fxn+2,fxn+1),d(gxn+2,gxn+1),d(fxn+2,gxn+2), d(fxn+1,gxn+1),d(gxn+2,fxn+1),d(gxn+1,fxn+2) ) ≤ 0, or φ ( d(yn+2,yn+1),d(yn+1,yn),d(yn+2,yn+1), d(yn+1,yn),d(yn+1,yn+1),d(yn,yn+2) ) ≤ 0, or φ ( d(yn+1,yn+2),d(yn,yn+1),d(yn+1,yn+2), d(yn,yn+1),0,d(yn,yn+1) + d(yn+1,yn+2) ) ≤ 0. in view of (φ2b), we have d(yn+1,yn+2) ≤ hd(yn,yn+1) where h ∈ (0,1). for all n = 0,1,2, . . . we get d(yn+1,yn+2) ≤ hd(yn,yn+1). moreover, for every integer p > 0, we obtain d(yn,yn+p) ≤ d(yn,yn+1) + d(yn+1,yn+2) + · · · + d(yn+p−1,yn+p) ≤ (1 + h + · · · + hp−1)d(yn,yn+1) ≤ ( 1 1 − h ) hnd(y0,y1), i.e., d(yn,yn+p) → 0 as n → ∞. therefore {yn} is a cauchy sequence in x. � utilizing the recently introduced notion of weak reciprocal continuity due to pant et al. [23], we prove a common fixed point theorem in a complete metric space for variants of compatible (g-compatible and f-compatible) and r-weakly commuting (r-weakly commuting of type (ag) and r-weakly commuting of type (af )) mappings via an implicit function studied by popa [27] which is a slightly refined version of some known implicit relations. some related results are also derived besides furnishing illustrative examples which establish the superiority of our results over some of the known results especially the ones contained in pant et al. [23]. theorem 3.3. let f and g be weakly reciprocally continuous self-mappings of a complete metric space (x,d) satisfying conditions (3.1) (for some φ ∈ φ′) and (3.2). if the mappings f and g are either compatible or g-compatible or f-compatible or r-weakly commuting or r-weakly commuting of type (ag) or r-weakly commuting of type (af ), then f and g have a unique common fixed point in x. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 72 unified common fixed point theorems... proof. let {yn} be a sequence defined by (3.3). by lemma 3.2, {yn} is a cauchy sequence in x. since x is complete, there exists a point z in x such that lim n→∞ yn = lim n→∞ fxn = lim n→∞ gxn+1 = z. case i: suppose that the mappings f and g are compatible. the weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) let gfxn → gz as n → ∞. the compatibility of f and g yields lim n→∞ d(fgxn,gfxn) = 0, hence fgxn → gz as n → ∞. by (3.2), this yields fgxn+1 = ffxn → gz as n → ∞. on using (3.1) with x = z and y = fxn, we get φ(d(fz,ffxn),d(gz,gfxn),d(fz,gz),d(ffxn,gfxn),d(gz,ffxn),d(gfxn,fz)) ≤ 0. taking the limit as n → ∞, we have φ(d(fz,gz),d(gz,gz),d(fz,gz),d(gz,gz),d(gz,gz),d(gz,fz)) ≤ 0, or, equivalently, φ(d(fz,gz),0,d(fz,gz),0,0,d(gz,fz)) ≤ 0, implying thereby d(fz,gz) = 0 (due to (φ2b)). hence fz = gz which shows that z is a coincidence point of the mappings f and g. also compatibility of f and g implies commutativity at a coincidence point. hence ffz = fgz = gfz = ggz. now we assert that fz is a common fixed point of f and g. to accomplish this, using (3.1) with x = z and y = fz, we have φ(d(fz,ffz),d(gz,gfz),d(fz,gz),d(ffz,gfz),d(gz,ffz),d(gfz,fz)) ≤ 0. and so φ(d(fz,ffz),d(fz,ffz),0,0,d(fz,ffz),d(ffz,fz)) ≤ 0, yielding thereby d(fz,ffz) = 0 (due to (φ3)). therefore, fz = ffz = gfz which shows that fz is a common fixed point of the mappings f and g. (ii) assume next that fgxn → fz as n → ∞. by virtue of (3.2), we have fz = gt for some t ∈ x hence fgxn → gt as n → ∞. the compatibility of the mappings f and g implies lim n→∞ d(fgxn,gfxn) = 0, hence gfxn → gt as n → ∞. again by (3.2), fgxn+1 = ffxn → gt as n → ∞. on using (3.1) with x = t and y = fxn, we get φ(d(ft,ffxn),d(gt,gfxn),d(ft,gt),d(ffxn,gfxn),d(gt,ffxn),d(gfxn,ft)) ≤ 0. taking the limit as n → ∞, we have φ(d(ft,gt),0,d(ft,gt),0,0,d(gt,ft)) ≤ 0, which implies d(ft,gt) = 0 (due to (φ2b)). hence ft = gt which shows that t is a coincidence point of the mappings f and g. also, compatibility of f and g implies commutativity at a coincidence point. hence fft = fgt = gft = ggt. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 73 z. kadelburg, m. imdad and s. chauhan we show that ft is a common fixed point of f and g. to assert this, using (3.1) with x = t and y = ft, we have φ(d(ft,fft),d(gt,gft),d(ft,gt),d(fft,gft),d(gt,fft),d(gft,ft)) ≤ 0, or, equivalently, φ(d(ft,fft),d(ft,fft),0,0,d(ft,fft),d(fft,ft)) ≤ 0, implying thereby d(ft,fft) = 0 (due to (φ3)). therefore, ft = fft = gft which shows that ft is a common fixed point of the mappings f and g. case ii: let the mappings f and g be g-compatible. the weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) suppose that gfxn → gz as n → ∞. then g-compatibility of f and g yields lim n→∞ d(ffxn,gfxn) = 0, hence ffxn → gz as n → ∞. on using (3.1) with x = z and y = fxn, φ(d(fz,ffxn),d(gz,gfxn),d(fz,gz),d(ffxn,gfxn),d(gz,ffxn),d(gfxn,fz)) ≤ 0. taking the limit as n → ∞, we get fz = gz (due to (φ2b)) which shows that z is a coincidence point of the mappings f and g. since g-compatibility of f and g implies commutativity at a coincidence point, we have ffz = fgz = gfz = ggz. now, we assert that fz is a common fixed point of f and g. on using (3.1) with x = z and y = fz, we have φ(d(fz,ffz),d(gz,gfz),d(fz,gz),d(ffz,gfz),d(gz,ffz),d(gfz,fz)) ≤ 0, implying thereby fz = ffz = gfz (due to (φ3)). hence fz is a common fixed point of the mappings f and g. (ii) assume that fgxn → fz as n → ∞. by (3.2), we have fz = gt for some t ∈ x. thus fgxn → gt as n → ∞. also (3.2) implies fgxn+1 = ffxn → gt as n → ∞. the g-compatibility of the mappings f and g implies lim n→∞ d(ffxn,gfxn) = 0, hence gfxn → gt as n → ∞. on using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (due to (φ2b)). hence t is a coincidence point of the mappings f and g. also, gcompatibility of f and g implies commutativity at a coincidence point. hence fft = fgt = gft = ggt. we assert that ft is a common fixed point of f and g. on using (3.1) with x = t, y = ft we obtain ft = fft = gft (due to (φ3)). therefore, ft is a common fixed point of the mappings f and g. case iii: suppose that the pair (f,g) is f-compatible. the weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) let us consider gfxn → gz as n → ∞. then f-compatibility of f and g implies lim n→∞ d(fgxn,ggxn) = 0. by (3.2), we get gfxn = ggxn+1 → gz as n → ∞ and so fgxn+1 = ffxn → gz as n → ∞. putting x = z and y = fxn in (3.1) and taking the limit as n → ∞, we get fz = gz (in view of (φ2b)). hence z is a coincidence point of the mappings f and g. since fcompatibility of f and g implies commutativity at a coincidence point, we have ffz = fgz = gfz = ggz. now we show that fz is a common fixed point of f c© agt, upv, 2014 appl. gen. topol. 15, no. 1 74 unified common fixed point theorems... and g. putting x = z and y = fz in (3.1), we have fz = ffz = gfz (due to (φ3)). thus fz is a common fixed point of the mappings f and g. (ii) assume that fgxn → fz as n → ∞. by (3.2), we have fz = gt for some t ∈ x. thus fgxn → gt as n → ∞. then f-compatibility of the mappings f and g implies lim n→∞ d(fgxn,ggxn) = 0, hence ggxn → gt as n → ∞. by (3.2), we get fgxn+1 = ffxn → gt as n → ∞ and so gfxn = ggxn+1 → gt as n → ∞. on using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we obtain ft = gt (in view of (φ2b)). thus t is a coincidence point of the mappings f and g. also, f-compatibility of f and g implies commutativity at a coincidence point. hence fft = fgt = gft = ggt. by putting x = t, y = ft in (3.1), we get ft = fft = gft (due to (φ3)). hence ft is a common fixed point of the mappings f and g. case iv: let the pair (f,g) be r-weakly commuting. the weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) first we assume that gfxn → gz as n → ∞. then r-weak commutativity of the pair (f,g) implies d(fgxn,gfxn) ≤ rd(fxn,gxn). taking the limit as n → ∞, we have fgxn → gz as n → ∞. by (3.2), we get fgxn+1 = ffxn → gz as n → ∞. on using (3.1) with x = z, y = fxn and taking the limit as n → ∞, we get fz = gz (in view of (φ2b)), hence z is a coincidence point of the mappings f and g. it is obvious that r-weakly commuting mappings commute at their coincidence points, i.e., d(fgz,gfz) ≤ rd(fz,gz), implying fgz = gfz and so ffz = fgz = gfz = ggz. now we assert that fz is a common fixed point of f and g. on using (3.1) x = z, y = fz, we have fz = ffz = gfz (due to (φ3)) and fz is a common fixed point of the mappings f and g. (ii) suppose that fgxn → fz as n → ∞. by (3.2), we have fz = gt for some t ∈ x. thus fgxn → gt as n → ∞. then r-weak commutativity of the pair (f,g), i.e., d(fgxn,gfxn) ≤ rd(fxn,gxn) implies gfxn → gt as n → ∞. by (3.2), we get fgxn+1 = ffxn → gt as n → ∞. on using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (in view of (φ2b)) which shows that t is a coincidence point of the mappings f and g. also, r-weak commutativity of the pair (f,g), d(fgt,gft) ≤ rd(ft,gt) implies fft = fgt = gft = ggt. by putting x = t, y = ft in (3.1), we can easily obtain ft = fft = gft (due to (φ3)). hence ft is a common fixed point of the mappings f and g. case v: assume that the pair (f,g) is r-weakly commuting of type (ag). the weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) let us assume that gfxn → gz as n → ∞. then r-weak commutativity of type (ag) of the pair (f,g) implies d(ffxn,gfxn) ≤ rd(fxn,gxn). on letting n → ∞, we have ffxn → gz as n → ∞. on using (3.1) with x = z, y = fxn and taking the limit as n → ∞, we have fz = gz (in view of (φ2b)) which shows that z is a coincidence point of the mappings f and g. r-weak commutativity of type (ag), d(ffz,gfz) ≤ rd(fz,gz) yields ffz = gfz and so ffz = fgz = gfz = ggz. now we show that fz is a common fixed point of c© agt, upv, 2014 appl. gen. topol. 15, no. 1 75 z. kadelburg, m. imdad and s. chauhan f and g. on using (3.1) x = z, y = fz, we have fz = ffz = gfz (due to (φ3)) and fz is a common fixed point of the mappings f and g. (ii) suppose that fgxn → fz as n → ∞. in view of (3.2), fz = gt for some t ∈ x. thus fgxn → gt as n → ∞. by (3.2), fgxn+1 = ffxn → gt as n → ∞. then r-weak commutativity of type (ag) of the pair (f,g), d(ffxn,gfxn) ≤ rd(fxn,gxn) implies gfxn → gt as n → ∞. on using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (in view of (φ2b)), hence t is a coincidence point of the mappings f and g. also, r-weak commutativity of type (ag) of the pair (f,g), d(fft,gft) ≤ rd(ft,gt) implies fft = fgt = gft = ggt. again using (3.1) with x = t, y = ft, we get ft = fft = gft (due to (φ3)). thus ft is a common fixed point of the mappings f and g. case vi: finally, let us consider that the pair (f,g) be r-weakly commuting of type (af ). the weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) suppose that gfxn → gz as n → ∞. by (3.2), we get gfxn = ggxn+1 → gz as n → ∞. then r-weak commutativity of type (af ) of the pair (f,g) implies d(fgxn,ggxn) ≤ rd(fxn,gxn). this yields fgxn → gz as n → ∞. in view of (3.2), fgxn+1 = ffxn → gz as n → ∞. putting x = z and y = fxn in (3.1) with limit as n → ∞, we get fz = gz (in view of (φ2b)). thus z is a coincidence point of the mappings f and g. since r-weak commutativity of type (af ), d(fgz,ggz) ≤ rd(fz,gz) yields fgz = ggz and so ffz = fgz = gfz = ggz. now we assert that fz is a common fixed point of f and g. on using (3.1) x = z, y = fz, we have fz = ffz = gfz (due to (φ3)). hence fz is a common fixed point of the mappings f and g. (ii) assume that fgxn → fz as n → ∞. in view of (3.2), fz = gt for some t ∈ x. thus fgxn → gt as n → ∞. then r-weak commutativity of type (af ) of the pair (f,g), d(fgxn,ggxn) ≤ rd(fxn,gxn) implies ggxn → gt as n → ∞. by (3.2), we get gfxn = ggxn+1 → gt and fgxn+1 = ffxn → gt as n → ∞. on using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (in view of (φ2b)) which shows that t is a coincidence point of the mappings f and g. also, r-weak commutativity of type (af ) of the pair (f,g), d(fgt,ggt) ≤ rd(ft,gt) implies fft = fgt = gft = ggt. again using (3.1) with x = t, y = ft, we get ft = fft = gft (due to (φ3)). thus ft is a common fixed point of the mappings f and g. in view of proposition 3.1, z is the unique common fixed point of the mappings f and g. � obviously, theorem 1.1 follows as a special case of the obtained result (using φ(t1, t2, t3, t4, t5, t6) = t1 − at2 − bt3 − ct4, with a,b,c ≥ 0, a + b + c < 1). we state some other corollaries. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 76 unified common fixed point theorems... corollary 3.4. the conclusions of proposition 3.1, lemma 3.2 and theorem 3.3 remain true if condition (3.1) is replaced by one of the following contraction conditions (each inequality is supposed to hold for all x,y ∈ x). (3.4) d(fx,fy) ≤ k max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 [d(gx,fy) + d(gy,fx)] } , where k ∈ (0,1). d2(fx,fy) ≤ d(fx,fy)[αd(gx,gy) + βd(fx,gx) + γd(fy,gy)](3.5) +ηd(gx,fy)d(gy,fx), where α > 0, β,γ,η ≥ 0, α + β + γ < 1 and α + η < 1. d3(fx,fy) ≤ αd2(fx,fy)d(gx,gy) + βd(fx,fy)d(fx,gx)d(fy,gy)(3.6) + γd(gx,fy)2d(gy,fx) + ηd(gx,fy)d(gy,fx)2, where α > 0, β,γ,η ≥ 0, α + β < 1 and α + γ + η < 1. (3.7) d3(fx,fy) ≤ kd 2(fx,gx)d2(fy,gy) + d2(gx,fy)d2(gy,fx) 1 + d(gx,gy) + d(fx,gx) + d(fy,gy) , where k ∈ (0,1). (3.8) d2(fx,fy) ≤ αd2(gx,gy) + β d(gx,fy)d(gy,fx) 1 + d2(fx,gx) + d2(fy,gy) , where α > 0, β ≥ 0 and α + β < 1. d2(fx,fy) ≤ αmax { d2(gx,gy),d2(fx,gx),d2(fy,gy) } (3.9) + β max{d(fx,gx)d(gx,fy),d(fy,gy)d(gy,fx)} + γd(gx,fy)d(gy,fx), where α > 0, β,γ ≥ 0, α + 2β < 1 and α + γ < 1. (3.10) d(fx,fy) ≤ k max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 d(gx,fy), 1 2 d(gy,fx) } , where k ∈ (0,1). (3.11) d(fx,fy) ≤ k max { d(gx,gy), 1 2 [d(fx,gx) + d(fy,gy)], 1 2 [d(gx,fy) + d(gy,fx)] } , where k ∈ (0,1). (3.12) d(fx,fy) ≤ αd(gx,gy) + βd(fx,gx) + γd(fy,gy) + ηd(gx,fy) + λd(gy,fx), where α + β + γ + η + λ < 1. (3.13) d(fx,fy) ≤ k 2 max {d(gx,gy),d(fx,gx),d(fy,gy),d(gx,fy),d(gy,fx)} , c© agt, upv, 2014 appl. gen. topol. 15, no. 1 77 z. kadelburg, m. imdad and s. chauhan where k ∈ (0,1). (3.14) d(fx,fy) ≤ αd(gx,gy) + βd(fx,gx) + γd(fy,gy) + η[d(gx,fy) + d(gy,fx)], where α + β + γ + 2η < 1. (3.15) d(fx,fy) ≤      a1 d 2(fx,gx)+d2(fy,gy) d(fx,gx)+d(fy,gy) + a2d(gx,gy) +a3[d(gx,fy) + d(gy,fx)], if d(fx,gx) + d(fy,gy) 6= 0; 0, if d(fx,gx) + d(fy,gy) = 0, where ai ≥ 0 with at least one ai non-zero and a1 + a2 + 2a3 < 1. (3.16) d(fx,fy) ≤      a1d(gx,gy) + a2d(fx,gx)d(fy,gy)+a3d(gx,fy)d(gy,fx) d(fx,gx)+d(fy,gy) , if d(fx,gx) + d(fy,gy) 6= 0; 0, if d(fx,gx) + d(fy,gy) = 0, d(fx,fy) ≤ a1[a2 max{d(gx,gy),d(fx,gx),d(fy,gy),(d(gx,fy) + d(gy,fx))/2} (3.17) + (1 − a2)(max{d2(gx,gy),d(fx,gx)d(fy,gy),d(gx,fy)d(gy,fx), (d(fx,gx)d(gy,fx))/2,(d(fy,gy)d(gx,fy))/2}) 12 ], where a1 ∈ (0,1) and 0 ≤ a2 ≤ 1. d2(fx,fy) ≤ a1 max{d2(gx,gy),d2(fx,gx),d2(fy,gy)} (3.18) + a2 max { 1 2 [d(fx,gx)d(gx,fy)], 1 2 [d(fy,gy)d(gy,fx)] } + a3d(gx,fy)d(gy,fx), where a1,a2,a3 ≥ 0 and a1 + a2 + a3 < 1. (3.19) d(fx,fy) ≤ ψ ( max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 [d(gx,fy) + d(gy,fx)] }) , where ψ : r+ → r+ is an increasing upper semi-continuous function with ψ(0) = 0 and ψ(t) < t for each t > 0. (3.20) d(fx,fy) ≤ ψ (max {d(gx,gy),d(fx,gx),d(fy,gy),d(gx,fy),d(gy,fx)}) , where ψ : r5+ → r+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. (3.21) d2(fx,fy) ≤ ψ ( max { d2(gx,gy),d(fx,gx)d(fy,gy),d(gx,fy)d(gy,fx), d(fx,gx)d(gy,fx),d(fy,gy)d(gx,fy) }) , c© agt, upv, 2014 appl. gen. topol. 15, no. 1 78 unified common fixed point theorems... where ψ : r5+ → r+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. proof. the proof of each case (3.4)–(3.21) easily follows from theorem 3.3 in view of examples 2.1–2.18. � the following example demonstrates the genuineness of theorem 1.1 over theorem 3.3. example 3.5. let x = [1,5.5] (with the standard metric d) and consider the mappings f,g : x → x given by fx =      1, for x = 1, 3, for 1 < x ≤ 2.5, 1, for 2.5 < x ≤ 5.5; gx =          1, for x = 1, 4, for 1 < x ≤ 2.5, 2x+1 6 , for 2.5 < x < 5.5, 3, for x = 5.5. firstly, we show that theorem 1.1 does not work in the context of this example. to accomplish this, we show that the contraction condition (ii) of theorem 1.1 is not satisfied. to accomplish this, take 1 < x ≤ 2.5 and 2.5 < y ≤ 5.5 so that d(fx,fy) = 2, and ad(gx,gy) + bd(fx,gx) + cd(fy,gy) = a ( 4 − 2y + 1 6 ) + b + c ( 2y + 1 6 − 1 ) . the right-hand side of the last equality is equal to 2a + b + c for y = 5.5 and tends to 3a + b for y → 2.5. hence, in order that conditon (ii) of theorem 1.1 holds, one must simultaneously require a,b,c ≥ 0, a + b + c < 1, 2a + b + c ≥ 2 and 3a + b ≥ 2 which is indeed impossible. next, we show that all the conditions of theorem 3.3 are fulfilled with the function φ taken from example 2.9. first of all, the mappings f and g are r-weakly commuting of type (ag) ( the condition d(ffx,gfx) ≤ rd(fx,gx) can be easily checked). also, f(x) = {1,3} ⊆ [1,2) ∪ {3,4} = g(x). in order to show that f and g are weakly reciprocally continuous, suppose that {xn} is a sequence in x such that fxn → t and gxn → t yielding thereby t = 1. then, we are left with two possibilities: either xn = 1 for almost each n, or xn = 2.5 + αn where αn → +0 as n → ∞. in the first case, obviously, fgxn → 1 = f1 and gfxn → 1 = g1 as n → ∞. in the second case gxn = 1 + 1 3 αn and fgxn = 3 6= f1, but fxn = 1 and gfxn = 1 = g1. hence, f and g are weakly reciprocally continuous which is not reciprocally continuous as well as non-compatible. finally, we check the contractive condition (3.12) of corollary 3.4 with η = 2 3 , λ = 2 9 and α = β = γ = 0 (note that α + β + γ + η + λ < 1) so that (3.22) d(fx,fy) ≤ 2 3 d(gx,fy) + 2 9 d(fx,gy). without loss of generality, we can assume that x ≤ y. consider the following cases: 1◦ x = y = 1; 2◦ x = 1, 1 < y ≤ 2.5; 3◦ x = 1, y > 2.5; 4◦ 1 < c© agt, upv, 2014 appl. gen. topol. 15, no. 1 79 z. kadelburg, m. imdad and s. chauhan x,y ≤ 2.5; 5◦ 1 < x ≤ 2.5, y > 2.5; 6◦ x,y > 2.5. the cases 1◦, 3◦, 4◦ and 6◦ are trivial as d(fx,fy) = 0. in the case 2◦ we have d(fx,fy) = 2 and 2 3 d(gx,fy) + 2 9 d(fx,gy) = 2 3 · 2 + 2 9 · 3 = 2, while in case 5◦ we have d(fx,fy) = 2 and 2 3 d(gx,fy) + 2 9 d(fx,gy) ≥ 2 3 · 3 + 2 9 · 0 = 2. therefore, condition (3.22) is satisfied. thus all the conditions of theorem 3.3 are satisfied and the mappings f and g have a unique common fixed point at z = 1. remark 3.6. notice that the preceding example confirms the importance of weak reciprocal continuity instead of reciprocal continuity when the given pair of mappings is not even compatible. in the following proposition, we notice that a non-compatible reciprocally continuous pair of self-mappings cannot have a common fixed point under certain contractive assumptions (see also [23, remark 1]). proposition 3.7. let (x,d) be a metric space and let f,g : x → x satisfy the following assumptions: (1) f and g are noncompatible. (2) f and g are reciprocally continuous. (3) f and g satisfy the condition (3.1) for some φ ∈ φ′. then f and g cannot have a common fixed point. proof. suppose, to the contrary, that there is a point z ∈ x such that (3.23) fz = gz = z. by the assumption (1), there exists a sequence {xn} in x such that lim n→∞ fxn = lim n→∞ gxn = t, but lim n→∞ d(fgxn,gfxn) is non-zero or non-existent. the assumption (2) implies that fgxn → ft and gfxn → gt as n → ∞, which means that ft 6= gt. now, put x = z and y = xn in the condition (3.1) to obtain φ ( d(fz,fxn),d(gz,gxn),d(fz,gz), d(fxn,gxn),d(gz,fxn),d(fz,gxn) ) ≤ 0, or φ ( d(z,fxn),d(z,gxn),d(z,z), d(fxn,gxn),d(z,fxn),d(z,gxn) ) ≤ 0. on taking the limit as n → ∞, we get that φ ( d(z,t),d(z,t),0,0,d(z,t),d(z,t) ) ≤ 0, which implies that d(z,t) = 0 (by (φ3)) so that z = t, which is a contradiction to (3.23) as ft 6= gt. � 4. results without continuity a critical examination of the recent literature reveals the fact that more natural results can be proved if we replace the completeness of the entire space by completeness of suitable subspaces (e.g. imdad et al. [12]) under relatively tighter conditions. concretely speaking, a common fixed point theorem under c© agt, upv, 2014 appl. gen. topol. 15, no. 1 80 unified common fixed point theorems... entire space completeness condition requires five conditions while the similar results can be proved under four conditions if the completeness of suitable subspaces are assumed. on the lines of the results proved in imdad et al. [12], we state and prove our next theorem which is an improvement over theorem 3.3 in the following three respects: (1) to relax the weak reciprocal continuity of the involved mappings, (2) to reduce the commutativity requirements of the mappings to coincidence points, (3) to replace the completeness of the whole space by the completeness of any one of the underlying subspaces. theorem 4.1. let f and g be two self-mappings of a metric space (x,d) satisfying conditions (3.1) (for some φ ∈ φ′) and (3.2). if one of f(x) and g(x) is a complete subspace of x, then f and g have a coincidence point. moreover, f and g have a unique common fixed point provided the pair (f,g) is coincidentally commuting. proof. let {yn} be a sequence defined by (3.3). by lemma 3.2, {yn} is a cauchy sequence in x. now assume that g(x) is a complete subspace of x, then the subsequence fx2n+1 = gx2n which is contained in g(x) must has a limit z ∈ g(x). as {yn} is a cauchy sequence containing a convergent subsequence {y2n+1}, therefore {yn} also converges implying thereby the convergence of the subsequence {y2n}, i.e. lim n→∞ fx2n = lim n→∞ gx2n+1 = z. let u ∈ g−1(z) such that gu = z. by using inequality (3.1) with x = x2n, y = u, we have φ ( d(fx2n,fu),d(gx2n,gu),d(fx2n,gx2n), d(fu,gu),d(gx2n,fu),d(gu,fx2n) ) ≤ 0, which on making n → ∞, gives rise φ(d(z,fu),d(z,z),d(z,z),d(fu,z),d(z,fu),d(z,z)) ≤ 0, φ(d(z,fu),0,0,d(fu,z),d(z,fu) + 0,0) ≤ 0, implying thereby d(z,fu) = 0 (due to (φ2a)). hence z = fu = su which shows that u is a coincidence point of the mappings f and g. if one assumes that f(x) is a complete subspace of x, then z ∈ f(x) ⊆ g(x). similarly, we can easily show that f and g have a coincidence point. since f and g are weakly compatible and fu = gu = z, then fgu = gfu which implies fz = gz. by using inequality (3.1) with x = z, y = u, we have φ(d(fz,fu),d(gz,gu),d(fz,gz),d(fu,gu),d(gz,fu),d(gu,fz)) ≤ 0, or, equivalently, φ(d(fz,z),d(gz,z),0,0,d(z,fu),d(z,fz)) ≤ 0, c© agt, upv, 2014 appl. gen. topol. 15, no. 1 81 z. kadelburg, m. imdad and s. chauhan a contradiction to (φ3) if d(z,fz) > 0. thus z = fz = gz which shows that z is a common fixed point of the mappings f and g. in view of proposition 3.1, z is the unique common fixed point of the mappings f and g. � remark 4.2. the conclusion of theorem 4.1 remains valid if we replace condition (3.1) by anyone of the contraction conditions (3.4)-(3.21) shown in corollary 3.4 (each inequality is supposed to hold for all x,y ∈ x). example 4.3. let x = [0,5] (with the standard metric d) and consider the mappings f,g : x → x given by f(x) = { 0, if x = 0; 1, if 0 < x ≤ 5. f(x) =    0, if x = 0; 5, if 0 < x < 5; 1, if x = 5. notice that the mappings f and g are discontinuous at ‘0’ which is also their common fixed point. also the pair (f,g) is coincidentally commuting with f(x) = {0,1} ⊆ {0,1,5} = g(x). define φ : r6+ → r as (4.1) φ(t1, t2, t3, t4, t5, t6) = t1 − ψ ( max{t2, t3, t4, 1 2 (t5 + t6)} ) , where ψ : r+ → r+ is an increasing upper semi-continuous function with ψ(0) = 0 and ψ(t) < t for each t > 0. in view of (4.1), condition (3.1) implies (for all x,y ∈ x) (4.2) d(fx,fy) ≤ ψ ( max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 [d(gx,fy) + d(gy,fx)] }) . by a routine calculation one can verify that implicit contraction condition (4.2) is satisfied if we choose ψ(s) = √ s. thus, all the conditions of theorem 4.1 are satisfied. notice that 0 is the unique common fixed point of the involved mappings f and g. as an application of theorem 4.1, we have the following interesting result involving two finite families of self mappings. corollary 4.4. let {fi}pi=1 and {gj} q j=1 be two finite families of self-mappings of a metric space (x,d) with f = f1f2 . . .fp and g = g1g2 . . .gq satisfying conditions (3.1) (for some φ ∈ φ′) and (3.2). then (f,g) has a point of coincidence. moreover, {fi}pi=1 and {gj} q j=1 have a unique common fixed point if the families ({fi},{gj}) commutes pairwise wherein i ∈ {1,2, . . . ,p} and j ∈ {1,2, . . . ,q}. proof. the proof of this theorem can be completed on the lines of theorem 2.2 of imdad et al. [12]. � by setting f1 = f2 = . . . = fp = f and g1 = g2 = . . . = gq = g in corollary 4.4, one deduces the following corollary which is a slight but partial generalization of theorem 4.1 as the commutativity requirements are slightly stronger as compared to theorem 4.1. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 82 unified common fixed point theorems... corollary 4.5. let f and g be two self-mappings of a metric space (x,d) satisfying (4.3) fp(x) ⊆ gq(x), (4.4) φ(d(fpx,fpy),d(gqx,gqy),d(fpx,gqx),d(fpy,gqy),d(gqx,fpy),d(gqy,fpx)) ≤ 0, for all x,y ∈ x, some φ ∈ φ′ and p,q are fixed positive integers. if one of fp(x) and gq(x) is a complete subspace of x, then the pair (fp,gq) has a coincidence point. moreover, the mappings f and g have a unique common fixed point provided that the pair (f,g) commutes. conclusion theorem 3.3 extends a result of pant et al. 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[31] s. l. singh and a. tomar, weaker forms of commuting maps and existence of fixed points, j. korea soc. math. educ. ser. b pure appl. math. 10, no. 3 (2003), 145–161. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 84 dmmenaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 245-252 uniformly discrete hit-and-miss hypertopology a missing link in hypertopologies giuseppe di maio, enrico meccariello† and somashekhar naimpally abstract. recently it was shown that the lower hausdorff metric (uniform) topology is generated by families of uniformly discrete sets as hit sets. this result leads to a new hypertopology which is the join of the above topology and the upper vietoris topology. this uniformly discrete hit-and-miss hypertopology is coarser than the locally finite hypertopology and finer than both hausdorff metric (uniform) topology and vietoris topology. in this paper this new hypertopology is studied. here is a hasse diagram in which each arrow goes from a coarser topology to a finer one and equality follows uc or tb as indicated. the diagram clearly shows that the new (underlined) topology provides the missing link. uc proximal locally finite −→ locally finite ↑ uc ↑ uc uc hausdorff metric −→ uniformly discrete ↑ tb ↑ tb uc proximal −→ vietoris 2000 ams classification: primary: 54b20. secondary: 54e35, 54e05, 54e15. keywords: hypertopology, hausdorff metric, vietoris topology, proximal topology, proximal locally finite, hit-and-miss hypertopology, ε-discrete subset, uniformly discrete, uniformly isolated, locally finite family. †in memory of professor enrico meccariello who made a considerable contribution to this work and who suddendly passed away before his time. 246 g. di maio, e. meccariello and s. naimpally 1. introduction and preliminaries during the early part of the last century there were two main studies on the hyperspace cl(x) of all non empty closed subsets of a topological (uniform, metric) space x. vietoris introduced a topology consisting of two parts (a) the lower finite hit part and (b) the upper miss part. hausdorff defined a metric on the hyperspace of a metric space, and the resulting topology depends on the metric rather than on the topology of the base space. two equivalent metrics on a metrizable space induce equivalent hausdorff metric topologies if, and only if, they are uniformly equivalent. recently, it was shown that all known hypertopologies are variations of the prototype: vietoris hit-and-miss topology, and so they can be described as the joins of lower and upper parts ([11]). in particular, the hausdorff metric topology coincides with the join τ (u)−∨σ(δ+), where (a) τ (u)− is the lower uniformly discrete hit part, and (b) σ(δ+) is the upper proximal miss part ([11]). a new hypertopology arises formally but, in fact, is indeed quite natural. it is the uniformly discrete hit-and-miss hypertopology τ (u), which is the join of the upper vietoris and the lower uniformly discrete topologies. this hypertopology, which is well placed in the family of hypertopologies, is finer than both the vietoris τ (v ) and the hausdorff metric τ (hd) topologies, but is coarser than the locally finite topology. the locally finite hypertopology τ (l) plays a key role (see [2], [8] and [14]). given a metric space (x, d), τ (l) is the sup of all hausdorff metric topologies {τ (h̺)} induced by all metrics ̺ topologically equivalent to d. the uniformly discrete hit-and-miss hypertopology τ (u(d)), or simply τ (u), has a decomposition which sheds more light on the nature of the hausdorff metric topology and on the relations among these classical hypertopologies. from this representation and the knowledge that lower and upper parts act separately in any comparison, it is convenient to study the two parts separately. we will show that the two parts, in spite of the definition, play a symmetric role. in fact, they coincide with the corresponding parts of the hausdorff metric topology if, and only if, the underlying metric space (x, d) is uc (a metric space (x, d) is uc if, and only if, each continuous real valued function defined on x is uniformly continuous). more surprising, the coincidence of either the lower or the upper parts forces the coincidence of the hypertopologies (this phenomenon is due to the symmetric nature of the hausdorff metric topology). we will focus our attention on metric spaces, but we point out that many results hold in uniform spaces too. in the second section we introduce the uniformly discrete hit-an-miss hypertopology τ (u) and state its basic, but important properties. the third section is devoted to comparisons and equalities: the role of the uniformly discrete hit-and-miss hypertopology τ (u) in the lattice of all hypertopologies on cl(x) is carefully investigated. uniformly discrete hit-and-miss ... 247 2. the uniformly discrete hit-and-miss hypertopology let (x, d), or simply x, be a metric space and cl(x) the family of all non empty closed subsets of x. for a ∈ cl(x), ε > 0, the set {sε(x) − : x ∈ q ⊂ a, where q is ε-discrete} is the (discrete) ε-screen of a centered at q and is denoted by sε(q, a) −. lε(a) − is the family of all discrete screens of a, namely {sε(q, a) − : q ⊂ a, where q is ε-discrete, ε > 0}. lemma 2.1. for a ∈ cl(x), ε1 > 0, ε2 > 0, let q1 be ε1-discrete and q2 be ε2-discrete subsets of a. then a ∈ sε(q, a) − ⊂ sε1 (q1, a) − ∩ sε2 (q2, a) −, for ε = 1 3 min{ε1, ε2}. definition 2.2. given a metric space (x, d), the lower uniformly discrete topology τ (u(d))− on cl(x), briefly the lower discrete topology, has the family {lε(a) − : ε > 0} as a local base at a ∈ cl(x). proposition 2.3. [11] let (x, d) be a metric space. the lower discrete topology τ (u(d))− on cl(x) equals the lower hausdorff metric topology τ (h− d ). definition 2.4. given a metric space (x, d), the uniformly discrete hit-andmiss topology τ (u(d)), or briefly τ (u), on cl(x) is the join of the lower (uniformly) discrete topology and the upper vietoris topology, i.e. τ (u(d)) = τ (u(d))− ∨ τ (v +). we call τ (u(d)) the ud-topology. we first study the countability properties of τ (u(d))− and τ (u(d)) = τ (u(d))− ∨ τ (v +). the first countability of the lower discrete topology τ (u(d))− on cl(x) is obvious since it is induced by a pseudometric (proposition 2.3), but we would like to offer a different proof. proposition 2.5. the lower discrete topology τ (u(d))− on cl(x) is first countable. proof. given a ∈ cl(x), let qn be a 1 n -maximal discrete subset of a. then the family l(a)− = {s 1 n (x)− : x ∈ qn : n ∈ n}, is a countable local base at a. � proposition 2.6. let (x, d) be a metric space. the following are equivalent. (1) (cl(x), τ (u(d))) is first countable; (2) (cl(x), τ (v +)) is first countable; (3) (x, d) is topologically uc. proof. (1) ⇒ (2) use the same argument as in theorem 1.1 in [5]. (2) ⇔ (3) is corollary 1.9 in [5] (a metric space (x, d) is topologically uc if, and only if, admits a compatible metric ̺ such that (x, ̺) is uc). (2) ⇒ (1) τ (u(d)) = τ (u(d))− ∨ τ (v +) is first countable since it is the join of two first countable topologies. � 248 g. di maio, e. meccariello and s. naimpally proposition 2.7. let (x, d) be a metric space. the following are equivalent: (1) the lower discrete topology τ (u(d))− on cl(x) is second countable; (2) x is totally bounded (t b). proof. (2) ⇒ (1) if x is t b, then τ (u(d))− = τ (v −) and the claim since x is also second countable (see [3] and [5]). (1) ⇒ (2) suppose that x fails to be t b. so it admits for some positive ε an infinite ε-discrete subset d = {xn : n ∈ n}. let d be the uncountable set of all nonempty subsets of d. let d1, d2 ∈ d with d1 6= d2 and d ⋆ ∈ d1 \ d2. then d2 ∈ {sε(d) − : d ∈ d2}, but d2 6∈ {sε(d) − : d ∈ d1} and the claim. � proposition 2.8. let (x, d) be a metric space. the following are equivalent: (1) the ud-topology τ (u(d)) on cl(x) is second countable; (2) τ (v +) is second countable; (3) x is compact; (4) the ud-topology τ (u(d)) on cl(x) is compact and metrizable. proof. see [3] and [5], use theorem 4.9.7 in [9] and observe that if x is compact all the involved hypertopologies coincide with the vietoris topology τ (v ) (see section 3). � we omit the proof of the following technical lemma. lemma 2.9. let (x, d) be a metric space and a ∈ cl(x). then (cl(a), τ (u(d | a))) = (cl(a), τ (u(d)) | cl(a)). proposition 2.10. let (x, d) be a metric space. the following are equivalent: (1) τ (u(d)) is normal; (2) x is compact; (3) τ (u(d)) is compact and metrizable. proof. we show (1) ⇒ (2). suppose that x fails to be compact. then there exists a countable closed discrete set a. now use lemma 2.9 and ivanova argument (see [6] or [7]). � we state the following proposition without a proof. proposition 2.11. let x be metrizable and d and e compatible metrics. then τ (u(d)) = τ (u(e)) if, and only if, d and e are uniformly equivalent. 3. comparisons the following proposition can be easily deduced from the definitions. nevertheless, it is important since it is useful to locate in a natural way the vietoris and hausdorff metric topologies in the lattice of all hypertopologies on cl(x). we recall that the lower locally finite topology τ (l−) has as a basis: the family of all sets of the form {e− : e ∈ l, l is locally finite}. proposition 3.1. let (x, d) be a metric space. the following inclusions hold on cl(x): uniformly discrete hit-and-miss ... 249 τ (u(d))− ⊂ τ (l−), τ (v −) ⊂ τ (u(d))−. hence, the following hasse diagram can be drawn: it is formed by two rectangles (upper/lower) with a common side (recall that the proximal locally finite topology σ(l) has been studied in [2]). each arrow goes from a coarser topology to a finer one and equality follows uc (see below) or t b (totally bounded) as indicated. the diagram clearly shows that the new topology provides the missing link. uc proximal locally finite −→ locally finite ↑ uc ↑ uc uc hausdorff metric −→ uniformly discrete ↑ tb ↑ tb uc proximal −→ vietoris observe that all the topologies in the first (resp. second) column have the same upper part: the proximal upper topology σ(v +) (resp. the upper vietoris topology τ (v +)); similarly all topologies in the first (resp. second, third) row have the same lower part: lower locally finite τ (l−) (resp. lower uniformly discrete τ (u(d))−, lower vietoris τ (v −)). to state comparisons the notion of uc metric is essential. again, we recall that a metric space (x, d) is uc if each real valued continuous function on (x, d) is uniformly continuous. this class of spaces has been investigated since 1950 and intensely in the last decades (see [4] where further references can be found). we use the following characterization of a uc space due to atsuji ([1]). a metric space (x, d) is uc if each sequence (xn : n ∈ n) without accumulation points is finally uniformly isolated, i.e. there exists a positive ε and an index n0 such that for all n ≥ n0, s(xn, ε) ∩ x = {xn}. the following is an unexpected result. proposition 3.2. let (x, d) be a metric space. the following are equivalent: (1) τ (l−) = τ (u(d))−; (2) (x, d) is uc. proof. (1) ⇒ (2) by contradiction, suppose (x, d) fails to be uc. then, passing to a subsequence if necessary, there is a sequence (xn : n ∈ n) without accumulation points such that for each n there is yn ∈ x such that d(xn, yn) ≤ 1 n . since (xn : n ∈ n) is without accumulation points, the same occurs to (yn : n ∈ n). set rn = d(xn, yn) and consider {s(xn, rn) : n ∈ n}. since x is metric (paracompact), we may suppose that {s(xn, rn) : n ∈ n} is a locally finite family. let a = (xn : n ∈ n) and an = {zk : k ∈ n, with zk = xk for k ≤ n and zk = yk for k > n}. the sequence an clearly converges to a w.r.t. τ (u(d)) −, but 250 g. di maio, e. meccariello and s. naimpally an fails to converge to a w.r.t. τ (l −). in fact, a ∈ {s(xn, rn) : n ∈ n} −, but for no n, an ∈ {s(xn, rn) : n ∈ n} −. (2) ⇒ (1) suppose that (x, d) is uc. let v = {s(xi, ri) : i ∈ i} be a discrete family. for each i ∈ i there exists a continuous function gi : x → [0, 1] such that gi(xi) = 1, gi(s(xi, ri)) ⊆ [0, 1], and gi(x) = 0 for x 6∈ s(xi, ri). set ̺(x, y) = d(x, y) + ∑ | gi(x) − gi(y) |. by a routine argument ̺ is a compatible metric. since the identity map id : (x, d) → (x, ̺) is continuous, it must be uniformly continuous. so, there exists ε such that d(x, y) < ε implies ̺(x, y) < 1. thus, the set {xi : i ∈ i} is ε-discrete and the claim τ (l−) = τ (u(d))−. � we recall the following visual characterization of uc spaces due to nagata ([10]): a metric space (x, d) is uc if, and only if, disjoint closed sets are a positive distance apart. thus, the corresponding results for the upper parts are easily stated. proposition 3.3. let (x, d) be a metric space. the following are equivalent: (1) τ (l+) = σ(l+); (2) τ (l) = σ(l); (3) τ (u(d))+ = τ (h+ d ); (4) τ (u(d)) = τ (hd); (5) τ (v +) = σ(v +); (6) τ (v ) = σ(v ); (7) (x, d) is uc. proof. note that the hypertopologies τ (l+) and τ (u(d))+ have as upper part the upper vietoris topoloy τ (v +), whereas the upper part of σ(l+) and τ (h+ d ) is the upper proximal topology σ(v +). clearly, according to the above mentioned characterization of uc spaces, τ (v +) and σ(v +) coincide if, and only if, the metric space (x, d) is uc. � proposition 3.4. let (x, d) be a metric space. the following are equivalent: (1) τ (v −) = τ (u(d))−; (2) τ (v ) = τ (u(d)); (3) τ (h− d ) = σ(v −); (4) τ (hd) = σ(v ); (5) (x, d) is totally bounded (t b). proof. see [11]. � observe that in the upper rectangle horizontal and vertical equalities are described by the same single property, i.e. uc. theorem 3.5. let (x, d) be a metric space. the following are equivalent: (1) τ (l) = σ(l) = τ (hd) = τ (u(d)); (2) (x, d) is uc. uniformly discrete hit-and-miss ... 251 proof. use propositions 3.2 and 3.3 or the following conceptual argument which uses the full power of the whole hypertopologies. the locally finite hypertopology τ (l) equals the hausdorff-boubaki hypertopology τ (w) induced by the fine uniformity w on (x, d) (theorem 2.2 in [12]). recall that (x, d) is uc if, and only if, the metric uniformity v(d) is the fine uniformity w (see [10], [12] and [13]). observe that the hausdorff-boubaki hypertopology τ (v(d)) induced by the metric uniformity v(d) is the hausdorff metric hypertopology τ (hd). thus, (x, d) is uc if, and only if, τ (l) = τ (w) = τ (hd) (corollary 2.2 in [12]). it follows easily that (x, d) is uc if, and only if, τ (l) = σ(l) = τ (hd) = τ (u(d)). � note that in the lower rectangle the horizontal (resp. vertical) equalities occur when the space is uc (resp. t b). to squeeze the lower rectangle to a point we need two properties, namely t b and uc. consequently, due to the uc property, also the upper rectangle reduces to a point. but, t b + uc is equivalent to compactness. so, we have that compactness is equivalent to the equality of all the six studied hypertopologies. corollary 3.6. let (x, d) be a metric space. the following are equivalent: (1) σ(v ) = τ (v ) = τ (u(d)) = τ (hd) = τ (l) = σ(l); (2) (x, d) is compact. acknowledgements. sn is grateful to gdm for organizing his visit to sun, caserta, italy. references [1] m. atsuji, uniform continuity of continuous functions of metric spaces, pacific j. math. 8 (1958), 11–16. [2] a. di concilio, s. naimpally and p. l. sharma, proximal hypertopologies, sixth brazilian topology meeting, campinas, brazil (1988) [unpublished]. [3] g. di maio and ľ. holá, on hit-and-miss hyperspace topologies, rend. acc. sci. fis. mat. napoli, (4) 62 (1995), 103–124. [4] g. di maio, e. meccariello and s. naimpally, decomposition of uc spaces, questions and answers in genaral topology 22 (2004), 13–22. [5] ľ. holá and s. levi, decomposition properties of hyperspace topologies, set-valued analysis 5 (1997), 309–321. [6] v. m. ivanova, on the theory of spaces of subsets, dokl. akad. nauk. sssr 101 (1955), 601–603. [7] j. keesling, normality and properties related to compactness in hyperspaces, proc. amer. math. soc. 24 (1970), 760–766. [8] m. marjanovic, topologies on collections of closed subsets, publ. inst. math. (beograd) 20 (1966), 125–130. 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[14] f. wattenberg, topologies on the set of closed subsets, pacific j. math. 68 (1977), 537–551. received june 2005 accepted july 2005 giuseppe di maio (giuseppe.dimaio@unina2.it) dipartimento di matematica, seconda università di napoli, via vivaldi 43, 81100 caserta, italy. enrico meccariello (meccariello@unisannio.it) facoltà di ingegneria, università del sannio, palazzo b. lucarelli, piazza roma, 82100 benevento, italy. somashekhar naimpally (somnaimpally@yahoo.ca) 96 dewson street, toronto, ontario, m6h 1h3, canada. @ appl. gen. topol. 23, no. 2 (2022), 377-390 doi:10.4995/agt.2022.17359 © agt, upv, 2022 fixed point theorems for a new class of nonexpansive mappings rajendra pant a and rahul shukla b a department of mathematics & applied mathematics, university of johannesburg kingsway campus, auckland park 2006, south africa (pant.rajendra@gmail.com, rpant@uj.ac.za) b department of mathematical sciences and computing, walter sisulu university, mthatha 5117, south africa (rshukla.vnit@gmail.com, rshukla@wsu.ac.za) communicated by s. romaguera abstract we consider a new class of nonlinear mappings that generalizes two well-known classes of nonexpansive type mappings and extends some other classes of mappings. we present some existence and convergence results for this class of mappings. some illustrative examples presented herein show the generality of the obtained results. 2020 msc: 47h10; 54h25. keywords: α-nonexpansive; opial property; condition (c). 1. introduction let k be a nonempty subset of x of a banach space (x ,‖.‖). a self-mapping ψ : k→k is 1-lipschitz or nonexpansive if ‖ψ(σ) − ψ(υ)‖≤‖σ −υ‖ for all σ,υ ∈ k. a fixed point σ of the mapping ψ is the point at which the mapping is invariant, that is, ψ(σ) = σ. in 1965, browder [6, 7], göhde [9] and kirk [10] initiated the existence theory for fixed points of nonexpansive mapping, independently (cf. [8]). in general nonexpansive mapping are uniformly continuous on their domains. to generalize, extend and accommodate discontinuous nonexpansive type mappings, many authors considered various received 16 march 2022 – accepted 12 may 2022 http://dx.doi.org/10.4995/agt.2022.17359 r. pant and r. shukla classes of mappings [18, 11, 19, 2, 3, 14, 1, 8] for more details, see [15]. in 2008, suzuki [18] considered a more general class of nonexpansive mappings (also known as suzuki type generalized nonexpansive mapping) and presented some interesting results for these mappings: definition 1.1 ([18]). assume that k is a nonempty subset of a banach space x . a mapping ψ : k→k is said to satisfy condition (c) if 1 2 ‖σ − ψ(σ)‖≤‖σ −υ‖ implies ‖ψ(σ) − ψ(υ)‖≤‖σ −υ‖ for all σ,υ ∈k. in 2011, aoyama and kohsaka [3] introduced another class of nonexpansive type mappings (called as α-nonexpansive mappings). this class of mappings generalizes several classes of mappings including λ-hybrid and nonspreading mappings. for more details one may refer to [11, 19, 2]. definition 1.2. let k be a nonempty subset of a banach space x and ψ : k →k a self-mapping. then ψ is an α-nonexpansive if there exists an α < 1 such that ‖ψ(σ) − ψ(υ)‖2 ≤ α‖ψ(σ) −υ‖2 + α‖ψ(υ) −σ‖2 +(1 − 2α)‖σ −υ‖2(1.1) for all σ,υ ∈k. remark 1.3. even though, the class of α-nonexpansive mappings was considered in [3] for any real number α < 1, but ariza-ruiz et al. [4] pointed out that for α < 0, this concept is trivial (see also [17]). we note that α-nonexpansive and mappings satisfying the condition (c) are independent, and need not be continuous on their domains of definitions, unlike nonexpansive mappings. a couple of examples below illustrate these facts. example 1.4. let k = [0, 5] ⊂ r with the usual norm on r. assume that ψ : k→k is a self-mapping defined as: ψ(σ) =   1 −σ, if σ ∈ [0, 1] 0, if σ ∈ (1, 5) 1, if σ = 5. if σ < υ and (σ,υ) ∈ ([0, 5] × [0, 5])\((4, 5) ×{5}), then it can be easily seen that ‖ψ(σ) − ψ(υ)‖≤‖σ −υ‖ holds. if σ ∈ (4, 5) and υ = 5, then 1 2 ‖σ − ψ(σ)‖ = σ 2 > 1 > ‖σ −υ‖ and 1 2 ‖υ − ψ(υ)‖ = 2 > ‖σ −υ‖. hence ψ satisfies condition (c). contrarily, at σ = 0 and υ = 1 α‖ψ(σ) −υ‖2 + α‖ψ(υ) −σ‖2 + (1 − 2α)‖σ −υ‖2 = 1 − 2α ≤ 1 = ‖ψ(σ) − ψ(υ)‖2 © agt, upv, 2022 appl. gen. topol. 23, no. 2 378 fixed point theorems for a new class of nonexpansive mappings in banach spaces holds only if α = 0. but for α = 0, and σ = 9 2 , υ = 5, we get ‖ψ(σ) − ψ(υ)‖ = 1 > 1 2 = ‖σ −υ‖. therefore, ψ is not an α-nonexpansive mapping for any α ∈ [0, 1). example 1.5. [14]. let k = [0, 4] ⊂ r endowed with the usual norm. define ψ : k→k as follows: ψ(σ) = { 0, if σ 6= 4, 2, if σ = 4. then it can be easily verified that ψ is α-nonexpansive mapping for α ≥ 1 2 . however ψ is not a mpping satisfying the condition (c) for σ ∈ (2, 3] and υ = 4. in [14], we introduced the following class of mappings: definition 1.6. suppose k is a nonempty subset of a banach space x , and ψ : k → k a self-mapping. then ψ is called a generalized α-nonexpansive mapping if there exists an α ∈ [0, 1) such that 1 2 ‖σ − ψ(σ)‖ ≤ ‖σ −υ‖ implies ‖ψ(σ) − ψ(υ)‖ ≤ α‖ψ(σ) −υ‖ + α‖ψ(υ) −σ‖ + (1 − 2α)‖σ −υ‖(1.2) for all σ,υ ∈k. the implication in inequality (1.2) is more restrictive than in (1.1), and therefore the above mapping does not contain α-nonexpansive mapping, properly. the present paper deals with this problem. indeed, we consider a class of mappings which properly contains the class of α-nonexpansive mappings. to show the generality of the class of mappings considered herein, we present some illustrative examples. we also obtain the demi-closedness principle in banach spaces. further, we employ a three step iterative method to approximate the fixed point of mapping considered herein. 2. preliminaries now onwards, r denotes the set of real numbers and n the set of natural numbers. definition 2.1. assume that k is a nonempty subset of a banach space x . a self-mapping ψ : k→k is a quasinonexpansive mapping if ‖ψ(σ) −w†‖≤‖σ −w†‖ for all σ ∈k and w† ∈ f(ψ). a banach space x is said to be uniformly convex, for each ε ∈ (0, 2], there exists δ > 0 such that the following holds: for each σ,υ ∈x ‖σ‖≤ 1 ‖υ‖≤ 1 ‖σ −υ‖≥ ε   ⇒ ∥∥∥∥σ + υ2 ∥∥∥∥ ≤ 1 − δ. © agt, upv, 2022 appl. gen. topol. 23, no. 2 379 r. pant and r. shukla definition 2.2 ([16]). let x be a normed space and k nonempty subset of x . a mapping ψ : k→k is said to satisfy condition (i) if there exists a function f : [0,∞) → [0,∞) with the following properties: • f is nondecreasing; • f(r) > 0 for all r ∈ (0,∞) and f(0) = 0; • ‖σ − ψ(σ)‖≥ f(d(σ,f(ψ))) for all σ ∈k, where d(x,f(ψ)) denotes distance of x from f(ψ). a banach x satisfies the opial conditions [13] if for each weakly convergent sequence {σn}⊂x having weak limit σ, we have lim inf n→∞ ‖σn −σ‖ < lim inf n→∞ ‖σn −υ‖ for all υ ∈ x , σ 6= υ. it can be easily seen that on passing through appropriate subsequences, the lower limit can be replaced with upper limits in opial property. the sequence {σn} is an approximate fixed point sequence for ψ (in short, a.f.p.s.) if lim n→∞ ‖σn − ψ(σn)‖ = 0. 3. c-α nonexpansive mapping we introduce the following notion of c-α nonexpansive mapping definition 3.1. suppose k is a nonempty subset of a banach space x and ψ : k→k a self-mapping. we say ψ is a c-α nonexpansive mapping if 1 2 ‖σ − ψ(σ)‖ ≤ ‖σ −υ‖ implies ‖ψ(σ) − ψ(υ)‖2 ≤ α‖ψ(σ) −υ‖2 + α‖ψ(υ) −σ‖2 + (1 − 2α)‖σ −υ‖2(3.1) for all σ,υ ∈k, where α ∈ [0, 1). we discuss some fundamental properties of c-α nonexpansive mapping. proposition 3.2. let ψ : k → k be a mapping satisfying the condition (c). then ψ is a c-α nonexpansive mapping. in the next example we show that the reverse implication is not true, in general. example 3.3. let (`2,‖.‖2) be the banach space of square-summable sequences endowed with its standard norm. assume that {en} is the canonical basis of `2. define k := conv{e1,e2} = {µe1 + (1 −µ)e2 : µ ∈ [0, 1]}, where conv{e1,e2} denotes the convex closure of {e1,e2}. now, define ψ : k→ k as follows: ψ(µe1 + (1 −µ)e2) =  e1, if µ = 0,(e1 + e2) 2 , otherwise. © agt, upv, 2022 appl. gen. topol. 23, no. 2 380 fixed point theorems for a new class of nonexpansive mappings in banach spaces then ψ is c1 3 nonexpansive mapping. indeed, if σ := e2 and υ := µe1 + (1 − µ)e2, µ ∈ (0, 1], we have ‖ψ(σ) − ψ(υ)‖2 = ∥∥∥∥e1 − ( e1 + e2 2 )∥∥∥∥ 2 = ∥∥∥∥e1 −e22 ∥∥∥∥ 2 = √ 2 2 , ‖ψ(υ) −σ‖2 = ∥∥∥∥e1 + e22 −e2 ∥∥∥∥ 2 = ∥∥∥∥e1 −e22 ∥∥∥∥ 2 = √ 2 2 , ‖ψ(σ) −υ‖2 = ‖e1 − (µe1 + (1 −µ)e2)‖2 = (1 −µ) √ 2 ‖σ −υ‖2 = ‖e2 − (µe1 + (1 −µ)e2)‖2 = µ √ 2. therefore, for α = 1 3 1 3 ‖ψ(σ) −υ‖2 + 1 3 ‖ψ(υ) −σ‖2 + ( 1 − 2 3 ) ‖σ −υ‖2 = 1 3 (1 −µ) √ 2 + 1 3 √ 2 2 + 1 3 µ √ 2 = √ 2 2 = ‖ψ(σ) − ψ(υ)‖2. by the convexity of function t 7→ t2, we obtain (‖ψ(σ)−ψ(υ)‖2)2 ≤ 1 3 (‖ψ(σ)−υ‖2)2 + 1 3 (‖ψ(υ)−σ‖2)2 + ( 1 − 2 3 ) (‖σ−υ‖2)2. contrarily, if σ := e2 and υ := 1 3 e1 + 2 3 e2, then 1 2 ‖υ − ψ(υ)‖2 = 1 2 ∥∥∥∥13e1 + 23e2 − (e1 + e2)2 ∥∥∥∥ 2 = 1 12 ‖e1 −e2‖2 = 1 12 √ 2 ≤ 1 3 √ 2 = ‖σ −υ‖2 and ‖ψ(σ) − ψ(υ)‖2 = √ 2 2 > 1 3 √ 2 = ‖σ − υ‖2. therefore, ψ does not satisfy the criterion of condition (c). note that (e1 + e2) 2 is a fixed point of ψ. a generalized α-nonexpansive mapping is c-α nonexpansive mapping but the reverse implication is not true (see example 3.5 below). proposition 3.4. assume that k is a nonempty subset of a banach space x and ψ : k →k a generalized α-nonexpansive mapping for all α ∈ [0, 1 2 ]. then ψ is c-α nonexpansive mapping for α ∈ [0, 1 2 ]. proof. let σ,υ ∈k and α ∈ [0, 1 2 ]. note that 1−2α ≥ 0. since ψ a generalized α-nonexpansive mapping, by implication in (1.2), we have ‖ψ(σ) − ψ(υ)‖≤ α‖ψ(σ) −υ‖ + α‖ψ(υ) −σ‖ + (1 − 2α)‖σ −υ‖. considering the convexity of function t 7→ t2, we conclude that ‖ψ(σ) − ψ(υ)‖2 ≤‖ψ(σ) −υ‖2 + ‖ψ(υ) −σ‖2 + (1 − 2α)‖σ −υ‖2. that is, ψ is c-α nonexpansive mapping for all α ∈ [0, 1 2 ]. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 381 r. pant and r. shukla example 3.5. let k = [0, 3] ⊂ r endowed with the usual norm in r. define a mapping ψ : k→k as follows: ψ(σ) = { σ 2 , if σ 6= 3, 5 2 , otherwise. then ψ is c-α nonexpansive mapping for α = 3 4 . indeed, if σ,υ 6= 3, then 3 4 |ψ(σ) −υ|2 + 3 4 |ψ(υ) −σ|2 + ( 1 − 2 × 3 4 ) |σ −υ|2 = 3 4 ∣∣∣σ 2 −υ ∣∣∣2 + 3 4 ∣∣∣υ 2 −σ ∣∣∣2 − 1 2 |σ −υ|2 = 7 16 σ2 + 7 16 υ2 − 1 2 συ = ( 1 4 σ2 + 1 4 υ2 − 1 2 συ ) + 3 16 σ2 + 3 16 υ2 ≥ 1 4 σ2 + 1 4 υ2 − 1 2 συ = ∣∣∣σ 2 − υ 2 ∣∣∣2 = |ψ(σ) − ψ(υ)|2. again if σ = 3 and υ 6= 3, then 3 4 |ψ(σ) −υ|2 + 3 4 |ψ(υ) −σ|2 + ( 1 − 2 × 3 4 ) |σ −υ|2 = 3 4 ∣∣∣∣52 −υ ∣∣∣∣2 + 34 ∣∣∣υ 2 − 3 ∣∣∣2 − 1 2 |3 −υ|2 = 7 16 υ2 − 12 4 υ + 111 16 = ( 1 4 υ2 − 10 4 υ + 100 16 ) + 3 16 υ2 − 1 2 υ + 11 16 . since 3 16 υ2 − 1 2 υ + 11 16 ≥ 0 for all υ ∈ [0, 3], we have 3 4 |σ − ψ(υ)|2 + 3 4 |ψ(σ) −υ|2 + ( 1 − 2 × 3 4 ) |σ −υ|2 ≥ 1 4 υ2 − 10 4 υ + 100 16 = ∣∣∣∣52 − υ2 ∣∣∣∣2 = |ψ(σ) − ψ(υ)|2. contrarily at σ = 3 and υ = 2, we get 1 2 |σ − ψ(σ) = 1 2 ∣∣∣∣3 − 52 ∣∣∣∣ = 14 ≤ 1 = |3 − 2| = |σ −υ| © agt, upv, 2022 appl. gen. topol. 23, no. 2 382 fixed point theorems for a new class of nonexpansive mappings in banach spaces and α|ψ(3) − 2| + α|ψ(2) − 3| + (1 − 2α)|3 − 2| = = α ∣∣∣∣52 − 2 ∣∣∣∣ + α|1 − 3| + 1 − 2α = 1 2 α + 2α + 1 − 2α = 1 + 1 2 α < 3 2 = ∣∣∣∣52 − 22 ∣∣∣∣ = |ψ(σ) − ψ(υ)|. hence ψ is not a generalized α-nonexpansive mapping for any value of α ∈ [0, 1). proposition 3.6. every α-nonexpansive is c-α nonexpansive mapping, but the converse is not true. example 3.7. let (`∞,‖.‖∞) be the banach space of all bounded real sequences endowed with the supremum norm. assume that {en} is the canonical basis of `∞. define k := {µe1 : µ ∈ [0, 1]} define ψ : k→k as follows: ψ(µe1) = { 0, if µ 6= 1, e1 3 , if µ = 1. then ψ is a c1 10 nonexpansive mapping. indeed, if σ = µ1e1, υ = µ2e1, where µ1,µ2 ∈ [0, 1) then ‖ψ(σ)−ψ(υ)‖2∞ = 0 ≤ 1 10 ‖ψ(σ)−υ‖2∞+ 1 10 ‖ψ(υ)−σ‖2∞+ ( 1 − 2 × 1 10 ) ‖σ−υ‖2∞. again if, σ = µ1e1, where µ1 ∈ [0, 23 ] and υ = e1, then 1 10 ‖ψ(σ) −υ‖2∞ + 1 10 ‖ψ(υ) −σ‖2∞ + ( 1 − 2 × 1 10 ) ‖σ −υ‖2∞ = 1 10 ‖e1‖2∞ + 1 10 ∥∥∥e1 3 −σ ∥∥∥2 ∞ + 4 5 ‖e1 −σ‖2∞ ≥ 1 10 + 4 5 × 1 9 = 17 90 > 1 9 = ‖ψ(σ) − ψ(υ)‖2∞. if σ = µ1e1, where µ1 ∈ ( 2 3 , 1 ) and υ = e1, then 1 2 ‖σ−ψ(σ)‖∞ = 1 2 ‖σ‖∞ > ‖e1−σ‖∞ and 1 2 ‖υ−ψ(υ)‖∞ = 1 2 ∥∥∥e1 − e1 3 ∥∥∥ ∞ = 1 3 > ‖e1−σ‖∞. on the other hand, at σ = 9 10 e1 and υ = e1, 1 10 ∥∥∥∥ψ ( 9 10 e1 ) −e1 ∥∥∥∥2 ∞ + 1 10 ∥∥∥∥ψ(e1) − 910e1 ∥∥∥∥2 ∞ + ( 1 − 2 × 1 10 )∥∥∥∥ 910e1 −e1 ∥∥∥∥2 ∞ = 1 10 + 289 9000 + 8 1000 = 397 9000 < 1 9 = ∥∥∥∥ψ ( 9 10 e1 ) − ψ(e1) ∥∥∥∥2 ∞ . © agt, upv, 2022 appl. gen. topol. 23, no. 2 383 r. pant and r. shukla thus ψ is not 1 10 -nonexpansive mapping. proposition 3.8. suppose that k is a nonempty subset a banach space x and ψ : k → k a c-α nonexpansive mapping and has a fixed point. then ψ is quasinonexpansive. proof. it follows from the proof of [14, proposition 3.5]. � lemma 3.9. suppose that k is a nonempty subset a banach space x and ψ : k → k a c-α nonexpansive mapping. then f(ψ) is closed. in addition, if k is convex and x is strictly convex, then f(ψ) is convex. proof. the proof is much similar to proof [14, lemma 3.6] � 4. main results proposition 4.1 (demiclosedness principle). assume that k is a nonempty subset of a banach space x which has the opial property and ψ : k → k be a c-α nonexpansive mapping. if {σn} converges weakly to a point σ and lim n→∞ ‖ψ(σn) − σn‖ = 0 then ψ(σ) = σ. that is, i − ψ is demiclosed at zero, where i is the identity mapping on x . proof. since the sequence {σn} is weakly convergent and lim n→∞ ‖ψ(σn)−σn‖ = 0, both sequences {σn} and {ψ(σn)} are bounded. first we assume that lim sup n→∞ ‖σn −σ‖ = 0. now by the triangle inequality, we get ‖σ − ψ(σ)‖ ≤ lim sup n→∞ ‖σn −σ‖ + lim sup n→∞ ‖σn − ψ(σ)‖ = lim sup n→∞ ‖σn − ψ(σ)‖. indeed, by opial property ‖σ − ψ(σ)‖≤ lim sup n→∞ ‖σn − ψ(σ)‖ < lim sup n→∞ ‖σn −σ‖ = 0. thus ψ(σ) = σ. if we assume that lim sup n→∞ ‖σn −σ‖ = r > 0. since lim n→∞ ‖ψ(σn) −σn‖ = 0, for large enough n, there exists a n0 ∈ n such that 1 2 ‖σn − ψ(σn)‖≤‖σn −σ‖ for all n ≥ n0. by (4.5), we have (4.1) ‖ψ(σn)−ψ(σ)‖2 ≤ α‖ψ(σn)−σ‖2 +α‖ψ(σ)−σn‖2 + (1−2α)‖σn−σ‖2. © agt, upv, 2022 appl. gen. topol. 23, no. 2 384 fixed point theorems for a new class of nonexpansive mappings in banach spaces now by the triangle inequality and (4.1), we have ‖σn − ψ(σ)‖2 ≤ (‖σn − ψ(σn)‖ + ‖ψ(σn) − ψ(σ)‖)2 ≤ ‖σn − ψ(σn)‖2 + ‖ψ(σn) − ψ(σ)‖2 + 2‖σn − ψ(σn)‖‖ψ(σn) − ψ(σ)‖ ≤ ‖σn − ψ(σn)‖2 + α‖ψ(σn) −σ‖2 + α‖ψ(σ) −σn‖2 +(1 − 2α)‖σn −σ‖2 + 2‖σn − ψ(σn)‖‖ψ(σn) − ψ(σ)‖ ≤ ‖σn − ψ(σn)‖2 + α(‖ψ(σn) −σn‖ + ‖σn −σ‖)2 + α‖ψ(σ) −σn‖2 +(1 − 2α)‖σn −σ‖2 + 2‖σn − ψ(σn)‖‖ψ(σn) − ψ(σ)‖ ≤ ‖σn − ψ(σn)‖2 + α‖ψ(σn) −σn‖2 + α‖σn −σ‖2 +2α‖ψ(σn) −σn‖‖σn −σ‖ + α‖ψ(σ) −σn‖2 +(1 − 2α)‖σn −σ‖2 + 2‖σn − ψ(σn)‖‖ψ(σn) − ψ(σ)‖. this implies that ‖σn − ψ(σ)‖2 ≤ (1 + α) (1 −α) ‖σn − ψ(σn)‖2 + 2 (1 −α) (α‖σn −σ‖ + ‖ψ(σn) − ψ(σ)‖) ‖ψ(σn) −σn‖ + ‖σn −σ‖2. therefore lim sup n→∞ ‖σn − ψ(σ)‖2 ≤ lim sup n→∞ ‖σn −σ‖2 as an application of opial property we conclude that ψ(σ) = σ. � theorem 4.2. suppose x is a banach space having the opial property. assume that k is a nonempty subset of x and ψ : k → k a c-α nonexpansive mapping such that ψ admits an a.f.p.s.. then ψ has a fixed point. proof. demiclosedness principle implies the conclusion. � noor [12] considered the following iterative process:  σ1 ∈k σn+1 = (1 − ζn)σn + ζnψ(υn) υn = (1 −γn)σn + γnψ(wn) wn = (1 −δn)σn + δnψ(σn), n ∈ n, (4.2) where {ζn}, {γn} and {δn} are sequences in [0, 1]. lemma 4.3 ([20, p.484]). assume that 0 < a ≤ ln ≤ b < 1 for all n ∈ n and x is a uniformly convex banach space. suppose {σn} and {υn} are sequences such that lim sup n→∞ ‖σn‖≤ r, lim sup n→∞ ‖υn‖≤ r and lim n→∞ ‖lnσn + (1 − ln)υn‖ = r hold for some r ≥ 0. then lim n→∞ ‖σn −υn‖ = 0. lemma 4.4. suppose k is a nonempty closed convex subset of a banach space x . let ψ : k → k be a c-α nonexpansive mapping. let {σn} be a sequence defined by (4.2). if f(ψ) 6= ∅, then the following postulation hold: (1): max{‖σn+1 −w†‖,‖υn−w†‖,‖wn−w†‖}≤‖σn−w†‖ for all n ∈ n and w† ∈ f(ψ); © agt, upv, 2022 appl. gen. topol. 23, no. 2 385 r. pant and r. shukla (2): lim n→∞ ‖σn −w†‖ exists; (3): lim n→∞ d(σn,f(ψ)) exists, where d(σ,f(ψ)) denotes the distance from σ to f(ψ). proof. in view (4.2) and proposition 3.8, we get ‖wn −w†‖ = ‖(1 −δn)σn + δnψ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖ψ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖σn −w†‖ = ‖σn −w†‖.(4.3) by (4.2), (4.3) and proposition 3.8, we have ‖υn −w†‖ = ‖(1 −γn)σn + γnψ(wn) −w†‖ ≤ (1 −γn)‖σn −w†‖ + γn‖ψ(wn) −w†‖ ≤ (1 −γn)‖σn −w†‖ + γn‖wn −w†‖ ≤ (1 −γn)‖σn −w†‖ + γn‖σn −w†‖ = ‖σn −w†‖.(4.4) using (4.2), (4.4) and proposition 3.8, we get ‖σn+1 −w†‖ = ‖(1 − ζn)σn + ζnψ(vn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖ψ(υn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖υn −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖σn −w†‖ = ‖σn −w†‖.(4.5) combining (4.3), (4.4) and (4.5) proves (1). also by (4.5) the sequence {‖σn− w†‖} is bounded and hence monotone decreasing. therefore lim n→∞ ‖σn−w†‖ exists and proves (2). now, since ‖σn+1−w†‖≤‖σn−w†‖ for each w† ∈ f(ψ) and for all n ∈ n, d(σn+1,f(ψ)) ≤ d(σn,f(ψ)) for all n ∈ n. thus {d(σn,f(ψ))} is a bounded sequence and monotone decreasing. hence, lim n→∞ d(σn,f(ψ)) exists. � theorem 4.5. let k, {σn} and ψ be same as in lemma 4.4. if f(ψ) 6= ∅ and x is a uniformly convex banach space. then lim n→∞ ‖ψ(σn) −σn‖ = 0. proof. let w† ∈ f(ψ). then from lemma 4.4, {σn} is bounded and lim n→∞ ‖σn− w†‖ exists. call it r. that is (4.6) lim n→∞ ‖σn −w†‖ = r. in view of (4.6) and proposition 3.8 it implies that (4.7) lim sup n→∞ ‖ψ(σn) −w†‖≤ r. © agt, upv, 2022 appl. gen. topol. 23, no. 2 386 fixed point theorems for a new class of nonexpansive mappings in banach spaces by (4.3) and (4.6), we get (4.8) lim sup n→∞ ‖wn −w†‖≤ lim n→∞ ‖σn −w†‖ = r. from (4.4) and (4.6), we get (4.9) lim sup n→∞ ‖υn −w†‖≤ r. in view of (4.9) and proposition 3.8 it follows that (4.10) lim sup n→∞ ‖ψ(υn) −w†‖≤ r. similarly, (4.11) lim sup n→∞ ‖ψ(wn) −w†‖≤ r. by (4.2), (4.4) and proposition 3.8, we have ‖σn+1 −w†‖ = ‖(1 − ζn)σn + ζnψ(υn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖ψ(υn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖υn −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖σn −w†‖ = ‖σn −w†‖, or ‖σn+1 −w†‖≤‖(1 − ζn)σn + ζnψ(υn) −w†)‖≤‖σn −w†‖, it implies that r ≤ lim n→∞ ‖(1 − ζn)σn + ζnψ(υn) −w†)‖≤ r. then, (4.12) lim n→∞ ‖(1 − ζn)σn + ζnψ(υn) −w†)‖ = r. by (4.10), (4.11), (4.12) and lemma 4.3, we get that (4.13) lim n→∞ ‖σn − ψ(υn)‖ = 0. in view of the triangle inequality and proposition 3.8, we obtain ‖σn −w†‖ ≤ ‖σn − ψ(υn)‖ + ‖ψ(υn) −w†‖ ≤ ‖σn − ψ(υn)‖ + ‖υn −w†‖ by (4.13), we have lim n→∞ ‖σn −w†‖ ≤ lim n→∞ ‖σn − ψ(υn)‖ + lim inf n→∞ ‖υn −w†‖ ≤ lim inf n→∞ ‖υn −w†‖. it follows that (4.14) r ≤ lim inf n→∞ ‖υn −w†‖. © agt, upv, 2022 appl. gen. topol. 23, no. 2 387 r. pant and r. shukla using (4.2), (4.9) and (4.14), we get (4.15) r = lim n→∞ ‖υn −w†‖ = ‖(1 −γn)(σn −w†) + γn(ψ(wn) −w†)‖. in view of lemma 4.3, and (4.6), (4.11), (4.15), we have (4.16) lim n→∞ ‖σn − ψ(wn)‖ = 0. by triangle inequality and proposition 3.8, we have ‖σn −w†‖ ≤ ‖σn − ψ(wn)‖ + ‖ψ(wn) −w†‖ ≤ ‖σn − ψ(wn)‖ + ‖wn −w†‖, using (4.16) it follows that (4.17) r ≤ lim inf n→∞ ‖wn −w†‖. combining (4.8) and (4.17) together we get (4.18) lim n→∞ ‖wn −w†‖ = 0. by (4.2) and proposition 3.8, we have ‖wn −w†‖ = ‖(1 −δn)σn + δnψ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖ψ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖σn −w†‖ = ‖σn −w†‖. this implies that r ≤ lim n→∞ ‖(1 − δn)(σn −w†) + δn(ψ(σn) −w†)‖≤ r. therefore, we get (4.19) lim n→∞ ‖(1 −δn)(σn −w†) + δn(ψ(σn) −w†)‖ = r. in view of lemma 4.3 and (4.6), (4.7), (4.19), it follows that lim n→∞ ‖ψ(σn) − σn‖ = 0. � theorem 4.6. let x be a uniformly convex banach space having the opial’s property, k, ψ and {σn} same as in theorem 4.5. if f(ψ) 6= ∅ then {σn} weakly converges to a fixed point of ψ. proof. this can be completed following [14, theorem 5.8]. � theorem 4.7. suppose that k, x , {σn} and ψ are same as in lemma 4.4. let f(ψ) 6= ∅ and lim inf n→∞ d(σn,f(ψ)) = 0. then the sequence {σn} strongly converges to a fixed point of ψ. proof. this can be completed following [14, theorem 5.9]. � © agt, upv, 2022 appl. gen. topol. 23, no. 2 388 fixed point theorems for a new class of nonexpansive mappings in banach spaces theorem 4.8. assume that k is a subset of a uniformly convex banach spacex . let ψ and {σn} are same as in theorem 4.5 and. let ψ satisfy condition (i) with f(ψ) 6= ∅. then the sequence {σn} strongly converges to a fixed point of ψ. proof. this can be completed following [14, theorem 5.10]. � acknowledgements. we are very much thankful to the reviewer for his/her constructive comments and suggestions which 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[20] e. zeidler, nonlinear functional analysis and its applications, i, fixed-point theorems, springer-verlag, new york, 1986. © agt, upv, 2022 appl. gen. topol. 23, no. 2 390 @ appl. gen. topol. 23, no. 2 (2022), 333-343 doi:10.4995/agt.2022.17427 © agt, upv, 2022 zariski topology on the spectrum of fuzzy classical primary submodules phakakorn panpho a and pairote yiarayong b a major of physics, faculty of science and technology, pibulsongkram rajabhat university, phitsanulok 65000, thailand (phakakorn.p@psru.ac.th) b department of mathematics, faculty of science and technology, pibulsongkram rajabhat university, phitsanulok 65000, thailand (pairote0027@hotmail.com) communicated by s. romaguera abstract let r be a commutative ring with identity and m a unitary r-module. the fuzzy classical primary spectrum fcp.spec(m) is the collection of all fuzzy classical primary submodules a of m, the recent generalization of fuzzy primary ideals and fuzzy classical prime submodules. in this paper, we topologize fm(m) with a topology having the fuzzy primary zariski topology on the fuzzy classical primary spectrum fcp.spec(m) as a subspace topology, and investigate the properties of this topological space. 2020 msc: 08a72; 54b35; 13c05; 13c13; 13c99; 16d10; 03e72; 16n80; 16w50. keywords: zariski topology; classical primary submodule; fuzzy classical primary submodule; fuzzy classical primary spectrum; fuzzy primary ideal. 1. introduction throughout this paper all rings are commutative with identity and all modules are unitary. the fuzzy classical primary submodule in module theory plays crucial role in algebra. the concept of fuzzy classical primary submodule which is a generalization of fuzzy primary ideals and fuzzy classical prime submodules. the zariski topology on the prime spectrum of an r-module have received 01 april 2022 – accepted 06 july 2022 http://dx.doi.org/10.4995/agt.2022.17427 https://orcid.org/0000-0003-1525-9536 p. panpho and p. yiarayong been introduced by lu [13] and these have been studied by several authors [1, 2, 6, 7, 8, 9, 15, 16]. in 2008, ameri and mahjoob [3] investigated some properties of the zarisky topology of prime l-submodules. as it is well known that ameri and mahjoob in 2009 [4] introduced the notion of the zarisky topology of prime fuzzy hyperideals. in 2013, darani and motmaen [10] introduced and studied the concept of the zariski topology on the spectrum of graded classical prime submodules and these have been studied by several authors [18]. the concept of zariski topology on prime fuzzy submodules was introduced ameri and mahjoob [5] in 2017. in 2021, goswami and saikia [11] gave the concept of the spectrum of weakly prime submodules and investigated related properties. in this paper, we rely on the fuzzy classical primary submodules, and then introduce and study a new topology on the fuzzy classical primary spectrum fcp.spec(m) is the collection of all fuzzy classical primary submodules a of m, which generalizes the zariski topology of fuzzy prime submodule, called fuzzy primary zariski topology and investigate several properties of the topology. 2. basic definitions and preliminary results in this section, a brief overview of the concepts of fuzzy sets and fuzzy modules are required in this study. a function µ : m → [0, 1] is called a fuzzy set [17] of a non empty set m. the concept of fuzzy ideals of a ring was introduced in [12] as a generalization of the notion of fuzzy subrings. definition 2.1 ([12]). a fuzzy set µ of a ring r is called a fuzzy ideal of r if (1) µ(ab) ≥ µ(a) ∨µ(b) for all a,b ∈ r; (2) µ(a− b) ≥ µ(a) ∧µ(b) for all a,b ∈ r. the set of all fuzzy sets ( fuzzy ideals) of r is denoted by fs(r) (fi(r)). let µ be a fuzzy set of a ring r. the radical of µ is denoted by <(µ) and is defined by (<(µ)) (r) = ∨ n∈n µ(rn) for every element r ∈ r. definition 2.2. let µ be a fuzzy ideal of a ring r. a fuzzy set µ is called a fuzzy primary ideal of r if for every fuzzy ideals ν and η of r with ν�η ≤ µ, then either ν ≤ µ or η ≤<(µ) the concept of fuzzy modules of an r-module m was introduced in [14] as a generalization of the notion of fuzzy ideals. definition 2.3 ([14]). a fuzzy set a of an r-module m is called a fuzzy module of m if (1) a(0) = 1; (2) a(am) ≥a(m) for all a ∈ r and m ∈ m; (3) a(m−n) ≥a(m) ∧a(n) for all m,n ∈ m. © agt, upv, 2022 appl. gen. topol. 23, no. 2 334 zariski topology on the spectrum of fuzzy classical primary submodules condition (3) of the above definition is equivalent to a(m + n) ≥ a(m) ∧ a(n), and a(m) = a(−m) for all m,n ∈ m. the set of all fuzzy sets (fuzzy modules) of an r-module m is denoted by fs(m) (fm(m)). theorem 2.4. let a be a fuzzy set over an r-module r such that a(0) = 1. then the following conditions are equivalent. (1) a is a fuzzy module of r. (2) a is a fuzzy ideal of r. proof. by definition 2.3 it suffices to prove that (2) implies (1). assume that (2) holds. let a and b be any elements of r. since a is a fuzzy ideal of r with a(0) = 1, we have µ(ab) ≥ µ(a)∨µ(b) ≥ µ(b) and µ(a−b) ≥ µ(a)∧µ(b). therefore, we obtain that a is a fuzzy module of r. � let n be a non empty subset of an r-module m. for each α ∈ [0, 1), a characteristic function of n is denoted by αcn and is defined as (αcn ) (m) = { 1 ; m ∈ n α ; otherwise. we note that the r-module m can be considered a bipolar fuzzy set of itself and we write m = αcm (r = αcr), i.e., m(m) = 1 for all m ∈ m. in the following theorem, we establish a relationship between bipolar fuzzy modules and submodules of an r-module. theorem 2.5. let α be any element of [0, 1) and let m be an r-module. then the following conditions are equivalent. (1) n is a submodule of m. (2) the characteristic function αcn of n is a fuzzy module over m. proof. first assume that n is a submodule of m. since 0 is an element of n, we have (αcn ) (0) = 1. let m and n be any elements of m and r ∈ r. if m,n ∈ n, then (αcn ) (m) = 1 = (αcn ) (n) and since m−n,rm ∈ n, we have (αcn ) (rm) = 1 = (αcn ) (m) and (αcn ) (m−n) = 1 = (αcn ) (m)∧(αcn ) (n). otherwise, if m 6∈ n or n 6∈ n, then (αcn ) (m) = α or (αcn ) (m) = α and so we have (αcn ) (m−n) ≥ α = (αcn ) (m) ∧ (αcn ) (n). it is obvious that (αcn ) (rm) ≥ α = (αcn ) (m). therefore αcn is a fuzzy module over m and hence (1) implies (2). conversely, assume that (2) holds. let m and n be any elements of m and r ∈ r such that m,n ∈ n. set x = rm and y = m − n. then (αcn ) (x) = (αcn ) (rm) ≥ (αcn ) (m) = 1 and (αcn ) (y) = (αcn ) (m−n) ≥ (αcn ) (m) ∧ (αcn ) (n) = 1 ∧ 1 = 1. hence we have (αcn ) (x) = 1 and (αcn ) (y) = 1, and so m − n,rm ∈ n. therefore n is a submodule of m and hence (1) implies (2). � from the above result, we have the following corollary: corollary 2.6. let m be an r-module. then the following conditions are equivalent. © agt, upv, 2022 appl. gen. topol. 23, no. 2 335 p. panpho and p. yiarayong (1) n is a submodule of m. (2) the characteristic function 0cn of n is a fuzzy module over m. let µ and a be a fuzzy set over a ring r and fuzzy set over an r-module m, respectively. define the composition µ�a, and product µa respectively as follows: (µ�a) (x) =   ∨ x=rm (µ(r) ∧a(m)) ; if x = rm for some r ∈ r,m ∈ m 0 ; otherwise, and (µa) (x) =   ∨ x= n∑ i rimi n∧ i=1 µ(ri) ∧ n∧ i=1 a(mi) ;x = n∑ i rimi∃ri ∈ r,mi ∈ m 0 ; otherwise. next, let x be an element of an r-module m and α ∈ (0, 1]. define the fuzzy set xα over m as follows: xα(a) = { α ; x = a 0 ; otherwise. then xα is called a fuzzy point or fuzzy singleton. let a be a fuzzy set over an r-module m. next, let 〈a〉 denote the intersection of all fuzzy modules over m which contain a. then 〈a〉 is a fuzzy module over m, called the fuzzy module generated by a. definition 2.7. let a and b be any fuzzy sets of an r-module m. for every fuzzy set µ of r define (a : b) and (a : µ), as follows: (a : b) = ∨ {µ ∈fs(r) : µ�b ≤a} and (a : µ) = ∨ {b ∈fs(m) : µ�b ≤a}. 3. topologies on fuzzy classical primary submodules the given definition of fuzzy classical primary submodule is a generalization of the notion of classical prime and classical primary submodules in module theory. definition 3.1. let a be a fuzzy submodule of an r-module m. a fuzzy set a is called a fuzzy classical primary submodule of m if for every elements a and b of r and every element x of m with aζbξxα ∈a, then either aζxα ∈a or bnζ xα ∈a for some positive integer n. we now present the following example satisfying above definition. © agt, upv, 2022 appl. gen. topol. 23, no. 2 336 zariski topology on the spectrum of fuzzy classical primary submodules example 3.2. let z be the set of all integers. suppose m = r = z is a commutative ring. define the fuzzy set a of z as follows: a(x) = { 1 ; if x ∈ 4z 0 ; if x 6∈ 4z. then it is easily seen that a is a fuzzy classical primary submodule of an an r-module m. let m be an r-module. in the sequel fcp.spec(m) denotes the set of all fuzzy classical primary submodules of an r-module m. we call fcp.spec(m), the fuzzy classical primary spectrum of m. for every fuzzy submodule a of m, the fuzzy classical variety of a is denoted by v(a), and is defined as the set of all fuzzy classical primary submodule containing a, i.e., v(a) = {b ∈ fcp.spec(m) : a≤b}. theorem 3.3. for any family of fuzzy submodules {ai}i∈i of an r-module m. then the following properties hold. (1) v(01) = fcp.spec(m) and v(m) = ∅. (2) ⋂ i∈i v (ai) = v (∑ i∈i ai ) . (3) v (a1) ∪v (a2) = v (a1 ∧a2). proof. (1). obvious. (2). let b be a fuzzy submodule of m such that b ∈ ⋂ i∈i v (ai). then we have b ∈v (ai) for all i ∈ i, i.e., ai ≤b. next let x be an element of m. we also consider (∑ i∈i ai ) (x) = ∨ x= ∑ i∈i xi   ∧ x= ∑ i∈i xi ai(xi)   ≤ ∨ x= ∑ i∈i xi   ∧ x= ∑ i∈i xi b(xi)   = ∨ x= ∑ i∈i xi b(xi) = b(x) © agt, upv, 2022 appl. gen. topol. 23, no. 2 337 p. panpho and p. yiarayong ans so ∑ i∈i ai ≤ b. thus we have b ∈ v (∑ i∈i ai ) , which implies that ⋂ i∈i v (ai) ⊆ v (∑ i∈i ai ) . on the other hand, let b be a fuzzy submodule of m such that b ∈ v (∑ i∈i ai ) . it is easy to see that ai ≤ ∑ i∈i ai ≤ b, i.e., b ∈ v (ai) for all i ∈ i. therefore v (∑ i∈i ai ) ⊆ ⋂ i∈i v (ai) and hence ⋂ i∈i v (ai) = v (∑ i∈i ai ) . (3). let b be a fuzzy submodule of m such that b ∈v (a1)∪v (a2). then we have a1 ≤b or a2 ≤b, it follows that a1∧a2 ≤b. thus b ∈v (a1 ∧a2) and so v (a1) ∪v (a2) ⊆ v (a1 ∧a2). on the other hand, let b be a fuzzy submodule of m such that b ∈ v (a1 ∧a2). this implies that a1 ∧a2 ≤ b, i.e., a1 ≤b or a2 ≤b. also, b ∈v (a1) ∪v (a2). therefore, we obtain that v (a1 ∧a2) ⊆v (a1) ∪v (a2) and hence v (a1) ∪v (a2) = v (a1 ∧a2). � corollary 3.4. let µ and ν be any fuzzy ideal of a ring r. then v (µ�m)∪ v (ν �m) = v (µ�ν �m). set x = fcp.spec(m). for every fuzzy submodule a of an r-module m we define e(a) and τ as follows: e(a) = x −v (a) and τ = {e(a) : a∈fm(m)}. in the next theorem we will show that the pair (x,τ) is a topological space. theorem 3.5. let m be an r-module. then the following statements hold: (1) the pair (x,τ) is a topological space. (2) x is a t0 topological space. proof. 1. since v(01) = x and v(m) = ∅, we have e(01) = x −x = ∅ and e(m) = x −∅ = x, .i.e., ∅,x ∈ τ. 2. let a and b be any fuzzy submodules of m. thus by theorem 3.3(3), we have e(a) ∩e(b) = (x −v (a)) ∩ (x −v (b)) = x ∩ ( v (a)′ ∩v (b)′ ) = x ∩ (v (a) ∪v (b))′ = x −v (a∧b) = e(a∧b). © agt, upv, 2022 appl. gen. topol. 23, no. 2 338 zariski topology on the spectrum of fuzzy classical primary submodules 3. for any family of fuzzy submodules {ai}i∈i of m. then by theorem 3.7(2), we have ⋃ i∈i e(ai) = ⋃ i∈i (x −v (ai)) = x − ⋂ i∈i v (ai) = x −v (∑ i∈i ai ) = e (∑ i∈i ai ) . 2 and 3 show that τ is closed under arbitrary union and finite intersection. thus the pair (x,τ) satisfies in axioms of a topological space. therefore we have (x,τ) is a topological space. (2) let a and b be two distinct points of x. if a 6≤ b, then obviously b ∈e(a) and a 6∈ e(a) showing that x is a t0 topological space. � in this case, the topology τ on x is called the fuzzy primary zariski topology. for every fuzzy submodule a of m, the set v∗(a) = { b ∈ fcp.spec(m) : √ (a : m) ≤ √ (b : m) } . then we have the following lemma. lemma 3.6. let a and b be any fuzzy submodules of an r-module m. if a≤b, then v∗ (b) ≤v∗ (a). proof. let c be a fuzzy submodule of m such that c ∈v∗ (b). then we have√ (b : m) ≤ √ (c : m). since a ≤ b, we have √ (a : m) ≤ √ (b : m), i.e.,√ (a : m) ≤ √ (c : m). therefore c ∈v∗ (a) and hence v∗ (b) ≤v∗ (a). � then we have the next results. theorem 3.7. for any family of fuzzy submodules {ai}i∈i of an r-module m. then the following properties hold. (1) v∗(01) = fcp.spec(m) and v∗(m) = ∅. (2) ⋂ i∈i v∗ (ai) = v∗ (∑ i∈i (ai : m) �m ) . (3) v∗ (a1) ∪v∗ (a2) = v∗ (a1 ∧a2). proof. (1). obvious. (2). let b be a fuzzy submodule of m such that b ∈ ⋂ i∈i v∗ (ai). then we have b ∈v∗ (ai) for all i ∈ i, i.e., √ (ai : m) ≤ √ (b : m). since (ai : m)� m ≤ √ (ai : m) � m ≤ √ (b : m) � m, we have ∑ i∈i (ai : m) � m ≤ © agt, upv, 2022 appl. gen. topol. 23, no. 2 339 p. panpho and p. yiarayong √ (b : m) �m, it follows that,√√√√((∑ i∈i (ai : m) �m ) : m ) ≤ √(√ (b : m) �m : m ) ≤ √√ (b : m) = √ (b : m). it is easy to see that b ∈ v∗ (∑ i∈i (ai : m) �m ) and so ⋂ i∈i v∗ (ai) ⊆ v∗ (∑ i∈i (ai : m) �m ) . on the other hand, let b be a fuzzy submodule of m such that b ∈v∗ (∑ i∈i (ai : m) �m ) . thus we have √√√√((∑ i∈i (ai : m) �m ) : m ) ≤ √ (b : m). clearly, we have (((ai : m) �m) : m) = (ai : m) for all i ∈ i. also for each i ∈ i, we obtain that√ (ai : m) = √ (((ai : m) �m) : m) ≤ √√√√((∑ i∈i (ai : m) �m ) : m ) ≤ √ (b : m) = √ (b : m). therefore we obtain that b ∈ ⋂ i∈i v∗ (ai) and hence v∗ (∑ i∈i (ai : m) �m ) ⊆⋂ i∈i v∗ (ai). (3). let b be a fuzzy submodule of m such that b ∈ v∗ (a1) ∪v∗ (a2). then we have √ (a1 : m) ≤ √ (b : m) or √ (a2 : m) ≤ √ (b : m). if√ (a1 : m) ≤ √ (b : m), then √ (a1 ∧a2 : m) ≤ √ (a1 : m) ≤ √ (b : m), it follows that, b ∈ v∗ (a1 ∧a2). similarly, if √ (a2 : m) ≤ √ (b : m), then b ∈ v∗ (a1 ∧a2). on the other hand, let b be a fuzzy submodule of m such that b ∈ v∗ (a1 ∧a2). then √ (a1 ∧a2 : m) ≤ √ (b : m). since a1∧a2 ≤a1 and a1∧a2 ≤a2, we have √ (a1 : m) ≤ √ (a1 ∧a2 : m) and√ (a2 : m) ≤ √ (a1 ∧a2 : m), which implies that,√ (a1 : m) � √ (a2 : m) ≤ √ (a1 ∧a2 : m). © agt, upv, 2022 appl. gen. topol. 23, no. 2 340 zariski topology on the spectrum of fuzzy classical primary submodules now since √ (b : m) is prime and √ (a1 : m) � √ (a2 : m) ≤ √ (b : m), it follows that √ (a1 : m) ≤ √ (b : m) or √ (a2 : m) ≤ √ (b : m). clearly, we have b ∈ v∗ (a1) or b ∈ v∗ (a2), i.e., b ∈ v∗ (a1) ∪v∗ (a2). therefore v∗ (a1 ∧a2) ≤v∗ (a1)∪v∗ (a2) and hence v∗ (a1)∪v∗ (a2) = v∗ (a1 ∧a2). � for every fuzzy submodule a of an r-module m we define e∗(a) and τ∗ as follows: e∗(a) = x −v∗ (a) and τ∗ = {e∗(a) : a∈fm(m)}. in the next theorem we will show that the pair (x,τ∗) is a topological space. theorem 3.8. let m be an r-module. then the following statements hold: (1) the pair (x,τ∗) is a topological space. (2) x is a t0 topological space. proof. the proof follows from theorem 3.5. � for any r-module m and a,b ∈fm(m) we have the next result. proposition 3.9. let a and b be any fuzzy submodules of an r-module m. if √ (a : m) = √ (b : m), then v∗ (a) = v∗ (b). moreover, the converse is true if both a and b are classical primary. proof. let a and b be any fuzzy submodules of m such that √ (a : m) =√ (b : m). next let c be a fuzzy submodule of m such that c ∈ v∗ (a). then we have √ (a : m) ≤ √ (c : m), i.e., √ (b : m) ≤ √ (c : m). thus c ∈ v∗ (b) and so v∗ (a) ⊆ v∗ (b). similarly, we obtain that v∗ (b) ⊆ v∗ (a). for the converse, suppose that a,b ∈ fm(m) is classical primary and v∗ (a) = v∗ (b). since a ∈ v∗ (a) ,b ∈ v∗ (b) and v∗ (a) = v∗ (b), we have √ (b : m) ≤ √ (a : m) and √ (a : m) ≤ √ (b : m). therefore, we obtain that √ (a : m) = √ (b : m). � for a fuzzy prime ideal p of r, by fcp.specp(m) we mean the set of all a∈fm(m) such that √ (a : m) = p. in other words fcp.specp(m) = { a∈ fcp.spec(m) : √ (a : m) = p } . theorem 3.10. let µ and a be any fuzzy ideal and any fuzzy submodule of r and m, respectively. then the following properties hold. (1) v∗(a) = ⋃ √ (a:m)≤p fcp.specp(m). (2) v∗ (µm �m) = v (µn �m) for some positive integers m,n. (3) v (√ (a : m) �m ) ⊆v∗ (a) ⊆v∗ ((a : m) �m). proof. (1). let b be a fuzzy submodule of m such that b ∈v∗ (a). then we have √ (a : m) ≤ √ (b : m) = p and so © agt, upv, 2022 appl. gen. topol. 23, no. 2 341 p. panpho and p. yiarayong b ∈ fcp.specp(m) ⊆ ⋃ √ (a:m)≤p fcp.specp(m). it is easy to see that v∗(a) ⊆ ⋃ √ (a:m)≤p fcp.specp(m). on the other hand, let b be a fuzzy submodule of m such that b ∈ ⋃ √ (a:m)≤p fcp.specp(m). thus there exists a fuzzy prime ideal p of r such that √ (a : m) ≤ p and b ∈ fcp.specp(m). clearly, we have √ (b : m) = p, i.e., √ (a : m) ≤ √ (b : m), it follows that, b ∈v∗ (a). therefore we obtain that ⋃ √ (a:m)≤p fcp.specp(m) ⊆ v∗(a) and hence v∗(a) = ⋃ √ (a:m)≤p fcp.specp(m). (2). let b be a fuzzy submodule of m such that b ∈ v (µn �m). then we have µn � m ≤ b, i.e., √ (µm �m : m) ≤ √ (b : m). this implies that b ∈ v∗ (µm �m) and so v (µn �m) ⊆ v∗ (µm �m). on the other hand, let b be a fuzzy submodule of m such that b ∈ v∗ (µm �m). thus√ (µm �m : m) ≤ √ (b : m). obviously, µm ≤ (µm �m : m). since√ (µm �m : m) ≤ √ (b : m) and µm ≤ (µn �m : m), we have µm ≤√ (b : m), which implies that, µmt � m ≤ b. it is easy to see that b ∈ v (µn �m). therefore v∗ (µm �m) ⊆v (µn �m) and hence v∗ (µm �m) = v (µn �m). (3). let b be a fuzzy submodule of m such that b ∈v∗ (a). then we have√ (a : m) ≤ √ (b : m). since (a : m) �m≤a, we have√ ((a : m) �m : m) ≤ √ (a : m) ≤ √ (b : m). this implies that b ∈v∗ ((a : m) �m) and so v∗ (a) ⊆v∗ ((a : m) �m). next, let b be a fuzzy submodule of m such that b ∈ v (√ (a : m) �m ) . thus √ (a : m)�m≤b. obviously, √ (a : m) ≤ (b : m). since (b : m) ≤√ (b : m), we have √ (a : m) ≤ √ (b : m), which implies that, b ∈ v∗ (a). therefore v (√ (a : m) �m ) ⊆ v∗ (a) and hence v (√ (a : m) �m ) ⊆ v∗ (a) ⊆v∗ ((a : m) �m). � acknowledgement this work (grant no. rgns 64-189) was financially supported by office of the permanent secretary, ministry of higher education, science, research and innovation. © agt, upv, 2022 appl. gen. topol. 23, no. 2 342 zariski topology on the spectrum of fuzzy classical primary submodules references [1] j. abuhlail, a zariski topology for modules, communications in algebra 39 (2011), 4163–4182. 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[18] k. a. zoubi and m. jaradat, the zariski topology on the graded classical prime spectrum of a graded module over a graded commutative ring, matematicki vesnik 70, no. 4 (2018), 303–313. © agt, upv, 2022 appl. gen. topol. 23, no. 2 343 gongreillyagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 267-272 on the order hereditary closure preserving sum theorem jianhua gong and ivan l. reilly abstract. the main purpose of this paper is to prove the following two theorems, an order hereditary closure preserving sum theorem and an hereditary theorem: (1) if a topological property p satisfies ( ∑′ ) and is closed hereditary, and if v is an order hereditary closure preserving open cover of x and each v ∈ v is elementary and possesses p, then x possesses p. (2) let a topological property p satisfy ( ∑′ ) and (β), and be closed hereditary. let x be a topological space which possesses p. if every open subset g of x can be written as an order hereditary closure preserving (in g) collection of elementary sets, then every subset of x possesses p. 2000 ams classification: 54d20 keywords: elementary set, order hereditary closure preserving, sum theorem. 1. introduction r. e. hodel [1] obtained sum theorems and an hereditary theorem for topological spaces. s. p. arya and m. k. singal [1, 2] and g. gao [4] have improved some of hodel’s sum theorems. we provide in this paper further improvements of these theorems. a topological property p is said to be hereditary (closed hereditary, open hereditary) if when p is possessed by a topological space x, it is also shared by every subspace (closed subspace, open subspace) of x. it is well known that covering properties such as paracompactness, subparacompactness, countable paracompactness, pointwise paracompactness, θ-refinement and collectionwise normality satisfy the following result which is denoted by (β). (β) : if every open subset of a space x has a property p, then every subset of x has the property p. 268 j. gong and i. l. reilly notice that x is an open subspace of itself, thus (β) states that open hereditary implies hereditary. y.k atuta [6] introduced the notion of an order locally finite family of subsets of a topological space. later g. gao [4] also introduced the notion of an order hereditary closure preserving family of subsets of a topological space. a family {aγ : γ ∈ γ} of subsets of a topological space x is called hereditary closure preserving relative to a subspace a of x if for any γ′ ⊂ γ and any eγ ⊂ aγ the following is true for all points in a. ⋃ γ∈γ′ eγ = ⋃ eγ . definition 1.1 (g. gao [4]). a family {aα : α < τ} (α and τ are ordinal numbers) is defined to be order hereditary closure preserving if for every ordinal number β < τ , the family {aα : α < β} is hereditary closure preserving relative to aβ. it is not difficult to see that the following implications are true for a family of subsets of a topological space. however, the converse implications are not true in general. proposition 1.2. given a family of subsets of a topological space, then locally f inite ⇒ hereditary closure preserving ⇓ ⇓ σ − locally f inite ⇒ σ − hereditary closure preserving ⇓ ⇓ order locally f inite ⇒ order hereditary closure preserving definition 1.3 (r. e. hodel [5]). let n be the set of all positive integers. an open subset v of a topological space is called an elementary set if v = ⋃ ∞ i=1 vi, where each vi is open and vi ⊂ v for all i ∈ n . the following two lemmas show that each open fσ set in a normal space is exactly an elementary set. lemma 1.4. every elementary set in a topological space is an open fσ set. proof. suppose the open subset v of a topological space is an elementary set, then v = ⋃ ∞ i=1 vi, vi is open and vi ⊂ v for all i ∈ n . hence ⋃ ∞ i=1 vi ⊂ v. on the other hand, vi ⊂ vi for all i ∈ n, so v = ⋃ ∞ i=1 vi ⊂ ⋃ ∞ i=1 vi. therefore, v = ⋃ ∞ i=1 vi, it follows that v is an open fσ set. � lemma 1.5. every open fσ subset of a normal space is an elementary set. proof. let v be an open fσ set of a normal space x, then v = ⋃ ∞ i=1 wi, wi is closed and wi ⊂ v for all i ∈ n . by the normality of x, for each wi there exists an open set vi such that wi ⊂ vi ⊂ vi ⊂ v . thus, v = ⋃ ∞ i=1 wi ⊂ ⋃ ∞ i=1 vi and ⋃ ∞ i=1 vi ⊂ v. that is v = ⋃ ∞ i=1 vi where each vi is open and vi ⊂ v for all i ∈ n. therefore v is an elementary set. � on the order hereditary closure preserving sum theorem 269 notice that an open fσ set may fail to be an elementary set in non-normal spaces, as the following example shows. example 1.6. let x be the set n of all positive integers with cofinite topology. then x is a t1 space which is not a normal space. take the set v = n/{1, 2, 3}, then v is an open set. furthermore, v = ⋃ ∞ i=4 {i}. since x is a t1 space, each singleton {i} is a closed subset, so that v is an open fσ set. for any subset s of x we have s = { s if s is finite, x if s is infinite. since every non-empty open subset s of x is infinite, for every open subset s of v , s = x 6⊂ v. so v is not an elementary set. we say that a topological property p satisfies the locally finite closed sum theorem if the following is satisfied and denote it by ( ∑ ). ( ∑ ) : let {fα : α ∈ a} be a locally finite closed cover of a topological space x and let each fα possess a property p, then x possesses the property p. we say that a topological property p satisfies the hereditary closure preserving closed sum theorem if the following is satisfied and denote it by ( ∑ ′ ). ( ∑ ′ ) : let {fα : α ∈ a} be an hereditary closure preserving closed cover of a topological space x and let each fα possess a property p, then x possesses the property p. observe from proposition 1.2 that ( ∑ ′ ) ⇒ ( ∑ ). for example, if the topological property p is one of paracompactness, subparacompactness, pointwise paracompactness, meso-compactness, θ-refinement, weak θ-refinement and ortho-compactness, then the property p satisfies ( ∑ ). if the topological property p is either paracompactness or t1 meso-compactness, then the property p satisfies ( ∑ ′ ). 2. a sum theorem in this section, we assume that the topological property p satisfies ( ∑ ′ ) (hence ( ∑ )) and is closed hereditary. theorem 2.1. let v = {vα : α < τ} be an order hereditary closure preserving open cover of a topological space x, and let each vα be an elementary set which possesses a topological property p. then x possesses the topological property p. 270 j. gong and i. l. reilly proof. since each vα is an elementary set and possesses the property p, (2.1) vα = ∞ ⋃ i=1 vα,i, vα,i ⊂ vα, α < τ, i ∈ n, where each vα,i is an open set. then the closed set vα,i possesses the property p by closed hereditary. for each i ∈ n, let vi = {vα,i : α < τ}. for each α < τ , let (2.2) f0,i = v0,i, fα,i = vα,i − ⋃ β<α vβ , 0 < α < τ. then each closed set fα,i possesses the property p. and we claim that the family {fα,i : α < τ} is an hereditary closure preserving collection. without loss of generality, for each α < τ , let aα,i ⊂ fα,i, we need to prove ⋃ α<τ aα,i = ⋃ α<τ aα,i. obviously, it is enough to prove (2.3) ⋃ α<τ aα,i ⊂ ⋃ α<τ aα,i. suppose x ∈ ⋃ α<τ aα,i, since v is a cover of x, we may assume x ∈ vβ0 . now the inequality (2.3) can be expressed in another way: (2.4)   ⋃ α<β0 aα,i   ∪ aβ0,i ∪   ⋃ β0<α<τ aα,i   ⊂   ⋃ α<β0 aα,i   ∪ aβ0,i ∪   ⋃ β0<α<τ aα,i   . according to (2.2), vβ0 ∩ fα,i = ∅, β0 < α < τ . so x − vβ0 ⊃ ⋃ β0<α<τ fα,i. since x−vβ0 is a closed set, then x − vβ0 ⊃ ⋃ β0<α<τ fα,i, that is x /∈ ⋃ β0<α<τ fα,i. therefore x /∈ ⋃ β0<α<τ aα,i. if x ∈ aβ0,i, the inequality (2.4) is satisfied. we may assume x ∈ ⋃ α<β0 aα,i. since v is order hereditary closure preserving, {vα : α < β0} is hereditary closure preserving at every point of vβ0 . notice that x ∈ vβ0 , thus x ∈ ⋃ α<β0 aα,i. so the inequality (2.2) is proved. let fi = ⋃ α<τ fα,i, then fi possesses the property p by applying ( ∑ ′ ), for all on the order hereditary closure preserving sum theorem 271 i ∈ n. for each i ∈ n, let v∗i = ⋃ α<τ {vα,i}, then v∗i ⊂ fi by the well order property. hence {v ∗ i } and {fi} are open covers and closed covers of the space x respectively. finally, let h1 = f1, hi = fi − i−1 ⋃ j=1 v∗j , i = 2, 3, ... then {hi} is a locally finite closed cover of x and each hi possesses the property p. it follows from ( ∑ ) that x possesses the property p. � apply proposition 1.2 to theorem 2.1, we can obtain the following two corollaries. corollary 2.2 (s. p. arya and m. k. singal [2]). let v be a σ-hereditary closure preserving cover of a topological space x and each v ∈ v be an elementary set which possesses a topological property p, then x possesses the property p. corollary 2.3 (r. e. hodel [5]). let v be a σ-locally finite cover of a topological space x and each v ∈ v be an elementary set which possesses a topological property p, then x possesses the property p. 3. two hereditary theorems we assume that the topological property p in this section satisfies ( ∑ ′ ) (hence ( ∑ )), (β) and is closed hereditary. theorem 3.1. let x be a topological space which possesses a topological property p. if every open subset g of x can be written as an order hereditary closure preserving (in g) collection of elementary sets, then every subset of x possesses the property p. proof. let v = {vα : α < τ} be order hereditary closure preserving at every point of g, and let v∗ = ⋃ α<τ vα = g, where each vα, α < τ is an elementary subset of x. we may assume vα = ∞ ⋃ i=1 vα,i, vα,i ⊂ vα, α < τ where each vα,i is an open set. let fα,1 = vα,1, fα,i = vα,i − ⋃ j b where k,α > 0. then (a,a,d) is a c∗-algebra valued quasi metric space. example 2.5. let a = [1,∞) and a = m2(c). define d : a×a →a by (for all a,b ∈ a): d(a,b) =   [ ln b− ln a 0 0 ln b− ln a ] if a ≤ b 1 3 [ ln a− ln b 0 0 ln a− ln b ] if a > b where ‘ ln′ is natural logarithmic function. then (a,a,d) is a c∗-algebra valued quasi metric space. © agt, upv, 2022 appl. gen. topol. 23, no. 2 289 m. asim, s. kumar, m. imdad and r. george now, we give some definitions of convergent, left-cauchy, right-cauchy and completeness of the quasi metric space as follows. definition 2.6. let (a,a,d) be a c∗-algebra valued quasi metric space and {an} a sequence in a. we say that (i) the sequence {an} is called convergent to a ∈ a, written lim n→∞ an = a, if lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0a. (ii) the sequence {an} is called left-cauchy if for each � � 0a there exists a positive integer n such that d(an,am) ≺ � for all n ≥ m ≥ n. (iii) the sequence {an} is called right-cauchy if for each � � 0a there exists a positive integer n such that d(an,am) ≺ � for all m ≥ n ≥ n. (iv) the sequence {an} is called cauchy if for each � � 0a there exists a positive integer n such that d(an,am) ≺ � for all m,n ≥ n, i.e., lim n,m→∞ d(an,am) = 0a. (v) the triplet (a,a,d) is left-complete if every left-cauchy sequence in (a,a,d) is convergent. (vi) the triplet (a,a,d) is right-complete if every right-cauchy sequence in (a,a,d) is convergent. (vii) the triplet (a,a,d) is complete if every cauchy sequence in (a,a,d) is convergent. remark 2.7. (1) every c∗-algebra valued metric space is c∗-algebra valued quasi metric space but the converse is not true in general. (2) in a c∗-algebra valued quasi metric space a sequence {an} is cauchy iff it is left-cauchy and right-cauchy. definition 2.8. let (a,a,d) c∗-algebra valued quasi metric space. the conjugate (or dual) c∗-algebra valued quasi metric space is denoted by dc and define by as follows: dc(a,b) = d(b,a), for all a,b ∈ a. a c∗-algebra valued quasi metric space is a c∗-algebra valued metric space iff it coincides with its conjugate dc, for all a,b ∈ a. let wc be a c ∗-algebra positive valued function on a. the quadruplet (a,a,d,wc) is called a c∗-algebra valued weighted quasi metric space, if for all a,b ∈ a d(a,b) + wc(a) = d(b,a) + wc(b). then (a,a,d,wc) is said to be c∗-algebra valued generalized weighted quasi metric space. © agt, upv, 2022 appl. gen. topol. 23, no. 2 290 c∗-avqms and fixed point results with an application proposition 2.9. let (a,a,d) c∗-algebra valued quasi metric space. the associated c∗-algebra valued metric ds is define by: ds(a,b) = 1 2 [d(a,b) + d(b,a)]. the associated c∗-algebra valued metric space ds is the smallest c∗-algebra valued metric space majorising d. proof. to verify condition (i), for each a,b ∈ a, we have d(a,b) < 0a. also ds(a,b) = 0a ⇔ 1 2 [d(a,b) + d(b,a)] = 0a ⇔ d(a,b) + d(b,a) = 0a ⇔ d(a,b) = d(b,a) = 0a ⇔ a = b. now, for condition (ii), for each a,b ∈ a, we have ds(a,b) = 1 2 [d(a,b) + d(b,a)] = 1 2 [d(b,a) + d(a,b)] = ds(b,a). finally, we show that condition (iii), for each a,b,c ∈ a, we have ds(a,b) = 1 2 [d(a,b) + d(b,a)] = 1 2 [d(a,c) + d(c,b) + d(b,c) + d(c,a)] = 1 2 [d(a,c) + d(c,a)] + 1 2 [d(b,c) + d(c,b)] = ds(a,b) + ds(b,a). thus, (a,a,ds) is c∗-algebra valued metric space. � definition 2.10. let (a,a,d) be a c∗-algebra valued quasi metric space, a ∈ a, n,m ⊆ a and 0a ≺ � ∈a. denoted by: • the diameter of set n diam(n) = sup{d(a,b) : a,b ∈ n}. • the left-open ball of radius � centered at a bl� (a) = {b ∈ a : d(a,b) ≺ �}. • the right-open ball of radius � centered at a br� (a) = {b ∈ a : d(b,a) ≺ �}. • the associated c∗-algebra valued quasi metric space open ball of radius � centered at a b� = {b ∈ a : ds(a,b) ≺ �}. © agt, upv, 2022 appl. gen. topol. 23, no. 2 291 m. asim, s. kumar, m. imdad and r. george • the left-distance from a to n distd(a,n) = inf{d(a,b) : b ∈ n}. • the right-distance from a to n distd(n,a) = inf{d(b,a) : b ∈ n}. • the left-�-neighbourhood of n nl� = inf{a ∈ a : distd(n,a) ≺ �}. • the right-�-neighbourhood of n nr� = inf{a ∈ a : distd(a,n) ≺ �}. • the associated metric �-neighbourhood of n nl� = inf{a ∈ a : dist s d(n,a) ≺ �}. • the distance between n and m d(n,m) = inf{d(a,b) : a ∈ n,b ∈ m}. proposition 2.11. let (a,a,d) be a c∗-algebra valued quasi metric space. then the collection of all open left-balls bl� (a) (right-balls b r � (a)) on a, ula = {b l � (a) : a ∈ a,� � 0a} forms a left-basis (right-basis) on a. proof. take a,b ∈ a and �1,�2 � 0a such that bl�1 (a) ∩ b l �2 (b) 6= ∅. now, choose c ∈ bl�1 (a) ∩b l �2 (b) and set �3 = min{�1 −d(a,c),�2 −d(b,c)}. observe that bl�3 (c) ⊆ b l �1 (a) ∩bl�2 (b). therefore, u l a forms a left-basis on a. similarly, the collection of all open right-balls bl� (a), ura = {b r � (a) : a ∈ a,� � 0a} forms a right-basis on a. � every c∗-algebra valued quasi metric space d naturally induces a t0 topology t la , where a set n is open if for each a ∈ n there exists � � 0a such that bl� (a) ⊆ n. similarly, the topology t ra can be define by using the right-balls (that is, br� (a)) as its base and hence a c ∗-algebra valued quasi metric space (a,a,d) can be naturally associated with a bi-topological space (a,a,t la ,t r a ). moreover, if the map d satisfies d(a,b) = 0 ⇔ a = b instead of condition (i) in definition (2.2) then d induces a t1 topology. proposition 2.12. let (a,a,d) be a c∗-algebra valued quasi metric space and associated (a,a,ds) a c∗-algebra valued metric space. then (1) a sequence {an} is convergent to a in (a,a,d) if and only if {an} is convergent to a in (a,a,ds). (2) a sequence {an} is cauchy in (a,a,d) if and only if {an} is cauchy in (a,a,ds). (3) the c∗-algebra valued quasi metric space (a,a,d) is complete if and only if c∗-algebra valued metric space (a,a,ds) is complete. © agt, upv, 2022 appl. gen. topol. 23, no. 2 292 c∗-avqms and fixed point results with an application proof. (1) suppose that {an} is convergent to a in (a,a,d), that is, lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0a. which is equivalent to lim n→∞ ds(an,a) = 1 2 [ lim n→∞ d(an,a) + lim n→∞ d(a,an)] = 0a. hence, the sequence {an} is convergent to a in (a,a,ds). (2) suppose that {an} is cauchy in (a,a,d), that is, lim n→∞ d(an,am) = lim n→∞ d(am,an) = 0a. which is equivalent to lim n→∞ ds(an,am) = 1 2 [ lim n→∞ d(an,am) + lim n→∞ d(am,an)] = 0a. therefore, the sequence {an} is cauchy in (a,a,ds). (3) it is a direct consequence of (1) and (2). � proposition 2.13. let (a,a,da) and (b,a,db) be two c∗-algebra valued quasi metric spaces. then (1) d(a,b) = da(a1,a2) + db(b1,b2), for all a = (a1,b1), b = (a2,b2) ∈ a×b is a c∗-algebra valued quasi metric on a×b. (2) lim n→∞ (an,bn) = (a,b) in (a × b,a,d) if and only if lim n→∞ an = a in (a,a,da) and lim n→∞ bn = b in (b,a,db). particularly, the topology induced by d coincides the product topology on a×b. (3) {(an,bn)} is a cauchy sequence in (a × b,a,d) if and only if {an} is a cauchy sequence in (a,a,da) and {bn} is a cauchy sequence in (b,a,db). (4) (a × b,a,d) is complete if and only if (a,a,da) and (b,a,db) are complete. proof. (1) assume that a = (a1,b1), b = (a2,b2), c = (c1,c2) ∈ a × b. then, we have d(a,b) = 0a if and only if da(a1,b1) + db(a2,b2) = 0a, that is, da(a1,b1) = db(a2,b2) = 0a, which implies that a1 = b1 and a2 = b2, that is, a = b. now, to show the triangular inequality, we have d(a,b) = da(a1,a2) + db(b1,b2) 4 da(a1,c1) + da(c1,a2) + db(b1,c2) + db(c2,b2) = da(a1,c1) + db(b1,c2) + da(c1,a2) + db(c2,b2) = d(a,c) + d(c,b). therefore, (a×b,a,d) is a c∗-algebra valued quasi metric space. (2) let lim n→∞ (an,bn) = (a,b) in (a×b,a,d) if and only if lim n→∞ d ( (an,bn), (a,b) ) = lim n→∞ [ da(an,a) + db(bn,b) ] = 0a and lim n→∞ d ( (a,b), (an,bn) ) = lim n→∞ [ da(a,an) + db(b,bn) ] = 0a © agt, upv, 2022 appl. gen. topol. 23, no. 2 293 m. asim, s. kumar, m. imdad and r. george which is equivalent to lim n→∞ da(an,a) = lim n→∞ db(bn,b) = lim n→∞ da(a,an) = lim n→∞ db(b,bn) = 0a. therefore, lim n→∞ an = a in (a,a,da) and lim n→∞ bn = b in (b,a,db). therefore, (3) suppose the sequence {(an,bn)} is a cauchy in (a×b,a,d) if and only if lim n,m→∞ d ( (an,bn), (am,bm) ) = lim n,m→∞ [ da(an,am) + db(bn,bm) ] = 0a which is equivalent to lim n,m→∞ da(an,am) = lim n,m→∞ db(bn,bm) = 0a. therefore, {an} is a cauchy sequence in (a,a,da) and {bn} is a cauchy sequence in (b,a,db). (4) it is a direct consequence of (2) and (3). � 3. fixed point results now, we present our main result as follows: theorem 3.1. let (a,a,d) be complete c∗-algebra valued quasi metric space and f : x → x a mapping satisfies the following (for λ ∈a with ‖λ‖ < 1): (3.1) d(fa,fb) 4 λ∗d(a,b)λ, ∀ a,b ∈ a. then f has a unique fixed point. proof. firstly, select a0 ∈ a and extract an iterative sequence {an} as: an = fan−1 = f na0, ∀ n ∈ n. now, we want to show that lim n,m→∞ d(an+1,an) = 0a. by choosing a = an+1 and b = an in 3.1, we wet d(an+1,an) = d(fan,fan−1) = λ ∗d(an,an−1)λ 4 (λ∗)2d(an−1,an−2)λ 2 4 . . . 4 (λ∗)nd(a1,a0)λ n. similarly, we can have d(an,an+1) 4 (λ ∗)nd(a0,a1)λ n. © agt, upv, 2022 appl. gen. topol. 23, no. 2 294 c∗-avqms and fixed point results with an application now, we assert that {an} is cauchy sequence. for any n,m ∈ n such that n + 1 > m, we have d(an+1,am) 4 d(an+1,an) + d(an,an−1) + · · · + d(am+1,am) 4 (λ∗)nd(a1,a0)λ n + · · · + (λ∗)md(a1,a0)λm = n∑ i=m (λ∗)id(a1,a0)λ i = n∑ i=m (λ∗)i(d(a1,a0)) 1 2 (d(a1,a0)) 1 2 λi = n∑ i=m ( (d(a1,a0)) 1 2 λi )∗( d(a1,a0) 1 2 λi ) = n∑ i=m ∣∣∣(d(a1,a0)) 12 λi∣∣∣2 4 ∥∥∥∥∥ n∑ i=m ∣∣∣(d(a1,a0)) 12 λi∣∣∣2 ∥∥∥∥∥i 4 n∑ i=m ∥∥∥(d(a1,a0)) 12 ∥∥∥2 ∥∥λi∥∥2 i 4 ‖d(a1,a0)‖ n∑ i=m ‖λ‖2i i 4 ‖d(a1,a0)‖ ‖λ‖2m 1 −‖λ‖ i → 0a (as m →∞). thus, {an} is left-cauchy sequence, that is lim m→∞ d(an,am) = 0a ∀ n ≥ m ≥ n. similarly, we can show that {an} is right-cauchy sequence, that is (for m > n) lim n→∞ d(am,an) = 0a ∀ m ≥ n ≥ n. therefore, {an} is cauchy sequence. since, (a,a,d) is complete c∗-algebra valued quasi metric space, then there exists a point a in a such that lim n→∞ an = a, that is, lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0a. then, we get d(fa,a) 4 d(fa,an+1) + d(an+1,a) 4 d(fa,fan) + d(an+1,a) 4 λ∗d(a,an)λ + d(an+1,a) 4 ‖λ2‖‖d(a,an)‖i + d(an+1,a). on taking limit as n →∞, we get fa = a. hence, a is fixed point of f. now, to show that the fixed point is unique, we assume that there are two fixed points, say a1,a2 ∈ a such that fa1 = a1 and fa2 = a2. then by using © agt, upv, 2022 appl. gen. topol. 23, no. 2 295 m. asim, s. kumar, m. imdad and r. george 3.1, we have ‖d(a1,a2)‖ = ‖d(fa1,fa2)‖ ≤ ‖λ∗d(a1,a2)λ‖ ≤ ‖λ∗‖‖d(a1,a2)‖‖λ‖ = ‖λ‖2‖(a1,a2)‖ deals a contradiction. hence, a1 = a2, that is, a1 is a unique fixed point of f. this completes the proof. � example 3.2. in the example 2.4, we define a self-mapping f : a → a by: fa = a 5 , ∀ a ∈ a. notice that, d(fa,fb) 4 λ∗d(a,b)λ, (for each a,b ∈ a) satisfies and λ = [√ 5 5 0 0 √ 5 5 ] ∈ a and ‖λ‖ = √ 5 5 = 1 √ 5 < 1. hence, all the assumptions of theorem 3.1 are fulfilled and f unique fixed point, namely a = 0a. before presenting the next theorem we recall the following lemma which is needed is the sequel. lemma 3.3. let a be a unital c∗-algebra with a unit i. we have (1) if a ∈ a+ with ‖a‖ < 12 , then i −a is invertible and ‖a(i −a) −1‖ < 1; (2) if a,b ∈ a+ with ab = ba, then ab ∈a+; (3) we denote a ′ = {a ∈ a : ab = ba, ∀b ∈ a}. let a ∈ a ′ , if b,c ∈ a with b < c < 0a and i − a ∈ a ′ + is an invertible operator, then (i −a)−1b < (i −a)−1c. theorem 3.4. let (a,a,d) be complete c∗-algebra valued quasi metric space and f : x → x a continuous mapping satisfies that the following (for λ ∈ a with ‖λ‖ < 1 2 ): (3.2) d(fa,fb) 4 λ[d(fa,b) + d(a,fb)], ∀ a,b ∈ a. then f has a unique fixed point. proof. firstly, select a0 ∈ a and extract an iterative sequence {an} as: an = fan−1 = f na0, ∀ n ∈ n. now, we want to show that lim n,m→∞ d(an+1,an) = 0a. by choosing a = an+1 and b = an in 3.2, we wet d(an+1,an) = d(fan,fan−1) = λ[d(fan,an−1) + d(an,fan−1)] = λ[d(an+1,an−1) + d(an,an)] 4 λ[d(an+1,an) + d(an,an−1)] = λd(an+1,an) + λd(an,an−1). © agt, upv, 2022 appl. gen. topol. 23, no. 2 296 c∗-avqms and fixed point results with an application thus, (i −λ)d(an+1,an) 4 λd(an,an−1). since, λ ∈a with ‖λ‖ < 1 2 , then we have (i−λ)−1 ∈a and also λ(i−λ)−1 ∈a with ‖λ(i −λ)−1‖ < 1 (by lemma 3.3). then, by assuming u = λ(i −λ)−1, we obtain d(an+1,an) 4 λ(i −λ)−1d(an,an−1) = ud(an,an−1). similarly, we can have d(an,an+1) 4 ud(an−1,an). now, we show that the sequence {an} is cauchy. suppose n+ 1 > m, ∀ n,m ∈ n, so we have d(an+1,am) 4 d(an+1,an) + d(an,an−1) + · · · + d(am+1,am) 4 (un + un−1 + · · · + um)d(a1,a0) = n∑ i=m u i 2 u i 2 (d(a1,a0)) 1 2 (d(a1,a0)) 1 2 = n∑ i=m (d(a1,a0)) 1 2 u i 2 u i 2 (d(a1,a0)) 1 2 λi = n∑ i=m ( u i 2 (d(a1,a0)) 1 2 )∗( u i 2 d(a1,a0) 1 2 ) = n∑ i=m ∣∣∣u i2 (d(a1,a0)) 12 ∣∣∣2 4 ∥∥∥∥∥ n∑ i=m ∣∣∣u i2 (d(a1,a0)) 12 ∣∣∣2 ∥∥∥∥∥i 4 n∑ i=m ∥∥∥(d(a1,a0)) 12 ∥∥∥2 ∥∥∥u i2 ∥∥∥2 i 4 ‖d(a1,a0)‖ n∑ i=m ∥∥∥u i2 ∥∥∥i i 4 ‖d(a1,a0)‖ ‖u‖m 1 −‖u‖ i → 0a (as m →∞). thus, {an} is left-cauchy sequence, that is lim n,m→∞ d(an,am) = 0a. similarly, we can have {an} is right-cauchy sequence, that is (for m > n) lim n,m→∞ d(am,an) = 0a. therefore, the sequence {an} is cauchy. since, (a,a,d) is complete c∗-algebra valued quasi metric space, then there exists a point a in a such that lim n→∞ an = © agt, upv, 2022 appl. gen. topol. 23, no. 2 297 m. asim, s. kumar, m. imdad and r. george a, that is, lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0a. now, by using the continuity of f, we have d(fa,a) 4 d(fa,an+1) + d(an+1,a) 4 d(fa,fan) + d(an+1,a) on taking limit as n →∞, we get fa = a. hence, a is fixed point of f. now, to show that the fixed point is unique, we assume that there are two fixed points, say a1,a2 ∈ a such that fa1 = a1 and fa2 = a2. then by employing 3.4, we have ‖d(a1,a2)‖ = ‖d(fa1,fa2)‖ ≤ ‖λ[d(fa1,a2) + d(a1,fa2)]‖ ≤ ‖λ‖‖d(a1,a2) + d(a1,a2)‖ ≤ ‖λ‖[‖d(a1,a2)‖ + ‖d(a1,a2)‖] = 2‖λ‖‖(a1,a2)‖ a contradiction (since 2‖λ‖ < 1). hence, a1 = a2, that is, a1 is a unique fixed point of f. this completes the proof. � now, we obtain following corollaries: remark 3.5. by taking d(a,b) = d(b,a), for all a,b ∈ a in theorem 3.1, we obtain theorem 2.1 of z. ma et al. [12]. remark 3.6. by taking d(a,b) = d(b,a), for all a,b ∈ a in theorem 3.4, we obtain theorem 2.3 of z. ma et al. [12]. 4. application to find the existence and uniqueness results of a contractive mapping on complete c∗-algebra valued metric space for the integral type equation is carried out by z. ma et al. [12] in 2014 whose lines are as under: example 4.1 ( [12]). consider the integral equation (4.1) a(ξ) = ∫ ∆ g(ξ,ω,a(ω))dω + h(ξ), ∀ ξ,ω ∈ ∆, where ∆ is a lebesgue measurable set. suppose that (1) h is an essentially bounded measurable function defined on ∆ and g : ∆2 ×r → r, (2) there exists a continuous function η : ∆ × ∆ → r and λ ∈ (0, 1) such that | g(ξ,ω,a(ω)) −g(ξ,ω,b(ω)) |≤ λ | η(ξ,ω) | (| a(ω) − b(ω) |) , for all ξ,ω ∈ ∆ and a,b ∈ r. (3) supξ∈∆ ∫ ∆ | η(ξ,ω) | dω ≤ 1. © agt, upv, 2022 appl. gen. topol. 23, no. 2 298 c∗-avqms and fixed point results with an application then the integral equation has a unique solution in a, where a stands for the space of essentially bounded measurable functions defined on ∆. now, we will utilize theorem 3.1 to find the solution of following integral equation: (4.2) a(ξ) = ∫ ∆ g(ξ,ω,a(ω))dω + h(ξ), ∀ ξ,ω ∈ ∆, where, ∆ is a lebesgue measurable set with m(∆) < ∞, g : ∆ × ∆ × r → r and h ∈ a. define d : a×a →a by (for all a,b ∈ a),: d(a,b) = { π|a−b|+|a| if a 6= b 0a if a = b. where l(h) = a, h stand for the set of square integrable functions defined on ∆, and πa : h → h is the multiplicative operator defined by: πa(θ) = a.θ, for all θ ∈ h. now, we present our following theorem: theorem 4.2. assume that (for all a,b ∈ a) (1) ∃ a continuous function η : ∆ × ∆ → r and λ ∈ (0, 1) such that | g(ξ,ω,a(ω)) −g(ξ,ω,b(ω)) |≤ λ | η(ξ,ω) | (| a(ω) − b(ω) | + | a(ω) |) , for all ξ,ω ∈ ∆. (2) supξ∈∆ ∫ ∆ | η(ξ,ω) | dω ≤ 1. then the integral equation (4.2) has a unique solution in a. proof. define f : a → a by: fa(ξ) = ∫ ∆ g(ξ,ω,a(ω))dω + h(ξ), ∀ ξ,ω ∈ ∆. © agt, upv, 2022 appl. gen. topol. 23, no. 2 299 m. asim, s. kumar, m. imdad and r. george set k = λi, then k ∈a and ‖k‖ = λ < 1. for any point u in h, we have ‖d(fa,fb)‖ = sup ‖u‖=1 (π|fa−fb|+|fa|(u),u) = sup ‖u‖=1 ∫ ∆ [∣∣∣∣ ∫ ∆ g(ξ,ω,a(ω)) −g(ξ,ω,b(ω))dω ∣∣∣∣ ] u(ξ) ¯u(ξ)dξ + sup ‖u‖=1 ∫ ∆ (∫ ∆ g(ξ,ω,a(ω)) ) u(ξ) ¯u(ξ)dξ ≤ sup ‖u‖=1 ∫ ∆ [∫ ∆ ∣∣g(ξ,ω,a(ω)) −g(ξ,ω,b(ω))∣∣dω]|u(ξ)|2dξ + sup ‖u‖=1 ∫ ∆ ∣∣∣∣ ∫ ∆ g(ξ,ω,a(ω)) ∣∣∣∣ |u(ξ)|2dξ ≤ sup ‖u‖=1 ∫ ∆ [∫ ∆ ∣∣λη(ξ,ω)(a(ω) − b(ω)+ | a(ω) |)∣∣dω]|u(ξ)|2dξ ≤ sup ‖u‖=1 ∫ ∆ [∫ ∆ ∣∣λ | η(ξ,ω) | (| a(ω) − b(ω) | + | a(ω) |)∣∣dω]|u(ξ)|2dξ ≤ λ sup ‖u‖=1 ∫ ∆ [∫ ∆ |η(ξ,ω)|dω ] |u(ξ)|2dξ‖a− b‖∞ ≤ λ sup ξ∈e ∫ ∆ |η(ξ,ω)|dω sup ‖u‖=1 ∫ ∆ |u(ξ)|2dξ‖a− b‖∞ ≤ λ‖a− b‖∞ = ‖k‖‖d(a,b)‖. since, ‖k‖ < 1, so one can easily seen that the mapping f satisfies all the assumptions of theorem 3.1. hence, (4.2) has a unique solution, means that f has a unique fixed point. � acknowledgements. the authors are thankful to the learned reviewer for his valuable comments. references [1] a. amini-harandi, metric-like spaces, partial metric spaces and fixed points, fixed point theory appl. 2012, article id 204. [2] m. asim and m. imdad, c∗-algebra valued extended b-metric spaces and fixed point results with an application, politehn. univ. bucharest sci. bull. ser. a appl. math. phys. 82, no. 1 (2020), 207–218. [3] m. asim and m. imdad, c∗-algebra valued symmetric spaces and fixed point results with an application, korean j. math. 28, no. 1 (2020), 17–30. © agt, upv, 2022 appl. gen. topol. 23, no. 2 300 c∗-avqms and fixed point results with an application [4] m. asim, m. imdad and s. radenovic, fixed point results in extended rectangular bmetric spaces with an application, politehn. univ. bucharest sci. bull. ser. a appl. math. phys. 81, no. 2 (2019), 43–50. [5] m. asim, a. r. khan and m. imdad, rectangular mb-metric spaces and fixed point results, journal of mathematical analysis 10, no. 1 (2019), 10–18. [6] m. asim, a. r. khan and m. imdad, fixed point results in partial symmetric spaces with an application, axioms 8, no. 1 (2019): 13. [7] i. a. bakhtin, the contraction mapping principle in almost metric spaces, funct. anal., gos. ped. inst. unianowsk 30 (1989), 26–37. [8] s. banach, sur les operations dans les ensembles abstraits et leur application aux equations integrals, fund. math. 3 (1922), 133–181. [9] a. branciari, a fixed point theorem of banachôçôcaccioppoli type on a class of generalized metric spaces, publ. math. 57 (2000), 31–37. [10] s. czerwik, contraction mappings in b-metric spaces, acta math. inform. univ. ostraviensis 1, no. 1 (1993), 5–11. [11] h. long-guang and z. xian, cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 332 (2007), 1468–1476. [12] z. h. ma, l. n. jiang and h. k. sun, c∗-algebra valued metric spaces and related fixed point theorems, fixed point theory appl. 2014, article id 206. [13] z. h. ma and l. n. jiang, c∗-algebra valued b-metric spaces and related fixed point theorems, fixed point theory appl. 2015, article id 222. [14] s. g. matthews, partial metric topology, annals of the new york academy of sciences 728 (1994), 183–197. [15] s. shukla, partial b-metric spaces and fixed point theorems, mediterranean journal of mathematics 11, no. 2 (2014), 703–711. [16] w. a. wilson, on quasi-metric spaces, american journal of mathematics 53, no. 3 (1931), 675–684. © agt, upv, 2022 appl. gen. topol. 23, no. 2 301 protasovaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 151-163 maximal balleans olga i. protasova abstract. a ballean is a set x endowed with some family of subsets of x which are called the balls. we postulate the properties of the family of balls in such a way that the balleans with the appropriate morphisms can be considered as the asymptotic counterparts of the uniform topological spaces. the purpose of the paper is to find and study the asymptotic counterparts for maximal topological spaces and maximal topological groups. 2000 ams classification: 22a05, 54e15, 54d35 keywords: ballean, ball structure, ballean envelope, group ideal 1. ball structures and balleans a ball structure is a triple b = (x,p,b), where x,p are nonempty sets and, for any x ∈ x and α ∈ p , b(x,α) is a subset of x which is called a ball of radius α around x. it is supposed that x ∈ b(x,α) for all x ∈ x, α ∈ p . the set x is called the support of b, p is called the set of radiuses. given any x ∈ x, a ⊆ x, α ∈ p , we put b∗(x,α) = {y ∈ x : x ∈ b(y,α)}, b(a,α) = ⋃ a∈a b(a,α). a ball structure b = (x,p,b) is called • lower symmetric if, for any α,β ∈ p , there exist α′,β′ ∈ p such that, for every x ∈ x, b∗(x,α′) ⊆ b(x,α), b(x,β′) ⊆ b∗(x,β); • upper symmetric if, for any α,β ∈ p , there exist α′,β′ ∈ p such that, for every x ∈ x, b(x,α) ⊆ b∗(x,α′), b∗(x,β) ⊆ b(x,β′); 152 o. i. protasova • lower multiplicative if, for any α, β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x,γ),γ) ⊆ b(x,α) ∩ b(x,β); • upper multiplicative if, for any α,β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x,α),β) ⊆ b(x,γ). let b = (x,p,b) be a lower symmetric, lower multiplicative ball structure. then the family { ⋃ x∈x b(x,α) × b(x,α) : α ∈ p} is a base of entourages for some (uniquely determined) uniformity on x. on the other hand, if u ⊆ x × x is a uniformity on x, then the ball structure (x,u,b) is lower symmetric and lower multiplicative, where b(x,u) = {y ∈ x : (x,y) ∈ u}. thus, the lower symmetric and lower multiplicative ball structures can be identified with the uniform topological spaces. a ball structure is said to be a ballean if it is upper symmetric and upper multiplicative. the balleans arouse independently in asymptotic topology [6], [19] under name of coarse structure and in combinatorics [7]. for good motivation to study the balleans related to metric spaces see the survey [3]. let b1 = (x1,p1,b1) and b2 = (x2,p2,b2) be balleans. a mapping f : x1 → x2 is called a ≺-mapping if, for every α ∈ p1, there exists β ∈ p2 such that, for every x ∈ x1, f(b1(x,α)) ⊆ b2(f(x),β). if f : x1 → x2 is a bijaction such that f and f −1 are the ≺-mappings, we say that the balleans b1 and b2 are asymorphic. if x1 = x2 and the identity mapping id : x1 → x2 is a ≺-mapping, we write b1 ⊆ b2. if b1 ⊆ b2 and b2 ⊆ b1, we write b1 = b2. if b1 ⊆ b2 but b1 6= b2, we write b1 ⊂ b2 and say that b2 is stronger than b1. by the definition, ≺-mappings can be considered as the asymptotic counterparts of the uniformly continuous mappings between the uniform topological spaces. the approach to study balleans with ≺-morphisms is reflected in the papers [11, 12, 13, 14]. to determine the subject of the paper we need some more definitions. let b = (x,p,b) be a ballean. a subset a of x is called bounded if there exist x ∈ x and α ∈ p such that a ⊆ b(x,α). a ballean b is called bounded if its support x is bounded. a ballean b is called connected if, for any x,y ∈ x, there exists α ∈ p such that y ∈ b(x,α). a ballean b is called proper if b is connected and unbounded. a proper ballean b = (x,p,b) is called maximal if every stronger ballean on x is bounded. by zorn lemma, every proper ballean can be strengthened to some maximal ballean. by [12, theorem 5.1], a ballean (x, r+,bd) of maximal balleans 153 unbounded metric space (x,d), where bd(x,r) = {y ∈ x : d(x,y) ≤ r}, is not maximal. for criterion of metrizability of balleans see [11]. 2. criterion of maximality given an arbitrary ball structure b = (x,p,b), we say that the ballean env b is a ballean envelope of b if env b is the smallest ballean on x such that b ⊆ env b. to describe env b more constructively, we consider the free semigroup f(p) in the alphabet p and take f(p) as the set of radiuses of env b. then, for every x ∈ x and every w ∈ f(p), we define the ball env b(x,w) inductively by the length of the word w. if w = α and α ∈ p , we put envb(x,w) = b(x,α) ⋃ b∗(x,α). if w = vα and α ∈ p , we put env b(x,w) = b(env b(x,v),α) ⋃ b∗(env b(x,v),α). note that y ∈ env b(x,w) if and only if x ∈ env b(y,w̃), where w̃ is the word w written in the reverse order, so the ball structure env b = (x,f(p), env b) is upper symmetric. since env b(x,uv) = env b(env b(x,u),v), then env b is upper multiplicative. hence, env b is a ballean. if b′ is a ballean on x such that b ⊆ b′, it follows from the upper symmetry and upper multiplicativity of b′, that env b ⊆ b′. let {bλ = (x,pλ,bλ) : λ ∈ λ} be a family of ball structures with common support x and pairwise disjoint family {pλ : λ ∈ λ} of radiuses. we put p = ⋃ {pλ : λ ∈ λ} and, for any x ∈ x and α ∈ p, α ∈ pλ, denote b(x,α) = bλ(x,α). then b = (x,p,b) is the smallest ball structure on x such that bλ ⊆ b for every λ ∈ λ, and env b is the smallest ballean on x such that bλ ⊆ env b for every λ ∈ λ. we say that a ball structure b = (x,p,b) is a monoball structure if its set of radiuses p is a singleton. theorem 2.1. a proper ballean b = (x,p,b) is maximal if and only if, for every monoball structure b′ on x, either b′ ⊆ b or the ballean envelope of b ⋃ b′ is bounded. proof. suppose that b is maximal and b′ * b. then b ⊂ b ⋃ b′ and the ballean env (b ⋃ b′) is stronger than b. it follows that env (b ⋃ b′) is bounded. assume that b is not maximal and choose a proper ballean b′′ = (x,p ′′,b′′) such that b ⊂ b′. then there exists β ∈ p ′′ such that, for every α ∈ p , there exists x(α) ∈ x such that b(x(α),α) * b′′(x(α),β). for every x ∈ x, we put b′(x,β) = b′′(x,β) and consider the monoball structure b′ = (x,{β},b′). by the choice of β, we have b′ * b. on the other hand, env (b ⋃ b′) ⊆ b′′, therefore env (b ⋃ b′) is unbounded. � example 2.2. let x be an infinite set of regular cardinality k. denote by f the family of all subsets of x of cardinality < k. let p be the set of all 154 o. i. protasova mappings f : x → f such that, for every x ∈ x, we have x ∈ f(x) and |{y ∈ x : x ∈ f(y)}| < k. given any x ∈ x and f ∈ p , we put b(x,f) = f(x) and consider the ball structure b = (x,p,b). since b∗(x,f) = {y ∈ x : x ∈ f(y)}, we have that b is upper symmetric. since k is regular, b is upper multiplicative. hence, b is a ballean. clearly, b is connected and unbounded, so b is proper. using theorem 2.1, we show that b is maximal. let b′ = (x,{1},b′) be a monoball structure on x such that b′ * b. then we have two possibilities. case 1. there exists x ∈ x such that b′(x, 1) = k. put y = x\b′(x, 1) and choose a subset z ⊆ b′(x, 1) such that |z| = |y |. fix an arbitrary bijection h : z → y and, for every x ∈ x, put f(x) = { {x}, if x /∈ z; {x,h(x)}, if x ∈ z. clearly, f ∈ p and b(b′(x, 1),f) = x. it means that env (b ⋃ b′) is bounded. case 2. there exists x ∈ x such that the set y = {y ∈ x : x /∈ b′(y, 1)} is of cardinality k. then y is a bounded subset in env (b ⋃ b′) and, repeating the argument of the case 1, we conclude that env (b ⋃ b′) is bounded. let x be an infinite set, ϕ be a filter on x. for any x ∈ x and f ∈ ϕ, we put bϕ(x,f) = { {x}, if x ∈ f ; x \ f, if x /∈ f ; and consider the ballean b(x,ϕ) = (x,ϕ,bϕ). clearly, b(x,ϕ) is unbounded and b(x,ϕ) is connected if and only if ⋂ ϕ = ∅. a ballean b = (x,p,b) is called pseudodiscrete if , for every α ∈ p , there exists a bounded subset v of x such that b(x,α) = {x} for every x ∈ x \ v . by [13], b is pseudodiscrete if and only if there exists a filter ϕ on x such that b = b(x,ϕ). if ϕ,ψ be filters on x such that ⋂ ϕ = ⋂ ψ = ∅ and ϕ ⊂ ψ, then b(x,ψ) is stronger than b(x,ϕ). hence, if ϕ is not ultrafilter, then b(x,ϕ) is not maximal. example 2.3. let x be an infinite set, ϕ be a free ultrafilter on x. using theorem 2.1, we show that the ballean b(x,ϕ) is maximal. let b′ = (x,{1},b′) be a monoball structure. we put y = {x ∈ x : b′(x, 1) 6= {x}} and consider two cases. case y ∈ ϕ. for very x ∈ y , we take an element f(x) ∈ b′(x, 1) such that x 6= f(x). for every x ∈ x \ y , we put f(x) = x. thus, we get the mapping f : x → x. endow x with the discrete topology and consider the stonečech extension fβ : βx → βx of f. we take the elements of βx to be the ultrafilters on x. since {x ∈ x : f(x) = x} /∈ ϕ, we have fβ (ϕ) 6= ϕ. choose z ∈ ϕ such that z ⊆ y and f(z) /∈ ϕ. then f(z) is bounded in b(x,ϕ) and z is bounded in env (b′ ⋃ b(x,ϕ)). hence, env (b′ ⋃ b(x,ϕ)) is bounded. case y /∈ ϕ. then y is bounded in b(x,ϕ). if b′(y, 1) ∈ ϕ, then env (b′ ⋃ b(x,ϕ)) is bounded. if b′(y, 1) /∈ ϕ, then b′ ⊆ b(x,ϕ). maximal balleans 155 3. subsets of maximal balleans let b = (x,p,b) be a ballean. given any a ⊆ x and α ∈ p , we put int(a,α) = {a ∈ a : b(a,α) ⊆ a}. we use the following classification of subsets of a ballean by their size from [7]. a subset a ⊆ x is called • large if there exists α ∈ p such that x = b(a,α); • small if x \ b(a,α) is large for every α ∈ p ; • piecewise large if there exists β ∈ p such that int(b(a,β),α) 6= ∅ for every α ∈ p ; • extralarge if int(a,α) is large for every α ∈ p . by [7, theorem 11.1], a subset a of x is small if and only if a is not piecewise large. some results from [7, section 11] witness that the large, extralarge and small subsets of a ballean can be considered as the asymptotic duplicates of dense, open dense and nowhere dense subsets of a topological space respectively. theorem 3.1. if a ballean b = (x,p,b) is maximal, then every unbounded subset y of x is large. proof. assume, otherwise, that y is unbounded but y is not large. for every x ∈ x, we put b′(x, 1) = { {x}, if x /∈ y ; y, if x ∈ y ; and consider the monoball structure b′ = (x,{1},b′). since y is unbounded in b, we have b′ ⊆ b. since y is not large, it follows from the above description of the ballean envelope, that env (b ⋃ b′) is unbounded, whence a contradiction to theorem 2.1. � now we give an example of a proper non-maximal ballean in which every unbounded subset is large, so the converse statement to theorem 3.1 is not true. example 3.2. let x be an infinite set, s be a group of all substitution of x, e be the identity substitution. we denote by f(s) the family of all finite subsets of s containing e. given any x ∈ x and f ∈ f(s), we put b(x,f) = {f(x) : f ∈ f} and consider the ballean b = (x,f(s),b). clearly, a subset y of x is bounded if and only if y is finite. by [18, theorem 1.2], a subset y is large if and only if |y | = |x|. now assume that x is countable. then every unbounded subset of x is large. we show that b is not maximal. partition x = ⋃ n∈n xn so that |xn| = n for every n ∈ n. then, for every x ∈ x, we choose n ∈ n such that x ∈ xn and put b ′(x, 1) = xn. in this way we get the monoball structure b′ = (x,{1},b′). since |b(x,f)| ≤ |f | and |xn| = n for every n ∈ ω, we have b′ * b. on the other hand, every ball in env (b ⋃ b′) is finite, so env (b ⋃ b′) is proper. by theorem 2.1, b is not maximal. 156 o. i. protasova let b = (x,p,b) be a proper ballean. a function h : x → [0, 1] is called slowly oscillating if, for every α ∈ p and every ε > 0, there exists a bounded subset v of x such that, for every x ∈ x \ v , diam h(b(x,α)) < ε, where diama = sup {|a − b| : a,b ∈ a}. we denote by x♯ the set of all ultrafilters ϕ on x such that every member of ϕ is unbounded and consider x♯ as a subspace of the stone-čech compactification βx of the discrete space x. given any ϕ,ψ ∈ x♯, we put ϕ ∼ ψ if and only if hβ(ϕ) = hβ(ψ) for every slowly oscillating function h : x → [0, 1]. then ∼ is a closed ( in x♯ × x♯) equivalence on x♯. the factor-space x♯/ ∼ is called the corona of b and is denote by ν(b). for coronas of balleans see [15]. theorem 3.3. let b = (x,p,b) be a proper ballean. if every unbounded subset of x is large, then (i) every small subset of x is bounded; (ii) every piecewise large subset of x is large; (iii) ν(b) is a singleton. proof. (i) assume, otherwise, that some small subset y of x is unbounded. then y is large, but by the definition a small subset can not be large. (ii) let y be a piecewise large subset of x. if y is unbounded, then y is large by the assumption. otherwise, y is bounded, but every bounded subset of a proper ballean is small, so y is not piecewise large. (iii) suppose that |ν(b)| > 1 and choose two distinct elements a,b ∈ ν(b). by the definition of ν(b), there exists a slowly oscillating function h : x → [0, 1] such that hβ(a) = 0, hβ(b) = 1. put a = {x ∈ x : h(x) ∈ [0, 1 3 ]}, b = {x ∈ x : h(x) ∈ [ 2 3 , 1]}. clearly, a,b are unbounded, but b(a,α) \b 6= ∅ for every α ∈ p , so a is not large. � the next three example show that every condition from (i), (ii), (iii) separately does not imply that every unbounded subset of x is large. example 3.4. let x be an infinite set and let ϕ be a filter of all cofinite subsets of x. then a subset y is small in the ballean b(x,ϕ) if and only if y is finite, so every small subset is bounded. on the other hand, a subset y is large if and only if x \ y is finite. thus, if y and x \ y are infinite, then y is unbounded, but y is not large. example 3.5. let x be an uncountable set, s be a group of all permutations of x, b = (x,f(x),b) be a ballean defined in example 3.2. by [18, theorem 1.2], a subset y of x is large if and only if |y | = |x|, a subset y is small if and only if |y | < |x|. hence, every piecewise large subset of x is large. on the other hand, a subset y of x is bounded if and only if y is finite. thus, if y is an infinite subset of x and |y | < |x|, then y is unbounded, but y is not large. maximal balleans 157 example 3.6. let g be an infinite group with the identity e. denote by f the family of all finite subsets of g containing e. given any g ∈ g and f ∈ f, we put bl(g,f) = gf, br(g,f) = fg, and consider the balleans bl(g) = (g,f,bl), br(g) = (g,f,br). clearly, bl(g) and br(g) are proper. now let g be an uncountable abelian group. by [15, theorem 4], ν(bl(g)) is a singleton. a subset y of g is large if and only if y is finite. if a subset y of g is large, then |y | = |g|. hence, if y is infinite and |y | < |g|, then y is unbounded, but y is not large. let b = (x,p,b) be a proper ballean satisfying the both conditions (i) and (ii) of theorem 3.3. then, clearly, every unbounded subset of x is large. it follows from example 3.2 and example 3.4, that the classes of proper balleans satisfying the conditions (i) and (ii) of theorem 3.3 respectively are not incident. 4. maximal balleans on groups let g be a group with the identity e, b = (g,p,b) be a ballean. following [17], we say that b is • left (resp. right) invariant if all the shifts x 7→ gx (resp. x 7→ xg) are ≺-mappings; • uniformly left (resp. right) invariant if, for every α ∈ p , there exists β ∈ p such that gb(x,α) ⊆ b(gx,β) (resp. b(x,α)g ⊆ b(xg,β)) for all x,g ∈ g; • a group ballean if it is uniformly left and right invariant. a family i of subsets of a set x is called an ideal if, for any subsets a,b ∈ i and a′ ⊆ a, we have a ⋃ b ∈ i and a′ ∈ i. a subset i′ ⊆ i is called a base for i if, for every a ∈ i, there exists a′ ∈ i′ such that a ⊆ a′. we say that an ideal i on a group g is a group ideal if, for any subsets a,b ∈ i, we have ab ∈ i and a−1 ∈ i. given a ballean b = (g,p,b), we say that a subset a ⊆ g is bounded from the identity if there exists α ∈ p such that a ⊆ b(e,α). for every uniformly left invariant ballean b = (g,p,b), the family i of all subsets of g bounded from the identity is a group ideal on g. moreover, the identity mapping id : g → g is an asymorphism between b and the ballean (g,i,bl), where bl(g,a) = ga ⋃ {g} for all g ∈ g,a ∈ i. on the other hand, for every group ideal i on g, (g,i,bl) is a uniformly left invariant ballean. thus, we get the natural correspondence between the family of all uniformly left invariant balleans on g and the family of all group ideals on g. following this correspondence, given an arbitrary group ideal i on g, we write (g,i) instead of (g,i,bl). we say that a group ideal is proper if (g,i) is a proper ballean. it is easy to see that a group ideal i is proper if and only if g /∈ i and f(g) ⊆ i, where f(g) is the ideal of all finite subsets of g. 158 o. i. protasova let g be an infinite group of regular cardinality. we put x = g and consider the ballean b on g defined in example 2.2. it is easy to see that b is left (and right) invariant. theorem 4.1. let g be an infinite group and let i be a proper group ideal on g. if the ballean b = (g,i) is maximal, then the subset {g2 : g ∈ g} is bounded in (g,i). proof. for every x ∈ g, we put b′(x, 1) = {x,x−1}. assume that b′ * b, where b′ is the monoball structure (g,{1},b′). by theorem 2.1, env (b ⋃ b′) is bounded. note that every bounded subset of env (b ⋃ b′) is bounded in b, so b is bounded, whence a contradiction. hence, b′ ⊆ b. pick a ∈ i such that {x,x−1} ∈ xa for every x ∈ g, so x−2 ∈ a and {g2 : g ∈ g} is bounded. � it follows from theorem 4.1, that if an abelian group g admits a maximal group ballean, then the factor group g/2g is infinite. in particular, there are no maximal group balleans on z. we show (example 4.3) that, under ch, there are maximal group ballean on the countable abelian group of exponent 2, but we begin with more simple example. example 4.2. let g be a countable abelian group of exponent 2. under ch we construct a proper group ideal j on g such that every unbounded subset of (g,j ) is large. we use the following auxiliary statement. if i is a proper group ideal with countable base on g and a is an unbounded subset of (g,i), then there exists a proper group ideal i′ with countable base on g such that i ⊆ i′ and a is large in (g,i′). let {bn : n ∈ ω} be a countable base for i such that 0 ∈ b0, bn ⊆ bn+1 for every n ∈ ω. let {gn : n ∈ ω} be a numeration of g. to construct i ′, we choose inductively an injective sequence (cn)n∈ω in g such that cn + gn ∈ a for each n ∈ ω, and the group ideal i′ = i ⋃ i(c) is proper, where c = {cn : n ∈ ω}, i(c) is the smallest group ideal containing c, i′ is the smallest group ideal such that i ⊆ i′,i(c) ⊆ i′. we fix an arbitrary sequence (xn)n∈ω in a going to infinity with respect to i (i.e. for every f ∈ i, there exists m ∈ ω such that xn /∈ f for every n ≥ m). clearly, the sequence (xn + gm)n∈ω is going to infinity with respect to i for every m ∈ ω. therefore we can choose inductively a subsequence (dn)n∈ω of (xn)n∈ω and a sequence (yn)n∈ω in g going to infinity with respect to i such that, for each n ∈ ω, we have yn /∈ bn + { ∑ i∈ω mi(di + gi) : mi ∈ {0, 1}, ∑ i∈ω mi ≤ n}. after that we put cn = dn + gn and note that, for every n ∈ ω, yn /∈ bn + c + · · · + c ︸︷︷︸ n , where c = {ci : i ∈ ω}. hence, i ′ = i ⋃ i(c) is a proper group ideal and a is large in (g,i′). maximal balleans 159 to construct the ideal j , we enumerate {aα : α < ω1} the family of all subset of g. fix an arbitrary proper group ideal i0 with countable base on g. if a0 is bounded in (g,i0), we put i ′ 0 = i0. otherwise, we choose a proper group ideal i′0 with countable base such that i0 ⊆ i ′ 0 and a0 is large in (g,i ′ 0). assume that, for some ordinal α < ω1, we have chosen the proper group ideals i′β, β < α with countable bases. put iα = ⋃ β<α i′β and note that iα is a proper group ideal with countable base. if aα is bounded in (g,iα), we put i′α = iα. otherwise, we choose a proper group ideal i ′ α with countable base such that iα ⊆ i ′ α and aα is large in (g,i ′ α). by the construction, the family {i′α : α < ω1} is well-ordered by inclusion. put j = ⋃ α<ω1 i′α. example 4.3. let g be a countable abelian group of exponent 2. under ch, we construct a proper group ideal j on g such that (g,j ) is maximal. we use the following auxiliary statement. let i be a proper group ideal with countable base on g, b′ = (g,{1},b′) be a monoball structure such that b′ * b. then there exists a proper group ideal i′ with countable base such that i ⊆ i′ and env (b′ ⋃ (g,i′)) is bounded. let f ∈ i, xf = ⋃ {b′(x, 1) : b′(x, 1) ⋂ f 6= ∅}. if xf is unbounded in (g,i), we take a proper group ideal i′ with countable base such that i ⊆ i′ and xf is large in (g,i ′) (see example 4.2). since xf is bounded in env(b′ ⋃ (g,i)), we have that env(b′ ⋃ (g,i′)) is bounded. assume that xf is bounded in (g,i) for every f ∈ i. let {bn : n ∈ ω} be a base for i such that 0 ∈ bn, bn ⊂ bn+1, n ∈ ω. then we can choose inductively a sequence (xn)n∈ω in g such that the family {xn + bn : n ∈ ω} is disjoint and (xn + bn) \ b ′(xn, 1) 6= ∅ for every n ∈ ω. for every n ∈ ω, pick yn ∈ (xn + bn) \b ′(xn, 1). since the sequence (yn)n∈ω is going to infinity with respect to i, we can choose inductively, using the arguments from example 4.2, an injective subsequence (ynk )k∈ω of (yn)n∈ω such that the ideal i ′′ with the base {bn + (y + · · · + y ) ︸︷︷︸ n : n ∈ ω} is proper, where y = {ynk : k ∈ ω} and the sequence (xnk )k∈ω is unbounded in (g,i′′). then we choose an ideal i′ with countable base such that i′′ ⊆ i′ and the set {xnk : k ∈ ω} is large in (g,i ′). since y is bounded in (g,i′′), {xnk : k ∈ ω} is bounded in env (b′ ⋃ (g,i′′)). since {xnk : k ∈ ω} is large in i ′ and i′′ ⊆ i′, we conclude that env (b′ ⋃ (g,i′)) is bounded. to construct the ideal j , we use ch to enumerate {bλ : λ < ω1} the set of all monoball structure on g. repeating the arguments from example 4.2, we get j such that, if b′λ * (g,i), then env (b ′ λ ⋃ (g,j )) is bounded. then we apply theorem 2.1. question 4.4. does there exist in zfc (without additional set-theoretic assumption) an infinite group g and a proper group ideal i on g such that every unbounded subset of (g,i) is large? is (g,i) maximal? 160 o. i. protasova question 4.5. assume ch. does every countable abelian group g admit a proper group ideal i such that every unbounded subset of (g,i) is large? to answer question 4.5 in affirmative, it suffices to prove the following statement. let g be a countable abelian group, i be a proper group ideal with countable base on i, a be an unbounded subset of (g,i). then there exists a proper group ideal i′ with countable base on g such that i ⊆ i′ and a is large in (g,i′). we have proved (in example 4.2) that this statement is true if g is a group of exponent 2 and this was crucial for the construction of the ideal j . unfortunately, this statement is not true in general as the following example shows. example 4.6. let g = ⊕ α<ω gα be a direct sum of cyclic groups gα of order 4. let i be the ideal of finite subsets of g. put a = 2g = {2g : g ∈ g} and note that a is unbounded in (g,i). assume that there exists a proper group ideal i′ on g such that a is large in (g,i′). choose f ∈ i′ such that g = f + a. then 2g = 2f so a is bounded subset of (g,i′). since a is large in (g,i′), we conclude that (g,i′) is bounded, whence a contradiction. question 4.7. does there exist in zfc a countable group g and a proper group ideal i on g such that every small subset of (g,i) is bounded? question 4.8. does there exist in zfc a countable group g and a proper group ideal i on g such that every piecewise large subset of (g,i) is large? in view of theorem 3.3 and example 4.2, under ch the answers to the questions 4.7 and 4.8 are positive. question 4.9. let g be an infinite group, i be a proper group ideal on g such that every small subset of (g,i) is bounded. is every piecewise large subset of (g,i) large? question 4.10. let g be an infinite group, i be a proper group ideal on g such that every piecewise large subset of (g,i) is large. is every small subset of (g,i) bounded? to answer question 4.9 positively, it suffices to give the affirmative answer to the following question. question 4.11. let g be an infinite group, i be a proper group ideal on g, a be a piecewise large subset of (g,i). does there exist a small subset s of (g,i) such that g = as? we show that, if g is abelian and i has a countable base, the question 4.11 has the positive answer. we use the following auxiliary statement. let g be an infinite group, i be a proper group ideal on g. let l be a subset of g such that, for every f ∈ i, there exists x ∈ l such that xf ⊆ l. then, for every f ∈ i, there exists x ∈ l such that xf ⊆ l and xf ⋂ f = ∅. to prove this statement, we may suppose that e ∈ f, f = f−1. choose h ∈ i such that f 4 ⊂ h and pick h ∈ h \ f 4. by assumption, there exists maximal balleans 161 x ∈ l such that xhf ⊆ l. assume that xf ⋂ f 6= ∅ and xhf ⋂ f 6= ∅. then x ∈ f 2, h ∈ x−1f 2, so h ∈ f 4. hence, either xf ⋂ f = ∅ or xhf ⋂ f = ∅. now assume that g is abelian and i has a countable base. let a be a piecewise large subset of (g,i). let {bn : n ∈ ω} be a base for i such that e ∈ b0, bn ⊂ bn+1 for each n ∈ ω. since a is piecewise large, there exists c ∈ i such that, for every f ∈ i, there exists x ∈ ac such that xf ⊆ ac. using the auxiliary statement with l = ac, we can choose inductively a sequence (xn)n∈ω in ac such that xnbn ⊆ ac and the family {xnbn : n ∈ ω} is disjoint. put x = {xn : n ∈ ω} and note that xmbn * x for every m ≥ n. it follows that x is small. since xnbn ⊆ ac for every n ∈ ω and ⋃ n∈ω bn = g, we have g = x−1ac. since g is abelian, g = a(x−1c). put s = x−1c. since x−1 is small and c is bounded, s is small. it should be mentioned that our consideration of maximal balleans on group was motivated by maximal topologies on groups. a topological space x without isolated points is called maximal if x has an isolated point in every stronger topology. every infinite group has a plenty of left invariant topologies that are maximal, among them there are even regular topologies [10]. on the other hand [5], every maximal topological group has a countable open abelian subgroup of exponent 2. under ma, the examples of maximal topological groups were constructed in [5], but the existence of a maximal topological group implies p-point in ω⋆ [8], so maximal topological groups can not be constructed in zfc without additional set-theoretic assumptions. 5. maximality and irresolvability a topological space x without isolated points is called irresolvable if x can not be partitioned into two dense subsets. for resolvability of topological spaces and topological groups see the surveys [1, 2, 9]. by analogy, a proper ballean b = (x,p,b) is called irresolvable if x can not be partitioned into two large subsets. for resolvability of balleans see [16]. let b = (x,p,b) be a proper ballean. we say that an ultrafilter ϕ on x is going to infinity in b if every member of ϕ is unbounded. theorem 5.1. let b = (x,p,b) be a proper ballean. then the following statements are equivalent: (i) there exists only one ultrafilter on x going to infinity in b; (ii) there exists an ultrafilter ϕ on x such that b = b(x,ϕ); (iii) b is maximal and irresolvable. proof. (i)⇒(ii). let ϕ be the ultrafilter on x going to infinity in b. for every α ∈ p , we put yα = {x ∈ x : b(x,α) = {x}} and show that yα ∈ ϕ. suppose, otherwise, that x \yα ∈ ϕ. for every x ∈ x \yα, we take an element f(x) ∈ b(x,α) such that f(x) 6= x. for every x ∈ yα, we put f(x) = x. thus, we get the mapping f : x → x. since {x ∈ x : f(x) = x} /∈ ϕ, we have fβ(ϕ) /∈ ϕ, where fβ is the stone-čech extension of f and x is endowed with the discrete topology. clearly, the ultrafilter fβ(ϕ) is going to infinity in b, 162 o. i. protasova a contradiction. hence, the family {yα : α ∈ p} is a base for some filter ψ on x and ψ ⊆ ϕ. if ϕ′ is an ultrafilter on x and ψ ⊆ ϕ′, then ϕ′ is going to infinity in b, so ϕ = ψ. if x \ yα is unbounded in b, then there exists an ultrafilter η on x going to infinity in b such that x \ yα ∈ η. hence, x \ yα is bounded for every α ∈ p . we show that b = b(x,ϕ). if α ∈ p and x ∈ yα, then bϕ(x,yα) = b(x,α). if α ∈ x \ yα, then we choose β ∈ p such that x \ yα ⊆ b(x,β) for every x ∈ x \ yα, so bϕ(x,yα) ⊆ b(x,β). hence, b(x,ϕ) ⊆ b. clearly, b ⊆ b(x,ϕ) and we have b = b(x,ϕ). (ii)⇒(iii). by [16, proposition 1], b(x,ϕ) is irresolvable, maximality of b(x,ϕ) follows from example 2.3. (iii)⇒(i). for every α ∈ p , we put yα = {x ∈ x : b(x,α) = {x}}. since b is irresolvable, the family {yα : α ∈ p} is a base for some filter ϕ on x. assume that there exists an unbounded subset z of x such that yα * z for every α ∈ p . we put b′(x, 1) = { {x}, if x /∈ z; z, if x ∈ z; and consider the monoball structure b′ = (x,{1},p ′). clearly, b′ * b. on the other hand, b(z,α) 6= x for every α ∈ p , so env (b′ ⋃ b) is unbounded. by theorem 2.1, b is not maximal. hence, ϕ is an ultrafilter and the only ultrafilter on x going to infinity in b. � let x be an infinite set, ϕ be a filter on x such that ⋂ ϕ = ∅. then the proper ballean b(x,ϕ) is irresolvable. if ϕ is not ultrafilter, then b(x,ϕ) is not maximal. on the other hand, the maximal ballean b from example 2.2 is resolvable by proposition 1 from [16]. thus, in contrast to topological situation, the classes of irresolvable and maximal balleans are not incident. let g be an infinite group and let i be a proper group ideal on g. pick f ∈ i such that e ∈ f and |f | > 1. then |gf | > 1 for every g ∈ g and, by proposition 1 from [16], the ballean (g,i) is resolvable. thus, every proper uniformly left invariant ballean on a group is resolvable. a filter ϕ on a group g is called left invariant if gf ∈ ϕ for any f ∈ ϕ and g ∈ g. if ϕ is a left invariant filter on g and ⋂ ϕ = ∅, then the ballean b(g,ϕ) is irresolvable and left invariant. by [4, theorem 6.42], for every infinite group g, there exist 22 |g| left invariant filters, so g admits a plenty of irresolvable left invariant balleans. let b = (g,p,b) be an irresolvable left invariant ballean. for every α ∈ p , we put yα = {g ∈ g : b(g,α) = {g}} and note that the family {yα : α ∈ p} is a base of some left invariant filter ϕ on g. clearly, b ⊆ b(g,ϕ), but the suspicion that always b = b(g,ϕ) does not hold. example 5.2. let g = z. for any z ∈ z and (m,n) ∈ n × n, we put b(z, (m,n)) = { {z}, if z ≥ n; {y ∈ z : |y − z| ≤ m}, if z < n; maximal balleans 163 and consider the ballean b = (z, n × n,b). clearly, b is irresolvable and left invariant. in this case, the filter ϕ has the base {fn : n ∈ ω}, where fn = {z ∈ z : z ≥ n}, but b(g,ϕ) is stronger than b. references [1] w. w. comfort, o. masaveau and h. zhou, resolvability in topology and in topological groups, annals of new york acad. of sciences, 767 (1995), 17-27. [2] w. w. comfort and s. garćıa-ferreira, resolvability: a selective survey and some new results, topology appl. 74 (1996), 149-167. [3] a. dranishnikov, asymptotic topology, russian math. survey 55 (2000), 71-116. [4] n. hidman and d. strauss, algebra in the stone-cech compactification, de grueter exposition in math., v.27, 1998. [5] v. i. malykhin, extremally disconnected and similar groups, soviet math. dokl. 16 (1975), 21-25. [6] p. d. mitchener, coarse homology theories, algebr. geom. topology, 1 (2001), 271294. [7] i. protasov and t. banakh, ball structures and colorings of groups and graphs, mat. stud. monogr. ser., v.11, 2003. [8] i. v. protasov, filters and topologies on groups, mat. stud. 3 (1994), 15-28. [9] i. v. protasov, resolvability of groups, mat. stud. 9 (1998), 130-148. [10] i. v. protasov, maximal topologies on groups, sib. math. j. 39 (1998), 1368-1381. [11] i. v. protasov, metrizable ball structures, algebra and discrete math., 2002, n1, 129141. [12] i. v. protasov, uniform ball structures, algebra and discrete math, 2003, n1, 93-102. [13] i. v. protasov, normal ball structures, mat. stud. 20 (2003), 3-16. [14] i. v. protasov, morphisms of ball structures of groups and graphs, ukr. math. j. 54 (2002), 847-855. [15] i. v. protasov, coronas of balleans, topology appl.149(2005), 149-160. [16] i. v. protasov, resolvability of ball structures, appl. gen. topology 5 (2004), 191-198. [17] i. v. protasov and o. i. protasova, sketch of group balleans, mat. stud. 22 (2004), 10-20. [18] o. i. protasova, ball structures of g-spaces, bulletin of the university of kiev, series: physics and math., 2004, 3, 54-69. [19] j. roe, lectures in coarse geometry, ams university lecture series, 31 (2004). received october 2004 accepted june 2005 o. i. protasova (polla@unicyb.kiev.ua) department of cybernetics, kyiv national university, volodimirska 64, kiev 01033, ukraine. mynardagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 171-185 relations that preserve compact filters frédéric mynard abstract. many classes of maps are characterized as (possibly multi-valued) maps preserving particular types of compact filters. 1. introduction a filter f on x is compact at a ⊂ x if every finer ultrafilter has a limit point in a. as a common generalization of compactness (in the case of a principal filter) and of convergence, it is not surprising that the notion turned out to be very useful in a variety of context (see for instance [5], [6], [2] under the name of compactoid filter, [16], [17], [18] under the name of total filter). the purpose of this paper is to build on the results of [5] and [6] to show that a large number of classes of single and multi-valued maps classically used in topology, analysis and optimization are instances of compact relation, that is, relation that preserves compactness of filters. it is well known (see for instance [5], [2]) that upper semi-continuous multivalued maps and compact valued upper semi-continuous maps are such instances. s. dolecki showed [6] that closed, countably perfect, inversely lindelöf and perfect maps are other examples of compact relations. in this paper, it is shown that continuous maps as well as various types of quotient maps (hereditarily quotient, countably biquotient, biquotient) are also compact relations. moreover, i show that maps among these variants of quotient and of perfect maps with ranges satisfying certain local topological properties (such as fréchetness, strong fréchetness and bisequentiality) can be directly characterized in similar terms. this requires to work in the category of convergence spaces rather than in the category of topological spaces. therefore, i recall basic facts on convergence spaces in the next section. the companion paper [15], which should be seen as a sequel to the present paper, uses these characterizations to present applications of product theorems for compact filters to theorems of stability under product of variants of compactness, of local topological properties (fréchetness and its variants, among others) and of the classes of maps discussed above. 172 f. mynard 2. terminology and basic facts 2.1. convergence spaces. by a convergence space (x, ξ) i mean a set x endowed with a relation ξ between points of x and filters on x, denoted x ∈ limξ f or f → ξ x, whenever x and f are in relation, and satisfying lim f ⊂ lim g whenever f ≤ g; {x}↑ → x (1) for every x ∈ x and lim (f ∧ g) = lim f ∩lim g for every filters f and g (2). a map f : (x, ξ) → (y, τ ) is continuous if f (limξ f) ⊂ limτ f (f). if ξ and τ are two convergences on x, we say that ξ is finer than τ, in symbols ξ ≥ τ, if idx : (x, ξ) → (x, τ ) is continuous. the category conv of convergence spaces and continuous maps is topological (3) and cartesian-closed (4). two families a and b of subsets of x mesh, in symbols a#b, if a ∩ b 6= ∅ whenever a ∈ a and b ∈ b. a subset a of x is ξ-closed if limξ f ⊂ a whenever f#a. the family of ξ-closed sets defines a topology t ξ on x called topological modification of ξ. the neighborhood filter of x ∈ x for this topology is denoted nξ(x) and the closure operator for this topology is denoted clξ . a convergence is a topology if x ∈ limξ nξ(x). by definition, the adherence of a filter (in a convergence space) is: (2.1) adhξ f = ⋃ g#f limξ g. in particular, the adherence of a subset a of x is the adherence of its principal filter {a}↑. the vicinity filter vξ(x) of x for ξ is the infimum of the filters converging to x for ξ. a convergence ξ is a pretopology if x ∈ limξ vξ(x). a convergence ξ is respectively a topology, a pretopology, a paratopology, a pseudotopology if x ∈ limξ f whenever x ∈ adhξ d, for every d-filter d#f where d is respectively, the class cl ♮ ξ (f1) of principal filters of ξ-closed sets ( 5), the class f1 of principal filters, the class fω of countably based filters, the class f of all filters. in other words, the map adhd [4] defined by (2.2) limadhd ξ f = ⋂ d�d#f adhξ d 1if a ⊂ 2x , a↑ = {b ⊂ x : ∃a ∈ a, a ⊂ b}. 2several different variants of these axioms have been used by various authors under the name convergence space. 3in other words, for every sink (fi : (xi, ξi) → x)i∈i , there exists a final convergence structure on x : the finest convergence on x making each fi continuous. equivalently, for every source (fi : x → (yi, τi))i∈i there exists an initial convergence: the coarsest convergence on x making each fi continuous. 4in other words, for any pair (x, ξ), (y, τ ) of convergence spaces, there exists the coarsest convergence [ξ, τ ] -called continuous convergenceon the set c(ξ, τ ) of continuous functions from x to y making the evaluation map ev : (x, ξ) × (c(ξ, τ ), [ξ, τ ]) → (y, τ ) (jointly) continuous. 5more generally, if o : 2x −→ 2x and f ⊂ 2x then o♮f denotes {o(f ) : f ∈ f} and if d is a class of filters (or of family of subsets) then o♮ (d) denotes {f : ∃d ∈ d, f = o♮d}. relations that preserve compact filters 173 is the (restriction to objects of the) reflector from conv onto the full subcategory of respectively topological, pretopological, paratopological and pseudotopological spaces when d is respectively, the class cl ♮ ξ (f1), f1, fω and f. a convergence space is first-countable if whenever x ∈ lim f, there exists a countably based filter h ≤ f such that x ∈ lim h. of course, a topological space is first-countable in the usual sense if and only if it is first-countable as a convergence space. analogously, a convergence space is called sequentially based if whenever x ∈ lim f, there exists a sequence (xn)n∈ω ≤ f ( 6) such that x ∈ lim(xn)n∈ω. a class of filters d (under mild conditions on d) defines a reflective subcategory of conv (and the associated reflector) via (2.2). dually, it also defines (under mild conditions on d) the coreflective subcategory of conv of d-based convergence spaces [4], and the associated (restriction to objects of the) coreflector based is (2.3) limbased ξ f = ⋃ d�d≤f limξ d. for instance, if d = fω is the class of countably based filter, then based is the coreflector on first-countable convergence spaces. if d is the class e of filters generated by sequences, then based is the coreflector on sequentially based convergences. 2.2. local properties and special classes of filters. recall that a topological space is fréchet (respectively, strongly fréchet) if whenever x is in the closure of a subset a (respectively, x is in the intersection of closures of elements of a decreasing sequence (an)n of subsets of x) there exists a sequence (xn)n∈ω of elements of a (respectively, such that xn ∈ an) such that x ∈ lim(xn)n∈ω. in other words, if x is in the adherence of a principal (resp. countably based) filter, then there exists a sequence meshing with that filter that converges to x. these are special cases of the following general notion, defined for convergence spaces. let d and j be two classes of filters. a convergence space (x, ξ) is called (j/d)-accessible if adhξ j ⊂ adhbased ξ j , for every j ∈ j. when d = fω and j is respectively the class f, fω and f1, then (j/d)-accessible topological spaces are respectively bisequential, strongly fréchet and fréchet spaces. analogously, if d is the class of filters generated by long sequences (of arbitrary length) and j = f1 then (j/d)-accessible topological spaces are radial spaces. we use the same names for these instances of (j/d)-accessible convergence spaces (see [4] for details). 6from the viewpoint of convergence, there is no reason to distinguish between a sequence and the filter generated by the family of its tails. therefore, in this paper, sequences are identified to their associated filter and i will freely treat sequences as filters. hence the notation (xn)n∈ω ≤ f. 174 f. mynard a filter f is called j to d meshable-refinable, in symbol f ∈ (j/d)#≥, if j ∈ j, j #f =⇒ ∃d ∈ d, d#j and d ≥ f. it follows immediately from the definitions that a topological space is (j/d)accessible if and only if every neighborhood filter is j to d meshable-refinable, and more generally that: theorem 2.1. let d and j be two classes of filters. (1) a convergence space (x, ξ) is (j/d)-accessible if and only if ξ ≥ adhj based ξ; (2) if ξ = base(j/d)#≥ ξ, then ξ is (j/d)-accessible. if moreover ξ is pretopological (in particular topological) then the converse is true. the following gathers the most common cases of (j/d)-accessible (topological) spaces and (j/d)#≥-filters when d = fω. denote by f∧ω the class of countably deep (7) filters. the names for (j/fω)#≥-filters come from the fact that a topological space is (j/fω)-accessible if and only if every neighborhood filter is a (j/fω)#≥-filter. class j (j/fω)-accessible space (j/fω)#≥-filter f bisequential [14] bisequential fω strongly fréchet or countably bisequential [14] strongly fréchet (fω/fω)#≥ productively fréchet[12] productively fréchet f∧ω weakly bisequential [1] weakly bisequential f1 fréchet [14] fréchet table 1 2.3. compactness. let d be a class of filters on a convergence space (x, ξ) and let a be a family of subsets of x. a filter f is d-compact at a (for ξ) [7] if (8) (2.4) d ∈d, d#f =⇒ adhξ d#a. notice that a subset k of a convergence space x (in particular of a topological space) is respectively compact, countably compact, lindelöf if {k}↑ is d-compact at {k} if d is respectively, the class f of all, fω of countably based, f∧ω of countably deep filters. on the other hand theorem 2.2. let d be a class of filters. a filter f is d-compact at {x} for ξ if and only if x ∈ limadhd ξ f. in particular, if ξ is a topology, then x ∈ lim f if and only if f is compact at {x} if and only if f is f1-compact at {x}. for a topological space x, a subset k is compact if and only if every open cover of k has a finite subcover of k, if and only if every filter on k has 7a filter f is countably deep if ⋂ a ∈ f whenever a is a countable subfamily of f. 8notice that (2.4) makes sense not only for a filter but for a general family f of subsets of x. such general compact families play an important role for instance in [8]. relations that preserve compact filters 175 adherent points in k. in contrast, for general convergence spaces, the definition of compactness in terms of covers (cover-compactness) and in terms of filters (compactness) are different. if (x, ξ) is a convergence space, a family s ⊂ 2x is a cover of k ⊂ x if every filter converging to a point of k contains an element of s. hence a subset k of a convergence space is called cover-(countably ) compact if every (countable) cover of k has a finite subcover. it is easy to see that a cover-compact convergence is compact, but in general not conversely. for instance, in a pseudotopological but not pretopological convergence, points are compact, but not cover-compact. notice that in this definition, we can assume the original cover s to be stable under finite union, in which case we call s an additive cover. the family sc of complements of elements of an additive cover s is a filter-base on x with empty adherence. hence k is cover-(countably) compact if every (countable) additive cover of k has an element that is a cover of k, or equivalently, if every (countable) filter-base with no adherence point in k has an element with no adherence point in k. in other words, k is cover-(countably) compact if every (countably based) filter whose every member has adherent points in k, has adherent points in k. more generally, we will need the following characterization of cover-compactness in terms of filters [6]. let d and j be two classes of filters. a filter f is (d/j)compact at b if d ∈ d, ∀j ∈ j, j ≤ d, adh j #f =⇒ adh d#b. it is clear than if f is (d/f1)-compact (at b), then it is d-compact (at b). more precisely, we have the following relationship between (d/f1)-compactness and d-compactness (which could be deduced from the results of [6, section 8]) proposition 2.3. let d be a class of filters on a convergence space (x, ξ). a filter f is (d/f1)-compact at b if and only if vξ(f) = ⋃ f ∈f ⋂ x∈f vξ(x) is d-compact at b. proof. by definition, f is (d/f1)-compact at b if and only if d ∈ d, ( adh ♮ ξ d ) #f =⇒ adh d#b. it is easy to verify that ( adh ♮ ξ d ) #f if and only if d#vξ(f), which concludes the proof. � calling a convergence ξ pretopologically diagonal, or p -diagonal, if limξ f ⊂ limξ vξ(f) for every filter f, we obtain the following result, which is a particular case of a combination of propositions 8.1 and 8.3 and of theorem 8.2 in [6], even though the assumption that adh ♮ ξ d ⊂ d seems to be erroneously missing in [6]. corollary 2.4. if ξ is p -diagonal (in particular if ξ is a topology) and if adh ♮ ξ d ⊂ d, then (d/f1)-compactness amounts to d-compactness for ξ. 176 f. mynard proof. assume that f is d-compact (at b). to show that it is (d/f1)-compact (at b), we only need to show that vξ(f) is d-compact (at b). but d#vξ(f) if and only if ( adh ♮ ξ d ) #f. therefore, adhξ(adh ♮ ξ d)#b because adh ♮ ξ d ∈ d. now, x ∈ adhξ(adh ♮ ξ d) if there exists a filter g# adh ♮ ξ d with x ∈ limξ g. note that vξ(g)#d and that, by p -diagonality, x ∈ limξ vξ(g). hence adhξ(adh ♮ ξ d) ⊂ adhξ d and adhξ d#b. � in some sense, the converse is true: proposition 2.5. if ξ = adhd ξ and d-compactness implies (d/f1)-compactness in ξ, then ξ is p -diagonal. proof. if x ∈ limξ f then f is d-compact at {x}, hence (d/f1)-compact at {x}. since ξ = adhd ξ, we only need to show that x ∈ adhξ d whenever d is a d-filter meshing with vξ(f). for any such d, we have adh ♮ ξ d#f so that x ∈ adhξ d because f is (d/f1)-compact at {x}. � 2.4. contour filters. if f is a filter on x and g : x → fx then the contour of g along f is the filter on x defined by ∫ f g = ∨ f ∈f ∧ x∈f g(x). this type of filters have been used in many situations, among others by froĺık under the name of sum of filters for a zfc proof of the non-homogeneity of the remainder of βn [11], by c. h. cook and h. r. fisher [3] under the name of compression operator of f relative to g, by h. j. kowalsky [13] under the name of diagonal filter, and after them by many other authors to characterize topologicity and regularity of convergence spaces. to generalize this construction, i need to reproduce basic facts on cascades and multifilters. detailed information on this topic can be found in [9]. if (w, ⊑) is an ordered set, then we write w (w) = {x ∈ w : w ⊑ x}. an ordered set (w, ⊑) is well-capped if its every non empty subset has a maximal point (9). each well-capped set admits the (upper) rank to the effect that r(w) = 0 if w ∈ max w , and for r(w) > 0, r(w) = rw (w) = sup v=w (r(v) + 1). a well-capped tree with least element is called a cascade; the least element of a cascade v is denoted by ∅ = ∅v and is called the estuary of v . the rank of a cascade is by definition the rank of its estuary. a cascade is a filter cascade if its every (non maximal) element is a filter on the set of its immediate successors. 9in other words, a well-capped ordered set is a well-founded ordered set for the inverse order. relations that preserve compact filters 177 a map φ : v \ {∅v } → x, where v is a cascade, is called a multifilter on x. we talk about a multifilter φ : v → x under the understanding that φ is not defined at ∅v . a couple (v, φ0) where v is a cascade and φ0 : max v → a is a called a perifilter on a. in the sequel we will consider v implicitly talking about a perifilter φ0. if φ|max v = φ0, then we say that the multifilter φ is an extension of the perifilter φ0. the rank of a multifilter (perifilter) is, by definition, the rank of the corresponding cascade. if d is a class of filters, we call d-multifilter a multifilter with a cascade of d-filters as domain. the contour of a multifilter φ : v → x depends entirely on the underlying cascade v and on the restriction of φ to max v , hence on the corresponding perifilter (v, φ|max v ). therefore we shall not distinguish between the contours of multifilters and of the corresponding perifilters. the contour of φ : w → x is defined by induction to the effect that ∫ φ = φ♮(∅w ) if r(φ) = 1, and ( 10) ∫ φ = ∫ ∅w ( ∫ φ|w (.) ) otherwise. with each class d of filters we associate the class ∫ d of all d-contour filters, i. e., the contours of d-multifilter. if d and j are two classes of filters, we say that j is d-composable if for every x and y, the (possibly degenerate) filter hf = {hf : h ∈ h, f ∈ f}↑ (11) belongs to j(y ) whenever f ∈ j(x) and h ∈ d(x × y ), with the convention that every class of filters contains the degenerate filter. if a class d is dcomposable, we simply say that d is composable. notice that (2.5) h# (f × g) ⇐⇒ hf#g ⇐⇒ h−g#f, where h−g={h−g = {x ∈ x : (x, y) ∈ h and y ∈ g} : h ∈ h, g ∈ g}↑. lemma 2.6. let d and j be two classes of filters. if d is a j-composable class of filters, then ∫ d is also j-composable. proof. we proceed by induction on the rank of a d-multifilter. the case of rank 1 is simply j-composability of d. assume that for each d-multifilter φ on x of rank β smaller than α and each j-filter j on x × y, the filter j ( ∫ φ) is the contour of some d-multifilter on y. consider now a d-multifilter (φ, v ) on x of rank α and a j-filter j on x × y. then ∫ φ = ∫ ∅v ( ∫ φ|v (.) ) = ∨ f ∈∅v ∧ v∈f ∫ φ|v (v), and j ( ∫ φ ) = ∨ f ∈∅v ∧ v∈f j ( ∫ φ|v (v) ) . 10φ(v) is the image by φ of v treated as a point of v , while φ♮(v) is the filter generated by {φ(f ) : f ∈ v}. 11hf = {y ∈ y : (x, y) ∈ h and x ∈ f }. 178 f. mynard as each φ|v (v) is a multifilter of rank smaller than α, each j (∫ φ|v (v) ) is a (∫ d ) -filter. moreover ∅v is a d-filter, so that j (∫ φ ) is a contour of (∫ d ) filters along a d-filter, hence a (∫ d ) -filter. � 3. compact relations a relation r : (x, ξ) ⇉ (y, τ ) is d-compact if for every subset a of x and every filter f that is d-compact at a, the filter rf is d-compact at ra. proposition 3.1. if d is f1-composable, then r : (x, ξ) ⇉ (y, τ ) is d-compact if and only if rf is d-compact at rx whenever x ∈ limξ f. proof. only the ”if” part needs a proof, so assume that rf is d-compact at rx whenever x ∈ limξ f, and consider a filter g on x which is d-compact at a. let d#rg be a d-filter on y . then r−d#g so that there exists x ∈ a∩adhξ r −d. therefore, there exists u#r−d such that x ∈ limξ u. by assumption, ru is d-compact at rx ⊂ ra. since d#ru, the filter d has adherent points in rx hence in ra. � corollary 3.2. let d be an f1-composable class of filters and let f : (x, ξ) → (y, τ ) with τ = adhd τ . the following are equivalent: (1) f is continuous; (2) f is a compact relation; (3) f is a d-compact relation. proof. (1 =⇒ 2). if x ∈ limξ f, then f (x) ∈ limτ f (f) so that f (f) is compact at f (x) and f is a compact relation by proposition 3.1. (2 =⇒ 3) is obvious and (3 =⇒ 1) follows from proposition 2.2. � in particular, f1-compact (equivalently compact) maps between pretopological spaces (in particular between topological spaces) are exactly the continuous ones. notice that when d contains the class of principal filters, then a d-compact relation r is f1-compact and rx is d-compact for each x in the domain of r, because {x}↑ is d-compact at {x}. when the cover and filter versions of compactness coincide (in particular, in a topological space), the converse is true: proposition 3.3. let d be an f1-composable class of filters. if r : (x, ξ) ⇉ (y, τ ) is an f1-compact relation and if rx is (d/f1)-compact in τ for every x ∈ x, then r is d-compact. proof. using proposition 3.1, we need to show that rf is d-compact at rx whenever x ∈ limξ f. consider a d-filter d#rf. then, adhτ d#rx for every d ∈ d so that adhτ d#rx, because rx is d f1 -compact. � in view of corollary 2.4, we obtain: relations that preserve compact filters 179 corollary 3.4. let d be an f1-composable class of filters such that adh ♮ τ d(τ ) ⊂ d(τ ) and let τ be a p -diagonal convergence (for instance a topology). then r : (x, ξ) ⇉ (y, τ ) is d-compact if and only if it is f1-compact and rx is dcompact in τ for every x ∈ x. an immediate corollary of [9, theorem 8.1] is that for a topology, d-compactness amounts to (∫ d ) -compactness, provided that d is a composable class of filters. however, the proof of [9, theorem 8.1] only uses f1-composability of d. consequently, corollary 3.5. let d be an f1-composable class of filters and let τ be a topology such that adh♮τ d(τ ) ⊂ d(τ ). let r : (x, ξ) ⇉ (y, τ ) be a relation. the following are equivalent: (1) r is d-compact; (2) r is f1-compact and rx is d-compact in τ for every x ∈ x; (3) r is f1-compact and rx is (∫ d ) -compact in τ for every x ∈ x; (4) r is (∫ d ) -compact. proof. (1 ⇐⇒ 2) and (3 ⇐⇒ 4) follow from corollary 3.4 and (1 ⇐⇒ 4) follows from [9, theorem 8.1]. � the observation that perfect, countably perfect and closed maps can be characterized as d-compact relations is due to s. dolecki [6, section 10]. recall that a surjection f : x → y between two topological spaces is closed if the image of a closed set is closed and perfect (resp. countably perfect, resp. inversely lindelöf ) if it is closed with compact (resp. countably compact, resp. lindelöf) fibers. once the concept of closed maps is extended to convergence spaces, all the other notions extend as well in the obvious way. as observed in [6, section 10], preservation of closed sets by a map f : (x, ξ) → (y, τ ) is equivalent to f1-compactness of the inverse map f − when (x, ξ) is topological, but not if ξ is a general convergence. more precisely, calling a map f : (x, ξ) → (y, τ ) adherent [6] if y ∈ adhτ f (h) =⇒ adhξ h ∩ f −y 6= ∅, we have: lemma 3.6. (1) a map f : (x, ξ) → (y, τ ) is adherent if and only if f − : (y, τ ) ⇉ (x, ξ) is an f1-compact relation; (2) if f : (x, ξ) → (y, τ ) is adherent, then it is closed; (3) if f : (x, ξ) → (y, τ ) is closed and if adherence of sets are closed in ξ (in particular if ξ is a topology), then f is adherent. proof. (1) follows from the definition and is observed in [6, section 10]. (2) if f (h) is not τ -closed, then there exists y ∈ adhτ f (h)�f (h). since f is adherent, there exists x ∈ adhξ h ∩ f −y. but x /∈ h because f (x) = y /∈ f (h).therefore h is not ξ-closed. (3) is proved in [6, proposition 10.2] even if this proposition is stated with a stronger assumption. � 180 f. mynard hence, a map f : (x, ξ) → (y, τ ) with a domain in which adherence of subsets are closed (in particular, a map with a topological domain) is adherent if and only if it is closed if and only if f − : (y, τ ) ⇉ (x, ξ) is an f1-compact relation. if the domain and range of a map are topological spaces, it is well known that closedness of the map amounts to upper semicontinuity of the inverse relation. it was observed (for instance in [5]) that a (multivalued) map is upper semicontinuous (usc) if and only if it is an f1-compact relation. a surjection f : x → y is d-perfect if it is adherent with d-compact fibers. in view of corollary 3.5, compact valued usc maps between topological spaces, known as usco maps, are compact relations. another direct consequence of lemma 1 and of corollary 3.5 is: theorem 3.7. let f : (x, ξ) → (y, τ ) be a surjection, let d be an f1composable class of filters, and let ξ be a topology such that adh ♮ ξ d ⊂ d. the following are equivalent: (1) f is d-perfect; (2) f − : y ⇉ x is d-compact; (3) f − : y ⇉ x is (∫ d ) -compact; (4) f is (∫ d ) -perfect. the equivalence between the first two points was first observed in [6, proposition 10.2] but erroneously stated for general convergences as domain and range. indeed, if f : (x, ξ) → (y, τ ) is a surjective map between two convergence spaces and if f − : (y, τ ) ⇉ (x, ξ) is d-compact, then f is adherent and has d-compact fibers; if on the other hand f is adherent and has (d/f1)compact fibers then f − : (y, τ ) ⇉ (x, ξ) is d-compact. hence, the two concepts are equivalent only when d-compact sets are (d/f1)-compact in ξ, for instance if adh ♮ ξ (d) ⊂ d and ξ is a p -diagonal convergence (in particular if ξ is a topology). s. dolecki offered in [4] a unified treatment of various classes of quotient maps and preservation theorems under such maps in the general context of convergences. he extended the usual notions of quotient maps to convergence spaces in the following way. a surjection f : (x, ξ) → (y, τ ) is d-quotient if (3.1) y ∈ adhτ h =⇒ f −(y) ∩ adhξ f −h 6= ∅, for every h ∈ d(y ). when d is the class of all (resp. countably based, principal, principal of closed sets) filters, then continuous d-quotient maps between topological spaces are exactly biquotient (resp. countably biquotient, hereditarily quotient, quotient) maps. now, i present a new characterization of d-quotient maps as d-compact relations, in this general context of convergence spaces. as mentioned before, the category of convergence spaces and continuous maps is topological, hence if f : (x, ξ) → y, there exists the finest convergence —called final convergence and denoted f ξ — on y making f continuous. analogously, if f : x → (y, τ ), there exists the coarsest convergence — called initial convergence and denoted f −τ — on x making f continuous. if τ is topological, so is f −τ. in contrast, f ξ can be non topological even when ξ is topological. relations that preserve compact filters 181 theorem 3.8. let d be an f1-composable class of filters. let f : (x, ξ) → (y, τ ) be a surjection. the following are equivalent: (1) f : (x, ξ) → (y, τ ) is d-quotient; (2) τ ≥ adhd f ξ; (3) f : (x, f −τ ) → (y, f ξ) is a d-compact relation. proof. the equivalence (1 ⇐⇒ 2) is [4, theorem 1.2]. (1 ⇐⇒ 3) . assume f is d-quotient and let x ∈ limf −τ f. then f (x) ∈ limτ f (f), so that f (x) ∈ adhτ d whenever d ∈d(y ) and d#f (f). by (3.1) , f −(f (x)) ∩ adhξ f −d 6= ∅ so that f (x) ∈ f (adhξ f −d) . in view of [6, lemma 2.1], f (x) ∈ adhf ξ d. conversely, assume that f : (x, f −τ ) → (y, f ξ) is d-compact and let y ∈ adhτ d. there exists g#d such that y ∈ limτ g. by definition of f −τ, the filter f −g is converging to every point of f −y for f −τ. in view of proposition 3.1, the filter f f −g is d-compact at {y} for f ξ. since f is surjective, f f −g ≈ g and g#d so that y ∈ adhf ξ d = f (adhξ f −d) , by [6, lemma 2.1]. therefore, f −(y) ∩ adhξ f −d 6= ∅. � notice that even if the map has topological range and domain, the notions need to be extended to convergence spaces to obtain such a characterization. 4. compactly meshable filters and relations in view of the characterizations above of various types of maps as d-compact relations, results of stability of d-compactness of filters under product would particularize to product theorems for d-compact spaces, but also for various types of quotient maps, for variants of perfect and closed maps, for usc and usco maps. product theorems for d-compact filters and their applications is the purpose of the companion paper [15]. a (complicated but extremely useful) notion fundamental to this study of products is the following: a filter f is m-compactly j to d meshable at a, or f is an m-compactly (j/d)#-filter at a, if j ∈ j, j #f =⇒ ∃d ∈ d, d#j and d is m-compact at a. while the importance of this concept will be best highlighted by how it is used in the companion paper [15], i show here that the notion of an mcompactly (j/d)#-filter is instrumental in characterizing a large number of classical concepts. the notion of total countable compactness was first introduced by z. froĺık [10] for a study of product of countably compact and pseudocompact spaces and rediscovered under various names by several authors (see [19, p. 212]). a topological space x is totally countably compact if every countably based filter has a finer (equivalently, meshes a) compact countably based filter. the name comes from total nets of pettis. obviously, a topological space is totally countably compact if and only if {x} is compactly fω to fω meshable. in [19], j. vaughan studied more generally under which condition a product of d-compact spaces is d-compact, under mild conditions on the class of filters 182 f. mynard d. he used in particular the concept of a totally d-compact space x, which amounts to {x} being a compactly (d/d)#-filter. on the other hand, theorem 2.1 can be completed by the following immediate rephrasing of the notion of m-compactly (j/d)#-filters relative to a singleton in convergence theoretic terms. proposition 4.1. let d, j and m be three classes of filters, and let ξ and θ be two convergences on x. the following are equivalent: (1) θ ≥ adhj based adhm ξ; (2) f is an m-compactly (j/d)#-filter at {x} in ξ whenever x ∈ limθ f. in particular, ξ = adhm ξ is (j/d)-accessible if and only if f is an m-compactly (j/d)#-filter at {x} whenever x ∈ lim f. in view of table 1, this applies to a variety of classical local topological properties. a relation r : (x, ξ) ⇉ (y, τ ) is m-compactly (j/d)-meshable if f → ξ x =⇒ r(f) is m-compactly (j/d) -meshable at rx in τ. theorem 4.2. let m ⊂ j, let τ = adhm τ and let f : (x, ξ) → (y, τ ) be a continuous surjection. the map f is m-quotient with (j/d)-accessible range if and only if f : (x, f −τ ) → (y, f ξ) is an m-compactly (j/d)-meshable relation. proof. assume that f is m-quotient with (j/d)-accessible range and let x ∈ limf −τ f. then y = f (x) ∈ limτ f (f). let j be a j-filter such that j #f (f). since y ∈ adhτ j and τ is (j/d)-accessible, there exists a d-filter d#j such that y ∈ limτ d. to show that f (f) is m-compactly (j/d)-meshable at y in f ξ, it remains to show that d is m-compact at {y} for f ξ, that is, that y ∈ limadhm f ξ d, which follows from the m-quotientness of f. conversely, assume that f : (x, f −τ ) → (y, f ξ) is an m-compactly (j/d)meshable relation, and let y ∈ limτ g. then f −(g) converges to any point x ∈ f −y for f −τ. therefore, f (f −g) is m-compactly (j/d)-meshable at {y} in f ξ. because f is a surjection, f (f −g) = g. consider m∈ m ⊂ j such that m#g. there exists a d-filter d#m which is m-compact at {y} in f ξ. hence, y ∈ adhf ξ m, so that y ∈ limadhm f ξ g. therefore, f is m-quotient. moreover, if y ∈ adhτ j for a j-filter j , then there exists g#j such that y ∈ limτ g. by the previous argument, g is m-compactly (j/d)-meshable at {y} in f ξ. in particular, there exists a d-filter d#j which is m-compact at {y} in f ξ. in other words, y ∈ limadhm f ξ d. since f : (x, ξ) → (y, τ ) is continuous, τ ≤ f ξ so that adhm τ = τ ≤ adhm f ξ. hence y ∈ limτ d and τ is (j/d)-accessible. � relations that preserve compact filters 183 m j d map f as in theorem 4.2 f1 f f1 hereditarily quotient with finitely generated range f1 f1 fω hereditarily quotient with fréchet range f1 fω fω hereditarily quotient with strongly fréchet range f1 f fω hereditarily quotient with bisequential range f1 f f hereditarily quotient fω fω f1 countably biquotient with finitely generated range fω fω fω countably biquotient with strongly fréchet range fω f fω countably biquotient with bisequential range fω f f countably biquotient f f f1 biquotient with finitely generated range f f fω biquotient with bisequential range f f f biquotient theorem 4.3. let m ⊂ j and d be three classes of filters, where j and d are f1-composable. let τ = adhm τ and let ξ be a p -diagonal convergence such that adh ♮ ξ(m) ⊂ m . let f : (x, ξ) → (y, τ ) be a continuous surjection. the map f is m-perfect with (j/d)-accessible range if and only if f − : (y, τ ) ⇉ (x, ξ) is an m-compactly (j/d)-meshable relation. proof. assume that f is m-perfect with (j/d)-accessible range and let y ∈ limτ g. consider a j-filter j #f −g. by f1-composability, f (j ) is a j-filter. moreover f (j )#g so that y ∈ adhτ f (j ). since τ is (j/d)-accessible, there exists a d-filter d#f (j ) such that y ∈ limτ d. in view of corollary 3.5, f − : (y, τ ) ⇉ (x, ξ) is m-compact because f is m-perfect. therefore, f −d is m -compact at f −y in ξ. moreover, f −d ∈ d because d is f1-composable and f −d#j . hence, f −g is m-compactly (j/d)-meshable at f −y in ξ. conversely, assume that f − : (y, τ ) ⇉ (x, ξ) is an m-compactly (j/d)meshable relation. it is in particular an m-compact relation because m ⊂ j. in view of corollary 3.5, f is m-perfect. now assume that y ∈ adhτ j where j ∈ j. there exists g#j such that y ∈ limτ g. therefore, f −g is m-compactly (j/d)-meshable at f −y in ξ. the filter f −j meshes with f −g because f is surjective, and is a j-filter because j is f1-composable. hence, there exists a d-filter d#f −j which is m-compact at f −y in ξ. by continuity of f , the filter f (d) is m-compact at {y} in τ (corollary 3.2). in view of proposition 2.2, y ∈ limadhm τ f (d). moreover, the filter f (d) meshes with j and is a d-filter, by f1-composability of d. since τ = adhm τ, we conclude that y ∈ limτ f (d) and τ is (j/d)-accessible. � 184 f. mynard m j d map f as in theorem 4.3 f1 f f1 closed with finitely generated range f1 f1 fω closed with fréchet range f1 fω fω closed with strongly fréchet range f1 f fω closed with bisequential range f1 f f closed fω fω f1 countably perfect with finitely generated range fω fω fω countably perfect with strongly fréchet range fω f fω countably perfect with bisequential range fω f f countably perfect f f f1 perfect with finitely generated range f f fω perfect with bisequential range f f f perfect references [1] a. v. arhangel’skii, bisequential spaces, tightness of products, and metrizability conditions in topological groups, trans. moscow math. soc. 55 (1994), 207–219. 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[18] j. e. vaughan, total nets and filters, topology (proc. ninth annual spring topology conf., memphis state univ., memphis, tenn., 1975), lecture notes in pure and appl. math., vol. 24 (new york), dekker, 1976, pp. 259–265. [19] , products of topological spaces, gen. topology appl. 8 (1978), 207–217. relations that preserve compact filters 185 received november 2005 accepted may 2006 frédéric mynard (fmynard@georgiasouthern.edu) dept. mathematical sciences, georgia southern university, po box 8093, statesboro, ga 30460-8093, u.s.a. linagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 103-107 tightness of function spaces shou lin ∗ abstract. the purpose of this paper is to give higher cardinality versions of countable fan tightness of function spaces obtained by a. arhangel’skǐı. let vet(x), ωh(x) and h(x) denote respectively the fan tightness, ω-hurewicz number and hurewicz number of a space x, then vet(c p (x)) = ωh(x) = sup{h(xn) : n ∈ n}. 2000 ams classification: 54c35; 54a25; 54d20; 54d99. keywords: function spaces; fan tightness; hurewicz spaces; cardinal functions. the general question in the theory of function spaces is to characterize topological properties of the space, c(x), of continuous real-valued functions on a topological space x. a study of some convergence properties in function spaces is an important task of general topology. it have been obtained interested results on some higher cardinal properties of first-countability, fréchet properties, tightness[2, 4, 6, 9]. arhangel’skǐı-pytkeev theorem[2] is a nice result about tightness of function spaces: t(cp(x)) = sup{l(x n) : n ∈ n} for any tychonoff space x. the following result on countable fan tightness of function spaces is shown by a. arhangel’skǐı[1]: cp(x) has countable fan tightness if and only if x n is a hurewicz space for each n ∈ n for an arbitrary space x. in this paper the higher cardinality versions of countable fan tightness of cp(x) are obtained. in this paper all spaces will be tychonoff spaces. let α be a network of compact subsets of a space x, which is closed under finite unions and closed subsets. then the space cα(x) is the set c(x) with the set-open topology as follows[9]: the subbasic open sets of the form [a, v ] = {f ∈ c(x) : f (a) ⊂ v }, where a ∈ α and v is open in r. then cα(x) is a topological vector space[9]. the family of all compact subsets of x generates the compact-open topology, ∗supported by the nnsf of china (10271026). 104 s. lin denoted by ck(x). also the family of all finite subsets of x generates the topology of pointwise convergence, denoted by cp(x). for each f ∈ c(x), a basic neighborhood of f in cp(x) can be expressed as w (f, k, ε) for each finite subset k of x and ε > 0, here w (f, k, ε) = {g ∈ c(x) : |f (x) − g(x)| < ε for each x ∈ k}. in this paper the alphabet λ is an infinite cardinal number, γ is an ordinal number, and i, m, n, j, k are natural numbers. the fan tightness of a space x is defined by vet(x) = sup{vet(x, x) : x ∈ x}, here vet(x, x) = ω + min{λ : for each family {aγ}γ<λ of subsets of x with x ∈ ⋂ γ<λ aγ there is a subset bγ ⊂ aγ with |bγ| < λ for each γ < λ such that x ∈ ⋃ γ<λ bγ}. a space x has countable fan tightness[1] if and only if vet(x) = ω. an α-cover of a space x is a family of subsets of x such that every member of α is contained in some member of this family. an α-cover is called a k-cover if α is the set of all compact subsets of x. also an α-cover is called an ω-cover if α is the set of all finite subsets of x. the α-hurewicz number of x is defined by αh(x) = ω + min{λ : for each family {uγ}γ<λ of open α-covers of x there is a subset bγ ⊂ uγ with |bγ| < λ for each γ < λ such that ⋃ γ<λ bγ is an α-cover of x}. the α-hurewicz number of x is called the hurewicz number of x and written h(x) if α consists of the singleton of x. a space x is hurewicz space[5] if and only if h(x) = ω. theorem 1. vet(cα(x)) = αh(x) for any space x. proof. let λ = vet(c α (x)), and let {uγ}γ<λ be any family of open α-covers of x. for each γ < λ, put aγ = {f ∈ cα(x) : there is u ∈ uγ such that f (x \ u ) ⊂ {0}}. then aγ is dense in cα(x). in fact, let ⋂ i≤m[ki, vi] be a non-empty basic open set of cα(x), fix f ∈ ⋂ i≤m[ki, vi]. there is u ∈ uγ such that ⋃ i≤m ki ⊂ u because uγ is an α-cover on x. since ⋃ i≤m ki is compact in tychonoff space x, there is g ∈ cα(x) such that g|∪i≤mki = f|∪i≤mki and g(x \ u ) ⊂ {0}. then g ∈ aγ ∩ ( ⋂ i≤m[ki, vi]), and aγ = cα(x). take f1 ∈ c(x) with f1(x) = {1}, then f1 ∈ ⋂ γ<λ aγ . for each γ < λ there is a subset bγ ⊂ aγ with |bγ| < λ such that f1 ∈ ⋃ γ<λ bγ by λ = vet(c α (x)). denote bγ = {fκ}κ∈φγ , here |φγ| < λ. there is uκ ∈ uγ such that fκ(x \ uκ) ⊂ {0} for each κ ∈ φγ . put u ′ γ = {uκ}κ∈φγ . then ⋃ γ<λ u ′ γ is an α-cover of x. in fact, for each a ∈ α, since f1 ∈ [a, (0, 2)], there are γ < λ and κ ∈ φγ such that fκ ∈ [a, (0, 2)], then a ⊂ uκ, so ⋃ γ<λ u′γ is an α-cover of x. this shows that αh(x) ≤ vet(c α (x)). to show the reverse inequality, let λ = αh(x). since cα(x) is a topological vector space, it is homogeneous. it suffices to show that vet(cα(x), f0) ≤ λ, here f0 ∈ c(x) with f0(x) = {0}. suppose that f0 ∈ ⋂ γ<λ aγ with each aγ ⊂ cα(x). for each γ < λ and n ∈ n, put uγ,n = {f −1(on) : f ∈ aγ}, here {on}n∈n is a decreasing local base of 0 in r. then uγ,n is an open α-cover of x. in fact, for each a ∈ α, f0 ∈ [a, on], there is f ∈ [a, on] ∩ aγ , thus a ⊂ f −1(on) ∈ uγ,n. tightness of function spaces 105 case 1. λ > ω. for each n ∈ n, since {uγ,n}γ<λ is a family of open αcovers of x, there is a subset u′γ,n ⊂ uγ,n with |u ′ γ,n| < λ for each γ < λ such that ⋃ γ<λ u′γ,n is an open α-cover of x. denote u ′ γ,n = {uτ }τ ∈φγ,n . there is fτ ∈ aγ such that uτ = f −1 τ (on) for each τ ∈ φγ,n. let bγ = {fτ : τ ∈ φγ,n, n ∈ n}. then bγ ⊂ aγ and |bγ| < λ. we show that f0 ∈ ⋃ γ<λ bγ . for arbitrary basic neighborhood [a, v ] of f0 in cα(x), there is n ∈ n such that on ⊂ v . since ⋃ γ<λ u′γ,n is an open α-cover of x, there are γ < λ and τ ∈ φγ,n such that a ⊂ uτ = f −1 τ (on), hence fτ (a) ⊂ v , i.e., fτ ∈ [a, v ], so f0 ∈ {fτ : τ ∈ φγ,n, n ∈ n, γ < λ} = ⋃ γ<λ bγ . case 2. λ = ω. put m = {n ∈ n : x ∈ un,n}. if m is infinite, there is m ∈ m such that om ⊂ v for arbitrary basic neighborhood [a, v ] of f0 in cα(x). by the definition of um,m, there is gm ∈ am such that x = g −1 m (om), then gm(x) ⊂ v , so gm ∈ [a, v ], thus the sequence {gm}m∈m converges to f0. if m is finite, there is n0 ∈ n such that for each m ≥ n0 and g ∈ am, g−1(om) 6= x. since {um,m}m≥n0 is a sequence of open α-covers of x, there is a finite subset u′m of um,m for each m ≥ n0 such that ⋃ m≥n0 u′m is an open α-cover of x. denote u′m = {um,j}j≤i(m). there is fm,j ∈ am such that um,j = f −1 m,j (om) for each m ≥ n0, j ≤ i(m). next, we shall show that f0 ∈ {fm,j : m ≥ n0, j ≤ i(m)}. for arbitrary basic neighborhood [a, v ] of f0 in cα(x), let f = {(m, j) ∈ n 2 : m ≥ n0, j ≤ i(m) and a ⊂ um,j}. obviously, f 6= ∅. if f is finite, take xm,j ∈ x \ um,j for each (m, j) ∈ f because um,j 6= x. there is k ∈ α with a ∪ {xm,j : (m, j) ∈ f } ⊂ k. then k is not contained by any element of ⋃ m≥n0 u′m, so ⋃ m≥n0 u′m is not an α-cover of x, a contradiction. hence f is infinite, and there are m ≥ n0 and j ≤ i(m) such that a ⊂ um,j = f −1 m,j(om) and om ⊂ v , so fm,j (a) ⊂ v , i.e., fm,j ∈ [a, v ]. thus f0 ∈ {fm,j : m ≥ n0, j ≤ i(m)}. this shows that vet(cα(x)) ≤ αh(x). � by theorem 1, cp(x) has countable fan tightness if and only if for each sequence {un} of open ω-covers of x there is a finite subset u ′ n ⊂ un for each n ∈ n such that ⋃ n∈n u ′ n is an ω-cover of x. theorem 2. vet(cp(x)) = sup{h(x n) : n ∈ n} for any space x. proof. let λ = vet(c p (x)) and n ∈ n. suppose that {uγ}γ<λ is a family of open covers of the space x n. for each γ < λ, a family v of subsets of x is called having a property pn,γ if for each {vi}i≤n ⊂ v there is u ∈ uγ such that∏ i≤n vi ⊂ u . denote by γn,γ the family of the all finite sets, which has the property pn,γ , of open sets in x. for each v ∈ γn,γ , let fv = {f ∈ cp(x) : f (x \ ⋃ v) ⊂ {0}}. we show that the set aγ = ⋃ {fv : v ∈ γn,γ} is dense in cp(x). let w (f, k, ε) be any basic neighborhood of f in cp(x). since k is finite, there is a finite family w of open subsets in x such that for any (x1, x2, ..., xn) ∈ k n there are u ∈ uγ and a finite subset {wi}i≤n ⊂ w such that (x1, x2, ..., xn) ∈ ∏ i≤n wi ⊂ u . then k ⊂ ⋃ w. for each x ∈ k, 106 s. lin put vx = ⋂ {w ∈ w : x ∈ w }, and v = {vx : x ∈ k}. then k ⊂ ⋃ v and the family v has the property pn,γ . in fact, take an arbitrary (x1, x2, ..., xn) ∈ k n, there are {wi}i≤n ⊂ w and u ∈ u such that (x1, x2, ..., xn) ∈ ∏ i≤n wi ⊂ u . since each vxi ⊂ wi, ∏ i≤n vxi ⊂ u . now, take g ∈ cp(x) such that f|k = g|k and g(x \ ⋃ v) = {0}, then g ∈ fv ⊂ aγ , so w (f, k, ε) ∩ aγ 6= ∅. thus aγ = cp(x). let f1 ∈ c(x) with f1(x) = {1}. then f1 ∈ ⋂ γ<λ aγ . there is a subset bγ ⊂ aγ with |bγ| < λ for each γ < λ such that f1 ∈ ⋃ γ<λ bγ . then there is a subset ∆n,γ ⊂ γn,γ with |∆n,γ| < λ such that bγ ⊂ ⋃ {fv : v ∈ ∆n,γ}. let v ∈ ∆n,γ . for each ξ = (v1, v2, ..., vn) ∈ v n, take gξ ∈ uγ such that∏ i≤n vi ⊂ gξ. put gγ = {gξ : ξ ∈ v n, v ∈ ∆n,γ}. clearly, |gγ| < λ and gγ ⊂ uγ . we show that ⋃ γ<λ gγ covers x. for an arbitrary (x1, x2, ..., xn) ∈ x n, let f = {f ∈ cp(x) : f (xi) > 0 for each i ≤ n}. then f is an open neighborhood of f1 in cp(x). since f1 ∈ ⋃ γ<λ bγ , there is γ < λ such that f ∩ bγ 6= ∅. then f ∩ fv 6= ∅ for some v ∈ ∆n,γ . let g ∈ f ∩ fv . then g(x \ ⋃ v) = 0 and g(xi) > 0 for each i ≤ n. take vi ∈ v such that xi ∈ vi for each i ≤ n, then there is gξ ∈ gγ such that (x1, x2, ..., xn) ∈ ∏ i≤n vi ⊂ gξ. so (x1, x2, ..., xn) ∈ ⋃ ( ⋃ γ<λ gγ ). hence h(x n) ≤ vet(c p (x)). conversely, suppose λ = sup{h(x n) : n ∈ n}. fix f ∈ cp(x) and a family {aγ}γ<λ of subsets in cp(x) such that f ∈ ⋂ γ<λ aγ . for each n ∈ n, γ < λ and x = (x1, x2, ..., xn) ∈ x n, there is gx,γ ∈ w (f, {x1, x2, ..., xn}, 1/n) ⋂ aγ . for each i ≤ n, since |gx,γ (xi) − f (xi)| < 1/n, by the continuity of f and gx,γ, there is an open neighborhood oi of xi in x such that |gx,γ (yi) − f (yi)| < 1/n if yi ∈ oi. the set ux,γ = ∏ i≤n oi is a neighborhood of x in x n. thus un,γ = {ux,γ : x ∈ x n} covers x n, and |gx,γ (yi) − f (yi)| < 1/n for each (y1, y2, ..., yn) ∈ ux,γ . case 1. λ > ω. since h(x n) ≤ λ, there is a family {sn,γ}γ<λ of subsets in x n with |sn,γ| < λ for each γ < λ such that ⋃ γ<λ sn,γ covers x n, here each sn,γ = {ux,γ : x ∈ sn,γ}. for each γ < λ, let bn,γ = {gx,γ : x ∈ sn,γ}, and bγ = ⋃ n∈n bn,γ . then bγ ⊂ aγ , |bγ| < λ, and f ∈ ⋃ γ<λ bγ . in fact, let w (f, {y1, y2, ..., yn}, ε) be a basic neighborhood of f in cp(x) with 1/n < ε. there is γ < λ such that (y1, y2, ..., yn) ∈ ⋃ sn,γ , thus there is x ∈ sn,γ such that (y1, y2, .., yn) ∈ ux,γ, so gx,γ ∈ bn,γ and |gx,γ (yi) − f (yi)| < 1/n < ε for each i ≤ n, hence gx,γ ∈ w (f, {y1, y2, ..., yn}, ε) ∩ bγ . this shows that f ∈ ⋃ γ<λ bγ . case 2. λ = ω. replace γ < λ by k ≥ n, and let bk = ⋃ n≤k bn,k in the proof of case 1, then bk is finite subset of ak and f ∈ ⋃ k∈n bk. in a word, vet(c p (x)) ≤ sup{h(x n) : n ∈ n}. � the following result obtained by a. arhangel’skǐı[1] is generalized: cp(x) has countable fan tightness if and only if x n is a hurewicz space for each n ∈ n. tightness of function spaces 107 references [1] a. arhangel’skǐı, hurewicz spaces, analytic sets, and fan tightness of functions, soviet math. dokl. 33(1986), 396-399. [2] a. arhangel’skǐı, topological function spaces (kluwer academic publishers, dordrecht, 1992). [3] r. engelking, general topology(revised and completed edition)(heldermann verlag, berlin, 1989). [4] j. gerlits, zs. nagy and z. szentmiklossy, some convergence properties in function space, in: general topology and its relations to modern analysis and algebra vi, proc. sixth prague topological symposium(heldermann verlag, berlin, 1988), 211-222. [5] w. hurewicz, über folgen stetiger fukktionen, fund. math. 9(1927), 193-204. [6] lj. d. kočinac, on radiality of function spaces, in: general topology and its relations to modern analysis and algebra vi, proc. sixth prague topological symposium(heldermann verlag, berlin, 1988), 337-344. [7] lj. d. kočinac, closure properties of function spaces, applied general topology 4(2003)(2), 255-261. [8] s. lin, c. liu and h. teng, fan tightness and strong fréchet property of ck(x), adv. math.(china) 23:3(1994), 234-237(in chinese). [9] r. a. mccoy, i. ntantu, topological properties of spaces of continuous functions, lecture notes in math., no. 1315(springer verlag, berlin, 1988). received august 2004 accepted may 2005 shou lin (linshou@public.ndptt.fj.cn) department of mathematics, zhangzhou teachers’ college, fujian 363000, p. r. china department of mathematics, ningde teachers’ college, fujian 352100,p. r. china () @ applied general topology c© universidad politécnica de valencia volume 12, no. 1, 2011 pp. 67-80 some remarks on stronger versions of the boundary problem for banach spaces jan-david hardtke abstract let x be a real banach space. a subset b of the dual unit sphere of x is said to be a boundary for x, if every element of x attains its norm on some functional in b. the well-known boundary problem originally posed by godefroy asks whether a bounded subset of x which is compact in the topology of pointwise convergence on b is already weakly compact. this problem was recently solved by pfitzner in the positive. in this note we collect some stronger versions of the solution to the boundary problem, most of which are restricted to special types of banach spaces. we shall use the results and techniques of pfitzner, cascales et al., moors and others. 2010 msc: 46a50; 46b50 keywords: boundary; weak compactness; convex hull; extreme points; εweakly relatively compact sets; ε-interchangeable double limits 1. introduction first we fix some notation: throughout this paper x denotes a real banach space, x∗ its dual, bx its closed unit ball and sx its unit sphere. for a subset b of x∗ we denote by σb the topology on x of pointwise convergence on b. if a ⊆ x, then co a stands for the convex hull of a and a τ for the closure of a in any topology τ on x, except for the norm closure, which we simply denote by a. also, we denote by ex c the set of extreme points of a convex subset c of x. now recall that a subset b of sx∗ is called a boundary for x, if for every x ∈ x there is some b ∈ b such that b(x) = ‖x‖. it easily follows from the krein-milman theorem that ex bx∗ is always a boundary for x. in 1980 68 j.-d. hardtke bourgain and talagrand proved in [4] that a bounded subset a of x is weakly compact if it is merely compact in the topology σe , where e = ex bx∗ . in [14] godefroy asked whether the same statement holds for an arbitrary boundary b, a question which has become known as the boundary problem. long since only partial positive answers were known, for example if x = c(k) for some compact hausdorff space k (cf. [5, proposition 3]) or x = ℓ1(i) for some set i (cf. [9, theorem 4.9]). in [24, theorem 1.1] the positive answer for l1-preduals is contained. moreover, the answer is positive if the set a is additionally assumed to be convex (cf. [15, p.44]). it was only in 2008 that the positive answer to the boundary problem was found in full generality by pfitzner in [20]. an important tool in the study of the boundary problem is the so called simons’ equality: theorem 1.1 (simons, cf. [23]). if b is a boundary for x, then (1.1) sup x∗∈b lim sup n→∞ x∗(xn) = sup x∗∈bx∗ lim sup n→∞ x∗(xn) holds for every bounded sequence (xn)n∈n in x. in particular, it follows from theorem 1.1 that the well-known rainwater’s theorem for the extreme points of the dual unit ball (cf. [21]) holds true for an arbitrary boundary: corollary 1.2 (simons, cf. [22] or [23]). if b is a boundary for x, then a bounded sequence (xn)n∈n in x is weakly convergent to x ∈ x iff it is σb convergent to x. pfitzner’s proof also uses simons’ equality, as well as a quantitative version of rosenthal’s ℓ1-theorem due to behrends (cf. [3]) and an ingenious variant of hagler-johnson’s construction. next we recall the following known characterization of weak compactness (compare [16, p.145-149], [12, theorem 5.5 and exercise 5.19] as well as the proof of [10, theorem v.6.2]). it is a strengthening of the usual eberleinšmulian theorem. theorem 1.3. let a be a bounded subset of x. then the following assertions are equivalent: (i) a is weakly relatively compact. (ii) for every sequence (xn)n∈n in a we have that ∞ ⋂ k=1 co {xn : n ≥ k} 6= ∅. (iii) for every sequence (xn)n∈n in a there is some x ∈ x such that x∗(x) ≤ lim sup n→∞ x∗(xn) ∀x ∗ ∈ x∗. in [18] moors proved a statement stronger than the equivalence of (i) and (ii), which also sharpens the result from [4]: some remarks on stronger versions of the boundary problem for banach spaces 69 theorem 1.4 (moors, cf. [18]). a bounded subset a of x is weakly relatively compact iff for every sequence (xn)n∈n in a we have that ∞ ⋂ k=1 coσe {xn : n ≥ k} 6= ∅, where e = ex bx∗ . in particular, a is weakly relatively compact if it is merely relatively countably compact in the topology σe . in fact, moors gets this theorem as a corollary to the following one: theorem 1.5 (moors, cf. [18]). let a be an infinite bounded subset of x. then there exists a countably infinite set f ⊆ a with coσe f = cof , where e = ex bx∗ . in particular, for each bounded sequence (xn)n∈n in x there is a subsequence (xnk )k∈n with co σe {xnk : k ∈ n} ⊆ co {xn : n ∈ n}. the object of this paper is to give some results related to theorem 1.4 in the more general context of boundaries. in particular, we shall see, by a very slight modification of the construction from [20], that a ‘non-relative’ version of 1.4 holds for any boundary b of x, see theorem 2.15. since we will also deal with some quantitative versions of theorem 1.4, it is necessary to introduce a bit more of terminology, which stems from [11]: given ε ≥ 0, a bounded subset a of x is said to be ε-weakly relatively compact (in short ε-wrc) provided that dist(x∗∗, x) ≤ ε for every element x∗∗ ∈ a w ∗ , where w∗ refers to the weak*-topology of x∗∗. for ε = 0 this is equivalent to the classical case of weak relative compactness. the authors of [11] used this notion to give a quantitative version of the well known theorem of krein (cf. [11, theorem 2]). in their proof they made use of double limit techniques in the spirit of grothendieck. more precisely, they worked with the following definition: let bounded subsets a of x, m of x∗ and ε ≥ 0 be given. then a is said to have ε-interchangeable double limits with m if for any two sequences (xn)n∈n in a and (x ∗ m)m∈n in m we have ∣ ∣ ∣ lim n→∞ lim m→∞ x∗m(xn) − lim m→∞ lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε, provided that all the limits involved exist. in this case we write a§ε§m . the connection to ε-wrc sets is given by the following proposition: proposition 1.6 (fabian et al., cf. [11]). let a ⊆ x be bounded and ε ≥ 0. then the following hold: (i) if a is ε-wrc, then a§2ε§bx∗. (ii) if a§ε§bx∗, then a is ε-wrc. in case ε = 0 this is the classical grothendieck double limit criterion. for various other quantitative results on weak compactness we refer the interested reader to [2], [6], [7] and [11]. for some related results on weak sequential completeness, see also [17]. 70 j.-d. hardtke we are now ready to formulate and prove our results. however, it should be added that all of them can easily be derived from already known results and techniques. 2. results and proofs we begin with a quantitative version of theorem 1.3. first we prove an easy lemma that generalizes the equivalence of (ii) and (iii) in said theorem (the proof is practically the same). lemma 2.1. let b be a subset of bx∗ that separates the points of x and let (xn)n∈n be a sequence in x as well as x ∈ x and ε ≥ 0. then the following assertions are equivalent: (i) x ∈ ⋂∞ k=1 coσb ({xn : n ≥ k} + εbx ). (ii) x∗(x) ≤ lim supn→∞ x ∗(xn) + ε ∀x ∗ ∈ bx∗ ∩ span b. proof. first we assume (i). it then directly follows that x∗(x) ∈ co ({x∗(xn) : n ≥ k} + [−ε, ε]) ∀k ∈ n ∀x ∗ ∈ bx∗ ∩ span b. thus we also have x∗(x) ≤ supn≥k x ∗(xn) + ε for all k ∈ n and all x ∗ ∈ bx∗ ∩ span b and the assertion (ii) follows. now we assume that (ii) holds and take k ∈ n arbitrary. suppose that x 6∈ coσb ({xn : n ≥ k} + εbx ) . then by the separation theorem we could find a functional x∗ ∈ (x, σb ) ′ = span b with ‖x∗‖ = 1 and a number α ∈ r such that x∗(y) ≤ α < x∗(x) ∀y ∈ coσb ({xn : n ≥ k} + εbx ) . it follows that lim sup n→∞ x∗(xn) + ε ≤ α < x ∗(x), a contradiction which ends the proof. � now we can give a quantitative version of the first equivalence in theorem 1.3. theorem 2.2. let a ⊆ x be bounded and ε ≥ 0. if for each sequence (xn)n∈n in a we have (2.1) ∞ ⋂ k=1 co ({xn : n ≥ k} + εbx ) 6= ∅, then a is 2ε-wrc. if a is ε-wrc, then (2.2) ∞ ⋂ k=1 co ({xn : n ≥ k} + rbx ) 6= ∅ holds for every sequence (xn)n∈n in a and every r > ε. some remarks on stronger versions of the boundary problem for banach spaces 71 proof. first we assume that (2.1) holds for every sequence in a. let (xn)n∈n and (x∗m)m∈n be sequences in a and bx∗ , respectively, such that the limits lim n→∞ lim m→∞ x∗m(xn) and lim m→∞ lim n→∞ x∗m(xn) exist. by assumption, we can pick an element x ∈ ∞ ⋂ k=1 co ({xn : n ≥ k} + εbx ) . from lemma 2.1 we conclude that (2.3) lim inf n→∞ x∗(xn) − ε ≤ x ∗(x) ≤ lim sup n→∞ x∗(xn) + ε ∀x ∗ ∈ bx∗ . it follows that (2.4) ∣ ∣ ∣ x∗m(x) − lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε ∀m ∈ n. now take a weak*-cluster point x∗ ∈ bx∗ of the sequence (x ∗ m)m∈n. then (2.5) lim n→∞ lim m→∞ x∗m(xn) = lim n→∞ x∗(xn). by (2.3) we have (2.6) ∣ ∣ ∣ x∗(x) − lim n→∞ x∗(xn) ∣ ∣ ∣ ≤ ε. since x∗(x)−limm→∞ limn→∞ x ∗ m(xn) is a cluster point of the sequence (x ∗ m(x)− limn→∞ x ∗ m(xn))m∈n it follows from (2.4) that (2.7) ∣ ∣ ∣ x∗(x) − lim m→∞ lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε. from (2.5), (2.6) and (2.7) we get ∣ ∣ ∣ lim m→∞ lim n→∞ x∗m(xn) − lim n→∞ lim m→∞ x∗m(xn) ∣ ∣ ∣ ≤ 2ε. thus we have proved a§2ε§bx∗ . hence, by proposition 1.6, a is 2ε-wrc. now assume that a is ε-wrc and take any sequence (xn)n∈n in a as well as r > ε. let x∗∗ ∈ a w ∗ be a weak*-cluster point of (xn)n∈n. since a is ε-wrc there is some x ∈ x such that ‖x∗∗ − x‖ ≤ r. for every x∗ ∈ bx∗ the number x ∗∗(x∗) is a cluster point of the sequence (x∗(xn))n∈n and thus x∗(x) ≤ ‖x − x∗∗‖ ‖x∗‖ + x∗∗(x∗) ≤ r + lim sup n→∞ x∗(xn). lemma 2.1 now yields x ∈ ∞ ⋂ k=1 co ({xn : n ≥ k} + rbx ) and the proof is finished. � as an immediate corollary we get 72 j.-d. hardtke corollary 2.3. if a ⊆ x is bounded and ε ≥ 0 such that ∞ ⋂ k=1 (co {xn : n ≥ k} + εbx ) 6= ∅ for every sequence (xn)n∈n in a, then a is 2ε-wrc. now we can also prove a quantitative version of theorem 1.4: corollary 2.4. let a ⊆ x be bounded, ε ≥ 0 and e = ex bx∗ . if for each sequence (xn)n∈n in a we have that ∞ ⋂ k=1 (coσe {xn : n ≥ k} + εbx ) 6= ∅, then a is 2ε-wrc. proof. let (xn)n∈n be a sequence in a. by means of theorem 1.5 and an easy diagonal argument we can find a subsequence (xnk )k∈n such that co σe {xnk : k ≥ l} ⊆ co {xn : n ≥ l} for all l (compare [18, corollary 0.2]). it then follows from our assumption that ∞ ⋂ l=1 (co {xn : n ≥ l} + εbx ) 6= ∅. hence, by corollary 2.3, a is 2ε-wrc. � next we observe that moors’ theorem 1.5 does not only work for the extreme points of bx∗ but also for any weak*-separable boundary. theorem 2.5. let b be a weak*-separable boundary for x and a a bounded infinite subset of x. then there is a countably infinite set f ⊆ a such that cof = coσb f . in particular, for every bounded sequence (xn)n∈n in x there exists a subsequence (xnk )k∈n with co σb {xnk : k ∈ n} ⊆ co {xn : n ∈ n}. proof. the proof is completely analogous to that of theorem 1.5 given in [18], in fact it is even simpler, so we shall only sketch it. arguing by contradiction, we suppose that for each countably infinite subset f of a there is an element z ∈ coσb f \ cof . then we can show exactly as in [18] (using the bishop-phelps theorem (cf. [13, theorem 5.5]) and the hahn-banach separation theorem) that for every sequence (xn)n∈n in a for which the set {xn : n ∈ n} is infinite, there is an element (2.8) x ∈ ∞ ⋂ k=1 coσb {xn : n ≥ k} \ co {xn : n ∈ n} . we remark that the weak*-separability of b is not needed for this step. next we fix a sequence (xn)n∈n in a whose members are distinct and a countable weak*-dense subset {x∗m : m ∈ n} of b. by the usual diagonal argument we may select a subsequence (again denoted by (xn)n∈n) such that limn→∞ x ∗ m(xn) exists for all m. some remarks on stronger versions of the boundary problem for banach spaces 73 we then choose an element x according to (2.8) and conclude that for each m ∈ n we have limn→∞ x ∗ m(xn) = x ∗ m(x). now let x∗ ∈ b be arbitrary. again as in [18] we will show that limn→∞ x ∗(xn) = x∗(x). suppose that this is not the case. then there is an ε > 0 such that |x∗(x) − x∗(xn)| > ε for infinitely many n ∈ n. let us assume x ∗(xn) > ε + x∗(x) for infinitely many n and arrange these indices in an increasing sequence (nk)k∈n. by (2.8) we can find z ∈ ∞ ⋂ l=1 coσb {xnk : k ≥ l} \ co {xnk : k ∈ n} . it follows that x∗m(z) = limk→∞ x ∗ m(xnk ) = x ∗ m(x) for all m and since {x ∗ m : m ∈ n} is weak*-dense in b this implies x∗(x) = x∗(z), whereas on the other hand x∗(z) ≥ ε + x∗(x), a contradiction. thus (xn)n∈n is σb -convergent to x and hence, by corollary 1.2 it is also weakly convergent to x, which in turn implies x ∈ co {xn : n ∈ n}, contradicting the choice of x. � note that the assumption of weak*-separability of b is fulfilled, in particular, if x is separable, for then the weak*-topology on bx∗ is metrizable. as an immediate corollary we get 2.4 for weak*-separable boundaries. corollary 2.6. let b be a boundary for x and a a bounded subset of x as well as ε ≥ 0. if b is weak*-separable (in particular, if x is separable) and for each sequence (xn)n∈n in a we have ∞ ⋂ k=1 (coσb {xn : n ≥ k} + εbx ) 6= ∅, then a is 2ε-wrc. proof. exactly as the proof of corollary 2.4. � let us now consider banach spaces of a certain type, namely the case x = c(k) for some compact hausdorff space k or x = ℓ1(i) for some index set i. in [5] respectively [9] cascales et al. found the positive solution to the boundary problem for these types of spaces. in fact, they even proved a stronger statement, namely that in the above cases the space (x, σb) is angelic1 for every boundary b of x. in order to get the statement of corollary 2.6 for arbitrary boundaries in c(k)and ℓ1(i)-spaces we shall need the following easy lemma. lemma 2.7. let t and s be subsets of x∗ such that for every countable set d ⊆ x and every x∗ ∈ t there is some y∗ ∈ s such that x∗(x) = y∗(x) for all x ∈ d. then for every countable set d ⊆ x we have coσs d ⊆ coσt d. 1see [5] or [12] for the definition and background. 74 j.-d. hardtke proof. let d ⊆ x be countable and take any x ∈ coσs d. further, fix x∗1, . . . , x ∗ n ∈ t and ε > 0. by assumption we can find y ∗ 1 , . . . , y ∗ n ∈ s such that x∗i (y) = y ∗ i (y) ∀y ∈ d ∪ {x} ∀i = 1, . . . , n. but then the same equality holds for every y ∈ co (d ∪ {x}) and since x ∈ coσs d we may select some y ∈ co d with |y∗i (x) − y ∗ i (y)| ≤ ε for all i = 1, . . . , n. it follows that |x∗i (x) − x ∗ i (y)| ≤ ε for i = 1, . . . , n and the proof is finished. � according to the results of cascales et al. the condition of lemma 2.7 is fulfilled if x = c(k) or x = ℓ1(i), t = ex bx∗ and s is any boundary for x (see [5, lemma 1] for x = c(k) and the proof of [9, theorem 4.9] for x = ℓ1(i)), thus we immediately get the following lemma. lemma 2.8. if x = c(k) for some compact hausdorff space k or x = ℓ1(i) for some index set i and b is any boundary for x, then for every countable set d ⊆ x we have coσb d ⊆ coσe d, where e = ex bx∗ . from lemma 2.8 and corollary 2.4 we now get the desired result. corollary 2.9. if x = c(k) for some compact hausdorff space k or x = ℓ1(i) for some set i and b is any boundary for x as well as a ⊆ x a bounded set and ε ≥ 0 such that for every sequence (xn)n∈n in a we have ∞ ⋂ k=1 (coσb {xn : n ≥ k} + εbx ) 6= ∅, then a is 2ε-wrc. next we turn to spaces not containing isomorphic copies of ℓ1. it is known that for such spaces one has coγ b = bx∗ for every boundary b of x, where we denote by γ the topology on x∗ of uniform convergence on bounded countable subsets of x (cf. [8, theorem 5.4]). we will also need two easy lemmas. lemma 2.10. let a ⊆ x and s ⊆ x∗ be bounded as well as ε ≥ 0 such that a§ε§s. then we also have a§ε§s γ . proof. let (xn)n∈n and (x ∗ m)m∈n be sequences in a and s γ , respectively, such that the limits lim n→∞ lim m→∞ x∗m(xn) and lim m→∞ lim n→∞ x∗m(xn) exist. for each m ∈ n we can pick a functional x̃∗m ∈ s with |x∗m(xn) − x̃ ∗ m(xn)| ≤ 1 m ∀n ∈ n. by the usual diagonal argument, choose a subsequence (xnk )k∈n such that limk→∞ x̃ ∗ m(xnk ) exists for all m. it then easily follows that lim m→∞ lim n→∞ x∗m(xn) = lim m→∞ lim k→∞ x̃∗m(xnk ) and lim n→∞ lim m→∞ x∗m(xn) = lim k→∞ lim m→∞ x̃∗m(xnk ). some remarks on stronger versions of the boundary problem for banach spaces 75 since a§ε§s, we conclude that ∣ ∣ ∣ lim n→∞ lim m→∞ x∗m(xn) − lim m→∞ lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε, finishing the proof. � lemma 2.11. let (xn)n∈n be a bounded sequence in x and b ⊆ bx∗ such coγ b = bx∗ . then coσb {xn : n ∈ n} = co {xn : n ∈ n} . proof. take x ∈ coσb {xn : n ∈ n} and let ε > 0 and x ∗ 1, . . . , x ∗ k ∈ bx∗ be arbitrary. by assumption, we can find x̃∗1, . . . , x̃ ∗ k ∈ co b such that for i = 1, . . . , k we have |x̃∗i (xn) − x ∗ i (xn)| ≤ ε ∀n ∈ n and |x̃ ∗ i (x) − x ∗ i (x)| ≤ ε. it follows that |x̃∗i (y) − x ∗ i (y)| ≤ ε ∀y ∈ co ({xn : n ∈ n} ∪ {x}) ∀i = 1, . . . , k. now take some element y ∈ co {xn : n ∈ n} with |x̃ ∗ i (y) − x̃ ∗ i (x)| ≤ ε for all i = 1, . . . , k. employing the triangle inequality we can deduce |x∗i (x) − x ∗ i (y)| ≤ 3ε, which ends the proof. � as an immediate consequence of lemma 2.11, corollary 2.3 and the aforementioned result [8, theorem 5.4] we get the following corollary. corollary 2.12. suppose ℓ1 6⊆ x and let b be a boundary for x. if a ⊆ x is bounded and ε ≥ 0 such that for each sequence (xn)n∈n in a we have ∞ ⋂ k=1 (coσb {xn : n ≥ k} + εbx ) 6= ∅, then a is 2ε-wrc. we can further get a kind of ‘boundary double limit criterion’. proposition 2.13. let b be a boundary for x as well as ε ≥ 0 and a ⊆ x be bounded such that a§ε§b. then a is 2ε-wrc. if ℓ1 6⊆ x, then a is even ε-wrc. proof. from [7, theorem 3.3] it follows that we also have a§ε§ co b. since b is a boundary for x the hahn-banach separation theorem implies bx∗ = cow ∗ b. therefore it follows from [2, lemma 3] that a§2ε§bx∗ . thus by (ii) of proposition 1.6 a is 2ε-wrc.2 moreover, if ℓ1 6⊆ x then we even have bx∗ = co γ b by the already cited [8, theorem 5.4]. hence a§ε§bx∗ by lemma 2.10, thus a is ε-wrc. � 2this proof also works under the weaker assumption that b is only norming for x, i.e. ‖x‖ = sup b∈b b(x) for all x ∈ x, because in this case we also have bx∗ = co w ∗ b by the separation theorem. 76 j.-d. hardtke our final aim in this note is to prove a ‘non-relative’ version of theorem 1.4 for arbitrary boundaries. to do so, we will use the techniques of pfitzner from [20]. more precisely, we can get the following slight generalization of the “in particular case” of [20, proposition 8]. recall that an ℓ1-sequence in x is simply a sequence equivalent to the canonical basis of ℓ1. proposition 2.14. let b be a boundary for x. if a ⊆ x is bounded and for every sequence (xn)n∈n in a we have (2.9) a ∩ ∞ ⋂ k=1 coσb {xn : n ≥ k} 6= ∅, then a does not contain an ℓ1-sequence. proof. the proof is completely analogous to that of [20, proposition 8], therefore we will only give a very brief sketch. we use the notation and definitions from [20]. arguing by contradiction, we assume that there is an ℓ1-sequence (xn)n∈n in a. by [20, lemma 2] we may assume that (xn)n∈n is δ-stable. we take a sequence (αk)k∈n of positive numbers decreasing to zero. by [20, lemma 7] we can find ε ≥ 1/2δ̃b(xn) = 1/2δ̃(xn) > 0, a sequence (bk)k∈n in b and a tree (ωσ)σ∈s such that for each k ∈ n and every σ, σ ′ ∈ sk with σk = 0 and σ′k = 1 we have bk(xn − xn′ ) ≥ 2ε(1 − αk) ∀n ∈ ωσ, n ′ ∈ ωσ′ . it follows that the same inequality holds for every x ∈ coσb {xn : n ∈ ωσ} and x′ ∈ coσb {xn′ : n ′ ∈ ωσ′}. now using our hypothesis we can proceed completely analogous to the proof of the claim in [20, proposition 8] to find a sequence (ym)m∈n in a∩ ⋂∞ k=1 coσb {xn : n ≥ k} such that bk(ym − ym′ ) ≥ 2ε(1 − αk) ∀m ≤ k < m ′. next we take an element y ∈ a ∩ ∞ ⋂ k=1 coσb {ym : m ≥ k} . as in the proof of [20, proposition 8] we put x = ∞ ∑ m=1 2−m(ym − y) and proceed again exactly as in the proof of [20, proposition 8] to show that ‖ym − y‖ ≤ 2ε for all m and ‖x‖ = 2ε. finally, taking a functional b ∈ b with b(x) = ‖x‖ we obtain b(y) = 2ε + b(y) and with this contradiction the proof is finished. � now we can get the final result. some remarks on stronger versions of the boundary problem for banach spaces 77 theorem 2.15. let b be a boundary for x and a ⊆ x be bounded. then the following assertions are equivalent: (i) a is countably compact in the topology σb . (ii) for every sequence (xn)n∈n in a we have a ∩ ∞ ⋂ k=1 coσb {xn : n ≥ k} 6= ∅. (iii) for every sequence (xn)n∈n in a there is some x ∈ a with x∗(x) ≤ lim sup n→∞ x∗(xn) ∀x ∗ ∈ span b. (iv) a is weakly compact. proof. the implications (i) ⇒ (ii) and (iv) ⇒ (i) are clear and the equivalence of (ii) and (iii) follows from lemma 2.1. it only remains to prove (ii) ⇒ (iv). let us assume that (ii) holds and take an arbitrary sequence (xn)n∈n in a. by proposition 2.14 no subsequence of (xn)n∈n is an ℓ 1-sequence and thus rosenthal’s theorem (cf. [3] or [1, theorem 10.2.1]) applies to yield a subsequence (xnk )k∈n which is weakly cauchy. now choose an element x ∈ a ∩ ∞ ⋂ l=1 coσb {xnk : k ≥ l} . it easily follows that limk→∞ b(xnk ) = b(x) for all b ∈ b. by corollary 1.2 (xnk )k∈n is weakly convergent to x. thus we have shown that a is weakly sequentially compact. hence it is also weakly compact, by the eberlein-šmulian theorem. � remark 2.1. it is proved in [11, remark 10] that for x = ℓ1 the statement bx§ε§bx∗ is false for every 0 < ε < 2. an alternative proof of this fact is given [6, example 5.2]. it is further proved in [11, remark 10] that every separable banach space x which contains an isomorphic copy of ℓ1 can be equivalently renormed such that, in this renorming, the statement bx§ε§bx∗ is false for every 0 < ε < 2. the proof makes use of the notion of octahedral norms. we wish to point out here that the argument from [6, example 5.2] can be carried over to arbitrary banach spaces containing a copy of ℓ1, precisely we have the following proposition. proposition 2.16. if x is a (not necessarily separable) banach space which contains ℓ1 than the statement bx§ε§bx∗ (in the original norm of x) is false for every 0 < ε < 2. proof. take 0 < ε < 2 arbitrary and fix 0 < δ < 1 such that 2(1 − δ) > ε. since x contains ℓ1 we may find, with the aid of james’ ℓ1-distortion theorem (cf. [1, theorem 10.3.1]), a sequence (xn)n∈n in the unit sphere of x such that t : ℓ1 → x defined by t y = ∞ ∑ k=1 αkxk ∀y = (αn)n∈n ∈ ℓ 1 78 j.-d. hardtke is an isomorphism (onto u = ran t ) with ∥ ∥t −1 ∥ ∥ ≤ (1 − δ)−1. consequently, the adjoint t ∗ : u ∗ → ℓ∞ is as well an isomorphism with ∥ ∥(t ∗)−1 ∥ ∥ ≤ (1−δ)−1. now we can define as in [6, example 5.2] for each n ∈ n a norm one functional y∗n ∈ ℓ ∞ by y∗n(m) = { 1, if m ≤ n −1, if m > n. put u∗n = (t ∗)−1y∗n for all n ∈ n. then ‖u ∗ n‖ ≤ (1 − δ) −1 and hence by the hahn-banach extension theorem we can find x∗n ∈ bx∗ with x ∗ n|u = (1 − δ)u ∗ n for all n ∈ n. it follows that ∣ ∣ ∣ lim n→∞ lim m→∞ x∗n(xm) − lim m→∞ lim n→∞ x∗n(xm) ∣ ∣ ∣ = 2(1 − δ) > ε and the proof is finished. � in the notation of [6] we have proved γ(bx ) = 2 for every banach space x containing an isomorphic copy of ℓ1, which implies that the value of bx under all other measures of weak non-compactness considered in [6] is equal to one (again compare [6, example 5.2]). so in a certain sense a banach space containing ℓ1 is ‘as non-reflexive as possible’. remark 2.2. shortly after the first version of this paper was published on the web, the author received a message from prof. warren b. moors, who kindly pointed out to him that the above lemma 2.8 probably also holds true if x is an l1-predual 3 (which includes all c(k)-spaces), refering to the paper [19]. indeed, from [19, theorem 3] one can easily get the following result: if b is a boundary for the l1-predual x and e = ex bx∗ , then coσb {xn : n ∈ n} ⊆ co σe {xn : n ∈ n} holds for every sequence (xn)n∈n in x. for the proof just apply [19, theorem 3] to the countable set coq {xn : n ∈ n} consisting of all convex combinations of the xn’s with rational coefficients. it follows that corollary 2.9 also carries over to arbitrary boundaries of l1-preduals. acknowledgements. the author wishes to express his gratitude to prof. warren b. moors for providing him with the important hint already mentioned in the remark above and to the anonymous referee for multiple comments and suggestions (in particular for proposing lemma 2.7) which improved the exposition of the results. 3recall that a banach space x is called an l1-predual if x ∗ is isometric to l1(µ) for some suitable measure µ. some remarks on stronger versions of the boundary problem for banach spaces 79 references [1] f. albiac and n. kalton, topics in banach space theory, springer graduate texts in mathematics vol.233, 2006 [2] c. angosto and b. cascales, measures of weak noncompactness in banach spaces, topology appl. 156, no. 7 (2009), 1412–1421. 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(received november 2010 – accepted january 2011) 80 j.-d. hardtke jan-david hardtke (hardtke@math.fu-berlin.de) department of mathematics, freie universität berlin, arnimallee 6, 14195 berlin, germany some remarks on stronger versions of the [3pt] boundary problem for banach spaces. by j.-d. hardtke casertawatsonagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 245-267 michael spaces and dowker planks agata caserta∗ and stephen watson abstract. we investigate the lindelöf property of dowker planks. in particular, we give necessary conditions such that the product of a dowker plank with the irrationals is not lindelöf. we also show that if there exists a michael space, then, under some conditions involving singular cardinals, there is one that is a dowker plank. 2000 ams classification: primary 54c50, 54d20, 54g15. keywords: michael space, michael function, nl property, haydon plank. 1. introduction in 1963 e. michael constructed, under the continuum hypothesis, a lindelöf space whose product with the irrationals is not normal (see [7]). such a space is known as a michael space. an open problem is to construct a michael space in zfc without additional axioms. the aim of this paper is to provide necessary conditions for the existence of a michael space, and to give some examples of michael spaces. our work is associated to the results in [8]. in this note, p stands for the set of the irrational numbers, and the cantor set c is viewed as a compactification of p obtained by adding a countable set qc . ordinal numbers are denoted by greek letters; when viewed as topological spaces, they are given the order topology. products of topological spaces are endowed with the standard product topology. the symbol [a]λ denotes the family of subsets of a having size exactly λ. the symbols [a]≤λ and [a]<λ have similar meaning. let ≤∗ be the quasi-order on a countable product of ordered sets that is associated to the coordinate-wise order on each set. thus f ≤∗ g stands for f (n) ≤ g(n) for all but finitely many n ∈ ω. a subset of ωω is unbounded if it is unbounded in (ωω, ≤∗). a dominating family is an unbounded set that is ∗corresponding author. 246 a. caserta and s. watson cofinal in (ω ω, ≤∗). a subset of ωω is a scale if it is a dominating family and is well-ordered by ≤∗. recall that p can be identified with ωω with the product topology. for each ξ ∈ <ωω = {η | η : [0, n] → ω for some n}, a basic open neighborhood of ξ in the product topology is {f ∈ ω ω : ξ ⊆ f}. for every g ∈ ωω, the sets {f ∈ ωω : f ≤ g} and {f ∈ ωω : f ≤∗ g} are respectively compact and σ-compact (see [2]). let x and y be topological spaces. a set a ⊆ x is y-analytic if it is a projection on x of a closed subset of x × y . in particular, a ⊆ x is analytic if it is p-analytic. given a function f : x → y , the small image of a ⊆ x is defined by f ♯(a) = {y ∈ y : f −1(y) ⊆ a}. sometimes we abuse of terminology and say that f ♯ is open, with the meaning that for each open subset a of x, f ♯(a) is an open subset of y . in most cases we will employ the notation used in [4] and [6]. 2. michael sequences and michael functions we start the section with the definition of a michael sequence. the first goal of this section is to show that michael sequences may be assumed to be continuous. definition 2.1. let {xξ}ξ≤θ be a decreasing sequence of sets. it is a continuous sequence if for any γ ≤ θ, with γ limit ordinal, xγ = ∩ξ<γ xξ. definition 2.2 (moore [8]). a decreasing sequence {xξ}ξ≤θ of subsets of a topological space z is said to be a k-michael sequence if the following conditions hold: (i) for each k compact subset of z \ xθ the ordinal δk = min{ξ ≤ θ : xξ ∩ k = ∅} does not have uncountable cofinality. in particular an f-michael sequence is a k-michael sequence satisfying the following additional condition: (ii)f for each f closed subset of z \ xθ the ordinal δf = min{ξ ≤ θ : xξ ∩ f = ∅} is either θ or does not have uncountable cofinality. also given a topological space y , an a(y )-michael sequence is a k-michael sequence satisfying the following additional condition: (ii)a for each a which is y -analytic in z \ xθ the ordinal δa = min{ξ ≤ θ : xξ ∩ a = ∅} is either θ or does not have uncountable cofinality. remark 2.3. in the definition of a k-michael sequence, we observe that the property of being a continuous sequence is partially satisfied . in other words, for every limit ordinal γ < θ with cfγ > ω it follows that xγ = ∩ξ<γ xξ. indeed, let x ∈ ∩ξ<γ xξ \ xγ . then {x} is a compact subset of z \ xθ, and δ{x} = γ, so that cfδ{x} > ω in contradiction with the definition of k-michael sequence. michael spaces and dowker planks 247 lemma 2.4. let θ be a cardinal and {xξ}ξ≤θ (strictly) decreasing sequence such that xγ = ∩ξ<γ xξ for every limit ordinal γ < θ with cfγ > ω. then there exists {yξ}ξ≤θ continuous (strictly) decreasing sequence, such that yα = xα for every α < θ with cfα 6= ω. proof. let {xξ}ξ≤θ be decreasing sequence. define {yξ}ξ≤θ such that yα = xα for every α < θ with cfα > ω, otherwise yα = ∩ξ≤αxξ. clearly yη ⊇ yξ for every η < ξ ≤ θ. moreover for every α < θ with cfα = ω, yα ⊇ xα. by construction, we have that {yξ}ξ≤θ is a continuous sequence. assume that all the subsets xξ ∈ {xξ}ξ≤θ are distinct. then yα ⊇ xα ⊃ xα+1 = yα+1 implies that yα’s are distinct. � in case we have two or more sequences of subsets of z of length θ + 1, having the same last element, and given h, we denote δh with respect the sequence {xξ}ξ≤θ with δ x̃ h . lemma 2.5. let θ be a cardinal with cfθ > ω, {xξ}ξ≤θ and {yξ}ξ≤θ two decreasing sequences of subsets of a topological space z, such that yα = xα for every α < θ with cfα 6= ω. then δx̃h < δ ỹ h ⇒ ( δ ỹ h = δ x̃ h + 1 ) ∧ ( cfδ x̃ h = ω ) with h ⊆ z. proof. from remark 2.3 it follows that for every α < θ with cfα > ω, xα = ∩ξ<αxξ, and xα = yα ⊇ ∩ξ<αyξ. we have also that for every α < θ with cfα > ω there exists a cofinal sequence (αη)η ω and (ii) cfδ ỹ h = ω. if (i) holds, then yδỹ h ∩ h = x δỹ h ∩ h = ∅, therefore δx̃h ≤ δ ỹ h . if δ x̃ h < δ ỹ h there exists αη, such that δ x̃ h < αη < δ ỹ h . then by minimality of δỹh we have yαη ∩ k 6= ∅ and yαη ∩ k ⊆ xαη ∩ k. moreover xαη ⊆ xδx̃ h , so x δx̃ h ∩ k 6= ∅ which is in contradiction with the definition of δk . thus δ x̃ h = δ ỹ h . for (ii), assume by contradiction, that δ x̃ h 6= δ ỹ h . since cfδx̃h = cfδ ỹ h = ω, there exists α successor ordinal such that δ ỹ h < α < δ x̃ h . then xα = yα, and so yδỹ h ⊇ yα = xα ⊇ xδx̃ h . therefore xα ∩ h = ∅ which is in contradiction with the minimality of δx̃h . thus δ x̃ h = δ ỹ h . � corollary 2.6. let θ be a cardinal with cfθ > ω, {xξ}ξ≤θ and {yξ}ξ≤θ two decreasing sequences of subsets of z, such that yα = xα for every α < θ with cfα 6= ω. let h ⊂ z, then δx̃h = δ ỹ h if either one has uncountable cofinality. corollary 2.7. let θ be a cardinal with cfθ > ω, {xξ}ξ≤θ and {yξ}ξ≤θ two decreasing sequences of subsets of a topological space z, such that yα = xα for every α < θ with cfα 6= ω. then {xξ}ξ≤θ is a k-michael (resp., f-michael or a(y )-michael) sequence if and only if {yξ}ξ≤θ is a k-michael (resp., f-michael or a(y )-michael) sequence. 248 a. caserta and s. watson proof. let {xξ}ξ≤θ be a k-michael (resp., f-michael or a(y )-michael) sequence. by hypothesis yθ = xθ. let h ⊆ (z \ xθ) compact (resp., closed or analytic). then cfδx̃h ≤ ω (resp., either δ x̃ h ≤ ω or δ x̃ h = θ). we want to check that cfδỹk ≤ ω (resp., either δ ỹ h ≤ ω or δ ỹ h = θ). assume not, i.e., cfδỹk > ω, (resp., ω < cfδ ỹ k < θ) corollary 2.6 implies that δ x̃ k = δ ỹ k , which is a contradiction. � corollary 2.8. let θ be a cardinal with cfθ > ω. the following are equivalent: (i) there exists {xξ}ξ≤θ which is k-michael (resp., f-michael or a(y )michael strictly decreasing) sequence; (ii) there exists {xξ}ξ≤θ continuous k-michael (resp., f-michael or a(y )michael strictly decreasing) sequence. next we introduce the definition of michael function and we analyze the relationship between michael functions and michael sequences. definition 2.9. let z be a topological space and f : z → θ + 1 an arbitrary function. then f is said to be a k-michael function if the following condition holds: (i) for each k compact subset of z \ f −1({θ}), supx∈kf(x) + 1 does not have uncountable cofinality. in particular an f-michael function is a k-michael function satisfying the following additional condition: (ii) for every f closed subset of z \ f −1({θ}), supx∈ff(x) + 1 is either θ or does not have uncountable cofinality. also given a topological space y , an a(y )-michael function is a k-michael function satisfying the following additional condition: (ii) for every a which is y -analytic in z \f −1({θ}), supx∈af(x)+ 1 is either θ or does not have uncountable cofinality. in the next proposition we will show the equivalence of continuous k-michael sequences with k-michael functions f : z → θ + 1. lemma 2.10. let z be a topological space, f : z → θ + 1 be an arbitrary function with θ cardinal. if xξ = {x ∈ z : f (x) ≥ ξ} for every ξ ∈ θ, then δh = supx∈hf(x) + 1 for every h ⊆ z \ xθ. proof. by definition we have that δh = min{ξ ≤ θ : k ∩ xξ = ∅} = min{ξ ≤ θ : ∀x ∈ k (x /∈ xξ)} = min{ξ ≤ θ : ∀x ∈ k (f (x) < ξ)} = sup{f(x) + 1 : x ∈ k}. � lemma 2.11. let θ be a cardinal, {xξ}ξ≤θ a continuous sequence of subsets of topological space z, and f : z → θ + 1 a function defined by f (x) = sup{γ ∈ θ + 1 : x ∈ xγ}. then we have: (i) xξ = {x ∈ z : f (x) ≥ ξ} for every ξ ∈ θ; (ii) f is surjective if and only if {xξ}ξ≤θ is strictly decreasing. michael spaces and dowker planks 249 proof. to show (i), we have that for every ξ ∈ θ, {x ∈ z : f (x) ≥ ξ} = {x ∈ z : sup{γ ∈ θ : x ∈ xγ} ≥ ξ}. from the continuity follow {x ∈ z : sup{γ ∈ θ : x ∈ xγ} ≥ ξ} = {x ∈ z : x ∈ xξ} = xξ. for (ii), first assume that f is surjective. by (i) we have that xξ = {x ∈ z : f (x) ≥ ξ} for every ξ ∈ θ, and so {xξ}ξ≤θ is a decreasing sequence. assume that there exist α, β ∈ θ with α < β such that xα = xβ. thus there exist ξ ∈ θ with α < ξ ≤ β and z ∈ z such that f (x) = ξ. hence x ∈ xβ but x /∈ xα, a contradiction. on the other hand, assume that {xξ}ξ≤θ is strictly decreasing, and f is not surjective. then there exists α < θ such that f (x) 6= α for any x ∈ z \ xθ, with xθ = f −1({θ}). let f (x) > α. from (i) it follows that there exists α < θ such that (z \ xθ) ⊆ xα. thus xβ = xα for any β ≤ α, which contradicts the fact that the sequence is strictly decreasing. if f (x) < α, follow that (z \ xθ) ∩ xα = ∅, which is a contradiction. � proposition 2.12. let z and y be two topological spaces, θ a cardinal with cfθ > ω. for every q ⊆ z, the following statements are equivalent: (i) there exists a continuous k-michael (resp., f-michael or amichael) sequence {xξ}ξ≤θ with q = xθ; (ii) there exists k-michael (resp., f-michael or a(y )-michael) function f : z → θ + 1, with q = f −1({θ}); (iii) there exists k-michael (resp., f-michael or a(y )-michael) sequence {xξ}ξ≤θ with q = xθ. proof. (i) ⇒ (ii). let {xξ}ξ≤θ be a continuous k-michael sequence. define the following map f : z → θ + 1 such that f (x) = sup{γ ∈ θ + 1 : x ∈ xγ}. clearly f −1({θ}) = xθ. now let h ⊂ (z \ q) be a compact (resp., closed or analytic) subset. by lemma 2.11, for any α ≤ θ, xα = {x ∈ z : f (x) ≥ α}. by lemma 2.10, supx∈hf(x) + 1) = δh. since cfδk ≤ ω (resp., either cfδh ≤ ω or cfδh = θ), then cf (supx∈kf(x) + 1)) ≤ ω (resp., either cf (supx∈hf(x) + 1) ≤ ω or cf (supx∈hf(x) + 1) = θ ). (ii) ⇒ (iii). let f : z → θ + 1 be a k-michael function with q = f −1({θ}). for any α ≤ θ define xα = {x ∈ z : f (x) ≥ α}. clearly xθ = f −1({θ}) and xξ ⊇ xη for any ξ < η ≤ θ. let now h ⊂ (z \q) be a compact (resp., closed or analytic) subset, we want to show that cfδh ≤ ω. by lemma 2.10, supx∈kf(x)+ 1 = δh. since cf (supx∈hf(x) + 1) ≤ ω (resp., either cf (supx∈hf(x) + 1) ≤ ω or cf (supx∈hf(x) + 1) = θ ), then cfδh ≤ ω (resp., either cfδh ≤ ω or cfδh = θ). (iii) ⇒ (i). follow from corollary 2.8. � corollary 2.13. let θ be a cardinal of uncountable cofinality, z and y two topological spaces. there exists a continuous k-michael (resp., f-michael or a(y )-michael) strictly decreasing sequence {xξ}ξ≤θ of subsets of z, if and only if there exists k-michael (resp., f-michael or a(y )-michael) function f : z → θ + 1 that is surjective. 250 a. caserta and s. watson 3. local properties of michael functions in this section we want to analyze and characterize the properties of being a michael function. first we need the following definition. definition 3.1. let z be a topological space, h : z → θ + 1 an arbitrary function with θ cardinal. for every α ≤ θ we say that h is michael at α if cfα > ω ⇒ (∀f ⊆ z closed (∀z ∈ f h(z) < α) ⇒ (supz∈fh(z) < α)). moreover h is σ-michael at α if cfα > ω ⇒ (∀c ⊆ z fσ-set (∀z ∈ c h(z) < α) ⇒ (supz∈ch(z) < α)). directly from the definition follow: lemma 3.2. let z be a topological space, h : z → θ + 1 an arbitrary function with θ cardinal. the following statements are equivalent: (i) h is michael at α, (ii) cfα > ω ⇒ (∀u ⊆ z open (h−1[α, θ] ⊆ u) ⇒ (supz∈z\uh(z) < α)). lemma 3.3. let z be a topological space, h : z → θ + 1 an arbitrary function with θ cardinal. the following statements are equivalent: (i) h is σ-michael at α, (ii) cfα > ω ⇒ (∀g ⊆ z gδ-set (h −1([α, θ]) ⊆ g) ⇒ (supz∈z\gh(z) < α)). lemma 3.4. let z be a topological space, h : z → θ + 1 an arbitrary function with θ cardinal. then h is σ-michael at α if and only if h is michael at α. proof. let cfα > ω, and c = ⋃ n∈ω fn with fn closed subset of z such that for every z ∈ c h(z) < α. then for every n ∈ ω and for every z ∈ fn, h(z) < α. let αn = supz∈fnh(z). since h is michael at α, follow αn < α for every n ∈ ω. then supz∈ch(z) = supn∈ωαn. from cfα > ω if follows that supn∈ωαn < α. � an arbitrary function h : z → θ+1, induces a new function ĥ : z×y → θ+1, defined by ĥ(x) = h(π1(x)) for every x ∈ z × y , where y is an arbitrary topological space, and π1 the projection of z × y onto its first coordinate space. clearly this raises the question whether ĥ is michael at some ordinal α ≤ θ. lemma 3.5. let z, y be two topological spaces, h : z → θ + 1 is an arbitrary function, ĥ : z × y → θ + 1 with θ cardinal. then the following statements are equivalent: (i) ĥ is michael at α, (ii) cfα > ω ⇒ (∀a ⊆ z y-analytic (∀z∈a h(z) < α) ⇒ supz∈ah(z) < α). moreover, if ĥ is michael at α for some α ≤ θ, then h is michael at the same ordinal. but the converse does not hold. now, given a function h : z → θ + 1, we want to characterize the property of being michael at some ordinal for h, in term of a michael function. michael spaces and dowker planks 251 proposition 3.6. let z be a topological space, h : z → θ + 1 a function with θ cardinal. then the following statements are equivalent: (i) h is a f-michael function; (ii) h is michael at α for every α ≤ θ. proof. assume that h is not a f-michael function. then there is a closed set f ⊆ (z \ h−1({θ})) such that cf(supz∈fh(z) + 1) > ω. let α = supz∈kh(z) + 1. note that h(z) < α for every z ∈ f . since h is michael at α, from lemma 3.2 follow that supz∈fh(z) < α, which is a contradiction. vice versa, assume that h is a f-michael function. let α ∈ θ such that cfα > ω. we want to show that h is michael at α. let u be an open set of z such that h−1([α, θ]) ⊂ u . then z \ u is such that h(z) < α for every z ∈ z. therefore cf(supz\uh(z) + 1) ≤ ω. since cfα > ω there exists β < α such that supz∈z\uh(z) + 1 ≤ β < α. � from the previous proof we can argue that if h is michael at α for every α ∈ θ, then h is f-michael function, and so k-michael function, but the vice versa does not hold. clearly it is true in case z is a compact space. moreover we have shown that if h is a f-michael function, then there exists an ordinal α such that h is michael at α. the vice versa does not hold, we needed the property of being michael to be satisfied at each ordinal into the codomain of h. proposition 3.7. let z, y be two topological spaces, h : z → θ + 1 and ĥ : z × y → θ + 1 functions with θ cardinal. then the following statements are equivalent: (i) h is a a(y )-michael function, (ii) ĥ is michael at α for each α ≤ θ. proof. assume that h is not a a(y )-michael function. then there is a set a ⊆ (z \h−1({θ})) which is the projection onto z of a closed subset f of z ×y , such that ω < cf(supz∈ah(z) + 1) < θ. let α = supz∈ah(z) + 1. since a is a y analytic subset of z and ĥ is michael at α it follows that supz∈ah(z) < α which is in contradiction with supz∈ah(z) = {β ≤ α : h(z) ≥ β for same z ∈ a} = α. assume that h is a a(y )-michael function. let α < θ with cfα > ω. let a be a y -analytic subset of z, i.e., a = π(f ) where f is a closed subset of z × y such that h(z) < α for every z ∈ a. then cf(supz∈ah(z) + 1) ≤ ω. since cfα > ω, it follows that supz∈ah(z) < α. � remark 3.8. note that by proposition 2.12 and proposition 3.6, it follows that if h : c → θ + 1 is such that qc = h −1({θ}), the property of being michael at α for every α ≤ θ is equivalent to the notion of k-michael sequence {xξ}ξ≤θ [m [8]], where for every ξ ∈ θ, xξ ⊆ c and xθ = qc . the next proposition give us conditions on the function h : z → θ + 1, so that the function ĥ is not michael at θ. 252 a. caserta and s. watson proposition 3.9. let θ be a cardinal with cfθ > ω, z a topological space. let h : z → θ + 1 be a function such that h(z) ∩ (α, θ) 6= ∅ for every α < θ. then ĥ is not michael at θ, where ĥ : z × (z \ h−1({θ})) → θ + 1. proof. set ∆ = {(z, z) : z ∈ z \ h−1({θ})}. then ∆ is a closed subset of z × (z \ h−1({θ})) such that for every z ∈ z \ h−1({θ}) we have h(z) < θ. but supz∈z\h−1({θ})h(z) = θ. if not, there exists α < θ such that supz∈z\h−1({θ})h(z) = α. since h(z) ∩ (α, θ) 6= ∅, there exists β with α < β < θ, z ∈ z \ h−1({θ}) such that h(z) = β which is a contradiction. � 4. nl property in this section we introduce the new definition of nl property at some ordinal, and we give examples of functions which have this property. definition 4.1. let x be a topological space, θ a cardinal and  : x → θ an arbitrary function. for each α ≤ θ with cfα > ω, we say that  has the property nl at α if for every a ⊆ x such that (a) is cofinal in α, a is not lindelöf remark 4.2. a banal case for the function  : x → θ + 1 to have the property nl at each α ≤ θ with cfα > ω, is for  = idθ+1. indeed every subset of α which is cofinal in α cannot be lindelöf another simple case in which  has the property nl at each α ≤ θ with cfα > ω is when −1(β) is open in x for every β < α. indeed, assume that a ⊆ x such that (a) is cofinal in α, and by contradiction a is lindelöf. then {−1(β)}β∈α is an open cover for a, therefore there exist β0 ∈ α countable such that a ⊆ ⋃ β∈β0 −1(β). thus a ⊆ −1(β0) which is a contradiction. other examples of function with the property nl are given. before we need the following definitions. definition 4.3. let θ be a cardinal and x a topological space. the family {aα}α∈θ is a special gδ family of x, if for every α ∈ θ, aα = ⋂ n∈ω a n α where each anα is open in x and for every n ∈ ω, {a n α}α∈θ is an increasing family. definition 4.4. let θ be a cardinal and x a topological space. the function  : x → θ + 1 is a special at α with α ≤ θ, if there exists a sequence of continuous functions (n)n∈ω with n : x → θ + 1 such that for every n ∈ ω, (i) −1(α) ⊆ −1n (α), (ii) (x) ≤ n(x) for every x ∈ x, (iii) {−1n (α)}α≤θ is an increasing family. lemma 4.5. let θ be a cardinal, x a topological space and  : x → θ + 1 a function. the following statements are equivalent: (i)  is special at each α ≤ θ (ii) {−1(α)}α≤θ is a special gδ family of x. proof. let α ≤ θ and  be special at α. let (n,α)n∈ω be a sequence of continuous functions n,α : x → θ + 1 satisfying properties in definition 4.4. by michael spaces and dowker planks 253 continuity of each n,α, the set  −1 n,α(α) is open in x for each n ∈ ω. since −1(α) ⊆ −1n,α(α), it follows that  −1(α) ⊆ ⋂ n∈ω  −1 n,α(α) for each α ∈ ω. we show that ⋂ n∈ω  −1 n,α(α) ⊆  −1(α). let x ∈ ⋂ n∈ω  −1 n,α(α), hence x ∈  −1 n,α(α) for each n ∈ ω, i.e., for each n, n,α(x) ∈ α. since (x) ≤ n,α(x) for all n ∈ ω and x ∈ x, we have that (x) ≤ α. thus x ∈ −1(α) and −1(α) = ⋂ n∈ω  −1 n,α) (α) for each α ≤ θ. moreover for every n ∈ ω, we have that {−1n,α(α)}α≤θ is an increasing family. vice versa, assume that {−1(α)}α≤θ is a special gδ family of x. let α ≤ θ. by hypothesis, −1(α) = ⋂ n∈ω a n α with the property that a n α is an open set and for every n the family {anα}α≤θ is increasing. define for each n ∈ ω, the function n : x → θ + 1 by n(x) = min{ξ ∈ θ + 1 : x ∈ a n ξ }. we have that for each α ≤ θ, −1n (α) = a n α. indeed, a n α ⊆  −1 n (α) and for each γ > α there is not y ∈ anγ \ a n α such that y ∈  −1 n (α). otherwise from y ∈  −1 n (α), it follows that y ∈ anα which is a contradiction. thus n is continuous for each n and the family {−1n (α)}α≤θ is an increasing. since  −1(α) = ⋂ n∈ω  −1 n (α), we have that for each n ∈ ω, −1(α) ⊆ −1n (α). let x ∈ x. it remains to prove that (x) ≤ n(x) for every n ∈ ω. let (x) = α. hence x ∈  −1(α) and x ∈ −1n (α) for every n ∈ ω, i.e., the point x is such that min{ξ ∈ θ + 1 : x ∈ anξ } = α for each n. therefore n(x) ≥ α for each n ∈ ω. � proposition 4.6. let x be a topological space, θ a cardinal and  : x → θ + 1 a function. if {−1(α)}α∈θ is a special gδ family, then  has the property nl for every α ≤ θ. proof. let a ⊆ x, α ≤ θ with cfα > ω and (a) is cofinal in α. for every β ∈ θ, we have −1(β) = ⋂ n∈ω g n β such that for every n ∈ ω, {g n β}β∈θ is an increasing family of open sets. since (a) is cofinal in α, for all β ∈ α a \ ⋂ n∈ω g n β 6= ∅, i.e., for all β ∈ α there exists n ∈ ω such that a \ gnβ 6= ∅. there exist n ∈ ω and (βξ)ξ∈cfα increasing sequence with βξ < α, such that a \ g n βξ 6= ∅. now, fixed n ∈ ω, we have that a ⊆ ⋃ ξ∈cfα g n βξ . therefore the family {gnβξ }ξ∈cfα is an open cover of a. if a was lindelöf, there should be β0 countable such that gnβ0 would cover a, which is a contradiction. � proposition 4.7. let x = ∏ n∈ω θ + 1,  : x → θ + 1 defined by (f ) = min{ξ ∈ θ + 1 : f ≤ fξ}, where {fα}α∈θ ⊆ ∏ n∈ω θ + 1 such that for every α < α ′ fα ≤ fα′ . then  has the property nl at every α ≤ θ. proof. by definition −1(α) = {f ∈ x : ∀n ∈ ω f (n) < fα(n)} = ⋂ n∈ω{f ∈ x : f (n) < fα(n)}. set g n α = {f ∈ x : f (n) < fα(n)}, then for every α ∈ θ+1, gnα is an open set in x, and moreover for every n ∈ ω, {g n α}α∈θ is an increasing family. hence {−1(α)}α∈θis a special gδ family of x. proposition 4.6 ends the proof. � corollary 4.8. let x = ∏ n∈ω θ + 1,  : x → θ + 1 defined by (f ) = min{ξ ∈ θ + 1 : f ≤ fξ}, where fξ is a constant function with value ξ for every ξ ≤ θ. then  has the property nl at every α ≤ θ. 254 a. caserta and s. watson remark 4.9. given x = ∏ n∈ω θ + 1, a family {fα}α∈θ ⊆ ∏ n∈ω θ + 1, a sequence of function n(f ) = min{ξ ∈ θ + 1 : f (n) ≤ fξ(n)} and a function (f ) = min{ξ ∈ θ + 1 : f ≤ fξ}, all of them defined in x with value in θ + 1. then  > supn, and the equality does not hold. indeed let f : ω → θ + 1 defined by f (n) = 0 for every n 6= 0 and f (0) = 2, and {fξ}ξ∈θ defined by fξ = ~ξ for every ξ ∈ θ with ξ 6= 2 and f2(n) = 0 for every n ∈ ω \ {0, 2} and f (0) = 2, f (2) = 0. then (f ) = 3 and supnn(f) = 2. there are examples of chain for countable product of ordered spaces, not considering the constant value function, which is a banal example. for example x = ∏ n∈ω\{0} ℵω·n. in (x, ≤) there exists a chain c such that ot(c) = ℵω·ω but not ot(c) = ℵω·ω+1. given {αn}n∈ω ordinals, what is the set of β such that there exists a function f : β →֒ παn? remark 4.10. let κ be a cardinal with cfκ > ω, x = ∏ n∈ω κ + 1 and {fξ}ξ∈κ ⊆ x such that fα ≤ fβ for every α < β < κ. let  : x → κ + 1, defined by (f ) = min{ξ ∈ κ : f ≤ fξ}. we have that the function  has the property nl at κ. let a ⊂ x such that (a) is cofinal in κ. then for every α ∈ κ a * −1(α), i.e., for every α ∈ κ and for every n ∈ ω a * {g ∈ x : g(n) ≤ fα(n)}. let v n, α = {g ∈ x : g(n) ≤ fα(n)}. then {vn,α}n,α is an uncountable open cover of a. if a was lindelöf, there should exist α0 ∈ κ countable such that {vn,α} n∈ω α∈α0 is a cover for a which is a contradiction with cfκ > ω. we give an example of function which has the property nl only at some ordinal. proposition 4.11. let x = ∏ n∈ω θn +1 with every θn cardinal with cfθn > ω, and  : x → κ + 1 defined by (f ) = min{ξ ∈ κ : f ≤∗ fξ} where κ is a cardinal with cfκ > ω and κ > θn for every n ∈ ω, {fξ}ξ∈κ a dominating family in ( ∏ n∈ω θn, ≤∗). then  has the property nl at κ. proof. let a ⊂ x such that (a) is cofinal in κ. then for every α ∈ κ a * −1(α), i.e., for every α ∈ κ a * {g ∈ x : g ≤∗ fα}. then there exists n ∈ ω such that {g(n) : g ∈ a} is unbounded in θn. if not, for every n ∈ ω {g(n) : g ∈ a} is bounded in θn, and since the family {fξ}ξ∈κ is an ≤∗dominating in ( ∏ n∈ω θn, ≤∗), there should exists ξ ∈ κ such that for every g ∈ a g ≤∗ fξ, which is a contradiction. thus there exist n ∈ ω such that for every α ∈ θn a * {g ∈ a : g(n) < α}. let vn,α = {g ∈ a : g(n) < α}. then {vn,α}n,α is an uncountable open cover of a. if a was lindelöf, there should exist α0 ∈ θn countable such that {vn,α} n∈ω α∈α0 is a cover for a which is a contradiction with cfθn > ω. � 5. closed mapping properties in this section we investigate different properties of the projection map, introducing two new definitions. michael spaces and dowker planks 255 let us recall that if f : x → y is a function and a ⊆ x, then the restriction of f to a, f↾a, is closed if the image of a closed subset of a is a closed subset of y . definition 5.1. given two arbitrary topological spaces x and y , we say that the function f : x → y is σ-closed if the image of a closed subset of x is an fσ subset of y . definition 5.2. let x, y be two topological spaces. f : x → y is strongly σ-closed if there exists (kn)n∈ω with kn’s closed subsets of x such that x =⋃ n∈ω kn and f↾kn is closed for every n ∈ ω. remark 5.3. we are dealing with three different properties of the function f : x → y . the following implications hold f closed ⇒ f strongly σ-closed ⇒ f σ-closed example 5.4. first note that for every countable topological space x which is t1, the map f : x → y is strongly σ-closed for every topological space y which is t1. therefore the map f : q → r with f = idq is strongly σ-closed, but it is not a closed map. example 5.5. [ac] under the axiom of choice, the set ω1 can be partitioned in ω stationary sets sn such that ω1 = ⋃ n∈ω sn. in other words,there exists a function f : ω1 → ω + 1 defined by f −1(n) = sn for every n ∈ ω. by definition of stationary set, it follows that for every n ∈ ω and for every club c in ω1 we have c ∩ f −1(n) 6= ∅. clearly f is σ-closed. we claim that f is not strongly σ-closed, which is equivalent to show that for every (kn)n∈ω with kn’s closed subsets of x such that x = ⋃ n∈ω kn there exists n0 ∈ ω such that the map f ↾ kn0 is not closed. indeed let (kn)n∈ω be any countable family of closed subsets of ω1 such that ω1 = ⋃ n∈ω kn. then there exist n0 ∈ ω such that |kn0| > ℵ0. then kn0 is a club in ω1, therefore kn0 ∩ f −1(n) 6= ∅ for every n ∈ ω. thus f (kn0 ) = ω, and so f↾kn0 is not closed, because the set ω is not closed in its compactification ω + 1. lemma 5.6. let x, z be topological spaces, such that x = ⋃ n∈ω kn. let f be a subset of x × z and fn = f ∩ (kn × z). let π : x × z → z be the projection map. then π(f ) = ⋃ n∈ω π↾(kn × z)(fn) proof. note that x × z = ⋃ n∈ω(kn × z), and for every n ∈ ω, fn is a subset of kn × z such that f = ⋃ n∈ω fn. let pn = π↾(kn × z). for every n ∈ ω, pn(fn) ⊆ π(f ). indeed if z ∈ pn(fn), there exists (x, z) ∈ fn such that pn(x, z) = z, therefore there exists (x, z) ∈ f such that π(x, z) = z. thus z ∈ π(f ). on the other side, if z ∈ π(f ), there exists (x, z) ∈ ⋃ n∈ω fn such that π(x, z) = z. therefore there exists n ∈ ω such that (x, z) ∈ fn such that pn(x, z) = z. � the kuratowski theorem is useful: theorem 5.7. given a compact hausdorff space x, the projection map π : x × z → z is a closed map, for every topological space z. 256 a. caserta and s. watson an application is given by: proposition 5.8. given an hausdorff space x and the projection map π : x × z → z, the following implications hold x σ-compact ⇒ π strongly σ-closed ⇒ π σ-closed proof. first we show that π is a strongly σ-closed map. from x σ−compact,let x = ⋃ n∈ω kn where kn’s are compact in x. therefore kn × z is closed in x × z. the kuratowski theorem assures that the projection map π↾kn × z is a closed map, for every topological space z. for the second implication, let x × z = ⋃ n∈ω kn where kn’s are closed. let f be a closed subset of x × z, and fn = f ∩ kn. then for every n ∈ ω fn is a closed subset of kn such that f = ⋃ n∈ω fn. from lemma 5.6, π(f ) = ⋃ n∈ω π↾kn(fn); moreover for every n ∈ ω π↾kn(fn) is closed. it follows that π(f ) is fσ in z. � the use of the small image of the projection map will recur often. so let us state an useful basic property: lemma 5.9. let x and y be two topological spaces and π : x × y → y a projection map. then for every a ⊆ y and b, k ⊆ x × y , (i) a ⊆ (π↾k)♯(b ∩ k) ⇔ (x × a) ∩ k ⊆ b; (ii) a ⊆ π♯(b) ⇔ x × a ⊆ b. proof. a ⊆ (π ↾k)♯(b ∩ k) = {y ∈ y : π−1(y) ∩ k ⊆ b ∩ k} ⇔ ∀y ∈ a π−1(y) ∩ k ⊆ b ∩ k ⇔ ∀y ∈ a {(x, y) ∈ k : x ∈ x ∧ π(x, y) = y} ⊆ b ⇔ {(x, y) ∈ k ∩ (x × a) : π(x, y) = y} ⊆ b ⇔ (x × a) ∩ k ⊆ b. � lemma 5.10. let x, y be two topological spaces, f : x → y an arbitrary function. if f is a closed map, then for every u open in x, f ♯(u ) is an open subset of y . moreover if f is σ-closed map, then f ♯(u ) is a gδ subset of y . proposition 5.11. let the projection π : k × z → z be σ-closed, and x ⊆ z. let u be an open subset in k × z which cover k × x, then there exists h ⊇ x which is a gδ in z such that u cover k × h. proof. set h = π♯(u ). then, since π is σ-closed, h is a gδ in z. by lemma 5.9 follow that k × h ⊆ u , and x ⊆ h. � proposition 5.12. let x be a subset of a topological space z, and k × z ⊆ ⋃ n∈ω kn with every kn lindelöf, and for every n ∈ ω π↾kn is closed, where π : k ×z → z is the projection map. if x is lindelöf, then k ×x is lindelöf. proof. let u be a cover of k × x made by open sets of k × z. without loss of generality we can assume that u is closed under countable unions. fix n ∈ ω, for each z ∈ x, kn ∩ (k × {z}) is lindelöf. for every z ∈ x, since u is closed under countable unions there exists uz ∈ u such that kn ∩ (k × {z}) ⊂ uz. set az,n = (π↾kn) ♯(uz ∩ kn). then az,n is an open subset of z containing z. from lemma 5.9 follow that (k × az,n) ∩ kn ⊆ uz. for a fixed n ∈ ω, {az,n}z∈x is a family of open sets in z which covers x. since x is lindelöf michael spaces and dowker planks 257 there exists countably many zni ’s such that {azni ,n}i∈ω cover x. moreover we have that for every n ∈ ω (k × azn i ,n) ∩ kn ⊆ uzn i . we claim that {uzn i }i,n∈ω covers k × x. indeed, let (k, z) ∈ k × x, then there exists n ∈ ω such that (k, z) ∈ (k × x) ∩ kn. fixed such n, there exists i ∈ ω such that z ∈ azn i ,n. thus (k, z) ∈ (k × azn i ,n) ∩ kn ⊆ uzn i . therefore we have that {uzn i }i,n∈ω is a countable family of open sets of u which covers k × x. � corollary 5.13. let x be a subset of a topological space z, and k = ⋃ n∈ω kn with every kn lindelöf, and for every n ∈ ω π↾kn × z is closed, where π : k × z → z is the projection map. if x is lindelöf, then k × x is lindelöf. remark 5.14. from the proof of lemma 5.12 we can also get that if k, x and π satisfy the assumptions, there exists u ∈ u such that it covers k × x, where u is an open cover of k × x made by open set in k × z closed under countable union. corollary 5.15. let k, z two topological spaces, π : k×z → z the projection map, and x ⊂ z. if (i) x is lindelöf, (ii) k is lindelöf, (iii) π is strongly σ-closed, then k × x is lindelöf. corollary 5.16. let k, z two topological spaces, π : k×z → z the projection map and x ⊂ z. let u be a family of open sets of k × z which covers k × x. if (i) x is lindelöf, (ii) k is lindelöf, (iii) π is strongly σ-closed, then there exists h ⊃ x which is a gδ in z and a countable subfamily of u which covers k × h. proof. without loss of generality we can assume that u is closed under countable unions. by corollary 5.15, k ×x is lindelöf, therefore there exists u0 ⊆ u countable such that cover k × x. set u0 = ⋃ u0. then u0 ∈ u. by proposition 5.11, there exists h ⊇ x which is a gδ in z, such that u ⊇ k × h. � lemma 5.17. let u be a family of open sets in k × z which covers k × x, with x ⊂ z. let π : k × z → z be the projection map. if (i) x is lindelöf, (ii) k is lindelöf, (iii) k = ⋃ n∈ω kn with kn closed and π↾kn × z is closed, then there exists a countable subfamily of u that covers k × x. corollary 5.18. let k be a σcompact space, x a lindelöf subset of a topological space z. let u be a family of open subsets in k × z which cover k × x, then there exists h ⊇ h which is a gδ in z, and u0 ⊆ u countable which cover k × h. 258 a. caserta and s. watson 6. lindelof haydon planks in this section we construct a dowker-style plank, i.e., a variation of dowker’s idea of 1955 in which we take the subspace of all points in the product lying below the graph of a function (see [3]). planks have been extensively studied by watson in [9]. definition 6.1. let x, z be topological spaces, θ a cardinal, h : z → θ + 1 an arbitrary function, and  : x → θ + 1 surjective. define the plank y,h = {(x, z) ∈ x × z : h(z) ≥ (x)} for every ξ ≤ γ ≤ θ denote y,h↾(ξ, ·) = {(x, z) ∈ y,h : (x) < ξ} and y,h↾(ξ, γ) = {(x, z) ∈ y,h : (x) < ξ ∧ h(z) < γ}. we investigate more in detail the relation between the plank and the functions. in the following, unless we state otherwise, we assume that the x, z and the function h and  are defined as in the definition 6.1 proposition 6.2. let α ≤ θ, if  has the property nl at α, then ( ∃b lindelöf : y,h↾(α, α) ⊆ b ⊆ y,h ⇒ h is m ichael at α). proof. let α ≤ θ with cfα > ω, and b lindelöf subset of y,h such that y,h ↾(α, α) ⊆ b. let f ⊂ z be closed such that for every z ∈ f, h(z) < α. then b ∩ (x × f ) is lindelöf. let a = πx (b ∩ (x × f )). thus a is a lindelöf subset of x, such that for every x ∈ a (x) < α. from  nl at α we have (a) is not cofinal in α, i.e, there exist β < α such that for every x ∈ a (x) ≤ β. since  is surjective, for every z ∈ f we can choose x ∈ x with (x) = h(z). then (x, z) ∈ y,h↾(α, α) ∩ (x × f ), therefore (x, z) ∈ b ∩ (x × f ). it follows that x ∈ a. hence for every z ∈ f there exists x ∈ a with (x) = h(z) ≤ β. thus supz∈fh(z) ≤ β < α, i.e., h is michael at α. � corollary 6.3. let θ be a cardinal. assume that  has the property nl at each α ≤ θ. if y,h is lindelöf then for each α ∈ θ, h is michael at α. proof. let α ∈ θ with cfα > ω. assume by contradiction that h is not michael at α. then y,h is lindelöf and y,h↾(α, α) ⊂ y,h. from proposition 6.2 follows that h is michael at α. � from proposition 3.9 and proposition 6.2 follow: corollary 6.4. let θ be a cardinal with cfθ > ω. if (i)  has the property nl at θ, (ii) for each α < θ, h(z) ∩ (α, θ) 6= ∅, then y,h × (z \ h −1({θ}) is not lindelöf. now we want to investigate when the plank y,h is lindelöf, and we give an inductive proof. first we need the following lemma. michael spaces and dowker planks 259 lemma 6.5. let u be a family of open sets in −1([0, α]) × z which covers y,h↾(α + 1, ·). let π :  −1([0, α]) × z → z be the projection map. if (i) there exists u0 ∈ u which covers  −1([0, α]) × h−1([α, θ]), (ii) h is michael at α, (iii) foe each ξ < α, y,h↾(ξ + 1, ·) is lindelöf;, (iv) π is σ-closed, then there exists a countable subfamily of u that covers y,h↾(α + 1, ·). proof. note that y,h↾(α+1, ·) = ( −1([0, α])×h−1([α, θ]))∪( ⋃ ξ<α y,h↾(ξ+1, ·). let u0 ∈ u that covers  −1([0, α]) × h−1([α, θ]). if cfα = ω, there exists an increasing sequence of ordinal (αn)n∈ω such that⋃ ξ<α y,h↾(ξ + 1, ·) = ⋃ n∈ω y,h↾(α + 1, ·), therefore there exists u1 ⊂ u which cover ⋃ ξ<α y,h↾(ξ + 1, ·). then u1 ∪ {u0} is a countable subcover of u that covers y,h↾(α + 1, ·). assume that cfα > ω. let u c0 = ( −1([0, α]) × z) \ u0. then u c 0 is closed in −1([0, α]) × z and for every (x, z) ∈ u c0 ∩ y,h↾(α + 1, ·) we have that (x) < α and h(z) < α. from (iv) follow that c = π(u c0 ) is an fσ subset of z, and for every z ∈ c h(z) < α. from (ii) follow that δ = supz∈ch(z) < α. we claim that y,h↾(α + 1, ·) \ u0 ⊆ y,h↾(δ + 1, ·). let (x, z) ∈ y,h↾(α + 1, ·) \ u0. from (x, z) ∈ u c0 , follow that z ∈ c; from h(z) < δ and (x) ≤ h(z), follow that (x, z) ∈ y,h↾(δ + 1, ·). by hypothesi, y,h↾(δ + 1, ·) is lindelöf, therefore there exists a countable subfamily u1 ⊂ u which is a cover for a y,h↾(δ + 1, ·). thus {u0} ∪ u1 is a countable subcover for u that covers y,h↾(α + 1, ·). � proposition 6.6. let z be a lindelöf space, θ a cardinal and α ≤ θ. if (i) h is michael at α, (ii) h−1([α, θ]) is lindelöf, (iii) −1([0, α]) is lindelöf, (iv) for each ξ < α, y,h↾(ξ + 1, ·) is lindelöf, (v) π : −1([0, α]) × z → z is strongly σ-closed, then y,h↾(α + 1, ·) is lindelöf. proof. let u be a cover of y,h↾(α + 1, ·) made by open sets of  −1([0, α]) × z, and without loss of generality we can assume that it is closed under countable union. note that y,h ↾ (α + 1, ·) = ( −1([0, α]) × h−1([α, θ])) ∪ ( ⋃ ξ<α y,h ↾ (ξ + 1, ·). from corollary 5.15, there exists u0 ⊂ u countable such that it covers −1([0, α]) × h−1([α, θ]). let u0 = ⋃ u0, then u0 ∈ u. lemma 6.5 ends the proof. � proposition 6.7. let z be a lindelöf space, θ a cardinal, and α ≤ θ. if for each β ≤ α (i) h is michael at β, (ii) h−1([β, θ]) is lindelöf, (iii) −1([0, β]) is lindelöf, (iv) π : −1([0, β]) × z → z is strongly σ-closed, then y,h↾(α + 1, ·) is lindelöf. 260 a. caserta and s. watson proof. assume that h is michael at β for every β ≤ α. from proposition 6.6, it remains to show that y,h↾(β + 1, ·) is lindelöf for every β < α. suppose not, there exists β < α such that y,h↾(β + 1, ·) is not lindelöf, and assume that β is the minimum ordinal with this property. then, for every γ < β, y,h↾(γ + 1, ·) is lindelöf, and for every γ < β, h is michael at γ. from proposition 6.6 follow that y,h↾(β + 1, ·) is lindelöf, a contradiction. � theorem 6.8. let z be a lindelöf space, θ a cardinal. if for each α ≤ θ (i) h is michael at α;, (ii) h−1([α, θ]) is lindelöf;, (iii) −1([0, α]) is lindelöf, (iv) π : −1([0, α]) × z → z is strongly σ-closed, then y,h is lindelöf. note that the problem to determinate when given an arbitrary topological space y , the product y,h × y is lindelöf becomes a problem to find condition on ĥ and  so that y ,ĥ is a lindelöf space where ĥ : z × y → θ + 1. simply applying proposition 6.2 and corollary 6.8 to ĥ the following corollary give us conditions to determinate when y,h × y is lindelöf. corollary 6.9. let z be a lindelöf space, x, y a topological spaces, θ a cardinal, ĥ : z × y → θ + 1. if for each α ≤ θ (i) ĥ is michael at α, (ii) ĥ−1([α, θ]) × y is lindelöf, (iii) −1([0, α]) is lindelöf, (iv) π : ̂−1([0, α]) × z × y → z × y is strongly σ-closed, then y,h × y is lindelöf. the next theorem give us a necessary condition to find a michael space: theorem 6.10. let x be a topological space, θ a cardinal with uncountable cofinality and h : c → θ + 1. if (i) qc = h −1({θ}), (ii) for each α ≤ θ, h is michael at α, (iii) for each α < θ, h(c) ∩ (α, θ) 6= ∅, (iv)  has the property nl at θ, (v) for each α ≤ θ, −1([0, α]) is lindelöf, (vi) for each α ≤ θ, π : −1([0, α]) × z → z is strongly σ-closed, then y,h is a michael space. 7. special cases one special case is obtained choosing x = θ+1 and the map  as the identity map on θ + 1. the plank y,h becomes yh = {(α, z) ∈ (θ + 1) × z : h(z) ≥ α} michael spaces and dowker planks 261 subset of (θ + 1) × z, and it is an haydon plank [(see [5]). for every α ∈ θ denote yh↾α = {(δ, z) ∈ yh : δ < α}. in this case, the plank is characterized as lindelöf, and it is also an example of michael space. theorem 7.1. let z be a lindelöf space, h : z → θ + 1 a function, θ a cardinal with cfθ > ω. then yh is lindelöf if and only if for every α ≤ θ (i) h is michael at α; (ii) h−1([α, θ]) is lindelöf. proof. let yh be lindelöf. then, for every α ∈ θ, yh ↾α + 1 is lindelöf. by proposition 6.2, h is michael at α for every α ∈ θ. moreover yh ∩ ({α} × z) ∼= h−1([α, θ]). theorem 6.8 ends the proof. � corollary 7.2. let θ be a cardinal with cfθ > ω, h : c → θ + 1 an arbitrary function. then yh is lindelöf if and only if for each α ≤ θ, h is michael at α. lemma 7.3. let y be a topological space, θ cardinal with cfθ > ω. if yh × y is lindelöf, then for every α≤θ, ĥ is michael at α, where ĥ : z × y → θ + 1. moreover, if h−1([α, θ]) × y is lindelöf for every α ≤ θ, then the converse holds. proof. follows from corollary 6.8 (applied to ĥ), proposition 6.2 and remark 4.2. � corollary 7.4. let y be a topological space, θ cardinal, h : z → θ + 1 a function such that for every α ≤ θ h−1([α, θ]) × y is lindelöf. then the following statements are equivalent: (i) h is a(y )-michael function, (ii) yh × y is lindelöf. proof. follows from theorem 7.3 and proposition 3.7. � corollary 7.5. let θ be a cardinal with uncountable cofinality, z a lindelöf space and h : z → θ + 1 a function such that for every α < θ h(z)∩(α, θ) 6= ∅. then yh × (z \ h −1({θ})) is not lindelöf proof. follows from corollary 6.4. � corollary 7.6. let θ be a cardinal with uncountable cofinality, z a lindelöf space and h : z → θ + 1 a function. if (i) for each α ≤ θ, h is michael at α, (ii) for each α < θ, h(z) ∩ (α, θ) 6= ∅, (iii) for each α ≤ θ, h−1([α, θ]) is lindelöf, then yh ⊆ (θ + 1) × z is a lindelöf space such that yh × (z \ h −1({θ})) is a non-lindelöf space. proof. follows from corollary 6.8 and corollary 7.5. � 262 a. caserta and s. watson theorem 7.7. let θ be a cardinal with uncountable cofinality and h a kmichael function defined on c such that (i) qc = h −1({θ}), (ii) h(c) ∩ (α, θ) 6= ∅ for every α < θ. then yh, subspace of (θ + 1) × c, is a michael space. we give some other examples of planks which are michael spaces. definition 7.8. a special plank is given by choosing x = ∏ n∈ω θ + 1 with θ of uncountable cofinality, and the map  : x → θ + 1 defined by (f ) = min{ξ ∈ θ + 1 : f ≤ fξ}, where fα is a constant function with value α for every α ≤ θ. we denote this plank y p,h. theorem 7.9. let θ be a cardinal with cfθ > ω and h a kmichael function defined on c such that (i) qc = h −1({θ}), (ii) for every α < θ, h(c) ∩ (α, θ) 6= ∅. then y p,h is a michael space. proof. by corollary 4.8, follow that the map  has the property nl at α for every α ≤ θ. moreover, for every α ≤ θ, −1([0, α]) = {f ∈ ∏ n∈ω θ + 1 : ∀n ∈ ω f (n) ≤ α} is a compact subset of ∏ n∈ω θ + 1. by lemma 5.8, the projection π : −1([0, α]) × z → z is strongly σ-closed. theorem 6.10 ends the proof. � another special plank is obtained for a particular choice of the map . definition 7.10. let x = ∏ n∈ω θn + 1 with every θn cardinal with uncountable cofinality, and  : x → κ + 1 defined by (f ) = min{ξ ∈ κ : f ≤∗ fξ} where κ is a cardinal with cfκ > ω and {fξ}ξ∈κ is a dominating family in ( ∏ n∈ω θn, ≤∗). we denote this plank y ∏ ,h . remark 7.11. the definition of a dominating family and the definition of the map  in the plank y ∏ ,h , imply that κ > θn for every n ∈ ω. indeed considering the special case in which the family {fξ}ξ∈κ is a family of constant functions, we need to have the function which assumes constant value θn. therefore κ > θn + 1 for every n ∈ ω. remark 7.12. let (x, ≤) be a partial order, f ⊆ x with f = {fξ}ξ∈κ a dominating family in x, (i.e. for all x ∈ x, there exists fξ ∈ f such that x ≤ fξ). define  : x → f by (x) = min{fξ ∈ f : x ≤ fξ}. we have that  is surjective if and only if fα � fβ for every α < β. remark 7.13. let x = ∏ n∈ω θn + 1 with every θn cardinal with uncountable cofinality, κ a cardinal with cfκ > ω and {fξ}ξ∈κ a dominating family in ( ∏ n∈ω θn, ≤∗). the map  : x → κ + 1 defined by (f ) = min{ξ ∈ κ : f ≤∗ fξ} might not be surjective. since f ′ = {fξ ∈ f : fξ ∈ (x)} is still a dominating family of x, when (x) has order type κ, we can assume without loss of generality that  is surjective. further,if the dominating family is a scale of x, michael spaces and dowker planks 263 we can consider f ′ , the dominating family of minimum cardinality which is a scale, i.e., |f ′ | = d. such a family is a dominating family with order type d and the map  ′ : x → f ′ defined by  ′ (x) = min{fξ ∈ f ′ : x ≤ fξ} is surjective. an example of y ∏ ,h -plank is given by cardinal of countable cofinality. indeed, from the theorem of shelah [b.m. [1]], given θ with cfθ = ω, there exists an increasing sequence of regular cardinals {θn}n∈ω cofinal in θ, and a scale {fξ}ξ∈θ+ on ( ∏ n∈ω θn, ≤∗). in this case choose x = ∏ n∈ω θn + 1 and the map  : x → θ+ + 1 defined by (f ) = min{ξ ∈ θ+ : f ≤∗ fξ}. then we have: theorem 7.14. let θ be a cardinal with cfθ > ω and h a kmichael function defined on c such that (i) qc = h −1({θ}), (ii) for every α < θ, h(c) ∩ (α, θ) 6= ∅. then y ∏ ,h is a michael space. proof. for every α ∈ κ, we have −1([0, α]) = {f ∈ x : f ≤∗ fα} = ⋃ f ∈[ω]<ω {f ∈ x : ∀n /∈ f f (n) ≤ fα(n)}. therefore for every α ∈ κ, the set  −1([0, α]) ⊂ x is σ-compact, hence by lemma 5.8, the projection map π : −1([0, α]) × z → z is strongly σ-closed. moreover from proposition 4.11, the map  has the property nl at κ. theorem 6.10 ends the proof. � 8. the cardinal l if x is a non-lindelöf space, l(x) denote the minimum cardinality of an uncountable open cover of x with no countable subcover, and if x is lindelöf, define l(x) = ∞. note that for a non-lindelöf space, l(x) ≤ w(x), where w(x) denote the weight of the topological space x, and l(x) is either a regular cardinal or has countable cofinality. the following lemma give us some relations between the l cardinals of related spaces. lemma 8.1. let x, y be topological spaces. the following properties hold: (i) if x is lindelöf and x × y is not lindelöf, then l(x × y ) ≤ |y |. (ii) if f ⊆ x is closed and not lindelöf space, then l(x) ≤ l(f ). (iii) if f is a continuous open map, such that f (x) is not lindelöf space, then l(f (x)) = l(x). proof. (i) let u be an open cover of x × y witnessing l(x × y ). for every y ∈ y , let u(y) = {un(y) : n ∈ ω} ⊂ u be a countable open subcover of x × {y}. thus v = {un(y) : n ∈ ω ∧ y ∈ y } ⊂ u is an open cover of x × y , such that |v| ≤ |y | with no countable subcover. therefore l(x × y ) ≤ |y |. (ii) let u be an open cover of f with |u| = l(f ) with no countable subcover. then u ∪ {f c} is an open cover for x of the same kind. 264 a. caserta and s. watson (iii) let u be an open cover of f (x) with |u| = l(f (x)) with no countable subcover. then f −1(u) is an open cover for x. thus l(x) ≤ l(f (x)). if v is an open cover of x with |v| = l(x) with no countable subcover. then f (v) is an open cover for f (x) of the same kind. � lemma 8.2. let x, y be topological with x lindelöf. for every f ⊆ y closed such that l(x × y ) > |f |, x × f is lindelöf. proof. if x × y is lindelöf, then l(x × y ) = ∞, and x × f is lindelöf. now, assume that x ×y and x ×f are not lindelöf. since x ×f is closed in x ×y , from lemma 8.1 we have that |f | < l(x × y ) ≤ l(x × f ). lemma 8.1 ends the proof. � corollary 8.3. let x, y be topological spaces with x lindelöf and l(x×y ) = |y |. then for every closed f ⊆ y with |f | < |y | follow that x × f is lindelöf. lemma 8.4. let x, y be topological spaces with x lindelöf and l(x × y ) = |y |, then y is not union of less than |y | many closed subsets of y with cardinality less than |y |. proof. let u = {uξ}ξ<|y | be an open cover of x ×y witnessing l(x ×y ). let κ cardinal with κ < |y |. assume by contradiction that y = ⋃ ξ∈κ yξ where for every ξ ∈ κ yξ are closed in y and |yξ| < |y |. therefore from corollary 8.3 follow that x × yξ is lindelöf for every ξ ∈ κ, and so there exists uξ ⊂ u countable subcover of x × yξ . set v = {uξ : ξ ∈ κ} ⊂ u. then v ⊆ u is an open cover of x × y of size κ. from l(x × y ) = |y | follow that there exist a countable subcover v ′ ⊂ v of x × y . then v ′ is also a countable subcover from u which is a contradiction. � let x, y be topological spaces, θ a cardinal, and p (x, y, θ) states that x is a lindelöf space such that x × y is not lindelöf space and l(x × y ) = θ. theorem 8.5. let x be a topological space and θ a cardinal. if y satisfies p (x, y, θ) and |y | < κ, with κ infinite cardinal, then there exists y ′ which satisfies p (x, y ′ , θ) and |y ′ | = κ. proof. let y ′ = y ⊕ αd(κ), where αd(κ) is the one-point compactification of a discrete set of cardinality κ. clearly |y ′ | = κ. since the space x × y is a closed subset of x × y ′ , it follows that x × y ′ is not lindelöf. it remains to show that l(x×y ′ ) = θ, assuming that l(x×y ) = θ. since the space x×y is a closed subset of x × y ′ , from lemma 8.1 follow that l(x × y ) ≤ l(x × y ′ ), and so l(x × y ′ ) ≥ θ. now, let u be an open cover for x × y of size θ with no countable subcover. we have that x × y ′ is homeomorphic to (x × y ) ⊕ (x × αd(κ)) hence it follows that u is an open family in x × y ′ such that u ∩ (x × αd(κ)) = ∅ for every u ∈ u . let v = u ∪ {x × αd(κ)}. then v is an open cover of y ′ of size θ with no countable subcover. � michael spaces and dowker planks 265 remark 8.6. in other words we have that for a fixed topological space x and a cardinal θ, if there exists y such that p (x, y, θ), then the set ax,θ = {κ : κ is cardinal ∧ ∃y p (x, y, θ) ∧ |y | = κ} is non empty and ax,θ = [min ax,θ, +∞). we conclude this work showing that if there is a michael space, then under some conditions involving singular cardinals, there must be one which is a haydon plank. theorem 8.7. let x be a lindelöf space, y a topological space such that x × y is not lindelöf, θ a cardinal with cfθ = ω and l(x × y ) = |y | = θ. let cy be any compactification of y . then there exists a function f : cy → θ + 1 such that (i) f −1({θ}) = cy \ y , (ii) for every α ≤ θ, f is michael at α, (iii) for every α < θ, f (cy ) ∩ (α, θ) 6= ∅. proof. let θ = l(x × y ), and u be an open cover of x × y witnessing l(x × y ). fix an enumeration {yξ}ξ<θ of y of order type θ. given y ∈ y , let u(y) = {un(y) : n ∈ ω} ⊂ u a countable open subcover of x × {y}. thus v = {un(y) : n ∈ ω ∧ y ∈ y } ⊂ u is an open cover of x ×y , such that |v| = θ. let cy be a compactification of y . define the function f : cy → θ + 1 as follows: for every y ∈y , f (y) = sup{γ ∈ θ : x ×{y} * ∪ξ<γ (∪n∈ωun(yξ))} and for every y ∈ cy \y f (y) = θ. then, by definition of v, there is not y ∈ y such that x × {y} * ∪ξ<α(∪n∈ωun(yξ)) for every α ≤ θ. thus f −1({θ}) = cy \ y . let α ∈ θ with cfα > ω, and f ⊂ cy closed such that f (y) < α for every y ∈ f . assume by contradiction that supy∈ff(y) = α. by definition of α we have that x × f ⊆ ∪ξ<α(∪n∈ω un(yξ)). then {un(yξ) : n ∈ ω ∧ ξ < α} is an uncountable cover of x × f with f compact. we want to show that it has no countable subcover which contradict x × f to be lindelöf. indeed if {um(yξn )}n,m∈ω was a countable subcover of x × f . let ν = supn∈ωξn. since cfα > ω, ν < α. by definition of ν there exists y ∈ f such that x × {y} * ∪n∈ωum(yξn ) which is a contradiction. now, by contradiction, there exists α ∈ θ such that f (cy ) ∩ (α, θ) = ∅, i.e., there exists α ∈ θ such that for every y ∈ y, f (y) < α. therefore for every y ∈ y, x × {y} ⊆ ∪ξ<α(∪n∈ωun(yξ)). thus {un(yξ) : n ∈ ω ∧ ξ < α} is an open cover of x × y with α < θ. by definition of l(x × y ) = θ, there exists a countable subcover for x × y from {un(yξ) : n ∈ ω ∧ ξ < α}, and therefore from u, which is a contradiction. � theorem 8.8. let x be a lindelöf space such that x × y is not lindelöf, θ a regular cardinal such that l(x × y ) = θ. let cy be any compactification of y . then there exists a function f : cy → θ + 1 such that (i) f −1({θ}) = cy \ y , (ii) for every α ≤ θ, f is michael at α, (iii) for every α < θ, f (cy ) ∩ (α, θ) 6= ∅. 266 a. caserta and s. watson proof. let θ = l(x × y ). fix an enumeration {uξ}ξ<θ of an open cover of x × y witnessing l(x × y ). let cy be a compactification of y . define the function f : cy → θ + 1 as follows: for every y ∈ y , f (y) = sup{γ ∈ θ : x × {y} * ∪ξ<γ uξ} and for every y ∈ cy \ y f (y) = θ. since θ is regular and {uξ}ξ<θ is an open cover of x × y , there is not y ∈ y such that for every α ≤ θ x × {y} * ∪ξ<αuξ. thus f −1({θ}) = cy \ y . let now α ∈ θ with cfα > ω, and f ⊂ cy closed such that f (y) < α for every y ∈ f . assume by contradiction that supy∈ff(y) = α. by definition of α we have that for every β ≥ α, x × {y} ⊆ ∪ξ<β uξ for every y ∈ f , therefore x × f ⊆ ∪ξ<αuξ. then {uξ}ξ<θ is an uncountable cover of x × f with f compact. we want to show that it has no countable subcover which contradict x × f to be lindelöf. indeed if {uξn}n∈ω was a countable subcover of x × f . let ν = supn∈ωξn. since cfα > ω, ν < α. by definition of ν there exists y ∈ f such that x × {y} * ∪n∈ωuξn , contradiction. now, by contradiction, there exists α ∈ θ such that f (cy ) ∩ (α, θ) = ∅, i.e., there exists α ∈ θ such that for every y ∈ y, f (y) < α. therefore for every y ∈ y, x × {y} ⊆ ∪ξ<αuξ. thus {uξ}ξ<α is an open cover of x × y with α < θ. by definition of l(x × y ) = θ, there exists a countable subcover for x ×y from {uξ}ξ<α, and therefore from {uξ}ξ<θ, which is a contradiction. � in the theorem 8.8, θ is a regular cardinal, we do not need any assumption about the cardinality of y (as in theorem 8.7) because the open cover{uξ}ξ<θ witnessing l(x × y ) = θ is never cofinal. this guarantee that f −1({θ}) = cy \ y . from theorem 7.7 it follows: theorem 8.9. let m be a michael space, θ a regula cardinal such that l(m × p) = θ. then there exists a function f and yf ⊆ (θ + 1) × c which is a michael space. the aim of proposition 8.7 and proposition 8.8 is to produce the following statement: given a lindelöf space x such that l(x × y ) = θ, there exists f and yf ⊆ (θ + 1) × cy such that yf is lindelöf and yf × y is not lindelöf, where cy is any compactification of y . we require the property that for all α ≤ θ, f −1([α, θ]) is lindelöf. clearly this is always true when y admits an hereditarily lindelöf compactification. when does the property hold? is there a function f : cy → θ + 1 such that satisfy the property of proposition 8.7 when x is a lindelöf space, y a topological space such that x × y is not lindelöf, θ a cardinal of countable 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[9] s. watson, the construction of topological spaces: planks and resolutions, in m. hus̄ek and j. van mill (eds.), recent progress in general topology, 673–757, northholland 1992. received january 2009 accepted june 2009 a. caserta (agata.caserta@unina2.it) dipartmento di matematica, seconda universitá degli studi di napoli, caserta 81100, italia s. watson (watson@hilbert.math.yorku.ca) department of mathematics and statistics, york university, toronto m3j1p3, canada panxuagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 41-50 criteria of strong nearest-cross points and strong best approximation pairs wenxi pan and jingshi xu ∗ abstract. the concept of strong nearest-cross point (strong n.c. point) is introduced, which is the generalization of strong uniqueness of best approximation from a single point. the relation connecting to localization is discussed. some criteria of strong n.c. points are given. the strong best approximation pairs are also studied. 2000 ams classification: 41a50, 41a28, 41a65. keywords: strong nearest-cross point, local strong nearest-cross point, strong best approximation pair. 1. introduction in [6], [5], [9] the first author of the paper studied the nearest cross points (in short, n.c. points) of two subsets of a normed space. more precisely, let g and f be two disjoint subsets of a normed space x. a point y0 ∈ g is called a n.c. point of g to f if ρ(y0, f ) = ρ(g, f ), where ρ(g, f ) = infy∈g,x∈f ρ(y, x), ρ(y, x) = ‖x − y‖ is the norm of x − y in the space x. moreover, if x0 ∈ f satisfies ρ(x0, y0) = ρ(f, g), we say that (x0, y0) is a best approximation pair of f and g. for details, one can see [4]. obviously, if (x0, y0) is a best approximation pair of f and g, then y0 is a n.c. point of g to f , and x0 is the best approximation of y0 from f . the analogous result for x0 also holds. however, the inverse is not true. if both n.c. points of f to g and g to f exist, a best approximation pair of f and g may not exist. but if n.c. points of g to f exist and f is a proximal set, then the best approximation pair of f and g exists. in [5], the author discussed the uniqueness of n.c. points (if it exists) and obtained that the n.c. point of g to f is unique if g is strict convex and f is convex. in this paper, we shall discuss a property which is stronger than being a n.c. point, which we will call a strong n.c. point. a strong n.c. point is ∗the corresponding author jingshi xu was supported by the nnsf (no. 60474070) of china 42 w. pan and j. xu the generalization of strong best approximation in a single best approximation problem. for strong best approximation, one can see [3],[7],[2] in detail. the organization of the paper is as follows. in section 2, we will give the definition of strong n.c. point. in section 3, we shall discuss the criteria of strong nearest cross point. in section 4, we shall discuss strong n.c. points and strong best approximation pairs by way of the concept of cusp. and we shall give more examples about the relation between strong best approximation pairs and strong n.c. points. finally, we declare that we will work in complex norm spaces in this paper and use the following notation. let x be a normed space. denote by x∗ the dual space of x. for a complex number u, we shall write re u to denote the real part of u. if f denotes a subset of a normed space x, then ‖f ‖ = supx∈f ‖x‖. 2. definition of strong n.c. point definition 2.1. let f and g be two disjoint sets, y0 ∈ g and a constant r, 0 < r < 1. if the condition (1) ρ(y, f ) − ρ(y0, f ) ≥ rρ(y, y0) holds for every y ∈ g, then y0 is called a strong n.c. point of g to f . notice that a strong nearest point is a n.c. point. in fact, since ρ(y, f ) − ρ(y0, f ) ≥ 0 for every y ∈ g, we have ρ(g, f ) = ρ(y0, f ). then, for every y ∈ g, y 6= y0, ρ(y, f ) > ρ(y0, f ), thus the n.c. point is unique. in definition 2.1, the constant r, r < 1 holds automatically because |ρ(y, f ) − ρ(y′, f )| ≤ ρ(y, y′) always holds. in fact, to say y0 is a strong n.c. point, it suffices to remark that it exists a sufficiently small r, such that (1) holds. if f is a singleton x0, y0 is a strong n.c. point of g to f, then y0 is the strong (unique) best approximation of x0 from g; see [3], [7], [2]. definition 2.2. consider f, g, r, y0 as in definition 2.1. if (1) holds only for y ∈ v0 ∩ g, v0 a neighborhood of y0, then we say y0 is a local strong n.c. point of g to f. obviously, if y0 is a strong n.c. point then y0 is a local strong n.c. point, but the converse does not hold in general, as the following example shows. example 2.3. in the euclidean space r2, let f = {(ξ, η) : (ξ−2)2+η2 = 1, ξ ≥ 2}, g = {(ξ, η) : ξ2 + η2 = 1, ξ ≤ 0}. if y0 = (0, 1), then y0 is a local strong n.c. point of g tof , since for every y ∈ g, y be near y0, ρ(y, f ) is equivalent to ρ(y, x0) = 2 + ρ(y, y0). x0 = (2, 1). so ρ(y, f ) − ρ(y0, f ) is equivalent to ρ(y, y0). but y0 is not a strong n.c. point. in fact, choose y ′ suffice to (0, −1), then ρ(y0, y ′) is to 2, ρ(y0, f ) = 2, thus ρ(y ′, f ) − ρ(y0, f ) converges to 0, so (1) does not hold. moreover, (2,-1) is a n.c. point of f to g, (0,1) is a nearest cross point of g to f , and (2,-1), (0,1) are not strong n.c. points. in the above example, f and g are not convex sets, but under convexity, we shall have different results. to state our results, we need the following lemma, which is well known. criteria of strong nearest-cross points 43 lemma 2.4. let (x, ρ) be a metric space and let f ⊂ x. then the function ρ(·, f ) is uniformly continuous. moreover if f is a convex set, then ρ(·, f ) is a convex function. theorem 2.5. let f , g be convex sets, and let y0 ∈ g. y0 is a strong nearest cross point of g to f if and only if y0 is a local strong n.c. point of g to f . proof. as above stated, we only need to show that if y0 is a local strong n.c. point, then y0 is a strong n.c. point. let v be a neighborhood of y0, where y0 is a strong n.c. point of g ∩ v to f . if y0 is not a strong n.c. point of g to f , then for every rn → 0, there exist yn ∈ g, n = 1, 2, . . . , such that ρ(yn, f ) − ρ(y0, f ) < rn‖yn − y0‖. in the segment [yn, y0], pick zn = λnyn + (1 − λn)y0, 0 < λn < 1, λn → 0, such that for n sufficiently large, zn ∈ g ∩ v. by lemma 2.4, ρ(y, f ) is a convex function. thus, ρ(zn, f ) ≤ λnρ(yn, f ) + (1 − λn)ρ(y0, f ) < λnrn‖yn − y0‖ + λnρ(y0, f ) + (1 − λn)ρ(y0, f ) = λnrn‖yn − y0‖ + ρ(y0, f ). so ρ(zn, f ) − ρ(y0, f ) < λnrn‖yn − y0‖. since ‖yn − y0‖ = 1/λn‖zn − y0‖, ρ(zn, f ) − ρ(y0, f ) ≤ rn‖zn − y0‖, zn ∈ g ∩ v. this contradicts the definition of a local strong n.c. point. this completes the proof. � in theorem 2.5 we suppose f and g are convex sets. if one of them is not convex, then the result does not hold. example 2.6. in the euclidean space r2, let g = {(ξ, η) : ξ2 + η2 = 1, ξ ≤ 0}. notice that g is not convex and f = {(2, 0)} is a singleton. then y0 = (0, 1) is a local strong n.c. point of g to f, but y0 is not a strong n.c. point of g to f . in fact, pick y = (0, −1). then ρ(y, f ) − ρ(y0, f ) = √ 5 − √ 5 = 0, but ‖y − y0‖ = 2. example 2.7. in the euclidean space r2, let g = {(ξ, η) : −3 ≤ ξ ≤ 2, η = 0} be a convex set, indeed, a segment, and f = {(ξ, η) : ξ2 + η2 = 25} a nonconvex set. then y0 = (2, 0) is a local strong n.c. point, but y0 is not a strong n.c. point, even if it is not a n.c. point. in fact, (-3,0) is a strong n.c. point. 3. kolmogorov type and differential type criteria of strong n.c. points in [6], the author gave sufficient and necessary conditions for a point to be a n.c. point by means of linear functions and differentials. following the same idea we first obtain a sufficient condition for strong n.c. points. theorem 3.1. let f , g be disjoint sets, y0 ∈ g, and r a constant, 0 < r < 1. if y0 satisfies one of the following conditions, then y0 is a strong n.c. point of g to f . (i) for every ǫ > 0, there exists f ǫ ∈ x∗, ‖f ǫ‖ = 1, such that inf x∈f re f ǫ(x−y) = ρ(y0, f ) and for every y ∈ g, re f ǫ(y0 − y) + ǫ ≥ r‖y0 − y‖ holds. 44 w. pan and j. xu (ii) for every ǫ > 0, and y ∈ g, there exists f y,ǫ ∈ x∗, ‖f y,ǫ‖ = 1, such that inf x∈f re f y,ǫ(x − y0) ≥ ρ(y0, f ), and re f y,ǫ(y0 − y) + ǫ ≥ r‖y0 − y‖. proof. obviously, (i) implies (ii). so we only need to show (ii). for this, given ǫ > 0, y ∈ g, and f y,ǫ as in (ii), we have r‖y0−y‖ ≤ re f y,ǫ(y0−y)+ǫ = re f y,ǫ(y0−x)+re f y,ǫ(x−y)+ǫ ≤ ‖x−y‖−ρ(y0, f )+ǫ. if in the right side of the above inequality, we take the infimum over x ∈ f, we have r‖y0 − y‖ ≤ ρ(y, f ) − ρ(y0, f ). thus y0 is the strong n.c. point of g to f . this completes the proof. � one may ask immediately if either (i) or (ii) are necessary conditions. to answer this question, we begin with directional derivatives of the distance function ρ(y, f ) (for details one can see [6]). from lemma 2.4, we know ρ(y, f ) is a convex function. furthermore, ρ′+(y, h, f ) = lim t→0+ ρ(y + th, f ) − ρ(y, f ) t exists for h 6= 0 ∈ x, and ρ′+(y, h, f ) is a subadditive homogenous function in the variable h. theorem 2.2 in [6] says that whenever y0 is a n.c. point of g to f, ρ′+(y0, y − y0) ≥ 0 for all y ∈ g. in the following, we shall obtain a necessary condition for strong n.c. points. theorem 3.2. let f , g be two disjoint convex sets of x. if y0 ∈ g is a strong n.c. point of g to f, then (ds) ρ ′ +(y0, y − y0, f ) ≥ r‖y − y0‖ for all y ∈ g, r > 0. proof. from definition 2.1, ρ(y, f ) − ρ(y0, f ) ≥ r‖y − y0‖. put h = y − y0, ρ(y0 + th, f ) − ρ(y0, f ) t ≥ r‖th‖ t = r‖h‖. set t towards to 0 from right of 0, then ρ′+(y0, h, f ) ≥ r‖y − y0‖. note that t > 0, t is sufficiently small and y0 + th ∈ g, since g is convex. this completes the proof. � we shall consider whether condition (ds) is sufficient. some lemmas are required. to state them we give first some notation. γ = {ϕ ∈ x∗ : ‖ϕ‖ = 1, re ϕ(u) ≤ ϕ(y0), for u ∈ h}, here, h = {u : ρ(u, f ) ≤ ρ(f, g)}. nf = {f ∈ x∗ : ‖f‖ = 1, inf x∈f re f (x) = inf u∈f ‖u‖}. the following lemma is lemma 2.3 and lemma 2.4 in [6]. lemma 3.3. for every h 6= 0 ∈ x, supϕ∈γ re ϕ(h) ϕ(y0) = ρ ′ + (y0,h,f ) ρ(y0,f ) , and −γ = nf −y0. criteria of strong nearest-cross points 45 theorem 3.4. let f be a subspace of x, g a convex subset of x, f ∩ g = ∅ and 0 < r < 1. if (ds) holds, then the following condition holds (ks) for every ǫ > 0, and every y ∈ g, there exists f0 (depend on ǫ, y) ∈ x∗ with ‖f0‖ = 1, such that re f0(y0 − y) + ǫ ≥ r‖y0 − y‖. proof. by the condition (ds), theorem 3.1 and lemma 3.3, we have sup f∈nf −y0 re f (y0 − y) re f (y0) ≥ r‖y − y0‖ ρ(y0, f ) . so for every ǫ, y ∈ g, there exists f0 ∈ nf −y0, such that re f0(y0−y) re f0(y0) + ǫ ρ(y0,f ) ≥ r‖y−y0‖ ρ(y0,f ) . since 0 ∈ f, the definition of nf −y0 , re f0(y0) ≥ ρ(y0). this implies that re f0(y0 −y)+ǫ ≥ r‖y0 −y‖, so (ks) holds. this completes the proof. � finally, we give a condition (bs) which is equivalent to (ks) for general disjoint sets f , g. before stating it, we require a notation. let f be a subset of the space x. denote qf = {u : re φ(u) ≤ ‖f ‖, for all φ ∈ nf }. notice that qf is a cone type set including f. for if z ∈ qf , x ∈ f, z′ = x + t(z − x), t > 0, then z′ ∈ qf . because for every φ ∈ nf , re φ(z) ≤ ‖f ‖, re φ(z′) = re φ(x) + tre φ(z −x) = (1−t)re φ(x) + tφ(z) ≤ (1−t)‖f ‖+ t‖f ‖ = ‖f ‖. specially, if f is a singleton x0, then qf is a cone including ball b(0, ‖x0‖); see [3]. theorem 3.5. let f , g be two disjoint sets, then (ks) is equivalent to (bs) qf −y0 ∩ cone(y0 − g) is bounded, cone(e) denotes the cone closure of e. proof. suppose (ks) holds, for every y ∈ g. then sup y∈nf −y0 re f (y0 − y) ≥ r‖y0 − y‖, 0 < r < 1. we should conclude that qf −y0 ∩ cone(y0 − g) ⊂ b(0, ‖f − y0‖/r). if this is not true, there exists t > 0, and some y ∈ g such that t(y0 − y) ∈ qf −y0 , ‖t(y0 − y)‖ > 1/r‖f − y0‖. from the definition of qf −y0 , for every f ∈ nf −y0 , |tre f (y0 − y)| ≤ ‖f − y0‖. if we take the supremum over all f ∈ nf −y0 , then ‖f − y0‖ ≥ tr‖y0 − y‖ > ‖f − y0‖, which leads us to a contradiction. now if (bs) holds, from the above statement, there exists a sufficient large number α, such that qf −y0 ∩cone(y0−g) ⊂ intb(0, α), α > 0. so for every y ∈ g, y0 −y ∈ y0 −g, and y0−y‖y0−y‖ ∈ cone(y0 −g). but since α y0−y ‖y0−y‖ /∈ intb(0, α), then α y0−y‖y0−y‖ /∈ qf −y0 . there exists f0 ∈ nf −y0 , such that re f0( α(y0−y) ‖y0−y‖ ) > ‖f −y0‖. it means that re f0(y0−y) > 1α ‖f −y0‖‖y0−y‖. if we put r = ‖f −y0‖ α and we take α large enough such that 0 < r < 1, then (ks) holds. � 46 w. pan and j. xu 4. cusp and strong best approximation pairs in this section, we shall discuss the case when either strong nearest cross point or strong best approximation pair involve a cusp. in the end of this section, we shall give three examples of strong best approximation pairs. let us begin with the definition of a cusp. definition 4.1. let g be a nonempty subset of x, and let ∂g be the boundary of g. given y0 ∈ g ∩ ∂g, a point y0 is called a cusp of g if there exists a hyperplane p supporting g at y0, and ρ(y,p ) (y,y0) > σ > 0 holds for every y ∈ g, where σ is a constant. obviously, every cusp is a strongly exposed point. we say that y0 is a strongly exposed point of g if there exists a hyperplane p supporting g at y0, x ∈ p, f (x) = c and such that if for every arbitrary ǫ > 0, there exists δ > 0, such that |f (y) − f (y0)| < δ for y ∈ g, then ρ(y, y0) < ǫ. in fact, ρ(y, p ) = |f (y)−c| ‖f‖ . without loss of generality, we assume ‖f‖ = 1, then ρ(y, p ) = |f (y) − f (y0)|. since y0 is a cusp of g, ρ(y,p )ρ(y,y0) > σ, ρ(y, y0) < |f (y) − f (y0)|/σ for y ∈ g. for exposed points and strongly exposed points, one can see [1], [8] and the references there in. let f , g be two nonempty sets with ρ(f, g) > 0. we say that two hyperplanes p , q regular separate f and g, if p , q are parallel, f and g are in two outer sides of p and q, and ρ(p, q) = ρ(f, g) = ρ(f, p ) = ρ(g, q). furthermore, if y0 ∈ g is such that ρ(y0, f ) = ρ(f, g), and ρ(y,p )ρ(y,y0) > δ > 0, we say that y0 is cusp of g to f . obviously, if y0 is a cusp of g to f, then y0 is a cusp of g. theorem 4.2. let f , g be two disjoint convex sets and let y0 ∈ g. if y0 is a cusp of g to f , then y0 is a strong n.c. point of g to f . proof. from the definition of cusp of g to f , there exist hyperplanes p , q separating f , g such that ρ(y,p ) ρ(y,y0) > δ for every y ∈ g. then, ρ(y, f ) − ρ(y0, f ) ≥ ρ(y, q) − ρ(y0, f ). note that since p is parallel to q, then ρ(y, q) = ρ(y, p ) + ρ(p, q) and ρ(p, q) = ρ(y0, f ). thus ρ(y, f ) − ρ(y0, f ) = ρ(y, p ) > δρ(y, y0). this means that y0 is a strong n.c. point. � let h be the hausdorff metric h(f0, f1) = max{△(f0, f1), △(f1, f0)}, where △(f0, f1) = sup x∈f0 inf x′∈f1 ‖x − x′‖. we have theorem 4.3. (freud type proposition) suppose y0 is a strong n.c. point of g to f. if y1 is a strong n.c. point of g to f1, then ‖y − y0‖ < 2/rh(f0, f1). proof. according to definition 2.1, there exists 0 < r < 1, such that r‖y1 − y0‖ ≤ △(f, g) + ρ(y1, f ) − ρ(y0, f ). it is easy to see that ρ(y, b) − ρ(y, a) ≤ (a, b) holds for every y. thus r‖y1 − y0‖ ≤ △(f1, f0) + ρ(y1, f1) − ρ(y0, f0) ≤ △(f1, f0) + ρ(y0, f1) − ρ(y0, f0) ≤ △(f1, f0) + △(f0, f1) ≤ 2h(f0, f1), criteria of strong nearest-cross points 47 (to obtain the second inequality, we used that y1 is a strong n.c. point of g to f ). this completes the proof. � in smooth normed spaces, f is a singleton and g is a normed subspace, then theorem 4.3 is the result of wulbert [3, page 95]. theorem 4.4. let f , g be two disjoint sets with f convex and g a linear subspace and y0 ∈ g. if ρ(y, f ) is gateaux differential at y0, then y0 is not a strong n.c. point of g to f . proof. suppose y0 is a strong n.c. point. by the gateaux differentiable of ρ(y, f ), we have ρ′ (y0, h, f ) + ρ ′ +(y0, −h, f ) = 0, for h 6= 0. by theorem 3.2, ρ′+(y0, y − y0, f ) ≥ r‖y − y0‖ holds for all y 6= y0, y ∈ g, where 0 < r < 1. if y − y0 is either h or −h, we have 0 ≥ 2r‖y − y0‖, which is a contradiction. the proof is complete. � definition 4.5. let f , g be two disjoint sets, x0 ∈ f, and y0 ∈ g. we say that (x0, y0) is a strong best approximation pair of f and g if there exist positive constants r, r′ such that ρ(y, y) − ρ(x0, y0) ≥ r‖x − x0‖ + r′‖y − y0‖ for all x ∈ f, y ∈ g. obviously, a strong best approximation pair of f and g is a best approximation pair of f and g; for best approximation pairs one can see [5] in detail. in the following, we shall discuss the connection between strong best approximation pairs and strong n.c. points. theorem 4.6. if f , g are two disjoint sets, then (x0, y0) is the strong best approximation pair of f and g, if and only if, y0 is a strong n.c. point of g to f , x0 is the strong n.c. point of f to g, and (x0, y0) is a best approximation pair of f and g. in this case, it is unique. proof. if (x0, y0) is the strong best approximation pair of f and g, by definition 4.5, ρ(x0, y0) = ρ(f, g) = ρ(y0, g) and ρ(x, y) − ρ(y0, f ) ≤ r′‖y − y0‖ for all x ∈ f. if we take the infimum over all x ∈ f, we have ρ(y, f ) − ρ(y,f ) ≥ r′‖y − y0‖. thus y0 is the strong n.c. point of g to f . similarly, x0 is a strong n.c. point of f to g. conversely, since (x0, y0) is a best approximation pair of f and g, then ρ(x0, y0) = ρ(f, g) = ρ(y0, f ) = ρ(x0, g). note that ρ(x, y) ≥ ρ(y, f ) and y0 is a strong n.c. point of g from f . so, ρ(x, y)−ρ(x0, y0) ≥ ρ(y, f )−ρ(y0, f ) ≥ r′‖y − y0‖. similarly, ρ(x, y)− ρ(x0, y0) ≥ r‖x− x0‖. thus, ρ(x, y)− ρ(x0, y0) ≥ r/2‖x − x0‖ + r′/2‖y − y0‖. therefore (x0, y0) is a strong best approximation pair. this completes the proof. � theorem 4.7. let f , g be two disjoint sets, ρ(f, g) > 0, and (x0, y0) a best approximation pair of f and g. if y0 is a cusp of g to f , and x0 is a cusp of f to g, then (x0, y0) is a strong best approximation pair. proof. by the definition, y0 is a cusp of g to f , and there exist parallel hyperplanes separating p , q, such that f and g are in the outer side of p and q, y0 ∈ p, and ρ(p, q) = ρ(f, g) = ρ(p, f ) = ρ(q, g) = ρ(y0, f ). obviously, 48 w. pan and j. xu ρ(x, y) ≥ ρ(y, q) for all x ∈ f, y ∈ g. since (x0, y0) is a best approximation pair of f and g, ρ(x0, y0) = ρ(y0, f ). so ρ(x, y)−ρ(x0, y0) ≥ ρ(y, q)−ρ(y0, f ). note that since ρ(y0, f ) = ρ(p, q), and ρ(y, q) = ρ(y, p ) + ρ(p, q), then ρ(x, y) − ρ(x0, y0) > ρ(y, p ) > σ′‖y − y0‖. thus ρ(x, y) − ρ(x0, y0) > σ/2‖x − x0‖ + σ′/2‖y − y0‖. this completes the proof. � finally, we shall give three examples. example 4.8. denote c[0,1] be all continuous function f on [0,1] with norm ‖f‖ = maxt∈[0,1] |f (t)|. in c[0,1], let f = {µt : −∞ < µ < ∞}, g = {λt2 :√ 2 + 1 ≤ λ ≤ 5}. we consider n.c. points, best approximation pairs, strong n.c. points and strong best approximation pairs between f and g. denote x(t) = µt, y(t) = λt2. then ρ(x, y) = ‖µt − λt2‖ = { λ − µ, for µ/λ ≤ √ 8 − 2 µ2/4λ, for µ/lz ≥ √ 8 − 2. first we compute inf −∞<µ<∞ ‖µt−λt2‖. for fixed λ, ‖µt−λt2‖ takes its infimum at µ = µλ. by the representation of ‖µt − λt2‖, µλ satisfies λ − µλ = µ2λ/4λ, so µλ = ( √ 8 − 2)λ. thus ρ(f, g) = inf√2+1≤λ≤5 ‖λt2 − ( √ 8 − 2)λt‖ = ( √ 2 + 1)‖t2 − ( √ 8 − 2)t‖ = ( √ 2 + 1)(3 − √ 8) = √ 2 − 1. from above we obtain that ‖x0 − y0‖ = ρ(f, g), when x0(t) = µ0t, y0(t) = λ0, λ0 = √ 2 + 1, µ0 = 2. we declare that y0(t) is a strong n.c. point of g to f, since for every y(t) = λt2 ∈ f, ρ(y, f ) = inf −∞<µ<∞ ‖λt2 − µt‖ = ‖λt2 − µλt‖ = λ‖t2 − ( √ 8 − 2)t‖ = (3 − √ 8)λ. therefore, ρ(y0, f ) = ρ0 = (3 − √ 8)λ0, ρ(y, f ) − ρ(y0, f ) = (3 − √ 8)(λ − λ0) = r′‖y − y0‖, r′ = 3 − √ 8. similarly, we have that x0 is a strong n.c. point of f to g, since for every x(t) = µt, ρ(x, g) = inf√ 2+1≤λ≤5 ‖λt2 − µt‖ = inf√ 2+1≤λ≤5 { λ − µ, forλ ≥ µ/ √ 8 − 2 µ2/4λ, forλ ≤ µ/ √ 8 − 2. so, ρ(x, g) takes the infimum at λµ = √ 2 + 1/2µ, and ρ(x, g) = λµ − µ =√ 2/2µ. thus, ρ(x, g) − ρ)x0, g) = √ 2 + 1/2µ − √ 2 − 1 = r‖x − x0‖, where r = √ 2 − 1/2. by theorem 3.4, we obtain that (x0, y0) is the strong best approximation pair of f and g. the following example shows that a n.c. point always exists, but strong n.c. points can fail to exist. example 4.9. denote ℓ21 = {(ξ, η) : ξ, η ∈ r, ‖(ξ, η)‖ = |ξ| + |η|}. in ℓ21 space, let f = {(ξ, η) : ξ = η}, g = {(ξ, η) : η = 0, 2 ≤ ξ ≤ 3}. it is easy to see that y0 = (2, 0) is a n.c. point of g to f . but the best approximation of (2,0) criteria of strong nearest-cross points 49 from f is not unique. thus the nearest cross points of g to f is not unique. therefore a strong n.c. point of g to f does not exist . at the end, we shall give an example, which shows that it is possible to find best approximation pairs which are not strong best approximation pairs. example 4.10. in the euclidean r2 space, set f = {(ξ, η) : ξ2 + η2 ≤ 1}, g = {(ξ, η) : ξ ≥ 2, −ξ + 2 ≥ η ≥ ξ − 2}. it is easy to see that x0 = (1, 0), y0 = (2, 0) is the unique best approximation pair of f and g. denote an arbitrary point of f as x = (cos θ, sin θ), 0 ≤ θ < 2π. put y = (2 + δ, δ) ∈ g, δ > 0. then ρ0 = ρ(x0, y0) = 1, ρ 2 = ρ2(x, y) = (2 + δ − cos θ)2 + (δ − sin θ)2. setting θ → 0, δ → 0, then ρ2 − ρ20 is asymptotic to 2(ρ − ρ0). ‖x − x0‖ = | sin θ|, ‖y − y0‖ = √ 2δ. since δ ρ2−ρ2 0 , and θ ρ2−ρ2 0 are not bounded, then ‖y−y0‖ ρ−ρ0 , and ‖x−x0‖ ρ−ρ0 are also not bounded. this means that (x0, y0) is not a strong best approximation pair. however, y0 is a strong n.c. point of g to f . acknowledgements. the authors would like to give their deep gratitude to the referee for his careful reading of the manuscript and his suggestions which made this article more readable. references [1] p. beneker and j. wiegerinck, strongly exposed points in uniform algebras, proc. amer. math. soc. 127 (1999), 1567-1570. [2] d. braess, nonlinear approximation theory, springer-verlag, berlin, heidelberg, 1986. [3] d. f. mah, strong uniqueness in nonlinear approximation, j. approx. theory 41 (1984), 91-99. [4] w. pan, the distance between sets and the properties of best approximation pairs, j. of math. (chinese), 14 (1994), 491-497. [5] w. pan, on the existence and uniqueness of proximity pairs and nearest-cross points, j. math. research and exposition (chinese), 15 (1995), 237-243. [6] w. pan, characterization of nearest-cross points in the problem of distance of two convex sets, j. jinan univ. (natural science) (chinese), 20(3) (1999), 1-7. [7] p. l. papini, approximation and strong approximation in normed spaces via tangent function, j. approx. theory 22 (1978), 111-118. [8] a. a. tolstonogov, strongly exposed points of decomposable sets in spaces of bochner integrable functions, mathematical notes 71 (2002), 267-275. [9] q. wang, w. pan, new concept of normal separation of two sets and its properties, acta sci. nat. univ. sunyatseni (chinese), 39(6) (2000), 20-25. 50 w. pan and j. xu received march 2004 accepted december 2005 wenxi pan department of mathematics, jinan university, 510632 guangdong, china. jingshi xu (jshixu@yahoo.com.cn) department of mathematics, hunan normal university, 510632 hunan, china claesagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 21-32 exponentiality for the construct of affine sets v. claes abstract. the topological construct sset of affine sets over the two-point set s contains many interesting topological subconstructs such as top, the construct of topological spaces, and cl, the construct of closure spaces. for this category and its subconstructs cartesian closedness is studied. we first give a classification of the subconstructs of sset according to their behaviour with respect to exponentiality. we formulate sufficient conditions implying that a subconstruct behaves similar to cl. on the other hand, we characterize a conglomerate of subconstructs with behaviour similar to top. finally, we construct the cartesian closed topological hull of sset. 2000 ams classification: 54a05, 54c35, 18d15 keywords: topological construct, affine space, cartesian closed category, cartesian closed topological hull, exponential object 1. introduction the lack of natural function spaces in a topological construct that is not cartesian closed, has long been recognized as an akward situation for various applications in homotopy theory and topological algebra and for use in infinite dimensional differential calculus. for references to the original sources one might consult [22, 11, 12, 18, 21]. topological constructs like top or the larger construct cl and several others that are commonly used by topologists, however are not cartesian closed. for the construct top of topological spaces this problem is extensively studied in the literature, see for example [17, 8, 4, 5, 19]. for the subconstruct cl of closure spaces, the author studied cartesian closedness in [6]. to remedy these facts, topologists have applied various methods. either they dealt with the local problem, the description of exponential objects and with the construction of cartesian closed subconstructs, or they looked for larger cartesian closed constructs. a topological construct is cartesian closed if and only if every object x is exponential in the sense that the functor x × − preserves coproducts and quotients. the reason for top not being cartesian 22 v. claes closed is that x ×− does not always preserve quotients, except for corecompact x. for cl it is just the other way around, x × − generally does not preserve coproducts, except for x being indiscrete. top as well as cl are fully embedded in the construct sset of affine spaces and affine maps which is a host for many other subconstructs that are important to topologists [13, 14] (see next section for the exact definitions). in this paper we investigate the problem of cartesian closedness for sset and we describe the exponential objects and deduce results for its subconstructs. we prove that for a non-indiscrete affine space x, the functor x × − does not preserve coproducts in sset and we describe a conglomerate of subconstructs of sset (to which cl belongs), in which this negative result goes through. on the other hand, we determine a large subconstruct of sset in all topological subconstructs of which (like for instance top) the functor x ×− does preserve coproducts. in the final section of the paper we describe the cartesian closed topological hull of sset. remark that, as was observed by e. giuli [13], our definition of affine spaces and maps, as we recall it in the next section, only differs slightly from the normal boolean chu spaces and continuous maps, as introduced by v. pratt to model concurrent computation. objects in sset have a structure containing constants. this assumption makes sset into a well-fibred topological construct in the sense of [1], which has the property that cartesian closedness is equivalent to the existence of ”nice” function spaces. 2. exponential objects in sset and in its subconstructs. an affine space x over the two point set s = {0, 1} is a structured set, where the structure on the underlying set x is a collection of subsets of x. the sets belonging to the structure are called the “open” sets of x. an affine map from x −→ y is a function f such that inverse images of open sets are open. an affine space can be isomorphically described in a functional way: an affine space (over s) (x, a) consists of a set x and a subset a of the powerset sx . an affine map f : (x, a) → (y, b) is a function f such that β ◦ f ∈ a for all β ∈ b. in this paper we will use the functional description. we will restrict ourselves to the affine spaces whose affine structure contains the constant functions 0 and 1. as in [13], the corresponding construct of affine spaces and affine maps will be denoted by sset. in that paper it was proved that sset is a well-fibred topological construct. an object x in a category with finite products is exponential if the functor x ×− has a right adjoint. in a well-fibred topological construct x, this notion can be characterized as follows: x is exponential in x iff for each x-object y the set homx(x, y ) can be supplied with the structure of a x-object a function space or a power object y x such that (1) the evaluation map ev: x × y x → y is a x-morphism (2) for each x-object z and each x-morphism f : x × z → y , the map f ∗ : z → y x defined by f ∗(z)(x) = f (x, z) is a x-morphism. exponentiality for the affine sets 23 it is well known that in the setting of a topological construct x, an object x is exponential in x iff x × − preserves final episinks [15], [16]. moreover, small fibredness of x ensures that this is equivalent to the condition that x ×− preserves quotients and coproducts. a well-fibred topological construct x is said to be cartesian closed (or to have function spaces) if every object is exponential. before characterizing the exponential objects in the subcategories of sset, we first prove some useful results for subconstructs of sset. we first recall the following result from [20]. proposition 2.1. [20] every topological subcategory x of sset is a bicoreflective subcategory of some full bireflective subcategory y of sset. the following propositions can be proved using techniques similar to those developed for top in [17]. proposition 2.2. every topological subcategory of sset is closed under the formation of retracts in sset. proposition 2.3. every non-trivial topological subcategory x of sset contains all complemented topological spaces. in order to investigate the interaction of products and coproducts in sset and in its subconstructs, we first look at the construction of coproducts in sset and in its subconstructs. let (xi, ai)i∈i be a family of affine sets, then one can easily verify that the coproduct in sset of these objects is given by (∐xi, a = { ⊔ i∈i αi : αi ∈ ai}) with ⊔ i∈i αi : ∐ i∈i xi → s : (xi, i) → αi(xi) proposition 2.4. let x be a non-trivial topological subconstruct of sset and (xi, ai)i∈i a family of x-objects. for every i ∈ i and every αi ∈ ai, the function ⊔ j∈i βj belongs to the affine structure a∐xi on the coproduct ∐ i∈i xi whenever βi = αi and βj = 0 for all j 6= i or βj = 1 for all j 6= i. proof. let y be the bireflective subcategory of sset such as in proposition 2.1 and let (xi, ai)i∈i be a family of x-objects. for r = (ri)i∈i ∈ π i∈i xi, we define the function fr : ∐ i∈i xi → π i∈i xi × i : (xi, i) → ((yj )j∈i , i) with yi = xi and yj = rj for j 6= i. let a be the initial affine structure for the source (fr : ∐ i∈i xi → π i∈i (xi, ai) × (i, s i ))r∈ π i∈i xi . then, a = a1 ∪ a2 with 24 v. claes a1 = {1j ◦ pri ◦ fr | j ⊂ i, r ∈ π i∈i xi} = ∪ j⊂i { ⊔ i∈i βi | βi = 1 if i ∈ j and βi = 0 if i /∈ j} a2 = {αi ◦ prxi ◦ fr | i ∈ i, αi ∈ ai, r ∈ π i∈i xi} = ∪ i∈i { ⊔ j∈i βj | βi = αi ∈ ai and βj = 1 if j 6= i} ∪ ∪ i∈i { ⊔ j∈i βj | βi = αi ∈ ai and βj = 0 if j 6= i} from the previous proposition follows that the categories x and y contain the discrete affine sets, and in particular (i, si ). since y is closed under the formation of initial structures in sset, it follows that ( ∐ i∈i xi, a) belongs to y. for all i ∈ i, the map ji : (xi, ai) → ( ∐ i∈i xi, a) is affine. hence, ( ∐ i∈i xi, a) is coarser than the coproduct ( ∐ i∈i xi, a∐yxi ) in the category y. since x is a bicoreflective subcategory of y, this implies that a ⊂ a∐xxi = a∐yxi . � hence, every non-trivial subcategory of sset for which the affine structures are closed under arbitrary suprema is closed under the formation of coproducts in sset. we will now recall a general method to construct hereditary bicoreflective subcategories of sset [9], [10] ,[13]. in order to define a subconstruct of sset we put an algebra structure on s = {0, 1}. recall that an algebra structure on the set s is a class of operations ω = {ωi : s ni → s | i ∈ i} of arbitrary arities. hence the ni are arbitrary cardinal numbers, and there is no condition on the size of the indexing system i. we assume that ω contains the constant operations. for every set x, by point-wise extension, the powerset sx carries an algebra structure. we denote by sset(ω) the subconstruct of sset consisting of those affine sets (x, a) for which a is a ω-subalgebra of the function algebra sx . the objects in sset(ω) are called affine sets over the algebra (s, ω). for example the category cl of closure spaces and the category top of topological spaces can be obtained this way. the construct obtained this way, by considering for ω the class containing the constant operations and the complementation¯: s → s defined by (̄0) = 1,̄ (1) = 0 will be denoted by sset(c). lemma 2.5. sset(ω) is a subcategory of sset(c) whenever ω contains an operation ωt : s t → s that satisfies the following condition: ∃ (xt)t∈t , (yt)t∈t ∈ s t such that ωt ((xt)t∈t ) = 0, ωt ((yt)t∈t ) = 1 and xt = 0 implies yt = 0 for all t ∈ t . exponentiality for the affine sets 25 proof. let (x, a) be an sset(ω)-object. for α ∈ a, define the serie functions (ft : x → s)t∈t as follows. ft =    0 if xt = 0 1 if yt = 1 α if xt = 1, yt = 0 since a contains all constant functions, we have that ft ∈ a for all t ∈ t . then, a contains the function ωt ◦ π t∈t ft : x → s : x → ωt (ft(x))t∈t . if α(x) = 1, then ωt ◦ π t∈t ft(x) = ωt ((xt)t∈t ) = 0 =¯◦ α(x). if α(x) = 0, then ωt ◦ π t∈t ft(x) = ωt ((yt)t∈t ) = 1 =¯◦ α(x). we can conclude that¯◦ α = ωt ◦ π t∈t ft ∈ a and thus (x, a) is an sset(c)-object. � lemma 2.6. if x is a non-trivial topological subcategory of sset and d2 is the two point discrete space, then for every non-indiscrete object (x, a) the following holds: (x, a) × d2 ∈ x ⇒ (x, a) is not an exponential object in x proof. for a non-constant function α ∈ a, 0 ⊔ α is an element of ax⊔x , while it is not contained in ax×d2 . � the previous negative result has important consequences with respect to exponential objects in sset and to cartesian closedness of topological subconstructs. corollary 2.7. if x is a topological subconstruct of sset which is finitely productive in sset, then the class of exponential objects in x coincides with the class of indiscrete spaces. we now characterize a conglomerate of subconstructs of sset, which are not finitely productive in sset, in which the class of exponential objects also coincides with the class of indiscrete objects. proposition 2.8. for every category sset(ω) that is not a subcategory of sset(min : s2 → s, max : s2 → s) the exponential objects are exactly the indiscrete affine sets. proof. suppose that sset(ω) has a non-indiscrete exponential object (x, a). then we have that (x, a)⊔(x, a) is isomorphic to (x, a)×d2. by proposition 2.4, it follows that 0 ⊔ α belongs to ax×d2 for every α ∈ a. the product ax×d2 is the smallest ω-subalgebra of s x×d2 containing b = { α ◦ prx | α ∈ a} ∪ { prd2 , prd2 : (x, a) → a}. hence, 0⊔α = ωα((γ ◦prx )γ∈a, prd2 , prd2 ) with ωα : s a∪s → s a composition of algebraic operations of ω. for a non-constant function α ∈ a, there exists x, y ∈ x such that α(x) = 1 and α(y) = 0. define (fγ : s × s → s)γ∈a as follows: • if γ(x) = γ(y), put fγ the constant function with value γ(x). • if γ(x) = 1 and γ(y) = 0, put fγ = pr1 : s × s → s : (a, b) → a 26 v. claes • if γ(x) = 0 and γ(y) = 1, put fγ =¯ ◦ pr1 : s × s → s : (a, b) → a then, for b ∈ s we have: • ωα ◦ ( π γ∈a fγ × pr2 × pr2)(0, b) = ωα((γ(y))γ∈a, b, b̄) = ωα((γ ◦ prx )γ∈a, prd2 , prd2 )(y, b) = 0 ⊔ α (y, b) = 0 = min(0, b). • ωα ◦ ( π γ∈a fγ × pr2 × pr2)(1, b) = ωα((γ(x))γ∈a, b, b̄) = ωα((γ ◦ prx )γ∈a, prd2 , prd2 )(x, b) = 0 ⊔ α (x, b) = b = min(1, b). this implies that min(a, b) = ωα ◦ ( π γ∈a fγ × pr2 × pr2)(a, b), which means that the operation min : s ×s → s can be written in terms of the operation ωα, the complementation ¯ and the constant functions. if sset(ω) is a subcategory of sset(c), it now follows that sset(ω) is a subcategory of sset(min : s2 → s). for the categories sset(ω) that are not embedded in sset(c), it follows from lemma 2.5 that: (1) ωα((γ(x))γ∈a, 0, 0) = 0, because ωα((γ(x))γ∈a, 0, 1) = 0 ⊔ α(x, 0) = 0 (2) ωα((0)γ∈a, 1, 0) = 0, because ωα((γ(y))γ∈a, 1, 0) = 0 ⊔ α(y, 1) = 0 (3) ωα((0)γ∈a, 0, 0) = 0 by defining fγ = pr1 : s × s → s : (a, b) → a if γ(x) = 1 and otherwise fγ = 0, we have that ωα ◦ ( π γ∈a fγ × pr2 × 0)(a, b) = min(a, b). indeed, for b ∈ s we have • ωα ◦ ( π γ∈a fγ × pr2 × 0)(0, b) = ωα((0)γ∈a, b, 0) = 0 • ωα ◦ ( π γ∈a fγ × pr2 × 0)(1, b) = ωα((γ(x))γ∈a, b, 0) = b where the last equation follows from previous observation (1) and the fact that ωα((γ(x))γ∈a, 1, 0) = 0 ⊔ α(x, 1) = 1 so in each category sset(ω) which has an non-indiscrete exponential object, a is closed under finite minima for every object (x, a). it can be proved in a similar way that a is closed under finite maxima. one can easily prove that the indiscrete affine sets are exponential. so we can conclude that the exponential objects of sset(ω) are exactly the indiscrete affine sets. � thus for the categories sset, cl and sset(c) the exponential objects are exactly the indiscrete objects. from the proof of previous proposition follows that in all the categories sset(ω) which are not embedded in sset(min : s2 → s, max : s2 → s), the functor x × − does not preserve coproducts for non-indiscrete objects x. in sset(min : s2 → s, max : s2 → s) itself, the functor x × − preserves coproducts for some non-indiscrete objects x. in the following proposition these objects are characterized. exponentiality for the affine sets 27 proposition 2.9. in the construct sset(min : s2 → s, max : s2 → s), we have that the functor (x, a)×− preserves coproducts if and only if a is a finite set. proof. it can be easily verified that for (x, a), with a a finite set, (x, a) × ∐ i∈i (yi, bi) is isomorphic to ∐ i∈i (x, a)×(yi, bi) for every collection sset(min : s2 → s, max : s2 → s)-objects (yi, bi)i∈i . suppose now that the functor (x, a) × − preserves coproducts, then we have that (x, a) × (a, sa) is isomorphic to ∐ α∈a (x, a). we consider the function ⊔ α∈a α : ∐ α∈a (x, a) → s : (x, α) → α(x) since sset(min : s2 → s, max : s2 → s) is a bicoreflective subcategory of sset, this function ⊔ α∈a α belongs to a ∐ α∈a (x,a) and thus ⊔ α∈a α belongs to ax×(a,sa). the product ax×(a,sa) is the smallest subalgebra containing b = { α ◦ prx | α ∈ a} ∪ { f ◦ pra | f ∈ s a}. hence, there exists a finite set i and for every i ∈ i there exist αi ∈ a, fi ∈ s a such that ⊔ α∈a α = max i∈i min(αi ◦ prx , fi ◦ pra). for β ∈ a, set iβ = {i ∈ i | fi(β) = 1}. for every x ∈ x, we have: β(x) = ⊔ α∈a α (x, β) = max i∈i min(αi(x), fi(β)) = max i∈iβ αi(x). since the set i is finite, this implies that a also shall be finite. � 3. subconstructs in which the functor x × − preserves coproducts the categories considered in the previous section fail to be cartesian closed because the functor x ×− does not preserve coproducts. from the last proposition, it follows that the condtion, sset(ω) is a subcategory of sset(min : s2 → s, max : s2 → s), is not a sufficient condition such that the functor x × − preserves coproducts for all objects x. in this section we formulate a sufficient condition. it is known [17] that in top and in all its topological subconstructs the functor x × − preserves coproducts for all objects x. we generalize these results to a larger subconstruct of sset. definition 3.1. let d be the full subcategory of sset with objects all affine sets (x, a) that satisfy the following two conditions. (d1) α ∈ a, β ∈ a ⇒ min(α, β) ∈ a (d2) {αi | i ∈ i} ⊂ a and min(αi, αj ) = 0 for each i 6= j ⇒ max i∈i αi ∈ a it is clear that top is a subcategory of this category d. for a collection b ⊂ sx we can define a d-structure a on x as follows. let c consist of all finite minima of elements of b ∪ {0, 1}. by adding to c the maxima of collections functions (αi)i∈i of c that satisfy the condition min(αi, αj ) = 0 for each i 6= j, we get a d-structure a. moreover, a is the smallest d-structure on x containing b. b is called the subbase of a and c the base of a. 28 v. claes proposition 3.2. d is a topological category. proof. for a family of functions (fi : x → (xi, ai))i∈i with (xi, ai) ∈ d the d-structure a generated by the subbase b = {αi ◦ fi | αi ∈ ai, i ∈ i} is the unique initial structure on x for the given source. � proposition 3.3. d is a bicoreflective subcategory of sset proof. for an affine set (x, a) the bicoreflection is given by 1x : (x, a ′) → (x, a) with a′ the d-structure generated by the subbase a. � remark that d is not a hereditary subcategory of sset as follows from the next example. example 3.4. let x = {0, 1, 2, 3} and a = {0, 1, 1{1,3}, 1{2,3}, 1{3}}, then (x, a) is a d-object. then (y, a|y ) = ({0, 1, 2}, {0, 1, 1{1}, 1{2}}) is the sset-subspace of (x, a) with underlying set {0, 1, 2}. min(1{1}, 1{2}) = 0 and max(1{1}, 1{2}) = 1{1,2} /∈ a|y , so (y, a|y ) does not belong to the category d. hence, there is no algebraic structure ω on s such that d has the form sset(ω). proposition 3.5. if sset(ω) is a subcategory of d, then it is a subcategory of top or a subcategory of sset(c). proof. for an arbitrary set i, take ∞ /∈ i and define for every i ∈ i the function αi : i ∪ {∞} → s with αi(i) = 1 and αi(x) = 0 for x 6= i. let a be the smallest ω-subalgebra of si∪{∞} containing {αi | i ∈ i}. then, (i ∪ {∞}, a) is an sset(ω)-object. since sset(ω) is a subcategory of d, we have that max i∈i αi belongs to a. hence, max i∈i αi = ωi (αi)i∈i , with ωi : s i → s a composition of algebraic operations of ω. this gives the following information about ωi : • ∀j ∈ i : ωi (αi(j))i∈i = max(αi(j))i∈i = 1 • ωi (0)i∈i = ωi (αi(∞)i∈i ) = max(αi(∞)i∈i ) = max(0)i∈i = 0 now two cases can arise: (1) ∀x 6= (0)i∈i ∈ s i : ωi (x) = 1. in this case ωi = max i∈i (2) there exists a x 6= (0)i∈i ∈ s i such that ωi (x) = 0. choose j ∈ i such that prj (x) 6= 0 and define (yi)i∈i ∈ s i with yj = 1 and yi = 0 for i 6= j. then we have ωi (x) = 0, ωi (yi)i∈i = ωi (αi(j))i∈i = 1 and pri(x) = 0 implies yi = 0. it then follows from lemma 2.5 that sset(ω) is a subcategory of sset(c). we can conclude that either for every set i, ωi = max i∈i and thus sset(ω) is a subcategory of top or sset(ω) is a subcategory of sset(c). � exponentiality for the affine sets 29 it can be verified that coproducts are universal in d, i.e. coproducts are preserved under pullbacks along arbitrary morphisms. from proposition 2.4 and the condition (d2) follows that d and its subconstructs are closed under the formation of coproducts in sset. combining this with 2.2, the following theorem can be proved. theorem 3.6. in every topological subcategory of d coproducts are preserved by the functor x × −. corollary 3.7. the exponential objects of a subcategory of d are the objects x for which the functor x × − preserves quotients. 4. cartesian closed topological hull of sset in section 2, we proved that for finitely productive subcategories of sset, products do not distribute over coproducts. if we want to work in a cartesian closed construct in which products are formed similarly to the ones in sset, we have to consider a larger scope. we will look for cartesian closed topological constructs that are larger than sset and in which sset is finally densely embedded. we know from [7] that quotients in sset are productive. in fact we have the same situation as for the category cl [6]. for cl a cartesian closed extension was constructed in [6] using the method presented by j. adámek and j. reiterman in [2]. we first look if this method is also applicable to sset. definition 4.1. for affine sets (x, a) and (y, b) we consider the collection of functions n = {γβ|β ∈ b} on hom(x, y ) with γβ : hom(x, y ) → s defined by γβ(f ) = 1 iff β ◦ f = 1. analogous to cl, we can prove the following result for this structure on the hom-sets of sset. proposition 4.2. if m ⊆ hom(x, y ) is a subset endowed with an affine structure m such that the evaluation map ev: (x, a) × (m, m) → (y, b) is an affine map, then the following conditions hold: (1) n|m ⊆ m (2) ev: x × (m, n|m ) → y is affine. this shows that sset is a type of category as considered in 4.3 of [2]. consider the following superconstruct k of sset. objects of k are triples (x, a, a) where x is a set, a is a cover of x such that u ′ ⊆ u, u ∈ a ⇒ u ′ ∈ a and a is an affine structure on x which is a-final in the sense that (i : (u, a|u ) → (x, a))u∈a is final in sset. the members of a are called generating sets. a morphism in k, f : (x, a, a) → (y, b, b) is a function that is affine (with respect to (x, a) and (y, b)) and preserves the generating sets: u ∈ a ⇒ f (u ) ∈ b. 30 v. claes sset is fully embedded in k by identifying (x, a) with (x, p(x), a). by the general theorem in [2] it follows that k is a cartesian closed topological category in which sset is finally densely embedded. a cartesian closed well-fibred topological construct y is called a cartesian closed topological hull (cct hull) of a construct x if y is a finally dense extension of x with the property that any finally dense embedding of x into a cartesian closed topological construct can be uniquely extended to y. starting from the cartesian closed extension k of sset, we will now construct the cartesian closed topological hull of sset. here again we will work as we did for cl. in particular, we apply the construction of the cartesian closed hull, using powerclosed collections, described by j. adámek, j. reiterman and g.e. strecker in ii.2 and ii.3 in [3]. we first recall from [3] some definitions and the construction of the cct-hull applied to categories with productive quotients. definition 4.3. let x be a construct and let h, k be x-objects and x a set. a function h : x × h → k is called a multimorphism if for each x ∈ x, h(x, −) : h → k defined by h(x, −)(y) = h(x, y) is a morphism. definition 4.4. let x be a well-fibred topological construct. a collection c of objects (a, u) with a ⊆ x is said to be power-closed in a set x provided that c contains each object (a0, u0) with a0 ⊆ x with the following property: given a multimorphism h : x × h → k such that for each (a, u) ∈ c the restriction h|a : (a, u) × h → k is a morphism, then the restriction h|a0 : (a0, u0) × h → k is also a morphism. we denote by pc(k) the category of power-closed collections. objects are pairs (x, c), where x is a set and c is a power-closed collection in x. morphisms f : (x, c) → (y, d) are functions from x to y such that for each (a, u) ∈ c the final object of the restriction fa : (a, u) → f (a) is in d. theorem 4.5. [3] any well-fibred topological construct with productive quotients has a cct hull. moreover, this hull is precisely the category of powerclosed collections. next we define a suitable subconstruct of the cartesian closed extension k of sset. definition 4.6. let k∗ be the full subconstruct of k whose objects are the k-objects (x, a, a) that satisfy the following condition: if v ⊂ x /∈ a, then there exists a set z ⊆ x with v ∩ z 6= ∅, v 6⊂ z and such that: ∀u ∈ a : u ∩ z = ∅ or u ⊆ z. in order to prove that k∗ is the cct hull of sset we establish an isomorphism between k∗ and the category of power-closed collections of sset. proposition 4.7. for each object (x, a, a) of k∗ the collection of affine spaces cx = {(u, b) | u ∈ a, a|u ⊆ b} is power-closed. exponentiality for the affine sets 31 proof. if (x0, a0) is an affine set with x0 ⊆ x and (x0, a0) /∈ cx , then either x0 /∈ a or a|x0 6⊆ a0. if a0 is not finer than a|x0 , there exists an α ∈ a such that α|x0 /∈ a0. for an arbitrary affine space h, the function α ◦ prx : x × h → s with s the sierpinski space is a multimorphism. for (u, b) ∈ cx the restriction h|u = α|u ◦ pru : (u, b) × h → s is affine and the restriction h|x0 = α|x0 ◦ prx0 : (x0, a0) × h → s is not affine. if x0 /∈ a, then there exists a subset z of x, not containing x0 and intersecting x0 such that for all u ∈ a we have that u ∩ z = ∅ or u ⊆ z. take an affine set h that has a non-constant function γ ∈ ah . the function h = min(1z ◦ prx , γ ◦ prh ) : x × h → s is a multimorphism. for (u, b) ∈ cx the restriction h|u is either the constant function 0 or γ ◦ prh . hence, the restrictions h|u : (u, b)×h → s are affine for all (u, b) ∈ cx . since x0∩z 6= ∅ and x0 6⊆ z, we have that h|x0 6= α ◦ prh and h|x0 6= β ◦ prx0 for all α ∈ ah and β ∈ ax0 . therefore the restriction h|x0 : (x0, a0) × h → s is not affine. � proposition 4.8. if c is a power-closed collection of sset-objects in a set x then there exists a unique k∗-object (x, a, a) such that c = cx . proof. for a power-closed collection c of sset-objects in x, we can prove analogously to cl [6] that (x, a, a) where a = {u ⊆ x | (u, b) ∈ c for someb} and a the final structure determined by the sink of inclusion maps (i : (u, b) → x)(u,b)∈c is a k ∗-object such that c = cx . � theorem 4.9. k∗ is the cct hull of sset. proof. it follows from the two previous propositions that the functor f : k∗ → pc(sset) defined by f (x f → x′) = (x, cx ) f → (x′, cx′ ) is bijective on objects. in a similar way as for cl in [6], one can prove that f is an isomorphism. from 4.5 it then follows that k∗ is the cct-hull of sset. � references [1] j. adámek, h. herrlich and g.e. strecker, abstract and concrete categories (wiley, new york, 1990). [2] j. adámek. and j. reiterman, cartesian closed hull of the category of uniform spaces, topology appl. 19 (1985), 261–276. [3] j. adámek, j. reitermann and g.e. strecker, realization of cartesian closed topological hulls, manuscripta math. 53 (1985), 1–33 . [4] p. antoine, extension minimale de la catégorie des espaces topologiques, c. r. acad. sci. paris sér. a-b 262 (1966), a1389–a1392. [5] g. bourdaud, espaces d’antoine et semi-espaces d’antoine, cahiers topologie géom. différentielle 16, no. 2 (1975), 107–133. [6] v. claes, e. lowen-colebunders and g. sonck, cartesian closed topological hull of the construct of closure spaces, theory appl. categories 8, no. 18 (2001), 481–489. [7] v. claes and e. lowen-colebunders, productivity of zariski-compactness for constructs of affine spaces, topology appl. 153, no. 5-6 (2005), 747–755. [8] b. j. day and g.m. kelly, on topological quotients preserved by pullback or products, proc. camb. phil. soc. 67 (1970), 553–558. [9] y. diers, categories of algebraic sets, appl. categ. structures 4 (1996), 329–341. 32 v. claes [10] y. diers, affine algebraic sets relative to an algebraic theory, j. geom. 65 (1999), 54–76. [11] e. dubuc and h. porta, convenient categories of topological algebras, and their duality theory, j. pure appl. algebra 1, no. 3 (1971), 281–316. [12] a. frölicher, cartesian closed categories and analysis of smooth maps, in categories in continuum physics (buffalo, n.y., 1982), volume 1174 of lecture notes in math., pages 43–51. (springer, berlin, 1986). [13] e. giuli, on classes of t0 spaces admitting completions, appl. gen. topol. 4, no. 1 (2003), 143–155. [14] e. giuli, the structure of affine algebraic sets, in categorical structures and their applications, pages 113–120 (world sci. publishing, river edge, nj, 2004). [15] h. herrlich, cartesian closed topological categories, math. colloq. univ. cape town 9 (1974), 1–16. [16] h. herrlich, categorical topology 1971-1981, sigma ser. pure math. 3, 279–383 (heldermann verlag, berlin, 1983). [17] h. herrlich, are there convenient subcategories of top?, topology appl. 15, no. 3 (1983), 263–271. [18] a. kriegl, a cartesian closed extension of the category of smooth banach manifolds, in categorical topology (toledo, ohio, 1983), volume 5 of sigma ser. pure math., pages 323–336 (heldermann, berlin, 1984). [19] a. machado, espaces d’antoine et pseudo-topologies, cahiers topologie géom. différentielle 14 (1973), 309–327. [20] h. müller, úber die vertauschbarkeit von reflexionen und corefectionen ( bielefeld, 1974). [21] l. d. nel, infinite-dimensional calculus allowing nonconvex domains with empty interior, monatsh. math. 110, no. 2 (1990), 145–166. [22] n. e. steenrod, a convenient category of topological spaces, michigan math. j. 14 (1967), 133–152. received july 2006 accepted december 2006 veerle claes (vclaes@vub.ac.be) wisk-ir, vrije universiteit brussel, pleinlaan 2, 1050 brussel, belgium bellaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 1, 2005 pp. 101-106 remarks on the finite derived set property angelo bella abstract. the finite derived set property asserts that any infinite subset of a space has an infinite subset with only finitely many accumulation points. among other things, we study this property in the case of a function space with the topology of pointwise convergence. 2000 ams classification: 54a25, 54a35, 54d55. keywords: accumulation points, urysohn spaces, product, function spaces. following [8], we say that a topological space x has the finite derived set (briefly fds) property if every infinite subset of x contains an infinite subset with only finitely many accumulation points. the class of spaces with the fds property obviously contains the class of sequentially compact hausdorff spaces. as sequential compactness, the validity of the fds property involves in some cases the cardinal characteristic of the continuum. an extensive investigation on the fds property was initiated in [1]. in this note we will collect a few more results. in particular, we will establish some sufficient conditions for a function space to have the fds property. all undefined notions can be found in [6]. a space x is said to be a sc space if every convergent sequence together with the limit point is a closed subset of x. in a sc space a convergent sequence must have a unique accumulation point. obviously, every sc space satisfies the separation axiom t1. as usual the formula a ⊆∗ b means that a \ b is finite. a set a is a pseudointersection of a collection s provided that a ⊆∗ s for every s ∈ s. the tower number t is the smallest cardinality of a collection of infinite subsets of ω which is well-ordered by ⊇∗ and has no infinite pseudointersection. two sets a and b are almost disjoint if a∩b is finite. a maximal family of infinite pairwise almost disjoint subsets of ω is briefly called a mad family. a 102 a. bella collection θ of mad families is said to be splitting if for any infinite subset a of ω there exists some b ∈ θ and b1, b2 ∈ b such that |a∩b1| = |a∩b2| = ω. h denotes the smallest cardinality of a splitting collection θ of mad families. s denotes the smallest cardinality of a splitting collection θ of mad families each of which consists of two sets. we have ω1 ≤ t ≤ h ≤ s ≤ c. theorem 1. every sc space of weight less than s has the fds property. proof. let x be a sc space of weight κ < s and let a be a countable infinite subset of x. if a contains an infinite subset without accumulation points in x then we are done. on the contrary, the definition of s and an argument analogous to the proof of theorem 6.1 in [5] sufficie to construct a convergent sequence s ⊆ a. since x is a sc space, the set s has only one accumulation point in x and again we are done. � as every hausdorff space of net-weight κ has a weaker hausdorff topology of weight κ and the fds property is maintained by passing to a stronger topology, we have: theorem 2. every hausdorff space of net-weight less than s has the fds property. corollary 1. every hausdorff space of cardinality less than s has the fds property. lemma 1. let x be a space with the fds property and a an infinite subset of x. if a contains no infinite subset without accumulation points then every infinite subset of a contains a convergent subsequence. proof. let a be an infinite subset of x containing no infinite subset without accumulation points and let a′ be any infinite subset of a. since x has the fds property, there exists a countable infinite set b ⊆ a′ whose set of accumulation points is {x0, . . . , xn}. if b is a sequence converging to x0, then we are done. if not, there exists an open neighbourhood u0 of x0 such that the set b1 = b \u0 is infinite. if b1 is a sequence converging to x1, then we are done. if not we may choose an open neighbourhood u1 of x1 such that the set b2 = b1\u1 is infinite. continuing this process, it is clear that for some k ≤ n the corresponding set bk must converge to xk. � the previous lemma immediately implies that any countably compact space with the fds property is sequentially compact. observe however that the same conclusion is no longer true for pseudocompact tychonoff spaces, even in the class of spaces of countable tightness. a possible counterexample is the space constructed assuming [ch] in [4], corollary 2. it is easy to realize that s is the smallest cardinal κ such that any product of κ non-trivial t1 spaces does not have the fds property. more accurate estimate is in the following: remarks on the finite derived set property 103 theorem 3. h is the smallest cardinal κ such that there is a family of κ sc spaces with the fds property whose product does not have the fds property. proof. let κ be the cardinal in the statement of the theorem. as there exists a family of h many compact sequentially compact hausdorff spaces whose product is not sequentially compact [7] and any compact hausdorff space with the fds property is sequentially compact, we immediately see that h ≤ κ. for the converse inequality, let λ < h and let {xα : α < λ} be a family of sc spaces with the fds property. put x = ∏ {xα : α < λ} and denote by πα : x → xα the usual projection mapping. fix a countable infinite set a ⊆ x. if there exists some α such that the set πα[a] contains an infinite subset b without accumulation points in xα, then we are done by considering the set a∩π −1 α (b). in the other case, with the help of lemma 1, we have that for each α every infinite subset of πα[a] contains a convergent subsequence. now, by arguing as in [7], we may construct a convergent sequence s ⊆ a. since any product of sc spaces is still a sc space, the set s has only one accumulation point and we are done. � remark 1. corollary 1 provides a consistent negative answer to question 1.9 in [1]. for this, it is enough to consider a model where ω1 = p < s. the next assertion improves theorem 1.6 in [1]. it should also be compared with corollary 1. theorem 4. any urysohn space of cardinality less than c has the fds property. proof. let x be a urysohn space without the fds property and fix an infinite set a such that every infinite subset of a has infinitely many accumulation points. let a∅ = a. for any s ∈ <ω2, we may select an infinite subset as of a according to the following rule: if t ∈ n2 then at⌢0 and at⌢1 are infinite subsets of at such that at⌢0 ∩at⌢1 = ∅. for any f ∈ ω2 we can fix an infinite set bf ⊆ a such that |bf \ af↾n| < ω for each n < ω. for any f ∈ ω2 let xf be an accumulation point of the set bf . it is easy to check that the map f → xf is injective and consequently |x| ≥ c. � the previous construction with some minor modifications provides a better result for lindelöf spaces. theorem 5. a lindelöf urysohn space of cardinality less than 2t has the fds property. proof. let x be a lindelöf urysohn space and assume that x does not have the fds property. so, we may fix a countable infinite set a ⊆ x such that every infinite subset of a has infinitely many accumulation points. for any α ∈ t and any f ∈ α2 we define an infinite set af ⊆ a in such a way that: (1) if β < α and f ∈ α2 then af ⊆ ∗ af↾β; (2) if f, g ∈ α2 and f 6= g then af ∩ ag is finite. 104 a. bella put a∅ = a and assume to have defined everything for each β < α. if α is a limit ordinal and f ∈ α2, then take as af any infinite pseudointersection of the family {af↾β : β < α}. if α = γ + 1 denote by f ′ and f′′ the only two elements in α2 such that f′ ↾ γ = f′′ ↾ γ = g for some g ∈ γ2 and define af′ and af′′ by selecting two infinite subsets of ag having disjoint closures. by the lindelöfness of x, for any f ∈ t2 we may pick a point xf ∈ ⋂ {a′f↾α : α ∈ t} (here a′ is the derived set of a). as the mapping f → xf is injective, we see that |x| ≥ 2t. � the next result is an attempt to weaken the separation axiom in the previous theorem. its proof mimics theorem 2.5 in [2]. theorem 6. any lindelöf sc space of cardinality not exceeding t has the fds property. proof. let x be a lindelöf sc space and assume that x = {xα : α ∈ t}. let a be a countable infinite subset of x. if a contains some non-trivial convergent sequence, then thanks to the sc property we obtain a subset of a with only one accumulation point. in the opposite case, for any α ∈ t we may select an open set uα and an infinite set aα ⊆ a in such a way that: (1) xα ∈ uα; (2) aα ∩ uα = ∅; (3) aα ⊆ ∗ aβ whenever β ≤ α. at the first step, since a does not converge to x0 there exists an open set u0 such that a \ u0 is infinite and we may put a0 = a \ u0. at step α let b be an infinite pseudointersection of the family {aβ : β < α} and select an open set uα such that xα ∈ uα and aα = b \ uα is infinite. since x is a lindelöf space, the family {uα : α ∈ t} has a countable subcover v. since t is a regular cardinal, we have v ⊆ {uα : α ≤ γ} for some γ ∈ t. to finish, observe that the set aγ has a finite intersection with each member of v and therefore it does not have any accumulation point in x. � corollary 2. every countable sc space has the fds property. recall that a space x is said to be ω-monolithic if any separable subspace of x has a countable net-work. lemma 2. every hausdorff ω-monolithic space has the fds property. proof. let x be a hausdorff ω-monolithic space and fix a countably infinite set a ⊆ x. assume that a is not closed and discrete and let x be an accumulation point of a. since the subspace y = a has a countable net-work, there exists a decreasing family {vn : n ∈ ω} of open neighbourhoods of x in y such that⋂ {vn : n ∈ ω} = {x}. for any n select a point an ∈ vn ∩ (a \ {x}) and let b be the set so obtained. all accumulation points of b must lye in vn for each n and therefore the set b has at most only x has accumulation point. � remarks on the finite derived set property 105 now, we look at the fds property in function spaces in the topology of pointwise convergence (see [3]). for this reason, in the sequel all spaces are assumed to be tychonoff. to begin, observe that a space of the form cp(x) may fail to have the fds property: it suffices to take as x any discrete space of cardinality at least s. a space x is said to be ω-stable if all continuous images of x that can be injectively mapped into a second countable space have a countable net-work. the class of ω-stable spaces contains all pseudocompact ([3], 2.6.2c), all lindelöf σ ([3], 2.6.21) and all lindelöf p spaces ([3], 2.6.28). thus, we have: corollary 3. if x is either a pseudocompact of a lindelöf σ or a lindelöf p space then cp(x) has the fds property. proof. the result follows from lemma 2 and the fact that a space x is ω-stable if and only if cp(x) is ω-monolithic ([3], 2.6.8). � corollary 4. every normed linear space in the weak topology has the fds property. proof. let x be a normed linear space and u∗ be the unit ball in the dual space x∗. the well known theorem of alaoglu says that u∗ is compact in the weak∗ topology on x∗. the result follows by observing that the space x in the weak topology is just a subspace of cp(u ∗). � theorem 7. the space cp(x) has the fds property in the following cases: (1) x has density less than s; (2) x has a dense set which is the union of less than h pseudocompact (or lindelöf σ, or lindelöf p) subspaces. proof. observe that if y is dense in x then there is an injective continuous mapping from cp(x) into cp(y ). therefore, if cp(y ) has the fds property then the same happens to cp(x). item 1 follows from theorem 1. if y = ⋃ {yα : α < κ} then cp(y ) is homeomorphic to a subspace of the space∏ {cp(yα) : α < κ}. taking into account theorem 3, item 2 follows from corollary 3. � corollary 5. if ξ is an ordinal then cp(ξ) has the fds property. looking again at corollary 3, one may wonder whether the second iteration of a function space, namely cp(cp(x)), on a compact space x has yet the fds property. this is not the case in general because the fds property of cp(cp(x)) implies that the compact space x must be sequentially compact. we do not know if the converse always holds, although it happens for sure in the class of compact ω-monolithic spaces. however, the fds property of cp(cp(x)) is far to imply either ω-monolithicity or ω-stability of the base space x. for instance, if x is a subspace of the sorgenfrey line of cardinality ω1 then x is neither ω-monolithic nor ω-stable, but by assuming ω1 < s cp(cp(x)) has the fds property. 106 a. bella of course, corollary 3 and theorem 7 leave open the general problem to characterize the spaces x for which cp(x) has the fds property. some specific questions are: 1. is it true that any hausdorff space of cardinality less than c has the fds property? 2. find a compact sequentially compact space x such that cp(cp(x)) does not have the fds property. references [1] o. t. alas, m. tkachenko, v. tkachuk and r. wilson the fds-property and the spaces in which compact sets are closed, sci. math. japan, to appear. [2] o. t. alas and r. g. wilson, when a compact space is sequentially compact?, preprint. [3] a. v. arhangel′skĭı, “topological function spaces”, kluwer academic publishers, dordrecht (1992) [4] a. bella and o. pavlov, embeddings into pseudocompact spaces of countable tightness, topology appl. 138 (2004), 161–166. [5] e. k. van douwen, “the integers and topology”, handbook of set-theoretic topology (k.kunen and j.e. vaughan editors), elsevier science publishers b.v., amsterdam (1984), 111 –167. [6] r. engelking, “general topology” heldermann-verlag, berlin (1989). [7] p. simon, product of sequentially compact spaces, rend. ist. mat. univ. trieste 25 (1994), 447–450. [8] d. shakmatov, m. tkachenko and r. wilson, transversal and t1-independent topologies, houston j. math. 30 (2004), 421–433. received december 2004 accepted january 2005 a. bella (bella@dmi.unict.it) dipartimento di matematica, cittá universitaria, viale a.doria 6, 95125, catania, italy. gutkubagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 239-242 sandwich-type characterization of completely regular spaces javier gutiérrez garćıa ∗,§ and tomasz kubiak ∗ abstract. all the higher separation axioms in topology, except for complete regularity, are known to have sandwich-type characterizations. this note provides a characterization of complete regularity in terms of inserting a continuous real-valued function. the known fact that each continuous real valued function on a compact subset of a tychonoff space has a continuous extension to the whole space is obtained as a corollary. 2000 ams classification: 54d15, 54d30, 54c30. keywords: insertion; sandwich theorem; insertion theorem; completely regular space; lower semicontinuous; upper semicontinuous; compact. 1. introduction all the higher separation axioms in general topology, except for complete regularity, are known to have sandwich-type (= insertion-type) characterizations. a canonical example is provided by the katětov-tong-hahn insertion theorem for normal spaces (see [6], [10], and [2]). a topological space is normal if, given two disjoint closed sets a and b, there exist two disjoint open sets u and v containing a and b respectively. also recall that, given a topological space x, a function f : x → r is lower [upper] semicontinuous if f −1(t, ∞) [f −1(−∞, t)] is open for each t ∈ r. ∗this research was supported by the ministry of education and science of spain and feder under grant mtm2006-14925-c02-02. §the first named author also acknowledges financial support from the university of the basque country under grant upv05/101. 240 j. gutiérrez garćıa and t. kubiak theorem (katětov-tong-hahn). let x be a topological space. then the following are equivalent: (1) x is normal. (2) if g, h : x → r, g is upper semicontinuous, h is lower semicontinuous, and g ≤ h, then there exists a continuous function f : x → r such that g ≤ f ≤ h. more examples can be seen in [8]. in this note we give an insertion-type characterization of completely regular spaces. we note that insertion theorems usually have urysohn-type lemmas and tietze-type extension theorems as corollaries, and so does the insertion theorem of this note. 2. sandwich-type characterization of completely regular spaces we need some notation. let c(x, i) [u sc(x, i)] be the set of all continuous [upper semicontinuous] functions from a topological space x to i = [0, 1]. we recall that x is completely regular (no lower separation axiom assumed) if, whenever k ⊂ x is closed and x ∈ x \ k, there exists an f ∈ c(x, i) such that f (x) = 1 and f (k) = {0}. equivalently, x is completely regular if and only if, given an open set u ⊂ x and x ∈ u , there is a continuous f : x → i such that 1{x} ≤ f ≤ 1u . here and elsewhere 1a denotes the characteristic function of a subset a ⊂ x. by using some ideas of fuzzy topology (cf. [4]) or point-free topology (cf. [5]), one has a yet more convenient formulation. statement a topological space x is completely regular if and only if, whenever u ⊂ x is open, there exists an open cover v of u with the property that for every v ∈ v there is an fv ∈ c(x, i) such that 1v ≤ fv ≤ 1u . proof. the only if part: by complete regularity, we have 1{x} ≤ gx ≤ 1u for each x ∈ u , where gx ∈ c(x, i). let vx = g −1 x ( 1 2 , 1] and fx = min(1, 2gx). then v = {vx}x∈u is an open cover of u and 1vx ≤ fx ≤ 1u . the if part is evident. � we recall that two disjoint subsets a and b of a topological space x are completely separated if there exists an f ∈ c(x, i) such that f = 1 on a and f = 0 on b. equivalently, if 1a ≤ f ≤ 1x\b. to state our insertion theorem, we need a “general” property of a function f : x → i holding, in particular, for 1{x}. the right choice is to require each [f ≥ t] to be compact for all t > 0. this can actually be taken as a definition, but we prefer to distinguish a class of maps for which this property becomes a characterization. in what follows, t stands for the constant map on x taking the value t ∈ i. all the infs and the sups of families of functions are pointwise. in particular, (inf k) (x) = inf{k(x) : k ∈ k}. definition given a topological space x, an f : x → i is called compact-like if, given a t ∈ i \ {0} and k ⊂ u sc(x, i) with min(f, inf k) < t, there exists a finite k0 ⊂ k such that min(f, inf k0) < t. sandwich-type characterization of completely regular spaces 241 properties let x be a topological space. the following hold: (1) f : x → i is compact-like iff [f ≥ t] is compact for all t ∈ i \ {0}. (2) a ⊂ x is compact iff 1a is compact-like. (3) if x is compact, then u sc(x, i) consists of compact-like functions. (4) if x is hausdorff and f : x → i is compact-like, then f ∈ u sc(x, i). proof. for (1): let u be an open cover of [f ≥ t] with t > 0, then min(f, inf{1x\u : u ∈ u}) < t. but then the finite subfamily u0 ⊂ u for which min(f, inf{1x\u : u ∈ u0}) < t yields [f ≥ t] ⊂ ⋃ u0. conversely, let min(f, inf k) < t with t > 0. then ∅ = [min(f, inf k) ≥ t] = [f ≥ t] ∩⋂ k∈k[k ≥ t]. by the finite intersection property, there exists a finite k0 ⊂ k with ∅ = [f ≥ t] ∩ ⋂ k∈k0 [k ≥ t] = [min(f, inf k0) ≥ t]. this translates into min(f, inf k0) < t and proves (1). finally, (2) follows from (1), while (3) and (4) are obvious. � the concept of a compact-like function shows that sandwich-type characterizations of higher separation axioms, viz.: perfect normality [9], complete normality [7], normality ([6] and [10]), continue to hold for the case of complete regularity. we shall need the following general insertion theorem. theorem 1 (blair [1], lane [8]). for x a topological space and two arbitrary functions g, h : x → i, the following statements are equivalent: (1) there exists a continuous function f : x → i such that g ≤ f ≤ h. (2) if s < t in i, then [g ≥ t] and [h ≤ s] are completely separated. the equivalence (1) ⇔ (2), in the theorem which follows, is well known (see 3.11(c) in [3]). we provide a short proof for completeness. theorem 2 for x a topological space, the following are equivalent: (1) x is completely regular. (2) [urysohn-type lemma] every two disjoint subsets of x, one of which is compact and the other is closed, are completely separated. (3) [insertion] if g, h : x → i, g is compact-like, h is lower semicontinuous, and g ≤ h, then there exists a continuous function f : x → i such that g ≤ f ≤ h. proof. (1) ⇒ (2): let a ∩ b = ∅, a being compact, b being closed. by complete regularity, there exist an open cover u of x \ b and a family {fu }u∈u ⊂ c(x, i) such that 1u ≤ fu ≤ 1x\b. since a is compact, a ⊂ ⋃ u0 for a finite u0 of u. then 1a ≤ g = sup{fu : u ∈ u0} ≤ 1x\b. the continuous g completely separates a and b. (2) ⇒ (3): let g ≤ h be as in (3). for any s, t ∈ i with s < t one has [g ≥ t] ∩ [h ≤ s] = ∅ where [g ≥ t] is compact and [h ≤ s] is closed. by theorem 1, there is a continuous f ∈ c(x, i) such that g ≤ f ≤ h. (3) ⇒ (1): this is obvious, for if x ∈ u with u open, then 1{x} ≤ 1u where 1{x} is compact-like and 1u is lower semicontinuous. � 242 j. gutiérrez garćıa and t. kubiak it is a heuristic principle that a sandwich-type theorem provides an extension theorem. this is the case of our sandwich theorem. in order to avoid speaking about compact-closed sets in a completely regular space, we shall assume x to be tychonoff (completely regular + hausdorff). corollary ([3], 3.11(c)). let x be a tychonoff space, let a ⊂ x be compact, and let f : a → r be continuous. then there exists a continuous function f : x → r such that f (x) = f (x) for all x ∈ a. proof. given a compact subset a ⊂ x and a continuous function f : a → r, the set f (a) is bounded and we can assume that f (a) ⊂ i. now, define g, h : x → i as follows: g = f = h on a, g = 0, and h = 1 on x \ a. since a is closed, h is lower semicontinuous. also, if t > 0, then [g ≥ t] = [f ≥ t] is closed in a, hence compact in x. by theorem 2, there exists a continuous f : x → i with g ≤ f ≤ h. clearly, f extends f to the whole of x. � references [1] r. l. blair, extension of lebesgue sets and real valued functions, czechoslovak math. j. 31 (1981) 63–74. [2] g. buskes and a. van rooij, topological spaces. from distance to neighborhood, springer, new york, 1997. [3] l. gillman and m. jerison, rings of continuous functions, springer-verlag, new york, 1976. [4] b. hutton, uniformities on fuzzy topological space, j. math. anal. appl. 58 (1977) 559–571. [5] p. t. johnstone, stone spaces, cambridge univ. press, cambridge, 1982. [6] m. katětov, on real-valued functions in topological spaces, fund. math. 38 (1951) 85– 91; correction: fund. math. 40 (1953) 203–205. [7] t. kubiak, a strengthening of the katětov-tong insertion theorem, comment. math. univ. carolinae 34 (1993) 357–362. [8] e. p. lane, insertion of a continuous function, top. proc. 4 (1979) 463–478. [9] e. michael, continuous selections i, ann. of math. 63 (1956) 361–382. [10] h. tong, some characterizations of normal and perfectlynormal spaces, duke j. math. 19 (1952) 289–292. received march 2006 accepted march 2007 javier gutiérrez garćıa (javier.gutierrezgarcia@ehu.es) departamento de matemáticas, upv-ehu, apdo. 644, 48080 bilbao, spain tomasz kubiak (tkubiak@amu.edu.pl) matematyki i informatyki, uniwersytet im. adama mickiewicza, ul. umultowska 87, 61-614 poznań, poland beluagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 281-292 scott-representability of some spaces of tall and mǐskin harold bennett and david lutzer ∗ abstract. in this paper we show that a variation of a technique of mǐskin and tall yields a cocompact completely regular moore space that is scott-domain-representable and has a closed gδ-subspace that is not scott-domain-representable. this clarifies the general topology of scott-domain-representable spaces and raises additional questions about scott-domain representability in moore spaces. 2000 ams classification: primary 54e30; secondary 54d70,06b35, 06f30, 54h12, 54d20 keywords: domain, scott-domain, scott-domain-representable space, moore space, complete moore space, cocompact, čech-complete, subcompact, choquet complete. 1. introduction a domain is a continuous poset (p, ⊑) in which each non-empty directed subset has a supremum. a scott domain is a domain in which each nonempty bounded set has a supremum. (for more details, see section 2.) representing mathematical objects as the set of maximal elements of a domain or of a scott domain is an idea that originated in theoretical computer science. every domain carries a natural topology, called the scott topology, and a topological space is said to be domain representable (respectively, scott-domainrepresentable) if it is homeomorphic to the set of maximal elements of a domain (respectively, a scott domain) with the relative scott topology. in recent years, topologists have come to see domain representability and scott-domain representability as strong completeness properties associated with the baire category theorem. for example, every subcompact regular space is domainrepresentable [4] and every domain-representable space is choquet complete [8], and therefore a baire space. (see section 2 for definitions.) ∗corresponding author. 282 h. bennett and d. lutzer the basic general topology of domain-representable spaces is fairly well understood. for example, while domain-representability is an open-hereditary property, it is not closed-hereditary (because if x is any completely regular space that is not domain-representable, then the space obtained from βx by isolating all points of βx − x is domain-representable [3] and contains x as a closed subspace). similarly, scott-domain-representability is open-hereditary and not closed-hereditary (as can be seen by applying the same βx construction described above). further, any gδ-subspace of a domain-representable space is domain-representable, as shown in [3], so it is natural to ask whether gδ-subspaces of scott-domain-representable spaces inherit scott-domain representability. among metrizable spaces, the answer is “yes,” because if x is a scott-domain representable metric space, then x is completely metrizable. let y be a gδ-subset of x. then y is also completely metrizable so that a recent result of kopperman, kunzi, and waszkiewicz [7] shows that y is scott-domain representable. the first goal of this paper is to show that, without metrizability, scott-domain-representability is not inherited by (closed) gδ-subspaces. furthermore, our example is a moore space, a particularly nicely-behaved type of generalized metric space. it was already known that the equivalence among metric spaces of essentially all strong completeness properties (complete metrizability, scott-domainrepresentability, čech-completeness, cocompactness, subcompactness, and domain-representability) breaks down outside of the metric space category. but there is still a rich theory of completeness in the wider class of moore spaces, and results due to k. martin, tall, rudin, bennett, lutzer, and reed show that • among moore spaces, domain-representability is equivalent to subcompactness [4] and is equivalent to rudin-completeness [2] which is strictly weaker than moore-completeness [6]; • for completely regular moore spaces, moore completeness is equivalent to čech-completeness [2]; • there is a completely regular moore space that is čech-complete but not cocompact [11] and not scott-domain-representable [5]; • if a moore space is scott-domain-representable, then it is completely regular and moore-complete, čech complete [8], and cocompact [7]; additional equivalents of domain-representability among moore spaces that involve the strong choquet game are given in [5]. the second goal of this paper is to explore the role of scott-domain representability in the class of completely regular moore spaces and we show that a certain moore space x0 (due to mǐskin [10]) is scott-domain-representable and contains a closed gδsubspace z (due to tall [11]) that is not scott-domain representable. this example raises a natural question about completeness and representability in moore spaces, namely: question 1.1. is scott-domain-representability equivalent to cocompactness among completely regular moore spaces? scott-representability of some spaces of tall and miškin 283 kopperman, kunzi, and waszkiewicz [7] have characterized scott-domainrepresentability in any completely regular space as being a combination of cocompactness and a bi-topological condition (“pairwise complete regularity”), but is not yet clear how to apply their characterization in the moore space context. a natural place to look for counterexamples to question 1.1 is in mǐskin’s construction of a cocompact moore space, mentioned above. in section 3 we show that some of mǐskin’s spaces are scott-domain-representable, but we do not know the answer to the following: question 1.2. is it true that each of mǐskin’s spaces in [10] is scott-domainrepresentable? in this paper we show that a certain čech-complete moore space constructed by tall embeds as a closed subspace of a scott-domain representable moore space. to what extent is this a general phenomenon? more precisely, we have: question 1.3. does each completely regular, čech-complete moore space x embed in a moore space y (x) that is scott-domain-representable? what if x is required to be a dense subspace of y (x)? what if x is required to be a closed subspace? basic definitions appear in section 2. section 3 gives the basic constructions due to tall and mǐskin, and shows that, with some additional restrictions, one of mǐskin’s spaces is scott-domain-representable and has a closed gδ-subspace that is not. throughout the paper, we reserve the symbols r, q, and p for the usual sets of real, rational, and irrational numbers. 2. basic definitions a space x is cocompact if it is t1 and has a collection c of closed subsets with the following two properties: a) if d is a centered 1 subcollection of c, then ⋂ d 6= ∅; b) if u is an open subset of x and x ∈ u , then some c ∈ c has x ∈ int(c) ⊆ c ⊆ u . note that the members of c might not be the closures of their interiors, even when the interiors are non-void. if one insists that members of c are the closures of their interiors, i.e., are regularly-closed sets, then one obtains a different notion called regular cocompactness. the sorgenfrey line, for example, is cocompact but not regularly cocompact [2]. cocompactness was introduced by de groot and his colleagues [1]. another strong completeness first studied by the amsterdam school is subcompactness, where we say that a space x is subcompact if x has a base b with the property that ⋂ f 6= ∅ whenever f ⊆ b has the property that if b1, b2 ∈ f, then some b3 ∈ f has cl(b3) ⊆ b1 ∩ b2. 1a collection d is centered if ⋂ {di : i ≤ n} 6= ∅ whenever {di : i ≤ n} is a finite subcollection of d. 284 h. bennett and d. lutzer to define domain-representability and scott-domain-representability, we begin with a poset (s, ⊑). a subset e ⊆ s is directed if for each e1, e2 ∈ e some e3 ∈ e has e1, e2 ⊑ e3. if sup(e) ∈ s whenever e is a nonempty directed subset of s, then s is a dcpo (“directed-complete partial order”). given a, b ∈ s, we write a ≪ b to mean that whenever e ⊆ s is a directed set with b ⊑ sup(e), then some e ∈ e has a ⊑ e. the set ⇓(b) is defined to be {a ∈ s : a ≪ b}. in case ⇓(b) is directed and has b as its supremum for each b ∈ s, we say that s is continuous. if s is a continuous dcpo, then we say that s is a domain. if the domain s has the additional property that every nonempty bounded subset of s has a supremum in s, then we say that s is a scott domain. among domains, scott domains are easily characterized: lemma 2.1. a domain (s, ⊑) is a scott domain if and only if sup({a, b}) exists whenever a, b ∈ s and a, b ⊑ c for some c ∈ s. proof. to prove the nontrivial half of the lemma, suppose e is a nonempty bounded subset of s. let f ∈ s be an upper bound for e. if e1, e2, e3 ∈ e, then sup({e1, e2}) ∈ s and f is an upper bound for {sup({e1, e2}), e3} in s so that sup(sup(e1, e2), e3) ∈ s. it is easy to show that sup(sup(e1, e2), e3) = sup(sup(ei, ej ), ek) for each permutation i, j, k of 1, 2, 3, so that the supremum of each three-element subset of the bounded set e is well-defined. similarly, sup(f ) is a well-defined point of s for each non-empty finite subset f ⊆ e. now let d := {sup(f ) : ∅ 6= f ⊆ e and |f | < ω}. then d is a directed subset of s so that, s being a domain, sup(d) ∈ s. clearly sup(d) = sup(e) as required. � every poset (s, ⊑) can be endowed with a special topology called the scott topology in which a set u is open if and only if it satisfies both (i) if x ⊑ y and x ∈ u , then y ∈ u , and (ii) if e ⊆ s is a nonempty directed set with sup(e) ∈ u , then e ∩ u 6= ∅. in a domain s, the collection of all sets ⇑(a) := {b ∈ s : a ≪ b} is a base for the scott topology on s. the set of maximal elements of a domain s is denoted by max(s). if a topological space x is homeomorphic to the subspace max(s) of some domain s with the relative scott topology, then we say that x is domain-representable. if s is a scott-domain and x is homeomorphic to max(s), then we say that x is scott-domain-representable. kopperman, kunzi, and waszkiewicz [7] have characterized scott-domainrepresentable spaces as being the cocompact spaces that also satisfy a certain bi-topological condition. a short, direct proof of cocompactness of any scottdomain-representable space is possible and we give it here. a central tool is the following interpolation lemma [9]. lemma 2.2. suppose a ≪ c in a domain s. then some b ∈ s has a ≪ b ≪ c. lemma 2.3. let s be a scott domain. for each p ∈ s, let ↑(p) = {q ∈ s : p ⊑ q}. then each set ↑(p) ∩ max(s) is a relatively closed subset of max(s). proof. suppose that x ∈ max(s) is a limit point of ↑(p) ∩ max(s). then for each q ≪ x, ⇑(q) ∩ ↑(p) 6= ∅. consequently p and q have a common extension, scott-representability of some spaces of tall and miškin 285 so that r(p, q) := sup{p, q} is in s. let e := {r(p, q) : q ≪ x}. we claim that e is a directed set. for suppose that r(p, q1), r(p, q2) ∈ e. because ⇓(x) is directed, some q3 ∈ ⇓(x) has q1, q2 ⊑ q3. then r(p, q3) ∈ e and r(p, qi) ⊑ r(p, q3) for i = 1, 2. because s is a dcpo, sup(e) ∈ s so that some z ∈ max(s) has sup(e) ⊑ z. recall that as a subspace of s, max(s) is a t1space. therefore, if z 6= x, then some q4 ≪ x has z 6∈ ⇑(q4). because q4 ≪ x, lemma 2.2 gives q5 ∈ s with q4 ≪ q5 ≪ x. but q5 ≪ x forces r(p, q5) ∈ e so that q4 ≪ q5 ⊑ r(p, q5) ⊑ sup(e) ⊑ z. therefore z ∈ ⇑(q5) ⊆ ⇑(q4), contrary to our choice of q4. therefore, p ⊑ sup(e) = z = x showing that x ∈ ↑(p) as required. � our next result appears in [7]. we present an easy direct proof. corollary 2.4. suppose s is a scott domain. then the subspace of maximal elements of s is cocompact. proof. first, the subspace max(s) of s is t1. second, let c = {max(s) ∩ ↑(p) : p ∈ s}. in the light of lemma 2.3, each member of c is a closed subset of max(s). to verify the first part of the cocompactness definition, suppose that d ⊆ c is a centered collection. write d = {max(s) ∩ ↑(a) : a ∈ a}. then, given any finite set f := {a1, · · · , ak} ⊆ a we know that ↑(a1) ∩ · · · ∩ ↑(ak) 6= ∅ because d is centered, so that sup(f ) ∈ s by lemma 2.1. let â := {sup(f ) : ∅ 6= f ⊆ a and |f | < ω}. then â is directed, so sup(â) ∈ s, say sup(â) = b ∈ s. then ∅ 6= ↑(b) ∩ max(s) ⊆ ⋂ d, as required. to verify the second part of the definition of cocompactness, it is enough to consider a point x in a basic open set max(s)∩⇑(q). the interpolation lemma 2.2 provides a point p ∈ s with q ≪ p ≪ x. then ⇑(p) is a neighborhood of x with ⇑(p) ⊆ ↑(p) ⊆ ⇑(q) so that x is in the relative interior of ↑(p) ∩ max(s) which is contained in the closed set ↑(p)∩max(s) ⊆ max(s)∩⇑(q), as required to show that max(s) is cocompact. � 3. a variation of spaces of tall and mǐskin tall and mǐskin began their constructions with a countable subset of the plane that had uncountably many limit points on the x-axis. we need more control and so we replace that countable set by a binary tree t with ω-many levels and use its branch space y in place of the limit points on the x-axis. this tree may be embedded in the upper half plane in such a way that its branch space corresponds in a natural way to an uncountable set (the cantor set) on the x-axis. therefore, the space we will construct is one of the spaces due to mǐskin. description of the space x and the subspace x0: a) the tree t : let t be a binary tree with ω-many levels. denote the unique minimal element of t by 0̄. the level of any d ∈ t in our tree is denoted by lv(d) and t (n) = {d ∈ t : lv(d) = n} so that t = ⋃ {t (n) : 0 ≤ n < ω}. 286 h. bennett and d. lutzer b) the branches of t : let y be the set of all branches of t , i.e., each y ∈ y is a maximal linearly ordered subset of t . we let e(y, n) denote the unique element of the branch y that lies at level n of the tree t . thus, for example, e(y, 0) = 0̄ for each y ∈ y and if y1, y2 ∈ y have e(y1, n) = e(y2, n), then e(y1, k) = e(y2, k) for each 0 ≤ k ≤ n. c) the space x: let t ∗ := {(d, s) : d ∈ t, ∅ 6= s ⊆ y }. the underlying set of our space is x = t ∗ ∪y and the set x is topologized by isolating each point of t ∗ and by using the sets n (y, n) = {y} ∪ {(d, s) ∈ t ∗ : lv(d) ≥ n, d ∈ y, and y ∈ s} as basic neighborhoods of y ∈ y . equivalently, n (y, n) = {y} ∪ {(e(y, k), s) ∈ t ∗ : n ≤ k < ω and y ∈ s}. d) the subspace x0 = x − {(0̄, y )}. this is a closed and open subspace of x and will be the space used in our example. however, x0 is not needed until the very end of section 3. as in [10], the space x is a cocompact, čech-complete, completely regular moore space and therefore so is its closed and open subspace x0. the subspace x0 contains a closed (and hence gδ) -subspace z := {(d, s) ∈ t ∗ : (d, s) 6= (0̄, y ), |s| < ω} ∪ y that is homeomorphic to one of the spaces constructed by tall in [11]. consequently, z is not cocompact and therefore is not scottdomain-representable (by corollary 2.4). what remains is to prove that x0 is scott-domain representable. our first step is to define the scott domain that comes very close to representing x, and then to work around the “almost” part of that statement to show that x0 is scott-domain representable. we begin by constructing our poset. the poset (s, ⊑) e) the sets i(b, k): let ∅ 6= b ⊆ y and let k ≥ 0. if |b| = 1, then let i(b, k) := n (y, k) where y is the unique point of b. if |b| ≥ 2, then let i(b, k) := {(d, s) ∈ t ∗ : lv(d) ≥ k, b ⊆ s, and ∀y ∈ b, d ∈ y}. note that the condition d ∈ y is equivalent to e(y, lv(d)) = d, a fact that will be used later. f) let s = {{t} : t ∈ x} ∪ {i(b, k) : i(b, k) 6= ∅, ∅ 6= b ⊆ y, 0 ≤ k < ω} and let ⊑ denote reverse inclusion. consequently, if t ∈ x, then i(b, k) ⊑ {t} means t ∈ i(b, k). remark 3.1. if |b| ≥ 2, one can prove that i(b, k) = ⋂ {n (y, k) : y ∈ b}, and that was the way we initially thought of the sets i(b, k). however, that fact is not really needed in our construction. the next example illustrates how the sets i(b, k) can behave. it introduces special notations, and parts c), d), and e) will be very important tools in the proofs of later lemmas in this section. example 3.2. let d′, d′′ be the two points of t (1) and recall that 0̄ is the unique point of t (0). let y′, y′′ ∈ y have e(y′, 1) = d′ and e(y′′, 1) = d′′ (so scott-representability of some spaces of tall and miškin 287 that y′, y′′ are two branches of t that disagree at level 1 of the tree). note that y′ ∩ y′′ = {(0̄, y )} = t (0). then a) a set of the form i(b, k) can be empty. for example, i({y′, y′′}, 1) = ∅ = i(y, 1) because if (d, s) ∈ i({y′, y′′}, 1) then lv(d) ≥ 1 and e(y′, lv(d)) = d = e(y′′, lv(d)). because t is a tree and lv(d) ≥ 1 we must have d′ = e(y′, 1) = e(y′′, 1) = d′′ so that y′ ∩ y′′ contains some element of t at or above level 1, which is false. b) i({y′, y′′}, 0) is the infinite set {(0̄, s) : y′, y′′ ∈ s ⊆ y } and i(y, 0) is the singleton set {(0̄, y )}. c) for any i(b, k), if (d, s) ∈ i(b, k) then (d̂, s) ∈ i(b, k) where d̂ is the unique predecessor of d in level k of the tree t . d) for any i(b, k), if (d, s) ∈ i(b, k) then (d, b) ∈ i(b, k) and (d, s′) ∈ i(b, k) whenever s ⊆ s′ ⊆ y . in particular (d, y ) ∈ i(b, k). e) from b), c) and d), the only way that |i(b, k)| = 1 is for k = 0 and b = y , and then i(y, 0) = {(0̄, y )}. lemma 3.3. for any b ⊆ y and any k ≥ 0, |i(b, k)∩y | ≤ 1. if |b| ≥ 2 then i(b, k) ⊆ t ∗ and π1[i(b, k)] is finite, where π1 : t ∗ → t is first coordinate projection. proof. the first two assertions follow directly from the definition of the sets i(b, k), so we prove only the final assertion. because |b| ≥ 2 we may choose distinct y1, y2 ∈ b. then there is some integer l such that e(y1, l) 6= e(y2, l) so that e(y1, j) 6= e(y2, j) for each j ≥ l. therefore, if (d, s) ∈ i(b, k), we know that d ∈ t and lv(d) < l, and there are only finitely many such points. � lemma 3.4. the maximal elements of s are the singleton sets {x} where x ∈ x. lemma 3.5. if ∅ 6= b1 ⊆ b2 ⊆ y with and k1 ≤ k2, then i(b1, k1) ⊑ i(b2, k2). furthermore if i(b1, k1) ⊑ i(b2, k2) 6= ∅, then b1 ⊆ b2 and k1 ≤ k2. proof. first suppose that b1 ⊆ b2 and k1 ≤ k2. if |b2| = 1 then b1 = b2. let y be the unique point of b2. then k1 ≤ k2 gives i(b2, k2) = n (y, k2) ⊆ n (y, k1) = i(b1, k1) and hence i(b1, k1) ⊑ i(b2, k2). in case b2 has at least two points, then i(b2, k2) ⊆ t ∗ so that each element of i(b2, k2) has the form (d, s) where lv(d) ≥ k2 and d ∈ b2. hence lv(d) ≥ k2 ≥ k1 and d ∈ y for each y ∈ b2. because b1 ⊆ b2, we have (d, s) ∈ i(b1, k1), as required. to prove the second claim, note that i(b1, k1) ⊑ i(b2, k2) gives i(b2, k2) ⊆ i(b1, k1) because ⊑ is reverse inclusion. now fix any (d, s) ∈ i(b2, k2). then lv(d) ≥ k2, b2 ⊆ s, and d ∈ y for all y ∈ b2. let d̂ be the unique predecessor of d at level k2 of the tree t . then (see example 3.2), (d̂, s) ∈ i(b2, k2) ⊆ i(b1, k1) so that k2 = lv(d̂) ≥ k1. thus k1 ≤ k2. next, example 3.2 shows that since (d, s) ∈ i(b2, k2), (d, b2) ∈ i(b2, k2) ⊆ i(b1, k1) so that b1 ⊆ b2, as required. � 288 h. bennett and d. lutzer lemma 3.6. let e := {i(bα, kα) : α ∈ a} be a directed subset of (s, ⊑) that contains no maximal element of itself. let c = ⋃ {bα : α ∈ a}. a) if |c| = 1 then the set {kα : α ∈ a} is unbounded, and sup(e) = {y} where y is the unique point of c. (note that in this case, y ∈ y .) b) if |c| ≥ 2, then {kα : α ∈ a} is bounded and sup(e) = i(c, l) where l = max{kα : α ∈ a}. proof. in case (a), it is clear that {y} is an upper bound for e, and that no other {z} for z ∈ y can be an upper bound for e. in addition, each bα = {y}. if the set {kα : α ∈ a} is bounded, let kβ be its largest member. then i(bβ , kβ ) is the maximal member of e, contrary to hypothesis. therefore {kα : α ∈ a} is unbounded, and now it is clear that sup e = {y}. to prove (b), fix distinct y1, y2 ∈ c and choose αi ∈ a with yi ∈ bαi for i = 1, 2. using directedness of e, find β ∈ a with i(bαi , kαi ) ⊑ i(bβ , kβ ). then i(bβ , kβ ) 6= ∅ so that by lemma 3.5 yi ∈ bαi ⊆ bβ . according to lemma 3.3, the set f := π1[i(bβ , kβ )] is finite. next, we claim that some d ∈ f has d ∈ π1[i(bα, kα)] for each α ∈ a. for contradiction, suppose that corresponding to each d ∈ f there is some γ(d) ∈ a with d 6∈ π1[i(bγ(d), kγ(d))]. directedness of e provides some η ∈ a with i(bβ , kβ ) ⊑ i(bη, kη) and such that i(bγ(d), kγ(d)) ⊑ i(bη, kη) for each of the finitely many d ∈ f . choose any (d̄, s) ∈ i(bη, kη). then i(bβ , kβ ) ⊑ i(bη, kη) yields i(bη, kη) ⊆ i(bβ , kβ) so that d̄ ∈ π1[i(bβ , kβ )] = f . because d̄ ∈ f we know that γ(d̄) is defined and d̄ 6∈ π1[i(bγ(d̄), kγ(d̄)]. because i(bγ(d̄), kγ(b̄)) ⊑ i(bη, kη) we have (d̄, s) ∈ i(bη, kη) ⊆ i(bγd̄ , kγb̄ ) and that is impossible because we know that d̄ 6∈ π1[i(bγ d̄ , kγ(d̄))]. at this stage of the argument, we know that there is some d0 ∈ f with d0 ∈ π1[i(bα, kα)] for each α ∈ a. then for some sα ⊆ y we have (d0, sα) ∈ i(bα, kα). because bα ⊆ c, part (c) of example 3.2 shows that (d0, c) ∈ i(bα, kα). consequently lv(d0) ≥ kα and we conclude that lv(d0) is an upper bound for the set {kα : α ∈ a}. let l be the largest member of the set {kα : α ∈ a}. note that lv(d0) ≥ l. next we claim that (d0, c) ∈ i(c, l). consider the membership criteria for i(c, l). we already know that lv(d0) ≥ l and obviously c ⊆ c, so all we must show is that d0 ∈ y for each y ∈ c. fix any y ∈ c. then there is some α ∈ a with y ∈ bα. from above we know that (d0, c) ∈ i(bα, kα) so that y ∈ bα gives d0 ∈ y as required. now we know that i(c, l) 6= ∅ so that i(c, l) ∈ s. according to lemma 3.5, i(c, l) is an upper bound for e. to complete the proof that i(c, l) = sup(e), we consider any upper bound g ∈ s for e and we will show that i(c, l) ⊑ g. with i(bβ , kβ ) as defined in the second paragraph of this proof, we have i(bβ , kβ ) ⊑ g so that g ⊆ i(bβ , kβ ). hence g ⊆ i(bβ , kβ ) ⊆ t ∗ so that either g has the form g = i(h, m) or else g = {(e, s)} ∈ max s. in the first case, lemma 3.5 shows that i(bα, kα) ⊑ i(h, m) implies bα ⊆ h and kα ≤ m for each α ∈ a, so that c ⊆ h and l = max{kα : α ∈ a} ≤ m. hence i(c, l) ⊑ i(h, m) = g, as claimed. in the second case, where g = {(e, s)}, we will show that (e, s) ∈ i(c, l). note scott-representability of some spaces of tall and miškin 289 that i(bα, kα) ⊑ g = {(e, s)} gives (e, s) ∈ i(bα, kα) so that lv(e) ≥ kα and bα ⊆ s for each α and therefore c ⊆ s and lv(e) ≥ max{kα : α ∈ a} = l. furthermore, if y ∈ c then y ∈ bα for some α ∈ a so that (e, s) ∈ i(bα, kα) guarantees that e ∈ y. therefore, i(c, l) ⊑ g, as required. to show that i(c, l) = sup(e). � lemma 3.7. in s, we have i(b1, k1) ≪ i(b2, k2) if and only if b1 is a finite set, b1 ⊆ b2, and k1 ≤ k2. proof. first suppose i(b1, k1) ≪ i(b2, k2). then i(b1, k1) ⊑ i(b2, k2) so that b1 ⊆ b2 and k1 ≤ k2. we let f be the collection of all finite subsets of b2. then e := {i(f, k2) : f ∈ f} is a directed subset of s and i(b2, k2) = sup e so that i(b1, k1) ≪ i(b2, k2) gives i(b1, k1) ⊑ i(f1, k2) for some f1 ∈ f, showing that b1 ⊆ f1. since f1 is finite, so is b1. for the converse, suppose that b1 is a finite set and b1 ⊆ b2 and k1 ≤ k2 (so that i(b1, k1) ⊑ i(b2, k2)), and suppose that e = {i(bα, kα) : α ∈ a} is a directed subset of s with i(b2, k2) ⊑ sup(e). if e contains a maximal element of itself, there is nothing to prove, so assume that e contains no maximal element. let c := ⋃ {bα : α ∈ a}. there are several cases to consider. in case |c| ≥ 2, lemma 3.6 gives i(b1, k1) ⊑ i(b2, k2) ⊑ sup e = i(c, l) where l is the largest member of the bounded set {kα : α ∈ a}, say l = kγ for some γ ∈ a. then i(b1, k1) ⊑ i(b2, k2) ⊑ i(c, l) gives b1 ⊆ b2 ⊆ c. therefore, each y in the finite set b1 is a point of c = ⋃ {bα : α ∈ a}, so we may find α(y) ∈ a with y ∈ bα(y). directedness of the collection e allows us to find β ∈ a with i(bα(y), kα(y)) ⊑ i(bβ , kβ) for each y in the finite set b1 and therefore y ∈ bα(y) ⊆ bβ . therefore b1 ⊆ bβ. once again using directedness, find δ ∈ a with i(bγ , kγ ), i(bβ , kβ ) ⊑ i(bδ, kδ). then b1 ⊆ bβ ⊆ bδ and k1 ≤ max{kα : α ∈ a} = l = kγ ≤ kδ ≤ l. therefore i(b1, k1) ⊑ i(bδ, kδ) ∈ e as required. the remaining case is where |c| = 1, say c = {z}. then bα = {z} for each α ∈ a. because e contains no maximal element of itself, lemma 3.6 shows that sup e = {z} and that {kα : α ∈ a} is unbounded. choose µ ∈ a with kµ > k1. then i(bµ, kµ) = n (z, kµ) ⊆ n (z, k1) = i(b1, k1) so that i(b1, k1) ⊑ i(bµ, kµ) ∈ e as required. � lemma 3.8. suppose s ∈ s and y ∈ y . then s ≪ {y} if and only if s = i({y}, k) for some k ≥ 0. proof. suppose s = i({y}, k). by lemma 3.7, i({y}, k) ≪ i({y}, k) ⊑ {y}, so we know that s = i({y}, k) ≪ {y}. for the converse, suppose s ∈ s has s ≪ {y}. then s ⊑ {y} so that y ∈ s. by lemma 3.3, either s = i({y}, k) or else s = {y}. if s = {y} let e := {i({y}, k) : k ≥ 0}. this is a directed set in s with sup e = {y} and yet no member i({y}, k) ∈ e has s = {y} ⊑ i({y}, k). therefore, s must have the form s = i({y}, k) as claimed. � 290 h. bennett and d. lutzer lemma 3.9. for t ∈ x − y, {t} ≪ {t} provided t 6= (0̄, y ). proof. write t = (d, s) with (d, s) 6= (0̄, y ). to show that {t} ≪ {t}, suppose {t} ⊑ sup e where e is a directed subset of s. maximality of {t} in s (see lemma 3.4) shows that sup(e) = {t}. if e contains a maximal member, there is nothing to prove, so for contradiction, suppose e contains no maximal member of itself. then the collection e must be of the form e = {i(bα, kα) : α ∈ a}. write c = ⋃ {bα : α ∈ a}. if |c| = 1, then c = {y} ⊆ y , so that lemma 3.6 shows sup e = {y} and hence {y} = {t}. that is impossible because y ∈ y and t ∈ x − y . therefore |c| ≥ 2. because |c| ≥ 2, from lemma 3.6 we know that the set {kα : α ∈ a} is bounded and sup e = i(c, l) where l is the maximal element of the bounded set {kα : α ∈ a}. then {t} = sup(e) = i(c, l) so that i(c, l) is a singleton. part (e) of example 3.2 shows that the set i(c, l) can be a singleton if and only if c = y and l = 0, and then i(c, l) = {(0̄, y )}, forcing us to conclude that t = (0̄, y ), which is false. this contradiction completes the proof of the lemma. � corollary 3.10. the poset (s, ⊑) is continuous. proof. consider any element s ∈ s. if s ≪ s, then s ∈ ⇓(s), so that ⇓(s) is directed with sup(⇓(s)) = s. so suppose s ≪ s is false. then lemmas 3.8 and 3.9 show that one of the following three statements must be true: (i) s = i(b, k) where b is infinite, or (ii) s = {y} for some y ∈ y , or (iii) s = {(0̄, y )}. if s = i(b, k) where b is infinite, let f be the collection of all finite subsets of b. then, by lemma 3.7, ⇓(i(b, k)) = {i(f, j) : j ≤ k, f ∈ f}, which is directed and has i(b, k) as its supremum, as required. in case s = {y} for some y ∈ y , then ⇓(s) = {i({y}, k) : k ≥ 1} which is also directed and has supremum s = {y}, as required. the case where s = {(0̄, y )} is actually a special case of item (i) because {(0̄, y )} = i(y, 0) as noted in example 3.2, above. � lemma 3.11. (s, ⊑) is a scott domain. proof. suppose u1, u2 ∈ s have a common extension. we may assume that neither ui is maximal in s (so that ui = i(bi, ki) for i = 1, 2) and that neither of u1, u2 is contained in the other. then there is some (d, s) ∈ i(b1, k1) ∩ i(b2, k2). let c = b1 ∪ b2. because neither of u1, u2 is contained in the other, |c| ≥ 2 and (d, s) ∈ i(c, max(k1, k2)) yields i(c, max(k1, k2)) 6= ∅ so that i(c, max(k1, k2)) ∈ s. clearly i(c, max(k1, k2)) is an upper bound for u1 and u2. to show that i(c, max(k1, k2)) is the least upper bound of u1 = i(b1, k1) and u2 = i(b2, k2), consider any upper bound u3 ∈ s for u1 and u2. from ui ⊑ u3 we obtain u3 ⊆ u1∩u2. because |c| ≥ 2 we know that u3 ⊆ u1∩u2 ⊆ scott-representability of some spaces of tall and miškin 291 x − y , so that u3 cannot have the form {y} for some y ∈ y . therefore either u3 = i(d, j) for some d and some j, or else u3 = {(d̂, ŝ)} ∈ max(s). in the first case bi ⊆ d and j ≥ ki for i = 1, 2 so that c ⊆ d and max(k1, k2) ≤ j and therefore (see lemma 3.5) i(c, max(k1, k2)) ⊑ u3. in the second case, where u3 = {(d̂, ŝ)} ∈ max(s), for i = 1, 2 we know that (d̂, ŝ) ∈ i(bi, ki) so that lv(d̂) ≥ ki, bi ⊆ ŝ, and that for each y ∈ bi, y ∈ d̂. hence i(c, max(k1, k2)) ⊑ u3. therefore i(c, max(k1, k2)) = sup(u1, u2) as required. � there is a natural-looking function that sends each x ∈ x to the element {x} ∈ s. this mapping is 1-1, onto, and continuous from x to max(s), and it is tempting to think that the function is an a homeomorphism from x onto max(s). unfortunately, it is not. the point (0̄, y ) ∈ x is isolated in x, but the point {(0̄, y )} is not an isolated point of max(s). we are lucky that (0̄, y ) is the only “bad” point for the natural mapping. recall that x0 = x − {(0̄, y )}. then we have: lemma 3.12. the function h : x0 → max(s)−{{(0̄, y )}} given by h(t) = {t} is a homeomorphism from x0 onto the open subspace max(s) − {{(0̄, y )}} of max(s) with the relative scott topology. proof. clearly the function h is 1-1 and h[x0] = max(s)−{{(0̄, y )}}. to prove that h is continuous, it is enough to consider what happens at non-isolated points of x0, i.e., at points y ∈ y . suppose h(y) ∈ ⇑(p) ∩ max(s) where p ∈ s. then lemma 3.8 guarantees that p = i({y}, k) = n (y, k) for some k. we claim that that h[n (y, k + 1)] ⊆ ⇑(p). apply lemma 3.9 to show that if (d, s) ∈ n (y, k+1) then (d, s) 6= (0̄, y ) so that h((d, s)) = {(d, s)} ≪ {(d, s)}. then note that p ⊑ {(d, s)} ≪ {(d, s)} so that h(d, s) ∈ ⇑(p) as required. to prove that h is an open mapping onto max(s) −{{(0̄, y )}}, the first step is to recall lemma 3.9 which shows that if t ∈ x − y with t 6= (0̄, y ), i.e., if t is an isolated point of x0, then in s, {t} ≪ {t} so that h(t) = {t} is an isolated point of max(s). second, consider any non-isolated point y ∈ x0 and note that for k ≥ 1, h[n (y, k)] = max(s) ∩ ⇑(i({y}, k). therefore h is an open mapping onto max(s) − {{(0̄, y )}} as required. � our next lemma shows that x0 is scott-domain-representable. lemma 3.13. the subspace x0 = x −{(0̄, y )} is scott-domain-representable. proof. because s is a scott domain, we know that its subspace max(s) is scott-domain-representable. it is easy to check that for any domain d, the subspace max(d) is t1. therefore we see that for our scott domain s, the set max(s)−{{(0̄, y )}} is an open subspace of the scott-domain-representable space max(s). now recall that any non-empty, relatively open subset of a scott-domain representable space is also scott-domain representable, and that completes the proof. � 292 h. bennett and d. lutzer references [1] j. aarts, j. degroot and r. mcdowell, cocompactness, nieuw archief voor wiskungid 36 (1970), 2–15. [2] j. aarts and d. lutzer, completeness properties designed for recognizing baire spaces, dissertationes mathematicae 116 (1974), 1–45. [3] h. bennett and d. lutzer, domain representable spaces, fundamenta mathematicae 189 (2006), 255–268. [4] h. bennett and d. lutzer, domain representability of certain complete spaces, houston j. math, to appear. [5] h. bennett, d. lutzer and g. m. reed, domain representability and the choquet game in moore and bco-spaces, topology and its applications, to appear. [6] m. e. estill, concerning abstract spaces, duke mathematics journal 17 (1950), 317–327. [7] r. kopperman, h. kunzi and p. waszkiewicz, bounded complete models of topological spaces, topology and its applications 139 (2004), 285–297. [8] k. martin, topological games in domain theory, topology and its applications 129 (2003), 177–186. [9] k. martin, m. mislove and g.m. reed, topology and domain theory, pp. 371-394 in recent progress in general topology ii, ed. by m husak and j. van mill, elsevier, amsterdam, 2002. [10] v. mǐskin, the amsterdam properties in moore spaces, colloq. math soc. janos bolyai 41 (1983), 427–439. [11] f. tall, a counterexample in the theories of compactness and metrization, indag. math. 35 (1973), 471–474. received july 2007 accepted october 2007 harold bennett (bennett@math.ttu.edu) texas tech university, lubbock, tx 79409, usa. david lutzer (lutzer@math.wm.edu) college of william & mary, williamsburg, va 23187, usa. () @ appl. gen. topol. 16, no. 1(2015), 15-17doi:10.4995/agt.2015.1826 c© agt, upv, 2015 contractibility of the digital n-space sayaka hamada department of mathematics, yatsushiro campus, national institute of technology, kumamoto college, 866-8501 japan. (hamada@kumamoto-nct.ac.jp) abstract the aim of this paper is to prove a known fact that the digital line is contractible. hence we have that the digital space (zn, κn) is also contractible where (zn, κn) is n products of the digital line (z, κ). this is a fundamental property of homotopy theory. 2010 msc: 14f35; 54b10. keywords: khalimsky topology; digital n-space; contractible; homotopy. 1. prelimarilies we consider an important property of homotopy theory for the digital nspace. the digital line (z, κ) is the set of the integers z equipped with the topology κ having {{2m − 1, 2m, 2m + 1} : m ∈ z} as a subbase. for x ∈ z, we set u(x) := { {2m − 1, 2m, 2m + 1} if x = 2m, {2m + 1} if x = 2m + 1. then {u(x)} is a fundamental neighborhood system at x. then it it obvious that {2m : m ∈ z} is closed and nowhere dense in z, {2m+ 1 : m ∈ z} is open and dense in z. u(x) is the minimal open set containing x for any x ∈ z. (see [1],[2],[4],[5]). the digital line (z, κ) was introduced by e. khalimsky in the late 1960’s and it was made use of studying topological properties of digital images. (see [3], [6]). received 4 november 2013 – accepted 23 september 2014 http://dx.doi.org/10.4995/agt.2015.1826 s. hamada the digital n-space (zn, κn) is the topological product of n copies of the digital line (z, κ). to investigate the digital n-space is very interesting for the application possibility. here we focus the contractibility of one. 2. contractibility of the digital line and digital n-space a space x is called contractible provided that there exists a homotopy h : x×i → x such that hx×{0} is the identity and hx×{1} is a constant function. the digital line is contractible as pointed out in remark 4.11 of [7]. we shall show by direct computation. theorem 2.1. the digital line is contractible. proof. defining h : z × i → z by h{0}×i ≡ 0 and for any n ∈ z\{0}, if n is an odd number, h(n, t) :=                    n if 0 ≤ t < 2−|n|, n − 1(if n > 0), n + 1(if n < 0) if 2−|n| ≤ t ≤ 2−(|n|−1), n − 2(if n > 0), n + 2(if n < 0) if 2−(|n|−1) < t < 2−(|n|−2), . . . 1(if n > 0), −1(if n < 0) if 2−2 < t < 2−1, 0 if 2−1 ≤ t, if n is an even number, h(n, t) :=                    n if 0 ≤ t ≤ 2−|n|, n − 1(if n > 0), n + 1(if n < 0) if 2−|n| < t < 2−(|n|−1), n − 2(if n > 0), n + 2(if n < 0) if 2−(|n|−1) ≤ t ≤ 2−(|n|−2), . . . 1(if n > 0), −1(if n < 0) if 2−2 < t < 2−1, 0 if 2−1 ≤ t, then we see that hz×{0} = idz and hz×{1} ≡ 0. since h is continuous, idz and the constant map (≡ 0) is homotopic. therefore we have (z, κ) is contractible. � since a contractible finite product of contractible spaces is contractible, we have the following. corollary 2.2. the digital n-space is contractible. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 16 contractibility of the digital n-space references [1] m. fujimoto, s. takigawa, j. dontchev, h. maki and t. noiri, the topological structures and groups of digital n-spaces, kochi j. math. 1(2006), 31–55. [2] s. hamada and t. hayashi, fuzzy topological structures of low dimensional digital spaces, journal of fuzzy mathematics 20, no. 1 (2012), 15–23. [3] e. d. khalimsky, on topologies of generalized segments, soviet math. doklady 10(1969) 1508–1511. [4] e. khalimsky, r. kopperman and p. r. meyer, computer graphics and connected topologies on finite ordered sets, topology appl, 36(1990), 1–17. [5] t. y. kong, r. kopperman and p. r. meyer, a topological approach to digital topology, am. math. monthly 98(1991), 901–917. [6] e. h. kronheimer, the topology of digital images, topology appl. 46(1992), 279–303. [7] g. raptis, homotopy theory of posets, homology, homotopy and applications 12, no. 2 (2010), 211–230. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 17 bareliagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 2, 2005 pp. 107-117 the language of topology: a turkish case study bill barton, frank lichtenberk and ivan reilly∗ abstract. topology has its own specialised language. where did this come from? what are the differences in the language of topology when it is expressed in english, spanish, mandarin, czech or turkish? does topology itself change when expressed in different languages? what effect has language had on the development of topology? does the language of the topologist make a difference to the mathematics? a research programme aimed at answering these questions has begun. this paper is the first in a series that provides a background to the research. topological discourse in various languages is being examined for its particular features, and possible influences on the concepts developed through these languages. data from turkish topologists and topological terminology are examined. they show why there is reason to suspect that language influences mathematical concept development. the data are also used to explore methodological issues for the research project. 2000 ams classification: 54a99, 00a35, 00a99 keywords: language and topology, open, connected 1. background this paper is part of the background to an investigation into the relationship between language and research level mathematics. general topology has been chosen as the context because it is one of the most abstract of all mathematical areas: it deals with basic and apparently highly defined concepts that are generally regarded as being universal amongst topologists. general topology ∗the research reported in this paper has been funded by the new zealand ministry of research, science & technology marsden fund and the university of auckland research committee the authors wish to record their thanks to their turkish consultants michael brown, yücel tiras, emin özçag, tüna yalvaç, and riza erturk, both for their academic contribution to this paper, and their hospitality in ankara. 108 b. barton, f. lichtenberk and i. reilly also has a large international research community in which several diverse languages are represented. it thus provides an ideal field in which to investigate the influence (if any) of different languages on the development of mathematical concepts. the central question of the project is whether the language of the research topologist affects the use and development of topological concepts in his/her research. if differences are found between language groups of topologists, then it is hoped to determine whether they are language-based, and how any differences evolved. the research study has been in progress for four years, during which time a questionnaire-type instrument has been developed and trialled ([2]). this instrument contains five tasks that request information about topological concepts in different ways. progress is currently being made towards the collection of data from about a dozen different language communities of research topologists: arabic, czech, english, greek, japanese, mandarin, polish, romanian, russian, spanish and turkish. respondents are being sought who have learned topology, and who teach and use it significantly, in the target language. the research instrument is completed in the target language. there are serious methodological issues involved in this research. one set of issues concerns the validity of translations of the instruments and the responses that are necessary to make cross-linguistic comparisons. another problem concerns the representativeness of the respondents, and a further difficulty refers to how it can be determined whether group conceptual differences exist. this paper is aimed at yet another set of problems. assuming that group differences are found, how can it be determined whether these differences are language-based? if linguistic issues are implicated, how can the interaction of language and mathematical concept development be investigated? in order to undertake such an investigation it is necessary to have a full understanding of the linguistic and social history of topological discourse in each of the languages of the study. this history includes the development of terminology in general topology, the network of influential people, the movement of topological knowledge between various communities of mathematicians, the possible external influences on scientific knowledge, general linguistic analysis, and the general relationships between the languages during the period of the development of general topology. in recent reviews both nagata [8] and rudin [11] refer to the set theoretic nature of the foundations of topology, where “topological properties were thought of as axioms” ([11], p. 566), and where the solution of problems depends upon set theoretic assumptions like zfc, martin’s axiom, or the negation of the continuum hypothesis ([8], p. 562). being reminded of these formalist beginnings leaves open the issue of the status of topological work—it certainly allows that individual topologists may operate with different ideas on this subject. our natural language is the medium through which we must strive to express our philosophical beliefs and which we use without generally being aware of the options that other languages provide. does it therefore influence our mathematical thoughts as we seek to describe fundamental properties? the language of topology 109 this paper builds on an examination of topological discourse in turkish, and is a first model of the work that is required. it is intended to be read as a stimulus to thinking about the link between topological concepts and language in any language familiar to the reader. 2. topology in the turkish language turkish is a member of the turkish branch of the altaic language family. among its closest relatives are azerbaijani and uzbek. mongolian is a more distant relative. an important event in the history of turkish was the language reform/revolution initiated by mustafa kemal atatürk in 1928 ([5]). called öz türkçe (pure turkish), its aims were the replacement of the arabic script by the latin alphabet (suitably modified) and “purification” of the language, ridding it of arabic and persian words. this reform continues through younger turkish speakers, who now have a poor knowledge of arabic grammar structures and tend not to use those arabic words that remain. the founder and leader of modern mathematics in turkey was cahit arf (19101997) ([7]). however much of general topology work in turkey was generated through l. michael brown, an english academic who arrived in ankara in 1968. his work at hacettepe university, both topological research and the encouragement and mentoring of new topologists there, has meant that this university has been a major centre of topological activity: it is the biggest community of topologists in turkey, and many of the other centres (for example, antalya, eskisehir, or mersin) contain graduates of hacettepe. the link with english topology remains, with the majority of graduate students who go overseas going to the united kingdom. a very few have gone to usa and to germany, but none have gone to the arab world or to the soviet union/russia (although a linguistic link between turkey and azerbaijan exists). at hacettepe, some undergraduate level courses are taught in turkish and some in english, but the topological ones are taught in turkish. at graduate level all courses are taught in turkish, and topological seminars are usually given in turkish. scientific publication in turkish journals is in english, occasionally together with turkish, and the turkish journal of mathematics contains english language articles exclusively. specialised topological vocabulary in turkish was developed mainly in hacettepe, partly as a result of the particular interest of professor brown. he was also a member of the team responsible for the collection and publication of a dictionary of mathematical words in turkish with english, german and russian equivalents ([4]). this focus in one place has resulted in some terminology that is peculiar to that university. for example, the word for ‘set’ is küme (meaning ‘heap, mound, pile, hill’) at hacettepe, but is cümle (meaning ‘sentence or clause’, but also ‘a whole, total, ensemble, group’) in ankara university (10 kilometres away). however, much terminology had already been developed by turkish professors at various universities in other fields of mathematics, and thus the topological discourse in turkish is both widely understood throughout turkey, and is consistent with other branches of mathematics. 110 b. barton, f. lichtenberk and i. reilly the development of topological vocabulary has reflected the development of turkish language in general. thus terms with arabic roots have been avoided, for example, the term for ‘field’ is alan (a general term for area) from old turkish rather than the arabic word saha (the more accurate translation of field in its agricultural sense). this may explain the cümle/küme change described above: cümle is an arabic word. (another reason might be the use of cümle in karacümle to mean ‘basic arithmetic’ and hence a double meaning within mathematical terminology). atatürk’s öz türkçe (pure turkish) did not just replace arabic and persian words, it used turkish roots to coin new words. the mathematical dizi (sequence) is an example, derived from the root diz meaning ‘to line up, arrange in a row, to string beads’. the gerund (noun form created from the verb) is dizen, but for the mathematical meaning a new word was created. the following explanation about topology in turkish was written by michael brown (personal communication, 2003): in general terms both [english and turkish] seem capable of expressing mathematical concepts and arguments with equal precision. but having said that i cannot help but feel that the structure of english is somewhat better suited to mathematics than that of turkish. one point . . . is the position of the verb at the end of the sentence. whereas in english one would write “there exists a continuous function f . . . ” which established from the beginning that it is the existence of something that is involved, in turkish one would say something like “having the property of continuity, a function f there is” giving the property (continuity) first, of what (the function) second, and its existence last. longer examples can have you describing quite complex properties of things before it comes clear what it is that has these properties. of course the end result is no less exact in an absolute sense, and one gets used to having things this way round, so perhaps it is just a question of what one is used to. however, there are ways of forcing a wordorder more similar to english by using an equivalent of “such that” (the result not being considered ‘good turkish’). [this is] often resorted to by speakers used to lecturing in english and (often) by research students, so perhaps the effort required to produce a well structured sentence in such cases is something that even native speakers of turkish find noticeable. turkish is quite an expressive language, and the use of suffixes means one can pack a lot of meaning into a single word, so it is often very economic. in some areas it is well supplied with synonyms, but not in all, so it is sometimes difficult to name new concepts similar, but not identical to, known ones. the language of topology 111 3. open sets and the issue of multiple meanings much topological terminology in all languages uses words that have general, everyday meanings. the general meanings are not only different from the specific technical meaning of topology, but also there may be more than one common meaning. an important question for the research project is to determine whether the general meaning of such terms interferes with the understanding and use of the mathematical term. where there are several general meanings, different languages often privilege different meanings, even when the words are regarded as being equivalent in translations. it has been hypothesised that such an influence is a likely source of differential linguistic effect across languages. ‘open’ is one such common word with many meanings, and its use in the key topological concept of ‘open set’ has been remarked on before ([2]). it was noted that different topologists had given each of four fundamentally different notions of ‘open’ as the one that applied to the use of this word in the technical term ‘open set’: • open as opposed to closed, i.e. simply an opposite; • open as in an open border, i.e. admits aliens or objects to pass through a boundary; • open as in an open door, i.e. the place of entry; • open as in an open field, i.e. without boundaries at all. in turkish, the word for open (açık ) is as versatile in general turkish as the word ‘open’ is in general english. indeed, in investigating this word, it became clear that there were more than these four categories of meaning for ‘open’ in both english and turkish that are available for interpretation mathematically. the meanings are not all common to both languages. what has been attempted in the table 1 below is a categorisation of the meanings of ‘open’ on a topological basis: ‘open’, in both languages, suggests several possible mathematical interpretations, and the research question is therefore whether topologists privilege one of these meanings when they are thinking about the topological concept. if so, is there a distinctive pattern of meanings amongst a group of topologists, or is it an individual phenomenon? if it is a group pattern, can this be related to the language of the group? 4. topological spaces and the issue of world views there are other terms where the word used for a topological concept has a distinct difference between languages in its general meaning. the term ‘topological space’ is a case in point. in english the word ‘space’, like ‘open’, has a variety of meanings. its general meanings can be both bounded (3-d: a room in a house, a place on a bookshelf; 2-d: a space in a carpark; 1-d: a typographical gap between words) or unbounded (3-d: the universe). the german term, raum, is similar, although less frequently used for unbounded outer space. 1 1 2 b . b a rto n , f . l ic h te n b e rk a n d i. r e illy table 1: open — açık meaning category 1 meaning category 2 english examples turkish example gateway (3-d) lets things in or out open bottle açık şişe lets things out open cage / open valve açık kafes lets things in / through open door açık kapu gateway (2-d) lets things through open border gateway (non-spacial) lets things in open mind açık fikir lets things out open mouth no border (3-d) not contained open fire / open air (no equivalent: ates, açmak) not restricted open day (public) no border (2-d) open sea / open field açık deniz no border (1-d) single direction open-ended açık birakılmıs, single direction increasing open auction açık kartirma (open auction – price can increase indefinitely) loose open weave (no equivalent) uncovered open eye açık göz uncompleted open order (for goods) other unanswered open question start open a conference ready for use open a shop welcoming open-faced açık yürekli clear in colour açık çay (open tea) clear weather açık hava (open air) clear in meaning açık mana (clear mathematical result) clever göz açık o n th e o th er h a n d , th e term in t u rk ish , u za y , ca n o n ly b e u sed in th e u n b o u n d ed sen se in ev ery d a y la n g u a g e. m a n d a rin , sim ila rly, u ses th e w o rd fo r u n iv erse. h o w ev er, th is w o rd is m a d e fro m tw o ch a ra cters, th e fi rst m ea n in g em p ty / n o n e/ n o th in g , th e seco n d m ea n in g b etw een o r so m eth in g -in -th e-m id d le. t h u s th ere is a sen se o f a n em p tin ess b etw een tw o b o u n d a ries. t h e resea rch q u estio n h ere is th e fo llo w in g . it ca n b e a ssu m ed th a t a lin g u istic p red isp o sitio n to a p a rticu la r m ea n in g ex ists, ev en fo r m a th em a ticia n s w h o k n o w clea rly th a t a m a th em a tica l ‘sp a ce’ is p recisely d efi n ed . d o es th is d isp o sitio n in fl u en ce to p o lo g y in a w a y th a t ca n b e id en tifi ed a m o n g st, sa y, t u rk ish o r g erm a n to p o lo g ists a s a g ro u p ? the language of topology 113 however the significance of the concept of topological space goes beyond the multiple meanings of the everyday use of the word ‘space’. it has been noted elsewhere ([1]) that different languages represent the world in different ways: indo-european languages represent it as empty space that gets filled with objects; navajo ([10]) and euskera ([1]), on the other hand, represent the universe as filled with ‘matter’ that takes on different forms at different times and in various places. a mathematical representation of the world has constructed topological space as the basic building block. for example nagata ([8]) declares his understanding of space as “an extensive vacancy, whose fundamental attribute consists of distance and dimension”. such a conception aligns more with the indo-european one, but is not exactly the same. the question of interest is whether world views of different languages interpret the concept of topological space in idiosyncratic ways. 5. field and the issue of historical antecedents the term ‘field’ also has several meanings, although they all derive from a common root meaning a piece of ground. thus ‘field of study’ is a metaphorical use of the idea of a large piece of land on which you might do something. the mathematical concept of field, however, was first referred to in german by dedekind in 1858 as zahlkörper (body of numbers) ([6]). the topological term in that language, körper, means a physical body. apart from the dimensional difference (2-d field versus 3-d body), and the ontological difference (field is a stretch of land on which objects might be placed or actions performed, a body is an object itself), topological ideas such as containment are differently represented. it is said that puritanical victorian english society did not allow the image of the naked human form to be used in mathematics hence a new word, field, was introduced in the 1890s. this entertaining hearsay is probably more a commentary on stereotyping than it is on mathematicians’ attitudes. nevertheless, it begs the question as to why the german was not directly translated into english as ‘body’ ? historical causes for differences in terminology in turkish have already been noted above, where the linguistic forces predisposing english/french over arabic have affected the topological language. another example of this is the term for connected. in the past the arabic word irtibatlı was used by analysts, however the topological term is based on the word bağlanmak meaning tied together (as shoelaces) or buttoned up (as the front of a coat). the spanish terms for the word ‘connected’, however, reveal another historical influence. this is one of several words that are different in castilian spanish from mexican spanish. in spain the word conexo is used, whereas in mexico connected is translated as conectado. the explanation is the differing origins of mathematical influence in spain and mexico. spanish topologists, like mathematicians in other branches, were originally influenced by french mathematicians ([9]). the french term for connected is the past participle connexé which was “spanified” by dropping an ‘n’ and changing the ending, to form a word that did not previously exist in spanish. mexican topologists, on the other 114 b. barton, f. lichtenberk and i. reilly hand, were influenced by american colleagues, and they directly translated the past participle to its normal spanish form conectado. for the purposes of this research study, it is possible that the new word will have a meaning more ‘pure’ mathematically since that is the only context of its use, whereas the regular past participle has everyday connotations that will affect its mathematical meaning. 6. neighbourhood and the issue of different common meanings a more subtle issue, but one that might be important mathematically, also surrounds the term ‘connected’. in english this is a general term, but usually has implications of a relatively permanent condition: for example, i am connected to someone through a genealogical relationship that will always exist. in turkish, the term bağlantili comes from bağlanmak, meaning tied together (as with shoelaces) or buttoned up (as the front of a coat). the difference, mathematically, is between a characteristic of something, a state that exists, or the effect of an action. the difference can be explained in english with the use of the word ‘connected’ with respect to telephones. if your telephone is connected, then it is in a permanent state of being available for use. you disconnect it when you move house. in the days of telephone exchanges, however, the operator would connect your call: this was a temporary state that was the result of an action. the mandarin word for connected is made up of two characters which, combined, carry both the above senses. the first character means ‘joined’ (as in touching) and the second means ‘connected by a route’ (as two towns might be). the way that mandarin can compound two or more ideas into one word appears to be a distinct advantage of this type of language. another place where this feature is apparent is the mandarin term for complete: it is again made from two characters, the first an abbreviation of the character for ‘perfect’, and the second for ‘prepared’, thus: ‘perfectly prepared’. how do these meanings play out in topology? a connected space is a special kind of space, some spaces are connected others are not—it is a fact of life for topological spaces, just as i have brown eyes. topologists check to see whether a candidate space has the (desirable) property of connectedness. thus the relatively permanent sense of ‘connected’ that implies a characteristic is indicated. it is possible to make a space which is not connected into a connected space (to connectify it)—the result, however, is a new space, it is not the same space with a new property. the same situation occurs with the term ‘neighbourhood’. in english the root word neighbour (from old english neah = near + bur = farmer) has two different extensions: neighbourhood, referring to the surrounding space; and neighbourliness, referring to the relationship between neighbours. it is the former of these, the geometrical meaning, that is adopted in topology. however the term in turkish komşuluk has the meaning of a relationship—a fundamentally different conception mathematically. the czech term is different again. okoli also has a geometric meaning, but it is based on the word for a circle and means ‘around’ in the sense that we the language of topology 115 might say someone lives around here, i.e. in any direction although reasonably close. this meaning mirrors the diagrammatic form: when neighbourhoods are drawn as part of explanations, they are usually drawn with small circles around a point, notwithstanding the definition that does not necessarily imply a circle nor a boundary. 7. topological discourse in addition to the actual vocabulary of topology, there is also the question of how things are phrased, how sentences are put together, habits of speaking, and so on. these discourse features are known to be different for different languages. a further complication is that, even within one language, the mathematical discourse is likely to be different from everyday discourse. for example, in english, mathematical discourse is generally more conceptually dense, the role of prepositions is heightened, there is a lack of redundancy, and an increased use of logical connectors ([3]). therefore important questions for this research study are whether distinct discourse features from the particular language being spoken are present in the mathematical context; and whether these affect the mathematics of the speakers of that language. for example, a feature of turkish grammar is that nouns are inflected for case. that is, a suffix is added to the noun to indicate the way it is being used in a sentence. thus, in the following phrases the words for ‘neighbourhood’ all have different forms: the neighbourhood of x is closed the point y is in the neighbourhood of x the set s is the neighbourhood of x all the points of the neighbourhood of x a function f from the neighbourhood of x another feature of turkish (and japanese) compared with english (or spanish) is that, in a sentence, all the qualifying clauses come before the main verb. thus it is not natural to say: “ the function f : (x, [nx]) → (y, [vx]) is continuous if for every x ∈ x and for every vf (x) there exists an nx such that f (nx) ⊂ vf (x). ” 8. conclusion after four years of preliminary investigation we have reason to believe that differences exist within the field of topology. we note the idiosyncratic approaches to topology between individuals, the way they will speak about their understanding of particular concepts (notwithstanding their analytic use of the same definitions). our question is whether there are also group differences in topological conceptions. at some level this is already noted within the community of topologists. rudin ([11], p. 565) writes: 116 b. barton, f. lichtenberk and i. reilly the difficulty is that topology is not, and never really has been, one subject. . . . the basic assumptions and definitions, the theorems which are considered classic and necessary for every student and educated mathematician to understand, the theorems which a particular topologist thinks are important or hopes to prove, the tools he expects to be used in proofs, the very meaning of the word topology, all vary so widely that large active groups of topologists can hardly speak to each other because their languages are so different. our concern is whether similar differences exist between topologists working in the same “active group” but using different natural languages to do their work. again, at one level, the answer is clear. for example, czech mathematicians use “mapping” for what is termed “function” in english (husek, personal communication). but such usages are known within the community of mathematicians and taken into account when publishing. however, the deeper nature of some of the differences between natural languages leads us to believe that this study is indeed warranted. the outcome remains open, however. the existence of differences is yet to be shown, and any differences need to be related to natural language features. in attempting this work we will be creating social histories of topology within particular language groups, analysing the topological discourse of different languages, and relating the content of topology to these social and historical features. such analyses will, we hope, be interesting of themselves, but cannot be completed without assistance from the international community of topologists. we therefore invite comment, correction, and critique of this, and subsequent, articles. references [1] b. barton and r. frank, mathematical ideas and indigenous languages: the extent to which culturally-specific mathematical thinking is carried through the language in which it takes place, in b. atweh, h. forgasz & b. nebres (eds) sociocultural research in mathematics education: an international perspective, mahwah, nj:lawrence erlbaum associates (2001), 135-149. [2] b. barton and i. reilly, topological concepts and language: a report of research in progress, notices of the south african mathematical society 30(2) (1999), 110-119. [3] t. dale and g. cuevas, integrating language and mathematics learning, in j. crandall (ed) esl through content-area instruction, englewood cliffs, nj: prentice hall regents (1987), 9-52. [4] h. hacisalihoglu, a. haciyev, v. kalantarov, a. sabuncuoglu, l. m. brown, e. ibikli and s. brown, matematik terimleri sözlügü, ankara: hacettepe university (2000). [5] g. lewis, the turkish language reform: a catastrophic success, oxford: oxford university press (1999). [6] j. miller, earliest known uses of some of the words of mathematics, website accessed 27.11.2003: http://members.aol.com/jeff570/mathword.html, (2003). [7] s. mardesic, topology in eastern europe 1900 – 1950, topology proceedings 25 (2000), 397-430. the language of topology 117 [8] j. nagata, looking back at modern general topology in the last century, in m. husek & j. van mill, recent progress in general topology ii, netherlands: elsevier science b. v. (2002), 561-564. [9] k. h. parshall and a. c. rice, mathematics unbound: the evolution of an international mathematical research community, 1800-1945, providence, ri: american mathematical society (2001). [10] r. pinxten, i. van dooren and f. harvey, the anthropology of space: explorations into the natural philosophy and semantics of the navajo, philadelphia: university of philadelphia press (1983). [11] m. e. rudin, topology in the 20th century, in m. husek & j. van mill, recent progress in general topology ii, netherlands: elsevier science b. v. (2002), 565-569. received july 2004 accepted february 2005 bill barton (b.barton@auckland.ac.nz) department of mathematics, the university of auckland, private bag 92019, auckland, new zealand. frank lichtenberk (f.lichtenberk@auckland.ac.nz) department of applied language studies and linguistics, the university of auckland, private bag 92019, auckland, new zealand. ivan reilly (i.reilly@auckland.ac.nz) department of mathematics, the university of auckland, private bag 92019, auckland, new zealand. huagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 203-209 generalized independent families and dense sets of box-product spaces wanjun hu abstract. a generalization of independent families on a set s is introduced, based on which various topologies on s can be defined. in fact, the set s with any such topology is homeomorphic to a dense subset of the corresponding box product space (theorem 2.2). from these results, a general version of the hewitt-marczewski-pondiczery theorem for box product spaces can be established. for any uncountable regular cardinal θ, the existence of maximal generalized independent families with some simple conditions, and hence the existence of irresolvable dense subsets of θ-box product spaces of discrete spaces of small sizes, implies the consistency of the existence of measurable cardinal (theorem 4.5). 2000 ams classification: 03e05, 05d05; secondary: 54a25. keywords: generalized independent family, box product. 1. introduction following notation in [5], a (θ, κ)-independent f amily on s is a subfamily i⊆ p(s) such that for any two disjoint subfamilies i0, i1 ⊆ i with |i0∪i1| < θ, the set ⋂ {a : a ∈ i0} ∩ ⋂ {s \ a : a ∈ i1} has cardinality κ. given a space 〈x, t 〉, it is irresolvable ([9]) if x does not have two disjoint dense subsets. following [3], let s(〈x, t 〉) be the smallest cardinal κ such that every family of pairwise disjoint nonempty open sets has size strictly less than κ. please refer to [10] about cardinals and ideals, and [6] and [3] for topological terminologies. the hewitt-marczewski-pondiczery theorem and hausdorff’s theorem (i.e., there are uniformly independent families of size 2κ on any set s of size κ. see [8], [6] for more details) are equivalent, since each separated (θ, κ)-independent family of size 2|s| on a set s induces a tychonoff topology on s which is homeomorphic to a dense subset of {0, 1}2 |s| . such kind of topologies induced 204 w. hu by independent families appeared in a different form in van douwen’s paper [4] and then the paper [5] by f.w. eckertson. on the other hand, kunen [11] established the equiconsistency between the existence of maximal σ-independent family and the existence of measurable cardinals. later kunen, szymanski and tall in [12] (see also [14]) studied the properties of the ideal of nowhere dense subsets of a λ-baire irresolvable space, and also gave a method to construct a λ-baire open-hereditarily irresolvable (the term ”strongly irresolvable” was used. we follow the notation in [4]) topology from a λ-complete ideal with a lifting. in [14], it was shown that a λ-complete ideal on λ with certain conditions has a lifting. in this paper, we study a generalization of independent families and its relation to box product spaces. we provide a generalization of the equivalence between hausdorff’s theorem and the hewitt-marczewski-pondiczery theorem to generalized independent families and dense subsets of box product spaces (theorem 3.2) (see also [7]). we show, in section 2, that various topologies can be defined on a set s by any generalized independent family on s, and any such topology is homeomorphic to a dense subset of the corresponding θ-box product spaces. this general equivalence enables us to obtain similar work (section 4) like that in [11] and [12] by substituting baire irresolvable dense subsets with irresolvable dense subsets of box product spaces. 2. generalized independent families and induced topologies an independent family can be viewed as a family of partitions on some set s, in which each partition consists of two subsets. in general, we can consider the following generalized version. definition 2.1. let i= {{iβα : β < λα} : α < τ} be a family of partitions on an infinite set s with each λα ≥ 2, and let κ, λ, θ ≥ ω be three cardinals. • if for any j ∈ [τ ]<θ and any f ∈ πα∈j λα the intersection ∩{i f (α) α : α ∈ j} has size at least κ, then i is called a “(θ, κ)generalized independent family” on s, and a “(θ, κ, λ)-generalized independent family” when λα = λ for all α < τ . • i is called “separated” if for any {x, y} ∈ [s]2, there exists an α < τ and β < λα such that x ∈ i β α and y /∈ i β α . a (θ, κ, 2)-generalized independent family is a (θ, κ)-independent family, and a σ-independent family defined in [11] is an (ω1, 1)-independent family. let i= {{iβα : β < λα} : α < τ} be a (θ, κ)-generalized independent family on some infinite set s, and let {〈xα, tα〉 : α < τ} be a family of topological spaces such that |xα| = λα for each α < τ . for each α < τ , index the α-th partition of i by {ixα : x ∈ xα}, and for each nonempty open subset u ∈ t α, define buα = ⋃ {ixα : x ∈ u}. set bα:= {b u α : ∅ 6= u ∈ tα}. the family bα is a sub-base for a topology on s. we denote it by sxα , and we use i{xα} to denote the topology generated by {sxα : α < τ}. when each 〈xα, tα〉 is discrete, the topology i{xα} is called “the simple topology” induced by i. generalized independent families and dense sets of box-product spaces 205 it is clear that 〈s, i{xα}〉 is a pθ-space whenever θ is regular. the space is hausdorff if i is separated and each 〈xα, tα〉 is hausdorff, and zero-dimensional if in addition each 〈xα, tα〉 is zero-dimensional. in the rest of this section, we only consider hausdorff spaces and separated families. theorem 2.2. let i and {〈xα, tα〉 : α < τ} be as above. any space 〈s, i{xα}〉 is homeomorphic to a dense subset of 2τθ 〈xα, t α〉 proof. for each α < τ , define fα : 〈s, i{xα}〉 → 〈xα, t α〉 such that fα(i x α) = x. by our definition of i{xα}, we know that fα is a continuous map. since i is separated, the family {fα : α < τ} separates points in 〈s, i{xα}〉. consider the map f = ∆α<τ fα : 〈s, i{xα}〉 → 2 τ θ 〈xα, t α〉 such that f (s) = {fα(s)}α<τ for all s ∈ s. certainly f is a one-one map, and f separates points. we need to show the following: (1) f is continuous; (2) the range of f is dense in 2τθ 〈xα, t α〉; (3) f separates points and closed sets. to see that f is continuous, it is enough to show that for any set a ∈ [τ ]<θ and any family {∅ 6= uα ∈ t α : α ∈ a} of nonempty open sets, the pre-image of the corresponding open set of 2α∈auα is open. by the definition of f , a point s ∈ s is in the pre-image of that open set if and only if fα(s) ∈ uα for all α ∈ a, and fα(s) ∈ uα if and only if there exists some x ∈ uα such that s ∈ ixα ⊆ b uα α . hence s ∈ ⋂ {buαα : α ∈ a} ∈ i {xα}. therefore the pre-image of 2α∈auα is ⋂ {buαα : α ∈ a}, which is open in i{xα}. for (2), we need to show that there exists a point s ∈ s such that f (s) is in the corresponding open set of 2α∈auα. using the same argument as above, it is enough to show that ⋂ {buαα : α ∈ a} is nonempty. since i is a θ-generalized independent family, it is clear that ⋂ {buαα : α ∈ a} 6= ∅. hence the range of f is dense in 2τθ 〈xα, t α〉. it remains to show that f separates points and closed sets. let s be a point and let f be a closed subset in 〈s, i{xα}〉 such that s /∈ f . since b is a base for i{xα}, for some set a ∈ [τ ] <θ and some family {∅ 6= uα ∈ t α : α ∈ a}, we have s ∈ ⋂ {buαα : α ∈ a} ⊆ f c. obviously f (s) is in the corresponding open set of 2α∈auα. we show that the corresponding open set of 2α∈auα is disjoint from f (f ). since the projection into any |a| < θ many products is open and continuous, it suffices to prove that 2α∈auα∩ ∆α∈afα(f ) = ∅. but this can be proved by the same argument used before: if fα(t) ∈ uα for some t ∈ f , then t ∈ ixα for some x ∈ uα and hence t ∈ b uα α , which implies that t ∈ ⋂ {buαα : α ∈ a} ⊆ f c, contradicting our early assumption. since {f} is continuous, separates points, and separates points and closed sets, it is a homeomorphism onto its range. it maps 〈s, i{xα}〉 onto a dense subset of 2τθ 〈xα, t α〉. � the following corollary is clear. corollary 2.3. let i= {{iβα : β < λα} : α < τ} be a (θ, κ)-generalized independent family on s, and let {〈xα, t α〉 : α < τ} be a family of topological spaces such that d(〈xα, t α〉) ≤ λα for all α < τ . then d(2 τ θ 〈xα, t α〉) ≤ |s|. 206 w. hu the converse of theorem 2.2 can be established for box product spaces of discrete spaces. theorem 2.4. for any dense subset d in 2τθ d(λα), there exists a (θ, 1)generalized independent family i on d. the set d is irresolvable if and only if i is a maximal (θ, 1)-generalized independent family. 3. the hewitt-marczewski-pondiczery theorem for box product spaces definition 3.1. let κ, θ, λ be two cardinals with κ, θ infinite. let s be an infinite set of size κ. the cardinal i(κ, θ, λ) is the smallest cardinal τ such that there are no (θ, 1, λ)-generalized independent families on s of size τ . the following generalizes the hewitt-marczewskipondiczery theorem. theorem 3.2. let s be a set and let θ, τ, λ be three cardinals with θ infinite. then the following are equivalent. • τ < i(|s|, θ, λ). • d(2τθ 〈xα, tα〉) ≤ |s| holds for any family of topological spaces {〈xα, tα〉 : α < τ} with each d(xα) ≤ λ. proof. (1)→(2). by corollary 2.3. (2)→ (1). let d be a dense subset of 2 τ θ d(λ) such that |d| = |s|. for each α < τ and β < λ, let i β α = d ∩{{xζ}ζ<τ ∈ 2τθ d(λ) : xα = β}. then the family i= {{i β α : β < λ} : α < τ} is a (θ, 1, λ)independent family on d. hence there is a (θ, 1, λ)-independent family of size τ on s, which implies τ < i(|s|, θ, λ). � comfort and negrepontis in [2] showed that |s|<θ = |s| is equivalent to the statement that there exists a subfamily of ss of size 2|s| that is of θ-large oscillation, which implies the existence of a (θ, 1, |s|)-independent family of size 2|s| on s. on the other hand, assuming there exists a (θ, 1, |s|)-independent family i of size 2|s| on s, for each β < 2|s| let fβ : s → s be such that fβ(i s β ) = s for each s ∈ s. then the family {fβ : β < 2 |s|} is a family of θ-large oscillation. hence, we have the following theorem. theorem 3.3. i(|s|, θ, |s|) = (2|s|)+ if and only if |s|<θ = |s|. we show in the following theorem that, in general, the cardinal i(|s|, θ, |s|) is regular. theorem 3.4. let θ, λ be two infinite cardinals such that θ ≤ λ. then i(λ, θ, λ) is regular. proof. let τ < i(λ, θ, λ) and let {τα : α < τ} be cardinals such that τα < i(λ, θ, λ). let also µ = sup{τα : α < τ}. by theorem 3.2, for each α < τ , the box product 2τα θ λ has density λ. by theorem 3.2 again, the space 2 µ θ λ = 2 τ θ (2 τα θ λ) has density λ. hence µ < i(λ, θ, λ) according to theorem 3.2. � it is clear that for any infinite set s and two cardinals λ1 ≤ λ2, we have (2|s|)+ ≥ i(|s|, θ, λ1) ≥ i(|s|, θ, λ2). when |s <θ| = |s|, we have i(|s|, θ, λ) = i(|s|, θ, |s|) = (2|s|)+ for any λ < |s|. generalized independent families and dense sets of box-product spaces 207 4. maximal generalized independent families the simple topology induced by a maximal (θ, 1)-generalized independent family is irresolvable. similarly, the simple topology induced by a maximal (θ, 1, λ)-independent family is λ-irresolvable. in this section, we show that for any uncountable regular cardinal θ, the existence of maximal (θ, 1)-generalized independent families with some simple conditions (equivalently, the existence of irresolvable dense subsets of θ-box product spaces with some simple conditions) implies the consistency of the existence of measurable cardinals. lemma 4.1. suppose 〈x, t 〉 is an open-hereditarily irresolvable space and t is a pθ-topology for some regular cardinal θ. let n denote the ideal of nowhere dense subsets, and let λ be the smallest cardinal such that n is not λ-complete. then for any γ < γ+ < λ and β < θ, n is (γβ )+-complete. proof. since the topology is open hereditarily irresolvable, n = {a ⊆ s : ao = ∅}. for a contradiction, let us assume that there exists yf ∈ n for each f ∈ γ β such that the yf are disjoint and ⋃ f yf ⊇ u for some nonempty open set u . we claim that there exists some member yg /∈ n . inductively define g : β → γ and a decreasing chain of non-empty basic open sets {u ζ : ζ < β} so that • u 0 = u , • u ζ = ⋂ {u η : η < ζ}, • u ζ+1 ⊆ u ζ and u ζ+1 ⊆ ⋃ {yf : f (ζ) = g(ζ)}. when ζ < θ is a limit, the set ⋂ {u η : η < ζ} defined in (2) is a nonempty open set, since t is a pθ-topology. for (3), we have γ-many disjoint sets {nα = ⋃ {yf : f (ζ) = α} : α ∈ γ}. the union of these sets contains u and hence u ζ . since the topology is open hereditarily irresolvable and n is γ+-complete, one of these sets {nα ∩ u ζ : α < γ}, say nα ∩ u ζ , has non-empty interior u ζ+1. set g(ζ + 1) = α. we have ⋂ {u ζ : ζ < β} ⊆ ⋂ ζ<β ⋃ {yf : f (ζ) = g(ζ)} = yg contradicting yg ∈ n . � in [2], comfort and negrepontis introduced the notion of strongly θ-inaccessible: a cardinal λ is called strongly θ-inaccessible if βγ < λ whenever β < λ and γ < θ. given a cardinal θ, denote by θin the smallest cardinal λ such that λ is strongly θ-inaccessible. lemma 4.2. let everything be as in lemma 4.1. then • λ is regular; • if λ is a successor cardinal, say λ = λ′+, then λ′ is strongly θ-inaccessible. • if λ is a limit cardinal, then λ is strongly θ-inaccessible proof. (1) is trivial. if λ = λ′+, then for any γ < λ′ and β < θ, (γβ)+ ≤ λ′ (lemma 4.1), and hence (γβ ) < λ′. this gives (2). (3) is trivial. � let everything be as in lemma 4.1. let us assume further that s(〈x, t 〉) ≤ λ with λ defined in lemma 4.1. then it is easy to see that the ideal n is 208 w. hu λ-saturated. under these assumption, there exists a λ-saturated (hence λ+saturated) λ-complete ideal over λ (using the proof of lemma 27.1 in [10]). lemma 35.10 and theorem 86 in [10] show that λ is a measurable cardinal in some model of zfc. in the following we show that for any uncountable regular cardinal θ, if there exists a maximal (θ, 1)-generalized independent family with some conditions, then the induced simple topology satisfies above conditions. theorem 4.3. let θ be a regular cardinal. suppose there exists a maximal (θ, 1)-generalized independent family i= {{iβα : β < λα} : α < τ} on a set s with each λα < θin. let n be the ideal of nowhere dense set of the simple topology induced by i and let λ be the smallest cardinal such that i is not λ-complete. then • there is a nonempty open set u of the simple topology such that u with the subspace topology satisfies all conditions in lemma 4.1 and the ideal iu of nowhere dense set of u is λ-saturated; and • 2<θ = θ proof. (i) let 〈s, t 〉 be the simple topology induced by i. since i is a maximal (θ, 1)-generalized independent family, it is irresolvable. using a standard argument ([9]), there is a nonempty basic open set u the subspace topology on which is hereditarily irresolvable. let nu be the set of all nowhere dense subsets in 〈u, t 〉, i.e, the set u with the subspace topology inherited from t . by lemma 4.2, if λ is a limit cardinal, then λ is strongly θ-inaccessible. if λ is a successor cardinal, say λ = λ′+, then λ′ is strongly θ-inaccessible. using theorem 2.3 in [2], we have that s(〈s, t 〉), and hence s(〈u, t 〉), is ≤ λ if λ is a limit cardinal, and < λ otherwise. hence nu is λ-saturated. (ii) the proof here uses a similar argument as that of lemma 1.4 in [11]. for each θ′ < θ we produce a map from θ onto 2θ ′ . index θ as a × b with a = {aη : η < θ} and b = {bζ : ζ < θ ′}. consider the family {i0α : α < θ} = {i 0 aη bζ : η < θ, ζ < θ′}. for each x ∈ x, define φx : θ → 2 θ ′ so that φx(η)(ζ) = 1 if and only if x ∈ i 0 aη bζ . for each f ∈ 2θ ′ , let rf be {x ∈ x : f /∈ range(φx)} = {x ∈ x : f 6= φx(η) for all η < θ}. we show that ⋂ f∈2θ ′ (x \ rf ) 6= ∅ by proving rf ∈ n and applying lemma 4.1, which shows that for some x, φx is onto. suppose that rf contains u = ⋂ {u σ(α) α : α ∈ a}, a non-empty basic open set, for some set a ∈ [τ ]<θ and some σ ∈ πα∈aλα. then there is an η < θ such that a ∩ ∪{(aη, bζ ) : ζ < θ ′} = ∅. now consider the open set u ′ = u ∩ {i0aη bζ : ζ < θ ′ and f (ζ) = 1} ∩{s \ i0aη bζ : ζ < θ ′ and f (ζ) = 0}. it is clear that u ′ 6= ∅ and u ′ ⊆ {s ∈ s : φs(η) = f} ∩ u ⊆ (s \ rf ) ∩ u , a contradiction. � generalized independent families and dense sets of box-product spaces 209 the following theorem is a direct corollary of theorem 4.3. corollary 4.4. for any uncountable regular cardinal θ, the existence of a maximal (θ, 1)-generalized independent family i= {{iβα : β < λα} : α < τ} on a set s with each λα < θin implies the consistency of the existence of measurable cardinals, and 2<θ = θ. the corresponding conclusion is about the existence of irresolvable dense subsets in a θ-box product space. theorem 4.5. let θ be an uncountable regular cardinal, and let {λα ≥ 2 : α < τ} be a family of cardinals with each λα < θin. if there exists an irresolvable dense subset s of the θ-box product space 2 τ θ d(λα), then • it is consistent that there exists a measurable cardinal; and • 2<θ = θ. references [1] w. w. comfort and w. hu, maximal independent families and a topological consequence, topology appl. 127 (2003), 343–354. [2] w. w. comfort and s.a. negrepontis, on families of large oscillation, fund. math. 75 (1972), 275–290. [3] w. w. comfort and s. a. negrepontis, the theory of ultrafilters, springer-verlag, 1974. [4] e. k. van douwen, applications of maximal topologies, topology appl. 51 (1993), 125– 139. [5] f. w. eckertson, resolvable, not maximally resolvable spaces, topology appl. 79 (1997), 1–11. [6] r. engelking, general topology, warszawa, 1977. [7] m. gotik and s. shelah, on densities of box products, topology appl. 88 (1998), 219– 237. [8] f. hausdorff, über zwei sätze von g. fichtenholz und l. kantorovitch, studia math. 6 (1936), 18–19. [9] e. hewitt, a problem of set-theoretic topology, duke math. j. 10 (1943), 309–333. [10] t. jech, set theory, second edition, springer-verlag, 1997. [11] k. kunen, maximal σ−independent families, fund. math. 117 (1983), 75–80. [12] k. kunen, a. szymanski and f. tall, baire resolvable spaces and ideal theory, prace nauk., ann. math. sil. 2(14) (1986), 98–107. [13] k. kunen and f. tall, on the consistency of the non-existence of baire irresolvable spaces, http://at.yorku.ca/v/a/a/a/27.htm, 1998. [14] s. shelah, baire irresolvable spaces and lifting for a layered ideal, topology appl. 33 (1989), 217–231. received may 2005 accepted july 2006 wanjun hu (wanjun.hu@asurams.edu) department of mathematics and computer science, albany state university, albany ga 31705. orospagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 2, 2005 pp. 185-194 the canonical partial metric and the uniform convexity on normed spaces s. oltra, s. romaguera and e. a. sánchez-pérez∗ abstract. in this paper we introduce the notion of canonical partial metric associated to a norm to study geometric properties of normed spaces. in particular, we characterize strict convexity and uniform convexity of normed spaces in terms of the canonical partial metric defined by its norm. we prove that these geometric properties can be considered, in this sense, as topological properties that appear when we compare the natural metric topology of the space with the non translation invariant topology induced by the canonical partial metric in the normed space. 2000 ams classification: 54e35, 46b04. keywords: partial metric, convexity, normed spaces. 1. introduction s. g. matthews introduced in [6] the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks, and obtained, among other results, a nice relationship between partial metric spaces and the so-called weightable quasi-metric spaces. partial metrics were also used in the context of the complexity analysis of algorithms and programs (see [6, 8, 7]). the domain of words which appears in a natural way by modelling the streams of information in g. kahn’s model of parallel computation provides a wellknown example of partial metric space (see [4] and [6]). other motivations for exploring partial metrics can be found in [6]. in this paper we present an application of the theory of partial metrics to the framework of the normed space theory. other applications in this direction can be found in [9]. let (x, ‖.‖) be a normed space. our aim is to show that it ∗the authors acknowledge the support of the generalidad valenciana, grant gv04b-371, the spanish ministry of science and technology, plan nacional i+d+i, grant bfm200302302, and feder, and the polytechnical university of valencia. 186 s. oltra, s. romaguera and e. a. sánchez-pérez is possible to define a partial metric p‖.‖ in x in a canonical way that implicitly contains the information about the convexity properties of the space. however, the topology that defines this partial metric is not standard at all, since it fails to satisfy some of the basic properties that topologies on topological linear spaces use to have. in particular, we obtain a topology τp with the following properties. (1) τp is not t1. (2) τp is not translation invariant. (3) τp coincides with the norm topology when restricted to the unit sphere of x. we present these results in three sections. section 2 is devoted to define and prove the basic properties of the canonical partial metric. in section 3 we characterize when (x, ‖.‖) is strictly convex in terms of the natural base of neighborhoods of (x, τp), providing also several examples. finally, in section 4 we present the results concerning uniform convexity of (x, ‖.‖) and the characterization of this property in terms of a particular class of neighborhoods of the elements of the unit sphere of (x, ‖.‖). in what follows we introduce the basic definitions and results on partial metrics. in this direction, our main references are [6, 7, 8]. each partial metric defines a quasi-metric that generates the same topology. therefore, we start by recalling several definitions on quasi-metrics. our basic references for quasimetric spaces are [3] and [4], and for general topological questions [2]. let us recall that a quasi-pseudo-metric on a set x is a nonnegative real valued function d on x × x such that for all x, y, z ∈ x, (i) d(x, x) = 0; (ii) d(x, y) ≤ d(x, z) + d(z, y). by a quasi-metric on a set x we mean a quasi-pseudo-metric d on x that satisfies also the following condition. (iii) d(x, y) = d(y, x) = 0 ⇔ x = y. a quasi-metric space is a pair (x, d) such that x is a (nonempty) set and d is a quasi-metric on x. each quasi-metric d on x generates a t0-topology t (d) on x which has as a base the family of open d-balls {bd(x, ε) : x ∈ x, ε > 0}, where bd(x, ε) = {y ∈ x : d(x, y) < ε} for all x ∈ x and ε > 0. we will write r+ for the set of nonnegative real numbers. a partial pseudometric on a (nonempty) set x is a function p : x × x → r+ such that for all x, y, z ∈ x, (i) x = y ⇒ p(x, x) = p(x, y) = p(y, y); (ii) p(x, x) ≤ p(x, y); (iii) p(x, y) = p(y, x); (iv) p(x, z) ≤ p(x, y) + p(y, z) − p(y, y). by a partial metric on a set x we mean a partial pseudo-metric p on x that satisfies the following condition, (i)′ x = y ⇔ p(x, x) = p(x, y) = p(y, y). the canonical partial metric and the uniform convexity on normed spaces 187 a partial (pseudo-)metric space is a pair (x, p) such that x is a (nonempty) set and p is a partial (pseudo-)metric on x. each partial metric p on x defines a quasi-metric dp on x by means of the formula, dp(x, y) := p(x, y) − p(x, x), x, y ∈ x, and the topology given by p is the one generated by dp. consequently, each partial metric generates a t0-topology t (p) on x and is the one given by the base b = {vε,p(x) : x ∈ x, ε > 0}, where vε,p(x) = {y ∈ x : p(x, y) < ε + p(x, x)} for all x ∈ x and ε > 0. the following lemma is a direct consequence of lemma 2.2 in [6] and gives information about the sets vε,p(x) that will be useful in this paper. lemma 1.1. let (x, p) be a partial metric space and x ∈ x. then b = {vε,p(x) : ε > 0} is a base of open neighborhoods of x for the topology t (p). let us finish this section by introducing several definitions of functional analysis. the basic references for these definitions are [10, 11, 5]. we use standard banach space notation. if (x, ‖.‖) is a normed space, we denote by sx the corresponding unit sphere {x ∈ x : ‖x‖ = 1}. if x ∈ x and ε > 0, we denote by bε,‖.‖(x) the set bε,‖.‖(x) := {y ∈ x : ‖x − y‖ < ε}. if x, y ∈ x, we denote by [x, y] the segment {z = θx + (1 − θ)y : 0 ≤ θ ≤ 1}. if x ∈ x, 〈x〉 denotes the linear span of x into the linear space x. although we present the results of the paper for normed spaces, completeness is not necessary. therefore, our main results are valid for normed spaces. the same holds for the following definitions. definition 1.2. let (x, ‖.‖) be a normed space. we say that the norm ‖.‖ (equivalently, the normed space (x, ‖.‖)) is strictly convex if for every pair of norm one elements x, y ∈ x, ‖x+y 2 ‖ = 1 implies x = y. it can be proved that this definition is equivalent to the following one. the norm ‖.‖ (equivalently, the normed space (x, ‖.‖)) is strictly convex if for every x, y ∈ x, if x 6= 0 and ‖x + y‖ = ‖x‖ + ‖y‖, then y ∈ 〈x〉. the proof of this equivalence can be found in proposition 1, part 3, ch.i of [1]. throughout the paper we will use both definitions. definition 1.3. let (x, ‖.‖) be a normed space. the norm ‖.‖ (equivalently, the normed space (x, ‖.‖)) is uniformly convex if for every ε > 0 there is a δ > 0 -only depending on εsuch that if ‖x‖ = 1 = ‖y‖ and ‖x − y‖ ≥ ε, then ‖x+y 2 ‖ ≤ 1 − δ, for every x, y ∈ x. we will use the following notation. let n ∈ n and consider the corresponding n-dimensional linear space rn. let x = (x1, ..., xn) ∈ r n. we will write ‖x‖2 for ‖x‖2 := ( n ∑ i=1 |xi| 2) 1 2 , 188 s. oltra, s. romaguera and e. a. sánchez-pérez and ‖x‖1 for ‖x‖1 := n ∑ i=1 |xi|. 2. partial metrics in normed spaces in this section we construct the canonical partial metric p‖.‖ associated to the norm of a normed space (x, ‖.‖), and we prove the main results that will be used in the following two sections. the relations between the elements of the base of neighborhoods given by lemma 1.1 of the topological space space (x, τp‖.‖) and the translation invariant topology associated to the norm ‖.‖ gives the characterization of the convexity properties of the normed space (x, ‖.‖). definition 2.1. let (x, ‖.‖) be a normed space. we define the nonnegative function p‖.‖ : x → r by the formula p‖.‖(x, y) := ‖x − y‖ + ‖x‖ + ‖y‖, x, y ∈ x. a related construction has been done at [6] definition 2.2. let τ be a topology on a linear space x. we say that a norm ‖.‖ is 0-compatible with the topology τ if the balls bε,‖.‖(0) = {x ∈ x : ‖x‖ < ε}, ε > 0, define a base of neighborhoods of 0 for the topology τ. proposition 2.3. if (x, ‖.‖) is a normed space, the function p‖.‖ is a partial metric that satisfies 1) p‖.‖(x + y, 0) ≤ p‖.‖(x, 0) + p‖.‖(y, 0), 2) p‖.‖(λx, λy) = |λ|p‖.‖(x, y) for every x, y ∈ x and λ ∈ r, 3) p‖.‖(x, y) = 0 if and only if x = y = 0, 4) the norm ‖.‖ is 0-compatible with the topology τp‖.‖. proof. the following calculations show that p‖.‖ is a partial metric. to prove the condition (i)′ in the definition of partial metric (section 1), let x, y ∈ x such that p‖.‖(x, x) = p‖.‖(x, y) = p‖.‖(y, y), hence 2‖x‖ = ‖x − y‖ + ‖x‖ + ‖y‖ = 2‖y‖ so that ‖x − y‖ + ‖x‖ − ‖y‖ = 0 and ‖x − y‖ − ‖x‖ + ‖y‖ = 0. this clearly implies ‖x − y‖ = 0, and then x = y. now suppose that x = y; the equalities above gives directly p‖.‖(x, x) = p‖.‖(x, y) = p‖.‖(y, y), since ‖x − y‖ = 0. now let us show that p‖.‖(x, x) ≤ p‖.‖(x, y) for every x, y ∈ x. but this is a direct consequence of the triangular inequality for the norm ‖.‖, since p‖.‖(x, x) = 2‖x‖ ≤ ‖x − y‖ + ‖x‖ + ‖y‖ = p‖.‖(x, y). the definition of p‖.‖ clearly gives p‖.‖(x, y) = p‖.‖(y, x) for every x, y ∈ x. to see the last condition of partial metric, consider x, y, z ∈ x. then p‖.‖(x, y) + p‖.‖(z, z) = ‖x − y‖ + ‖x‖ + ‖y‖ + 2‖z‖ the canonical partial metric and the uniform convexity on normed spaces 189 ≤ ‖x − z‖ + ‖x‖ + ‖z‖ + ‖z − y‖ + ‖y‖ + ‖z‖ = p‖.‖(x, z) + p‖.‖(y, z). now let us show 1). if x, y ∈ x, then p‖.‖(x + y, 0) = ‖x + y‖ + ‖x + y‖ ≤ 2‖x‖ + 2‖y‖ = p‖.‖(x, 0) + p‖.‖(y, 0). condition 2) is a consequence of the homogeneity of the norm. p‖.‖(λx, λy) = ‖λx−λy‖+‖λx‖+‖λy‖ = |λ|(‖x−y‖+‖x‖+‖y‖) = |λ|p‖.‖(x, y) for every x, y ∈ x and λ ∈ r. 3) is also given directly by the definition. if x ∈ x, obviously p‖.‖(x, y) = ‖x − y‖ + ‖x‖ + ‖y‖ = 0 if and only if x = y = 0. finally, let us show that ‖.‖ is 0-compatible with τp‖.‖. it is enough to write explicitly the basic neighborhoods vε,p‖.‖(x) for the case x = 0. vε,p‖.‖(0) = {x ∈ x : p‖.‖(x, 0) < ε+p(0, 0)} = {x ∈ x : 2‖x‖ < ε} = b ε2 ,‖.‖(0) � we will call the function p‖.‖ the canonical partial metric associated to ‖.‖. consider ε > 0 and x ∈ x. the basic neighborhood of x, vε,‖.‖(x) is given in this case by the particular expression vε,p‖.‖(x) := {y ∈ x : p‖.‖(x, y) < p‖.‖(x, x)+ε} = {y ∈ x : ‖x−y‖+‖y‖−‖x‖ < ε}. this description of the neighborhood vε,p‖.‖(x) will be useful in the following sections. 3. strict convexity and the canonical partial metric let (x, ‖.‖) be a normed space. in this section we characterize when the norm ‖.‖ is strictly convex in terms of the base of neighborhoods for the topology τp‖.‖ given by lemma 1.1 and described at the end of section 2 for the particular case of the canonical partial metric. lemma 3.1. let (x, ‖.‖) be a normed linear space. for every x ∈ x, [0, x] ⊂ ∩ε>0vε,p‖.‖ (x). proof. let y ∈ [0, x] and let ε > 0. then there exists an α such that 0 ≤ α ≤ 1 and y = αx. thus, p(x, y) = ‖x−y‖+‖y‖+‖x‖ = (1−α)‖x‖+α‖x‖+‖x‖ = 2‖x‖ = p(x, x) < p(x, x)+ε. this proves the lemma, since implies that y ∈ vε,p‖.‖(x) for every ε > 0. � note that the only point that satisfies that the intersection of all its neighborhoods is the same point is 0. as a direct consequence, we obtain that the topology generated by the canonical partial metric only satisfy the separation axiom t0. remark 3.2. the topology defined by the canonical partial metric in a non trivial normed space is not t1. to prove this, consider x ∈ x − {0} and define y = 1 2 x. suppose that there is a neighborhood of x, v such that y do not belong to v . since v is a neighborhood of x, by lemma 1.1 there exists ε > 0 such that vε,p‖.‖(x) ⊂ v . but y ∈ vε,p‖.‖ as a consequence of lemma 3.1, and then y ∈ v , a contradiction. 190 s. oltra, s. romaguera and e. a. sánchez-pérez let us discuss in what follows the situation for the converse inclusion that the one given in lemma 3.1. the first one shows that the 2-dimensional euclidean space satisfies also this inclusion. example 3.3. consider the 2-dimensional euclidean space r22 := (r 2, ‖.‖2), and let x0 ∈ r 2 2. then ⋂ ε>0 vε,p‖.‖(x0) = ⋂ ε>0 {y ∈ r2 : ‖x0 − y‖2 + ‖y‖2 < ε + ‖x0‖2} = = {y : ‖x0 − y‖2 + ‖y‖2 = ‖x0‖2}, since the inequality ‖x‖ ≤ ‖x − y‖ + ‖y‖ always holds for every x, y ∈ x and every norm. in terms of the euclidean distance d2 in r 2 2, the above condition can be written as d2(y, x0) + d2(0, y) = d2(0, x0), which only holds when y = αx0 for some α ∈ [0, 1]. therefore, in this case ⋂ ε>0 vε,p‖.‖(x0) = [0, x0]. example 3.4. consider the 2-dimensional space r21 := (r 2, ‖.‖1), and the element x0 := (1/2, 1/2). then vε,p‖.‖(x0) = {(y1, y2) : ‖(1/2−y1, 1/2−y2)‖1+‖(y1, y2)‖1 < ε+‖(1/2, 1/2)‖1}, and then ⋂ ε>0 vε,p‖.‖ (x0) = {(y1, y2) : ‖(1/2 − y1, 1/2 − y2)‖1 + ‖(y1, y2)‖1 = ‖(1/2, 1/2)‖1}. consider now any element (a, b) ∈ [0, 1/2] × [0, 1/2]. then the condition that appears in ⋂ ε>0 vε,p‖.‖(x0) can be written as | 1 2 − a| + | 1 2 − b| + |a| + |b| = 1 2 − a + 1 2 − b + a + b = 1 that obviously holds for every (a, b) ∈ [0, 1/2]×[0, 1/2]. then [0, 1/2]×[0, 1/2] ⊂ ⋂ ε>0 vε,p‖.‖(x0), which implies that ⋂ ε>0 vε,p‖.‖(x0) is not contained in [0, x0]. lemma 3.5. a normed space (x, ‖.‖) is strictly convex if and only if for every x, y ∈ sx, if ‖2x − y‖ = 1 then x = y. proof. suppose that (x, ‖.‖) is strictly convex, and consider two elements x, y ∈ sx such that ‖2x − y‖ = 1. then ‖2x − y‖ + ‖x + y‖ ≥ ‖3x‖ = 3, and so ‖x + y‖ ≥ 2. moreover, 2‖x + y‖ = ‖2x + 2y‖ ≤ ‖2x − y‖ + ‖3y‖ = 4, and then ‖x + y‖ ≤ 2. since (x, ‖.‖) is strictly convex, we obtain that x = y. conversely, suppose that the second property in the statement of the lemma holds, and consider two elements x, y ∈ sx satisfying ‖x + y‖ = 2. then we define z1 = x+y 2 and z2 = y, that obviously satisfy ‖z1‖ = ‖z2‖ = 1 and ‖2z1 − z2‖ = 1. thus the property gives z1 = z2, and then x+y 2 = y, which clearly implies x = y. � the canonical partial metric and the uniform convexity on normed spaces 191 theorem 3.6. a normed linear space (x, ‖.‖) is strictly convex if and only if ∩ε>0vε,p‖.‖ (x) = [0, x]. proof. first let us prove that ∩ε>0vε,p‖.‖(x) = [0, x] implies that the normed linear space is strictly convex. suppose that ‖x‖ = 1. then ∩ε>0vε,p‖.‖(2x) ∩ sx = [0, 2x] ∩ sx = {x}. since ∩ε>0vε,p‖.‖ (2x) ∩ sx = {y : ‖y‖ = 1, ‖2x − y‖ = 1}, we obtain the result as a consequence of lemma 3.5. conversely, we use the characterization given by proposition 1 (p. 175 of [1]) of the strict convexity of (x, ‖.‖) that we have referred in section 1. consider x ∈ sx and suppose that y ∈ ∩ε>0vε,p‖.‖(x) but y is not in [0, x]. then ‖x − y‖ + ‖y‖ = ‖x‖ = 1 and in particular ‖y‖ ≤ 1 and ‖x − y‖ ≤ 1. these two inequalities and the fact that y is not an element of [0, x] clearly imply that y is not a linear combination of x. if we write z1 = x − y and z2 = y, we have ‖z1‖ + ‖z2‖ = 1 = ‖z1 + z2‖, and then the proposition quoted above gives that there is a λ 6= 0 such that z1 = λz2, and then x − y = λy. thus x = (1 + λ)y, a contradiction. � 4. uniform convexity and the canonical partial metric strict convexity of the norm can be understood, in a certain sense, as a limit case of the uniform convexity. after the result obtained in the theorem above, we will prove in this section that it is also possible to give a characterization of the uniform convexity of a normed space (x, ‖.‖) in terms of a particular class of neighborhoods of the points of x for the topology τp‖.‖. let us fix a normed space (x, ‖.‖). let x ∈ sx and δ > 0. we define the set wδ,‖.‖(x) = {y ∈ x : ‖x‖ + ‖y‖ ≤ ‖x + y‖ + δ}. lemma 4.1. let (x, ‖.‖) be a linear normed space. for every x ∈ x and ε > 0, vε,p‖.‖(x) ⊂ wε,‖.‖(x). in particular, wε,‖.‖(x) is a neighborhood of x for the topology τp‖.‖. proof. fix x ∈ x and consider an element y ∈ vε,p‖.‖(x). then 2‖x‖ + ‖y‖ ≤ ‖x + y‖ + ‖x − y‖ + ‖y‖ ≤ ‖x + y‖ + ‖x‖ + ε, thus ‖x‖ + ‖y‖ < ‖x + y‖ + ε, and so y ∈ wε,‖.‖(x). � lemma 4.2. for every x ∈ sx and every ε > 0, vε,p‖.‖(x) ∩ sx = bε,‖.‖(x) ∩ sx. proof. if ‖x‖ = 1 and y ∈ vε,p‖.‖(x) ∩ sx, ‖x − y‖ + ‖y‖ = ‖x − y‖ + 1 ≤ ‖x‖ + ε = 1 + ε. then ‖x − y‖ < ε, and thus y ∈ bε,‖.‖(x). the same argument shows the opposite inclusion. � 192 s. oltra, s. romaguera and e. a. sánchez-pérez definition 4.3. let (x, τ) be a topological space. for every element x ∈ x, consider two sets of neighborhoods of x both of them indexed by ε ∈ r+; vx = {vε(x) : ε > 0} and wx = {wε(x) : ε > 0}. consider now the families of neighborhoods v = {vx : x ∈ x} and w = {wx : x ∈ x}. we say that v and w are uniformly equivalent if they verify the following relations: (i) for every ε > 0 there is a δ > 0 (only depending on ε) such that wδ(x) ⊂ vε(x) for all x ∈ x, and (ii) for every ε′ > 0 there is a δ′ > 0 (only depending on ε′) such that vε′ (x) ⊂ wδ′ (x) for all x ∈ x. for the following theorem, we consider the particular families of neighborhoods v = {vx : x ∈ sx}, w = {wx : x ∈ sx} and b = {bx : x ∈ sx}, given by vx = {vε,p‖.‖(x) ∩ sx : ε > 0}, wx = {wε,‖.‖(x) ∩ sx : ε > 0} and bx = {bε,‖.‖(x) ∩ sx}. theorem 4.4. let (x, ‖.‖) be a normed space. the following are equivalent. (1) (x, ‖.‖) is uniformly convex. (2) for every ε > 0 there is a δ > 0 (only depending on ε) such that wδ,‖.‖(x) ∩ sx ⊂ vε,p‖.‖(x), for all x ∈ sx. (3) w and v are uniformly equivalent families of neighborhoods. (4) w and b are uniformly equivalent families of neighborhoods. moreover, if any of the statements (1) to (4) holds, then the family wx defines a local base in the topological space (sx, τp‖.‖|sx ) (equivalently, in the topological space (sx, τ‖.‖|sx )) for every x ∈ sx. proof. let us prove first that (1) implies (2). suppose now that (x, ‖.‖) is uniformly convex. it follows that for all x, y ∈ sx and ε there is a δ > 0 such that if ‖x + y‖ ≥ 2 − 2δ it is true that ‖x − y‖ < ε. let us show that wδ,‖.‖(x) ∩ sx ⊂ vε,p‖.‖(x). to check this, consider y ∈ wδ,‖.‖(x) ∩ sx. note that wδ,‖.‖(x)∩sx = {y ∈ sx : ‖x‖+‖y‖ ≤ ‖x+y‖+δ} = {y ∈ sx : 2−δ ≤ ‖x+y‖}. hence 2 − 2δ ≤ 2 − δ ≤ ‖x + y‖, and since x is uniformly convex this implies that ‖x − y‖ < ε. consequently, it follows that ‖x − y‖ + ‖y‖ ≤ ‖x‖ + ε. this completes the proof of (1) implies (2). the canonical partial metric and the uniform convexity on normed spaces 193 to deduce (1) from (2), suppose that for every ε > 0 there is a δ > 0 such that wδ,‖.‖(x) ∩ sx ⊂ vε,p(x) for every x ∈ sx. consider ε > 0 and δ′ = δ 2 . if y ∈ wδ,‖.‖(x) ∩ sx = {y ∈ sx : 2 − δ ≤ ‖x + y‖}, we have that y ∈ vεp‖.‖ (x), and then ‖x − y‖ < ε. this condition is equivalent to the following one. for every x, y ∈ sx, if ‖x − y‖ ≥ ε, then 2 − 2δ′ ≥ ‖x + y‖. hence, for such x, y ∈ sx, we have ‖x + y‖ < 2 − δ = 2 − 2δ′, which can be written as ‖x+y‖ 2 < 1−δ′. this shows that x is uniformly convex. let us prove now the equivalence between (2) and (3). as a consequence of lemma 4.1, we only need to prove that for each ε > 0 there is a δ > 0 (only depending on ε) such that wδ,‖.‖(x) ∩ sx ⊂ vε,p‖.‖(x) for all x ∈ sx. but this is what the statement (2) of the theorem assures, so v and w are uniformly equivalent. the converse is obvious. it follows easily that (3) if and only if (4), as a consequence of lemma 4.2. finally, we have to check that {wδ,‖.‖(x) ∩ sx : δ > 0} defines for every x ∈ sx a local base for the topological space (sx, τp‖.‖|sx ). it is clear, using lemma 4.2, that this is equivalent to the fact that {wδ,‖.‖(x) ∩ sx : δ > 0} a local base for the topological space (sx, ‖.‖|sx ) for each x ∈ sx. obviously {wδ,‖.‖(x) ∩ sx : δ > 0} 6= φ since x ∈ {wδ,‖.‖(x) ∩ sx : δ > 0}. it is also clear that given wδ1,‖.‖(x) ∩ sx and wδ2,‖.‖(x) ∩ sx we obtain x ∈ wδ3,‖.‖(x) ∩ sx ⊂ ( wδ1,‖.‖(x) ∩ sx ) ∩ ( wδ2,‖.‖(x) ∩ sx ) , where δ3 := min{δ1, δ2}. consider wδ,‖.‖(x)∩sx. since as v and w are uniformly equivalent families of neighborhoods, there is ε > 0 such that vε,p‖.‖(x)∩sx ⊂ wδ,‖.‖(x)∩sx. since vε,p‖.‖(x)∩sx is a local base, then there is a vε1,p‖.‖(x)∩sx ⊂ vε,p‖.‖(x)∩sx such that for all y ∈ vε1,p‖.‖(x) ∩ sx there is an ε2 > 0 such that vε2,p‖.‖(y) ∩ sx ⊂ vε,p‖.‖ (x) ∩ sx. given vε1(x) we can find a δ1 > 0 such that it verifies that wδ1,‖.‖(x) ∩ sx ⊂ vε1,p‖.‖(x) ∩ sx ⊂ wδ,‖.‖(x) ∩ sx. now we have to show that for all y ∈ wδ1,‖.‖(x)∩sx there is a δ2 > 0 such that wδ2,‖.‖(y)∩sx ⊂ wδ,‖.‖(x)∩sx . we know that for all y ∈ wδ1,‖.‖(x) ∩ sx ⊂ vε1,p‖.‖(x) ∩ sx there is vε2,p‖.‖(y) ⊂ vε,p‖.‖ (x). then for each y ∈ wδ1,‖.‖(x) there exists vε2,p‖.‖(y) ⊂ vε(x), and given vε2,p‖.‖(y) there is δ2 such that wδ2,‖.‖(y) ⊂ vε2,p‖.‖(y) ⊂ wδ,‖.‖(x). therefore, we have proved that the family {wδ,‖.‖(x) ∩ sx : δ > 0} defines for every x ∈ sx a local base for the topological space (sx, τp‖.‖|sx ). this finishes the proof. � the ideas exposed in this paper allow to consider the convexity properties of normed spaces as sequential properties, since they can be characterized using the topology defined by the canonical partial metric. this provides a new framework for the study of certain geometric properties of normed spaces and in the particular case of the banach spaces. 194 s. oltra, s. romaguera and e. a. sánchez-pérez references [1] b. beauzamy, introduction to banach spaces and their geometry, north holland math. studies, amsterdam (1985). 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[9] s. oltra and e. a. sánchez-pérez, order properties and p-metrics on köthe function spaces, houston j. math., to appear. [10] w. rudin, functional analysis, mcgraw-hill, new york (1973). [11] p. wojtaszczyk, banach spaces for analysts, cambridge university press, cambridge (1991). received october 2004 accepted february 2005 s. oltra (soltra@mat.upv.es) escuela politécnica superior de alcoy, departamento de matemática aplicada, universidad politécnica de valencia, 03801 alcoy (alicante), spain. s. romaguera (sromague@mat.upv.es) escuela de caminos, departamento de matemática aplicada, universidad politécnica de valencia, 46071 valencia, spain. e. a. sánchez-pérez (easancpe@mat.upv.es) escuela de caminos, departamento de matemática aplicada, universidad politécnica de valencia, 46071 valencia, spain. () @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 143-162 core compactness and diagonality in spaces of open sets francis jordan and frédéric mynard abstract we investigate when the space ox of open subsets of a topological space x endowed with the scott topology is core compact. such conditions turn out to be related to infraconsonance of x, which in turn is characterized in terms of coincidence of the scott topology of ox ×ox with the product of the scott topologies of ox at (x, x). on the other hand, we characterize diagonality of ox endowed with the scott convergence and show that this space can be diagonal without being pretopological. new examples are provided to clarify the relationship between pretopologicity, topologicity and diagonality of this important convergence space. 2010 msc: 54a20, 54a10, 54b20, 54d45. keywords: scott convergence, scott topology, upper kuratowski convergence, upper kuratowski topology, core compact, diagonal convergence, pretopology, consonance, infraconsonance. 1. introduction definitions and notations concerning convergence structures follow [3] and are gathered as an appendix at the end of these notes (1). in particular, if x and y are two convergence spaces, the continuous convergence [x, y ] on the set c(x, y ) of continuous maps from x to y is the coarsest convergence making the evaluation jointly continuous. this is the canonical function space structure in the cartesian closed category of convergence spaces and continuous maps. this paper is concerned with certain properties of this canonical convergence on functions valued into the sierpiński space: 1terms and notations that are not defined in the text can be found in the appendix. 144 f. jordan and f. mynard let $0 and $1 denote two versions of the sierpiński space on {0, 1}: {0} is the only non-trivial open subset of $0, and {1} is the only non-trivial open subset of $1. let 1a : x → {0, 1} denote the indicator function of a subset a of a convergence space x defined by 1a(x) = 1 if and only if x ∈ a. with those conventions, a is an open subset of x if and only if 1a : x → $1 is continuous and closed if and only if 1a : x → $0 is continuous. therefore, c(x, $1) can be identified with the set ox of open subsets of x, and c(x, $0) can be identified with the set cx of closed subsets of x. if x is a topological space, the continuous convergences [x, $1] and [x, $0] turn out to be familiar convergences, on ox and cx respectively (see, e.g., [5]): u ∈ lim[x,$1] f ⇐⇒ u ⊆ ⋃ f ∈f int( ⋂ o∈f o)(1.1) c ∈ lim[x,$0] f ⇐⇒ ⋂ f ∈f cl( ⋃ a∈f a) ⊆ c.(1.2) both are instances of scott convergence (in the sense of, for instance, [11]), i.e., (1.3) x ∈ lim f ⇐⇒ ∨ f ∈f ∧ f ≥ x, in the complete lattices (ox, ⊆) and (cx, ⊇) respectively. however, (1.2) is usually called upper kuratowski convergence. the topological modification t [x, $0] is called upper kuratowski topology. the topological modification t [x, $1] is the scott topology, whose open sets are exactly compact families: families a of open subsets of x that are closed under open supersets and satisfy ⋃ i∈i oi ∈ a =⇒ ∃f ∈ [i] <∞ : ⋃ i∈f oi ∈ a for any collection {oi : i ∈ i} of open subsets of x, where [i] <∞ denotes the set of finite subsets of i. for instance, if k is a compact subset of x then o(k) := {o ∈ ox : k ⊆ o} is a compact family. the family of all sets o(k) where k ranges over compact subsets of x is a basis for a topology on ox = c(x, $1). we write ck(x, $1) for the corresponding topological space. of course, complementation c : ox → cx defined by c(u) = x \ u is an homeomorphism between [x, $1] and [x, $0] and we only need study one of these two convergences. we choose to focus on ox. hence, from now on, $ means $1 and we only formulate results on ox but they have counterparts for the upper kuratowski convergence and upper kuratowski topology on cx, which the interested reader can easily write out (see e.g., [8] for a study of [x, $0] and t [x, $0] on cx). it is clear from the definitions that [x, $] ≥ t [x, $] ≥ ck(x, $). core compactness and diagonality in spaces of open sets 145 a convergence space x is called t -dual if [x, $] = t [x, $] and consonant [5] if t [x, $] = ck(x, $). it is easily seen that ∩ : [x, $] × [x, $] → [x, $] (u, v ) 7→ u ∩ v is continuous for any convergence space x. to simplify the discussion, let us momentarily assume that x is a completely regular topological space. in this case, x is t -dual if and only if x is locally compact. moreover, ck(x, $) = [kx, $], where kx is the locally compact modification of x (2) [21, proposition 4.3]. therefore intersection is also jointly continuous for ck(x, $). additionally, infraconsonance in the sense of [9] can then be characterized in similar terms: x is infraconsonant if ∩ : t [x, $] × t [x, $] → t [x, $] is continuous. thus t -dual =⇒ consonant =⇒infraconsonant. the problem of characterizing t -dual topological spaces has long been settled (e.g., [13], [23]): a topological space x is t -dual if and only if it is core compact. recall that a topological space x is core compact if for every x and o ∈ o(x), there is u ∈ o(x) such that every open cover of o has a finite subfamily that covers u. in the case of a general convergence space x, the situation is more complicated. it is known (e.g., [24], [7]) that the following are equivalent: ∀y, t (x × y ) ≤ x × t y ;(1.4) ∀y = t y, [x, y ] = t [x, y ] t (x × [x, $]) ≤ x × t [x, $]; x is t -dual. moreover, it was shown in [7] that (1.5) x is core compact =⇒ x is t -dual =⇒ x is t -core compact, where a convergence space is called core compact if whenever x ∈ lim f, there is g ≤ f with x ∈ lim g and for every g ∈ g there is g′ ∈ g such that g′ is compact at g; and a convergence space is called t -core compact if whenever x ∈ lim f and u ∈ ox(x), there is f ∈ f that is compact at u. the three notions clearly coincide if x is topological. however, so far, it was not known whether they do in general. at the end of the paper, we provide an example (example 5.8) of a t -dual convergence that is not core-compact. section 2 examines when [x, $] and t [x, $] are t -dual. the latter question, while natural in itself, is motivated by its connection (established in section 3) with the (now recently solved [18]) problem [9, problem 1.2] of finding a 2with the abuse of notation that [kx, $] is identified with the convergence it induces on the subset c(x, $) of c(kx, $). 146 f. jordan and f. mynard completely regular infraconsonant topological space that is not consonant. we obtain that x is infraconsonant whenever t [x, $] is t -dual, and we prove more generally that x is infraconsonant if and only if the scott topology on ox ×ox for the product order coincides with the product of the scott topologies at the point (x, x) (theorem 4.2). infraconsonance was introduced while studying the isbell topology on the set of real-valued continuous functions over a topological space. in fact a completely regular space x is infraconsonant if and only if the isbell topology on the set of real-valued continuous functions on x is a group topology [6, corollary 4.6]. on the other hand, the fact that the scott topology on the product does not coincide in general with the product of the scott topologies has been at the origin of a number of errors, as pointed out for instance in [11, p.197]. therefore, theorem 4.2 provides new motivations to investigate infraconsonance. in [7], it is shown that a convergence space x is t -core compact if and only if [x, $] is pretopological. therefore, if x is topological, [x, $] is topological whenever it is pretopological. as topologies are exactly the diagonal (3) pretopologies, it raises the question of whether [x, $] is diagonal whenever x is topological. in section 5, diagonality of [x, $] is characterized in terms of a variant of core-compactness that do not need to coincide with core-compactness. as a result [x, $] does not need to be diagonal even if x is topological. 2. core-compactness of ox for a general convergence space x, the underlying set of [x, $] can still be identified with the collection ox of open subsets of x (or t x ), but the characterization (1.1) of convergence in [x, $] needs to be modified. a family s of subsets of a convergence space y is a cover of a ⊆ y if every filter on y converging to a point of a contains an element of the family s. then we have: u ∈ lim[x,$] f ⇐⇒ { ⋂ o∈f o : f ∈ f} is a cover of u. the space [[x, $], $] has as underlying set the set of scott-open subsets of ox, that is, if x is topological, the set κ(x) of openly isotone compact families on x. note that the family {u∼ := {a ∈ κ(x) : u ∈ a} : u ∈ ox} forms a subbase for a topology on κ(x), called stone topology. it is the analog on κ(x) of the standard topology on the set βx of ultrafilters on x. as observed in [10, proposition 5.2], when x is topological, the convergence [x, $] is based in filters of the form (2.1) o♮(p) := {o(p) : p ∈ p}, where p is an ideal subbase of open subsets of x, that is, such that there is p ∈ p with ⋃ q∈p0 q ⊆ p whenever p0 is a finite subfamily of p. more 3in the sense of e.g., [4]. see definition 5.1. core compactness and diagonality in spaces of open sets 147 precisely, for every filter f on [x, $] with u ∈ lim[x,$] f there is an open cover p of u that forms an ideal subbase, such that u ∈ lim[x,$] o ♮(p) and o♮(p) ≤ f. note also that (2.2) a ⊆ b =⇒ a ∈ lim[[x,$],$]{b} ↑, for every a and b in κ(x). in particular if o is [[x, $], $]-open, a ∈ o and a ⊆ b ∈ κ(x) then b ∈ o. it was observed in [12], as a consequence of a general theory, that if x is topological, then so is [[x, $], $]. we provide here an independent proof, that shows that [x, $] is then a core compact convergence. proposition 2.1. if x is topological, then [x, $] is core compact, so that [[x, $], $] is topological. more precisely, it is homeomorphic to κ(x) with the stone topology. proof. let u ∈ lim[x,$] o ♮(p) for an ideal subbase p of open subsets of x. then for each p ∈ p, the set o(p) is a compact subset of [x, $] because p ∈ lim[x,$] o(p). indeed, p = int (⋂ o∈o(p ) o ) . u∼ is [[x, $], $]-open for each u ∈ ox. indeed, if a ∈ u ∼ ∩ lim[[x,$],$] f then {⋂ b∈f b : f ∈ f } is a cover of a (in the sense of convergence) so that there is f ∈ f with ⋂ b∈f b ∈ {u}↑ because u ∈ lim[x,$]{u} ↑ ∩ a. in other words, f ⊆ u∼, so that u∼ ∈ f. conversely, if o is [[x, $], $]-open and a ∈ o, there is u ∈ a such that u∼ ⊆ o. otherwise, for each u ∈ a, there is b ∈ κ(x) with u ∈ b and b /∈ o. in that case, û := {b ∈ κ(x) : u ∈ b, b /∈ o} 6= ∅ for all u ∈ a. note also that in view of (2.2), bu ∩ bv ∈ û ∩ v̂ whenever bu ∈ û and bv ∈ v̂ . therefore {⋂ i∈i ûi : ui ∈ a : card i < ∞ } is a filter-base generating a filter f. this filter converges to a in [[x, $], $]. to show that, we need to see that{⋂ b∈û b : u ∈ a } is a cover of a for [x, $]. in view of the form (2.1) of a base for [x, $], it is enough to show that if u0 ∈ a and p is an ideal subbase of open subsets of x covering u0, then there is a ∈ a with ⋂ b∈â b ∈ o♮(p). because u0 ⊆ ⋃ p ∈p p and a is a compact family, there is a finite subfamily p0 of p such that ⋃ p ∈p0 p ∈ a. since p is an ideal subbase, there is p ∈ p ∩a. then o(p) ⊆ ⋂ b∈p̂ b, which concludes the proof that a ∈ lim[[x,$],$] f. on the other hand, o /∈ f, which contradicts the fact that o is open for [[x, $], $]. � remark 2.2. if x is a non topological convergence space, then by [8, corollary 16.3], the open subsets of [x, $] are the rigidly compact families: families a of open subsets of x, closed under open supersets, such that adhξ h#a whenever h is a filter such that for every h ∈ h there is a closed subset b of h with b ∈ a#. hence the underlying set of [[x, $], $] is no longer κ(x) but the larger set of rigidly compact families on x. we will see below (proposition 2.3) that the convergence [[x, $], $] fails to be topological in this case. we do not know 148 f. jordan and f. mynard whether t [[x, $], $] can be expressed as an analog of the stone topology on the set of rigidly compact families on x. in order to investigate when t [x, $] is core compact, we will need notions and results from [7]. the concrete endofunctor epit of the category of convergence spaces (and continuous maps) is defined (on objects) by epit x = i −[t [x, $], $] where i : x → [[x, $], $] is defined by i(x)(f) = f(x). in view of [7, theorem 3.1] (2.3) w ≥ epit x ⇐⇒ t [x, $] ≥ [w, $], where x ≥ w have the same underlying set. in particular, x is t -dual if and only if x ≥ epit x. a convergence space x is called epitopological if i : x → [[x, $], $] is initial (in the category conv of convergence spaces and continuous maps). epitopologies form a reflective subcategory epi of conv and the (concrete) reflector is given (on objects) by epi x = i−[[x, $], $]. because [epi x, $] = [x, $], it is enough to consider epitopologies in the study of dual convergences. observe that a topological space is epitopological. note that if [x, $] is t -dual, then epi x = x is topological. therefore, in contrast to proposition 2.1, [x, $] is not t -dual if x is not topological. proposition 2.3. let x be an epitopological space. then x is topological if and only if [x, $] is t -dual. note also that epi x ≤ epit x and that epit ◦ epi = epit , so that epit restricts to an expansive endofunctor of epi. by iterating this functor, we obtain the coreflector on t -dual epitopologies. more precisely, if f is an expansive concrete endofunctor of c, we define the transfinite sequence of functors f α by f 1 = f and f αx = f (∨ β<α f βx ) . for each epitopological space x, there is an ordinal α(x) such that epi α(x) t x = epi α(x)+1 t x := dt x. proposition 2.4. the class of t -dual epitopologies is concretely coreflective in epi and the coreflector is dt . while this proposition easily follows from general results in [7] or [20], and galois connections, we provide a self-contained proof. proof. the class of t -dual convergences is closed under infima because [ ∧ i∈i xi, z ] = ∨ i∈i [xi, z]. indeed, if each xi is t -dual, then [ ∧ i∈i xi, $ ] = ∨ i∈i [xi, $] = ∨ i∈i t [xi, $] ≤ t ( ∨ i∈i [xi, $] ) = t ([ ∧ i∈i xi, $ ]) , core compactness and diagonality in spaces of open sets 149 and ∧ i∈i xi is t -dual. the functor epit is expansive on epi and therefore, so is dt . moreover, dt x is t -dual for each epitopological space x because [dt x, $] = [epi α(x)+1 t x, $] ≤ t [epi α(x) t x, $] = t [dt x, $]. therefore, for each epitopological space x, there exists the coarsest t -dual convergence x finer than x. by definition x ≤ x ≤ dt x. then [x, $] ≤ [x, $] and [x, $] is topological, so that [x, $] ≤ t [x, $]. but epit x is the coarsest convergence with this property. therefore epit x ≤ x = epit x and dt x ≤ x. � proposition 2.5. if x is a core compact topological space, then [x, $] is also a core compact topological space. proof. [x, $] = t [x, $] because x is core compact, and [x, $] is t -dual by proposition 2.1, because x is topological. therefore t [x, $] is a core compact topology. � however, if x is a non-topological t -dual convergence space (4), then [x, $] = t [x, $] is not core compact, by proposition 2.3. in other words, we have: proposition 2.6. if [x, $] is topological then x is topological if and only if [x, $] is core compact. in particular, dt x is topological if and only if [dt x, $] is core compact. theorem 2.7. if x ≥ t dt x then t [x, $] is core compact if and only if x is a core compact topological space. proof. we already know that if x is a core compact topological space then [x, $] = t [x, $] and that [x, $] is core compact by proposition 2.1. conversely, if t [x, $] is core compact then [t [x, $], $] is topological, so that epit x is topological. under our assumptions, x ≥ t dt x ≥ t epit x = epit x, hence by (2.3), x is t -dual. therefore [x, $] = t [x, $] is core compact and, in view of proposition 2.3, x is topological, and t -dual, hence a core compact topological space. � remark 2.8. note that, at least among hausdorff topological spaces, theorem 2.7 generalizes [19, corollary 3.6] that states that if x is first countable, then x is core compact if and only if t [x, $] is core compact. indeed, the locally compact coreflection kx of a hausdorff topological space is t -dual so that dt x ≤ kx. moreover, [1] characterizes a number of topological properties in terms of functorial inequalities of the form x ≥ je(x), 4such convergences exist: take for a instance a non-locally compact hausdorff regular topological k-space. then x = tkhx but x < khx so that khx is non-topological. 150 f. jordan and f. mynard where j is a concrete reflector and e a concrete coreflector in the category of convergence spaces. for instance, it is observed that a (hausdorff) topological space x is a k-space if and only if (2.4) x ≥ t k x, so that (2.4) can be taken as a definition of a k-convergence. hence if x is a hausdorff topological k-space (in particular a first-countable space) then x ≥ t dt x. on the other hand, in view of the results of [1], if f : x → y is a quotient map (in the topological sense) and x is core compact (so that x = dt x) then y ≥ t dt y . we will see in the next section that similarly, if x is a consonant topological space, then t [x, $] is core compact if and only if x is locally compact. problem 2.9. are there completely regular non locally compact topological spaces x such that t [x, $] is core compact? of course, in view of remark 2.8, such a space cannot be a k-space or consonant. 3. core compact dual, consonance, and infraconsonance a topological space is consonant if t [x, $] = ck(x, $), that is, if every scott open subset a of ox is compactly generated, that is, there are compact subsets (ki)i∈i of x such that a = ⋃ i∈i o(ki) [5]. a space is infraconsonant [9] if for every scott open subset a of ox there is a scott open set c such that c ∨ c ⊆ a, where c ∨ c := {c ∩ d : c, d ∈ c}. the notion’s importance stems from theorem 3.1 below. if the set c(x, y ) of continuous functions from x to y is equipped with the isbell topology (5), we denote it cκ(x, y ), while ck(x, y ) denotes c(x, y ) endowed with the compact-open topology. note that cκ(x, $) = t [x, $]. theorem 3.1 ([6]). let x be a completely regular topological space. the following are equivalent: (1) x is infraconsonant; (2) addition is jointly continuous at the zero function in cκ(x, r); (3) cκ(x, r) is a topological vector space; (4) ∩ : t [x, $] × t [x, $] → t [x, $] is jointly continuous. on the other hand, if x is consonant then cκ(x, r) = ck(x, r) so that consonance provides an obvious sufficient condition for cκ(x, r) to be a topological vector space. 5whose sub-basic open sets are given by [a, u] := {f ∈ c(x, y ) : ∃a ∈ a, f(a) ⊆ u} , where a ranges over openly isotone compact families on x and u ranges over open subsets of y . core compactness and diagonality in spaces of open sets 151 hence theorem 3.1 becomes truly interesting if completely regular examples of infraconsonant non consonant spaces can be provided [9, problem 1.2]. the first author recently obtained the first example of this kind [18]. the following results show that a space answering positively problem 2.9 would necessarily be infraconsonant and non-consonant and might provide an avenue to construct new examples. theorem 3.2. if x is topological and t [x, $] is core compact then x is infraconsonant. proof. [9, lemma 3.3] shows the equivalence between (1) and (4) in theorem 3.1, and that the implication (4)=⇒(1) does not require any separation. therefore, it is enough to show that ∩ : t [x, $] × t [x, $] → t [x, $] is continuous. since x is topological, [x, $] is t -dual by proposition 2.1. in view of (1.4) t ([x, $] × [x, $]) ≤ [x, $] × t [x, $] so that t ([x, $] × [x, $]) ≤ t ([x, $] × t [x, $]). if t [x, $] is core compact, hence t -dual then t ([x, $] × t [x, $]) ≤ t [x, $] × t [x, $] so that (3.1) t ([x, $] × [x, $]) ≤ t [x, $] × t [x, $]. therefore the continuity of ∩ : [x, $] × [x, $] → [x, $] implies that of ∩ : t ([x, $] × [x, $]) → t [x, $] because t is a functor, and in view of (3.1), that of ∩ : t [x, $] × t [x, $] → t [x, $]. � recall that a basis for the topology of ck(x, $) is given by sets of the form o(k) where k ranges over compact subsets of x. theorem 3.3. let x be a topological space. if ck(x, $) is core compact then x is locally compact. proof. if x is not locally compact, then ck(x, $) � [x, $] (e.g., [23, 2.19]) so that there is u0 ∈ ox with u0 /∈ lim[x,$] nk(u0). therefore, there is x0 ∈ u0 such that x0 /∈ int( ⋂ v ∈o(k) v ) whenever k is a compact subset of x with k ⊆ u0. in other words, for each such k and for each u ∈ o(x0) there is vu ∈ o(k) and xu ∈ u \ vu . then ck(x, $) is not core compact at u0. indeed, there is u0 ∈ o(x0) such that for every compact set k with k ⊆ u0, the k-open set o(k) is not relatively compact in o(x0). to see that, consider the cover s := {o(xu ) : u ∈ o(x0)} of o(x0). no finite subfamily of s covers o(k) because for any finite choice of u1, . . . , un in o(x0), we have w := ∩i=ni=1 vui ∈ o(k) but w /∈ ∪ i=n i=1 o(xui). � note that a hausdorff topological space x is locally compact if and only if it is core compact, and that the scott open filter topology on ox then coincides with ck(x, $) (e.g., [11, lemma ii.1.19]). hence theorem 3.3 could also be deduced (for the hausdorff case) from [19, corollary 3.6]. corollary 3.4. if x is a consonant topological space such that t [x, $] is core compact, then x is locally compact. 152 f. jordan and f. mynard 4. scott topology of the product versus product of scott topologies we now turn to a new characterization of infraconsonance, which motivates further the systematic investigation of the notion. recall that in a complete lattice (l, ≤) the scott convergence is given by (1.3), and the scott topology is its topological modification. a subset a of l is scott-open if and only if it is upper-closed and satisfies ∨ d ∈ a =⇒ ∃d ∈ d ∩ a, for every directed supset d of l (e.g., [11]). a product of complete lattices is a complete lattice for the coordinatewise order, and we can therefore consider the scott topology on the product for the coordinatewise order, and compare it with the product of the scott topologies. proposition 4.1. t ([x, $]2) is the scott topology on ox × ox. theorem 4.2. a space x is infraconsonant if and only if the product t [x, $]× t [x, $] of the scott topologies and the scott topology t ([x, $] × [x, $]) on the product coincide at (x, x). lemma 4.3. a subset s of ox × ox is [x, $] 2-open if and only if (1) s = s↑, that is, if (u, v ) ∈ s and u ⊆ u′, v ⊆ v ′ then (u′, v ′) ∈ s; (2) s is coordinatewise compact, that is, ( ⋃ i∈i oi, ⋃ j∈j vj) ∈ s =⇒ ∃i0 ∈ [i] <ω, j0 ∈ [j] <ω : ( ⋃ i∈i0 oi, ⋃ j∈j0 vj) ∈ s proof. assume s is [x, $]2-open and let (u, v ) ∈ s and u ⊆ u′, v ⊆ v ′. then (u, v ) ∈ lim[x,$]2{(u ′, v ′)}↑ so that (u′, v ′) ∈ s. assume now that ( ⋃ i∈i oi, ⋃ j∈j vj) ∈ s. then {o( ⋃ i∈f oi) : f ∈ [i] <∞} is a filter-base for a filter γ on ox such that ⋃ i∈i oi ∈ lim[x,$] γ and {o( ⋃ j∈d vj) : d ∈ [j] <∞} is a filter-base for a filter η on ox such that ⋃ j∈j vj ∈ lim[x,$] η. hence s ∈ γ×η because s is [x, $]2-open. therefore, there are finite subsets i0 of i and j0 of j such that o( ⋃ i∈i0 oi) × o( ⋃ j∈j0 vj) ⊆ s, so that ( ⋃ i∈i0 oi, ⋃ j∈j0 vj) ∈ s. conversely, assume that s satisfies the two conditions of the lemma and (u, v ) ∈ s∩lim[x,$]2(γ×η). since u ⊆ ⋃ g∈γ int( ⋂ g) and v ⊆ ⋃ h∈η int( ⋂ h∈h h), we have, by the first condition, that ( ⋃ g∈γ int( ⋂ g∈g g), ⋃ h∈η int( ⋂ h∈h h)) ∈ s. by the second condition, there are g1, . . . , gk ∈ γ and h1, . . . , hn ∈ η such that ( k⋃ i=1 int( ⋂ g∈gi g), n⋃ j=1 int( ⋂ h∈hj h)) ∈ s. core compactness and diagonality in spaces of open sets 153 therefore (int( ⋂ g∈ ⋂ k i=1 gi g), int( ⋂ h∈ ⋂ n j=1 hj h)) ∈ s so that ( k⋂ i=1 gi, n⋂ j=1 hj) ⊆ s, and s ∈ γ × η. � proof of proposition 4.1. in view of lemma 4.3, every [x, $]2-open subset of ox ×ox is scott open. conversely, consider a scott open subset s of ox ×ox. we only have to check that s satisfies the second condition in lemma 4.3. let ( ⋃ i∈i oi, ⋃ j∈j vj) ∈ s. the set d := { (⋃ i∈i0 oi, ⋃ j∈j0 vj ) : i0 ∈ [i] <ω, j0 ∈ [j]<ω} is a directed subset of ox ×ox (for the coordinatewise inclusion order) whose supremum is ( ⋃ i∈i oi, ⋃ j∈j vj). as s is scott-open, there are finite subsets i0 of i and j0 of j such that (⋃ i∈i0 oi, ⋃ j∈j0 vj ) ∈ s. � lemma 4.4. if a ∈ κ(x) then sa := {(u, v ) ∈ ox × ox : u ∩ v ∈ a} ↑ is [x, $]2-open. proof. let ( ⋃ i∈i oi, ⋃ j∈j vj) ∈ sa. then ( ⋃ i∈i oi) ∩ ( ⋃ j∈j vj) = ⋃ (i,j)∈i×j oi ∩ vj ∈ a. by compactness of a, there is a finite subset i0 of i and a finite subset j0 of j such that ⋃ (i,j)∈i0×j0 oi ∩ vj ∈ a, so that ( ⋃ i∈i0 oi, ⋃ j∈j0 vj) ∈ sa. in view of lemma 4.3, sa is [x, $] 2-open. � lemma 4.5. if s is [x, $]2-open, then ↓ s := ox({u ∪ v : (u, v ) ∈ s}) is a compact family on x. proof. if u ∪ v ⊆ ⋃ i∈i oi for some (u, v ) ∈ s then (⋃ i∈i oi, ⋃ i∈i oi ) ∈ s so that, in view of lemma 4.3, there is a finite subset i0 of i such that(⋃ i∈i0 oi, ⋃ i∈i0 oi ) ∈ s. hence ⋃ i∈i0 oi ∈↓ s. � proof of theorem 4.2. suppose that x is infraconsonant. note that (t [x, $])2 ≤ t ([x, $]2) is always true, so that we only have to prove the reverse inequality at (x, x). consider an [x, $]2-open neighborhood s of (x, x). by lemma 4.5, the family ↓ s is compact. by infraconsonance, there is c ∈ κ(x) with c ∨ c ⊆↓ s. note that c × c ⊆ s, because if (c1, c2) ∈ c × c then c1 ∩ c2 ∈↓ s so that c1 ∩ c2 ⊇ u ∪ v for some (u, v ) ∈ s, and therefore (c1, c2) ∈ s. conversely, assume that n[x,$]2(x, x) = nt [x,$]2(x, x) and let a ∈ κ(x). by lemma 4.4, sa ∈ n[x,$]2(x, x) so that sa ∈ nt [x,$]2(x, x). in other words, there are families b and c in κ(x) such that b × c ⊆ sa. in particular 154 f. jordan and f. mynard d := b ∩ c belongs to κ(x) and satisfies d × d ⊆ sa. by definition of sa, we have that d ∨ d ⊆ a and x is infraconsonant. � 5. topologicity, pretopologicity and diagonality of [x, $] a selection for a convergence space x is a map s[·] : x → fx such that x ∈ limx s[x] for all x ∈ x. definition 5.1. a convergence space x is diagonal if for every selection s[·] and every filter f with x0 ∈ limx f the filter (5.1) s[f] := ⋃ f ∈f ⋂ x∈f s[x] converges to x0. if this property only holds when f is additionally principal, we say that x is f0-diagonal. of course, every topology is diagonal. in fact a convergence is topological if and only if it is both pretopological and diagonal (e.g., [4]). in order to compare our condition for diagonality of [x, $] with core-compactness, we first rephrase the latter. lemma 5.2. a topological space is core compact if and only if for every x ∈ x, every u ∈ o(x) and every family h of filters on x, we have (5.2) ∀h ∈ h : adh h ∩ u = ∅ =⇒ x /∈ adh ∧ h∈h h. proof. if x is core compact, then there is v ∈ o(x) which is relatively compact in u. if adh h∩u = ∅, then u ⊆ ⋃ h∈h (cl h)c so that, by relative compactness of v in u there is, for each h ∈ h, a set hh ∈ h with v ∩ cl hh = ∅. then⋃ h∈h hh ∈ ∧ h∈h h but ⋃ h∈h hh ∩ v = ∅ so that x /∈ adh ∧ h∈h h. conversely, if (5.2) is true, consider the family h := {h ∈ fx : adh h ∩ u = ∅}. in view of (5.2), x /∈ adh ∧ h∈h h so that there is v ∈ o(x) such that v /∈ (∧ h∈h h )# . now v is relatively compact in u because any filter than meshes with v cannot be in h and has therefore an adherence point in u. � recall that [x, $] = p [x, $] if and only if x is t -core compact, and that, if x is topological, [x, $] is topological whenever it is pretopological. while the latest is well-known, and follows for instance from the results of [7], it seems difficult to find an elementary argument in the literature, which is why we include the following proposition, which also illustrates the usefulness of lemma 5.2. proposition 5.3. if x is topological and [x, $] is pretopological, then [x, $] is topological. proof. we will show that under these assumptions, x satisfies (5.2). let x ∈ x and u ∈ o(x). let h be a family of filters satisfying the hypothesis of (5.2). let h ∈ h. consider the filter base h∗ := {o(x \ cl(h)): h ∈ h} on [x, $]. since adh(h)∩u = ∅, it follows that u ∈ lim h∗. since [x, $] is pretopological, core compactness and diagonality in spaces of open sets 155 u ∈ lim ∧ h∈h h∗. in particular, there exist, for each h ∈ h, a hh ∈ h such that x ∈ int( ⋂ ⋃ h∈h o(x \ cl hh)) = int( ⋂ h∈h (x \ cl hh)) = int(x \ ( ⋃ h∈h cl hh)) ⊆ x \ cl( ⋃ h∈h hh). thus, x /∈ adh( ∧ h∈h h). � in other words, if [x, $] is pretopological it is also diagonal, provided that x is topological. we will see that even if x is topological, [x, $] is not always diagonal. moreover it can be diagonal without being pretopological (examples 5.5 and 5.7). we call a topological space injectively core compact if for every x ∈ x and u ∈ o(x) the conclusion of (5.2) holds for every family h of filters such that there is an injection θ : h → o(u) satisfying adh h∩θ(h) = ∅ for each h ∈ h. as such a family h clearly satisfies the premise of (5.2), every core compact space is in particular injectively core compact. theorem 5.4. let x be a topological space. the following are equivalent: (1) x is injectively core compact; (2) [x, $] is diagonal; (3) [x, $] is f0-diagonal. proof. (1)=⇒(2): let s[�] : ox → fox be a selection for [x, $] and let u ∈ lim[x,$] f. if x ∈ u, there is f ∈ f such that x ∈ int (⋂ o∈f o ) := v . note that f ⊆ o(v ). for each o ∈ f , consider the filter ho on x generated by {clx (⋃ w∈s w c ) : s ∈ s[o]}. because o ∈ lim[x,$] s[o], we have that adhx ho ∩ o = ∅. choose g ⊆ f so that {ho : o ∈ g} = {ho : o ∈ f} and ho 6= hp for every two distinct o, p ∈ g. because x is injectively core compact and h := {ho : o ∈ g} satisfies the required condition (with θ(ho) = o ), we conclude that x /∈ adhx ∧ o∈g ho. by the way we chose g, we have ∧ o∈g ho = ∧ o∈f ho. so, x /∈ adhx ∧ o∈f ho. in other words, there is an h ∈ ∧ o∈f ho such that x /∈ clx h, that is, x ∈ intx h c. therefore, for each o ∈ f there is so ∈ s[o] such that x ∈ int( ⋂ o∈f int( ⋂ w∈so w)) ⊆ int( ⋂ w∈ ⋃ o∈f so w). in other words, there is f ∈ f and m ∈ ∧ o∈f s[o] such that x ∈ intx (⋂ w∈m w ) , that is, u ∈ lim[x,$] s[f]. (2)=⇒(3) is clear. (3)=⇒(1): suppose x is not injectively core compact. then there is x ∈ x, u ∈ o(x) and a family h of filters on x with an injective map θ : h → o(u) such that θ(h) ∩ adhx h = ∅ for each h ∈ h but x ∈ adhx ∧ h∈h h. define a relation ∼ on h by h1 ∼ h2 provided that 156 f. jordan and f. mynard the collections {cl(h): h ∈ h1} and {cl(h): h ∈ h2} both generate the same filter. clearly, ∼ is an equivalence relation. let h∗ ⊆ h be such that h∗ contains exactly one element of each equivalence class of ∼. for each h ∈ h∗ let h∗ be the filter with base {cl(h): h ∈ h}. let j = {h∗ : h ∈ h∗}. define θ∗ : j → o(u) so that θ∗(j ) = θ(h), where h ∈ h∗ is such that j = h∗. it is easily checked that θ∗ is injective. since adh(h∗) = adh(h) for every h ∈ h∗, we have θ∗(j ) ∩ adh(j ) = ∅. it is also easy to check that x ∈ adh (∧ j ∈j j ) . for each j ∈ j, the filter j̃ generated on ox by the filter-base {ox(x \ j) : j ∈ j } converges to θ∗(j ). consider now the subset θ∗(j) of o(u) ⊆ ox and the selection s[�] : ox → fox defined by s[θ(j )] = j̃ for each j ∈ j and s[o] = {o}↑ for o /∈ θ∗(j). this is indeed a well-defined selection because θ∗ is injective. notice that u ∈ lim[x,$] θ ∗(j) because θ∗(j) ⊆ o(u). let l ∈ s[θ∗(j)]. we may pick from each j ∈ j a closed set jj ∈ j such that ⋃ j ∈j ox(x\jj ) ⊆ l. let v be an open neighborhood of x. since x ∈ adhx ∧ j ∈j j and ⋃ j ∈j jj ∈∧ j ∈j j , there is an j0 ∈ j such that v ∩ jj0 6= ∅. since v 6⊆ x \ jj0 and x\jj0 ∈ ox(x\jj0), v 6⊆ ⋂ ox(x\jj0). since ox(x\jj0) ⊆ l, v 6⊆ ⋂ l. since v was an arbitrary neighborhood of x, x 6∈ int( ⋂ l). thus, u /∈ s[θ∗(j)]. therefore, [x, $] is not f0-diagonal. � a cardinal number κ is regular if a union of less than κ-many sets of cardinality less than κ has cardinality less than κ. a strong limit cardinal κ is a cardinal for which card(2a) < κ whenever card(a) < κ. a strongly inaccessible cardinal is a regular strong limit cardinal. uncountable strongly inaccessible cardinals cannot be proved to exist within zfc, though their existence is not known to be inconsistent with zfc. let us denote by (*) the assumption that such a cardinal exist. example 5.5 (a hausdorff space x such that [x, $] is diagonal but not pretopological under (*)). assume that κ is a (uncountable) strong limit cardinal. let x be the subspace of κ ∪ {κ} endowed with the order topology, obtained by removing all the limit ordinals but κ. since x is a non locally compact hausdorff topological space, [x, $] is not pretopological. to show that x is injectively core compact, we only need to consider x = κ and u ∈ o(κ) in the definition, because κ is the only non-isolated point of x. let h be a family of filters on x admitting an injective map θ : h → o(u) such that adh h ∩ θ(h) = ∅ for each h ∈ h. for each h ∈ h there is hh ∈ h such that κ /∈ cl(hh) so that card(hh) < κ. since u is a neighborhood of κ, there is a β < κ such that {ξ ∈ x : β ≤ ξ} ⊆ u. since v \ u ⊆ {ξ ∈ x : ξ < β} for every v ∈ o(u), we have cardh ≤ card o(u) ≤ 2β. since κ is a strong limit cardinal, cardh < κ. since κ is regular, card ⋃ h∈h hh < κ so that κ /∈ adh ∧ h∈h h. core compactness and diagonality in spaces of open sets 157 we do not know if the existence of large cardinals is necessary for the construction of a hausdorff space x such that [x, $] is diagonal and not pretopological, but, as the next proposition shows, such a space cannot be too small. let c denote the cardinality of the real numbers. proposition 5.6. let x be a hausdorff topological space. if x is a non locally compact space of character not exceeding c, then [x, $] is not diagonal. proof. let p ∈ x be such that x is not locally compact at p. since x is not compact, there is a neighborhood u of p such that x \ u is infinite. since x is hausdorff, there exists a countably infinite a ⊆ x \ u and mutually disjoint open sets {wa : a ∈ a} such that a ∈ wa for every a ∈ a. it follows that the collection {u ∪ ⋃ a∈e wa : e ⊆ a} is a collection of c-many distinct elements of o(u). since the character of x is at most c, there is a neighborhood base b at p with at most c-many elements. since x is not locally compact at p, there is for each b ∈ b a filter hb on b such that adh(hb) = ∅. let h = {hb : b ∈ b}. since card b ≤ card o(u), there is an injection θ : h → o(u). clearly, adh(hb) ∩ θ(hb) = ∅ for every b ∈ b. however, p ∈ adh (∧ b∈b hb ) . hence, x is not injectively core compact at p. thus, [x, $] is not diagonal. � on the other hand, we can construct in zfc a t0 space x such that [x, $] is diagonal and not pretopological. example 5.7 (a t0 space x such that [x, $] is diagonal but not pretopological). let z stand for integers and c+ be the cardinal successor of c. let ∞ be a point that is not in c+ × z and x = {∞} ∪ (c+ × z). for each (α, n) ∈ c+ × z define sα,n = {(β, k): α ≤ β and n ≤ k}. for each α ∈ c +, let tα = {(β, k): α ≤ β and k ∈ z} ∪ {∞}. topologize x by declaring all sets of the form tα and sα,n to be sub-basic open sets. we show that x is not core compact at ∞. let u be a neighborhood of ∞. there is an α such that tα ⊆ u. notice that tα+1 ∪ {s0,n : n ∈ z} is a cover of x but no finite subcollection covers tα. thus, x is not core compact at ∞. in particular, [x, $] is not pretopological. let (α, n) ∈ x \ {∞}. let u be an open neighborhood of (α, n). since (α, n) ∈ u it follows from the way we chose our sub-base that sα,n ⊆ u. since (α, n) has a minimal open neighborhood, x is core compact at (α, n). let v be an open neighborhood of ∞. there is an α such that tα ⊆ v . let u ⊆ x be an open superset of v . for every n ∈ z, u ∩ (c+ × {n}) 6= ∅. for each n ∈ z define αn = min{β : (β, n) ∈ u}. notice that {β : αn ≤ β} × {n} = u ∩ (c+ × {n}) and αn ≤ α. since each open superset of v will determine a unique sequence (αn)n∈z, it follows that the open supersets of v can injectively be mapped into the countable sequences on {β : β ≤ α} × z. since {β : β ≤ α} × z has cardinality at most c, {β : β ≤ α} × z has at most c-many countable sequences. thus, v has at most c-many supersets. let v be an open neighborhood of ∞, h be a collection of filters, and θ : h → ox(v ) be an injection such that adh(h)∩θ(h) = ∅ for every h ∈ h. since v 158 f. jordan and f. mynard has at most c-many open supersets, card h ≤ c. let h ∈ h. since ∞ /∈ adh h, there is an αh ∈ c + such that adh(h) ∩ tαh = ∅. let α = (suph∈h αh) + 1 < c +. it is easy to check that, adh (∧ h∈h h ) ∩ tα = ∅. thus, x is injectively core compact at ∞. since x is injectively core compact at each point, [x, $] is diagonal, by theorem 5.4. example 5.8 (a t -dual convergence space that is not core compact). consider a partition {an : n ∈ ω} of the set ω ∗ of free ultrafilters on ω satisfying the condition that for every infinite subset s of ω and every n ∈ ω, there is u ∈ an with s ∈ u. let m := {mn : n ∈ ω} be disjoint from ω and let x := ω ∪ m. define on x the finest convergence in which lim{mn} ↑ = m for all n ∈ ω, and each free ultrafilter u on ω converges to mn (and mn only), where n is defined by u ∈ an. claim. x is not core compact. proof. let mn ∈ m and u ∈ an. pick s ⊆ ω, s ∈ u, and k 6= n. for every u ∈ u there is w ∈ ak such that u ∈ w. but lim w = {mk} is disjoint from s. � claim. x is t -core compact, and therefore [x, $] is pretopological. proof. for each mn ∈ m, the set m is included in every open set containing mn because mn ∈ ⋂ k∈ω lim{mk} ↑. if u is a non-trivial convergent ultrafilter in x then lim u = {mn} for some n ∈ ω. for any s ∈ u, s ∩ ω is infinite and any free ultrafilter w on s ∩ ω belongs to one of the element ak of the partition, so that lim w = {mk} intersects m, and therefore any open set containing mn. � claim. [x, $] is diagonal. proof. let s[�] : ox → fox be a selection for [x, $] and let u ∈ lim[x,$] f. now, { ⋂ f : f ∈ f} is a (convergence) cover of u. let x ∈ u and d be a filter on x such that x ∈ lim d. there is an f ∈ f and a d ∈ d such that d ⊆ ⋂ f := v . assume x ∈ ω, in which case d = {x}↑. in particular, x ∈ o for every o ∈ f . for every o ∈ f there is a to ∈ s[o] such that x ∈ ⋂ to. now, x ∈ ⋂⋂ o∈f to ∈ s[f ]. so, ⋂⋂ o∈f to ∈ {x} ↑ = d. assume x ∈ m. in this case, m ∩ o 6= ∅ for all o ∈ f and, by definition of the convergence on x, m ⊆ o for all o ∈ f . since o ∈ lim[x,$] s[o] and m ⊆ o, there is s ∈ s[o] such that x ∈ ⋂ s, and, since each element of s is open, m ⊆ ⋂ s. if there is no s ∈ s[o] such that o ⊆ ⋂ s then the filter h generated by {(o ∩ ω) \ ⋂ s : s ∈ s[o]} is non degenerate. notice that it is not free, for otherwise there would be an n ∈ ω and u ∈ an with u ≥ h. but mn ∈ lim u ∩ o, and there would be s ∈ s[o] such that ⋂ s ∈ u, which is not possible. therefore, there is y ∈ ⋂ s∈s[o] ((o ∩ ω) \ ⋂ s) which core compactness and diagonality in spaces of open sets 159 contradicts o ∈ lim[x,$] s[o]. hence, there is so ∈ s[o] such that o ⊆ ⋂ so. now, d ⊆ ⋂ f ⊆ ⋂ o∈f ⋂ so. in particular, ⋂ o∈f ⋂ so ∈ d. thus, { ⋂ j : j ∈ s[f]} is a cover of u, and [x, $] is diagonal. � therefore [x, $] is pretopological and diagonal, hence topological, and x is t -dual. 6. appendix: convergence spaces a family a of subsets of a set x is called isotone if b ∈ a whenever a ∈ a and a ⊆ b. we denote by a↑ the smallest isotone family containing a, that is, the collection of subsets of x that contain an element of a. if a and b are two families of subsets of x we say that b is finer than a, in symbols a ≤ b, if for every a ∈ a there is b ∈ b such that b ⊆ a. of course, if a and b are isotone, then a ≤ b ⇐⇒ a ⊆ b. this defines a partial order on isotone families, in particular on the set fx of filters on x. every family (fα)α∈i of filters on x admits an infimum ∧ α∈i fα := ⋂ α∈i fα = { ⋃ α∈i fα : fα ∈ fα }↑ . on the other hand the supremum even of a pair of filters may fail to exist. we call grill of a the collection a# := {h ⊆ x : ∀a ∈ a, h ∩ a 6= ∅}. it is easy to see that a = a## if and only if a is isotone. in particular f = f## ⊆ f# if f is a filter. we say that two families a and b of subsets of x mesh, in symbols a#b, if a ⊆ b#, equivalently if b# ⊆ a. the supremum of two filters f and g exists if and only if they mesh, in which case f ∨ g = {f ∩ g : f ∈ f, g ∈ g} ↑ . an infinite family (fα)α∈i of filters has a supremum∨ α∈i fα if pairwise suprema exist and for every α, β ∈ i there is γ ∈ i with fγ ≥ fα ∨ fβ. a convergence ξ on a set x is a relation between x and the set fx of filters on x, denoted x ∈ limξ f whenever x and f are in relation, satisfying that x ∈ limξ{x} ↑ for every x ∈ x, and limξ f ⊆ limξ g whenever f ≤ g. the pair (x, ξ) is called a convergence space. a function f : (x, ξ) → (y, σ) between two convergence space is continuous if x ∈ limξ f =⇒ f(x) ∈ limσ f(f), where f(f) is the filter {f(f) : f ∈ f}↑ on y . if ξ and τ are two convergences on the same set x, we say that ξ is finer than τ, in symbols ξ ≥ τ, if limξ f ⊆ limτ f for every f ∈ fx. this defines a partial order on the set of convergence structures on x, which defines a complete lattice for which supremum ∨i∈iξi and infimum ∧i∈iξi of a family {ξi : i ∈ i} of convergences are defined by lim∨i∈iξi f = ⋂ i∈i limξi f, lim∧i∈iξi f = ⋃ i∈i limξi f. 160 f. jordan and f. mynard every topology can be identified with a convergence, in which x ∈ lim f if f ≥ n(x), where n(x) is the neighborhood filter of x for this topology. a convergence obtained this way is called topological. moreover, a function f : x → y between two topological spaces is continuous in the usual topological sense if and only if it is continuous in the sense of convergence. on the other hand, every convergence determines a topology in the following way: a subset c of a convergence space (x, ξ) is closed if limξ f ⊆ c for every filter f on x with c ∈ f. a subset o is open if its complement is closed, that is, if o ∈ f whenever limξ f ∩ o 6= ∅. the collection of open subsets for a convergence ξ is a topology t ξ on x, called topological modification of ξ. the topology t ξ is the finest topological convergence coarser than ξ. if f : (x, ξ) → (y, τ) is continuous, so is f : (x, t ξ) → (y, t τ). in other words, t is a concrete endofunctor of the category conv of convergence spaces and continuous maps. continuity induces canonical notions of subspace convergence, product convergence, and quotient convergence. namely, if f : x → y and y carries a convergence τ, there is the coarsest convergence on x making f continuous (to (y, τ)). it is denoted f−τ and called initial convergence for f and τ. for instance if s ⊆ x and (x, ξ) is a convergence space, the induced convergence by ξ on s is by definition i−ξ where i is the inclusion map of s into x. similarly, if {(xi, ξi) : i ∈ i} is a family of convergence space, then the product convergence πi∈iξi on the cartesian product πi∈ixi is the coarsest convergence making each projection pj : πi∈ixi → xj continuous. in other words, πi∈iξi = ∨i∈ip − i ξi. in the case of a product of two factors (x, ξ) and (y, τ), we write ξ × τ for the product convergence on x × y . dually, if f : x → y and (x, ξ) is a convergence space, there is the finest convergence on y making f continuous (from (x, ξ)). it is denoted fξ and called final convergence for f and ξ. if f : (x, ξ) → y is a surjection, the associated quotient convergence on y is fξ. note that if ξ is a topology, the quotient topology is not fξ but t fξ. the functor t is a reflector. in other words, the subcategory top of conv formed by topological spaces and continuous maps is closed under initial constructions. note however that the functor t does not commute with initial constructions. in particular t ξ × t τ ≤ t (ξ × τ) but the reverse inequality is generally not true. similarly, if i : s → (x, ξ) is an inclusion map, i−(t ξ) ≤ t (i−ξ) but the reverse inequality may not hold. a convergence ξ is pretopological or a pretopology if limξ ∧ α∈i fα = ⋂ α∈i limξ fα. of course, every topology is a pretopology, but not conversely. for any convergence ξ there is the finest pretopology pξ coarser than ξ. moreover, x ∈ limp ξ f if and only if f ≥ vξ(x) where vξ(x) := ∧ x∈limξ f f is called vicinity filter of x. the subcategory prtop of conv formed by pretopological spaces and continuous maps is reflective (closed under initial constructions). moreover, in contrast with topologies, the reflector p commutes with subspaces. however, like t, it does not commute with products. core compactness and diagonality in spaces of open sets 161 the adherence adhξ f of a filter f on a convergence space (x, ξ) is by definition adhξ f := ⋃ h#f limξ h = ⋃ u∈u(f) limξ u, where ux denotes the set of ultrafilters on x and u(f) denotes the set of ultrafilters on x finer than the filter f. we write adhξ a for adhξ{a} ↑. note that in a convergence space x, adhξ may not be idempotent on subsets of x. in fact a pretopology is a topology if and only if adh is idempotent on subsets. we reserve the notations cl and int to topological closure and interior operators. a family a of subsets of x is compact at a family b for ξ if f#a =⇒ adhξ f#b. we call a family compact if it is compact at itself. in particular, a subset a of x is compact if {a} is compact, and compact at b ⊆ x if {a} is compact at {b}. given a class d of filters, a convergence is called based in d or d-based if for every convergent filter f, say x ∈ lim f, there is a filter d ∈ d with d ≤ f and x ∈ lim d. a convergence is called locally compact if every convergent filter contains a compact set, and hereditarily locally compact if it is based in filters with a filter-base composed of compact sets. for every convergence, there is the coarsest locally compact convergence kξ that is finer than ξ and the coarsest hereditarily locally compact convergence khξ that is finer than ξ. both k and kh are concrete endofunctors of conv that are also coreflectors. if a ⊆ x and (x, ξ) is a convergence space, then o(a) denotes the collection of open subsets of x that contain a and if a is a family of subsets of x then o(a) := ⋃ a∈a o(a). a family is called openly isotone if a = o(a). note that in a topological space x, an openly isotone family a of open subsets of x is compact if and only if, whenever ⋃ i∈i oi ∈ a and each oi is open, there is a finite subset j of i such that ⋃ i∈j oi ∈ a. if (x, ξ) and (y, σ) are two convergence spaces, c(x, y ) or c(ξ, σ) denote the set of continuous maps from x to y . the coarsest convergence on c(x, y ) making the evaluation map e : x × c(x, y ) → y , e(x, f) = f(x), jointly continuous is called continuous convergence and denoted [x, y ] or [ξ, σ]. explicitly, f ∈ lim[x,y ] f ⇐⇒ ∀x∈x∀g∈fx:x∈limξ g f(x) ∈ limσ e (g × f) . references [1] s. dolecki, 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(received december 2010 – accepted june 2011) francis jordan (fejord@hotmail.com) queensborough community college, queens, ny, usa frédéric mynard (fmynard@georgiasouthern.edu) georgia southern university, statesboro, ga30460, usa core compactness and diagonality in spaces of open sets. by f. jordan and f. mynard singhagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 293-300 cl-supercontinuous functions d. singh abstract. basic properties of cl-supercontinuity, a strong variant of continuity, due to reilly and vamanamurthy [indian j. pure appl. math., 14 (1983), 767–772], who call such maps clopen continuous, are studied. sufficient conditions on domain or range for a continuous function to be cl-supercontinuous are observed. direct and inverse transfer of certain topological properties under cl-supercontinuous functions are studied and existence or nonexistence of certain cl-supercontinuous function with specified domain or range is outlined. 2000 ams classification: 54c08, 54c10, 54d10, 54d20, 54d30. keywords: clopen continuous function, supercontinuous function, perfectly continuous function, strongly continuous function, zero-dimensional space, ultrahausdorff space. 1. introduction strong variants of continuity are of considerable significance and arise in many branches of mathematics including topology, complex analysis and functional analysis. reilly and vamanamurthy [12] call a function f : x → y clopen continuous if for each open set v containing f (x) there exists a clopen ( closed and open) set u containing x such that f (u ) ⊂ v . in this paper we elaborate on the properties of these mappings introduced by reilly and vamanamurthy. however, in the topological folklore the phrase “clopen map” is used for the functions which map clopen sets to open sets. so in this paper we rename “clopen continuous maps” as cl-supercontinuous functions which appears to be a better nomenclature, since it is a strong form of supercontinuity introduced by munshi and basan [9]. the class of cl-supercontinuous functions strictly contains the class of perfectly continuous functions of noiri [11] which in turn properly include all strongly continuous functions of levine [8]. furthermore, the class of cl-supercontinuous functions is properly contained in the class of z-supercontinuous functions [3] which in its turn is contained in the class of supercontinuous functions [9]. 294 d. singh in section 2, several characterizations of cl-supercontinuity are obtained and it is shown that cl-supercontinuity is preserved under restrictions, compositions, products, and passage to the graph function. the notions of cl-quotient topology and cl-quotient space are introduced in section 3. section 4 is devoted to the study of the behavior of separation axioms under cl-supercontinuous functions. in section 5 we conclude with alternative proofs of certain results of preceding sections. lastly, we mention some possible application of cl-supercontinuity to topology and analysis. 2. basic properties of cl-supercontinuous functions definition 2.1. a set g in a topological space x is said to be cl-open if for each x ∈ g, there exist a clopen set h such that x ∈ h ⊆ g, equivalently g is the union of clopen sets. the complement of a cl-open set will be referred to as cl-closed set. theorem 2.2. for a function f : x → y , the following statements are equivalent. (a) f is cl-supercontinuous. (b) inverse image of every open subset of y is a cl-open in x. (c) inverse image of every closed subset of y is a cl-closed in x. proof of theorem 2.2 is routine and hence omitted. remark 2.3. if either of the spaces x and y is zero-dimensional, then any continuous function from x to y is cl-supercontinuous. definition 2.4. let x be a topological space and let a ⊂ x. a point x ∈ x is said to be a cl-adherent of a if every clopen set containing x intersects a. let [a]cl denote the set of all cl-adherent points of a. then the set a is cl-closed if and only if [a]cl = a. theorem 2.5. for a function f : x → y the following statements are equivalent. (a) f is cl-supercontinuous. (b) f ([a]cl) ⊂ f (a) for every set a ⊂ x. (c) [f −1(b)]cl ⊂ f −1(b̄) for every b ⊂ y . proof. (a) ⇒ (b). since f (a) is closed in y , by theorem 2.2 f −1(f (a)) is a clclosed in x. again, since a ⊂ f −1(f (a)), [a]cl ⊂ [f −1(f (a))]cl = f −1(f (a)) and so f ([a]cl) ⊂ f (f −1(f (a)) ⊂ f (a). (b) ⇒ (c). let b ⊂ y . then by (b), f ([f −1(b)]cl) ⊂ f (f −1(b)) ⊂ b̄ and so it follows that [f −1(b)]cl ⊂ f −1(b̄). (c) ⇒ (a). let f be any closed set in y . then [f −1(f )]cl ⊂ f −1(f̄ ) = f −1(f ). again, since f −1(f ) ⊂ (f −1(f )) ⊂ [f −1(f )]cl, f −1(f ) = [f −1(f )]cl which in turn implies that f −1(f ) is cl-closed and so in view of theorem 2.2 f is cl-supercontinuous. � cl-supercontinuous functions 295 definition 2.6. a filter base t− is said to cl-converge to a point x written as t− cl −→ x if every clopen set containing x contains a member of t−. theorem 2.7. a function f : x → y is cl-supercontinuous if and only if for each x ∈ x and each filter base t− that cl-converges to x, f (t−) → f (x). proof. assume that f is cl-supercontinuous and let t− cl −→ x. let w be an open set containing f (x). then x ∈ f −1(w ) and f −1(w ) is cl-open. let h be a clopen set such that x ∈ h ⊂ f −1(w ) and so f (h) ⊂ w . since t− cl −→ x, there exists a u ∈ t− such that u ⊂ h and hence f (u ) ⊂ f (h) ⊂ w . thus f (t−) → f (x). � conversely, let w be an open subset of y containing f (x). now the filter base n x consisting of all clopen sets containing x cl-converges to x and so by hypothesis f (n x) → f (x). hence there exists a member f (n ) of f (n x) such that f (n ) ⊂ w . since n ∈ n x, n is an clopen set containing x. thus f is cl-supercontinuous. it is routine to verify that cl-supercontinuity is invariant under restriction and composition of functions and enlargement of range. moreover, the composition is cl-supercontinuous whenever f : x → y is cl-supercontinuous and g : y → z is continuous. remark 2.8. in general cl-supercontinuity of g ◦ f need not imply even continuity of f . for example, let x be the real line with cofinite topology, y = {0, 1} be the two point sierpinski space [16] and let f : x → y be defined by f (x) = { 0, if x is irrational 1, if x is rational let z = {0, 1} be endowed with the indiscrete topology and let g : y → z be the identity map. then g ◦ f and g are cl-supercontinuous, however, f is not continuous. definition 2.9. a function f : x → y is said to be cl-open (cl-closed) if f (a) is open (closed) in y for every clopen set a in x. theorem 2.10. let f : x → y be a cl-open, cl-supercontinuous surjection and g : y → z be any function. then g ◦ f is cl-supercontinuous if and only if g is continuous. theorem 2.11. let f : x → y be any function. if {uα : α ∈ ∆} is a cl-open cover of x and for each α, fα = f | uα : uα → y is cl-supercontinuous, then f is cl-supercontinuous. proof. let v be a clopen subset of y . then f −1(v ) = ∪ {f −1α (v ) : α ∈ ∆} and since each fα is cl-supercontinuous, each f −1 α (v ) is cl-open in uα and hence in x. thus f −1(v ) being the union of cl-open sets is cl-open. � 296 d. singh theorem 2.12. let {fα : x → xα | α ∈ λ} be a family of functions and let f : x → ∏ α∈λ xα be defined by f (x) = (fα(x)). then f is cl-supercontinuous if and only if each fα : x → xα is cl-supercontinuous. proof. let f : x → ∏ α∈λ xα be cl-supercontinuous. then the composition fα = pα ◦ f , where pα denotes the projection of ∏ α∈λ xα onto αth-coordinate space xα, is cl-supercontinuous for each α. � conversely, suppose that each fα : x → xα is cl-supercontinuous. to show that the function f is cl-supercontinuous, in view of theorem 2.2 it is sufficient to show that f −1(u ) is cl-open for each open set u in the product space ∏ α∈λ xα. since the finite intersections and arbitrary unions of cl-open sets is cl-open, it suffices to prove that f −1 (s) is cl-open for every subbasic open set s in the product space ∏ α∈λ xα. let uβ × ∏ α6=β xα be a subbasic open set in ∏ α∈λ xα. then f −1(uβ × ∏ α6=β xα) = f −1(p−1 β (uβ )) = f −1 β (uβ ) is cl-open in x. hence f is cl-supercontinuous. theorem 2.13. for each α ∈ ∆, let fα : xα → yα be a mapping and let f : ∏ xα → ∏ yα be a mapping defined by f ((xα)) = (fα(xα)) for each (xα) in ∏ xα. then f is cl-supercontinuous if and only if fα is cl-supercontinuous for each α ∈ ∆. proof. let f : ∏ xα → ∏ yα be cl-supercontinuous. let vβ be an open subset of yβ . then vβ × ( ∏ α6=β yα) is a subbasic open subset of the product space ∏ yα. since f is cl-supercontinuous, f −1(vβ × ( ∏ α6=β yα)) = f −1 β (vβ ) × ( ∏ α6=β xα) is cl-open in ∏ xα. consequently, f −1 β (vβ ) is a cl-open set in xβ and hence fβ is cl-supercontinuous. � conversely, suppose that each fα : xα → yα is cl-supercontinuous. let v = vβ × ( ∏ α6=β yα) be a subbasic open set in ∏ yα. since each fα is cl-supercontinuous and since f −1(v ) = f −1(vβ × ( ∏ α6=β yα)) = f −1 β (vβ ) × ( ∏ α6=β xα), f −1(v ) is cl-open, and so f is cl-supercontinuous. theorem 2.14. let f : x → y be a function and g : x → x × y , defined by g(x) = (x, f (x)) for each x ∈ x, be the graph function. then g is cl-supercontinuous if and only if f is cl-supercontinuous and x is zerodimensional. proof. to prove necessity, suppose that g is cl-supercontinuous. then the composition f = py ◦ g is cl-supercontinuous, where py is the projection from x × y onto y . let u be any open set in x and let x ∈ u . then u × y is an open set containing g(x). since g is cl-supercontinuous, there exists a clopen set w containing x such that g(w ) ⊂ u × y . thus x ∈ w ⊂ u , which shows that u is a cl-open and so x is zero-dimensional. � to prove sufficiency, let x ∈ x and let w be an open set containing g(x). there exist open sets u ⊂ x and v ⊂ y such that (x, f (x)) ∈ u × v ⊂ w . since x is zero-dimensional, there exists a clopen set g1 in x containing x cl-supercontinuous functions 297 such that x ∈ g1 ⊂ u . since f is cl-supercontinuous, there exists a clopen set g2 in x containing x such that f (g2) ⊂ v . let g = g1 ∩ g2. then g is a clopen set containing x and g(g) ⊂ u × v ⊂ w , which implies that g is cl-supercontinuous. definition 2.15. a function f : x → y is said to be slightly continuous [1] if f −1(a) is open in x for every clopen set a in y . lemma 2.16. for a function f : x → y , the following statements are equivalent. (a) f is slightly continuous. (b) f (ā) ⊆ [f (a)]cl for all a ⊆ x. (c) (f −1(b)) ⊆ f −1([b]cl) for all b ⊆ y . (d) inverse image of every cl-closed set is closed. (e) inverse image of every cl-open set is open. proof. (a) ⇒ (b): let y ∈ f (ā). choose x ∈ ā such that f (x) = y. let v be a clopen set containing y. since f is slightly continuous, f −1(v ) is an open set containing x. this gives f −1(v ) ∩ a 6= φ which in turn implies that v ∩ f (a) 6= φ and consequently y ∈ [f (a)]cl. hence f (ā) ⊂ [f (a)]cl. (b) ⇒ (c): let b be any subset of y . then f (f −1(b)) ⊆ [f (f −1(b))]cl and consequently (f −1(b)) ⊆ f −1([b]cl). (c) ⇒ (d): since a set a is cl-closed if and only if a = [a]cl, therefore the implication (c) ⇒ (d) is obvious. (d) ⇒ (e): obvious. (e) ⇒ (a): this is immediate since every clopen set is cl-open and since a function is slightly continuous if and only if the inverse image of every clopen set is open. � theorem 2.17. let x, y and z be topological spaces and let the function f : x → y be slightly continuous and g : y → z be cl-supercontinuous. then gof is continuous. proof. it is immediate in view of the above lemma and theorem 2.2. however, if f : x → y is slightly continuous and gof : x → z is continuous, the function g : y → z may not be cl-supercontinuous. � example 2.18. let x = {a, b} endowed with discrete topology. let y = {c, d}, τ = {φ, y, {c}}. let f : x → y be defined by f (a) = c, f (b) = d. let z = {e, f}, ℑ = {φ, z, {e}}. let g : y → z be defined by g(c) = e, g(d) = f . then f : x → y is slightly continuous and g ◦ f : x → z is continuous but g : y → z is not cl-supercontinuous. 3. cl-quotient topology and cl-quotient spaces let f : x → y be a surjection from a topological space x onto a set y . the quotient topology on y is the largest topology on y , which makes f continuous. analogously, the largest topology on y for which f satisfies a strong variant 298 d. singh of continuity yields a variant of quotient topology which in general is coarser than quotient topology. such variants of quotient topology are dealt with in ([3] [4] [5] [9] and [13]) and interrelations among these are outlined in [7]. in the same spirit we define cl-quotient topology on y as the finest topology on y for which f is cl-supercontinuous. in this case the map f is called a cl-quotient map. theorem 3.1. let f : x →y be a cl-quotient map. then a function g : y →z is continuous if and only if g ◦ f is cl-supercontinuous. 4. topological properties and cl-supercontinuity theorem 4.1. let f : x → y be a cl-supercontinuous open bijection. then x and y are homeomorphic zero-dimensional spaces. proof. let x ∈ x and let u be an open set containing f (x). since f is an open map, f (u ) is an open set containing f (x). in view of cl-supercontinuity of f , there exists a clopen set v containing x such that f (v ) ⊂ f (u ). this implies x ∈ f −1(f (v )) ⊂ f −1(f (u )). since f is a bijection, f −1(f (v )) = v and f −1(f (u )) = u , so x ∈ v ⊂ u . thus the space x has a base of clopen sets and so it is zero-dimensional. since zero-dimensionality is a topological property and f is a homeomorphism, y is also zero-dimensional. � definition 4.2. a function f : x → y is said to be a cl-homeomorphism if f is a bijection such that both f and f −1 are cl-supercontinuous. theorem 4.3. let f : x → y be a cl-homeomorphism from a space x onto a space y . then both x and y are homeomorphic zero-dimensional spaces. definition 4.4. a topological space x is said to be ultra-hausdorff [15] if each pair of distinct points are contained in disjoint clopen sets. theorem 4.5. let f : x → y be a cl-supercontinuous injection.if y is a t0-space, then x is ultra-hausdorff. proof. let x1 and x2 be two distinct points in x. then f (x1) 6= f (x2). since y is t0-space, there exists an open set v containing one of the points f (x1) or f (x2) but not the other. for definiteness assume that f (x1) ∈ v . since f cl-supercontinuous, in view of theorem 2.2 f −1(v ) is cl-open containing x1 but not x2. hence, there exists a clopen set u ⊂ f −1(v ) containing x1 but not x2. then u and x \ u are disjoint clopen sets containing x1 and x2 respectively. hence x is ultra-hausdorff. � definition 4.6. a space x is called mildly compact [15] if every clopen cover of x has a finite subcover. in [14] sostak calls mildly compact spaces as clustered spaces. theorem 4.7. let f : x → y be a cl-supercontinuous function from a clustered space x onto y . then y is compact. further, if y is hausdorff, then f is a cl-closed function. cl-supercontinuous functions 299 corollary 4.8. let f : x → y be a cl-supercontinuous surjection from a connected space x onto y . then y is a connected, compact space. the following result shows that there exists no nonconstant cl-supercontinuous function from a connected space into a t0-space. theorem 4.9. let f : x → y be a non constant cl-supercontinuous function. if y is a t0-space, then x is disconnected. 5. change of topology if in the domain of a cl-supercontinuous functions f , it is defined another topology in an appropriate way, then f is simply a continuous function. let (x, τ ) be a topological space, and let β denote the collection of all clopen subsets of (x, τ ). since the intersection of two clopen sets is a clopen set, the collection β is a base for a topology τ ∗ on x. clearly τ ∗ ⊂ τ . the space (x, τ ) is zero-dimensional if and only if τ ∗ = τ . throughout the section, the symbol τ ∗ will have the same meaning as in the above paragraph. theorem 5.1. a function f : (x, τ ) → (y, ℑ) is cl-supercontinuous if and only if f : (x, τ ∗) → (y, ℑ) is continuous. many of the results studied in preceding sections follow now from above theorem and the corresponding standard properties of continuous functions. theorem 5.2. let (x, τ ) be a topological space. then the following statements are equivalent. (a) (x, τ ) is zero-dimensional. (b) every continuous function from (x, τ ) into a space (y, ℑ) is cl-supercontinuous. proof. (a) ⇒ (b) is obvious. (b) ⇒ (a): take (y, ℑ) = (x, τ ). then the identity function 1x on x is continuous, and hence cl-supercontinuous. hence by theorem 5.1, 1x : (x, τ ∗) → (x, τ ) is continuous. since u ∈ τ implies 1−1x (u ) = u ∈ τ ∗, τ ⊂ τ ∗. therefore τ ∗ = τ , and so (x, τ ) is a zero-dimensional. � theorem 5.3. let f : (x, τ ) → (y, ℑ) be a function. then (a) f is slightly continuous if and only if f : (x, τ ) → (y, ℑ∗) is continuous. (b) f is cl-open if and only if f : (x, τ ∗) → (y, ℑ) is open. in the light of theorems 5.1 and 5.3 theorem 2.10 can be restated as follows. if f : (x, τ ∗) → (y, ℑ) is a continuous open surjection and g : (y, ℑ) → (z, ν) is a function, then g is continuous if and only if g ◦f is continuous and theorem 2.17 is simply the result that the composition g ◦ f of the continuous functions f : (x, τ ) → (y, ℑ∗) and g : (y, ℑ∗) → (z, ν) is continuous. moreover, cl-quotient topology on y determined by f : (x, τ ) → y in section 3 coincides with usual quotient topology on y determined by f : (x, τ ∗) → y . finally we point out that in certain situations, in contrast to continuous functions, the set l of all cl-supercontinuous functions is closed in the topology of 300 d. singh pointwise convergence (see [6], [10]). for example, if x is sum connected [2] (e.g. connected or locally connected) and y is hausdorff, then the set l(x, y ) of all cl-supercontinuous functions is closed in y x in the topology of pointwise convergence. in particular, if x is connected (or locally connected) and y is hausdorff, then the pointwise limit of a sequence of cl-supercontinuous functions is cl-supercontinuous. acknowledgements. the author would like to thank the referee for helpful suggestions. this led to an improvement of an earlier version of the paper. references [1] r. c. jain, the role of regularly open sets in general topology, ph.d. thesis, meerut univ., institute of advanced studies, meerut, india (1980). [2] j. k. kohli, a class of spaces containing all connected and all locally connected spaces, math. nachricten 82 (1978), 121–129. [3] j. k. kohli and r. kumar, z-supercontinuous functions, indian j. pure appl. math. 33 (7) (2002), 1097–1108. [4] j. k. kohli and d. singh, d-supercontinuous functions, indian j. pure appl. math. 32 (2) (2001), 227–235. [5] j. k. kohli and d. singh, dδ-supercontinuous functions, indian j. pure appl. math. 34 (7) (2003), 1089–1100. [6] j. k. kohli and d. singh, function spaces and strong variants of continuity, applied gen. top. (accepted). [7] j. k. kohli, d. singh and rajesh kumar, some properties of strongly θ-continuous functions, communicated. [8] n. levine, strongly continuity in topological spaces, amer. math. monthly 67 (1960), 269. [9] b. m. munshi and d.s. bassan, super-continuous mappings, indian j. pure appl. math. 13 (1982), 229–236. [10] s. a. naimpally, on strongly continuous functions, amer. math. monthly 74 (1967), 166–168. [11] t. noiri, supercontinuity and some strong forms of continuity, indian j. pure. appl. math. 15 (3) (1984), 241–250. [12] i. l. reilly and m. k. vamanamurthy, on super-continuous mappings, indian j. pure. appl. math. 14 (6) (1983), 767–772. [13] d. singh, d∗-supercontinuous functions, bull. cal. math. soc. 94 (2) (2002), 67–76. [14] a. sostak, on a class of topological spaces containing all bicompact and connected spaces, general topology and its relation to modern analysis and algebra iv: proceedings of the 4th prague topological symposium, (1976), part b, 445–451. [15] r. staum, the algebra of bounded continuous functions into a nonarchimedean field, pac. j. math. 50 (1) (1974), 169–185. [16] l. a. steen and j. a. seeback, jr., counter examples in topology, springer verlag, new york, 1978. received may 2006 accepted february 2007 d. singh (dstopology@rediffmail.com) department of mathematics, sri aurobindo college, university of delhi – south campus, delhi 110 017, india gariveraagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 165-170 extension of compact operators from df-spaces to c(k) spaces fernando garibay bonales and rigoberto vera mendoza abstract. it is proved that every compact operator from a dfspace, closed subspace of another df-space, into the space c(k) of continuous functions on a compact hausdorff space k can be extended to a compact operator of the total df-space. 2000 ams classification: primary 46a04, 46a20; secondary 46b25. keywords: topological vector spaces, df-spaces and c(k) the spaces. 1. introduction let e and x be topological vector spaces with e a closed subspace of x. we are interested in finding out when a continuous operator t : e → c(k) has an extension t̃ : x → c(k), where c(k) is the space of continuous real functions on a compact hausdorff space k and c(k) has the norm of the supremum. when this is the case we will say that (e,x) has the extension property. several advances have been made in this direction, a basic resume and bibliography for this problem can be found in [5]. in this work we will focus in the case when the operator t is a compact operator. in [4], p.23 , it is proved that (e,x) has the extension property when e and x are banach spaces and t : e → c(k) is a compact operator. in this paper we extend this result to the case when e and x are df-spaces (to be defined below), for this, we use basic tools from topological vector spaces. 2. notation and basic results in df-spaces. we will use basic duality theory of topological vector spaces. for concepts in topological vector spaces see [3] or [2]. all the topological vector spaces in this work are hausdorff and locally convex. let (x,t) be a topological vector space and e < x be a closed vector subspace. let x′ = (x,t)′, e′ = (e,t)′ be the topological duals of x and e respectively. 166 f. garibay and r. vera a topological vector space (x, t) possesses a a fundamental sequence of bounded sets if there exists a sequence b1 ⊂ b2 ⊂ · · · of bounded sets in (x, t), such that every bounded set b is contained in some bk. we take the following definition from [3], p. 396. definition 2.1. a locally convex topological vector space (x,t) is said to be a df-space if (1) it has a fundamental sequence of bounded sets, and (2) every strongly bounded subset m of x′ which is the union of countably many equicontinuous sets is also equicontinuous a quasi-barrelled locally convex topological vector space with a fundamental sequence of bounded set is always a df-space. thus every normed space is a df-space. later we will mention topological vector spaces which are df-spaces but they are not normed spaces. first, we state some theorems to be used in the proof of the main result. if k is a compact hausdorff topological space, we define, for each k ∈ k the injective evaluation map k̂ : c(k) → r , k̂(f) = f(k) which is linear and continuous, that is k̂ ∈ c(k)′. let k̂ = {k̂ |k ∈ k} ⊂ c(k)′ and cch(k̂) the balanced, closed and convex hull of k̂ (which is bounded). theorem 2.2. with the notation above we have (1) k̂ is σ(c(k)′,c(k))-compact and k is homeomorphic to (k̂,σ(c(k)′,c(k)) . here σ(c(k)′,c(k)) denotes the weak-* topology on c(k)′. (2) if t : e → c(k) is a compact operator then a = t ′(cch(k̂)) β is β(e′,e)-compact. here β(e′,e) is the strong topology on e′, this topology is generated by the polars sets of all bounded sets of (e, t). proof. see [1], p. 490. � theorem 2.3. if (x, t) is a df-space then (x′, β(x′,x) ) is a frechet space. proof. see [3], p. 397 � theorem 2.4. let m be paracompact, z a banach space, n ⊂ z convex and closed, and ϕ : m → f(n) lower semicontinuous (l.s.c.) then ϕ has a selection. proof. see [6] � in the above theorem, f(n) = {s ⊂ n : s 6= ∅, s closed in n and convex}; ϕ : m → f(n) is l.s.c. if {m ∈ m : ϕ(m)∩v 6= ∅} is open in m for every open v in n, and f : m → n is a selection for ϕ if f is continuous and f(m) ∈ ϕ(m) for every m ∈ m. theorem above remains true if z is only a complete, metrizable, locally convex topological vector space (see [7]). extension of compact operators 167 3. main results lemma 3.1. let a ⊂ e′. if there is a continuous map f : (a,σ(e′,e)) → (x′,τ′), σ(x′,x) ≤ τ′ ≤ β(x′,x) such that (1) f(a)|e = a and (2) f(a) is an equicontinuous subset of x′. then every linear and continuous map t : e → c(k) has a linear and continuous extension t̃ : x → c(k). proof. let us define t̃ : x → c(k) in the following way: for each x ∈ x , t̃(x) : k → r is given by t̃(x)(k) = f(t ′(k̂))(x). here, k̂ is the injective evaluation map defined before theorem 2.2. it is easy to check that t̃ is linear and extends t . first, let us show that t̃(x) ∈ c(k) for each x ∈ x. for this let o ⊂ r be an open set. we have that t̃(x)−1(o) = t ′−1(f−1(x−1(o) ) ). since x : x′[σ(x′,x)] → r , f and t ′ are all continuous maps with the weak∗ topology, t̃(x)−1(o) is open in k. this proves that t̃(x) ∈ c(k). let us check that t̃ is continuous. let {xλ}λ t → 0 in x, we need to show that {t̃(xλ)} ||·||c(k) −→ 0 . for this, let ǫ > 0. by hypothesis f(a) is a equicontinuous subset of x′, so that, ǫf(a)◦ ⊂ x is a t-neighborhood of 0. here f(a)◦ denotes de polar set of f(a). hence, there is λ0 ∈ λ such that xλ ∈ ǫf(a) ◦ for all λ ≥ λ0. from part 2 of theorem 2.2 we have t ′(k̂) ⊂ a, hence |t̃(xλ)(k̂)| = |f(t ′(k̂))(xλ)| ≤ ǫ for all λ ≥ λ0 this implies that ||t̃(xλ)||c(k) = sup{ |f(t ′(k̂))(xλ)|/k ∈ k} ≤ ǫ for all λ ≥ λ0 this proves that {t̃(xλ)} ||·||c(k) −→ 0 . � let i : e → x be the inclusion map and i′ : x′ → e′ the dual map of i, that is, if y ∈ x′, i′(y) = y|e . let p(x′) = {y | y 6= ∅, y ⊂ x′} and define ψ : e′ → p(x′) by ψ(e′) = { extensions of e′ to x}. notice that y ∈ ψ(i′(y)) for all y ∈ x′ and ψ(e′) ∈ f(x′). with this notation, we have proposition 3.2. let (e, t) and (x, t) be df-spaces, with e < x a closed subspace. if o ⊂ x′ is a β(x′, x)-open set then the set uo = {z ∈ e ′ |ψ(z) ∩ o 6= ∅} is an open set in (e′, β(e′, e) ). proof. notice that uo = {z ∈ e ′ | ψ(z) ∩ o 6= ∅} = i′(o). by theorem 2.3 (x′,β(x′, x) ) and (e′,β(e′, e) ) are frechet spaces. by the banach-schauder theorem (see [3], p. 166), the map i′ : (x′, β(x′, x) ) → (e′, β(e′, e) ) is an open map. since i′(o) is open in e′, uo is also open. � 168 f. garibay and r. vera corollary 3.3. let (e, t) and (x, t) be df-spaces, with e < x a closed subspace. let a = t ′(cch(k̂)) β be as in part 2 of theorem (2.2) then ϕ : (a,β(e′,e)) → p(x′) given by ϕ = ψ|a is a lower semicontinuous function, x′ provided with the strong topology β(x′, x). proof. it follows from {z ∈ a | ϕ(z) ∩ o 6= ∅} = {z ∈ e′ | ψ(z) ∩ o 6= ∅} ∩ a and proposition 3.2. � with the notation in corollary 3.3, we have proposition 3.4. if (x,t) is a df-space then ϕ : (a,β(e′,e)) → p(x′) admits a selection, that is, there is a continuous function f : (a,β(e′,e)) → (x′,β(x′,x)) such that f(a) ∈ ϕ(a). proof. from theorem 2.3, (x,t) df-space implies (x′,β(x′,x)) frechet. from theorem 2.2, part 2, a is β(e′,e)-compact, hence a is a paracompact set. by corollary 3.3, ϕ is a lower semi continuous function, therefore, by theorem 2.4, ϕ admits a selection. � theorem 3.5. if (x,t) and the closed subspace e are df-spaces then every compact operator t : e → c(k) has a compact extension t̃ : x → c(k). proof. let a be as in proposition 3.4 and f : (a,β(e′,e)) → (x′,β(x′,x)) a selection function. since a is β(e′,e)-compact and f is continuous, f(a) is compact, hence f(a) is an equicontinuous set. let t̃ be the linear extension of t given in lemma 3.1. let us prove that t̃ is a compact operator. for this, we need to show that there is a t-neighborhood v such that t̃(v ) is a relatively compact set. since f(a) ⊂ x′ is an equicontinuous set and x is a df space, [2] (p. 260 and p. 214) tells us that there is v ⊂ x a balanced, closed and convex t-zero-neighborhood such that f(a) ⊂ v ◦ and the topologies β(x′, x) and ρv ◦ coincide on f(a). here ρv ◦ is the minkowski functional of v ◦. in this case ρv ◦ is a norm and (x ′ v ◦, ρv ◦ ) is a banach space. by using the arzela-ascoli theorem, we will show that t̃(v ) ⊂ c(k) is relatively compact. first, t̃ (v ) is pointwise bounded because, for each x ∈ v and k ∈ k , |t̃(x)(k)| = |f(t ′(k̂))(x)| ≤ 1 since f(a) ⊂ v ◦. now let us prove that t̃(v ) is equicontinuous in c(k). choose and fix k0 ∈ k and ǫ > 0. since the chain of functions k−̂→k̂ t ′ −→ (a,β(x′,x)) f −→ (f(a),β(x′,x)) is continuous, given a β-neighborhood w of f(t ′(k̂0)) on f(a) , there exists o ⊂ k neighborhood of k0 such that k ∈ o ⇒ f(t ′(k̂)) ∈ w . since ρv ◦|f (a) = β(x ′, x)|f (a), we can say that k ∈ o ⇒ ρv ◦ ( f(t ′(k̂)) − f(t ′(k̂0)) ) < ǫ extension of compact operators 169 for each x ∈ x , x : (x′v ◦, ρv ◦ ) → r is linear and continuous, moreover, |x′(x)| ≤ ||x||ρv ◦ ρv ◦ (x ′) for all x′ ∈ x′ , where ||x||ρv ◦ = sup{|x ′(x)| | x′ ∈ v ◦} if x ∈ v , ||x||ρv ◦ ≤ 1. therefore, for every k ∈ o and every x ∈ v ∣ ∣ ∣ (f(t ′(k̂)) − f(t ′(k̂0))(x) ∣ ∣ ∣ ≤ ||x||ρv ◦ ρv ◦ ( f(t ′(k̂)) − f(t ′(k̂0)) ) ≤ (1)(ǫ) this proves that t̃(v ) is equicontinuous in c(k) and, by the arzela-ascoli theorem, t̃(v ) is relatively compact which means that t̃ is a compact operator. � in [3] (p. 402) it is shown that the topological inductive limit of a sequence of df-spaces is a df-space. in particular, if (en) is a sequence of banach spaces such that en is a proper subspace of en+1, its inductive limit is dfspace. this inductive limit is not metrizable (see [8] p. 291). for this kind of spaces, theorem 3.5 can be applied, i.e., given a fixed n, a compact operator t : en → c(k) can be extended to a compact operator of the inductive limit. acknowledgements. the research of the authors was supported by the coordinación de la investigación cient́ıfica de la umsnh. references [1] n. dunford and j. schwartz, linear operators, vol. i., wiley interscience, new york, 1957. [2] h. jarchow, locally convex spaces, b. g. teubener stuttgart, 1981. [3] g. kothe, topological vector spaces, vol. i., springer verlag, new york, 1969. [4] w. b. johnson and j. lindenstrauss, basic concepts in the geometry of banach spaces. preprint. ( 2001). [5] w. b. johnson and m. zippin, extension of operators from weak*-closed subspaces of l1 to c(k), studia mathematica 117 (1) (1995), 43–45. [6] e. michael, continuous selections i, annals of mathematics 63 (2) (1956), 361–382. [7] e. michael, some problems, open problems in topology, elsevier, amsterdam, 1990. [8] l. narici and e. beckenstein, topological vector spaces, marcel dekker, inc., new york, 1985. received november 2004 accepted june 2005 170 f. garibay and r. vera f. garibay bonales (fgaribay@zeus.umich.mx) facultad de ingenieŕıa qúımica, universidad michoacana de san nicolás de hidalgo, morelia, michoacán 58060, méxico. r. vera mendoza (rvera@zeus.umich.mx) facultad de f́ısico-matemáticas, universidad michoacana de san nicolás de hidalgo, morelia, michoacán 58060, méxico. boumikurichagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 2, 2009 pp. 173-186 convergence semigroup actions: generalized quotients h. boustique, p. mikusiński and g. richardson abstract. continuous actions of a convergence semigroup are investigated in the category of convergence spaces. invariance properties of actions as well as properties of a generalized quotient space are presented 2000 ams classification: 54a20, 54b15. keywords: continuous action, convergence space, quotient map, semigroup. 1. introduction the notion of a topological group acting continuously on a topological space has been the subject of numerous research articles. park [8, 9] and rath [10] studied these concepts in the larger category of convergence spaces. this is a more natural category to work in since the homeomorphism group on a space can be equipped with a coarsest convergence structure making the group operations continuous. moreover, unlike in the topological context, quotient maps are productive in the category of all convergence spaces with continuous maps as morphisms. this property plays a key role in the proof of several results contained herein; for example, theorem 4.11. given a topological semigroup acting on a topological space, burzyk et al. [1] introduced a ”generalized quotient space.” elements of this space are equivalence classes determined by an abstraction of the method used to construct the rationals from the integers. general quotient spaces are used in the study of generalized functions [5, 6, 7]. generalized quotients in the category of convergence spaces are studied in section 4. first, invariance properties of continuous actions of convergence semigroups on convergence spaces are investigated in section 3. 174 h. boustique, p. mikusiński and g. richardson 2. preliminaries basic definitions and concepts needed in the area of convergence spaces are given in this section. let x be a set, 2x the power set of x, and let f(x) denote the set of all filters on x. recall that b ⊆ 2x is a base for a filter on x provided b 6= ∅, ∅ /∈ b, and b1,b2 ∈ b implies that there exists b3 ∈ b such that b3 ⊆ b1 ∩ b2. moreover, [b] denotes the filter on x whose base is b; that is, [b] = {a ⊆ x : b ⊆ a for some b ∈ b}. fix x ∈ x, define ẋ to be the filter whose base is b = {{x}}. if f : x → y and f ∈ f(x), then f→f denotes the image filter on y whose base is {f(f) : f ∈ f}. a convergence structure on x is a function q : f(x) → 2x obeying : (cs1) x ∈ q(ẋ) for each x ∈ x (cs2) x ∈ q(f) implies that x ∈ q(g) whenever f ⊆ g. the pair (x,q) is called a convergence space. the more intuitive notation f q → x is used for x ∈ q(f). a map f : (x,q) → (y,p) between two convergence spaces is called continuous whenever f q → x implies that f→f p → f(x). let conv denote the category whose objects consist of all the convergence spaces, and whose morphisms are all the continuous maps between objects. the collection of all objects in conv is denoted by |conv|. if p and q are two convergence structures on x, then p ≤ q means that f p → x whenever f q → x. in this case, p(q) is said to be coarser(finer) than q(p), respectively. also, for f,g ∈ f(x), f ≤ g means that f ⊆ g, and f(g) is called coarser(finer) than g(f), respectively. it is well-known that conv possesses initial and final convergence structures. in particular, if (xj,qj ) ∈ |conv| for each j ∈ j, then the product convergence structure r on x = × j∈j xj is given by h r → x = (xj ) iff π→j h qj → xj for each j ∈ j, where πj denotes the j th projection map. also, if f : (x,q) → y is a surjection, then the quotient convergence structure σ on y is given by h σ → y iff there exists x ∈ f−1(y) and f q → x such that f→f = h. in this case, σ is the finest convergence structure on y making f : (x,q) → (y,σ) continuous. unlike the category of all topological spaces, conv is cartesian closed and thus has suitable function spaces. in particular, let (x,q), (y,p) ∈ |conv| and let c(x,y ) denote the set of all continuous functions from x to y . define ω : (x,q)×c(x,y ) → (y,p) to be the evaluation map given by ω(x,f) = f(x). there exists a coarsest convergence structure c on c(x,y ) such that w is jointly continuous. more precisely, c is defined by : φ c → f iff w→(f × φ) p → f(x) whenever f q → x. this compatibility between (x,q) and (c(x,y ),c) is an example of a continuous action in conv discussed in section 3. continuous actions which are invariant with respect to a convergence space property p are studied in section 3. choices for p include : locally compact, locally bounded, regular, choquet(pseudotopological), and first-countable. convergence semigroup actions: generalized quotients 175 an object (x,q) ∈ |conv| is said to be locally compact (locally bounded) if f q → x implies that f contains a compact (bounded) subset of x, respectively. a subset b of x is bounded provided that each ultrafilter containing b q-converges in x. further, (x,q) is called regular (choquet) provided clqf q → x (f q → x) whenever f q → x (each ultrafilter containing f q-converges to x), respectively. here clqf denotes the filter on x whose base is {clqf : f ∈ f}. some authors use the term ”pseudotopological space” for a choquet space. finally, (x,q) is said to be first-countable whenever f q → x implies the existence of a coarser filter on x having a countable base and q-converging to x. let sg denote the category whose objects consist of all the semigroups (with an identity element), and whose morphisms are all the homomorphisms between objects. further, (s,.,p) is said to be a convergence semigroup provided : (s,.) ∈ |sg|, (s,p) ∈ |conv|, and γ : (s,p) × (s,p) → (s,p) is continuous, where γ(x,y) = x.y. let csg be the category whose objects consist of all the convergence semigroups, and whose morphisms are all the continuous homomorphisms between objects. 3. continuous actions an action of a semigroups on a topological space is used to define ”generalized quotients” in [1]. below is rath’s [10] definition of an action in the convergence space context. let (x,q) ∈ |conv|, (s,.,p) ∈ |csg|, λ : x × s → x, and consider the following conditions : (a1) λ(x,e) = x for each x ∈ x (e is the identity element) (a2) λ(λ(x,g),h) = λ(x,g.h) for each x ∈ x, g,h ∈ s (a3) λ : (x,q) × (s,.,p) → (x,q) is continuous. then (s,.)((s,.,p)) is said to act(act continuously) on (x,q) whenever a1a2 (a1-a3) are satisfied and, in this case, λ is called the action (continuous action), respectively. for sake of brevity, (x,s) ∈ a(ac) denotes the fact that (s,.,p) ∈ |csg|) acts (acts continuously) on (x,q) ∈ |conv|, respectively. moreover, (x, s, λ) ∈ a indicates that the action is λ. the notion of ”generalized quotients” determined by commutative semigroup acting on a topological space is investigated in [1]. elements of the semigroup in [1] are assumed to be injections on the given topological space. lemma 3.1 ([1]). suppose that (s,x,λ) ∈ a, (s,.) is commutative and λ(.,g) : x → x is an injection, for each g ∈ s. define (x,g) ∼ (y,h) on x × s iff λ(x,h) = λ(y,g). then ∼ is an equivalence relation on x × s. in the context of lemma 3.1, let 〈(x,g)〉 be the equivalence class containing (x,g), b(x, s) denote the quotient set (x×s)/ ∼, and define ϕ : (x×s,r) → b(x,s) to be the canonical map, where r = q × p is the product convergence structure. equip b(x,s) with the convergence quotient structure σ. then 176 h. boustique, p. mikusiński and g. richardson k σ → 〈(y,h)〉 iff there exist (x,g) ∼ (y,h) and h r → (x,g) such that ϕ→h = k. the space (b(x,s),σ) is investigated in section 4. remark 3.2. fix a set x. the set of all convergence structures on x with the ordering p ≤ q defined in section 2 is a complete lattice. indeed, if (x,qj ) ∈ |conv|, j ∈ j, then sup j∈j qj = q 1 is given by f q 1 → x iff f qj → x, for each j ∈ j. dually, inf j∈j qj = q 0 is defined by f q 0 → x iff f qj → x, for some j ∈ j. it is easily verified that if ((x,qj ), (s,.,p),λ) ∈ ac for each j ∈ j, then both ((x,q1), (s,.,p),λ) and ((x,q0), (s,.,p),λ) belong to ac. theorem 3.3. assume that ((x,q), (s,.,p),λ) ∈ ac. then (a) there exists a finest convergence structure qf on x such that ((x,qf ), (s,.,p),λ) ∈ ac (b) there exists a coarsest convergence structure pc on s for which ((x,q), (s,.,pc),λ) ∈ ac (c) ((b(x,s),σ), (s,.,p)) ∈ ac provided (s,.) is commutative and λ(.,g) is an injection, for each g ∈ s. proof. (a): define qf as follows: f q f → x iff there exist z ∈ x, g p → g such that x = λ(z,g) and f ≥ λ→(ż × g). then (x,qf ) ∈ |conv|. indeed, ẋ q f → x since x = λ(x,e) and ẋ = λ→(ẋ × ė). hence (cs1) is satisfied. clearly (cs2) is valid, and (x,qf ) ∈ |conv|. it is shown that λ : (x,qf ) × (s,p) → (x,qf ) is continuous. suppose that f q f → x and h p → h; then there exist z ∈ x, g p → g such that x = λ(z,g) and f ≥ λ→(ż ×g). hence, f ×h ≥ λ→(ż ×g) ×h, and employing (a2), λ→(f × h) ≥ λ→(λ→(ż × g) × h) = [{λ({z} ×g.h) : g ∈ g,h ∈ h}] = λ→(ż × g.h). since g.h p → g.h and λ(z,g.h) = λ(λ(z,g),h) = λ(x,h), it follows from the definition of qf that λ→(f × h) q f → λ(x,h). hence ((x,qf ), (s,.,p),λ) ∈ ac. assume that ((x,s), (s,.,p),λ) ∈ ac. it is shown that s ≤ qf . suppose that f q f → x; then there exist z ∈ x, g p → g such that x = λ(z,g) and f ≥ λ→(ż×g). since λ→(ż × g) s → λ(z,g), it follows that f s → x and thus s ≤ qf . hence qf is the finest convergence structure on x such that ((x,qf ), (s,.,p),λ) ∈ ac. (b): define pc as follows: g p c → g iff for each f q → x, λ→(f × g) q → λ(x,g). then (s,pc) ∈ |conv|. first, it is shown that (s,.,pc) ∈ |csg|; that is, if g p c → g and h p c → h, then g.h p c → g.h. assume that f q → x; then using (a2), λ→(f × g.h) = [{λ(f × g.h) : f ∈ f,g ∈ g,h ∈ h}] = [{λ(λ(f × g) × h) : f ∈ f,g ∈ g,h ∈ h}] = λ→(λ→(f × g) × h). it follows from the definition of pc that λ→(f × g) q → λ(x,g), and thus λ→(λ→(f × g) × h) q → λ(λ(x,g),h) = λ(x,g.h). hence g.h p c → g.h, and convergence semigroup actions: generalized quotients 177 thus (s,.,pc) ∈ |csg|. according to the construction, pc is the coarsest convergence structure on s such that λ : (x,q) × (s,pc) → (x,q) is continuous. (c): define λb : (b(x,s),σ) × (s,.,p) → (b(x,s),σ) by λb (〈(x,g)〉,h) = 〈(x,g.h)〉. it is shown that λb is a continuous action. indeed, λb(〈(x,g)〉,e) = 〈(x,g)〉, and λb (λb (〈(x,g)〉,h),k) = λb (〈(x,g.h)〉,k) = 〈(x,g.h.k)〉 = λb(〈(x,g)〉,h.k). hence λb is an action. it remains to show that λb is continuous. suppose that k σ → 〈(x,g)〉 and l p → l. since ϕ is a quotient map in conv, there exists h r → (x1,g1) ∼ (x,g) such that ϕ →h = k. then λ→b (k × l) = λ → b (ϕ →h × l). let k ∈ k and l ∈ l, and note that λb (ϕ(h) × l) ⊆ λb(ϕ(π1(h) × π2(h)) × l) = ϕ(π1(h) × π2(h).l). hence λ→b (ϕ →h × l) ≥ ϕ→(π→1 h × π → 2 h.l) σ → ϕ(x1,g1.l) = 〈(x1,g1.l)〉 = λb(〈(x1,g1)〉, l) = λb (〈(x,g)〉, l). therefore (b(x,s),s,λb ) ∈ ac. � remark 3.4. let (x,q) ∈ |conv| and let (c(x,x),c) denote the space defined in section 2. since c is the coarsest convergence structure for which the evaluation map ω : (x,q) × (c(x,x),c) → (x,q) is continuous, this is a particular case of theorem 3.3(b), where λ = ω, (s,.,pc) = (c(x,x), .,c), and the group operation is composition. moreover, it is well-known that, in general, there fails to exist a coarsest topology on c(x,x) for which ω : (x,q)× c(x,x) → (x,q) is jointly continuous (even when q is a topology). assume that (x,s,λ) ∈ a; then λ is said to distinguish elements in s whenever λ(x,g) = λ(x,h) for all x ∈ x implies that g = h. in this case, define θ : s → c(x,x) by θ(g)(x) = λ(x,g), for each x ∈ x. note that θ is an injection iff λ separates elements in s. moreover, θ is a homomorphism whenever the operation in c(x,x) is k.l = l ◦ k is composition. theorem 3.5. suppose that ((x,q), (s,.,p),λ) ∈ ac, and assume that λ distinguishes elements in s. then the following are equivalent: (a) θ : (s,p) → (c(x,x),c) is an embedding (b) p = pc (c) if g p 6→ g, then there exists f q → x such that λ→(f × g) q 6→ λ(x,g). proof. (a) ⇒ (b): assume that θ : (s,p) → (c(x,x),c) is an embedding. according to theorem 3.3(b), pc ≤ p. suppose that g p c → g; then if f q → x, λ→(f × g) q → λ(x,g). it is shown that θ→g c → θ(g). indeed, note that ω→(f × θ→g) = [{ω(f × θ(g)) : f ∈ f,g ∈ g}] = [{λ(f × g) : f ∈ f,g ∈ g}] = λ→(f ×g) q → λ(x,g) = ω(x,θ(g)). hence θ→g c → θ(g), and thus g p → g. therefore p = pc. (b) ⇒ (c): verification follows directly from the definition of pc. (c) ⇒ (a): suppose that g p → g and f q → x. since λ : (x,q) × (s,p) → (x,q) is continuous, λ→(f × g) q → λ(x,g). hence ω→(f × θ→g) = λ→(f × g) q → 178 h. boustique, p. mikusiński and g. richardson λ(x,g) = ω(x,θ(g)), and thus θ→g c → θ(g). conversely, if g ∈ f(s) such that θ→g c → θ(g), then the hypothesis implies that g p → g. hence θ : (s,p) → (c(x,x),c) is an embedding. � remark 3.6. the map θ given in theorem 3.5 is called a continuous representation of (s,.,p) on (x,q). rath [10] discusses this concept in the context of a group with (c(x,x), .,c) replaced by (h(x), .,γ), where (h(x), .) is the group of all homeomorphisms on x with composition as the group operation, and γ is the coarsest convergence structure making the operations of composition and inversion continuous. quite often it is desirable to consider modifications of convergence structures. for example, given (x,q) ∈ |conv|, there exists a finest regular convergence structure on x which is coarser than q [4]. the notation p q denotes the p-modification of q. generally, p represents a convergence space property; however, it is convenient to include the case whenever pq = q. let pconv denote the full subcategory of conv consisting of all the objects in conv that satisfy condition p . condition p is said to be finitely productive(productive) provided that for each collection (xj,qj ) ∈ |conv|, j ∈ j, p( × j∈j qj ) = × j∈j pqj whenever j is a finite (arbitrary) set, respectively. theorem 3.7. assume that fp : conv → pconv is a functor obeying fp (x,q) = (x,pq), fp (f) = f, and suppose that p is finitely productive. if ((x,q), (s,.,p),λ) ∈ ac and h : (t,.,ξ) → (s,.,p) is a continuous homomorphism in csg, then ((x,pq), (t,.,pξ)) ∈ ac; in particular, ((x,pq), (s,.,pp),λ) ∈ ac. proof. given that ((x,q), (s,.,p),λ) ∈ ac, define λ : (x,q) × (t,ξ) → (x,q) by λ(x,t) = λ(x,h(t)). clearly λ is an action; moreover, λ is continuous. indeed, suppose that f q → x and g ξ → t; then λ→(f × g) = [{λ(f × g) : f ∈ f,g ∈ g}] = [{λ(f × h(g)) : f ∈ f,g ∈ g}] = λ→(f × h→g) q → λ(x,h(t)) = λ(x,t). therefore λ is continuous. since fp is a functor and p is finitely productive, continuity of the operation γ : (t,.,ξ) × (t,.,ξ) → (t,.,ξ), defined by γ(t1, t2) = t1.t2, implies continuity of γ : (t,.,pξ)×(t,.,pξ) → (t,.,pξ). hence (t,.,pξ) ∈ |csg|. likewise, λ : (x,pq) × (t,pξ) → (x,pq) is continuous, and thus ((x,pq), (t,.,pξ), λ) ∈ ac. � let (sj, .,pj ) ∈ |csg|, j ∈ j, and denote the product by (s,.,p) = × j∈j (sj, .,pj ). the direct sum of (sj, .), j ∈ j, is the subsemigroup of (s,.) defined by ⊕j∈jsj = {(gj) ∈ s : gj = ej for all but finitely many j ∈ j}. denote θj : sj → ⊕j∈jsj to be the map θj (g) = (gk), where gj = g and gk = ek whenever k 6= j, and let θ : ⊕j∈jsj → × j∈j sj be the inclusion map. define h η → (gj ) in ⊕j∈jsj iff h ≥ θ → k1 g1.θ → k2 g2...θ → kn gn, where gj pkj → gkj in (skj , .,pkj ) and convergence semigroup actions: generalized quotients 179 n ≥ 1. then (⊕j∈jsj, .,η) ∈ |csg|, and θ : (⊕j∈jsj, .,η) → (s,.,p) is a continuous homomorphism. theorem 3.8. suppose that fp : conv → pconv is a functor satisfying fp (x,q) = (x,pq), fp (f) = f, and p is productive. assume that ((xj,qj ), (sj, .,pj ),λj ) ∈ ac for each j ∈ j. then (a) ( × j∈j (xj,pqj ), × j∈j (sj, .,ppj )) ∈ ac (b) ( × j∈j (xj,pqj ), (⊕j∈jsj, .,pη)) ∈ ac. proof. (a): denote (x,q) = × j∈j (xj,qj ), (s,.,p) = × j∈j (sj, .,pj ), and define λ : (x,q) × (s,p) → (x,q) by λ((xj ), (gj )) = (λj (xj,gj)). clearly λ is an action. then, according to theorem 3.7 and the assumption that p is productive, it suffices to show that ((x,q), (s,p),λ) ∈ ac. the latter follows from a routine argument, and thus ( × j∈j (xj,pqj ), × j∈j (sj, .,ppj ),λ) ∈ ac. (b): since θ : (⊕sj, .,η) → (s,.,p) is a continuous homomorphism in csg and p is productive, it follows from theorem 3.7 that ( × j∈j (xj,pqj), (⊕sj, .,pη)) ∈ ac. � corollary 3.9. assume that fp : conv → pconv is a functor satisfying fp (x,q) = (x,pq), fp (f) = f, and p is finitely productive. suppose that ((xj,qj ), (sj, .,pj )) ∈ ac for each j ∈ j. denote (x,q) = × j∈j (xj,qj ) and (s,.,p) = × j∈j (sj, .,pj). then (a) ((x,pq), (s,.,pp)) ∈ ac (b) ((x,pq), (⊕j∈jsj, .,pη)) ∈ ac. verification of corollary 3.9 follows the proof of theorem 3.8 with the exception that since p is only finitely productive, (x,pq) and × j∈j (xj,pqj), as well as (s,.,pp) and × j∈j (sj, .,ppj ), may differ. of course equality holds whenever the index set is finite. choices of p that are finitely productive, and preserve continuity when taking p-modifications include: locally compact, locally bounded, regular, and first-countable. the property of being choquet is productive, and continuity is preserved under taking choquet modifications. 4. generalized quotients recall that if ((x,q), (s,.,p),λ) ∈ ac, (s,.) is commutative, λ(.,g) is an injection, then by lemma 3.1, (x,g) ∼ (y,h) iff λ(x,h) = λ(y,g) is an equivalence relation. denote r = {((x,g), (y,h)) : (x,g) ∼ (y,h)}, r = q × p, and ϕ : (x × s,r) → ((x × s)/ ∼,σ) the convergence quotient map defined by ϕ(x,g) = 〈(x,g)〉. then (b(x, s), σ):= ((x × s)/ ∼,σ) is called the generalized quotient space. convergence space properties of (b(x,s),σ) are 180 h. boustique, p. mikusiński and g. richardson investigated in this section. for ease of exposition, ((x,q), (s,.,p),λ) ∈ gq denotes that ((x,q), (s,.,p),λ) ∈ ac, (s,.) is commutative, and λ(.,g) is an injection, for each g ∈ s. the generalized quotient space (b(x,s),σ) exists whenever ((x,q), (s,.,p),λ) ∈ gq. theorem 4.1. assume that ((x,q), (s,.,p),λ) ∈ gq. then the following are equivalent: (a) (x,q) is hausdorff (b) r is closed in ((x × s) × (x × s),r × r) (c) (b(x,s),σ) is hausdorff. proof. (a) ⇒ (b): let πij denote the projection map defined by : πij : (x × s) × (x × s) → x × s where πij (((x,g), (y,h))) = (x,g) when i,j = 1, 2 and πij (((x,g), (y,h))) = (y,h) when i,j = 3, 4. suppose that h r×r → ((x,g), (y,h)) and r ∈ h. let h ∈ h; then h∩r 6= ∅, and thus there exists ((x1,g1), (y1,h1)) ∈ h ∩ r. hence λ(x1,h1) = λ(y1,g1), and consequently λ((π1 ◦ π12)(h) × (π2 ◦ π34)(h)) ∩ λ((π1 ◦ π34)(h) × (π2 ◦ π12)(h)) 6= ∅, for each h ∈ h. it follows that k := λ→((π1 ◦π12) →h×(π2 ◦π34) →h)∨λ→((π1 ◦π34) →h×(π2 ◦π12) →h) exists. however, (π1 ◦ π12) →h q → x, (π2 ◦ π34) →h p → h, (π1 ◦ π34) →h q → y, (π2 ◦ π12) →h p → g, and thus k q → λ(x,h),λ(y,g). since (x,q) is hausdorff, λ(x,h) = λ(y,g) and thus (x,g) ∼ (y,h). therefore, ((x,g), (y,h)) ∈ r, and thus r is closed. (b) ⇒ (c): assume that k σ → 〈(yi,hi)〉, i = 1, 2. since ϕ : (x × s,r) → (b(x,s),σ) is a quotient map in conv, there exist (xi,gi) ∼ (yi,hi) and hi r → (xi,gi) such that ϕ →hi = k, i = 1, 2. then for each hi ∈ hi, ϕ(h1) ∩ ϕ(h2) 6= ∅ and thus there exists (si, ti) ∈ hi such that (s1, t1) ∼ (s2, t2), i = 1, 2. hence the least upper bound filter l := (h1 × h2) ∨ ṙ exists, and l r×r → ((x1,g1), (x2,g2)). since r is closed, (x1,g1) ∼ (x2,g2) and thus 〈(y1,h1)〉 = 〈(y2,h2)〉. therefore (b(x,s),σ) is hausdorff. (c) ⇒ (a): suppose that (b(x,s),σ) is hausdorff and f q → x,y. then ϕ→(f ×ė) σ → 〈(x,e)〉,〈(y,e)〉, and thus (x,e) ∼ (y,e). therefore, x = λ(x,e) = λ(y,e) = y, and thus (x,q) is hausdorff. � conditions for which (b(x,s),σ) is t1 are given below. in the topological setting, sufficient conditions in order for the generalized quotient space to be t2 are given in [1] whenever (s,.) is equipped with the discrete topology. theorem 4.2. suppose that ((x,q), (s,.,p),λ) ∈ gq. then (b(x,s),σ) is t1 iff ϕ −1(〈(y,h)〉) is closed in (x × s,r), for each (y,h) ∈ x × s. proof. the ”only if” is clear since {〈(y,h)〉} is closed and ϕ is continuous. conversely, assume that ϕ−1(〈(y,h)〉) is closed, for each (y,h) ∈ x × s, and convergence semigroup actions: generalized quotients 181 suppose that ˙〈(x,g)〉 σ → 〈(y,h)〉. since ϕ is a quotient map in conv, there exist (s,t) ∼ (y,h) and h r → (s,t) such that ϕ→h = ˙〈(x,g)〉. then ϕ−1(〈(x,g)〉) ∈ h, and thus (s,t) ∈ clrϕ −1(〈(x,g)〉) = ϕ−1(〈(x,g)〉). hence (x,g) ∼ (s,t) ∼ (y,h), and thus 〈(x,g)〉 = 〈(y,h)〉. therefore (b(x,s),σ) is t1. � corollary 4.3. assume that ((x,q), (s,.,p),λ) ∈ gq, and let p denote the discrete topology. then (b(x,s),σ) is t1 iff (x,q) is t1. proof. suppose that (b(x,s),σ) is t1 and ẋ q → y. then ˙(x,e) r → (y,e), and thus ˙〈(x,e)〉 = ϕ→( ˙(x,e)) σ → 〈(y,e)〉. it follows that 〈(x,e)〉 = 〈(y,e)〉 and hence x = y. therefore (x,q) is t1. conversely, assume that (x,q) is t1 and (y,h) ∈ clrϕ −1(〈(x,g)〉). then there exists h r → (y,h) such that ϕ−1(〈(x,g)〉) ∈ h, π→1 h q → y, π→2 h p → h, and since p is the discrete topology, choose h ∈ h for which π2(h) = {h} and ϕ(h) = {〈(x,g)〉}. if (s,t) ∈ h, then (s,t) ∼ (x,g), t = h, and thus λ(s,g) = λ(x,h). hence λ(π1(h) × {g}) = {λ(x,h)}, and thus ˙λ(x,h) = λ→(π→1 h × ġ) q → λ(y,g). then λ(x,h) = λ(y,g), (x,g) ∼ (y,h), and thus ϕ−1(〈(x,g)〉) is r-closed. hence it follows from theorem 4.2 that (b(x,s),σ) is t1. � corollary 4.4 ([1]). suppose that the hypotheses of corollary 4.3 are satisfied with the exception that (x,q) is a topological space and b(x,s) is equipped with the quotient topology τ. then (b(x,s),τ) is t1 iff (x,q) is t1. proof. it follows from theorem 2 [2] that since ϕ : (x × s,r) → (b(x,s),σ) is a quotient map in conv, ϕ : (x × s,r) → (b(x,s), tσ) is a topological quotient map, where tσ is the largest topology on x × s which is coarser than σ. moreover, τ = tσ , and a ⊆ b(x,s) is σ-closed iff it is τ-closed. hence the desired conclusion follows from corollary 4.3. � an illustration is given to show that the generalized quotient space may fail to be t1 even though (x,q) is a t1 topological space. example 4.5. denote x = (0, 1), q the cofinite topology on x, and define f : x → x by f(x) = ax, where 0 < a < 1 is fixed. let s = {fn : n ≥ 0}, where f0 = idx and f n denotes the n-fold composition of f with itself. then (s,.) ∈ |sg| is commutative with composition as the operation. also equip (s,.) with the cofinite topology p. it is shown that the operation γ : (s,p) × (s,p) → (s,p) defined by γ(g,h) = g.h := h ◦ g is continuous at (fm,fn). define c = {fk : k ≥ k0}; then {f m+n} ∪ c is a basic p-neighborhood of fm+n, where k0 ≥ 0. observe that if a = {f m} ∪ c and b = {fn} ∪ c, then γ(a × b) ⊆ c ∪ {fm+n}. therefore γ is continuous, and (s,.,p) ∈ |csg|. define λ : x × s → x by λ(x,g) = g(x), for each x ∈ x, g ∈ s, and note that λ is an action. it is shown that λ : (x,q) × (s,p) → (x,q) is continuous at (x0,f n) in x × s. a basic q-neighborhood of λ(x0,f n) = fn(x0) is of the 182 h. boustique, p. mikusiński and g. richardson form w = x − f , where fn(x0) /∈ f and f is a finite subset of x. let y0 be the smallest member of f , and choose k0 to be a natural number such that ak0 < y0. then for each k ≥ k0, f k(x) = akx < y0 for each x ∈ x. since f n is injective, f0 = (f n)−1(f) is a finite subset of x. then u = x − f0 is a q-neighborhood of x0, v = {f n}∪{fk : k ≥ k0} is a p-neighborhood of f n, and λ(u × v ) ⊆ w . indeed, if x ∈ u and k ≥ k0, then λ(x,f k) = fk(x) < y0, and thus fk(x) ∈ w . further, if x ∈ u, then fn(x) /∈ f , and hence fn(x) ∈ w . it follows that λ(u × v ) ⊆ w , and thus λ is a continuous action. it is shown that ϕ−1(〈(x0, idx )〉) is not closed in (x × s,r). note that (x,fn) ∈ ϕ−1(〈(x0, idx )〉) iff idx (x) = f n(x0). hence ϕ −1(〈(x0, idx )〉) = {(fn(x0),f n) : n ≥ 0}. since idx = f 0 > f1 > f2 > ..., it easily follows that clrϕ −1(〈(x0, idx )〉) = x ×s, and thus ϕ −1(〈(x0, idx )〉) is not r-closed. it follows from theorem 4.2 that (b(x,s),σ) is not t1 even though both (x,q) and (s,p) are t1 topological spaces. a continuous surjection f : (x,q) → (y,p) in conv is said to be proper map provided that for each ultrafilter f on x, f→f p → y implies that f q → x, for some x ∈ f−1(y). proper maps in conv are discussed in [3]; in particular, proper maps preserve closures. a proper convergence quotient map is called a perfect map [4]. remark 4.6. assume that ((x,q), (s,.,p),λ) ∈ gq, (x,q) and (s,p) are regular, and ϕ : (x×s,r) → ((b(x,s),σ) is a perfect map. then (b(x,s),σ) is also regular. indeed, suppose that h ∈ f(b(x,s)) such that h σ → 〈(y,h)〉. since ϕ is a quotient map in conv, there exists (x,g) ∼ (y,h) and k r → (x,g) such that ϕ→k = h. moreover, the regularity of (x × s,r) implies that clrk r → (x,g). since ϕ is a proper map and thus preserves closures, ϕ→(clrk) = clσϕ →k = clσh σ → 〈(y,h)〉. hence (b(x,s),σ) is regular. the proof of the following result is straightforward to verify. lemma 4.7. suppose that (s,.,p) ∈ |csg| and (t,.) ∈ |sg|. assume that f : (s,.,p) → (t,.,σ) is both a homomorphism and a quotient map in conv. then (t,.,σ) ∈ |csg|. assume that ((x,q), (s,.,p),λ) ∈ ac. recall that λ distinguishes elements in s whenever λ(x,g) = λ(x,h) for each x ∈ x implies g = h. this property was needed in the verification of theorem 3.5. in the event that λ fails to distinguish elements in s, define g ∼ h iff λ(x,g) = λ(x,h) for each x ∈ x. then ∼ is an equivalence relation on s; denote s1 = s/ ∼= {[g] : g ∈ s}, and define the operation [g].[h] = [g.h], for each g,h ∈ s. the operation is well defined and (s1, .) ∈ |sg|. let p1 denote the quotient convergence structure on s1 determined by ρ : (s,p) → s1, where ρ(g) = [g]. then ρ : (s,.) → (s1, .) is a homomorphism, and it follows from lemma 4.7 that (s1, .,p1) ∈ |csg|. define λ1 : x × s1 → x by λ1(x, [g]) = λ(x,g). convergence semigroup actions: generalized quotients 183 theorem 4.8. assume ((x,q), (s,.,p),λ ∈) gq, λ fails to distinguish elements in s, and let (b(x × s),σ), (b(x × s1),σ1) denote the generalized quotient spaces corresponding to (x × s,r) and (x × s1,r1), where r = q × p and r1 = q × p1. then (a) λ1 : (x × s1,r1) → (x,q) is a continuous action (b) λ1 separates elements in s1 (c) (b(x,s),σ) and (b(x,s1),σ1) are homeomorphic. proof. (a): it is routine to verify that λ1 is an action. let us show that λ1 is continuous. suppose that f q → x and g p1 → [g]; then since p1 is a quotient structure in conv, there exists g1 p → g1 ∼ g such that ρ →g1 = g. hence λ→1 (f × g) = λ → 1 (f × ρ →g1) = [{λ1(f × ρ(g1)) : f ∈ f,g1 ∈ g1}] = [{λ(f × g1) : f ∈ f,g1 ∈ g1}] = λ →(f × g1) q → λ(x,g1) = λ1(x, [g]), and thus λ1 is continuous. (b): suppose that λ1(x, [g]) = λ1(x, [h]) for each x ∈ x. then λ(x,g) = λ(x,h) for each x ∈ x, and thus [g] = [h]. hence λ1 distinguishes elements in s1. (c): it easily follows that the diagram below is commutative: x × s ϕ1 b(x,s) x × s1 ψ1 ? ϕ2 b(x,s1) ψ2 ? where ϕ1, ϕ2 are quotient maps, ψ1(x,g) = (x, [g]), and ψ2(〈x,g〉) = 〈(x, [g])〉. moreover, ψ2 is an injection. indeed, assume that 〈(x, [g])〉 = ψ2(〈(x,g)〉) = ψ2(〈(y,h)〉) = 〈(y, [h])〉; then λ1(x, [h]) = λ1(y, [g]) and thus λ(x,h) = λ(y,g). hence 〈(x,g)〉 = 〈(y,h)〉 and ψ2 is an injection. clearly ψ2 is a surjection. it is shown that ψ2 is continuous. indeed, suppose that h σ → 〈(y,h)〉; then there exist (x,g) ∼ (y,h) and k r → (x,g) such that ϕ→1 k = h. since the diagram above commutes with ψ1 and ϕ2 continuous, it follows that ψ → 2 h = (ψ2 ◦ϕ1) →k = (ϕ2 ◦ψ1) →k σ1 → (ϕ2 ◦ψ1)(x,g) = (ψ2 ◦ϕ1)(x,g) = ψ2(〈(x,g)〉) = ψ2(〈(y,h)〉). hence ψ2 is continuous. finally, let us show that ψ−12 is continuous. assume that h σ1 → 〈(y, [h])〉. since ϕ2 is a quotient map, there exist (x, [g]) ∼ (y, [h]) and k r1 → (x, [g]) such that ϕ→2 k = h. in particular, f = π → 1 k q → x and g = π→2 k p1 → [g]. since ρ : (s,p) → (s1,p1) is a quotient map, there exist g1 ∼ g and g1 p → g1 such that ρ→g1 = g. then f×g1 r → (x,g1), and thus ψ → 1 (f×g1) = f×ρ →g1 = f×g ≤ k. hence (ϕ2 ◦ ψ1) →(f × g1) ≤ ϕ → 2 k = h, and since the diagram commutes, ψ←2 h ≥ (ψ −1 2 ◦ ϕ2 ◦ ψ1) →(f × g1) = ϕ → 1 (f × g1) σ → 〈(x,g)〉 = ψ−12 (〈(y, [h])〉). therefore ψ2 is a homeomorphism. � 184 h. boustique, p. mikusiński and g. richardson sufficient conditions in order for (x,q) to be embedded in (b(x,s),σ) are presented below. theorem 4.9. suppose that ((x,q), (s,.,p),λ) ∈ gq. define β : (x,q) → (b(x,s),σ) by β(x) = 〈(x,e)〉, for each x ∈ x. then (a) β is a continuous injection (b) β is an embedding provided that (x,q) is a choquet space, p is discrete, and λ is a proper map. proof. (a): clearly β is an injection. next, assume that f q → x; then β→f = [{β(f) : f ∈ f}] = [{ϕ(f × {e}) : f ∈ f}] = ϕ→(f × ė) σ → ϕ(x,e) = β(x). therefore β is continuous. (b): first, suppose that f is an ultrafilter on x such that β→f σ → β(x) = 〈(x,e)〉. since ϕ : (x × s,r) → (b(x,s),σ) is a quotient map in conv, there exist (y,g) ∼ (x,e) and k r → (y,g) such that ϕ→k = β→f. denote f1 = π → 1 k q → y and g1 = π → 2 k p → g. since p is the discrete topology, g1 = ġ, and thus k ≥ π→1 k×π → 2 k = f1 × ġ. let f1 ∈ f1; then ϕ →(f1 × ġ) ≤ ϕ →k = β→f implies that there exists f ∈ f such that β(f) ⊆ ϕ(f1 × {g}). if z ∈ f , then β(z) = 〈(z,e)〉 = 〈(z1,g)〉, for some z1 ∈ f1, and thus λ(z,g) = λ(z1,e) = z1 ∈ f1. it follows that λ(f × {g}) ⊆ f1, and thus λ →(f × ġ) ≥ f1 q → y. since f × ġ is an ultrafilter on x × s and λ is a proper map, f × ġ r → (s,t), for some (s,t) ∈ λ−1(y). then f q → s and g = t since p is discrete. it follows that λ(y,e) = y = λ(s,t) = λ(s,g), and thus (s,e) ∼ (y,g). as shown above, (y,g) ∼ (x,e), and thus (x,e) ∼ (s,e). therefore x = s, and f q → x. finally, let f be any filter on x such that β→f σ → β(x). if h is any ultrafilter on x containing f, then β→h σ → β(x), and from the previous case, h q → x. since (x,q) is a choquet space, f q → x and hence β is an embedding. � assume that ((x,q), (s,.,p),λ) ∈ gq, (x,q̄) is the finest choquet space such that q̄ ≤ q, r̄ = q̄ ×p, and let σ̄ denote the quotient convergence structure on b(x,s) determined by ϕ : (x × s, r̄) → b(x,s). corollary 4.10. assume ((x,q), (s,.,p),λ) ∈ gq, p is discrete, and λ is a proper map. then, using the above notations, β : (x,q̄) → (b(x,s), σ̄) is an embedding. proof. it follows from theorem 3.7 that ((x,q̄), (s,.,p),λ) ∈ ac. since q and q̄ agree on ultrafilter convergence, λ : (x,q̄) × (s,p) → (x,q̄) is also a proper map, and (x,q̄) is a choquet space. then according to theorem 4.9, β : (x,q̄) → (b(x × s), σ̄) is an embedding. � convergence semigroup actions: generalized quotients 185 let us conclude by showing that the generalized quotient of a product is homeomorphic to the product of the generalized quotients. assume that ((xj,qj ), (sj, .,pj ),λj ) ∈ gq, for each j ∈ j. let (x,q) = × j∈j (xj,qj ) and (s,.,p) = × j∈j (sj, .,pj) denote the product spaces, and define λ : x × s → x by λ((xj ), (gj )) = (λj (xj,gj )). according to corollary 3.9, ((x,q), (s,.,p),λ) ∈ ac. moreover, since each (sj, .,pj ) is commutative and λj (.,g) is an injection for each j ∈ j, (s,.,p) is commutative and λ(.,g) is an injection. hence ((x,q), (s,.,p),λ) ∈ gq. let ϕj : (xj,qj) × (sj, .,pj) → (b(xj,sj),σj ) denote the convergence quotient map, rj = qj × pj, ϕ = × j∈j ϕj , for each j ∈ j. since the product of quotient maps in conv is again a quotient map, ϕ : × j∈j (xi × sj,rj ) → × j∈j (b(xj,sj ),σj ) is also a quotient map. denote σ = × j∈j σj . define ((xj ), (gj )) ∼ ((yj ), (hj )) in x × s iff λ((xj ), (hj )) = λ((yj ), (gj )). this is an equivalence relation on x × s, and it follows from the definition of λ that ((xj ), (gj )) ∼ ((yj ), (hj )) iff (xj,gj ) ∼ (yj,hj ), for each j ∈ j. let (b(x,s), σ) denote the corresponding generalized quotient space, where φ : (x × s,r) → (b(x,s), σ) is the quotient map and r = × j∈j rj . theorem 4.11. suppose that ((xj,qj ), (sj, .,pj),λj ) ∈ gq, for each j ∈ j. then, employing the notations defined above, × j∈j (b(xj,sj ),σj ) and (b(x,s), σ) are homeomorphic. proof. consider the following diagram: × j∈j (xj × sj,rj ) δ (x × s,r) × j∈j (b(xj,sj ),σj ) ϕ ? ∆ (b(x,s), σ), φ ? where δ(((xj,gj)j )) = ((xj ), (gj )) and ∆((〈(xj,gj )〉j )) = 〈((xj ), (gj ))〉. then δ is a homeomorphism, and the diagram commutes. note that ∆ is a bijection. indeed, if ∆((〈(xj,gj )〉j )) = ∆((〈(yj,hj )〉j )), then ((xj ), (gj )) ∼ ((yj ), (hj )) and thus (xj,gj) ∼ (yj,hj), for each j ∈ j. hence 〈(xj,gj)〉j = 〈yj,gj〉j for each j ∈ j, and thus ∆ is an injection. clearly ∆ is a surjection. it is shown that ∆ is continuous. assume that h σ → (〈(yj,hj )〉j ); then since ϕ is a quotient map, there exist ((xj ), (gj )) ∼ ((yj ), (hj )) and k r → ((xj,gj )j ) such that ϕ→k = h. however, the diagram commutes, and thus ∆→h = (∆◦ϕ)→k = (φ◦δ)→k σ → φ((xj ), (gj )) = φ((yj ), (hj )) = 〈((yj ), (hj ))〉. hence ∆ is continuous. conversely, suppose that h σ → 〈((yj ), (hj ))〉; then since φ is a quotient map, 186 h. boustique, p. mikusiński and g. richardson there exist ((xj ), (gj )) ∼ ((yj ), (hj )) and k r → ((xj ), (gj )) such that φ →k = h. using the fact that δ is a homeomorphism and that the diagram commutes, ∆←h = (ϕ ◦ δ−1)→k σ → ϕ((xj,gj )j ) = ϕ((yj,hj )j ) = (〈(yj,hj )〉j ), and thus ∆−1 is continuous. therefore ∆ is a homeomorphism. � remark 4.12. in general, quotient maps are not productive in the category of all topological spaces with the continuous maps as morphisms. whether or not theorem 4.11 is valid in the topological context is unknown to the authors. references [1] j. burzyk, c. ferens and p. mikusiński, on the topology of generalized quotients, applied gen. top. 9 (2008), 205–212. [2] d. kent, convergence quotient maps, fund. math. 65 (1969), 197–205. [3] d. kent and g. richardson, open and proper maps between convergence spaces, czech. math. j. 23(1973), 15–23. [4] d. kent and g. richardson, the regularity series of a convergence space, bull. austral. math. soc. 13 (1975), 21–44. [5] m. khosravi, pseudoquotients: construction, applications, and their fourier transform, ph.d. dissertation, univ. of central florida, orlando, fl, 2008. [6] p. mikusiński, boehmians and generalized functions, acta math. hung. 51 (1988), 271– 281. [7] p. mikusiński, generalized quotients with applications in analysis, methods and applications of anal. 10 (2003), 377–386. [8] w. park, convergence structures on homeomorphism groups, math. ann. 199 (1972), 45–54. [9] w. park, a note on the homeomorphism group of the rational numbers, proc. amer. math. soc. 42 (1974), 625–626. [10] n. rath, action of convergence groups, topology proceedings 27 (2003), 601–612. received november 2008 accepted november 2009 gary richardson (garyr@mail.ucf.edu) department of mathematics, university of central florida,orlando, fl 32816, usa, fax: (407) 823-6253, tel: (407) 823-2753 kosikuagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 239-251 generalizations of z-supercontinuous functions and dδ-supercontinuous functions j. k. kohli, d. singh and rajesh kumar abstract. two new classes of functions, called ‘almost zsupercontinuous functions’ and ‘almost dδ -supercontinuous functions’ are introduced. the class of almost z-supercontinuous functions properly includes the class of z-supercontinuous functions (indian j. pure appl. math. 33(7), (2002), 1097-1108) as well as the class of almost clopen maps due to ekici (acta. math. hungar. 107(3), (2005), 193-206) and is properly contained in the class of almost dδsupercontinuous functions which in turn constitutes a proper subclass of the class of almost strongly θ-continuous functions due to noiri and kang (indian j. pure appl. math. 15(1), (1984), 1-8) and which in its turn include all δ-continuous functions of noiri (j. korean math. soc. 16 (1980), 161-166). characterizations and basic properties of almost z-supercontinuous functions and almost dδ-supercontinuous functions are discussed and their place in the hierarchy of variants of continuity is elaborated. moreover, properties of almost strongly θ-continuous functions are investigated and sufficient conditions for almost strongly θ-continuous functions to have uθ -closed (θ-closed) graph are formulated. 2000 ams classification: primary: 54c05, 54c08, 54c10; secondary: 54d10, 54d15, 54d20. keywords: (almost) z-supercontinuous function, (almost) dδ-supercontinuous function, (almost) strongly θ-continuous function, almost continuous function, δ-continuous function, faintly continuous function, , uθ-closed graph , θ-closed graph, uθ-limit point, θ-limit point, z-convergence. 240 j. k. kohli, d. singh and r. kumar 1. introduction among several of the variants of continuity in the literature, some are stronger than continuity and some are weaker than continuity and yet others are independent of continuity. in this paper we introduce two new variants of continuity which represent generalizations of the notions of z-supercontinuity and dδ -supercontinuity and are independent of continuity and coincide with zsupercontinuity and dδ-supercontinuity, respectively if the range is a semiregular space. the class of almost z-supercontinuous functions besides containing the class of z-supercontinuos functions contains the class of almost clopen (≡ almost cl-supercontinuous [34]) functions defined by ekici [4]. characterizations and basic properties of almost z-supercontinuous (almost dδ-supercontinuous) functions are elaborated in section 3 and their place in the hierarchy of variants of continuity is discussed. section 4 is devoted to the study of the behaviour of separation axioms under almost z-supercontinuous (almost dδ-supercontinuous) functions. in section 5, characterizations and properties of almost strongly θ-continuous functions are elaborated. section 6 is devoted to separation axioms and sufficient conditions for almost strongly θ-continuous functions to have uθ-closed (θ-closed) graphs are obtained. 2. preliminaries and basic definitions a subset s of a space x is said to be an h-set [36] or quasi h-closed relative to x [28] (respectively n -closed relative to x [1]) if for every cover { uα|α∈λ} of s by open sets of x, there exists a finite subset λo of λ such that s⊂∪{u α|α∈λo} (respectively s⊂∪{(u α) o|α∈λo}). a space x is said to be quasi h-closed [28] (respectively nearly compact [32]) if the set x is quasi h-closed relative to x (respectively n -closed relative to x). a space x is said to be quasicompact [5] if every cover of x by cozero sets admits a finite subcover. a space x is said to be δ-completely regular [13] (almost completely regular [31]) if for each regular gδ -set (regularly closed set) f and a point x not in f there exists a continuous function f : x→[0, 1] such that f (x) = 0 and f (f ) = 1. a subset a of a space x is called a regular gδ-set [21] if a is an intersection of a sequence of closed sets whose interiors contain a, i.e., if a = ∞⋂ n=1 fn = ∞⋂ n=1 f on , where each fn is a closed subset of x. the complement of a regular gδ-set is called a regular fσ-set. a space x is called a dδ-completely regular ([15], [16]) if it has a base of regular fσ-sets. definition 2.1. a function f : x→y from a topological space x into a topological space y is said to be almost z-supercontinuous (almost dδ-supercontinuous) if for each x∈x and each open set v containing f (x), there exists a cozero set (regular fσ-set) u containing x such that f (u )⊂(v ) o. generalizations of z-supercontinuous functions and dδ -supercontinuous functions 241 definition 2.2. a set g is said to be δ-open [36] (dδ-open [13], z-open [12]) if for each x∈g, there exists a regular open set (regular fσ-set, cozero set) h such that x∈h⊂g, or equivalently, g can be obtained as an arbitrary union of regular open sets (regular fσ-sets, cozero sets). the complement of a δ -open (dδ-open, z-open) set will be referred to as a δ-closed (dδ-closed, z-closed) set. definition 2.3. let x be a topological space and let a⊂x. a point x∈x is called a δ-adherent [36] ( θ-adherent [36], uθ-adherent ([9], [10]), dδ-adherent [13], z-adherent [12]) point of a if every regular open set (closed neighborhood, θ-open set, regular fσ-set, cozero set) containing x has non-empty intersection with a. let aδ denote the set of all δ-adherent points (clθa the set of all θ-adherent points, auθ the set of all uθ-adherent points, [a]dδ the set of all dδadherent points, az the set of all z-adherent points) of a set a. the set a is δ-closed (θ-closed, dδ-closed, z-closed) if a = aδ (a = clθa or a = auθ , a = [a]dδ , a = az ). lemma 2.4 ([8], [11]). a subset a of a topological space x is θ-open if and only if for each x∈a, there is an open set u such that x∈u⊂u⊂a. definition 2.5. a space x is called θ-compact [10] ( dδ-compact [14]) if every θ-open cover (cover by regular fσ -sets) of x has a finite subcover. definitions 2.6. a function f : x→y from a topological space x into a topological space y is said to be (a) strongly continuous [18] if f (a) ⊂ f (a) for each subset a of x. (b) perfectly continuous( [25], [26]) if f −1(v ) is clopen in x for every open set v ⊂y . (c) almost perfectly continuous (≡ regular set connected [3]) if f −1(v ) is clopen for every regular open set v in y . (d) cl-supercontinuous [34] (≡ clopen continuous [29]) if for each open set v containing f (x) there is a clopen set u containing x such that f (u )⊂v . (e) almost cl-supercontinuous[17] (≡ almost clopen continuous[4]) if for each x∈x and each regular open set v containing f (x) there is a clopen set u containing x such that f (u )⊂v . (f) z-supercontinuous [12] if for each x∈x and for each open set v containing f (x), there exists a cozero set u containing x such that f (u )⊂v . (g) strongly θ-continuous [24] if for each x∈x and for each open set v containing f (x), there exists an open set u containing x such that f (u )⊂v . (h) supercontinuous [22] if for each x∈x and for each open set v containing f (x), there exists an open set u containing x such that f (u )o⊂v . (i) almost strongly θ-continuous [27] if for each x∈x and for each open set v containing f (x), there exists an open set u containing x such that f (u )⊂(v )o. (j) δ-continuous [24] if for each x∈x and for each open set v containing f (x), there exists an open set u containing x such that f (u )o⊂(v )o. (k) almost continuous [33] if for each x∈x and for each open set v containing f (x), there exists an open set u containing x such that f (u )⊂(v )o. 242 j. k. kohli, d. singh and r. kumar (l) faintly continuous [20] if for each x∈x and for each θ-open set v containing f (x), there exists an open set u containing x such that f (u )⊂v . (m) dδ-supercontinuous [13] if for each x∈x and for each open set v containing f (x), there exists a regular fσ set u containing x such that f (u )⊂v . the following diagram well illustrates the relationships that exist among almost z-supercontinuous functions, almost dδ -supercontinuous functions and various variants of continuity defined above. however, none of the above implications in general is reversible. kohli and kumar [12] showed that a strongly θ-continuous function need not be z-supercontinuous function. noiri and kang [27] gave examples to show that a δ-continuous function need not be almost strongly θ-continuous and that almost strongly θ-continuous function need not be strongly θ-continuous. moreover, noiri [24] showed that an almost continuous function need not be δ-continuous. example 2.7. let x = n = y be the set of positive integers equipped with cofinite topology. the identity function on x is almost z-supercontinuous but not dδ-supercontinuous. example 2.8. let x = y be the mountain chain space due to heldermann [6] which is a regular space. the identity map from x onto y is strongly θ-continuous but not almost dδ-supercontinuous. example 2.9. let x = {x1, x2, x3, x4} and γ = {x, φ, {x3}, {x1, x2}, {x1, x2, x3}} let y = {y1, y2, y3, y4} and σ = {y, φ, {y1}, {y3}, {y1, y2}, {y1, y3}, {y1, y2, y3}, {y1, y3, y4}} define a function f : (x, γ) → (y, σ) as follows: f (x1) = f (x2) = y2 and f (x3) = f (x4) = y1 then f is an almost z-supercontinuous functions but not continuous. generalizations of z-supercontinuous functions and dδ -supercontinuous functions 243 example 2.10. let a = k∪{a+,a−} be the space due to hewitt [7] which is dδ-completely regular. the identity function defined on a is dδ-supercontinuous but not almost z-supercontinuous. example 2.11. let x denote the real line endowed with usual topology. the identity function defined on x is almost z-supercontinuous but not almost clsupercontinuous (=almost clopen). examples 2.8 and 2.9 show that the notions of almost z-supercontinuous function (almost dδ-supercontinuous function) and continuous function are independent of each other. 3. characterizations and basic properties of almost z-supercontinuous and dδ-supercontinuous functions proposition 3.1. for a function f : x→y from a topological space x into a topological space y , the following statements are equivalent: (a) f is almost z-supercontinuous (almost dδ-supercontinuous). (b) the inverse image of every regular open subset of y is z-open (dδ-open) in x. (c) the inverse image of every regular closed subset of y is z-closed (dδ-closed) in x. (d) the inverse image of every δ-open subset of y is z-open (dδ-open) in x. (e) the inverse image of every δ-closed subset of y is z-closed (dδ-closed) in x. proof. it is easy using definitions. � theorem 3.2. for a function f : x→y the following statement are equivalent. (a) f is almost z-supercontinuous. (b) f (az )⊂(f (a))δ for every a⊂x. (c) (f −1(b))z⊂f −1(bδ) for every b⊂y . proof. (a)⇒(b). let y = f (x) for some x∈az. to show that f (x)∈(f (a))δ , let v be any regular open set containing f (x). then there exists a cozero set u containing x such that f (u )⊂v . since x∈az , u∩a6=φ and so f (u∩a)6=φ which in turn implies that f (u )∩f (a)6=φ and hence v ∩f (a)6=φ. thus f (x)∈(f (a))δ. hence f (az )⊂(f (a))δ for every a⊂x. (b)⇒(c). let b⊂y . then f ((f −1(b))z )⊂(f (f −1(b)))δ⊂bδ and so it follows that (f −1(b))z⊂f −1(bδ). (c)⇒(a). let f be any δ-closed set in y . then (f −1(f ))z⊂f −1(fδ) = f −1(f ). since f −1(f )⊂(f −1(f ))δ⊂(f −1(f ))z , so f −1(f ) = (f −1(f ))z which in turn implies that f is almost z-supercontinuous. � theorem 3.3. for a function f : x→y the following statement are equivalent. (a) f is almost dδ-supercontinuous. (b) f ([a]dδ )⊂(f (a))δ for every a⊂x. (c) [f −1(b)]dδ ⊂f −1(bδ) for every b⊂y . 244 j. k. kohli, d. singh and r. kumar proof. (a)⇒(b). let y = f (x) for some x∈[a]dδ . to show that y∈(f (a))δ , let v be a regular open set containing f (x). since f is almost dδ-supercontinuous, there is a regular fσ-set u containing x such that f (u )⊂v . since x∈[a]dδ , u∩a6=φ and hence f (u∩a)6=φ which in turn implies that f (u )∩f (a)6=φ. thus v ∩f (a)6=φ and so y∈(f (a))δ for every a⊂x. (b)⇒(c). let b⊂y . then f ([f −1(b)]dδ )⊂(f (f −1(b)))δ⊂bδ and so it follows that [f −1(b)]dδ ⊂f −1(bδ). (c)⇒(a). let f be any δ-closed set in y . then [f −1(f )]dδ ⊂f −1(fδ) = f −1(f ). since f −1(f )⊂[f −1(f )]dδ , f −1(f ) = [f −1(f )]dδ and so f −1(f ) is dδ-closed. it follows that f is almost dδ-supercontinuous. � definition 3.4. a filterbase f is said to z-converge[12] ( dδ-converge[13], δconverge[36]) to a point x, written as f z → x(f dδ → x, f δ → x), if every cozero set (regular fσ-set, regular open set) containing x contains a member of f. theorem 3.5. a function f : x→y is almost z-supercontinuous (almost dδsupercontinuous) if and only if f (f) δ → f (x) for each x∈x and each filter f in x that z-converges (dδ-converges) to x. proof. we shall prove the result in the case of almost z-supercontinuous functions only. suppose that f is almost z-supercontinuous and let f be a filter in x that z-converges to x. let w be a regular open set containing f (x). then x∈f −1(w ) and f −1(w ) is z-open. let h be a cozero set such that x∈h⊂f −1(w ) and so f (h)⊂w . since f z-converges to x, there exists u∈f such that u⊂h and hence f (u )⊂f (h)⊂w . thus, f (f) δ → f (x). conversely, let w be a regular open set containing f (x). now, the filter f generated by the filterbase bx consisting of cozero sets containing x, z-converges to x. since by hypothesis f (f) δ → f (x), there exists a member f (f ) of f (f) such that f (f )⊂w . choose b∈bx such that b⊂f . since b is a cozero set containing x and since f (b)⊂f (f )⊂w, f is almost z-supercontinuous. � remark 3.6. it is routine to verify that almost z-supercontinuity (almost dδ-supercontinuity) is invariant under restrictions and composition of functions and enlargement of range. moreover, the composition gof is almost z-supercontinuous whenever f : x→y is almost z-supercontinuous and g : y →z is δ-continuous. furthermore, if gof is almost z-supercontinuous and f is a surjection which maps z-open sets to z-open sets, then g is almost zsupercontinuous. the following lemma is due to singal and singal [33] and will be used in the sequel. lemma 3.7 (singal and singal [33]). let {xα : α∈i} be a family of spaces and let x = ∏ xα be the product space. if x = (xα)∈x and v is a regular open subset of x containing x, then there exists a basic regular open set πvα such that x∈πvα⊂v , where vα is regular open in xα for each α∈i and vα = xα for all α∈i except for a finite number of indices αi, i = 1, 2, . . . , n. generalizations of z-supercontinuous functions and dδ -supercontinuous functions 245 theorem 3.8. let {fα : xα→yα} be a family of almost z-supercontinuous (almost dδ -supercontinuous) functions. let x = πxα and y = πyα. then f : x→y defined by f ((xα)) = (fα(xα)) for each (xα)∈x is almost z-supercontinuous ( almost dδ-supercontinuous). proof. let (xα)∈x and w be a regular open set in y containing f ((xα)). by lemma 3.7 there exists a basic regular open set v = πvα such that f (x)∈v ⊂w , where each vα is a regular open set in yα and vα = yα for α∈∆ except for α = α1, α2, . . . , αn. for each i = 1, 2, . . . , n, in view of almost z-supercontinuity (almost dδ-supercontinuity) of fαi there exists a cozero set (regular fδ-set) uαi containing xαi such that fαi (uαi )⊂vαi . let u = ∏ uα, where uα = xα for α 6= αi, (i = 1, 2, . . . , n). then u is a cozero set (regular fδ-set) in x such that (xα)∈u and f (u )⊂v ⊂w . thus f is almost zsupercontinuous (almost dδ -supercontinuous). � theorem 3.9. let f : x→y be any function. if {uα : α∈∆} is a cover of x by cozero sets (regular fδ-sets) and for each α, fα = f|uα : uα→y is almost z-supercontinuous (almost dδ-supercontinuous), then f is almost zsupercontinuous (almost dδ-supercontinuous). proof. let v be a regular open set in y . then f −1(v ) = ∪{f −1α (v ) : α∈∆} and since each fα is almost z-supercontinuous (almost dδ-supercontinuous), each f −1α (v ) is z-open (dδ-open) in uα and hence in x. thus f −1(v ) being the union of z-open (dδ -open) sets is z-open (dδ-open). thus f is almost z-supercontinuous (almost dδ-supercontinuous). � theorem 3.10. let f : x→y be a function and g : x→x×y , defined by g(x) = (x, f (x)) for each x∈x, be the graph function. then g is almost zsupercontinuous if and only if f is almost z-supercontinuous and x is an almost completely regular space. proof. suppose that g is almost z-supercontinuous. let v be a regular open set in y . then p−1y (v ) = x×v is a regular open set in x×y , where py is the projection from x×y onto y . therefore f −1(v ) = (pyog) −1(v ) = g−1(p−1y (v )) = g −1(x×v ) is z-open and so f is almost z-supercontinuous. to prove that x is an almost completely regular space, let f be a regular closed set and suppose that x /∈f . then x∈x \ f and g(x)∈(x\f )×y which is a regularly open set in x×y . so there exists a cozero set w in x such that g(w )⊂(x\f )×y . hence x∈w ⊂x\f . thus x is an almost completely regular space. to prove sufficiency, let x∈x and let w be a regular open set containing g(x). by lemma 3.7 there exist regular open sets u⊂x and v ⊂y such that (x, f (x))∈u×v ⊂w . since x is almost completely regular, there exists a cozero set g1 in x containing x such that x∈g1⊂u . since f is almost z-supercontinuous, there exists a cozero set g2 in x containing x such that f (g2)⊂v . let g = g1 ∩ g2. then g is a cozero set containing x and g(g)⊂u×v ⊂w . this proves that g is almost z-supercontinuous. � 246 j. k. kohli, d. singh and r. kumar proposition 3.11. let f : x→y be a function defined on a δ-completely regular space x. then the graph function g(f ) is almost dδ-supercontinuous if and only if f is almost dδ-upercontinuous. proof. it is easy using definitions. � 4. separation axioms related to almost z-supercontinuous functions and dδ-supercontinuous functions theorem 4.1. an almost z-supercontinuous (almost dδ -supercontinuous) image of a quasicompact (dδ-compact) space is nearly compact. proof. let f : x→y be an almost z-supercontinuous (almost dδ -supercontinuous) surjection from a quasicompact (dδ -compact) space x onto a space y . let v = {vα : α∈λ} be a cover of y by regularly open sets (regular fδ-sets) in y . since f is almost z-supercontinuous (almost dδ -supercontinuous), each f −1(vα) is z-open (dδ-open) in x and so is a union of cozero sets (regular fδ-sets). this in turn yields a cover g of x consisting of cozero sets (regular fδ-sets). since x is quasicompact (dδ-compact), there is a finite subcollection {c1, . . . , cn} of g which covers x. suppose ci⊂f −1(vαi ) for some αi∈λ(i = 1, . . . , n). then {vα1 , . . . , vαn } is a finite subcollection of v which covers y . thus y is nearly compact. � corollary 4.2. let f : x→y be an almost z-supercontinuous (almost dδ supercontinuous) surjection from a quasicompact (dδ -compact) space onto a semiregular space y . then y is compact. proof. a semiregular nearly compact space is compact. � definition 4.3 ([30]). a space x is said to be almost regular if for each regular closed set a and each point x /∈a, there exist disjoint open sets u and v such that x∈u, a⊂v . theorem 4.4. let f : x→y be an almost dδ-supercontinuous open bijection onto a space y . then y is an almost regular space. further, if y is a semiregular space, then y is a regular space. proof. let b be any regularly closed set in y and let y /∈b. then f −1(b)∩f −1(y) = φ. since f is almost dδ-supercontinuous, by proposition 3.1 f −1(b) is dδ-closed and so f −1(b) = ⋂ α∈λ zα, where each zα is a regular gδ-set. since f is one-one, f −1(y) is a singleton and so there exists αo∈λ, such that f −1(y) /∈zαo . since zαo is a regular gδ-set, zαo = ∞⋂ i=1 hi = ∞⋂ i=1 hoi , where each hi is a closed set. so there exists an integer j such that f −1(y) /∈ hj . then x \ hj and h o j are disjoint open sets containing f −1(y) and f −1(b), respectively. since f is an open bijection, f (x\hj ) and f (h o j ) are disjoint open sets containing y and b, respectively. thus y is an almost regular space. since a semiregular almost regular space is regular, the last assertion is immediate. � generalizations of z-supercontinuous functions and dδ -supercontinuous functions 247 5. characterizations and some basic properties of almost strongly θ-continuous functions proposition 5.1. a function f : x→y is almost strongly θ-continuous if and only if for each x∈x and each regular open set v containing f (x), there exists a θ-open set u containing x such that f (u )⊂v . proof. it is easy using definitions. � theorem 5.2. for a function f : x→y the following statement are equivalent. (1) f is almost strongly θ-continuous. (2) f (auθ )⊂(f (a))δ for each a⊂x. (3) (f −1(b))uθ ⊂f −1(bδ) for every b⊂y . proof. (a)⇒(b). since (f (a))δ is δ-closed in y , by [27, theorem 3.1, (f )], f −1((f (a))δ ) is θ-closed in x. again, since a⊂f −1((f (a))δ), auθ ⊂(f −1(f (a))δ)uθ = f −1((f (a))δ ) and so f (auθ )⊂(f (a))δ. (b)⇒(c). let b⊂y . then, by hypothesis f ((f −1(b))uθ )⊂(f (f −1(b))δ⊂bδ and so it follows that (f −1(b))uθ ⊂f −1(bδ). (c)⇒(a). let f be any δ-closed set in y . then (f −1(f ))uθ ⊂f −1(fδ) = f −1(f ) which implies that f −1(f ) = (f −1(f ))uθ and so f −1(f ) is θ-closed. this proves that f is almost strongly θ-continuous. � definition 5.3. ( [9], [10] ): a filter f is said to uθ-converge to a point x, written as f uθ → x, if every θ-open set containing x contains a member of f. theorem 5.4. a function f : x→y is almost strongly θ-continuous if and only if f (f) δ → f (x) for each x∈x and each filter in x which uθ-converges to a point x. proof. suppose that f is almost strongly θ-continuous and let f uθ → x. let w be a regular open set in y containing f (x). then by proposition 5.1, f −1(w ) is a θ-open set in x. since f uθ → x, there exists f ∈f such that f ⊂f −1(w ) and so f (f )⊂w . this shows that f (f) δ → f (x). conversely, let v be a regular open subset of y containing f (x). now let f be the filter generated by the filterbase vx consisting of all θ-open sets containing x. by hypothesis f (f) δ → f (x) and so there exists a member f (f ) of f (vx) such that f (f )⊂v . choose u∈vx such that u⊂f which implies that f (u )⊂f (f ) and f (f )⊂v and so f (u )⊂v . hence f is almost strongly θ-continuous. � theorem 5.5. if f : x→y is faintly continuous and g : y →z be almost strongly θ-continuous. then gof is almost continuous. proof. let v be a regular open set in z. by almost strongly θ-continuity of g, g−1(v ) is θ-open in y . so (gof )−1(v ) = f −1(g−1(v )) is open in x, since f is faintly continuous. hence gof is almost continuous. � theorem 5.6. let f : x→y be an almost continuous function defined on a completely regular space x. then f is almost z-supercontinuous. 248 j. k. kohli, d. singh and r. kumar proof. let v be a regular open set containing f (x). since f is almost continuous, f −1(v ) is open. again since x is completely regular space, f −1(v ) is z-open. hence f is almost z-supercontinuous. � corollary 5.7. if f : x→y is a δ-continuous function defined on a completely regular space x, then f is almost z-supercontinuous. proof. a δ-continuous function is almost continuous. � corollary 5.8. let f : x→y be an almost strongly θ-continuous function defined on a completely regular space x, then f is almost z-supercontinuous. proof. an almost strongly θ-continuous function is a δ -continuous function and hence almost continuous. � 6. separation axioms and almost strongly θ-continuous functions definition 6.1 ([2], [10]). a subset s of a space x is said to be θ-set if for every cover {uα|α∈λ} of s by θ-open subsets of x, there exists a finite subset λo of λ such that s⊂∪{uα|α∈λo}. theorem 6.2. if f : x→y is almost strongly θ-continuous and a is a θ-set in x, then f (a) is n -closed relative to y . proof. let {uα : α∈λ} be a cover of f (a) by regular open sets in y . since f is almost strongly θ-continuous, {f −1(uα) : α∈λ} is a cover of a, by θ-open sets in x. since a is θ-set in x, so a⊂∪{f −1(uα) : α∈λo} for some finite subset λo of λ. thus f (a)⊂∪{uα : α∈λo}. hence f (a) is n -closed relative to y . � corollary 6.3. an almost strongly θ-continuous image of a θ-compact space is nearly compact. corollary 6.4. an almost strongly θ-continuous image of an almost compact space is nearly compact. definition 6.5 ([2], [35]). a topological space x is said to be θ-hausdorff if each pair of distinct points are contained in disjoint θ-open sets. theorem 6.6. let f : x→y be an almost strongly θ-continuous injection into a hausdorff space y . then x is θ-hausdorff. proof. let x6=y be two points in x. since f is one-one, f (x)6=f (y). since y is hausdorff, there exist disjoint open sets u and v containing f (x) and f (y), respectively. now, u∩v = φ which implies that u∩v = φ and so (u )o∩v = φ which in turn implies that (u )o∩v = φ and thus, (u )o∩(v )o = φ. let v1 = (u ) o and v2 = (v ) o, which are regular open sets such that v1∩v2 = φ. by almost strongly θ-continuity of f, f −1(v1) and f −1(v2) are disjoint θ-open sets containing x and y, respectively. hence x is θ-hausdorff. � definition 6.7. a space x is said to be a δt0-space [17] if for each pair of distinct points x and y in x there exists a regular open set containing one of the points x and y but not the other. generalizations of z-supercontinuous functions and dδ -supercontinuous functions 249 theorem 6.8. let f : x→y be an almost strongly θ-continuous injection into a δt0-space. then x is a hausdorff space. proof. let x1 and x2 be two distinct points in x. then f (x1) 6= f (x2). since y is a δt0-space, there exists a regular open set v containing one of the points f (x1) or f (x2) but not the other. to be precise, assume that f (x1)∈v . since any union of θ-open sets is θ-open, in view of proposition 5.1 it follows that f −1(v ) is a θ-open set containing x1. by lemma 2.4 there exists an open set u such that x1∈u ⊂ u ⊂ f −1(v ). then u and x\u are disjoint open sets containing x1 and x2, respectively and so x is hausdorff. � functions with closed graphs are important in functional analysis and several other areas of mathematics. several variants of closed graphs occur in literature (see for example [19], [23]). definition 6.9 ([19]). the graph g(f ) of f : x→y is called θ-closed with respect to x if for each (x, y) /∈g(f ) there exist open sets u and v containing x and y, respectively such that (u × v )∩g(f ) = φ. definition 6.10 ([19]). the graph g(f ) of f : x→y is called θ-closed with respect to x × y if for each (x, y) /∈g(f ), there exist open sets u and v containing x and y, respectively such that (u × v )∩g(f ) = φ definition 6.11. the graph g(f ) of f : x→y is called uθ-closed with respect to x × y if for each (x, y) /∈g(f ), there exist θ-open sets u and v containing x and y, respectively such that (u × v )∩g(f ) = φ. theorem 6.12. let f : x→y be a function whose graph is uθ-closed with respect to x × y . if k is a θ-set in y , then f −1(k) is θ-closed in x. proof. let f : x→y be a function whose graph g(f ) is uθ-closed with respect to x × y . let x∈x\f −1(k). for each y∈k, (x, y) /∈g(f ), there exist θ-open sets uy and vy containing x and y, respectively such that f (uy)∩vy = φ. the family {vy|y∈k} is a cover of k by θ-open sets of y . since k is a θ-set, so k⊂∪{vy|y∈ko} for some finite subset ko of k. let u = ∩{uy|y∈ko}. then u is θ-open set containing x and f (u )∩k = φ which implies that u∩f −1(k) = φ and hence x /∈(f −1(k))uθ . this shows that f −1(k) is θ-closed in x. � corollary 6.13 ([27]). let f : x→y be a function whose graph is θ-closed with respect to x × y . if k is quasi h-closed relative to y , then f −1(k) is θ-closed in x. proof. since k is quasi h-closed relative to y , it is a θ-set in y (see [10]). � theorem 6.14. if f : x→y is an almost strongly θ-continuous function and y is a hausdorff space, then g(f ), the graph of f is θ-closed with respect to x × y . proof. let x∈x and let y 6=f (x). since y is hausdorff, there exist disjoint open sets v and w containing y and f (x), respectively. so v and (w )o are disjoint sets containing y and f (x), respectively. since f is almost strongly 250 j. k. kohli, d. singh and r. kumar θ-continuous, so there is an open set u containing x such that f (u )⊂(w )o. then f (u )⊂(w )o⊂y \v . consequently, u × v contains no point of g(f ). hence g(f ) is θ-closed with respect to x × y . � corollary 6.15. if f : x→y is an almost strongly θ-continuous function and y is hausdorff, then g(f ), the graph of f is θ-closed with respect to x. theorem 6.16. if f : x→y is an almost strongly θ-continuous function and y is an almost regular hausdorff space, then g(f ), the graph of f is uθ-closed with respect to x × y . proof. let x∈x and let y 6=f (x). since y is hausdorff, there exist disjoint open sets v1 and w1 containing y and f (x), respectively. thus, there exist disjoint regular open sets v = (v 1) o and w = (w 1) o containing y and f (x), respectively. since f is almost strongly θ-continuous, by proposition 5.1, there exists a θ-open set u containing x such that f (u )⊂w and so f (u )⊂w ⊂y \v . thus u ×v contains no point of g(f ). since y is almost regular, v is a θ-open set. thus u × v is a θ-open set and (u × v )∩g(f ) = φ. hence g(f ) is uθ -closed with 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[36] n. k. veličko, h-closed topological spaces, amer. math. soc. transl. 78, no. 2 (1968), 103–118. received may 2007 accepted january 2008 j. k. kohli (jk kohli@yahoo.com) dep. of mathematics, hindu college, university of delhi, delhi 110 007, india d. singh (dstopology@rediffmail.com) dep. of mathematics, sri aurobindo college, university of delhi-south campus, delhi 110 017, india rajesh kumar (rkumar2704@yahoo.co.in) dep. of mathematics, rajdhani college, university of delhi, delhi 110 015, india sharmanagaragt.dvi @ applied general topology c© universidad politécnica de valencia volume 11, no. 1, 2010 pp. 1-19 topological dynamics on hyperspaces puneet sharma and anima nagar ∗ abstract. in this paper we wish to relate the dynamics of the base map to the dynamics of the induced map. in the process, we obtain conditions on the endowed hyperspace topology under which the chaotic behaviour of the map on the base space is inherited by the induced map on the hyperspace. several of the known results come up as corollaries to our results. we also discuss some metric related dynamical properties on the hyperspace that cannot be deduced for the base dynamics. 2000 ams classification: 54b20, 54c05 keywords: hyperspace, hit and miss topology, hit and far-miss topology, induced map, transitivity, mixing, horseshoe, equicontinuity, scrambled set 1. introduction for past many years, use of topological methods to study the chaotic nature embroiled in dynamical systems has been of wide interest. also, most of the dynamics observed seems to be collective phenomenon emerging out of many segregated components. this leads to the belief that most of these systems are collective (set valued) dynamics of many units of individual systems. hence, arises the need of a topological treatment of such collective dynamics. some recent studies of dynamical systems, in branches of engineering and physical sciences, have revealed that the underlying dynamics is set valued or collective, instead of the normal individual kind which is usually studied ( c.f. [10, 16, 18, 19]). with these illustrations of collective dynamics, some natural questions arise. what is the significance of the underlying topology for any kind of join of dynamics? if the individual dynamics of each unit as well as the topological details are known, what kind of collective dynamics will the combination display? given a topological structure, how can the dynamical ∗the first author thanks csir and the second author thanks dst for financial support. 2 p. sharma and a. nagar behaviour of a unit influence the collective behaviour? similarly given a collective behaviour, under proper topological framework, what can be concluded about the individual dynamics of a specific unit? so far, each of these questions is open. however, in this paper, we try to answer a part of the last two questions. we try to investigate the relation between individual dynamics and the induced dynamics on the “hyperspace” (i.e. set valued dynamics), considering all possible topological framework. we derive relation between properties like dense periodicity, topological transitivity, weakly mixing and topologically mixing, existence of horseshoe, of the map on the base space with that of the induced map on the hyperspace. we also derive conditions on the topology of the hyperspace for these properties to be equivalent at both places. we also discuss the relation between some of the metric dependent properties like equicontinuity, sensitivity, expansivity, existence of scrambled sets etc. on the base space and the hyperspace. since our work is a convolution of ‘topological dynamics’ and ‘hyperspace theory’ we separately give some preliminaries on both these topics which we shall subsequently use, before a brief survey of the work done till now in this direction and thence our contribution. 1.1. dynamical systems. by a dynamical system, we mean a pair (x, f ) where x is a topological (metric) space and f is any continuous self map on x. we study the behavior of each point x ∈ x under repeated actions of f . a point x ∈ x is called periodic if f n(x) = x for some positive integer n, where f n = f ◦ f ◦ f ◦ . . . ◦ f (n times). the least such n is called the period of the point x. a map f is called transitive if for any pair of non-empty open sets u, v in x, there exist a positive integer n such that f n(u ) ⋂ v 6= φ. a map f is called weakly mixing if for any pairs of non-empty open sets u1, u2 and v1, v2 in x, there exists n ∈ n such that f n(ui) ⋂ vi 6= φ for i = 1, 2. it is known that for any continuous self map f , if f is weakly mixing and u1, u2, . . . un, v1, v2, . . . vn are non-empty open sets, then there exists a k ≥ 1 such that f k(ui) ⋂ vi 6= φ for i = 1, 2, . . . n. a map f is called mixing or topologically mixing if for each pair of non-empty open sets u, v in x, there exists a positive integer k such that f n(u ) ⋂ v 6= φ for all n ≥ k. we now define the notion of topological entropy. let x be a compact space and let u be an open cover of x. then u has a finite subcover. let l be the collection of all finite subcovers and let u∗ be the subcover with minimum cardinality, say nu . define h(u) = lognu . then h(u) is defined as the entropy associated with the open cover u. if u and v are two open covers of x, define, u ∨ v = {u ⋂ v : u ∈ u, v ∈ v}. for a self map f on x, f −1(u) = {f −1(u ) : u ∈ u} is also an open cover of x. define, hf,u = lim n→∞ h(u ∨ f −1(u) ∨ f −2(u) ∨ . . . ∨ f −n+1(u)) n . then sup hf,u , where u runs over all possible open covers of x is known as the topological entropy of the map f and is denoted by h(f ). topological dynamics on hyperspaces 3 we say that x contains a topological horseshoe if there is a compact set q in x such that f (q) = q and f|q factors over the shift on m symbols for some m > 1. in other words, we say that f has a m-horseshoe in x for m > 1; if we have a compact set q with q = m ⋃ i=1 qi such that qi ⋂ qj = φ and m ⋃ i=1 qi ⊆ m ⋂ i=1 f (qi) with each qi compact. a self map f on a metric space (x, d) is said to be equicontinuous at a point x ∈ x if for each ǫ > 0, there exists η > 0 such that d(x, y) < η implies d(f n(x), f n(y)) < ǫ for all n ∈ n, y ∈ x. the map f is called equicontinuous if it is equicontinuous at every point of x. a map f is said to be uniformly equicontinuous if for each ǫ > 0, there exists η > 0 such that d(x, y) < η implies d(f n(x), f n(y)) < ǫ for all n ∈ n, x, y ∈ x. for x ∈ x, if there exists a δ > 0 such that for each ǫ > 0 there exists y ∈ x and a positive integer n such that d(x, y) < ǫ and d(f n(x), f n(y)) > δ, then f is said to be sensitive at x. if f is sensitive at each point x ∈ x, f has sensitive dependence on initial conditions or is simply called sensitive. a map f is called δ-expansive if for any pair of distinct elements x, y ∈ x, there exists k ∈ z+ such that d(f k(x), f k(y)) > δ. a set s is called scrambled for f if for any x, y ∈ s, lim inf n→∞ d(f n(x), f n(y)) = 0 but lim sup n→∞ d(f n(x), f n(y)) > 0. a system (x, f ) is called li-yorke chaotic if there exists an uncountable scrambled set. see [4, 5, 6, 9, 12] for details. while defining different dynamical notions, we observe that each such notion defined involves the topology of the space x. we shall consider only those topologies on x for which the self map f remains continuous. as the topology is made finer(or coarser), various dynamical properties behave differently. some properties evolve as the topology is made finer(coarser), while some vanish. for example, if the periodic points are dense for the dynamical system (x, f ) (when x is given topology τ ), then they are dense when x is endowed with any coarser topology. however, denseness of periodic points may not be preserved when x is endowed with a finer topology. similarly, if (x, f ) is transitive, weakly mixing or topologically mixing, then the properties are preserved in any coarser topology. however, like previously, clouds of uncertainty prevail if the topology is made finer. one of the properties which may be preserved if we make the topology finer is the topological entropy. if we take a finer topology, we increase the number of open sets and hence the number of open covers. thus, the topological entropy increases as the supremum is taken over a larger set. however, for metrizable spaces, some dynamical properties like sensitive dependence on initial conditions, li-yorke sensitivity, existence of a li-yorke pair etc. depend only on the underlying metric, rather than the generated topology and hence are not preserved under some other equivalent metric. 4 p. sharma and a. nagar 1.2. hyperspace topologies. for a hausdorff space (x, τ ), a hyperspace (ψ, ∆) comprises of a subfamily ψ of all non-empty closed subsets of x endowed with the topology ∆, where the topology ∆ is generated using the topology τ of x. the set ψ may either comprise of all compact subsets of x, or all compact and connected subsets of x or all closed subsets of x. a hyperspace topology is called admissible if the map x → {x} is continuous. the topology ∆ can be generated in several ways, however, we are interested in only those topologies ∆ that are admissible. more generally, once ψ and ∆ are fixed, the space (ψ, ∆) is called the hyperspace of the space (x, τ ). let, cl(x) = {e ⊆ x : e is closed and non empty } f(x) ={e ∈ cl(x) : e is finite } fn(x) ={e ∈ cl(x) : |e| = n } k(x) = {e ∈ cl(x) : e is compact in x } kc(x) = {e ∈ cl(x) : e is compact and connected in x } e− = {a ∈ cl(x) : a ⋂ e 6= φ} e+ = {a ∈ cl(x) : a ⊆ e} e++ = {a ∈ cl(x) : ∃ ǫ ≥ 0 and sǫ(a) ⊆ e} where sǫ(a) = ⋃ a∈a s(a, ǫ), where s(a, ǫ) = {x ∈ x : d(a, x) < ǫ} some of the standard hyperspace topologies are: let i be a finite index set and for all such i, let {ui : i ∈ i} be a collection of open subsets of x. define for each such collection of open sets, 〈ui〉i∈i = {e ∈ cl(x) : e ⊆ ⋃ i∈i ui and e ⋂ ui 6= φ ∀i} the topology generated by such collections is known as the vietoris topology. let (x, d) be a metric space. for any two closed subsets a, b of x, define, dh (a, b) = inf{ǫ > 0 : a ⊆ sǫ(b) and b ⊆ sǫ(a)} it is easily seen that dh defined above is a metric on cl(x) and is called hausdorff metric on cl(x). this metric preserves the metric on x, i.e. dh ({x}, {y}) = d(x, y) for all x, y ∈ x. the topology generated by this metric is known as the hausdorff metric topology on cl(x) with respect to the metric d on x. it is known that the hausdorff metric topology equals the vietoris topology if and only if the space x is compact. let φ be a subfamily of the collection of all non-empty closed subsets of x. the hit and miss topology determined by the collection φ is the topology having subbasic open sets of the form u − where u is open in x and (ec)+ with e ∈ φ. as a terminology, u is called the hit set and any member e of φ is referred as the miss set. it has been proved that any topology on the hyperspace is of this type [14]. a typical member of the base for the lower vietoris topology on the hyperspace cl(x) consists of the set, each of whose elements intersect or hit finitely many open sets u , i.e. a typical basic open set is the intersection of finitely many u −. the lower vietoris topology is the smallest topology on the hyperspace containing all the sets u − where u is open in x. topological dynamics on hyperspaces 5 a typical basic open set for the upper vietoris topology on the hyperspace cl(x) is of the form u + where u is open in x. thus, given a closed set c, a typical member of the base in the upper vietoris topology is the set whose elements are the elements of the hyperspace disjoint from the closed set c. the vietoris topology equals the join of upper vietoris and lower vietoris topology, and is infact an example of a hit and miss topology. let (x, d) be a metric space. for each element x in x we define a function dx as: dx : cl(x) −→ r such that dx(a) = d(x, a) the wijsman topology determined on cl(x) is the weak topology determined by the family {dx : x ∈ x}, i.e. the smallest topology on cl(x) for which the family of above defined functions is continuous. it is the topology generated by the sets of the form {a ∈ cl(x) : d(x, a) < α} and {a ∈ cl(x) : d(x, a) > α}, where x varies in x and α ≥ 0 varies in r. for a metric space (x, d) and a given collection φ of closed subsets of x, the hit and far miss topology determined by the collection φ is the topology having subbasic open sets of the form u − where u is open in x and (ec)++ with e ∈ φ. here the collection hits each open set u and far misses the complement of each member of φ and hence forms a hit and far miss topology. if we replace the family {uk} of hit sets in lower vietoris topology by pairwise disjoint family of open balls s(xk, ǫ), then the topology thus obtained is known as the lower discrete topology. it can be seen that lower discrete topology is coarser than the hausdorff metric topology. further, the topology determines the hit sets for the hausdorff metric topology. infact, the hausdorff metric topology is the join of upper vietoris topology and lower discrete topology. the ball proximal topology on cl(x) is defined by the collection of sets v − and (bc)++ where v is an open subset of x and b is a closed ball. in most cases, the wijsman topology is same as the ball proximal topology. in short, each topology ∆ on the hyperspace is either hit and miss or hit and far miss type. and for some of the main hyperspace topologies, this can be briefly summarized as: topology hit sets miss sets far miss sets lower vietoris open sets upper vietoris closed sets vietoris open sets closed sets fell open sets compact sets wijsman open sets closed balls hausdorff metric discrete open balls closed sets it may be noted that if x is compact, each of the topologies wijsman, vietoris, fell, hausdorff and ball proximal coincide and hence are equal. 6 p. sharma and a. nagar again, as shown in [14], every hyperspace topology is of this type only, hence all topologies ∆ on ψ ⊆ cl(x) can be obtained in this way. so for any ψ ⊆ cl(x) we can thus talk of the hyperspace (ψ, ∆). let f : x → x be a continuous function. then for such an f , there is a naturally induced map f̂ on the hyperspace of all nonempty closed subsets of x defined as, f̂ : cl(x) → cl(x) such that f̂ (k) = f (k) = {f (k) : k ∈ k} where a denotes the closure of the set a. if f is a closed map then this induced map can also be given as, f : cl(x) → cl(x) such that f (k) = f (k) = {f (k) : k ∈ k} however, in each of the above cases, the continuity of the induced maps f or f̂ is not guaranteed by the continuity of the map f . it is very well known that if cl(x) is assigned the vietoris topology or the hausdorff metric topology and f is a continuous self map on x, then f̂ : cl(x) → cl(x) is always continuous. it can be seen that, for a closed map f , f −1 (u −) = {f −1 (a) : a ∈ u −} = {b ∈ cl(x) : f (b) = a, a ∩ u 6= φ} = {b ∈ cl(x) : f (b) ∩ u 6= φ} = {b ∈ cl(x) : b ∩ f −1(u ) 6= φ} = (f −1(u ))− f −1 (u +) = {f −1 (a) : a ∈ u +} = {b ∈ cl(x) : f (b) = a, a ⊂ u} = {b ∈ cl(x) : f (b) ⊂ u} = {b ∈ cl(x) : b ⊂ f −1(u )} = (f −1(u ))+ this gives rise to the following theorem: theorem 1.1. let x and y be topological spaces and let cl(x) and cl(y )) be the induced hyperspaces with collections of miss sets as ∆x and ∆y respectively. a continuous and closed map f : x → y induces a mapping f : cl(x) → cl(y ). then f is continuous if and only if f −1(a) ∈ ∆x for all a ∈ ∆y . however, in general, f −1 (u ++) 6= (f −1(u ))++, as shown in the below example. example 1.2. define f on each interval of the form [2n − 1, 2n] as, f (x) =            4 n x + n + 3 n − 8, 2n − 1 ≤ x ≤ 2n − 7 8 ; n − 1 2n , 2n − 7 8 ≤ x ≤ 2n − 3 4 ; 2 n x + n + 1 n − 4, 2n − 3 4 ≤ x ≤ 2n − 1 4 ; n + 1 2n , 2n − 1 4 ≤ x ≤ 2n − 1 8 ; 4 n x + n + 1 n − 8, 2n − 1 8 ≤ x ≤ 2n; topological dynamics on hyperspaces 7 define f in a suitable way on [2n, 2n + 1] so that the extended function remains continuous. let g denote the extended function. then g : r → r is continuous. let a = ∞ ⋃ n=1 [2n − 3 4 , 2n − 1 4 ] and u = ∞ ⋃ n=1 (n − 1 n , n + 1 n ) it is clear that images of a and s 1 16 (a) are same. also, these images are contained in u . thus, a ∈ (g−1(u ))++. however, a /∈ g−1(u ++) as g(a) = ∞ ⋃ n=1 (n − 1 2n , n + 1 2n ) and any δ ball around g(a) cannot be contained in u . on the other hand, a similar result does not hold when we consider the induced map f̂ . it can be seen that, f̂ −1(u −) = {f̂ −1(a) : a ∈ u −} = {b ∈ cl(x) : f (b) = a, a ∩ u 6= φ} = {b ∈ cl(x) : f (b) ∩ u 6= φ} = {b ∈ cl(x) : f (b) ∩ u 6= φ} = {b ∈ cl(x) : b ∩ f −1(u ) 6= φ} = (f −1(u ))− f̂ −1(u +) = {f̂ −1(a) : a ∈ u +} = {b ∈ ψ : f (b) = a, a ⊂ u} = {b ∈ ψ : f (b) ⊂ u} ⊂ {b ∈ ψ : b ⊂ f −1(u )} = (f −1(u ))+ also, by giving an example similar to the previous one, it can be shown that, f̂ −1(u ++) 6= (f −1(u ))++ in general. thus, the relations, f̂ −1(u +) = (f −1(u ))+ and f̂ −1(u ++) = (f −1(u ))++ may not hold in general. conditions under which the induced map f̂ can be continuous is still an open problem for investigation. however, some such conditions ensuring the continuity of f̂ for various topologies on cl(x) has been beautifully described in [7]. in this article, we shall consider only those hyperspace topologies under which the induced functions remain continuous. see [2, 7, 8, 13, 14] for details. 1.3. recent results. in recent years, some relations between the individual dynamics of a system and its collective dynamics have been studied. in [15], heriberto roman-flores proved that for a metric space (x, d), for the hyperspace k(x), under hausdorff metric topology, transitivity of the map f on the hyperspace implies transitivity of the map f on the base space. further, he demonstrated by an illustration that transitivity of f need not imply transitivity of f . in [1], banks improved the result by proving that if the hyperspace k(x) is endowed with vietoris topology, the transitivity of the map f is infact equivalent to weakly mixing of the map f . in [11], dominik kwietnaik and 8 p. sharma and a. nagar piotr oprocha proved that positive topological entropy of the map f need not imply positive topological entropy for the map f . we briefly summarize their results. proposition 1.3 ([15]). for continuous f : x → x and f : k(x) → k(x), f transitive implies f transitive. proposition 1.4 ([15]). let f : x → x and f : k(x) → k(x) be continuous maps. then the following conditions are equivalent 1. f is transitive in (x, d). 2. f is transitive in the we topology. remark 1.5. we note here that the we topology mentioned in [15] is actually the upper vietoris topology. let h ⊆ cl(x) and let f be a continuous self map admissible with h. then f induces another self map on h, denoted by f h. let f = f h proposition 1.6 ([1]). suppose h is dense in cl(x). then the following are equivalent. 1. f is weakly mixing. 2. f is weakly mixing. 3. f is transitive. proposition 1.7 ([1]). suppose h is dense in cl(x). then f is mixing if and only if f is mixing. proposition 1.8 ([1]). suppose f(x) ⊆ h. if f has a dense set of periodic points, then so does f . example 1.9 ([11]). let ∑ 2 be the sequence space of all bi-infinite sequences of two symbols 0 and 1. for any two sequences x = (xi) and y = (yi), define d(x, y) = ∞ ∑ i=−∞ |xi−yi| 2|i| it is easily seen that the metric d generates the product topology on ∑ 2 . let s ⊂ ∑ 2 be the set of all bi-infinite sequences for which the symbol 1 occurs at most once. σ : ∑ 2 → ∑ 2 σ(. . . x−2x−1.x0x1 . . .) = . . . x−2x−1x0.x1x2x3 . . . the map σ is known as the shift map and is continuous with respect to the metric d defined. denote σs for σ|s , the restriction of the shift map to s. let σs denote the induced map on k(s). then, topological entropy of σs , h(σs ) = 0 but h(σs ) = log2. 2. main results let (x, f ) be a dynamical system. let ψ ⊆ cl(x) be a collection admissible with the map f , i.e. f (ψ) ⊆ ψ. the topology ∆ on ψ is either a hit and miss topological dynamics on hyperspaces 9 or a hit and far miss topology. we note that (x, d) is a metric space in case we consider a hit and far miss topology. further, we consider only those topologies ∆ on ψ with respect to which f : ψ → ψ is continuous. now the original dynamical system (x, f ) induces another dynamical system (ψ, f ). we will be dealing with the question: given a topological framework for a base space x and its associated hyperspace ψ, how does the dynamics on one space effect the dynamics of the other. that is, under what conditions properties like dense set of periodic points transitivity, weakly mixing etc. in one space imply the same in the other. as the mentioned properties depend on the topology on ψ, we shall derive suitable conditions on the topology ∆ on ψ for this to happen. proposition 2.1. let f(x) ⊆ ψ and ψ be endowed with any admissible hyperspace topology ∆. if (x, f ) has dense set of periodic points, then so does (ψ, f ). proof. let the topology ∆ on the hyperspace ψ be the hit and miss ( or hit and far miss) topology determined by the collection c. let u be a non empty basic open set for the hyperspace (ψ, ∆). then u hits finitely many open sets, say w1, w2, . . . , wn and misses(far misses) finitely many elements of c say, t1, t2, . . . , tm. let t = ⋃ tj . thus each vi = wi ⋂ t c is non-empty, open in x. as periodic points of f are dense in x, each vi contains a periodic point xi of period ki. as each xi is periodic of period ki, the set {x1, x2, ...xn} is periodic of period r = lcm{k1, k2, ...kn}. thus the point {x1, x2, ...xn} ∈ u is a periodic point for f . hence the result holds. � remark 2.2. it can be noted that the above result holds for any hit and miss or hit and far miss topology. however, using denseness of periodic points, a periodic point generated in the hyperspace is a finite set and hence the condition f(x) ⊆ ψ cannot be relaxed in the above proof. again, as f(x) * kc (x), the condition f(x) ⊆ ψ is not satisfied and hence the result is not true in this case, as noted in [1]. also, the converse is not true. (ψ, f ) may have a dense set of periodic points with (x, f ) having no periodic point. an illustration for the same is given in [1]. proposition 2.3. if there exists a base β for the topology on x such that u + is non empty and u + ∈ ∆ for every u ∈ β, then f transitive on ψ implies that f is transitive on x. proof. let u and v be any two non-empty open sets in x. as β forms a base for topology on x, there exists u1, v1 ∈ β such that u1 ⊆ u and v1 ⊆ v . by given hypothesis, u +1 and v + 1 are non empty open in the hyperspace ψ. as f is transitive, there exists n ∈ n such that f n (u +1 ) ⋂ v +1 6= φ. as u1 ⊆ u and v1 ⊆ v , f is transitive. � 10 p. sharma and a. nagar remark 2.4. for the above result to hold, it is firstly necessary that the set u + in (ψ, ∆) is non-empty for any member u of the base. since if u + = φ for some u ∈ β, the transitivity of f cannot be used to establish the transitivity of f . secondly, it is sufficient to have sets of the form u + to be open in the hyperspace, for u open in x. this is basically the property of upper vietoris topology. thus, if the hyperspace ψ contains all singletons and is endowed with any topology finer than the upper vietoris topology, the transitivity of the induced map on the hyperspace ensures the transitivity of the base map on the space x. since upper vietoris topology is indeed coarser than topologies like hausdorff metric topology and vietoris topology, the result holds good in each of these topologies. however as known, when topologies like wijsman topology, ball proximal topology or fell topology become finer than upper vietoris topology, they actually coincide with the vietoris topology. hence the result may not hold for them in general. remark 2.5. again, it can be seen that for the converse, the transitivity of f can guarantee transitivity of f when the hyperspace is endowed with upper vietoris topology or any coarser hyperspace topology. thus, if the hyperspace is endowed with upper vietoris topology, the result holds in both directions and the transitivity of f is in fact equivalent to transitivity of f as also proved in [15]. proposition 2.6. let f(x) ⊆ ψ. if f is weakly mixing, then so is f. the converse holds if there exists a base β for topology on x such that u + ∈ ∆ for every u ∈ β. proof. let u1, u2, v1, v2 be non-empty open sets in the hyperspace such that u1, u2, v1, v2 hits the open sets w11, w21, . . . wn11; w12, w22, . . . wr12; r11, r21, . . . rn11 and r12, r22, . . . rr12 and misses(far misses) the closed sets t11, t21, . . . tm11; t12, t22, . . . ts12; s11, s21, . . . sm11 and s12, s22, . . . ss12 respectively. let ti = ⋃ j tji. and si = ⋃ j sji. let m ij = wji ⋂ t ci and let n i j = rji ⋂ sci . now, each of m ij , n i j are open sets and as f is weakly mixing, there exists k ∈ n such that f k(m ij ) ⋂ n ij 6= φ, ∀ i, j. let x i j ∈ m i j such that f k(xij ) ∈ n i j . then xi = {x i j}j ∈ ui such that f k (xi) ∈ vi. hence f is weakly mixing. conversely, let u1, u2, v1, v2 be open in x. as β is the base for the topology on x, ∃ u11, u22, v11, v22 ∈ β such that uii ⊆ ui and vii ⊆ vi for i = 1, 2. by given hypothesis, u +ii and v + ii are non-empty open in the hyperspace ψ. hence, there exists n ∈ n such that f n (u +ii ) ⋂ v +ii 6= φ. thus, f is weakly mixing. � proposition 2.7. let f(x) ⊆ ψ. if f is topologically mixing, then so is f . the converse holds if there exists a base β for topology on x such that u + ∈ ∆ for every u ∈ β. topological dynamics on hyperspaces 11 proof. let u, v be two non-empty open sets in the hyperspace (ψ, ∆). let u and v hit w1, w2, . . . wr; r1, r2, . . . rr and miss t1, t2, . . . tm; s1, s2, . . . sm respectively. let t = ⋃ j tj and ui = wi ⋂ t c. let s = ⋃ j sj and vi = ri ⋂ sc. it may be noted that as u, v are non-empty, each ui, vi are also non-empty. now as f is topological mixing, for each pair of non empty open sets ui, vi, we obtain ni ∈ n such that f k(ui) ⋂ vi 6= φ, ∀ k ≥ ni. let n = max {ni : i = 1, 2, . . . r}. then f k (u) ⋂ v 6= φ ∀k ≥ n. thus, f is topological mixing. conversely, let f be topological mixing. let u and v be non empty open subsets of x. as β is the base for the topology on x, there exists u1, v1 ∈ β such that u1 ⊆ u and v1 ⊆ v . by given hypothesis, u + 1 and v + 1 are open. as f is topological mixing, ∃ n ∈ n such that f k (u +1 ) ⋂ v +1 6= φ, ∀k ≥ n which implies that f is topological mixing. � remark 2.8. for both the above results, we note that in the forward part, we need f(x) ⊆ ψ. once again, for the converse part, we need the sets u + to be non-empty open for any member u of the base. remark 2.9. it is clear from the above proof that topological mixing of f is equivalent to the topological mixing of f when the hyperspace ψ contains f(x) and is endowed with upper vietoris topology or any of the finer hyperspace topologies. remark 2.10. in each the above proofs, to prove the existence of a dynamical property on the hyperspace, a finite set has been generated. as for any finite set a, if f̂ is continuous, since f (a) = f̂ (a), therefore the above results hold good for f̂ also. proposition 2.11. let(x, f ) be a dynamical system and let (ψ, f ) be the induced dynamical system on the hyperspace. then, the system (ψ, f ) has a positive topological entropy need not imply the same for the system (x, f ). proof. we demonstrate the proof by giving a counterexample. we prove the result on the same lines as done in [11] for vietoris topology. let ∑ 2 be the sequence space of all bi-infinite sequences of two symbols 0 and 1. let s ⊂ ∑ 2 be the set of all bi-infinite sequences for which the symbol 1 occurs atmost once. denote σs for σ|s , the restriction of the shift map to s. let σs denote the induced map on k(s). let ψ = k(s) be endowed with any hyperspace topology such that σs is continuous. then, h(σs ) = 0, but, h(σs ) = log2. as s is countable, h(σs ) = 0. further, define, φ : ∑ 2 → k(s) φ(x = . . . x−2x−1.x0x1 . . .) = {an : n ∈ ax} where ax = {n ∈ z : xn = 1}. 12 p. sharma and a. nagar then, φ is a continuous, onto function. further, any k ∈ k(s) has atmost two preimages. thus, φ is a uniformly finite-to-one semiconjugacy between the systems ( ∑ 2 , σ) and (k(s), σs ). thus, h(σs ) = h(σ) = log2 (by [11]). � remark 2.12. in the above example, we correct the uniformly finite-to-one semiconjugacy given in [11]. further, we establish the result for a general hit and miss(hit and far-miss) topology on the hyperspace and hence generalize the result given in [11]. proposition 2.13. let f1(x) ⊆ ψ. if (x, f ) has a m-horseshoe, then so does (ψ, f ). proof. let s1, s2, . . . sm constitute a m-horseshoe for the system (x, f ). then, k(s1) ⋂ ψ, k(s2) ⋂ ψ, . . . k(sm ) ⋂ ψ constitutes a m-horseshoe for the system (ψ, f ). � we now give an example to show that the converse is not true, i.e. existence of a horseshoe for the induced system need not guarantee the same for the original system. example 2.14. let ∑ 2 be the sequence space of one sided infinite sequences of two symbols 0 and 1. for any two sequences x = (xi) and y = (yi), define d(x, y) = ∞ ∑ n=1 |xi−yi| 2i the metric d generates the product topology on ∑ 2 . let s = {an : n ∈ n} ⋃ {a0}, where an ∈ ∑ 2 such that an is sequence with all 0 and a 1 only at the n-th place and a0 is the sequence of all 0. as any two infinite subsets of s intersect, there cannot exist a horseshoe for any self map f on s. we now define a self map f on s such that (k(s), f ) has a 2-horseshoe. define, f (ak) =        a2n−1, k = 4n − 1; a2n, k = 4n + 1; an−1, k = 2n, n 6= 0; a0, k = 0; we claim that the induced system (k(s), f ) has a 2-horseshoe. let s1 = {a1, a3, a5, . . . a2n−1, . . .} and s2 = {a2, a4, a6, . . . a2n, . . .}. let j1 be a subset of the hyperspace with elements of the form {ank : k ∈ n} ⋃ {a0} where n1 = 1 and each ank ∈ s1. similarly, let j2 be a subset of the hyperspace with elements of the form {ank : k ∈ n} ⋃ {a0} where n1 = 2 and each ank ∈ s2. let k1 = j1 and let k2 = j2. then we claim that k1, k2 constitute a 2-horseshoe for the system (k(s), f ). any element {a2nk−1 : k ∈ n} ⋃ {a0} ∈ k1 is image of {a1} ⋃ {a4nk−1 : k ∈ n} ⋃ {a0} ∈ k1 and {a2} ⋃ {a4nk : k ∈ n} ⋃ {a0} ∈ k2. again, any element {a2nk : k ∈ n} ⋃ {a0} ∈ k2 is image of {a1} ⋃ {a4nk+1 : k ∈ n} ⋃ {a0} ∈ k1 and {a2} ⋃ {a4nk+2 : k ∈ n} ⋃ {a0} ∈ k2. topological dynamics on hyperspaces 13 also any other element of k1 is of the form {a0, a1, a2r1−1, a2r2−1, . . . a2rk−1} which is image of {a0, a1, a4r1−1, a4r2−1, . . . a4rk−1} ∈ k1 and {a0, a2, a4r1 , a4r2 , . . . a4rk } ∈ k2. lastly, any other element of k2 is of the form {a0, a2, a2r1 , a2r2 , . . . a2rk } which is image of {a0, a1, a4r1+1, a4r2+1, . . . a4rk+1} ∈ k1 and {a0, a2, a4r1+2, a4r2+2, . . . a4rk+2} ∈ k2. hence k1 ⋃ k2 ⊆ f (k1) ⋂ f (k2). hence, k1, k2 is a 2-horseshoe for the system (k(s), f ). hence, existence of a horseshoe for the induced system does not imply the same for the original system. for a hyperspace ψ ⊂ cl(x), and a topology ∆ finer than the upper vietoris topology, we have the following lemma. lemma 2.15. let (x, τ ) be a topological space and let (ψ, ∆) be the induced hyperspace such that ψ ⊆ k(x). let ∆ be finer than the upper vietoris topology. then, b ∈ k(ψ, ∆) ⇒ ( ⋃ e∈b e) ∈ k(x) proof. let a = ⋃ e∈b e. let ⋃ i∈i ui be an open cover of a. let e ∈ b. then, as each e ∈ b is also contained in a, ⋃ i∈i ui is also an open cover of e. thus there exists u e1 , u e 2 , . . . u e ne such that e ⊆ ne ⋃ i=1 u ene and thus e ∈< ne ⋃ i=1 u ene >. thus ⋃ e∈b < ne ⋃ i=1 u ene > is an open cover of b. as b is compact, there exists e1, e2, . . . ek such that b ⊆ k ⋃ i=1 < nei ⋃ j=1 u eij >. hence a ⊆ k ⋃ i=1 nei ⋃ j=1 u eij and thus has a finite subcover. hence a is compact. � proposition 2.16. let f1(x) ⊆ ψ ⊆ k(x) and let ∆ be finer than the upper vietoris topology. then the following are equivalent: (1) there exists compact sets s1, s2, . . . sm constituting a horseshoe for (x, f ). (2) there exists compact sets of the hyperspace j1, j2, . . . jm constituting a m-horseshoe on the hyperspace such that the sets qi = ⋃ k∈ji k are pairwise disjoint. proof. the proof of (1) =⇒ (2) follows from result 2.13. for (2) =⇒ (1) let j1, j2, . . . jm constitute a m-horseshoe on the hyperspace. then, the sets qi = ⋃ k∈ji k are compact by lemma 2.15 and thus form a horseshoe for the system (x, f ). � 14 p. sharma and a. nagar we now deal with metric related properties. as observed in the previous section, since these properties are highly metric dependent, we can only think of discussing these properties on the hyperspace ψ with the hausdorff metric, dh . we recall here that the metric dh on ψ preserves the metric d on x. proposition 2.17. let f1(x) ⊆ ψ. then, (ψ, f ) is equicontinuous ⇒ (x, f ) is equicontinuous. proof. if f is equicontinuous, corresponding to ǫ > 0 and {x} ∈ ψ, there exists η > 0 such that dh ({x}, k) < η implies dh (f n ({x}), f n (k)) < ǫ for all n ∈ n, k ∈ ψ. thus, whenever d(x, y) = dh ({x}, {y}) < η, dh (f n ({x}), f n ({y})) = d(f n(x), f n(y)) < ǫ for all n ∈ n and hence the result holds. � proposition 2.18. let f(x) ⊆ ψ ⊆ k(x). (x, f ) is uniformly equicontinuous if and only if (ψ, f ) is uniformly equicontinuous. proof. let ǫ > 0 be given. as f is uniformly equicontinuous, corresponding to ǫ 2 > 0, there exists η > 0 such that d(x, y) < η implies d(f n(x), f n(y)) < ǫ 2 for all n ∈ n, x, y ∈ x. let k ∈ ψ be a non-empty compact subset of x. as k is compact, there exists a finite subset {x1, x2, . . . xr} such that k ⊆ r ⋃ i=1 s η 4 (xi). now, if dh ({x1, x2, . . . xr}, a) < η, then it is clear that dh (f n({x1, x2, . . . xr}), f n(a)) < ǫ 2 for all n ∈ n and a ∈ ψ. thus, by triangle inequality, for any compact set k∗ such that dh (k, k ∗) < η 4 , dh ({x1, x2, . . . xr}, k ∗) < η 2 < η and hence dh (f n({x1, x2, . . . xr}), f n(k∗)) < ǫ 2 for all n ∈ n. thus, dh (f n(k), f n(k∗)) < ǫ for all n ∈ n. conversely, when f is uniformly equicontinuous, corresponding to ǫ > 0, there exists η > 0, such that dh ({x}, {y}) = d(x, y) < η implies dh (f n ({x}), f n ({y})) = d(f n(x), f n(y)) < ǫ and hence the result holds. � proposition 2.19. (x, f ) is uniformly equicontinuous if and only if (cl(x), f ) is uniformly equicontinuous. proof. let ǫ > 0 be given. then, as f is uniformly equicontinuous, corresponding to ǫ 2 > 0, there exists η > 0 such that d(x, y) < η implies d(f n(x), f n(y)) < ǫ 2 for all n ∈ n, x, y ∈ x. let k be a non-empty closed subset of x. for α > 0, an α-discrete subset of x is the set such that for any two distinct elements x, y in the set, d(x, y) ≥ α. let a be a η 4 discrete subset such that dh (k, a) < η 4 . now, for any non-empty closed set b with dh (a, b) < η, dh (f n(a), f n(b)) < ǫ 2 for all n ∈ n. thus, for any closed set k∗ such that dh (k, k ∗) < η 4 , dh (a, k ∗) < η 2 < η and hence dh (f n(a), f n(k∗)) < ǫ 2 for all n ∈ n. hence, dh (f n(k), f n(k∗)) < ǫ for all n ∈ n. conversely, if f is uniformly equicontinuous, corresponding to ǫ > 0, there exists η > 0, such that dh ({x}, {y}) = d(x, y) < η implies dh (f n ({x}), f n ({y})) = d(f n(x), f n(y)) < ǫ and hence the result holds. � topological dynamics on hyperspaces 15 remark 2.20. the equivalence of the dynamical property of uniform equicontinuity on the base space and the hyperspace holds for any ψ ⊆ k(x) and for cl(x). however, the result may not hold for any ψ ⊂ cl(x) as the existence of an appropriate α-discrete set in the desired neighborhood cannot be guaranteed. proposition 2.21. let f(x) ⊆ ψ ⊆ k(x). (x, f ) is equicontinuous ⇒ (ψ, f ) is almost equicontinuous. proof. let (x, f ) be equicontinuous and let ǫ > 0 be given. to establish our claim, we prove that every finite set is a point of equicontinuity for the map f . let a = {x1, x2, . . . , xr} be a finite set in the hyperspace. thus, corresponding to ǫ > 0, by equicontinuity of f at xi, there exists ηi > 0 such that d(xi, y) < ηi implies d(f n(xi), f n(y)) < ǫ for all n ∈ n, y ∈ x. let η =min {ηi : i = 1, 2, . . . , r}. then, for k ∈ ψ with dh (a, k) < η, dh (f n(a), f n(k∗)) < ǫ for all n ∈ n. hence the system (ψ, f ) is almost equicontinuous. � remark 2.22. from the above result, we can infer that, if (x, f ) has no points of sensitivity, then, ψ(as above) has a dense set on which f is not sensitive. however, the case of sensitivity is a bit involved. it is observed that sensitivity of f on (x, d) implies sensitivity of f for some (ψ, dh ) but not for any ψ in general. the converse also holds true for some restricted cases. such results are discussed in [17]. stronger than sensitivity is the property of expansivity. proposition 2.23. let f1(x) ⊆ ψ ⊆ cl(x). then, (ψ, f ) is δ-expansive implies (x, f ) is δ-expansive. proof. let (ψ, f ) be δ-expansive and let x, y ∈ x. then, as f is expansive, for {x}, {y} ∈ ψ, there exists k ∈ z+ such that d(f k(x), f k(y)) = dh (f k ({x}), f k ({y})) ≥ δ. thus, (x, f ) is also δ-expansive. � the converse does not hold true in general. we provide an example to show that the converse is not true. example 2.24. let ∑ 2 be the sequence space of two symbols 0 and 1 and let k( ∑ 2) be the hyperspace of all non empty compact subsets of ∑ 2. it can be easily observed that ( ∑ 2 , σ) is expansive with expansivity constant 1 2 . however, we prove that the system (k( ∑ 2), σ) is not expansive. let if possible, (k( ∑ 2 ), σ) be expansive with expansivity constant δ. let n ∈ n such that 1 2n < δ. let s1 = {0 k1r0∞ : k ≥ 0, r ≤ n} and let s2 = {0k1r0∞ : k ≥ 0, r ≤ n + 1}. then, dh (s1, s2) = 1 2n+1 , also, σ(si) = si, i = 1, 2. thus, for any k ∈ n, dh ((σ k(s1), σ k(s2) = dh (s1, s2) = 1 2n+1 < δ which is a contradiction. thus, the system (k( ∑ 2 ), σ) is not expansive. 16 p. sharma and a. nagar proposition 2.25. let ψ contain the set of all singletons. if (x, f ) is liyorke chaotic, so is (ψ, f ). proof. let {aλ : λ ∈ λ} be an uncountable scrambled set in x. then, {{aλ} : λ ∈ λ} is the desired scrambled set in the hyperspace. � as observed in [11], there is a dynamical system (x, f ) with zero topological entropy but (k(x), f ) has positive topological entropy. as proved in [3], if any system (x, f ) has positive topological entropy, then it is li-yorke chaotic. thus, it can be conduced that there can exist a system (x, f ) which is not li-yorke chaotic but its induced counterpart on some hyperspace is li-yorke chaotic. taking an example similar to that of [11], we now show that the existence of a scrambled set for the map f on the hyperspace need not guarantee the same for the map f on the base space x. example 2.26. let ∑ 2 be the sequence space of one sided infinite sequences of two symbols 0 and 1. let s = {an : n ∈ n} ⋃ {a}, where an ∈ ∑ 2 such that an is a sequence with 1 only at the n-th place and 0 at all other places and let a be the sequence of all 0’s. it can be seen that the set s is compact in ∑ 2 and is invariant under σ. hence we can talk of σ : s → s. it can be seen that under iterative application of the map σ, any point an reaches a in finitely many steps. for any n1, n2 ∈ n, if n = max {n1, n2}, then f n(an1 ) = f n(an2 ) = a. thus, there exists no scrambled set for the map σ on s. we, however show the existence of an uncountable scrambled set for (k(s), σ). let (an) = (2, 3, 5, 9, 17, . . .) = (2 n−1+1)n∈n and (bn) = (3, 4, 6, 10, 18, . . .) = (2n−1 + 2)n∈n be two fixed sequences of natural numbers. consider the collection p of all subsets of ∑ such that any two distinct sequences in any set in p differs at infinitely many places. then, the collection p of all such sets is a poset under the usual set inclusion. let a be its maximal element. we show that a is uncountable. any sequence (xn) ∈ a c eventually coincides with some sequence in a. if not so, then the sequence (xn) differs from every sequence in a at infinitely many places. thus, a ⋃ {(xn)} violates the maximality of a and thus any sequence in ac eventually coincides with some sequence in a. for any sequence (yn), the number of sequences eventually coinciding with (yn) are countable. thus, if a were countable, its complement will also be countable. this would imply that ∑ 2 is countable, which is a contradiction. thus, a is an uncountable set whose each element is itself a sequence. for any sequence z = (zn) ∈ a, define a sequence (b z n) of natural numbers as, bzn = { an, if zn = 0 bn, if zn = 1 let b be the set of all sequences thus generated. then, b is a collection of sequences of natural numbers. as any two sequences in a differ at infinitely topological dynamics on hyperspaces 17 many places, and so any two sequences in b will also differ at infinitely many places. also, as a is uncountable, b is uncountable. for any s = (sn) ∈ b, define ks = {asn : n ∈ n} ⋃ {a}. then ks is an element of k(s). as b is uncountable, we now have an uncountable subset of k(s) which we claim to be a scrambled set for σ. let this set be denoted by d. let kr, ks ∈ d. then, ks = {asn : n ∈ n} ⋃ {a}, kr = {arn : n ∈ n} ⋃ {a}. for sk 6= rk, dh (σ 2 k−2 +2(ks), σ 2 k−2 +2(kr)) = 1 22 k−2−1 and dh (σ 2 k−1 (ks), σ 2 k−1 (kr)) = 1 2 . as sk and rk differ for infinitely many k, the above relation also hold for infinitely many k. so lim inf n→∞ dh (σ n(kr), σ n(ks)) = 0 but lim sup n→∞ dh (σ n(kr), σ n(ks)) ≥ 1 2 . thus, d is an uncountable scrambled set in the hyperspace k(s). thus, existence of a scrambled set in the hyperspace does not guarantee the same in the base space. 3. conclusion in section 2, we have studied relations between the dynamical properties of the system (x, f ) and the induced system (ψ, f ), where ψ ⊆ cl(x) is any hyperspace endowed with some topology ∆. each of these topologies ∆ is of the form hit and miss or hit and far miss. we have seen that whenever ψ contains all finite subsets of x, the property of dense periodicity is preserved in (ψ, f ) for any topology ∆ on ψ. however, for any ψ ⊆ cl(x) with any topology ∆ on ψ, such property in (ψ, f ) need not conduce the same on (x, f ). however, the property of transitivity is equivalent for both (x, f ) and (ψ, f ) whenever ψ is large enough and is endowed with the upper vietoris topology. also it is guaranteed that transitivity will be preserved in (ψ, f ) when ψ is endowed with a topology coarser than the upper vietoris topology. again, transitivity in (ψ, f ) ensures the transitivity in (x, f ) when ψ is large enough and is endowed with the upper vietoris topology or any finer topology. if ψ contains all finite subsets of x, then given any topology ∆, the properties of weakly mixing and mixing are preserved in (ψ, f ). however, such properties on (ψ, f ) conduce the same on (x, f ) only when the topology ∆ is endowed with the upper vietoris topology or any finer topology. also such properties on (ψ, f ) are preserved if ψ is endowed with any topology coarser than the upper vietoris topology. thus, we can conclude that if the collection ψ contains all singletons and the topology ∆ on ψ is atleast upper vietoris, then these properties are equivalent on both (x, f ) and (ψ, f ). this generalizes the results in [1, 15] where ψ is either k(x) or cl(x) and ∆ is either the hausdorff metric topology or the vietoris topology. 18 p. sharma and a. nagar however, if topologies like wijsman topology, ball proximal topology or fell topology become finer than the upper vietoris topology, then they actually coincide with the vietoris topology. since such topologies in general are not finer than the upper vietoris topology, hence most of our observations cannot be established when ψ is endowed with any of these topologies. again when ψ is large enough to contain f1(x) then the existence of the horseshoe on the base space implies the same on the hyperspace. the converse need not be true. but when f1(x) ⊆ ψ ⊆ k(x) then for any topology finer than the upper vietoris topology the property ‘existence of horseshoe’ is equivalent for both the individual dynamics and the induced dynamics under some conditions. for the metric dependent properties, the comparison in the dynamic behaviour of the base map and the induced map is valid only when the hyperspace is endowed with the hausdorff metric topology. since the hausdorff metric preserves the metric on the base space. here again ‘uniform equicontinuity’ is preserved whenever the hyperspace ψ ⊂ k(x). however, if ψ is big enough to contain k(x), then an exact equivalence does not hold true. and finally, for li-yorke chaos the implication holds only in one direction. all these 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[19] z. yang, y. satoshi and c. guanhua, reduced density matrix and combined dynamics of electron and nuclei, journal of chemical physics 13, no. 10 (2000), 4016–4027. received july 2008 accepted january 2009 puneet sharma (puneet.iitd@yahoo.com) department of mathematics, indian institute of technology delhi, hauz khas, new delhi 110016, india. anima nagar (anima@maths.iitd.ac.in) department of mathematics, indian institute of technology delhi, hauz khas, new delhi 110016, india. nitkaiiagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 1, 2005 pp. 43-56 the character of free topological groups ii peter nickolas and mikhail tkachenko 1 abstract. a systematic analysis is made of the character of the free and free abelian topological groups on metrizable spaces and compact spaces, and on certain other closely related spaces. in the first case, it is shown that the characters of the free and the free abelian topological groups on x are both equal to the “small cardinal” d if x is compact and metrizable, but also, more generally, if x is a nondiscrete kω-space all of whose compact subsets are metrizable, or if x is a non-discrete polish space. an example is given of a zero-dimensional separable metric space for which both characters are equal to the cardinal of the continuum. in the case of a compact space x, an explicit formula is derived for the character of the free topological group on x involving no cardinal invariant of x other than its weight; in particular the character is fully determined by the weight in the compact case. this paper is a sequel to a paper by the same authors in which the characters of the free groups were analysed under less restrictive topological assumptions. 2000 ams classification: primary 22a05, 54h11, 54a25; secondary 54d30, 54d45 keywords: free (abelian) topological group, entourage of the diagonal, character, compact, locally compact, pseudocompact, metrizable, kω-space, dominating family 1the first author wishes to thank the second author, and his department, for hospitality extended during the course of this work. the second author was supported by mexican national council of sciences and technology (conacyt), grant no. 400200-5-28411-e. 44 p. nickolas and m. tkachenko 1. introduction in a previous paper [11], we investigated the topological character of free and free abelian topological groups. the results obtained were for the free groups on uniform spaces, with applications to the free groups on topological spaces deduced as appropriate. also, the principal results were obtained without the imposition of strong uniform or topological conditions on the given spaces, though numerous corollaries were derived at various points for metrizable spaces, compact spaces and other classes of spaces. in this sequel to [11], we specifically investigate the characters of the free and free abelian topological groups on metrizable spaces and on compact spaces, and on certain closely related spaces, obtaining more detailed information in both cases than was available in [11]. in the metrizable case, we show that the equality χ(a(x)) = χ(f(x)) = d holds if x is a compact metrizable space (as was already observed in [11]), but also if x is a non-discrete kω-space all compact subsets of which are metrizable (theorem 2.9), or if x is a non-discrete polish space (corollary 2.12). on the other hand, there exists a zero-dimensional separable metric space x such that χ(a(x)) = χ(f(x)) = c (example 2.18). if x is a metrizable space in which the subset of all non-isolated points is compact and non-empty, then χ(a(x)) = d (theorem 2.7), but under the same hypotheses the character χ(f(x)) may be arbitrarily large (example 2.8). in the case of a compact space x, our main result gives an explicit formula for χ(f(x)) involving no cardinal invariant of x other than the weight (theorem 3.5), showing in particular that the character is fully determined by the weight in the compact case. if the weight w(x) of x is at least c, then our result implies that χ(a(x)) = χ(f(x)) = w(x)ℵ0. our notation and terminology here are as in [11]. from time to time, results from [11] will be used here, and again the reader is referred to the source for these, though on occasion we quote them here for convenience. 2. free groups on metrizable spaces as usual, w(x,u) denotes the weight of the hausdorff uniform space (x,u), where by the weight we mean in all cases the least cardinal of a base of u, so that w(x,u) = ℵ0 implies in particular that the family of all uniform entourages of the diagonal in x2 does not have a minimal element. (a similar convention applies to our usage of other cardinal invariants.) the principal results of [11] on the characters of the free groups on (pseudo) metrizable spaces are the following three (see theorem 2.21 and corollaries 2.22 and 3.16, respectively). theorem 2.1. let (x,u) be an arbitrary uniform space with w(x,u) = ℵ0. then χ(a(x,u)) = d. corollary 2.2. if x is an infinite compact metrizable space, then χ(a(x)) = d. the character of free topological groups ii 45 following [11], we call a space x ω-narrow (equivalently, pseudo-ω1-compact) if every locally finite family of open sets in x is countable. it is clear that all lindelöf and all separable spaces are ω-narrow. corollary 2.3. χ(a(x)) = χ(f(x)) for every ω-narrow space x. from corollaries 2.2 and 2.3, we have: corollary 2.4. the equalities χ(f(x)) = χ(a(x)) = d hold for every infinite compact metrizable space x. brief comments were made in [11] about the character of a free topological group when equipped with the graev topology rather than the free topology. we make one further such observation. recall that if x is a topological space, then fg(x) denotes the abstract free group fa(x) over x topologized with graev’s topology, that is, the finest invariant group topology on fa(x) coarser than the topology of f(x). from corollary 2.6 of [11] it follows that χ(fg(x)) = χ(a(x)), for every space x. this equality combined with our corollary 2.3 implies the following result. theorem 2.5. if x is an ω-narrow space, then χ(fg(x)) = χ(f(x)). a further result from [11] is the following (theorem 2.23). in it, we use χ∆(x) to denote the character of the diagonal ∆ in x × x. theorem 2.6. if a tychonoff space x satisfies χ∆(x) ≤ ℵ0, then either x and a(x) are discrete or χ(a(x)) = d. starting from the above results, we develop here a sequence of new results which give more detailed information on the characters of the free and free abelian topological groups on metrizable spaces and certain spaces closely related to metrizable spaces. theorem 2.7. if x is a metrizable space and the set x′ of all non-isolated points of x is compact and non-empty, then χ(a(x)) = d. proof. we claim that χ∆(x) = ℵ0. indeed, let d be a metric on x which induces the topology of x. for every x ∈ x and ε > 0, denote by b(x,ε) the open ball with center at x and radius ε with respect to d. if n ∈ n, we put un = ∆ ∪ ⋃ {b(x,1/n) × b(x,1/n) : x ∈ x′}, where ∆ is the diagonal in x × x. it is easy to see that the sets un form a base at the diagonal ∆ in x × x, which proves our claim. now the desired conclusion follows from theorem 2.6. � it is interesting to note that the non-abelian analog of theorem 2.7 fails, as the next example shows. example 2.8. the character of the free topological group f(x) on a metrizable space x with a single non-isolated point can be arbitrarily large (while the character of a(x) is equal to d by theorem 2.7). 46 p. nickolas and m. tkachenko indeed, let x = c ⊕ d be the topological sum of a non-trivial convergent sequence c with limit point x0 ∈ c and a discrete space d of an infinite cardinality τ. for every a ∈ d, put ca = a −1x−10 ca, and consider the subspace y = ⋃ a∈d ca of f(x). as is shown in [1], y is homeomorphic to the fréchet–urysohn fan v (τ) of cardinality τ with vertex at the identity e of the group f(x). a straightforward diagonal argument shows that τ < χ(e,y ) ≤ χ(f(x)). it turns out that corollary 2.4 remains valid in a more general case. let us say that a space x with a kω-decomposition x = ⋃ n∈ω xn is a kmω-space if each xn is metrizable. equivalently, a kmω-space is a kω-space all compact subsets of which are metrizable. theorem 2.9. let x be a non-discrete kmω-space. then χ(a(x)) = d = χ(f(x)). proof. by assumption, there exists a kω-decomposition x = ⋃ n∈ω xn, where each xn is compact and metrizable. clearly xn is non-discrete for some n ∈ ω, for otherwise x would be discrete. therefore, x contains infinite compact subsets (convergent sequences), so corollary 2.18 of [11] and our corollary 2.3 together imply that d ≤ χ(a(x)) = χ(f(x)). it remains to verify that χ(f(x)) ≤ d. denote by y the one-point compactification of the topological sum x′ = ⊕n∈ωx ′ n, where x ′ n = xn × {n} for each n ∈ ω. it is easy to see that the infinite compact space y is metrizable, and so we have χ(f(y )) = d by corollary 2.4. choose an element a ∈ y and put z = ⋃ n∈ω an · x′n ⊆ f(y ). then z ∩ fn+1(y ) = ⋃n k=0 ak · x′k, so the intersection z ∩ fn+1(y ) is closed in fn+1(y ) for each n ∈ ω. by graev’s theorem in [4], ⋃ n∈ω fn(y ) is a kωdecomposition of the group f(y ), so z is closed in f(y ). therefore, f(y ) contains a subgroup topologically isomorphic to f(z) (see [8, th. 1]), and hence χ(f(z)) ≤ χ(f(y )). let f : x′ → x be the mapping defined by f(y,n) = y for all y ∈ xn, n ∈ ω. since x = ⋃ n∈ω xn is a kω-decomposition of x, the mapping f is quotient. clearly, f(x′) = x. define a mapping g : x′ → z by g(x) = anx for each x ∈ x′n, n ∈ ω. it is easy to see that g is a homeomorphism, so that the mapping h = f ◦ g−1 : z → x is a quotient. hence the extension of h to a homomorphism ĥ: f(z) → f(x) is continuous and open. we therefore conclude that χ(f(x)) ≤ χ(f(z)) ≤ χ(f(y )) = d. � by corollary 2.3, χ(f(x)) = χ(a(x)) for each separable metrizable space x. our next task is to calculate the values χ(a(q)) = χ(f(q)) and χ(a(rω)) = χ(f(rω)). here we show that all these cardinals are equal to d. this will follow from a more general result: if x is a non-discrete separable metrizable space which is absolutely gδ, fσ or gδσ, then χ(a(x)) = χ(f(x)) = d (see theorem 2.11). in particular, χ(a(x)) = χ(f(x)) = d for every non-discrete polish space x. lemma 2.10. let x be a non-discrete separable metrizable space such that x × ω ∼= x. then χ(a(x)) = χ(f(x)) = χ∆(x). the character of free topological groups ii 47 proof. since x is lindelöf and x × ω ∼= x, corollary 3.21 of [11] gives us χ(f(x)) = χ(a(x)) = χ∆(x × ω) = χ∆(x), as required. � we recall that a separable metrizable space x is called absolutely gδ, fσ or gδσ if x is of type gδ, fσ or gδσ, respectively, in some (equivalently, every) metrizable compactification of x. theorem 2.11. let x be a non-discrete separable metrizable space. if x is absolutely gδ, fσ or gδσ, then χ(a(x)) = χ(f(x)) = d. proof. since χ(f(x)) = χ(a(x)) by corollary 2.3, it suffices to verify that χ(a(x)) = d. if x is absolutely gδ, then it is a perfect image of a closed subspace k of the irrationals p ∼= nω, by (c) on page 144 of [2]. let f : k → x be the corresponding perfect mapping. then f extends to a continuous open homomorphism f̂ : a(k) → a(x), so that χ(a(x)) ≤ χ(a(k)). since k is closed in the separable metrizable space p, every continuous (pseudo)metric on k extends to a continuous (pseudo)metric on p, and theorem 1.2.9 of [10] and lemma 4 of [14] imply that a(k) is topologically isomorphic to a subgroup of a(p). hence χ(a(k)) ≤ χ(a(p)). note that p ∼= p × ω. since p is absolutely gδ, theorem 8.13 of [2] implies that χ∆(p) = d, and hence χ(a(p)) = d by lemma 2.10. since x is non-discrete (hence contains infinite compact subsets), from corollary 2.18 of [11] it follows that d ≤ χ(a(x)) ≤ χ(a(k)) ≤ χ(a(p)) = d. similarly, if x is absolutely fσ or gδσ, then so is the product x ×ω, and [2, th. 8.13] implies that χ∆(x ×ω) = d. since x ×ω ×ω ∼= x ×ω, we can apply lemma 2.10 to conclude that χ(a(x×ω)) = d. clearly, x is a continuous open image of x ×ω, which immediately implies that χ(a(x)) ≤ χ(a(x × ω)) = d. since x is non-discrete, an application of corollary 2.18 of [11] finishes the proof. � the above theorem implies, in particular, that χ(f(q)) = χ(f(k ×q)) = d for every compact metrizable space k. since complete separable metrizable (≡ polish) spaces are absolutely gδ, we obtain the following. corollary 2.12. if x is a non-discrete polish space, then d = χ(a(x)) = χ(f(x)). we therefore have, for example, χ(f(p)) = χ(f(rω)) = d. however, our results leave the following open problems. problem 2.13. does the inequality χ(f(x)) ≤ d hold for any absolutely borel (analytic) separable metrizable space x? problem 2.14. let x be an ω-narrow space such that w(x,u) ≤ d, where u is the fine uniformity of x. is then χ(f(x)) ≤ d? what if x × ω ∼= x? since the group a(x) on a separable metrizable space x is separable, its character does not exceed c. on the other hand, χ(a(x)) ≥ d for a non-discrete metrizable space x by corollary 2.18 of [11]. our aim is to show that there 48 p. nickolas and m. tkachenko exists in zfc a separable metrizable space x satisfying χ(a(x)) = χ(f(x)) = c. first, we study the character of x in its čech–stone compactification βx, and relate it with the character of the diagonal ∆x in the product x ×x. the straightforward proof of the next lemma is left to the reader. lemma 2.15. let bx be an arbitrary compactification of a tychonoff space x. then χ(x,bx) = χ(x,βx). the following result generalizes corollary 15 of [12]. lemma 2.16. if a space x is paracompact, then χ(x,βx) ≤ χ∆(x). proof. let b be a base for the diagonal ∆x in x 2 such that |b| = χ∆(x). for every u ∈ b, choose an open cover γ = γu of x such that ⋃ {v × v : v ∈ γ} ⊆ u. for every open set v in x, put ṽ = βx \ clβx(x \ v ). it is clear that ṽ is open in βx and ṽ ∩ x = v . in particular, v is dense in ṽ , and hence ṽ ⊆ clβx(v ). if u ∈ b, consider the family γ̃u = {ṽ : v ∈ γu } and the set wu = ⋃ γ̃u . then wu is open in βx and x ⊆ wu for each u ∈ b. we claim that the family λ = {wu : u ∈ b} is a base for x in βx. let w be an arbitrary open neighborhood of x in βx. put f = βx \ w and consider the closed subset p = x × f of x × βx. denote by ∆βx the diagonal in (βx)2. evidently, ∆x = (x × βx) ∩ ∆βx is closed in x × βx and p ∩ ∆x = ∅. since the product x × βx is normal (see [3, th. 5.1.38]), we can find disjoint open sets o and o′ in x × βx such that ∆x ⊆ o and p ⊆ o′. then there exists u ∈ b such that u ⊆ o ∩ (x × x). take an arbitrary element v ∈ γu and pick a point x ∈ v . since v × v ⊆ u ⊆ o, we have {x} ×v ⊆ o, and hence {x} ×clβx(v ) ⊆ clx×βx(o). by our choice, the sets o and o′ are disjoint and p = x × f ⊆ o′. therefore, clβx(v ) ∩ f = ∅. since ṽ ⊆ clβx(v ) and f = βx \w , we conclude that ṽ ⊆ w . this inclusion holds for each v ∈ γu , so wu = ⋃ γ̃u ⊆ w . this proves our claim. finally, from our definition of λ it follows follows that |λ| ≤ |b|, and hence χ(x,βx) ≤ χ∆(x). � let x be a paracompact space. then every open neighborhood of the diagonal ∆x in x 2 belongs to the fine uniformity u on x. this implies, in our notation, that w(x,u) = χ∆(x). since the free abelian topological group on x is precisely the free abelian topological group on the uniform space (x,u), corollary 2.11 of [11] implies the following result. corollary 2.17. let x be a paracompact space. then χ∆(x) ≤ χ(a(x)). by corollary 2.4, the character of the groups f(x) and a(x) on every infinite compact metrizable space x is equal to the cardinal d, which is consistently less than c (see [2, 15]). the equalities χ(f(x)) = χ(a(x)) = d remain valid for every non-discrete polish space x (see corollary 2.12). in the general case, the situation is different. the character of free topological groups ii 49 example 2.18. there exists a zero-dimensional separable metric space x such that the groups f(x) and a(x) both have character equal to c. indeed, let x and y be disjoint bernstein subsets of the real line r such that r = x ∪ y and |x| = |y | = c. then compact subsets of x and y are at most countable and both x and y are dense in r. hence x is a zero-dimensional separable metric space. clearly, the groups f(x) and a(x) are also separable, so their respective characters do not exceed c. since χ(a(x)) = χ(f(x)), by corollary 2.3, it suffices to verify that χ(a(x)) = c. denote by z the one-point compactification of r. then z is also a compactification of x. therefore, χ(x,z) = χ(x,βx) ≤ χ∆(x), by lemmas 2.15 and 2.16. since y is a bernstein subset of r, we have |r \ u| ≤ ℵ0 for every open set u in r containing x. in addition, the cardinality of y is equal to c, so we have c ≤ ψ(x,r) = ψ(x,z) ≤ χ(x,z) ≤ χ∆(x), where we use ψ to denote the pseudocharacter. this chain of inequalities and corollary 2.17 enable us to conclude that c ≤ χ∆(x) ≤ χ(a(x)) ≤ c. thus we have χ(f(x)) = χ(a(x)) = c. it may be worth remarking that if u′ is the natural metric uniformity on x inherited from r, then (x,u′) is an ω-narrow uniform space of countably infinite weight, and theorem 2.1 together with corollary 2.3 imply that χ(f(x,u′)) = χ(a(x,u′)) = d. 3. free groups on compact spaces let x be a compact hausdorff space. then by combining corollary 2.12 of [11] and our corollary 2.3, we obtain the inequality w(x) ≤ χ(a(x)) = χ(f(x)) ≤ w(x)ℵ0, which constitutes one of the main facts about the characters of the free groups on compact hausdorff spaces derived in [11]. in this section, we apply different methods to derive a great deal more detailed information about these characters. as just noted, the difference between f(x) and a(x) mentioned in example 2.8 disappears in the case when x is compact. in fact, we will show that the character of these groups depends only on the weight of x in the compact case. our proof of this fact requires several auxiliary results. first, we recall some definitions and notation used in [11]. a pair (p,≤) is a quasi-ordered set if ≤ is a reflexive transitive relation on the set p . if (p,≤) has the additional property of antisymmetry, then it is a partially ordered set. a set d ⊆ p is called dominating or cofinal in the quasi-ordered set (p,≤) if for every p ∈ p there exists q ∈ d such that p ≤ q. similarly, a subset e of p is said to be dense in (p,≤) if for every p ∈ p there exists q ∈ e with q ≤ p. the minimal cardinality of a dominating family in (p,≤) is denoted by d(p,≤) while we use d(p,≤) for the minimal cardinality 50 p. nickolas and m. tkachenko of a dense set in (p,≤). the notions of dominating and dense sets are dual: if a set s is dense in (p,≤), then it is dominating in (p,≥) and vice versa. therefore, d(p,≤) = d(p,≥) and d(p,≤) = d(p,≥). for a space x, denote by mx the family of all continuous mappings of x onto separable metrizable spaces. equivalently, since every separable metrizable space is homeomorphic to a subspace of iω, where i = [0,1], we can consider mx as a family of continuous mappings of x to i ω. if f : x → y and g : x → z are elements of mx, we say that f refines g or, in symbols, f ≺ g if there exists a continuous mapping ϕ: y → z such that g = ϕ◦f. also, following [11], denote by px the family of all continuous pseudometrics on the space x bounded by 1. for d1,d2 ∈ px, we write d1 ≤ d2 if d1(x,y) ≤ d2(x,y) for all x,y ∈ x. this gives us the quasi-ordered set (mx,≺) and the partially ordered set (px,≤). lemma 3.1. the equality d(px,≤) = d ·d(mx,≺) is valid for every infinite compact hausdorff space x. proof. by corollary 2.4 of [11] and our corollary 2.2, we have d(py ,≤) = χ(a(y )) ≤ d for every compact metrizable space y . let d be a continuous pseudometric on a given compact space x, with d ≤ 1. there exists a continuous mapping f : x → y onto a compact metrizable space y and a continuous metric ̺ on y such that d(x,y) = ̺(f(x),f(y)) for all x,y ∈ x. since d(py ,≤) ≤ d, we can find a dominating family df in (py ,≤) satisfying |df| ≤ d. for every κ ∈ df , define a continuous pseudometric κ̃ on x by κ̃(x,y) = κ(f(x),f(y)) for all x,y ∈ x. then d̃f = {κ̃ : κ ∈ df } ⊆ px for each f ∈ mx. let n be a dense subset of (mx,≺) satisfying |n| = d(mx,≺). it is easy to see that the family d = ⋃ {d̃f : f ∈ n} is dominating in (px,≤), so that d(px,≤) ≤ |d| ≤ d · |n| = d · d(mx,≺). conversely, let d be a dominating family in px such that |d| = d(px,≤). since x is compact, for every d ∈ d we can find a continuous mapping f = fd of x onto a compact metrizable space y and a continuous metric ̺ on y such that d(x,y) = ̺(f(x),f(y)) for all x,y ∈ x. then the set {fd : d ∈ d} is dense in (mx,≺). indeed, let g ∈ mx be arbitrary. then the image z = g(x) is a compact metrizable space. choose a metric ̺ ∈ pz which generates the topology of z and define a continuous pseudometric ˜̺ on x by ˜̺(x,y) = ̺(g(x),g(y)) for all x,y ∈ x. clearly ˜̺ ∈ px, so there exists d ∈ d such that ˜̺ ≤ d. an easy verification shows that fd ≺ g, and hence the family {fd : d ∈ d} is dense in (mx,≺). we conclude therefore that d(mx,≺) ≤ |d| = d(px,≤). finally, since d ≤ χ(a(x)) = d(px,≤) by corollaries 2.4 and 2.18 of [11], we apply the inequalities just proved to deduce that d(px,≤) ≤ d · d(mx,≺) ≤ d · d(px,≤) = d(px,≤), which implies the required equality. � now we present a theorem which summarizes several results established earlier. the character of free topological groups ii 51 theorem 3.2. d ≤ χ(a(x)) = χ(f(x)) = d(px,≤) = d · d(mx,≺) for every infinite compact space x. proof. combining corollaries 2.4 and 2.18 of [11] and lemma 3.1 of the present paper, we obtain d ≤ χ(a(x)) = d(px,≤) = d · d(mx,≺). since χ(a(x)) = χ(f(x)) by corollary 2.3, this proves the theorem. � on occasion, the exact calculation of d(px,≤) or d(mx,≺) for a compact space x can be a non-trivial task. in theorem 3.5, we give an explicit value for the character of the groups f(x) and a(x) on an infinite compact space x which avoids any reference to the quasi-ordered sets (px,≤) or (mx,≺). nevertheless, our proof of theorem 3.5 will involve the set (mx,≺) in an essential way, as well as the family cz(x) of all cozero-sets in x. as usual, we denote by [cz(x)]≤ω the collection of all countable subfamilies of cz(x). given γ,λ ∈ [cz(x)]≤ω, we write γ ≪ λ if every element of λ is the union of a subfamily of γ. this gives rise to the quasi-ordered set ([cz(x)]≤ω,≪). lemma 3.3. if x is compact hausdorff, then d([cz(x)]≤ω,≪) = d(mx,≺). proof. if w(x) ≤ ℵ0, then x is metrizable, so that d([cz(x)] ≤ω,≪) = 1 = d(mx,≺). suppose therefore that w(x) > ℵ0. for every u ∈ cz(x), fix a continuous function fu : x → i such that x \ u = f−1u (0). given a countably infinite subfamily γ = {un : n ∈ ω} of cz(x), we consider the corresponding diagonal product fγ = △n∈ωfun : x → iω. we also consider the analogously defined diagonal product into ik for some k ∈ n corresponding to any given finite subfamily of cz(x). this correspondence defines a mapping φ from [cz(x)]≤ω to mx, where φ is defined by φ(γ) = fγ, if we agree to restrict the range space of fγ to fγ(x). choose a dense set γ in ([cz(x)]≤ω,≪) of the minimal cardinality. we claim that the image φ(γ) is dense in the quasi-ordered set (mx,≺). indeed, let g be a continuous mapping of x onto a second countable space y . choose a countable base b for y and put λ = {g−1(v ) : v ∈ b}. then λ ∈ [cz(x)]≤ω, so we can find γ ∈ γ with γ ≪ λ. let us show that fγ = φ(γ) satisfies fγ ≺ g. suppose that x,y ∈ x and g(x) 6= g(y). then g(x) ∈ v 6∋ g(y) for some v ∈ b. since x ∈ g−1(v ) ∈ λ and γ ≪ λ, there exists u ∈ γ such that x ∈ u ⊆ g−1(v ). then y /∈ u. clearly fu(x) 6= fu(y), and from u ∈ γ it follows that fγ(x) 6= fγ(y). therefore, fγ(x) = fγ(y) always implies g(x) = g(y). this fact enables us to define a mapping h: fγ(x) → y such that g = h ◦ fγ. since g,fγ are continuous mappings and fγ is closed, we conclude that h is also continuous. hence fγ ≺ g. this proves our claim, and hence d(mx,≺) ≤ |φ(γ)| ≤ |γ| = d([cz(x)] ≤ω,≪). conversely, let n be a dense set in (mx,≺) of the minimal cardinality. choose a countable base b for iω and put γf = {f −1(v ) : v ∈ b} for each f ∈ mx. evidently, γf ∈ [cz(x)] ≤ω. let us verify that the set γ = {γf : f ∈ n} is dense in ([cz(x)]≤ω,≪). consider an arbitrary element λ = {un : n ∈ ω} 52 p. nickolas and m. tkachenko of [cz(x)]≤ω. as in the first part of the proof, for every n ∈ ω take the function gn = fun and put g = △n∈ωgn. by our choice of n , there exists f ∈ n with f ≺ g. then f ≺ gn, and hence un = f −1f(un) for each n ∈ ω. since f is a closed mapping, the sets f(un) are open in f(x). for n ∈ ω, apply the fact that b is a base for iω to choose a family µn ⊆ b such that f(un) = f(x) ∩ ⋃ µn. it follows that un = f −1f(un) = ⋃ {f−1(v ) : v ∈ µn}, and since {f−1(v ) : v ∈ µn} ⊆ γf for each n ∈ ω, we conclude that γf ≪ λ. this proves that γ is dense in ([cz(x)]≤ω,≪), whence d([cz(x)]≤ω,≪) ≤ |γ| ≤ |n| = d(mx,≺). the lemma is proved. � the use of the quasi-ordered set ([cz(x)]≤ω,≪) enables us to calculate the character of the group f(x) on a compact space x in purely set-theoretical terms. the result of this calculation turns out to be somewhat unexpected: χ(f(x)) = χ(f(y )) whenever the compact spaces x and y have the same weight (see corollary 3.6). let τ be an infinite cardinal. a subset y = {xα : α < τ} of a space x is called right-separated [7] if the set {xβ : β < α} is open in y for each α < τ. the next fact is well known in the folklore, but is proved here for the reader’s convenience. lemma 3.4. if x is compact, then x2 contains a right-separated subset of cardinality τ = w(x). proof. denote by ∆ the diagonal in x × x. then χ∆(x) = χ(∆,x 2) = w(x) = τ. let γ be an open cover of x2 \ ∆ such that the closure of each u ∈ γ does not intersect ∆. since ψ(∆,x2) = χ(∆,x2) = τ, the set x2 \ ∆ cannot be covered by less than τ elements of γ. therefore, we can construct by recursion a subset y = {xα : α < τ} of x 2 \ ∆ and a subfamily {uα : α < τ} of γ such that xα ∈ uα and xβ /∈ uα whenever α < β < τ. then the set y is as required. � theorem 3.5. if x is an infinite compact space of weight τ, then χ(f(x)) = χ(a(x)) = d · d([τ]≤ω,⊆). proof. note that d([ω]≤ω,⊆) = 1, so if w(x) = ℵ0, the required conclusion follows from corollary 2.4. hence we assume that w(x) = τ > ℵ0. put κ = d([τ]≤ω,⊆). first we show that χ(f(x)) ≤ d·κ. let 2 = {0,1} be the discrete doubleton. since w(x) = τ, we can find a closed subspace y of the cantor cube z = 2τ and a continuous onto mapping f : y → x. extend f to a continuous homomorphism f̂ : f(y ) → f(x). since f is a closed mapping, the homomorphism f̂ is open. therefore, χ(f(x)) ≤ χ(f(y )). in addition, y is compact, so f(y ) is topologically isomorphic to the subgroup f(y,z) of f(z) generated by y [4, §12], and hence χ(f(y )) ≤ χ(f(z)). from theorem 3.2 it follows that χ(f(z)) = d·d(mz,≺), so it suffices to verify that d(mz,≺) ≤ κ. for a non-empty a ⊆ τ, denote by πa the projection of z = 2 τ onto 2a. as is well known, every continuous mapping h: z → m to a metrizable space m depends on at most countably many coordinates [9, 6]. in other words, the character of free topological groups ii 53 there exists a countable set a ⊆ τ such that if x,y ∈ z and πa(x) = πa(y), then h(x) = h(y). hence we can define a mapping g : 2a → m satisfying g ◦ πa = h. since πa is an open mapping, we conclude that g is continuous. therefore, πa ≺ h. this means that the family {πa : a ∈ [τ] ≤ω} is dense in (mz,≺). it is clear, further, that if a set a ⊆ [τ] ≤ω is dominating in ([τ]≤ω,⊆), then the family {πa : a ∈ a} is dense in (mz,≺). this proves that d(mz,≺) ≤ d([τ] ≤ω,⊆) = κ, so that χ(f(x)) ≤ χ(f(z)) ≤ d · κ. to show that χ(f(x)) ≥ d·κ, we argue as follows. the group f(x) contains a closed subspace homeomorphic to x2, so f(x) also contains a subgroup topologically isomorphic to f(x2) [8]. hence χ(f(x2)) ≤ χ(f(x)). put y = x2. then χ(f(y )) = d·d([cz(y )]≤ω,≪) by theorem 3.2 and lemma 3.3. therefore, all we need to prove is that d([cz(y )]≤ω,≪) ≥ κ. let d be a dense set in ([cz(y )]≤ω,≪) of the minimal cardinality. it follows from lemma 3.4 that the space y = x2 contains a right-separated subset {xα : α < τ}. for every α < τ, choose a cozero set uα in y such that xα ∈ uα and xβ /∈ uα if α < β < τ. if u is a non-empty element of cz(y ), we define αu as the maximal element of the set {β < τ : xβ ∈ u} in the case when it exists, and αu = 0 otherwise. given a countable subfamily µ of cz(y ), put aµ = {αu : u ∈ µ}. we claim that the family a = {aµ : µ ∈ d} is dominating in ([τ]≤ω,⊆). indeed, let a be a countable subset of τ. then γ = {uα : α ∈ a} is an element of [cz(y )] ≤ω, so there exists µ ∈ d such that µ ≪ γ. by definition of the quasi-order ≪, for every α ∈ a there exists a subfamily µα ⊆ µ such that uα = ⋃ µα. hence µα contains an element v such that xα ∈ v ⊆ uα. in particular, α = αv , so that a ⊆ aµ. this proves that a is dominating in ([τ]≤ω,⊆). therefore, we have κ = d([τ]≤ω,⊆) ≤ |a| ≤ |d| = d([cz(y )]≤ω,≪). this finishes the proof. � corollary 3.6. if infinite compact spaces x and y satisfy w(x) = w(y ), then χ(a(x)) = χ(f(x)) = χ(f(y )) = χ(a(y )). proof. let g(z) be either a(z) or f(z), where z ∈ {x,y }. if w(x) = w(y ) = ℵ0, then χ(g(x)) = d = χ(g(y )) by corollary 2.4. if w(x) = w(y ) = τ > ℵ0, then we apply theorem 3.5 to conclude that χ(g(x)) = χ(g(y )) = d · d([τ]≤ω,⊆). � finally, one applies lemma 4.1 of the next section and theorem 3.5 to deduce the following two corollaries. corollary 3.7. let x be an infinite compact space satisfying w(x) < ℵω. then χ(a(x)) = χ(f(x)) = d · w(x). corollary 3.8. if a compact space x satisfies w(x) ≥ c, then χ(a(x)) = χ(f(x)) = w(x)ℵ0 . it is worth noting that if x and y are infinite compact spaces satisfying w(x) = ℵ0 and w(y ) = ℵ1, then nevertheless χ(f(x)) = χ(f(y )) = d. this follows easily from corollary 3.7 and the fact that d ≥ ℵ1. 54 p. nickolas and m. tkachenko 4. the possible values of the character it is of interest to discover which cardinal values the characters of free and free abelian topological groups can assume. by corollary 2.16 of [11], we know that the character of the groups f(x) and a(x) on a non-p-space x is at least d. as usual, we say that x is a p-space if every gδ-set in x is open. one can enquire whether there exists in zfc a space x such that χ(a(x)) = ℵ1 or χ(a(x)) = ℵ2, etc. we show below that the answer is affirmative. since the place of the cardinal d in the line of alephs is undefined in zfc, such a space x has necessarily to be a p-space. we start with a simple auxiliary fact, the very beginning of the pcf theory founded by shelah [13]. lemma 4.1. let τ be a cardinal. then: (a) d([τ]≤ω,⊆) = τ if τ = ℵn for some integer n ≥ 1; (b) if τ ≥ c, then d([τ]≤ω,⊆) = τω. proof. first we note that τ ≤ d([τ]≤ω,⊆) for every τ > ℵ0, because τ can be partitioned into τ disjoint countably infinite subsets, and these cannot be covered by any collection of fewer than τ countable subsets. (a) it suffices to show that d([ℵn] ≤ω,⊆) ≤ ℵn. if n = 1, then the required dominating family in ([ℵ1] ≤ω,⊆) is {α : α < ω1}. suppose that the lemma holds for some integer n ≥ 1. by assumption, for every uncountable ordinal α < ℵn+1 there exists a dominating family γα in ([α] ≤ω,⊆) satisfying |γα| = |α| ≤ ℵn. put γ = ⋃ {γα : ω1 ≤ α < ℵn+1}. then |γ| ≤ ℵn+1, and it is easy to see that γ is dominating in ([ℵn+1] ≤ω,⊆). indeed, if a is a countable subset of ℵn+1, then a ⊆ α for some uncountable α < ℵn+1, and hence there exists b ∈ γα with a ⊆ b. since γα ⊆ γ, this proves that γ is dominating in ([ℵn+1] ≤ω,⊆). therefore, d([ℵn+1] ≤ω,⊆) ≤ ℵn+1. (b) the case τ = c is trivial, so we assume that τ > c. suppose that γ = {ti : i ∈ i} is a dominating subset of ([τ] ≤ω,⊆) of the minimal cardinality. it is clear that the number of elements t of [τ]≤ω such that t ⊆ ti for any fixed i ∈ i is at most c, and that the cardinality of [τ]≤ω is τω. therefore, we have the inequality τω ≤ c · |i|. but using the assumption that τ > c, we have τω ≥ τ > c, and hence c < c · |i|. it follows that |i| > c, and therefore that c · |i| = |i|, from which we have |i| ≥ τω. since we know already that |i| ≤ τω, we finally have |i| = τω, as required. � proposition 4.2. let p∗ be the one-point lindelöfication of a discrete space p of cardinality τ > ℵ0. then χ(f(p ∗)) = χ(a(p∗)) = d([τ]≤ω,⊆). proof. put κ = d([p ]≤ω,⊆) = d([τ]≤ω,⊆). suppose that p∗ = p ∪ {x∗}, where x∗ is the unique non-isolated point in p∗. first, we consider the group f(p∗). for every countable subset k of p , denote by uk the minimal normal subgroup of f(p∗) containing the set p∗ \ k. by lemma 2.9 of [5], the family {uk : k ∈ [p ] ≤ω} is a base at the identity in f(p∗). note that if k,l ∈ [p ]≤ω and k ⊆ l, then ul ⊆ uk. choose a dominating family γ in ([p ] ≤ω,⊆) with the character of free topological groups ii 55 |γ| = κ. then {uk : k ∈ γ} is again a base at the identity in f(p ∗), and hence χ(f(p∗)) ≤ |γ| = κ. in addition, p∗ is a subspace of f(p∗), so κ = d([p ]≤ω,⊆)) = χ(x∗,p∗) ≤ χ(f(p∗)). we have thus proved that χ(f(p∗)) = κ. since the lindelöf space p∗ is ω-narrow, corollary 2.3 implies that χ(a(p∗)) = κ. � combining lemma 4.1 and proposition 4.2, we obtain: corollary 4.3. let p∗ be the one-point lindelöfication of a discrete space p of cardinality τ > ℵ0. then χ(f(p ∗)) = χ(a(p∗)) = τ if ℵ1 ≤ τ < ℵω, and χ(f(p∗)) = χ(a(p∗)) = τω if τ ≥ c. note that it follows in particular that when the (uncountable) cardinality of a discrete space p is sufficiently small, then it is consistent with zfc that p∗, the one-point lindelöfication of p , satisfies χ(f(p∗)) = τ < τω, but that the situation differs markedly if p has sufficiently large cardinality. references [1] a. v. arhangel’skii, o. g. okunev and v. g. pestov, free topological groups over metrizable spaces, topology appl. 33 (1989), 63–76. [2] e. k. van douwen, the integers and topology, in: handbook of set-theoretic topology, k. kunen and j.e. vaughan, eds., elsevier science publ. b. v., 1984, pp. 111–167. [3] r. engelking, general topology, heldermann verlag, berlin, 1989. [4] m. i. graev, free topological groups, in: topology and topological algebra, translations series 1, vol. 8 (1962), pp. 305–364, american mathematical society. russian original in: izvestiya akad. nauk sssr ser. mat. 12 (1948), 279–323. [5] c. hernández, d. robbie and m. tkachenko, some properties of o-bounded and strictly o-bounded groups, applied general topology 1(1) (2000), 29–43. [6] m. hušek, continuous mappings on subspaces of products, symp. math. 17 (1976), 25–41. [7] i. juhász, cardinal functions in topology, math. centre tracts 34, amsterdam 1971. [8] e. katz, s. a. morris and p. nickolas, characterization of bases of subgroups of free topological groups, j. london math. soc. 27(2) (1983), 421–426. [9] y. mibu, on baire functions on infinite product spaces, proc. japan acad. 20 (1944), 661–663. [10] p. nickolas, free topological groups and free products of topological groups, phd thesis, university of new south wales, australia, 1976. [11] p. nickolas and m. g. tkachenko, the character of free topological groups i, this volume. [12] v. i. ponomarev and v. v. tkachuk, countable character of x in βx in comparison with countable character of the diagonal of x in x × x, vestnik mosk. univ. no. 5 (1987), 16–19 (in russian). [13] s. shelah, cardinal arithmetic, oxford logic guides, 29, the clarendon press, oxford university press, new york, 1994. [14] m. g. tkachenko, on the completeness of free abelian topological groups, soviet math. dokl. 27 (1983), 341–345. russian original in: dokl. an sssr 269 (1983), 299–303. [15] j. e. vaughan, small uncountable cardinals and topology, in: problems in topology, j. van mill and g.m. reed, eds., north-holland publ. co., amsterdam (1990), 195–218. 56 p. nickolas and m. tkachenko received november 2002 accepted june 2004 p. nickolas (peter−nickolas@uow.edu.au) department of mathematics and applied statistics, university of wollongong, nsw 2522, australia m. tkachenko (mich@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, av. san rafael atlixco 186, col. vicentina, del. iztapalapa, c.p. 09340, méxico, d.f. alastamarizagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 67-76 the čech number of cp(x) when x is an ordinal space ofelia t. alas and ángel tamariz-mascarúa ∗ abstract. the čech number of a space z, č(z), is the pseudocharacter of z in βz. in this article we obtain, in zf c and assuming sch, some upper and lower bounds of the čech number of spaces cp(x) of realvalued continuous functions defined on an ordinal space x with the pointwise convergence topology. 2000 ams classification: 54c35, 54a25, 54f05 keywords: spaces of continuous functions, topology of pointwise convergence, čech number, ordinal space 1. notations and basic results in this article, every space x is a tychonoff space. the symbols ω (or n), r, i, q and p stand for the set of natural numbers, the real numbers, the closed interval [0, 1], the rational numbers and the irrational numbers, respectively. given two spaces x and y , we denote by c(x, y ) the set of all continuous functions from x to y , and cp(x, y ) stands for c(x, y ) equipped with the topology of pointwise convergence, that is, the topology in c(x, y ) of subspace of the tychonoff product y x . the space cp(x, r) is denoted by cp(x). the restriction of a function f with domain x to a ⊂ x is denoted by f ↾ a. for a space x, βx is its stone-čech compactification. recall that for x ⊂ y , the pseudocharacter of x in y is defined as ψ(x, y ) = min{|u| : u is a family of open sets in y and x = ⋂ u}. definition 1.1. (1) the čech number of a space z is č(z) = ψ(z, βz). (2) the k-covering number of a space z is kcov(z) = min{|k| : k is a compact cover of z}. ∗research supported by fapesp, conacyt and unam. 68 o. t. alas and á. tamariz-mascarúa we have that (see section 1 in [8]): č(z) = 1 if and only if z is locally compact; č(z) ≤ ω if and only if z is čech-complete; č(z) = kcov(βz \ z); if y is a closed subset of z, then kcov(y ) ≤ kcov(z) and č(y ) ≤ č(z); if f : z → y is an onto continuous function, then kcov(y ) ≤ kcov(z); if f : z → y is perfect and onto, then kcov(y ) = kcov(z) and č(y ) = č(z); if bz is a compactification of z, then č(z) = ψ(z, bz). we know that č(cp(x)) ≤ ℵ0 if and only if x is countable and discrete ([7]), and č(cp(x, i)) ≤ ℵ0 if and only if x is discrete ([9]). for a space x, ec(x) (the essential cardinality of x) is the smallest cardinality of a closed and open subspace y of x such that x \ y is discrete. observe that, for such a subspace y of x, č(cp(x, i)) = č(cp(y, i)). in [8] it was pointed out that ec(x) ≤ č(cp(x, i)) and č(cp(x)) = |x| · č(cp(x, i)) always hold. so, if x is discrete, č(cp(x)) = |x|, and if |x| = ec(x), č(cp(x)) = č(cp(x, i)). consider in the set of functions from ω to ω, ωω, the partial order ≤∗ defined by f ≤∗ g if f (n) ≤ g(n) for all but finitely many n ∈ ω. a collection d of (ωω, ≤∗) is dominating if for every h ∈ ωω there is f ∈ d such that h ≤∗ f . as usual, we denote by d the cardinal number min{|d| : d is a dominating subset of ωω}. it is known that d = kcov(p) (see [3]); so d = č(q). moreover, ω1 ≤ d ≤ c, where c denotes the cardinality of r. we will denote a cardinal number τ with the discrete topology simply as τ ; so, the space τ κ is the tychonoff product of κ copies of the discrete space τ . the cardinal number τ with the order topology will be symbolized by [0, τ ). in this article we will obtain some upper and lower bounds of č(cp(x, i)) when x is an ordinal space; so this article continues the efforts made in [1] and [8] in order to clarify the behavior of the number č(cp(x, i)) for several classes of spaces x. for notions and concepts not defined here the reader can consult [2] and [4]. 2. the čech number of cp(x) when x is an ordinal space for an ordinal number α, let us denote by [0, α) and [0, α] the set of ordinals < α and the set of ordinals ≤ α, respectively, with its order topology. observe that for every ordinal number α ≤ ω, [0, α) is a discrete space, so, in this case, č(cp([0, α), i)) = 1. if ω < α < ω1, then [0, α) is a countable metrizable space, hence, by theorem 7.4 in [1], č(cp([0, α), i)) = d. we will analyze the number č(cp([0, α), i)) for an arbitrary ordinal number α. we are going to use the following symbols: notations 2.1. for each n < ω, we will denote as en the collection of intervals [0, 1/2n+1), (1/2n+2, 3/2n+2), (1/2n+1, 2/2n+1), (3/2n+2, 5/2n+2), ... ..., ((2n+2 − 2)/2n+2, (2n+2 − 1)/2n+2), ((2n+1 − 1)/2n+1, 1]. observe that en is an irreducible open cover of [0, 1] and each element in en has diameter = 1/2n+1. for a set s and a point y ∈ s, we will use the symbol [ys]<ω in order to denote the collection of finite subsets of s containing y. the čech number of cp(x) when x is an ordinal space 69 moreover, if γ and α are ordinal numbers with γ ≤ α, [γ, α] is the set of ordinal numbers λ which satisfy γ ≤ λ ≤ α. the expression α0 < α1 < ... < αn < ... ր γ will mean that the sequence (αn)n<ω of ordinal numbers is strictly increasing and converges to γ. lemma 2.2. let γ be an ordinal number such that there is ω < α0 < α1 < ... < αn < ... ր γ. then č(cp([0, γ], i) ≤ č(cp([0, γ), i) · kcov(|γ| ω). proof. for m < ω, f ∈ [γ[αm, γ]] <ω = {m ⊂ [αm, γ] : |m| < ℵ0 and γ ∈ m} and n < ω, define b(m, f, n) = ⋃ e∈en b(m, f, e) where b(m, f, e) = ∏ x∈[0,γ] jx with jx = e if x ∈ f , and jx = i otherwise. (so, b(m, f, n) is open in i[0,γ].) define b(m, n) = ⋂ {b(m, f, n) : f ∈ [γ[αm, γ]] <ω}. observe that b(m, n) is the intersection of at most |γ| open sets b(m, f, n). define g(n) = ⋃ m<ω b(m, n), and g = ⋂ n<ω g(n). claim: g is the set of all functions g : [0, γ] → [0, 1] which are continuous at γ. proof of the claim: let g : [0, γ] → [0, 1] be continuous at γ. given n < ω there is e ∈ en such that g(γ) ∈ e. since g is continuous at γ, there is β < γ so that g(t) ∈ e if t ∈ [β, γ]. fix m < ω so that β < αm. for every f ∈ [γ[αm, γ]] <ω we have that g ∈ b(m, f, e) ⊂ b(m, f, n); hence, g ∈ b(m, n) ⊂ g(n). we conclude that g belongs to g. now, let h ∈ g. we are going to prove that h is continuous at γ. assume the contrary, that is, there exist ǫ > 0 and a sequence t0 < t1 < ... < tn < ... ր γ such that (1) |f (tj ) − f (γ)| ≥ ǫ, for every j < ω. fix n < ω such that 1/2n+1 < ǫ. since h ∈ g, then h ∈ g(n) and there is m ≥ 0 such that h ∈ b(m, n). choose tnp > αm and take f = {tnp , γ}. thus h ∈ b(m, f, n), but if e ∈ en and h(γ) ∈ e, then h(tnp ) 6∈ e, which is a contradiction. so, the claim has been proved. now, we have i[0,γ] \ g = ⋃ n<ω (i[0,γ] \ g(n)), and i[0,γ] \ g(n) = ⋂ m<ω ⋃ f ∈γ[αm,γ]ω (i[0,γ] \ b(m, f, n)). so i[0,γ] \ g(n)) is an f|γ|δ-set. by corollary 3.4 in [8], kcov(i [0,γ] \ g(n)) ≤ kcov(|γ|ω). hence, č(g) = kcov(i[0,γ] \ g) ≤ ℵ0 · kcov(|γ| ω). thus, it follows that č(cp([0, γ], i) ≤ č(cp([0, γ), i) · kcov(|γ| ω ). � 70 o. t. alas and á. tamariz-mascarúa lemma 2.3. if γ < α, then č(cp([0, γ), i)) ≤ č(cp([0, α), i)). proof. first case: γ = β + 1. in this case, [0, γ) = [0, β] and the function φ : [0, α) → [0, β] defined by φ(x) = x if x ≤ β and φ(x) = β if x > β is a quotient. so, φ# : cp([0, β], i) → cp([0, α), i) defined by φ #(f ) = f ◦φ, is a homeomorphism between cp([0, β], i) and a closed subset of cp([0, α), i) (see [2], pages 13,14). then, in this case, č(cp([0, γ), i)) ≤ č(cp([0, α), i)). now, in order to finish the proof of this lemma, it is enough to show that for every limit ordinal number α, č(cp([0, α), i)) ≤ č(cp([0, α], i)). let κ = cof (α), and α0 < α1 < ... < αλ < ... ր α with λ < κ. for each of these λ, we know, because of the proof of the first case, that κλ = č(cp([0, αλ], i)) ≤ č(cp([0, α], i)). let, for each λ < κ, {v λ ξ : ξ < κλ} be a collection of open subsets of i[0,αλ] such that cp([0, αλ], i) = ⋂ ξ<κλ v λξ . for each λ < κ and each ξ < κλ, we take w λ ξ = v λ ξ × i (αλ,α). each w λξ is open in i[0,α) and ⋂ λ<κ ⋂ ξ<κλ w λξ = cp([0, α), i)). therefore, č(cp([0, α), i)) ≤ κ · sup{κλ : λ < κ} ≤ κ · č(cp([0, α], i)). but κ ≤ |α| = ec([0, α]) ≤ č(cp([0, α], i)). then, č(cp([0, α), i)) ≤ č(cp([0, α], i)). � lemma 2.4. let α be a limit ordinal number > ω. then č(cp([0, α), i)) = |α| · supγ<αč(cp([0, γ), i)). in particular, č(cp([0, α), i)) = supγ<αč(cp([0, γ), i)) if cof (α) < α. proof. by lemma 2.3, supγ<αč(cp([0, γ), i)) ≤ č(cp([0, α), i)), and, by corollary 4.8 in [8], |α| ≤ č(cp([0, α), i)). for each γ < α, we write κγ instead of č(cp([0, γ), i)). let {v γ λ : λ < κγ} be a collection of open sets in iγ such that cp([0, γ), i) = ⋂ λ<κγ v γ λ . now we put w γ λ = v γ λ × i[γ,α)]. we have that w γ λ is open for every γ < α and every λ < γ, and cp([0, α), i) = ⋂ γ<α ⋂ λ<κγ w γ λ . so, č(cp([0, α), i)) = |α| · supγ<αč(cp([0, γ), i)). � in order to prove the following result it is enough to mimic the prove of 5.12.(c) in [5]. lemma 2.5. if α is an ordinal number with cof (α) > ω and f ∈ cp([0, α), i)), then there is γ0 < α for which f ↾ [γ0, α) is a constant function. lemma 2.6. if α is an ordinal number with cofinality > ω, then č(cp([0, α], i)) = č(cp([0, α), i)). proof. let κ = č(cp([0, α), i)). there are open sets vλ (λ < κ) in i [0,α) such that cp([0, α), i) = ⋂ λ<κ vλ. for each λ < κ, we take wλ = vλ × i {α}. each wλ is open in i [0,α] and ⋂ λ<κ wλ = {f : [0, α] → i | f ↾ [0, α) ∈ cp([0, α), i)}. for each (γ, ξ, e) ∈ α×α×en, we take b(γ, ξ, e) = ∏ λ≤α jλ where jλ = e if λ ∈ {ξ + γ, α}, and jλ = i otherwise. let b(γ, ξ, n) = ⋃ e∈en b(γ, ξ, e). the čech number of cp(x) when x is an ordinal space 71 finally, we define b(γ) = ⋃ ξ<α b(γ, ξ, n), which is an open subset of i[0,α]. we denote by m the set ⋂ λ<κ wλ ∩ ⋂ γ<α b(γ). we are going to prove that cp([0, α], i) = m . let f ∈ cp([0, α], i). we know that f ∈ ⋂ λ<κ wλ, so we only have to prove that f ∈ ⋂ γ<α b(γ). for n < ω, there is e ∈ en such that f (α) ∈ e. since f ∈ c([0, α], i), there are γ0 < α and r0 ∈ i such that f (λ) = r0 if γ0 ≤ λ < α. let χ < α such that χ + γ ≥ γ0. thus, f ∈ b(γ, χ, n) ⊂ b(γ). therefore, cp([0, α], i) ⊂ m . take an element f of m . since f ∈ ⋂ λ<α wλ, f is continuous at every γ < α, thus f ↾ [γ0, α) = r0 for a γ0 < α and an r0 ∈ i. for each n < ω, and each γ ≥ γ0, f ∈ b(γ, ξ, n) for some ξ < α. then, |r0 − f (α)| = |f (γ + ξ) − f (α)| < 1/2 n. but, these relations hold for every n. so, f (α) must be equal to r0, and this means that f is continuous at every point. therefore, č(cp([0, α], i)) ≤ |α| · č(cp([0, α), i)). since č(cp([0, α), i)) ≥ ec([0, α)) = |α|, č(cp([0, α], i)) ≤ č(cp([0, α), i)). finally, lemma 2.3 gives us the inequality č(cp([0, α), i)) ≤ č(cp([0, α], i)). � theorem 2.7. for every ordinal number α > ω, |α| · d ≤ č(cp([0, α), i)) ≤ kcov(|α| ω ). proof. because of theorem 7.4 in [1], corollary 4.8 in [8] and lemma 2.3 above, |α| · d ≤ č(cp([0, α), i)). now, if ω < α < ω1, we have that č(cp([0, α), i)) ≤ kcov(|α| ω ) because of corollary 4.2 in [1]. we are going to finish the proof by induction. assume that the inequality č(cp([0, γ), i)) ≤ kcov(|γ| ω ) holds for every ω < γ < α. by lemma 2.4 and inductive hypothesis, if α is a limit ordinal, then č(cp([0, α), i)) ≤ |α| · supγ<αkcov(|γ| ω) ≤ kcov(|α|ω ). if α = γ0+2, then č(cp([0, α), i)) = č(cp([0, γ0+1), i)) ≤ kcov(|γ0+1| ω) = kcov(|α|ω ). now assume that α = γ0 + 1, γ0 is a limit and cof (γ0) = ω. we know by lemma 2.2 that č(cp([0, γ0 + 1), i)) ≤ č(cp([0, γ0), i) · kcov(|γ0| ω). so, by inductive hypothesis we obtain what is required. the last possible case: α = γ0 + 1, γ0 is limit and cof (γ0) > ω. by lemma 2.6, we have č(cp([0, γ0 + 1), i)) = |α| · č(cp([0, γ0), i). by inductive hypothesis, č(cp([0, γ0), i) ≤ kcov(|α| ω ). since |α| ≤ kcov(|α|ω ), we conclude that č(cp([0, α), i)) ≤ kcov(|α| ω ). � as a consequence of proposition 3.6 in [8] (see proposition 2.11, below) and the previous theorem, we obtain: corollary 2.8. for an ordinal number ω < α < ωω, č(cp([0, α), i)) = |α| · d. 72 o. t. alas and á. tamariz-mascarúa in particular, we have: corollary 2.9. č(cp([0, ω1), i)) = č(cp([0, ω1], i)) = d. by using similar techniques to those used throughout this section we can also prove the following result. corollary 2.10. for every ordinal number α > ω and every 1 ≤ n < ω, |α| · d ≤ č(cp([0, α) n, i)) ≤ kcov(|α|ω ). for a generalized linearly ordered topological space x, χ(x) ≤ ec(x), so χ(x) ≤ č(cp(x, i)), where χ(x) is the character of x. this is not the case for every topological space, even if x is a countable eg-space, as was pointed out by o. okunev to the authors. indeed, let x be a countable dense subset of cp(i). we have that χ(x) = χ(cp(i)) = c and č(cp(x, i)) = d. so, it is consistent with zf c that there is a countable eg-space x with χ(x) > č(cp(x, i)). one is tempted to think that for every linearly ordered space x, the relation č(cp(x, i)) ≤ kcov(χ(x) ω) is plausible. but this illusion vanishes quickly; in fact, when d < 2ω and x is the doble arrow, then x has countable character and ec(x) = |x| = 2ω. hence, č(cp(x, i)) ≥ 2 ω > d = kcov(χ(x)ω) (compare with theorem 2.7, above, and corollary 7.7 in [1]). in [8] the following was remarked: proposition 2.11. (1) for every cardinal number ω ≤ τ < ωω, kcov(τ ω ) = τ · d, (2) for every cardinal τ ≥ λ, kcov((τ +)λ) = τ + · kcov(τ λ), and, (3) if cf (τ ) > λ, then kcov(τ λ) = τ · sup{kcov(µλ) : µ < τ}. lemma 2.12. for every cardinal number κ with cof (κ) = ω, we have that kcov(κω) > κ. proof. let {kλ : λ < κ} be a collection of compact subsets of κ ω. let α0 < α1 < ... < αn < ... be an strictly increasing sequence of cardinal numbers converging to κ. we are going to prove that ⋃ λ<κ kλ is a proper subset of κω. denote by πn : κ ω → κ the n-projection. since πn is continuous and kλ is compact, πn(kλ) is a compact subset of the discrete space κ, so, it is finite. thus, we have that | ⋃ λ<αn πn(kλ)| ≤ αn < κ for each n < ω. hence, for every n < ω, we can take ξn ∈ κ \ ⋃ λ<αn πn(kλ). consider the point ξ = (ξn)n<ω of κω. we claim that ξ 6∈ ⋃ λ<κ kλ. indeed, assume that ξ ∈ kλ0 . there is n < ω such that λ0 < αn. so, ξn ∈ ⋃ λ<αn πn(kλ) which is not possible. � recall that the singular cardinals hypothesis (sch) is the assertion: for every singular cardinal number κ, if 2cof (κ) < κ, then κcof (κ) = κ+. a proposition, apparently weaker than sch, is: “for every cardinal number κ with cof (κ) = ω, if 2ω < κ, then κω = κ+.” but this last assertion is equivalent to sch as was settled by silver (see [6], theorem 23). the čech number of cp(x) when x is an ordinal space 73 proposition 2.13. if we assume sch and c ≤ (ωω) +, and if τ is an infinite cardinal number, then (∗) kcov(τ ω ) =    τ · d if ω ≤ τ < ωω τ if τ > ωω and cof (τ ) > ω τ + if τ > ω and cof (τ ) = ω proof. our proposition is true for every ω ≤ τ < ωω because of (1) in proposition 2.11. assume now that κ ≥ ωω and that (∗) holds for every τ < κ. we are going to prove the assertion for κ. case 1: cof (κ) = ω. by lemma 2.12, kcov(κω) > κ. on the other hand, kcov(κω) ≤ κω. first two subcases: either c < ωω or κ > ωω. in both subcases, we can apply sch and conclude that kcov(κω) = κ+. third subcase: c = (ωω) + and κ = ωω. in this case we have kcov((ωω ) ω) ≤ (ωω) ω ≤ cω = c = (ωω) +. moreover, by lemma 2.12, (ωω) + ≤ kcov((ωω ) ω). therefore, kcov((ωω ) ω) = (ωω) +. case 2: cof (κ) > ω. by proposition 2.11 (3), kcov(κω) = κ · sup{kcov(µω) : ω ≤ µ < κ}. by inductive hypothesis we have that for each µ < κ (∗∗) kcov(µω ) =    µ · d if ω ≤ µ < ωω µ if µ > ωω and cof (µ) > ω µ+ if µ > ω and cof (µ) = ω first subcase: κ is a limit cardinal. for every µ < κ, kcov(µω ) < κ (because of (∗∗) and because we assumed that κ > (ωω) + ≥ c ≥ d); and so sup{kcov(µω) : µ < κ} = κ. thus, kcov(κω ) = κ. second subcase: assume now that κ = µ+0 . in this case, by proposition 2.11, kcov(κω) = κ·kcov(µω0 ). because of (∗∗) and because µ0 ≥ ωω, kcov(µ0) ω ≤ κ. we conclude that kcov(κω) = κ. � proposition 2.14. let κ be a cardinal number with cof (κ) = ω. then č(cp([0, κ], i)) > κ. proof. let 0 = α0 < α1 < · · · < αn < . . . be a strictly increasing sequence of cardinal numbers converging to κ. assume that {vλ : λ < κ} is a collection of open sets in i[0,κ] which satisfies cp([0, κ], i) ⊂ ⋂ λ<κ vλ. we are going to prove that ⋂ λ<κ vλ contains a function h : [0, κ] → i which is not continuous. in order to construct h, we are going to define, by induction, the following sequences: 74 o. t. alas and á. tamariz-mascarúa (i) elements t0, . . . , tn, ... which belong to [0, κ] such that (1) 0 = t0 < t1 < · · · < tn < . . . , (2) ti ≥ αi for each 0 ≤ i < ω, (3) each ti is an isolated ordinal, and (4) κ = lim(tn); (ii) subsets g0, ..., gn, ... ⊂ [0, κ] with |gi| ≤ αi for every i < ω, and such that each function which equals 0 in gi and 1 in {t0, ..., ti} belongs to ⋂ λ<αi vλ for every 0 ≤ i < ω and ( ⋃ n gn) ∩ {t0, ..., tn, ...} = ∅; (iii) functions f0, f1, ..., fn, ... such that f0 ≡ 0, and fi is the characteristic function defined by {t0, ..., ti−1} for each 0 < i < ω. let f0 be the constant function equal to 0. assume that we have already defined t0, ..., ts−1, g0, ..., gs−1 and f0, ..., fs−1. we now choose an isolated point ts ∈ [αs, κ] \ g0 ∪ ... ∪ gs−1 (this is possible because |g0 ∪ ... ∪ gs−1| < κ). consider the characteristic function defined by {t0, ..., ts−1, ts}, fs. this function is continuous, so it belongs to ⋂ λ<αs vλ. for each λ < αs, there is a canonical open set asλ of the form [fs; x s 1, ..., x s ns(λ) ; 1/ms(λ)] = {f ∈ i[0,κ] : |fs(x s i ) − f (x s i )| < 1/m s(λ) ∀ 1 ≤ i ≤ ns(λ)} satisfying fs ∈ a s λ ⊂ vλ. for each λ < αs we take f s λ = {x s 1, ..., x s ns(λ) }. put gs = ⋃ λ<αs f sλ \ {t0, ..., ts}. it happens that {f ∈ i[0,κ] : f (x) = 0 ∀ x ∈ gs and f (ti) = 1 ∀ 0 ≤ i ≤ s} is a subset of ⋂ λ<αs vλ. this finishes the inductive construction of the required sequences. now, consider the function h : [0, κ] → [0, 1] defined by h(x) = 0 if x 6∈ {t0, ..., tn, ...}, and h(tn) = 1 for every n < ω. this function h is not continuous at κ because h(κ) = 0, κ = lim(tn), and h(tn) = 1 for all n < ω. now, take λ0 ∈ κ. there exists l < ω such that λ0 < αl. since h is equal to 0 in gl and 1 in {t0, ..., tl}, then h ∈ ⋂ λ<αl vλ. therefore, h ∈ vλ0 . so, cp([0, κ], i) is not equal to ⋂ λ<κ vλ. this means that č(cp([0, κ], i)) > κ. � theorem 2.15. sch + c ≤ (ωω) + implies: č(cp([0, α), i)) =                    1 if α ≤ ω |α| · d if α > ω and ω ≤ |α| < ωω |α| if |α| > ωω and cof (|α|) > ω |α| if cof (|α|) = ω and α is a cardinal number > ωω |α| if |α| = ωω and d < (ωω) + |α|+ if cof (|α|) = ω, |α| > ωω, α is not a cardinal number |α|+ if |α| = ωω and d = (ωω) + proof. if α ≤ ω, cp([0, α), i) = i [0,α), so č(cp([0, α), i)) = 1. if α > ω and ω ≤ |α| < ωω, we obtain our result because of theorem 2.7 and proposition 2.13. if |α| > ωω and cof (|α|) > ω, by theorem 2.7 and proposition 2.13, |α| · d = |α| ≤ č(cp([0, α), i)) ≤ kcov(|α| ω ) = |α|. the čech number of cp(x) when x is an ordinal space 75 if cof (|α|) = ω and α is a cardinal number > ωω, by lemma 2.4, č(cp([0, α), i)) = |α| · supγ<αč(cp([0, γ), i)). the number α is a limit ordinal and for every γ < α, č(cp([0, γ), i)) ≤ |γ| + · d. since d ≤ (ωω) + < |α|, then č(cp([0, α), i)) = |α|. by lemma 2.4 and theorem 2.7, if |α| = ωω, then ωω · d ≤ č(cp([0, α), i)) = |α| · supγ<αč(cp([0, γ), i)) ≤ |α| · supγ<α(|γ| + · d). thus, if |α| = ωω and d < (ωω) +, č(cp([0, α), i)) = |α|. assume now that cof (|α|) = ω, |α| > ωω and α is not a cardinal number. there exists a cardinal number κ such that κ = |α| and [0, α) = [0, κ]⊕[κ+1, α). so, č(cp([0, α), i)) = č(cp([0, κ], i)) · č(cp([κ + 1, α), i)) = č(cp([0, κ], i)) (see proposition 1.10 in [8] and lemma 2.3). by theorem 2.7 and proposition 2.14, κ · d ≤ č(cp([0, κ], i)) ≤ κ +. being κ a cardinal number > ωω with cofinality ω, it must be > (ωω) +; so κ > d and, then, κ ≤ č(cp([0, κ], i)) ≤ κ +. now we use proposition 2.14, and conclude that č(cp([0, α), i)) = κ + = |α|+. finally, assume that |α| = ωω and d = (ωω) +. by theorems 2.7 and proposition 2.13 we have |α| · d ≤ č(cp([0, α), i)) ≤ kcov(|α| ω ) = (ωω ) +. and we conclude: č(cp([0, α), i)) = |α| +. � references [1] o. t. alas and á. tamariz-mascarúa, on the c̆ech number of cp(x), ii q & a in general topology 24 (2006), 31–49. [2] a. v. arkhangel’skii, topological function spaces, kluwer academic publishers, 1992. [3] e. van douwen, the integers and topology, in handbook of set-theoretic topology, north-holland, 1984, amsterdam–new-york–oxford, 111–167. [4] r. engelking, general topology, pwn, warszawa, 1977. [5] l. gillman and m. jerison, rings of continuous functions, springer-verlag, new-yorkheidelberg-berlin, 1976. [6] t. jech, set theory, academic press, new-york-san francisco-london, 1978. [7] d. j. lutzer and r. a. mccoy, category in function spaces, i, pacific j. math. 90 (1980), 145–168. [8] o. okunev and a. tamariz-mascarúa, on the c̆ech number of cp(x), topology appl. 137 (2004), 237–249. [9] v. v. tkachuk, decomposition of cp(x) into a countable union of subspaces with “good” properties implies “good” properties of cp(x), trans. moscow math. soc. 55 (1994), 239–248. 76 o. t. alas and á. tamariz-mascarúa received august 2006 accepted february 2007 ofelia t. alas (alas@ime.usp.br) universidade de são paulo, caixa postal 66281, cep 05311-970, são paulo, brasil. ángel tamariz-mascarúa (atamariz@servidor.unam.mx) departamento de matemáticas, facultad de ciencias, universidad nacional autónoma de méxico, ciudad universitaria, méxico d.f. 04510, méxico. () @ applied general topology c© universidad politécnica de valencia volume 11, no. 2, 2010 pp. 95-115 closed injective systems and its fundamental limit spaces marcio colombo fenille abstract. in this article we introduce the concept of limit space and fundamental limit space for the so-called closed injected systems of topological spaces. we present the main results on existence and uniqueness of limit spaces and several concrete examples. in the main section of the text, we show that the closed injective system can be considered as objects of a category whose morphisms are the so-called cismorphisms. moreover, the transition to fundamental limit space can be considered a functor from this category into the category of topological spaces and continuous maps. later, we show results about properties on functors and counter-functors for inductive closed injective system and fundamental limit spaces. we finish with the presentation of some results of characterization of fundamental limit spaces for some special systems and the study of the so-called perfect properties. 2000 ams classification: 18a05, 18a30, 18b30, 54a20, 54b17. keywords: closed injective system, limit space, category, functoriality, compatibility of limits, perfect property. 1. introduction the purpose of this article is to introduce and study what we call the category of closed injective systems and cis-morphisms and the concept of limit spaces of such systems. we start by defining the so-called closed injective systems (cis to shorten), and the concepts of limit space for such systems. we have particular interest in a special type of limit space, those we call fundamental limit space. in section 3 we introduce this concept and we demonstrate theorems of existence and uniqueness of fundamental limit spaces. in section 4 we present some very illustrative examples. 96 m. c. fenille section 5 is one of the most important and interesting for us. there we show that a closed injective system can be considered as a object of a category, whose morphisms are the so-called cis-morphisms, which we define in this occasion. furthermore, we prove that this category is complete with respect to direct limits, that is, all inductive system of cis’s and cis-morphisms has a direct limit. in section 6, we prove that the transition to the fundamental limit can be considered as a functor from the category of cis’s and cis-morphisms into the category of topological spaces and continuous maps. in section 7, we show that the transition to the direct limit on the category of cis’s and cis-morphisms is compatible (in a way) to transition to the fundamental limit space. in section 8, we study a class of special cis’s called inductive closed injective systems. in the two following sections, we study the action of functors and counter-functors, respectively, in such systems, and present some simple applications of the results demonstrated. we finish with the presentation of some results of characterization of fundamental limit space for some special systems, the so-called finitely-semicomponible and stationary systems, and the study of the so-called perfect properties over topological spaces of a system and over its fundamental limit spaces. 2. closed injective systems and limit spaces let {xi} ∞ i=0 be a countable collection of nonempty topological spaces. for each i ∈ n, let yi be a closed subspace of xi. assume that, for each i ∈ n, there exists a closed injective continuous map fi : yi → xi+1. this structure is called closed injective system, or cis, to shorten. we write {xi,yi,fi} to represent this system. moreover, by injection we mean a injective continuous map. we say that two injection fi and fi+1 are semicomponible if fi(yi)∩yi+1 6= ∅. in this case, we can define a new injection fi,i+1 : f −1 i (yi+1) → xi+2 by fi,i+1(y) = (fi+1 ◦ fi)(y), for all y ∈ f −1 i (yi+1). for convenience, we put fi,i = fi. moreover, we say that fi is always semicomponible with itself. also, we write fi,i−1 to be the natural inclusion of yi into xi for each i ∈ n. given i,j ∈ n, j > i + 1, we say that fi and fj are semicomponible if fi,k and fk+1 are semicomponible for all i + 1 ≤ k ≤ j − 1, where fi,k : f −1 i,k−1(yk) → xk+1 is defined inductively. to facilitate the notations, if fi and fj are semicomponible, we write yi,j = f −1 i,j−1(yj ), cis’s and its fundamental limit spaces 97 that is, yi,j is the domain of the injection fi,j . according to the agreement fi,i = fi, we have yi,i = yi. lemma 2.1. if fi and fj are semicomponible, i < j, then fk and fl are semicomponible, for any integers k,l with i ≤ k ≤ l ≤ j. if fi and fj are not semicomponible, then fi and fk are not semicomponible, for any integers k > j. lemma 2.2. if fi and fj are semicomponible, with i < j, then yi,j = (fj−1 ◦ · · · ◦ fi) −1(yj ) and fi,j (yi,j ) = (fj ◦ fi,j−1)(yi,j−1). the proofs of above results are omitted. henceforth, since products of maps do not appear in this paper, we can sometimes omit the symbol ◦ in the composition of maps. definition 2.3. let {xi,yi,fi} be a cis. a limit space for this system is a topological space x and a collection of continuous maps φi : xi → x satisfying the conditions: l.1. x = ⋃∞ i=0 φi(xi); l.2. each φi : xi → x is a imbedding; l.3. φi(xi) ∩φj (xj ) . = φjfi,j−1(yi,j−1) if i < j and fi and fj are semicomponible; l.4. φi(xi) ∩ φj (xj ) = ∅ if fi and fj are not semicomponible; where . = indicates, besides the equality of sets, the following: if x ∈ φi(xi) ∩ φj (xj ), say x = φi(xi) = φj (xj ), with xi ∈ xi and xj ∈ xj , then we have necessarily xi ∈ yi,j−1 and xj = fi,j−1(xi). remark 2.4. the “pointwise identity” indicated by . = in l.3 reduced to identity of sets indicates only that φi(xi) ∩ φj (xj ) = φi(yi,j−1) ∩ φjfi,j−1(yi,j−1). the existence of different interpretations of condition l.3 is very important. furthermore, equivalent conditions to those of the definition can be very useful. the next results give us some practical interpretations and equivalences. lemma 2.5. let {x,φi} be a limit space for the cis {xi,yi,fi} and suppose that fi and fj are semicomponible, i < j. then φjfi,j−1(yi) = φi(yi) for yi ∈ yi,j−1. proof. let yi ∈ yi,j−1 be a point. by condition l.3 we have φjfi,j−1(yi) ∈ φi(xi), that is, φjfi,j−1(yi) = φi(xi) for some xi ∈ xi. again by condition l.3, xi ∈ yi,j−1 and fi,j−1(xi) = fi,j−1(yi). since each fk is injective, also fi,j−1 is injective. therefore xi = yi, which implies φjfi,j−1(yi) = φi(yi). � lemma 2.6. let {x,φi} be a limit space for the cis {xi,yi,fi} and suppose that fi and fj are semicomponible, with i < j. then φi(xi − yi,j−1) ∩ φj (xj − fi,j−1(yi,j−1)) = ∅. 98 m. c. fenille proof. it is obvious that if x ∈ φi(xi − yi,j−1) ∩ φj (xj − fi,j−1(yi,j−1)) then x ∈ φi(xi) ∩ φj (xj ) . = φjfi,j−1(yi,j−1). but this is a contradiction, since φj is an imbedding, and so φj (xj − fi,j−1(yi,j−1)) = φj (xj ) − φjfi,j−1(yi,j−1). � proposition 2.7. let {xi,yi,fi} be an arbitrary cis and let φi : xi → x be imbedding into a topological space x = ∪∞i=0φi(xi) satisfying the following properties: l.4. φi(xi) ∩ φj (xj ) = ∅ always that fi and fj are not semicomponible; l.5. φjfi,j−1(yi) = φi(yi) for every yi ∈ yi,j−1, always that fi and fj are semicomponible, with i < j; l.6. φi(xi − yi,j−1) ∩ φj (xj − fi,j−1(yi,j−1)) = ∅, always that fi and fj are semicomponible, with i < j. then {x,φi} is a limit space for the cis {xi,yi,fi}. proof. we will prove that condition l.3 is true. suppose that fi and fj are semicomponible, with i < j. by condition l.5, the sets φi(xi) ∩ φj (xj ) and φjfi,j−1(yi,j−1) are nonempty. we will prove that they are pointwise equal. let x ∈ φi(xi)∩φj (xj ), say x = φi(xi) = φj (xj ) with xi ∈ xi and xj ∈ xj . suppose, by contradiction, that xi /∈ yi,j−1. then φi(xi) ∈ φi(xi − yi,j−1). by condition l.6 we must have φj (xj ) = φi(xi) /∈ φj (xj − fi,j−1(yi,j−1)), that is, φj (xj ) ∈ φjfi,j−1(yi,j−1). so xj ∈ fi,j−1(yi,j−1). thus, there is yi ∈ yi,j−1 such that fi,j−1(yi) = xj . by condition l.5, φi(yi) = φjfi,j−1(yi) = φj (xj ). however, φj (xj ) = φi(xi). it follows that φi(yi) = φi(xi), and so xi = yi ∈ yi,j−1, which is a contradiction. therefore xi ∈ yi,j−1. in order to prove the remaining, take x ∈ φi(xi) ∩ φj (xj ), x = φi(yi) = φj (xj ), with yi ∈ yi,j−1 and xj ∈ xj . we must prove that xj = fi,j−1(yi). by condition l.5, φjfi,j−1(yi) = φi(yi) = φj (xj ). thus, the desired identity is obtained by injectivity. this proves that φi(xi) ∩ φj (xj ) . = φjfi,j−1(yi,j−1) and so that {x,φi} is a limit space for {xi,yi,fi}. � corollary 2.8. condition l.3 can be replaced by both together l.5 and l.6. proof. lemmas 2.5 e 2.6 and proposition 2.7 implies that. � theorem 2.9. let {xi,yi,fi} be a cis. assume that {x,φi} and {z,ψi} are two limit spaces for this cis. then there is a unique bijection (not necessarily continuous) β : x → z such that ψi = β ◦ φi for every i ∈ n. proof. define β : x → z in the follow way: for each x ∈ x, we have x = φi(xi), for some xi ∈ xi. then, we define β(x) = ψi(xi). we have: • β is well defined. let x ∈ x be a point with x = φi(xi) = φj (xj ), where xi ∈ xi, xj ∈ xj and i < j. then x ∈ φi(xi) ∩ φj (xj ) . = φjfi,j−1(yi,j−1) and xj = fi,j−1(xi) by condition l.3. thus ψj (xj ) = ψjfi,j−1(xi) = ψi(xi), where the latter identity follows from condition l.3. • β is injective. suppose that β(x) = β(y), x,y ∈ x. consider x = φi(xi) and y = φj (yj ), xi ∈ xi, yj ∈ xj, i < j (the case where j < i is symmetrical and the case where i = j is trivial). then ψi(xi) = β(x) = β(y) = ψj (yj ). cis’s and its fundamental limit spaces 99 it follows that ψi(xi) = ψj (yj ) ∈ ψi(xi) ∩ ψj (xj ) . = ψjfi,j−1(yi,j−1). by the condition l.3, xi ∈ yi,j−1 and yj = fi,j−1(xi). by condition l.5, it follows that φi(xi) = φjfi,j−1(xi) = φj (yj ). therefore x = y. • β is surjective. let z ∈ z be an arbitrary point. then z = ψi(xi) for some xi ∈ xi. take x = φi(xi) and we have β(x) = z. the uniqueness is trivial. � 3. fundamental limit space in this section, we define the main concept of this paper, namely, the fundamental limit space for a closed injective system. definition 3.1. let {x,φi} be a limit space for the cis {xi,yi,fi}. we say that x has the weak topology (induced by the collection {φi}i∈n) if the following sentence is true: a ⊂ x is closed in x ⇔ φ−1i (a) is closed in xi for every i ∈ n. when this occurs, we say that {x,φi} is a fundamental limit space for {xi,yi,fi}. proposition 3.2. let {x,φi} be fundamental limit space for the cis {xi,yi,fi}. then φi(xi) is closed in x for every i ∈ n. proof. we will prove that φ−1j (φi(xi)) is closed in xj for any i,j ∈ n. we have φ−1j (φi(xi)) =    xi if i = j ∅ if i < j, fi and fj not semicomponible ∅ if i > j, fj and fi not semicomponible fi,j−1(yi,j−1) if i < j and fi and fj are semicomponible fj,i−1(yj,i−1) if i > j and fj and fi are semicomponible in the first three cases it is obvious that φ−1j (φi(xi)) is closed in xj . in the fourth case we have the following: if j = i + 1, then fi,j−1(yi,j−1) = fi(yi), which is closed in xi+1, since fi is a closed map. for j > i + 1, since fi is continuous and yi+1 is closed in xi+1, then yi,i+1 = f −1 i (yi+1) is closed in xi. thus, since fi is closed, lemma 2.2 shows that fi,i+1(yi,i+1) = fi+1fi(yi,i) = fi+1fi(yi), which is closed in xi+1. again by lemma 2.2 we have fi,j−1(yi,j−1) = fj−1fi,j−2(yi,j−2). thus, by induction it follows that fi,j−1(yi,j−1) is closed in xj . the fifth case is similar to the fourth. � corollary 3.3. let {x,φi} be a fundamental limit space for the cis {xi,yi,fi}. if x is compact, then each xi is compact. proof. each xi is homeomorphic to the closed subspace φi(xi) of x. � proposition 3.4. let {x,φi} and {z,ψi} be two limit spaces for the cis {xi,yi,fi}. if {x,φi} is a fundamental limit space for {xi,yi,fi}, then the bijection β : x → z in theorem 2.9 is continuous. proof. let a be a closed subset of z. we have β−1(a) = ∪∞i=0φi(ψ −1 i (a)) and φ−1j (β −1(a)) = ψ−1j (a). since ψj is continuous and x has the weak topology, we have that β−1(a) is closed in x. � 100 m. c. fenille theorem 3.5 (uniqueness of the fundamental limit space). let {x,φi} and {z,ψi} be two fundamental limit spaces for the cis {xi,yi,fi}. then, the bijection β : x → z in theorem 2.9 is a homeomorphism. moreover, β is the unique homeomorphism from x onto z such that ψi = β ◦ φi for every i ∈ n. proof. let β′ : z → x be the inverse map of the bijection β. by preceding proposition, β and β′ are both continuous maps. therefore β is a homeomorphism. the uniqueness is the same of theorem 2.9. � theorem 3.6 (existence of fundamental limit space). every closed injective system has a fundamental limit space. proof. let {xi,yi,fi} be an arbitrary cis. define x̃ = x0∪f0 x1 ∪f1 x2 ∪f2 · · · to be the quotient space obtained of the coproduct (or topological sum) ∐∞ i=0 xi by identifying each yi ⊂ xi with fi(yi) ⊂ xi+1. define each ϕ̃i : xi → x̃ to be the projection from xi into the quotient space x̃. then {x̃,ϕ̃i} is a fundamental limit space for the given cis {xi,yi,fi}. � the latter two theorems implies that every cis has, up to homeomorphisms, a unique fundamental limit space. this will be remembered and used many times in the article. 4. examples of cis’s and limit spaces in this section we show some interesting examples of limit spaces. the first example is very simple and the second shows the existence of a limit space which is not a fundamental limit space. this example will be highlighted in the last section of the article. the other examples show known spaces as fundamental limit spaces. example 4.1 (identity limit space). let {xi,yi,fi} be the cis with yi = xi = x and fi = idx for every i ∈ n, where x is an arbitrary topological space and idx : x → x is the identity map of x. it is easy to see that {x,idx} is a fundamental limit space for {xi,yi,fi}. example 4.2 (existence of limit space which is not a fundamental limit space). assume x0 = [0, 1) and y0 = {0}. take xi = yi = [0, 1] for each i ≥ 1. let f0 : y0 → x1 be the inclusion f(0) = 0 and fi = identity for each i ≥ 1. consider the sphere s1 as a subspace of r2. define the maps φ0 : x0 → s 1 by φ0(t) = (cos πt,− sin πt) and φi : xi → s 1 by φi(t) = (cos πt, sin πt), for each i ≥ 1. it is easy to see that s1 = ⋃∞ i=0 φi(xi) and each φi is an imbedding onto its image. moreover, φi(xi) ∩φj (xj ) . = φjfi,j−1(yi), which implies condition l.3. therefore, {s1,φi} is a limit space for the cis {xi,yi,fi}. however, this limit space is not a fundamental limit space, since φ0(x0) is not closed in s 1, (or again, since s1 is compact though x0 is not). (see figure 1 below). cis’s and its fundamental limit spaces 101 0 x 1 x 2 xf 0 1 f id 1 s figure 1. limit space (not fundamental) 0 x 1 x 2 x id x 1 y 0 y figure 2. fundamental limit space now, we consider the subspace x = {(x, 0) ∈ r2 : 0 ≤ x ≤ 1} ∪ {(0,y) ∈ r 2 : 0 ≤ y < 1} of r2. define the maps ψ0 : x0 → x by ψ0(t) = (0, t) and ψi : xi → x by ψi(t) = (t, 0), for each i ≥ 1. we have x = ⋃∞ i=0 ψi(xi), where each φi is an imbedding onto its image, such that ψi(xi) is closed in x. moreover, since ψi(xi) ∩ ψj (xj ) . = ψjfi,j−1(yi), it follows that {x,ψi} is a fundamental limit space for the cis {xi,yi,fi}. (see figure 2 above). note that the bijection β : s 1 → x of theorem 2.9 is not continuous here. example 4.3 (the infinite-dimensional sphere s∞). for each n ∈ n, we consider the n-dimensional sphere sn = {(x1, . . . ,xn+1) ∈ r n+1 : x21 + · · · + x 2 n+1 = 1}, and the “equatorial inclusions” fn : s n → sn+1, defined by fn(x1, . . . ,xn+1) = (x1, . . . ,xn+1, 0). then {s n,sn,fn} is a cis. its fundamental limit space is {s∞,φn}, where s ∞ is the infinite-dimensional sphere and, for each n ∈ n, the imbedding φn : s n → s∞ is the natural “equatorial inclusion”. example 4.4 (the infinite-dimensional torus t ∞). for each n ≥ 1, we consider the n-dimensional torus t n = ∏n i=1 s 1 and the closed injections fn : t n → t n+1 given by fn(x1, . . . ,xn) = (x1, . . . ,xn, (1, 0)), where each xi ∈ s1. then {t n,t n,fn} is a cis, whose fundamental limit space is {t ∞,φn}, where t ∞ = ∏∞ i=1 s 1 is the infinite-dimensional torus and, for each n ∈ n, the imbedding φn : t n → t ∞ is the natural inclusion φn(x1, . . . ,xn) = (x1, . . . ,xn, (1, 0), (1, 0), . . .). example 4.3 is a particular case of the following one: example 4.5 (cw-complexes as fundamental limit spaces for its skeletons). let k be an arbitrary cw-complex. for each n ∈ n, let kn be the n-skeleton of k and consider the natural inclusions ln : k n → kn+1 of the n-skeleton into the (n + 1)-skeleton. if the dimension dim(k) of k is finite, then we put km = k and lm : k m → km+1 to be the identity map, for every m ≥ dim(k). it is well known that a cw-complex has the weak topology with respect to their skeletons, that is, a subset a ⊂ k is closed in k if and only if a∩kn is closed 102 m. c. fenille in kn for every n. thus, {kn,kn, ln} is a cis, whose fundamental limit space is {k,φn}, where each φn : k n → k is the natural inclusions of the n-skeleton kn into k. for details of the cw-complex theory see [2] or [6]. the example below is a consequence of the previous one. example 4.6 (the infinite-dimensional projective space rp∞). there is always a natural inclusion fn : rp n → rpn+1, which is a closed injective continuous map. (the n-dimensional projective space rpn is the n-skeleton of (n + 1)-dimensional projective space rpn+1). it follows that {rpn, rpn,fn} is a cis. the fundamental limit space for this cis is the infinite-dimensional projective space rp∞. for details about infinite-dimensional sphere and projective plane see [2]. 5. the category of cis’s and cis-morphisms let x = {xi,yi,fi}i and z = {zi,wi,gi}i be two closed injective systems. by a cis-morphism h : x → z we mean a collection h = {hi : xi → zi}i of closed continuous maps checking the following conditions: 1. hi(yi) ⊂ wi for every i ∈ n. 2. hi+1 ◦ fi = gi ◦ hi|yi for every i ∈ n. this latter condition is equivalent to commutativity of the following diagram for each i ∈ n: yi fi �� hi|yi // wi gi �� xi+1 hi+1 // zi+1 we say that a cis-morphism h : x → z is a cis-isomorphism if each map hi : xi → zi is a homeomorphism which carries yi homeomorphicaly onto wi. for each arbitrary cis, say x = {xi,yi,fi}i, there is an identity cismorphism 1 : x → x given by 1i : xi → xi equal to identity map for each i ∈ n. moreover, if h : x(1) → x(2) and k : x(2) → x(3) are two cis-morphisms, then it is clear that its natural composition k ◦ h : x(1) → x(3) is a cis-morphism from x(1) into x(3). also, it is easy to check that associativity of compositions holds whenever possible: if h : x(1) → x(2), k : x(2) → x(3) and r : x(3) → x(4), then r ◦ (k ◦ h) = (r ◦ k) ◦ h. this shows that the closed injective system and the cis-morphisms form a category, which we denote by cis. (see [3] for details on basic category theory). cis’s and its fundamental limit spaces 103 theorem 5.1. all inductive systems on category cis admit limit. proof. let {x(n), h(mn)}m,n be an inductive system of closed injective system and cis-morphisms. each x(n) is of the form x(n) = {x (n) i ,y (n) i ,f (n) i }i and each h(mn) : x(m) → x(n) is a cis-morphism and, moreover, h(pq) ◦ h(qr) = h(pr) for every p,q,r ∈ n. for each m ∈ n, we write h(m) to be h(mn) when m = n + 1. for each i ∈ n, we have the inductive system {x (n) i ,h (mn) i }m,n, that is, the injective system of the topological spaces x (1) i ,x (2) i , . . . and all continuous maps h (mn) i : x (m) i → x (n) i , m,n ∈ n, of the collection h (mn). now, each inductive system {x (n) i ,h (mn) i }m,n can be consider as the closed injective system {x (n) i ,x (n) i ,h (n) i }n. let {xi,ξ (n) i }n be a fundamental limit space for {x (n) i ,x (n) i ,h (n) i }n. xi h (qm) i // x (m) i h (mn) i // ξ (m) i 44iiiiiiiiiiiiiiiiiiiiii x (n) i h (np) i // ξ (n) i == {{{{{{{{ then, each ξ (n) i : x (n) i → xi is an imbedding and we have ξ (m) i = φ (n) i ◦h (mn) i for all m < n. moreover, xi has a weak topology induced by the collection {ξ (n) i }n. for any m,n ∈ n, with m ≤ n, we have ξ (m) i (y (m) i ) = ξ (n) i ◦ h (mn) i (y (m) i ) ⊂ ξ (n) i (y (n) i ), by condition 1 of the definition of cis-morphism. moreover, each ξ (n) i (y (n) i ) is closed in xi, since each ξ (n) i is an imbedding. for each i ∈ n, we define yi = ⋃ n∈n ξ (n) i (y (n) i ). then, by preceding paragraph, yi is a union of linked closed sets, that is, yi is the union of the closed sets of the ascendent chain ξ (1) i (y (1) i ) ⊂ ξ (2) i (y (2) i ) ⊂ · · · ⊂ ξ (m) i (y (m) i ) ⊂ ξ (m+1) i (y (m+1) i ) ⊂ · · · now, since {xi,ξ (n) i }n is a fundamental limit space for {x (n) i ,y (n) i ,h (n) i }n, for each m ∈ n, we have (ξ (m) i ) −1(yi) = (ξ (m) i ) −1(∪n∈nξ (n) i (y (n) i )) = y m i which is closed in x (m) i . therefore, since xi has the weak topology induced by the collection {ξ (n) i }n, it follows that yi is closed in xi. now, we will build, for each i ∈ n, an injection fi : yi → xi+1 making {xi,yi,fi}i a closed injective system. for each i ∈ n, we have the diagram shown below. 104 m. c. fenille y (n) i f (n) i �� ξ (n) i // ξ (n) i (y (n) i ) fi �� x (n) i+1 ξ (n) i+1 // xi+1 for each x ∈ ξ (n) i (y (n) i ) ⊂ xi, there is a unique y ∈ y (n) i such that ξ (n) i (y) = x. then, we define fi(x) = (ξ (n) i+1 ◦ f (n) i )(y). it is clear that each fi : ξ (n) i (y (n) i ) → xi+1 is a closed injective continuous map, since each ξi and f (n) i are closed injective continuous maps. now, we define fi : yi → xi+1 in the following way: for each x ∈ yi, there is an integer n ∈ n such that x ∈ ξ (n) i (y n i ). then, there is a unique y ∈ y (n) i such that ξ (n) i (y) = x. we define fi(x) = (ξ (n) i+1 ◦ f (n) i )(y). each fi : yi → xi+1 is well defined. in fact: suppose that x belong to ξ (m) i (y m i ) ∩ ξ (n) i (y n i ). suppose, without loss of generality, that m < n. there are unique ym ∈ y m i and yn ∈ y n i such that ξ (m) i (ym) = y = φ (n) i (yn). then, yn = h (mn) i (ym). thus, ξ (n) i+1◦f (n) i (yn) = ξ (n) i+1◦f (n) i ◦h (mn) i (ym) = ξ (n) i+1◦h (mn) i+1 ◦f (m) i (ym) = ξ (m) i+1 ◦f (m) i (ym). now, since each fi : yi → xi+1 is obtained from a collection of closed injective continuous maps which coincides on closed sets, it follows that each fi is a closed injective continuous map. this proves that {xi,yi,fi}i is a closed injective system. denote it by x. for each n ∈ n, let e(n) : x(n) → x be the collection e(n) = {ξ (n) i : x (n) i → xi}i. it is clear by the construction that e(n) is a cis-morphism from x(n) into x. moreover, we have e(m) = h(mn) ◦ e(n). therefore, {x,e(n)}n is a direct limit for the inductive system {x(n), h(mn)}m,n. � 6. the transition to fundamental limit space as a functor henceforth, we will write top to denote the category of the topological spaces and continuous maps. for each cis x = {xi,yi,fi}i, we will denote its fundamental limit space by £(x). the passage to the fundamental limit defines a function £ : cis −→ top which associates to each cis x its fundamental limit space £(x) = {x,φi}. theorem 6.1. let h : x → z be a cis-morphism between closed injective systems and let £(x) = {x,φi}i and £(z) = {z,ψi}i be the fundamental limit spaces for x and z, respectively. then, there is a unique closed continuous map £h : x → z such that £h ◦ φi = ψi ◦ hi for every i ∈ n. cis’s and its fundamental limit spaces 105 proof. write h = {hi : xi → zi}i. we define the map £h : x → z as follows: first, consider £(x) = {x,φi} and £(z) = {z,ψi}. for each x ∈ x, there is xi ∈ xi, for some i ∈ n, such that x = φi(xi). then, we define £h(x) = ψi ◦ hi(xi). this map is well defined. in fact, if x = φi(xi) = φj (xj ), with i < j, then x ∈ φi(xi) ∩ φj (xj ) . = φjfi,j−1(yi,j−1) and xj = fi,j−1(xi). thus, ψj ◦ hj(xj ) = ψj ◦ hj ◦ fi,j−1(xi) = ψj ◦ gi,j−1 ◦ hi(xi) = ψi ◦ hi(xi). now, since £h is obtained from a collection of closed continuous maps which coincide on closed sets, £h is a closed continuous map. moreover, it is easy to see that £h is the unique continuous map from x into z which verifies, for each i ∈ n, the commutativity £h ◦ φi = ψi ◦ hi. � sometimes, we write £h : £(x) → £(z) instead £h : x → y . this map is called the fundamental map induced by h. corollary 6.2. the transition to the fundamental limit space is a functor from the category cis into the category top. for details on functors see [3]. corollary 6.3. if h : x → z is a cis-isomorphism, then the fundamental map £h : £(x) → £(z) is a homeomorphism. this implies that isomorphic closed injective systems have homeomorphic fundamental limit spaces. 7. compatibility of limits in this section, given a cis x = {xi,yi,fi} with fundamental limit space {x,φi}, sometimes we write £(x) to denote only the topological space x. this is clear in the context. theorem 7.1. let {x(n), h(mn)}m,n be an inductive system on the category cis and let {x,e(n)}n its direct limit. then {£(x (n)), £h(mn)}m,n is an inductive system on the category top, which admits £(x) as its directed limit homeomorphic. proof. by uniqueness of the direct limit, we can assume that {x, φ(n)}n is the direct limit constructed in the proof of theorem 5.1. then, we have e(n) : x(n) → x given by e(n) = {ξ (n) i : x (n) i → xi}i, where {xi,ξ (n) i }n is a fundamental limit space for {x (n) i ,x (n) i ,h (n) i }n. by theorem 6.1, {£(x(n)), £h(mn)}m,n is a inductive system. for each n ∈ n, write x(n) = {x (n) i ,y (n) i ,f (n) i }i and £(x n) = {x(n),φ (n) i }i. moreover, write x = {xi,yi,fi}i and £(x) = {x,φi}i. the inductive system {£(x(n)), £h(mn)}m,n can be write as {x (n), £h(mn)}m,n. 106 m. c. fenille we need to show that there is a collection of maps {ϑ(n) : x(n) → x}n such that {x,ϑ(n)}n is a direct limit for the system {x (n), £h(mn)}m,n. for each x ∈ x(n), there is a point xi ∈ x (n) i , for some i ∈ n, such that x = φ (n) i (xi). we define ϑ (n) : x(n) → x by ϑ(n)(x) = φi ◦ ξ (n) i (xi). the map ϑ(n) is well defined. in fact: if x = φ (n) i (xi) = φ (n) j (xj ), with i ≤ j, then we have x ∈ φ (n) i (x (n) i ) ∩ φ (n) j (x (n) j ) . = φ (n) j f (n) i,j−1(y (n) i,j−1) and, moreover, xj = f (n) i,j−1(xi) and xi ∈ yi,j ⊂ xi. now, in the diagram below, the two triangles and the big square are commutative. in it, we write ξ (n) i | and φ (n) i | to denote the obvious restriction. it follows that φj ◦ ξ (n) j (xj ) = φj ◦ ξ (n) j ◦ fi,j−1(n)(xi) = φj ◦ fi,j−1 ◦ ξ (n) i (xi) = φi ◦ ξ (n) i (xi). this is sufficient to prove that the map ϑ(n) is well defined. moreover, note that this map makes the diagram below in a commutative diagram. xi fi,j−1 // φi ##g gg gg gg gg xj φj{{ww ww ww ww w x x(n) ϑ (n) oo y (n) i,j ξ (n) i | oo φ (n) i | << zzzzzzzz f (n) i,j−1 // x (n) j φ (n) j bbeeeeeeee ξ (n) j oo now, by theorem 6.1 we have £h(mn) ◦ φ (m) i = φ (n) i ◦ h (n) i for all integers m < n, since £(xn) = {x(n),φ (n) i }i. let x ∈ x(m) be an arbitrary point. then, there is xi ∈ x (m) i such that x = φ (m) i (xi). also, for each n ∈ n with m < n, we have £h (mn)(x) = φ (n) i ◦ h (mn) i (xi). thus, we have, ϑ(n) ◦ £h(mn)(x) = φi ◦ ξ (n) i (h (mn) i (xi)) = φi ◦ ξ (m)(xi) = ϑ (m)(x). this shows that ϑ(n) ◦ £h(mn) = ϑ(m) for all integers m < n. let a be a closed subset of x. then it is clear that (φi ◦ξ (n) i ) −1(a) is closed in x (n) i , since φi and ξ (n) i are continuous maps. now, we have (ϑ (n))−1(a) = φ (n) i ((φi ◦ ξ (n) i ) −1(a)). then, since φ (n) i is an imbedding (and so a closed map), it follows that (ϑ(n))−1(a) is a closed subset of x(n). therefore, ϑ(n) is continuous. now, it is not difficult to prove that {x,ϑ(n)}n satisfies the universal mapping problem (see [3]). this concludes the proof. � cis’s and its fundamental limit spaces 107 8. inductive closed injective systems in this section, we will study a particular kind of closed injective systems, which has some interesting properties. more specifically, we study the cis’s of the form {xi,xi,fi}, which are called inductive closed injective system, or an inductive cis, to shorten. in an inductive cis {xi,xi,fi}, any two injections fi and fj, with i < j, are componible, that is, the composition fi,j = fj ◦ · · · ◦ fi is always defined throughout domain xi of fi. hence, fixing i ∈ n, for each j > i we have a closed injection fi,j : xi → xj+1. because this, we define, for each i < j ∈ n, fii = idxi : xi → xi and f j i = fi,j−1 : xi → xj by this definition, it follows that fki = f k j ◦ f j i , for all i ≤ j ≤ k. therefore, {xi,f j i } is an inductive system on the category top. we will construct a direct limit for this inductive system. let ∐ xi = ∐∞ i=0 xi be the coproduct (or topological sum) of the spaces xi. consider the canonical inclusions ϕi : xi → ∐ xi. it is obvious that each ϕi is a homeomorphism onto its image. over the space ∐ xi consider the relation ∼ defined by: x ∼ y ⇔ { ∃ xi ∈ xi,yj ∈ xj with x = ϕi(xi) e y = ϕj (yj ), such that yj = f j i (xi) if i ≤ j and xi = f i j (yj ) if j < i. . lemma 8.1. the relation ∼ is an equivalence relation over ∐ xi. proof. we will check the veracity of the properties reflexive, symmetric and transitive. reflexive: let x ∈ x be a point. there is xi ∈ xi such that x = ψi(xi), for some i ∈ n. we have xi = f i i (xi). therefore x ∼ x. symmetric: it is obvious by definition of the relation ∼. transitive: assume that x ∼ y and y ∼ z. suppose that x = ϕi(xi) and y = ϕj (yj ) with yj = f j i (xi). in this case, i ≤ j. (the other case is analogous and is omitted). since y ∼ z, we can have: case 1 : y = ϕj (y ′ j ) and z = ϕk(zk) with j ≤ k and zk = f k j (y ′ j ). then ϕj (yj ) = y = ϕj (y ′ j ), and so yj = y ′ j . since i ≤ j ≤ k, we have zk = f k j (yj ) = fkj f j i (xi) = f k i (xi). therefore x ∼ z. case 2: y = ϕj (y ′ j ) and z = ϕk(zk) with k < j and y ′ j = f j k (zk). then yj = y ′ j , as before. now, we have again two possibility: (a) if i ≤ k < j, then f j k (zk) = yj = f j i (xi) = f j k fki (xi). thus zk = f k i (xi) and x ∼ z. (b) if k < i ≤ j, then f j i (xi) = yj = f j k (zk) = f j i f i k(zk). thus xi = f i k(zk) and x ∼ z. � let x̃ = ( ∐ xi)/ ∼ be the quotient space obtained of ∐ xi by the equivalence relation ∼, and for each i ∈ n, let ϕ̃i : xi → x̃ be the composition 108 m. c. fenille ϕ̃i = ρ ◦ ϕi, where ρ : ∐ xi → x̃ is the quotient projection. ϕ̃i : xi ϕi // ∐ xi ρ // x̃ note that, since x̃ has the quotient topology induced by projection ρ, a subset a ⊂ x̃ is closed in x̃ if and only if ϕ̃i −1 (a) is close in xi for each i ∈ n. given x,y ∈ ∐ xi with x,y ∈ xi, then x ∼ y ⇔ x = y. thus, each ϕ̃i is one-to-one fashion onto ϕ̃i(xi). moreover, it is obvious that x̃ = ∪ ∞ i=0ϕ̃i(xi). these observations suffice to conclude the following: theorem 8.2. {x̃,ϕ̃i} is a fundamental limit space for the inductive cis {xi,xi,fi}. moreover, {x̃,ϕ̃i} is a direct limit for the inductive system {xi,f j i }. for details on direct limit see [3]. remark 8.3. if we consider an arbitrary cis {xi,yi,fi}, then the relation ∼ is again an equivalence relation over the coproduct ∐ xi. moreover, in this circumstances, if ϕi(xi) = x ∼ y = ϕj (yj ), then we must have: (a) if i = j, then x = y. (b) if i < j, then fi and fj−1 are semicomponible and xi ∈ yi,j−1; (c) if i > j, then fj and fi−1 are semicomponible and yj ∈ yj,i−1. therefore, it follows that the space x̃ = ( ∐ xi)/ ∼ is exactly the attaching space x0 ∪f0 x1 ∪f1 x2 ∪f2 · · · , and the maps ϕ̃i are the projections from xi into x̃, as in theorem 3.6. 9. functoriality on fundamental limit spaces let f : top → m be a functor of the category top into a complete category m (a category in which every direct (inductive) or inverse system has a limit). let {xi,xi,fi} be an arbitrary inductive cis, and consider the inductive system {xi,f j i } constructed in the previous section. the functor f turns this system into the inductive system {fxi, ff j i } on the category m. theorem 9.1 (of the functorial invariance). let {x,φi} be a fundamental limit space for the inductive cis {xi,xi,fi} and let {m,ψi} be a direct limit for {fxi, ff j i }. then, there is a unique isomorphism h : fx → m such that ψi = h ◦ fφi for every i ∈ n. proof. by theorem 8.2 and by uniqueness of fundamental limit space, there is a unique homeomorphism β : x → x̃ such that ϕ̃i = β ◦ φi for every i ∈ n. hence, fβ : fx → fx̃ is the unique r-isomorphism such that fϕ̃i = fβ◦fφi. since {x̃,ϕ̃i} is a direct limit for the inductive system {xi,f j i } on the category top, it follows that {fx̃, fϕi} is a direct limit of the system {fxi, ff j i } on the category m. by universal property of direct limit, there is a unique isomorphism ω : fx̃ → m such that ψi = ω ◦ fϕ̃i. then, we take h : fx → m to be the composition h = ω ◦ fβ. � cis’s and its fundamental limit spaces 109 the universal property of direct limits among others properties can be found, for example, in chapter 2 of [3]. now, we describe some basic applications of theorem 9.1. we write mod to denote the (complete) category of r-modules and r-homomorphisms, where r is a commutative ring with identity element. example 9.2. let k be a cw-complex and let {kn,kn, ln} be the cis as in example 4.5. it is clear that this cis is an inductive cis. let f : top → mod be an arbitrary functor. given m < n in n, write lnm to denote the composition ln−1 ◦ · · · ◦ lm : k m → kn. then, {fkn, flnm} is an inductive system on the category mod. by theorem 9.1, its direct limit is isomorphic to fk. example 9.3 (homology of the sphere s∞). let {sn,sn,fn} be the cis of example 4.3. its fundamental limit space is the infinite-dimensional sphere s∞. let p > 0 be an arbitrary integer. by previous example, hp(s ∞) is isomorphic to direct limit of inductive system {hp(s n),hp(f n m)}, where f n m = fn−1 ◦ · · · ◦ fm : s m → sn, for m ≤ n. now, since hp(s n) = 0 for n > p, it follows that hp(s ∞) = 0 for each p > 0. details on homology theory can be found in [1], [2] and [5]. example 9.4 (the infinite projective space rp∞ is a k(z2, 1) space). we know that π1(rp n) ≈ z2 for all n ≥ 2 and π1(rp 1) ≈ z. moreover, for integers m < n, the natural inclusion fnm : rp m →֒ rpn induces a isomorphism (fnm)# : π1(rp m) ≈ π1(rp n). for details see [2]. the fundamental limit space for the cis {rpn, rpn,fn} of example 4.6 is the infinite projective space rp∞. by example 9.2, we have that π1(rp ∞) is isomorphic to direct limit for the inductive system {π1(rp n), (fnm)#}. then, by previous arguments it is easy to check that π1(rp ∞) ≈ z2. on the other hand, for each r > 1, we have πr(rp n) ≈ πr(s n) for every n ∈ n (see [2]). then, πr(s n) = 0 always that 1 < r < n. thus, it is easy to check that πr(rp ∞) = 0 for each r > 1. for details on homotopy theory and k(π, 1) spaces see [2] and [6]. example 9.5 (the homotopy groups of s∞). since πr(s n) = 0 for all integers r < n, it is very easy to prove that πr (s ∞) = 0 for every r ≥ 1. example 9.6. the homology of the torus t ∞. some arguments very simple and similar to above can be used to prove that h0(t ∞) ≈ r and hp(t ∞) ≈ ⊕∞ i=1 r for every p > 0. 10. counter-funtoriality on fundamental limit spaces let g : top → m be a counter-functor from the category top into a complete category m (a category in which every direct (inductive) or inverse system has a limit). let {xi,xi,fi} be an arbitrary inductive cis and consider the inductive system {xi,f j i } as before. the counter-functor g turns this inductive system on the category top into the inverse system {gxi, gf j i } on the category m. 110 m. c. fenille theorem 10.1 (of the counter-functorial invariance). let {x,φi} be a fundamental limit space for the inductive cis {xi,xi,fi} and let {m,ψi} be an inverse limit for {gxi, gf j i }. then, there is a unique isomorphism h : m → gx such that ψi = gφi ◦ h for every i ∈ n. proof. by theorem 8.2 and by uniqueness of fundamental limit space, there is a unique homeomorphism β : x → x̃ such that ϕ̃i = β ◦ φi, for all i ∈ n. hence, gβ : gx̃ → gx is the unique isomorphism such that gϕ̃i = gφi◦gβ. since {x̃,ϕ̃i} is a direct limit for the inductive system {xi,f j i } on the category top, it follows that {gx̃, gϕi} is an inverse limit for the inverse system {gxi, gf j i } on the category m. by universal property of inverse limit, there is a unique isomorphism ω : m → gx̃ such that ψi = gϕ̃i ◦ ω. then, we take h : m → gx to be the composition h = gβ ◦ ω. � the property of the inverse limit can be found in [3]. now, we describe some basic applications of theorem 10.1. example 10.2 (cohomology of the sphere s∞). since hp(sn; r) ≈ hp(s n; r) for all p,n ∈ z, it follows by theorem 10.1 and example 9.3 that h0(s∞; r) ≈ r and hp(s∞; r) = 0 for everyp > 0. example 10.3 (the cohomology of the torus t ∞). since the homology and cohomology modules of a finite product of spheres are isomorphic, it follows by theorem 10.1 and example 9.6 that h0(t ∞) ≈ r and hp(t ∞) ≈ ⊕∞ i=1 r for every p > 0. 11. finitely semicomponible and stationary cis’s we say that a cis {xi,yi,fi} is finitely semicomponible if, for each i ∈ n, there is only a finite number of indices j ∈ n such that fi and fj (or fj and fi) are semicomponible, that is, there is not an infinity sequence {fk}k≥i0 of semicomponible maps. obviously, {xi,yi,fi} is finitely semicomponible if and only if for some (so for all) limit space {x,φi} for {xi,yi,fi}, the collection {φi(xi)}i is a pointwise finite cover of x (that is, each point of x belongs to only a finite number of φi(xi) ′s). we say that a cis {xi,yi,fi} is stationary if there is a nonnegative integer n0 such that, for all n ≥ n0, we have yn = yn0 = xn0 = xn and fn = identity map. this section of the text is devoted to the study and characterization of the limit space of these two special types of cis’s. theorem 11.1. let {x,φi} be an arbitrary limit space for the cis {xi,yi,fi}. if the collection {φi(xi)}i is a locally finite cover of x, then {xi,yi,fi} is finitely semicomponible. the reciprocal is true if {x,φi} is a fundamental limit space. proof. the first part is trivial, since if the collection {φi(xi)}i is a locally finite cover of x, then it is a pointwise finite cover of x. cis’s and its fundamental limit spaces 111 suppose that {x,φi} is a fundamental limit space for the finitely semicomponible cis {xi,yi,fi}. let x ∈ x be an arbitrary point. then, there are nonnegative integers n0 ≤ n1 such that φ −1 i ({x}) 6= ∅ ⇔ n0 ≤ i ≤ n1. for each n0 ≤ i ≤ n1, write xi to be the single point of xi such that x = φi(xi). it follows that xi ∈ yni for n0 ≤ i ≤ n1 − 1, but xn1 /∈ yn1 and xn0 /∈ fn0−1(yn0−1). since fn0−1(yn0−1) is closed in xn0 and xn0 /∈ fn0−1(yn0−1), we can choose an open neighborhood vn0 of xn0 in xn0 such that vn0 ∩ fn0−1(yn0−1) = ∅. similarly, since xn1 /∈ yn1 and yn1 is closed in xn1 , we can choose an open neighborhood vn1 of xn1 in xn1 such that vn1 ∩ yn+1 = ∅. define v = φn0 (vn0 ) ∪ φn0+1(xn0+1) ∪ · · · ∪ φn1−1(xn1−1) ∪ φn1 (vn1 ). it is clear that x ∈ v ⊂ x and v ∩ φj (xj ) = ∅ for all j /∈ {n0, . . . ,n1}. moreover, we have φ−1j (x − v ) =    xn0 − vn0 if j = n0 xn1 − vn1 if j = n1 ∅ if n0 < j < n1 xj otherwise . in all cases, we see that φ−1j (x − v ) is closed in xj . thus, x − v is closed in x. therefore, we obtain an open neighborhood v of x which intersects only a finite number of φi(xi) ′s. � the reciprocal of the previous proposition is not true, in general, when {x,φi} is not a fundamental limit space. in fact, we have the following example in which the above reciprocal fails. example 11.2. consider the topological subspaces x0 = [1, 2] and xn = [ 1 n+1 , 1 n ], for n ≥ 1, of the real line r, and take y0 = {1} and yn = { 1 n+1 } for each n ≥ 1. define fn : yn → xn+1 to be the natural inclusion, for all n ∈ n. it is clear that the cis {xn,yn,fn} is finitely semicomponible, and its fundamental limit space is, up to homeomorphism, the subspace x = (0, 2] of the real line, together the collection of natural inclusions φn : xn → x. it is also obvious that the collection {φi(xi)}i is a locally finite cover of x. on the other hand, take z = ((0, 1] × {0}) ∪ {(1 + cos(πt − π), sin(πt − π)) ∈ r2 : t ∈ [1, 2]}. consider z as a subspace of r2. then z is homeomorphic to the sphere s1. consider the maps ψ0 : x0 → z given by ψ0(t) = (1 +cos(πt−π), sin(πt−π)), and ψn : xn → z given by ψn(t) = (t, 0), for each n ≥ 1. it is easy to see that {z,ψn} is a limit space for the cis {xn,yn,fn}. now, note that the point (0, 0) ∈ z has no open neighborhood intercepting only a finite number of ψn(xn) ′s. theorem 11.3. let {x,φi} be a limit space for the cis {xi,yi,fi} and suppose that the collection {φi(xi)}i is a locally finite closed cover of x. then {x,φi} is a fundamental limit space. proof. we need to prove that a subset a of x is closed in x if and only if φ−1i (a) is closed in xi for every i ∈ n. 112 m. c. fenille if a ⊂ x is closed in x, then it is clear that φ−1i (a) is closed in xi for each i ∈ n, since each φi is a continuous map. now, let a be a subset of x such that φ−1i (a) is closed in xi for every i ∈ n. then, since each φi is a imbedding, it follows that φi(φ −1 i (a)) = a ∩ φi(xi) is closed in φi(xi). but by hypothesis, φi(xi) is closed in x. therefore a∩φi(xi) is closed in x for each i ∈ n. let x ∈ x − a be an arbitrary point and choose an open neighborhood v of x in x such that v ∩ φi(xi) 6= ∅ ⇔ i ∈ λ, where λ ⊂ n is a finite subset of indices. it follows that v ∩ a = ⋃ i∈λ v ∩ a ∩ φi(xi). now, since each a∩φi(xi) is closed in x and x /∈ a∩φi(xi), we can choose, for each i ∈ λ, an open neighborhood vi ⊂ v of x, such that vi∩a∩φi(xi) = ∅. take v ′ = ⋂ i∈λ vi. then v ′ is an open neighborhood of x in x and v ′∩a = ∅. therefore, a is closed in x. � corollary 11.4. let {x,φi} be a limit space for the cis {xi,yi,fi} in which each xi is a compact space. if x is hausdorff and {φi(xi)}i is a locally finite cover of x, then {x,φi} is a fundamental limit space. proof. each φi(xi) is a compact subset of the hausdorff space x. therefore, each φi(xi) is closed in x. the result follows from the previous theorem. � corollary 11.5. let {x,φi} be a limit space for the finitely semicomponible cis {xi,yi,fi}. then, {x,φi} is a fundamental limit space if and only if the collection {φi(xi)}i is a locally finite closed cover of x. proof. proposition 3.2 and theorems 11.1 and 11.3. � let f : z → w be a continuous map between topological spaces. we say that f is a perfect map if it is closed, surjective and, for each w ∈ w , the subset f−1(w) ⊂ z is compact. (see [4]). let p be a property of topological spaces. we say that p is a perfect property if always that p is true for a space z and there is a perfect map f : z → w , we have p true for w . again, we say that a property p is countable-perfect if p is perfect and always that p is true for a countable collection of spaces {zn}n, we have p true for the coproduct ∐∞ n=0 zn. we say that p is finite-perfect if the previous sentence is true for finite collections {zn} n0 n=0 of topological spaces. every countable-perfect property is also a finite-perfect property. the reciprocal is not true. every perfect property is a topological invariant. example 11.6. the follows one are examples of countable-prefect properties: hausdorff axiom, regularity, normality, local compactness, second axiom of countability and lindelöf axiom. the compactness is a finite-perfect property which is not countable-perfect. (for details see [4]). cis’s and its fundamental limit spaces 113 theorem 11.7. let {x,φi} be a fundamental limit space for the finitely semicomponible cis {xi,yi,fi}, in which each xi has the countable-perfect property p. then x has p. proof. let {x,φi} be a fundamental limit space for {xi,yi,fi}. by theorems 8.2 and 3.5, there is a unique homeomorphism β : x̃ → x such that φi = β◦ϕ̃i for every i ∈ n. then, simply to prove that x̃ has the property p, where x̃ = ( ∐ xi)/ ∼ is the quotient space constructed in section 8 (remember remark 8.3). consider the quotient map ρ : ∐ xi → x̃. it is continuous and surjective. moreover, since the cis {xi,yi,fi} is finitely semicomponible, it is obvious that for x ∈ x̃ we have that ρ−1(x) is a finite subset, and so a compact subset, of ∐ xi. therefore, simply to prove that ρ is a closed map, since this is enough to conclude that ρ is a perfect map and, therefore, the truth of the theorem. let e ⊂ ∐ xi be an arbitrary closed subset of ∐ xi. we need to prove that ρ(e) is closed in x̃, that is, that ρ−1(ρ(e)) ∩xi is closed in xi for each i ∈ n. but note that ρ−1(ρ(e)) ∩ xi = (e ∩ xi) ∪ i−1⋃ j=0 fj,i−1(e ∩ yj,i−1) ∪ ∞⋃ j=i f−1i,j (e ∩ xj+1), where each term of the total union is closed. now, since the given cis is finitely semicomponible, there is on the union ⋃∞ j=i f −1 i,j (e ∩xj+1) only a finite nonempty terms. thus, ρ−1(ρ(e)) ∩ xi can be rewritten as a finite union of closed subsets. therefore ρ−1(ρ(e)) ∩ xi is closed. � the quotient map ρ : ∐ xi → x̃ is not closed, in general. to illustrate this fact, we introduce the following example: example 11.8. consider the inductive cis {sn,sn,fn} as in example 4.3, starting at n = 1. consider the sequence of real numbers (an)n, where an = 1/n, n ≥ 1. let a = {an}n≥2 be the set of points of the sequence (an)n starting at n = 2. then, the image of a by the map γ : [0, 1] → s1 given by γ(t) = (cos t, sin t) is a sequence (bn)n≥2 in s 1 such that the point b = (1, 0) ∈ s1 is not in γ(a) and (bn)n converge to b. it follows that the subset b = γ(a) of s1 is not closed in s1. now, for each n ≥ 2, let en be the closed (n− 1)-dimensional half-sphere imbedded as the meridian into sn going by point f1,n−1(bn). it is easy to see that e n is closed in sn for each n ≥ 2. let e = ⊔∞ n=2 e n be the disjoint union of the closed half-spheres en. then, for each n ≥ 2, e ∩ sn = en and e ∩ s1 = ∅. thus, e is a closed subset of coproduct space ∐∞ n=1 s n. however, ρ−1(ρ(e)) ∩ s1 = b is not closed in s1. hence ρ(e) is not closed in the sphere s∞. therefore, the projection ρ : ∐ sn → (( ∐ sn)/ ∼) ∼= s∞ is not a closed map. 114 m. c. fenille now, we will prove the result of the previous theorem in the case of stationary cis’s. in this case the result is stronger and applies to properties finitely perfect. we started with the following preliminary result, whose proof is obvious and therefore will be omitted (left to the reader). lemma 11.9. let {x,φi} be a fundamental limit space for the stationary cis {xi,yi,fi}. suppose that this cis park in the index n0 ∈ n. then φi = φn0 for every i ≥ n0 and x ∼= ∪ n0 i=0φi(xi). moreover, the composition ρn0 : ∐n0 i=0 xi inc. // ∐∞ i=0 xi ρ // x̃ is a continuous surjection, where inc. indicates the natural inclusion. theorem 11.10. let {x,φi} be a fundamental limit space for the stationary cis {xi,yi,fi} in which each xi has the finite-perfect property p. then x has p. proof. as in theorem 11.7, simply to prove that x̃ = ( ∐ xi)/ ∼ has p. suppose that the cis {xi,yi,fi} parks in the index n0 ∈ n. by the previous lemma, the map ρn0 : ∐n0 i=0 xi → x̃ is continuous and surjective. thus, simply to prove that ρn0 is a perfect map. in order to prove this, it rests only to prove that ρn0 is a closed map and ρ −1 n0 (x) is a compact subset of ∐n0 i=0 xi, for each x ∈ x̃. this latter fact is trivial, since each subset ρ−1n0 (x) is finite. in order to prove that ρn0 is a closed map, let e be an arbitrary closed subset of ∐n0 i=0 xi. we need to prove that ρ −1(ρn0 (e)) ∩ xi is closed in xi for each i ∈ n. but note that, as before, we have ρ−1(ρn0 (e)) ∩ xi = (e ∩ xi) ∪ i−1⋃ j=0 fj,i−1(e ∩ yj,i−1) ∪ ∞⋃ j=i f−1i,j (e ∩ xj+1), where each term of this union is closed. now, since e ⊂ ∐n0 i=0 xi, we have e ∩ xj+1 = ∅ for all j ≥ n0. thus, the subsets f −1 i,j (e ∩ xj+1) which are in the last part of the union are empty for all j ≥ n0. hence, ρ −1(ρn0 (e)) ∩ xi is a finite union of closed subsets. therefore, ρ−1(ρn0 (e)) ∩ xi is closed. � references [1] m. j. greenberg and j. r. harper, algebraic topology, a first course, benjamin/cummings publishing company, london, 1981. [2] a. hatcher, algebraic topology, cambridge university press, 2002. [3] j. j. hotman, an introduction to homological algebra, academic press, inc., 1979. [4] j. r. munkres, topology, prentice-hall, 1975. [5] e. h. spanier, algebraic topology, springer-verlag new york, inc. 1966. [6] g. w. whitehead, elements of homotopy theory, springer-verlag new york, inc. 1978. cis’s and its fundamental limit spaces 115 received january 2010 accepted july 2010 m. c. fenille (mcfenille@gmail.com) instituto de ciências exatas universidade federal de itajubá, av. bps 1303, pinheirinho, cep 37500-903, itajubá, mg, brazil. closed injective systems and its fundamental[8pt] limit spaces. by m. c. fenille gutevagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 7178 some problems on selections for hyperspace topologies valentin gutev and tsugunori nogura abstract. the theory of hyperspaces has attracted the attention of many mathematicians who have found a large variety of its applications during the last decades. the theory has taken also its natural course and has yielded lots of problems which, besides their independent inner beauty, provide ties with numerous classical fields of mathematics. in the present note we are concerned with some open problems about selections for hyperspace topologies which have been in the scope of our recent research interests. 2000 ams classification: 54b20, 54c65, 54f05, 54e50 keywords: selections, hyperspaces, ordered spaces, complete metric spaces. 1. the concept of a τ-continuous selection let (x, t ) be a t1-space, where t is the topology of x, and let f(x, t ) be the set of all non-empty closed (with respect to t ) subsets of x. let us stress the reader’s attention that f(x, t ) is different for different topologies t on x, while x is always a subset of f(x, t ) because we may identify each point x ∈ x with the corresponding singleton {x} ∈ f(x, t ). definition 1.1. a topology τ on f(x, t ) is called admissible (see [18]) if its restriction on the set of all singletons {{x} : x ∈ x} of x coincides with the topology t . in the light of definition 1.1, we may look at (f(x, t ), τ) as a topological extension of the topological space (x, t ) provided τ is an admissible topology. it should be mentioned that the concept of an admissible topology may refer also to some additional structures on x, see [18]. the second basic concept of this paper is related to a selection for a hyperspace topology. let d ⊂ f(x, t ). definition 1.2. a map f : d → x is a selection for d if f(s) ∈ s for every s ∈ d. definition 1.3. if τ is a topology on f(x, t ), then a map f : d → x is a τcontinuous selection for d if it is a selection for d which is continuous with respect to the relative topology on d as a subspace of (f(x, t ), τ). so far, one of the best known admissible topologies on f(x, t ) is the vietoris one τv (t ). let us recall that all collections of the form 〈v〉 = { s ∈ f(x, t ) : s ⊂ ⋃ v and s ∩ v 6= ∅, whenever v ∈ v } , 72 valentin gutev and tsugunori nogura where v runs over the finite subsets of t , provide a base for the topology τv (t ). any selection has the following property with respect to the vietoris topology, it appeared in several papers in an explicit or implicit way. proposition 1.4. if f : f(x, t ) → x is a selection for f(x, t ), then f is τv (t )continuous at {x} for every x ∈ x. 2. selections and orderability in what follows, all spaces are assumed to be at least hausdorff. for a space (x, t ) and 0 < n < ω, we let fn(x) = {s ⊂ x : 0 < |s| ≤ n}. note that f1(x) is the set of all singletons of x, and always fn(x) ⊂ f(x, t ). let τ be a topology on f(x, t ), and let seℓτ(x, t ) be the set of all τ-continuous selections for f(x, t ). also, let seℓ(τ,n)(x, t ), n > 1, be the set of all τcontinuous selections for fn(x), and seℓn(x) that of all selections (not necessarily τ-continuous) for fn(x). any selection f ∈ seℓ2(x) naturally defines an order-like relation ≺f on x [18] by letting for x 6= y that x ≺f y iff f({x, y}) = x. however, in general, ≺f fails to be a linear order on x. let us denote by tf the topology generated by all possible “open” ≺f -intervals. it is easy to observe that tf is also a hausdorff topology, see [16]. theorem 2.1 ([18]). let (x, t ) be a space, and let f ∈ seℓ(τv (t ),2)(x, t ). then, (a) tf ⊂ t . if, in addition, (x, t ) is connected, then we also have that (b) “≺f” is a proper linear order on x, (c) (x, tf ) is connected, (d) f ∈ seℓ(τv (tf ),2) (x, tf ). theorem 2.2 ([20]). let (x, t ) be a compact space, with seℓ(τv (t ),2)(x, t ) 6= ∅. then, (a) (x, t ) is a linear ordered topological space (in particular, ind(x, t ) ≤ 1), (b) tf = t for every f ∈ seℓ(τv (t ),2)(x, t ), (c) seℓτv (t )(x, t ) 6= ∅. here, ind(x, t ) means the small inductive dimension of (x, t ). in view of theorem 2.1, it makes some sense to investigate the topology tf . for instance, the following simple observation was obtained in [16]. proposition 2.3 ([16]). let (x, t0) be a space, and let f ∈ seℓ(τv (t0),2) (x, t0). then, f ∈ seℓ(τv (t ),2)(x, t ) for every topology t on x which is finer than t0. consider the natural partial order on all hausdorff topologies on a set x defined by t1 ≪ t2 provided t2 is finer than t1, i.e. t1 ⊂ t2. then, by proposition 2.3, f ∈ seℓ(τv (t0),2) (x, t0) implies f ∈ seℓ(τv (t ),2)(x, t ) for every hausdorff topology t on x, with t0 ≪ t . thus, we have the following natural question about a possible ≪-minimal topology t on a set x such that a given selection f ∈ seℓ2(x) is τv (t )-continuous. namely, problem 2.4 ([16]). let x be a set, and let f ∈ seℓ2(x). does there exist a topology t on x which is ≪-minimal with respect to the property “f ∈ seℓ(τv (t ),2)(x, t )”? some problems on selections for hyperspace topologies 73 related to this question, let us observe that, by theorem 2.1, f ∈ seℓ(τv (t ),2)(x, t ) implies tf ≪ t . so, tf is a possible candidate for a ≪-minimal topology in that sense. however, we have the following recent example. example 2.5 ([16]). there exists a set x and σ ∈ seℓ2(x) such that σ is not τv (tσ)-continuous. by theorem 2.2, if (x, t ) is a compact space, then seℓ(τv (t ),2)(x, t ) 6= ∅ if and only if seℓτv (t )(x, t ) 6= ∅. on the other hand, seℓ(τv (te),2)(r, te) 6= ∅, while seℓτv (te)(r, te) = ∅ (see [8]), where r are the real numbers and te is the usual euclidean topology on r. thus, in view of theorem 2.1, we get the following natural question. problem 2.6 ([16]). let (x, t ) be a connected space, and let f ∈ seℓ(τv (t ),2)(x, t ). is it true that seℓτv (t )(x, t ) 6= ∅ if and only if seℓτv (tf )(x, tf ) 6= ∅? in general, the answer is “no” which was provided by the following example. example 2.7 ([16]). there exists a separable, connected and metrizable space (x, t ) such that (i) seℓ(τv (t ),2)(x, t ) 6= ∅, (ii) seℓτv (tf ) (x, tf ) 6= ∅, for every f ∈ seℓ(τv (t ),2)(x, t ), (iii) seℓτv (t )(x, t ) = ∅. in contrast to this, theorem 2.1 implies that seℓ(τv (t ),n)(x, t ) = seℓ(τv (t ),2)(x, t ), for every n ≥ 2, provided (x, t ) is a connected space. on this base, we have also the following question. problem 2.8. does there exist a space (x, t ) such that seℓ(τv (t ),2)(x, t ) 6= ∅ but seℓ(τv (t ),n)(x, t ) = ∅ for some n > 2? suppose that (x, t ) is connected, and f ∈ seℓ(τv (t ),2)(x, t ). then, by theorem 2.1, the space (x, tf ) will be locally compact as a connected linear ordered space. hence, a possible common point of view to theorems 2.1 and 2.2 is suggested by the following question. problem 2.9. let (x, t ) be a locally compact space, with seℓ(τv (t ),2)(x, t ) 6= ∅. does there exist a topology t∗ ≪ t on x such that (x, t∗) is a linear ordered topological space? it should be mentioned that all known selection constructions are based on some extreme principle related to “orderability”. hence, it seems natural to expect that some dimension-like function might be bounded. this is, in fact, the motivation for our next question. problem 2.10. does there exist a space (x, t ) such that seℓ(τv (t ),2)(x, t ) 6= ∅ and ind(x, t ) > 1? for some related results and open questions we refer the interested reader to [2, 9, 11]. 3. on the cardinality of seℓτv (t )(x, t ) the cardinality of seℓτv (t )(x, t ) may provide some information for (x, t ) but mainly when it is finite. theorem 3.1. for a space (x, t ), with seℓτv (t )(x, t ) 6= ∅, the following holds: (a) if (x, t ) is connected, then |seℓτv (t )(x, t )| ≤ 2, [18]. (b) seℓτv (t )(x, t ) is finite if and only if (x, t ) has finitely many connected components, [22]. 74 valentin gutev and tsugunori nogura (c) if (x, t ) is infinite and connected, then |seℓτv (t )(x, t )| = 2 if and only if (x, t ) is compact, [21]. for some other relations between |seℓτv (t )(x, t )| and (x, t ), the interested reader is refer to [10, 21, 22]. 4. on the variety of seℓτv (t )(x, t ) as it was mentioned above, all known selection constructions are based on some extreme principle, so our knowledge about particular members of seℓτv (t )(x, t ) is mainly related to this. here are some result about “extreme-like” members of seℓτv (t )(x, t ). theorem 4.1 ([17]). let (x, t ) be a space, with seℓτv (t )(x, t ) 6= ∅. then, the set {f(x) : f ∈ seℓτv (t )(x, t )} is dense in (x, t ) provided (x, t ) is zero-dimensional, while (x, t ) is totally disconnected provided {f(x) : f ∈ seℓτv (t )(x, t )} is dense in (x, t ). here, as usual, a space (x, t ) is zero-dimensional if it has a base of clopen sets, i.e. if ind(x, t ) = 0. problem 4.2 ([17]). does there exist a space (x, t ) which is not zero-dimensional but {f(x) : f ∈ seℓτv (t )(x, t )} is dense in (x, t )? problem 4.3. let (x, t ) be a totally disconnected space, with seℓτv (t )(x, t ) 6= ∅. is the set {f(x) : f ∈ seℓτv (t )(x, t )} dense in (x, t )? some other results about extreme-like selections are summarized below. theorem 4.4. for a space (x, t ), with seℓτv (t )(x, t ) 6= ∅, the following holds: (a) (x, t ) is zero-dimensional provided for every point x ∈ x there exists an fx ∈ seℓτv (t )(x, t ), with f −1 x (x) = {s ∈ f(x, t ) : x ∈ s}, [17]. (b) if (x, t ) is first countable and zero-dimensional, then for every point x ∈ x there exists an fx ∈ seℓτv (t )(x, t ), with f −1 x (x) = {s ∈ f(x, t ) : x ∈ s}, [17]. (c) if (x, t ) is separable, then it is zero-dimensional and first countable if and only if for every point x ∈ x there exists an fx ∈ seℓτv (t )(x, t ), with f−1x (x) = {s ∈ f(x, t ) : x ∈ s}, [10]. 5. more about the selection problem for topologically generated hyperspace topologies suppose that “r” is a rule by which for any space (x, t ) we may assign a topology τr(t ) on f(x, t ) depending only on the topological structure t of x. we consider the class seℓr of those spaces (x, t ) which admit a τr(t )-continuous selection for their hyperspaces f(x, t ) of closed subsets, i.e. (x, t ) ∈ seℓr if and only if f(x, t ) has a τr(t )-continuous selection. note that if “v ” is the rule by which we assign the vietoris topology τv (t ) on f(x, t ), then (x, t ) ∈ seℓv if and only if seℓτv (t )(x, t ) 6= ∅. in what follows, let us recall that, for a space (x, t ), the fell topology τf (t ) on f(x, t ) is defined by all basic vietoris neighbourhoods 〈v〉 such that x \ ⋃ v is compact. as it becomes clear, we will use “f” to denote the rule that assigns the fell topology. under this terminology, some of the known results can be summarized as follows. theorem 5.1. let (x, t ) be a strongly zero-dimensional metrizable space. then, (a) (x, t ) ∈ seℓv if and only if (x, t ) is completely metrizable, [6, 8, 19]. some problems on selections for hyperspace topologies 75 (b) (x, t ) ∈ seℓf if and only if (x, t ) is locally compact and separable, [15]. the statement (b) of theorem 5.1 is not surprising since the fell topology τf (t ) on f(x, t ) is, in general, not admissible. related to this, let us recall that a space (x, t ) is topologically well-orderable [8] if there exists a linear order ≺ on x such that (x, t ) is a linear ordered space with respect to “≺”, and every non-empty closed subset of (x, t ) has a “≺”-minimal element. for instance, a strongly zerodimensional metrizable space (x, t ) is topologically well-orderable if and only if it is locally compact and separable, [8]. theorem 5.2 ([14]). a space (x, t ) is topologically well-orderable if and only if (x, t ) ∈ seℓf . a further generalization of theorem 5.2 based on its proof was obtained in [1, 13]. 6. selections in metrizable spaces theorem 6.1 ([6, 8]). let (x, t ) be a completely metrizable space such that dim(x, t ) = 0. then, there exists a τv (t )-continuous selection for f(x, t ). here, dim(x, t ) means the covering dimension of (x, t ). most of the hypotheses in theorem 6.1 are the best possible. a metrizable space (x, t ) is completely metrizable provided there exists a τv (t )-continuous selection for f(x, t ) [19] (see theorem 5.1); the assumption dim(x, t ) = 0 cannot be dropped or even weakened to dim(x, t ) ≤ 1 [8, 20]. related to this, the following question seems to be open. problem 6.2. does there exist a zero-dimensional metrizable space (x, t ) such that f(x, t ) has a τv (t )-continuous selection but dim(x, t ) > 0? 7. more continuous selections for metric-generated hyperspace topologies the continuity of a selection f ∈ seℓτv (t )(x, t ) can be improved in several directions involving hyperspace topologies weaker than the vietoris one. towards this end, let us briefly recall some of the most important admissible hyperspace topologies on a metric space (x, d). in what follows, we use td to denote the topology on x generated by a metric d on x. the hausdorff topology τh(d) on f(x, td) depends essentially on the metric d on x. it is the topology on f(x, td) generated by the hausdorff distance h(d) associated to d. let us recall that h(d) is defined by h(d)(s, t ) = sup {d(s, x) + d(x, t ) : x ∈ s ∪ t } , s, t ∈ f(x, td). it is well-known that τv (td) coincides with τh(d) if and only if x is compact [18] while, in general, these two topologies are not comparable. in view of that, we need also some hyperspace topologies which are coarser than both τv (td) and τh(d). a very interesting such topology is the d-proximal topology τδ(d) on f(x, td) [4]. a base for τδ(d) is defined by all collections of the form 〈〈v〉〉d = { s ∈ 〈v〉 : d ( s, x\ ⋃ v ) > 0 } , where v is again a finite family of open subsets of (x, td). here, and in the sequel, we assume that d(s, ∅) = +∞ for every s ∈ f(x, td). another topology of this type is the d-ball proximal topology τδb(d) on f(x, td). a base for τδb(d) is defined by all collections of the form 〈〈v〉〉d, where v is a finite family of open subsets of (x, td) such that x\ ⋃ v is a finite union of closed balls of (x, d). a very similar to the d-ball proximal topology is the d-ball topology τb(d) on f(x, td) generated by all collections of the form 〈v〉, where v runs over the finite 76 valentin gutev and tsugunori nogura families of open subsets of (x, td) such that x\ ⋃ v is a finite union of closed balls of (x, d). finally, we need also the wijsman topology τw(d) which is the weakest topology on f(x, td) such that all distance functionals d(x, ·) : f(x, td) → r, x ∈ x, are continuous. it should be mentioned that τδ(d), τδb(d), τb(d) and τw(d) also depend on the metric d on x. however, they are metrizable only under additional conditions on the metric space (x, d). on the other hand, we always have the following (usually strong) inclusions τw(d) ⊂ τδb(d) ⊂ τδ(d) ⊂ τv (td) ⋂ τh(d), and τδb(d) ⊂ τb(d) ⊂ τv (td). for these and other properties of the above hyperspace topologies, we refer the interested reader to [3] and [4]. for a metrizable space (x, t ), let m(x, t ) denote the set of all metrics d on x compatible with the topology of x, i.e. for which td = t . concerning hyperspace topologies which are “mixed” – where the definition includes a topological part from the topological space (x, t ) and a metric part from a compatible metric on x, there arise at least three different points of view given by how useful selections for these hyperspace topologies are. let τr be such a class of hyperspace topologies which are generated by the compatible metrics on x, i.e. for every d ∈ m(x, t ) we have a corresponding topology τr(d) on f(x, t ). for convenience, we will restrict our attention only to strongly zero-dimensional metrizable spaces considering the following: (s)w the class w-seℓr of those strongly zero-dimensional metrizable spaces (x, t ) which have the weak τr-selection property defined by (x, t ) ∈ w-seℓr if and only if there exists a τr(d)-continuous selection for f(x, t ) for some d ∈ m(x, t ). (s) the class seℓr of those strongly zero-dimensional metrizable spaces (x, t ) which have the τr-selection property defined by (x, t ) ∈ seℓr if and only if f(x, t ) has a τr(d)-continuous selection for every d ∈ m(x, t ). (s)s the class s-seℓr of those strongly zero-dimensional metrizable spaces (x, t ) which have the strong τr-selection property defined by (x, t ) ∈ s-seℓr if and only if f(x, t ) has a selection which is τr(d)-continuous for every d ∈ m(x, t ). obviously, we always have s-seℓr ⊂ seℓr ⊂ w-seℓr. however, in general, no one of these inclusions is invertible, see [15]. to become more specific, we will use r = w for the wijsman topology; r = δb for the ball proximal topology; r = b for the ball topology; and r = δ for the proximal topology. theorem 7.1 ([15]). in the class of strongly zero-dimensional metrizable spaces, the following holds: (a) seℓf = s-seℓw = seℓw $ w-seℓw . (b) seℓf = s-seℓδb = seℓδb $ w-seℓδb. (c) seℓf = s-seℓb $ seℓb ⊂ w-seℓb. (d) s-seℓδ $ seℓδ $ w-seℓδ. related the the above theorem, the following two questions are of interest. problem 7.2 ([15]). does there exist a strongly zero-dimensional non-separable metrizable space (x, t ) such that x ∈ w-seℓr for some r ∈ {w, δb, b}? some problems on selections for hyperspace topologies 77 problem 7.3 ([15]). does there exist a strongly zero-dimensional metrizable space (x, t ) such that x ∈ w-seℓb\seℓb? finally, we have also the following two general questions: problem 7.4 ([7, 15]). let r ∈ {w, δb, b, δ}, (x, t ) be a strongly zero-dimensional metrizable space, and let d ∈ m(x, t ) be a compatible metric. does there exist a topological property p such that f(x, t ) has a τr(d)-continuous selection if and only if (f(x, t ), τr(d)) ∈ p? problem 7.5 ([7, 15]). let r ∈ {w, δb, b, δ}, (x, t ) be a strongly zero-dimensional metrizable space, and let d ∈ m(x, t ) be a compatible metric. does there exist a metric property d such that f(x, t ) has a τr(d)-continuous selection if and only if d ∈ d? the interested reader is referred to [5, 7, 12, 15] for some additional discussion on the topic. 8. special metrics and selections let (x, d) be a metric space. a subset a ⊂ x is called d-clopen if d(a, x\a) > 0, [7, 15]. every d-clopen set is clopen but the converse fails. for more information about this concept, see [7, 15]. we shall say that a metric space (x, d) is totally disconnected with respect to d, or totally d-disconnected, if every singleton of x is an intersection of d-clopen subsets of (x, d), [7]. example 8.1 ([7]). there exists a metric space (x, d) with only two non-isolated points which is not totally d-disconnected. in view of this example, the following questions about the selection problem for the d-proximal topology are still open. problem 8.2 ([7]). let (x, t ) be a (strongly zero-dimensional) completely metrizable space, and let d ∈ m(x, t ) be such that (x, d) is totally d-disconnected. does there exist a τδ(d)-continuous selection for f(x, t )? problem 8.3 ([7]). let x be a metrizable scattered space, and let d ∈ m(x, t ) be such that (x, d) is totally d-disconnected. does there exist a τδ(d)-continuous selection for f(x, t )? problem 8.4 ([7]). let (x, t ) be a metrizable scattered space, and d ∈ m(x, t ). does there exist a τδ(d)-continuous selection for f(x, t )? the above question is open even in the special case when (x, t ) has only two non-isolated points, the answer is “yes” if (x, t ) has only one non-isolated point [7, theorem 5.5]. finally, the following further question seems to be also interesting. problem 8.5. let (x, t ) be a metrizable space which is scattered with respect to compact subsets, i.e. every non-empty closed subset of (x, t ) contains a non-empty compact and relatively open subset. also, let d ∈ m(x, t ). does there exist a τδ(d)continuous selection for f(x, t )? references [1] g. artico and u. marconi, selections and topologically well-ordered spaces, topology appl. 115 (2001), 299–303. [2] g. artico, u. marconi, j. pelant, l. rotter, and m. tkachenko, selections and suborderability, fund. math. 175 (2002), no. 1, 1–33. 78 valentin gutev and tsugunori nogura [3] g. beer, topologies on closed and closed convex sets, mathematics and its applications, vol. 268, kluwer academic publishers, the netherlands, 1993. [4] g. beer, a. lechicki, s. levi, and s. naimpally, distance functionals and suprema of hyperspace topologies, ann. mat. pure appl. 162 (1992), 367–381. [5] d. bertacchi and c. costantini, existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, topology appl. 88 (1998), 179–197. [6] m. choban, many-valued mappings and borel sets. i, trans. moscow math. soc. 22 (1970), 258– 280. [7] c.costantini and v. gutev, recognizing special metrics by topological properties of the “metric”proximal hyperspace, tsukuba j. math. 26 (2002), no. 1, 145–169. [8] r. engelking, r. w. heath, and e. michael, topological well-ordering and continuous selections, invent. math. 6 (1968), 150–158. [9] s. fujii and t. nogura, characterizations of compact ordinal spaces via continuous selections, topology appl. 91 (1999), 65–69. [10] s. garćıa-ferreira, v. gutev, t. nogura, m. sanchis, and a. tomita, extreme selections for hyperspaces of topological spaces, topology appl. 122 (2002), 157–181. [11] s. garćıa-ferreira and m. sanchis, weak selections and pseudocompactness, preprint, 2001. [12] v. gutev, selections and hyperspace topologies via special metrics, topology appl. 70 (1996), 147–153. [13] , fell continuous selections and topologically well-orderable spaces ii, proceedings of the ninth prague topological symposium (2001), topology atlas, toronto, 2002, pp. 157–163 (electronic). [14] v. gutev and t. nogura, fell continuous selections and topologically well-orderable spaces, internal report no. 13/99, university of natal. [15] , selections for vietoris-like hyperspace topologies, proc. london math. soc. 80 (2000), no. 3, 235–256. [16] , selections and order-like relations, applied general topology 2 (2001), 205–218. [17] , vietoris continuous selections and disconnectedness-like properties, proc. amer. math. soc. 129 (2001), 2809–2815. [18] e. michael, topologies on spaces of subsets, trans. amer. math. soc. 71 (1951), 152–182. [19] j. van mill, j. pelant, and r. pol, selections that characterize topological completeness, fund. math. 149 (1996), 127–141. [20] j. van mill and e. wattel, selections and orderability, proc. amer. math. soc. 83 (1981), no. 3, 601–605. [21] t. nogura and d. shakhmatov, characterizations of intervals via continuous selections, rendiconti del circolo matematico di palermo, serie ii, 46 (1997), 317–328. [22] , spaces which have finitely many continuous selections, bollettino dell’unione matematica italiana 11-a (1997), no. 7, 723–729. received october 2002 accepted december 2002 valentin gutev (gutev@nu.ac.za) school of mathematical and statistical sciences, faculty of science, university of natal, king george v avenue, durban 4041, south africa tsugunori nogura (nogura@ehimegw.dpc.ehime-u.ac.jp) department of mathematics, faculty of science, ehime university, matsuyama, 790-8577, japan @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 71–77 dense sδ-diagonals and linearly ordered extensions masami hosobuchi abstract. the notion of the sδ-diagonal was introduced by h. r. bennett to study the quasi-developability of linearly ordered spaces. in an earlier paper, we obtained a characterization of topological spaces with an sδ-diagonal and we showed that the sδ-diagonal property is stronger than the quasi-gδ-diagonal property. in this paper, we define a dense sδ-diagonal of a space and show that two linearly ordered extensions of a generalized ordered space x have dense sδ-diagonals if the sets of right and left looking points are countable. keywords: sδ-diagonal, dense sδ-diagonal, linearly ordered space (lots), generalized ordered space (go-space), linearly ordered extension. 2000 ams classification: 54f05. 1. sδ-diagonals we review in this section the definitions of sδ-set and sδ-diagonal, and state our results obtained in [5]. the following definition is a generalization of a gδ-set and was introduced by h. r. bennett [2] to study the quasi-developability of linearly ordered (topological) spaces. definition 1.1. let x be a topological space. a subset a of x is called an sδ-set if there exists a countable collection {u(1),u(2), . . .} of open subsets of x such that, for two points p ∈ a and q ∈ x \a, there exists an n such that p ∈ u(n) and q /∈ u(n). it is easy to see that a gδ-set is an sδ-set. hence the notion of sδ-set is a generalization of gδ-set. see [3] for a description of s-normal spaces whose closed subsets are sδ-sets. definition 1.2. let x be a topological space. x has an sδ-diagonal if the diagonal subset ∆x of x × x is an sδ-set of x × x, where ∆x denotes the 72 m. hosobuchi diagonal set {(x,x) : x ∈ x} in the cartesian product x×x. the symbol ( , ) is used to stand for a point of x ×x. it is useful to show the following lemma that relates to the property (∗) given in [2]. n denotes the set of natural numbers. lemma 1.3. [5] let x be a topological space. let {g(n) : n ∈ n} be a family of countable collections of open subsets of x. suppose that, for any three points x,y and z with y 6= z, there exists an m ∈ n such that x ∈ ⋃ g(m) and that no element of g(m) contains the set {y,z}, where ⋃ g(m) denotes⋃ {u : u ∈ g(m)}. then, there exists a family {f(n) : n ∈ n} of countable collections of open subsets of x such that, for such three points above, there exists an m ∈ n such that x ∈ ⋃ f(m) and any two distinct points of {x,y,z} do not belong to the same member of f(m). theorem 1.4. [5] let x be a topological space. x has an sδ-diagonal if and only if there exists a family {g(n) : n ∈ n} of countable collections of open subsets of x such that, for three points x,y and z with y 6= z, there exists an m ∈ n such that x ∈ ⋃ g(m) and any two distinct points of {x,y,z} do not belong to the same member of g(m). 2. two linearly ordered extensions and notation recall that a generalized ordered space (go-space) is a triple (x,τ,<), where < is a linear ordering of the set x and τ a hausdorff topology on x having a base of order-convex sets. we will denote by λ the order topology on (x,<). it is known that λ ⊂ τ. a space of the form (x,λ,<) is called a linearly ordered topological space (lots). every lots is a go-space, but not conversely. in fact it is known that the class of go-spaces coincides with the class of subspaces of lots. given a go-space x there are two well-known linearly ordered extensions of x. one of these is x∗ and was defined by d. j. lutzer [7]. the other one is l(x) and was studied in [8]. we review here the definitions of those linearly ordered extensions. the intervals in a go-space or a lots are written in the form [a,b], [a,b[, ]a,b] and ]a,b[. for example, [a,b] = {x : a ≤ x ≤ b}, [a,b[ = {x : a ≤ x < b} and so on. for a go-space x, we set r = {x ∈ x : [x,→ [ ∈ τ −λ} and l = {x ∈ x : ]← ,x] ∈ τ −λ}, where λ denotes the order topology as mentioned above. r (resp. l) is called the set of right (resp. left ) looking points. then x∗ is defined as follows: x∗ = (x ×{0}) ∪{(x,k) : x ∈ r,k < 0,k ∈ z}∪{(x,k) : x ∈ l,k > 0,k ∈ z} ⊂ x ×z, where z denotes the set of integers. on the other hand, l(x) is defined as follows: l(x) = (x ×{0}) ∪{(x,−1) : x ∈ r}∪{(x, 1) : x ∈ l}⊂ x ×{−1, 0, 1}. x∗ and l(x) are linearly ordered topological spaces equipped with the lexicographic order topologies. we, furthermore, need some technical notation dense sδ-diagonals 73 for the proof of the theorems in section 5. for a convex open subset u of a go-space x, we define a convex open subset ũ of e(x), where e(x) denotes either x∗ or l(x). then eight cases can occur. in the following, the intervals must be considered in e(x). (1) if a is the minimum point of u, then we define ũ1 = [(a, 0),→ [ ⊂ e(x). (2) let a = inf u and a ∈ x \ u. if e(x) = x∗, then ũ1 = {(x,k) ∈ x∗ : a < x} = ](a, +∞),→ [ ⊂ x∗, where (a, +∞) ∈ x × (z∪{+∞}) and the interval is taken in x∗. likewise, if e(x) = l(x), then ũ1 = {(x,k) ∈ l(x) : a < x} = ](a, 1),→ [ ⊂ l(x). note that (a, 1) may not belong to l(x). (3) if there is a gap u = (a,b) such that u is the left end-point of u, then we define ũ1 =](u, 0),→ [ ⊂ e(x). (4) if none of cases 1 − 3 occurs, then we define ũ1 = e(x). (5) if b is the maximum point of u, then we define ũ2 = ] ← , (b, 0)] ⊂ e(x). (6) let b = sup u and b ∈ x \ u. if e(x) = x∗, then ũ2 = {(x,k) ∈ x∗ : x < b} = ] ← , (b,−∞)[ ⊂ x∗ (cf. (2)). if e(x) = l(x), then ũ2 = {(x,k) ∈ l(x) : x < b} = ] ← , (b,−1)[ ⊂ l(x). note that (b,−1) may not belong to l(x). (7) if there is a gap v = (a,b) such that v is the right end-point of u, then we define ũ2 = ]← , (v, 0)[ ⊂ e(x). (8) if none of cases 5 − 7 occurs, then we define ũ2 = e(x). we set ũ = ũ1 ∩ ũ2. ũ is called the convex open set associated with u. let u be an open set of a go-space x. then u is decomposed into a union of open convex subsets {uα : α ∈ a}. in this case, we define ũ = ⋃ {ũα : α ∈ a}, where ũα is the open set associated with uα. then ũ is an open subset of e(x), and called the open set associated with u. 3. sδ-diagonals in linearly ordered extensions the following theorems are proved in our paper [5]. let x be a go-space. it is easily seen that x∗ contains x as a closed subset and l(x) contains x as a dense subset. see [7, 8] for further information about x∗ and l(x). in both cases, x and x ×{0} are identified by the correspondence of x to (x, 0). theorem 3.1. [5] let x be a generalized ordered space with an sδ-diagonal. if r∪l is countable, then x∗ has an sδ-diagonal. to prove a similar theorem concerning l(x), it is necessary to assume the existence of sequences in x that witnesses first-countability for points of r∪l: theorem 3.2. [5] let x be a go-space with an sδ-diagonal. assume that, for every point s ∈ l, there exists a decreasing sequence {x(s,n) : n ∈ n} in x such that inf{x(s,n)} = s and, for every point s ∈ r, there exists an increasing sequence {y(s,n) : n ∈ n} in x such that sup{y(s,n)} = s. if r∪l is countable, then l(x) has an sδ-diagonal. 74 m. hosobuchi 4. dense sδ-diagonals the following definition gives an analogy to the dense gδ-diagonal in [1]. definition 4.1. a hausdorff space x has a dense sδ-diagonal if there exists a dense subset d of ∆x such that d is an sδ-subset of x × x, where ∆x denotes the diagonal subset of the cartesian product x ×x. we show the following theorem that is analogous to a result concerning spaces that have a dense gδ-diagonal [1]. theorem 4.2. let x be a hausdorff space. then x has a dense sδ-diagonal if and only if x has a dense subset y such that y is an sδ-subset of x and y has an sδ-diagonal. proof. if d ⊂ ∆x is a dense sδ-set in x×x, then d∩∆x is a dense sδ-set in ∆x. now the map h : ∆x → x defined by h(x,x) = x is a homeomorphism, and the homeomorphic image of a dense sδ-set is a dense sδ-set. conversely, suppose y is a dense sδ-subset of x. then h−1(y ) is a dense sδ-subset of ∆x. the rest is easily verified. � 5. theorems concerning dense sδ-diagonals of linearly ordered extensions theorem 5.1. let x = (x,τ) be a go-space with a dense sδ-diagonal. if r∪l is countable, then x∗ has a dense sδ-diagonal. we first show the following lemma. lemma 5.2. let x be a go-space and x∗ the linearly ordered extension of x. for a subspace y of x, set z = y ∪ (x∗ \x). if y is dense in x and an sδ-subset of x, then z is a dense subspace of x∗ and an sδ-subset of x∗. proof. to see that z is dense in x∗, let x ∈ x∗ \ z = x \ y and v be a neighborhood of x in x∗, where x is identified with x ×{0} as usual. since v ∩ x is a neighborhood of x in x, it follows that v ∩ x ∩ y 6= ∅. since v ∩x ∩y ⊂ v ∩z, it follows that v ∩z 6= ∅. hence z is a dense subspace of x∗. to show the last part, let {u(n) : n ∈ n} be a countable collection of open subsets of x such that, for y ∈ y and x ∈ x \ y, there exists an m ∈ n such that y ∈ u(m) and x 6∈ u(m). for every n ∈ n, let ũ(n) be the open subset associated with u(n) as in section 3. set ũ(0) = x∗ \ x. then it is obvious that ũ(0) is an open subset of x∗. we show that the countable collection {ũ(n) : n ≥ 0} of open subsets of x∗ assures that z is an sδ-subset of x∗. let z ∈ z and x ∈ x∗ \z = x \y. case 1. let z ∈ x∗\x. then it is easy to see that z ∈ ũ(0) and x 6∈ ũ(0). case 2. let z ∈ y. since x ∈ x \ y, there exists an m ∈ n such that z ∈ u(m) and x 6∈ u(m). by the definition of ũ(m), it follows that z ∈ ũ(m) and x 6∈ ũ(m). this completes the proof. � dense sδ-diagonals 75 now we shall prove theorem 5.1. proof of theorem 5.1. by theorem 4.2, there exists a dense subspace y of x such that y is an sδ-subset of x and that y has an sδ-diagonal. let {g(n) : n ∈ n} be a family of countable collections of open subsets of y such that, for three points x,y and z of y with y 6= z, there exists an m ∈ n such that x ∈ ⋃ g(m) and no element of g(m) contains {y,z}. the existence of the above family is guaranteed by theorem 1.4. for an open subset v of y, there exists an open set vx of x such that vx ∩y = v. let ṽx be the open subset of x∗ associated with vx as explained in section 2. set vz = ṽx ∩z, where z is as mentioned in lemma 5.2. it is clear that vz is open in z and that vz ∩y = v . for every n ∈ n, set g̃(n) = {vz : v ∈ g(n)} and g̃(0) = {{x} : x ∈ x∗ \ x}. let s = r ∪ l = {si : i ∈ n} be an enumeration of the countable set s. let (si,k) ∈ x∗\x. set g̃+(si,k) = {](si,k),→ [ ∩z} and g̃−(si,k) = {]← , (si,k)[∩z}, where these intervals are considered in x∗. by virtue of lemma 5.2 and theorem 4.2, it is sufficient to show that a family of those countable collections of open subsets of z witnesses the sδ-diagonal of z. to see this, let x,y and z be three points of z with y 6= z. we may assume without loss of generality that y < z. case 1. if {x,y,z} ⊂ y, then there exists an m ∈ n such that x ∈ ⋃ g(m) and {y,z} 6⊂ v for any v ∈g(m). hence it follows that x ∈ ⋃ g̃(m) and that {y,z} 6⊂ vz for any vz ∈ g̃(m). case 2. let x ∈ y and y or z belong to z \y. (i) we assume that y ∈ z \ y . then we can write y = (si,k), where k 6= 0. if x < y, then x ∈ ]← ,y[∩z and {y,z} 6⊂ ]← ,y[. hence, by the definition, it follows that x ∈ ⋃ g̃−(si,k) and that {y,z} 6⊂ v for v ∈ g̃−(si,k). if y < x, then x ∈ ]y,→ [∩z and {y,z} 6⊂ ]y,→ [. hence it follows that x ∈ ⋃ g̃+(si,k) and that {y,z} 6⊂ v for v ∈ g̃+(si,k). (ii) let z ∈ z \y. then the proof is analogous to (i). case 3. let x ∈ z \ y. then it follows that x ∈ ⋃ g̃(0) and {y,z} 6⊂ v for any v ∈ g̃(0). therefore, by virtue of lemma 1.3 and theorem 1.4, x∗ has a dense sδ-diagonal. this completes the proof. � theorem 5.3. let x = (x,τ) be a go-space with a dense sδ-diagonal. if r∪l is countable, then l(x) has a dense sδ-diagonal. proof. by theorem 4.2, there exists a dense subspace y of x such that y is an sδ-subset of x and y has an sδ-diagonal. since x is dense in l(x), it follows that y is a dense subspace of l(x). to prove that l(x) has a dense sδ-diagonal, it is sufficient to show, by theorem 4.2, that y is an sδ-subset of l(x). let {u(n) : n ∈ n} be a countable collection of open subsets of x such that, for y ∈ y and x ∈ x \y, there exists an m ∈ n such that y ∈ u(m) and x 6∈ u(m). for every n ∈ n, let ũ(n) be the open subset of l(x) associated with u(n). for si ∈ s = r ∪ l and ε ∈ {−1, 1}, set ũ+(si,ε) = ](si,ε),→ [ and ũ−(si,ε) = ]← , (si,ε)[, where the intervals are considered in l(x). the 76 m. hosobuchi countable collection {ũ(n), ũ+(si,ε), ũ−(si,ε) : n ∈ n, i ∈ n, ε = ±1} of open subsets of l(x) guarantees that y is an sδ-subset of l(x). to see this, let y ∈ y and z ∈ l(x) \y. case 1. let z ∈ x \y. then there exists an m ∈ n such that y ∈ u(m) and z 6∈ u(m). hence it follows that y ∈ ũ(m) and z 6∈ ũ(m). case 2. let z ∈ l(x) \ x. we can write z = (si,ε), where ε ∈ {−1, 1}. if y < z, then it follows that y ∈ ũ−(si,ε) and z 6∈ ũ−(si,ε). if z < y, then it follows that y ∈ ũ+(si,ε) and z 6∈ ũ+(si,ε). hence y is an sδ-subset of l(x). this completes the proof of theorem 5.3. � 6. examples example 6.1. theorems 3.1 and 3.2 do not hold without the assumption of the countability of the set r∪l. let us consider the sorgenfrey line x = (r,s). in this case, the right looking points r = r is uncountable. since x has a gδdiagonal, x has an sδ-diagonal. however, x∗ does not have an sδ-diagonal. to prove this, it is sufficient to see that x∗ does not have a quasi-gδ-diagonal [5]. we easily see that there does not exist a family of countable collections of open subsets of x∗ that separates two points of the form (x, 0) and (x, 1), where x ∈ x. example 6.2. theorem 3.2 does not hold without the existence of the sequences for points of r ∪ l. to show a counterexample, let y be the set of countable ordinals [0,ω1[ with the discrete topology. the right looking points of y comprise the set of limit ordinals. let y ∗ be the linear extension of y defined in section 2. let x = y ∗∪{(ω1, 0)}, where x is ordered as (ω1, 0) > α for all α ∈ y ∗, and given the discrete topology. then r = {(ω1, 0)} is a singleton and l(x) = y ∗ ∪{(ω1,−1)}∪{(ω1, 0)}, where α < (ω1,−1) < (ω1, 0) for all α ∈ y ∗. there does not exist an increasing sequence in y ∗ that converges to (ω1,−1). furthermore, l(x) does not have a quasi-gδ-diagonal, because the points of x and the point {(ω1,−1)} are not separated by a family of countable collections of open subsets of l(x). hence l(x) does not have an sδ-diagonal. example 6.3. a generalized ordered space does not necessarily have a dense sδ-diagonal. to show this, consider the linearly ordered space z that was constructed by h. r. bennett and d. j. lutzer [4]. they proved that z is not first-countable at any point. z is defined as follows: z = {(α1,α2, . . . ,αn,ω1,ω1, . . .) : αi < ω1, 1 ≤ i ≤ n, αi = ω1, i > n, n ≥ 1}, with the lexicographic order. since z is densely-ordered, a dense subset y of z is a lots. if y has a quasi-gδ-diagonal, y is quasi-developable. since a quasi-developable space is first-countable, y does not have a quasi-gδ-diagonal. therefore, z does not have a dense sδ-diagonal. dense sδ-diagonals 77 acknowledgements. the author is grateful to the referees for their helpful comments. references [1] a. v. arhangel’skĭı and lj. d. kočinac, on a dense gδ-diagonal, publ. l’institut math. 47 (61) (1990), 121–126. [2] h. r. bennett, lots with sδ-diagonals, topology proc. 12 (1987), 211–216. [3] h. r. bennett, m. hosobuchi and d. j. lutzer, a note on perfect generalized ordered spaces, rocky mountain j. math. 29 (4) (1999), 1195–1207. [4] h. r. bennett and d. j. lutzer, point countability in generalized ordered spaces, topology appl. 71 (1996), 149–165. [5] m. hosobuchi, sδ-diagonals and generalized ordered spaces, j. tokyo kasei gakuin univ. (nat. sci. tech.) 41 (2001), 1–7. [6] d. j. lutzer, a metrization theorem for linearly orderable spaces, proc. amer. math. soc. 22 (1969), 557–558. [7] d. j. lutzer, on generalized ordered spaces, dissertationes math. 89 (1971), 1–32. [8] t. miwa and n. kemoto, linearly ordered extensions of go spaces, topology appl. 54 (1993), 133–140. received january 2002 revised december 2002 masami hosobuchi tokyo kasei gakuin university, department of housing and planning, 2600 aihara, machida, tokyo 194-0292, japan. e-mail address : mhsbc@kasei-gakuin.ac.jp dense s-diagonals and linearly ordered extensions. by m. hosobuchi caogreenagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 253-264 the ideal generated by σ-nowhere dense sets jiling cao and sina greenwood ∗ abstract. in this paper, we consider the ideal iσ generated by all σ-nowhere dense sets in a topological space. properties of this ideal and its relations with the volterra property are explored. we show that iσ is compatible with the topology for any given topological space, an analogue to the banach category theorem. some applications of this result and the banach category theorem are also given. 2000 ams classification: 26a15, 28a05, 54c05, 54e52. keywords: compatible, ideal, resolvable, σ-nowhere dense, volterra, weakly volterra. 1. introduction let (x, τ ) be a topological space. an ideal i on x is a family of subsets of x such that (i) b ∈ i , if b ⊂ a and a ∈ i ; (ii) a ∪ b ∈ i , if a, b ∈ i . if (ii) is replaced by (ii)’ ⋃ n<ω an ∈ i for any sequence 〈an : n < ω〉 in i , then i is called a σ-ideal. for any given ideal i on x, the minimal σ-ideal containing i shall be called the σ-extension of i . an ideal is said to be proper if it is not equal to the power set p(x) of x. all these notions come from the algebra of p(x) if some appropriate operations are introduced. ideals in general topological spaces were considered in [12], and a more modern study can be found in [7]. one connection between an ideal and the topology on a given topological space arises through the concept of the local function of a subset with respect to the ideal. ∗the first author was supported by the foundation for research, science and technology of new zealand under project number uoax0240. 254 j. cao and s. greenwood definition 1.1 ([3, 7]). let (x, τ ) be a topological space, and let i be an ideal on x. a∗(i ) = {x ∈ x : a ∩ n 6∈ i for every n ∈ n (x)} is called the local function of a ⊂ x with respect to i , where n (x) denotes the collection of all neighbourhoods of x in (x, τ ). the local function operator was used in [3] in the investigation of ideal resolvability. observe that cl∗(a) = a ∪ a∗(i ) defines a kuratowski closure operator, which generates a new topology τ ∗(i ) on x finer than τ . it can be easily checked that b(i ) = {u r i : u ∈ τ and i ∈ i } is a base for the topology τ ∗(i ). for general properties of the local function operator and τ ∗(i ), we refer readers to [7]. ideals have been frequently used in fields closely related to topology, such as, real analysis, measure theory, and descriptive set theory. the following ideals have been of particular interest: in – the ideal of all nowhere dense sets in (x, τ ), im – the σ-ideal of all meager sets in (x, τ ), ib – some σ-ideal consisting of boundary sets in (x, τ ), i0 – the σ-ideal of all lebesgue measure zero sets in r n. for example, by using ib, semadeni [14] established a purely topological generalization of the carathéodory characterization of functions which are equal to riemann-integrable functions almost everywhere. the crucial fact that semadeni required is the following: if a set a is locally in ib (that is, for each x ∈ a, there is a neighbourhood of x in the subspace a which is a member of ib) then a is a member of ib. requirements similar to this one have been used in many other places. for instance, in bourbaki’s integration theory in locally compact spaces, it is required that a set which is locally negligible is of measure zero. another interesting result is that if a set is locally in im then it is a member of im. this is the banach category theorem, first proved by banach for metric spaces in [1]. in [2] and [5], cao, gauld, greenwood and piotrowski studied the volterra property. the class of volterra spaces is closely related to the class of baire spaces. cao and gauld [2] proved an analogue to the oxtoby’s banach category theorem stated in [9], namely, the union of any family of non-weakly volterra open subspaces is still non-weakly volterra. since the ideal im plays an important role in the study of baire and other related properties, and in particular the banach category theorem can be formulated by using the σ-ideal im, it is natural to consider which ideal might play a similar role in the study of the volterra property, and if there exists a result analogous to the banach category theorem for the volterra property in terms of ideals. in the following we show the ideal iσ is such an ideal. after discussing some basic properties of iσ in section 2, relations between the volterra property and iσ are investigated in section 3, and the compatibility of iσ with the topology the ideal generated by σ-nowhere dense sets 255 of any given topological space is established in section 4. in the last section, we shall consider some applications of the banach category theorem and its analogue. 2. preliminaries in this section, we discuss some basic properties of the ideal generated by all σ-nowhere dense sets in a topological space, where σ-nowhere dense sets are defined as follows. definition 2.1. a subset of a topological space (x, τ ) is called σ-nowhere dense if it is an fσ-set with empty interior. note that any subset with empty interior is also called a boundary set. in general, the family of σ-nowhere dense sets in a topological space (x, τ ) is not an ideal. the smallest ideal on x that contains all σ-nowhere dense sets in (x, τ ) will be denoted by iσ. sometimes, iσ is also named as the ideal generated by the family of σ-nowhere dense sets. it is clear that in ⊂ iσ ⊂ im, and that the σ-extension of iσ is precisely im. the following two examples show that iσ may be distinct from in and im in a general topological space (x, τ ), and the topology τ ∗(iσ) may be strictly between τ and the discrete topology on x. example 2.2. let (x, τ ) be the real line r with the usual topology. since q ∈ iσ rin, we have in 6= iσ. it follows that τ ∗(iσ) 6= τ , since q is τ ∗(iσ)closed, but not τ -closed. suppose that {x} is τ ∗(iσ)-open for some x ∈ x, then there exists an open set u ∈ τ and an i ∈ iσ such that {x} = u r i. but i is a subset of a countable union of nowhere dense sets, and since x is baire, this gives a contradiction. hence, we conclude that τ ∗(iσ) is not the discrete topology on x. � example 2.3. if x is any countably infinite set with the cofinite topology, then any open set is meager but not in iσ. � in example 3.8 below, we shall give a tychonoff space (x, τ ) in which iσ 6= im, and in which τ ∗(iσ) is neither τ nor discrete. lemma 2.4. let (x, τ ) be a topological space, and m ∈ n. for each family {ei : i < m} of σ-nowhere dense sets, there is another family {gi : i < m} of σ-nowhere dense sets such that gi ⊂ ei for each i < m, ⋃ ik gi because for each i > k, gi ⊂ ei r nk ⊂ ei r ( intτ ( ⋃ j≤k gj ) ∩ intτ ( ⋃ j≥k gj )) . it follows from x ∈ intτ ( ⋃ i≥k gi) that x ∈ gk. � the following theorem and its corollaries are useful in the sequel. theorem 2.5. in a topological space (x, τ ), any subset of x, that is the union of finitely many σ-nowhere dense sets, can be expressed as the union of exactly two σ-nowhere dense subsets. proof. suppose that a = ⋃ i (1 2 , 1)-starcompact (1, 1)-starcompact l-starcompact 11 2 -starcompact (11 2 , 1)-starcompact 2-starcompact diagram 2 in this section, we shall first provide some examples to show the difference among concepts in diagram 2. lemma 2.1. [6] if a regular space x contains a closed discrete subspace y such that |y | = |x| ≥ ω, then x is not 11 2 -starcompact. example 2.2. there is a 2-starcompact, l-starcompact tychonoff space which is not (11 2 , 1)-starcompact. let r be a maximal almost disjoint family of infinite subsets of ω with |r| = c. it is proved that the isbell-mrówka space ψ = ω ∪r is 2-starcompact in [1]. since ψ is separable, it is l-starcompact. note that every 11 2 -starcompact subspace of ψ is compact. for, if there exists a 11 2 starcompact non-compact subspace x ⊆ ψ, then |x ∩ r| < |x| ≤ ω by lemma 2.1. it follows from |x ∩ r| = |{r1, · · · , rn}| < ω that there exists a ⊆ x ∩ ω such that |a| = ω and a ∩ ⋃n i=1 ri = ø. this implies that x is not pseudocompact, which is a contradiction. enumerate r as {rβ : β < c}. since the intersection of every compact subspace of ψ with r is finite, we can enumerate all compact subsets of ψ as k = {fα : α < c}. for each α < c, choose βα > α such that |rβα ∩ fα| < ω. in addition, we may requre βα < βα′ whenever α < α′. choose an open neighborhood o(rβα) of rβα such that o(rβα) ∩ fα = ø. let i = {βα : α < c}. then u = {{n} : n ∈ ω} ∪ {{rα} ∪ rα : α ∈ c r i} ∪ {o(rβα) : α ∈ c} is an open cover of ψ. let k be any compact subspace of ψ. then k = fα for some α < c. by the construction of u, rβα 6∈ st(k, u). therefore, ψ is not (11 2 , 1)-starcompact. � example 2.3. there is a (1, 1)-starcompact tychonoff space which is neither 11 2 -starcompact nor l-starcompact. let τ be a regular cardinal with τ ≥ ω1. let d be the discrete space with |d| = τ and let d∗ = d∪{∞} be the one-point 4 junhui kim compactification of d. consider x = (d∗ × (τ + 1)) r {〈∞, τ〉} as a subspace of the usual product space d∗ × (τ + 1). since d∗ × τ is a countably compact dense subspace of x, x is (1, 1)-starcompact. but x is not 11 2 -starcompact, since |x| = |d| and d × {τ} is a closed discrete subspace of x. now, we will show that x is not l-starcompact. enumerate d as {dα : α < τ}. for each α < τ, choose an open set uα = {dα} × (α, τ]. then u = {uα : α < τ} ∪ {d ∗ × τ} is an open cover of x. let l be any lindelöf subspace of x. then l∩(d×{τ}) must be countable. let l′ = lr ⋃ {l∩uα : 〈dα, τ〉 ∈ l ∩ (d × {τ})}. without loss of generality, we may assume l ′ 6= ø. since l′ is closed in l, l′ is lindelöf. note that l′ ⊆ d∗×τ. let π : d∗×τ → τ be the projection. then π(l′) is a lindelöf subspace of the countably compact space τ. therefore there exists κ0 < τ which is greater than all elements of π(l′), i.e., uα ∩ l ′ = ø for all α ≥ κ0. since τ is a regular cardinal with τ ≥ ω1 and l ∩ (d × {τ}) is countable, there exists some κ < τ such that κ0 < κ and uκ ∩ l = ø. because uκ is the only one element of u containing dκ, dκ 6∈ st(l, u). therefore, x is not l-starcompact. � example 2.4. there is an l-starcompact and (1, 1)-starcompact tychonoff space x which is not 11 2 -starcompact. let ψ = ω ∪ r be the isbell-mrówka space, where r is a maximal almost disjoint family of infinite subsets of ω with |r| = c and let d be the discrete space such that |d| = |r| and d ∩ r = ø. let y = (d∗ × (ω1 + 1)) r {〈∞, ω1〉}, where d ∗ = d ∪ {∞} is the one-point compactification of d. take a bijection i : r → d×{ω1}. let x be a quotient space of ψ ∪ y and π : ψ ∪ y → x a quotient mapping which identifies r with i(r) for each r ∈ r. then x = π(ω) ∪ π(y ) = π(ψ) ∪ π(d∗ × ω1). since x is locally compact hausdorff, it is tychonoff. first, we will show that x is (1, 1)-starcompact. let u be an open cover of x. note that a = π(d∗ × ω1) is a countably compact dense subset of π(y ). hence π(y ) ⊆ st(a, u). since π(ω) is relatively countably compact in π(ψ), b = ω r st(a, u) is finite. thus st(a ∪ b, u) = x and a ∪ b is a countably compact subspace of x. now, we will show that x is l-starcompact. since a is countably compact, there is a finite subset f of a such that a ⊆ st(f, u). moreover, π(ω) is a countable dense subset of π(ψ), thus we have st(f∪π(ω), u) = x and f∪π(ω) is a lindelöf subspace of x. but π(r) is closed and discrete in x and |π(r)| = |x|. therefore, x is not 11 2 -starcompact. � example 2.5. there is a 11 2 -starcompact, l-starcompact hausdorff space which is not (1, 1)-starcompact. let x = [0, 1] and let τ0 be the euclidean topology on x. define τ1 = {u r f : u ∈ τ0, f is a countable subset of x}. then (x, τ1) is hausdorff. we will show that (x, τ1) is 1 1 2 -starcompact. let u be a basic open cover of (x, τ1). for each u ∈ u, select an open subset v (u) of (x, τ0) and a countable subset f(u) of x such that u = v (u) r f(u). then v = {v (u) : u ∈ u} is an open cover of (x, τ0). since (x, τ0) is compact, v has a finite subcover v0. let u0 = {u ∈ u : v (u) ∈ v0}. then |x r ⋃ u0| ≤ ω. since every neighborhood of each point of x r ⋃ u0 meets⋃ u0, st( ⋃ u0, u) = x. it is easy to prove that (x, τ1) is lindelöf. note that iterated starcompact topological spaces 5 every countable subset is closed and discrete in (x, τ1). so every countably compact subspace is finite. since (x, τ1) is not countably compact (i.e., not 1-starcompact), it is not (1, 1)-starcompact. � a space x is said to be meta-lindelöf (para-lindelöf ) if every open cover of x has a point (locally) countable open refinement. it is well-known that every pseudocompact para-lindelöf tychonoff space is compact. theorem 2.6. let x be a meta-lindelöf t1 space. if x is (1, 1)-starcompact, then it is 11 2 -starcompact. proof. let u be an open cover of x. since x is meta-lindelöf, we may assume that u is point countable. since x is (1, 1)-starcompact, there exists a countably compact subspace a of x such that st(a, u) = x. we may assume a ∩ u 6= ø for all u ∈ u. now, we will show that some finite subcollection v of u covers a. therefore st( ⋃ v, u) = x. suppose that it is not true, and pick an arbitrary point x0 ∈ a. denote by vx0 the subcollection {v ∈ u : x0 ∈ v } of u. since a is countably compact and vx0 is countable, a r ⋃ vx0 6= ø (otherwise, we have a ⊆ ⋃ vx0, and thus there exists a finite subfamily of vx0 which covers a). inductively, we can choose an infinite sequence {xn : n ∈ ω} such that xn ∈ a r ⋃ i0 = {x ∈ f : x > 0} such that for all x ∈ x, |f(x)| > ǫ. since f ∈ c∗(x, f), if and only if, f has a pre-compact co-domain, and ·−1 : f6=0 ≃ −→ f6=0 as topological spaces, where f6=0 = {x ∈ f : x 6= 0}, it follows that f ∈ c ∗(x, f) is a unit of c∗(x, f), if and only if, f does not vanish anywhere on x and is bounded away from 0. remark 2.2. note that a subset u of a topological space is said to be precompact, if and only if, clx(u) is a compact subset of x. the following result will be required in the sequel : theorem 2.3. if f, g ∈ c(x, f) such that zx,f (f) ⊆ intx(zx,f (g)) ⊆ zx,f (g) then f divides g. proof. consider the function h(x) =    g(x) f(x) , if x 6∈ intx(zx,f (g)) 0, otherwise . clearly h ∈ c(x, f) and consequently, g = hf, proving the theorem. � we shall require some regularity properties for the topological spaces under consideration, see [4] for details. definition 2.4. (1) a, b ⊆ x are said to be completely f separated, if and only if, there exists some f ∈ c(x, f) such that f(x) = 0 on a and f(x) = 1 on b. (2) x is said to be completely f regular, if and only if, for every closed subset a of x and every x ∈ x \ a, the sets {x} and a are completely f separated. the property of being completely f regular is clearly the analogue of the tychonoff property when f = r, and the following statements strengthen the similarity. theorem 2.5. for any topological space x the following are equivalent : (1) x is completely f regular; 106 s. k. acharyya, k. c. chattopadhyaya and p. p. ghosh (2) z(x, f) is a base for the closed subsets of x; (3) x has the weak topology induced by c(x, f); (4) x has the weak topology induced by a subset of c(x, f); (5) b(x, f) separates points and closed subsets of x; (6) c∗(x, f) separates points and closed subsets of x. the following provide examples of completely f regular topological spaces : theorem 2.6. (1) f is completely f regular. (2) the property of complete f regularity is productive and hereditary. (3) for f 6= r, a topological space x is completely f regular, if and only if, it is zero dimensional. (4) a topological space x is completely f regular, if and only if, it is homeomorphic to a subspace of a product of f. theorem 2.6(4) equates the class of completely f regular topological spaces to the f-completely regular spaces of mroẃka, see [2]. theorem 2.6(3) is an immediate consequence of the equivalent formulations in theorem 2.5 and the fact that an ordered field f is either connected, in which case it is just isomorphic to r, or else is zero dimensional, see [1]. the next theorem settles our choice of spaces, and is the analogue of the celebrated theorem of stone : theorem 2.7. for any topological space x there exists a completely f regular topological space y and an isomorphism σ : c(x, f) ≃ −→ c(y, f) of lattice ordered commutative rings with unity, such that σ restricted to b(x, f) and c∗(x, f) also produce isomorphisms. in other words, σ ↿b(x, f) : b(x, f) ≃ −→ b(y, f) as well as the restriction to c∗(x, f), σ ↿c∗(x, f) : c ∗(x, f) ≃ −→ c∗(y, f). we will often deal with ideals of the rings. an ideal i of c(x, f) is called a fixed ideal, if and only if, the intersection of the zero sets in x with respect to f of the members of i be non-empty, i.e., the formula (∃x ∈ x)(∀f ∈ i)(f(x) = 0) is true. an ideal is said to be free, if and only if, it is not a fixed ideal. in §4 we shall develop an entity like the structure space of a commutative ring with unity, and for ease of reference when we compare it with our construction we include the required definitions and results. given a commutative ring k with unity, let the set of all its maximal ideals be denoted by m, and let for any a ∈ k, ma = {m ∈ m : a ∈ m}. it is easy to see that {ma : a ∈ k} makes a base for the closed subsets of some unique topology on m, often called the structure space of the commutative ring k, and this topology is sometimes referred to as the stone topology or as the hull kernel topology on m. it is easy to see that : (1) for any x ⊆ m, clm(x) = {m ∈ m : m ⊇ ⋂ x}. indeed, it is this fact from which the name hull kernel topology is derived. (2) for any x ⊆ m, x is dense in m, if and only if, ⋂ x = ⋂ m. (3) m is a compact t1 space. continuous functions with compact support 107 (4) m is hausdorff, if and only if, for every pair of distinct maximal ideals m and n of k there exist points a, b ∈ k such that a 6∈ m, b 6∈ n and ab ∈ ⋂ m. thus, the structure space of z is not hausdorff, while if x is completely f regular then the structure space of c(x, f) is hausdorff. for a much more detailed account of structure spaces see [3, ex. 7a (page 108), ex. m and ex. n (page 111)]. we will fix some notations for the rest of the paper. henceforth, in this paper f shall always refer to a fixed ordered field equipped with its order topology; topological spaces shall always be completely f regular, unless mentioned to the contrary, and for any topological space x and x ∈ x, n xx will refer to the neighborhood filter at the point x. 3. functions with compact support we first clarify the position of ck(x, f) in the hierarchy of subsystems of c(x, f). theorem 3.1. for any topological space x, ck(x, f) ⊆ c ∗(x, f), and if x is non-compact then ck(x, f) is a proper ideal of both c ∗(x, f) and c(x, f). proof. if x is a compact topological space, then clearly ck(x, f) = c(x, f). let x be not compact. then c(x, f) ⊃ ck(x, f) and 1 6∈ ck(x, f); indeed no unit of any of the function rings c(x, f), b(x, f) or c∗(x, f) then belongs to ck(x, f). finally, for any f ∈ ck(x, f), as suppx(f) is compact and x = zx,f (f) ∪ suppx(f) ⇒ f(x) = {0} ∪ f(suppx(f)), so that f(x) is a compact subset of f, implying thereby that f ∈ c∗(x, f). thus we have ck(x, f) ⊆ c∗(x, f). we shall show that ck(x, f) is a proper ideal of c ∗(x, f). obvious modifications of this argument will show that ck(x, f) is a proper ideal of c(x, f), too. for f, g ∈ ck(x, f), since both suppx(f) and suppx(g) are compact, it follows from suppx(f + g) ⊆ suppx(f) ∪ suppx(g), that f + g ∈ ck(x, f). similarly, if h ∈ c∗(x, f), then since suppx(fh) ⊆ suppx(f) ∩ suppx(h), it follows that fh ∈ ck(x, f), too. consequently ck(x, f) is a proper ideal of c∗(x, f). � indeed, not only ck(x, f) is a proper ideal of c ∗(x, f) or c(x, f), but more strongly we have : theorem 3.2. for any completely f regular topological space x, the subring ck(x, f) is contained in every free ideal of c(x, f) or c ∗(x, f). the proof depends on the lemma : lemma 3.3. an ideal i in c(x, f) or c∗(x, f) is free, if and only if, for any compact subset a ⊆ x there is an f ∈ i such that zx,f (f) ∩ a = ∅. proof. 108 s. k. acharyya, k. c. chattopadhyaya and p. p. ghosh proof of the sufficiency part: suppose that i is a free ideal of a, where a is any of the rings c(x, f) or c∗(x, f). let a ⊆ x be compact. since i is free, for every x ∈ a, there exists an fx ∈ i with fx(x) 6= 0. then the set h = {coz(fx) : x ∈ a} is an open cover of a and therefore from compactness of a, there exists a finite sub-cover h1 = {fx1, fx2, . . . , fxn} of a. let f = ∑n i=1 f 2 xi . then f ∈ i and zx,f (f) ∩ a = ∅. proof of the necessity part: trivial. � we now prove theorem 3.2 : proof. if i is a free ideal of a, where a is any one of c(x, f) or c∗(x, f), and f ∈ ck(x, f), then from lemma 3.3 it follows that there exists a g ∈ i such that zx,f (g) ∩ suppx(f) = ∅. consequently, zx,f (g) ⊆ x \ suppx(f) ⊆ x \coz(f) = zx,f (f), so that zx,f (f) is a zero set neighbourhood of zx,f (g), and therefore by theorem 2.3 it follows that g divides f. consequently, f ∈ i as g ∈ i. this proves that ck(x, f) ⊆ i. � the main objective in this section is to show that in the class of completely f regular topological spaces both locally compact spaces and nowhere locally compact spaces can be characterised in terms of the ring ck(x, f). theorem 3.4. a non-compact completely f regular topological space x is locally compact, if and only if, ck(x, f) is a free ideal in both the rings c(x, f) and c∗(x, f). proof. in view of theorem 3.1 it is enough to consider any of the rings c(x, f) or c∗(x, f). the non-compactness of the space x is necessary and sufficient to ensure the inequality ck(x, f) ⊂ c ∗(x, f). proof of the sufficiency part: let x ∈ x. since x is locally compact it follows that there exists an open neighbourhood v ∈ nxx so that clx(v ) is compact. since x is completely f regular, c ∗(x, f) separates points and closed subsets of x, so that there exists some f ∈ c∗(x, f) such that f(x) = 1 and x \ v ⊆ zx,f (f). thus coz(f) ⊆ v ⇒ suppx(f) ⊆ clx(v ) , so that suppx(f) is a compact set, entailing thereby that f ∈ ck(x, f). thus there exists some f ∈ ck(x, f) so that f(x) 6= 0; and as this holds for any x ∈ x, it follows that ck(x, f) is a free ideal of any of the function rings c(x, f) or c∗(x, f). proof of the necessity part: if ck(x, f) is a free ideal of c ∗(x, f) then for any x ∈ x there exists a g ∈ ck(x, f) so that g(x) 6= 0. since suppx(g) is a compact neighbourhood of x, it follows that every point of x has a pre-compact neighbourhood. hence x is locally compact. � continuous functions with compact support 109 theorem 3.5. a completely f regular topological space x is nowhere locally compact, if and only if, ck(x, f) = {0}. proof. proof of the sufficiency part: if x is nowhere locally compact then for any non-zero f ∈ c(x, f), coz(f) is a non-empty open set in x with suppx(f) non-compact — for otherwise suppx(f) will be a compact neighbourhood for every point of coz(f), contradicting the fact that x is nowhere locally compact. consequently, f 6∈ ck(x, f). this implies that ck(x, f) = {0}. proof of the necessity part: it is enough to show that for any x ∈ x there cannot exist any open u ∈ nxx with clx(u) compact. choose and fix any x ∈ x and any open u ∈ n xx . then since x is completely f regular there exists an f ∈ c(x, f) such that f(x) = 1 and x \ u ⊆ zx,f (f). consequently, coz(f) ⊆ u, and thus if clx(u) is compact then suppx(f) will be compact implying thereby that ck(x, f) contains a non-zero member, namely f, contradicting the hypothesis ck(x, f) = {0}. hence clx(u) is non-compact. � 4. structure space of ck(x, f) with any commutative ring with unity one can associate a topological space, called its structure space, [3, ex. 7a (page 108), ex. m and ex. n (page 111)]. the result that in a commutative ring with unity every maximal ideal is prime is a crucial tool in showing the set of all sets of maximal ideals that contain a given point of the ring is a base for the topology of the structure space. since ck(x, f) does not have units and a maximal ideal need not be prime, the classical construction fails. we show here that the set of those maximal ideals which are also prime is the right analogue for building the structure space. we will use the term prime maximal ideal for a maximal ideal which is prime. theorem 4.1. if x is locally compact then the set b = {zx,f (f) : f ∈ ck(x, f)} is a base for the closed subsets of x. proof. let a be a closed subset of x with x ∈ x \ a. since x is locally compact there exists an open set v ∈ nxx so that clx(v ) is compact and that clx(v ) ⊆ x \ a. since v is open and x 6∈ x \ v, it follows from complete f regularity of x that there exists an f ∈ c(x, f) such that f(x) = 1 and x \ v ⊆ zx,f (f). consequently, coz(f) ⊆ v ⇒ suppx(f) ⊆ clx(v ) , and thus suppx(f) is a compact subset of x, i.e., f ∈ ck(x, f). thus we have f ∈ ck(x, f) such that x 6∈ zx,f (f) ⊇ a. hence the assertion is proved. � theorem 4.2. mkx = {f ∈ ck(x, f) : f(x) = 0}, x ∈ x, are precisely the fixed maximal ideals of ck(x, f). 110 s. k. acharyya, k. c. chattopadhyaya and p. p. ghosh furthermore, if x, y ∈ x, x 6= y then mkx 6= m k y . proof. it is clear from theorem 3.4 that for each x ∈ x, mkx is a proper ideal of ck(x, f) and it remains to show that these are maximal. choose and fix one x ∈ x and let g ∈ ck(x, f)\m k x. it suffices to show that the ideal (mkx, g) generated by m k x and g in ck(x, f) is the whole of ck(x, f). for this purpose let h ∈ ck(x, f) \ m k x. from theorem 4.1 it follows that there exists an s ∈ ck(x, f) such that s(x) = 1. consequently, if f = s h(x) g(x) ∈ ck(x, f), then f(x) = h(x) g(x) . hence, h − gf ∈ ck(x, f) with (h − gf)(x) = 0, so that h − gf ∈ mkx, implying thereby h ∈ (m k x, g). hence, m k x is a maximal ideal of ck(x, f). if m is a fixed maximal ideal of ck(x, f) then there exists an x ∈ x such that m ⊆ mkx, and then the maximality of m ensures m = m k x. the last part regarding the one-to-oneness of the map x 7→ mkx follows from theorem 4.1. � theorem 4.3. every proper prime ideal of ck(x, f) is also a proper prime ideal of c(x, f). proof. it is enough to show that if p is a proper prime ideal in ck(x, f) then it is a proper ideal of c(x, f). therefore, we show : if h ∈ p and g ∈ c(x, f) \ ck(x, f), then gh ∈ p. as ck(x, f) is an ideal of c(x, f) it follows that gh, g 2h ∈ ck(x, f) which implies that (g2h)h = g2h2 ∈ p. as p is prime in ck(x, f), it follows that gh ∈ p. � theorem 4.4. there does not exist any proper free prime ideal in ck(x, f). proof. if possible, let p be a proper free prime ideal in ck(x, f). then by theorem 4.3 it follows that p is a prime ideal of c(x, f); as p is a free ideal of c(x, f) it follows from theorem 3.2 that ck(x, f) ⊆ p — contradicting the fact that p is a proper ideal of ck(x, f). this proves the proposition. � theorem 4.5. the entire family of prime maximal ideals in the ring ck(x, f) is {mkx : x ∈ x}. (we state again for emphasis : a maximal ideal in a commutative ring without identity need not be a prime). proof. the assertion follows from theorem 4.2 and theorem 4.4. � we set mkx,f to be the set of all prime maximal ideals of the ring ck(x, f). for f ∈ ck(x, f), let m k x,f (f) = {m ∈ m k x,f : f ∈ m}. then : theorem 4.6. {mkx,f (f) : f ∈ ck(x, f)} is a base for the closed subsets of some topology on mkx,f . continuous functions with compact support 111 proof. since for f, g ∈ ck(x, f) and m ∈ m k x,f : m ∈ mkx,f (fg) ⇔ fg ∈ m ⇔ f ∈ m or g ∈ m (since m is prime) ⇔ m ∈ mkx,f (f) ∪ m k x,f (g), so that mkx,f (fg) = m k x,f (f) ∪ m k x,f (g). the remaining statements follow exactly as in the case of a commutative ring with unity. � definition 4.7. we shall call the set mkx,f equipped with the topology described in theorem 4.6 the structure space of the ring ck(x, f). theorem 4.8. if x is a locally compact non-compact topological space then pkx,f : x ≃ −→ mkx,f , where p k x,f : x 7→ m k x. proof. it follows from theorem 4.2 and theorem 4.5 that x pkx,f −−−−→ mkx,f is a bijection from x onto mkx,f . furthermore, for f ∈ ck(x, f) and x ∈ x, x ∈ zx,f (f) ⇔ f(x) = 0 ⇔ f ∈ m k x ⇔ m k x ∈ m k x,f (f), so that using theorem 4.1 and theorem 4.6 the map pkx,f establishes a one-to-one map between the basic closed sets in the two spaces concerned and therefore pkx,f : x ≃ −→ mkx,f . � it is well known that the structure space of a commutative ring with identity is a compact topological space. the proof for this assertion heavily depends on the existence of the identity element in the ring. theorem 4.8 sounds something contrary in this regard — the structure space of a commutative ring without identity may fail to be compact — indeed for a locally compact non-compact topological space x the structure space mkx,f of ck(x, f) is locally compact without being compact. nevertheless we are now in a position to formulate the principal theorem of this paper. theorem 4.9. suppose that x and y are two locally compact non-compact hausdorff topological spaces. then : (1) x ≃ −→ y as topological spaces, if and only if, ck(x, r) ≃ −→ ck(y, r), as lattice ordered commutative rings. (2) if further, x and y are zero dimensional and f, g be two ordered fields then x ≃ −→ y , as topological spaces, if ck(x, f) ≃ −→ ck(y, f), as lattice ordered commutative rings. proof. 112 s. k. acharyya, k. c. chattopadhyaya and p. p. ghosh (1) if x ≃ −→ y then it is trivial that ck(x, r) ≃ −→ ck(y, r). conversely, ck(x, r) ≃ −→ ck(y, r) implies that m k x,r ≃ −→ mk y,r , since the structure spaces of two isomorphic rings are homeomorphic, so that from theorem 4.8 it follows that x ≃ −→ y . (2) by theorem 2.6(3), for f 6= r the completely f regular topological spaces are precisely the zero dimensional topological spaces; also any zero dimensional hausdorff topological space is completely regular, i.e., completely r regular in our terminology. thus, ck(x, f) ≃ −→ ck(y, g) implies that m k x,f ≃ −→ mky,g, since the structure spaces of two isomorphic rings are homeomorphic, so that from theorem 4.8 it follows that x ≃ −→ y . � acknowledgements. the authors are thankful to professor hans peter künzi, department of mathematics and applied mathematics, university of cape town, for his valuable suggestions regarding the paper. the last author is thankful to the support extended by dept. of math and appl. math., university of cape twon, during his tenure as a post doctoral student there, and further, acknowledges his teacher professor v. saroja, without whose constant support the present work would not have been possible. references [1] w. wieslaw, topological fields, marcell dekker (1978). [2] s. mroẃka and r. engelking, on e-compact spaces, bull. acad. polon. sci. ser. sci. math. astronom. phys. 6 (1958), 429–435. [3] l. gillman and m. jerison, rings of continuous functions, van nostrand reinhold company, edited by m. h. stone, l. nirenberg and s. s. chern (1960). [4] s. k. acharyya, k. c. chattopadhyaya and p. p. ghosh, constructing banaschewski compactification without dedekind completeness axiom, to appear in international journal for mathematics and mathematical sciences. [5] s. k. acharyya, k. c. chattopadhyaya and p. p. ghosh, the rings ck(x) and c∞(x), some remarks, kyungpook journal of mathematics, 43 (2003), 363 369. received february 2003 accepted may 2003 continuous functions with compact support 113 s. k. acharyya (department of pure mathematics, university of calcutta, 35, ballygaunge circular road, calcutta 700019, west bengal, india.) k. c. chattopadhyaya (department of mathematics, university of burdwan, burdwan, west bengal, india) partha pratim ghosh (vsatsxc@cal.vsnl.net.in, pghosh@maths.uct.ac.za) department of mathematics, st. xavier’s college 30, park street, calcutta 700016, india. current address (department of mathematics and applied mathematics, university of cape town, rondebosch 7701, cape town, south africa) @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 217–221 transitivity of hereditarily metacompact spaces hans-peter a. künzi ∗ dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. we prove that each regular hereditarily metacompact (monotonic) β-space has the property that the third power of any neighbornet belongs to its point-finite quasi-uniformity. 2000 ams classification: 54e15, 54d20, 54e18, 54e25. keywords: point-finite quasi-uniformity, transitive quasi-uniformity, transitive topological space, β-space, monotonic β-space, hereditarily metacompact, choquet game. 1. introduction. junnila [6, corollary 4.13] showed (see also [4, theorem 6.21]) that in a semistratifiable metacompact space the third power of each neighbornet belongs to the point-finite quasi-uniformity. similarly, in [7] it was proved that each regular hereditarily metacompact compact space possesses the latter property. junnila’s result and the techniques used in [7] suggested that it should be possible to generalize the latter result beyond (local) compactness using methods known from the theory of monotonic properties (compare [2]). in this note we verify this conjecture by presenting a proof which shows that each regular hereditarily metacompact (monotonic) β-space satisfies the condition that the third power of any neighbornet belongs to its point-finite quasi-uniformity. recall that a topological space is called transitive (see e.g. [4]) provided that its finest compatible quasi-uniformity has a base consisting of transitive entourages. hence in particular our result implies that each regular hereditarily metacompact (monotonic) β-space is transitive. for basic facts about quasi-uniformities we refer the reader to [4]. ∗the author acknowledges support by the swiss national science foundation (under grant 20-63402.00) during stays at theuniversity of berne, switzerland. 218 h.-p.a. künzi 2. main result. let us first mention some pertinent definitions and recall a few well-known facts. a regular topological space x is said to be a monotonic β-space [1] if, for each point x ∈ x, there exists a decreasing sequence 〈bn(x)〉n∈ω of open neighborhood bases of x at the point x such that if bn ∈ bn(xn) and bn+1 ⊆ bn whenever n ∈ ω and if ⋂ n∈ω bn is nonempty, then the sequence 〈xn〉n∈ω has a cluster point. the family {〈bn(x)〉n∈ω : x ∈ x} is called a monotonic β-system of x. the following results are known to hold (in the class of regular (t1-)spaces): each β-space is a monotonic β-space. every monotonic p-space is a monotonic β-space [1, proposition 1.7]. furthermore, every submetacompact monotonic p-space is a p-space [2, theorem 2.8(b)]. recall also that each submetacompact space is a p-space if and only if it is a w∆-space [5, theorem 3.19]. we shall find it convenient to work with the following class of (regular) topological spaces x that is defined in terms of a game g(x) in x, which is a modification of certain games introduced in [3] and was suggested to us by prof. j. chaber. the game g(x) is similar to the strong game of choquet. player i starts the game by choosing a nonempty open set v0 and a point x0 ∈ v0. after player i has chosen his nonempty open set vn and xn ∈ vn in his nth move where n ∈ ω, player ii replies with an open set wn ⊆ vn containing xn and player i, in the next move, has to pick vn+1 inside wn. player ii wins if either ∩n∈ωwn = ∅ or the sequence 〈xn〉n∈ω has a cluster point in x. a winning strategy for player ii is a function s into the topology of x defined on all finite sequences of moves of player i so that player ii always wins when using the function s to determine his next move. it is readily seen that for each (regular) monotonic β-space x, player ii has a winning strategy for the game g(x). indeed in his nth move he will choose as wn some (fixed) member of bn(xn) contained in vn. a scattered partition (see e.g. [8, definition 2.4]) of a topological space x is a cover {lα : α < γ} of x by pairwise disjoint sets such that the set sβ = ⋃ {lα : α < β} is open for each β ≤ γ. a binary relation n on a topological space x is called a neighbornet of x if n(x) = {y ∈ x : (x,y) ∈ n} is a neighborhood at x whenever x ∈ x. for any interior-preserving open cover c of a topological space x, we define the neighbornet dc of x by setting dc(x) = ⋂ {c ∈ c : x ∈ c} whenever x ∈ x. we recall that the filter on x ×x generated by the subbase {dc : c is a point-finite open cover of x} is called the point-finite quasi-uniformity of x (see e.g. [4]). lemma 2.1. suppose that x is a hereditarily metacompact space and let o be a neighbornet of x such that o(x) is open whenever x ∈ x. then there is a point-finite open cover g(x) of x such that for each member h ∈g(x) there is xh ∈ x such that xh ∈ h ⊆ o(xh). proof. choose inductively a possibly transfinite sequence 〈xα〉α of points in x such that xα ∈ x\ ⋃ β<α o(xβ) as long as possible, say whenever α < γ. then transitivity 219 {o(xα) \ ⋃ β<α o(xβ) : α < γ} is a scattered partition of x. according to [8, theorem 6.3] a topological space x is hereditarily metacompact if and only if every scattered partition of x has a point-finite open expansion. hence there is a point-finite open collection {pα : α < γ} of x such that [o(xα) \ ⋃ β<α o(xβ)] ⊆ pα whenever α < γ. it follows that ⋃ {pα ∩ o(xα) : α < γ} = x and xα ∈ pα∩o(xα) ⊆ o(xα) whenever α < γ. therefore we can set g(x) = {pα ∩o(xα) : α < γ}. � theorem 2.2. let o be a neighbornet of a regular hereditarily metacompact space x. if player ii has a winning strategy in the game g(x) described above, then there exists a point-finite open family u of x such that du ⊆ o3. proof. without loss of generality we suppose that o(x) is open whenever x ∈ x. inductively for each n ∈ ω we shall define a point-finite open family un of x. by lemma 2.1 there is a point-finite open cover u0 of x which has the property that for each u0 ∈ u0 there is some point pu0 ∈ x such that pu0 ∈ u0 ⊆ o(pu0 ). let n ∈ ω. suppose that we have defined the point-finite open family uk+1 as the union of families gk+1(u) where u runs through a subfamily of uk whenever k < n. (in the following we distinguish between members in different families gk+1(u) or in u0 that denote the same set; in this way each member arises on a well-defined level of the construction and each member v belonging to the level uk+1 has a unique element u in the level uk preceding it in the sense that v ∈gk+1(u).) furthermore suppose that each un where uk+1 ∈gk+1(uk) whenever k < n determines the sequence (u0 \ o−1(pu0 ),pu1,w0,u1 \ o−1(pu1 ),pu2,w1, . . . , un−1 \ o−1(pun−1 ),pun,wn−1) which describes the moves k < n of a welldefined instance of the game g(x) in the sense that (1) player i has used uk \ o−1(puk) and some well-defined point puk+1 of uk \o−1(puk) in his k th-move whenever k < n, and (2) player ii has chosen the set wk according to his winning strategy in his kth move whenever k < n. in particular note that each wk is determined by the preceding moves of player i and that each uk \o−1(puk) 6= ∅ whenever k < n. call a member un of un suitable (in un) if un 6⊆ o−1(pun). assume now that un is a suitable member of un. suppose that player i continues the beginning of the game g(x) associated with un by choosing un\o−1(pun) and any x ∈ un\o−1(pun) in his nth move. then player ii finds wn(. . . ,un,x) according to his winning strategy such that x ∈ wn(. . . ,un,x) ⊆ un \o−1(pun). by hereditary metacompactness and regularity of x there exists a pointfinite open cover vn+1(un) of un\o−1(pun) such that the closures of its members are all contained in un \o−1(pun). consider the neighbornet of the subspace un \o−1(pun) of x determined by the neighborhoods wn(. . . ,un,x) ∩ 220 h.-p.a. künzi⋂ {e ∈ vn+1(un) : x ∈ e}∩ o(x) whenever x ∈ un \ o−1(pun). by lemma 2.1 there exists a point-finite open cover gn+1(un) of un \o−1(pun) such that for each un+1 ∈ gn+1(un) there is some point pun+1 ∈ un \ o−1(pun) satisfying pun+1 ∈ un+1 ⊆ wn(. . . ,un,pun+1 ) ∩ ⋂ {e ∈ vn+1(un) : pun+1 ∈ e}∩o(pun+1 ). set un+1 = ⋃ {gn+1(un) : un is a suitable member of un}. note that un+1 is a point-finite open family of x. observe also that for each suitable un of un the closures of all members of gn+1(un) are contained in un \ o−1(pun) because vn+1(un) had the latter property. furthermore by the construction above it is readily checked that for each member un+1 ∈ gn+1(un) we have constructed the moves k < n + 1 of the instance of the game g(x) associated with un+1 by adding to the (unique) sequence of moves associated with un the nth moves (un\o−1(pun),pun+1,wn) of player i and player ii, respectively, where wn = wn(. . . ,un,pun+1 ). claim 2.3. there exists a point-finite family u of open sets of x such that the family h = {u ∩o−1(pu ) : u ∈u} covers x. we shall show that our claim holds for the family u = ⋃ n∈ω un: suppose that for some x ∈ x there are infinitely many sets in u containing x. consider the family s of all sets in u containing x. since each family un is point-finite, we conclude by könig’s lemma [9] and the definition of the families un+1 that in s there exists a sequence 〈un〉n∈ω such that for each n ∈ ω, un+1 ∈ gn+1(un). we shall show next that such a sequence does not exist. note first that the sequence 〈un \o−1(pun),pun+1〉n∈ω yields the moves of player i in an instance of the game g(x) where player ii uses his winning strategy to find the sets wn = wn(. . . ,un,pun+1 ) whenever n ∈ ω. by the construction of the family gn+1(un), un+1 ⊆ un+1 ⊆ un\o−1(pun) and pun+1 ∈ un+1 ⊆ wn(. . . ,un,pun+1 ) ⊆ un \o−1(pun) whenever n ∈ ω. since x ∈ ∩n∈ωun and thus x ∈ ∩n∈ωwn(. . . ,un,pun+1 ), we conclude that 〈pun〉n∈ω has a cluster point z in x. thus pun ∈ o(z) for infinitely many n ∈ ω. but also z ∈ un+1 whenever n ∈ ω, because for each n ∈ ω a tail of the sequence 〈pun〉n∈ω is contained in un+1. since un+1 ∩ o−1(pun) = ∅ whenever n ∈ ω, we see that z 6∈ o−1(pun) whenever n ∈ ω —a contradiction. we conclude that the family u is point-finite. suppose that some point x ∈ x is not contained in any set u ∩ o−1(pu ) where u ∈ u. since u0 is a cover of x, there exists u0 ∈ u0 such that x ∈ u0. suppose that n ∈ ω and sets uk (k ≤ n) have inductively been defined such that x ∈ uk+1 ∈ gk+1(uk) (k < n). by our assumption, we have that x ∈ un \ o−1(pun). in particular, un is suitable in un. since gn+1(un) covers un\o−1(pun), there exists un+1 ∈gn+1(un) such that x ∈ un+1. this concludes the induction. of course, x ∈ ∩n∈ωun. but as we just noted above such a sequence 〈un〉n∈ω cannot exist. hence h is a cover of x. transitivity 221 finally we show that du ⊆ o3. let x ∈ x. by the claim verified above there exists u ∈ u such that x ∈ u ∩ o−1(pu ). furthermore, we see that du(x) = ⋂ {v ∈ u : x ∈ v} ⊆ u ⊆ o(pu ) by the selection of the sets u belonging to u. since we have that x ∈ o−1(pu ), there exists a point y ∈ o(x) ∩ o−1(pu ). we now conclude that y ∈ o(x) and pu ∈ o(y). it follows that pu ∈ o2(x) and, furthermore, that o(pu ) ⊆ o3(x). as a consequence, we see that du(x) ⊆ o(pu ) ⊆ o3(x), which confirms the assertion. � acknowledgements. we would like to thank prof. j. chaber for suggestions that led to several improvements of this note. references [1] j. chaber, on point-countable collections and monotonic properties, fund. math. 94 (1977), 209–219. [2] j. chaber, m.m. čoban and k. nagami, on monotonic generalizations of moore spaces, čech complete spaces and p-spaces, fund. math. 84 (1974), 107–119. [3] j. chaber and r. pol, on hereditarily baire spaces and σ-fragmentability of mappings and namioka property, preprint. [4] p. fletcher and w.f. lindgren, quasi-uniform spaces, lecture notes pure appl. math. 77, dekker, new york, 1982. [5] g. gruenhage, generalized metric spaces, in: handbook of set-theoretic topology, k. kunen and j.e. vaughan, eds., north-holland amsterdam, 1984, pp. 423–501. [6] h.j.k. junnila, neighbornets, pacific j. math. 76 (1978), 83–108. [7] h.j.k. junnila, h.p.a. künzi and s. watson, on a class of hereditarily paracompact spaces, topology proc. 25 summer (2000), 271–289; russian translation in: fundam. prikl. mat. 4 (1998), 141–154. [8] h.j.k. junnila, j.c. smith and r. telgársky, closure-preserving covers by small sets, topology appl. 23 (1986), 237–262. [9] d. könig, sur les correspondances multivoques des ensembles, fund. math. 8 (1926), 114–134. received september 2001 revised july 2002 hans-peter a. künzi dept. math. and appl. math., university of cape town, rondebosch 7701, south africa e-mail address : kunzi@maths.uct.ac.za kosiagagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 69-83 f-supercontinuous functions j. k. kohli, d. singh∗ and jeetendra aggarwal abstract. a strong variant of continuity called ‘f -supercontinuity’ is introduced. the class of f -supercontinuous functions strictly contains the class of z-supercontinuous functions (indian j. pure appl. math. 33 (7) (2002), 1097–1108) which in turn properly contains the class of cl-supercontinuous functions (≡ clopen maps) (appl. gen. topology 8 (2) (2007), 293–300; indian j. pure appl. math. 14 (6) (1983), 762–772). further, the class of f -supercontinuous functions is properly contained in the class of r-supercontinuous functions which in turn is strictly contained in the class of continuous functions. basic properties of f -supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity, which already exist in the mathematical literature, is elaborated. if either domain or range is a functionally regular space (indagationes math. 15 (1951), 359–368; 38 (1976), 281–288), then the notions of continuity, f-supercontinuity and r-supercontinuity coincide. 2000 ams classification: 54c08, 54c10, 54d10, 54d20, 54d30 keywords: z-supercontinuous function, f-supercontinuous function, functionally regular space, functionally hausdorff space, f-completely regular space, f-quotient topology 1. introduction several strong variants of continuity occur in the lore of mathematical literature which arise in many branches of mathematics and applications of mathematics. in many situations in topology, analysis and other disciplines continuity is not sufficient and a strong form of continuity is required to meet the demand of a particular situation. the strong variants of continuity with which we shall be dealing in this paper include, among others, are strongly continuous functions [16], perfectly continuous functions [20], clopen maps [21] (≡ cl-supercontinuous ∗this research was partially supported by university grants commission, india. 70 j. k. kohli, d. singh and j. aggarwal functions [23]), z-supercontinuous functions [8], d-supercontinuous functions [10], d∗-supercontinuous functions [22], dδ-supercontinuous functions [11], strongly θ-continuous functions [19] and supercontinuous functions [18]. the main purpose of this paper is to introduce a new class of functions called ‘f supercontinuous functions’, study their basic properties and discuss their place in the hierarchy of strong variants of continuity that already exist in the mathematical literature. the notion of f -supercontinuous functions arise naturally in case either domain or range is a functionally regular space ([1], [2]). it turns out that the class of f -supercontinuous functions properly includes the class of z-supercontinuous functions [8] and is strictly contained in the class of rsupercontinuous functions [14] which in turn is properly contained in the class of continuous functions. further if either domain or range is a functionally regular space ([1], [2]) then all the three classes of (i) f -supercontinuous functions (ii) r-supercontinuous functions, and (iii) continuous functions coincide. moreover, if either x or y is a completely regular space, then all these three classes of functions are identical with the class of z-supercontinuous functions [8]. furthermore, if either domain or range is zero dimensional space, then all the four above classes of functions coincide with the class of cl-supercontinuous functions ([21], [23]). section 2 is devoted to the preliminaries and basic definitions. in section 3, we introduce the notion of ‘f -supercontinuous function’ and elaborate on its place in the hierarchy of strong variants of continuity which already exist in the literature. basic properties of f -supercontinuous functions are studied in section 4, while properties of graph of an f -supercontinuous function are discussed in section 5. interplay of topological properties and f-supercontinuous functions is investigated in section 6 and the notion of f -quotient topology is formulated in section 7. change of topology of a topological space (x, τ ) into a functionally regular topology τf and a completely regular topology τz are considered in section 8 wherein interrelations betweenτ , τf and τz are elaborated and alternative proofs of certain results in the preceding sections are suggested. 2. basic definitions and preliminaries a collection β of subsets of a space x is called as open complementary system [6] if β consists of open sets such that for every b ∈ β, there exist b1, b2, . . . ∈ β with b = ∪{x \ bi : i ∈ n}. a subset a of a space x is called a strongly open fσ-set [6] if there exists a countable open complementary system β(a) with a ∈ β(a). the complement of a strongly open fσ -set is called strongly closed gδ-set. a subset a of a space x is called a regular gδ-set [17] if a is an intersection of a sequence of closed sets whose interiors contain a, i.e., if a = ∞⋂ n=1 fn = ∞⋂ n=1 f ◦n , where each fn is a closed subset of x. the complement of a regular gδ-set is called a regular fσ-set. an open subset a of a space x is said to be r-open [14] if it is expressible as a union of closed sets. f-supercontinuous functions 71 definition 2.1. a function f : x → y from a topological space x into a topological space y is said to be (a) strongly continuous [16] if f (ā) ⊂ f (a) for each subset a of x. (b) perfectly continuous [20] if f −1(v ) is clopen in x for every open set v ⊂ y . (c) cl-supercontinuous [23] (≡ clopen map [21]) if for each x ∈ x and each open set v containing f (x) there is a clopen set u containing x such that f (u ) ⊂ v . (d) z-supercontinuous [8] if for each x ∈ x and for each open set v containing f (x), there exists a cozero set u containing x such that f (u ) ⊂ v . (e) dδ-supercontinuous [11] if for each x ∈ x and for each open set v containing f (x), there exists a regular fσ set u containing x such that f (u ) ⊂ v . (f) d-supercontinuous [10] if for each x ∈ x and each open set u containing f (x) there exists an open fσ-set v containing x such that f (v ) ⊂ u . (g) d∗-supercontinuous [22] if for each x ∈ x and each open set u containing f (x) there exists a strongly open fσ-set v containing x such that f (v ) ⊂ u . (h) strongly θ-continuous [19] if for each x ∈ x and for each open set v containing f (x), there exists an open set u containing x such that f (ū ) ⊂ v . (i) r-supercontinuous [14] if for each x ∈ x and each open set u containing f (x) there exists an r-open set v containing x such that f (v ) ⊂ u . (j) supercontinuous [18] if for each x ∈ x and for each open set v containing f (x), there exists a regular open set u containing x such that f (u ) ⊂ v . definition 2.2. a topological space x is said to be (i) functionally regular ([1], [2]) if for each closed set a and a point x /∈ a there exists a continuous real-valued function f defined on x such that f (x) /∈ f (a); or equivalently for each x ∈ x and each open set u containing x there exists a zero set z such that x ∈ z ⊂ u . (ii) d-regular space [6] if it has a base of open fσ-sets. (iii) functionally hausdorff [25] if for x, y ∈ x, x 6= y there exists a continuous function f : x → [0, 1] such that f (x) 6= f (y). (iv) countably h-closed if it is hausdorff and every countable open cover of x has a finite subcollection whose union is dense in x. (v) semiregular if it has a base of regular open sets. in order to systematize the study of separation by continuous real-valued functions, van est and freudenthal [25] introduced the notion of a functionally regular space and showed its distinctiveness from the standard separation axioms and other separation axioms defined by them. further properties of functionally regular spaces have been studied by aull ([1], [2]). 72 j. k. kohli, d. singh and j. aggarwal 3. f -supercontinuous functions an open set u in a space x is said to be f -open if for each x ∈ u , there exists a zero set z in x such that x ∈ z ⊂ u , or equivalently, u is expressible as a union of zero sets. the complement of an f -open set will be referred to as an f -closed set. definition 3.1. a function f : x → y from a topological space x into a topological space y is said to be f -supercontinuous if for each x ∈ x and each open set u containing f (x) there exists an f -open set v containing x such that f (v ) ⊂ u . the following diagram reflects upon the place of f -supercontinuous functions in the hierarchy of strong variants of continuity that already exist in the literature. the implications are either well known or immediately follow from definitions. however, none of the above implications is reversible which is either well known or follows from the following observations and examples. observations and examples 3.2 if either x or y is a functionally regular space, then every continuous function f : x → y is f -supercontinuous and hence r-supercontinuous. 3.3 if either x or y is a completely regular space, then every continuous function f : x → y is z-supercontinuous. f-supercontinuous functions 73 3.4 let x = y be the regular space due to hewitt [7] on which every continuous real valued function is constant and let f denote the identity map defined on x. then f is strongly θ-continuous and so r-supercontinuous but it is not f -supercontinuous. 3.5 let x be a functionally regular space which is not completely regular and let y = x. then the identity mapping defined on x is f -supercontinuous but not z-supercontinuous. 3.6 let x be the space of [3, exercise 24, p. 139]. then x is a hausdorff semiregular space which is not regular. further, aull pointed out that the space x is a functionally regular space (see [1, example 3]). let f denote the identity mapping defined on x. then the function f is supercontinuous as well as f -supercontinuous but not strongly θ-continuous. 3.7 let us denote by x the space of arens square [24, example 80, p. 98]. then x is a hausdorff space which is not functionally hausdorff and hence not a functionally regular space. also, x is semiregular but not regular. so the identity mapping defined on x is supercontinuous but not f -supercontinuous. 3.8 let us denote by x the space of irregular lattice topology [24, example 79, p. 97]. then x is a functionally hausdorff lindelöf space which is not a semiregular space. in view of [1, theorem 3], the space x is a functionally regular space. let f denote the identity mapping defined on x. then f is an f -supercontinuous function but not supercontinuous. 3.9 let us denote by x the real line with the smallest topology generated by the euclidean topology and the cocountable topology on x. the space x is a functionally regular space, since it is a functionally hausdorff lindelöf space (see [1, theorem 3]). the space x is not d-regular, since x is not a subparacompact space and every lindelöf, d-regular space is subparacompact (see [4, theorem 2]). then the identity mapping defined on x is an f -supercontinuous function but not a d-supercontinuous function. proposition 3.10. let f : x → y be a continuous function, defined on a functionally hausdorff lindelöf space x. then f is f -supercontinuous. proof. since a functionally hausdorff lindelöf space is functionally regular (see [1]) and since every continuous function defined on a functionally regular space is f -supercontinuous, f is f -supercontinuous. � proposition 3.11. let f : x → y be a continuous function. if x is a countably paracompact functionally regular space, then f is f -supercontinuous as well as strongly θ-continuous. proof. since every continuous function defined on a functionally regular space is f -supercontinuous, so is f . again, since every countably paracompact functionally regular space is regular (see [1]), and since every continuous function defined on a regular space is strongly θ-continuous, f is strongly θ-continuous. � 74 j. k. kohli, d. singh and j. aggarwal proposition 3.12. let f : x → y be a continuous function defined on a countably compact functionally regular space x. then f is z-supercontinuous. proof. this is immediate in view of the fact that every countably compact functionally regular space is completely regular (see [1]), and every continuous function defined on a completely regular space is z-supercontinuous. � proposition 3.13. let f : x → y be a continuous function defined on a countably h-closed, semiregular, functionally regular space x. then f is zsupercontinuous. proof. this is immediate from the fact that a countably h-closed, semiregular, functionally regular space x is completely regular (see [1]). � 4. basic properties of f-supercontinuous functions theorem 4.1. for a function f : x → y from a topological space x into a topological space y , the following statements are equivalent: (a) f is f -supercontinuous. (b) the inverse image of every open subset of y is f -open in x. (c) the inverse image of every closed subset of y is f -closed in x. (d) the inverse image of every subbasic open subset of y is f -open in x. proof. it is easy using definitions. � definition 4.2. let x be a topological space and let a ⊂ x. a point x ∈ x is said to be an f-adherent point of the set a if every f -open set containing x has non-empty intersection with a. let af denote the set of all f -adherent points of the set a. the set a is f -closed if and only if a = af . theorem 4.3. for a function f : x → y the following statement are equivalent. (a) f is f -supercontinuous. (b) f (af ) ⊂ f (a) for every a ⊂ x. (c) (f −1(b))f ⊂ f −1(b) for every b ⊂ y . proof. (a) ⇒ (b). since f (a) is closed in y , by theorem 4.1 f −1(f (a)) is an f closed set in x. again, since a ⊂ f −1(f (a)), af ⊂ [f −1(f (a))]f = f −1(f (a)) and so f (af ) ⊂ f (f −1(f (a))) ⊂ f (a). (b) ⇒ (c). let b ⊂ y . then f ((f −1(b))f ) ⊂ f (f −1(b)) ⊂ b̄ and so it follows that (f −1(b))f ⊂ f −1(b̄). (c) ⇒ (a). let b be any closed set in y . then (f −1(b))f ⊂ f −1(b). since f −1(b) ⊂ f −1(b) ⊂ (f −1(b))f , f −1(b) = (f −1(b))f which in turn implies that f is f -supercontinuous. � definition 4.4. a filterbase f is said to f-converge to a point x, written as f f → x if every f -open set containing x contains a member of f. theorem 4.5. a function f : x → y is f -supercontinuous if and only if for each x ∈ x and each filter base f in x that f -converges to x, f (f) → f (x). f-supercontinuous functions 75 proof. suppose that f is f -supercontinuous and let f be a filter base in x that f -converges to x. to show that the filter base f (f) converges to f (x), let w be any open set containing f (x). then x ∈ f −1(w ) and f −1(w ) is f -open. since the filter base f converge to x, there exists f ∈ f such that f ⊂ f −1(w ). then f (f ) ⊂ f (f −1(w )) ⊂ w and so f (f) → f (x). conversely, let w be an open set containing f (x). now, the filter f generated by the filterbase bx consisting of f -open sets containing x, f -converges to x. since by hypothesis f (f) → f (x), there exists a member f (f ) of f (f) such that f (f ) ⊂ w . choose b ∈ bx such that b ⊂ f . since b is an f -open set containing x and since f (b) ⊂ f (f ) ⊂ w , f is f -supercontinuous. � theorem 4.6. if f : x → y is f -supercontinuous and g : y → z is continuous, then the composition g ◦ f is f -supercontinuous. in particular, the composition of f -supercontinuous functions is f -supercontinuous. in general f -supercontinuity of g ◦ f need not imply even continuity of f . for example, let x be the real line with cofinite topology, y be the real line with cocompact topology and z be the real line with indiscrete topology. let f : x → y and g : y → z be the identity mappings. then g ◦ f and g are f -supercontinuous. however, f is not continuous. it is routine to verify that f -supercontinuity is invariant under restrictions and enlargement of range. definition 4.7. a function f : x → y is said to be f-open (f-closed) if f (a) is open (closed) in y for every f -open (f -closed) set a in x. theorem 4.8. let f : x → y be an f -open (f -closed), f -supercontinuous surjection and g : y → z be any function. then the composition g ◦ f is f supercontinuous if and only if g is continuous. further, if in addition f maps f open (f -closed) set to f -open (f -closed) set, then g is an f -supercontinuous function. proof. suppose that g ◦ f is f -supercontinuous. to show that g is continuous, let w be an open (closed) subset of z. then by theorem 4.1 (g ◦ f )−1(w ) = f −1(g−1(w )) is f -open (f -closed) in x. since f is an f -open (f -closed) surjection f (f −1(g−1(w ))) = g−1(w ) is open (closed) in y and so g is continuous. conversely, suppose that g is continuous and let w be an open (closed) set in z. then g−1(w ) is open (closed) in y . since f is f -supercontinuous, f −1(g−1(w )) = (g ◦ f )−1(w ) is f -open (f -closed) in x and so g ◦ f is f supercontinuous. � for the last assertion we need only note that f -openness (f -closedness) of g−1(w ) ensures the f -supercontinuity of g. theorem 4.9. let f : x → y be any function. then the following statements are true. (a) if {uα : α ∈ λ} is an f -open cover of x and for each α, fα = f|uα is f -supercontinuous, then f is f -supercontinuous. 76 j. k. kohli, d. singh and j. aggarwal (b) if {fi : i = 1, . . . , n} is an f -closed cover of x and fi = f|fi is f supercontinuous, then f is f -supercontinuous. proof. (a) let v be any f -open set in y . then f −1(v ) = ∪{f −1α (v ) : α ∈ λ}. since each fα is f -supercontinuous, in view of theorem 4.1 each f −1 α (v ) is f open in uα and hence in x. since any union of f -open sets is f -open, f −1(v ) is f -open. (b) let b be any f -closed set in y . then f −1(b) = n⋃ i=1 f −1i (b). since each fi is f -supercontinuous, by theorem 4.1 each f −1 i (b) is f -closed in fi and hence in x. then f −1(b) being a finite union of f -closed sets is f -closed. so f is f -supercontinuous. � theorem 4.10. let {fα : x → xα : α ∈ λ} be a family of functions and let f : x → ∏ α∈λ xα be defined by f (x) = (fα(x)) for each x ∈ x. then f is f -supercontinuous if and only if each fα : x → xα is f -supercontinuous. proof. let f : x → ∏ α∈λ xα be f -supercontinuous. then the composition pα ◦ f = fα, where pα denotes the projection of ∏ α∈λ xα onto α th-coordinate space xα. so in view of theorem 4.6 each fα is f -supercontinuous. conversely, suppose that each fα : x → xα is f -supercontinuous. to show that the function f is f -supercontinuous, it is sufficient to show that f −1(v ) is f -open for each open set v in the product space ∏ α∈λ xα. since arbitrary unions and finite intersections of f -open sets is f -open, it suffices to prove that f −1(s) is f -open for every subbasic open set s in the product space ∏ α∈λ xα. let vβ × ∏ α6=β xα be a subbasic open set in ∏ α∈λ xα. then f −1(vβ × ∏ α6=β xα) = f −1(p−1 β (vβ )) = f −1 β (vβ ) is f -open in x. hence f is f -supercontinuous. � theorem 4.11. for each α ∈ ∆, let fα : xα → yα be a mapping and let f : ∏ xα → ∏ yα be a mapping defined by f ((xα)) = (fα(xα)) for each (xα) in ∏ xα. then f is f -supercontinuous if and only if fα is f -supercontinuous for each α ∈ ∆. proof. let f : ∏ xα → ∏ yα be f -supercontinuous. let vβ be an open subset of yβ . then vβ × ( ∏ α6=β yα) is a subbasic open subset of the product space ∏ yα. since f is f -supercontinuous, f −1(vβ × ∏ α6=β yα) = f −1 β (vβ ) × ( ∏ α6=β xα) is f -open in ∏ xα. consequently, f −1 β (vβ ) is a f -open set in xβ and hence fβ is a f -supercontinuous. conversely, suppose that each fα : xα → yα is f -supercontinuous. let v = vβ × ( ∏ α6=β yα) be a subbasic open set in ∏ yα. since each fα is f supercontinuous, and since f −1(v ) = f −1(vβ × ( ∏ α6=β yα)) = f −1 β (vβ ) × ( ∏ α6=β xα), f −1(v ) is f -open, and so f is f -supercontinuous. � f-supercontinuous functions 77 theorem 4.12. let f : x → y be a function and g : x → x × y , defined by g(x) = (x, f (x)) for each x ∈ x, be the graph function. then g is f supercontinuous if and only if f is f -supercontinuous and x is functionally regular. proof. to prove necessity, suppose that g is f -supercontinuous. then the composition f = py ◦ g is f -supercontinuous, where py is the projection from x × y onto y . let u be any open set in x and let x ∈ u . then u × y is an open set containing g(x). since g is f -supercontinuous, there exists an f -open set wx containing x such that g(wx) ⊂ u × y . thus x ∈ wx ⊂ u and since u is a union of f -open sets, then it is f -open and so x is functionally regular. to prove sufficiency, let x ∈ x and let w be an open set containing g(x). there exist open sets u ⊂ x and v ⊂ y such that (x, f (x)) ∈ u × v ⊂ w . since x is functionally regular, there exists an f -open set g1 in x containing x such that x ∈ g1 ⊂ u . since f is f -supercontinuous, there exists an f -open set g2 in x containing x such that f (g2) ⊂ v . let g = g1 ∩ g2. then g is an f -open set containing x and g(g) ⊂ u × v ⊂ w , which implies that g is f -supercontinuous. � the following example shows that the hypothesis that ‘x is functionally regular’ in theorem 4.12 cannot be omitted. example 4.13. let x = y = {a, b, c, d}. let the topology on x be given by τ = {φ, x, {a, b}, {d}, {a, b, d}} and let y be equipped with indiscrete topology. let f : x → y be the constant function which takes the value b. then f is f -supercontinuous but the graph function g : x → x × y is not f supercontinuous. theorem 4.14. let f, g : x → y be f -supercontinuous functions from x into a hausdorff space y . then the equalizer e = {x ∈ x : f (x) = g(x)} of the functions f and g is an f -closed set in x. proof. to show that e is f -closed, we shall show that its complement x \ e is an f -open subset of x. let x ∈ x\e. then f (x) 6= g(x). since y is hausdorff, there exist disjoint open sets v and w containing f (x) and g(x), respectively. since f and g are f -supercontinuous, f −1(v ) and g−1(w ) are f -open sets containing x. then u = f −1(v )∩g−1(w ) is an f -open set containing x which is contained in x \ e and so x \ e is f -open. � corollary 4.15. let x be a hausdorff space. then the set of fixed points of every f -supercontinuous function f : x → x is an f -closed set. definition 4.16. a space x is said to be f -completely regular if for every f -closed set a and a point x outside a there exists a continuous function f : x → [0, 1] such that f (x) = 0 and f (a) = 1. theorem 4.17. let f : x → y be an f -supercontinuous function. if x is f -completely regular, then f is z-supercontinuous. 78 j. k. kohli, d. singh and j. aggarwal proof. let x ∈ x and let v be an open set containing f (x). since f is f supercontinuous, there exists an f -open set u containing x such that f (u ) ⊂ v . since x is a f -completely regular space, there exists a continuous function h : x → [0, 1] such that h(x) = 0 and h(x \ u ) = 1. then h−1[0, 1) is a cozero set containing x and contained in u and so it is mapped into v by f . this shows that f is z-supercontinuous. � 5. properties of the graph of an f -supercontinuous function let f : x → y be an f -supercontinuous function. since every f -supercontinuous function is continuous, the family {1x, f}, where 1x denotes the identity mapping on x, separates points and separates points from closed sets. therefore, the mapping g : x → x × y defined by g(x) = (x, f (x)) is an embedding of x into x × y . thus x is homeomorphic to its graph g(f ) = g(x) and so every topological property enjoyed by x is also enjoyed by its graph g(f ). the next two notions reflect upon the fact that how the graph g(f ) of an f -supercontinuous function f : x → y is situated in the product space x × y . definition 5.1. let f : x → y be a function from a topological space x into a topological space y . the graph g(f ) of f is said to be (i) f-closed with respect to x if for each (x, y) /∈ g(f ), there exist open sets u and v containing x and y, respectively such that u is f -open and (u × v ) ∩ g(f ) = φ. (ii) f-closed with respect to x × y if for each (x, y) /∈ g(f ), there exist f -open sets u and v containing x and y, respectively such that (u × v )∩ g(f ) = φ. proposition 5.2. for a topological space x the following are equivalent. (a) x is functionally hausdorff. (b) every pair of distinct points in x are contained in disjoint cozero sets. (c) every pair of distinct points in x are contained in disjoint f -open sets. proof. the implication (a)⇒(b)⇒(c) are trivial. to prove (c)⇒(a), let x, y ∈ x, x 6= y and let u and v be disjoint f -open sets containing x and y, respectively. let a and b be the zero sets in x such that x ∈ a ⊂ u and y ∈ b ⊂ v . let f, g be the real-valued functions defined on x such that z(f ) = a and z(g) = b, where z(f ) and z(g) denote the zero sets of f and g, respectively. let h : x → r be the function defined by h(t) = f (t)/[f (t)+g(t)], for t ∈ x. then h is a continuous function defined on x such that h(x) = 0 and h(y) = 1. � theorem 5.3. if f : x → y is f -supercontinuous and y is functionally hausdorff, then g(f ), the graph of f is f -closed with respect to x × y . proof. let x ∈ x and let y 6= f (x). since y is functionally hausdorff, there exist disjoint f -open sets v and w containing y and f (x), respectively. by f -supercontinuity of f there exists an f -open set u containing x such that f-supercontinuous functions 79 f (u ) ⊂ w ⊂ y \ v̄ . consequently, u × v contains no point of g(f ). hence g(f ) is f -closed with respect to x × y . � the following result is immediate. proposition 5.4. if f : x → y is f -supercontinuous and y is hausdorff, then g(f ), the graph of f is f -closed with respect to x. 6. topological properties and f -supercontinuity theorem 6.1. let f : x → y be an f -supercontinuous open bijection. then x and y are homeomorphic functionally regular spaces. proof. let u be an open set in x and let x ∈ u . since f is an open map, f (u ) is an open set containing f (x). since f is f -supercontinuous, there exists an f -open set g containing x such that f (g) ⊂ f (u ). now, x ∈ f −1(f (g)) ⊂ f −1(f (u )). since f is a bijection, f −1(f (g)) = g and f −1(f (u )) = u . thus x ∈ g ⊂ u . so u being a union of f -open sets is f open. thus x is a functionally regular space. since f is a homeomorphism and functional regularity is a topological property, y is functionally regular. � theorem 6.2. let f : x → y be an f -supercontinuous injection into a t0space, then x is a functionally hausdorff space. proof. let x and y be two distinct points in x. then f (x) 6= f (y). since y is a t0-space, there exists an open set v containing one of the points f (x) or f (y) but not the other. to be precise, assume that f (x) ∈ v . since f is f -supercontinuous, f −1(v ) is an f -open set containing x but not y. so there exists a zero set zx such that x ∈ zx ⊂ f −1(v ). let ϕ : x → [0, 1] be the continuous function such that z(ϕ) = zx the zero set of ϕ. then ϕ(x) = 0 6= ϕ(y). let g and h be disjoint open sets in [0,1] containing ϕ(x) and ϕ(y) respectively. it follows that ϕ−1(g) and ϕ−1(h) are disjoint cozero sets in x containing x and y, respectively. so x is a functionally hausdorff space. � corollary 6.3. let f : x → y be a z-supercontinuous injection into a t0space, then x is a functionally hausdorff space. definition 6.4 ([13]). a space x is said to be f -compact if every f -open cover of x has a finite subcover. theorem 6.5. if f : x → y is an f -supercontinuous surjection from an f -compact space x onto y , then y is compact. proof. let β = {vα|α ∈ ∆} be an open cover of y . in view of f -supercontinuity of f , {f −1(vα) : α ∈ ∆} is an f -open cover of x. since x is f -compact, there exists a finite subset {α1, . . . , αn} of ∆ such that n⋃ i=1 f −1(vαi ) = x. since f is surjection, {vα1 , . . . , vαn } is a finite subcover of y . � 80 j. k. kohli, d. singh and j. aggarwal definition 6.6 ([13]). a space x is said to be weakly f -normal if every pair of disjoint f -closed sets are contained in disjoint open sets. theorem 6.7 ([13]). let f : x → y be an f -supercontinuous closed surjection. if x is a weakly f -normal space, then y is a normal space. 7. f -quotient topology and f -quotient spaces let f : x → y be a surjection from a topological space x onto a set y . the quotient topology on y is the finest topology on y , which makes f continuous. several variants of quotient topology have been defined in the literature (see [8, 10, 11, 15, 22, and 23]) which in general are weaker than quotient topology and coincide with the quotient topology if the domain is suitably augmented. for interrelations among these variants of quotient topology we refer the interested reader to [15]. in this section we introduce the notion of f -quotient topology which in general lies strictly between the quotient topology and the z-quotient topology [8]. we may recall that a set u in a space x is said to be z-open if it is expressible as a union of cozero sets in x. definition 7.1. let p : x → y be a surjection from a topological space x onto a set y . (i) the collection τ of all subsets a ⊂ y such that p−1(a) is z-open in x is a topology on y and is called z-quotient topology [8] and the map p is called the z-quotient map. (ii) the collection τ of all subsets a ⊂ y such that p−1(a) is f -open in x is a topology on y and is called f -quotient topology and the map p is called the f -quotient map. clearly, z-quotient topology ⊂ f -quotient topology ⊂ quotient topology. however, none of the above inclusions are reversible as is well exhibited by the following examples. example 7.2. let (x, τ ) be the space of example 3.8. then x is a functionally regular space which is not a completely regular space. let y = x and let p denote the identity map defined on x. then p is an f -supercontinuous function which is not z-supercontinuous. the f -quotient topology on y is identical with τ while z-quotient topology is strictly coarser than τ . example 7.3. let x be the space of all positive integers endowed with the prime integer topology σ [24, example 61, p. 82]. then x is a hausdorff space which is not a functionally hausdorff space and hence not a functionally regular space. let y = x and let p denote the identity map defined on x. then the quotient topology on y is same as prime integer topology σ but f -quotient topology on y is strictly coarser than σ. theorem 7.4. let p : x → y be a surjection from a topological space x onto a topological space (y, τ ), where τ is the f -quotient topology on y . then p f-supercontinuous functions 81 is f -supercontinuous. moreover, τ is the finest topology on y which makes p : x → y, f -supercontinuous. proof. f -supercontinuity of p is an immediate consequence of the definition of f -quotient topology. now let τ1 be a topology on y such that p : x → (y, τ1) is f -supercontinuous. let g be a τ1 open set in y . by f -supercontinuity of p, p−1(g) is f -open in x. now by the definition of f -quotient topology, g is τ -open and hence τ1 ⊂ τ . � in contrast with the quotient space, the following result shows that a function out of an f -quotient space is continuous if and only if its composition with the f -quotient map is f -supercontinuous. theorem 7.5. let p : x → y be an f -quotient map. then a function g : y → z is continuous if and only if g ◦ p is f -supercontinuous. proof. let u be an open set in z and g ◦ p is f -supercontinuous, then (g ◦ p)−1(u ) = p−1(g−1(u )), is f -open in x. since p is an f -quotient map, g−1(u ) is open in y . hence, g is continuous. the converse is immediate. � 8. change of topology if the topology of domain of an f -supercontinuous function is changed in an appropriate way, then f is simply a continuous function. for, let (x, τ ) be a topological space, and let β denote the collection of all f -open subsets of (x, τ ). since the intersection of two f -open sets is f -open, the collection β is a base for a topology τf on x. indeed β = τf and τf ⊂ τ . the space (x, τ ) is functionally regular if and only if τf = τ . further, if bz denotes the collection of all cozero subsets of (x, τ ), then bz is a base for a topology τz on x. it is immediate that τz ⊂ τf ⊂ τ and the space x is completely regular if and only if τ = τz . thus for a completely regular space τz = τf = τ . throughout the section, the symbol τf will have the same meaning as in the above paragraph. remark 8.1. any topological property which is invariant under continuous bijection will be transferred from (x, τ ) to (x, τf ). the list of such properties is fairly long. in particular, if (x, τ ) is compact lindelöf or countably compact, pseudocompact or quasicompact [5], d-compact or d∗-compact or dδ-compact [12], separable, connected or pathwise connected, then so is (x, τf ). theorem 8.2. a function f : (x, τ ) → (y, ℑ) is f -supercontinuous if and only if f : (x, τf ) → (y, ℑ) is continuous. many of the results studied in preceding sections follow now from above theorem and the corresponding standard properties of continuous functions. theorem 8.3. let (x, τ ) be a topological space. then the following statements are equivalent. 82 j. k. kohli, d. singh and j. aggarwal (a) (x, τ ) is functionally regular. (b) every continuous function from (x, τ ) into a space (y, ℑ) is f -supercontinuous. proof. (a) ⇒(b) is obvious. (b)⇒(a): take (y, ℑ) = (x, τ ). then the identity function 1x on x is continuous, and hence f -supercontinuous. hence by theorem 8.2, 1x : (x, τf ) → (x, τ ) is continuous. since u ∈ τ implies 1−1x (u ) = u ∈ τf , therefore τ ⊂ τf . thus it follows that τ = τf , and so (x, τ ) is a functionally regular. � definition 8.4 ([9]). a function f : x → y from a topological space x into a topological space y is said to be f -continuous if for each x ∈ x and each f -open set u containing f (x) there exists an open set v containing x such that f (v ) ⊂ u . theorem 8.5. let f : (x, τ ) → (y, ℑ) be a function. then (a) f is f -continuous if and only if f : (x, τ ) → (y, ℑf ) is continuous. (b) f is f -open if and only if f : (x, τf ) → (y, ℑ) is open. in view of theorems 8.2 and 8.3, theorem 4.8 can be restated as follows. if f : (x, τf ) → (y, ℑ) is a continuous open surjection and g : (y, ℑ) → (z, v) is a function, then g is continuous if and only if gof is continuous. moreover, f -quotient topology on y determined by the surjection f : (x, τ ) → y in section 7 coincides with usual quotient topology on y determined by f : (x, τf ) → y . references [1] c. e. aull, notes on separation by continuous functions, indag. math. 31 (1969), 458– 461. [2] c. e. aull, functionally regular spaces, indag. math. 38 (1976), 281–288. [3] n. bourbaki, elements of general topology part i, hermann, addison-wesley, 1966. [4] h. brandenburg, on spaces with gδ-basis, arch. math. 35 (1980), 544–547. [5] z. froli’k generalization of compact and lindelöf spaces, czechoslovak. math. j. 13 (84) (1959), 172–217 (russian). 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[23] d. singh, cl-supercontinuous functions, applied general topology 8, no. 2 (2007), 293– 300. [24] l. a. steen and j. a. seeback, jr., counter examples in topology, springer verlag, new york, 1978. [25] w. t. van est and h. freudenthal, trennung durch stetige funktionen in topologischen raümen, indagationes math. 15 (1951), 359–368. received june 2008 accepted january 2009 j. k. kohli (jk kohli@yahoo.com) department of mathematics, hindu college, university of delhi, delhi 110 007, india. d. singh (dstopology@rediffmail.com) department of mathematics, sri aurobindo college, university of delhi-south campus, delhi 110 017, india. jeetendra aggarwal (jitenaggarwal@gmail.com) department of mathematics, university of delhi, delhi 110 007, india. @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 15–24 the quasitopos hull of the construct of closure spaces v. claes and g. sonck abstract. in the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. the topological construct cls of closure spaces and continuous maps is not a quasitopos. in this article we give an explicit description of the quasitopos topological hull of cls using a method of f. schwarz: we first describe the extensional topological hull of cls and of this hull we construct the cartesian closed topological hull. 2000 ams classification: 18a35, 18b25, 18d15, 54a05, 54c35 keywords: topological construct, closure space, extensional topological construct, quasitopos, cartesian closed category, cartesian closed topological hull. 1. introduction cartesian closedness is an interesting property for topological constructs. it guarantees the existence of nice function spaces in the construct. this property has been studied extensively in the literature. a closure space is a set x endowed with a closure operator, i.e. a map cl : p(x) → p(x) satisfying the following conditions: cl ∅ = ∅, a ⊂ cl a, a ⊂ b ⇒ cl a ⊂ cl b and cl (cl a) = cl a. one notes that the closure is allowed to be non-additive. a map f : x → y is said to be continuous if f(cl a) ⊂ cl (f(a)) for all subsets a ⊂ x. the construct of closure spaces and continuous maps is denoted by cls. cls is known to be a well-fibred topological construct. it is an interesting construct since non-additive closures arise in different fields of mathematics, in particular in algebra, geometry and analysis. perhaps the best known example is the convex hull in vector spaces. other examples are listed in the introductory chapter of [6]. there is also a strong relation between closures and complete lattices [4]. in recent years 16 v. claes and g. sonck closures have even been used in connection with quantum logic and in the representation theory of physical systems [13, 14, 19]. the construct cls is not cartesian closed. in [3] the cartesian closed topological hull of cls was described. in this paper we will describe an even more convenient hull: the quasitopos hull. a topological construct is a quasitopos if it is both extensional and cartesian closed. the property of extensionality ensures the existence of one-point extensions in the topological construct, or, equivalently, that quotients and coproducts are preserved by pullbacks along embeddings. this property is extensively treated in [20]. the stronger concept topos is not interesting for topological categories, since the only topological topoi are isomorphic to the category set [1, 22]. quasitopoi (also called topological universes for topological categories) were introduced by j. penon [18] as a generalization of topoi. this generalization is broad enough to allow topological examples, but not too broad for losing most useful properties of topoi. topological universes are used for functional analysis and differential calculus (see e.g. [15, 16, 17]) and for a theory of holomorphic maps (see e.g. [5]). also for a topological construct being a quasitopos is equivalent to being locally cartesian closed i.e. all comma-categories being cartesian closed. it is well known that the construct top of topological spaces and continuous maps is not a quasitopos: it is neither extensional nor cartesian closed. schwarz [21] showed that the topological quasitopos hull of a construct –if it exists– can be described as the cartesian closed topological hull of the extensional topological hull. therefore we first construct the extensional topological hull of cls and then the cartesian closed topological hull of this extensional hull. categorical terminology follows [1]. we will only consider constructs that are well-fibred. 2. the extensional topological hull of cls in this section, we construct the extensional topological hull of cls. we do this in the same way as has been done for top: we weaken the axioms of the closure operator. definition 2.1. [10, 11, 20] a topological construct a is called extensional if it has representable partial morphisms to all a-objects, where • a partial morphism from x to y is a morphism f : z → y whose domain z is a subspace of x. • partial morphisms to y are representable provided y can be embedded via the addition of a single point ∞y into an object y ] with the property that for every partial morphism f : z → y from x to y , the map fx : x → y ], defined by fx(x) = f(x) if x ∈ z, fx(x) = ∞y if x ∈ x \ z, is a morphism. the object y ] is called the one-point extension of y . extensional was called hereditary in [10] and [20]. in [20] schwarz proved that the one-point extension of an object y (if it exists) carries the smallest the quasitopos hull of cls 17 (i.e. coarsest) structure that makes y a subspace. thus the one-point extension is unique (up to isomorphism). there is no smallest closure on {0, 1,∞} which makes the sierpinski space 2 a subspace, so the construct cls can not be extensional. we shall now obtain an extensional supercategory of cls by dropping the idempotency of the closure operator. definition 2.2. a preclosure space (x, cl) is a set x structured by a preclosure operator cl : p(x) →p(x) satisfying the following conditions. for a,b ⊂ x : (c1) cl ∅ = ∅, (c2) a ⊂ cl a, (c3) a ⊂ b ⇒ cl a ⊂ cl b. a function f : (x, clx) → (y, cly ) between preclosure spaces is continuous iff f(clxa) ⊂ cly f(a) for all a ⊂ x. the construct of preclosure spaces and continuous maps is denoted by prcls. a preclosure space can also be described in an isomorphic way using neighborhoods. for a preclosure space (x, cl) and x ∈ x, the collection of neighborhoods v(x) = {v ⊂ x | x /∈ cl (x\v )} satisfies the following three conditions: (v1) x ∈v(x), (v2) ∀v ∈v(x) : x ∈ v , (v3) ∀v ∈v(x) : v ⊂ w ⇒ w ∈v(x). conversely, if the family (v(x))x∈x satisfies the conditions (v1), (v2) and (v3) for each x ∈ x, then a unique preclosure operator cl on x exists, such that for each x ∈ x, v(x) are the neighborhoods of x. this closure operator is defined by: cl a = {x ∈ x | ∀v ∈ v(x) : v ∩ a 6= ∅} for all a ⊂ x. a function f : (x,µ) → (y,η) is continuous iff f−1(η(f(x))) ⊂ µ(x) for every x ∈ x. from the definition of prcls follows immediately that cls is a full subcategory of prcls. proposition 2.3. prcls is a topological construct. proof. for a structured source (fi : x → (yi, cli))i∈i in prcls, the initial preclosure operator on x is defined by: cla = ⋂ i∈i f−1i (cli(fi(a))) for each a ⊂ x or with neighborhoods: v(x) = {v ⊂ x | ∃i ∈ i, ∃w ∈ vi(fi(x)) : f−1i (w) ⊂ v}. � it is clear that cls is closed under formation of initial structures in prcls, and thus we have: proposition 2.4. cls is a bireflective subconstruct of prcls. theorem 2.5. prcls is extensional. proof. analogously as for prtop [11]. � definition 2.6. [9, 11] an extensional topological construct b is called an extensional topological hull of a construct a if b is a finally dense extension of a with the property that any finally dense embedding of a into an extensional topological construct can be uniquely extended to b. 18 v. claes and g. sonck the extensional topological hull of a construct – if it exists – is unique up to isomorphism. theorem 2.7. [10, 11] the extensional topological hull b of a construct a is characterized by the following properties: (1) b is an extensional topological construct. (2) a is finally dense in b. (3) {y ] | y ∈ |a|} is initially dense in b. the proofs of the following propositions are similar to those in [11]. proposition 2.8. cls is a finally dense subcategory of prcls. definition 2.9. the one-point extension 2] of the sierpinski space is the preclosure space 3 with underlying set {0, 1, 2} (we denote ∞ by 2) and neighborhoods: v(0) = v(2) = 0̇ ∩ 1̇ ∩ 2̇ and v(1) = 1̇ ∩ 2̇. proposition 2.10. {3} is initially dense in prcls. now from theorem 2.5, proposition 2.8, proposition 2.10 and theorem 2.7 we have the following result: theorem 2.11. prcls is the extensional topological hull of cls. 3. the cartesian closed hull of prcls an object x in a category with finite products is exponential if x ×− has a right adjoint. in a well-fibred topological construct d, this notion can be characterized as follows: x is exponential in d iff for each d-object y the set homd(x,y ) can be supplied with the structure of a d-object – a function space or a power object y x – such that (1) the evaluation map ev : x ×y x → y is a d-morphism, and (2) for each d-object z and each d-morphism f : x × z → y the map f∗ : z → y x defined by f∗(z)(x) = f(x,z) is a d-morphism. it is well known that in the setting of a topological construct d, an object x is exponential in d iff x × − preserves final episinks [7, 8]. moreover, small fibredness of d ensures that this is equivalent to the condition that x ×− preserves quotients and coproducts. a well-fibred topological construct d is said to be cartesian closed (or to have function spaces) if every object is exponential. it was shown in [3] that cls is not cartesian closed. in fact, the class of exponential objects consists precisely of all indiscrete closure spaces. we show that exponential objects are unchanged if we replace cls by prcls. proposition 3.1. a preclosure space x is an exponential object in prcls if and only if x is indiscrete. proof. suppose µ is an admissible prcls-structure on prcls(x, 3). take x ∈ x and v ∈vx(x). then f : x → 3 defined by f−1(1) = {x},f−1({1, 2}) = v is a prcls-morphism. by continuity of the evaluation map ev in (x,f) there exists b ∈vµ(f) such that ev(x ×b) ⊂{1, 2}. this implies x = f−1({1, 2}) the quasitopos hull of cls 19 and so v = x. if x is indiscrete, then the prcls-structure µ on prcls(x, 3) given by vµ(f) = {prcls(x, 3)} if f−1(1) = ∅ and vµ(f) = stack{{g ∈ prcls(x, 3); g(x) ⊂{1, 2}}} if f−1(1) 6= ∅ is admissible and proper. � we first recall the definitions of cct hull, multimorphism, strictly dense subcategory, power-closed collection and the construction of the cct hull presented by j. adámek, j. reiterman and g. e. strecker [2]. then we use this method to construct the cct hull of prcls. definition 3.2. [12] a cartesian closed topological construct b is called a cartesian closed topological hull (cct hull) of a construct a if b is a finally dense extension of a with the property that any finally dense embedding of a into a cartesian closed topological construct can be uniquely extended to b. definition 3.3. [2] let k be a construct and let h,k be k-objects and x a set. a function h : x ×h → k is called a multimorphism if for each x ∈ x, h(x,−) : h → k defined by h(x,−)(y) = h(x,y) is a morphism. definition 3.4. [2] let k be a construct with quotients and finite products. a full subcategory h of k is said to be strictly dense in k provided that : (1) for each object k ∈ |k| there exists a productively final sink (hi hi→ k)i∈i with hi ∈ |h|, i.e., a final sink such that for each l ∈ |k| the sink (hi ×l hi×1l−→ k ×l)i∈i is final as well. (2) h is well-fibred, closed under quotients, and has productive quotients (i.e., for each quotient e : a → b with a ∈ |h|, we have b ∈ |h| and e× 1h : a×h → b ×h is a quotient for each h ∈ |h|). definition 3.5. [2] let k be a construct with quotients and finite products and let h be strictly dense in k. a collection a of h-objects (a,α) with a ⊂ x is said to be power-closed in x provided that a contains each h-object (a0,α0), a0 ⊂ x, with the following property: given a multimorphism h : x ×h → k with h ∈ |h| and k ∈ |k| such that for each (a,α) ∈ a the restriction h|a : (a,α) ×h → k is a morphism, then the restriction h|a0 : (a0,α0) ×h → k is also a morphism. we denote by pch(k) the category of power-closed collections in h. objects are pairs (x, a), where x is a set and a is a power-closed collection of h-objects in x. morphisms f : (x, a) → (y, b) are functions from x to y such that for each (a,α) ∈ a the final object of the restriction fa : (a,α) → f(a) is in b. if h = k then we simply write pc(k). theorem 3.6. [2] any construct k which has quotients and finite products that are preserved by the forgetful functor, and which has a strictly dense subcategory h, has a cct hull. moreover, this hull is precisely the category of power-closed collections in h. proposition 3.7. in prcls arbitrary products of quotients are quotients. 20 v. claes and g. sonck proof. let xi fi→ yi be a quotient in prcls for any i ∈ i, which means that for every ai ⊂ yi : clyiai = fi(clxi(f −1 i (ai))). let f = ∏ i∈i fi : ∏ i∈i xi → ∏ i∈i yi and a ⊂ ∏ i∈i yi, (yi)i∈i ∈ cla. since all fi : xi → yi are quotients, we have: yi ∈ clyi(pryia) = fi(clxif −1 i (pryia)) = fi(clxiprxi(f −1(a)) for all i ∈ i. this implies: (yi)i∈i ∈ f(clf−1(a)). � from the previous proposition follows that the construct prcls is strictly dense in itself. our aim is to give an explicit description of the objects of the construct pc(prcls) in terms of preclosures. proposition 3.8. if x is a set and c is a power-closed collection in x, then there exists a unique collection a⊂p(x) and for all a ∈a a unique prclsstructure αa on a such that c = {(a,β) ∈ |prcls| : a ∈a, αa 6 β}. proof. for a we take the set of underlying sets of objects in c and for a ∈a we set αa the final prcls-structure on a for the sink (1a : (a,β) → a)(a,β)∈c. it remains to prove that (a,αa) ∈ c for all a ∈ a. take a ∈ a and take a multimorphism h : x × (h,γ) → 3, with (h,γ) ∈ |prcls|, such that for all (c,δ) ∈ c the restriction h|c : (c,δ) × (h,γ) → 3 is a prcls-morphism. then h|a : (a,β) × (h,γ) → 3 is a prcls-morphism for all (a,β) ∈ c with underlying set a. so if (a,y) ∈ a × h satisfies h(a,y) = 1, (h|a)−1({1, 2}) is a neighborhood of (a,y) in (a,β) × (h,γ) for all (a,β) ∈ c, and one of the following cases arises: (1) ∃(a,β) ∈ c, ∃w ∈vγ(y) with a×w ⊂ (h|a)−1({1, 2}). (2) ∀(a,β) ∈ c, ∃vβ ∈vγ(a) with vβ ×h ⊂ (h|a)−1({1, 2}). in case (1), a×w is a neighborhood of (a,y) in (a,αa) × (h,γ). in case (2), v = ⋃ {vβ | (a,β) ∈ c} is a neighborhood of a in αa and v ×h ⊂ (h|a)−1({1, 2}). we conclude that h|a : (a,αa) × (h,γ) → 3 is a prcls-morphism. � proposition 3.9. with the notation of proposition 3.8 the collection a has the properties (a1) ∀x ∈ x, {x}∈a, (a2) a′ ⊂ a ∈a =⇒ a′ ∈a, and the prcls-structures αa satisfy (b1) a′ ⊂ a ∈a =⇒ αa′ 6 αa|a′, (b2) ∀a ∈a, ∀x ∈ a, ∀v ∈vαa(x), ∃t ∈vαa(x) with t ⊂ v and {x}∈vα(a\t)∪{x}(x). proof. for the difficult part take x ∈ a ∈ a and v ∈ vαa(x) and suppose v 6= a (otherwise we can take t = v = a). a prcls-structure µ on a is given by vµ(x) = vαa(x) \{w ∈p(a) | w ⊂ v}, vµ(y) = vαa(y) if y 6= x. the quasitopos hull of cls 21 clearly (a,µ) does not belong to c and so there exists a multimorphism h : x × (h,γ) → 3, with (h,γ) ∈ |prcls|, such that for all (c,δ) ∈ c, h|c : (c,δ) × (h,γ) → 3 is a prcls-morphism and h|a : (a,µ) × (h,γ) → 3 is not a prcls-morphism. there exists y ∈ h such that h(x,y) = 1 and h|a : (a,µ)×(h,γ) → 3 is not continuous at (x,y). since h|a : (a,αa)×(h,γ) → 3 is continuous one of the following cases arises: (1) ∃w ∈vγ(y) with a×w ⊂ (h|a)−1({1, 2}). (2) ∃t ′ ∈vαa(x) with t ′ ×h ⊂ (h|a)−1({1, 2}). since h|a : (a,µ) × (h,γ) → 3 is not continuous at (x,y) we can suppose (2). then t = {t ∈ a | {t}×h ⊂ (h|a)−1({1, 2})} belongs to vαa(x) and satisfies t ×h ⊂ (h|a)−1({1, 2}) and t ⊂ v . now h|(a\t)∪{x} : ((a\t) ∪{x},α(a\t)∪{x}) × (h,γ) → 3 is continuous at (x,y) and so one of the following cases arises: (1) ∃w ′ ∈vγ(y) with ((a\t) ∪{x}) ×w ′ ⊂ (h|(a\t)∪{x})−1({1, 2}). (2) ∃t ′′ ∈vα(a\t)∪{x}(x) with t ′′ ×h ⊂ (h|(a\t)∪{x})−1({1, 2}). in the first case a×w ′ would be a neighborhood of (x,y) in (a,µ) × (h,γ) with a×w ′ ⊂ (h|a)−1({1, 2}), so we can suppose (2). for t ′′ we have t ′′ ⊂ (a\t) ∪{x} as well as t ′′ ⊂ t , so t ′′ = {x}. � proposition 3.10. if x is a set, a⊂p(x) a collection of subsets satisfying the conditions (a1 ) and (a2 ); and if for each a ∈ a a prcls-structure αa is given such that (b1 ) and (b2 ) are satisfied, then the set c = {(a,β) ∈ |prcls| : a ∈a, αa 6 β} is a power-closed collection in x. proof. take a prcls-object (a0,α0) with a0 ⊂ x that does not belong to c. then either a0 /∈ a or both a0 ∈ a and αa0 66 α0. in both cases we give a multimorphism h : x × (h,γ) → 3, with (h,γ) ∈ |prcls|, such that for all (a,β) ∈ c, h|a : (a,β) × (h,γ) → 3 is a morphism and h|a0 : (a0,α0) × (h,γ) → 3 is not a morphism. first suppose a0 /∈ a and take x0 ∈ a0. the prcls-object (h,γ) and multimorphism h : x × (h,γ) → 3 are defined as follows: h = a∪{∞} (∞ /∈a), vγ(y) = {h} if y ∈a, vγ(∞) = stackh{{∞,a} | x0 ∈ a ∈a}, h : x ×h → 3 : (x,y) 7→   1 if (x,y) = (x0,∞), 2 if (x,y) ∈ ⋃ x0∈a∈a a×{a,∞}\{(x0,∞)}, 0 in all other cases. for all (a,β) ∈ c, h|a : (a,β) × (h,γ) → 3 is continuous since if x0 ∈ a the neighborhood a×{a,∞} of (x0,∞) in (a,β) × (h,γ) satisfies a×{a,∞}⊂ (h|a)−1({1, 2}). however h|a0 : (a0,α0) × (h,γ) → 3 is not continuous at (x0,∞) since for v ∈ vα0 (x0) we have v × h 6⊂ (h|a0 )−1({1, 2}) (because (x0,φ) ∈ v ×h but h(x0,φ) = 0) and for x0 ∈ a ∈a we have a0 ×{a,∞} 6⊂ (h|a0 )−1({1, 2}) (for x ∈ a0 \a the pair (x,a) satisfies (x,a) ∈ a0 ×{a,∞} and h(x,a) = 0). now suppose a0 ∈ a and αa0 66 α0. then there exist 22 v. claes and g. sonck x0 ∈ a0 and v ∈ vαa0 (x0) \vα0 (x0). take t ∈ vαa0 (x0), t ⊂ v such that {x0} ∈ vαa0\t∪{x0}(x0). if (a0 \t) ∪{x0} ⊂ a ∈ a, then using (b1) we can choose va ∈vαa(x0) with va∩((a0 \t)∪{x0}) = {x0}. so we can define the map h : x ×h → 3 (with (h,γ) as in the case a0 /∈a) as follows: h(x,y) =   1 if (x,y) = (x0,∞), 2 if (x,y) ∈ ⋃ {va ×h | (a0 \t) ∪{x0}⊂ a ∈a}∪⋃ {a×{a,∞}\{(x0,∞)} | x0 ∈ a ∈a, (a0 \t) ∪{x0} 6⊂ a}, 0 in all other cases. then h : x × (h,γ) → 3 is a multimorphism. now for all (a,β) ∈ c, h|a : (a,β)×(h,γ) → 3 is a prcls-morphism: if (a0\t)∪{x0}⊂ a then va×h is a neighborhood of (x0,∞) in (a,β)×(h,γ) and va×h ⊂ (h|a)−1({1, 2}) and if (a0\t)∪{x0} 6⊂a and x0 ∈ a then a×{a,∞} is a neighborhood of (x0,∞) in (a,β)×(h,γ) which is contained in (h|a)−1({1, 2}). now we only have to prove that h|a0 : (a0,α0) × (h,γ) → 3 is not continuous at (x0,∞). therefore we show that for x0 ∈ a ∈a, a0×{a,∞} 6⊂ (h|a0 )−1({1, 2}) and for w ∈vα0 (x0) we have w ×h 6⊂ (h|a0 )−1({1, 2}). take x0 ∈ a ∈a. if (a0 \t) ∪{x0}⊂ a then for x ∈ a0\t we have h(x,a) = 0. if (a0\t)∪{x0} 6⊂ a then h(x,a) = 0 for x ∈ ((a0 \t) ∪{x0}) \a. finally take w ∈vα0 (x0). then h(x,{x0}) = 0 for x ∈ w \t . � definition 3.11. [21] a quasitopos b is called a quasitopos hull of a construct a if b is a finally dense extension of a with the property that any finally dense embedding of a into a quasitopos can be uniquely extended to b. f. schwarz [21] proved that the quasitopos hull of a construct (if it exists) can be described as the cartesian closed topological hull of the extensional topological hull. using this we have the following proposition. proposition 3.12. the quasitopos hull of cls is the construct which has as objects the pairs (x,{(a,β) ∈ |prcls| : a ∈ a,αa 6 β}), where x is a set, a ⊂ p(x) is a collection of subsets of x satisfying (a1 ) and (a2 ) (see proposition 3.9 ), and for each a ∈a, αa is a prcls-structure on a such that the properties (b1 ) and (b2 ) (see proposition 3.9 ) are fulfilled. morphisms f : (x, a) → (y, b) are functions f : x → y such that for each (a,β) ∈ a the final prcls-object of the restriction f|a : (a,β) → f(a) is in b. in [3] we constructed the cartesian closed topological hull of cls. this construct was denoted by l∗. the following diagram shows the bireflective inclusions into the three hulls of cls under discussion. prcls is not a subcategory of l∗, otherwise l∗ would be the cct hull of prcls. there is no concrete embedding from l∗ into prcls. the quasitopos hull of cls 23 pc(prcls) r tt tt tt tt tt r mm mm mm mm mm l∗ r kk kk kk kk kk prcls r ppp ppp ppp pp cls references [1] j. adámek, h. herrlich and g. e. strecker, abstract and concrete categories (wiley, new york, 1990). 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[15] l. d. nel, topological universes and smooth gelfand-naimark duality, in: j. w. gray (ed.), mathematical applications of category theory (proc. denver 1983), contemporary math. 30 (amer. math. soc., providence, ri, 1984), 244–276. [16] l. d. nel, upgrading functional analytic categories, in: h. l. bentley et al. (eds.), categorical topology (proc. toledo 1983), (heldermann, berlin, 1984), 408–424. [17] l. d. nel, enriched locally convex structures, differential calculus and riesz representation, j. pure appl. algebra 42 (2) (1986), 165–184. [18] j. penon, quasi-topos, c. r. acad. sc. paris sér. a 276 (1973), 237–240. [19] c. piron, mécanique quantique. bases et applications (presses polytechniques et universitaires romandes, lausanne, second edition 1998). [20] f. schwarz, hereditary topological categories and topological universes, quaest. math. 10 (1986), 197–216. 24 v. claes and g. sonck [21] f. schwarz, description of the topological universe hull, in: h. ehrig et al. (eds.), categorical methods in computer science with aspects from topology (proc. berlin 1988), lecture notes computer science 393 (springer, berlin, 1989), 325–332. [22] o. wyler, are there topoi in topology?, in: e. binz and h. herrlich (eds.), categorical topology (proc. mannheim 1975), lecture notes math. 540 (springer, berlin, 1976), 699–719. received december 2001 revised december 2002 v. claes and g. sonck departement wiskunde, vrije universiteit brussel, pleinlaan 2, 1050 brussel, belgium. e-mail address : vclaes@vub.ac.be ggsonck@vub.ac.be the quasitopos hull of the construct of closure spaces. by v. claes and g. sonck @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 193–200 injective locales over perfect embeddings and algebras of the upper powerlocale monad mart́ın escardó abstract. we show that the locales which are injective over perfect sublocale embeddings coincide with the underlying objects of the algebras of the upper powerlocale monad, and we characterize them as those whose frames of opens enjoy a property analogous to stable supercontinuity. 2000 ams classification: 06d22, 54c20, 54d80, 06b35 keywords: injective locale, perfect embedding, powerlocale, free frame, kock–zöberlein monad, stably supercontinuous lattice. 1. introduction an object d of a category is said to be injective over a map j : x → y if for every map f : x → d there is at least one map f̄ : y → d with f̄ ◦j = f. the injectives over subspace embeddings in the category of t0 topological spaces are known to be precisely the continuous lattices under the scott topology [24]. although the extension f̄ of f is far from unique in general, in this case there is a canonical choice, the greatest extension in the pointwise specialization order, denoted by f/j. it is natural to ask whether, for x and y exponentiable topological spaces and for d injective over subspace embeddings, the greatestextension operator f 7→ f/j : dx → dy is continuous [24]. it was shown in [6] that, excluding the trivial situation in which d is the one-point space, this is the case if and only if the embedding j : x → y is a perfect map. here a continuous map g : x → y of topological spaces is called perfect if the right adjoint of the frame homomorphism g−1 : oy →ox preserves directed joins, where ox and oy are the frames of open sets of x and y . hofmann and lawson [12] showed that, for x and y sober, this is equivalent to saying that (1) for every closed set c ⊆ x, the lower set of the direct image g[c] in the specialization order of y is closed and (2) the inverse image g−1(q) of every compact saturated set q ⊆ y is compact (saturated). hofmann [11] showed that, under the additional assumption of local compactness, the first condition 194 m. escardó follows automatically from the second. notice that the perfect maps are the continuous maps satisfying half of the definition of proper map of locales [27], where the missing half is the frobenius condition. topologically, the proper maps are known to be precisely the closed continuous maps that reflect compact saturated sets. perfect maps, under various names and guises, arise frequently in topology and locale theory. it is a folkloric fact that the category of perfect maps of stably locally compact spaces is equivalent to nachbin’s category [20] of monotone continuous maps of compact order-hausdorff spaces – see e.g. [9]. this is an extension of the equivalence of the category of perfect maps of spectral spaces with that of monotone continuous maps of ordered stone spaces, previously established by priestley [21]. localic versions and variations of this can be found in [2, 26, 7] – see also section 4 below. coming back to the subject of the first paragraph, the second natural question is what are the injective spaces over perfect embeddings. it was also shown in [6] that they coincide with the algebras of the upper powerspace monad, but an intrinsic characterization of the algebras was left as an open problem. this paper solves this problem in the localic case and shows that the characterization of the injectives as algebras also holds in this setting. notice that the notion of perfect map as defined for topological spaces makes sense for locales: a continuous map of locales is called perfect if the right adjoint of its defining frame homomorphism preserves directed joins. for a definition of the upper powerlocale monad and its basic properties, see section 2 below. if u,v ∈ ox are opens of a locale x, we write u ≺ v to mean that every scott closed subset c of ox with v ≤ ∨ c has u as a member. theorem 1.1. the following are equivalent for any locale x. (1) x is injective over perfect sublocale embeddings. (2) x is the underlying locale of an algebra of the upper powerlocale monad. (3) (a) every open v ∈ox is the join of the set {u ∈ox | u ≺ v}, and (b) 1 ≺ 1, and u ≺ v and u ≺ w together imply u ≺ v ∧w. schalk [22, 23] characterized the algebras of the restriction of the monad to the category of locally compact locales using a different criterion. as already pointed out by her, the monad on the whole category is of the kock–zöberlein type [18]. convenient references for our purposes are [5] or [8]. by a general result established in [6, theorem 4.2.2], the underlying objects of the algebras of this monad coincide with the locales which are injective over perfect embeddings, using the characterization of perfect maps given in vickers [28]. moreover, it also follows from [6, theorem 4.2.2] that the greatest-extension property discussed above also holds for injectives over perfect embeddings. our main tool for identifying the algebras, and hence the injectives, is kock’s criterion: an object is the underlying object of an algebra if and only if its unit has a right adjoint, which is then its unique structure map – see section 2 below. injective locales over perfect embeddings 195 we emphasize that the proof of theorem 1.1, given in section 3 below, is constructive in the sense of topos logic. reinhold heckmann is gratefully acknowledged for a careful reading of a previous version of this paper. 2. preliminaries we take the basic notions concerning locales and frames for granted [14] – we just emphasize that the category loc of continuous maps of locales is defined to be the opposite of the category frm of homomorphisms of frames. the topology, or frame of opens, of a locale x is denoted by ox and is ranged over by the letters u,v,w. the defining frame homomorphism of a continuous map f : x → y of locales is denoted by f∗ : oy →ox. as a map of posets, this has a right adjoint, which is denoted by f∗ : ox →oy . for the sake of completeness, we recall the definition of the upper powerlocale monad and show that it is of the kock–zöberlein type. a preframe is a poset with finite meets and directed joins in which the former distribute over the latter, and a preframe homomorphism is a map that preserves both operations. the forgetful functor g: frm → prfrm into the category of preframes has a left adjoint f : prfrm → frm. by banaschewski [1], and independently heckmann [10], for any preframe l, the free frame fl can be constructed as the set of scott closed subsets of l ordered by inclusion, with insertion of generators given by principal ideals: �: l → fl u 7→ ↓u. if a is any frame and h: l → a is any preframe homomorphism, the unique frame homomorphism h̄: fl → a with h̄(�u) = h(u) is given by h̄(c) = ∨ {h(u) | u ∈ c}. this induces a monad on prfrm and a comonad on frm, and hence a monad (u,η,µ) on loc, with oux = fgox, known as the upper powerlocale monad. (notice that, by virtue of the freeness property, the global points 1 → ux are in bijection with the preframe homomorphisms ox → o 1 and hence with the scott open filters of ox. thus, by the localic hofmann–mislove theorem [15, 29], they are in bijection with the compact saturated sublocales of x). explicitly, the upper powerlocale monad is constructed as follows. for any continuous map f : x → y , the continuous map uf : ux →uy is given by (uf)∗ : ouy → oux �v 7→ �f∗(v). the unit ηx : x →ux of a locale x is given by η∗x : oux → ox �u 7→ u. 196 m. escardó in both cases, we are using the fact that the opens of the form �u freely generate the frame of opens of the upper powerlocale, as explained above. explicitly, the unit is given by η∗x(c) = ∨ c, a fact used in the proof of lemma 3.1 below. multiplication is given by µ∗x : oux → ouux �u 7→ � � u, but this is not needed for our considerations. the category of locales is poset-enriched (and, in fact, dcpo-enriched) with f ≤ g in loc(x,y ) if and only if f∗(v) ≤ g∗(v) for all v ∈oy . the following lemma implicitly refers to this enrichment, which is known as the specialization ordering. a monad (t,η,µ) on a poset-enriched category c is said to be of the kock– zöberlein type if the functor t is order-preserving and, for every object x, the inequality ηtx ≤ tηx holds (the original, official definition has the inequality in the opposite direction, but the convention that we adopt is the one which is convenient for our purposes). in the presence of order preservation, the inequality is equivalent to saying that structure maps α: tx → x are right adjoint to units ηx : x → tx (and hence uniquely determined by x when they exist), a fact which is also used in the proof of lemma 3.1 below. here, by an adjunction f a g in the hom-poset c(x,y ) it is meant a pair of maps f : x → y (the left adjoint) and g : y → x (the right adjoint) with f◦g ≤ idy and g ◦ f ≥ idx. the adjunction f a g is said to be reflective if g ◦ f = idx. once again for use in lemma 3.1 below, observe that, by definition of structure map [19], a right adjoint to a unit has to be reflective in order to be a structure map. notice that, because loc = frmop, we have that f a g holds in loc(x,y ) with respect to the specialization order if and only if g∗ a f∗ holds in frm(oy,ox) with respect to the pointwise order. lemma 2.1. the upper powerlocale monad is of the kock–zöberlein type. proof. the functor is monotone: assume that f ≤ g in loc(x,y ) and let v ∈ oy . then �f∗(v) ≤ �g∗(v) by monotonicity of �, which amounts to (uf)∗(�v) ≤ (ug)∗(�v). this completes the proof of monotonicity, because the opens of the form �v form a subbase of the topology of uy . the inequality ηux ≤ uηx holds in loc(ux,uux): for c ∈ oux, we have that η∗ux(�c) = c ⊆↓ ∨ c = �η∗x(c) = (uηx) ∗(�c). the result then follows again from the fact that the opens of the form �c with c ∈ oux form a subbase of the topology of uux. � 3. proof of the theorem recall the definition of the relation ≺ on the opens of a locale formulated in the paragraph preceding theorem 1.1. injective locales over perfect embeddings 197 lemma 3.1. a locale is the underlying object of an algebra of the upper powerlocale monad if and only if the following two conditions hold. (1) every open v is the join of the opens u ≺ v, and (2) 1 ≺ 1, and u ≺ v and u ≺ w together imply u ≺ v ∧w. proof. by the kock–zöberlein property, a locale x is the underlying object of an algebra if and only if its unit ηx : x → ux has a reflective right adjoint α: ux → x. this amounts to saying that α∗◦η∗x ≤ id and η ∗ x ◦α ∗ = id hold in the category of frames. because η∗x(c) = ∨ c, the adjoint-functor theorem shows that the inequalities α∗ ◦ η∗x ≤ id and η ∗ x ◦ α ∗ ≥ id are equivalent to the equation α∗(v) = ⋂ {c ∈ oux | v ≤ ∨ c}. it follows that u ∈ α∗(v) if and only if u belongs to every c ∈ oux with v ≤ ∨ c, which amounts to saying that u ≺ v; that is, α∗(v) = {u ∈ ox | u ≺ v}. hence the equation η∗x ◦ α ∗ = id amounts to condition 1. being a left adjoint, α∗ preserves all joins. preservation of finite meets amounts to condition 2. � vickers showed that a continuous map f : x → y of locales is perfect if and only if the continuous map uf : ux → uy has a left adjoint [28, proposition 5.6]. a slight modification of his proof establishes the following. lemma 3.2. a continuous map j : x → y of locales is a perfect sublocale embedding if and only if uj : ux →uy has a reflective left adjoint. proof. assume that j : x → y is a perfect map. being a right adjoint, j∗ : ox → oy preserves meets and hence is a preframe homomorphism by perfectness of j. by freeness of the frame oux, there is a unique continuous map u∗j : uy → ux such that (u∗j)∗(�u) = �j∗(u) for all u ∈ ox. then (uj)∗ ◦ (u∗j)∗(�u) = (uj)∗(�j∗(u)) = �j∗j∗(u) ≤ �u, and a similar calculation shows that (u∗j)∗ ◦ (uj)∗(�v) ≥ �v. since each inequality holds for all opens of a subbase, they hold for all opens, which establishes the desired adjunction u∗j a uj. reflectivity holds if and only if (uj)∗ ◦ (u∗j)∗ = id if and only if �j∗j∗(u) = �u for all u ∈ ox if and only if j∗j∗(u) = u for all u ∈ox if and only if j is an embedding. conversely, assume that uj : ux →uy has a (reflective) left adjoint u∗j : uy →ux and define ox r oy = ox � oux (u∗j)∗ ouy η∗yoy. being a composition of maps that preserve directed joins, r itself preserves directed joins. we show that j∗ a r (reflectively), so that j∗ = r and hence j is a perfect map (embedding): j∗ ◦r(u) = j∗ ◦η∗y ◦ (u ∗j)∗(�u) by definition of r = (ηy ◦ j) ∗ ◦ (u∗j)∗(�u) by contravariance of (−)∗ = (uj ◦ηx) ∗ ◦ (u∗j)∗(�u) by naturality of η = η∗x ◦ (uj) ∗ ◦ (u∗j)∗(�u) by contravariance of (−)∗ = η∗x ◦ (u ∗j ◦uj)∗(�u) by contravariance of (−)∗ ≤ η∗x(�u) because u ∗j auj, = u, 198 m. escardó where the inequality is an equality if the adjunction u∗j auj is reflective, and r ◦ j∗(v) = η∗y ◦ (u ∗j)∗(�j∗(v)) by definition of r = η∗y ◦ (u ∗j)∗ ◦ (uj)∗(�v) because (uj)∗(�v) = �j∗(v) = η∗y ◦ (uj ◦u ∗j)∗(�v) by contravariance of (−)∗ ⊇ η∗y (�v) because u ∗j auj = v. � corollary 3.3. a locale is the underlying object of an algebra of the upper powerlocale monad if and only if it is injective over perfect sublocale embeddings. proof. by [6, theorem 4.2.2], for any kock–zöberlein monad t on a posetenriched category, the objects which are injective over maps j : x → y such that tj : tx → ty has a reflective left adjoint coincide with the underlying objects of t-algebras. � this concludes the proof of theorem 1.1. notice that lemma 3.2 shows that the unit ηx : x → ux of any locale x is a perfect sublocale embedding, because the map tηx : tx → ttx has µx : ttx → tx as a reflective left adjoint for any kock–zöberlein monad (t,η,µ) on a poset-enriched category [5]. hence, by corollary 3.3, every locale can be perfectly embedded into a perfectly injective locale, because ux is a free algebra. 4. remarks the (independent and unpublished) work of ho weng kin and zhao dongsheng [17] characterizes the lattices of scott closed sets of various kinds of posets. when applied to frames, their results imply that the free algebras of the upper powerlocale monad are precisely the locales such that every open v is the join of the opens u ≤ v with u ≺ u and such that the relation ≺ satisfies the stability condition (3b) of theorem 1.1. they also consider lattices in which every element v is the join of the elements u ≺ v, with a different motivation in mind, and establish a number of interesting results for them. as far as we know, the characterization of the injectives over perfect embeddings given here is new. however, as shown in [6, 5, 8], several known injectivity results can be established by proofs following the above pattern, provided the monads under consideration are of the kock–zöberlein type. an example mentioned in [8] is that of stably locally compact locales. these coincide with the injectives over flat sublocale embeddings [13, 16], and with the algebras of the monad on locales that arises from the forgetful functor from frames into distributive lattices, whose left adjoint is ideal completion. in connection with a discussion started in the introduction, we observe that the algebra homomorphisms are precisely the perfect maps, so that there is a further equivalence with nachbin’s category of compact order-hausdorff spaces. this is related to the work of simmons [25]. an example not mentioned in [8] is that of injectives over arbitrary sublocale embeddings; these coincide with the algebras of injective locales over perfect embeddings 199 the monad on locales that arises from the forgetful functor from frames into meet-semilattices. let’s develop this second example in more detail. as in the main example discussed in this paper, an explicit construction of the free frame is known: it is the set of down sets of the meet-semilattice, ordered by inclusion. again, the insertion of generators is given by principal ideals. thus, one is led to consider the relation u ≺ v redefined to mean that every cover of v which is a down set has u as a member, or, equivalently, every cover of v has a member that covers u, and a theorem analogous to the one proved in this paper holds with an analogous proof. the frames which satisfy the third condition of the theorem, with the relation redefined as above, are called stably supercontinuous and are known to coincide with the scott topologies of continuous lattices. part of what is described here is proved in [3]. by the freeness property, the global points of the locale whose topology is the free frame over a frame qua semilattice coincide with the meet-semilattice homomorphisms from the frame into the initial frame, which in turn coincide with the filters of the frame. thus, this monad is the localic version of the filter monad on the category of t0 topological spaces discussed in [5] and previously by a. day [4] and wyler [30]. notice that, in the main example developed in this paper and the ones discussed in this section, it is crucial to know an explicit construction of the free frame and of the insertion of generators. for instance, we are unable to apply our technique to the lower powerlocale monad, because we lack a sufficiently explicit description of the left adjoint of the forgetful functor from frames into sup-lattices. references [1] b. banaschewski, another look at the localic tychonoff theorem, comment. math. univ. carolin. 29 (4) (1988), 647–656. 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[9] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. mislove, and d. s. scott, continuous lattices and domains (cambridge university press, 2002). [10] r. heckmann, lower and upper power domain constructions commute on all cpos, information processing letters 40 (1) (1991), 7–11. 200 m. escardó [11] k. h. hofmann, stably continuous frames, and their topological manifestations, in categorical topology, pages 282–307. (heldermann, 1984). [12] k. h. hofmann and j. d. lawson, on the order theoretical foundation of a theory of quasicompactly generated spaces without separation axiom, volume 27 of mathematikarbeitspapiere, pages 143–160. (university of bremen, 1982). [13] p. t. johnstone, the gleason cover of a topos, ii, j. pure appl. algebra 22 (1981), 229–247. [14] p. t. johnstone, stone spaces (cambridge university press, 1982). [15] p. t. johnstone, vietoris locales and localic semilattices, in continuous lattices and their applications, pages 155–180. (dekker, new york, 1985). [16] p. t. johnstone, sketches of an elephant: a topos theory compendium, number 43–44 in oxford logic guides (oxford science publications, 2002). [17] ho weng kin and zhao dongsheng, on the characterization of scott-closed set lattices, preprint. mathematics department, national institute of education university, singapore, 2002. [18] a. kock, monads for which structures are adjoint to units (version 3), j. pure appl. algebra 104 (1995), 41–59. [19] s. mac lane, categories for the working mathematician (springer-verlag, 1971). [20] l. nachbin, topologia e ordem (university of chicago press, 1950). in portuguese. english translation published 1965 by van nostrand, princeton, as topology and order. [21] h. a. priestley, ordered topological spaces and the representation of distributive lattices, proc. london math. soc. (3) 24 (1972), 507–530. [22] a. schalk, algebras for generalized power constructions, phd thesis, mathematics department, technische hochschule darmstadt, july 1993. available at ftp://ftp.cl.cam.ac.uk/papers/as213/diss.dvi.gz. [23] a. schalk, domains arising as algebras for powerspace constructions, j. pure appl. algebra 89 (3) (1993), 305–328. [24] d. s. scott, continuous lattices, in f. w. lawvere, editor, toposes, algebraic geometry and logic, volume 274 of lecture notes in mathematics, pages 97–136, 1972. [25] h. simmons, a couple of triples, topology appl. 13 (1982), 201–223. [26] c. f. townsend, localic priestley duality, j. pure appl. algebra 116 (1-3) (1997), 323– 335. [27] j. j. c. vermeulen, proper maps of locales, j. pure appl. algebra 92 (1994), 79–107. [28] s. j. vickers, locales are not pointless, in c. hankin, i. mackie, and r. nagarajan, editors, theory and formal methods 1994, (ic press, 1995). [29] s. j. vickers, constructive points of powerlocales, math. proc. cambridge philos. soc. 122 (2) (1997), 207–222. [30] o. wyler, algebraic theories for continuous semilattices, archive for rational mechanics and analysis 90 (2) (1985), 99–113. received august 2002 revised december 2002 mart́ın h. escardó school of computer science, university of birmingham, birmingham b15 2tt, england. e-mail address : m.escardo@cs.bham.ac.uk http://www.cs.bham.ac.uk/~mhe/ http://www.cs.bham.ac.uk/~mhe/ injective locales over perfect embeddings and algebras of the upper powerlocale monad. by m. escardó songagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 2, 2008 pp. 293-299 on σ-starcompact spaces yan-kui song ∗ abstract. a space x is σ-starcompact if for every open cover u of x, there exists a σ-compact subset c of x such that st(c, u) = x. we investigate the relations between σ-starcompact spaces and other related spaces, and also study topological properties of σ-starcompact spaces. 2000 ams classification: 54d20, 54b10, 54d55. keywords: lindelöf, σ-starcompact, l-starcompact. 1. introduction by a space, we mean a topological space. let us recall that a space x is countably compact if every countable open cover of x has a finite subcover. fleischman [3] defined a space x to be starcompact if for every open cover u of x, there exists a finite subset f of x such that st(f, u) = x, where st(f, u) = ⋃ {u ∈ u : u ∩ f 6= ∅}, and he proved that every countably compact space is starcompact. conversely, van douwen-reed-roscoe-tree [1] proved that every hausdorff starcompact space is countably compact, but this does not hold for t1-space (see [7]). as generalizations of starcompactness, the following classes of spaces were given: definition 1.1 ([1, 6]). a space x is star-lindelöf if for every open cover u of x, there exists a countable subset f of x such that st(f, u) = x. definition 1.2. a space x is σ-starcompact if for every open cover u of x, there exists a σ-compact subset c of x such that st(c, u) = x. definition 1.3 ([3, 6, 8]). a space x is l-starcompact if for every open cover u of x, there exists a lindelöf subset l of x such that st(l, u) = x. ∗the author acknowledges support from the nsf of china(grants 10571081) and the national science foundation of jiangsu higher education institutions of china (grant no 07kjb110055) 294 y.-k. song in [1], a star-lindelöf space is called strong star-lindelöf, in [3], l-starcompactness is called slc property. from the above definitions, we have the following diagram: star-lindelöf ⇒ σ-starcompact ⇒ l-starcompact. in the following section, we give examples showing that the converses in the above diagram do not hold. thorough this paper, the symbol β(x) means the čech-stone compactification of a tychonoff space x. the cardinality of a set a is denoted by |a|. let ω be the first infinite cardinal, ω1 the first uncountable cardinal and c the cardinality of the set of all real numbers. as usual, a cardinal is the initial ordinal ordinals. for each ordinals α, β with α < β, we write (α, β) = {γ : α < γ < β}, (α, β] = {γ : α < γ ≤ β} and [α, β] = {γ : α ≤ γ ≤ β}. every cardinal is often viewed as a space with the usual order topology. other terms and symbols follow [2]. 2. σ-starcompact spaces and related spaces in this section, we give two examples which show the converses in the above diagram in the section 1 do not hold. example 2.1. there exists a tychonoff σ-starcompact space which is not star-lindelöf. proof. let d be a discrete space of the cardinality c. define x = (β(d) × (ω + 1)) \ ((β(d) \ d) × {ω}). then, x is σ-starcompact, since β(d) × ω is a σ-compact dense subset of x. next, we show that x is not star-lindelöf. since |d| = c, then we can enumerate d as {dα : α < c}. for each α < c, let uα = {dα} × [0, ω]. then uα ∩ uα′ = ∅ for α 6= α ′. let us consider the open cover u = {uα : α < c} ∪ {β(d) × ω}. of x. let f be a countable subset of x. then, there exists a α0 < c such that f ∩ uα0 = ∅. since uα0 is the only element of u containing the point 〈dα0 , ω〉 and uα0 ∩ f = ∅, then 〈dα0 , ω〉 /∈ st(f, u), which shows that x is not star-lindelöf. � example 2.2. there exists a tychonoff l-starcompact space which is not σ-star-compact. proof. let d = {dα : α < c} be a discrete space of the cardinality c and let y = d ∪ {∞}, where ∞ /∈ d be the one-point lindelöfication of d. then, every compact subset of y is finite by the construction of the topology of y . hence, y is not σ-compact. define x = (y × (ω + 1)) \ (〈∞, ω〉). then, x is l-starcompact, since y × ω is a lindelöf dense subset of x. on σ-starcompact spaces 295 now, we show that x is not σ-starcompact. for each α < c, let uα = {dα} × [0, ω]. then uα ∩ uα′ = ∅ for α 6= α ′. let us consider the open cover u = {uα : α < c} ∪ {y × {n} : n ∈ ω}. of x. let c be σ-compact subset of x. then, c ∩ (d × {ω}) is countable, since d × {ω} is discrete closed in x. on the other hand, for each n ∈ ω, c ∩ (y × {n}) is countable in y × {n}, since y × {n} is open and close in x. thus, c is a countable subset of x. since c is countable, then {α : c∩uα 6= ∅} is countable, hence, there exists a αω ∈ c such that c ∩ uα = ∅ for each α > αω. if we pick α′ > αω. then, 〈dα′ , ω〉 /∈ st(c, u), since uα′ is the only element of u containing 〈dα′ , ω〉 and uα′ ∩ c = ∅, which shows that x is not σstarcompact. � remark 2.3. the author does not know if there exists a normal l-starcompact which is not σ-starcompact and a normal σ-starcompact space which is not star-lindelöf. 3. properties of σ-starcompact spaces in example 2.1, the closed subset d ×{ω} of x is not σ-starcompact, which shows that a closed subset of a σ-starcompact space need not be σ-starcompact. in the following, we construct an example which shows that a regular-closed gδ-subspace of a σ-starcompact space need not be σ-starcompact. example 3.1. there exists a star-lindelöf (hence, σ-starcompact) tychonoff space having a regular-closed gδ-subspace which is not σ-starcompact. proof. let s1 = (y × (ω + 1)) \ (〈∞, ω〉). be the same space as the space x in the proof of example 2.2. as we prove above, s1 is not σ-starcompact. let s2 = ω ∪ r be the isbell-mrówka space [7], where r is a maximal almost disjoint family of infinite subsets of ω with |r| = c. then, s2 is star-lindelöf, since ω is a countable dense subset of s2. hence, it is σ-starcompact. we assume s1 ∩ s2 = ∅. let π : d × {ω} → r be a bijection and let x be the quotient image of the disjoint sum s1 ⊕ s2 by identifying 〈dα, ω〉 of s1 with π(〈dα, ω〉) of s2 for each 〈dα, ω〉 of d × {ω}. let ϕ : s1 ⊕ s2 → x be the quotient map. then, ϕ(s1) is a regular-closed gδ-subspace of x which is not σ-starcompact. we shall show that x is star-lindelöf. to this end, let u be an open cover of x. since ϕ(ω) is a countable dense subset of π(s2), then ϕ(s2) ⊆ st(ϕ(ω), u). on the other hand, since ϕ(y × ω) is lindelöf there exists a countable subset f1 of ϕ(y × ω) such that ϕ(y × ω) ⊆ st(f1, u). let f = ϕ(ω) ∪ f1. then, x = st(f, u). hence, x is star-lindelöf, which completes the proof. � 296 y.-k. song we give a positive result: theorem 3.2. an open fδ-subset of a σ-starcompact space is σ-starcompact. proof. let x be an σ-starcompact space and let y = ∪{hn : n ∈ ω} be an open fδ-subset of x, where the set hn is closed in x for each n ∈ ω. to show that y is σ-starcompact, let u be an open cover of y . we have to find a σ-compact subset c of y such that st(c, u) = y . for each n ∈ ω, consider the open cover un = u ∪ {x \ hn} of x. since x is σ-starcompact, there exists a σ-compact subset cn of x such that st(cn, un) = x. let dn = cn ∩ y . since y is a fδ-set, dn is σ-compact, and clearly hn ⊆ st(dn, u). thus, if we put c = ∪{dn : n ∈ ω}, then c is a σ-compact subset of y and st(c, u) = y . hence, y is σ-starcompact. � a cozero-set in a space x is a set of the form f −1(r \ {0}) for some realvalued continuous function f on x. since a cozero-set is an open fσ-set, we have the following corollary: corollary 3.3. a cozero-set of a σ-starcompact space is σ-starcompact. since a continuous image of a σ-compact space is σ-compact, then it is not difficult to show the following result. theorem 3.4. a continuous image of a σ-starcompact space is σ-starcompact. next, we turn to consider preimages. to show that the preimage of a σ-starcompact space under a closed 2-to-1 continuous map need not be σstarcompact we use the alexandorff duplicate a(x) of a space x. the underlying set of a(x) is x ×{0, 1}; each point of x ×{1} is isolated and a basic neighborhood of a point 〈x, 0〉 ∈ x×{0} is of the from (u×{0})∪((u×{1})\{〈x, 1〉}), where u is a neighborhood of x in x. example 3.5. there exists a closed 2-to-1 continuous map f : x → y such that y is a σ-starcompact space, but x is not σ-starcompact. proof. let y be the space x in the proof of example 2.1. then y is σstarcompact and has the infinite discrete closed subset f = d × {ω}. let x be the alexandroff duplicate a(y ) of y . then, x is not σ-starcompact, since f × {1} is an infinite discrete, open and closed set in x. let f : x → y be the natural map. then, f is a closed 2-to-1 continuous map, which completes the proof. � now, we give a positive result: theorem 3.6. let f be an open perfect map from a space x to a σ-starcompact space y . then, x is σ-starcompact proof. since f (x) is open and closed in y , we may assume that f (x) = y . let u be an open cover of x and let y ∈ y . since f −1(y) is compact, there exists a finite subcollection uy of u such that f −1(y) ⊆ ∪uy and u ∩ f −1(y) 6= ∅ for on σ-starcompact spaces 297 each u ∈ uy. pick an open neighbourhood vy of y in y such that f −1(vy ) ⊆ ∪{u : u ∈ uy}, and we can assume that (1) vy ⊆ ∩{f (u ) : u ∈ uy} because f is open. taking such open set vy for each y ∈ y , we have an open cover v = {vy : y ∈ y } of y . hence, there exists a σ-compact subset c of y such that st(c, v) = y , since y is σ-compact. since f is perfect, the set f −1(c) is a σ-compact subset of x. to show that st(f −1(c), v) = x, let x ∈ x. then, there exists y ∈ y such that f (x) ∈ vy and vy ∩ c 6= ∅. since x ∈ f −1(vy ) ⊆ ∪{u : u ∈ uy}, we can choose u ∈ uy with x ∈ u . then vy ⊆ f (u ) by (1), and hence u ∩ f −1(c) 6= ∅. therefore, x ∈ st(f −1(c), u). consequently , we have that st(f −1(c), u) = x. � by theorem 3.6, we have the following corollary 3.7. corollary 3.7. let x be a σ-starcompact space and y a compact space. then, x × y is c-starcompact. the following theorem is a generalization of corollary 3.7. theorem 3.8. let x be a σ-starcompact space and y a locally compact, lindelöf space. then, x × y is σ-starcompact. proof. let u be an open cover of x × y . for each y ∈ y , there exists an open neighbourhood vy of y in y such that cly vy is compact. by the corollary 3.7, the subspace x × cly vy is σ-starcompact. thus, there exists a σ-compact subset cy ⊆ x × cly vy such that x × cly vy ⊆ st(cy, u). since y is lindelöf, there exists a countable cover {vyi : i ∈ ω} of y . let c = ∪{cyi : i ∈ ω}. then, c is a σ-compact subset of x × y such that st(c, u) = x × y . hence, x × y is σ-starcompact. � in the following, we give an example showing that the condition of the locally compact space in theorem 3.8 is necessary. example 3.9. there exist a countably compact space x and a lindelöf space y such that x × y is not σ-starcompact. proof. let x = ω1 with the usual order topology. y = ω1+1 with the following topology. each point α with α < ω1 is isolated and a set u containing ω1 is open if and only if y \ u is countable. then, x is countably compact and y is lindelöf. now, we show that x × y is not σ-starcompact. for each α < ω1, let uα = [0, α] × [α, ω1], and vα = [α, ω1) × {α}. consider the open cover u = {uα : α < ω1} ∪ {vα : α < ω1} 298 y.-k. song of x × y and let c be a σ-compact subset of x × y . then, πx (c) is a σcompact subset of x, where πx : x × y → x is the projection. thus, there exists β < ω1 such that πx (c) ∩ (β, ω1) = ∅ by the definition of the topology of x. pick α0 with α0 > β. then, vα0 ∩c = ∅. if we pick α′ > α0, then 〈α ′, α0〉 /∈ st(c, u) since vα0 is the only element of u containing 〈α′, α0〉. hence, x × y is not σ-starcompact, which completes the proof. � the theorem 3.9 also shows the product of two σ-starcompact spaces need not be σ-starcompact. next, we give a well-known example showing that the product of two countably compact spaces need not be σ-starcompact. we give the proof roughly for the sake of completeness. example 3.10. there exist two countably compact spaces x and y such that x × y is not σ-starcompact. proof. let d be a discrete space of the cardinality c. we can define x = ∪α<ω1 eα, y = ∪α<ω1 fα, where eα and fα are the subsets of β(d) which are defined inductively so as to satisfy the following conditions (1), (2) and (3): (1) eα ∩ fβ = d if α 6= β; (2) |eα| ≤ c and |fα| ≤ c; (3) every infinite subset of eα (resp. fα) has an accumulation point in eα+1 (resp. fα+1). those sets eα and fα are well-defined since every infinite closed set in β(d) has the cardinality 2c (see [5]). then, x × y is not σ-starcompact, because the diagonal {〈d, d〉 : d ∈ d} is a discrete open and closed subset of x × y with the cardinality c and σ-starcompactness is preserved by open and closed subsets. � example 3.11. there exist a separable space x and a lindelöf space y such that x × y is not σ-starcompact. proof. let x = y be the same space y in the proof of example 2.2. then, y is lindelöf, however is not σ-starcompact. let y = ω ∪r be the isbell-mrówka space [7], where r is a maximal almost disjoint family of infinite subsets of ω with |r| = c. then, y is separable. since |r| = c, then we can enumerate r as {rα : α < c}. to show that x × y is not σ-starcompact. for each α < c, let uα = {dα}×y and vα = (x \{dα})×({rα}∪rα). for n ∈ ω, let wn = x ×{n}. we consider the open cover u = {uα : α < c} ∪ {vα : α < c} ∪ {wn : n ∈ ω} of x ×y . let c be a σ-compact subset of x ×y . then, πx (c) is a σ-compact subset of x, where πx : x × y → x is the projection. thus, there exists α < c such that c ∩ uα = ∅. hence, 〈dα, rα〉 /∈ st(c, u) since uα is the only element of u containing 〈dα, rα〉. hence, x × y is not σ-starcompact. which completes the proof. � on σ-starcompact spaces 299 theorem 3.12. every tychonoff space can be embedded in a σ-starcompact tychonoff space as a closed gδ-subspace. proof. let x be a tychonoff space. if we put z = (β(x) × (ω + 1)) \ ((β(x) \ x) × {ω}), then x × {ω} is a closed subset of z, which is homeomorphic to x. since β(d) × ω is a σ-compact dense subset of z, then z is σ-starcompact, which completes the proof. � acknowledgements. the author is most grateful to the referee for his kind help and valuable suggestions references [1] e. k. van douwen, g. m. reed, a. w. roscoe and i. j. tree, star covering properties, topology appl. 39 (1991), 71–103. [2] r. engelking, general topology, revised and completed edition, heldermann verlag., 1989. (1991), 255–271. [3] g. r. hiremath, on star with lindelöf center property, j. indian math. soc. 59 (1993), 227–242. [4] w. m. fleischman, a new extension of countable compactness, fund. math. 67 (1971), 1–9. [5] r. c. walker, the stone-čech compactification, berlin, 1974. [6] m. v. matveev, a survey on star covering properties, topology atlas., preprint no. 330, 1998. [7] s. mrówka, on completely regular spaces, fund. math. 41 (1954), 105–106. [8] y.-k. song, on l-starcompact spaces, czech. math. j. 56 (2006), 781–788. received september 2007 accepted november 2007 yan-kui song (songyankui@njnu.edu.cn) department of mathematics, nanjing normal university, nanjing 210097, p. r. china. dmmenaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 109-132 symmetric bombay topology giuseppe di maio, enrico meccariello∗ and somashekhar naimpally abstract. the subject of hyperspace topologies on closed or closed and compact subsets of a topological space x began in the early part of the last century with the discoveries of hausdorff metric and vietoris hit-and-miss topology. in course of time, several hyperspace topologies were discovered either for solving some problems in applied or pure mathematics or as natural generalizations of the existing ones. each hyperspace topology can be split into a lower and an upper part. in the upper part the original set inclusion of vietoris was generalized to proximal set inclusion. then the topologization of the wijsman topology led to the upper bombay topology which involves two proximities. in all these developments the lower topology, involving intersection of finitely many open sets, was generalized to locally finite families but intersection was left unchanged. recently the authors studied symmetric proximal topology in which proximity was used for the first time in the lower part replacing intersection with its generalization: nearness. in this paper we use two proximities also in the lower part and we obtain the lower bombay hypertopology. consequently, a new hypertopology arises in a natural way: the symmetric bombay topology which is the join of a lower and an upper bombay topology. 2000 ams classification: 54b20, 54e05, 54e15, 54e35. 1. introduction and preliminaries. given a topological space, a topological vector space or a banach space x, frequently it is necessary to study a family of closed or compact (convex) subsets of x, called a hyperset of x, in (a) optimization (b) measure theory (c) function space topologies ( each function f : x → y , as a graph, is a subset of x × y ) (d) geometric functional analysis (e) image processing (f ) convex analysis etc. so there is a need to put an appropriate topology on the hyperset ∗we regret to announce the sad demise of our friend and collaborator enrico. 110 g. di maio, e. meccariello and s. naimpally and so we construct an hyperspace. two early discoveries were the hausdorff metric topology (1914) [12] [earlier studied by pompeiu (1905)] when x is a metric space and the vietoris hit-and-miss topology (1922) [26] when x is a t1 space. since then many hyperspace topologies have been studied (see [20]). all hyperspace topologies have a lower and an upper part. a typical member of the lower hyperspace topology consists of members which hit a finite or a locally finite family of open sets. the authors showed that all upper hyperspace topologies known until last year can be expressed with the use of two proximities ([7]) and called the resulting upper hyperspace topology, the upper bombay topology. in this project the lower part was left unchanged using the hit sets. recently the authors radically changed the lower part by replacing the hit sets by near sets and thus getting symmetric hyperspaces [8]. in this paper we generalize hyperspace topologies by using two proximities in the lower part obtaining the lower bombay topology. combining the lower and the upper bombay topologies, we have the symmetric bombay topology. henceforth, (x, τ ), or x, denotes a t1 space. for any e ⊂ x, cle, inte and ec stand for the closure, interior and complement of e in x, respectively. a binary relation δ on the power set of x is a basic proximity iff (i) aδb implies bδa; (ii) aδ(b ∪ c) implies aδb or aδc; (iii) aδb implies a 6= ∅, b 6= ∅; (iv) a ∩ b 6= ∅ implies aδb. a basic proximity δ is a lo-proximity iff it satisfies (lo) aδb and bδc for every b ∈ b together imply aδc. a basic proximity δ is an r-proximity iff it satisfies (r) xδa ( where δ means the negation of δ) implies there exists e ⊂ x such that xδe and ecδa. moreover, a proximity δ which is both lo and r is called a lr-proximity. a basic proximity δ is an ef-proximity iff it satisfies (ef) aδb implies there exists e ⊂ x such that aδe and ecδb. note that each ef-proximity is a lr-proximity, but, in general, the converse does not hold. if δ is a lo-proximity, then for each a ⊂ x, we denote aδ = {x ∈ x : xδa}. then τ (δ) is the topology on x induced by the kuratowski closure operator a → aδ. the proximity δ is compatible with the topology τ iff τ = τ (δ) ( see [10], [21], [25] or [28]). a t1 space x admits a compatible lo-proximity. a space x has a compatible lr( respectively, ef-) proximity iff it is t3 ( respectively, tychonoff). moreover, δ is a compatible lr-proximity iff (lr) for each xδa, there is a closed nbhd. w of x such that w δa. symmetric bombay topology 111 if aδb, then we say a is δ-near b; if aδb we say a is δ-far from b. a ≪δ b stands for aδb c and a is said to be strongly δ-contained in b whereas a≪δb stands for its negation, i.e. aδb c. in the sequel η1, η2, η or γ1, γ2, γ denote (compatible) proximities on x. we recall that η2 is coarser than η1 (or equivalently η1 is finer than η2), written η2 ≤ η1, iff aη 2 b implies aη 1 b. the most important and well studied proximity is the wallman or fine loproximity η0 given by aη0b ⇔ cla ∩ clb 6= ∅. the wallman proximity η0 is the finest compatible lo-proximity on a t1 space x. we note that η0 is a lr-proximity iff x is regular (see [13] lemma 2). moreover, η0 is an ef-proximity iff x is normal (urysohn’s lemma). if x is a metric space with metric d, the metric proximity η is given by aηb iff d(a, b) = inf{d(a, b), a ∈ a, b ∈ b} = 0. another useful proximity is the discrete proximity η⋆ given by aη⋆b ⇔ a ∩ b 6= ∅. we note that η⋆ is the finest proximity on x, but is not a compatible one, unless (x, τ ) is discrete. we point out that in this paper η⋆ is the only proximity that might be non compatible. u , v denote open sets. cl(x) is the family of all nonempty closed subsets of x. for any set e in x we use the notation: e++γ = {f ∈ cl(x) : f ≪γ e or equivalently f γe c}. e++γ0 = e + = {f ∈ cl(x) : f ≪γ0 e or equivalently f ⊂ inte}. e ++ η∗ = {f ∈ cl(x) : f ≪η∗ e or equivalently f ⊂ e}. e−η = {f ∈ cl(x) : f ηe}. e − η⋆ = {f ∈ cl(x) : f η ⋆e or equivalently f ∩ e 6= ∅}. now, we define the lower bombay topology. let η1, η2 be two proximities on a t1 space x with η2 ≤ η1. a typical nbhd. of a ∈ cl(x) in the lower bombay topology σ(η2, η1) − consists of the sets of the form u −η2 with aη1u , i.e. {e ∈ cl(x) : eη2u, where u is open and aη1u}. we stress that in the description of the lower bombay topology σ(η2, η1) − the order in which the coordinate proximities are written (η2, η1) emphasizes the fact that the first coordinate proximity η2 is coarser than the second one η1, 112 g. di maio, e. meccariello and s. naimpally i.e. η2 ≤ η1. furthermore, the proximity η1 selects the open subsets u which delineate the nbhds of a (u fulfills the property aη1u ), whereas η2 describes the nbhd. labelled by u , namely u −η2 . if η2 = η1 = η, then we have the lower η-proximal topology (cf. [8]) denoted by σ(η−) = σ(η, η)−. as for the upper part we have two compatible lo-proximities γ1, γ2 with γ1 ≤ γ2, and ∆ ⊂ cl(x) a cobase, i.e. ∆ is closed under finite unions and contains singletons. a typical nbhd. of a ∈ cl(x) in the upper bombay topology σ(γ1, γ2; ∆) + consists of the sets of the form u ++γ2 , where u c ∈ ∆ and a ≪γ1 u , i.e. {e ∈ cl(x) : e ≪γ2 u, where u c ∈ ∆ and a ≪γ1 u}. similarly, in the description of the upper bombay topology σ(γ1, γ2; ∆) +, the order in which the coordinate proximities are written (γ1, γ2) stresses the fact that the first coordinate proximity γ1 is coarser than the second one γ2, i.e. γ1 ≤ γ2. furthermore, the proximity γ1 and the cobase ∆ ⊂ cl(x) together select the open subsets u which delineate the nbhds of a (u fulfills the property a ≪γ1 u and u c ∈ ∆), whereas γ2 describes the nbhd. indexed by u , namely u ++γ2 . if γ2 = γ1 = γ, then we have the upper γ-∆-proximal topology (cf. [8]) σ(γ+; ∆) = σ(γ, γ; ∆)+. for futher details on proximities and hyperspaces see [7], [8], [25]. 2. basic results on lower bombay topology. since we have already studied the upper bombay topology in our previous paper, we investigate here the salient properties of the lower bombay topology. first, we recall that if (x, τ ) is a t1 space, then a topology τ ′ on cl(x) is declared admissible if the map i : (x, τ ) → (cl(x), τ ′), defined by i(x) = {x}, is an embedding. we point out that all the classical hypertopologies, namely, lower vietoris topology, upper vietoris topology, wijsman topology, fell topology, hausdorff topology etc. are admissible. on the contrary, if the involved proximities η1, η2 are different from the discrete proximity η⋆, then the map i : (x, τ ) → (cl(x), σ(η2, η1) −) is, in general, not even continuous as the following example shows. example 2.1. let x = [−1, 1] with the metric proximity η, η0 the wallman proximity. let a = {0}, an = { 1 n } for all n ∈ n. then 1 n converges to 0 in x, but an does not converge to a in the topology σ(η, η0) −, because if u = (−1, 0), then 0η0u , but anηu for all n. hence the map i : (x, τ ) → (cl(x), σ(η, η0) −), where i(x) = {x}, is not an embedding. symmetric bombay topology 113 lemma 2.2. let x be a t1 space, η1, η2, lo-proximities on x with η2 ≤ η1 and η⋆ the discrete proximity. the following results hold: (a) σ(η⋆−) = τ (v −), the lower vietoris topology. (b) σ(η2, η1) − ⊂ σ(η−1 ) ∩ σ(η − 2 ). example 2.3. we give examples to show that, in general, σ(η2, η1) − is not comparable with the lower vietoris topology τ (v −). (a) let x = r with the usual metric and η the metric proximity, a = [0, 1], an = [0, 1− 1 n ], n ∈ n. then an → a in τ (v −), but does not converge in the topology σ(η, η0) −, since for u = (1, 2) aη0u , but anηu for all n. hence σ(η, η0) − 6⊂ τ (v −) ( see also example 2.1). (b) let l2 denote the space of all square summable sequences. x = b(θ, 1) ∪ {(1 + 1 k )ek : k ∈ n} ⊂ l2, endowed with the alexandroff proximity ηa (i.e. eηaf iff cle ∩ clf 6= ∅ or both cle, clf are noncompact) and the wallman proximity η0. then an = {(1 + 1 k )ek : k ≥ n} converges to a = {θ} in the topology σ(ηa, η0) −, but not in τ (v −). hence τ (v −) 6⊂ σ(ηa, η0) −. theorem 2.4. let (x, τ ) be a t1 regular space and η1, η2 lo-proximities on x. if η2 is also a compatible lr-proximity on x, then τ (v −) ⊂ σ(η2, η1) −. proof. suppose that the net (aλ) of closed sets converges to a closed set a in the topology σ(η2, η1) −. if a ∈ u −, where u ∈ τ , then there is an a ∈ a ∩ u . since η2 is a compatible lr-proximity, there is an open set v such that a ∈ v and clv η 2 u c by (lr) axiom. therefore, a ∈ v ⊂ clv ⊂ u and clv η 2 u c. we claim that eventually aλ intersects u . for if not, then frequently aλ ⊂ u c and so frequently aλη 2 v ; a contradiction. � corollary 2.5. if η is the metric proximity on a metric space (x, d), then (a) τ (v −) ⊂ σ(η, η0) − ⊂ σ(η−). (b) τ (v −) ⊂ σ(η, η0) − ⊂ σ(η−0 ). remark 2.6. example 2.3 (b) points out that the assumption η2 is a compatible lr-proximity on x cannot be dropped in theorem 2.4. example 2.3 (a) shows that the inclusion in theorem 2.4 might be strict even in nice spaces. we note that the base space x is one of the best possible spaces and the sets involved are also compact. theorem 2.7. if η is the metric proximity on a metric space (x, d), then the following are equivalent: (a) x is a uc space; (b) σ(η−) ⊂ σ(η, η0) −; (c) σ(η−) = σ(η, η0) −. 114 g. di maio, e. meccariello and s. naimpally proof. we need prove only (b) ⇒ (a). suppose x is not a uc space. then, there are two disjoint closed sets of distinct points a = {an : n ∈ n}, b = {bn : n ∈ n} such that d(an+1, bn+1) < d(an, bn) → 0. then, an = {ak : k ≤ n} converges to a in the topology σ(η, η0) −, but not in σ(η−). in fact, for each natural number n choose 0 < εn < ( 1 4 )n d(an, bn) and for n 6= m, sd(bn, εn) ∩ sd(bm, εm) = ∅ (where sd(x, r) is the open sphere centered at x with radius r). let u = ⋃ n∈n sd(bn, εn). then, u is open and aηu (in fact b ⊂ u and d(an, bn) → 0). but, anηu for each n ∈ n (in fact, it is easy to check that for each n ∈ n, 0 < εn < inf{d(ak, u) : ak ∈ an and u ∈ u}). � let (x, u) be a t2 uniform space. we recall that for u ∈ u u (x) = {y ∈ x : (x, y) ∈ u} and for any subset e of x, u (e) = ⋃ e∈e u (e). moreover, e is declared u-discrete if u (e) ∩ e = {e} for each e ∈ e. definition 2.8 (cf. [12] or [1]). let (x, u) be a t2 uniform space. (a) a typical nbhd. of a ∈ cl(x) in the lower hausdorff-bourbaki or lower h-b topology τ (h−) consists of the sets of the form {b ∈ cl(x) : a ⊂ u (b)}, where u ∈ u. (b) a typical nbhd. of a ∈ cl(x) in the upper hausdorff-bourbaki or upper h-b topology τ (h+) consists of the sets of the form {b ∈ cl(x) : b ⊂ u (a)}, where u ∈ u. note that the lower h-b uniform topology τ (h−) is a topology of locally finite type as naimpally first proved in [19]. more precisely, naimpally showed that the topology τ (h−) is generated by hit sets lu = the collection of families of the form {u (x) : x ∈ q ⊂ a}, a ∈ cl(x), u ∈ u and q is u -discrete. note also that the upper h-b uniform topology τ (h+) is a topology of upper proximal type, i.e. τ (h+) = σ(δ+) (see [5]), where δ is the ef-proximity associated to u (see [21] or [10]). now, we give examples to show that the lower hausdorff-bourbaki or h-b uniform topology τ (h−) is not comparable with σ(η2, η1) −. example 2.9. (a) again, let x = r endowed with the euclidean metric and η the metric proximity, a = [0, 1], an = [0, 1 − 1 n ], n ∈ n. then, an → a in τ (h−), but it does not converge in the topology σ(η, η0) −. in fact, if u = (1, 2), then aη0u , but anηu for all n. hence σ(η, η0) − 6⊂ τ (h−). (b) let x = n = a with the usual metric, an = {1, 2, . . . , n}. here η = η0 = η ⋆. moreover, the sequence (an) converges to a with respect the topology σ(η, η0) − = σ(η⋆−) = τ (v −), but (an) does not converge to a with respect to the lower h-b uniform topology τ (h−). therefore, τ (h−) 6⊂ σ(η, η0) −. symmetric bombay topology 115 let η1, η2 be lo-proximities on x with η2 ≤ η1, a ∈ cl(x) and e a locally finite family of open sets such that aη1u for all u ∈ e. then e − η2 is the set {b ∈ cl(x) : bη2u for all u ∈ e}. definition 2.10. let η1, η2 be lo-proximities on x with η2 ≤ η1. given a collection l of locally finite families of open sets, la denotes the subcollection of l such that if e is a family of l verifying aη1u for all u ∈ e, then e ∈ la. suppose l is a collection of locally finite families of open sets satisfying the filter condition: (♯) for each a ∈ cl(x) whenever e, f ∈ la implies there exists a g ∈ la such that g−η2 ⊂ e − η2 ∩ f−η2 . under the above condition, the topology σ(η2, η1; l) − on cl(x) is the topology which has as a basic nbhds system of a ∈ cl(x) all sets of the form {b ∈ cl(x) : bη2u for each u ∈ e}, where e ∈ la. it is obvious that if l is a collection of locally finite families of open sets satisfying the above filter condition (♯) and η⋆ is the discrete proximity, then τ (l−) = σ(η⋆−; l) = σ(η⋆, η⋆; l)−. we say that a collection l of locally finite families of open sets is stable under locally finite families if e ∈ l and f is a locally finite family of open sets such that for all v ∈ f there exists u ∈ e with v ⊆ u , then f ∈ l. theorem 2.11. let x be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1 and l a collection of locally finite families of open sets which satisfies the filter condition (♯) and is stable under locally finite families. if η2 is a compatible lr-proximity on x, then τ (l−) ⊂ σ(η2, η1; l) −. proof. suppose η2 is a lr-proximity and a ∈ e − where e ∈ l. then for each u ∈ e there exists au ∈ u ∩ a. so, for each u ∈ e there is an open set vu such that au ∈ vu ≪η2 u . let f = {vu : au ∈ vu ≪η2 u and u ∈ e}. hence, f ∈ la and w = {b ∈ cl(x) : bη2vu for each vu ∈ f} is a σ(η2, η1; l) − nbhd. of a which is contained in e−. � corollary 2.12. let (x, d) be a metric space and η the associated metric proximity. let l be the collection of families of open sets of the form {sd(x, 1 n ) : x ∈ q ⊂ a, where a ∈ cl(x), n ∈ n, q is 1 n -discrete}. then we have the following: σ(η, η0) − τ (v −) ⊂ ⊂ σ(η, η0; l) − τ (h−) = σ(η⋆; l)− furthermore, in general, σ(η, η0) − and τ (h−) are not comparable. 116 g. di maio, e. meccariello and s. naimpally theorem 2.13. let (x, d) be a locally compact metric space and η the associated metric proximity. then τ (v −) ⊂ σ(η, η0) − = σ(η−0 ) ⊂ σ(η −). proof. we only prove σ(η, η0) − = σ(η−0 ). it suffices to show σ(η − 0 ) ⊂ σ(η, η0) −, since by corollary 2.5 (b) σ(η, η0) − ⊂ σ(η−0 ). so, suppose that the net (aλ) of closed sets converges to a closed set a in the topology σ(η, η0) −. let v be an open set and let a ∈ v −η0 . then, there is an a ∈ a ∩ clv . let u be a compact nbhd. of a and set w = u ∩v . note that clw is compact and that a closed set is η-near a compact set iff it is η0-near. as a result, eventually aλ ∈ w − η ⊂ v − η0 and the claim holds. � remark 2.14. more generally, the equality σ(η, η0) − = σ(η−0 ) in the above theorem 2.13 holds if one of the following conditions is satisfied: (i) the base space x is compact; (ii) the involved proximities η, η0 are lr, the net of closed sets (aλ) is eventually locally finite and converges to a in the topology σ(η, η0) −. now, we compare two different lower bombay topologies σ(γ2, γ1) −, σ(η2, η1) −. if η is a lo-proximity on x, then for a ⊂ x, η(a) = {e ⊂ x : eηa}. theorem 2.15 (main theorem). let (x, τ ) be a t1 space with lo-proximities γ1, γ2, η1, η2 with γ2 ≤ γ1 and η2 ≤ η1. if γ2 and η2 are compatible, then the following are equivalent: (a) σ(γ2, γ1) − ⊂ σ(η2, η1) −; (b) whenever a ∈ cl(x) and u ∈ τ with aγ1u , there exists a v ∈ τ such that: (i) aη1v , and (ii) η2(v ) ⊂ γ2(u ). proof. (a) ⇒ (b). let a ∈ cl(x) and u ∈ τ with aγ1u . then u − γ2 is a σ(γ2, γ1) −-nbhd. of a. by assumption there exists a σ(η2, η1) −-nbhd. l of a such that a ∈ l ⊂ u −γ2 . l has the form: n⋂ k=1 {(vk) − η2 with vk ∈ τ and aη1vk for each k ∈ {1, . . . , n}}. we claim that there exists k0 ∈ {1, . . . , n} such that η2(vk0 ) ⊂ γ2(u ). assume not. then, for each k ∈ {1, . . . , n} there is a closed set fk with fk ∈ η2(vk) \ γ2(u ). set f = ⋃n k=1 fk. then f ∈ l, but f 6∈ u − γ2 ; a contradiction. (b) ⇒ (a). let a ∈ cl(x) and u −γ2 (u ∈ τ and aγ1u ) be a subbasic σ(γ2, γ1) −-nbhd. of a. by assumption, there is an open subset v with aη1v and η2(v ) ⊂ γ2(u ). it follows that v − η2 is a σ(η2, η1) −-nbhd. of a with v −η2 ⊂ u −γ2 . � corollary 2.16. let (x, τ ) be a t1 space with a compatible lo-proximity η. if η⋆ is the discrete proximity, then the following are equivalent: (a) τ (v −) = σ(η⋆−) ⊂ σ(η, η⋆)−; (b) whenever a ∈ cl(x) and u ∈ τ with a ∩ u 6= ∅, there exists a v ∈ τ such that a ∩ v 6= ∅ and η(v ) ⊂ η⋆(u ). symmetric bombay topology 117 3. first and second countability of lower and upper bombay topologies. lemma 3.1. let (x, τ ) be a t1 space with a compatible lo-proximity η and w , v open subsets of x. the following are equivalent: (a) clw ⊂ clv ; (b) η(w ) ⊂ η(v ). proof. (a) ⇒ (b). let f ∈ η(w ). then f ηw . since clw ⊂ clv , we have f ηv and hence f ∈ η(v ). (b) ⇒ (a). assume not. then, there exists an x ∈ clw \ clv . set f = {x}. we have f ∈ η(w ), but f 6∈ η(v ); a contradiction. � definition 3.2 (see [8]). let (x, τ ) be a t1 space, η a compatible lo-proximity on x and a ∈ cl(x). a family na of open subsets of x is an external proximal local base of a with respect to η (or, briefly a η-external proximal local base of a) if for any u open subset of x with aηu , there exists v ∈ na satisfying aηv and clv ⊂ clu . the external proximal character of a with respect to η ( or, briefly the ηexternal proximal character of a) is defined as the smallest (infinite) cardinal number of the form |na|, where na is a η-external proximal local base of a, and it is denoted by eχ(a, η). the external proximal character of cl(x) with respect to η (or, briefly the η-external proximal character ) is defined as the supremum of all cardinal numbers eχ(a, η), where a ∈ cl(x) and is denoted by eχ(cl(x), η). note that if η = η⋆, then a family na of open subsets of x is an external local base of a if for any u , an open subset of x with aη⋆u ( i.e. a ∩ u 6= ∅), there exists a v ∈ na satisying aη ⋆v (i.e. a ∩ v 6= ∅) and v ⊂ u (cf. [2]). in a similar way, we can define the external character eχ(a) of a and the external character eχ(cl(x)) of cl(x) (cf. [2]). remark 3.3. obviously, if the external character eχ(cl(x)) is countable, then the base space x is first countable and each closed subset of x is separable. on the other hand, if we consider the proximal case, x might not be separable even if the η-external character eχ(cl(x), η) is countable. however, we have the following: if x is a t3 space, η is a compatible lr-proximity and the η-external proximal character eχ(cl(x), η) is countable, then x is separable. we now consider the first countability of the lower bombay topology σ(η2, η1) −. if η2 = η1 = η ⋆, then σ(η2, η1) − is the lower vietoris topology τ (v −). its first countability has been studied since 1971 ( [2], see also [6] and [15]) and holds if and only if x is first countable and each closed subset of x is separable. hence, we investigate the case σ(η2, η1) − 6= τ (v −), i.e. η2 6= η ⋆. first, we study the case η2 ≤ η1 6= η ⋆. 118 g. di maio, e. meccariello and s. naimpally theorem 3.4. let (x, τ ) be a t1 space with compatible lo-proximities η1, η2, η2 ≤ η1 and η1 6= η ⋆. the following are equivalent: (a) (cl(x), σ(η2, η1) −) is first countable; (b) the η1-external proximal character eχ(cl(x), η1) is countable; (c) (cl(x), σ(η−1 )) is first countable. proof. (a) ⇒ (b). it suffices to show that for each a ∈ cl(x) there exists a countable family na of open subsets of x which is a η1-external proximal local base of a. so, let a ∈ cl(x) and z be a countable σ(η2, η1) −-nbhd. system of a. then, z = {ln : n ∈ n}, where each ln has the form ⋂ k∈in {(vk) − η2 with vk ∈ τ, aη1vk for every k ∈ in and in finite subset of n}. set na = {vk : k ∈ in, n ∈ n}. na is a countable family and if vk ∈ na, then aη1vk (by construction). we claim that for each open set u with aη1u , there is a vk ∈ na with clvk ⊂ clu . so, let u be open with aη1u and consider u − η2 (which is a σ(η2, η1) −-nbhd. of a). by assumption, there is some ln ∈ z with ln ⊂ u − η2 . since ln has the form ⋂ k∈in {(vk) − η2 with vk ∈ τ , aη1vk for every k ∈ in and in finite subset of n}, we have that ⋂ k∈in (vk) − η2 ⊂ u −η2 . we claim that there exists some k0 ∈ in such that clvk0 ⊂ clu . assume not and for each k ∈ in let xk ∈ clvk \ clu . set e = ⋃ k∈in {xk}. then e ∈ cl(x) as well as e ∈ ln, but e 6∈ u − η2 ; a contradiction. (b) ⇒ (a). let a ∈ cl(x) and na = {vn : n ∈ n} be a countable η1external proximal local base of a. obviously, z = {(vn) − η2 : vn ∈ na} is a countable σ(η2, η1) −-subbasic nbhds system of a. (b) ⇔ (c). theorem 2.11 in [8]. � the case η2 ≤ η1 = η ⋆ can be handled similarly. so, we have theorem 3.5. let (x, τ ) be a t1 space with a compatible lo-proximity η, and the discrete proximity η⋆ on x. the following are equivalent: (a) (cl(x), σ(η, η⋆)−) is first countable; (b) the external character eχ(cl(x)) is countable; (c) (cl(x), τ (v −)) is first countable. remark 3.6. from the above discussion the following unexpected, but natural result holds. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1. the following are equivalent: (a) (cl(x), σ(η2, η1) −) is first countable; (b) (cl(x), σ(η−1 )) is first countable. now, we study the first countability of the upper bombay ∆ topology σ(γ1, γ2; ∆) +. first, we need the following definition. symmetric bombay topology 119 definition 3.7. let (x, τ ) be a t1 space with a compatible lo-proximity γ, a ∈ cl(x) and ∆ ⊂ cl(x) a cobase. a family la of open nbhds. of a is a local proximal ∆ base with respect to γ ( or, briefly a γ-local proximal ∆ base of a) if for any open subset u of x with u c ∈ ∆ and a ≪γ u , there exists v ∈ la with v c ∈ ∆ and a ≪γ v ⊂ u . the γ-proximal ∆ character of a is defined as the smallest ( infinite) cardinal number of the form |la|, where la is a γ-local proximal ∆ base of a, and it is denoted by χ(a, γ, ∆). the γ-proximal ∆ character of cl(x) is defined as the supremum of all cardinal numbers χ(a, γ, ∆), where a ∈ cl(x), and is denoted by χ(cl(x), γ, ∆). theorem 3.8. let (x, τ ) be a t1 space with compatible lo-proximities γ1, γ2, γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. the following are equivalent: (a) (cl(x), σ(γ1, γ2; ∆) +) is first countable; (b) the γ1-proximal ∆ character χ(cl(x), γ1, ∆) of cl(x) is countable; (c) (cl(x), σ(γ+1 ; ∆)) is first countable. proof. (a) ⇒ (b). let a ∈ cl(x) and ua be a countable σ(γ1, γ2; ∆) +-nbhd. system of a. then, ua = {on : n ∈ n}, where on = v ++ γ2 with v c ∈ ∆ and a ≪γ1 v . let la = {v : a ≪γ1 v and v ++ γ2 = on for some on ∈ ua}. by construction la is countable. we claim that la is a γ1-proximal local ∆ base of a. so, let u be an open subset of x with a ≪γ1 u and u c ∈ ∆. then, u ++γ2 is a σ(γ1, γ2; ∆) +-nbhd. of a. by assumption, there is on ∈ ua with a ∈ on ⊂ u ++ γ2 . note that on = v ++ γ2 , where v c ∈ ∆ and a ≪γ1 v . we claim v ⊂ u . assume not. let x ∈ v \ u and set f = {x}. then, f ∈ on, but f 6∈ u ++γ2 ; a contradiction. (b) ⇒ (a). let a ∈ cl(x) and la be a countable γ1-proximal local ∆ base of a. set ua = {v ++ γ2 : v ∈ la}. by construction ua is a countable family of open σ(γ1, γ2; ∆) +-nbhd. of a. we claim that ua is a σ(γ1, γ2; ∆) +-nbhd. system of a. so, let b be a σ(γ1, γ2; ∆) +-nbhd. of a. then, b has the form {u ++γ2 : u c ∈ ∆ and a ≪γ1 u}. let v be an open subset with v ∈ la and a ≪γ1 v ⊂ u and consider v ++ γ2 . then v ++γ2 ∈ ua and since v ⊂ u we have v ++γ2 ⊂ u ++ γ2 . (b) ⇔ (c). it is straightforward. � remark 3.9. let (x, τ ) be a t1 space with compatible lo-proximities γ1, γ2, γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. the following are equivalent: (a) (cl(x), σ(γ1, γ2; ∆) +) is first countable; (b) (cl(x), σ(γ+1 ; ∆)) is first countable. definition 3.10 (cf. [8]). let (x, τ ) be a t1 space and η a compatible loproximity on x. a family n of open subsets of x is an external proximal base with respect to η (or, briefly a η-external proximal base) if for any a ∈ cl(x) and any u ∈ τ with aηu , there exists v ∈ n satisfying aηv and clv ⊂ clu . 120 g. di maio, e. meccariello and s. naimpally the external proximal weight of cl(x) with respect to η (or, briefly the η-external proximal weight of cl(x)) is the smallest (infinite) cardinality of its η-external proximal bases and it is denoted by ew (cl(x), η). note that if η = η⋆, then a family n of open subsets of x is an external base if for any a ∈ cl(x) and any u ∈ τ with aη⋆u (i.e. a ∩ u 6= ∅), there exists a v ∈ n satisfying aη⋆v (i.e. a ∩ v 6= ∅) and v ⊂ u ( see [2]). the external character eχ(a) of a and the external weight ew (cl(x)) of cl(x) can be defined similarly (see [2]). now, we study the second countability of the lower bombay topology σ(η2, η1) −. if η2 = η1 = η ⋆, then σ(η2, η1) − is the lower vietoris topology τ (v −). its second countability has been studied by [6] and [15] and holds if and only if x is second countable. hence we investigate the case σ(η2, η1) − 6= τ (v −), i.e. η2 6= η ⋆. first, the case η2 ≤ η1 6= η ⋆. theorem 3.11. let (x, τ ) be a t1 space with compatible lo-proximities η1, η2, η2 ≤ η1 and η1 6= η ⋆. the following are equivalent: (a) (cl(x), σ(η2, η1) −) is second countable; (b) the η1-external proximal weight ew (cl(x), η1) of cl(x) is countable; (c) (cl(x), σ(η−1 )) is second countable. now, we discuss the case η2 ≤ η1 = η ⋆. theorem 3.12. let (x, τ ) be a t1 space with a compatible lo-proximity η and η⋆ the discrete proximity on x. the following are equivalent: (a) (cl(x), σ(η, η⋆)−) is second countable; (b) the external proximal weight ew (cl(x)) of cl(x) is countable; (c) (cl(x), τ (v −)) is second countable. remark 3.13. let (x, τ ) be a t1 space and η1, η2 lo-proximities on x with η2 ≤ η1. the following are equivalent: (a) (cl(x), σ(η2, η1) −) is second countable; (b) (cl(x), σ(η−1 )) is second countable. definition 3.14. let (x, τ ) be a t1 space with a compatible lo-proximity γ and ∆ ⊂ cl(x) a cobase. a family b of open subsets of x is a γ-proximal base with respect to ∆ if whenever a ≪γ u with u c ∈ ∆, there exists v ∈ b with v c ∈ ∆ and a ≪γ v ⊂ u . the γ-proximal weight of cl(x) with respect to ∆ (or, briefly the γ-proximal weight with respect to ∆) is the smallest (infinite) cardinality of its γ-proximal bases with respect to ∆ and it is denoted by w (cl(x), γ, ∆). symmetric bombay topology 121 theorem 3.15. let (x, τ ) be a t1 space with compatible lo-proximities γ1, γ2, γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. the following are equivalent: (a) (cl(x), σ(γ1, γ2; ∆) +) is second countable; (b) the γ1-proximal weight w (cl(x), γ1, ∆) of cl(x) with respect to ∆ is countable; (c) (cl(x), σ(γ+1 ; ∆)) is second countable. remark 3.16. let (x, τ ) be a t1 space with compatible lo-proximities γ1, γ2, γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. the following are equivalent: (a) (cl(x), σ(γ1, γ2; ∆) +) is second countable; (b) (cl(x), σ(γ+1 ; ∆)) is second countable. 4. symmetric bombay topology and some of its properties. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. the lower bombay topology σ(η2, η1) − combined with the upper one σ(γ1, γ2; ∆) + yields a new hypertopology, namely the ∆-symmetric bombay topology with respect to η2, η1, γ1, γ2 denoted by π(η2, η1, γ1, γ2; ∆) = σ(η2, η1) − ∨ σ(γ1, γ2; ∆) +. if η2 = η1 = η ⋆, then we have the standard γ1-γ2-∆-bombay topology σ(γ1, γ2; ∆) = π(η ⋆, η⋆, γ1, γ2; ∆) = τ (v −) ∨ σ(γ1, γ2; ∆) +, investigated in [7]. if η2 = η1 = η and γ1 = γ2 = γ, then we have the η-γ-∆-symmetric proximal topology π(η, γ; ∆) = σ(η, η)− ∨σ(γ, γ; ∆)+ = σ(η−)∨σ(γ+; ∆), studied in [8]. we now consider some basic properties of π(η2, η1, γ1, γ2; ∆). in general, the space x is not embedded in (cl(x), π(η2, η1, γ1, γ2; ∆)) (cf. example 2.1) and so π(η2, η1, γ1, γ2; ∆) is not an admissible topology. lemma 4.1. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2, ∆ ⊂ cl(x) a cobase and a ∈ cl(x). if γ1 ≤ η1, then a base for the nbhd. system of a with respect to the π(η2, η1, γ1, γ2; ∆) topology consists of all sets of the form: v ++γ2 ∩ ⋂ j∈j (sj ) − η2 , with a ≪γ1 v , v c ∈ ∆, aη1sj , sj ∈ τ for each j ∈ j, j finite and ⋃ j∈j sj ⊂ v . proof. let a ∈ v ++γ2 ∩ ⋂ j∈j (sj ) − η2 , where a ≪γ1 v , v c ∈ ∆, aη1sj , sj ∈ τ for each j ∈ j and j finite. we may replace each sj with sj ∩ v . in fact, from γ1 ≤ η1 we have aη 1 v c and thus aη1sj iff aη1(sj ∩ v ). � remark 4.2. the condition γ1 ≤ η1 in the above lemma 4.1 is indeed a natural one. in fact, in the presentation v ++γ2 ∩ ⋂ j∈j (sj ) − η2 we may assume that⋃ j∈j sj ⊂ v as in the classic vietoris topology. when γ1 ≤ η1, they associated symmetric bombay topology π(η2, η1, γ1, γ2; ∆) is called standard, otherwise abstract. we will see that the most significant result hold for standard symmetric bombay topologies. often, we will omit the term standard. 122 g. di maio, e. meccariello and s. naimpally we point out that all the symmetric bombay topologies investigated in this section are standard. remark 4.3. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2, ∆ ⊂ cl(x) a cobase, γ1 ≤ η1, i.e. π(η2, η1, γ1, γ2; ∆) is standard. if d is a dense subset of x, then the family of all finite subsets of d is dense in (cl(x), π(η2, η1, γ1, γ2; ∆)). theorem 4.4. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2, ∆ ⊂ cl(x) a cobase and γ1 ≤ η1. the following are equivalent: (a) (cl(x), π(η2, η1, γ1, γ2; ∆)) is first countable; (b) (cl(x), σ(η2, η1) −) and (cl(x), σ(γ1, γ2; ∆) +) are both first countable. proof. (b) ⇒ (a) is clear, since π(η2, η1, γ1, γ2; ∆) = σ(η2, η1) − ∨ σ(γ1, γ2; ∆) +. (a) ⇒ (b). let a ∈ cl(x). assume z = {l = v ++γ2 ∩ ⋂ j∈j (sj ) − η2 , with a ≪γ1 v , v c ∈ ∆, sj open, aη1sj and j finite } is a countable local base of a with respect to the topology π(η2, η1, γ1, γ2; ∆). we claim that the family z+ = {v ++γ2 : v ++ γ2 occurs in some l ∈ z} ⋃ {cl(x)} forms a local base of a with respect to the topology σ(γ1, γ2; ∆) +. indeed, if there is no open subset u with a ≪γ1 u , u c ∈ ∆, then cl(x) is the only open set in σ(γ1, γ2; ∆) + containing a. if there is u with a ≪γ1 u , u c ∈ ∆, then u ++γ2 is a π(η2, η1, γ1, γ2; ∆)-nbhd. of a. therefore, there exists l ∈ z with a ∈ l ⊂ u ++γ2 . note that l cannot be of the form ⋂ j∈j (sj ) − η2 , otherwise by setting f = a ∪ u c, we have f ∈ l, but f 6∈ u ++γ2 ; a contradiction. thus, l has the form v ++γ2 ∩ ⋂ j∈j (sj ) − η2 . we claim that v ++γ2 ⊂ u ++ γ2 . assume not and let e ∈ {v ++γ2 , with a ≪γ1 v , v c ∈ ∆} \ u ++γ2 . set f = e ∪ a, we have f ∈ l \ u ++γ2 ; a contradiction. now, we show that there is a countable local base of a with respect to the topology σ(η2, η1) −. without any loss of generality, we may assume that in the expression of every element from z the family of index set j is non-empty. in fact {v ++γ2 : a ≪γ1 v, v c ∈ ∆} = {v ++γ2 : a ≪γ1 v, v c ∈ ∆} ∩ v −η2 . moreover, by lemma 4.1, if l ∈ z then l = v ++γ2 ∩ ⋂ j∈j (sj ) − η2 , where a ≪γ1 v , v c ∈ ∆, aη1sj , sj ∈ τ for each j ∈ j, ⋃ j∈j sj ⊂ v and j finite. set z− = {(sj) − η2 : (sj ) − η2 occurs in some l ∈ z}. we claim that the family z− is a local subbase of a with respect to the topology σ(η2, η1) −. take s open with aη1s. then, s − η2 is a π(η2, η1, γ1, γ2; ∆)-nbhd. of a. hence, there exists l ∈ z with a ∈ l = v ++γ2 ∩ ⋂ j∈j (sj ) − η2 ⊂ s−η2 . we claim that there exists a j0 ∈ j such that (sj0 ) − η2 ⊂ s−η2 . it suffices to show that there exists a j0 ∈ j such that sj0 ⊂ s η2 (where sη2 = {x ∈ x : xη2s}). assume not and for each j ∈ j let xj ∈ sj \ s η2 . the set f = ⋃ j∈j {xj} ∈ l \ s − η2 ; a contradiction. � symmetric bombay topology 123 by theorems 3.4, 3.8 and 4.3 we have corollary 4.5. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2, ∆ ⊂ cl(x) a cobase and γ1 ≤ η1. the following are equivalent: (a) (cl(x), π(η2, η1, γ1, γ2; ∆)) is first countable; (b) the η1-external proximal character eχ(cl(x), η1) of cl(x) and the γ1-proximal ∆ character χ(cl(x), γ1, ∆) of cl(x) are both countable; (c) the η1-γ1-∆-symmetric proximal topology π(η1, γ1; ∆) on cl(x) is first countable. proof. note that (b) ⇔ (c) follows from theorem 4.9 in [8]. � theorem 4.6. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2, ∆ ⊂ cl(x) a cobase and γ1 ≤ η1. the following are equivalent: (a) (cl(x), π(η2, η1, γ1, γ2; ∆)) is second countable; (b) (cl(x), σ(η2, η1) −) and (cl(x), σ(γ1, γ2; ∆) +) are both second countable. the next corollary follows from theorems 3.11, 3.15 and corollary 4.5. corollary 4.7. let (x, τ ) be a t1 space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2, ∆ ⊂ cl(x) a cobase and γ1 ≤ η1. the following are equivalent: (a) (cl(x), π(η2, η1, γ1, γ2; ∆)) is second countable; (b) the η1-external proximal weight ew (cl(x), η1) of cl(x) and the γ1proximal weight w (cl(x), γ1, ∆) of cl(x) with respect to ∆ are both countable; (c) the η1-γ1-∆-symmetric proximal topology π(η1, γ1; ∆) on cl(x) is second countable. proof. note that (b) ⇔ (c) follows from theorem 4.12 in [8]. � theorem 4.8. let (x, τ ) be a tychonoff space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2, ∆ ⊂ cl(x) a cobase, γ1 ≤ η1. if η1 is a compatible lr-proximity, γ1 a compatible efproximity, then the following are equivalent: (a) (cl(x), π(η2, η1, γ1, γ2; ∆)) is metrizable; (b) (cl(x), π(η2, η1, γ1, γ2; ∆)) is second countable and uniformizable; (c) (cl(x), π(η1, γ1; ∆)) is metrizable. proof. (b) ⇒ (a). it follows from the urysohn’s metrization theorem. (a) ⇒ (b). since (cl(x), π(η2, η1, γ1, γ2; ∆)) is first countable, x is separable ( use theorem 4.4 and remark 3.3). thus, (cl(x), π(η2, η1, γ1, γ2; ∆)) is second countable ( see remark 4.3). (b) ⇔ (c). use corollary 4.7. � 124 g. di maio, e. meccariello and s. naimpally now, we compare two symmetric standard bombay topologies. theorem 4.9. let (x, τ ) be a t1 space; η1, η2, α1, α2 lo-proximities on x with η2 ≤ η1, α2 ≤ α1; γ1, γ2, δ1, δ2 compatible lo-proximities on x with γ1 ≤ γ2, δ1 ≤ δ2 and ∆ and λ ⊂ cl(x) cobases. if η2, α2 are compatible and γ1 ≤ η1 as well as δ1 ≤ α1, then the following are equivalent: (a) π(η2, η1, γ1, γ2; ∆) ⊂ π(α2, α1, δ1, δ2; λ); (b) (1) for each f ∈ cl(x) and u ∈ τ with f η1u there are w ∈ τ and l ∈ λ such that (1i) f ∈ [α1(w ) \ δ1(l)], and (1ii) [α2(w ) \ δ2(l)] ⊂ η2(u ); (2) for each b ∈ ∆ and w ∈ τ , w 6= x with b ≪γ1 w there exists m ∈ λ such that (2i) m ≪δ1 w , and (2ii) γ2(b) ⊂ δ2(m ). proof. (a) ⇒ (b). we start by showing (1). so, let f ∈ cl(x) and u ∈ τ with f η1u . then u − η2 is a π(η2, η1, γ1, γ2; ∆)-nbhd. of f . by assumption there is a π(α2, α1, δ1, δ2; λ)-nbhd. w of f such that w ⊂ u − η2 . w = w ++ δ2 ∩ ⋂n i=1 (wi) − α2 , f α1wi, wi ∈ τ , i ∈ {1, . . . , n}, ⋃n i=1 wi ⊂ w , w c ∈ λ and f ≪δ1 w . set l = w c. by construction f δ1l as well as f α1wi, i.e. f ∈ [α1(wi) \ δ1(l)], for i ∈ {1, . . . , n}. we claim that there exists i0 ∈ {1, . . . , n} such that [α2(wi0 ) \ δ2(l)] ⊂ η2(u ). assume not. then, for each i ∈ {1, . . . , n} there exists ti ∈ cl(x) with ti ∈ [α2(wi) \ δ2(l)] and ti 6∈ η2(u ), i.e. tiα2wi, ti ≪δ2 w = l c and tiη 2 u . set t = ⋃n i=1 ti. t ∈ cl(x), t ∈ w = w ++ δ2 ∩ ⋂n i=1 (wi) − α2 and t 6∈ u −η2 which contradicts w ⊂ u −η2 . now, we show (2). so, let b ∈ ∆ and w ∈ τ , w 6= x with b ≪γ1 w . set a = w c. then, a ∈ cl(x) and a ∈ (bc)++γ2 ∈ π(η2, η1, γ1, γ2; ∆). thus, there exists a π(α2, α1, δ1, δ2; λ)-nbhd. o of a such that o ⊂ (b c)++γ2 . o = o++ δ2 ∩ ⋂n i=1(oi) − α2 , aα1oi, oi ∈ τ for each i ∈ {1, . . . , n}, ⋃n i=1 oi ⊂ o, oc ∈ λ and a ≪δ1 o. set m = o c. by construction aδ1m . therefore, m ≪δ1 w = a c. we claim that γ2(b) ⊂ δ2(m ). assume not and let e ∈ [γ2(b) \ δ2(m )] with e ∈ cl(x). set f = a ∪ e. then f ∈ cl(x), f ∈ o = o++ δ2 ∩ ⋂n i=1 (oi) − α2 and f 6∈ (bc)++γ2 ; a contradiction. (b) ⇒ (a). let f ∈ cl(x), u = u ++γ2 ∩ ⋂n i=1 (ui) − η2 be a π(η2, η1, γ1, γ2; ∆)nbhd. of f . then, f η1ui, ui ∈ τ , i ∈ {1, . . . , n}, ⋃n i=1 ui ⊂ u , f ≪γ1 u and b = u c ∈ ∆. by (1) for each i ∈ {1, . . . , n} there exist wi ∈ τ and li ∈ λ such that f ∈ [α1(wi) \ δ1(li)] and [α2(wi) \ δ2(li)] ⊂ η2(ui). by (2), there exists m ∈ λ such that m ≪δ1 f c ( i.e. m δ1f ) and γ2(b) ⊂ δ2(m ). set n = ⋃n i=1 li ∪ m ∈ λ, o = n c and for each i ∈ {1, . . . , n} oi = wi \ n . note that f δ1n and by construction oi ⊂ o for each i ∈ {1, . . . , n}. thus⋃n i=1 oi ⊂ o. we claim that f α1oi for each i ∈ {1, . . . , n}. assume not. since, f α1n and f α1wi0 ∩ n c, then f α1n ∪ (wi0 ∩ n c), and hence f ≪α1 n c ∩ w ci0 ⊆ w c i0 and hence f α1wi0 . but f α1wi0 , a contradiction. it follows that o = o++ δ2 ∩ ⋂n i=1 (oi) − α2 is a π(α2, α1, δ1, δ2; λ)-nbhd. of f . we claim that o ⊂ u. assume not. then there exists e ∈ o, but e 6∈ u. hence either (♦⋆) eη2ui for some i or (♦ ⋆♦⋆) eγ2u c. symmetric bombay topology 125 if (♦⋆) occurs, then since eα2oi, oi ⊂ wi, e ≪δ2 o = n c and li ⊂ n we have e ∈ [α2(wi) \ δ2(li)] \ η2(ui), and hence e ∈ [α2(wi) \ δ2(li)] 6⊂ η2(ui) which contradicts (1ii). if (♦⋆♦⋆) occurs, then since eγ2b = u c, e ≪δ2 o = n c and m ⊂ n we have e ∈ γ2(b) \ δ2(m ), i.e. γ2(b) 6⊂ δ2(m ); which contradicts (2ii). � 5. uniformizable symmetric abstract bombay topologies. this section is devoted to find conditions which guarantee the uniformizability of a ∆-symmetric abstract bombay topology π(η2, η1, γ1, γ2; ∆). first we need the following definitions. definition 5.1 (cf. [8]). let (x, τ ) be a t1 space, δ a compatible lo-proximity on x and ∆ ⊂ cl(x). (a) ∆ is δ-urysohn iff whenever d ∈ ∆ and a ∈ cl(x) are δ-far, there exists an e ∈ ∆ with d ≪δ e ≪δ a c (see also [9], [3]). (b) ∆ is urysohn iff (a) above is true w.r.t. the lo-proximity δ0, i.e. whenever d ∈ ∆ and a ∈ cl(x) are disjoint, there exists e ∈ ∆ with d ⊂ inte ⊂ e ⊂ ac. lemma 5.2 (cf. theorem 1.6 in [9]). let (x, τ ) be a tychonoff space, γ a compatible lo-proximity on x and ∆ ⊂ cl(x) a cobase. if ∆ is γ-urysohn, then the relation δ defined on the power set of x by (∗) aδb iff claγclb and either cla ∈ ∆ or clb ∈ ∆ is a compatible ef-proximity on x. moreover, δ ≤ γ and ∆ is γ-urysohn iff ∆ is δ-urysohn. we recall that if (x, τ ) is a tychonoff space with a compatible ef-proximity δ, then a uniformity u on x is compatible w.r.t. δ if the proximity relation δ(u) defined by aδ(u)b iff a ∩ u (b) 6= ∅ for each u ∈ u equals δ (see [21] or [10]). we point out that δ admits a unique compatible totally bounded uniformity uw(δ) ([21], [10]). we will omit reference to δ if this is clear from the context. let u be a compatible uniformity on x and ∆ a cobase. for each d ∈ ∆ and u ∈ u set [d, u ] = {(a1, a2) ∈ cl(x)×cl(x) : a1∩d ⊂ u (a2) and a2 ∩d ⊂ u (a1)}. the family {[d, u ] : d ∈ ∆ and u ∈ u} is a base for a filter u∆ on cl(x) called the ∆-attouch-wets filter . u∆ induces the topology τ (u∆) called the ∆-attouch-wets topology (cf. [1] and [9]). we recall that a cobase ∆ is a cover on x iff it is closed hereditary (cf. [9]). the following theorem is given in [9]. 126 g. di maio, e. meccariello and s. naimpally theorem 5.3 (cf. theorem 2.1 in [9]). let (x, τ ) be a tychonoff space with a compatible ef-proximity δ, uw the unique totally bounded uniformity which induces δ and ∆ ⊆ cl(x) a cover of x. then the following are equivalent: (a) ∆ is δ-urysohn; (b) 1) the ∆-attouch-wets filter uw∆ is a hausdorff uniformity, and 2) the proximal ∆-topology σ(δ, ∆) equals τ (uw∆). lemma 5.4 (cf. theorem 2.2 in [9]). let (x, τ ) be a tychonoff space, γ1, γ2, compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cover of x. if ∆ is γ1-urysohn, δ the compatible ef-proximity on x defined by (∗) aδb iff claγ 1 clb and either cla ∈ ∆ or clb ∈ ∆ and uw the unique totally bounded uniformity on x compatible w.r.t. δ, then the bombay topology σ(γ1, γ2; ∆), the proximal ∆-topology σ(δ; ∆) and the topology τ (uw∆) induced by the ∆-attouch-wets uniformity uw∆ all coincide. thus the bombay topology σ(γ1, γ2; ∆) is tychonoff. proof. we omit the proof that is similar to that of theorem 2.2 in [9]. � by theorem 2.4 and lemma 2.2 (b) we know that if η2 is a compatible lr-proximity on x, then τ (v −) ⊂ σ(η2, η1) − ⊂ σ(η−2 ) ∩ σ(η − 1 ). so, in order to get σ(η2, η1) − we have to augment a typical entourage [d, u ] ∈ uw∆ by adding sets of the type p{vk} = {(a, b) ∈ cl(x) × cl(x) : aη1vk and bη1vk} and q{vk} = {(a, b) ∈ cl(x) × cl(x) : aη2vk and bη2vk} for a finite family of open sets {vk}. then, we have theorem 5.5. let (x, τ ) be a tychonoff space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cover of x. if η2 is a compatible lr-proximity, ∆ γ1-urysohn, δ the compatible ef-proximity defined by (∗) aδb iff claγ 1 clb and either cla ∈ ∆ or clb ∈ ∆ and uw the unique totally bounded uniformity on x compatible w.r.t. δ, then the family s = uw∆ ∪ {[d, u ] ∩ p{vk} : [d, u ] ∈ uw∆, {vk} finite family of open sets } ∪{[d, u ] ∩ q{vk} : [d, u ] ∈ uw∆, {vk} finite family of open sets} defines a compatible uniformity on (cl(x), π(η2, η1, γ1, γ2; ∆)). proof. it is easy to show that the above family s is a base for a uniformity on cl(x). nevertherless, it is indeed tricky to prove the compatibility, i.e. that τ (s) equals π(η2, η1, γ1, γ2; ∆) on cl(x). so, let (aλ) be a net converging to a w.r.t. τ (s). if a ∈ v −η1 , where v ∈ τ , then v − η1 ∈ τ (s) and eventually aλ ∈ v − η1 . but v −η1 ⊂ v − η2 ( because η2 ≤ η1). thus, eventually aλ ∈ v − η2 . if a ∈ v ++γ1 , where v c ∈ ∆, then v c ≪δ a c, where d = v c and δ is the symmetric bombay topology 127 compatible ef-proximity defined in (∗). by lemma 5.2 there is s ∈ ∆ such that d ≪δ s ≪δ a c. hence, there is w ∈ uw such that w (a) ∩ s = ∅. eventually aλ ∈ [s, w ](a) ⊂ v ++ γ1 . so, eventually aλ ∈ v ++ γ2 (because γ1 ≤ γ2). thus, π(η2, η1, γ1, γ2; ∆) ⊂ τ (s). on the other hand, let (aλ) be a net converging to a w.r.t. π(η2, η1, γ1, γ2; ∆), d ∈ ∆ and u ∈ uw. let w ∈ uw, w symmetric, be such that w ◦ w ⊂ u . by lemma 5.4 two cases arise: i) a ∈ (dc)++ δ . then eventually aλ ∈ (d c)++ δ and obviously, aλ ∩ d = ∅ ⊂ w (a) and a ∩ d = ∅ ⊂ w (aλ). ii) a 6∈ (dc)++ δ . then w (a) ∩ d 6= ∅. since w is totally bounded, there are xk ∈ a, k ∈ {1, . . . , n}, such that a ⊂ ⋃n k=1 w (xk) ⊂ w 2(a). note that since η2 is a compatible lr-proximity we have τ (v −) ⊂ σ(η2, η1) − ( cf. theorem 2.4). now, a ∩ w (xk) 6= ∅ for k ∈ {1, . . . , n}. hence, for each k there is a vk with xk ∈ vk and clvkη 2 [w (xk)] c. but, eventually aλη2vk and since clvkη 2 [w (xk)] c we have that eventually aλ ∩ w (xk) 6= ∅. therefore, eventually xk ∈ w (aλ). hence, eventually a ∩ d ⊂⋃n k=1 w (xk) ⊂ w 2(aλ) ⊂ u (aλ). furthermore, note that [d ∩ (w (a)) c] ∈ ∆ and a ∈ [dc∪w (a)]++ δ . so, eventually aλ ∈ [d c∪w (a)]++ δ . thus, eventually aλ ∩ d = [aλ ∩ d ∩ w (a)] ⊂ u (a), i.e. eventually aλ ∈ [d, u ](a). therefore, τ (s) ⊂ π(η2, η1, γ1, γ2; ∆). combining the earlier part we get τ (s) = π(η2, η1, γ1, γ2; ∆). � 6. appendix (admissibility). it is a well known fact that if (x, τ ) is a t1 space, then the lower vietoris topology τ (v −) is an admissible topology, i.e. the map i : (x, τ ) → (cl(x), τ (v −)), defined by i(x) = {x}, is an embedding. on the other hand ( as observed in example 2.1), if the involved proximities η1, η2 are different from the discrete proximity η ⋆, then the map i : (x, τ ) → (cl(x), σ(η2, η1) −) is, in general, not even continuous. so, we study the behaviour of i : (x, τ ) → (cl(x), σ(η2, η1) −), when η2 ≤ η1 and η1 6= η ⋆. first, we state the following lemma. lemma 6.1. let (x, τ ) be a t1 space, u ∈ τ with clu 6= x and v = (clu ) c. if z ∈ clu ∩ clv , then there exists a net (zλ) τ -converging to z such that for all λ either i) zλ ∈ u and zλ 6= z, or ii) zλ ∈ v and zλ 6= z. proof. let n (z) be the filter of open neighbouhoods of z. for each i ∈ n (z), select wi ∈ i ∩ v and yi ∈ i ∩ u . then, the net (wi ) τ -converges to z and (wi ) ⊂ v as well as the net (yi ) τ -converges to z and (yi ) ⊂ u . we claim that for all i ∈ n (z) either wi 6= z or yi 6= z. assume not. then there exist i and j ∈ n (z) such that yi = z and wj = z. as a result, z ∈ u ∩ v ⊂ clu ∩ v = ∅; a contradiction. � 128 g. di maio, e. meccariello and s. naimpally recall that a hausdorff space x is extremally disconnected if for every open set u ⊂ x, clu is open in x (see [10] page 368). proposition 6.2. let (x, τ ) be a hausdorff space and η1, η2 two compatible lo-proximities on x with η2 ≤ η1 and η1 6= η ⋆. then the following are equivalent: (a) x is extremally disconnected; (b) the map i : (x, τ ) → (cl(x), σ(η2, η1) −), defined by i(x) = {x}, is continuous. proof. (a) ⇒ (b). let x ∈ x and (xλ) a net τ -converging to x. let v ⊂ x with v open and {x}η1v . since {x}η1v and η2 ≤ η1, then {x}η2v and so x ∈ clv . by assumption clv is an open subset of x and the net (xλ) τ -converges to x. thus, eventually xλ ∈ clv . (b) ⇒ (a). by contradiction, suppose (a) fails. then, there exists an open set u ⊂ x such that clu is not open in x. then, clu 6= x. set v = (clu )c. v is non-empty and open in x. we claim that clu ∩ clv 6= ∅. assume not, i.e. clu ∩ clv = ∅. then, clu ⊂ (clv )c ⊂ v c = clu . thus, clu = (clv )c, i.e. clu is open; a contradiction. let z ∈ clu ∩ clv . by lemma 6.1, there exists a net (zλ) τ -converging to z such that for all λ either (1) zλ ∈ u and zλ 6= z or (2) zλ ∈ v and zλ 6= z. in both cases, there exists an open subset w such that z ∈ clw and zλ 6∈ clw for all λ. in fact if (1) holds, then set w = v , otherwise set w = u . thus, the net (zλ) τ -converges to z, but there exists an open subset w such that {z}η1w ( because z ∈ clw and η1 is a compatible lo-proximity on x) as well as {zλ}η 2 w ( again because zλ 6∈ clw and η2 is a compatible lo-proximity on x) for all λ. hence, the map i : (x, τ ) → (cl(x), σ(η2, η1) −) fails to be continuous. � remark 6.3. if η1 = η ⋆, then the map i : (x, τ ) → (cl(x), σ(η2, η1) −) is always continuous. definition 6.4 (cf. definition 6.3 in [8]). a t1 space (x, τ ) is nearly regular iff whenever x ∈ u with u ∈ τ there exists v ∈ τ with x ∈ clv ⊂ u . proposition 6.5. let (x, τ ) be t1 space and η1, η2 lo-proximities on x with η2 ≤ η1. if η2 is a compatible lo-proximity, then the following are equivalent: (a) (x, τ ) is nearly regular; (b) the map i : (x, τ ) → (cl(x), σ(η2, η1) −) is open. proof. left to the reader. � note that if (x, τ ) is a t1 space, γ1, γ2 are compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase, then the map i : (x, τ ) → (cl(x), σ(γ1, γ2; ∆) +) is always continuous. so, we have: symmetric bombay topology 129 proposition 6.6. let (x, τ ) be a t1 space, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. the following are equivalent: (a) the map i : (x, τ ) → (cl(x), σ(γ1, γ2; ∆) +), defined by i(x) = {x}, is an embedding; (b) the map i : (x, τ ) → (cl(x), σ(γ1, γ2; ∆) +), defined by i(x) = {x}, is an open map; (c) whenever u ∈ τ and x ∈ u , there exists a b ∈ ∆ such that x ∈ bc ⊂ u . finally, we have the following results dealing with admissibility of the symmetric standard bombay ∆ topology π(η2, η1, γ1, γ2; ∆). obviously, we investigate just the significant case η2 6= η ⋆ ( the standard bombay ∆ topology σ(γ1, γ2; ∆) is always admissible). we have to distinguish the two subcases (1) η1 6= η ⋆, (2) η1 = η ⋆. proposition 6.7. let (x, τ ) be a regular hausdorff space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. suppose that η1 6= η ⋆ and γ1 ≤ η1. then the map i : (x, τ ) → (cl(x), π(η2, η1, γ1, γ2; ∆)) is an embedding if and only if the following three conditions are fulfilled: (a) i : (x, τ ) → (cl(x), σ(η2, η1) −) is continuous; (b) i : (x, τ ) → (cl(x), σ(η2, η1) −) is open; (c) i : (x, τ ) → (cl(x), σ(γ1, γ2; ∆) +) is open. proof. only necessity requires a proof. namely just (b) and (c). now, we show that i : (x, τ ) → (cl(x), σ(η2, η1) −) is open. assume not. then there exist x ∈ x and u ∈ n (x), where n (x) is the filter of open nhoods at x such that i(u ) 6∈ σ(η2, η1) − ∩ i(x). so, there is y ∈ u such that for each w −η2 with w ⊂ x open and yη1w , we have w −η2 ∩ i(x) 6⊂ i(u ). but i : (x, τ ) → (cl(x), π(η2, η1, γ1, γ2; ∆)) is an embedding. thus, for each u ∈ n (x), i(u ) ∈ π(η2, η1, γ1, γ2; ∆) ∩ i(x). hence there exists o ∈ π(η2, η1, γ1, γ2; ∆) such that {y} ∈ o ∩ i(x) ⊂ i(u ). note that o has the form o++γ2 ∩ ⋂n j=1 (oj ) − η2 with yη1oj , oj ∈ τ , o c ∈ ∆ and y ≪γ1 o. furthermore, since π(η2, η1, γ1, γ2; ∆) is standard we may assume that ∪nj=1oj ⊂ o. now, since i : (x, τ ) → (cl(x), σ(η2, η1) −) is continuous, there exists a vj ∈ n (y) such that i(vj ) ⊂ (oj ) − η2 for each j ∈ {1, . . . , n}. because x is regular, there exists lj, j ∈ {1, . . . , n}, lj ∈ n (y) such that lj ⊂ cllj ⊂ vj . since x is regular select w ∈ n (y) with w = ∩nj=1lj and clw ⊂ o. it follows i(w ) ⊂ i(clw ) = w −η2 ∩ i(x) ⊂ i(u ). thus i(u ) ∈ σ(η2, η1) − ∩ i(x), a contradiction. now we prove that i : (x, τ ) → (cl(x), σ(γ1, γ2; ∆) +) is open. assume not. so, there exists x′ ∈ x and v ∈ n (x′), such that i(v ) 6∈ σ(γ1, γ2; ∆) + ∩ i(x). so there is y′ ∈ v such that for each w ++γ2 with w c ∈ ∆ and y′ ≪γ1 w , we have w ++γ2 ∩ i(x) 6⊂ i(v ). but i : (x, τ ) → (cl(x), π(η2, η1, γ1, γ2; ∆)) is an embedding. thus, for each v ∈ n (x′), i(v ) ∈ π(η2, η1, γ1, γ2; ∆) ∩ i(x). 130 g. di maio, e. meccariello and s. naimpally hence there exists t ∈ π(η2, η1, γ1, γ2; ∆) such that {y ′} ∈ t ∩ i(x) ⊂ i(v ). note that t has the form t ++γ2 ∩ ⋂n j=1 (tj ) − η2 with y′η1tj, tj ∈ τ , t c ∈ ∆ and y′ ≪γ1 t . moreover, since π(η2, η1, γ1, γ2; ∆) is standard, we may assume that tj ⊂ t for each j ∈ {1, . . . , n}. again, since i : (x, τ ) → (cl(x), σ(η2, η1) −) is continuous, there exists a sj ∈ n (y ′) such that i(sj ) ⊂ (tj) − η2 for each j ∈ {1, . . . , n}. select mj ∈ n (y ′), j ∈ {1, . . . , n} such that mj ⊂ clmj ⊂ sj . set w = t ∩ ⋂n j=1 mj . it follows that w ∈ n (y ′). again, since i : (x, τ ) → (cl(x), π(η2, η1, γ1, γ2; ∆)) is an embedding, there exists o ∈ π(η2, η1, γ1, γ2; ∆) such that {y′} ∈ o∩i(x) ⊂ i(w ). note that o has the form o++γ2 ∩ ⋂m k=1 (ok) − η2 with y′η1ok, ok ∈ τ , o c ∈ ∆ and y′ ≪γ1 o. we may assume ∪ n k=1ok ⊂ o. moreover, from o∩i(x) ⊂ i(w ) we have o ⊂ w (otherwise, select z ∈ o\w and zk ∈ ok; the set f = {z} ∪ ⋃n k=1 zk ∈ o \ t, a contradiction). as a result, i(o) = i(x) ∩ o++γ2 ⊂ t ⊂ i(v ). thus i(v ) ∈ σ(γ1, γ2; ∆) + ∩ i(x); a contradiction. � theorem 6.8. let (x, τ ) be a regular hausdorff space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. if η1 6= η ⋆ and γ1 ≤ η1, then the map i : (x, τ ) → (cl(x), π(η2, η1, γ1, γ2; ∆)) is an embedding if and only if the following conditions are fulfilled: (a) x is extremally disconnected; (b) whenever u ∈ τ and x ∈ u , there exists a b ∈ ∆ such that x ∈ bc ⊂ u . theorem 6.9. let (x, τ ) be a regular hausdorff space, η1, η2 lo-proximities on x with η2 ≤ η1, γ1, γ2 compatible lo-proximities on x with γ1 ≤ γ2 and ∆ ⊂ cl(x) a cobase. if η1 = η ⋆ and η2 is a compatible lo-proximity, then the following are equivalent: (a) the map i : (x, τ ) → (cl(x), π(η2, η1, γ1, γ2; ∆)) is an embedding; (b) whenever u ∈ τ and x ∈ u , there exists a b ∈ ∆ such that x ∈ bc ⊂ u . acknowledgements. the authors thank the referee for a careful reading of the manuscript and a number of valuable comments and suggestions, which led to an improvement of the paper. references [1] g. beer, topologies on closed and closed convex sets, kluwer academic publishers, 1993. 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[28] s. willard, general topology, addison-wesley, 1970. received november 2006 accepted december 2007 132 g. di maio, e. meccariello and s. naimpally giuseppe di maio (giuseppe.dimaio@unina2.it) seconda università degli studi di napoli, facoltà di scienze, dipartimento di matematica, via vivaldi 43, 81100 caserta, italia somashekhar naimpally (somnaimpally@yahoo.ca) 96 dewson street, toronto, ontario, m6h 1h3, canada cazaculawsonagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 1, 2007 pp. 1-33 quasicontinuous functions, domains, and extended calculus rodica cazacu and jimmie d. lawson abstract. one of the aims of domain theory is the construction of an embedding of a given structure or data type as the maximal or “ideal” elements of an enveloping domain of “approximations,” sometimes called a domain environment. typically the goal is to provide a computational model or framework for recursive and algorithmic reasoning about the original structure. in this paper we consider the function space of (natural equivalence classes of) quasicontinuous functions from a locally compact space x into l, an n-fold product of the extended reals [−∞, ∞] (more generally, into a bicontinuous lattice). we show that the domain of all “approximate maps” that assign to each point of x an order interval of l is a domain environment for the quasicontinuous function space. we rely upon the theory of domain environments to introduce an interesting and useful function space topology on the quasicontinuous function space. we then apply this machinery to define an extended differential calculus in the quasicontinuous function space, and draw connections with viscosity solutions of hamiltonian equations. the theory depends heavily on topological properties of quasicontinuous functions that have been recently uncovered that involve dense sets of points of continuity and sections of closed relations and usco maps. these and other basic results about quasicontinuous functions are surveyed and presented in the early sections. 2000 ams classification: 54c08, 06b35, 26b05, 49l25, 54c60 keywords: quasicontinuous functions, usco maps, domain theory, bicontinuous lattices, generalized calculus, hamiltonian equations, viscosity solutions 2 r. cazacu and j. d. lawson 1. introduction recall that a function f : x → y between topological spaces is quasicontinuous if the inverse image of every open set is quasi-open, that is, has dense interior. although such maps have been considered for some time [11], there has been a recent revival of interest in their topological study (e.g. the works of borśık [2],[3]) and in their study in a variety of applications such as selection theorems for set-valued maps [4], [7], the dynamics of quasicontinuous functions under iteration [7], and viscosity solutions of certain partial differential equations [19], [20]. the primary purpose of this paper is to introduce the tools of domain theory to the study of quasicontinuous function spaces and point toward applications in nonsmooth analysis. an important aim of computational domain theory is to develop computationally useful mathematical models of data types (e.g., booleans, integers, reals, and the higher types derived from them) that incorporate both the data types and computationally realizable approximations. these models can be useful for providing a theoretical computational framework for studying computational issues and questions, for investigating and developing a theory of computability, and in some cases for suggesting computational algorithms or approaches. the mathematical models considered here are continuous domains, a special class of partially ordered sets that are • directed complete: each directed set has a supremum, and • continuous: each element is a directed supremum of its (“finitary”) approximations, where x (finitarily) approximates y, written x ≪ y, if y ≤ sup d, d a directed set, implies x ≤ d for some d ∈ d. the intuition is that directed sets represent partial states of knowledge or stages in a computation, and x ≪ y if any computation of y reveals x at some stage of the computation. for some of the more important classes of quasicontinuous function spaces, we shall see that domain theory provides a nice and natural approach for their study. in addition, domain theory suggests a useful function space topology for these classes of functions, a function space topology that is built up from one that has been well-studied in domain theory. this approach provides a significant advantage for the study of quasicontinuous functions, since this function space topology has no clear counterpart for general sets of quasicontinuous functions, yet has many useful properties, as we shall see. as an application of the quasicontinuous function space and its domain setting, we close with a brief foray into a generalized differential calculus that employs quasicontinuous functions and suggest connections with viscosity solutions of partial differential equations. for another approach to differential calculus and analysis via domain theory we refer the reader to recent work of abbas edalat, see for example [9], [8], and related works. there are certainly significant overlaps (and significant differences) between his notion and theory of a domain derivative and the generalized derivative treated in this paper. keye martin has introduced quasicontinuous functions 3 an alternative approach via the “informatic derivative,” which again seeks a domain-based approach to differentiation and generalizations thereof [13],[14]. the following second and third sections of the paper survey and extend a variety of recent developments concerning topological aspects of quasicontinuous functions. in particular, we review and develop in our own framework the work of crannell, frantz, and lemasurier [7] and others regarding equivalence classes of quasicontinuous functions arising from their graph closures and related selection theorems. in the third section we derive the important equivalence between classes of quasicontinuous maps and minimal usco maps. the fourth section develops an approach to quasicontinuous functions from the viewpoint of domain theory. we show how quasicontinuous function classes arise naturally in this context and can be fruitfully treated using basic ideas and results of domain theory. in particular domain theory suggests a useful function space topology for the quasicontinuous function space, which we explore in sections 5 and 6, and this is a main motivation for introducing domain theory. in the seventh section we consider extensions of the differential calculus to certain classes of quasicontinuous functions and in the eighth section explore samborski’s ideas and results concerning the use of quasicontinuous functions as viscosity solutions of certain partial differential equations, particularly those arising in hamiliton-jacobi theory [19], [20]. indeed it was his work that was a major inspiration for this paper, and significant portions of this paper constitute a survey and amplification of his work, with the twist of a domain-theoretic perspective. much of the material in the latter portions of the paper are drawn from the dissertation of the first author [6]. 2. quasicontinuous functions a function f : x → y is quasicontinuous at x if for any open set v containing f (x) and any u open containing x, there exists a nonempty open set w ⊆ u such that f (w ) ⊆ v . it is quasicontinuous if it is quasicontinuous at every point. for an overview of the theory of quasicontinuous functions together with a rather extensive bibliography, we recommend the survey article of t. neubrunn [17]. call a set quasi-open (or semi-open) if it is contained in the closure of its interior. then f : x → y is quasicontinuous if and only if the inverse of every open set is quasi-open. it then follows that g ◦ f is quasicontinuous whenever g is continuous and f is quasicontinuous. some basic examples of quasicontinuous functions are the doubling function d : [0, 1) → [0, 1) defined by d(x) = 2x (mod 1), the floor function from r to r defined by ⌊x⌋ = max{n ∈ z : n ≤ x}, 4 r. cazacu and j. d. lawson and the extended sin(1/x) function f : r → r defined by f (x) = { 0 if x = 0, sin( 1 x ) otherwise. the doubling map is a basic example in the study of the dynamics of realvalued functions, and indicates why there is interest in a general theory of the dynamics of quasicontinuous functions. 2.1. graph closures. definition 2.1. let x, y be topological spaces and f : x → y a function. as usual, we identify f with its graph, f = {(x, y) ∈ x×y : y = f (x)}, and define the graph closure f as the closure cl(f ) of f in x × y . we define f (x) := {y ∈ y : (x, y) ∈ f}, f (a) := ⋃ x∈a f (x). the same construction of the graph closure h, closure taken in x ×y , extends to partial functions h : x ⇀ y, functions defined from a subset d of x into y . note that the domain πx (h) of the relation h may be strictly larger than that of h. for partial functions h, h(x) and h(a) are defined by the same formulas, but now the possibility exists that they may be empty. finally if f : x → y is a function and d ⊆ x, then we denote by f ↾ d the partial function that arises by restricting f to d and by f ↾ d the closure of this partial function in x × y . if f : x → y is continuous and y is hausdorff, then the graph closure of f is again f . several elementary facts about continuous functions extend to quasicontinuous functions if we work with graph closures instead of the quasicontinuous functions themselves. for example, the fact that a continuous function on a dense subset has at most one continuous extension to the whole space becomes lemma 2.2. if f : x → y is quasicontinuous and d ⊆ x is dense, then f = f ↾ d. proof. it suffices to show that f ⊆ f ↾ d, and then take closures. let y = f (x) and let x ∈ u , y ∈ v , where u is open in x and v is open in y . by quasicontinuity, there exists w open in x such that w ⊆ u and f (w ) ⊆ v . there exists w ∈ d ∩ w , and then (w, f (w)) ∈ u × v . it follows that f ⊆ f ↾ d. � our focus will be more on equivalence classes determined by graph closures of quasicontinuous functions than on the individual functions themselves (see [7]). definition 2.3. two (arbitrary) functions f, g : x → y are said to be closed graph equivalent or simply equivalent if f = g. we write f ∼ g if f = g, and denote the equivalence class of f by [f ]. quasicontinuous functions 5 corollary 2.4. if f, g : x → y agree on a dense subset of x and f is quasicontinuous, then f ⊆ g. hence f ∼ g if both are quasicontinuous. proof. we have from lemma 2.2 for some dense subset d that f = f ↾ d = g ↾ d ⊆ g, and dually if g is also quasicontinuous. � there is also a converse of sorts to the preceding corollary, but first we need a small lemma. lemma 2.5. for a partial function f : x ⇀ y , suppose that f (u ) ⊆ v for some u open in x. then f (u ) ⊆ v −. proof. the inclusion f ⊆ u × v ∪ u c × y , where u c = x \ u , implies f ⊆ u − × v − ∪ u c × y = u × v − ∪ u c × y, where the last equality follows from (u − \ u ) × v − ⊆ u c × y . it follows that if u ∈ u , (u, w) ∈ f , then (u, w) ∈ u × v −, hence w ∈ v −, and thus f (u ) ⊆ v −. � proposition 2.6. suppose that f, g : x → y , f is quasicontinuous, y is regular, and g ⊆ f . then g is quasicontinuous. proof. let y = g(x) ∈ o and x ∈ u , where o is open in y and u is open in x. pick v open in y such that y ∈ v ⊆ v − ⊆ o. since g ⊆ f , we have (x, y) ∈ f , and hence (u, f (u)) ∈ u × v for some u ∈ u . by quasicontinuity of f , there exists some nonempty open set w ⊆ u such that f (w ) ⊆ v . by lemma 2.5 f (w ) ⊆ v − ⊆ o. then g(w ) ⊆ g(w ) ⊆ f (w ) ⊆ o. � note 1. observe that the preceding proposition shows that the equivalence class [f ] of a quasicontinuous function consists of quasicontinuous functions only. 2.2. points of continuity. in this section we consider sets of points of continuity for quasicontinuous functions. note 2. we denote the set of points of continuity of a function f : x → y between two topological spaces by c(f ). points of continuity enjoy a type of “extended continuity.” (the following lemma slightly generalizes parts of [5, theorem 3.1].) lemma 2.7. let d be a subspace of a topological space x, let f : d → y be a function that is continuous at some x ∈ d. (i) if f (x) ∈ v , an open subset of y , then there exists u open in x containing x such that f (u ) ⊆ v −, and in the case y is regular, such that f (u ) ⊆ v . (ii) if y is hausdorff, then f (x) = {f (x)}. 6 r. cazacu and j. d. lawson proof. (i) let f (x) ∈ v , an open set. pick u open in x such that x ∈ u and f (u ∩ d) ⊆ v . then by lemma 2.5, f (u ) ⊆ v −. if x is regular, then we may pick w open such that y ∈ w ⊂ w − ⊆ v and u such that x ∈ u , f (u ) ⊆ w − ⊆ v . (ii) if y 6= f (x), then there exists an open set v containing f (x) and an open set w containing y such that v ∩ w = ∅. by part (i) we may pick u open containing x such that f (u ) ⊆ v − ⊆ y \ w . thus y /∈ f (x). assertion (ii) follows. � corollary 2.8. if f ∼ g for f, g : x → y and y is regular hausdorff, then c(f ) = c(g) and f and g agree on this set. proof. suppose that x ∈ c(f ). by lemma 2.7(ii), g(x) ∈ g(x) = f (x) = {f (x)}, so g(x) = f (x). let v be an open set containing g(x) = f (x). then by lemma 2.7(i), there exists an open set u containing x such that g(u ) ⊆ g(u ) = f (u ) ⊆ v . thus x ∈ c(g). since the argument is symmetric, c(f ) = c(g) and f and g agree on this set. � recall that a baire space is one in which in which every countable intersection of dense open sets is dense. the next proposition shows that under rather general hypotheses c(f ) is large for quasicontinuous functions. proposition 2.9. if x is a baire space, y is a metric space, and f : x → y is quasicontinuous, then c(f ) is a dense gδ-set. proof. recall that the oscillation of f at x is defined by osc(f )(x) = inf{diamf (u ) : u open, x ∈ u}. then it is standard and straightforward to verify that (i) f is continuous at x if and only if osc(f ) = 0, and (ii) for 0 < ε, oε := {x : osc(f )(x) < ε} is open. it follows easily from the quasicontinuity of f that each oε is dense. thus⋂ n o1/n is a dense gδ-set, and is precisely the set of points of continuity. � corollary 2.10. for f : x → y , consider the following conditions: (1) f is quasicontinuous; (2) f = f ↾ c(f ). then (2) implies (1) and the converse holds if c(f ) is dense, in particular if x is baire and y is metric. proof. (1)⇒(2): an immediate consequence of lemma 2.2 and proposition 2.9. (2)⇒(1): let y = f (x), x ∈ u , y ∈ v , where u is open in x, v is open in y . by hypothesis there exists w ∈ c(f ) such that (w, f (w)) ∈ u × v . since w ∈ c(f ), there exists an open set w containing w such that f (w ) ⊂ v , and w ∩ u is the desired neighborhood to establish quasicontinuity at x. � a selection function of the graph closure f is a function σ : x → y whose graph is contained in f , i.e., σ(x) ∈ f (x) for all x ∈ x. note that the function σ is a selection function of f if and only if σ ⊆ f . quasicontinuous functions 7 theorem 2.11. let f, g : x → y where f is quasicontinuous, c(f ) is dense, and y is regular hausdorff. the following are equivalent: (1) g ∼ f . (2) g is quasicontinuous and agrees with f on a dense subset. (3) g is a selection function for f. proof. (2)⇒(1): corollary 2.4. (1)⇒(3): always g is a selection function for g, hence for f if the graph closures are equal. (3)⇒(2): let g : x → y be a selection function of f . then g ⊆ f and by proposition 2.6 g is quasicontinuous. by lemma 2.7(ii), g(x) ∈ f (x) = {f (x)}, so g(x) = f (x) for all x ∈ c(f ), which is dense. � 3. usco maps let f : x ⇉ y be a set-valued map (also called a multifunction). we say that f is compact-valued if f (x) is a nonempty compact subset of y for each x ∈ x, that f is upper semicontinuous at x ∈ x (usc at x) if f (x) ⊆ v , v open in y , implies there exists an open neighborhood u of x such that f (u ) = ⋃ u∈u f (u) ⊆ v , and that f is upper semicontinuous (usc) if it is upper semicontinuous at each x ∈ x. if f is both usc and compact-valued, then is is said to be a usco map. if f : x ⇉ y , we identify f with its graph {(x, y) : y ∈ f (x)}. thus multifunctions can alternatively be viewed as relations. we again let f denote the closure in x × y . lemma 3.1. let f : x ⇉ y be usco and let r ⊆ f be a closed subset. then the projection πx (r), the domain of r, is closed in x. in particular if the domain of r is dense, it is all of x. proof. suppose that x /∈ πx (r). then for each y ∈ f (x), there exist open sets py containing x and qy containing y such that (py × qy) ∩ r = ∅. since f (x) is compact, finitely many of the {qy : y ∈ f (x)} cover f (x). let q be their union and p the corresponding intersection of the finite subcollection of the {py}. since f is usco, there exists an open set u containing x such that f (u ) ⊆ q, and by intersecting with p if necessary, we may assume that u ⊆ p . then (u × y ) ∩ r ⊆ (u × y ) ∩ f ⊆ u × f (u ) ⊆ p × q. it follows that (u × y ) ∩ r ⊆ (p × q) ∩ r = ∅, and thus u is an open set containing x and missing πx (r). � proposition 3.2. if f : x ⇉ y is usco, y is hausdorff, and g ⊆ f , then g is usco if and only if g = g and the domain of g, πx (g), equals x. proof. assume that g is closed and the domain of g is x. then the intersection g∩ ({x}×f (x)) is the intersection of a closed set and a compact set, hence a compact set. thus its projection under πy , which is g(x), is also compact. let 8 r. cazacu and j. d. lawson v be an open set containing g(x). by lemma 3.2, πx (g∩(x ×v c)) is closed in x, and it follows from a straightforward verification that its complement u in x is an open set containing x and satisfying f (u ) ⊆ v . hence f is also usc. conversely assume that g is usco. then by definition the domain of g is x. suppose y /∈ g(x). using the hausdorffness of y and the compactness of g(x), one finds disjoint open sets v and w such that y ∈ v and g(x) ⊆ w . for u open containing x such that g(u ) ⊆ w , we have u ×v ∩g = ∅, so the complement of g is open. � remark 3.3. from the preceding proposition we see that the theory of usco maps is a generalization to more general spaces of the theory of closed relations r ⊆ x ×y with πx (r) = x for the case of y compact hausdorff (since in this case f = x × y is a usco map that contains all closed relations). given a fixed usco map f : x ⇉ y , we freely view any closed relation with domain x contained in f as a usco map and vice-versa, as convenient. the theory of quasicontinuous functions provides useful and important techniques for constructing special selections for usco maps f , functions σ such that σ(x) ∈ f (x) for all x. the following is a very general recent result of cao and moors [4]. theorem 3.4. let x be a baire space, y a regular hausdorff space, and f : x ⇉ y a usco map. then f admits a quasicontinuous selection. the theorem is actually more general, and holds for compact-valued multifunctions that are “upper baire continuous”: for each pair of open sets u, w with x ∈ u and f (x) ⊆ w , there exist a nonempty open set v ⊆ u and a residual (recall that a set is “residual” if it contains a countable intersection of dense open subsets.) set r ⊆ v such that f (z) ⊆ w for all z ∈ r. this work generalizes earlier work of matejdes [15], who introduced the notion of upper baire continuous. we call a usco map minimal if, interpreted as a graph, it contains no strictly smaller usco map. the next result establishes an equivalence between quasicontinuous selections and minimal selections of usco maps (where a selection f is minimal if f is a minimal usco map). observe that a selection is minimal if and only if it is a selection function for some minimal usco map. corollary 3.5. let f : x ⇉ y be a usco map, where x is a baire space and y is a metric space, and let f : x → y be a selection function. then the graph closure of f is a minimal usco map iff f is quasicontinuous. furthermore, any minimal usco map is the graph closure of any selection function, and these are all quasicontinuous. proof. assume that f is a minimal usco map. by theorem 3.4, f has a quasicontinuous selection g. then g ⊆ f , and thus the two are equal by minimality of f and proposition 3.2. then f is quasicontinuous by proposition 2.6. quasicontinuous functions 9 conversely, suppose that f is a quasicontinuous selection function for f . then f is a usco map (proposition 3.2). let h : x ⇉ y be a usco map such that h ⊆ f . if h is any selection function for h, then h is also a selection function for f . by theorem 2.11 h ∼ f , and hence h = f . thus h = h = f and f is minimal. the last assertion follows along the lines of the first paragraph of the proof. � not surprisingly, stronger conclusions are available if we strengthen the hypotheses. lemma 3.6. let f : x ⇉ r be usco, where x is a baire space. then f admits a lower semicontinuous quasicontinuous selection function. proof. by theorem 3.4 f admits a quasicontinuous selection g : x → r. then g is usco, and g ⊆ f . define h : x → r by h(x) = inf g(x). that h is a lower semicontinuous selection of g follows directly from the fact that g is usco. from corollary 3.5 g is a minimal usco map and its selection h is quasicontinuous. � the following is a theorem of crannell, frantz, and lemasurier [7] , which builds on ideas of w. miller and e. akin [16]. theorem 3.7. let r ⊆ x × y be a closed relation, where projx (r) = x, x is a baire space, and y is compact metric. then r admits a quasicontinuous borel selection function f : x → y . proof. let k be the standard cantor set in r and let g : k → y be a continuous surjective map, which is possible since y is compact metric. treating r as a usco map r : x ⇉ y (see remark 3.3), we see that g := g−1 ◦ r : x → k ⊂ r is a usco map. by the preceding lemma g admits a lower semicontinuous, quasicontinuous selection σ, and g ◦ σ is a quasicontinuous, borel selection for r. � note that the selection function is actually the composition of a lower semicontinuous, quasicontinuous real-valued function with a continuous function. corollary 3.8. let f : x ⇉ y be usco, where x is a baire space and y is (separable) metrizable. then f admits a (borel) quasicontinuous selection function that is continuous at a dense set of points. proof. if y is separable metrizable, we can embed y in z, a countable product of the interval [0, 1]. we can then extend the codomain of f from y to z; note that f : x ⇉ z is still usco. treating f equivalently as a closed relation, we can obtain a quasicontinuous selection function f for f by theorem 3.7, and the range of f is contained in y , since it is a selection function for f . by proposition 2.9, f is continuous at a dense gδ-set of points. if y is only metrizable, then f again admits a quasicontinuous selection function f by theorem 3.4, which has a dense gδ-set of points of continuity, again by proposition 2.9. � 10 r. cazacu and j. d. lawson remark 3.9. recall that a regular hausdorff space y is called a stegall space if whenever x is a baire space and f : x ⇉ y is a minimal usco map, there exists a residual set d of x such that f (x) is a singleton for all x ∈ d. it follows from the upper semicontinuity of f that any selection function f will be continuous at any point of d and then from corollary 2.10 that f will be quasicontinuous. conversely, suppose that the regular hausdorff space y has the property that whenever x is a baire space and f : x ⇉ y is a minimal usco map, then any selection function f has a residual set of points of continuity. then by lemma 2.7(ii) f = f (by minimality) has a residual set of points for which f (x) is a singleton. putting together the previous remark with corollary 3.5, we obtain corollary 3.10. a regular hausdorff space y is a stegall space if and only if c(f ) is residual for every quasicontinuous selection function f for any usco map f : x ⇉ y from a baire space x. let x be a space and y be a compact metric space. if for some dense subset d of x, f : d → y is a continuous map, then the closed relation f is called a densely continuous form. the relation f is contained in the usco map x×y , and hence by lemma 3.1 the projection πx (f ) is a closed set containing d, hence equal to x. by proposition 3.2 f is a usco map. clearly for any selection function h of f , h ⊆ f . by lemma 2.7(ii) any h must agree with f on d, and hence the reverse inclusion holds, thus h = f . it follows that f is a minimal usco map. corollary 3.11. let x be a baire space and y a compact metric space. then f : x ⇉ y is a minimal usco map iff it is a densely continuous form. the graph closure of a map f : x → y is a densely continuous form if and only if f is quasicontinuous, and the correspondence [f ] ↔ f is a one-to-one correspondence between the equivalence classes of quasicontinuous functions and the densely continuous forms (resp. the minimal usco maps). proof. by the preceding comments a densely defined form is a minimal usco map. the converse follows from corollary 3.8, since by minimality f will agree with f ↾ d, where f is a selection function continuous on a dense subset d. the remaining assertions follows from the first and corollary 3.5. � 4. domains as pointed out in the introduction, a central goal of this paper is the development and study of a quasicontinuous function space from the perspective of domain theory. to this task we now turn. 4.1. basic domain theory. in this section we quickly recall basic notions concerning continuous domains (see [10]). a nonempty subset d of a partially ordered set (x,≤) is directed if given x, y ∈ d, there exists z ∈ d such that x, y ≤ z. a directed complete partially quasicontinuous functions 11 ordered set or dcpo is a partially ordered set (x,≤) such that every directed subset of x has a least upper bound in x. let x, y ∈ x where x is a dcpo. then we say x approximates y, denoted by x ≪ y, if for every directed set d with y ≤ sup d we have x ≤ d for some d ∈ d. for y ∈ x we define ⇓y = {x ∈ x : x ≪ y}. then we say a dcpo is continuous if • y = sup ⇓y for all y ∈ x and • each ⇓y is a directed set. a base for a continuous dcpo is a set b ⊆ x such that for all x ∈ x, x = sup{⇓x ∩ b}, and the supremum is taken over a directed set. a continuous domain, or domain for short, is a continuous dcpo and an ω-continuous domain is a domain with a countable base. for a dcpo x, we can define the scott topology as follows: a subset o ⊆ x is scott-open if • o is an upper set, i.e., if x ≤ y and x ∈ o, then y ∈ o. • o is inaccessible by least upper bounds of directed sets, i.e., if sup d ∈ o for a directed set d, then d ∈ o for some d ∈ d. one of the unusual features of the scott topology is that it is only t0, not hausdorff, as long as the order on x is non-trivial. we will henceforth use freely the fact (see [10, chapter ii.1]) that in a continuous domain the scott topology has a basis of open sets of the form ⇑z := {y ∈ x : z ≪ y}. a function between dcpos x and y is scott continuous if it is monotone and preserves directed suprema. equivalently a scott continuous function is continuous with respect to the scott topologies on x and y . example 4.1. consider the extended real numbers r = [−∞,∞] equipped with the usual order. then the scott topology consists of [−∞,∞] and all open right rays (x,∞]. a function f : x → r from a topological space x is scott continuous if and only if it is lower semicontinuous in the usual sense. the upper sets of the form ↑x := {y : x ≤ y}, sometimes called principal filters, form a subbasis for the closed sets of another topology, commonly called the lower topology. its join with the scott topology (the smallest topology containing both) gives the lawson topology. for continuous domains, the lawson topology is a hausdorff topology. it is finer than the interval topology, which has as subbasis for the closed sets all closed order intervals [a, b] = {x : a ≤ x ≤ b}. for any topology defined from the order of a partially ordered set l, one can define the dual topology that arises by reversing the order and defining the topology for that order, i.e., defining the topology on lop. for example the dual scott topology on r consists of all open left rays [−∞, x), −∞ < x. the biscott 12 r. cazacu and j. d. lawson topology is the join of the scott topology and the dual scott topology. on r both it and the lawson topology agree with usual topology of the extended reals. 4.2. bicontinuous lattices. a partially ordered set is a lattice if any two points have a least upper bound and a greatest lower bound and a complete lattice if every subset has a least upper bound and a greatest lower bound. a continuous lattice is a continuous domain that is also a complete lattice. definition 4.2. a complete lattice l is linked bicontinuous, or simply bicontinuous for short, if it satisfies: (1) l and lop are continuous domains; (2) l is a complete lattice; (3) the interval, biscott, lawson, and dual lawson topologies all agree. remark 4.3. a variety of equivalent conditions for being a bicontinuous lattice appear in [10] proposition vii-2.9, for example the following : (i) (l,∨,∧) is a compact topological lattice with a basis of open sets that are sublattices. in this case the topology must be the biscott. (ii) a complete distributive lattice l is bicontinuous if and only if it is completely distributive, that is, arbitrary joins distribute over arbitrary meets and vice-versa. note 3. when we are working in the context of bicontinuous lattices we have a notion of approximation in both directions, so there is potential for confusion in the notation. we adopt the conventions • a ≪ b means any directed sup exceeding b must have some member exceeding a; • ⇑a = {b : a ≪ b}; • a ≫ b means any directed inf preceding b must have some member preceding a; • ⇓a = {b : a ≫ b}. in what follows we will primarily restrict our attention to those bicontinuous lattices l that are ω-continuous and these are called ω-bicontinuous lattices. this is equivalent to assuming that the biscott topology is metrizable, and hence equivalent to the dual lop being ω-continuous (see [12, proposition 7.1]). for applications our focus will not be on general ω-bicontinuous lattices, but instead on the following example, and those less familiar with domain may basically restrict their attention to this example. however, even for this specific example, domain theory provides a convenient tool and framework for our considerations. primary example. for r = [−∞,∞], the extended reals, we form r n extended n-dimensional euclidian space. observe that r n is a product of completely distributive lattices, hence completely distributive with respect to the coordinatewise order: (x1, · · · , xn) ≤ (y1, · · · , yn) ⇔∀i, xi ≤ yi, quasicontinuous functions 13 and thus a bicontinuous lattice. we observe that the scott open sets are the open sets u =↑u , the open upper sets. the biscott topology is the usual product topology, which is metrizable, so r n is ω-continuous. the preceding observations remain valid for r n , a countable product of extended reals. the latter is convenient to keep in mind for generalizations, since any separable metrizable space can be embedded in it. 4.3. domain environments. one of the aims of domain theory is to provide semantic or computational models for structures that include approximations to members of the structure. often members of the structure are modeled as maximal “ideal” members of the model and elements below are thought of as approximations. this is often thought of as an “information ordering,” the higher the element the more nearly it approximates ideal elements at the top. definition 4.4. a domain environment for a topological space x is a homeomorphic embedding x →֒ m ax(d) onto the set of maximal points of a continuous domain d equipped with the relative scott topology. remark 4.5. a natural domain environment l for a bicontinuous lattice l (always endowed with the biscott=lawson topology) consists of all nonempty order intervals [u, v] := {x ∈ l | u ≤ x ≤ v}, where the order intervals are ordered by reverse inclusion, the “information order,” and l embeds as the degenerate intervals. lemma 4.6. let l be a bicontinuous lattice, and l the set of all order intervals. let a1, a2, b1, b2 ∈ l. the following are equivalent: (i) [a1, b1] ≪ [a2, b2]; (ii) [a2, b2] ⊆ int[a1, b1], the topological interior; (iii) a1 ≪ a2 in l, and b1 ≪ b2 in l op, written b2 ≫ b1. proof. a directed subset d ⊆ l of closed intervals has supremum (equal intersection) contained in [a2, b2] if and only if the lower endpoints have directed supremum greater than or equal to a2 and the upper endpoints have directed infimum less than or equal to b2. from this observation the equivalence of (i) and (iii) readily follows. by [10, proposition ii-1.10] a1 ≪ a2 if and only if a2 ∈ int ↑a1, where the interior is taken in the scott topology. this statement and its dual yield the equivalence of (ii) and (iii). � a bounded complete domain is a domain that is also a complete (meet)semilattice, a partially ordered set in which every nonempty subset has an infimum. theorem 4.7. the set l is a bounded complete domain. proof. any nonempty family a ⊆ l has supremum the closed interval obtained by taking the infimum (resp. supremum) of all lower (resp. upper) endpoints of members of a for its lower (resp. upper) endpoint. thus l is a complete 14 r. cazacu and j. d. lawson semilattice. directed suprema are formed in an analogous way, but now taking the supremum (resp. infimum) of the lower (resp. upper) endpoints. the fact that every element is a supremum of approximating elements follows readily from the preceding lemma. � theorem 4.8. the map u 7→ [u, u] : l −→ l is a homeomorphic embedding, hence a domain environment for (l, biscott), representing l as the degenerate intervals [u, u]. proof. consider the map u 7→ [u, u] : l −→ l. we want to show that the map is one-to-one, continuous, open and its image is the set of maximal elements of l. if x, y ∈ l, x 6= y, then [x, x] = {x} 6= {y} = [y, y], and so the map is one-to-one. let u =⇑ [a, b] be a basic scott-open set in l, [x, x] ∈ u . by lemma 4.6, x ∈⇑a∩ ⇓ b, an open set in l. let c be in this open set. then a ≪ c, and b ≫ c. using lemma 4.6 we can conclude that [a, b] ≪ [c, c], and so [c, c] ∈ u . to see that the embedding is an open map onto its image, it suffices to show that images of a subbasis of open sets are again open. using lemma 4.6, we see that the image of ⇑a (resp. ⇓b) is the intersection of the maximal elements with ⇑[a,⊤] (resp. ⇑[⊥, b]), where ⊤ (resp. ⊥) is the top (resp. bottom) element of l. � 5. function spaces our goal in this section is to define and study a natural domain environment for the equivalence classes of quasicontinuous functions that we introduced earlier. 5.1. approximate functions. intuitively an “approximate” or “fuzzy” function is one for which we have incomplete information. one way of modelling such functions is to assume that we know f (x) only up to an interval of values. definition 5.1. an approximate function f from a topological space x into a bicontinuous lattice l is a function f : x −→ l. the approximate function f is scott-continuous if it is continuous into the scott topology of l. since each f (x) is an order interval, we can write f (x) as f (x) = [f ∧(x), f ∨(x)], where f ∧, f ∨ : x −→ l. in this case we write the interval function f = [f ∧, f ∨]. theorem 5.2. let f : x → l be an approximate function, f = [f ∧, f ∨]. the following are equivalent: (1) the approximate function f is scott-continuous. (2) viewed as a multifunction, f : x ⇉ l is upper semicontinuous, and hence a usco map. quasicontinuous functions 15 (3) f ∧ : x → l is scott-continuous (also called lower semicontinuous) and f ∨ is dually scott-continuous (or upper semicontinuous). in particular, as a relation f is a closed subset of x × l. proof. (1)⇒(2): suppose that f is scott-continuous. we first observe that the lawson topology on the bicontinous, hence continuous, lattice l is compact hausdorff and each order interval [a, b] is closed, hence compact (see, for example, [10, chapter iii.1]). let x ∈ x, and let u be open in l and contain f (x). we observe that ũ := {ξ ∈ l : ξ ⊆ u} is a scott-open set in l since it is closed under subsets (hence an upper set) and any directed intersection (equal supremum) of closed, hence compact, order intervals with intersection a member of ũ , hence contained in u , must have some member contained in u . by scott-continuity of f , there exists some open set w containing x such that f (w ) ⊆ ũ , that is, f (w) ⊆ u for each w ∈ w . (2)⇒(3): let f (x) = [f ∧(x), f ∨(x)] for each x. let x ∈ x and let z ≪ f ∧(x). then ⇑z is a basic scott-open set containing f ∧(x) and hence contains [f ∧(x), f ∨(x)]. therefore there exists an open set w containing x such that f (w) ⊆⇑z for each w ∈ w . it follows that z ≪ f ∧(w) for each w ∈ w , and hence that f ∧ is scott-continuous. (3)⇒(1): let f (x) = [f ∧(x), f ∨(x)] and let ⇑[c, d] be a basic scott-open set containing f (x) in l. then c ≪ f ∧(x) by lemma 4.6. by scott-continuity of f ∧, there exists w1 open containing x such that f ∧(w1) ⊆⇑c. similarly there exists w2 open containing x such that f ∨(w2) ⊆⇓d, and then by lemma 4.6 f (w) ∈⇑[c, d] for all w ∈ w = w1 ∩ w2. the last assertion follows from proposition 3.2. � note 4. in light of the preceding, we henceforth refer to scott-continuous approximate functions as usc approximate functions. 5.2. the domain of approximate functions. in this subsection x denotes a locally compact hausdorff space, l a bicontinuous lattice, and l its domain environment of closed order intervals. proposition 5.3. the set of all usc approximate functions from a locally compact space x to a bicontinuous lattice l ordered by the pointwise order is a bounded complete domain [x −→ l], called the domain of approximate functions. proof. from theorem 4.7 we know that l is a bounded complete domain, and this makes the set of usc approximate functions to be one. see [10, proposition ii-4.6]. � we have additionally the space of lower semicontinuous functions, denoted by (lsc(x, l),≤) and the space of upper semicontinuous functions denoted by (usc(x, l),≤op=≥), where the order for both of them is the pointwise order. these are each bounded complete domains, again by [10, proposition ii-4.6]. we define l̂x = {(f, g) ∈ lsc(x, l) × usc(x, l) : f ≤ g}. 16 r. cazacu and j. d. lawson for lsc(x, l) × usc(x, l) we consider the order given by (f1, g1) ≤ (f2, g2) ⇔ f1 ≤ f2 in lsc(x, l) and g1 ≤op g2 in usc(x, l). proposition 5.4. the set l̂x is a scott closed bounded complete subdomain of lsc(x, l)×usc(x, l), and it is homeomorphic to the domain of approximate functions, [x → l] under the identification (f, g) ↔ [f, g]. proof. the set l̂x ⊆ lsc(x, l)×usc(x, l) is closed under directed sups and arbitrary infs, so, by [10] theorem i-2.6, is a scott closed bounded complete subdomain of the domain lsc(x, l) × usc(x, l). for the second part of the proposition let o : [x → l] → l̂x be defined by o([f, g]) = (f, g) for any [f, g] ∈ [x → l]. since [f, g] ∈ [x → l] then f ∈ lsc(x, l), g ∈ usc(x, l) and f ≤ g, which makes our application well defined. if (f, g) ∈ l̂x then it is clear that [f, g] ∈ [x → l], so o is surjective, and it is immediate that it is injective. one sees directly that this one-to-one correspondence is an order isomorphism, hence a homeomorphism for the scott and lawson topologies. the inclusion of l̂x into lsc(x, l)×usc(x, l) preserves directed sups and arbitrary nonempty infs, so is continuous for the lawson topologies. since both are compact t2 in the lawson topology, it follows that the lawson topology of l̂x agrees with the relative lawson topology from lsc(x, l) × usc(x, l) (see chapter iii.1 of [10] for these facts about the lawson topology). using the compactness, one sees that if a is a scott-closed subset of l̂x , then ↓a is lawson-compact, hence scott-closed in lsc(x, l) × usc(x, l), and thus is scott-closed. since a =↓a ∩ l̂x , we conclude that the scott topology on l̂x agrees with the relative scott topology. � remark 5.5. proposition 5.4 is important because it allows us to study the topology of the domain of usc approximate functions in terms of the function spaces lsc(x, l) and usc(x, l). the latter function spaces have been objects of serious investigation in the theory of domains (see, for example, section ii.4 of [10]) and much is already understood about them. in particular it is important to note that in light of the previous proposition we have a net fα = [fα, gα] → f = [f, g] in the scott (resp. lawson topology) of the domain of approximate functions if and only if fα → f in the scott (resp. lawson) topology of lsc(x, l) and gα → g in the scott (resp. lawson) topology of usc(x, l). (note that the convergence need not be directed convergence, only convergence in the respective topologies.) theorem 5.6. the function e : [x → l] ×x → l defined by e(f, x) = f (x) is continuous, where we assume that [x → l] and l are equipped with the scott topology. hence e satisfies the following joint continuity condition: if e(f, x) = f (x) = [f ∧(x), f ∨(x)] ⊆ w , where w is open in l, then there exist scott-open sets u1 in lsc(x, l) and u2 in usc(x, l) containing f ∧ and quasicontinuous functions 17 f ∨ resp. and v open containing x such that if γ ∈ u1, δ ∈ u2, γ ≤ δ and y ∈ v , then [γ(y), δ(y)] ⊆ w . proof. by [10, proposition ii-4.10(7)] the map e is continuous if [x → l] is endowed with the standard isbell function space topology, and by [10, proposition ii-4.6] the isbell and scott topology agree since l is a bounded complete domain. thus the first assertion follows. we have [f ∧(x), f ∨(x)] = ⋂ {[a, b] : a ≪ α(x), b ≫ β(x)}. since the intersection on the right is a directed intersection of compact subsets, it follows that [f ∧(x), f ∨(x)] ⊆⇑a∩⇓b ⊆↑a∩↓b = [a, b] ⊆ w for some a ≪ f ∧(x), b ≫ f ∨(x). then [a, b] ≪ [f ∧(x), f ∨(x)], i.e., [f ∧(x), f ∨(x)] ∈⇑[a, b] ⊆ w . since ⇑[a, b] is scott-open in l, the joint continuity condition follows from the previous paragraph and the equivalence of proposition 5.4. � corollary 5.7. the multifunction e : [x → l]×x ⇉ l defined by e(f, x) = f (x) is a usco map. proof. this follows directly from theorem 5.6, lemma 4.6, and the equivalence of proposition 5.4, since the order intervals of l are compact. � definition 5.8. we define the extended compact-open topology on [x → l] as the topology that has a subbasis of open sets of the form n (k, u ) := {f ∈ [x → l] : f (k) = ⋃ x∈k f (x) ⊆ u}, where k is compact in x and u is open in l. note 5. note that when restricted to the continuous functions from x to l the extended compact-open topology is the compact-open topology. proposition 5.9. the extended compact-open topology on [x → l] is equal to the scott topology. proof. it follows easily from corollary 5.7 that the subbasic open sets n (k, u ) are open in the scott topology of [x → l]. the scott topology of the function space lsc(x, l) is equal to the isbell topology [10, proposition ii-4.6], which in turn is equal to the the compact-open topology from x into lσ, l equipped with the scott-topology [10, lemma ii4.2(i)]. for f = [f ∧, f ∨], k compact in x, and w open in lσ, we have f ∧(k) ⊆ w if and only if f (k) = ∪x∈k [f ∧(x), f ∨(x)] ⊆ w , since w = ↑w . thus under the correspondence of proposition 5.4, n (k, w ) corresponds to l̂x ∩ ({g ∈ lsc(x, l) : g(k) ⊆ w}× usc(x, l)). clearly a dual argument is valid for usc(x, l). it then follows from the equivalence of proposition 5.4 that the scott topology of [x → l] is contained in the extended compact-open topology. � 5.3. maximal approximate functions. we turn to a common construction in domain theory and its basic properties (see, for example, [10, exercise ii3.19]). 18 r. cazacu and j. d. lawson definition 5.10. for any function f : x → l, we define f∗(x) := sup{inf f (u ) : x ∈ u, u is open} and f ∗(x) := inf{sup f (u ) : x ∈ u, u is open}. lemma 5.11. let d be a dense subset of x, let f : d → l, and set f∗(x) := sup{inf f (u ∩ d) : x ∈ u, u is open}. then f∗ : x → l ∈ lsc(x, l), and satisfies the following: (i) f∗(x) ≤ f (x) for all x ∈ d, (ii) for x ∈ d, f∗(x) = f (x) ⇔ f is lower semicontinuous at x; (iii) if g : x → l, g ≤ f on d, and g is lower semicontinuous at x ∈ x, then g(x) ≤ f∗(x). proof. let x ∈ x and let v be a scott-open set containing f∗(x). pick z ∈ v such that z ≪ f∗(x) = sup{inf f (u∩d) : x ∈ u , open}. since this is a directed sup, z ≤ inf f (u∩d) for some u open, x ∈ u . if follows that z ≤ inf f (u∩d) ≤ f∗(w) for all w ∈ u , i.e., f∗(u ) ⊆↑z ⊆ v . hence f∗ ∈ lsc(x, l). since for x ∈ d, inf f (u ∩ d) ≤ x for each u open containing x, property (i) follows. since for any x ∈ x, f∗ is lower semicontinous at x and f∗ ≤ f on d by (i), it follows immediately that f is lower semicontinuous at any x ∈ d where f (x) = f∗(x). conversely suppose that f is lower semicontinuous at x ∈ d and let z ≪ f (x). then there exists u open containing x such that f (u ∩d) ⊆↑z, and hence z ≤ f∗(x). since f (x) = sup{z : z ≪ f (x)}, it follows that f (x) ≤ f∗(x) and hence from (i) f (x) = f∗(x). (iii) if g is lower semicontinuous at x ∈ x, then g(x) ≤ sup{inf g(u ∩ d) : x ∈ u open} ≤ sup{inf f (u ∩ d) : x ∈ u open} = f∗(x). � the next proposition follows in a straightforward fashion from the preceding lemma. proposition 5.12. let f : x → l be a function, and f∗, f ∗ be defined as in definition 5.10. the following are true: (i) f∗ ≤ f ≤ f ∗; (ii) f∗ is lower semicontinuous and f ∗ is upper semicontinuous; (iii) f is lower semicontinuous if and only if f = f∗; (iv) f upper semicontinuous if and only if f = f ∗; (v) f is continuous if and only if f = f ∗ = f∗; (vi) f∗ is the largest lower semicontinuous function such that f∗ ≤ f ; (vii) f ∗ is the smallest upper semicontinuous function such that f ≤ f ∗. quasicontinuous functions 19 for any continuous function f : x → l, the approximate function f = [f, f ] is clearly maximal in the domain of approximate functions. there are, however, additional maximal elements. proposition 5.13. the maximal elements in the domain [x → l] of approximate functions have the form f (x) = [α(x), β(x)], where α∗ = β and β∗ = α. these include the continuous functions. proof. we know that the elements of the domain [x → l] are usc approximate functions, which, by theorem 5.2, means that α is lower semicontinuous, and β is upper semicontinuous. that is, α∗ = α ≤ α ∗ and β∗ ≤ β = β ∗. let f be a maximal element of the domain [x → l]. since α ≤ β we have that α∗ ≤ β∗ = β. thus [α(x), α∗(x)] ⊆ [α(x), β(x)], which means that [α(x), β(x)] ≤ [α(x), α∗(x)]. if f is maximal then we must have [α(x), β(x)] = [α(x), α∗(x)], and that gives us α∗ = β. a similar proof yields that α = β∗ if f is maximal in the domain. now suppose that f = [α, β] is such that α∗ = β and β∗ = α. suppose that f ≤ g = [α1, β1]. then α ≤ α1 ≤ β1 ≤ β implies β = α ∗ ≤ α∗1 ≤ β ∗ 1 = β1 ≤ β, so α∗1 = β1 and β = β1. similarly α1 = (β1)∗ and α = α1. that means f = g, so f is maximal in the domain. � 6. quasicontinuous function spaces we assume in this section as a standing hypothesis that x is a locally compact hausdorff space and l is a bicontinuous lattice, although we will often require even stronger hypotheses than this. we return to our consideration of quasicontinuous functions and their graph closure equivalence classes. the domain-theoretic setting allows us to define a useful function space topology on these classes. lemma 6.1. if f : x → l is quasicontinuous, then f := [f∗, f ∗] is maximal in the domain of approximate functions, and every maximal element arises in this way. proof. suppose that f ≤ g = [α, β]. if it were the case that α ≤ f ≤ β, then by proposition 5.12, α ≤ f∗ ≤ f ∗ ≤ β, yielding g ≤ f , so f = g, implying that f is maximal. thus it suffices to show that f ⊆ g = g, where the last equality follows from theorem 5.2. let (x, f (x)) ∈ u × v , a basic open set in x × l. pick b, c ∈ l such that f (x) ∈⇑b∩ ⇓c ⊆ v . by quasicontinuity there exists a nonempty open set w ⊆ u such that f (w ) ⊆⇑b∩⇓c. it follows that b ≤ f∗(w) ≤ α(w) ≤ β(w) ≤ f ∗(w) ≤ c for all w ∈ w . in particular g∩(u×v ) 6= ∅. it follows that (x, f (x)) ∈ g = g and hence α ≤ f ≤ β. conversely let g = [α, β] be a maximal approximate function and let f : x → l be a quasicontinuous selection function (theorem 3.4). by proposition 5.12, α ≤ f∗ ≤ f ∗ ≤ β, so by maximality g = [f∗, f ∗]. � 20 r. cazacu and j. d. lawson the next proposition extends the equivalences for two quasicontinuous functions to be closed graph equivalent. proposition 6.2. let f, g : x → l be quasicontinuous functions such that c(f ) is dense. the following are equivalent: (1) f, g agree on a dense set. (2) f ∗ = g∗, f∗ = g∗. (3) f ∼ g. in particular, these all hold for l ω-bicontinuous. proof. items (1) and (3) are equivalent by theorem 2.11. assume (3). set f = [f∗, f ∗], a closed relation (theorem 5.2). thus g ⊆ g = f ⊆ f. it follows that f∗ ≤ g ≤ f ∗, and hence that [f∗, f ∗] ≤ [g∗, g ∗]. interchanging f and g yields item (2). conversely assume (2). then f∗ and f ∗ agree with f on the dense set c(f ) by lemma 5.11, and thus so do g∗ and g ∗ by hypothesis. since g∗ ≤ g ≤ g ∗, g also agrees with f on c(f ). hence f ∼ g by theorem 2.11. in the case the l is ω-bicontinuous, it is separable metrizable, so by proposition 2.9 quasicontinuous functions have a dense gδ-set of points of continuity. � note 6. for l ω-bicontinuous, we denote by q(x, l) the space of equivalence classes of quasicontinuous functions, and denote the class of f by [f ]. note that a continuous function has a singleton equivalence class. we define for [f ] ∈ q(x, l), [f ](x) = [f∗(x), f ∗(x)] and [f ](a) = ⋃ x∈a [f ](x). note that in light of proposition 6.2, these definitions are well-defined. theorem 6.3. for l an ω-bicontinuous lattice, the association [f ] ←→ [f∗, f ∗] is a one-to-one correspondence between the classes of quasicontinuous functions, q(x, l), and the maximal elements of the domain [x → l] of approximate maps from x to l. proof. the theorem follows readily from lemma 6.1 and proposition 6.2. � theorem 6.3 suggests a natural topology for the quasicontinuous equivalence classes, namely the scott topology, on the domain of approximate functions restricted to the quasicontinuous equivalence classes, which we identify with the maximal approximate functions. definition 6.4. let x be a locally compact hausdorff space, let l be an ωbicontinuous lattice, and let the function space [x → l] be equipped with the scott topology. the topology on q(x, l), the set of classes of quasicontinuous functions, that makes the injection [f ] ↔ [f∗, f ∗] of theorem 6.3 a topological embedding is called the quasiorder topology or qo-topology for short. quasicontinuous functions 21 remark 6.5. the qo-topology is defined in such a way that the domain of approximate functions forms a domain environment for the quasicontinuous equivalence classes. by proposition 5.4 the scott topology in [x → l] agrees with the one arising from simultaneous scott-convergence in the lsc and usc variables. by proposition 5.9 this scott topology on the space of approximate functions is equal to the extended compact-open topology, so the qo-topology may also be considered to be the restriction of the extended compact-open topology. theorem 6.6. the lawson and scott topologies agree on the set of maximal elements of the domain of approximate functions [x → l], and hence this topology is completely regular and hausdorff. if we restrict to the case that x is locally compact and separable metrizable and l is ω-continuous, then we may identify this space with q(x, l) with the qo-topology, which makes the space a polish space. in particular, we may restrict function space convergence to sequences in studying continuity, closedness, compactness, etc. proof. to show that the lawson and scott relative topologies agree on the maximal elements of a domain d it suffices to show that for any p ∈ d, ↑p ∩ maxd = a ∩ maxd for some scott-closed set a [10, definition v-6.1]. in the case of a bounded complete domain l this is always satisfied, since ↑p is closed in the compact hausdorff lawson topology, hence compact, and thus a := ↓(↑p) is scott-closed and satisfies ↑p ∩ maxd = a ∩ maxd. since any subspace of the compact hausdorff space l is completely regular and hausdorff, the first assertion is satisfied. the fact that [x → l] is ω-continuous if x is locally compact and separable metrizable and l is ω-bicontinuous follows from the identification of proposition 5.4 and a standard theorem that gives the cardinality of a basis for the function space lsc(x, l) from those of the domain and codomain [10, corollary iii-4.10]. the space of maximal points of the ω-continuous domain is polish by [10, theorem v-6.6]. the identification with q(x, l) and its topology comes from theorem 6.3 and definition 6.4. since polish spaces are metrizable the last assertion of the theorem follows. � proposition 6.7. let l be an ω-bicontinuous lattice, and let f, g : x → l be quasicontinuous maps such that [f ] 6= [g]. then there exist a nonempty open set u ⊆ x and a, b ∈ l, b a such that for any x ∈ u , [f ](x) ⊆⇓a and [g](x) ⊆⇑b (or vice-versa). proof. since l is separable metrizable, by theorem 2.9 and proposition 6.2 there exists x ∈ c(f ) ∩ c(g) such that f (x) 6= g(x), say g(x) f (x). since l is bicontinuous, we can find a, b ∈ l such that b a and f (x) ∈⇓a, g(x) ∈⇑b. since f and g are continuous at x we have that [f ](x) = [f (x), f (x)] and [g](x) = [g(x), g(x)]. by theorem 5.6 there exists an open set u containing x such that [f ](u ) ⊆⇓a and [g](u ) ⊆⇑b. � 22 r. cazacu and j. d. lawson the following gives some equivalent characterizations of convergence in q(x, l). proposition 6.8. let ([fn])n ⊆ q(x, l) and [f ] ∈ q(x, l), where we assume that x is locally compact and separable metrizable and l is ω-bicontinuous. the following are equivalent: (1) (fn)∗ −→ f∗ in the scott topology, and (fn) ∗ −→ f ∗ in the dual-scott topology; (2) f∗ = sup n ( inf n≤m , (fm)∗ ) ∗ = lim(fn)∗f ∗ = inf n ( sup n≤m (fm) ∗ )∗ = lim(fn) ∗; (3) [(fn)∗, (fn) ∗] → [f∗, f ∗] in the relative scott topology of the set of maximal elements of the domain [x → l]; (4) there exist an increasing sequence (gn)n ⊆ lsc(x, l) and a decreasing sequence (hn)n ⊆ u sc(x, l) such that f∗ = supn gn, f ∗ = infn hn and gn ≤ (fn)∗ ≤ (fn) ∗ ≤ hn, for each n. proof. (1) ⇔ (2). from the definition of scott convergence we have that (fn)∗ −→ f∗ if and only if f∗ ≤ lim(fn)∗, and similarly for the dual scott convergence. the only thing that must be proved is that f∗ = sup n ( inf n≤m (fm)∗ ) ∗ and f ∗ = inf n ( sup n≤m (fm) ∗ )∗ , where the inequalities follow from [10, proposition iii-3.12]. we have that [ sup n ( inf m≤n (fn)∗ ) ∗ , inf n ( sup m≤n (fn) ∗ )∗] ∈ [x → l], and [f∗, f ∗] ≤ [ sup n ( inf m≤n (fn)∗ ) ∗ , inf n ( sup m≤n (fn) ∗ )∗] in [x → l]. since [f∗, f ∗] is a maximal element of the domain [x → l], we must have the equality of the two intervals, therefore the equalities we want. (1) ⇔ (3). this equivalence follows directly from proposition 5.4. (2) ⇔ (4). suppose that (2) is true. for each n ≥ 1 let gn = ( inf n≤m (fm)∗ ) ∗ and hn = ( sup n≤m (fm) ∗ )∗ . it is clear that each gn is lower semicontinuous and each hn is upper semicontinuous. since n1 ≤ n2 implies inf n1≤m (fm)∗ ≤ inf n2≤m (fm)∗ quasicontinuous functions 23 and sup n1≤m (fm) ∗ ≥ sup n2≤m (fm) ∗, we have gn1 ≤ gn2 and hn1 ≥ hn2 , which means (gn)n is increasing and (hn)n is decreasing. it is also clear that we have gn ≤ (fn)∗ ≤ (fn) (∗) ≤ hn for each n > 0. for the other implication, let (gn)n and (hn)n like in (3). since (gn)n is increasing, we have gn ≤ gm ≤ fm for every m ≥ n, which implies gn ≤ inf n≤m (fm)∗, and, since gn ∈ lsc(x, l), gn ≤ ( inf n≤m (fm)∗ ) ∗ . therefore f∗ ≤ sup n ( inf n≤m (fm)∗ ) ∗ . similarly we get f ∗ = inf n ( sup n≤m (fm) ∗ )∗ = lim(fn) ∗, and because f ∈ q(x, l), [f∗, f ∗] is a maximal element of the domain [x → l], hence we have (2). � 7. generalized derivatives in this section we will restrict our attention to the bicontinuous lattices r and r n . we make the standing assumption that x is a locally compact, locally convex subset of rm and consider functions from x to r or r n . we adopt what will be a convenient convention of identifying two quasicontinuous functions f, g if they belong to the same equivalence class, in much the same way that we identify two functions in measure theory if they differ on set of measure 0. since in this section we are only considering functions from a locally compact subset x of rn into r m , this means that the two agree on their common set of points of continuity (corollary 2.8), a dense gδ-set (proposition 2.9), and f (x) ∈ g(x) and vice-versa otherwise. thus f is uniquely defined on c(f ) and is ambiguous up to f (x) otherwise (with no ambiguity for continuous functions, their class consisting of one element). occasionally it will also be convenient to treat (equivalence classes of) quasicontinuous functions as maximal elements of the domain of approximate functions, or as minimal usco maps f : x ⇉ rn (via the identification of the previous sections) such that each f (x) is an order interval [f∗(x), f ∗(x)], where the latter is independent of the representative of the equivalence class (proposition 6.2). furthermore, we don’t distinguish between points of r n and degenerate order intervals. in particular, singleton-valued usco maps from x to r n are, for us, the same as continuous maps from x to r n . 24 r. cazacu and j. d. lawson we work mostly with finite-valued functions. we denote by q(x, r) the members f of q(x, r) with [f∗(x), f ∗(x)] ⊆ r for all x ∈ x and employ a similar convention for q(x, rn). in this section we extend results of samborski [19] using the machinery that we have developed in earlier sections. we consider the partial derivative operator (7.1) ∂ ∂xk : c1(x, r) −→ c0(x, r) ⊆ q(x, r) and the gradient operator ∇ : c1(x, r) −→ c0(x, rm) ⊆ q(x, rm),(7.2) ∇ = ( ∂ ∂x1 , ∂ ∂x2 , · · · , ∂ ∂xm ) . lemma 7.1. suppose fn → f in q(x, r), fn ∈ c 1(x, r) ⊆ q(x, r), and ∇fn → f in q(x, r m). then f ∈ q(x, r) is a locally lipschitz function, and fn converges to f in the compact-open topology. proof. let x ∈ x. since f (x) ⊆ rm, we may pick a, b ∈ rm such that b ≫ f ∗(x) and a ≪ f∗(x). it follows from theorem 5.6 that there exists n > 0 and u open containing x such that (7.3) ∇fn(u) ∈⇓b∩⇑a for each u ∈ u and each n ≥ n. let m > 0 such that [a, b] ⊆ bm (0), the open ball in r m around 0 of radius m . then (7.4) ‖∇fn(u)‖≤ m for each u ∈ u and each n ≥ n. we can choose u such that u is also convex, so that we can apply the mean value theorem for differentiable functions on rm. therefore for each n > n and each u, v ∈ u there exists 0 < tn < 1 such that fn(u) − fn(v) = 〈∇fn(ξn), u − v〉 , where ξn = tnu + (1 − tn)v ∈ u , and by (7.4) we get ‖fn(u) − fn(v)‖≤‖∇fn(ξn)‖‖u − v‖≤ m‖u − v‖, for each n ≥ n , which means that f = {fn|u : n ≥ n} is an equicontinuous family of functions. using the same arguments that we used for ∇fn to find (7.4), we can find u0 open containing x, n0 > 0, m0 > 0 such that {fn(y) : n ≥ n0}⊆ (−m0, m0) ⊆ r for each y ∈ v = u0 ∩ u , which makes the closure of {fn(y) : n ≥ n0} compact in r. thus we are in the setting of ascoli’s theorem [18], so we obtain a subsequence of {fn|v : n ≥ n}, (fnk ), which converges pointwise to a continuous function g, the convergence being uniform on each compact subset of v . indeed since all fn are m -lipschitz on u , then g is m -lipschitz on v also. equivalently, we can say that (fnk ) → g in the compact-open topology, so in the qo-topology (see note 5 and proposition 5.9). quasicontinuous functions 25 the convergence fn → f in q(x, r) makes fnk → f in q(x, r), and since q(x, r) is hausdorff, f|v = g|v in q(x, r), so f is a locally lipschitz function. � example 7.2. consider the absolute value function on the interval x = [−1, 1]. it admits an extended derivative [g] that is the sign function, with either the value 1 or −1 at 0, i.e., [g](0) = {−1, 1}. recall that the strong derivative of a function f : u ⊆ r → r is given by lim u,v→x u6=v f (u) − f (v) u − v , if the limit exists. definition 7.3. let u ⊆ rm be locally compact with dense interior, and let f : u → rn. and x ∈ u . we will say that f is strongly differentiable at x if there exists a linear operator l : rm → rn such that for all u, v ∈ u , f (u) − f (v) = l(u − v) + r(u, v) where lim u,v→x u6=v ‖r(u, v)‖ ‖u − v‖ = 0. the operator l, if it exists, is unique and is called the strong derivative at x and denoted df (x). theorem 7.4. let u ⊂ rm be locally compact, locally convex with dense interior, and fn ∈ c 1(u, r) ⊆ q(u, r) such that fn → f in q(x, r) and ∇fn → g in q(u, r m). then the strong derivative of f exists and is equal to g on a dense gδ-set d ⊆ x. in particular ∇f = g on d so we can say that the gradient of f is given by ∇f = [(∇f )∗, (∇f ) ∗] = [g∗, g ∗], where (7.5) (∇f )∗(x) = sup x∈u open inf {∇f (y) : y ∈ u ∩ d} , and (7.6) (∇f )∗(x) = inf x∈u open sup{∇f (y) : y ∈ u ∩ d} . furthermore, [∇f ] = [g]. proof. by the previous lemma f is a locally lipschitz function. by proposition 2.9, the set d of points of continuity of g is a dense gδ-set. let x ∈ int(u )∩c(g), and let u, v ∈ u . since each fi is differentiable and u is locally convex, we can apply the mean value theorem on rm for each fi for u 6= v close to x. therefore, there exists ξi = (1 − t)u + tv for some 0 < t < 1 such that fi(u) − fi(v) = 〈∇fi(ξi), u − v〉 . 26 r. cazacu and j. d. lawson then we have |f (u) − f (v) −〈g(x), u − v〉 ‖u − v‖ ≤ |f (u) − f (v) − (fi(u) − fi(v)) ‖u − v‖ + |fi(u) − fi(v) −〈∇fi(ξi), u − v〉 | ‖u − v‖ + | 〈∇fi(ξi) − g(x), u − v〉| ‖u − v‖ . the middle term of the right-hand side of the inequality is zero. since ξi → x as u, v → x, and since ∇fn → g in q(u, r m), by corollary 5.7 for any ε > 0 there exists n1 > 0 and v open and convex containing x such that ‖∇fi(ξi) − g(x)‖ < ε 2 for i ≥ n1 and u, v ∈ v , u 6= v. then we have ‖〈∇fi(ξi) − g(x), u − v〉‖ ‖u − v‖ ≤ ‖∇fi(ξi) − g(x)‖‖u − v‖ ‖u − v‖ = ‖∇fi(ξi) − g(x)‖ < ε 2 . by the previous lemma fn converges to f in the compact-open topology, so in particular fn(y) → f (y) for any y ∈ u . thus there exists for any distinct u, v ∈ v , an n2 > 0 such that |f (u) − f (v) − (fi(u) − fi(v))| ‖u − v‖ < ε 2 for every i ≥ n2. putting this all together, we conclude that |f (u) − f (v) −〈g(x), u − v〉 | ‖u − v‖ < ε 2 + ε 2 = ε. therefore lim u,v→x u6=v f (u) − f (v) ‖u − v‖ = 〈g(x), u − v〉 ‖u − v‖ , so the strong derivative of f exists for x a continuity point for g, and such points form a dense gδ-set d. for x ∈ d we have also ∇f (x) = g(x), and since d is dense we can define (∇f )∗ = (x →∇f (x)|x ∈ d ⊆ x)∗ and (∇f )∗ = (x →∇f (x)|x ∈ d ⊆ x)∗. hence, by lemma 5.11(iii), we have g∗ ≤ (∇f )∗ and (∇f ) ∗ ≤ g∗, so by minimality of g ∇f = [(∇f )∗, (∇f ) ∗] = g in q(x, r), and the theorem is proved. � quasicontinuous functions 27 the preceding theorem easily extends to the case of general functions from rm to rn. let d : c1(u, rn) → c0(u, rm×n) be defined by df (x) is the jacobian matrix of f at x, where u is a locally compact subset of rm with dense interior. we consider the closure of the set {(f, df ) : f ∈ c1(u, rn), df ∈ c0(u, rm×n} in q(x, rn)×q(x, rm×n), where the latter is endowed with the product of the qo-topologies. corollary 7.5. (i) closing up the differentiation operator d : c1(x, rn) → c0(x, rm×n) in q(x, rn) × q(x, rm×n) yields an extended operator d. we denote the domain of the extended operator by q1(x, rn) and call df the generalized derivative of f for f ∈ q1(x, rn). (ii) each member f ∈ q1(x, rn) is a locally lipschitz map from x to rn and is strongly differentiable at a dense subset of points of x. the image df in q(x, rm×n) is the closure of the densely defined mapping on x sending x to the strong derivative dsf (x). proof. it follows from the preceding theorem that the theorem is true in each of the n-coordinate functions and hence true overall. � we remark that the generalized derivatives considered in [9] are intervalvalued, i.e., approximate functions in our sense. we consider only the special case of maximal approximate functions identified with quasicontinuous functions. thus we only consider derivatives for which the intervals are degenerate (or single-valued) for a dense subset and the resulting function is continuous on this dense subset. 8. hamiltonian equations in this section we recall ideas of samborski [19], [20] for applying the theory of quaisicontinuous functions to the study of viscosity solutions of hamiltonian equations. let x be a locally compact subset of rn that has dense interior, and let h : x ×r×rn → r be a function convex in the last argument. in this section we consider solutions of the hamiltonian (8.1) h(x, y(x),∇y(x)) = h(x). 8.1. continuous hamiltonians. recall that if α ∈ lsc(x, r) and ∂−α(x) = {ζ ∈ rm : α(y) ≥ α(x) + 〈ζ, y − x〉− σ‖y − x‖2, for some σ > 0 and y close enough to x} is the subgradient of α at x, then the subset ∂−α(x) 6= ∅ for x in a dense subset of x. the same is true for β ∈ usc(x, r) and its supergradient ∂+β. we shall need the following proposition. proposition 8.1. let (un)n ⊆ lsc(x, r) and u ∈ lsc(x, r). then un → u in lscλ(x, r), where λ denotes the lawson topology, if and only if the following are true: (1) if xn → x ∈ x, then u (x) ≤ lim infn un(xn); (2) for x ∈ x there exists zn → x such that un(zn) → u (x). 28 r. cazacu and j. d. lawson proof. (⇒): suppose un → u in lscλ(x, r̄). then (1) is a consequence of the scott convergence and the continuity of the evaluation function e : lscσ(x, r̄) × x → r̄σ. (2) for each n, set βn = inf{d(y, x) + d(un(y), u(x)) : y ∈ x}. we will prove that βn → 0. let ε > 0. pick b open in lscσ(x, r̄) containing u and 0 < δ < ε 2 such that e(b×bδ(x)) ⊆ (u (x)− ε 2 ,∞]. define q : x → r̄ by q(bδ(x)) = u (x)+ ε 2 , q(y) = −∞ otherwise. then u /∈↑q since u (x) < q(x). thus there exists n such that un /∈↑q and un ∈ b for n ≥ n . then for n ≥ n , u (x) − ε 2 < un(z) < u (x) + ε 2 for some z ∈ bδ(x). thus d(z, x) + d(un(z), u (x)) < ε 2 + ε 2 = ε, and hence βn < ε for n ≥ n , thus βn → 0. now choose for each n, a point zn such that d(zn, x) + d(un(zn), u (x)) < βn + 1 n . it follows that zn → x and un(zn) → u (x). (⇐). suppose (un)n ⊆ lsc(x, r) and u ∈ lsc(x, r) such that we have (1) and (2). it is clear that (1) implies un → u in lscσ(x, r). let f ∈ lsc(x, r) such that u ∈ lsc(x, r) \↑f , a basic open set in the λ−topology. therefore f u , or equivalently, there exists x ∈ x such that f (x) u (x) in r, which means u (x) < f (x). then there exists a ∈ r such that u (x) < a < f (x). by (2) there exists (zn)n ⊆ r such that zn → x and un(zn) → u (x). since u (x) ∈ [−∞, a) ⊆ r is open, there exists n1 > 0 such that for every n ≥ n1 un(zn) ∈ [−∞, a). since f is lower semicontinuous and f (x) ∈ (a,∞], there exists an open w ⊆ x, x ∈ w such that f (w ) ⊆ (a,∞], and since zn → x there exists n2 > 0 such that zn ∈ w for any n ≥ n2. then for every n ≥ n = max(n1, n2) we have f (zn) un(zn), which implies that for any n ≥ n f un, or, equivalently, un ∈ lsc(x, r) \↑f . therefore we have un → u in lscλ(x, r̄). � proposition 8.2. let h : x × r × rn → r be continuous. for f ∈ q(x, r) let d1 = {x : ∂−f∗(x) 6= ∅} and d2 = {x : ∂+f ∗(x) 6= ∅}, which are known to be dense subsets of x. we define d− : q(x, r) → lsc(x, r) by (8.2) d−f =   x → inf a∈∂−f∗(x) ∂−f∗(x) 6=∅ {h(x, f∗(x), a) | x ∈ d1 ⊂ x}    ∗ , quasicontinuous functions 29 and d+ : q(x, r) → usc(x, r) by (8.3) d+f =   x → sup b∈∂+f ∗(x) ∂+f ∗(x) 6=∅ {h(x, f ∗(x), b) | x ∈ d2 ⊆ x}    ∗ . let ∆ ⊆ q(x, r), ∆ = {f ∈ q(x, r) : df = [d−f,d+f ] ∈ q(x, r)}. then d is a closed operator in q(x, r) with domain ∆. for proving this proposition we will need the next result. lemma 8.3. let u ⊆ rn be locally compact, f : u → r be lower semicontinuous, x ∈ u such that ∂−f (x) 6= ∅ and a ∈ ∂−f (x). suppose also that (fi) ∈ lsc(x, r) is a sequence such that fi → f in lscλ(x, r). then there exists x′i ∈ u , ∂−fi(x ′ i) 6= ∅ and a ′ i ∈ ∂−fi(x ′ i) such that (8.4) x′i → x, fi(x ′ i) → f (x) and a ′ i → a. proof. this is a particular case of proposition 8.1 from [1], applied to lower semicontinuous functions. by our proposition 8.1 from earlier in this section the lawson convergence in lsc(x, r) is equivalent with the conditions assumed in proposition 8.1 from [1] for the lower semicontinuous case. � proof. (of proposition 8.2) let (fi) ⊆ ∆ such that fi → f , f ∈ ∆, dfi → f in q(x, r). we will show that f = df in q(x, r). suppose f 6= df . by proposition 6.7 there exist a nonempty open u ⊆ x, b1, b2 ∈ r, b1 < b2 such that df (x) ⊆ [−∞, b1) and f (x) ⊆ (b2,∞] for any x ∈ u or vice versa. therefore in u we have f∗ > b2 and d+f < b1. let x ∈ u . then f∗(x) > b2 and d+f (x) < b1. using the continuity of the evaluation map e : lscσ(x, r) ×x → rσ we find o ⊆ lsc(x, r) open, f∗ ∈ o and u1 ⊆ x open, x ∈ u1 such that e(o × u1) ∈ (b2,∞]. since dfi → f in q(x, r) then d−fi → f∗ in lscσ(x, r). therefore there exists n1 > 0 such that d−fi(y) > b2, for each i ≥ n1, y ∈ u1. for every y ∈ u we have d+f (y) < b1. let w = u ∩ u1. thus d−fi(y) > b2 and d+f (y) < b1, each i ≥ n1, y ∈ w, which implies that for every y, y′ ∈ w and n ≥ n1 we have d−fi(y ′) −d+f (y) > b2 − b1 = c. because df ∈ q(x, r) we have d+f (y) ≥d−f (y) for any y ∈ w , so we get d−fi(y ′) −d−f (y) > c, for each y, y ′ ∈ w and i ≥ n1. 30 r. cazacu and j. d. lawson let µ = inf d−f (w ). it follows from the fact that inf(f∗(w )) = inf(f (w )) for any open set w and any function f and the definition of d−f that inf{h(x, f∗(x), a) : x ∈ w, a ∈ ∂−f∗(x)} = µ. thus there exist z ∈ w , a ∈ ∂−f∗(z) 6= ∅ such that d−fi(y ′) − h(z, f∗(z), a) > c/2, for every y ′ ∈ w and i ≥ n1. from the definition of d−(fi)∗, for any x ′ ∈ w for which ∂−(fi)∗(x ′) 6= ∅, we have h(x′, (fi)∗(x ′), a′) ≥d−fi(x ′), for every a′ ∈ ∂−(fi)∗(x ′). therefore, from the last two inequalities we conclude: statement 1. for any i ≥ n1, there exists (z, a) ∈ w × r n, a ∈ ∂−f∗(z) 6= ∅ such that for every (x′, a′) ∈ w × rn, a′ ∈ ∂−(fi)∗(x ′) 6= ∅ we have (8.5) h(x′, (fi)∗(x ′), a′) − h(z, f∗(z), a) > c/2. we now apply lemma 8.3, knowing from theorem 6.6 that the lawson topology and the scott topology agree on the set maximal elements of l̂ (which are the ones whose coordinates come from quasicontinuous functions). statement 2. for any ε > 0, there exists n2 > 0 such that for every i ≥ n2, there exists (x′, a′) ∈ w × rn where a′ ∈ ∂−(fi)∗(x ′) with the property (8.6) ‖z − x′‖ < ε, |f∗(z) − (fi)∗(x ′)| < ε, ‖a − a′‖ < ε. the continuity of h implies that for any η > 0 there exists ε > 0 such that for any i > 0 max{‖z − x′‖, |f∗(z) − (fi)∗(x ′)|,‖a − a′‖} < ε implies |h(z, f∗(z), a) − h(x ′, (fi)∗(x ′), a′)| < η. choosing η < c/2 we obtain an ε = ε(η), and for this ε, using statement 2, we can find an n > 0 for which there exists (x′, a′) ∈ w × rn, where a′ ∈ ∂−(fi)∗(x ′) such that we have (8.6), which implies (8.7) |h(z, f∗(z), a) − h(x ′, (fi)∗(x ′), a′)| < c/2. we can observe that (8.7) is in contradiction with (8.5), and that means the operator d is closed. � remark 8.4. in the begining of the proof of proposition 8.2 we considered only one case of proposition 6.7. for the other case the proof is similar to this one, only we work in usc(x, r), and we use the exact form of proposition 8.1 from [1]. quasicontinuous functions 31 9. viscosity functions definition 9.1. a function ϕ : x → r is a (discontinuous) viscosity solution of h(x, f,∇f ) = g(x) if for any x ∈ x such that ∂−ϕ∗(x) 6= ∅, for any a ∈ ∂−ϕ∗(x) the inequality (9.1) h∗(x, ϕ∗(x), a) ≥ g∗(x) is true, and for any x ∈ x such that ∂+ϕ ∗(x) 6= ∅, for any b ∈ ∂+ϕ ∗(x) the inequality (9.2) h∗(x, ϕ∗(x), b) ≤ g∗(x) is true. we will call such a function a viscosity function. remark 9.2. if f ∈ q(x, r), then either none or all representatives of the class of f are viscosity solutions of the equation (8.1). proposition 9.3. let h : x × r × rn → r be a continuous function, d be the operator in q(x, r) with the domain ∆ defined in proposition 8.2, and g ∈ q(x, r).then every solution f ∈ ∆ of the equation dy = g is a viscosity solution of the equation (9.3) h(x, y(x),∇y(x)) = g(x). proof. let f ∈ ∆ be such that df = g. then we have d−f = g∗ and d+f = g ∗. by definition of d− in proposition 8.2, for any x ∈ x such that δ−f∗(x) 6= ∅ we have that g∗(x) = d−f (x) ≤ inf a∈∂−f∗(x) h(x, f∗(x), a), which implies that for any a ∈ ∂−f∗(x) h(x, f∗(x), a) ≥ g∗(x). since h is continuous, h = h∗, which implies that inequality (9.1) from definition 9.1 is true. similarly we can obtain inequality (9.2), therefore f is a viscosity solution. � 10. future directions in regard to generalized derivatives, we would like to extend the definition of q1(x, rn) ⊆ q(x, rn) to include those quasicontinuous functions that have a strong derivative at a dense set of points. one would then like to work out in more detail the calculus of such functions. we have only indicated an approach to connecting quasicontinuous functions with the study of hamiltonian equations. although samborski [19] has carried out some work in this direction, it appears that much remains to be done. in 32 r. cazacu and j. d. lawson particular, we would like to see if domain theoretic ideas can contribute to this investigation. acknowledgements. the authors would like to thank frederick mynard for pointing out reference [5] and michelle lemasurier for supplying a preprint of [7], both of which led in turn to other useful references. the authors also benefitted from the careful reading and insightful suggestions of the referee. references [1] m. bardi, m. g. crandall, l. c. evans, h. m. soner and p. e. souganidis, viscosity solutions and applications, lectures notes in mathematics 1660, springer-verlag, berlin, heidelberg, 1997 [2] j. borśık, products of simply continuous and quasicontinuous functions, math. slovaca 45 (1995), 445–452. [3] j. borśık, maxima and minima of simply continuous and quasicontinuous functions, math. slovaca 46 (1996), 261–268. [4] j. cao and w. moors, quasicontinuous selections of upper continuous set-valued mappings, real anal. exchange 31 (2005), 63–72. [5] b. cascales and l. oncina, compactoid filters and usco maps, j. math. anal appl. 283 (2003), 826–845. [6] r. cazacu, quasicontinuous derivatives and viscosity functions, dissertation, louisiana state university, 2005. [7] a. crannell, m. frantz and m. lemasurier, closed relations and equivalence classes of quasicontinuous functions, real anal. exchange 31 (2006), 409–424. [8] a. edalat and a. lieutier, domain theory and differential calculus (functions of one variable), mathematical structures in computer science 14 (2004), 771–802. [9] a. edalat, a. lieutier, and d. pattinson, a computational model for multi-variable differential calculus, proceedings of fossacs 2005, 26 pages. [10] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m.w. mislove and d.s. scott, continuous lattices and domains, cambridge university press, 2003. [11] s. kempisty, sur les fonctions quasicontinues, fund. math. 19 (1932), 184–197. [12] j. lawson, encounters between topology and domain theory, domains and processes, kluwer academic publishers, netherlands, 2001, 1–32. [13] k. martin, the informatic derivative at a compact element, proc. fossacs02, springer lncs 2303 (2002), 310–325. [14] k. martin and j. ouaknine, informatic vs. classical differentation on the real line, electronic notes in theor. comp. sci. 73 (2003), 8 pages. url: http://www.elsevier.nl/locate/entcs/volume73.html [15] m. matejdes, sur les sélecteurs des multifonction, math. slovaca 37 (1987), 111–124. [16] w. miller and e. akin, invariant measure for set-valued dynamical systems, trans. amer. math. soc. 351:3 (1999) 1203-1225. [17] t. neubrunn, quasi-continuity, real anal. exchange 14 (1988/89), 259–306. [18] h.l. royden, real analysis, macmillan, new york, 1965. [19] s. samborski, a new function space and extension of partial differential operators in it, preprint. [20] s. samborski, expansions of differential operators and nonsmooth solutions of differential equations, cybern. syst. anal. 38 (3) (2002), 453–466. quasicontinuous functions 33 rodica cazacu (cazacu2@math.lsu.edu) department of mathematics, university of arkansas-ft. smith, ft. smith, ar 72913, usa jimmie lawson (lawson@math.lsu.edu) department of mathematics, louisiana state university, baton rouge, la 70803, usa varelagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 155-171 on separation axioms of uniform bundles and sheaves clara m. neira and januario varela 1 abstract. in the context of the theory of uniform bundles in the sense of j. dauns and k. h. hofmann, the topology of the fiber space of a uniform bundle depends on the assumption of upper semicontinuity of its defining set of pseudometrics when composed with local sections. in this paper we show that the additional hypothesis of lower semicontinuity of these functions secures that the fiber space of the uniform bundle is hausdorff, regular or completely regular provided that the base space has the corresponding separation axiom. similar results for the particular important case of sheaves of sets follow suit. 2000 ams classification: primary 55r65, secondary 54e15, 54b40 keywords: uniform bundle, sheaf of sets, lower semicontinuity, upper semicontinuity, separation axioms. 1. introduction in the general theory of uniform bundles laid down by k. h. hofmann and j. dauns [1], theorem i, page 23, the topology of the fiber space e of a uniform bundle (e,p,b), where b is the base space and p : e −→ b is a surjection, is constructed in terms of the data provided by a family of local selections (functions from a variable open subset s of b, that when composed with p give the identity of s) and a uniformity on e. this construction require some conditions to be carried out successfully, conditions that in the simplest case on bundles of metric spaces (when the uniformity is associated with a metric on e) amount to the requirement that the distance functions s 7−→ d(σ(s),τ(s)) : s −→ r are upper semicontinuous, where σ and τ are local selections. a basic question that has been pending in this theory is the significance of the additional hypothesis of lower semicontinuity and consequently of the more 1the first author acknowledges the financial support by the fundación mazda para el arte y la ciencia. 156 clara m. neira and januario varela stringent condition of continuity of these distance functions. the aim of this paper is to answer this question in a general manner. in the section of preliminaries we establish notation and recall an existence theorem of uniform bundles indispensable in the different examples presented in the paper. in the third section on separation axioms appear the main results of the paper: theorem 3.10 establishes that, under the ground assumption of lower semicontinuity of the distance functions, if the base space b is hausdorff, then the fiber space e is also hausdorff, theorems 3.15 and 3.19 contain similar results in the case of regularity and complete regularity respectively. the concept of sheaf of sets can be regarded as a particular case of the concept of uniform bundle by the simple expedient of considering each fiber equipped with the discrete metric. this allows us to examine the upper and lower semicontinuity of the distance functions. as in the general case, the first condition determines the topology of e, while the second has to do with the separation axioms of the fiber space. it is well known that the fiber space of the sheaf of germs of holomorphic functions is a tychonoff space. as an application of the results presented in this paper we obtain an alternative proof of this property. 2. preliminaries let e and b be topological spaces, p : e −→ b be a surjective function. for each t ∈ b, the set et = p −1(t) = {a ∈ e : p(a) = t} is called the fiber above t. note that e is the disjoint union ∐ t∈b et of the family (et)t∈b. a local selection for p is a function σ : q −→ e such that q ⊂ b is an open set and p◦σ is the identity map idq of q. a local section for p is by definition a continuous local selection. let γq(p) := {σ : q −→ e | q ⊂ b is open, σ is continuous and p◦σ = idq}. if q = b, then σ is a global section and we write γ(p) instead of γb(p). a set σ of local sections is called full if for every x ∈ e there exists σ ∈ σ such that σ(p(x)) = x. let e ∨ e := {(u,v) ∈ e × e : p(u) = p(v)}. the function d : e ∨ e −→ r is called a pseudometric for p provided that the restriction of d to et × et is a pseudometric on et, for each t ∈ b. a family (di)i∈i of pseudometrics for p is directed if for each pair i1, i2 ∈ i there exists i ∈ i such that di1(u,v) ≤ di(u,v) and di2(u,v) ≤ di(u,v), for every (u,v) ∈ e ∨ e. definition 2.1. let (di)i∈i be a directed family of pseudometrics for p and consider a local selection σ, i ∈ i and ǫ > 0. the set t iǫ (σ) = {u ∈ e : di(u,σ(p(u))) < ǫ} is called the ǫ-tube around σ with respect to di. definition 2.2. let e and b be topological spaces, p : e −→ b a surjective function and (di)i∈i a family of pseudometrics for p. the triple (e,p,b) is called a bundle of uniform spaces or, for short, a uniform bundle, provided that: 1. for every u ∈ e, every ǫ > 0 and every i ∈ i, there exists a local section σ such that u ∈ t iǫ (σ). separation axioms of bundles and sheaves 157 2. the tubes around all the local sections for p form a base for the topology of e. the space b is called the base space and the space e is called the bundle space. note that if (e,p,b) is a uniform bundle, then the function p is continuous and open. from definition 2.2 we obtain the upper semicontinuity of the distance functions s 7−→ d(σ(s),τ(s)) : q −→ r+, q being an open subset of domσ∩domτ and σ and τ arbitrary local sections for p. the following theorem on existence of uniform bundles will be used freely when required and without previous notice in the constructions outlined in the examples presented in this work. we sketch its proof since the reference [4], containing it, is not easily accesible. theorem 2.3. let b be a topological space and p : e −→ b be a surjective function. denote by σ a set of local selections for p and let (di)i∈i be a directed family of pseudometrics for p. we make the following assumptions: a) for every u ∈ e, every i ∈ i and every ǫ > 0, there exists α ∈ σ such that u ∈ t iǫ (α). b) for every i ∈ i and every (α,β) ∈ σ×σ the function s 7−→ di(α(s),β(s)) : domα ∩ domβ −→ r is upper semicontinuous. then e can be equipped with a topology s such that: 1) s has a base consisting of the sets of the form t iǫ (αq), where i ∈ i, ǫ > 0, α ∈ σ, q is an open subset of domα and αq denotes the restriction of α to q. 2) each α ∈ σ is a section. 3) (e,p,b) is a uniform bundle. proof. we first show that the collection of all sets t iǫ (αq), with the specifications given in conclusion 1), is a base for a topology s in e. given two such tubes t iǫ (αq) and t j δ (βp ) and u ∈ t i ǫ (αq) ∩ t j δ (βp ) let ρ = min{1 4 (ǫ − di(u,α(p(u)))), 1 4 (δ − dj(u,β(p(u))))}. let k ∈ i such that di(u1,u2) ≤ dk(u1,u2) and dj(u1,u2) ≤ dk(u1,u2) for every (u1,u2) ∈ e ∨ e and let ξ ∈ σ such that u ∈ t kρ (ξ) = {v ∈ e : dk(v,ξ(p(v))) < ρ}, then p(u) ∈ {s ∈ b : di(ξ(s),α(s)) < ǫi}, where ǫi = 1 2 (di(u,α(p(u))) + ǫ), in fact, since u ∈ t iǫ (αq) it follows that di(u,α(p(u))) < ǫ and thus di(u,α(p(u))) < 3 4 di(u,α(p(u))) + 1 4 ǫ. on the other hand, the relation u ∈ t kρ (ξ) implies di(u,ξ(p(u))) < 1 4 (ǫ − di(u,α(p(u)))) and therefore di(ξ(p(u)),α(p(u))) < ǫi. similarly, p(u) ∈ {s ∈ b : dj(ξ(s),β(s)) < δj} where δj = 1 2 (dj(u,β(p(u))) + δ). by the semicontinuity hypothesis the sets {s ∈ b : di(ξ(s),α(s)) < ǫi} and {s ∈ b : dj(ξ(s),β(s)) < δj} are open, it follows that s = p ∩ q ∩ {s ∈ b : di(ξ(s),α(s)) < ǫi} ∩ {s ∈ b : dj(ξ(s),β(s)) < δj} is a neighborhood of p(u) in the space b and t kρ (ξs) ⊂ t i ǫ (α), indeed, the relation 158 clara m. neira and januario varela v ∈ t kρ (ξs) implies di(v,ξ(p(v))) < ρ < 1 2 (ǫ − di(u,α(p(u)))), but p(v) ∈ s, then di(ξ(p(v)),α(p(v))) < 1 2 (di(u,α(p(u))) + ǫ), thus di(v,α(p(v))) < ǫ and therefore v int iǫ (αq). the inclusion t k ρ (ξs) ⊂ t j δ (βp ) is obtained in the same manner. 2) let α ∈ σ and t ∈ domα. a fundamental neighborhood of α(t) in e is of the form t iǫ (βq), where β ∈ σ, q ⊂ domα is open in b, ǫ > 0, i ∈ i and α(t) ∈ t iǫ (βq). by hypothesis b), the set α −1(t iǫ (βq)) = {s ∈ q : di(α(s),β(s)) < ǫ} is open in b, therefore α is a section. 3) the tubes around arbitrary local sections are open, in fact, let u ∈ e and let σ be a local section for p (not necessarily in σ) such that u ∈ t iǫ (σ). to prove that (e,p,b) is a uniform bundle, we must exhibit η > 0 and α ∈ σ such that u ∈ t iη (α) and t i η (αp ) ⊂ t i ǫ (σ) for some neighborhood p of p(u) in b. let η = 1 4 (ǫ−di(u,σ(p(u)))) and let α ∈ σ be such that u ∈ t i η (α). since u ∈ t iǫ (σ) we have di(u,σ(p(u))) < ǫ and thus di(u,σ(p(u))) < 3 4 di(u,σ(p(u))) + 1 4 ǫ. on the other hand, the relation u ∈ t iη (α) implies di(u,α(p(u))) < η = 1 4 (ǫ−di(u,σ(p(u)))), therefore di(σ(p(u)),α(p(u))) < 1 2 di(u,σ(p(u)))+ 1 2 ǫ, then p(u) ∈ σ−1(t iǫi(α)), where ǫi = 1 2 (di(u,σ(p(u))) + ǫ). since σ is continuous, σ−1(t iǫi(α)) is an open neighborhood p of p(u), then v ∈ t i η (αp ) implies p(v) ∈ p and hence di(α(p(v)),σ(p(v))) < 1 2 (di(u,σ(p(u))) + ǫ), we also have di(v,α(p(v))) < η < 1 2 (ǫ − di(u,σ(p(u)))), thus di(v,σ(p(v))) < ǫ, that is v ∈ t i ǫ (σ). � definition 2.4. let e and b be topological spaces and let p : e −→ b be a surjective function. a triple (e,p,b) is said to be a sheaf of sets provided that p is a local homeomorphism, that is, each point a ∈ e has an open neighborhood which is mapped homeomorphically by p onto an open subset of b. recall that if (e,p,b) is a sheaf of sets we have: (1) the ranges of the local sections for p form a base of the topology of e. (2) if two local sections intersect at a point t, they agree on an open neighborhood of t. (3) the discrete metric d : e ∨ e −→ r defined by d(m,n) = { 0 if m = n 1 if m 6= n is in particular a pseudometric for p, and it can be seen that the sheaf (e,p,b), with the family of pseudometrics reduced to the single discrete pseudometric, becomes a uniform bundle, indeed: i. consider m ∈ e, ǫ > 0 and an open neighborhood m of m in e such that p ↾m is a homeomorphism from m onto an open separation axioms of bundles and sheaves 159 set of b. it is clear that (p ↾m ) −1 is a local section such that m ∈ tǫ((p ↾m ) −1) = {n ∈ e : d(n,(p ↾m ) −1(p(n))) < ǫ} because m = (p ↾m ) −1(p(m)). ii. if σ : q −→ e is a local section for p and ǫ > 0, then tǫ(σ) = {m ∈ e : d(m,σ(p(m))) < ǫ} is the range of σ provided that ǫ ≤ 1, but tǫ(σ) = p −1(q) if ǫ > 1. hence the tubes around all the local sections for p form a base for the topology of e. conversely, if (e,p,b) is a uniform bundle with the directed family of pseudometrics (di)i∈i generating the discrete uniform structure on e, that is, if the diagonal ∆e = {(m,m) : m ∈ e} belongs to the uniform structure, then there exist j ∈ i and ǫ > 0 such that ∆e is precisely the set u j ǫ = {(m,n) : dj(m,n) < ǫ}. it is apparent that for each local section σ for p, we have t jǫ (σ) = ranσ. given m ∈ e, since (e,p,b) is a uniform bundle there exists a local section σ for p such that m ∈ t jǫ (σ), then σ(p(m)) = m, it follows that t jǫ (σ) is an open neighborhood of m and that the restriction q of p to t j ǫ (σ) is a homeomorphism from t jǫ (σ) onto the domain of σ whose inverse is σ. then p is a local homeomorphism and thus (e,p,b) is a sheaf of sets. in particular, if the family generating the uniformity for p reduces to the one pseudometric whose restriction to each fiber is the discrete metric, then (e,p,b) is a sheaf of sets. the following example shows that the requirement that each fiber et of a uniform bundle (e,p,b) has the discrete topology does not guarantee it to be a sheaf of sets. example 2.5. let b = r with the usual topology, let e be the subset of the euclidean plane defined by e = {(x,y) : y = x or y = −x} and let p : e −→ b be the map such that p(x,y) = x. consider the family of pseudometrics for p reduced to the pseudometric d defined on each fiber by d((x,y1),(x,y2)) = |y1 − y2|. let σ = {σ1, σ2} be the full set of global selections for p defined by σ1(x) = (x,x) and σ2(x) = (x,−x). the function ϕ : b −→ r defined by ϕ(x) = d(σ1(x),σ2(x)) is upper semicontinuous, thus the tubes tǫ(σ), where ǫ > 0 and σ is the restriction of any one of the elements of σ to an open set of b, form a base for a topology on e that gives to the triple (e,p,b) the structure of a uniform bundle in which σ1 and σ2 are sections. each fiber ex is discrete, but (e,p,b) is not a sheaf of sets since σ1(0) = σ2(0) but for each open interval j containing 0, σ1(x) 6= σ2(x) if x ∈ j and x 6= 0. 3. separation axioms the proofs of the next two elementary lemmas are straighforward and are omitted. 160 clara m. neira and januario varela lemma 3.1. let e, b be topological spaces and p : e −→ b be a continuous map. assume that b is a t0 (resp. t1) space and that for each t ∈ b the subspace p−1(t) is t0 (resp. t1), then e is also a t0 (resp. t1) space. lemma 3.2. let e, b be topological spaces, p : e −→ b be a continuous function and σ : b −→ e be a global section for p, then σ is an embedding. if in addition e is supposed to be a t2 space, then σ is a closed embedding. proposition 3.3. let (e,p,b) be a uniform bundle whose uniformity is given by the directed family (di)i∈i of pseudometrics. if the space e is t0, then the space b is also t0. proof. let t and t′ be two different points of b. choose a point u on the fiber et and i ∈ i. given ǫ > 0, there exists a local section σ such that u ∈ t i ǫ (σ). if t′ /∈ domσ, then domσ is a neighborhood of t which does not contain t′, but if t′ ∈ domσ, then σ(t) and σ(t′) are different points of e and thus there exists an open set m ⊂ e containing only one of the two points either σ(t) or σ(t′), but not both, then σ−1(m) is an open subset of b containing one of the points, either t or t′ but not both. it follows that b is a t0 space. � the next proposition is a direct consequence of lemma 3.1. proposition 3.4. let (e,p,b) be a uniform bundle. if the fiber et is a t0 space for each t ∈ b and the space b is t0, then the space e is also t0. since the fibers in a bundle of metric spaces (that is, a bundle whose uniformity is given by a metric), particularly in a sheaf of sets, are hausdorff spaces, we have the following corollary. corollary 3.5. let (e,p,b) be a bundle of metric spaces (resp. a sheaf of sets). the space b is t0 if and only if e is t0. in the next two propositions we examine how the property of being t1 is inherited from e to b and vice versa. proposition 3.6. let (e,p,b) be a uniform bundle whose uniformity is given by the directed family (di)i∈i of pseudometrics. if the space e is t1, then the space b is also t1. proof. let t and t′ be two different points of b. choose a point u on the fiber et above t and take i ∈ i. for ǫ > 0, let σ be a local section such that u ∈ t iǫ (σ). if t and t ′ belong to σ−1(t iǫ (σ)) we have that σ(t) and σ(t ′) are two different points of e and thus there exists an open neighborhood v of σ(t) with σ(t′) /∈ v and there exists an open neighborhood w of σ(t′) with σ(t) /∈ w . therefore σ−1(v ) is an open neighborhood of t such that t′ /∈ σ−1(v ) and σ−1(w) is an open neighborhood of t′ such that t /∈ σ−1(w). if t′ /∈ σ−1(t iǫ (σ)) we choose a point v ∈ et′ and a local section τ such that v ∈ t i ǫ (τ). if t and t′ belong to τ−1(t iǫ (τ)) we are in the previous case, but if t does not belong to τ−1(t iǫ (τ)), then σ −1(t iǫ (σ)) is an open neighborhood of t which does not contain t′ and τ−1(t iǫ (τ)) is an open neighborhood of t ′ that does not contain t. then b is a t1 space. � separation axioms of bundles and sheaves 161 conversely, the following property follows directly from lemma 3.1. proposition 3.7. let (e,p,b) be a uniform bundle. if the fiber et is a t1 space for each t ∈ b and if the space b is t1, then the space e is t1. in the context of bundles of metric spaces (resp. sheaves of sets) we have the following result. corollary 3.8. let (e,p,b) be a bundle of metric spaces (resp. a sheaf of sets). the space b is t1 if and only if e is t1. in the next two statements it is shown how the property of being a hausdorff space is transferred from the bundle space to the base space and vice versa. proposition 3.9. let (e,p,b) be a uniform bundle. if e is a hausdorff space and if there exists a global section for p, then b is also a hausdorff space. proof. it follows from lemma 3.2. � theorem 3.10. let (e,p,b) be a uniform bundle, (di)i∈i be a directed family of pseudometrics inducing the uniformity for p and suppose that the fiber et is hausdorff for every t ∈ b. if b is a hausdorff space and if the functions t 7−→ di(σ(t),τ(t)) : q −→ r are continuous for each open set q ⊂ b, each i ∈ i and each pair σ, τ ∈ γq(p), then e is also a hausdorff space. proof. let u, v ∈ e such that u 6= v. if p(u) 6= p(v), then there exist disjoint open neighborhoods q and r of p(u) and p(v) respectively. since p is a continuous function, p−1(q) is an open neighborhood of u, p−1(r) is an open neighborhood of v and p−1(q) ∩ p−1(r) = ∅, but if p(u) = p(v) = t one can find i ∈ i such that di(u,v) > 0. for ǫ, δ > 0 such that ǫ + δ < di(u,v) 2 , there exist two sections σ and τ with domain p , where p is an open neighborhood of p(u), such that u ∈ t iǫ (σ) and v ∈ t i δ (τ). it then follows that 2(ǫ + δ) < di(u,v) ≤ di(u,σ(t)) + di(σ(t),v) ≤ di(u,σ(t)) + di(σ(t),τ(t)) + di(τ(t),v) < ǫ + δ + di(σ(t),τ(t)). therefore di(σ(t),τ(t)) > ǫ + δ, and since the function ϕ : q −→ r, s 7−→ di(σ(s),τ(s)) is continuous, there exists an open neighborhood s of t such that ϕ(s) > ǫ + δ, for each s ∈ s. let σ′ := σ ↾s and τ ′ := τ ↾s, one has that u ∈ t iǫ (σ ′), v ∈ t iδ (τ ′) and these tubes are disjoint because if z ∈ t iǫ (σ ′) ∩ t iδ (τ ′), then ǫ + δ < di(σ ′(p(z),τ′(p(z)))) ≤ di(σ ′(p(z)),z) + di(z,τ ′(p(z))) < ǫ + δ, which is a contradiction. � in the special case of sheaves of sets we have: corollary 3.11. let (e,p,b) a sheaf of sets and assume that b is a hausdorff space. the space e is hausdorff if and only if the functions t 7−→ d(σ(t),τ(t)) : q −→ r are continuous for each open subset q of b and each σ, τ ∈ γq(p), d being the discrete metric on each fiber. 162 clara m. neira and januario varela proof. it remains to prove that if e is a hausdorff space, then the functions t 7−→ d(σ(t),τ(t)) : q −→ r are lower semicontinuous. to this end, consider a > 0 such that d(σ(t),τ(t)) > a. it follows that a < 1 and σ(t) 6= τ(t). since (e,p,b) is a sheaf of sets and e is hausdorff there exist an open neighborhood r of t and local sections σ1 and τ1 such that σ(r) = σ1(r) and τ(r) = τ1(r) for each r ∈ r and ranσ1 ∩ ranτ1 = ∅. then d(σ(r),τ(r)) = 1 > a for each r ∈ r. � the following example shows that, in the general case of uniform bundles, the continuity hypothesis made on the functions t 7−→ d(σ(t),τ(t)), where σ and τ are local sections defined in an open set q ⊂ b, is a sufficient but not a necessary condition for e to be a hausdorff space. example 3.12. let b = r with the usual topology, let e be the subset of the euclidean plane defined by e = {(x,y) : x 6= 0 and (y = 1 2 or y = 1)} ∪ {(0, 1 4 ),(0, 5 4 )} and let p : e −→ b be the map such that p(x,y) = x. consider the family of pseudometrics for p reduced to the pseudometric d defined on each fiber by d((x,y1),(x,y2)) = |y1 − y2|. let σ = {σ1, σ2} be the full set of global selections for p defined by σ1(x) = { (x,1) if x 6= 0 (0, 5 4 ) if x = 0 σ2(x) = { (x, 1 2 ) if x 6= 0 (0, 1 4 ) if x = 0. the function ϕ : b −→ r defined by ϕ(t) = d(σ1(t),σ2(t)) is upper semicontinuous. in fact ϕ(t) = { 1 2 if t 6= 0 1 if t = 0 thus the tubes tǫ(σ), where ǫ > 0 and σ is the restriction of any one of the elements of σ to an open set of b, form a base for a topology on e that gives to the triple (e,p,b) the structure of a uniform bundle, actually of a sheaf of sets, in which σ1 and σ2 are sections. although the space e is hausdorff ϕ fails to be continuous. it is interesting to remark that the chosen metric is not the discrete metric despite the topology of the fibers being the discrete one. if we had constructed the bundle by means of the discrete metric on the fibers, we had obtained that the function ϕ would be constant (equal to 1) and consequently continuous. in the following example the space b is hausdorff while the space e is not. example 3.13. let b = r with the usual topology, let e be the subset of the euclidean plane e = {(x,0) : x < 0} ∪ {(x,y) : x ≥ 0 and y = ±1} and let p : e −→ b be the function defined by p((x,y)) = x. separation axioms of bundles and sheaves 163 consider the family of pseudometrics for p with the only pseudometric d that is the discrete metric on each fiber. let σ = {σ1, σ2} be the set of global selections for p given by σ1(x) = { (x,0) if x < 0 (x,1) if x ≥ 0 σ2(x) = { (x,0) if x < 0 (x,−1) if x ≥ 0. so we have that if (x,y) ∈ e, then either (x,y) = σ1(x) or (x,y) = σ2(x). the function ϕ : b −→ r, t 7−→ d(σ1(t),σ2(t)) is upper semicontinuous, in fact, ϕ(t) = { 0 if t < 0 1 if t ≥ 0. therefore the tubes tǫ(σ), where ǫ > 0 and σ is a restriction of an element of σ to an open set of b, form a base for a topology on e such that (e,p,b) is a uniform bundle, actually a sheaf of sets, and σ is a set of sections for p. the points (0,1) and (0,−1) of the space e can not be separated by disjoint open sets because every open set in e containing one of these points contains the set {(x,0) : ξ < x < 0} for some ξ < 0; therefore e is not a hausdorff space. now we recall that a topological space x is said to be regular if for every closed subset k of x and every x ∈ x r k, there are open subsets v and w of x such that k ⊂ v , x ∈ w and v ∩ w = ∅. equivalently, x is regular if and only if for every open subset a of x and every x ∈ a there is an open neighborhood v of x such that v ⊂ a. the following proposition is a direct consequence of lemma 3.2. proposition 3.14. let (e,p,b) be a uniform bundle. if e is a regular space and if there exists a global section for p, then b is also a regular space. the converse of the above proposition is not so trivial and gives rise to following result. theorem 3.15. let (e,p,b) be a uniform bundle and let (di)i∈i be a directed family of pseudometrics inducing the uniformity for p. if b is a regular space and the functions t 7−→ di(σ(t),τ(t)) : q −→ r are continuous for each open set q ⊂ b, each i ∈ i and each pair σ, τ ∈ γq(p), then e is also a regular space. proof. let m be an open subset of e and u ∈ m. there exist a local section σ for p, i ∈ i and ǫ > 0 such that u ∈ t iǫ (σ) and t i ǫ (σ) ⊂ m. let r := domσ. since r is an open subset of b containing p(u) and b is by hypothesis a regular space, there exists an open neighborhood s ⊂ b of p(u) such that s ⊂ r. (1) 164 clara m. neira and januario varela let δ ∈ r be such that δ > 0 and di(u,σ(p(u))) < δ < ǫ. if σ ′ := σ ↾s, then t iδ (σ ′) is an open neighborhood of u. to show that t δi(σ′) ⊂ t i ǫ (σ) we give the following indirect argument: suppose that z /∈ t iǫ (σ), then either di(z,σ(p(z))) ≥ ǫ or p(z) /∈ r. if di(z,σ(p(z))) ≥ ǫ we choose ξ > 0 with ξ ≤ di(z,σ(p(z))) − δ 2 and a local section τ such that z ∈ t iξ (τ). then di(z,σ(p(z))) ≤ di(z,τ(p(z))) + di(τ(p(z)),σ(p(z))) < ξ + di(τ(p(z)),σ(p(z))). therefore di(τ(p(z)),σ(p(z))) > di(z,σ(p(z))) − ξ ≥ di(z,σ(p(z))) − di(z,σ(p(z))) − δ 2 = di(z,σ(p(z))) + δ 2 = di(z,σ(p(z))) − δ 2 + δ ≥ δ + ξ. on the other hand, by assumption the function t 7−→ di(σ(t),τ(t)) : domσ ∩ domτ −→ r is continuous, thus there exists an open neighborhood p of p(z) such that if s ∈ p , then di(τ(s),σ(s)) > δ + ξ. (2) let τ′ = τ ↾p . the tubes t i ξ (τ ′) and t iδ (σ) are disjoint since y ∈ t i ξ (τ ′)∩t iδ (σ) implies p(y) ∈ domτ′ = p and di(τ(p(y)),σ(p(y))) = di(τ ′(p(y)),σ(p(y))) ≤ di(τ ′(p(y)),y) + di(y,σ(p(y))) < ξ + δ that contradicts (2). taking into account that t iξ (τ ′) is a neighborhood of z, it follows that z /∈ t i δ (σ′). if p(z) /∈ r, then, from (1), p(z) /∈ s. using again the regularity of b, we find two disjoint open sets q1 and q2 in b such that p(z) ∈ q1 and s ⊂ q2. let ρ be a local section with domρ ⊂ q1 such that z ∈ t iδ (ρ). since domρ ∩ domσ ′ = ∅, then t iδ (ρ) ∩ t i δ (σ ′) = ∅ and consequently z /∈ t i δ (σ′). this proves the theorem. � if there are two sections σ and τ in γq(p) and i ∈ i such that ϕστ : q −→ r, t 7−→ di(σ(t),τ(t)) is not continuous, then the space e could fail to be regular even if b is regular. in example 3.13 we have that b = r is a regular space, k = {(x,1) : x ≥ 0} being the complement of t 1 2 (σ2) is a closed subset of e, (0,−1) /∈ k and if v and w are open subsets of e such that k ⊂ v and (0,−1) ∈ w , then there exist ǫ, ξ, δ > 0 and sections σ, τ which are defined in the interval (−δ,δ) such that tǫ(σ) ⊂ v and tξ(τ) ⊂ w . separation axioms of bundles and sheaves 165 then the points (x,0) with −δ < x < 0 belong to v ∩ w and therefore v and w are not disjoints. in the particular case of sheaves of sets we have the following result. corollary 3.16. let (e,p,b) be a sheaf of sets and suppose that b is a regular space. the space e is a regular space if and only if the functions t 7−→ d(σ(t),τ(t)) : q −→ r are continuous, for each open set q ⊂ b and each pair σ, τ ∈ γq(p), d being the discrete metric on each fiber. proof. it remains to prove that if e is a regular space the functions t 7−→ d(σ(t),τ(t)) : q −→ r are lower semicontinuous, for each open set q ⊂ b and each pair σ, τ ∈ γq(p). to this end, let t ∈ b, a > 0 and suppose that d(σ(t),τ(t)) > a, then a < 1 and σ(t) 6= τ(t). since ranσ is an open neighborhood of σ(t) there exist an open neighborhood r of t and local sections σ1 and τ1 such that σ(r) = σ1(r) for each r ∈ r, τ(r) = τ1(r) for each r ∈ r, ranσ1 ⊂ ranσ and ranτ1 ∩ ranσ1 = ∅, the last identity holds since ranσ1 is a closed set and τ(t) /∈ ranσ implies τ(t) /∈ ranσ1. then d(σ(r),τ(r)) = 1 > a, for each r ∈ r. � recall that a topological space x is completely regular if for every closed subset k of x and every x0 ∈ xrk there is a continuous function f : x −→ r such that f(k) = 0 and f(x0) = 1. from lemma 3.2 it follows: proposition 3.17. let (e,p,b) be a uniform bundle. if e is completely regular and there exists a global section for p, then b is also completely regular. the following result plays a crucial role in establishing the upcoming theorem 3.19 on complete regularity. lemma 3.18. let (e,p,b) be a uniform bundle and (di)i∈i a family of pseudometrics that induces the uniformity of e. let i ∈ i, q be an open subset of b and σ ∈ γq(p) be a fixed local section for p. the function ϕ : q −→ r given by ϕ(t) = di(σ(t),τ(t)) is continuous for each τ ∈ γq(p) if and only if the function ψ : p−1(q) −→ r, defined by ψ(x) = di(x,σ(p(x))) is continuous. proof. let i ∈ i, q be an open set of b, σ ∈ γq(p) be a fixed local section for p and suppose that for each τ ∈ γq(p) the function ϕ is continuous. the function ψ is upper semicontinuous, indeed, if a > 0, then {x ∈ p−1(q) : di(x,σ(p(x)) < a} = t ia (σ) is an open set. to see that ψ is lower semicontinuous, let a ∈ r with a > 0 and let u ∈ p−1(q) such that di(u,σ(p(u)) > a. choose b ∈ r such that di(u,σ(p(u))) > b > a and let δ = b − a 2 . there exists an open neighborhood p of p(u) in b and a local section τ with domain p such that di(u,τ(p(u))) < δ. therefore di(u,σ(p(u))) ≤ di(u,τ(p(u))) + di(σ(p(u)),τ(p(u))) < δ + di(σ(p(u)),τ(p(u))), 166 clara m. neira and januario varela thus di(σ(p(u)),τ(p(u))) > b − δ = b + a 2 . let s be an open neighborhood of p(u) contained in q ∩ p such that if t ∈ s, then di(σ(t),τ(t)) > b + a 2 . the existence of such an s is secured by the lower semicontinuity of ϕ. for y ∈ t iδ (τ ↾s) one has di(y,σ(p(y))) ≥ di(σ(p(y)),τ(p(y))) − di(y,τ(p(y))) > b + a 2 − b − a 2 = a. then ψ is lower semicontinuous. conversely, suppose now that the function ψ is continuous and let τ be a section in γq(p). since (e,p,b) is a uniform bundle, ϕ is upper semicontinuous. it remains to show that ϕ is lower semicontinuous. let a > 0 and t0 ∈ q such that di(σ(t0),τ(t0)) > a. since the function u 7−→ di(u,σ(p(u))) : p−1(q) −→ r is continuous at τ(t0) there exists an open neighborhood m of τ(t0) in p −1(q) such that di(u,σ(p(u))) > a for each u ∈ m. since τ is continuous there exists an open neighborhood r of t0 such that τ(t) ∈ m if t ∈ r. therefore di(τ(t),σ(t)) > a for each t ∈ r. this establishes the lower semicontinuity of ϕ and proves the lemma. � theorem 3.19. let (e,p,b) be a uniform bundle whose uniformity is given by the directed family (di)i∈i of pseudometrics. if b is a completely regular space and if the functions ϕ : q −→ r defined by ϕ(t) = di(σ(t),τ(t)) are continuous, for each open set q ⊂ b, each pair σ, τ ∈ γq(p) of local sections and each i ∈ i, then e is also a completely regular space. proof. let k be a closed subset of e and let z0 be a point of e such that z0 /∈ k. there exist ǫ > 0, i ∈ i and a local section σ such that z0 ∈ t i ǫ (σ) and t iǫ (σ) ⊂ e r k. let q := domσ and take a = ǫ ǫ − di(z0,σ(p(z0))) . since b rq is a closed set, p(z0) /∈ b rq and b is completely regular, there exists a continuous function f : b −→ [0,a] such that f(p(z0)) = a and f(b r q) = 0. consider the function g : [0,+∞[−→ [0,1] defined by g(t) =    ǫ − t ǫ if t < ǫ 0 if t ≥ ǫ and the function h : p−1(q) −→ r given by h(u) = di(u,σ(p(u))), whose continuity was established in lemma 3.18. define ζ : e −→ [0,1] by ζ(u) = { g(h(u))f(p(u)) when u ∈ p−1(q) 0 otherwise. the function ζ is continuous in p−1(q) ∪ (e r p−1(q))◦. it remains to show the continuity of ζ at the points of the boundary of p−1(q). let y ∈ p−1(q) r p−1(q). taking into account that f(p(y)) = 0, since f is separation axioms of bundles and sheaves 167 continuous, for a given δ > 0 there exists an open neighborhood p of p(y) such that |f(t)| < δ for each t ∈ p . moreover, from the continuity of p we have that p−1(p) is an open neighborhood of y. it remains to see that |ζ(w)| < δ for every w ∈ p−1(p). for such a w consider two cases: w /∈ p−1(q) and w ∈ p−1(q). if w /∈ p−1(q), then ζ(w) = 0 and if w ∈ p−1(q) and h(w) ≥ ǫ, then g(h(w)) = 0 and ζ(w) = 0, but if h(w) < ǫ, then g(h(w)) = ǫ − h(w) ǫ ≤ 1 and thus |ζ(w)| = |g(h(w))f(p(w))| ≤ |f(p(w))| < δ. it follows that ζ : e −→ [0,1] is continuous, ζ(z0) = 1 and ζ(k) = 0 since ζ(e r t i ǫ (σ)) = 0. hence e is also a completely regular space. � in the case of sheaves of sets we have the following corollary. corollary 3.20. let (e,p,b) be a sheaf of sets and suppose that b is a completely regular space. the space e is a completely regular space if and only if the functions t 7−→ d(σ(t),τ(t)) : q −→ r are continuous for each open set q ⊂ b and each pair σ, τ ∈ γq(p), d being the discrete metric on each fiber. proof. since every completely regular space is a regular space, the corollary follows from theorem 3.19 and corollary 3.16. � remark 3.21. from proposition 3.6 and proposition 3.17 follows that if e is a tychonoff space, that is, completely regular and t1, and if there exists a global section, then b is also a tychonoff space. from proposition 3.7 and theorem 3.19 follows that if et is a t1 space for each t ∈ b, if the functions t 7−→ di(σ(t),τ(t)) : q −→ r are continuous for each i ∈ i and each pair σ, τ ∈ γq(p) of local sections and if b is a tychonoff space, then e is also a tychonoff space. recall that a topological space x is normal if for each pair k, l of closed subsets in x there are disjoint open subsets v and w of x such that k ⊂ v and l ⊂ w . proposition 3.22. let (e,p,b) be a uniform bundle whose uniformity is given by the directed family (di)i∈i of pseudometrics. if e is a normal space and if there exists a global section for p, then b is a normal space. proof. let σ : b −→ e be a global section and let k and l be two disjoint closed subsets of b, then p−1(k) and p−1(l) are closed subsets of e without common points, thus there exist open and disjoint subsets v and w of e such that p−1(k) ⊂ v and p−1(l) ⊂ w . it follows that the sets σ−1(v ) and σ−1(w) are open and disjoint subsets of b, k ⊂ σ−1(v ) and l ⊂ σ−1(w). � remark 3.23. in the absence of a global section in the uniform bundle (e,p,b), as assumed in proposition 3.9, one may suppose that given two distinct points of the base space, there exists a local section whose domain contains them, the conclusion, that the base space satisfies the hausdorff axiom, still holds. in propositions 3.14 and 3.17 such hypothesis can be replaced by the assumption that given a point and a closed subset of b that does not contain the 168 clara m. neira and januario varela point, there exists a local section whose domain contains both the point and the closed subset, one can still conclude the regularity and completely regularity respectively. the arguments rest on the fact that a local section is a homeomorphism from its domain onto its range, as stated in lemma 3.2, and that the subspaces of a topological space inherit the properties of being hausdorff, regular or completely regular. regarding proposition 3.22, the authors do not know if a similar or somewhat weaker assumption could replace the condition of existence of a global section and still secures the normality of the base space. the normality of b by itself does not guarantee the normality of e. in example 3.13 the space b = r is a normal space, k = {(x,1) : x ≥ 0} and l = {(x,−1) : x ≥ 0} are closed subsets of e without common points, and if v and w are open subsets of e such that k ⊂ v and l ⊂ w , then there exist points (x,0) with x < 0 belonging to v ∩ w . the next example exibits a uniform bundle (e,p,b) whose uniformity for p is given by the directed family (di)i∈i of pseudometrics, such that b is a normal space, the functions ϕ : q −→ r given by ϕ(t) = di(σ(t),τ(t)) are continuous for each i ∈ i, each open set q ⊂ b and each pair σ, τ ∈ γq(p) of local section, but e fails to be a normal space. we cite first the following result of f. b. jones [2] required in the example. lemma 3.24. if x contains a dense set d and a closed, relatively discrete subspace s such that |s| ≥ 2|d|, then x is not normal. example 3.25. let e = {(x,y) ∈ r2 : y ≥ 0} be the moore plane. here the basic neighborhoods of each point (x,y) ∈ e with y > 0, are the intersections of e with the open disks in r2 that have the center at (x,y) and if (x,y) ∈ e and y = 0, its basic neighborhoods are the sets {(x,y)} ∪ a, where a is an open disk in the upper half plane, tangent to the x-axis at (x,y). the space e is completely regular (hence uniformizable [3], corollary 17, page 188, [5], theorem 38.2, page 256 and t1 [5], examples 14.5, page 93.) let (di)i∈i be the caliber of e, that is, the collection of all finite uniformly continuous pseudometrics of e. consider the topological space b = {t} and the map p : e −→ b defined by p(x,y) = t for each (x,y) ∈ e. every local section σ for p can be identified with the point σ(t) in e and the triple (e,p,b) is a uniform bundle. for each i ∈ i and each pair σ, τ ∈ γq(p), the map ϕ : q −→ r defined by ϕ(t) = di(σ(t),τ(t)) is continuous. on the other hand, from lemma 3.24, the space e is not normal, because if s = {(x,0) : x ∈ r} and d = {(x,y) ∈ e : x, y ∈ q}, then s turns out to be a closed, relatively discrete subspace of e, d is dense in e and |s| ≥ 2|d| on account of d being contable and |s| = c. remark 3.26. let (e,p,b) be a sheaf of sets and σ and τ be local sections for p defined in an open set q of b. the upper semicontinuity of the function ϕ : q −→ r given by ϕ(t) = d(σ(t),τ(t)), d being the discrete metric on each fiber, amounts to the assertion that the set {t ∈ q : σ(t) = τ(t)} is open and the additional hypothesis that ϕ is lower semicontinuous amounts to the separation axioms of bundles and sheaves 169 assertion that {t ∈ q : σ(t) 6= τ(t)} is also open. under the assumption that ϕ is continuous, if σ and τ agree at a point t they agree on the whole connected component of t in q. example 3.27. (the sheaf of germs of holomorphic functions.) let c be the field of complex numbers endowed with the usual topology, let q ⊂ c be an open set, f : q −→ c and z ∈ q. the function f is holomorphic (or regular) at the point z provided that f is complex differenciable in an open disk d(z,ǫ) ⊂ q, with center z and radius ǫ > 0. for every complex number z denote by az the set of all holomorphic functions at z. in az define the equivalence relation rz by f rz g if and only if f and g coincide in an open disk with center z. the class [f]z of f module rz is called the germ of the holomorphic function f at z. the set of germs of holomorphic functions at z, that is, the quotient set ez = az/rz is identified to the set of all sequences (an)n∈n of complex numbers such that lim sup |an| 1 n < ∞, indeed, every [f]z ∈ ez determines the sequence ( f(n)(z) n! ) n with that property, and conversely every such a sequence determines the class module rz of the holomorphic function defined by the power series ∑∞ n=0 an(w − z) n in the open disk with center z and radius 1 lim sup |an| 1 n . let ê = ∐ z∈c ez be the disjoint union of the family {ez : z ∈ c} and let p̂ : ê −→ c be the function defined by p̂(z, [f]z) = z and consider each fiber ez equipped with the discrete metric. for every holomorphic function f in an open set q define f̂ : q −→ ê by f̂(z) = (z, [f]z). the function f̂ is a local selection for p̂ and if σ̂ = {f̂ : f is holomorphic in some open set of c} and d denotes the pseudometric whose restriction to each fiber is the discrete metric, then, by theorem 2.3, the triple (ê, p̂,c) is a sheaf of sets and every element of σ̂ is a local section for p̂. actually, the set of all local sections of this sheaf coincides with the set σ̂, in fact, let σ : q −→ ê be a local section for p̂ and for each z ∈ q let fz be a holomorphic function at z such that [fz]z = σ(z). consider the map f : q −→ c defined by f(z) = fz(z). if z ∈ q and g is a holomorphic function at z, the relation fz rz g implies that fz and g coincide in an open disk with center z, in particular fz(z) = g(z), hence f is a well defined function. since fz is holomorphic at z, there exist 0 < ǫ < 1 and a power series ∑∞ n=0 an(w−z) n convergent in the disk d(z,ǫ) such that fz(w) =∑∞ n=0 an(w −z) n for every w ∈ d(z,ǫ). since σ is a continuous function, there exists 0 < δ < ǫ such that if w ∈ d(z,δ), then σ(w) = [fw]w ∈ tǫ(f̂z), that is, d([fw]w, [fz]w) < ǫ < 1, therefore d([fw]w, [fz]w) = 0, thus [fw]w = [fz]w and f(w) = fw(w) = fz(w) = ∑∞ n=0 an(w − z) n. it follows that f is holomorphic at z. from this argument it also follows that f and fz coincide in an open disk with center z, then [f]z = [fz]z and therefore f̂ = σ. suppose that f̂, ĝ ∈ σ̂, then the map ϕ : domf̂ ∩ domĝ −→ r, z 7−→ d([f]z, [g]z) is continuous. 170 clara m. neira and januario varela to obtain the lower semicontinuity of this map observe that [f]z 6= [g]z implies that there is an n such that f(n)(z) 6= g(n)(z), then there is an open disk s with center z such that f(n)(w) 6= g(n)(w) and therefore [f]w 6= [g]w for each w ∈ s. we conclude that z 7−→ d([f]z, [g]z) is also lower semicontinuous. corollaries 3.11, 3.16 and 3.20 guarantee that the space ê is hausdorff, regular and completely regular and consequently a tychonoff space as it is well known in the literature. the following example shows a sheaf of sets where local sections σ, τ can be found, such that the function t 7−→ d(σ(t),τ(t)) fails to be continuous. example 3.28. for every complex number z denote by cz the set of all continuous complex valued functions defined in some open set of the complex plane containing z. in cz define the equivalence relation rz by f rz g if and only if f and g coincide in an open disk with center z and denote by [f]z the class of f module rz. let ez = {[f]z : f is continuous at z}, ê = ∐ z∈c ez and p̂ : ê −→ c the function defined by p̂(z, [f]z) = z. each fiber êz is considered to be endowed with the discrete metric d. for every continuous function f in the open set q define f̂ : q −→ ê by f̂(z) = (z, [f]z). the function f̂ is a local selection for p̂ and if σ̂ = {f̂ : f is continuous in some open set of c}, then the triple (ê, p̂,c) is a uniform bundle, even more, it is a sheaf of sets in which every f̂ ∈ σ̂ is a local section. we claim that each local section of this sheaf belongs to σ̂, to this effect, let σ : q −→ ê be a local section for p̂ and for each z ∈ q, let fz be a continuous function defined in an open set containing z such that σ(z) = [fz]z. it is apparent that the map f : q −→ c defined by f(z) = fz(z) is a well defined function and once the continuity of f has been established, the relation σ = f̂ and the claim will follow. consider z ∈ q and ǫ > 0. taking into account that fz is continuous at z, that σ and f̂z are local sections and that σ(z) = f̂z(z), there exists δ > 0 such that fz(w) ∈ d(fz(z),ǫ) and σ(w) = f̂z(w) for every w ∈ d(z,δ). then [fw]w = [fz]w for each w ∈ d(z,δ), in particular fw(w) = fz(w) for each w ∈ d(z,δ), thus f(w) = fw(w) ∈ d(fz(z),ǫ) for each w ∈ d(z,δ) and therefore f is continuous. consider the continuous functions f, g : c −→ c defined by f(z) = { z if |z| ≤ 1 z |z| if |z| > 1 and g(z) = z. if |z| < 1, then [f]z = [g]z and if |z| ≥ 1, then [f]z 6= [g]z. thus d([f]z, [g]z) = { 0 if |z| < 1 1 if |z| ≥ 1 and the function z 7−→ d([f]z, [g]z) : c −→ r is not continuous. corollaries 3.11, 3.16 and 3.20 back up the assertion that the space ê is neither separation axioms of bundles and sheaves 171 hausdorff nor regular nor completely regular and then that it is not a tychonoff space either. references [1] j. dauns, k. h. hofmann, representations of rings by sections, mem. amer. math. soc. 83 (1968). [2] f. b. jones, concerning normal and completely normal spaces, bull. amer. math. soc. 43 (1937) 671-677. [3] j. l. kelley, general topology, d. van nostrand company, inc. , (canada, 1955). [4] j. varela, existence of uniform bundles, rev. colombiana mat. 18 (1984) 1-8. [5] s. willard, general topology, addison wesley, (1970). received june 2002 accepted january 2003 clara m. neira (cmneirau@unal.edu.co) dep. de matemáticas, universidad nacional de colombia, bogotá, colombia januario varela (jvarelab13@yahoo.com) dep. de matemáticas, universidad nacional de colombia, bogotá, colombia induagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2005 pp. 213-230 continuous representability of interval orders j. c. candeal, e. induráin, and m. zudaire1 abstract. in the framework of the analysis of orderings whose associated indifference relation is not necessarily transitive, we study the structure of an interval order, and its representability through a pair of continuous real-valued functions. inspired in recent characterizations of the representability of interval orders, we obtain new results concerning the existence of continuous real-valued representations. classical results are also restated in a unified framework. 2000 ams classification: 54f05, 06a06. keywords: orderings on a set. interval orders. numerical representations of orderings. continuous representations of orderings. 1. introduction dealing with different classes of orderings ≺ defined on a nonempty set x, the concept of an interval order was introduced by peter c. fishburn [17] in contexts of economic theory, in order to build models of preferences whose associated indifference may fail to be transitive. (see also fishburn [18], or bosi and isler [4]). a very complete and informative study of interval orders appears in chapter 6 of bridges and mehta [6]. an interval order ≺ defined on a set x is representable if there exists a pair of real-valued functions u, v : x −→ r such that x ≺ y ⇐⇒ v(x) < u(y) (x, y ∈ x). the question of finding a complete characterization of the representability of interval orders was solved by fishburn [20] (see also fishburn [21], theorem 5 on p. 135). a different characterization was obtained by doignon et al. [14]. furthermore, a new alternative solution using only one ordinal condition has been recently obtained by olóriz et al. [27]. some further characterizations have appeared in bosi et al. [3]. 1a substantial part of this work comes from the ph. d. of m. zudaire made under the advisement of j. c. candeal and e. induráin. 214 j. c. candeal, e. induráin and m. zudaire to deal with numerical representations of interval orders three main techniques have been used in the literature: i) the first technique (see e.g. bridges and mehta [6], pp. 88 and ff.), associates to each element of the set x where the interval order ≺ has been defined, two suitable subsets a(x) and b(x) of natural numbers in a way that x ≺ y ⇐⇒ b(x) ( a(y). ii) the second technique, used in bosi and isler [4], is based on measure theory. the functions u, v that represent the interval order are related to the measures of lower and upper contour sets, in some orderings related to ≺, of the elements of x. iii) the third technique, introduced in olóriz et al. [27], is based on the theory of functional equations. the idea is to construct a bivariate map f : x × x −→ r such that, for instance, x ≺ y ⇐⇒ f(x, y) > 0. in the case of an interval order such functions f should also verify a characteristic functional equation, namely the functional equation of separability: f(x, y) + f(y, z) = f(x, z) + f(y, y) (x, y, z ∈ x). paying attention to continuity we will be looking for representations (u, v) of an interval ordered structure (x, ≺) such that u and v are continuous (with respect to a given topology τ on x and the usual euclidean topology on r). in chateauneuf [11] a characterization of the continuous representability of an interval order was given for the particular case of a connected topological space x. the purpose of the present paper is twofold: our first task is to introduce new characterizations of the continuous representability of an interval order, comparing our results with previous ones existing in the literature (see chateauneuf [11], bridges and mehta [6]), and looking for a standard and unified notation. our second task is to adapt the main techniques that have been considered to get numerical representations of interval orders to obtain characterizations of the continuous representability. 2. previous concepts let x be a nonempty set. in what follows “≺” will denote an asymmetric binary relation defined on a nonempty set x. associated to ≺ we will also consider the binary relations “-” and ”∼”, respectively defined as x y ⇐⇒ ¬(y ≺ x) and x ∼ y ⇐⇒ x y , y x. the relation ≺ is usually called strict preference. the relation is said to be the weak preference, and ∼ is called the indifference, associated to ≺. the opposite ordering ≺op of ≺ is defined by x ≺op y ⇐⇒ y ≺ x. given x ∈ x, the sets l(x) = {y ∈ x : y ≺ x} and u(x) = {y ∈ x : x ≺ y} are called, respectively, the lower contour set and the upper contour set relative to ≺. continuous representability of interval orders 215 definition 2.1. the binary relation ≺ is said to be an interval order if (x ≺ y , a ≺ b) =⇒ either x ≺ b , or else a ≺ y (or both). an interval order ≺ is said to be a semiorder if in addition a ≺ b ≺ c =⇒ a ≺ d or else d ≺ c (a, b, c, d ∈ x). observe that because ≺ is asymmetric, if it is an interval order then it must be transitive. notice also that ≺ being an interval order, the associated relations and ∼ may fail to be transitive. an example is the relation ≺ defined on the real line r as x ≺ y ⇐⇒ x + 1 < y. however an interval order ≺ is always pseudotransitive: that is x ≺ y z ≺ t =⇒ x ≺ t (x, y, z, t ∈ x). it is straightforward to see that: for an asymmetric binary relation ≺ on x, pseudotransitivity is equivalent to the fact of ≺ being an interval order. moreover: an asymmetric binary relation ≺ defined on x is a semiorder if and only if it satisfies the condition of generalized pseudotransitivity, namely, for every x, y, z, t ∈ x the following three conditions hold true: 1. x ≺ y z ≺ t =⇒ x ≺ t, 2. x y ≺ z ≺ t =⇒ x ≺ t, 3. x ≺ y ≺ z t =⇒ x ≺ t. (see gensemer [24] for details). an interval order ≺ defined on x is said to be representable if there exist two real-valued functions u, v : x −→ r such that x ≺ y ⇐⇒ v(x) < u(y) (x, y ∈ x). since ≺ is asymmetric, this is equivalent to associate to each element x ∈ x a real-interval (that eventually may collapse to a single point), ix = [u(x), v(x)]. thus x ≺ y if and only if ix is located on the left of iy, and ix does not meet iy. this kind of “interval-representation” gave raise to the nomenclature of interval order. however not every interval order is representable (see e.g. bosi et al. [3] for details). a semiorder ≺ defined on x is said to be representable if there exist a real-valued function u : x −→ r and a non-negative real number α (called threshold) such that x ≺ y ⇐⇒ u(x) + α < u(y). observe that this is a particular case of representation of an interval order, in which v(x) can be defined as u(x) + α (x ∈ x). observe also that this is equivalent to associate to each element x ∈ x a real-interval ix = [u(x), u(x) + α]. in this case, all the intervals have the same length α. obviously all them will collapse to a single point if and only if α = 0. (see fishburn [20], gensemer [22, 23, 24, 25] or candeal et al. [10] for more details). 216 j. c. candeal, e. induráin and m. zudaire following fishburn [17, 19] we shall associate to an interval order ≺ two new binary relations, respectively denoted by ≺∗ and ≺∗∗ and defined by x ≺∗ y ⇐⇒ x ≺ z y for some z ∈ x (x, y ∈ x), and, similarly, x ≺∗∗ y ⇐⇒ x z ≺ y for some z ∈ x (x, y ∈ x). define the relation x -∗ y ⇐⇒ ¬(y ≺∗ x) and x -∗∗ y ⇐⇒ ¬(y ≺∗∗ x) (x, y ∈ x). then it is straightforward to see that x -∗ y ⇐⇒ (y ≺ z =⇒ x ≺ z (z ∈ x)) and similarly x -∗∗ y ⇐⇒ (z ≺ x =⇒ z ≺ y (z ∈ x)). observe also that in terms of contour sets, it follows that x -∗ y ⇐⇒ u(y) ⊆ u(x) and also x -∗∗ y ⇐⇒ l(x) ⊆ l(y) (x, y ∈ x). a preorder defined on a nonempty set x is a reflexive and transitive binary relation defined on x. if it is also complete (i.e., either x y or else y x for every x, y ∈ x), is said to be a total preorder. a total preorder is said to be representable if there exists a real-valued function, usually called utility function, u : x −→ r such that x y ⇐⇒ u(x) ≤ u(y). if x ≺ y ⇐⇒ ¬(y x) and x ∼ y ⇐⇒ x y x (x, y ∈ x), then a representable total preorder clearly corresponds to a representation of the weak preference associated to a semiorder ≺ for the special case in which the threshold α equals zero. an antisymmetric total preorder on a set x (i.e. x y x =⇒ x = y (x, y ∈ x)) is said to be a total order. let be a total preorder on a set x, and let x ≺ y (x, y ∈ x). we say that the pair (x, y) defines a jump if there is no z ∈ x such that x ≺ z ≺ y. a subset y ⊆ x is said to be cofinal if for every x ∈ x there exists y ∈ y such that x ≺ y. similarly y is said to be coinitial if for every x ∈ x there exists y ∈ y such that y ≺ x. an element z ∈ x is said to be minimal (respectively: maximal ) with respect to if z x (respectively: x z) for every x ∈ x. consider now the quotient space x/ ∼ of x through the equivalence relation ∼. the ordering is compatible with this quotient, so that x/ ∼ becomes a totally ordered set. the total preorder is said to be dedekind complete if in x/ ∼ every subset c ⊆ x/ ∼ that is bounded above with respect to ≺ has a supremum (i.e.: smallest upper bound) sup c in x/ ∼. when dealing with a total order defined on x we can endow x with the order topology whose subbasis is defined by the lower and upper contour sets relative to ≺. it can be proved that: lemma 2.2. i) (see gillman and jerison [26] p. 3 , birkhoff [2], p. 200, or else candeal and induráin [9]) : every total preorder has a dedekind complete extension without jumps that has neither minimal nor maximal elements. this extension is essentially unique. continuous representability of interval orders 217 ii) (see e.g. birkhoff [2], p. 243) : the order topology relative to a total order on a set x is connected if and only if is dedekind complete and has no jumps. remark 2.3. the extension corresponding to lemma 2.2 (i) may produce a set x̄ that is much bigger than the given set x. notice that if we start with a single x = {x}, with the trivial total ordering, then we arrive to an extension that is isotonic to the real line r. coming back to the study of interval orders and semiorders, it is well-known (see e.g. proposition 2.1 in bridges [5]) that: lemma 2.4. let ≺ be an asymmetric binary relation defined on a nonempty set x. then the following statements are equivalent: i) ≺ is an interval order, ii) -∗ is a total preorder, iii) -∗∗ is a total preorder. in addition, is transitive if and only if , -∗, and -∗∗ coincide. in what concerns semiorders among interval orders, suppose that an interval order ≺ has been defined on a set x, and define the following new binary relation, introduced in fishburn [18]: x ≺0 y ⇐⇒ x ≺∗ y or else x ≺∗∗ y (x, y ∈ x). in bosi and isler [4] it is proved the following fact: lemma 2.5. the interval order ≺ is actually a semiorder if and only if ≺0 is asymmetric. a key concept, used in olóriz et al. [27] to get a characterization of the representability of interval orders, is that of interval order separability (henceforward i.o.-separability). an interval order ≺ on a set x is said to be i.o.-separable if there exists a countable subset d ⊆ x such that for every x, y ∈ x with x ≺ y there exists an element d in d such that x ≺ d -∗∗ y. the nub result on the representability of interval orders is in order now: lemma 2.6 (olóriz et al. [27]). let x be a nonempty set endowed with an interval order ≺. then, the following statements are equivalent: i) ≺ is i.o.-separable, ii) there exists a bivariate map f : x × x −→ r such that x y ⇐⇒ f(x, y) ≥ 0 and f(x, y)+f(y, z) = f(x, z)+f(y, y) for every x, y, z ∈ x, iii) ≺ is representable. 3. continuous representations of interval orders let x be a nonempty set endowed with an interval order ≺ and a topology τ. now we consider the possibility of finding a representation (u, v) for ≺ such that both u and v are continuous when considering on x the given topology τ and on the real line r the usual euclidean topology. to do so, we need to introduce some previous concepts and results. 218 j. c. candeal, e. induráin and m. zudaire definition 3.1. let (x, τ) be a topological space endowed with an interval order ≺ . we say that ≺ is τ-continuous if for each a ∈ x the upper and lower contour sets l(a) and u(a) are τ-open sets. let ≺ be an interval-order, and ≺∗ and ≺∗∗ its associated relations as defined in section 2. the τ-continuity of ≺∗ (respectively: ≺∗∗) is defined similarly, now considering lower and upper contours relative to ≺∗ (respectively: ≺∗∗). definition 3.2. let (x, τ) a topological space endowed with an interval order ≺ . the topology τ is said to be natural for the interval order ≺ if ≺, ≺∗ and ≺∗∗ are all τ-continuous. observe that a topology τ is natural for the interval order ≺ if and only if it is finer than the topology θ, a subbasis of which is given by the family: {u(a) : a ∈ x} ∪ {l(b) : b ∈ x} ⋃ {u≺∗(c) : c ∈ x} ∪ {l≺∗(d) : d ∈ x} ⋃ {u≺∗∗(e) : e ∈ x} ∪ {l≺∗∗(f) : f ∈ x}. the existence of continuous representations for an interval order will lean, of course, on the continuity of ≺, ≺∗, ≺∗∗ with respect to the given topology τ. also, it will lean on some ordinal condition of separability or density. definition 3.3. let x be a nonempty set endowed with an interval order ≺ . we say that ≺ is i) strongly separable if there exists a countable subset d ⊆ x such that for every x, y ∈ x with x ≺ y, there exist a, b ∈ d such that x ≺ a b ≺ y, ii) full if for every x, y ∈ x with x ≺ y, there exist a, b ∈ x such that x ≺ a b ≺ y. (it is obvious that strongly separable implies full). the main well-known result on the continuous representability of an interval order ≺ on a topological space (x, τ) was introduced in chateauneuf [11] for the case in which (x, τ) is a connected topological space. theorem 3.4 (chateauneuf [11]). an interval order ≺ defined on a connected topological space (x, τ) admits a representation (u, v) with u and v continuous if and only if ≺ is strongly separable and in addition ≺∗ and ≺∗∗ are both τcontinuous. moreover, in that case there is also a continuous representation (u, v ) such that u is a representation for the total preorder -∗∗ and v is a representation for the total preorder -∗. we can now improve chateauneuf’s theorem with more equivalences, having in mind the concept of i.o.-separability, that is weaker than strong separability. theorem 3.5. let ≺ be an interval order defined on a connected topological space (x, τ). the following conditions are equivalent: continuous representability of interval orders 219 i) (x, ≺) has a continuous representation, ii) ≺ is strongly separable and τ is a natural topology, iii) (x, ≺) has a representation (u, v) such that u and v are τ-upper semicontinuous, and ≺∗, ≺∗∗ are τ-continuous, iv) (x, ≺) has a representation (u, v) such that u and v are τ-lower semicontinuous, and ≺∗, ≺∗∗ are τ-continuous, v) ≺ is strongly separable and ≺∗, ≺∗∗ are τ-continuous, vi) ≺ is i.o.-separable and full, and ≺∗, ≺∗∗ are τ-continuous, vii) the topology θ is separable (i.e: there exists a countable subset that meets every nonempty θ-open set) and coarser than τ, viii) ≺ is i.o-separable and τ is a natural topology, ix) ≺ is i.o.-separable, and ≺∗, ≺∗∗ are τ-continuous x) (x, ≺) has a continuous representation (u, v) such that u is a representation for ≺∗∗ and v is a representation for ≺∗, xi) there exists a bivariate map f : x × x −→ r that is continuous with respect to the product (τ × τ) topology on x × x and the euclidean topology on r such that x y ⇐⇒ f(x, y) ≥ 0 and f(x, y)+f(y, z) = f(x, z) + f(y, y) for every x, y, z ∈ x, xii) the topology θ is second countable (i.e.: there exists a countable basis for such topology) and coarser than τ. proof. the proof will follow the scheme : i) ⇐⇒ ii), i) ⇐⇒ iii), i) ⇐⇒ iv), ii) =⇒ v) =⇒ i), v) ⇐⇒ vi), vi) ⇐⇒ viii), viii) ⇐⇒ ix), viii) =⇒ x) =⇒ i), i) ⇐⇒ xi), vii) ⇐⇒ ii), xii) =⇒ vii) and finally i) =⇒ xii). i) =⇒ ii) see proposition 6.2.3, lemma 6.5.1 and lemma 6.5.3 in bridges and mehta [6]. ii) =⇒ i) see theorem 6.5.5 in bridges and mehta [6]. i) =⇒ iii) it is obvious that (x, ≺) has a representation (u, v) such that u and v are τupper semicontinuous. the τ-continuity of ≺∗, ≺∗∗ follows from lemma 6.5.1 in bridges and mehta [6]. iii) =⇒ i) by proposition 6.2.3 in bridges and mehta [6], ≺ is continuous. moreover, it is i.o.-separable by lemma 4, since it is representable. the proof of theorem 6.5.5 in bridges and mehta [6] gives now a construction of a continuous representation for the interval order ≺. 220 j. c. candeal, e. induráin and m. zudaire i) =⇒ iv) =⇒ i) this is analogous to i) =⇒ iii) =⇒ i). notice also that an interval order is representable if and only if the opposite interval order ≺op is representable. if (u, v) is a representation for ≺, then (−v, −u) is a representation for ≺op. finally, a map f : x −→ r is lower-semicontinuous if and only if −f is uppersemicontinuous. ii) =⇒ v) this is obvious. v) =⇒ i) see the proof of theorem 6.5.5 in bridges and mehta [6]. v) =⇒ vi) =⇒ v) it follows from the equivalence, proved in bosi et al. [3], that states that strongly separable is the same as i.o.-separable plus full. vi) =⇒ viii) notice that vi) ⇐⇒ v) ⇐⇒ ii) and obviously ii) =⇒ viii). viii) =⇒ vi) this is obvious. viii) =⇒ ix) this is immediate. ix) =⇒ viii) following the proof of theorem 6.5.5 in bridges and mehta [6] we first obtain a representation (u, v) for ≺ such that u and v are τ-upper semicontinuous. now proposition 6.2.3 in bridges and mehta [6] proves that ≺ is continuous. viii) =⇒ x) the proof of theorem 6.5.5 in bridges and mehta [6] furnishes not only a continuous representation (u, v) for ≺. it also states that u is a continuous representation for ≺∗∗ and v is a continuous representation for ≺∗. x) =⇒ i) this is obvious. i) =⇒ xi) let (u, v) be a continuous representation for ≺, and let f : x × x −→ r be defined as follows: f(x, y) = v(y) − u(x) (x, y ∈ x). f is continuous because u and v are. a final checking shows that x y ⇐⇒ f(x, y) ≥ 0 and f(x, y) + f(y, z) = f(x, z) + f(y, y) for every x, y, z ∈ x. continuous representability of interval orders 221 xi) =⇒ i) suppose f is given in the conditions of xi). following lemma 1 in olóriz et al. [27], fix an element x0 ∈ x and call u(x) = −f(x, x0) ; v(y) = f(y, y) + u(y) (x, y ∈ x). it is straightforward to see now that (u, v) is a continuous representation for ≺. vii) =⇒ ii) because θ is coarser than τ, we have that τ is a natural topology and ≺ is θconnected. the connectedness and separability of θ imply, following the proof of corollary 6.5.6 in bridges and mehta [6], that ≺ is strongly separable. thus we arrive to ii). ii) =⇒ vii) since τ is a natural topology, it is, by definition, finer than the topology θ. moreover, the τ-connectedness of x implies the connectedness of x in any topology coarser than τ. also, ≺ is by hypothesis strongly separable. therefore, as shown in bosi et al. [3], the following conditions are equivalent: 1. ≺ is strongly separable. 2. there exists a countable subset d ⊆ x such that for every x, y ∈ x with x ≺ y there exists d ∈ d such that x ≺∗ d ≺ y, 3. there exists a countable subset d ⊆ x such that for every x, y ∈ x with x ≺ y there exists d ∈ d such that x ≺ d ≺∗∗ y. let us prove now that θ is separable: to do so, we consider the subbasis {u(a) : a ∈ x} ∪ {l(b) : b ∈ x} ⋃ {u≺∗(c) : c ∈ x} ∪ {l≺∗(d) : d ∈ x} ⋃ {u≺∗∗(e) : e ∈ x} ∪ {l≺∗∗(f) : f ∈ x} for such topology θ so that our task consists in proving the existence of a countable subset c ⊆ x meeting any nonempty basic θ-open set. a basis for θ appears as the collection of all finite intersections of elements in a subbasis. also, since condition ii) is equivalent to condition x), it follows that the interval order ≺ and the total preorders -∗ and -∗∗ are representable. in particular, their corresponding order topologies are separable. (see candeal and induráin [7] for the case of a total preorder, and bosi et al. [3] for the case of an interval order). consequently, it is enough to check the following possibilities: case 1. u≺(a) ∩ u≺∗(b): if there exists z ∈ u≺(a) ∩ u≺∗(b), then a ≺ z and b ≺ ∗ z. take a countable subset d ⊆ x corresponding to the strong separability of ≺. take also a countable subset r ⊆ x corresponding to the cantor separability of the preorder ≺∗, that is, being x, y ∈ x with x ≺∗ y, there exists r ∈ r such that x ≺∗ r ≺∗ y. (the existence of such r follows from the connectedness of the order topology of the representable preorder ≺∗. for more details, consult 222 j. c. candeal, e. induráin and m. zudaire candeal and induráin [7].) the family {u≺(d) ∩ u≺∗(r) : d ∈ d , r ∈ r} is obviously countable. for any nonempty set s of this family, we choose s ∈ s. this new set of elements, say c, is also countable. now consider d ∈ d with a ≺∗ d ≺ z and r ∈ r such that b ≺∗ r ≺∗ z. it follows that z ∈ u≺(d)∩u≺∗(r). take s ∈ c ∩ u≺(d) ∩ u≺∗ (r). it follows easily that s ∈ u≺(a) ∩ u≺∗(b) and we are done. case 2. u≺(a) ∩ u≺∗∗(b): since u≺∗∗(x) = ⋃ {η∈x, x-η} u≺(η), this case follows from the separability of the order topology corresponding to ≺, a subbasis for which is the collection {u(a) : a ∈ x} ∪ {l(b) : b ∈ x}. case 3. u≺∗(a) ∩ u≺∗∗(b): if there exists z ∈ u∗≺(a) ∩ u≺∗∗(b), then a ≺ ∗ z and b ≺∗∗ z. take a countable subset r ⊆ x corresponding to the cantor separability of the preorder ≺∗. take also a countable subset s ⊆ x corresponding to the cantor separability of the preorder ≺∗∗. the family {u≺∗(r)∩u≺∗∗ (s) : r ∈ r , s ∈ s} is obviously countable. for any nonempty set p of this family, we choose p ∈ p . this new set of elements, say c, is also countable. now consider r ∈ r with a ≺∗ r ≺∗ z and s ∈ s such that b ≺∗∗ s ≺∗∗ z. it is plain that z ∈ u≺∗(r) ∩ u≺∗∗(s). take c ∈ c ∩ u≺∗(r) ∩ u≺∗∗(s). it follows easily that c ∈ u≺∗(a) ∩ u≺∗∗(b) and we are done. case 4. l≺(a) ∩ l≺∗(b): since l≺∗(x) = ⋃ {ξ∈x, ξ-x} l≺(ξ), this case follows from the separability of the order topology corresponding to ≺. case 5. l≺(a) ∩ l≺∗∗(b): this case is analogous to case 1. actually, if there exists z ∈ l≺(a) ∩ l≺∗∗(b), then z ≺ a and z ≺ ∗∗ b. take a countable subset d ⊆ x corresponding to the strong separability of ≺. take also a countable subset r ⊆ x corresponding to the cantor separability of the preorder ≺∗∗, that is, being x, y ∈ x with x ≺∗∗ y, there exists r ∈ r such that x ≺∗∗ r ≺∗∗ y. the family {l≺(d) ∩ l≺∗∗(r) : d ∈ d , r ∈ r} is obviously countable. for any nonempty set s of this family, we choose s ∈ s. this new set of elements, say c, is also countable. now consider d ∈ d with z ≺ d ≺∗∗ a and r ∈ r such that z ≺∗∗ r ≺∗∗ b. it follows that z ∈ l≺(d) ∩ l≺∗∗(r). take s ∈ c ∩ l≺(d) ∩ l≺∗∗(r). it follows easily that s ∈ l≺(a) ∩ l≺∗∗(b) and we are done. case 6. l≺∗(a) ∩ l≺∗∗(b): this case is analogous to case 3. case 7. u≺(a) ∩ l≺∗(b): continuous representability of interval orders 223 since l≺∗(x) = ⋃ {ξ∈x, ξ-x} l≺(ξ), this case follows from the separability of the order topology corresponding to ≺. case 8. u≺(a) ∩ l≺∗∗(b): if there exists z ∈ u≺(a) ∩ l≺∗∗(b), then a ≺ z and z ≺ ∗∗ b. take a countable subset d ⊆ x corresponding to the strong separability of ≺. take also a countable subset r ⊆ x corresponding to the cantor separability of the preorder ≺∗∗. the family {u≺(d) ∩ l≺∗∗(r) : d ∈ d , r ∈ r} is obviously countable. for any nonempty set s of this family, we choose s ∈ s. this new set of elements, say c, is also countable. now consider d ∈ d with a ≺∗ d ≺ z and r ∈ r such that z ≺∗∗ r ≺∗∗ b. it follows that z ∈ u≺(d) ∩ l≺∗∗(r). take s ∈ c ∩ u≺(d) ∩ l≺∗∗(r). it follows easily that s ∈ u≺(a) ∩ l≺∗∗(b). case 9. u≺∗(a) ∩ l≺∗∗(b): if there exists z ∈ u≺∗(a) ∩ l≺∗∗(b), then a ≺ ∗ z and z ≺∗∗ b. take a countable subset d ⊆ x corresponding to the cantor separability of the preorder ≺∗. take also a countable subset r ⊆ x corresponding to the cantor separability of the preorder ≺∗∗. the family {u≺∗(d) ∩ l≺∗∗(r) : d ∈ d , r ∈ r} is obviously countable. for any nonempty set s of this family, we choose s ∈ s. this new set of elements, say c, is also countable. now consider d ∈ d with a ≺∗ d ≺∗ z and r ∈ r such that z ≺∗∗ r ≺∗∗ b. it follows that z ∈ u≺∗(d) ∩ l≺∗∗(r). take s ∈ c ∩ u≺∗(d) ∩ l≺∗∗(r). it follows easily that s ∈ u≺∗(a) ∩ l≺∗∗(b). case 10. l≺(a) ∩ u≺∗(b): this case is analogous to case 8. case 11. l≺(a) ∩ u≺∗∗(b): this case is analogous to case 7. case 12. l≺∗(a) ∩ u≺∗∗(b): since u≺∗∗(x) = ⋃ {η∈x, x-η} u≺(η) and l≺∗(x) = ⋃ {ξ∈x, ξ-x} l≺(ξ), this case follows from the separability of the order topology corresponding to ≺. the union of the countable subsets used along the cases considered above is of course countable and meets any nonempty basic θ-open set. therefore θ is separable. xii) =⇒ vii) just notice that a second countable topology is always separable. (for aspects concerning general topology, consult dugundji [15]). i) =⇒ xii) 224 j. c. candeal, e. induráin and m. zudaire the topologies θ1, θ2 and θ3 whose respective subbasis are σ1 = {u(a) : a ∈ x} ∪ {l(b) : b ∈ x} σ2 = {u≺∗(c) : c ∈ x} ∪ {l≺∗(d) : d ∈ x} σ3 = {u≺∗∗(e) : e ∈ x} ∪ {l≺∗∗(f) : f ∈ x} are second countable because ≺, -∗ and -∗ are representable. (see bosi et al. [3] or candeal and induráin [7] for more details). then the topology θ whose subbasis is σ1 ∪ σ2 ∪ σ3 is also second countable. this concludes the proof. � remark 3.6. i) the version of chateauneuf’s theorem that appears on pp. 106-107 of bridges and mehta [6] corresponds to the equivalence i) ⇐⇒ v) but actually it is proved x) (a condition that is stronger than i) ). nothing is said about i.o.-separability, a condition less restrictive than strong separability, because such concept of interval order separability appeared later in the literature (in olóriz et al. [27]). ii) several intermediate steps in the proof have appeared in previous works, usually as necessary lemmata to introduce the main chateauneuf’s theorem. so, for instance, proposition 6.2.3 in bridges and mehta [6] shows that if a structure of interval order (x, ≺) endowed with a topology τ has a representation (u, v) such that u and v are τ-upper semicontinuous then ≺ is τ-continuous. also, lemma 6.5.1 and lemma 6.5.3 in the same work state that if (x, ≺) has a continuous representation and x is connected, then ≺∗, ≺∗∗ are τ-continuous and ≺ is strongly separable. moreover, lemma 6.5.4 in bridges and mehta [6] states that if ≺∗, ≺∗∗ are τ-continuous and ≺ is full, then ≺ is continuous. finally, in bosi et al. [3] it has been proved that strongly separable is the same as i.o.-separable plus full. iii) an immediate corollary follows. this is corollary 6.5.6 in bridges and mehta [6] and can be considered as an extension to the context of interval orders of the classical result (see eilenberg [16]) that states that a continuous total preorder defined on a connected and separable topological space has a continuous numerical representation by means of a utility function. in this new context of interval orders, from the equivalent condition vii) in the statement of theorem 3.5, it follows that: << an interval order ≺ on a connected and separable topological space (x, τ) has a continuous representation if and only if τ is a natural topology (in other words: if and only if ≺, ≺∗ and ≺∗∗ are all τcontinuous). >> iv) some method of proof of chateauneuf’s theorem, whose main ideas have been used to state some equivalences of theorem 3.5 above, reminds us the classical debreu’s open gap lemma (see debreu [12, 13] or continuous representability of interval orders 225 else bridges and mehta [6], ch. 3) that states the possibility of getting a continuous utility function representing a total preorder, if it is the case in which a utility function (continuous or not) is available. in the context of an interval order ≺ on a connected topological space (x, τ) such that ≺∗ and ≺∗∗ are continuous, a technique to get a continuous representation (u, v ) starts by taking a utility function v for the total preorder -∗. this utility function may or may not be continuous, but using debreu’s open gap lemma the continuity of v can be assumed without loss of generality. from v and following the proof of chateauneuf’s theorem that appears in bridges and mehta [6], p. 107, we can construct a real-valued function u that is upper semicontinuous and such that (u, v) represents ≺. from proposition 6.2.3 in bridges and mehta [6] it follows now that ≺ is continuous. then lemma 6.5.3 in bridges and mehta [6] states that ≺ is strongly separable, and finally the proof of theorem 6.5.5 in bridges and mehta [6] furnishes a continuous representation (u, v ) for ≺. a final glance to this process tells us that from just a utility funtion for -∗, even discontinuous, we are able to get a continuous representation (u, v ) for the interval order ≺. this is the kind of ideas underlying in the classical debreu’s open gap lemma. v) the equation f(x, y) + f(y, z) = f(x, z) + f(y, y) is known as the functional equation of separability because it corresponds to maps f defined in x × x such that f can be separated as the sum of two functions of only one variable each, that is f(x, y) = g(x)+h(y) ; g, h : x −→ r. (for more details, consult aczél and dhombres [1]). vi) in theorem 3.5 connectedness cannot be ruled out, nor even in the particular case of a total preorder on a set x on which we consider the order topology. for instance, it is not difficult to see that the order topology given by the lexicographic ordering on r×{0, 1} is separable, but this ordering is not representable. vii) similarly to eilenberg’s theorem, the classical debreu’s theorem on second countability (see debreu [12, 13] or else ch. 3 in bridges and mehta [6]) states that a continuous total preorder always admit a continuous utility function that represents it. looking for an extension of this result to the context of interval orders, we must observe that the equivalent condition xii) that appears in the statement of theorem 3.5 proves such a fact for the particular case of connected topological spaces. also, the continuity here is considered with respect to a natural topology. it is an open question to prove or disprove the existence of continuous representations for interval orders defined on a nonempty set x on which we consider either the topology θ, or more generally a natural topology, for which x is second countable. that is: the task would consist in proving the equivalence i) ⇐⇒ xii) in theorem 3.5 226 j. c. candeal, e. induráin and m. zudaire making no use of connectedness, or else disprove it by an adequate counterexample. viii) some steps in the proof of theorem 3.5 could be shortened if ≺ is a semiorder instead of just an interval order. for instance, if x is a nonempty set endowed with a semiorder ≺, even without assuming connectedness then, it is not difficult to prove that the strong separability of ≺ carries the topological separability of θ1, the topology on x whose subbasis is σ1 = {u(a) : a ∈ x} ∪ {l(b) : b ∈ x}. indeed, if x, y in x are such that x ≺ y and u≺(x) ∩ l≺(y) 6= ∅, we have that there exists z ∈ x such that x ≺ z ≺ y. let d ⊆ x be a countable subset that furnishes the strong separability of ≺ . then there exist d1, d2 ∈ d such that x ≺ d1 ≺ ∗∗ z ≺ d2 ≺ ∗∗ y. by definition of ≺∗∗, there exist a, b ∈ x such that x ≺ d1 a ≺ z ≺ d2 b ≺ y. by the generalized pseudotransitivity of the semiorder ≺, where in particular the fact “-≺≺” implies “≺”, it follows now that x ≺ d1 ≺ d2 b ≺ y =⇒ x ≺ d1 ≺ y, so that θ1 is separable. a remarkable question that we must consider now concerns the existence of continuous representations for interval orders in the general case (i.e.: not necessarily connected). in the case of a total preorder, the crucial debreu’s open gap theorem provides continuous representations once we have a utility representation. but, as far as we know, the validity or not of a similar result for interval orders is still an open problem. anyways, we can study some cases different from the connected case. definition 3.7. an interval order ≺ on a nonempty set x is said to be densein-itself if for every x, y ∈ x with x ≺ y there exists z ∈ x such that x ≺ z ≺ y. proposition 3.8. let x be a nonempty set endowed with an interval order ≺ and a natural topology τ for which x is connected. then it holds that either the interval order ≺ is dense-in-itself or there exist elements x, y, z ∈ x such that x ≺ y z x. in particular, if is a total preorder, ≺ is dense-in-itself. proof. actually, if x, y ∈ x are such that x ≺ y and there is no z ∈ x such that y z x then it must exist an element a ∈ x with x ≺ a ≺ y since otherwise the open and nonempty subsets l≺(y) and u≺(x) would give raise to a disconnection of the set x, in contradiction with the hypothesis of connectedness. in the particular case of a total preorder, the situation x ≺ y z x would imply x ≺ x, against the irreflexivity of the binary relation ≺. � an important fact that we can prove at this point, states that on suitable topologies, representable interval-orders that are dense-in-itself admit a continuous representation. continuous representability of interval orders 227 theorem 3.9. a dense-in-itself and representable interval order ≺ defined on a nonempty set x endowed with a natural topology τ has a continuous representation. proof. it suffices to prove the existence of a continuous representations when the topology considered on x is θ. let a, b ∈ x with a ≺∗ b. by definition of ≺∗ there is an element c ∈ x such that a ≺ c b. since ≺ is dense-in-itself, there also exists an element d ∈ x such that a ≺ d ≺ c b. thus a ≺ d ≺∗ b which implies a ≺∗ d ≺∗ b. therefore ≺∗ is also dense-in-itself. in a completely analogous way we can prove that ≺∗∗ is dense-in-itself, too. since ≺ is representable, the associated total preorders -∗ and -∗∗ are representable. (see bosi et al. [3]). since ≺∗ is dense-in-itself, standard techniques of construction of utility functions for total preorders allow us to consider a utility function v for ≺∗ with the additional property that (0, 1) ∩ q ⊆ v(x) ⊆ [0, 1], so that v(x) is dense in [0, 1] with respect to the euclidean topology on r. (see, e.g., birkhoff [2], p. 200 or else candeal and induráin [7, 8, 9]). the proof concludes as in bridges and mehta [6], p.107, proof of the continuity of the representation in chateauneuf’s theorem. � definition 3.10. an interval order ≺ on a nonempty set x endowed with a topology τ is said to be τ-locally non-satiated (see, e.g. bridges and mehta [6], pp. 33 and 93) if for every x ∈ x it holds that u≺(x) meets every τneighbourhood of x. if τ is a natural topology, ≺ is τ-locally non-satiated, and a, b ∈ x are such that a ≺ b, then since l≺(b) is a neighbourhood of a there must exist an element c ∈ u≺(a) ∩ l≺(b). in particular, ≺ is dense-in-itself. corollary 3.11. a representable and τ-locally non-satiated interval order ≺ defined on a nonempty set x endowed with a natural topology τ has a continuous representation. a deep analysis of the proof of theorem 6.5.5 in bridges and mehta [6] tell us that the key to finally get a continuous representation (u, v ) for an interval order ≺ defined on a nonempty set x which we shall consider endowed with a natural topology τ is nothing else but the fact that the associated total preorder ≺∗ is dense-in-itself (dually, it is also true that under the same conditions for x, when the associated total preorder ≺∗∗ is dense-in-itself, the interval order ≺ admit a continuous representation (u, v ) ). this fact implies important consequences: theorem 3.12. let x be a nonempty set endowed with an interval order ≺ and a natural topology τ. then if ≺ is strongly separable it admits a continuous representation (u, v ). proof. by the previous comments, it is enough to prove that ≺∗ is dense-initself. let x, y ∈ x be such that x ≺∗ y. it follows, by definition of ≺∗, that there exists an element z ∈ x such that x ≺ z y. since ≺ is strongly 228 j. c. candeal, e. induráin and m. zudaire separable, there exists now an element a ∈ x such that x ≺∗ a ≺ z y. therefore x ≺∗ a ≺∗ y, and we are done. � remark 3.13. i) it is an open problem to prove whether or not theorem 3.12 remains valid when the condition of strong separability is substituted for the weaker one of i.o.-separability. ii) the converse of theorem 3.12 is not true: a clear example is the set of natural numbers n endowed with the usual euclidean total ordering and the discrete topology. considered as a particular case of interval order, such ordering is i.o.-separable, but not strongly separable. and it is plain that it admits a continuous representation. in what concerns the techniques used to obtain continuous representations of structures of interval order, the first impression is that they are different from the main techniques used to just get representations (not necessarily continuous). let us explain why: the first technique, based on constructions using the auxiliar sets a(x), b(x), leans on numerical series. (see e.g. bridges and mehta [6], pp. 88 and ff. , or else bosi et al. [3]). however, it is well known that functions defined through series fail to be continuous in a countable set of points (for instance, if (qn) ∞ n=1 is an enumeration of the set q of rational numbers, the function f : r −→ r given by f(x) = ∑ {k∈n : qk≤x} 2−k fails to be continuous at each rational number). consequently, the first technique is so good to obtain representations of interval orders, but it is not so good to get continuous representations. the third technique, introduced in olóriz et al. [27] and based on the theory of functional equations presents similar problems than the first technique, because the typical constructions of the bivariate map f : x × x −→ r that satisfies x ≺ y ⇐⇒ f(x, y) > 0 and is a solution of the functional equation of separability f(x, y) + f(y, z) = f(x, z) + f(y, y) (x, y ∈ x), are also based on suitable numerical series that could lead to the discontinuity of f . fortunately, the second technique, based on measure theoretical constructions may lead to a continuous representation, as proved in bosi and isler [4]. on the other hand, it is clear that the techniques used to prove theorem 3.5 above, understood as a generalization of chateauneuf’s theorem (as proved e.g. in bridges and mehta [6], pp. 106-107) are different from the three techniques, just cited, used to get representations (continuous or not) of interval-orders. the key now is obtaining a continuous representation v for ≺∗ (making use of debreu’s open gap lemma, eventually) and then use v to get a continuous representation u for ≺∗∗ in such a way that (u, v) is a continuous representation for ≺. continuous representability of interval orders 229 acknowledgements. this work has been partially supported by the research project pb98-551 “estructuras ordenadas y aplicaciones”. ministerio de educación y cultura. españa. 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[27] e. olóriz, j. c. candeal and e. induráin, representability of interval orders, journal of economic theory 78 (1) (1998), 219-227. received may 2003 accepted august 2003 j.c. candeal (candeal@posta.unizar.es) departamento de análisis económico, facultad de ciencias económicas y empresariales, c/doctor cerrada 1-3, 50005 zaragoza, spain e. induráin (steiner@si.unavarra.es) departamento de matemática e informática, universidad pública de navarra, campus arrosad́ıa, edificio “las encinas”, 31006 pamplona (navarra), spain m. zudaire (mzudair2@pnte.cfnavarra.es) instituto de educación secundaria barañain, (gobierno de navarra), avda. central 3, 31010 barañáin (navarra), spain benkafadaragt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 1, 2005 pp. 87-100 a generalized coincidence point index n. m. benkafadar and m. c. benkara-mostefa 1 abstract. the paper is devoted to build for some pairs of continuous single-valued maps a coincidence point index. the class of pairs (f, g) satisfies the condition that f induces an epimorphism of the ∨ cech homology groups with compact supports and coefficients in the field of rational numbers q. using this concept one defines for a class of multi-valued mappings a fixed point degree. the main theorem states that if the general coincidence point index is different from {0}, then the pair (f, g) admits at least a coincidence point. the results may be considered as a generalization of the above eilenberg-montgomery theorems [12], they include also, known fixed-point and coincidence-point theorems for single-valued maps and multi-valued transformations. 2000 ams classification: 54c60, 54h25, 55m20. 58c06. keywords: fixed point, concidence point, index, degree, multi-valued mapping. 1. introduction let f, g : x −→ y be two continuous single valued maps of hausdorff topological spaces. the coincidence problem, which is a generalization of the fixed point problem, is concerned with conditions which guarantees the existence of a solution for the equation f(x) = g(x). a such point x ∈ x is called a coincidence point of the pair of maps (f,g). the study of this problem has been treated first in 1946 by eilenberg-montgomery [12]. note that the eilenbergmontgomery theorem is a natural generalization of the lefschetz fixed point theorem, it implies also, the fixed point theorems of kakutani [21] and wallace [30]. topological invariants for different classes of pairs of maps have been studied by many authors [9], [14], [15], [20], [22], [23], [27] and others. the purpose of this note is to describe a generalized coincidence point index for a new class 1the authors acknowledge the support of a.n.d.r.u., (contract no 03/06 code cu 19905) and m.e.r.s., (project no b*2501/04/04), laboratory m.m.e.r.e. 88 n. m. benkafadar and m. c. benkara-mostefa of pairs of continuous maps (f,g) which satisfy the condition that f induces a r-homomorphism [3], [4] for homology with compact carries. moreover, one gives several applications of the general coincidence point index in fixed point theory for multi-valued mappings. one uses the dold’s fundamental class around a compact of a finite euclidean space en [10], h denotes the ∨ cech homology functor with compact carries and coefficients in the field of rational numbers q, from the category top(2) of hausdorff topological pairs and continuous maps to the category lg of graded vector spaces over the set of rational numbers q and linear maps of degree zero [13], [18], [29]. 2. maps n-decomposing. let g1 and g2 be two additive abelian groups, τ : g1 −→ g2 be a homomorphism. definition 2.1 ([3]). a homomorphism τ is a called a r-homomorphism if τ admits a right-inverse homomorphism. the definition signifies, since τ : g1 −→ g2 is a r-homomorphism then there exists a homomorphism σ : g2 −→ g1 such that τ ◦σ = idg2, where idg2 is the automorphism identity on g2. the following properties are satisfied. proposition 2.2. a homomorphism τ : g1 −→ g2 is a r-homomorphism if and only if the following conditions are satisfied : (1) τ is an epimorphism; (2) g1 = kerτ ⊕g , where g is a subgroup of g1. proposition 2.3. if g1 and g2 are two modules over a field k and if τ : g1 −→ g2 is an epimorphism then τ is a r-homomorphism. proposition 2.4 ([3]). let τ1 : g1 −→ g2 and τ2 : g2 −→ g3 be two rhomomorphisms then their composition τ = τ2 ◦ τ1 : g1 −→ g3 is also a r-homomorphism. the notion of r-homomorphisms has been introduced by borsuk and kosinsk [3], [4]. let (x,a) and (y,b) be two objects of the category top(2) of hausdorff topological pairs and continuous maps and f : (x,a) −→ (y,b) be a morphism from the hausdorff pair (x,a) into an other hausdorff pair (y,b). let h be the ∨ cech homology functor with compact carries and coefficients in the field of rational numbers q, from the category top(2) of hausdorff topological pairs and continuous maps to the category lg of graded vector spaces over the set of rational numbers q and linear maps of degree zero [13], [18], [29]. a generalized coincidence point index 89 definition 2.5. a continuous single-valued map f : (x,a) −→ (y,b) is said to be n-decomposing in the rank n ≥ 0 on the hausdorff pair (y,b) if the homomorphism f∗ : hn(x,a) → hn(y,b) induced by f, is a r-homomorphism. the set of the right-inverse homomorphisms of f∗ on (y,b) will be denoted by ω(f∗;y,b). the following propositions and corollaries, prove that the class of n-decomposing maps is vast. definition 2.6 ([3]). a continuous single-valued map f : (x,a) −→ (y,b) is called a r-map if f admits a continuous right inverse. proposition 2.7. let f : (x,a) −→ (y,b) be a single-valued map which is a r-map, then f is n-decomposing on (y,b) for every rank n ≥ 0. corollary 2.8. a retraction r of a pair (x,a) onto (x′,a′) is n-decomposing on the retract (x′,a′) of (x,a). definition 2.9 ([3]). a continuous single-valued map f : (x,a) −→ (y,b) is said to be a h-map if there exists a continuous single-valued g : (y,b) −→ (x,a) such that their composition f◦g and the identity map id(y,b) : (y,b) −→ (y,b) are homotopic. proposition 2.10. if f : (x,a) −→ (y,b) is a h-map, then f is n-decomposing on (y,b) for every n ≥ 0. corollary 2.11. a lower retraction r : (x,a) −→ (x′,a′) is n-decomposing on each lower retract (x′,a′) of (x,a). proposition 2.12. let f : (x,a) −→ (y,b) be a continuous single-valued map. if there exists a continuous single-valued map g : (z,c) −→ (x,a) such that their composition f ◦ g is n-decomposing on (y,b), then f is also n-decomposing on (y,b). corollary 2.13. let f : (x,a) −→ (y,b) be a continuous single-valued map and (z,c) ⊆ (x,a). if the restriction of f on (z,c) is n-decomposing on (y,b), then f is also n-decomposing on (y,b). proposition 2.14. let f : (x,a) −→ (y,b) be a n-decomposing on (y,b) and g : (y,b) −→ (z,c) be a n-decomposing on (z,c), then their composition g ◦f is n-decomposing on (z,c). definition 2.15 ([5]). a space x is q-acyclic provided: (i) x is non-empty, (ii) hq(x) = 0 for all q > 1 and (iii) h0(x) ≈ q. proposition 2.16. let f : (x,a) −→ (y,b) be a continuous single-valued map such that: (1) f is proper and surjective; (2) f−1(b) = a; (3) f−1(y) is q-acyclic for every y ∈ y. then the map f is n-decomposing on (y,b) for every n ≥ 0. 90 n. m. benkafadar and m. c. benkara-mostefa proposition 2.17. let u be an open subset of an euclidean space en and k be a compact subset of u, then the injection i : (u,u\k) −→ (en,en\k) is n-decomposing on (en,en\k). 3. generalized coincidence point index let u be an open subset of an euclidean vector space en which has a fixed orientation. let (f,g) be a pair of continuous single-valued maps defining as follows: (3.1) u f ←− x g −→ en where x is an arbitrary hausdorff topological space. definition 3.1. an element x ∈ x is said to be a coincidence point of the pair (f,g) if f(x) = g(x). let s(f,g) be the set of all coincidence points of the pair (f,g) and f(f,g) be the subset of u defined as follows: f(f,g) = {u ∈ u | u ∈ g(f−1(u)}. lemma 3.2. one has the equality f(s(f,g)) = f(f,g). proof. the proof is obvious. � let k be a compact subset of u which contains f(f,g). thus, one obtains the following diagram: (3.2) (u,u\k) f ←− (x,x\f−1(k)) f−g −→ (en,en\{θ}) definition 3.3. a pair of continuous single-valued maps (f,g) as above defined, is called n-admissible on (u,u\k) if f is n-decomposing on (u,u\k). the set of all n-admissible pairs on (u,u\k) is denoted pd(u,u\k). let (f,g) ∈ pd(u,u \k), then if σ ∈ ω(f∗;u,u\k) the diagram (3.2) induces the following diagram: (3.3) hn(u,u\k) f∗ ←− hn(x,x\f −1(k)) (f−g)∗ −→ hn(e n,en\{θ}) σ ց m hn(x,x\f −1(k)) let ok ∈ hn(u,u\k) be the image of 1 under the composite map: z = hn(s n) −→ hn(s n,sn\k) ≅ hn(u,u\k) and o{θ} ∈ hn(e n,en\{θ}) be the image of 1 under the composition map: z = hn(s n) −→ hn(s n,sn\{θ}) ≅ hn(e n,en\{θ}) where sn = en ∪{∞}. a generalized coincidence point index 91 the elements ok and o{θ} are called the fundamental classes around the compacts k and {θ} respectively [9], [10]. definition 3.4. let (f,g) be a n-admissible pair on (u,u\k). the generalized coincidence point index of (f,g) relatively σ ∈ ω(f∗;u,u\k) is defined as being the rational number iσ(f,g) which verifies the equality (f − g)∗ ◦ σ(ok) = iσ(f,g) ·o{θ}. definition 3.5. let (f,g) be a n-admissible pair on (u,u\k). the generalized coincidence point index of (f,g) is defined as being the set of rational numbers i(f,g) = {iσ(f,g) | σ ∈ ω(f∗;u,u\k)}. proposition 3.6. if the single-valued map f : (x,x\f−1(k)) −→ (u,u\k) verifies the conditions of the proposition (2.16), then the pair (f,g) is nadmissible on (u,u\k) and i(f,g) = {i(f∗)−1(f,g)}. proof. the single-valued map f : (x,x\f−1(k)) −→ (u,u\k) induces an isomorphism f∗ : hn(x,x\f −1(k)) −→ hn(u,u\k) therefore ω(f∗;u,u\k) = {(f∗) −1}. � proposition 3.7. if f(f,g) = ∅, then i(f,g) = {0}. proof. suppose that f(f,g) = ∅ then using lemma (3.2) one deduces that s(f,g) = ∅. this equality means that f(x) 6= g(x) for each x ∈ x. therefore for every σ ∈ ω(f∗;u,u\k) we have the following commutative diagram: hn(u,u\k) σ −→ hn(x,x\f −1(k)) (f−g)∗ −→ hn(e n,en\{θ}) (f −g)∗ ց m hn(e n\{θ},en\{θ}) where (f −g) = f −g. one concludes the proof remarking that (f −g)∗ is the trivial homomorphism. � corollary 3.8. if i(f,g) 6= {0}, then the pair (f,g) admits at least a coincidence point. proof. this is a consequence of lemma (3.2). � let g : u −→ en be a continuous single-valued map defined from an open subset u of an euclidean vector space en and k be a compact subset of u which contains fix(g) = {x ∈ u | x = g(x)}. the fixed point index of g defined in [9] is the rational ig which verifies the equality: (i−g)n∗ (ok) = ig ·o{θ}, where i : u −→ en is the natural injection. proposition 3.9. the generalized coincidence point index of the pair (i,g) is defined and equal to the fixed point index of g. 92 n. m. benkafadar and m. c. benkara-mostefa proof. first note that f(i,g) = fix(g) = {x ∈ u | x = g(x)}. let k be a compact subset of en which contains f(i,g) = fix(g). so, one has the diagram: hn(e n,en\k) i∗ ←− hn (u,u\k) (i−g)∗ −→ hn(e n,en\{θ}). therefore, i(i,g) ·o{θ} = (i−g)∗ ◦ i −1 ∗ (ok) = (i−g)∗(ok) = ig ·o{θ}. � corollary 3.10. if i(i,g) 6= {0} then g admits at least a fixed point. let (f,g) and (f1,g1) be two pairs of continuous single-valued maps defining as follows: (3.4) u f ←− x g −→ en and (3.5) v f1 ←− x1 g1 −→ en, where u and v are two open subsets of en and x and x1 are two hausdorff topological spaces. let k and k1 be two compact subsets of e n which contain f(f,g) and f(f1,g1) respectively and such that k ⊂ k1 ⊂ v ⊂ v ⊂ u. for instance, one obtains the following diagrams: (3.6) (u,u\k) f ←− (x,x\f−1(k)) f−g −→ (en,en\{θ}) and (3.7) (v,v\k1) f1 ←− (x1,x1\f −1 1 (k1)) f1−g1 −→ (en,en\{θ}). proposition 3.11. under the above hypotheses, assume that h : (x1,x1\f −1 1 (k1)) −→ (x,x\f−1(k)) is a continuous single-valued map such that the following diagram is commutative: (v,v\k1) f1 ←− (x1,x1\f −1 1 (k1)) f1−g1 −→ (en,en\{θ}) i ↓ ↓ h m (u,u\k) f ←− (x,x\f−1(k)) f−g −→ (en,en\{θ}) where i is the natural injection. then if the pair (f1,g1) ∈ pd(v,v\k1) one can infer that (f,g) ∈pd(u,u \k) and i(f1,g1) ⊂ i(f,g). proof. of course, i induces an isomorphism i∗ : hn(v,v\k1) −→ hn(u,u\k) which takes ok1 in ok. moreover, if σ ∈ ω(f1∗,v,v\k1), then h∗ ◦σ ◦ i −1 ∗ ∈ ω(f∗,u,u\k). � let (f,g) be a pair of continuous single-valued maps such that: u f ←− x g −→ en and h : x1 −→ x be a continuous single-valued map defined between two hausdorff topological spaces x1 and x. a generalized coincidence point index 93 proposition 3.12. if h : (x1,x1\(f ◦ h) −1(k)) −→ (x,x\f−1(k)) is ndecomposing on (x,x\f−1(k)) and the pair (f,g) is n-admissible on (u,u\k), then (f ◦h,g ◦h) ∈pd(u,u \k) and i(f ◦h,g ◦h) ⊂ i(f,g). proof. note that f(f ◦ h,g ◦ h) ⊆ f(f,g) ⊆ k, the composition f ◦ h is ndecomposing on (u,u\k) (see proposition 2.14), and one has the following diagram: (u,u\k) f◦h ←− (x1,x1\(f ◦h) −1(k)) f◦h−g◦h −→ (en,en\{θ}) let k ∈ i(f ◦ h,g ◦ h), then there exists σ ∈ ω((f ◦h)∗ ,u,u\k) such that (f ◦ h − g ◦ h)∗ ◦ σ(ok) = k · oθ, therefore (f − g)∗ ◦ h∗ ◦ σ(ok) = k · oθ. because h∗ ◦σ ∈ ω(f∗;u,u\k), one deduces that k ∈ i(f,g). � definition 3.13. two pairs of continuous single-valued maps defined as follows: u fi ←− x gi −→ en, i = 0,1, are called equivariant on a compact k ⊂ en if there exist: (1) a hausdorff pair (x,x\x′) such that: (u,u\k) fi ←− (x,x\x′) fi−gi −→ (en,en\{θ}), i = 0,1, (2) a pair of continuous maps (ϕ,ψ) n-admissible on (u,u\k) such that: (u,u\k) ϕ ←− (x,x\ϕ−1(k)) ϕ−ψ −→ (en,en\{θ}) (3) a single-valued map h : (x,x\ϕ−1(k)) −→ (x,x\x′) n-decomposing on (x,x\x′) such that the following diagram is commutative: (u,u\k) f0 ←− (x,x\x′) f0−g0 −→ (en,en\{θ}) m ↑ h m (u,u\k) ϕ ←− (x,x\ϕ−1(k)) ϕ−ψ −→ (en,en\{θ}) m ↓ h m (u,u\k) f1 ←− (x,x\x′) f1−g1 −→ (en,en\{θ}) proposition 3.14. if (fi,gi), i = 0,1 are two equivariant pairs on a compact k ⊂ en, then (fi,gi) ∈pd(u,u \k), i = 0,1, and i(f0,g0) = i(f1,g1). proof. assume (f0,g0) and (f1,g1) are equivariant, then f0∗ ◦ h∗ = ϕ∗ = f1∗ ◦h∗ therefore f0∗, f1∗ are both n-decomposing on (u,u\k) and f0∗ = f1∗. moreover, (f0−g0)∗◦h∗ = (ϕ−ψ)∗ = (f1−g1)∗◦h∗ so (f0−g0)∗ = (f1−g1)∗. � definition 3.15. two pairs (fi,gi), i = 0,1 defined as follows: (u,u\k) fi ←− (x,x\x′) fi−gi −→ (en,en\{θ}), i = 0,1, are called homotopic on a compact k ⊂ en if the following conditions are verified: 94 n. m. benkafadar and m. c. benkara-mostefa (1) there exists a pair of single-valued maps (ϕ,ψ) n-admissible on (u,u\k)× [0,1] such that: (u,u\k)× [0,1] ϕ ←− (x,x\ϕ−1(k × [0,1])) ϕ−ψ −→ (en,en\{θ}), (2) there exists a single valued map h : (x,x\x′) −→ (x,x\ϕ−1(k × [0,1])), n-decomposing on (x,x\ϕ−1(k × [0,1])), (3) the following diagram is commutative: (u,u\k) f0 ←− (x,x\x′) f0−g0 −→ (en,en\{θ}) χ0 ↓ h ↓ m (u,u\k)× [0,1] ϕ ←− (x,x\ϕ−1(k × [0,1])) ϕ−ψ −→ (en,en\{θ}) χ1 ↑ h ↑ m (u,u\k) f1 ←− (x,x\x′) f1−g1 −→ (en,en\{θ}) where χi(x) = (x,i), for every x ∈ u and i = 0,1. proposition 3.16. if (f0,g0) and (f1,g1) are homotopic on a compact k ⊂ e n then (fi,gi) ∈pd(u,u \k), i = 0,1 and i(f0,g0) = i(f1,g1). proof. of course, χ0∗ and χ1∗ are both isomorphisms and are equal, so f0∗ = f1∗. one deduces also that f0∗ and f1∗ are both n-decomposing on (u,u\k). in an other hand, from the commutativity of the diagram one obtains that (f0−g0)∗◦h∗ = (f1−g1)∗◦h∗ = (ϕ−ψ)∗ therefore (f0−g0)∗ = (f0−g0)∗. � let (f,g) and (f′,g′) be two pairs defined by the following way: u f ←− x g −→ en and u′ f ′ ←− x g ′ −→ em where u and u′ are two open subsets of en and em respectively. let k be a compact subset of en which contains f(f,g) and k′ be a compact subset e which contains f(f′,g′). proposition 3.17. if the pairs (f,g) and (f′,g′) are n-admissible on (u,u\k) and (u′,u′\k′) respectively then the pair (f ×f′,g×g′) is (n + m)-admissible on (u ×u′,u ×u′\k ×k′) and i(f ×f′,g ×g′) ⊃ i(f,g) ·i(f′,g′). proof. one has the following equalities: f(f ×f′,g ×g′) = f(f,g) ×f(f′,g′), ok×k′ = ok ×ok′ ∈ hn+m [(u,u\k)× (u ′,u′\k′)] = hn+m(u ×u ′,u ×u′\k ×k′) and the inclusion: k ×k′ ⊃ f(f,g) ×f(f′,g′). therefore, if (σ,σ′) ∈ ω(f∗,u,k) × ω(f ′ ∗,u ′,k′) one obtains the equalities: (f ×f′ −g ×g′)∗ ◦ (σ ×σ ′)(ok×k′) = a generalized coincidence point index 95 [(f −g)∗ ◦σ × (f ′ −g′)∗ ◦σ ′] (ok ×ok′) = (f −g)∗ ◦σ(ok) × (f ′ −g′)∗ ◦σ ′(ok) = [iσ(f,g) ·iσ′ (f ′,g′)]o{θ}. � 4. generalized fixed point degree of multi-valued mappings let x and y be two hausdorff topological spaces. a multi-valued mapping taking x to y is a relation f which associates to each element x ∈ x a non empty subset f(x) ⊂ y. let k(y ) be the collection of all non empty compact subsets of y and f : x −→ k(y ) be a multi-valued mapping. the subset: γx(f) = {(x,y) ∈ x ×y | y ∈ f(x)} , of x ×y is called the graph of the multi-valued mapping f on x. in this case one could define two natural projectors: tf : γx(f) −→ x and rf : γx(f) −→ y such tf (x,y) = x, rf (x,y) = y for every (x,y) ∈ γx(f). for each element x ∈ x one has the equality f(x) = rf (t −1 f (x)). the quintuple [x,y,γx(f), tf ,rf ] is called the canonical representation of the multivalued f : x −→ k(y ). let [x1,x2,x0,f1,f2] be a quintuple constituted of hausdorff topological spaces xi, i = 0,1,2 and continuous maps fj : x0 −→ xi, j = 1,2 and such that f1 is onto and the inverse image of each element x ∈ x1 is compact, then the equality f(x) = g ◦ f−1(x) defines a multi-valued mapping f : x1 −→ k(x2). in this case the quintuple [x1,x2,x0,f1,f2] is called a representation of f : x1 −→ k(x2). two quintuples [x1,x2,x0,f1,f2], [x1,x2,x0,g1,g2] are called equivalents if g1 ◦f −1 1 (x) = f(x) = g2 ◦f −1 2 (x) for each x ∈ x1. a multi-valued mapping f : x −→ k(y ) is called upper semi continuous if f−1+ (v ) = {x ∈ x | f(x) ⊂ v} is an open subset of y for every open subset v of x. a multi-valued g : x −→ k(y ) is said to be a selector of f : x −→ k(y ) if g(x) ⊆ f(x) for every element x ∈ x. let h be the ∨ cech homology functor with compact carries and coefficient in the set of rational numbers q. a multi-valued mapping f : x −→ k(y ) is called to be q-acyclic provided the image f(x) is q-acyclic for every element x ∈ x, f is said to be compact provided f(x) is contained in a compact subset of y . more properties on multi-valued mappings can be found in [24]. let f : u −→ k(en) be a multi-valued mapping and k be a compact subset of u ⊆ en. in this case f(tf ,rf ) = {x ∈ u | x ∈ rf (t −1 f )(x)} = {x ∈ u | x ∈ f(x)} = fix(f). 96 n. m. benkafadar and m. c. benkara-mostefa definition 4.1. a multi-valued mapping f : u −→ k(en) is called n-admissible on (u,u\k) if the pair (tf ,rf ) of projectors: u tf ←− γu(f) rf −→ en satisfies the following conditions: (1) k ⊃ fix(f) = {x ∈ u | x ∈ f(x)}; (2) the pair (tf ,rf ) is n-admissible on (u,u\k). lemma 4.2. let f : u −→ k(en) be a multi-valued mapping n-admissible on (u,u\k), then one has the following diagram: hn(u,u\k) (tf ) ∗ ←− hn(γu(f),γu\k(f)) (tf −rf ) ∗ −→ hn(e n,en\{θ}) proof. the proof is obvious. � definition 4.3. the generalized fixed point degree of a n-admissible multivalued mapping f on (u,u\k) is defined as the following set of rational numbers: i(f ;u,k) = i(tf ,rf ) = {iσ(tf ,rf ) | σ ∈ ω((tf )∗ ;u,u\k)} let us describe some properties of this generalized fixed point degree. theorem 4.4. if i(f ;u,k) 6= {0} then f admits at least a fixed point i.e. a point x ∈ u such that x ∈ f(x). proof. this is a consequence of corollary (3.8). � definition 4.5. a representation ρ = [u,en,z,f,g] of a multi-valued mapping f : u −→ k(en) is called n-admissible on (u,u\k) if the pair (f,g) is nadmissible on (u,u\k) and {x ∈ u | x ∈ f(x)}⊆ k. let u and v be two open subsets of en, k and k1 be two compact subsets of en such that k ⊂ k1 ⊂ v ⊂ v ⊂ u. if the restriction ∼ f : v −→ k(en) of f : u −→ k(en) defined by the rule ∼ f(x) = f(x) for every x ∈ v admits a representation ρ = [v,en,z,f,g] n-admissible on (v,v\k1), so one can consider the following diagram: (4.8) (v,v\k1) f ←− (z,z\f−1(k1)) f−g −→ (en,en\{θ}) let ω(f∗;v,v\k1) be the set of the right inverse homomorphisms of: f∗ : hn(z,z\f −1(k1)) −→ hn(v,v\k1). in this case one can define: iρ( ∼ f ;v,k1) = {iσ(f,g) | σ ∈ ω(f∗;v,v\k1)}. proposition 4.6. if a multi-valued mapping f : u −→ k(en) has a restriction ∼ f : v −→ k(en) which admits a representation ρ = [v,en,z,f,g] n-admissible on (v,v\k1) then the multi-valued mapping f is n-admissible on (u,u\k) and iρ( ∼ f ;v,k1) ⊂i(f ;u,k). a generalized coincidence point index 97 proof. the proof is a consequence of proposition (3.11) and the following commutative diagram: hn(v,v\k1) f∗ ←− hn(z,z\f −1(k1)) (f−g)∗ −→ hn(e n,en\{θ}) i∗ ↓ ↓ α∗ m hn(u,u\k) (tf )∗ ←− hn(γu(f),γu\k(f)) (tf −rf )∗ −→ hn(e n,en\{θ}) where α(z) = (f(z),g(z)) for each z ∈ z. � corollary 4.7. if a multi-valued mapping f : u −→ k(en) admits a representation ρ = [v,en,z,f,g] n-admissible on (v,v\k1), then f is n-admissible on (u,u\k) and iρ(f ;v,k1) ⊂i(f ;u,k). proposition 4.8. let f : u −→ k(en) be a multi-valued mapping and φ : u −→ k(en) be a selector of f, then if φ is a multi-valued mapping n-admissible on (u,u\k) the multi-valued mapping f is also n-admissible on (u,u\k) and i(φ;u,k) ⊂i(f ;u,k). proof. the proof is a consequence of the following commutative diagram: hn(u,u\k) (tφ)∗ ←− hn(γu(φ),γu\k(φ)) (tφ−rφ)∗ −→ hn(e n,en\{θ}) m i∗ ↓ m hn(u,u\k) ←− (tf )∗ hn(γu(φ),γu\k(φ)) −→ (tf −rf )∗ hn(e n,en\{θ}) where i is the canonical injection. � definition 4.9. a continuous single-valued map λ : [0,1]×u ×en −→ en is said to be a distortion of en if for each element x ∈ u the single-valued map λ(0,x, .) : en −→ en is the map identity. definition 4.10. a multi-valued f : u −→ k(en) n-admissible on (u,u\k) distorts into the multi-valued g : u −→ k(en) if there exists a distortion of en such that : (1) λ(1,x,f(x)) = g(x) for every x ∈ u; (2) x /∈ λ(t,x,f(x)) for every t ∈ [0,1] and x ∈ (u\k) . proposition 4.11. if a multi-valued f : u −→ k(en) n-admissible on (u,u\k) distorts into the multi-valued g : u −→ k(en), then g is n– admissible on (u,u\k) and i(f ;u,k) ⊂i(g;u,k). proof. consider ξ : (γu(f),γu\k(f)) −→ (γu(g),γu\k(g)) defined by the rule ξ(x,u) = (x,λ(1,x,u)) for every (x,u) ∈ γu(f). form the equality tf = tg ◦ ξ one deduces that g is n-admissible on (u,u\k) . in an other hand, the continuous single-valued maps (tf −rf ), (tg−rg)◦ξ : (γu(f),γu\k(f)) −→ (en,en\{θ}) are homotopic by the homotopy h(t,(x,u) = x − λ(t,x,u) for every t ∈ [0,1] and (x,u) ∈ γu(f). let σ ∈ ω ((tf )∗) , then ξ∗ ◦σ ∈ ω ((tg)∗) and one has the equalities: iσ ·o{θ} = (tf −rf )∗ ◦σ(ok) = (tg −rg)∗ ◦ ξ∗ ◦ σ(ok) = iξ∗◦σ ·o{θ}, which means that i(f ;u,k) ⊂i(g;u,k). � 98 n. m. benkafadar and m. c. benkara-mostefa assume that u and v are two open subsets of en, k and k1 are two compact subsets of en such that k ⊂ k1 ⊂ v ⊂ v ⊂ u. proposition 4.12. let f : u −→ k(en) be a multi-valued mapping upper semi continuous compact and q-acyclic. if g : u −→ k(en) is a selector of f and n-admissible on (v,v\k1), then i(g;v,k1) = i(f ;u,k) = {k}, where k is the rational number which verifies the equality (tf −rf )∗ ◦ (tf ) −1 ∗ (ok) = k ·o{θ}. proof. the proof is a consequence of the vietoris maps theorems [12], proposition (4.8) and the following commutative diagram: hn(v,v\k1) tg∗ ←− hn(γv (g),γv \k1(g)) (tg−rg)∗ −→ hn(e n,en\{θ}) i∗ ↓ j∗ ↓ m hn(u,u\k) ←− tf ∗ hn(γu(f),γu\k(f)) −→ (tf −rf )∗ hn(e n,en\{θ}) where i : (v,v\k1) −→ (u,u\k) and j : (γv (g),γv \k1(g)) −→ (γu(f),γu\k(f)) are the natural injections. � proposition 4.13. let k be a compact q-acyclic subset of en and f : u −→ k(en) be a multi-valued mapping such that f(u) ⊂ k, then f is n-admissible on (u,u\k) and i(f ;u,k) = {1}. proof. consider x0 ∈ k and let f : u −→ k(e n) be the map defined by the rule f(x) = {x0} for each x ∈ u. the quintuple ρ = [u,e n,u,idu,f] is a representation n-admissible on (u,u\k) of f. consider the following commutative diagram: hn(u,u\k) (idu )∗ ←− hn(u,u\k) (idu −f)∗ −→ hn(e n,en\{θ}) j∗ ↓ m hn(e n,en\{x0}) −→ (iden−f)∗ hn(e n,en\{θ}) where j∗ is an isomorphism induced by the natural injection and (iden −f)∗ is the isomorphism induced by the homeomorphism (iden −f) : (e n,en\{x0}) −→ (en,en\{θ}) defined by the rule (iden − f)(x) = x − x0 for every x ∈ e n. for instance, one deduces that iρ(f;u,k) = {1}. in an other hand, consider the following commutative diagram: hn(u,u\k) (idu ) ∗ ←− hn(u,u\k) (idu −f) ∗ −→ hn(e n,en\{θ}) m ↓ µ∗ m hn(u,u\k) ←− (tf ) ∗ hn(γu(i−r),γu\k(i−r)) −→ (tf −rf ) ∗ hn(e n,en\{θ}) where µ(x) = (x,f(x)), for each x ∈ u. the multi-valued mapping f is nadmissible on (u,u\k) because (idu)∗ is an isomorphisms. from the propositions (3.9), (4.7) and the commutativity of the above diagram one infers iρ(f;u,k) ⊂ i(f ;u,k). the multi-valued mapping f : u −→ k(e n) is a selector of the upper 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[31] j. warga, optimal control of differential and functional equations, (acad. press, new york and london, 1975). received may 2004 accepted december 2004 n. m. benkafadar (benkafadar@caramail.com) department of mathematics, faculty of sciences, university of constantine, road of ain el bey 25000, constantine, algeria m. c. benkara-mostefa (karamos@yahoo.fr) department of mathematics, faculty of sciences, university of constantine, road of ain el bey 25000, constantine, algeria karuganeagt.dvi @ applied general topology c© universidad politécnica de valencia volume 11, no. 1, 2010 pp. 21-27 one point compactification for generalized quotient spaces v. karunakaran and c. ganesan ∗ abstract. the concept of generalized function spaces which were introduced and studied by zemanian are further generalized as boehmian spaces or as generalized quotient spaces in the recent literature. their topological structure, notions of convergence in these spaces are also investigated. some sufficient conditions for the metrizability are also obtained. in this paper we shall assume that a generalized quotient space is non-compact and realize its one point compactification as a quotient space. 2000 ams classification: 46f30, 46a19, 22a30. keywords: generalized quotient space, compact, locally compact and hausdorff, one point compactification. 1. introduction schwartz distribution spaces are generalized in different ways in the literature. some of these are “generalized function spaces” (introduced and studied in detail by zemanian see [8]), “boehmian spaces” (motivated by the concept of regular operator introduced by boehme (see [1]) and studied in [3, 4, 5]) and most recently “the generalized quotient spaces” (see [6]) in [2] the authors introduce the concepts of δ-convergence and ∆-convergence in these generalized quotient spaces and investigate the behavior of convergence sequences and the topological properties of these spaces under the quotient topology. suitable conditions for metrizability of these spaces are also obtained. it turns out that these generalized function spaces, in general, are not compact. further it is also difficult to find out suitable conditions under which these spaces (under the canonical quotient topology) are locally compact and ∗the research of the second author is supported by a “university grants commission research fellowship in sciences for meritorious students”, india. 22 v. karunakaran and c. ganesan hausdorff. thus the problem of realizing the one point compactifications of these spaces assumes significance. in this paper we shall identify the one point compactification of a non-compact generalized quotient space as another quotient space. the results in this paper are also motivated by a desire to find an analogue of the following classical result which can be easily proved. let x and y be locally compact non-compact hausdorff spaces and p : x → y be a quotient map. let p∗ : x∗ → y ∗ be the natural extension of p to their respective one point compactifications. then p∗ is continuous and hence a quotient map if and only if p−1(k) is compact in x for every compact k in y . thus under certain conditions the one point compactification of a quotient space becomes another quotient space. the analogue of this result in the context of a generalized quotient space will be studied here. in section 2 we shall develop the required preliminaries and in section 3 we shall state and prove the main theorem. the conditions under which the one point compactifications of these generalized quotient spaces can be realized as generalized quotient spaces also guarantee that the original generalized quotient spaces are locally compact and hausdorff. 2. preliminaries we shall briefly recall the concept of generalized quotient spaces as described in [6]. let x be a non-empty set and let g be a commutative semi-group acting on x injectively. this means that to every g ∈ g there corresponds an injective map g : x → x such that (g1g2)(x) = g1(g2(x)) for all g1,g2 ∈ g and x ∈ x. for g ∈ g and x ∈ x, g(x) denotes the action of g on x in x. let a = x ×g. for (x,f), (y,g) ∈ a we write (x,f) ∽ (y,g) if g(x) = f(y). then ∽ is an equivalence relation in a. we define the space of generalized quotients as b = b(x,g) = a/ ∽. we denote the equivalence class containing (x,f) by [ x f ] . suppose g fails to act injectively on x then we proceed as follows: let i be a non-empty index set and let ∆ ⊂ gi be a semi-group (this only means that ∆ is closed for the canonical semi-group operation available in gi which is defined as follows: if α,β ∈ gi then (αβ)(i) = α(i)β(i) for all i ∈ i). for α ∈ ∆ and x ∈ x define αx ∈ xi by (αx)(i) = α(i)(x), so that each α gives rise to a mapping from x in to xi . we assume that these maps are injective. for α ∈ ∆ and ψ ∈ xi we also define (αψ)(i) = α(i)(ψ(i)) so that αψ defines an element of xi . suppose χ ⊂ xi satisfies the following conditions: a: αx ∈ χ for all α ∈ ∆ and all x ∈ x. b: αψ ∈ χ for all α ∈ ∆ and all ψ ∈ χ. let a = {(ξ,α)/ξ ∈ χ,α ∈ ∆ and α(i)(ξ(j)) = α(j)(ξ(i)), i,j ∈ i} . one point compactification for generalized quotient spaces 23 for (f,φ), (g,ψ) ∈ a we write (f,φ) ∽ (g,ψ) if φ(i)(g(j)) = ψ(j)(f(i)), for all i,j ∈ i. then ∽ is an equivalence relation on a. we define the space of generalized quotients as b = b(χ, ∆) = a/ ∽ and we shall denote the equivalence class containing (f,φ) by [ f (i) φ(i) ] . we shall assume that the reader is familiar with the above construction of generalized quotient spaces. further we shall assume the following: (1) x is a non-compact locally compact hausdorff space, g is a commutative semi group acting continuously on x (but not necessarily injectively) equipped with a hausdorff topology. (2) the mapping λ : x × g → x defined by λ(x,g) = g(x) is continuous and that λ−1(k) is compact in x × g for each compact k in x. (3) χ ⊂ xi is closed in xi . (4) ∆ ⊂ gi is compact. with these assumptions, the constructed generalized quotient space will be denoted by b. we explicitly assume that such a b is non-compact. we shall now give an example to show that the above conditions are realizable. let x = (−∞,−2] ∪ [2,∞) considered as a subspace of the real line under the usual topology. let g = z \ {0} (the set of all non-zero integers) with discrete topology. note that g is a commutative semi-group under usual multiplication. let i = n (the set of natural numbers). we shall allow g to act continuously on x by g(x) = x|n| where g = n ∈ g. note that even though g acts continuously on x the action is not injective for any even integer n. it is now easy to prove the following points. (1) λ : x × g → x defined by λ(x,n) = x|n| is continuous. (note that xj → x0 and nj → n0 as j → ∞ imply that sequence nj is eventually a constant (= n0, say) and hence λ(xj,nj ) → λ(x0,n0) as j → ∞). (2) for any compact (and hence bounded) set k in x, λ−1(k) ⊂ a × b with a compact in x and b is a finite subset of g and hence λ−1(k) is compact. we shall now take χ = xi so that χ is closed in xi . we shall also take ∆ = { (αn) ∈ g n/ αn = 1 for odd n and αn = ±1 for even n } ⊂ sn where s = {−1, 1}. it is now easy to see that ∆ is a semi-group, is closed in sn and that it is compact because sn is compact. further each β ∈ ∆ induces an injective map from x to xn given by (βx)(i) = β(i)(x) as required for the construction of a generalized quotient space. we shall need the following lemmas. lemma 2.1. let x be locally compact non-compact hausdorff space and g a hausdorff space. if λ : x × g → x is continuous and λ−1(k) is compact in x × g for each k compact in x then g : x → x is continuous and g−1(k) is compact in x for each k compact in x. 24 v. karunakaran and c. ganesan proof. fix g ∈ g. then g(x) = λ(x,g) = λg(x) is continuous in the variable x. let k be any compact set in x. define a mapping f : x → x × g by f(x) = (x,g). then we have (λ ◦ f)(x) = g(x) ∀ x ∈ x. now g−1(k) = {x ∈ x/g(x) ∈ k} = (λ ◦ f)−1(k) = f−1(λ−1(k)) = f−1(h) (where h = λ−1(k) ⊂ x × g is compact) = {x ∈ x/f(x) ∈ h} = {x ∈ x/(x,g) ∈ h} = π(h ∩ (x × {g})) where π : x × g → x is defined by π(x,h) = x since x × g is hausdorff and h is compact subset of x × g, h is closed in x × g. it is clear that x × {g} is closed in x × g. hence h ∩ (x × {g}) is closed in x × g. but h ∩ (x × {g}) ⊂ h and h is closed in x × g implies that h ∩ (x × {g}) is closed in h and hence h ∩ (x × {g}) is compact in h. thus h ∩ (x × {g}) is compact in x × g. now the continuity of π will show that g−1(k) = π(h ∩ (x × {g})) is compact in x. � note that the condition on λ already implies that g acts continuously on x, a fact which we have explicitly assumed. lemma 2.2. let x, g, λ and g be as in lemma 2.1. let x∗ be the one point compactification of x and let g∗ : x∗ → x∗ be the natural extension of g to x∗ ie., g∗|x = g and g ∗(∞) = ∞. then g∗ is continuous. proof. follows easily using lemma 2.1 and is left to the reader. � lemma 2.3. let x, g and λ be as in lemma 2.1. the mapping λ∗ : x∗×g → x∗ defined by λ∗(x,g) = g∗(x) = { λ(x,g) if x ∈ x g∗(∞) = ∞ if x = ∞ is continuous proof. follows easily using the property of λ and is left to the reader. � 3. construction of a new generalized quotient space in this section we shall define a new generalized quotient space b∗ which will be shown to be the one point compactification of the generalized quotient space b constructed in section 2. let a∗ = {(f,α) ∈ χ∗ × ∆/ α(i)(f(j)) = α(j)(f(i)) ∀ i,j ∈ i}, where χ∗ = [closure of χ in x∗ i ] ∪ {f∞} with f∞ : i → x ∗ is defined by f∞(i) = ∞ ∀ i ∈ i. define a relation ∼ on a∗ as follows: (f,α) ∼ (g,β) if α(i)(g(j)) = β(j)(f(i)) ∀ i,j ∈ i. it is clear that this relation ∼ is an equivalence relation in a∗ (note that each element α ∈ ∆ gives raise to a map α∗ : x∗ → x∗ i defined by α∗|x = α and α ∗(∞) = f∞ which is easily seen to be injective. one point compactification for generalized quotient spaces 25 this observation is indeed crucial to the proof of the fact that ∼ is transitive in the same way as in the proof of the transitivity of ∽ in a). we now observe the following properties of χ∗. a: αx ∈ χ∗ for all α ∈ ∆ and all x ∈ x∗. b: αψ ∈ χ∗ for all α ∈ ∆ and all ψ ∈ χ∗. indeed property (a) can be easily proved where as property (b) can be proved using the properties of net convergence in x∗ and x∗ i . now in a canonical manner we can define the generalized function space b∗ by b∗ = a∗|∼. we shall also give the quotient topology to b ∗ given by the map p∗ : a∗ → b∗ defined by p∗((f,α)) = [ f (j) α(j) ] . lemma 3.1. a∗ = a ∪ {(f∞,α)/ α ∈ ∆}. proof. it is clear that a ∪ {(f∞,α)/ α ∈ ∆} ⊂ a ∗. let (f,α) ∈ a∗. if f = f∞ then there is nothing to prove. therefore assume f(i0) 6= ∞ for some i0 ∈ i. then α(i)(f(j)) = α(j)(f(i)) ∀ i,j ∈ i will imply that f(i) 6= ∞ ∀ i ∈ i. hence f ∈ xi . but (f,α) ∈ χ∗ × ∆ will imply that f ∈ χ∗ ie., f is in the closure of χ in x∗ i . this implies that f is in the closure of χ in xi (indeed if w = vα1 × vα2 × · · ·vαn × ∏ x is any basic open set of f in xi then w∗ = vα1 × vα2 × · · ·vαn × ∏ x∗ is a basic open set of f in x∗ i . hence w ∩ χ = w∗ ∩ χ 6= φ). now χ is closed in xi implies that (f,α) ∈ χ × ∆ and as α(i)(f(j)) = α(j)(f(i)) ∀ i,j ∈ i, (f,α) ∈ a. thus a∗ ⊂ a ∪ {(f∞,α)/ α ∈ ∆}. this completes the proof. � lemma 3.2. a is open in a∗. proof. equivalently we shall show that the set d = {(f∞,α)/ α ∈ ∆} is closed in a∗. let (f,α) ∈ a∗ be a limit point of d. suppose f(i0) 6= ∞ for some i0 ∈ i. then there are open sets u and v in x ∗ containing f(i0) and ∞ respectively such that u ∩ v = φ. then it is clear that w1 = u × ∏ i6=i0 x∗ is a basic open neighbourhood of f in x∗ i such that f∞ 6∈ w1. if w2 is any open set in gi containing α then w1 × w2 is an open set containing (f,α) in x∗ i × gi but not containing any element of the form (f∞,β), β ∈ ∆. this shows that (f,α) is not a limit point of d in a∗ which is a contradiction. hence f(i) = ∞ ∀ i ∈ i and hence (f,α) = (f∞,α) ∈ d. thus d is closed in a ∗. � lemma 3.3. a∗ is a compact hausdorff space. proof. since x∗ i × gi is hausdorff and a subspace of a hausdorff space is hausdorff, a∗ is hausdorff. since a∗ ⊂ χ∗ × ∆ which is compact (note that χ∗ is a closed subset of the compact space x∗ i and ∆ is compact), we merely show that a∗ is closed in χ∗ × ∆. let (f,α) ∈ χ∗ × ∆ be a limit point of a∗. then there is a net (fγ,αγ ) of points from a∗ such that (fγ,αγ ) → (f,α) in χ ∗ × ∆. from this we have 26 v. karunakaran and c. ganesan fγ (i) → f(i) in x ∗ and αγ (i) → α(i) in g for each i ∈ i. since (fγ,αγ ) ∈ a ∗ we have , αγ (i)(fγ (j)) = αγ (j)(fγ (i)) ∀ i,j ∈ i. using lemma 2.3 we now have α(i)(f(j)) = α(j)(f(i)) for each i,j ∈ i. hence (f,α) ∈ a∗. this completes the proof. � lemma 3.4. the set kerp∗ = {((f,α), (g,β)) ∈ a∗ × a∗/ (f,α) ∼ (g,β)} is closed in a∗ × a∗. proof. follows using net convergences and arguments similar to the one given in lemma 3.3 and is left to the reader. � let us now recall the following theorem (see [7] , pp 183). theorem 3.5. let x be a non-compact topological space. then x is locally compact and hausdorff if and only if there exists a topological space y satisfying the following conditions (1) x is a subspace of y . (2) the set y \ x consists on a single point. (3) y is a compact hausdorff space. we now make the following observations. (1) (f,α) ∼ (g,β) in a∗ if and only if (f,α), (g,β) ∈ a and (f,α) ∼ (g,β) or (f,α) = (f∞,α), (g,β) = (f∞,β). in particular b ∗ = b∪ {[ f∞(j) α(j) ]} . (2) since x is a subspace of x∗ (ie., the original topology of x is the same as the subspace topology of x in x∗), the product space xi × gi is a subspace of the product space x∗ i ×gi . in particular a is a subspace of a∗ (in the above sense) and hence b is a subspace of b∗. (3) b∗ = p∗(a∗) ⇒ b∗ is compact (note that a∗ is compact and p∗ is continuous). (4) since kerp∗ is closed (lemma 3.4) and a∗ is a compact hausdorff space it follows that b∗ is hausdorff. (5) since b∗ \ b is a singleton and b is non-compact we have b = b∗. using all the above observations together with theorem 3.5 we now get the following main theorem. theorem 3.6. b is locally compact, hausdorff and its one point compactification is b∗. references [1] t. k. boehme, the support of mikusinski operators, trans. amer. math. soc. 176 (1973), 319–334. [2] v. karunakaran and c. ganesan, topology and the notion of convergence on generalized quotient spaces, int. j. pure appl. math. 44, no. 5 (2008), 797–808. [3] j. mikusinski and p. mikusinski, quotients de suites et leurs applications dans l’analyse fonctionnelle, comptes rendus 293, serie i (1981), 463–464. one point compactification for generalized quotient spaces 27 [4] j. mikusinski and p. mikusinski, quotients of sequences, proc. of the ii conference on convergence szezyrk (1981), 39–45. [5] p. mikusinski, convergence of boehmians, japan j. math 9 (1983), 159–179. [6] p. mikusinski, generalized quotients with applications in analysis, methods appl. anal. 10 (2004), 377–386. [7] j. r. munkres, topology, second edition, prentice-hall of india, private limited, new delhi (2003). [8] a. h. zemanian, generalized integral transformation, john wiley and sons, inc., new york, (1968). received june 2009 accepted march 2010 v. karunakaran (vkarun−mku@yahoo.co.in) senior professor, school of mathematics, madurai kamaraj university, tamil nadu, madurai 625 021, india. c. ganesan (mkuganc@yahoo.com) research scholar, school of mathematics, madurai kamaraj university, tamil nadu, madurai 625 021, india. richkunagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 2, 2005 pp. 207-216 ti-ordered reflections hans-peter a. künzi and thomas a. richmond ∗ abstract. we present a construction which shows that the tiordered reflection (i ∈ {0, 1, 2}) of a partially ordered topological space (x, τ, ≤) exists and is an ordered quotient of (x, τ, ≤). we give an explicit construction of the t0-ordered reflection of an ordered topological space (x, τ, ≤), and characterize ordered topological spaces whose t0-ordered reflection is t1-ordered. 2000 ams classification: 54f05, 18b30, 54g20, 54b15, 54c99, 06f30 keywords: ordered topological space, t2-ordered, t1-ordered, t0-ordered, ordered reflection, ordered quotient 1. introduction the t0-, t1-, and t2-reflections of a topological space have long been of interest to categorical topologists. methods of constructions of these in the category top are described in the references given in [3] (see p. 302). here, we consider the corresponding concepts of ti-ordered reflections in the category ordtop of partially ordered topological spaces with continuous increasing functions as morphisms. in section 2, we construct the ti-ordered reflection (i = 0, 1, 2, s2) of a partially ordered topological space (x, τ, ≤). the construction is extrinsic, occurring in the category preordtop of preordered topological spaces with continuous increasing functions as morphisms, which contains ordtop as a subcategory. in section 3, we give an intrinsic construction of the t0-ordered reflection of a partially ordered space (x, τ, ≤) and examine some properties of this reflection. a preordered topological space (x, τ, �) is a set x with a topology τ and a preorder �. following the notation of nachbin ([7]), for a ⊆ x, the increasing hull of a is i(a) = {y ∈ x : ∃a ∈ a with a � y}. a set a is an increasing set ∗the first author would like to thank the south african research foundation for partial financial support under grant number 2068799. the first version of the article was completed in germany during the dagstuhl-seminar 04351 on spatial representation: discrete vs. continuous computational models. 208 h.-p. a. künzi and t. a. richmond if a = i(a). the closed increasing hull i(a) of a ⊆ x is the smallest closed increasing set containing a. decreasing sets, decreasing hulls d(a), and closed decreasing hulls d(a) are defined dually. a set is monotone if it is increasing or decreasing. there are many compatibility conditions between the topology and order of a preordered topological space which one may stipulate. these include the convex topology condition (τ has a subbase of monotone open sets) or the ordered separation axioms, some of which are defined below. our preordered spaces need not satisfy any of these compatibility conditions. suppose (x, τ, �) is a preordered topological space. it is well-known that the preorder � induces an equivalence relation ∼ = g(�) ∩ g(�−1) on x defined by x ∼ y if and only if x � y and y � x. the preordered topological space (x, τ, �) is said to be t0-preordered if any of the following equivalent statements holds. (a) x 6∼ y ⇒ [i(x) 6= i(y) or d(x) 6= d(y)] (b) if i(x) = i(y) and d(x) = d(y), then x ∼ y. (c) if x 6∼ y, there exist a monotone open neighborhood of one of the points which does not contain the other point. observe that if � is a partial order, then the relation ∼ is equality. a preordered topological space (x, τ, �) is t1-preordered if i(x) and d(x) are closed for every x ∈ x, or equivalently, if x 6� y in x implies there exists an open increasing neighborhood of x which does not contain y and there exists an open decreasing neighborhood of y which does not contain x. a preordered topological space (x, τ, �) is t2-preordered if there is an increasing neighborhood of x disjoint from some decreasing neighborhood of y whenever x 6� y. equivalently, (x, τ, �) is t2-preordered if the preorder � is closed in (x, τ) × (x, τ). a preordered topological space (x, τ, �) is strongly t2-preordered, or for notational convenience, ts2-preordered, if there is an increasing open neighborhood of x disjoint from some decreasing open neighborhood of y whenever x 6� y. if � is a partial order, then (x, τ, �) is a partially ordered topological space, or simply an ordered topological space. if the preorder of a ti-preordered topological space (x, τ, ≤) is a partial order, we will say (x, τ, ≤) is ti-ordered. we will typically denote preorders by � and partial orders by ≤. to avoid confusion when indicating inclusions, we may represent a preorder ⊑ by its graph g(⊑). 2. existence of ti-ordered reflections a special quotient. the definition of a t0-preordered topological space (x, τ, �) involved the equivalence relation ∼ on x defined by a ∼ b if and only if a � b and b � a. for any preordered topological space (x, τ, �), we obtain a partially ordered topological space by giving x/ ∼ the quotient topology τ/ ∼ and the order ≤ defined by [a] ≤ [b] if and only if a � b. the following properties of this quotient construction are easily verified. ti-ordered reflections 209 proposition 2.1. suppose � is a preorder on a set x, ∼ is the equivalence relation g(�) ∩ g(�−1) on x, and ≤ is the partial order on x/ ∼ defined by [a] ≤ [b] if and only if a � b. (a) any �-increasing or �-decreasing set is ∼-saturated. (b) the quotient map q : x → x/ ∼ carries increasing (decreasing) sets to increasing (decreasing) sets. specifically, q(i�(a)) = i≤(q(a)) for any subset a ⊆ x, and dually. (c) if �∗ is a preorder on x with g(�) ⊆ g(�∗) and ∼∗ is defined from � ∗ as ∼ is defined from �, then the ∼∗-equivalence classes are ∼-saturated. preorders induced by functions. any continuous increasing function f : (x, τ, ≤) → (y, γ, ⊑) between two partially ordered topological spaces induces a preorder �f on x defined by a �f b if and only if f(a) ⊑ f(b). furthermore, g(≤) ⊆ g(�f ). suppose (y, γ, ⊑) is ti-ordered for some i ∈ {0, 1, 2, s2}. now g(�f ) = (f−1×f−1)(⊑). noting that f−1 and f−1×f−1 carry open (respectively, closed, ⊑-increasing, ⊑-decreasing, disjoint) sets to open (respectively, closed, �f increasing, �f-decreasing, disjoint) sets and that the ti-(pre)ordered properties (i ∈ {0, 1, 2, s2}) are defined in terms of open/closed/increasing/decreasing/ disjoint sets, it follows that (x, τ, �f ) is ti-preordered. also, if ∼f is the equivalence relation g(�f ) ∩ g(� −1 f ), observe that the ∼f-equivalence class of a ∈ x is [a] = f −1(f(a)), a fiber of the map f. thus, �f is a partial order if and only if f is injective. we now apply the special quotient construction described above to the preorder �f induced by a function f. suppose f : (x, τ, ≤) → (y, γ, ⊑) is a continuous increasing function between partially ordered spaces (x, τ, ≤) and (y, γ, ⊑), �f is the preorder on x defined by a �f b if and only if f(a) ⊑ f(b), ∼f is the equivalence relation on x defined by a ∼f b if and only if a �f b and b �f a, and ≤f is the partial order on x/ ∼f defined by [a] ≤f [b] if and only if a �f b. we now show that if (y, γ, ⊑) is ti-ordered for some i ∈ {0, 1, 2, s2}, then the partially ordered space (x/ ∼f , τ/ ∼f , ≤f) is also ti-ordered. define h : (x/ ∼f, τ/ ∼f , ≤f) → (y, γ, ⊑) by h([a]) = h(f −1f(a)) = f(a), that is, h([a]) = fq−1[a] where q : x → x/ ∼f is the natural quotient map. now h is continuous, for if u is open in y , then h−1(u) = qf−1(u) which is open since f is continuous and the quotient map carries saturated open sets to open sets. the definitions of ≤f, �f , h, and �h respectively imply the following implications: [a] ≤f [b] ⇐⇒ a �f b ⇐⇒ f(a) ⊑ f(b) ⇐⇒ h([a]) ⊑ h([b]) ⇐⇒ [a] �h [b]. thus, h is increasing and �h=≤f. by the remarks of the second paragraph after proposition 2.1, it follows that (x/ ∼f , τ/ ∼f , ≤f) is ti-ordered. 210 h.-p. a. künzi and t. a. richmond proposition 2.2. suppose (x, τ, �) is a preordered topological space, ∼ is the equivalence relation on x defined by a ∼ b if and only if a � b and b � a, and ≤ is the partial order on x/ ∼ defined by [a] ≤ [b] if and only if a � b. then for i ∈ {0, 1, s2}, (x, τ, �) is ti-preordered if and only if (x/ ∼, τ/ ∼, ≤) is ti-ordered. proof. suppose i ∈ {0, 1, s2} and (x/ ∼, τ/ ∼, ≤) is ti-ordered. the two paragraphs following proposition 2.1 remain valid for a function f from a preordered space to a partially ordered space, and if f is taken to be the quotient map q from (x, τ, �) to (x/ ∼, τ/ ∼, ≤), then �f =�. thus, (x, τ, �) is ti-preordered. for the converse, first suppose that (x, τ, �) is t0-preordered. if [a] 6= [b] in x/ ∼, then there is a �-monotone open neighborhood n of one of the points a or b which does not contain the other. now q(n) is a ≤-monotone open neighborhood of one of the points [a] or [b] in x/ ∼ which does not contain the other, so x/ ∼ is t0-ordered. now suppose (x, τ, �) is t1-preordered. for [x] ∈ x/ ∼, we have i≤([x]) = i≤(q(x)) = q(i�(x)) by proposition 2.1 (b). since i�(x) is closed and saturated, q(i�(x)) will be closed in x/ ∼. with the dual argument, this shows that x/ ∼ is t1-ordered. finally, suppose (x, τ, �) is ts2preordered. if [a] 6≤ [b] in x/ ∼, then a 6� b in x, so there exist a �-increasing τ-open neighborhood na of a and a �-decreasing τ-open neighborhood nb of b in x which are disjoint. by proposition 2.1 (a) and (b), it follows that q(na) and q(nb) are the required ≤-monotone τ/ ∼-open neighborhoods separating [a] and [b] in x/ ∼. � the result of proposition 2.2 does not hold for i = 2. while the reasoning of the first paragraph of the proof shows that if (x/ ∼, τ/ ∼, ≤) is t2-ordered then (x, τ, �) is t2-preordered, the example below shows that the converse fails. example 2.3. if (x, τ, �) is a t2-preordered space, ∼ is the equivalence relation on x defined by a ∼ b if and only if a � b and b � a, and ≤ is the partial order on x/ ∼ defined by [a] ≤ [b] if and only if a � b, then (x/ ∼, τ/ ∼, ≤) need not be t2-ordered. let γ be the euclidean topology on r. define a topology τ on x = r as follows: each point of q\{0} is isolated. for x ∈ (r\q)∪{0}, a τ-neighborhood of x is {x} ∪ (u ∩ q) where u is a γ-neighborhood of x. define a preorder � on x by a � b if and only if a = b or {a, b} ⊆ r\q. (in fact, � is already an equivalence relation, so ∼ = �.) the graph of � is (r \ q)2 ∪ ∆x. because each γ-neighborhood of x ∈ x contains a τ-neighborhood of x, it follows that (x, τ) is t2, and thus ∆x is closed in x ×x. observe that q is a neighborhood of each of its points, so r \ q is closed in x. it follows that g(�) = (r \ q)2 ∪ ∆x is closed in x × x, so � is t2-preordered. to see that (x/ ∼, τ/ ∼, ≤) is not t2-ordered, suppose to the contrary that it is. now π 6� 0 in x, so [π] 6≤ [0] in x/ ∼, and thus there exist disjoint sets m, n, where m is a ≤-increasing τ/ ∼-neighborhood of [π] and n is a ≤-decreasing τ/∼-neighborhood of [0]. if q : x → x/ ∼ is the quotient map, ti-ordered reflections 211 then q−1(n) contains a τ-neighborhood {0}∪(b ∩q) of 0, where b is a γ-open neighborhood of 0. for any b ∈ b \ q, we have b ∈ r \ q = [π] ⊆ q−1(m). now q−1(m) is a �-increasing τ-neighborhood of b. but a �-increasing τneighborhood of b ∈ r \ q has form (r \ q) ∪ u where u is a γ-neighborhood of b. now u ∩ b 6= ∅ ⇒ u ∩ (b ∩ q) 6= ∅ ⇒ q−1(m) ∩ q−1(n) 6= ∅, contrary to m ∩ n = ∅. thus, (x/ ∼, τ/ ∼, ≤) is not t2-ordered. the existence construction. suppose (x, τ, ≤) is a given partially ordered topological space. for i ∈ {0, 1, 2, s2}, let pi = {� ∗ : �∗ is a preorder on x, g(≤) ⊆ g(�∗), and (x/ ∼∗, τ/ ∼∗, ≤ ∗) is ti−ordered}. note that pi 6= ∅ since x×x belongs to it. if i ∈ {0, 1, s2}, by proposition 2.2, we have pi = {� ∗: g(≤) ⊆ g(�∗) and (x, τ, �∗) is ti−preordered}. let g(�i) = ⋂ pi. as an intersection of preorders containing g(≤), � i is also a preorder containing g(≤). proposition 2.4. for i ∈ {0, 1, 2, s2}, (x/ ∼i, τ/ ∼i, ≤ i) is ti-ordered. proof. i = 0: suppose [x] 6= [y] in (x/ ∼0, τ/ ∼0, ≤ 0). then x 6∼0 y in x, so for some �∗∈ p0, there exists a � ∗-monotone open neighborhood n of a which does not contain b, where {a, b} = {x, y}. applying proposition 2.1 (b), we obtain a ≤∗-monotone open neighborhood n′ of [a]∼∗ which does not contain [b]∼∗. since g(≤ 0) ⊆ g(�∗), proposition 2.1 (c) implies the existence of a natural increasing quotient map q : x/ ∼0→ x/ ∼∗, and q −1(n′) is a ≤0monotone open neighborhood of [a]∼0 which does not contain [b]∼0. thus, (x/ ∼0, τ/ ∼0, ≤ 0) is t0-ordered. i = 1: because the increasing hull of x in �1= ⋂ p1 is the intersection of the increasing hulls of x in each �∗ in p1, and each of these latter increasing hulls is closed, it follows that i�1(x) is closed for any x ∈ x. with the dual argument, we have (x, τ, �1) is t1-preordered. by proposition 2.2, (x/ ∼1, τ/ ∼1, ≤ 1) is t1-ordered. i = 2: suppose [a] 6≤ [b] in (x/ ∼2, τ/ ∼2, ≤ 2). then there exists �∗∈ p2 such that [a]∼∗ 6≤ ∗ [b]∼∗ in the t2-ordered space (x/ ∼∗, τ/ ∼∗, ≤ ∗). let na and nb be disjoint τ/ ∼∗ neighborhoods of [a]∼∗ and [b]∼∗ respectively, with na being ≤ ∗-increasing and nb being ≤ ∗-decreasing. since g(≤2) ⊆ g(�∗), the natural quotient map q from x/ ∼2 to x/ ∼∗ yields q −1(na) and q −1(nb) as oppositely directed monotone neighborhoods separating [a] and [b] in x/ ∼2. thus, (x/ ∼2, τ/ ∼2, ≤ 2) is t2-ordered. taking na and nb above to be open sets proves the case i = s2. � we are now ready for the main result of this section. theorem 2.5. suppose (x, τ, ≤) is a partially ordered topological space and i ∈ {0, 1, 2, s2}. then the ti-ordered reflection of (x, τ, ≤) is (x/ ∼i, τ/ ∼i, ≤i). 212 h.-p. a. künzi and t. a. richmond proof. suppose i ∈ {0, 1, 2, s2} is given, (y, γ, ⊑) is ti-ordered, and f : (x, τ, ≤) → (y, γ, ⊑) is continuous and increasing. from the paragraph preceding proposition 2.2, it follows that (x/ ∼f , τ/ ∼f, ≤f ) is a ti-ordered space with g(≤) ⊆ g(�f ). from the definition of � i, we have g(�i) ⊆ g(�f ). from proposition 2.1 (c), there is a natural continuous increasing quotient map q : (x/ ∼i, τ/ ∼i, ≤ i) → (x/ ∼f , τ/ ∼f , ≤f) which carries [a]∼i to [a]∼f . we have shown above that there is a continuous increasing function h : (x/ ∼f , τ/ ∼f, ≤f) → (y, γ, ⊑). now hq : x/ ∼i→ y is continuous and increasing. thus, each continuous increasing function f : x → y can be lifted through x/ ∼i, and from the construction, this lifting is unique. thus, x/ ∼i is the ti-ordered reflection of x. � it is easy to verify that this construction gives the property q reflection of (x, τ, ≤) as a quotient for any property q for which (a) (x/ ∼q, τ/ ∼q, ≤ q) has property q where a ∼q b if and only if a �q b and b �q a; g(�q) = ⋂ {g(�∗) : g(≤) ⊆ g(�∗) and (x/ ∼∗, τ/ ∼∗, ≤∗) is an ordered space with property q}; and [a] ≤q [b] in x/ ∼q if and only if a �q b in x. (b) if (y, γ, ⊑) has property q and f : (x, τ, ≤) → (y, γ, ⊑) is continuous and increasing, then (x/ ∼f , τ/ ∼f , ≤f) has property q. furthermore, the methods of this section can be used to find the ti reflection (i = 0, 1, 2) of a topological space (x, τ) by considering (x, τ) as a discretely ordered topological space (x, τ, =) and taking all preorders to be equivalence relations, so that the resulting quotients are discretely ordered. 3. the t0-ordered reflection the construction of the ti-ordered reflections in the previous section was an extrinsic construction—working from outside the space (x, τ, ≤)—which produced the ti-ordered reflection as a quotient based on the intersection of all suitable preorders on x for which the indicated quotient construction would yield a ti-ordered space. in this section, we present an intrinsic construction of the t0-ordered reflection and discuss some other properties of the t0-ordered reflection. intrinsic constructions of the other ti-ordered reflections (i > 0) studied in the previous section appear to be much more complicated. in a t0-ordered space, d(x) = d(y) and i(x) = i(y) would imply x = y. if our space is not t0-ordered, then there may be distinct elements x and y with d(x) = d(y) and i(x) = i(y). our strategy will be to say two such points are equivalent and mod out by this equivalence relation. suppose (x, τ, ≤) is an ordered topological space. for x, y ∈ x, define x ≈ y if and only if d(x) = d(y) and i(x) = i(y). order the set x/ ≈ of ≈-equivalence classes by the finite step order: [z0] ≤ 0 [zn] ⇐⇒ ∃[z1], [z2], . . . , [zn−1] and ∃z ′ i, z ∗ i ∈ [zi](i = 0, 1, . . . , n) with z′i ≤ z ∗ i+1 ∀i = 0, 1, . . . , n − 1. ti-ordered reflections 213 first note that this is indeed antisymmetric and therefore is a partial order: suppose [z0] ≤ 0 [zn] and [zn] ≤ 0 [z0]. then there exist [zi] (i = 1, . . . , n, . . . , m) with [z0] = [zm] and there exist z ′ i, z ∗ i ∈ [zi] such that z ′ i ≤ z ∗ i+1 for all i = 0, . . . , m − 1. to show [z0] = [zn], suppose not. then either d(z0) 6= d(zn) or i(z0) 6= i(zn). now z′i ≤ z ∗ i+1 ⇒ z ∗ i+1 ∈ i(z ′ i) ⇒ i(z ∗ i+1) ⊆ i(z ′ i) ⇒ i(zi+1) ⊆ i(zi). applying this for i = 0, . . . , m − 1 gives i(z0) ⊇ i(z1) ⊇ · · · ⊇ i(zm) = i(z0). thus, i(zi) = i(z0) ∀i ∈ {1, . . . , m}. dually, d(zi) = d(z0) ∀i ∈ {1, . . . , m}. it follows that zi ≈ z0 ∀i ∈ {1, . . . , m}, so [z0] = [zn], and ≤ is antisymmetric. in fact, the argument above shows that [x] ≤0 [y] ⇒ i(y) ⊆ i(x) and d(x) ⊆ d(y). at this point, one can verify that the equivalence relation ≈ agrees with ∼0 introduced in the previous section and that the finite step order described above agrees with the order ≤0 defined in the previous section by [a] ≤0 [b] if and only if a �0 b where g(�0) = ⋂ p0, and thus the t0-ordered reflection of (x, τ, ≤) is (x/ ≈, τ/ ≈, ≤0). however, we will continue our intrinsic approach and prove this directly. it is easy to show that any closed or open monotone set in x is ≈-saturated and that the quotient map f : x → x/ ≈ carries closed increasing sets to closed increasing sets and open increasing sets to open increasing sets. the dual statement (obtained by replacing “increasing” by “decreasing”) also holds. it follows that f is an ordered quotient map as defined in definition 6 of [6]. it is easily verified that if d = {f−1(y) : y ∈ x/ ≈} is the decomposition of x associated with the quotient map f : x → x/ ≈, then for each [x] ∈ d and each increasing (decreasing) open set u containing [x], there exists a saturated increasing (decreasing) open set containing [x] which is contained in u. we have a is closed and increasing in x if and only if f(a) is closed and increasing in x/ ≈, and b is closed and increasing in x/ ≈ if and only if f−1(b) is closed and increasing in x. furthermore, because i(x) = ⋂ c where c is the collection of closed increasing sets containing x and f( ⋂ c) = ⋂ f(c) for any collection c of ≈-saturated sets, it follows that f(i(x)) = ix/≈([x]). dually, f(d(x)) = dx/≈([x]). theorem 3.1. suppose (x, τ, ≤) is a partially ordered topological space, and a ≈ b if and only if d(a) = d(b) and i(a) = i(b). then x/ ≈ with the quotient topology and the finite-step order is the t0-ordered reflection of x. proof. first we will show that x/ ≈ is t0-ordered. suppose ix/≈([x]) = ix/≈([y]) and dx/≈([x]) = dx/≈([y]). if f : x → x/ ≈ is the natural ordered quotient map, then we have f(i(x)) = f(i(y)) and f(d(x)) = f(d(y)). applying f−1 to the equalities above and recalling that i(x) and d(x) are saturated, we have i(x) = i(y) and d(x) = d(y), which implies [x] = [y]. thus, x/ ≈ is t0-ordered. now suppose y is any t0-ordered space and g : x → y is continuous and increasing. we will show that {g−1(y) : y ∈ y } is saturated with respect 214 h.-p. a. künzi and t. a. richmond to d = {f−1([x]) : [x] ∈ x/ ≈}. suppose to the contrary that there exists y ∈ y such that g−1(y) is not d-saturated. then there exist b ∈ g−1(y) and a ∈ x \ g−1(y) such that [a] = [b] (that is, f(a) = f(b)). now g−1(iy (g(b)) is a closed increasing set in x which contains g −1(g(b)) and therefore contains b. but [a] = [b] ⇒ i(a) = i(b) ⇒ a is an element of every closed increasing set containing b ⇒ a ∈ g−1(iy (g(b)) ⇒ g(a) ∈ iy (g(b)) ⇒ iy (g(a)) ⊆ iy (g(b)). repeating the argument of the last paragraph with a and b interchanged shows the reverse inclusion, so iy (g(a)) = iy (g(b)). the dual argument shows that dy (g(a)) = dy (g(b)). since y is t0-ordered, this implies g(a) = g(b), contrary to a ∈ x \ g−1(y) and b ∈ g−1(y). now since {g−1(y) : y ∈ y } is d-saturated, there is a natural quotient map h from x/d = x/ ≈ to y , and hf = g. from the definition of the finite step order on x/ ≈, it is clear that h is increasing, and h is clearly unique from the construction. thus, x/ ≈ is the t0-ordered reflection of x. � the theorem below characterizes those spaces whose t0-ordered reflections are t1-ordered. similar results for the non-ordered setting can be found in [1], where a t(i,j)-space is defined to be one whose ti-reflection satisfies the tj separation axiom (0 ≤ i < j ≤ 2). comparing theorem 3.5(iv) of [1] with theorem 2(b) of [2], we note that t(0,1)-spaces have been studied by davis and others subsequently under the name of r0-spaces. theorem 3.2. the following are equivalent. (a) the t0-ordered reflection x/ ≈ of x is t1-ordered. (b) [x] 6≤0 [y] in x/ ≈ implies there exists an open increasing neighborhood of x not containing y and there exists an open decreasing neighborhood of y not containing x. (c) i([x]) = ⋂ {n : n is an open increasing neighborhood of x} for any x ∈ x, and d([x]) = ⋂ {n : n is an open decreasing neighborhood of x} for any x ∈ x. proof. (a) ⇒ (c): because closed or open increasing sets are saturated, we have i([x]) ⊆ ⋂ {n : n is an open increasing neighborhood of x}. suppose m = ⋂ {n : n is an open increasing neighborhood of x} 6⊆ i([x]). then there exists y ∈ m \ i([x]), and since m is saturated, [y] 6⊆ i([x]). in particular, [x] 6≤0 [y] in the t1-ordered space x/ ≈, so there exists an increasing open neighborhood j of [x] in x/ ≈ disjoint from [y]. now if f : x → x/ ≈ is the quotient map, f−1(j) is an open increasing neighborhood of x disjoint from y. this contradicts y ∈ m. this proves that i([x]) = ⋂ {n : n is an open increasing neighborhood of x} for any x ∈ x. the other statement is proved dually. ti-ordered reflections 215 (c) ⇒ (a): suppose (c). if x/ ≈ is not t1-ordered, then there exist [x] 6≤ 0 [y] such that either (i) every increasing open neighborhood of [x] in x/ ≈ contains [y], or (ii) every decreasing open neighborhood of [y] in x/ ≈ contains [x]. if (i) holds, then [y] ∈ ⋂ {n : n is an open increasing neighborhood of x} = i([x]), contrary to [x] 6≤0 [y]. if (ii) holds, then [x] ∈ ⋂ {n : n is an open decreasing neighborhood of y} = d([y]), contrary to [x] 6≤0 [y]. (a) ⇒ (b): suppose (a). now [x] 6≤0 [y] in x/ ≈ implies there exists an open increasing (respectively, decreasing) neighborhood of [x] (respectively, [y]) not containing [y] (respectively, [x]). taking f−1 of these neighborhoods gives the desired neighborhoods in x. (b) ⇒ (a): if [x] 6≤0 [y] in x/ ≈, then by (b) there exists an open increasing neighborhood n of x not containing y and there exists an open decreasing neighborhood m of y not containing x. now m and n are saturated, and since f is an ordered quotient map, f(m) and f(n) are monotone open neighborhoods of [y] and [x], respectively, which show that x/ ≈ is t1-ordered. � a set a which satisfies a = i(a) ∩ d(a) is called a c-set. in [4], maximal filters of c-sets are used to construct the wallman ordered compactification of an ordered space with convex topology. the wallman ordered compactification w0x is a universal compact t1 extension. in [5], conditions involving c-sets are given to insure w0x is t1-ordered. thus, one might expect c-sets to play a role in the t1-ordered or even t0-ordered reflection. let c(a) = i(a) ∩ d(a), that is, let c(a) be the smallest c-set containing a. proposition 3.3. suppose (x, τ, ≤) is an ordered topological space and let ≈ be the equivalence relation on x defined by x ≈ y if and only if d(x) = d(y) and i(x) = i(y). then x ≈ y if and only if c(x) = c(y). proof. suppose c(x) = c(y). then x ∈ c(y) ⊆ i(y), so i(x) ⊆ i(y). interchanging x and y shows that i(y) ⊆ i(x), so i(x) = i(y). dually, d(x) = d(y), so x ≈ y. the converse is immediate. � thus, the equivalence classes of the t0-ordered reflection are determined by the closure operator c(·). if x has a convex topology, this closure operator is especially nice. theorem 3.4. if the ordered topological space (x, τ, ≤) has a convex topology, then the topological space (x′, τ′) underlying its t0-ordered reflection (x ′, τ′, ≤′) is simply the t0 reflection of (x, τ). proof. suppose x has a convex topology. we will show that c(x) = i(x) ∩ d(x) = cl{x}. clearly y ∈ cl{x} ⇒ y ∈ i(x)∩d(x). for the converse, suppose y 6∈ cl{x}. then there exist an increasing open neighborhood ny of y and a decreasing open neighborhood my of y with x 6∈ ny ∩my. thus, either x 6∈ ny or x 6∈ my, and taking complements shows that y 6∈ d(x) or y 6∈ i(x), that is, y 6∈ i(x) ∩ d(x), as needed. by proposition 3.3, x ≈ y if and only if cl{x} = cl{y}. it is well-known that the t0 reflection of (x, τ) is given by the quotient topology on the quotient set x/ ≃ where x ≃ y if and only if cl{x} = cl{y}. � 216 h.-p. a. künzi and t. a. richmond references [1] k. belaid, o. echi, and s. lazaar, t(α,β)-spaces and the wallman compactification, internat. j. math. & math. sci. 2004 (68) (2004), 3717–3735. [2] a. s. davis, indexed systems of neighborhoods for general topological spaces, am. math. monthly 68 (9) (1961), 886–893. [3] h. herrlich and g. strecker, “categorical topology—its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971”, in handbook of the history of general topology, c.e. aull and r. lowen (eds.), volume 1, kluwer academic publishers, 1997, 255–341. [4] d. c. kent, on the wallman order compactification, pacific j. math. 118 (1985), 159– 163. [5] d. c. kent and t. a. richmond, separation properties of the wallman ordered compactification, internat. j. math. & math. sci. 13 (2) (1990), 209–222. [6] d. d. mooney and t. a. richmond, ordered quotients and the semilattice of ordered compactifications, proceedings of the tennessee topology conference, p. r. misra and m. rajagopalan (eds.), world scientific inc., 1997, 141–155. [7] l. nachbin, “topology and order”, van nostrand math. studies 4, princeton, n.j., (1965). received january 2005 accepted january 2005 hans-peter a. künzi (kunzi@maths.uct.ac.za) department of mathematics and applied mathematics, university of cape town, rondebosch 7701, south africa. thomas a. richmond (tom.richmond@wku.edu) department of mathematics, western kentucky university, bowling green, ky 42101, usa. zvinaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 51-66 on i-topological spaces: generalization of the concept of a topological space via ideals irina zvina abstract. the aim of this paper is to generalize the structure of a topological space, preserving its certain topological properties. the main idea is to consider the union and intersection of sets modulo “small” sets which are defined via ideals. developing the concept of an i-topological space and studying structures with compatible ideals, we are concerned to clarify the necessary and sufficient conditions for a new space to be homeomorphic, in some certain sense, to a topological space. 2000 ams classification: 54a05, 54e99. keywords: compatible ideal, generalization, topological space. 1. introduction the use of the ideals in general topology historically developed along two main lines. the first line is concerned with the study of the local properties of topological spaces that may be extended to the global properties [7, 12]. the central concept in these investigations is the compatibility of an ideal with a topology. for example, the well-known banach category theorem is an immediate corollary from the σ-extension theorem [7]. the works of the second line use ideals to generalize the certain properties of topological spaces, such as a compactness[9, 10, 11] and the separation axioms[1]. the aim of this paper is to generalize, via ideals, the concept of a topological space itself, thus it differs from the works of two mentioned lines. however, we make use of the ideas, developed in those works. in the first section, we define an i-topological space, provide basic examples, introduce the notions of a subspace, an i-continuous mapping, a scattered set and a nowhere dense set for such spaces and study properties of these notions. we propose a sufficient condition for the existence of a supratopological space [8] that is i-homeomorphic to a given i-topological space. in the second 52 i. zvina section, we introduce the concept of a unified i-topological space and discuss two important examples. in the last section, we study some properties of i-topological spaces with compatible ideals. we propose a construction which shows that for every such space there exists an i-homeomorphic supratopological space such that for every two its i-open subsets u and v there is an element a in the ideal for which (u ∩ v ) \ a is i-open. on the other hand, we show that the compatibility is necessary for the existence of a topological space that is i-homeomorphic to a given i-topological space. we introduce a sufficient condition for the existence of such a space and leave as an open question whether there exists an i-topological space with a compatible ideal which is not i-homeomorphic to any topological space. we refer a reader to the papers [3]-[6] by dragan janković and t.r. hamlett for the survey on the use of ideals in topological spaces. these works also provide the historical background and contain many references to the related papers. 2. general notions of i-topological spaces 2.1. i-topological spaces. subspaces. a nonempty family i of subsets of a set x is called an ideal on x iff x /∈ i, a,b ∈ i implies a ∪ b ∈ i, and a ∈ i and b ⊆ a imply b ∈ i. we define the relations ≤ and ≈ on p(x) as follows: (1) a ≤ b iff a \ b ∈ i, (2) a ≈ b iff (a \ b) ∪ (b \ a) ∈ i, where a,b ⊆ x. one can easily check that ≤ and ≈ are a preorder and an equivalence on p(x), respectively. definition 2.1. let x be a set and i be an ideal on x. then an i-topology on x is a family t of subsets of x that satisfies the following conditions: (t1) ∅,x ∈ t ; (t2) for any u ⊆ t there exists u ∈ t such that ⋃ u ≈ u; (t3) for any v,w ∈ t there exists u ∈ t such that v ∩w ≈ u; (t4) t ∩ i = {∅}. the triple (x,t ,i) is called an i-topological space and the elements of t are called i-open sets. we will use the notation t (x) = { u ∈ t | x ∈ u } for any x ∈ x. let us give some basic examples. example 2.2. suppose we are given a topological space (x,t ). then (x,t ,{∅}) and (x,tr,in) are i-topological spaces, where tr is the family of all regular open sets and in is the family of all nowhere dense sets. for the next example we need the following lemma. on i-topological spaces 53 lemma 2.3. let (x,t ) be a topological space and c ⊆ x. then there exist a,b ⊆ x and an open set u such that c = a ∪ b, the set a is nowhere dense and cl(b) = cl(u). proof. it is sufficient to take u = int(cl(c)), b = c∩u and a = c\b. then a is nowhere dense, since a = c \ b = c \ (c ∩ int(cl(c))) = c \ int(cl(c)). to complete the proof, it remains to show that ux ∩ b 6= ∅ for each x ∈ u and any its open neighborhood ux. observe that ux ∩ u is also an open neighbourhood of x. since x belongs to the closure of c, it follows that ux ∩ u ∩ c 6= ∅ and hence ux ∩ b 6= ∅. � example 2.4. consider the real line r with the usual topology tu and the family in of all nowhere dense sets. then the triple (r,d,in) is an i-topological space, where d = { a ⊆ r | there exists u ∈ tu such that cl(a) = cl(u) }. obviously, this can be generalized to any topological space. however, in our paper we will only use to the example based on the real line with the usual topology. as we know the intersection of a family of ideals on the same set is an ideal. we provide an example where a family t is an i-topology with respect to two different ideals, but is not an i-topology with respect to their intersection. example 2.5. let x be the union of x1 = r × {1} and x2 = r × {2}. we construct a family t of subsets of x as follows: t = {∅,x} ∪ t1 ∪ t2, where t1 = { {x} ∪ b | x ∈ x1 and b ⊆ x2 } , t2 = {a ∪ {y} | a ⊆ x1 and y ∈ x2 } . then t is an i-topology on x with respect to the ideals p(x1) and p(x2). however, t is not an i-topology with respect to {∅}. now, our aim is to define the notion of a subspace. suppose we are given an i-topological space (x,t ,i) and a subset m ⊆ x. we will use the following notations: t |m = { m ∩ u | u ∈ t and m ∩ u /∈ i } ∪ {∅,m}, if m /∈ i then i|m = { m ∩ a | a ∈ i } and if m ∈ i then i|m = {∅}. the triple (m,t |m,i|m ) is called a subspace of the i-topological space (x,t ,i). proposition 2.6. every subspace of any i-topological space is an i-topological space. 54 i. zvina proof. fix an i-topological space (x,t ,i) and its subspace (m,t |m,i|m ). let us prove that i|m is an ideal. if a,b ∈ i then (m ∩a)∪(m ∩b) = m ∩ (a∪b) ∈ i|m . hence, we have finite additivity. now, if a ∈ i and b ⊆ m ∩a then b ⊆ a and by heredity b ∈ i. hence, b = m ∩ b ∈ i|m and we have heredity. let us prove that (m,t |m,i|m ) is an i-topological space. it follows from the definition, that ∅,m ∈ t |m . hence, we have (t1). suppose that c,d ⊆ x and c ≈ d. then m ∩ c ≈i|m m ∩ d. indeed, there exist a,b ∈ i such that d = (c ∪a)\b, since c ≈ d. hence, m ∩d = m ∩ ((c ∪ a) \ b) = ((m ∩ c) ∪ (m ∩ a)) \ (m ∩ b) ≈i|m m ∩ c. it follows from the definition of subspace, that for any family v = {vs}s∈s of the elements of t |m there exists a family u = {us}s∈s of the elements of t such that vs = m ∩ us for each vs ∈ v. take u from t such that ⋃ u ≈ u. then ⋃ v = m∩( ⋃ u ) and by the previous paragraph m∩( ⋃ u ) ≈i|m m∩u. therefore, ⋃ v ≈i|m m ∩ u. if v and u are finite, we take w ∈ t such that w ≈ ⋂ u and in the same way show that ⋂ u ≈i|m m ∩ w . thus, we have (t2) and (t3). to complete the proof, it remains to show that t |m and i|m satisfy (t4). suppose that v ∈ t |m ∩ i|m . then it follows from the definition of subspace, that v = ∅, since i|m ⊆ i. � 2.2. i-continuous mappings. suppose we are given two i-topological spaces (x,tx,ix) and (y,ty ,iy ). we say that a mapping f : x → y is i-continuous if it satisfies the following conditions: (n1) for every family {vs}s∈s of i-open subsets of y satisfying vs ∩ f(x) /∈ iy for each s ∈ s there exists a family {us}s∈s of nonempty i-open subsets of x such that us ≈ f −1(vs) holds for each s ∈ s and ⋃ s∈s us ≈ f −1 ( ⋃ s∈s vs ) ; (n2) f−1iy ⊆ ix. notice that our definition generalizes the usual definition of continuous mapping. indeed, if we consider topological spaces with the ideal {∅} and in (n1) examine families which consist of only one i-open set, we obtain the usual definition. the mapping f is called i-homeomorphism if f is a bijection and both f and f−1 are i-continuous. two spaces are called i-homeomorphic if there exists an i-homeomorphism of one space to the other. the following proposition is a natural consequence from the definition of i-continuous mapping. on i-topological spaces 55 proposition 2.7. let f : x → y and g : y → z be i-continuous mappings of i-topological spaces. assume that for any i-open set in y its intersection with f(x) does not lie in the ideal of the space y . then the composition g ◦ f is an i-continuous mapping. suppose we are given a family f = {(x,ts,is)}s∈s of i-topological spaces. an i-topological space (x,tx,ix) is called minimal in the family f if it is an element of f and the identity mapping id: (x,ts,is) → (x,tx,ix) is i-continuous for each s ∈ s. let fs : x → ys and (ys,ts,is) be a mapping and an i-topological space for each s ∈ s, respectively. then (x,tx,ix) is called an initial i-topological space generated by {fs}s∈s if it is minimal in the following family of i-topological spaces { (x,t ,i) | fs : x → ys is i-continuous for each s ∈ s }. the next proposition shows that our definition of continuity generalizes the usual definition in the sense of initial topology. we omit the proof. proposition 2.8. let (y,t ,i) be an i-topological space and f : x → y be a mapping. consider families tx = f −1t |f(x) and ix = f −1i|f(x). then (x,tx,ix) is an initial i-topological space generated by {f}. 2.3. scattered and nowhere dense sets. first, let us recall the definition of the set operator ψ [3]. the various properties of this operator are to be found in [3]. suppose we are given a topological space (x,t ) and an ideal i on x. then ψ : p(x) → p(x) is defined as follows. for any a ⊆ x, ψ(a) = { x ∈ x | there exists u ∈ t (x) such that u \ a ∈ i }. we use this definition for i-topological spaces. one can easily check that ψ(a) = ⋃ { u ∈ t | u ≤ a} holds for any a ⊆ x. if there is a chance for confusion, we will use indexes to specify the ideal or i-topology in the way we do it in proposition2.10 and lemma 2.12. now, let (x,t ,i) be an i-topological space and a be a subset of x. we say that x ∈ x is an isolated point of a if there exists u ∈ t (x) such that a ∩ u \ {x} ∈ i. we say that a is scattered if every its subset contains an isolated point. we say that a ⊆ x is nowhere dense if ψ(a ∪ b) = ∅ for each b ⊆ x such that ψ(b) = ∅. in what follows we will use the notations s(i,t ) = { a ⊆ x | a is scattered } and n(i,t ) = { a ⊆ x | a is nowhere dense }. clearly, a subset is scattered or nowhere dense with respect to a certain i-topology and ideal. however, in case there is no chance for confusion, we will simply say scattered or nowhere dense and write s(i) and n(i) instead of s(i,t ) and n(i,t ). what is stated in the following proposition is an easy corollary from the definition of a scattered set. the result of proposition2.10 will be used later in the next section. proposition 2.9. let (x,t ,i) be an i-topological space. then s({∅}) ⊆ s(i) and i ⊆ s(i). 56 i. zvina proposition 2.10. let (x,t1,i1) and (x,t2,i2) be i-topological spaces. assume that, for any subset a ⊆ x, it holds that ψ1(a) = ∅ iff ψ2(a) = ∅ holds. then n(i1,t1) = n(i2,t2). proof. take b ∈ n(i1,t1) and a ⊆ x such that ψ2(a) = ∅. then by our assumption ψ1(a) = ∅ and hence ψ1(a ∪ b) = ∅, since b ∈ n(i1,t1). thus, under the assumption ψ2(a∪b) = ∅ and we have b ∈ n(i2,t2). in the similar way we can show that b ∈ n(i2,t2) implies b ∈ n(i1,t1) for each b ⊆ x. � the converse statement of the previous lemma does not hold. indeed, consider the i-topological spaces (r,tu,in) and (r,d,in) from example 2.2 and example 2.4. it is not difficult to check that in = n(in,tu) = n(in,d). however, ψtu(q) = ∅ and ψd(q) 6= ∅. the next proposition shows that in an i-topological space the ideal can be extended over the family of all nowhere dense sets such that together with the given i-topology it will satisfy the conditions of an i-topological space. moreover, this process is finite. proposition 2.11. let (x,t ,i) be an i-topological space. then (i) i ⊆ n(i); (ii) n(i) is an ideal; (iii) n(i) ∩ t = {∅}; (iv) n(n(i)) = n(i). proof. statement (i) is obvious. let us prove (ii). suppose that a, b and c are subsets of x such that a and b are nowhere dense and ψ(c) = ∅. then ψ(b∪ c) = ∅ and ψ(a ∪ (b ∪ c)) = ψ((a ∪ b) ∪ c) = ∅. hence, a ∪ b is nowhere dense. on the other hand, a subset of a nowhere dense set is nowhere dense. hence, we have finite additivity and heredity. suppose that u ∈ n(i) ∩ t . then ψ(u ∪ ∅) = ψ(u) = ∅, since ψ(∅) = ∅. on the other hand, ψ(v ) 6= ∅ for each nonempty v ∈ t . hence, u = ∅ and we have (iii). the last statement follows from lemma 2.12. � lemma 2.12. let (x,t ,i) be an i-topological space and a ⊆ x. then ψi(a) = ∅ iff ψn(i)(a) = ∅ for each a ⊆ x. proof. suppose that a is a subset of x, ψi(a) = ∅ and u is an i-open set such that u ⊆ ψn(i)(a). then there exists b ∈ n(i) such that u ⊆ a ∪ b. therefore, u ⊆ ψi(a∪b). on the other hand, ψi(a∪b) = ∅, since ψi(a) = ∅ and b is nowhere dense. thus, u = ∅ and ψn(i)(a) = ∅. now, notice that ψi(a) ⊆ ψn(i)(a) follows from i ⊆ n(i) for any subset a ⊆ x. hence, ψn(i)(a) = ∅ implies ψi(a) = ∅ for any a ⊆ x. � 2.4. existence of a supratopological space that is i-homeomorphic to a given i-topological space. let us recall that (x,t ), where x is a set and t is a family of subsets of x, is called supratopological space if x ∈ t and ⋃ u ∈ t for each u ⊆ t [8]. on i-topological spaces 57 suppose we are given an i-topological space (x,t ,i). then an operation α: p(t ) → t such that α(u) ≈ ⋃ u holds for each u ⊆ t is called associative if it satisfies α({ α( us ) | s ∈ s }) ≈ α ( ⋃ s∈s us ) for each collection {us}s∈s of families of i-open sets. to simplify our notations, we write αs∈s(α(us)) instead of α({ α( us ) | s ∈ s }). in what follows, we will use the notation t ∪ = { ⋃ u | u ⊆ t }. proposition 2.13. let (x,t ,i) be an i-topological space. then (x,t ∪,i) is an i-topological space. if there exists an associative operation α: p(t ) → t such that α(u) ≈ ⋃ u for each u ⊆ t then the spaces (x,t ,i) and (x,t ∪,i) are i-homeomorphic. proof. obviously, (t1) and (t4) are satisfied for the triple (x,t ∪,i) and (t2) is satisfied, since the family t ∪ is closed under the arbitrary unions. let us prove (t3). suppose that u,v ∈ t ∪. then there exist u,v ⊆ t and u1,v1 ∈ t such that u = ⋃ u ≈ u1 and v = ⋃ v ≈ v1. take w ∈ t satisfying w ≈ u1 ∩ v1. then w ≈ u ∩ v and (t3) is proved. to prove that the spaces (x,t ,i) and (x,t ∪,i) are i-homeomorphic under the assumption that there exists an associative operation α: p(t ) → t , such that α(u) ≈ ⋃ u for each u ⊆ t , we have to show that the identity mapping id: x → x satisfies (n1). fix some u ⊆ t ∪. by the definition, there exists a family vu ⊆ t , for each u ∈ u, such that u = ⋃ vu. consider the union v = ⋃ u∈u vu of all these families. clearly, v ⊆ t . then u ≈ α(vu ), for each u ∈ u, and ⋃ u ≈ ⋃ v ≈ α(v) ≈ αu∈u(α(vu )) ≈ ⋃ u∈u α(vu ). the proof is complete. � 3. unified i-topological spaces in this section, we will make use of the notion of a compatible ideal. let us recall the definition [6]. in a topological space (x,t ) an ideal i of subsets of x is said to be compatible with t , denote i ∼ t , if it satisfies the following condition for every subset a of x and every subfamily u of t if a ⊆ ⋃ u and u ∩ a ∈ i holds for every u ∈ u then a ∈ i. we preserve this definition for i-topologies: in i-topological space, the ideal is compatible with the i-topology if the above property is satisfied. an i-topological space (x,t ,i) is called unified if u ≈ v implies u = v for each i-open u and v . a triple (x,t1,i) is called a unification of (x,t ,i) if x ∈ t1 and for any u ∈ t the set { v ∈ t1 | u ≈ v } contains exactly one element. we omit the proofs of the results from the following lemma. the statement about the nowhere dense sets is an easy consequence from proposition2.10. 58 i. zvina proposition 3.1. let (x,t ,i) be an i-topological space and (x,t1,i) be its unification. then the following statements hold: (i) (x,t1,i) is a unified i-topological space; (ii) n(i,t ) = n(i,t1). if in additional i ∼ t then: (iii) i ∼ t1; (iv) (x,t ,i) and (x,t1,i) are i-homeomorphic. the easiest way to obtain an i-homeomorphic unified i-topological space for a given i-topological space (x,t ,i) with a compatible ideal is to choose one element from each class of equivalence which are considered with the respect to the equivalence relation ≈. notice that we can construct such i-topology without using the axiom of choice. indeed, we take ⋃ { v ∈ t | u ≈ v } for any u ∈ t . the compatibility of the ideal implies that the union of any class of equivalence is equivalent, in the sense of ≈, to each its member. in the last section of this paper we will propose another way how to obtain such i-topology without using the axiom of choice (see proposition4.12). example 3.2. consider the real line with the usual topology. clearly, the partial orders ⊆ and ≤, where ≤ is generated by the ideal of all nowhere dense sets, are not equivalent. indeed, a ≤ b does not imply a ⊆ b for each a,b ⊆ r. moreover, we will show that there does not exist family f of subsets of r such that f contains at least one element from each equivalence class generated by ≤, i.e. f ∩ [a ] = { b ⊆ r | a ≈ b } 6= ∅, and the condition a ≤ b implies a ⊆ b for each a,b ∈ f. in the process of proving of this statement we will show that in any unification of the i-topological space (r,d,in) from example 2.4 the ideal is not compatible with the i-topology. first, let us prove the following lemmas. lemma 3.3. consider the real line with the usual topology and the family d from example 2.4. for any nonempty a ⊆ r, whose closure is equal to the closure of some nonempty open set, there exist non-empty b,c ∈ d such that cl(a) = cl(b) = cl(c), b ∪ c ⊆ a and b ∩ c = ∅. proof. take u ∈ tu such that cl(a) = cl(u). to simplify the proof, we suppose that u = (0,1). in all other cases, the proof will be similar. let us define the sets {bn}n∈n and {cn}n∈n as follows. put b0 = ∅ and c0 = ∅. then bn = bn−1 ∪ ( ⋃2n−1 k=1 {bnk} ) and cn = cn−1 ∪ ( ⋃2n−1 k=1 {cnk} ) where bnk ∈ i n k ∩ (a1 \ (bn−1 ∪ cn−1)), c n k ∈ i n k ∩ (a1 \ (bn ∪ cn−1)), ink = ( k − 1 2n , k + 1 2n ) , a1 = a \ {0,1} and k ∈ {1, . . . ,2 n−1}. clearly, a1 is dense in u. thus, a complement in a1 of a union of two finite sets is dense in u. the sets bn and cn are finite for any n ∈ n. hence, on i-topological spaces 59 the intersections in the middle line above are not empty for any valid n and k. therefore, we can take points bn+1 k and cn+1 k for any k ∈ {1, . . . ,2n−1}. we define the sets b and c as follows: b = ⋃ n∈n bn and c = ⋃ n∈n cn. clearly, b ∪ c ⊆ a and b ∩ c = ∅. to complete the proof, it remains to show that b and c are dense in u. fix x ∈ u and ε > 0. we have to show that there exist n,k ∈ n such that in k ⊆ (x − ε,x + ε). take n ∈ n such that 1 2n−1 < ε. now, we can choose the necessary k, since intervals in k cover u, their lengths are equal to 1 2n−1 and length of two intervals is less than 2ε. � lemma 3.4. consider the i-topological space (r,d,in) from example 2.4. then in any its unification the ideal is not compatible with an i-topology. proof. suppose that (r,d1,in) is a unification of (r,d,in). take u = (0,1). define the sets {bn}n∈n and {c n}n∈n as follows. put c 0 = u. then take bn and cn such that they are dense in u, bn ∩ cn = ∅ and bn ∪ cn ⊆ cn−1 for each n ∈ n. we can choose such bn and cn, since the previous lemma holds. now, take b̂n ∈ d1 such that b̂ n ≈ bn for each n ∈ n. then b̂n∪cn ≤ cn−1. define a as follows: a = ⋃ n∈n an, where a0 = ∅, an = ⋃2n−1 k=1 {ank}, ank ∈ i n k ∩ ( b̂n \ ( ⋃n−1 k=1 b̂k )) and k ∈ {1, . . . ,2n−1}. then a is dense in u and hence a /∈ i. clearly, a ⊆ ⋃ n∈n b̂ n and a∩b̂n ∈ i for each n ∈ n. thus, in ≁ d1. � lemma 3.5. consider the i-topological space (r,d,in) from example 2.4. assume that (r,d1,in) is its unification and u ≤ v implies u ⊆ v for each u,v ∈ d1. then in ∼ d1. proof. fix u ∈ d1 and v ⊆ d1 such that u ∩v ∈ in for each v ∈ v. it follows from lemma 2.3, that there exists w ∈ d1 such that u ∩w ∈ i and v ≤ w for each v ∈ v. hence, ⋃ v ⊆ w and ( ⋃ v ) ∩ u ∈ in. observe that in = n(in). then in ∼ d1 is an immediate corollary from lemma 4.4. � comparing the results of lemma 3.4 and lemma 3.5, we conclude that in any unification (r,d1,in) of the i-topological space (r,d,in) the condition u ≤ v does not imply u ⊆ v for each u,v ∈ d1. to complete the proof that there does not exist family f of subsets of r such that f contains at least one element from each equivalence class generated by ≤ and the condition a ≤ b implies a ⊆ b for each a,b ∈ f it remains to observe that for any f there exists a unification of (r,d,in) such that its i-topology is a subfamily of f. thus, by the result from the previous paragraph such family f does not exist. example 3.6. consider the real line r with the usual topology tu. clearly, an i-topological space (r,tu,i) is unified iff i = {∅}. now, let us assume that i is an ideal of subsets of r such that ⋃ i = r and there exists another 60 i. zvina topology t on the real line, which is a unified i-topology with respect to i, and the spaces (r,tu,i) and (r,t ,i) are i-homeomorphic. then (r,t ,i) possesses the following properties. (i) u ≤ v implies u ⊆ v for each u,v ∈ t . (ii) for any x ∈ r, there exist u,v ∈ t such that u ≈ (−∞,x), v ≈ (x,+∞), x ∈ u ∪ v and x /∈ u ∩ v . (iii) define two mappings f : r → t and g : r → {0,1} as follows: for any x ∈ r, f(x) ≈ (−∞,x), and g(x) = 0, if x /∈ f(x), or g(x) = 1, if x ∈ f(x). the both mappings are defined correctly. (iv) for any x ∈ r satisfying g(x) = 1, there exists δ(x) > 0 such that x ∈ f(y) for each y ∈ (x −δ(x),x) and x /∈ f(y) for each y ∈ (−∞,x − δ(x)). we allow δ(x) to be equal to +∞. (v) at least one of the two sets a = { x ∈ r | g(x) = 1 } and b = { x ∈ r | g(x) = 0 } is dense. (vi) if a is dense then for any x ∈ r satisfying g(x) = 1 there exist y ∈ (x−δ(x),x) and a decreasing sequence (yn)n∈n ⊆ (y,x) such that (yn)n∈n converges to y and (yn)n∈n ⊆ f(y). however, we leave as an open question whether there exists such topological space (r,t ,i). 4. i-topological spaces with compatible ideals in this section, we concentrate our attention on i-topological spaces with compatible ideals. the corresponding definition is to be found at the beginning of the second section. 4.1. some general properties. the propositions of this subsection provide some properties of i-topological spaces with compatible ideals that are wellknown for topological spaces[2]. suppose we are given an i-topological space (x,t ,i). then an operation β : p(t ) → t such that β(u) ≈ ⋂ u for each finite u ⊆ t is called distributive over the union operation if it satisfies ⋃ v ∈v β({u,v }) ≈ u ∩ ( ⋃ v ) for each i-open set u and each collection of i-open sets v. proposition 4.1. let (x,t ,i) be an i-topological space satisfying i ∼ t . then: (i) ⋃ u ≈ ⋃ v holds for any two families of i-open sets u = {us}s∈s and v = {vs}s∈s which satisfy us ≈ vs for each s ∈ s; (ii) any operation α: p(t ) → t which satisfies α(u) ≈ ⋃ u for each u ⊆ t is associative; on i-topological spaces 61 (iii) any operation β : p(t ) → t which satisfies β(u) ≈ ⋂ u for each finite u ⊆ t is distributive over the union operation. proof. the proofs of all statements are not difficult, so we demonstrate just one of them. let us prove (iii). suppose we are given an i-open set u and a family of i-open sets v. clearly, β({u,v }) ≤ u and β({u,v }) ≤ ⋃ v for each v ∈ v. since i is compatible with t , it follows that ⋃ v ∈v β({u,v }) ≤ u and ⋃ v ∈v β({u,v }) ≤ ⋃ v. therefore, ⋃ v ∈v β({u,v }) ≤ u ∩ ( ⋃ v). on the other hand, if we take a = (u ∩( ⋃ v))\( ⋃ v ∈v β({u,v })) then a ⊆ ⋃ v and v ∩ a ∈ i for each v ∈ v. since i is compatible with t , it follows that a belongs to the ideal. thus, ⋃ v ∈v β({u,v }) ≤ u ∩ ( ⋃ v) and we complete the proof. � a family b ⊆ t is called a base for the i-topology t if for any u ∈ t there exists u ⊆ b such that u ≈ ⋃ u. a family c ⊆ t is called a subbase for the i-topology t if there exists b ⊆ t such that b is a base for t and for any v ∈ b there is a finite family v ⊆ c such that v ≈ ⋂ v. we omit the proof of the following proposition. proposition 4.2. let (x,tx,ix) and (y,ty ,iy ) be i-topological spaces with compatible ideals and f : x → y be a mapping such that fix = iy . then the following conditions are equivalent: (i) f is i-continuous; (ii) inverse image of any member of a subbase c for ty such that its intersection with f(x) does not lie in iy is equivalent to some nonempty i-open set from tx; (iii) inverse image of any member of a base b for ty such that its intersection with f(x) does not lie in iy is equivalent to some nonempty i-open set from tx; (iv) inverse image of any i-open set from ty such that its intersection with f(x) does not lie in iy is equivalent to some nonempty i-open set from tx. 4.2. scattered and nowhere dense sets. the aim of this subsection is to show some properties of families of scattered and nowhere dense sets in i-topological spaces with compatible ideals. proposition 4.3. let (x,t ,i) be an i-topological space such that ⋃ i = x and i ∼ t . then i = s(i). proof. the inclusion i ⊆ s(i) is obvious. let us prove s(i) ⊆ i. consider a ∈ s(i). let b ⊆ a be a set of all points of a which are isolated from a. then b ∈ i. by the definition of scattered set there exists a point x ∈ a \ b which is isolated from a \ b. since a ≈ a \ b, it follows that any point isolated from a \ b is isolated from a, too. then a \ b = ∅ and hence a ∈ i. � 62 i. zvina lemma 4.4. let (x,t ,i) be an i-topological space. then the following statements are equivalent: (i) if a ⊆ x and there exists u ⊆ t such that a ⊆ ⋃ u and u ∩ a ∈ i for each u ∈ u then a is nowhere dense; (ii) if v ∈ t and there exists u ⊆ t such that u ∩ v ∈ i for each u ∈ u then ( ⋃ u ) ∩ v is nowhere dense; (iii) if a ⊆ x and there exists u ⊆ t such that a ⊆ ⋃ u and u ∩ a is nowhere dense for each u ∈ u then a is nowhere dense. proof. to prove that (i) implies (ii), it is sufficient to observe that the condition from (i) is satisfied for ⋃ u∈u(v ∩ u). let us prove that (ii) implies (iii). suppose that b ⊆ x and ψ(b) = ∅. then ψ((a ∩ u) ∪ b) = ∅ for each u ∈ u. take v ∈ t such that v ≤ a ∪ b. then u ∩ v ≤ (a ∩ u) ∪ b and hence u ∩ v ∈ i. it follows form (ii), that ( ⋃ u ) ∩ v is nowhere dense. under the assumption that ψ(b) = ∅ we conclude that v = ∅, since v ≤ ( ( ⋃ u ) ∩ v ) ∪ b. the fact that (iii) implies (i) is obvious, since every element of an ideal is nowhere dense. � the next corollary is an immediate consequence from the previous lemma. corollary 4.5. let (x,t ,i) be an i-topological space. then n(i) ∼ t if one of the following conditions is satisfied: (1) i ∼ t ; (2) u ∩ v ∈ i implies u ∩ v = ∅ for each u,v ∈ t ; (3) there exists an operation β : p(t ) → t such that β(u) ≈ ⋂ u for each finite u ⊆ t which is distributive over the union operation. 4.3. existence of a topological space that is i-homeomorphic to a given i-topological space. in this subsection, we provide necessary and sufficient conditions for the existence of a topological space that is i-homeomorphic to a given i-topological space. in what follows, we will need the next lemma. lemma 4.6. let (x,tx,ix) and (y,t ,i) be i-topological spaces, f : x → y be a surjective i-continuous mapping such that fix = i. assume that ix ∼ tx. then i ∼ t . proof. consider a subset a ⊆ y and a family v = {vs}s∈s of i-open subsets of y such that a ⊆ ⋃ v and a ∩ vs ∈ i for each vs ∈ v. then there exists a family u = {us}s∈s of i-open subsets of x such that f −1( ⋃ v) ≈ ⋃ u and f−1(vs) ≈ us for each us ∈ u. clearly, f −1(a) ∩ us lies in the ideal ix for every s ∈ s. then it follows from ix ∼ tx, that f −1(a) ∩ ( ⋃ u) ∈ ix. on the other hand, f −1(a) ⊆ f−1( ⋃ v) and f−1( ⋃ v) ≈ ⋃ u. consequently, f−1(a) ∈ ix and by the assumption a ∈ i. � on i-topological spaces 63 now, we refer the reader to the paper by dragan janković and t.r. hamlett [6]. let (x,t ) be a topological space and i be an ideal on x satisfying t ∩i = {∅}. then there exists an extension ĩ of the ideal i such that t ∩ ĩ = {∅} and the ideal ĩ is compatible with t . this is an immediate corollary from theorems 3.1. and 3.5. in [6]. we use this result to prove the following proposition. proposition 4.7. let (x,t ,i) be an i-topological space. assume that there is a topological space, i-homeomorphic to (x,t ,i). then there exists a compatible extension ĩ of the ideal i such that ĩ ∩ t = {∅}. proof. let (y,ty ,iy ) and h: x → y be the corresponding topological space and i-homeomorphism, respectively. take ĩ = h−1ĩy , where ĩy is a compatible extension of the ideal iy defined in [6]. the statements that ĩ is an ideal, i ⊆ ĩ and ĩ ∩ t = {∅} are the consequences from the facts that ĩy is an ideal satisfying ĩy ∩ ty = {∅} and h is an i-homeomorphism. the fact that ĩ is compatible with t is an immediate corollary from lemma 4.6. � given an i-topological space (x,t ,i), we say that a family a ⊆ p(x) covers a set a ⊆ x if a ≤ ⋃ a. the following corollary is a natural consequence from the previous proposition. corollary 4.8. let (x,t ,i) be an i-topological space with a compatible ideal. assume that there is a nonempty i-open set u such that it can be covered with i-open sets for which the intersection of u and each element of the cover lies in the ideal. then there does not exist a topological space, i-homeomorphic to (x,t ,i). consider (r,d,in) from example 2.4. the family every element of which is of the form (r \ q) ∪ {x}, x ∈ q, consists of open sets. clearly, this family covers q and every intersection of q and an element of the family belongs to the ideal. thus, by the previous corollary (r,d,in) is not i-homeomorphic to any topological space. now, we provide one particular construction for i-topological spaces with compatible ideals and prove a sufficient condition for the existence of a topological space that is i-homeomorphic to a given i-topological space. first, let us recall two related notions [5]. for a topological space (x,t ) and an ideal i on x, an operator ∗ : p(x) → p(x) is said to be a local function of i with respect to t iff a∗ = { x ∈ x | u ∩ a /∈ i for each u ∈ t (x) } for any a ⊆ x. the various properties of this operator are to be found in [5]. an extension t ∗ of the topology t is defined as follows: t ∗ = { u \ a | u ∈ t and a ∈ i }. we preserve these definitions for i-topological spaces. we will exploit an operator ˜: p(x) → p(x) satisfying ã = ψ(a) ∩ a∗ for each a ⊆ x and use the notation t̃ = { ũ | u ∈ t }. notice that our use of tilde differs from the notation ĩ for ideals in [6]. we hope that there will be no chance for confusion. lemma 4.9. let (x,t ,i) be an i-topological space satisfying i ∼ t . then the following statements hold for any i-open u,v,w and u ⊆ t̃ : 64 i. zvina (i) u ≈ ũ; (ii) if ũ ≤ ṽ then ũ ⊆ ṽ ; (iii) if ũ ≈ ṽ then ũ = ṽ ; (iv) if ũ ∩ ṽ ∈ i then ũ ∩ ṽ = ∅; (v) if ũ ≈ ṽ ∩ w̃ then ũ ⊆ ṽ ∩ w̃; (vi) if ũ ≈ ⋃ u then ⋃ u ⊆ ũ. proof. statement (i) is obvious, since u ∈ ψ(u), ψ(u) \ u ∈ i and u \ u∗ ∈ i for each i-open u. first, we show that ũ ⊆ ṽ follows from u ≤ v . clearly, wx ∩ u /∈ i implies wx ∩ v /∈ i for each x ∈ u ∗ and any its i-open neighbourhood wx. hence, u∗ ⊆ v ∗. on the other hand, for any x ∈ ψ(u) there exists its i-open neighbourhood wx such that wx ≤ u. then wx ≤ v for each such wx and hence ψ(u) ⊆ ψ(v ). thus, we have ũ ⊆ ṽ . it follows from (i), that ũ ≤ ṽ implies u ≤ v . thus, by the previous paragraph we have (ii) and hence (iii). let us prove (iv). suppose that ũ ∩ ṽ ∈ i and x ∈ ũ ∩ ṽ . then there is an i-open neighbourhood wx of x such that wx ≤ u. it follows from (i), that wx ≤ ũ. hence, wx ∩ṽ ∈ i and wx ∩v ∈ i. then by the definition x /∈ v ∗ and hence x /∈ ṽ . the last statement contradicts our assumption, thus ũ ∩ ṽ = ∅. now, let us prove (v). suppose that ũ ≈ ṽ ∩ w̃ . then it follows from (i), that u ≈ v ∩ w and hence u ≤ v and u ≤ w . by the second paragraph of this proof we conclude that ũ ⊆ ṽ and ũ ⊆ w̃ . therefore, ũ ⊆ ṽ ∩ w̃ . finally, let us prove (vi). suppose ũ ≈ ⋃ u. then ṽ ≤ ũ for each ṽ ∈ u. it follows from (ii), that ṽ ⊆ ũ for each ṽ ∈ u. thus, ⋃ u ⊆ ũ. � proposition 4.10. let (x,t ,i) be an i-topological space satisfying i ∼ t . then ( x, t̃ ,i ) is a unified i-topological space such that i ∼ t̃ and the spaces (x,t ,i) and ( x, t̃ ,i ) are i-homeomorphic. proof. the facts that (t1), (t3) and (t4) hold for ( x, t̃ ,i ) are obvious. let us prove (t2). suppose that u ⊆ t and ũ = { ũ | u ∈ u }. then ( ⋃ u )\ ( ⋃ ũ ) belongs to the ideal i, since u \ ũ ∈ i for each u ∈ u and i is compatible with t . on the other hand, for any x ∈ ũ \ u there is an i-open neighbourhood ux such that ux ≤ u. take ux = { ux | x ∈ ũ and u ∈ u } and a = ( ⋃ ũ ) \ ( ⋃ u ). it follows that ux ∩ a ∈ i for each ux ∈ ux and, since i is compatible with t , ( ⋃ ux ) ∩ a belongs to i. thus, we have (t2). the fact that (n2) holds for the identity mapping id: x → x is obvious and (n1) follows from the previous paragraph of this proof. then it follows from lemmas 4.9 and 4.6, respectively, that ( x, t̃ ,i ) is unified and the ideal i is compatible with t̃ . � on i-topological spaces 65 notice that ˜̃ t = t̃ holds for each i-topological space (x,t ,i) with a compatible ideal. this is an immediate consequence from the definition of t̃ , the previous proposition and lemma 4.9. proposition 4.11. let (x,t ,i) be an i-topological space satisfying i ∼ t . then (x,t ∗,i) is an i-topological space such that i ∼ t ∗ and the spaces (x,t ,i) and (x,t ∗,i) are i-homeomorphic. proof. suppose that u = {us}s∈s ⊆ t ∗. then there are an i-open set u and a collection a = {as}s∈s of the elements of the ideal such that us ∪ as ∈ t for each s ∈ s and u ≈ ⋃ s∈s(us ∪ as). it follows from i ∼ t , that ⋃ a \ ⋃ u belongs to i. hence, ⋃ u ≈ ⋃ s∈s(us ∪ as) ≈ u. observe that u = u \ ∅ is an element from t ∗. thus, we have (t2). all other facts which it remains to show to complete the proof are obvious. � the next proposition is a natural corollary from propositions 4.12 and 4.11. in what follows, all results hold if we replace ( t̃ )∗ with ( t̃ )∪ . notice that by lemma 4.9 ( t̃ )∪ ⊆ ( t̃ )∗ . proposition 4.12. let (x,t ,i) be an i-topological satisfying i ∼ t . assume that p = ( t̃ )∗ . then the following statements hold: (i) (x,p,i) is an i-topological space with a compatible ideal; (ii) (x,t ,i) and (x,p,i) are i-homeomorphic; (iii) u ∩ v ∈ i implies u ∩ v = ∅ for each u,v ∈ p; (iv) ⋃ u ∈ p for each u ⊆ p; (v) for any u,v ∈ p there is a ∈ i such that (u ∩v )\a ∈ p. the following proposition gives us a sufficient condition for the existence of a topological space i-homeomorphic to a given i-topological space. notice that this condition implies compatibility of an ideal and it holds for any topological space. proposition 4.13. let (x,t ,i) be an i-topological space. assume that i = n(i) and for any x ∈ x and any u,v ∈ t (x) there is w ∈ t (x) such that w ≤ u ∩ v . then there exists a topological space, i-homeomorphic to (x,t ,i). proof. it follows from corollary4.5, that i ∼ t . first, let us show that ũ1 ∩ ũ2 ∈ t̃ for each ũ1, ũ2 ∈ t̃ . take x ∈ ũ1 ∩ ũ2. then there are two i-open neighbourhoods w1 and w2 of x satisfying w1 ≤ u1 and w2 ≤ u2. by the assumption there is the third i-open neighbourhood w of x such that w ≤ w1 ∩ w1. then w ≤ v , where v is an i-open set such that ṽ ≈ ũ1 ∩ ũ2. thus, x ∈ ψ(v ). on the other hand, wx ∩ w 6∈ i and hence wx ∩ v /∈ i for each i-open neighbourhood wx of x, since x ∈ u ∗ 1 . hence, x ∈ v ∗ and we have x ∈ ṽ . the point x is an arbitrary point from ũ1 ∩ ũ2, thus ũ1 ∩ ũ2 ⊆ ṽ and by lemma 4.9 ṽ = ũ1 ∩ ũ2. 66 i. zvina we observe that u ∩ v = w implies (u \ a) ∩ (v \ a) = w \ a for each subsets u,v,w and a. suppose that p = ( t̃ )∗ . then for any u,v ∈ p there exists w ∈ p such that u ∩ v = w . finally, it follows from proposition4.12 that (x,p,i) is a topological space, i-homeomorphic to (x,t ,i). � we leave as an open question whether there exists an i-topological space with a compatible ideal that is not i-homeomorphic to any topological space. references [1] f. g. arenas, j. dontchev, m. l. puertas, idealization of some weak separation axioms, acta math. hung. 89 (1-2) (2000), 47–53. [2] r. engelking, general topology, warszawa, 1977. [3] t. r. hamlett and d. janković, ideals in topological spaces and the set operator ψ, bollettino u.m.i. 7 (1990), 863–874. [4] t. r. hamlett and d. janković, ideals in general topology, general topology and applications, (middletown, ct, 1988), 115 – 125; se: lecture notes in pure & appl. math., 123 1990, dekker, new york. [5] d. janković and t. r. hamlett, new topologies from old via ideals, amer. math. monthly 97 (1990), 295–310. [6] d. janković and t. r. hamlett, compatible extensions of ideals, bollettino u.m.i. 7 (1992), 453–465. [7] d. janković, t. r. hamlett and ch. konstadilaki, local-to-global topological properties, mathematica japonica 52 (1) (2000), 79–81. [8] a. s. mashhour, a. a. allam, f. s. mahmoud, f. h. khedr, on supratopological spaces, indian j. pure appl. math. 14 (1983), 502–510. [9] r. l. newcomb, topologies which are compact modulo an ideal, ph.d. dissertation, univ. of cal. and santa barbara, 13 (1) 1972, 193–197. [10] d. v. rančin, compactness modulo an ideal, soviet math. dokl. 13 (1) (1972), 193–197. [11] s. solovjovs, topological spaces with a countable compactness defect (in latvian), bachelor thesis, univ. of latvia, riga, 1999. [12] r. vaidyanathaswamy, the localization theory in set-topology, proc. indian acad. sci. math sci. 20 (1945), 51–61. received may 2004 accepted april 2005 irina zvina (irinazvina@inbox.lv) institute of mathematics of latvian academy of sciences and university of latvia, akademijas laukums 1, riga, lv-1050, latvia. coratriagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 109-124 making group topologies with, and without, convergent sequences∗ w. w. comfort, s. u. raczkowski and f. j. trigos-arrieta abstract. (1) every infinite, abelian compact (hausdorff) group k admits 2|k|many dense, non-haar-measurable subgroups of cardinality |k|. when k is nonmetrizable, these may be chosen to be pseudocompact. (2) every infinite abelian group g admits a family a of 22 |g| -many pairwise nonhomeomorphic totally bounded group topologies such that no nontrivial sequence in g converges in any of the topologies t ∈ a. (for some g one may arrange w(g, t ) < 2|g| for some t ∈ a.) (3) every infinite abelian group g admits a family b of 22 |g| -many pairwise nonhomeomorphic totally bounded group topologies, with w(g, t ) = 2|g| for all t ∈ b, such that some fixed faithfully indexed sequence in g converges to 0g in each t ∈ b. 2000 ams classification: primary: 22a10, 22b99, 22c05, 43a40, 54h11. secondary: 03e35, 03e50, 54d30, 54e35. keywords: haar measure, dual group, character, pseudocompact group, totally bounded group, maximal topology, convergent sequence, torsion-free group, torsion group, torsion-free rank, p-rank, p-adic integers. 1. introduction 1.1. historical background. not long after e. čech and m. h. stone associated with each tychonoff space x its maximal compactification β(x) (the socalled stone-čech compactification), it was noted, denoting by ω the countably infinite discrete space, that β(ω) contains no nontrivial convergent sequence. this observation stimulated efimov [10] to pose in 1969 a question which in its full generality remains unsolved today: does every compact hausdorff space ∗ portions of this paper were presented by the first-listed author at the 2004 annual meeting of the american mathematical society (phoenix, january, 2004). 110 w. w. comfort, s. u. raczkowski and f. j. trigos-arrieta contain either a copy of β(ω) or a nontrivial convergent sequence? (in models of ⋄ the answer is negative [12]. see [34] for several additional relevant references.) the present paper is concerned with topological groups. in that context, a correct and natural companion to efimov’s question is this: given a class c of topological groups, does every group in c contain a nontrivial convergent sequence? there is an extensive literature on questions of this form. here are some samples of both positive and negative results. positive (a) according to a result to which šapirovskĭı, gerlits and efimov have contributed (see [35] for historical details and for an “elementary” proof), every infinite compact group k contains topologically a copy of the generalized cantor space {0, 1}w(k), hence contains a convergent sequence; (b) assuming gch, malykhin and shapiro [27] showed that every totally bounded group g with w(g) < (w(g))ω contains a nontrivial convergent sequence; (c) raczkowski [30], [31] and others [1], [29], [44] have shown that for every suitably fast-growing sequence xn ∈ z there is a totally bounded group topology on z with respect to which xn → 0. negative. (a) glicksberg [16] showed that when a locally compact abelian group (g,t ) is given its associated bohr topology (that is, the weak topology induced on g by (̂g,t )), no new compact sets are created; in particular, as shown earlier by leptin [26], the topology induced on a (discrete) abelian group by hom(g, t) has no infinite compact subsets, in particular has no nontrivial convergent sequence; (b) there are infinite pseudocompact topological groups containing no nontrivial convergent sequence [36]; see also the more recent papers [14] and [15] and the literature cited there for results in the same vein. perhaps the most celebrated unsolved question in this area of mathematics is this: does there exist in zfc a countably compact topological group with no nontrivial convergent sequence? (many examples are known in augmented axiom systems. see for example the constructions of van douwen [9], of hart and van mill [20], and of tomita [40], [41], and see also [8] for a characterization, in a forcing model of zfc + ch with 2c “arbitrarily large”, of those abelian groups which admit a hereditarily separable pseudocompact (alternatively, countably compact) group topology with no infinite compact subsets.) 1.2. outline. it is well known and easily proved that every uncountable abelian group k admits a subgroup h of index exactly ω; if k has a compact (hausdorff) group topology, such h is necessarily nonmeasurable with respect to the associated haar measure. it has been noticed by stromberg [39] that r contains a nonmeasurable subgroup of index | r|. in this spirit, we show in sections 2 and 3 that every infinite compactly generated abelian group k has a family of 2|k|-many dense, nonmeasurable subgroups of index |k|. we deduce from this that every infinite abelian group g admits a family of 22 |g| -many totally bounded group topologies with no nontrivial convergent sequence (theorem 4.1). we show also by very different methods that such g has the same number of totally bounded group topologies in each of which some nontrivial sequence (fixed, and chosen in advance) does converge (theorem 5.5). in the making group topologies with, and without, convergent sequences 111 obvious sense, these results are clearly optimal. an elementary cardinality argument shows that the various topological groups (g,t ) may be chosen to be pairwise nonhomeomorphic as topological spaces. 1.3. notation. the symbols κ and α denote infinite cardinals; ω is the least infinite cardinal, and c := 2ω. for s a set we write [s]κ := {a ⊆ s : |a| = κ}; the symbol [s]<κ is defined analogously. z and r denote respectively the group of integers and the group of real numbers, often with their usual (metrizable) topologies, and t := r/z. general groups g are written multiplicatively, with identity 1 = 1g, but groups g known or hypothesized to be abelian are written additively, with identity 0 = 0g. we identify each finite cyclic group with its copy in t; in particular for 0 < n < ω we write z(n) := { k n : 0 ≤ k < n} ⊆ t. we denote by p the set of primes. we work exclusively with tychonoff spaces, i.e., with completely regular, hausdorff spaces. a topological group which is a hausdorff space is necessarily a tychonoff space [22](8.4). a topological group (g,t ) is said to be totally bounded if for ∅ 6= u ∈ t there is f ∈ [g]<ω such that g = fu. it is a theorem of weil [42] that a topological group is totally bounded if and only if g embeds densely into a compact group; this latter group, unique in an obvious sense, is called the weil completion of g and is denoted g. given a topological group g = (g,t ), the symbol ĝ = (̂g,t ) denotes the set of continuous homomorphisms from g to the (compact) group t. a group g with its discrete topology is written gd, so ĝd = hom(g, t); this is a closed subgroup of the compact group tg. it is easily seen, as in [6](1.9), that for a subgroup h of hom(g, t) these conditions are equivalent: (a) h separates points of g; (b) h is dense in the compact group hom(g, t). for g abelian we denote by s(g) the set of point-separating subgroups of hom(g, t), and by t(g) the set of (hausdorff) totally bounded group topologies on g. theorem 1.2 below describes, for each infinite abelian group g, a useful order-preserving bijection between s(g) and t(g). the rank, the torsion-free rank, and (for p ∈ p) the p-rank of an abelian group g are denoted r(g), r0(g), and rp(g), respectively. we have r(g) = r0(g) + σp∈p rp(g), and when |g| > ω we have |g| = r(g) (cf. [13](§16) or [22](appendix a)). following those sources we write gp := ∪k<ω {x ∈ g : p k · x = 0}. we write x =h y if x and y are homeomorphic topological spaces, and we write g ≃ h if g and h are isomorphic groups; it is important to note that x =h y conveys no information about the algebraic structure of x or y (if any), and g ≃ h conveys no information about the topological structure of g or h (if any). we assume familiarity on the reader’s part with the essentials of haar measure λ = λk on a locally compact group k. the set of borel sets, and the set of λ-measurable sets, are denoted b(k) and m(k), respectively. of course b(k) ⊆ m(k). our convention is that haar measure is complete in the sense that if s ∈ m(k) with λ(s) = 0, then each a ⊆ s satisfies a ∈ m(k) (with 112 w. w. comfort, s. u. raczkowski and f. j. trigos-arrieta λ(a) = 0). we assume for simplicity that if k is compact then λ is normalized in the sense that λ(k) = 1. the following three theorems are essential to our argument. therem 1.1(a) is due to steinhaus [37] when g = r and to weil [43](p. 50) in the general case; see stromberg [38] for a pleasing, efficient proof. parts (b) and (c) are the immediate consequences which we use here frequently. theorem 1.1 ([37, 43]). let k be a locally compact group and let s be a λ-measurable subset of k with λ(s) > 0. then (a) the difference set ss−1 contains a neighborhood of 1k; (b) if s is a subgroup of k then s is open and closed in k; and (c) if s is a dense subgroup of k then s = k. theorem 1.2 ([6]). let g be an abelian group. (a) for every h ∈ s(g), the topology th induced on g by h is a (hausdorff ) totally bounded group topology such that w(g,th ) = |h|; (b) if (g,t ) is totally bounded then t = th with h := (̂g,t ) ∈ s(g). it is clear with g as in theorem 1.2 that distinct h0,h1 ∈ s(g) induce distinct topologies th0,th1 ∈ t(g)—indeed ̂(g,th0 ) = h0 6= h1 = ̂(g,th1 ); thus the bijection t(g) ↔ s(g) given by th ↔ h is indeed order-preserving. theorem 1.3 ([7]). let g be an infinite abelian group with k := hom(g, t) = ĝd, let (xn)n be a faithfully indexed sequence in g, and let a := {h ∈ k : h(xn) → 0}. then a is a subgroup of k such that a ∈ b(k) and λ(a) = 0. proof. [outline]. that a is a subgroup of k is obvious. according to the duality theorem of pontrjagin [28] and van kampen [25], the map gd ։ ̂̂ gd = ̂hom(g, t) given by x → x̂ (with x̂(h) = h(x) for x ∈ g, h ∈ k) is a bijection. writing an,m := {h ∈ k : |x̂n(h) − 1| < 1 m }, the relation a = ∩m<ω ∪n≥m ∩n≥n an,m expresses a as a gδσδ-subset of the compact group k, so a ∈ b(k). if g is torsion-free, a condition equivalent to the condition that hom(g, t) is connected (cf. [22](24.25)), then a reference to theorem 1.1 completes the proof: the condition λ(a) > 0 would imply a = hom(g, t), so that xn → 0 in the bohr topology of gd, contrary to the theorem of leptin and glicksberg cited above. we refer the reader to [7] for the proof (for general abelian g) that λ(a) = 0. � in what follows we will frequently invoke this simple algebraic fact. theorem 1.4. let k be an abelian group, let h be a subgroup of k of index α > ω, and let h := {s : h ⊆ s ⊆ k,s is a proper subgroup of k, |s| = |k|}. then |h| = 2α. making group topologies with, and without, convergent sequences 113 proof. the inequality ≤ is obvious. we have |k/h| = r(k/h) = α > ω, so algebraically k/h ⊇ ⊕ξ<α cξ with each cξ cyclic. let φ : k ։ k/h be the canonical homomorphism, and for a ∈ [α]α\{α} set ha := φ −1(⊕ξ∈a cξ × {0ξ}ξ∈α\a). the map [α] α\{α} → h given by a → ha is an injection, so |h| ≥ |[α]α| = 2α, as asserted. � theorem 1.3 explains our interest in the existence of (many) point-separating nonmeasurable subgroups of compact abelian groups k (of the form k = hom(g, t)): each such h ∈ s(g) will induce on g a totally bounded group topology without nontrivial convergent sequences. in order to show that such k admit 2|k|-many such subgroups, we find it convenient to treat separately the metrizable case (that is, w(k) = ω) and the nonmetrizable case (w(k) > ω). we do this in sections 2 and 3 respectively. 2. many nonmeasurable subgroups: the metrizable case for simplicity, and because it suffices for our applications, we take the groups k and m in lemma 2.1 and theorem 2.2 to be compact; the reader may notice that this hypothesis can be significantly relaxed. indeed both groups are abelian and m is metrizable in our applications, but since those hypotheses save no labor we omit them for now. lemma 2.1. let k and m be compact groups with haar measures λ and µ respectively, and let φ : k ։ m be a continuous surjective homomorphism. then µ(e) = λ(φ−1[e]) for every e ∈ b(m). proof. define m : b(m) → [0, 1] by m(e) := λ(φ−1[e]). according to [22](15.8), and using the numbering system there, it is enough to show that (iv) m(c) < ∞ for compact c ∈ b(m); (v) m(u) > 0 for some open u ∈ b(m); (vi) m(a + f) = m(f) for all a ∈ m, f ∈ b(m); and (vii) m(u) = sup{m(f) : f ⊆ u,f is compact} for open u ⊆ m, and m(e) = inf{m(u) : e ⊆ u,u is open} for e ∈ b(m). the verifications are routine and will not be reproduced here. in addition to [22], the reader seeking hints might consult [17](63c and 64h, or 52g and 52h). � theorem 2.2. let k and m be compact groups with haar measures λ and µ respectively, and let φ : k ։ m be a continuous, surjective homomorphism. if d is a dense, non-µ-measurable subgroup of m, then h := φ−1(d) is a dense, non-λ-measurable subgroup of k. proof. φ is an open map [22](5.29), so h is dense in k. suppose now that h ∈ m(k), so that either λ(h) > 0 or λ(h) = 0. if λ(h) > 0 then h = k by theorem 1.1(c) so d = φ[k] = m, a contradiction. if (h ∈ m(k) and) λ(h) = 0 then since λ is (inner-) regular there is a sequence kn (n < ω) of compact subsets of k\h such that λ(∪n kn) = λ(k\h) = 1. we write mn := φ[kn] and k̃n := φ −1(mn). then kn ⊆ k̃n ⊆ k\h and from lemma 2.1 we have 114 w. w. comfort, s. u. raczkowski and f. j. trigos-arrieta µ(∪n mn) = λ(∪n k̃n) ≥ λ(∪n kn) = 1, so µ(∪n mn) = 1 and hence d ∈ m(m) with µ(d) = 0, a contradiction. � our goal is to show that every (infinite) compact abelian metrizable group contains a dense, nonmeasurable subgroup of index c. we treat some special cases first. in what follows we denote the torsion subgroup of an abelian group k by t(k), for s ∈ k and 0 6= n ∈ z we write [ s n ] = {x ∈ k : nx = s}, and for a subgroup s of k we set ∆(s) := ∪{[ s n ] : s ∈ s, 0 6= n ∈ z}. when [ s n ] 6= ∅ we choose sn ∈ [ s n ], and we write λ(s) := {sn : s ∈ s, 0 6= n ∈ z} ∪ {0g}. then |λ(s)| ≤ |s| · ℵ0, and ∆(s) = λ(s) + t(k). lemma 2.3. let m be an abelian group such that |m| = κ > ω and let s ∈ [m]<κ, e ∈ [m]κ with s a subgroup. if either (i) |t(m)| < κ, or (ii) there is p ∈ p such that p · m = {0}, then there is x ∈ e such that 〈x〉 ∩ s = {0}. in case (i), x may be chosen in m\t(m). proof. (i) from ∆(s) = λ(s) + t(m) follows |∆(s)| < κ, and any x ∈ e\∆(s) ⊆ e\t(m) is as required. (ii) since m ≃ ⊕κ z(p) = ⊕ξ<κ z(p)ξ, there is a ∈ [κ] <κ such that s ⊆ ⊕ξ∈a z(p)ξ. any x ∈ e such that 0 6= x /∈ ⊕ξ∈a z(p)ξ is as required. � theorem 2.4. let m be an infinite, compact, metrizable, abelian group such that either (i) |t(m)| < c or (ii) there is p ∈ p such that p · m = {0}. then m has a dense, nonmeasurable subgroup d such that |m/d| = c. in case (i) one may arrange d ≃ ⊕c z, in case (ii) one may arrange d ≃ ⊕c z(p). proof. let {fξ : ξ < c} be an enumeration of all uncountable, closed subsets of m, and define eξ := fξ\t(m) in case (i), eξ := fξ\{0} in case (ii). it is a theorem of cantor [2](page 488) that each |fξ| = c (see [21](viii §9 ii) or [11](4.5.5(b)) for more modern treatments); hence each |eξ| = c. there is x0 ∈ e0, and by lemma 2.3 there is y0 ∈ e0 such that 〈x0〉 ∩ 〈y0〉 = {0}. now let ξ < c, suppose that xη, yη have been chosen for all η < ξ, and apply lemma 2.3 twice to choose xξ,yξ ∈ eξ such that 〈{xξ}〉 ∩ 〈{xη : η < ξ} ∪ {yη : η < ξ}〉 = {0}, and 〈{yξ}〉 ∩ 〈{xη : η ≤ ξ} ∪ {yη : η < ξ}〉 = {0}. thus xξ,yξ are defined for all ξ < c. we define d := 〈{xξ : ξ < c}〉. clearly d = ⊕ξ 0. then xξ ∈ eξ ∩ d ⊆ fξ ∩ d. thus d is dense in k. if d ∈ m(k) with λ(d) > 0 then d = k by theorem 1.1(c), contrary to the relation |k/d| = c. if d ∈ m(k) with λ(d) = 0 then λ(k\d) = 1 and there is fξ ⊆ k\d such that λ(fξ ) > 0; then xξ ∈ eξ ∩ d ⊆ (k\d) ∩ d = ∅, a contradiction. � corollary 2.5. let m be a (compact, abelian, metrizable) group of one of these types. (i) m = t; (ii) m = ∆p (p ∈ p), the group of p-adic integers; (iii) m = πk<ω z(pk), pk ∈ p, (pk)k faithfully indexed; (iv) m = (z(p))ω (p ∈ p). then m has a dense, nonmeasurable subgroup d such that |m/d| = c. proof. surely |t(t)| = ω, and t(πk<ω z(pk)) is the countable group ⊕k<ω z(pk). if 0 6= h ∈ ∆p = hom(z(p ∞), t), then h[z(p∞)] ≃ z(p∞)/ ker(h) ≃ z(p∞) since | ker(h)| < ω, so h[z(p∞)] is not of bounded order; thus t(∆p) = {0}. it follows for m as in (i), (ii) and (iii) that |t(m)| = 1 < c or |t(m)| = ω < c, so theorem 2.4(i) applies. for m as in (iv) surely p·m = {0}, so theorem 2.4(ii) applies. � theorem 2.6. let k be an infinite, compact, abelian metrizable group. then (|k| = c and) k has a dense, nonmeasurable subgroup h such that |h| = c and |k/h| = c. proof. the (discrete) dual group g = k̂ satisfies |g| = w(k) = ω. as with any countably infinite abelian group, g must satisfy (at least) one of these conditions: (i) r0(g) > 0: (ii) |gp| = ω with rp(g) < ω for some p ∈ p; (iii) 0 < rp(g) < ω for infinitely many p ∈ p; (iv) rp(g) = ω for some p ∈ p. according as (i), (ii), (iii) or (iv) holds we have, respectively, g ⊇ z, g ⊇ z(p∞), g ⊇ ⊕k<ω z(pk), or g ⊇ ⊕ω z(p), so taking adjoints we have a continuous surjection φ from k onto a group m of the form ẑ = t, ẑ(p∞) = ∆p, ̂⊕k<ω z(pk) = πk<ωz(pk), or ⊕̂ω z(p) = (z(p)) ω . according to corollary 2.5 the group m has a dense, nonmeasurable subgroup d such that |m/d| = c, and then by theorem 2.2 with h := φ−1(d) the group h is dense and nonmeasurable in k. if a,b ∈ k with a + h = b + h then φ(a) + d = φ(b) + d, so c = |k| ≥ |k/h| ≥ |m/d| = c. � theorem 2.7. let k be an infinite, compact, abelian, metrizable group. then k admits a family of 2|k|-many dense, nonmeasurable subgroups, each of cardinality c. 116 w. w. comfort, s. u. raczkowski and f. j. trigos-arrieta proof. let h be as given in theorem 2.6 and let h := {s : h ⊆ s ⊆ k,s is a proper subgroup of k}. then |h| = 2c by theorem 1.4. theorem 1.1(c) shows for s ∈ h that s ∈ m(k) with λ(s) > 0 is impossible, and if s ∈ h with λ(s) = 0 then λ(h) = 0, a contradiction. � remark 2.8. for our application in theorem 4.1 below we do not require that |k/s| = c, but in fact that condition does hold for 2c-many s ∈ h. 3. many nonmeasurable subgroups: the nonmetrizable case we turn now to the case w(k) = κ > ω. again, our goal is to show that such a compact abelian group k contains 2|k|-many dense, nonmeasurable subgroups of cardinality |k|. we find it convenient to show a bit more, namely that k admits a family of 2|k|-many subgroups each of which is gδ-dense in k (and hence pseudocompact). in the transition, we will invoke the following lemma. lemma 3.1. a proper, gδ-dense subgroup h of a compact group k is nonmeasurable. proof. as usual, using theorem 1.1(c), h ∈ m(k) with λ(h) > 0 is impossible; it suffices then to show that λ(h) = 0 is also impossible. the following argument is from [24] and [23], as exposed by halmos [17]. if λ(k\h) = 1 > 0 there are a compact set c and a baire set f of k such that f ⊆ c ⊆ k\h and λ(f) = λ(c) > 0 ([17](64h and p. 230)). as with any nonempty baire set, f has the form f = xb for a suitably chosen compact baire subgroup b of k and x ⊆ k ([17](64e)). since every compact baire set is a gδ-set ([17](51d)), each x ∈ x has xb a gδ-set. from xb ∩ h = ∅ it follows that h is not gδ-dense in k, a contradiction. � theorem 3.2. let k be a compact, abelian group such that w(k) = κ > ω. then k has a family of 2|k|-many gδ-dense subgroups of cardinality |k|. proof. let g = k̂, so that |g| = w(k) = κ, and let κ0 = r0(g) and κp = rp(g) for p ∈ p. since |g| = κ = κ0 + σp∈p κp (with perhaps κi = 0 for certain i ∈ p ∪ {0}), we have algebraically g ⊇ ⊕κ0 z ⊕ ⊕p∈p ⊕κp z(p). (*) the map ψ adjoint to the inclusion map in (*) is a continuous, surjective homomorphism ψ : k = ĝ ։ m := tκ0 × πp∈p (z(p)) κp = πξ<κ nξ (**) (with, again, perhaps κi = 0 for certain i ∈ p ∪ {0}), each nξ a nondegenerate compact metric group; here w(m) = κ, |m| = 2κ. now using κ = κ · ω+ let {aη : η < κ} be a partition of κ into disjoint sets of cardinality ω +, and rewrite (**) in the form ψ : k ։ m = πη<κ mη making group topologies with, and without, convergent sequences 117 with mη = πξ∈aη nξ a compact group of weight ω +. each group mη has a proper gδ-dense subgroup (for example, the σ-product), and that in turn extends to a proper subgroup dη of mη of finite or countably infinite index. then |dη| = |mη| = 2 (ω+), so d := πη<κ dη is gδ-dense in m with |d| = (2(ω +))κ = 2κ and 2κ = |m| ≥ |m/d| = πη<κ |mη/dη| ≥ 2 κ. now let h := ψ−1(d). then 2κ = |k| ≥ |h| ≥ |d| = 2κ, and |k/h| = 2κ. as noted in [3](2.2) the image under ψ of each nonempty gδ-subset of k contains a nonempty gδ-subset of m, so h is gδ-dense in k. by theorem 1.4 there are 22 κ -many proper subgroups of k containing h, each necessarily gδdense, as required. � we note in passing that in general not every gδ-dense subgroup d of a compact, nonmetrizable group k satisfies |d| = |k|. see in this connection remark 6.4 below. corollary 3.3 is now immediate from lemma 3.1 and theorem 3.2; and theorem 3.4, which is (1) of our abstract, is the conjunction of theorem 2.7 and corollary 3.3. corollary 3.3. let k be a compact, abelian group such that w(k) > ω. then k admits a family of 2|k|-many dense, nonmeasurable subgroups, each of cardinality |k|. theorem 3.4. every infinite, abelian compact (hausdorff ) group k admits 2|k|-many dense, non-haar-measurable subgroups of cardinality |k|. when k is nonmetrizable, these may be chosen to be pseudocompact. remark 3.5. (a) in an earlier version of this manuscript privately circulated to colleagues we expanded the scope of arguments from [5] to associate with each uniform ultrafilter q over κ a homomorphism hq : k ։ f with f = t or f = z(p) in such a way that mq := ker(hq) is gδ-dense in k. (the cases cf(κ) > ω, cf(κ) = ω required separate treatment.) the present alternative proof of theorem 3.2, contributed anonymously, seems briefer and conceptually simpler. (b) it is known [39] that r contains a dense non-measurable subgroup of both cardinality and index c. arguing as in theorem 2.7, we see then that r has 2c-many dense non-measurable subgroups of both cardinality and index c. we say as usual that a compactly generated group is a topological group generated, in the algebraic sense, by a compact subset. by [22] (9.8) for every (hausdorff) locally compact abelian compactly generated group g there are non-negative integers m and n and a compact abelian group k such that g is of the form rm × k × zn. if g is not discrete, then either m > 0 or k is not discrete, so w(g) = ω+w(k). it follows that a non-discrete (hausdorff ) locally compact abelian compactly generated group g has 2|g|-many dense non-haarmeasurable subgroups of both cardinality and index |g|. the result cannot be generalized to arbitrary locally compact abelian groups since such groups may fail to have a proper dense subgroup [32]. 118 w. w. comfort, s. u. raczkowski and f. j. trigos-arrieta note added in proof. we are grateful to professor james d. reid for calling to our attention this theorem of w. r. scott [proc. amer. math. soc. 5 (1) (1954), 19–22], which is closely related to our arguments in sections 2 and 3: if g is an abelian group with |g| = α > ω, and if κ is a cardinal satisfying α ≥ κ ≥ ω, then g has 2α-many subgroups of index κ. it is clear (using theorem 1.1) that if in addition g has a compact group topology then those subgroups of index ω are necessarily non-measurable; but it is not obvious to us whether simple additional algebraic arguments show that they must be, or may be chosen to be, dense in g (or gδ-dense, in the case that g is non-metrizable). 4. group topologies without convergent sequences here we pull together the threads of sections 2 and 3. theorem 4.1. every infinite abelian group g admits a family a of totally bounded group topologies, with |a| = 22 |g| , such that no nontrivial sequence in g converges in any of the topologies in a. one may arrange in addition that (i) w(g,t ) = 2|g| for each t ∈ a, and (ii) for distinct t0, t1 ∈ a the spaces (g,t0) and (g,t1) are not homeomorphic. proof. by theorem 2.7 when |g| = ω, and by corollary 3.3 when |g| > ω, the compact group k := hom(g, t) = ĝd admits a family h of dense, nonmeasurable subgroups such that |h| = 22 |g| and |h| = |k| for each h ∈ h. according to theorems 1.2 and 1.3 the family a := {th : h ∈ h} satisfies all requirements except (perhaps) (ii). a homeomorphism between two of the spaces (g,t0), (g,t1) with ti ∈ a is realized by a permutation of g, and there are just 2|g|-many such functions, so for each t ∈ a there are at most 2|g|-many t ′ ∈ a such that (g,t ) =h (g,t ′). statement (ii) then follows (with a replaced if necessary by a suitably chosen subfamily of cardinality 2|k| = 22 |g| ). � remark 4.2. the case g = z of theorem 4.1 is not new. see in this connection [30] and [31], which were the motivation for much of the present paper. 5. topologies with convergent sequences we turn now to the complementary or opposing problem, that of finding on an arbitrary infinite abelian group g the maximal number (that is, 22 |g| ) of totally bounded group topologies in which some nontrivial sequence converges. it was shown in [29], [44] concerning the group z that if 0 < n ∈ z with xn+1/xn → ∞ then there is a topology in t(z) with respect to which xn → 0; independently raczkowski [30], [31] proved that if xn+1/xn ≥ n + 1 then there are 2c-many such topologies in t(z); later the authors of [1] obtained the same conclusion assuming only that xn+1/xn → ∞. to handle the case of general abelian g, our strategy is to show first that certain “basic” countable groups accept 2c many such topologies. we begin making group topologies with, and without, convergent sequences 119 with technical results concerning groups of the form ⊕k<ω z(p rk k ) and of the form z(p∞). we remark for emphasis that in theorem 5.1 the given sequence (pk)k in p is not necessarily faithfully indexed. indeed the case pk = p ∈ p (a constant sequence) is not excluded. for x ∈ a = ⊕k<ω z(p rk k ) we write x = (x(k))k<ω . theorem 5.1. let pk ∈ p and a = ⊕k<ω z(p rk k ) with 0 < rk < ω, and let (xn)n<ω be a faithfully indexed sequence in a such that (i) there is s ∈ [ω]ω such that xn(k) = 0 for all n < ω, k ∈ s, and (ii) |{n < ω : xn(k) 6= 0}| < ω for all k < ω. let {aξ : ξ < c} enumerate p(s) ∪ [ω] <ω, and for ξ < c define hξ ∈ hom(a, t) by hξ(x) = σk∈aξ x(k). then (a) the set {hξ : ξ < c} is faithfully indexed; (b) the set {hξ : ξ < c} separates points of a; and (c) hξ(xn) → 0 for each ξ < c. proof. (a) if ξ,ξ′ < c with ξ 6= ξ′, say k ∈ aξ\aξ′ , then any x ∈ a such that 0 6= x(k) ∈ z(p rk k ) and x(m) = 0 for k 6= m < ω satisfies hξ(x) = x(k) 6= 0 = hξ′ (x). (b) let x,x′ ∈ a with, say, x(k) 6= x′(k). there is ξ < c such that {k} = aξ, and then hξ(x) = x(k) 6= x ′(k) = hξ(x ′). (c) if aξ ∈ p(s) then hξ(xn) = 0 for all n by (i), so hξ(xn) → 0. if aξ ∈ [ω] <ω then by (ii) there is n < ω such that hξ(xn) = 0 for all n > n, so again hξ(xn) → 0. � next, following [22] and [13], we identify the elements of the compact group ∆p = hom(z(p ∞), t) with those sequences h = (h(k))k<ω of integers such that 0 ≤ h(k) ≤ p − 1 for all k < ω. for a pn ∈ z(p∞) (with 0 ≤ a ≤ pn − 1) we have h ( a pn ) = a · h ( 1 pn ) = a · σn−1k=0 h(k) pn−k = a pn · σn−1k=0 h(k) · p k (mod 1). in what follows we write fac := {n! : n < ω}. theorem 5.2. let (an)n<ω be a sequence of integers such that 0 ≤ an ≤ p− 1 for all n < ω, and let xn = an pn! ∈ z(p∞). let {aξ : ξ < c} enumerate p(fac) ∪ [ω]<ω, and for ξ < c define hξ = (hξ(k))k<ω ∈ ∆p by hξ(k) = { 1 if k ∈ aξ, 0 otherwise. then (a) the set {hξ : ξ < c} is faithfully indexed; (b) the set {hξ : ξ < c} separates points of a; (c) if an > 0 for all but finitely many n, then the sequence (xn)n<ω is faithfully indexed; and (d) hξ(xn) → 0 for each ξ < c. 120 w. w. comfort, s. u. raczkowski and f. j. trigos-arrieta proof. the proofs of (a) and (b) closely parallel their analogues in theorem 5.1, and (c) is obvious. we prove (d). if aξ ∈ [ω] <ω there is n < ω such that hξ(xn) = an pn! · σn!−1k=0 hξ(k) · p k ≤ p−1 pn! · σnk=0 p k for all n > n, so hξ(xn) → 0. if aξ ∈ p(fac) then hξ(xn) = an pn! · σn!−1k=0 hξ(k) · p k = an pn! · σk∈fac,0≤k ω we write a(ĝ,a) := {k ∈ hom(g, t) : k ≡ 0 on a} ⊆ h∗; then |a(ĝ,a)| = 2|g/a| = 2|g| since algebraically a(ĝ,a) = hom(g/a, t) and |g| = |g/a|, so |h∗| = 2|g| in this case also. � we have arrived at the final result of this section, which we view as a companion or “echo” to theorem 4.1. theorem 5.5. every infinite abelian group g admits a family b of totally bounded group topologies, with |b| = 22 |g| , such that some (fixed) nontrivial sequence in g converges in each of the topologies in b. one may arrange in addition that (i) w(g,t ) = 2|g| for each t ∈ b, and (ii) for distinct t0, t1 ∈ b the spaces (g,t0) and (g,t1) are not homeomorphic. making group topologies with, and without, convergent sequences 121 proof. let h∗ be a subgroup of hom(g, t) such that |h∗| = 2|g| and some nontrivial sequence (xn)n<ω in g satisfies xn → 0 in (g,th∗ ). there is a subgroup h of h∗ such that h separates points of g and |h| = |g|, and since |h∗/h| = 2|g| there is by theorem 1.4 a faithfully indexed family {hξ : ξ < 22 |g| } of (point-separating) groups such that h ⊆ hξ ⊆ h ∗ and |hξ| = |h∗| = 2|g| for each ξ < 22 |g| . then from theorem 1.2(a) we have w(g,thξ ) = |hξ| = 2 |g| for each ξ < 22 |g| , and the family b := {thξ : ξ < 2 2|g|} satisfies (i); condition (ii) is then achieved as in the final sentences of the proof of theorem 4.1. � remark 5.6. (a) if a and b are as in theorems 4.1 and 5.5, and if t0,t1 ∈ a∪b with t0 6= t1, then the spaces (g,t0) and (g,t1) are not homeomorphic. for if both ti ∈ a or both ti ∈ b this is already proved, while if (say) t0 ∈ a and t1 ∈ b then (g,t1) has a nontrivial convergent sequence and (g,t0) does not. (b) we emphasize that for an infinite abelian group g, the algebraic structure of a point-separating subgroup h ⊆ hom(g, t) by no means determines the topology th on g. it is noted explicitly in [30], [31] that when g = z then every one of the topologies in the families a and b (as in theorems 4.1 and 5.5) can be chosen of the form th with h ⊆ t = hom(g, t) and with h ≃ ⊕ξ ω.) this discussion suggests a question. question 6.5. given an infinite abelian group g, what is the minimal weight of a topology in t ∈ t(g) such that no nontrivial sequence converges in (g,t )? for which g is this 2|g|? acknowledgements. the second listed author acknowledges partial support from the university research council at csu bakersfield. she also wishes to thank mrs. mary connie comfort for her encouragement, without which this paper would never see the daylight. thank you. references [1] g. barbieri, d. dikranjan, c. milan and h. weber, answer to raczkowski’s questions on convergent sequences of integers, topology appl. 132 (1) (2003), 89–101. 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[deuxième édition #1145, 1951.] [44] y. zelenyuk, topologies on abelian groups, candidate of sciences dissertation, kyiv university, 1990. received september 2004 accepted february 2005 w. w. comfort (wcomfort@wesleyan.edu) department of mathematics, wesleyan university, middletown, ct 06459. s. u. raczkowski (racz@csub.edu) department of mathematics, california state university, bakersfield, bakersfield, ca, 93311-1099. f. j. trigos-arrieta (jtrigos@csub.edu) department of mathematics, california state university, bakersfield, bakersfield, ca, 93311-1099. @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 509–512 density topology and pointwise convergence w ladys law wilczyński dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. we shall show that the space of all approximately continuous functions with the topology of pointwise convergence is not homeomorphic to its category analogue. 2000 ams classification: 54c35, 54c30, 26a15. keywords: approximately continuous functions, i-approximately continuous functions, topology of pointwise convergence. 1. density topology and pointwise convergence. let s be a σ-algebra of lebesgue measurable subsets of the real line r, l ⊂ s – a σ-ideal of null sets, b – a σ-algebra of subsets of r posessing a property of baire and i ⊂ b – a σ-ideal of sets of the first category. the sets of the first and the second category are considered only with respect to the natural topology. recall that a point x0 ∈ r is a density point of the set a ∈ s if and only if lim h→0+ λ(a∩ [x0 −h,x0 + h]) 2h = 1, where λ stands for the lebesgue measure on s. let φ(a) be a set of all density points of a ∈ s. if we denote a ∼ b in the case when a4b ∈l then we have (compare [6], th. 22.2): theorem 1.1. (1) φ(a) ∼ a for each a ∈ s (lebesgue density theorem), (2) if a,b ∈ s and a ∼ b, then φ(a) = φ(b), (3) φ(∅) = ∅, φ(r) = r, (4) φ(a∩b) = φ(a) ∩ φ(b) for each a,b ∈ s. observe that from 1 it follows immediately that φ(a) ∈ s for each a ∈ s. the function φ : s → s is usually called a lower density operator. 510 w ladys law wilczyński theorem 1.2. ([6], th. 22.5) a family td = {a ∈ s : a ⊂ φ(a)} = {φ(e) \ p : e ∈ s and p ∈l} is a topology on the real line stronger than the natural topology. the topology td is usually called the density topology. for further properties of td see, for example, [4] or [9]. a real function of a real variable is called approximately continuous if it is continuous when the domain is equipped with the density topology and the range – with the natural topology. since (r,td) is a tikhonov (completely regular) topological space ([4]), the density topology is the coarsest topology for the class of all approximately continuous functions. observe that the following conditions are equivalent (see [8]) for a set a ∈ s : 1) 0 is a density point of a, 2) limn→∞ λ(a∩(− 1 n , 1 n )) 2 n = 1, 3) limn→∞λ((n ·a) ∩ (−1, 1)) = 2 (where n ·a = {nx : x ∈ a}, 4) {χn·a∩(−1,1)}n∈n converges to χ(−1,1) in measure, 5) for each increasing sequence {nm}m∈n of positive integers there exists a subsequence {nmp}p∈n such that lim p→∞ χ(nmp·a)∩(−1,1) = χ(−1,1) almost everywhere. the equivalence of 1)-4) is immediate, while the equivalence of 4) and 5) follows from a well known theorem of riesz. the above observation was a starting point to study a category analogue of a density point, density topology and approximate continuity. definition 1.3. ([8]) we say that 0 is an i-density point of a set a ∈b if and only if for each increasing sequence {nm}m∈n of positive integers there exists a subsequence {nmp}p∈n such that lim p→∞ χ(nmp·a)∩(−1,1) = χ(−1,1) except on a set of the first category (in abbr. i-a.e.). we say that x0 is an i-density point of a ∈b if and only if 0 is an i-density point of a set a−x0 = = {x−x0 : x ∈ a}. let φi(a) be a set of all i-density points of a ∈b. if we denote now a ∼ b in the case when a4b ∈i, then we have theorem 1.4. ([8]) (1) ψi(a) ∼ a for each a ∈b, (2) if a,b ∈b and a ∼ b, then φi(a) = φi(b), (3) φi(∅) = ∅, φi(r) = r, (4) φi(a∩b) = φi(a) ∩ φi(b) for each a,b ∈b. theorem 1.5. ([8]) a family ti = {a ∈b : a ⊂ φi(a)} = {φi(e)\p : e ∈ b and p ∈i} is a topology on the real line stronger than the natural topology. density topology and pointwise convergence 511 the topology ti is called the i-density topology. for further properties of ti see, for example, [8] or [3]. a real function of a real variable is called iapproximately continuous if it is continuous when the domain is equipped with the i-density topology and the range – with the natural topology. unfortunately, (r,ti) is a not a tikhonov topological space ([8]). however, the coarsest topology for i-approximately continuous functions, which must be completely regular, is studied in details in [5] and [7]. we shall trace a description of such a topology (called the deep i-density topology) after [3]. definition 1.6. a point x0 ∈ r is called a deep i-density point of a ∈ b if there exists a closed (in the natural topology) set f ⊂ a∪{x0} such that x0 is an i-density point of f. let φid(a) be a set of all deep i-density points of a ∈b. theorem 1.7. a family tid = {a ∈b : a ⊂ φid(a)} is a topology stronger than the natural topology and weaker than the i-density topology. moreover tid is a completely regular topology. from the above theorem it follows immediately that the class of i-approximately continuous functions is equal to the class of deeply i-approximately continuous real functions of a real variable (i.e. the class of functions which are continuous when the domain is equipped with the deep i-density topology and the range with the natural topology). since both spaces (r,td) and (r,tid) are tikhonov topological spaces, it is reasonable to consider the spaces cp(rd) and cp(ri) of all approximately continuous and all i-approximately continuous functions with the topology of pointwise convergence. the question: are cp(rd) and cp(ri) homeomorphic seems to be interesting. in this note we shall try to find an answer. first of all, observe that the problem is not trivial by virtue of the following theorem: theorem 1.8. the spaces (r,td) and (r,tid) are not homeomorphic. proof. suppose that h : r → onto r is a homeomorphism between (r,td) and (r,tid). if e ⊂ r is td-connected set, then h(e) is tid-connected set. from [4] it follows that the family of all td-connected sets coincides with the family of all sets connected in the natural topology (i.e. with the family of all intervals — open, half-open, closed, bounded or unbounded). the same holds for the topology ti (see [8]). since tdi is between ti and the natural topology, it has the same family of connected sets. so for h we see that the image of an arbitrary interval is an interval. from this it is easy to conclude that h is a strictly monotone and continuous (in the sense that both the domain and the range are equipped with the natural topology) function, in fact h is a homeomorphism from (r, nat) to (r, nat). let e ∈ td be a set which is nowhere dense in the natural topology (for example e = c∩φ(c), where c is a nowhere dense cantor set of positive measure). then h(e) is also nowhere dense in the natural topology. but from the definition of deep i-density point 512 w ladys law wilczyński and deep i-density topology it follows immediately that each set in tid is of the second category in fact, it must contain a nondegenerate closed interval (see the proof of th. 2 below). so h(e) /∈tid – a contradiction. � theorem 1.9. the spaces cp(rd) and cp(ri) are not homeomorphic. proof. suppose that cp(rd) and cp(ri) are homeomorphic. then d(r,td) = d(r,tid) ([1], p. 26), where d denotes the density of the topological space, i.e. the smallest cardinal number of dense subsets of this space. but from the definition of deep i-density topology it follows that each tid-open set includes a (nondegenerate) closed interval, because tid-open set includes a closed set (in the natural topology) of the second category. hence the set e ⊂ r is dense in the topology tid if and only if it is dense in the natural topology and d(r,tdi) = ℵ0. simultaneously if the set e is dense in the topology td, then λ∗(e) > 0 (in fact, λ∗(e ∩ (a,b)) = b − a for each interval (a,b)). so d(r,td) > ℵ0 – a contradiction. observe that the exact value of d(r,td) depends essentially of the system of axioms (see [2]). � references [1] a.v. arkhangel’skǐı, prostranstwa funktsǐı v topologii potochechnǒı skhodimosti (spaces of functions in the topology of pointwise convergence), general topology, spaces of functions and dimension, mgu moskva 1985 (in russian). [2] t. bartoszyński, h. judah, set theory. on the structure of the real line, a.k. peters, wellesley 1995. [3] k. ciesielski, l. larson, k. ostaszewski, i-density continuous functions, memoirs of ams 515(1994). [4] c. goffman, c.j. neugebauer, t. nishiura, density topology and approximate continuity, duke math. j. 28 ( 1961), 497–505. [5] e. lazarow, the coarsest topology of i-approximately continuous functions, comm. math. univ. carolinae 27 (1986), 695–704. [6] j.c. oxtoby measure and category, springer verlag, new york-heidelberg-berlin, (1980). [7] w. poreda, e. wagner-bojakowska, the topology of i-approximately continuous functions, radovi mat. 2 (1986), 263–277. [8] w. poreda, e. wagner-bojakowska, w. wilczyński, a category analogue of the density topology fund. math. 125 (1985), 167–173. [9] f.d. tall, the density topology pacific j. math. 62 (1976), 275–284. received january 2002 revised september 2002 w ladys law wilczyński faculty of mathematics, chair of real functions, lódź university, stefana banacha 22, 90-238 lódź, poland e-mail address : wwil@krysia.uni.lodz.pl dikproagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 265-268 every infinite group can be generated by p-small subset dikran dikranjan and igor protasov abstract. for every infinite group g and every set of generators s of g, we construct a system of generators in s which is small in the sense of prodanov. 2000 ams classification: 20d30, 20f05. keywords: group, large set, small set, p-small set. a subset b of a group g is called large if g = f · b = b · f for some finite subset f of g. a subset s of a group g is called small if the subset g\f · s · f is large for every finite subset f of g. v. malykhin and r. moresco [4] posed the following question: can ever infinite group by generated by small subset? this question was answered positively in [6] (see also [7, theorem 13.1], some partial results were obtain also in [2]). following [2, §2.1] we call a subset s of a group g left small in the sense of prodanov (briefly left p-small) if there exist an injective sequence (an)n<ω such that the family {an · s : n < ω} consists of pairwise disjoint subsets. analogously, right small in the sense of prodanov (briefly right p-small) is introduced. the set s is called p-small when it is both left p-small and right p-small. clearly, all these notions coincide in the abelian case. that was the case considered by prodanov [5], who introduced the notion by noticing that if for a subset a of an abelian group g the difference set a − a is not not large, then a is p-small. by [3, theorem 4.2], every p-small subset of abelian group is small, but there are small subsets of abelian groups which are not p-small. on the other hand, the free group of rank 2 contains p-small subsets which are not small. it was proved in [2, theorem 3.6] that every abelian group has a p-small set of generators. furthermore, every free group (more generally, every group admitting an infinite abelian quotient) and every infinite symmetric group admit 266 d. dikranjan and i. protasov a p-small set of generators [2, proposition 3.7, theorem 3.11]. in this paper we offer a common generalization of all preceding results in our theorem below by proving that every set of generators of an infinite group contains a p-small subset of generators. for a subset a of a group g we denote by 〈a〉 the subgroup generated by a. theorem 1. let g be an infinite group, a ⊆ g, g = 〈a〉. then there exists a small and p-small subset x of g such that 〈x〉 = g and x ⊆ a. proof. if g is finitely generated, the statement is trivial since every set of generators of g contains a finite set of generators. we can take an arbitrary finite system x, x ⊆ a of generators of g and choose inductively the sequences (yn)n<ω, (zn)n<ω such that yn · x ∩ ym · x = ∅, x · zn ∩ x · zm = ∅ for all n, m such that n < m < ω. assume that g is not finitely generated and fix some minimal well-ordering {gα : α < κ} of a ∪ {e}, g0 = e, e is the identity of g. put g0 = {e} and x0 = g1. suppose that, for some ordinal λ < κ, the elements {xα : α < λ} and the subgroup {gα : α < λ} have been chosen. if λ is a limit ordinal, we put gλ = ⋃ α<λ gα, take the first element gβ such that gβ /∈ gλ and put xλ = gβ . if λ is a non-limit ordinal, we denote by gλ the subgroup generated by gλ−1 ∪{xλ−1}, take the first element gβ such that gβ /∈ gλ and put xλ = gβ. after κ steps we get the subset x = {xα : α < κ} and the properly increasing chain {gα : α < κ} of subgroups of g such that x ⊆ a, g = 〈x〉 and xα ∈ dα := gα+1 \ gα for every α < κ. by [5, theorem 13.1], x is small. to show that x is p-small, we build a sequence sequences (yn)n<ω of elements of g such that yn · x ∩ yi · x = ∅ (1) for every i < n. to this end we use the following easy to see properties of the sets dα: (a) g = ⋃ α<κ dα is a partition with dα ∩ gλ = ∅ whenever λ ≤ α < κ; (b) gα · dα = dα · gα = dα for every α < κ; (c) |dm| ≥ |gm| ≥ 2 m, for all m < ω. for every m < ω let xm = {x0, x1, ..., xm}. put y0 = e. suppose that, for some natural number n, the elements y0, y1, ..., yn−1 have been chosen so that {y0, y1, ..., yn−1} ⊂ gω and yi · x ∩ yj · x = ∅ for all i, j such that i < j ≤ n − 1. to determine yn, we take a natural number m such that {y0, y1, ..., yn−1} ⊂ gm and 2m > n(m + 1)2. every infinite group can be generated by p-small subset 267 by (c) and by the inequality |{y0, y1, ..., yn−1} · xm · x −1 m | ≤ n(m + 1) 2 we can take the element yn ∈ dm such that {y0, y1, ..., yn−1} · xm ∩ yn · xm = ∅. by the choice of yn, we have ynxm ∩ yi · xm = ∅ for every i < n. if k, l < ω, k > m, then yj xk ∈ dk for every j ≤ n. hence ynxk = yj xl with k, l > m yields k = l and n = j. now assume that yixk = yjxl holds with k > m, i, j ≤ n and l ≤ m. then according to (a) and (b) this is not possible as yn · xk ∈ dk, while yj · xl ∈ gm+1. analogously, yn · xk = yj · xl is not possible with k ≤ m and l > m. this proves that yn · x ∩ yi · x = ∅ for every i < n. after ω steps we get the sequence (yn)n<ω such that the family {yn · x : n < ω} consists of pairwise disjoint subsets. applying these arguments to the set x−1, we get the sequence (zn)n∈ω such that the family {x · zn : n ∈ ω} consists of pairwise disjoint subsets. hence, x is p -small. � question 2. let g be an infinite group of cardinality κ. does there exist a subset x of g and a κ-sequence (yα)α<κ such that the family {yα · x : α ∈ κ} consists of pairwise disjoint subsets and g = 〈x〉? if g is abelian the answer is positive (see the proof of theorem 3.6 from [2]). finally, we offer also the following question 3. (a) let x be a subset of g such that, for every natural number n there exits a subset yn of g such that |yn| = n and the family {y · x : y ∈ yn} is disjoint. is x left p -small ? (b) by [7, theorem 12.10], every infinite group can be partitioned into countably many small subsets. can every infinite group be partitioned into countably many p -small subsets? (c) let g be an infinite group. does there exist a system s of generators of g such that g 6= (s · s−1)n for every natural number n? note added in november 2006. recently t. banakh and n. lyaskovska answered negatively item (a) of question 3. 268 d. dikranjan and i. protasov references [1] a. bella and v. malykhin, s mall, large and other subsets of a group, questions and answers in general topology 17 (1967), 183–197. [2] d. dikranjan, u. marconi and r. moresco, groups with small set of generators, applied general topology 4 (2) (2003), 327–350. [3] r. gusso, large and small sets with respect to homomorphisms and products of groups, applied general topology 3 (2) (2002), 133–143. [4] v. malykhin and r. moresco, s mall generated groups, questions and answers in general topology 19 (1) (2001), 47–53. [5] iv. prodanov, s ome minimal group topologies are precompact, math.ann. 227 (1977), 117–125. [6] i. protasov, e very infinite group can be generated by small subset, in: third intern. algebraic conf. in ukraine, sumy, (2001), 92–94. [7] i. protasov and t. banakh, ball structures and colorings of graphs and groups, matem. stud. monogr. series, vol 11, lviv, 2003. received august 2005 accepted january 2006 dikran dikranjan (dikranja@dimi.uniud.it) dipartimento di matematica e informatica, università di udine, via delle scienza 206, 33100 udine, italy igor protasov (kseniya@profit.net.ua) department of cybernetics, kyiv national university, volodimirska 64, kiev 01033, ukraine @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 115–131 effective representations of the space of linear bounded operators vasco brattka ∗ abstract. representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of turing machines. from the computer science point of view such representations can be considered as data structures of topological spaces. formally, a representation of a topological space is a surjective mapping from cantor space onto the corresponding space. typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. we discuss a number of representations of the space of linear bounded operators on a banach space. since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. however, other topologies, like the compact open topology and the fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. these representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. we investigate the sublattice of these representations with respect to continuous and computable reducibility. certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn. 2000 ams classification: 03f60, 03d45, 26e40, 46s30, 68q05. keywords: computable functional analysis, effective representations. ∗work partially supported by dfg grant br 1807/4-1 116 vasco brattka 1. preliminaries from computable analysis in this paper we will study representations of the set of linear bounded operators on banach spaces from the point of view of computable analysis, which is the turing machine based theory of computability on real numbers and other topological spaces. pioneering work on this theory has been presented by turing [23], banach and mazur [1], lacombe [14] and grzegorczyk [10]. recent monographs have been published by pour-el and richards [18], ko [11] and weihrauch [26]. certain aspects of computable functional analysis have already been studied by several authors, see for instance [16, 9, 24, 27, 28]. in this section we briefly summarize some notions from computable analysis. for details the reader is referred to [26]. the basic idea of the representation based approach to computable analysis is to represent infinite objects like real numbers, functions or sets, by infinite strings over some alphabet σ (which should at least contain the symbols 0 and 1). thus, a representation of a set x is a surjective mapping δ :⊆ σω → x and in this situation we will call (x,δ) a represented space. here σω denotes the set of infinite sequences over σ and the inclusion symbol is used to indicate that the mapping might be partial. if we have two represented spaces, then we can define the notion of a computable function. definition 1.1 (computable function). let (x,δ) and (y,δ′) be represented spaces. a function f :⊆ x → y is called (δ,δ′)–computable, if there exists some computable function f :⊆ σω → σω such that δ′f(p) = fδ(p) for all p ∈ dom(fδ). of course, we have to define computability of functions f :⊆ σω → σω to make this definition complete, but this can be done via turing machines: f is computable if there exists some turing machine, which computes infinitely long and transforms each sequence p, written on the input tape, into the corresponding sequence f(p), written on the one-way output tape. later on, we will also need computable multi-valued operations f :⊆ x ⇒ y , which are defined analogously to computable functions by substituting δ′f(p) ∈ fδ(p) for the equation in definition 1.1 above. if the represented spaces are fixed or clear from the context, then we will simply call a function or operation f computable. for the comparison of representations it will be useful to have the notion of reducibility of representations. if δ,δ′ are both representations of a set x, then δ is called reducible to δ′, δ ≤ δ′ in symbols, if there exists a computable function f :⊆ σω → σω such that δ(p) = δ′f(p) for all p ∈ dom(δ). obviously, δ ≤ δ′ holds, if and only if the identity id : x → x is (δ,δ′)–computable. moreover, δ and δ′ are called equivalent, δ ≡ δ′ in symbols, if δ ≤ δ′ and δ′ ≤ δ. analogously to the notion of computability we can define the notion of (δ,δ′)–continuity for single and multi-valued operations, by substituting a continuous function f :⊆ σω → σω for the computable function f in the definitions above. on σω we use the cantor topology, which is simply the product effective representations 117 topology of the discrete topology on σ. the corresponding reducibility will be called continuous reducibility and we will use the symbols ≤t and ≡t in this case. again we will simply say that the corresponding function is continuous, if the representations are fixed or clear from the context. if not mentioned otherwise, we will always assume that a represented space is endowed with the final topology induced by its representation. this will lead to no confusion with the ordinary topological notion of continuity, as long as we are dealing with admissible representations. a representation δ of a topological space x is called admissible, if δ is maximal among all continuous representations δ′ of x, i.e. if δ′ ≤t δ holds for all continuous representations δ′ of x. if δ,δ′ are admissible representations of t0–spaces x, y with countable bases, then a function f :⊆ x → y is (δ,δ′)–continuous, if and only if it is continuous in the ordinary topological sense. for an extension of these notions to larger classes of spaces cf. [20, 21]. given a represented space (x,δ), we will occasionally use the notions of a computable sequence and a computable point. a computable sequence is a computable function f : n → x, where we assume that n = {0, 1, 2, . . .} is represented by δn(1n0ω) := n and a point x ∈ x is called computable, if there is a constant computable function with value x. given two represented spaces (x,δ) and (y,δ′), there is a canonical representation [δ,δ′] of x × y and a representation [δ → δ′] of certain functions f : x → y . if δ,δ′ are admissible representations of t0–spaces with countable bases, then [δ → δ′] is actually a representation of the set c(x,y ) of continuous functions f : x → y . if y = r, then we write for short c(x) := c(x,r). the function space representation can be characterized by the fact that it admits evaluation and type conversion. proposition 1.2 (evaluation and type conversion). let (x,δ), (y,δ′) be admissibly represented t0–spaces with countable bases and let (z,δ′′) be a represented space. then: (1) (evaluation) ev : c(x,y )×x → y, (f,x) 7→ f(x) is ([[δ → δ′],δ],δ′)– computable, (2) (type conversion) f : z ×x → y , is ([δ′′,δ],δ′)–computable, if and only if the function f̌ : z → c(x,y ), defined by f̌(z)(x) := f(z,x) is (δ′′, [δ → δ′])–computable. the proof of this proposition is based on a version of the smn– and utm– theorems, and can be found in [26]. if (x,δ), (y,δ′) are admissibly represented t0–spaces with countable bases, then in the following we will always assume that c(x,y ) is represented by [δ → δ′]. it is known that the computable points in (c(x,y ), [δ → δ′]) are just the (δ,δ′)–computable functions f : x → y [26]. if (x,δ) is a represented space, then we will always assume that the set of sequences xn is represented by δn := [δn → δ]. the computable points in (xn,δn) are just the computable sequences in (x,δ). moreover, we assume 118 vasco brattka that xn is always represented by δn, which can be defined inductively by δ1 := δ and δn+1 := [δn,δ]. to make this paper as self-contained as possible, we will discuss some basic facts on computable metric spaces and computable banach spaces in the following section. section 3 will be devoted to a short introduction into hyperspace representations and, finally, section 4 presents our results on representations of the set of linear bounded operators. 2. computable metric and banach spaces in this section we will briefly discuss computable metric spaces and computable banach spaces. the notion of a computable banach space will be the central notion for all following results. computable metric spaces have been used in the literature at least since lacombe [15]. restricted to computable points they have also been studied by various authors [8, 12, 17, 22]. we consider computable metric spaces as special separable metric spaces, but on all points and not only restricted to computable points [25]. pour-el and richards have introduced a closely related axiomatic characterization of sequential computability structures for banach spaces [18] which has been extended to metric spaces by mori, tsujii, and yasugi [27]. before we start with the definition of computable metric spaces we just mention that we will denote the open balls of a metric space (x,d) by b(x,ε) := {y ∈ x : d(x,y) < ε} for all x ∈ x, ε > 0 and correspondingly the closed balls by b(x,ε) := {y ∈ x : d(x,y) ≤ ε}. occasionally, we denote complements of sets a ⊆ x by ac := x \a. definition 2.1 (computable metric space). a tuple (x,d,α) is called a computable metric space, if (1) d : x ×x → r is a metric on x, (2) α : n → x is a sequence which is dense in x, (3) d◦ (α×α) : n2 → r is a computable (double) sequence in r. here, we tacitly assume that the reader is familiar with the notion of a computable sequence of reals, but we will come back to that point below. occasionally, we will say for short that x is a computable metric space. obviously, a computable metric space is especially separable. given a computable metric space (x,d,α), its cauchy representation δx :⊆ σω → x can be defined by δx(01 n0+101n1+101n2+1 . . .) := lim i→∞ α(ni) for all ni such that (α(ni))i∈n converges and d(α(ni),α(nj)) ≤ 2−i for all j > i (and undefined for all other input sequences). in the following we tacitly assume that computable metric spaces are represented by their cauchy representations. if x is a computable metric space, then it is easy to see that d : x ×x → r becomes computable (see proposition 3.2 in [3]). all cauchy representations are admissible with respect to the corresponding metric topology. effective representations 119 an important computable metric space is (r,dr,αr) with the euclidean metric dr(x,y) := |x−y| and some numbering of the rational numbers q, as αr〈i,j,k〉 := (i − j)/(k + 1). here, 〈i,j〉 := 1/2(i + j)(i + j + 1) + j denotes cantor pairs and this definition is extended inductively to finite tuples. for short we will occasionally write k := αr(k). in the following we assume that r is endowed with the cauchy representation δr induced by the computable metric space given above. this representation of r can also be defined, if (r,dr,αr) just fulfills (1) and (2) of the definition above and this leads to a definition of computable real number sequences without circularity. occasionally, we will also use the represented space (q,δq) of rational numbers with δq(1n0ω) := αr(n) = n. many important representations can be deduced from computable metric spaces, but we will also need some differently defined representations. for instance, we will use two further representations ρ< , ρ> of the real numbers, which correspond to weaker information on the represented real numbers. here ρ<(01 n0+101n1+101n2+1 . . .) = x : ⇐⇒ {q ∈ q : q < x} = {ni : i ∈ n} and ρ< is undefined for all other sequences. thus, ρ<(p) = x, if p is a list of all rational numbers smaller than x. analogously, ρ> is defined with “>” instead of “<”. we write r< = (r,ρ<) and r> = (r,ρ>) for the corresponding represented spaces. the computable numbers in r< are called left-computable real numbers and the computable numbers in r> right-computable real numbers. the representations ρ< and ρ> are admissible with respect to the lower and upper topology on r, which are induced by the open intervals (q,∞) and (−∞,q), respectively. computationally, we do not have to distinguish the complex numbers c from r2. thus, we can directly define a representation of c by δc := δ2r. if z = a + ib ∈ c, then we denote by z := a − ib ∈ c the conjugate complex number and by |z| := √ a2 + b2 the absolute value of z. alternatively to this ad hoc definition of δc, we could consider δc as the cauchy representation of a computable metric space (c,dc,αc), where αc is a numbering of q[i], defined by αc〈n,k〉 := n + ki and dc(w,z) := |w − z| is the euclidean metric on c. the corresponding cauchy representation is equivalent to δ2 r . in the following we will consider vector spaces over r, as well as over c. we will use the notation f for a field which always might be replaced by both, r or c. correspondingly, we use the notation (f,df,αf) for a computable metric space which can be replaced by either of the computable metric spaces (r,dr,αr), (c,dc,αc) defined above. we will also use the notation qf = range(αf), i.e. qr = q and qc = q[i]. for the definition of a computable banach space it is helpful to have the notion of a computable vector space, which we will define next. definition 2.2 (computable vector space). a represented space (x,δ) is called a computable vector space (over f), if (x, + , · , 0) is a vector space over f such that the following conditions hold: 120 vasco brattka (1) + : x ×x → x, (x,y) 7→ x + y is computable, (2) · : f×x → x, (a,x) 7→ a ·x is computable, (3) 0 ∈ x is a computable point. here, (f,δf) is a computable vector space and if (x,δ) is a computable vector space over f, then (xn,δn) and (xn,δn) are computable vector spaces over f. if, additionally, (x,δ), (y,δ′) are admissibly represented second countable t0–spaces, then (c(y,x), [δ′ → δ]) is a computable vector space over f. here we tacitly assume that the vector space operations on product, sequence and function spaces are defined componentwise. the proof for the function space is a straightforward application of evaluation and type conversion. the central definition for the present investigation will be the notion of a computable normed space. definition 2.3 (computable normed space). a tuple (x,‖ ‖,e) is called a computable normed space, if (1) ‖ ‖ : x → r is a norm on x, (2) e : n → x is a fundamental sequence, i.e. its linear span is dense in x, (3) (x,d,αe), with d(x,y) := ‖ x−y ‖ and αe〈k,〈n0, . . . ,nk〉〉 := ∑k i=0 αf(ni)ei, is a computable metric space with cauchy representation δx, (4) (x,δx) is a computable vector space over f. if, in the situation of the definition, the underlying space (x,‖ ‖) is actually a banach space, i.e. if (x,d) is a complete metric space, then (x,‖ ‖,e) is called a computable banach space. if the norm and the fundamental sequence are clear from the context, or locally irrelevant, we will say for short that x is a computable normed space or a computable banach space. we will always assume that computable normed spaces are represented by their cauchy representations, which are admissible with respect to the norm topology. if x is a computable normed space, then ‖ ‖: x → r is a computable function. of course, all computable banach spaces are separable. in the following proposition a number of computable banach spaces are defined. proposition 2.4 (computable banach spaces). let p ∈ r be a computable real number with 1 ≤ p < ∞ and let a < b be computable real numbers. the following spaces are computable banach spaces over f. (1) (fn,‖ ‖p,e) and (fn,‖ ‖∞,e) with • ‖ (x1,x2, . . . ,xn) ‖p := p √ n∑ k=1 |xk|p and ‖ (x1,x2, . . . ,xn) ‖∞ := max k=1,...,n |xk|, • ei = e(i) = (ei1,ei2, . . . ,ein) with eik := { 1 if (∃j) i = jn + k, 0 otherwise. (2) (`p,‖ ‖p,e) with • `p := {x ∈ fn : ‖ x ‖p< ∞}, effective representations 121 • ‖ (xk)k∈n ‖p := p √ ∞∑ k=0 |xk|p, • ei = e(i) = (eik)k∈n with eik := { 1 if i = k, 0 otherwise. (3) (c[a,b],‖ ‖,e) with • c[a,b] := {f : [a,b] → f | f continuous}, • ‖ f ‖ := max t∈[a,b] |f(t)|, • ei(t) = e(i)(t) = ti. we leave it to the reader to check that these spaces are actually computable banach spaces. unless stated otherwise, we will assume that (fn,‖ ‖) is endowed with the maximum norm ‖ ‖ = ‖ ‖∞. it is known that the cauchy representation δc[a,b] of c[a,b] = c([a,b],r) is equivalent to [δ[a,b] → δr], where δ[a,b] denotes the restriction of δr to [a,b] (cf. lemma 6.1.10 in [26]). in the following we will occasionally utilize the sequence spaces `p to construct counterexamples. we close this section with a brief discussion of product spaces of computable normed spaces. proposition 2.5 (product spaces). if (x,‖ ‖,e), (y,‖ ‖′,e′) are computable normed spaces, then the product space (x ×y,‖ ‖′′,e′′), defined by ‖ (x,y) ‖′′ := max{‖ x ‖,‖ y ‖′} and e′′〈i,j〉 := (e(i),e′(j)), is a computable normed space too and the canonical projections of the product space pr1 : x ×y → x and pr2 : x ×y → y are computable. 3. hyperspaces of closed subsets since we want to use the hyperspace a(x) of closed subsets a ⊆ x in order to represent linear bounded operators, we have to discuss representations of hyperspaces. such representations have been studied in the euclidean case in [6, 26] and for the metric case in [5]. definition 3.1 (hyperspace of closed subsets). let (x,d,α) be a computable metric space. we endow the hyperspace a(x) := {a ⊆ x : a closed} of closed subsets with the representation δa(x), defined by δ>a(x)(01 〈n0,k0〉+101〈n1,k1〉+101〈n2,k2〉+1 . . .) := x \ ∞⋃ i=0 b(α(ni),ki). whenever we have two representations δ,δ′ of some set, we can define the infimum δ u δ′ of δ and δ′ by (δ u δ′)〈p,q〉 = x : ⇐⇒ δ(p) = x and δ′(q) = x. we use the short notations a< = a<(x) = (a<(x),δ = a>(x) = (a>(x),δ>a(x)) and a = a(x) = (a(x),δ < a(x) uδ > a(x)) for the corresponding 122 vasco brattka represented spaces. for the computable points of these spaces special names are used. definition 3.2 (recursively enumerable and recursive sets). let x be a computable metric space and let a ⊆ x be a closed subset. (1) a is called r.e. closed, if a is a computable point in a<(x), (2) a is called co-r.e. closed, if a is a computable point in a>(x), (3) a is called recursive closed, if a is a computable point in a(x). these definitions generalize the classical notions of r.e. and recursive sets, since a set a ⊆ n, considered as a closed subset of r, is r.e., co-r.e., recursive closed, if and only if it has the same property in the classical sense as a subset of n [6]. we close this section with a helpful result on hyperspaces, which follows directly from results in [5] and which can be considered as an effective version of the statement that closed subsets of separable metric spaces are separable. here and in the following, a denotes the topological closure of a subset a ⊆ x of some topological space x. proposition 3.3 (separable closed subsets). let x be a computable metric space. the mapping range : xn → a<(x), (xn)n∈n 7→ {xn : n ∈ n} is computable and if x is complete, then it admits a computable multi-valued partial right-inverse ⊆a<(x) ⇒ xn, defined for all non-empty closed subsets. 4. representations of the operator space in this section we will define and compare several representations of the set b(x,y ) of linear bounded operators t : x → y on computable normed spaces x and y . definition 4.1 (representations of the operator space). let (x,‖ ‖,e) and y be computable normed spaces. we define representations of b(x,y ): (1) δev(p) = t : ⇐⇒ [δx → δy ](p) = t , (2) δgraph(p) = t : ⇐⇒ δa(x×y )(p) = graph(t), (3) δ of b(x,y ) by (1) δ=〈p,q〉 = t : ⇐⇒ δ(p) = t and δr(q) = ‖ t ‖, (2) δ>〈p,q〉 = t : ⇐⇒ δ(p) = t and δr(q) ≥‖ t ‖. all the representations introduced are admissible with respect to certain topologies on the space of linear bounded operators (see the brief discussion in the conclusion). while these representations separate into several distinct topological and computational equivalence classes, the corresponding computable effective representations 123 linear bounded operators do essentially coincide (see corollary 4.6). this follows from the following characterization which we have proved in [3]. theorem 4.2 (computable linear operators). let x,y be computable banach spaces, let (ei)i∈n be a computable sequence in x whose linear span is dense in x and let t : x → y be a linear operator. then the following conditions are equivalent: (1) t : x → y is computable, (2) t : x → y is bounded and maps computable sequences in x to computable sequences in y, (3) t : x → y is bounded and (tek)k∈n is a computable sequence in y , (4) graph(t) is an r.e. closed subset of x ×y , (5) graph(t) is a recursive closed subset of x ×y . the equivalence in this theorem is “non-constructive” in the sense that nothing is said about the possibility of converting one type of information into another type effectively. one of the main purposes of this paper is to study these possibilities. in this sense the following result includes a uniform version of the previous theorem. theorem 4.3 (representations of the operator space). let x and y be computable banach spaces. then the following reductions for representations of b(x,y ) hold: (1) δ=ev ≤ δev ≤ δseq ≤ δ < graph and δev ≤ δgraph ≤ δ < graph, (2) δev ≡ δ>ev ≡ δ>seq ≡ δ <> graph ≡ δ > graph, (3) δ=ev ≡ δ=seq ≡ δ <= graph ≡ δ = graph. proof. (1) “δ=ev ≤ δev” holds obviously and “δev ≤ δseq” follows by the evaluation property. “δseq ≤ δev” follows from theorem 9.10 in [3] (see also theorem 4.1 in [4]) which states that for every t ∈ b(x,y ), given with respect to δev, we can effectively find some upper bound s ≥‖ t ‖. “δ>ev ≤ δev” obviously holds. “δ>ev ≤ δ>seq ≤ δ <> graph” follows directly from (1). “δ<>graph ≤ δ > ev” given the graph(t) ∈ a<(x × y ) of some linear bounded operator t : x → y , we can effectively find a sequence f : n → x × y such that range(f) is dense in graph(t) by proposition 3.3. by proposition 2.5 we can effectively find the projections f1 : n → x and f2 : n → y of f too. given x ∈ x, a real number s ≥ ‖ t ‖ and a precision m ∈ n we can effectively find n,k ∈ n and numbers q0, . . . ,qn ∈ qf such that s ≤ 2k and ‖ ∑n i=0 qif1(i) −x ‖ < 2 −m−k−1 since range(f1) is dense in x. it follows that∣∣∣∣∣ ∣∣∣∣∣t ( n∑ i=0 qif1(i) ) −t(x) ∣∣∣∣∣ ∣∣∣∣∣ ≤ ‖ t ‖ · ∣∣∣∣∣ ∣∣∣∣∣ n∑ i=0 qif1(i) −x ∣∣∣∣∣ ∣∣∣∣∣ < 2−m−1. by linearity of t we obtain t( ∑n i=0 qif1(i)) = ∑n i=0 qitf1(i) = ∑n i=0 qif2(i) and thus we can evaluate t effectively up to any given precision m. applying this idea, we can effectively find a cauchy sequence which rapidly converges to t(x). using type conversion we obtain the desired reducibility. “δ>ev ≤ δ > graph” and “δ > graph ≤ δ <> graph” follow from (1). (3) this follows directly from (2). � it should be mentioned that we have used completeness only for the reduction “δ<>graph ≤ δ > ev” (implicitly, by application of proposition 3.3) and thus for the results in (2) and (3), while the statement of (1) remains true for computable normed spaces x,y . in case of a finite-dimensional y = fm, one can prove that the inverse graph−1 of the graph map graph : c(x,fm) → a(x × fm) is (δa(x×fm), [δx → δfm])–computable, see theorem 14.6 in [3]. as a direct consequence we obtain the following corollary. corollary 4.4. let x be a computable normed space and m ≥ 1 a natural number. then δev ≡ δgraph for the corresponding representations of b(x,fm). now the question arises as to which of the reductions given in theorem 4.3(1) are strict reductions. at least for certain spaces all of these reductions are strict as the following theorem shows. figure 1 summarizes the results. δ=ev δev δgraph δseq δ 1. “δev 6≤t δ=ev” in order to prove this, it suffices to show that the operator norm mapping ‖ ‖: b(`p,f) → r, defined for linear bounded t : `p → f, is not (δev,δr)–continuous. let q be such that 1p + 1 q = 1 and q = ∞ if p = 1. for any sequence a = (ak)k∈n ∈ `q we define the functional λa : `p → f, (xk)k∈n 7→ ∑∞ k=0 akxk. in the following we will use the map l :⊆ f n×r → b(`p,f), (a,s) 7→ λa with dom(l) := {(a,s) ∈ `q ×r : ‖ a ‖q≤ s} and with q such that 1 p + 1 q = 1 and q = ∞ if p = 1. for all (a,s) ∈ dom(l) we obtain ‖ l(a,s) ‖= ‖ λa ‖= ‖ a ‖q≤ s and thus l is ([δnf ,δr],δev)–continuous. if we assume that the operator norm mapping ‖ ‖ is (δev,δr)–continuous, then this implies that ‖ ‖ ◦l :⊆ fn ×r → r is ([δn f ,δr],δr)–continuous. in particular, this implies that n :⊆ fn → r,a 7→‖ a ‖q is continuous for all a with ‖ a ‖q≤ 1. but this is obviously not the case (in general, no finite prefix of the sequence a determines the value ‖ a ‖q up to a given precision). “δseq 6≤t δev” let us assume that δseq ≤t δev. let l, λ and q be as defined above. since λaei = ai for all a = (an)n∈n ∈ `q, it follows by the evaluation property and theorem 9.10 in [3] (cf. theorem 4.1 in [4]) that l admits a (δev, [δnf ,δr])–continuous multi-valued right-inverse l −1 :⊆b(`p,f) ⇒ fn×r. we consider the function λ :⊆ rn →b(`p,f), a 7→ λa with dom(λ) := rn∩`q. since λaei = ai for all a = (an)n∈n ∈ `q, it also follows that λ is (δnr,δseq)– computable and thus (δn r ,δev)–continuous by assumption. thus, l−1 ◦ λ :⊆ r n ⇒ fn ×r is (δn r , [δn f ,δr])–continuous too and hence the projection s :⊆ r n ⇒ r on the second component is continuous in the ordinary sense too. thus s is a continuous operation such that there exists some s ∈ s(a) for all a ∈ rn with ‖ a ‖q< ∞ and for all such s the inequality ‖ a ‖q≤ s holds. such a continuous operation can obviously not exist (since it would have to determine an upper bound of ‖ a ‖q, knowing only a finite prefix of the sequence a). a contradiction! “δseq 6≤t δgraph” this follows from δseq 6≤t δev since δgraph ≡ δev in the case of b(`p,f) by corollary 4.4. “δev) does not lead to a strictly stronger representation while theorem 4.5 shows that adding the operator bound itself actually leads to a stronger type of information. representing linear bounded operators by a sequence of values on a fundamental sequence (i.e. δseq) and representing operators by positive and negative information on their graphs (i.e. δgraph) both lead to mutually incomparable but weaker representations than δev. but both types of information are strictly stronger than just positive information on the operator graph (i.e. δ ac(x) or λ + a(x) > 1 (see [10]). a fuzzy set a is said to be quasi-coincident with b, denoted aqb, if and only if there exists x ∈ x such that a(x) > bc(x) or a(x) + b(x) > 1 (see [10]). if a does not quasi-coincident with b, then we write a q6 b. let f be a function from x to y . then (see for example [1], [2], [3], [8], [11], [12], [13], [16], and [17]): (1) f−1(bc) = (f−1(b))c, for any fuzzy set b in y . (2) f(f−1(b)) ≤ b, for any fuzzy set b in y . (3) a ≤ f−1(f(a)), for any fuzzy set a in x. (4) let p be a fuzzy point of x, a be a fuzzy set in x and b be a fuzzy set in y . then, we have: (i) if f(p) q b, then p q f−1(b). (ii) if p q a, then f(p) q f(a). (5) let a and b be fuzzy sets in x and y , respectively and p be a fuzzy point in x. then we have: (i) p ∈ f−1(b) if f(p) ∈ b. (ii) f(p) ∈ f(a) if p ∈ a. let λ be a directed set and x be an ordinary set. the function s : λ → x is called a fuzzy net in x. for every λ ∈ λ, s(λ) is often denoted by sλ and hence a net s is often denoted by {sλ, λ ∈ λ} (see [10]). on some applications of fuzzy points 121 let {an, n ∈ n} be a net of fuzzy sets in a fuzzy topological space x. then by f − lim n (an), we denote the fuzzy upper limit of the net {an, n ∈ n} in ix, that is, the fuzzy set which is the union of all fuzzy points pλx in x such that for every n0 ∈ n and for every fuzzy open q−neighborhood u of p λ x in x there exists an element n ∈ n for which n ≥ n0 and anqu. in other cases we set f − lim n (an) = 0. for the notions of fuzzy upper limit and fuzzy lower limit see [6]. recall that a fuzzy subset a of a fuzzy topological space x is called fuzzy preopen (see [5] and [14]) if a ≤ int(cl(a)), where int and cl denoted the interior and closure operators. a is called fuzzy preclosed if cl(int(a)) ≤ a. we denote the family of all fuzzy preopen (respectively, fuzzy preclosed) sets of x by fpo(x) (respectively, fpc(x)). also the intersection of all fuzzy preclosed sets containing a is called fuzzy preclosure of a, denoted by pcl(a), that is pcl(a) = inf{k : a ≤ k, k ∈ fpc(x)}. similar the fuzzy preinterior of a, denoted by pint(a), is defined as follows: pint(a) = sup{u : u ≤ a, u ∈ fpo(x)}. let a be a fuzzy preopen (respectively, preclosed) set of a fuzzy space x. then, by theorem 3.7 of [14], pint(a) = a (respectively, pcl(a) = a). also, by theorem 3.6 of [14], we have pcl(ac) = 1̄ − pint(a) = 1̄ − a = ac (respectively, pint(ac) = 1̄ − pcl(a) = 1̄ − a = ac). thus, the fuzzy set ac is fuzzy preclosed (respectively, preopen). 2. fuzzy points, preclosed sets and separations axioms definition 2.1. a fuzzy set a in a fuzzy space x is called a fuzzy preneighborhood of a fuzzy point pλx if there exists a v ∈ fpo(x) such that pλx ∈ v ≤ a. a fuzzy pre-neighborhood a is said to be preopen if a ∈ fpo(x). definition 2.2. a fuzzy set a in a fuzzy space x is called a fuzzy q−preneighborhood of pλx if there exists b ∈ fpo(x) such that p λ xqb and b ≤ a. remark 2.3. a fuzzy q−pre-neighborhood of a fuzzy point generally does not contain the point itself. in what follows by nq−p−n(p λ x) we denote the family of all fuzzy preopen q−pre-neighborhoods of the fuzzy point pλx in x. the set nq−p−n(p λ x) with the relation ≤ ∗ (that is, u1 ≤ ∗ u2 if and only if u2 ≤ u1) form a directed set. proposition 2.4. let a be a fuzzy set of a fuzzy space x. then, a fuzzy point pλx ∈ pcl(a) if and only if for every u ∈ fpo(x) for which p λ xqu we have uqa. proof. the fuzzy point pλx ∈ pcl(a) if and only if p λ x ∈ f , for every fuzzy preclosed set f of x for which a ≤ f . equivalently pλx ∈ pcl(a) if and only if λ ≤ 1 − u(x), for every fuzzy preopen set u for which a ≤ 1̄ − u. thus 122 m. ganster, d. n. georgiou, s. jafari and s. p. moshokoa pλx ∈ pcl(a) if and only if u(x) ≤ 1 − λ, for every fuzzy preopen set u for which u ≤ 1̄ − a. so, pλx ∈ pcl(a) if and only if for every fuzzy preopen set u of x such that u(x) > 1 − λ we have u 6≤ 1̄ − a. therefore by proposition 2.1 of [10], pλx ∈ pcl(a) if and only if for every fuzzy preopen set u of x such that u(x) + λ > 1 we have uqa. thus, pλx ∈ pcl(a) if and only if for every fuzzy preopen set u of x such that pλxqu we have uqa. � definition 2.5. let a be a fuzzy set of a fuzzy space x. a fuzzy point pλx is called a pre-boundary point of a fuzzy set a if and only if pλx ∈ pcl(a) ∧ (1̄ − pcl(a)). by pbd(a) we denote the fuzzy set pcl(a) ∧ (1̄ − pcl(a)). proposition 2.6. let a be a fuzzy set of a fuzzy space x. then a ∨ pbd(a) ≤ pcl(a). proof. let pλx ∈ a ∨ pbd(a). then p λ x ∈ a or p λ x ∈ pbd(a). clearly, if pλx ∈ pbd(a), then p λ x ∈ pcl(a). let us suppose that p λ x ∈ a. we have pcl(a) = ∧{f : f ∈ ix, f is preclosed and a ≤ f}. so, if pλx ∈ a, then p λ x ∈ f , for every fuzzy preclosed set f of x for which a ≤ f and therefore pλx ∈ pcl(a). � example 2.7. let (x, τ) be a fuzzy space such that x = {x, y} and τ = {0̄, 1̄, p 1 2 x }. the family of all fuzzy preclosed sets of x contains the following fuzzy sets a of x: i) a ∈ ix such that a(x) ∈ [0, 1 2 ) and a(y) ∈ [0, 1]. indeed, cl(int(a)) = cl(0̄) = 0̄ ≤ a. ii) a ∈ ix such that a(x) ∈ [1 2 , 1] and a(y) = 1. indeed, cl(int(a)) = cl(p 1 2 x ) ≤ (p 1 2 x ) c ≤ a. also, the family of all fuzzy preopen sets of x are the following fuzzy sets u of x: i) u ∈ ix such that u(x) ∈ [0, 1 2 ] and u(y) = 0. indeed, int(cl(u)) = int((p 1 2 x ) c) = p 1 2 x ≥ u. ii) u ∈ ix such that u(x) ∈ (1 2 , 1] and u(y) ∈ [0, 1]. indeed, int(cl(u)) = int(1̄) = 1̄ ≥ u. we consider the fuzzy set b ∈ ix such that b = p 2 3 x . by the above we have: pcl(b) = (p 1 3 x ) c, where (p 1 3 x ) c(z) = 2 3 , if z = x and (p 1 3 x ) c(z) = 1, if z = y. on some applications of fuzzy points 123 also, we have 1̄ − pcl(b) = p 1 3 x and pbd(b) = pcl(b) ∧ (1̄ − pcl(b)) = p 1 3 x . thus b ∨ pbd(b) = b 6= pcl(b). definition 2.8. a fuzzy space x is called pre-t0 if for every two fuzzy points pλx and p µ y such that p λ x 6= p µ y , either p λ x 6∈ pcl(p µ y) or p µ y 6∈ pcl(p λ x). definition 2.9. a fuzzy space x is called pre-t1 if every fuzzy point is fuzzy preclosed. remark 2.10. clearly, every pre-t1 fuzzy space is pre-t0. proposition 2.11. a fuzzy space x is pre-t1 if and only if for each x ∈ x and each λ ∈ [0, 1] there exists a fuzzy preopen set a such that a(x) = 1 − λ and a(y) = 1 for y 6= x. proof. ⇒) let λ = 0. we set a = 1̄. then a is fuzzy preopen set such that a(x) = 1 − 0 and a(y) = 1 for y 6= x. now, let λ ∈ (0, 1] and x ∈ x. we set a = (pλx) c. the set a is fuzzy preopen such that a(x) = 1 − λ and a(y) = 1 for y 6= x. ⇐) let pλx be an arbitrary fuzzy point of x. we prove that the fuzzy point pλx is fuzzy preclosed. by assumption there exists a fuzzy preopen set a such that a(x) = 1 − λ and a(y) = 1 for y 6= x. clearly, ac = pλx. thus the fuzzy point pλx is fuzzy preclosed and therefore the fuzzy space x is pre-t1. � definition 2.12. a fuzzy space x is called a pre-hausdorff space if for any fuzzy points pλx and p µ y for which supp(p λ x) = x 6= supp(p µ y) = y, there exist two fuzzy preopen q-pre-neighbourhoods u and v of pλx and p µ y, respectively, such that u ∧ v = 0̄. example 2.13. let (x, τ) be a fuzzy space such that x = {x, y} and τ = {0̄, 1̄, p 1 2 x }. the fuzzy point p 1 2 x is not fuzzy preclosed. indeed, we have: cl(int(p 1 2 x )) = cl(p 1 2 x ) = (p 1 2 x ) c 6≤ p 1 2 x . thus the fuzzy space x is not pre-t1. also, it is clear that the fuzzy space x is pre-t0. example 2.14. let (x, τ) be a fuzzy space such that x = {x, y} and τ = {0̄, 1̄}. we observe that every fuzzy point pλx is fuzzy preclosed. indeed, we have cl(int(pλx)) = 0̄ ≤ p λ x. thus the fuzzy space x is pre-t1 and therefore is pre-t0. also, it is clear that the fuzzy space x is pre-hausdorff. 124 m. ganster, d. n. georgiou, s. jafari and s. p. moshokoa it is not difficult to see that the fuzzy space x is not t0, t1 and hausdorff. for the definitions of t0, t1 and hausdorff fuzzy spaces see [10]. definition 2.15. a fuzzy space x is called a pre-regular space if for any fuzzy point pλx and a fuzzy preclosed set f not containing p λ x, there exist u, v ∈ fpo(x) such that pλx ∈ u, f ≤ v and u ∧ v = 0̄. example 2.16. let (x, τ) be a fuzzy space such that x = {x, y} and τ = {0̄, 1̄}. the fuzzy space x is pre-hausdorff but it is not pre-regular. we prove only that the fuzzy space x is not pre-regular. we consider the fuzzy point p 1 3 x and the fuzzy set a of x such that a(x) = 1 4 and a(y) = 1. for the fuzzy set a we have cl(int(a)) = 0̄ ≤ a. thus the fuzzy set a is fuzzy preclosed. also, we have p 1 3 x 6∈ a. if u and v are two arbitrary fuzzy preopen sets such that p 1 3 x ∈ u and a ≤ v , then (u ∧ v )(x) ≥ 1 4 and therefore u ∧ v 6= 0̄. thus the fuzzy space x is not pre-regular. definition 2.17. a fuzzy space x is called a quasi pre-t1 if for any fuzzy points pλx and p µ y for which supp(p λ x) = x 6= supp(p µ y) = y, there exists a fuzzy preopen set u such that pλx ∈ u and p µ y 6∈ u and another v such that p λ x 6∈ v and pµy ∈ v . example 2.18. let (x, τ) be a fuzzy space such that x = {x, y} and τ = {0̄, 1̄, p 1 2 x }. the fuzzy space x is quasi pre-t1 but it is not pre-t1. definition 2.19. (see [7]) a fuzzy point pλx is called weak (respectively, strong) if λ ≤ 1 2 (respectively, λ > 1 2 ). definition 2.20. a fuzzy set a of a fuzzy space x is called pre-generalized closed (briefly fpg-closed) if pcl(a) ≤ u whenever a ≤ u and u fuzzy preopen set of x. proposition 2.21. let x be a fuzzy space x. suppose that pλx and p µ y are weak and strong fuzzy points, respectively. if pλx is pre-generalized closed, then pµy ∈ pcl(p λ x) ⇒ p λ x ∈ pcl(p µ y). proof. suppose that pµy ∈ pcl(p λ x) and p λ x 6∈ pcl(p µ y). then pcl(p µ y)(x) < λ. also λ ≤ 1 2 . thus pcl(pµy)(x) ≤ 1 − λ and therefore λ ≤ 1 − pcl(p µ y)(x). so pλx ∈ (pcl(p µ y )) c. but pλx is pre-generalized closed and (pcl(p µ y )) c is fuzzy preopen. thus pcl(pλx) ≤ (pcl(p µ y)) c . by assumption we have pµy ∈ pcl(p λ x). thus pµy ∈ (pcl(p µ y)) c. on some applications of fuzzy points 125 we prove that this is a contradiction. indeed, we have µ ≤ 1 − pcl(pµy )(y) or pcl(pµy)(y) ≤ 1 − µ. also pµy ∈ pcl(p µ y). thus µ ≤ 1 − µ. but pµy is a strong fuzzy point, that is µ > 1 2 . so the above relation µ ≤ 1−µ is a contradiction. thus pλx ∈ pcl(p µ y). � proposition 2.22. if x is a quasi pre-t1 fuzzy space and p λ x a weak fuzzy point in x, then (pλx) c is a fuzzy pre-neighborhood of each fuzzy point pµy with y 6= x. proof. let y 6= x and pµy be a fuzzy point of x. since the space x is a quasi pre-t1 there exists a fuzzy preopen u of x such that p µ y ∈ u and p λ x 6∈ u. this implies that λ > u(x). also, λ ≤ 1 2 . thus u(x) ≤ 1 − λ. therefore u(y) ≤ 1 = (pλx) c(y), for every y ∈ x \ {x}. so u ≤ (pλx) c. therefore the fuzzy point pλx is a pre-neighborhood of p µ y . � proposition 2.23. if x is a pre-regular fuzzy space, then for any strong fuzzy point pλx and any fuzzy preopen set u containing p λ x, there exists a fuzzy preopen set w containing pλx such that pcl(w) ≤ u. proof. suppose that pλx is any strong fuzzy point contained in u ∈ fpo(x). then 1 2 < λ ≤ u(x). thus the complement of u, that is the fuzzy set uc, is a fuzzy preclosed set to which does not belong the fuzzy point pλx. thus, there exist w, v ∈ fpo(x) such that pλx ∈ w and u c ≤ v with w ∧ v = 0̄. hence, we have w ≤ v c and by theorem 3.8 of [14] pcl(w) ≤ pcl(v c) = v c. now uc ≤ v implies v c ≤ u. this means that pcl(w) ≤ u which completes the proof. � proposition 2.24. if x is a fuzzy pre-regular space, then the strong fuzzy points in x are fpg-closed. proof. let pλx be any strong fuzzy point in x and u be a fuzzy open set such that pλx ∈ u. by proposition 2.23 there exists a w ∈ fpo(x) such that pλx ∈ w and pcl(w) ≤ u. by theorem 3.8 of [14], we have pcl(pλx) ≤ pcl(w) ≤ u. thus the fuzzy point pλx is fpg-closed. � definition 2.25. a fuzzy space x is called a weakly pre-regular space if for any weak fuzzy point pλx and a fuzzy preclosed set f not containing p λ x, there exist u, v ∈ fpo(x) such that pλx ∈ u, f ≤ v and u ∧ v = 0̄. observe that every pre-regular fuzzy space is weakly pre-regular. 126 m. ganster, d. n. georgiou, s. jafari and s. p. moshokoa definition 2.26. let x be a fuzzy space. a fuzzy set u in x is said to be fuzzy pre-nearly crisp if pcl(u) ∧ (pcl(u))c = 0̄. proposition 2.27. let x be a fuzzy space. if for any weak fuzzy point pλx and any u ∈ fpo(x) containing pλx, there exists a fuzzy preopen and pre-nearly crisp fuzzy set w containing pλx such that pcl(w) ≤ u, then x is fuzzy weakly pre-regular. proof. assume that f is a fuzzy preclosed set not containing the weak fuzzy point pλx. then f c is a fuzzy preopen set containing pλx. by hypothesis, there exists a fuzzy preopen and pre-nearly crisp fuzzy set w such that pλx ∈ w and pcl(w) ≤ f c. we set n = pint(pcl(w)) and m = 1 − pcl(w). then n is fuzzy preopen, pλx ∈ n and f ≤ m. we are going to prove that m ∧ n = 0̄. now assume that there exists y ∈ x such that (n ∧ m)(y) = µ 6= 0̄. then pµy ∈ n ∧m. hence, p µ y ∈ pcl(w) and p µ y ∈ (̄pcl(w)) c. this is a contradiction since w is pre-nearly crisp. thus the fuzzy space x is weakly pre-regular. � definition 2.28. let x be a fuzzy space. a fuzzy point pλx in x is said to be well-preclosed if there exists pµy ∈ pcl(p λ x) such that supp(p λ x) 6= supp(p µ y). proposition 2.29. if x is a fuzzy space and pλx is a fpg-closed, well-preclosed fuzzy point, then x is not quasi pre-t1 space. proof. let x be a fuzzy quasi pre-t1 space. by the fact p λ x is well-preclosed, there exists a fuzzy point pµy with supp(p λ x) 6= supp(p µ y) such that p µ y ∈ pcl(p λ x). then there exists u ∈ fpo(x) such that pλx ∈ u and p µ y 6∈ u. therefore pcl(pλx) ≤ u and p µ y ∈ u. but this is a contradiction and hence x can not be quasi pre-t1 space. � definition 2.30. let x be a fuzzy space. a fuzzy point pλx is said to be justpreclosed if the fuzzy set pcl(pλx) is again fuzzy point. clearly, in a fuzzy pre-t1 space every fuzzy point is just-preclosed. proposition 2.31. let x be a fuzzy space. if pλx and p µ x are two fuzzy points such that λ < µ and pµx is fuzzy preopen, then p λ x is just-preclosed if it is fpgclosed. proof. we prove that the fuzzy set pcl(pλx) is again a fuzzy point. we have pλx ∈ p µ x and the fuzzy set p µ x is fuzzy preopen. since p λ x is fpg-closed we have pcl(pλx) ≤ p µ x. thus pcl(p λ x)(x) ≤ µ and pcl(p λ x)(z) ≤ 0, for every z ∈ x \{x}. so the fuzzy set pcl(pλx) is a fuzzy point. � 3. fuzzy pre-convergence and fuzzy points definition 3.1. let {an, n ∈ n} be a net of fuzzy sets in a fuzzy space x. then by f − pre − lim n (an), we denote the fuzzy pre-upper limit of the net {an, n ∈ n} in i x, that is, the fuzzy set which is the union of all fuzzy on some applications of fuzzy points 127 points pλx in x such that for every n0 ∈ n and for every fuzzy preopen q−preneighborhood u of pλx in x there exists an element n ∈ n for which n ≥ n0 and anqu. in other cases we set f − pre − lim n (an) = 0. example 3.2. let (x, τ) be a fuzzy space such that x = {x, y} and τ = {0̄, 1̄, p 1 2 x }. also let {an, n ∈ n} be a net of fuzzy sets of x such that an(x) = {0.5} for every n ∈ n. the fuzzy point p 1 2 x ∈ f − lim n (an). indeed, for every n0 ∈ n and for the only fuzzy open q-neighborhood u = 1̄ of p 1 2 x there exists an element n ∈ n for which n ≥ n0 and anqu. the fuzzy point p 1 2 x 6∈ f − pre − lim n (an). indeed, for every n0 ∈ n and for the fuzzy preopen q-pre-neighborhood u = p 2 3 x of p 1 2 x does not exist any element n ∈ n such that n ≥ n0 and anqu. by the above we have f − lim n (an) 6= f − pre − lim n (an). definition 3.3. let {an, n ∈ n} be a net of fuzzy sets in a fuzzy space x. then by f − pre − lim n (an), we denote the fuzzy pre-lower limit of the net {an, n ∈ n} in i x, that is, the fuzzy set which is the union of all fuzzy points pλx in x such that for every fuzzy preopen q−pre-neighborhood u of p λ x in x there exists an element n0 ∈ n such that anqu, for every n ∈ n, n ≥ n0. in other cases we set f − pre − lim n (an) = 0. definition 3.4. a net {an, n ∈ n} of fuzzy sets in a fuzzy topological space x is said to be fuzzy pre-convergent to the fuzzy set a if f − pre − lim n (an) = f − pre − lim n (an) = a. we then write f − pre − lim n (an) = a. proposition 3.5. let {an, n ∈ n} and {bn, n ∈ n} be two nets of fuzzy sets in x. then the following statements are true: (1) the fuzzy pre-upper limit is preclosed. (2) f − pre − lim n (an) =f − lim n (pcl(an)). (3) if an = a for every n ∈ n, then f − pre − lim n (an) = pcl(a) (4) the fuzzy upper limit is not affected by changing a finite number of the an. (5) f − pre − lim n (an) ≤ pcl(∨{an : n ∈ n}). (6) if an ≤ bn for every n ∈ n, then f −pre−lim n (an) ≤f −pre−lim n (bn). (7) f − pre − lim n (an ∨ bn) =f − pre − lim n (an)∨f − pre − lim n (bn). (8) f − pre − lim n (an ∧ bn) ≤f − pre − lim n (an)∧f − pre − lim n (bn). proof. we prove only the statements (1)-(5). (1) it is sufficient to prove that 128 m. ganster, d. n. georgiou, s. jafari and s. p. moshokoa pcl(f − pre − lim n (an)) ≤ f − pre − lim n (an). let prx ∈ pcl(f − pre − lim n (an)) and let u be an arbitrary fuzzy preopen q−pre-neighborhood of pry. then, we have: uqf − pre − lim n (an). hence, there exists an element x′ ∈ x such that u(x′) + f − pre − lim n (an)(x ′) > 1. let f − pre − lim n (an)(y ′) = k. then, for the fuzzy point pkx′ in x we have p k x′ q u and p k x′ ∈ f − pre − lim n (an). thus, for every element n0 ∈ n there exists n ≥ n0, n ∈ n such that anqu. this means that prx ∈ f − pre − lim n (an). (2) clearly, it is sufficient to prove that for every fuzzy preopen set u the condition uqan is equivalent to uqpcl(an). let uqan. then there exists an element x ∈ x such that u(y)+an(x) > 1. since an ≤ pcl(an) we have u(x)+pcl(an)(x) > 1 and therefore uqpcl(an). conversely, let uqpcl(an). then there exists an element x ∈ x such that u(x) + pcl(an)(x) > 1. let pcl(an)(x) = r. then p r x ∈ pcl(an) and the fuzzy preopen set u is a fuzzy preopen q−pre-neighborhood of prx. thus uqan. (3) it follows by proposition 2.4 and the definition of the fuzzy pre-upper limit. (4) it follows by definition of the fuzzy pre-upper limit. (5) let prx ∈f−pre−lim n (an) and u be a fuzzy preopen q−pre-neighborhood of prx in x. then for every n0 ∈ n there exists n ∈ n, n ≥ n0 such that anqu and therefore ∨{an : n ∈ n}qu. thus, p r x ∈ pcl(∨{an : n ∈ n}). � proposition 3.6. let {an, n ∈ n} and {bn, n ∈ n} be two nets of fuzzy sets in y . then the following statements are true: (1) the fuzzy pre-lower limit is preclosed. (2) f − pre − lim n (an) =f − pre − lim n (pcl(an)). (3) if an = a for every n ∈ n, then f − pre − lim n (an) = pcl(a) (4) the fuzzy upper limit is not affected by changing a finite number of the an. (5) ∧{an : n ∈ n} ≤f − pre − lim n (an). (6) f − pre − lim n (an) ≤ pcl(∨{an : n ∈ n}). (7) if an ≤ bn for every n ∈ n, then f −pre−lim n (an) ≤f −pre−lim n (bn). (8) f − pre − lim n (an ∨ bn) ≥f − pre − lim n (an)∨f − pre − lim n (bn). (9) f − pre − lim n (an ∧ bn) ≤f − pre − lim n (an)∧f − pre − lim n (bn). proof. the proof is similar to the proof of proposition 3.5. � on some applications of fuzzy points 129 proposition 3.7. for the fuzzy upper and lower limit we have the relation f − pre − lim n (an) ≤ f − pre − lim n (an). proof. it is a consequence of definitions of fuzzy pre-upper and fuzzy pre-lower limits. � proposition 3.8. let {an, n ∈ n} and {bn, n ∈ n} be two nets of fuzzy sets in a fuzzy space y . then the following propositions are true (in the following properties the nets {an, n ∈ n} and {bn, n ∈ n} are supposed to be fuzzy pre-convergent): (1) pcl(f − pre− lim n (an)) = f − pre− lim n (an) = f − pre− lim n (pcl(an)). (2) if an = a for every n ∈ n, then f − pre − lim n (an) = pcl(a) (3) if an ≤ bn for every n ∈ n, then f −pre−lim n (an) ≤f −pre−lim n (bn). (4) f − pre − lim n (an ∨ bn) =f − pre − lim n (an)∨f − pre − lim n (bn). proof. the proof of this proposition follows by propositions 3.5 and 3.6. � 4. fuzzy pre-continuous functions, fuzzy pre-continuous convergence and fuzzy points definition 4.1. a function f from a fuzzy space y into a fuzzy space z is called fuzzy pre-continuous if for every fuzzy point pλx in y and every fuzzy preopen q−pre-neighborhood v of f(pλx), there exists a fuzzy preopen q−preneighborhood u of pλx such that f(u) ≤ v . let y and z be two fuzzy spaces. then by fpc(y, z) we denote the set of all fuzzy pre-continuous maps of y into z. example 4.2. let (y, τ1) and (y, τ2) be two fuzzy spaces such that y = {x, y}, τ1 = {0̄, 1̄} and τ2 = {0̄, 1̄, p 1 2 x }. we consider the map i : (y, τ1) → (y, τ2) for which i(z) = z for every z ∈ y . we prove that the map i is not fuzzy continuous at the fuzzy point p0.8x but it is fuzzy pre-continuous at the fuzzy point p0.8x . indeed, for the fuzzy open q-neighborhood v = p 1 2 x of i(p 0.8 x ) = p 0.8 x does not exist a fuzzy open q-neighborhood u of p0.8x such that i(u) ≤ v . the only fuzzy open q-neighborhood u of p0.8x in (y, τ1) is the fuzzy set 1̄ and i(1̄) 6≤ v . now, we prove that the map i is fuzzy pre-continuous at the fuzzy point p0.8x . let v be an arbitrary fuzzy preopen q−pre-neighborhood v of i(p0.8x ) = p 0.8 x . the family of all fuzzy preopen sets of (y, τ2) are the following fuzzy sets v of y : i) v ∈ iy such that v (x) ∈ [0, 1 2 ] and v (y) = 0. ii) v ∈ iy such that v (x) ∈ (1 2 , 1] and v (y) ∈ [0, 1]. the above fuzzy sets v (cases i) and ii)) are also fuzzy preopen sets of (y, τ1). so for every fuzzy preopen q−pre-neighborhood v of i(p0.8x ) in (y, τ2) there exists the fuzzy preopen q−pre-neighborhood u = v of p0.8x in (y, τ1) such that i(u) ≤ v . 130 m. ganster, d. n. georgiou, s. jafari and s. p. moshokoa definition 4.3. a fuzzy net s = {sλ, λ ∈ λ} in a fuzzy space (x, τ) is said to be p-convergent to a fuzzy point e in x relative to τ and write p lim sλ = e if for every fuzzy preopen q-pre-neighborhood u of e and for every λ ∈ λ there exists m ∈ λ such that uqsm and m ≥ λ. proposition 4.4. a function f from a fuzzy space x into a fuzzy space y is fuzzy pre-continuous if and only if for every fuzzy net s = {sλ, λ ∈ λ}, s pconverges to p, then f ◦ s = {f(sλ), λ ∈ λ} is a fuzzy net in y and p-converges to f(p). proof. it is obvious. � proposition 4.5. let f : y → z be a fuzzy pre-continuous map, p be a fuzzy point in y and u, v be fuzzy preopen q−neighborhoods of p and f(p), respectively such that f(u) 6≤ v . then there exists a fuzzy point p1 in y such that p1qu and f(p1) q6 v . proof. since f(u) 6≤ v . we have u 6≤ f−1(v ). thus there exists x ∈ y such that u(x) > f−1(v )(x) or u(x) − f−1(v )(x) > 0 and therefore u(x) + 1 − f−1(v )(x) > 1 or u(x) + (f−1(v ))c(x) > 1. let (f−1(v ))c(x) = r. clearly, for the fuzzy point prx we have p r xqu and p r x ∈ (f −1(v ))c. hence for the fuzzy point p1 ≡ p r x we have p1qu and f(p1) q6 v . � definition 4.6. a net {fµ, µ ∈ m} in fpc(y, z) fuzzy pre-continuously converges to f ∈ fpc(y, z) if for every fuzzy net {pλ, λ ∈ λ} in y which p-converges to a fuzzy point p in y we have that the fuzzy net {fµ(pλ), (λ, µ) ∈ λ × m} p-converges to the fuzzy point f(p) in z. proposition 4.7. a net {fµ, µ ∈ m} in fpc(y, z) fuzzy pre-continuously converges to f ∈ fc(y, z) if and only if for every fuzzy point p in y and for every fuzzy preopen q−pre-neighborhood v of f(p) in z there exist an element µ0 ∈ m and a fuzzy preopen q−pre-neighborhood u of p in y such that fµ(u) ≤ v, for every µ ≥ µ0, µ ∈ m. proof. let p be a fuzzy point in y and v be a fuzzy preopen q−pre-neighborhood of f(p) in z such that for every µ ∈ m and for every fuzzy preopen q−preneighborhood u of p in y there exists µ′ ≥ µ such that fµ′(u) 6≤ v. then for every fuzzy preopen q−neighborhood u of p in y we can choose a fuzzy point pu in y (see proposition 4.5) such that pu q u and fµ′(pu ) 6q v. clearly, the fuzzy net {pu, u ∈ nq−p−n(p)} p-converges to p, but the fuzzy net {fµ(pu ), (u, µ) ∈ nq−p−n(p) × m} does not p-converge to f(p) in z. on some applications of fuzzy points 131 conversely, let {pλ, λ ∈ λ} be a fuzzy net in fpc(y, z) which p-converges to the fuzzy point p in y and let v be an arbitrary fuzzy preopen q−preneighborhood of f(p) in z. by assumption there exist a fuzzy preopen q−preneighborhood u of p in y and an element µ0 ∈ m such that fµ(u) ≤ v , for every µ ≥ µ0, µ ∈ m. since the fuzzy net {pλ, λ ∈ λ} p-converges to p in y . there exists λ0 ∈ λ such that pλqu, for every λ ∈ λ, λ ≥ λ0. let (λ0, µ0) ∈ λ × m. then for every (λ, µ) ∈ λ × m, (λ, µ) ≥ (λ0, µ0) we have fµ(pλ) q fµ(u) ≤ v , that is fµ(pλ) q v . thus the net {fµ(pλ), (λ, µ) ∈ λ×m} p-converges to f(p) and the net {fµ, µ ∈ m} fuzzy pre-continuously converges to f. � proposition 4.8. a net {fλ, λ ∈ λ} in fpc(y, z) fuzzy pre-continuously converges to f ∈ fpc(y, z) if and only if f − pre − lim λ (f−1 λ (k)) ≤ f−1(k), (1) for every fuzzy preclosed subset k of z. proof. let {fλ, λ ∈ λ} be a net in fpc(y, z), which fuzzy pre-continuously converges to f and let k be an arbitrary fuzzy preclosed subset of z. let p ∈f− pre − lim λ (f−1 λ (k)) and w be an arbitrary fuzzy preopen q−pre-neighborhood of f(p) in z. since the net {fλ, λ ∈ λ} fuzzy pre-continuously converges to f, there exist a fuzzy preopen q−pre-neighborhood v of p in y and an element λ0 ∈ λ such that fλ(v ) ≤ w , for every λ ∈ λ, λ ≥ λ0. on the other hand, there exists an element λ ≥ λ0 such that v qf −1 λ (k). hence, fλ(v )qk and therefore wqk. this means that f(p) ∈ pcl(k) = k. thus, p ∈ f−1(k). conversely, let {fλ, λ ∈ λ} be a net in fpc(y, z) and f ∈ fpc(y, z) such that the relation (1) holds for every fuzzy preclosed subset k of z. we prove that the net {fλ, λ ∈ λ} fuzzy continuously converges to f. let p be a fuzzy point of y and w be a fuzzy preopen q−pre-neighborhood of f(p) in z. since p 6∈ f−1(k), where k = w c we have p 6∈ f − pre − lim λ (f−1 λ (k)). this means that there exist an element λ0 ∈ λ and a fuzzy preopen q−preneighborhood v of p in y such that f−1 λ (k) q6 v , for every λ ∈ λ, λ ≥ λ0. then we have v ≤ (f−1 λ (k))c = f−1 λ (kc) = f−1 λ (w). therefore, fλ(v ) ≤ w , for every λ ∈ λ, λ ≥ λ0, that is the net {fλ, λ ∈ λ} fuzzy pre-continuously converges to f. � proposition 4.9. the following statements are true: (1) if {fλ, λ ∈ λ} is a net in fpc(y, z) such that fλ = f, for every λ ∈ λ, then the {fλ, λ ∈ λ} fuzzy pre-continuously converges to f ∈ fpc(y, z). (2) if {fλ, λ ∈ λ} is a net in fpc(y, z), which fuzzy pre-continuously converges to f ∈ fpc(y, z) and {gµ, µ ∈ m} is a subnet of {fλ, λ ∈ λ}, then the net {gµ, µ ∈ m} fuzzy pre-continuously converges to f. 132 m. ganster, d. n. georgiou, s. jafari and s. p. moshokoa acknowledgements. s. p. moshokoa has been supported by the south african national research foundation under grant number 2053847. also, the authors thank the referee for making several suggestions which improved the quality of this paper. references [1] k. k. azad, on fuzzy semi continuity, fuzzy almost continuity, and fuzzy weakly continuity, j. math. anal. appl. 82 (1981), 14–32. [2] naseem ajmal and b. k. tyagi, on fuzzy almost continuous functions, fuzzy sets and systems 41 (1991), 221–232. [3] c. l. chang, fuzzy topological spaces, j. math. anal. appl. 24 (1968), 182–190. [4] h. corson and e. michael, metrizability of certain countable unions, ilinois j. math. 8 (1964), 351–360. [5] m. a. fath alla, on fuzzy topological spaces, ph.d. thesis, assuit univ. sohag, egypt (1984). [6] d. n. georgiou and b. k. papadopoulos, convergences in fuzzy topological spaces, fuzzy sets and systems, 101 (1999), no. 3, 495–504. [7] t. p. johnson and s. c. mathew, on fuzzy point in topological spaces, far east j. math. sci. (2000), part i (geometry and topology), 75–86. [8] hu cheng-ming, fuzzy topological spaces, j. math. anal. appl. 110 (1985), 141–178. [9] a. s. mashhour, m. e. abd el monsef and s. n. el-deeb, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. egypt 53 (1982), 47–53. [10] pu. pao-ming and liu ying-ming, fuzzy topology. i. neighbourhood structure of a fuzzy point and moore-smith convergence, j. math. anal. appl. 76 (1980), 571–599. [11] pu. pao-ming and liu ying-ming, fuzzy topology. ii. product and quotient spaces, j. math. anal. appl. 77 (1980), 20–37. [12] m. n. mukherjee and s. p. sinha, on some near-fuzzy continuous functions between fuzzy topological spaces, fuzzy sets and systems 34 (1990) 245–254. [13] s. saha, fuzzy δ−continuous mappings, j. math. anal. appl. 126 (1987), 130–142. [14] m. k. singal and niti prakash, fuzzy preopen sets and fuzzy preseparation axioms, fuzzy sets and systems 44 (1991), no. 2, 273–281. [15] c. k. wong, fuzzy points and local properties of fuzzy topology, j. math. anal. appl. 46 (1974), 316–328. [16] c. k. wong, fuzzy topology, fuzzy sets and their applications to cognitive and decision processes (proc. u. s.-japan sem., univ. calif., berkeley, calif., 1974), 171–190. academic press, new york, 1975 [17] tuna hatice yalvac, fuzzy sets and functions on fuzzy spaces, j. math. anal. appl. 126 (1987), 409–423. [18] l. a. zadeh, fuzzy sets, inform. control 8 (1965), 338–353. received september 2004 accepted january 2005 on some applications of fuzzy points 133 m. ganster (ganster@weyl.math.tu-graz.ac.at) department of mathematics, graz university of technology, steyrergasse 30 a-8010 graz, austria. d. n. georgiou (georgiou@math.upatras.gr) department of mathematics, university of patras, 265 00 patras, greece. s. jafari (sjafari@ruc.dk) department of mathematics and physics, roskilde university, postbox 260, 4000 roskilde, denmark. s. p. moshokoa (moshosp@unisa.ac.za) department of mathematics, applied mathematics and astronomy, p. o. box 392, pretoria, 0003, south africa. cawaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 2, 2007 pp. 187-205 the alexandroff duplicate and its subspaces agata caserta ∗ and stephen watson abstract. we study some topological properties of the class of the alexandroff duplicates and their subspaces. we give a characterization of metrizability and lindelöf properties of subspaces of the alexandroff duplicate. this characterization clarifies the potential for finding michael spaces among the subspaces of alexandroff duplicates. 2000 ams classification: 54b05, 54b10. keywords: resolution, alexandroff duplicate, lindelöf property, michaeltype line. in their famous 1922 memoir on compact spaces [1], alexandroff and urysohn defined a topological space that has become known as the alexandroff double circle or the alexandroff duplicate. in this paper we study several version of the alexandroff duplicate by viewing it as particular resolution by constant maps. alexandroff duplicates have been studied and used by many topologists. in particular, michael’s 1963 example of a michael spaces is subspace of an alexandroff duplicate. this paper is an organized study of the topological properties of the class of the alexandroff duplicates and of their subspaces. in particular, we characterize when subspaces of alexandroff duplicates have the lindelöf property. this suggests that the potential for finding michael spaces among the subspaces of alexandroff duplicates is not high. in this note, p and q denote the set of the irrational and rational numbers, respectively. ordinal numbers are denoted by greek letters; when viewed as topological spaces, they are given the order topology. products of topological spaces are endowed with the standard product topology. the symbol [a]λ denotes the family of subsets of a having size exactly λ. the symbols [a]≤λ and [a]<λ have similar meaning. ∗corresponding author. 188 a. caserta and s. watson let ≤∗ be the quasi-order on a countable product of ordered sets that is associated to the coordinate-wise order on each set. thus f ≤∗ g stands for f(n) ≤ g(n) for all but finitely many n ∈ ω. a subset of ωω is unbounded if it is unbounded in (ωω,≤∗). a dominating family is an unbounded set that is cofinal in (ωω,≤∗). a subset of ωω is said to be a scale if it is a dominating family and is well-ordered by ≤∗. recall that p can be identified with ωω with the product topology. for each ξ ∈ <ωω = {η | η : [0,n] → ω for some n}, a basic open neighborhood of ξ in the product topology is {f ∈ ωω : ξ ⊆ f}. for every g ∈ ωω, the sets {f ∈ ωω : f ≤ g} and {f ∈ ωω : f ≤∗ g} are respectively compact and σ-compact (see [14]). let x and y be topological spaces. a set a ⊆ x is y-analytic if it is a projection on x of a closed subset of x × y . in particular, a ⊆ x is analytic if it is p-analytic. given a function f : x → y , the small image of a ⊆ x is defined by f♯(a) = {y ∈ y : f−1(y) ⊆ a}. sometimes we abuse of terminology and say that f♯ is open, with the meaning that for each open subset a of x, f♯(a) is an open subset of y . in most cases we will employ the notation used in [6] and [9]. 1. basic definitions and preliminary results we begin with the definition of the alexandroff double circle as a resolution by constant map. definition 1.1. let (x,τ) be a topological space. for any u ∈ τ, denote û = u × 2. define a base for a topology on y = x × 2 by b = b0 ∪ b1, where b0 is the family of all subsets û \ (f × {1}) of y , with u ∈ τ, and f ∈ [x] <ω, and b1 = {(x, 1) : x ∈ x}. this topological space is the resolution of x at each point into the two point space by the constant zero function (see [15] and [7]). however we use the notation y = x ×ad 2 (the subscript ad stands for alexandroff duplicate). for each x ∈ x, we denote τ(x) = {u ∈ τ : x ∈ u} and b(x) = {(x, 1)} ∪ {û \ {(x, 1)} : u ∈ τ(x)}. further, let b ′ = ⋃ x∈x b(x). lemma 1.2. if x is a t1 space, then b ′ is a base, such that b ′ ⊂ b, and b(x) is a local base at each x ∈ x. moreover, if z = (a×{1})∪(b×{0}) ⊆ x ×ad 2 and u is a base in x at x ∈ b \ a, then {û ∩ z : u ∈ u} is a base at (x, 0) in z. proof. let (x, 0) ∈ û \ (f × {1}). set f ′ = f \ {x} and v = u \ f ′ . then v̂ \ {(x, 1)} ⊂ û \ (f × {1}) and v̂ \ {(x, 1)} ∈ b(x). � in this paper, unless otherwise stated, topological spaces are considered t1. furthermore, if u = {ui : i ∈ i}, let û = {ûi | ui ∈ u} and u ∗ = {ûi \ (f × {1}) : f ∈ [x] <ω,ui ∈ u}. the alexandroff duplicate and its subspaces 189 remark 1.3. the space x is homeomorphic to the subset z = (i×{1})∪((x\ i) × {0}) of its duplicate x ×ad 2. indeed, the function φ : x → z defined by φ(x) = (idx (x),χi (x)) is clearly a bijection. now, let u ∗ = (û \(fu ×{1}))∩z be a basic open set in z. then φ−1(u∗) = u \ (fu ∩ i) which is open in x. moreover, if u ∈ τ, then φ(u) = (u ∩ i) × {1} ∪ (u ∩ (x \i)) × {0} is an open set in z. given the topological space x ×ad 2, we consider the following functions: • r : x ×ad 2 → x × {0} such that for each x ∈ x, r(x, 1) = (x, 0), and r ↾ x × {0} = idx×{0}; • π0 : x ×ad 2 → x such that π0(x,i) = x for each x ∈ x and i = 0, 1; • ι : x → x × {0} ⊂ x ×ad 2 such that for each x ∈ x, ι(x) = (x, 0); and its inverse map π0 ↾ (x × {0}) = ι̂. note that the projection map π0 is continuous. in the following we show some properties of this functions. lemma 1.4. let x be a topological space and x × {0} ⊂ x ×ad 2, then x ∼= x × {0}. further, ι and ι̂ are homeomorphisms. proof. let (û \ (f × {1})) ∩ (x × {0}) = u × {0} be an open set in the subspace x × {0}. hence ι−1(û \ (f × {1})) = u, therefore ι is continuous. the continuity of ι̂ follows from the continuity of π0. since ι̂◦ ι = idx and ι◦ ι̂ = idx×{0}, both ι and ι̂ are homeomorphisms, i.e., ι −1 = ι̂. � definition 1.5. let y ⊂ x and f : x → y be a function. then f is called a retraction of x onto y (y is the retract of x) if it is continuous and f ↾ y = idy . if f is continuous only at the points of y , then it is called a weak retraction of x onto y . lemma 1.6. the map r is a retraction of x ×ad 2 onto x × {0}. moreover r is a closed map. proof. from r = î◦π0, it follows that r is a continuous map. since r ↾ x ×{0} = idx × {0}, r is a retraction of x ×ad 2 onto x × {0}. we show that r # is open on b. let û \ (f ×{1}) ∈ b. then r#(û \ (f ×{1})) = u ×{0}\f ×{0}, which is an open set in x × {0}. thus r is a closed map. � lemma 1.7. let z = (a × {1}) ∪ (b × {0}) ⊆ x ×ad 2 and suppose φ : z → b × {0} is a continuous map. define φ ′ : z → b × {0} by φ ′ (x,i) = { φ(x, 0) if x ∈ a ∩ b ∧ i = 1 φ(x,i) otherwise. then φ ′ is continuous too. moreover if φ is a retraction, then φ ′ is a retraction too. proof. let h = z \((a∩b)×{1}) and k = ((a∩b)×{1})∪(b×{0}) be two closed sets of z. let ψ = φ ↾ h and θ = r ↾ k two continuous maps, such that ψ ↾ k∩h = θ ↾ k∩h. then, by the pasting lemma, φ ′ = ψ∪θ is a continuous map. since φ ↾ b × {0} = idb×{0}, then φ ′ ↾ b × {0} = idb×{0}. � 190 a. caserta and s. watson definition 1.8. let x and y be two topological spaces, and f : x → y be a continuous map. then f is hereditarily closed if f ↾ z : z → f(z) is closed, for each subset z of x. the function f is hereditarily perfect if it is hereditarily closed and all fibers (f ↾ z)−1(y) are compact subsets of z respectively, for each subset z of x. remark 1.9. observe that for an hausdorff space x, the function f : x → y is hereditarily perfect if and only if f is hereditarily closed and all fibers are finite. indeed, assume that |(f ↾ z)−1(y)| ≥ ℵ0. let {zn}n be an infinite subset of (f ↾ z)−1(y). then {zn}n has a cluster point z which is an element of (f ↾ z)−1(y), because it is a closed subset of z. hence, {f(zn)}n does not have accumulation point in f(z) which is a contradiction. note that for each z ⊆ x ×ad 2 the function r ↾ z has finite fibers. the following example shows that in general the retraction is not a hereditarily closed map, hence not hereditarily perfect. example 1.10. let r : [0, 1] ×ad 2 → [0, 1] × {0} be a retraction. let z = [0, 1) × 2 ∪ {(1, 1)} ⊆ [0, 1] ×ad 2, and f = [0, 1) × 2. then f is a closed subset of z such that r ↾ z(f) = [0, 1)×{0}, which is not closed in r(z) = [0, 1]×{0}. next we show a characterization for a retraction of a subspace of the alexandroff duplicate. before we need the following definitions. definition 1.11. let a and b be subsets of a topological space x. we say that a is closed in b if for each b ∈ b such that b ∈ a, then b ∈ a. definition 1.12. let {aα}α∈κ, b be subsets of a topological space x. the family {aα}α∈κ is locally finite in b if for each point b of b there exists a neighborhood of b in x which intersects finitely many elements of {aα}α∈κ. lemma 1.13. let x and y be topological spaces, b ⊂ x and {aα}α∈κ a family of sets closed in b which is locally finite in b. let f : b → y and for each α ∈ κ, gα : aα → y are continuous maps which are compatible with f and each other. then h = f ∪ ( ⋃ α∈κ gα) is a continuous map at each point of b. proof. since h is the extension of each gα and f, it remains to check the continuity of h at points of b. let (xσ)σ∈σ a net, with b ∈ b such that xσ → b. then for each neighborhood of b, ix (b), there exists σ0 ∈ σ such that for each σ ≥ σ0, xσ ∈ ix (b). by hypothesis the family {aα}α∈κ is locally finite in b, hence for each σ ≥ σ0, ix (b) ∩ aαi = ∅ for i /∈ {1, ...,n}. set σ′ = {σ ∈ σ : σ ≥ σ0}. then (xσ)σ∈σ′ ⊆ b ∪ ( ⋃n i=1 aαi ) and xσ → b. since every net has a subnet which is an ultranet , we have that if a is a finite cover of x, there exists a ∈ a and subnet (xσλ )λ∈λ such that (xσλ )λ∈λ ⊂ a. then assume, without loss of generality, that (xσλ )λ∈λ ∈ b or (xσλ )λ∈λ ∈ aαi for some i ∈ {1, ...,n}. first assume that (xσλ )λ∈λ ∈ b. since xσλ → b and (h(xσλ ))λ∈λ = (f(xσλ ))λ∈λ, by continuity of f, follow that h(xσλ ) → h(b). let (xσλ )λ∈λ ∈ aαi for some i ∈ {1, ...,n}. from aαi closed in b and b ∈ aαi ∩b, the alexandroff duplicate and its subspaces 191 it follows that b ∈ aαi . by continuity of gαi we have gαi (xσλ ) → gαi (x). thus h(xσλ ) → h(b). � note that in the previous lemma, if the family of aα’s is finite, we only need that all sets aα are closed in b. definition 1.14. let a, b be subsets of a topological space x with a ⊆ b. we say that a is discrete in b if for each b ∈ b there exists a neighborhood of b in x which intersects a at most in one point. lemma 1.15. let x, y be topological spaces and b ⊂ x such that the points of x \ b ∩ b can be separated by a disjoint family of open sets in x. then there exists {aα : α ∈ κ}, with κ = |x \ b ∩ b|, a family of sets closed in b such that: (i) x = ( ⋃ α∈κ aα) ∪ b, (ii) |aα ∩ b| = 1 for each α ∈ κ, (iii) aα ∩ aβ = ∅ for each α and β distinct. moreover, under the additional assumption that x \ b ∩ b is discrete in b, it follows that the family {aα : α ∈ κ} is locally finite in b. proof. let {xα : α ∈ κ} = x \ b ∩ b and let {uα : α ∈ κ} be a disjoint family of open sets, such that xα ∈ uα for each α ∈ κ. let {aα : α ∈ κ} be defined by: aα = (uα \b)∪{xα} for each α ∈ κ and a0 = (x \ ( ⋃ α∈κ uα ∪b))∪{xκ}. each aα with α ∈ κ is closed in b, because the only accumulation points of x \b in b are xα’s and {xα} = aα ∩ b. moreover, since xα’s are distinct and uα’s are disjoint, {aα : α ∈ κ} is disjoint. assume that {xα : α ∈ κ} is discrete in b, we prove that{aα : α ∈ κ} is locally finite in b. note that, for all α ∈ κ, aα ∩ (b \ {xα : α ∈ κ}) = ∅. if b ∈ b \ {xα : α ∈ κ}, then b /∈ x \ b. thus there exists a neighborhood ix (b) of b such that ix (b) ∩ (x \ b) = ∅. since x \ b ∩ b is discrete in b, there exists a neighborhood i′x (b) of b such that |i ′ x (b) ∩ {xα : α ∈ κ}| ≤ 1. let i′x (b) ∩ {xα : α ∈ κ} = xα with α ∈ κ. assume that (ix (b) ∩ i ′ x (b)) ∩ aβ 6= ∅ for β ∈ κ and β 6= α. let x ∈ (ix (b) ∩ i ′ x (b)) ∩ aβ , then either x ∈ x \ b hence x /∈ b, or x /∈ b\{xα : α ∈ κ}, hence x = xβ , a contradiction. therefore ix (b) ∩ i ′ x (b) intersects {aα : α ∈ κ} in xα. now, if b ∈ {xα : α ∈ κ}, then b = xα for α ∈ κ. since, by construction, uα ∩uβ = ∅ for α 6= β and aβ ⊂ uβ, it follows that uα ∩aβ = ∅ for β 6= α except for β = 0. therefore uα intersects finitely many elements of {aα : α ∈ κ}. � lemma 1.16. let x, y be topological spaces and b ⊂ x such that x \ b ∩b is discrete in b and its points can be separated by a disjoint family of open sets in x. then any f : b → y continuous map can be extended to x, so that it remains continuous at points of b. proof. let κ be a cardinal with |x \ b ∩ b| = κ. by lemma 1.15, there exists {aα : α ≤ κ} a disjoint family of closed sets in b such that x = ( ⋃ α∈κ aα)∪b and aα ∩ b = {xα} for some xα. let gα : aα → y given by gα(x) = f(xα). 192 a. caserta and s. watson then gα’s are continuous maps. by lemma 1.13, there exists h = f ∪ ⋃ α∈κ gα, which extends f to x and is continuous at b. � note that in the previous lemma, if |x \ b ∩ b| < ℵ0, we only need the space x to be hausdorff. corollary 1.17. let b ⊆ x. if x \ b ∩ b is locally discrete in b and its points can be separated by a disjoint family of open sets in x, then there is a weak retraction of x onto b. proof. let idb be the identity map on b. apply lemma 1.16 with y = b. � with reference to definition 1.5, the following proposition gives a characterization for a retraction of a subspace z of x ×ad 2. proposition 1.18. let z = (a × {1}) ∪ (b × {0}) ⊆ x ×ad 2. then z can be retracted onto b × {0} if and only if a ∪ b can be weakly retracted onto b. proof. let φ be a retraction of z onto b × {0}, and assume, without loss of generality, by lemma 1.7, that for each x ∈ a ∩ b, φ(x, 1) = φ(x, 0) = (x, 0). for each x ∈ a ∪ b choose ix ∈ {0, 1} such that (x,ix) ∈ z. since φ(x, 1) = φ(x, 0) for each x ∈ a∪b the choice of ix does not matter. define f : a∪b → b by f(x) = ι−1 ◦ φ(x,ix). for each x ∈ b, f(x) = ι −1 ◦ φ(x, 0) = x. it remains to prove the continuity of f at the points of b. let (xσ)σ∈σ a net in x, x 6= xσ for each σ ∈ σ and x ∈ b, such that xσ → x. choose iσ ∈ {0, 1} where (xσ, iσ) ∈ z. then (xσ, iσ) → (x, 0). since ι −1 ◦ φ is a continuous function, ι−1 ◦ φ(xσ, iσ) → ι ◦ φ(x, 0). then f(xσ) → f(x). vice versa, let f : a ∪ b → b be a weak retraction and (r ↾ z) : z → (a ∪ b) × {0} the retraction of z onto b × {0}. define φ : z → b × {0} such that φ(x,i) = ι ◦ f ◦ ι−1 ◦ (r ↾ z)(x,i). then φ is a continuous map at the points of b × {0}. since a× {1} is a discrete subset of x ×ad 2, it follows that φ is continuous. moreover φ ↾ b × {0} = idb×{0}. � lemma 1.19. let x be a complete metrizable space, y ⊆ x such that y is a countable dense set with no isolated points. then there is no weak retraction of x onto y . proof. by contradiction, let f : x → y be a weak retraction and a ⊆ x the set of all points in which f is continuous. then y ⊆ a and a is a gδ set in x, hence a is a complete metrizable space. observe that a = ⋃ y∈y (f ↾ a) −1({y}), i.e., a is an fσ in x. by baire category theorem, there exists y ∈ y such that (f ↾ a)−1({y}) is not nowhere dense, hence (f ↾ a)−1({y}) contains u, an open set in a. since u ∩ y 6= {y}, we have a contradiction. � corollary 1.20. let z = (p × {1}) ∪ (q × {0}) ⊂ [0, 1] ×ad 2. then z cannot be retracted onto q × {0}. proof. by lemma 1.19, there is not a weakly retraction of r onto q. proposition 1.18 ends the proof. � the alexandroff duplicate and its subspaces 193 2. properties preserved by the alexandroff duplicate and its subspaces as a special case of the the fundamental theorem of resolutions, we have that if x is a compact space, then x ×ad 2 is also compact. next we show that many properties of x are preserved by its duplicate. lemma 2.1. x is tychonoff space if and only if x ×ad 2 is tychonoff. proof. first we show that t1 is preserved. any isolated point (x, 1) ∈ x ×ad 2 is clopen. moreover, since x is t1, and x ∼= x × {0} is a closed subspace of x ×ad 2, it follows that every point is closed in x ×ad 2. observe that for the points of x × {1}, since they are isolated points, there exists always a continuous map f : x ×ad 2 → i such that f((x, 1)) = 0 and f((x,i)) = 1 for any (x,i) 6= (x, 1). let (x0, 0) ∈ x×ad 2 and u ∗ = û \{(x0, 1)} a neighborhood of (x0, 0). since x is tychonoff, x0 ∈ x and u ∈ τx , there exists g : x → i continuous map such that g(x0) = 0 and g(u c) = 1. we define f : x ×ad 2 → i such that f((x,i)) = 1 for (x,i) = (x0, 1) and f((x,i) = g(x) otherwise. then f is continuous map such that f((x0, 0)) = 0 and f((x,i)) = 1 for any (x,i) ∈ (x ×ad 2) \ u ∗. since x ∼= x × {0}, the vice versa holds as well. � lemma 2.2. if x is normal, then also x ×ad 2 is normal. proof. let k and c be two closed disjoint subsets of x×ad2. let k∩x×{0} = k1 and c∩x ×{0} = c1 closed disjoint subsets in x ×{0}, by normality of x, there exists u1 and v1 open sets in x ×{0} such that k1 ⊆ u1and c1 ⊆ v1 and u1∩v1 = ∅. then v = (v̂1 \k)∪(c\x×{0}) and u = (û1\c)∪(k\x×{0}) are open disjoint subsets in x ×ad 2 containing c and k respectively. then x ×ad 2 is normal. � let a and b subsets of x and z = (a × {1}) ∩ (b × {0}) ⊆ x ×ad 2. henceforth, unless we state otherwise, we denote a subset of x ×ad 2 simply with z. in general a subspace z ⊆ x ×ad 2 need not to be normal even if x is normal. indeed, if x is not hereditarily normal, there exists a ⊆ x that is not normal. take z = a × {0}. then z is not normal as a subset of that normal space x ×ad 2. lemma 2.3. let x be a normal space and b a closed subset of x. then z ⊆ x ×ad 2 is normal. proof. let k = k ′ ∩z and c = c ′ ∩z be two closed disjoint subsets of z, with k ′ and c ′ closed subsets of x×ad2. let k∩b×{0} = k1 and c∩b×{0} = c1 be closed disjoint subsets in b × {0}. since b is a closed subset of x, and b × {0} is normal, there exists u1 = u ∩ b and v1 = v ∩ b, with u ,v ∈ τ, which are disjoint open sets in b × {0} such that k1 ⊆ u1 and c1 ⊆ v1. take v ∗ = ((v̂1 ∩z)\k)∪(c\b×{0}) and u ∗ = ((û1∩z)\c)∪(k\b×{0}). then v ∗ and u∗ are open disjoint subsets in z containing c and k respectively. � 194 a. caserta and s. watson lemma 2.4. for any topological space x, the following hold: (i) l(x) = l(x ×ad 2), (ii) if |x| ≥ ℵ0, c(x ×ad 2) = |x|, (iii) χ(x) = χ(x ×ad 2). proof. (i). observe that l(x) = l(x × {0}) ≤ l(x ×ad 2), because x × {0} is closed. it is sufficient to show that l(x ×ad 2) ≤ l(x). let u be an open cover for x ×ad 2, and assume, without loss of generality, that u ⊆ b. thus, u∩b0 = {ûi\(fui ×{1}) : i ∈ i∧fui ∈ [x] <ω}. since l(x) = κ, there exists an open refinement of {ui}i∈i , {vi}i∈j which cover x, such that |j| ≤ κ. for each vi ∈ {vi}i∈j choose ui ∈ {ui}i∈i , such that vi ⊆ ui, and define fui = fvi . so {v̂i \ (fvi × {1}) : i ∈ j ∧ fvi ∈ [x] <ω}, leaves uncovered ⋃ i∈j fvi × {1} which can be covered by |j| many open sets. thus l(x ×ad 2) ≤ κ. (ii). let |x| = κ ≥ ℵ0. the set x ×{1} ⊂ x ×ad 2 is a set of isolated points of size κ , so c(x × {1}) = κ. since x × {1} is an open subset of x ×ad 2, c(x × {1}) ≤ c(x ×ad 2). thus c(x ×ad 2) ≥ κ. from c(x × {0}) ≤ |x ×ad 2|, if follows that c(x ×ad 2) ≤ κ. (iii). for each a = (x, 1) ∈ x×{1}, {a} is a local base at a. thus χ(a,x×ad 2) =1. next, let a = (x, 0) ∈ x × {0} and χ(x) = k. let u(x) be a local base at each x ∈ x. then {û \ {(x, 1)} : u ∈ u(x)} is a local base at a = (x, 0) ∈ x × {0}, which has the same size as u(x) . then χ(x × {0}) = χ(x ×ad 2) = χ(x). � lemma 2.5. if u is a locally finite family in x, then û is a locally finite family in x ×ad 2. proof. let z = (x,i) ∈ x ×ad 2, observe that if v is a neighborhood of x in x, then v̂ is a neighborhood of z in x ×ad 2. now, since u is a locally finite family of x, there exists a neighborhood vx of x which intersects only finitely many elements of u. thus v̂x is a neighborhood of z in x ×ad 2, which intersect only finitely many elements of û. � remark 2.6. furthermore, if u is a locally finite family of open subsets of x, for any map f : u → [x × 2]<ω such that u → fu , the set {û \ fu : u ∈ u} is locally finite family of open subsets of x ×ad 2. proposition 2.7. if x is a paracompact space, then also x ×ad 2 is a paracompact space. proof. let u be an open cover for x ×ad 2, and assume, without loss of generality, u ⊂ b. so u ∩ b0 = {û \ (fu × {1}) : u ∈ u ′ ∧ fu ∈ [x] <ω} where u ′ is a cover of x. by paracompactness of x, there exist an open refinement v ′ of u ′ which is locally finite. for each v ∈ v ′ , choose u ∈ u ′ such that v ⊆ u. define fv = fu . then v0 = {v̂ \ fv × {1} : fv ∈ [x] <ω,v ∈ v ′ } is also a locally finite family of x ×ad 2 and it is an open refinement of u ∩ b0. let v1 = {{(x, 1)} : (x, 1) 6∈ ∪v0}. then v1 is a discrete family of open sets, since x × {0} ⊆ ∪v0. thus v = v0 ∪ v1 is an open refinement of u which is locally finite. � the alexandroff duplicate and its subspaces 195 proposition 2.8. if x is an hereditarily paracompact space, then also x ×ad 2 is an hereditarily paracompact space. proof. it is sufficient to show that any open subspace of x×ad 2 is paracompact (see [4]). let a be an open subspace of x ×ad 2 and u an open cover of a, without loss of generality, assume that u ⊂ b. then u = u0 ∪ u1 where u0 = {û \ (fu × {1}) : u ∈ u ′ ∧ fu ∈ [x] <ω} and u ′ is an open covering of π0(a ∩ x × {0}), and u1 ⊆ {{(x, 1)} : (x, 1) ∈ a}. since x is hereditarily paracompact, there exists v ′ open refinement of u ′ which is locally finite at the points of π0(a ∩ x × {0}) . now, for each v ∈ v ′ , choose u ∈ u ′ such that v ⊆ u and define fv = fu . let v0 = {v̂ \ (fv × {1}) : v ∈ v ′ } and v1 = {{(x, 1)} : (x, 1) 6∈ ∪v0} ∩ u1. then v1 is a discrete family of open sets and v = v0 ∪ v1 is an open refinement of u which is locally finite. then a is paracompact in x ×ad 2. � corollary 2.9 (alexandroff, urysohn [1]). [0, 1]×ad 2 is a first countable compact hausdorff, hereditarily paracompact space which has an uncountable disjoint family of open sets. from the definition of menger-urysohn dimension, it follows that indx ≤ n with n ≥ 0 if and only if there exists a base b of x such that ind∂b ≤ n− 1 for each b ∈ b. in particular indx = 0 if and only if the space is 0-dimensional and for every subspace y ⊂ x, and we have that indy ≤ indx. in order to calculate the small inductive dimension of the alexandroff duplicate and its subspaces, we need the following lemmas. lemma 2.10. let (x,τ) be a topological space, and u,v ∈ τ. then (i) v̂ ⊆ v̂ and v̂ \ v̂ = ∂v × {1} (ii) if v ⊆ u, then v̂ ⊆ û proof. observe that v × {0} = v × {0} and v × {1} = v × {1} ∪ v ′ × {0}. then v̂ = v × {0} ∪ v × {1} = v × {0} ∪ v × {1} ⊆ v × {0} ∪ v × {1} = v̂ . moreover v̂ \v̂ = (v ×{0}∪v ×{1})\(v ×{0}∪v × {1}) = v ×{1}\v ×{1} = ∂v × {1}. if v ⊆ u, by (i), v̂ ⊆ v̂ ⊆ û. � lemma 2.11. let z ⊆ x ×ad 2. if w ⊆ z, then clz (w) ∩ (a × {1}) = w ∩ (a × {1}) = intz (w) ∩ (a × {1}) proof. for each (x, 1) ∈ a×{1}, (x, 1) ∈ clz (w) if and only if (x, 1) ∈ w , and (x, 1) ∈ intz (w) if and only if (x, 1) ∈ w . � lemma 2.12. let z ⊆ x ×ad 2 and d ⊆ b. then (i) clz (d̂) ∩ (b × {0}) = clb(d) × {0}; (ii) intz (d̂) ∩ (b × {0})= intb(d) × {0}; (iii) ∂z (d̂ ∩ (a × {1})) = ((d ∩ a) × {0}) ′ ; (iv) clz (d̂ ∩ z) = (clb(d) × {0}) ∪ (d ∩ (a × {1})); 196 a. caserta and s. watson (v) intz (d̂ ∩ z) = (intb(d) × {0}) ∪ (d ∩ (a × {1})); (vi) ∂b×{0}(d × {0}) = ∂z (d̂ ∩ z). proof. (i). the point (x, 0) belongs to clz (d̂)∩(b×{0}) if and only if for each u∗ ∈ τz (x, 0), u ∗ = (û \{(x, 1)})∩z , u∗∩(d̂) 6= ∅ hence u∗∩(d×{0}) 6= ∅. thus (x, 0) ∈ clz (d × {0}) ∩ b × {0} = clb(d) × {0}. (ii). clearly if (x, 0) ∈ intx (d ∩ b) × {0} = intb(d) × {0}, then (x, 0) ∈ intz (d̂). moreover, if (x, 0) ∈ intz (d̂), there exists u ∗ ∈ τz (x, 0), u ∗ = (û \ {(x, 1)}) ∩ z, u∗ ⊆ d̂, so (u ∩ b) × {0} ⊆ d × {0}. hence (x, 0) ∈ intx ((d ∩ b) × {0}) = intb (d) × {0}. (iii). since d̂ ∩ (a × {1}) is an open subset of z, ∂z (d̂ ∩ a × {1}) = d̂ ∩ (a × {1}) \ (d̂ ∩ (a × {1})). moreover, for each (x, 1) ∈ z \ (d̂ ∩ a × {1}), (x, 1) /∈ d̂ ∩ (a × {1}), thus for each (x, 1) ∈ z \ (d̂ ∩ (a × {1})), (x, 1) /∈ ∂z (d̂ ∩ (a × {1})). for each (x, 0) ∈ z \ (d̂ ∩ (a × {1})), (x, 0) ∈ (d̂ ∩ (a × {1})) if and only if (x, 0) ∈ (d ∩ a) × {1} ∪ ((d ∩ a) × {0}) ′ , since (d̂ ∩ (a × {1})) = (d∩a)×{1}∪((d∩a)×{0}) ′ . then ∂z (d̂ ∩ (a × {1})) = ((d ∩ a) × {0}) ′ . (iv). from (i), it follows that clz (d̂ ∩ z) ⊆ clb (d) × {0} ∪ (d ∩ (a × {1})). on the other hand, clz (d̂∩z) ⊇ clz (d×{0}) = clb(d)×{0} and clz (d̂∩z) ⊇ (d̂ ∩ z) ⊇ (d ∩ (a × {1})). (v). from (ii), it follows that intz (d̂∩z) ⊆ intb (d)×{0}∪(d∩(a×{1})). on the other hand, we have that intz (d̂ ∩ z) ⊇ intz (d̂ ∩ z) ∩ (a × {1}) = (d∩a)×{1} and intz (d̂∩z) ⊇ intz (întb∩z) = întb∩z ⊆ (întb∩z)∩(b×{0}) = intb(d) × {0}. � theorem 2.13. let z ⊆ x ×ad 2 with b 6= ∅, then indz = indb. proof. since b ≃ b ×{0} ⊂ z, then indb ≤ indz. it is sufficient to show that indz ≤ indb, using induction on indb. for indb = 0, there exists a base g for b, such that for each g ∈ g, g is clopen in b, then for each g ∈ g, also ĝ ∩ z is clopen in z. now, for each g ∈ g and f ∈ [a]<ω, (ĝ \ f × {1}) ∩ z is a basic open sets in z. they are also a clopen sets, because the boundaries never contain isolated points. then {(a, 1)}a∈a ∪ {(ĝ \ f × {1}) ∩ z : f ∈ [a] <ω,g ∈ g} is a base of clopen sets for z and indz = 0. let n ∈ ω and indb = n, we want to prove that indz ≤ n. since indb = n there exists a base g for b, such that for each g ∈ g, ind∂bg ≤ n − 1. thus ∂b×{0}g × {0} ≤ n − 1. let (ĝ \ f × {1}) ∩ z the corresponding basic open set in z, then from lemma 2.12 we have that ∂z ((ĝ \ f × {1}) ∩ z) = ∂z (ĝ ∩ z) = ∂b×{0}g × {0}, then it follows that ind∂z ((ĝ \ f × {1}) ∩ z) = ind∂b×{0}g × {0} ≤ n − 1. thus indz ≤ n − 1. � the alexandroff duplicate and its subspaces 197 we conclude this section analyzing the baire property of the alexandroff duplicate and its subspaces. we first characterize dense sets of subspaces of the alexandroff duplicate. lemma 2.14. a topological space (x,τ) has a dense set of isolated points, if and only if any {di}i∈i arbitrary family of dense subsets of x is such that⋂ i∈i di is dense too. proof. let d a dense set in x consisting of isolated points. notice that if di is a dense set in x, then d ⊆ di, otherwise there exist x ∈ d \ di and {x} is an open set which does not intersect di, contradiction. therefore d ⊆ ⋂ i∈i di. viceversa, if x does not have a dense set of isolated points, then there exists an open set u such that u has no isolated point. now, for each x ∈ u, x \ {x} is dense, so ⋂ x∈u x \ {x} is dense too. we have a contradiction, because x \ u = ⋂ x∈u x \ {x} is not dense in x. � lemma 2.15. let (x,τ) be a topological space. if {di}i∈i is an arbitrary family of dense subsets of x ×ad 2, then ⋂ i∈i di is dense too. proof. the set x × {1} ⊆ x ×ad 2 is a dense set of isolated points. the result follows from lemma 2.14. � corollary 2.16. x ×ad 2 is baire for each topological space x. lemma 2.17. let z ⊆ x ×ad 2. an open set u is a dense subset of z if and only if there exists d dense open in b \ a ′ and u = (a × {1}) ∪ (d × {0}). proof. let u = (a × {1}) ∪ (d × {0}) be a subset of z. since d is dense in b \ a ′ , we have d × {0} = d × {0} = (b \ a ′ ) × {0}. then u = (a × {1}) ∪ (d × {0}) = (a×{1}) ∪d × {0} = a×{1}∪ (b ∩a ′ ) ×{0}∪ (b \ a ′ ) × {0} = z. since b \a ′ is an open set we have that u is an open set in z. let u be a dense open set in z, then a×{1} ⊆ u. assume by contradiction that for every dense open in b\a ′ we have that either u ( (a×{1})∪(d×{0}) or u ) (a × {1}) ∪ (d × {0}). let d = u ∩ (b \ a ′ ). since (b \ a ′ ) × {0} is open in z, i.e., d is dense open in b \ a ′ . then v = (a × {1}) ∪ (d × {0}) is dense open set in z such that v ⊆ u. if u ) (a × {1}) ∪ (d × {0}) then v ) u which is a contradiction. if u ( (a × {1}) ∪ (d × {0}) we have that u ( v ⊆ u, a contradiction. � lemma 2.18. let {dα}α∈κ be a family of dense open sets in b \ a ′ . then⋂ α∈κ a × {1} ∪ dα × {0} is dense in z if and only if ⋂ α∈κ dα is dense in b \ a ′ . proof. ⋂ α∈κ(a × {1}) ∪ (dα × {0}) = z if and only if a × {1}∪ ⋂ α∈κ dα × {0} ⊇ z if and only if a × {1} ∪ (b ∩ a ′ ) × {0} ∪ ⋂ α∈κ dα × {0} ⊇ z if and only if⋂ α∈κ dα × {0} ⊇ (b \ a ′ ) × {0}, i.e., ⋂ α∈κ dα is dense in b \ a ′ . � definition 2.19. a topological space x is κ-baire if the intersection of less then κ dense open sets is dense; in this case, we write baire(x) = κ. further, 198 a. caserta and s. watson we set baire(x) = ∞ if for all dense open sets their intersection is dense, and assume that baire(∅) = ∞. theorem 2.20. let z ⊆ x ×ad 2. then baire(z)= baire(b \ a ′ ). proof. to prove the equality, we first observe that if (b \ a ′ ) × {0} = ∅, then a ′ ⊇ b, i.e., a × {1} = z. thus z has a dense set of isolated points and so baire(z) = baire((b \ a ′ ) × {0}) = ∞. assume that (b \ a ′ ) × {0} 6= ∅ and baire(z) = κ. since z has κ many dense open sets with non-dense intersection, by lemma 2.17, without loss of generality, dense open sets have form a×{1}∪dα with α ∈ κ, and dα‘s are dense open sets in (b\a ′ )×{0}. by lemma 2.17, ⋂ α∈k dα × {0} is not dense in (b \ a ′ ) × {0}. so (b \ a ′ ) × {0} has κ many dense open sets with non-dense intersection. viceversa, assume that baire((b \a ′ ) ×{0}) = κ, so b \a ′ has κ many dense open sets {dα}α∈k with non-dense intersection. by lemma 2.17 and lemma 2.18, {a×{1}∪dα × {0}}α∈κ is a family of dense open sets with non-dense intersection. thus z has κ many dense open sets with non-dense intersection, i.e., baire(z) ≤ κ. � 3. metrizability and lindelöf property before extracting the michael line and its relatives from [0, 1]×ad 2, we prove some useful characterizations of those subspaces of x ×ad 2 that are metrizable or lindelöf. definition 3.1. let (x,τ) be a topological space and a and b subsets of x. a is κ-discrete in a∪b, if a is the union of κ many sets with no accumulation points in b. lemma 3.2. let z ⊆ x ×ad 2. if every closed set of z is gκ, then a is κ-discrete in a ∪ b. proof. since a × {1} is open set in z, then it is an fκ set. let a = ∪{kα : α ∈ κ} where each kα × {1} is closed in z. by contradiction, if kα had an accumulation point b ∈ b, then kα × {1} would have an accumulation point at (b, 0) ∈ z. � lemma 3.3. let z ⊆ x ×ad 2 and k ⊆ a such that k has no accumulation points in b. then k × {1} is closed in z. proof. since k has no accumulation points in b, for each b ∈ b there exists u ∈ τx (b) such that u ∩ k ⊆ {b}. for each (b, 0) ∈ b × {0} there exists uz = (û \ {(b, 1)}) ∩ z such that uz ∩ (k × {1}) = ∅. then k × {1} has no accumulation points in b×{0}. on the other hand, since every point in a×{1} is an isolated point, k ×{1} cannot have accumulation points in a×{1}. thus k × {1} has no accumulation points, so k × {1} is closed in z. � definition 3.4. let (x,τ) be a topological space and a ⊆ x. we say that b is a base for the points of a in x, if each b ∈ b is an open set in x and for each x ∈ a and u ∈ τx (x) there exist b ∈ b such that x ∈ b ⊆ u. the alexandroff duplicate and its subspaces 199 theorem 3.5. let z ⊆ x ×ad 2 and κ be an infinite cardinal. then z has a κ-discrete base if and only if b has a κ-discrete base in x and a is κdiscrete in a ∪ b. furthermore we have: (i) if b is a κ-discrete base for b in x and a = ⋃ β<κ kβ with each kβ having no accumulation points in b, then b∗ = {v̂ \( ⋃ β∈f kβ ×{1}) : f ∈ [κ]<ω,v ∈ b} is a κ-discrete base for b × {0} in z. (ii) if b∗ = ⋃ γ∈κ bγ ∗ is a κ discrete base for z, then bb×{0} = ⋃ γ∈κ{u ∩ (b × {0}) : u ∈ b∗γ} is a κ-discrete base for b × {0} in z and a =⋃ γ∈κ{a : {(a, 1)} ∈ b ∗ γ}. proof. assume that b is a κ-discrete base for b in x and a = ⋃ α∈κ kα where each kα has no accumulation points in b. then kα = {{(k, 1)} : k ∈ kα} is a discrete family of open sets in z and k = ∪{kα : α ∈ κ} is a κ-discrete open family which is a base for a × {1}. next we want to find a κ-discrete base for b×{0} in z. let (b, 0) ∈ b×{0}, and (û \ {(b, 1)}) ∩z a neighborhood of (b, 0) in z. if b ∈ a, there exists γ ∈ κ such that (b, 1) ∈ kγ ×{1} which is closed in z by lemma 3.3. moreover there exists v ∈ b such that b ∈ v ⊆ u. let f = {γ} hence (b, 0) ∈ v̂ \ ( ⋃ β∈f kβ × {1}) ∩ z ⊆ v̂ \ {(b, 1)} ∩ z ⊆ û \ {(b, 1)} ∩ z. then b∗ is a base for b × {0} in z. since b is a κ-discrete base for b, we have that b = ⋃ γ∈κ bγ . then b∗ = ⋃ γ∈κ bγ ∗ where bγ ∗ = {v̂ \ ( ⋃ β∈f kβ × {1}) : f ∈ [κ] <ω,v ∈ bγ}. it remains to prove that for each γ ∈ κ, bγ ∗ is a κ-discrete family in z. since for every γ ∈ κ, bγ is discrete family, then {v̂ : v ∈ bγ} is discrete too. let γ ∈ κ and f ∈ [κ]<ω be fixed, then{v̂ \ ( ⋃ ξ∈f kξ × {1}) : v ∈ bγ} ⊆ {v̂ : v ∈ bγ} is a discrete family. hence b∗ = ⋃ γ∈κ( ⋃ {v̂ \( ⋃ ξ∈f kξ ×{1}) : f ∈ [κ] <ω,v ∈ bγ}) is κ-discrete too. further, b ∗ ∪ k is a κ-discrete base for z. assume now that z has a κ-discrete base b∗ = ⋃ γ∈κ b ∗ γ . set bb×{0} =⋃ γ∈κ{u ∩ (b × {0}) : u ∈ b ∗ γ}. then bb×{0} is a κ-discrete base for b × {0} in z. since a × {1} is a set of isolated points in z, then for each a ∈ a, the set {(a, 1)} ∈ b∗. hence kγ = {a : {(a, 1)} ∈ b ∗ γ} is a discrete family of points such that a = ⋃ γ<κ kγ . clearly every kγ has no accumulation points in b, otherwise kγ would not be discrete. � definition 3.6. let (x,τ) be a regular topological space and a ⊆ x. we say that a is metrizable in x if there exists a σ-discrete open family in x which is a base for the points of a in x. corollary 3.7. let x be a regular space. the subspace z ⊆ x ×ad 2 is metrizable if and only if b has a countable base in x, and a is the union of countable many sets with no accumulation points in b. proof. since x ×ad 2 is regular, it follows from theorem 3.5. � in the following we characterize the lindelöf property of a subspace z ⊆ x ×ad 2. first we need the following lemma. 200 a. caserta and s. watson lemma 3.8. if z ⊆ x ×ad 2 is lindelöf, then every uncountable subset k of a, with cf|k| > ℵ0, has a complete accumulation point in b. proof. assume that z is lindelöf . by contradiction, assume that there exists k ⊂ a such that cf|k| > ℵ0 and for each b ∈ b there exists ub ∈ τx containing b such that |ub ∩ k| < |k|. let ub ∗ = ûb \ {(b, 1)} ∩z the corresponding open set in z. then {ub ∗}b∈b is an open cover of b × {0}, which is a closed subset of a lindelöf space. by lindelöfness of b × {0} we can find a countable open subcover {ubi ∗}i∈ω of b × {0}. let u ∗ = ⋃ i∈ω ubi ∗. then b × {0} ⊂ u∗ and |u∗ ∩ k × {1}| = | ⋃ i∈ω (ubi ∗ ∩ k × {1})| ≤ σi∈ω|ubi ∗ ∩ k × {1}|. since |ubi ∗∩k×{1}| < |k| and cf|k| > ℵ0, it follows that |u ∗∩k×{1}| < |k|. now, denote ã = {a ∈ k : (a, 1) /∈ u∗} and let u = {u∗} ∪ {(a, 1) : a ∈ ã} be a cover of z. since {(a, 1) : a ∈ ã} contain an uncountable subset of isolated points we cannot find a countable subcover of u in contradiction with the lindelöfness of z . � theorem 3.9. let z ⊆ x ×ad 2. the following statements are equivalent: (i) z is lindelöf; (ii) b is lindelöf and every uncountable subset k of a, with cf|k| > ℵ0, has a complete accumulation point in b; (iii) b is lindelöf and every uncountable subset k of a, with |k| regular cardinal has a complete accumulation point in b; (iv) b is lindelöf and every uncountable subset k of a, with |k| = ℵ1 has a complete accumulation point in b; (v) b is lindelöf and every uncountable subset k of a has an accumulation point in b; (vi) b is lindelöf and for every open subset u in x containing b, we have |a \ u| ≤ ℵ0; (vii) b is lindelöf and for every closed subset f in x that misses b, we have |a ∩ f | ≤ ℵ0. proof. assume that (i) holds. since b × {0} is closed subspace of z, it follows that b × {0} and thus b is lindelöf. hence (ii) follows from lemma 3.8. the implications (ii) ⇒ (iii) and (iii) ⇒ (iv) are immediate. for (iv) ⇒ (v), let k be uncountable subset of a. for |k| = ℵ1 is immediate. assume that |k| > ℵ1. assume that k has no accumulation points in b. then there exists k ′ ⊂ k such that |k ′ | = ℵ1 with no complete accumulation points in b, contradiction. now, assume that (v) holds, we want to prove (i). let u be an open cover of z, and assume, without loss of generality, that it consists of basic open sets. since b × {0} is lindelöf, we can cover it with countably many open sets from the cover u. denote such a cover ub×{0}. let k ⊆ a such that k × {1} = z \ ⋃ ub×{0} where ⋃ ub×{0} ⊇ b ×{0}. we claim that k ×{1}∪ub×{0} is a countable subcover of u for z. indeed, k × {1} ∩ ( ⋃ ub×{0}) = ∅, so k × {1} the alexandroff duplicate and its subspaces 201 has no accumulation points in b × {0} and so k has no accumulation points in b. for (i) ⇒ (vi), assume that z is lindelöf. let u open set containing b. if |a \ u| > ℵ0, let ub = (û ∩ z) ⊇ b × {0} be an open set in z. then ub with the points of (a × {1}) \ ub has no countable subcover, a contradiction. assume that (vi) holds, we want to prove (v). let k be uncountable subset of a with no accumulation points in b, then there exists an open set u ⊃ b × {0} which misses k × {1}. then |a \ u| > ℵ0. the equivalence between (vi) and (vii) is immediate. � corollary 3.10. let x be hereditarily lindelöf and z ⊆ x ×ad 2. the following statements are equivalent: (i) z is lindelöf; (ii) every uncountable subset k of a, with cf|k| > ℵ0, has a complete accumulation point in b; (iii) every uncountable subset k of a, with |k| regular cardinal has a complete accumulation point in b; (iv) every uncountable subset k of a, with |k| = ℵ1 has a complete accumulation point in b; (v) every uncountable subset k of a has an accumulation point in b; (vi) for every open subset u in x containing b, we have |a \ u| ≤ ℵ0 (vii) for every closed subset f in x that misses b, we have |a ∩ f | ≤ ℵ0. corollary 3.11. let z ⊆ x ×ad 2. if (i) b is lindelöf and (ii) for every closed subset f of x that misses b, we have |a ∩ f | ≤ ℵ0, then z is lindelöf. moreover, under the additional assumption that a is uncountable, z is not metrizable. proof. from theorem 3.9 it follows that z is lindelöf, and since a is uncountable, any uncountable subset of a has accumulation points in b. hence a is not σ-discrete in a∪b. by proposition 3.5, it follows that z is not metrizable. � the following example shows that the restriction on the cofinality is needed. example 3.12 (watson [15]). let x = [0, 1] and assume that 2ℵ0 > ℵω. consider subsets of [0, 1], an’s, such that |an| = ℵn and an ⊂ [ 1 n+1 , 1 n ]. take a = ⋃ n∈ω an, b = (0, 1] and z = (a × {1}) ∪ (b × {0}) ⊂ [0, 1] ×ad 2. let zn = z ∩ ([ 1 n , 1] ×{2}). then for every n ∈ ω, zn is compact, so z = ⋃ n∈ω zn is σ-compact, thus lindelöf. however a has no complete accumulation points in b, but every uncountable subset of a has accumulation points in b. corollary 3.13 (michael, corson [3] (see olso [8])). z ⊆ [0, 1]×ad2 is lindelöf if and only if every uncountable subset of a has an accumulation point in b. the michael line and its relatives are subspaces of [0, 1] ×ad 2 of a certain kind: 202 a. caserta and s. watson definition 3.14. if z ⊆ x ×ad 2 where a and b are disjoint, then we say that z is a michael-type line. assume x = [0, 1] unless stated otherwise. corollary 3.7 gives us a necessary and sufficient condition for the metrizability of a subspace of alexandroff duplicate. in the next result we give a different condition for metrizability, which takes into account only the size of the subspace a. proposition 3.15. under maℵ1 , if a,b ⊂ [0, 1] such that |a| ≤ ℵ1, then the michael-type line is metrizable. proof. maℵ1 implies that every subset of the reals of cardinality at most ℵ1 is a q-set. thus a is the union of countably many subsets closed in a ∪ b, i.e., a = ⋃ n∈ω cn. since a ∩ b = ∅ and cn’s are closed in a ∪ b, we have that for each n ∈ ω, cn has no accumulation point in b. by corollary 3.7 it follows that z is metrizable. � example 3.16 (michael, corson [12]). let a = p and b = q. call this michael-type line the michael line lnmic. now we show some property of the michael-type line lnmic. lemma 3.17. let z ⊆ x ×ad 2. if (i) x hereditarily paracompact, (ii) indb = 0, (iii) a is not σ-discrete in a ∪ b, then z is zero-dimensional, hereditarily paracompact but not metrizable. proof. since indb = 0 applying theorem 2.13, follow that z is zero-dimensional. moreover a is not a countable union of sets with no accumulation points in b. by proposition 3.5, follow that z is not metrizable. from proposition 2.8 it follows that z is hereditarily paracompact. � corollary 3.18. lnmic is zero-dimensional, hereditarily paracompact but not metrizable. note that corollary 1.20 can be restate as follows: corollary 3.19 (wille [16]). lnmic cannot be retracted onto q × {0}. example 3.20 (michael [12]). choose a bernstein partition [0, 1] = a ∪ b (that is, neither a nor b contains an uncountable compact subset). call this michael-type line lnmcb, the michael-bernstein line corollary 3.21 (see also tanaka [13]). lnmcb is lindelöf space but not metrizable. proof. let f be a closed subset of x such that f ∩ b = ∅. since f is a compact contained in a, then |f | ≤ ℵ0. lemma 3.11 ends the proof. � corollary 3.22 (dow [5]). under maℵ1 , ln mcb is a lindelöf first countable space which is not metrizable but all of whose subspaces of cardinality ℵ1 are metrizable. the alexandroff duplicate and its subspaces 203 proof. apply proposition 2.4, corollary 3.15 and corollary 3.21. � next we provide a sufficient condition for z ⊆ x ×ad 2 to have a product with the irrationals that fails to be normal. in the following, unless stated otherwise, we assume that the x is tychonoff. proposition 3.23. let a, b and c subsets of a topological space x. if (i) c is separable, (ii) a ⊆ c and b ∩ c = ∅, (iii) a is not union of countably many subsets with no accumulation points in b, then z not metrizable, and the product of the michael-type line z with c is not normal. proof. let k = {((a, 1),a) : a ∈ a} and l = (b×{0})×c be subsets in z×c. first of all we prove that kc is an open set in z ×c. indeed, let b ∈ b, c ∈ c, and ((b, 0),c) ∈ kc. since c ∩ b = ∅, then b 6= c, and so, there exists u and v disjoint open sets in x such that b ∈ u and c ∈ v . hence (û ∩z) × (v ∩c) is an open set in z × c contained in kc. now, let a ∈ a, c ∈ c \ {a}, and ((a, 1),c) ∈ kc. then there exists u ∈ τx such that c ∈ u and a 6∈ u. then {(a, 1)} × (u ∩ c) is an open set in z × c containing ((a, 1),c) and contained in kc. thus k is closed in z × c, k and l are disjoint closed sets in z × c. suppose these sets were separated by disjoint open sets u and v respectively. each element ((a, 1),a) has a neighborhood of the form {(a, 1)} × ua ⊆ u with ua open set in c containing a. let d be the countable dense set in c. take da ∈ ua ∩ d. now, a is not the union of countably many subsets with no accumulation points in b, then we can find a net {aσ : σ ∈ σ} ⊂ a accumulating to b0 ∈ b and d ∈ d so that daσ = d for every σ ∈ σ. then ((aσ, 1),d) accumulates to ((b0, 0),d) with ((aσ, 1),d) ∈ u and ((b0, 0),d) ∈ v . thus we have found ((b0, 0),d) ∈ u ∩ v which is impossible. � corollary 3.24. let a, b and c be subsets of a topological space x. if (i) c is separable, (ii) a ⊆ c and b ∩ c = ∅, (iii) every uncountable subset k of a has an accumulation point in b, (iv) a is uncountable (v) b is lindelöf, then the michael-type line z is lindelöf, not metrizable, and z × c is not normal. proof. it follows directly from theorem 3.9 and proposition 3.23. � corollary 3.25. let a and b be subsets of a topological space x. if x \ b is separable and every uncountable subset k of a has an accumulation point in b, then z × (x \ b) is not normal. moreover, if x is compact, a is uncountable and x \b čech-complete, then the michael-type line z is also lindelöf and not metrizable. 204 a. caserta and s. watson proof. since x\b is a čech-complete in x, it is a gδ in x. thus b is an fσ of a compact space, hence lindelöf. the statement follows from corollary 3.24. � corollary 3.26 (based on michael [11]). let a and b be subsets of [0, 1]. if a is contained in a copy of the irrationals disjoint from b, and a is not the union of countably many subsets with no accumulation points in b, then the product of the michael-type line with the irrational is not normal. corollary 3.27 (michael [12]). lnmic × p is not normal. corollary 3.28 (michael [12]). lnmcb × a is not normal. in order to give another example of a lindelöf space whose product with the irrationals is not normal, we first recall another class of michael-type lines. example 3.29. (michael [12]; burke, davis [2]; van douwen [14]; lawrence [10]) under b = ω1, let a ⊂ ωω be unbounded and well-ordered in type ω1 by < ∗. let b = q. call the michael-type line lnmbd. lemma 3.30. under b = ω1, let c = ωω ⊆ x with x compact and a ⊆ ωω be unbounded and well-ordered in type ω1 by < ∗, b = x \ c. then every uncountable subset of a has accumulation points in x \ c. proof. let f be the standard homeomorphism between p and ωω. the set a is a fσ in a∪(x\c) ⊂ x if and only if a ⊂ ⋃ n∈ω kn where kn’s are closed in x and ( ⋃ n∈ω kn)∩(x \c) = ∅, i.e., kn is a compact subset of p for each n ∈ ω. since every compact subset of ωω is bounded in ωω, then for each kn ⊂ p, there exists fn ∈ ω ω such that f(kn) ≤ ∗ fn. since b = ω1, take g ∈ ω ω such that fn ≤ ∗ g for each n ∈ ω . then g bounds ⋃ n∈ω f(kn) = f( ⋃ n∈ω kn) which contains f(a). then f(a) ∼= a ⊂ ωω is bounded. observe that any uncountable k ⊂ a is unbounded and then suck k cannot be compact, i.e., k cannot be closed in (x\c)∪k. thus any uncountable k ⊂ a has accumulation points in x \ c. � corollary 3.31. (michael [12]; burke, davis [2]; van douwen [14]; lawrence [10]) lnmbd is a lindelöf space whose product with the irrationals is not normal. proof. follows from lemma 3.30, corollary 3.25. � problem 3.32 (implicit in michael [12]). is there a lindelöf space whose product with the irrationals is not normal? references [1] p. s. alexandroff and p. s. urysohn, mémoire sur les espaces topologiques compacts, verh. akad. wetensch. amsterdam, 14 (1929). [2] d. k. burke and s. w. davis, subsets of ωω and generalized metric spaces, pacific j. math. 110(2) (1984), 273–281. [3] h. h. corson and e. michael, metrizability of certain countable unions, illinois j. math. 8 (1964), 351–360. the alexandroff duplicate and its subspaces 205 [4] j. dieudonné, une généralisation des espace compact, j. de math. pures et appl. 23 (1944), 65–76. [5] a. dow, an empty class of nonmetric spaces, proc. amer. math. soc. 104(3) (1988), 999–1001. [6] r. engelking, general topology (heldermann verlag, berlin 1989). [7] v. v. fedorc̆uk, bicompacta with noncoinciding dimensionalities, soviet math. doklady 9(5) (1968), 1148–1150. [8] r. f. gittings, finite-to-one open maps of generalized metric spaces, pacific j. math. 59(1) (1975), 33–41. [9] k. kunen, set theory. an introduction to independence proofs, (north-holland, amsterdam 1980). [10] l. b. lawrence, the influence of a small cardinal on the product of a lindelöf space and the irrationals, proc. amer. math. soc. 110(2) (1990), 535–542. [11] e. michael, paracompactness and the lindelöf property in finite and countable cartesian products, compositio math. 23(2) (1971), 199–214. [12] e. michael, the product of a normal space and a metric space need not be normal, bull. amer. math. soc. 69 (1963), 375–376. [13] y. tanaka, decompositions of spaces determined by compact subsets, proc. amer. math. soc. 97(3) (1986), 549–555. [14] e. k. van douwen, the integers and topology, in k. kunen and j. vaughan, (eds.), handbook of set-theoretic topology 111–169 (north-holland, amsterdam, 1984). [15] s. watson, the construction of topological spaces: planks and resolutions, in m. hus̄ek and j. van mill (eds.), recent progress in general topology, 673–757 ( northholland 1992). [16] r. j. wille, sur les espaces faiblement rétractiles, ned. akad. weten. proc. 57 (1954), 527–532. received november 2005 accepted june 2006 a. caserta (agata.caserta@unina2.it) dipartimento di matematica, seconda università degli studi di napoli, caserta 81100, italia s. watson (watson@hilbert.math.yorku.ca) department of mathematics and statistics, york university, toronto m3j1p3, canada hoskova.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 1, 2005 pp. 57-65 abelization of join spaces of affine transformations of ordered field with proximity šárka hošková abstract. using groups of affine transformations of linearly ordered fields a certain construction of non-commutative join hypergroups is presented based on the criterion of reproducibility of semi-hypergroups which are determined by ordered semigroups. the aim of this paper is to construct the abelization of the non-commutative join space of affine transformations of ordered fields. a construction of commutative weakly associative hypergroup (hν-group) is made and a proximity is defined on this structure. 2000 ams classification: 20f60, 20n20 keywords: transformation group, join space, abelization, hyperoperation, hyperstructures, weak associativity transformation groups which represent the classical and developing discipline are situated in the intersection of several parts of mathematical structures. transformation groups in the discrete approach create in a very natural way commutative hyperstuctures, which are in addition join spaces. on the other hand, groups of affine transformations of the field create naturally a transformation group on the supporting set of this field. more precisely: let (f, +, ·) be a field, a(f) be a group of affine transformations of f of the form ϕa,b(x) = ax + b; x ∈ f , where the coefficient a ∈ f is different from zero. then we construct a discrete transformation group (x, t, π) by putting x = f ; t = a(f) and the action π we define by π(x; ϕa,b) = ϕa,b(x). clearly the identity axiom and homomorphism axiom are satisfied. in this paper hyperstructures on groups of affine transformations of ordered fields are constructed. these affine transformations are represented by ordered pairs of elements of a given field. during the construction of final hyperstructures upper ends of the products of pairs of elements of a given field are used. this idea is adopted from the functorial assignment of a commutative hypergroup to an arbitrary transformation (discrete) group. we will describe this 58 š. hošková construction in more details. let g = (x, t, π) be a transformation group, (i.e. x-phase set, t -phase group, πaction (projection): x × t → x). for any pair x, y ∈ x we define x ∗g y = π(x, t ) ∪ π(y, t ) = {π(x, t); t ∈ t } ∪ {π(y, t); t ∈ t }. it is easy to show that (x, ∗g) is an extensive commutative hypergroup— moreover a join space (see the definition below). in the paper [1] the authors studied the non-commutative join hypergroups of affine transformations of ordered fields. the aim of this paper is to abelize this structure using the construction described in papers [7, 14]. recall that an ordering on a field (f, +, . ) (with the zero element 0 f and the unit 1 f ) is given by the choice of a set p ⊆ f (called the positive cone of the ordering) which satisfies the following axioms: (1) p + p ⊆ p , (2) p . p ⊆ p , (3) p ∪ (−p) = f , (4) p ∩ (−p) = {0 f } . as usual, we define a binary relation ≦p on the set f by x ≦p y (resp. x < p y) if y − x ∈ p(resp. y − x ∈ ṗ , where ṗ = p r {0 f }). the relation ≦p results in a total ordering of the elements of f . in detail, the relation ≦p on f is reflexive, anti symmetrical and transitive, compatible in the usual sense with commutative binary operations addition “+” and multiplication “ . ”, and satisfies the law of trichotomy (for any pair x, y ∈ f exactly one of the following possibilities occurs: x = y or x < p y or y < p x). note that an ordered field (f, +, . , ≦p ) has necessarily the characteristic equal to zero. as usual, p ∗(s) denotes the system of all nonempty subsets of s. let (f, +, . ) be a field of the characteristic zero. an affine transformation f : f → f of the form f(x) = a . x + b, a, b ∈ f is uniquely represented by an ordered pair of its coefficients [a, b] ∈ f × f . we shall consider non-constant transformations only, i.e., transformations f(x) = a.x + b satisfying the condition a 6= 0 f . so, let us denote a(f) = (f r {0 f }) × f and define a binary operation “·” on the set a(f) by the rule [a, b] · [c, d] = [a.c, a.d + b] , which corresponds to the usual composition of affine transformations of f . it is easy to see that ( a(f), · ) is a non-commutative group with the identity [1 f , 0 f ] and inverse elements of the form [a, b]−1 = [a−1, −a−1.b]. denote by k the subset {[a, b]; a, b ∈ f, a > p 0 f } of a(f) = (f r{0 f })×f . it is easy to see that (k, ·) is a subgroup of the group ( a(f), · ) . define a binary relation “≦” on the set k by the rule: [a, b] ≦ [c, d] for [a, b], [c, d] ∈ k whenever a = c and b ≦p d. then evidently “≦” is an ordering on k and we have proposition 1 ([1]). if k = {[a, b]; a, b ∈ f, a >p 0f }, then (k, ·) is an ordered group such that it is a normal subgroup of the group ( a(f), · ) . abelization of join spaces 59 recall that a hypergroupoid is a pair (h, ·) where h is a (nonempty) set and · : h × h → p∗(h) is a binary hyperoperation on the set h. if a · (b · c) = (a · b) · c for all a, b, c ∈ h , (associativity) , then (h, ·) is called a semihypergroup. a semihypergroup (h, ·) is said to be a hypergroup if the following axiom a · h = h = h · a for all a ∈ h , (reproduction axiom) , is satisfied—see e.g. [3],[4]. here, for a, b ⊆ h, a 6= ∅ 6= b we define a · b = ⋃ {a.b; a ∈ a, b ∈ b}. moreover, for subset a and b of h, it becomes convenient to use the relational notation a ≈ b (read a meets b) to assert that a and b have an element in common, that is, that a ∩ b 6= ∅ ([16], p. 79). a hypergroup (h, ·) is called a transposition hypergroup if it satisfies the transposition axiom: for all a, b, c, d ∈ h the relation b\a ≈ c/d implies a · d ≈ b · c , where b\a = {x ∈ h; a ∈ b · x}, c/d = {x ∈ h; c ∈ x · d}. a commutative transposition hypergroup (h, ∗) is called a join space ([15]). the hyperoperation ⋆: h × h → p∗(h) is called weakly associative hyperoperation if ( a ⋆ (b ⋆ c) ) ∩ ( (a ⋆ b) ⋆ c ) 6= ∅ for any triad a, b, c ∈ h. a weak semihypergroup (hν-semigroup) is a set h (h 6= ∅) equipped with a weakly associative hyperoperation. a hν-semigroup is called a weak hypergroup (hν-group) if moreover the reproduction axiom is satisfied for any a ∈ h ([18]). proposition 2 ([5] theorem 1). let (s, ·, ≦) be an ordered semigroup. a binary operation ∗ : s × s → p∗(s) defined by x ∗ y = [x . y)≦ (= {t ∈ s; x . y ≦ t}) for any pair x, y ∈ s is associative. then we have 1◦ the semi-hypergroup (s, ∗) is commutative if and only if the semigroup (s, ·) is commutative. 2◦ for the ordered semigroup (s, ·, ≦) the following conditions are equivalent: (i) for any pair of elements x, y ∈ s there exists a pair z, z′ ∈ s such that y · z ≦ x, z′ · y ≦ x, (ii) the semihypergroup (s, ∗) satisfies the reproduction condition (i.e. t∗s = s = s ∗ t for any t ∈ s), hence it is a hypergroup. lemma 1 ([7, 14]). let (h, ·) be a hypergroupoid. define a hyperoperation “⋆” on the diagonal ∆ h as follows: [x, x] ⋆ [y, y] = d(x · y ∪ y · x) = { [u, u]; u ∈ x · y ∪ y · x } for any pair [x, x], [y, y] ∈ ∆ h . then the following assertions hold: 1o for any hypergroupoid (h, ·) we have that (∆ h , ⋆) is a commutative hypergroupoid. 60 š. hošková 2o if (h, ·) is a weakly associative hypergroupoid, then the hypergroupoid (∆ h , ⋆) is weakly associative, as well. 3o if (h, ·) is a quasi-hypergroup, the hypergroupoid (∆ h , ⋆) also satisfies the reproduction law, i.e., it is a quasi-hypergroup. define a hyperoperation ∗: k × k → p∗(k) by [a, b] ∗ [c, d] = { [x, y]; [a, b] · [c, d] ≦ [x, y] } = { [a.c, y]; a.d + b ≦p y } . from proposition 2 it is evident, that the hypergroupoid (k, ∗) is non-commutative hypergroup. the hypergroup (k, ∗) will be called to be determined by the ordered group (k, ·, ≦). the hypergroup (k, ∗) is non-commutative, therefore we will abelize it. let us define the set ∆ k = { [ [a, b], [a, b] ] ; [a, b] ∈ k } and a hyperoperation ⋆ : ∆ k × ∆ k → p∗(∆ k ) by [ [a, b], [a, b] ] ⋆ [ [c, d], [c, d] ] = (0.1) = { [ [x, y], [x, y] ] ; [x, y] ∈ ([a, b] ∗ [c, d]) ∪ ([c, d] ∗ [a, b]) } = { [ [x, y], [x, y] ] ; [x, y] ∈ { [a.c, u]; a.d + b ≦p u } ∪ { [c.a, v]; c.b + d ≦p v } } = { [ [x, y], [x, y] ] ; x = a.c ∧ z ≦ y) } , where z = min{a.d + b; c.b + d}. theorem 1. the hyperstructure (∆ k , ⋆) is commutative weakly associative hypergroup, simply hν-group. proof. recall that the field f is commutative, so the multiplication is commutative, thus a.c = c.a. it is evident, that (∆ k , ⋆) ∼= (m, ⋄), where m = { [a, b, a, b]; [a, b] ∈ k } and [a1, b1, a1, b1] ⋄ [a2, b2, a2, b2] = { [a, b, a, b]; a = a1 · a2, z ≦p b } , z = min{a1 · b2, +b1, a2, ·b1 + b2, }. due to lemma 1 it is evident that (∆k, ⋆) is the commutative weakly associative hypergroup—simply abelian hν-group. to investigate the associativity law in more detail let us prove the weak associativity law using the concrete form of the structure (k, ∗). we want to verify that ( [ [a, b], [a, b] ] ⋆ [ [c, d], [c, d] ] ) ⋆ [ [p, q], [p, q] ] ∩ [ [a, b], [a, b] ] ⋆ ( [ [c, d], [c, d] ] ⋆ [ [p, q], [p, q] ] ) 6= ∅. abelization of join spaces 61 for any [a, b], [c, d], [p, q] ∈ k. for the next computation let us denote a = ( [ [a, b], [a, b] ] ⋆ [ [c, d], [c, d] ] ) ⋆ [ [p, q], [p, q] ] , b = [ [a, b], [a, b] ] ⋆ ( [ [c, d], [c, d] ] ⋆ [ [p, q], [p, q] ] ) , z1 = min{a.d + b; c.b + d}, z2 = min{c.q + d; p.d + q}. we get a = { [ [x, y], [x, y] ] ; x = a.c ∧ z1 ≦p y } ⋆ [ [p, q], [p, q] ] = = ⋃ z 1 ≦ p y [ [a.c, y], [a.c, y] ] ⋆ [ [p, q], [p, q] ] = { [ [u, v], [u, v] ] ; u = a.c.p ∧ w1 ≦p v } , where w1 = min{a.c.q + y, p.y + q}. as p ∈ f, p > p 0 f , we have w1 = min { a.c.q + y, p. min{a.d + b, c.b + d} + q } = = min { a.c.q + min{a.d + b, c.b + d}, min{p.a.d + p.b, p.c.b + p.d} + q } = = min { min{a.c.q + a.d + b, a.c.q + c.b + d}, min{p.a.d + p.b + q, p.c.b + p.d + q} } = = min{a.c.q + a.d + b, a.c.q + c.b + d, p.a.d + p.b + q, p.c.b + p.d + q}. so, the set a= { [ [u, v], [u, v] ] ; u = a.c.p ∧ w1 ≦p v } . on the other hand b = [ [a, b], [a, b] ] ⋆ { [ [x, y], [x, y] ] ; x = c.p ∧ z2 ≦p y } = ⋃ z 2 ≦ p y [ [a, b], [a, b] ] ⋆ [ [x, y], [x, y] ] = { [ [u, v], [u, v] ] ; u = a.c.p ∧ w2 ≦p v } , where w2 = min{a.y + b, x.b + y}. similarly, because a ∈ f, a > p 0f w2 = min { a. min{c.q + d, p.d + q} + b, x.b + min{c.q + d; p.d + q} } = = min { min{a.c.q + a.d, a.p.d + a.q} + b, min{x.b + c.q + d, x.b + p.d + q} } = = min { min{a.c.q + a.d + b, a.p.d + a.q + b}, min{c.p.b + c.q + d, c.p.b + p.d + q} } = = min{a.c.q + a.d + b, a.p.d + a.q + b, c.p.b + c.q + d, c.p.b + p.d + q}. so, the set b= { [ [u, v], [u, v] ] ; u = a.c.p ∧ w2 ≦p v } . choose [ [u0, v0], [u0, v0] ] ∈ ∆ k , such that u0 = a.c.p and v0 ≧p max{w1, w2}. evidently, this pair of pairs belongs to a ∩ b. � 62 š. hošková remark 1. using the previous calculations it is easy to check that the structure (∆ k , ⋆) is never associative. to see it let us first consider f = q. necessarily, q+ ⊂ p. choose, for example a = 1, b = 2, c = 3, d = −4, p = 5, q = −7. from the above notations we have w1 = min{−23, −19, −17, 3} = −23, w2 = min{−23, −25, 5, 3} = −25. thus the triple [a, b], [c, d], [p, q] fulfils only the weak associativity law. as it was mentioned earlier each ordered field f contains in itself a copy of the set q. thus each time it is possible to find a triple [a, b], [c, d], [p, q] ∈ q+ × q such that w1 6= w2 and therefore the structure (∆k, ⋆) is only weakly associative. remark 2. in the sense of the paper [14] it is possible to define a proximity on this structure, for example, in this way. let a, b ⊂ k, then a p b if and only if [a)≦ ∩ [b)≦ 6= ∅. where for m ⊂ k we define [m)≦ = ⋃ m∈m [m)≦. we mean the proximity in the sense of the čech monograph [2]: a relation p on the family of all subsets of the set h is called a proximity on the set h if p satisfies the following conditions: p1. ∅ non p h p2. the relation p is symmetric, i.e., a, b ⊂ h, a p b implies b p a . p3. for any pair of subset a, b ⊂ h, a ∩ b 6= ∅ implies a p b. p4. if a, b, c are subsets of h then (a∪b) p c if and only if either a p c or b p c. recall that a triad (h, ·, p h ) such that (h, ·) is a hypergroupoid and (h, p h ) is a proximity space will be called a hypergroupoid with a proximity. if for any triad of elements x, y, z ∈ h ( x · (y · z) ) p h ( (x · y) · z ) is valid, then the hyperoperation “·” is called proximally weakly associative–see e.g. [9],[14]. in fact, axioms p 1, p 2 and p 3 from the definition of proximity are obvious. it remains to show that the axiom p 4 is satisfied too. first let us prove the equality [a∪b)≦ = [a)≦ ∪[b)≦, which will be helpful in the next calculations. 1. x ∈ l ⇒ ∃ u ∈ a ∪ b : u ≦ x ⇒ ∃ u ∈ a: u ≦ x or ∃ u ∈ b : u ≦ x ⇒ x ∈ [a)≦ or x ∈ [b)≦ 2. x ∈ p ⇒ x ∈ [a)≦ or x ∈ [b)≦ ⇒ x ∈ [a ∪ b)≦ (by l we mean the left hand side of the equation and by p the right hand side.) now we can prove the axiom p 4: if a, b, c are subsets of k, then (a∪b) p c if and only if either a p c or b p c. abelization of join spaces 63 “⇒” (a ∪ b) p c ⇒ [a ∪ b)≦ ∩ [c)≦ 6= ∅ ⇒ ( [a)≦ ∪ [b)≦ ) ∩ [c)≦ 6= ∅ ⇒ a p c or b p c “⇐” a p c or b p c ⇒ [a)≦ ∩[c)≦ 6= ∅ or [b)≦ ∩[c)≦ 6= ∅. since [a)≦ ⊆ [a∪b)≦ and [b)≦ ⊆ [a ∪ b)≦, we have [a ∪ b)≦ ∩ [c)≦ 6= ∅, i.e., (a ∩ b) p c. thus we obtain theorem 2. the hypergroup (k, ∗, p) is a hypergroup with proximity. theorem 3. the hypergroupoid (∆ k , ⋆) is a weakly associative and commutative transposition hypergroup, i.e., a weakly associative join space. proof. first we will show that the reproduction axiom is fulfilled. due to theorem 2 the structure (k, ∗) is the hypergroup, thus [a, b]∗k = k = k∗[a, b] for any [a, b] ∈ k. if we define [a, b] ⋆© [c, d] = { [x, y]; [x, y] ∈ ([a, b] ∗ [c, d]) ∪ ([c, d] ∗ [a, b]) } = { [x, y]; [x, y] ∈ {[a.c, u]; a.d + b ≦p u} ∪ {[c.a, v]; c.b + d ≦p v} } = { [x, y]; x = a.c ∧ z ≦ y) } , where z = min{a.d+b; c.b+d}, evidently [a, b] ⋆©[c, d] ⊃ [a, b]∗[c, d]. therefore k ⊃ [a, b] ⋆© k ⊃ [a, b] ∗ k = k, which implies that k ⋆© [a, b] = k and similarly [a, b] ⋆© k = k. from this we obtain that reproduction axiom holds in (∆ k , ⋆). second we will verify the transposition axiom. with respect to the definition of join space and (0.1) we get a = [ [b1, b2], [b1, b2] ]∖[ [a1, a2], [a1, a2] ] = {[ [x, y], [x, y] ] ; [a1, a2] ∈ [b1, b2] ⋆© [x, y] } = {[ [x, y], [x, y] ] ; a1 = b1, .x ∧ a2 ≧p min{b1.y + b2b2.x + y} } = = {[ [x, y], [x, y] ] ; x = a1.b −1 1 ∧ y ≦p max{(a2 − b2).b −1 1 , a2 − b2.a1.b −1 1 } } , b = [ [c1, c2], [c1, c2] ]/[ [d1, d2], [d1, d2] ] = {[ [x, y], [x, y] ] ; [c1, c2] ∈ [x, y] ⋆© [d1, d2] } = {[ [x, y], [x, y] ] ; c1 = d1.x ∧ c2 ≧p min{d1.y + d2, d2.x + y} } = = {[ [x, y], [x, y] ] ; x = c1.d −1 1 ∧ y ≦p max{(c2 − d2).d −1 1 , c2 − d2.c1.d −1 1 } } , c = [ [a1, a2], [a1, a2] ] ⋆ [ [d1, d2], [d1, d2] ] = = {[ [u, v], [u, v] ] ; u = a1.d1 ∧ v ≧p min{a1.d2 + a2, a2.d1 + d2} } , d = [ [b1, b2], [b1, b2] ] ⋆ [ [c1, c2], [c1, c2] ] = = {[ [u, v], [u, v] ] ; u = b1.c1 ∧ v ≧p min{b1.c2 + b2, b2.c1 + c2} } . 64 š. hošková we have [ [x, y], [x, y] ] ∈ a ∩ b, i.e., a ≈ b, if and only if x = a1.b −1 1 = c1.d −1 1 and y ≦p min { max{(a2 − b2).b −1 1 , a2 − b2.a1.b −1 1 }, max{(c2 − d2).d −1 1 , c2 − d2.c1.d −1 1 } } . if a ≈ b, then necessarily a1.d1 = b1.c1. let us denote u0 = a1.d1. for any v0 such that v0 ≧p max { min{a1.d2 + a2, a2.d1 + d2}, min{b1.c2 + b2, b2.c1 + c2} } we obtain [ [u0, v0], [u0, v0] ] ∈ c ∩ d which proves the transposition axiom. � remark 3. it is easy to verify that under the assumption of the previous theorem even the following equivalence holds: b\a ≈ c/d if and only if a . d ≈ b . c . in fact, one implication follows from theorem 2. to obtain the converse one (using the notation from the proof of the mentioned theorem) suppose c ≈ d. thus a1.d1 = b1.c1. let us denote x0 = a1.b −1 1 = c1.d −1 1 . then for an arbitrary y0 such that y0 ≦p min { max{(a2 − b2).b −1 1 , a2 − b2.a1.b −1 1 }, max{(c2 − d2).d −1 1 , c2 − d2.c1.d −1 1 } } . we have [ [x0, y0], [x0, y0] ] ∈ a ∩ b, i.e., a ≈ b. references [1] j. beránek, j. chvalina, noncommutative join hypergroups of affine transformations of ordered fields, dept.math.report series 10, univ.of south bohemia, (2002), 15–22. [2] e. čech, topological spaces, revised by z. froĺık and m. katětov, (academia, praha 1966). [3] p. corsini, prolegomena of hypergroup theory, (aviani editore tricesimo 1993). [4] p. corsini, v. leoreanu, applications of hypergroup theory, ( kluwer academic publishers, dordrecht, hardbound, 2003). [5] j. chvalina, functional graphs, quasi-ordered sets and commutative hypergroups, mu brno (1995), (in czech). [6] j. chvalina, l. chvalinová, transposition hypergroups formed by transformation operators on rings of differentiable functions, ital. j. pure and appl. math., 13 pp. in print. [7] j. chvalina, š. hošková, abelization of weakly associative hyperstructures based on their direct squares, acta mathematica et informatica universitatis ostraviensis, volume 11/2003, no. 1, 11–25. [8] j. chvalina, š. hošková, join space of first-order linear partial differential operators with compatible proximity induced by a congruence on their group, proc. of mathematical and computer modelling in science and engineering, prague (2003), 166-170. [9] j. chvalina, š. hošková, modelling of join spaces with proximities by first-order linear partial differential operators, submitted to applications of mathematics, 13p. abelization of join spaces 65 [10] w. h. gottschalk, g. a. hedlund, topological dynamics, ams colloq. publ. vol.xxxvi, providence, rhode island, (1974) usa. [11] š. hošková, examples of abelization of hypergroups based on their direct products, sborńık va, part b, 2, 7–19, (2002). [12] š. hošková, abelization of differential rings, proc. of the 1st international mathematical workshop fast vut brno, 2p., (2002). [13] š. hošková, abelization of a certain representation of non-commutative join space, proc. of international conference aplimat 2003, bratislava, slovakia, (2003), 365–368. [14] š. hošková, j. chvalina, abelization of proximal hν-rings using graphs of good homomorphisms and diagonals of direct squares of hyperstructures, proceedings of 8th internat. congress on aha, samothraki, greece (2002), 147–159. [15] j. jantosciak, transposition in hypergroups, algebraic hyperstructures and appl. proc. sixth internat. congress prague 1996, democritus univ. of thrace press, alexandroupolis (1997), 77–84. [16] j. jantosciak, transposition hypergroups: noncommutative join spaces, j. algebra 187 (1997), 97–119. [17] v. leoreanu, on the hart of join spaces and of regular hypergroups, riv. mat. pura appl. no. 17 (1995), 133–142. [18] t. vougiouklis, hyperstructures and their representations, hadronic press monographs in mathematics, (palm harbor florida 1994). received september 2003 accepted january 2005 šárka hošková (sarka.hoskova@seznam.cz) department of mathematic, university of defence brno, kounicova 65, 612 00 brno, czech republic. beerseguraagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 131-157 well-posedness, bornologies, and the structure of metric spaces gerald beer and manuel segura ∗ abstract. given a continuous nonnegative functional λ that makes sense defined on an arbitrary metric space 〈x, d〉, one may consider those spaces in which each sequence 〈xn〉 for which limn→∞λ(xn) = 0 clusters. the compact metric spaces, the complete metric spaces, the cofinally complete metric spaces, and the uc-spaces all arise in this way. starting with a general continuous nonnegative functional λ defined on 〈x, d〉, we study the bornology bλ of all subsets a of x on which limn→∞λ(an) = 0 ⇒ 〈an〉 clusters, treating the possibility x ∈ bλ as a special case. we characterize those bornologies that can be expressed as bλ for some λ, as well as those that can be so induced by a uniformly continuous λ. 2000 ams classification: primary 54c50, 49k40; secondary 46a17, 54b20, 54e35, 54a10 keywords: well-posed problem, bornology, uc-space, cofinally complete space, strong uniform continuity, bornological convergence, shielded from closed sets 1. introduction in a first course in analysis, one is introduced to two important classes of metric spaces as those in which certain sequences have cluster points: a metric space 〈x, d〉 is called compact if each sequence 〈xn〉 in x has a cluster point, whereas 〈x, d〉 is called complete if each cauchy sequence in x has a cluster point. a cauchy sequence is one of course for which there exists for each ε > 0 a residual set of indices whose terms are pairwise ε-close. if we replace residual by cofinal in the definition, we get a so-called cofinally cauchy sequence and the metric spaces x in which each cofinally cauchy sequence has a cluster point are called cofinally complete [10, 13, 18, 24, 36]. these are a well-studied class of spaces lying between the compact spaces and the complete ones. notably, ∗this research was supported by the following grant: nih marc u*star gm08228. 132 g. beer and m. segura these are the metric spaces that are uniformly paracompact [13, 23, 24, 35] and also those on which each continuous function with values in a metric space is uniformly locally bounded [10]. lying between the compact spaces and the cofinally complete spaces is the class of uc-spaces, also known as atsuji spaces, which are those metric spaces on which each continuous function with values in a metric space is uniformly continuous [1, 5, 6, 7, 8, 27, 34, 38]. these are also called the lebesgue spaces [32], as they are those metric spaces 〈x, d〉 for which each open cover has a lebesgue number [1, 8]. these uc-spaces, too, can be characterized sequentially, as observed by toader [37]: 〈x, d〉 is a uc space if and only if each pseudo-cauchy sequence in x with distinct terms clusters, where 〈xn〉 is called pseudo-cauchy [8, p. 59] if for each ε > 0 and n ∈ n, there exists k > j > n with d(xj , xk) < ε. it seems worthwhile to study in some organized way classes of metric spaces on which prescribed sequences have cluster points. one program could be to look at other modifications of the definition of cauchy sequence, but this approach is limited in scope and is not our purpose here. instead, given some continuous nonnegative extended real-valued functional λ that makes sense defined on an arbitrary metric space, we look at the class of ”λ-spaces”, i.e., the class of metric spaces 〈x, d〉 such that each sequence 〈xn〉 in x with limn→∞λ(xn) = 0 has a cluster point. in terms of the language of optimization theory, a space is in this class if either inf{λ(x) : x ∈ x} > 0 or the functional λ is tychonoff well-posed in the generalized sense [20, 31]. all of the classes mentioned in the first paragraph fall within this framework. for the compact spaces, the zero functional does the job. for the uc-spaces, the measure of isolation functional i(x) = d(x, x\{x}) is characteristic [1, 8, 27]. for the cofinally complete spaces, it is the measure of local compactness functional [10, 13] defined by ν(x) = { sup{α > 0 : cl(sα(x)) is compact} if x is a point of local compactness 0 otherwise . for the complete metric spaces, and paralleling the cofinally complete spaces as we will see in section 5 infra, it is it is the measure of local completeness functional defined by β(x) = { sup{α > 0 : cl(sα(x)) is complete} if x has a complete neighborhood 0 otherwise . in each case discussed above, unless identically equal to ∞, the functional is uniformly continuous; but we do not restrict ourselves in this way, nor do we insist that our metric spaces be complete. we find it advantageous to first study more primitively the ”λ-subsets” of an arbitrary metric space 〈x, d〉: those nonempty subsets a such that each sequence 〈an〉 within satisfying λ(an) → 0 clusters. in general these form a bornology with closed base. as a major result, we characterize those bornologies that arise in this way. well-posedness, bornologies, and the structure of metric spaces 133 2. preliminaries all metric spaces are assumed to contain at least two points. we denote the closure, set of limit points and interior of a subset a of a metric space 〈x, d〉 by cl(a), a′ and int(a), respectively. we denote the power set of a by p(a) and the nonempty subsets of a by p0(a). we denote the set of all closed and nonempty subsets of x by c0(x), and the set of all closed subsets by c(x). we call a ∈ p0(x) uniformly discrete if ∃ε > 0 such that whenever a1, a2 are in a and a1 6= a2, then d(a1, a2) ≥ ε. if 〈y, ρ〉 is a second metric space, we denote the continuous functions from x to y by c(x, y ). if x0 ∈ x and ε > 0, we write sε(x0) for the open ε-ball with center x0. if a is a nonempty subset of x, we write d(x0, a) for the distance from x0 to a, and if a = ∅ we agree that d(x0, a) = ∞. with d(x, a) now defined, we denote for ε > 0 the ε-enlargement of a ∈ p(x) by sε(a), i.e., sε(a) := {x ∈ x : d(x, a) < ε} = ⋃ x∈a sε(x). if a ∈ p0(x) and b ∈ p(x), we define the gap between them by dd(a, b) := inf {d(a, b) : a ∈ a}. we can define the hausdorff distance [8, 28] between two nonempty subsets a and b in terms of enlargements: hd(a, b) := inf {ε > 0 : a ⊆ sε(b) and b ⊆ sε(a)}. hausdorff distance so defined is an extended real-valued pseudometric on p0(x) which when restricted to the nonempty bounded sets is finite valued, and which when restricted to c0(x) is an extended real-valued metric. hausdorff distance restricted to c0(x) preserves the following properties of the underlying space (see, e.g., [8, thm 3.2.4]). proposition 2.1. let 〈x, d〉 be a metric space. the following are true: (1) 〈c0(x), hd〉 is complete if and only if 〈x, d〉 is complete; (2) 〈c0(x), hd〉 is totally bounded if and only if 〈x, d〉 is totally bounded; (3) 〈c0(x), hd〉 is compact if and only if 〈x, d〉 is compact. a weaker form of convergence for sequences of closed sets than convergence with respect to hausdorff distance is kuratowski convergence [8, 29, 28]. given a sequence 〈an〉 in c0(x), we define li an := {x ∈ x : ∀ε > 0, sε(x) ∩ an 6= ∅ residually} and ls an := {x ∈ x : ∀ε > 0, sε(x) ∩ an 6= ∅ cofinally}. we say 〈an〉 is kuratowski convergent to a and write k-lim an = a if a = li an = ls an. the following facts are well-known. 134 g. beer and m. segura proposition 2.2. let 〈an〉 be a sequence in c0(x). then the following are true: (1) li an and ls an are both closed (but perhaps empty); (2) li an ⊆ ls an; (3) if an = {an}, then li an = {lim an} if lim an exists and li an = ∅ if not; (4) if an = {an}, then ls an = {x : x is a cluster point of 〈an〉} = ⋂∞ n∈n cl({ak : k ≥ n}); (5) lim hd(an, a) = 0 ⇒ k-lim an = a; (6) if 〈an〉 is decreasing, then k-lim an = ⋂∞ n=1 an. we can also define the hausdorff measure of noncompactness [4] of a nonempty subset a in terms of enlargements: α(a) = inf {ε > 0 : a ⊆ sε(f ), where f is a nonempty finite subset of x}. clearly, α(a) = ∞ if and only if a is unbounded. the functional α behaves as follows: (1) if a ⊆ b, then α(a) ≤ α(b); (2) α(cl(a)) = α(a); (3) α(a) = 0 if and only if a is totally bounded; (4) α(a ∪ b) = max {α(a), α(b)}; (5) if lim hd(an, a) = 0 then lim α(an) = α(a). a famous theorem concerning the hausdorff measure of noncompactness is kuratowski’s theorem [4, 29], proved in a novel way below; but first, we state and prove a useful lemma: lemma 2.3. let 〈x, d〉 be a metric space. suppose 〈an〉 is a decreasing sequence in c0(x) which is not hd-cauchy. then ∃ n1 < n2 < n3 < · · · and xnk ∈ ank such that {xnk : k ∈ n} is uniformly discrete. proof. let 〈an〉 be a decreasing sequence in c0(x) that is not hd-cauchy. then ∃ε > 0 such that ∀n0 ∈ n, ∃m > n > n0 such that hd(am, an) > ε. choose m1, m2 with m2 > m1 > 1 such that hd(am1 , am2 ) > ε. then let i1 > m2, and choose m3, m4 with m4 > m3 > i1 such that hd(am3 , am4 ) > ε. then let i2 > m4, and choose m5, m6 with m6 > m5 > i2 such that hd(am5 , am6 ) > ε. continuing, we have m1 < m2 < m3 < · · · such that hd(am2j−1 , am2j ) > ε where j > 1. now for i > 1, pick xni ∈ am2i−1 with d(xni , am2i ) > ε. then for i < k, d(xni , xnk ) > d(xni , am2k−1 ) > d(xni , am2k−2 ) > d(xni , am2i ) > ε. we have shown 〈xni 〉 has distinct terms and is a uniformly discrete sequence. � well-posedness, bornologies, and the structure of metric spaces 135 here is our novel proof (in one direction) of kuratowski’s theorem based on completeness of 〈c0(x), hd〉. theorem 2.4 (kuratowski’s theorem on completeness). let 〈x, d〉 be a metric space. then 〈x, d〉 is complete if and only if whenever 〈an〉 is a decreasing sequence in c0(x) with lim α(an) = 0, then a := ⋂ n∈n an 6= ∅. proof. suppose 〈x, d〉 is complete and 〈an〉 is a decreasing sequence in c0(x) with lim α(an) = 0. suppose 〈an〉 is not hd-cauchy. then ∃n1 < n2 < n3 < ... and xnk ∈ ank such that 〈xnk 〉 is a uniformly discrete sequence with distinct terms. then ∃ε > 0 such that d(xni , xnj ) > ε where i 6= j. hence ∀n ∈ n, α(an) ≥ ε 2 ⇒ lim α(an) ≥ ε 2 , which is a contradiction. thus 〈an〉 must be hd-cauchy. since 〈x, d〉 is complete, by proposition 2.1(1) 〈c0(x), hd〉 is complete, so ∃b ∈ c0(x) with b = hd − lim〈an〉. since lim hd(an, b) = 0, by proposition 2.2(5) b = k − liman, . since 〈an〉 is decreasing in c0(x), by proposition 2.2(6) k-lim an = ⋂∞ n=1 an. hence b = a ⇒ a 6= ∅. conversely, suppose whenever 〈an〉 is a decreasing sequence in c0(x) with lim α(an) = 0, then a 6= ∅. then if 〈xn〉 is a cauchy sequence, we have ⋂∞ n∈n cl{xk : k ≥ n} nonempty, so by proposition 2.2(4), 〈xk〉 has a cluster point. hence 〈x, d〉 is complete. � let f : x → [0, ∞]. then by saying f is lower semi-continuous at a point x0 ∈ x, we mean whenever α < f (x0), α ∈ r, then ∃δ > 0 such that ∀x ∈ sδ(x0), f (x) > α. by saying f is upper semi-continuous at a point x0 ∈ x, we mean whenever α > f (x0), α ∈ r, then ∃δ > 0 such that ∀x ∈ sδ(x0), f (x) < α. note that f (x0) = ∞ ⇒ f is upper semi-continuous at x0, and f (x0) = 0 ⇒ f is lower semi-continuous at x0. if f is both upper and lower semi-continuous at a point x0 ∈ x, then we say f is continuous at x0. 3. notes on nonnegative continuous functionals we first discuss a framework in which many important nonnegative continuous functionals arise. let p be an hereditary property of open subsets of 〈x, d〉: if v, w are open sets where w ⊆ v then p (v ) ⇒ p (w ). define λp : x → [0, ∞] by λp (x) = { sup{α > 0 : p (sα(x))} if ∃α > 0 where p (sα(x)); 0 otherwise. example 3.1. consider p (v ) := v contains at most one point. then the resulting λp is the measure of isolation functional λp (x) = i(x) := d(x, x \ {x}). example 3.2. next consider p (v ) := v ∩ e = ∅ where e ⊆ x. then the induced λp gives the distance from a variable point of x to the set e. example 3.3. now consider the case when p (v ) := cl(v ) is compact. then the resulting λp is the measure of local compactness functional ν giving the supremum of the radii of the closed balls with center x that are compact. 136 g. beer and m. segura example 3.4. a final example of an hereditary property p is p (v ) := v is countable. of particular importance is the kernel of the metric space 〈x, d〉 with respect to a continuous function λ : x → [0, ∞], which we define as ker(λ) := {x ∈ x : λ(x) = 0}. if we consider the resulting λp from example 3.1, then ker(λp )= x ′, the set of limit points of x. for λp from example 3.2, we get ker(λp )= cl(e). for λp from example 3.3, ker(λp ) equals the points of non-local compactness of x. finally, if we consider the corresponding λp for example 3.4, then ker(λp ) equals the set of condensation points of x. proposition 3.5. let p be an hereditary property of open sets in 〈x, d〉. if λp (x0) = ∞ for some x0 ∈ x, then λp (x) = ∞ for all x ∈ x. otherwise if λp is finite valued, then λp is 1-lipschitz. proof. suppose λp (x0) = ∞ for some x0 ∈ x. let x ∈ x where x0 6= x, and let α > 0 be arbitrary. since sup{µ > 0 : p (sµ(x0))} = ∞, ∃α0 > 0 such that p (sα0 (x0)) and sα(x) ⊆ sα0 (x0), so that p (sα(x)). this shows that λp (x) = ∞ for all x ∈ x. otherwise, λp is finite valued. if λp fails to be 1-lipschitz, there exist x, w ∈ x with λp (x) > λp (w) + d(x, w). take an α > 0 where λp (x) > α > λp (w) + d(x, w), so that p (sα(x)). then sα−d(x,w)(w) ⊆ sα(x), so p (sα−d(x,w)(w)). however, α − d(x, w) > λp (w), which is a contradiction. hence, λp is 1-lipschitz. � we next introduce the induced set functional λ : p0(x) → [0, ∞] that we will use to characterize λ-spaces in section 5: λ(a) := sup{λ(a) : a ∈ a}, where λ : x → [0, ∞] is a continuous functional. the following proposition lists obvious properties of the set functional λ. proposition 3.6. let 〈x, d〉 be a metric space and let λ : p0(x) → [0, ∞] be as defined above. then the following are true for nonempty subsets a, b: (1) λ(a ∪ b) = max{λ(a), λ(b)}; (2) λ(cl(a)) = λ(a); (3) λ(a) = 0 if and only if a ⊆ ker(λ). it is now useful to introduce a strengthening of uniform continuity of a function restricted to a subset of x as considered in [14, 15]. definition 3.7. let 〈x, d〉 and 〈y, ρ〉 be metric spaces and let a be a subset of x. we say that a function f : x → y is strongly uniformly continuous on a if ∀ε > 0 ∃δ > 0 such that if d(x, w) < δ and {x, w} ∩ a 6= ∅, then ρ(f (x), f (w)) < ε. note that strong uniform continuity on a = {x0} means simply that f is continuous at x0. strong uniform continuity on a = x is uniform continuity. a continuous function on x is strongly uniformly continuous on each nonempty well-posedness, bornologies, and the structure of metric spaces 137 compact subset, not merely uniformly continuous when restricted to such a subset. lemma 3.8. let λ : x → [0, ∞] be continuous. if λ is finite-valued and strongly uniformly continuous on a ∈ p0(x) then λ is hd-continuous at a. proof. we show that λ is lower and upper semi-continuous at a, respectively. for lower semi-continuity, we have nothing to show if λ(a) = 0. otherwise, fix α0 > 0 and suppose α0 < λ(a). then ∃a0 ∈ a such that λ(a0) > α0 + ε0, where ε0 > 0. choose by strong uniform continuity of λ on a δ0 > 0 such that if a ∈ a, x ∈ x and d(a, x) < δ0, then |λ(a) − λ(x)| < ε0. now suppose hd(a, b) < δ0; choose b ∈ b such that d(a0, b) < δ0. then |λ(a0) − λ(b)| < ε0 ⇒ λ(b) > α0 ⇒ λ(b) > α0. for upper semi-continuity, we have nothing to show if λ(a) = ∞. otherwise, fix α1 > 0 with λ(a) < α1. fix ε1 > 0 so that ∀a ∈ a, λ(a) < α1 − ε1 2 . let δ1 > 0 be such that if a ∈ a, x ∈ x and d(a, x) < δ1 then |λ(a) − λ(x)| < ε1 3 . suppose hd(a, b) < δ1 and let b ∈ b be arbitrary. choose a ∈ a such that d(a, b) < δ1. then |λ(a) − λ(b)| < ε1 3 ⇒ −ε1 3 < λ(a) − λ(b) < α1 − ε1 2 − λ(b) ⇒ λ(b) < α1 − ε1 6 . since b ∈ b was arbitrary, λ(b) < α1. � the next counterexample shows that when λ is not strongly uniformly continuous on a, it is not guaranteed that the λ functional is hd-continuous at a. example 3.9. let x = [0, ∞)×[0, ∞) and define λ : x → [0, ∞) by λ(x, y) = xy. let a = {(0, y) : y ∈ [0, ∞)}. obviously, λ is not strongly uniformly continuous on a, since one can take ε = 1 and for any δ > 0, if n > 2 δ we have d((0, n), ( 2 n , n)) < δ, but |λ(0, n)−λ( 2 n , n)| = 2 > ε. if we let an = { 1 n }×[0, ∞), then 〈an〉 hd → a. but for all n, λ(an) = ∞ while λ(a) = 0, showing λ is not hd-continuous at a. 4. λ-subsets definition 4.1. let 〈x, d〉 be a metric space, and λ : x → [0, ∞] be continuous. we say a ∈ p0(x) is a λ-subset of x if whenever 〈an〉 is a sequence in a and λ(an) → 0, then 〈an〉 has a cluster point in x. when x is itself a λ-subset, then 〈x, d〉 is called a λ-space. we denote the family of λ-subsets by bλ. note that bλ is not altered by replacing λ by min{λ, 1}, if one is bothered by functionals that naturally assume values of ∞. we now provide some examples. example 4.2. if 〈x, d〉 is any metric space, and λ(x) ≡ 1, then bλ = p0(x). example 4.3. if 〈x, d〉 is a metric space, then the family of nonempty subsets with compact closure k0(x) is bλ for the zero functional λ on x. 138 g. beer and m. segura example 4.4. if 〈x, d〉 is an unbounded metric space and x0 ∈ x, then the family of nonempty d-bounded subsets bd(x) is bλ for the continuous functional on x defined by λ(x) = 1 1 + d(x, x0) notice here that while inf λ(x) = 0, we have ker(λ) = ∅. we shall see presently that bλ for all such λ-functionals arises in this way (see theorem 4.17 infra). example 4.5. the λ-subsets of a metric space corresponding to the measure of isolation functional i(x) = d(x, x\{x}) are called the uc-subsets, as studied in [15]. the λ-subsets of a metric space corresponding to the measure of local compactness functional ν are called the cofinally complete subsets, as studied in [13]. definition 4.6. let x be a topological space. we call a family of nonempty subsets a of x a bornology [9, 14, 22, 30] provided (1) ⋃ a = x; (2) {a1, a2, a3, ..., an} ⊆ a ⇒ ⋃n i=1 ai ∈ a; (3) a ∈ a and ∅ 6= b ⊆ a ⇒ b ∈ a. we will of course be focusing on bornologies in a metric space 〈x, d〉. the largest bornology is p0(x) and the smallest is the set of nonempty finite subsets f0(x). the bornologies k0(x) and bd(x) lie between these extremes. of importance in the sequel are functional bornologies, that is, bornologies arising as the family of subsets on which a real-valued function with domain x is bounded. the proof of the next proposition is left to the reader, and it implies that k0(x) is the smallest possible bλ. proposition 4.7. let λ : x → [0, ∞] be continuous. then bλ forms a bornology containing the nonempty compact subsets. by a base for a bornology, we mean a subfamily that is cofinal in the bornology with respect to inclusion. for example, a countable base for the metrically bounded subsets of 〈x, d〉 consists of all balls with a fixed center and integral radius. the next result says that bλ has a closed base, that is, a base that consists of closed sets. proposition 4.8. let λ : x → [0, ∞] be continuous, and let a be a λ-set. then cl(a) is also a λ-set. proof. let 〈xn〉 be a sequence in cl(a) where λ(xn) → 0. we may assume ∀n ∈ n that λ(xn) < ∞. by the continuity of λ, ∃ a sequence 〈an〉 in a where ∀n ∈ n, d(xn, an) < 1 n and λ(an) < λ(xn) + 1 n . then since 〈an〉 has a cluster point, 〈xn〉 must have one also. � well-posedness, bornologies, and the structure of metric spaces 139 the following elementary proposition was not noticed for either the bornology of uc-subsets or the bornology of cofinally complete subsets. it will be used to characterize those bornologies that are bλ for some λ ∈ c(x, [0, ∞)). proposition 4.9. let 〈x, d〉 be a metric space and let λ : x → [0, ∞] be continuous. suppose b is a nonempty closed subset of x. then b is a λsubset if and only if b ∩ ker(λ) is compact, and whenever a is a nonempty closed subset of b with a ∩ ker(λ) = ∅, then inf λ(a) > 0. proof. suppose first that b is a λ-set. then each sequence in b ∩ ker(λ) is a minimizing sequence and since b ∩ ker(λ) is closed, the sequence clusters to a point of b ∩ ker(λ). suppose next that a ∈ c0(x) ∩ p0(b) does not intersect ker(λ), yet inf λ(a) = 0. then λ has a minimizing sequence in a that clusters to a point of a which by continuity also must be in ker(λ) , contradicting a ∩ ker(λ) = ∅. conversely, suppose b satisfies the two conditions, and 〈bn〉 is a sequence in b with λ(bn) → 0 but that does not cluster. by the assumed compactness of b ∩ ker(λ), and by passing to a subsequence, we may assume that ∀n, bn /∈ ker(λ). but then with a = {bn : n ∈ n}, the second condition is violated. � proposition 4.10. let λ : x → [0, ∞] be continuous. then a λ-set a is compact if and only if ∀ε > 0, bε := {a ∈ a : λ(a) ≥ ε} is compact. proof. let a be compact λ-set. since λ is a continuous function, {x : λ(x) ≥ ε} is closed. since bε = a ∩ {x : λ(x) ≥ ε}, bε is compact. conversely, suppose 〈an〉 is an arbitrary sequence in a. if λ(an) → 0, then the sequence clusters because a is a λ-set. otherwise, ∃ε > 0 and an infinite subset n1 of n such that ∀n ∈ n1, λ(an) ≥ ε. hence 〈an〉n∈n1 is a sequence in the compact set bε. thus, the sequence 〈an〉 clusters, and a is compact. � our next proposition involves λ-subsets and strong uniform continuity. proposition 4.11. let λ : x → [0, ∞) be continuous. (1) if a is a λ-subset, λ is strongly uniformly continuous on a, and 〈xn〉 is a sequence in x with lim d(xn, a) = 0 and lim λ(xn) = 0, then 〈xn〉 clusters. (2) strong uniform continuity of λ on each member of bλ coincides with global uniform continuity. proof. we prove statement (2), leaving (1) to the reader. suppose λ fails to be globally uniformly continuous. then for some ε > 0, there exist sequences 〈xn〉 and 〈wn〉 in x such that for each n, d(xn, wn) < 1 n yet f (xn) + ε < f (wn). while b := {wn : n ∈ n} is in bλ, λ is not strongly uniformly continuous on b. � example 4.12. for a counterexample to proposition 4.11(1), let us revisit the metric space x and the functional λ of example 3.9. then a := {(x, y) : xy = 1} ∪ {(x, y) : x = y and x ≤ 1}, as shown in figure 1, is a λ-set. if 140 g. beer and m. segura xn = (0, n), then lim d(xn, a) = 0 and lim λ(xn) = 0, but the sequence 〈xn〉 does not cluster. figure 1 the next result is anticipated by a decomposition theorem for spaces on which a continuous function that is tychonoff well-posed in the generalized sense is defined [31, prop 10.1.7]. it is also anticipated by particular decomposition theorems in the special cases of the bornology of uc-subsets and the bornology of cofinally complete subsets [13, 15] (see previously for uc spaces and cofinally complete spaces [8, 10, 23]). theorem 4.13. let 〈x, d〉 be a metric space and suppose λ ∈ c(x, [0, ∞]). then a ∈ p0(x) is a λ-subset if and only if cl(a) ∩ ker(λ) is compact and ∀δ > 0, ∃ε > 0 such that a ∈ a \ sδ(cl(a) ∩ ker(λ)) ⇒ λ(a) > ε. proof. first, suppose cl(a) ∩ ker(λ) is not compact, and therefore nonempty. choose a sequence 〈an〉 in cl(a) ∩ ker(λ) with no cluster point. then λ(an) → 0, but 〈an〉 has no cluster point ⇒ cl(a) is not a λ-set ⇒ a is not a λ-set. suppose now that for some δ > 0 that inf{λ(a) : a ∈ a \ sδ(cl(a) ∩ ker(λ))} = 0. select an ∈ a \ sδ(cl(a) ∩ ker(λ)) with λ(an) < 1 n . there can be no possible cluster point p for 〈an〉 as by continuity λ(p) = 0 must hold, while d(p, cl(a) ∩ ker(λ)) ≥ δ. again, a is not a λ-set. conversely, suppose cl(a) ∩ ker(λ) is compact, and ∀δ > 0, ∃εδ > 0 such that a ∈ a \ sδ(cl(a) ∩ ker(λ)) ⇒ λ(a) > εδ. let 〈an〉 be a sequence in a where λ(an) → 0. if cl(a) ∩ ker(λ) = ∅, then a = a \ sδ(cl(a) ∩ ker(λ)) for each δ. so then given δ > 0, ∀n λ(an) > εδ, which is a contradiction. we conclude that cl(a) ∩ ker(λ) 6= ∅. then given δ > 0, λ(an) ≤ εδ eventually ⇒ an ∈ sδ(cl(a) ∩ ker(λ)) eventually ⇒ 〈an〉 has a cluster point by the compactness of cl(a) ∩ ker(λ). � we now address a basic question: what are necessary and sufficient conditions on a bornology b in a metric space 〈x, d〉 such that b = bλ for some well-posedness, bornologies, and the structure of metric spaces 141 λ ∈ c(x, [0, ∞))? the key tools in answering this question are proposition 4.9 and an important lemma of s.-t. hu ([25, thm 13.2] or [26, p. 189]), proved using the urysohn lemma. lemma 4.14 (hu’s lemma). let b 6= p0(x) be a bornology on a normal topological space x having a countable base {bn : n ∈ n} such that ∀n ∈ n, cl(bn) ⊆ int(bn+1). then there exists an unbounded f ∈ c(x, [0, ∞)) such that b = {a : f (a) is a bounded set of reals}. it is easy to see that the conditions of the lemma are satisfied if and only if (1) ∀b ∈ b, b 6= x; (2) b has a countable base; (3) b has an open base; and (4) b has a closed base. to obtain our characterization, we break our λ-functionals into two classes: those for which ker(λ) = ∅, and those for which ker(λ) 6= ∅. we need an immediate consequence of theorem 4.13 to deal with the first situation that we record as a lemma. lemma 4.15. let λ ∈ c(x, [0, ∞)) have no minimum value, yet inf λ(x) = 0. then bλ = {a ∈ p0(x) : inf λ(a) > 0}. theorem 4.16. let b be a bornology on 〈x, d〉. then b = bλ for some λ ∈ c(x, [0, ∞)) with ker(λ) = ∅ if and only if b has a countable base {bn : n ∈ n} such that ∀n ∈ n, cl(bn) ⊆ int(bn+1). proof. for sufficiency, if x ∈ b, we can put λ(x) ≡ 1. otherwise, applying hu’s lemma to generate an unbounded f ∈ c(x, [0, ∞)), put λ(x) := (1 + f (x))−1. noting that λ is bounded away from zero on a subset of x if and only if f is bounded above on the subset, we see by lemma 4.15 that λ does the job. for necessity, if b = bλ where ker(λ) = ∅, then by lemma 4.15, b ∈ b ⇔ inf λ(b) > 0. by the continuity of λ, {λ−1([ 1 n , ∞)) : n ∈ n} is the desired countable base. � theorem 4.17. let b be a bornology on 〈x, d〉. the following conditions are equivalent: (1) b = bλ for some λ ∈ c(x, [0, ∞)) with ker(λ) = ∅; (2) b = bρ(x) for some metric ρ equivalent to d. proof. (2) ⇒ (1). if ρ is a bounded metric, take λ(x) ≡ 1. otherwise, we invoke theorem 4.16 for bρ(x), putting bn := {x : ρ(x, x0) ≤ n} where x0 ∈ x is fixed. (1) ⇒ (2). if bλ = p0(x), take ρ = min{1, d}. otherwise, with bn = λ−1([ 1 n , ∞)) 6= x, apply hu’s lemma to once again generate an unbounded f . the metric ρ(x, w) := min{1, d(x, w)} + |f (x) − f (w)| satisfies bλ = bρ(x) and is equivalent to d. � 142 g. beer and m. segura we now come to the harder part. theorem 4.18. let b be a bornology on 〈x, d〉. the following conditions are equivalent: (1) b = bλ for some λ ∈ c(x, [0, ∞)) with ker(λ) 6= ∅; (2) b has a closed base, and ∃c ∈ c0(x) with open neighborhoods {vn : n ∈ n} satisfying ∩∞n=1vn = c and ∀n ∈ n, cl(vn+1) ⊆ vn such that ∀b ∈ c0(x), b ∈ b ⇔ b ∩ c is compact, and whenever a is a nonempty closed subset of b disjoint from c, then for some n, a ∩ vn = ∅. proof. (1) ⇒ (2). by proposition 4.8, b has a closed base, and by proposition 4.9, we can take c = ker(λ) and vn = λ −1([0, 1 n )). (2) ⇒ (1). we consider several cases for the set c. first if c = x, then a nonempty closed set b is in b if and only if b is compact, and since the bornology has a closed base, it is the bornology k0(x) of nonempty subsets with compact closure and with λ(x) ≡ 0, we get b = bλ. a second possibility is that c = vn ⊂ x for some n. since {c, x\c} forms a nontrivial separation of x, the function λ assigning 0 to each point of c and 1 to each point of x\c is continuous. we intend to show that b = bλ. since both bornologies have closed bases, it suffices to show closed members of one belong to the other. if b ∈ b ∩ c0(x), then any minimizing sequence in b lies eventually in c, and since b ∩ c is compact, it clusters. this shows b ∈ bλ. for the reverse inclusion, if b ∈ bλ is closed, then b ∩ ker(λ) is compact, that is, b ∩ c is compact. also if a is a closed subset of b disjoint from c, then a ∩ vn = ∅ without any consideration of λ. in the remaining case we may assume without loss of generality that ∀n ∈ n, c ⊂ vn ⊂ x. we now apply hu’s lemma to the metric subspace x\c with respect to the bornology having the closed base {x\vn : n ∈ n}. we produce an unbounded continuous f : x\c → [0, ∞) such that ∀a ∈ p0(x\c), f (a) is bounded if and only if for some n, a ⊆ x\vn. we next define our function λ by λ(x) = { 0 if x ∈ c 1 1+f (x) otherwise . evidently λ is continuous restricted to the open set x\c. given ε ∈ (0, 1), choose n ∈ n with {x ∈ x\c : f (x) ≤ 1−ε ε } ⊆ x\vn. it follows that ∀x ∈ vn, we have λ(x) < ε, establishing global continuity of λ. again we must show that b ∩ c0(x) = bλ ∩ c0(x). for a closed set b, b ∩ ker(λ) is compact if and only if b ∩ c is compact because by construction ker(λ) = c. if b ∈ c0(x) and a is a nonempty closed subset with a ∩ c = a ∩ ker(λ) = ∅ then ∃n with a ∩ vn = ∅ ⇔ ∃n with a ⊆ x\vn ⇔ f is bounded above on a ⇔ inf λ(a) > 0. the result now follows from proposition 4.9. � well-posedness, bornologies, and the structure of metric spaces 143 we next show that that there are bornologies with closed base that fail to be a bornology of λ-subsets. example 4.19. consider r with the zero-one metric and and let b be the bornology of countable nonempty subsets. since r is uncountable, b fails to have a countable base. by theorem 4.16, it remains to show that b cannot be bλ for any λ with nonempty kernel. we show that condition (2) of theorem 4.18 cannot hold. suppose to the contrary that such a c with neighborhoods {vn : n ∈ n} existed. since the intersection of c with each countable set must be compact, we conclude c is finite. for each n, put bn := x\vn. clearly, bn ∩ c is compact as it is empty. also each (closed) subset of bn is trivially disjoint from vn. by condition (2) of theorem 4.18, bn must be countable, and since x\c = ∪∞n=1bn, it too must be countable. this is a contradiction, and so the bornology of countable subsets cannot be a bornology of λ-subsets. here is a natural follow-up question: when is a bornology b a bornology of λ-subsets for some uniformly continuous λ : x → [0, ∞)? in our analysis, strong uniform continuity of a function on members of a bornology plays a key role. we first obtain an analog of hu’s lemma, which is implicit in the proof of [14, thm. 3.18]. lemma 4.20. suppose b is a bornology on a metric space 〈x, d〉 that does not contain x. suppose b has a countable base {bn : n ∈ n} such that ∀n ∈ n, ∃δn > 0 with sδn (bn) ⊆ bn+1. then there exists an unbounded f ∈ c(x, [0, ∞)) such that f is strongly uniformly continuous on each bn and such that b = {a : f (a) is a bounded set of reals}. proof. for each n ∈ n let fn : x → [0, 1] be the uniformly continuous function defined by fn(x) = min{1, 1 δn d(x, bn)}. the values of fn all lie in [0, 1], and fn(bn) = {0} and fn(x\bn+1) = {1}. put f = f1 + f2 + f3 + · · · . first note that the restriction of f to each bn agrees with f1 + f2 + f3 + · · · + fn−1 so that (1) ∀n, f restricted to bn is uniformly continuous; (2) ∀n, f (bn) ⊆ [0, n − 1]. by (1) f is strongly uniformly continuous on each bn because f is uniformly continuous restricted to bn+1 and this larger set contains an enlargement of bn. by (2) ∀n, f (bn) is bounded, so f restricted to each member of b is bounded because {bn : n ∈ n} is a base. finally, if f (a) is bounded, then for some n, a ⊆ bn because x /∈ bn+1 ⇒ f (x) ≥ n . � we note that the function f in the lemma 4.20 is strongly uniformly continuous on each member of b. more generally, the sets on which a continuous real function g is strongly uniformly continuous always form a bornology containing the uc-subsets; in fact, the uc-subsets form the largest common bornology as g runs over c(x, r) [15]. we also note that if δn can be chosen independent of 144 g. beer and m. segura n in the statement of lemma 4.20, one can construct a uniformly continuous function f , but the proof is a little more delicate [9, thm. 4.2]. we will need the following fact about strong uniform continuity. proposition 4.21. suppose g : 〈x, d〉 → (0, ∞) is strongly uniformly continuous on a nonempty subset a of x, and g is bounded away from zero in some enlargement of a. then λ(x) := 1 g(x) is strongly uniformly continuous on a. proof. suppose ∀x ∈ sδ(a), we have g(x) ≥ α > 0. given ε > 0, ∃δε ∈ (0, δ) such that if a ∈ a and x ∈ x and d(a, x) < δε, then |g(x) − g(a)| < εα 2. we compute |λ(x) − λ(a)| = ∣ ∣ ∣ ∣ 1 g(x) − 1 g(a) ∣ ∣ ∣ ∣ = |g(a) − g(x)| |g(x)g(a)| , and since {a, x} ⊆ sδε (a) ⊆ sδ(a), we further have |g(a) − g(x)| |g(x)g(a)| < εα2 |g(x)g(a)| ≤ εα2 α2 = ε, and this yields |λ(x) − λ(a)| < ε. � theorem 4.22. let b be a bornology on 〈x, d〉. the following conditions are equivalent: (1) b = bλ for some uniformly continuous λ : x → [0, ∞) with ker(λ) = ∅; (2) b has a countable base {bn : n ∈ n} such that ∀n ∈ n, ∃δn > 0 with sδn (bn) ⊆ bn+1. proof. (1) ⇒ (2). if inf λ(x) > 0, then x ∈ b and we can put bn := x for each n ∈ n. otherwise, put bn = λ −1([ 1 n , ∞)) ∈ b; choose by uniform continuity of λ a positive δn such that d(x, w) < δn ⇒ |f (x) − f (w)| < 1 n − 1 n + 1 . then we have ∀n ∈ n, sδn (bn) ⊆ bn+1. (2) ⇒ (1) the case x ∈ b, that is b = p0(x), is of course trivial. otherwise, we take f as guaranteed by lemma 4.20 and as expected put λ(x) = (1 + f (x))−1. we use proposition 4.11(2) to establish uniform continuity. fix n ∈ n. we know g(x) := 1 + f (x) is strongly uniformly continuous on bn and that g is bounded below by 1 on all of x. taking the reciprocal, by proposition 4.21, we see that λ is strongly uniformly continuous on each bn and thus on each b ∈ b, as required. � as expected, the bornologies that fulfill the conditions of theorem 4.22 are metric boundedness structures [9], that is, they are of the form bρ for certain ρ equivalent to d. in turns out that the metrics ρ are those for which the identity well-posedness, bornologies, and the structure of metric spaces 145 id : 〈x, d〉 → 〈x, ρ〉 is strongly uniformly continuous on each ρ-bounded subset. we leave this as an exercise to the interested reader, following the proof of theorem 4.17 (see also [14]). theorem 4.23. let b be a bornology on 〈x, d〉. the following conditions are equivalent: (1) b = bλ for some uniformly continuous λ : x → [0, ∞) with ker(λ) 6= ∅; (2) b has a closed base, and ∃c ∈ c0(x) with open neighborhoods {vn : n ∈ n} satisfying ∩∞n=1vn = c and ∀n ∈ n, ∃δn > 0 with sδn (vn+1) ⊆ vn such that ∀b ∈ c0(x), b ∈ b ⇔ b ∩ c is compact, and whenever a is a nonempty closed subset of b disjoint from c, then for some n, a ∩ vn = ∅. proof. (1) ⇒ (2). let λ satisfy condition (1), and put c = ker(λ). if c = x, ∀n ∈ n, put vn = x. otherwise, ∃k ∈ n and x ∈ x with λ(x) > 1 k . in this case ∀n ∈ n, put vn := {x ∈ x : λ(x) < 1 n+k }. by uniform continuity of λ, ∃δn > 0 with d(x, w) < δn ⇒ |λ(x) − λ(w)| < 1 n + k − 1 n + k + 1 which means that sδn (vn+1) ⊆ vn. by proposition 4.8 and proposition 4.9, bλ satisfies the conditions on a bornology b listed in (2). (2) ⇒ (1). we handle this implication by modifying the proof of (2) ⇒ (1) in theorem 4.18. the case c = x is handled in exactly the same manner. in the case that c = vn ⊂ x for some n, we define a uniformly continuous function λ on x by λ(x) = min{ 1 δn d(x, c), 1}. since sδn (c) ⊆ vn, we see that λ maps each point of x\c to 1 and each point of c to 0. verification that b = bλ proceeds exactly as in the proof of theorem 4.18. in the remaining case we can assume for each n ∈ n that c ⊂ vn ⊂ x. by condition (2), ∀n ∈ n, we have sδn (x\vn) ⊆ x\vn+1. we now apply lemma 4.20 to the space x\c equipped with the bornology with base {x\vn : n ∈ n} to produce an unbounded f : x\c → [0, ∞) that is strongly uniformly continuous on each set x\vn and such that f (a) is bounded if and only if a is a subset of some x\vn. we now define λ : x → [0, ∞) by λ(x) = { 0 if x ∈ c 1 1+f (x) otherwise . the proof of theorem 4.22 shows that the restriction of λ to x\c is uniformly continuous, so if λ fails to be globally uniformly continuous, ∃ε > 0 such that 146 g. beer and m. segura ∀k ∈ n, ∃ck ∈ c and xk ∈ x\c such that d(ck, xk) < 1 k while λ(xk) > ε. now as λ is bounded below by ε on {xk : k ∈ n}, f is bounded above so restricted. it follows that for some n0 ∈ n, we have {xk : k ∈ n} ∩ vn0 = ∅. but choosing 1 k < δn0 , by condition (2) d(ck, xk) < 1 k ⇒ ck ∈ x\vn0+1. this is a contradiction because x\vn0+1 ∩ c = ∅. this contradiction establishes global uniform continuity, and agreement of the bornologies is argued as before. � to end this section, we note that convergence in hausdorff distance need not preserve λ-sets, even when the λ-functional is uniformly continuous. example 4.24. let λ : r2 → [0, ∞), where λ(x, y) = y, and for each positive integer n put an := {(x, 0) : x ∈ [0, n]} ∪ {(x, y) : y = 1 n (x − n), x ∈ [n, n + 1]} ∪ {(x, y) : y = 1 n , x ≥ n + 1}, as shown in figure 2. then λ is uniformly continuous and 〈an〉 is a sequence of closed λ-sets converging in hausdorff distance to a, where a := {(x, y) : y = 0, x ≥ 0}. but a is not a λ-set. figure 2 5. λ-spaces given a continuous nonnegative function λ on a metric space 〈x, d〉, recall that x is called a λ-space provided each sequence 〈xn〉 in x with lim λ(xn) = 0 has a cluster point. as noted in the introduction, if λ is defined appropriately, the λ-spaces include the compact metric spaces, the uc-spaces and the cofinally complete metric spaces. we now show that they include the complete metric spaces. well-posedness, bornologies, and the structure of metric spaces 147 proposition 5.1. let 〈x, d〉 be a metric space, and let p (v ) mean cl(v ) is a complete subspace equipped with the metric d. put β := λp , so that β(x) = { sup{α > 0 : cl(sα(x)) is complete} if ∃α > 0 with cl(sα(x)) complete; 0 otherwise. then 〈x, d〉 is a complete metric space if and only if 〈x, d〉 is a β-space. proof. proving this is straightforward. first suppose 〈x, d〉 is complete, so ∀x ∈ x, β(x) = ∞. each sequence 〈xn〉 with lim β(xn) = 0 has a cluster point as this is true vacuously. hence 〈x, d〉 is a β-space. to see the converse, suppose 〈x, d〉 is a β-space and 〈xn〉 is a cauchy sequence. there are two possibilities: (1) lim β(xn) = 0, and (2) lim sup β(xn) > 0. if lim β(xn) = 0, then there exists a cluster point by the definition of a β-space. otherwise ∃ε > 0 and and infinite subset n1 of n such that ∀n ∈ n1, β(xn) > ε. choose k ∈ n such that if n > m > k, then d(xn, xm) < ε. if n1 ∈ n1 and n1 > k, then {x : d(x, xn1 ) ≤ ε} contains a tail of 〈xn〉 that is also cauchy. since β(xn1 ) > ε, {x : d(x, xn1 ) ≤ ε} is complete, which implies the tail has a cluster point, so 〈xn〉 has a cluster point also. hence 〈x, d〉 is complete. � proposition 4.9 and theorem 4.13 provide characterizations of λ-spaces, which we now list. theorem 5.2. let 〈x, d〉 be a metric space, and let λ : x → [0, ∞] be continuous. the following are equivalent: (1) 〈x, d〉 is a λ-space; (2) ker(λ) is compact, and if a ∈ c0(x) with a ∩ ker(λ) = ∅, then inf λ(a) > 0; (3) ker(λ) is compact, and ∀δ > 0, ∃ε > 0 such that d(x, ker(λ)) > δ ⇒ λ(x) > ε. although all λ-spaces must have a compact kernel, it is easy to produce examples showing that this alone is not sufficient (see, e.g., [31, ex. 10.1.3]). the following proposition shows how normal pathology is in this regard. proposition 5.3. let 〈x, d〉 be a noncompact metric space and let c be an arbitrary compact subset. then there exists λ ∈ c(x, [0, ∞)) with ker(λ) = c for which x is not a λ-space. proof. pick distinct points x1, x2, x3, . . . in x\c such that 〈xn〉 has no cluster point. note that a := {xn : n ∈ n} is a closed discrete set. if c = ∅, choose by the tietze extension theorem [21, p. 149] f ∈ c(x, [0, ∞)) satisfying f (xn) = n, and clearly λ(x) = (1 + f (x)) −1 does the job. when c is nonempty, by the tietze extension theorem, there is a nonnegative continuous function λ1 on x mapping c to 0 such that ∀n, λ1(xn) = 1 n . the desired λ is defined by λ(x) = λ1(x) + d(x, a ∪ c). � 148 g. beer and m. segura the last result of course shows that whenever c is a nonempty compact subset of a metric space 〈x, d〉, then there is a function having c as its set of minimizers that fails to be tychonoff well-posed in the generalized sense. the next result characterizes λ-spaces in terms of a general cantor-type theorem. as its validity is known in the most important special cases (see [6, 10]), it comes as no surprise. theorem 5.4. let λ : 〈x, d〉 → [0, ∞] be a continuous function. then 〈x, d〉 is a λ-space if and only if whenever 〈an〉 is a decreasing sequence in c0(x) with λ(an) → 0 then ⋂ n∈n an is nonempty. proof. suppose 〈x, d〉 is a λ-space and 〈an〉 is decreasing in c0(x) with λ(an) → 0. for each n ∈ n, pick xn ∈ an arbitrarily. we have 0 ≤ λ(xn) ≤ sup{λ(a) : a ∈ an}. as λ(an) → 0, we have λ(xn) → 0, so 〈xn〉 must have a cluster point, say p. then given ε > 0 and n0 ∈ n, ∃k ≥ n0 such that d(xk, p) < ε ⇒ xk ∈ sε(p) ⇒ p ∈ cl({xj : j ≥ n0}) ⊆ cl   ∞ ⋃ j=n0 aj   ⊆ an0 , because 〈an〉 is a decreasing sequence and an0 is closed. hence, p ∈ ∩n∈nan. conversely, let 〈yn〉 be a sequence in 〈x, d〉 where lim λ(yn) = 0. for each n ∈ n, put an := cl({yk : k ≥ n}). fix ε > 0; ∃n0 ∈ n such that n ≥ n0 ⇒ λ(yn) < ε. as a result, ∀n ≥ n0, sup{λ(a) : a ∈ an} ≤ ε ⇒ lim λ(an) = 0. hence ⋂∞ n=1 cl({yk : k ≥ n}) 6= ∅, and 〈yn〉 has a cluster point. � lemma 5.5. let 〈x, d〉 be a λ-space. suppose 〈an〉 is a decreasing sequence in c0(x) with lim λ(an) = 0. then a := ⋂ n∈n an is nonempty and compact and lim hd(an, a) = 0. proof. the set a is nonempty by theorem 5.4. choose an arbitrary sequence x1, x2, x3, ... in a. since λ is monotone and lim λ(an) = 0, we have λ(a) = 0. hence ∀n ∈ n, λ(xn) = 0 ⇒ 〈xn〉 has a cluster point in a because a is closed. thus, a is compact. now we show lim hd(an, a) = 0. suppose this does not hold; then ∃ε > 0 such that ∀n0 ∈ n, ∃k ≥ n0 with hd(ak, a) > ε. since an0 ⊇ ak, clearly an0 * sε(a). pick ∀n ∈ n xn ∈ an \ sε(a). since lim λ(xn) = 0, 〈xn〉 must have a cluster point, say p. hence p ∈ ⋂ k∈n cl({xn : n ≥ k}) ⊆ ⋂ k∈n ak = a. but ∀n ∈ n, d(xn, p) ≥ d(xn, a) ≥ ε, which is a contradiction. thus, 〈an〉 converges to a in hausdorff distance. � well-posedness, bornologies, and the structure of metric spaces 149 theorem 5.6. if 〈x, d〉 is complete, then the following statements are equivalent: (1) 〈x, d〉 is a λ-space; (2) the measure of noncompactness functional α is continuous with respect to λ on c0(x) : ∀ε > 0, ∃δ > 0 such that a ∈ c0(x) and λ(a) < δ ⇒ α(a) < ε. proof. (2) ⇒ (1). let 〈an〉 be a decreasing sequence in c0(x) with lim λ(an) = 0. fix ε > 0; ∃δ > 0 such that λ(an) < δ ⇒ α(an) < ε. since lim λ(an) = 0, we have lim α(an) = 0. since x is complete, by kuratowski’s theorem, ⋂ n∈n an 6= ∅. hence, by theorem 5.4, x is a λ-space. (1) ⇒ (2). assume (1) holds but (2) fails, i.e., ∃ε > 0 such that given n ∈ n, ∃bn ∈ c0(x) with λ(bn) ≤ 1 n but α(bn) ≥ ε. let an := {x : λ(x) ≤ 1 n } and put a := ⋂ n∈n an which by lemma 5.5 is nonempty and compact and lim hd(an, a) = 0. since an ⊇ bn, by continuity of α with respect to hausdorff distance, ∀n ∈ n, α(an) ≥ ε ⇒ α(a) ≥ ε. but α(a) = 0 as a is compact; thus we have a contradiction. � given an hereditary property p of open subsets of a metrizable space x, the induced functional λp depends on the nature of the balls of the particular metric chosen. with one choice, we might obtain a λp -space but with another, not so. example 5.7. let x = {0} ∪ { 1 n : n ∈ n} ∪ {4 − 1 n : n ∈ n} as a topological subspace of r, and let p (v ) be the property that v contains at most one point. for a particular compatible metric d, the associated functional λdp is of course the measure of isolation functional. when d is the euclidean metric, the resulting space is not a λp -space, as λ d p (4 − 1 n ) = 1 n2+n while 〈4 − 1 n 〉 fails to cluster in x. on the other hand the mapping g : x → r defined by g(x) = { 2n if x = 4 − 1 n for some n x otherwise is a topological embedding, and this yields a metric ρ on x defined by ρ(x, w) = |g(x) − g(w)| for which 〈x, ρ〉 is a λp -space. the next result, in the special case of uc-spaces, appears in the first john rainwater paper [34], a pseudonym used by mathematicians associated with the university of washington. in the special case of cofinally complete spaces, it is due to s. romaguera [36]. theorem 5.8. let x be a metrizable topological space, and let p be an hereditary property of open sets. the following conditions are equivalent: (1) x has a compatible metric d such that 〈x, d〉 is a λp -space; (2) ker(λp ) is compact. 150 g. beer and m. segura proof. if d is a compatible metric, let us write for the purposes of this proof sdα(x) for the open d-ball with center x and radius α, and λ d p for the induced functional. note that the set {x ∈ x : λp (x) = 0} is well-defined, i.e., it does not depend on the particular metric chosen, for if ρ is another compatible metric, then at each x, ∀α > 0, ¬p (sdα(x)) if and only if ∀α > 0, ¬p (s ρ α(x)). let us denote this well-defined set by ker(λp ). with this in mind, it follows from theorem 5.2 that (2) is necessary for (1). for the sufficiency of (2) for (1), we use this technical fact about open covers: if x is metrizable and {ωk : k ∈ n} is a family of open covers of x, then there exists a compatible metric d for x such that ∀k ∈ n, {sd 1/k (x) : x ∈ x} refines ωk [21, p. 196]. it is possible that while compact, ker(λp ) is empty. then each x ∈ x has an open neighborhood vx such that p (vx). by the just-stated refinement result, there exists a compatible metric d such that {sd1 (x) : x ∈ x} refines {vx : x ∈ x}. since p is hereditary, ∀x, λ d p (x) = sup{α > 0 : p (s d α(x))} ≥ 1, and so 〈x, d〉 is a λp -space. otherwise, ker(λp ) is nonempty and compact and so there is a countable family of open neighborhoods {wk : k ∈ n} of ker(λp ) such that whenever v is open and ker(λp ) ⊆ v , ∃k ∈ n with wk ⊆ v . again, for each x /∈ ker(λp ), let vx be an open neighborhood of x with p (vx). for each k ∈ n, define an open cover ωk of x as follows: ωk := {vx : x /∈ wk} ∪ {wk}. choose a compatible metric d such that for each k, {sd 1/k (x) : x ∈ x} refines ωk. now let 〈xn〉 satisfy limn→∞λ d p (xn) = 0. for each k, wk contains a tail of 〈xn〉, specifically xn ∈ wk when λ d p (xn) < 1 k . since {wk : k ∈ n} forms a base for the neighborhoods of ker(λp ), ∀ε > 0, ∃nε ∈ n ∀n ≥ nε, xn ∈ s d ε (ker(λp ). since ker(λp ) is compact, 〈xn〉 has a cluster point and 〈x, d〉 is a λp -space in this second case, too. � with respect to product spaces equipped with the box metric, if we consider again an hereditary property of open sets p , we can write a formula for λp if the property p ”factors”, as it does in the case of the measure of isolation functional and the measure of local compactness functional. proposition 5.9. let p1, p2 be hereditary properties of open sets in x1, x2 respectively, and p be a property of open sets in x1 × x2 such that p (u × v ) if and only if both p1(u ) and p2(v ). then λp : x1 × x2 → [0, ∞] can be expressed by λp (x, y) = min{λp1 (x), λp2 (y)}. proof. let x ∈ x1 and y ∈ x2. suppose α < min{λp1 (x), λp2 (y)}. then p1(sα(x)) ∧ p2(sα(y)) ⇒ p (sα(x, y)), and so λp (x, y) ≥ α. as a result, λp (x, y) ≥ min{λp1 (x), λp2 (y)}. suppose β < λp (x, y). then p (sβ (x, y)) ⇒ well-posedness, bornologies, and the structure of metric spaces 151 p1(sβ (x))∧p2(sβ (y)) ⇒ λp1 (x) ≥ β∧λp2 (y) ≥ β, so min{λp1 (x), λp2 (y)} ≥ β. hence min{λp1 (x), λp2 (y)} ≥ λp (x, y). � proposition 5.10. suppose λ(x1, x2) = min{λ1(x1), λ2(x2)} where λ1 and λ2 are continuous, nonnegative extended real-valued functions on x1 and x2, respectively. then λ is continuous and nonnegative, and ker(λ) = [ker(λ1) × x2] ∪ [x1 × ker(λ2)]. the next result is hinted at by a result of hohti [23, thm. 2.2.1] for cofinally complete metric spaces. theorem 5.11. let 〈x1, d1〉 and 〈x2, d2〉 be metric spaces, where λ1 : x1 → [0, ∞) and λ2 : x2 → [0, ∞) are continuous. consider the metric space 〈x1 × x2, d〉, where d is the box metric, and λ(x1, x2) = min{λ1(x1), λ2(x2)}. the following are equivalent: (1) x1 × x2 is a λ-space; (2) x1 is a λ1-space, x2 is a λ2-space, and additionally both (i) ker(λ1) 6= ∅ ⇒ x2 is compact, and (ii) ker(λ2) 6= ∅ ⇒ x1 is compact. proof. (1)⇒(2): to show that x1 is a λ1-space, let 〈an〉 be a sequence in x1 where λ1(an) → 0. consider 〈(an, c)〉 as a sequence in x1 ×x2, where c ∈ x2 is fixed arbitrarily. then λ(an, c) = min{λ1(an), λ2(c)} → 0 because λ(an) → 0. as a result, 〈(an, c)〉 must have a cluster point (p1, c). hence, 〈an〉 clusters. in a similar manner, it can be shown that x2 is a λ2-space. suppose now ker(λ1)6= ∅. then ∃x ∈ x1 with λ1(x) = 0. let 〈bn〉 be an arbitrary sequence in x2. we can then let 〈(x, bn)〉 be a sequence in x1 × x2. then λ(x, bn) = min{λ1(x), λ2(bn)} → 0 so 〈(x, bn)〉 has a cluster point (x, p3). hence 〈bn〉 clusters ⇒ x2 compact. similarly, it can be shown that if ker(λ2)6= ∅, then x1 is compact. (2)⇒(1): to show x1×x2 is a λ-space, let 〈(an, bn)〉 be a sequence in x1×x2 with λ(an, bn) → 0. consider the case where there exists a subsequence of 〈an〉, say 〈an1〉n1∈n1 with n1 ⊆ n, where λ1(an1 ) → 0. then ∃n2 ⊆ n1 where 〈an2〉n2∈n2 converges to a point of ker(λ1). since 〈bn〉 is in x2, which must be compact, then ∃n3 ⊆ n2 such that 〈bn3〉n3∈n3 converges and 〈an3〉n3∈n3 converges. hence, 〈(an3 , bn3 )〉n3∈n3 converges, which implies 〈(an, bn)〉 clusters. in the case where there exists a subsequence of 〈bn〉, say 〈bn1〉n1∈n1 with n1 ⊆ n, where λ2(bn1 ) → 0, it can be similarly shown that 〈(an, bn)〉 clusters, and this is left to the reader. � remark 5.12. proposition 5.10 gives an alternate justification that conditions (2i) and (2ii) are necessary in theorem 5.11. example 5.13. in the case that λ1 = λ2 = the measure of local completeness functional, when both x1 and x2 are complete, it is clear that ker(λ1) = 152 g. beer and m. segura ker(λ1) = ∅, so that x1 × x2 is complete if and only if x1 and x2 are complete, as we all know. example 5.14. in the case that λ1 = λ2 = the measure of isolation functional, condition (2) becomes x1 and x2 are both uc-spaces, and if either space has limit points, the other must be compact. what is most interesting about this result emerges after we take a closer look at statement (2) of theorem 5.11 from the perspective of mathematical logic. formally, statement (2) is of the form p ∧ (q ⇒ s) ∧ (r ⇒ t ), which is logically equivalent to [(p ∧ ¬q) ∨ (p ∧ s)] ∧ [(p ∧ ¬r) ∨ (p ∧ t )]. since conjunction is distributive over disjunction, the following four-part disjunction is equivalent to (2): inf{λ1(x) : x ∈ x1} > 0 and inf{λ2(x) : x ∈ x2} > 0, or x2 is an λ2-space, x1 is compact, and inf{λ1(x) : x ∈ x1} > 0, or x1 is an λ1-space, x2 is compact, and inf{λ2(x) : x ∈ x2} > 0, or both x1 and x2 are compact. thus, all factor spaces that would yield a product space that is a λ-space, where λ is as defined in theorem 5.11, must fall into one of these four categories. example 5.15. in the case that λ1 = λ2 = the measure of local compactness functional, when x1 (resp. x2) is compact, then automatically λ1(x) ≡ ∞ (resp. λ2(x) ≡ ∞). thus, the final three statements of the four just listed can be condensed down to one statement: either x1 or x2 is compact, while the other is cofinally complete. the disjunction of this statement with the first, which in this context says that both x1 and x2 are uniformly locally compact, can be seen to be equivalent to hohti’s formulation [23]. 6. λ-subsets and bornological convergence over the last few years, there has been intense interest in bornological convergence of nets of sets in a metric space [12, 14, 15, 16, 30]. this was first described for nets of closed sets by borwein and vanderweff [17] as follows. definition 6.1. let b be a bornology in metric space 〈x, d〉. we declare a net 〈ai〉i∈i of closed subsets of x b-convergent to a closed subset a of x if for each b ∈ b and each ε > 0, we have eventually both ai ∩ b ⊆ sε(a) and a ∩ b ⊆ sε(ai). well-posedness, bornologies, and the structure of metric spaces 153 notice that convergence to the empty set means that eventually the net lies outside each set in the bornology. when b = p0(x), we obtain restricting our attention to c0(x) convergence in hausdorff distance because x ∈ p0(x). when b is the bornology of nonempty bounded subsets, we obtain attouch-wets convergence [2, 3, 8], also called bounded-hausdorff convergence [33]. when b is the bornology of nonempty subsets with compact closure, we obtain convergence with respect to the fell topology [8, theorem 5.1.6], also called the topology of closed convergence [28], which for sequences of closed sets reduces to classical kuratowski convergence [8, theorem 5.2.10]. recently it has been shown that convergence of linear transformations with respect to standard topologies of uniform convergence can be understood as bornological convergence of their associated graphs [11]. each of the bornological convergences just listed above are topological; in fact, the first two are compatible with metrizable topologies on c(x). as shown in [12], those bornologies for which b-convergence is topological on c(x) are those that are shielded from closed sets, according to the following definition. definition 6.2. let b be a bornology on a metric space 〈x, d〉. we say that b1 ∈ b is a shield for b ∈ b provided b ⊆ b1 and whenever c ∈ c0(x) is disjoint from b1, we have dd(b, c) > 0. we say b is shielded from closed sets provided each b in b has a shield in the bornology. in terms of open sets, b is shielded from closed sets if and only if given b ∈ b ∃b1 ∈ b such that b ⊆ b1 and each neighborhood of b1 contains some ε-enlargement of b. hence, a bornology having the property that b ∈ b ⇒ ∃ε > 0 with sε(b) ∈ b is obviously shielded from closed sets. so is a bornology having a base of compact sets, as then for each b ∈ b, the compact set cl(b) serves as shield for b. more generally, whenever b is shielded from closed sets, then ∀b ∈ b, cl(b) ∈ b. a wealth of additional information about this concept can be found in [12]. theorem 6.3. let λ ∈ c(x, [0, ∞)) be strongly uniformly continuous on some b ∈ bλ. then b has a shield in bλ. proof. without loss of generality, we may assume x is not a λ-space and b is a closed λ-set. by strong uniform continuity of λ on b, ∀n ∈ n, ∃δn ∈ (0, 1 n ) such that ∀b ∈ b, ∀x ∈ x, d(x, b) < δn ⇒ |λ(b) − λ(x)| < 1 n . we may also assume that 〈δn〉 is decreasing. let b ∈ b \ ker(λ). there exists a smallest nb ∈ n such that 1 nb < λ(b). if x ∈ x satisfies d(x, b) < δ2nb , then 1 2nb < λ(x) < λ(b) + 1 2nb . also note that λ(b) ≤ 1 nb−1 , whenever nb 6= 1. set δ(b) = δ2nb . we claim b1 := (ker(λ) ∩ b) ∪ ⋃ b∈b\ker(λ) sδ(b)(b) is a shield for b which lies in bλ. 154 g. beer and m. segura we first show b1 is λ-set. let 〈xk〉 be a sequence in b1 with λ(xk) → 0. if infinitely many terms of 〈xk〉 are contained in ker(λ) ∩ b, then 〈xk〉 must cluster by the compactness of ker(λ) ∩ b. otherwise, by passing to a subsequence we can assume ∀k ∈ n, xk ∈ ⋃ b∈b\ker(λ) sδ(b)(b) and λ(xk) < 1 2 . pick bk ∈ b \ ker(λ) with xk ∈ sδ(bk )(bk). fix k and let’s for the moment write n := nbk . we know that 1 2n < λ(xk), so n ≥ 2 and λ(bk) ≤ 1 n−1 . note also 1 n−1 ≤ 2 n , so λ(bk) ≤ 2 n = 4 · 1 2n < 4λ(xk). hence λ(bk) → 0, so 〈bk〉 has a cluster point p. thus, p is a cluster point of 〈xk〉 because δ(bk) → 0. now we must show whenever c ∈ c0(x) with c ∩b1 = ∅, then dd(c, b) > 0. by the compactness of ker(λ) ∩ b, we find µ > 0 such that dd(c, ker(λ) ∩ b) > 2µ. put t1 := b ∩ sµ(ker(λ) ∩ b) and t2 := b \ sµ(ker(λ) ∩ b), so that t1 ∪ t2 = b. then dd(c, t1) ≥ µ > 0. by theorem 4.13, there exists ε > 0 such that ∀b ∈ t2, λ(b) > ε. let k ∈ n satisfy 1 k < ε. if b ∈ t2, then λ(b) > 1 k so δ2k ≤ δ(b). hence, ⋃ b∈t2 sδ2k (b) ⊆ ⋃ b∈t2 sδ(b)(b) ⊆ b1. as a result, c ∩ ⋃ b∈t2 sδ2k (b) = ∅. then dd(c, t2) ≥ δ2k > 0. thus, dd(c, b) = dd(c, t1 ∪ t2) = min{dd(c, t1), dd(c, t2)} > 0. � corollary 6.4. let λ : x → [0, ∞) be uniformly continuous. then bλ is shielded from closed sets. example 6.5. consider for a counterexample [0, ∞) × [0, ∞) equipped with the usual metric. if λ : [0, ∞) × [0, ∞) → [0, ∞), where λ(x, y) = xy, then b := {(x, y) : xy = 1} ∪ {(x, y) : x = y and x ≤ 1} is a λ-set. suppose b1 were a shield for b. as a result of b1 being a λ-set, ∃n ∈ n such that b1 ∩ {(x, 0) : x ≥ 0} ⊆ [0, n] × {0}. then c := [2n, ∞) × {0} is closed and disjoint from b1, but dd(c, b) = 0. this is a contradiction. note of course that λ is not strongly uniformly continuous on b. bornological convergence of a net 〈ai〉i∈i of closed sets to a closed set a as determined by a bornology b can obviously be broken into two conditions, the first of which is called upper b-convergence, and the second lower b-convergence [30]: (i) ∀b ∈ b, ∀ε > 0 eventually ai ∩ b ⊆ sε(a), and (ii) ∀b ∈ b, ∀ε > 0 eventually a ∩ b ⊆ sε(ai). as bornologies are hereditary, evidently, (ii) is in general equivalent to (ii′) ∀b ∈ b, b ⊆ a ⇒ ∀ε > 0, b ⊆ sε(ai) eventually. well-posedness, bornologies, and the structure of metric spaces 155 as shown in [12], when the two-sided convergence is topological, condition (i) can be replaced by the following condition: (i′) ∀b ∈ b, if dd(b, a) > 0, then eventually dd(b, ai) > 0. from condition (i′), the topology t+ b of upper b-convergence is generated by all sets of the form {a ∈ c(x) : dd(a, b) > 0} (b ∈ b), called the upper b-proximal topology in the literature [19]. the topology t− b of lower b-convergence is not so transparent. in the case that b is the bornology of cofinally complete subsets, this was executed in [13]. here we show that the description obtained for t− bλ when λ is the measure of local compactness extends naturally to the case when λ is a general uniformly continuous nonnegative functional. our proof here is based on condition (ii′) rather than on condition (ii) as it was in the particular case addressed in [13] and seems simpler to us. to describe a set of generators for the topology, we employ notation used in [13]: if v is a nonempty open subset of x, put v − := {a ∈ c(x) : a∩v 6= ∅}, and if w is a family of nonempty open subsets of x, put w −− := {a ∈ c(x) : ∃ε > 0 ∀w ∈ w, ∃aw ∈ a with sε(aw ) ⊆ w }. note that for a nonempty open subset v, {v }−− = v −. theorem 6.6. let λ be a nonnegative uniformly continuous real-valued function on a metric space 〈x, d〉 . then the topology t− bλ of lower bλ-convergence on the closed subsets of x is generated by all sets of the form v − where v is a nonempty open subset of x plus all sets of the form w−− where w is a family of nonempty open sets with inf {λ(x) : x ∈ ∪w} > 0. proof. first suppose 〈ai〉i∈i is a net in c(x) that is lower bλ-convergent to a. suppose a ∈ v − where v is open. pick a ∈ a and ε > 0 with sε(a) ⊆ v . since {a} ⊆ a, applying condition (ii′) with b = {a} ∈ bλ gives eventually ai ∩ sε(a) 6= ∅, so eventually ai ∩ v 6= ∅. next suppose a ∈ w −− where inf{λ(x) : x ∈ ∪w} = µ > 0. choose α > 0 such that ∀w ∈ w, ∃aw ∈ a with sα(aw ) ⊆ w. since b = {aw : w ∈ w} ∈ bλ and b ⊆ a, by (ii ′) ∃i0 ∈ i ∀i � i0, b ⊆ s α 2 (ai). fix i � i0; ∀w ∈ w, s α 2 (aw ) ∩ ai 6= ∅, and we conclude ai ∈ {sα(aw ) : w ∈ w} −− ⊆ w−−. for the converse, suppose 〈ai〉i∈i converges to a in the topology with the prescribed set of generators. let b1 be a fixed λ-set with b1 ⊆ a and let ε > 0 be arbitrary. put b := cl(b1) ⊆ a; it suffices to show that eventually b ⊆ sε(ai). we first consider two extreme cases for b: (1) b is compact, and (2) inf {λ(b) : b ∈ b} = µ > 0. 156 g. beer and m. segura in case (1), by compactness ∃{b1, b2, b3, . . . , bn} in b such that b ⊆ ∪ n j=1s ε2 (bj ). as {b1, b2, b3, . . . , bn} ⊆ a, ∀j ≤ n we have a ∈ s ε 2 (bj ) −, and so eventually ai ∈ ∩ n j=1s ε2 (bj ) −. it follows that {b1, b2, b3, . . . , bn} ⊆ s ε 2 (ai) eventually and so b ⊆ sε(ai) eventually. in case (2) by uniform continuity there exists δ ∈ 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[38] w. waterhouse, on uc spaces, amer. math. monthly 72 (1965), 634–635. received october 2008 accepted january 2009 gerald beer (gbeer@cslanet.calstatela.edu) department of mathematics, california state university los angeles, 5151 state university drive, los angeles, california 90032, usa manuel segura (msegura4@calstatela.edu) department of mathematics, california state university los angeles, 5151 state university drive, los angeles, california 90032, usa kohpraagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 177-189 fuzzy uniformities on function spaces j. k. kohli and a. r. prasannan ∗ abstract. we study several uniformities on a function space and show that the fuzzy topology associated with the fuzzy uniformity of uniform convergence is jointly fuzzy continuous on cf (x, y ) , the collection of all fuzzy continuous functions from a fuzzy topological space x into a fuzzy uniform space y . we define fuzzy uniformity of uniform convergence on starplus-compacta and show that its corresponding fuzzy topology is the starplus-compact open fuzzy topology. moreover, we introduce the notion of fuzzy equicontinuity and fuzzy uniform equicontinuity on fuzzy subsets of a function space and study their properties. 2000 ams classification: 03e72, 04a72, 54a40, 54c35, 54d30, 54e15. keywords: starplus-compact open fuzzy topology, fuzzy uniformity of uniform convergence, jointly fuzzy continuous fuzzy topology, fuzzy uniformity of uniform convergence on starplus-compacta, fuzzy equicontinuity, fuzzy uniform equicontinuity. 1. introduction the notion of a uniform space was introduced by andre weil [18] in 1937. the first systematic exposition of the theory of uniform spaces was given by bourbaki [4] in 1940. weil elaborated the topology associated with a uniformity and proved that a topological space is uniformizable if and only if it is completely regular. he extended the notion of uniform continuity and uniform isomorphisms to the framework of uniform spaces and obtained the uniform space version of alexandroff-uryshon metrization theorem that a uniform space is metrizable if and only if its uniformity has a countable base. the concept of function space was evolved around the close of the nineteenth century and the study of function spaces began with the work of ascoli [3], arzelà [2] and hadamard [8]. the uniformity of pointwise convergence and uniform ∗corresponding author. 178 j. k. kohli and a. r. prasannan convergence were first defined and studied by fox [7]. the term function space is introduced much earlier in connection with questions of a topological nature about sets of functions. the study of topologies on function spaces is an active area of research and, besides their multifaceted applications, forms a well-established and sound body of knowledge. the study of useful fuzzy topologies and uniformities on function spaces, besides their intrinsic interest, is important from applications viewpoint. the first effort in this direction was made by peng [17] in 1984. subsequently, alderton [1] studied the problem from categorical viewpoint and utilized the well-developed theory of cartesian closedness of monotopological categories to the fuzzy topologies on a function space. burton [6] defined analogues of the uniformities of pointwise convergence and uniform convergence and obtained conditions for completeness and compactness of fuzzy subsets of a function space. jäger considered fuzzy uniform convergence and equicontinuity in [9]. in [13], we defined three different fuzzy topologies on a function space, which are analogues of the topology of pointwise convergence, compact-open topology and the topology of joint continuity in general topology. in this paper, we elaborate on the pointwise fuzzy uniformity, fuzzy uniformity of uniform convergence and fuzzy uniformity of uniform convergence on starplus compacta and their associated fuzzy topologies on a function space. it turns out that the fuzzy topology associated with the fuzzy uniformity of uniform convergence is jointly continuous; and that the fuzzy topology of uniform convergence on starplus compacta is the starplus-compact open fuzzy topology [13]. further, we study the notion of fuzzy equicontinuity on fuzzy subsets of a family of functions from a fts/fuzzy uniform space to a fuzzy uniform space. 2. preliminaries throughout the paper the closed unit interval [0, 1] will be denoted by i. the symbols and i0 and i1 will stand for the intervals (0, 1] and [0, 1), respectively. definition 2.1. for a fuzzyy set µ in x, the set µα = {x ∈ x : µ(x) > α} and µα = {x ∈ x : µ(x) ≥ α} are called the strong α -level set of µ and the weak α-level set of µ , respectively. the set {x ∈ x : µ(x) > 0} is called the support of µ and is denoted by suppµ . definition 2.2 ([13]). a fuzzy set µ in a fts (x, τ ) is said to be starpluscompact if µα is compact in (x, iα(τ )) for each α ∈ i1. the fts (x, τ ) is said to be starplus-compact if (x, iα(τ )) is compact for each α ∈ i1. let x be a non-empty set and let (x, τ ) be a fuzzy topological space. let y x denote the collection of all functions from x into y and let ℑ be a nonempty subset of y x . definition 2.3 ([12]). for each x ∈ x, let the map ex : ℑ −→ y be defined by ex(f ) = f (x). we call ex the evaluation map at x ∈ x . the initial fuzzy topology on ℑ generated by the collection of maps {ex : x ∈ x} is called the fuzzy function spaces 179 pointwise fuzzy topology on ℑ and is denoted by τp. the pair (ℑ, τp) is called the pointwise fuzzy function space. the pointwise fuzzy topology on ℑ concides with the subspace fuzzy topolology it inherits as a subspace of the product fuzzy topology on y x . . definition 2.4 ([13]). let (x, τ ) and (y, σ) be fts and let ℑ be a nonempty subset of y x . for each starplus-compact fuzzy set κ in x and each open fuzzy set µ in y , define a fuzzy set κµ on ℑ by κµ = ∧ x∈suppκ e−1x (µ). the collection of all fuzzy sets κµ, where κ is a starplus-compact fuzzy set in x and µ is an open fuzzy set in y , forms a subbase for a fuzzy topology τ + ∗c on ℑ called the starplus-compact open fuzzy topology on ℑ. the pair (ℑ, τ + ∗c ) is referred to as a starplus-compact open fuzzy function space. proposition 2.5 ([13]). the starplus-compact open fuzzy topology on ℑ is stronger than the pointwise fuzzy topology τp on ℑ. theorem 2.6 ([13]). let (x, τx ) be a topologically generated fts and let (y, τy ) be a fts. then a fuzzy topology τ is a starplus-compact open fuzzy topology on ℑ if and only if iα(τ ) = t α c for each α ∈ i1, where t α c denotes the compact open topology on ℑ and x is endowed with the topology i0(τx ) and y is equipped with the topology iα(τy ). definition 2.7 ([13]). a fuzzy topology τ on ℑ such that the map φ : ℑ×x −→ y defined by φ(f, x) = f (x) is fuzzy continuous, where ℑ × x is endowed with the product fuzzy topology, is called a jointly fuzzy continuous fuzzy topology on ℑ. theorem 2.8 ([13]). the fuzzy topology of joint fuzzy continuity is a good extension. definition 2.9 ([15]). a subset f ⊂ ix is called a prefilter if and only if f 6= φ, and i) for all µ, ν ∈ f we have µ ∧ ν ∈ f. ii) if µ ≥ ν and ν ∈ f, then µ ∈ f. iii) 0 6∈ f. definition 2.10 ([15]). a subset b ⊂ ix is a base for a prefilter if and only if b 6= φ, and i) for all µ, ν ∈ b there exists a ξ ∈ b such that ξ ≤ µ ∧ ν. ii) 0 6∈ b. definition 2.11 ([15]). a prefilter generated by a prefilter base b is denoted as 〈b〉 and 〈b〉 = {µ ∈ ix : there exists a ν ∈ b such that µ ≥ ν}. if b is a prefilter base, then b̂ = {sup ǫ∈i0 (βǫ − ǫ) : (βǫ)ǫ∈i0 ∈ b i0}. proposition 2.12 ([15]). if b is a prefilter base, then 〈b̂〉= 〈b〉. we shall denote by b̃ the prefilter 〈b̂〉 = 〈b〉. 180 j. k. kohli and a. r. prasannan definition 2.13 ([15]). a prefilter f is called prime if µ ∨ ν ∈ f implies µ ∈ f or ν ∈ f. definition 2.14 ([15]). if f is a prefilter on x, then we define the following: p (f) = {g : g is a prime prefilter and f ⊂ g} and pm(f) = {g : g ∈ p (f) and g is minimal}. definition 2.15 ([15]). for a prefilter f the characteristic of f is defined by c(f) = inf ν∈f sup ν. for a prefilter f the lower characteristic of f is defined by c(f) = inf g∈pm(f) c(g). if f is a prime, then c(f) = c(f). definition 2.16 ([16]). if x is a set, µ ∈ ix and ν ∈ ix×x , then the section of ν over µ is defined by ν〈µ〉(x) = sup y∈x µ(y) ∧ ν(y, x) for all x ∈ x. definition 2.17 ([16]). if µ, ν ∈ ix×x , then the composition µ◦ν is defined by µ ◦ ν(x, y) = sup z∈x ν(x, z) ∧ µ(z, y) for all (x, y) ∈ x × x. definition 2.18 ([16]). if ν ∈ ix×x , then its symmetric sν ∈ i x×x is defined by sν(x, y) = ν(y, x) for all (x, y) ∈ x × x. throughout this paper we follow the terminology and notions of a fuzzy uniformity as defined by lowen [16]. definition 2.19 ([16]). a fuzzy uniformity on x is a subset u ⊂ ix×x , which satisfies the following conditions: i) u is a prefilter. ii) û = u, i.e., for every family (νǫ)ǫ∈i0 ∈ u i0 =⇒ sup ǫ∈i0 (νǫ − ǫ) ∈ u. iii) for all ν ∈ u and for all x ∈ x, ν(x, x) = 1. iv) for all ν ∈ u, sν ∈ u. v) for all ν ∈ u and for all ǫ ∈ i0 there exists νǫ ∈ u such that νǫ◦νǫ−ǫ ≤ ν. the pair (x, u) is called a fuzzy uniform space. definition 2.20 ([16]). a subset b ⊂ ix×x is called a base for a fuzzy uniformity if and only if the following conditions hold: i) b is a prefilter basis. ii) for all β ∈ b and for all x ∈ x, β(x, x) = 1. iii) for all β ∈ b and for all ǫ ∈ i0, there exists βǫ ∈ b such that βǫ − ǫ ≤ sβ. iv) for all β ∈ b and for all ǫ ∈ i0, there exists βǫ ∈ b such that βǫ ◦ βǫ − ǫ ≤ β. definition 2.21. if u is a fuzzy uniformity on x then b ⊂ ix×x is a basis for u iff b is a prefilter basis and b̃ = u. proposition 2.22. if u is a fuzzy uniformity on x, then the family of symmetric fuzzy entourages su = {ν ∈ u : sν = ν} is a basis for u. fuzzy function spaces 181 definition 2.23. let (x, τ ) be a fts. then the closure µ of a fuzzy set µ of x is defined as µ = inf{ν : µ ≤ ν, 1 − ν ∈ τ}. definition 2.24 ([14]). a fuzzy closure operator on a fts x is a map ¯: ix −→ ix which satisfies the following conditions: i) α = α, for all α ∈ i. ii) µ ≥ µ, for all µ ∈ ix . iii) µ ∨ ν = µ ∨ ν, for all µ, ν ∈ ix . iv) µ = µ, for all µ ∈ ix . proposition 2.25 ([16]). let (x, u ) be a fuzzy uniform space. the map¯: ix −→ ix defined by µ= inf ν∈u ν〈µ〉 is a fuzzy closure operator. definition 2.26 ([5]). if f is a prefilter on (x, u), then adhf and lim f are fuzzy sets in x and is defined by adhf = inf ν∈f ν and lim f = inf g∈pm(f) adhg. if µ ∈ ix , we say that f is u-convergent in µ iff c(f) 6 sup µ ∧ lim f and f is u-convergent iff c(f) 6 sup lim f . definition 2.27. let (x, u) be a fuzzy uniform space and f is a prefilter on x. then f is u-cauchy iff c(f) 6 inf σ∈u sup inf g∈pm(f) inf ν∈g σ〈ν〉. definition 2.28. let (x, u) and (y, u1) be fuzzy uniform spaces. then a map f : x −→ y is said to be fuzzy uniformly continuous if for each ν ∈ u1, (f × f )−1(ν) ∈ u. proposition 2.29. if (x, u) and (y, u1) are fuzzy uniform spaces, b and b1 are basis for u and u1, respectively and f : x −→ y , then f is fuzzy uniformly continuous iff for all β1 ∈ b1 and for all ε ∈ i0 there exist β ∈ b such that β − ε ≤ (f × f )−1(β1). theorem 2.30. if (x, u) and (y, u1) are fuzzy uniform space and f : x −→ y is uniformly continuous, then f is fuzzy continuous. throughout this paper uniformity on a nonempty set x is denoted by u and a fuzzy uniformity by u. we denote the topology associated with a uniformity u by t (u) and the fuzzy topology associated with a fuzzy uniformity u by τ (u) [16], where τ (u) is the fuzzy topology whose fuzzy closure operator is defined in proposition 2.25. definition 2.31 ([16]). let x be a non empty set and let {fj : x → (yj , uj ), j ∈ j} be a family of functions from x into the family of fuzzy uniform spaces {(yj , uj), j ∈ j}. then the coarsest fuzzy uniformity u on x making each fj, j ∈ j is fuzzy uniformly continuous is called the initial fuzzy uniformity on x and is denoted by sup j∈j (fj × fj) −1(uj ). let unif denote the category of uniform spaces and uniformly continuous functions and funif denote the category of fuzzy uniform spaces and fuzzy uniformly continuous functions. then the functors ωu : unif → funif and 182 j. k. kohli and a. r. prasannan iu : funif → unif are defined as follows and are the uniform analogues of the functors ω and i introduced in [14]. for each (x, u) ∈ unif, ωu(x, u) = (x, ωu(u)), where ωu(u) = {µ ∈ ix×x : µ−1(α, 1] ∈ u, ∀α ∈ i1} and for each (x, u) ∈ funif, iu(x, u) = (x, iu(u)), where iu(u) = {µ −1(α, 1] : µ ∈ u, α ∈ i1} . theorem 2.32 ([16]). if u is a uniformity on x and u a fuzzy uniformity on x. then, 1) ωu(u) is a fuzzy uniformity on x. 2) iu(u) is a uniformity on x. 3) iu(ωu(u)) = u. 4) ωu(iu(u)) is the coarsest fuzzy uniformity generated by a uniformity and is finer than u. we denote ωu(iu(u)) by u. 5) τ (ωu(u))= ω(t (u)). 6) t (iu(u)) = i(τ (u). 7) τ (u) = τ (u). we shall call a notion in funif a good extension of a notion in unif if it reduces to the standard notion in case of the fuzzy uniformity u = ω(u). definition 2.33 ([10]). if u is a fuzzy uniformity on a nonempty set x, then its α-level uniformity, for 0 ≤ α ≤ 1 is defined by iu,α(u) = {µ β ∈ 2x×x : µ ∈ u, β ∈ [0, 1 − α)} . the functor iu,α : funif −→ unif is the uniform analogue of the functor iα discussed in [16]. also, iu = iu,0 . theorem 2.34 ([10]). the topology t (iu,α(u)) on x, induced by the α-level uniformity iu,α(u) of the fuzzy uniform space (x, u), coincides with the α-level topology iα(τ (u)). 3. fuzzy uniformities on function spaces let x be a non-empty set and let (y, u) be a fuzzy uniform space. let y x denote the collection of maps from x into y . let ℑ be a nonempty subset of y x . in this section we study the pointwise fuzzy uniformity and fuzzy uniformity of uniform convergence and their associated fuzzy topologies. it is shown that the fuzzy topology associated with the fuzzy uniformity of uniform convergence is jointly fuzzy continuous on cf (x, y ). definition 3.1. the initial fuzzy uniformity up on ℑ generated by the collection of maps {ex : x ∈ x} is called the pointwise fuzzy uniformity or fuzzy uniformity of pointwise convergence on ℑ. the pair (ℑ, up) is called the pointwise fuzzy uniform space. remark 3.2. the above definition of pointwise fuzzy uniformity on ℑ coincides with the definitions of pointwise fuzzy uniformity given in [6, 9]. fuzzy function spaces 183 the following theorem of lowen [16] reflects upon the relationship that exists between the initial fuzzy topology and the fuzzy topology generated by the initial fuzzy uniformity. theorem 3.3 ([16]). let x be a set and let (y, uj ), j ∈ j be a family of fuzzy uniform spaces. if {fj : x −→ yj , j ∈ j} is a family of functions, then τ (sup j∈j (fj × fj ) −1(uj )) = sup j∈j f −1 j (τ (uj )). in view of definitions 2.3, 3.1 and theorem 3.3, the following result is immediate. theorem 3.4. the fuzzy topology associated with the fuzzy uniformity up of pointwise fuzzy uniformity is the pointwise fuzzy topology τ p . theorem 3.5. let x be a non-empty set and let (y, u) be a uniform space. let up denote the pointwise uniformity on ℑ and let up be the pointwise fuzzy uniformity on ℑ, where y is endowed with the fuzzy uniformity ωu(u). then ωu(up) = up. proof. let n∧ i=1 (ex × ex) −1(νi) be a basic element for the fuzzy uniformity up, where y is endowed with the fuzzy uniformity ωu(u). since νi ∈ ωu(u), ν α i ∈ u for each α ∈ i1 and hence [ n⋂ i=1 (ex × ex) −1(ναi )] ∈ up. this shows that [ n∧ i=1 (ex × ex) −1(νi] α ∈ up and so n∧ i=1 (ex × ex) −1(νi) ∈ ωu(up). thus we have ωu(up) ⊇ up. the proof of the opposite inclusion, ωu(up) ⊆ up, is similar to the one given above. � proposition 3.6. let x be a set and let (y, u) be a fuzzy uniform space. let up be the fuzzy uniformity of pointwise convergence on ℑ . then for each α ∈ i1, the α-level uniformity iu,α(up) is the uniformity of pointwise convergence on ℑ with respect to iu,α(u) on y . proof. let (ex × ex) −1(ν) be a subbasic element in up. then for each α ∈ i1, [(ex × ex) −1(ν)]β = {(f, g) ∈ ℑ × ℑ : ν(f (x), g(x)) > β}, β ∈ [0, 1 − α) = {(f, g) ∈ ℑ × ℑ : (f (x), g(x)) ∈ νβ} = (ex × ex) −1(νβ ), which is a subbasic element in the pointwise uniformity on ℑ, where y is endowed with the uniformity iu,α(u). � theorem 3.7 ([6]). let f be a prefilter in the pointwise fuzzy uniform space (y x , up). then f is a cauchy prefilter if and only if for each x ∈ x, ex(f) is a cauchy prefilter in (y, uy ). 184 j. k. kohli and a. r. prasannan fuzzy uniformity of uniform convergence. in this subsection we study the notion of fuzzy uniformity of uniform convergence on a function space ℑ ⊂ y x (which was initiated by burton [6]), where x is a nonempty set and (y, u) is a fuzzy uniform space. definition 3.8 ([6]). for each ν ∈ u, the fuzzy set wν = ∧ x∈x (ex × ex) −1(ν) in ℑ × ℑ, where ex : ℑ −→ y is the evaluation map, is defined by, wν (f, g) = ∧ x∈x ν(f (x), g(x)). let b be the collection of all wν , where ν varies over u. proposition 3.9 ([6]). the collection b is a base for a fuzzy uniformity on ℑ. example 3.10. if µ ∈ iℑ, then the section of wν over µ is defined by wν〈µ〉(f ) = ∧ x∈x ν〈ex(µ)〉(f (x)) = inf x∈x {sup g∈ℑ (ex(µ)(g(x)) ∧ ν(g(x), f (x)))} for each f ∈ ℑ. definition 3.11 ([6]). the fuzzy uniformity uu generated by b is called the fuzzy uniformity of uniform convergence. the fuzzy topology associated with uu is called the fuzzy topology of uniform convergence and it is denoted by τu. in the following results we give a short description of the concepts that burton [6] uses relative to the fuzzy uniform convergence. theorem 3.12 ([6]). let f be a prefilter in y x with c(f) = c(f) . then f is uu-cauchy, α ≤ c, lim up (f)(f ) ≥ α =⇒ lim uu (f)(f ) ≥ α . theorem 3.13 ([6]). if (x, u) is complete then (y x , uu) is complete. corollary 3.14 ([6]). if µ : y x −→ i is up-closed and (y, u) is complete, then µ is uu-complete. theorem 3.15 ([6]). if µ is a closed fuzzy set in (y x , up) and for all x ∈ x, ex(µ) is complete in (x, u) , then the fuzzy set µ is complete in (y x , uu). proposition 3.16. if wνi , 1 ≤ i ≤ n are members of uu. then the following hold: i) n∧ i=1 wνi = w ( n v i=1 νi) and ii) w α ν = wνα , for α ∈ i1. fuzzy function spaces 185 proof. i) for each i, wνi = ∧ x∈x (ex × ex) −1(νi). hence, n∧ i=1 wνi = n∧ i=1 { ∧ x∈x (ex × ex) −1(νi)} = ∧ x∈x [(ex × ex) −1( n∧ i=1 νi) = w ( n v i=1 νi) . ii) for each α ∈ i1, wνα = ⋂ x∈x {(f, g) : (f (x), g(x)) ∈ να} = ⋂ x∈x {(f, g) : (ex × ex)(f, g) ∈ ν α} = ⋂ x∈x {(f, g) : (f, g) ∈ (ex × ex) −1(να)} = ⋂ x∈x {(f, g) : (f, g) ∈ [(ex × ex) −1(ν)]α} = ⋂ x∈x [(ex × ex) −1(ν)]α = [ ∧ x∈x (ex × ex) −1(ν)]α = (wν ) α. � the following theorem shows that for each α ∈ i1, the α-level uniformity of the fuzzy uniformity of uniform convergence uu on the function space ℑ coincides with the uniformity of uniform convergence on ℑ when y is endowed with the uniformity iu,α(u). theorem 3.17. for each α ∈ i1, iu,α(uu) is the uniformity of uniform convergence on ℑ, where y is endowed with the uniformity iu,α(u). proof. let wν be a basic element for the fuzzy uniformity of uniform convergence uu, where ν ∈ u. then w α ν is a basic element for the uniformity iu,α(uu). by proposition 3.16(ii), wνα = w α ν and the fact that wνα is a basic element for the uniformity of uniform convergence on ℑ, where y is endowed with the uniformity iu,α(uu), the theorem follows. � jäger in [9] showed that the notion of fuzzy uniformity of uniform convergence is a good extension. 186 j. k. kohli and a. r. prasannan theorem 3.18. let c(x, y ) denote the collection of all continuous functions from a topological space x into a uniform space y and let uu denote the uniformity of uniform convergence on c(x, y ). then the fuzzy topology associated with the fuzzy uniformity ωu(uu) is jointly fuzzy continuous. proof. the uniform topology t (uu) on c(x, y ) is jointly continuous. since in view of theorem 2.8 the fuzzy topology of joint fuzzy continuity is a good extension, the fuzzy topology ω(t (uu)) is jointly fuzzy continuous. by theorem 2.32, τ (ωu(uu)) = ω(t (uu)). this shows that the fuzzy topology τ (ωu(uu)) associated with the fuzzy uniformity ωu(uu) is jointly fuzzy continuous. � corollary 3.19. let x be a topological space and let y be a uniform space with the uniformity uy . let cf (x, y ) denote the collection of all fuzzy continuous maps from the topologically generated fts ω(x) into the topologically generated fts ω(y ). then the fuzzy topology of uniform convergence on cf (x, y ) is jointly fuzzy continuous. proof. since x and y are topologically generated fts, cf (x, y ) and c(x, y ) are equal as sets. since the uniformity of uniform convergence is a good extension, the fuzzy uniformity of uniform convergence uu is same as ωu(uu), where uu is the uniformity of uniform convergence on c(x, y ). hence in view of theorem 3.18, the fuzzy topology associated with the fuzzy uniformity of uniform convergence uu is jointly fuzzy continuous. � definition 3.20. let x be a fuzzy topological space and let (y, u) be a fuzzy uniform space. let s be the collection of all starplus-compact fuzzy sets in x. for each κ ∈ s and ν ∈ u, define a fuzzy set w(κ, ν) : ℑ × ℑ −→ i by w(κ, ν)(f, g) = ∧ x∈suppκ (ex × ex) −1(ν)(f, g) = ∧ x∈suppκ ν(f (x), g(x)). then the collection of all fuzzy sets {w(κ, ν) : κ ∈ s, ν ∈ u} is a base for a fuzzy uniformity on ℑ and is called the fuzzy uniformity of uniform convergence on starplus-compacta. theorem 3.21. let ℑ be the set of all fuzzy continuous maps from a topologically generated fts (x, τx ) into a fuzzy uniform space (y, u). then the fuzzy topology of fuzzy uniform convergence on starplus-compacta is the starpluscompact open fuzzy topology. proof. let u+ ∗c be the fuzzy uniformity of uniform convergence on starpluscompacta. since x is a topologically generated fts, a subset k of x is compact in (x, i0(τx )) if and only if χk is starplus-compact in (x, τx ). hence iuα(u + ∗c ) = uuc , where uuc denotes the uniformity of uniform convergence on compacta, where x is endowed with the topology i0(τx ) and y is equipped with the uniformity iuα(u). so by [11, theorem 7.11], t (iuα(u + ∗c )) is the compact open topology for each α ∈ i1. by theorem 2.6, t (iuα(u + ∗c )) = iα(τ + ∗c ) fuzzy function spaces 187 for each α ∈ i1. again by theorem 2.34, iα(τ (u + ∗c )) = t (iuα(u + ∗c )) = iα(τ + ∗c ) for each α ∈ i1. this completes the proof. � 4. fuzzy equicontinuity in this section we introduce the notion of fuzzy equicontinuity and fuzzy uniform equicontinuity on fuzzy subsets of ℑ and obtain results, which show that if a fuzzy subset κ of y x is fuzzy equicontinuous (respectively, fuzzy uniformly equicontinuous), then each f ∈ suppκ is fuzzy continuous (respectively, fuzzy uniformly continuous). definition 4.1. let (x, τ ) be a fts and let (y, u) be a fuzzy uniform space. then a fuzzy subset κ of y x is said to be fuzzy equicontinuous at a fuzzy point xα of x if for each µ ∈ u, there is a τ -neighbourhood η of xα such that f (η) ≤ µ〈f (xα)〉 for each f ∈ suppκ. we say that the fuzzy subset κ of y x is fuzzy equicontinuous on a fuzzy subset θ of x if κ is fuzzy equicontinuous at each fuzzy point xα in θ. definition 4.2. a fuzzy subset κ of y x , where (x, ux ) and (y, uy ) are fuzzy uniform spaces, is said to be fuzzy uniformly equicontinuous if ∧ f∈suppκ (f × f )−1(µ) ∈ ux for each µ ∈ uy . the following is a characterization of fuzzy equicontinuity. theorem 4.3. a fuzzy subset κ of y x is fuzzy equicontinuous if and only if for each µ ∈ uy , the fuzzy set ∧ f∈suppκ f −1(µ〈f (xα)〉) is a neighbourhood of xα. proof. suppose that a fuzzy subset κ of y x is fuzzy equicontinuous at a fuzzy point xα of x. then for each µ ∈ uy , there is a τ -neighbourhood η of xα such that f (η) ≤ µ〈f (xα)〉 for each f ∈ suppκ. so η ≤ ∧ f∈suppκ f −1(µ〈f (xα)〉) and hence ∧ f∈suppκ f −1(µ〈f (xα)〉) is a neighbourhood of xα. conversely, suppose that ∧ f∈suppκ f −1(µ〈f (xα)〉) is a neighbourhood of the fuzzy point xα for each µ ∈ uy . let η = ∧ f∈suppκ f −1(µ〈f (xα)〉). then clearly g(η) ≤ µ〈g(xα)〉 for each g ∈ suppκ and so κ is fuzzy equicontinuous at the fuzzy point xα. � theorem 4.4. if a fuzzy set κ in y x is fuzzy equicontinuous, then suppκ is equicontinuous, where x is endowed with the topology i0(τx ) and y is equipped with the uniformity iu,α(uy ). proof. let κ be fuzzy equicontinuous. then for each fuzzy point xλ in x and for each µ ∈ uy , there exist a neighbourhood η of xλ in x such that f (η) ≤ µ〈f (xλ)〉 for each f ∈ suppκ. hence in particular, f (η) ≤ µ〈f (x1)〉 with λ = 1. this shows that f (ηβ ) ⊂ µβ〈f (x)〉, β ∈ [0, 1 − α). thus for each 188 j. k. kohli and a. r. prasannan x ∈ x and µβ ∈ iu,α(uy ), there exist a neighbourhood η β of x such that f (ηβ) ⊂ µβ〈f (x)〉, for each f ∈ suppκ. hence suppκ is equicontinuous. � theorem 4.5. if a fuzzy subset κ in y x is fuzzy equicontinuous, then κα is equicontinuous for each α ∈ i1, where y is equipped with the uniformity iu,α(uy ). first we prove the following lemma. lemma 4.6. if ν ≤ κ and κ is fuzzy equicontinuous then ν is also fuzzy equicontinuous. proof. since ν ≤ κ, suppν ⊂ suppκ. hence ∧ f∈suppκ f −1(µ〈f (xα)〉) ≤ ∧ f∈suppν f −1(µ〈f (xα)〉) and so the result follows. � proof of theorem 4.5. since κα ⊂ suppκ for each α ∈ i1 and since κ is fuzzy equicontinuous then by theorem 4.4 and lemma 4.6, κα is equicontinuous.2. theorem 4.7. if a fuzzy subset κ of y x is fuzzy equicontinuous, then each f ∈ suppκ is fuzzy continuous. proof. since ∧ f∈suppκ f −1(µ〈f (xα)〉) ≤ f −1(µ〈f (xα)〉) for each fuzzy point xα of x, f −1(µ〈f (xα)〉) is a neighbourhood of xα for each µ ∈ uy and so f is fuzzy continuous at each xα. hence f is fuzzy continuous on x. � theorem 4.8. if κ is fuzzy uniformly equicontinuous, then each f ∈ suppκ is fuzzy uniformly continuous. proof. suppose that κ is fuzzy uniformly equicontinuous. then for each µ ∈ uy , ∧ f∈suppκ (f × f )−1(µ) ∈ ux . this implies that for each µ ∈ uy , (f × f )−1(µ) ∈ ux . hence f is fuzzy uniformly continuous. � acknowledgements. the authors are thankful to the referee for helpful suggestions. references [1] i. w. alderton, function spaces in fuzzy topology, fuzzy sets and systems 32 (1989), 115-124. [2] c. arzelà, funzioni di linee, atti della reale accademia dei lincei, rendiconti 5 (1889), 342–348. [3] g. ascoli, le curve limite di una varieta data di curve, mem. acad. lincei (3) 18(1883), 512–586. fuzzy function spaces 189 [4] n. bourbaki, topologe géneralé, ch.i et ii, paris, 1940. [5] m. h. burton, cauchy filters and prefilters, fuzzy sets and systems 54 (1993), 317–331. [6] m. h. burton, the fuzzy uniformisation of function spaces, quaestiones math. 20 (1997), 283–290. [7] r. h. fox,on topologies for function spaces, bull. amer. math. soc. 51 (1945), 429-432. [8] j. hadamard, sur certaines applications possibles de la théorie des ensembles, verhandlungen des ersten internationalen der mathematiker-kongresses, b.g.teubner, leipzig (1898). [9] g. jäger, fuzzy uniform convergence and equicontinuity, fuzzy sets and systems 109 (2000), 187–198. [10] a. kandil, khaled a. hashem and nehad n. morsi, a level topologies criterian for lowen fuzzy uniformity, fuzzy sets and systems 62 (1994), 211–226. [11] j. l. kelley, general topology, van nosterand, new york, 1955. [12] j. k. kohli and a. r. prasannan, fuzzy topologies on function spaces, fuzzy sets and systems 116 (3)(2000), 415–420. [13] j. k. kohli and a. r. prasannan, starplus-compactness in fuzzy topological spaces and starplus-compact open fuzzy topologies on function spaces, j. math. anal. appl. 254 (2001), 87–100. [14] r. lowen, fuzzy topological spaces and fuzzy compactness, j. math. anal. appl. 56 (1976), 621–633. [15] r. lowen, convergence in fuzzy topological spaces, gen. top. appl. 10 (1979), 147–160. [16] r. lowen, fuzzy uniform spaces, j. math. anal. appl. 82 (1981), 370–385. [17] y. w. peng, topological structure of a fuzzy function spacethe pointwise convergent topology and compact open topology, kexue tongbao (english ed.) 29 (1984), 289–292. [18] a.weil, sur les espaces à structure uniform et sur la topologie générale, act.sci.et ind. 551, hermann, paris (1937). received april 2005 accepted march 2006 j. k. kohli department of mathematics, hindu college, university of delhi, delhi 110 007, india. a. r. prasannan (arprasannan@yahoo.co.in) department of mathematics, maharaja agrasen college, university of delhi, pocket iv, mayur vihar i, delhi 110 091, india. () @ applied general topology c© universidad politécnica de valencia volume 12, no. 2, 2011 pp. 163-173 on a type of generalized open sets bishwambhar roy ∗ abstract in this paper, a new class of sets called µ-generalized closed (briefly µg-closed) sets in generalized topological spaces are introduced and studied. the class of all µg-closed sets is strictly larger than the class of all µ-closed sets (in the sense of á. császár). furthermore, g-closed sets (in the sense of n. levine) is a special type of µg-closed sets in a topological space. some of their properties are investigated here. finally, some characterizations of µ-regular and µ-normal spaces have been given. 2010 msc: 54d10, 54d15, 54c08, 54c10. keywords: µ-open set, µg-closed set, µ-regular space, µ-normal space. 1. introduction in the past few years, different forms of open sets have been studied. recently, a significant contribution to the theory of generalized open sets, was extended by a. császár. especially, the author defined some basic operators on generalized topological spaces. it is observed that a large number of papers is devoted to the study of generalized open like sets of a topological space, containing the class of open sets and possessing properties more or less similar to those of open sets. for example, [22] has introduced g-open sets, [4, 30, 2] sg-open sets, [25] pg-open sets, [27, 28] gα-open sets, [13] δg∗-open sets, [21, 17] bg-open sets. owing to the fact that corresponding definitions have many features in common, it is quite natural to conjecture that they can be obtained and a considerable part of the properties of generalized open sets can be deduced from suitable more general definitions. the purpose of this paper is to point ∗the author acknowledges the financial support from ugc, new delhi. 164 b. roy out extremely elementary character of the proofs and to get many unknown results by special choice of the generalized topology. we recall some notions defined in [9]. let x be a non-empty set, expx denotes the power set of x. we call a class µ ⊆ expx a generalized topology [9], (briefly, gt) if ∅ ∈ µ and union of elements of µ belongs to µ. a set x, with a gt µ on it is said to be a generalized topological space (briefly, gts) and is denoted by (x, µ). the θ-closure [35] (resp. δ-closure [35]) of a subset a of a topological space (x, τ) is defined by {x ∈ x : clu ∩ a 6= ∅ for all u ∈ τ with x ∈ u} (resp. {x ∈ x : a ∩ u 6= ∅ for all regular open sets u containing x}, where a subset a is called regular open if a = int(cl(a))). a is called δ-closed [35] (resp. θ-closed [35]) if a = clδa (resp. a = clθa) and the complement of a δ-closed set (resp. θ-closed) set is known as a δ-open (resp. θopen) set. a subset a of a topological space (x, τ) is called preopen [29] (resp. semiopen [23], δ-preopen [33], δ-semiopen [32], α-open [27], β-open [1], b-open [21]) if a ⊆ int(cl(a)) (resp. a ⊆ cl(int(a)),a ⊆ int(clδa), a ⊆ cl(intδa), a ⊆ int(cl(int(a))), a ⊆ cl(int(cla)), a ⊆ cl(int(a)) ∪ int(cl(a))). we note that for any topological space (x, τ), the collection of all open sets denoted by τ (preopen sets denoted by po(x), semi-open sets denoted by so(x), δopen sets denoted by δo(x), δ-preopen sets denoted by δ-po(x), δ-semiopen sets denoted by δ-so(x), α-open sets denoted by αo(x), β-open sets denoted by βo(x), θ-open sets denoted by θo(x), b-open sets denoted by bo(x) or γo(x)) forms a gt. for a gts (x, µ), the elements of µ are called µ-open sets and the complement of µ-open sets are called µ-closed sets. for a ⊆ x, we denote by cµ(a) the intersection of all µ-closed sets containing a, i.e., the smallest µ-closed set containing a; and by iµ(a) the union of all µ-open sets contained in a, i.e., the largest µ-open set contained in a (see [9, 10]). obviously in a topological space (x, τ), if one takes τ as the gt, then cµ becomes equivalent to the usual closure operator. similarly, cµ becomes pcl, scl, clδ, pclδ, sclδ, clα, clβ, bcl if µ stands for po(x) (resp. so(x), δo(x), δ-po(x), δ-so(x), αo(x), βo(x), bo(x) or γo(x)). it is easy to observe that iµ and cµ are idempotent and monotonic, where γ : expx → expx is said to be idempotent iff a ⊆ b ⊆ x implies γ(γ(a)) = γ(a) and monotonic iff γ(a) ⊆ γ(b). it is also well known from [10, 11] that if g is a gt on x and a ⊆ x, x ∈ x, then x ∈ cµ(a) iff x ∈ m ∈ µ ⇒ m ∩ a 6= ∅ and cµ(x \ a) = x \ iµ(a). in this paper we introduce the concepts of µg-closed sets and µg-open sets. it is shown that many results in previous papers can be considered as special cases of our results. 2. properties of µg-closed sets definition 2.1. let (x, µ) be a gts. then a subset a of x is called a µgeneralized closed set (or in short, µg-closed set) iff cµ(a) ⊆ u whenever a ⊆ u where u is µ-open in x. the complement of a µg-closed set is called a µg-open set. on a type of generalized open sets 165 remark 2.2. (i) if (x, τ) is a topological space, the definition of g-open set [22] (resp. sg-open set [4, 2], pg-open set [25], gα-open set [27], δg∗-open set [13], bg-open set [21] or γg-open set [17]) can be obtained by taking µ = τ (resp. so(x), po(x), αo(x), δo(x), γo(x)). (ii) every µ-open set in a gts (x, µ) is µg-open. in fact, if a is a µ-open set in (x, µ), then x \ a is a µ-closed set. let x \ a ⊆ u ∈ µ. then cµ(x \ a) = x \ a ⊆ u. thus x \ a is a µg-closed set and hence a is a µg-open set. the converse of remark 2.2(ii) is not true as seen from the next example : example 2.3. let x = {a, b, c} and µ = {∅, x, {a}, {b, c}, {a, c}}. then (x, µ) is a gts. it is easy to verify that {c} is µg-open in (x, µ) but not µ-open. the next two examples show that the union (intersection) of two µg-open sets is not in general µg-open. example 2.4. (a) let x = {a, b, c} and µ = {∅, x, {a}}. then (x, µ) is a gts. it can be shown that if a = {b} and b = {c}, then a and b are two µg-open sets but a ∪ b = {b, c} is not a µg-open set. (b) let x = {a, b, c, d} and µ = {∅, x, {a, b}, {a, c, d}, {a, b, d}, {b, c, d}}. then (x, µ) is a gts. it follows from remark 2.2(ii) that {a, b} and {a, c, d} are two µg-open sets but it is easy to check that their intersection {a} is not µg-open. theorem 2.5. a subset a of a gts (x, µ) is µg-closed iff cµ(a)\ a contains no non-empty µ-closed set. proof. let f be a µ-closed subset of cµ(a) \ a. then a ⊆ f c (where f c denotes as usual the complement of f). hence by µg-closedness of a, we have cµ(a) ⊆ f c or f ⊆ (cµ(a)) c. thus f ⊆ cµ(a) ∩ (cµ(a)) c = ∅, i.e., f = ∅. conversely, suppose that a ⊆ u where u is µ-open. if cµ(a) * u, then cµ(a) ∩ u c (6= ∅) is a µ-closed subset of cµ(a) \ a a contradiction. hence cµ(a) ⊆ u. � theorem 2.6. if a µg-closed subset a of a gts (x, µ) be such that cµ(a)\ a is µ-closed, then a is µ-closed. proof. let a be a µg-closed subset such that cµ(a) \ a is µ-closed. then cµ(a) \ a is a µ-closed subset of itself. then by theorem 2.5, cµ(a) \ a = ∅ and hence cµ(a) = a, showing a to be a µ-closed set. � that the converse is false follows from the following example. example 2.7. let x = {a, b, c} and µ = {∅, {a}, {a, b}}. then (x, µ) is a gts. it is easy to observe that {b, c} is µ-closed and hence a µg-closed set (by remark 2.2), but cµ(a) \ a = ∅, which is not µ-closed. 166 b. roy theorem 2.8. let a be a µg-closed set in a gts (x, µ) and a ⊆ b ⊆ cµ(a). then b is µg-closed. proof. let b ⊆ u, where u is µ-open in (x, µ). since a is µg-closed and a ⊆ u, cµ(a) ⊆ u. now, b ⊆ cµ(a) ⇒ cµ(b) ⊆ cµ(a). so cµ(b) ⊆ u. � theorem 2.9. in a gts (x, µ), µ = ω (the collection of all µ-closed sets) iff every subset of x is µg-closed. proof. suppose µ = ω and a (⊆ x) be such that a ⊆ u ∈ µ. then cµ(a) ⊆ cµ(u) = u and hence a is µg-closed. conversely, suppose that every subset of x is µg-closed. let u ∈ µ. then u ⊆ u and by µg-closedness of u, we have cµ(u) ⊆ u, i.e., u ∈ ω. thus µ ⊆ ω. now, if f ∈ ω then f c ∈ µ, so f c ∈ ω (as µ ⊆ ω), i.e., f ∈ µ. � theorem 2.10. a subset a of a gts (x, µ) is µg-open iff f ⊆ iµ(a), whenever f is µ-closed and f ⊆ a. proof. obvious and hence omitted. � theorem 2.11. a set a is µg-open in a gts (x, µ) iff u = x whenever u is µ-open and iµ(a) ∪ a c ⊆ u. proof. suppose u is µ-open and iµ(a) ∪ a c ⊆ u. now, uc ⊆ (iµ(a)) c ∩ a = cµ(x \ a) \ (x \ a). since u c is µ-closed and x \ a is µg-closed, by theorem 2.5, uc = ∅, i.e., u = x. conversely, let f be a µ-closed set and f ⊆ a. then by theorem 2.10, it is enough to show that f ⊆ iµ(a). now, iµ(a) ∪ a c ⊆ iµ(a) ∪ f c, where iµ(a) ∪ f c is µ-open. hence by the given condition, iµ(a) ∪ f c = x, i.e., f ⊆ iµ(a). � theorem 2.12. a subset a of a gts (x, µ) is µg-closed iff cµ(a) \ a is µg-open. proof. suppose a is µg-closed and f ⊆ cµ(a)\a, where f is a µ-closed subset of x. then by theorem 2.5, f = ∅ and hence f ⊆ iµ[cµ(a) \ a]. then by theorem 2.10, cµ(a) \ a is µg-open. conversely, suppose that a ⊆ u where u is µ-open. now, cµ(a) ∩ u c ⊆ cµ(a) ∩ a c = cµ(a) \ a. since cµ(a) ∩ u c is µ-closed and cµ(a) \ a is µg-open, cµ(a) ∩ u c = ∅ (by theorem 2.5). thus cµ(a) ⊆ u, i.e., a is µg-closed. � definition 2.13. a gts (x, µ) is said to be (i) µ-t0 [34] iff x, y ∈ x, x 6= y implies the existence of k ∈ µ containing precisely one of x and y. (ii) µ-t1 [34] iff x, y ∈ x, x 6= y implies the existence of k, k 1 ∈ µ such that x ∈ k, y 6∈ k and x 6∈ k1, y ∈ k1. (iii) µ-t1/2 iff every µg-closed set is µ-closed. on a type of generalized open sets 167 remark 2.14. a topological space (x, τ) is ti [16] (resp. semi-ti [4], pre-ti [25], α-ti [28], δ-ti [13], b-ti [21]) for i = 0, 1/2, 1 by taking µ = τ (resp. so(x), po(x), αo(x), δo(x), bo(x) or γo(x)). theorem 2.15. if a gts (x, µ) is µ-t1/2 then it is µ-t0. proof. suppose that (x, µ) is not a µ-t0 space. then there exist distinct points x and y in x such that cµ({x}) = cµ({y}). let a = cµ({x}) ∩ {x} c. we shall show that a is µg-closed but not µ-closed. suppose that a ⊆ v ∈ µ. we have to show that cµ(a) ⊆ v . thus it is enough to show that cµ({x}) ⊆ v (as a ⊆ cµ({x})). again, since cµ({x}) ∩ {x} c = a ⊆ v , we need only to show that x ∈ v . in fact, if x 6∈ v , then y ∈ cµ({x}) ⊆ v c (as v c is µ-closed). so y ∈ a ⊆ v c and hence y ∈ v ∩ v c a contradiction. if x ∈ u ∈ µ, then u ∩ a ⊇ {y} 6= ∅, and hence x ∈ cµ(a). clearly, x 6∈ a and thus a is not µ-closed. � example 2.16. let x = {a, b, c, d} and µ = {∅, x, {a, b}, {a, c, d}, {a, b, d}, {b, c, d}}. then (x, µ) is a gts. clearly, this gts is µ-t0 and it can be shown that the collection of all µg-open sets are {∅, x, {d}, {a, b}, {a, c, d}, {a, b, d}, {b, c, d}}. thus this space is not µ-t1/2. theorem 2.17. if a gts (x, µ) is µ-t1 then it is µ-t1/2. proof. suppose that a is a subset of x which is not µ-closed. take x ∈ cµ(a) \ a. then {x} ⊆ cµ(a) \ a and {x} is µ-closed (as (x, µ) is µ-t1). thus by theorem 2.5, a is not µg-closed. � example 2.18. let x = {a, b, c, d} and µ = {∅, x, {d}, {a, b}, {b, c, d}, {a, c, d}, {a, b, d}}. then (x, µ) is a gts. it is easy to verify that (x, µ) is µ-t1/2 but not µ-t1. definition 2.19. a gts (x, µ) is said to be µ-symmetric iff for each x, y ∈ x, x ∈ cµ({y}) ⇒ y ∈ cµ({x}). remark 2.20. it is easy to check that the above definition of a µ-symmetric space gt unifies the existing definitions of δ-symmetric space [8], (δ, p)-symmetric space [5], α-symmetric [6], δ-semi symmetric space [7] if (x, τ) is a topological space and µ = δo(x), δ-po(x), αo(x), δ-so(x) respectively. theorem 2.21. a gts (x, µ) is µ-symmetric iff {x} is µg-closed for each x ∈ x. proof. let {x} ⊆ u ∈ µ and (x, µ) be µ-symmetric but cµ({x}) * u. then cµ({x}) ∩ u c 6= ∅. let y ∈ cµ({x}) ∩ u c. then x ∈ cµ({y}) ⊆ u c ⇒ x 6∈ u a contradiction. conversely, let for each x ∈ x, {x} is µg-closed and x ∈ cµ({y}) ⊆ (cµ({x})) c (as {y} is µg-closed). thus x ∈ (cµ({x})) c a contradiction. � corollary 2.22. if a gts (x, µ) is µ-t1 then it is µ-symmetric. example 2.23. let x = {a, b} and µ = {∅, x}. then (x, µ) is a µsymmetric space which is not µ-t1. 168 b. roy theorem 2.24. a gts (x, µ) is µ-symmetric and µ-t0 iff (x, µ) is µ-t1. proof. if (x, µ) is µ-t1 then it is µ-symmetric (by corollary 2.22) and µ-t0 (by definition 2.13). conversely, let (x, µ) be µ-symmetric and µ-t0. we shall show that (x, µ) is µ-t1. let x, y ∈ x and x 6= y. then by µ-t0-ness of (x, µ), there exists u ∈ µ such that x ∈ u ⊆ {y}c. then x 6∈ cµ({y}) and hence y 6∈ cµ({x}). thus there exists v ∈ µ such that y ∈ v and x 6∈ v . thus (x, µ) is µ-t1. � theorem 2.25. if (x, µ) is µ-symmetric, then (x, µ) is µ-t0 iff (x, µ) is µ-t1/2 iff (x, µ) is µ-t1. proof. follows from theorem 2.24 and the fact that µ-t1 ⇒ µ-t1/2 ⇒ µ-t0. � 3. preservation of µg-closed sets definition 3.1. let (x, µ 1 ) and (y, µ 2 ) be two gts’s. a mapping f : (x, µ 1 ) → (y, µ 2 ) is said to be (i) (µ 1 , µ 2 ) continuous [9] iff f−1(g2) ∈ µ1 for each g2 ∈ µ2; (ii) (µ 1 , µ 2 )-closed iff for any µ 1 -closed subset a of x, f(a) is µ 2 -closed in y . theorem 3.2. let (x, µ 1 ) and (y, µ 2 ) be two gts’s and f : (x, µ 1 ) → (y, µ 2 ) be (µ 1 , µ 2 )-continuous and (µ 1 , µ 2 )-closed mapping. if a is µ 1 g-closed in x then f(a) is µ 2 g-closed in y . proof. let f(a) ⊆ g2, where g2 is a µ2-open set in y . then a ⊆ f −1(g2), where f−1(g2) is a µ1-open set in x. thus by µ1g-closedness of a, cµ1 (a) ⊆ f−1(g2). thus f(cµ1(a)) ⊆ g2 and f(cµ 1 (a)) is µ 2 -closed in y . it thus follows that cµ 2 (f(a)) ⊆ cµ 2 (f(cµ1(a))) = f(cµ 1 (a)) ⊆ g2. thus f(a) is µ2g-closed in y . � theorem 3.3. let (x, µ 1 ) and (y, µ 2 ) be two gts’s and f : (x, µ 1 ) → (y, µ 2 ) be a (µ1, µ2)-continuous and (µ1, µ2)-closed mapping. if b is a µ2g-closed set in y , then f−1(b) is µ 1 g-closed in x. proof. suppose that b is a µ 2 g-closed set in y and f−1(b) ⊆ g1, where g1 is µ 1 -open in x. we shall show that cµ1(f −1(b)) ⊆ g1. now f[cµ 1 (f−1(b)) ∩ gc1] ⊆ cµ 2 (b) \ b and by theorem 2.5, f[cµ 1 (f−1(b)) ∩ gc1] = ∅. thus cµ 1 (f−1(b)) ∩ gc1 = ∅. thus cµ 1 (f−1(b)) ⊆ g1 and hence f −1(b) is µ 1 gclosed in x. � next two examples show that (µ 1 , µ 2 )-continuity and (µ 1 , µ 2 )-closedness in both of the above theorems are essential. example 3.4. let x = {a, b, c, d}, µ 1 = {∅, x, {a, b}, {c, d}, {a, c, d}, {a, b, d}} and µ 2 = {∅, x, {a, b}, {c, d}, {a, c, d}}. then (x, µ 1 ) and (x, µ 2 ) are two gts’s. consider the identity mapping f : (x, µ 1 ) → (x, µ 2 ). it is easy to see on a type of generalized open sets 169 that f is a (µ 1 , µ 2 )-continuous mapping which is not (µ 1 , µ 2 )-closed. the families of µ 1 g-open and µ 2 g-open sets are respectively {∅, x, {a}, {d}, {c, d}, {a, d}, {a, b}, {a, c, d}, {a, b, d}} and {∅, x, {a}, {c}, {d}, {c, d}, {a, d}, {a, b}, {a, c}, {a, c, d}, {a, b, d}, {a, b, c}}. we note that {d} is gµ 2 -closed but f−1({d}) is not gµ 1 -closed. again, the identity map h defined by h : (x, µ 2 ) → (x, µ 1 ) is not a (µ2, µ1)continuous mapping but it is (µ 2 , µ 1 )-closed. clearly, {d} is a µ 2 g-closed set but h({d}) is not a µ 1 g-closed set. example 3.5. let x = {a, b, c, d} , µ 1 = {∅, x, {a, b}, {c, d}, {a, c, d}, {a, b, d}} and µ 2 = {∅, x, {a, b}, {a, b, d}, {a, c, d}}. then (x, µ 1 ) and (x, µ 2 ) are gts’s. now, consider the identity map f : (x, µ 1 ) → (x, µ 2 ). it is easy to verify that f is a (µ 1 , µ 2 )-continuous mapping which is not (µ 1 , µ 2 )-closed. the family of gµ 1 -open and gµ 2 -open sets are respectively {∅, x, {a}, {d}, {c, d}, {a, d}, {a, b}, {a, c, d}, {a, b, d}} and {∅, x, {a}, {d}, {a, b}, {a, d}, {a, b, d}, {a, c, d}}. we note that {a, b} is µ 1 g-closed but f({a, b}) is not µ 2 g-closed. again, consider the identity map h : (x, µ 2 ) → (x, µ 1 ). then, clearly h is a (µ 2 , µ 1 )-closed map which is not (µ 2 , µ 1 )-continuous. clearly, {a, b} is µ 1 g-closed but h−1({a, b}) is not a µ 2 g-closed set. 4. properties of µ-regular and µ-normal spaces definition 4.1. a gts (x, µ) is said to be µ-regular if for each µ-closed set f of x not containing x, there exist disjoint µ-open set u and v such that x ∈ u and f ⊆ v . remark 4.2. regular space, pre-regular space, semi-regular space, β-regular space, α-regular space are defined and studied in [16, 31, 15, 19, 20] respectively. the above definition gives a unified version of all these definitions if µ takes the role of τ, po(x), so(x), βo(x), αo(x) respectively. theorem 4.3. for a gts (x, µ) the followings are equivalent: (a) x is µ-regular. (b) for each x ∈ x and each u ∈ µ containing x, there exists v ∈ µ such that x ∈ v ⊆ cµ(v ) ⊆ u. (c) for each µ-closed set f of x, ∩{cµ(v ) : f ⊆ v ∈ µ} = f. (d) for each subset a of x and each u ∈ µ with a ∩ u 6= ∅, there exists a v ∈ µ such that a ∩ v 6= ∅ and cµ(v ) ⊆ u. (e) for each non-empty subset a of x and each µ-closed subset f of x with a ∩ f = ∅, there exist u, v ∈ µ such that a ∩ v 6= ∅, f ⊆ w and w ∩ v = ∅. (f) for each µ-closed set f with x 6∈ f there exist u ∈ µ and a µg-open set v such that x ∈ u, f ⊆ v and u ∩ v = ∅. (g) for each a ⊆ x and each µ-closed set f with a ∩ f = ∅ there exist a u ∈ µ and a µg-open set v such that a ∩ u 6= ∅, f ⊆ v and u ∩ v = ∅. (h) for each µ-closed set f of x, f = ∩{cµ(v ) : f ⊆ v, v is µg-open}. 170 b. roy proof. (a) ⇒ (b) : let u be a µ-open set containing x. then x 6∈ x\u, where x \ u is µ-closed. then by (a) there exist g, v ∈ µ such that x \ u ⊆ g and x ∈ v and g ∩ v = ∅. thus v ⊆ x \ g and so x ∈ v ⊆ cµ(v ) ⊆ x \ g ⊆ u. (b) ⇒ (c) : let x \ f ∈ µ be such that x 6∈ f . then by (b) there exists u ∈ µ such that x ∈ u ⊆ cµ(u) ⊆ x \ f . so, f ⊆ x \ cµ(u) = v (say)∈ µ and u ∩ v = ∅. thus x 6∈ cµ(v ). hence f ⊇ ∩{cµ(v ) : f ⊆ v ∈ µ}. (c) ⇒ (d) : let u ∈ µ with x ∈ u ∩ a. then x 6∈ x \ u and hence by (c) there exists a µ-open set w such that x \ u ⊆ w and x 6∈ cµ(w). we put v = x \ cµ(w), which is a µ-open set containing x and hence a ∩ v 6= ∅ (as x ∈ a ∩ v ). now v ⊆ x \ w and so cµ(v ) ⊆ x \ w ⊆ u. (d) ⇒ (e) : let f be a µ-closed set as in the hypothesis of (e). then x \ f is a µ-open set and (x \ f) ∩ a 6= ∅. then there exists v ∈ µ such that a ∩ v 6= ∅ and cµ(v ) ⊆ x \ f . if we put w = x \ cµ(v ), then f ⊆ w and w ∩ v = ∅. (e) ⇒ (a) : let f be a µ-closed set not containing x. then by (e), there exist w, v ∈ µ such that f ⊆ w and x ∈ v and w ∩ v = ∅. (a) ⇒ (f) : obvious as every µ-open set is µg-open (by remark 2.2). (f) ⇒ (g) : let f be a µ-closed set such that a ∩ f = ∅ for any subset a of x. thus for a ∈ a, a 6∈ f and hence by (f), there exist a u ∈ µ and a µg-open set v such that a ∈ u, f ⊆ v and u ∩ v = ∅. so a ∩ u 6= ∅. (g) ⇒ (a) : let x 6∈ f , where f is µ-closed. since {x}∩f = ∅, by (g) there exist a u ∈ µ and a µg-open set w such that x ∈ u, f ⊆ w and u ∩ w = ∅. now put v = iµ(w). then f ⊆ v (by theorem 2.10) and u ∩ v = ∅. (c) ⇒ (h) : we have f ⊆ ∩{cµ(v ) : f ⊆ v and v is µg-open} ⊆ ∩{cµ(v ) : f ⊆ v and v is µ-open} = f . (h) ⇒ (a) : let f be a µ-closed set in x not containing x. then by (h) there exists a µg-open set w such that f ⊆ w and x ∈ x \ cµ(w). since f is µ-closed and w is µg-open, f ⊆ iµ(w) (by theorem 2.10). take v = iµ(w). then f ⊆ v , x ∈ x\cµ(v ) = u (say) (as (x\f)∩v = ∅) and u ∩v = ∅. � definition 4.4. a gts (x, µ) is µ-normal [12] if for any pair of disjoint µclosed subsets a and b of x, there exist disjoint µ-open sets u and v such that a ⊆ u and b ⊆ v . remark 4.5. normal space, pre-normal space, semi-normal space, α-normal space, β-normal space, γ-normal space are defined and studied in [16, 31, 2, on a type of generalized open sets 171 20, 19, 17] respectively. the above definition gives a unified version of all these definitions if µ takes the role of τ, po(x), so(x), αo(x), βo(x) respectively. theorem 4.6. for a gts (x, µ) the followings are equivalent: (a) x is µ-normal; (b) for any pair of disjoint µ-closed sets a and b, there exist disjoint µg-open sets u and v such that a ⊆ u and b ⊆ v ; (c) for every µ-closed set a and µ-open set b containing a, there exists a µg-open set u such that a ⊆ u ⊆ cµ(u) ⊆ b; (d) for every µ-closed set a and every µg-open set b containing a, there exists a µ-open set u such that a ⊆ u ⊆ cµ(u) ⊆ iµ(b); (e) for every µg-closed set a and every µ-open set b containing a, there exists a µ-open set u such that a ⊆ cµ(a) ⊆ u ⊆ cµ(u) ⊆ b. proof. (a) ⇒ (b) : let a and b be two disjoint µ-closed subsets of x. then by µ-normality of x, there exist disjoint µ-open sets u and v such that a ⊆ u and b ⊆ v . then u and v are µg-open by remark 2.2. (b) ⇒ (c) : let a be a µ-closed set and b be a µ-open set containing a. then a and bc are two disjoint µ-closed sets in x. then by (b), there exist disjoint µg-open sets u and v such that a ⊆ u and bc ⊆ v . thus a ⊆ u ⊆ x \ v ⊆ b. again, since b is µ-open and x \ v is µg-closed, cµ(x \ v ) ⊆ b. hence a ⊆ u ⊆ cµ(u) ⊆ b. (c) ⇒ (d) : let a be a µ-closed subset of x and b be a µg-open set containing a. since b is a µg-open set containing a and a is µ-closed, by theorem 2.10, a ⊆ iµ(b). thus by (c) there exists a µg-open set u such that a ⊆ u ⊆ cµ(u) ⊆ iµ(b). (d) ⇒ (e) : let a be a µg-closed set and b be a µ-open set in x containing a. a ⊆ b implies cµ(a) ⊆ b, where cµ(a) is µ-closed and b is µg-open (as b is µ-open). then by (d), there exists a µ-open set u such that a ⊆ cµ(a) ⊆ u ⊆ cµ(u) ⊆ iµ(b). thus a ⊆ cµ(a) ⊆ u ⊆ cµ(u) ⊆ b. (e) ⇒ (a) : let a and b be two disjoint µ-closed subsets of x. then a is µg-closed and a ⊆ x \ b, where x \ b is µ-open. thus by (e), there exists a µ-open set u such that a ⊆ cµ(a) ⊆ u ⊆ cµ(u) ⊆ x \ b. thus a ⊆ u, b ⊆ x \ cµ(u) and u ∩ (x \ cµ(u)) = ∅. hence x is µ-normal. � remark 4.7. 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(received december 2010 – accepted july 2011) bishwambhar roy (bishwambhar roy@yahoo.co.in) department of mathematics, women’s christian college, 6 greek church row, kolkata-700026, india. on a type of generalized open sets. by b. roy @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 301–316 bounded point evaluations for cyclic hilbert space operators a. bourhim ∗ dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. in this talk, to be given at a conference at seconda università degli studi di napoli in september 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator t on a hilbert space h and shall answer some questions due to l. r. williams. 2000 ams classification: primary 47a10; secondary 47b20. keywords: cyclic operator, bounded point evaluation, single-valued extension property, bishop’s property (β). 1. introduction. throughout this paper, l(h) will denote the algebra of all linear bounded operators on an infinite–dimensional separable complex hilbert space h. let t ∈ l(h) be a cyclic operator on h with cyclic vector x ∈ h i.e., the linear subspace {p(t)x : p polynomial} is dense in h. a complex number λ ∈ c is said to be a bounded point evaluation for t if there is a positive constant m such that for every polynomial p, |p(λ)| ≤ m‖p(t)x‖; equivalently, if λ induces a continuous linear functional on h which maps p(t)x to p(λ) for every polynomial p. therefore, it follows from the riesz representation theorem that a complex number λ ∈ c is a bounded point evaluation for t if and only if there is a unique vector k(λ) ∈h such that (1.1) p(λ) = 〈p(t)x,k(λ)〉 for every polynomial p. the set of all bounded point evaluations for t will be denoted by b(t). a point λ ∈ b(t) is called an analytic bounded point evaluation for t if there is ∗this research is supported in part by the abdus salam ictp, trieste, italy. 302 a. bourhim an open neighborhood o of λ contained in b(t) such that for every y ∈h, the complex function ŷ defined on b(t) by ŷ(λ) := 〈y,k(λ)〉, is analytic on o. the set of all analytic bounded point evaluations for t will be denoted by ba(t). an operator t ∈l(h) is said to be subnormal if it has a normal extension, i.e., if there is a normal operator n on a hilbert space k, containing h, such that h is a closed invariant subspace of n and the restriction n|h coincides with t . the operator t is said to be hyponormal if ‖t∗x‖ ≤ ‖tx‖ for every x ∈h, where t∗ denotes the adjoint of t . note that every subnormal operator is hyponormal, with the converse false (see [8] and also example 5.6). using bran’s theorem [5] and the maximum modulus principle for analytic functions, tavan t. trent proved in [20] that for every cyclic subnormal operator t ∈ l(h), we have ba(t) = γ(t)\σap(t). here, σap(t) denotes the approximate point spectrum of t , that is the set of complex numbers λ for which there is a sequence (xn)n of elements of the unit sphere of h such that lim n→+∞ ‖(t − λ)xn‖ = 0, and γ(t) denotes the compression spectrum of t , that is the set of complex numbers λ such that the range of (t − λ) is not dense in h. more informations about bounded point evaluations for cyclic subnormal operators can be found in [8]. in [21], l. r. williams followed trent’s method to shown that (1.2) γ(t)\σap(t) ⊂ ba(t) for every cyclic operator t ∈l(h) and posed the following question. question 1.1. let t ∈l(h) be a cyclic operator. is ba(t) = γ(t)\σap(t)? note that, in general, the basic spectral properties of subnormal operators remain valid for hyponormal operators. thus we pose the following weaker question. question 1.2. is ba(t) = γ(t)\σap(t) for every cyclic hyponormal operator t ∈l(h)? in this paper, we shall explain more about bounded point evaluations for cyclic hilbert space operators from the point of view of local spectral theory and shall answer the above questions. in section 2, we give a complete description of the set of analytic bounded point evaluations for arbitrary cyclic operators and derive some consequences from it. in section 3, we give a necessary and sufficient condition for unilateral weighted shift operators to satisfy trent’s result, and exhibit some operators which provide a negative answer to question 1.1. in section 4, we generalize a result of l. yang which allows us to give a positive answer to question 1.2. as a corollary, we get that two quasisimilar cyclic hyponormal operators have equal approximate point spectra; this result is a generalization of theorem 4 of [16]. in section 5, we show that if t ∈l(h) is a cyclic operator for which the span of the eigenvectors of t∗ associated with bounded point evaluations for . . . 303 a connected component of ba(t) is dense in h, then t is without eigenvalues and has a connected spectrum. some related examples are given. before going further, we need to introduce some notations and recall some basic notions concerning local spectral theory; we refer to the monographs [7] and [13] for further informations. for an operator t ∈ l(h), we denote as usual by σ(t) := {λ ∈ c : t −λ is not invertible}, ρ(t) := c\σ(t), σp(t) := {λ ∈ c : t − λ is not injective}, ker t , and rant the spectrum, the resolvent set, the point spectrum, the kernel, and the range of t , respectively. for a subset m of h, we use cl(m), and ∨ m to denote the closure of m, and the closed linear subspace generated by m, respectively. for a subset f of c, we use f, and fr(f) to denote the complex conjugates of the points in f, and the boundary of f, respectively. for an open subset u of c, we let o(u,h) denote the space of analytic h−valued functions on u. it is a fréchet space when equipped with the topology of uniform convergence on compact subsets of u and the space h may be viewed as simply the constants in o(u,h). one says that an operator t ∈ l(h) possesses bishop’s property (β) if for each open subset u of c, the multiplication operator tu : o(u,h) −→o(u,h), f 7−→ (t −z)f is injective with closed range. m. putinar [14] has shown that hyponormal operators have bishop’s property (β). recall that t is said to have the single– valued extension property provided that, for every open subset u of c, the only analytic h−valued solution f of the equation (t −λ)f(λ) = 0, (λ ∈ u), is the identically zero function f ≡ 0 on u. equivalently, if for every open subset u of c, the mapping tu is injective. a localized version of this property dates back to j. k. finch [10] and can be defined as follows (see [1], and [10]). an operator t ∈ l(h) is said to have the single–valued extension property at a point λ ∈ c if for every open disk u centered at λ, the mapping tu is injective. let <(t) := { λ ∈ c : t does not have the single-valued extension property at λ } . obviously, <(t) is an open subset contained in σp(t) and is empty precisely when t has the single–valued extension property. the local resolvent set of t at a vector y ∈ h, denoted by ρ t (y), is the union of all open subsets u of c for which y ∈ rantu . the local spectrum of t at y is σt (y) := c\ρt (y); it is a closed subset of σ(t). in the sequel, t ∈ l(h) will be a cyclic operator with cyclic vector x ∈ h; and for λ ∈ b(t), k(λ) will denote the vector of h as defined in (1.1). 2. description of b(t) and ba(t). the proofs of proposition 2.1 and lemma 2.2 are similar to the ones for the cyclic subnormal operators (see [2], and [8]); we include them for completeness. a complete description of b(t) is given by the following result. 304 a. bourhim proposition 2.1. let λ ∈ c; the following statements are equivalent. (i) λ ∈ b(t). (ii) λ ∈ γ(t). (iii) ker ( (t −λ)∗ ) is one dimensional. proof. first, note that if (t − λ)∗u = 0 for some u ∈ h, then for every polynomial p, we have (2.3) 〈p(t)x,u〉 = p(λ)〈x,u〉. next, we mention that it suffices to prove the implications (ii)⇒(i) and (i)⇒(iii), since the implication (iii)⇒(ii) can be deduced trivially from the fact that γ(t) = σp(t∗). let λ ∈ b(t); it is clear that k(λ) 6= 0 since 〈x,k(λ)〉 = 1. on the other hand, for every polynomial p, we have 0 = 〈(t −λ)p(t)x,k(λ)〉 = 〈p(t)x, (t −λ)∗k(λ)〉. since x is a cyclic vector for t , we have (t −λ)∗k(λ) = 0. hence, λ ∈ σp(t∗). now, let u ∈h be such that (t −λ)∗u = 0. it follows then from equation (2.3) that u = 〈x , u〉k(λ). therefore, (i)⇒(iii). let λ ∈ γ(t). then there is a non-zero vector u ∈h such that (t−λ)∗u = 0. since x is a cyclic vector of t , it follows from equation (2.3) that 〈x,u〉 6= 0. hence, p(λ) = 〈p(t)x, u 〈x,u〉〉 for every polynomial p. therefore, λ ∈ b(t) and k(λ) = u 〈x,u〉 , which proves that (ii)⇒(i). � lemma 2.2. an open subset o of c is contained in ba(t) if and only if it is contained in b(t) and the function λ 7−→ ‖k(λ)‖ is bounded on compact subsets of o. proof. assume that o ⊂ ba(t) and let k be a compact subset of o. for every y ∈h, the function ŷ is analytic on o; in particular, sup λ∈k |〈y,k(λ)〉| < +∞. so it follows from the uniform boundedness principle that sup λ∈k ‖k(λ)‖ < +∞. conversely, suppose that o ⊂ b(t) and the function λ 7−→ ‖k(λ)‖ is bounded on compact subsets of o. let y ∈ h; then there is a sequence of polynomials (pn)n such that lim n→+∞ ‖pn(t)x−y‖ = 0. and so, for every compact subset k of o, it follows from the cauchy-schwartz inequality that, sup λ∈k |pn(λ) − ŷ(λ)| ≤ sup λ∈k ‖k(λ)‖‖pn(t)x−y‖. hence, the function ŷ is analytic on o. therefore, o ⊂ ba(t). � the following gives a complete description of ba(t). bounded point evaluations for . . . 305 theorem 2.3. ba(t) = <(t∗). proof. suppose that λ ∈ <(t∗). then there is a non-zero analytic h–valued function φ on some open disk v centered at λ such that (t −µ)φ(µ) = 0 for all µ ∈v. using the fact that a non-zero analytic h–valued function has isolated zeros, one can assume that the function φ has no zero in v. hence, v ⊂ σp(t∗) = b(t). as before, it follows from (2.3) that k(µ) = φ(µ) 〈x,φ(µ)〉 for every µ ∈v. therefore, the function k : v → h is continuous; in particular, the function µ 7−→‖k(µ)‖ is bounded on compact subsets of v. by lemma 2.2, v ⊂ ba(t). thus <(t∗) ⊂ ba(t). conversely, set o = ba(t) and consider the following h–valued function φ defined on o by φ(λ) := k(λ), λ ∈ o. first, we show that φ is analytic on o. indeed, for every y ∈h and for every λ0 ∈ o, we have lim λ→λ0 〈φ(λ),y〉−〈φ(λ0),y〉 λ−λ0 = lim λ→λ0 〈k(λ),y〉−〈k(λ0),y〉 λ−λ0 = lim λ→λ0 ŷ(λ) − ŷ(λ0) λ−λ0 = ŷ′(λ0). hence, for every y ∈ h, the function λ 7−→ 〈φ(λ),y〉 is differentiable on o; therefore, φ is analytic on o. on the other hand, φ has no zeros on o and satisfies the following equation (t∗ −λ)φ(λ) = 0 for every λ ∈ o. this gives o = ba(t) ⊂<(t∗), and the proof is complete. � corollary 2.4. the following holds: ba(t) = {λ ∈ γ(t) : σ t∗−λ (k(λ)) = ∅} = {λ ∈ γ(t) : σ t∗ (k(λ)) = ∅}. proof. since, for every λ ∈ ba(t), λ is a simple eigenvalue of t∗ with corresponding eigenvector k(λ), the proof follows by combining theorem 2.3 and theorem 1.9 of [1]. � remark 2.5. (i) note that proposition 2.1 and proposition 2.3 show that both b(t) and ba(t) are independent of the choice of cyclic vector for t (see proposition 1.4 of [21]). (ii) using theorem 2.3 and theorem 2.6 of [1], one can easily prove (1.2). 306 a. bourhim 3. resolution of question 1.1. the weighted shift operators have proven to be an interesting rich collection of operators providing examples and counterexamples to illustrate many properties of operators. the allen shields’s excellent survey [18] contains their basic facts and properties concerning their spectral theory (see also [2]). throughout this section, s will denote a unilateral weighted shift operator on h with a positive bounded weight sequence (ωn)n≥0, that is sen = ωnen+1, n ≥ 0, where (en)n≥0 is an orthonormal basis of h. let (βn)n≥0 be the following sequence given by: βn =   ω0...ωn−1 if n > 0 1 if n = 0 set r1(s) = lim n→∞ [ inf k≥0 βn+k βk ] 1 n , r2(s) = lim inf n→∞ [ βn ] 1 n and r(s) = lim n→∞ [ sup k≥0 βn+k βk ] 1 n ; and note that, r1(s) ≤ r2(s) ≤ r(s) ≤‖s‖. the following gives a necessary and sufficient condition for the weighted shift s to answer affirmatively question 1.1. theorem 3.1. the following are equivalent. (i) ba(s) = γ(s)\σap(s). (ii) r1(s) = r2(s). proof. since σp(s∗) ⊂{λ ∈ c : |λ| ≤ r2(s)} (see theorem 9 of [18]), we have <(s∗) ⊂ o := {λ ∈ c : |λ| < r2(s)}. conversely, consider the following non-zero analytic h–valued function φ defined on o by φ(λ) := +∞∑ n=0 λn βn en. it is easy to see that (s∗ −λ)φ(λ) = 0 for every λ ∈ o. hence, o = {λ ∈ c : |λ| < r2(s)}⊂<(s∗). therefore, <(s∗) = {λ ∈ c : |λ| < r2(s)}; by theorem 2.3, (3.4) ba(s) = {λ ∈ c : |λ| < r2(s)}. on the other hand, by [18, theorems 4 and 6], the spectrum and the approximate point spectrum of s are given, respectively, by σ(s) = {λ ∈ c : |λ| ≤ r(s)} and σap(s) = {λ ∈ c : r1(s) ≤ |λ| ≤ r(s)}. bounded point evaluations for . . . 307 hence, (3.5) γ(s)\σap(s) = σ(s)\σap(s) = {λ ∈ c : |λ| < r1(s)}. and so the proof follows from (3.4) and (3.5). � now, to give a counterexample to question 1.1, we only need to produce a weight sequence (ωn)n≥0 for which the corresponding weighted shift s satisfies r1(s) < r2(s). example 3.2. let (ck)k≥0 be the sequence of successive disjoint segments covering the set of non-negative integers n such that each segment ck contains k2 + k elements. let r > 1 be a real number and let k ∈ n; for every n ∈ ck we set, ωn =   r if n lies in the first k2 terms of ck 1 otherwise hence, the unilateral weighted shift s corresponding to the weight (ωn)n≥0 is bounded and satisfies ‖s‖ = r and r1(s) = 1. on the other hand for every n ≥ 3, there is k(n) ≥ 2 such that n ∈ ck(n). hence, r k(n)−1∑ s=1 s2 ≤ βn and n ≤ k(n)∑ s=1 (s2 + s). therefore, r (2k(n)−1)(k(n)−1) 2(k(n)+1)(k(n)+2) ≤ [ βn ] 1 n . since lim n→+∞ k(n) = +∞, it follows that, r ≤ lim inf n→+∞ [ βn ] 1 n . we deduce that r1(s) = 1 and r2(s) = ‖s‖ = r. thus, γ(s)\σap(s) = {λ ∈ c : |λ| < 1}$ ba(s) = {λ ∈ c : |λ| < r}. the original idea of this construction is due to w. c. ridge [17]. for other example see [3], where the authors constructed a unilateral weighted shift s for which γ(s)\σap(s) = ∅ and ba(s) = {λ ∈ c : |λ| < 1}. 4. resolution of question 1.2. if t possesses bishop’s property (β), then we obtain the following. theorem 4.1. if t possesses bishop’s property (β), then the following are equivalent. (i) ba(t) = γ(t)\σap(t). (ii) ba(t) ∩σp(t) = ∅. 308 a. bourhim proof. if ba(t) = γ(t)\σap(t) then it is clear that ba(t) ∩ σp(t) = ∅ since σp(t) ⊂ σap(t). conversely, suppose that ba(t) ∩ σp(t) = ∅. since γ(t)\σap(t) ⊂ ba(t) ⊂ γ(t), it suffices to prove that ba(t) ∩ σap(t) = ∅. suppose that there is λ ∈ ba(t) ∩σap(t). it then follows that ran(t −λ) is not closed. let y ∈ cl ( ran(t −λ) ) \ran(t −λ); then y 6∈ ran(t −λ) and 〈y,k(λ)〉 = 0. therefore, there is a sequence of polynomials (pn)n vanishing at λ such that lim n→+∞ ‖pn(t)x−y‖ = 0. define on u := ba(t) the following analytic h–valued functions f and fn by f(µ) = y− ŷ(µ)x and fn(µ) = pn(t)x−pn(µ)x, n ≥ 0. since f(λ) = y 6∈ ran(t −λ), then f 6∈ ran(tu ). on the other hand, it is easy to see that fn ∈ ran(tu ) for every n ≥ 0. now, let k be a compact subset of u; we have sup µ∈k ‖fn(µ) −f(µ)‖ ≤ ‖pn(t)x−y‖ + sup µ∈k ‖ [ pn(µ) − ŷ(µ) ] x‖ ≤ ‖pn(t)x−y‖ + ‖x‖ sup µ∈k |pn(µ) − ŷ(µ)| ≤ [ 1 + ‖x‖ sup µ∈k ‖k(µ)‖ ] ‖pn(t)x−y‖. therefore, fn −→ f in o(u,h). thus ran(tu ) is not closed which contradicts the fact that t possess bishop’s property (β). the proof is complete. � the following is immediate (see theorem 3.1 of [23]). corollary 4.2. suppose that t possesses bishop’s property (β). if σp(t) = ∅, then ba(t) = γ(t)\σap(t). the following gives a positive answer to question 1.2. theorem 4.3. if t is hyponormal, then ba(t) = γ(t)\σap(t). proof. since t is hyponormal, then for every λ ∈ c, we have t −λ is hyponormal i.e., (4.6) ‖(t −λ)∗y‖≤‖(t −λ)y‖, ∀y ∈h. now, let λ ∈ ba(t) and suppose that there is y ∈ h such that ty = λy. for every µ ∈ ba(t), we have, λŷ(µ) = 〈ty , k(µ)〉 = 〈y , t∗k(µ)〉 = 〈y , µk(µ)〉 = µŷ(µ). hence, the analytic function ŷ is identically zero on ba(t). on the other hand, it follows from (4.6) and proposition 2.1 that y = αk(λ) for some α ∈ c. therefore, ŷ(λ) = α‖k(λ)‖2 = 0 gives y = 0. thus, ba(t) ∩ σp(t) = ∅. by theorem 4.1, the proof is complete. � bounded point evaluations for . . . 309 in [16], m. raphael has shown that two quasisimilar cyclic subnormal operators have equal approximate point spectra. the following generalizes m. raphael’s result to cyclic hyponormal operators. suppose that h1 and h2 are hilbert spaces. recall that two operators t1 ∈ l(h1) and t2 ∈ l(h2) are said to be quasisimilar if there exist two bounded linear transformations x : h1 → h2 and y : h2 → h1 injectives and having dense range such that xt1 = t2x and t1y = y t2. corollary 4.4. suppose that h1 and h2 are hilbert spaces, and let t1 ∈l(h1) and t2 ∈ l(h2). if t1 and t2 are quasisimilar cyclic hyponormal operators, then σap(t1) = σap(t2). proof. in view of theorem 1.5 of [21] and theorem 4.3, we have γ(t1)\σap(t1) = γ(t2)\σap(t2). hence, σ(t1)\σap(t1) = σ(t2)\σap(t2). since σ(t1) = σ(t2) (see theorem 2 of [6], and also corollary 2.2 of [23]), we deduce that σap(t1) = σap(t2). � remark 4.5. theorem 4.3 and corollary 4.4 can be extended with no extra effort to the class of operators satisfying the following. (i) t possesses bishop’s property (β). (ii) ker(t −λ) ⊂ ker(t −λ)∗ for every λ ∈ c. immediate other examples of operators satisfying (i) and (ii) are provided by m–hyponormal and p–hyponormal operators (see [14] and [9]). 5. examples and comments. in this section, we start by proving the following result that we will need throughout. proposition 5.1. if h = ∨ {k(λ) : λ ∈ ba(t)}, then σp(t) = ∅. moreover, if h = ∨ {k(λ) : λ ∈ g} for some connected component g of ba(t), then the following hold. (i) cl(g) ⊂ σ t (y) for every non-zero element y ∈h. (ii) for every y ∈h, σ t (y) is a connected set. (iii) σ(t) is a connected set. proof. suppose that h = ∨ {k(λ) : λ ∈ ba(t)}, and ty = λy for some λ ∈ c and y ∈h. for every µ ∈ ba(t)\{λ}, we have, λŷ(µ) = 〈ty,k(µ)〉 = 〈y,t∗k(µ)〉 = µŷ(µ). hence, the analytic function ŷ is identically zero on ba(t); this means that 〈y,k(λ)〉 = 0 for every λ ∈ ba(t). since h = ∨ {k(λ) : λ ∈ ba(t)}, we have 310 a. bourhim y = 0. thus σp(t) = ∅. now, suppose that h = ∨ {k(λ) : λ ∈ g} for some connected component g of ba(t). (i) suppose that there is an element y ∈h such that g∩ρ t (y) 6= ∅. then there is an analytic h–valued function f such that (t −λ)f(λ) = y for λ ∈ v, where v = g∩ρ t (y). hence, for every λ ∈ v , we have ŷ(λ) = 〈y,k(λ)〉 = 〈(t −λ)f(λ),k(λ)〉 = 〈f(λ), (t −λ)∗k(λ)〉 = 0. therefore, ŷ ≡ 0 on g. since h = ∨ {k(λ) : λ ∈ g}, it follows that y = 0. thus the first statement (i) holds. (ii) suppose that there is a non-zero element y ∈ h for which σ t (y) is disconnected, then σ t (y) = σ1 ∪ σ2, where σ1 and σ2 are two non-empty disjoint compact subsets of c. since σp(t) = ∅, we note that t has the single–valued extension property. and so, using the local version of riesz’s functional calculus, one shows that there are two non-zero elements y1 and y2 of h such that y = y1 + y2 and σt (yi) ⊂ σi, i = 1, 2 which contradicts (i). (iii) by (i), we have ⋂ y∈h\{0} σ t (y) 6= ∅, and σ t (y) is connected for every y ∈h\{0}. since t has the single–valued extension property, we have σ(t) =⋃ y∈h\{0} σ t (y) (see proposition 1.3.2 of [13]). the result follows. � corollary 5.2. suppose that t possesses bishop’s property (β). if h =∨ {k(λ) : λ ∈ ba(t)}, then ba(t) = γ(t)\σap(t). we shall also need the following result due to l. r. williams [21]. recall that t is said to be pure if t does not have a non-zero reducing subspace m for which t is normal when restricted to m. theorem 5.3. if t is a pure cyclic hyponormal operator so that σap(t) has zero planar lebesgue measure, then h = ∨ {k(λ) : λ ∈ ba(t)}. example 5.4. let d be the open unit disk in the complex plane c, and let h = l2a(d) be the bergman space, consisting of those analytic functions on d that are square integrable on d with respect to area measure. it is a hilbert space when equipped with the inner product given by the formula 〈f,g〉 = ∫ d f(z)g(z)dm(z), f, g ∈ l2a(d), bounded point evaluations for . . . 311 where dm denotes planar area measure, normalized so that d has total mass 1. the bergman operator s for d is the operator multiplication by z on l2a(d); i.e., (sf)(z) = zf(z) for f ∈ l2a(d) and z ∈ d. it is a pure cyclic subnormal operator, with cyclic vector the constant function 1, with σ(s) = cl(d) and σap(s) = {λ ∈ c : |λ| = 1} (see [8]). in particular, ba(s) = σ(s)\σap(s) = d 6= ∅. by theorem 5.3, we have l2a(d) = ∨ {k(λ) : λ ∈ ba(s)}. this fact can be proved without using theorem 5.3 and the vectors k(λ) can be given explicitely. to do this, fix λ ∈ d and consider the power series expansion (5.7) k(λ)(z) = +∞∑ n=0 anz n. fixing a non-negative integer n, and taking the polynomial p(z) = zn, we have λn = 〈p(s)1,k(λ)〉 = 〈zn,k(λ)〉 = ∫ d zn [ +∞∑ i=0 aizi ] dm(z) = π an n + 1 . so, by (5.7), k(λ) should be given by the formula (5.8) k(λ)(z) = 1 π +∞∑ n=0 (n + 1)(λz)n = 1 π (1 −λz)−2. the above infinite sum is evaluated by letting ω = λz and noting that +∞∑ n=0 (n + 1)ωn = [ +∞∑ n=0 ωn+1 ]′ = [ ω 1 −ω ]′ = 1 (1 −ω)2 . now, let f ∈ l2a(d) with a power series expansion f(z) = +∞∑ n=0 anz n, z ∈ d. it is easy to verify that the monomials (zn)n≥0 form an orthogonal basis for l2a(d), hence the partial sums of the above series converge to f in l 2 a(d). thus, for every λ ∈ d, we have f̂(λ) = 〈f,k(λ)〉 = lim n→+∞ 〈 n∑ i=0 aiz i,k(λ)〉 = lim n→+∞ n∑ i=0 aiλ i = f(λ). so, if f is orthogonal to ∨ {k(λ) : λ ∈ ba(s)}, then f must be identically zero. therefore, l2a(d) = ∨ {k(λ) : λ ∈ ba(s)}. 312 a. bourhim the following example shows that ∨ {k(λ) : λ ∈ ba(s)} needs not always be equal to h. it also shows that the purity of t is necessary condition in theorem 5.3. example 5.5. let δ be the σ–finite dirac measure on c at 2; i.e., for every subset a of c, δ(a) =   1 if 2 ∈ a 0 otherwise the normal operator nδ : l2(δ) → l2(δ) defined by nδf(z) = zf(z) for all f ∈ l2(δ), is cyclic and satisfies σ(nδ) = σap(nδ) = σp(nδ) = {2}. now, let h = l2a(d) ⊕l2(δ) and let t = s ⊕nδ, where s : l2a(d) → l2a(d) is the bergman operator for d. by proposition 1-viii of [12], t is cyclic subnormal operator with cyclic vector 1 ⊕ 1. on the other hand, we have, σ(t) = cl(d) ∪{2}, σap(t) = {λ ∈ c : |λ| = 1}∪{2} and σp(t) = {2}. in particular, ba(t) = σ(t)\σap(t) = d 6= ∅. since σp(t) 6= ∅, it then follows from proposition 5.1 that∨ {k(λ) : λ ∈ ba(t)}$h. note that all the examples considered in the present paper are given either by weighted shift operators or by subnormal operators. so, it would be interesting to give other examples. we begin by recalling that an operator r ∈l(h) is said to be fredholm if ranr is closed, and ker r and ker r∗ are finite dimensional. the essential spectrum σe(r) of r is the set of all λ ∈ c such that r − λ is not fredholm; in fact, σe(r) is exactly the spectrum σ ( π(r) ) in the calkin algebra l(h)/c of π(r), where π is the natural quotient map of l(h) onto l(h)/c; here, c denotes the ideal of all compact operators in l(h). note that if r ∈l(h) is a cyclic operator such that σp(r) = ∅, then σap(r) = σe(r) = {λ ∈ c : ran ( r−λ ) is not closed }. example 5.6. let s be the unweighted shift on h i.e., sen = en+1 for every non-negative integer n, where (en)n≥0 is an orthonormal basis of h. it is shown in [11] that the operator t := s∗ + 2s is hyponormal, but t 2 is not. therefore, t is not subnormal operator since every power of a subnormal operator is subnormal. on the other hand, it is easy to see that t is a cyclic operator. indeed, by induction, one can show that ek = pk(t)e0 for every k ≥ 0, where (5.9)   p0(z) = 1, p1(z) = 12z pk+1(z) = 12 [ zpk(z) −pk−1(z) ] , ∀k ≥ 1. bounded point evaluations for . . . 313 hence, e0 is a cyclic vector for the operator t . moreover, the following properties hold. (i) t is a pure cyclic hyponormal operator. (ii) σap(t) = {a + ib ∈ c : (a3 ) 2 + b2 = 1}. (iii) σ(t) = {a + ib ∈ c : (a 3 )2 + b2 ≤ 1}. hence, t is not a weighted shift operator since its spectrum is not a disc. (iv) ba(t) = {a + ib ∈ c : (a3 ) 2 + b2 < 1} and for every λ ∈ ba(t), k(λ) = +∞∑ n=0 pn(λ)en, where the polynomials pn are given by (5.9). (v) σ t (y) = σ(t) for every non-zero element y ∈h. (i) let m be a reducing subspace of t . since s = 2t −t∗ 3 , then m is a reducing subspace of s. and so m = {0} or m = h. hence t is pure operator. (ii) since, σ ( π(s) ) = σe(s) = t, where t is the unit circle of c, and π(s) is a normal element in the calkin algebra, it follows from the spectral mapping theorem that σ ( π(t) ) = {λ + 2λ : λ ∈ t} = {a + ib ∈ c : ( a 3 )2 + b2 = 1}. since σap(t) = σe(t) = σ ( π(t) ) , the desired result holds. (iii) note that the operator t is a toeplitz operator tφ with associated function φ = z + 2z. it then follows from theorem 5 of [19] that { a + ib ∈ c : (a 3 )2 + b2 > 1 } ⊂ ρ(t), and either{ a + ib ∈ c : (a 3 )2 + b2 < 1 } ⊂ σ(t) or { a + ib ∈ c : (a 3 )2 + b2 < 1 } ⊂ ρ(t). in view of theorem 1 of [15], the last inclusion is impossible since t is not normal. hence, σ(t) = {a + ib ∈ c : (a 3 )2 + b2 ≤ 1}. (iv) by theorem 4.3, we have ba(t) = σ(t)\σap(t) = { a + ib ∈ c : (a 3 )2 + b2 < 1 } . 314 a. bourhim now, let λ ∈ ba(t); then we have k(λ) = +∞∑ n=0 〈k(λ),en〉en = +∞∑ n=0 〈k(λ),pn(t)e0〉en = +∞∑ n=0 pn(λ)en. (v) by theorem 5.3, we have h = ∨ {k(λ) : λ ∈ ba(t)}. and so, by proposition 5.1, σ(t) = cl ( ba(t) ) ⊂ σ t (y) for every non-zero element y ∈h. the statement (v) is proved. the following example shows that h = ∨ {k(λ) : λ ∈ ba(t)} is not sufficient condition for t to have a connected spectrum, and gives a negative answer to question b of [21] (see also [22]). the following lemma is needed. lemma 5.7. suppose that r ∈ l(h). if σ r (y) = σ(r) for every non-zero element y ∈h, then σ(r) is connected. proof. the proof is similar to the one of proposition 5.1-(ii). � example 5.8. in considering the operator t given in example 5.6, we let t̃ := (t − 4) ⊕ (t + 4) ∈l(h̃), where h̃ = h⊕h. if we set g+ = { a+ib ∈ c : (a + 4 3 )2 +b2 < 1 } and g− = { a+ib ∈ c : (a− 4 3 )2 +b2 < 1 } , then the following hold. (i) t̃ is a pure cyclic hyponormal operator with cyclic vector x = e0 ⊕e0. (ii) σap(t̃) = fr(g−) ∪ fr(g+). hence, by theorem 5.3, h̃ = ∨ {k t̃ (λ) : λ ∈ ba(t̃)}. (iii) σ(t̃) = cl(g−) ∪ cl(g+). hence, σ(t̃) is disconnected; by lemma 5.7, there is a non-zero element y ∈ h̃ for which σ t̃ (y) $ σ(t̃). (iv) ba(t̃) = g− ∪g+. what we need to show here is that x = e0 ⊕e0 is a cyclic vector for t̃ . indeed, let b+ = { λ ∈ c : |λ + 4| < 7 2 } and b− = { λ ∈ c : |λ− 4| < 7 2 } . bounded point evaluations for . . . 315 observe that b− ∩ b+ = ∅, and consider the following analytic function on b := b− ∪b+ f(z) =   1 if z ∈ b+ 0 if z ∈ b−. since c\b is connected, runge’s theorem shows that there is a sequence of polynomials (pn)n which converges uniformly to f on compact subsets of b. hence, pn(t̃)x → f(t̃)x = e0 ⊕ 0. since e0 is cyclic vector for t , for every y ∈h, we have y ⊕ 0 ∈ ∨{ p(t̃)x : p polynomial } . similarly, for every y ∈h, we have 0 ⊕y ∈ ∨{ p(t̃)x : p polynomial } . therefore, h̃ = ∨{ p(t̃)x : p polynomial } . so, x = e0 ⊕e0 is cyclic vector for t̃ . acknowledgements. part of this material is contained in the author’s ph.d thesis, [4], written at the mohammed v university, rabat-morocco. the author expresses his gratitude to professor o. el-fallah for his encouragement and helpful discussion. he also acknowledges the stimulating atmosphere of the international conference on function spaces, proximities and quasiuniformities in caserta, september 2001, and expresses his warmest thanks to the organizers especially giuseppe di maio. references [1] p. aiena and o. monsalve, operators which do not have the single valued extension property, j. math. anal. appl. 250 (2000), 435-448. [2] a. bourhim, bounded point evaluations and local spectral theory, 1999-2000 dictp diploma thesis, (available on: http:/www.ictp.trieste.it/˜pub−off (ic/2000/118)). [3] a. bourhim, c. e. chidume and e. h. zerouali, bounded point evaluations for cyclic operators and local spectra, proc. amer. math. soc. 130 (2002), 543-548. [4] a. bourhim, points d’évaluations bornées des opérateurs cycliques et comportement radial des fonctions dans la classe de nevalinna ph.d. thesis, université mohammed v, morocco, 2001. [5] j. bran, subnormal operators, duke math. j. 22 (1955), 75-94. [6] s. clary, equality of spectra of quasisimilar hyponormal operators, proc. amer. math. soc., 53 (1975), 88-90. [7] i. colojoara and c. foias, theory of generalized spectral operators, gordon and breach, new york, 1968. 316 a. bourhim [8] j. b. conway, the theory of subnormal operators, volume 36 of mathematical surveys and monographs. american mathematical society, providence, r.i, 1991. [9] b. p. duggal, p-hyponormal operators satisfy bishop’s condition (β), integral equations operator theory 40 (2001), 436–440. [10] j. k. finch, the single valued extension property on a banach space, pacific j. math. 58 (1975), 61–69. [11] p. r. halmos, a hilbert space problem book, springer-verlag, new york, 1982. [12] d. e. herrero, on multicyclic operators, integral equations and operator theory, 1/1 (1978), 57-102. [13] k. b. laursen and m. m. neumann, an introduction to local spectral theory, london mathematical society monograph new series 20 (2000). [14] m. putinar, hyponormal operators are subscalar, j. operator theory, 12 (1984), 385-395. [15] c. r. putnam, an inequality for the area of hyponormal spectra, math. z. 116 (1970), 323-330. [16] m. raphael, quasisimilarity and essential spectra for subnormal operators, indiana univ. math. j. 31 (1982), 243-246. [17] w. c. ridge, approximate point spectrum of a weighted shift, trans. amer. math. soc. 147 (1970), 349-356. [18] a. l. shields, weighted shift operators and analytic function theory, in topics in operator theory, mathematical surveys, n0 13 (ed. c. pearcy), pp. 49-128. american mathematical society, providence, rhode island, 1974. [19] j. g. stampfli, on hyponormal and toeplitz operators, math. ann. 183 (1969), 328-336. [20] t. t. trent, h2(µ) spaces and bounded point evaluations, pac. j. math., 80 (1979), 279-292. [21] l. r. williams, bounded point evaluations and local spectra of cyclic hyponormal operators, dynamic systems and applications 3 (1994), 103-112. [22] l. r. williams, subdecomposable operators and rationally invariant subspaces, operator theory: adv. appl., 115 (2000), 297-309. [23] l. yang, hyponormal and subdecomposable operators, j. functional anal. 112 (1993), 204217. received november 2001 revised september 2002 a. bourhim the abdus salam international centre for theoretical physics, trieste, italy e-mail address : bourhim@ictp.trieste.it current address: a. bourhim département de mathématiques, université mohamed v, b.p. 1014, rabat, morocco e-mail address : abourhim@fsr.ac.ma @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 263–279 locally convex approach spaces m. sioen and s. verwulgen dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. we continue the investigation of suitable structures for quantified functional analysis, by looking at the notion of local convexity in the setting of approach vector spaces as introduced in [6]. we prove that the locally convex objects are exactly the ones generated (in the usual approach sense) by collections of seminorms. furthermore, we construct a quantified version of the projective tensor product and show that the locally convex objects admitting a decent exponential law with respect to it are precisely the seminormed spaces. 2000 ams classification: 18b99, 18d15, 46a03, 46b04, 46m05, 46m15. keywords: approach vector space, topological vector space, locally convex space, locally convex approach space, minkowski functional, minkowski system, projective tensor product. 1. introduction and preliminaries. in the setting of contemporary functional analysis the notion of a topological vector space plays a central role and over the last decennia, a great deal of effort has been done in studying their properties and relating them to different other fields in mathematics. however the totality of all topological vector spaces seems too large too handle at times to further develop and fine-tune this theory, as a lot of deep and interesting results (such as e.g hahn-banach type theorems, or suitable duality) only work well or are valid for certain subclasses of topological vector spaces. certainly one of the most important better-behaved types of topological vector spaces are the locally convex ones, not in the least since they allow for the construction of several notions of tensor products. on the other hand, it is well-known from the classical theory that a topological vector space x is locally convex if and only if it can be generated by a collection 264 m. sioen and s. verwulgen of seminorms, in the sense that there exists a set p of continuous seminorms on x such that {{x ∈ x | p(x) < ε} | p ∈p,ε > 0} is a base for the neighborhood system of 0. it is also a well-known fact that this exactly means that x as a topological space can be embedded in the (topological !) product of the family {(x,tµ) | µ ∈p} of semi-normable topological vector spaces (where tµ denotes the topology generated by µ) in the following way: x ↪→ ∏ µ∈p (x,tµ) : x 7→ (µ(x))µ∈p. here another aspect, which is the central incentive for approach theory as introduced by r. lowen in [8, 9] appears: in (semi)normed spaces, canonical numerical information is intrinsically present in terms of the norm itself, but when we want to consider more general structures on vector spaces, essentially involving the construction of arbitrarily large set-indexed products,we classically automatically end up with topologies. so it seems that all the numerical information has been lost somewhere along the way: this also reflects in the fact that, for a given vector space x and a given set p of seminorms on x, both p and {αµ | α ∈ r+0 ,µ ∈p} generate the same locally convex topology, although for each µ ∈p, α 6= 1, the seminorms µ and αµ are essentially different from the numerical point of view ! in the setting of topological vs. (pseudo-quasi-)metric spaces, a general solution to this problem was proposed by r. lowen in [8, 9] through the introduction of the category ap of approach spaces and contractions, in which arbitrary (set-indexed) products of metric objects can be formed yielding a canonical numerification of the product of the topologies underlying these objects. we refer to [8, 9] for any further detailed information. on the other hand, r. lowen together with the second named author introduced the notion of an approach vector space in [6] and it follows from the results proved there and also from [5] that the category apvec of approach vector spaces and linear contractions is the right framework for quantified functional analysis, into which both the categories topvec (of topological vector spaces and continuous maps) and pmetvec (of vector pseudometric vector spaces (in the sense of [16])(see also [6]) and linear non-expansive maps) are simultaneously embedded as full subcategories! informally, one can say that apvec solves the quantifying problem sketched above for pmetvec and topvec, as ap does for pmet and top. locally convex topological spaces are not only prominent within the realm of topological vector spaces because they allow for stronger theorems, but also because some crucial constructions behave better under the assumption of local convexity. one such construction, having paramount applications, as can be seen e.g. in [16], is the projective tensor product of locally convex topological spaces locally convex approach spaces 265 which reaches deeply into other subbranches such as e.g. the theory of nuclear operators [16] and vector valued integration [15]. the purpose of this paper is twofold: we want to put local convexity in the right perspective in the setting of approach vector spaces and then we want to use this locally convex approach structure (as we will call it) to create a canonical numerified counterpart of the classical topological projective tensor product, overlying the latter in a natural way. the main result of this paper is theorem 3.9, where we exactly characterize the exponential behaviour of this quantified projective tensor product by showing that only seminormed spaces exhibit a categorically good notion of exponentiability. first, let us mention that all vector spaces considered in this paper are vector spaces over the field of reals. we take [1] as our blanket reference for all facts of categorical nature and we refer the reader to [8, 9] for a detailed account on approach theory, together with the basic definitions, terminology and notations. throughout the present paper, we will use the same terminology and notational conventions concerning approach vector spaces as defined in [6]. we briefly recall the most needed basic definitions and results, to make the text more self-contained. let x be a vector space. for each d ∈ [0, +∞]x×x, we define d[x] : x → [0, +∞] : y 7→ d(x,y) and whenever d ⊂ [0, +∞]x×x, we let d[x] = {d[x] | d ∈d}. conversely, for φ : x → [0, +∞] we put φ(2) : x ×x → [0, +∞] : (x,y) 7→ φ(y −x) and whenever c ⊂ [0, +∞]x, we put c(2) = {φ(2) | φ ∈c}. throughout the whole framework of approach theory, the following concept of saturatedness, together with the obvious notion of saturation, for a collection of functionals plays a prominent role: we call b ⊂ c ⊂ [0, +∞]x saturated in c if and only if the following condition holds: ∀φ ∈c : (∀� > 0, ∀ω < ∞ ∃ψ ∈b : φ∧ω ≤ ψ + �) ⇒ φ ∈b. if c = [0,∞]x, we simply talk about saturatedness. we write 〈b〉 for the saturation of b within [0,∞]x, i.e. 〈b〉 := {φ ∈ [0,∞]x | ∀� > 0, ∀ω < ∞ ∃ψ ∈b : φ∧ω ≤ ψ + �}, which obviously is the smallest saturated subset of [0,∞]x containing b. furthermore we call a functional φ ∈ [0, +∞]x sub–additive if ∀x,y ∈ x : φ(x + y) ≤ φ(x) + φ(y) and we say that φ is balanced if ∀x ∈ x,∀λ ∈ [−1, 1] : φ(λx) ≤ φ(x). finally φ is said to be absorbing if ∀x ∈ x,∀� > 0 : ∃δ > 0 : ∀λ ∈ [−δ,δ] : φ(λx) ≤ �. 266 m. sioen and s. verwulgen a sub–additive, balanced and absorbing functional φ ∈ [0, +∞]x then is called a prenorm on x and φ(2) is called a vector pseudometric on x. (also note that, as remarked in [6], prenorms only take finite values.) definition 1.1. [6] a pair (x,a) consisting of a vector space x and an approach system a = (a(x))x∈x is called an approach vector space if the following assertions hold (av1) for all x ∈ x we have that a(x) = {y 7→ φ(y −x) | φ ∈a(0)}. (av2) for all φ ∈a(0), for all � > 0 and for all ω < ∞, there exists ψ ∈a(0) such that for all x,y ∈ x φ(x + y) ∧ω ≤ ψ(x) + ψ(y) + �. (av3) for all φ ∈a(0), for all � > 0 and for all ω < ∞, there exists ψ ∈a(0) such that for all x ∈ x and for all λ ∈ r with |λ| ≤ 1 φ(λx) ∧ω ≤ ψ(x) + �. (av4) every φ ∈a(0) is absorbing. note that (av1) together with (av2) exactly mean that (x,a) is an approach group in the sense of [7]. a morphism between two approach vector spaces (x,ax) and (y,ay ) is a linear map f : x → y such that ∀φ ∈ay (0) : φ◦f ∈ax(0), meaning that it is a contraction in the sense of [9]. approach vector spaces and linear contractions then form a topological category over the category of vector spaces, as was shown in [6]. it was also proved in [6] that for a vector space x and an approach system a = (a(x))x∈x the following assertions are equivalent (1) (x,a) is an approach vector space (2) a(0) has a base of prenorms and ∀x ∈ x : a(x) = {y 7→ φ(y −x) | φ ∈a(0)}. if g is the gauge corresponding to a (see [9] for additional information) these assertions are moreover equivalent to saying that g has a base of vector pseudometrics. 2. locally convex approach spaces. note that, with x a vector space, a functional φ ∈ [0,∞]x is called convex if ∀x,y ∈ x,∀λ ∈ [0, 1] : ϕ(λx + (1 −λ)y) ≤ λφ(x) + (1 −λ)φ(y), which obviously is equivalent to stating that, whenever we take a finite number of vectors x1, . . . ,xn and real numbers λ1, . . . ,λn ∈ [0, 1] with ∑n i=1 λi = 1, we have φ ( n∑ i=1 λixi ) ≤ n∑ i=1 λiφ(xi). locally convex approach spaces 267 the following basic lemma will be the key tool for proving the main result (2.2) of this section: it provides a notion of a minkowski-like functional associated with a given convex, balanced and absorbing functional, rather than with a given convex, balanced and absorbing set as suffices in the general topological vector space setting. lemma 2.1. let x be a vector space, φ ∈ [0,∞]x be a balanced, absorbing and convex functional and take 0 < ω < ∞. now define a new functional ηωϕ ∈ [0,∞]x by ηωφ (x) := inf{λ > 0 | φ(λ −1ωx) ≤ ω}, x ∈ x. then the following assertions hold: (1) ηωφ takes finite values and is a seminorm. (2) ηωφ ≤ φ on {φ ≥ ω}. (3) φ ≤ ηωφ on {φ ≤ ω}. (4) ηωφ ≤ φ∨ω ≤ φ + ω. (5) φ∧ω ≤ ηωφ . proof. (1) the set {x ∈ x | φ(ωx) ≤ ω} is absorbing, balanced and convex and ηωφ is the minkowski functional of this set. (2) take x ∈ x such that φ(x) ≥ ω. from the convexity of φ we have φ(φ(x)−1ωx) ≤ φ(x)−1ωφ(x) = ω. thus φ(x) ∈ {λ > 0 | φ(λ−1ωx) ≤ ω} and so ηωφ (x) ≤ φ(x). (3) let x ∈ x such that φ(x) ≤ ω. suppose ηωφ (x) < φ(x). thus there exists λ ∈ r such that 0 < λ < φ(x) and φ(λ−1ωx) ≤ ω. because obviously also λω−1 ≤ 1, we would obtain that φ(x) = φ ( λω−1λ−1ωx ) ≤ λω−1φ(λ−1ωx) < φ(x)ω−1φ(λ−1ωx) ≤ φ(x), which is impossible. (4) if φ(x) ≥ ω the inequality follows from 2. above. if φ(x) < ω note that φ(x) = φ(ω−1ωx) ≤ ω and hence ηωφ (x) ≤ ω. (5) we only have to show the inequality for φ(x) > ω. then for all 0 < λ ≤ ω we have that ω < φ( λω λω x) ≤ φ(λ−1ωx) and hence ηωφ (x) ≥ ω. � as addressed in the introduction, it is a well-known fact that locally convex spaces (i.e. topological vector spaces which admit a base for the neighborhoods 268 m. sioen and s. verwulgen of 0 consisting of convex sets), can be generated by a so-called “gauge” of (continuous) seminorms. the following theorem is the quantified counterpart of it: it states that these approach vector spaces for which the local approach system (which is a lattice-theoretic ideal of functionals on x !) at 0 is generated by convex functions (in the sense of saturation as defined above), can be generated in the same sense from seminorms. compare this with [6] theorem 5 (which was already quoted in the preliminaries), stating that any approach vector space has a generating set of prenorms. theorem 2.2. let (x,a) be an approach vector space. then the following expressions are equivalent. (1) a(0) has a base of seminorms, meaning there exists a set p of seminorms on x which is an ideal base in the lattice [0,∞]x and for which a(0) = 〈p〉, (2) a(0) has a base of balanced, absorbing and convex functionals, meaning there exists a set b of absorbing and convex functionals on x which is an ideal base in the lattice [0,∞]x and for which a(0) = 〈b〉. proof. 1 ⇒ 2. this is clear, since any seminorm is a balanced, absorbing and convex function. 2 ⇒ 1. let φ be a balanced, absorbing convex function in a(0) and fix 0 < ω < ∞. from lemma 2.1 we deduce that ηωφ is a seminorm and that φ∧ω ≤ ηωφ . in order to show η ω φ ∈ a(0), pick � > 0 and ω ′ < ∞. let m ∈n such that ω m ≤ �. since (x,a) is an approach group ([6]), there exists ψ ∈a(0) such that for all x1, . . . ,xm ∈ x : φ(x1 + · · · + xm) ∧mω′ ≤ ψ(x1) + · · · + ψ(xm). from lemma 2.1 we have ηωφ ≤ φ+ω, hence for all x ∈ x we have that ηωφ (mx) ∧mω ′ ≤ φ(mx) ∧mω′ + ω ≤ mψ(x) + ω and therefore ηωφ ∧ω ′ ≤ ψ + �. because a(0) has a base b of balanced, absorbing and convex functions, it follows that a(0) = 〈{ηωϕ | ϕ ∈b, 0 < ω < +∞〉, and therefore automatically that a(0) = 〈{∨ni=1η ωi ϕi | n ∈ n0,ϕ1, . . . ,ϕn ∈b, 0 < ω1, . . . ,ωn < +∞}〉, where the generating functionals in the last step still are seminorms. � this now justifies the following definition. definition 2.3. an approach vector space satisfying the properties of theorem 2.2 is called a locally convex approach space. if we take locally convex approach spaces as objects and linear contractions (in the sense of [8, 9, 6] as morphisms) we clearly obtain a full subcategory of apvec, which we denote lcapvec. locally convex approach spaces 269 whenever x is a vector space, we call a set m of seminorms on x a minkowski system (on x) if it is a saturated ideal in the lattice of all seminorms on x. if (x,a) is a locally convex approach space, then ma := {η ∈a(0) | η is a seminorm} is a minkowski system. every minkowski system is obtained in this way. indeed, let m be a minkowski system on x. then we have, with (2.1) am(x) =< m(2)[x] >, that (x,am) is a locally convex approach space such that m = mam (since m is a saturated ideal of seminorms). moreover by theorem 2.2, the approach system a of a locally convex approach space is derived from a minkowski system—in the sense of (2.1) because we have a = ama. this even shows that, given a vector space x, there is a one-to-one correspondence between locally convex approach structures on x and minkowski systems on x. we will therefore also use notations like (x,m) with m a minkowski system on x to denote lcapvec objects. note that if m is a minkowski system, then the collection {η(2) |η ∈ m} is a gauge base for the gauge of the approach structure derived from m. corollary 2.4. let (x1,a1) and (x2,a2) be locally convex approach spaces with corresponding minkowski systems ma1 and ma2 respectively. a linear map f : x1 → x2 is a morphism in lcapvec if and only if ∀η ∈ma2 : η ◦f ∈ma1. to simplify the language, let us agree upon the following convention: in the sequel initial and final structures in (lc)apvec (resp. (lc)topvec) are understood to be taken with respect to the forgetful functor from (lc)apvec (resp. (lc)topvec) to vec. theorem 2.5. lcapvec is initially closed in apvec. therefore lcapvec is topological over vec. proof. consider a class indexed source (fi : x → (xi,ai))i∈i in lcapvec and let a be the initial apvec-structure on x for this source, viewed as a source in apvec. for each i ∈ i, let mi be the minkowski system of (xi,ai). it was shown in [6] that the initial structures in apvec are just the initial approach structures, and therefore it follows that { n sup j=1 ηij ◦fij | n ∈n0, ∀j ∈{1, . . . ,n} : ij ∈ i, ηij ∈mij} is a base for a(0) which consists of seminorms, yielding that (x,a) is a locally convex approach space. hence (fi : (x,a) → (xi,ai))i∈i is initial in lcapvec. � corollary 2.6. the category lcapvec is concretely reflective in apvec. 270 m. sioen and s. verwulgen remark that if (x,a) is an approach vector space then the set {η | η is a seminorm in a(0)} can be taken as the minkowski system of an lcapvec structure a′ on x. clearly (x,a′) is the reflection of (x,a) in lcapvec. let (x,η) be a seminormed space. then η(2) is a pseudometric and (x,aη(2) ) is a locally convex approach space. in the sequel we will make no distinction between a seminorm and the associated approach structure. theorem 2.7. (1) a locally convex approach space which is at the same time a vector pseudometric space is a seminormed space. (2) the category snorm, consisting of seminormed spaces and linear nonexpansive maps between them, is embedded as a full subcategory of lcapvec. moreover if (x,a) is a locally convex approach space for which the approach structure is metric, then (x,a) is a seminormed space. proof. (1) let d be a vector pseudometric such that ad(0) has a base of seminorms. then the prenorm d[0] is the supremum of seminorms and hence is a seminorm. (2) it is easy to see that associating to every seminormed space (x,η) the locally convex approach space (x,aη(2) ) defines a concrete full embedding of snorm into lcapvec. to prove the second assertion, let (x,a) be a locally convex approach space which at the same time is a metric object in ap. this means that the approach gauge corresponding to a (see [9] for a definition) contains a largest metric d which at the same time can be written as the pointwise supremum of a set of vector pseudometrics. repeating the proof of [6] theorem 16 now yields that d itself is a vector pseudometric, hence d[0] is a prenorm on x for which a(0) = 〈{d[0]}〉. applying 1 now finishes the proof. � theorem 2.8. snorm is initially dense in lcapvec. proof. this is a consequence of theorem 2.2: if (x,a) is a locally convex approach space, then the source (idx : (x,a) → (x,η))η∈ma is initial in lcapvec. � theorem 2.9. (1) lctopvec is embedded as a full subcategory of lcapvec locally convex approach spaces 271 (2) lctopvec is concretely coreflective in lcapvec. moreover we have the commutation of lcapvec // �� lctopvec �� uap // creg where the horizontal arrows are the concrete coreflectors and the vertical arrows the forgetful functors. (3) if (x,a) is a locally convex approach space such that the underlying approach system is topological then (x,a) is a locally convex topological space. proof. (1) a locally convex topological space is also an approach vector space ([6]) and since the characteristic function of a convex set is a convex function we know the approach system of 0 has a base of convex functions. (2) let (x,a) be a locally convex approach space and let t be the topological coreflection of (x,a) in ap. from [6], theorem 21, we know that (x,t ) is a topological vector space. since ma is a base for a(0), we know that {{n ≤ �} | � > 0, n ∈m} is a base of convex sets for the neighborhood system of 0. (3) let (x,a) be a topological locally convex approach space. then by 2. above, (x,a) equals its lctopvec coreflection and is thus a locally convex topological space. � proposition 2.10. let m be an ideal of seminorms. then m is the minkowski system of a locally convex topological space if and only if (2.2) ∀λ ∈ r+, ∀η ∈m : λη ∈m. proof. the minkowski system of a locally convex topological space is just the set of minkowski functionals of the balanced, convex and absorbing open sets and hence it satisfies (2.2). conversely m is saturated if it satisfies (2.2). moreover, in this case the associated approach system is topological and the result follows from theorem 2.9. � corollary 2.11. lctopvec is initially closed in lcapvec. corollary 2.12. lctopvec is initially closed in topvec ([3]) and hence lctopvec is concretely reflective in topvec. corollary 2.13. lctopvec is finally closed in lcapvec corollary 2.14. let (x,m) be a locally convex approach space. then the set {λη | λ ∈ r+, η ∈m} is the minkowski system of the topological coreflection of (x,m). 272 m. sioen and s. verwulgen in order to sketch the general results we extend the summarizing diagram of [6] to the following commutative diagram: snorm � � // �� n ��: :: :: :: :: :: :: :: :: :: :: :: :: :: ,, lcapvec �� l r ��6 66 66 66 66 66 66 66 66 66 66 66 66 6 ++ snormtopvec � � // � n ��: :: :: :: :: :: :: :: :: :: :: :: :: :: ,, lctopvec� l r ��6 66 66 66 66 66 66 66 66 66 66 66 66 6 ++ pmet � � // �� uap �� pmettop � � // creg pmetvec � � // �� >> apvec �� aa pmettopvec � � // >> topvec aa here the categories in the top layer are the approach versions of those in the bottom layer. the horizontal embeddings are initially dense and the dashed and vertical arrows are the usual forgetful functors. since lctopvec is not finally closed in topvec ([3]) and the latter category is finally closed in apvec ([6], corollary 22), by theorem 2.9 we have lcapvec is not finally closed in apvec. note that normable topological vector spaces are not the result of imposing the convexity condition on the metrizable topological vector spaces; there’s a subtle difference with theorem 2.7: see [14] for a locally convex topological vector space for which the topology is metrizable but not normable. 3. the projective tensor product and bicontractions. in the sequel x, y and z are presumed to be locally convex approach vector spaces with approach and minkowski systems ax, ay , az and mx, my , mz respectively. every locally convex topology being defined by a family of seminorms, in [4] the projective tensor product of two locally convex topological vector spaces is introduced using these seminorms. recall from [16] that, if µ and ν are seminorms on respectively x and y , the map µ⊗ν : x ⊗y → r+ : u 7→ µ⊗ν(u) := inf{ n∑ i=1 µ(xi)ν(yi) | n∑ i=1 xi ⊗yi = u} is a seminorm on the algebraic tensor product x ⊗y of the vector spaces x and y which, moreover, satisfies the following identity: ∀x ∈ x,∀y ∈ y : µ⊗ν(x⊗y) = µ(x)ν(y). we now in the obvious way can define a quantified version of the projective tensor product in the realm of locally convex approach spaces. locally convex approach spaces 273 definition 3.1. the projective tensor product x⊗y is the algebraic tensor product of x and y endowed with mx⊗y , the minkowski system generated by the set {µ⊗ν | µ ∈mx, ν ∈my}. note that {µ⊗ν | µ ∈ mx, ν ∈ my} indeed is a base for an ideal in the set of all seminorms on x ⊗y , because for all µ,µ′ ∈mx, ν,ν′ ∈my clearly (µ⊗ν) ∨ (µ′ ⊗ν′) ≤ (µ∨µ′) ⊗ (ν ∨ν′) and mx,my are closed for taking finite suprema. theorem 3.2. the locally convex topological coreflection of (x ⊗y,mx⊗y ) is the projective tensor product of the locally convex topological coreflections of x and y . proof. this follows from [16]. � corollary 3.3. the projective tensor product of topological locally convex approach spaces is again topological. moreover if we restrict the projective tensor product to lctopvec we obtain the classical projective tensor product. the following very useful lemma, tells us that the saturation condition we have used throughout the whole paper for the approach system a(0) within [0,∞]x, corresponds to a more elegant and especially more manageable form when restricted to the case of the associated minkowski system m within the set of all seminorms on the vector space x. lemma 3.4. let m be a minkowski system on a vector space w . for a seminorm η on w , the following equivalence holds: η ∈m⇔∀� > 0 : ∃µ ∈m : η ≤ µ(1 + �). proof. suppose η ∈ m. let � > 0. since m is saturated there exists µ ∈ m such that η∧1 ≤ µ + �′, where �′ = 1− 1 1+� . then for all x in w we have (with n(x) 6= 0) η( x η(x) ) ≤ µ( x η(x) ) + �′. therefore η(x)(1 − �′) ≤ µ(x), from which η(x) ≤ µ(x) 1−�′ = µ(x)(1 + �) follows, for all x in w . conversely let � > 0, let 0 < ω < ∞ and let µ in m such that η ≤ (1 + �′)µ, where 0 < �′ < min{1, � ω } is fixed. note that 1 ≥ 1 −�′2 = (1 −�′)(1 + �′) > 0. hence η ≤ µ(1 + �′) ≤ µ 1−�′ . therefore η ≤ µ + � ′η. now take x ∈ w . if η(x) ≤ ω, that is η( x ω ) ≤ 1, then η( x ω ) ≤ µ( x ω ) + �′η( x ω ) ≤ µ( x ω ) + �′. thus η(x) ≤ µ(x) + �′ω ≤ µ(x) + �. now suppose that η(x) > ω and at the same time that µ(x) + � < ω. then 1 > µ( x ω ) + � ω > µ( x ω ) + �′. since this implies that µ( x ω ) ≤ 1 we also obtain 1 > µ( x ω )(1 + �′) ≤ η( x ω ), which is impossible. we thus have proved η ∧ω ≤ µ + �. � let (µi)i∈i be a collection of seminorms on a vector space x. note that the infimum sinfi∈iµi in the set of all seminorms on x exists. definition 3.5. a binorm b on the vector space x ×y is a function b : x ×y → r+ such that, for all λ ∈ r, for all x, x′ ∈ x and y, y′ ∈ y : 274 m. sioen and s. verwulgen (1) b(λx,y) = |λ|b(x,y) = b(x,λy), (2) b(x + x′,y) ≤ b(x,y) + b(x′,y), (3) b(x,y + y′) ≤ b(x,y) + b(x,y′). let (bi)i∈i be a collection of binorms on x×y . then the infimum binfi∈ibi exists in the set of all binorms on x ×y . moreover we have, for (µi)i∈i and (νj)j∈j a collection of seminorms on x respectively y , that ((µi ◦ prx)(νj ◦ pry ))(i,j)∈i×j is a collection of binorms on x ×y and ((sinfi∈iµi)◦textrmprx)((sinfj∈jνj)◦pry ) = binf(i,j)∈i×j(µi◦prx)(νj ◦pry ). if f : x ×y → z is a bilinear map, we write f̄ : x ⊗y → z for its obvious, linear factorization over the tensor product x ⊗ y , which is well-defined by putting f̄(x⊗y) = f(x,y) for all x ∈ x and y ∈ y . proposition 3.6. let f : x × y → z be a bilinear map. the following are equivalent. (1) the linear lift f̄ : (x ⊗y,mx⊗y ) → (z,mz) is a contraction. (2) ∀η ∈mz,∀ω < ∞,∀� > 0 : ∃µ ∈mx : ∃ν ∈my : ∀x ∈ x,∀y ∈ y : η(f(x,y)) ∧ω ≤ µ(x)ν(y) + �. (3) ∀η ∈mz,∀� > 0 : ∃µ ∈mx : ∃ν ∈my : ∀x ∈ x,∀y ∈ y : η(f(x,y)) ≤ µ(x)ν(y)(1 + �). (4) ∀η ∈mz : ∃µ ∈mx : ∃ν ∈my : ∀x ∈ x,∀y ∈ y : η(f(x,y)) ≤ µ(x)ν(y). proof. 1 ⇔ 2 ⇔ 3. straightforward. 3 ⇒ 4. take η ∈ mz and choose, for all n ∈ n0, µ′n ∈ mx and ν′n ∈ my such that for all x ∈ x and y ∈ y η(f(x,y)) ≤ µ′n(x)ν ′ n(y)(1 + 1/n). let µn = supk≤n µ ′ k and let νn = supk≤n ν ′ k. define µ = sinfn∈n0µn(1 + 1/n) and ν = sinfn∈n0νn(1 + 1/n). then µ is in mx and µ is in my and we have (µ◦ prx)(ν ◦ pry ) = ((sinfm∈n0µm(1 + 1/m)) ◦ prx) ((sinfn∈n0νn(1 + 1/n)) ◦ pry ) = binf(n,m)∈n20 (µm ◦ prx)(νn ◦ pry )(1 + 1/m)(1 + 1/n) ≥ binf(n,m)∈n20 (µmin(m,n) ◦ prx)(νmin(m,n) ◦ pry )(1 + 1/min(m,n)) = binfn∈n0 (µn ◦ prx)(νn ◦ pry )(1 + 1/n) ≥ binfn∈n0η ◦f = η ◦f. the implication 4 ⇒ 1 is obvious. � locally convex approach spaces 275 definition 3.7. if f satisfies (one of) the above properties it is called a bicontraction. example 3.8. (1) consider r, equipped with |·|. then scalar multiplication m : r×x → x is a bicontraction. moreover m : r ⊗ x → x is an isomorphism between locally convex approach spaces. (2) the map t : x×y → x⊗y : (x,y) 7→ x⊗y is a bicontraction. hence for all x we have a functor −⊗x : lcapvec → lcapvec. note that it is clear from the definition that for seminormed spaces (x,η) and (y,µ), their projective tensor product in lcapvec corresponds exactly to their “classical” seminormed tensor product (x⊗y,η⊗µ), so that no notational ambiguity can arise. from the thoroughly studied tensor product in snorm (see e.g. [15]) we know however that there is much more to say in this case: the projective tensor product defines a symmetric bifunctor −⊗− : snorm × snorm → snorm which determines a so–called symmetric monoidal closed structure. this means that for every seminormed space x = (x,η) the functor −⊗x : snorm → snorm has a right adjoint, a so–called inner hom functor (−)x : snorm → snorm, which is completely determined by putting for every seminormed space y = (y,µ) y x to be the vector space of all linear continuous functions from x to y , equipped with the seminorm µη, given by µη(f) = sup η(x)≤1 µ(f(x)). alternatively, this means that for all x,y,z ∈ |snorm| (3.3) zx⊗y ' (zy )x, where the isomorphism is natural in x,y,z. for the locally convex topological case, the projective tensor product in lctopvec (to which our quantified version reduces according to theorem 3.2) determines a symmetric bifunctor −⊗− : lctopvec × lctopvec → lctopvec which however no longer gives rise to a symmetric monoidal closed structure! (note that as can be seen e.g. from [17] other interesting symmetric monoidal closed structures on categories of barreled–like locally convex topological spaces exist.) also in the case of locally convex approach spaces, the projective tensorproduct we introduced gives rise to a bifunctor −⊗− : lcapvec × lcapvec → lcapvec. 276 m. sioen and s. verwulgen our main theorem of this section exactly shows how far this bifunctor is away from determining a symmetric monoidal closed structure on lcapvec, by classifying those locally convex approach spaces x for which −⊗ x has a right adjoint. we recall the nice exponential law (3.3) as a simple corollary to this theorem. theorem 3.9. the functor −⊗ x has a right adjoint if and only if x is a seminormed space. proof. to begin with, suppose −⊗x has a right adjoint. we now proceed in several steps. (1) fix y and let w be a locally convex approach space and � : w⊗x → y be a linear contraction such that for all z and for all f : z ⊗x → y linear contraction there exists a unique linear contraction f̂ : z → w such that the following diagram is commutative: w ⊗x � // y z ⊗x f̂⊗id oo f ;;wwwwwwwww define σ : w → l(x,y ) by σ(w)(x) = �(w⊗x), where l(x,y ) is the set of all linear maps from x to y . let y x be the subspace σ(w) of l(x,y ). note that, for all ν ∈ mw , σ(ν), defined by σ(ν)(a) = infσ(w)=a ν(w), is a seminorm on y x. let my x be the minkowski system generated by {σ(ν) | ν ∈mw}. then σ is a linear contraction. for further reference, my x is called the function space structure. (2) we claim that ev : y x ×x → y : (a,x) 7→ a(x) is a bicontraction. to prove this let η ∈ my and let � > 0. let µ ∈ mx and ν ∈ mw such that for all w in w and x in x we have η(�(w⊗x)) ≤ ν(w)µ(x)(1+�). fix w in w . let w′ in σ−1(σ(w)). then η(ev(σ(w),x)) = η(ev((σ(w′),x)) = η(�(w′ ⊗ x)) ≤ ν(w′)µ(x)(1 + �). hence η(ev(σ(w),x)) ≤ infw′∈σ−1(σ(w)) ν(w′)µ(x)(1 + �) = σ(ν)(σ(w)) µ(x)(1 + �). (3) note that the triangle y x ⊗x ev ##h hh hh hh hh w ⊗x σ⊗id oo � // y is commutative. take f : z ⊗x → y a linear contraction, let f̂ : z → w be such that f = �◦(f̂ ⊗ id). then we have following commutative locally convex approach spaces 277 diagram y x ⊗x ev ##h hh hh hh hh w ⊗x σ⊗id oo � // y z ⊗x. (σ◦f̂)⊗id :: f̂⊗id oo f ::vvvvvvvvv so we have f = ev ◦ ((σ ◦ f̂) ⊗ id). let g : z → y x be a linear contraction such that f = ev ◦g ⊗ id. then for all x in x and z in z we have g(z)(x) = ev ◦ (g × id) (z,x) = ev ◦ (g ⊗ id)(z ⊗x) = f(z ⊗x) = �(f̂(z) ⊗x) = σ(f̂(z))(x) = (σ ◦ f̂)(z)(x). this means g = σ ◦ f̂. (*) for all linear contractions f : z ⊗x → y , there exists a unique linear contraction f̃ : z → y x such that f = ev ◦ (f̃ ⊗ id). (4) since ev : y x ×x → y is a bicontraction, we know that y x contains only continuous linear maps from x to y . let a : x → y be a linear continuous map. then the map a′ : rt ×x → y : (λ,x) 7→ λa(x) is a linear contraction, where rt is endowed with the euclidean topology. then we have that ã′(1) = a. this shows that y x is the set of all continuous linear maps. (5) in the above, let y = r equipped with the absolute value. we write x′ for the space of linear continuous maps from x to r, let mx′ be the function space structure. define, for all η in mx and a in x′, |a|η = sup η(x)≤1 |a(x)|. so we have a map | · |η : x′ → [0,∞]. take ξ ∈ mrx and η ∈ mx such that, for all a ∈ x′ and x ∈ x we have |a(x)| ≤ ξ(a)η(x). then it follows (3.4) | · |η ≤ ξ take µ ∈mx. note that µ∨η ≥ η, hence (3.5) | · |µ∨η ≤ | · |η so, from (3.4), we see |·|η is a seminorm on x. note also that |a(x)| ≤ |a|µ∨η(µ ∨ η(x)). so if we consider x′ equipped with | · |η∨µ then ev : x′ ×x → r is a bicontraction. using the universal property (*) 278 m. sioen and s. verwulgen we obtain that ẽv = id : (x′, | · |µ∨η) → (x′,mx′) is a contraction. therefore ξ ≤ | · |η∨µ. now it follows from (3.4) and (3.5) that | · |η = | · |η∨µ. from the hahn–banach theorem it follows η = η ∨µ, that is, µ ≤ η. we have therefore shown that x is a seminormed space. conversely let (x,η) be a seminormed space. fix y ∈ |lcapvec|. for all µ ∈ my we have that the map µη : y x → r+ : a 7→ supη(x)≤1 µ(a(x)) is well defined and a seminorm, where y x is the set of all linear continuous maps from x to y . let my x be the minkowski system generated by the base {µη | µ ∈ my}. then ev : y x × x → y is a bicontraction; let ev be the linear lift. take f : z ⊗ x → y a linear contraction. for all z ∈ z the map f̂(z) : x → y : x 7→ f(z ⊗ x) is linear and continuous. moreover f̂ : z → y x is a linear contraction: let ξ ∈ my x , pick µ ∈ my such that ξ ≤ µη. there exists ν ∈mz such that µ(f(z⊗x)) ≤ ν(z)η(x). thus we have ξ(f̂(z)) ≤ µη(f̂(z)) ≤ ν(z). it is straightforward to verify that f̂ is unique with the property that f = ev ◦ (f̂ ⊗ id). � the projective tensor product does not define a monoidal closed structure on lcapvec. however since the projective tensor product of seminormed spaces is again a seminormed space, we obtain the following well–known fact as a simple corollary: corollary 3.10. the projective tensor product restricted to snorm defines a symmetrical monoidal closed structure. if (x,η) and (y,µ) are seminormed spaces, the seminorm on y x is given by the well known seminorm µη = sup η(x)≤1 µ(f(x)), where f is a continuous linear function from x to y . in particular, when y = (r, | · |) in the above, we obtain the strong dual of x. references [1] adámek j., herrlich h. and strecker g., abstract and concrete categories, j. wiley and sons, 1990. [2] borceux f. handbook of categorical algebra, vol. i,ii,iii, encyclopedia of math. and its appl., cambridge university press, 1996. [3] bourbaki n. eléments de mathématique, livre v, espaces vectoriels topologiques, hermann, paris 1964. [4] grothendieck a. produits tensoriels topologiques et espaces nucléaires, mem. amer. math soc. 16,1955. [5] lowen r. and sioen m. approximations in functional analysis, result. math. 37 (2000) 345–372. [6] lowen r. and verwulgen s. approach vector spaces, submitted. [7] lowen r. and windels b. approach groups, rocky mountain j. of math 30(3) (2000) 1057–1074. [8] lowen r. approach spaces a common supercategory of top and met, math. nachr. 141 (1989) 183–226. locally convex approach spaces 279 [9] lowen r. approach spaces: the missing link in the topology–uniformity–metric triad, oxford mathematical monographs, oxford university press, 1997. [10] mac lane s. categories for the working mathematician, springer 1997. [11] pumplün d. and röhrl h. banach spaces and totally convex spaces i, communications in algebra, 12(8) (1984) 953–1019. [12] pumplün d. and röhrl h. banach spaces and totally convex spaces ii, communications in algebra 13(5) (1985) 1047–1113. [13] pumplun d. eilenberg–moore algebras revisited, seminarberichte, fb mathematik und informatic, fernuniversität 29 (1988) 57–144. [14] rudin w. functional analysis, intern. series in pure and appl. math., mcgraw–hill, 1991. [15] ryan r.a. introduction to tensor product of banach spaces, springer monographs in math., springer verlag (london), 2002. [16] schaefer h.h. topological vector spaces, graduate texts in mathematics, springer, 1999. [17] sydow w. on hom–functors and tensor products of topological vector spaces, lecture notes in math. 962 (1982) 292–301. [18] schatten r. a theory of cross spaces, annals of math. studies 26, princeton university press, 1950. [19] von neumann j. on infinite direct products, compositio math. 6 (1938) 1–77. received november 2001 revised september 2002 m. sioen departement wiskunde, vrije universiteit brussel, pleinlaan 2, 1050 brussel, belgium e-mail address : msioen@vub.ac.be s. verwulgen departement wiskunde–informatica, universiteit antwerpen, middelheimlaan 1, 2020 antwerpen, belgium e-mail address : stijn.verwulgen@ua.ac.be sthuragt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 7989 star-hurewicz and related properties m. bonanzinga, f. cammaroto and lj.d.r. kočinac ∗ abstract. we continue the investigation of star selection principles first considered in [9]. we are concentrated onto star versions of the hurewicz covering property and star selection principles related to the classes of open covers which have been recently introduced. 2000 ams classification: 54d20. keywords: selection principles, (strongly) star-menger, (strongly) starrothberger, (strongly) star-hurewicz, groupability, weak groupability, ω-cover, γ-cover. 1. introduction a number of the results in the literature show that many topological properties can be described and characterized in terms of star covering properties (see [3], [13], [2], [12]). the method of stars has been used to study the problem of metrization of topological spaces, and for definitions of several important classical topological notions. we use here such a method in investigation of selection principles for topological spaces. let a and b be collections of open covers of a topological space x. the symbol s1(a, b) denotes the selection hypothesis that for each sequence (un : n ∈ n) of elements of a there exists a sequence (un : n ∈ n) such that for each n, un ∈ un and {un : n ∈ n} ∈ b [18]. the symbol sfin(a, b) denotes the selection hypothesis that for each sequence (un : n ∈ n) of elements of a there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un and ⋃ n∈n vn is an element of b [18]. in [9], kočinac introduced star selection principles in the following way. ∗the first and the second authors were supported by murst pra 2000. the third author (corresponding author) was supported by msrs, grant n0 1233. 80 m. bonanzinga, f. cammaroto, lj.d.r. kočinac definition 1.1. let a and b be collections of open covers of a space x. then: (a) the symbol s∗1(a, b) denotes the selection hypothesis: for each sequence (un : n ∈ n) of elements of a there exists a sequence (un : n ∈ n) such that for each n, un ∈ un and {st(un, un) : n ∈ n} is an element of b; (b) the symbol s∗fin(a, b) denotes the selection hypothesis: for each sequence (un : n ∈ n) of elements of a there is a sequence (vn : n ∈ n) such that for each n ∈ n, vn is a finite subset of un, and ⋃ n∈n{st(v, un) : v ∈ vn} ∈ b; (c) by u∗fin(a, b) we denote the selection hypothesis: for every sequence (un : n ∈ n) of members of a there exists a sequence (vn : n ∈ n) such that for every n, vn is a finite subset of un and {st(∪vn, un) : n ∈ n} ∈ b or there is some n ∈ n such that st(∪vn, un) = x. definition 1.2. let a and b be collections of open covers of a space x and let k be a family of subsets of x. then we say that x belongs to the class ss ∗ k(a, b) if x satisfies the following selection hypothesis: for every sequence (un : n ∈ n) of elements of a there exists a sequence (kn : n ∈ n) of elements of k such that {st(kn, un) : n ∈ n} ∈ b. when k is the collection of all one-point [resp., finite, compact] subspaces of x we write ss∗1(a, b) [resp., ss ∗ fin(a, b), ss ∗ comp(a, b)] instead of ss ∗ k(a, b). here, as usual, for a subset a of a space x and a collection p of subsets of x, st(a, p) denotes the star of a with respect to p, that is the set ∪{p ∈ p : a ∩ p 6= ∅}; for a = {x}, x ∈ x, we write st(x, p) instead of st({x}, p). in [10] it was explained that selection principles in uniform spaces are actually a kind of star selection principles. let x be a space. if u and v are families of subsets of x we denote by u ∧v the set {u ∩ v : u ∈ u, v ∈ v}. the symbols [x]<ω and [x]≤n denote the collection of all finite subsets of x and all subsets of x having ≤ n elements, respectively. in this paper all spaces will be hausdorff. a and b will be collections of the following open covers of a space x: o: the collection of all open covers of x; ω: the collection of ω-covers of x. an open cover u of x is an ω-cover [5] if x does not belong to u and every finite subset of x is contained in an element of u; γ: the collection of γ-covers of x. an open cover u of x is a γ-cover [5] if it is infinite and each x ∈ x belongs to all but finitely many elements of u. ogp: the collection of groupable open covers. an open cover u of x is groupable [11] if it can be expressed as a countable union of finite, pairwise disjoint subfamilies un, n ∈ n, such that each x ∈ x belongs to ∪un for all but finitely many n; owgp: the collection of weakly groupable open covers. a cover u of x is a weakly groupable [1] if it is a countable union of finite, pairwise disjoint sets un, n ∈ n, such that for each finite set f ⊂ x we have f ⊂ ∪un for some n. star-hurewicz and related properties 81 we consider only spaces x whose each ω-cover contains a countable ωsubcover (or equivalently, for each n ∈ n, every open cover of xn has a countable subcover). thus all considered covers are assumed to be countable. recall that a space x is said to have the menger property [15], [6], [16], [8] (resp. the rothberger property [17], [16], [18] if the selection hypothesis sfin(o, o) (resp. s1(o, o)) is true for x. the following terminology was introduced in [9]. a space x is said to have: 1. the star-rothberger property sr, 2. the star-menger property sm, 3. the strongly star-rothberger property ssr, 4. the strongly star-menger property ssm, if it satisfies the selection hypothesis: 1. s∗1(o, o), 2. s∗fin(o, o) (or, equivalently, u ∗ fin(o, o)), 3. ss∗1(o, o), 4. ss∗fin(o, o), respectively. in 1925 in [6] (see also [7]), w. hurewicz introduced the hurewicz covering property for a space x in the following way: h: for each sequence (un : n ∈ n) of open covers of x there is a sequence (vn : n ∈ n) of finite sets such that for each n vn ⊂ un, and for each x ∈ x, for all but finitely many n, x ∈ ∪vn. two star versions of this property are: sh: a space x satisfies the star-hurewicz property if for each sequence (un : n ∈ n) of open covers of x there is a sequence (vn : n ∈ n) such that for each n ∈ n vn is a finite subset of un and each x ∈ x belongs to st(∪vn, un) for all but finitely many n. ssh: a space x satisfies the strongly star-hurewicz property if for each sequence (un : n ∈ n) of open covers of x there is a sequence (an : n ∈ n) of finite subsets of x such that each x ∈ x belongs to st(an, un) for all but finitely many n (i.e. if x satisfies ss∗fin(o, γ)). in this paper we study some properties of these spaces. we also consider sm and ssm spaces, in particular in connection with new classes of covers that appeared recently in the literature groupable and weakly groupable covers. 2. spaces related to sm spaces theorem 2.1. if each finite power of a space x is sm, then x satisfies u∗fin(o, ω). proof. let (un : n ∈ n) be a sequence of open covers of x and let n = n1 ∪ n2 ∪ · · · be a partition of n into infinitely many infinite subsets. for each k and each m ∈ nk let wm = {u1 × · · · × uk : u1, · · · , uk ∈ um}. then 82 m. bonanzinga, f. cammaroto, lj.d.r. kočinac (wm : m ∈ nk) is a sequence of open covers of x k, and since xk is a starmenger space, one can choose a sequence (hm : m ∈ nk) such that for each m, hm ∈ [wm] <ω and ⋃ m∈nk {st(h, wm) : h ∈ hm} is an open cover of x k. for every m ∈ nk and every h ∈ hm we have h = u1(h) × · · · × uk(h), where ui(h) ∈ um for every i ≤ k. put vm = {ui(h) : i ≤ k, h ∈ hm}. then for each m ∈ nk vm is a finite subset of um. we claim that {st(∪vn, un) : n ∈ n} is an ω-cover of x. let f = {x1, · · · , xs} be a finite subset of x. then x = (x1, · · · , xs) ∈ x s so that there is an n ∈ ns such that x ∈st(h, wn) for some h ∈ hn. but h = u1(h) × · · · × us(h), where u1(h), · · · , us(h) ∈ vn. the point x belongs to some w ∈ wn of the form v1 × · · · × vs, vi ∈ un for each i ≤ s, which meets ui(h)×· · ·×us(h). this means that for each i ≤ s we have xi ∈ st(ui(h), un) ⊂ st(∪vn, un), i.e. f ⊂ st(∪vn, un). so, x satisfies u∗fin(o, ω). � now we shall see that the previous theorem can be given in another form. theorem 2.2. for a space x the following are equivalent: (1) x satisfies u∗fin(o, ω); (2) x satisfies u∗fin(o, o wgp). proof. because each countable ω-cover is weakly groupable, (1) implies (2) is trivial, so that we have to prove only (2) ⇒ (1). let (un : n ∈ n) be a sequence of open covers of x. let for each n, hn := ∧ i≤n ui. apply (2) to the sequence (hn; n ∈ n). there is a sequence (wn : n ∈ n) such that for each n wn ∈ [hn] <ω and {st(∪wn, hn) : n ∈ n} is a weakly groupable cover of x. there is, therefore, a sequence n1 < n2 < · · · in n such that for each finite set f in x one has f ⊂ ∪{st(∪wi, hi) : nk ≤ i < nk+1} for some k. consider the sequence (vn : n ∈ n) defined in the following way: vn = ⋃ i k0. since st(ti, vi) ⊂ st(si, ui) for all i with nk ≤ i < nk+1, we have that for each k > k0, x ∈ st(sk, uk), i.e. {st(sn, un) : n ∈ n} is a γ-cover of x. � the previous theorem suggests to consider also the selection principle ss∗1(o, o gp) that is naturally related to the ssh property. we have the following result. theorem 5.3. let a space x satisfies the following condition: for each sequence (un : n ∈ n) of open covers of x there is a sequence (an : n ∈ n) of subsets of x such that for each n |an| ≤ n and {st(an, un) : n ∈ n} is a γ-cover of x. then x satisfies ss∗1(o, o gp). proof. let (un : n ∈ n) be a sequence of open covers of x. for each n let vn = ∧ (n−1)n/2 n0. for each n write an as an = {xi : (n − 1)n/2 < i ≤ n(n + 1)/2}. then {st(xi, ui) : i ∈ n} is an open groupable cover of x. indeed, consider the sequence n1 < n2 < · · · < nk < · · · of natural numbers defined by nk = k(k − 1)/2. then for each point x ∈ x we have x ∈ ⋃ nk fa(n) for every n ≥ na. further, consider for each n ∈ n the finite set an := {1, 2, · · · , f(n)} subset of n. we claim that the sequence (an : n ∈ n) witnesses for (un : n ∈ n) that a is relatively ssh in ψ(a). indeed, for each a ∈ a the intersection 88 m. bonanzinga, f. cammaroto, lj.d.r. kočinac un(a) ∩ an 6= ∅ (because fa(n) ∈ an ∩ un(a)) for each n ≥ na, i.e. each point a ∈ a belongs to all but finitely many sets st(an, un). on the other hand, the subspace a of ψ(a) is the discrete space of cardinality b and thus it can not be ssh. let us remark that according to a result from [14] this space ψ(a) is ssm. acknowledgements. the third author thanks indam for the support and the dipartimento di matematica of the università di messina and f. cammaroto for the hospitality he enjoyed during his visit in november/december 2002. references [1] l. babinkostova, lj.d.r. kočinac and m. scheepers, combinatorics of open covers (viii), topology appl. (to appear). [2] m. bonanzinga, star-lindelöf and absolutely star-lindelöf spaces, q & a in gen. topology 16 (1998), 79–104. [3] e.k. van douwen, g.m. reed, a.w. roscoe and i.j. tree, star covering properties, topology appl. 39 (1991), 71–103. [4] r. engelking, general topology (pwn, warszawa, 1977). [5] j. gerlits and zs. nagy, some properties of c(x), i, topology appl. 14 (1982), 151–161. [6] w. hurewicz, über eine verallgemeinerung des borelschen theorems, math. z. 24 (1925), 401–421. [7] w. hurewicz, über folgen stetiger funktionen, fund. math. 9 (1927), 193–204. [8] w. just, a.w. miller, m. scheepers and p.j. szeptycki, the combinatorics of open covers ii, topology appl. 73 (1996), 241–266. [9] lj.d. kočinac, star-menger and related spaces, publ. math. debrecen 55 (1999), 421–431. [10] lj.d.r. kočinac, selection principles in uniform spaces, note di matematica (to appear). [11] lj.d.r. kočinac and m. scheepers, combinatorics of open covers (vii): groupability, fund. math. 179 (2003), 131–155. [12] m.v. matveev, on properties similar to countable compactness and pseudocompactness, vestnik mgu, ser. mat. mekh., (1984), n0 2, 24–27 (in russian). [13] m.v. matveev, a survey on star covering properties, topology atlas, preprint n0 330, 1998. [14] m.v. matveev, on the extent of ssm spaces, preprint. [15] k. menger, einige überdeckungssätze der punktmengenlehre, sitzungsberischte abt. 2a, mathematik, astronomie, physik, meteorologie und mechanik (wiener akademie, wien) 133 (1924), 421–444. [16] a.w. miller and d.h. fremlin, on some properties of hurewicz, menger and rothberger, fund. math. 129 (1988), 17–33. [17] f. rothberger, eine verschärfung der eigenschaft c, fund. math. 30 (1938), 50–55. [18] m. scheepers, combinatorics of open covers i: ramsey theory, topology appl. 69 (1996), 31–62. received december 2002 accepted february 2004 star-hurewicz and related properties 89 m. bonanzinga, f. cammaroto (milena@dipmat.unime.it, camfil@unime.it) dipartimento di matematica, università di messina, 98166, messina, italia lj.d.r. kočinac (lkocinac@ptt.yu) faculty of sciences and mathematics, university of nǐs, 18000, nǐs, serbia @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 361–376 orderability and continuous selections for wijsman and vietoris hyperspaces debora di caprio and stephen watson dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. bertacchi and costantini obtained some conditions equivalent to the existence of continuous selections for the wijsman hyperspace of ultrametric polish spaces. we introduce a new class of hypertopologies, the macro-topologies. both the wijsman topology and the vietoris topology belong to this class. we show that subject to natural conditions, the base space admits a closed order such that the minimum map is a continuous selection for every macro-topology. in the setting of polish spaces, these conditions are substantially weaker than the ones given by bertacchi and costantini. in particular, we conclude that polish spaces satisfying these conditions can be endowed with a compatible order and that the minimum function is a continuous selection for the wijsman topology, just as it is for [0, 1]. this also solves a problem implicitely raised in bertacchi and costantini’s paper. 2000 ams classification: 54b20, 54a10, 54d15, 54e35. keywords: selection, vietoris topology, wijsman topology, macro-topology, ∆topology, ordered space, compatible order, sub-compatible order, extra-dense set, lexor, complete lexor, polish space, star-set, n-coordinated-function, ncoordinated-set. 1. introduction. we define a “macro-topology” to be an admissible hyperspace topology, finer than the lower vietoris topology. this new class of hypertopologies contains the wijsman topology, the vietoris topology and a rich subclass of ∆-topologies. with the help of a new object called “lexor” and using the “extra-dense” sets (new objects as well), we construct a suitable order on a generic topological space. we give conditions under which this order is closed, showing its “subcompatibility” under some further, but natural hypotheses. the properties of 362 d. di caprio and s. watson this order make it possible to prove the continuity of the minimum function a → min(a) from a “macro-hyperspace” to the base space. when the base space is metrizable and complete, in particular, we find a sufficient condition for the existence of such an order, and consequently, for the minimum map to be a continuous selection for the wijsman hyperspace, just as happens in the setting of compact orderable metric spaces, such as [0, 1]. the result about wijsman hyperspaces leads to a deeper study of the conditions that in [3] the authors assign on a metric space (x,d) in order to obtain a continuous selection for the associated wijsman hyperspace. as a final result, they prove that every separable complete “ultrametric” space has a wijsman continuous selection if and only if a further condition, called “condition (])”, holds at each point. we focus our attention on a particular subfamily of r p(x)\{∅}, the collection of all the real-valued functions defined on the set of all nonempty subsets of x, whose elements are determined by a finite number of points and real numbers. we call “n-coordinated-functions” the elements of this family determined by n points and n real numbers. starting on them, we introduce the notions of “n-coordinated-set” and “star-set” (when n = 1), showing the existence of a natural relationship between both of them and the wijsman basic and subbasic open subsets, respectively. having a base of starsets is proved to be both a weaker condition than condition (]) given in [3], and a stronger condition than the one developed to prove the existence of the order. 2. preliminaries. let x be a nonempty set. we denote with (x,τ) a topological space and with (x,d) a metric space, which is understood to be endowed with the topology induced by the metric d. given a topology on x, let cl(x) be the collection of all nonempty closed subsets of x and c(x,r) the set of all the real-valued continuous functions on x. for every e ⊆ x, e and ec stand for the closure and the complement of e in x, respectively. we also set: e− = {a ∈ cl(x) : a∩e 6= ∅}, e+ = {a ∈ cl(x) : a ⊆ e} = {a ∈ cl(x) : a∩ec = ∅}. for every v ⊆ x, cl(x)\(v c)+ = v −. let (x,τ) be a topological space and ∆ be a nonempty subfamily of cl(x). the ∆-topology τ∆ on cl(x) has as a subbase all sets of the form u−, where u is an open set, plus all the sets of the form (bc)+, where b ∈ ∆ (see [13]). when ∆ = cl(x), the corresponding ∆-topology is the well-known vietoris topology. let: orderability and continuous selections 363 v− = {u− : u open in x}, v+ = {u+ : u open in x}. the family v− forms a subbase for the lower vietoris topology τ−v on cl(x); while the family v+ determines a subbase for the upper vietoris topology τ+v on cl(x). the supremum of these two hypertopologies is the vietoris topology : τv = τ + v ∨ τ−v . in general, given ∆ ⊆ cl(x), τ∆ = τ+∆ ∨ τ−v , where τ + ∆ denotes the upper ∆-topology on cl(x). let (x,d) be a metric space. the open ball with center x ∈ x and radius � > 0 is given by sd(x,�) = {y ∈ x : d(x,y) < �}. the diameter of a nonempty subset a of x and the distance from x ∈ x to a are expressed by the familiar formulas: diam(a) = sup{d(a,b) : a,b ∈ a} and d(x,a) = inf{d(x,a) : a ∈ a}. for every x ∈ x, d(x,−) denotes the distance functional from cl(x) to r defined by d(x,−)(a) = d(x,a) for every a ∈ cl(x). the wijsman topology τwd on cl(x) is the weak topology determined by the family of distance functionals {d(x,−) : x ∈ x}. equivalently, the wijsman topology on cl(x) can be defined by having as a subbase all the sets of the form: a−(x,α) = {a ∈ cl(x) : d(x,a) < α} and a+(x,α) = {a ∈ cl(x) : d(x,a) > α}, where x ∈ x and α > 0. a net of closed subsets of x, {aλ}λ∈λ, τwd-converges to a ∈ cl(x) if for every x ∈ x, limλ d(x,aλ) = d(x,a), i.e. (cl(x),τwd) can be embedded in c(x,r), equipped with the topology of the pointwise convergence, under the identification map a → d(−,a) (cf. sections 1.2 and 2.1 in [1]). we can also present the wijsman topology as split in two halves (section 4.2 in [1]; see also [5], [8], [11]): τwd = τ + wd ∨ τ−wd. a subbase for τ + wd consists of all the sets of the form a+(x,α) (x ∈ x and α > 0), whereas τ−wd, coinciding with the lower vietoris topology, has as a subbase all the sets of the form u− (u open in x). a map f : cl(x) → x is a selection for cl(x) if f(c) ∈ c for every c ∈ cl(x). by continuous selection we mean a selection f : cl(x) → x also continuous with respect to the hypertopology on cl(x). if x is a set linearly ordered by a relation <, we denote by τ< the order topology induced by < on x, i.e. the topology having as a subbase all the rays (←,x) and (x,→), where x ∈ x. in particular, all the intervals of the form 364 d. di caprio and s. watson (a,b), where a < b, are open with respect to τ<. given a topology τ on x, the order relation < on x is called compatible w.r.t. τ if τ = τ< and closed if the set (x × x)\ ≤ is open in the product topology on x × x (≤ is the partial order induced by <). every compatible order is a closed order. we denote by 〈x,y〉 a point in x×x. we will also use the following terminology: <-interval for any of the possible intervals (a,b), [a,b), (a,b] or [a,b], where a ≤ b; right <-ray for any ray of the form (a,→) or [a,→); left <-ray for any ray of the form (←,a) or (←,a]. 3. admissibility and macro-topologies. let (x,τ) be a topological space. a hyperspace topology on cl(x) is called admissible if the relative topology induced on x by the identification map x →{x}, coincides with the initial topology on x ([12]). it is understood that x must satisfy some separation properties in order to get the admissibility of the corresponding hyperspace. if (x,τ) is a t1-space, then τ∆ is admissible (remark 5.1 in [7]): in particular, the vietoris topology is admissible, whenever the base space is t1. if (x,d) is a metric space, then the wijsman topology on cl(x) is admissible (lemma 2.1.4 in [1]). definition 3.1. let (x,τ) be a topological space. a topology ω on cl(x) is called a macro-topology if it is admissible and τ−v < ω. if (x,τ) is a t1-space, then every ∆-topology on cl(x) is a macro-topology: in particular, the vietoris topology is a macro-topology. it is also clear that, given a metric space (x,d), the relative wijsman topology is a macro-topology. the following will turn out to be a useful result. lemma 3.2. let (x,τ) be a t1-space and h any hypertopology on cl(x) such that the identification map x →{x} is continuous. let b ⊆ x. if b− is a h-closed subset of cl(x), then b is a closed subset of x. proof. suppose b is not closed in x, i.e. there exists x ∈ b\b. then there exists a net {xα}α∈λ of points of b converging to x. since the identification map is continuous, {{xα}}α∈λ converges to {x} with respect to h. for every α ∈ λ, {xα}∈ b−, hence {x} must be in clh(b−) = b−, so that x ∈ b. � corollary 3.3. let (x,τ) be a topological space, ω a macro-topology on cl(x) and b ⊆ x. if b− is a ω-closed subset of cl(x), then b is a closed subset of x. corollary 3.4. let (x,τ) be a t1-space and b ⊆ x. if b− is a τ∆-closed subset of cl(x), then b is a closed subset of x. corollary 3.5. let (x,d) be a metric space and b ⊆ x. if b− is a τwd-closed subset of cl(x), then b is a closed subset of x. remark 3.6. let (x,τ) be a topological space and ω a macro-topology on cl(x). if a ∈ cl(x) and u = ac, then the following two chains of implications are equivalent: orderability and continuous selections 365 • a− is ω-closed ⇒ a is closed in x ⇒ a+ is ω-closed; • u+ is ω-open ⇒ u is open in x ⇒ u− is ω-open. the implication u is open in x ⇒ u− is ω-open follows from τ−v < ω; while a− is ω-closed ⇒ a is closed in x has just been shown (corollary 3.3). 4. when the minimum map is a continuous selection for macro-hyperspaces. proposition 4.1. let (x,τ) be a topological space, h be any hypertopology on cl(x) and < be a linear order on x such that τ is generated by a family r of right <-rays. if: (i) for every a ∈ cl(x), min(a) exists; (ii) for every r ∈r, r+ is a h-open subset of cl(x). then the mapping a → min(a) is a continuous selection from (cl(x),h) to (x,τ). proof. only the continuity of the mapping a → min(a) needs to be proved. let a ∈ cl(x) and r be a basic open right <-ray containing min(a). then r+ is h-open neighbourhood of a such that for every c ∈ r+, min(c) ∈ r. � the dual of proposition 4.1 is also true: we leave the proof to the reader. proposition 4.2. let (x,τ) be a topological space, h be any hypertopology on cl(x) and < be a linear order on x such that τ is generated by a family l of left <-rays. if: (i) for every a ∈ cl(x), min(a) exists; (ii) for every l ∈l, l− is a h-open subset of cl(x). then the mapping a → min(a) is a continuous selection from (cl(x),h) to (x,τ). definition 4.3. let (x,τ) be a topological space. a linear order < on x is called sub-compatible w.r.t. τ if τ< ≤ τ and τ has a subbase consisting of right and left <-rays. a topological space (x,τ) is sub-orderable (see 14.b.13 and 15.a.14 in [4]) if there exists a linear order < sub-compatible w.r.t. τ. it is well-known that the class of sub-orderable space concides with the one of subspaces of orderable spaces (17.a.22 in [4]). obviously, every compatible order is sub-compatible, just as every ordarable space is also sub-orderable. a combination of proposition 4.1 and proposition 4.2 leads to the following: proposition 4.4. let (x,τ) be a topological space, h be any hypertopology on cl(x) and < be sub-compatible w.r.t. τ. if: (i) for every a ∈ cl(x), min(a) exists; (ii) for every subbasic open right <-ray r, r+ is a h-open subset of cl(x); (iii) for every subbasic open left <-ray l, l− is a h-open subset of cl(x). 366 d. di caprio and s. watson then the mapping a → min(a) is a continuous selection from (cl(x),h) to (x,τ). given a sub-orderable space, conditions (ii) and (iii) of the previous proposition indeed quite often can be verified: for instance if h is any ∆-topology, where ∆ contains all the left rays which are complement of subbasic right rays (see lemma 3.8 in [6]); or, more concretely, if h is the vietoris topology. after section 6 and the main result, it will be clear that (ii) holds true for the wijsman topology if x is a complete metric space having a countable base with a peculiar property (see corollary 7.2). in case the base space is orderable, the subbasic open rays involved are of the form (←,x) and (x,→), where x ∈ x. moreover, if h is finer than the lower vietoris topology, condition (iii) is always satisfied. therefore, the following are immediate consequences of proposition 4.4. corollary 4.5. let (x,τ) be a topological space, ω be a macro-topology on cl(x) and < be a compatible linear order on x. suppose that: (i) for every a ∈ cl(x), min(a) exists; (ii) for every x ∈ x, (x,→)+ is a ω-open subset of cl(x). then the mapping a → min(a) is a continuous selection from (cl(x), ω) to (x,τ). corollary 4.6. let (x,d) be a metric space whose topology admits a compatible linear order < with respect to which each a ∈ cl(x) has a smallest element. if for every x ∈ x, (x,→)+ is a τwd-open subset of cl(x), then the mapping a → min(a) is a continuous selection from (cl(x),τwd) to (x,d). another consequence of proposition 4.4 is a classical and well-known result about the existence of continuous selections for the vietoris hyperspace of orderable spaces ([9], [10]). corollary 4.7. let (x,τ) be a topological space whose topology admits a compatible order < with respect to which each closed subset has a smallest element. then the mapping a → min(a) is a continuous selection from (cl(x),τv ) to (x,τ). the following includes an important result of beer, lechicki, levi and naimpally (corollary 5.6 in [2]): we give here a more direct and easier proof. proposition 4.8. let (x,d) be a metric space. the following are equivalent: (1) x is compact; (2) τwd = τv on cl(x); (3) τ+wd = τ + v on cl(x). proof. (2) ⇔ (3). it follows from the fact that τ−wd = τ − v and τ + wd ≤ τ+v always happens. (1) ⇒ (3). let u be open in x and a ∈ u+. uc is closed in x, and hence compact. also a∩uc = ∅. since a and uc are closed, d(x,a) > 0, whenever x ∈ uc. for every x ∈ uc, choose rx such that 0 < rx < d(x,a). the family orderability and continuous selections 367 {sd(x,rx) : x ∈ uc} is an open cover for the compact uc. then there exist x1, · · · ,xn ∈ x and r1, · · · ,rn > 0 (n ∈ ω) such that uc ⊆ ⋃n i=1 sd(xi,ri) and a∩sd(xi,ri) = ∅ for every i = 1, · · · ,n. then, a ∈ ⋂n i=1 a +(xi,ri) ⊆ u+. (3) ⇒ (1). if τ+wd = τ + v , x is separable (see following remark) and (cl(x),τwd) = (cl(x),τv ) is metrizable (see theorem 2.1.5 in [1]). by theorem 4.6 in [12], x is compact. � remark 4.9. let (x,d) be a metric space and τδ(d) be the topology on cl(x) generated by all the sets of the form v − and v ++ = {f ∈ cl(x) : d(f,v c) > 0}, where v runs over the open subsets of x. we can rappresent this topology as splitted in two parts: τδ(d) = τ + δ(d) ∨ τ−v : it is called d-proximal topology (see [1], [2], [8] among the others). it is known, but it is also easy to verify, that τ+wd ≤ τ + δ(d) ≤ τ+v . so, if τ + wd = τ+v , then τ + wd = τ+ δ(d) . by lemma 5.4 in [2] (if (x,d) is a metric space which is not second countable, then τwd 6= τδ(d).), x is separable. corollary 4.10. let (x,d) be a compact orderable metric space. the following are equivalent: (a) for every x ∈ x, (x,→)+ is a τv -open subset of cl(x); (b) for every x ∈ x, (x,→)+ is a τwd-open subset of cl(x). remark 4.11. it follows immediately from corollary 4.10 that corollary 4.6 and corollary 4.7 are equivalent formulations of the same result for compact orderable metric spaces. corollary 4.6 (or corollary 4.7) and proposition 4.8 yield: corollary 4.12. let (x,d) be a compact orderable metric space. then the mapping a → min(a) is a continuous selection from (cl(x),τwd) to (x,d). corollary 4.13. the mapping a → min(a) is a continuous selection from (cl([0, 1]),τwd) to ([0, 1],d), where d denotes the restriction to [0, 1] of the euclidean metric on r. 5. introducing a closed linear order on x: lexors and extra-dense sets. we have just shown (in section 4) how the existence of a sub-compatible order on a topological space (x,τ) with well specified properties ((i), (ii) and (iii) of proposition 4.4), is a sufficient condition for the minimum map from cl(x) to x to be a continuous selection when cl(x) is endowed with any hypertopology. but, when does this order exist? answering this question begins our main goal. this is why the present section, which focuses on the construction of such an order, can actually be considered as the heart of the paper. we start with some useful definitions. 368 d. di caprio and s. watson definition 5.1. let x be a set. if {un}n∈ω is a family of (arbitrary) covers of x satisfying the property: (i) if {an}n∈ω is a family of subsets of x such that an ∈ un for every n ∈ ω, then | ⋂ n∈ω an| ≤ 1, and for every n ∈ ω, 0 there exist δ,θ ∈ r, with 0 < δ < θ ≤ �, such that sd(x,δ) = sd(x,θ). a polish space is a separable completely metrizable space. the main result of bertacchi and costantini (theorem 3 in [3]) can be written as follows: proposition 8.2. let (x,d) be a polish ultrametric space. the following are equivalent: (a) condition (]) holds at each x ∈ x; (b) there exists a continuous selection from (cl(x),τwd) to (x,d). in this section, we give some conditions weaker than condition (]): under such conditions each polish space is proved to have a countable base satisfying the requirement of corollary 7.2. we can then conclude, without caring about ultrametric properties, that a continuous selection not only always exists for wijsman hyperspaces, but that there is a very natural map which is a selection, namely, the minimum map. we introduce the following notion: definition 8.3. let (x,d) be a metric space, x1, · · · ,xn ∈ x and r1, · · · ,rn > 0 (n ∈ ω). the n-coordinated-function determined by x1, · · · ,xn, r1, · · · ,rn is the function frixi : p(x)\{∅} → r defined by f ri xi (a) = max(ri −d(xi,a)), for every nonempty a ⊆ x. we write coordinated-function instead of 1coordinated-function. definition 8.4. let (x,d) be a metric space and v be a proper subset of x. v is a n-coordinated-set if there exist x1, · · · ,xn ∈ x and r1, · · · ,rn > 0 (n ∈ ω) such that orderability and continuous selections 373 (i) frixi (a) ≥ 0 for every a ∈ v ; (ii) frixi (v c) < 0. a coordinated-set is a 1-coordinated-set. for the next definition we need the notion of “excess”. given a metric space (x,d) and a,b ⊆ x, the excess of a over b with respect to d is defined by the formula ed(a,b) = sup{d(a,b) : a ∈ a}. excess may assume value +∞ and is not symmetric (see section 1.5 in [1] for more details and examples). in particular, ed(a,x) will denote the excess of the set a over the singleton {x}. notice that ed(x,a) = d(x,a), while ed(a,x) is quite different. definition 8.5. let (x,d) be a metric space and v be a proper subset of x. v is a star-set if there exists xv ∈ x, such that: (∗) ed(v,xv ) < d(xv ,v c). we say that v is a star-set around x. lemma 8.6. let (x,d) be a metric space, x ∈ x and v be a proper subset of x. v is a coordinated-set if and only if it is a star-set. proof. if v is a coordinated-set, there exist x ∈ x and r > 0 such that frx(a) ≥ 0 for every a ∈ v and frx(v c) < 0, i.e. r ≥ d(x,a) if a ∈ v and r < d(x,v c). hence d(a,x) ≤ r < d(x,v c) whenever a ∈ v . therefore, ed(v,x) = sup{d(a,x) : a ∈ v} < d(x,v c). suppose now that v is a star-set and that (∗) holds for some x. let r = ed(v,x). then, d(x,a) ≤ ed(v,x) = r < d(x,v c) for every a ∈ v . the coordinated-function frx satisfies (i) and (ii) of the definition of coordinatedset. � lemma 8.7. let (x,d) be a metric space and x ∈ x. the following are equivalent: (a) condition (]) holds at x; (b) for every � > 0 there exists a positive δ < � such that sd(x,δ) is a star-set around x; (c) inf{r > 0 : sd(x,r) is a star-set around x} = 0. proof. (a) ⇒ (b). given � > 0, there exist δ, θ ∈ r such that 0 < δ < θ ≤ � and sd(x,δ) = sd(x,θ). it is easy to check that ed(sd(x,δ),x) < d(x,sd(x,δ) c), so that (∗) holds at x. (b) ⇒ (c) and (c) ⇒ (a) are easy to check. � corollary 8.8. let (x,d) be a metric space. if condition (]) holds at each x ∈ x, then x has a base b of star-sets of the form sd(x,δ). 374 d. di caprio and s. watson proof. suppose condition (]) holds at each point. by lemma 8.7, for every x ∈ x and � > 0, there exists δ� < � such that sd(x,δ�) is a star-set. since {sd(x,�) : x ∈ x,� > 0} is a base for the topology on x, so is the family b = {sd(x,δ�) : x ∈ x,� > 0}. � proposition 8.9. let (x,d) be a separable metric space. if condition (]) holds at each x ∈ x, then x has a countable base b consisting of star-sets. proof. by corollary 8.8, x has a base a consisting of star-sets. since x is separable, it also has a countable base. hence, there exists a countable subfamily b ⊆ a, which is still a base for x (in general, if x is a regular second countable space and b is a base for x, then there exists a countable subcollection b′ ⊆b which is again a base for x). � lemma 8.10. let (x,d) be a metric space, v an open subset of x and n ∈ ω. the following are equivalent: (1) v is an n-coordinated-set; (2) (v c)+ = ⋂n i=1 a +(xi,ri), where xi ∈ x and ri > 0 for every i. proof. let x1, · · · ,xn ∈ x and r1, · · · ,rn > 0. the following are equivalent: • frixi (a) ≥ 0 for every a ∈ v and f ri xi (v c) < 0; • for every a ∈ v there exists i such that d(xi,a) ≤ ri, and d(xi,v c) > ri for every i; • for every a ∈ cl(x), a∩v = ∅ if and only if d(xi,a) > ri for all i; • (v c)+ = ⋂n i=1 a +(xi,ri). � let (x,τ) be a topological space and a ⊆ x. a is a basic closed subset of x if ac is a basic open subset. corollary 8.11. let (x,d) be a metric space and v an open subset of x. then: (i) v is a star-set if and only if (v c)+ is a τ+wd-subbasic open subset of cl(x); (ii) v is a n-coordinated-set, for some n ∈ ω, if and only if (v c)+ is a τ+wd-basic open subset of cl(x); (iii) v is a n-coordinated-set, for some n ∈ ω, if and only if v − is τ+wd-basic closed subset of cl(x). in particular, (iv) if v is a n-coordinated-set, for some n ∈ ω, then (v c)+ and v − are τwd-clopen subsets of cl(x). proof. recall that v − = cl(x)\(v c)+ (see preliminaries) and v − is open in the wijsman topology, if v is open (remark 3.6). � remark 8.12. from corollary 8.11(iv), it follows immediately that if (x,d) has a base b consisting of n-coordinated-sets (n ∈ ω), then (bc)+ and b− are τwd-clopen subsets of cl(x), for every b ∈b. orderability and continuous selections 375 we close with the preannounced result. proposition 8.13. let (x,d) be a polish space having a countable base of star-sets. then there exists a sub-compatible order on x such that the mapping a → min(a) is a continuous selection from (cl(x),τwd) to (x,d). proof. let b be the given base of star-sets. by corollary 8.11(iv), b− is τwdclopen, whenever b ∈b. apply corollary 7.3. � the fact that condition (]) is sufficient for the wijsman hyperspace to admit a continuous selection, stated by bertacchi and costantini in the setting of ultrametric spaces (see (a) ⇒ (b) of proposition 8.2), follows now as an easy consequence and without requiring d to be an ultrametric. this also solves the problem which is implicitely raised in the last two lines of [3]. corollary 8.14. let (x,d) be a polish space such that condition (]) holds at each point of x. then there exists a continuous selection from (cl(x),τwd) to (x,d). proof. by proposition 8.9, x has a countable base of star-sets. apply proposition 8.13. � notice that the converse of proposition 8.13 holds if (x,d) is an ultrametric space: use (b) ⇒ (a) of proposition 8.2 and proposition 8.9. this yields to the following result, which completes the main one of [3] and shows that the converse of proposition 8.9 is also valid provided that d is an ultrametric. proposition 8.15. let (x,d) be a polish ultrametric space. the following are equivalent: (a) x has a countable base of star-sets; (b) there exists a sub-compatible order on x such that the minimum map a → min(a) is a continuous selection from (cl(x),τwd) to (x,d); (c) there exists a continuous selection from (cl(x),τwd) to (x,d); (d) condition (]) holds at each x ∈ x. proof. (a) ⇒ (b) follows from propositon 8.13; (b) ⇒ (c) is trivial; (c) ⇒ (d) is (b) ⇒ (a) of proposition 8.2; (d) ⇒ (a) follows from proposition 8.9. � acknowledgements. the authors wish to thank c. costantini for remarking that corollary 8.14 actually solves the problem left open in [3] and suggesting proposition 8.15, as well as the comments before it. 376 d. di caprio and s. watson references [1] g. beer, topologies on closed and closed convex sets (kluwer academic publishers, 1993). [2] g. beer, a. lechicki, s. levi, s.a. naimpally, distance functionals and suprema of hyperspace topologies, ann. mat. pura ed appl. 162 (1992), 367–381. [3] d. bertacchi, c. costantini, existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, topology and its applications 88 (1998), 179–197. [4] e. čech, topological spaces (wiley, new york, 1966). [5] i. del prete, b. lignola, on convergence of closedvalued multifunctions, boll. un. mat. ital. 6-b (1983), 819–834. [6] d. di caprio, e. meccariello, notes on separation axioms in hyperspaces, q. & a. in general topology 18 (2000), 65–86. [7] g. di maio, ľ. holá, on hit-and-miss topologies, rend. acc. sc. fis. mat. napoli lxii (1995), 103–124. [8] g. di maio, s.a. naimpally, comparison of hypertopologies, rend. ist. mat. univ. trieste 22 (1990), 140–161. [9] r. engelking, r.v. heath, e. michael, topological well-ordering and continuous selections, invent. math. 6 (1968), 150–158. [10] w. fleischman (ed.), set-valued mappings, selections, and topological properties of 2x, lecture notes in mathematics 171 (springer-verlag, new york, 1970). [11] ľ. holá, r. lucchetti, equivalence among hypertopologies, setvalued analysis 3 (1995), 339–350. [12] e. michael, topologies on spaces of subsets, trans. amer. math. soc. 71 (1951), 152–182. [13] h. poppe, eine bemerkung über trennungsaxiome im raum der abgeschlossenen teilmengen eines topologischen raumes, arch. math. 16 (1965), 197–199. received february 2002 revised october 2002 debora di caprio department of mathematics and statistics, york university, 4700 keele street, north york, ontario, canada, m3j 1p3 e-mail address : dicaper@mathstat.yorku.ca stephen watson department of mathematics and statistics, york university, 4700 keele street, north york, ontario, canada, m3j 1p3 e-mail address : watson@mathstat.yorku.ca @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 79–89 holonomy, extendibility, and the star universal cover of a topological groupoid osman mucuk and i̇lhan i̇çen abstract. let g be a groupoid and w be a subset of g which contains all the identities and has a topology. with some conditions on g and w, the pair (g,w) is called a locally topological groupoid. we explain a criterion for a locally topological groupoid to be extendible to a topological groupoid. in this paper we apply this result to get a topology on the monodromy groupoid mg which is the union of the universal covers of gx’s. 2000 ams classification: 22a05, 55m99, 55r15. keywords: locally topological groupoids, holonomy groupoid, extendibility. 1. introduction a groupoid is a small category in which each morphism is an isomorphism. thus a group is a particular example of a groupoid. there is considerable evidence (see for example [9]) that the extension from groups to groupoids is widely useful in mathematics, and is one way of encoding many of the intuitions and methods of sophus lie which are difficult to encode in the language purely of group theory. for this encoding, we need the notion of topological (and of lie groupoid) and so it is important to examine the extent to which standard constructions on topological groups are available for topological groupoids. the book [9] gives considerable information on this. in this paper we give an exposition of the construction of an analogue of the classical universal cover of a connected topological group, and which we call the monodromy groupoid, following pradines [12]. the ideas for this are taken from [10,6] but we use a result from [6] to give a more direct proof of the construction than in [6], although in this way we do lose some power, notably the monodromy principle as given in [6]. we again emphasise the use of the holonomy groupoid construction, as first developed by pradines in [12], which however contains no details. full details 80 o. mucuk and i̇. i̇çen were first given in [1], as explained there. we feel it important to stress the construction of pradines as expressing well the intuitive idea of non trivial holonomy as dealing with an ‘iteration of local procedures which returns to the starting position but not the starting value’. thus we see the concept of groupoid as adding to the concept of group an extra notion of ‘position’, through the set or space of objects, and of ‘transition’, through the arrows between objects. this extension has proved to be generally powerful. let g be a groupoid and w a subset of g containing all the identities in g. suppose that w has a topology. for certain conditions on w the pair (g,w) is called a locally topological groupoid. the topology on w does not in general extend to a topological groupoid structure on g which restricts to that on w , but there is a topological groupoid h, called the holonomy groupoid, with a morphism h → g such that h contains w as a subspace and h has a universal property. the full details of this result are given by aof and brown in [1]. a locally topological groupoid (g,w) is called extendible if there is a topology on g such that g is a topological groupoid with this topology and w is open in g. a locally topological groupoid is not in general extendible. it is proven by brown and mucuk in [7] that the charts of a foliated manifold may be chosen so that they give rise to a locally topological groupoid which in general is not extendible. we have also examples of locally topological groupoids, due to pradines and explained in [1], which are not extendible. a full account of the monodromy groupoids was given in [10] and published in [6]. let g be a topological groupoid in which each star gx has a universal cover. then the monodromy groupoid gm is constructed by mackenzie in [9] as the union over x in og of the universal covers based at 1x of the stars gx. in the locally trivial case in [9], the groupoid mg is given a topology such that mg is a topological groupoid with this topology. let g be a locally sectionable topological groupoid and w an open subset containing all the identities. in this paper we use a criterion obtained from holonomy to prove that the monodromy groupoid mg has a structure of topological groupoid such that each star (mg)x is a universal cover of gx. in [6] the groupoid associated with a pregroupoid is used to verify a monodromy property for mg, namely extendibility to mg of a local morphism on g. 2. groupoids and topological groupoids a groupoid g on og is a small category in which each morphism is an isomorphism. thus g has a set of morphisms, which we call just elements of g, a set og of objects together with functions α,β : g → og, �: og → g such that α� = β� = 1og, the identity map. the functions α, β are called initial and final point maps respectively. if a,b ∈ g and β(a) = α(b), then the product or composite ba exists such that α(ba) = α(a) and β(ba) = β(b). further, this composite is associative, for x ∈ og the element �(x) denoted by 1x acts as the identity, and each element a has an inverse a−1 such that α(a−1) = β(a), holonomy and extendibility 81 β(a−1) = α(a), aa−1 = (�β)(a), a−1a = (�α)(a). the map g → g, a 7→ a−1, is called the inversion. in a groupoid g for x,y ∈ og we write g(x,y) for the set of all morphisms with initial point x and final point y. for x ∈ og we denote the star {a ∈ g: α(a) = x} of x by gx and the costar {a ∈ g: β(a) = x} of x by gx. in g the set og is mapped bijectively to the set of identities by �: og → g. so we sometimes write og for the set of identities. let g be a groupoid and w a subset of g such that og ⊆ w . we say g is generated by w if each element of g may be written as a product of elements of w . let g be a groupoid. a subgroupoid of g is a pair of subsets h ⊆ g and oh ⊆ og such that α(h) ⊆ oh, β(h) ⊆ oh, 1x ∈ h for each x ∈ oh and h is closed under the partial multiplication and the inversion in g. a morphism of groupoids h and g is a functor, that is, it consists of a pair of functions f : h → g and of : oh → og preserving all the structures. definition 2.1. a topological groupoid is a groupoid g on og, together with topologies on g and og, such that the maps which define the groupoid structure are continuous, namely the initial and final point maps α,β : g → og, the object inclusion map �: og → g, x 7→ �(x), the inversion g → g, a 7→ a−1 and the partial multiplication gα ×β g → g, (b,a) 7→ ba, where the pullback gα ×β g = {(b,a) ∈ g×g: α(b) = β(a)} has the subspace topology from g×g. a morphism of topological groupoids f : h → g is a morphism of groupoids in which both maps f : h → g and of : oh → og are continuous. note that in this definition the partial multiplication gα×β g → g, (b,a) 7→ ba and the inversion map g → g,a 7→ a−1 are continuous if and only if the map δ : g ×α g → g, (b,a) 7→ ba−1, called the groupoid difference map, is continuous, where the pullback g×α g = {(b,a) ∈ g×g: α(b) = α(a)} has the subspace topology from g×g. again if one of the maps α,β and the inversion are continuous, then the other map is continuous. let x be a topological space. then g = x × x is a topological groupoid on x, in which each pair (y,x) is a morphism from x to y and the groupoid composite is defined by (z,y)(y,x) = (z,x). the inverse of (y,x) is (x,y) and the identity at 1x is the pair (x,x). note that in a topological groupoid, g, for each a ∈ g(x,y) right translation ra : gy → gx,b 7→ ba and left translation la : gx → gy, b 7→ ab are homeomorphisms. a groupoid g in which each star gx has a topology such that for each a ∈ g(x,y) the right translation ra : gy → gx,b 7→ ba (and hence also the left translation la : gx → gy,b 7→ ab) is a homeomorphism, is called a star topological groupoid 82 o. mucuk and i̇. i̇çen 3. locally topological groupoids and extendibility the following definition is due to ehresmann [8]. definition 3.1. let g be a groupoid and let x = og be a topological space. an admissible local section of g is a function σ : u → g from an open set in x such that ασ(x) = x for all x ∈ u; βσ(u) is open in x, and βσ maps u homeomorphically to βσ(u). let w be a subset of g containing og and let w have the structure of a topological space. we say that (α,β,w) is locally sectionable if for each w ∈ w there is an admissible local section σ : u → g of g such that (i) σα(w) = w, (ii) σ(u) ⊆ w and (iii) σ is continuous as a function from u to w . such a σ is called a continuous admissible local section. the following definition is due to pradines [12] under the name “morceau de groupoide différentiables”. definition 3.2. a locally topological groupoid is a pair (g,w) consisting of a groupoid g and a topological space w such that: i) og ⊆ w ⊆ g; ii) w = w−1; iii) the set w(δ) = (w ×α w) ∩ δ−1(w) is open in w ×α w and the restriction of δ to w(δ) is continuous; iv) the restrictions to w of the source and target maps α and β are continuous, and the triple (α,β,w) is locally sectionable; v) w generates g as a groupoid. note that in this definition, g is a groupoid but does not need to have a topology. the locally topological groupoid (g,w) is said to be extendible if there can be found a topology on g making it a topological groupoid and for which w is an open subset. in general, (g,w) is not extendible, but there is a holonomy groupoid hol(g,w) and a morphism ψ : hol(g,w) → g such that hol(g,w) admits the structure of topological groupoid and is the “minimal” such overgroupoid of g. the construction is given in detail in [1] and is outlined below. it is easiest to picture locally topological groupoids (g,w) for groupoids g such that α = β, so that g is just a bundle of groups. here is a specific such example of a locally topological groupoid [1], which is not extendible. example 3.3. let f be the bundle of groups p : r × r → r, where r is the set of real numbers and p is the first projection. the usual topology on r×r gives f the structure of a topological groupoid in which each p−1(x) is isomorphic as an additive group to r. let n be the subbundle of f given by the union of the sets {(x, 0)} if x < 0 and {x}× z if x ≥ 0, where z is the set of integers. let g be the quotient bundle f/n and let q : f → g be the quotient morphism. then the source map α : g → r has α−1(x) isomorphic to r for x < 0 and to r/z for holonomy and extendibility 83 x ≥ 0. let w ′ be the subset r× (−1 4 , 1 4 ) of f. then q maps w ′ bijectively to w = q(w ′); let w have the topology in which this map is a homeomorphism. it is easily checked that (g,w) is a locally topological groupoid. suppose this locally topological groupoid is extended to a topological groupoid structure on g. let s′ be the section of p in which x 7→ (x, 1 8 ), and let s = qs′. then s is an admissible section of α but t = 9s is not. however t(0) = q(0, 1 8 ). let u be an open neighbourhood of ( 1 8 , 0) in r2 such that u is contained in w ′. then p(u) is contained in w and is a neighbourhood of t(0). but t−1q(u) is contained in [0,∞), so that t is not continuous. this gives a contradiction, and shows that the locally topological groupoid (g,w) is not extendible. by contrast, if we proceed as before but replace n by n1, which is the union of the sets {(x, 0)} for x ≤ 0 and {x}×z for x > 0, then the resulting locally topological groupoid (g1,w1) is extendible. example 3.4. there is a variant of the last example in which f is as before, but this time n is the union of the groups {x}×(1 + |x|)z for all x ∈ r. if one takes w ′ as before, and w is the image of w ′ in g = f/n, then the locally topological groupoid (g,w) can be extended to give a topological groupoid structure on g. however, now consider w as a differential manifold. the differential structure cannot be extended to make g a differential groupoid with w as submanifold. the reason is analogous to that given in the previous example, namely that such a differential structure would entail the existence of a local differentiable admissible section whose sum with itself is not differentiable, thus giving a contradiction. example 3.5. ([7]) let x be a paracompact foliated manifold. then there is an equivalence relation, written rf , on x determined by the leaves. so rf is a subspace of x ×x and becomes a topological groupoid on x with the usual multiplication (z,y)(y,x) = (z,x), for (y,x), (z,y) ∈ rf . for any subset u of x we write rf (u) for the equivalence relation on u whose classes are the plaques of u. if λ = {(uλ,φλ)} is a foliated atlas for x, we write w(λ) for the union of the sets rf (uλ) for all domains uλ of charts of λ. let w(λ) have its topology as a subspace of rf and so of x ×x. in [7] it is proved that the pair (rf ,w ′), where w ′ derives from a refinement of λ, is a locally topological groupoid. some special foliated manifolds are given in which the locally topological groupoid (rf ,w ) is not extendible. there is a main globalisation theorem for a locally topological groupoid due to aof-brown [1], and a lie version of this is stated in brown-mucuk [6]; it shows how a locally topological groupoid gives rise to its holonomy groupoid, which is a topological groupoid satisfying a universal property. this theorem gives a full statement and proof of a part of théorème 1 of [12]. theorem 3.6. (globalisation theorem) let (g,w) be a locally topological groupoid. then there is a topological groupoid h, a morphism φ : h → g of groupoids, and an embedding i : w → h of w to an open neighbourhood of oh, such that: 84 o. mucuk and i̇. i̇çen i) φ is the identity on objects, φi = idw , φ−1(w) is open in h, and the restriction φw : φ−1(w) → w of φ is continuous; ii) (universal property) if a is a topological groupoid and ζ : a → g is a morphism of groupoids such that: a) ζ is the identity on objects; b) the restriction ζw : ζ(w) → w of ζ is continuous and ζ−1(w) is open in a and generates a; c) the triple (αa,βa,a) has enough continuous admissible local sections, then there is a unique morphism ζ′ : a → h of topological groupoids such that φζ′ = ζ and ζ′a = iζa for a ∈ ζ−1(w). the groupoid h is called the holonomy groupoid hol(g,w) of the locally topological groupoid (g,w); its essential uniqueness follows from the condition (ii) above. we outline the proof of which full details are given in [1]. some details of part of the construction are needed for proposition 3.7. proof. (outline) let γ(g) be the set of all admissible local sections of g. define a product on γ(g) by (ts)x = (tβsx)(sx) for two admissible local sections s and t. if s is an admissible local section then write s−1 for the admissible local section βsd(s) → g,βsx 7→ (sx)−1. with this product γ(g) becomes an inverse semigroup. let γc(w) be the subset of γ(g) consisting of admissible local sections which have values in w and are continuous. let γc(g,w) be the subsemigroup of γ(g) generated by γc(w). then γc(g,w) is again an inverse semigroup. intuitively, it contains information on the iteration of local procedures. let j(g) be the sheaf of germs of admissible local sections of g. thus the elements of j(g) are the equivalence classes of pairs (x,s) such that s ∈ γ(g),x ∈d(s), and (x,s) is equivalent to (y,t) if and only if x = y and s and t agree on a neighbourhood of x. the equivalence class of (x,s) is written [s]x. the product structure on γ(g) induces a groupoid structure on j(g) with x as the set of objects, and source and target maps [s]x 7→ x, [s]x 7→ βsx. let jc(g,w) be the subsheaf of j(g) of germs of elements of γc(g,w). then jc(g,w) is generated as a subgroupoid of j(g) by the sheaf jc(w) of germs of elements of γc(w). thus an element of jc(g,w) is of the form [s]x = [sn]xn · · · [s1]x1 where s = sn · · ·s1 with [si]xi ∈ jc(w),xi+1 = βsixi, i = 1, . . . ,n and x1 = x ∈d(s). let ψ : j(g) → g be the final map defined by ψ([s]x) = s(x), where s is an admissible local section. then ψ(jc(g,w)) = g. let j0 = jc(w) ∩ ker ψ. then j0 is a normal subgroupoid of jc(g,w); the proof is in [1] lemma 2.2. the holonomy groupoid hol = hol(g,w) is defined to be the quotient holonomy and extendibility 85 jc(g,w)/j0. let p : jc(g,w) → hol be the quotient morphism and let p([s]x) be denoted by 〈s〉x. since j0 ⊆ ker ψ there is a surjective morphism φ : hol → g such that φp = ψ. the topology on the holonomy groupoid hol such that hol with this topology is a topological groupoid, is constructed as follows. let s ∈ γc(g,w). a partial function σs : w → hol is defined as follows. the domain of σs is the set of w ∈ w such that βw ∈d(s). a continuous admissible local section f through w is chosen and the value σsw is defined to be σsw = 〈s〉βw〈f〉αw = 〈sf〉αw. it is proven that σsw is independent of the choice of the local section f and that these σs form a set of charts. then the initial topology with respect to the charts σs is imposed on hol. with this topology hol becomes a topological groupoid. the proof is in aof-brown [1]. � from the construction of the holonomy groupoid we easily obtain the following extendibility condition, which is proved in [6]. proposition 3.7. the locally topological groupoid (g,w) is extendible to a topological groupoid structure on g if and only if the following condition holds: (1) if x ∈ og, and s is a product sn · · ·s1 of local sections about x such that each si lies in γc(w) and s(x) = 1x, then there is a restriction s′ of s to a neighbourhood of x such that s′ has its image in w and is continuous, i.e. s′ ∈ γc(w). proof. the canonical morphism φ : h → g is an isomorphism if and only if ker ψ ∩ jc(w) = ker ψ. this is equivalent to ker ψ ⊆ jc(w). we now show that ker ψ ⊆ jc(w) if and only if the condition (1) is satisfied. suppose ker ψ ⊆ jc(w). let s = sn · · ·s1 be a product of admissible local sections about x ∈ og with si ∈ γc(w) and x ∈ds such that s(x) = 1x. then [s]x ∈ jc(g,w) and ψ([s]x) = s(x) = 1x. so [s]x ∈ ker ψ, so that [s]x ∈ jc(w). so there is a neighbourhood u of x such that the restriction s | u ∈ γc(w). suppose the condition (1) is satisfied. let [s]x ∈ ker ψ. since [s]x ∈ jc(g,w), then [s]x = [sn]xn · · · [s1]x1 where s = sn · · ·s1 and [si]xi ∈ jc(w), xi+1 = βsixi, i = 1, . . . ,n and x1 = x ∈ d(s). since s(x) = 1x, then by (1), [s]x ∈ jc(w). � in effect, proposition 3.7 states that the non-extendibility of (g,w) arises from the holonomically non trivial elements of jc(g,w). intuitively, such an element h is an iteration of local procedures (i.e. of elements of jc(w)) such that h returns to the starting point (i.e. αh = βh) but h does not return to the starting value (i.e. ψh 6= 1). the following result, which is given as corollary 4.6 in [6], gives a circumstance in which this extendibility condition is easily seen to apply. 86 o. mucuk and i̇. i̇çen corollary 3.8. let q be a topological groupoid and let p : m → q be a morphism of groupoids such that p : om → oq is the identity. let w be an open subset of q such that (1) oq ⊆ w ; (2) w = w−1; (3) w generates q; (4) (αw ,βw ,w ) is continuously locally sectionable; and suppose that ı̃ : w → m is given such that pı̃ = i : w → q is the inclusion and w ′ = ı̃(w) generates m. then m admits a unique structure of topological groupoid such that w ′ is an open subset and p : m → q is a morphism of topological groupoids mapping w ′ homeomorphically to w . proof. it is easy to check that (m,w ′) is a locally topological groupoid. we prove that condition (1) in proposition 3.7 is satisfied (with (g,w) replaced by (m,w ′)). suppose we are given the data of (1). clearly, ps = psn · · ·ps1, and so ps is continuous, since g is a topological groupoid. since s(x) = 1x, there is a restriction s′ of s to a neighbourhood of x such that im(ps) ⊆ w . since p maps w ′ homeomorphically to w , then s′ is continuous and has its image contained in w . so (1) holds, and by proposition 3.7, the topology on w ′ is extendible to make m a topological groupoid. � remark 3.9. it may seem unnecessary to construct the holonomy groupoid in order to verify extendibility under condition (1) of proposition 3.7. however the construction of the continuous structure on m in the last corollary, and the proof that this yields a topological groupoid, would have to follow more or less the steps given in aof and brown [1] as sketched above. thus it is more sensible to rely on the general result. as corollary 3.8 shows, the utility of (1) is that it is a checkable condition, both positively or negatively, and so gives clear proof of the non-existence or existence of non-trivial holonomy. 4. the star universal cover of a topological groupoid let x be a topological space and suppose that each path component of x admits a simply connected covering space. it is standard that if π1x is the fundamental groupoid of x, topologised as in brown and danish-naruie [3] and x ∈ x, then the target map β : (π1x)x → x is the universal covering map of x based at x. let g be a topological groupoid. the groupoid mg is defined as follows. as a set, mg is the union of the stars (π1gx)1x. the object set is the same as that of g. the initial point map α: mg → x maps all of (π1gx)1x to x, while the final point map β : mg → x is on (π1gx)1x the composition of the two target maps (π1gx)1x β→ gx β→ x. holonomy and extendibility 87 as explained in [9] there is a groupoid multiplication on mg defined by concatenation, i.e. [b] ◦ [a] = [ba(1) + a] where the + inside the bracket denotes the usual composition of the paths. here a is assumed to be a path in gx from 1x to a(1), where β(a(1)) = y, say, so that b is a path in gx, and for each t ∈ [0, 1], the product b(t)a(1) is defined in g, yielding a path b(a(1)) from a(1) to b(1)a(1). it is straightforward to prove that in this way mg becomes a groupoid, and that the final maps of paths induces a morphism of groupoids p: mg → g. if each gx admits a simply connected cover at 1x then we may topologise each (mg)x so that it is the universal cover of gx based at 1x, and then mg becomes a star topological groupoid, which means each star (mg)x has a topology such that each right translation (and hence each left translation) is a homeomorphism we call mg the star universal cover of g. if x is a topological space which has a simply connected cover and g = x×x, then mg = π1(x). if g is a topological group, then mg is a universal cover of g. theorem 4.1. let g be a locally sectionable topological groupoid in which each star gx is path connected and has a simply connected cover. let w be an open subset of g containing og such that w = w−1 and w generates g. suppose that each star wx = w ∩ gx is connected and simply connected. then the groupoid mg constructed above may be given a structure of topological groupoid such that each star (mg)x is a universal cover of gx and w is isomorphic to an open subset w̃ of mg. proof. to get a topology on mg as required we use corollary 3.8. for this we first define a map ı̃: w → mg as follows: let u ∈ w(x,y), where w(x,y) = w ∩ g(x,y), then u ∈ wx. since wx is path connected, there is a path a in wx from 1x to u. here note that 1x ∈ wx since og ⊆ w . define ı̃(u) to be the unique homotopy class of a in wx. note that since wx is simply connected, ı̃ is well defined. the map ı̃: w → mg is injective. for if u,v ∈ w such that ı̃(u) = ı̃(v), then we have p̃ı(u) = p̃ı(v) and so u = v. let w̃ denote the image of w under the map ı̃: w → mg. thus w̃ has a topology such that the map ı̃: w → w̃ is a homeomorphism. note that by assumption the pair (g,w) satisfies the conditions 1-4 of corollary 3.8. so to apply corollary 3.8 to the pair (mg,w̃), we only need to prove that the subset w̃ generates mg as a groupoid. we prove this in the following lemma. lemma 4.2. the subset w̃ generates mg as a groupoid. proof. for this let [a] ∈ mg(x,y). so a is a path from 1x to g ∈ g(x,y). let s ⊆ [0, 1] be the set of s ∈ [0, 1] such that as = a|[0,s], the restriction of a to [0,s], can be written as = an ◦ · · · ◦ a1 for some n and im ai ⊆ w . since 88 o. mucuk and i̇. i̇çen s ⊆ [0, 1], s is bounded above by 1, and so u = sup s exists. then we prove the following: i) u ∈ s ii) u = 1. to prove (i), let a(u) ∈ g(x,xu). then the map f : [0, 1] → gxu defined by f(t) = a(t)(a(u))−1 is continuous and f(u) = 1xu ∈ w . hence there is an � > 0 such that f([u− �,u + �]) ⊆ w . hence the composition map δw ◦ (f ×f) : [u− �,u + �] × [u− �,u + �] → w ×α w → g (t1, t2) 7→ a(t1)(a(t2))−1 is continuous, where δw is the restriction to w ×α w → g of the groupoid difference map δ : g ×α g → g, (b,a) 7→ ba−1. hence there is an �′ > 0 such that �′ < � and δw (f ×f)([u− �′,u + �′] × [u− �′,u + �′]) ⊆ w (?) since u = sup s, there is an element s ∈ s such that u − �′ < s. hence as can be written as an ◦ · · · ◦ a1 for n with im ai ⊆ w and so we have that au = an+1 ◦·· ·◦a1 where an+1(t) = a(t)(a(s))−1 for t ∈ [s,u]. by (?) we have that im an+1 ⊆ w . hence u ∈ s. to prove (ii) suppose that u < 1. since u ∈ s, we have au = an ◦ · · · ◦ a1 for some n such that im ai ⊆ w . let ai(1) = gi ∈ g(xi−1,x) with x0 = x and xn = y. hence we have a(u) = gn ◦ ·· · ◦g1 and the path a can be divided into small paths as a = a(u + �) + a(u) + (an ◦ · · · ◦a1) where im ai ⊆ w . since the map [u, 1] → gxn, t 7→ a(t)(a(u)) −1 is continuous there is an � > 0 such that a(t)(a(u))−1 ∈ w for t ∈ [u,u + �]. hence au+� = an+1 ◦ (an · · ·a1) with an+1(t) = a(t)((a(u))−1 for t ∈ [u,u + �]. hence we have that au+� ∈ s, which is a contradiction. this proves that u = 1. � hence by corollary 3.8, the groupoid mg has a unique structure of topological groupoid such that w̃ is open in mg and p: mg → g is a morphism of the topological groupoids. � acknowledgements. we are grateful to ronald brown for introducing us to this area and for his helpful encouragement. we would also like to thank the referee for several helpful comments. holonomy and extendibility 89 references [1] m. e. -s. a. -f. aof and r. brown, the holonomy groupoid of a locally topological groupoid, topology appl. 47 (1992), 97–113. [2] r. brown, topology; a geometric account of general topology, homotopy types and the fundamental groupoid (ellis horwood, chichester, 1988). [3] r. brown and g. danesh-naruie, the fundamental groupoid as a topological groupoid, proc. edinburgh math. soc. 19 (2) (1975), 237–244. [4] r. brown and i̇. i̇çen, lie local subgroupoids and their lie holonomy and monodromy groupoids, topology appl. 115 (2001), 125–138. [5] r. brown, i̇. i̇çen and o. mucuk, local subgroupoids ii: examples and properties, topology appl. 127 (2003), 393–408. [6] r. brown and o. mucuk, the monodromy groupoid of a lie groupoid, cah. top. géom. diff. cat. 36 (1995), 345–369. [7] r. brown and o. mucuk, foliations, locally lie groupoids and holonomy, cah. top. géom. diff. cat. 37 (1996), 61–71. [8] c. ehresmann, catégories topologiques et catégories différentiables, coll. géom. diff. glob. bruxelles (1959), 137–150. [9] k. c. h. mackenzie, lie groupoids and lie algebroids in differential geometry, london math. soc. lecture note series 124, (cambridge university press, 1987). [10] o. mucuk, covering groups of non-connected topological groups and the monodromy groupoid of a topological groupoid, phd thesis, university of wales, 1993. [11] o. mucuk, locally topological groupoid, turkish journal of mathematics 21-2 (1997), 235–243. [12] j. pradines, théorie de lie pour les groupoides différentiables, relation entre propriétés locales et globales, comptes rendus acad. sci. paris 263 (1966), 907–910. received january 2002 revised september 2002 o. mucuk erciyes university, faculty of science and art, department of mathematics, kayseri, turkey. e-mail address : mucuk@erciyes.edu.tr i̇. i̇çen i̇nönü university, faculty of science and art, department of mathematics, malatya, turkey. e-mail address : iicen@inonu.edu.tr holonomy, extendibility, and the star universal cover of a topological groupoid. by o. mucuk and i. içen shahdasagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 125-130 on nearly hausdorff compactifications sejal shah and t. k. das abstract. we introduce and study here the notion of nearly hausdorffness, a separation axiom, stronger than t 1 but weaker than t 2. for a space x, from a subfamily of the family of nearly hausdorff spaces, we construct a compact nearly hausdorff space rx containing x as a densely c*-embedded subspace. finally, we discuss when rx is βx. 2000 ams classification: primary 54c45, secondary 54d35. keywords: regular closed set, filter, compactification, wallman base. 1. introduction a closed subset f in a topological space x is called a regular closed set if f = cl(intf ). we denote the family of all regular closed subsets of x by r(x). observe that r(x) is closed under finite union. also, if f ∈ r(x), then cl(x − f ) = x − intf ∈ r(x). in section 2, we define and study the notion of a nearly hausdorff space (nh-space). we introduce a topological property π and note that a space with property π is an nh-space if and only if it is urysohn. a flow diagram showing various implications about separation axioms supported by necessary counter examples is included in this section. a map f :x→y is called a density preserving map (dp-map) if for a ⊆ x, int(clf (a)) 6= φ whenever inta 6= φ [2]. we provide here an example showing that the nh-property is not preserved even under continuous dp-maps. note that if x is an nh-space then r(x) forms a base for closed sets in x. in section 3, we obtain a ’βx like’ compactification of an nh-space x with property π. since r(x) need not be closed under finite intersections, we form a new collection rf (x), of all possible finite intersections of members of r(x). we observe that for an nh-space x with the property π, the set rx = {α ⊆ rf (x) | α is an r-ultrafilter} with the natural topology, is a nearly hausdorff compact space which contains x as a dense c*-embedded subspace. the natural question when rx = βx is discussed in section 4. we observe that 126 s. shah and t. k. das an nh-space x for which rf (x) is a wallman base, is a completely regular hausdorff space and hence for such a space x, rx = βx, the stone-čech compactification of x. in particular, if x is normal or zero−dimensional then rx = βx. the problem whether rx = βx for any tychonoff space x is still open. 2. nearly hausdorff spaces definition 2.1. distinct points x and y in a topological space x are said to be separated by subsets a and b of x if x ∈ a−b and y ∈ b−a. definition 2.2. a topological space x is called nearly hausdorff (nh-) if for every pair of distinct points of x there exists a pair of regular closed sets separating them. definition 2.3. a space x is said to have property π if for every f ∈ r(x) and x 6∈ f there exists an h ∈ r(x) such that x ∈ inth and h ∩ f = φ. the symbol x(π) denotes a space x having property π. remark 2.4. henceforth all our regular spaces are hausdorff. recall that a space x is urysohn [5] if for each pair of distinct points of x, we can find disjoint regular closed sets of x containing the points in their respective interiors. we have following implications: regular ⇒ urysohn (π) ⇔ nearly hausdorff (π) ⇒ urysohn ⇓ t1 ⇐ nearly hausdorff ⇐ hausdorff examples given below (refer [4, 5]) justify that unidirectional implications in the above flow diagram need not be revertible. in addition, example 2.5(b) shows that nearly hausdorffness is not a closed hereditary property. example 2.5. (a) an infinite cofinite space is a t 1 space but not an nh-space. the onepoint compactification of the space x in our note 2 is a non-hausdorff compact nh-space. (b) consider n, the set of natural numbers with cofinite topology and i = [0, 1] with the usual topology. let x = n×i and define a topology on x as follows: neighborhoods of (n, y), y 6= 0 will be usual neighborhoods {(n, z) | y − ǫ < z < y + ǫ} in in = {n}×i for small positive ǫ; neighborhoods of (n, 0) will have the form {(m, z) | m ∈ u , 0 ≤ z < ǫm}, where u is a neighborhood of n in n and ǫm is a small positive number for each m ∈ u . the resulting space x is a non hausdorff, nh-space without property π. it is easy to observe that the subspace n of x is closed but is a non-nh, t 1 space. (c) let a be the linearly ordered set {1, 2, 3, ...., ω, ...., -3, -2, -1} with the interval topology and let n be the set of natural numbers with the discrete topology. define x to be a×n together with two distinct on nearly hausdorff compactifications 127 points say a and −a which are not in a×n. the topology ℑ on x is determined by the product topology on a×n together with basic neighborhoods m n(a) = {a} ∪ {(i, j) | i < ω, j > n} and m n(−a) = {−a} ∪ {(i, j) | i > ω, j > n} about a and −a respectively. resulting space x is a non-urysohn hausdorff space without property π. in fact, there does not exist any regular closed set containing a and disjoint from mn(−a). this example also justifies that a hausdorff space need not have property π. (d) let s be the set of rational lattice points in the interior of the unit square except those whose x-coordinate is 1 2 . define x to be s ∪ {(0, 0)} ∪ {(1, 0)} ∪ {( 1 2 , r √ 2) | r ∈ q, 0 < r √ 2 < 1}. topologize x as follows: local basis for points in x from the interior of unit square are same as those inherited from the euclidean topology and for other points following local bases are taken: u n(0, 0) = {(x, y) ∈ s | 0 < x < 14 , 0 < y < 1 n } ∪ {(0, 0)}, u n(1, 0) = {(x, y) ∈ s | 3 4 < x < 1, 0 < y < 1 n } ∪ {(1, 0)}, u n( 12 , r √ 2) = {(x, y) ∈ s | 1 4 < x < 3 4 , |y−r √ 2| < 1 n }. the resulting space x is a urysohn space without property π. (e) let x be the set of real numbers with neighborhoods of non-zero points as in the usual topology, while neighborhoods of 0 will have the form u − a, where u is a neighborhood of 0 in the usual topology and a = { 1 n | n ∈ n}. note that x is a non regular urysohn space with property π. theorem 2.6. a nonempty product of an nh-space is an nh-space if and only if each factor is an nh-space proof. let {x γ}γ∈λ be a family of nh-spaces, λ 6= φ and let x, y ∈ x =∏ γ∈λ xγ , x 6= y. then xγ 6= yγ for some γ ∈ λ. since each xγ is an nh-space, there exist regular closed sets fx and fy separating xγ and yγ . define u =∏ β∈λ uβ and v = ∏ β∈λ vβ , where vβ = uβ = xβ, for β 6= γ and uγ = intfx, vγ = intfy. the regular closed sets clu and clv in x separate x and y. proof of the converse is similar. � lemma 2.7. let x be an nh-space and let f :x→y be a dp-epimorphism. then for a regular closed subset h of y we have clf (clf −1(inth)) = h and hence r(y ) = {clf (f ) | f ∈ r(x)}. proof. clearly for h ∈ r(y ), clf (clf −1(inth)) ⊆ h. for the reverse containment, if y ∈ h−clf (clf −1(inth)) then there exists an open set u containing y satisfying f −1(u∩inth) = φ which contradicts y∈h = clinth. � note 1. lemma 2.7 is stated in note 2.2 of [2] for a regular space. further, observe that the first projection of the space n×i in example 2.5 (b) shows that continuous image of an nh-space need not be an nh-space. on the other hand, if we consider second projection of n×i on [0, 1] with cofinite topology then we get that even a continuous density preserving image of an nh-space need not be an nh-space. 128 s. shah and t. k. das 3. the space rx for an nh-space x, a filter α ⊆ rf (x)−{φ} is called an r-filter. a maximal r-filter is called an r-ultrafilter. the family of all r-ultrafilters in x is denoted by rx. observe that for x ∈ x, there exists a unique r-ultrafilter αx in rx such that ∩αx = {x}. further, if x is compact then each r-ultrafilter in x is fixed. the converse is also true: if c is an open cover of x then b = {f ∈ r(x)|x − u ⊂ f , for some u ∈ c} does not have finite intersection property for otherwise b will generate a fixed r-ultrafilter which will contradict that c is a cover of x. hence c has a finite subcover. topologize the set rx by taking b = {f | f ∈ r(x)} as a base for closed sets in rx, where f = {α ∈ rx | f ∈ α} and f ∈ r(x). the map r:x→rx defined by r(x) = αx, where αx = {f ∈ rf (x) | x ∈ f } is an embedding. lemma 3.1. let x be an nh-space with property π. then the space rx of all r-ultrafilters in x is a compact nh-space which contains x as a dense subspace. proof. clearly αx = {f ∈ rf (x)|x ∈ f } is an r-filter. for maximality of αx, suppose a = ∩ni=1ai in rf (x) be such that a ∩ f 6= φ, for each f in αx. if possible suppose for some i, ai 6∈ αx. then x 6∈ ai. by the property π, there exists an h in r(x) such that x ∈ inth and h ∩ ai = φ. therefore h ∈ αx and hence h ∩ a 6= φ. but this implies φ 6= h ∩ a ⊂ h ∩ ai = φ, a contradiction. further clrx r(f ) = f for all f ∈ r(x) implies r is a dense embedding. � note 2. a compactification of a non-urysohn space without property π may also be an nh-space. for example, consider the subspace y = {( 1 n , 1 m ) | n ∈ n, |m| ∈ n} ∪ {( 1 n , 0) | n ∈ n} of the usual space r2. take x = y ∪ {p+, p−}; p+, p− 6∈ y and topologize it by taking sets open in y as open in x and a set u containing p+ (respectively p−) to be open in x if for some r ∈ n, {( 1 n , 1 m ) | n ≥r, m ∈ n} ⊆ u (respectively {( 1 n , 1 m ) | n ≥ r, −m ∈ n} ⊆ u ). the resulting space x is a non-urysohn hausdorff space without property π and its one point compactification is an nh-space. proposition 3.2. let the space x and rx be as in lemma 3.1. then x is c*-embedded in rx. proof. let f ∈ c*(x). suppose range of f ⊆ [0, 1] = i. for α in rx, define f ♯(α) = {h1 ∪ h2 ∈ r(i) | clx f −1(inth1 ∪ inth2) ∈ α}. note that if h1 ∪ h2 ∈ f ♯(α) then either h1 ∈ f ♯(α) or h2 ∈ f ♯(α). also f ♯(α) satisfies finite intersection property. thus ∩f ♯(α) 6= φ. we assert that ∩f ♯(α) = {t}, for some t ∈ i. assuming the assertion in hand, we define rf : rx→i by rf (α) = ∩f ♯(α). clearly rf restricted to x is f . we now establish continuity of rf . let α ∈ rx. then choose an open set g of i such that t ∈ g, where rf (α)=t. using on nearly hausdorff compactifications 129 regularity of i successively we obtain open sets g1, g2 such that t ∈ g1 ⊆ clg1 ⊆ g2 ⊆ clg2 ⊆ g. set ft = clg2 and ht = cl(i − clg1). since intft ∪ intht = i. we have ft ∪ ht ∈ f ♯(α) and as t 6∈ ht, ft ∈ f ♯(α) and ht 6∈ f ♯ (α). if lt = clx f −1(intht), then α 6∈ lt and the open set rx−lt contains α. finally the containment rf (rx− lt) ⊆ g establishes the continuity. for the assertion, one may use the above technique to note that {f ∈ r(i) | t ∈ intf } ⊆ f ♯ (α), for each t ∈ f ♯(α). � theorem 3.3. let x be an nh-space with property π. then there exists a compact nh-space rx in which x is densely c*-embedded. proof. follows from lemma 3.1 and proposition 3.2. � corollary 3.4. if x is a regular space, then it is densely c∗-embedded in rx. 4. when rx = βx? let x be an nh-space such that rf (x) is a wallman base. then by 19l(7) in [5], x is a completely regular space. therefore by corollary 3.4, x is c*embedded in rx. further if x is an nh-space such that rf (x) forms a wallman base then by 19l(5) in [5], rx is hausdorff. hence we have the following result: theorem 4.1. let x be an nh-space such that rf (x) is a wallman base. then rx = βx. corollary 4.2. if x is normal or zero-dimensional then rx = βx. question: is rx = βx when x is a tychonoff space? acknowledgements. we thank the referee for his/her valuable suggestions. references [1] e. čech, topological spaces, (john wiley and sons ltd., 1966). [2] t. k. das, on projective lift and orbit space, bull. austral. math. soc. 50 (1994), 445-449. [3] j. r. porter and r. g. woods, extensions and absolutes of hausdorff spaces, (springerverlag, 1988). [4] l. a. steen and j. a. seebach, jr., counterexamples in topology, (springer-verlag, 1978). [5] s. willard, general topology, (addition-wesley pub. comp., 1970). 130 s. shah and t. k. das received november 2004 accepted july 2005 sejal shah department of mathematics, faculty of science, the m. s. university of baroda, vadodara, india. t. k. das (tarunkd@yahoo.com) department of mathematics, faculty of science, the m. s. university of baroda, vadodara, india. @ appl. gen. topol. 15, no. 1 (2014), 1-9doi:10.4995/agt.2014.2019 c© agt, upv, 2014 hausdorff connectifications solai ramkumar department of mathematics, alagappa university, karaikudi 630 003, sivagangai district, tamil nadu, india. (ramkumarsolai@gmail.com) abstract disconnectedness in topological space is analyzed to obtain hausdorff connectifications of that topological space. hausdorff connectifications are obtained by some direct constructions and by some partitions of connectifications. also lattice structure is included in the collection of all hausdorff connectifications. 2010 msc: 54d35; 54d05; 54d40. keywords: h-closed sets; cut points, n-disconnected set; pointly connected mapping; connectifications; remainders of connectifications; lattice isomorphism. 1. introduction an extension of a topological space x is a topological space that contains x as a dense subspace. if an extension is a connected space, then that extension is called a connectification of that space. there is a book of j. r. porter and r. grandwoods that is devoted for hausdorff extensions. this paper also studies only hausdorff spaces and hausdorff connected extensions. it is easy to see that if x has a proper compact (or h-closed) open subset, then x has no hausdorff connectification. there are spaces which can not have connectification. porter and grandwoods gave some nice examples of hausdorff spaces in [7], that can not be densely embedded in a connected hausdorff space. several papers have been devoted to connectifications (see: [1], [2], [3], [9]). fedeli and le donne in [4] proved that a t1-space can be densely embedded in a pathwise connected t1-space if and only if it has no isolated points. charatonik in [3] considered generalized linear graphs to obtain hausdorff connectifications and characterized the one point hausdorff connectification of a subspace of a generalized received december 2011 – accepted may 2013 http://dx.doi.org/10.4995/agt.2014.2019 s. ramkumar linear graph in [2]. section 2 contains some basic ideas about disconnectedness of topological spaces. section 3 presents some direct constructions to obtain hausdorff connectifications of a topological space. we also obtain hausdorff connectifications through remainders in section 4. final section proves that if f is a continuous and connected mapping from x onto y such that f seperates every pair of disjoint regular open subsets of x, then the lattice c(x) is isomorphic to c(y ). all spaces under consideration are hausdorff topological spaces. 2. some disconnected spaces definition 2.1. a subset a of a space x is n-disconnected if a has exactly n + 1 no. of clopen subsets in a except ∅ and a. definition 2.2. a subset a of a space x is countably infinite disconnected if a has only countably infinite number of clopen subsets in a. a is countably disconnected if it is either n-disconnected or countably infinite disconnected. definition 2.3. a subset a of a space x is uncountably disconnected if a is not countably disconnected. example 2.4. (i) (0, 1) ∪(1, 2) is 1-disconnected. (ii) n ∪ k=1 (k, k + 1) is (2n − 2)-disconnected. (iii) set of all irrationals is uncountably disconnected subset of r. theorem 2.5. let f : x(⊆ z1) → z be a continuous mapping such that f(x) = y ⊆ z. if y is n-disconnected subset of z, then x is atleast a ndisconnected subset of z1. also, if a is a component in y , then f −1(a) is a component in x. proof. let y be a n-disconnected subset of a space z. then y has n + 1 no of clopen subsets. let them be {a1, a2, · · · · · · an+1}. since f is continuous, x has atleast n+ 1 no of clopen subsets namely, {b1, b2, · · · · · · bn+1}. also, if a is a component in y , then f−1(a) is a component in x. if not, then there is a connected subset c of x containing f−1(a). then f(c) is a connected subset of y containing a, which is a contradiction to the maximality of a. � theorem 2.6. let f : x → z be an one to one and open mapping. if y = f(x) and if x is a n-disconnected subset of z1, then y is atleast a ndisconnected subset of z. also, image of a component under the mapping f is a component in y . proof. if f is an one to one and open mapping, then f−1 : y → x is a continuous mapping. by the previous theorem 2.5, x is a n-disconnected subset of z1. � a subspace of a n-disconnected space need not be a n-disconnected space. consider a subspace a = [0, 1] ∪[2, 3] of [0, 1] ∪[2, 3] ∪[4, 5], for some fixed n. then a is 1-disconnected subspace of a 2-disconnected space [0, 1] ∪[2, 3] ∪[4, 5]. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 2 hausdorff connectifications theorem 2.7. let a be subset of a n-disconnected space x. then a is ndisconnected if a is a connected dense subspace of x. definition 2.8. a point x ∈ y is a cut point of x ⊆ y if there is a clopen subset a of x such that a ∪{x} ∪(x\a) is a connected subset of y . theorem 2.9. if x(⊆ y ) has n cut points, then x is atleast a n + 1disconnected subspace of y . proof. if x has n cut points (say) p1, p2, · · · · · · pn, then there are n-clopen subsets {ai : i = 1, 2, · · · · · · n} such that ai ∪(x\ai) ∪{pi} is a connected subset of y . thus x has 2n clopen subsets. also unions and intersections of clopen subsets are clopen subsets which increases the no of clopen subsets of x. thus x is atleast a n + 1-disconnected subspace of y . � there may be a clopen subset ak of x such that ak ∪(x\ak) ∪ e is a connected subset of y , where e is a subset of y and e contains more than one point. this also increases the no of clopen subsets of x. theorem 2.10. let x be a subspace of a space y . if x is n-disconnected subspace of y , then x has atmost n cut points. theorem 2.11. a point x ∈ y is a cut point of x ⊆ y if and only if there is a connected subset c of y such that c ∩ x = a ∪(x\a) and y \(c ∩ x) = {x}, where a is a clopen subset of x. proof. proof follows directly from the definition of cut point. � 3. some connectifications theorem 3.1. let x be a space having no isolated points. if x has finite number of distinct clopen subsets(2n-disconnected space) such that each clopen subset is neither h-closed nor compact, then there is an extension y of x such that y is connected and hence y is a connectification of x. proof. let x be a space having 2n distinct clopen subsets of x. let them be {a1, a2, · · · · · · a2n}. let a11 = a1. since x\a1 is a clopen subset of x, x\a1 is one of the member in {a2, · · · · · · a2n}. let it be a12. let a21 ∈ {a1, a2, · · · · · · a2n}\{a11, a12}. then x\a21 ∈ {a1, a2, · · · · · · a2n}\{a11, a12, a21}. let it be a22. thus we can arrange the clopen subsets {a1, a2, · · · · · · a2n} as {a11, a12, a21, a22 · · · · · · an1, an2}. let y = x ∪{p1, p2, · · · · · · pn}, where pi /∈ x for every i ∈ {1, 2, · · · · · · n}. define the open neighborhoods of pi be ui1 ∪ ui2 ∪ pi, where uij ∈ uij and uij is a decreasing sequence of nonempty open subsets of aij’s. then y is an extension of x. if y is not connected, then there exists a clopen subset c of y such that c ∩ x is a clopen subset of x. by construction, there are some ai1 and ai2 such that c ∩ x = ai1 or c ∩ x = ai2. then there is an element pi ∈ y such that every open neighborhood of pi intersects c ∩ x and x\c. thus pi ∈ cly (c ∩ x) and pi ∈ cly (x −c). that is pi ∈ cly (c) = c and pi ∈ cly (y −c) = y −c. that c© agt, upv, 2014 appl. gen. topol. 15, no. 1 3 s. ramkumar is pi ∈ c ∩(y − c), which is a contradiction. it remains only to prove that y is a hausdorff space. let x, y ∈ y and x 6= y. case: 1. let x, y ∈ y \x. then there exists pi and pj such that x = pi and y = pj. we can find nonempty open subsets ui1, ui2, vj1 and vj2 of ai1, ai2, aj1 and aji2, respectively such that ui1 ∪ ui2 ∪{pi} and vj1 ∪ vj2 ∪{pj} are open sets in y containing x and y, respectively. let x1 ∈ ui1, x2 ∈ ui2, x3 ∈ vj1 and x4 ∈ vj2. find disjoint open neighborhoods wx1, wx2, wx3, wx4 for x1, x2, x3, x4 such that wx1 ⊆ ui1, wx2 ⊆ ui2, wx3 ⊆ vj1 and wx4 ⊆ vj2. let u = wx1 ∪ wx2 ∪{pi} and v = wx3 ∪ wx4 ∪{pj}. thus u and v are disjoint open subsets of pi and pj, respectively. case: 2. let x ∈ x and y ∈ y \x. then there exists a clopen subset ai in x and pi ∈ y such that y = pi and x ∈ ai1 or x ∈ ai2. let us assume that x ∈ ai1. choose any open neighborhood u1x ∪ u2x ∪{pi} of pi and x0 ∈ u1x such that x 6= x0. then there are disjoint open sets vx ⊆ u1x and vx0 ⊆ u1x such that x ∈ vx and x0 ∈ vx0. let w = vx0 ∪ wx0 ∪{pi}, where wx0 is any nonempty open subset in ai2. then ux and w are disjoint open neighborhoods of x and y, respectively. case: 3. let x, y ∈ x. then there are disjoint open neighborhoods for x and y in x. by our construction, they are also open in y . � if x is finitely disconnected and has no isolated points, then we can give a one point connectification of x. we can attach only one point, whose open neighborhoods are of the form u1 ∪ u2 ∪ · · · · · · un, where ui ∈ ui and ui is decreasing sequence of non empty open subsets of the clopen subset ai of x. corollary 3.2. if x is a space having countable number of distinct clopen subsets of x(countably disconnected space) such that each clopen subset is neither h-closed nor compact and having no isolated points, then there is an extension y of x such that y is connected. corollary 3.3. if x is a space having uncountable number of distinct clopen subsets of x(uncountably disconnected space) such that each clopen subset is neither h-closed nor compact and having no isolated points, then there is an extension y of x such that y is connected. definition 3.4. let f : x → y be a mapping from x to y . then f is said to separates disjoint subsets a and b of x if f(a) ∩ f(b) = φ. theorem 3.5. let f : x → y be a continuous mapping from x onto y such that f seperates every pair of disjoint regular open subsets of x. if ξx is any connectification of x, then there is a connectification ζy for y . proof. let ζy = y ∪ k, where k = ξx\x. let u be an open set in y , then f−1(u) is open in x. find an open set v in ξx such that v ∩ x = f−1(u). let w = v ∩ x = f−1(u) and w1 = v ∩ k. define a topology on ζy by the collection of all sets of the form f(w) ∪ w1, where w = v ∩ x, w1 = v ∩ k and v = f−1(u) is an open subset of ξx. we prove that ζy with this topology is a connectification of y . c© agt, upv, 2014 appl. gen. topol. 15, no. 1 4 hausdorff connectifications let v be any open set in ζy . then there exist an open set u in ξx such that v = f(u ∩ x) ∪(u\x). we may choose u = f−1(v ) ∩ y . since x is dense in ξx, u ∩ x 6= φ. thus f(u ∩ x) ∩ y 6= φ and hence v ∩ y 6= φ. this implies that y is dense in ζy . if ζy is not connected, there exists a nonempty clopen subset u of ζy such that u ⊂ ζy and u 6= ζy . then, there exists an open set v (= f−1(u ∩ y )) such that f(v ) = u ∩ y . then v ∪ h is an open set in ξx, where h = u\y . similarly, since y \u is an open set, there exists an open set v1(= f−1((y \u) ∩ y )) such that f(v1) = y \u. then v1 ∪ h1 is an open set in ξx, where h1 = (y \u)\y . trivially, ξx\(v ∪ h) = v1 ∪ h1. also v ∪ h and v1 ∪ h1 are both open sets. this proves that ζy is connected. it remains only to prove that ζy is a hausdorff space. let x, y ∈ y and x 6= y. if x is in k, let x1 = x and if y is in k, let y1 = y. otherwise choose x1 ∈ f −1(x) and y1 ∈ f −1(y). then there are disjoint open sets u and v in ξx such that x1 ∈ u, y1 ∈ v and int(cl(u)) ∩ int(cl(v )) = φ. let u1 = int(cl(u)) ∩ x, u2 = int(cl(u))\x, v1 = int(cl(v )) ∩ x and v2 = int(cl(v ))\x. since f seperates every pair of disjoint regular open subsets in x, we have f(u1) ∩ f(v1) = φ. then f(u1) ∪ u2 and f(v1) ∪ v2 are disjoint open subsets of ζy containing x and y, respectively. � remark 3.6. in the above theorem 3.5, the condition “f seperates every pair of disjoint regular open subsets of x” used only for hausdorffness of a connectification. without that assumption, we can get a non hausdorff connectification of a space y . definition 3.7 ([8]). a mapping f : x → y is a said to be compact mapping if f−1(x) is a compact subset of x, for everey x ∈ x. theorem 3.8. let f : x → y be a continuous and compact mapping from x onto y . if ξx is any connectification of x, then there is a connectification ζy for y proof. proof for “ζx is a connected extension of x” follows from theorem 3.5. hausdorffness of ζx is to be proved. let x, y ∈ y and x 6= y. k = ξx\x. if x is in k, let c1 = {x} and if y is in k, let c2 = {y}. since f is a compact maping, f−1(x) = c1 and f −1(y) = c2 are compact subsets of x. then there are disjoint open sets u and v in ξx such that c1 ⊆ u and c2 ⊆ v . then f(u ∩ x) ∪(u\x) and f(v ∩ x) ∪(v \x) are disjoint open subsets of ζx containing x and y respectively. � definition 3.9. a mapping f : x → y is said to be a pointly connected mapping if f−1(x) is a connected subset of x, for every x ∈ y . the following examples show that the restriction of a pointly connected mapping need not be a pointly connected mapping and the composition of pointly connected mappings need not be a pointly connected mapping. example 3.10. let f : (r+ − z+) → z+ ∪{0} be a mapping defined by f(x) = [x], largest integer less than or equal to x. then f is a pointly connected c© agt, upv, 2014 appl. gen. topol. 15, no. 1 5 s. ramkumar mapping. let a = r+ − (z+ ∪{2n+1 2 : n = 0, 1, 2, · · · · · · }). then f|−1 a (m) = (m, 2m+1 2 ) ∪(2m+1 2 , m + 1), which is not a connected subset of a. example 3.11. let f : (r − z) → z be a mapping defined by f(x) = [x], for every x ∈ (r−z)and g : r−(z ∪{2n+1 2 : n = 0, 1, 2, · · · · · · }) → r−z be a mapping defined by f(x) = x, for every x ∈ r − (z ∪{2n+1 2 : n = 0, 1, 2, · · · · · · }). here, f and g are pointly connected mappings. but f ◦ g is not a pointly connected mapping, because (f ◦g)−1(m) = (m, 2m+1 2 ) ∪(2m+1 2 , m+1), for m > 0, which is not a connected subset of r − (z ∪{2n+1 2 : n = 0, 1, 2, · · · · · · }). definition 3.12. a mapping f : x → y is said to be a connected mapping if f−1(a) is a connected subset of x, whenever a is a connected subset of y . remark 3.13. (1) restriction of a connected mapping need not be a connected mapping. this can be verified from example 3.10 (2) composition of connected mappings is a connected mapping. (3) every connected mapping is a pointly connected mapping. the converse need not be true. (4) let f : x → y and g : y → z be two pointly connected mappings. then g◦f is a pointly connected mapping if g is an one to one mapping or f is a connected mapping. (5) let f : x → y be an one to one mapping from x onto y . if f is an open mapping, then f is a connected mapping. definition 3.14 ([6]). let f : x → y be a mapping from x to y and u ⊆ x. then f is said to be saturated on u, if there is a subset v of y such that u = f−1(v ) theorem 3.15. let f : x → y be a mapping from x to y . if f is an open mapping on every saturated subset of x and f seperates open subsets of x, then f is a connected mapping. proof. let u be an connected subset of y . if f−1(u) is not connected, we can find a clopen subset v of x such that v ∪x\v = f−1(u). since f is an open mapping on every saturated subset of x, f(v ) and f(x\v ) are open subsets of y . also f(v )∪f(x\v ) = u, because f seperates open sets in x, which is a seperation on u. thus f−1(u) is connected � the converse of the above remark 3.13 need not be true. the following example shows it. example 3.16. let x be the subspace [0, 1] ∪(2, 3] of r and let y be the subspace [0, 2] of r. define a map f : x → y by f(x) = { x if x ∈ [0, 1] x − 1 if x ∈ (2, 3] . then f is a continuous onto mapping. also f is a connected mapping and seperates open subsets of x, but not an open mapping. theorem 3.17. let f : x → y be a continuous and connected mapping from x onto y . then x is n-connected if and only if y is n-connected. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 6 hausdorff connectifications proof. if f is a continuous and connected mapping, then f preserves connected subset of x onto a connected subset of y . � 4. remainders of connectifications definition 4.1 ([8]). let ξ1x, ξ2x be two connectifications of a space x. then, we write ξ1x ≥ ξ2x, if there is a continuous function f : ξ1x → ξ2x such that f(x) = x, for all x ∈ x. theorem 4.2. let ξx be any connectification of a space x and {ki : i = 1, 2, · · ·n} be a collection of mutually disjoint nonempty compact subsets of ξx\x. choose n distinct points {pi : i = 1, 2, · · ·n} not in ξx and define a mapping h : ξx → ζx by h(p) = { p if p ∈ ξx\( n ∪ i=1 ki) pi if p ∈ ki , where ζx = (ξx\ n ∪ i=1 ki) ∪{p1, p2 · · · · · · pn}. let γx have the quotient topology induced by h. then ζx is a connectification for x such that ξx ≥ ζx. proof. proof of this theorem is similar to the proof of lemma: 2 in [5] � theorem 4.3. let ξx be any connectification of a regular space x. let {ki : i ∈ i} be a collection of mutually disjoint nonempty compact subsets of ξx\x and they are locally finite in ξx. let {pi : i ∈ i} be such that pi /∈ x, for every i ∈ i. then there is a connectification ζx for x such that ξx ≥ ζx. proof. let ζx = (ξx\ ∪ i∈i ki)∪{pi : i ∈ i} and y = (ξx\ ∪ i∈i ki) where pi are distinct, and pi /∈ ξx. define a map h : ξx → ζx by h(x) = { x if x ∈ y pi if x ∈ ki . let ζx have the quotient topology induced by the map h. since ξx is connected, ζx is connected. let u be an open set in ζx. then h−1(u) is an open set in ξx which intersects x so that h(h−1(u)) = u intersects h(x) = x. hence x is dense in ζx. consider two distinct elements y1, y2 in ζx. let a = h −1(y1) and b = h−1(y2). for any x ∈ a, there is an open set ux of x in ξx such that ux intersects only finite number of ki’s and ux ∩ b = φ. define an open set wx = ux\( n ∪ i=1 ki) in ξx containing x. find an open set vx of x such that cl(vx) ⊆ wx then {vx : x ∈ a} is an open cover of a. since f −1(x) is compact, for every x ∈ x. this open cover has a finite subcover {vx1, vx2, · · · vxm}, say. let u = m ∪ i=1 vxi. then u is an open set such that a ⊆ u and clu ∩ b = φ. similarly, we can find another open set v such that b ⊆ v and u ∩ v = φ. then f(u) and f(v ) are disjoint open sets in ζx such that f(a) = y1 ∈ f(u) and f(b) = y2 ∈ f(v ). this proves the hausdorffness of ζx. � remark 4.4. in the above theorem 4.3, regularity of x is used only for hausdorffness of ζx. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 7 s. ramkumar definition 4.5 ([8]). a space x is said to be a p-space if every gδ set is an open set in x. theorem 4.6. let x be a regular p-space having no isolated points and ξx be any connectification of x. let {ki : i ∈ i} be a collection of mutually disjoint nonempty compact subsets of ξx\x and they are locally countable in ξx. let {pi : i ∈ i} be such that pi /∈ x, for every i ∈ i. then there is a connectification ζx for x such that ξx ≥ ζx. proof. proof for connected extension is same as theorem: 4.3. hausdorffness of connectification has some difficulties. consider two distinct elements y1, y2 in ζx. let a = f−1(y1) and b = f −1(y2). for any x ∈ a, find an open set ux of x such that ux ∩ ki 6= φ, for every i ∈ n1 ⊆ n, the set of all natural numbers. define an open set wx = ux\( ∞ ∪ i=1 ki) in ξx containing x. find an open set vx of x such thatclvx ⊆ wx. since f −1(x) is compact, for every x ∈ x, {vx : x ∈ a} has a finite subcollection {vx1, vx2, · · · vxm}, such that a ⊆ u = m ∪ i=1 vxi. similarly, we can find another open set v such that b ⊆ v and u ∩ v = φ. then f(u) and f(v ) are disjoint open sets in ζx such that f(a) = y1 ∈ f(u) and f(b) = y2 ∈ f(v ). this proves the hausdorffness of ζx. � 5. lattices of connectifications theorem 5.1. if y is a dense subspace of a space x, then the lattice c(x) of all connectifications of x can be embedded into the lattice c(y ) of all connectifications of y by an order preserving map which also preserves join. proof. since y is dense in x, every connectification of x is also a connectification of y . � theorem 5.2. let x and y be two spaces having no isolated points. let f : x → y be a continuous and connected mapping from x onto y such that f seperates every pair of disjoint regular open subsets of x. then the lattice c(x) is isomorphic to c(y ). proof. if ξx is any connectification of x, then by theorem 3.5, we can find a connectification ζy for y with remainder ξx\x. similarly, if ζy is any connectification of y , then there is a connectification ξx for x with remainder ζy \y . thus we have a one to one correspondence from c(x) onto c(y ). let ξ1x and ξ2x be two connectifications of x such that ξ1x ≤ ξ2x, then there are two connectifications ζ1y and ζ2y of y with remainders ξ1x\x and ξ2x\x such that ξ1x 7→ ζ1y and ξ2x 7→ ζ2y . by our construction in theorem 3.5, we have ζ1y ≤ ζ2y . this completes the proof of this theorem. � acknowledgements. i am grateful to prof. c. ganesa moorthy, alagappa university, karaikudi for his valuable suggessions to improve this article c© agt, upv, 2014 appl. gen. topol. 15, no. 1 8 hausdorff connectifications references [1] o. t. alas, m. g. tkačenko, v. v. tkachuk and r. g. wilson, connectifying some spaces, topology appl. 71, (1996), 203–215. [2] j. j. charatonik, one point connectifications of subspaces of generalized graphs, kyungpook math. j. 41 (2001), 335–340. [3] j. j. charatonik, on one point connectifications of spaces, kyungpook math. j. 43 (2003), 149–156. [4] a. fedeli and a. le donne, dense embeddings in pathwise connected spaces, topology appl. 96, (1999), 15–22. [5] k. d. jr. magill, the lattice of compactifications of a locally compact space, proc. london math. soc. 18, (1968), 231–244. [6] j. r. munkres, topology, second edi., prentice hall of india, new delhi, 2000. [7] j. r. porter and r. grandwoods, subspaces of connected spaces, topology appl. 68 (1996), 113–131. [8] j. r. porter and r. grandwoods, extensions and absolutes of hausdorff spaces, springerverlag, new york, 1988. [9] s. w. watson and r .g. wilson, embeddings in connected spaces, houston j. math. 19, no. 3 (1993), 469–481. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 9 gicawaagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 211-231 on resolutions of linearly ordered spaces agata caserta, alfio giarlotta∗ and stephen watson abstract. we define an extended notion of resolution of topological spaces, where the resolving maps are partial instead of total. to show the usefulness of this notion, we give some examples and list several properties of resolutions by partial maps. in particular, we focus our attention on order resolutions of linearly ordered sets. let x be a set endowed with a hausdorff topology τ and a (not necessarily related) linear order �. a unification of x is a pair (y, ı), where y is a lots and ı : x →֒ y is an injective, order-preserving and open-in-the-range function. we exhibit a canonical unification (y, ı) of (x, �, τ ) such that y is an order resolution of a go-space (x, �, τ ∗), whose topology τ ∗ refines τ . we prove that (y, ı) is the unique minimum unification of x. further, we explicitly describe the canonical unification of an order resolution. 2000 ams classification: 54f05, 06f30, 46a40, 54a10 keywords: resolution, lexicographic ordering, go-space, linearly ordered topological space, pseudo-jump, to-embedding, unification. 1. resolutions by partial maps let (x,τ) be a topological space; if the topology is understood, we denote it by x. by neighborhood of a point we mean open neighborhood. for each point x ∈ x, the set of all neighborhoods of x is denoted by τ(x). similarly, if b is a base for x, then b(x) denotes the set of all basic neighborhoods of x. in this paper topological spaces are assumed to be hausdorff. a chain is a linearly ordered set. if (x,�) is a chain, then its reverse chain is (x,�∗), where x �∗ y if and only if y � x for each x,y ∈ x; to simplify notation, we denote this chain by x∗. if a and b are subchains of (x,�), the notation a ≺ b stands for a ≺ b for each a ∈ a and b ∈ b; in particular, if a = {a}, we simplify notation and write a ≺ b. ∗corresponding author. 212 a. caserta, a. giarlotta and s. watson a topological space (x,τ) is orderable if there exists a linear order � on x such that the order topology τ� coincides with τ. a linearly ordered topological space (for short, a lots ) is a chain endowed with the order topology; we denote a lots by (x,�,τ�). a topological space (x,τ) is suborderable if there exists an orderable topological space (y,σ) such that (x,τ) embeds homeomorphically into (y,σ). it is known that a topological space (x,τ) is suborderable if and only if there exists a total order on x (called a compatible order) such that (i) the original topology is finer than the order topology, and (ii) each point of x has a local base consisting of (possibly degenerate) intervals. a generalized ordered space (for short, a go-space) is a suborderable space endowed with a compatible order. the class of go-spaces is known to coincide with the class of topological subspaces of lots. in the sequel we assume without loss of generality that a go-space x is a subspace of the lots in which it embeds. for a recent survey on lots and go-spaces, see [3] (section f-7) and references therein. now we define the notion of a resolution of a family of topological spaces. this elegant and fundamental idea was introduced by fedorc̆uk in 1968 (see [2]) and extensively studied by watson in 1992 (see [4]). definition 1.1. let (x,τ) be a topological space, (yx,τx)x∈x a family of topological spaces and (fx : x(x) → yx)x∈x a family of continuous maps, where x(x) is an open subset of x \{x}. we endow the set ⋃ x∈x{x}×yx with a topology τ⊗ induced by a base b. for each point x ∈ x, neighborhood u ⊆ x of x and open set v ⊆ yx, we define basic open sets u ⊗x v in b as follows: u ⊗x v := {x}×v ∪ ⋃ { {x′}×yx′ : x ′ ∈ (u ∩f−1x v ) } . the topological space ( ⋃ x∈x{x}×yx,τ⊗ ) is called the resolution of x at each x ∈ x into yx by the map fx; we denote it by ⊗ x∈x (yx,fx). the space x is the global space and the yx’s are the local spaces. without loss of generality, we assume that the global space x is non-trivial; in fact, if x = {x}, then the resolution is homeomorphic to yx. our definition slightly extends the classical notion of resolution, in which the resolving functions fx are defined on the whole set x \ {x}. for sake of clarity, we refer to the classical notion as a resolution by total functions. on the other hand, the resolving functions fx described in definition 1.1 can be thought of as partial functions fx : x \{x}→ yx; we call the associated topological space a resolution by partial functions or simply a resolution. in this section we give some examples and list several properties of resolutions. we start by showing that ⊗ x∈x (yx,fx) is a well-defined topological space. lemma 1.2. the family b is a base for a topology on the set ⋃ x∈x{x}×yx. we need a technical result, which we prove first. lemma 1.3. for each (x′,y′) ∈ u ⊗x v , there exists u ′⊗x′ v ′ ∈b such that (x′,y′) ∈ u′ ⊗x′ v ′ ⊆ u ⊗x v . on resolutions of linearly ordered spaces 213 proof. the result is obvious if x′ = x. assume that x′ 6= x. then (x′,y′) ∈ u ⊗x v implies that (x ′,y′) ∈ ⋃ { {w}×yw : w ∈ (u ∩f −1 x v ) } , whence x′ ∈ u ∩ f−1x v and y ′ ∈ yx′ . set u ′ := u ∩ f−1x v and v ′ := yx′ . note that (x′,y′) ∈ u′⊗x′ v ′ ∈b, because fx is continuous on an open subset of x\{x} and so u′ is a neighborhood of x′. furthermore, we have: u′⊗x′ v ′ ⊆ ⋃ { {w}×yw : w ∈ u ∩f −1 x v } ⊆ u ⊗x v. thus u′⊗x′ v ′ satisfies the claim. � by lemma 1.3, we can assume without loss of generality that a basic neighborhood of (x,y) is of the type u ⊗x v , where u ⊆ x is a neighborhood of x and v ⊆ yx is an open set. proof of lemma 1.2. it suffices to show that for any two basic open sets u1 ⊗x1 v1 and u2 ⊗x2 v2 in b, if (x ′,y′) ∈ (u1 ⊗x1 v1)∩(u2 ⊗x2 v2), then there exists u′⊗x′ v ′ ∈b such that (x′,y′) ∈ u′⊗x′ v ′ ⊆ (u1 ⊗x1 v1)∩(u2 ⊗x2 v2). for each i ∈ {1, 2}, if (x′,y′) ∈ ui ⊗xi vi, then lemma 1.3 implies that there exists u′i ⊗x′ v ′ i such that (x ′,y′) ∈ u′i ⊗x′ v ′ i ⊆ ui ⊗xi vi. set u ′ := u′1 ∩u ′ 2 and v ′ := v ′1 ∩v ′ 2 . then, we obtain: (x′,y′) ∈ u′ ⊗x′ v ′ = (u′1 ⊗x′ v ′ 1 ) ∩ (u ′ 2 ⊗x′ v ′ 2 ) ⊆ (u1 ⊗x1 v1) ∩ (u2 ⊗x2 v2). this proves the claim. � a smaller base for the resolution topology τ⊗ is the following. lemma 1.4. let (x,τ) and (yx,τx)x∈x be topological spaces. assume that bx is a base for τ and for each x ∈ x, bx is a base for τx. the family b⊗ := {u ⊗x v : x ∈ x ∧ u ∈bx (x) ∧ v ∈bx} is a base for ⊗ x∈x (yx,fx). proof. let u ⊗x v be a basic open set in the resolution topology and z a point of u ⊗x v . by lemma 1.3, we can assume that z = (x,y) for some y ∈ v . select u′ ∈bx and v ′ ∈bx such that x ∈ u ′ ⊆ u and y ∈ v ′ ⊆ v . then the set u′⊗x v ′ ∈b⊗ is such that (x,y) ∈ u ′⊗x v ′ ⊆ u ⊗x v . it follows that b⊗ is a base for ⊗ x∈x (yx,fx). � some topological spaces can viewed as a resolution of other spaces only if we use partial resolving functions. in the next example we show that the closed unit interval is a resolution by partial maps of the unit circle. example 1.5. let s1 be the unit circle (having the origin (0, 0) as its center). resolve each point s ∈ s1\{(1, 0)} into the one-point space {ys} by the constant function fs. note that in order to resolve the point t = (1, 0) into a two-point discrete space {y−t ,y + t }, we cannot use a total function to map continuously s1 \ {t} onto {y−t ,y + t }. let t ∗ = (−1, 0) be the antipodal point of t. define ft : s 1 \ {t,t∗} → {y−t ,y + t } by ft(x,y) := y − t if y < 0 and ft(x,y) := y + t if y > 0. the resolution is (homeomorphic to) the unit interval [0, 1]. 214 a. caserta, a. giarlotta and s. watson another advantage of this extended notion of resolution is that a wide class of subspaces of a resolution by total functions can be seen as a resolution by partial functions. lemma 1.6. let z := ⊗ x∈x (yx,gx) be the resolution of x at each x ∈ x into yx by a total map gx. assume that w is a subspace of z with the property that for each x ∈ x, there exists an open set vx ⊆ yx such that w ∩ ({x}×yx) = {x}×vx. then w is a resolution ⊗ x∈x (yx,fx) by partial maps fx. proof. for each x ∈ x, let vx be an open subset of yx such that w ∩ ({x}× yx) = {x}×vx. continuity of the total function gx : x\{x}→ yx implies that the set x(x) := g−1x vx is open in x \{x}. denote by fx the partial maps on x \{x}, defined as gx ↾x(x). then w = ⊗ x∈x (yx,fx). � in the next examples, the symbol n denotes the discrete lots with exactly n elements, i.e., n = {0, 1, . . . ,n− 1}. example 1.7. the double arrow space is the lexicographic product r ×lex 2 endowed with the order topology. this space can be seen as the resolution of r at each point x into the discrete space 2 by the function fx : r \{x} → 2, defined by fx(x ′) = 0 for x′ < x and fx(x ′′) = 1 for x′′ > x. the sorgenfrey line is the subspace s = {(x, 1) : x ∈ r} of the double arrow space. the sorgenfrey line can be trivially seen as a resolution by partial maps: the global space is r, the local spaces are all equal to 1 and the resolving functions are the constant maps gx : (x,→) → 1. example 1.8. the alexandroff duplicate is the space r×2, whose topology is such that the subspace r×{0} is homeomorphic to r and the subspace r×{1} is made of isolated points. the michael line m is the usual space r with each irrational isolated. the michael line is homeomorphic to the subspace (p ×{1}) ∪ (q ×{0}) of the alexandroff duplicate. the space m can also be viewed as a subspace of the following resolution of r. resolve each rational into the space 1. further, resolve each irrational x into the discrete space 3 by the function fx : r \ {x} → 3, defined by fx(x ′) = 0 for x′ < x and fx(x ′′) = 2 for x′′ > x. the michael line is the subspace {(x, 1) : x ∈ r} of this resolution. according to lemma 1.6, we can view the michael line m as a resolution by partial maps. resolve the global space r into 1 at each rational x by the constant (total) map fx : r \{x}→ 1. further, resolve r at each irrational x into 2 by the empty map. next we list some simple properties of resolutions; their proof is straightforward and is omitted. lemma 1.9. (monotonicity) let u,u1,u2 be neighborhoods of x ∈ x and v,v1,v2 open sets in yx. we have: (i) if u1 ⊆ u2, then u1 ⊗x v ⊆ u2 ⊗x v ; on resolutions of linearly ordered spaces 215 (ii) if v1 ⊆ v2, then u ⊗x v1 ⊆ u ⊗x v2; (iii) if u1 ⊆ u2 and v1 ⊆ v2, then u1 ⊗x v1 ⊆ u2 ⊗x v2. lemma 1.10. (distributivity) let (ui)i∈i be a family of neighborhoods of x ∈ x and (vj )j∈j a family of open sets in yx. for each h ∈ i and k ∈ j, we have: (i) ( ⋂ i∈i ui ) ⊗x vk = ⋂ i∈i ( ui ⊗x vk ) ; (ii) uh ⊗x ( ⋂ j∈j vj ) = ⋂ j∈j ( uh ⊗x vj ) ; (iii) ( ⋂ i∈i ui ) ⊗x ( ⋂ j∈j vj ) = ⋂ i∈i ⋂ j∈j ( ui ⊗x vj ) ; (i’) ( ⋃ i∈i ui ) ⊗x vk = ⋃ i∈i ( ui ⊗x vk ) ; (ii’) uh ⊗x ( ⋃ j∈j vj ) = ⋃ j∈j ( uh ⊗x vj ) ; (iii’) ( ⋃ i∈i ui ) ⊗x ( ⋃ j∈j vj ) = ⋃ i∈i ⋃ j∈j ( ui ⊗x vj ) . lemma 1.11. (decomposability) let u be a neighborhood of x ∈ x and v an open set in yx. we have: u ⊗x v = (u ⊗x yx) ∩ (x ⊗x v ) . lemma 1.12. for each x ∈ x, we have: x ⊗x yx = ⋃ {{x′}×yx′ : x ′ ∈ domfx ∪{x}} . in particular, if domfx = x \{x}, then x ⊗x yx = ⋃ x∈x{x}×yx. now we define the projections of the resolution on the global and the local spaces. definition 1.13. the global projection is the function π : ⋃ x∈x{x}×yx → x defined by π(x,y) := x for each (x,y) ∈ domπ. further, for each x ∈ x, the local projection at x is the function πx : ⋃ x′∈x(x)∪{x}{x ′}×yx′ → yx defined as follows for each (x′,y′) ∈ domπx: πx(x ′,y′) : = { y′ if x′ = x fx(x ′) if x′ ∈ x(x). note that for each x ∈ x such that domfx = x \ {x}, the domain of πx is⋃ x∈x{x}×yx. if the resolution functions fx are total maps, then the global and local projections are continuous (see [4], theorem 6). the same holds also in the case that the resolution functions are partial maps and their domain is endowed with the subspace topology of τ⊗. the next lemma summarizes some related facts; its proof is easy and is omitted. lemma 1.14. for each neighborhood u ⊆ x of x and open set v ⊆ yx, we have: (i) π−1u = (u ⊗x yx) ∪ ⋃ x′∈u\domfx {x′}×yx′ ; (ii) if u \{x}⊆ domfx, then π −1u = u ⊗x yx; (iii) π−1x v = x ⊗x v . corollary 1.15. the global and the local projections are continuous functions. 216 a. caserta, a. giarlotta and s. watson let u ⊆ x be a neighborhood of x and v ⊆ yx an open set. the global section and the local section of u ⊗x v are defined, respectively, by globx(u,v ) := π −1u and locx(u,v ) := π −1 x v . each basic open set is the intersection of its global and local section. corollary 1.16. for each neighborhood u ⊆ x of x and open set v ⊆ yx, we have u ⊗x v = π −1u ∩π−1x v = globx(u,v ) ∩ locx(u,v ). proof. since ⋃ {{x′}×yx′ : x ′ ∈ u \ domfx} ∩ (x ⊗x v ) = {x} × v , the claim follows from lemmas 1.11 and 1.14. � 2. order resolutions in this section we focus our attention on particular types of resolutions, in which both the global space (x,�,τ�) and the local spaces (yx,�x,τ�x )x∈x are lots. if the spaces yx are compact, there is a standard way to define the resolving functions fx (called order maps in this setting); the resulting topological space is called an order resolution. in our definition we allow the resolving functions fx to be partially defined, in order to deal also with the cases in which some of the local spaces yx have no maximum and/or no minimum. we use the following notation for intervals in x (rays are considered as particular intervals): • i is the family of all intervals in x (including x); further, i(x) := {i ∈i : x ∈ i}; • −→ i := {(x′,→) : x′ ∈ x}∪{x} and ←− i := {(←,x′′) : x′′ ∈ x}∪{x}; • −−→ i(x) := {i ∈ −→ i : x ∈ i} and ←−− i(x) := {i ∈ ←− i : x ∈ i}. similarly, intervals in yx are denoted as follows: • ix: the family of all intervals in yx (including yx); • −→ ix := {(y ′,→) : y′ ∈ yx} and ←− ix := {(←,y ′′) : y′′ ∈ yx}. definition 2.1. for each x ∈ x, let x(x) be the following subset of x \{x}: x(x) : =    x \{x} if ∃ minyx ∧ ∃ maxyx (←,x) if ∃ minyx ∧ ∄ maxyx (x,→) if ∄ minyx ∧ ∃ maxyx ∅ if ∄ minyx ∧ ∄ maxyx. the order map fx : x(x) → yx is defined as follows for each x ′ ∈ x(x) (if any): fx(x ′) : = { min yx if x ′ ≺ x max yx if x ′ ≻ x. the space ( ⋃ x∈x{x}×yx,τ⊗ ) is called the resolution of x at each x ∈ x into yx by the order map fx and is denoted by ⊗ord x∈x yx or simply by ⊗ x∈x yx. also, we denote by b⊗ the following base (cf. lemma 1.4) for the resolution topology τ⊗: b⊗ := {u ⊗x v : x ∈ x ∧ u ∈i(x) ∧ v ∈ix} . on resolutions of linearly ordered spaces 217 example 2.2. resolve each ordinal α in ω1 into the half-open interval [0, 1) by the order map. denote this resolution by ⊗ α∈ω1 [0, 1). for each successor ordinal α, a neighborhood base is given by sets of the type {α}×(a,b) and {α}× [0,b). further, if α is a limit ordinal, then a neighborhood base is composed of all sets of the type {α}×(a,b) and {α}× [0,b)∪ (⋃ β∈(γ,α){β}× [0, 1) ) , where γ < α. note that the resolution space described above has the same underlying set of the lexicographic product ω1 ×lex [0, 1), but its topology τ⊗ is finer than the order topology τ�lex . in fact, neighborhood bases at limit ordinals are the same for the two topologies, but at successor ordinals neighborhood bases for the resolution topology are strictly finer than for the order topology (cf. example 2.10). next we compute u ⊗x v in the case that both u and v are rays. the notation ⋃ (x,→){w} × yw stands for ⋃ w∈(x,→){w} × yw; a similar meaning have the other symbols. lemma 2.3. let (x′,→) and (←,x′′) be open rays in x containing x, and (y′,→) and (←,y′′) open rays in yx. the following equalities hold: (x′,→) ⊗x (y ′,→) =    {x}× (y′,→) if ∄maxyx {x}× (y′,→) ∪ ⋃ (x,→) {w}×yw if ∃maxyx (←,x′′) ⊗x (y ′,→) =    {x}× (y′,→) if ∄maxyx {x}× (y′,→) ∪ ⋃ (x,x′′) {w}×yw if ∃maxyx (x′,→) ⊗x (←,y ′′) =    {x}× (←,y′′) if ∄minyx {x}× (←,y′′) ∪ ⋃ (x′,x) {w}×yw if ∃minyx (←,x′′) ⊗x (←,y ′′) =    {x}× (←,y′′) if ∄minyx {x}× (←,y′′) ∪ ⋃ (←,x) {w}×yw if ∃minyx x ⊗x (y ′,→) = (x′ →) ⊗x (y ′,→) x ⊗x (←,y ′′) = (←,x′′) ⊗x (←,y ′′) 218 a. caserta, a. giarlotta and s. watson (x′,→) ⊗x yx =    {x}×yx if ∄minyx ∧ ∄maxyx ⋃ (x′,x] {w}×yw if ∃minyx ∧ ∄maxyx ⋃ [x,→) {w}×yw if ∄minyx ∧∃maxyx ⋃ (x′,→) {w}×yw if ∃minyx ∧∃maxyx (←,x′′) ⊗x yx =    {x}×yx if ∄minyx ∧ ∄maxyx ⋃ (←,x] {w}×yw if ∃minyx ∧ ∄maxyx ⋃ [x,x′′) {w}×yw if ∄minyx ∧∃maxyx ⋃ (←,x′′) {w}×yw if ∃minyx ∧∃maxyx and x ⊗x yx =    {x}×yx if ∄minyx ∧ ∄maxyx ⋃ (←,x] {w}×yw if ∃minyx ∧ ∄maxyx ⋃ [x,→) {w}×yw if ∄minyx ∧∃maxyx ⋃ x {w}×yw if ∃minyx ∧∃maxyx. proof. straightforward from definition. � we use rays to define a subfamily s⊗ ⊆b⊗, which is a subbase for τ⊗. set • −→ s := {u ⊗x v : x ∈ x ∧ u ∈ −−→ i(x) ∧ v ∈ −→ ix}, • ←− s := {u ⊗x v : x ∈ x ∧ u ∈ ←−− i(x) ∧ v ∈ ←− ix}, • s⊗ := −→ s ∪ ←− s . lemma 2.4. for each u1 ∈ −−→ i(x), u2 ∈ ←−− i(x), v1 ∈ −→ ix\{yx} and v2 ∈ ←− ix\{yx}, we have: (i) u2 ⊗x v1 = (x ⊗x v1) ∩ (u2 ⊗x yx) ⊆ x ⊗x v1 = u1 ⊗x v1; (ii) u1 ⊗x v2 = (u1 ⊗x yx) ∩ (x ⊗x v2) ⊆ x ⊗x v2 = u2 ⊗x v2; on resolutions of linearly ordered spaces 219 (iii) (u1 ∩ u2) ⊗x (v1 ∩ v2) = (u1 ⊗x v2) ∩ (u2 ⊗x v1) = (u1 ⊗x v1) ∩ (u2 ⊗x v2) = {x}× (v1 ∩v2); (iv) (u1 ∩u2) ⊗x yx = (u1 ⊗x yx) ∩ (u2 ⊗x yx). proof. parts (i) and (ii), as well as the second and third equality in (iii) follow from lemma 2.3. the first equality in (iii) follows from part (i), part (ii) and lemma 1.10 (iii). finally, part (iv) is an immediate consequence of lemma 1.10 (i). � corollary 2.5. for each x ∈ x and u ⊗x v ∈b⊗, there exist u1 ⊗x v1 ∈ −→ s and u2 ⊗x v2 ∈ ←− s such that u ⊗x v = (u1 ⊗x v1) ∩ (u2 ⊗x v2). in particular, s⊗ is a subbase for τ⊗. proof. let x ∈ x. the open interval u ⊆ x containing x can have the following forms: (i) u = x; (ii) u = (x′,→), with x′ ≺ x; (iii) u = (←,x′′), with x ≺ x′′; (iv) u = (x′,x′′), with x′ ≺ x ≺ x′′. similarly, we have four possibilities for the open interval v ⊆ yx, namely: (i) v = yx; (ii) v = (y ′,→); (iii) v = (←,y′′); (iv) v = (y′,y′′). in all sixteen cases, the claim follows from lemma 2.4. � since both the global space x and the local spaces yx are lots, we can obtain another topological space as follows. consider the reverse of all chains and endow them with the relative order topology. by applying the resolution operator to the lots x∗ and (y ∗x )x∈x , we obtain a new topological space, the reverse order resolution ⊗ord x∈x∗ y ∗ x . to simplify notation, we denote this space by ⊗∗ x∈x yx. we shall show that ⊗∗ x∈x yx = ⊗ x∈x yx (see corollary 2.8). note that if i = (y,z) is an open interval in yx, then i ∗ ⊆ y ∗x is the open interval (y,z)∗ = (z,y). fix x ∈ x. assume that yx has a minimum element yx but no maximum element. the order map fx and the reverse order map f ∗ x are defined, respectively, as follows: fx : (←,x) → yx , w 7−→ yx f∗x : (x,→) → y ∗ x , w 7−→ yx. thus, if i is the open interval [yx,y ′′) ⊆ yx, then (fx) −1i = (←,x) ⊆ x and (f∗x ) −1 i∗ = (x,→) ⊆ x∗ whence (f∗x ) −1 i∗ = ( fx −1i )∗ . similar considerations can be done in the case that yx has a maximum but not a minimum, or has both a minimum and a maximum. the following lemma summarizes the above results. lemma 2.6. let x ∈ x and assume that yx has a minimum (respectively, maximum) element yx. if i ⊆ yx is the open interval [yx,y) (respectively, (y,yx]), then (f ∗ x ) −1 i∗ = ( fx −1i )∗ . order resolution and reverse order resolution have the same basic open sets. 220 a. caserta, a. giarlotta and s. watson theorem 2.7. each basic open set u ⊗x v in ⊗ x∈x yx is equal to the basic open set u∗ ⊗∗x v ∗ in ⊗∗ x∈x yx. proof. several cases have to be considered: (i) x = minx; (ii) x = maxx; (iii) x 6= minx and x 6= maxx. we examine only case (iii), since the others are similar. let u ⊗x v be a basic open set in ⊗ x∈x yx; without loss of generality, let u = (x′,x′′), where x′ ≺ x ≺ x′′. for v ⊆ yx, we can assume that one of the following cases occurs: (a) v = [y,y′′), with y = minyx; (b) v = (y ′,y], with y = maxyx; (c) v = (y ′,y′′). in case (a), lemma 2.6 yields u ⊗x v = (x ′,x′′) ⊗x [y,y ′′) = {x}× [y,y′′) ∪ ⋃ {{w}×yw : w ∈ (x ′,x)} and u∗⊗∗xv ∗ = (x′,x′′)∗⊗∗x[y,y ′′)∗ = {x}×(y′′,y] ∪ ⋃ {{w}×y ∗w : w ∈ (x,x ′)} . therefore, u ⊗xv = u ∗⊗∗xv ∗, as claimed. case (b) is similar to (a). for case (c), we have: u ⊗x v = (x ′,x′′) ⊗x (y ′,y′′) = {x}× (y′,y′′) and u∗⊗∗x v ∗ = (x′,x′′)∗ ⊗∗x (y ′,y′′)∗ = {x}× (y′′,y′) whence u ⊗xv = u ∗⊗∗xv ∗ also in this case. � corollary 2.8. ⊗ x∈x yx = ⊗∗ x∈x yx. proof. since the underlying set is the same for both spaces, it suffices to show that their topologies coincide. theorem 2.7 yields u ⊗x v = u ∗ ⊗∗x v ∗ for any basic open set u ⊗x v in ⊗ x∈x yx. by duality, we obtain u ∗ ⊗∗x v ∗ = u∗∗ ⊗∗∗x v ∗∗ = u ⊗x v for any basic open set u ∗ ⊗∗x v ∗ in ⊗∗ x∈x yx. the result follows. � in the last part of this section we study the relationship between two natural topologies defined on the set ⋃ x∈x{x}× yx. namely, we compare the order resolution ⊗ x∈x yx and the chain ∑ x∈x yx endowed with the order topology τς. lemma 2.9. the resolution topology τ⊗ on ⊗ x∈x yx is finer than the order topology τς on ∑ x∈x yx. proof. let i be an open ray in ∑ x∈x yx and (x,y) a point in i. to prove the claim we exhibit a neighborhood w ∈ τ⊗(x,y) such that w ⊆ i. by duality (see theorem 2.7), it suffices to examine the case i = (←, (x′′,y′′)), where x � x′′. first assume that x ≺ x′′. select u ∈ τ(x) such that x′′ /∈ u and set w := π−1u. continuity of the global projection π (see corollary 1.15) implies that w is a neighborhood of (x,y) in ⊗ x∈x yx, which does not contain the point (x′′,y′′). thus (x,y) ∈ w ⊆ i, as claimed. on resolutions of linearly ordered spaces 221 next, let x = x′′ and y ≺x y ′′. set u := x and v := (←,y′′). note that fx −1v is equal either to the empty set (if x = minx or yx has no minimum element) or to the open ray (←,x) (if x 6= minx and yx has a minimum element). thus we obtain: (x,y) ∈ u⊗xv = {x}×(←,y ′′)∪ ⋃ { {x′}×yx′ : x ′ ∈ ( fx −1(←,y′′) ∩x )} ⊆ i. the open set w := u ⊗x v satisfies the claim. � the converse of lemma 2.9 does not hold in general. in particular, the order resolution of a lots into lots is a go-space that is not necessarily a lots. example 2.10. let x be the discrete lots 2 = {0, 1} and y0 = y1 the half-open interval [a,b) ⊆ r. we show that the resolution topology τ⊗ on⊗ x∈x yx = ⊗ i∈2[a,b) is strictly finer than the order topology τς on the chain∑ x∈x yx = 2 ×lex [a,b). consider the point (1,a) ∈ 2 × [a,b), the open neighborhood {1} of 1 and the open set [a,b) = y1. observe that f0 is undefined and f1 : x(1) → [a,b) is defined by f1(0) = a. thus, the basic open set {1}⊗1 [a,b) is the half-open interval [(1,a), (1,b)). on the other hand, any interval i ∈ τς satisfying (1,a) ∈ i ⊆ {1}⊗1 [a,b) must contain a subinterval of the type i ′ = ((0,y1), (1,y2)), where y1,y2 ∈ [a,b). thus, {1}⊗1 [a,b) is not open in 2 ×lex [a,b). the global projection π : ⋃ x∈x{x}× yx → x is continuous whenever its domain is endowed with the resolution topology. on the other hand, continuity of π is not ensured if its domain is endowed with the order topology. for example, π : (2 ×lex [a,b),τς) → 2 fails to be continuous, because (2 ×lex [a,b),τς) is homeomorphic to the connected interval [a,b). the local projections πx : ⋃ x′∈x(x)∪{x}{x ′}× yx′ → yx, with x ∈ x, are continuous if their domain is endowed with the subspace topology of the resolution (see corollary 1.15). next we show that they are continuous also if we endow their domain with the subspace topology of the lots (∑ x∈x yx,τς ) . before proving this fact, we mention a technical lemma. lemma 2.11. let v ⊆ yx be an open set. the set {x} × v is open in(∑ x∈x yx,τς ) if both of the following conditions are verified: (a) if v contains maxyx, then either x = maxx or x has an immediate successor x′′ and yx′′ has a minimum; (b) if v contains minyx, then either x = minx or x has an immediate predecessor x′ and yx′ has a maximum. in particular, {x}×v is open in (∑ x∈x yx,τς ) whenever v contains neither maxyx nor minyx. proof. straightforward from definition. � lemma 2.12. for each x ∈ x, the function πx is continuous with respect to the subspace topology of τς. 222 a. caserta, a. giarlotta and s. watson proof. let v ⊆ yx be an open set. then π −1 x v = x ⊗x v , using lemma 1.14. to prove the result, it suffices to show that x ⊗x v is open (in the subspace topology of τς) for v = (←,y). if yx has no minimum element, then x ⊗x (← ,y) = {x}× (←,y) is open by lemma 2.11. on the other hand, if yx has a minimum element ymin, then π −1 x (←,y) = x ⊗x [ymin,y) = (←, (x,y)). � next we introduce the notion of pseudojump. definition 2.13. a pseudojump in the chain ∑ x∈x yx is a jump (x ′,x′′) in x (i.e., a pair of consecutive points of x) with the property that either (a) ∃maxyx′ and ∄ minyx′′ , or (b) ∄ maxyx′ and ∃minyx′′ . the notion of pseudojump of a chain l = ∑ x∈x yx obviously depends on the chosen representation of l as a sum of other chains. consider, e.g., the isomorphic chains (0, 1) (which lacks pseudojumps), (0, 1/2) ⊕ [1/2, 1) (which has exactly one pseudojumps) and ∑ n∈ω ( 1 n+3 , 1 n+2 ] ⊕ (1/2, 1) (which has countably many pseudojumps). on the other hand, if we endow the chains with additional structure, then the notion becomes significant. the next theorem characterizes the order resolution as a lots. theorem 2.14. let (x,�,τ�) and {(yx,�x,τ�x )}x∈x be lots. the following statements are equivalent: (i) the order topology τς on ∑ x∈x yx is equal to the resolution topology τ⊗ on ⊗ x∈x yx, i.e., ⊗ x∈x yx is a lots; (ii) the global projection π : (∑ x∈x yx,τς ) → (x,τ�) is continuous; (iii) the chain ∑ x∈x yx has no pseudojumps. proof. (i) ⇒ (ii). this implication follows from corollary 1.15. (ii) ⇒ (i). by lemma 2.9, it suffices to show that τ⊗ ⊆ τς. let u ⊗x v be a basic open set for τ⊗. since u ⊗x v = ( π−1u ) ∩ ( π−1x v ) by corollary 1.16, the claim follows from hypothesis and lemma 2.12. (ii) ⇒ (iii). we prove the contrapositive. without loss of generality, assume that there exists a jump (x′,x′′) in x such that yx′ has no maximum and yx′′ has a minimum y ′′ min. consider the nonempty open ray (x ′,→) ⊆ x. since the set π−1(x′,→) = [(x′′,y′′min),→) is not open in (∑ x∈xyx,τς ) (cf. example 2.10), it follows that π is not continuous with respect to τς. (iii) ⇒ (ii). assume that ∑ x∈x yx has no pseudojumps. it suffices to prove that π−1(x′,→) is open in τς for each x ′ ∈ x. let (x,y) ∈ π−1(x′,→). we find an interval i ⊆ ∑ x∈x yx such that (x,y) ∈ i ⊆ π −1(x′,→). if y 6= minyx, then there exists t ∈ yx such that t ≺x y. set i := ((x,t),→). next assume that y = minyx. if (x ′,x) is a jump in x, then by hypothesis there exists y′max := maxyx′ . thus i := ((x ′,y′max),→) satisfies the claim. on the other hand, if (x′,x) is not a jump in x, then we can select w ∈ (x′,x) and t ∈ yw, and set i := ((w,t),→). � on resolutions of linearly ordered spaces 223 3. unifications and resolutions by to-space we mean a triple (x,�,τ) such that x is a nonempty set, � is a linear order on x and τ is a hausdorff topology on x (not necessarily related to the order �). we describe how a to-space can be canonically embedded into a lots that is an order resolution. definition 3.1. let (x,�,τ) and (y,≤,σ) be to-spaces. a function f : x → y is a to-homomorphism if f : (x,τ) → (y,σ) is open in the range (i.e., open sets of x are mapped into open sets of f(x)) and f : (x,�) → (y,≤) is orderpreserving. in particular, a to-embedding (respectively, a to-isomorphism) is an injective (respectively, bijective) to-homomorphism. next we list some simple but useful properties of order-preserving maps between chains. their proof is easy and is omitted. lemma 3.2. let f : x → y be an order-preserving map between chains. we have: (i) the f-preimage of a convex subset of y is a convex subset of x; (ii) if f is surjective, then the f-preimage of an open [closed, half-open] interval is an open [closed, half-open] interval; (iii) if x is a go-space, y is a lots and f is surjective, then f is continuous; (iv) if x is a lots, y is a go-space and f is injective, then f is a toembedding; (v) if x and y are lots and f is bijective, then f is a homeomorphism. as the next example shows, the hypothesis that y is a lots is necessary in (iii) and (v). in particular, a to-isomorphism may fail to be a homeomorphism. example 3.3. let x be the unit interval [0, 1] (with the order topology) and y the topological sum [0, 1) ⊕{1} (with the natural order). the identity map is a to-isomorphism but is not continuous. note that x is a lots, whereas y is a go-space but not a lots. let x be a to-space. observe that if f : x → y is a to-embedding of x into a lots, then its (partial) inverse f−1 : f(x) → x is a continuous and order-preserving map of a go-space onto x. vice versa, assume that g : z → x is a continuous and order-preserving map of a go-space onto x. let y be a lots such that z ⊆ y . choose zx ∈ z is such that g(zx) = x. the function h: x → y , defined by h(x) := zx for each x ∈ x, is a to-embedding of x into a lots. definition 3.4. let (x,�,τ) be a to-space and ı: x →֒ y is a to-embedding of x into a lots (y,≤,τ≤). the pair ((y,≤,τ≤) , ı) is called a unification of (the topology and the order of) x; if there is no risk of confusion, we simplify the notation and write (y,ı). without loss of generality, we assume that a unification of a to-space x is a pair (y,ı) such that x is a subchain of y and ı is the canonical inclusion. (note that x and ı(x) have different topologies, in general.) 224 a. caserta, a. giarlotta and s. watson a unification (y,ı) of x is continuous if the to-embedding ı is a continuous map. further, (y,ı) is a minimum unification if for any other unification (z,ϕ) of x, there exists a to-embedding ψ : y →֒ z such that ϕ = ψ ◦ ı. example 3.5. let x = 2 ×lex [0, 1) = [0, 1) ⊕ [0, 1) be a chain endowed with the resolution topology. the following lots (together with the canonical inclusions ıy , ıw and ız , respectively) are examples of unifications of x: y = [0, 1] ⊕ [0, 1), w = [0, 2] ⊕ [0, 1) and z = [0, 1) ⊕ ( 1 n+1 )n∈ω ⊕ [0, 1). observe that (y,ıy ) is a continuous minimum unification of x, whereas (w,ıw ) and (z,ız ) are continuous unifications of x, which fail to be minimum. note also that despite y and z are homeomorphic lots, the unifications (y,ıy ) and (z,ız ) are different. example 3.6. define on the set x = ω ∪{ω + 1}, endowed with the natural order ≤, a topology σ as follows. let u be an ultrafilter on ω containing the cofinite filter. we define σ as the topology on x such that all natural numbers are isolated points and a system of σ-neighborhoods for the point ω + 1 is given by {u ∪{ω + 1} : u ∈u}. then (x,≤,σ) is a to-space that fails to be a gospace, because its character χ(x,σ) is uncountable but its pseudo-character ψ(x,σ) is countable (see [1], problem 3.12.4). a (minimum) unification for x is given by (y,ı), where y is the lots (ω + 2,≤,τ≤) and ı is the canonical embedding. minimum unifications of a to-space are essentially unique. to prove uniqueness, we first define the so-called canonical unification of a to-space and then show that this is (up to a relabeling) its unique minimum unification. the canonical unification of a to-space (x,�,τ) is defined in two steps: (i) we obtain a minimum (in the sense of definition 3.7) refinement τ∗ of the topology τ such that (x,�,τ∗) is a go-space; (ii) we embed the go-space (x,�,τ∗) into a minimum (in the sense of definition 3.9) lots that is an order resolution ⊗ x∈x yx. definition 3.7. let (x,�,τ) be a to-space. the go-cone above x is the (nonempty) family of all go-spaces (x,�,σ) such that σ refines τ. the goextension of x, denoted by (x,�,τ∗), is the minimum of the go-cone above x, in the sense that if (x,�,σ) is another go-space such that σ refines τ, then σ refines also τ∗. the go-extension of x is well-defined. lemma 3.8. for each to-space (x,�,τ), the go-cone above x has a minimum (x,�,τ∗). proof. we define τ∗. let x ∈ x. for each τ-neighborhood u of x, denote by c(x,u) the union of all �-convex subsets of u containing x. let bx := {c(x,u) : u ∈ τ(x)}. then b∗ := ⋃ x∈x bx is a base for a topology τ ∗ on x. the topology τ∗ refines both the original topology τ and the order topology τ� (because (x,τ) is hausdorff). further, (x,�,τ ∗) is a go-space. on resolutions of linearly ordered spaces 225 next we prove that (x,�,τ∗) is the minimum of the go-cone above x. let (x,�,σ) be a go-space such that σ refines τ. let b be a base for σ composed of convex sets and c(x,u) a basic open set in τ∗. for each x′ ∈ c(x,u), there exists b ∈b such that x′ ∈ b ⊆ u. thus x′ ∈ b ⊆ c(x,u), hence c(x,u) is open in σ. this shows that σ refines τ∗. � definition 3.9. let (x,�,τ) be a go-space. a completion of x is a pair (y,ı), where y is a lots and ı: x →֒ y is an order-preserving homeomorphic embedding. a completion (y,ı) is minimum if for any other completion (z,ϕ), there exists an order-preserving homeomorphic embedding ψ(ı,ϕ) : y →֒ z such that ϕ = ψ(ı,ϕ) ◦ ı. note that if x is a go-space, then both notions of unification of x (as a tospace) and completion of x make sense. we show that the unique minimum unification of x is indeed its unique minimum completion. theorem 3.10. let (x,�,σ) be a go-space. (i) there exists a minimum completion ( ⊗ x∈x yx, ı) of x such that the space ⊗ x∈x yx is the order resolution of x at each point x into a chain yx with either one, two or three points. (ii) the image ı(x) is topologically dense in ⊗ x∈x yx. (iii) if (w,η) is another minimum completion of x, then the maps ψ(ı,η) and ψ(η,ı) are order-preserving homeomorphisms between ⊗ x∈x yx and w such that ψ(ı,η) = ψ(η,ı)−1. proof. for each x ∈ x, we define a chain yx with either one, two or three points as follows. if x has a σ-neighborhood base consisting of open intervals, then set yx := {yx}. on the other hand, if x has no neighborhood base consisting of open intervals, then we define yx according to cases (i)-(vii) described below. let x be a non-endpoint of x. if (i) x has an immediate predecessor and no immediate successor, set yx := {yx,y + x }. if (ii) x has an immediate successor and no immediate predecessor, set yx := {y − x ,yx}. further, if x has neither immediate predecessor nor immediate successor, it follows that a σ-neighborhood base at x has one of the following forms: (iii) the family of all half-open intervals (a,x] containing a point different from x; (iv) the family of all half-open intervals [x,b) containing a point different from x; (v) the singleton {x}. let yx be the chain {yx,y + x } in case (iii), {y − x ,yx} in case (iv), and {y − x ,yx,y + x } in case (v). finally, if (vi) x is the minimum point of x and has no immediate successor, or (vii) x is the maximum point of x and has no immediate predecessor, set yx := {yx,y + x } in case (vi), and yx := {y − x ,yx} in case (vii). let ⊗ x∈x yx be the order resolution of the lots (x,�,τ�) at each point x into the lots yx defined as above. by theorem 2.14, the space ⊗ x∈x yx is a lots. further, the correspondence x 7→ (x,yx) gives a order-preserving homeomorphic embedding ı: (x,�,σ) → ⊗ x∈x yx. thus ( ⊗ x∈x yx, ı) is a completion of x. next we prove that it is minimum. assume that ϕ: x →֒ z is an order-preserving homeomorphic embedding of x into a lots. we define an order-preserving homeomorphic embedding 226 a. caserta, a. giarlotta and s. watson ψ = ψ(ı,ϕ) : ⊗ x∈x yx →֒ z such that ϕ = ψ ◦ ı. for each x ∈ x, denote zx := ϕ(x) and let ψ(x,yx) := zx. to define ψ for the points of ⊗ x∈x yx that are not in the range of ı, we carry a case by case analysis. since ϕ is a to-embedding into a lots, in cases (i) and (vi) the point zx is isolated in the range of ϕ, whereas in case (iii) the interval (←,zx] is open in the range of ϕ. it follows that there exists z+x ∈ z that is the immediate successor of zx. set ψ(x,y + x ) := z + x . further, cases (ii), (iv) and (vii) are dual to, respectively, (i), (iii) and (vi). thus, we set ψ(x,y−x ) := z − x , where z−x is the immediate predecessor of zx in z. finally, in case (v), a combination of the arguments given above yields that there exist z−x ,z + x ∈ z, which are, respectively, the immediate predecessor and successor of zx. set ψ(x,y − x ) := z − x and ψ(x,y+x ) := z + x . by construction, ψ is an injective order-preserving map such that ϕ = ψ ◦ ı. thus ψ is a to-embedding by lemma 3.2. next we prove continuity of ψ using continuity of ϕ. it suffices to show that for any open ray (←,z) ⊆ z, the preimage ψ−1(←,z) is open in ⊗ x∈x yx. if z belongs to the image of ψ, then ψ−1(←,z) = (←, ψ−1(z)) is open in ⊗ x∈x yx. now let (←,z) be such that z is not in the image of ψ. without loss of generality, assume that ϕ−1(←,z) 6= ∅. let (x,y) ∈ ψ−1(←,z). we claim that either (x,y) has an immediate successor (x′′,y′′), or there exists (x′′,y′′) ∈⊗ x∈x yx such that (x,y) ≺ (x ′′,y′′) and ψ(x′′,y′′) � z. continuity of ψ follows from the claim, since (x,y) ∈ (←, (x′′,y′′)) ⊆ ψ−1(←,z). to prove the claim, assume by contradiction that (x,y) has no immediate successor and for each (x′′,y′′) ∈ ⊗ x∈x yx, if (x,y) ≺ (x ′′,y′′) then z ≺ ψ(x′′,y′′). then x has no immediate successor and ı(x) = (x,yx) is such that yx is the maximum of yx. it follows that ϕ −1(←,z) = (←,x] is not open in x, which contradicts the continuity of ϕ. this finishes the proof of (i). to prove (ii), observe that any two points in ⊗ x∈x yx \ ı(x) cannot be consecutive. furthermore, it is easy to show that all points in ⊗ x∈x yx \ ı(x) are not isolated in τ⊗. it follows that ı(x) = ⊗ x∈x yx. finally, assume that (w,η) is another minimum completion of x. we show that the compositions ψ(η,ı)◦ψ(ı,η) and ψ(ı,η)◦ψ(η,ı) are the identity maps on ⊗ x∈x yx and w , respectively. this will prove (iii). by hypothesis, there exist order-preserving homeomorphic embedding ψ(ı,η) and ψ(η,ı) such that η = ψ(ı,η) ◦ ı and ı = ψ(η,ı) ◦η. thus the composition ψ(η,ı) ◦ ψ(ı,η) : ⊗ x∈x yx →֒ ⊗ x∈x yx is an order-preserving homeomorphic embedding, whose restriction to ı(x) is the canonical inclusion of ı(x) into⊗ x∈x yx. by (ii), it follows that ψ(η,ı) ◦ ψ(ı,η) is the identity on ⊗ x∈x yx. to prove that ψ(ı,η) ◦ ψ(η,ı) is the identity on w , we first show that η(x) is topologically dense in w . by way of contradiction, assume that there exists w ∈ w and an open interval (a,b) ⊆ w such that w ∈ (a,b) and (a,b) ∩ η(x) = ∅. it follows that (ψ(a), ψ(b)) is an open interval containing ψ(w), which does not intersect ı(x). this contradicts property (ii). now the equality ψ(ı,η) ◦ ψ(η,ı) = idw follows by an argument similar to that given above. � on resolutions of linearly ordered spaces 227 note that the topology of ⊗ x∈x yx is the same as its order topology (see theorem 2.14). an immediate consequence of theorem 3.10 is corollary 3.11. for each go-space (x,�,σ), the pair ( ⊗ x∈x yx, ı) is the unique (up to a relabeling) minimum completion. remark 3.12. for any go-space (x,�,σ), a completion (y,ı) of x is also a unification of x. the proof of theorem 3.10 yields that the unique (up to a relabeling) minimum completion ( ⊗ x∈x yx, ı) of x has the property that for each to-embedding ϕ: x →֒ z of x into a lots, there exists a to-embedding ψ(ı,ϕ) : ⊗ x∈x yx →֒ z such that ϕ = ψ(ı,ϕ) ◦ ı. thus the minimum completion of x is also a minimum unification of x (indeed, the minimum unification of x, cf. corollary 3.15). definition 3.13. let (x,�,τ) be a to-space. using lemma 3.8 and theorem 3.10, we define the canonical unification ( ⊗̂ x∈xyx, ı̂) of x as follows. the lots ⊗̂ x∈xyx is an order resolution obtained as in the proof of theorem 3.10: the global space is the go-extension (x,�,τ∗) of (x,�,τ), whereas the local spaces yx are the discrete lots with either one, two or three points. the toembedding ı̂ is the composition of the identity idτ ∗ τ : (x,�,τ) → (x,�,τ ∗) and the homeomorphic embedding ı: (x,�,τ∗) → ⊗̂ x∈xyx defined in the proof of theorem 3.10. theorem 3.14. let (x,�,τ) be a to-space. the canonical unification of x is a minimum unification. further, for any other minimum unification (w,η̂) of x, there exist order-preserving homeomorphisms ψ : ⊗̂ x∈xyx → w and χ: w → ⊗̂ x∈xyx such that ψ ◦ ı̂ = χ, χ◦ ı̂ = ψ and ψ −1 = χ. proof. first we show that ( ⊗̂ x∈xyx, ı̂) is a minimum unification. let (z,ϕ̂) be a unification of x. we define a to-embedding ψ : ⊗̂ x∈xyx → z such that ϕ̂ = ψ ◦ ı̂. note that ϕ̂(x) ⊆ z is a go-space. let σ be the topology on x such that ϕ̂ gives a homeomorphism between (x,σ) and ϕ̂(x). thus (x,σ) is an element of the go-cone above (x,τ). further, we have ϕ̂ = ϕ ◦ idστ , where id σ τ : (x,τ) → (x,σ) is the identity map and ϕ: (x,σ) → z is the homeomorphic embedding defined as ϕ̂. by definition of go-extension of (x,�,τ), the identities id τ ∗ τ : (x,τ) → (x,τ ∗), id σ τ ∗ : (x,τ ∗) → (x,σ) and idστ : (x,τ) → (x,σ) are order-preserving open maps. set ϕ′ := ϕ ◦ idστ ∗ . then ϕ ′ : (x,�,τ∗) →֒ z is a to-embedding. by theorem 3.10, there exists a to-embedding ψ(ı,ϕ′) : ⊗̂ x∈xyx → z such that ϕ′ = ψ(ı,ϕ′) ◦ ı. it follows that ϕ̂ = ϕ′ ◦ idτ ∗ τ = ψ(ı,ϕ ′) ◦ ı̂. thus ψ := ψ(ı,ϕ′) satisfies the claim. now let (w,η̂) be another minimum unification of x. using the same notation as above (and as in theorem 3.10), one can show that the maps ψ := ψ(ı,η′) : ⊗̂ x∈x yx → w and χ := ψ(η,ı ′) : w → ⊗̂ x∈x yx 228 a. caserta, a. giarlotta and s. watson are to-embeddings such that ψ◦ ı̂ = χ and χ◦ ı̂ = ψ. theorem 3.10 (iii) yields that ψ and χ are order-preserving homeomorphisms such that ψ−1 = χ. � remark 3.12 can now be restated as follows. corollary 3.15. the canonical unification of a go-space is its unique (up to a relabeling) minimum unification and minimum completion. we conclude the paper by describing explicitly the minimum unification (and completion) of an order resolution. let (x,�) and (yx,�x)x∈x be chains. endow the set ⋃ x∈x{x}×yx with the lexicographic order �lex and the order resolution topology τ⊗. we denote this go-space by (⊗ x∈x yx,�lex ) . definition 3.16. for each x ∈ x, define a chain (y +x ,� + x ) such that yx ⊆ y + x and �+x extends �x in the following way: (i) if x has an immediate predecessor x′ and an immediate successor x′′ such that ∃maxyx′ , ∄ minyx, ∄ maxyx and ∃minyx′′ , then set y + x := yx ∪{y0,y1} and y0 ≺ + x y ≺ + x y1 for all y ∈ yx; (ii) if (i) does not hold and x has an immediate predecessor x′ such that ∃maxyx′ but ∄ minyx, then set y + x := yx ∪{y0} and y0 ≺ + x y for all y ∈ yx; (iii) if (i) does not hold and x has an immediate successor x′′ such that ∃minyx′′ but ∄ maxyx, then set y + x := yx ∪{y1} and y ≺ + x y1 for all y ∈ yx; (iv) in all other cases, set y +x := yx. endow the chain ∑ x∈x y + x with the order topology τ + σ . we call the lots(∑ x∈x y + x ,τ + σ ) the lexicographic completion of the family (yx,�x)x∈x . open intervals in this lots are denoted by ((x′,y′), (x′′,y′′))+; a similar notation is used for the other types of intervals and for rays. (note that the lexicographic completion of a family of chains is obtained by inserting a jump per pseudojump, thus eliminating all pseudojumps in the representation of the sum.) the set ⋃ x∈x{x}×yx can be endowed with two topologies (apart from the resolution topology): the order topology τς and the subspace topology of τ + σ (inherited from the lexicographic completion of the family (yx)x∈x ). note that∑ x∈xyx is an open dense subspace of (∑ x∈xy + x ,τ + σ ) . in the next result we list some sets that are always open in the subspace topology (but possibly fail to be open in the order topology). lemma 3.17. for each x ∈ x, the following sets are open in the space(⋃ x∈x{x}×yx,τ + σ ) : (i) a = ⋃ w∈(x,→){w}×yw; (ii) b = ⋃ w∈[x,→){w}× yw, if either ∄minyx, or ∃minyx and x has an immediate predecessor; (iii) c = ⋃ w∈[x,x′′){w}× yw, if either ∄minyx, or ∃minyx and x has an immediate predecessor. (if x = maxx, then [x,x′′) = {x}.) on resolutions of linearly ordered spaces 229 proof. for (i), note that a is always open in τς (and hence in τ + σ ), except for the following case: yx has no maximum, x has an immediate successor x ′′ and yx′′ has a minimum y ′′ min. in this case, there exists ymax := maxy + x ∈ y + x \yx. it follows that a = [(x′′,y′′min),→) = ((x,ymax),→) + ∩ ( ⋃ x∈x {x}×yx ) is open in τ+σ . to prove (ii), first assume that yx has no minimum and let (s,t) be a point of b. if s = x, then we can select y ∈ yx such that y ≺x t and so (s,t) ∈ ((s,y),→) ⊆ b. if s 6= x, then (s,t) ∈ ((x,y),→) ⊆ b, where y is an arbitrary point of yx. thus b is open in this case. next assume that yx has a minimum ymin and x has an immediate predecessor x′. if yx′ has no maximum, then there exists y ′ max := maxy + x′ ∈ y + x′ \yx′ . since b = [(x,ymin),→) = ((x ′,y′max),→) + ∩ ( ⋃ x∈x {x}×yx ) it follows that b is open. on the other hand, if yx′ has a maximum, then b is open in the order topology τς and thus the same holds in the subspace topology of τ+σ . to prove (iii), observe that c = a∩b, where a := ⋃ w∈(←,x′′){w}×yw and b := ⋃ w∈[x,→){w}× yw. therefore, the claim follows from (the dual of) (i) and (ii). � lemma 3.18. the inclusion ı⊗ : (⊗ x∈x yx,�lex ) →֒ (∑ x∈xy + x ,τ + σ ) is an order-preserving homeomorphic embedding. proof. it suffices to prove that ı⊗ is a homeomorphic embedding. first we show that ı⊗ is continuous. let s = (←, (x,y)) + ∩ (⋃ x∈x{x}×yx ) be a subbasic open set in ı⊗ (⊗ x∈x yx ) ; we show that ı−1⊗ (s) = s is open in ⊗ x∈x yx. if (x,y) is such that y ∈ yx, then lemma 2.9 implies that s = (←, (x,y)) is open in τ⊗. next, assume that y ∈ y + x \ yx; thus, either y = miny + x or y = maxy +x . if y = miny + x , then x has an immediate predecessor x ′ and yx′ has a maximum y′max. by lemma ??, s = (←, (x ′,y′max)] = π −1(←,x) is open in τ⊗. on the other hand, if y = miny + x , then s is the open ray (←, (x ′′,y′′min)), where x′′ is the immediate successor of x and y′′min is the minimum of yx′′ . next we show that ı−1⊗ : ı⊗ (⊗ x∈x yx ) → ⊗ x∈x yx is continuous. let u⊗x v be a subbasic open set in s⊗; we prove that u⊗xv is open in ı⊗ (⊗ x∈x yx ) ⊆∑ x∈x y + x . by lemma 2.3 and duality, it suffices to examine the following two cases: (a) u ⊗x v = x ⊗x (y,→) for some y ∈ yx; (b) u ⊗x v = (x ′,→)⊗x yx for some x′ ≺ x. for (a), lemma 2.3 yields that x ⊗x (y,→) is equal to either {x}× (y,→) (if yx has no maximum) or ((x,y),→) (if yx has a maximum); in both cases the claim holds. for (b), the claim follows from lemma 2.3, lemma 2.11 and lemma 3.17. � 230 a. caserta, a. giarlotta and s. watson theorem 3.19. ((∑ x∈xy + x ,τ + σ ) , ı⊗ ) is the minimum unification of the gospace (⊗ x∈x yx,�lex ) . proof. since lemma 3.18 implies that ((∑ x∈x y + x ,τ + σ ) , ı⊗ ) is a unification of (⊗ x∈x yx,�lex ) , it suffices to show that it is minimum. let (z,ϕ) be a unification of ⊗ x∈x yx. we define a to-embedding ψ : ∑ x∈xy + x → z such that ϕ = ψ ◦ ı⊗. let (x,y) ∈ ∑ x∈xy + x . if (x,y) ∈ ⋃ x∈x{x}× yx, then let ψ(x,y) := ϕ(ı−1⊗ (x,y)). if (x,y) ∈ ∑ x∈xy + x \ (⋃ x∈x{x}×yx ) , then either (i) x has an immediate predecessor x′, yx′ has a maximum and yx has no minimum, or (ii) x has an immediate successor x′′, yx′′ has a minimum and yx has no maximum. we define ψ in case (ii) only, since (i) is dual to (ii). observe that y ∈ y +x \ yx is the maximum of y +x . if we denote y ′′ min := minyx′′ , then ((x,y),→) + = [(x′′,y′′min),→) + is an open ray in ∑ x∈xy + x . thus the set [(x′′,y′′min),→) = ı −1 ⊗ ( [(x′′,y′′min),→) + ∩ ı⊗ (⊗x∈xyx) ) is open in ⊗ x∈x yx. since ϕ is a to-embedding, the sets a := ϕ(←, (x ′′,y′′min)) and b := ϕ[(x′′,y′′min),→) are open in ϕ (⊗ x∈x yx ) ; further, a ≺ minb = ϕ(x′′,y′′min). note that there exists z0 ∈ z such that a ≺ z0 ≺ minb, since otherwise b would fail to be open in ϕ (⊗ x∈x yx ) ⊆ z. select such a z0 ∈ z and define ψ(x,y) := z0. this completes the definition of ψ in case (ii). it is immediate to check that ψ is injective and order-preserving. thus ψ is a to-embedding by lemma 3.2. � references [1] r. engelking, general topology (heldermann verlag, berlin, 1989). [2] v. v. fedorc̆uk, bicompacta with noncoinciding dimensionalities, soviet math. doklady, 9/5 (1968), 1148–1150. [3] k. p. hart, j. nagata and j.e. vaughan (eds.), encyclopedia of general topology (north-holland, amsterdam, 2004). [4] s. watson, the construction of topological spaces: planks and resolutions, in m. hus̄ek and j. van mill (eds.), recent progress in general topology, 673–757 (northholland, amsterdam, 1992). received may 2005 accepted june 2006 a. caserta (agata.caserta@unina2.it) department of mathematics, seconda università degli studi di napoli, caserta 81100, italia on resolutions of linearly ordered spaces 231 a. giarlotta (giarlott@unict.it) department of economics and quantitative methods, università di catania, catania 95129, italia s. watson (watson@hilbert.math.yorku.ca) department of mathematics and statistics, york university, toronto m3j1p3, canada @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 25–34 on complete objects in the category of t0 closure spaces d. deses ∗ , e. giuli and e. lowen-colebunders abstract. in this paper we present an example in the setting of closure spaces that fits in the general theory on ‘complete objects’ as developed by g. c. l. brümmer and e. giuli. for v the class of epimorphic embeddings in the construct cl0 of t0 closure spaces we prove that the class of v-injective objects is the unique firmly v-reflective subconstruct of cl0. we present an internal characterization of the vinjective objects as ‘complete’ ones and it turns out that this notion of completeness, when applied to the topological setting is much stronger than sobriety. an external characterization of completeness is obtained making use of the well known natural correspondence of closures with complete lattices. we prove that the construct of complete t0 closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins. 2000 ams classification: primary: 54a05, 54b30, 18g05; secondary: 54d10 keywords: complete object, firm, injective, complete lattice, t0 object, closure space. 1. introduction a closure space (x,c) is a pair, where x is a set and c is a subset of the power set p(x) satisfying the conditions that x and ∅ belong to c and that c is closed for arbitrary unions. the sets in c are called open sets. a function f : (x,c) → (y,d) between closure spaces (x,c) and (y,d) is said to be continuous if f−1(d) ∈c whenever d ∈d. cl is the construct of closure spaces as objects and continuous maps as morphisms. some isomorphic descriptions of cl are often used e.g. by giving the collection of all closed sets (the so called moore family [4]) where, as usual, the closed sets are the complements of the open ones and continuity is defined accordingly. another isomorphic ∗the first author is ‘aspirant’ of the f.w.o.-vlaanderen. 26 d. deses, e. giuli and e. lowen-colebunders description is obtained by means of a closure operator [4]. the closure operation cl : p(x) →p(x) associated with a closure space (x,c) is defined in the usual way by x ∈ cl a ⇐⇒ (∀c ∈ c : x ∈ c ⇒ c ∩ a 6= ∅) where a ⊂ x and x ∈ x. this closure need not be finitely additive, but it does satisfy the conditions cl ∅ = ∅, (a ⊂ b ⇒ cl a ⊂ cl b), a ⊂ cl a and cl (cl a) = cl a whenever a and b are subsets of x. continuity is then characterized in the usual way. finally closure spaces can also be equivalently described by means of neighborhood collections of the points. these neighborhood collections satisfy the usual axioms, except for the fact that the collections need not be filters. so in a closure space (x,c) the neighborhood collection of a point x is a non empty stack (in the sense that with every v ∈n(x) also every w with v ⊂ w belongs to n(x)), where every v ∈ n(x) contains x and n(x) satisfies the open kernel condition. in the sequel we will just write x for a closure space and we’ll choose the most convenient form for its explicit structure. motivations for considering closure spaces can be found in g. birkhoff’s book [4] where he associates closures to binary relations in a natural way. similar ideas appeared in g. aumann’s work on contact relations with applications to social sciences [3] or in a more recent work of b. ganter and r. wille on formal contexts with applications in data analysis and knowledge representation [11]. in recent years closures have also been used in connection with quantum logic and in the representation theory of physical systems, see e.g. [2] or [16]. in these applications the t0 axiom we are dealing with plays a key role [20]. in 1940 g. birkhoff’s motivation for considering closures also came from the observation that the collection of closed sets of a closure space forms a complete lattice. the interrelation between closures and complete lattices has been investigated by many authors and a general discussion of this subject can be found in m. erné’s paper [10]. in the last section of our paper further investigation of the correspondence with complete lattices leads to an external characterization of the complete objects we are studying. for all categorical terminology we refer the reader to the books [1, 13] or [18]. 2. the construct of t0 spaces 2.1. as is well known [9] cl is a topological construct in the sense of [1]. cl0 is the subconstruct consisting of its t0-objects. applying marny’s definition [15] we say that a closure space x is a t0-object in cl if and only if every morphism from the indiscrete object i2 on the two point set {0, 1} to x is constant. this equivalently means that for every pair of different points in x there is a neighborhood of one of the points not containing the other one. cl0 is an extremally epireflective subconstruct of cl [15] and as such it is initially structured in the sense of [17, 18]. in particular cl0 is complete and cocomplete and well-powered, it is an (epi extremal mono) category and an (extremal epi mono) category [13]. also from the general setting it follows that monomorphisms in cl0 are exactly the injective continuous maps and the category of t0 closure spaces 27 a morphism in cl0 is an extremal epimorphism if and only if it is a regular epimorphism if and only if it is surjective and final. 2.2. in order to describe the epimorphisms and the extremal monomorphisms in cl0 we need the regular closure operator determined by cl0 as introduced in [8, 9]. given a closure space x and a subset m ⊂ x one defines the regular closure of m in x as follows. a point x of x is in the closure of m if and only if (i) for every t0 closure space z and every pair of morphisms f,g : x → z, m ⊂{f = g} =⇒ f(x) = g(x). using the fact that cl0 is the epireflective hull in cl of the two point sierpinski space s2, we obtain the following equivalent description. (ii) for every pair of morphisms f,g : x → s2, m ⊂{f = g} =⇒ f(x) = g(x). quite similar to the topological situation one can prove yet another equivalent formulation. (iii) for every neighborhood v of x: v ∩ cl{x}∩m 6= ∅ in each of the equivalent cases we’ll write x ∈ clxb m. it was shown in [9] that the regular closure clb = {clxb : p(x) →p(x)}x∈|cl| defines a closure operator on cl. by the equivalent description (ii) clb coincides with the zariski closure operator as considered in [7] and [12]. the equivalent formulation (iii) is the formula for the b-closure (or front closure) in top. for this reason we will also call clb the b-closure operator on cl. it follows from theorem 2.8 in [9] that the epimorphisms in cl0 are the bdense continuous maps. so in fact the inclusion functor top0 ↪→ cl0 preserves epimorphisms. one observes that this is not so for the inclusion functor from top0 to the construct prtop0 of t0 pretopological spaces. using arguments analogous to the ones used in the topological case, one proves that cl0 is cowell-powered. the closure operator clb is idempotent and grounded and is easily seen to be hereditary in the sense that for a closure space y , a subspace x and m ⊂ x ⊂ y we have clxb m = cl y b m ∩x using this fact one can prove that a morphism in cl0 is an extremal monomorphism if and only if it is a regular monomorphism if and only if it is a b-closed embedding. explicit proofs of the previous statements have been worked out in [19]. 28 d. deses, e. giuli and e. lowen-colebunders 3. injective objects in cl0 and firmness in this paragraph we consider a particular class of morphisms in cl0. let v be the class of epimorphic embeddings, i.e. the class of all b-dense embeddings. this class v satisfies the following conditions: (α) closedness under composition, (β) closedness under composition with isomorphisms on both sides. (α) and (β) are standing assumptions made in [5] and enable us to apply to cl0 the theory developed in that paper. a t0 closure space b is v-injective if for each v : x → y in v and f : x → b there exists f′ : y → b such that f′◦v = f. in this case f′ is called an extension of f along v. inj v denotes the full subcategory of all v-injective objects in cl0. proposition 3.1. the two point sierpinski space s2 is v-injective in cl0. next consider rcl0 ({s2}), the epireflective hull of s2 in cl0. in view of the properties of cl0 listed in paragraph 2, this hull consists of all b-closed subspaces of powers of s2. recall that a reflective subcategory is v-reflective if the reflection morphisms all belong to v. proposition 3.2. a t0 closure space is v-injective if and only if it is a b-closed subspace of some power of s2. proof. in view of theorem 37.1 in [13] inj v is epireflective in cl0 and since it contains s2 we immediately have rcl0 ({s2}) ⊂ inj v. moreover rcl0 ({s2}) clearly is v-reflective, so if b is v-injective, the reflection morphism v : b → rb belongs to v. we have f′◦v = 1b where f′ is the extension of 1b : b → b along v. then clearly v is an isomorphism and therefore b ∈ |rcl0 ({s2})|. � remark 3.3. the notion of v-injectivity in cl0 differs from injectivity related to the class of all embeddings. v-injectivity is a strictly weaker condition as is shown by the example b = {(0, 0, 1), (0, 1, 1), (0, 1, 0), (1, 1, 1)} which is a b-closed subspace of s32 and hence is v-injective by proposition 3.2. however b is not injective in cl0 with respect to embeddings. we use the terminology of [5] (which slightly differs from [6]). a class u of morphisms in a category x (satisfying the standing assumptions (α) and (β)) is said to be (i) a subfirm class: if there exists a u-reflective subcategory with reflector r such that rf is an isomorphism whenever f is in u. (ii) a firm class: if there exists a u-reflective subcategory with reflector r such that rf is an isomorphism if and only if f is in u. in these cases the corresponding subcategory is said to be (sub-)firmly ureflective and it coincides with inj u [5]. again we consider the particular class v of b-dense embeddings in cl0. in view of the equivalent descriptions given in 2.2 and the fact that rcl0 ({s2}) = inj v, the class inj v is v-reflective. so we can apply theorems 1.4 and 1.14 in [5] to formulate the following result. the category of t0 closure spaces 29 proposition 3.4. the class v of b-dense embeddings is a firm class of morphisms in cl0 and inj v is the unique firmly v-reflective subcategory of cl0. in the context of an epireflective subcatgory x of a topological category, with s the class of embeddings in x and v the class of epimorphic embeddings, the notion of v-injective object can be linked to a few others, as discussed in [6]. an object x in x is said to be s-saturated if an x-morphism f : x → y is an isomorphism whenever f is in v. x is said to be absolutely s-closed if an x-morphism f : x → y is a regular monomorphism whenever f ∈s. in the particular situation where moreover in x extremal monomorphisms coincide with regular monomorphisms and where inj v is (sub-)firmly v-reflective, it was shown in [6] that for an object x in x one has x is v-injective ⇐⇒ x is s-saturated ⇐⇒ x is absolutely s-closed. from the results in paragraph 2 and from proposition 3.4 we can conclude that the v-injective objects of cl0 coincide with the s-saturated or equivalently with the absolutely s-closed t0 objects. these properties have also been considered by diers [7] in the setting of t-sets and the objects fulfilling the equivalent conditions were called algebraic t-sets. our example also fits in that context. 4. internal characterizations via complete objects the results displayed so far in paragraph 3 are quite similar to the well known topological situation on v-injective objects in top0. in that setting these objects can be internally characterized as t0 topological spaces for which every nonempty irreducible closed set is the closure of a point, i.e. as sober spaces [14, 6]. in this paragraph we give an internal characterization of the v-injective objects in cl0. this description for cl0, when applied to top0 will turn out to deal with a notion much stronger than sobriety. definition 4.1. let x be a closure space. for a⊂p(x) we write stacka = {b ⊂ x | ∃a ∈a : a ⊂ b} and a is said to be a stack if a = stacka. a nonempty stack is said to be open based if a = stack{g ∈ a | g open}. a proper open based stack a is said to be fundamental if a contains a member of every open cover of every element of a, i.e. whenever a ∈a and g is an open cover of a then ∃g ∈g : g ∈a. more briefly a fundamental nonempty open based stack is called an o-stack. as an easy example we note that in every closure space the neighborhood collection n(x) of a point x is an o-stack. proposition 4.2. on a closure space x and for a⊂p(x) we have: a is an o-stack if and only if there exists a (closed ) nonempty set f ⊂ x such that a = stack{g ⊂ x | g open, g∩f 6= ∅}. 30 d. deses, e. giuli and e. lowen-colebunders proof. for f nonempty it is clear that a = stack{g | g open, g∩f 6= ∅} = stack{g | g open, g∩ cl f 6= ∅} is an o-stack. conversely let a be an o-stack. let f = {x ∈ x | n(x) ⊂ a}. f clearly is nonempty since otherwise there would exist an open cover of x of which all members are not in a. if g is open and g∩f 6= ∅ then n(x) ⊂ a for some point x ∈ g. so g ∈a. on the other hand, if g is open and belongs to a then g has to intersect f. if not, there would exist an open cover of g of which all members are not in a. so finally we can conclude that a = stack{g | g open, g∩f 6= ∅}. remark that the set f = {x ∈ x | n(x) ⊂a} is in fact closed. � definition 4.3. a t0 closure space x is called complete if every o-stack is a neighborhood collection n(x) for some (unique) point x ∈ x. the uniqueness of the point follows from the t0 condition. in view of proposition 4.2 we get the following equivalent description. proposition 4.4. a t0 closure space x is complete if and only if every nonempty closed set is the closure of a (unique) point. proof. if f is closed and nonempty then there is a point x ∈ x such that stack{g | g open, g∩f 6= ∅} = n(x). then clearly f = cl{x}. conversely if a is an o-stack, as in the proof of proposition 4.2, let f be the nonempty closed set f = {x ∈ x | n(x) ⊂a}. now f = cl{x} implies a = n(x). � let ccl0 be the full subconstruct of cl0 consisting of the complete objects. proposition 4.5. complete t0 objects are absolutely s-closed. proof. let x be a complete t0 space and let f : x → y be an embedding in cl0. we prove that f(x) is b-closed in y . let a ∈ y \ f(x). either cly {a}∩f(x) = ∅ or cly {a}∩f(x) = clf(x) {f(z)} for some z ∈ x. in the latter case, let u = y \ cly {f(z)}, then by the t0 condition on y we have a ∈ u. moreover u ∩ cly {a}∩f(x) = ∅. so in both cases we can conclude that a 6∈ clyb f(x). � proposition 4.6. ccl0 is v-reflective in cl0. proof. first we construct the reflector r : cl0 → ccl0. let x be a t0 closure space and let x̂ = {a |a is a o-stack on x}. on x̂ we define a closure space as follows. for g ⊂ x open let ĝ = {a o-stack | g ∈a} the category of t0 closure spaces 31 then {ĝ | g ⊂ x open} defines a t0 closure structure on x̂. if a is an o-stack on x then â = stack{ĝ | g ⊂ x, g open, g ∈a} is its neighborhood collection in x̂. if ψ is an o-stack on x̂ then ψ̌ = stack{g | g ⊂ x, g open, ĝ ∈ ψ} is an o-stack on x. it follows that x̂ is a complete t0 closure space. let rx : x → x̂ be the natural injective map sending x ∈ x to n(x) ∈ x̂. clearly for g ⊂ x open, we have r−1x (ĝ) = g and hence rx is an embedding (in cl0). this embedding is b-dense since for an o-stack a and g ∈ a there exists x ∈ g such that n(x) ⊂a and therefore ĝ∩ cl x̂ {a} ∩rx(x) 6= ∅. now let b be a complete t0 closure space and f : x → b a continuous map. by proposition 3.4 inj v is v-reflective, so that the reflection map, say sb, belongs to v. hence sb is a b-dense embedding and by proposition 4.5 sb is also b-closed, therefore it is an isomorphism. thus b is v-injective. so there is an extension f′ of f along rx : x → x̂. since rx is an epimorphism in cl0 this extension moreover is unique. � corollary 4.7. ccl0 is v firmly reflective in cl0 and therefore coincides with the class of all v-injective t0 objects. remark 4.8. the previous conclusion combined with the characterization of ccl0 given in proposition 4.4 and the remarks at the end of paragraph 3, imply the result stated in example 9.7 in [7] that algebraic t0 closure spaces are those for which every nonempty closed set is the closure of a point. 5. an external characterization via the natural correspondence with complete lattices in the topological counterpart on complete t0 objects the duality between sober topological spaces and spatial frames leads to an external characterization of ‘completeness’. in this paragraph we base our external characterization on the correspondence between closure spaces and complete lattices. let clat∨,1 be the category whose objects are complete lattices and whose morphisms are maps preserving arbitrary joins and the top element. the dual category will be denoted clatop∨,1. to every closure space x we associate the lattice o(x) of its open subsets. with f : x → y we associate the map o(y ) → o(x) : g 7→ f−1(g). this correspondence defines a functor ωc : cl → clatop∨,1. 32 d. deses, e. giuli and e. lowen-colebunders in order to define an adjoint for ωc, let l be a complete lattice. a point of l is a surjective clat∨,1 morphism l → 2 where 2 = {0, 1} is the two point complete lattice. in the sequel we’ll use pts(l) to denote the set of points of l and for u ∈ l we’ll write σu = {ξ ∈ pts(l) | ξ(u) = 1}. observe that in contrast to the topological and frame counterpart, for objects u and v in l, we always have u 6= v ⇒ σu 6= σv. with this notation we can describe the functor σc : clatop∨,1 → cl sending a lattice l to the set pts(l), endowed with the closure structure {σu | u ∈ l}, and f : m → l to σcm → σcl : ξ 7→ ξ ◦f. proposition 5.1. for a complete lattice l, σcl is a complete t0 closure space. proof. consider two distinct ξ1,ξ2 ∈ pts(l) of σcl. there exist a u ∈ l such that ξ1(u) 6= ξ2(u). hence either ξ1 ∈ σu and ξ2 6∈ σu or ξ1 6∈ σu and ξ2 ∈ σu. so σc is t0. to prove the completeness we choose an o-stack a in σcl and consider v =∨ {u ∈ l | σu 6∈ a}. next we define the point ξ : l → 2 : u 7→ { 1 u 6≤ v 0 u ≤ v . we have that ξ(v) = 0, hence (σu 6∈ a ⇒ ξ(u) = 0). conversely, if σu ∈ a then u 6≤ v since a is an o-stack. thus (σu ∈ a ⇒ ξ(u) = 1). finally σu ∈a ⇐⇒ ξ(u) = 1. therefore a = n(ξ) in σcl. � proposition 5.2. the restrictions ωc : ccl0 → clat op ∨,1 σc : clatop∨,1 → ccl0 define an equivalence of categories. proof. the proof consists of three parts. (1) let l be a complete lattice then l ' ωcσcl. choose the isomorphism as follows: �l : ω cσcl → l : σu 7→ u this is a well defined clatop∨,1-isomorphism. (2) let x be a complete t0 closure space then x ' σcωcx. define the following map ηx : x → pts(ωcx) : x 7→ ξx where ξx(a) = { 1 x ∈ a 0 x 6∈ a , for all open sets a. ηx is injective since by the t0 property we have for x 6= y an open subset a such that ξx(a) 6= ξy(a). therefore ξx 6= ξy. to show that ηx is surjective we choose a point ξ, and consider stack ξ−1(1). this is obviously a stack with an open basis, so that if the category of t0 closure spaces 33⋃ i∈i ai ∈ stack ξ −1(1) then there exists b ∈ ξ−1(1) : b ⊂ ⋃ i∈i ai. so we get 1 = ξ(b) ≤ ξ( ⋃ i∈i ai) = ∨ i∈i ξ(ai). hence there exists an i ∈ i with ai ∈ ξ−1(1) and so stack ξ−1(1) is an o-stack. therefore there is an x ∈ x such that n(x) = stack ξ−1(1). we now have ξx(a) = 1 ⇔ a ∈n(x) ⇔∃b ∈ ξ−1(1) : b ⊂ a ⇔ ξ(a) = 1. hence ξ = ξx. moreover ηx and η −1 x are both continuous. this follows from η−1x (σa) = {x ∈ x | ξx(a) = 1} = a, ηx(a) = {ξx | x ∈ a} = {ξx | ξx(a) = 1} = σa, where a is open. (3) to see the naturality of η, consider continuous f : x → y where x and y are complete t0 closure spaces. we have the following compositions: (σc(ωc(f))◦ηx)(x) = (σc(f−1))(ξx) = ξx◦f−1 = ξf(x) and ηy ◦f(x) = ξf(x). hence η = (ηx)x∈|ccl0| is a natural isomorphism η : 1ccl0 ' σcωc. the naturality of � follows since if h : l → m is a clat∨,1morphism, we have the compositions (�m ◦ ωc(σc(h)))(σu) = �m ((σc(h))−1(σu)) = �m ({ξ ∈ pts(m) | ξ ◦h ∈ σu}) = �m ({ξ ∈ pts(m) | ξ ◦h(u) = 1}) = �m (σh(u)) = h(u) and h ◦ �l(σu) = h(u). therefore � = (�l)l∈|clat∨,1| is a natural isomorphism � : ωcσc ' 1clat∨,1 . hence we have proven the above equivalence. � references [1] j. adámek, h. herrlich and g. strecker, abstract and concrete categories (wiley, new york, 1990). [2] d. aerts, foundations of quantum physics: a general realistic and operational approach, internat. j. theoret. phys. 38 (1) (1999), 289–358. [3] g. aumann, kontaktrelationen, bayer. akad. wiss. math.-nat. kl. sitzungsber (1970), 67–77. [4] g. birkhoff, lattice theory (american mathematical society, providence, rhode island, 1940). [5] g. c. l. brümmer and e. giuli, a categorical concept of completion of objects, comment. math. univ. carolin. 33 (1) (1992), 131–147. [6] g. c. l. brümmer, e. giuli and h. herrlich, epireflections which are completions, cahiers topologie géom. diff. catég. 33 (1) (1992), 71–93. [7] y. diers, categories of algebraic sets, appl. categ. structures 4 (2-3) (1996), 329–341. [8] d. dikranjan and e. giuli, closure operators. i. topology appl. 27 (1987), 129–143. [9] d. dikranjan, e. giuli and a. tozzi, topological categories and closure operators, quaestiones math. 11 (3) (1988), 323–337. 34 d. deses, e. giuli and e. lowen-colebunders [10] m. erné, lattice representations for categories of closure spaces, categorical topology (heldermann verlag, berlin 1983), 197–222. [11] b. ganter and r. wille, formal concept analysis (springer verlag, berlin, 1998). [12] e. giuli, zariski closure, completeness and compactness, mathematik arbeitspapiere 54 (2000), universität bremen, proceedings of catmat 2000, h. herrlich and h. porst editors, 207–216. [13] h. herrlich and g. strecker, category theory, sigma series in pure mathematics (heldermann verlag, berlin, 1979). [14] r. -e. hoffmann, topological functors admitting generalized cauchy-completions, in categorical topology, lecture notes in math. 540 (1976), 286–344. [15] th. marny, on epireflective subcategories of topological categories, gen. topology appl. 10 (2) (1979), 175–181. [16] d. j. moore, categories of representations of physical systems, helv. phys. acta 68 (1995), 658–678. [17] l. d. nel, initially structured categories and cartesian closedness, canad. j. math. 27 (6) (1975), 1361–1377. [18] g. preuss, theory of topological structures (d. reidel publishing company, dordrecht, 1988). [19] a. van der voorde, separation axioms in extensiontheory for closure spaces and their relevance to state property systems, phd thesis vrije universiteit brussel, july 2001. [20] b. van steirteghem, t0 separation in axiomatic quantum mechanics, internat. j. theoret. phys. 39 (3) (2000), 955–962. received december 2001 revised november 2002 d. deses and e. lowen-colebunders department of mathematics, vrije universiteit brussel, 1050 brussels, belgium. e-mail address : diddesen@vub.ac.be evacoleb@vub.ac.be e. giuli dipartimento di matematica pura ed applicata, università di l’aquila, 67100 l’aquila, italy. e-mail address : giuli@univaq.it on complete objects in the category of t0 closure spaces. by d. deses, e. giuli and e. lowen-colebunders arhuspagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 67-72 topological groups: local versus global a. v. arhangelskii and v. v. uspenskij abstract. it is well known that locally compact groups are paracompact. we observe that this theorem can be generalized as follows: every locally paracompact group is paracompact. we prove a more general version of this statement using quotients. similar ‘local implies global’ theorems hold also for many other properties, such as normality, metacompactness, stratifiability, etc. 2000 ams classification: primary: 54h11. secondary: 54020, 54d60, 54e15 keywords: topological group, paracompact, lindelöf, local properties 1. introduction it is well known that every locally compact group is paracompact 1. recall the easy proof. let g be a locally compact group, and let v be a symmetric compact neighborhood of the neutral element. then h = ∪nv n is an open σ-compact subgroup of g, and g is the disjoint union of closed-and-open σcompact subsets – the left cosets with respect to h. it follows that g is strongly paracompact [3, corollary 5.3.11]. recall that a space is strongly paracompact if every open cover has a star-finite open refinement. the aim of the present paper is to generalize the above assertion in several directions. in particular, we observe that every locally paracompact group is paracompact (corollary 1.2), and every locally lindelöf group has an open lindelöf subgroup and hence is strongly paracompact. let p be a class of topological spaces. we say that p is closed hereditary if every closed subspace of every x ∈ p is in p. the notion of a class that is closed under locally finite unions of closed sets is self-explanatory. note that every class which is closed under disjoint sums and (continuous) closed mappings is closed under locally finite unions of closed sets. indeed, if x = ∪γ 1all spaces that we consider are assumed to be tychonoff. for topological groups, tychonoff is equivalent to hausdorff. 68 a. v. arhangelskii and v. v. uspenskij and γ is a locally finite closed cover, then the natural map ⊕γ → x is closed and onto. a space x is locally in p if every point in x has a (not necessarily open) neighborhood which belongs to p. theorem 1.1. let p be any class of topological spaces which is closed hereditary and closed under locally finite unions of closed sets. then every topological group which is locally in p is in p. corollary 1.2. every locally paracompact topological group is paracompact. theorem 1.1 can be similarly applied in many other cases. for example, we have the following two results: corollary 1.3. every locally normal topological group is normal. corollary 1.4. every locally stratifiable topological group is stratifiable. similarly, every locally metacompact group is metacompact, every locally subparacompact group is subparacompact; etc. however, it is not clear if the analogue of corollary 1.2 is true for strongly paracompact groups. question 1.5. is every locally strongly paracompact topological group strongly paracompact? the conclusion of theorem 1.1 remains valid for some classes p which are not closed under locally finite (or just finite) unions. for example, the union of two closed dieudonné-complete subspaces need not be dieudonné-complete [5] (recall that dieudonné-complete is equivalent to realcompact for spaces of cardinality less than the first measurable cardinal). however, the counterpart of theorem 1.1 holds for this class: theorem 1.6. let p be the class of all dieudonné-complete spaces. then every topological group which is locally in p is in p. a locally lindelöf group need not be lindelöf (consider a discrete uncountable group). however, we have the following: theorem 1.7. every locally lindelöf group contains an open lindelöf subgroup and is strongly paracompact. note that the easy argument that worked for locally compact groups does not work in the more general setting of locally lindelöf groups, since the lindelöf property is not preserved by products. 2. quotients g/h, where h is locally compact we give some applications of the above results to quotients of topological groups. a mapping f : x → y is locally perfect if every x ∈ x has a closed neighborhood n such that f|n : n → y is perfect (in particlular, f(n) is closed in y ). it was established in [2] that, for any locally compact subgroup h of topological groups: local versus global 69 arbitrary topological group g, the natural quotient mapping π of g onto the quotient space g/h is locally perfect. this turns out to be a key result in the proof that a number of topological properties are transfered from the quotient space g/h to the topological group g, provided that h is locally compact. one of such statements is the classical result of j.-p. serre that if h and g/h are locally compact, then the topological group g also is locally compact [7]. for example, it was shown in [2] in this way that if g/h is čech-complete (or feathered), and the subgroup h is locally compact, then g is čech-complete (respectively, feathered) as well. now we have a tool to prove similar statements for paracompactness, normality, dieudonné completeness, and for several other properties. indeed, the following general statement holds [2]: lemma 2.1. let p be a topological property preserved by preimages of spaces under perfect mappings (in the class of tychonoff spaces) and also inherited by closed sets. suppose further that g is a topological group and h is a locally compact subgroup of g such that the quotient space g/h has the property p. then there exists an open neighbourhood u of the neutral element e such that u has the property p. the next theorem immediately follows from corollary 1.2 and from lemma 2.1. theorem 2.2. suppose that g is a topological group, and h is a locally compact subgroup of g such that the quotient space g/h is paracompact. then the space g is paracompact. similarly, the results obtained above imply the following statements: theorem 2.3. suppose that g is a topological group, and h is a locally compact subgroup of g such that the quotient space g/h is dieudonné complete. then the space g is dieudonné complete. theorem 2.4. suppose that g is a topological group, and h is a locally compact subgroup of g such that the quotient space g/h is metacompact (subparacompact). then the space g is metacompact (subparacompact). question 2.5. suppose that g is a topological group, and that h is a locally compact subgroup of g such that the quotient space g/h is strongly paracompact. is then the space g strongly paracompact? clearly, this question is closely related to question 1.5. note that if g/h in question 2.5 is assumed to be lindelöf, then g is locally lindelöf (lemma 2.1) and hence strongly paracompact (therorem 1.7). note that for locally connected groups the answer for questions 1.5 and 2.5 is in the affirmative, since connected strongly paracompact spaces are lindelöf. question 2.6. suppose that g is a topological group, and that h is a locally compact metrizable subgroup of g such that the quotient space g/h is stratifiable. is then the space g stratifiable? 70 a. v. arhangelskii and v. v. uspenskij 3. proofs proposition 3.1. let g be a topological group with the neutral element e, and let u be a neighborhood of e. there exist a metric space (m, d) on which g acts continuously and transitively by isometries and a neighborhood o of a = p(e) in m such that p−1(o) ⊂ u, where p : g → m is the map defined by p(g) = ga. proof. this follows from the fundamental fact that every topological group can be embedded in the group of isometries of a metric space, see e.g. [8]. see [6] for the history of this assertion that was rediscovered many times by various authors. there also is an easy direct argument for the proposition. there exists a continuous left-invariant pseudometric ρ on g such that {x ∈ g : ρ(x, e) < 1} ⊂ u. this follows from the fact that every topological group has natural uniform structures [3, example 8.1.17]. let (m, d) be the metric space associated with the pseudometric space (g, ρ). � proof of theorem 1.1. let p be a class which is closed hereditary and closed under locally finite unions of closed sets. let g be a topological group which is locally in p. we must prove that g is in p. let u be a neighborhood of e such that u ∈ p. in virtue of proposition 3.1, there exists a metric space (m, d) on which g acts continuously and transitively by isometries, a point a ∈ m and an open neighborhood o of a in m such that p−1(o) ⊂ u, where p : g → m is the map defined by p(g) = ga. since m is paracompact, there exists a locally finite closed cover f of m which refines the open cover {go : g ∈ g}. consider the closed cover h = p−1(f) = {p−1(f) : f ∈ f} of g. each element of this cover is a closed subset of the set of the form gu. since gu is homeomorphic to u, u ∈ p, and p is closed hereditary, it follows that h ⊂ p. since h is locally finite and p is closed under locally finite unions of closed sets, we have g ∈ p. � to prove theorem 1.6, we need some preparations. for a space x we denote by ux the fine uniformity on x, that is, the finest compatible uniformity. a space x is dieudonné-complete iff the uniform space (x, ux) is complete. lemma 3.2. let a ⊂ b be subsets of a topological space x such that a and x \ b are functionally separated. then the uniformities ux and ub induce the same uniformity on a. proof. it suffices to prove that for every continuous pseudometric d on b there exists a continuous pseudometric d1 on x such that d(x, y) = d1(x, y) for all x, y ∈ a. there exists a normed linear space e and a map p : b → e such that d(x, y) = ‖p(x) − p(y)‖. let f : x → [0, 1] be a function such that f = 1 on a and f = 0 on a neighborhood of x \ b. define q : x → e as follows: q(x) = f(x)p(x) if x ∈ b; q(x) = 0 if x ∈ x \ b. then q is continuous and q = p on a. the pseudometric d1 on x defined by d1(x, y) = ‖q(x) − q(y)‖ is as required. � topological groups: local versus global 71 proposition 3.3. let f : x → y be an onto map. suppose that y is dieudonné-complete, and every point in y has a neighbourhood n such that f−1(n) is dieudonné-complete. then x is dieudonné-complete. proof. let f be a cauchy filter on (x, ux). we must prove that f converges. since f(f) is a cauchy filter on the complete space (y, uy ), it has a limit y ∈ y . let n be a neighborhood of y such that b = f−1(n) is dieudonné-complete. find a closed neighborhood k of y such that k ⊂ n and k and y \ n are functionally separated. let a = f−1(k). according to lemma 3.2, ux|a = ub|a. since a is closed in the complete space (b, ub), the uniform space (a, ux|a) = (a, ub|a) is complete. every member of f meets a, because every member of f(f) meets k. it follows that f induces a cauchy filter on the complete space (a, ux|a) and hence converges. � proof of theorem 1.6. let g be a topological group such that some neighborhood u of the neutral element e is dieudonné-complete. we prove that g is dieudonné-complete. in virtue of proposition 3.1, there exists a metric space m on which g transitively acts and a closed neighborhood n of a = p(e) in m such that p−1(n) ⊂ u, where p : g → m is the map defined by p(g) = ga. every point x ∈ m has a neighbourhood k such that p−1(k) is dieudonné-complete. indeed, write x = ga for some g ∈ g, and put k = gn. then p−1(k) = gp−1(n) is dieudonné-complete, since p−1(n) is closed in u. proposition 3.3 implies that g is dieudonné-complete. � proof of theorem 1.7. it suffices to note that every topological group g generated by a lindelöf neighborhood of its neutral element is lindelöf. this follows from guran’s theorem [4]: if g is generated by a lindelöf set, then for every neighborhood u of e the cover {gu : g ∈ g} has a countable subcover. see e.g. [8] for the proof. � references [1] a. v. arhangel’skĭı, classes of topological groups, russian math. surveys 36:3 (1981), 151–174. [2] a. v. arhangel’skĭı, quotients with respect to locally compact subgroups, submitted, december 2002. [3] r. engelking, general topology (pwn, warszawa, 1977). [4] i. guran, on topological groups close to being lindelof, soviet math. dokl. 23 (1981), 173–175. [5] a. mysior, a union of realcompact spaces, bull. acad. polon. sci. sér. sci. math. 29 (1981), no. 3-4, 169–172. [6] v. pestov, topological groups: where to from here?, topology proc. 24 (1999), 421–502. e-print: math.gn/9910144 [7] j.-p. serre, compacité locale des espaces fibré, c. r. acad. paris 229 (1949), 1295–1297. [=œuvres, vol. 1] [8] v. v. uspenskij, why compact groups are dyadic, in: general topology and its relations to modern analysis and algebra vi: proc. of the 6th prague topological symposium 1986, edited by z. frolik (heldermann, berlin, 1988), 601-610. 72 a. v. arhangelskii and v. v. uspenskij received june 2004 accepted january 2005 a. v. arhangelskii (arhangel@math.ohiou.edu) department of mathematics, 321 morton hall, ohio university, athens, ohio 45701, usa. v. v. uspenskij (uspensk@math.ohiou.edu) department of mathematics, 321 morton hall, ohio university, athens, ohio 45701, usa. beeragt.dvi @ applied general topology c© universidad politécnica de valencia volume 9, no. 1, 2008 pp. 133-142 product metrics and boundedness gerald beer ∗ abstract. this paper looks at some possible ways of equipping a countable product of unbounded metric spaces with a metric that acknowledges the boundedness characteristics of the factors. 2000 ams classification: primary 54e35; secondary 46a17. keywords: product metric, metric of uniform convergence, bornology, convergence to infinity. 1. introduction let 〈x, d〉 be an unbounded metric space. a net 〈xλ〉λ∈λ in x based on a directed set λ is called convergent to infinity in distance if eventually 〈xλ〉 stays outside of each d-bounded set: whenever b is contained in some d-ball, there exists λ0 ∈ λ such that λ > λ0 ⇒ xλ /∈ b. with sα(x) representing the open ball of radius α and center x, this condition can be reformulated in any of these equivalent ways: (1) ∀x ∈ x and α > 0, 〈xλ〉 is eventually outside of sα(x); (2) ∀x ∈ x we have limλd(xλ, x) = ∞; (3) ∃x0 ∈ x with limλd(xλ, x0) = ∞. now if {〈xn, dn〉 : n 6 n0} is a finite family of metric spaces, there are a number of standard ways to give the product ∏n0 n=1 xn a metric compatible with the product topology, the most familiar of which are these [11, pg. 111]: (1) ρ1(x, w) = max {dn(πn(x), πn(w)) : n 6 n0}; (2) ρ2(x, w) = ∑n0 n=1 dn(πn(x), πn(w)); (3) ρ3(x, w) = √ ∑n0 n=1 dn(πn(x), πn(w))2. ∗the author thanks richard katz for useful comments that were the genesis of this note. 134 g. beer all three of the metrics determine the same class of unbounded sets, and a net 〈xλ〉 in the product is convergent to infinity in ρi-distance for each i if and only if (i) ∀x ∈ n0 ∏ n=1 xn we have limλ maxn6n0 dn(πn(xλ), πn(x)) = ∞, or equivalently, (ii) ∃x0 ∈ n0 ∏ n=1 xn with limλ maxn6n0 dn(πn(xλ), πn(x0)) = ∞. for example, in r2 equipped with any of the standard metrics, the sequence 〈(j, 0)〉 is deemed convergent to infinity even though the second coordinate sequence is constant. thus, while convergence in the product with respect to each of the standard product metrics to a finite point amounts to convergence in each coordinate, this is not the case with respect to convergence to infinity in distance. this lack of symmetry is a little odd. something entirely different occurs when considering a countably infinite family of unbounded metric spaces {〈xn, dn〉 : n ∈ n}. the standard way to define a metric on ∏ ∞ n=1 xn equipped with the product topology is this [11, 8]: ρ∞(x, w) = ∞ ∑ n=1 2−n min{1, dn(πn(x), πn(w))}. the standard product metric is of course a bounded metric and all the boundedness features of the coordinate spaces are obliterated. in particular, no sequence in the countable product can converge to infinity in ρ∞-distance. while one can dispense with the weights in finitely many factors and permit convergence to infinity in a restricted setting, this construction, while having the desirable local comportment, is myopic, speaking both figuratively and literally. for a product metric expressed as a supremum but with the same limitations, see [7, pg. 190]. it is natural to consider, in the case of countably infinitely many coordinates, the natural analogs of conditions (i) and (ii) above, namely, (i′) ∀x ∈ ∞ ∏ n=1 xn we have limλ supn∈n dn(πn(xλ), πn(x)) = ∞, (ii′) ∃x0 ∈ ∞ ∏ n=1 xn with limλ supn∈n dn(πn(xλ), πn(x0)) = ∞. as in the case of finitely many factors, the existence of some coordinate for which 〈πn(xλ)〉 converges to infinity in dn-distance is sufficient but not necessary for convergence in the of sense (i′). actually, it is easier to understand what it means for (i′) to fail than for it to hold. product metrics and boundedness 135 proposition 1.1. let 〈xλ〉λ∈λ be a net in a product of unbounded metric spaces ∏ ∞ n=1 xn. the following conditions are equivalent: (1) condition (i′) does not hold for 〈xλ〉λ∈λ; (2) there exists a cofinal subset λ0 of λ and α > 0 such that ∀n ∈ n we have diam ({πn(xλ) : λ ∈ λ0}) < α. proof. suppose (i′) does not hold; pick x in the product and α > 0 such that lim infλ supn∈n dn(πn(xλ), πn(x)) < α 2 . we can then find λ0 cofinal in λ such that ∀n ∈ n ∀λ ∈ λ0 we have dn(πn(xλ), πn(x)) < α 2 , and so ∀n ∈ n ∀λ ∈ λ0 πn(xλ) ∈ s α 2 (πn(x)), from which (2) follows. conversely, if (2) holds, fix λ ∈ λ0 and set x1 = xλ0 . then ∀λ ∈ λ0 we have supn∈n dn(πn(xλ), πn(x1)) 6 α, and as a result, lim infλ supn∈n dn(πn(xλ), πn(x1)) 6 α, so that condition (i′) fails. � on the other hand, condition (ii′) is much too weak to be useful, for if x0 is a given point of the product and x1 is a second point satisfying supn∈n dn(πn(x1), πn(x0)) = ∞, then the constant sequence each of whose terms is x1 obviously satisfies (ii ′) but not (i′). there are two main objectives of this note. first, while (i′) may be worthy of study as a generalization of convergence to infinity with respect to the ℓ∞-metric, we intend to show that no metric exists on ∏ ∞ n=1 xn compatible with the product topology or otherwise with respect to which convergence in the sense of (i′) corresponds to convergence to infinity in distance. in other words, it is impossible to find a metric compatible with any metrizable topology on the product of a countably infinite collection of unbounded metric spaces such that convergence of nets to infinity in distance generalizes what occurs with respect our standard metrics when there are only finitely many factors. second, we display an unbounded metric compatible with the product topology with respect to which convergence to infinity in distance means convergence to infinity in distance in all coordinates. 136 g. beer 2. an alternate product metric we address our objectives in reverse order. to construct our metric, we use a standard device [8, pg. 347] : if 〈x, d〉 is a metric space and f is a continuous real-valued function on x then df : x → [0, ∞) defined by df (x, w) = d(x, w) + |f (x) − f (w)| is a metric on x equivalent to d. theorem 2.1. let {〈xn, dn〉 : n ∈ n} be a family of unbounded metric spaces. then there exists an unbounded metric ρ on ∏ ∞ n=1 xn compatible with the product topology such that a net in the product is convergent to infinity in ρ-distance if and only if it is convergent to infinity coordinatewise with respect to each of the coordinate metrics dn. proof. we start with the standard bounded metric ρ∞ on the product and modify it by a continuous real-valued function f as indicated above. formally, we define f to be an infinite sum of nonnegative continuous functions {fk : k ∈ n} each defined on ∏ ∞ n=1 xn. fix x0 ∈ ∏ ∞ n=1 xn, and for each k, we define fk by the formula fk(x) = minn6k min{1, dn(πn(x), sk(πn(x0))}. the following three properties of fk are evident from the definition: (1) fk is a continuous function with respect to the product topology; (2) ∀x ∈ ∏ ∞ n=1 xn ∀k ∈ n, 0 6 fk+1(x) 6 fk(x) 6 1; (3) if ∃n 6 k such that πn(x) ∈ sk(πn(x0)), then fk(x) = 0. in addition we have the following key property: (4) ∀x ∈ ∏ ∞ n=1 xn ∃k0 ∈ n such that fk0 vanishes on some neighborhood of x. to establish property (4), choose k0 such that π1(x) ∈ sk0 (π1(x0)). then for each w ∈ π−1 1 [sk0 (π1(x0)], a product neighborhood of x, we have fk0 (w) = 0. from properties (2) and (4), setting ek = {x : fk(x) > 0} (k ∈ n), we see that the family {ek : k ∈ n} is locally finite. it now follows that f : ∏ ∞ n=1 xn → [0, ∞) defined by f = f1 + f2 + f3 + · · · is real-valued and continuous. we are now ready to define the desired metric ρ on the product: ρ(x, w) := ρ∞(x, w) + |f (x) − f (w)|. as we indicated earlier, ρ is compatible with the product topology. now a net 〈xλ〉λ∈λ in the product converges to infinity in ρ-distance if and only if limλρ(xλ, x0) = ∞, and since for all λ ρ∞(xλ, x0) 6 1, this occurs if and only limλf (xλ) = ∞. we first show, assuming limλf (xλ) = ∞, that for each n ∈ n, 〈πn(xλ)〉 converges to infinity in dn-distance. to this end, fix n ∈ n, product metrics and boundedness 137 say n = n0. we will show that if k0 is an arbitrary positive integer, then for all λ sufficiently large, πn0 (xλ) /∈ sk0 (πn0 (x0)) there is no loss in generality in assuming k0 > n0. pick λ0 ∈ λ such that λ > λ0 ⇒ f (xλ) > k0 − 1. fix λ > λ0; if πn0 (xλ) ∈ sk0 (πn0 (x0)) were true, then by properties (2) and (3) ∀k > k0 we have fk(xλ) = 0. as a result, we have f (xλ) = ∞ ∑ k=1 fk(xλ) = k0−1 ∑ k=1 fk(xλ) 6 k0 − 1 this contradiction shows that for all λ > λ0 we have πn0 (xλ) /∈ sk0 (πn0 (x0)) as required. conversely, suppose ∀n that 〈πn(xλ)〉 converges to infinity in dn-distance. again fix k0 ∈ n; we intend to show that eventually f (xλ) > k0. pick λ0 ∈ λ such that condition (∗) below holds: (∗) ∀λ > λ0 ∀n 6 k0 + 1, πn(xλ) /∈ sk0+1(πn(x0)). now fix λ > λ0. by (∗), ∀k 6 k0 ∀n 6 k we have dn(πn(xλ), sk(πn(x0)) > 1, and as a result ∀k 6 k0 we have fk(xλ) = 1. we conclude that f (xλ) > k0 ∑ k=1 fk(xλ) = k0 as required. � the proof presented above goes through in the case that the product is finite, say, ∏n0 n=1 xn, by slightly altering the definition of each fk as follows: fk(x) = minn6n0 { min{1, dn(πn(x), sk(πn(x0))} } . when the finite product is rn0 , the author’s metric of choice is the following one: ρ(x, w) = min{1, maxn6n0|πn(x)−πn(w)|}+|minn6n0|πn(x)|−minn6n0|πn(w)||. 3. convergence to infinity in distance and bornologies there is another way to approach the question of the existence of the metric that theorem 2.1 provides, following an axiomatic approach to boundedness developed by s.-t. hu [10, 11] over 50 years ago. hu discovered that the family bd of bounded sets determined by an unbounded metric d on a metrizable space x had certain characteristic properties. first, the bounded sets form a bornology [9, 1, 3, 4, 12]; that is, they form of a cover of x that is closed under taking finite unions and subsets. second, x is not itself in the bornology. third, bd has a countable base {bn : n ∈ n}, i.e., each bounded set is contained in some bn. finally, for each element b ∈ bd, there exists b ′ in the bornology with cl(b) ⊆ int(b′). conversely, if a is a bornology with a countable base 138 g. beer on a noncompact metrizable space x, x /∈ a , and ∀a ∈ a ∃a′ ∈ a with cl(a) ⊆ int(a′), then there exists a compatible unbounded metric d such that a = bd. a bornology that satisfies hu’s axioms or coincides with the power set p(x) of x (the bornology of a bounded metric) is called a metric bornology[3]. for example, the bornology consisting of the subsets of x with compact closure is a metric bornology if and only if x is locally compact and separable [14]. as is well-known, x is compact if and only if there is exactly one metrizable bornology, namely p(x). it can be shown [1] that if x is noncompact and metrizable, there is actually an uncountable family of compatible metrics {d : d ∈ d} whose associated metric bornologies {bd : d ∈ d} are distinct. in particular, the usual metric on the real line r is just one of many (in terms of boundedness) compatible with the usual topology. with hu’s result in mind, let’s return to the context of a product of a family {〈xn, dn〉 : n ∈ n} of unbounded metric spaces. again fixing x0 in the product, consider the bornology on ∏ ∞ n=1 xn having as a countable base all sets of the form △(k, f ) := {x : ∃n ∈ f πn(x) ∈ sk(πn(x0)} = ⋃ n∈f π−1n (sk(πn(x0))) (k ∈ n, f a finite subset of n). it is easy to verify that hu’s axioms all are verified, and in particular that relative to the product topology, one has cl(△(k, f )) ⊆ int(△(k + 1, f )) = △(k + 1, f ). now if ρ is an unbounded metric whose bounded sets coincide with this bornology, and a net 〈xλ〉λ∈λ converges to infinity in ρ-distance, then for each n and k, the net is outside △(k, {n}) eventually which means that 〈πn(xλ)〉 converges to infinity in dn-distance. on the other hand, if for each fixed k and n, 〈πn(xλ)〉 is outside sk(πn(x0)) eventually, then for any finite set of integers f , eventually 〈xλ〉 is outside of △(k, f ), and so 〈xλ〉 converges to infinity in ρ-distance. by definition a net in 〈x, d〉 is convergent to infinity in d-distance if it is eventually outside of each element of bd. abstracting from this, given a bornology b on x, we say 〈xλ〉λ∈λ is convergent to infinity with respect to b if for each b ∈ b there exists λ0 ∈ λ such that λ > λ0 ⇒ xλ /∈ b. observe, that there is no loss of generality in defining this notion for bornologies rather than for covers, and that nets cannot simultaneously converge to infinity with respect to a bornology and to a finite point if and only if x is locally bounded [10]: each x ∈ x has a neighborhood in the bornology (see also [6, proposition 2.7]). local boundedness of course implies that each compact set is in the bornology [10]. this all leads naturally to an investigation of extensions of the space and their relation to bornologies that is outside the scope of this paper (see [2, 5, 6]). to show that convergence to infinity with respect to a bornology is more generally a useful notion, we offer three simple propositions. product metrics and boundedness 139 proposition 3.1. let x be a metrizable space and let f be the bornology of finite subsets of x. then a sequence 〈xj〉 is convergent to infinity with respect to f if and only if 〈xj〉 has no constant subsequence. proposition 3.2. let x be a metrizable space and let b be the bornology of subsets of x with compact closure. then a sequence 〈xj〉 is convergent to infinity with respect to b if and only if 〈xj〉 has no convergent subsequence. proof. if 〈xj〉 has a convergent subsequence 〈xjn 〉 to a point p, then the original sequence is not eventually outside the compact set {p, xj1 , xj2 , xj3 , . . .}. sufficiency is obvious. � proposition 3.3. let x be a normed linear space and let b be the bornology of weakly bounded subsets of x. then a net 〈xλ〉λ∈λ is convergent to infinity with respect to b if and only if ∀α > 0 ∃λ0 ∈ λ such that λ > λ0 ⇒ ||xλ|| > α. proof. recall that a ⊆ x is weakly bounded if ∀f ∈ x∗, f (a) is a bounded set of scalars. evidently, the weakly bounded sets so defined also form a bornology. now the uniform boundedness principle of functional analysis [13], when applied to the banach space x∗ equipped with the usual operator norm, says that each weakly bounded subset of x is norm bounded. as the converse is obviously true, the bornology of weakly bounded sets coincides with the metric bornology determined by the norm. � the next proposition and its corollary show that the relative size of two bornologies is determined by the set of nets that converge to infinity with respect to them. proposition 3.4. let b1 and b2 be bornologies on a metrizable space x. the following conditions are equivalent: (1) b2 ⊆ b1; (2) whenever a net 〈xλ〉λ∈λ converges to infinity with respect to b1, then 〈xλ〉λ∈λ converges to infinity with respect to b2. proof. only the implication (2) ⇒ (1) requires proof. suppose (1) fails; then there exists b2 ∈ b2 that is not a subset of any element of b1. now since b1 is closed under finite unions, it is directed by inclusion. for each b ∈ b1, pick xb ∈ b2 ∩ b c. then the net 〈xb〉b∈b1 converges to infinity with respect to b1 but not with respect to b2. � corollary 3.5. let b1 and b2 be bornologies on a metrizable space x. the following conditions are equivalent: (1) b1 = b2; (2) b1 and b2 determine the same nets convergent to infinity. the next example shows that the same set of sequences can converge to infinity for distinct bornologies that do not have countable bases. 140 g. beer example 3.6. in the real line r, consider these two bornologies: b1 = {a ∪ f : a is a countable subset of n c and f is finite}, b2 = {a ∪ e ∪ f : a is a countable subset of n c, e ⊆ (0, 1), and f is finite}. observe that neither has a countable base. while b2 properly contains b1, the bornologies determine the same sequences convergent to infinity. specifically, 〈xj〉 converges to infinity with respect to either if and only if 〈xj〉 has no constant subsequence and eventually is in n. using hu’s axioms and corollary 3.5, we can directly verify that convergence to infinity as described by condition (i′) in the introduction is not convergence to infinity with respect to any metric on the product. now convergence of nets in this sense is obviously convergence to infinity for a bornology b on ∏ ∞ n=1 xn having as a base all finite unions of sets of the form b(w, k) := {x : ∀n ∈ n πn(x) ∈ sk(πn(w))} = ∞ ∏ n=1 sk(πn(w)) where k runs over n and w runs over ∏ ∞ n=1 xn (note the family of all sets of the form ∏ ∞ n=1 sk(πn(w)) is not directed by inclusion). we claim that this b does not have a countable base. if it did we could find a sequence of the form 〈(wj , kj )〉 such that {∪nj=1b(wj , kj ) : n ∈ n} forms a countable base for b. in fact, no such countable family even forms a cover of the product. to see this, take x ∈ ∏ ∞ n=1 xn where πn(x) /∈ ∪nj=1skj (πn(wj )). a stronger notion than convergence to infinity in distance coordinatewise is that the convergence be uniform coordinatewise, according to the following definition. definition 3.7. a net 〈xλ〉λ∈λ in a product ∏ ∞ n=1 xn of unbounded metric spaces is said to converge coordinatewise to infinity in distance uniformly with respect to x0 ∈ ∏ ∞ n=1 xn if ∀α > 0 ∃λ0 ∈ λ such that whenever λ > λ0, we have infn∈n dn(πn(xλ), πn(x0)) > α. definition 3.7 was formulated for a countably infinite product only because for a finite product, the concept is no stronger than convergence to infinity in distance coordinatewise which we have already discussed. example 3.8. a sequence 〈xj〉 in a countably infinite product can converge to infinity in distance coordinatewise but not uniformly with respect to any x0 in the space. for our product take n∞, that is, the product of countably many copies of the positive integers each equipped with the usual metric of the line. for each j define xj ∈ n ∞ by πn(xj ) = { j − n + 1 if n 6 j 1 if n > j. product metrics and boundedness 141 as a particular case, 〈x4〉 is the sequence 4, 3, 2, 1, 1, 1, . . . . for each coordinate index n we have limj→∞πn(xj ) = limj→∞j−n+1 = ∞, establishing coordinatewise convergence to infinity with respect to the usual metric. we claim that assuming that the convergence is uniform with respect to some x0 leads to a contradiction. if this occurs, then in particular ∃j0 ∈ n such that whenever j > j0, we have (∗) infn∈n|πn(xj ) − πn(x0)| > 1, and in particular, |πj0 (xj0 ) − πj0 (x0)| > 1. now πj0 (xj0 ) = 1, and so πj0 (x0) ∈ {3, 4, 5, . . .}. set k = πj0 (x0) − 1; then j0 + k > j0 and we compute |πj0 (xj0+k) − πj0 (x0)| = |(j0 + k) − j0 + 1 − (k + 1)| = 0 and a contradiction to (∗) is obtained as claimed. if a net 〈xλ〉λ∈λ in a product ∏ ∞ n=1 xn of unbounded metric spaces converges coordinatewise to infinity in distance uniformly with respect to x0, then this is also true if we replace x0 by any x with supn∈n dn(πn(x), πn(x0)) < ∞. on the other hand, given any sequence 〈xj〉 in the product, coordinatewise convergence to infinity uniformly with respect to all points w in the product is impossible: for example, a ”bad point” w with respect to 〈xj〉 is defined by πj (w) = πj (xj ) for j = 1, 2, 3, . . . . hopefully, this discussion will give the reader a feeling for the nature of the dependence of this mode of convergence on x0. now convergence of 〈xλ〉λ∈λ to infinity as described by definition 3.7 is clearly convergence to infinity with respect to a bornology b on ∏ ∞ n=1 xn having a countable base consisting of those product open sets of the form bk := {x : ∃n ∈ n πn(x) ∈ sk(πn(x0))} = ∪n∈n π −1 n (sk(πn(x0))) (k ∈ n). since each bk is in fact dense with respect to the product topology, hu’s axioms are not satisfied, so that by corollary 3.5 there is no metric ρ compatible with the product topology such that convergence to infinity in ρ-distance equates with convergence as described by definition 3.7. but the situation is salvageable, provided we are willing to relinquish the product topology in favor of a stronger metrizable one, namely, the topology determined by the bounded metric ρuc(x, w) = min { 1, sup{dn(πn(x), πn(w)) : n ∈ n} } . 142 g. beer when all 〈xn, dn〉 are the same unbounded metric space 〈x, d〉, so that our product is of the form x∞, this is the metric of uniform convergence for sequences in 〈x, d〉. using the formula cluc(bk) = {x : infn∈n dn(πn(x), sk(πn(x0))) = 0} and keeping in mind that the product topology is coarser than the ρuc-topology, is easy to check that the bornology on ∏ ∞ n=1 xn with base {bk : k ∈ n} satisfies hu’s axioms with respect to the ρuc-topology. thus, we can remetrize the product equipped with this stronger topology in a way that convergence to infinity in distance for the metric equates with definition 3.7. we leave it to the imagination of the reader to come up with possible formulas for such a metric. references [1] g. beer, on metric boundedness structures, set-valued anal. 7 (1999), 195-208. [2] g. beer, on convergence to infinity, monat. math. 129 (2000), 267-280. [3] g. beer, metric bornologies and kuratowski-painlevé convergence to the empty set, j. convex anal. 8 (2001), 279-289. [4] j. borwein, m. fabian, and j. vanderwerff, locally lipsschitz functions and bornological derivatives, cecm report no. 93:012. [5] a. caterino and s. guazzone, extensions of unbounded topological spaces, rend. sem. mat. univ. padova 100 (1998), 123-135. [6] a. caterino, t. panduri, and m. vipera, boundedness, one-pont extensions, and bextensions, math. slovaca 58, no. 1 (2008), 101–114. [7] j. dugundji, topology, allyn and bacon, boston, 1966. [8] r. engelking, general topology, polish scientific publishers, warsaw, 1977. [9] h. hogbe-nlend, bornologies and functional analysis, north-holland, amsterdam, 1977. [10] s.-t. hu, boundedness in a topological space, j. math pures appl. 228 (1949), 287-320. [11] s.-t. hu, introduction to general topology, holden-day, san francisco, 1966. [12] a. lechicki, s. levi, and a. spakowski, bornological convergences, j. math. anal. appl. 297 (2004), 751-770. [13] a. taylor and d. lay, introduction to functional analysis, wiley, new york, 1980. [14] h. vaughan, on locally compact metrizable spaces, bull. amer. math. soc. 43 (1937),532-535. received november 2006 accepted january 2007 gerald beer (gbeer@cslanet.calstatela.edu) department of mathematics, california state university los angeles, 5151 state university drive, los angeles, california 90032 usa induragt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 1123 representations of ordered semigroups and the physical concept of entropy juan c. candeal, juan r. de miguel∗, esteban induráin† and ghanshyam b. mehta abstract. the abstract concept of entropy is interpreted through the concept of numerical representation of a totally preordered set so that the concept of composition of systems or additivity of entropy can be analyzed through the study of additive representations of totally ordered semigroups. 2000 ams classification: 80a10, 06f05, 91b16. keywords: topological ordered sets, utility functions, entropy, semigroups. 1. introduction in an interesting paper cooper [?] studies the foundations of thermodynamics and the existence of entropy functions on state spaces of thermodynamic systems. three different formulations of the second law of thermodynamics due to clausius, kelvin and caratheodory are considered and it is proved that caratheodory’ s axiom is not sufficient for the existence of an entropy function even in simple spaces. cooper studies this problem by formulating a concept of an accesibility relation on the state space s of a thermodynamical system. this accessibility relation on s is a total preorder and an entropy function is defined to be a real-valued function on s that preserves the accessibility relation. if, in addition, the state space s has an additive structure then an entropy function is also required to preserve the algebraic structure, i.e. it is an order-preserving function which is also an (algebraic) homomorphism. moreover if the phase space of the thermodynamic system is a topological space and ∗corresponding author. †the participation of candeal, de miguel and induráin has been partially supported by the research project pb98-0551 “estructuras ordenadas y aplicaciones” (m.e.c.–spain). also, the research of coauthors candeal and induráin has been supported by the “integrated action of research hi2000-0116 (spain-italy)”. 12 candeal, de miguel, induráin and mehta the accessibility relation is continuous then the problem of the existence of a continuous entropy function is equivalent to that of proving the existence of a continuous order-preserving real-valued function defined on a topological space equipped with a continuous total preorder. originally the problem of the existence of an order-preserving real-valued function defined on a totally ordered set was posed and solved by cantor (see [?], [?]). subsequent generalizations were made by birkhoff (see [?]) and debreu (see [?], [?]) among others, including those generalizations that deal with continuity. for more recent discussions concerning the new contributions to this framework, see [?]. for an account of the mathematical aspects and foundations of thermodynamics the reader is referred to the book [?] which appeared before the paper of cooper. see also the recent account of the second law of thermodynamics in [?]. cooper’s paper has the merit of independently discovering or anticipating, at least implicitly, some deep concepts and results that are to be found in the vast literature on the existence of order-preserving functions in mathematics, mathematical economics, measurement theory and other related fields. in particular, this is true for the concept of additivity of entropy. cooper’s ideas about additive entropy closely parallel the modern theory of order-preserving functions on ordered semigroups and algebraic utility theory. one of the objectives of this paper is to establish this fact. it is a remarkable fact that, with the exception of a few quotations (see, e. g., [?]), cooper’s paper appears to have gone largely unnoticed by researchers in mathematical economics, in the theory of measurement, or in the representation theory of totally ordered sets and semigroups by real valued functions. it transpires that there are some mathematical mistakes in cooper’s paper. in the present paper we pay special attention to the algebraic aspects, and show how these errors may be rectified or extenuated. we feel strongly that cooper’s paper should be highly commended for introducing several crucial ideas, enabling us to discover the astonishing similarity between the structure of the entropy representation problem and that of the utility representation problem. to this end we propose certain mathematical interpretations of cooper’s modelling of entropy. the article is organized as follows: section ?? contains definitions, notations, and necessary background. in section ?? the concept of entropy is compared with the concept of utility function. in section ?? the algebraical aspects of entropy functions are considered. 2. definitions and previous results let x be a nonempty set. a binary relation “-” defined on x is a total preorder if it is reflexive, transitive and total. if in addition “-” is antisymmetric, then it is said to be a total order. associated to “-” we define the strict preference and the indifference relations, respectively denoted by “≺” and “∼”, given by x ≺ y ⇐⇒ ¬(y x) and x ∼ y ⇐⇒ x y, y x (x, y ∈ x). ordered semigroups and entropy 13 a total preorder “-” on x is said to be representable (respectively: pseudorepresentable) if there is a real-valued function u : x −→ r such that x y ⇐⇒ u(x) ≤ u(y) (respectively: x y =⇒ u(x) ≤ u(y)) (x, y ∈ x). such a function is said to be an order-preserving function or a strictly isotone function. if the set x is a set of alternatives of some economic agent on which is defined a preference relation then such an order-preserving function is said to be a utility (respectively: pseudoutility) function in the economics literature. on a totally ordered set x it is possible to define a natural topology, called the order topology, a subbasis of which is given by the family of subsets: (−∞, a) = {x ∈ x : x ≺ a}, (b, +∞) = {y ∈ x : b ≺ y} (a, b ∈ x). let (x, τ) be a topological space endowed with a total preorder “-”. then “-” is said to be τ-continuous if for every x ∈ x the subets (−∞, x) and (x, +∞) are τ-open. a total preorder “-” on x is said to be separable in the sense of debreu if there is a countable subset c ⊆ x such that for every x, y ∈ x with x ≺ y, there exists c ∈ c with x c y. it turns out that this property characterizes the representability of a total preorder “-” by means of an order-preserving function. (see, e.g., [?], pp. 14 and ff.) in general an order-preserving function may or may not be continuous with respect to the order topology of x and the euclidean topology of r. the problem of the existence of a continuous order-preserving function on an ordered topological space was solved by debreu (see [?], [?]) in two classical papers. to that end, debreu introduced the concept of a gap. definition 2.1. let r̄ denote the extended real line. a degenerate set in r̄ is one having at most one element. a gap of a subset s of r̄ is a maximal nondegenerate interval disjoint from s and with a lower bound and an upper bound in s. an interval of r̄ of the form (a, b] or [a, b) is said to be half-open half-closed. theorem 2.2. (debreu’ s open gap lemma): if s is a subset of r̄, there is an increasing function g : s −→ r such that all the gaps of g(s) are open. theorem 2.3. (debreu’ s representation theorem): if there is a real-valued order-preserving function on a totally preordered topological space x then there is a continuous real-valued order-preserving function on x. finally, we mention that the abstract study of the relationship between order and topology was initiated by nachbin (see [?]). 3. entropy and utility theory in this section we present the approach given by cooper in [?], relative to the existence of a continuous utility function on a totally preordered topological space. first we recall some nomenclature and notations used in cooper’s work. (see [?]). 14 candeal, de miguel, induráin and mehta the state space s of a thermodynamic system is a separable topological space. there is a relationship, called accesibility relation and denoted “→” among the elements of s. the fact s1→s2 may be read “a transition from s1 to s2 is possible”. we write s1 9 s2 for the negation of the statement s1→s2, and we write s1 ⇄ s2 if both s1→s2 and s2→s1 hold. for s1 → s2 we shall understand s1→s2 and s2 9 s1. a function f : s −→ r is an entropy function for an accesibility relation “→” whenever s1→s2 ⇐⇒ f(s1) ≤ f(s2) for every s1, s2 ∈ s. remark 3.1. observe the analogy with the classical framework coming from economics in which a consumer defines a preference relation “-” on a nonempty set of goods x, usually called consumption set. as defined above, in this context a utility function is a map u : x −→ r such that x y ⇐⇒ u(x) ≤ u(y) for every x, y ∈ x. utility functions and entropy functions (and other similar concepts such as scale in measurement theory) are examples of order-preserving functions. henceforth we will not use cooper’s notation and refer to all order-preserving functions as utility functions. coming again to cooper’s work, theorem 1 in [?] can be stated as follows: theorem 3.2. (cooper’s theorem 1): let (x, τ) be a separable topological space equipped with a continuous total preorder “-”. then there is a continuous utility function for “-”. unfortunately cooper’s result is not correct in the general case. this can be easily seen in the next example. example 3.3. let x = [0, 1] × {0, 1} ⊆ r2 endowed with the lexicographic order “-l” given by (x, y) -l (a, b) ⇐⇒ x < a, or x = a , y ≤ b. let τ be the order topology on x relative to “-l”. observe that (q ∩ [0, 1]) × {0, 1} is topologically dense in x. however x is not order separable in the sense of debreu because x has uncountably many jumps. (see [?], proposition 1.6.11 on p. 23). therefore, there is no utility function for “-l”. (see, e.g., [?], pp. 14-15). here a jump in x is defined as a pair of points x, y ∈ x with x ≺ y such that there is no z ∈ x with x ≺ z ≺ y. remark 3.4. in view of this example some additional condition must be added to separability in order to get the desired result. a possible such condition is connectedness. since cooper’s paper deals with problems in physics, perhaps it was taken for granted that the state spaces that do arise in thermodynamics satisfy the connectedness assumption. in cooper’s words: <> we can interpret here that the existence of one of the “barriers” there mentioned would carry a disconnection of the space. so that if “the barriers were removed”, the space would become connected. here we should recall newton’s words: “natura non facit saltum”. debreu’s theorem (see [?], [?]) states that being (x, τ) a second countable space, every τ-continuous total preoder “-” defined on x is continuously representable. in the particular case of x being a metric space, separability and second countability are equivalent conditions. so, another way to correct cooper’s theorem 1 is assuming that the system space is metric. 4. entropy and the theory of ordered semigroups section 4 in cooper’s paper [?] is devoted to the study of “composition of systems: additivity of entropy”. having a positive perspective in mind we only want here to give a possible interpretation of cooper’s arguments, and to establish the results in a more rigorous setting. in our opinion, the ideas contained in section 4 of [?] are very rich and deep, and have clear analogies with powerful items concerning the representation of totally ordered semigroups through additive utility functions. a glance at the beginning of section 4 in [?] shows that the main idea object of study is the possibility of finding thermodynamic systems that interact to get a new system. consequently, a natural question arises: “what relationships, if any, appear between the entropy of the new system after interaction, and the entropies before the interaction of the systems involved?” cooper establishes some axioms to deal with this kind of problem. such axioms lean on additivity properties. the objective is finding entropy functions that are unique up to linear transformations. as in the previous sections in [?], the validity of several lemmata and theorems in section 4 of cooper’s paper require additional assumptions and considerations for the physical models studied that are not explicitly mentioned by cooper. for instance, if we understand the composition of systems as a binary operation defined on the set of all possible systems, it is natural that this binary operation be associative and commutative as well. cooper does not mention the above conditions in [?], in spite that in cooper’s arguments such properties seem to be implicitly assumed. with such interpretations, our framework will be the theory of totally ordered semigroups. useful references here for further reading are [?], [?], [?], [?], [?]. remark 4.1. (1) despite it is greatly at variance with the notation common in semigroup theory, along this section ?? we will keep additive notation, much more 16 candeal, de miguel, induráin and mehta familiar to reserchers on algebraic utility. we have already chosen that notation in previous works as [?], [?] and [?], and it fits better with the notation used in section ?? of the present paper. (2) there are interpretations of several axioms encountered in other mathematical theories that could have some similarities with cooper’s axioms and ideas in [?]. the reader could investigate analogies with expected utility theory (see [?]) and axiomatical treatment of statistical means (see [?], [?]). in order to deal with the algebraic setting used in cooper’s work (see [?]), we introduce some previous concepts about ordered semigroups. a semigroup (s, +) is a set s endowed with a binary operation + that is associative. a semigroup s having a null element e such that x+e = x = e+x for every x ∈ s is said to be a monoid. if each element x of a monoid s has a converse −x such that x + (−x) = (−x) + x = e then s is said to be a group. a semigroup (s, +) endowed with a total ordering is said to be a totally ordered semigroup if the ordering is translation-invariant (i.e.: x y ⇐⇒ x + z y + z ⇐⇒ z + x z + y for every x, y, z ∈ s). in particular, a totally ordered semigroup s is always cancellative, i.e.: x + z = y + z ⇐⇒ x = y ⇐⇒ z + x = z + y for every x, y, z ∈ s. given a totally ordered semigroup (s, +, -), an element x ∈ s is said to be positive (respectively: negative) when y ≺ x+y and also y ≺ y+x (respectively: when x + y ≺ y and also y + x ≺ y) for every y ∈ s. notice that an element x ∈ s is positive (respectively: negative) if and only if x ≺ x + x (respectively: x + x ≺ x). the set of positive (respectively: negative) elements of s is said to be the positive cone of s, denoted s+ (respectively: s−). a simple exercise shows that these cones are stable in the following sense: if x, y ∈ s+ (respectively: s−) then x + y, y + x ∈ s+ (respectively: s−). notice also that s may have an element e that is neither positive nor negative. in this case e must be the null element for the operation +, and s is, a fortiori, a monoid. moreover, in this case it is clear that an element x is positive (respectively: negative) if and only if e ≺ x (respectively: x ≺ e). a totally ordered semigroup (s, +, -) is said to be: (1) positive (respectively: negative) if it consists only of positive (respectively: negative) elements. (2) additively representable (respectively: pseudo-representable) if there exists a utility (respectively: pseudo-utility) function u for that is an homomorphism (i.e.: u(x + y) = u(x) + u(y), for every x, y ∈ s). the associated function u is said to be an additive utility (respectively: pseudo-utility) function. a positive semigroup (s, +, -) is said to be: (1) archimedean if for every x, y ∈ s with x ≺ y, there exists n ∈ n such that y ≺ n · x, (n · x = n times ︷︸︸︷ x + · · · + x ), ordered semigroups and entropy 17 (2) super-archimedean if for every x, y ∈ s such that x ≺ y there exists n ∈ n such that (n + 1) · x ≺ n · y. a totally ordered group is said to be archimedean if its positive cone is archimedean. a totally ordered semigroup (s, +, -) is said to be super-archimedean if its positive cone (s+, +, -) is super-archimedean and also the negative cone (s−, + -op), endowed with the converse ordering “-op” defined by x -op y ⇐⇒ y x (x, y ∈ s), is super-archimedean. in the case of totally ordered groups, the archimedean condition is equivalent to the existence of an additive representation. this is a key result stated by hölder early in 1901. (see [?], or [?], p. 300.) remark 4.2. (1) even in the case of positive semigroups archimedeaness is not good enough to guarantee the additive representability. an example is the strictly positive cone (0, ∞) × (0, ∞) of the lexicographic plane (r2, +, -l) where the sum + is defined coordinatewise and the ordering -l is given by (a, b) -l (c, d) if a < c or else a = c , b ≤ d. it is well-known that this ordered set does not admit a utility representation, even non-additive. (see [?], pp. 200-201). (2) in [?] it is proved that super-archimedean implies archimedean: for instance, in the particular case of s being a commutative and positive semigroup, it holds that if x, y are positive, then y ≺ x + y, so that (n + 1) · y ≺ n · (x + y) =⇒ y ≺ n · x. the converse is not true: in the strictly positive cone of the lexicographic plane we have that (1, 1) ≺ (1, 2) but, for any positive n ∈ n, (n + 1, n + 1) ≺ (n, 2n), so that it is not super-archimedean. (3) as also shown in [?], archimedean totally ordered groups are superarchimedean: for instance, if g is abelian and e ≺ x ≺ y, we have that y − x ≺ y. thus if y ≺ n · (y − x) =⇒ (n + 1) · x ≺ n · y. in this framework of semigroups, there is also a characterization of additive representability: theorem 4.3. (1) the following statements are equivalent for a positive totally ordered semigroup (s, +, -): (i) (s, +, -) is additively representable, (ii) (s, +, -) is super-archimedean. (2) a semigroup (s, +, -) is additively representable if and only if its positive and negative cones are additively representable. proof. the proof may be seen in [?]. for the sake of completeness let us see its main ideas: in order to prove the key implication (ii) =⇒ (i) of part (1), fix an element x0 ∈ s and, given x ∈ s, set u(x) = sup{m/n : m, n ∈ n , m · x0 ≺ n·x}. following the proof of hölder’s theorem that appears in [?], pp. 300-301, we obtain that u is an additive pseudo-utility. so, it only remains to check the 18 candeal, de miguel, induráin and mehta injectivity of u, and this comes from the fact of s being super-archimedean: observe that being x, y ∈ s such that x ≺ y, there exists n ∈ n for which (n + 1) · x ≺ n · y =⇒ (n + 1) · u(x) ≤ n · u(y) =⇒ u(x) ≤ (n/(n + 1)) · u(y) < u(y). to prove part (2) we must give a construction of a global utility function u : s −→ r from partial utility functions u+, u− : s+, s− −→ r. the key step consists in proving that for any x ∈ s one can find an element y ∈ s+, that depends on x, such that x + y ∈ s+. then set u(x) = u+(x + y) − u+(y), and test that u is the required additive utility function. � hölder’s main result can be improved taking into account the continuity of the additive utilities involved. proposition 4.4. let (g, +, -) be a totally ordered group. then (g, +, -) is representable through a continuous and additive utility function if and only if (g, +, -) is archimedean. proof. see theorem 1 in [?]. � remark 4.5. (1) one may expect that the key property of archimedeaness established in proposition ??, that in the case of totally ordered groups guarantees the existence of a continuous additive utility function, will be maintained for semigroups. unfortunately things are no longer the same in this case. it follows from remark ?? that archimedeaness is not good enough to obtain additive utility representations (continuous or not) for totally ordered semigroups. actually, we have that: even being representable by an additive utility function, a semigroup could not admit a continuous additive utility representation. an example is the semigroup s = [2, 3) ∪ [4, ∞) with the usual addition and ordering of the reals. the crux for the non-existence of a continuous and additive utility function in this example, is the discontinuity as regards the order topology of the algebraic operation +. in other words, s is not a “topological” totally ordered semigroup in the sense that the algebraic operation is not continuous as regards the order topology. (2) the result established in debreu’s open gap lemma cannot be extended to the framework of additive utility functions on semigroups. the last example shows that there exists positive semigroups that admit additive utility functions, but none of such additive utility representations is continuous. of course, by theorem ??, any such ordered semigroup will admit continuous utility representations, but now none of those continuous utilities is additive. let (s, +, -) be a totally ordered semigroup. first we might notice that there is no topology given a priori on s, except maybe the order topology. but, even endowed with the order topology, we do not know whether (s, +, -) is a topological semigroup or not, in the sense of the following definition. ordered semigroups and entropy 19 definition 4.6. a topological semigroup (s, +, τ) is a semigroup (s, +) endowed with a topology “τ” that makes continuous the binary operation “+”: (x, y) ∈ s × s 7−→ x + y ∈ s. a totally ordered semigroup (s, +, -) is said to be a topological totally ordered semigroup if the binary operation “+” is continuous with respect to the order topology on s. similarly, a topological group (g, +, τ) is a group (g, +) endowed with a topology “τ” that makes continuous the binary operations “+” : g × g −→ g, and “inv” : g −→ g, given by inv(x) = −x, for every x ∈ g. so a topological totally ordered group is a totally ordered group (g, +, -) such that “+” and “inv” are both continuous as regards the order topology. remark 4.7. it is known that totally ordered groups are topological as regards the order topology (see [?]), so that theorem ?? can be extended to the framework of totally ordered groups. as was pointed out in remark ??, the above property is no longer true for totally ordered semigroups. the condition of being topological will be necessary for the existence of a continuous additive representation on totally ordered semigroups. in addition, the following main question arises now : let (s, +, -) be a super-archimedean topological totally ordered semigroup. is s representable by a continuous utility function? the answer is positive, as next result states. theorem 4.8. (1) let (s, +, -) be a totally ordered semigroup, additively representable by a continuous utility function u : s −→ r. then (s, +, -) is a topological semigroup as regards the order topology. (2) let (s, +, -) be a super-archimedean topological totally ordered semigroup. then s is representable by a continuous additive utility function. proof. see proposition 1 and theorem 2 in [?]. � coming back to cooper’s work, we observe that the nub of the reasoning in section 4 of [?] seems to be in the lemma 1, in which cooper justifies the existence of an state that is the mid point between two given states that interact. cooper’s proof is essentially based on the connectedness of the system and the continuity of the relation “→”. cooper’s result presents an evident analogy with the following result that appears in [?]. proposition 4.9. let (s, +, -) be a topological totally ordered semigroup that is positive and connected. then given u, v ∈ s there exists s ∈ s such that s + s = u + v. (such point s is said to be the mid point between u and v). proof. just consider the sets a = {z ∈ s : z + z ≺ u + v} and b = {y ∈ s : u + v = y + y} and use an standard argument of connectedness. (for details, see lemma 6 in [?]). � one of the keys in proposition ?? is the condition of translation-invariance. cooper, in section 4 of [?], uses a very similar condition, denoted “int(a)”. in cooper’s words: 20 candeal, de miguel, induráin and mehta << a system ξ is called the composition of the systems ξ1, ξ2, . . . ξn and is written ξ = {ξ1, ξ2, . . . ξn} if there is a homeomorphism of the product space s1×s2×. . .×sn onto the state space of s which is such that if {s1, s2, . . . sn} is the state corresponding to (s1, s2, . . . sn) then {s1, s2, . . . sr−1, sr1, s r+1, . . . , sn} → {s1, s2, . . . sr−1, sr2, s r+1, . . . , sn} if and only if sr1 ≺ s r 2. >> observe that if we understand the composition of systems as being commutative, and denote a+b the composition of states a and b (i.e.: a+b is {a, b} in cooper’s notation) then by int(a) we have that s1 → s2 ⇐⇒ s1 + t → s2 + t ( ⇐⇒ t + s1 → t + s2, by commutativity of the composition), for any states s1, s2 and t. thus we recover the translation-invariance of the operation “+”. moreover, using henceforward the usual notation “≺” instead of cooper’s “→”, it follows from int(a) that p ≺ s ≺ q =⇒ p + p ≺ s + s ≺ q + q for any states p, s, q. this fact is used by cooper to justify the uniquenesss of the mid point. as a matter of fact, in cooper’s arguments the property int(a) is not used in such justification. apparently cooper only uses int(a) for compositions in which at least four states are involved. let us analyze now the topological condition that is imposed on the semigroup in the statement of proposition ??. observe that the binary operation “+” is required to be continuous as regards the order topology. this requirement is essential to obtain the desired result. coming back to the example introduced in remark ??, we notice that “+” is not continuous there: indeed (7, 9) is a neighbourhood of 8 = 4 + 4, but every neighbourhood of 4 as regards the order topology must contain some element α smaller than 3 so that α + α /∈ (7, 9). in particular, the existence of a mid point fails to be true in such example because there is no s ∈ s such that s + s = 2 + 5. therefore we may assert that cooper should have noticed that not only a continuous order is necessary, but also the composition of systems must be continuous. in cooper’s proof it is said that, given two states u and v, the sets l(u, v) = {s : s + s ≺ u + v} and r(u, v) = {s : u + v ≺ s + s} are open, due to the continuity of “-”. but this is not true in general: in the example given in remark ??, we see that l(2, 5) = [2, 3) is open, but r(2, 5) = [4, +∞) fails to be open. this anomalous behaviour cannot appear when the operation “+” is continuous. the result stated in proposition ?? is used in [?] to prove that, under such conditions, if there exists an additive utility function that represents (s, +, -), then it must be continuous. later in [?] it is proved that an additive utility must always exist, as a consequence of the connectedness and the fact of (s, +, -) being topological. actually it is proved that, under the conditions of proposition ??, connected implies super-archimedean. this fact is used to get an additive utility function, in view of theorem ??. cooper’s reasoning in section 4 of [?] follows a different path: cooper starts by assuming that there is a continuous entropy (constructed in [?], theorem 1). then cooper tries to achieve a new entropy, now continuous and additive, by modifying the (not necessarily additive) original entropy. in cooper’s method, given any entropy for the system, the set of states can be embedded in a segment ordered semigroups and entropy 21 of the real line, since it is the continuous image of a connected set. so any other entropy will be the composition of the given entropy with a strictly increasing function from the real line into itself. interpreting cooper’s arguments, it seems that any two states s0 and s1, such that s0 → s1, are identified with the real numbers 0 and 1. then cooper applies the lemma 1 in [?] again and again, to obtain all the states that are in correspondence with a dyadic number in (0, 1). by continuity, and density of the dyadic numbers in (0, 1), cooper obtains the converse of an additive entropy whose range is the whole [0, 1]. it seems that cooper is arguing that, being u an entropy, the expression u−1(u(a) + u(b)) taken as the converse entropy of the composition a + b, defines an additive entropy v such that v(a + b) = u−1(u(a) + u(b)), a, b being any two systems. such construction is not clear, however: actually the so defined function v may or may not be additive. indeed, the original system could fail to have a null element as regards “+”, whereas v−1(0) should act as a null element. all along this construction, the only true fact concerning additivity is that the original entropy u is additive as regards a new binary operation “∗” defined by a ∗ b = u−1(u(a) + u(b)), a, b being any two states. unfortunately, this new composition of states, “∗”, could have no connection with the original one “+”. anyways, cooper’s arguments in section 4 of [?] are by no means worthless: on the one hand in cooper’s proof of lemma 1 the density of the dyadic numbers in (0, 1) is considered in order to get a suitable entropy. this argument has been used to analyze the structure of nontrivial totally ordered connected topological semigroups (s, +, -), and prove that they are homeomorphic, algebraically isomorphic and isotonic to some unlimited interval of the totally ordered group of additive real numbers endowed with the usual euclidean topology. (for further details see corollary 1 and theorem 5 in [?]). on the other hand, it is also noticeable that in theorem 2 in section 4 of [?], cooper says that: << the entropy function is uniquely determined for any one system by its values for two particular systems and, when so defined for one system, is defined uniquely for any other system by its value for one state of that system. any two possible choices of the entropy function are related linearly.>> cooper’s argument follows from lemma 1 in [?] whose proof is not clear as we have already mentioned. however, we feel that while cooper’s argument is not completely rigorous from a mathematical point of view, it does demonstrate that cooper’s intuition was correct and provides a basis for some deep results in fields that are apparently far removed from concepts and theories of thermodynamics such as, for example, algebraic utility theory. proposition 4.10. given two additive pseudo-utilities u, v defined on a totally ordered semigroup (s, +, -) there exists a positive constant α such that v = α·u. proof. let us see a proof for the particular case of positive semigroups and u, v being pseudoutilies that take values in (0, +∞). 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[22] d. h. krantz, r. d. luce, p. suppes and a. tversky, foundations of measurement, (academic press, new york and london, 1971). ordered semigroups and entropy 23 [23] e. h. lieb and j. yngvason, a guide to entropy and the second law of thermodynamics, notices of the american mathematical society 5 (1975), 195–204. [24] a. a. j. marley, abstract one-parameter families of commutative learning operators, journal of mathematical psychology 4 (1967), 414–429. [25] l. nachbin, topologia e ordem, (university of chicago press, 1950). [26] p. j. nyikos and h. c. reichel, topologically orderable groups, general topology and applications 45 (5) (1998), 571–581. received may 2002 accepted december 2002 juan c. candeal (candeal@posta.unizar.es) facultad de ciencias económicas y empresariales. departamento de análisis económico. universidad de zaragoza. doctor cerrada 1-3. e-50005. zaragoza. spain. juan r. de miguel (jrmiguel@si.unavarra.es) departamento de matemática e informática. universidad pública de navarra. campus de arrosad́ıa. e-31006. pamplona. spain. esteban induráin (steiner@si.unavarra.es) departamento de matemática e informática. universidad pública de navarra. campus de arrosad́ıa. e-31006. pamplona. spain. ghanshyam b. mehta (g.mehta@economics.uq.edu.au) departament of economics. university of queensland. 4072 brisbane, queensland. australia. @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 445–465 graph topologies on closed multifunctions giuseppe di maio, enrico meccariello and somashekhar naimpally dedicated by the first two authors to professor s. naimpally on the occasion of his 70th birthday. abstract. in this paper we study function space topologies on closed multifunctions, i.e. closed relations on x ×y using various hypertopologies. the hypertopologies are in essence, graph topologies i.e topologies on functions considered as graphs which are subsets of x ×y . we also study several topologies, including one that is derived from the attouch-wets filter on the range. we state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions. 2000 ams classification: 54b20, 54c25, 54c35, 54c60, 54e05, 54e15. keywords: hyperspaces, function spaces, graph topologies, vietoris topology, fell topology, uniform convergence on compacta, u-topology, ∆-topology, proximal ∆-topology, ∆u-topology, proximal ∆u-topology, hausdorff-bourbaki uniformity, ∆-attouch-wets filter. 1. introduction. recently mccoy [24] studied relations among four hyperspace topologies (viz. fell topology, fell uniform topology, vietoris topology and hausdorffbourbaki topology) and the corresponding topologies on set valued maps. in this paper we plan to study the subject comprehensively in more general situations. we recall that for a topological space z the hyperspace, 2z, of closed subsets of z has a number of natural topologies on it obtained from the topology on z. in our setting (x,τ1) and (y,τ2) denote hausdorff topological spaces and z the product space x×y equipped with the product topology τ = τ1×τ2. if δ1 and δ2 are compatible proximities on x and y respectively, then on z is assigned the product proximity δ = δ1 × δ2. the hyperspace 2z = 2x×y can be considered as the space f of all set valued maps on x to 2y taking points 446 g. di maio, e. meccariello and s. naimpally of x to (possibly empty) closed subsets of y . we do not distinguish between a function f ∈ f and its graph {(x,f(x)) : x ∈ x} ⊂ z = x ×y . thus our study includes topologies on the spaces of partial maps studied first in 1936 and which are being studied intensively in recent times ([1], [2], [3], [13], [19], [20], [23], [27], [33], [34]). given a hausdorff topological space z, for each subset e of z, clze, inte and ec stand for the closure, interior and complement of e in z. moreover e− = {a ∈ 2z : a∩e 6= ∅}; e+ = {a ∈ 2z : a ⊂ e}. futhermore, if δ is a compatible proximity on z (for details see [30]), we set e++δ = {a ∈ 2 z : a �δ e}. (note: a �δ e iff a6δec where 6δ denotes the negation of δ). we omit δ if it is clear from the context and write e++δ simply as e ++. we recall that the set of all compatible proximities on z is partially ordered as follows: δ1 ≤ δ2 iff whenever a, b ⊂ z and a6δ1b, then a6δ2b (see [30]). some special cases of δ are: δ0 the fine lo-proxmity on z given by aδ0b iff clza∩ clzb 6= ∅. δ0 is called the wallman proximity. it is well known that δ0 is, by far, the most important compatible loproximity on z, and that δ0 is ef iff z is normal (urysohn’s lemma). if z is tychonoff and v is a compatible uniformity on z, then δ(v) denotes the ef-proximity on z given by aδ(v)b iff v (a) ∩b 6= ∅ for each v ∈v. δ(v) is called the uniform proximity (induced by v). if z is a metrizable space with metric d, then δ(d) is the ef-proximity on z given by aδ(d)b iff dd(a,b) = inf{d(a,b) : a ∈ a,b ∈ b} = 0. δ(d) is called the metric proximity (induced by d). for z = x ×y , we use the symbol ∆ (resp. ∆1, ∆2) to denote a subfamily of cl(z) = 2z \{∅} (resp. of cl(x) = 2x \{∅}, cl(y ) = 2y \{∅}) which is a cobase, i.e. (a) is closed under finite unions; and (b) contains the singletons. a cover is a cobase which is also closed hereditary. moreover, we assume (c) ∆1 × ∆2 ⊂ ∆. in some cases, in addition to the above condition, we suppose graph topologies on closed multifunctions 447 (d) p1(∆) ⊂ ∆1 and p2(∆) ⊂ ∆2, where p1 and p2 are projections from z to x and y respectively. a typical and important example of a cover is ∆ = k(z), the family of all nonempty compact subsets of z, ∆1 = k(x), ∆2 = k(y ). moreover, in this case (c) − (d) also hold. in what follows, unless explicitly stated, we assume always that ∆ ⊂ cl(z) (resp. ∆1 ⊂ cl(x), ∆2 ⊂ cl(y )) is a cobase. we now describe some hypertopologies on 2z (for details see [4]). suppose δ is a compatible lo-proximity on z. the lower vietoris topology τ−v on 2 z has a subbase {w− : w ∈ τ}. the upper ∆-topology τ(∆)+ on 2z has a base {w + : wc ∈ ∆}. the upper proximal ∆-topology σ(∆,δ)+ on 2z has a base {w ++ : wc ∈ ∆}. the ∆ topology τ(∆) on 2z equals τ−v ∨ τ(∆) +. the proximal ∆ topology σ(∆,δ) on 2z equals τ−v ∨σ(∆,δ) +. the upper ∆u-topology τ(∆u)+ on 2z has a base {w + : wc ∈ ∆ or clw ∈ ∆}. the upper proximal ∆u topology σ(∆u,δ)+ on 2z has a base {w ++ : wc ∈ ∆ or clw ∈ ∆}. the ∆u-topology τ(∆u) on 2z equals τ−v ∨ τ(∆u) + ([8] and [16]). the proximal ∆u-topology σ(∆u,δ) on 2z equals τ−v ∨σ(∆u,δ) +. special cases: (1) vietoris and proximal topologies: when ∆ = cl(z), the upper vietoris topology τ(v )+ = τ(cl(z))+; the vietoris topology τ(v ) = τ(cl(z)); the upper proximal topology σ(δ)+ = σ(cl(z),δ)+ and the proximal topology σ(δ) = σ(cl(z),δ) = σ, if δ is understood. the paper [7] deals with only metric proximities, and [18] remains unpublished. it is not widely known that proximal hypertopologies can be studied in more general situations and not merely in metric spaces. (however, see the recent papers [11], [12] and [16]). we note that the vietoris topology is itself a proximal topology, i.e. τv = σ(δ0). (2) fell topology: when ∆ = k(z), the upper fell topology (also called the co-compact topology) τ(f)+ = τ(k(z))+; the fell topology τ(f) = τ(k(z)); 448 g. di maio, e. meccariello and s. naimpally the u-topology τ(u) = τ(k(z)u) (see [8]). when ∆ = k(z) and δ is ef, we have τ(f) = τ(k(z)) = σ(k(z),δ) = σ(f,δ) and τ(u) = τ(k(z)u) = σ(k(z)u,δ) = σ(u,δ). in this case the fell topology equals the proximal fell topology and this explains the reason for several beautiful results. in generalizing results concerning fell topology to ∆-topologies, we find that some are true in τ(∆) while others are true in σ(∆)! (also see below about weak topologies). (3) ball and proximal ball topologies: when z is a metric space, ∆ = b is the cobase generated by all finite unions of all closed balls of nonnegative radii and δ is the metric proximity, we have the ball topology τ(b) = τ(∆) ([4]); the proximal ball topology σ(b) = σ(∆,δ) ([17]). the proximal ball topology is very close to the wjisman topology. in fact, the two are equal in metric spaces satisfying some simple conditions that are present in a normed linear space ([17]). for z = x×y , when we wish to refer to hypertopologies on 2y , we use the suffix 2 e.g. τ2(v ) denotes the vietoris topology on 2y ; τ2(f) denotes the fell topology on 2y ; σ2(δ2) = σ2 denotes the proximal topology w.r.t. δ2 on 2y etc.. (4) weak topologies: if z = x×y , then for each of the topologies involving ∆ described above, we also have an associated weak topology wherein ∆ is replaced by ∆1×∆2 (see [31]) and we attach the letter ”w”. thus τ(w∆) = τ(∆1 × ∆2) and important special examples are: the weak vietoris topology τ(wv ) = τ(cl(x) × cl(y )) ⊂ τ(v ) = τ(cl(z)); the weak fell topology τ(wf) = τ(k(x) ×k(y )) ⊂ τ(f) = τ(k(z)). for single-valued functions with closed graphs, it was shown in [21] that τ(wf) = τ(f). the proof also works for f and combining this result with the fact that when the proximity δ is ef, τ(f) = σ(f,δ) we have τ(wf) = τ(f) = σ(wf,δ) = σ(f,δ). graph topologies on closed multifunctions 449 in generalizing mccoy’s results involving fell topology we find that our generalizations hold if we replace an appropriate member from the above four. (5) hausdorff-bourbaki and attouch-wets topologies: definition 1.1. let y be a tychonoff space, v a compatible uniformity and ∆2 ⊂ cl(y ). (i) for each v ∈v set: vh = {(a,b) ∈ 2y × 2y : a ⊂ v (b) and b ⊂ v (a)}. the family {vh : v ∈v} is a base for a uniformity vh on 2y called the hausdorff-bourbaki uniformity (cf. [4]) (or the hb-uniformity for short). (ii) whereas for each d ∈ ∆2 and v ∈v set: [d,v ] = {(a,b) ∈2y ×2y : a∩d ⊂v (b) and b ∩d ⊂v (a)}. the family {[d,v ] : d ∈ ∆2 and v ∈ v} is a base for a filter v∆2 on 2y called the ∆2-attouch-wets filter (cf. [5], [6] and [16]) (or the ∆2-aw filter for short). remark 1.2. let y be a locally compact space, v a compatible uniformity and ∆2 = k(y ). then the corresponding ∆2-aw filter v∆2 on 2y is a uniformity (see [4] or [5]) and it will be denoted with u(f). moreover, if 2y is equipped with the fell topology τ2(f), it is known that u(f) is compatible with τ2(f) (see [4] and [10]). observe that in this case (2y ,τ2(f)) is a compact hausdorff space and thus u(f) is the unique uniformity on 2y corresponding to the fell topology and it is generated by all τ2(f) × τ2(f) open neighbourhoods of the diagonal in 2y × 2y . thus, if y is a tychonoff space with a compatible uniformity v, ∆2 ⊂ cl(y ) and vh and v∆2 the associated hb-uniformity and ∆2-aw filter on 2y respectively, then on the space f: (a) a typical basic open set in the hb-uniform convergence topology τ(uc∆1,vh) on ∆1 is of the form < f,a,vh >= {g ∈ f : for all x ∈ a, (f(x),g(x)) ∈ vh}, where f ∈ f, a ∈ ∆1 and vh ∈vh. if ∆1 = k(x), we get one of the most important topologies, namely the hb-uniform convergence topology on compacta τ(ucc,vh). if we replace a by x, we get another important topology: the hbuniform convergence topology τ(uc,vh). if ∆1 is the family of all finite subsets of x, then we have the pointwise hb-convergence topology τp(vh). when vh is understood, we may omit it and just write τ(ucc) and τ(uc). (b) the topology on f generated by 450 g. di maio, e. meccariello and s. naimpally {< f,a,m >: f ∈ f,a ∈ ∆1 and m ∈v∆2} is called the ∆2-aw convergence topology τ(uc∆1,v∆2 ) on ∆1. if a = x, we have the ∆2-aw convergence topology τ(uc,v∆2 ). as before, we replace ∆1 by c for ”compacta”. if ∆1 = k(x), we obtain the ∆2-aw convergence topology on compacta τ(ucc,v∆2 ). if ∆1 is the family of all finite subsets of x, then we have the pointwise ∆2-aw convergence topology τp(v∆2 ). by remark 1.2 it follows that whenever y is a locally compact space and 2y is equipped with the fell topology τ2(f), then the corresponding ∆2-aw filter u(f) is a uniformity which is independent of the uniformity v chosen on y . note that the topology τ(ucc,u(f)) on f is just what mccoy calls ”fell uniform topology (on compact sets)” (see [24]). (6) pseudo uniform topologies: in [22] and [25] function space topologies akin to uniform topologies were studied. the range space was not necessarily uniformizable. here we introduce a similar concept. let w be a symmetric neighbourhood of the diagonal in (2y × 2y ,τ2 × τ2). for each f ∈ f and a ∈ ∆1 we set w?(f,a) = {g ∈ f : for all x ∈ a, (f(x),g(x)) ∈ w}. the topology on f generated by {w?(f,a) : f ∈ f,a ∈ ∆1 and w a symmetric τ2×τ2 neighbourhood of the diagonal in 2y ×2y} is the τ2-pseudo uniform topology on ∆1: ps(τ(uc∆1,τ2)). if a = x, we have the pseudo τ2-uniform topology ps(τ(uc,τ2)). as before, if ∆1 = k(x), we replace ∆1 by c for ”compacta” and we have the pseudo τ2-uniform topology on compacta ps(τ(ucc,τ2)). in case (2y ,τ2) is uniformizable and we restrict w ’s to symmetric entourages, we do get a uniform topology. this is true as in (5) above or in (6) when y is a locally compact space and 2y is equipped with the fell topology τ2(f) on 2y and in this case ps(τ(uc∆1,τ2(f)) = τ(uc∆1,u(f)) (cf. above remark 1.2). although mccoy got his results in uniform setting, we find that some of his results do not need uniformity at all! finally, if τ2 is a given hypertopology on 2y , then τp(τ2) is the corresponding τ2-pointwise convergence topology on f which agrees with the pseudo τ2 uniform topology on ∆1 ps(τ(uc∆1,τ2)) when ∆1 is the family of all finite subsets of x. graph topologies on closed multifunctions 451 those interested in more details are referred to [4] for hypertopologies, [30] for proximities, [26] and [28] for function space topologies, [9], [10] and [24] for uniform topologies and convergences on spaces of multifunctions. 2. basic results. one of the most valuable result in function space topologies is the embedding of the range space in the function space (cf. theorem 2.1.1, page 15 in [26]). in this section we prove similar results for multifunctions which are of fundamental importance in our work. we need to introduce ”upper” hypertopologies that are specially meant for the family c = {x ×e : e ∈ cl(y )} of constant multifunctions. these topologies depend on ∆2 alone, unlike other hypertopologies which depend on either ∆ or ∆1 ×∆2. we use the suffix r (for range) for such topologies. on cl(z), we have the upper r-∆2-topology τ(r∆2)+ which is generated by the basis {(x ×v )+ : v c ∈ ∆2}∪{cl(z)}. similarly, we have the upper r-∆2u-topology τ(r∆2u)+ which is generated by the basis {(x ×v )+ : v c ∈ ∆2 or clv ∈ ∆2}∪{cl(z)}. if δ2 is a lo-proximity on y , then we define an associated ”proximity” rδ2 on c by (x ×e) � (x ×v ) w.r.t. rδ2 iff e � v w.r.t. δ2. if δ2 is an ef-proximity on y , then it is easy to see that rδ2 on c is also ef. naturally we also have the proximal versions: the upper proximal r-∆2-topology σ(r∆2,rδ2)+; the upper proximal r-∆2u-topology σ(r∆2u,rδ2)+. we have on cl(z) the r-∆2-topology τ(r∆2) = τ(r∆2)+ ∨ τ−v . similarly the analogues: the r-∆2u-topology τ(r∆2u) = τ(r∆2u)+ ∨ τ−v ; the proximal r-∆2-topology σ(r∆2,rδ2) = σ(r∆2,rδ2)+ ∨ τ−v and the proximal r-∆2u-topology σ(r∆2u,rδ2) = σ(r∆2u,rδ2)+ ∨ τ−v . let p(y ) and p(z) denote the set of all subsets of y and z, respectively. consider the map j : p(y ) ↪→ p(z) defined by j(e) = (x × e) ∈ p(z) . obviously, (a) j : cl(y ) ↪→c is a bijection; 452 g. di maio, e. meccariello and s. naimpally (b) j(v +) ⊂ [j(v )]+; (c) j(v ++δ2 ) ⊂ [j(v )] ++ rδ2 ; (d) j(v −) ⊂ [j(v )]−; (e) j(e) ∈ w− and w ∈ τ together imply e ∈ [p2(w)]−, where p2 : z → y is the projection. (f) let ∆1 ⊂ cl(x), d ∈ ∆1, u a filter on 2y , m a symmetric member of u, a ∈ cl(y ) and f = j(a) = x ×a. then: < f,x,m > ∩c =< f,d,m > ∩c =< f,x,m > ∩c = j(m(a)) for each x ∈ x. so, we have the following results. theorem 2.1. let x and y be hausdorff spaces with compatible lo-proximities. the following are embeddings: (a) j : (cl(y ),τ2(∆2)+) ↪→ (cl(z),τ(r∆2)+); (b) j : (cl(y ),τ2(∆2u)+) ↪→ (cl(z),τ(r∆2u)+); (c) j : (cl(y ),σ2(∆2,δ2)+) ↪→ (cl(z),σ(r∆2,rδ2)+); (d) j : (cl(y ),σ2(∆2u,δ2)+) ↪→ (cl(z),σ(r∆2u,rδ2)+); (e) j : (cl(y ),τ2(∆2)) ↪→ (cl(z),τ(r∆2)); (f) j : (cl(y ),τ2(∆2u)) ↪→ (cl(z),τ(r∆2u)); (g) j : (cl(y ),σ2(∆2,δ2)) ↪→ (cl(z),σ(r∆2,rδ2)); (h) j : (cl(y ),σ2(∆2u,δ2)) ↪→ (cl(z),σ(r∆2u,rδ2)). remark 2.2. let y be a tychonoff space, v a compatible uniformity on y , ∆1 ⊂ cl(x) and ∆2 ⊂ cl(y ). if vh and v∆2 are respectively the associated hb-uniformity and ∆2-aw filter on 2y , then on c: (1) τ(uc,vh) = τ(uc∆1,vh) = τp(vh); (2) τ(uc,v∆2 ) = τ(uc∆1,v∆2 ) = τp(v∆2 ). thus the following are embeddings: (1a) j : (cl(y ),τ2(vh)) ↪→ (cl(z),τ(uc,vh)); (1b) j : (cl(y ),τ2(vh)) ↪→ (cl(z),τ(uc∆1,vh)). (2a) j : (cl(y ),v∆2 ) ↪→ (cl(z),τ(uc,v∆2 )); (2b) j : (cl(y ),v∆2 ) ↪→ (cl(z),τ(uc∆1,v∆2 )). similarly, if y is a hausdorff space, τ2 a given hypertopology on 2y and ∆1 ⊂ cl(x), then on c: (3) ps(τ(uc,τ2)) = ps(τ(uc∆1,τ2)) = τp(τ2). thus the following are embeddings: (3a) j : (cl(y ),τ2) ↪→ (cl(z),ps(τ(uc,τ2))); (3b) j : (cl(y ),τ2) ↪→ (cl(z),ps(τ(uc∆1,τ2))). lemma 2.3. let x and y be hausdorff spaces. then, on the family c of constant multifunctions: (a) τ(w∆)+ ≤ τ(r∆2)+ ≤ τ(r∆2u)+ ≤ τ(v )+; and if p2(∆) ⊂ ∆2, then τ(w∆)+ ≤ τ(∆)+ ≤ τ(r∆2)+ ≤ τ(r∆2u)+ ≤ τ(v )+. graph topologies on closed multifunctions 453 (b) τ(w∆)+ ≤ τ(∆)+ ≤ τ(∆u)+ ≤ τ(v )+; and if p2(∆) ⊂ ∆, then τ(w∆)+ ≤ τ(∆)+ ≤ τ(∆u)+ ≤ τ(r∆2u)+ ≤ τ(v )+. (c) τ(w∆) ≤ τ(r∆2) ≤ τ(r∆2u) ≤ τ(v ); and if p2(∆) ⊂ ∆2, then τ(w∆) ≤ τ(∆) ≤ τ(r∆2) ≤ τ(r∆2u) ≤ τ(v ). (d) τ(w∆) ≤ τ(∆) ≤ τ(∆u) ≤ τ(v ); and if p2(∆) ⊂ ∆2, then τ(w∆) ≤ τ(∆) ≤ τ(∆u) ≤ τ(r∆2u) ≤ τ(v ). lemma 2.4. let x and y be hausdorff spaces with compatible lo-proximities. then, on the family c of constant multifunctions: (a) σ(w∆)+ ≤ σ(r∆2)+ ≤ σ(r∆2u)+ ≤ σ+; and if p2(∆) ⊂ ∆2, then σ(w∆)+ ≤ σ(∆)+ ≤ σ(r∆2)+ ≤ σ(r∆2u)+ ≤ σ+. (b) σ(w∆)+ ≤ σ(∆)+ ≤ σ(∆u)+ ≤ σ+; and if p2(∆) ⊂ ∆2, then σ(w∆)+ ≤ σ(∆)+ ≤ σ(∆u)+ ≤ σ(r∆2u)+ ≤ σ+. (c) σ(w∆) ≤ σ(r∆2) ≤ σ(r∆2u) ≤ σ; and if p2(∆) ⊂ ∆2, then σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2u) ≤ σ. (d) σ(w∆) ≤ σ(∆) ≤ σ(∆u) ≤ σ; and if p2(∆) ⊂ ∆2, then σ(w∆) ≤ σ(∆) ≤ σ(∆u) ≤ σ(r∆2u) ≤ σ. we say that z is locally ∆ iff for each z ∈ z with z ∈ v ∈ τ, there is d ∈ ∆ with z ∈ intd ⊂ d ⊂ v . (note that this is a generalization of local compactness in which case ∆ = k(z)). lemma 2.5. let x and y be hausdorff spaces with compatible lo-proximities, z = x ×y and ∆ ⊂ cl(z) a cover. if z is locally ∆, then: (a) τ(∆)+ = τ(∆u)+ if and only if z ∈ ∆ i.e. ∆ = cl(z). (b) σ(∆)+ = σ(∆u)+ if and only if z ∈ ∆ i.e. ∆ = cl(z). proof. we prove only (a). it suffices to show that τ(∆u)+ ≤ τ(∆)+ implies z ∈ ∆. suppose u is a nonempty subset of z with a ⊂ u and clu ∈ ∆ (note that such u exists since z is locally ∆). then there is an open subset v in z with v c ∈ ∆ such that a ⊂ v ⊂ u. clearly z = clu ∪v c ∈ ∆. � corollary 2.6. let x and y be hausdorff spaces with compatible lo-proximities, z = x ×y and ∆ ⊂ cl(z) a cover. if z is locally ∆, then: (a) τ(∆) = τ(∆u) if and only if z ∈ ∆ i.e. ∆ = cl(z) (cf. [14], theorem 3.2). (b) σ(∆) = σ(∆u) if and only if z ∈ ∆ i.e. ∆ = cl(z). (c) when ∆ = k(z), we have τ(f) = τ(u) if and only if x is compact. remark 2.7. in the following relations, vertical lines show embeddings: 454 g. di maio, e. meccariello and s. naimpally (a) cl(y ): τ2(∆2) ≤ τ2(∆2u) ≤ τ2(v ) ↓ j ↓ ↓ ↓ c ⊂ f: τ(w∆) ≤ τ(r∆2) ≤ τ(r∆2u) ≤ τ(rcl(y )) ≤ τ(v ). (b) cl(y ): τ2(∆2) ≤ τ2(∆2u) ≤ τ2(v ) ↓ j ↓ ↓ c ⊂ f: τ(w∆) ≤ τ(∆) ≤ τ(∆u) ≤ τ(r∆2u) ≤ τ(rcl(y )) ≤ τ(v ). (c) cl(y ): σ2(∆2) ≤ σ2(∆2u) ≤ σ2 ≤ τ2(vh) ↓ j ↓ ↓ ↓ ↓ c ⊂ f: σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2u) ≤ σ(rcl(y )) ≤ τ(vh). (d) cl(y ): σ2(∆2) ≤ σ2(∆2u) ≤ σ2 ≤ τ2(vh) ↓ j ↓ ↓ ↓ c ⊂ f: σ(w∆) ≤ σ(∆) ≤ σ(∆u) ≤ σ(r∆2u) ≤ σ(rcl(y )) ≤ τ(vh). (e) if the proximities involved are ef, then cl(y ): σ2(∆2) ≤ σ2(∆2u) ≤ σ2 ≤ τ2(v ) ↓ j ↓ ↓ ↓ ↓ c ⊂ f : σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2u) ≤ σ(rcl(y )) ≤ τ(rcl(y )) ≤ τ(v ). (f) cl(y ): σ2(∆2) ≤ σ2(∆2u) ≤ σ2 ≤ τ2(v ) ↓ j ↓ ↓ ↓ c ⊂ f : σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2u) ≤ σ(rcl(y )) ≤ τ(rcl(y )) ≤ τ(v ). (g) cl(y ): τ2(∆2) ≤ τ2(v ) ↓ j ↓ ↓ c ⊂ f: ps(τ(uc∆1,τ2(∆2))) ≤ τ(rcl(y )) ≤ τ(v ). 3. generalization of mccoy’s results. in this section we begin comparing some of the topologies defined in the previous section and find conditions for their pairwise equivalence. we study some simple ones which are analogues of those in mccoy’s paper and state mccoy’s results just below the analogues. again, we recall that (x,τ1) and (y,τ2) are hausdorff spaces. we set z = x×y and assign the product topology τ = τ1 × τ2. if δ1 and δ2 are compatible proximities on x and y respectively, then on z is assigned the product proximity δ = δ1 × δ2. the family 2z of graph topologies on closed multifunctions 455 closed subsets of z can be identified with the space f of all set valued maps on x to 2y taking points of x to closed (possibly empty) subsets of y . other assumptions will be stated at the places where they are needed. mccoy assumed that both x and y are locally compact spaces and y is a non-trivial complete metric space. he then studied four topologies: τ(f), τ(v ), τ(ucc,u(f)) and τ(vh). in this section we pursue τ(∆), τ(w∆), σ(∆), σ(w∆), τ(v ), τ(vh), τ(uc∆1,vh) and τ(uc,vh) as well as τ(uc∆1,v∆2 ) and τ(uc,v∆2 ). moreover, we also consider ps(τ(uc,τ2)) and ps(τ(uc∆1,τ2)) for an arbitrary topology τ2 on 2y . theorem 3.1. let x and y be hausdorff spaces with compatible lo-proximities δ1 and δ2, respectively. then on f: (a) τ(w∆)+ = τ(∆1 × ∆2)+ ≤ τ(∆)+ ≤ τ(∆u)+ ≤ τ(v )+. (b) τ(w∆) ≤ τ(∆) ≤ τ(∆u) ≤ τ(v ). moreover, if δ = δ1 × δ2 is ef , then: (c) σ(w∆,δ)+ ≤ σ(∆,δ)+ ≤ σ(∆u,δ)+ ≤ σ(δ)+ ≤ τ(v )+. (d) σ(w∆,δ) ≤ σ(∆,δ) ≤ σ(∆u,δ) ≤ σ(δ) ≤ τ(v ). (cf. [24] prop. 4.1) if x and y are hausdorff spaces, then τ(f) ≤ τ(u) ≤ τ(v ). the following results and lemmas play a key role. lemma 3.2. let x be a hausdorff space, y a tychonoff space, u and v compatible uniformities on y with u ⊂ v and uh and vh the corresponding hb-uniformities on 2y associated to u and v, respectively. then on f: (a) τ(uc∆1,uh) ≤ τ(uc∆1,vh); (b) τ(uc,uh) ≤ τ(uc,vh). furthermore, if ∆2 ⊂ cl(y ) and u∆2 and v∆2 are the corresponding ∆2aw filters on 2y associated to u and v, respectively, then: (c) τ(uc∆1,u∆2 ) ≤ τ(uc∆1,v∆2 ); (d) τ(uc,u∆2 ) ≤ τ(uc,v∆2 ). proof. (a) and (b) (resp. (c) and (d)) follow from the fact that if u ⊂v on y , then uh ⊂vh (resp. u∆2 ⊂v∆2 ) on 2y . � lemma 3.3. let y be a tychonoff space, v a compatible uniformity on y , ∆2 ⊂ cl(y ), vh and v∆2 the corresponding hb-uniformity and ∆2-aw filter on 2y , respectively. then, on 2y , v∆2 ⊂vh. proof. let [d,v ] be basic element of v∆2 where d ∈ ∆ and v ∈ v. we claim that the corresponding element vh ∈ vh is such that vh ⊂ [d,v ]. assume not. then there exists (a,b) ∈ 2y × 2y such that (a,b) ∈ vh but (a,b) 6∈ [d,v ]. thus either (i) a ∩ d 6⊂ v (b) or (ii) b ∩ d 6⊂ v (a). if (i) occurs, then there exists y ∈ (a∩d)\v (b); a contradiction because a ⊂ v (b). similarly, if (ii) occurs. � 456 g. di maio, e. meccariello and s. naimpally corollary 3.4. let x be a hausdorff space, y a tychonoff space, v a compatible uniformity on y , ∆1 ⊂ cl(x), ∆2 ⊂ cl(y ), vh and v∆2 the corresponding hb-uniformity and ∆2-aw filter on 2y . then on f: (a) τ(uc∆1,v∆2 ) ≤ τ(uc∆1,vh); (b) τ(uc,v∆2 ) ≤ τ(uc,vh). proof. (a) and (b) follow from above lemma 3.3. � we recall that if (z,τ) is a tychonoff space with a compatible ef-proximity δ, then a uniformity u on z is called compatible w.r.t. δ iff the uniform proximity δ(u) induced by u equals δ (see section 1 and [30]). δ admits a unique compatible totally bounded uniformity uw ([30]). theorem 3.5. (cf. [18]) let y be a tychonoff space and δ2 a compatible ef -proximity on y . the corresponding proximal topology σ2(δ2) on 2y is always uniformizable. in fact, it is the topology induced on 2y by the hbuniformity uhw which is derived from the unique totally bounded uniformity uw on y compatible w.r.t. δ2. lemma 3.6. (cf. theorem 2.1 in [16]) let y be a tychonoff space, δ2 a compatible ef -proximity on y and ∆2 ⊂ cl(y ) a cover. if the proximal ∆2 topology σ2(∆2,δ2) is uniformizable, then the proximal ∆2-topology σ2(∆2,δ2) equals the topology τ(u∆w ) induced by the ∆-aw filter u∆2w , where uw is the unique totally bounded uniformity on y compatible w.r.t. δ2. proof. let uw be the unique totally bounded uniformity on y compatible with δ2. without loss of generality we assume that all entourages w ∈ uw are open and symmetric. first, let {aλ : λ ∈ λ} ⊂ 2y be a net τ(u∆2w )-converging to a ∈ 2y . we claim that the net {aλ : λ ∈ λ} σ2(∆2,δ2)-converges to a. (i) if a ∈ v − where v ∈ τ, then there exist a ∈ a∩v and a w ∈uw such that w(a) ⊂ v . since a ∈ [{a},w ](a) ⊂ v −, eventually aλ ∈ [{a},w ](a) ⊂ v −. (ii) if a ∈ (dc)++δ2 where d ∈ ∆2, then d �δ2 a c. since uw is compatible w.r.t. δ2 and by assumption σ2(∆2,δ2) is uniformizable there are s ∈ ∆2 and w ∈uw such that d ⊂ w(d) ⊂ s ⊂ w(s) ⊂ ac (see theorem 4.4.5, lemma 4.4.3 and definition 4.4.2 in [4]). since a ∈ [s,w](a), w(a) ∩ s = ∅ and eventually aλ ∈ [s,w ](a), then eventually aλ ∈ (dc)++δ2 . thus σ2(∆2,δ2) ≤ τ(u∆2w ). on the other hand, let {aλ : λ ∈ λ} ⊂ 2y be a net σ2(∆2,δ2)-converging to a ∈ 2y . we claim that the net {aλ : λ ∈ λ} τ(u∆2w )-converges to a. so, let [w,d](a) a τ(u∆w )-neigbourhood at a where d ∈ ∆2 and w ∈ uw. let v ∈uw be such that v 2 ⊂ w . we have two cases: (i) a ∈ (dc)++δ2 . then eventually aλ ∈ (d c)++δ2 and obviously, ∅ = aλ∩d ⊂ w(a) and ∅ = a∩d ⊂ w(aλ), i.e eventually aλ ∈ [w,d][a]. graph topologies on closed multifunctions 457 (ii) a 6∈ (dc)++δ2 . then v (a) ∩ d 6= ∅. since v is totally bounded, there are xj ∈ a, 1 ≤ j ≤ n such that a ⊂ n⋃ j=1 v (xj) ⊂ v 2(a). since a∩v (xj) 6= ∅ for each j, eventually aλ ∩ v (xj) 6= ∅ and so xj ∈ v (aλ). hence, eventually a ∩ d ⊂ n⋃ j=1 v (xj) ⊂ v 2(aλ) ⊂ w(aλ). note that (d ∩ v (a)c) ∈ ∆2 and a ∈ (dc ∪ v (a))++δ2 ∈ σ2(∆,δ2). so eventually, aλ ∈ (d c ∪ v (a))++δ2 . thus eventually aλ ∩ d = [aλ ∩ (d ∩ v (a))] ⊂ w(a), i.e. eventually aλ ∈ [w,d](a). thus τ(u∆w ) ≤ σ2(∆2,δ2). combining the earlier part we get τ(u∆2w ) = σ2(∆2,δ2). � remark 3.7. (a) in [15] it is shown that if τ2(∆2) is uniformizable, then there is a compatible ef-proximity δ2 on y such that τ2(∆2) = σ2(∆2,δ2) (see lemma 2.2 in [15]). (b) above lemma and remark 3.7 (a) show that the appropriate extension of u(f) (see remark 1.2) for uniformizable ∆2and proximal ∆2topologies are the ∆2-aw filters induced by compatible totally bounded uniformities uw on y . thus, as in the definition of u(f) (see remark 1.2), we reserve the symbol u(τ2) to denote the corresponding compatible aw filter associated with the totally bounded uniformity uw on y whenever τ2 is a uniformizable (proximal) ∆2-topology on 2y . theorem 3.8. let x be a hausdorff space, y a tychonoff space with a compatible ef -proximity δ2 and uw the unique totally bounded uniformity associated to δ2, ∆1 ⊂ cl(x), ∆2 ⊂ cl(y ) a cover and 2y equipped with the proximal ∆2-topology σ2(∆2) induced by δ2. if σ2(∆2) is uniformizable and u(σ2(∆2)) and uh are respectively the corresponding ∆2-aw filter compatible w.r.t. σ(∆2) and hb-uniformity on 2y associated to uw, then on f: (a) τ(uc∆1,u(σ2(∆2))) ≤ τ(uc∆1,uh); (b) τ(uc,u(σ2(∆2))) ≤ τ(uc,uh). proof. (a) and (b) follow by corollary 3.4 and lemma 3.6. � next theorem generalizes propositions 4.2 and 4.5 and shows that the assumption of local compactness on the base space x it is not nedded (see also proposition 4.1 in [10]). theorem 3.9. let x be a hausdorff space, y a tychonoff space, v a compatible uniformity on y and vh the corresponding hb-uniformity on 2y . suppose δ2 is an ef -proximity on y with δ2 ≤ δ2(v), 2y equipped with the proximal topology σ2 induced by δ2. if u(σ2) is the corresponding compatible hbuniformity associated with uw, the unique totally bounded uniformity compatible w.r.t. δ2, then on f: (a) τ(uc∆1,u(σ2)) ≤ τ(uc∆1,vh). 458 g. di maio, e. meccariello and s. naimpally (b) τ(uc,u(σ2)) ≤ τ(uc,vh). (c) (cf. prop. 4.2 in [24] and prop. 4.1 in [10]) if x is a hausdorff space, y a locally compact space and τ2(f) the fell topology on 2y , then τ(ucc,u(f)) ≤ τ(ucc,vh). (d) furthermore, there is equality either in (a) or in (b) if and only if v is totally bounded. (e) (cf. prop. 4.5 in [24]) if x is a hausdorff space, y a locally compact and completely metrizable space with metric d, v the metric uniformity associated with d and τ2(f) the fell topology on 2y , then τ(ucc,u(f)) = τ(ucc,vh) if and only if y is compact. proof. let uw and u′w be the totally bounded uniformities on y compatible with δ2 and δ2(v), respectively. from theorems 12.7 and 12.14 in [30] uw ⊂u′w and u′w ⊂v and hence uw ⊂v. thus (a) and (b) follow from theorem 3.5 and (a) and (b) in lemma 3.2. (d) it follows from the fact that equality either in (a) or in (b) is equivalent to σ2(δ2) = τ2(vh), which in turn, is equivalent to the total boundedness of v. whereas (c) and (e) follow from above (a) and (d) respectively when ∆1 = k(x), remark 1.2 and the well-known relation τ2(f) = σ2(f,δ2) ≤ σ2(δ2). � next theorem shows that in propositions 4.6 and 4.8 in [24] the assumption of local compactness on the base space x can be dropped. first we give the following remark. remark 3.10. it is well known (see [32]) that on fuction spaces the lower vietoris topology is coarser than the topology of pointwise convergence. thus, whenever τ2 is a given topology on 2y , we have: (?) τ(v −) ≤ τp(τ2). theorem 3.11. let x be a hausdorff space with a compatible loproximity δ1, y a tychonoff space with a compatible ef -proximity δ2, ∆1 ⊂ cl(x), ∆2 ⊂ cl(y ) a cover, z = x×y equipped with the product proximity δ = δ1×δ2 and 2y equipped with the proximal ∆2-topology σ2(∆2) induced by δ2. if σ2(∆2) is uniformizable and u(σ2(∆2)) is the corresponding compatible ∆2-aw filter associated to uw, the unique totally bounded uniformity compatible w.r.t. δ2, then on f: (a) σ(w∆,δ) ≤ τ(uc∆1,u(σ2(∆2))) ≤ τ(uc,u(σ2(∆2))). (b) furthermore, under the conditions of theorem (3.9) we have σ(w∆,δ) ≤ τ(uc∆1,u(σ2(∆2)) ≤ τ(uc∆1,vh). (cf. [24] prop. 4.3 and prop. 4.4) if x is a hausdorff space, y a locally compact space and v a compatible uniformity on y , then τ(f) ≤ τ(ucc,u(f)) ≤ τ(ucc,vh). proof. first we show (a). so, let m = u × v , u ∈ τ1, v ∈ τ2 and f ∈ m−. hence, there is a point (x,y) ∈ f ∩m. then by (?) in the above remark there exists a τp(σ2(∆2))-neighbourhood h of f such that h⊂ m− and clearly h is also a τ(uc∆1,u(σ2(∆2))-neighbourhood of f. graph topologies on closed multifunctions 459 next, suppose d = a × b where a ∈ ∆1, b ∈ ∆2 and f �δ dc. since δ = δ1 × δ2 and δ2 is ef, there is an open set v in y with b �δ2 v and f �δ (a×v c). let uw be the unique totally bounded uniformity on y compatible with δ2. thus, there is a w ∈uw such that b ⊂ w(b) ⊂ w 2(b) ⊂ v (see [30]). clearly, < f,a; [b,w ] > is τ(uc∆1,u(σ2(∆2))-neighbourhood of f. we claim < f,a; [b,w ] >⊂ (dc)++δ . in fact, if g ∈< f,a; [b,w ] >, then g(x) ∈ [b,w ] for each x ∈ a. now, w 2(b) ∩ v c = ∅ together with g(x) ∈ [b,w ] for each x ∈ a imply g �δ a×v c. hence g ∈ (dc)++δ . so the first inclusion follows. the second one is trivial. (b) it follows from (a) above and corollary 3.4. � remark 3.12. by (a) in remark 3.7 we give statements and proofs only for the τ(uc∆1,u(σ2(∆2))) topology. similar ones for the τ(uc∆1,u(τ2(∆2))) topology, when τ2(∆2) is uniformizable and ∆2 is a cover, are left to the reader. theorem 3.13. let x be a hausdorff space with a compatible lo-proximity δ1, y a tychonoff space with a compatible ef -proximity δ2, z = x × y equipped with the product proximity δ = δ1 × δ2, ∆2 ⊂ cl(y ) a cover and 2y equipped with the proximal ∆2-topology σ2(∆2) induced by δ2. let σ2(∆2) be uniformizable and u(σ2(∆2)) the corresponding compatible ∆2-aw filter. then on f: (a) if ∆1 is the family of all finite subsets of x, then τ(uc∆1,u(σ2(∆2))) ≤ σ(w∆). (b) if τ(uc∆1,u(σ2(∆2))) ≤ σ(w∆), then x is discrete. (cf. [24] prop. 4.6 and prop. 4.8) if x is a hausdorff space and y a locally compact space, then the following are equivalent: (α) x is discrete; (β) τ(f) = τ(ucc,u(f)); (γ) τ(ucc,u(f)) ≤ τ(f) ≤ τ(ucc,vh). (c) let y be a tychonoff space, v a compatible uniformity on y and vh the corresponding hb-uniformity on 2y . then the hbuniform topology on ∆1 τ(uc∆1,vh) equals the weak proximal ∆ topology σ(w∆) if and only if each member of ∆1 is finite and v is totally bounded. ([24] prop. 4.7) if y is a completely metrizable space with metric d and v is the d-metric uniformity, then τ(ucc,vh) = τ(f) if and only if x is discrete and y is compact. proof. (a) it suffices to observe that if ∆1 is the family of all finite subsets of x, then τ(uc∆1,u(σ2(∆2))) = τp(σ2(∆2)) and clearly τp(σ2(∆2)) ≤ σ(w∆). (b) suppose x is not discrete. then there exists a point x0 in x which is not isolated. denote by n(x0) the family of all open neighbourhoods of x0 and let y0, y1 be two distinct points in y . for u ∈n(x0) define fu (x) = {y0,y1} for x 6∈ u and fu (x) = {y0} for x ∈ u. it is easy to verify that fu ∈ f and that the net {fu : u ∈n(x0)} σ(w∆)-converges to a multifunction f defined by f(x) = {y0,y1} for x ∈ x. since {fu (x0) : u ∈ n(x0)} does not τ2(v −) converge to 460 g. di maio, e. meccariello and s. naimpally f(x0), it follows that {fu : u ∈n(x0)} cannot τ(uc∆1,u(σ2(∆2))) converge to f. (c) it follows from the above and the fact that the equality is equivalent to σ2(∆2) = τ2(vh), which in turn, is equivalent to the total boundedness of v. � theorem 3.14. let x be a hausdorff space, y a tychonoff space, v a compatible uniformity on y and vh the corresponding hb-uniformity on 2y . then on c: (a) if v is totally bounded, then τ(uc,vh) ≤ τ(rcl(y )) ≤ τ(v ). (b) if v is not totally bounded, then on c, τ(uc,vh)6≤τ(v ). thus, if on f τ(uc,vh) ≤ τ(v ), then v is totally bounded. ([24] prop. 4.9) if x is a locally compact space, y a locally compact completely metrizable space with metric d and v the d-metric uniformity, then on f τ(ucc,vh) ≤ τ(v ) if and only if x is discrete and y is compact. proof. (a) let u ∈ v be open and symmetric and f = (x × e) ∈ c. total boundedness of v implies there is a finite set {yk : 1 ≤ k ≤ n}) ⊂ y such that e ⊂ n⋃ k=1 u(yk). consider a typical τ(uc,vh)neighbourhood of f, i.e. < f,x,u2 >. it is easy to see that [x × u(e)]+ ∩ n⋂ k=1 u(yk) − is a τ(wv )neighbourhood of f which is contained in < f,x,u2 > . (b) if v is not totally bounded there is an open u ∈ v and a sequence {yk : k ∈ in} ⊂ y such that y 6⊂ n⋃ k=1 u(yk) for each n ∈ in. then it is clear that fc = {f = x ×e : e ⊂ y is finite} is a subset of c which is not dense in (c,τ(uc,vh)) but which is dense in (c,τ(v )). � proposition 3.15. let x be a hausdorff space, y a tychonoff space, v a compatible uniformity on y and vh the corresponding hbuniformity on 2y . if on f τ(v ) ≤ τ(uc∆1,vh), then x ∈ ∆1 and y is atsuji (i.e. ∆1 = cl(x) and every real-valued continuous function on y is uniformly continuous). proof. first we show x ∈ ∆1. assume not and let y1, y2 be distinct points of y . define f = x ×{y1} and d = x ×{y2}. then f ∈ (dc)+ but for any a ∈ ∆1 and any v ∈ v, < f,a; vh > is not contained in (dc)+. in fact, choose x′ ∈ (x \ a) (which exists since we are assumming x 6∈ ∆1) and set g = f ∪ {(x′,y2)} then g ∈< f,a; vh >, but g 6∈ (dc)+ because (x′,y2) ∈ g(x′) ∩ d; a contradiction. then, the result follows from the fact that on c τ(v ) ≤ τ(uc,vh) if and only if τ2(v ) ≤ τ2(vh) on cl(y ) which in turn is equivalent to y being atsuji. � graph topologies on closed multifunctions 461 next example suggested by ľubica holá shows that the converse is not in general true. example 3.16. let x = [1, +∞) and y = [0, 1] subspaces of the real line and vh the hausdorff metric uniformity on 2y . set f = x ×{0} and for each natural number n define fn = x ×{ 1 n }. then the sequence {fn : n ∈ in} τ(uc,vh)-converges to f but it fails to τ(v )-converge to f. in fact, take g = {(x,y) ∈ x ×y : y < 1 x }. then f ∈ g but fn 6∈ g, for each n ∈ in. however, if ∆1 = k(x) we have the next result. proposition 3.17. let x be a hausdorff space, y a tychonoff and locally ∆2 space, v a compatible uniformity on y , then on f τ(v ) ≤ τ(ucc,vh) if and only if x is compact and y is atsuji. (cf. [24] prop. 4.10) if x is a hausdorff space, y a locally compact and completely metrizable space with metric d and v the d-metric uniformity, then on f τ(v ) ≤ τ(ucc,vh) if and only if x is compact and y is a topological sum of a compact space and a discrete space. proof. it is known from [29] that on c(x,y ), the family of all continuous functions on x to y , the graph topology equals the vietoris topology. moreover, observe that τ(ucc,vh) on c(x,y ) equals the compact open topology τk. thus, from a result analogous to 2(d) page 14 of [26] it follows that on c(x,y ), τ(v ) ≤ τk if and only if x ∈ k(x). the statement then follows from the known result that y is atsuji if and only if τ2(v ) ≤ τ2(vh). � the proof of the next proposition is left to the reader. proposition 3.18. let x and y be hausdorff spaces. then on f τ(∆) = τ(v ) if and only if z ∈ ∆ i.e. ∆ = cl(z). ([24] prop. 4.11) τ(f) = τ(v ) if and only if z is compact i.e. x and y are both compact. to study comparisons between the pseudo uniform topologies with some other topologies we give the following lemma. lemma 3.19. let x be a hausdorff space, y a tychonoff and locally ∆2 space, δ1 a compatible lo-proximity on x, δ2 a compatible ef -proximity on y , z = x × y equipped with the product proximity δ = δ1 × δ2, τ2(v −) and σ2(∆2) respectively the lower vietoris topology and the proximal ∆2-topology on 2y . then on f: (a) τp(τ2(v −)) ≤ ps(τ(uc∆1,σ2(∆2))) ≤ ps(τ(uc,σ2(∆2))). if y is a hausdorff and locally ∆2 space and 2y is equipped with the ∆2topology τ2(∆2), then: (b) τp(τ2(v −)) ≤ ps(τ(uc∆1,τ2(∆2))) ≤ ps(τ(uc,τ2(∆2))). 462 g. di maio, e. meccariello and s. naimpally proof. we prove only (a). to show (b) few changes are needed. suppose < f,{x},v − > is a τp(τ2(v −)) neighbourhood of f, where v ∈ τ2 and f ∈ f. so there is a point y ∈ f(x) ∩ v . since y is locally ∆2, there is a d ∈ ∆2 such that y ∈ intd ⊂ d ⊂ v . since the proximity δ2 is ef we also have y ∈ intd ⊂ d � v . then w = (v − × v −) ∪ [(dc)++ × (dc)++] is a symmetric neighbourhood of the diagonal in (2y × 2y ,σ2 ×σ2). clearly, f ∈ w?(f,{x}) ⊂ v −. � theorem 3.20. let x be a hausdorff space with a compatible lo-proximity δ1, y a tychonoff space with a compatible ef -proximity δ2, z = x × y equipped with the product proximity δ = δ1 × δ2 and 2y equipped with the proximal ∆2-topology σ2(∆2) induced by δ2. if y is locally ∆2, then on f: (a) σ(w∆,δ) ≤ ps(τ(uc∆1,σ2(∆2))) ≤ ps(τ(uc,σ2(∆2))). if y is a hausdorff and locally ∆2 space and 2y is equipped with the ∆2 topology τ2(∆2), then: (b) τ(w∆) ≤ ps(τ(uc∆1,τ2(∆2))) ≤ ps(τ(uc,τ2(∆2))). proof. again it suffices to show (a). by above lemma and remark 3.10 it suffices to show that on f σ(w∆,δ)+ ≤ ps(τ(uc∆1,σ2(∆2))) ≤ ps(τ(uc,σ2(∆2))). thus, suppose d = a × b where a ∈ ∆1, b ∈ ∆2 and f � dc. there is an open set v in y with b �δ2 v and such that f � a × v (because δ2 is ef). set s = (v −×v −)∪ [(bc)++ ×(bc)++] a symmetric neighbourhood of the diagonal in (2y × 2y ,σ2 ×σ2). then f ∈ s?(f,a) ⊂ (dc)++. so the first inclusion follows. the second one is trivial. clearly (b) follows from above with obvious changes. � theorem 3.21. let x and y be hausdorff spaces with y locally ∆2, δ1 a compatible lo-proximity on x, δ2 a compatible lo-proximity on y and δ = δ1×δ2 the product proximity on z = x×y . let σ2 and τ2 denote the proximal ∆2-topology and the ∆2-topology on 2y , respectively. then on f: (a) if ∆1 is the family of all finite subsets of x, then ps(τ(uc∆1,σ2)) ≤ σ(w∆) and ps(τ(uc∆1,τ2)) ≤ τ(w∆). (b) if ps(τ(uc∆1,σ2)) ≤ σ(w∆) or ps(τ(uc∆1,σ2)) ≤ τ(w∆), then x is discrete. proof. to check (a) observe that if ∆1 is the family of all finite subsets of x, then ps(τ(uc∆1,σ2(∆2))) = τp(σ2(∆2)) as well as ps(τ(uc∆1,τ2(∆2))) = τp(τ2(∆2)) and clearly τp(σ2(∆2)) ≤ σ(w∆) as well as τp(τ2(∆2)) ≤ τ(w∆). (b) we prove only the second part, i.e if ps(uc∆1,τ2(∆2)) ≤ τ(w∆), then x is discrete. assume not. then there exists a point x0 in x wich is not isolated. denote by n(x0) the family of all open neighbourhoods of x0 and let y0, y1 be two different points in y . for u ∈n(x0) define fu (x) = {y0,y1} for x 6∈ u and fu (x) = {y0} for x ∈ u. it is easy to verify that fu ∈ f and that the net {fu : u ∈ n(x0)} τ(w∆)-converges to f defined by f(x) = {y0,y1} graph topologies on closed multifunctions 463 for x ∈ x. since {fu (x0) : u ∈n(x0)} does not τ2(v −) converge to f(x0), it follows that {fu : u ∈n(x0)} cannot ps(τ(uc∆1,τ2(∆2))) converge to f. � theorem 3.22. let x and y be hausdorff spaces. if y is locally ∆2, then on f τ(wv ) ≤ ps(τ(uc∆1,τ2(∆2))) if and only if x ∈ ∆1 and y ∈ ∆2 (i.e. ∆1 = cl(x) and ∆2 = cl(y )). (cf. [24] prop. 4.12) if x is a hausdorff space and y is a locally compact space, then τ(v ) ≤ τ(ucc,u(f)) if and only if z is compact i.e. x and y are both compact. proof. first observe that from remark 2.8 (g) it follows that on c τ(wv ) ≤ ps(τ(uc∆1,τ2(∆2))) ⇔ τ2(v ) ≤ τ2(∆2) ⇔ y ∈ ∆2 i.e. ∆2 = cl(y ). next, by (b) in theorem 3.20 to show that τ(wv ) ≤ ps(τ(uc∆1,τ(v ))) is equivalent to x ∈ ∆1 it suffices to prove that the inequality τ(wv ) ≤ ps(τ(uc∆1,τ2(v )) implies x ∈ ∆1. assume not. let y1, y2 be distinct points of y . set f = x ×{y1} and d = x ×{y2}. clearly, d ∈ cl(x) ×cl(y ) and f ∈ (dc)+. we claim that for each ps(τ(uc∆1,τ2(v ))) neighbourhood w?(f,a) of f there exists g ∈ w?(f,a) such that g 6∈ (dc)+. in fact, since a 6= x there exists some x′ ∈ (x \a). set g = f ∪{(x′,y2)}. then g ∈ w?(f,a) but g 6∈ (dc)+ showing thereby that x must be in ∆1. � the following example, due to ľubica holá, shows that the uniform hausdorff convergence topology τ(uc,vh) is in general not finer than the pseudo proximal uniform topology ps(τ(uc,σ2)). example 3.23. in the real line with the usual metric d, set x = ⋃ n∈ in ( 1 n + 1 , 1 n ). let y = x, v the metric uniformity on y associated to d, vn = ( 1 n + 1 , 1 n ), ηn = 1 n − 1 n + 1 and yn ∈ vn be fixed for each n ∈ in. let f : x → y defined by f(vn) = yn. let w = ⋃ n∈in (v −n ×v − n ) ∪{∅,∅}. then w is a symmetric σ2 ×σ2 open neigbourhood of the diagonal of 2y ×2y such that the corresponding w?(f,x) = {g ∈ f : ∀x ∈ x(f(x),g(x)) ∈ w} 6∈ τ(uc,vh). assume not, i.e. w?(f,x) ∈ τ(uc,vh). then there exists a positive real ε such that < f,x; ε >⊂ w?(f,x), where < f,x; ε >= {g ∈ f : hd(f(x),g(x)) < ε ∀x ∈ x} (here hd denotes the hausdorff distance associated to d). let η < ε. for each x ∈ x, set g(x) = f(x) + η and let n ∈ in be such that ηn < η. of course g ∈< f,x; ε >, but g 6∈ w?(f,x). in fact choose xn = yn ∈ vn. then f(xn) ∈ vn, but g(xn) 6∈ vn. acknowledgements. the authors are grateful to ľ. holá for her detailed criticism, advice and friendship. 464 g. di maio, e. meccariello and s. naimpally references [1] a.m. abd-allah and r. brown, a compact-open topology on partial maps with open domains, j. london math. soc. 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[34] s.k. zaremba, sur certaines familles de courbes en relation avec la theorie des equations differentielles, rocznik polskiego tow. matemat. 15 (1936), 83–100. received february 2002 revised august 2002 giuseppe di maio seconda università degli studi di napoli, facoltà di scienze, dipartimento di matematica, via vivaldi 43, 81100 caserta, italia e-mail address : giuseppe.dimaio@unina2.it enrico meccariello università del sannio, facoltà di ingegneria, piazza roma, palazzo b. lucarelli, 82100 benevento, italia e-mail address : meccariello@unisannio.it somashekhar naimpally 96 dewson street, toronto, ontario, m6h 1h3, canada e-mail address : sudha@accglobal.net @ appl. gen. topol. 15, no. 1 (2014), 25-32doi:10.4995/agt.2014.2049 c© agt, upv, 2014 near metrizability via a new approach d. mandal ∗,a and m. n. mukherjee a a department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata–700019, india (dmandal.cu@gmail.com, mukherjeemn@yahoo.co.in) abstract the present article deals with near metrizability, initiated in an earlier paper [7], with a new orientation and approach. the notions of nearly regular and uniform pseudo-bases are introduced and analogues of some results concerning metrizability and paracompactness are obtained for near metrizability and near paracompactness respectively via the proposed approach, suitably formulated. 2010 msc: 54d20; 54e99. keywords: nearly paracompact space; regular open set; nearly regular and uniform pseudo-bases; nearly metrizable. 1. introduction the idea of near paracompactness, a well known weaker form of paracompactness, was initiated by singal and arya [9], followed by an extensive study of the concept by many topologists from different perspectives and with different applications (for instance see [3], [4], [5], [6], [8]). now, in [7] we introduced a neighbouring form of metrizability, termed near metrizability, which plays the same role with regard to near paracompactness as is done by metrizability visa-vis paracompactness. it was shown in [7] that there exist nearly metrizable, non-metrizable spaces that are not paracompact, moreover some other facts were established in [7]. ∗the author is thankful to the university grants commission, new delhi110002, india for sponsoring this work under minor research project vide letter no. f. no. 411388/2012(sr). received november 2012 – accepted july 2013 http://dx.doi.org/10.4995/agt.2014.2049 d. mandal and m. n. mukherjee the intent of the present article is to do a further study of nearly mertizable spaces from an altogether new approach. the notion of pseudo-base was introduced and studied in [7], and here, we define regular and uniform pseudo-bases, and ultimately achieve analogues of two well known results on metrizability in our setting. at the outset we recall a few definitions which may be found in [1, 2]. a base b for a topological space x is called regular if for each x ∈ x and any neighbourhood u of x, there exists a neighbourhood o of x such that the set of all members of b that meet both o and x \ u, is finite; and a base b is called a uniform base if for each x ∈ x and every neighbourhood u of x, the set of all members of b that contain x and meet x \ u, is finite. it is clear that every regular base is a uniform base. the next two metrization theorems are known (see [1, 2]), which have been formulated in terms of the above special base. theorem 1.1. (a). a t3-paracompact space x with a uniform base b is metrizable. (b). every t1-space x with a regular base b is metrizable. as already proposed, our principal aim in this paper is to achieve analogous versions of the results in theorem 1.1 for near metrizability with accessories formulated suitably. in what follows, by a space x we shall mean a topological space x endowed with a topology τ(say). the notations ‘cla’, ‘inta’ and ‘|a|’ will respectively stand for the closure, interior and cardinality of a set a of a space x. a set a(⊆ x) is called regular open if a = intcla, and the complement of a regular open set is called regular closed. the set of all regular open (resp. closed) sets of a space x will be denoted by ro(x)(resp. rc(x)). we shall sometimes write a∗ for intcla for a subset a of x and c# = {a∗ : a ∈ c}, for any open cover c of a space x. singal and arya formulated the following definitions which are quite well known by now. definition 1.2 ([10]). a topological space x is called nearly paracompact if every regular open cover of x has a locally finite open refinement. definition 1.3 ([9]). a topological space x is said to be almost regular, if for any regular closed set a and any x ∈ x \ a, there exist disjoint open sets u and v in x such that x ∈ u and a ⊆ v . 2. main results we start by recalling a few definitions from [7] as follows: definition 2.1. if x and y are two topological spaces, then a continuous, injective map f : x → y is called a pseudo-embedding of x into y , if for any a ∈ ro(x), f(a) is open. if there is a pseudo-embedding f of x into y , then we say that x is pseudoembeddable in y . if a pseudo-embedding f : x → y is surjective, we say that f is a pseudo-embedding of x onto y . c© agt, upv, 2014 appl. gen. topol. 15, no. 1 26 near metrizability it is known [7] that every embedding is a pseudo-embedding; but the converse is false. definition 2.2 ([7]). a space x is called nearly metrizable if it is pseudoembeddable in a metric space y . definition 2.3 ([7]). suppose b is a family of open subsets of x. we say that b is a pseudo-base in x if for any a ∈ ro(x), there is a subfamily b0 of b such that a = ⋃ {b : b ∈ b0}. we now define a family b of open subsets of x to be a pseudo-base at a point x ∈ x if for each u ∈ ro(x) containing x, there exists a b ∈ b such that x ∈ b ⊆ u. clearly, a family b of open subsets of a space x is pseudo-base for x if and only if it is so at each x ∈ x. we shall call a pseudo-base b σ-locally finite if b can be expressed as b = ∞⋃ n=1 bn, where bn is locally finite, for each n ∈ n. we now define another type of bases as follows: definition 2.4. let (x, τ) be a topological space. (a) a family b of subsets of x is called nearly regular if for each u ∈ b and any point x ∈ u, there exists a regular open set ox containing x such that the set {v ∈ b : v ⋂ ox 6= φ and v ⋂ (x \ u) 6= φ} is finite. (b) a pseudo-base b for x is called nearly regular if for each x ∈ x and any regular open set ox containing x, there exists a regular open set gx containing x such that the set {u ∈ b : u ⋂ gx 6= φ and u ⋂ (x \ ox) 6= φ} is finite. remark 2.5. it is clear from the above definition that a subfamily of a nearly regular family is a nearly regular family. proposition 2.6. if b is a nearly regular pseudo-base for a space x, then so is b# = {b∗ : b ∈ b}. proof. first let x ∈ x and u a regular open set in x such that x ∈ u. as b is a pseudo-base for x, there exists b ∈ b such that x ∈ b ⊆ u. then x ∈ b∗ ⊆ u∗ = u, and hence b# is a pseudo-base for x. next, let x ∈ x and ox be any regular open set in x containing x. as b is a nearly regular pseudo-base, there exists a regular open set gx containing x such that the set {b ∈ b : b ⋂ gx 6= φ 6= b ⋂ (x \ ox)} is finite. it suffices to show that {b∗ ∈ b# : b∗ ⋂ gx 6= φ 6= b ∗ ⋂ (x \ ox)} is finite, for which we need only to show that {b∗ ∈ b# : b∗ ⋂ gx 6= φ 6= b ∗ ⋂ (x \ ox)} ⊆ {b ∈ b : b ⋂ gx 6= φ 6= b ⋂ (x \ ox)}. in fact, b ⋂ gx = φ ⇔ intclb ⋂ intclgx = φ ⇔ b ∗ ⋂ gx = φ, and b ⋂ (x \ ox) = φ ⇒ b ⊆ ox ⇒ b∗ ⊆ intclox = ox ⇒ b ∗ ⋂ (x \ ox) = φ. � we shall call a space x to be an almost t3-space if it is almost regular and hausdorff. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 27 d. mandal and m. n. mukherjee theorem 2.7. a t2-space x, possessing a nearly regular pseudo-base b is an almost t3-space. proof. let f be a regular closed set and x ∈ x \f . then there exists a regular open set ox containing x such that ox ⋂ f = φ, i.e., f ⊆ x \ ox. by hypothesis, there exists a regular open set gx containing x such that the family u = {u ∈ b : u ⋂ gx 6= φ and u ⋂ (x \ ox) 6= φ} is finite. put o = ox ⋂ gx. then o is a regular open set containing x such that o ⋂ f = φ. consider the family c = {u ∈ b : u ⋂ o 6= φ and u ⋂ f 6= φ}. since f ⊆ x \ ox, c is finite. now for each u ∈ c, |u| ≥ 2 as o ⋂ f = φ. let b′ = b \ c. we show that b′ is a pseudo-base for x. in fact, let p ∈ x and w a regular open set containing p. let us enumerate c as {w1, w2, ..., wn} and let x1, x2,..., xn be points from w1, w2,..., wn respectively different from p. since x is t2, each {xi} is regular closed and so x \{x1, x2, ..., xn} is a regular open set containing p and hence there exists a b1 ∈ b such that p ∈ b1 ⊆ x \ {x1, x2, ..., xn}. again there exists b2 ∈ b such that p ∈ b2 ⊆ w . thus there exists b3 ∈ b such that p ∈ b3 ⊆ b1 ⋂ b2 ⊆ w i.e., p ∈ b3 ⊆ w where b3 6∈ c. this shows that b ′ is a pseudo-base for x. put g = {u ∈ b′ : u ⋂ f 6= φ} and g = ⋃ {u : u ∈ g}. then f ⊆ g and g ⋂ o = φ with x ∈ o (since for u ∈ g, if u ⋂ o 6= φ then u ∈ c, a contradiction). this shows that f and x are strongly separated. thus x is almost regular and consequently x is an almost t3-space. � definition 2.8 ([2]). let x be a topological space and b a family of subsets of x. an element u of b is called a maximal element of b if it is not contained in any element of b other than u. we denote by m(b), the set of all maximal elements of b and call m(b) the surface of b. theorem 2.9. let b be a nearly regular family which is a cover of x. then the surface m(b) of b is a cover of x and is locally finite. proof. let x ∈ x be taken arbitrarily and kept fixed, and let u ∈ b such that x ∈ u. if u 6∈ m(b), then the family λu = {v ∈ b : v % u} is finite. in fact, by definition of b, there exists a regular open set ox containing x such that the collection d = {v ∈ b : v ⋂ ox 6= φ and v ⋂ (x \ u) 6= φ} is finite. clearly, λu ⊆ d and therefore λu is finite (note that x ∈ v ⋂ ox). consequently λu has a maximal element v ′(say). again x ∈ v ′ and v ′ ∈ m(b). hence m(b) is a cover of x. we now show that m(b) is locally finite. as m(b) ⊆ b and b is nearly regular, m(b) is nearly regular. again every element of m(b) is maximal in m(b) (because it is maximal in b and m(b) ⊆ b). let x ∈ x. then there exists a u ∈ m(b) such that x ∈ u. since m(b) is nearly regular, there exists a regular open set ox containing x such that the family b ′ = {v ∈ m(b) : v ⋂ ox 6= φ and v ⋂ (x \ u) 6= φ} is finite. but v \ u 6= φ for all v ∈ m(b) with v 6= u c© agt, upv, 2014 appl. gen. topol. 15, no. 1 28 near metrizability (because every element v in m(b) is maximal, there is no set l ∈ m(b) which properly contains v ). thus {v ∈ m(b) : v ⋂ ox 6= φ} = b ′ ⋃ {u} is a finite set and hence m(b) is locally finite. � theorem 2.10. a space possessing a nearly regular pseudo-base b is nearly paracompact. proof. let g be any regular open cover of x and let gb = {u ∈ b : ∃g ∈ g with u ⊆ g}. we check that gb is a pseudo-base for x. in fact, let x ∈ x and g be any regular open set containing x. now g being a cover, there exists g1 ∈ g such that x ∈ g1. thus g ⋂ g1 is a regular open set containing x. since b is a pseudo-base for x, there exists u ∈ b such that x ∈ u ⊆ g ⋂ g1 ⊆ g1 ∈ g ⇒ u ∈ gb with x ∈ u ⊆ g ⇒ gb is a pseudo-base for x. since b is nearly regular and gb ⊆ b, gb is nearly regular. thus by theorem 2.9, m(gb) is an open cover of x and locally finite. also clearly m(gb) is an open refinement of g. hence x is nearly paracompact. � analogous to the concept of uniform base, we now define a special type of base as follows: definition 2.11. a pseudo-base b for a space x is called a uniform pseudobase if for each x ∈ x and each regular open set ox containing x, uox = {u ∈ b : x ∈ u and u ⋂ (x \ ox) 6= φ} is finite. lemma 2.12. let b be a family of open sets of a space x such that b# is a uniform pseudo-base for x. then the surface m(b#) is a point finite regular open cover of x. proof. let x ∈ x. then there exists u∗ ∈ b# (where u ∈ b) such that x ∈ u∗. if u∗ 6∈ m(b#) then the set λu∗ = {v ∈ b # : v ⊇ u∗} is finite. in fact, u∗ is a regular open set containing x and hence the family v = {v ∈ b# : x ∈ v and v ⋂ (x \u#) 6= φ} is finite and λu∗ ⊆ v ⋃ {u∗}. then λu∗ has a maximal element m(λu# ) which is also a maximal element of b # and which also contains x. hence m(b#) is a regular open cover of x. we now show that m(b#) is point finite. if possible let x ∈ x be such that x belongs to an infinite collection d of members of m(b#). then we claim that d is a pseudo-base for x at x. if d is not a pseudo-base for x at x, there exists a regular open set w containing x such that x ∈ d ⊆ w holds for no d ∈ d, i.e., for all d ∈ d, d ⋂ (x\w) 6= φ. but {b ∈ d : b ⋂ (x \w) 6= φ} is finite as b# is a uniform pseudo-base. hence d is a pseudo-base for x at x. next let, u and v be two distinct (and hence non comparable) elements of d. since x ∈ u ⋂ v and u ⋂ v is a regular open set, there exists a w ∈ d such that x ∈ w $ u ⋂ v (note that u ⋂ v 6∈ d, since otherwise u ⋂ v $ u would contradict the maximality of u ⋂ v ), i.e., x ∈ w $ u and hence w is not a c© agt, upv, 2014 appl. gen. topol. 15, no. 1 29 d. mandal and m. n. mukherjee maximal element of d although d ⊆ m(b#), a contradiction. hence m(b#) is a point finite regular open cover of x. � lemma 2.13. let b be a family of open sets of a t2-space x such that b # is a uniform pseudo-base. then there exists a countable family of point finite regular open covers which taken together is a pseudo-base for x. proof. let b # 1 = b # and b # 2 = b # 1 \ m ∗(b # 1 ), where m ∗(b # 1 ) is the collection of all maximal elements of b # 1 each of which contains at least two points. we first show that b # 2 is a pseudo-base for x. in fact, let x ∈ x. then by lemma 2.12, x belongs to only finitely many members u1, u2,..., un (say) of m ∗(b # 1 ). let xi ∈ ui with x 6= xi for i = 1, 2, ..., n. since x is t2, x \ {x1, x2, ..., xn} is a regular open set containing x and so there exists b in b# such that x ∈ b ⊆ x \ {x1, x2, ..., xn}. let w be any regular open set containing x. then there exists a b′ ∈ b# such that x ∈ b′ ⊆ w . again there exists b1 ∈ b # such that x ∈ b1 ⊆ b ⋂ b′ ⇒ x ∈ b1 ⊆ w and b1 6∈ m ∗(b # 1 ) [b1 ∈ m ∗(b # 1 ) ⇒ b1 = ui for some i = 1, 2, ..., n ⇒ xi ∈ b1 but (xi 6∈ b) ⇒ b1 6⊆ b, a contradiction]. therefore, x ∈ b1 ⊆ w and b1 ∈ b # 2 . again b # 2 ⊆ b # 1 and b # 1 is a uniform pseudo-base ⇒ b # 2 is a uniform pseudo-base. now proceed by induction, if b # k is already defined then put b # k+1 = b # k \ m∗(b # k ) and as above, b # k+1 is a uniform pseudo-base for x. then for each n ∈ n, b#n is a uniform pseudo-base for x and so m(b # n ) is a point finite regular open cover of x (by lemma 2.12). consider an arbitrary x ∈ x. for each n ∈ n, choose un ∈ m(b#n ) such that x ∈ un. if there is n ∈ n satisfying |un| = 1 then {un : n ∈ n} is a pseudo-base at x. if |un| ≥ 2 for all n ∈ n then by definition of b#n , un 6= um for n 6= m. hence l = {un : n ∈ n} is an infinite set of elements of the uniform pseudo-base b#n , each containing x. we claim that l is a pseudo-base for x at x. if not, then for some regular open set d containing x, there does not exist any c ∈ l such that x ∈ c ⊆ d holds, i.e., for all c ∈ l, c ⋂ (x \ d) 6= φ. but since l ⊆ b#, {v ∈ b# : x ∈ u and u ⋂ (x \ d) 6= φ} is finite, a contradiction. consequently, l is a pseudo-base for x at x. hence {m(b#n ) : n ∈ n} is the required family. � definition 2.14 ([11]). let a be a family of subsets of a space x. the star of a point x ∈ x in a, denoted by st(x, a), is defined by the union of all members of a which contain x. a family a of subsets of a space x is said to be a star refinement of another family b of subsets of x if the family of all stars of points of x in a forms a covering of x which refines b. theorem 2.15 ([10]). an almost regular space x is nearly paracompact if and only if every regular open covering of x has a regular open star refinement. definition 2.16. let x be a topological space and γ a family of covers of x. we call γ refined if for any point x ∈ x and any regular open set ox containing c© agt, upv, 2014 appl. gen. topol. 15, no. 1 30 near metrizability x, there exists b ∈ γ such that st(x, b) ⊆ ox. if all the members of γ are regular open covers, then we say that γ is a refined family of regular open covers. theorem 2.17. let b be a family of open sets of an almost t3 nearly paracompact space x such that b# is a uniform pseudo-base for x. then x has a countable refined family of regular open covers. proof. by lemma 2.13, there exists a countable family of point finite regular open covers bn, which taken together is a pseudo-base for x. since x is almost regular and nearly paracompact, by theorem 2.15, each bn has a regular open star refinement un. now fix x ∈ x, and for each n ∈ n, choose bn ∈ bn so that st(x, un) ⊆ bn. then {bn : n = 1, 2, ...} is a pseudo-base for x at x. let u be a regular open set containing x. then there exists bk(say) such that x ∈ bk ⊆ u and then x ∈ st(x, uk) ⊆ bk ⊆ u. thus {un : n = 1, 2, ...} is a countable refined family of regular open covers. � theorem 2.18 ([7]). a space x is nearly metrizable if and only if it is almost t3 and possesses a σ-locally finite pseudo-base. theorem 2.19. let x be an almost t3 nearly paracompact space such that x has a countable refined family {ui} ∞ i=1 of regular open covers. then x is nearly metrizable. proof. since x is nearly paracompact, each ui has a locally finite open refinement bi. let b = ∞⋃ i=1 bi. we show that b is a pseudo-base for x. in fact, let x ∈ x and u be any regular open set containing x. then since {ui} ∞ i=1 is a refined family of covers there exists k ∈ n such that x ∈ st(x, uk) ⊆ u. but bk being a cover of x, there exists bk ∈ bk such that x ∈ bk and bk is contained in some member of uk containing x and hence is contained in st(x, uk). thus x ∈ bk ⊆ u. hence b is a σ-locally finite pseudo-base for x and hence by theorem 2.18, x is nearly metrizable. � theorem 2.20. let b be a family of open sets of an almost t3 nearly paracompact space x such that b# is a uniform pseudo-base for x. then x is nearly metrizable. proof. follows from theorems 2.17 and 2.19. � theorem 2.21. every almost t3-space x with a nearly regular pseudo-base b is nearly mertizable. proof. by theorem 2.10, x is nearly paracompact. again by proposition 2.6, b# is a nearly regular pseudo-base. since every nearly regular pseudo-base is a uniform pseudo-base, b# is a uniform pseudo-base for x, and then by theorem 2.20, it follows that x is nearly metrizable. � c© agt, upv, 2014 appl. gen. topol. 15, no. 1 31 d. mandal and m. n. mukherjee acknowledgements. the authors are grateful to the referee for some suggestions towards certain improvement of the paper. references [1] a. v. arhangel’skii and v. i. ponomarev, fundamentals of general topology: problems and exercises, hindustan publishing corporation(india), 1984. 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[11] j. w. tukey, convergence and uniformity in topology, princeton university press, princeton, n. j. 1940. ix+90 pp. transl. (2), 78 (1968), 103–118. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 32 applied general topology c© universidad politécnica de valencia volume 1, no. 3, 2002 pp. 13–23 a contribution to fuzzy subspaces miguel alamar and vicente d. estruch ∗ abstract. we give a new concept of fuzzy topological subspace, which extends the usual one, and study in it the related concepts of interior, closure and conectedness. 2000 ams classification: 54a40 keywords: fuzzy connectedness, fuzzy topology, q-neighborhood. 1. introduction a simplest and at the same time a very important operation of general topology is transition to a subspace because it let us consider hereditary and local properties, completation and compactification of subspaces, etc. when a is a subspace of a topological space x, the following assertion is satisfied: f is closed in a iff f = a∩t for a closed set t of x. this is not true, in general, in considering a similar problem for fuzzy topological spaces (fts). for this reason, in fuzzy research the consideration of subspaces in a fts (x,t ) is restricted by authors only to ordinary subsets of x. in this paper we extend the concept of a fuzzy topological subspace of x, to fuzzy sets a of x for which the above assertion is satisfied, and then we will be able to extend some concepts and results of the (fuzzy) topological spaces to these subspaces. the structure of the paper is as follows: after preliminaries, in section 3 we define and study the concept of (fuzzy) subspace, and in sections 4–5 we study in it the concepts of interior, closure and connectedness. 2. preliminaries throughout this paper, i will denote the unit real interval [0, 1]. for a non-empty set x, ix denotes the collection of all mappings from x into i. a member b of ix is called a fuzzy set of x. the set {x ∈ x : b(x) > 0} is called the support of b and is denoted by supp b. if b takes only the values 0, ∗while working on this paper the authors have been partially supported by a grant from upv: ”incentivo a la investigación/99” 14 m. alamar and v. d. estruch 1, b is called a crisp set in x. from now on, we shall not differentiate between a crisp set b in x and (the ordinary subset of x) supp b. nevertheless, the crisp set which always takes the value 1 (respectively, 0) on x is denoted by 1 (respectively, 0). the union and intersection of a family of fuzzy sets {ai}i of x is ∨ i ai and∧ i ai, respectively. the complement of a ∈ i x, denoted by a′, is defined by the formula a′(x) = 1−a(x), x ∈ x. for a,b ∈ ix we write a ⊂ b or b ⊃ a if a(x) ≤ b(x), for each x ∈ x. the fuzzy set xλ of x given by xλ(y) = 0 if y 6= x, and xλ(x) = λ (λ ∈]0, 1]) is called a fuzzy point of x with support x [5]. the fuzzy point xλ is said to be contained in a fuzzy set a or belonging to a, denoted by xλ ∈ a, if λ ≤ a(x). a family t of fuzzy sets of x containing 0 and 1, is called a fuzzy topology on x [1] if it is closed under arbitrary unions and finite intersections. the pair (x,t ) is called a fuzzy topological space (fts). each member of t is called an open (fuzzy) set. the complement of an open set is called a closed (fuzzy) set. t c denotes the family of all closed sets of (x,t ). 3. fuzzy subspaces definition 3.1. let (x,t ) be a fts and y ∈ ix. the pair (y,ty ) is called a fuzzy topological subspace of (x,t ) if the family ty = {g∩y : g ∈t} satisfies the following conditions: (c1) for each h ∈ty there exist fh ∈t c such that y −h = fh ∩y . (c2) for each f ∈t c there exist gf ∈t such that y −(f ∩y ) = gf ∩y . in this case, the members of ty will be called ty -open. if f ⊂ y and y −f ∈ ty , f will be called a ty -closed (fuzzy) set. for shortness we will say y is a subspace of x. notice that ty is closed under finite intersections and arbitrary unions and that 0 and y (instead of x) are both ty -open and ty -closed. also, if y is an ordinary subspace of the topological space x, y is a subspace of x in the sense of definition 3.1. remark 3.2. the above conditions (c1) and (c2) establish that f is ty -closed iff f = t ∩y for a t -closed set t of the fts (x,t ). example 3.3. (a) (ordinary subsets of a fts x are subspaces of x). let (x,t ) be a fts and let y an ordinary subset of x. if h ∈ ty then h = g∩y for some g ∈t , thus y −h = (1 −g) ∩a and (c1) is satisfied. if f ∈ t c then y − (f ∩y ) = (1 −f) ∩y , where 1 −f ∈ t , and (c2) is satisfied. so, y is a subspace of x. (b) let x 6= ∅. we will denote by fc the constant function on x given by fc(x) = c, for each x ∈ x, with 0 ≤ c ≤ 1. now, consider the lowen indiscrete fuzzy topology t = {fc : c ∈ [0, 1]}, on x. (notice that t = t c.) a contribution to fuzzy subspaces 15 fix a real number k ∈]0, 1[, and choose a non-empty ordinary subset b of x. now consider y ∈ ix given by y (x) = k if x ∈ b, and y (x) = 0 if x /∈ b. we will see that y is a subspace of (x,t ). in fact, h ∈ ty iff h(x) = m, for each x ∈ b and h(x) = 0 elsewhere, for some m ∈ [0,k]. then y − h is given by (y − h)(x) = k − m if x ∈ b and (y − h)(x) = 0 if x /∈ b. so, y − h = fk−m ∩ y and (c1) is satisfied since 1 −fk−m = f1−(k−m) ∈t . now suppose f ∈ t c. then, also f ∈ t , and thus f = fm, for some m ∈ [0, 1]. hence, y − (f ∩y ) is given by (y − (f ∩y ))(x) = k− (m∧k) for x ∈ b and (y − (f ∩y ))(x) = 0 if x /∈ b, and clearly (c2) is satisfied. (c) (construction of fuzzy topological subspaces). fix a real number γ ∈]0, 1 2 [, and let l be a fuzzy topology on x such that g(x) ≤ γ, for each x ∈ x and g ∈l ∼ {1}. take a ∈ ix such that a(x) ≤ 1 2 , for x ∈ x. clearly, if l contains at least two proper open sets then a is not a subspace of (x,l). now, denote l∗ = {g∗ : g∗ = 1 − (a− (a∩g)), g ∈l}. we will see that l∗ ∪{0} is a fuzzy topology on x: consider a family {g∗i : i ∈ i} of elements of l ∗, and suppose g∗i = 1 − (a− (a∩gi)), where gi ∈l, i ∈ i. we will see that ⋃ i g ∗ i ∈l ∗. we have ⋃ i g ∗ i = ⋃ i(1 − (a− (a∩gi)) = (1 −a) + ⋃ i(a∩gi) = (1 −a) + (a∩ ( ⋃ i gi)) = (1 −a) + (a∩g) where g = ⋃ i gi ∈l, and so ⋃ i g ∗ i ∈l ∗. now, we will see l∗ is closed under finite intersection. take g∗i = 1 − (a− (a∩gi)), where gi ∈l, i = 1, 2. we have g∗1 ∩g ∗ 2 = (1 − (a− (a∩g1))) ∩ (1 − (a− (a∩g2))) = (1 −a) + ((a∩g1) ∩ (a∩g2)) = (1 −a) + (a∩ (g1 ∩g2)) = 1 −a + (a∩g) where g = g1 ∩g2 ∈t , and thus g∗1 ∩g∗2 ∈l∗. clearly 1 ∈l∗ and then l∗ ∪{0} is a fuzzy topology on x. now, if g ∈l with g 6= 1, then g(x) < 1 2 for each x ∈ x, and if g∗ ∈ l∗ with g∗ 6= 0 we have g∗(x) ≥ 1 2 , and from these facts it is easy to verify that t = l∪l∗ is a fuzzy topology on x. now we will see that a is subspace of (x,t ): clearly, ta = la ∪{a}. first we will see condition (c1) is satisfied. let h ∈ta and suppose h = g∩a with g ∈t . we distinguish two possibilities: (i) g ∈l∗ ∪{1}. in this case h = g∩a = a and a−h = 0 = 0 ∩a. 16 m. alamar and v. d. estruch (ii) g = l ∼ {1}. in this case a−h = a− (g∩a) = (a− (g∩a)) ∩a since a− (g∩a) ⊂ a, with 1 − (a− (g∩a)) ∈l∗ ⊂t . now, we will prove that condition (c2) is satisfied. suppose f ∈ t c. we distinguish two possibilities: (i) 1 −f ∈l∗. in this case f = a− (a∩g), with g ∈l. so, a− [(a− (a∩g)) ∩a] = a− (a− (a∩g)) = a∩g ∈ta (ii) 1 − f ∈ l ∼ {1}. in this case f(x) ≥ 1 2 for each x ∈ x, and thus f ⊃ a. now, a− (f ∩a) = 0 ∈ta. definition 3.4. let y be a subspace of the fts (x,t ), and suppose b ⊂ y . the pair (b,tb) is called a fuzzy topological subspace of (y,ty ) if the family tb = {g∩b : g ∈ty} satisfy the following conditions (c1)’ for each h ∈tb it exists fh ∈t cy such that b −h = fh ∩b. (c2)’ for each f ∈t cy it exists gf ∈ty such that b − (f ∩b) = gf ∩b. the terminology tb (instead of tyb ) is justified in proposition 3.6. otherwise, the elements of tb are called tb-open. if f ⊂ b and b−f ∈tb, f is called tb-closed. for shortness we will say b is a subspace of y . remark 3.5. the above conditions (c1)’ and (c2)’ establish that f is tbclosed iff f = b ∩t for a ty -closed set t of y . proposition 3.6. suppose b is a subspace of y and y is a subspace of the fts (x,t ). then, b is a subspace of (x,t ). proof. it is an immediate consequence of remarks 3.2 and 3.5. � proposition 3.7. let y be a subspace of the fts (x,t ). if y is t -open (respectively, t -closed) then g ∈ ty iff g ∈ t (respectively, f ∈ t cy iff f ∈ t c). proposition 3.8. let b and y the two subspaces of the fts (x,t ). if b ⊂ y , then b is a subspace of y . proof. we have h ∈ tb iff there exists g∗ ∈ t such that h = g∗ ∩b. now, g∩b = g∩y ∩b, and since g = g∗∩y ∈ ty , then tb = {g∩b : g ∈ty}. we will see that the family tb satisfies (c1)’ and (c2)’. let h ∈tb; then there exists fh ∈t c such that b −h = fh ∩b = (fh ∩y ) ∩b, where fh ∩y ∈t cy , and so (c1)’ is satisfied. now, let f ∈t cy ; then there exists gf ∈t such that b − (f ∩b) = gf ∩b = (gf ∩y ) ∩b, where gf ∩y ∈ty , and so (c2)’ is satisfied. � a contribution to fuzzy subspaces 17 lemma 3.9. let m, n and p be fuzzy sets of x. then (m ∩n) − (p ∩ (m ∩n)) = (m − (p ∩m)) ∩ (n − (p ∩n)) proof. let x ∈ x and suppose m(x) ≥ n(x). we distinguish three possibilities: (1) n(x) ≤ p(x) ≤ m(x). in this case, the left hand of the above inequality becomes n(x) − n(x) = 0, and the right hand becomes (m(x) −p(x)) ∧ (n(x) −n(x)) = 0. (2) p(x) ≤ n(x) ≤ m(x). now the left hand of the inequality becomes n(x) − p(x), and the right hand becomes (m(x) − p(x)) ∧ (n(x) − p(x)) = n(x) −p(x). (3) n(x) ≤ m(x) ≤ p(x). now, the left hand of the inequality becomes n(x)−n(x) = 0, and the right hand becomes (m(x)−m(x))∧(n(x)− n(x)) = 0. since the announced equality is symmetric respect m and n, the same argument is valid for n(x) ≥ m(x), and then the equality is established. � proposition 3.10. let a and b be two subspaces of the fts (x,t ). then a∩b is subspace of (x,t ). proof. consider the family ta∩b = {g∩ (a∩b) : g ∈ t}. we will see that ta∩b satisfies (c1) and (c2). let h ∈ta∩b; then h = g∩ (a∩b) with g ∈t . now, by lemma 3.9, (a∩b) −h = (a∩b) − (g∩ (a∩b)) = (a− (g∩a)) ∩ (b − (g∩b)). but a−(g∩a) is ta-closed and hence a−(g∩a) = fa∩a for some fa ∈t c. also b − (g∩b) = fb ∩b for some fb ∈t c and therefore (a∩b) −h = (fa ∩a) ∩ (fb ∩b) = (fa ∩fb) ∩ (a∩b) and (c1) is satisfied since fa ∩fb is t -closed. now, let f ∈t c. by lemma 3.9, (a∩b) − (f ∩ (a∩b)) = (a− (f ∩a)) ∩ (b − (f ∩b)) = ga ∩gb where ga = a−(f ∩a) ∈ta and gb = b−(f ∩b) ∈tb. so, there are g1, g2 ∈t such that ga = g1 ∩a and gb = g2 ∩b and therefore (a∩b) − (f ∩ (a∩b)) = (g1 ∩a) ∩ (g2 ∩b) = (g1 ∩g2) ∩ (a∩b) = g∩ (a∩b) where g = g1 ∩g2 ∈t , and (c2) is satisfied. � since ordinary subsets in a fts (x,t ) are subspaces, we have the following corollary. 18 m. alamar and v. d. estruch corollary 3.11. let a be a subspace of the fts (x,t ) and y an ordinary subset of x. then a∩y is a subspace of (x,t ). if a and b are two subspaces of x, in general a ∪ b is not a subspace of x, even if a∩b = 0, as shows the following example. example 3.12. let x = [0, 1] and choose three the real numbers a, b and c, such that 0 < c < a < b < 1. consider x endowed with the lowen indiscrete topology t of example 3.3 (b). consider the fuzzy sets a and b of x given by a(x) = { a 0 ≤ x ≤ 1 2 0 1 2 < x ≤ 1 b(x) = { 0 0 ≤ x ≤ 1 2 b 1 2 < x ≤ 1 by (b) of the example 3.3, a and b are subspaces of x. obviously a∩b = 0 but we will see a∪b is not a subspace of x. consider the constant function fc on x defined by fc(x) = c, x ∈ x. we have that fc ∈t c and since (a∪b − (fc ∩ (a∪b)))(x) = (a∪b −fc)(x) = { a− c 0 ≤ x ≤ 1 2 b− c 1 2 < x ≤ 1 it is obvious that condition (c2) cannot be satisfied. 4. interior and closure in this section, y will be a subspace of the fts (x,t ). according with [1], the ty -interior, denoted by inty a, of a fuzzy set a contained in y is the largest ty -open (fuzzy) set contained in a, and the ty closure, denoted by cly a, is the smallest ty -closed (fuzzy) set containing a. proposition 4.1. let a ⊂ y . then (i) intx a = inty a∩ intx y . (ii) cly a = clx a∩y . proof. it is similar to the classic case. � through the notion of fuzzy point, it is possible to study the concepts of interior and cluster (adherence) point. according with [5], we give the following definition. definition 4.2. a fuzzy set a in (x,ty ) is called a ty -neighborhood of the fuzzy point xλ if there exists b ∈ty such that xλ ∈ b ⊂ a. we also say that xλ is ty -interior of a. then, xλ ∈ inty a iff xλ is ty -interior of a. in [3], is given the following definition. definition 4.3. the fuzzy set point xλ is said to belong to b, written xλ ∼ ∈ b, iff b(x) > λ. according with [3], the following is a distinct definition of an interior point. a contribution to fuzzy subspaces 19 definition 4.4. a fuzzy set a in (y,ty ) is called a t ∼y -neighborhood of the fuzzy point xλ if there exists b ∈ty such that xλ ∼ ∈ b ⊂ a. also, xλ is called a t ∼y -interior point of a. then, xλ ∼ ∈ inty a iff xλ is a t ∼y -interior pointof a. notice that if (inty a)(x) > 0 then x(inty a)(x) is ty -interior of a, but it is not t ∼y -interior of a. nevertheless, inty a = ⋃ {xλ : xλ is ty -interior of a} = ⋃ {xλ : xλ is t ∼y -interior of a}. by (i) of proposition 4.1, we have the following corollary. corollary 4.5. let a be contained in y . then, a fuzzy point xλ ∈ intx y (respectively, xλ ∼ ∈ intx y ) is a ty -interior (respectively, a t ∼y -interior) point of a iff it is a t -interior point of a. the following definitions and results are obvious generalizations of the ones given in [5]. definition 4.6. a fuzzy point xλ is said to be y -quasi-coincident with the fuzzy set a of x, denoted by xλ qy a, if λ + a(x) > y (x). definition 4.7. let a,b ⊂ y . a is said to be y -quasi-coincident with b, denoted by a qy b, if there exists x ∈ x such that a(x) + b(x) > y (x). definition 4.8. a fuzzy set a in(x,ty ) is called a qy -neighborhood of xλ if there exists b ∈ty , b ⊂ a, such that xλ qy b. proposition 4.9. let a,b ⊂ g. then a ⊂ b iff a and y − b are not y -quasi-coincident; particularly xλ ∈ a iff xλ is not y -quasi-coincident with y −a. theorem 4.10. a fuzzy point xλ ∈ cly a iff each qy -neighborhood of xλ is y -quasi-coincident with a. definition 4.11. a fuzzy point xλ is called a ty -adherence point of a fuzzy set a if every qy -neighborhood of xλ is y -quasi-coincident with a. according with [3] we give the following definition. definition 4.12. the fuzzy point xλ is called a ty -cluster point of a if for each g ∈ty such that xy (x)−λ ∼ ∈ g implies g 6⊂ y −a. proposition 4.13. let a ⊂ y . the fuzzy point xλ is ty -cluster point of a iff it is a ty -adherence point of a. proof. suppose xλ is ty -cluster point of a. let g ∈ ty a q-neighborhood of xλ. then, λ + g(x) > y (x) and thus xy (x)−λ ∼ ∈ g, and g 6⊂ y − a since xλ is a ty -cluster point of a. therefore, there exists x ∈ x such that g(x) > y (x) −a(x), and so g is y -quasi-coincident with a. suppose xλ is a ty -adherence fuzzy point of a. let g ∈ ty such that xy (x)−λ ∼ ∈ g. then y (x) − λ < g(x) and thus g is a neighborhood of xλ. so, g is y -quasi-coincident with a. and then there exists x ∈ x such that g(x) + a(x) > y (x); therefore g 6⊂ y −a and xλ is ty -cluster point of a. � 20 m. alamar and v. d. estruch by (ii) of proposition 4.1 we have the following corollary. corollary 4.14. let a be contained in y . then, a fuzzy point xλ ∈ y is a ty -adherence (cluster) point of a iff it is a t -adherence (cluster) point of a. 5. connectedness we will use the concepts of connectedness due to pu and liu [5], [6], but with terminology of [7]. definition 5.1. a fuzzy set d in the fts (x,t ) is called c-disconnected (respectively o-disconnected) if there are a,b ∈ t c (respectively, a,b ∈ t ) such that a ∩ d 6= 0, b ∩ d 6= 0, a ∩ b ∩ d = 0 and a ∪ b ⊃ d. a fuzzy set is called c-connected (respectively o-connected) if it is not c-disconnected (respectively o-disconnected). in contrast to general topology the use of closed and open fuzzy sets in definitions of connectedness of fuzzy sets results in two distinct concepts. nevertheless, we will see that these concepts agree in subspaces. according with definition 5.1, we give the following definition. definition 5.2. a subspace (y,ty ) of the fts x will be called c-disconnected (respectively o-disconnected) if there are two non-empty ty -closed (respectively ty -open) sets a and b such that a∩b = 0 and y = a∪b. the following proposition shows that connected properties are absolute properties in subspaces. proposition 5.3. if y is a subspace of the fts x, then the fuzzy set y is c-connected (respectively o-connected) iff the subspace (y,ty ) is c-connected (respectively o-connected). proof. it is straightforward. � it is clear that a fts (x,t ) is o-connected iff it is c-connected. now this fact is extendable to subspaces in the next proposition. proposition 5.4. let y be a subspace of the fts (x,t ). then y is cconnected iff it is o-connected. proof. suppose y is not o-connected. then there are two sets g,h ∈t such that (1) g∩y 6= 0, h ∩y 6= 0, g∩h ∩y 6= 0 and (2) y = (g∩y ) ∪ (h ∩y ). now, by (1), for x ∈ x, (g ∩ y )(x) 6= 0 iff (h ∩ y )(x) = 0 and also (h∩y )(x) 6= 0 iff (g∩y )(x) = 0, and thus y −(g∩y ) 6= 0, y −(h∩y ) 6= 0, and (y − (g∩y )) ∪ (y − (h ∩y )) = y . also, by (2), for x ∈ x if g(x) < y (x) then h(x) ≥ y (x), and if h(x) < y (x) then g(x) ≥ y (x), and thus (y − (g ∩ y )) ∩ (y − (h ∩ y )) = 0, and then (y,ty ) is not c-connected. the converse is showed with a similar argument. � a contribution to fuzzy subspaces 21 definition 5.5 (p1). two fuzzy sets a1 and a2 in a fts (x,t ) are said to be q-separated if there exist two t -closed sets hi (i = 1, 2) such that hi ⊃ ai (i = 1, 2), and h1 ∩ a2 = h2 ∩ a1 = 0. it is obvious that a1 and a2 are q-separated iff cl a1 ∩a2 = cl a2 ∩a1 = 0. note 5.6. a fuzzy set d is c-disconnected [5] iff there exist two non-empty sets a and b, both two contained in supp d, such that a and b are q-separated, and d = a∪b. according with this, we give the following definition. definition 5.7. a fuzzy subspace (y,ty ) of x is here called disconnected if there exists two non-empty sets, both two contained in y , such that a and b are q-separated and y = a∪b. y is called connected if it is not disconnected. theorem 5.8. let y be a subspace of (x,t ). they are equivalent: (i) y is connected. (ii) y is c-connected. (iii) y is o-connected. proof. by proposition 5.4 we only have to prove that (i) and (ii) are equivalent. suppose y is disconnected. then there exist two sets a,b ⊂ y such that clx a∩b = 0, clx b ∩a = 0 and y = a∪b. now, by (ii) of proposition 4.1, we have cly a = y ∩ clxa = (a∪b) ∩ clxa = (a∩ clxa) ∪ (b ∩ clxa) = a, and hence a is ty -closed. similarly, b is ty -closed and then y is c-disconnected. suppose now that y is c-disconnected. then, there are two non-empty ty closed sets a and b, such that a∩b = 0 and y = a∪b. now, otherwise by (ii) of proposition 4.1, we have a∩ clxb = (a∩y ) ∩ clxb = a∩ cly b = a∩b = 0. similarly, b ∩ clx a = 0, and then a and b are q-separated. � lemma 5.9. if a and b are q-separated in the fts x and y is a connected subspace of x, with y ⊂ a∪b, then y ⊂ a or y ⊂ b. proof. if a and b are q-separated in the fts x, then a∩y , and b∩y are also q-separated in x, and y = (a∩y ) ∪ (b ∩y ), then a∩y = 0 or b ∩y = 0, i.e., y ⊂ b or y ⊂ a. � as in the classic case, theorem 5.8 and lemma 5.9 provide with some neat ways of proving a given space x is connected. theorem 5.10. let x be a fts. (a) if x = ⋃ α xα, where each xα is a connected subspace of x and⋂ α xα 6= 0, then x is connected. (b) if each pair p,q of fuzzy points of x lies in some connected subspace ep,q of x, then x is connected. 22 m. alamar and v. d. estruch (c) if x = ⋃∞ n=1 xn where each xn is a connected subspace of x and xn−1 ∩xn 6= 0, for each n ≥ 2, then x is connected. proof. the proofs are slight modifications of the classic cases. � next theorem is a generalization of theorem 10.1 of [5]. theorem 5.11. let y be a subspace of the fts (x,t ) and let d be a cconnected fuzzy set in (x,t ). if d ⊂ y then cly d is also c-connected. proof. suppose cly d is c-disconnected. then, there are two t -closed sets a and b such that a∩cly d 6= 0, b∩cly d 6= 0, a∩b∩y = 0 and a∪b ⊃ cly d. by the connectedness of d, we may assume that a∩d = 0, that is d ⊂ b. it follows that cly d ⊂ b and thus a∩ cly d = 0, which is a contradiction. � definition 5.12. two fuzzy sets a1 and a2 contained in a subspace (y,ty ) of the fts x is said q-separated in y if there exist ty -closed sets hi (i = 1, 2) such that hi ⊃ ai (i = 1, 2) and h1 ∩a2 = h2 ∩a1 = 0. theorem 5.13. let a be a family of c-connected fuzzy sets in fts x. suppose⋃ a is a fuzzy subspace of x. if no two members of a are q-separated in ⋃ a, then ⋃ a is connected. proof. the same proof as theorem 10.2 of [5], but replacing supp ⋃ a by ⋃ a, proves that ⋃ a is c-connected, and by theorem 5.8 ⋃ a is connected. � 5.14 final considerations. one can extend in a natural way the ti-fuzzy separation axioms of [5] in fts to subspaces, in such manner that were hereditary properties. notice that there are many definitions of t2-fts in the literature (see [2]), but are particularly interesting the fuzzy separation axioms given in [4], through the concept of r-neighborhood. references [1] c.l. chang, fuzzy topological spaces, j. math. anal. appl. 24 (1968), 182–190 [2] d. r. cuttler, i.l. reilly, a comparison of some haussdorf notions in fuzzy topological spaces, computers math. applic., vol. 19, n. 11, (1990) 97–104 [3] z. deng, fuzzy pseudo-metric spaces, j. math. anal. appl. 86 (1982), 74–95 [4] w. guojun, a new fuzzy compactness defined by fuzzy sets, j. math. anal. appl. 94 (1983), 1–23 [5] p.m. pu, y.m. liu, fuzzy topology. i: neighborhood stucture of a fuzzy point and moore-smith convergence, j. math. anal. appl. 76 (1980), 571–599 [6] p.m. pu, y.m. liu, fuzzy topology. ii: product and quotient spaces, j. math. anal. appl. 77 (1980), 20–37 [7] a.p. shostak, two decades of fuzzy topology: basic ideas, notions and results, russian math. surveys, 44: 6 (1989), 125–186 received march 2001 a contribution to fuzzy subspaces 23 m. alamar and v.d. estruch dep. de matemática aplicada escuela politécnica superior de gandia universidad politécnica de valencia 46022 valencia spain e-mail address : malamar@mat.upv.es, vdestruc@mat.upv.es kopperag.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 115127 partial metrizability in value quantales r. kopperman∗, s. matthews and h. pajoohesh abstract. partial metrics are metrics except that the distance from a point to itself need not be 0. these are useful in modelling partially defined information, which often appears in computer science. we generalize this notion to study “partial metrics” whose values lie in a value quantale which may be other than the reals. then each topology arises from such a generalized metric, and for each continuous poset, there is such a generalized metric whose topology is the scott topology, and whose dual topology is the lower topology. these are both corollaries to our result that a bitopological space is pairwise completely regular if and only if there is such a generalized metric whose topology is the first topology, and whose dual topology is the second. 2000 ams classification: 06b35, 06d10, 06f30, 54e35, 54e55 keywords: value lattice, partial metric, quasimetric, completely regular bitopological space, value quantale, well above, auxiliary relation 1. introduction a partial metric [7] is a generalised metric for modelling partially defined information. for example, if a person is told to visit london in the uk they would instantly ask where in london. a more precise instruction might be to visit london’s hyde park, yet again they would ask, where in hyde park. a sufficiently precise instruction might be to visit the prince albert memorial, or the serpentine gallery in hyde park. let the relation l1 ≤ l2 on such ∗this author wishes to acknowledge support for this research from the epsrc of the united kingdom (grant gr/s07117/01), and from the city university of new york (psccuny grant 64472-00 33). 116 r. kopperman, s. matthews and h. pajoohesh locations be defined by l2 is a place in l1. then this is a partial ordering. for example, london ≤    hyde park ≤ { serpentine gallery prince albert memorial trafalgar square ≤ nelson′s column a partial metric is a means of extending the notion of metric, such as the euclidean distance between the serpentine gallery and nelson’s column, to posets. to see how and, most importantly, why this is useful a brief discussion of the programming problem in computer science which led to partial metrics is now given. a concurrent program is a computer program consisting of two or more processes to be executed in parallel, and also to communicate with each other in some way. a problem arises when two processes are each waiting upon the other for a communication before each itself can communicate. this is a deadly embrace situation known as deadlock where two processes remain alive yet doing nothing as each waits for a communication that can never arrive. in general it is not decidable whether or not an arbitrary concurrent program may deadlock at some time in its execution. however, in a concurrent program for a safety critical application, such as for use in a hospital’s intensive care unit, it is essential that there be some means of proving that deadlock can never occur. the best that can be done is to consider certain concurrent programs where it can be proven that deadlock can never arise. deadlock is a problem between two processes which are directly or indirectly connected by some path of communication, and so a consideration of such paths provides an analysis of possible deadlock. a cycle is a set of processes p1, p2, . . . , pn where each pi communicates by sending messages to p(i+1) mod n, and so receiving messages from p(i−1) mod n. for each pi let ci be the largest possible difference at any time between the number of messages sent minus the number received. for example, consider the case for n = 2. suppose c1 = 5, then at any time p1 has sent at least five more messages than it has received. suppose also that c2 = −3, then p2 has sent at least −3 more (i.e. at most 3 fewer) messages than it has received. this cycle of two processes cannot deadlock as at any time there is a net surplus of at least c1 +c2 = 5+(−3) = 2 messages being produced by the cycle than being received. if the so-called cycle sum ∑n i=1 ci > 0 then this cycle of processes can never deadlock. to prove that a concurrent program will not deadlock it is thus sufficient to prove that each and every cycle sum is positive. this is the cycle sum theorem [10], later extended to a more sophisticated model of concurrent programming as the cycle contraction mapping theorem [6]. the virtue of these theorems is that they are an extensional treatment of deadlock, one which does not require a detailed understanding of exactly how programs are executed. they prove that a deadlock free computation is necessarily the only possible behaviour for a concurrent program. this is in contrast to the more usual (but more difficult) procedure of constructing the partial metrizability in value quantales 117 sequence of all intermediate states of an execution, and demonstrating that the limit is a deadlock free computation. the behaviour of a cycle can be studied as the fixed point of a function. the purpose of the cycle sum and cycle contraction mapping theorems is to firstly prove that the fixed point is unique, and secondly that this point is, in a desired sense, totally defined. being an extensional treatment of deadlock we require not the details of how programs are executed, only the distinction between the so-called total (the word total is used here in place of complete as in [7]) computations (i.e. the desirable deadlock free executions) and the partial, that is, those initial parts of a total computation. to appreciate the distinction between total and partial objects we return to the analogy of a visitor to london. a reference to london is only partially informative, as it does not refer to a more specific place of interest such as hyde park or trafalgar square. london is thus a partialization of hyde park, trafalgar square, etc. where the (extent of) totalness of a place can be measured by the area of ground upon which it stands. london is a partial approximation to hyde park as the latter stands upon a smaller space and contained within that of london. the cycle sum and cycle contraction mapping theorems are in essence inductive results to prove that eventually a total result must follow from the inductive hypothesis that for each step in a computation the next step will steadily increase the (extent of) totalness. to formulate such an hypothesis requires a function to measure the (extent of) totalness of a computation. for example, a tourist might ‘visit’ the partial places london, trafalgar square, nelson’s column. the (extent of) totalness of each partial place, as measured by its area, becomes increasingly precise at each step. in the theorems under discussion the property of deadlock free for programs is partialized precisely to the extent to which it applies to each initial part of a deadlock free computation. this is a downward approach in which a partial object is viewed as a partialized total one. this is in contrast to the established scott-strachey least fixed point semantics [9] where the behaviour of a program is viewed as the limit of a chain of partial approximations, an upward approach where a total object (if the notion exists) is a completion of partial objects. while the upward view is necessary to define the semantics for an arbitrary program, the downward view is sufficient to reason about well behaved programs such as those which are deadlock free. the general problem arising from these deadlock studies [10, 6] is how to partialize theories. given a theory of (now to be known as) total objects how can additional (to be known as) partial objects be introduced and the theory extended yet weakened to apply to them? in each of the above deadlock studies the total objects form a metric space of infinite sequences. for a set x let, d : xω × xω → ℜ+ be the metric such that d(x,y) = 2−sup{n|∀m 0 note that a pmetric restricted to the total objects is a metric. for all x,y ∈ x, x < y ⇒ |x| > |y|, and so |·| can be used to measure the (extent of) totalness of each member of x. the banach contraction mapping theorem can be extended to partial metrics. a contraction is a function f : x → x for which there exists a 0 < c < 1 such that ∀ x,y ∈ x, p(f(x), f(y)) ≤ c × p(x,y). a cauchy sequence is an x ∈ xω such that there exists a > 0 such that for each ǫ > 0 there exists k ∈ ω such that for all n,m > k, |p(xn,xm) − a| < ǫ. a sequence x ∈ xω converges if there exists a ∈ x such that for each ǫ > 0 there exists k > 0 such that for all n > k, p(xn,a) − p(a,a) < ǫ. p is complete if every cauchy sequence converges. the partial metric contraction mapping theorem [6] is that each contraction for a complete partial metric has a unique fixed point, and this point is total. although originally developed as a partialized theory for extensional reasoning about properties of programs such as deadlock, partial metrics have since been developed in computer science as a theory of partiality for studying continuous lattices, using the induced ordering x ≤ y iff p(x,x) = p(x,y). a partial metric p : x × x → ℜ+ generalises the theory of metric spaces by dropping the requirement that self distance always be zero, and in so doing opens up the study of t0 spaces to a symmetric (in contrast to quasimetrics) metric style treatment, which in addition incorporates a weight function | · | : x → ℜ+. the present paper takes the process of generalisation further by replacing the range ℜ+ of a pmetric by a value quantale. 2. p-metrics and qmetrics in value lattices definition 2.1. a value lattice is a poset (v,≤), whose least element is denoted 0 and largest is ∞, such that (v,≥) is a continuous lattice; together with an associative, commutative operation + : v × v → v such that 0 is an identity and for each r,s ⊇ v, ( ∧ r) + ( ∧ s) = ∧ {r + s|r ∈ r,s ∈ s}, where partial metrizability in value quantales 119 ∧ denotes inf. here are some simple but useful consequences of this infinite distributive law: • (s1) for all p ∈ v , p + ∞ = ∞. • (s2) for all p,q,r,s ∈ v , p ≥ q and r ≥ s implies p + r ≥ q + s. a value lattice v, is boolean if for each a ∈ v, a + a = a. like any map preserving arbitrary infima, for each p, the map +p, +p(q) = p + q has a right adjoint (see [4], chap. 0.3), −̇ p defined by −̇ p(q) = ∧ {r ∈ v |p + r ≥ q}(= q −̇ p). the properties of −̇ (see [2]) include for any p,q,r ∈ v, s ⊆ v : • (d1) p + r ≥ q iff r ≥ q −̇ p. • (d2) q −̇ p = 0 iff p ≥ q; • (d3) p + (q −̇ p) ≥ q; the reversal of order above – that is, the requirement that (v,≥), rather than the expected (v,≤) be a continuous lattice – is due our need to maintain the traditional way of writing axioms for metrics, in order to allow easy comparison between metric spaces and our structures. the reader must be careful when looking at references (except [2]), which use the traditional order. notice that products of value lattices are again value lattices, and that with the way-above relation denoted ≫, that in a product ∏ i vi, a ≫ b if and only if each ai ≫ bi and {i|ai 6= ∞} is finite. when we write ≪, we mean the inverse of ≫. key examples of value lattices include the extended nonnegative reals, ie = [0,∞], the unit interval ii = [0,1], and the two-point set, ib = {0,∞}, together with the usual ≤,+, except that in ii we truncate addition, via a+b = min{a+u b,1}, where +u denotes the usual sum. by a qmetric from a set x into a value quantale v, we mean a map q : x × x → v such that for each x,y,z ∈ x, q(x,z) ≤ q(x,y) + q(y,z), and q(x,x) = 0. definition 2.2. a v-pseudopmetric space is a pair (x,p), consisting of a set x and a function p : x × x → v satisfying the following conditions: p1) for every x,y ∈ x, p(x,y) ≥ p(x,x), p2) for every x,y ∈ x, p(x,y) = p(y,x), p3) for every x,y,z ∈ x, p(x,z) ≤ p(x,y) + (p(y,z) −̇ p(y,y)). its associated qmetric is qp : x×x → v, defined by qp(x,y) = p(x,y) −̇ p(y,y). the dual of any qmetric is the qmetric defined by q∗(x,y) = q(y,x) (so q∗p(x,y) = p(x,y) −̇ p(x,x)). a v-pmetric is a v-pseudopmetric which also satisfies: p4) for every x,y ∈ x, x = y iff p(x,y) = p(x,x) = p(y,y). definition 2.3. given a v-qmetric q : x × x → v, the ball (or closed ball) about x ∈ x of radius r ∈ v for is the set nr(x) = {y ∈ x|q(x,y) ≤ r}; also, n∗r (x) = {y ∈ x|q ∗(x,y) ≤ r}, and the open ball about x of radius r is br(x) = {y|q(x,y) ≪ r}. 120 r. kopperman, s. matthews and h. pajoohesh a subset u of x is open in x if for each x ∈ u there is an r ≫ 0 such that nr(x) ⊆ u. we write τq, for the collection of all open subsets of a v-qmetric space (x,q), τp = τqp and τp∗ = τ(qp)∗ for a v-pseudopmetric space. theorem 2.4. let (v,≥,+) be a value lattice. (a) if p : x × x → v is a v-pseudopmetric, then qp is a v-qmetric. (b) for each v-qmetric, τq is a topology. for each x ∈ x, {br(x)|r ≫ 0} and {nr(x)|r ≫ 0}, are neighborhood bases for τq at x. for each x ∈ x,r ∈ v, n∗r (x) is a closed set in τq. also, if r ≫ 0 then br(x) is an open set in τq, so in particular, the set of all open balls is a base for the topology τq. proof. (a) certainly, if p : x × x → v is a v-pseudopmetric and q(x,y) = p(x,y) −̇ p(y,y), then q(x,x) = 0. next, we show that if v is a value lattice and a,b,c ∈ v then (a+b) −̇ c ≤ (a −̇ c)+b. by definition of −̇ , a ≤ (a −̇ c)+c. thus for every b ∈ v, a+b ≤ (a −̇ c)+c+b. therefore (a+b) −̇ c ≤ (a −̇ c)+b. now by the above for every x,y,z ∈ x, q(x,z) = p(x,z) −̇ p(z,z) ≤ (p(z,y) + (p(x,y) −̇ p(y,y))) −̇ p(z,z) ≤ (p(x,y) −̇ p(y,y)) + (p(y,z) −̇ p(z,z)) = q(x,y) + q(y,z). (b) these proofs are left to the reader. that τq is a topology, is straightforward (or see [5]). the others use facts about the continuous lattice (v,≥), which generalize those about ii: a ≪ r,s iff r ∧ s ≫ a and r,s ≥ a iff r ∧ s ≥ a. ≫ is interpolative, so if r ≫ 0 then for some s, r ≫ s ≫ 0. by scott continuity of +, if q(x,y) ≪ r then ∧ {q(x,y)+s|s ≫ 0} = q(x,y)+ ∧ {s ≫ 0} = q(x,y) ≪ r so for some s ≫ 0, q(x,y) + s ≪ r, thus if z ∈ ns(y) then z ∈ br(x), so br(x) is open. � theorem 2.5. in any v-qmetric space, x ∈ cl(y) if and only if q(x,y) = 0. in any v-pseudopmetric space, x ∈ cl(y) if and only if p(x,y) = p(y,y). proof. by theorem 2.4, n∗0 (y) is closed, so cl(y) ⊆ n ∗ 0 (y). but if x 6∈ cl(y) then for some open t , x ∈ t and y 6∈ t . by theorem 2.4, nǫ(x) ⊆ t for some ǫ ≫ 0, thus in particular, q(x,y) 6≤ ǫ, so q(x,y) 6= 0. this shows the reverse inclusion, n∗0 (y) ⊆ cl(y). the assertion about v-pseudopmetric spaces is seen by noting that by the above, x ∈ cl(y) if and only if qp(x,y) = 0, and certainly this happens if and only if p(y,y) ≥ p(x,y), so they are equal by p1). � corollary 2.6. for a v-qmetric, τq is t0 if and only if for each x,y ∈ x, q(x,y) = q(y,x) = 0 ⇒ x = y. a v-pseudopmetric p is a v-pmetric if and only if τp is t0. proof. it is known that a topology is t0 if and only if, for each x,y ∈ x, x ∈ cl(y)&y ∈ cl(x) ⇒ x = y. thus by theorem 2.5, τq is t0 if and only if q(x,y) = q(y,x) = 0 ⇒ x = y, and so τp is t0 if and only if p(x,y) = p(y,y) = p(x,x) ⇒ x = y. � lemma 2.7. p(x,y) = max(x,y) is an ii-pmetric on ii and is also a ib-pmetric on ib. also, τp is the scott (or upper) topology, σ = {(x,1]|x ∈ (0,1)}∪{∅,ii}, and τp∗ is the lower topology, ω = {[0,x)|x ∈ (0,1)} ∪ {∅,ii}. partial metrizability in value quantales 121 proof. since for every x,y ∈ ii, max(y,x) = max(x,y) ≥ max(x,x), p satisfies p1 and p2; also p4 is clear. for p3, if y = max(x,y,z) then p(x,z) ≤ y + p(y,z) − p(y,y) = p(x,y) + p(y,z) − p(y,y); if z = max(x,y,z) then p(x,z) ≤ p(x,y) + z − p(y,y) = p(x,y) + p(y,z) − p(y,y) and the case x = max(x,y,z) is similar. also if p(x,x) = p(y,y), then x = y. thus p is a ii-pmetric on ii. now we show that τp = σ and τp∗ = ω: let a ∈ τp. then for each x ∈ a there exists r > 0 (notice that here r ≫ 0 if and only if r > 0), such that nr(x) ⊆ a. thus {y|x − r ≤ y} = nr(x) ⊆ a, hence ↑ (x − r) ⊆ a. also, if ∨ d ∈ a and d is directed, then there is some r > 0 such that nr( ∨ d) ⊆ a, and by properties of ∨ , for some d ∈ d we have ∨ d −̇ r ≤ d, showing a ∈ σ. if x ∈ a ∈ σ, then ↑ x ⊆ a. but x = ∨ (x − 1/n) ∈ a, n ∈ in and {x− 1/n|n ∈ in} is a directed set, thus there is m ∈ in such that x− 1/m ∈ a. so ↑ (x − 1/m) ⊆ a. therefore n1/m(x) ⊆ a and hence a ∈ τp. now let a ∈ τp∗; then for every x ∈ a there is r > 0 such that n ∗ r (x) ⊆ a. thus {y|y < x + r/2} ⊆ {y|y − x ≤ r} = n∗r (x) ⊆ a. so (x− ↑ (x + r/2)) ⊆ a. therefore a ∈ ω. for the reverse inclusion, assume a ∈ ω. then for every x ∈ a there is a ∈ x such that x ∈ (x− ↑ a) ⊆ a. thus x < a and hence r = (a − x)/2 > 0 and x ∈ n∗r (x) ⊆ a. for the assertions about ib, since ib ⊆ ii, p is a ib-pmetric on ib. note that n0(∞) = {∞} and n0(0) = n∞(x) = ib for x ∈ ib, so τp = σ, and a similar proof shows τp∗ = ω. � lemma 2.8. if f : x → y and p is a v-pseudopmetric on y , then pf (x,y) = p(f(x),f(y)) defines a v-pseudopmetric on x. also, f is continuous from (x,τ) to (y,τp), if and only if τpf ⊆ τ. proof. since p is a v-pseudopmetric on y , pf is on x. we distinguish open qpballs in y from open qpf -balls in x, denoting the former by b y r (y), the latter by bxr (x). by definition of pf, b x r (x) = {y|p(f(x),f(y)) −̇ p(f(y),f(y)) ≪ r} = f−1[byr (f(x))]. of course, f is continuous if and only if the inverse image of each set in the base of open qp-balls is open, that is, if and only if each open qpf -ball is open; the latter occurs if and only if τpf ⊆ τ. � recall that a bitopological space (x,τ, τ∗) is completely regular if whenever x ∈ t ∈ τ then there is a pairwise continuous f : (x,τ,τ∗) → (ii,σ,ω), such that f(x) = 1 and f is 0 off t ; it is zero-dimensional if whenever x ∈ t ∈ τ then there is a pairwise continuous f : (x,τ,τ∗) → (ib,σ,ω) such that f(x) = ∞ and f is 0 off t . a bitopological space (x,τ,τ∗) is said to have a property pairwise, if both (x,τ,τ∗) and its dual, (x,τ∗,τ) have the property. theorem 2.9. if (x,τ,τ∗) is completely regular then there is a value lattice v and a v-pseudopmetric such that τ = τp and τ ∗ ⊇ τp∗ . further, if (x,τ,τ∗) is pairwise completely regular then there is a value lattice v and a v-pseudopmetric such that τ = τp and τ ∗ = τp∗ . the analogous result holds for zero-dimensionality in place of complete regularity, with boolean value lattices. 122 r. kopperman, s. matthews and h. pajoohesh conversely, if there is a value lattice and a v-pseudopmetric such that τ = τp and τ∗ ⊇ τp∗, then (x,τ,τ ∗) is completely regular, and converses also hold in the other three cases. throughout the above, p is a v-pmetric if and only if τ is t0. proof. let pc(x,ii) be the collection of all pairwise continuous functions from (x,τ,τ∗) to (ii,σ,ω), and define a = iip c(x,ii). with the pointwise order, a is a value lattice and for φ,ψ ∈ a, φ ≫ ψ if and only if φ(f) > ψ(f), for every f ∈ pc(x,ii) and {f|φ(f) 6= 1} is finite. now define p : x ×x → a such that p(x,y)(f) = max{f(x),f(y)} for every f ∈ pc(x,ii). a coordinatewise proof then shows that p is an a-pseudopmetric on x. now we show that τ = τp. let t ∈ τ. if x ∈ t then there is a pairwise continuous function g : (x,τ,τ∗) → (ii,σ,ω) such that g(x) = 1 and g is 0 off t . now take r ∈ a such that r(g) = 1/2 and r(f) = 1 for f 6= g. then nr(x) = {y|g(y) ≥ 1/2} ⊆ t . thus τ ⊆ τp. now assume that t ∈ τp. for each x ∈ t , there exists r ≫ 0 such that nr(x) ⊆ t . let i = {f ∈ pc(x,ii)|r(f) 6= 1}. then nr(x) = ⋂ {y|fi(x) − fi(y) ≤ r(fi)}, where i ∈ i. hence x ∈ ⋂ {y|fi(x)−fi(y) < r(fi)/2} ⊆ nr(x) ⊆ t , where i ∈ i. therefore x ∈ ⋂ {y|y ∈ f−1i ((fi(x) − r(fi)/2,1])}. since every fi is continuous, {y|y ∈ f −1 i ((fi(x) − r(fi)/2,1])} ∈ τ and since i is finite, ⋂ {y|y ∈ f−1i ((fi(x) − r(fi)/2,1])} ∈ τ. thus t ∈ τ and by the above we now have τ = τp. similarly τp∗ ⊆ τ ∗. for zero-dimensionality replace a by ibp c(x,ib) and proceed as in the completely regular case. in the “pairwise” cases, the above p satisfies τ = τq and τp∗ = τ ∗. for the converses, consider the relation ⊳p on subsets of x, defined by s ⊳p t ⇔ (∃r ≫ 0)(nr(s) ⊆ t), where nr(s) is defined to be ⋃ x∈s nr(x). then ⊳p can easily be seen to satisfy the properties (a1)-(a3) of an auxiliary relation given below (where (a3) results from the interpolation property of the waybelow relation ≫ in the continuous lattice (v,≥)). further, it is clear that r,s ⊳p t ⇒ r ∪ s ⊳p t , r ⊳p s,t ⇒ r ⊳p s ∩ t , ∅ ⊳p ∅ and x ⊳p x. these are the defining properties of a quasiproximity (see [3]). each quasiproximity ⊳ has a dual, ⊳∗, defined by s ⊳∗ t ⇔ x \ t ⊳x \ s, and gives rise to a topology, τ⊳ = {t |x ∈ t ⇒ {x} ⊳ t}. certainly, τp = τ⊳p and τp∗ = τ⊳∗p. in the reference just mentioned, urysohn’s lemma is shown for quasiproximities; thus, if s ⊳ t then there is a pairwise continuous function f : (x,τ⊳,τ⊳∗) → ([0,1],σ,ω) such that f[s] = {1}, f[x \ t ] = {0}, and as a result, if x ∈ t ∈ τp then, letting s = {x}, there is a pairwise continuous function f : (x,τp,τp∗ ) → ([0,1],σ,ω) such that f(x) = 1 and f[x \ t ] = {0}; the same then holds for (x,τp∗,τp) using ⊳∗. this yields the results for complete regularity and pairwise complete regularity, and those involving 0-dimensionality are simpler, since in this case, for each x ∈ x, r ≫ 0, the function defined by f(y) = { 1 y ∈ nr(x) 0 y 6∈ nr(x) , is pairwise continuous from (x,τp,τp∗) to ({0,1},σ,ω). the last statement is immediate from corollary 2.6. � if ≤ is a transitive, reflexive relation on x (= a pre-order) then the alexandroff topology is α(≤) = {t ⊆ x|x ∈ t & x ≤ y ⇒ y ∈ t}. partial metrizability in value quantales 123 theorem 2.10. (a) for any topology τ, (x,τ,α(≥τ )) is pairwise 0-dimensional. thus there is a boolean value lattice v and a v-pseudopmetric such that τ = τp. (b) for each continuous bounded dcpo, there is a value lattice v and a vpmetric such that its scott topology, σ, is τp and its lower topology, ω, is τp∗ . if the dcpo is algebraic as well, then v can be assumed boolean. proof. (a) notice that u ∈ α(≥τ) if and only if x ∈ u and y ∈ cl({x}) imply that y ∈ u. now consider x ∈ t ∈ τ then define f : (x,τ,α(≥τ)) → (ib,σ,ω) such that f = ∞ on t and f = 0 on x \t . since t ∈ τ implies x \t ∈ α(≥τ ), f is pairwise continuous. now let x ∈ u ∈ α(≥τ ). since (x \ cl({x})) ∈ τ, cl({x}) ∈ α(≥τ). define f : (x,α(≥τ),τ) → (ib,σ,ω) by f = ∞ on cl({x}) and f = 0 on x \ cl({x}). by the construction, f is pairwise continuous. (b) for each continuous bounded dcpo, (p,σ,ω) is pairwise completely regular and σ is t0 (see [3]); additionally if p is algebraic, then (p,σ,ω) is pairwise 0-dimensional, using the fact that for compact x, ↑ x =⇑ x, so a base for the open sets in σ is a subbase for the closed sets in ω. � 3. value quantales first, we recall the definition and a few basic properties of value quantales from [2]. assume v is a complete lattice. then v is completely distributive if for any family {xi,j | j ∈ j,k ∈ kj} of elements of v , ∧ j∈j ∨ k∈kj xj,k = ∨ f∈m ∧ j∈j xj,f(j), where m = ∏ j∈j kj. assume v is a complete lattice and p,q ∈ v . then q is well above p, denoted by q ≻ p, iff for any subset s ⊆ v , if p ≥ ∧ s, then for some r ∈ s, q ≥ r. raney ([8]) has shown that a complete lattice is completely distributive if and only if each p = ∧ {q | q ≻ p }. this is the criterion we shall use below. the “way-above” relation ≫ for a continuous lattice (l,≥) and ≻ on a completely distributive lattice (v,≥) differ in that for the former (unlike the latter), the set s must be directed (by ≥). like ≫, ≻ satisfies most of the axioms for an auxiliary relation, ⊲ on a poset (p,≥): if p,q,r,s ∈ p and s ⊆ p , • (a1) q ⊲ p implies q ≥ p; • (a2) s ≥ q, q ⊲ p and p ≥ r implies s ⊲ r; and • (a3) (interpolation property) if q ⊲ p, then for some r, q ⊲ r and r ⊲ p. property (a3) is more special, and holds for ≫ in any continuous lattice (l,≥) ([4]), and for ≻ in any completely distributive lattice (v,≥) ([8]). the definitions of completely distributive lattice (for ≻) and continuous lattice (for ≫) amount to the statement that the relation is approximating: • (a4) each p is the inf, ∧ {r|r ⊲ p}. for an arbitrary set in a completely distributive lattice, q ≻ ∧ s iff for some r ∈ s, q ≥ r, and for directed set d in a continuous lattice, that q ≫ ∧ d iff for some r ∈ d, q ≥ r. 124 r. kopperman, s. matthews and h. pajoohesh however, many auxiliary relations, like ≫, are subdirecting : for each x ∈ x, {y|y ⊲ x} is directed by ≥. this need not hold for ≻; for example, in ibf r ≻ 0 if and only if, there is at most one f such that r(f) < ∞, and this collection is clearly not directed. but we need this property for 0: a value distributive lattice is a completely distributive lattice v satisfying the following two conditions: • (v1) ∞ 6= 0. • (v2) if p ≻ 0 and q ≻ 0, then p ∧ q ≻ 0. a value quantale v =< v,≤,+ > consists of a value distributive lattice < v,≤> and an operation + on v satisfying definition 1. we now describe a special case of the construction of value quantale from [2]. assume (l,≥) is a continuous lattice in which ⊥6= ⊤, and let a = a(l) = {x ∈ l | x ≫ ⊥}, where ≫ denotes the way-above relation on l. then ⊤ ∈ a and if a,b ∈ a, then a∧b ∈ a. by a round upper set in l, we mean a nonempty i ⊆ a for which: (r1) j ≥ k ∈ i ⇒ j ∈ i, and (r2) (∀j ∈ i) ⇓ j ∩ i 6= ∅. (note in particular, that we do not require that i be directed by ≥.) let r = (r[l],⊇) denote the poset of round upper sets in l, with reverse set inclusion. since r is an inf-closed subset of (2a,⊇)(∼= (ib a,≥)), r is a complete lattice with ∧ s = ⋃ s ∪ {⊤}. for a ∈ l, define θ(a) = {x ∈ a|x ≫ a}. then θ(a) ∈ r and for a ∈ a, and i ∈ r, notice that a ∈ i ⇒ θ(a) ≻ i, since if a ∈ i ≥ ∧ s then a ∈ i ⊆ ⋃ s so for some j ∈ s, a ∈ j, showing θ(a) ⊆ j, thus θ(a) ≥ j. thus i = ⋃ {θ(a)|a ∈ i} ⊆ ⋃ {j ≻ i} ⊆ ⋃ {j ⊆ i} = i, so in particular, each i = ∧ {j ≻ i}, thus r is completely distributive by raney’s result. also note that if j ≻ i then for some a ∈ i, j ≥ θ(a), that is, j ⊆ θ(a); this with the previous paragraph shows j ≻ i ⇔ (∃a ∈ i)j ⊆ θ(a). here are some other properties of θ that we need later: (θ1) θ preserves direct inf: for let d be directed by ≥. then for x ∈ a, x ∈ θ( ∧ d) ⇔ x ≫ ∧ d ⇔ for some y ∈ l, x ≫ y ≫ ∧ d ⇔ for some z ∈ l there is a d ∈ d such that x ≫ z ≥ d ⇔ there is a d ∈ d such that x ≫ d ⇔ x ∈ ⋃ d∈d θ(d) = ∧ θ[d]. hence ∧ θ[d] = θ( ∧ d) as required. (θ2) for each a,b ∈ l, a ≥ b ⇔ θ(a) ≥ θ(b): if a ≥ b then b = ∧ {a,b} a directed set, so by θ1, θ(b) = ∧ {θ(a),θ(b)} ≤ θ(a). conversely, if θ(a) ≥ θ(b), then {x|x ≫ a} ⊆ {x|x ≫ b}, so ∧ {x|x ≫ a} ≥ ∧ {x|x ≫ b}. but since l is a continuous lattice, a = ∧ {x|x ≫ a} and b = ∧ {x|x ≫ b}. hence a ≥ b. in r, clearly the smallest element, called 0, is a and the largest, ∞, is {⊤}. the two differ, since ⊥6= ⊤, so by (a4), for some a 6= ⊤, a ≫⊥; thus a ∈ a so 0 6= {⊤} = ∞. note also that if i,j ≻ 0 then for some a,b ∈ a we have i ⊆ θ(a), j ⊆ θ(b) so i,j ⊆ θ(a∧b), and a∧b ∈ a, showing that i,j ≥ θ(a∧b) ≻ 0. thus {i ≻ 0} is ≥-directed, so that r is a value distributive lattice in the terminology of [2]. partial metrizability in value quantales 125 now suppose that ⋆ : l × l → l is a binary operation on l such that (l,⋆,⊥) is a commutative monoid and for any a ∈ l, the function a⋆ : l → l preserves infs and the way above relation; that is, any indexed family {bi}i∈i in l, a⋆ ∧ i∈i bi = ∧ i∈i(a⋆bi), and whenever b ≫ b ′, then a⋆b ≫ a⋆b′. then ⋆ : a × a → a. for i,j ∈ r, define i + j = {a|(∃x ∈ i,y ∈ j)a ≥ x ⋆ y}. clearly, + is associative, commutative and monotone. also 0 is a unit for +, because of the following: if a ∈ i + a then for some x ∈ i and y ∈ a, a ≥ x ⋆ y. thus a ≥ x ⋆ y ≥ x⋆ ⊥= x, and since i is an upper set, a ∈ i. for the reverse set inclusion, if a ∈ i then by (r2) there is m ∈ i such that a ≫ m. thus a ≫ m = m⋆ ⊥= m ⋆ ∧ k≫⊥ k = ∧ k≫⊥ m ⋆ k. thus there is k0 ≫⊥ such that a ≥ m ⋆ k0. hence a ∈ i + a. thus i + a = i for every i. note also that for s ⊆ r and i ∈ r, i + ∧ s = i + ⋃ (s ∪ {⊤}) = {a|(∃x ∈ i,y ∈ j ∈ s ∪ {⊤})(a ≥ x ⋆ y)} = ⋃ j∈s∪{⊤}{a|(∃x ∈ i,y ∈ j)(a ≥ x ⋆ y)} = ⋃ j∈s∪{⊤}(i + j) = ∧ j∈s(i + j), so (r,+) is a value quantale. further: (θ3) each θ(a ⋆ b) = θ(a) + θ(b). for if t ∈ θ(a) + θ(b) then (∃x ∈ θ(a),y ∈ θ(b))t ≥ x⋆y thus t ≥ x⋆y ≫ a⋆b, hence t ∈ θ(a⋆b). but if t ∈ θ(a⋆b). then t ≫ a ⋆ b = ∧ {x ⋆ y|x ≫ a,y ≫ b}. thus by definition of ≫ there are w ≫ a and v ≫ b such that t ≥ w ⋆ v. therefore t ∈ θ(a) + θ(b). as a result of this and θ2, we also have: (θ4) for each a,b,c ∈ l, a − ⋆b ≤ c ⇔ θ(a) −̇ θ(b) ≤ θ(c), since a −⋆b ≤ c ⇔ a ≤ b ⋆ c ⇔ θ(a) ≤ θ(b ⋆ c) = θ(b) + θ(c) ⇔ θ(a) −̇ θ(b) ≤ θ(c) we have two special cases in mind: assume k is a nonempty set and let if be (ii,≤,+) (with truncated addition + as introduced preceding definition 1) or (ib,≤,+). let k be any nonempty set. then as a product of continuous lattices, l = ifk is also a continuous lattice with the pointwise order, and in it, a ≫⊥ if and only if for each i ∈ k, a(i) ≫⊥i and for all but a finite number of i, a(i) = ⊤i. thus in particular, for if = ib, a = {r ∈ ib k|r−1[{∞}] is cofinite}, and for if = ii, a = {r ∈ [0,1)k|r−1[{1}] is cofinite}. let ⋆ be pointwise addition; certainly ⋆ is in both cases an associative, commutative operation preserving ≤ and ≫, for which 0 is the unit, since these all hold coordinatewise. thus ⋆ obeys the assumptions made of it, so (r[ifk],+) is a value quantale. following [2], we call the r[ibk] the value quantales of subsets, and denote them by γ(k), and we call the r[iik] the value quantales of fuzzy subsets, and denote them by λ(k). one special property of γ(k) worth noting is that (r2) is trivial, since for r ∈ a, it is easy to see that r ≫ r; for λ(k) it is worth noticing for (r2) that for r,s ∈ a, r ≫ s if and only if r(i) > s(i) whenever r(i) 6= 1. theorem 3.1. in theorem 2.9 and its corollaries, “value lattice” can be improved to “value quantale”. proof. by theorem 2.9, for k = pc(x,if) there is a k-pseudopmetric p such that τ = τp and τ ∗ ⊇ τp∗ . let θ : k → r[if k] be as defined above. in addition to the properties already established, notice that θ(0) = 0 and θ(∞) = ∞. 126 r. kopperman, s. matthews and h. pajoohesh we finish the proof by showing that for the r[ifk]-pseudopmetric d = θ ◦ p, τd = τp and τd∗ = τp∗ . for suppose t ∈ τp; if x ∈ t then for some r ≫ 0, nr(x) ⊆ t . but r ∈ a(ifk) so θ(r) ≻ a(ifk), the 0 of r[ifk]. further, if y ∈ nθ(r)(x), we have qd(x,y) ≤ θ(r) so θ(p(x,y)) −̇ θ(p(y,y)) ≤ θ(r), thus by (θ4), p(x,y) −̇ p(y,y) = qp(x,y) ≤ r, showing y ∈ t . this shows nθ(r)(x) ⊆ t , so t ∈ τd. conversely, suppose t ∈ τd; if x ∈ t then for some s ≻ 0, s ∈ r[if k], ns(x) ⊆ t . by the beginning of the discussion of θ there is an r ∈ k, r ≫ 0, so that θ(r) ≤ s. if qp(x,y) ≤ r then p(x,y) −̇ p(y,y) ≤ r, thus θ(p(x,y)) −̇ θ(p(y,y)) ≤ θ(r) ≤ s, so y ∈ t . this shows nr(x) ⊆ t , so t ∈ τp. the above show that τp = τd, and a similar proof shows that τp∗ = τd∗. this completes the proof that throughout theorem 2.9, the value lattice ifk and pseudopmetric p can be replaced by the value quantale r[ifk] and pseudopmetric d. � references [1] k. ciesielski, r.c. flagg, and r.d. kopperman, characterizing topologies with bounded complete computational models, electron. notes theor. comput. sci. 20 (1999), 11 pages. [2] r.c. flagg and r.d. kopperman, continuity spaces: reconciling domains and metric spaces, theor. comput. sci. 177 (1997), 111-138. [3] r.c. flagg and r.d. kopperman, tychonoff poset structures and auxiliary relations, ann. new york acad. sci. 767 (andima et. al., eds.) (1995), 45–61. [4] g.k. gierz, k.h. hofmann, k. keimel, j.d. lawson, m.w. mislove and d.s. scott, a compendium of continuous lattices, springer-verlag, berlin, 1980. [5] r. d. kopperman, all topologies come from generalized metrics, am. math. monthly 95 (1988), 89–97. [6] matthews, s.g., an extensional treatment of lazy data flow deadlock, theoretical computer science, 151, (1995), 195–205. [7] matthews, s.g., partial metric topology, proc. 8th summer conference on topology and its applications, ed s. andima et al., annals of the new york academy of sciences, new york, 728, (1994) 183–197. [8] g.n. raney, completely distributive lattices, proc. amer. math. soc., 3 (1952), 677– 680. [9] stoy, joseph e., denotational semantics: the scott-strachey approach to programming language theory, the mit press, cambridge, massachusetts, and london, england, 1977 [10] wadge, w.w., an extensional treatment of dataflow deadlock, theoretical computer science, 13(1), (1981) 3–15. received march 2003 accepted june 2003 r. d. kopperman (rdkcc@cunyvm.cuny.edu) partial metrizability in value quantales 127 department of mathematics, city college, city university of new york, new york, ny 10031, usa. department of computer science, university of birmingham, uk. s. g. matthews (sgm@dcs.warwick.ac.uk) department of computer science, university of warwick coventry, cv4 7al, uk. h. pajoohesh (h pajoohesh@yahoo.com) department of mathematics shahid beheshti university teheran 19839, iran. bcri and ceol, university college cork, (unit 2200, cork airport business park, kinsale road, co. cork), ireland department of computer science, university of birmingham, uk. @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 223–241 extensions of closure spaces d. deses ∗ , a. de groot-van der voorde, e. lowen-colebunders dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. a closure space x is a set endowed with a closure operator p(x) →p(x), satisfying the usual topological axioms, except finite additivity. a t1 closure extension y of a closure space x induces a structure γ on x satisfying the smallness axioms introduced by h. herrlich [?], except the one on finite unions of collections. we’ll use the word seminearness for a smallness structure of this type, i.e. satisfying the conditions (s1),(s2),(s3) and (s5) from [?]. in this paper we show that every t1 seminearness structure γ on x can in fact be induced by a t1 closure extension. this result is quite different from its topological counterpart which was treated by s.a. naimpally and j.h.m. whitfield in [?]. also in the topological setting the existence of (strict) extensions satisfying higher separation conditions such as t2 and t3 has been completely characterized by means of concreteness, separatedness and regularity [?]. in the closure setting these conditions will appear to be too weak to ensure the existence of suitable (strict) extensions. in this paper we introduce stronger alternatives in order to present internal characterizations of the existence of (strict) t2 or strict regular closure extensions. 2000 ams classification: 54a05, 54d35, 54e15, 54e17, 54d10. keywords: closure space, seminearness, separation, regularity, (strict) extension, minimal small stack. 1. introduction. the structures we will be dealing with, can be defined in various equivalent ways, from which we shall use frequently two particular descriptions, namely by small collections and by uniform covers. ∗the first author is ‘aspirant’ of the f.w.o.-vlaanderen. 224 d. deses, a. de groot-van der voorde, e. lowen-colebunders 1.1. let x be a set. a nonempty collection a ⊂ p(x), not containing ∅ is said to be a stack if a ∈ a whenever there exists b ∈ a with b ⊂ a. if a⊂p(x) is an arbitrary collection of nonempty subsets then we put stack a = {a ⊂ x|∃b ∈a : b ⊂ a} and sec a = {a ⊂ x|∀b ∈a : a∩b 6= ∅} we write ẋ for stack {{x}}. in [?] h. herrlich considered the following smallness axioms, which can be expressed in terms of stacks in the following way. for γ ⊂p2(x), a collection of stacks, consider the conditions (s1) if a⊂b and a∈ γ then b ∈ γ (s2) ∀x ∈ x : ẋ ∈ γ (s3) γ 6= p2(x) (s4) if (a∪b) ∈ γ then a∈ γ or b ∈ γ (s5) if sec{cl a|a ∈a}∈ γ then seca∈ γ where cl a = {x ∈ x|sec {a,{x}}∈ γ} a structure γ satisfying (s1),(s2),(s3) is called a prenearness structure, if (s4) is added γ is a merotopic structure and if γ satisfies all five of the conditions then it is a nearness structure. we will not be dealing with axiom (s4) but we will assume that γ satisfies (s1), (s2), (s3) and (s5). then γ is called a seminearness structure and (x,γ) is a seminearness space. as in [?] a function f : (x,γ) → (x′,γ′) between seminearness spaces is said to be uniformly continuous if it preserves smallness in the sense that a∈ γ ⇒ stack {f(a)|a ∈a}∈ γ′ seminearness spaces and uniformly continuous maps form a topological construct in the sense of [?]. we refer to the original papers [?], [?] for a systematic study of prenearness and nearness spaces. on the latter a selfcontained textbook ”uniforme raüme” appeared [?]. another textbook by g. preuss [?] also contains an introduction to nearness spaces and to some of the more general structures such as prenearness and merotopic structures. in [?] however the latter are called seminearness spaces, so that the terminology used in [?] differs from the one we use here. 1.2. in [?] an equivalent way of describing the structure was presented in terms of uniform covers. if x is a set then the following conditions on µ ⊂p2(x) are considered (u1) if u ≺ v and u ∈ µ then v ∈ µ (where ≺ denotes the classical refinement relation) (u2) if u ∈ µ then u is a cover of x (u3) ∅ 6= µ 6= p2(x) (u4) if u ∈ µ and v ∈ µ then {u ∩v |u ∈u,v ∈v}∈ µ (u5) if u ∈ µ then {intµu|u ∈u}∈ µ where intµu = {x ∈ x|{u,x −{x}}∈ µ} the covers in µ are called uniform covers. in our setting we will not be dealing with (u4), so our covering structures µ satisfy (u1), (u2), (u3) and extensions of closure spaces 225 (u5). (x,µ) then forms an equivalent way for the description of a seminearness space and the translation between (x,γ) and (x,µ) is as usual: u ∈ µ ⇐⇒ ∀a∈ γ : u ∩a 6= ∅ a∈ γ ⇐⇒ ∀u ∈ µ : u ∩a 6= ∅ 1.3. some of the examples we will construct in the last section of the paper, satisfy even stronger conditions than the seminearness axioms. a uniform space, described in terms of covers satisfies (u1), (u2), (u3), (u4) and the condition (u5’), saying that every uniform cover has a uniform star refinement, which is in fact stronger than (u5). if we leave out (u4), as we did before, and retain (u1), (u2), (u3) and (u5’) then we still have a (covering) seminearness space. in this case we will say that the seminearness space is a uniform seminearness space . a special case of this situation is the following. let x be a set and let {ui|i ∈ i} be any collection of partitions of x then u ∈ µ ⇐⇒ ∃i ∈ i : ui ≺u defines a (covering) uniform seminearness structure on x. it is said to be zero dimensional since it is generated by a collection of partitions. these structures are investigated in more detail in [?]. 1.4. a closure space (x,c) is a pair, where x is a set and c is a subset of the power set p(x) satisfying the conditions that x belongs to c and that c is closed for arbitrary unions. the sets in c are called open sets. a function f : (x,c) → (y,d) between closure spaces (x,c) and (y,d) is said to be continuous if f−1(d) ∈c whenever d ∈d. cl is the construct with closure spaces as objects and continuous maps as morphisms. some isomorphic descriptions of cl are often used f.i. by giving the collection of all closed sets (the so called moore family [?]) where, as usual, the closed sets are the complements of the open ones and continuity is defined accordingly. another isomorphic description is obtained by means of a closure operator [?]. the closure operation cl : p(x) → p(x) associated with a closure space (x,c) is defined in the usual way by x ∈ cl a ⇐⇒ (∀c ∈ c : x ∈ c ⇒ c ∩ a 6= ∅) where a ⊂ x and x ∈ x. this closure need not be finitely additive, but it does satisfy the conditions cl ∅ = ∅, (a ⊂ b ⇒ cl a ⊂ cl b),a ⊂ cl a and cl(cl a) = cl a whenever a and b are subsets of x. continuity is then characterized in the usual way. finally closure spaces can also be equivalently described by means of neighborhood collections of the points. these neighborhood collections satisfy the usual axioms, except for the fact that the collections need not to be filters. so in a closure space the neighborhood collection v(x) of a point x is a stack, where every v ∈v(x) contains x and v(x) satisfies the open kernel condition. in the sequel we will just write x for a closure space and we’ll choose the most convenient form for its explicit structure. 226 d. deses, a. de groot-van der voorde, e. lowen-colebunders motivations for considering closure spaces can be found in several applications. we refer to [?] and [?] for applications in geometry, to [?] for applications in lattice theory, to [?], [?] and [?] for the use of closures in the development of representations of physical systems, to [?] for the use in social sciences and to [?] for applications in the context of knowledge representation. the introduction of [?] contains some more details on motivation. a closure space satisfies the r0 symmetry axiom if x ∈ cl {y} ⇐⇒ y ∈ cl {x}, ∀x,y ∈ x and it satisfies t1 if {x} is closed for every x ∈ x. if (x,γ) is a seminearness space then the closure defined in paragraph 1.1 by cl a = {x ∈ x|sec {a,{x}}∈ γ} is an r0 closure in our sense. this closure is the underlying closure of (x,γ) and we also say that it is compatible with (x,γ). whenever we consider neighborhood collections vγ(x), convergence or open sets for a seminearness space (x,γ), we are in fact referring to the underlying closure. as for nearness spaces we have in this more general context that the neighborhood collections vγ(x) are minimal small stacks (where minimality refers to the inclusion order). next we further illustrate the relation between r0 closure spaces and seminearness spaces. 1.5. let y be an r0 closure space and define a seminearness structure by a∈ γ ⇐⇒ ∃y ∈ y : v(y) ⊂a remark that the underlying closure of the seminearness γ coincides with the given closure on y . the construct of r0 closure spaces is bicoreflectively embedded in the construct of seminearness spaces, cfr. [?], [?]. 1.6. let y be an r0 closure space and let x be a subset of y . the closure structure of y induces a seminearness structure on x as follows a∈ γ ⇐⇒ ∃y ∈ y : v(y) ⊂ stackya the underlying closure of γ on x coincides with the closure structure induced by y on x. if x is dense in y (cly x = y ) then y is said to be a closure extension of x and we say that (x,γ) is induced by the extension y of x. x is said to be strictly dense in y if {cly b|b ⊂ x} is a base for the closed subsets of y , in the sense that every closed set of y can be obtained by intersecting sets from the base. in that case y is said to be a strict extension of x and (x,γ) is said to be induced by a strict extension. the meaning of 1.6 is that, given an r0 closure extension y of a closure space x, a seminearness structure γ is induced on x which is compatible with the given closure on x. the first question we will be dealing with in this paper is, whether every seminearness γ compatible with x as a closure space, can be induced by some r0 closure extension y of x. extensions of closure spaces 227 the parallel question in the setting of topological spaces is whether every compatible nearness space can be induced by some r0 topological extension. this question was answered negatively by s.a. naimpally and j.h.m. whitfield in [?]. a thorough study on extensions of topological spaces was later carried out by h.l. bentley and h. herrlich in [?], in particular giving internal characterizations for nearness spaces to be induced by t1, t2 or t3 (strict) extensions. in this paper we’ll deal with the closure counterparts of such questions. 2. extensions. in this section, starting from a seminearness space (x,γ) we construct two types of enlargements, one type are the so called ”loose” enlargements and the other type is a strict one. 2.1. construction of a loose enlargement. let (x,γ) be a seminearness space and let {ya|a∈ α} be a collection of points, not belonging to x and in one to one correspondence to a collection α of nonconvergent small stacks with an open base. let x′ = x ∪{ya|a∈ α}. on x′ we define a closure structure cl′ by determining the neighborhood collections of the points as follows: v′(x) = stackx′vγ(x) for x ∈ x v′(ya) = stackx′a∩ ẏa for a∈ α clearly (x′,cl′) is an r0 closure space of which x is a dense subset. moreover if d is a stack on x and stackx′d converges in x′, then d is small in (x,γ). it is clear that in order to obtain an extension of (x,γ) in the sense of 1.6 the condition d ∈ γ ⇒∃a∈ α : a⊂d has to be fulfilled. the following proposition is relevant in this respect since it shows that in fact openbased stacks determine the structure. proposition 2.1. if (x,γ) is a seminearness space then for every a∈ γ there exists a b ∈ γ such that b has an open base and such that b ⊂a. proof. for a ∈ γ let b = stack {b ⊂ x|b open,b ∈ a}. if u ∈ µ then so is {int u|u ∈u}. so finally b∩u 6= ∅. � proposition 2.2. let (x,γ) be a seminearness space and let α = {a∈p2(x)|a is a small openbased nonconvergent stack} then (x′,γ′) is an r0 closure extension of (x,γ). we conclude from this fact that every seminearness space can be induced by an r0 closure extension. remark that by exactly the same construction one has that every t1 seminearness space is induced by some t1 extension. remark also that these results deviate from their well known topological counterparts, cfr. [?], [?], [?]. in general, given a t1 space (x,γ) there can be many different t1 extensions inducing γ. on the other hand, as we will see, strict t1 extensions need not exist. however, if there exist t1 strict extensions, then they are essentially 228 d. deses, a. de groot-van der voorde, e. lowen-colebunders unique. the reason for this is explained in the next results on minimal small stacks, where again minimality refers to the inclusion order on stacks. the following result is quite parallel to its topological counterpart developed in [?]. proposition 2.3. (1) if (x,γ) is a seminearness space induced by an r0 extension y of x, then every minimal small stack is a trace v(y)|x for some y ∈ y . (2) if (x,γ) is a seminearness space induced by a strict r0 extension y of x then {v(y)|x|y ∈ y} is the collection of all minimal small stacks. proof. (1) if m is minimal small and stackym converges to y ∈ y then v(y)|x ⊂ m and hence v(y)|x = m. (2) let y ∈ y be some point of a strict extension y , and suppose that a ⊂ v(y)|x and that a is small. suppose that stackya converges to z, i.e. v(z)|x ⊂a. it follows that z ∈ cly{y}. indeed otherwise there would exist a subset b ⊂ x such that y ∈ cly b and z 6∈ cly b. and this is impossible. hence we can conclude that v(y) = v(z) and so a = v(y)|x. � corollary 2.4. if (x,γ) is a seminearness space and y is a strict t1 extension then the points of y are in one to one correspondence to the minimal small stacks in γ. also the following construction is quite similar to its topological counterpart [?],[?]. 2.2. construction of a strict enlargement. let (x,γ) be a seminearness space and let x̂ = x ∪{ym|m nonconvergent minimal small} where again different points ym are chosen to be outside of x and in one to one correspondence with the minimal small nonconvergent stacks. for a ⊂ x put o(a) = int a∪{ym|a ∈m} and let ĉl on x̂ be the closure having as an open base {o(a)|a ⊂ x}. remark that {ĉl k|k ⊂ x} is a base for the closed sets where ĉl k = cl k ∪{ym|k ∈ sec m}. clearly (x̂, ĉl) is an r0 closure space and it contains x as a strictly dense subset. moreover if d is a stack on x and stack x̂ d converges in x̂ then: either v x̂ (ym) ⊂ stack d for m minimal small and not convergent, then we extensions of closure spaces 229 have m ⊂ d. or v x̂ (x) ⊂ stack d and then vx(x) ⊂ d. so in any case d ∈ γ. in order to obtain an extension of (x,γ) we need to impose the following condition (cfr. [?]). definition 2.5. a seminearness space is concrete if the minimal small stacks determine the structure in the following sense ∀a∈ γ : ∃m minimal small m⊂a proposition 2.6. (x,γ) is induced by a strict r0 extension if and only if it is a concrete seminearness space. proof. if (x,γ) is a concrete seminearness space we make the construction developed in paragraph 2.2 and we prove that (x̂, ĉl) is an extension. so it remains to show that if d is a small stack on x it converges in (x̂, ĉl). choose m⊂d minimal small. either m = v(x) for some x ∈ x and then for a ⊂ x with x ∈ int a we have int a ∈d and hence o(a) ∈d. so we have v x̂ (x) ⊂d. or m does not converge. then we prove that stack x̂ d ⊃v x̂ (ym) let a ⊂ x such that ym ∈ o(a), i.e. a ∈ m. in view of proposition 2.1 the stack m has an open base. then also int a ∈ m and finally int a ∈ d. so again we can conclude that o(a) ∈ stack x̂ d. conversely, suppose that (x,γ) has a strict r0 extension (y,cly ). let d be small in (x,γ) then stackyd converges to some y ∈ y . then clearly d ⊃vy (y)|x and in view of proposition 2.3 we have that vy (y)|x is a minimal small stack. hence (x,γ) is concrete. � remark that using exactly the same construction one has that (x,γ) is induced by a strict t1 closure extension if and only if it is t1 and concrete. remark that if y is a strict t1 extension of (x,γ) then y is unique up to an isomorphism leaving x pointwise fixed. it can easily be seen that the function φ : y → x̂ mapping y ∈ y to ym with m = vy (y)|x if y 6∈ x and mapping x ∈ x to x, is bijective and satisfies φ(cly b) = ĉl b, for every b ⊂ x. therefore we also have φ(cly z) = ĉl(φ(z)) for every z ⊂ y . the previous results on r0 and t1 strict extensions are completely analogous to their topological counterparts. in the next section, where higher separation is considered, the parallelism with the topological situation does not go through. 3. separation and extensions. in this section we introduce higher separation conditions for seminearness spaces. the notion ”separatedness” was introduced in [?] in the setting of prenearness spaces and it proved to be very useful in the study of topological extensions. however, in our setting, in order to produce hausdorff closure 230 d. deses, a. de groot-van der voorde, e. lowen-colebunders extensions, ”separatedness” will no longer be strong enough. we briefly recall some definitions and results. if (x,γ) is a seminearness space then a stack a is said to be near if sec a is small. for instance, if ⋂ a∈aclγa 6= ∅ then a is near. a stack a is said to be concentrated if it is small and near. for example, the neighborhood collections in (x,γ) are concentrated. small filters are also always concentrated. definition 3.1. [?] a seminearness space (x,γ) is separated if for every concentrated stack a also b = {b ⊂ x|stackx{b}∪a near} is near. the proof of the following proposition is similar to the one of proposition 10.5 in [?] and can be found in [?]. proposition 3.2. for a seminearness (x,γ) the following are equivalent (1) (x,γ) is separated (2) every concentrated stack contains a unique minimal small stack proposition 3.3. if (x,γ) is separated, m is a minimal small concentrated stack and m6= v(x) then ∃a ∈v(x) : ∃m ∈m : m ∩a = ∅ proof. if on the other hand every a ∈ v(x) intersects every m ∈ m then v(x) ∪m would be concentrated and then v(x) = m in view of the previous proposition. � corollary 3.4. if (x,γ) is separated and t1 then in the underlying closure, distinct points have disjoint neighborhoods. in particular a closure space (considered as a seminearness space) is separated and t1 if and only if distinct points have disjoint neighborhoods. we use the label ”hausdorff” or t2 for this property. the following conditions (i) and (ii) clearly are strengthening those formulated in proposition 3.3. proposition 3.5. for a seminearness space (x,γ) the following are equivalent (i) for m and n minimal small concentrated stacks, m6= n then ∃m ∈m : ∃n ∈n : n ∩m = ∅ (ii) for m and n minimal small stacks, m6= n then ∃m ∈m : ∃n ∈n : n ∩m = ∅ proof. that (i) implies (ii) follows from the observation that when n is small but not concentrated, then sec n is not small. so if m is small we have m6⊂ sec n . therefore m and n contain disjoint sets. � definition 3.6. a seminearness space (x,γ) satisfies (s) if it fulfills one (and hence both) of the conditions formulated in proposition 3.5 extensions of closure spaces 231 remark that if (x,γ) satisfies (s) and m and n are different minimal small stacks, m ∈m and n ∈n satisfying (ii) can be taken to be disjoint and open. hence in that case we have ∀x ∈ x : m 6∈ v(x) or n 6∈ v(x) next we generalize these ideas in order to introduce an even stronger separation condition. let (x,γ) be a seminearness space and let σ = {ẋ|x ∈ x}∪{m| minimal small nonconvergent} definition 3.7. subsets d and b are said to be γ-disjoint if ∀p ∈ σ : d 6∈ p or b 6∈ p clearly d and b are γ-disjoint if and only if (i) d ∩b = ∅ (ii) for every m minimal small nonconvergent stack, d 6∈m or b 6∈m. definition 3.8. a seminearness space (x,γ) is said to satisfy (t) if minimal small stacks m6= n contain sets m ∈m and n ∈n that are γ-disjoint. conditions (s) and (t) will play an important role in the investigation of hausdorff closure extensions. first we discuss the relation between the various separation conditions. proposition 3.9. (1) in a seminearness space we have (t) ⇒ (s) (2) in a concrete seminearness space we have (t) ⇒ (s) ⇒ separated proof. (1) let m and n be (concentrated) minimal small, choose γ-disjoint sets m ∈m and n ∈n . then we have m ∩n = ∅. (2) (x,γ) is concrete and satisfies (s). let a be concentrated and let m be a minimal small stack, m⊂a. consider sec a which is small and a minimal small stack n ⊂ sec a. it follows that m = n . finally by proposition 3.2 the space (x,γ) is separated. � from the proof of (2) we immediately have the following. corollary 3.10. if (x,γ) is concrete and satisfies (s) then every minimal small concentrated stack m satisfies m⊂ sec m i.e. m is a linked system in the sense of [?]. 232 d. deses, a. de groot-van der voorde, e. lowen-colebunders no other implications between the conditions (t), (s) and ”separated” are true in general (except those obtained by transitivity). we refer to section ?? for the summarizing diagrams. example ?? provides a separated concrete seminearness space which does not satisfy (s). example ?? is a concrete seminearness space satisfying (s) but not (t). example 3 in [?] satisfies (t) but it is not separated (and not concrete). remark that this example moreover is a nearness space. in the case of a nearness space however some other implications become true. proposition 3.11. (1) for a nearness space we have separated ⇒ (s) ⇔ (t) (2) for a concrete nearness space we have separated ⇔ (s) ⇔ (t) proof. (1) let (x,γ) be a separated nearness space. if m and n are minimal small concentrated stacks then m and n are filters [?], [?]. if every m ∈m would intersect every n ∈n then stack {m∩n|m ∈m,n ∈ n} would be a filter too and so it would be concentrated. hence m = n . so (x,γ) satisfies (s). next suppose (x,γ) is a nearness space satisfying (s). let m and n be minimal small. since m and n are filters, both are concentrated and so (s) implies that ∃m ∈ m,∃n ∈ n : m ∩n = ∅. now every minimal small p is a filter too, so m and n can not be both in p. (2) follows immediately from (1) in combination with proposition 3.9 (2). � that (s) 6⇒ separated, even in the nearness case follows from example 3 in [?]. next we discuss the impact of the separation conditions on extensions. proposition 3.12. a seminearness space is induced by a hausdorff closure extension if and only if it is separated, t1 and satisfies (s). proof. suppose (y,cly ) is a hausdorff extension of (x,γ) then if m and n are different minimal small stacks, by proposition 2.3 there are y and z in y such that m = vy (y)|x and n = vy (z)|x. it follows that y 6= z and then disjoint sets can be chosen using the hausdorff property of (y,cly ). so (x,γ) satisfies (s). that (x,γ) is separated and t1 follows from the fact that these properties are hereditary. conversely, suppose (x,γ) is separated, t1 and satisfies (s). consider α = {m|m minimal small, concentrated, not convergent} ∪ {a|a small, openbased, not concentrated, not convergent} extensions of closure spaces 233 and construct the loose enlargement on x′ = x ∪{yp|p ∈ α} in view of paragraph 2.1, in order to conclude that (x′,cl′) is an extension, it suffices to prove that if d ∈ γ is openbased and not convergent in x then stackx′d converges to some point in x′. either d is concentrated and then d ⊃ m for a unique minimal small concentrated stack m. in this case stackx′d ⊃ stackx′m∩ ẏm for ym ∈ x′. or d is not concentrated and then stackx′d converges to yd ∈ x′. finally we prove that this loose extension is a hausdorff closure space. consider two different points in x′. if each of them corresponds to a concentrated minimal small stack, then (s) implies that disjoint neighborhoods can be found. if at least one of the points corresponds to some openbased small stack a which is not concentrated and not convergent, then sec a is not small and then the argument developed in the proof of proposition 3.5 (ii) can be used to obtain disjoint neighborhoods. � proposition 3.13. a seminearness space (x,γ) is induced by a strict hausdorff closure extension if and only if (x,γ) is concrete, t1 and satisfies (t). proof. suppose (x,γ) is induced by a strict hausdorff closure extension (y,cly ). then by proposition 2.6 we already know that (x,γ) is t1 and concrete. in paragraph 2.2 we remarked that (y,cly ) is unique up to isomorphism. so we have that (x̂, ĉl) is hausdorff. in order to prove (t), let m 6= n be minimal small stacks. by corollary 2.4 they correspond to different points y and z in x̂. consider disjoint basic neighborhoods o(a) and o(b) of y and z, respectively where a,b ⊂ x. then clearly int a ∈m and int b ∈n , so int a and int b are γ-disjoint. conversely, suppose (x,γ) is t1, concrete and satisfies (t). then we already know that (x̂, ĉl) is a strict t1 closure extension. in order to prove that (x̂, ĉl) is hausdorff let y and z be different points. these points correspond to different minimal small stacks in (x,γ) which therefore contain γ-disjoint sets a and b. it follows that o(a) and o(b) are disjoint and belong to the respective neighborhood collections vy (y) and vy (z). � remark that again our situation differs fundamentally from its topological counterpart. the existence of a topological hausdorff extension inducing (x,γ) implies the existence of a strict topological hausdorff extension [?]. whereas here for the existence of a hausdorff closure extension only separated and (s) are needed on (x,γ), and for the existence of a strict hausdorff closure extension concreteness and (t) are involved and as announced in 3.10 these conditions are not equivalent. in section ?? examples are listed showing that even in the concrete case (s) plus separated does not imply (t). 234 d. deses, a. de groot-van der voorde, e. lowen-colebunders 4. regularity and extensions. regularity was introduced in the context of prenearness spaces in [?]. so we can apply the definition to our setting of seminearness spaces. if a and b are subsets in (x,γ) one puts b <µ a ⇐⇒ {a,x −b}∈ µ where µ is the covering structure associated with γ. for a stack a one puts a<µ = {a|∃b ∈a : b <µ a} and for covers u and v one writes v <µ u ⇐⇒ ∀v ∈v : ∃u ∈u : v <µ u using this notation one has the equivalence of the following statements (i) ∀a∈ γ also a<µ ∈ γ (ii) ∀u ∈ µ : ∃v ∈ µ : v <µ u a seminearness space is said to be regular if it satisfies the previous equivalent statements [?]. next we introduce a stronger version of regularity by strengthening ”disjointness” as we did before. using γ-disjointness instead of disjointness we again obtain equivalent statements. proposition 4.1. let (x,γ) be a seminearness space and µ be the associated covering structure. let a and b be subsets of x. the following are equivalent: (i) {a}∪{d|d and b γ-disjoint}∈ µ (ii) ∀a small, if for every d ∈a the sets d and b are not γ-disjoint, then a ∈a we write b <<µ a if one and then both conditions stated in the previous proposition hold. in view of the fact that {a}∪{d|d and b γ-disjoint}≺{a,x −b} we have that b <<µ a ⇒ b <µ a for a stack a let a<<µ = {a|∃b ∈a : b <<µ a} for covers u and v we write v <<µ u ⇐⇒ ∀v ∈v : ∃u ∈u : v <<µ u using this notation we obtain the following equivalent statements. the proof of the equivalence is quite similar to the equivalence based on <µ instead of <<µ. proposition 4.2. the following are equivalent: (i) ∀a∈ γ we have a<<µ ∈ γ (ii) ∀u ∈ µ : ∃v ∈ µ : v <<µ u extensions of closure spaces 235 definition 4.3. a seminearness space satisfies (r) if it fulfills one (and then both) of the previous statements. clearly (r) implies regularity. moreover as for nearness spaces every uniform seminearness space is regular. however not every uniform seminearness space satisfies (r), example ?? serves as a counterexample. in general we have the following implications. proposition 4.4. if (x,γ) is a seminearness space then we have (1) regular implies separated and regular implies (s) (2) (r) implies regularity and (r) implies (t) proof. (1) that a regular seminearness space is separated is proved analogously to the nearness case. a regular seminearness space also satisfies (s). indeed: let m and n be small stacks and suppose ∀m ∈m,∀n ∈n : m ∩n 6= ∅. consider the stack m∩n and let u ∈ µ. take v ∈ µ such that v <µ u. further let v ∈v∩m and let u ∈u be such that v <µ u. now since n is small we have n ∩{u,x −v} 6= ∅ clearly x − v 6∈ n and so finally we have u ∈ m∩n . so we can conclude that m∩n is small and in case m and n are minimal small this implies m = n . (2) (r) implies (t). let m and n be minimal small stacks. suppose m6= n and assume that m and n do not contain γ-disjoint sets. we prove that m∩n is small. let u ∈ µ and consider v ∈ µ such that v <<µ u. since m is small we can take v ∈v ∩m and then u ∈u such that v <<µ u. consider the uniform cover w = {u}∪{d|d and v γ-disjoint} then n ∩w 6= ∅. now by assumption on m and n , u must belong to n . so finally u ∈ m∩n and the rest follows as in the previous part. � in fact no other implications (except for those obtained by transitivity) hold. counterexamples for the nonvalid ones can be found in section ??. proposition 4.5. let (x,γ) be a seminearness space. if m⊂ sec m for all minimal small stacks m that do not converge, then we have (i) d and b are γ-disjoint if and only if d and b are disjoint (ii) b <<µ a if and only if b <µ a (iii) (x,γ) has (r) if and only if it is regular proof. we only need to show (i). clearly if m ⊂ sec m then disjoint sets d and b can not both belong to m � 236 d. deses, a. de groot-van der voorde, e. lowen-colebunders in every nearness space minimal small stacks are filters and so the statements in the previous proposition hold, in particular for nearness spaces, regularity is equivalent to (r). applying corollary 3.10 we obtain that in every concrete seminearness space in which all minimal small stacks are concentrated and in which (s) holds, also m⊂ secm holds for every minimal small stack. it follows that every regular concrete seminearness space in which all minimal small stacks are concentrated satisfies (r). in particular a closure (seminearness) space is regular if and only if it satisfies (r). it can be easily seen that analogously to the topological case, a closure (seminearness) space x is regular if and only if ∀x ∈ x : v(x) has a closed base however even on a nearness space the condition that ”clγa is small whenever a is small” (called weakly regular in [?]) is strictly weaker than regularity. next we investigate the regularity of the strict extension of a concrete seminearness space. proposition 4.6. let (x,γ) be a concrete t1 seminearness space, then it is induced by a strict regular extension if and only if (x,γ) satisfies (r). proof. suppose (x,γ) satisfies (r). we prove that the strict t1 extension (x̂, ĉl) is regular. let ym ∈ x̂ where m is a minimal small stack that is not convergent (or alternatively let x ∈ x). let o(a) be a basic open set containing ym (or x) where a is some subset of x. so a ∈m (or a ∈v(x)). since m<<µ = m (v(x)<<µ = v(x)) we have a ∈m<<µ (a ∈v(x)<<µ) and so we can find b ∈m (b ∈v(x)) such that b <<µ a. now we prove that ĉl o(b) ⊂ o(a) let yn ∈ ĉl o(b) where n is minimal small, not convergent. then for every d ∈n we have o(d) ∩o(b) 6= ∅ it follows that for every d ∈ n the sets d and b are not γ-disjoint. hence a ∈n and then yn ∈ o(a). moreover if x ∈ ĉl o(b) then for every d ∈v(x) we have o(d)∩o(b) 6= ∅. again it follows that for every d ∈ v(x) the sets d and b are not γ-disjoint and therefore a ∈v(x) and x ∈ o(a). conversely suppose that (x,γ) is induced by a strict regular t1 extension. by proposition 2.6 this means that (x̂, ĉl) is regular. we prove that (x,γ) satisfies (r). let a be small. since (x,γ) is concrete we can take a minimal small stack m such that m⊂a. we prove that m⊂m<<µ. let a ∈m. if m is not convergent we have ym ∈ o(a) (or if m = v(x) we have x ∈ o(a)). since o(a) is open in (x̂, ĉl) it contains a closed neighborhood and so we can find b ⊂ x such that ym ∈ o(b) (or x ∈ o(b)) and such that o(b) ⊂ ĉl o(b) ⊂ o(a) extensions of closure spaces 237 but then we have b ∈m (or b ∈v(x)) and it is clear that b can be assumed to be open. now b <<µ a. indeed let d be small such that for every d ∈d the sets d and b are not γ-disjoint. let n be minimal small with n ⊂ d. then also for every n ∈ n the sets n and b (which can be considered to be open) are not γ-disjoint. it follows that o(n) ∩o(b) 6= ∅,∀n ∈n and finally that yn ∈ ĉl o(b) if n is not convergent, or that x ∈ ĉl o(b) if n = v(x). so we have yn ∈ o(a) or x ∈ o(a), respectively and we can conclude that a ∈n . � it is known that for a topological extension, regularity of the extension implies strictness. in our setting of closure extensions this is no longer true. a seminearness space can be induced by a regular t1 extension without having a strict regular t1 extension. in examples ?? and ?? we’ll prove that the structure is induced by a regular t1 extension. however neither ?? nor ?? satisfies (r). 5. summarizing diagrams and examples. 5.1. for closure spaces and for concrete nearness spaces we have that several notions coincide, namely that ”t ⇐⇒ s ⇐⇒ separated” and that ”r ⇐⇒ regularity”. 5.2. for arbitrary nearness spaces we have the implications of figure ??. r separated t regular s uniform figure 1. implications for nearness spaces the only valid implications are those indicated and those obtained by transitivity. 5.3. as an example of a nonconcrete nearness space satisfying (s) but that is not separated, one can consider example 3 in [?]. that this example indeed satisfies condition (s) follows from the fact that the only concentrated minimal small stacks are the pointfilters. 238 d. deses, a. de groot-van der voorde, e. lowen-colebunders r regular separated t s uniform 5.7 5.7 5.8 5.6 5.6 5.8 5.8 5.7 figure 2. implications for concrete seminearness spaces 5.4. for concrete seminearness spaces we proved the implications in figure ??. again no other implications hold except for those obtained by transitivity. the numbers refer to the examples presented below and these are counterexamples for the reversed arrows. 5.5. for general seminearness spaces the diagram is as in figure ??. r separated t regular s uniform figure 3. implications for seminearness spaces as a counterexample showing that (s) does not imply separated in the nonconcrete case, we again refer to example 3 in [?]. 5.6. a concrete and separated t1 seminearness space which does not satisfy (s). therefore it is nonregular, nonuniform and satisfies neither (r) nor (t). let x = r2 and define m = stack {pr−11 (c)|c ∈ r} n = stack {pr−12 (c)|c ∈ r} extensions of closure spaces 239 for a stack a we define a∈ γ ⇐⇒ m⊂a or n ⊂a or ẋ ⊂a for some x ∈ x the underlying closure is the discrete one. clearly every small stack contains a unique minimal small stack, but m and n do not contain disjoint sets and so (x,γ) does not satisfy (s). 5.7. a concrete t1 seminearness space which is uniform (and even zerodimensional) and so it is regular and satisfies (s) and is separated. however it satisfies neither (t) nor (r). let a,b,c be three pairwise disjoint sets with more than one point and x = a∪b ∪c. in order to define γ on x consider the following stacks m = stack {a,b} n = stack {b,c} p = stack {a,c} define a stack a to be small if and only if m⊂a or n ⊂a or p ⊂a or ẋ ⊂a for some x ∈ x then the pointfilters ẋ for x ∈ x are the only concentrated minimal stacks, and the other minimal stacks m,n ,p are not concentrated. (x,γ) does not satisfy (t) since for instance for m and n neither of the disjoint sets a and b, a and c or b and c are γ-disjoint. it follows that (x,γ) does not satisfy (r). however (x,γ) is uniform since its collection µ of uniform covers is generated by the following collection µ′ = {u1,u2,u3} of partitions u1 = {a,b}∪{{x}|x ∈ c} u2 = {b,c}∪{{x}|x ∈ a} u3 = {a,c}∪{{x}|x ∈ b} it follows that (x,γ) is regular, also separated and satisfies (s). so the strict extension (x̂, ĉl) is a t1 extension that is not hausdorff and not regular. remark however that the loose extension constructed by adding different points for the minimal small stacks that do not converge, is regular and t1. 5.8. a concrete t1 seminearness space which is uniform (and even zerodimensional) and therefore is regular. it satisfies (t) and hence also (s) and it is separated. however it does not satisfy (r). let a,a′,p,p ′,q and q′ be pairwise disjoint sets with more than one point and let x = a∪a′ ∪p ∪p ′ ∪q∪q′ 240 d. deses, a. de groot-van der voorde, e. lowen-colebunders in order to define γ on x consider the following stacks a = stack {a,a′} p = stack {p,p ′,a} q = stack {q,q′,a} b = stack {p,q} define a stack s to be small if and only if a⊂s,p ⊂s,q⊂s,b ⊂s or ẋ ⊂s for some x ∈ x clearly (x,γ) is concrete and t1. use the fact that a′,p ′,q′ are sets belonging to just one nonconvergent minimal small stack to see that (x,γ) satisfies (t). again (x,γ) is uniform and in fact (x,µ) is generated by a collection of partitions. so (x,γ) is regular. however (r) is not satisfied. let’s concentrate on a and consider a ∈ a. the sets p and a are not γ-disjoint since they both belong to p. also q and a are not γ-disjoint. it follows that for u = {a}∪{d|d and a γ-disjoint} we have u∩b = ∅. so u 6∈ µ and therefore a <6<µ a. clearly this implies that a<<µ 6∈ γ. it follows that the strict extension (x̂, ĉl) of (x,γ) is a hausdorff closure space that is not regular. remark however that the loose extension constructed by adding different points for the minimal small stacks that do not converge, is a regular t1 extension. references [1] j. adámek, h. herrlich and g. strecker, abstract and concrete categories, wiley and sons, new york (1990). [2] d. aerts, foundations of quantum physics: a general realistic and operational approach, internat. j. theoret. phys., 38(1) (1999), 289-358. [3] g. aumann, kontaktrelationen, bayer. akad. wiss. math. nat. kl. sitzungber. (1970), 67-77. [4] m.k. bennett, affine and projective geometry, wiley and sons, new york (1995). [5] h.l. bentley, nearness spaces and extensions of topological spaces, studies in topology, academic press (1975), 47-66. [6] h.l. bentley and h. herrlich, extensions of topological spaces, in topology, proc. memphis state univ. conf., marcel decker (1976), 129-184. [7] h.l. bentley, h. herrlich, and e. lowen-colebunders, convergence, j. of pure and appl. algebra, 68 (1990), 27-45. [8] h.l. bentley and e. lowen-colebunders, completely regular spaces, commentat. math. univ. carol., 32(1) (1991), 129-153. [9] g. birkhoff, lattice theory, american mathematical society, providence, rhode island (1940). [10] v. claes, e. lowen-colebunders and g. sonck, cartesian closed topological hull of the construct of closure spaces, theory appl. categ., 8 (2001), 481-489. [11] d. deses and e. lowen-colebunders, on completeness in a non-archimedean setting, via firm reflections, accepted for publication, bulletin belgian math. soc. extensions of closure spaces 241 [12] m. erné, lattice representations for categories of closure spaces, categorical topology, heldermann verlag, berlin (1984), 197-222. [13] c.a. faure and a. frölicher, modern projective geometry, kluwer academic publishers, dordrecht (2000). [14] b. ganter and r. wille, formal concept analysis, springer verlag, berlin (1998). [15] j. de groot, g.a. jensen and a. verbeek, superextensions, math. centrum, amsterdam (1968). [16] a. de groot van der voorde, separation axioms in extension theory for closure spaces and their relevance to state property systems, phd thesis, vrije universiteit brussel, july 2001. [17] h. herrlich, a concept of nearness, gen. topol. appl., 4 (1974), 191-212. [18] h. herrlich, topological structures, math. centre tracts, 52 (1974), 59-122. [19] h. herrlich, topologie ii: uniforme räume, heldermann verlag, berlin (1987). [20] s.a. naimpally and j.h.m. whitfield, not every near family is contained in a clan, proc. of the amer. math. soc., 47 (1975), 237-238. [21] d.j. moore, categories of representations of physical systems, helv. phys. acta, 68 (1995), 658-678. [22] c. piron, recent developments in quantum mechanics, helv. phys. acta, 62 (1989), 82-90. [23] g. preuss, theory of topological structures, d. reidel publishing company, dordrecht (1989). received november 2001 revised october 2002 d. deses, a. de groot-van der voorde, e. lowen-colebunders department of mathematics, vrije universiteit brussel, 1050 brussels, belgium e-mail address : diddesen@vub.ac.be, avdvoord@vub.ac.be, evacoleb@vub.ac.be garciarsanchezpagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 2, 2005 pp. 135-142 compactness properties of bounded subsets of spaces of vector measure integrable functions and factorization of operators l. m. garćıa-raffi and e. a. sánchez-pérez ∗ abstract. using compactness properties of bounded subsets of spaces of vector measure integrable functions and a representation theorem for q-convex banach lattices, we prove a domination theorem for operators between banach lattices. we generalize in this way several classical factorization results for operators between these spaces, as psumming operators. 2000 ams classification: 46g10, 54d30. keywords: compactness, vector measures, integration, factorization. 1. introduction compactness of the unit ball of banach spaces is a useful tool in the theory of operators between these spaces. one of the basic arguments that provides important applications in this field uses ky fan´s lemma with a family of functions on the unit ball of a banach space that are continuous with respect to the weak* topology. this argument can be found in the proof of the pietsch domination theorem for p-summing operators, the characterization of the p, qdominated operators or the maurey-rosenthal theorem for factorization of operators through lp-spaces (see for instance [12, 15, 5]). roughly speaking, weak* compactness of the unit ball of a banach space is one of the keys to relate vector valued norm inequalities and domination/factorization theorems for operators. ∗the authors acknowledge the support of the generalitat valenciana, spain, grant gv04b-371, the spanish ministry of science and technology, plan nacional i+d+i , grant bfm2003-02302; and the support of the universidad politécnica de valencia, under grant 2003-4114 for interdisciplinary research projects. 136 l. m. garćıa raffi and e.a. sánchez-pérez in the context of the spaces lq(m) of q-integrable functions with respect to the (countably additive) vector measure m, it is possible to obtain more compactness results with respect to topologies that are defined using the properties of the integration map that appears in a natural way in this framework. in particular, we will use the fact that for reflexive sapces lq(m) the unit ball of lq(m), 1 < q < ∞, is compact for the m-weak topology (see proposition 13 in [13] for the λ-weak topology, assuming lq(m) is reflexive). a characterization of the compactness of the unit ball of such spaces with respect to other different topology can also be found in [13] (see theorem 14 for the λ-topology). more compactness results for the integration operator have been recently obtained in [11] (see also [10]). in this paper we present a domination theorem for operators that satisfy a p-summing type vector norm inequality. for its proof, we use the compactness of bounded sets in one of the topologies quoted above on spaces of q-integrable functions with respect to a vector measure. every q-convex banach lattice with order continuous norm and weak order unit can be represented as a space of integrable functions with respect to a vector measure (see proposition 2.4 in [7]). we use these representations of the banach lattices and the compactness with respect to the m-weak topology of their unit balls to prove a (representationdepending) general domination theorem for operators on q-convex banach lattices. let e be an order continuous q-convex banach lattice with weak order unit, 1 ≤ q < ∞. we will say that e is q-represented by the vector measure m : σ → x, where x is a banach space, if e is order isomorphic to lq(m). as a direct consequence of the proposition quoted above, such a representation always exists for every such a banach lattice e. we use standard banach lattice concepts and notation (see [8, 15]). if 1 ≤ p ≤ ∞, we write p′ for the extended real number that satisfies 1/p + 1/p′ = 1. we will write r for the set of real numbers. let e be a banach lattice and 1 ≤ r < ∞. it is said that e is r-convex if there is a constant c > 0 such that for every finite sequence x1, ..., xn ∈ e, ‖( n∑ k=1 |xk| r) 1 r ‖ ≤ c( n∑ k=1 ‖xk‖ r) 1 r . the real number m(r)(e) defined as the best constant c in the inequality above is called the r-convex constant of e. let x, y be a pair of banach spaces, 1 ≤ p < ∞, and consider an operator t : x → y . t is p-summing (p-absolutely summing in [12]) if there is a constant c > 0 such that for every finite set x1, ..., xn ∈ x, the inequality ( n∑ i=1 ‖t (xi)‖ p) 1 p ≤ c sup x′∈bx′ ( n∑ i=1 | < xi, x ′ > |p) 1 p holds (see e.g. [12, 5]). the pietsch domination theorem establishes that an operator t : x → y is p-summing if and only if there is a (regular borel) probability measure µ on compactness properties of bounded subsets of spaces of vector measure... 137 the weak* compact set bx′ and a positive constant c such that ‖t (x)‖ ≤ c( ∫ bx′ | < x, x′ > |pdµ) 1 p , x ∈ x. in this paper we provide a new version of this result. we complete in this way the results of [13] that relates compactness properties of the unit ball of the spaces lq(m) of a vector measure with domination/factorization theorems (see also [9]). this is the reason we assume through all the paper that the spaces lq(m) involved are reflexive. let x be a banach space and let (ω, σ) be a measurable space. consider a countably additive vector measure m : σ → x. we say that a measurable function f : ω → r is integrable with respect to m if it is scalarly integrable (i.e. it is integrable with respect to every scalar measure mx′, x ′ ∈ x′, given by mx′(a) :=< m(a), x ′ >, a ∈ σ), and there is an element ∫ ω fdm ∈ x such that for every x′ ∈ x′, < ∫ ω fdm, x′ >= ∫ ω fdmx′ (see for instance [1]). a rybakov measure for m is a measure defined by the variation |mx′| of a measure mx′ that controls m (see [6]). the space l1(m) of integrable functions with respect to m is the banach space of all the classes of mx′-a.e. equal functions, where mx′ is a rybakov measure for m. endowed with the norm ‖f‖l1(m) = supx′∈bx′ ∫ ω |f|d|mx′| and the |mx′|-a.e. order, it is a köthe function space over |mx′| with weak unit. the reader can see [1, 2] for the fundamental facts about these spaces. if 1 < q < ∞, we say that a measurable function f is q-integrable with respect to m if |f|q ∈ l1(m). the construction of the space lq(m) follows in the same way that in the case of l1(m). it is also a köthe function space over |mx′| and the norm is given by ‖f‖lq(m) = sup x′∈bx′ ( ∫ ω |f|qd|mx′|) 1 q f ∈ lq(m), (see [13, 7]). this space is q-convex when considered as a banach lattice. 2. extensions of operators defined on spaces of integrable functions with respect to a vector measure let 1 ≤ q < ∞ and consider an element x′ ∈ x′ such that the measure mx′ is positive. it is easy to see that the operator ix′ : lq(m) → lq(mx′) defined by ix′(f) := f, f ∈ lq(m) is well-defined and continuous. moreover, ‖ix′‖ ≤ 1. however, note that we can assure that ix′ is an injection only if mx′ is a rybakov measure for m. definition 2.1. consider two banach spaces x, y , a family of banach spaces b = {xi : i ∈ i}, and an operator t : x → y . we say that t can be uniformly extended to b if the identity map ixi : x → xi is defined, continuous and ‖ixi‖ ≤ 1, for every i ∈ i, and there is a constant c > 0 such that all the extensions ti : xi → y of the operator t (i.e. ti ◦ ixi(x) = t (x), x ∈ x) are defined, continuous and ‖ti‖ ≤ c. 138 l. m. garćıa raffi and e.a. sánchez-pérez proposition 2.2. let m : σ → x be a countably additive vector measure, y a banach space and 1 ≤ q < ∞, and consider an operator t : lq(m) → y . suppose that there is a subset s ⊂ x′ such that for every x′ ∈ s, ‖x′‖ = 1 and mx′ is a positive measure. then the following conditions are equivalent. (1) there is a constant c > 0 such that for every x′ ∈ s, ‖t (f)‖ ≤ c( ∫ ω |f|qdmx′) 1 q , f ∈ lq(m). (2) the operator t can be uniformly extended to all the spaces lq(mx′), x′ ∈ s. proof. let us show (1) → (2). first note that the inequality of (1) provides the way of extending t to every space lq(mx′), x ′ ∈ s. let us write [f]x′ for the equivalence class of the function f ∈ lq(mx′) (only for the aim of this proof, in the rest of the paper we will simply write f). suppose that f1 6= f2 as elements of lq(m) but [f1]x′ = [f2]x′. then, (1) gives ‖t (f1 − f2)‖ ≤ c( ∫ ω |f1 − f2| qdmx′) 1 q = 0, and thus t (f1) = t (f2). now, let us show that the argument above is enough to prove that the operator t is well-defined. the simple functions are dense in the spaces lq(m) for every countably additive vector measure m (see [13]). then, for every x′ ∈ s the operator tx′ : lq(mx′) → y given by tx′(f) := t (f) for every simple function f and extended to all lq(mx′) by continuity is well-defined. moreover, we directly obtain ‖tx′‖ ≤ c as a consequence of the inequality (1). since this argument does not depend on x′ ∈ s, we obtain (2). the converse is obvious. � the theorem above provides a family of factorization theorems through lqspaces (indexed by s). indeed, since the identity map ix′ : lq(m) → lq(mx′) is continuous, we directly obtain the following corollary 2.3. let e be a q-convex banach lattice that can be q-represented by the vector measure m. consider an operator t : e → y that satisfies (1) or (2) in proposition 2.2 for a subset s ⊂ x′ satisfying the conditions in this proposition. then for every x′ ∈ s, t can be factorized as follows. lq(m) t ✲ y ix′ ❍❍❍❍❍❍❥ lq(mx′) ✟✟ ✟✟ ✟✟✯ tx′ moreover, ‖ix′‖‖tx′‖ ≤ ck for every x ′ ∈ s, where c is the constant given in proposition 2.2 and k is the corresponding constant of the equivalence of norms between ‖.‖e and ‖.‖lq(m). compactness properties of bounded subsets of spaces of vector measure... 139 a particular straightforward application of this result -that provides also the canonical situation of this extension theoremis the case when s contains only one element x′. in this case, we obtain directly an extension/factorization theorem through an lq-space. using the representation theorem for banach lattices given by proposition 2.4 in [7] quoted in section 1, we obtain a maureyrosenthal type factorization for an operator t whenever it satisfies an inequality as the one given by theorem 2.2. moreover, in this case the multiplication operator that defines the factorization is simply the identity. 3. a pietsch type domination theorem for operators on spaces of p-integrable functions with respect to a vector measure in this section we provide a domination theorem for operators that satisfy a vector valued norm inequality involving strong and weak convergent sequences. we obtain in this way a pietsch type domination theorem for operators on reflexive q-convex banach lattices, and complete the research that we started in [13]. in this paper, we obtained a factorization theorem through spaces of bochner integrable functions and we characterized this situation by means of a vector valued norm inequality, whenever a certain compactness property for the integration operator was fulfilled (theorem 17 in [13]). the key for the proof of this factorization result is the requirement of compactness of the unit ball of the space lq(m), where m is a countably additive vector measure, with respect to the m-topology (the λ-topology in [13]). the theorem of this section gives the weak version of this result. however, no compactness requirement is needed in this case, since the unit ball of a reflexive space lq(m) is always compact with respect to the m-weak topology. these compactness properties of the unit ball of q-convex banach lattices represented by lq spaces of a vector measure (proposition 13 and theorem 14 in [13]), can be generalized to all bounded subsets under the (obvious) adequate requirements. first, let us write the definition of the m-weak topology for the space lq(m), where m : σ → x is a countably additive vector measure and q > 1. this is the topology that has as a basis of neighborhoods of an element g0 ∈ lq(m) the following sets. let ǫ > 0, n ∈ n, x′1, ..., n ′ n ∈ x ′ and f1, f2, ..., fn ∈ lq′(m). we define the set ξǫ,f1,...,fn,x′1,...,x′n(g0) := {g ∈ lq(m) : | < ∫ ω fi(g − g0)dm, x ′ i > | < ǫ, ∀i = 1, ..., n}. the m-weak topology is the topology which has as a basis of neighborhoods the family of sets ξǫ,f1,...,fn,x′1,...,x′n(g0). it is easy to prove that this topology is a well-defined hausdorff locally convex topology on lq(m). the reader can find more information about it in [13]. theorem 3.1. let e be a q-convex banach lattice that can be q-represented by the vector measure m : σ → x, where x is a banach space and 1 < q < ∞. suppose that lq′(m) is reflexive. let 1 ≤ p < ∞. consider an operator 140 l. m. garćıa raffi and e.a. sánchez-pérez t : e → x, and suppose that there is a subset s ⊂ x′ such that for every x′ ∈ s, ‖x′‖ = 1 and mx′ is a positive measure. then the following conditions are equivalent. (1) there is a constant c > 0 such that for every pair of finite families x′1, ..., x ′ n ∈ s and f1, ..., fn ∈ lq(m) ( n∑ i=1 ‖t (fi)‖ p) 1 p ≤ c sup g∈bl q′ (m) ( n∑ i=1 | < ∫ ω figdm, x ′ i > | p) 1 p . (2) there is a constant c > 0 and a regular borel probability measure µ over the compact hausdorff space blq′ (m) endowed with the m-weak topology such that ‖t (f)‖ ≤ c inf x′∈s ( ∫ bl q′ (m) | < ∫ ω fgdm, x′ > |pdµ(g)) 1 p for every f ∈ lq(m). moreover, the infimum of all the constants c that satisfy (1) coincides with the infimum of all the constants c in (2). proof. the conditions on e and m allow us to consider that the operator t is directly defined on lq(m). for the proof of this result we adapt the argument that proves the pietsch domination theorem for p-summing operators. a direct calculation gives (2) → (1). for the converse, consider the m-weak compact (convex and hausdorff) set blq′ (m) and the space c(blq′ (m)) of continuous functions on blq′ (m), with respect to the m-weak topology. consider its dual, the space of regular borel measures m, and the (compact and convex) subset of probability measures p. for every pair of finite families x′1, ..., x ′ n ∈ s and f1, ..., fn ∈ lq(m) we define the function φx′1,...,x′n,f1,...,fn : p → r, φx′1,...,x′n,f1,...,fn(µ) := ( n∑ i=1 ‖t (fi)‖ p) − cp ∫ bl p′ n∑ i=1 | < ∫ ω figdm, x ′ i > | pdµ. note that the inequality given in (1) provides an element g0 ∈ blq′ (m) such that n∑ i=1 ‖t (fi)‖ p ≤ cp n∑ i=1 | < ∫ ω fig0dm, x ′ i > | p. thus, for each function φx′1,...,x′n,f1,...,fn, there is a probability measure (the dirac measure at the point g0, δg0) such that φx′1,...,x′n,f1,...,fn(δg0) ≤ 0. it is easy to see that the set of all the functions as φx′1,...,x′n,f1,...,fn is concave. in fact, it is clear that the sum of two such functions gives other function of the family. moreover, the product of a function like this and a positive scalar is also other function of the family (it is enough to consider the product of the same functions fi that define the function of the family by the scalar to the power 1/p to define the new function). thus we can apply ky fan´s lemma to obtain an element of p that satisfies the inequalities of the type of (2) for compactness properties of bounded subsets of spaces of vector measure... 141 all functions φx′ 1 ,...,x′ n ,f1,...,fn. thus, there is a probability measure µ ∈ p such that ‖t (f)‖ ≤ c( ∫ bl q′ (m) | < ∫ ω fgdm, x′ > |pdµ(g)) 1 p for every f ∈ lq(m) and each x ′ ∈ s. this gives the result. � the canonical situation that generalizes this theorem is the case of a psumming operator on lq(ν) of a scalar measure ν. in this case, we obtain a factorization through the identity operator i : c(blp′ ) → lq(blp′ , µ), as can be obtained as a direct application of the pietsch domination theorem. the set s contains only one element (formally s = {1}), since the range of ν is a subset of r. in the general case, an operator t that satisfies the conditions of theorem 3.1 verifies a family of factorizations indexed by the same set s. note that the conditions of proposition 3.1 imply in particular the ones of theorem 2.2. moreover, (2) of theorem 3.1 implies a p-summing inequality for each extension to an lq(mx′)-space. thus, if x ′ ∈ s and t : e → y satisfies (1) of the theorem, there is a probability measure νx′ such that t can be factorized as c(blq′ (mx′ )) e ✲ lq(mx′) id ❄ t1 ✲ iq rg(ip) ⊂ lp(blq′ , νx′) y✲ t x′ ✻ where t x′ is the extension of the operator t given in theorem 2.2, e is included continuously in lq(mx′), id and iq are inclusion operators, t1 is a continuous map and rg(ip) is the (norm) closure of ip(c(blq′ )) in lp(blq′ ). therefore, theorem 3.1 provides a family of mixed factorization schemes. we can obtain a factorization of the operator t through an lq-space, a c(k)space and an lp-space for each x ′ ∈ s. the general theory of operator ideals and its applications in the theory of banach spaces can then be used to relate this result with well-known properties of operators and banach spaces (see [4]). for instance, we directly obtain that the conditions of our theorem imply that it is (p, q′)-factorable (see theorem 19.4.6 in [12]). the results of this section complete in this way the domination/factorization results given in [13] (see also [9]); all of them can be obtained using the compactness properties of the unit ball of lq(m)-spaces with respect to different topologies defined by means of the integration operator associated to m. 142 l. m. garćıa raffi and e.a. sánchez-pérez references [1] g. p. curbera, operators into l1 of a vector measure and applications to banach lattices, math. ann. 293 (1992), 317-330. [2] g. p. curbera, banach space properties of l1 of a vector measure, proc. amer. math. soc. 123 (1995), 3797-3806. [3] a. defant, variants of the maurey-rosenthal theorem for quasi köthe function spaces, positivity 5 (2001), 153-175. [4] a. defant and k. floret, “tensor norms and operator ideals”, north holland, amsterdam (1993). [5] j. diestel, h. jarchow and a. tonge, “absolutely summing operators”, cambridge studies in advanced mathematics 43, cambridge (1995). [6] j. diestel and j. j. uhl, vector measures, math. surveys, 15, amer. math. soc., providence, ri. 1977. [7] a. fernández, f. mayoral, f. naranjo, f. sáez and e. a. sánchez-pérez, spaces of p-integrable functions with respect to a vector measure, positivity, to appear. [8] j. lindenstrauss and l. tzafriri, “classical banach spaces i and ii”, springer, berlin (1996). [9] f. mart́ınez-giménez and e. a. sánchez-pérez, vector measure range duality and factorizations of (d, p)-summing operators from banach function spaces, bull. braz. math. soc., new series 35(1)(2004), 51-69. [10] s. okada and w. j. ricker, the range of the integration map of a vector measure, arch. math. 64(1995), 512-522. [11] s. okada, w. j. ricker and l. rodŕıguez-piazza, compactness of the integration operator associated with a vector measure, studia math. 150(2) (2002), 133-149. [12] a. pietsch, “operator ideals”, north-holland, amsterdam (1980). [13] e. a. sánchez-pérez, compactness arguments for spaces of p-integrable functions with respect to a vector measure and factorization of operators through lebesgue-bochner spaces, illinois j. math. 45(3) (2001), 907-923. [14] e. a. sánchez-pérez, spaces of integrable functions with respect to vector measures of convex range and factorization of operators from lp-spaces, pacific j. math. 207 (2) (2002), 489-495. [15] p. wojtaszczyk, “banach spaces for analysts”, cambridge university press. cambridge. 1991. received september 2004 accepted september 2004 luis m. garćıa-raffi (lmgarcia@mat.upv.es) e.t.s.i. caminos, canales y puertos. departamento de matemática aplicada. universidad politécnica de valencia. 46071 valencia, spain. e. a. sánchez-pérez (easancpe@mat.upv.es) e.t.s.i. caminos, canales y puertos. departamento de matemática aplicada. universidad politécnica de valencia. 46071 valencia, spain. @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 133–142 skew compact semigroups ralph kopperman and desmond robbie abstract. skew compact spaces are the best behaving generalization of compact hausdorff spaces to non-hausdorff spaces. they are those (x,τ) such that there is another topology τ∗ on x for which τ ∨τ∗ is compact and (x,τ,τ∗) is pairwise hausdorff; under these conditions, τ uniquely determines τ∗, and (x,τ∗) is also skew compact. much of the theory of compact t2 semigroups extends to this wider class. we show: a continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ2 → τ. each of these contains a unique minimal ideal which is an upper set with respect to the specialization order. a skew compact semigroup which is a continuous semigroup with respect to both topologies is called a de groot semigroup. given one of these, we show: it is a compact hausdorff group if either the operation is cancellative, or there is a unique idempotent and s2 = s. its topology arises from its subinvariant quasimetrics. each *-closed ideal 6= s is contained in a proper open ideal. 2000 ams classification: 54d30, 54e55, 22a25. keywords: continuity space, de groot (cocompact) dual, de groot map, de groot skew compact semigroup, order-hausdorff space, skew compact space, saturated set, specialization order of a topology. 1. introduction skew compact spaces have been studied in some guise since at least 1948, when nachbin introduced them as “compact ordered spaces”; his work is most conveniently found in [10]. they have recently become important in computer science, as well as topology, since they are the spaces needed to approximate compact hausdorff spaces with finite t0 (rarely t2) spaces. the purpose of this paper is to show that many basic concepts of the theory of compact (hausdorff) semigroups can be extended to these spaces with relatively little change. in the 134 r. kopperman and d. robbie next section we give basic results, motivation, and references on skew compact spaces. in section 3 we extend the classically-known fact that compact (hausdorff) semigroup topologies arise from subinvarant pseudometrics by observing that skew compact semigroup topologies arise from subinvarant quasimetrics. the central result of section 4 is that skew compact cancellative semigroups with de groot continuous operations, are compact hausdorff groups; related results can also be found there. the final section shows that much of the basic structure theory of compact (hausdorff) semigroup can be extended to this situation. in all cases, a key difference between the traditional hausdorff situation and this non-hausdorff situation is the need to pay attention to the specialization order ≤τ . for it, and any binary relation ≤, we adopt the conventions ↑[s] = {y | (∃x ∈ s)(x ≤τ y)}, and ↓[s] = {y | (∃x ∈ s)(x ≥τ y)} (sometimes we may decorate the notation to indicate which relation we have in mind, e.g. ↑≤ [s]). definition 1.1. for any topology, τ, the (alexandroff ) specialization (order ), ≤τ is defined by x ≤τ y if x ∈ cl(y). thus, cl(x) =↓(x). the saturation of a set s ⊆ x is ↑[s], and a set s is ≤τ -saturated if it is a ≤τ −upper set, that is, if ↑[s] ⊆ s. it is easy to see that x ≤τ y if and only if cl(x) ⊆ cl(y), and so ≤τ is transitive and reflexive; it is a partial order if and only if τ is t0, and equality if and only if τ is t1. the study of spaces in which ≤τ is not assumed symmetric is called asymmetric topology. example 1.2. the upper topology on the unit interval i = [0, 1] is u = {(a, 1] | 0 ≤ a}∪{i}. notice that: • ≤u is the usual order on i, • the saturated sets are the upper sets, • the compact sets are those with a least element. 2. skew compact spaces except as noted below, in this section our notation and results are from [7]. it is characteristic of asymmetric topology that we must construct and study auxiliary topologies on the same space. our terminology is adapted as follows; topological terms (eg. open, cl(osure), continuous) when not modified, refer to τ; we use these notations with decorations to refer to auxiliary topologies (eg. *-open means open in τ∗, cls means the closure in the topology τs = τ ∨ τ∗, and *-continuous means continuous from (x,τ∗x) to (y,τ ∗ y )). definition 2.1. a t0 topological space (x,τ) is skew compact if there is a topology τ∗ on x such that: ≤τ∗ = ≤−1τ (that is, y ∈ cl(x) ⇔ x ∈ cl ∗(y)), (x,τ,τ∗) is pseudohausdorff, that is: whenever x 6∈ cl(y) then there are disjoint open t and *-open t∗ such that x ∈ t and y ∈ t∗, skew compact semigroups 135 τ ∨ τ∗ is compact. theorem 2.2. this second topology, τ∗ is uniquely determined by τ. it is its de groot dual: the topology τg whose closed sets are generated by the saturations of the τ-compact subsets of x. as a result of our discussion in 1.2, ug = l, that is, {[0,a) | 1 ≥ a}∪{i}. notice that for topological spaces (x,τx), (y,τy ), a function f : x → y is continuous from (x,τgx ) to (y,τ g y ) if and only if whenever f(x) 6∈ k, k compact saturated, then there is a compact, saturated l such that x 6∈ l and f−1[k] ⊆ l. a function is de groot if it is continuous with respect to both the original and de groot dual topologies. there are several ways of saying that only symmetry has been sacrificed in definition 2.1: theorem 2.3. a topological space (x,τ) is compact hausdorff if and only if it is skew compact, and any of the following equivalent conditions hold: (a) ≤τ is a symmetric relation, (b) ≤τ is equality (that is, τ is t1), (c) the second topology is equal to the first. what follows is a special case of the definition of continuity space in [6], which suffices for our uses: definition 2.4. a continuity space, is a set x together with two other sets, a,p and a function d : x ×x → a, where for some index set j: a ⊆ [0,∞]j contains 0,∞, and is closed under these pointwise operations: +, truncated −, multiplication by 1 2 , arbitrary ∨ , ∧ , d(x,x) = 0 and d obeys the triangle inequality, and finally, p ⊆ a is an upper set closed under finite ∧, multiplication by 1 2 , and such that if for a,b ∈ a, a ≤ b + r for each r ∈ p, then a ≤ b (p is called the set of positives ). from a continuity space, we get an induced topology, τd defined by t ∈ τd whenever for each x ∈ t there is an r ∈ p such that nr(x) = {y | d(x,y) ≤ r}⊆ t , and an induced quasiuniformity, qd = {u ⊆ x ×x | for some r ∈ p, nr ⊆ u}, where nr = {(x,y) | d(x,y) ≤ r}. we also get a dual, (x,d∗,a,p), where d∗(x,y) = d(y,x), and a symmetrization (x,ds,a,p), where ds = d + d∗. a continuity space is symmetric if for all x,y ∈ x,d(x,y) = d(y,x). topological notions for continuity spaces are defined in terms of the induced topologies, and uniform notions for them are defined in terms of the induced quasiuniformities; however, it can be shown that these are equivalent to extensions of the usual metric notions (but replace d(x,y) < r by d(x,y) ≤ r and 0 < r by r ∈ p). note that a quasiuniformity q also has a dual, q∗ = {u−1 | u ∈q} and a symmetrization, the join qs = q∨q∗. 136 r. kopperman and d. robbie theorem 2.5. a t0 topological space (x,τ) is skew compact if and only if any of the following (equivalent ) conditions occurs: (a) τ arises from a continuity space whose symmetrization is complete and totally bounded. any two such continuity spaces are uniformly equivalent. (b) τ arises from a quasiuniformity with complete, totally bounded symmetrization. there is exactly one such quasiuniformity. as is well-known, a topology is compact hausdorff if and only if, it comes from a uniformity which is complete and totally bounded (and, like each uniformity, is its own symmetrization). a similar equivalent is that it arises from a symmetric, complete and totally bounded continuity space. another useful characterization is given in topological ordered space terms: definition 2.6. recall from nachbin [10] that an order-hausdorff space, (x,τ, ≤), is a topological space together with a partial order closed in its square, x ×x. for such, τ≤ denotes the topology of open upper sets. if (x,τ,≤) is order-hausdorff then τ is hausdorff. theorem 2.7. a topological space (x,τ) is skew compact if and only if either of the following (equivalent ) conditions occurs: (a) there is a compact topology τs and a partial order ≤, both on x, such that (x,τs,≤) is order-hausdorff and τ = (τs)≤. (this topology and partial order are uniquely determined by τ: τs = τ ∨τg, and ≤=≤τ ). (b) (x,τ ∨ τg) is a compact hausdorff topological space. again, the space is compact hausdoff if and only if further, in (a), ≤ is equality, and in (b), τ = τg. 3. lengths on skew-compact semigroups definition 3.1. a length space is a continuity space (s,d,a,p) together with a semigroup operation on s for which d is subinvariant; that is, whenever a,b ∈ s∪{1} (1 an identity element added to s unless s has one already) and x,y ∈ x then d(axb,ayb) ≤ d(x,y). the above results from the following theorem in [8]. for it we need for any r ⊆ s ∪{1} the notation ir for the diagonal of r, {(r,r) | r ∈ r}. theorem 3.2. let · be an associative operation on s, q a quasiuniformity on s, and l,r ⊆ s closed under · be such that if u ∈q there are v,w ∈q such that ilw∪v ir ⊆ u. then there is a continuity space structure on s for which q is the induced quasiuniformity and such that whenever a ∈ l,b ∈ r,x,y ∈ s, we have d(axb,ayb) ≤ d(x,y). if further, q is a uniformity, then d may be chosen symmetric. skew compact semigroups 137 proof. (sketch; details in [8]): usual (quasi)uniform techniques yield for each u ∈ q a v ∈ q (symmetric in the uniform case) such that v ◦ v ◦ v ⊆ u and ilv ∪ v ir ⊆ v . as in kelley, p. 185, there is a quasimetric q on s (a pseudometric in the uniform case) such that u ∈ qq ⊆ q and whenever a ∈ l,b ∈ r,x,y ∈ s, we have q(axb,ayb) ≤ q(x,y). let k be a set of such quasimetrics containing one having the above property for each u ∈ q. then take our continuity space distance to be the product of the collection of quasimetric spaces so obtained. � the theory of hausdorff topological semigroups could have several possible generalizations to the asymmetric case. the two topologies τ, and τg are equal in the compact hausdorff case, so the statement that · : τ × τ → τ is continuous could be generalized by allowing g to appear on any of the τ’s. but mixed forms, such as · : τ × τg → τ are usually trivial since then · : {1}× (s,τg) → (s,τ), so τ ⊆ τg, and this forces τ = τg for skew compact topologies, so it puts us back into the hausdorff case. in our definition and below, if (s,τ, ·) is a skew compact space with semigroup operation, then we use the abbreviations s for (s,τ, ·) and sg for (s,τg, ·). definition 3.3. a skew compact (semi )group is a skew compact space (s,τ) with a (semi)group operation · : s ×s → s. a skew compact (semi)group is: continuous if · : s ×s → s is continuous; de groot if s and sg are continuous semigroups ( · is a de groot map). left continuous if the translation y → xy is continuous for each x ∈ s. left de groot if s and sg are left continuous (each y → xy is a de groot map). corollary 3.4. (a) the associative operation · on s, is uniformly continuous with respect to the quasiuniformity q on s, if and only if whenever u ∈ q there are v,w ∈q such that is∪{1}w ∪v is∪{1} ⊆ u. (b) de groot semigroup topologies are induced by length spaces. proof. (a) is a special case of theorem 3.2, while (b) results from the fact that de groot maps on skew compact spaces are uniformly continuous ([7], 3.8). � 4. cancellative semigroups some asymmetric topological semigroups in which the indicated operation is continuous, are: (a) (r, + ,u ), u the upper topology, {(a,∞) | −∞≤ a ≤∞}, (b) the circle group (t, ·); the indiscrete topology strictly weakens the usual compact hausdorff topology on this group. (c) the circle group (t, ·); the sorgenfrey topology strictly strengthens the usual compact hausdorff topology on this group. its symmetrization is the locally compact discrete topology. 138 r. kopperman and d. robbie theorem 4.1. the topology of a left continuous skew compact group is compact t2. proof. it will do to show that ≤τ is equality, since then the topology is compact t2. toward this end, notice that any continuous function is specializationpreserving: if x ≤τ y and f : (x,τ) → (y,τ′) then f(x) ∈ f(cl(y)) ⊆ cl′(f(y)), so f(x) ≤τ′ f(y). thus each left translation, y → xy, is specialization-preserving. now consider cl(e) = ↓≤τ e ; as a compact set, it contains a specialization-minimal element, g (in fact, this is the only element in the intersection of a maximal chain of closures of points). we show that g = e : otherwise, gg ≤τ ge = g ≤τ e, and · is cancellative, ruling out equality and contradicting the minimality of g. thus ≤τ is equality: for if k ≤τ h then h−1k ≤τ h−1h = e, so h−1k = e, thus k = h, as required. but then τ is compact t2, since it is skew compact, and its specialization is equality. � theorem 4.2. (a) any continuous skew compact group is a compact t2 topological group. (b) any de groot (two-sided ) cancellative semigroup is a compact t2 group. proof. (a) is immediate from theorem 4.1. for (b), note that de groot maps are pairwise continuous, thus s-continuous, and the symmetrization topology τs is compact t2. each (two-sided) cancellative compact t2 semigroup is wellknown to be a topological group; in particular, (s, · ,τs) is a topological group, so s is a group. thus by (a), (s, · ,τ) is a compact t2 topological group. � comments 4.3. (a) the argument in 4.2 can be used to show that algebraic facts about compact hausdorff semigroups hold about de groot semigroups, since the operation is continuous with respect to the compact hausdorff symmetrization topology. in particular, each de groot semigroup has an idempotent, and has maximal and minimal ideals. (b) if x is a de groot semigroup, x2 = x, and x has a unique idempotent, then x is a group. also, the operation is continuous, so by theorem 4.2 (a), it is a compact hausdorff topological group. 5. some structure results below, all semigroups will be assumed skew compact unless stated otherwise. some implications hold with weaker assumptions: eg. 5.1 (c) and (d) do not require any compactness assumption. also, each compact semigroup has minimal closed subsemigroups, left ideals, and right ideals, and a minimum closed ideal (simply adapt the first two paragraphs of the proof of 5.3 below). but ((0, .9], · ,l) is a compact space with continuous abelian semigroup operation, and has no idempotent. skew compact semigroups 139 comments 5.1. (a) compact hausdorff topological semigroups are de groot since · is assumed continuous and τg = τ. (b) let s = [0, 1] with the upper topology, u, and define ⊗ : s ×s → s by x⊗y = { 0 if xy = 0, 1 otherwise. then ⊗ is continuous with respect to the upper topology, because the one nontrivial open set in the subspace {0, 1} is {1}, and ⊗−1[{1}] = (0, 1]×(0, 1], an open set. but ⊗ is not continuous with respect to ug = l, for if so, it would be continuous with respect to their join, the usual topology, and it clearly is not. (c) closures of subsemigroups (resp. left, right, two-sided ideals) in continuous semigroups are subsemigroups (resp. left, right, two-sided ideals). here is the argument for subsemigroups, typical of the four: suppose that g is a subsemigroup, and x,y ∈ cl(g). if xy ∈ t , t open, then for some open u,v , x ∈ u, y ∈ v , and uv = ·[u × v ] ⊆ t . but since x,y ∈ cl(g), there are u ∈ g∩u, v ∈ g∩v , and uv ∈ (g∩u) · (g∩v ) ⊆ g∩uv ⊆ g∩t , so the latter is nonempty. this shows that xy ∈ cl(g). (d) if · is specialization-preserving, upper sets of subsemigroups (resp. left, right, two-sided ideals) in these semigroups are subsemigroups (resp. left, right, two-sided ideals). this is a special case of (c), since specialization-preserving functions are precisely ≥-alexandroff-continuous functions, and ≥-alexandroff closures are exactly upper sets. further, whenever s (or sg) is continuous, · is specialization-preserving. definition 5.2. an upper left (resp. right, two-sided ) ideal is an upper set which is also a left (resp. right, two-sided) ideal. by an ideal we mean a twosided ideal. theorem 5.3. if s is a continuous skew compact semigroup, then s contains a unique minimal upper ideal, and it is *-closed. further, it contains at least one minimal upper left ideal and at least one minimal upper right ideal; these are also *-closed. all minimal upper left ideals and all minimal upper right ideals are subsets of the minimal upper ideal. proof. x is a *-closed ideal of x. if m1, . . . ,mn is a finite number of *-closed ideals in x, then ∅ 6= m1 · · ·mn ⊆ m1∩·· ·∩mn; thus the set of *-closed ideals has the finite intersection property, and so has nonempty intersection in the compact (x,τ∗). this intersection is a *-closed, thus upper ideal, which we call m; it is clearly the smallest *-closed ideal. now let i ⊆ m be an upper ideal (not assumed *-closed). let m ∈ i; by the continuity of s, x{m}x is a compact ideal, so clg(x{m}x) = ↑(x{m}x) is a *-closed ideal which is a subset of i, thus of m. since m is minimal among *-closed ideals, clg(x{m}x) = m, so i = m. so m is the smallest upper ideal of x, and it is *-closed. by zorn’s lemma, there are minimal *-closed upper left and right ideals (not necessarily unique). by an argument similar to that ending the last paragraph, 140 r. kopperman and d. robbie these are minimal among the not necessarily *-closed upper left and upper right ideals, respectively. let l be a minimal *-closed left ideal of x. the saturated ↑(lm) is a subset of l, and by continuity it is also compact, so it is a *-closed left ideal, showing ↑ (lm) = l. but ↑ (lm) ⊆ m, since m is an upper (two-sided) ideal; thus l ⊆ m. a similar argument works for right ideals. � comments and examples 5.4. (a) in the hausdorff case, the ≤τ -upper ideals are simply the ideals; and so given a de groot semigroup s, the compact hausdorff topological semigroup (s, · ,τs) has a smallest ideal. further, if m is this smallest ideal, then ↑m is the smallest upper ideal of s: for if i is any upper ideal, then the ideal m ⊆ i, thus ↑m ⊆ i. also, ↑m is an ideal (by comments 5.1 (d)), and surely an upper set, so it is the smallest such. (b) let s = ([0, 1],u,×), where a×b = b. then s is de groot, its upper left ideals are the upper intervals, the *-closed ones are the closed upper intervals, and the minimal *-closed left ideal is {1}. but its only right or two-sided ideal is [0, 1]. thus in particular, the minimal upper ideal is not the disjoint union of the minimal *-closed left ideals. (c) the following is an example of a compact hausdorff semigroup (whose operation is not continuous) with no idempotent. let (n, + ) be the positive integers with the usual addition, and let τ be the topology in which each nontrivial sequence in n\{1} has 1 as a cluster point (that is, use the map n → { 1/(n− 1) if n > 1, 0 if n = 1, to identify n with {1/n | n ∈ n}∪{0}). since ≤τ is equality for hausdorff spaces, the operation is specialization-preserving. no translation is continuous since limk→1(n + k) = 1 6= n + limk→1 k. further, the only (*-) closed subsemigroup is the whole space, since if n is in our subsemigroup s, then each kn ∈ s, so 1 = limk→1 kn ∈ s, and so each k = k1 ∈ s. in [7] it is pointed out that for each skew compact topological space, the bitopological space (x,τ,τ∗) (τ∗ as in definition 2.1) is normal if c ⊆ t , where t ∈ τ and c is τ∗-closed, then for some u ∈ τ and τ∗-closed d, c ⊆ u ⊆ d ⊆ t . theorem 5.5. (a) if a ⊆ x, b ⊆ y are compact and a×b ⊆ t , t open, then for some open ta ⊇ a, tb ⊇ b, ta ×tb ⊆ t . (b) suppose s is a de groot skew compact semigroup. then each proper ∗-closed (left, right, two-sided ) ideal in s is contained in a proper open (left, right, two-sided ) ideal. proof. (a) this is a result of a. d. wallace (see [4], page 142). (b) we prove the two-sided case. for a ⊆ s let j(a) = cl∗(a∪sa∪as ∪ sas), certainly this is the smallest ∗-closed ideal containing a. now let i = i0 skew compact semigroups 141 be a proper ∗-closed ideal. if x 6∈ i then t = s \ cl(x) is a proper open subset of s containing i. this t will be kept fixed throughout the proof. by the normality of (s,τ,τ∗), find v ∈ τ such that i0 ⊆ v and cl∗(v ) ⊆ t . by compactness of s thus (a), for each y ∈ i there is an open uy such that y ∈ uy and j(uy) ⊆ cl∗(v ) ⊆ t ; by compactness of i0, there is a finite subcover of i0, ⋃ y∈f uy. let i1 = ⋃ y∈f j(uy). thus i0 ⊆ ⋃ y∈f uy ⊆ int(i1) and i1 is a ∗-closed ideal contained in t . proceed in this manner to obtain i2, . . . ; clearly ⋃∞ 0 in is an open upper ideal containing i which is contained in t , so is a proper subset of s. the other cases are similar. � references [1] p. fletcher and w. f. lindgren, quasi-uniform spaces, (dekker, new york, 1982). [2] j. de groot, an isomorphism principle in general topology bull. amer. math. soc. 73 (1967), 465–467. [3] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. mislove and d. s. scott, a compendium of continuous lattices, (springer-verlag, berlin, 1980). [4] j. l. kelley, general topology, van nostrand, new york, 1955. [5] j. c. kelly, bitopological spaces, proc. london math. soc. 13 (1963), 71–89. [6] r. d. kopperman, all topologies come from generalized metrics, amer. math. monthly 95 (1988), 89–97. [7] r. d. kopperman, asymmetry and duality in topology, topology and appl. 66 (1995), 1–39. [8] r. d. kopperman, lengths on semigroups and groups, semigroup forum 25 (1984), 345–360. [9] j. d. lawson, order and strongly sober compactifications, topology and category theory in computer science, g. m. reed, a. w. roscoe and r. f. wachter, eds. (oxford university press, 1991), 179–205. [10] l. nachbin, topology and order, van nostrand, 1965. [11] d. robbie and s. svetlichny, an answer to a. d. wallace’s question about countably compact cancellative semigroups, proc. amer. math. soc. 124 (1) (1996), 325–330. [12] s. salbany, bitopological spaces, compactifications and completions, math. monographs 1 (university of cape town, 1974). received march 2002 revised january 2003 r. d. kopperman department of mathematics, city college, city university of new york, new york, ny 10031. e-mail address : rdkcc@cunyvm.cuny.edu 142 r. kopperman and d. robbie d. robbie department of mathematics and statistics, the university of melbourne, parkville, vic 3010 australia. e-mail address : darobbie@unimelb.edu.au skew compact semigroups. by r. kopperman and d. robbie alaswilsonagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 231-242 which topologies can have immediate successors in the lattice of t 1-topologies? ofelia t. alas and richard g. wilson ∗ abstract. we give a new characterization of those topologies which have an immediate successor or cover in the lattice of t1-topologies on a set and show that certain classes of compact and countably compact topologies do not have covers. keywords: lattice of t1-topologies, maximal point, kc-space, cover of a topology, upper topology, lower topology 2000 ams classification: primary 54a10; secondary 06a06 1. introduction and preliminary results the lattice l1(x) of t1-topologies on a set x has been studied, with differing emphasis, by many authors. most articles on the subject have considered complementation in the lattice and properties of mutually complementary spaces (see for example, [12, 13, 15]. here we consider a facet of the order structure of the lattice l1(x), specifically the problem of when a jump can occur in the order; that is to say, when there exist topologies τ and τ+ on a set x such that whenever µ is a topology on x such that τ ⊆ µ ⊆ τ+ then µ = τ or µ = τ+. the existence of jumps in l1(x) has been studied in [10] and [16] where the immediate successor τ+ was said to be a cover of (or simply to cover) τ. we prefer order-theoretic terminology and then call τ a lower topology for x; the topology τ+ will then be termed an upper topology. that such topologies exist was noted in both of the previously mentioned papers the best known example is the topology of the space σ of problem 4m of [7], called an ultratopology in [10] and [16], which is an anti-atom of l1(ω) (for lattice-theoretic terminology we refer the reader to the survey paper [9]). however, topologies which are dense-in-themselves can also be lower, a tychonoff example is the maximal space constructed in [4]. a characterization of lower topologies was given in ∗research supported by consejo nacional de ciencia y tecnoloǵıa (méxico), grant 38164e and fundação de amparo a pesquisa do estado de são paulo (brasil) 232 o. t. alas and r. g. wilson theorem 1 of [10] in terms of equivalence classes of locally equal sets. here we give a somewhat different characterization of lower topologies which we find much simpler and easier to work with, but we note that lemma 1.2 and corollary 1.3 can, with a little effort, be derived from section 2 of [10]. corollary 1 to theorem 10 in [16] states that no first countable hausdorff topology on x has a cover in l1(x), thus generalizing example 2 of [10] where this same result was proved for the space of real numbers with the usual metric topology. however, these papers do not address further the problem: which topologies can have, and which classes of topologies do not have, covers in l1(x)? this is the problem we study below. all spaces considered below are (at least) t1 and undefined topological notation and terminology can be found in [6]. the closure (respectively, interior) of a set a in a topological space (x,τ) will be denoted by clτ (a) (respectively, intτ(a)) or simply by cl(a) (respectively int(a)) if no confusion is possible. if s is a family of sets, then we use the notation < s > to denote the topology generated by s as a subbase and the symbol is used to denote proper containment. we call a point p a maximal point of a topological space (x,τ) if it is not isolated and whenever u is open in x and p ∈ clτ(u), then u ∪ {p} ∈ τ. it is clear that if p is a maximal point of x and p is an accumulation point of a ⊆ x, then p is a maximal point of a ∪ {p}. using the well-known fact (see for example problem 12g of [18]) that an open filter f is an ultrafilter if and only if whenever u ∈ τ, then u ∈ f or x \ clτ (u) ∈ f, it is an easy exercise to check that p is a maximal point of x if and only if the trace of the open neighbourhood system at p on x \ {p} is an open ultrafilter in the open sets of this latter space. we formulate this fact as: proposition 1.1. a point p ∈ x is a maximal point if and only if the trace of the open neighbourhood filter at p on the subspace x \{p} is an open ultrafilter. we note that a maximal point is also a 1-point in the sense of [17] but not vice versa. it is also true that in a space which is maximal in the sense of [4], that is to say, is maximal with respect to being dense-in-itself, every point is a maximal point, but again the converse is false (the cofinite topology on a countable set is the requisite counterexample). our first task is to give a new characterization of lower topologies in terms of maximal points. lemma 1.2. let (x,τ) be a t1-space; if τ has an immediate successor, which we denote by τ+, then there is p ∈ x and u ∈ τ such that u ∪ {p} 6∈ τ and τ+ =< τ ∪ {u ∪ {p}} > proof. first note that if τ σ, then we can find v ∈ σ \ τ and hence the topology τ∗ generated by τ ∪ {v } is such that τ τ∗ ⊆ σ. thus if τ+ exists it must have the form < τ ∪ {v } > for some v 6∈ τ. we now claim that the set v defined in the previous paragraph must have the form u ∪ {p} for some p ∈ x \u and u ∈ τ; again let σ =< τ ∪ {v } >. to prove our claim, let u = intτ(v ); then if u = v , it follows that v ∈ τ and hence topologies with immediate successors 233 σ = τ. thus we suppose u v . if |v \ u| ≥ 2, then we can choose distinct points p,q ∈ v \u and since p 6∈ intτ(v ) it is clear that v \{q} 6∈ τ. we consider the topology τ∗ =< τ ∪{v \{q}} >; since x is t1 and v \{q} = v ∩(x \{q}), it follows that v \ {q} ∈ σ. however, since v 6∈ τ∗ we have that τ τ∗ σ, showing that σ is not an immediate successor of τ. thus |v \ u| = 1 and we are done. � to simplify the notation somewhat, we denote the topology < τ∪{u∪{p}} > by τu(p) or simply by τu whenever p is understood to be fixed. corollary 1.3. a topology τ on x is a lower topology if and only if there is p ∈ x and u ∈ τ such that whenever v ∈ τ is such that τu = τu∩v then either τv = τu or τv = τ. proof. suppose that τ is a lower topology, whose immediate successor we again denote by τ+. by lemma 1.2, there exist p ∈ x and u ∈ τ such that τ+ = τu(p)(= τu ). but then, if v ∈ τ is such that τu∩v = τv , then τ ⊆ τv and since v ∪ {p} = v ∪ [(u ∩ v ) ∪ {p}] it follows that τv ⊆ τu∩v = τu and hence τv = τu or τv = τ. conversely, suppose that τ is not a lower topology and let p ∈ x be fixed. if u ∈ τ, then there is some topology σ on x such that τ σ τu . it is clear that there is then some v ∈ τ such that σ = τv and then since v ∪ {p} ∈ τu , it follows that (u ∩ v ) ∪ {p} ∈ τu and hence τu∩v = τu, a contradiction. � let d be a directed set and f : d → x a net which is finally in u ∪ v . if f is cofinally in both u and v and du = {α ∈ d : f(α) ∈ u} and dv = {α ∈ d : f(α) ∈ v } then both du and dv are directed sets and if the nets fu = f|du and fv = f|dv both converge to the same point p, then f also converges to p. we will make use of this simple result in the next theorem. theorem 1.4. a topology τ is a lower topology on x if and only if (x,τ) has a closed subspace with a maximal point. proof. for the necessity, we suppose that no closed subspace of (x,τ) has a maximal point. then for all w ∈ τ and p ∈ x \w , p is not a maximal point of x \ w . however, if τ were a lower topology, then there would exist u ∈ τ and p ∈ x such that τ+ = τu(p) 6= τ. for the rest of this paragraph, we consider p fixed and write τu(p) = τu . since p is not a maximal point of x \ u, there is some open set v ′ ∈ τ|(x \ u) such that v ′ ∪ {p} 6∈ τ|(x \ u) and p ∈ clτ(v ′); then p ∈ clτ[(x \ u) \ (v ′ ∪ {p})]. let v ∈ τ be such that v ∩ (x \ u) = v ′. we claim that τ τu∪v τu . that τ ⊆ τu∪v ⊆ τu is clear and to see that (ı) τ 6= τu∪v , note that p 6∈ clτu∪v [(x \ u) \ (v ∪ {p})]. (ıı) τu∪v 6= τu , note that p is not isolated in τu∪v |(x \ u). for the converse, suppose that p is a maximal point of the closed subspace c ⊆ x and let x \ c = u ∈ τ. let v ∈ τ be such that τu∩v = τu , and hence τu = τu∩v ⊇ τv ⊇ τ; then by corollary 1.3, it suffices to show that τv = τ or τv = τu . there are two cases to consider. 234 o. t. alas and r. g. wilson (1) if p 6∈ clτ (v ∩ c), then a net f that converges to p in the topology τv is finally in each τ-open set containing p and (a) finally in v ∪ {p}, and (b) finally outside of v ∩ c = v \ u since c \ clτ(v ∩ c) is an open set in c which contains p. thus the net f is finally in (v ∩ u) ∪ {p} and in each τ-neighbourhood of p and hence converges to p in τu∩v = τu . since τu and τv coincide on x \ {p}, it follows that τu = τv . (2) if p ∈ clτ (v ∩c), then since p is a maximal point of c, (v ∩c)∪{p} ∈ τ|c and hence there is w ∈ τ such that p ∈ w and w ∩ c = (v ∩ c) ∪ {p}, that is, (v \ u) ∪ {p} = w \ u. but then, p ∈ w ⊆ (u ∪ v ) ∪ {p} and hence (u ∪ v ) ∪ {p} ∈ τ. since τ and τu∪v coincide on x \ {p}, it follows that τ = τu∪v . thus a net f which converges to p in τ is finally in each τ-neighbourhood of p, (including (u ∪ v ) ∪ {p}) and then either, (c) it is finally in u ∪ {p} and hence converges in τu and then also in τv since τv ⊆ τu , or (d) it is finally in v ∪ {p} and hence converges in τv , or (e) it is cofinally in both u ∪ {p} and v ∪ {p}. using the notation introduced in the paragraph prior to this theorem, the net fu converges to p in τu and hence also in τv and the net fv converges to p in τv . thus the net f converges to p in τv and so τ = τv . � on the other hand, τ can be a lower topology on x but have no maximal point. let p be a free ultrafilter on ω and let σ = ω ∪ {p} be the space of problem 4m of [7]. let (x,τ) be the quotient space obtained by identifying the point p ∈ σ with the unique accumulation point of a convergent sequence. that τ is a lower topology follows from theorem 1.4 and the fact that p is a maximal point of the closed subspace σ, but the space (x,τ) has no 1-point. corollary 1.5. if a topology σ on a set x has the property that for some infinite closed subspace d ⊆ x, σ|d is a lower (respectively, upper) topology on d, then σ is a lower (respectively upper) topology on x. at this point, having mentioned upper topologies for the first time, the reader might wonder why we have fixed our attention so exclusively on lower topologies since to every lower topology there corresponds at least one upper topology. however, as we shall see, unlike lower topologies, upper topologies are abundant. recall from [11] that a space is hyperconnected if it contains no disjoint non-empty open sets, or equivalently, if every non-empty open set is dense. clearly no hausdorff space with at least two points is hyperconnected. if σ is the cofinite topology on a set x, then (x,σ) is hyperconnected, but other examples of hyperconnected spaces are easy to construct. however, we have the following simple result which we feel sure must be known. lemma 1.6. a t1-topology σ on a set x is the cofinite topology if and only if every infinite closed subspace of (x,σ) is hyperconnected. topologies with immediate successors 235 proof. the necessity is clear. for the sufficiency, suppose that σ is not the cofinite topology on x; then there is some infinite closed proper subset c x. let p ∈ x \c, then the infinite closed subspace c ∪{p} is not hyperconnected. � proposition 1.7. a topology σ is not an upper topology on an infinite set x if and only if σ is the cofinite topology. proof. the sufficiency is clear, since the cofinite topology is the minimal t1topology on a set. for the necessity, suppose first that there is a non-empty regular closed set c in (x,σ) whose complement is infinite and let p ∈ intσ(c). let u be an open ultrafilter on the infinite open subspace x \ c such that ∩u = ∅ and define a topology τ on x by v ∈ τ if and only if v ∈ σ and whenever p ∈ v , then there exists u ∈ u such that u ⊆ v . clearly σ|(x \ {p}) = τ|(x \ {p}) and p is a maximal point (in the topology τ) of the τ-closed set (x \intσ(c))∪{p} and hence it follows from theorem 1.4 that τ is a lower topology. furthermore, if we denote by w the set intσ(c)\{p}, then w ∈ τ and we see from the proof of theorem 1.4 that σ = τ+ = τw(p). thus if σ is not an upper topology, then the closure of every non-empty element of σ has finite complement. now suppose that ∅ 6= u ∈ σ; if the finite open set x \ clσ(u) 6= ∅ for some u ∈ σ, each point q ∈ x \ clσ(u) is isolated and so {q} is a regular closed set with infinite complement, a contradiction. thus clσ(u) = x, that is u is dense in x and so x is hyperconnected. however, from corollary 1.5, we have that if σ is not an upper topology on x, then it is not an upper topology on any infinite closed subset of x. hence every infinite closed subspace of x is hyperconnected. the result now follows from lemma 1.6. � it is not hard to see that in proposition 1.7, if (x,σ) is hausdorff but not h-closed, then τ can be chosen to be hausdorff as well. 2. topologies which cannot be lower it is now clear that if p is a topological property which is inherited by closed subspaces, then the class of topologies with property p can contain a lower topology if and only if there is a member of the class with a maximal point. we now consider the problem of which classes of topologies do not contain a lower topology. as mentioned earlier, the best known examples of lower topologies, σ and van douwen’s maximal space are far from being compact. that this is not a coincidence is shown in the next theorem. recall that a topological space is a kc-space if all compact subsets are closed. hausdorff spaces are kc and kc-spaces are necessarily t1. our main results below show that kc-spaces with “nice” covering and convergence properties cannot be lower topologies. however, these results fail in case the kc separation axiom is weakened to t1. 236 o. t. alas and r. g. wilson theorem 2.1. a compact kc-space cannot have a maximal point. proof. let (x,τ) be a compact kc-space; we assume to begin with that x is dense-in-itself. suppose to the contrary that p is a maximal point of x and let k = ∩{cl(v ) : p ∈ v ∈ τ}. clearly, k is compact and p ∈ k. let λ be the minimal cardinal such that k = ∩{cl(vα) : α ∈ λ, p ∈ vα ∈ τ} and let fβ = ∩{cl(vα) : α ≤ β}. proceeding as in lemma 2.3 of [1], we can find a discrete subset d in x such that p 6∈ d and cl(d) ∩ fα 6= ∅ for each α ∈ λ. since x is compact, it follows that cl(d) ∩ k 6= ∅. there are now two possibilities: (ı) k = {p} and hence p ∈ cl(d). since x has no isolated points it follows that cl(d) is nowhere dense and hence u = x \ cl(d) is a dense open set. since p is maximal, u ∪ {p} is open, which contradicts the fact that p ∈ cl(d). (ıı) if k \ {p} 6= ∅, then choose q ∈ k \ {p}. if f is an open covering of x \ {p}, then there is u ∈ f such that q ∈ u and since p and q do not have disjoint neighbourhoods, p ∈ cl(u). since p is maximal, u ∪ {p} is open and hence f ∪ {u ∪ {p}} is an open covering of the compact space x, which then has a finite subcovering. since u ∈ f, this clearly induces a finite subcovering of f of x \ {p}, showing that this latter space is compact. since x is kc, p must be an isolated point of x, a contradiction. now suppose that x is an arbitrary compact kc-space, p ∈ x and let d be the set of isolated points of x. if p 6∈ cl(d), then there is some neighbourhood u of p which is dense-in-itself and hence p is a maximal point of cl(u) which is a dense-in-itself compact kc-space. that p is not maximal now follows from the previous argument. if p ∈ cl(d), then if p were a maximal point of x it would also be a maximal point of cl(d) and hence without loss of generality we assume that x = cl(d). since p is maximal, it follows that d ∪ {p} is open. let k = ∩{cl(v ) : v is an open neighbourhood of p}. if k \ {p} 6= ∅, then as in (ıı) above, p is an isolated point of x, a contradiction. if on the other hand k = {p}, then p can be separated from every point q ∈ x by disjoint open sets. since p is an isolated point of x \ d, it follows that (x \ d) \ {p} is compact and a standard argument now shows that p and (x \ d) \ {p} can be separated by open sets. thus there is some closed, hence compact neighbourhood v of p which misses (x \ d) \ {p}. thus v is a compact kc-space whose only accumulation point is a maximal point. this is clearly seen to be impossible and we are done. � corollary 2.2. a compact kc-topology is not a lower topology. in contrast with corollary 2.2, the topology τ of the one-point compactification x of the space σ is a lower topology and the cofinite topology on ω is even a first countable compact t1-topology in which every point is a maximal point. topologies with immediate successors 237 if (x,τ) is a hausdorff space, then for each p ∈ x, set ψc(p,x) = min{|u| : u ⊆ τ, p ∈ u for all u ∈ u and ⋂ {cl(u) : u ∈ u} = {p}}; as in [8], we then define the closed pseudocharacter of x, ψc(x) = sup{ψc(p,x) : p ∈ x}. recall that if λ is a cardinal, then a space x is said to be initially λ-compact if every open cover of size at most λ has a finite subcover. it is a trivial exercise to show that if x is initially λ-compact, then every decreasing sequence of length ≤ λ of closed sets in x has non-empty intersection. furthermore, an initially λ-compact hausdorff space x with ψc(x) ≤ λ is regular (see theorems 2.2 and 3.4 of [14]). the next theorem has a proof very similar to that of theorem 2.1, we mention only the minor points of difference. theorem 2.3. an initially λ-compact hausdorff space x with ψc(x) ≤ λ cannot have a maximal point. proof. again we begin by assuming that x is dense-in-itself, p is a maximal point of x and κ ≤ λ is the minimal cardinal such that there exists a family {vα : α ∈ κ} of open sets such that {p} = ∩{cl(vα) : α ∈ κ}. using exactly the same notation as that of theorem 2.1 and applying lemma 2.3 of [1], we can find a discrete set d ⊆ x \ {p} such that cl(d) ∩ fα 6= ∅ for each α ∈ κ. we note that since x is initially λ-compact, it is κ-compact and hence ∅ 6= ⋂ {cl(d) ∩ fα : α ∈ κ} ⊆ {p} and so p ∈ cl(d). the rest of the proof proceeds exactly as in theorem 2.1, after noting that x is, in fact, a t3-space. � a space x is said to be weakly discretely generated if whenever a x is not closed, then there is some discrete subset d ⊆ a such that cl(d) \ a 6= ∅. the space x is discretely generated if whenever x ∈ cl(a) there is a discrete subspace d ⊆ a such that x ∈ cl(d); being discretely generated is a hereditary property. clearly, a discretely generated space is weakly discretely generated and it was shown in proposition 3.1 of [5] that every compact hausdorff space is weakly discretely generated. a similar proof applying lemma 2.3 of [1] can be used to show that compact kc-spaces are weakly discretely generated. it was further shown in [2] (and implicitly earlier in [3]) that a regular countably compact space with countable tightness is discretely generated. our next lemma generalizes this last result and has a similar proof. we say that a space x is locally countably compact if each point p ∈ x has a local base of countably compact neighbourhoods. a countably compact t3-space is locally countably compact, but this is not the case for hausdorff spaces as we shall see later. it is easy to see that being locally countably compact is both open and closed hereditary. lemma 2.4. a locally countably compact hausdorff space x with countable tightness is discretely generated. proof. let a ⊆ x be a set which is not closed and y ∈ cl(a) \ a. since x has countable tightness, we may assume that a = {an : n ∈ ω} is countable. our aim is to construct a discrete subset d ⊆ a such that y ∈ cl(d). since 238 o. t. alas and r. g. wilson x is hausdorff, for each n ∈ ω there is an open neighbourhood un of y such that an 6∈ cl(un). we assume that un+1 ⊆ un for each n ∈ ω and clearly⋂ {cl(un) : n ∈ ω} ∩ a = ∅. now let s be the set of all cluster points of sets of the form {zn : n ∈ ω} where zn ∈ un ∩ a; we claim that y ∈ cl(s). to see this, let w be a countably compact neighbourhood of y and for each n ∈ ω, pick zn ∈ w ∩ un ∩ a. each accumulation point of {zn : n ∈ ω} lies in s, and since w is countably compact at least one of them also lies in w ; hence w ∩s 6= ∅. again since x has countable tightness, there is a countable subset {sn : n ∈ ω} ⊆ s such that y ∈ cl({sn : n ∈ ω}) and for each n ∈ ω we choose a sequence {zkn : n ∈ ω} ⊆ a which witnesses the fact that sk ∈ s, that is to say, zkn ∈ un ∩a and sk ∈ cl({z k n : n ∈ ω}). now let d = {z k n : k,n ∈ ω and k ≤ n}. it is clear that sk ∈ cl(d) for each k ∈ ω and hence y ∈ cl(d). to see that d is discrete, fix k,n ∈ ω; then there is some m ≥ n such that zkn 6∈ cl(um) and since all but finitely many elements of d lie in um, z k n is not an accumulation point of d. � theorem 2.5. a locally countably compact hausdorff topology of countable tightness is not a lower topology. proof. let (x,τ) be a locally countably compact hausdorff space and suppose that τ has an immediate successor which we denote by σ. then there is some u ∈ τ and p ∈ x such that σ =< τ ∪ {u ∪ {p}} >. furthermore, since the property of being locally countably compact is inherited by closed subspaces, we may assume that p is a maximal point of x. let s = x \ u and note that p is an accumulation point of the closed subspace s, for otherwise we have u ∪ {p} ∈ τ and hence σ = τ. furthermore, by the previous lemma, s is discretely generated. there are two possibilities to consider. 1) the point p is an accumulation point of some dense-in-itself closed subset c of s. since x is discretely generated, there is some discrete subset d in c \ {p} such that p ∈ clτ(d). furthermore, since c is dense-in-itself, t = clτ(d) is nowhere dense in c, hence also in s, and so p ∈ clτ(s \t). now define τ # =< τ ∪ {(x \ t) ∪ {p} >; since p ∈ clτ(t \ {p}) it follows immediately that t \ {p} is not closed in τ, but is closed in τ# and hence τ τ#. furthermore, since (x\t)∪{p} = ((x\s)∪{p})∪(x\t) it follows that (x\t)∪{p} ∈ σ, showing that τ# ⊆ σ. finally, since each τ#-neighbourhood of p meets s \ t , but the σ-neighbourhood u ∪{p} of p misses s \t , we have p ∈ clτ#(s \t)\ clσ(s \t) and it then follows that τ# 6= σ, contradicting the supposition that σ is the immediate successor of τ. 2) suppose now that p is not in the closure of any dense-in-itself subset of s. if s is not scattered, then let g be the family of all dense-in-themselves subsets of s ordered by set inclusion. clearly the union of any chain of elements of g is dense-in-itself, and hence g has a maximal element g say, which clearly must be closed and p 6∈ g. since s is closed in x, it is locally countably compact and so there is a countably compact neighbourhood w of p in s missing g. then p is an accumulation point of the countably compact scattered space w ⊆ s. suppose that the scattering length of w is κ and for each α < κ, let wα denote topologies with immediate successors 239 the set of points of scattering order α; since p is not an isolated point of w , p ∈ wα for some α > 0. there are two subcases to consider: a) if p ∈ wα where α > 1, then let t = w \ w0. it is clear that t is a closed subspace of w and hence t = clτ (t)∩w , p is an accumulation point of t and w \ t is dense in w . now define τ# =< τ ∪ {(x \ clτ(t)) ∪ {p} >. obviously clτ(t) \ {p} is closed in τ # but not in τ and hence τ τ#. furthermore, (x \ clτ (t)) ∪ {p} = (x \ s) ∪ {p} ∪ (x \ clτ (t)) ∈ τ and so τ # ⊆ σ. to show that σ 6= τ# we will show that p ∈ clτ#(s\t)\clσ(s\t). that p 6∈ clσ(s\t) is clear since u ∪{p} ∈ σ is a neighbourhood of p and (s\t)∩(u ∪{p}) = ∅. on the other hand, an open τ#-neighbourhood of p is of the form (v \clτ (t))∪{p} where p ∈ v ∈ τ and a short calculation now shows that (v \ clτ (t)) ∪ {p} ∩ (s \t) = v \ (u ∪ clτ(t)). this latter set is nonempty since if v ⊆ u ∪ clτ (t), then v ∩ w ⊆ clτ (t) ∩ w = t which is a contradiction, since each open set in w meets w0. thus we have again shown that σ is not the immediate successor of τ. b) if p ∈ w1, then w0 ∪ w1 is an open neighbourhood of p in w and since w is a neighbourhood of p in s, we have p ∈ intτ|s(w0 ∪ w1) and this latter set, being an open subset of s, is locally countably compact. thus there is some countably compact neighbourhood v of p in which p is the only accumulation point. clearly v is compact and it then follows from theorem 2.1 that p is not a maximal point of v , a contradiction. � question 2.6. can a countably compact hausdorff topology with countable tightness (or a countably compact regular topology) be a lower topology? we note however that a countably compact hausdorff topology may be a lower topology and need not be locally countably compact. let p ∈ βω \ ω and let v be the filter of open neighbourhoods of p. denoting by τ the usual topology of βω, we define σ =< τ ∪ {(u ∩ ω) ∪ {p} : u ∈ v} >. it is straightforward to check that p is a maximal point of the countably compact urysohn space (βω,σ). this latter space is even h-closed (see 3.12.5 of [6]) and its semiregularization is (βω,τ). our next theorem generalizes corollary 1 to theorem 10 of [16]. theorem 2.7. a sequential kc-space is not a lower topology. proof. suppose (x,τ) is a sequential kc-space and τ σ. then there is some set v ⊆ x which is closed in (x,σ) but not closed in (x,τ). since this latter space is sequential, there is a sequence s = {bn}n∈ω in v which converges to b 6∈ v . since (x,τ) is kc, s∪{b} is τ-closed and hence σ-closed and so s is σclosed and discrete. let µ =< τ ∪{x \{b2n : n ∈ ω}} >. then τ µ σ since {b2n+1}n∈ω converges in (x,µ) but not in (x,σ), while {b2n}n∈ω converges in (x,τ) but not in (x,µ). � corollary 2.8. between any two distinct comparable sequential kc-topologies on a set x, there are an infinite number of topologies. 240 o. t. alas and r. g. wilson we note in passing that all we have used in theorem 2.7 is that sequences have unique limits. on the other hand, as with corollary 2.2, both the theorem and its corollary are false if kc is replaced by the t1 separation axiom: if τ is the cofinite topology on a countably infinite set x, then (x,τ) is second countable, but each point p ∈ x is maximal and (x,τ) is a lower topology. it is also easy to see that any successor topology to τ is also second countable. however, the next theorem shows that the cofinite topology is crucial in the construction of a first countable lower topology. theorem 2.9. a sequential t1-space with a maximal point contains an infinite subspace whose relative topology is cofinite. proof. suppose (x,τ) is a sequential t1-space with a maximal point p. since p is not isolated, we can choose a sequence of distinct points s = {xn}n∈ω ⊆ x \ {p} converging to p and hence p is an accumulation point of the subspace s ∪ {p}. if the set of isolated points i of (s,τ|s) is infinite, then p ∈ cl(i) and if i = {dn : n ∈ ω} is an enumeration of i, then p ∈ cl({d2n : n ∈ ω}) ∩ cl({d2n+1 : n ∈ ω}) showing that p is not a 1-point, thus is not maximal in i ∪ {p}, hence not maximal in x, which contradicts our hypothesis. thus i is finite and by replacing s with s \ i we may, without loss of generality, assume that s has no isolated points. suppose that u ∈ τ|s; since s has no isolated points, u is infinite and hence p ∈ cl(u). since p is a maximal point of s ∪ {p}, it follows that u ∪ {p} ∈ τ|(s ∪ {p}). if d = {n : xn ∈ s \ u} is infinite then {xn : n ∈ d} is a subsequence of s which does not converge to p, again a contradiction. thus s \ u is finite, showing that s has the cofinite topology. � corollary 2.10. if τ is a first countable lower t1-topology on x then there is an infinite subset s ⊆ x such that τ|s is the cofinite topology. as a partial converse to theorem 2.9, it follows from theorem 1.4, that if a space has an infinite closed subspace with the cofinite topology, then its topology is lower. however, a first countable t1-space may have an infinite subset with the cofinite topology and still not be lower as the following example shows: let µ denote the usual metric topology on the set of reals r and define a new topology σ on r as follows: u ∈ σ if and only if u ∈ µ and there is some ǫ > 0 such that u ⊇ (n − ǫ,n + ǫ) for all but finitely many n ∈ n. clearly σ is a first countable t1 topology on r and σ|n is the cofinite topology on n. further note that if a is a bounded set in r, then σ|a = µ|a. to show that (r,σ) is not a lower topology, it suffices to show that no closed subspace has a maximal point. let c be a closed subset of (r,σ) and p ∈ c; if c ∩n is finite, then for some m ∈ n, p ∈ a = c ∩ (−m,m) and σ|a = µ|a. thus by theorem 2.7, p is not a maximal point of a; now a little thought shows that p is an accumulation point of a if and only if it is an accumulation point of c and it then follows that p is not a maximal point of c. if c ∩ n is topologies with immediate successors 241 infinite, then c is dense in (r,σ) and hence c = r. however, if p ∈ r, then we define u = (p,p + 1 2 ) ∪ ⋃ {(n − 1 2 ,n + 1 2 ) : n ∈ n \ (p − 1,p + 1)}. that σ is not a lower topology now follows since u ∈ σ, p ∈ clσ(u), but u ∪ {p} 6∈ σ showing that p is not a maximal point of (r,σ). recall that a space x is radial if whenever a ⊆ x and x ∈ cl(a) there is a well-ordered net in a converging to x. a space x is pseudoradial if whenever a x is not closed there is a well-ordered net in a which converges to a point of x \ a. every first countable space is radial and both sequential and radial spaces are pseudoradial; clearly, being radial is a hereditary property. theorem 2.11. a radial hausdorff space cannot have a maximal point. proof. suppose that (x,τ) is a radial hausdorff space; by theorem 4.3 of [1], x is discretely generated. suppose that p ∈ x is not isolated. since x is discretely generated, there is some discrete set d ⊆ x\{p} such that p ∈ cl(d). since x is radial, there is a well-ordered net {dα}α∈κ in d converging to p. let p = {dα : α ∈ κ}, s = {dα : α = γ + 2n for some limit ordinal γ ∈ κ and n ∈ ω} and t = {dα : α = γ + 2n + 1 for some limit ordinal γ ∈ κ and n ∈ ω}. clearly both s and t are disjoint open subsets of the subspace p ∪ {p} and p ∈ cl(s) ∩ cl(t) showing that p is not a 1-point, thus is not maximal in the space p ∪ {p} and hence is not maximal in x. � theorem 2.11 cannot be extended to kc-spaces since a set of size ω1 with the cocountable topology is a radial kc-space, each point of which is maximal. (this space even has the stronger property that each point has a nested local base.) the question then arises: question 2.12. can a pseudoradial t2-space have a maximal point? preceding theorem 2.7, we gave an example of an h-closed space with a maximal point. thus the following question arises: question 2.13. can a minimal hausdorff topology be a lower topology? references [1] o. t. alas, v. v. tkachuk and r. g. wilson, closures of discrete sets often reflect global properties, topology proceedings 25 (2000), 27-44. [2] a. bella, few remarks on spaces which are generated by discrete sets, to appear. [3] a. bella and v. i. malykhin, f-points in countably compact spaces, applied general topology 2 no. 1 (2001), 33-37. [4] e. van douwen, applications of maximal topologies, topology and its applications 51 (1993), 125-139. [5] a. dow, m. g. tkachenko, v. v. tkachuk and r. g. wilson, topologies generated by discrete subspaces, glasnik mat. ser. iii, 37 (57) no. 1 (2002), 187-210. [6] r. engelking, general topology, (heldermann verlag, berlin, 1989). [7] l. gillman and m. jerison, rings of continuous functions, (van nostrand, princeton, nj, 1960). [8] i. juhász, cardinal functions in topology ten years later, mathematical centre tracts 123, (amsterdam, 1980). 242 o. t. alas and r. g. wilson [9] r. e. larson and s. andima, the lattice of topologies; a survey, rocky mountain j. of math. 5 no. 2 (1975), 177-198. [10] r. e. larson and w. j. thron, covering relations in the lattice of t1-topologies, transactions of the american mathematical society 168 (1972), 101-111. [11] l. a. steen and j. a. seebach, counterexamples in topology, (springer verlag, new york, 1978). [12] a. k. steiner, complementation in the lattice of t1-topologies, proceedings of the american mathematical society 17 (1966), 884-885. [13] e. f. steiner and a. k. steiner, topologies with t1-complements, fundamenta mathematicae 61 (1967), 23-28. [14] r. m. stephenson, initially κ-compact and related spaces, in handbook of set-theoretic topology, k. kunen and j. e. vaughan, (eds.), (north holland, amsterdam, 1984). [15] m. g. tkachenko, v. v. tkachuk, r. g. wilson and i. v. yaschenko, no submaximal topology on a countable set is t1-complementary, proceedings of the american mathematical society 128 no. 1 (1999), 287-297. [16] r. valent and r. e. larson, basic intervals in the lattice of topologies, duke math. j. 39 (1972), 401-411. [17] r. c. walker, the stone-čech compactification, (springer-verlag, new york, 1974). [18] s. willard, general topology, (addison wesley, reading, mass., 1970). received may 2003 accepted november 2003 o. t. alas (alas@ime.usp.br) instituto de matemática e estat́ıstica, universidade de são paulo, caixa postal 66281, 05311-970 são paulo, brasil. r. g. wilson (rgw@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, unidad iztapalapa, avenida san rafael atlixco, #186, apartado postal 55-532, 09340, méxico, d.f., méxico. caogreistagt04.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 1, 2005 pp. 79-86 δ-closure, θ-closure and generalized closed sets j. cao, m. ganster, i. reilly and m. steiner abstract. we study some new classes of generalized closed sets (in the sense of n. levine) in a topological space via the associated δ-closure and θ-closure. the relationships among these new classes and existing classes of generalized closed sets are investigated. in the last section we provide an extensive and more or less complete survey on separation axioms characterized via singletons. 2000 ams classification: 54a05, 54a10, 54d10, 54f65. keywords: δ-closed, θ-closed, qr-closed, separation properties. 1. introduction and preliminaries let (x, τ) be a topological space. recall that a point x ∈ x is said to be in the δ-closure (resp. θ-closure) of a subset a ⊆ x (see [15]) if for each open neighbourhood u of x we have int(cl(u)) ∩ a 6= ∅ (resp. cl(u) ∩ a 6= ∅) . we shall denote the δ-closure (resp. θ-closure) of a by clδ(a) (resp. clθ(a)). a subset a ⊆ x is called δ-closed (resp. θ-closed) if a = clδ(a) (resp. a = clθ(a)). the complement of a δ-closed (resp. θ-closed) set is called δ-open (resp. θ-open). it is very well known that the families of all δ-open (resp. θopen) subsets of (x, τ) are topologies on x which we shall denote by τδ (resp. τθ). from the definitions it follows immediately that τθ ⊆ τδ ⊆ τ . the space (x, τδ) is also called the semi-regularization of (x, τ) . a space (x, τ) is said to be semi-regular if τδ = τ . (x, τ) is regular if and only if τθ = τ . it should be noted that clδ(a) is the closure of a with respect to (x, τδ). in general, clθ(a) will not be the closure of a with respect to (x, τθ). it is easily seen that one always has a ⊆ cl(a) ⊆ clδ(a) ⊆ clθ(a) ⊆ a θ where a θ denotes the closure of a with respect to (x, τθ). definition 1.1. a subset a of a space (x, τ) is called (i) α-closed if cl(int(cl(a))) ⊆ a , (ii) α-open if x \ a is α-closed, or equivalently, if a ⊆ int(cl(int(a))), 80 j. cao, m. ganster, i. reilly and m. steiner (iii) semi-closed if int(cl(a)) ⊆ a , (iv) semi-open if x \a is semi-closed, or equivalently, if a ⊆ cl(int(a)), (v) preclosed if cl(int(a)) ⊆ a, (vi) preopen if x \ a is preclosed, or equivalently, if a ⊆ int(cl(a)), (vii) β-closed if int(cl(int(a))) ⊆ a, (viii) β-open if x \ a is β-closed, or equivalently, if a ⊆ cl(int(cl(a))). for a subset a of (x, τ) the α-closure (resp. semi-closure, preclosure, β-closure) of a is the smallest α-closed (resp. semi-closed, preclosed, β-closed) set containing a. these closures are denoted by clα(a), cls(a), clp(a) and clβ(a), respectively. it is known that clα(a) = a ∪ cl(int(cl(a))), cls(a) = a ∪ int(cl(a)), clp(a) = a ∪ cl(int(a)) and clβ(a) = a ∪ int(cl(int(a))). for the sake of completeness, a subset a of (x, τ) is called regular open (resp. regular closed, nowhere dense) if a = int(cl(a)) (resp. a = cl(int(a)), int(cl(a)) = ∅). it is well known that the family of regular open subsets of (x, τ) form a base for τδ . in 1970, n. levine [11] defined a subset a of a space (x, τ) to be generalized closed (briefly, g-closed) if cl(a) ⊆ u whenever a ⊆ u and u ∈ τ . by considering other generalized closures or classes of generalized open sets, numerous additional notions analogous to levine’s g-closed sets have been introduced. we refer the reader to [1] for further details. in 2001, cao, greenwood and reilly [3] provided a general framework to deal with these notions by introducing the concept of a qr-closed set. for convenience it is useful to denote closed (resp. semi-closed, preclosed) by τ-closed (resp. s-closed, p-closed), and cl(a) by clτ(a) for a subset a ⊆ x . similarly, open (resp. semi-open, preopen) are denoted by τ-open (resp. s-open, p-open). if p = {τ, α, s, p, β} and q, r ∈ p then a subset a ⊆ x is called qr-closed if clq(a) ⊆ u whenever a ⊆ u and u is r-open. using this notation, a set a is g-closed if and only if it is ττ-closed, and most types of generalized closed sets can be captured within this notation. one basic result (theorem 2.5 in [3]) says that if q, r ∈ p then every qr-closed subset of (x, τ) is q-closed if and only if each singleton of x is either q-open or r-closed. the aim of this paper is to continue the discussion initiated in [3] by considering the expanded family p∗ = {τ, α, s, p, β, δ, θ} . it is easily observed that theorem 2.5 in [3] still remains valid. remark 1.2. if q, r ∈ p∗ then every qr-closed subset of (x, τ) is q-closed if and only if each singleton of x is either q-open or r-closed. so far we are aware of three relevant notions of generalized closed sets that have appeared in the literature for the δ-topology. a subset a of a space (x, τ) is called (i) δg-closed [6] if clδ(a) ⊆ u whenever a ⊆ u and u ∈ τ , (ii) gδ-closed [5] if clτ(a) ⊆ u whenever a ⊆ u and u ∈ τδ , (iii) δg∗-closed [5] if clδ(a) ⊆ u δ-closure, θ-closure and generalized closed sets 81 whenever a ⊆ u and u ∈ τδ . in terms of the qr-closed notation of [3], (i) is equivalent to δτ-closed, (ii) is equivalent to τδ-closed and (iii) is equivalent to δδ-closed. the most important notion of generalized closed set involving the θ-topology is due to dontchev and maki. a ⊆ x is said to be (iv) θg-closed [7] if clθ(a) ⊆ u whenever a ⊆ u and u ∈ τ . clearly, (iv) is equivalent to θτ-closed. 2. δθ-closed sets and θδ-closed sets we shall investigate what happens when we mix δ and θ in the context of generalized closed sets. definition 2.1. a subset a of a space (x, τ) is called (i) δθ-closed, if clδ(a) ⊆ u whenever a ⊆ u and u ∈ τθ, (ii) θδ-closed, if clθ(a) ⊆ u whenever a ⊆ u and u ∈ τδ. remark 2.2. obviously every δ-closed (resp. θ-closed) set is δθ-closed (resp. θδ-closed). since τθ ⊆ τδ ⊆ τ , every θδ-closed set is δθ-closed. if x ∈ u and u ∈ τθ then there is v ∈ τ such that x ∈ v ⊆ cl(v ) ⊆ u . since cl(v ) is δ-closed we have clδ({x}) ⊆ u, i.e. every singleton in any space is always δθ-closed. now let x be an infinite set and p ∈ x . let τ be the topology on x consisting of x and all subsets of x not containing p . if x 6= p then {x} is δ-open and cl({x}) = {x, p} ⊆ clθ({x}) . thus {x} is δθ-closed but fails to be θδ-closed. clearly every δg-closed subset is δg∗-closed, and every δg∗-closed subset is δθ-closed. moreover, every θg-closed subset is θδ-closed. consider the space (x, τ) in remark 2.2. if x 6= p then {x} is δθ-closed but obviously not δg∗closed. now let x be an infinite set and τ be the cofinite topology on x. then τθ = τδ = {∅, x} hence every subset of x is θδ-closed. if a is a proper cofinite subset of x then a is θδ-closed but not θg-closed. remark 1.2 suggests the consideration of the following properties as candidates for possibly new separation properties. definition 2.3. a space (x, τ) satisfies property (i) a if every δθ-closed set is δ-closed, i.e. each singleton is either δ-open or θ-closed, (ii) b if every θδ-closed set is θ-closed, i.e. each singleton is either θ-open or δ-closed. we shall see, however, that we do not obtain any new separation axioms. recall that a space (x, τ) is said to be t1/2 [8] if each singleton is either open or closed. (x, τ) is called weakly hausdorff [13] (resp. almost weakly hausdorff [6]) if (x, τδ) is t1 (resp. t1/2). we also mention the folklore result that a space (x, τ) is hausdorff if and only if (x, τθ) is t1 if and only if (x, τθ) is t1/2 if and only if (x, τθ) is t0. 82 j. cao, m. ganster, i. reilly and m. steiner theorem 2.4. for a space (x, τ) the following are equivalent: (a) (x, τ) is hausdorff, (b) (x, τ) satisfies a, (c) (x, τ) is almost weakly hausdorff and δ-closed singletons are θ-closed. proof. (a) ⇒ (b): if (x, τ) is hausdorff then (x, τθ) is t1, i.e. singletons are θ-closed. thus (x, τ) satisfies a. (b) ⇒ (a): if (x, τ) satisfies a then, by remark 2.2, each singleton is either δ-clopen or θ-closed. hence (x, τθ) is t1 and thus (x, τ) is hausdorff. (b) ⇒ (c): suppose that (x, τ) satisfies a. then each singleton is clearly either δ-open or δ-closed, i.e. (x, τ) is almost weakly hausdorff. if {x} is δ-closed then {x} is either δ-clopen or θ-closed, hence always θ-closed. (c) ⇒ (b): this is obvious. � theorem 2.5 ([14]). for a space (x, τ) the following are equivalent: (a) (x, τ) is weakly hausdorff, (b) (x, τ) satisfies b. proof. (a) ⇒ (b): if (x, τ) is weakly hausdorff then each singleton is δ-closed. hence (x, τ) satisfies b. (b) ⇒ (a): this follows from the fact that each θ-open singleton must be clopen. � 3. separation axioms characterized via singletons remark 1.2 suggests to characterize the topological spaces in which each singleton is either q-open or r-closed, where q, r ∈ p∗ = {τ, α, s, p, β, δ, θ} . we shall first present what is already known and then answer the remaining cases. to do this we need some further preparation. observation 3.1. let (x, τ) be a space and let x ∈ x . (a) {x} is either preopen or nowhere dense [10], (b) {x} is either open or preclosed, (c) {x} is open ⇔ {x} is α-open ⇔ {x} is semi-open, (d) {x} is preopen ⇔ {x} is β-open, (e) {x} is nowhere dense ⇒ {x} is α-closed and thus semi-closed, preclosed and β-closed, (f) {x} is semi-closed ⇔ {x} is nowhere dense or regular open. definition 3.2. a space (x, τ) is said to be (i) semi-t1 (resp. pre-t1, β-t1) if each singleton is semi-closed (resp. preclosed, β-closed), (ii) a t3/4 space [6] if each singleton is either δ-open or closed, (iii) semi-t1/2 if each singleton is either semi-open or semi-closed, (iv) feebly t1 [12] if each singleton is either nowhere dense or clopen, (v) tgs [2] if each singleton is either preopen or closed. δ-closure, θ-closure and generalized closed sets 83 the table below exhibits what is already known in the literature and what has been obtained in section 2. it has to be read in the following way: each column represents a q-open set and each row represents a r-closed set. according to observation 3.1 we only need to consider q ∈ {p, τ, δ, θ} . a separation axiom (p) in a cell means that a space (x, τ) satisfies (p) if and only if each singleton is either q-open or r-closed. the symbol ” √ ” in a cell means that in any space each singleton is either q-open or r-closed. finally, the symbol ”?” in a cell means that the property in question has yet to be determined. we also observe that ”a.w. t2” (resp. ”w. t2”) means almost weakly hausdorff (resp. weakly hausdorff). the present entries in our table can easily be verified by observation 3.1 and definition 3.2. the result that (x, τ) is almost weakly hausdorff if and only if each singleton is either open or δ-closed can be found in [5]. preopen open δ-open θ-open β-closed √ √ ? ? preclosed √ √ ? ? semi-closed √ semi-t1/2 ? ? α-closed √ ? ? ? closed tgs t1/2 t3/4 ? δ-closed ? a.w. t2 a.w. t2 w. t2 θ-closed ? t2 t2 t2 as an immediate consequence of observation 3.1 we note that a space (x, τ) is semi-t1/2 if and only if each singleton is either α-closed or open. proposition 3.3 ([14]). for a space (x, τ) the following are equivalent: (a) (x, τ) is semi-t1 , (b) each singleton is either θ-open or semi-closed, (c) each singleton is either δ-open or semi-closed, (d) each singleton is either δ-open or α-closed. proof. (a) ⇒ (b) ⇒ (c) is obvious. (c) ⇒ (d) follows from observation 3.1 and (d) ⇒ (a) is clear. � by observing that a θ-open singleton must be clopen and observation 3.1 we have that a space (x, τ) is feebly t1 if and only if each singleton is either θopen or α-closed. by a similar argument, (x, τ) is pre-t1 if and only if each singleton is either θ-open or preclosed. in addition, (x, τ) is t1 if and only if each singleton is either closed or θ-open. proposition 3.4 ([14]). for a space (x, τ) the following are equivalent: (a) (x, τ) is β-t1 , (b) each singleton is either θ-open or β-closed, (c) each singleton is either δ-open or β-closed, (d) each singleton is either δ-open or preclosed. 84 j. cao, m. ganster, i. reilly and m. steiner proof. (a) ⇒ (b) ⇒ (c) is obvious. to show that (c) ⇒ (d) let x ∈ x such that {x} is β-closed. if int({x}) = ∅ then {x} is preclosed. otherwise, {x} is open and β-closed and so regular open, i.e. δ-open. (d) ⇒ (a) is clear. � definition 3.5. a space (x, τ) is called (i) r1 if two points x and y have disjoint neighbourhoods whenever cl({x}) 6= cl({y}) , (ii) subweakly t2 [4] if clδ({x}) = cl({x}) for each x ∈ x , (iii) pointwise semi-regular (briefly p-semi-regular) [5] if each closed singleton is δ-closed, (iv) pointwise regular [14] if each closed singleton is θ-closed. proposition 3.6. let (x, τ) be a space. (a) if a ⊆ x is preopen then cl(a) = clθ(a) , (b) (x, τ) is r1 if and only if cl({x}) = clθ({x}) for each x ∈ x. proof. (a) the proof is straightforward, hence it is omitted. the proof of (b) is due to jankovic [9]. � theorem 3.7. for a space (x, τ) the following are equivalent: (a) each singleton is either θ-closed or preopen, (b) (x, τ) is tgs and r1, (c) (x, τ) is tgs and pointwise regular. proof. (a) ⇒ (b) : suppose that each singleton is either θ-closed or preopen. then (x, τ) clearly is tgs. let x ∈ x. if {x} is preopen then cl({x}) = clθ({x}) by proposition 3.6. if {x} is θ-closed then {x} = clθ({x}) = cl({x}). hence (x, τ) is r1. (b) ⇒ (c) : this follows immediately from proposition 3.6. (c) ⇒ (a) : follows straightforward from the definitions. � theorem 3.8. for a space (x, τ) the following are equivalent: (a) each singleton is either δ-closed or preopen, (b) (x, τ) is tgs and subweakly t2, (c) (x, τ) is tgs and p-semi-regular. proof. (a) ⇒ (b) : suppose that each singleton is either δ-closed or preopen. then (x, τ) clearly is tgs . let x ∈ x . if {x} is preopen then cl({x}) = cl(int(cl({x}))) , i.e. cl({x}) is regular closed and so cl({x}) = clδ({x}). if {x} is δ-closed then obviously we have cl({x}) = clδ({x}). thus (x, τ) is subweakly t2. (b) ⇒ (c) ⇒ (a) is clear. � δ-closure, θ-closure and generalized closed sets 85 as our final result we are now able to present the complete table. preopen open δ-open θ-open β-closed √ √ β-t1 β-t1 preclosed √ √ β-t1 pre-t1 semi-closed √ semi-t1/2 semi-t1 semi-t1 α-closed √ semi-t1/2 semi-t1 feebly t1 closed tgs t1/2 t3/4 t1 δ-closed tgs + subweakly t2 a.w. t2 a.w. t2 w. t2 θ-closed tgs + r1 t2 t2 t2 references [1] j. cao, m. ganster and i. reilly, on generalized closed sets, topology & appl. 123 (2002), 37-46. [2] j. cao, m. ganster and i. reilly, submaximality, extremal disconnectedness and generalized closed sets, houston j. math. 24 (1998), 681-688. [3] j. cao, s. greenwood and i.reilly, generalized closed sets: a unified approach, applied general topology 2 (2001), 179-189. [4] k. dlaska and m. ganster, s-sets and co-s-closed topologies, indian j. pure appl. math. 23 (1992), 731-737. [5] j. dontchev, i. arokiarani and k. balachandran, on generalized δ-closed sets and almost weakly hausdorff spaces, q & a in general topology 18 (2000), 17-30. [6] j. dontchev and m. ganster, on δ-generalized closed sets and t3/4 spaces, mem. fac. sci. kochi univ. ser. a math. 17 (1996), 15-31. [7] j. dontchev and h. maki, on θ-generalized closed sets, internat. j. math. & math. sci. 22 (1999), 239-249. [8] w. dunham, t1/2-spaces, kyungpook math. j. 17 (1977), 161-169. [9] d. jankovic, on some separation axioms and θ-closure, mat. vesnik 32 (4) (1980), 439-449. [10] d. jankovic and i. reilly, on semi-separation properties, indian j. pure appl. math. 16 (1985), 957-964. [11] n. levine, generalized closed sets in topological spaces, rend. circ. mat. palermo 19 (1970), 89-96. [12] s. n. maheshwari and u. tapi, feebly t1-spaces, an. univ. timisoara ser. stiint. mat 16 (1978), no.2, 173-177. [13] t. soundararajan, weakly hausdorff spaces and the cardinality of topological spaces, general topology and its relations to modern analysis and algebra iii, proc. conf. kanpur 1968, academia, prague (1971), 301-306. [14] m. steiner, verallgemeinerte abgeschlossene mengen in topologischen räumen, master thesis, graz university of technology, 2003. [15] n.v. veličko, h-closed topological spaces, amer. math. soc. transl. 78 (2) (1968), 103118. received may 2004 accepted september 2004 86 j. cao, m. ganster, i. reilly and m. steiner j. cao (cao@math.auckland.ac.nz ) department of mathematics, university of auckland, private bag 92019, auckland, new zealand m. ganster (ganster@weyl.math.tu-graz.ac.at) department of mathematics, graz university of technology, steyrergasse 30, a-8010 graz, austria i. reilly (i.reilly@auckland.ac.nz ) department of mathematics, university of auckland, private bag 92019, auckland, new zealand m. steiner (msteiner@sbox.tugraz.at) department of mathematics, graz university of technology, steyrergasse 30, a-8010 graz, austria applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 55–64 minimal tud spaces a. e. mccluskey and w. s. watson abstract. a topological space is tud if the derived set of each point is the union of disjoint closed sets. we show that there is a minimal tud space which is not just the alexandroff topology on a linear order. indeed the structure of the underlying partial order of a minimal tud space can be quite complex. this contrasts sharply with the known results on minimality for weak separation axioms. 2000 ams classification: 54d10, 06a10, 54a10, 54g20. keywords: minimal topologies, weak separation axioms. 1. introduction definition 1.1. [2] a topological space is said to be tud if the derived set of each point is the (possibly empty) union of disjoint closed sets. in this introduction, we provide a complete brief survey and bibliography of minimality. the family lt(x) of all topologies definable for an infinite set x is a complete atomic and complemented lattice (under set inclusion). if t and s are two members of lt(x) with s ⊆ t , then s is said to be weaker than t . given a topological invariant p, a member t of lt(x) is said to be minimal p if and only if t possesses property p but no weaker member of lt(x) possesses property p. the concept of minimal topologies was first introduced in 1939 by parhomenko [27] when he showed that compact hausdorff spaces are minimal hausdorff. motivation for such an investigation is provided by realising that it is in seeking to identify those members of lt(x) which minimally satisfy an invariant that we are, in a very real sense, examining the topological essence of the invariant. given a topological space (x,t ), (x,t ) is minimal hausdorff if and only if it is hausdorff and every open filterbase which has a unique adherent point is convergent to this point (see [5], [9], [10], [27], [31], [32], and [36]) minimal t1 if and only if t is the cofinite topology c on x 56 a. e. mccluskey and w. s. watson minimal regular if and only if it is regular and every regular filter-base which has a unique adherent point is convergent ([4], [8]) minimal completely regular if and only if it is compact and hausdorff ([4], [5]) minimal normal if and only if it is compact and hausdorff ([5]) minimal urysohn if and only if it is urysohn and every filter with a unique adherence point converges to this point ([11], [34]) minimal (locally compact and hausdorff ) if and only if it is compact and hausdorff ([5], [4]) minimal paracompact if and only if it is compact and hausdorff ([35]) minimal metric only if it is compact and hausdorff ([35]) minimal completely normal only if it is compact and hausdorff ([35]) minimal completely hausdorff only if it is compact and hausdorff ([35]) minimal t0 if and only if it is t0, nested and generated by the family {x \{x} : x ∈ x}∪{∅,x} ([1], [12], [19], [22], [26]) minimal td if and only if it is td and nested ([1], [12], [19], [22], [26]) minimal tδ if and only if it is tδ and nested ([1], [22]) minimal tξ if and only if it is tξ and nested ([1]) minimal ta if and only if it is ta and partially nested ([22]) minimal tes if and only if either t = c or t = e(x \y )∪(c∩i(y )) for some non-empty proper subset y of x ([21]) minimal tef if and only if t = c or t = i(x) or t = e(x) for some x ∈ x ([21]) minimal tff if and only if there exists x ∈ x such that either t = c∩ i(x) or t = c∩e(x) ([15]) minimal tf if and only if either there exists x ∈ x such that t = c∩ i(x) or there exists a non-empty proper non-singleton subset y of x such that t = d(y ) ([15]) minimal ty s if and only if t = w(p)∨(c∩i(k)) for some subset k of x and some partition p of x such that p is simply associated with k and is associated with x \k. ([16]) minimal tdd if and only if t = wk(p) ∨ (c∩i(k)) for some subset k of x and partition p of x such that p is simply associated with k and associated with x \k ([16]) minimal ty y if and only if t = mp(p) ∨ (c∩i(k)) for some p ∈ x, subset k of x \{p} such that p is simply associated with k ([23]) minimal ty if and only if t = w(f)∨(c∩i(k)) for some degenerate k-cover f of x ([23]) minimal tsa if and only if t = e(x\b)∨sk(p)∨(c∩i(k∪b)) for some disjoint subsets b and k of x such that k 6= ∅ and k ∪ b 6= x, and partition p of x \ b such that p is simply associated with x \ (k ∪b) and associated with k. ([24]) minimal tsd if and only if t = sk(p) ∨ (c ∩i(k)) for some nonempty proper subset k of x and partition p of x such that p is simply associated with x \k and associated with k ([24]) minimal tud spaces 57 minimal tfa if and only if either (x,t ) is minimal tes with at least one isolated point and at least two closed points or (x,t ) is minimal tsd or t = e(x \b)∨sk(p)∨d(b∪k)) for some non-empty, disjoint subsets b, k of x such that b ∪ k is a proper, infinite subset of x with |x \ (b ∪ k)| > 1, a subset g of x \ (b ∪ k) and a partition p of x \ (b ∪g) such that p is simply associated with x \k and associated with k. 2. constructing the partial order we need an axiom for partial orders which implies the tud axiom for topological spaces. definition 2.1. a partial order (x, �) is said to be t +ud if there is a family {y (x) : x ∈ x} of subsets of x such that • (∀y ∈ y (x))y � x∧y 6= x • (∀z � x)z 6= x ⇒ (∃y ∈ y (x))z � y • (∀y,y′ ∈ y (x))(∀z ∈ x)(z � y ∧z � y′) ⇒ y = y′ we shall write yx(x) if the underlying partial order is ambiguous. a few comments: we conjecture that the minimal tud topologies must be the weak topologies on a minimal t +ud partial order. we conjecture that the minimal t +ud partial orders are just the t + ud and suitable partial orders. this would provide a characterization which requires for each pair of elements an infinite set which satisfies a first order formula. maybe those weak separation axioms which have simpler minimality characterizations do so because of logical considerations, i.e., must all first order weak separation axioms have minimalities which are weak topologies for partial orders either without infinite chains or without infinite antichains? next we describe a way in which two t +ud partial orders can be combined and yet preserve t +ud. definition 2.2. if x0 ⊂ x1 are partial orders, where x0 has the order induced by x1, then we say that x0 is a simple subset of x1 if there are distinct x0,x1 ∈ x0 and w ∈ x1 −x0and a ⊂ x1 −x0 such that • x1 � w 6∈ a • x0 and x1 are incomparable in x0 • a is the set of all elements of x1 −x0 strictly below x0 • each element of a is minimal in x1 • (∀x ∈ x0)(∀y ∈ x1 −x0)(x � y ⇒ x � x1 � y = w) • (∀x ∈ x0)(∀y ∈ x1 −x0)(y � x ⇒ y � x0 � x) proposition 2.3. if x0 is a simple subset of x1 and both x0 and x1 − x0 are t +ud, then x1 is also t + ud. moreover, we can get yx1 (x) ∩x0 = yx0 (x) for each x ∈ x0. 58 a. e. mccluskey and w. s. watson proof. let x0,x1 ∈ x0 and w ∈ x1 − x0 and a ⊂ x1 − x0 be as in definition 2.2. suppose x ∈ x1. we must define y (x) as in definition 2.1. we do this by cases. (1) if x ∈ x0 and x 6= x0, then we let y (x) = yx0 (x). (2) if x ∈ x1 −x0 and x 6= w, then we let y (x) = yx1−x0 (x). (3) if x = x0, then we let y (x) = yx0 (x) ∪a. (4) if x = w, then we let y (x) = yx1−x0 (x) ∪{x1}. it suffices to show that definition 2.1 is satisfied by {y (x) : x ∈ x1}. verifying the first condition requires us to use only the facts (∀a ∈ a)a � x0 ∧a 6= x0 and x1 � w ∧x1 6= w. verifying the second condition requires examination of the same four cases. (1) if y � x and y ∈ x1 −x0 then simplicity says that y � x0 � x. now, since x 6= x0 ∈ x0, we know that (∃s ∈ yx0 (x))x0 � s and thus that y � s. (2) if y � x and y ∈ x0 then x = w which is impossible. (3) if x = x0 and y � x and y ∈ x1 −x0 then y ∈ a which suffices. (4) if x = w and y � x and y ∈ x0 then y � x1 which suffices. verifying the third condition also requires the examination of these same four cases. (1) suppose y0,y1 are distinct elements of y (x) and z � y0,y1. then z ∈ x1−x0 so that x0 �y0 and x0 �y1 by the sixth condition of simplicity— clearly a contradiction. (2) suppose y0,y1 are distinct elements of y (x) and z�y0,y1. then z ∈ x0 so that, by the fifth condition of simplicity, y0 = w = y1! (3) the first case for yx0 (x) and the fact that a is a set of minimal points in x1 suffices. (4) the second case for yx1−x0 (x) leaves the possibility that there is z�x1 and z � y ∈ yx1−x0 (x). if z ∈ x1 − x0, then z � x0 � x1 which is impossible. if z ∈ x0, then z � x1 � y = w—yet y ∈ yx1−x0 (w)! the proof is complete. � next, we describe when two incomparable elements of a t +ud partial order cannot be made comparable in a given “direction” without destroying t +ud. moreover, since a tud-topology may induce an order which is not t + ud, we stipulate a condition to ensure that the resulting order has no compatible tudtopology. definition 2.4. if x0 ⊂ x1 are partial orders, where x0 has the order induced by x1, and x0,x1 ∈ x0 are incomparable, then we say that x0 is a suitable subset in x1 with respect to (x0,x1), if there are, in x1, elements w, {yi : i ∈ ω} and {zi : i ∈ ω} all distinct from each other and from x0 and x1 such that • (∀i ∈ ω)zi � yi � w • (∀i ∈ ω)zi � x0 • x1 � w minimal tud spaces 59 • (∀f ∈ [x1]<ω)(((∀f ∈ f)w 6�f) ⇒ ((∃i ∈ ω)(∀f ∈ f)yi 6�f)) note that this definition applies also when x0 = x1. indeed, we can “make” a t +ud partial order suitable for two incomparable elements in a “simple” way. proposition 2.5. if x is any t +ud partial order and x0,x1 ∈ x are incomparable, then there is a partial order y ⊃ x such that • x is a simple subset of y • x is a suitable subset of y with respect to (x0,x1) • y −x is countable and t +ud proof. we let y = x ∪ {yi,zi : i ∈ ω} ∪ {w} where all these elements are distinct and not in x. we declare • (∀i ∈ ω)zi � x0 • (∀i ∈ ω)zi � yi � w • x1 � w and close off under transitivity. to check that x is a simple subset of y , define a = {zi : i ∈ ω}. since nothing is defined to be below any zi, we know that each zi is minimal in y . thus we have conditions 1, 2 and 4 in definition 2.2. further, clearly w cannot be below x0 nor can any yi be below x0, so that condition 3 is satisfied. if x ∈ x, y ∈ y −x and x�y, then x1 �w must be a step in the calculation. since nothing is defined to be above w, w is maximal in y and so w = y as required. thus condition 5 is satisfied. if x ∈ x, y ∈ y −x and y �x, then zi �x0 must be a step in the calculation as required. thus condition 6 is satisfied. to check that x is a suitable subset of y with respect to (x0,x1), suppose that there exists finite f ⊂ y such that (∀f ∈ f)w 6= f and (∀i ∈ ω)(∃f ∈ f)yi �f. if yi �f and yi 6= f, then some step in the calculation must be yi �w. since w is maximal in y , we must have f = w which is impossible. thus we know that (∀i ∈ ω)(∃f ∈ f)yi = f and thus f ⊃{yi : i ∈ ω}! to check that y − x is t +ud, let y (w) = {yi : i ∈ ω}, y (yi) = {zi} and y (zi) = ∅. � definition 2.6. a partial order (x, �) is said to be suitable if, for each x0,x1 ∈ x which are incomparable, x is suitable in itself with respect to (x0,x1). suitability can be obtained in a “simple” increasing sequence if suitability with respect to each incomparable pair is accomplished along the way. proposition 2.7. if {xi : i ∈ ω} is an increasing sequence of (partially ordered) subsets of a partial order x such that • each xi is a simple subset of xi+1 • ⋃ {xi : i ∈ ω} = x • (∀ incomparable x0,x1 ∈ x)(∃i ∈ ω)xi is a suitable subset of xi+1 with respect to (x0,x1) 60 a. e. mccluskey and w. s. watson then x is suitable. proof. given any incomparable elements x0, x1 in x, we must check that x is suitable in itself with respect to (x0,x1). now suppose that xi is a suitable subset of xi+1 with respect to (x0,x1). this gives us distinct w, {yi : i ∈ ω} and {zi : i ∈ ω} as in definition 2.4. thus the first three conditions of definition 2.4 are satisfied. we need to check the fourth condition. suppose f ∈ [x]<ω and (∀f ∈ f)w 6�f and (∀i ∈ ω)(∃f ∈ f)yi � f. we shall argue that no such f can exist by mathematical induction. find such an f with j∗ = minimum{max[j ∈ ω : f ∩ (xj+1 − xj) 6= ∅]} and furthermore such that f ∩ (xj∗+1 −xj∗) has minimum cardinality. let f ′ = {f ∈ f : (∃i ∈ ω)yi � f}. we shall prove that j∗ ≤ i. suppose j∗ > i and choose f ∈ f ′ ∩ (xj∗+1 − xj∗)—such a choice is possible because of the minimum nature of j∗. we know that for certain i ∈ ω, yi � f. each such yi is an element of xi+1 ⊂ xj∗. let the fact that xj∗ is a simple subset of xj∗+1 be witnessed by x∗0,x ∗ 1,w ∗. the fifth condition of simplicity gives us that yi �x∗1 �f = w ∗ so that f ′∩(xj∗+1−xj∗) = {w∗} = {f}. thus f can be replaced by x∗1 ∈ xj∗, giving a subset f∗ = (f −{f})∪{x∗1} of f which again contradicts the minimum nature of j∗. thus j∗ ≤ i. it follows that f ⊂ xi+1, contradicting the suitability of xi in xi+1 with respect to (x0,x1). � finally we can accomplish our aim of making a t +ud partial order suitable without destroying t +ud. proposition 2.8. any countable t +ud partial order can be embedded in a suitable t +ud partial order. proof. first, we define a partition {pi : i ∈ ω} of ω. given pi ⊂ ω for i < n, define pn to be any infinite, co-infinite subset of ω− ⋃ i ∆(x) and a ∩ b is nowhere dense whenever a and b are distinct elements of d. in [11] v. i. malykhin proved that every countably infinite totally bounded group is extraresolvable (through an almost disjoint family of cardinality c). recall that a family d of infinite subsets of a set is said to be almost disjoint if a∩b is finite whenever a and b are distinct elements of d. below we give a small survey with different proofs of extraresolvability for countably infinite spaces and, in particular, for totally bounded groups. each proof presents different techniques and results which may be useful in studying this topic. each section title is given according to the main tool of resolvability used in that section. the neutral element of a group will always be denoted by e and every group topology is assumed to be t0 (hence completely regular). furthermore, we work with groups which admit a totally bounded hausdorff topology (e.g., every abelian group). 2. discrete subsets. a subset m of a topological space x is said to be strongly discrete if for every x ∈ m there exists an open neighborhood v (x) of x such that v (a)∩v (b) = ∅ whenever a and b are different points of m. definition 2.1. a point z of x is called an lsd-point if there exists a strongly discrete subset m such that z ∈ m′. the proof of the following propositions is straightforward. proposition 2.2. a point z is an lsd-point iff there exists a set m, points of which have a disjoint system of open neighborhoods θ = {v (x) : x ∈ m} such that z 6∈ ∪θ and z ∈ m. proposition 2.3. let m be a strongly discrete subset of a regular space x. if z 6∈ m then m ∪{z} is strongly discrete. proposition 2.4. let y be a subset of an hausdorff space x. if y ′ 6= ∅, then y contains an infinite strongly discrete subset. in [16] p.l. sharma and s. sharma proved that if every point of a t1 space is an lsd-point, then the space is ℵ0-resolvable. in the following theorem we use some ideas of their construction. theorem 2.5. let x be a countably infinite regular space. if every point of x is an lsd-point, then x is extraresolvable through a collection of cardinality c. extraresolvability 319 proof. let {zn : n ∈ ω} be a one to one numeration of the space x. since z0 is an lsd-point, then there exist m and θ = {v (x) : x ∈ m} as in proposition 2.2. put m0 = m and θ0 = θ. by proposition 2.3, it is not restrictive to assume that z1 ∈ m0. now we are going to describe the next step of the inductive construction, which is very familiar with the general step. for each x ∈ m0, still choose mx and θx as in proposition 2.2. we can assume that ⋃ θx ⊆ v (x) for each x ∈ m0. let m1 = ⋃ {mx : x ∈ m0}. the set m1 is strongly discrete with disjoint system of open neighborhoods θ1 = ⋃ {θx : x ∈ m1}. notice that θ1 refines θ0. still by proposition 2.3, it is not restrictive to assume that z2 ∈ m0 ∪m1. in the general case, by repeating the process for every x ∈ mn, we get sequences {mn} and {θn} satisfying the following: (1) zn+1 ∈ m0 ∪ . . .∪mn, (2) mn is strongly discrete with disjoint systems of open neighborhoods θn, (3) mn ⊆ m′n+1, (4) θn+1 refines θn. by (2) and (3), we get that ( ⋃ θn+1)∩mn = ∅; consequently, by (4), the sets mn are mutually disjoint. for every infinite subset a ⊆ ω put x(a) = ⋃ n∈a mn. by (1) and (3), x(a) is dense in x. since the sets mn are mutually disjoint, then x(a) ∩x(b) = x(a ∩ b). consequently x(a) ∩ x(b) is nowhere dense whenever a ∩ b is finite (proposition 1.1). thus if a is an almost disjoint family of cardinality c of infinite subsets of ω, then x(a) = {x(a) : a ∈ a} is a collection of cardinality c which ensures the extraresolvability of x. � theorem 2.6. every countably infinite totally bounded hausdorff group is extraresolvable. proof. i. v. protasov [13] constructed a strongly discrete subset d such that e ∈ d′ in every totally bounded group topology. consequently the identity (hence every element) is an lsd-point and the conclusion follows from theorem 2.5. � 3. weak sequences. a weak sequence on x is a countably infinite disjoint family f of finite subsets of x. we say that f converges to a point x if {f ∈ f : f ∩v = ∅} is finite for each neighborhood v of x. proposition 3.1. let x be a countably infinite space such that every point admits a weak sequence converging to it. then there exist a weak sequence converging to every point of x. proof. let x = {zn : n ∈ ω} be a one to one numeration of x and for each n let fn be a weak sequence converging to zn. we shall construct a new weak sequence {km : m ∈ ω} which converges to each element of the space. by induction, suppose that k0, . . . ,km−1 have been already defined. the set 320 g. artico, v. i. malykhin and u. marconi t = ⋃ i ∆(x) and |a∩b| < nwd(x) whenever a and b are distinct elements of a. w. w. comfort and s. garcia-ferreira proved that if d(x) = |g| ≥ ω, then g is strongly extraresolvable [5]. the previous theorem 3.5 proves that a countably infinite totally bounded group is strongly extraresolvable through a family of cardinality c. 4. talagrand’s theorem. in this section we shortly present the proof of theorem 3.5 given in [11]. we identify a subset a ⊆ x with its characteristic function χa ∈ 2x, where 2x has the product topology. if f denotes a family of subsets of x, then the subspace b(f) = {χf : f ∈f}⊆ 2x is called the binary space of f. m. talagrand [18] proved that for a free filter f on a set x the following conditions are equivalent: (a) b(f) is meager (as a subset of 2x). (b) there exists a sequence kn of mutually disjoint finite subsets of x such that the set {n : kn ∩f = ∅} is finite for each f ∈f. an accurate reading of the proof of talagrand’s theorem shows that it suffices to assume that the family f of non-empty subsets of x satisfy the following condition: if a ∈f and b ⊇ a then b ∈f. a subset l of an infinite group g is called large if there exists a finite subset k ⊆ g such that kl = lk = g. in [1] the authors proved that the binary space b(l) of all large subsets of a countably infinite group g is meager. so, according to the general form of talagrand’s theorem, there exists a sequence {kn} of mutually disjoint finite subsets of g such that {n : kn ∩ l = ∅} is finite for every large subset l of g. as each non-empty open subset of a totally bounded group is large, we deduce the following proposition. proposition 4.1. let g be a countably infinite group. there exists a weak sequence {kn} which converges at each point of g with respect to any hausdorff totally bounded group topology. 322 g. artico, v. i. malykhin and u. marconi the third proof of theorem 2.6 follows as in theorem 3.2 by considering the family of sets g(a) = ⋃ n∈a kn, where a ranges over an almost disjoint family of cardinality c. 5. quadrosequences. the proof given here uses some ideas which are already present in section 3. one interesting point is the result provided in proposition 5.1. we denote by xm = {x1, . . . ,xm} a one to one numeration of a set with m elements of a group g and by d(xm) the set {xix−1k : i < k, k ≤ m}. proposition 5.1. let g = {gn : n ∈ ω} be a one to one numeration of a countably infinite group g. there exist infinite subsets xn, n ∈ ω, such that the subsets tn = {g1, . . . ,gn} ·d(xn) are mutually disjoint. proof. let us consider a general step of a pyramidal inductive construction. let us assume that some initial parts xmkk = {xk,1, . . . ,xk,mk} have already been defined for each k ≤ n in such a way that the sets t mk k = {g1, . . . ,gk} ·d(x mk k ) are pairwise disjoint. the n + 1th step consists in adding a new point to every x mk k and starting with the first two points of xn+1. • adding a new point to some xmkk , for k ≤ n. in this case, an added point x = xmk+1 must satisfy to the following conditions: giyx −1 6∈ tmrr , i ≤ k, r 6= k, r ≤ n, y ∈ x mk k . the sets tmrr vary during the process of adding these points. such a point x does exist since the number of excluded conditions is finite. • forming a new set x2n+1 = {a,b}. in this case the elements a = xn+1,1 and b = xn+1,2 must satisfy to the following conditions: giab −1 6∈ tmkk , i ≤ n + 1, k ≤ n. this construction ends the proof. � arguing as in theorem 3.4, item (2), one obtains the following: lemma 5.2. let f be a finite subset of g and let x be a countably infinite subset of g. there exists an infinite subset y of x such that the sets f ·dm(y ) are mutually disjoint. now we deduce theorem 3.5 from proposition 5.1. proof. by theorem 3.4, item (1), g ∈ g · ⋃ {dm(xn) : m ∈ j} for every infinite subset j of ω. consequently, if a is a subset of ω, the set t (a) = ⋃ n∈a ( ⋃ m∈a {g1, . . . ,gn} ·dm(xn) ) extraresolvability 323 is dense whenever a is infinite. by lemma 5.2, in proposition 5.1 it is not restrictive to assume that, for each n ∈ ω, the family {{g1, . . . ,gn} ·dm(xn) : m ∈ ω} is disjoint. since the sets tn are mutually disjoint too, the set t (a) ∩t (b) = t (a∩b) = ⋃ n∈a∩b ( ⋃ m∈a∩b {g1, . . . ,gn} ·dm(xn) ) is finite whenever a∩b is finite. the required collection of sets is obtained by considering the almost disjoint family {t (a)}, where a ranges over an almost disjoint family of ω of cardinality c. � 6. protasov method. in this section we use a method due to protasov, by applying his argument to the countable case [14, 12]. as in section 4, a subset l of a group g is said to be large if there exists a finite subset k of g such that kl = lk = g (e.g., see [1]). sets of the form gk and kg are called right and left circles of radius k and center g, respectively. the following criterium of [1] will be useful in the sequel. proposition 6.1. the set g\s fails to be large if and only if s contains (right or left) circles of any finite radius. proposition 6.2. let g be a countably infinite group. there exists a weak sequence f = {fn} such that, whenever a ⊆ ω is infinite and ω\a is infinite, both sets f(a) = ⋃ n∈a fn and g\f(a) fail to be large. proof. let g = {gn} be a one to one numeration of g and let gn = {gk : k < n}, for each n ∈ ω. arguing by induction, we shall construct a weak sequence {fn} in such a way that fn contains a circle of radius gn for each n. let us assume that f0, . . . ,fn−1 have been already defined. since t = ⋃ i 0} where bd(x, r) = {y ∈ x : d(x, y) < r}. we remark that the topology t (d) is t0. moreover, if condition (i) above is replaced by (i′) d(x, y) = 0 ⇔ x = y, then t (d) is a t1 topology. a quasi-metric d is said to be bicomplete if d s is a complete metric. for more information about quasi-metric spaces see [4] and [8]. following [7], a cone is a triple (x, +, ·) such that (x, +) is an abelian semigroup with neutral element 0 and · is a function from r+ × x into x which satisfies for all a, b ∈ r+ and x, y ∈ x: (i) a·(b·x) = (ab)·x, (ii) (a+b)·x = (a·x)+(b·x), (iii) a·(x+y) = (a·x)+(a·y) and (iv) 1 · x = x. a quasi-norm on a cone (x, +, ·) is a function ‖ · ‖ : x → r+ such that for all x, y ∈ x and r ∈ r+: (i) x = 0 if and only if there is −x ∈ x and ‖x‖ = 0 = ‖ − x‖, (ii) ‖r · x‖ = r‖x‖, and (iii) ‖x + y‖ ≤ ‖x‖ + ‖y‖. if the quasi-norm q satisfies: (i′) ‖x‖ = 0 if and only if x = 0, then q is called a norm on the cone (x, +, ·). a (quasi-)normed cone is a pair (x, ‖ · ‖) such that x is a cone and ‖ · ‖ is a (quasi-)norm on x. if (x, +, ·) is a linear space and ‖ · ‖ is a quasi-norm on x, then the pair (x, ‖ · ‖) is called a quasi-normed linear space. note that in this case, the function ‖ · ‖−1 : x → r+ given by ‖x‖−1 = ‖ − x‖ is also a quasi-norm on x and the function ‖ · ‖s : x → r+ given by ‖x‖s = ‖x‖ ∨ ‖x‖−1 is a norm on x. 2. on the structure of the set of semi-lipschitz functions let (x, d), (y, q) be a quasi-metric space and a quasi-normed space respectively. a function f : x −→ y is said to be a semi-lipschitz function if there exists k ≥ 0 such that q(f(x)−f(y)) ≤ kd(x, y) for all x, y ∈ x. the number k is called a semi-lipschitz constant for f. on semi-lipschitz functions with values in a quasi-normed linear space 219 a function f on a quasi-metric space (x, d) with values in a quasi-normed linear space (y, q) is called ≤(d,q)-increasing if q(f(x) − f(y)) = 0 whenever d(x, y) = 0. by y x (d,q) we shall denote the set of all ≤(d,q)-increasing functions from (x, d) to (y, q). it is clear that if (x, d) is a t1 quasi-metric space, then every function from x to y is ≤(d,q) -increasing. if for each f, g ∈ y x (d,q) and a ∈ r + we define f + g and af in the usual way, then it is a routine to show that (y x(d,q), +, ·) is a cone. example 2.1. let x = z3. let d be the quasi-metric on x given by d(x, y) = { 1 if x > y, 0 if x ≤ y. let y = r, q(x) = x ∨ 0 and take f such that f(0) = 0, f(1) = 1 and f(−1) = −2. it is easy to see that f ∈ y x (d,q) but −f /∈ y x (d,q) . thus y x (d,q) is not a linear space. a simple but interesting example of a semi-lipschitz function is the following: example 2.2. let (n, d) be a quasi-metric space where: d(x, y) = { 1 if y > x, 0 if y ≤ x. then, the dual complexity space, is the quasi-normed space (b∗, q), with b ∗ = {f : ω → r/ ∞ ∑ n=0 2−n(f(n) ∨ 0) < ∞} and q(f) = ∞ ∑ n=0 2−n(f(n) ∨ 0). let now f : (n, d) → (b∗, q) be the function defined by: f(0) = 0, f(n) = fn such that n < m implies fn > fm, where the order is given by fn > fm if and only if fn(x) > fm(x) for all x ∈ ω. clearly f is a semi-lipschitz function. given a quasi-metric space (x, d) and quasi-normed space (y, q), fix x0 ∈ x and put sl0(d, q) = {f ∈ y x (d,q) : sup d(x,y) 6=0 q(f(x) − f(y)) d(x, y) < ∞ , f(x0) = 0}. then sl0(d, q) is exactly the set of all semi-lipschitz functions that vanishes at x0, and it is clear that (sl0(d, q), +, ·) is a subcone of (y x (d,q) , +, ·). now let ρ(d,q) : sl0(d, q) × sl0(d, q) −→ [0, ∞] defined by ρ(d,q)(f, g) = sup d(x,y) 6=0 q((f − g)(x) − (f − g)(y)) d(x, y) for all f, g ∈ sl0(d, q). then ρ(d,q) is a quasi-distance on sl0(d, q). however ρ(d,q) is not a quasi-metric in general, as example 1.1 of [11] shows. 220 j. m. sánchez-álvarez furthermore, it is clear that for each f, g, h ∈ sl0(d, q) and each r > 0, ρ(d,q)(f + h, g + h) = ρ(d,q)(f, g) and ρ(d,q)(rf, rg) = rρ(d,q)(f, g) i.e., ρ(d,q) is an invariant quasi-distance. moreover, it is easy to check that ρ(d,q)(f, 0) = 0 if and only if f = 0, where by 0 we denote the function that vanishes at every x ∈ x. moreover, we can see that, by example 2.1, there exists f ∈ sl0(d, q) such that ρ(d,q)(0, f) = 0 but f 6= 0. consequently, the nonnegative function ‖ · ‖(d,q) defined on sl0(d, q) by ‖f‖(d,q) = ρ(d,q)(f, 0) is a norm on sl0(d, q). therefore (sl0(d, q), ‖ · ‖(d,q)) is a normed cone. example 1.1 of [11] provides an instance of a t1 quasi-metric space (x, d) such that (sl0(d, q), +) is not a group for some x0 ∈ x. this example suggests the question of characterizing when (sl0(d, q), +) is a group. in order to give an answer to this question note that if x0 is a fixed point in the quasi-metric space (x, d), then the set sl0(d −1, q) = {f ∈ y x(d−1,q) : sup d(y,x) 6=0 q(f(x) − f(y)) d(y, x) < ∞ , f(x0) = 0} has also a structure of a cone and (sl0(d −1, q), ‖ · ‖(d−1,q)) is a normed cone, where ‖f‖(d−1,q) = ρ(d−1,q)(f, 0), i.e., ‖f‖(d−1,q) = sup d(y,x) 6=0 q(f(x) − f(y)) d(y, x) for all f ∈ sl0(d −1, q). proposition 2.3. let (x,d), (y ,q) be a quasi-metric space and a quasinormed space respectively. then f ∈ sl0(d, q) if and only if −f ∈ sl0(d −1, q). proof. let f ∈ sl0(d, q) then there exists k ∈ r + such that q(f(x) − f(y)) ≤ kd(x, y) for all x, y ∈ x. we change x by y hence q(f(y) − f(x)) ≤ kd(y, x) and q(−f(x)−(−f(y))) ≤ kd−1(x, y) then −f ∈ sl0(d −1, q). the converse is analogous. � corollary 2.4. let (x,d), (y ,q) be a quasi-metric space and a quasi-normed space respectively.then (sl0(d, q) ∩ sl0(d −1, q), +, ·) is a linear space. proof. it follows from proposition 2.3 that f ∈ sl0(d, q) ∩ sl0(d −1, q) if and only if −f ∈ sl0(d, q) ∩ sl0(d −1, q). � remark 2.5. note that for each f ∈ sl0(d, q), ‖f‖(d,q) = ‖ − f‖(d−1,q). thus the normed cones (sl0(d, q), ‖ · ‖(d,q)) and (sl0(d −1, q), ‖ · ‖(d−1,q)) are isometrically isomorphic by the bijective map φ : sl0(d, q) −→ sl0(d −1, q) defined by φ(f) = −f. furthermore, we have sl0(d, q) ∩ sl0(d −1, q) = {f ∈ y x(d,q) ∩ y x (d−1,q) : on semi-lipschitz functions with values in a quasi-normed linear space 221 sup d(x,y) 6=0 q(f(x) − f(y)) ∨ q(f(y) − f(x)) d(x, y) < ∞, f(x0) = 0}. hence (sl0(d, q)∩sl0(d −1, q), ‖ ·‖b) is a normed linear space, where ‖ ·‖b is the norm defined by ‖f‖b = sup d(x,y) 6=0 q(f(x) − f(y)) ∨ q(f(y) − f(x)) d(x, y) , for all f ∈ sl0(d, q) ∩ sl0(d −1, q). observe that ‖ · ‖b = ‖ · ‖(d,q) ∨ ‖ · ‖(d−1,q) on sl0(d, q) ∩ sl0(d −1, q). the next result, whose proof is very easy, is a characterization that will be useful. proposition 2.6. f ∈ sl0(d, q) ∩ sl0(d −1, q) if and only if f(x0) = 0 and there exists k ≥ 0 such that qs(f(x) − f(y)) ≤ kd(x, y). remark 2.7. it is straightforward to see that f : (x, d) −→ (y, q) belongs to y x (d,q) ∩ y x (d−1,q) if and only if f(x) = f(y) whenever d(x, y) = 0. example 2.8. let (x, d), (y, q) be a quasi-metric and a quasi-normed space such that there is x0 ∈ x satisfying d(x, x0) ∧ d(x0, x) = 0 for all x ∈ x. then sl0(d, q) ∩ sl0(d −1, q) = {0}. example 2.9. let x = [0, 1] and let d be the quasi-metric on x given by d(x, y) = y−x if x ≤ y and d(x, y) = 1 otherwise. clearly t (d) is the restriction of the sorgenfrey topology to [0, 1]. let (y, q) be a quasi-normed space and put x0 = 0. then, a function f : x −→ y satisfies f ∈ sl0(d, q) ∩ sl0(d −1, q) if and only if there is k ≥ 0 such that q(f(x) − f(y)) ∨ q(f(y) − f(x)) ≤ k(d(x, y) ∧ d(y, x)) for all x, y ∈ x. theorem 2.10. let (x,d), (y ,q) be a quasi-metric and a quasi-normed space respectively. then the following assertions are equivalent: (1) sl0(d, q) = sl0(d −1, q). (2) sl0(d, q) is a group. (3) sl0(d −1, q) is a group. (4) sl0(d, q) ⊂ sl0(d −1, q). (5) sl0(d −1, q) ⊂ sl0(d, q). proof. (1) ⇒ (2) by corollary 2.4 (sl0(d, q) ∩ sl0(d −1, q), +, ·) is a linear space. if sl0(d, q) = sl0(d −1, q) then (sl0(d, q), +) is a group. (2) ⇒ (3) let f ∈ sl0(d −1, q). by proposition 2.3 −f ∈ sl0(d, q), since sl0(d, q) is a group, f ∈ sl0(d, q), by proposition 2.3 −f ∈ sl0(d −1, q). (3) ⇒ (4) the proof is similar to the proof of (2) ⇒ (3). (4) ⇒ (5) let f ∈ sl0(d −1, q). then −f ∈ sl0(d, q) ⊂ sl0(d −1, q) hence −f ∈ sl0(d −1, q).thus f ∈ sl0(d, q). (5) ⇒ (1) is the same that (4) ⇒ (5). � proposition 2.11. let (x,d), (y ,q) be a quasi-metric and a quasi-normed space respectively. if there exists x0 ∈ x such that sl0(d, q) = sl0(d −1, q), then sl1(d, q) = sl1(d −1, q) for each x1 ∈ x. 222 j. m. sánchez-álvarez proof. let f ∈ sl1(d, q). define a function g on x by g(x) = f(x) − f(x0) for all x ∈ x. it easy to check that g ∈ sl0(d, q). thus, g ∈ sl0(d −1, q). since g(x) − g(y) = f(x) − f(y) for all x, y ∈ x we obtain that f ∈ sl1(d −1, q). � 3. completeness properties in this section, we discuss the completeness properties of the semi-lipschitz function space. the following result allows us to prove that if (y , q) is a bibanach space then (sl0(d, q) ∩ sl0(d −1, q), ‖ · ‖b) is a banach space. theorem 3.1. let (x,d), (y ,q) be a quasi-metric and a quasi-normed bicomplete space respectively. then (sl0(d, q) ∩ sl0(d −1, q), ‖ · ‖b) is a banach space. proof. let {fn} be a cauchy sequence in (sl0(d, q)∩sl0(d −1, q), ‖·‖b). then, given ε ≥ 0 there is n0 ∈ n such that (∗) sup d(x,y) 6=0 qs((fn − fm)(x) − (fn − fm)(y)) d(x, y) < ε for all n, m ≥ n0. if x = x0 then fn(x) = 0 for all n ∈ n. let x 6= x0. we consider the following cases: case 1. d(x, x0) 6= 0. then, we deduce from (∗) that given ε d(x,x0) there exists n′0 ∈ n such that if n, m ≥ n ′ 0 then q s(fn(x) − fm(x)) < ε. therefore, fn(x) is a cauchy sequence in (y, q s). case 2. d(x, x0) = 0. then from remark 2.7 fn(x) = fn(x0) and fm(x) = fm(x0). therefore q s(fn(x) − fm(x)) = 0 < ε. consequently, fn(x) is a cauchy sequence in (y, q s) and {fn(x)} converges to an element f(x) in (y, qs) for all x ∈ x. moreover, {fn} converges to f in sl0(d, q) ∩ sl0(d −1, q). indeed, given ε, since {fn(x)} converges to f(x) for all x ∈ x, for each x, y there exists n′ such that if m′ ≥ n′ then qs(f(x) − f′m(x) − (f(y) − f ′ m(y))) d(x, y) < ε 2 . since {fn} is a cauchy sequence, we can also find n0 such that if n, m ≥ n0 then qs(fn(x) − fm(x) − (fn(y) − fm(y))) d(x, y) < ε 2 for all x, y ∈ x. thus we have ε > qs(f(x) − f′m(x) − (f(y) − f ′ m(y))) d(x, y) ≥ qs(f(x) − fn(x) − (f(y) − fn(y))) d(x, y) − qs(f′m(x) − fn(x) − (f ′ m(y) − fn(y))) d(x, y) and hence qs(f(x) − fn(x) − (f(y) − fn(y))) d(x, y) ≤ ε 2 + ε 2 = ε. on semi-lipschitz functions with values in a quasi-normed linear space 223 since n0 is independent of x, y, we obtain sup d(x,y) 6=0 qs(f(x) − fn(x) − (f(y) − fn(y))) d(x, y) < ε, for all n ≥ n0. � corollary 3.2. let {fn} be a cauchy sequence in (sl0(d, q) ∩ (sl0(d −1, q), ‖ · ‖b). then there exists a convergent sequence {kn} in (r, tu) such that kn is a semi-lipschitz constant for fn, where tu is the usual topology. theorem 3.3. let (y, q) be a bi-banach space. (1) (x, d) is a metric space. (2) sl0(d, q) = sl0(d −1 , q), ‖ · ‖(d,q) = ‖ · ‖(d−1,q). (3) (sl0(d, q), ‖ · ‖(d,q)) is a banach space. then: (1) ⇒ (2) ⇔ (3) proof. (1) ⇒ (2) trivial. (2) ⇒ (3) trivial. (3) ⇒ (2) suppose that (sl0(d, q), ‖ · ‖(d,q)) is a banach space. then sl0(d, q) is a group, so sl0(d, q) = sl0(d −1, q). moreover ‖ · ‖(d,q) is a norm on sl0(d, q), so that ‖f‖(d,q) = ‖ − f‖(d,q) for all f ∈ sl0(d, q). since −f ∈ sl0(d, q) it follows that ‖ − f‖(d,q) = ‖f‖(d−1,q). we conclude that ‖ · ‖(d,q) = ‖ · ‖(d−1,q) on (sl0(d, q). � to see that in general (3) ⇒ (1) is not true we take y = 0 for a quasi-metric space that is not a metric space. corollary 3.4. if q is a bicomplete quasi-norm on y then ρ(d,q) is a bicomplete quasi-metric in sl0(d, q) ∩ sl0(d −1, q). the following result allow us to prove that if (y, q) is a bicomplete space then (sl0(d, q), ρ(d,q)) is a bicomplete space: theorem 3.5. let (x, d), (y, q) be a quasi-metric and a quasi-normed bicomplete spaces respectively. then (sl0(d, q), ρ(d,q)) is a bicomplete space. proof. let {fn} be a cauchy sequence in (sl0(d, q), ρ(d,q)). then, given ε ≥ 0 there is n0 ∈ n such that (∗) sup d(x,y) 6=0 q((fn − fm)(x) − (fn − fm)(y)) d(x, y) < ε for all n, m ≥ n0. if x = x0 then fn(x) = 0 for all n ∈ n. let x 6= x0.we consider the following cases. case 1. d(x, x0) 6= 0. then, we deduce from (∗) that given ε d(x,x0) there exists n′0 ∈ n such that if n, m ≥ n ′ 0 then q(fn(x) − fm(x)) < ε and if we 224 j. m. sánchez-álvarez change n and m q(fm(x) − fn(x)) < ε. therefore, fn(x) is a cauchy sequence in (y, qs). case 2. d(x, x0) = 0. then d(x0, x) 6= 0 so q(fm(x) − fn(x)) < ε and q(fn(x) − fm(x)) < ε. consequently, fn(x) is a cauchy sequence in (y, q s), thus {fn(x)} converges in (y, qs) and we define f such that {fn(x)} −→ f(x) in (y, q). moreover, {fn} converges to f in (sl0(d, q), ρ(d,q)). indeed, given ε, since fn(x) converges to f(x) for all x ∈ x, for each x, y there exists n′ such that if m′ ≥ n′ then qs(f(x) − fm′(x) − (f(y) − fm′(y))) d(x, y) < ε 2 and since {fn} is a cauchy sequence, we can also find n0 such that if m ′, n ≥ n0 then qs(fm′(x) − fn(x) − (fm′(y) − fn(y))) d(x, y) < ε 2 for all x, y ∈ x. thus we have ε > qs(f(x) − fm′(x) − (f(y) − fm′(y))) d(x, y) ≥ qs(f(x) − fn(x) − (f(y) − fn(y))) d(x, y) − qs(f′m(x) − fn(x) − (f ′ m(y) − fn(y))) d(x, y) and hence qs(f(x) − fn(x) − (f(y) − fn(y))) d(x, y) ≤ ε 2 + ε 2 = ε. since n0 is independent of x, y sup d(x,y) 6=0 qs(f(x) − fn(x) − (f(y) − fn(y))) d(x, y) < ε, for all n ≥ n0. consequently (sl0(d, q), ρ(d,q)) is a bicomplete space. � corollary 3.6. let {fn} be a cauchy sequence in (sl0(d, q), ‖ · ‖(d,q)) there exists a convergent sequence {kn} in (r, tu) such that kn is a semi-lipschitz constant for fn. 4. another completeness properties in this section, we discuss another completeness properties of the semilipschitz function space. let us recall that right k-completeness and left k-completeness constitute very useful extensions of the notion of completeness to the nonsymmetric context. in fact, they have been successfully applied to different fields from hyperspaces and function spaces to topological algebra and theoretical computer science. on semi-lipschitz functions with values in a quasi-normed linear space 225 let (x, d) be a quasi-metric space. a net {xδ} ⊂ x, δ ∈ λ, is called left k-cauchy provided that for each ε > 0 there is δ0 such that d(xδ2, xδ1) < ε for all δ1 ≥ δ2 ≥ δ0, and {xδ} ⊂ x is called right k-cauchy provided that for each ε > 0 there exists δ0 such that d(xδ1, xδ2 ) < ε for all δ1 ≥ δ2 ≥ δ0. a quasi-metric ρ is called left k-complete (resp. right k-complete) if each left k-cauchy net (resp. right k-cauchy net) converges. the following result allow us to prove that if (y, q) is a bibanach and finite dimensional space then (sl0(d, q) ρ(d,q)) is a right k-complete space: theorem 4.1. let (x, d), (y, q) be a quasi-metric and a quasi-normed bicomplete finite dimensional space respectively. then ρ(d,q) is right k-complete. proof. let {fδ} be a right k-cauchy net in (sl0(d, q), ρ(d,q)). then, given ε ≥ 0 there is δ0 such that (⋆) sup d(x,y) 6=0 q((fδ1 − fδ2)(x) − (fδ1 − fδ2)(y)) d(x, y) < ε for all δ1 ≥ δ2 ≥ δ0. let x 6= x0.we consider the following cases. case 1. d(x, x0) 6= 0 and d(x0, x) 6= 0. then, we deduce from (⋆) that given ε d(x,x0) and ε d(x0,x) respectively there exists δ′0 such that if δ1 ≥ δ2 ≥ δ ′ 0 then qs(fδ1(x) − fδ2(x)) < ε. therefore{fδ(x)} is a cauchy net in (y, q). case 2. d(x0, x) = 0 and d(x, x0) 6= 0. then q(fδ1(x)) − q(fδ2(x)) ≤ q(fδ1(x) − fδ2(x)) ≤ εd(x, x0) for all δ1 ≥ δ2 ≥ δ0 and q(−fδ(x)) = 0, since q s(fδ1(x)) ≤ εd(x, x0) + q(fn0(x)) thus {fn(x)} is a bounded net in the finite dimensional space (y, q), there exists a convergent subnet {fδ′(x)} in (y, q). given ε > 0 there exists δ0 ∈ λ such that if δ1 ≥ δ ′ 2 ≥ δ0 then qs(fδ1(x) − f(x)) = q s(fδ1(x) − fδ′ 2 (x) + fδ′ 2 (x)) ≤ qs(fδ1(x) − fδ′ 2 (x)) + qs(fδ′ 2 (x) − f(x)). now qs(fδ′ 2 (x)−f(x)) < ε 2 because {fδ′ 1 (x)} is a convergent net. given δ′1 ≥ δ1, such that fδ′ 1 is in the subnet then q(fδ′ 2 (x) − fδ1(x)) = q(fδ′ 2 (x) − fδ′ 1 (x) + fδ′ 1 (x) − fδ1(x)) ≤ q(fδ′ 2 (x) − fδ′ 1 (x)) + q(fδ′ 1 (x) − fδ1(x)) < ε 2 since {fδ′ 1 (x)} converges and {fδ} is right k-cauchy. on the other hand q(fδ1(x) − fδ′2(x)) < ε 2 because {fδ2} is a right k-cauchy net. thus {fδ(x)} is a convergent net. case 3. d(x0, x) 6= 0 and d(x, x0) = 0. then q(−fδ1(x)) ≤ εd(x0, x) + q(−fδ0(x)) and q(fδ(x)) = 0 respectively, since {fδ} is a bounded sequence on the finite dimensional space (y, q). 226 j. m. sánchez-álvarez consequently {fδ(x)} is a convergent net in (y, q s), and we define f such that {fn(x)} converges to f(x) for each x ∈ x. let {fδ} a right k-cauchy net in (sl0(d, q), ρ(d,q)). let us see that { q(fδ(x)−fδ(y)) d(x,y) } converges to q(f(x)−f(y)) d(x,y) for all x, y ∈ x such that d(x, y) 6= 0. since {fδ} is a right k-cauchy net, given ε > 0 there exists δ0 such that if δ1 ≥ δ2 ≥ δ0 then sup d(x,y) 6=0 q((fδ1 − fδ2)(x) − (fδ1 − fδ2)(y)) d(x, y) < ε 2 . since fn(x) converges to f(x)∀x ∈ x, for each x, y there exists δ ′ 0 such that if δ′1 ≥ δ ′ 0 then qs(f(x) − fδ′ 1 (x) − (f(y) − fδ′ 1 (y))) d(x, y) < ε 2 . thus given ε > 0, for all δ′ ≥ δ0 and for each x, y ∈ x : d(x, y) 6= 0 and we take δ1 ≥ (δ ′ ∨ δ′0) q(f(x) − fδ′(x) − (f(y) − fδ′(y))) d(x, y) = q(f(x) − fδ′(x) − fδ1(x) + fδ1(x) − (f(y) − fδ′(y) − fδ1(y) + fδ1(y))) d(x, y) ≤ q(f(x) − fδ1(x) − (f(y) + fδ1(y))) d(x, y) + q(fδ1(x) − fδ′(x) − (fδ′(y) + fδ1(y))) d(x, y) < ε. for all x, y such that d(x, y) 6= 0 sup d(x,y) 6=0 q(f(x) − fδ′(x) − (f(y) − fδ′(y))) d(x, y) < ε, for all δ′n ≥ δ0. � corollary 4.2. let (x, d), (y, q) be a quasi-metric and a quasi-normed space respectively. let {fδ} be a right k-cauchy in (sl0(d, q), ρ(d,q)). if for each x ∈ x {fδ(x)} converges to f(x) in (y, qs) then {fδ} converges to f in (sl0(d, q), ρ(d,q)). theorem 4.3. let (x, d), (y, q) be a quasi-metric t1 and a quasi-normed bicomplete space respectively. then ρ(d,q) is right k-complete. proof. let {fδ} be a right k-cauchy net in (sl0(d, q), ρ(d,q)). then, given ε ≥ 0 there is δ0 such that (♦) sup d(x,y) 6=0 q((fδ1 − fδ2)(x) − (fδ1 − fδ2)(y))) d(x, y) < ε for all δ1 ≥ δ2 ≥ δ0. let x 6= x0. on semi-lipschitz functions with values in a quasi-normed linear space 227 since (x, d) is t1, d(x, x0) 6= 0 and d(x0, x) 6= 0 then, we deduce from (♦) that given ε d(x,x0) and ε d(x0,x) there exists δ′0 such that if δ1 ≥ δ2 ≥ δ ′ 0 then qs(fδ1(x) − fδ2(x)) < ε. thereforefn(x) is a cauchy net in (y, q) for all x ∈ x. thus {fδ(x)} is a convergent net in (y, q s) and we define f such that {fn(x)} converges to f(x) for each x ∈ x. let {fδ} a right k-cauchy net in (sl0(d, q), ρ(d,q)). let us see that { q(fδ(x)−fδ(y)) d(x,y) } converges to q(f(x)−f(y)) d(x,y) for all x, y ∈ x such that d(x, y) 6= 0. since {fδ} is a right k-cauchy net, given ε > 0 there exists δ0 such that if δ1 ≥ δ2 ≥ δ0 then sup d(x,y) 6=0 q((fδ1 − fδ2)(x) − (fδ1 − fδ2)(y)) d(x, y) < ε 2 . since fn(x) converges to f(x)∀x ∈ x, for each x, y there exists δ ′ 0 such that if δ′1 ≥ δ ′ 0 then qs(f(x) − fδ′ 1 (x) − (f(y) − fδ′ 1 (y))) d(x, y) < ε 2 . thus given ε > 0, for all δ′ ≥ δ0 and for each x, y ∈ x : d(x, y) 6= 0 and we take δ1 ≥ (δ ′ ∨ δ′0) q(f(x) − fδ′(x) − (f(y) − fδ′(y))) d(x, y) = q(f(x) − fδ′(x) − fδ1(x) + fδ1(x) − (f(y) − fδ′(y) − fδ1(y) + fδ1(y))) d(x, y) ≤ q(f(x) − fδ1(x) − (f(y) + fδ1(y))) d(x, y) + q(fδ1(x) − fδ′(x) − (fδ′(y) + fδ1(y))) d(x, y) < ε. for all x, y such that d(x, y) 6= 0 sup d(x,y) 6=0 q(f(x) − fδ′(x) − (f(y) − fδ′(y))) d(x, y) < ε, for all δ′n ≥ δ0. � references [1] j. deák, a bitopological view of quasi-uniform completeness, i, studia sci. math. hungar. 30 (1995), 389–409; ii 30 (1995), 411-431; 31 (1996), 385–404. 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[8] h. p. a. künzi, nonsymmetric distances and their associated topologies: about the origin of basic ideas in the area of asymmetric topology, handbook of the history of general topology, ed. by c. e. aull and r. lowen, vol 3, hist. topol. 3, kluwer acad. publ., dordrecht, (2001), 853–968. [9] h. p. a. künzi, nonsymmetric topology, in topology with applications, bolyai soc. math. studies 4, pp. 303–338, szekszard, hungary, (1993). [10] s. romaguera, left k-completeness in quasi-metric spaces, math. nachr. 157 (1992), 15–23. [11] s. romaguera and m. sanchis, properties of the normed cone of semi-lipschitz functions, acta math. hungar. 108 (1-2) (2005), 55–70. [12] s. romaguera and m. sanchis, on semi-lipschitz functions and best approximation in quasi-metric spaces, j. approximation theory 283 (2000), 292–301. [13] s. romaguera and m. sanchis, applications of utility functions defined on quasi-metric spaces, j. math. anal. appl. 283 (2003), 219–235. 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[19] ph. sünderhauf, quasi-uniform completeness in terms of cauchy nets, acta math. hungar. 69 (1995), 47–54. received january 2005 accepted may 2005 josé manuel sánchez-álvarez (jossnclv@doctor.upv.es) escuela de caminos, departamento de matemática aplicada, universidad politécnica de valencia, 46071 valencia, spain. kohliagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 1, 2005 pp. 1-14 on functionally θ-normal spaces j. k. kohli and a. k. das abstract. characterizations of functionally θ-normal spaces including the one that of urysohn’s type lemma, are obtained. interrelations among (functionally) θ-normal spaces and certain generalizations of normal spaces are discussed. it is shown that every almost regular (or mildly normal ≡ κ-normal) θ-normal space is functionally θ-normal. moreover, it is shown that every almost regular weakly θ-normal space is mildly normal. a factorization of functionally θ-normal space is given. a tietze’s type theorem for weakly functionally θ-normal space is obtained. a variety of situations in mathematical literature wherein the spaces encountered are (functionally) θ-normal but not normal are illustrated. 2000 ams classification: primary: 54d10, 54d15, 54d20; secondary: 14a10. keywords: θ-closed (open) set, regularly closed (open) set, zero set, regular gδ-set, (weakly) (functionally) θ-normal space, (weakly) θ-regular space, almost regular space, mildly normal (≡ κ-normal) space, almost normal space, δ-normal space, δ-normally separated space, zariski topology, (distributive) lattice, complete lattice, affine algebraic variety, projective variety. 1. introduction normality is an important topological property and hence it is of significance both from intrinsic interest as well as from applications view point to obtain factorizations of normality in terms of weaker topological properties. first step in this direction was taken by viglino [18] who defined seminormal spaces, followed by the work of singal and arya [13] who introduced the class of almost normal spaces and proved that a space is normal if and only if it is both a seminormal space and an almost normal space. a search for another decomposition of normality motivated us to introduce in [6] the class of (weakly) θ-normal spaces and (weakly) functionally θ-normal spaces. these weak forms of normality serve as a necessary ingredient towards a decomposition of normality. 2 j. k. kohli and a. k. das in [6] functionally θ-normal spaces are defined in terms of the existence of certain continuous real-valued functions. in this paper, in analogy with the normal spaces, we obtain a characterization of functionally θ-normal spaces in terms of separation of certain closed sets by θ-open sets. the resulting characterizations are then used to investigate the interrelations that exist among certain generalizations of normal spaces such as (weakly) (functionally) θ-normal spaces, δ-normal spaces defined by mack [10], δ-normally separated spaces initiated by mack [10] and zenor [19], mildly normal spaces (≡ κ-normal spaces) studied by stchepin [15], and singal and singal [14], and almost normal spaces studied by singal and arya [13]. moreover, we obtain a decomposition of functional θ-normality in terms of weak functional θ-normality. in the process we obtain improvements of several known results in the literature. furthermore, a tietze’s type theorem for weakly functionally θ-normal spaces is obtained. section 2 is devoted to basic definitions and preliminaries. characterizations of functionally θ-normal spaces are obtained in section 3. a tietze’s type extension theorem is included in section 4. interrelations among (weakly) (functionally) θ-normal spaces and certain other weak variants of normality are investigated in section 5. moreover, a factorization of functional θ-normality in terms of weak functional θ-normality is obtained in section 5. finally, an appendix (section 6) is included which exhibits a wide variety of situations encountered in mathematical literature wherein a space may be (functionally) θ-normal but not normal. throughout the paper, no separation axioms are assumed unless explicitly stated otherwise. the closure of a set a will be denoted by a and interior by inta. 2. basic definitions and preliminaries definition 2.1 ([17]). let x be a topological space and let a ⊂ x. a point x ∈ x is called a θ-limit point of a if every closed neighbourhood of x intersects a. let aθ denote the set of all θ-limit points of a. the set a is called θ-closed if a = aθ. the complement of a θ-closed set is referred to as a θ-open set. lemma 2.2 ([17]). an arbitrary intersection of θ-closed sets is θ-closed and the union of finitely many θ-closed sets is θ-closed. in general the θ-closure operator is not a kuratowski closure operator, since θ-closure of a set may not be θ-closed (see [4]). however, the following modification yields a kuratowski closure operator. definition 2.3. let x be a topological space and let a ⊂ x. a point x ∈ xis called a uθ-limit point of a if every θ-open set u containing x intersects a. let auθ denote the set of all uθ-limit points of a. lemma 2.4. the correspondence a → auθ is a kuratowski closure operator. it turns out that the set auθ is the smallest θ-closed set containing a. on functionally θ-normal spaces 3 definition 2.5 ([2]). a function f : x → y is said to be θ-continuous if for each x ∈ x and each open set u containing f (x), there exists an open set v containing x such that f (v ) ⊂ u . the concept of θ-continuity is a slight generalization of continuity and is useful in studying spaces which are not regular. lemma 2.6. let f : x → y be a θ-continuous function and let u be a θ-open set in y. then f −1(u ) is θ-open in x. lemma 2.7 ([6, 7]). a subset a of a topological space x is θ-open if and only if for each x ∈ a there exists an open set u such that x ∈ u ⊂ u ⊂ a. definition 2.8. a subset a of a topological space x is called regular gδ-set if a is the intersection of countably many closed sets whose interiors contain a. the following lemma seems to be known and is easily verified. lemma 2.9. in a topological space, every zero set is a regular gδ-set and every regular gδ-set is θ-closed. definition 2.10 ([13]). a topological space x is said to be almost normal if every pair of disjoint closed sets one of which is regularly closed are contained in disjoint open sets. definition 2.11 ([14, 15]). a topological space x is said to be mildly normal (≡ κ-normal) if every pair of disjoint regularly closed sets are contained in disjoint open sets. 3. functionally θ-normal spaces definition 3.1 ([6]). a topological space xis said to be (i) θ-normal if every pair of disjoint closed sets one of which is θ-closed are contained in disjoint open sets; (ii) weakly θ-normal if every pair of disjoint θ-closed sets are contained in disjoint open sets; (iii) functionally θ-normal if for every pair of disjoint closed sets a and b one of which is θ-closed there exists a continuous function f : x →[0,1] such that f (a) = 0 and f (b)=1; (iv) weakly functionally θ-normal (wf θ-normal) if for every pair of disjoint θ-closed sets a and b there exists a continuous function f : x → [0,1] such that f (a) = 0 and f (b)= 1; and (v) θ-regular if for each closed set f and each open set u containing f , there exists a θ-open set v such that f ⊂ v ⊂ u . definition 3.2. a topological space x is said to be (i) δ-normal [10] if every pair of disjoint closed sets one of which is a regular gδ-set are contained in disjoint open sets; and 4 j. k. kohli and a. k. das (ii) δ-normally separated [10, 19] if for every pair of disjoint closed sets a and b one of which is a zero set there exists a continuous function f : x → [0,1] such that f (a)=0 and f (b) = 1. the following implications are immediate in view of definitions and lemma 2.9 and well illustrate the interrelations that exist among generalizations of normality outlined in definitions 3.1 and 3.2. none of the above implications is reversible as is shown by examples 3.6, 3.7, 3.8 in [6], examples [10, p. 267, p.270] and example 3.4 in the sequel. it is shown in [6] that in the class of hausdorff spaces, the notions of θnormality and functional θ-normality coincide with normality and that in the class of θ-regular spaces all the four variants of θ-normality characterize normality. furthermore, it is shown in [6] that every lindelöf space as well as every almost compact space is weakly θ-normal. in contrast the class of functionally θ-normal space is much larger than the class of normal spaces (see section 6). our next result shows that a urysohn type lemma holds for functionally θ-normal spaces. theorem 3.3. for a topological space x, the following statements are equivalent. (a) x is functionally θ-normal. (b) for every pair of disjoint closed sets one of which is θ-closed are contained in disjoint θ-open sets. (c) for every θ-closed set a and every open set u containing a there exists a θ-open set v such that a ⊂ v ⊂ vuθ ⊂ u . (d) for every closed set a and every θ-open set u containing a there exists a θ-open set v such that a ⊂ v ⊂ vuθ ⊂ u . (e) for every pair of disjoint closed sets a and b, one of which is θclosed there exist θ-open sets u and v such that a ⊂ u , b ⊂ v and uuθ ∩ vuθ = φ. on functionally θ-normal spaces 5 proof. to prove the assertion (a) ⇒ (b), let x be a functionally θ-normal space and let a, b be disjoint closed sets in x, where b is θ-closed. by functional θ-normality of x there exists a continuous function f : x →[0,1] such that f (a) = 0 and f (b) = 1. since every continuous function is θ-continuous, by lemma 2.6, f −1[0, 1/2) and f −1 (1/2, 1] are disjoint θ-open sets containing a and b respectively. to prove (b) ⇒ (c), let a be a θ-closed set in x and let u be an open set containing a. since a and x −u are disjoint, by hypothesis there exist disjoint θ-open sets v and w such that a ⊂ v and x −u ⊂ w . so a ⊂ v ⊂ x −w ⊂ u . since x − w is θ-closed and vuθ is the smallest θ-closed set containing v , a ⊂ v ⊂ vuθ ⊂ u . to prove (c) ⇒ (d), let a be a closed set contained in a θ-open set u . then x − u is a θ-closed set contained in the open set x − a. by hypothesis, there exists a θ-open set w such that x −u ⊂ w ⊂ wuθ ⊂ x −a. let v = x −wuθ. then a ⊂ v ⊂ x − w ⊂ u . since x − w is θ-closed and vuθ is the smallest θ-closed set containing v , a ⊂ v ⊂ vuθ ⊂ u . to prove (d) ⇒ (e), let a be a closed set disjoint from a θ-closed set b. then x − b is a θ-open set containing a. so there exists a θ-open set w such that a ⊂ w ⊂ wuθ ⊂ x − b. again by hypothesis there exists a θ-open set u such that a ⊂ u ⊂ uuθ ⊂ w ⊂ wuθ ⊂ x − b. let v = x − wuθ, then u and v are θ-open sets containing a and b respectively and uuθ ∩ vuθ = ∅. the assertion (e) ⇒ (c) is easily verified. finally to prove the implication (c) ⇒ (a), let a be a θ-closed set disjoint from a closed set b. then a ⊂ x − b = u1 (say). since u1 is open, there exists a θ-open set u1/2 such that a ⊂ u1/2 ⊂ (u1/2)uθ ⊂ u1. again, since (u1/2)uθ is a θ-closed set, there exist θ-open sets u1/4 and u3/4 such that a ⊂ u1/4 ⊂ (u1/4)uθ ⊂ u1/2 and (u1/2)uθ ⊂ u3/4 ⊂ (u3/4)uθ ⊂ u1. continuing the above process, we obtain for each dyadic rational r, a θ-open set ur satisfying r < s implies (ur)uθ ⊂ us. let us define a mapping f : x →[0,1] by f (x) = { inf { x : x ∈ ur } if x belongs to some ur, 1 if x does not belongs to any ur. clearly f is well defined and f (a) = 0, f (b) = 1. now it remains to prove that f is continuous. to this end we first observe that if x ∈ ur, then f (x) ≤ r. similarly f (x) ≥ r if x /∈ (ur)uθ. to prove continuity, let x ∈ x and (a, b) be an open interval containing f (x). now choose two dyadic rationals p and q such that a < p < f (x) < q < b. let u = uq − (up)uθ. then u is an open set containing x. now for y ∈ u , y ∈ uq. so f (y) ≤ q. also as y ∈ u , y /∈ (up)uθ. thus f (y) ≥ q. and so f (y) ∈ [p, q]. therefore f (u ) ⊂ [p, q] ⊂ (a, b). hence f is continuous. � example 3.4. a θ-normal space which is not functionally θ-normal. let x be the set of positive integers. define a topology t on x, where every odd integer is open and a set u is open if for every even integer p ∈ u , the successor and the predecessor of p also belongs to u . let y = x∪{∞} be the one point compactification of the space x. since every paracompact space is θ-normal 6 j. k. kohli and a. k. das [6, theorem 3.12], y is θ-normal. the space y is not functionally θ-normal since the θ-closed set {∞} and the closed set {1,2} cannot be separated by disjoint θ-open sets in y . the above example shows that even a compact θ-normal space need not be functionally θ-normal, and so fills a gap left in [6]. 4. a tietze type theorem in this section we formulate a tietze’s type extension theorem for weakly functionally θ-normal spaces. first we introduce the notion of a θ-embedded subset which is instrumental in the formulation of tietze type theorem. definition 4.1. a subset a of a topological space x is said to be θ-embedded in x if every θ-closed set in the subspace topology of a is the intersection of a with a θ-closed set in x; equivalently a ⊂ xis θ-embedded in x if every θ-open set in the subspace topology of a is the intersection of a with a θ-open set in x. remark 4.2. a θ-closed subset of a topological space x need not be θembedded in x. for example, let us consider the closed unit interval x = [0,1] with smirnov’s deleted sequence topology [16, p. 86]. let a = k∪{0}, where k = {1/n : n ∈ n}. here a is a θ-closed subset of x which is not θ-embedded in x. theorem 4.3. let x be a weakly functionally θ-normal space and let a be a θ-closed, θ-embedded subset of x. then (a) every continuous function f : a → [0, 1] defined on the set a can be extended to a continuous function g : x → [0, 1]. (b) every continuous function f : a → r can be extended to a continuous function g : x → r. proof. we shall prove the theorem only in case (a). let x be a weakly functionally θ-normal space and let a be a θ-closed set in x. let a1 = {x ∈ a : f (x) ≥ 1/3} and b1 = {x ∈ a : f (x) ≤ −1/3}. then a1 and b1 are two disjoint θ-closed sets in a. since a is θ-embedded in x, a1 = f1 ∩ a and b1 = f2 ∩ a, where f1 and f2 are θ-closed sets in x. since a is θ-closed in x, by lemma 2.2, a1 and b1 are disjoint θ-closed sets in x. by weak functional θ-normality of x, there exists a continuous function f1 : x → [−1/3, 1/3] such that f1(a1) = 1/3 and f1(b1) = −1/3. now, for each x ∈ a, it is clear that |f (x) − f1(x)| ≤ 2/3. so f − f1 is a mapping of a into [-2/3, 2/3]. let g1 = f − f1, then a2 = {x ∈ a : g1(x) ≥ 2/9} and b2 = {x ∈ a : g1(x) ≤ −2/9} are two disjoint θ-closed sets in a. as argued earlier, a2 and b2 are θ-closed sets in x. again, by weak functional θ-normality of x there exists a continuous function f2 : x → [−2/9, 2/9] such that f2(a2) = 2/9 and f2(b2) = −2/9. clearly, |(f − f1) − f2| ≤ (2/3) 2 on a. continuing this process, we obtain a sequence of continuous functions {fn} defined on a such that |f − n∑ i=1 fi| ≤ (2/3) n on a. it is routine to verify that on functionally θ-normal spaces 7 the function g : x → r defined by g(x) = ∞∑ i=1 fi(x) for every x ∈ x, is the desired continuous extension of f . � corollary 4.4. let x be a functionally θ-normal space. then every continuous function f : a → [0, 1] (f : a → r) defined on a θ-closed, θ-embedded subset a of x can be extended to a continuous function g : x → [0, 1] (g : x → r). 5. interrelations in this section we exhibit the relationships that exists among (functionally) θ-normal spaces, mildly normal spaces and δ-normally separated spaces etc. in the presence of additional mild conditions. this in turn yields improvements of certain known results in the literature. definition 5.1 ([12]). a topological space x is said to be almost regular if every regularly closed set and a point out side it are contained in disjoint open sets. the following characterization of almost regular spaces obtained in [8] will be useful in the sequel. theorem 5.2 ([8]). a topological space x is almost regular if and only if, for every open set u in x, int u is θ-open. theorem 5.3 ([6]). a topological space x is θ-normal if and only if for every pair of disjoint closed sets a, b one of which is θ-closed, there exist disjoint open sets u and v such that a ⊂ u , b ⊂ v and u ∩ v = ∅. theorem 5.4. an almost regular, θ-normal space is functionally θ-normal. proof. let x be an almost regular, θ-normal space. let a be a θ-closed set disjoint from a closed set b. since x is θ-normal, by theorem 5.3, there exist disjoint open sets u and v such that a ⊂ u , b ⊂ v and u ∩ v = ∅. now a ⊂ u ⊂ intu ⊂ u and b ⊂ v ⊂ intv ⊂ v . since x is almost regular, by theorem 5.2, intu and intv are disjoint θ-open sets containing a and b respectively. so in view of theorem 3.3, the space x is functionally θ-normal. � corollary 5.5. an almost regular, θ-normal space is δ-normally separated. proof. every functionally θ-normal space is δ-normally separated. � corollary 5.6. an almost regular paracompact space is functionally θ-normal. proof. it is shown in [6] that, every paracompact space is θ-normal. hence the result follows by theorem 5.4. � theorem 5.7. a mildly normal, θ-normal space is functionally θ-normal. 8 j. k. kohli and a. k. das proof. suppose x is a mildly normal, θ-normal space and let a and b be disjoint closed sets in x, where a is θ-closed . since x is θ-normal, by theorem 5.3, there exists disjoint open sets u and v such that a ⊂ u , b ⊂ v and u ∩ v = ∅. then u and v are disjoint regularly closed sets in x. since x is mildly normal, by [13, theorem 3], there exists a continuous function f : x → [0, 1] such that f (u ) = 0 and f (v ) = 1. consequently f (a) = 0 and f (b) = 1 and so x is a functionally θ-normal space. � corollary 5.8. a mildly normal paracompact space is functionally θ-normal. proof. every paracompact space is θ-normal [6]. � corollary 5.9. a mildly normal, θ-normal space is δ-normally separated. proof. every functionally θ-normal space is δ-normally separated. � remark 5.10. the example of smirnov’s deleted sequence topology [16, p.86] shows that the hypothesis of θ-normal space in theorem 5.7 can not be weakened to even “weakly functionally θ-normal space”. remark 5.11. example 3.4 shows that the hypothesis of almost regularity in theorem 5.4 can not be omitted. the same example also shows that the hypothesis of mild normality in theorem 5.7 can not be deleted. moreover, simple examples can be given to show that even an almost regular compact space need not be normal. theorem 5.12. an almost regular weakly θ-normal space is mildly normal. proof. suppose x is an almost regular weakly θ-normal space and let a and b be disjoint regularly closed sets in x. since a is regularly closed, x − a is regularly open and so x − a = int(x − a). by theorem 5.2, x − a is θ-open and hence a is θ-closed. similarly, b is θ-closed. since x is weakly θ-normal, there exist disjoint open sets u and v containing a and b, respectively. � corollary 5.13 ([14]). an almost regular almost compact space is mildly normal. proof. it is observed in [6] that every almost compact space is weakly θ-normal. � corollary 5.14 ([14]). an almost regular lindelöf space is mildly normal. proof. every lindelöf space is weakly θ-normal [6]. � corollary 5.15 ([11]). an almost compact urysohn space is mildly normal. proof. every almost compact urysohn space is almost regular [11]. � theorem 5.16. an almost regular θ-normal space is almost normal. on functionally θ-normal spaces 9 proof. suppose x is an almost regular, θ-normal space. let a and b be disjoint closed sets in x such that a is regularly closed. in view of theorem 5.2 (as also shown in the proof of theorem 5.12), a is θ-closed. so by θ-normality, there exist disjoint open sets u and v containing a and b, respectively. thus x is almost normal. � corollary 5.17 ([13]). an almost regular paracompact space is almost normal. proof. every paracompact space is θ-normal [6]. � theorem 5.18. an almost regular weakly θ-normal space is weakly functionally θ-normal. proof. let x be an almost regular weakly θ-normal space and let a and b be disjoint θ-closed sets in x. then there exist disjoint open sets u and v containing a and b, respectively. it is easily verified that the sets intu and intv are disjoint. by theorem 5.2, each of the sets intu and intv is θ-open. thus every pair of disjoint θ-closed sets in xare separated by disjoint θ-open sets and so by [7, theorem 8], x is weakly functionally θ-normal. � corollary 5.19. an almost regular almost compact space is weakly functionally θ-normal. corollary 5.20. an almost regular lindelöf space is weakly functionally θnormal. next we give a factorization of functionally θ-normal space in terms of weakly functionally θ-normal space. definition 5.21. a topological space x is said to be weakly θ-regular if for each θ-closed set f and each open set u containing f , there exists a θ-open set v such that f ⊂ v ⊂ u . every θ-regular space is weakly θ-regular. however, the cofinite topology on an infinite set yields a weakly θ-regular space (vacuously) which is not θ-regular. theorem 5.22. a topological space x is functionally θ-normal if and only if it is both a weakly θ-regular space and a wf θ-normal space. proof. necessity is immediate in view of theorem 3.3 and the fact that every functionally θ-normal space is wf θ-normal. to prove sufficiency suppose that x is a weakly θ-regular space and a wf θ-normal space. let a and b be disjoint closed sets in x such that a is θ-closed. then a ⊂ x − b and x − b is open. so by weakly θ-regularity of x, there exists a θ-open set u such that a ⊂ u ⊂ x − b. then a and x − u are disjoint θ-closed sets and b ⊂ x − u . since x is wf θ-normal, there exists a continuous function f : x → [0, 1] such that f (a) =0 and f (x − u ) = 1. since b ⊂ x − u , x is functionally θ-normal. � theorem 5.23. a topological space x which is both a weakly θ-regular space and a weakly θ-normal space is θ-normal. 10 j. k. kohli and a. k. das remark 5.24. example 3.4 is a wf θ-normal space which is not weakly θregular. similarly, the smirnov’s deleted sequence topology [16, p.86] is a weakly θ-normal space which is not weakly θ-regular. we conclude this section with the following characterizations of normality in hausdorff spaces. theorem 5.25. for a hausdorff space x, the following statements are equivalent. (a) x is normal. (b) x is functionally θ-normal. (c) x is θ-normal. (d) x is weakly θ-regular and wf θ-normal. (e) x is weakly θ-regular and weakly θ-normal. proof. equivalence of (a)-(c) is given in [6] and the equivalence of (a), (d) and (e) is immediate in view of theorems 5.22 and 5.23. � 6. appendix (examples) theorem 5.25 shows that the notions of θ-normality and functional θ-normality assume significance in non-hausdorff spaces. modern applications of topology in algebraic geometry, spectral theory of commutative rings and lattices, and theoretical computer science have advanced the point of view that interesting topological spaces need not be hausdorff. in this section we exhibit a few situations arising in spectral theory of commutative rings, topologies on partially ordered sets and lattices and in algebraic geometry wherein the spaces involved are (functionally) θ-normal but not necessarily normal. • every finite topological space is functionally θ-normal which need not be normal. • cofinite topology on an infinite set as well as the co-countable topology on an uncountable set is a functionally θ-normal space which is not normal. • one point compactification of a t1-space which is not a locally compact hausdorff space is a θ-normal space which is not normal [5]. • the wallman compactification of a non-normal t1-space is a θ-normal space which is not normal [5]. 6.1. prime spectrum of a ring. let r be a commutative ring with unity. let x = spec(r) be the set of all prime ideals of r. for each ideal i of r, let v (i) = {p ∈ spec(r) : i ⊂ p }. the collection {spec(r)\v (i) : i is an ideal of r} is a topology on x and the collection of all sets xf = {p ∈ spec(r) : f /∈ p }, f ∈ r constitutes a base for this topology. the topology on specr described above is called the zariski topology. spec(r) endowed with zariski topology is called prime spectrum of the ring r and is always a compact t0-space. so by [6, theorem 3.12], spec(r) is a θ-normal space which is not necessarily a normal space. on functionally θ-normal spaces 11 for example, the spectrum of the ring z of integers is homeomorphic to a countably infinite space x in which, apart from x, only finite subsets are closed. this space is a compact t1-space which is not hausdorff and in which any two nonempty open sets intersects. hence x is a functionally θ-normal space which is not normal. 6.2. topological representation of lattices. details of the definitions and results quoted here may be found in [1]. let l be a distributive lattice. let p (l) denotes the set of all prime ideals of l. for each x ∈ i, x̂ = {i ∈ p (l) : x /∈ i}. then (1) x̂ ∪ ŷ = x̂ + y for all x, y ∈ l. (2) x̂ ∩ ŷ = x̂y for all x, y ∈ l. definition 6.1. a stone space is a topological space x satisfying: (i) xis a t0-space. (ii) compact open subsets of x constitute a base for x and ring of sets. (iii) if (xs)s∈s and (yt)t∈t are nonempty families of nonempty compact open sets and ⋂ s∈s xs ⊂ ⋂ t∈t yt, then there exist finite subsets s’ and t ’ of s and t respectively, such that ⋂ s∈s′ xs ⊂ ⋂ t∈t ′ yt. for a distributive lattice l, let r(l) denotes the representation space of l, whose points are members of p (l) with the topology induced by taking the collection {∅}∪{ ∧ x : x ∈ l} as a base. then the space r(l) is a stone space. • let l be a distributive lattice with 1. then its representation space r(l) is a compact space and so it is a θ-normal space which need not be normal (see [1, p.78]). • a distributive lattice is relatively complemented if and only if its representation space r(l) is a t1-space. hence if l is a relatively complemented distributive lattice with 1 which is not a boolean algebra, then its representation space r(l) is a compact t1-space which is not hausdorff (see [1, p.78]). so r(l) is a θ-normal space which is not normal. 6.3. topologies on lattices and posets. details of the results and definitions quoted below may be found in [3]. definition 6.2. let l be a lattice. the topology generated by the complements l\ ↑ x of all filters is called the lower topology on l and is denoted by w(l). a base for the lower topology w(l) is given by {l\ ↑ f : f is a finite subset of l}. definition 6.3. let l be a lattice. the upper topology on l is generated by the collection of all sets l\ ↑ x and is denoted by v(l). definition 6.4. let l be a poset closed under directed sups. a topology on l is said to be order consistent if 12 j. k. kohli and a. k. das (i) {x} =↓ x for all x ∈ l (ii) if x = sup i for an ideal, then x = lim i. definition 6.5. let l be a complete lattice. a subset u of l is said to be scott open if (i) u =↑ u , i.e., u is an upper set. (ii) sup d ∈ u implies d ∩ u 6= ∅ for all directed d ⊆ l. the collection of all scott open subsets of l constitutes a topology on l called scott topology and is denoted by σ(l). definition 6.6. let l be a complete lattice. then the common refinement of scott topology and the lower topology is called the lawson topology and is denoted by λ(l). definition 6.7. a topology on a poset is called compatible if the directed nets converge to their sups and filtered nets converge to their infs. • for a complete lattice l, the space (l, λ(l)) is a compact t1-space and hence a θ-normal space which may fail to be normal (unless it is hausdorff) (see [3, p.146]). • let (x, ≤) be a complete lattice equipped with a topology such that the relation ≤ is lower semicontinuous (i.e. each ↓ x is closed in x). then the set of open upper sets is an order consistent topology on x and hence x is a compact space [3, p.307]. so x is a θ-normal space. 6.4. affine algebraic varieties. details of the definitions and results quoted below may be found in [9]. let an(l) be n-dimensional affine space over a field l and let k ⊂ l be a subfield. let k[x1, . . . , xn ] be the polynomial ring in n variables over k. definition 6.8. a subset v ⊂ an(l) is called an affine algebraic k-variety if there are polynomials f1, . . . , fm ∈ k[x1,. . . xn] such that v is the solution set of the equations fi (x1, . . . , xn)=0 ( i =1,. . . ,m). a k-variety is called irreducible if v = v1 ∪ v2 with k-varieties v1, v2, then v = v1 or v = v2. the finite unions and arbitrary intersections of k-varieties in an(l) are kvarieties. thus k-varieties form the closed sets of a topology on an(l) called the zariski topology with respect to k. if v ⊂ an(l) is a k-variety, then v carries the relative zariski topology on v . definition 6.9. [16] a topological space x is said to be hyperconnected if every nonempty open set is dense in x. definition 6.10. a topological space x is called noetherian if every descending chain f1 ⊃ f2 ⊃. . . .. of closed subsets fi ⊂ x is stationary. • an algebraic k-variety endowed with zariski topology is a hyperconnected space if and only if it is irreducible, and every hyperconnected space is functionally θ-normal (which need not be normal). for example, co-finite topology on an infinite set is a hyperconnected space on functionally θ-normal spaces 13 which is not normal. thus it turns out that every irreducible algebraic k-variety is a functionally θ-normal space which need not be normal. • it turns out that every open subset of a noetherian topological space is compact and hence θ-normal by [6, theorem 3.12]. since every algebraic k-variety v endowed with zariski topology is a noetherian topological space, so every open subset of v is a θ-normal space which is not necessarily normal. 6.5. projective varieties. the n-dimensional projective space p n(l) over a field l is the set of all lines through the origin in ln+1. a point x ∈ p n(l) can be represented by an (n+1)-tuple (x0,. . . .,xn) 6=(0,. . . ,0) in ln+1 and (x0’,. . . ,xn’)∈l n+1 defines the same point if and only if there is λ ∈l, with (x0,. . . ,xn)=(λx0’,. . . ,λxn’). an (n+1)-tuple representing x is called the system of homogeneous co-ordinates of x. we write x = < x0,. . . .,xn >. if k is a subfield of l and f ∈ k[x0,. . . ,xn], then x ∈ p n(l) is called a zero of f if f (x0,. . . , xn)=0 for every system (x0,. . . ,xn) of homogeneous co-ordinates of x. definition 6.11. a subset v ⊂ p n(l) is called a projective algebraic kvariety if there are homogeneous polynomials f1,. . . ,fm ∈ k[x0,. . . ,xn] such that v is the set of all common zeros of fi in p n(l). the finite unions and arbitrary intersections of projective k-varieties in p n(l) are projective k-varieties. thus projective k-varieties in p n(l) constitute the closed sets of a topology on p n(l), called the k-zariski topology. it turns out p n(l) (and hence any projective k-variety) endowed with k-zariski topology is a noetherian topological space and so every open subset of p n(l) is a θ-normal space which need not be normal. moreover, an irreducible projective k-variety endowed with k-zariski topology is a hyperconnected space and so it is a functionally θ-normal space which need not be normal. references [1] r. balbes and p. dwinger, distributive lattices, university of missouri press, missouri (1974). [2] s. fomin, extensions of topological spaces, ann. of math. 44 (1943), 471–480. [3] g. gierz, k. h. hoffman, k. keimel, j. d. lawson, m. mislov and d. s. scott, a compendium of continuous lattices, springer verlag, berlin (1980). [4] j. e. joseph, θ-closure and θ-subclosed graphs, math. chron. 8(1979), 99–117. [5] j. l. kelley, general topology, van nostrand, new york, (1955). [6] j. k. kohli and a. k. das, new normality axioms and decompositions of normality, glasnik mat. 37(57) (2002), 163–173. [7] j. k. kohli, a. k. das and r. kumar, weakly functionally θ-normal spaces, θ−shrinking of covers and partition of unity, note di matematica 19(2) (1999), 293–297. [8] j. k. kohli and a. k. das, a class of spaces containing all almost compact spaces (preprint). [9] ernst kunz, introduction to commutative algebra and algebraic geometry, birkhäuser, boston, (1985). [10] j. mack, countable paracompactness and weak normality properties, trans. amer. math. soc. 148 (1970), 265–272. 14 j. k. kohli and a. k. das [11] p. papić sur les espaces h-fermes, glasnik mat. -fiz astr. 14 (1959) 135–141. [12] m. k. singal and s. p. arya, on almost regular spaces, glasnik mat. 4(24) (1969), 89–99. [13] m. k. singal and s. p. arya, on almost normal and almost completely regular spaces, glasnik mat. 5(25) (1970), 141–152. [14] m. k. singal and a. r. singal, mildly normal spaces, kyungpook math j. 13 (1973), 27–31. [15] e. v. stchepin, real valued functions and spaces close to normal, sib. j. math. 13:5 (1972), 1182–1196. [16] l. a. steen and j. a. seeback, counter examples in topology, springer verlag, new york, (1978). [17] n. v. veličko h-closed topological spaces, amer. math. soc, transl. 78(2), (1968), 103– 118. [18] g. vigilino, seminormal and c-compact spaces, duke j. math. 38 (1971), 57–61. [19] p. zenor, a note on z-mappings and wz-mappings, proc. amer. math. soc. 23 (1969), 273–275. received may 2002 accepted september 2002 j. k. kohli department of mathematics, hindu college, university of delhi, delhi-110007, india a. k. das (akdas@du.ac.in, ak das@lycos.com) department of mathematics, university of delhi, delhi-110007, india carvalag.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 9196 homeomorphisms of r and the davey space sheila carter and f. j. craveiro de carvalho abstract. up to homeomorphism, there are 9 topologies on a three point set {a, b, c} [4]. among the resulting topological spaces we have the so called davey space, where the only non-trivial open set is, let us say, {a}. this is an interesting topological space to the extent that every topological space can be embedded in a product of davey spaces [3]. in this note we will consider the problem of obtaining the davey space as a quotient r/g, where g is a suitable homeomorphism group. the present work can be regarded as a follow-up to some previous work done by one of the authors and bernd wegner [1]. 2000 ams classification: 54f65. keywords: davey space, homeomorphism group, cantor set. 1. r/g as the davey space -necessary conditions we will take the topological space ({a, b, c}, τ), with τ = {∅, {a}, {a, b, c}}, as a model for the davey space and the real line will be denoted by r. our purpose is to obtain a group g of homeomorphisms of r whose natural action on r gives rise to the davey space and we start by establishing a number of observations which guided our quest. below we assume that the homeomorphism group g is such that r/g is the davey space and π will stand for the projection from r to r/g. proposition 1.1. g is not finite. proof. if g were finite then, for instance, π−1(b) would be finite and, consequently, {a, c} would be open. � proposition 1.2. π−1(a) is bounded neither above nor below. proof. assume that π−1(a) is bounded above and let x be its supremum. then π((x, +∞)) is open in r/g and, consequently must contain a. � 92 sheila carter and f. j. craveiro de carvalho proposition 1.3. π−1({b, c}) is bounded neither above nor below. proof. assume that x is the supremum of π−1({b, c}). since this set is closed in r, x belongs to it. let us suppose that π(x) = b. as we will see below it then follows that | π−1(b) |≤ 2 which, as remarked above, is impossible. let y, z be points in π−1(b) with y < z < x. there is a homeomorphism f in g such that f(z) = x. if f were increasing then f(x) > x. therefore f must be decreasing and, since y < z, f(y) > x which, again, is impossible. � proposition 1.4. π−1(b), π−1(c) are bounded neither above nor below. proof. assume that π−1(b) is bounded above and let x be its supremum. then π((x, +∞)) must be {a} and x is an upper bound for π−1(c). consequently π−1({b, c}) would be bounded above. � we are now in a position which allows us to conclude theorem 1.5. the action of g is not free. proof. let π−1(a) = ⋃ i∈i ci, where the ci’s are the connected components. from above it follows that, for each i, ci = (ai, bi). fix an i and choose x, y distinct in ci. there is an f ∈ g such that f(x) = y. since f maps [ai, bi] into itself, it must have a fixed point. � it is also clear that π−1({b, c}) is totally disconnected and that every point in it is a limit point of that set. proposition 1.6. π−1({b, c}) is uncountable. proof. write π−1(a) = ⋃ i∈i ci and choose x, y in different components, with x < y. then π−1({b, c}) ⋂ [x, y] is a compact, hausdorff space having all its elements as limits points. therefore it is uncountable [4]. � 2. an example this section is devoted to the construction of an example of a group g such that r/g is the davey space. since they are homeomorphic spaces we will use the open interval (0, 1) instead of r. let c denote the intersection of the cantor set [2], [5] with (0, 1) and consider the partition (0, 1) = a∪b∪c, where a = (0, 1)\c is a union of open intervals, the “middle thirds”, b is the set of end-points of the open intervals in a and c = c \ b. the cantor set can be described in terms of ternary expansions. we then have, for x ∈ (0, 1), that homeomorphisms of r and the davey space 93 x ∈ a if and only if there is n ∈ n such that x = ∞∑ i=1 xi 3i , where, for i < n, xi = 0 or 2, xn = 1 and 0 < ∞∑ i=n+1 xi 3i < 1 3n , x ∈ b if and only if there is n ∈ n such that x = n∑ i=1 xi 3i , where, for i < n, xi = 0 or 2, xn = 1 or 2. x ∈ c if and only if x = ∞∑ i=1 xi 3i , with xi = 0 or 2, and there are arbitrarily large i and j for which xi = 0, xj = 2. proposition 2.1. the quotient topological space originated by the partition (0, 1) = a ∪ b ∪ c is the davey space. proof. let x = {a, b, c} be the quotient space obtained by identifying a, b, c to points a, b, c, respectively. since a is open in (0, 1) it follows that {a} is open in x. let now x ∈ b and suppose that x = n−1∑ i=1 xi 3i + 1 3n , where xi = 0 or 2. for k ≥ n + 1, define yk ∈ c by yk = n−1∑ i=1 xi 3i + k∑ i=n+1 2 3i + ∞∑ j=1 2 3k+2j . then the sequence (yk) converges to n−1∑ i=1 xi 3i + ∞∑ i=n+1 2 3i , which is x. similarly if x = n−1∑ i=1 xi 3i + 2 3n , where xi = 0 or 2, for k ≥ n + 1, define yk = x + ∞∑ j=1 2 3k+2j . this sequence also converges to x. thus every element of b belongs to the closure of c and, since it is an end-point of an open interval in a, it also lies in the closure of a. hence every open set in x containing b also contains a and c and the only such open set is x itself. next consider x ∈ c, say x = ∞∑ i=1 xi 3i , where xi = 0 or 2. there exists an arbitrarily large i for which xi = 2. let xl be the first nonzero term and, for k ≥ l, define yk ∈ b by yk = k∑ i=1 xi 3i . the sequence (yk) converges to x. thus x lies in the closure of b and, as each yk is in the closure of a, it also lies in the closure of a. so every open set in x containing c also contains a and b and the only such open set is x itself. therefore x is the davey space. � 94 sheila carter and f. j. craveiro de carvalho let g = {h : (0, 1) → (0, 1) | h is a homeomorphism and h(a) = a}. if h ∈ g then h takes an open interval in a to an open interval in a and, consequently, the end-points to end-points. so h(b) = b and then h(c) = c. if we prove that g acts transitively on a, b and c we may conclude that those subsets are the orbits of the natural action of g on (0, 1) and, by proposition 2.1, (0, 1)/g is the davey space. proposition 2.2. g acts transitively on a. proof. we start by observing that, given any open interval (α, β) in (0, 1) and x, y ∈ (α, β), there exists a homeomorphism h : (0, 1) → (0, 1) such that h((α, β)) = (α, β), h(x) = y and h | (0, 1) \ (α, β) is the identity function. to prove transitivity on a it is therefore enough to show that, for any open interval (α, β) in a, with α, β ∈ b, there is h ∈ g such that h((α, β)) = (1 3 , 2 3 ). assume α = 2 3i1 +. . .+ 2 3ik + 1 3n , 1 ≤ i1 < i2 < . . . < ik < n. so β = α+ 1 3n . let j1, . . . , jl be such that 1 ≤ j1 < . . . jl < n, {j1, . . . , jl}∪{i1, i2, . . . , ik} = {1, 2, . . . , n − 1}, l + k = n − 1. hence α + 2 3j1 + . . . + 2 3jl + 1 3n = n∑ i=1 2 3i = 1 − 1 3n and, in the construction of h, we will use 1 − ( 2 3j1 + . . . + 2 3jl + 1 3n + s 3n ) = α + t 3n , where s = 1 − t, t ∈ [0, 1]. we define h : (0, 1) → (0, 1) as follows: (0, 2 3i1 ] is mapped to (0, 2 32 ] by h( 2t 3i1 ) = 2t 32 , with t ∈ (0, 1], for r = 2, . . . , k, [ 2 3i1 + . . . + 2 3ir−1 , 2 3i1 + . . . + 2 3ir ] is mapped to [ 2 32 + . . . + 2 3r , 2 32 + . . . + 2 3r+1 ] by h( 2 3i1 + . . . + 2 3ir−1 + 2t 3ir ) = 2 32 + . . . + 2 3r + 2t 3r+1 , with t ∈ [0, 1], [ 2 3i1 +. . .+ 2 3ik , α] is mapped to [ 2 32 +. . .+ 2 3k+1 , 1 3 ] by h( 2 3i1 +. . .+ 2 3ik + t 3n ) = 2 32 + . . . + 2 3k+1 + t 3k+1 , with t ∈ [0, 1], [α, α + 1 3n ] is mapped to [1 3 , 2 3 ] by h(α + t 3n ) = 1 3 + t 3 , with t ∈ [0, 1], [α + 1 3n , α + 2 3n ] is mapped to [2 3 , 2 3 + 1 3l+1 ] by h(1 − ( 2 3j1 + . . . + 2 3jl + s 3n )) = 1 − ( 2 32 + . . . + 2 3l+1 + s 3l+1 ), with s ∈ [0, 1], for r = 2, . . . , l, [1 − ( 2 3j1 + . . . + 2 3jr ), 1 − ( 2 3j1 + . . . + 2 3jr−1 )] is mapped to [1 − ( 2 32 + . . . + 2 3r+1 ), 1 − ( 2 32 + . . . + 2 3r )] by h(1 − ( 2 3j1 + . . . + 2 3 jr−1 + 2s 3jr )) = 1 − ( 2 32 + . . . + 2 3r + 2s 3r+1 ), with s ∈ [0, 1], [1 − 2 3j1 , 1) is mapped to [1 − 2 32 , 1) by h(1 − 2s 3j1 ) = 1 − 2s 32 , with s ∈ (0, 1]. we have then a homeomorphism h : (0, 1) → (0, 1) such that h((α, β)) = (1 3 , 2 3 ). on each interval of its definition, h is of the form h(x) = λx + µ, for some λ, µ ∈ r. hence it takes middle thirds in (0, 1 3i1 ) to middle terms in homeomorphisms of r and the davey space 95 (0, 1 32 ), middle thirds in ( 2 3i1 + . . . + 2 3 ir−1 , 2 3i1 + . . . + 2 3 ir−1 + 1 3r ) to middle thirds in ( 2 32 + . . . + 2 3r , 2 32 + . . . + 2 3r + 1 3r+1 ), for r = 2, . . . , k, and so on for the other intervals. so h(a) = a and h ∈ g as required. � proposition 2.3. g acts transitively on b. proof. the homeomorphism h constructed above maps [α, β] to [1 3 , 2 3 ], with h(α) = 1 3 , h(β) = 2 3 and every element in b is such an α or β. composing h with the reflection of (0, 1) that sends x to 1 − x gives an element of g that takes β to 1 3 . hence, for α ∈ b, there exists g ∈ g with g(α) = 1 3 . therefore g acts transitively on b. � proposition 2.4. g acts transitively on c. proof. since 1 4 ∈ c, it suffices to show that, for γ ∈ c, there is h ∈ g such that h(γ) = ∞∑ n=1 2 32n = 1 4 . let γ = ∞∑ n=1 2 3in , i = {i1, i2, . . .}, j = n \ i = {j1, j2, . . .}. define h : (0, 1) → (0, 1) as follows: (0, 2 3i1 ] is mapped to (0, 2 32 ] by h( 2t 3i1 ) = 2t 32 , with t ∈ (0, 1], for n = 2, . . ., [ 2 3i1 +. . .+ 2 3in−1 , 2 3i1 +. . .+ 2 3in ] is mapped to [ 2 32 + 2 34 +. . .+ 2 32n−2 , 2 32 +. . .+ 2 32n ] by h( 2 3i1 +. . .+ 2 3 in−1 + 2t 3in ) = 2 32 + 2 34 +. . .+ 2 32n−2 + 2t 32n , with t ∈ [0, 1], h( ∞∑ n=1 2 3in ) = ∞∑ n=1 2 32n , for n = 2, . . ., [1 − ( 2 3j1 + . . . + 2 3jn ), 1 − ( 2 3j1 + . . . + 2 3jn−1 )] is mapped to [1 − (2 3 + 2 33 + . . . + 2 32n−1 ), 1 − (2 3 + 2 33 + . . . + 2 32n−3 )] by h(1 − ( 2 3j1 + . . . + 2 3jn−1 + 2s 3jn )) = 1 − (2 3 + 2 33 + . . . + 2 32n−3 + 2s 32n−1 ), with s ∈ [0, 1], [1 − 2 3j1 , 1) is mapped to [1 3 , 1) by h(1 − 2s 3j1 ) = 1 − 2s 3 , with s ∈ (0, 1]. we have therefore defined an h ∈ g with h(γ) = 1 4 as required. � we can now conclude with our main result. theorem 2.5. there is a group g of homeomorphisms of r such that the quotient space r/g is the davey space. 96 sheila carter and f. j. craveiro de carvalho acknowledgements. the authors are very grateful to alan west for discussions on the topic of this paper which have led to a complete proof of theorem 2.5. references [1] f. j. craveiro de carvalho and bernd wegner, locally sierpinski spaces as interval quotients, kyungpook math. j. 42 (2002), 165-169. [2] ryszard engelking, general topology, heldermann verlag, 1989. [3] sidney a. morris, are finite topological spaces worthy of study?, austral. math. soc. gazette 11 (1984), 563-564. [4] james r. munkres, topology, a first course, prentice-hall, inc., 1975. [5] stephen willard, general topology, addison-wesley, inc., 1970. received december 2002 accepted april 2003 sheila carter (s.carter@leeds.ac.uk) school of mathematics, university of leeds, leeds ls2 9jt, u. k. f. j. craveiro de carvalho (fjcc@mat.uc.pt) departamento de matemática, universidade de coimbra, 3001 454 coimbra, portugal @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 91–97 functorial approach structures g. c. l. brümmer and m. sioen abstract. we show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful functor t : ap → top (where ap denotes the topological construct of approach spaces and contractions as introduced by r. lowen). there is however a great difference in nature of these functorial approach structures when compared to the quasi-uniform paradigm which has been extensively studied by the first author: whereas it is well-known from [2] that a large class of epireflective subcategories of top0 can be ‘parametrized’ using the interaction of functorial quasi-uniformities with the quasi-uniform bicompletion, we show that using functorial approach structures together with the approach bicompletion developed in [10], only top0 itself can be retrieved in this way. 2000 ams classification: 18b30, 18b99, 54b30, 54e15, 54e99. keywords: approach space, (approach) bicompleteness, epireflective subcategory, functorial approach structure, spanning, topological space. 1. introduction and preliminaries in [2, 3, 4, 5, 7, 8, 9] so-called functorial (quasi-)uniformities were extensively studied. a functorial quasi-uniformity, is a functor f : top → qu (where top, resp. qu, stands for the topological construct of topological spaces and continuous maps, resp. of quasi-uniform spaces and uniformly continuous maps) which is a section for the usual forgetful functor tqu : qu → top, i.e. such that tquf = 1top. first of all, let us recall from [3, 9] that there is a one-to-one correspondence between functorial quasi-uniformities in the above sense, and functorial quasi-uniformities f : top0 → qu0 in the t0 case. we refer to [1] as our blanket reference for categorical material and to [11] for all information about quasi-uniformities. let us only mention that for the order in the fibres of a topological construct a, we take the opposite convention to the one taken in [1]: if a,b are two objects on the same underlying set, we call a finer than b (or b coarser than a), and write a ≥ b, iff the identity map on the underlying set becomes a morphism a → b. then all the fibres 92 g. c. l. brümmer and m. sioen become complete lattices. one of the most important results about functorial quasi-uniformities, is their interplay with the quasi-uniform bicompletion (cf. [2]). it was shown by the first named author that functorial quasi-uniformities can e.g. be used to classify epireflective subcategories of top0, in the sense that for every (full) epireflective subcategory e of top0 with |sob| ⊆ |e| ⊆ |topbicompl0| (where sob stands for the subcategory of top0 formed by all sober objects and topbicompl0 for the one formed by all topologically bicomplete t0-spaces (i.e. those t0 topological spaces admitting a bicomplete quasi-uniformity)), there exists a functorial quasi-uniformity f : top0 → qu0 such that (1.1) e = {x ∈ top0 | fx bicomplete}. we refer to [2] for a survey on this topic. in [13], the topological construct ap of approach spaces and contractions was introduced by r. lowen as a quantified supercategory of top and it has been proved since then that approach spaces are an interesting framework for quantitative topology (see e.g. [14, 15, 16, 19] for applications of approach theory to hyperspaces or topological vector spaces). for a detailed motivation and more information about ap, we refer the reader to [13]. for an approach space x, we will write x for its underlying set and δx for its approach distance. in this context, top can be proved to be concretely bicoreflectively embedded into ap, and the corresponding concrete bicoreflector t : ap → top plays the role of forgetful functor in this setting. recall that it was shown in [17] that the t0-objects in the sense of marny [18], in the setting of approach spaces, are exactly those approach spaces with t0 topological coreflection. we denote by ap0 the corresponding subcategory of ap. recently, in [10], the present authors derived a notion of symmetry for approach spaces, and together with it a categorically satisfactory (i.e. sub-firmly epireflective in the sense of [6]) notion of approach bicompleteness and bicompletion, which has a totally different behavior compared to the behavior in the quasi-uniform case. this different behavior again highlights the structural difference between approach spaces and quasi-uniform spaces: although both of them can be described using pseudo-quasi-metrics, approach spaces simultaneously quantify topological and (pseudo-quasi)-metric spaces, so with regard to concepts such as ‘bicompleteness’ or ‘cauchy filters’, a very different paradigm compared to the quasi-uniform one is to be expected. let us now for the moment only recall that the category pqmet∞ of extended pseudo-quasi-metric (or ∞pq-metric) spaces can be fully and concretely bicoreflectively embedded into ap, and that a t0 approach space x = (x,δx) is approach bicomplete iff the ∞pq-metric space (x,dδx ) is bicomplete in the usual sense, where dδx (x,y) := δx(x,{y}), x,y ∈ x. functorial approach structures 93 we now want to address the question, whether or not, functorial approach structures and approach bicompleteness can be used to capture (preferably more) epireflective subcategories of top0. 2. results in all that follows, t : ap → top denotes the topological bicoreflector, which in approach theory serves as the underlying functor. we refer to [13] for a detailed description of t . a functor f : top → ap which is a section for t , i.e. for which tf = 1top, will be called a functorial approach structure. we first need an obvious lemma characterizing all t-sections. the notation prtop stands for the topological construct of pre-topological spaces and continuous maps. for any pre-topological space, we use x for its underlying set and clx for the corresponding closure operator on x. lemma 2.1. f : top → ap is a section for t : ap → top if and only if f can be written as an approach tower (fε : top → prtop)ε≥0 of concrete functors with (1) f0 is the embedding of top into prtop, which we denote by 1top, (2) f∞ is the indiscrete functor, which we denote by i and which equips each set with the coarsest possible topology, (3) ∀x ∈ |top|, ∀ε ≥ 0 : fεx = ∨ ε<γ fγx, (4) ∀x ∈ |top|, ∀ε,γ ≥ 0 : clfγx ◦ clfεx ≥ clfγ+εx. proof. this immediately follows from the description of approach spaces in terms of towers (see [13]) and the fact that f0 = tf . � next we show that, like in the quasi-uniform paradigm, there are “enough” functorial approach structures. the proof however becomes more intricate. theorem 2.2. the conglomerate of t -sections is at least a proper class. proof. suppose that the conglomerate of all different t-sections would be in one-to-one correspondence with a set of cardinality κ. then consider the cardinal number 2κ > κ. note that 2κ also is an (initial) ordinal number and that γ := {α | α ordinal number and α < 2κ}, equipped with the inclusion ⊆ is a lattice without top element. it was proved in [12] that (γ,⊆) has a latticeisomorphic representation within the large lattice of bireflective subcategories 94 g. c. l. brümmer and m. sioen of top (with the natural order defined there). this entails that we can find a class r of mutually different bireflective subcategories of top which is in one-to-one correspondence with the set 2κ. (note that we make no distinction between a bireflective subcategory and its corresponding bireflector, and that we consider such a bireflector as an endofunctor on top). for every r ∈ r, we define: frε :=   i ε = ∞, r ε ∈ [1,∞ [ (viewed as a functor into prtop), 1top ε ∈ [0, 1 [ (viewed as a functor into prtop). then according to lemma 2.1, it is clear that fr := (frε )ε≥0 is a t-section, and that {fr | r ∈ r} is a class of mutually different t-sections, being in one-to-one correspondence with the set 2κ, yielding a contradiction with the definition of κ. � finally, we come to proving our main theorem showing that “locally around the 0-level”, all functorial approach structures however become trivial. first note that, with exactly the same proof as in the (quasi)-uniform cases treated in [7, 8, 2], we can prove that every t-section can be obtained through the spanning construction as defined by the first author. this means that given a t-section f, there exists a class a⊂ |ap| for which ta := {ta | a ∈a} is initially dense in top and such that for all x ∈ |top| the source (f : fx → a)a∈a,f∈top(x,ta) is ap-initial. to summarize this we write f = 〈a〉 and we say that “a spans f”. theorem 2.3. for every t -section f , there exists γ > 0 such that ∀ε ∈ [0,γ [ : fε = 1top. proof. take an arbitrary t-section f : top → ap. according to lemma 2.1, we can write f as a tower (fε)ε≥0 of functors subject to the conditions listed there. on the other hand we know from the general spanning construction that there exists a⊂ |ap| such that ta is initially dense in top and f = 〈a〉. in particular this means for the sierpinski space $, that the source (f : $ → ta)a∈a, f∈top($,ta) is initial in top. this clearly can only happen when there exists b ∈ a such that $ can be embedded into tb as a topological subspace. this means that we can find x,y ∈ b with δb(x,{y}) = 0 and γ := δb(y,{x}) > 0. now fix ε ∈ ] 0,γ [. then obviously the previous implies that $ still is a pretopological subspace of bε := (b, tbε ) (here t b ε is the pretopological closure on functorial approach structures 95 the level ε in the approach tower corresponding to δb, i.e. for all y ⊂ b, tbε (y ) := {b ∈ b | δb(b,y ) ≤ ε}). fix x ∈ |top|. because f = 〈a〉, the source (f : fx → c)c∈a,f∈top(x,tc) is initial in ap. therefore, fεx has to be finer than the initial pre-topological structure on x for the prtop-source (2.2) (f : x → cε)c∈a, f∈top(x,tc). the initial prtop-structure being the coarsest one on x making all functions in the source (2.2) above continuous, it certainly is coarser than clx. because $ is a pretopological subspace of bε, it is clear on the other hand that the initial structure for the source (2.2) above at the same time has to be finer than the initial prtop-structure for the source (f : x → $)f∈prtop(x,$). because top is fully and concretely embedded as an initially closed subcategory in prtop, and because $ is initially dense in top, the initial prtop-structure for the latter source is clx. so the initial prtop-structure for (2.2) is clx, whence fεx ≥ x and since automatically fεx ≤ f0x = x, we are done. � let us now recall from [3] that, with the same argument as in the quasiuniform case used in [9], it follows from the universality of ap in the sense of [18] (meaning that ap is the bireflective = initial hull of its t0-objects), which was proved in [17], that there is a one-to-one correspondence between t-sections, and sections to the functor t |ap0 : ap0 → top0. in one direction this correspondence is simply given by restriction of t-sections to top0. we therefore immediately have the analogue of theorem 2.3 in this setting, providing a description of all t |ap0 -sections. this yields that we automatically also obtain: theorem 2.4. for every t |ap0 -section f , there exists γ > 0 such that ∀ε ∈ [0,γ [ : fε = 1top0. now take an arbitrary t |ap0 -section f : top0 → ap0 and let γ > 0 be as in the theorem above. then for all x ∈ |top0|, the ∞pq-metric dfx only takes values in {0}∪ [γ, +∞] and therefore obviously is a bicomplete metric on x, whence fx is automatically approach bicomplete. this answers the question posed at the end of the first paragraph in the negative, again showing a drastically different behaviour 96 g. c. l. brümmer and m. sioen of the approach setting in comparison to the quasi-uniform one: the only epireflective subcategory of top0 we can retrieve via functorial approach structures is top0 itself. references [1] j. adámek, h. herrlich and g. strecker, abstract and concrete categories (wiley, new york, 1990). [2] g. c. l. brümmer, categorical aspects of the theory of quasi-uniform spaces, rend. istit. mat. univ. trieste 30 (suppl.) (1999), 45–74. [3] g. c. l. brümmer, extending constructions from the t0-spaces to all topological spaces, in preparation. [4] g. c. l. brümmer, functorial transitive quasi-uniformities, categorical topology (proc. conf. toledo, ohio, 1983), (heldermann verlag, berlin, 1984), 163–184. [5] g. c. l. brümmer, completions of functorial topological structures, recent developments of general topology and its applications (proc. conf. berlin 1992), (akademie verlag, berlin, 1992), 60–71. [6] g. c. l. brümmer and e. giuli, a categorical concept of completion of objects, math. univ. carolinae 33 (1992), 131–147. [7] g. c. l. brümmer and a. w. hager, completion-true functorial uniformities, seminarberichte fachber. math. inf. fernuniv. hagen 19 (1984), 95–104. [8] g. c. l. brümmer and a. w. hager, functorial uniformization of topological spaces, topology appl. 27 (1987), 113–127. [9] g. c. l. brümmer and h.-p künzi, bicompletion and the samuel bicompactification, appl. categ. struct. 10 (2002), 317–330. [10] g. c. l. brümmer and m. sioen, approach bicompleteness and bicompletion, in preparation. [11] p. fletcher and w. f. lindgren, quasi-uniform spaces, (marcel dekker, new york and basel, 1982). [12] m. hušek, applications of category theory to uniform structures, lecture notes math. 962 (springer, berlin, 1982), 138–144. [13] r. lowen, approach spaces. the missing link in the topology-uniformity-metric triad, (clarendon press, oxford, 1997). [14] r. lowen and m. sioen, proximal hypertopologies revisited, set-valued analysis 6 (1998), 1–19. [15] r. lowen and m. sioen, approximations in functional analysis, results in mathematics 37 (2000), 345–372. [16] r. lowen and m. sioen, weak representations of quantified hyperspace structures, top. appl. 104 (2000), 169–179. [17] r. lowen and m. sioen, a short note on separation in ap, appl. gen. top., to appear. [18] t. marny, on epireflective subcategories of topological categories, gen. top. appl. 10 (1979), 175–181. [19] m. sioen and s. verwulgen, locally convex approach spaces, appl. gen. top., to appear. received january 2002 revised december 2002 functorial approach structures 97 guillaume c. l. brümmer department of mathematics and applied mathematics, university of cape town, rondebosch 7701, south africa. e-mail address : gclb@maths.uct.ac.za mark sioen department of mathematics, free university of brussels, pleinlaan 2, b-1020 brussel, belgium. e-mail address : msioen@vub.ac.be functorial approach structures. by g. c. l. brümmer and m. sioen cardagtlatex.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 173-190 on the cardinality of indifference classes gerhard herden and andreas pallack abstract. let “ ” be a continuous total preorder on some topological space (x, t). then the cardinality or at least lower and upper bounds of the cardinality of the indifference (equivalence) classes of “ ” will be computed. in addition, the relevance of these bounds in mathematical utility theory and the theory of orderable topological spaces will be discussed. 2000 ams classification: 54f05, 91b16, 06a05. keywords: indifference potency, calculable set, degree of connectedness, path rank, separation rank. 1. introduction let (x, t) be a topological space that is endowed with a total preorder “ ”. the reader may recall that a total preorder “ ” on x is a reflexive and transitive relation on x that satisfies the additional property that for all pairs (x, y) ∈ x2 at least one of the relations x y or y x holds. the relation x ∼ y ⇐⇒ x y ∧ y x defines an equivalence relation on x. the corresponding eqivalence classes are the indifference classes of “ ”. the order topology ton x that is induced by “ ” is generated by the sets l(x) := {y ∈ x|y ≺ x} and k(x) := {z ∈ x|x ≺ z} where x runs through x. in mathematical utility theory usually continuous total preorders “ ” on x are considered, which means that the order topology tis coarser than t. ideally, the indifference classes of a continuous total preorder “ ” on (x, t) only consist of single points. this means from the viewpoint of mathematical utility theory that different alternatives are perfectly distinguishable by preferences (utilities) and from the viewpoint of pure mathematics that the given topology t on x is orderable. the reader may notice that in our terminology orderability only means that t⊂ t. if also t ⊂ tthen we speak of a strictly orderable topology t on x (cf. kok [12]). of course, in general, indifference 174 g. gerden and a. pallack classes are thick. this means that, in general, one cannot escape from the fact that one has to deal with a preorder. in general, a topology t on x is not orderable. indeed, let n > 1 be any natural number. then the natural topology tnat on r n is not orderable (cf. section 2 and the nice results in beardon [2], candeal and induráin [6] and candeal, induráin and mehta [7]). clearly, a topology t on x is not orderable if and only if every continuous total preorder “ ” on x has at least one indifference class that contains more than one point. in this case we may select the greatest of all these big indifference classes. the smallest of these greatest classes then represents in some sense the degree up to which t is orderable or not. in this way the concept of an orderable topology t on x seems to be generalizable in a natural way. in addition, by strengthening this idea we are able to measure in a precise sense up to which degree alternatives can or cannot be distinguished by preferences (utilities). indeed, the fundamental concepts that are introduced and discussed in section 3 are based upon this idea. in order to compute or to measure how big an indifference class [x] of “ ” can be two possibilities seem to be natural. indeed, let a be an appropriate σ-field in x that contains t and let µ be an appropriate measure on a. then for every indifference class [x] of “ ” the measure µ ([x]) might be computed. the reader may recall that the continuity of “ ” implies that every indifference class [x] of “ ” is a closed subset of x. since t ⊂ a, therefore, µ ([x]) is defined. in the arrow-hahn or euclidean distance approach to mathematical utility theory an indifference class [x] of “ ” is considered as being thick if it contains a non-empty open subset of x, which means that its lebesgue measure is nonzero. in order to rule out thick indifference classes “ ”, thus, is required to be locally non-satiated, which means that for every point x ∈ x and every neighborhood u of x there exists some point y ∈ u such that x ≺ y. besides the original arrow-hahn approach [1] the reader may consult, in particular, the book of bridges and mehta [5]. on the other hand, one simply could compute the cardinality |[x]| of every indifference class [x] of “ ”. let n ≥ 1 be a natural number. usually on rn the lebesgue measure λ is considered. one verifies immediately that there exist total preorders “ ” on rn that are continuous with respect to tnat and only have indifference classes [x] the lebesgue measure λ ([x]) of which is zero. for any two points x = (x1, ..., xn), y = (y1, ..., yn) ∈ r n one may set, for instance, x y ⇐⇒ x1 ≤ y1. with the exception of n = 1 in this case the indifference classes are hyperplanes of rn which still look very big. in addition, although for n > 1 the natural topology tnat on r n is not orderable (cf. section 2) this example guarantees, however, the existence of continuous total preorders “ ” on rn which, following the spirit of the first possibility, only have thin or small indifference classes. this means that the first possibility is not appropriate if one is interested in measuring or computing the degree of orderability of a cardinality of indifference classes 175 topology t or its potency of distinguishing between different alternatives by preferences (utilities). hence, in the remainder of this paper we mainly study the second possibility and, thus, compute or at least estimate the cardinality of indifference classes. after introducing in section 3 the basic concepts of this paper in section 4 we, therefore, concentrate on the computation of the cardinality or at least lower and upper bounds of the cardinality of indifference classes. meanwhile hilbert spaces or banach spaces or even general convex spaces are commonly encountered in mathematical utility theory (cf. [3], [11], [13], [14]). thus, in particular, the degree of orderability or the potency of distinguishing between different alternatives by preferences (utilities) in these spaces will be computed (proposition 4.6, corollary 4.7 and proposition 4.9). in addition, in remark 4.10 the (possible) relevance of these results in mathematical utility theory will be discussed. finally, also a generalization of well known results on orderable connected topologies will be proved (proposition 4.11). 2. a first approach a well known result of candeal and induráin [6, theorem 4] states that for n ≥ 1 a closed and convex subset c of rn can be endowed with a continuous total order if and only if c is homeomorphic and isotonic to a connected subset of r. an immediate and often quoted consequence of this theorem of candeal and induráin says that rn for n > 1 cannot be endowed with a continuous total order. related but more general results also can be found in beardon [2] and candeal, induráin and mehta [7]. the quoted consequence of the theorem of candeal and induráin also follows from the remarkable fact that for n > 1 each indifference class [x] of a continuous total preorder “ ” on rn such that x is neither a first nor a last element of “ ”contains exactly 2ℵ0 elements. this result is interesting. indeed, let n > 1 and let s be a countable subset of rn. then a well known result of dugundji [8, chapter 5, theorem 2.2] states that rn\s is connected. as we shall soon see this result of dugundji also implies the consequence of the candeal-induráin theorem. on the other hand, if ℵ1 < 2 ℵ0 it does not imply that any continuous total preorder on rn contains at least one indifference class that consists of 2ℵ0 elements. in order to be more precise we are going to outline a proof of the fact that for n > 1 each indifference class [x] of a continuous total preorder “ ” on rn such that x neither is a first nor a last element of “ ”contains exactly 2ℵ0 elements that relies on standard arguments from topology. then it follows, in particular, that the result of dugundji does not necessarily imply that any continuous total preorder on rn contains at least one indifference class that consists of 2ℵ0 elements. let n > 1 and a, b two different points of rn. then a family {fi}i∈i of paths fi : [0, 1] −→ r n that connect a and b is said to be separated if for any pair i, j of different indexes of i the meets of the images imfi and imfj of fi 176 g. gerden and a. pallack and fj respectively only consist of the points a and b. now a straightforward consideration (cf. the general construction in proposition 4.6) allows us to construct a separated family {fi}i∈i of paths fi : [0, 1] −→ r n that connect a and b and the cardinality of which is 2ℵ0. this means that the result of dugundji can be improved. indeed, let n > 1 and let s be a subset of rn such that |s| < 2ℵ0. then for every pair of different points a, b ∈ rn\s there exists at least one index i ∈ i such that imfi ∩ s = ∅, and we may conclude that rn\s is path connected. in case that x is neither a first nor a last element of “ ” it follows that rn\ [x] = {y ∈ rn|y ≺ x} ∪ {z ∈ rn|z ≻ x} is not connected. hence, we may conclude that |[x]| = 2ℵ0. obviously, the original result of dugundji only implies that ℵ0 < |[x]| ≤ 2 ℵ0. in section 4 the aforepresented observations will be generalized to arbitrary real or complex convex spaces (cf. proposition 4.6, corollary 4.7 and corollary 4.8). let s be a non-empty subset of rn. then we choose an arbitrary vector v ∈ s and postulate for the moment the dimension of s to be the dimension of the linear subspace of rn that is generated by s − {v} := {w − v|w ∈ s}. the reader may verify that the definition of the dimension of s is independent of any particular chosen vector v ∈ s. let tnat be the natural topology on r n. with the help of our definition of the dimension of s the indifference dimension of a continuous total preorder “ ” on (rn, tnat) or shortly r n is defined as the maximum of all dimensions of all indifference classes [(x)] of “ ”. then the indifference dimension of rn is defined as the minimum of all dimensions of relations “ ” where “ ” runs through all continuous total preorders on rn. of course, these concepts can be generalized to arbitrary real or complex convex spaces (cf. remark 4.10). in proposition 4.9 these types of indifference dimensions on rn (at least implicitly) will be computed. in remark 4.10 (possible) consequences in mathematical utility theory will be discussed. on the other hand, these concepts of measuring the size or (thickness) of indifference classes in rn hardly seem to be generalizable to arbitrary topological spaces. hence, in general topology we restrict on computing the cardinality of indifference classes. 3. fundamental concepts and inequalities let (x, t) be some arbitrarily but fixed chosen non-trivial topological space. non-trivial means that x contains at least two (different) elements. according to our considerations in the introduction for every continuous total preorder “ ” on x the weak indifference potency of “ ” is defined by winpot(-) :=        0 if “ ” is an order on x sup{| [x] | | [x] is an indifference class of “ ”, otherwise . then the weak indifference potency of x is defined by cardinality of indifference classes 177 winpot(x) := min{winpot(-)|“ ” is a continuous total preorder on (x, t)}. in section 2 the cardinality of an indifference class has been computed with the help of its potency to separate a (connected) component of x. let, therefore, c be the set of all (connected) components of x. we consider some continuous total preorder “ ” on x and choose a component c ∈ c. now an indifference class [x] of “ ” is said to be calculable with respect to “ -|c ” if c ⊂ [x] or there exist points y, z ∈ c such that y ≺ v ≺ z for every point v ∈ c ∩ [x].this notation allows us to define the indifference potency inpot(-) of “ ” by “0” if “ ” is an order on x and by supc∈cmin{|[x]| [x] is a calculable set with respect to “ -|c ”}, otherwise. then the indifference potency of x is defined by inpot(x) := min{inpot(-)|“ ” is a continuous total preorder on (x, t)}. finally, a subset s of x is said to be calculable if there exists a component c of x and some continuous total preorder “ ” on x such that s is an indifference class of “ ” and c ⊂ s or there exist points y, z ∈ c such that y ≺ x ≺ z for every point x ∈ c ∩ s. let now for every component c ∈ c the set ca(c) consist of all subsets s of x that are indifference classes of continuous total preorders “ ” on x such that c ⊂ s or there exist points y, z ∈ c such that y ≺ x ≺ z for every point x ∈ c ∩ s. then we still may define the strong indifference potency of x by stinpot(x) := { 0, if t is orderable supc∈c min {|s| |s ∈ ca(c)} otherwise . the above definitions imply that the assertions winpot(x) = inpot(x) = stinpot(x) = 0 and t is orderable are equivalent. the following lemma justifies the concept of a calculable set. lemma 3.1. let s be a subset of x. then the following assertions are equivalent: (i) s is calculable. (ii) there exists a continuous total preorder “ ” on x and some component c of x such that s is an indifference class of “ ” and c ⊂ s or c\s is not a connected subset of x with respect to t-. proof. (i) =⇒ (ii): let s = [x] and let there exist points y, z ∈ c such that y ≺ x ≺ z. it suffices to verify that c\ [x] is not connected with respect to t-. let, therefore, l(x) := {u ∈ x|u ≺ x} and k(x) := {v ∈ x|x ≺ v}. then l(x) and k(x) are two disjoint open subsets of ( x, t) such that l(x) ∩ c\s 6= ∅ 178 g. gerden and a. pallack and k(x) ∩ c\s 6= ∅ and c\s ⊂ l(x) ∪ k(x), which implies that c\s is not connected with respect to t-. (ii) =⇒ (i): let “ ” be some continuous total preorder on x such that s = [x] for an indifference class [x] of “ ”. then we assume the existence of a component c of x such that c\ [x] is not connected in the order topology ton x that is induced by “ ”. because of the definition of a calculable set it is sufficient to prove that there exist points y, z ∈ c such that y ≺ x ≺ z. since t⊂ t and c is a connected subset of x with respect to t it follows that c is also a connected subset of x with respect to t-. in addition, the definition of timplies that every indifference class [u] of “ ” is a connected subset of x with respect to t-. hence, i := {[v] c ∩ [v] 6= ∅} is a connected subset of x with respect to t-, which means that i is an interval of (x, -) that is connected with respect to t-. furthermore, the reader may recall that a totally preordered set (z, .) is connected with respect to its order topology if and only if it is order dense and dedekind complete. order dense means that for any two points a < b ∈ z there exists some point c ∈ z such that a < c < b. dedekind complete means that every bounded subset d of z has a least upper and a greatest lower bound. these results are well known for chains. the reader may consult, for instance, the famous book of birkhoff [4] on lattice theory. by considering on the set of indifference classes the induced total ordering these results easily can be generalized to total preorders. since c\ [x] is not connected with respect to twe may summarize our considerations for concluding that x neither can be a first nor a last element of (i, -). therefore, the definition of i guarantees the existence of points y, z ∈ c such that y ≺ x ≺ z, and the proof is complete. � remark and example 3.2. let “ ” be a continuous and total preorder on x and let c be a component of x. with the help of the proof of lemma 3.1 it follows that an indifference class [x] of “ ” is calculable with respect to ” -|c ” if and only if c ⊂ [x] or c\ [x] is not a connected subset of x with respect to t-. in assertion (ii) of lemma 3.1 the condition that c\s is not a connected subset of x with respect to tcannot be replaced by the condition that c\s is not a connected subset of x with respect to t. indeed, let x := [0, 1] be the real unit interval. we set u := [0, 1)∩q and v := [0, 1)\q and consider the coarsest topology t on x that contains tnat and both sets u and v . then {1} is an indifference class of the natural total ordering “ ≤=≤|x ” on [0, 1]. in addition, the definition of t imples that t≤ = tnat ⊂ t. furthermore, since 1 neither is contained in u nor in v it follows that (x, t) is a connected space. moreover, we may conclude with the help of the definitions of u and v that x\ {1} is not connected with respect to t. since x\{1} is connected with respect to t≤ it follows that {1} cannot be calculable with respect to ” ≤|x ” = ” ≤ ”. now the connectedness of t and the particular form of the neighborhoods of 1 also imply that 1 is either a first or a last element of (x, -) for every continuous total preorder “ ” on x. hence, there cannot exist any other continuous cardinality of indifference classes 179 total preorder “ ” on x such that x\ {1} is not connected with respect to t-, which means that {1} cannot be a calculable set. in order to prepare the main result of this section the reader may recall that (x, t) is said to be hereditarily lindelöf if for every collection {oi}i∈i of open subsets of x there exists a countable subset j of i such that ⋃ i∈i oi = ⋃ j∈j oj. in addition, we define for every component c ∈ c the degree of connectedness of c by dcon(c) := { min {|z| |z ∈ t (c)} , if t (c) 6= ∅ |c| , otherwise . with the help of this definition we may define the degree of connectedness of x by dcon(x) := { 0, if t is orderable supc∈cdcon(c), otherwise . the reader may compare the definitions of dcon(c) and dcon(x) with assertion (ii) of lemma 3.1 and the above given example in order to understand the particular relevance of these concepts within our approach. now we still consider the set s(c) of all components c of x such that dcon(c) < |c| and prove the following proposition that in combination with remark and example 3.4 will help us to clarify in some degree the interrelations between the afore-introduced concepts. proposition 3.3. the following assertions hold: (i) winpot(x) ≥ inpot(x) ≥ stinpot(x) ≥ dcon(x). (ii) let every component c of x be open and closed and let every component c of x be metrizable or regular and hereditarily lindelöf. then inpot(x) = stinpot(x) = dcon(x). (iii) let every component c of x be open and closed and let s(c) = ∅. then winpot(x) = inpot(x) = stinpot(x) = dcon(x). proof. (i): because of the definitions of winpot(x), inpot(x), stinpot(x) and dcon(x) it suffices to verify that stinpot(x) ≥ dcon(x). let, therefore, some component c of x and a continuous total preorder “ ” on x be arbitrarily chosen. then there exists an indifference class [x] of “ ” such that c∩[x] 6= ∅. now we distinguish between the following two cases: case 1: c ⊂ [x]. in this case [x] is a calculable subset of x such that |[x]| ≥ dcon(c). case 2: c\ [x] 6= ∅. since c is connected we may conclude that i := {v ∈ x|c ∩ [v] 6= ∅} is a connected interval of (x, -) (cf. the corresponding part in the proof of lemma 3.1). hence, we may choose some point z ∈ i that neither is a first nor a last element of i and the indifference class [z] which 180 g. gerden and a. pallack has minimal cardinality with respect to any other indifference class of an inner point of i. it follows from the definition of a calculable set with respect to “ -|c ” that [z] is a calculable subset of x with respect to “ -|c ”. since c * [z] the first part of remark and example 3.2 implies that c\ [z] is not connected with respect to t-, which, in particular, means that c\ [z] is not connected with respect to t. therefore, [z] is a calculable subset of x such that |[z]| ≥ dcon(c). since c and “ ” have been arbitrarily chosen we now may conclude with the help of the definitions of stinpot(x) and dcon(x) that the desired inequality stinpot(x) ≥ dcon(x) actually holds. (ii): let every component c of x be open and closed and let, in addition, every component c of x be metrizable or regular and hereditarily lindelöf. because of assertion (i) it remains to verify that dcon(c) ≥ inpot(x). therefore, we choose for every component c ∈ s(c) some set z ∈ t (c) that has minimal cardinality and construct in three steps a continuous total preorder “ ” on x such that dcon (x) ≥ inpot (-). then the definition of inpot (x) implies that the desired inequality dcon(x) ≥ inpot(x) actually holds. let, therefore, some component c ∈ s (c) and a set z ∈ t (c) that has minimal cardinality with respect to any other set z ′ ∈ t (c) be arbitrarily but fixed chosen. 1. in the first step we want to show that, without loss of generality, z may be assumed to be a closed subset of x. indeed, since c\z is not connected and c is an open and closed subset of x, there exist open subsets u and v of c such that u ∩ v ∩ c\z = ∅ and u ∩ c\z 6= ∅ and v ∩ c\z 6= ∅ and c\z ⊂ u ∪ v . now we distinguish between the following two cases: case 1: u ∩ v = ∅. in this case we set z ′ := c\(u ∪ v ). the inclusion z ′ ⊂ z allows us to conclude that u ∩ c\z ′ 6= ∅ and v ∩ c\z ′ 6= ∅. since c\z ′ = u ∪ v it, thus, follows that c\z ′ is not connected. z ′ is a closed subset of x by construction. hence, in the first case we may assume that z is closed. case 2: u ∩v 6= ∅. the relation u ∩v ∩c\z = ∅ implies that u ∩v ⊂ z. let x ∈ u ∩ v be some arbitrarily chosen point. since the relativized topology t|c on c is metrizable or regular and (hereditarily) lindelöf it follows that ( c, t|c ) is a normal and, therefore, in particular a completely regular space. hence, there exists some continuous function f from c into the real interval [0, 1] such that f(x) = 1 and f(c\(u ∩ v )) = {0}. let z ′ := f−1 ({ 1 2 }) . the continuity of f allows us to conclude that z ′ is a closed subset of x. in addition, it follows that z ′ ⊂ u ∩ v ⊂ z. furthermore, the sets o := { y ∈ c|f(y) < 1 2 } and w := { z ∈ c|f(z) > 1 2 } are two disjoint non-empty open subsets of c\z ′ the union of which is c\z ′ , which means that c\z ′ is not connected. therefore, also in the second case z may be assumed to be a closed subset of x. 2. in the second step we want to construct a continuous total preorder “ -c ” on c such that z is an indifference class of “ -c ”. because of the first cardinality of indifference classes 181 step we may assume that z is a closed subset of x. in addition, we already know that ( c, t|c ) is a metrizable or competely regular hereditarily lindelöf space. therefore, we distinguish between the following two cases: case 1: ( c, t|c ) is metrizable. in this case we choose some metric d on c that induces t|c. then we consider the continuous function g : c −→ r ≥0 that is defined for all points x ∈ c by setting g(x) := d(z, x). since g−1({0}) = z we may conclude that z is an indifference class of the continuous total preorder ” -c ” on c that is defined by y -c z ⇐⇒ g(y) ≤ g(z). case 2: ( c, t|c ) is a completely regular hereditarily lindelöf space. now there exists for every point x ∈ c\z a continuous function fx : c −→ [0, 1] such that fx(x) = 1 and fx(z) = {0}. moreover, since ( c, t|c ) is a hereditarily lindelöf space countably many points x1, x2, ..., xn, ... ∈ c\z can be chosen such that c\z = ⋃ x∈c\z f−1x ((0, 1]) = ⋃ n∈n\{0} f−1xn ((0, 1]), which implies that z = ⋂ n∈n\{0} f−1xn ({0}). therefore, we set g := ∑ n∈n\{0} 1 2n fxn. then g : c −→ [0, 1] is a continuous function. hence, in the same way as in the first case, we may consider the continuous total preorder “ -c ” on c that is defined by y -c z ⇐⇒ g(y) ≤ g(z). since the definition of g implies that g−1 ({0}) = z it follows that z is an indifference class of “ -c ”. 3. in the third step we define the continuous total preorder “ ” on x such that dcon(x) ≥ inpot(-). let, therefore, x ∈ x be an arbitrarily chosen point. then there exists a uniquely determined component c of x that contains x, and we set d(x) := c. now we choose an arbitrary total ordering “ � ” on c and set y z ⇐⇒        y -c z, if there exists some c ∈ s (c) that contains both point y and z d(y) � d(z), otherwise for every pair of points y, z ∈ x. since every component of x is open and closed and since every total preorder “ -c ” that has been defined is continuous we may conclude that “ ” is a continuous total preorder on x. because of the definition of s (c) and the second step it follows with the help of the definitions of inpot(-) and dcon(x), in addition, that dcon(x) ≥ inpot(-) (indeed, the equality dcon(x) = inpot(-) holds), which settles assertion (ii). (iii): let s(c) = ∅. then the reader may notice that the definition of “ ” in the third step of the proof of assertion (ii) also implies that winpot(x) = dcon(c). this observation finishes the proof of the proposition. � remark and example 3.4. the proof of assertion (ii) of the above proposition motivates the problem of determining all topologies t on x for which every non-empty closed subset z of x is the indifference class of an appropriate continuous total preorder “ ” on x. meanwhile the first author has obtained some partial results on the characterization of hausdorff-topologies t 182 g. gerden and a. pallack on x that have the property that every non-empty closed subset of x is an indifference class of some continuous total preorder ”-” on (x, t) (cf. [10]). clearly, these topologies must be completely regular. if t is countably compact it follows, in addition, that t must satisfy ccc (countable chain condition). the authors think that these results somewhat justify the assumptions of assertion (ii) of the proposition. one may verify that, in general, winpot(x) > inpot(x) ≥ stinpot(x) > dcon(x). indeed, let [0, 1] be the real unit interval. then we substitute 0 by a copy r(0) of the reals and every real r ∈ (0, 1] by a copy q(r) of the rationals in order to set x := r(0) ∪ ⋃ r∈(0,1] q(r). now it follows for every x ∈ x that x ∈ r(0) or x ∈ q(r) for some uniquely determined real r ∈ (0, 1]. in this way a canonical function g : x −→ [0, 1] is given, and we may define a total preorder “ . ” on x by setting x . y ⇐⇒ g(x) ≤ g(y). we proceed by considering two arbitrarily but fixed chosen subsets u and v of x such that u ∩ r(0) 6= ∅ and v ∩ r(0) 6= ∅ and u ∩ q(r) 6= ∅ and v ∩ q(r) 6= ∅ for every real r ∈ (0, 1] and u ∪ v = x and u ∩ v is a non-empty finite set. then we consider the topology t on x that is generated by the order topology t. and the sets u and v . because of the definition of t we may conclude that t is connected and that for every continuous total preorder “ ” on x either the inclusion . ⊂ or & ⊂ holds. it, thus follows that winpot(x) = winpot(.) = winpot(&) = 2ℵ0 and that inpot(x) = stinpot(x) = inpot(.) = inpot(&) = ℵ0. in addition, since u ∪ v = x and u ∩ v is a non-empty finite set we may conclude that dcon(x) = |u ∩ v | < ℵ0, which settles the example. on the other hand, it seems to be difficult to construct a topological space (x, t) such that inpot(x) > stinpot(x). the authors conjecture, instead, that inpot(x) = stinpot(x) for every topological space (x, t). indeed, the set ct p(x) of all continuous total preorders “ ” on (x, t) is naturally ordered by set inclusion. an application of the lemma of zorn now implies that (ct p(x), ⊂) contains minimal elements “ . ”. then with the help of some uniqueness argument for ” .|c ” on every component c of x that is well known for orders (cf., for instance, eilenberg [9] or any other book or paper on connected orderable spaces) it should be possible to prove that for any minimal element “ . ” of (ct p(x), ⊂) the equalities inpot(.) = inpot(x) = stinpot(x) hold. but at present the authors are not sure about some crucial technical details of the argument. in order to also generalize the approach that has been sketched in section 2 we still introduce the following concepts. let x, y ∈ x be different points. then we denote by p(x, y) the set of all paths f : [0, 1] −→ x that connect x and y and define the path rank of the pair (x, y) by cardinality of indifference classes 183 prank(x, y) :=        min{|s| |s ⊂ x\ {x, y} and imf ∩ s 6= ∅ for every path f ∈ p(x, y)}, if p(x, y) 6= ∅ 0, otherwise . this definition allows us to define for an arbitrarily chosen set d ⊂ x the path rank of d by prank(d) := min{prank(x, y)|x, y ∈ d, x 6= y}. moreover, we define the separation rank of the pair (x, y) by srank(x, y) :=            min{|f| |f is with respect to set inclusion a maximal separated subset of p(x, y)}, if p(x, y) 6= ∅ 0, otherwise , in order to finally denote for every set d ⊂ x the separation rank of d by srank(d) := min{srank(x, y)|x, y ∈ d, x 6= y}. 4. lower and upper bounds and a topological characterization of real intervals as in the previous section we consider some fixed chosen non-trivial topological space (x, t) . with the help of the proof of assertion (i) of proposition 3.3 the following proposition follows immediately. proposition 4.1. let “ ” be some continuous total preorder on x. then for every component c of x there exists at least one indifference class [x] of “ ” such that |[x]| ≥ dcon(x). let now “ ” be an arbitrarily chosen continuous total preorder on x. then we consider an indifference class [x] of “ ” for which the set c ([x]) of all components c of x such that [x] is a calculable set with respect to ” -|c ” is non-empty. now the following proposition holds. proposition 4.2. |[x]| ≥ ∑ c∈c([x]) dcon(c). proof. let some component c ∈ c ([x]) be arbitrarily chosen. then we may conclude with the help of the first part of remark and example 3.2 that c ⊂ [x] or c ∩ [x] ∈ t (c). hence, it follows for every component c ∈ c ([x]) that |c ∩ [x]| ≥ dcon(c). since the components c ∈ c ([x]) are pairwise disjoint we, thus, may conclude that the desired inequality holds. � now we want to generalize the situation that has been considered in section 2. let, therefore, two different points x, y ∈ x be arbitrarily chosen. then the following proposition holds. 184 g. gerden and a. pallack proposition 4.3. srank(x, y) ≤ prank(x, y) ≤ 2ℵ0 · srank(x, y). proof. the validity of the inequality srank(x, y) ≤ prank(x, y) is a straightforward consequence of the definitions of srank(x, y) and prank(x, y) respectively. hence, only the inequality prank(x, y) ≤ 2ℵ0 · srank(x, y) has to be verified. the definitions of srank(x, y) and prank(x, y) respectively allow us to assume without loss of generality that p(x, y) 6= ∅. let, therefore, f be some separated subset of p(x, y) that is maximal with respect to set inclusion. then we set p(f) := ⋃ f∈f (imf\{x, y}). now the maximality of f implies that p(f)∩imf 6= ∅ for every path f ∈ p(x, y). since p(f) ⊂x\{x, y}, thus, the inequality prank(x, y) ≤ |p(f)| holds. in addition, it follows from the definition of p(f) that |p(f)| ≤ 2ℵ0 ·|f|. summarizing these considerations we may conclude that prank(x, y) ≤ 2ℵ0 · srank(x, y), which still was to be shown. � let us now assume, that every component c of x is path connected or locally path connected which means that the components of x with respect to path connectedness coincide with the components c of x. then the following lemma is an immediate consequence of the definitions of dcon(c) and prank(c) respectively. lemma 4.4. dcon(c) = prank(c) for every component c of x. because of proposition 4.3 and lemma 4.4 also the following proposition holds. proposition 4.5. supc∈csrank(c) ≤ supc∈cprank(c) = dcon(x) ≤ 2 ℵ0 · supc∈csrank(c). let (v, t) be some real or complex convex space the dimension dim v of which is greater than 1. then our considerations also imply the following proposition and corollaries (cf. section 2). proposition 4.6. winpot(v ) = inpot(v ) = stinpot(v ) = dcon(v ) = |v | . proof. let x, y ∈ v be arbitrarily chosen different points. because of proposition 4.3 and lemma 4.4 and the inequalities dcon(v ) ≤ stinpot(v ) ≤ inpot(v ) ≤ winpot(v ) ≤ |v | it suffices to prove that |v | ≤ srank(x, y). therefore, we consider some (closed) hyperplane h of v that separates the points x and y. then we define for every point z ∈ h some path fz ∈ p(x, y) by setting fz(λ) := { x + 2λ(z − x), if λ ∈ [0, 1 2 ) z + (2λ − 1)(y − z), if λ ∈ [1 2 , 1] for every real λ ∈ [0, 1]. a routine argument that uses for different points z, cardinality of indifference classes 185 z ′ ∈ h the linear independency of the vectors z−x and z ′ −x, respectively y−z and y − z ′ , implies that the collection f := {fz}z∈h is a separated family of paths fz ∈ p(x, y). since |h| = |v | the definition of srank(x, y), thus, implies that |v | = |f| ≤ srank(x, y), and the desired inequality follows. � corollary 4.7. |[x]| = |v | for every continuous total preorder “ ” on v and every indifference class [x] of “ ” such that x neither is a first nor a last element of “ ”. proof. if x neither is a first nor a last element of “ ” then l(x) and k(x) are two non-empty disjoint open subsets of v the union of which is v \ [x]. hence, |v | ≥ |[x]| ≥ dcon(v ) and the desired conclusion follows from proposition 4.6. � the following corollary strengthens and generalizes dugundji’s result that has been quoted in section 2 to arbitrary real or complex convex spaces. corollary 4.8. let s be a subset of v such that |s| < |v |. then v \s is path connected. proof. in the proof of proposition 4.6 it has been shown that srank(x, y) = |v | for every pair of different points x, y ∈ v . if |s| < |v | this means, in particular, that for every pair of points x, y ∈ v \s there exists some path f : [0, 1] −→ v \s that connects x and y (cf. the corresponding argument in section 2). � let now n > 1 be a natural number. according to the last paragraph of section 2 we define for every non-empty subset s of rn the dimension dims of s by choosing an arbitrary vector v ∈ s in order to then identify dims with the dimension of the linear subspace of rn that is generated by s − {v}. the reader may recall from section 2 that the definition of dims is independent of any particular chosen vector v ∈ s. then the following proposition holds. proposition 4.9. let “ ” be a continuous total preorder on rn. then the following assertions are valid: (i) dim[x] ≥ n − 1 for every indifference class [x] of “ ” such that x neither is a first nor a last element of “ ”. (ii) [x] is a hyperplane for every indifference class [x] of “ ” such that dim[x] = n − 1 and x neither is a first nor a last element of “ ”. proof. (i): let [x] be an indifference class of “ ” such that x neither is a first nor a last element of “ ”. by considering instead of “ ” the continuous total preorder “ -x ” on r n that is defined by u − x -x v − x ⇐⇒ u v we may assume without loss of generality that [x] contains the zero vector “0”. therefore, we may consider the linear subspace v[x] of r n that is generated by [x]. then we assume, in contrast, that dim v[x] ≤ n − 2. because of this assumption there exists a linear subspace w of rn that is generated by two linearly independent vectors y and z such that v[x] ∩w = {0}. for every vector c ∈ rn we now set lcy := {c + λ(y − c)|λ ∈ r} = {y + γ(c − y)|γ ∈ r} and 186 g. gerden and a. pallack lcz := {c+λ(z−c)|λ ∈ r} = {z+γ(c−z)|γ ∈ r}. then we choose two arbitrary vectors a, b ∈ rn\v[x]. since v[x] ∩ w = {0} it follows by direct computation that is based upon contraposition that lay ∩ v[x] = ∅ or l a z ∩ v[x] = ∅ and lby ∩ v[x] = ∅ or l b z ∩ v[x] = ∅. we may assume without loss of generality that lay ∩ v[x] = ∅ and l b z ∩ v[x] = ∅. the remaining possibilities can be settled by analogous arguments. the equations lay ∩ v[x] = ∅ and v[x] ∩ w = {0} and lbz ∩ v[x] = ∅ allow us to define a path f : [0, 1] −→ r n\v[x] by setting f(λ) :=    a + 3λ(y − a), if λ ∈ [0, 1 3 ) y + (3λ − 1)(z − y), if λ ∈ [1 3 , 2 3 ) z + (3λ − 2)(b − z), if λ ∈ [2 3 , 1] for every real λ ∈ [0, 1]. since the vectors a, b ∈ rn\v[x] have been arbitrarily chosen we may conclude that rn\v[x] is path connected and, therefore, connected. now the connectedness of rn\v[x] implies with the help of a straightforward indirect argument that is based upon the fact that every non-empty open subset of rn contains a base of linearly independent vectors of rn that also rn\ [x] is connected, which contradicts the proof of corollary 4.7 and, thus, finishes the proof of assertion (i). (ii): let x ∈ rn be any point that neither is a first nor a last element of “ ” such that dim [x] = n − 1. in order to prove that [x] is a hyperplane of rn we may assume, as in the proof of assertion (i), that 0 ∈ [x]. using the notation of the proof of assertion (i) it suffices to verify that v[x] ⊂ [x]. we assume, in contrast, that there exists some non-empty subset s of v[x] that is not contained in [x]. we may assume without loss of generality that |s| = 1. indeed, the last argument in the proof of assertion (i) shows in which way the case that |s| > 1 can be reduced to the case that |s| = 1. let, therefore, s = {v} for some vector v ∈ v[x]. then we set v − [x] := v[x]\{v} and prove that rn\v − [x] is connected. as in the proof of assertion (i) it, thus, follows that also rn\ [x] is connected, which contradicts the proof of corollary 4.7. hence, if we are able to verify that rn\v − [x] is connected nothing remains to be shown. in order to prove that rn\v − [x] is connected we choose two arbitrary points a, b ∈ rn\v − [x] and distinguish between the cases that a = v or b = v and that neither a = v nor b = v. case 1: a = v or b = v. in this case, we may assume without loss of generality, that a = v and that b 6= v. since a = v and b /∈ v[x] it follows that {a + λ(b − a)| λ ∈ r} ∩ v − [x] = lab ∩ v − [x] = ∅. this means that f : [0, 1] −→ rn\v − [x] defined by f (λ) := a + λ(b − a) for every real λ ∈ [0, 1] is a path that connects a and b. case 2: a 6= v and b 6= v. now it follows that {a + λ(v − a)|λ ∈ r} ∩ v − [x] = lav ∩ v − [x] = ∅ and that {v + γ(b − v)|γ ∈ r} ∩ v − [x] = lv b ∩ v − [x] = ∅. hence, f : [0, 1] −→ rn\v − [x] defined by cardinality of indifference classes 187 f (λ) := { a + 2λ(v − a), if λ ∈ [0, 1 2 ) v + (2λ − 1)(b − v), if λ ∈ [1 2 , 1] for every real λ ∈ [0, 1] is a path that connects a and b (cf. the definition of fz in the proof of proposition 4.6). since a, b ∈ r n\v − [x] have been arbitrarily chosen we may summarize our considerations for concluding that rn\v − [x] is path connected and, therefore, also connected. thus, the proof of assertion (ii) is complete. � remark 4.10. an analysis of the proof of proposition 4.9 implies that proposition 4.9 can be generalized to arbitrary real or complex convex spaces v the dimension dim v of which is greater than 1. but then the conclusion dim [x] ≥ n−1 has to be replaced by the condition that in case that 0 /∈ [x] the linear subspace of v that is generated by [x] coincides with v and that in case that 0 ∈ [x] the linear subspace of v that is generated by [x] coincides with v or is a maximal linear subspace of v . in combination with proposition 4.6 and corollary 4.7 this additional result, thus, provides a rather complete survey on the size of indifference classes of a continuous total preorder “ ” on a real or complex convex space. in this sense proposition 4.9 completes proposition 4.6 and corollary 4.7. indeed, in mathematical utility theory a particular chosen set of coordinates or a particular chosen base of linearly independent vectors of a real or complex convex space v can be interpreted as the collection of those (latent) factors or dimensions that influence preferences between alternatives. let, for the moment, an indifference class of a continuous total preference relation “ ” on v for which x neither is a first nor a last element of “ ” said to be an inner indifference class of “ ”. then we learn from proposition 4.6, corollary 4.7 and proposition 4.9 that inner indifference classes not only have maximal cardinality but that, in addition, preferences between alternatives y and z, the corresponding indifference classes of which are inner, can be assumed to be determined by different expressions of exactly one coordinate or different expressions of the same coordinates or different expressions of nearly the same coordinates, which means that one of the given alternatives y or z can be influenced by one more coordinate than the other alternative. indeed, if [y] and [z] are hyperplanes then the disjointness of [y] and [z] implies that the preference between [y] and [z] can be assumed to be influenced by exactly one coordinate (cf. the corresponding example in the introduction). otherwise, at least one of the indifference classes [y] or [z] is influenced by a maximal set of coordinates, which means that [y] or [z] can be assumed to be influenced by at most one more coordinate than any other inner indifference class of “ ”. in order to complete these considerations we still consider a continuous total preference relation “ ” on a real or complex convex space v that is locally non-satiated (cf. the introduction) and has the additional property that all its inner indifference classes are path connected. in this case the authors conjecture that there exists a homeomorphism h : v −→ v such that every inner 188 g. gerden and a. pallack indifference class of the continuous total preorder ” -h ” on x that is defined by h(x) -h h(y) ⇐⇒ x y is a hyperplane of v . if, however, an arbitrary abstract topological space is given an appropriate generalization of proposition 4.9 hardly seems to be possible, and the cardinality approach gains additional importance. we conclude by proving the following proposition that is a slight generalization of well known results in general topology (cf. eilenberg [9] or willard [15, section 28] or kok [12, chapter ii]). it provides a new proof of these classical results and includes for every natural number n ≥ 1 the family of all connected and convex subspaces of the euclidean space (rn, tnat). therefore, it also generalizes theorem 4 of candeal and induráin [6] and is closely related to generalizations of this theorem by beardon [2] and candeal, induráin and mehta [7] (cf., in particular, [7, theorem 4]), where necessary and sufficient conditions for the agreement of the euclidean and the order topology on totally ordered subsets of rn are discussed. proposition 4.11. let (x, t) be a separable, connected and locally connected hausdorff space. then the following assertions are equivalent: (i) every non-trivial connected subspace ( y, t|y ) of (x, t) contains some point y such that y \{y} splits into two components. (ii) there exists some continuous total order “ � ” on x. (iii) there exists some real interval i such that (x, t) is homeomorphic to (i, tnat). proof. since the proof of the implication “(iii)=⇒(i)” is straightforward it suffices to verify the validity of the implications “(i)=⇒(ii)” and “(ii)=⇒(iii)”. (i)=⇒(ii): we construct by transfinite induction a continuous total order ” � ” on x. α = 0 : we set -0:= x × x. 0 < α is not a limit ordinal: let iα−1 := {[x] | [x] is a non − trivial indifference class of ” -α−1 ”}. in case that iα−1 = ∅ we set �:= -α−1 in order to then finish the induction process. otherwise, we choose an arbitrary indifference class [x] ∈ iα−1. because of the induction hypothesis we may assume that ( ([x] , t|[x] ) is a connected (and locally connected) hausdorff space and that [x] = x or there exists a single point s ∈ x such that s = max{u ∈ x|u ≺α−1 x} or there exists a single point t ∈ x such that t = min{v ∈ x|x ≺α−1 v}. we abbreviate these assumptions on [x] by (*). in addition, the induction hypothesis allows us to assume that the sets dα−1(s) := {u ∈ x|u -α−1 s} and iα−1(s) := {u ∈ x|s -α−1 u} or dα−1(t) := {v ∈ x|v -α−1 t} and iα−1(t) := {v ∈ x|t -α−1 v} are closed subsets of x. let us abbreviate these last assumptions by (**). since the afore-presented cases can be settled by cardinality of indifference classes 189 analogous arguments we may concentrate without loss of generality on the case that there exist single points s, t ∈ x such that s =max{u ∈ x|u ≺α−1 x} and t =min{v ∈ x|x ≺α−1 v}. assertion (i) implies the existence of some point z ∈ [x] such that [x] \{z} splits into two (non-empty) connected and then, obviously, locally connected components cα1 and c α 2 . the assumptions (**) imply that [x] is an open subset of x. since (x, t) is a locally connected hausdorff space we, thus, may conclude that also cα1 and c α 2 are open subsets of x. furthermore, these assumptions on (x, t) allows us to assume without loss of generality that the (topological) closure cα1 of c α 1 is c α 1 ∪ {s, z} and that the (topological) closure cα2 of c α 2 is c α 1 ∪ {z, t}. hence, we set -α:=-α−1 \{(u, v) ∈ x × x|u ∈ c α 2 ∪ {z} and v ∈ c α 1 or u ∈ c α 2 and v ∈ cα1 ∪ {z}}. summarizing the afore-considered arguments it follows that both sets cα1 and c α 2 are indifference classes of ” -α ” and that the assumptions (*) and (**) are also satisfied with respect to ” -α ”. 0 < α is a limit ordinal: in this case we set -α:= ⋂ β<α -β. the reader may verify that in this way, actually, a total order ” � ” on x has been constructed. with the help of the assumptions (**) we may conclude, in addition, that ” � ” is a continuous total order on x. (ii)=⇒(iii): let ” � ” be a continuous total order on x. the general assumptions of the proposition imply with the help of eilenberg’s well known re-presentation theorem [9] that there exists some real interval i such that ( x, t� ) is homeomorphic to (i, tnat). it, thus, suffices to show that t � = t. the continuity of “ � ” allows us to conclude that t� ⊂ t and it remains to verify that also t ⊂ t�. since (x, t) is locally connected we may choose some connected open subset o of x. then we must prove that o ∈ t�. because of the assumption that “ � ” is a continuous total order on x it follows from the connectedness of o that o is an interval of (x, �). since o is an open subset of x the connectedness of (x, t) and the continuity of “ � ” imply, moreover, that o must be an open interval of (x, �). hence, o ∈ t� and nothing remains to be shown. � acknowledgements. the authors are grateful to two anonymous referees for a very careful reading of an earlier version of this paper and for several valuable comments and suggestions that have resulted in improvements. references [1] k. arrow and f. hahn, general competetive analysis, (oliver and boyd, edinburgh, 1971). [2] a.f. beardon, totally ordered subsets of euclidean space, j. math. econom. 23 (1994), 391-393. 190 g. gerden and a. pallack [3] t. bewley, existence of equilibria in economics with infinitely many commodities, j. of econom. theory 4 (1972), 514-540. [4] g. birkhoff, lattice theory, (american mathematical society, rhode island, 1940 (first edition), 1967 (third edition)). [5] d. bridges and g.b. mehta, representation of preference orderings, (springer, new york, 1995). [6] j.c. candeal and e. induráin, utility functions on chains, j. math. econom. 22 (1993), 161-168. [7] j.c. candeal, e. induráin and g.b. mehta, further remarks on totally ordered representable subsets of euclidean space, j. math. econom. 25 (1996), 381-390. [8] j. dugundji, topology, (allyn and bacon, boston, 1966). [9] s. eilenberg, ordered topological spaces, amer. j. math. 24 (1941), 305-309. [10] g. herden, on the semicontinuous and continuous analogue of the szpilrajn theorem, in preparation, university of essen, 2003. [11] l. jones, a competetive model of product differentiation, econometrica 52 (1984), 507530. [12] h. kok, connected orderable spaces, (mathematical centre tracts, am-sterdam, 1973). [13] a. mas-colell, the price equilibrium existence problem in topological vector lattices, econometrica 54 (1986), 1039-1053. [14] g.b. mehta, infinite dimensional arrow-hahn theorem, preprint, university of brisbane, 1989. [15] s. willard, general topology, (addison-wesley, mass.-london-don mills, 1970). received july 2002 accepted june 2003 gerhard herden (gerhard.herden@uni-essen.de) universität duisburg-essen, fachbereich 6 (mathematik), universitaets-strasse 2, d-45117 essen, germany andreas pallack (andreas.pallack@uni-essen.de) universität duisburg-essen, fachbereich 6 (mathematik), universitaets-strasse 2, d-45117 essen, germany songagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 131-138 on spaces with the property (wa) yan-kui song ∗ abstract. a space x has the property (wa) (or is a space with the property (wa)) if for every open cover u of x and every dense subspace d of x, there exists a discrete subspace f ⊆ d such that st(f, u) = x. in this paper, we give an example of a tychonoff space without the property (wa), and also study topological properties of spaces with the property (wa) by using the example. 2000 ams classification: 54d20, 54b10, 54d55 keywords: property (a), property (wa) 1. introduction by a space, we mean a topological space. matveev [2] defined a space x to have the property (a) if for every open cover u of x and every dense subspace d of x, there exists a discrete closed subspace f ⊆ d such that st(f, u) = x, where st(f, u) = ⋃ {u ∈ u : u ∩ f 6= ∅}. as a way to weaken the above definition, he also gave the following definition: definition 1.1 ([2]). a space x has the property (wa) if for every open cover u of x and every dense subspace d of x, there exists a discrete subspace f ⊆ d such that st(f, u) = x. a space having the property (wa) is also called a space with the property (wa). from the above definitions, it is not difficult to see that every space with the property (a) is a space with the property (wa). the purpose of this paper is to give an example of a tychonoff space without the property (wa) and to study topological properties of spaces with the property (wa) by using the example. as usual, r, p and q denote the set of all real numbers, all irrational numbers and all rational numbers, respectively. for a set a, |a| denotes the cardinality ∗the author is supported by nsfc project 10271056. 132 y.-k. song of a. for a cardinal κ, κ+ denotes the smallest cardinal greater than κ. in particular, let ω denote the first infinite cardinal, ω1 = ω + and c the cardinality of the continuum. as usual, a cardinal is the initial cardinal and an ordinal is the set of smaller ordinals. when viewed as a space, every cardinal has the usual order topology. for each ordinal α, β with α < β, we write (α, β) = {γ : α < γ < β}, [α, β) = {γ : α ≤ γ < β} and (α, β] = {γ : α < γ ≤ β}. other terms and symbols that we do not define will be used as in [1]. 2. a tychonoff space without the property (wa) matveev [2] gave an example of a t1-space without the property (wa) and he asked if there exists a t2 (t3, tychonoff) space without the property (wa). yang [8] constructed a t2 space without the property (wa). in this section, we give an example of a tychonoff space without the property (wa). we omit the easy proof of the following lemma. lemma 2.1. let r be endowed with the usual topology and a a discrete subspace of r. then, |a| ≤ ω and clra is nowhere dense in r. example 2.2. there exists a 0-dimensional, first countable, tychonoff space without the property (wa). proof. let a = ⋃ n∈n an, where an = q × {1/n} and let a = {s : s is a discrete subspace of a}. then, we have: claim 2.3. |a| = c. proof. since |a| = ω, |a| ≤ c. let s = {〈n, 1〉 : n ∈ n} ⊆ a. since every subset of s is discrete, {f : f ⊆ s} ⊆ a. hence, |a| ≥ |{f : f ⊆ s}| = c. � since |a| = c, we can enumerate the family a as {sα : α < c}. for each α < c and each n ∈ n , put sα,n = {q ∈ q : 〈q, 1/n〉 ∈ sα}. claim 2.4. for each α < c, |r \ ⋃ n∈n clrsα,n| = c. proof. for each α < c, let xα = r \ ⋃ n∈n clr sα,n. since xα is a gδ-set in r, xα is a complete metric space. to show that xα is dense in itself, suppose that xα has an isolated point x. then, there exists ε > 0 such that (x − ε, x + ε) ∩ xα = {x}. let i = (x, x + ε), then, i ⊂ r \ xα ⊂ ⋃ n∈n clr sα,n. moreover, since i is open in r, clr sα,n ∩ i ⊆ clr(sα,n ∩ i). hence, (6) i = ( ⋃ n∈n clr sα,n) ∩ i = ⋃ n∈n (clr sα,n ∩ i) ⊆ ⋃ n∈n clr(sα,n ∩ i). on spaces with the property (wa) 133 by lemma 2.1, each clr(sα,n ∩ i) is nowhere dense in r. thus, (6) contradicts the baire category theorem. hence, xα is dense in itself. it is known ([1, 4.5.5]) that every dense in itself complete metric space includes a cantor set. hence, |xα| = c. � claim 2.5. there exists a sequence {pα : α < c} satisfying the following conditions: (1) for each α < c, pα ∈ p. (2) for any α, β < c, if α 6= β, then pα 6= pβ . (3) for each α < c, pα /∈ ⋃ n∈n clrsα,n. proof. by transfinite induction, we define a sequence {pα : α < c} as follows: there is p0 ∈ p such that p0 6∈ ⋃ n∈n cl s0,n by claim 2.4. let 0 < α < c and assume that pβ has been defined for all β < α. by claim 2.4, |r \ ⋃ n∈n clr sα,n| = c. hence, we can choose a point pα ∈ (p \ ⋃ n∈n clr sα,n) \ {pβ : β < α}. now, we have completed the induction. then, the sequence {pα : α < c} satisfies the conditions (1) (2) and (3). � claim 2.6. for each α < c, there exists a sequence {εα,n : n ∈ n} in q satisfying the following conditions: (1) for each n ∈ n, (pα − εα,n, pα + εα,n) ∩ sα,n = ∅. (2) for each n ∈ n, εα,n ≥ εα,n+1. (3) limn→∞ εα,n = 0. proof. let α < c. for n = 1, since pα 6∈ clr sα,1, there exists a rational εα,1 > 0 such that (pα − εα,1, pα + εα,1) ∩ sα,1 = ∅. let n > 1 and assume that we have defined {εα,m : m < n} satisfying that εα,1 > εα,2 > · · · > εα,n−1. since pα 6∈ clr sα,n, there exists a rational ε ′ α,n such that (pα − ε ′ α,n, pα + ε ′ α,n) ∩ sα,n = ∅. put εα,n = n −1 min{εα,n−1, ε ′ α,n}. now, we have completed the induction. then, the sequence {εα,n : n ∈ n} satisfies (1) (2) and (3). � define x = a ∪ b, where b = {〈pα, 0〉 : α < c}. topologize x as follows: a basic neighborhood of a point in a is a neighborhood induced from the usual topology on the plane. for each α < c, a neighborhood base {un〈pα, 0〉 : n ∈ ω} of 〈pα, 0〉 ∈ b is defined by un〈pα, 0〉 = {〈pα, 0〉} ∪ ( ⋃ i≥n {((pα − εα,i, pα + εα,i) ∩ q) × {1/i}}). 134 y.-k. song for each n ∈ ω. then, x is a first countable t2-space. for each α < c and each n ∈ ω. un〈pα, 0〉 is open and closed in x, because pα ± εα,i 6∈ q for each i ∈ ω. it follows that x is 0-dimensional, and hence, a tychonoff space. claim 2.7. the space x has not the property (wa). proof. let u = {a} ∪ {u1〈pα, 0〉 : α < c}. then, u is an open cover of x and a is a dense subspace of x. for each discrete subset f of a, there exists α < c such that f = sα. since u1〈pα, 0〉 ∩ sα = ∅, 〈pα, 0〉 6∈ st(f, u). this shows that x does not have the property (wa). � � remark 2.8. the above example was announced in [6]. the author does not know if there exists a normal space without the property (wa). remark 2.9. just, matveev and szeptycki [5] constructed an example that has similar properties as example 2.2, but the construction of our example seems to be simpler than their example. 3. some topological properties of spaces with the property (wa) first, we give an example showing that a continuous image of a space with the property (wa) need not be a space with the property (wa). example 3.1. there exists a continuous bijection f : x → y from a tychonoff space x with the property (wa) to a tychonoff space y without the property (wa). proof. we define the space x by changing the topology of the space of example 2.2 by the discrete space. then, the space x is a space with property (wa). let y be the space of example 2.2 as in the proof of example 2.2. then, the space y is a tychonoff space without property (wa). let f : x → y be the identity map. clearly f is continuous, which completes the proof. � let us recall that a mapping f : x → y is varpseudocompact if int(f (u )) 6= ∅ for every non-empty set u of x. theorem 3.2. let x be a space with the property (wa) and f : x → y be a varpseudocompact continuous closed mapping. then, y is a space with the property (wa). proof. let f : x → y be a varpseudocompact continuous closed mapping. let u be an open cover of y and d a dense subspace of y . then, u0 = {f −1(u ) : u ∈ u} is an open cover of x and d0 = f −1(d) is dense in x since f is varpseudocompact. then, there is a discrete subset b ⊆ d0 such that st(b, u0) = x, since x is a space with property (wa). let f = f (b). then, f is a discrete subset of d since f is closed, and st(f, u) = y , which completes the proof. � on spaces with the property (wa) 135 in the following, we give an example to show that a regular-closed subset of a space with the property (a) (hence, (wa)) need not be a space with the property (wa). recall [3] that a space x is absolutely countably compact if for every open cover u of x and every dense subspace d of x, there exists a finite subset f ⊆ d such that st(f, u) = x. it is known that every absolutely countably compact t2 space is countably compact and has the property (a) (see [2, 3]). moreover, vaughan [7] proved that every countably compact go-space is absolutely countably compact. thus, every cardinality with uncountable cofinality is absolutely countably compact. example 3.3. there exists a tychonoff space x with the property (a) (hence,(wa)) having a regular-closed subspace without the property (wa). proof. let x = a ∪ b be as in the proof of example 2.2. let s1 = (c + × a) ∪ b. we topologize s1 as follows: c + × b has the usual product topology and is an open subspace of s1. for each α < c, a basic neighbourhood of 〈pα, 0〉 takes the form gβ,n(〈pα, 0〉) = {〈pα, 0〉} ∪ ({α : β < α < c +} × (un〈pα, 0〉 \ {〈pα, 0〉})). for β < c+ and n ∈ n , where un〈pα, 0〉 is defined in example 2.2. then, the space s1 is tychonoff. now, we show that s1 has the property (a). for this end, let u be an open cover of s1. let d0 be the set of all isolated points of c + and let d = d0 × a. then, d is dense in s1 and every dense subspace of s1 contains d. thus, it suffices to show that there exists a subset f ⊆ d such that f is discrete closed in s1 and st(f, u) = s1. for each q ∈ q and each n ∈ n , since c+ × {〈q, 1/n〉} is absolutely countably compact, there exists a finite subset fq,n ⊆ d0 × {〈q, 1/n〉} such that c + × {〈q, 1/n〉} ⊆ st(fq,n, u). let f ′ = ⋃ {fq,n : q ∈ q and n ∈ ω}. then, c + × a ⊆ st(f ′, u). for each α < c, take uα ∈ u with 〈pα, 0〉 ∈ uα, and fix βα < c + and nα ∈ n such that {〈α, 〈pα, 0〉〉 : βα < α < c +} ⊆ uα. for each n ∈ n , let bn = {α < c : nα = n} and choose βn ∈ s with βn > sup{βα : α ∈ bn}. then, bn ⊆ st(〈βn, n〉, u). thus, if we put f ′′ = {〈βn, n〉 : n ∈ n}. then b ⊆ st(f ′′, u). let f = f ′ ∪ f ′′. then, f is a countable subset of d such that s1 = st(f, u). since f ∩ (c + × {〈q, n〉}) is finite for each q ∈ q 136 y.-k. song and each n < ω, f is discrete and closed in s1, which shows that s1 has the property (a). let s2 be the same space x as in example 2.2. then, the space s2 is a tychonoff space without the property (wa). we assume that s1 ∩ s2 = ∅. let ϕ : b → b be the identify map. let x be the quotient space obtained from the discrete sum s1 ⊕ s2 by identifying 〈pα, 0〉 with ϕ(〈pα, 0〉) for each α < c. let π : s1 ⊕ s2 → x be the quotient map. it is easy to check that π(s2) is a regular-closed subset of x, however, it is not a subspace of x with the property (wa), since it is homeomorphic to s2. next, we show that x has the property (a). for this end, let u be an open cover of x. let s = π(a ∪ d) then, s is dense in x and every dense subspace of x contains s, since each point of s is a isolated point of x. thus, it suffices to show that there exists a subset c of s such that c is discrete closed in x and x = st(c, u). since π(s1) is homeomorphic to the space s1, then there exists a discrete closed subset c0 ⊆ π(d) such that π(s1) ⊆ st(c0, u). since π(s1) is closed in x, then c0 is closed in x. let c1 = x \ st(π(c0, u). then, c1 ⊆ s. if we put c = c0 ∪ c1, then x = st(c, u). since c ⊆ s and c is a discrete closed subset of x, then x has the property (a), which completes the proof. � considering other types of subspaces, we arrive to the following result, which is rather unexpected even thought the lindelöf property is preserved by arbitrary fσ-subspaces, and which is a minor improvement of theorem 84 from [4]. recall that a space is a p -space if every gδ-set is open. theorem 3.4. an open fσ-subset of a p -space with the property (wa) has the property (wa). proof. let x be a p -space with the property (wa) and let y = ⋃ {hn : n ∈ ω} be an open fσ-subset in x (each hn is closed in x). let u be an open cover of y and let d be a dense subset of y . we have to find a discrete set f ⊆ d such that st(f, u) = y . for each n ∈ ω, let us consider the open cover un = u ∪ {x \ hn} of x and the dense subset d ∪ (x \ y ) of x. since x has the property (wa), there is a discrete subset bn ⊆ d ∪ (x \ y ) such that st(bn, un) = x. put an = bn ∩ d. it is clear that hn ⊆ st(an, u). put f = ⋃ {an : n ∈ ω}. then f is a discrete subset of d, since x is a p -space and st(f, u) = y , which completes the proof. � since a cozero-set is open fσ-set, thus we have the following corollary. corollary 3.5. a cozero-set of a p -space with the property (wa) has the property (wa). on spaces with the property (wa) 137 recall that the alexandorff duplicate a(x) of a space x is constructed as follows. the underlying set of a(x) is x × {0, 1}; each point of x × {1} is isolated and a basic neighborhood of a point 〈x, 0〉 ∈ x × {0} is the set of the form (u × {0}) ∪ ((u × {1}) \ {〈x, 1〉}), where u is a neighborhood of x in x. theorem 3.6. let x be any space. then, a(x) is a space with the property (wa). proof. let d0 be the set of all isolated points of x. if we put d = d0 ∪ (x × {1}), then d is a dense subset of a(x). since each point of x × {1} is a isolated point of a(x), then every dense subset of a(x) contains d. we show that a(x) is a space with the property (wa). for this end, let u be an open cover of a(x). it suffices to show that there exists a discrete subspace f of d such that st(f, u) = a(x). since d is dense in a(x) and each point of d is isolated. taking f = d, then d is discrete and st(d, u) = a(x), since d is dense in a(x),which completes the proof. � the following corollary follows directly from theorem 3.6: corollary 3.7. every space can be embedded as a closed subset into a space with the property (wa). just, matveev and szeptycki proved in theorem 16 of [5] that the product of a countably paracompact (a)-space and a compact metrizable space is a (a)-space. in a similar way, we may prove the following: theorem 3.8. let x be a countably paracompact space with the property (wa) and y a compact metric space.then, x × y is a space with the property (wa). remark 3.9. the author does not know if the assumption that x is countably paracompact can be removed. acknowledgements. the paper was written, while prof. ohta and doctor jiling cao visited department of mathematics of nanjing normal university. the author would like to thank prof. ohta and doctor jiling cao for their valuable suggestions and comments. references [1] r. engelking, general topology, revised and completed edition heldermann, berlin (1989). [2] m. v. matveev, some questions on property (a), quest. answers gen. topology 15 (1997), 103–111. [3] m. v. matveev, absolutely countably compact spaces, topology appl. 58 (1994), 81–92. [4] m. v. matveev, a survey on star-covering properties, topological atlas, preprint no. 330 (1998). 138 y.-k. song [5] w. just, m. v. matveev and p. j. szeptycki, some results on property (a), topology appl. 100 (2000), 67–83. [6] y. song, absolutely countably compact spaces and related spaces, general and geometric topology (in japan) kyoto. 1074 (1999), 55–60. [7] j. e. vaughan, on the product of a compact space with an absolutely countably compact spac, annals of the new york acad. sci. 788 (1996), 203–208. [8] z. yang, a method constructing hausdorff spaces without property (wa), quest. answers gen. topology 18 (2000), 113–116. received november 2004 accepted april 2005 yan-kui song (songyankui@njnu.edu.cn) department of mathematics, nanjing normal university, nanjing, 210097 p.r of china. georgiouagt.dvi @ applied general topology c© universidad politécnica de valencia volume 10, no. 1, 2009 pp. 159-171 topologies on function spaces and hyperspaces d. n. georgiou ∗ abstract. let y and z be two fixed topological spaces, o(z) the family of all open subsets of z, c(y, z) the set of all continuous maps from y to z, and oz (y ) the set {f −1(u ) : f ∈ c(y, z) and u ∈ o(z)}. in this paper, we give and study new topologies on the sets c(y, z) and oz (y ) calling (a, a0)-splitting and (a, a0)-admissible, where a and a0 families of spaces. 2000 ams classification: 54c35 keywords: function space, hyperspace, splitting topology, admissible topology. 1. preliminaries let y and z be two fixed topological spaces. by c(y, z) we denote the set of all continuous maps from y to z. if t is a topology on the set c(y, z), then the corresponding topological space is denoted by ct(y, z). let x be a space. to each map g : x × y → z which is continuous in y ∈ y for each fixed x ∈ x, we associate the map g∗ : x → c(y, z) defined as follows: for every x ∈ x, g∗(x) is the map from y to z such that g∗(x)(y) = g(x, y), y ∈ y . obviously, for a given map h : x → c(y, z), the map h⋄ : x × y → z defined by h⋄(x, y) = h(x)(y), (x, y) ∈ x × y , satisfies (h⋄) ∗ = h and is continuous in y for each fixed x ∈ x. thus, the above association (defined in [7]) between the mappings from x × y to z that are continuous in y for each fixed x ∈ x, and the mappings from x to c(y, z) is one-to-one. in 1946 r. arens [1] introduced the notion of an admissible topology: a topology t on c(y, z) is called admissible if the map e : ct(y, z) × y → z, called evaluation map, defined by e(f, y) = f (y), is continuous. in 1951 r. arens and j. dugundji [2] introduced the notion of a splitting topology: a topology t on c(y, z) is called splitting if for every space x, the continuity of a map g : x × y → z implies the continuity of the map ∗work supported by the caratheodory programme of the university of patras. 160 d. n. georgiou g∗ : x → ct(y, z). on the set c(y, z) there exists the greatest splitting topology, denoted here by tgs (see [2]). they also proved that a topology t on c(y, z) is admissible if and only if for every space x, the continuity of a map h : x → ct(y, z) implies that of the map h ⋄ : x × y → z if in the above definitions it is assumed that the space x belongs to a fixed class a of topological spaces, then the topology t is called a-splitting or a-admissible, respectively (see [8]). in the case where a = {x} we write x-splitting (respectively, x-admissible) instead of {x}-splitting (respectively, {x}-admissible). let x be a space. in what follows by o(x) we denote the family of all open subsets of x. also, for two fixed topological spaces y and z we denote by oz (y ) the set {f −1(u ) : f ∈ c(y, z) and u ∈ o(z)}. the scott topology ω(y ) on o(y ) (see, for example, [11]) is defined as follows: a subset ih of o(y ) belongs to ω(y ) if: (α) the conditions u ∈ ih, v ∈ o(y ), and u ⊆ v imply v ∈ ih, and (β) for every collection of open sets of y , whose union belongs to ih, there are finitely many elements of this collection whose union also belongs to ih. the strong scott topology ωs(y ) on o(y ) (see [12]) is defined as follows: a subset ih of o(y ) belongs to ωs(y ) if: (α) the conditions u ∈ ih, v ∈ o(y ), and u ⊆ v imply v ∈ ih, and (β) for every open cover of y there are finitely many elements of this cover whose union also belongs to ih. the isbell topology tis (respectively, strong isbell topology tsis) on c(y, z) (see, for example, [13] and [12]) is the topology, which has as a subbasis the family of all sets of the form: (ih, u ) = {f ∈ c(y, z) : f −1(u ) ∈ ih}, where ih ∈ ω(y ) (respectively, ih ∈ ωs(y )) and u ∈ o(z). the compact open topology (see [7]) on c(y, z), denoted here by tco, is the topology for which the family of all sets of the form (k, u ) = {f ∈ c(y, z) : f (k) ⊆ u}, where k is a compact subset of y and u is an open subset of z, form a subbase. it is known that tco ⊆ tis (see, for example, [13]). a subset k of a space x is said to be bounded if every open cover of x has a finite subcover for k (see [12]). a space x is called corecompact (see [11]) if for every x ∈ x and for every open neighborhood u of x, there exists an open neighborhood v of x such that the subset v is bounded in the space u (see [11]). topologies on function spaces 161 below, we give some well known results: (1) the isbell topology and, hence, the compact open topology, and the point open topology (denoted here by tpo) on c(y, z) are always splitting (see, for example, [2], [3], and [13]). (2) the compact open topology on c(y, z) is admissible if y is a regular locally compact space. in this case the compact open topology is also the greatest splitting topology (see [2]). (3) the isbell topology on c(y, z) is admissible if y is a corecompact space. in this case the isbell topology is also the greatest splitting topology (see, for example, [12] and [14]). (4) a topology larger than a admissible topology is also admissible (see [2]). (5) a topology smaller than a splitting topology is also splitting (see [2]). (6) the strong isbell topology on c(y, z) is admissible if y is a locally bounded space (see [12]). for a summary of all the above results and some open problems on function spaces see [10]. also, [4] and [5] are other papers related to this area. in what follows if ϕ : x → y is a map and x0 ⊆ x, then by ϕ|x0 : x0 → y we denote the restriction of the map ϕ on the set x0. also, if h : x × y → z is a map and x0 ⊆ x, then by h|x0×y we denote the restriction of the map h on the set x0 × y . in sections 2 and 3 we give and study new topologies on the sets c(y, z) and oz (y ) calling (a, a0)-splitting and (a, a0)-admissible, where a and a0 families of spaces. 2. (a, a0)-splitting and (a, a0)-admissible topologies on the set c(y, z) note 1. let a be a family of topological spaces. for every x ∈ a we denote by x0 a subspace of x and by a0 the family of all such subspaces x0. in all paper by (a, a0) we denote the family of all pairs (x, x0) such that x ∈ a, x0 ∈ a0, and x0 is a subspace of x. definition 2.1. a topology t on c(y, z) is called (a, a0)-splitting if for every pair (x, x0) ∈ (a, a0), the continuity of a map g : x × y → z implies the continuity of the map g∗|x0 : x0 → ct(y, z), where g ∗ : x → ct(y, z) the map which is defined in preliminaries. a topology t on c(y, z) is called (a, a0)-admissible if for every pair (x, x0) ∈ (a, a0), the continuity of a map h : x → ct(y, z) implies that of the map h⋄|x0×y : x0 × y → z, where h ⋄ : x × y → z the map which is defined in preliminaries. in the case where a = {x} and a0 = {x0}, where x0 is a subspace of x, we write (x, x0)-splitting (respectively, (x, x0)-admissible) instead of ({x}, {x0})-splitting (respectively, ({x}, {x0})-admissible). 162 d. n. georgiou clearly, the following theorem is true. theorem 2.2. the following statements are true: (1) every splitting (respectively, admissible) topology on c(y, z) is (a, a0)splitting (respectively, (a, a0)-admissible), where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. (2) every a-splitting (respectively, a-admissible) topology on c(y, z) is (a, a0)-splitting (respectively, (a, a0)-admissible), where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. example 2.3. (1) the point-open, the compact open, and the isbell topologies are (a, a0)splitting, where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. (2) if y is a regular locally compact space, then the compact-open topology is (a, a0)-admissible, where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. (3) if y is a corecompact space, then the isbell topology is (a, a0)-admissible, where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. (4) if y is a locally bounded space, then the strong isbell topology is (a, a0)-admissible, where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. (5) let x be a space, x0 ∈ x, x0 the subspace {x0} of x, and t an arbitrary topology on c(y, z) which it is not x-splitting. then, the topology t is (x, x0)-splitting. it is clear that this topology t is not splitting. (6) let x be a space, x0 ∈ x, x0 the subspace {x0} of x, and t an arbitrary topology on c(y, z) which it is not x-admissible. then, the topology t is (x, x0)-admissible. it is clear that this topology t is not admissible. theorem 2.4. the following statements are true: (1) a topology smaller than an (a, a0)-splitting topology is also (a, a0)splitting. (2) a topology larger than an (a, a0)-admissible topology is also (a, a0)admissible. proof. we prove only the statement (1). the proof of (2) is similar. let t1 be an (a, a0)-splitting topology on c(y, z) and t2 a topology on c(y, z) such that t2 ⊆ t1. we prove that the topology t2 is a (a, a0)-splitting topology. indeed, let (x, x0) ∈ (a, a0) and let g : x × y → z be a continuous map. since the topology t1 is (a, a0)-splitting, the map g ∗|x0 : x0 → ct1 (y, z) is continuous. also, since t2 ⊆ t1, the identical map id : ct1 (y, z) → ct2 (y, z) is topologies on function spaces 163 continuous. so, the map g∗|x0 : x0 → ct2 (y, z) is continuous as a composition of continuous maps. thus, the topology t2 is (a, a0)-splitting. � definition 2.5. let (a1, a10) and (a 2, a20) two pairs of spaces, where a 1 (respectively, a2) and a10 (respectively, a 2 0) are arbitrary families of spaces such that every element x0 ∈ a 1 0 (respectively, every element x0 ∈ a 2 0) is a subspace of an element x ∈ a1 (respectively, of an element x ∈ a2). we say that the pairs (a1, a10) and (a 2, a20) are equivalent if a topology t on c(y, z) is (a1, a10)-splitting if and only if t is (a 2, a20)-splitting, and t is (a 1, a10)admissible if and only if t is (a2, a20)-admissible. in this case we write (a1, a10) ∼ (a 2 , a20). theorem 2.6. for every pair (a, a0), where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a, there exists a pair (x(a), x(a0)), where x(a) is a space and x(a0) is a subspace of x(a) such that (a, a0) ∼ (x(a), x(a0)). proof. let t csp be the set of all topologies on c(y, z) which are not (a, a0)splitting and let t cad the set of all topologies on c(y, z) which are not (a, a0)admissible. for each t ∈ t csp there exists in (a, a0) a pair (x sp t , x sp t,0) such that t is not (x sp t , x sp t,0)-splitting. similarly, for each t ∈ t c ad there exists in (a, a0) a pair (x adt , x ad t,0) such that t is not (x ad t , x ad t,0)-admissible. let a′ = {x sp t : t ∈ t c sp} ∪ {x ad t : t ∈ t c ad} and a′0 = {x sp t,0 : t ∈ t c sp} ∪ {x ad t,0 : t ∈ t c ad}. of course, we can suppose that the spaces from a′ and a′0 are pair-wise disjoint. let x(a) and x(a0) be the free union of all the spaces from a ′ and a′0, respectively. we prove that the pair (x(a), x(a0)) is the required pair. let t be an (a, a0)-splitting topology on c(y, z). we prove that this topology is (x(a), x(a0))-splitting. indeed, let g : x(a) × y → z be a continuous map. it suffices to prove that the map g∗|x(a0) : x(a0) → ct(y, z) is continuous. let x ∈ a′ ⊆ a. then, the restriction g|x×y of the map g on x × y ⊆ x(a) × y is also a continuous map and, therefore, since the topology t is (a, a0)-splitting we have that the map (g|x×y ) ∗|x0 : x0 → ct(y, z) is continuous. since x(a0) is the free union of all the spaces from a ′ 0 and (g|x×y ) ∗|x0 = (g ∗|x(a0))|x0 , it follows that the map g ∗|x(a0) : x(a0) → ct(y, z) is continuous. thus, the topology t on c(y, z) is (x(a), x(a0))splitting. now, let t be an (x(a), x(a0))-splitting topology on c(y, z). we prove that t is (a, a0)-splitting. we suppose that t is not (a, a0)-splitting. then, t ∈ t csp and, therefore, t is not (x sp t , x sp t,0)-splitting for some pair (x sp t , x sp t,0) ∈ (a, a0). thus, there exists a continuous map g : x sp t × y → z such that the 164 d. n. georgiou map g∗|xsp t,0 : x sp t,0 → ct(y, z) is not continuous. since the space x(a) is the free union of all the spaces from the family a′, the map g can be extended to a continuous map g1 : x(a) × y → z. since the map g ∗|xsp t,0 is not continuous, x sp t,0 ∈ a ′ 0, and the space x(a0) is the free union of all spaces from a ′ 0 we have that the map g∗|x(a0) : x(a0) → ct(y, z) is not continuous, which contradicts our assumption that t is a (x(a), x(a0))splitting topology. thus, a topology t on c(y, z) is (a, a0)-splitting if and only if it is (x(a), x(a0))-splitting. similarly, a topology t on c(y, z) is (a, a0)-admissible if and only if is (x(a), x(a0))-admissible. hence, (a, a0) ∼ (x(a), x(a0)). � theorem 2.7. there exists the greatest (a, a0)-splitting topology, where a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. proof. let {ti : i ∈ i} be the family of all (a, a0)-splitting topologies on c(y, z). we consider the topology t = ∨{ti : i ∈ i}. clearly, t is (a, a0)splitting and ti ⊆ t, for every i ∈ i. thus, t is the greatest (a, a0)-splitting topology. � note 2. in what follows we denote by t(a, a0) the greatest (a, a0)-splitting topology on c(y, z), theorem 2.8. the following statements are true: (1) if (a, a0) = ∪{(a i, ai0) : i ∈ i}, then t(a, a0) = ∩{t(a i , ai0) : i ∈ i}. (2) t(a, a0) = ∩{t(x, x0) : (x, x0) ∈ (a, a0)}. (3) if (a, a0) = ∩{(a i, ai0) : i ∈ i}, then ∨{t(ai, ai0) : i ∈ i} ⊆ t(a, a0). proof. (1) since (a, a0) = ∪{(a i, ai0) : i ∈ i} we have that every topology which is (a, a0)-splitting is also (a i, ai0)-splitting, for every i ∈ i. thus, the topology t(a, a0) is (a i, ai0)-splitting and, therefore, t(a, a0) ⊆ t(a i, ai0), for every i ∈ i. so, we have t(a, a0) ⊆ ∩{t(a i, ai0) : i ∈ i}. now, we prove the converse relation, that is ∩{t(ai, ai0) : i ∈ i} ⊆ t(a, a0). topologies on function spaces 165 for the above relation it suffices to prove that the topology ∩{t(ai, ai0) : i ∈ i} is (a, a0)-splitting. let (x, x0) ∈ (a, a0) and let g : x × y → z be a continuous map. we prove that the map g ∗|x0 : x0 → c∩{t(ai,ai 0 ):i∈i}(y, z) is continuous. since (x, x0) ∈ (a, a0), there exists i ∈ i such that (x, x0) ∈ (ai, ai0). this means that the map g∗|x0 : x0 → ct(ai,ai 0 )(y, z) is continuous. also, since ∩{t(ai, ai0) : i ∈ i} ⊆ t(a i, ai0), the identical map id : ct(ai,ai 0 )(y, z) → c∩{t(ai,ai 0 ):i∈i}(y, z) is continuous. so, the map g∗|x0 : x0 → c∩{t(ai,ai 0 ):i∈i}(y, z) is continuous as a composition of continuous maps. thus, the topology ∩{t(ai, ai0) : i ∈ i} is (a, a0)-splitting. (2) the proof of this is a corollary of the statement (1). (3) the proof of this follows by the fact that the topology ∨{t(ai, ai0) : i ∈ i} is (a, a0)-splitting. � theorem 2.9. let t be an (a, a0)-admissible topology on c(y, z). if (ct(y, z), ct(y, z)) ∈ (a, a0), then t is admissible and t(a, a0) ⊆ t. proof. let id ≡ h : ct(y, z) → ct(y, z) be the identical map. clearly, this map is continuous. since (ct(y, z), ct(y, z)) ∈ (a, a0) and t is (a, a0)-admissible, the map h ⋄|ct(y,z) ≡ h ⋄ : ct(y, z) × y → z is continuous. hence, the topology t is admissible. now, since the map h⋄ ≡ g : ct(y, z) × y → z is continuous, (ct(y, z), ct(y, z)) ∈ (a, a0), and the topology t(a, a0) is (a, a0)-splitting, the map g∗|ct(y,z) = id : ct(y, z) → ct(a,a0)(y, z) is also continuous. thus, t(a, a0) ⊆ t. � corollary 2.10. let t be an (a, a0)-splitting and (a, a0)-admissible topology on c(y, z). if (ct(y, z), ct(y, z)) ∈ (a, a0), then t(a, a0) = t. proof. by theorem 2.9, t(a, a0) ⊆ t. also, since the topology t is (a, a0)splitting, t ⊆ t(a, a0). thus, t(a, a0) = t. � 166 d. n. georgiou theorem 2.11. let y be a regular locally compact space, a the family of all tispaces, i = 0, 1, 2, 3, 3 1 2 , a0 an arbitrary family of spaces containing subspaces of spaces of a, ctco (y, z) ∈ a0, and z ∈ a. then, we have t(a, a0) = tco = tis. proof. since y is a regular locally compact space, the compact open topology coincides with the isbell topology on c(y, z) and it is admissible. hence, tco is (a, a0)-admissible. also, the topology tco is splitting and, therefore, tco is (a, a0)-splitting. since z ∈ a, we have that ctco (y, z) ∈ a (see preliminaries) and, therefore, (ctco (y, z), ctco (y, z)) ∈ (a, a0). thus, by corollary 2.10 we have that t(a, a0) = tco. � theorem 2.12. let y be a regular locally compact space, a the family of all topological spaces whose weight is not greater than a certain fixed infinite cardinal, a0 an arbitrary family of spaces containing subspaces of spaces of a, ctco (y, z) ∈ a0, and y, z ∈ a. then, we have t(a, a0) = tco = tis. proof. the proof of this theorem is similar to the proof of theorem 2.11 and follows by corollary 2.10 and theorem 3.4.16 of [6]. � theorem 2.13. let y be a regular second-countable locally compact space, a the family of all metrizable spaces, a0 an arbitrary family of spaces containing subspaces of spaces of a, ctco (y, z) ∈ a0, and z ∈ a. then, we have t(a, a0) = tco = tis. proof. the proof of this theorem is similar to the proof of theorem 2.11 and follows by corollary 2.10 and exercices 4.2.h and 3.4.e(c) of [6]. � theorem 2.14. let y be a regular locally compact lindelöf space, a the family of all completely metrizable spaces, a0 an arbitrary family of spaces containing subspaces of spaces of a, ctco (y, z) ∈ a0, and z ∈ a. then, we have t(a, a0) = tco = tis. proof. the proof of this theorem is similar to the proof of theorem 2.11 and follows by corollary 2.10 and exercice 4.3.f(a) of [6]. � theorem 2.15. let y be a corecompact space, a the family of all ti-spaces, where i = 0, 1, 2, a0 an arbitrary family of spaces containing subspaces of spaces of a, ctis (y, z) ∈ a0, and z ∈ a. then, we have t(a, a0) = tis. proof. since y is corecompact, the isbell topology tis on c(y, z) is admissible. hence the topology tis is (a, a0)-admissible. also, the topology tis is splitting and, therefore, tis is (a, a0)-splitting. since z ∈ a, we have that ctis (y, z) ∈ a (see preliminaries) and, therefore, (ctis (y, z), ctis (y, z)) ∈ (a, a0). thus, by corollary 2.10 we have that t(a, a0) = tis. � theorem 2.16. let y be a corecompact space, a the family of all secondcountable spaces, a0 an arbitrary family of spaces containing subspaces of spaces of a, ctis (y, z) ∈ a0, and y, z ∈ a. then, we have t(a, a0) = tis. topologies on function spaces 167 proof. the proof of this theorem is similar to the proof of theorem 2.15 and follows by corollary 2.10 and the fact that ctis (y, z) ∈ a (see [12]). � 3. on dual topologies note 3. let y and z be two fixed topological spaces. by oz (y ) we denote the set {f −1(u ) : f ∈ c(y, z) and u ∈ o(z)}. let ih ⊆ oz (y ), h ⊆ c(y, z), and u ∈ o(z). we set (ih, u ) = {f ∈ c(y, z) : f −1(u ) ∈ ih} and (h, u ) = {f −1(u ) : f ∈ h}. definition 3.1. (see [9]) let τ be a topology on oz (y ). the topology on c(y, z), for which the set {(ih, u ) : ih ∈ τ, u ∈ o(z)} is a subbasis, is called dual to τ and is denoted by t(τ ). now, let t be a topology on c(y, z). the topology on oz (y ), for which the set {(h, u ) : h ∈ t, u ∈ o(z)} is a subbasis, is called dual to t and is denoted by τ (t). we observe that if τ is a topology on oz (y ) and σ a subbasis for τ , then the set {(ih, u ) : ih ∈ σ, u ∈ o(z)} is a subbasis for t(τ ) (see lemma 2.5 in [9]). also, if t is a topology on c(y, z) and s a subbasis for t, then the set {(h, u ) : h ∈ s, u ∈ o(z)} is a subbasis for τ (t) (see lemma 2.6 in [9]). note 4. let x be a space and g : x ×y → z a continuous map. if gx : y → z is the map for which gx(y) = g(x, y), for every y ∈ y , then by g we denote the map of x × o(z) into oz (y ), for which g(x, u ) = g −1 x (u ) for every x ∈ x and u ∈ o(z). now, let h : x → c(y, z) be a map. by h we denote the map of x × o(z) into oz (y ), for which h(x, u ) = (h(x)) −1(u ) for every x ∈ x and u ∈ o(z). definition 3.2. let τ be a topology on oz (y ). we say that a map m : x × o(z) → oz (y ) is continuous with respect to the first variable if for every fixed element u of o(z), the map mu : x → (oz (y ), τ ), for which mu (x) = m (x, u ) for every x ∈ x, is continuous. definition 3.3. a topology τ on oz (y ) is called (a, a0)-splitting if for every (x, x0) ∈ (a, a0) the continuity of a map g : x × y → z implies the continuity with respect to the first variable of the map g|x0×o(z) : x0 × o(z) → (oz (y ), τ ). a topology τ on oz (y ) is called (a, a0)-admissible if for every (x, x0) ∈ (a, a0) and for every map h : x → c(y, z) the continuity with respect to the first variable of the map h : x × o(z) → (oz (y ), τ ) implies the continuity of 168 d. n. georgiou the map h⋄|x0×y : x0 × y → z defined by h ⋄|x0×y (x, y) = h(x)(y), (x, y) ∈ x0 × y . theorem 3.4. a topology τ on oz (y ) is (a, a0)-splitting if and only if the topology t(τ ) on c(y, z) is (a, a0)-splitting. proof. suppose that the topology τ on oz (y ) is (a, a0)-splitting, that is for every pair (x, x0) ∈ (a, a0) the continuity of a map g : x × y → z implies the continuity with respect to the first variable of the map g|x0×o(z) : x0 × o(z) → (oz (y ), τ ). we prove that the topology t(τ ) on c(y, z) is (a, a0)-splitting. let (x, x0) ∈ (a, a0) and g : x × y → z be a continuous map. we need to prove that g∗|x0 : x0 → ct(τ )(y, z) is a continuous map. let x ∈ x0 and (ih, u ) be an open neighborhood of (g ∗|x0 )(x) in ct(τ )(y, z). we must find an open neighborhood v of x in x0 such that (g ∗|x0 )(v ) ⊆ (ih, u ). we have that ((g∗|x0 )(x)) −1(u ) ∈ ih. since (g∗|x0 )(x) = gx, we have g−1x (u ) ∈ ih, that is, g(x, u ) ∈ ih. since the map g|x0×o(z) : x0 × o(z) → (oz (y ), τ ). is continuous with respect to the first variable, the map (g|x0×o(z))u : x0 → (oz (y ), τ ) is continuous. also, (g|x0×o(z))u (x) ∈ ih. thus, there exists an open neighborhood v of x in x0 such that (g|x0×o(z))u (v ) ⊆ ih. let x′ ∈ v . then, (g|x0×o(z))u (x ′) ∈ ih, that is, g−1 x′ (u ) ∈ ih or (g∗|x0 )(x ′) ∈ (ih, u ). thus, (g∗|x0 )(v ) ⊆ (ih, u ), which means that the map g∗|x0 is continuous. conversely, suppose that t(τ ) is (a, a0)-splitting. we prove that τ is (a, a0)splitting. let (x, x0) be an element of (a, a0) and g : x ×y → z a continuous map. it is sufficient to prove that g|x0×o(z) : x0 × o(z) → (oz (y ), τ ) is continuous with respect to the first variable. let u be a fixed element of o(z). consider the map (g|x0×o(z))u : x0 → (oz (y ), τ ). let x ∈ x0, ih ∈ τ , and (g|x0×o(z))u (x) = g −1 x (u ) ∈ ih. we need to find an open neighborhood v of x in x0 such that (g|x0×o(z))u (v ) ⊆ ih. consider the open set (ih, u ) of the space ct(τ )(y, z). since (g|x0×o(z))u (x) = g −1 x (u ) ∈ ih, we have gx ∈ (ih, u ). since t(τ ) is (a, a0)-splitting, the map g ∗|x0 : x0 → ct(τ )(y, z) is continuous. hence, there exists an open neighborhood v of x in x0 such that (g ∗|x0 )(v ) ⊆ (ih, u ). let x′ ∈ v . then, (g∗|x0 )(x ′) = gx′ ∈ (ih, u ), that is, g −1 x′ (u ) ∈ ih or (g|x0×o(z))u (x ′) ∈ ih. thus, (g|x0×o(z))u (v ) ⊆ ih, which means that the map (g|x0×o(z))u is continuous. � theorem 3.5. a topology t on c(y, z) is (a, a0)-splitting if and only if the topology τ (t) on oz (y ) is (a, a0)-splitting. proof. the proof of this theorem is similar to the proof of theorem 3.4. � topologies on function spaces 169 example 3.6. (1) the topologies τ (tco) and τ (tis) are (a, a0)-splitting for every pair (a, a0). this follows by the fact that the topologies tco and tis are splitting and, therefore, (a, a0)-splitting. (2) let z be the sierpinski space, ω(y ) the scott topology, and ωz (y ) the relative topology of ω(y ) on oz (y ). then, the topology t(ωz (y )) coincides with the isbell topology on c(y, z). hence, the topology t(ωz (y )) is splitting and, therefore, (a, a0)-splitting. thus, the topology τ (t(ωz (y ))) on oz (y ) is (a, a0)-splitting. theorem 3.7. a topology τ on oz (y ) is (a, a0)-admissible if and only if the topology t(τ ) on c(y, z) is (a, a0)-admissible. proof. suppose that the topology τ on oz (y ) is (a, a0)-admissible, that is for every space (x, x0) ∈ (a, a0) and for every map h : x → c(y, z) the continuity with respect to the first variable of the map h : x × o(z) → (oz (y ), τ ) implies the continuity of the map h ⋄|x0×y : x0 ×y → z. we prove that t(τ ) is (a, a0)-admissible. let (x, x0) ∈ (a, a0) and h : x → ct(τ )(y, z) be a continuous map. it is sufficient to prove that the map h⋄|x0×y : x0 ×y → z is continuous. clearly, it suffices to prove that the map h : x × o(z) → (oz (y ), τ ) is continuous with respect to the first variable. let x ∈ x, u ∈ o(z) and ih ∈ τ such that hu (x) = h(x, u ) = (h(x)) −1(u ) ∈ ih. we prove that there exists an open neighborhood v of x in x such that hu (v ) ⊆ ih. consider the open set (ih, u ) of the space ct(τ )(y, z). then, h(x) ∈ (ih, u ). since the map h : x → ct(τ )(y, z) is continuous, there exists an open neighborhood v of x in x such that h(v ) ⊆ (ih, u ). let x′ ∈ v . then h(x′) ∈ (ih, u ), that is (h(x′))−1(u ) ∈ ih or hu (x ′) = h(x′, u ) ∈ ih. thus, hu (v ) ⊆ ih, which means that hu is continuous. conversely, suppose that the topology t(τ ) is (a, a0)-admissible. we prove that the topology τ is (a, a0)-admissible. let (x, x0) be a pair of (a, a0) and h : x → c(y, z) a map such that h : x × o(z) → (oz (y ), τ ) is continuous with respect to the first variable. we need to prove that the map h⋄|x0×y : x0 × y → z is continuous. since t(τ ) is (a, a0)-admissible, it is sufficient to prove that the map h : x → ct(τ )(y, z) is continuous. let x ∈ x, u ∈ o(z), and ih ∈ τ such that h(x) ∈ (ih, u ). then, (h(x))−1(u ) ∈ ih. since the map hu : x → (oz (y ), τ ) is continuous, there exists an open neighborhood v of x in x such that hu (v ) ⊆ ih. let x′ ∈ v . then, hu (x ′) = (h(x′))−1(u ) ∈ ih or h(x′) ∈ (ih, u ). thus, h(v ) ⊆ (ih, u ), which means that the map h is continuous. � theorem 3.8. a topology t on c(y, z) is (a, a0)-admissible if and only if the topology τ (t) on oz (y ) is (a, a0)-admissible. proof. the proof of this theorem is similar to the proof of theorem 3.7. � 170 d. n. georgiou example 3.9. (1) if y is a regular locally compact space, then the topology τ (tco) is (a, a0)-admissible for every pair (a, a0). (2) if y is a corecompact space, then the topology τ (tis) is (a, a0)-admissible for every pair (a, a0). (3) if y is a locally bounded space, then the topology τ (tsis) is (a, a0)admissible for every pair (a, a0). (4) let ω(y ) be the scott topology on o(y ). by ωz (y ) we denote the relative topology of ω(y ) on ωz (y ). if y is corecompact, then the topology ωz (y ) is admissible (see corollary 3.12 of [9]) and, therefore, it is (a, a0)-admissible. thus, the topology t(ωz (y )) on c(y, z) is (a, a0)-admissible. theorem 3.10. let a and a0 are arbitrary families of spaces such that every element x0 ∈ a0 is a subspace of an element x ∈ a. then in the set oz (y ) there exists the greatest (a, a0)-splitting topology. proof. let {τi : i ∈ i} be the set of all (a, a0)-splitting topologies on oz (y ). we consider the topology τ = ∨{τi : i ∈ i}. it is not difficult to prove that this topology is (a, a0)-splitting. by this fact we have that this topology is the required greatest (a, a0)-splitting topology. � references [1] r. arens, a topology of spaces of transformations, annals of math. 47 (1946), 480–495. [2] r. arens and j. dugundji, topologies for function spaces, pacific j. math. 1 (1951), 5–31. [3] j. dugundji, topology, allyn and bacon, inc. boston 1968. [4] g. di maio, l. holá, d. holý and r. mccoy, topologies on the set space of continuous functions, topology appl. 86 (1998), no. 2, 105–122. [5] g. di maio, e. meccariello and s. naimpally, hyper-continuous convergence in function spaces, quest. answers gen. topology 22 (2004), no. 2, 157–162. [6] r. engelking, general topology, warszawa 1977. [7] r. h. fox, on topologies for function spaces, bull. amer. math. soc. 51 (1945), 429-432. [8] d. n. georgiou, s. d. iliadis and b. k. papadopoulos, topologies on function spaces, studies in topology vii, zap. nauchn. sem. s.-peterburg otdel. mat. inst. steklov (pomi) 208(1992), 82-97 (russian). translated in: j. math. sci., new york 81, (1996), no. 2, 2506–2514. [9] d. n. georgiou, s. d. iliadis and b. k. papadopoulos, on dual topologies, topology appl. 140 (2004), 57–68. [10] d. n. georgiou, s.d. iliadis and f. mynard, function space topologies, open problems in topology 2 (elsevier), 15–23, 2007. [11] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. mislove and d.s. scott, a compendium of continuous lattices, springer, berlin-heidelberg-new york 1980. [12] p. lambrinos and b. k. papadopoulos, the (strong) isbell topology and (weakly) continuous lattices, continuous lattices and applications, lecture notes in pure and appl. math. no. 101, marcel dekker, new york 1984, 191–211. [13] r. mccoy and i. ntantu, topological properties of spaces of continuous functions, lecture notes in mathematics, 1315, springer verlang. topologies on function spaces 171 [14] f. schwarz and s. weck, scott topology, isbell topology, and continuous convergence, lecture notes in pure and appl. math. no.101, marcel dekker, new york 1984, 251-271. received february 2009 accepted march 2009 d. n. georgiou (georgiou@math.upatras.gr) department of mathematics, university of patras, 265 04 patras, greece @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 201–209 multivalued function spaces and atsuji spaces som naimpally dedicated to professor john g. hocking, a great teacher and researcher who initiated me into the beauty and excitement of mathematical research. abstract. in this paper we present two themes. the first one describes a transparent treatment of some of the recent results in graph topologies on multi-valued functions. the study includes vietoris topology, fell topology, fell uniform topology on compacta and uniform topology on compacta. the second theme concerns when continuity is equivalent to proximal continuity or uniform continuity. 2000 ams classification: 54c35, 54b20, 54c05, 54e05, 54e15, 54c60. keywords: graph topology, multi-valued functions, vietoris topology, fell topology, uniform topology, fell uniform topology, atsuji spaces, uniform continuity, proximal continuity. 1. graph topologies on multifunctions. suppose (x,t1) and (y,t2) are hausdorff spaces. we set z = x ×y and assign the product topology t = t1×t2. 2z denotes the family of closed subsets of z and can be considered as a space f of all set valued maps on x to 2y taking points of x to (possibly empty) closed subsets of y. we don’t distinguish between a function f ∈f and its graph {(x,f(x) : x ∈ x}⊂ z = x×y. thus our study includes topologies on the spaces of partial maps. for each subset e of z and a compatible lo-proximity δ on z [21], set e− = {a ∈ 2z : a∩e 6= ∅} e+ = {a ∈ 2z : a ⊂ e} e++ = {a ∈ 2z : a � e} w. r. t. δ (note: a � e iff aδec where δ denotes the negation of δ) cl(z) denotes the family of all nonempty closed subsets of z. k(z) denotes the family of all nonempty compact subsets of z. 202 s. naimpally 1.1. the vietoris and fell topologies on cl(z). the vietoris topology τ(v ) is generated by {e+ : ec ∈ cl(z)}∪{e− : e ∈ t}. the proximal topology σ(δ) is generated by {e++ : ec ∈ cl(z)}∪{e− : e ∈ t}. the fell topology τ(f) is generated by {e+ : ec ∈ k(z)}∪{e− : e ∈ t}, and if δ is ef or r, then τ(f) is also generated by {e++ : ec ∈ k(z)}∪ {e− : e ∈ t}. thus when δ is ef or r, the fell topology equals the proximal fell topology and this explains the reason for several beautiful results for this topology ! the paper [5] deals with only metric proximities, and [10] remains unpublished. it is not widely known that proximal hypertopologies can be studied in more general situations and not merely in metric spaces as one usually finds in the literature. (however, see the recent papers [7, 8]). we note that the vietoris topology is itself a proximal topology i. e. τ(v ) = σ(δ0), where the fine lo-proximity δ0 is given by aδ0b iff cla∩ clb 6= ∅. the well known urysohn lemma says that δ0 is ef iff z is normal. when we wish to refer to hypertopologies on cl(y ), we use the suffix 2 e. g. t2(v ) denotes the vietoris topology on cl(y ) t2(f) denotes the fell topology on cl(y ) σ2(δ2) denotes the proximal topology w. r. t. δ2 on cl(y ) etc. 1.2. weak topologies on cl(z). for each of the topologies described above, we also have an associated • weak topology wherein cl(z) (respectively k(z)) is replaced by cl(x)×cl(y ) ( respectively k(x)×k(y )) (see [22]) and we attach the letter “w”. • the weak vietoris topology τ(wv ) is generated by {e+ : ec ∈ cl(x) ×cl(y )}∪{e− : e ∈ t}. • the weak fell topology τ(wf) is generated by {e+ : ec ∈ k(x) ×k(y )}∪{e− : e ∈ t}. for the family f it can be proven easily that τ(wf) = τ(f). combining this result with the fact that when the proximity is ef or r, τ(f) = σ(f) we have τ(wf) = τ(f) = σ(wf) = σ(f). 1.3. uniform topologies. (a) suppose y has a compatible uniformity v and vh denotes the corresponding hausdorff uniformity (also called bourbaki-hausdorff uniformity) on cl(y ). a typical basic open set in the uniform topology on compacta, τ(ucc,vh) on f is of the form < f,a,m >= {g ∈f : for all x ∈ a, (f(x),g(x)) ∈ m}, where f ∈f, a ∈ k(x) and m ∈vh. multivalued function spaces and atsuji spaces 203 (b) in [12] and [16] function space topologies akin to uniform topologies were studied. the range space was not necessarily uniformizable. here we introduce a similar concept. suppose cl(y ) is assigned some hypertopology τ2. suppose w is a symmetric nbhd. of the diagonal in (cl(y ) ×cl(y ),τ2 × τ2). for each f ∈f and a ∈k(x) we set w∗(f,a) = {g ∈f : for all x ∈ a, (f(x),g(x)) ∈ w}. the topology on f generated by {w∗(f,a) :f ∈f,a ∈k(x) and w a symmetric τ2 × τ2 nbhd. of the diagonal in cl(y ) ×cl(y )} is called the τ2-uniform topology on compacta τ(ucc,τ2). in case (cl(y ),τ2) is uniformizable and we restrict w ’s to symmetric entourages, we do get a uniform topology. this is true as in (a) above or in the case of a locally compact space y with the fell topology τ2(f) on cl(y ). in this case (2y ,τ2(f)) is compact hausdorff and has a unique compatible uniformity uf . mccoy calls τ(ucc,τ2(f)) = τ(ucc,uf ) “the fell uniform topology (on compact sets)”. we use u to denote the restriction of uf to y. theorem 1.1 (cf. [15]). suppose x and y are hausdorff. then (a) τ(f) ⊂ τ(wv ) ⊂ τ(v ). if further, y is locally compact then, (b) τ(f) ⊂ τ(ucc,uf ) ⊂ τ(ucc,vh). those interested in more details are referred to [3] for hypertopologies, [21] for proximities, [17] and [19] for function space topologies. 2. embedding theorems and applications. one of the most valuable results in function space topologies for single valued functions is the embedding of the range space in the function space via the constant functions. (cf. theorem 2.1.1, page 15 in [17]) in this section we prove similar results for multifunctions which are of fundamental importance in our work. theorem 2.1. suppose c = {x×e : e ∈ cl(y )} is the family of all constant multifunctions. the map j : 2y → 2z defined by j(e) = (x × e) ∈ c, is a bijection and it is easy to show that the following are embeddings: (a) j : (2y ,τ2(f)) → (2z,τ(f)). (b) j : (2y ,τ2(v )) → (2z,τ(wv )). (c) j : (2y ,uf ) → (2z,τ(ucc,uf )). (d) j : (2y ,vh) → (2z,τ(ucc,vh)). theorems 1.1 and 2.1 give us the following in which vertical lines show embeddings: 204 s. naimpally theorem 2.2. (a) if x and y are hausdorff, then cl(y ) : τ2(f) ⊂ τ2(v ) j c ⊆ f : τ(f) ⊂ τ(wv ) ⊂ τ(v ) (b) if x is hausdorff and y is locally compact hausdorff, then cl(y ) : τ2(f) = τ2(uf ) ⊂ τ2(vh) j c ⊂f : τ(f) ⊂ τ(ucc,uf ) ⊂ τ(ucc,vh) now we show how 2.2 enables us to prove some of mccoy’s results in a simple manner without any further work. theorem 2.3 (cf. [15], proposition 4.5). τ(ucc,vh) ⊂ τ(ucc,uf ) if and only if y is compact. moreover, u = v. (we note that mccoy assumes that x is also locally compact and y is completely metrizable.) proof. the result follows from 2.2(b) and the known facts that τ2(f) = τ2(vh) iff y is compact and a compact hausdorff space has a unique compatible uniformity. � mccoy shows that τ(f) = τ(ucc,uf ) if and only if x is discrete. ([15, proposition 4.6]) we have not yet found a “transparent” proof. theorem 2.4 (cf. [15], proposition 4.9). τ(ucc,vh) ⊂ τ(v ) if and only if x is discrete and (y,v) is totally bounded. proof. the result follows from 2.2 and the known fact that τ2(vh) ⊂ τ2(v ) iff (y,v) is totally bounded. (we note that compactness of y follows since mccoy assumes that x is completely metrizable.) � theorem 2.5 (cf. [15], proposition 4.10). τ(v ) ⊂ τ(ucc,vh) if and only if x is compact and (y,v) is atsuji. proof. we note that on c(x,y ), the space of single-valued continuous functions, τ(v ) equals the graph topology ([20]) and is finer than τ(ucc,vh) which equals the compact-open topology. they are equal iff x is compact ([18]). the result then follows from the above result and the known fact : τ2(v ) ⊂ τ2(vh) iff (y,v) is atsuji. (we note that a locally compact metric space is atsuji iff it is a topological sum of a compact space and a discrete space) � theorem 2.6 (cf. [15], proposition 4.12). τ(v ) ⊂ τ(ucc,uf ) if and only if x and y are compact. proof. from 2.2 it follows that on c, τ(wv ) ⊂ τ(ucc,uf ) ⇔ τ2(v ) ⊂ τ2(f) ⇔ y is compact. the result then follows from 2.5. � multivalued function spaces and atsuji spaces 205 3. atsuji spaces. in the first course of analysis we learn that a continuous function from a compact metric space to an arbitrary metric space is uniformly continuous. example of a non-compact domain such as an infinite discrete space, shows that equivalence of continuity and uniform continuity does not characterize compactness. there are analogous results in uniform and proximity spaces. there is a considerable literature dealing separately with the three cases of atsuji spaces i. e. metric, uniform and proximity spaces in which continuity is equivalent to uniform or proximal continuity. here we tackle the three cases together, compare them and describe the results briefly. (for details see [9]) from now onwards we suppose that (x,t1) and (y,t2) are tychonoff spaces. depending on the situation, the topology t1 is induced by a metric d, or a uniformity u, or an ef-proximity δ. we also suppose that y has a compatible uniformity and an ef-proximity. we use the notation: c(x,y ) is the set of all continuous functions from x to y. u(x,y ) is the set of all uniformly continuous functions from x to y. p(x,y ) is the set of all proximally continuous functions from x to y. when y = r, we use the standard notations c(x),c∗(x),p(x),p∗(x), etc. x′ denotes the set of all limit points of x. 4. proximity. suppose (an) and (bn) are sequences of sets in an ef-proximity space (x,δ). (a) (an) is proximally discretely separated by (bn) iff for each n ∈ n, an � bn w. r. t. δ and (bn) is discrete. (b) in (a) if δ = δf then we use the term “(an) is discretely normally separated by (bn)”. (c) (an) is functionally (respectively proximally) discrete iff there is an f ∈ c(x) (respectively f ∈ p(x)) such that for each n ∈ n, f(an) = n. (d) a sequence (xn) is pseudo-cauchy iff for each m ∈ n, there are disjoint sets a,b of n beyond m such that {xn : n ∈ a}δ{xn : n ∈ b}. lemma 4.1. if (an) is discretely normally separated by an open family (bn), then (an) is functionally separated by (bn) i. e. there is an f ∈ c(x) such that f(an) = n and f(x −∪bn) = 0. lemma 4.2. x is normal if and only if every discrete sequence of subsets of x is functionally discrete. lemma 4.3. every proximally discrete sequence of sets (an) in (x,δ1) is uniformly discrete w. r. t. some compatible uniformity u i. e. there exists a u ∈u such that u(an) ∩am = ∅ 206 s. naimpally for m 6= n. we now state the main result in atsuji spaces : theorem 4.4. for an ef-proximity space (x,δ) the following are equivalent: (a) c(x,y ) = p(x,y ) for each ef-proximity space y. (b) c(x) = p(x). (c) if (an) is discretely normally separated by an open family (bn), then (an) is proximally separated by (bn) i. e. there is an f ∈ p(x) such that f(an) = n and f(x −∪bn) = 0. (d) each functionally discrete sequence of sets is proximally discrete. (e) each functionally discrete sequence of sets is uniformly discrete w. r. t. some compatible uniformity. (f) for each pair of disjoint zero sets a, b there exists an f ∈ p∗(x) such that f(a) = 0 and f(b) = 1. (g) each pair of disjoint zero sets are far w. r. t. δ. (h) δ = δf . (i) c∗(x) = p∗(x). 4.1. normal space. it is not widely known that many results in normal spaces are true in tychonoff spaces with compatible ef-proximities. urysohn’s lemma of normal spaces is available in an ef-proximity space if we replace disjoint closed sets by sets that are far ! previous authors have unnecessarily assumed normality which we have shown above is not needed. now we show how the earlier results follow from ours given above. theorem 4.5. for a normal proximity space (x,d) the following are equivalent: (a) c(x,y ) = p(x,y ) for each ef-proximity space y. (b) c(x) = p(x). (c) if (bn) is a sequence of pairwise disjoint open sets, clan ⊂ bn for each n, and ∪clan is closed, then (an) is proximally separated by (bn). (d) each discrete sequence of sets is proximally discrete. (e) each discrete sequence of sets is uniformly discrete w. r. t. some compatible uniformity. (f) for each pair of disjoint closed sets a, b there exists an f ∈ p∗(x) such that f(a) = 0 and f(b) = 1. (g) each pair of disjoint closed sets are far w. r. t. δ. (h) δ = δ0. (aδ0b iff cla∩ clb 6= ∅) (i) c∗(x) = p∗(x). the space of all ordinals less than the first uncountable ordinal shows that even with normality, x′ need not be compact when c(x) = p(x) for every compatible ef-proximity on x. this space has a unique compatible uniformity or ef-proximity. 4.2. uniformity. uniform case is a bit tricky since c(x) = u(x) does not imply that c(x,y ) = u(x,y ) multivalued function spaces and atsuji spaces 207 for each uniform space y and c∗(x) = p∗(x) does not imply c(x) = u(x)! so we have three cases to consider: (1) the strong atsuji viz. c(x,y ) = u(x,y ) for each uniform space y, (2) the atsuji viz. c(x) = u(x), (3) the cech-atsuji viz. c∗(x) = p∗(x). the strong atsuji case has a simple solution : u must be the fine uniformity. also the cech-atsuji case is equivalent to δ = δf and this is a proximal property. so the only non-trivial case to study is the atsuji one. ([2]). theorem 4.6. for a uniform space (x,u) the following are equivalent: (a) c(x) = u(x), (b) if (an) is discretely normally separated by (bn), then (an) is uniformly separated by (bn) i. e. there is u ∈u such that u(an) ⊂ bn for each n ∈ n. (c) every functionally discrete sequence of sets is uniformly discrete. theorem 4.7. for a uniform normal space (x,u) the following are equivalent: (a) c(x) = u(x), (b) if clan ⊂ intbn for each n ∈ n, (bn) is pairwise disjoint and ∪clan is closed, then there is a u ∈u such that u(an) ⊂ bn for each n ∈ n. (c) every discrete sequence of sets is uniformly discrete. we now consider several statements which are analogues in the uniform case of the well known equivalent statements in the metric case. theorem 4.8. consider the following statements concerning a uniform space (x,u). (a) u is fine. (b) c(x,y ) = u(x,y ) for each uniform space y. (c) c(x) = u(x), (d) every functionally discrete sequence of sets is uniformly discrete. (e) each pair of disjoint zero sets can be separated by a uniformly continuous function. (f) each pair of disjoint zero sets can be separated by an entourage. (g) δ = δf . (h) c∗(x) = u∗(x). then (a) ⇔ (b) ⇒ (c) ⇔ (d) ⇒ (e) ⇔ (f) ⇔ (g) ⇔ (h) and none of the arrows can be reversed. theorem 4.9. consider the following statements concerning a normal uniform space (x,u). (a) x′ is compact and for each entourage u ∈ u, there is an entourage v ∈u such that u(x)c is v -discrete. (b) u is a lebesgue uniformity. (c) u is fine. 208 s. naimpally (d) c(x,y ) = u(x,y ) for each uniform space y. (e) c(x) = u(x), (f) every discrete sequence of sets is uniformly discrete. (g) c∗(x) = u∗(x), (h) δ = δ0. (i) disjoint closed sets are uniformly separated. (j) τ(v ) ⊂ τ(uh) on cl(x). (k) every pseudo-cauchy sequence of distinct points has a cluster point. then (a) ⇒ (b) ⇒ (c) ⇔ (d) ⇒ (e) ⇔ (f) ⇒ (g) ⇔ (h) ⇔ (i) ⇔ (j) ⇒ (k). (j) 6= (k) is open and none of the arrows can be reversed. 4.3. metric space. finally we put all our results together and get several known characterizations of metric atsuji spaces. theorem 4.10. suppose (x,d) is a metric space, u is the metric uniformity and δ is the metric proximity. the following are equivalent: (a) x′ is compact and for each ε > 0, there is a η > 0 such that [s(x,ε)]c is ηdiscrete. (b) every open cover of x is lebesgue. (c) u is fine. (d) c(x,y ) = u(x,y ) for each uniform space y. (e) c(x) = u(x), (f) c∗(x) = u∗(x), (g) δ = δ0. (h) disjoint closed sets are at a positive distance apart. (i) τ(v ) ⊂ τ(uh) on cl(x). (j) every pseudo-cauchy sequence of distinct points has a cluster point. acknowledgements. i am most grateful to peppe (giuseppe di maio) for his friendship and collaboration since 1984 and for arranging this conference. i also thank anna di concilio for initiating my italian visits and inviting me even after retirement. i value my friendship with enrico meccariello. the first part of this talk is based upon recent work with peppe and enrico; the second part was done quite sometime back with anna. references [1] m. atsuji uniform continuity of continuous functions of metric spaces, pacific j. math. 8 (1958), 11–16. [2] m. atsuji, uniform continuity of continuous functions of uniform spaces, canadian j. math. 13 (1961), 657–663. [3] g. beer, topologies on closed and closed convex sets, kluwer publ., holland (1993). [4] g. beer, metric spaces in which continuous functions are uniformly continuous and hausdorff distance, proc. amer. math. soc. 95 (1985), 653–658. multivalued function spaces and atsuji spaces 209 [5] g. beer, a. lechicki, s. levi and s. naimpally, distance functionals and suprema of hyperspace topologies, ann. mat. pura appl. 162 (1992), 715–726. [6] n. bourbaki, general topology, part 2, addison-wesley publishing company, reading, ma, (1966). [7] d. di caprio and e. meccariello, notes on separation axioms in hyperspaces, q & a in general topology 18 (2000), 65–86. [8] d. di caprio and e. meccariello, g-uniformities lr-uniformities and hypertopologies, acta math. hungarica 88 (2000), 73–93. [9] a. di concilio and s. naimpally, atsuji spaces: continuity versus uniform continuity, sixth brazilian topology meeting, campinas, brazil (1988) (unpublished). [10] a. di concilio, s. naimpally and p. sharma, proximal hypertopologies, sixth brazilian topology meeting, campinas, brazil (1988) (unpublished). [11] j. d. hansard, function space topologies, pacific j. math. 35 (1970), 381–388. [12] a. irudayanathan, cover-close topologies for function spaces, gen. top. and appl. 10 (1979), 275–282. [13] j. l. kelly, general topology, d. van nostrand company, princeton, nj (1960). [14] k. kuratowski, sur l’espaces des fonctions partielles, ann. di mat. pura ed appl. 40 (1955), 61–67. [15] r. a. mccoy, comparison of hyperspace and function space topologies, recent progress in function spaces, quaderni di matematica, vol. 3, editors: giuseppe di maio and lubica hola, seconda universita di napoli, aracne, (1998), 241–258. [16] r. a. mccoy, the open-cover topology for function spaces, fund. math. 104 (1979), 69–73. [17] r. a. mccoy and i. ntantu, topological properties of spaces of continuous functions, lecture notes in mathematics # 1315, springer-verlag, berlin (1988). [18] s. a. naimpally, graph topology for function spaces, trans. amer. math. soc. 123 (1966), 267–272. [19] s. a. naimpally, a brief survey of topologies on function spaces, recent progress in function spaces, quaderni di matematica, vol. 3,editors: giuseppe di maio and lubica hola, seconda universita di napoli, aracne, (1998), 259–283. [20] s. a. naimpally and c. m. pareek, graph topologies for function spaces ii, ann. soc. math. pol. series i 13 (1970), 222–231. [21] s. a. naimpally and b. d. warrack, proximity spaces, cambridge tract # 59 (1970). [22] h. poppe, uber graphentopologien fur abbildungsraume i, bull. acad. pol. sci. ser. sci. math. astron. phy. 15 (1967), 71–80. received september 2001 revised august 2002 s. naimpally professor emeritus, lakehead university, 96 dewson street, toronto, ontario, m6h 1h3 canada e-mail address : sudha@accglobal.net @ appl. gen. topol. 15, no. 1 (2014), 11-24doi:10.4995/agt.2014.2032 c© agt, upv, 2014 quadruple fixed point theorems for nonlinear contractions on partial metric spaces erdal karapınar∗,a and kenan tas b a department of mathematics, atilim university 06836, i̇ncek, ankara, turkey (ekarapinar@atilim.edu.tr, erdalkarapinar@yahoo.com) b çankaya university, department of mathematics and computer science, ankara, turkey (kenan@cankaya.edu.tr) abstract the notion of coupled fixed point was introduced by guo and laksmikantham [12]. later gnana bhaskar and lakshmikantham in [11] investigated the coupled fixed points in the setting of partially ordered set by defining the notion of mixed monotone property. very recently, the concept of tripled fixed point was introduced by berinde and borcut [7]. following this trend, karapınar[19] defined the quadruple fixed point. in this manuscript, quadruple fixed point is discussed and some new fixed point theorems are obtained on partial metric spaces. 2010 msc: 47h10; 54h25; 46j10; 46j15. keywords: fixed point theorems; nonlinear contraction; partial metric space; partially ordered set; quadruple fixed point. 1. introduction and preliminaries the existence of fixed points in partially ordered metric spaces was considered first by ran and reurings [37]. after this remarkable paper, several authors have studied such problems (see e.g. [32, 33, 34, 11, 29, 30, 45, 9, 8] ). the notion of coupled fixed point was introduced by guo and laksmikantham [12]. after the interesting paper of gnana bhaskar and lakshmikantham [11], many authors focused on coupled fixed point theory and proved several results (see e.g. [29, 30, 45, 9, 8, 18, 17]). ∗corresponding author received march 2012 – accepted september 2012 http://dx.doi.org/10.4995/agt.2014.2032 e. karapınar and k. tas we recall the basic definitions and results from which our quadruple fixed point is inspired. the triple (x, d, ≤) is called a partially ordered metric spaces if (x, ≤) is a partially ordered set and (x, d) is a metric space. further, if (x, d) is a complete metric space, then the triple (x, d, ≤) is called partially ordered complete metric spaces. definition 1.1 (see [11]). let (x, ≤) be a partially ordered set and f : x × x → x. we say that f has mixed monotone property if f(x, y) is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any x, y ∈ x, x1 ≤ x2 ⇒ f(x1, y) ≤ f(x2, y), for x1, x2 ∈ x, and y1 ≤ y2 ⇒ f(x, y2) ≤ f(x, y1), for y1, y2 ∈ x. definition 1.2 (see [11]). an element (x, y) ∈ x × x is said to be a couple fixed point of the mapping f : x × x → x if f(x, y) = x and f(y, x) = y. we endow the product space x × x with the following partial order: (1.1) (u, v) ≤ (x, y) ⇔ u ≤ x, y ≤ v; for all (x, y), (u, v) ∈ x × x. two results of bhaskar and lakshmikantham [11] can be unified as follows: theorem 1.3. let (x, ≤) be a partially ordered set endowed with a metric d on x such that (x, d) is a complete metric spaces. let f : x × x → x have the mixed monotone property on x. assume that there exists a k ∈ [0, 1) with (1.2) d(f(x, y), f(u, v)) ≤ k 2 [d(x, u) + d(y, v)] , for all u ≤ x, y ≤ v. suppose either f is continuous or x has the following properties: (i) if a non-decreasing sequence {xn} → x, then xn ≤ x, ∀n; (i) if a non-increasing sequence {yn} → y, then y ≤ yn, ∀n. if, in addition, there are x0, y0 ∈ x such that x0 ≤ f(x0, y0) and f(y0, x0) ≤ y0, then, there exists x, y ∈ x such that x = f(x, y) and y = f(y, x). we notice that theorem 1.3 was extended to class of cone metric spaces in [17]. inspired by definition 1.1, berinde and borcut [7] introduced the following definition: (1.3) (u, v, w) ≤ (x, y, z) if and only if x ≥ u, y ≤ v, z ≥ w, where (u, v, w), (x, y, z) ∈ x3. definition 1.4 (see [7]). let (x, ≤) be a partially ordered set and f : x × x × x → x. the mapping f is said to has the mixed monotone property if for any x, y, z ∈ x x1, x2 ∈ x, x1 ≤ x2 =⇒ f(x1, y, z) ≤ f(x2, y, z), c© agt, upv, 2014 appl. gen. topol. 15, no. 1 12 quadruple fixed point theorems y1, y2 ∈ x, y1 ≤ y2 =⇒ f(x, y1, z) ≥ f(x, y2, z), z1, z2 ∈ x, z1 ≤ z2 =⇒ f(x, y, z1) ≤ f(x, y, z2), the following is the main tripled fixed point result of berinde and borcut [7]. theorem 1.5. let (x, ≤) be partially ordered set and (x, d) be a complete metric space. let f : x × x × x → x be a continuous mapping having the mixed monotone property on x. assume that there exist constants a, b, c ∈ [0, 1) such that a + b + c < 1 for which (1.4) d(f(x, y, z), f(u, v, w)) ≤ ad(x, u) + bd(y, v) + cd(z, w) for all x ≥ u, y ≤ v, z ≥ w. if there exist x0, y0, z0 ∈ x such that x0 ≤ f(x0, y0, z0), y0 ≥ f(y0, x0, y0), z0 ≤ f(x0, y0, z0) then there exist x, y, z ∈ x such that f(x, y, z) = x and f(y, z, y) = x and f(z, y, x) = z the notion of metric space was introduced by maurice rené fréchet [10] in 1906. pseudometric space, quasimetric space, semimetric space, partial metric space are some examples of the generalizations of metric space. in this manuscript, we discuss partial metric space, introduced by matthews (see e.g. [31]). the concept of the metric space started to apply to computer science around 1970. by using baire metric, g. khan [16] modeled a parallel computation. it consists of a set computing via sending unending streams of information by using infinite sequences. hence, with this paper, reservoir of the theory of metric space started to be used in the branches of computer science, such as, domain theory and semantics. the handicap of this approaches is, in computer science, infinite sequence corresponding to unterminated programs. but, in computer science, unterminated program is bad. this un-solicited status solved by matthews with his suggestion of non-zero self distance in metric construction. in the last decade, on partial metric spaces remarkable number of papers were reported (see e.g. [1]-[6],[13]-[15],[24]-[28],[39]-[55]) a mapping p : x × x → [0, ∞) is called partial metric (see e.g.[31]) on a nonempty set x if the following conditions are satisfied: (pm1) p(x, y) = p(y, x) (symmetry) (pm2) if p(x, x) = p(x, y) = p(y, y) then x = y (equality) (pm3) p(x, x) ≤ p(x, y) (small self-distances) (pm4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z) (triangularity) the pair (x, p) is called a partial metric space (pms). additionally, a triple (x, p, ≤) is called a partially ordered partial metric space if (x, p) is a partial metric space and (x, ≤) is a partially ordered set. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 13 e. karapınar and k. tas for a partial metric p on x, the functions dp, dm : x × x → r + given by (1.5) dp(x, y) = 2p(x, y) − p(x, x) − p(y, y) and (1.6) dm(x, y) = max{p(x, y) − p(x, x), p(x, y) − p(y, y)} are (usual) metrics on x. it is clear that dp and dm are equivalent. moreover, (1.7) lim n→∞ dp(x, xn) = 0 ⇔ p(x, x) = lim n→∞ p(x, xn) = lim n,m→∞ p(xn, xm) each partial metric p on x generates a t0 topology τp on x with a base of the family of open p-balls {bp(x, ε) : x ∈ x, ε > 0}, where bp(x, ε) = {y ∈ x : p(x, y) < p(x, x) + ε} for all x ∈ x and ε > 0. example 1.6 (see e.g. [31, 24, 3]). consider x = [0, ∞) with p(x, y) = max{x, y}. then (x, p) is a partial metric space. it is clear that p is not a (usual) metric. note that in this case dm(x, y) = |x − y| and dp(x, y) = 1 2 |x − y|. example 1.7 (see [31]). let x = {[a, b] : a, b, ∈ r, a ≤ b} and define p([a, b], [c, d]) = max{b, d} − min{a, c}. then (x, p) is a partial metric spaces. example 1.8 (see [31]). let x := [0, 1] ∪ [2, 3] and define p : x × x → [0, ∞) by p(x, y) = { max{x, y} if {x, y} ∩ [2, 3] 6= ∅, |x − y| if {x, y} ⊂ [0, 1]. then (x, p) is a partial metric space. example 1.9 (see [31]). let s be a non-empty set. by sω, we denote the set of all infinite sequence x = {x0, x1, · · · } over s. for all such sequences x, y ∈ sω define ds(x, y) = 2 −k, where k is the largest number (possibly ∞) such that xi = yi for each i < k, that is, ds(x, y) = 2 − sup{n|∀i 0. remark 1.14. since dp and dm are equivalent, we can take dm instead of dp in the above lemma. karapınar [19] introduced the concept of quadruple fixed point and proved some quadruple fixed point theorems in partially ordered metric spaces (see also [20][23]). the aim of this paper is introduce the concept of quadruple fixed point and prove the related fixed point theorems in the context of partially ordered partial metric spaces. 2. quadruple fixed point theorems let (x, p, ≤) be a partially ordered partial metric spaces. we consider the following partial order on the product space x4 = x × x × x × x: (2.1) (u, v, r, t) ≤ (x, y, z, w) if and only if x ≥ u, y ≤ v, z ≥ r, t ≤ w where (u, v, r, t), (x, y, z, w) ∈ x4. regarding this partial order, we state the definition of the following mapping. definition 2.1. let (x, ≤) be partially ordered set and f : x4 → x. we say that f has the mixed monotone property if f(x, y, z, w) is monotone nondecreasing in x and z, and it is monotone non-increasing in y and w, that is, for any x, y, z, w ∈ x (2.2) x1, x2 ∈ x, x1 ≤ x2 ⇒ f(x1, y, z, w) ≤ f(x2, y, z, w), y1, y2 ∈ x, y1 ≤ y2 ⇒ f(x, y1, z, w) ≥ f(x, y2, z, w), z1, z2 ∈ x, z1 ≤ z2 ⇒ f(x, y, z1, w) ≤ f(x, y, z2, w), w1, w2 ∈ x, w1 ≤ w2 ⇒ f(x, y, z, w1) ≥ f(x, y, z, w2). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 15 e. karapınar and k. tas definition 2.2. an element (x, y, z) ∈ x4 is called a quadruple fixed point of f : x4 → x if (2.3) f(x, y, z, w) = x and f(y, z, w, x) = y and f(z, w, x, y) = z and f(w, x, y, z) = w for a metric space (x, d), the function ρ : x4 → [0, ∞), given by, ρ((x, y, z, w), (u, v, r, t)) := d(x, u) + d(y, v) + d(z, r) + d(w, t) is a metric space on x4, that is, (x4, ρ) is a metric induced by (x, d). the aim of this paper is to prove the following theorem. theorem 2.3. let (x, ≤) be partially ordered set and (x, p) be a complete partial metric space. let f : x4 → x be a mapping having the mixed monotone property on x. assume that there exists a constant k ∈ [0, 1) such that (2.4) p(f(x, y, z, w), f(u, v, r, t)) ≤ k 4 [p(x, u) + p(y, v) + p(z, r) + p(w, t)] for all x ≥ u, y ≤ v, z ≥ r, w ≤ t. suppose there exist x0, y0, z0, w0 ∈ x such that x0 ≤ f(x0, y0, z0, w0), y0 ≥ f(y0, z0, w0, x0), z0 ≤ f(z0, w0, x0, y0), w0 ≥ f(w0, x0, y0, z0). suppose either (a) f is continuous, or (b) x has the following property: (i) if {xn} is a non-decreasing sequence xn → x (respectively, zn → z), then xn ≤ x (respectively, zn ≤ z) for all n, (ii) if {yn} is a non-increasing sequence yn → y(respectively, wn → w), then yn ≥ y (respectively, wn ≥ w) for all n, then there exist x, y, z, w ∈ x such that f(x, y, z, w) = x, f(y, z, w, x) = y, f(z, w, x, y) = z, f(w, x, y, z) = w. proof. we construct a sequence {(xn, yn, zn, wn)} in the following way: set x1 = f(x0, y0, z0, w0) ≥ x0, y1 = f(y0, z0, w0, x0) ≤ y0, z1 = f(z0, w0, x0, y0) ≥ z0, w1 = f(w0, x0, y0, z0) ≤ w0, and by the mixed monotone property of f , for n ≥ 1, inductively we get (2.5) xn = f(xn−1, yn−1, zn−1, wn−1) ≥ xn−1 ≥ · · · ≥ x0, yn = f(yn−1, zn−1, wn−1, xn−1) ≤ yn−1 ≤ · · · ≤ y0, zn = f(zn−1, wn−1, xn−1, yn−1) ≥ zn−1 ≥ · · · ≥ z0, wn = f(wn−1, xn−1, yn−1, zn−1) ≤ wn−1 ≤ · · · ≤ w0, c© agt, upv, 2014 appl. gen. topol. 15, no. 1 16 quadruple fixed point theorems due to (2.4) and (2.5), we have (2.6) p(x1, x2) = p(f(x0, y0, z0, w0), f(x1, y1, z1, w1)) ≤ k 4 [p(x0, x1) + p(y0, y1) + p(z0, z1) + p(w0, w1)] (2.7) p(y1, y2) = p(f(y0, z0, w0, x0), f(y1, z1, w1, x1)) ≤ k 4 [p(y0, y1) + p(z0, z1) + p(w0, w1) + p(x0, x1)] (2.8) p(z1, z2) = p(f(z0, w0, x0, y0), f(z1, w1, x1, y1)) ≤ k 4 [p(z0, z1) + p(w0, w1) + p(x0, x1) + p(y0, y1)] (2.9) p(w1, w2) = p(f(w0, x0, y0, z0), f(w1, x1, y1, z1)) ≤ k 4 [p(w0, w1) + p(x0, x1) + p(y0, y1) + p(z0, z1)] regarding (2.4) together with (2.6),(2.7),(2.8) we have (2.10) p(x2, x3) = p(f(x1, y1, z1, w1), f(x2, y2, z2, w2)) ≤ k 4 [p(x1, x2) + p(y1, y2) + p(z1, z2) + p(w1, w2)] (2.11) p(y2, y3) = p(f(y1, z1, w1, x1), f(y2, z2, w2, x2)) ≤ k 4 [p(y1, y2) + p(z1, z2) + p(w1, w2) + p(x1, x2)] (2.12) p(z2, z3) = p(f(z1, w1, x1, y1), f(z2, w2, x2, y2)) ≤ k 4 [p(z1, z2) + p(w1, w2) + p(x1, x2) + p(y1, y2)] (2.13) p(w2, w3) = p(f(w1, x1, y2, z1), f(w2, x2, y2, z2)) ≤ k 4 [p(w1, w2) + p(x1, x2) + p(y1, y2) + p(z1, z2)] recursively we have (2.14) p(xn+1, xn+2) = p(f(xn, yn, zn, wn), f(xn+1, yn+1, zn+1, wn+1)) ≤ k 4 [p(xn, xn+1) + p(yn, yn+1) + p(zn, zn+1) + p(wn, wn+1)] (2.15) p(yn+1, yn+2) = p(f(yn, zn, wn, xn), f(yn+1, zn+1, wn+1), xn+1) ≤ k 4 [p(yn, yn+1) + p(zn, zn+1) + p(wn, wn+1) + p(xn, xn+1)] (2.16) p(zn+1, zn+2) = p(f(zn, wn, xn, yn), f(zn+1, wn+1, xn+1, yn+1)) ≤ k 4 [p(zn, zn+1) + p(wn, wn+1) + p(xn, xn+1) + p(yn, yn+1)] (2.17) p(wn+1, wn+2) = p(f(wn, xn, yn, zn), f(wn+1, xn+1, yn+1, zn+1)) ≤ k 4 [p(wn, wn+1) + p(xn, xn+1) + p(yn, yn+1) + p(zn, zn+1)] c© agt, upv, 2014 appl. gen. topol. 15, no. 1 17 e. karapınar and k. tas for simplicity, we can use the matrix notation as follow. set m =     1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4     , dn =     p(xn+1, xn) p(yn+1, yn) p(zn+1, zn) p(wn+1, wn)     and r = ( 1 4 1 4 1 4 1 4 ) . notice that (2.18) rm = r and mn = m for all n ∈ n. so we have, (2.19) d1 ≤ kd0, (2.20) d2 ≤ kmd1 ≤ k 2 m 2 d0 = k 2 md0, and, inductively (2.21) dn ≤ kmdn−1 ≤ k n md0. (2.22) p(xn+1, xn+2) ≤ krdn     p(xn, xn+1) p(yn, yn+1) p(zn, zn+1) p(wn, wn+1)     hence, by (2.18),(2.4) and (2.5), we have (2.23) p(xn+1, xn+2) = p(f(xn, yn, zn, wn), f(xn+1, yn+1, zn+1, wn+1)) ≤ k 4 [p(xn, xn+1) + p(yn, yn+1) + p(zn, zn+1) + p(wn, wn+1)] ≤ krdn ≤ k n+1rmd0 ≤ k n+1rd0. we shall show the sequences {xn} are cauchy easily by using (2.14)-(2.21). without loss of generality, we may assume that m > n. by using (2.14)-(2.21) together with triangle inequality, we obtain that (2.24) p(xm, xn) ≤ p(xm, xm−1) + p(xm−1, xm−2) + · · · + p(xn+1, xn) ≤ km−1rd0 + · · · + k nrd0 ≤ kn(1 + · · · + km−n−1)rd0 ≤ kn 1 1−k rd0 letting n → ∞ in (2.24) and recalling that k ∈ [0, 1), we get that lim n→∞ p(xn, xm) = 0. by definition, dp(xn, xm) = 2p(xn, xm) − p(xn, xn) − p(xm, xm) ≤ 2p(xn, xm). thus, we have (2.25) lim n→∞ dp(xn, xm) = 0. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 18 quadruple fixed point theorems since (x, p) is a complete partial metric space, then by lemma 1.11, (x, dp) is a complete metric space. thus, {xn} converges in (x, dp), say x. again by 1.11, we have (2.26) p(x, x) = lim n→∞ p(xn, xm) = lim n→∞ p(xn, x) = 0. analogously, one can prove that {yn}, {zn} and {wn} are cauchy sequences. since (x, dp) is complete metric space, there exists x, y, z, w ∈ x such that (2.27) p(y, y) = lim n→∞ p(yn, ym) = lim n→∞ p(yn, y) = 0, p(z, z) = lim n→∞ p(zn, zm) = lim n→∞ p(zn, z) = 0, p(w, w) = lim n→∞ p(wn, wm) = lim n→∞ p(wn, w) = 0. suppose now the assumption (a) holds. then by (2.26) and (2.27), we have (2.28) x = lim n→∞ xn = lim n→∞ f(xn−1, yn−1, zn−1, wn−1) = f( lim n→∞ xn−1, lim n→∞ yn−1, lim n→∞ zn−1, lim n→∞ wn−1) = f(x, y, z, w) analogously, we also observe that (2.29) y = lim n→∞ yn = lim n→∞ f(xn−1, wn−1, zn−1, yn−1) = f(x, w, z, y) z = lim n→∞ zn = lim n→∞ f(zn−1, yn−1, xn−1, wn−1) = f(z, y, x, w) w = lim n→∞ wn = lim n→∞ f(zn−1, wn−1, xn−1, yn−1) = f(z, w, x, y) thus, we have f(x, y, z, w) = x, f(x, w, z, y) = y, f(z, y, x, w) = z, f(z, w, x, y) = w. suppose now the assumption (b) holds. since {xn}, {zn} are non-decreasing and xn → x, zn → z and also {yn}, {wn} are non-increasing and yn → y, wn → w, then by assumption (b) we have xn ≥ x, yn ≤ y, zn ≥ z, wn ≤ w for all n. due to (2.26) and (2.27), we have (2.30) p(f(x, y, z, w), f(x, y, z, w)) ≤ k 4 [p(x, x) + p(y, y) + p(z, z) + p(w, w)] = 0. consider now, (2.31) p(xn, f(x, y, z, w)) = p(f(xn−1, yn−1, zn−1, wn−1), f(x, y, z, w)) ≤ k 4 [p(xn−1, x) + p(yn−1, y) + p(zn−1, z) + p(wn−1, w)] letting n → ∞ in (2.31), by lemma 1.12 we get (2.32) p(x, f(x, y, z, w)) ≤ k 4 [p(x, x) + p(y, y) + p(z, z) + p(w, w)] c© agt, upv, 2014 appl. gen. topol. 15, no. 1 19 e. karapınar and k. tas regarding (2.26) and (2.27), we conclude that p(x, f(x, y, z, w)) = 0. hence, by (2.26),(2.30),(2.32) and definiton (2.33) dp(x, f(x, y, z, w)) = 2p(x, f(x, y, z, w))−p(f(x, y, z, w), f(x, y, z, w))−p(x, x) = 0. thus, we have x = f(x, y, z, w). analogously we we get f(y, z, w, x) = y, f(z, w, x, y) = z, f(w, x, y, z) = w. thus, we proved that f has a quadruple fixed point. � 3. uniqueness of quadruple fixed point in this section we shall prove the uniqueness of quadruple fixe point. for a product x4 of a partial ordered set (x, ≤) we define a partial ordering in the following way: for all (x, y, z, t), (u, v, r, t) ∈ x4 (3.1) (x, y, z, w) ≤ (u, v, r, t) ⇔ x ≤ u, y ≥ v, z ≤ r, w ≥ r. we say that (x, y, z, w) is equal (u, v, r, t) if and only if x = u, y = v, z = r and w = t. theorem 3.1. in addition to hypothesis of theorem 2.3, suppose that for all (x, y, z, t), (u, v, r, t) ∈ x ×x ×x ×x, there exists (a, b, c, d) ∈ x ×x ×x ×x that is comparable to (x, y, z, t) and (u, v, r, t), then f has a unique quadruple fixed point. proof. the set of quadruple fixed point of f is not empty due to theorem 2.3. assume, now, (x, y, z, t) and (u, v, r, t) are the quadruple fixed point of f , that is, f(x, y, z, w) = x, f(u, v, r, t) = u, f(y, z, w, x) = y, f(v, r, t, u) = v, f(z, w, x, y) = z, f(r, t, u, v) = r, f(w, x, y, z) = w, f(t, u, v, r) = t, we shall show that (x, y, z, w) and (u, v, r, t) are equal. by assumption, there exists (a, b, c, d) ∈ x×x×x×x that is comparable to (x, y, z, t) and (u, v, r, t). define sequences {an}, {bn}, {cn} and {dn} such that a = a0, b = b0, c = c0, d = d0 and (3.2) an = f(an−1, bn−1, zn−1, dn−1), bn = f(bn−1, cn−1, dn−1, an−1), cn = f(cn−1, dn−1, an−1, bn−1), dn = f(dn−1, an−1, bn−1, cn−1). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 20 quadruple fixed point theorems for all n. since (x, y, z, w) is comparable with (a, b, c, d), we may assume that (x, y, z, w) ≥ (a, b, c, d) = (a0, b0, c0, d0). recursively, we get that (3.3) (x, y, z, w) ≥ (an, bn, cn, dn) for all n. by (3.3) and (2.4), we have (3.4) p(x, an+1) = p(f(x, y, z, w), f(an, bn, cn, dn)) ≤ k 4 [p(x, an) + p(y, bn) + p(z, cn) + p(w, dn)] (3.5) p(bn+1, y) = p(f(bn, cn, dn, an), f(y, z, w, x)) ≤ k 4 [p(bn, y) + p(cn, z) + p(dn, w) + p(an, x)] (3.6) p(z, cn+1) = p(f(z, w, x, y), f(cn, dn, an, bn)) ≤ k 4 [p(z, cn) + p(w, dn) + p(x, an) + p(y, bn)] (3.7) p(dn+1, w) = p(f(cn, dn, an, bn), f((w, x, y, z))) ≤ k 4 [p(dn, w) + p(an, x) + p(bn, y) + p(cn, z)] set γn = p(x, an) + p(y, bn) + p(z, cn) + p(w, dn). then, due to (3.7)-(3.7), we have (3.8) γn+1 ≤ kγn ≤ k nγ0, for all n. � since 0 ≤ k < 1, the sequence {γn} is decreasing and bounded below. thus, there exists γ ≥ 0 such that lim n→∞ γn = γ. now, we shall show that γ = 0. letting n → ∞ in (3.8), and having mind 0 ≤ k < 1, we obtain that γ ≤ 0. therefore, γ = 0. that is, lim n→∞ γn = 0. consequently, we have (3.9) limn→∞ p(x, an) = 0, limn→∞ p(y, bn) = 0, limn→∞ p(z, cn) = 0, limn→∞ p(w, dn) = 0. analogously, we show that (3.10) limn→∞ p(u, an) = 0, limn→∞ p(v, bn) = 0, limn→∞ p(r, cn) = 0, limn→∞ p(s, dn) = 0. combining (3.9) and (3.10) yield ,by uniqueness of the limit, that (x, y, z, w) and (u, v, r, t) are equal. now, in the following example neither the continuity of the mapping f is satisfied nor the 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[53] p. waszkiewicz, quantitative continuous domains, applied categorical structures, 11 (2003), 41–67. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 24 @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 377–390 paths in hyperspaces camillo costantini and wies law kubís ∗ dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. we prove that the hyperspace of closed bounded sets with the hausdorff topology, over an almost convex metric space, is an absolute retract. dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. we give some necessary conditions for the path-wise connectedness of the hausdorff metric topology on closed bounded sets. finally, we describe properties of a separable metric space, under which its hyperspace with the wijsman topology is path-wise connected. 2000 ams classification: 54b20, 54c55, 54d05. keywords: hyperspace, wijsman topology, hausdorff metric, path-wise connectedness, absolute retract. 1. introduction. given a metric space (x,d), let cl(x) denote the hyperspace of closed nonempty subsets of x. we are interested in path-wise connectedness and related properties of hyperspace topologies on cl(x), mainly the hausdorff metric topology and the wijsman topology. these two topologies come from identifying a closed set a ⊆ x with its distance functional x 7→ dist(x,a), so that cl(x) can be regarded as a subspace of c(x,r), the space of all continuous real functions on x. under this identification, the hausdorff and wijsman topologies are the topologies of uniform and pointwise convergence, respectively. both are different from the well known vietoris topology (unless x is compact). the advantage of these topologies is metrizability: the hausdorff topology on bounded sets is always metrizable and the wijsman one is metrizable provided the base space is separable. the vietoris topology on closed sets is metrizable only if the base space is compact. for a general reference concerning hyperspace topologies see beer’s book [3]. ∗the research was supported by kbn grant no. 5p03a04420. 378 camillo costantini and wies law kubís global and local path-wise connectedness of the vietoris topology on compact sets has been studied since 1930s. borsuk and mazurkiewicz [4] showed in 1931 that both k(x), the hyperspace of compact subsets of x and c(x), the hyperspace of subcontinua of x, are path-wise connected provided x is a metrizable continuum. the case of non-metrizable spaces was investigated by mcwaters [14] and ward [13]: they obtained results on generalized pathwise connectedness of the vietoris hyperspace of compact sets. local path-wise connectedness of k(x) and c(x), for a compact metric space x, were first characterized by wojdys lawski [15] in 1939. namely, k(x) is locally path-wise connected iff x is locally connected. the same is true for c(x) and this property is equivalent to the fact that k(x) (or c(x)) is an absolute neighborhood retract. the case of all metric spaces is due to curtis [5]: k(x) is locally pathwise connected (equivalently: k(x) ∈ anr) iff x is locally continuum-wise connected, i.e. for every p ∈ x and every neighborhood v of it there is another neighborhood u of p such that any two points of u lie in a subcontinuum of v . a famous result of curtis and schori [7] says that k(x) is homeomorphic to the hilbert cube iff x is a locally connected, nondegenerate, metric continuum (for other results in this spirit see e.g. [5, 6]). let us also mention a useful result of curtis and nguyen to nhu [6]: the vietoris hyperspace of finite sets over a locally path-wise connected metric space is an anr. for a study of topological properties of compact vietoris hyperspaces we refer to nadler’s book [11] or to a recent one by illanes and nadler [8]. concerning other hyperspace topologies, not much is known. antosiewicz and cellina [1] showed that the hyperspace of closed bounded sets with the hausdorff metric topology, over a convex subspace of a normed linear space, is an absolute retract. sakai and yang [12] proved that cl(x) with the fell topology is homeomorphic to the hilbert cube minus a point iff x is a locally compact, locally connected, separable metrizable space with no compact components. banakh, kurihara and sakai [2] showed that for a normed linear space x, cl(x), k(x) and some other subspaces of cl(x), equipped with the attouch-wets topology, are absolute retracts; in case where x is a banach space, cl(x) is homeomorphic to a hilbert space. finally, sakai, yaguchi and the second author [9] gave general conditions for the anr property of cl(x) with the wijsman topology. in [9] it is also proved that cl(x) with the wijsman topology is homeomorphic to the separable hilbert space, provided x is a separable banach space. we give several results on path-wise connectedness and absolute neighborhood retract property for some hyperspace topologies. using well-known results for compact hyperspaces, we characterize path-wise connectednes of the vietoris topology on closed sets over a metrizable space; we apply this result for the wijsman topology. we note that for a noncompact metrizable space x, cl(x) with the vietoris topology is not locally connected. we prove that the hyperspace of closed bounded sets endowed with the hausdorff topology is an absolute retract, provided the base space is almost convex (see the definitions below). this improves the above-mentioned result of antosiewicz and cellina paths in hyperspaces 379 [1]. we give some necessary conditions for the path-wise connectedness of the hausdorff topology on closed bounded sets. finally, we discuss the path-wise connectedness of the wijsman topology. we show, among other results, that cl(x) with the wijsman topology is path-wise connected if (x,d) is separable and path-wise connected at infinity or continuum-wise connected. notation. all topological spaces are assumed to be t1. for a given topological (metric) space x, we denote by cl(x), k(x), c(x) and clb(x) the hyperspace of closed, compact, compact connected and closed bounded nonempty subsets of x, respectively. for any set x, we denote by fin(x) the collection of all nonempty finite subsets of x. ω denotes the set of all nonnegative integers. let (x,d) be a metric space. we will denote by b(a,r) and b(a,r) the open and closed ball centered at a ⊆ x and with radius r > 0, respectively. the hausdorff metric on clb(x) is defined by dh(a,b) = inf{r > 0 : a ⊆ b(b,r) & b ⊆ b(a,r)}. we can define dh(a,b) also for unbounded sets, setting dh(a,b) = +∞ if there is no r > 0 with a ⊆ b(b,r) and b ⊆ b(a,r). the topology induced by the hausdorff metric is called the hausdorff topology. it is reasonable to consider the hausdorff topology on all closed subsets of x, because dh is locally a metric. however, clb(x) is clopen in cl(x) and hence cl(x) is not connected for an unbounded metric space (x,d). the hausdorff topology on k(x) agrees with the vietoris topology. the wijsman topology on cl(x) is the least topology t such that for every x ∈ x the function a 7→ dist(x,a) is continuous. the formula dh(a,b) = sup x∈x ∣∣dist(x,a) − dist(x,b)∣∣ implies that the wijsman topology is weaker than the hausdorff topology. for a noncompact metric space, the wijsman topology is strictly weaker than the vietoris one (even on finite sets). we will denote by tv , th and tw the vietoris, hausdorff and wijsman topology, respectively (the latter two depend on the metric). a metric space (x,d) is almost convex if for every x,y ∈ x and for every s,t > 0 with d(x,y) < s + t, there exists z ∈ x such that d(x,z) < s and d(z,y) < t. for example, a dense subspace of a normed linear space (or, more generally, of a convex metric space) is almost convex. a path in a space x is a continuous map γ : j → x where j ⊆ r is a closed interval (usually j = [0, 1]). we denote by bk+1 and sk the standard k + 1dimensional closed ball and the standard k-dimensional sphere (which is the boundary of bk+1), respectively. a topological space x is k-connected (k ∈ ω) if every continuous map f : sk → x has a continuous extension f : bk+1 → x. in particular, ”0-connected” means ”path-wise connected”. x is homotopically trivial if it is k-connected for every k ∈ ω. local versions of k-connectedness are defined in the obvious way. a metric space (x,d) is path-wise connected at infinity if for every x ∈ x there exists a continuous map f : [0, +∞) → 380 camillo costantini and wies law kubís x such that f(0) = x and limt→+∞d(x,f(t)) = +∞. a topological space x is continuum-wise connected if every two points of x are contained in a subcontinuum of x (i.e. a compact connected subspace of x). an absolute neighborhood retract (briefly anr) is a metrizable space x such that for every metric space y , every continuous map f : a → x defined on a closed set a ⊆ y , has a continuous extension f : u → x, for some open set u with a ⊆ u ⊆ y . if, under these assumptions, u = y then x is an absolute retract (briefly ar). it is well-known that an absolute neighborhood retract is locally k-connected for every k ∈ ω and a homotopically trivial anr is an absolute retract. a (join) semilattice is a commutative semigroup (l,∨) such that a∨a = a for every a ∈ l. a semilattice comes from a partially ordered set (l,6) such that every two elements of l have a supremum. specifically, setting a∨b = sup{a,b}, (l,6) becomes a semilattice. conversely, if (l,∨) is a semilattice then defining a 6 b iff a ∨ b = b, we get a partial order on l such that a ∨ b = sup{a,b}. a lawson semilattice [10] is a topological semilattice (l,∨) (i.e. a semilattice equipped with the topology such that ∨: l×l → l is continuous) which has a neighborhood base consisting of subsemilattices. most hyperspaces are lawson semilattices with respect to ∪ (see section 2.2). 2. general results. 2.1. on path-wise connectedness. fix a topological (metric) space x and let t be the vietoris topology or the wijsman topology on cl(x). observe that (cl(x),t ) has the following properties: (i) if {an}n∈ω converges to a then {c ∪an}n∈ω converges to c ∪a for each c ∈ cl(x). (ii) if {an}n∈ω is increasing and such that ⋃ n∈ω an is dense in x then {an}n∈ω converges to x. most hyperspace topologies satisfy a stronger condition than (i), namely the union operator ∪: cl(x) × cl(x) → cl(x) is continuous. on the other hand, for a bounded metric space x the hausdorff topology on cl(x) does not necessarily satisfy (ii). proposition 2.1. let x be a separable topological space and let t be a topology on cl(x) satisfying conditions (i), (ii) above. then the following conditions are equivalent: (a) (cl(x),t ) is path-wise connected. (b) for each a,b ∈ x there is a continuous map γ : [0, 1] → (cl(x),t ) such that γ(0) = {a} and b ∈ γ(1). proof. it is enough to show that (b) =⇒ (a). fix c ∈ cl(x). we show that there exists a path joining c to x. fix a countable dense set {dn : n ∈ ω}⊆ x with d0 ∈ c. for each n ∈ ω choose a continuous map γn : [0, 1] → (cl(x),t ) such that γn(0) = {dn} and dn+1 ∈ γn(1). define φn : [n,n + 1] → cl(x) by paths in hyperspaces 381 setting φn(t) = c ∪ ⋃ k 0. proof. let f : sk → l be a continuous map. then f extends naturally to a vietoris continuous map f : fin(sk) → fin(l). as k > 0, fin(sk) is an ar, so there is a map j : bk+1 → fin(sk) such that j(x) = {x} for x ∈ sk (in fact, there is a straightforward formula for j, see [6]). now setting f = rfj, where r: fin(l) → l is the retraction defined above, we get a continuous extension of f over bk+1. � to apply the above results to hyperspaces we need to know that they are lawson semilattices. proposition 2.4. the vietoris, wijsman and hausdorff hyperspaces are lawson semilattices with respect to ∪. proof. let x be a topological (metric) space and let t ∈ {tv ,th,tw}. clearly, t has a base consisting of subsemilattices. we need to show the continuity of the union. first, let t = tv and fix (a0,b0) ∈ cl(x)×cl(x). if a0∪b0 ∈ u+ then (a0,b0) ∈ u+×u+ and ∪[u+×u+] ⊆ u+, where ∪[m] is the image of m ⊆ cl(x)×cl(x) under ∪: cl(x)×cl(x) → cl(x). if a0∪b0 ∈ u− then (a0,b0) ∈ w , where w = (u−×cl(x))∪(cl(x)×u−), 382 camillo costantini and wies law kubís and we have ∪[w ] ⊆ u−. thus, ∪ is continuous with respect to tv . for t = tw and t = th the continuity of ∪ follows from the formulae: dist(x,a∪b) = min{dist(x,a), dist(x,b)}, dh(a∪b,a′ ∪b′) 6 max{dh(a,a′),dh(b,b′)}. � 3. the vietoris topology. in this section we note some results on path-wise connectedness of the vietoris topology. recall that the theorem of borsuk and mazurkiewicz [4] says that both k(x) and c(x) are path-wise connected whenever x is a metrizable continuum. using this result, we are able to investigate the case of noncompact metric spaces. we use the following fact: if γ : [0, 1] → k(x) is a path and γ(0) is connected then ⋃ γ[0, 1] is a subcontinuum of x (see, e.g., nadler [11]). theorem 3.1. for a metrizable space x the following conditions are equivalent: (a) every compact subset of x is contained in a continuum. (b) (k(x),tv ) is path-wise connected. proof. (a) =⇒ (b) fix a,b ∈ k(x). let c ⊆ x be a continuum such that a ∪ b ⊆ c. then a,b ∈ k(c) and hence, by the theorem of borsukmazurkiewicz, there exists a path γ : [0, 1] → k(c) such that γ(0) = a and γ(1) = b. (b) =⇒ (a) fix a ∈ k(x) and pick an a ∈ x. let γ : [0, 1] → k(x) be a path joining a to {a}. then d = ⋃ γ[0, 1] is a continuum containing a. � example 3.2. an example of a path-wise connected subspace of the plane r2 which does not satisfy (a) above. consider x = ({0}× [0, +∞)) ∪s ∪t, where s = {(x, |sin(π/x)|/x) : x ∈ (0, 1]} and t = {(x,y) ∈ r2 : (x− 1/2)2 + y2 = 1/4 and y < 0}. observe that x is path-wise connected. let a = ({0}∪{1/n: n ∈ ω}) ×{0}. then a ∈ k(x) but each closed connected subset of x containing a also contains {0}×[0, +∞) and therefore is not compact. theorem 3.3. for a metrizable space x the following conditions are equivalent: (a) (c(x),tv ) is path-wise connected. (b) there exists g ⊆ k(x) containing all singletons of x, such that (g,tv ) is path-wise connected. (c) x is continuum-wise connected, i.e. each two points of x lie in a subcontinuum of x. paths in hyperspaces 383 proof. (a) =⇒ (b) is obvious. (b) =⇒ (c) fix a,b ∈ x and let γ : [0, 1] →g be a path joining {a} and {b}. then ⋃ γ[0, 1] is a subcontinuum of x containing a,b. (c) =⇒ (a) fix a,b ∈ c(x). let g be a subcontinuum of x intersecting both a and b. then a,b ∈ c(d), where d = a∪b ∪g. by the theorem of borsuk-mazurkiewicz, there exists a path in c(d) joining a and b. � using the above result and proposition 2.1 we obtain the following. corollary 3.4. let x be a separable topological space. if x is path-wise connected or x is continuum-wise connected and metrizable then (cl(x),tv ) is path-wise connected. let x be a metrizable space. a theorem of curtis [5] says that (k(x),tv ) is locally path-wise connected (equivalently: (k(x),tv ) ∈ anr) iff x is locally continuum-wise connected. concerning cl(x), we have the following negative result. theorem 3.5. if x is a noncompact metrizable space then (cl(x),tv ) is not locally connected. proof. let c = {xn : n ∈ ω} be a (one-to-one) sequence in x having no cluster point (by the noncompactness of x); then c ∈ cl(x). choose a disjoint family {un}n∈ω of open sets such that xn ∈ un for n ∈ ω. let u = ⋃ n∈ω un. then u+ is a neighborhood of c. let v ∈tv be any neighborhood of c such that v ⊆ u+. then v contains a basic neighborhood w = v +∩v −0 ∩·· ·∩v − m−1 of c, with vi ⊆ v and v,v0, . . . ,vm−1 open in x. this implies, in particular, that c ⊆ v and that for every i < m there is an n(i) ∈ ω with xn(i) ∈ vi. let f = {xn(0), . . . ,xn(m−1)}, so that f ∈ w, and fix k > max{n(i) : i < m}. we will prove that s = v ∩ u−k is clopen in v, and this will imply that v is disconnected, because c ∈s while f ∈w \s ⊆v \s. clearly, s is open in v because u−k is open in cl(x). on the other hand, we may observe that s = v ∩ [cl uk]−: indeed, every element of v is a subset of u and hence it cannot contain any point of (cl uk)\uk (because the sets ui are pairwise disjoint). therefore, s is also closed in v. � 4. the hausdorff metric topology. in this section we consider clb(x) endowed with the hausdorff metric topology. the hausdorff metric is actually defined on cl(x) but one can easily observe that clb(x) is clopen in cl(x) so cl(x) is not connected unless x is bounded. if x is not compact then there is an unbounded metric on x; on the other hand if d is an unbounded metric on x then ρ(x,y) = min{1,d(x,y)} defines a bounded, uniformly equivalent metric, so the hausdorff metric induced by ρ is equivalent to the one induced by d. it follows that a noncompact metrizable space admits a metric for which the hausdorff hyperspace of closed bounded sets is disconnected. 384 camillo costantini and wies law kubís observe that if γ : [0, 1] → clb(x) is a path then the map γ : [0, 1] → clb(x) defined by the formula γ(t) = cl (⋃ s6t γ(s) ) is also a path in clb(x). such a map will be called an order arc in clb(x). lemma 4.1. for a metric space (x,d) the following conditions are equivalent: (a) (clb(x),th) is path-wise connected. (b) for each p ∈ x and for each n ∈ ω there exists a path γ : [0, 1] → clb(x) such that γ(0) = {p} and b(p,n) ⊆ ⋃ t∈[0,1] γ(t). proof. we need to show that (b) =⇒ (a). fix c,d ∈ clb(x). fix n ∈ ω with n > max{dh({c},d),dh(c,{d})}, where c ∈ c and d ∈ d are fixed arbitrarily. by (b) there exist paths f,g : [0, 1] → clb(x) with f(0) = {c}, g(0) = {d}, b(c,n) ⊆ ⋃ t∈[0,1] f(t) and b(d,n) ⊆ ⋃ t∈[0,1] g(t). we may assume that f,g are order arcs, thus b(c,n) ⊆ f(1) and b(d,n) ⊆ g(1). set f = f(1) ∪ g(1). then f ∈ clb(x) and c ∪d ⊆ b(d,n) ∪ b(c,n) ⊆ f. define γ : [0, 2] → clb(x) by setting γ(t) = { c ∪f(t) if t ∈ [0, 1], c ∪f(1) ∪g(t− 1) if t ∈ [1, 2]. observe that γ is well-defined, continuous and γ(0) = c, γ(2) = f. it follows that c and f can be joined by a path. similarly, there is a path joining d to f. � 4.1. almost convex metric spaces. recall that a metric space (x,d) is almost convex if for each a,b ∈ x and for each s,t > 0 such that d(a,b) < s + t there exists x ∈ x with d(a,x) < s and d(x,b) < t. clearly, every convex metric space is almost convex and a dense subspace of an almost convex metric space is almost convex. lemma 4.2. a metric space (x,d) is almost convex iff for each a ⊆ x and for each s,t > 0 we have b(b(a,s), t) = b(a,s + t). proof. if (x,d) satisfies the above condition then for a,b ∈ x and s,t > 0 with d(a,b) < s + t we have b ∈ b(a,s + t) = b(b(a,s), t) and hence there is x ∈ b(a,s) with d(x,b) < t. thus (x,d) is almost convex. assume now that (x,d) is almost convex and fix a ⊆ x and s,t > 0. clearly b(b(a,s), t) ⊆ b(a,s + t). fix p ∈ b(a,s + t). let a ∈ a be such that d(a,p) < s + t. there exists x ∈ x with d(a,x) < s and d(x,p) < t. thus p ∈ b(b(a,s), t). � lemma 4.3. let (x,d) be an almost convex metric space and let c ∈ cl(x). then the map γ : [0, +∞) → cl(x) defined by the formula γ(t) = b(c,t) paths in hyperspaces 385 is a constant 1 lipschitz map with respect to the hausdorff metric on cl(x) and the standard metric on [0, +∞). proof. first observe that cl b(a,r) = b(a,r) for every a ⊆ x and r > 0. thus, by lemma 4.2 we have γ(t + r) ⊆ b(γ(t),r) for every t,r > 0. it follows that dh(γ(t),γ(t + r)) 6 r. � theorem 4.4. let (x,d) be an almost convex metric space. then clb(x) with the hausdorff metric topology is an absolute retract. proof. clb(x) is path-wise connected by lemma 4.1, but we need to show that it is locally path-wise connected. fix c,d ∈ clb(x) and r > dh(c,d). let a = b(c,r) ∪ b(d,r). then c ∪ d ⊆ a and a ∈ clb(x). define γ : [0, 2r] → clb(x) by γ(t) = { b(c,t), if t 6 r, b(c,r) ∪ b(d,t−r), if r 6 t 6 2r. by lemma 4.3, γ is continuous with respect to the hausdorff metric. observe that dh(γ(t),c) 6 2r. thus c and a can be joined by a path contained in the open ball centered at c and with radius 2r. by symmetry, the same applies to d and a. this proves that, given the neighbourhood bdh (c, 3r) of c in (clb(x),dh), bdh (c,r) in another neighbourhood of c such that every element of it may be joined to c by a path lying in bdh (c, 3r). therefore, (clb(x),dh) is locally path-wise connected . as clb(x) is a lawson semilattice, by theorem 2.2 it is an anr. on the other hand, clb(x) is homotopically trivial (proposition 2.3), so it is an ar. � corollary 4.5. let x be a dense subset of a convex subset of a normed linear space, endowed with the metric induced by the norm. then (clb(x),th) is an absolute retract. the above result in the case of convex subsets of normed spaces was proved, using elementary although complicated methods, by antosiewicz and cellina [1]. 4.2. c-connectedness. we investigate necessary conditions for the path-wise connectedness of (clb(x),dh) and we present some counterexamples. let us call a metric space (x,d) c-connected (or connected in cantor’s sense) if for each a,b ∈ x and for each ε > 0 there exist x0, . . . ,xn ∈ x such that x0 = a, xn = b and d(xi,xi+1) < ε for i < n. clearly every connected space is c-connected and every compact c-connected space is connected. a sequence (x0, . . . ,xn) with d(xi,xi+1) < ε for i < n will be called an ε-sequence of size n joining x0,xn. call a metric space (x,d) uniformly c-connected if for each bounded set b ⊆ x and for each ε > 0 there exists k ∈ ω such that for each x,y ∈ b there exists an ε-sequence in x of size at most k joining x,y. observe that the closure of a uniformly c-connected subset of x is also uniformly c-connected. 386 camillo costantini and wies law kubís proposition 4.6. let (x,d) be a metric space. then (clb(x),dh) is c-connected if and only if (x,d) is uniformly c-connected. proof. suppose that (clb(x),dh) is c-connected. fix a closed bounded set b ⊆ x and pick a p ∈ b. fix ε > 0 and let (a0, . . . ,ak) be an ε-sequence in clb(x) with a0 = {p} and ak = b. fix x ∈ b. we can find xk−1 ∈ ak−1 such that d(xk−1,x) < ε, because dh(ak−1,b) < ε. inductively, we find xi ∈ ai such that d(xi,xi+1) < ε. then x0 = {p} and (x0, . . . ,xk−1,x) is an ε-sequence of size k joining p,x. it follows that every two points of b are joined by an ε-sequence of size at most 2k. thus (x,d) is uniformly c-connected. suppose now that (x,d) is uniformly c-connected and fix b ∈ clb(x). fix p ∈ b. we show that {p} and b can be joined by an ε-sequence for every ε > 0. as (x,d) is c-connected and is isometrically embedded in clb(x), it then follows that clb(x) is c-connected. fix ε > 0. let k be such that for each x ∈ b there exists an ε/2sequence (y0(x), . . . ,yk(x)) such that y0(x) = p and yk(x) = x. define ai = cl{yi(x) : x ∈ b}. observe that ai ∈ clb(x) and dh(ai,ai+1) 6 ε/2 < ε for i < k. thus (a0, . . . ,ak) is an ε-sequence in clb(x) joining a0 = {p} to ak = b. � consider the following metric properties: c1: every bounded subset of x is contained in a uniformly c-connected bounded subset of x. c2: for each p ∈ x and for each r > s > 0 such that b(p,r) \ b(p,s) 6= ∅ there exists a uniformly c-connected set s ⊆ b(p,r) such that p ∈ s and s ∩ b(p,r) \ b(p,s) 6= ∅. proposition 4.7. let (x,d) be a metric space such that (clb(x),th) is path-wise connected. then (x,d) has properties c1, c2. proof. fix p ∈ x and r > 0. let f : [0, 1] → clb(x) be a hausdorff continuous order arc with f(0) = {p} and f(1) ⊇ b(p,r). fix ε > 0. let k ∈ ω be such that |t − s| 6 1/k implies dh(f(t),f(s)) < ε. fix x0 ∈ b(p,r). as dh(f(1),f(1 − 1/k)) < ε, we can find x1 ∈ f(1 − 1/k) with d(x0,x1) < ε. inductively, we find xi ∈ f(1− i/k) with d(xi−1,xi) < ε. finally xk = p which means that (x0, . . . ,xk) is an ε-sequence of size k joining p,x0. this shows that f(1) is uniformly c-connected and consequently (x,d) has property c1. by the same argument, f(t) is c-connected for each t ∈ [0, 1]. now let 0 < s < r be such that b(p,r) \ b(p,s) 6= ∅. then dh({p},f(1)) ≥ s. let t0 ∈ [0, 1] be such that dh({p},f(t0)) ∈ [s,r). then s = f(t0) is a uniformly c-connected set included in b(p,r) with p ∈ s and s ∩ b(p,r) \ b(p,s) 6= ∅. this shows property c2. � we now describe an example of a path-wise connected space with path-wise disconnected hausdorff hyperspace, and an example of a connected, path-wise disconnected hausdorff hyperspaces. paths in hyperspaces 387 example 4.8. let (x,ρ) be an unbounded metric space and define a bounded metric on x by the formula d(x,y) = min{ρ(x,y), 1}. then (cl(x),dh) is not c-connected since (x,d) is not uniformly c-connected. indeed, (x,d) is bounded, but for every k we can find two points of x which cannot be joined by a 1-sequence of size k. this shows that if a metrizable space x is not compact then there exists a metric d on x such that (clb(x),dh) is not c-connected. example 4.9. let a = ⋃ n∈ω(3 −n, 2 · 3−n) and b = ⋃ n∈ω[3 −n, 2 · 3−n]. consider x = {(x,y) ∈ [0, 1]2 : χa(x) 6 y 6 χb(x)} with the topology inherited from the plane. then for any compatible metric d on x, clb(x) is not path-wise connected since (x,d) does not have property c2. indeed, if u is a neighborhood of p = (0, 0) contained in [0, 1] × [0, 1/2) then the only c-connected subset of u containing p is {p}. on the other hand, if d is the euclidean metric on x then (x,d) has property c1. to show that (clb(x),dh) is connected, one first proves that clb(x) \ {{(0, 0)}} is connected (actually, path-wise connected, as it is homeomorphic to clb(x)\{(0, 0)}); and then one observes that {(0, 0)} is in the dh-closure of clb(x) \{{(0, 0)}}. we omit details. problem 4.10. do properties c1, c2 characterize path-wise connectedness of the hausdorff topology? 5. the wijsman topology. let (x,d) be a metric space. recall that the wijsman topology is weaker than the vietoris one; on clb(x) it is also weaker than the hausdorff metric topology. (cl(x),tw ) is completely regular and, it is metrizable iff x is separable. see beer’s book [3] for details. applying corollary 3.4 and lemma 4.3 together with proposition 2.1 we obtain the following. corollary 5.1. if (x,d) is a separable continuum-wise connected metric space then (cl(x),tw ) is path-wise connected. theorem 5.2. if (x,d) is an almost convex metric space then (cl(x),tw ) is path-wise connected. proof. if (x,d) is separable, this follows from proposition 2.1 and lemma 4.3. however in general, by lemma 4.3, the formula γ(t) = { b(a,t) if t ∈ [0, +∞), x if t = +∞. defines a wijsman continuous path γ : [0, +∞] → cl(x) joining a to x, for each a ∈ cl(x). � the next result describes different situations. it appears that (cl(x),tw ) may be path-wise connected even if x is far from being connected. 388 camillo costantini and wies law kubís theorem 5.3. let (x,d) be a separable metric space such that for each a,b ∈ x either there exists a uniformly continuous map f : q∩[0, 1] → x with f(0) = a and f(1) = b, or else there exists a map g : q∩[0, +∞) → x such that g(0) = b, limt→+∞d(g(t),b) = +∞ and g � (q ∩ [0,n]) is uniformly continuous for every n ∈ ω. then cl(x) with the wijsman topology is path-wise connected. proof. we use proposition 2.1. fix a,b ∈ x. assume first that there exists a uniformly continuous map f : q∩ [0, 1] → x with f(0) = a and f(1) = b. define a path γ : [0, 1] → cl(x) by setting γ(t) = cl{f(q) : q ∈ q∩ [0, t]}. clearly, γ(0) = {a} and b ∈ γ(1). we need to show that γ is continuous. fix ε > 0. let δ > 0 be such that d(f(q0),f(q1)) < ε whenever |q0 − q1| < δ, q0,q1 ∈ q∩[0, 1]. suppose to have any t0, t1 ∈ [0, 1] with |t0−t1| < δ. consider x = f(q0), where q0 ∈ q∩[0, t0]. choose q1 ∈ q∩[0, t1] such that |q0 −q1| < δ. then d(x,f(q1)) < ε and f(q1) ∈ γ(t1). it follows that γ(t0) ⊆ b(γ(t1),ε). by symmetry we get dh(γ(t0),γ(t1)) 6 ε. it follows that γ is uniformly continuous with respect to the hausdorff metric on cl(x). hence, γ is also continuous with respect to the wijsman topology. now assume that there exists a map g : q∩[0, +∞) → x such that g(0) = b, limt→+∞d(g(t),b) = +∞ and g � (q∩ [0,n]) is uniformly continuous for each n ∈ ω. define γ : [0, +∞] → cl(x) by setting γ(t) = {a}∪ cl{g(q) : q ∈ q∩ [t, +∞)} for t ∈ [0, +∞) and γ(+∞) = {a}. clearly, b ∈ γ(0). observe that γ � [0,n] can be represented in the form γ(t) = η(t) ∪bn, where bn = {a}∪ cl{g(q) : q ∈ [n, +∞)} and η(t) = cl{g(q) : q ∈ q∩ [t,n]}. thus, by the previous argument, γ � [0,n] is continuous with respect to the wijsman topology. it remains to show that γ is continuous at +∞. fix x ∈ x. then dist(x,γ(+∞)) = d(x,a). on the other hand, d(g(q),x) > d(g(q),b) − d(x,b) so there exists n0 ∈ ω such that d(g(q),x) > d(x,a) for q ∈ q∩[n0, +∞). hence dist(x,γ(t)) = d(x,a) = dist(x,γ(+∞)) for t > n0. � corollary 5.4. let (x,d) be a separable metric space which is path-wise connected at infinity. then (cl(x),tw ) is path-wise connected. let (x,d) be a separable, locally path-wise connected metric space. in [9], sakai, yaguchi and the second author proved that (cl(x),tw ) is an absolute neighborhood retract provided x\ ⋃ b has finitely many components, for every finite family b consisting of closed balls in (x,d). we give an example of an almost convex, locally path-wise connected, separable metric space, for which the wijsman hyperspace is not locally connected. paths in hyperspaces 389 example 5.5. let (x,d) be the separable hedgehog space, i.e. x = {θ}∪ ⋃ n∈ω (0, 1] ×{n}, where d(θ, (t,n)) = t, d((t,n), (s,n)) = |t−s| and d((t,n), (s,m)) = t+s for n 6= m. then (x,d) is an almost convex metric space and it is an absolute retract. we claim that (cl(x),tw ) is not locally connected. let a0 = {(1,n) : n ∈ ω} and let u = {a ∈ cl(x) : dist(θ,a) > 1/2}. then u is a neighborhood of a0. for each n ∈ ω define v−n = {a ∈ cl(x) : dist((1,n),a) < 1/2}, v+n = {a ∈ cl(x) : dist((1,n),a) > 1}. clearly v−n ∩v+n = ∅ and v−n ,v+n ∈ tw . observe that u ⊆ v−n ∪v+n . also, v−n is a neighborhood of a0. now we claim that for every neighborhood v of a0 with v ⊆ u, there is n ∈ ω such that v+n ∩v 6= ∅, i.e. v+n disconnects v. indeed, observe that a0 = limn→∞an, where an = {(1,k) : k < n}. thus, for every open v ⊆ u with a0 ∈ v there exists n ∈ ω such that an ∈ v. on the other hand dist((1,n),an) = 2 so an ∈v+n . references [1] h.a. antosiewicz and a. cellina, continuous extensions of multifunctions, ann. polon. math. 34 (1977) 107–111. [2] t. banakh, m. kurihara and k. sakai, hyperspaces of normed linear spaces with the attouch-wets topology , preprint. [3] g. beer, topologies on closed and closed convex sets, kluwer academic publishers 1993. [4] k. borsuk and s. mazurkiewicz, sur l’hyperespace d’un continu, c. r. soc. sc. varsovie 24 (1931) 149–152. [5] d.w. curtis, hyperspaces of noncompact metric spaces, comp. math. 40 (1980) 139– 152. [6] d. curtis and nguyen to nhu, hyperspaces of finite subsets which are homeomorphic to ℵ0-dimensional linear metric spaces, topology appl. 19 (1985) 251–260. [7] d.w. curtis and r.m. schori, hyperspaces of peano continua are hilbert cubes, fund. math. 101 (1978) 19–38. [8] a. illanes and s. nadler, hyperspaces, marcel-dekker, new york 1999. [9] w. kubís, k. sakai and m. yaguchi, hyperspaces of separable banach spaces with the wijsman topology , preprint. [10] j.d. lawson, topological semilattices with small subsemilattices, j. london math. soc. (2) 1 (1969), 719–724. [11] s. nadler, hyperspaces of sets, marcel-dekker 1978. [12] k. sakai and z. yang, hyperspaces of non-compact metrizable spaces which are homeomorphic to the hilbert cube, preprint. [13] l.e. ward, jr., arcs in hyperspaces which are not compact , proc. amer. math. soc. 28 (1971) 254–258. [14] m.m. mcwaters, arcs, semigroups and hyperspaces, can. j. math. 20 (1968) 1207– 1210. [15] m. wojdys lawski, retractes absolus et hyperespaces des continus, fund. math. 32 (1939) 184–192. 390 camillo costantini and wies law kubís received february 2002 revised september 2002 camillo costantini department of mathematics, university of torino, via carlo alberto, 10 10123 torino, italy e-mail address : costanti@dm.unito.it wies law kubís institute of mathematics, university of silesia, ul. bankowa 14 40-007 katowice, poland e-mail address : kubis@ux2.math.us.edu.pl @ appl. gen. topol. 16, no. 2(2015), 89-98doi:10.4995/agt.2015.1874 c© agt, upv, 2015 free paratopological groups ali sayed r. elfard school of mathematics and applied statistics, university of wollongong, australia (a.elfard@yahoo.com) abstract let fp(x) be the free paratopological group on a topological space x in the sense of markov. in this paper, we study the group fp(x) on a pα-space x where α is an infinite cardinal and then we prove that the group fp(x) is an alexandroff space if x is an alexandroff space. moreover, we introduce a neighborhood base at the identity of the group fp(x) when the space x is alexandroff and then we give some properties of this neighborhood base. as applications of these, we prove that the group fp(x) is t0 if x is t0, we characterize the spaces x for which the group fp(x) is a topological group and then we give a class of spaces x for which the group fp(x) has the inductive limit property. 2010 msc: primary 22a30; secondary 54d10; 54e99; 54h99. keywords: topological group; paratopological group; free paratopological group; alexandroff space; partition space, neighborhood base at the identity. 1. introduction let fp(x) and ap(x) be the free paratopological group and the free abelian paratopological group, respectively, on a topological space x in the sense of markov. the group fp(x) is the abstract free group fa(x) on x with the strongest paratopological group topology on fa(x) that induces the original topology on x and the abelian group ap(x) is the abstract free abelian group aa(x) on x with the strongest paratopological group topology on aa(x) that induces the original topology on x. for more information about free paratopological groups, see ([11], [7], [3], [4], [5]). received 13 november 2013 – accepted 8 march 2015 http://dx.doi.org/10.4995/agt.2015.1874 a. elfard in 1937, p. alexandroff [10] introduced a class of topological spaces under the name of diskrete räume (discrete space) which is a space in which an arbitrary intersections of open sets is open. now the name has been changed to alexandroff space since the discrete space is a space in which every singleton set is open. recently, researchers have shown an increased interest in studying alexandroff spaces. this may be due to the important applications of alexandroff spaces in some areas of mathematical sciences such as the field of computer science. in this paper, we study the groups fp(x) and ap(x) on a pα-space x, where α is an infinite cardinal (a topological space x is a pα-space, where α is an infinite cardinal if the set ⋂ c is open in x for each family c of open subsets of x with |c | < α). then in theorem 4.1, we prove that the groups fp(x) and ap(x) are alexandroff spaces if the space x is alexandroff and in theorem 4.4, we introduce simple neighborhood bases at the identities of the groups fp(x) and ap(x) for their topologies. moreover, we study the groups fp(x) and ap(x) in the case where x is a partition space and in another case where x is a t0 alexandroff space. as applications of these results, in theorem 5.1, we characterize the spaces x for which the paratopological groups fp(x) and ap(x) are topological groups and in theorem 5.6, we prove that the group fp(x) is t0 if the space x is t0. finally, in theorem 5.7, we give a class of spaces x for which the groups fp(x) and ap(x) have the inductive limit property. the content of this paper is adapted from the author’s thesis [5], chapter 3. we remark that the results in theorem 5.1 and theorem 5.6 were found independently by the author in his thesis [5]. however, similar to these results were found by pyrch ([8], [9]). 2. definitions and preliminaries a paratopological group is a pair (g, t ), where g is a group and t is a topology on g such that the mapping (x, y) 7→ xy of g×g into g is continuous. if in addition, the mapping x 7→ x−1 of g into g is continuous, then (g, t ) is a topological group. if (g, t ) is a paratopological group, then simply we denote it by g. marin and romaguera [6] described a complete neighborhood base at the identity of any paratopological group as follows: proposition 2.1. let g be a group and let n be a collection of subsets of g, where each member of n contains the identity element e of g. then the collection n is a base at e for a paratopological group topology on g if and only if the following conditions are satisfied: (1) for all u, v ∈ n , there exists w ∈ n such that w ⊆ u ∩ v ; (2) for each u ∈ n , there exists v ∈ n such that v 2 ⊆ u; (3) for each u ∈ n and for each x ∈ u, there exists v ∈ n such that xv ⊆ u and v x ⊆ u; and c© agt, upv, 2015 appl. gen. topol. 16, no. 2 90 free paratopological groups (4) for each u ∈ n and each x ∈ g, there exists v ∈ n such that xv x−1 ⊆ u. definition 2.2 ([3]). let x be a subspace of a paratopological group g. suppose that (1) the set x generates g algebraically, that is, 〈x〉 = g and (2) every continuous mapping f : x → h of x to an arbitrary paratopological group h extends to a continuous homomorphism f̂ : g → h. then g is called the markov free paratopological group on x, and is denoted by fp(x). by substituting “abelian paratopological group” for each occurrence of “paratopological group” above we obtain the definition of the markov free abelian paratopological group on x and we denote it by ap(x). remark 2.3. we denote the free topology of fp(x) by tf p and the free topology of ap(x) by tap and we note that the topologies tf p and tap are the strongest paratopological group topologies on the underlying sets of fp(x) and ap(x), respectively, that induce the original topology on x. 3. pα-spaces let x be a topological space and α be an infinite cardinal. we say that x is a pα-space if the set ⋂ c is open in x for each family c of open subsets of x with |c | < α. let τ be the topology of x. then we define the topology τα to be the intersection of all topologies o on x where τ ⊆ o and (x, o) is a pα-space. since the discrete topology on x contains τ and is a pα-space, τα exists and (x, τα) is a pα-space. we call the topology τα the pα-modification of τ. we note that if x is a pα-space, then x is a pα+-space, where α + is the successor cardinal of α. for the remain of this section we assume that α is a fixed infinite cardinal unless we say otherwise. theorem 3.1. let (x, τ) be a topological space and let α+ be the infinite successor cardinal of α. then the collection of all sets which are the intersection of fewer than β open subsets of x is a base for the topology τα on x, where β = α if α is regular and β = α+ if α is singular. proof. let τ = {ui}i∈i. we show that the collection b = { ⋂ d∈d ud : d ⊆ i and |d| < β} of subsets of x is a base for the topology τα on x, where β as defined in the statement of the theorem. it is well known that every infinite successor cardinal is regular, so in both cases, β is regular. we show first that b is a base for some topology τ∗ on x. if x ∈ x, then there exists i0 ∈ i where x ∈ ui0 and such that ui0 ∈ b. let b1, b2 ∈ b and let x ∈ b1 ∩b2. assume that b1 = ⋂ d∈d ud and b2 = ⋂ t∈t ut, where d, t ⊆ c© agt, upv, 2015 appl. gen. topol. 16, no. 2 91 a. elfard i and |d|, |t | < β. let r = d ∪ t . so |r| < β. hence b3 = ⋂ r∈r ur ∈ b and x ∈ b3 ⊆ b1 ∩ b2. therefore, b is a base for some topology τ ∗ on x. we show second that (x, τ∗) is a pα-space. let τ ∗ = {vj}j∈j and let m ⊆ j where |m| < β. then we have ⋂ m∈m vm = ⋂ m∈m ⋃ i∈im bm,i = ⋃ f : m→ ⋃ m∈m im,f(m)∈im∀m∈m ( ⋂ m∈m bm,f(m)) ∈ τ ∗, where im is an index set and bm,i ∈ b for all m ∈ m and i ∈ im. thus τ ∗ contains τ and (x, τ∗) is a pβ-space, which implies that in both cases of β, (x, τ∗) is a pα-space. now let τ̂ be a topology on x containing τ such that (x, τ̂) is a pα-space. then in the case where α is regular, we have b ⊆ τ̂ and in the case where α is singular, by the argument above, we have (x, τ̂) is a pα+-space, which implies that b ⊆ τ̂. thus τ∗ ⊆ τ̂ and hence τ∗ is the smallest topology on x containing τ such that (x, τ∗) is a pα-space. therefore, τ ∗ = τα. � proposition 3.2. let (g, τ) be a paratopological group. then (g, τα) is a paratopological group. proof. let g1, g2 ∈ g and let u ∈ τα contains g1g2. by theorem 3.1, there is a set λ, where |λ| < β and β is as in the theorem such that g1g2 ∈ ⋂ λ∈λ uλ ⊆ u where uλ ∈ τ for all λ ∈ λ. thus g1g2 ∈ uλ for all λ ∈ λ. since τ is a paratopological group topology on g, for each λ ∈ λ, there are v (λ), w(λ) ∈ τ containing g1, g2, respectively, such that v (λ)w(λ) ⊆ uλ. let u1 = ⋂ λ∈λ v (λ) and u2 = ⋂ λ∈λ w(λ). then u1u2 ⊆ uλ for all λ ∈ λ. hence, u1, u2 ∈ τα and u1u2 ⊆ ⋂ λ∈λ uλ ⊆ u. therefore, τα is a paratopological group topology on g. � proposition 3.3. let x be a topological space. then the group fp(x) is a pα-space if and only if the space x is a pα-space. proof. =⇒: it is easy to prove that x is a pα-space. ⇐=: let τ be the topology of x and let tf p be the free topology of fp(x). we show that (tf p )α = tf p . by proposition 3.2, (tf p )α is a paratopological group topology on fa(x) and it is stronger than tf p . however, tf p is the free paratopological group topology on fa(x), which is the strongest paratopological group topology on fa(x) inducing the original topology τ on x. since (tf p )α|x = (tf p |x)α and (tf p |x)α = (τ)α = τ, (tf p )α induces the topology τ of x. thus we have (tf p )α = tf p and therefore, fp(x) is a pα-space. � the same result of proposition 3.3 is true for ap(x). 4. free paratopological groups on alexandroff spaces a topological space x is said to be alexandroff [1] if the intersection of every family of open subsets of x is open in x. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 92 free paratopological groups we note that a topological space x is alexandroff if and only if x is a pαspace for every infinite cardinal α. thus by using proposition 3.3, we get the next result. theorem 4.1. the group fp(x) (ap(x)) on a space x is an alexandroff space if and only if x is an alexandroff space. let g be a group and let h be a submonoid of g. then we say that h is a normal submonoid of g if ghg−1 ∈ h for all h ∈ h and g ∈ g. proposition 4.2. if h is a normal submonoid of a group g, then {h} is a neighborhood base at the identity of g for a paratopological group topology on g. proof. let h be a normal submonoid of g. then {h} satisfies the conditions of proposition 2.1, and therefore, {h} is a neighborhood base at the identity of g for a paratopological group topology on g. � let x be a topological space. for each x ∈ x, let u(x) = ⋂ {u : x ∈ u and u is open}. then it is easy to see that the space x is alexandroff if and only if the set u(x) is open in x for each x ∈ x. let x be an alexandroff space and let fp(x) and ap(x) be the free paratopological group and the free abelian paratopological group, respectively, on x. let ua = ⋃ x∈x(u(x) − x) ⊆ ap(x). then we define na to be the smallest submonoid of ap(x) containing the set ua. so na is of the form na = {y1 − x1 + y2 − x2 + · · · + yn − xn : xi ∈ x, yi ∈ u(xi) for all i = 1, 2, . . . , n, n ∈ n}. or simply, we write na = 〈ua〉. since every submonoid of an abelian group is normal, na is normal. however, in this case, we will omit the word normal and say submonoid. let na = {na}. since na is a submonoid of ap(x), by proposition 4.2, na is a neighborhood base at the identity 0a of ap(x) for a paratopological group topology oa on aa(x). now for the group fp(x), let uf = ⋃ x∈x x −1u(x) ⊆ fp(x) and then we define nf to be the smallest normal submonoid of fp(x) containing the set uf . the normal submonoid nf consists exactly of the set of all elements of the form, w = g1x −1 1 y1g −1 1 · g2x −1 2 y2g −1 2 · · · gnx −1 n yng −1 n where n ∈ n, g1, g2, . . . , gn is an arbitrary finite system of elements of fa(x) and x−11 y1, x −1 2 y2, . . . , x −1 n yn is an arbitrary finite system of elements of uf . define nf = {nf }. by proposition 4.2, nf is a neighborhood base at the identity e of fp(x) for a paratopological group topology of on the free group fa(x). proposition 4.3. the topologies of and oa induce topologies coarser than the original topology on x. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 93 a. elfard theorem 4.4. the collection nf (na) is a neighborhood base at e (0a) for the free topology of fp(x) (ap(x)). proof. we prove the theorem for nf , since the proof for na is the same. we show first that the topology of is finer than the free topology tf p of fp(x). let ξ : x → g be a continuous mapping of the space x into an arbitrary paratopological group g. then ξ extends to a homomorphism ξ̂ : fa(x) → g. we show that ξ̂ is continuous with respect to the topology of . let v be a neighborhood of ξ̂(e) = eg in g. fix x ∈ x. then ξ(x)v is a neighborhood of ξ(x) in g. since ξ is continuous at x, ξ(u(x)) ⊆ ξ(x)v and since ξ̂|x = ξ, ξ̂(u(x)) ⊆ ξ̂(x)v . because ξ̂ is a homomorphism, ξ̂ ( x−1u(x) ) ⊆ v . since x is any point in x, we have (4.1) ξ̂ ( ⋃ x∈x x−1u(x) ) ⊆ v. fix n ∈ n. then there exists a neighborhood u of eg in g such that u n ⊆ v and also, for all g ∈ fa(x), there exists a neighborhood w of eg in g such that ξ̂(g)w ( ξ̂(g) )−1 ⊆ u. since v is any neighborhood of eg in g, from (4.1), we have ξ̂ ( ⋃ x∈x x −1u(x) ) ⊆ w . fix g ∈ fa(x). so we have ξ̂(g)ξ̂ ( ⋃ x∈x x−1u(x) )( ξ̂(g) )−1 ⊆ ξ̂(g)wξ̂(g)−1. since ξ̂ is a homomorphism, (4.2) ξ̂ ( ⋃ x∈x gx−1u(x)g−1 ) ⊆ ξ̂(g)wξ̂(g)−1 ⊆ u. since (4.2) holds for every g ∈ fa(x), we have ξ̂ ( ⋃ g∈fa(x) ⋃ x∈x gx−1u(x)g−1 ) ⊆ u. thus we have ξ̂ ( ( ⋃ g∈fa(x) ⋃ x∈x gx−1u(x)g−1 )n ) ⊆ un ⊆ v. since n is any element of n, ξ̂ ( ⋃ n∈n ( ⋃ g∈fa(x) ⋃ x∈x gx−1u(x)g−1 )n ) ⊆ v. since nf = ⋃ n∈n ( ⋃ g∈fa(x) ⋃ x∈x gx −1u(x)g−1 )n , we have ξ̂(nf ) ⊆ v . thus ξ̂ is continuous with respect to the topology of and therefore, of is finer than tf p . by proposition 4.3, of |x is coarser than the original topology on x. since of is finer that tf p , of |x induces the original topology on x. thus we satisfied the conditions of definition 2.2, which implies that of = tf p . therefore, nf is a neighborhood base at e of the group fp(x). � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 94 free paratopological groups now let hf = {gnf : g ∈ fa(x)} and let ha = {g + na : g ∈ aa(x)}. if g1, g2 ∈ fp(x) such that g1 ∈ g2nf , then we have g1nf ⊆ g2nf nf = g2nf . a similar result is true for ha. let x be a set. then for all k ∈ z, we define zk(x) = {x ǫ1 1 x ǫ2 2 . . . x ǫn n ∈ fa(x) : ∑n i=1 ǫi = k} and z a k (x) = {ǫ1x1 + ǫ2x2 + · · · + ǫnxn ∈ aa(x) : ∑n i=1 ǫi = k}. for every k1, k2 ∈ z and k1 6= k2, the sets zk1(x) and zk2(x) are disjoint and the sets zak1(x) and z a k2 (x) are disjoint. the set z0(x) is the smallest normal subgroup of fa(x) containing the set zf = ⋃ x∈x x −1x and the set za0 (x) is the smallest subgroup of aa(x) containing the set za = ⋃ x∈x x − x. let x be a topological space and let i : x → ap(x) be the identity mapping of the space x to the abelian group ap(x). then we extend i to the continuous homomorphism mapping î : fp(x) → ap(x). we call the mapping î the canonical mapping. theorem 4.5. let x be an alexandroff space. then the following are equivalent. (1) the space x is indiscrete. (2) nf = z0(x) in fp(x). (3) na = z a 0 (x) in ap(x). proof. (1)⇒(2): assume that x is indiscrete. then u(x) = x for all x ∈ x and so uf = zf , where zf is the generating set for z0(x). therefore, nf = z0(x). (2)⇒(3): assume that nf = z0(x). let î : fp(x) → ap(x) be the canonical mapping. thus î(nf ) = î(z0(x)). since î(nf ) = na and î(z0(x)) = za0 (x), so na = z a 0 (x). (3)⇒(1): assume that na = z a 0 (x). thus z a k (x) is open in ap(x) for each k ∈ z. since za1 (x) ∩ x = x and z a k (x) ∩ x = ∅ for all k ∈ z \ {1}, we have x is indiscrete. � we call a space x a partition space if x has a base which is a partition of x. clearly, every partition space is an alexandroff space. it is easy to see that if x is a partition space, then the collection {u(x)}x∈x is a partition on x. theorem 4.6. if x is a partition space, then the free paratopological groups fp(x) and ap(x) are partition spaces. proof. let x be a partition space. then nf and na are normal subgroups of fp(x) and ap(x), respectively. thus the collections hf and ha as defined above are partitions of fp(x) and ap(x), respectively. therefore the result follows. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 95 a. elfard 5. applications let ta be the topology of the subspace x −1 of fp(x), where x be any topological space. then by theorem 4.2 of [3], the topology ta has as an open base the collection {c−1 : c closed in x}. in this topology, the intersection of every collection of open subsets is open, and the space x−1 a = (x−1, ta) is therefore an alexandroff space. theorem 5.1. let x be a topological space. then the group fp(x) on x is a topological group if and only if x is a partition space. proof. =⇒: assume that fp(x) is a topological group. let u be an open set in x. by the argument above, the topology on the subspace x−1 a of fp(x) has the collection {c−1 : c is closed in x} as a base. thus (uc)−1 is open in x−1 a . since the inversion mapping of x−1 to x is a homeomorphism, uc is open in x. so u is closed in x and therefore, x is a partition space. ⇐=: if x is a partition space, then nf is a subgroup of fp(x). therefore, fp(x) is a topological group. the same proof works for ap(x). � proposition 5.2. let x be an alexandroff space and let fp(x) be the free paratopological group on x. then the space x is t0 if and only if for each w ∈ nf and w 6= e we have î(w) 6= 0a. proof. =⇒: suppose that there exists w ∈ nf and w 6= e such that î(w) = 0a, where w = g1x −1 1 y1g −1 1 g2x −1 2 y2g −1 2 · · · gnx −1 n yng −1 n for some n ∈ n, yi 6= xi and yi ∈ u(xi) for all i = 1, 2, . . . , n. then we have î(w) = y1 − x1 + y2 − x2 + · · · + yn − xn = 0a. if n = 1, then x1 = y1, which gives a contradiction. assume that n > 1. since î(w) = 0a, for each i ∈ a = {1, 2, . . . , n}, there exists ji ∈ a, where ji 6= i such that xi = yji. define σ : a → a by setting σ(i) = ji for all i ∈ a. clearly σ is a permutation on a. since any permutation can be written as product of cycles, there are m ∈ n, where 2 ≤ m ≤ n and distinct i1, i2, . . . , im ∈ a such that σ(i1) = i2, σ(i2) = i3, . . . , σ(im−1) = im, σ(im) = i1 and such that xik = yσ(ik) for k = 1, 2, . . . , m. thus xi1 = yi2, xi2 = yi3, . . . , xim−1 = yim, xim = yi1 and hence u(yi1) ⊆ u(xi1) = u(yi2) ⊆ u(xi2 ) = u(xi3) ⊆ · · · ⊆ u(xim−1) = u(yim) ⊆ u(xim) = u(yi1), which implies that u(yi1) = u(xi1 ) = u(yi2) = u(xi2 ) = · · · = u(xim−1 ) = u(yim). thus we can not separate the points yi1, xi1, yi2, xi2, . . . , xim−1, yim. therefore, x is not a t0 space. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 96 free paratopological groups ⇐=: assume that x is not t0. then there are x, y ∈ x such that x 6= y and u(x) = u(y), which implies that x ∈ u(y) and y ∈ u(x). hence w = x−1yxy−1 = (x−1y)(x(y−1x)x−1) ∈ nf and w 6= e, but î(w) = 0a. therefore, the space x is t0. � we note that a corollary of this result is that if x is an alexandroff t0 space, then î has the property that ker î ∩ nf = {e}. the following result is easy to prove. proposition 5.3. let g be a paratopological group. then g is a t0 space if and only if for all a ∈ g such that a 6= e, there exists a neighborhood u of e such that either a /∈u or a−1 /∈ u. proposition 5.4. let x be an alexandroff space. then fp(x) is a t0 space if and only if x is a t0 space. proof. =⇒: since x is a subspace of fp(x), the result follows. ⇐=: assume that x is t0. we claim that fp(x) is t0. in fact, if fp(x) is not t0, then by proposition 5.3, there exists w ∈ fp(x), w 6= e such that w, w−1 ∈ nf . hence by proposition 5.2, î(w) 6= 0a and it is easy to see that î(w), −î(w) ∈ na, which implies that î(w), −î(w) are in every neighborhood of 0a. once again by proposition 5.3, ap(x) is not t0 and so by proposition 3.4 of [7] (which says that if a space x is t0, then ap(x) is t0), x is not a t0 space, which gives a contradiction. therefore, fp(x) is t0. � fix n ∈ n and let rn = {1, 2, . . . , n} ⊆ n. for i = 0, 1, . . . , n, define rn,i = {1, 2, . . . , i} and τn = {rn,i : i = 0, . . . , n}. then it is easy to see that τn is a topology on rn. let m, k ∈ rn, where m 6= k and assume that m < k. then m ∈ rn,m and k /∈ rn,m. therefore, (rn, τn) is a t0 space. proposition 5.5. let x be a t0 space and let x1, x2, . . . , xn be distinct elements of x. then there exists a continuous mapping µ: x → rn such that µ|{x1,x2,...,xn} is one-to-one. theorem 5.6. let x be a topological space. then the free paratopological group fp(x) on x is t0 if and only if the space x is t0. proof. =⇒: it is clear. ⇐=: let w = xǫ11 x ǫ2 2 · · · x ǫm m ∈ fp(x) for some m ∈ n such that w 6= e. choose indices i1, i2, . . . , in for some n ≤ m such that xi1 , xi2, . . . , xin are the distinct letters among x1, x2, . . . , xm. then by proposition 5.5, there exists a continuous mapping µ : x → rn such that µ|{xi1 ,xi2 ,...,xin } is one-toone, where rn is the space defined above. then we extend µ to a continuous homomorphism µ̂ : fp(x) → fp(rn). since µ|{xi1 ,...,xin } is one-to-one, µ̂(w) = [µ̂(x1)] ǫ1[µ̂(x2)] ǫ2 · · · [µ̂(xn)] ǫn 6= e∗, where e∗ is the identity of fp(rn). by proposition 5.4, we have fp(rn) is a t0 space. so there is an open set u in fp(rn), which contains one of e ∗ or µ̂(w) and does not contain the other. say e∗ ∈ u and µ̂(w) /∈ u. since µ̂ is continuous, µ̂−1(u) is an open set in c© agt, upv, 2015 appl. gen. topol. 16, no. 2 97 a. elfard fp(x) such that e ∈ µ̂−1(u) and w /∈ µ̂−1(u). similarly for the other case. therefore, the free paratopological group fp(x) is t0. � a topological space x is said to be the inductive limit of a cover c of x if a subset v of x is open whenever v ∩ u is open in u for each u ∈ c . a parallel result of the next theorem was proved in proposition 7.4.8 of [2] in the case of free topological groups. theorem 5.7. let x be a t1 p-space. then the free paratopological group fp(x) (ap(x)) is the inductive limit of the collection {fpn(x): n ∈ n} ({apn(x): n ∈ n}). proof. we prove the statement for fp(x), since the proof for ap(x) is similar. let c be a subset of fp(x) such that c ∩ fpn(x) is closed in fpn(x) for all n ∈ n. by theorem 4.1.3 of [3], the sets fpn(x) are closed in fp(x) for all n ∈ n. thus the sets c ∩ fpn(x) are closed in fp(x) for all n ∈ n, which implies that c is a countable union of closed sets in fp(x). since the group fp(x) is a p-space, c is closed in fp(x) and then fp(x) is the inductive limit of the collection {fpn(x): n ∈ n}. � acknowledgements. the author wishes to thank associate professor peter nickolas for helpful comments. references [1] f. g. arenas, alexandroff spaces, acta math. univ. comenian. (n.s.) 68 (1999), 17–25. [2] a. v. arhangel’skii and m. g. tkachenko, topological groups and related structures, atlantis studies in mathematics, vol. 1, atlantis press, paris, 2008. [3] a. s. elfard and p. nickolas, on the topology of free paratopological groups, bulletin of the london mathematical society 44, no. 6 (2012), 1103–1115. [4] a. s. elfard and p. nickolas, on the topology of free paratopological groups. ii, topology appl. 160, no. 1 (2013), 220–229. [5] a. s. elfard, free paratopological groups, phd thesis, university of wollongong, australia (2012). [6] j. maŕın and s. romaguera, a bitopological view of quasi-topological groups, indian j. pure appl. math. 27 (1996),393–405. [7] n. m. pyrch and o. v. ravsky, on free paratopological groups, mat. stud. 25 (2006), 115–125. [8] n. m. pyrch, on isomorphisms of the free paratopological groups and free homogeneous spaces i, visnyk liviv univ. ser. mech-math. 63 (2005), 224–232. [9] n. m. pyrch, on isomorphisms of the free paratopological groups and free homogeneous spaces ii, visnyk liviv univ. textbf71 (2009), 191–203. [10] p. alexandroff, diskrete räume, mat. sb. (n.s.) 2 (1937), 501–û518. [11] s. romaguera, m. sanchis and m. tkačenko, free paratopological groups, proceedings of the 17th summer conference on topology and its applications 27 (2003), 613–640. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 98 geoili.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 2, 2005 pp. 143-148 a note on locally ν-bounded spaces d. n. georgiou and s. d. iliadis abstract. in this paper, on the family o(y ) of all open subsets of a space y (actually on a complete lattice) we define the so called strong ν-scott topology, denoted by τs ν , where ν is an infinite cardinal. this topology defines on the set c(y, z) of all continuous functions on the space y to a space z a topology ts ν . the topology ts ν , is always larger than or equal to the strong isbell topology (see [8]). we study the topology ts ν in the case where y is a locally ν-bounded space. 2000 ams classification: 54c35 keywords: strong scott topology, strong isbell topology, function space, admissible topology. 1. basic notions let x be a space and g a map of x into c(y, z). by g̃ we denote the map of x × y to z such that g̃(x, y) = g(x)(y) for every (x, y) ∈ x × y . a topology t on c(y, z) is called admissible if for every space x, the continuity of a map g : x → ct(y, z) implies that of the map g̃ : x × y → z. equivalently, a topology t on c(y, z) is admissible if the evaluation map e : ct(y, z) × y → z defined by relation e(f, y) = f(y), (f, y) ∈ c(y, z) × y , is continuous (see [1]). let l be a poset. the scott topology τω (see, for example, [5]) is the family of all subsets ih of l such that: (α) ih =↑ ih, where ↑ ih = {y ∈ l : (∃x ∈ ih) x ≤ y}, and (β) for every directed subset d of l with sup d ∈ ih, d ∩ ih 6= ∅. below, we consider the poset o(y ) of all open subsets of the space y on which the inclusion is considered as the order. the isbell topology tω on c(y, z) (see, for example, [8], [11] and [9]) is the topology for which the family of all sets of the form (ih, u) = {f ∈ c(y, z) : f−1(u) ∈ ih}, 144 d. n. georgiou and s. d. iliadis where u ∈ o(z) and ih ∈ τω, constitute a subbasis for this topology. the notion of a bounded subset was introduced in [3] and the notion of a locally bounded space in [7]. some generalizations of locally bounded spaces are given in [10]. the notion of the strong scott topology (defined on a complete lattice) was given in [8]. this topology determines on the set c(y, z) a topology called the strong isbell topology (see [8]). it is proved that a space y is locally bounded if and only if the strong isbell topology on c(y, 2), where 2 is the sierpinski space, is admissible. in the case, where y is locally bounded and z is an arbitrary space, it is proved that the strong isbell topology on c(y, z) is admissible. in this paper we denote by ν a fixed infinite cardinal. a subset d of a poset l is called ν-directed if every subset of d with cardinality less than ν has an upper bound in d (see [4]). suppose that l is a complete lattice. we say that x is ν-way below y and write x <<ν y (see [4]) if for every ν-directed subset d of l the relation y ≤ sup d implies the existence of d ∈ d with x ≤ d. in particular, for two elements u and v of the complete lattice o(y ) we have: u <<ν v if for every open cover {wi : i ∈ i} of v there is a subcollection {wi : i ∈ j ⊆ i} of this cover such that |j| < ν and u ⊆ ∪{wi : i ∈ j}. it is clear that if u ⊆ v <<ν y , then u <<ν y . 2. other notions definition 2.1. a subset b of y is called ν-bounded if every open cover of y contains a cover of b of cardinality less than that of ν. (for the related notion of an (m, n)-bounded subset see [6].) a space is called locally ν-bounded if it has a basis for the open subsets consisting of ν-bounded sets. (for the related notion of a local p-space see [10].) definition 2.2. let (l, ≤) be a fixed complete lattice and 1 the maximal element of l. by τsν we denote the family of all subsets ih of l such that: (α) ih =↑ ih, where ↑ ih = {y ∈ l : (∃x ∈ ih) x ≤ y}, and (β) for every ν-directed subset d of l with sup d = 1 we have d ∩ ih 6= ∅. it is clear that, the family τsν is a t0 topology on l called the strong ν-scott topology. in the case, where l = o(y ), a subset ih of o(y ) belongs to the strong ν-scott topology if the following properties are true: property (α). the conditions u ∈ ih, v ∈ o(y ), and u ⊆ v imply v ∈ ih. property (β). for every open cover {ui : i ∈ i} of y there exists a subset j of i of cardinality less than ν such that ∪{ui : i ∈ j} ∈ ih. remark 2.3. if µ is an infinite cardinal such that µ ≤ ν, then τsω ⊆ τ s µ ⊆ τ s ν , where ω is the first infinite cardinal. definition 2.4. let l be a complete lattice. an element x ∈ l is called ν-bounded if x <<ν 1. on locally ν-bounded spaces 145 the lattice l is called weakly ν-continuous if for all x ∈ l x = sup{u ∈ l : u ≤ x and u <<ν 1}. in the case, where l = o(y ), a set u ∈ o(y ) is ν-bounded if u <<ν y . notation. we denote by tsν the topology on the set c(y, z) for which the sets of the form: (ih, u) = {f ∈ c(y, z) : f−1(u) ∈ ih}, where u ∈ o(z) and ih ∈ τsν , compose a subbasis. obviously, if ω ≤ µ ≤ ν, then tsω ⊆ t s µ ⊆ t s ν. remark 2.5. for ν = ω the notions of an ω-bounded subset, a locally ωbounded space, and a weakly ω-continuous lattice coincide with the notions of a bounded subset, a locally bounded space, and a weakly continuous lattice, respectively. also, the topologies τsω and t s ω coincide with the strong scott topology and the strong isbell topology, respectively. 3. the results proposition 3.1. if y is locally ν-bounded, then the topology tsν on c(y, z) is admissible. proof. it is sufficient to prove that the evaluation map e : cts ν (y, z) × y → z is continuous. let (f, y) ∈ cts ν (y, z) × y , w ∈ o(z), and e(f, y) = f(y) ∈ w. we need to prove that there exist ih ∈ τsν , u ∈ o(z), and an open neighborhood v of y in y such that f ∈ (ih, u) and e((ih, u) × v ) ⊆ w. since y is locally ν-bounded and y ∈ f−1(w) there exists an open νbounded set v such that: y ∈ v ⊆ f−1(w). we consider the set ih = {p ∈ o(y ) : v ⊆ p} and prove that ih ∈ τsν , that is ih satisfies properties (α) and (β). property (α) is clear. property (β). let {ui : i ∈ i} be an open cover of y . since v is νbounded there exists a subset j of i of cardinality less than of ν such that v ⊆ ⋃ {ui : i ∈ j}. by the definition of ih we have ⋃ {ui : i ∈ j} ∈ ih. since v ⊆ f−1(w) we have f−1(w) ∈ ih and therefore f ∈ (ih, w). thus, the subset (ih, w) × v is a neighborhood of (f, y) in cτs ν (y, z) × y . now, we prove that e((ih, w) × v ) ⊆ w . let (g, z) ∈ (ih, w) × v . then g−1(w) ∈ ih, z ∈ v , and v ⊆ g−1(w). therefore e((g, z)) = g(z) ∈ w. 146 d. n. georgiou and s. d. iliadis thus, the map e is continuous which means that tsν is admissible. � proposition 3.2. for the space y the following statements are equivalent: (1) y is locally ν-bounded. (2) for every space z the evaluation map e : cts ν (y, z) × y → z is continuous. (3) the evaluation map e : cts ν (y, 2) × y → 2 is continuous. (4) for every open neighborhood v of a point y of y there is an open set ih ∈ τsν such that v ∈ ih and the set ∩{p : p ∈ ih} is a neighborhood of y in y . (5) the lattice o(y ) is weakly ν-continuous. proof. (1) =⇒ (2) follows by proposition 3.1. (2) =⇒ (3) it is obvious. (3) =⇒ (4) let v be an open neighborhood of y in y . consider the sets o(y ) and c(y, 2). we identify each element u of o(y ) with the element fu of c(y, 2) for which fu(u) ⊆ {0} and fu (y \ u) ⊆ {1}. then, each topology on one of the above sets can be considered as a topology on the other. in this case tsν = τ s ν and the map e : o(y ) × y → 2 is continuous. since e(v, y) = e(fv , y) = fv (y) = 0, the continuity of e implies that for the open neighborhood {0} of e(v, y) in 2 there exist an open neighborhood ih ∈ τsν of v in o(y ) and an open neighborhood v ′ of y in y such that e(ih ×v ′) ⊆ {0}. obviously, v ∈ ih. we need to prove that the relation v ′ ⊆ ∩{p : p ∈ ih} is true. indeed, in the opposite case, there exist z ∈ v ′ and p ∈ ih such that z 6∈ p . then, e(p, z) = e(fp , z) = fp (z) = 1 which contradicts the fact that e(ih × v ′) ⊆ {0}. thus, the set ∩{p : p ∈ ih} is a neighborhood of y in y . (4) =⇒ (5) let v be an open subset of y . it suffices to prove that for every y ∈ v there exists an open ν-bounded neighborhood u of y such that u ⊆ v . by assumption there exists a set ih ∈ τsν such that v ∈ ih and ∩{p : p ∈ ih} ≡ q is a neighborhood of y in y . we prove that the set q is ν-bounded. let {ui : i ∈ i} be an open cover of y . since ih ∈ τ s ν , by the definition of τsν there exists a subset j of i of cardinality less than of ν such that ∪{ui : i ∈ j} ∈ ih and therefore q ⊆ ∪{ui : i ∈ j}, which means that q is ν-bounded. the required open neighborhood of y is an open subset u of y such that y ∈ u ⊆ q. (5) =⇒ (1) let y ∈ y and v be an open neighborhood of y. since o(y ) is weakly ν-continuous we have v = ∪{u ∈ o(y ) : u ⊆ v and u <<ν y } and therefore there exists an open ν-bounded subset u of y such that y ∈ u ⊆ v. � on locally ν-bounded spaces 147 proposition 3.3. if y is ν-locally bounded, then the usual compositions operations (see [2]) i) t : ctco(x, y ) × cts ν (y, z) → ctco(x, z) and ii) t : ctω (x, y ) × cts ν (y, z) → ctω (x, z), where tco and tω is the compact open and the isbell topology, respectively, are continuous for arbitrary spaces x and z. proof. we prove only the statement ii). the proof of the case i) is similar. let (f, g) ∈ ctω (x, y ) × cts ν (y, z), ih a scott open subset of x, and u ∈ o(z) such that t (f, g) = g ◦ f ∈ (ih, u). it suffices to prove that there exist open neighborhoods ih1 and ih2 of f and g in ctω (x, y ) and cts ν (y, z), respectively, such that t (ih1 × ih2) ⊆ (ih, u). we consider the open set g−1(u) of y . by locally ν-boundedness of y , for each point y ∈ g−1(u) ∈ o(y ), there is an open set vy of y such that y ∈ vy ⊆ g −1(u) and vy <<ν y . therefore g−1(u) = ∪{vy : y ∈ g −1(u)} and f−1(g−1(u)) = f−1(∪{vy : y ∈ g −1(u)}) or (g ◦ f)−1(u) = ∪{f−1(vy) : y ∈ g −1(u)}. since g ◦ f ∈ (ih, u) we have (g ◦ f)−1(u) ∈ ih or ∪{f−1(vy) : y ∈ g −1(u)} ∈ ih. thus there exists a finite subset j of g−1(u) such that ∪{f−1(vy) : y ∈ j} ∈ ih. let v = ∪{vy : y ∈ j}. then f −1(v ) ∈ ih and v is a ν-bounded open set of y . the set ih(v ) = {w ∈ o(y ) : v ⊆ w} is strong ν-scott open (see the proof of proposition 3.1). since v = ∪{vy : y ∈ j ⊆ g −1(u)} and g −1(u) = ∪{vy : y ∈ g −1(u)} we have that v ⊆ g−1(u) and therefore g−1(u) ∈ ih(v ). setting ih1 = (ih, v ) and ih2 = (ih(v ), u) we have that the set ih1 × ih2 = (ih, v ) × (ih(v ), u) is an open neighborhood of (f, g) in ctω (x, y ) × cts ν (y, z). finally, we prove that t ((ih, v ) × (ih(v ), u)) ⊆ (ih, u). let (p, q) ∈ (ih, v ) × (ih(v ), u). then, p−1(v ) ∈ ih and q−1(u) ∈ ih(v ). therefore v ⊆ q−1(u). thus, p−1(v ) ⊆ p−1(q−1(u)) = (q ◦ p)−1(u). since p−1(v ) ∈ ih, (q ◦ p)−1(u) ∈ ih, and therefore q ◦ p ∈ (ih, u). � 148 d. n. georgiou and s. d. iliadis references [1] r. arens and j. dugundji, topologies for function spaces, pacific j. math. 1(1951), 5-31. [2] j. dugundji, topology, allyn and bacon, inc., boston, mass. 1966. [3] s. gagola and m. gemignani, absolutely bounded sets, mathematica japonicae, vol. 13, no. 2 (1968), 129-132. [4] d. n. georgiou and s. d. iliadis, a generalization of core compact spaces, (v iberoamerican conference of topology and its applications) topology and its applications. [5] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. mislove and d. s. scott, a compendium of continuous lattices, springer, berlin-heidelberg-new york 1980. [6] p. lambrinos, subsets (m, n)-bounded in a topological space, mathematica balkanica, 4(1974), 391-397. [7] p. lambrinos, locally bounded spaces, proceedings of the edinburgh mathematical society, vol. 19 (series ii) (1975), 321-325. [8] p. lambrinos and b. k. papadopoulos, the (strong) isbell topology and (weakly) continuous lattices, continuous lattices and applications, lecture notes in pure and appl. math. no. 101, marcel dekker, new york 1984, 191-211. [9] r. mccoy and i. ntantu, topological properties of spaces of continuous functions, lecture notes in mathematics, 1315, springer verlang (1988). [10] h. poppe, on locally defined topological notions, q and a in general topology, vol. 13 (1995), 39-53. [11] f. schwarz and s. weck, scott topology, isbell topology and continuous convergence, lecture notes in pure and appl. math. no. 101, marcel dekker, new york 1984, 251271. received october 2004 accepted january 2005 d. n. georgiou (georgiou@math.upatras.gr) department of mathematics, university of patras, 265 00 patras, greece. s. d. iliadis (iliadis@math.upatras.gr) department of mathematics, university of patras, 265 00 patras, greece. @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 467–474 n-tuple relations and topologies on function spaces d. n. georgiou, s. d. iliadis and b. k. papadopoulos dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. in [7] some results concerning s-splitting, s-jointly continuous, d-splitting and d-jointly continuous topologies are considered, where s and d are the sierpinski space and the double-point space, respectively. here we generalize these results replacing the spaces s and d by any finite space. 2000 ams classification: 54c35. keywords: function space, a-splitting topology, a-jointly continuous topology. 1. introduction. by y and z we denote two fixed topological spaces and by tz the topology of z. by c(y,z) we denote the set of all continuous maps of y into z. if τ is a topology on the set c(y,z), then the corresponding topological space is denoted by cτ (y,z). let x be a space and f : x ×y → z be a continuous map. by fx, where x ∈ x, we denote the continuous map of y into z, for which fx(y) = f(x,y), for every y ∈ y . by f̂ we denote the map of x into the set c(y,z), for which f̂(x) = fx for every x ∈ x. let g be a map of the space x into the set c(y,z). by g̃ we denote the map of the space x × y into the space z, for which g̃(x,y) = g(x)(y) for every (x,y) ∈ x ×y . a topology t on c(y,z) is called splitting if for every space x, the continuity of a map f : x×y → z implies that of the map f̂ : x → ct(y,z). a topology t on c(y,z) is called jointly continuous if for every space x, the continuity of a map g : x → ct(y,z) implies that of the map g̃ : x ×y → z (see [5], [1], [2] and [3]). if in the above definitions it is assumed that the space x belongs to a given family a of spaces, then the topology τ is called a−splitting (respectively, 468 d. n. georgiou, s. d. iliadis and b. k. papadopoulos a−jointly continuous) (see [6]). in the present paper we shall considered only the case a = {f}, where f is a space, and instead of a-splitting and a-jointly continuous we write f-splitting and f-jointly continuous. let x be a space with a topology τ. we denote (see, for example, [8]) by ≤ τ (respectively, by ∼ τ ) a preorder (respectively, an equivalence relation) on x defined as follows: if x,y ∈ x, then we write x ≤ τ y (respectively, x∼ τ y) if and only if x ∈ clx({y}) (respectively, x ∈ clx({y}) and y ∈ clx({x})). (by clx(q) we denote the closure of a set q in the space x). on the set c(y,z) we denote a preorder ≤ (respectively, an equivalence relation ∼ ) as follows: if g,f ∈ c(y,z), then we write g ≤ f (respectively, g ∼ f) if g(y) ≤ τz f(y) (respectively, g(y) ∼ τz f(y)) for every y ∈ y (see, for example, [7]). by s we denote the sierpinski space, that is, the set {0, 1} equipped with the topology τ(s) ≡ {∅,{0, 1},{1}}, and by d the set {0, 1} with the trivial topology. in [7] the notions of s-splitting and s-jointly continuous (respectively, d-splitting and d-jointly continuous) topologies are characterized by the above preorders (respectively, equivalence relations) on c(y,z). by the trivial topology on a set x we mean the topology {∅,x}. let u be a quasi-uniformity on the space z (see, for example, [4]). this quasi-uniformity defines on the set c(y,z) a quasi-uniformity q(u) as follows (see [11]): the set of all subsets of c(y,z) of the form (y,u) = {(f,g) ∈ c(y,z) ×c(y,z) : (f(y),g(y)) ∈ u, for every y ∈ y}, where u ∈ u, is a basis for the quasi-uniformity q(u). we denote by τq(u) (see [11]) the topology on c(y,z), which is defined by the quasi-uniformity q(u), that is: the subbasic neighborhoods of an arbitrary element f ∈ c(y,z) in τq(u) are of the form: (y,u)[f] = {g ∈ c(y,z) : (f,g) ∈ (y,u)}, where u ∈ u. in this case we shall say also that τq(u) is generated by the quasiuniformity u. let o(y ) be the family of all open sets of the space y . the scott topology on o(y ) (see, for example, [8]) is defined as follows: a subset ih of o(y ) is open if: (α) the conditions u ∈ ih, v ∈o(y ), and u ⊆ v imply v ∈ ih, and (β) for every collection of open sets of y , whose union belongs to ih, there are finitely many elements of this collection whose union also belongs to ih. the isbell topology τis on c(y,z) (see [9] and [10]) is the topology for which the family of all sets of the form (ih,u) = {f ∈ c(y,z) : f−1(u) ∈ ih}, where ih is scott open in o(y ) and u ∈o(z), is a subbasis. the pointwise topology (see, for example, [3]) τp on c(y,z) is the topology for which the family of all sets of the form n-tuple relations and topologies on function spaces 469 ({y},u) = {f ∈ c(y,z) : f(y) ∈ u}, where y ∈ y and u ∈o(z), is a subbasis. the compact open (see [5]) topology τc on c(y,z) is the topology for which the family of all sets of the form (k,u) = {f ∈ c(y,z) : f(k) ⊆ u}, where k is a compact subset of y and u ∈o(z), is a subbasis. below, we recall some well known results: (1) the pointwise topology, the compact open topology and the isbell topology on c(y,z) are always splitting (see, for example, [1], [2], [3], [5], [9] and [10]). (2) the compact open topology on c(y,z) is jointly continuous if y is locally compact (see [5] and [2]). (3) the isbell topology on c(y,z) is jointly continuous if y is corecompact (see, for example, [9]). (4) the topology τq(u) is jointly continuous (see [11]). 2. f-splitting and f-jointly continuous topologies. in the paper we denote by f a non-discrete space which is the set {0, 1, ...,n}, n > 0, equipped with an arbitrary fixed topology. by uj, j = 0, 1, ...,n, we denote the intersection of all open neighborhoods of j in f. it is clear that if f is the discrete space, then every topology τ on c(y,z) is f-splitting and f-jointly continuous. theorem 2.1. the trivial topology and, hence, every topology on the set c(y,z) is f-jointly continuous if and only if the topology of z is trivial. proof. suppose that the topology of z is trivial. then for any topology τ on c(y,z) and any continuous map g : f → cτ (y,z), the map g̃ : f ×y → z is trivially continuous, that is τ is f-jointly continuous. conversely, suppose that the trivial topology τ on c(y,z) is f-jointly continuous. we prove that the topology of z is trivial. indeed, in the opposite case, there exist two distinct elements z1, z2 of z and an open subset u of z such that z1 ∈ u and z2 6∈ u. we consider the maps f,g ∈ c(y,z) such that f(y ) = {z1} and g(y ) = {z2}. denote by i, the element of f such that ui 6= {i}. let g : f → cτ (y,z) be a map such that g(i) = f and g(j) = g, for every j ∈ f \ {i}. since τ is trivial, the map g is continuous. since τ is f-jointly continuous, the map g̃ : f × y → z is also continuous. by the definition of g̃, g̃(i,y) = g(i)(y) = f(y) = z1 ∈ u, y ∈ y . therefore for a fixed y ∈ y there exists an open neighborhood vy such that g̃(ui ×vy) ⊆ u. let j ∈ ui \{i}. then, we have g̃(j,y) = g(j)(y) = g(y) = z2 6∈ u which is a contradiction. thus the topology of z is trivial. � theorem 2.2. if the discrete topology, and hence, every topology on c(y,z) is f-splitting, then z is a t0 space. 470 d. n. georgiou, s. d. iliadis and b. k. papadopoulos proof. suppose that the discrete topology τ on c(y,z) is f-splitting and z is not t0 space. we shall construct a continuous map f : f ×y → z such that f̂ is not continuous, which will be a contradiction. there exist two distinct elements z1, z2 of z such that either z1,z2 ∈ v or z1,z2 6∈ v for every open subset v of z. let i be an element of f such that ui 6= {i}. we consider the map f : f × y → z such that f(i,y) = z1 for every y ∈ y , and f(j,y) = z2 for every j ∈ f \{i} and y ∈ y . let v be an open subset of z. then, either f−1(v ) = f×y or f−1(v ) = ∅, which means that f is continuous. by the definition of f̂ : f → cτ (y,z) we have f̂(i)(y ) = {z1}, and f̂(j)(y ) = {z2} for every j ∈ f \{i}. let j ∈ ui \{i}. then f̂(j) 6∈ {f̂(i)}, that is, f̂(ui) 6⊆ {f̂(i)}, which means that f̂ is not continuous. � theorem 2.3. let z be a t1 space. then, the discrete topology, and hence, every topology on c(y,z) is f-splitting. proof. let τ be the discrete topology on c(y,z) and f : f × y → z a continuous map. we prove that the map f̂ : f → cτ (y,z) is continuous. let i ∈ f and f̂(i) = f. then f ∈ {f} ∈ τ. it is suffices to prove that f̂(ui) ⊆ {f}, that is, f̂(j) = f for every j ∈ ui. let j ∈ ui and y be an arbitrary point of y . we need to prove that f̂(j)(y) = f(y). let u be an arbitrary open neighborhood of f(y) = f̂(i)(y) = f(i,y) in z. since the map f is continuous there exists an open neighborhood vy of y in y such that f(ui × vy) ⊆ u. therefore, f(j,y) = f̂(j)(y) ∈ u, which means that f(y) ∈ clz({f̂(j)(y)}). since z is a t1 space, f(y) = f̂(j)(y). hence, f̂(j) = f. thus, the map f̂ : f → cτ (y,z) is continuous and therefore the topology τ on c(y,z) is f-splitting. � theorem 2.4. the pointwise topology τp, the compact-open topology τc, and the isbell topology τis on c(y,z) are f-splitting and f-jointly continuous. proof. first, we prove that τp is f-jointly continuous. let g : f → cτp(y,z) be a continuous map. we need to prove that the map g̃ : f × y → z is continuous. let (i,y) ∈ f × y and u be an arbitrary open neighborhood of g̃(i,y) = g(i)(y) in z. then g(i) ∈ ({y},u). since g is continuous, g(ui) ⊆ ({y},u). also, since the map g(j), j ∈ ui, is continuous and g(j)(y) ∈ u there exists an open neighborhood v jy of y in y such that g(j)(v j y ) ⊆ u. let vy = ∩{v jy : j ∈ ui}. then, g̃(ui × vy) ⊆ u. thus, the map g̃ is continuous and the therefore the topology τp is f-jointly continuous. since τp ⊆ τc and τp ⊆ τis (see [10]) the topologies τc and τis are also f-jointly continuous. finally, since the topologies τp, τc and τis are splitting, they are also fsplitting. � n-tuple relations and topologies on function spaces 471 theorem 2.5. the topology τq(u) on the set c(y,z) generated by a quasiuniformity u on the space z is f-splitting and f-jointly continuous. proof. let u be a quasi-uniformity on the space z. since τq(u) is jointly continuous (see [11]), this topology is also f-jointly continuous. we prove that τq(u) is f-splitting. let f : f × y → z be a continuous map. we need to prove that f̂ : f → cτq(u) (y,z) is continuous. let i ∈ f and f̂(i) = fi. the set (y,u)[fi] = {h ∈ c(y,z) : (fi,h) ∈ (y,u)}, where u is an element of u, is an open neighborhood of fi in cτq(u) (y,z). we prove that f̂(ui) ⊆ (y,u)[fi]. let j ∈ ui. it is suffices to prove that f̂(j) = fj ∈ (y,u)[fi], that is fj ∈ (y,u)[fi] or (fi(y),fj(y)) ∈ u for every y ∈ y . let y ∈ y and u[fi(y)] = {z ∈ z : (fi(y),z) ∈ u}. since f is continuous there exists an open neighborhood vy of y in y such that f(ui × vy) ⊆ u[fi(y)]. so, for the element (j,y) of ui × vy we have f(j,y) = fj(y) ∈ u[fi(y)] or (fi(y),fj(y)) ∈ u. thus, the map f̂ is continuous and therefore the topology τq(u) is f-splitting. � definition 2.6. for every space x with a topology t we define an (n + 1)tuple relation denoted by rt in x as follows: an (n + 1)-tuple (x0,x1, ...,xn) of elements of x belongs to rt if for every i,j ∈ f, xi ∈ clx({xj}) provided that i ∈ clf({j}). we observe that if t1, t2 are two topologies on a set x such that t1 ⊆ t2, then rt2 ⊆ rt1 . definition 2.7. on the set c(y,z) we define an (n+1)-tuple relation denoted by r as follows: an (n + 1)-tuple (f0,f1, ...,fn) of elements of c(y,z) belongs to r if (f0(y),f1(y), ...,fn(y)) ∈ rtz for every y ∈ y . below we give necessary and sufficient conditions for an arbitrary topology τ on c(y,z) to be f-splitting or f-jointly continuous. theorem 2.8. a topology τ on c(y,z) is f-splitting if and only if r ⊆ rτ . proof. let τ be an f-splitting topology on c(y,z). suppose that (f0,f1, ...,fn) ∈ r. we need to prove that (f0,f1, ...,fn) ∈ rτ . let f : f×y → z be a map for which f(i,y) = fi(y), for every i ∈ f and y ∈ y . this map is continuous. indeed, let u be an open neighborhood of fi(y) in z. since fi is continuous, the set f −1 i (u) is open neighborhood of y in y . therefore it is sufficient to prove that: f(ui ×f−1i (u)) ⊆ u. let (j,y′) ∈ ui × f−1i (u). by the definition of f, f(j,y ′) = fj(y′). since j ∈ ui we have i ∈ clf({j}). also, by the definition of the (n + 1)-tuple relation r we have fi(y′) ∈ clz(fj(y′)). since fi(y′) ∈ u we have fj(y′) ∈ u. thus, f(ui ×f−1i (u)) ⊆ u, that is f is continuous. furthermore, since τ is f-splitting, the map f̂ : f→ cτ (y,z) is continuous. 472 d. n. georgiou, s. d. iliadis and b. k. papadopoulos now, we prove that (f0,f1, ...,fn) ∈ rτ . let i,j ∈ f such that i ∈ clf({j}). we need to prove that fi ∈ clcτ (y,z)({fj}). let w be an open neighborhood of fi in cτ (y,z). then, f̂−1(w) is an open neighborhood of i in f and therefore j ∈ f̂−1(w). this means that f̂(j) = fj ∈ w and therefore fi ∈ clcτ (y,z)({fj}). hence, (f0,f1, ...,fn) ∈ r τ . conversely, let τ be a topology on c(y,z) such that the condition (f0,f1, ..,fn) ∈ r implies (f0,f1, ...,fn) ∈ rτ . we prove that τ is f-splitting. let f : f×y → z be a continuous map. consider the map f̂ : f→ cτ (y,z) and let f̂(i) = fi, i ∈ f. first, we prove that (f0,f1, ...,fn) ∈ r. indeed, let y ∈ y . consider the (n + 1)-tuple (f0(y),f1(y), ...,fn(y)) and suppose that i ∈ clf({j}). let u be an open neighborhood of fi(y) in z. since f(i,y) = fi(y) and f is continuous, the set f−1(u) is an open neighborhood of (i,y) in f×y . therefore there exist open sets v and w of f and y , respectively, such that (i,y) ∈ v × w ⊆ f−1(u). this means that j ∈ v and f(j,y) = fj(y) ∈ u and therefore fi(y) ∈ clz(fj(y)), that is (f0(y),f1(y), ...,fn(y)) ∈ rtz . hence (f0,f1, ...,fn) ∈ r. by the assumption, (f0,f1, ...,fn) ∈ rτ . now, we prove that f̂ is continuous. let f̂(i) = fi and h be an open neighborhood of fi in cτ (y,z). it suffices to prove that f̂(ui) ⊆ h. let j ∈ ui. then i ∈ clf({j}). since (f0,f1, ...,fn) ∈ r we have (f0(y),f1(y) , ...,fn(y)) ∈ rtz for every y ∈ y . therefore fi(y) ∈ clz({fj(y)}) for every y ∈ y , that is fi ∈ clcτ (y,z)({fj}) which means that f̂(j) = fj ∈ h. hence the map f̂ is continuous and the topology τ is f-splitting. � the next corollary follows by the fact that for f=s (respectively, for f=d) then the 2-tuple relations r and rτ on c(y,z) coincide with the relations ≤ and ≤ τ (respectively, with the relations ∼ and ∼ τ ). corollary 2.9. the following (see [7]) are true: (1) a topology τ on c(y,z) is s-splitting if and only if the condition f ≤ g implies f ≤ τ g. (2) a topology τ on c(y,z) is d-splitting if and only if the condition f ∼ g implies f ∼ τ g. theorem 2.10. a topology τ on c(y,z) is f-jointly continuous if and only if rτ ⊆ r. proof. let τ be an f-jointly continuous topology on c(y,z). suppose that (f0,f1, ...,fn) ∈ rτ . we need to prove that (f0,f1, ...,fn) ∈ r. let g : f→ cτ (y,z) be a map for which g(i) = fi for every i ∈ f. we prove that g is continuous. let h be an open subset of cτ (y,z) such that fi ∈ h. it is suffices to prove that g(ui) ⊆ h. let j ∈ ui. since, i ∈ clf({j}). and (f0,f1, ...,fn) ∈ rτ we have fi ∈ clcτ (y,z)({fj}). therefore g(j) = fj ∈ h, that is the map g is continuous. n-tuple relations and topologies on function spaces 473 moreover, since τ is f-jointly continuous, the map g̃ : f×y → z is also continuous. now, we prove that (f0,f1, ..,fn) ∈ r. let y ∈ y . consider the (n+1)-tuple (f0(y),f1(y), ...,fn(y)) and let i ∈ clf({j}). it is suffices to prove prove that fi(y) ∈ clz({fj(y)}). let u be an open neighborhood of fi(y) in z. since g̃(i,y) = fi(y) we have g̃−1(u) is an open subset of f×y containing the point (i,y). there exist an open neighborhood v of i in f and an open neighborhood w of y in y such that v × w ⊆ g̃−1(u). since i ∈ clf({j}) we have that j ∈ v and therefore (j,y) ∈ g̃−1(u), which means that g̃(j,y) = fj(y) ∈ u. thus, fi(y) ∈ clz({fj(y)}). hence, (f0,f1, ...,fn) ∈ r. conversely, let τ be a topology on c(y,z) such that the condition (f0,f1, ...,fn) ∈ rτ implies (f0,f1, ...,fn) ∈ r. we prove that τ is f-jointly continuous. let g : f→ cτ (y,z) be a continuous map such that g(i) = fi for every i ∈ f. then the (n+1)-tuple (f0,f1, ...,fn) belongs to rτ . indeed, let i ∈ clf({j}) and h be an open neighborhood of fi in cτ (y,z). since g is continuous, the set g−1(h) is an open subset of f containing the point i. hence, j ∈ g−1(h) and, therefore, g(j) = fj ∈ h, which means that fi ∈ clcτ (y,z)({fj}). thus, (f0,f1, ...,fn) ∈ rτ . now, we consider the map g̃ : f×y → z and prove that this map is continuous. let (i,y) ∈ f×y . suppose that u is an open subset of z such that g̃(i,y) = g(i)(y) = fi(y) ∈ u. since the map fi is continuous and fi(y) ∈ u, there exists an open neighborhood w of y in y such that fi(w) ⊆ u. to prove that g̃ is continuous it is suffices to prove that g̃(ui ×w) ⊆ u. indeed, let (j,y′) ∈ ui × w . then j ∈ ui, that is i ∈ clf({j}). by the above fi ∈ clcτ (y,z)({fj}). since (f0,f1, ...,fn) ∈ r τ , by assumption we have (f0,f1, ...,fn) ∈ r. thus fi(y) ∈ clz({fj(y)}) for every y ∈ y and therefore fi(y′) ∈ clz({fj(y′)}). hence g̃(j,y′) = g(j)(y′) = fj(y′) ∈ u. thus, g̃ is continuous and therefore τ is an f-jointly continuous topology. � corollary 2.11. the following (see [7]) are true: (1) a topology τ on c(y,z) is s-jointly continuous if and only if the condition g ≤ τ f implies g ≤ f. (2) a topology τ on c(y,z) is d-jointly continuous if and only if the condition f ∼ τ g implies f ∼ g. remark 2.12. the first five theorems of this paper can be obtained by the last two theorems provided that: (1) for the trivial topology, and hence, for every topology τ on the set c(y,z) we have rτ ⊆ r if and only if the topology of z is trivial. (2) if for the discrete topology, and hence, for every topology τ on c(y,z) we have r ⊆ rτ , then z is t0 space. 474 d. n. georgiou, s. d. iliadis and b. k. papadopoulos (3) let z be a t1 space. then, for the discrete topology, and hence, for every topology τ on c(y,z) we have r ⊆ rτ . (4) for the pointwise topology, for the compact open topology, and for the isbell topology τ on c(y,z) we have rτ = r. (5) for the topology τq(u) on the set c(y,z) which generated by a quasiuniformity u we have r = rτq(u) . the above statements can be easily proved. references [1] r. arens, a topology of spaces of transformations, annals of math., 47(1946), 480-495. [2] r. arens and j. dugundji, topologies for function spaces, pacific j. math. 1(1951), 5-31. [3] j. dugundji, topology, (allyn and bacon, inc., boston 1968). [4] p. fletcher and w. lindgren, quasi-uniform spaces, (lecture notes in pure and applied mathematics; vol. 77 (1982)). [5] r. h. fox, on topologies for function spaces, bull. amer. math. soc. 51(1945), 429-432. [6] d. n. georgiou, s.d. iliadis and b. k. papadopoulos, topologies on function spaces, studies in topology, vii, zap. nauchn. sem. s.-peterburg otdel. mat. inst. steklov (pomi), 208(1992), 82-97. j. math. sci., new york 81(1996), no. 2, 2506-2514. [7] d. n. georgiou, s.d. iliadis and b. k. papadopoulos, topologies and orders on function spaces, publ. math. debrecen, 46/ 1-2 (1995), 1-10. [8] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. mislove and d. s. scott, a compendium of continuous lattices, (springer, berlin-heidelberg-new york 1980). [9] p. lambrinos and b. k. papadopoulos, the (strong) isbell topology and (weakly) continuous lattices, continuous lattices and applications, lecture notes in pure and appl. math. no. 101, marcel dekker, new york 1984, 191-211. [10] r. mccoy and i. ntantu, topological properties of spaces of continuous functions, (lecture notes in mathematics, 1315, springer verlang). [11] m. g. murdershwar and s. a. naimpaly, quasi uniform spaces, (noordhoff, 1966). received january 2002 revised september 2002 d. n. georgiou department of mathematics, university of patras, 265 00 patras, greece e-mail address : georgiou@math.upatras.gr s. d. iliadis department of mathematics, university of patras, 265 00 patras, greece e-mail address : iliadis@math.upatras.gr b. k. papadopoulos department of civil engineering, democritus university of thrace, 67100 xanthi, greece e-mail address : papadob@civil.duth.gr applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 25–32 some results on best proximity pair theorems p. s. srinivasan and p. veeramani abstract. best proximity pair theorems are considered to expound the sufficient conditions that ensure the existence of an element x◦ ∈ a, such that d(x◦,tx◦) = d(a,b) where t : a → 2b is a multifunction defined on suitable subsets a and b of a normed linear space e. the purpose of this paper is to obtain best proximity pair theorems directly without using any multivalued fixed point theorem. in fact, the well known kakutani’s fixed point theorem is obtained as a corollary to the main result of this paper. 2000 ams classification: primary 47h10; secondary 54h25. keywords: best proximity pairs, kakutani multifunctions, simplicial approximations, best approximations 1. introduction let t be a multifunction from a to b where a and b are non-empty subsets of a normed linear space. best proximity pair theorem analyzes the conditions under which the problem of minimizing the real valued function x → d(x,tx) has a solution. it is evident that d(x,tx) ≥ d(a,b) for all x ∈ a. therefore, a nice solution to the above optimization problem will be one for which the value d(a,b) is attained. consider the fixed point equation tx = x where t is a non-self operator. if this equation does not have a solution then the next attempt is to find an element x in a suitable space such that x is close to tx in some sense. in fact, the “best approximation pair theorems” and “best proximity pair theorems” are pertinent to be explored in this direction. in the setting of a normed linear space e if t is a mapping with domain a, then a best approximation theorem provides sufficient conditions that ascertain the existence of an element x◦, known as best approximant, such that d(x◦,tx◦) = d(tx◦,a) 26 p. s. srinivasan and p. veeramani where d(x,y ) := inf{‖x−y‖ : x ∈ x and y ∈ y} for any non-empty subsets x and y of the space e. a classical best approximation theorem, due to ky fan [3], states that if k is a non-empty compact convex subset of a hausdorff locally convex topological vector space e with a continuous seminorm p and t : k −→ e is a single valued continuous map, then there exists an element x◦ ∈ k such that p(x◦ −tx◦) = d(tx◦,k) later, reich [7] has weakened the compactness condition and thereby obtained the generalization of the above theorem. this result has been further extended by sehgal and singh [11], to the one for continuous multifunctions. further, they have also proved a generalization of a best approximation theorem due to prolla [6]. the authors vetrivel, veeramani and bhattacharyya [15] have established the existence of a best approximant for continuous kakutani factorizable multifunctions which unifies and generalizes the known results on best approximations. on the other hand, even though a best approximation theorem guarantees the existence of an approximate solution, it is contemplated to find an approximate solution which is optimal. the best proximity pair theorem sheds light in this direction. indeed, a best proximity pair theorem due to sadiq basha and veeramani [8] provides sufficient conditions that ensure the existence of an element x◦ ∈ a, such that d(x◦,tx◦) = d(a,b) where t : a −→ 2b is a kakutani factorizable multifunction defined on suitable subsets a and b of a locally convex topological vector space e. the pair (x◦,tx◦) is called a best proximity pair of t . because of the fact that d(x, tx) ≥ d(tx, a) ≥ d(a, b), for all x ∈ a, an element xo satisfying the conclusion of a best proximity pair theorem is a best approximant but the refinement of the closeness between xo and its image txo is demanded. apart from the purpose of seeking an approximate solution which is optimal, these best proximity pair theorems also blend the results on best approximations and proximal points of a pair of sets considered by the authors beer and pai [1], sahney and singh [10], and xu [16], of providing sufficient conditions for the non-emptiness of the set prox(a,b) := {(a,b) ∈ a×b : d(a,b) = d(a,b)}. best proximity pair theorems obtained by sadiq basha and veeramani [8], [9] hinges on lassonde’s multivalued fixed point theorem. the purpose of this paper is to elicit best proximity pair theorems without using lassonde’s theorem. indeed, the main best proximity pair theorem obtained in this paper does not employ any multivalued fixed point theorems in the proof. some results on best proximity pair theorems 27 2. preliminaries let x and y be non-empty sets. a multivalued map or multifunction t from x to y denoted by t : x → 2y , is defined to be a function which assigns to each element of x ∈ x, a non-empty subset tx of y . fixed points of t : x → 2x will be the points x ∈ x such that x ∈ tx. for further discussion, let x and y be any two normed linear spaces. let t : x → 2y be a multivalued map. t is said to be upper semi-continuous (resp. lower semi-continuous ) if t−1(a) = {x ∈ x : t(x) ∩a 6= ∅} is closed (resp. open) in x whenever a is a closed (resp. open) subset of y . also, t is said to be continuous if it is both lower semi-continuous and upper semicontinuous. a multifunction t : x → 2y is said to have compact and convex values if for each x ∈ x, t(x) is a compact and convex subset of y . also, t is said to be a compact multifunction if t(x) is a compact subset of y . a multifunction t : x → 2y is said to be closed if gr(t) := {(x,y) : x ∈ x and y ∈ tx} is a closed subset of x ×y . it is known that if t is an upper semi-continuous multifunction with compact values, then t(k) is compact whenever k is a compact subset of x. it is a noteworthy fact that the composition of two convex valued maps need not be convex valued. a multivalued map t : x −→ 2y is said to be a kakutani multifunction[5] if the following conditions are satisfied (i) t is upper semi-continuous and (ii) either t is singleton or for each x ∈ x tx is non-empty, compact and convex (in which case y is assumed to be a convex set). the collection of all kakutani multifunctions from x to y is denoted by k(x, y ). a multifunction is said to be a kakutani factorizable multifunction [5] if the multifunction can be expressed as a composition of finitely many kakutani multifunctions. the class of all kakutani factorizable multifunctions from x to y is denoted by kc(x,y ). a non-empty set a of x is said to be approximately compact if for each y in x and each sequence {xn} in a satisfying the condition that ‖xn −y‖−→ d(y,a), there is a subsequence of {xn} converging to an element of a. let a be any non-empty subset of x. then pa : x −→ 2a defined by pa(x) = {a ∈ a : ‖a−x‖ = d(x,a)} is the set of all best approximations in a to any element x ∈ x. it is known that if a is an approximately compact convex subset of x, pa(x) is a non-empty compact convex subset of a and the multivalued map pa is upper semi-continuous on x. 28 p. s. srinivasan and p. veeramani let a and b be any two non-empty subsets of a normed linear space. the following notions are also recalled. d(a,b) := inf{d(a,b) : a ∈ a and b ∈ b} a◦ := {a ∈ a : d(a,b) = d(a,b) for some b ∈ b} b◦ := {b ∈ b : d(a,b) = d(a,b) for some a ∈ a} if a = {x}, then d(a,b) is written as d(x,b). also, if a = {x} and b = {y}, then d(x,y) denotes d(a,b) which is precisely ‖x−y‖. 3. main results this section is devoted to the principal results on best proximity pair theorems. the proof of the main theorem requires the following lemma: lemma 3.1. let a, b, c and d be non-empty closed subsets of a normed linear space e. let t : a → 2b be an upper semi-continuous compact multifunction with closed values. further, let f : c → 2d be a compact and closed multifunction. then the set k = {(x,y) ∈ a×c : d(tx,fy) = d(b,d)} is closed in a×c proof. let (xn,yn) ∈ k such that xn → x and yn → y. this implies that d(txn,fyn) = d(b,d). as the multifunctions t and f are compact valued, it is possible to choose for every n, an ∈ txn and bn ∈ fyn such that d(an,bn) = d(txn,fyn). hence d(an,bn) = d(b,d). since the multifunctions t and f are compact, without loss of generality, it may be assumed that an → a and bn → b. from the above facts, it is easy to see that d(a,b) = d(b,d). it follows from the upper semi-continuity of t , that a ∈ tx. also, as f is a closed multifunction, b ∈ fy. now, d(tx,fy) ≤ d(a,b) = d(b,d). therefore, (x,y) ∈ k and hence k is closed. � the following theorem is the first in the sequel of obtaining best proximity pair theorems. the proof does not subsume any multivalued fixed point theorem. the crux in the proof is the technique of simplicial approximation carried out in the lines of [5]. some results on best proximity pair theorems 29 theorem 3.2. let x be a simplex and let y and z be non-empty closed convex sets in a finite dimensional space e. let λ : x → 2z be a compact and closed multifunction. then the following statements are equivalent. (i) for every f ∈ c(x,y ), there exists x ∈ x such that d(fx, λx) = d(y,z). (ii) for every f ∈ k(x,y ), there exists x ∈ x such that d(fx, λx) = d(y,z). proof. suppose that (i) holds. it is proved that (ii) holds using the same construction as that of lassonde [5]. for p = 1, 2, . . ., let σp be a simplicial subdivision of x of mesh size lower than 1/p. let apo, · · · ,apmp be the vertices of σ p. also, let λpo, · · · ,λpmp be the co-ordinate maps associated with those vertices. it follows that, each point x ∈ x can be uniquely written as x = ∑mp i=0 λ p i (x)a p i . for each vertex a p i of σp a point bpi in fa p i is chosen. with this choice, let a single valued function fp : x → y be defined as fp(x) = ∑mp i=0 λ p i (x)b p i . evidently, f p is continuous. now, by property (i), it follows that (3.1) for every p, there exists xp ∈ x such that d(fpxp, λxp) = d(y,z). let the dimension of the simplex x be n. let b denote the unit ball of the euclidean space spanned by x. let apio, · · · ,a p in be the vertices of any n-simplex of σp containing xp. then, apik ∈ x p + (1/p)b for each k = 0, · · · ,n, λ p i (x p) = 0 for all i /∈ {io, · · · , in} and λp(xp) := ( λ p io (xp), · · · ,λpin(x p) ) ∈ λn, where λn = { (λo, · · · ,λn) ∈ rn+1 : λi ≥ 0 for all i and n∑ i=0 λi = 1 } . let the multifunction t : λn ×xn+1 → 2y be defined as t(λ,ao, · · · ,an) = n∑ i=0 λifai. the following fact is immediate (3.2) fp(xp) = n∑ k=0 λ p ik (xp)bpik ∈ t ( λp(xp),apio,, · · · ,a p in ) . now, from 3.1 for every p, there exist xp in x, λp ∈ λn and (apo, · · · ,apn) ∈ xn+1 such that api ∈ x p + (1/p)b for each i = 0, · · · ,n and d(t (λp,apo, · · · ,a p n) , λx p) ≤ d(fpxp, λxp) (by 3.2) = d(y,z). (by 3.1) as t (λp,apo, · · · ,apn) ⊆ y and λxp ⊆ z, d(y,z) ≤ d(t (λp,apo, · · · ,a p n) , λx p). 30 p. s. srinivasan and p. veeramani therefore, for every p, there exists xp in x, λp ∈ λn and (apo, · · · ,apn) ∈ xn+1 such that api ∈ x p + (1/p)b for each i = 0, · · · ,n and (3.3) d (t (λp,apo, · · · ,a p n) , λx p) = d(y,z). now, t is compact-valued upper semi-continuous, since it can be written as gg, where the multifunction g: λn ×xn+1 → 2λn×y n+1 defined as g(λ,ao, · · · ,an) = {λ}×fao ×···×fan is compact and upper semi-continuous with closed values and the single valued function g : λn ×y n+1 → y given by g(λ,bo, · · · ,bn) = n∑ i=o λibi is continuous since y is convex. consider the following set h := {(λ,ao, · · · ,an,x) ∈ λn×xn+1×x : d (t(λ,ao, · · · ,an), λx) = d(y,z)}. by lemma 3.1, h is closed in the compact space λn ×xn+1 ×x. without loss of generality, assume that the sequence {xp} converges to a point x̂ ∈ x and the sequence {λp} to a point λ̂ ∈ λn as p tends to infinity. from 3.3, it follows that the sequence {api} converges to x̂ for each i = 0, · · · ,n. since for every p, (λp,apo, · · · ,apn,xp) ∈ h, and h is closed, (λ̂, x̂, · · · , x̂) ∈ h. further, this implies that d(t(λ̂, x̂, · · · , x̂), λx̂) = d(y,z). but, t(λ̂, x̂, · · · , x̂) = n∑ i=0 λifx̂ ⊆ fx̂, since fx̂ is convex. therefore, d(fx̂, λx̂) ≤ d(t(λ̂, x̂, · · · , x̂), λx̂) = d(y,z). the condition (ii) implying (i) is immediate as c(x,y ) ⊆k(x,y ). � by choosing z = x and λ to be the identity function in theorem 3.2, the following corollary is immediate. corollary 3.3. let x be a simplex and y be a non-empty compact convex set in a finite dimensional space e. then the following statements are equivalent. (i) for every f ∈ c(x,y ), there exists x ∈ x such that d(x,fx) = d(x,y ). (ii) for every f ∈ k(x,y ), there exists x ∈ x such that d(x,fx) = d(x,y ). some results on best proximity pair theorems 31 remark 3.4. the above corollary contains the multivalued fixed point theorem due to kakutani [4]. this follows by choosing y = x in the above corollary and noting the fact that condition (i) always holds by brouwer’s fixed point theorem [2]. the following corollary contains simplex analogue of a best proximity pair theorem due to sadiq basha and veeramani [8]. corollary 3.5. let a be a simplex and b be a non-empty compact convex set in a finite dimensional space e such that a0 is also a simplex. then the two following equivalent statements hold. (i) for every f ∈ c(a,b) with f(a0) ⊆ b0, there exists x ∈ a such that d(x,fx) = d(a,b). (ii) for every f ∈k(a,b) with f(a0) ⊆ b0, there exists x ∈ a such that d(x,fx) = d(a,b). proof. choosing x = a0 and y = b0 in the above corollary and noting the fact d(a,b) = d(a0,b0), the proof of (i) being equivalent to (ii) follows. it remains to show the validity of the statement (i). for this, consider the multifunction pa ◦ f : a0 → 2a0 , where f ∈ c(a,b) with f(a0) ⊆ b0. as pa ◦f ∈k(a0,a0), by remark 3.4 (or kakutani’s fixed point theorem) there exists x ∈ a0 such that x ∈ (pa ◦ f)x. that is d(x,fx) = d(a,fx). since f(a0) ⊆ b0 it is clear that d(a,fx) = d(a,b). therefore d(x,fx) = d(a,b). therefore statement (i) holds. since (i) is equivalent to (ii), this completes the proof of the corollary. � acknowledgements. both authors are thankful to referees for the careful reading and helpful suggestions for the improvement of this paper. references [1] g. beer and d. v. pai, proximal maps, prox maps and coincidence points, numer. funct. anal. and opt. 11 (1990), 429–448. [2] l. e. j. brouwer, über abbildungen von mannigfaltigkeiten, math. ann. 71 (1912), 97–115. [3] k. fan, extensions of two fixed point theorems of f.e.browder, math. z. 112 (1969), 234–240. [4] s. kakutani, a generalization of brouwer’s fixed point theorem, duke math. j. 8 (1941), 457–459. [5] m. lassonde, fixed points for kakutani factorizable multifunctions, j. math. anal. appls. 152 (1990), 46–60. [6] j. b. prolla, fixed point theorems for set valued mappings and existence of best approximations, numer. funct. anal. and optimiz. 5 (1982-1983), 449–455. [7] s. reich, approximate selections, best approximations, fixed points and invariant sets, j. math. anal. appl. 62 (1978), 104–113. 32 p. s. srinivasan and p. veeramani [8] s. sadiq basha and p. veeramani, best proximity pairs and best approximations, acta sci. math. szeged. 63 (1997), 289–300. [9] s. sadiq basha and p. veeramani, best proximity pair theorems for multifunctions with open fibres, j. approx. theory 103 (2000), 119–129. [10] b. n. sahney and s. p. singh, on best simultaneous approximation, approximation theory iii (e. w. cheney, ed.) (1980), 783–789. [11] v. m. sehgal and s. p. singh, a generalization to multifunctions of fan’s best approximation theorem, proc. amer. math. soc. 102 (1988), 534–537. [12] v. m. sehgal and s. p. singh, a theorem on best approximations, numer. funct. anal. and optimiz. 10 (1989), 181–184. [13] v. m. sehgal, a simple proof of a theorem of ky fan, proc. amer. math. soc. 63 (1977), 368–369. [14] i. singer, best approximation in normed linear spaces by elements of linear spaces, springer-verlag, new york, 1970. [15] v. vetrivel, p. veeramani and p. bhattacharyya, some extensions of fan’s best approximation theorem, numer. funct. anal. and optimiz. 13 (1992), 397–402. [16] x. xu, a result on best proximity pair of two sets, j. approx. theory 54 (1988), 322–325. received april 2001 revised june 2001 p.s. srinivasan and p. veeramani department of mathematics indian institute of technology madras, chennai 600 036, india e-mail address : pvmani@iitm.ac.in kohdasagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 2, 2006 pp. 233-244 a class of spaces containing all generalized absolutely closed (almost compact) spaces j. k. kohli and a. k. das abstract. the class of θ-compact spaces is introduced which properly contains the class of almost compact (generalized absolutely closed) spaces and is strictly contained in the class of quasicompact spaces. in the realm of almost regular spaces, the class of θ-compact spaces coincides with the class of nearly compact spaces. moreover, an almost regular θ-compact space is mildly normal (= κ-normal). a θ-closed, θ-embedded subset of a θ-compact space is θ-compact and the product of two θ-compact space is θ-compact if one of them is compact. a (strongly) θ-continuous image of a θ-compact space is θ-compact (compact). a space is compact if and only if it is θ-compact and θ-point paracompact. 2000 ams classification: primary: 54d10, 54d20, 54d30, secondary: 54c08, 54c10. keywords: θ-compact space, almost compact (generalized absolutely closed) space, nearly compact space, quasicompact space, θ-point paracompact space, θ-closed (θ-open) set, θ-limit point, almost regular space, mildly normal(κnormal) space, almost normal space, (strongly) θ-continuous function, θ-map, θ-closed function, θ-limit point. 1. introduction and preliminaries compactness plays a prominent role in topology, analysis and many other branches of mathematics. several weak variants of compactness occur in the literature which capture partial features of compactness and are thus useful in the topological/ analytical situations where the full force of compactness is not required (see [25] [26]). the variants of compactness with which we shall be dealing in this paper include almost compactness (= generalized closedness) (see [18], [17], [8], [15]); near compactness [21]; and quasicompactness (see [8], [1] and [25]). 234 j. k. kohli and a. k. das in section 2, we introduce the class of θ-compact spaces and establish their place in the hierarchy of compactness and certain of its variants. the class of θ-compact spaces lies strictly between the class of almost compact spaces [3] and quasicompact spaces. in the realm of almost regular spaces the notions of θ-compactness and near compactness are equivalent. interrelations between θ-compactness and certain weak variants of normality are investigated. it is shown that an almost regular θ-compact space is mildly normal and that a hausdorff almost regular θ-compact space is almost normal. presevation under mappings and products of θ-compact spaces are considered in section 4. it is shown that a (strongly) θ-continuous image of a θ-compact space is θ-compact (compact). in section 5, we discuss characterizations of θ-compact spaces and conclude with a factorization theorem that a space is compact if and only if it is both a θ-compact space and a θ-point paracompact space. let x be a topological space and let a ⊂ x. throughout the present paper the closure of a set a will be denoted by a and the interior by inta. a point x ∈ x is called a θ-limit point [28] of a if every closed neighbourhood of x intersects a. let clθa denotes the set of all θ-limit point of a. the set a is called θ-closed if a = clθa. the complement of a θ-closed set will be referred to as a θ-open set. it is easily verified that in a topological space every cozero set is θ-open. a set u ⊂ x is said to be regularly open [14] if u = intu . the complement of a regularly open set is called regularly closed. a space x is said to be almost regular [19] if every regularly closed set and a point out side it are contained in disjoint open sets. a space x is said to be mildly normal [22] ( or κ-normal [24]) if every pair of disjoint regularly closed sets are contained in disjoint open sets and a space is called almost normal [20] if every pair of disjoint closed sets one of which is regularly closed are contained in disjoint open sets. a space x is said to be nearly compact [21] if every open covering of x admits a finite subcollection the interiors of the closures of whose members cover x. a space x is said to be almost compact [3] if every open covering of x has a finite subcollection the closures of whose members covers x. almost compact spaces have been referred to as h(i) spaces by scarborough and stone [18] and are called generalized absolutely closed spaces by liu [15], while porter and thomas [17] call them quasi-h-closed spaces. a hausdorff almost compact space is called an h-closed space. h-closed spaces have many properties similar to that of compact hausdorff spaces. a space x is said to be quasicompact [8] if every covering of x by cozero sets admits a finite subcollection which covers x. functionally hausdorff, quasicompact spaces are precisely the spaces in which stone-weierstrass theorem holds ([25], [26]). lemma 1.1 ([11, 13]). a subset a of a topological space x is θ-open if and only if for each x ∈ a, there is an open set u such that x ∈ u ⊂ u ⊂ a. lemma 1.2. ([6, 2.4]). a space x is regular if and only if every closed set in x is θ-closed. a class of spaces ... 235 2. θ-compact spaces definition 2.1. a space x is said to be θ-compact if every open covering of x by θ-open sets has a finite subcollection that covers x. a subset a of x is said to be θ-compact if it is θ-compact with respect to the topology it inherits as a subspace of x. a subset a of x is said to be a θ-set in x if every covering of a by θ-open sets in x has a finite subcollection that covers a. remark 2.2. a θ-set in a topological space need not be θ-compact. for let x = [0, 1] with every point having usual euclidean neighbourhood except 0. a basic neighbourhood of 0 is of the form u − k, where u is an euclidean neighbourhood of 0 and k = {1/n : n ∈ n}. let a = {0} ∪k. then a is a θ-set in x which is not θ-compact. the following implications are immediate from the definitions. compact ⇒ nearly compact ⇒ almost compact ⇒ θ-compact ⇒ quasicompact. however, none of the above implications is reversible. example 2.3. a θ-compact space which is not almost compact. let x = n, the set of positive integers. define a topology on x by taking every odd integer to be open and a set u is open if for every even integer p ∈ u , the predecessor and the successor of p are also in u . now the collection u = {{2k-1, 2k, 2k+1} : k ∈ n} is an open covering of x which does not possess a finite subcollection whose closures covers x. thus x is not almost compact. however, it is θ-compact. remark 2.4. hewitt’s example [9] of a t1-regular space on which every continuous real-valued function is constant is a quasicompact space which is not θ-compact. singal and mathur [21] gave an example of a nearly compact space which is not compact and an example of an almost compact space which is not nearly compact. the following characterization of almost regular spaces besides being useful in the sequel has been extensively used in [12]. theorem 2.5. a space x is almost regular if and only if for every open set u in x, intu is θ-open. proof. suppose that u is an open set in x. if intu = x, we are through. in case intu 6= x and x ∈ intu , then x is not in the regularly closed set x −intu. by almost regularity of x, there exist disjoint open sets v and w containing x and x − intu , respectively. then v ∩ w = ∅ and so v ⊂ x − w ⊂ intu . in view of lemma 1.1, it follows that intu is θ-open. to prove the converse, let f be a regularly closed set in x and let x be a point in x outside f . then x ∈ x − f . since x − f is regularly open, x − f = intx − f , which is θ-open. so by lemma 1.1, there exists an open set u containing x such that u ⊂ x − f . thus u and x − u are disjoint open sets containing x and f , respectively. consequently, x is almost regular. � theorem 2.6. an almost regular θ-compact space is nearly compact. 236 j. k. kohli and a. k. das proof. let u be an open covering of x. by theorem 2.5, for each u ∈ u, intu is a θ-open set containing u and hence the collection {intu : u ∈ u} is a θ-open covering of x. since x is θ-compact, there exist, a finite subcollection {u1,... , un} of u such that n⋃ i=1 intui = x, and so x is a nearly compact space. � corollary 2.7. ([21, theorem 2.3]). an almost regular almost compact space is nearly compact. theorem 2.8. an almost regular θ-compact space is mildly normal ( = κnormal). proof. let x be an almost regular θ-compact space. let a and b be any two disjoint regularly closed subsets of x. since x is almost regular, for each b ∈ b, there exist disjoint open sets ub and vb containing a and b, respectively. then ub ∩vb = ∅. so by theorem 2.5, intvb is a θ-open set containing b which is disjoint from ub. thus the collection { intvb : b ∈ b} consists of θ-open sets and covers b. let w = ⋃ b∈b intvb. then b ⊂ w . let c = x − w . since b is a regularly closed set disjoint from c, by almost regularity of x, for each c ∈ c, there exist disjoint open sets xc and wc containing b and c, respectively. again, in view of theorem 2.5, intwc is a θ-open set which is disjoint from b. hence the collection c = { intvb : b ∈ b} ∪ {intwc: c ∈ c} is a covering of x by θ-open sets. since x is θ-compact, there exists a finite subcollection u of c which covers x. let g denote the members of u which intersect b. since for each c ∈ c, intwc ∩ b = ∅, each member of g is of the form intvb for some b ∈ b. suppose g = { intvbi : i = 1, .... n}. let u = n⋂ i=1 ubi and v = n⋃ i=1 intvbi . then u and v are disjoint open sets containing a and b, respectively. hence x is a mildly normal space. � corollary 2.9. an almost regular almost compact space is mildly normal (κnormal). with the additional hypothesis of hausdorffness theorem 2.8 is strengthened as follows. theorem 2.10. a hausdorff θ-compact space is almost normal if and only if it is almost regular. proof. since a t1-almost normal space is almost regular, necessity is obvious. to prove the sufficiency, let x be a hausdorff almost regular, θ-compact space. by theorem 2.6, x is nearly compact and hence almost compact. since an almost regular hausdorff space is urysohn [19] and since an almost compact urysohn space is almost normal [20], x is almost normal. � we may recall that a space xis an r0-space [5] if every open set in x is the union of closed sets. in [3], r0-spaces are referred to as s1-spaces. it is shown a class of spaces ... 237 in [3, p. 196] that in an r0-space the closure of every singleton is compact. for θ-closure we have the following. theorem 2.11. in a topological space x the θ-closure of every singleton is a θ-set. proof. let u be a covering of clθ{x} by θ-open sets in x. let x ∈ u ∈ u. by lemma 1.1, there exists an open set v containing x such that v ⊂ u . in view of [28, lemma 2], clθ{x} ⊂ clθv = v ⊂ u . hence clθ{x} is a θ-set. � remark 2.12. theorem 2.11 cannot be strengthened to read “θ-compact” instead of “θ-set”. for example, let x = {a}∪b ∪ c with pairwise disjoint members and infinite sets b and c. let the topology τ be defined by the base {{c}, ({a}∪u ), ({b}∪u ) : b ∈ b, c ∈ c, u ⊂ c is cofinite in c}. clearly, cl θ({a}) = {a}∪b which is discrete and infinite in the subspace topology. 3. subspaces the following formulation of the notion of a θ-embedded set is useful in studying subspaces of θ-compact spaces. definition 3.1. a subset y of a topological space x is said to be θ-embedded in x if every θ-closed set in the subspace topology of y is the intersection of y with a θ-closed set in x. remark 3.2. let x be the same space as in remark 2.2. the set a therein is a θ-closed subset of x which is not θ-embedded in x. the same example also shows that a θ-closed subset of a θ-compact space need not be θ-compact. however, the following is true. theorem 3.3. a θ-closed, θ-embedded subset of a θ-compact space is θ-compact. proof. let a be a θ-closed, θ-embedded subset of a θ-compact space x. let u be a covering of a by sets θ-open in a. since a is θ-embedded in x, for each uα ∈ u there exists a θ-open set vα in x such that uα = a ∩ vα. then the collection v = {vα : uα ∈ u} ∪ {x − a} is a θ-open covering of x. by θ-compactness of x, there is a finite subcollection {vα1 ,...,vαn } of v which covers x. then the collection {vα1 ∩ a, ..., vαn ∩ a} is a finite subcollection of u which covers a and so a is θ-compact. � corollary 3.4. every clopen subset of a θ-compact space is θ-compact. a topological space x is said to be θ-hausdorff [23] if any two distinct points in x are contained in disjoint θ-open sets. a space x is called a locally θ-space [4] if each x ∈ x has a neighbourhood which is a θ-set. to conclude this section we quote the following theorem from [4]. theorem 3.5. a non θ-compact, θ-hausdorff, locally θ-space x has a one point hausdorff θ-compactification, i.e., it is a dense open subspace of a θcompact hausdorff space x∗ such that x∗ − x is a singleton. 238 j. k. kohli and a. k. das 4. direct and inverse preservation under mappings and products. definition 4.1. a function f : x → y is said to be (i) θ-continuous [7] if for each x ∈ x and each open set u containing f (x) there exists an open set v containing x such that f (v ) ⊂ u , and (ii) strongly θ-continuous [16] if for each x ∈ x and each open set u containing f (x) there exists an open set v containing x such that f (v ) ⊂ u . we say that a function f : x → y is a θ-map if for every θ-open set u in y , f −1(u ) is θ-open in x. theorem 4.2. every θ-continuous function is a θ-map. proof. let f : x → y be a θ-continuous function and let u be a θ-open set in y . let x ∈ f −1(u ). then, f (x) ∈ u . since u is θ-open, by lemma 1.1, there exists an open set v in y such that f (x) ∈ v ⊂ v ⊂ u . by θ-continuity of f , there exists an open set w in x containing x such that f (w ) ⊂ v ⊂ u . thus x ∈ w ⊂ w ⊂ f −1(u ). so in view of lemma 1.1, f −1(u ) is θ-open and hence f is a θ-map. � remark 4.3. the converse of theorem 4.2 is not true. for let x = y be the set of positive integers. let x be endowed with the cofinite topology and let y be equipped with the topology as defined in example 2.3. then the identity mapping of x onto y is a θ-map which is not θ-continuous. theorem 4.4. let f : x → y be a θ-map from a θ-compact space x onto y . then y is θ-compact. proof. let v be a θ-open covering of y . then since f is a θ-map, the collection u={f −1(u ) : u ∈ v} is a θ-open covering of x. since x is θ-compact, there exists a finite subcollection {f −1(ui) : i = 1, ..., n} of u which covers x. now since f is onto, {ui : i = 1, ..., n} is a finite subcollection of v which covers y . hence y is a θ-compact space. � corollary 4.5. every θ-continuous image of a θ-compact space is θ-compact. theorem 4.6. a strongly θ-continuous image of a θ-compact space is compact. proof. suppose x is a θ-compact space and let f : x → y be a strongly θcontinuous surjection. let v be an open covering of y . let v ∈ v. we show that f −1(v ) is θ-open. if x ∈ f −1(v ), then f (x) ∈ v . since f is strongly θ-continuous, there exists an open set u containing x such that f (u ) ⊂ v . then x ∈ u ⊂ u ⊂ f −1(v ) and so in view of lemma 1.1, f −1(u ) is θ-open. thus the collection u={f −1(v ) : v ∈ v} is a θ-open covering of x and so there is a finite subcollection {f −1(vi) : i = 1, 2, . . . n} of u which covers x. hence {vi : i = 1, 2, . . . , n} is a finite subcollection of v which covers y and so y is compact. � a class of spaces ... 239 we say that a function f : x → y is said to be θ-closed if each θ-closed set f in x, f (f ) is θ-closed. the following characterization of θ-closed functions will be used in the sequel and seems to be of interest in itself. theorem 4.7. a function f : x → y is θ-closed if and only if for each set b ⊂ y and for each θ-open set u containing f −1(b), there is a θ-open set v containing b such that f −1(v ) ⊂ u . proof. necessity. since u is θ-open, x − u is θ-closed and so f (x − u ) is θ-closed in y . now, v = y − f (x − u ) is θ-open, b ⊂ v and f −1(v ) = f −1(y − f (x − u )) = x − f −1(f (x − u )) ⊂ x − (x − u ) = u . to prove sufficiency, let a be a θ-closed set in x. to prove that f (a) is θ-closed, we shall show that y − f (a) is θ-open. let y ∈ y − f (a). then f −1(y) ∩ f −1(f (a)) = φ and so f −1(y) ⊂ x − f −1(f (a)) ⊂ x − a. by hypothesis there exists a θ-open set v containing y such that f −1(v ) ⊂ x − a. so a ⊂ x − f −1(v ) and hence f (a) ⊂ f (x − f −1(v )) = y − v . thus v ⊂ y − f (a) and so the set y − f (a) being the union of θ-open sets is θ-open. � theorem 4.8. let f : x → y be a θ-closed surjection such that for each y ∈ y , f −1(y) is a θ-compact subset of x. if y is θ-compact, then so is x. proof. let u = {uα : α ∈ λ} be a θ-open covering of x. since for each y ∈ y , f −1(y) is a θ-compact subset of x, we can choose a finite subset λy of λ such that {uβ : β ∈ λy} is a covering of f −1(y). now, by theorem 4.7, there exists a θ-open set vy containing y such that f −1(vy ) ⊂ ∪ {uβ : β ∈ λy}. the collection v ={vy : y ∈ y } is a θ-open covering of y . in view of θ-compactness of y there exists a finite subcollection {vy1 , ..., vyn } of v which covers y . then the finite subcollection {uβ : β ∈ λyi , i = 1, ..., n} of u covers x. hence x is a θ-compact space. � lemma 4.9. let x be a compact space. then the projection map py : x ×y → y is a θ-closed surjection for any space y . proof. let f be a θ-closed subset of x × y . to show that py(f ) is θ-closed, we prove that y − py(f ) is θ-open. let y ∈ y − py(f ). then (x × {y}) ∩ f = ∅. since (x × y ) − f is θ-open, for each (x, y), there is a basic open set u (x) × vy(x) containing (x, y) such that u (x) × vy (x) is disjoint from f . the collection {u (x) × vy(x) : x ∈ x} is an open covering of the compact set x × {y} and so has a finite subcovering {u (xi) × vy(xi) : i = 1, ..., n}. then v = n⋂ i=1 vy (xi)is an open set containing y such that v ⊂ y − py(f ). in view of lemma 1.1, y − py(f ) is θ-open and hence py is a θ-closed surjection. � the following corollary concerning θ-open sets in the product space x ×y is analogous to its counterpart for open sets which is widely used in applications. 240 j. k. kohli and a. k. das corollary 4.10. let y be a compact space and let a ⊂ x be arbitrary. let u be a θ-open set in x × y containing a × y . then there is a θ-open set v containing a such that v × y ⊂ u . proof. by lemma 4.9, the projection map px : x × y → x is a θ-closed function. now, p−1x (a) = a × y ⊂ u . so an application of theorem 4.7, yields the desired θ-open set v containing a such that p−1x (v ) = v × y ⊂ u . � theorem 4.11. let x and y be θ-compact spaces such that x is compact. then x × y is a θ-compact space. proof. by lemma 4.9, the projection map py : x × y → y is a θ-closed surjection. again, for each y ∈ y , the fiber p−1y (y) is compact and hence θ-compact. so the result is immediate in view of theorem 4.8. � 5. characterizations and a factorization of compactness. in this section we obtain characterizations of θ-compactness analogous to that of compactness. in general θ-closure operator is not a kuratowski closure operator, since θ-closure of a set may not be θ-closed (see [10]). however, the following modification yields a kuratowski closure operator. definition 5.1. let x be a topological space and let a ⊂ x. a point x ∈ x is called a uθ-limit point of a if every θ-open set u containing x intersects a. the set of all uθ-limit points of a is either denoted by auθ or by cluθa. the set a is called uθ-closed if a = auθ. lemma 5.2. the operator a → auθ is a kuratowski closure operator. proof. clearly a ⊂ auθ, for every subset a of x. clearly, (auθ ∪ buθ) ⊂ (a ∪ b)uθ. to prove the reverse inequality (a∪b)uθ ⊂ auθ ∪buθ, let x ∈ (a∪b)uθ. suppose x /∈ auθ ∪ buθ. then x /∈ auθ and x /∈ buθ. so there exist θ-open sets u and v containing x such that (u ∩ a) = ∅ = (v ∩ b). then u ∩ v is a θ-open set containing x such that (u ∩ v ) ∩ (a ∪ b) = ∅, which contradicts the fact that x ∈ (a ∪ b)uθ. consequently, auθ ∪ buθ = (a ∪ b)uθ. to prove (auθ)uθ ⊂ auθ, let x ∈ (auθ )uθ. then for every θ-open set u containing x, u ∩ auθ 6= ∅. choose y ∈ u ∩ auθ. since u is a θ-open set containing y, u ∩ a 6= ∅ and so x ∈ auθ. hence (auθ)uθ = auθ. this completes the proof that the correspondence a → auθ is a kuratowski closure operator. � remark 5.3. for any set a ⊂ x, a ⊂ a ⊂ aθ ⊂ auθ and auθ is the smallest θ-closed set containing a. definition 5.4. let ℑ be a filter on x. a point x ∈ x is said to be uθ-cluster point of ℑ if every θ-open set containing x intersects every member of the filter ℑ. the filter ℑ is said to uθ-converges to x if every θ-open set containing x belongs to ℑ. in symbols ℑ uθ −→ x. a class of spaces ... 241 definition 5.5. let {xα}α∈λ be a net in x. a point x is said to be a uθcluster point of {xα} if for every θ-open set u containing x and for each α0 ∈ λ there is an α ∈ λ such that α > α0 and xα ∈ u . the net {xα} is said to be uθ-converges to x if for each θ-open set containing x, there is some λ0 ∈ λ such that xλ ∈ u for all λ ≥ λ0. proofs of following four lemmas are not so very different from those in the classical case of convergence and cluster points of nets and filters and hence omitted. lemma 5.6. a point x is a uθ-cluster point of a filter ℑ if and only if there is a filter ℜ finer than ℑ and ℜ uθ −→ x. lemma 5.7. an ultranet uθ-converges to each of its uθ-cluster points. lemma 5.8. a point x ∈ x is a uθ-cluster point of a net {xα} if and only if x is a uθ-cluster point of the filter generated by the net {xα}. lemma 5.9. a filter ℑ uθ-converges to x if and only if the net based on ℑ uθ-converges to x. theorem 5.10. for a topological space x, the following statements are equivalent. (a) x is θ-compact. (b) every family of θ-closed sets with finite intersection property has a nonempty intersection. (c) each filter on x has a uθ-cluster point in x. (d) each net in x has a uθ-cluster point in x. (e) each ultranet in x is uθ-convergent. (f) each ultrafilter on x is uθ-convergent. proof. the assertion of (a) ⇒ (b) is easy. to prove (b) ⇒ (c), let ℑ be a filter on x. suppose that ℑ has no uθ-cluster point in x. then for each x ∈ x, there exists a θ-open set ux containing x such that ux ∩ fx = ∅ for some fx in ℑ and so x /∈ cluθfx. thus the collection {cluθf : f ∈ ℑ} is a family of θ-closed sets with finite intersection property which has empty intersection. the implication (c) ⇒ (d) is immediate in view of lemma 5.8 and (d) ⇒ (e) follows from lemma 5.7. similarly, the assertion (e) ⇒ (f) is clear from lemma 5.9. to prove (f) ⇒ (a), suppose that each ultrafilter on x is uθ-convergent. let v be a covering of x by θ-open sets. if v has no finite subcovering of x, then the collection {x − v : v ∈ v} is a filter base and so is contained in an ultrafilter g. by hypothesis the ultrafilter g is uθ-convergent to x, say. let v ∈ v be a θ-open set containing x. since g uθ −→ x, v ∈ g. but x − v ∈ g. this contradiction proves that v has a finite subcovering. � a space x is said to be point paracompact [2] if for every open covering u of x and each x ∈ x there is an open refinement v of u such that v is 242 j. k. kohli and a. k. das locally finite at x. we say that a space x is θ-point paracompact if for each open covering u of x and each x ∈ x there is an open refinement v of u and a θ-open set u containing x which intersects only finitely many members of v. it is clear from the definitions that every compact space is θ-point paracompact and that a θ-point paracompact space is point paracompact. the space x in example 2.3 is a point paracompact space which is not θ-point paracompact. theorem 5.11. a topological space x is θ-point paracompact if and only if every open covering of x has, for each x ∈ x, a finite subset whose union contains the closure of some θ-open set containing x. proof. suppose x is θ-point paracompact space and let u = {uα : α ∈ λ} be an open covering of x. suppose x ∈ x. then there exists an open refinement {vλ : λ ∈ λx} of u and a θ-open set n (x) containing x such that n (x)∩vλ 6= ∅ for only finitely many λ ∈ λx. it follows that there exists a finite subset {uα(k) : k = 1, . . . , n} of u such that n (x) ⊂ n⋃ k=1 uα(k). conversely, let {uα : α ∈ λ} be an open covering of x and let x ∈ x. so by hypothesis there exists a θ-open set n (x) containing x and a finite subset {uα(i) : i = 1, . . . , n} of u such that n (x) ⊂ n⋃ i=1 uα(i). let v = { (x − n (x) ∩ uα : α ∈ λ } ∪{uα(i) : i = 1, ..., n}. clearly, v is a refinement of u with n (x) intersecting only finitely many members of v. so x is θ-point paracompact. � corollary 5.12. every regular space is θ-point paracompact. corollary 5.13 ([2]). every regular space is point paracompact. theorem 5.14. if x is a θ-point paracompact space then every filter with a uθ-cluster point has a cluster point. proof. let x be θ-point paracompact space and let ℑ be a filter with no cluster point. let x ∈ x, then u = {x − f : f ∈ ℑ} is a directed open covering of x. by the hypothesis, there exists an open refinement v of u which is θ-locally finite at x. so there exists a θ-open set v containing x which intersects only finitely many members of v. let w = ∪{u : u ∈ v, u ∩ v = ∅ }. we have v ∩ w = ∅. now x − w ⊆ ∪ {u : u ∈ v; u ∩ v 6= ∅}. since v is a refinement of u there exist finitely many oi’s in u such that ∪{u ∈ v; u ∩ v 6= ∅}⊆ n⋃ i=1 {oi : oi ∈ u }. as u is a directed cover, there is some f in ℑ such that n⋃ i=1 {oi : oi ∈ u}⊆ x − f . thus x − w ⊆ x − f . i.e., f ⊆ w . hence v is a θ-open set containing x and v ∩ f = ∅. therefore x cannot be a uθ-limit point of ℑ. � a combination of theorems 5.10(c) and 5.14, yields the following factorization of compactness. a class of spaces ... 243 theorem 5.15. a topological space x is compact if and only if it is both a θ-compact space and a θ-point paracompact space. corollary 5.16. a regular θ-compact space is compact. corollary 5.17 ([3, 27]). a regular almost compact space is compact. references [1] a. j. d’aristotle, quasicompactness and functionally hausdorff spaces, j. austral. math. soc. 15 (1973), 319–324. [2] j. m. boyte, point (countable) paracompactness, j. austral. math. soc.,15 (1973), 138– 144. [3] á. császár, general topology, adam higler ltd, bristol, 1978. [4] a. k. das, a note on θ-hausdorff spaces, bull. cal. math. soc. 97(1) (2005), 15–20. [5] a. s. davis, indexed systems of neighbourhoods for general topological spaces, amer. math. monthly 68 (1961), 886–893. [6] r. f. dickman and j. r. porter, θ-perfect and θ-absolutely closed functions, illinois j. math. 21 (1977), 42–60. [7] s. fomin, extensions of topological spaces, ann. math. 44 (1943), 471–480. [8] z. frolik, generalizations of compact and lindelöf spaces, czechoslovak math j. 13 (84) (1959), 172–217 (russian) mr 21 3821. [9] e. hewitt, on two problems of urysohn, ann. math. (2) 47 (1946), 503–509. [10] j. e. joseph, θ-closure and θ-subclosed graphs, math. chron. 8 (1979), 99–117. [11] j. k. kohli and a. k. das, new normality axioms and decompositions of normality, glasnik mat. 37(57)(2002), 163–173. [12] j. k. kohli and a. k. das, on functionally θ-normal spaces, applied general topology 6(1) (2005), 1–14. [13] j. k. kohli, a. k. das and r. kumar, weakly functionally θ-normal spaces, θ-shrinking of covers and partition of unity, note di matematica, 19(2)(1999), 293–297. [14] c. kuratowski, topologie i, hafner, new york, 1958. [15] c. t. liu, absolutely closed spaces, trans. amer. math. soc. 130 (1968), 68–104. [16] t. noiri, δ-continuous functions, j. korean math. soc. 16 (1980), 161–166. [17] j. r. porter and j. thomas, on h-closed spaces and minimal hausdorff spaces, trans. amer. math. soc. 138 (1969), 159–170. [18] c. t. scarborough and a. h. stone, products of nearly compact spaces, trans. amer. math. soc. 124 (1966), 131–147. [19] m. k. singal and s. p. arya, on almost regular spaces, glasnik mat. 4 (24) (1969), 89–99. [20] m. k. singal and s. p. arya, on almost normal and almost completely regular spaces, glasnik mat. 5(25)(1970), 141–152. [21] m. k. singal and a. mathur, on nearly compact spaces, boll. u.m.i. 2(4) (1969), 702– 710. [22] m. k. singal and a. r. singal, mildly normal spaces, kyungpook math. j. 13(1) (1973), 27–31. [23] s. sinharoy and b. bandopadhyay, on θ-completely regular and locally θ-h closed spaces, bull. cal. math. soc. 87 (1995), 19–28. [24] e. v. stchepin, real valued functions and spaces close to normal, sib. j. math. 13:5 (1972), 1182–1196. [25] r. m. stephenson, jr., spaces for which the stone-weierstrass theorem holds, trans. amer. math. soc. 133 (1968), 537–546. [26] r. m. stephenson, jr., product spaces for which stone-weierstrass theorem holds, proc. amer. math. soc. 21 (1969), 284–288. [27] w. j. thron, topological structures, holt, rinehart and winston, new york (1966). 244 j. k. kohli and a. k. das [28] n. v. veličko, h-closed topological spaces, amer. math. soc. transl. 2, 78 (1968), 103–118. received may 2005 accepted november 2005 j. k. kohli department of mathematics, hindu college, university of delhi, delhi-110007, india a. k. das (ak das@lycos.com) department of mathematics, bhim rao ambedkar college, university of delhi, yamuna vihar, delhi-110094, india @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 35–46 on ϕ1,2-countable compactness and filters t. h. yalvaç ∗ abstract. in this work the author investigates some relations between ϕ1,2-countable compactness, filters, sequences and ϕ1,2-closure operators. 2000 ams classification: 54a20, 54d30. keywords: countable compactness, filter, convergence, operation, unification. 1. introduction many generalizations of the notion of compact space have been defined in the literature, including those of quasi h-closed space, s-closed space, rs-compact space, feebly compact space, countably s-closed space, countably rs-compact space, and many more. some of these concepts have been characterized in terms of filters and nets, and this has lead to such notions as r-convergence, rcconvergence, sr-convergence, r-accumulation point, rc-accumulation point and sr-accumulation point of filters and filterbases. the notion of an operation on a topological space is a useful tool when attempting to unify such concepts, and in earlier studies we have defined ϕ1,2-countably compact sets, ϕ1,2-convergence of a filter and ϕ1,2-accumulation points of a filter, and used these to obtain some such unifications. in the present work we will study the relations between ϕ1,2-countable compactness, filters, sequences and ϕ1,2-closure operators. there are several different definitions of an operation in the literature. we have used the one first given in [12] for fuzzy topological spaces. in a topological space (x,τ), int , cl , scl , pcl etc. will stand for the interior, closure, semi-closure, pre-closure operations, and so on. for a subset a of x, ao, ā will also be used to denote the interior and closure of a, respectively. ∗dedicated to the memory of professor doğan çoker. 36 t. h. yalvaç definition 1.1. let (x,τ) be a topological space. a mapping ϕ : p(x) → p(x) is called an operation on (x,τ) if ϕ(∅) = ∅ and ao ⊆ ϕ(a), ∀a ∈ p(x). the class of all operations on a topological space (x,τ) will be denoted by o(x,τ). for ϕ1,ϕ2 ∈ o(x,τ) we set ϕ1 ≤ ϕ2 ⇐⇒ ϕ1(a) ⊆ ϕ2(a), ∀a ∈ p(x). the operations ϕ, ϕ̃ are dual if ϕ̃(a) = x \ϕ(x \a), ∀a ∈ p(x). an operation ϕ ∈ o(x,τ) is called monotonous if ϕ(a) ⊆ ϕ(b) whenever a ⊆ b (a,b ∈ p(x)). definition 1.2. let ϕ ∈ o(x,τ). then a ⊆ x is called ϕ-open if a ⊆ ϕ(a). dually, b ⊆ x is called ϕ-closed if x \b is ϕ-open. clearly, x and ∅ are both ϕ-open and ϕ-closed, while each open set is a ϕ-open set for any ϕ ∈ o(x,τ). if (x,τ) is a topological space, ϕ ∈ o(x,τ), then ϕo(x), ϕc(x) will denote respectively the set of ϕ-open, ϕ-closed subsets of x. for x ∈ x we set ϕo(x,x) = {u ∈ ϕo(x) | x ∈ u}. for ϕ2,ϕ1 ∈ ϕo(x) sufficient, generally not necessarily, conditions for ϕ1o(x) ⊆ ϕ2o(x) are ϕ2 ≥ ϕ1 or ϕ2 ≥ ı [21]. here ı is the identity operation. definition 1.3. for the operations ϕ1, ϕ2 ∈ o(x,τ), ϕ2 is called regular with respect to ϕ1o(x) if for each x ∈ x and u,v ∈ ϕ1o(x,x), there exists a w ∈ ϕ1o(x,x) such that ϕ2(w) ⊆ ϕ2(u) ∩ϕ2(v ). clearly, if ϕ1o(x) is closed under finite intersection and ϕ2 is monotonous, then ϕ2 is regular w.r.t. ϕ1o(x). definition 1.4. let ϕ1,ϕ2 ∈ o(x,τ), a ⊆ x, x ∈ x. then: (a) x ∈ ϕ1,2int a iff there exists a u ∈ ϕ1o(x,x) such that ϕ2(u) ⊆ a. (b) x ∈ ϕ1,2cl a ⇐⇒ ϕ2(u) ∩a 6= ∅ for each u ∈ ϕ1o(x,x). (c) a is ϕ1,2-open ⇐⇒ a ⊆ ϕ1,2int a. (d) a is ϕ1,2-closed ⇐⇒ ϕ1,2cl a ⊆ a. for any set a we have x \ϕ1,2int a = ϕ1,2cl (x \a) and a is ϕ1,2-open iff x \a is ϕ1,2-closed. definition 1.5. [1] a subfamily u of the power set of a non-empty set x is called a supratopology on x if ∅,x ∈u and u is closed under arbitrary unions. if u is a supratopology on x, then the pair (x,u) is called a supratopological space. the notions of base, first and second countablility for a supratopology may be defined as for topological spaces [2]. if the operation ϕ ∈ o(x,τ) is monotonous, then ϕo(x) is a supratopology. theorem 1.6. [22] let ϕ1,ϕ2 ∈ o(x,τ). then: on ϕ1,2-countable compactness 37 (a) ϕ1,2o(x), the family of all ϕ1,2-open subsets of x, is a supratopology on x. (b) if ϕ2 is regular w.r.t. ϕ1o(x), then the operator ϕ1,2cl defines the topology τϕ1,2 = {t | t ⊆ x, ϕ1,2cl (x \t) ⊆ x \t} = ϕ1,2o(x). (c) if ϕ2 is regular w.r.t. ϕ1o(x) and ϕ1o(x) ⊆ ϕ2o(x), then the operator ϕ1,2cl defines the topology τϕ1,2 = {t | t ⊆ x,ϕ1,2cl (x \t) = x \t} = ϕ1,2o(x). (d) if ϕ2 is regular w.r.t. ϕ1o(x), ϕ1o(x) ⊆ ϕ2o(x), and ϕ2(u) ∈ ϕ1,2o(x) for each u ∈ ϕ1o(x), then the operator ϕ1,2cl is a kuratowski closure operator and ϕ1,2cl a = τϕ1,2 cl a, ∀a ⊆ x. clearly if ϕ1 ∈ o(x,τ) is monotonous and ϕ2 = ı then ϕ1,2o(x) = ϕ1o(x) and ϕ1,2c(x) = ϕ1c(x). the following example illustrates the wide range of well known concepts covered by the notions defined above. example 1.7. for the operations ϕ1 = int , ϕ2 = cl ◦ int , ϕ3 = cl , ϕ4 = scl , ϕ5 = ı, ϕ6 = int ◦ cl , defined on a topological space we have: • ϕ1 ≤ ϕ2 ≤ ϕ3 and ϕ1 ≤ ϕ6 ≤ ϕ4 ≤ ϕ3. • ϕ1o(x) = τ, • ϕ2o(x) = so(x) = the family of semi-open sets. • ϕ3o(x) = ϕ5o(x) = p(x) = the power set of x. • ϕ6 = po(x) = the family of pre-open sets. • ϕ1,3o(x) = τθ = the topology of all θ-open sets. • ϕ2,4o(x) = sθo(x) = the family of semi-θ-open sets. • ϕ1,6o(x) = τs = the semi regularization topology of x. • ϕ2,3o(x) = θso(x) = the family of all θ-semi-open sets. • the operations ϕ1,ϕ3 and ϕ2,ϕ6 are dual to one another. all these operations are regular w.r.t. ϕ1o(x). 2. ϕ1,2–countable compactness definition 2.1. [21] let ϕ1,ϕ2 ∈ o(x,τ), x ∈ a ⊆ p(x) and a ∈ p(x). then: (a) if each countable a-cover u of a has a finite subfamily u′ such that a ⊆ ⋃ {ϕ2(u) | u ∈ u′}, then we say that a is (a ϕ2)-countably compact relative to x (for short, a (a ϕ2)-c.c. set). (b) we call a (a ı)-c.c. set a a-c.c. set. (c) if we take a = ϕ1o(x) in (a) we say that a is a ϕ1,2-c.c. set. if we take a = ϕ1,2o(x) in (b) we say that a is a ϕ1,2o(x)-c.c. set. if x is ϕ1,2-c.c. (ϕ1,2o(x)-c.c.) relative to itself, then x will be called a ϕ1,2-c.c. (ϕ1,2o(x)-c.c.) space. we remark that the condition x ∈ a is added here, and in our earlier papers, to guarantee the existence of an a-cover or of a countable a-cover of a subset of x. however, all the results still hold without this condition. 38 t. h. yalvaç one may define ϕ1,2-compact, a-compact, ϕ1,2-lindelöf and a-lindelöf sets in a similar way [20, 23]. we assume that all the operations ϕi, i = 1, 2, . . . are defined on (x,τ) whenever they are used. example 2.2. let a ⊆ x. (1) if ϕ1 = int , ϕ2 = ı, then a is a ϕ1,2-c.c. set iff a is countably compact. (2) if ϕ1 = int , ϕ2 = cl , then a is a ϕ1,2-c.c. set iff a is feebly compact relative to x [16], and x is ϕ1,2-c.c. iff x is feebly compact (or, equivalently, lightly compact). x is h(1)-closed [16] iff it is a hausdorff first countable ϕ1,2-c.c. space with respect to these operations. (3) if ϕ1 = cl ◦int , ϕ2 = cl , then x is ϕ1,2-c.c. iff it is countably s-closed [6]. (4) if ϕ1 = int , ϕ2 = int ◦ cl , then x is strongly h(1)-closed [19] iff it is a hausdorff first countable ϕ1,2-c.c. space. (5) if ϕ1 = cl ◦ int , ϕ2 = scl , then x is ϕ1,2-c.c. iff it is countably rs-compact [7]. (6) for ϕ1 = int ◦ cl ◦ int , ϕ2 = ı, we have ϕ1o(x) = ϕ1,2o(x) = τα. hence, x is countably α-compact [13] iff it is ϕ1,2-c.c. iff it is ϕ1,2o(x)-c.c. iff it is ϕ1o(x)-c.c. definition 2.3. let f be a filter (or filterbase) on x, (xn) a sequence in x and a ∈ x. we say that: (a) f, ϕ1,2-accumulates to a, if a ∈ ⋂ {ϕ1,2cl f | f ∈f} [20]. (b) f, ϕ1,2-converges to a, if for each u ∈ ϕ1o(x,a), there exists f ∈ f such that f ⊆ ϕ2(u) [20]. (c) (xn), ϕ1,2-accumulates to a, if for each u ∈ ϕ1o(x,a) and for each n, there exists an n0 such that n0 ≥ n and xn0 ∈ ϕ2(u). (d) (xn), ϕ1,2-converges to a, if for each u ∈ ϕ1o(x,a), there exists an n0 such that for each n (n ≥ n0), xn ∈ ϕ2(u). example 2.4. let f be a filter (or filterbase) on x and a ∈ x. (1) if ϕ1 = int , ϕ2 = ı, then f, ϕ1,2-converges to a iff f converges to a in (x,τ) and f, ϕ1,2-accumulates to a iff f accumulates to a (or a is an adherent point of f) in (x,τ). (2) if ϕ1 = int , ϕ2 = cl , then f, ϕ1,2-converges to a iff f, r-converges [10] (or equivalently θ-converges [9], almost converges [3]) to a, and f, ϕ1,2-accumulates to a iff a is an r-accumulation point [10] (or an almost adherent point [3]) of f. (3) for ϕ1 = cl ◦ int , ϕ2 = cl , it can be seen that, f, ϕ1,2-converges (ϕ1,2-accumulates) to a iff f, rc-converges (rc-accumulates) to a [9], since {v̄ | v ∈ τ, a ∈ v̄} = {ū | u ∈ so(x), a ∈ u}. at the same time, f, ϕ1,2-converges (ϕ1,2-accumulates) to a iff f, s-converges (s-accumulates) to a [4]. (4) if ϕ1 = int ◦cl ◦int , ϕ2 = ı, then f, ϕ1,2-converges (ϕ1,2-accumulates) to a iff f, α-converges (α-accumulates) to a [14]. on ϕ1,2-countable compactness 39 (5) if ϕ1 = cl ◦ int , ϕ2 = scl , it can be easily seen that f, ϕ1,2-converges (ϕ1,2-accumulates) to a iff f, sr-converges (sr-accumulates) to a [5]. (6) for ϕ1 = cl ◦ int , ϕ2 = int ◦ scl , then we see that f, ϕ1,2-converges (ϕ1,2-accumulates) to a iff f, rs-converges (rs-accumulates) to a [15]. (7) if ϕ1 = int , ϕ2 = int ◦cl then f, ϕ1,2-converges (ϕ1,2-accumulates) to a iff f, δ-converges (δ-accumulates) to a [19]. similar characterizations of the various notions of convergence and accumulation point for sequences and nets given in the literature can be easily given, and we omit the details. theorem 2.5. let a ⊆ x and f = {fn | n ∈ n} be a countable filterbase which meets a. if some sequence satisfying xn ∈ ( ⋂n i=1 fi) ∩ a for each n, ϕ1,2-accumulates to some point a ∈ x, then the filterbase f, ϕ1,2-accumulates to a. conversely if for any sequence (xn) in a the countable filterbase f = {{xm | m ≥ n} | n ∈ n} which consists of the tails of the sequence (xn), ϕ1,2accumulates to some point a ∈ x, then the sequence (xn), ϕ1,2-accumulates to a. proof. let f = {fn | n ∈ n} be a countable filterbase which meets a. then f′ = { ⋂n i=1 fi | n ∈ n} is a decreasing countable filterbase which meets a and generates the same filter as f. take xn ∈ ( ⋂n i=1 fi) ∩ a for each n, and let (xn), ϕ1,2-accumulate to a. then, for each u ∈ ϕ1o(x,a) and for each n, ∅ 6= ϕ2(u) ∩ ( ⋂n i=1 fi) ∩a ⊆ ϕ2(u) ∩ ( ⋂n i=1 fi), hence ϕ2(u) ∩fn 6= ∅. so, f, ϕ1,2-accumulates to a. conversely let (xn) be a sequence in a, and let f = {tn | n ∈ n} be the countable filterbase consisting of the tails of (xn), which ϕ1,2-accumulate to some point a and meets a. then for each u ∈ ϕ1o(x,a) and for each n, ϕ2(u) ∩tn 6= ∅. this means that a is a ϕ1,2-accumulation point of (xn). � corollary 2.6. let a ⊆ x. each countable filterbase which meets a, ϕ1,2accumulates to some point of a iff each sequence in a, ϕ1,2-accumulates to some point of a. theorem 2.7. let a ⊆ x. if each countable filterbase which meets a, ϕ1,2accumulates to some point of a, then a is a ϕ1,2-c.c. set. proof. let a ⊆ ⋃ u, u = {un | n ∈ i}, i countable and un ∈ ϕ1o(x). assume that for each finite subset j of i we have a 6⊆ ⋃ i∈j ϕ2(ui). then a∩(x\ ⋃ i∈j ϕ2(ui)) 6= ∅. the family f = {x\ ⋃ i∈j ϕ2(ui) | j ⊆ i, j finite} is a countable filterbase which meets a. so, a∩ ( ⋂ {ϕ1,2cl f | f ∈ f}) 6= ∅. let f, ϕ1,2-accumulate to a ∈ a. there exists an i0 ∈ i such that a ∈ ui0 . now x \ϕ2(ui0 ) ∈ f, ϕ2(ui0 ) ∩ (x \ϕ2(ui0 )) 6= ∅. this contradiction completes the proof. � however, the converse of the above theorem need not hold. for operations ϕ1 = int , ϕ2 = cl in (x,τ), each countable filterbase ϕ1,2-accumulates in 40 t. h. yalvaç (x,τ) iff (x,τ) is sq-closed [18]. also, (x,τ) is ϕ1,2-c.c. iff it is a feebly compact space. herrington [11] gave an example, occurring in [8], of a regular, feebly compact but not countably compact space. since this space is regular, a ϕ1,2-accumulation point is the same as an accumulation point of a sequence (filterbase) in (x,τ), so there is a sequence (countable filterbase) which does not ϕ1,2-accumulate to any point in x. clearly any ϕ1,2-compact set is a ϕ1,2-lindelöf set and a ϕ1,2-c.c. set. a set is a ϕ1,2o(x)-compact set iff it is a ϕ1,2o(x)-lindelöf set and a ϕ1,2o(x)c.c. set. if ϕ1,2o(x) has a countable base then each ϕ1,2o(x)-c.c. set is a ϕ1,2o(x)-compact set. we will define conditions (∗) and (∗∗) on the operations ϕ1 and ϕ2 in the following way: (∗) ϕ2 ≥ ϕ1 or ϕ2 ≥ ı, (∗∗) ϕ2(u) ∈ ϕ1o(x) and ϕ2(ϕ2(u)) ⊆ ϕ2(u), for each u ∈ ϕ1o(x). example 2.8. (1) if ϕ1 = int , ϕ2 = cl , then the condition (∗) is satisfied. (2) if ϕ1 = cl ◦int , ϕ2 = scl , then the conditions (∗) and (∗∗) are satisfied. (3) if ϕ1 = int , ϕ2 = int ◦cl , then the conditions (∗) and (∗∗) are satisfied. (4) if ϕ1 = cl ◦ int , ϕ2 = cl , then the conditions (∗) and (∗∗) are satisfied. if the condition (∗∗) is satisfied then a set is ϕ1,2-compact set iff it is both a ϕ1,2-lindelöf set and a ϕ1,2-c.c. set. theorem 2.9. let ϕ1 be monotonous, (x,ϕ1o(x)) be a second countable supratopological space and a ⊆ x. if a is a ϕ1,2-c.c. set then each filterbase which meets a, ϕ1,2-accumulates to some point of a. proof. let the supratopology ϕ1o(x) have a countable base, a be a ϕ1,2-c.c. set and f a filterbase which meets a. assume that a∩ ( ⋂ {ϕ1,2cl f | f ∈f}) = ∅. for any x ∈ a, there exists a ux ∈ ϕ1o(x,x) and an fx ∈ f such that ϕ2(ux) ∩fx = ∅. now, u = {ux | x ∈ a} is a ϕ1-open open cover of a. since ϕ1o(x) has a countable base, u has a countable subfamily which covers a. since a is a ϕ1,2-c.c. set, there exists a finite subfamily {ux1,ux2, . . . ,uxn} of u such that a ⊆ ⋃n i=1 ϕ2(uxi). now ( ⋃n i=1 ϕ2(uxi)) ∩ ( ⋂n i=1 fxi) = ∅, so a ∩ ( ⋂n i=1 fxi) = ∅. this contradiction completes the proof. � corollary 2.10. under the assumptions of theorem 2.9., the following are equivalent. (a) a is a ϕ1,2-c.c. set. (b) a is a ϕ1,2-compact set. (c) each countable filterbase which meets a, ϕ1,2-accumulates to some point of a. proof. in [20], it is shown that a is a ϕ1,2-compact set iff each filterbase which meets a, ϕ1,2-accumulates to some point of a. since each ϕ1,2-compact set is a ϕ1,2-c.c. set, the proof is now clear from theorem 2.7. � on ϕ1,2-countable compactness 41 theorem 2.11. let ϕ1, ϕ2 be monotonous and suppose that the conditions (∗) and (∗∗) hold. if the supratopology ϕ1o(x) has a countable base b(ϕ1o(x)), then b′ = {ϕ2(u) | u ∈b(ϕ1o(x))} is a countable base for the supratopology ϕ1,2o(x). proof. under the given conditions, b = {ϕ2(u) | u ∈ ϕ1o(x)} is a base for the supratopology ϕ1,2o(x) and b′ ⊆ b ⊆ ϕ1,2o(x). let v ∈ ϕ1,2o(x) and x ∈ v . there exists a u ∈ ϕ1o(x,x) such that ϕ2(u) ⊆ v . hence, x ∈ u ⊆ ϕ2(u) ⊆ v . there exists a u ′ ∈ b(ϕ1o(x)) such that x ∈ u ′ ⊆ u. hence, we have x ∈ ϕ2(u′) ⊆ ϕ2(u) ⊆ v and ϕ2(u′) ∈b′. � theorem 2.12. let (∗) and (∗∗) hold and let b = {ϕ2(u) | u ∈ ϕ1o(x)}. then the following are equivalent for any subset a of x. (a) a is a ϕ1,2-compact set. (b) a is a b-compact set. (c) a is both a ϕ1,2-lindelöf set and a ϕ1,2-c.c. set. (d) a is both a b-lindelöf set and a b-c.c. set. proof. under the given conditions, a is a ϕ1,2-compact set iff it is b-compact set [20], a is a ϕ1,2-lindelöf set iff it is a b-lindelöf set [23], a is a ϕ1,2-c.c. set iff it is a b-c.c. set [22]. hence (b) ⇐⇒ (d) is now clear, as are the others. � theorem 2.13. let ϕ1, ϕ2 be monotonous and suppose that the conditions (∗) and (∗∗) hold. if the supratopology ϕ1o(x) has a countable base b(ϕ1o(x)), or if b = {ϕ2(u) | u ∈ ϕ1o(x)} is countable, then the following are equivalent. (a) a is a ϕ1,2-c.c. set. (b) a is a ϕ1,2o(x)-c.c. set. (c) a is a b-c.c. set. (d) a is a ϕ1,2-compact set. (e) a is a ϕ1,2o(x)-compact set. (f) a is a b-compact set. proof. under the conditions (∗) and (∗∗), (a) ⇐⇒ (c), (b) =⇒(c) and (d) ⇐⇒ (e) ⇐⇒ (f) are given in [22] and [20] respectively. if b is a countable base of ϕ1,2o(x), then (c) =⇒(b) is clear. in the other case, b ′ = {ϕ2(u) | u ∈ b(ϕ1o(x))} is a countable base of ϕ1,2o(x) and b ′ ⊆b ⊆ ϕ1,2o(x). hence, a b-c.c. set will be a b ′ -c.c. set and a b ′ -c.c. set will be a ϕ1,2o(x)-c.c. set, so we have again (c) =⇒(b). in each case (b) ⇐⇒ (e) is clear. � theorem 2.14. let ϕ1 be monotonous and let a ∈ x have a countable local base cϕ1 (a) in the supratopological space (x,ϕ1o(x)). (1) if ϕ2 is monotonous and regular w.r.t. ϕ1o(x), then the family f = {ϕ2(u) | u ∈ cϕ1 (a)} is a countable filterbase and ϕ1,2-converges to a. (2) if ϕ1o(x) is a topology and ϕ1o(x) ⊆ ϕ2o(x), then cϕ1 (a) is a countable filterbase which ϕ1,2-converges to a. 42 t. h. yalvaç proof. (1) for u,u′ ∈ cϕ1 (a), a ∈ u ∩ u′ and u,u′ ∈ ϕ1o(x). since ϕ2 is regular w.r.t. ϕ1o(x), there exists a v ∈ ϕ1o(x,a) such that ϕ2(v ) ⊆ ϕ2(u) ∩ ϕ2(u′). there exists a vc ∈ cϕ1 (a) such that vc ⊆ v . since ϕ2 is monotonous, we have ϕ2(vc) ⊆ ϕ2(v ) ⊆ ϕ2(u) ∩ ϕ2(u′). hence f is a countable filterbase. let u ∈ ϕ1o(x,a). there exists a uc ∈ cϕ1 (a) such that uc ⊆ u. ϕ2(uc) ∈ f and, since ϕ2 is monotonous ϕ2(uc) ⊆ ϕ2(u). so, f is ϕ1,2-convergent to a. (2) for u,u′ ∈ cϕ1 (a), a ∈ u∩u′ ∈ ϕ1o(x,a). there exists a uc ∈ cϕ1 (a) such that uc ⊆ u ∩u′. hence cϕ1 (a) is a countable filterbase. now, let v ∈ ϕ1o(x,a). there exists a vc ∈ cϕ1 (a) such that vc ⊆ v . since ϕ1o(x) ⊆ ϕ2o(x), we have vc ⊆ v ⊆ ϕ2(v ). hence cϕ1 (a), ϕ1,2-converges to a. � theorem 2.15. let ϕ1, ϕ2 be monotonous, let a ∈ x have a countable local base cϕ1 (a) in (x,ϕ1o(x)) and also let ϕ2 be regular w.r.t. ϕ1o(x). for a ⊆ x, a ∈ ϕ1,2cl a iff there exists a filter which contains a, has a countable base and ϕ1,2-converges to a. proof. let a ∈ ϕ1,2cl a. then for each u ∈ ϕ1o(x,a), ϕ2(u) ∩ a 6= ∅. as in the proof of theorem 2.14.(1), it is easly seen that fb = {ϕ2(v ) ∩a | v ∈ cϕ1 (a)} is a countable filterbase. the filter f generated by fb contains a, and {ϕ2(v ) | v ∈ cϕ1 (a)}⊆f. clearly f is ϕ1,2-convergent to a. the other part of the proof is clear from corollary 3.4. in [20]. � theorem 2.16. let ϕ1, ϕ2 be monotonous, (x,ϕ1o(x)) be a first countable supratopological space, and define cl ∗ : p(x) −→ p(x) by cl ∗(a) = {x | there exists a filter that contains a, has a countable base and ϕ1,2-converges to x}, for each a ∈ p(x). (1) if ϕ2 is regular w.r.t. ϕ1o(x), then cl ∗(a) = ϕ1,2cl a for each a ∈ p(x), and cl ∗ defines the topology τ∗ = {u ⊆ x | (x\u)∗ ⊆ x\u} = ϕ1,2o(x). (2) if ϕ2 is regular w.r.t. ϕ1o(x) and ϕ1o(x) ⊆ ϕ2o(x), then cl ∗ defines the topology τ∗ = {u ⊆ x | (x \u)∗ = x \u} = ϕ1,2o(x). (3) if ϕ2 is regular w.r.t. ϕ1o(x), ϕ1o(x) ⊆ ϕ2o(x), and ϕ2(u) ∈ ϕ1,2o(x) for each u ∈ ϕ1o(x), then the operator cl ∗ is a kuratowski closure operator defining τ∗ = {u ⊆ x | (x \u)∗ = x \u} = ϕ1,2o(x). hence, if ϕ1,ϕ2 are monotonous and (x,ϕ1o(x)) is a first countable topological space, then the ϕ1,2-closure operator and the topology τϕ1,2 = {u ⊆ x | ϕ1,2cl (x\u) ⊆ x\u} = ϕ1,2o(x) can be defined using filters with countable bases. proposition 2.17. if ϕ1o(x) ⊆ ϕ2o(x) (hence, if ϕ2 ≥ ϕ1 or ϕ2 ≥ ı), then a ⊆ ϕ1,2cl a for each a ∈ p(x). on ϕ1,2-countable compactness 43 proposition 2.18. if (∗∗) holds, then ϕ2(u) ⊆ ϕ1,2int (ϕ2(u)) (i.e., ϕ2(u) ∈ ϕ1,2o(x)) for each u ∈ ϕ1o(x). proof. let u ∈ ϕ1o(x) and x ∈ ϕ2(u). then x ∈ ϕ2(u) ∈ ϕ1o(x) and ϕ2(ϕ2(u)) ⊆ ϕ2(u). so x ∈ ϕ1,2int (ϕ2(u)). � corollary 2.19. (a) under the condition (∗∗), we have, ϕ1,2cl (x \ϕ2(u)) ⊆ x \ϕ2(u) for each u ∈ ϕ1o(x). (b) if ϕ1o(x) ⊆ ϕ2o(x) and (∗∗) holds, then ϕ1,2cl (x \ ϕ2(u)) = x \ ϕ2(u) for each u ∈ ϕ1o(x). remark 2.20. a) if ϕ̃2 is the dual operation of ϕ2, then {x \ ϕ2(u) | u ∈ ϕ1o(x)} = {ϕ̃2(x \u) | u ∈ ϕ1o(x)} = {ϕ̃2(k) | k ∈ ϕ1c(x)}. b) if ϕ1 is monotonous (in which case ϕ1o(x) is a supratopology), and ϕ2(u ∪ v ) = ϕ2(u) ∪ ϕ2(v ) for each u,v ∈ ϕ1o(x), then for each finite subfamily {u1,u2, . . . ,un} of ϕ1o(x), ⋃n i=1 ui ∈ ϕ1o(x) and ϕ2( ⋃n i=1 ui) =⋃n i=1 ϕ2(ui). theorem 2.21. consider the following statements: (i) ϕ1 is monotonous. (ii) ϕ2 is monotonous. (iii) ϕ2 ≥ ı or ϕ2 ≥ ϕ1 (i.e. (∗)), (iv) ∀u ∈ ϕ1o(x), ϕ2(u) ∈ ϕ1o(x) and ϕ2(ϕ2(u)) ⊆ ϕ2(u) (i.e. (∗∗)). (v) for each u,v ∈ ϕ1o(x), ϕ2(u ∪v ) = ϕ2(u) ∪ϕ2(v ), (vi) ϕ̃2 is the dual of ϕ2. and (a) a is a ϕ1,2-c.c. set. (b) each countable filterbase f ⊆{x \ϕ2(u) | u ∈ ϕ1o(x)} which meets a, ϕ1,2-accumulates to some point of a. (c) for each countable filterbase f ⊆ {x \ ϕ2(u) | u ∈ ϕ1o(x)} which meets a, we have a∩ ( ⋂ f) 6= ∅. (d) for each decreasing countable filterbase f ⊆{x\ϕ2(u) | u ∈ ϕ1o(x)} which meets a, we have a∩ ( ⋂ {ϕ1,2clf | f ∈f}) 6= ∅. (e) for each decreasing countable filterbase f ⊆{x\ϕ2(u) | u ∈ ϕ1o(x)} which meets a, we have a∩ ( ⋂ f) 6= ∅. (f) if φ is any decreasing sequence of countable non-empty ϕ1-closed sets such that for each f ∈ φ, a∩ ϕ̃2(f) 6= ∅, then a∩ ( ⋂ φ) 6= ∅. then, (1) (b) =⇒ (d) and (c) =⇒ (e). (2) if (iii) holds, then (c) =⇒ (b) and (e) =⇒ (d). (3) if (iii) and (iv) hold, then (c) ⇐⇒ (b) and (e) ⇐⇒ (d). (4) if (iv) holds, then (a) =⇒ (c). (5) if (i) and (v) hold, then (d) =⇒ (b) and (b) =⇒ (a). (6) if (ii) and (vi) hold, then (a) =⇒ (f). (7) if (i), (iii), (v) and (vi) hold, then (f) =⇒ (a). 44 t. h. yalvaç proof. (1) immediate. 2) clear from proposition 2.17. (3) clear from corollary 2.19. (4) let a be a ϕ1,2-c.c. set, and f = {x \ϕ2(ui) | i ∈ i}, ui ∈ ϕ1o(x), be a countable filterbase which meets a. assume that a ∩ ( ⋂ f) = ∅ and a ⊆ ⋃ i∈i ϕ2(ui). since, ϕ2(u) ∈ ϕ1o(x), ϕ2(ϕ2(u)) ⊆ ϕ2(u), for each u ∈ ϕ1o(x), and a is a ϕ1,2-c.c. set, there exists a finite subset j of i such that, a ⊆ ⋃ i∈j ϕ2(ϕ2(ui)) ⊆ ⋃ i∈j ϕ2(ui). we have a ∩ ( ⋂ i∈j(x \ ϕ2(ui))) = ∅. this contradiction completes the proof. (5) let f ⊆ {x \ ϕ2(u) | u ∈ ϕ1o(x)} be a countable filterbase which meets a. then f = {fn | n ∈ n}, where fn = x \ ϕ2(un), n ∈ n and un ∈ ϕ1o(x). let f ′n = ⋂n i=1 fi for each n. then f ′ = {f ′n | n ∈ n} is a decreasing countable filterbase, and f ′n = ⋂n i=1 fi = ⋂n i=1(x \ ϕ2(ui)) = x \ ⋃n i=1 ϕ2(ui) = x \ϕ2( ⋃n i=1 ui). hence, f ′ ⊆{x \ϕ2(u) | u ∈ ϕ1o(x)}. if we assume that (d) holds then a∩ ( ⋂ {ϕ1,2cl f ′n | f ′n ∈ f′}) 6= ∅. since f ′n ⊆ fn for each n, we have ϕ1,2cl f ′n ⊆ ϕ1,2cl fn. so a∩ ( ⋂ {ϕ1,2cl fn | fn ∈ f}) 6= ∅. now, let us verify that (b) =⇒ (a). let a ⊆ ⋃ u, u ⊆ ϕ1o(x) and u = {ui | i ∈ i} be countable. assume that for each finite subset j of i, a 6⊆ ⋃ i∈j ϕ2(ui). then, a ∩ (x \ ⋃ i∈j ϕ2(ui)) 6= ∅. from our hypotheses, ⋃ i∈j ui ∈ ϕ1o(x) and ϕ2( ⋃ i∈j ui) = ⋃ i∈j ϕ2(ui). so, for each finite subset j of i, we have a∩(x\ϕ2( ⋃ i∈j ui)) 6= ∅. let f = {x\ϕ2( ⋃ i∈j ui) | j ⊆ i,j finite}. then f ⊆ {x \ϕ2(u) | u ∈ ϕ1o(x)} and f is a countable filterbase which meets a. there exists an a ∈ a such that a ∈ ⋂ {ϕ1,2cl f | f ∈ f} and a ua ∈ u such that a ∈ ua. now, x \ϕ2(ua) ∈f and ϕ2(ua)∩(x \ϕ2(ua)) = ∅. this contradiction completes the proof. (6) let φ be a countable decreasing sequence of nonempty ϕ1-closed sets such that for each f ∈ φ, a∩ ϕ̃2(f) 6= ∅. assume that a∩( ⋂ φ) = ∅. then, a ⊆ ⋃ {x\f | f ∈ φ}. since for each f ∈ φ, x\f ∈ ϕ1o(x), and a is a ϕ1,2c.c. set, there exists a finite subfamily φ′ of φ such that a ⊆ ⋃ {ϕ2(x \f) | f ∈ φ′}. since ϕ2 is monotonous, a ⊆ ϕ2( ⋃ f∈φ′(x \ f)). there exists an f ′ ∈ φ′ such that ⋃ f∈φ′(x\f) = x\f ′. then a ⊆ ϕ2(x\f ′) = x\ϕ̃2(f ′), so a∩ ϕ̃2(f ′) = ∅. this contradiction completes the proof. (7) let u = {un | n ∈ n} be a countable ϕ1-open cover of a. assume that for each finite subset j of n, a 6⊆ ⋃ i∈j ϕ2(ui). in this case, for each finite subset j of n, x 6= ⋃ i∈j ui since, otherwise, we would have a ⊆ ⋃ i∈j ui ⊆⋃ i∈j ϕ2(ui) for a finite subset j of n. let fn = x \ ⋃n i=1 ui for each n. for each n, fn 6= ∅, fn ∈ ϕ1c(x) and a∩ (x \ ⋃n i=1 ϕ2(ui)) 6= ∅. now on ϕ1,2-countable compactness 45 a∩ (x \ n⋃ i=1 ϕ2(ui)) = a∩ (x \ϕ2( n⋃ i=1 ui)) = a∩ (ϕ̃2(x \ n⋃ i=1 ui)) = a∩ ϕ̃2(fn) 6= ∅. hence, a∩( ⋂∞ n=1 fn) 6= ∅. but ⋂∞ n=1 fn = x\( ⋃∞ n=1 un) and we obtain that a∩ (x \ ⋃∞ n=1 un) = ∅. this contradiction completes the proof. � example 2.22. (1) if ϕ1 = int , ϕ2 = cl , then ϕ̃2 = int and the conditions (i), (ii), (iii), (v) and (vi) are satisfied. (2) if ϕ1 = cl ◦ int , ϕ2 = scl , then conditions (i), (ii), (iii), (iv) and (vi) are satisfied, and ϕ̃2 = semi-interior is the dual of ϕ2. 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[15] t. noiri, on rs-compact spaces, j. korean math. soc. 22 (1) (1985), 19–34. [16] t. g. raghavan, on h(1)-closed spaces-ii, bull. cal. math. soc. 77 (1985), 171–180. [17] r. m. stephenson jr., pseudocompact spaces, trans. amer. math. soc. 134 (1968), 437– 448. [18] t. thompson, sq-closed spaces, math. japonica 22 (4) (1977), 491–495. [19] d. thanapalan and t. g. raghavan, on strongly h(1)-closed spaces, bull. cal. math. soc. 76 (1984), 370–383. [20] t. h. yalvaç, a unified approach to compactness and filters, hacettepe bull. nat. sci. eng., series b 29 (2000), 63–75. [21] t. h. yalvaç, on some unifications (presented at the first turkish international conference on topology and its applications, istanbul, 2000), hacettepe bull. nat. sci. eng., series b 30 (2001), 27–38. [22] t. h. yalvaç, a unified theory on some basic topological concepts, international conference on topology and its applications, macedonia, (2000). [23] t. h. yalvaç, unifications of some concepts related to the lindelöf property, submitted. received december 2001 revised february 2003 t. h. yalvaç hacettepe university, faculty of science, department of mathematics, 06532 beytepe, ankara, turkey. e-mail address : hayal@hacettepe.edu.tr on 1,2-countable compactness and filters. by t.h. yalvaç salvagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 129136 fuzzy quasi-metric spaces valent́ın gregori and salvador romaguera∗ abstract. we generalize the notions of fuzzy metric by kramosil and michalek, and by george and veeramani to the quasi-metric setting. we show that every quasi-metric induces a fuzzy quasi-metric and, conversely, every fuzzy quasi-metric space generates a quasi-metrizable topology. other basic properties are discussed. 2000 ams classification: 54a40, 54e35, 54e15. keywords: fuzzy quasi-metric space; quasi-metric; quasi-uniformity; bicomplete; isometry. 1. introduction in [9], kramosil and michalek introduced and studied an interesting notion of fuzzy metric space which is closely related to a class of probabilistic metric spaces, the so-called (generalized) menger spaces. later on, george and veeramani started, in [3] (see also [5]), the study of a stronger form of metric fuzziness. in particular, it is well known that every metric induces a fuzzy metric in the sense of george and veeramani, and, conversely, every fuzzy metric space in the sense of george and veeramani (and also of kramosil and michalek) generates a metrizable topology ([4], [6], [9], [11], [13]). on the other hand, it is also well known that quasi-metric spaces constitute an efficient tool to discuss and solve several problems in topological algebra, approximation theory, theoretical computer science, etc. (see [10]). in this paper, we introduce two notions of fuzzy quasi-metric space that generalize the corresponding notions of fuzzy metric space by kramosil and michalek, and by george and veeramani to the quasi-metric context. several basic properties of these spaces are obtained. we show that every quasi-metric induces a fuzzy quasi-metric and, conversely, every fuzzy quasi-metric generates a quasi-metrizable topology. with the help of these results one can easily derive many properties of fuzzy quasi-metric spaces. ∗the authors acknowledge the support of generalitat valenciana, grant grupos 03/027. 130 v. gregori and s. romaguera our basic references for quasi-uniform and quasi-metric spaces are [2] and [10]. let us recall that a quasi-pseudo-metric on a set x is a nonnegative real valued function d on x × x such that for all x, y, z ∈ x : (i) d(x, x) = 0; (ii) d(x, z) ≤ d(x, y) + d(y, z). following the modern terminology (see section 11 of [10]), by a quasi-metric on x we mean a quasi-pseudo-metric d on x that satisfies the following condition: d(x, y) = d(y, x) = 0 if and only if x = y. if the quasi-pseudo-metric d satisfies: d(x, y) = 0 if and only if x = y, then we say that d is a t1 quasi-metric on x. a quasi-(pseudo-)metric space is a pair (x, d) such that x is a (nonempty) set and d is a quasi-(pseudo-)metric on x. the notion of a t1 quasi-metric space is defined in the obvious manner. each quasi-pseudo-metric d on x generates a topology τd on x which has as a base the family of open d-balls {bd(x, r) : x ∈ x, r > 0}, where bd(x, r) = {y ∈ x : d(x, y) < r} for all x ∈ x and r > 0. observe that if d is a quasi-metric, then τd is a t0 topology, and if d is a t1 quasi-metric, then τd is a t1 topology. a topological space (x, τ) is said to be quasi-metrizable if there is a quasimetric d on x such that τ = τd. in this case, we say that d is compatible with τ, and that τ is a quasi-metrizable topology. given a quasi-(pseudo-)metric d on x, then the function d−1 defined on x × x by d−1(x, y) = d(y, x), is also a quasi-(pseudo-)metric on x, called the conjugate of d. finally, the function ds defined on x × x by ds(x, y) = max{d(x, y), d−1(x, y)} is a (pseudo-)metric on x. 2. definitions and basic results according to [13], a binary operation ∗ : [0, 1]×[0, 1] → [0, 1] is a continuous t-norm if ∗ satisfies the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous; (iii) a ∗ 1 = a for every a ∈ [0, 1]; (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, with a, b, c, d ∈ [0, 1]. definition 2.1. a km-fuzzy quasi-pseudo-metric on a set x is a pair (m, ∗) such that ∗ is a continuous t-norm and m is a fuzzy set in x × x × [0, +∞) such that for all x, y, z ∈ x : (km1) m(x, y, 0) = 0; (km2) m(x, x, t) = 1 for all t > 0; (km3) m(x, z, t + s) ≥ m(x, y, t) ∗ m(y, z, s) for all t, s ≥ 0; (km4) m(x, y, ) : [0, +∞) → [0, 1] is left continuous. definition 2.2. a km-fuzzy quasi-metric on x is a km-fuzzy quasi-pseudometric (m, ∗) on x that satisfies the following condition: (km2’) x = y if and only if m(x, y, t) = m(y, x, t) = 1 for all t > 0. if (m, ∗) is a km-fuzzy quasi-pseudo-metric on x satisfying: (km2”) x = y if and only if m(x, y, t) = 1 for all t > 0, fuzzy quasi-metric spaces 131 we say that (m, ∗) is a t1 km-fuzzy quasi-metric on x. definition 2.3. a km-fuzzy (pseudo-)metric on x is a km-fuzzy quasi(pseudo-)metric (m, ∗) on x such that for each x, y ∈ x : (km5) m(x, y, t) = m(y, x, t) for all t > 0. remark 2.4. it is clear that every km-fuzzy metric is a t1 km-fuzzy quasimetric; every t1 km-fuzzy quasi-metric is a km-fuzzy quasi-metric, and every km-fuzzy quasi-metric is a km-fuzzy quasi-pseudo-metric. definition 2.5. a km-fuzzy quasi-(pseudo-)metric space is a triple (x, m, ∗) such that x is a (nonempty) set and (m, ∗) is a km-fuzzy quasi-(pseudo)metric on x. the notions of a t 1 km-fuzzy quasi-metric space and of a km-fuzzy (pseudo)metric space are defined in the obvious manner. note that the km-fuzzy metric spaces are exactly the fuzzy metric spaces in the sense of kramosil and michalek. if (m, ∗) is a km-fuzzy quasi-(pseudo-)metric on a set x, it is immediate to show that (m−1, ∗) is also a km-fuzzy quasi-(pseudo-)metric on x, where m−1 is the fuzzy set in x × x × [0, +∞) defined by m−1(x, y, t) = m(y, x, t). moreover, if we denote by mi the fuzzy set in x × x × [0, +∞) given by mi(x, y, t) = min{m(x, y, t), m−1(x, y, t)}, then (mi, ∗) is, clearly, a km-fuzzy (pseudo-)metric on x. proposition 2.6. let (x, m, ∗) be a km-fuzzy quasi-pseudo-metric space. then, for each x, y ∈ x the function m(x, y, ) is nondecreasing. proof. let x, y ∈ x and 0 ≤ t < s. then m(x, y, s) ≥ m(x, x, s − t) ∗ m(x, y, t) = m(x, y, t). � given a km-fuzzy quasi-pseudo-metric space (x, m, ∗) we define the open ball bm (x, r, t), for x ∈ x, 0 < r < 1, and t > 0, as the set bm (x, r, t) := {y ∈ x : m(x, y, t) > 1 − r}. obviously, x ∈ bm (x, r, t). by proposition 2.6, it immediately follows that for each x ∈ x, 0 < r1 ≤ r2 < 1 and 0 < t1 ≤ t2, we have bm(x, r1, t1) ⊆ bm(x, r2, t2). consequently, we may define a topology τm on x as τm := {a ⊆ x : for each x ∈ a there are r ∈ (0, 1), t > 0, with bm (x, r, t) ⊆ a}. moreover, for each x ∈ x the collection of open balls {bm(x, 1/n, 1/n) : n = 2, 3, ...}, is a local base at x with respect to τm . it is clear, that if (x, m, ∗) is a km-fuzzy quasi-metric (respectively, a t1 km-fuzzy quasi-metric, a km-fuzzy metric), then τm is a t0 (respectively, a t1, a hausdorff) topology. the topology τm is called the topology generated by the km-fuzzy quasipseudo-metric space (x, m, ∗). similarly to the proof of result 3.2 and theorem 3.11 of [3], one can show the following results. 132 v. gregori and s. romaguera proposition 2.7. let (x, m, ∗) be a km-fuzzy quasi-pseudo-metric space. then, each open ball bm (x, r, t) is an open set for the topology τm . proposition 2.8. a sequence (xn)n in a km-fuzzy quasi-pseudo-metric space (x, m, ∗) converges to a point x ∈ x with respect to τm if and only if limn m(x, xn, t) = 1 for all t > 0. definition 2.9. a gv-fuzzy quasi-pseudo-metric on a set x is a pair (m, ∗) such that ∗ is a continuous t-norm and m is a fuzzy set in x × x × (0, +∞) such that for all x, y, z ∈ x, t, s > 0 : (gv1) m(x, y, t) > 0; (gv2) m(x, x, t) = 1; (gv3) m(x, z, t + s) ≥ m(x, y, t) ∗ m(y, z, s); (gv4) m(x, y, ) : (0, +∞) → (0, 1] is continuous. definition 2.10. a gv-fuzzy quasi-metric on x is a gv-fuzzy quasi-pseudometric (m, ∗) on x such that for all t > 0: (gv2’) x = y if and only if m(x, y, t) = m(y, x, t) = 1. if (m, ∗) is a gv-fuzzy quasi-pseudo-metric on x such that for all t > 0: (gv2”) x = y if and only if m(x, y, t) = 1, we say that (m, ∗) is a t1 km-fuzzy quasi-metric on x. definition 2.11. a gv-fuzzy (pseudo-)metric on x is a gv-fuzzy quasi(pseudo-)metric (m, ∗) on x such that for all x, y ∈ x, t > 0 : (km5) m(x, y, t) = m(y, x, t). remark 2.12. it is clear that every gv-fuzzy metric is a t1 gv-fuzzy quasimetric; every t1 gv-fuzzy quasi-metric is a gv-fuzzy quasi-metric, and every gv-fuzzy quasi-metric is a gv-fuzzy quasi-pseudo-metric. definition 2.13. a gv-fuzzy quasi-(pseudo-)metric space is a triple (x, m, ∗) such that x is a (nonempty) set and (m, ∗) is a gv-fuzzy quasi-(pseudo-)metric on x. the notions of a t 1 gv-fuzzy quasi-metric space and of a gv-fuzzy metric space are defined in the obvious manner. note that the gv-fuzzy metric spaces are exactly the fuzzy metric spaces in the sense of george and veeramani. remark 2.14. note that if (m, ∗) is a gv-fuzzy quasi-(pseudo-)metric on x, then the fuzzy sets in x × x × (0, +∞), m−1 and mi given by m−1(x, y, t) = m(y, x, t) and mi(x, y, t) = min{m(x, y, t), m−1(x, y, t)}, are, as in the kmcase, a gv-fuzzy quasi-(pseudo-)metric and a gv-fuzzy (pseudo-)metric on x, respectively. thus, condition (gv2’) above is equivalent to the following: m(x, x, t) = 1 for all x ∈ x and t > 0, and mi(x, y, t) < 1 for all x 6= y and t > 0. remark 2.15. obviously, each gv-fuzzy quasi-(pseudo-)metric (m, ∗) can be considered as a km-fuzzy quasi-(pseudo-)metric by defining m(x, y, 0) = 0 for fuzzy quasi-metric spaces 133 all x, y ∈ x. therefore, each gv-fuzzy quasi-pseudo-metric space generates a topology τm defined as in the km-case, and propositions 2.6, 2.7 and 2.8 above remain valid for gv-fuzzy quasi-pseudo-metric spaces. example 2.16 (compare example 2.9 of [3]). álet (x, d) be a quasi-metric space. denote by a · b the usual multiplication for every a, b ∈ [0, 1], and let md be the function defined on x × x × (0, +∞) by md(x, y, t) = t t + d(x, y) . then (x, md, ·) is a gv-fuzzy quasi-metric space called standard fuzzy quasimetric space and (md, ·) is the fuzzy quasi-metric induced by d. furthermore, it is easy to check that (md) −1 = md−1 and (md) i = mds. finally, from proposition 2.8 and remark 2.15, it follows that the topology τd, generated by d, coincides with the topology τmd generated by the induced fuzzy quasi-metric (md, ·). definition 2.17. we say that a topological space (x, τ) admits a compatible km (resp. gv)-fuzzy quasi-metric if there is a km (resp. gv)-fuzzy quasimetric (m, ∗) on x such that τ = τm . it follows from example 2.16 that every quasi-metrizable topological space admits a compatible gv-fuzzy quasi-metric. in section 3 we shall establish that, conversely, the topology generated by a km-fuzzy quasi-metric space is quasi-metrizable. 3. quasi-metrizability of the topology of a fuzzy quasi-metric space a slight modification of the proof of theorem 1 of [6], permits us to show the following result. lemma 3.1. let (x, m, ∗) be a km-fuzzy quasi-metric space. then {un : n=2, 3, ...} is a base for a quasi-uniformity um on x compatible with τm , where un = {(x, y) ∈ x × x : m(x, y, 1/n) > 1 − 1/n}, for n = 2, 3, ... moreover the conjugate quasi-uniformity (um ) −1 coincides with um−1 and it is compatible with τm−1. from example 2.16, lemma 3.1 and the well-known result that the topology generated by a quasi-uniformity with a countable base is quasi-pseudometrizable ([2]), we immediately deduce the following. theorem 3.2. for a topological space (x, τ) the following are equivalent. (1) (x, τ) is quasi-metrizable. (2) (x, τ) admits a compatible gv-fuzzy quasi-metric. (3) (x, τ) admits a compatible km-fuzzy quasi-metric. remark 3.3. it is almost obvious that the uniformity umi coincides with the uniformity (um ) s := um ∨ (um ) −1 134 v. gregori and s. romaguera 4. bicomplete fuzzy quasi-metric spaces there exist many different notions of quasi-uniform and quasi-metric completeness in the literature (see [10]). then, by lemma 3.1 and remark 3.3, one can define in a natural way the corresponding notions of completeness in a fuzzy setting and easily deduce several properties taking into account the well-known completeness properties of quasi-uniform and quasi-metric spaces (compare with [6], where these ideas are used to study completeness in the fuzzy metric case). in this section we only consider the notion of bicompleteness because it provides a satisfactory theory of quasi-uniform and quasi-metric completeness. let us recall that a quasi-metric space (x, d) is bicomplete provided that (x, ds) is a complete metric space. in this case we say that d is a bicomplete quasi-metric on x. a metrizable topological space (x, τ) is said to be completely metrizable if it admits a compatible complete metric. on the other hand, a fuzzy metric space (x, m, ∗) is called complete ([5]) if every cauchy sequence is convergent, where a sequence (xn)n is cauchy provided that for each r ∈ (0, 1) and each t > 0, there exists an n0 such that m(xn, xm, t) > 1 − r for every n, m ≥ n0. if (x, m, ∗) is a complete fuzzy metric space, we say that (m, ∗) is a complete fuzzy metric on x. it was proved in [6] that a topological space is completely metrizable if and only if it admits a compatible complete fuzzy metric. definition 4.1. a km (resp. gv)-fuzzy quasi-metric space (x, m, ∗) is called bicomplete if (x, mi, ∗) is a complete fuzzy metric space. in this case, we say that (m, ∗) is a bicomplete km (resp. gv)-fuzzy quasi-metric on x. proposition 4.2. (a) let (x, m, ∗) be a bicomplete km-fuzzy quasi-metric space. then (x, τm ) admits a compatible bicomplete quasi-metric. (b) let (x, d) be a bicomplete quasi-metric space. then (x, md, ·) is a bicomplete gv-fuzzy quasi-metric space. proof. (a) let d be a quasi-metric on x inducing the quasi-uniformity um. then d is compatible with τm . now let (xn)n be a cauchy sequence in (x, d s). clearly (xn)n is a cauchy sequence in the fuzzy metric space (x, m i, ∗). so it converges to a point y ∈ x with respect to τmi . hence (xn)n converges to y with respect to τds. consequently d is bicomplete. (b) this part is almost obvious because (md) i = mds (see example 2.16), and thus each cauchy sequence in (x, (md) i, ·) is clearly a cauchy sequence in (x, ds). � extending the classical metric theorem, it was independently proved in [1] and [12], that every quasi-metric space admits a (quasi-metric) bicompletion which is unique up to isometry. although the problem of completion of fuzzy metric spaces in the sense of kramosil and michalek has a satisfactory solution fuzzy quasi-metric spaces 135 ([14]), the corresponding situation for fuzzy metric spaces in the sense of george and veeramani is quite different. in fact, it was obtained in [7] an example of a fuzzy metric space (x, m, ∗) that does not admit completion, i.e. there no exist any complete fuzzy metric space having a dense subspace isometric to (x, m, ∗). a characterization of those fuzzy metric spaces (in the sense of george and veeramani) that admit a fuzzy metric completion has recently been obtained in [8]. although the problem of bicompletion for gv-fuzzy quasi-metric spaces will be discussed elsewhere, we next present some concepts and facts that are basic in solving this problem. definition 4.3. let (x, m, ∗) and (y, n, ⋆) be two km (resp. gv)-fuzzy quasimetric spaces. then (a) a mapping f from x to y is called an isometry if for each x, y ∈ x and each t > 0, m(x, y, t) = n(f(x), f(y), t). (b) (x, m, ∗) and (y, n, ⋆) are called isometric if there is an isometry from x onto y. definition 4.4. let (x, m, ∗) be a km (resp. gv)-fuzzy quasi-metric space. a km (resp. gv)-fuzzy quasi-metric bicompletion of (x, m, ∗) is a bicomplete km (resp. gv)-fuzzy quasi-metric space (y, n, ⋆) such that (x, m, ∗) is isometric to a τni-dense subspace of y . proposition 4.5. álet (x, m, ∗) be a km-fuzzy quasi-metric space and (y, n, ⋆) a bicomplete km-fuzzy quasi-metric space. if there is a τmi-dense subset a of x and an isometry f : (a, m, ∗) → (y, n, ⋆), then there exists a unique isometry f : (x, m, ∗) → (y, n, ⋆) such that f |a= f. proof. áit is clear that f is a quasi-uniformly continuous mapping from the quasi-uniform space (a, um |a×a) to the quasi-uniform space (y, un). by theorem 3.29 of [2], f has a unique quasi-uniformly continuous extension f : (x, um) → (y, un). we shall show that actually f is an isometry from (x, m, ∗) to (y, n, ⋆). indeed, let x, y ∈ x and t > 0. then, there exist two sequences (xn)n and (yn)n in a such that xn → x and yn → y with respect to τmi. thus f(xn) → f(x) and f(yn) → f(y) with respect to τni. choose ε ∈ (0, 1) with ε < t. therefore, there is nε such that for n ≥ nε, m(x, xn, ε/2) > 1 − ε, m(yn, y, ε/2) > 1 − ε, n(f(xn), f(x), ε/2) > 1 − ε, n(f(y), f(yn), ε/2) > 1 − ε. thus m(x, y, t) ≥ m(x, xn, ε/2) ∗ m(xn, yn, t − ε) ∗ m(yn, y, ε/2) ≥ (1 − ε) ∗ n(f(xn), f(yn), t − ε) ∗ (1 − ε) ≥ (1 − ε) ∗ [(1 − ε) ⋆ n(f(x), f(y), t − 2ε) ⋆ (1 − ε)] ∗ (1 − ε). by continuity of ∗ and ⋆ and by left continuity of n(f(x), f(y), ) it follows that m(x, y, t) ≥ n(f(x), f(y), t). similarly we show that n(f(x), f(y), t) ≥ m(x, y, t). consequently f is an isometry from (x, m, ∗) to (y, n, ⋆). � 136 v. gregori and s. romaguera corollary 4.6. let (x, m, ∗) be a gv-fuzzy quasi-metric space and (y, n, ⋆) a bicomplete gv-fuzzy quasi-metric space. if there is a τmi-dense subset a of x and an isometry f : (a, m, ∗) → (y, n, ⋆), then there exists a unique isometry f : (x, m, ∗) → (y, n, ⋆) such that f |a= f. references [1] a. di concilio, spazi quasimetrici e topologie ad essi associate, rend. accad. sci. fis. mat. napoli 38 (1971), 113-130. 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[14] h. sherwood, on the completion of probabilistic metric spaces, z. wahrsch. verw. geb. 6 (1966), 62-64. received february 2003 accepted june 2003 v. gregori (vgregori@mat.upv.es) escuela politécnica superior de gandia, universidad politécnica de valencia, 46730 grau de gandia, valencia, spain. s. romaguera (sromague@mat.upv.es) escuela de caminos, departamento de matemática aplicada, universidad politécnica de valencia, 46071 valencia, spain. tomitagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 97101 a countably compact free abelian group whose size has countable cofinality i. castro pereira and a. h. tomita ∗ abstract. based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. compactness here is sometimes viewed splitting into ”κ-lindelöfness” and ”κcompactness” for cardinals κ. to focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result of som naimpally, to who this article is hearty dedicated. 2000 ams classification: primary 54h15: secondary 22b99, 54d30. keywords: forcing, countably compact group, convergence, continuum hypothesis, countable cofinality, size 1. introduction in 1990, tkachenko showed that there exists a countably compact group topology on the free abelian group of size c, assuming the continuum hypothesis (ch). dikranjan and shakmatov (see [1]) proved that there is no countably compact group topology on any free group and asked the following question: for which cardinals κ can the free abelian group of size κ be endowed with a countably compact group topology?. in [3], it was shown that it is consistent that 2c is such a cardinal, and as a consequence, any infinite cardinal κ ≤ 2c with κ = κω. for cardinals that do not satisfy κ = κω, a natural question due to van douwen [2] needs to be addressed: 1.5 question. if x is an infinite group (or homogeneous space) which is countably compact, is |x|ω = |x|? is at least cf(|x|) 6= ω? it was shown in [2] that the answer to van douwen’s question is yes under gch; recently in [6], question 1.5 above was answered in the negative. however, the example contains convergent sequences and all its elements have order 2. in this note, we obtain the following: example 1.1. it is consistent that 2c is ‘arbitrarily large’ and that there exists a countably compact group topology on the free abelian group of size λ for any λ ∈ [c,2c]. in particular, it is consistent that there are countably compact group topologies on a free abelian group whose size has countable cofinality. ∗the research in this paper was partially conducted while the second author was visiting professor t. nogura at ehime university with the financial support of the ministry of education of japan. 98 i. castro pereira and a. h. tomita 2. preliminaries we will construct via forcing a free set x = {xβ : β < κ} such that for every γ ∈ [c,κ), the group generated by {xβ : β < γ} is countably compact. an element of the group generated is given by ∑ ξ∈dom f f(ξ)xξ, where f is a function whose domain is a finite subset of κ and the range is z. a sequence {gn : n ∈ ω} in the group g generated by x can be coded as {f(n) : n ∈ ω}, where dom f(n) ∈ [κ]<ω and rng f(n) ⊆ z. so gn = ∑ ξ∈domf(n) f(n)(ξ)xξ = xf(n). for the properties of denseness that will be required later on, it will be useful to work with particular sequences (see [5]): definition 2.1. let f : ω −→ ⋃ e∈[κ]<ω\{∅}(z \ {0}) e, where f is 1 − 1. we say that f is a sequence of type i if {dom f(n) : n ∈ ω} is faithfully indexed. we say that f is a sequence of type ii if there exists e such that dom f(n) = e for every n ∈ ω and there exists µ ∈ e such that {f(n)(µ) : n ∈ ω} is faithfully indexed. lemma 2.2. let {an : n ∈ ω} be a sequence in g = 〈{xβ : β < κ}〉 that does not contain a constant subsequence. then there exists a sequence f of type i or type ii such that {xf(n) : n ∈ ω} ⊆ {an : n ∈ ω} proof. easy exercise � thus, from the lemma above, it suffices to show that every sequence in g coded by a sequence of type i or type ii has an accumulation point. let t be the set of opens subarcs in t. we say that φ : dp −→ t has finite support if {ξ ∈ dp : φ(ξ) 6= t} is finite. given two such functions φ and φ ∗, we say that φ ≤ φ∗ if φ(ξ) ⊆ φ∗(ξ) for each ξ ∈ dp. lemma 2.3. if f is of type i or type ii and φ∗ is a function of finite support as above then given an open set w of t there exists m ∈ dom f and φ ≤ φ∗ of finite support such that ∑ ξ∈dom f(m) f(m)(ξ)φ(ξ) ⊆ w. proof. see [5]. � definition 2.4. let {fβ : β < κ} be an enumeration of all sequences of type i or type ii in definition 2.1. 3. the partial order all the background information on forcing required in this paper can be found in [4]. definition 3.1. let p be the family of all element p = (αp,{xp,ξ : ξ ∈ dp},{ap,ζ : ζ ∈ ep) satisfying the following conditions: i) αp ∈ ω1; ii) dp ∈ [κ] ω; iii) xp,ξ ∈ t αp; iv) ep ∈ [κ] ω; v) ap,ζ ⊆ dp ∩ c; vi) dom fζ(n) ⊆ dp for each n ∈ ω, ζ ∈ ep; vii) xp,ξ is an accumulation point of {xp,fζ(n) : n ∈ ω} for each ξ ∈ ap,ζ and each ζ ∈ ep; viii) {xp,ξ : ξ ∈ dp} is a faithfully indexed free set. given p,q ∈ p, p extends q if a) αp ≥ αq; free abelian groups whose size has countable cofinality 99 b) dp ⊇ dq; c) xp,ξ|αq = xq,ξ for all ξ ∈ dq; d) ep ⊇ eq; e) ap,ζ ⊇ aq,ζ for all ζ ∈ eq. the following results will be proven in the next section: lemma 3.2. (ch) the partial order p is countably closed and ω2−cc. lemma 3.3. the set dα,ξ,ζ,µ = {p ∈ p : αp ≥ α,ξ ∈ dp,ζ ∈ ep ∧ ap,ζ \ µ 6= ∅} is dense in p, for each α,µ < ω1 and ξ,ζ < κ. we are ready to prove the main result: proof. (of example 1.1) start with a model of gch and let g be a generic set for the partial order p. the forcing is cardinal preserving by lemma 3.2, no new countable subsets of the ground model are added, ch holds and κ = 2c. for each ξ,ζ ∈ κ, let xξ = ⋃ p∈g∧ξ∈dp xp,ξ and aζ = ⋃ p∈g∧ζ∈ep ap,ζ. by denseness of the sets in lemma 3.3, each xξ is defined and is a function in 2 c and the set {xβ : β < κ} is free. also, each aζ is defined and has size c. clearly, xµ is an accumulation point of {xfζ(n) : n ∈ ω} for each µ ∈ aζ. fix λ ∈ [c,κ]. clearly the group generated by {xβ : β < λ} is free abelian and from lemma 2.2 it will be countably compact as well. � 4. some proofs we start by proving an auxiliary lemma which will be used in the proofs of lemmas 3.2 and 3.3. lemma 4.1. let r = (αr,{xr,ξ : ξ ∈ dr},{ar,ζ : ζ ∈ er}) satisfying all conditions in definition 3.1 with the exception of condition viii). then there exists a condition p ∈ p such that p ‘extends’ r, that is, conditions a) − e) are satisfied. proof. we shall define p such that αp = αr + ω, dp = dr, ep = er, ap,ζ = ar,ζ for all ζ ∈ ep and xp,ξ|αr = xr,ξ for all ξ ∈ dp. list ⋃ e∈[dp]<ω (z \ {0})e in length ω as {fn : n ∈ ω}. we shall define by induction xp,ξ(α + n) for each ξ ∈ dp and each n ∈ ω so that conditions vii) is satisfied (all other conditions are trivially satisfied) and xp,fn (α + n) 6= 0 ∈ t. suppose that condition vii) is satisfied for {xp,ξ|αr+n : ξ ∈ dp}. for each ζ ∈ ep and µ ∈ ap,ζ let bζ,µ be an infinite subset of ω such that {xp,fζ(m)|αr+n : m ∈ bζ,µ} converges to xp,µ|αr+n. partition each bζ,µ into {bζ,µ,k : k ∈ ω} each of infinite size and let {wl : l ∈ ω} be a basis for t. enumerate all possible pairs (bζ,µ,k,wl) as {(bζt,µt,kt,wlt) : k ∈ ω}. we will define by induction a decreasing family φt : ω −→ t for t ∈ ω of finite support as defined prior to lemma 2.3. start with φ0 such that the support of φ0 contains domfn and 0 /∈ ∑ ξ∈domfn fn(ξ)φ0(ξ). since fζ|bζ0,µ0,k0 is of type i or ii, from lemma 2.3, there exists m0 ∈ bζ0,µ0,k0 such that φ1 ≤ φ0 of finite support such that ∑ ξ∈dom fζ0 (m0) fζ(m0)(ξ)φ1(ξ) ⊆ w0. by induction, using lemma 2.3, define {φt : t ∈ ω} and {mt : t ∈ ω} such that φt+1 ≤ φt and ∑ ξ∈dom fζt (mt) fζt(mt)(ξ)φt+1(ξ) ⊆ wt. for each ξ ∈ dp define xp,ξ(α + n) ∈ ⋂ t∈ω φt(ξ). then condition vii) is satisfied by {xp,ξ|α+n+1 : ξ ∈ dp} and ∑ ξ∈dom fn fn(ξ)xp,ξ(α + n) 6= 0. � 100 i. castro pereira and a. h. tomita proof. (of lemma 3.2) it is straightforward to see that the partial order is countably closed. indeed, if pn+1 ≤ pn for each n ∈ ω, define pω = (αpω,{xpω,ξ : ξ ∈ dpω },{apω,ζ : ζ ∈ epω }), where αω = sup{αpn : n ∈ ω}, dpω = ⋃ n∈ω dpn, xpω,ξ = ⋃ ξ∈dpn xpn,ξ, epω = ⋃ n∈ω epn, apω,ζ = ⋃ apn,ζ. then pω ≤ pn for each n ∈ ω. we will now check that p is ω2-cc. let {pβ : β < ω2} be a subset of p. from ch and the ∆-system lemma, we conclude that there exists i ∈ [ω2] ω2 and d ∈ [κ]≤ω such that dpβ ∩ dpγ = d for any pair {β,γ} ∈ [i] 2. without loss of generality, we can assume that there exists α < ω1 such that αpβ = α for every β ∈ i. also we can assume that |{xpβ,ξ : β ∈ i}| = 1 for each ξ ∈ d. fix β,γ ∈ i. let dr = dpβ ∪ dpγ . define xr,ξ as xpβ,ξ if ξ ∈ dpβ or xpγ,ξ if ξ ∈ dpγ \ d. set er = epβ ∪ epγ and define ar,ζ as apβ,ζ if ζ ∈ epβ \ epγ ; apγ,ζ if ζ ∈ epγ \ epβ or apβ,ζ ∪ apγ,ζ if ζ ∈ epβ ∩ epγ . note that r = (α,{xr,ξ : ξ ∈ dr},{ar,ζ : ζ ∈ er}) may not be a condition in p but it can be extended to a condition p by applying lemma 4.1. the condition p extends pγ and pβ. therefore, there are no antichains of size ω2. � proof. (of lemma 3.3) let q be an arbitrary element of p. if ζ ∈ eq, define er = eq. let θ ∈ (µ,c) such that θ /∈ dq ∪ {ξ}, set dr = dq ∪ {θ}∪{ξ} and define xq,θ as an accumulation point of the sequence {xq,fζ(n) : n ∈ ω}. if ξ ∈ dr \ dq define xq,ξ = 0 ∈ t αq . if ζ /∈ eq, choose θ ∈ (µ,c) \ (dq ∪ ⋃ n∈ω dom fζ(n) ∪ {ξ}) and set dr = dq ∪ ( ⋃ n∈ω dom fζ(n)) ∪ {θ,ξ}. if ψ ∈ dr \ (dq ∪ {θ}), define xq,ψ = 0 ∈ t αq . define xq,θ as an accumulation point of {xq,fζ(n) : n ∈ ω}. in either case, define ar,ρ = aq,ρ for each ρ ∈ er \ {ζ} and ar,ζ = aq,ζ ∪ {θ}. if αq ≥ α, let xr,η = xq,η for each η ∈ dr; otherwise, let xr,η = xq,η ∪ {(β,0) : αq ≤ β < α} for each η ∈ dr. set αr = max{α,αq}. the set r = (αr,{xr,η : η ∈ dr},{ar,ρ : ρ ∈ er}) satisfies all the conditions to be an element of p with the possible exception of condition viii). applying the lemma 4.1, there exists a condition p ‘below’ r. such condition p will be an element of dα,ξ,ζ,µ and below q. � note: independently from this note, d. dikranjan and d. shakmatov produced a model of zfc + ch with 2c ”arbitrarily large” and, in this model they obtained a characterization of abelian groups of size κ with κ ≤ 2c wich admit a countably compact group topology without non–trivial convergent sequences. using forcing they constructed a group monomorphism π : q(κ) ⊕(q/z)(κ) −→ tω1 such that for every almost n– torsion subset e from q(κ) ⊕ (q/z)(κ), π(e) is hfd (hereditarily finally dense) in t[n]ω1 and has a cluster point in π(q(κ) ⊕ (q/z)(κ)). references [1] w. w. comfort, k. h. hofmann and d. remus, topological groups and semigroups, recent progress in general topology (prague, 1991), 57–144, north-holland, amsterdam, 1992. [2] e. k. van douwen, the weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, proc. amer. math. soc. 80 (1980), 678–682. [3] p. b. koszmider, a. h. tomita and s. watson, forcing countably compact group topologies on a larger free abelian group, topology proc. 25 (summer 2000), 563–574. [4] k. kunen, set theory. an introduction to independence proofs, studies in logic and the foundations of mathematics, 102. north-holland publishing co., amsterdam, 1980. xvi+313. [5] m. g. tkachenko, countably compact and pseudocompact topologies on free abelian groups, izvestia vuz. matematika 34 (1990), 68–75. [6] a. h. tomita, two countably compact groups: one of size ℵω and the other of weight ℵω without non-trivial convergent sequences, to appear in proc. amer. math. soc. free abelian groups whose size has countable cofinality 101 received december 2002 accepted february 2003 i. castro pereira, a. h. tomita (castro@ime.usp.br, tomita@ime.usp.br) departamento de matemática, instituto de matemática e estat́ıstica, universidade de são paulo, caixa postal 66281, cep 05315-970, são paulo, brasil applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 91–112 duality and quasi-normability for complexity spaces salvador romaguera ∗ and michel schellekens abstract. the complexity (quasi-metric) space was introduced in [23] to study complexity analysis of programs. recently, it was introduced in [22] the dual complexity (quasi-metric) space, as a subspace of the function space [0, +∞)ω. several quasi-metric properties of the complexity space were obtained via the analysis of its dual. we here show that the structure of a quasi-normed semilinear space provides a suitable setting to carry out an analysis of the dual complexity space. we show that if (e,‖.‖) is a bibanach space (i.e., a quasi-normed space whose induced quasi-metric is bicomplete), then the function space (b∗e,‖.‖b∗) is bibanach, where b ∗ e = {f : ω → e |∑∞ n=0 2−n(‖f(n)‖∨‖−f(n)‖) < +∞}, and ‖f‖b∗ = ∑∞ n=0 2−n‖f(n)‖. we deduce that the dual complexity space admits a structure of quasinormed semlinear space such that the induced quasi-metric space is order-convex, upper weightable and smyth complete, not only in the case that this dual is a subspace of [0,+∞)ω but also in the general case that it is a subspace of fω where f is any bibanach normweightable space. we also prove that for a large class of dual complexity (sub)spaces, lower boundedness implies total boundedness. finally, we investigate completeness of the quasi-metric of uniform convergence and of the hausdorff quasi-pseudo-metric for the dual complexity space, in the context of function spaces and hyperspaces, respectively. 2000 ams classification: 54e50, 54e15, 54c35, 46e15. keywords: complexity space, quasi-norm, quasi-metric, bibanach space, smyth complete. 1. introduction and preliminaries throughout this paper the letters r, r+, ω and n will denote the set of all real numbers, of all nonnegative real numbers, of all nonnegative integer ∗the first-listed author acknowledges the support of the spanish ministry of science and technology, grant bfm2000-1111 92 s. romaguera and m. schellekens numbers and of all positive integer numbers, respectively. our basic references for quasi-metric spaces are [7] and [12]. let us recall that a quasi-pseudometric on a (nonempty) set x is a nonnegative real-valued function d on x × x such that for all x,y,z ∈ x: (i) d(x,x) = 0 and (ii) d(x,y) ≤ d(x,z) + d(z,y). in our context a quasi-metric on x is a quasi-pseudometric d on x which satisfies: (iii) d(x,y) = d(y,x) = 0 ⇔ x = y. if d is a quasi-(pseudo)metric on x, then the function ds defined on x ×x by ds(x,y) = max{d(x,y),d(y,x)} is a (pseudo)metric on x. a quasi-(pseudo)metric space is a pair (x,d) such that x is a (nonempty) set and d is a quasi-(pseudo)metric on x. the function u defined on r×r by u(x,y) = (y−x)∨0 for all x,y ∈ r, is an interesting example of a quasi-metric, where, as usual, ∨ denotes the maximum of y −x and 0. note that us is exactly the euclidean metric on r. the function u−1 defined on (0, +∞] × (0, +∞] by u−1(x,y) = ( 1y − 1 x ) ∨ 0 for all x,y ∈ (0, +∞] also provides an interesting example of a quasi-metric, in our context, where we adopt the convention that 1∞ = 0. each quasi-pseudometric d on x generates a topology t(d) on x which has as a base the family of open balls {sd(x,r) | x ∈ x, r > 0}, where sd(x,r) = {y ∈ x | d(x,y) < r} for all x ∈ x and r > 0. it is clear, that d is a quasi-metric if and only if t(d) is a t0 topology. a quasi-(pseudo)metric d on x is called bicomplete [7] if ds is a complete (pseudo)metric on x. in this case, (x,d) is said to be a bicomplete quasi(pseudo)metric space. in [27] and [28], smyth presented a topological framework for denotational semantics based on the theory of complete (and totally bounded) quasi-uniform and quasi-metric spaces. sünderhauf continued this work in the setting of topological quasi-uniform spaces [29]. künzi characterized in [12] both smyth completable and smyth complete quasi-uniform spaces in terms of left k-cauchy filters as discussed in [19]. we shall formulate these characterizations in the special case of quasi-(pseudo)metric spaces: a quasi-(pseudo)metric space (x,d) is smyth completable if and only if every left k-cauchy filter on (x,d) is a cauchy filter on the (pseudo)metric space (x,ds) [12], where a filter f on (x,d) is left k-cauchy provided that for each ε > 0 there is an fε ∈ f such that sd(x,ε) ∈ f for all x ∈ fε [18]. a quasi-metric space (x,d) is smyth complete if and only if every left k-cauchy filter on (x,d) is convergent with respect to the metric topology t(ds) [12]. therefore, every bicomplete smyth completable quasi-metric space is smyth complete. smyth completable quasi-pseudometric spaces also can be studied in terms of left k-cauchy sequences. in fact, it is proved in [25] that a quasi-pseudometric space (x,d) is smyth completable if and only if every left k-cauchy sequence in (x,d) is a cauchy sequence in the pseudometric space (x,ds), where a sequence (xn)n∈n in (x,d) is left k-cauchy if for each ε > 0 there is an nε ∈ n such that d(xn,xm) < ε whenever m ≥ n ≥ nε [17] (equivalently, (xn)n∈n is duality and quasi-normability for complexity spaces 93 left k-cauchy if and only the filter that generates is left k-cauchy [18, lemma 2]). the weightable quasi-metric spaces, or the equivalent partial metric spaces, were introduced by matthews [16], as a part of the study of denotational semantics of dataflow networks. excellent topological results on this class of spaces may be found in [12] and in [15]. let us recall that a quasi-metric space (x,d) is called weightable if there is a function w: x → r+, such that w(x) + d(x,y) = w(y) + d(y,x) for all x,y ∈ x. the function w is said to be a weighting function for (x,d). it was proved in [12] that every weightable quasi-metric space is smyth completable. hence, every weightable bicomplete quasi-metric space is smyth complete. the upper weightable quasi-metric spaces were introduced in [24], in the context of the development of a topological foundation for complexity analysis. a quasi-metric space (x,d) is called upper weightable if it is functionally bounded by a weighting function w, where (x,d) is functionally bounded provided that there is a function f : x → r+, such that d(x,y) ≤ f(y) for all x,y ∈ x. as usual, the associated preorder ≤d of a quasi-pseudometric space (x,d) is defined by x ≤d y ⇔ d(x,y) = 0. a quasi-pseudometric space has a maximum if the associated preorder has a maximum. a join semilattice is a partially ordered set (x,≤) such that every two elements x,y ∈ x have a supremum xty. according to [24] a quasi-pseudometric join semilattice is a quasi-pseudometric space which is a join semilattice for its associated preorder. an optimal quasipseudometric join semilattice is a quasi-pseudometric join semilattice (x,d) such that d(x t y,y) = d(x,y) for all x,y ∈ x. a quasi-pseudometric space (x,d) is called order-convex if d(x,z) = d(x,y)+d(y,z) whenever z ≤d≤ y ≤d x (see [24]). the theory of complexity (quasi-metric) spaces, introduced in [23], provides a topological foundation for the complexity analysis of algorithms. this theory constitutes a part of the research in theoretical computer science and topology and is developed in the setting of the smyth completion of quasi-metric spaces. applications of this theory to the complexity analysis of divide & conquer algorithms have been discussed in [23]. let us recall that the complexity space (with values in (0, +∞]) is the pair (c,dc), where c = {f : ω → (0, +∞] | ∑∞ n=0 2 −n 1 f(n) < +∞}, and dc is the quasi-metric defined on c by dc(f,g) = ∑∞ n=0 2 −n[( 1 g(n) − 1 f(n) )∨0], whenever f,g ∈ c. dc is called in [23] “the complexity distance”, and intuitively it measures relative improvements in the complexity of programs. the dual complexity space (with values in (r+)ω) is introduced in [22] as a pair (c∗,dc∗), where c∗ = {f : ω → r+ | ∑∞ n=0 2 −nf(n) < +∞}, and dc∗ is the quasi-metric defined on c∗ by dc∗(f,g) = ∑∞ n=0 2 −n[(g(n) −f(n)) ∨ 0], whenever f,g ∈ c∗. (c,dc) is isometric to (c∗,dc∗) by the isometry ψ : c∗ → c, defined by ψ(f) = 1/f (see [22]). thus, via the analysis of its dual, several quasi-metric 94 s. romaguera and m. schellekens properties of (c,dc), in particular smtyh completeness and total boundedness, are studied in [22]. a motivation for the use of the dual instead of the original complexity space is the fact that the dual is mathematically somewhat more appealing, since dc∗ is ”derived” from the restriction to r+ of the quasi-metric u defined above. consequently, the presentation of the proofs becomes somewhat more elegant. furthermore, it is possible to carry out the complexity analysis of algorithms based on the dual complexity space. in fact, the dual complexity space has the advantage that it respects the interpretation usually given to the minimum ⊥ in semantic domains (see [22, section 4]). the complexity of a given program is frequently obtained by a summation of complexity functions or by a product of a complexity function by a constant, where these operations intuitively correspond to operations carried out by the program on data structures. in order to obtain an appropriate structure both for realizing these operations and for developing a consistent theory for the analysis of the dual complexity space we introduce, in section 2, the notion of a bibanach space and study a kind of bibanach function space for which the dual complexity space is a quasi-normed semilinear subspace whose induced quasi-metric is upper weightable, order-convex and smyth complete, even in the general case that the dual is a subspace of fω, where f is any bibanach norm-weightable space (see section 3). we also show, among other things, that a dual complexity space having a lower bound is totally bounded whenever the induced quasi-metric on the range space is linear. finally, in section 4, we study completeness of the quasi-metric of uniform convergence and of the hausdorff quasi-pseudometric for the dual complexity space, in the context of function spaces and hyperspaces, respectively. 2. bibanach function spaces we start this section giving the definitions of a quasi-norm and of a quasinormed space in the sense of [5], [6] and [21] (see [4] for the related notion of a nonsymmetric norm). let (e, +, ·) be a linear space on r. a quasi-norm on e is a nonnegative real-valued function ‖.‖ on e such that for all x,y ∈ e and a ∈ r+: (i ) ‖x‖ = ‖−x‖ = 0 ⇔ x = e (where e denotes the neutral element of (e, +)); (ii ) ‖ax‖ = a‖x‖; (iii ) ‖x + y‖≤‖x‖ + ‖y‖. the pair (e,‖.‖) is then called a quasi-normed space. (note that the function ‖.‖s defined on e by ‖x‖s = max{‖x‖ ,‖−x‖}, for all x ∈ e, is a norm on e.) the quasi-norm ‖.‖ induces, in a natural way, a quasi-metric d‖.‖ on e, defined by d‖.‖(x,y) = ‖y −x‖ for all x,y ∈ e. if the quasi-metric d‖.‖ is bicomplete, we say that (e,‖.‖) is a bibanach space. duality and quasi-normability for complexity spaces 95 example 2.1. let (r, +, ·) be the (usual) euclidean linear space. for each x ∈ r define ‖x‖ = max{x, 0}. then ‖.‖ is a quasi-norm on r such that ‖.‖s is the euclidean norm. therefore, (r,‖.‖) is a bibanach space. example 2.2. let (e,‖.‖) be a quasi-normed space. define b∗e = {f : ω → e | ∞∑ n=0 2−n‖f(n)‖s < +∞}. if for each f,g ∈b∗e and each a ∈ r we define f + g and a ·f in the natural way, then it easily follows that (b∗e, +, ·) is a linear space (on r) because, clearly, − f ∈ b∗e whenever f ∈ b ∗ e. we then deduce that (b ∗ e,‖.‖b∗) is a quasi-normed space, where ‖f‖b∗ = ∞∑ n=0 2−n‖f(n)‖ for all f ∈b∗e. remark 2.3. the definition of the space b∗e may seem somewhat surprising at first because it could be considered more natural to define this space as {f : ω → e | ∑∞ n=0 2 −n‖f(n)‖ < +∞}. however, the following simple example justifies our selection: consider the bibanach space (r,‖.‖) of example 2.1. define f : ω → r by f(n) = −2n for all n ∈ ω. then, ∑∞ n=0 2 −n‖f(n)‖ = 0, but ∑∞ n=0 2 −n‖−f(n)‖ = +∞, so for the possible alternative definition mentioned above, b∗e would not be a group. in the rest of this section we focus our attention on the quasi-normed space (b∗e,‖.‖b∗), because the dual complexity space will be a closed semilinear subspace of it (see section 3 for the definition of a semilinear space). theorem 2.4. let (e,‖.‖) be a bibanach space. then (b∗e,‖.‖b∗) is a bibanach space. proof. let (fk)k∈ω be a cauchy sequence in the normed space (b∗e, (‖.‖b∗) s). define a quasi-metric p on b∗e by p(f,g) = ∞∑ n=0 2−n min{‖g(n) −f(n)‖ , 1} for all f,g ∈b∗e. then p induces the topology of pointwise convergence on b ∗ e. since p ≤ d‖.‖b∗ , (fk)k∈ω is a cauchy sequence in the metric space (b ∗ e,p s). then, for each n ∈ ω, the sequence (fk(n))k∈ω is a cauchy sequence in the banach space (e,‖.‖s), so it is convergent to a point xn ∈ e with respect to the topology induced by the norm ‖.‖s on e. define a function g : ω → e, by g(n) = xn for all n ∈ ω. we first prove that g ∈b∗e : 96 s. romaguera and m. schellekens indeed, assume the contrary. then, for each j ∈ ω there is an mj ∈ ω such that (2.1) j < mj∑ n=0 2−n‖g(n)‖s . on the other hand, since (fk)k∈ω is a cauchy sequence in (b∗e, (‖.‖b∗) s), there exists a k1 ∈ ω such that for each k ≥ k1, (2.2) ∞∑ n=0 2−n‖fk(n) −fk1 (n)‖ s < 1. thus, (2.2′) mj∑ n=0 2−n‖fk(n) −fk1 (n)‖ s < 1 whenever k ≥ k1. let j ∈ ω. then there exists a k0 ≥ k1 such that ‖g(n) − fk0 (n)‖s < 2−j, for n = 0, 1, . . . ,mj. hence, (2.3) mj∑ n=0 2−n‖g(n) −fk0 (n)‖ s < 2−j mj∑ n=0 2−n < 2−(j−1). by (2.1), (2.3) and (2.2′), we obtain, (2.4) j < 2−(j−1) + mj∑ n=0 2−n‖fk0 (n)‖ s < 2 + mj∑ n=0 2−n‖fk1 (n)‖ s , which implies that ∞∑ n=0 2−n‖fk1 (n)‖ s = +∞, a contradiction. we conclude that g ∈b∗e. finally, we prove that (‖g −fk‖b∗)s → 0 as k → +∞. (compare [22, proof of theorem 3]): let j ∈ ω. then there exists a k(j) ∈ ω such that for every k,m ≥ k(j), (2.5) ∞∑ n=0 2−n‖fk(n) −fm(n)‖ s < 2−3j. since both fk(j) and g are in b∗e, there exists n0 ∈ ω (depending on j) such that n0 > 1 and 2−(n0−1)n0 < 2 −3j,(2.6) ∞∑ n=n0 2−n ∥∥fk(j)(n)∥∥s < 2−3j, and ∞∑ n=n0 2−n‖g(n)‖s < 2−3j. duality and quasi-normability for complexity spaces 97 moreover, there exists a kj ≥ k(j) such that for every k,m ≥ kj, (2.7) ∞∑ n=0 2−n‖fk(n) −fm(n)‖ s < 2−n0. choose any k ≥ kj. then for each n ∈ ω with 0 ≤ n ≤ n0 − 1, there exists an mn ≥ k such that ‖g(n) − fmn(n)‖s < 2−n0 . by (2.7) and the triangle inequality, ‖g(n) −fk(n)‖ s < 2−n0 (1 + 2n) whenever 0 ≤ n ≤ n0−1. therefore n0−1∑ n=0 2−n‖g(n) −fk(n)‖ s < 2−n0 · 2n0 < 2−3j. moreover, it follows from (2.5) and (2.6), that ∞∑ n=n0 2−n‖g(n) −fk(n)‖ s ≤ ∞∑ n=n0 2−n‖g(n)‖s + ∞∑ n=n0 2−n‖fk(n)‖ s < 2−3j + ∞∑ n=n0 2−n ∥∥fk(j)(n)∥∥s + 2−3j < 3 · 2−3j for every k ≥ kj. thus we have shown that for each j ∈ ω there is a kj ∈ ω such that ∞∑ n=0 2−n‖g(n) −fk(n)‖ s < 4 · 2−3j ≤ 2−j whenever k ≥ kj. we conclude that (b∗e,‖.‖b∗) is a bibanach space. � remark 2.5. note that if (e,‖.‖) is a quasi-normed space, then (‖.‖b∗) s and (‖.‖s)b∗ are equivalent norms on b∗e. we finish this section with a result on the preservation of order-convexity which will be used later on. a quasi-normed space (e,‖.‖) is called order-convex if the quasi-metric space (e,d‖.‖) is order-convex. proposition 2.6. let (e,‖.‖) be an order-convex quasi-normed space. then (b∗e,‖.‖b∗) is order-convex. proof. let f,g,h ∈b∗e be such that f ≤d‖.‖b∗ g ≤d‖.‖b∗ h, where d‖.‖b∗ denotes the quasi-metric induced on b∗e by ‖.‖b∗. then f(n) ≤d‖.‖ g(n) ≤d‖.‖ h(n) for all n ∈ ω. since (e,‖.‖) is order convex, ‖f(n) −h(n)‖ = ‖f(n) −g(n)‖ + ‖g(n) −h(n)‖ for all n ∈ ω. but ∞∑ n=0 2−n‖f(n) −g(n)‖ < +∞ and ∞∑ n=0 2−n‖g(n) −h(n)‖ < +∞. 98 s. romaguera and m. schellekens hence, ∞∑ n=0 2−n‖f(n) −g(n)‖ + ∞∑ n=0 2−n‖g(n) −h(n)‖ = ∞∑ n=0 2−n[‖f(n) −g(n)‖ + ‖g(n) −h(n)‖] = ∞∑ n=0 2−n‖f(n) −h(n)‖ . so, ‖f −h‖b∗ = ‖f −g‖b∗ +‖g −h‖b∗. we conclude that (b ∗ e,‖.‖b∗) is orderconvex. � 3. the dual complexity space in our context a semilinear space on r+ is an ordered triple (e, +, ·), such that (e, +) is an abelian semigroup with neutral element e, and · is a function from r+ ×e to e such that for all x,y ∈ e and a,b ∈ r+: a ·(b ·x) = (ab) ·x, (a + b) ·x = (a ·x) + (b ·x), a · (x + y) = (a ·x) + (a ·y), and 1 ·x = x. let us recall that every semilinear space is a cone in the sense of keimel and roth [10]. definition 3.1. a quasi-normed semilinear space is a pair (f,‖.‖f ) such that f is a nonempty subset of a quasi-normed space (e,‖.‖), ‖.‖f denotes the restriction of the quasi-norm ‖.‖ to f, and (f, + |f , · |f ) is a semilinear space on r+. if (f,‖.‖f ) is a quasi-normed semilinear space, then the restriction to f of the quasi-metric d‖.‖, induced on e by the quasi-norm ‖.‖, will be denoted by d‖.‖f . definition 3.2. a bibanach semilinear space is a pair (f,‖.‖f ) such that f is a nonempty subset of a bibanach space (e,‖.‖), f is closed in the banach space (e,‖.‖s), and (f,‖.‖f ) is a quasi-normed semilinear space. if in addition, the following condition is satisfied: (i) (f,d‖.‖ f ) is an order-convex optimal quasi-metric join semilattice having a maximum, then, (f,‖.‖f ) is called a bibanach norm-weightable space. the terminology “norm-weightable” is justified by corollary 3.8 below. remark 3.3. note that if (f,‖.‖f ) is a bibanach semilinear space, then d‖.‖f is a bicomplete quasi-metric on f. lemma 3.4. let (f,‖.‖f ) be a bibanach semilinear space such that (f,d‖.‖f ) has a maximum. then the neutral element e is the (unique) maximum of (f,d‖.‖f ). duality and quasi-normability for complexity spaces 99 proof. let x0 be the maximum for (f,d‖.‖f ). then e ≤d‖.‖f x0, so ‖x0‖f = d‖.‖f (e,x0) = 0. moreover, 2 ·x0 ≤d‖.‖f x0, so, ‖−x0‖ = d‖.‖f (2 ·x0,x0) = 0. we conclude that x0 = e. � corollary 3.5. let (f,‖.‖f ) be a bibanach semilinear space such that (f,d‖.‖f ) has a maximum. then, ‖x−y‖≤‖x‖f , for all x,y ∈ f . proof. let x,y ∈ f. then ‖x−y‖ = d‖.‖ f (y,x) ≤ d‖.‖ f (y,e) + d‖.‖ f (e,x). by lemma 3.4, d‖.‖f (y,e) = 0. hence, ‖x−y‖≤ d‖.‖f (e,x) = ‖x‖f . � example 3.6. consider the bibanach space (r,‖.‖) of example 2.1. it is straightforward to show that (r+,‖.‖ r + ) is a bibanach norm-weightable space. of course, (r+, + |r+ ) is not a group. next we give an auxiliary lemma on quasi-metric join semilattices, which, joint with its corollaries, will be useful later on. it can be derived from [24, theorem 15] which states that an optimal quasi-metric join semilattice (x,d) is weigthable if and only if it is it is functionally bounded and order-convex. however, in order to help the reader we shall give a direct proof. lemma 3.7. let (x,d) be an order-convex optimal quasi-metric join semilattice having a maximum element x0. then, (x,d) is upper weightable by the weighting function w: x → r+ defined by w(x) = d(x0,x) for all x ∈ x. proof. choose any pair of points x,y ∈ x. since (x,d) is a join semilattice and x0 is its maximum, we obtain x ≤d x t y ≤d x0, and y ≤d x t y ≤d x0. therefore, w(x) = d(x0,x) = d(x0,xty) + d(xty,x) = d(x0,xty) + d(y,x) and w(y) = d(x0,y) = d(x0,xty) + d(xty,y) = d(x0,xty) + d(x,y). hence, w(x) + d(x,y) = d(x0,xty) + d(y,x) + d(x,y) = w(y) + d(y,x). finally, d(x,y) ≤ d(x,x0) + d(x0,y) = w(y), since d(x,x0) = 0. we conclude that (x,d) is upper weightable with weighting function w(x) = d(x0,x) for all x ∈ x. � from lemmas 3.4 and 3.7, we deduce the following result. corollary 3.8. let (f,‖.‖f ) be a bibanach norm-weightable space. then the quasi-metric space (f,d‖.‖ f ) is upper weightable by the weighting function w: f → r+ defined by w(x) = ‖x‖f for all x ∈ f . since every (upper) weightable bicomplete quasi-metric space is smyth complete [12], we deduce from lemma 3.7 and remark 3.3 the following corollary 3.9. let (f,‖.‖f ) be a bibanach norm-weightable space. then the quasi-metric space (f,d‖.‖f ) is smyth complete. 100 s. romaguera and m. schellekens let (f,‖.‖f ) be a bibanach semilinear space. then, by definition 3.2, there exists a bibanach space (e,‖.‖) such that f is a (nonempty) closed subset of the banach space (e,‖.‖s), ‖.‖f denotes the restriction of ‖.‖ to f and (f,‖.‖f ) is a quasi-normed semilinear space. now define c∗f = {f : ω → f | ∞∑ n=0 2−n‖f(n)‖s < +∞}, and, for each f ∈c∗f , ‖f‖c∗ = ∞∑ n=0 2−n‖f(n)‖f . obviously, c∗f ⊆b ∗ e. the following proposition should be compared with remark 2.3. proposition 3.10. let (f,‖.‖f ) be a bibanach norm-weightable space. then c∗f = {f : ω → f | ∞∑ n=0 2−n‖f(n)‖f < +∞}. proof. let f : ω → f be such that ∑∞ n=0 2 −n‖f(n)‖f < +∞. since e ∈ f, it follows from corollary 3.5 that ‖−f(n)‖ = ‖e‖ = 0 for all n ∈ ω. hence∑∞ n=0 2 −n‖f(n)‖s = ∑∞ n=0 2 −n‖f(n)‖f < +∞. we conclude that f ∈c ∗ f . � proposition 3.11. let (f,‖.‖f ) be a bibanach semilinear space. then (c∗f ,‖.‖c∗) is a bibanach semilinear space. proof. let (e,‖.‖) be the bibanach space for which (f,‖.‖f ) is a bibanach semilinear space. from the semilinearity of (f, + |f , · |f ) it immediately follows that (c∗f , +, .) is a semilinear space on r + for the natural addition and multiplication. on the other hand, (b∗e,‖.‖b∗) is a bibanach space by theorem 2.4. we shall prove that c∗f is a closed subset of the banach space (b∗e, (‖.‖b∗) s). indeed: let f ∈ b∗e be such that (‖f −fk‖b∗) s → 0, where (fk)k∈ω is a sequence of elements of c∗f . then ‖f(n) −fk(n)‖ s → 0 whenever n ∈ ω. since f is closed in (e,‖.‖s), we deduce that f(n) ∈ f for all n ∈ ω. thus, f ∈c∗f . we conclude that (c∗f ,‖.‖c∗) is a bibanach semilinear space. � it follows from the preceding result that the quasi-metric d‖.‖c∗ defined on c∗f by d‖.‖c∗ (f,g) = ‖g −f‖b∗ is bicomplete. definition 3.12. let (f,‖.‖f ) be a bibanach norm-weightable space. then, the quasi-metric space (c∗f , d‖.‖c∗ ) is called the dual complexity space (of (f,‖.‖f )). any subspace of (c∗f ,d‖.‖c∗ ) is also called a dual complexity space. lemma 3.13 ([26]). a quasi-pseudometric join semilattice (x,d) is optimal if and only if for all x,y,z ∈ x, d(xtz,y tz) ≤ d(x,y). duality and quasi-normability for complexity spaces 101 it is interesting to note that the equivalent condition to optimality, in the preceding lemma, is exactly the more familiar notion of t-invariance as discussed in [8]. proposition 3.14. the dual complexity space (c∗f ,d‖.‖c∗ ) is an order-convex optimal quasi-metric join semilattice and it has a maximum. proof. we first show that (c∗f ,d‖.‖c∗ ) is a quasi-metric join semilattice. let f,g ∈ c∗f . since (f,d‖.‖f ) is a quasi-metric join semilattice, for each n ∈ ω there is a supremum f(n) t g(n) ∈ f of f(n) and g(n). define a function f tg : ω → f by (f tg)(n) = f(n) tg(n) for all n ∈ ω. by lemma 3.13 and the fact that e is the maximum of (f,d‖.‖ f ), we have: ‖f(n) tg(n)‖f = d‖.‖f (e,f(n) tg(n)) = d‖.‖f (etg(n),f(n) tg(n)) ≤ d‖.‖ f (e,f(n)) = ‖f(n)‖f , for all n ∈ ω. therefore, ∑∞ n=0 2 −n‖(f tg)(n)‖f ≤ ∑∞ n=0 2 −n‖f(n)‖f < +∞, so f tg ∈c∗f by proposition 3.10. on the other hand, since f(n) ≤d‖.‖f f(n) t g(n) and g(n) ≤d‖.‖f f(n) t g(n) for all n ∈ ω, we deduce that ‖(f tg)(n) −f(n)‖ = ‖(f tg)(n) −g(n)‖ = 0 for all n ∈ ω, so d‖.‖c∗ (f,f t g) = d‖.‖c∗ (g,f t g) = 0, and, hence, f ≤d‖.‖c∗ f t g and g ≤d‖.‖c∗ f t g. furthermore, if h ∈c∗f satisfies f ≤d‖.‖c∗ h and g ≤d‖.‖c∗ h, we deduce, by the definition of supremum, that (f t g)(n) ≤d‖.‖f h(n) for all n ∈ ω. therefore, f tg ≤d‖.‖c∗ h. thus, we have shown that the function f tg, is the (unique) supremum of f and g in (c∗,d‖.‖c∗ ). hence, (c ∗,d‖.‖c∗ ) is a quasi-metric join semilattice. next we show that the quasi-metric join semilattice (c∗,d‖.‖c∗ ) is optimal. let f,g ∈c∗f . then, by the optimality of (f,d‖.‖f ), we have d‖.‖c∗ (f tg,g) = ∞∑ n=0 2−nd‖.‖ f (f(n) tg(n),g(n)) = ∞∑ n=0 2−nd‖.‖ f (f(n),g(n)) = ‖g −f‖b∗ = d‖.‖c∗ (f,g). hence, (c∗,d‖.‖c∗ ) is optimal. on the other hand, the argument used in the proof of proposition 2.6 shows that (c∗,d‖.‖c∗ ) is also order-convex. finally, the function fe : ω → f defined by fe(n) = e for all n ∈ ω, is the (unique) maximum of (c∗,d‖.‖c∗ ) because d‖.‖c∗ (f,fe) = 0 for all f ∈c ∗ f . this completes the proof. � 102 s. romaguera and m. schellekens from propositions 3.11 and 3.14, we immediately deduce the following theorem 3.15. let (f,‖.‖f ) be a bibanach norm-weightable space. then, (c∗f ,‖.‖c∗) is a bibanach norm-weightable space. corollary 3.16. the dual complexity space (c∗f ,d‖.‖c∗ ) is an upper weightable smyth complete quasi-metric space. proof. by theorem 3.15 and corollary 3.8, (c∗f ,d‖.‖c∗ ) is upper weightable with weighting function w: c∗f → r + given by w(f) = ‖f‖c∗ for all f ∈ c ∗ f . finally, by theorem 3.15 and corollary 3.9, (c∗f ,d‖.‖c∗ ) is a smyth complete quasi-metric space. � remark 3.17. in proposition 3.14, we have shown that for any dual complexity space (c∗f ,d‖.‖c∗ ), d‖.‖c∗ (f,fe) = 0 whenever f ∈ c ∗ f , where fe(n) = e for all n ∈ ω. from this fact, it easily follows that the dual complexity space also is a baire space. example 3.18. let (e,‖.‖) be a bibanach space. by theorem 2.4, the function space (b∗e,‖.‖b∗) is also a bibanach space. now let f be a nonempty subset of e such that (f,‖.‖f ) is a bibanach norm-weightable space. thus, (c∗f ,‖.‖c∗) is a bibanach norm-weightable space by theorem 3.15. define b∗b∗ e = {f : ω →b∗e | ∞∑ n=0 2−n(‖f(n)‖b∗) s < +∞} and c∗c∗ f = {f : ω →c∗f | ∞∑ n=0 2−n‖f(n)‖c∗ < +∞}. by theorem 2.4, (b∗b∗ e ,‖.‖b∗b∗ ) is a bibanach space and, by proposition 3.10 and theorem 3.15, (c∗c∗ f ,‖.‖c∗c∗ ) is a bibanach norm-weightable space. let (c∗f ,d‖.‖c∗ ) be the dual complexity space (of the bibanach norm-weightable space (f,‖.‖f )) and let f ⊆c ∗ f . then, the restriction of d‖.‖c∗ to f will be also denoted by d‖.‖c∗ . definition 3.19. a dual complexity space (f,d‖.‖c∗ ), where f ⊆c ∗ f , has an upper bound u ∈ c∗f if for each f ∈ f, f ≤d‖.‖c∗ u. similarly, (f,d‖.‖c∗ ) has a lower bound l ∈c∗f if for each f ∈f, l ≤d‖.‖c∗ f. for each u ∈c∗f , we define (c ∗ f ) u = {f ∈c∗f | u is an upper bound for f}. the following easy example shows that the structure of a semilinear space of c∗f is not preserved by (c ∗ f ) u , in general. example 3.20. let (r+,‖.‖ r + ) be the bibanach norm-weightable space of example 3.6. let u ∈ c∗ r + defined by u(n) = 1 for all n ∈ ω. then the function f defined on ω by f(n) = 1 2 u(n) for all n ∈ ω is in c∗f \ (c ∗ f ) u . duality and quasi-normability for complexity spaces 103 however, the complexity (sub)space ((c∗f ) u,d‖.‖c∗ ) inherits the quasi-metric properties of (c∗f ,d‖.‖c∗ ), obtained in proposition 3.14 and corollary 3.16, as follows. theorem 3.21. for each u ∈ c∗f , ((c ∗ f ) u,d‖.‖c∗ ) is an upper weightable smyth complete order-convex optimal quasi-metric join semilattice and it has a maximum. proof. let u ∈ c∗f . let f,g ∈ (c ∗ f ) u . by proposition 3.14, there is f t g ∈ c∗f . by the definition of supremum, f t g ≤d‖.‖c∗ u, so f t g ∈ (c ∗ f ) u . hence, ((c∗f ) u,d‖.‖c∗ ) is a quasi-metric join semilattice. moreover, it is upper weightable, order-convex and optimal, because these properties are hereditary, and, obviously, u is its maximum. finally, in order to show that ((c∗f ) u,d‖.‖c∗ ) is smyth complete, it suffices to show that (c∗f ) u is a closed subset of the metric space (c∗f , (d‖.‖c∗ ) s), and, then, apply corollary 3.16. assume the contrary. then there is f ∈c∗f \ (c ∗ f ) u such that (d‖.‖c∗ ) s(f,fk) → 0 for some sequence (fk)k∈ω in (c∗f ) u . furthermore, there is n0 ∈ ω such that d‖.‖f (f(n0),u(n0)) = δ > 0. therefore, δ ≤ d‖.‖ f (f(n0),fk(n0))+d‖.‖ f (fk(n0),u(n0)) = d‖.‖ f (f(n0), fk(n0)) for all k ∈ ω. but, d‖.‖f (f(n0),fk(n0)) → 0 as k → +∞. so, we have reached a contradiction. the proof is complete. � let us recall [24] that a linear quasi-metric space is a quasi-metric space (x,d) such that is associated order ≤d is linear. it is known [24] that every linear quasi-metric space is an optimal quasi-metric join semilattice. note that (r+,d‖.‖ r+ ) is a linear quasi-metric space. a quasi-metric space (x,d) is called precompact if for each ε > 0 there is a finite subset a of x such that x = ⋃ a∈a sd(a,ε). (x,d) is called totally bounded if (x,ds) is a totally bounded metric space (see, for instance, [7]). it is well known that every totally bounded quasi-metric space is precompact but the converse implication is not true in general. it follows from a result of künzi [12, proposition 12] that every hereditarly precompact weightable quasi-metric space is totally bounded. by using this result we shall prove the following theorem 3.22. let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear and let f ⊆c ∗ f . if (f, d‖.‖c∗ ) has a lower bound, then it is totally bounded. proof. since (c∗f ,d‖.‖c∗ ) is (upper) weightable and weightability is a hereditary property it suffices to show that (f,d‖.‖c∗ ) is hereditarily precompact by künzi’s proposition cited above. actually, it is enough to prove that any subspace (g, d‖.‖c∗ ) of (c ∗ f ,d‖.‖c∗ ) which has a lower bound is precompact, since all subspaces of (f,d‖.‖c∗ ) are of this kind. hence, let (g,d‖.‖c∗ ) be a subspace of (c ∗ f ,d‖.‖c∗ ) which has a lower bound, say l ∈c∗f and let ε > 0. 104 s. romaguera and m. schellekens since (f,‖.‖f ) is a bibanach norm-weightable space, it follows from lemma 3.4 and corollary 3.8, that d‖.‖f (x,e) = 0 for all x ∈ f, and that the quasimetric space (f,d‖.‖ f ) is (upper) weightable by the function w defined on f by w(x) = ‖x‖f for all x ∈ f. (let us also recall that f is a subset of a bibanach space (e,‖.‖) such that ‖.‖f denotes the restriction of ‖.‖ to f.) since l ∈c∗f , ∑∞ n=0 2 −n(‖l(n)‖f ) s < +∞. hence, there is k ∈ ω such that∑∞ n=k+1 2 −n‖l(n)‖f < ε/2. moreover, for each f ∈ g and each n ∈ ω, we have d‖.‖f (e,f(n)) ≤ d‖.‖f (e,l(n)) +d‖.‖f (l(n),f(n)) = d‖.‖f (e,l(n)). thus, ‖f(n)‖f ≤‖l(n)‖f . therefore, for each f ∈g, ∞∑ n=k+1 2−n‖f(n)‖f < ε/2, and ‖f(n)‖f ≤ b for all n ≤ k, where b = max{‖l(n)‖f | n ≤ k}. now consider the set of functions gk obtained from g by restricting each function of g to the domain {0, . . . ,k}. fix an m ∈ ω such that m ≥ 1 and b m+1 < ε 4 . define a partition of the real interval [0,b] consisting of the intervals bm0 , . . . ,b m m, where bm0 = [0, b m+1 ], and for all j ∈{1, . . . ,m}, bmj = (j b m+1 , (j + 1) b m+1 ]. take the quotient of the set gk, given by the equivalence relation ∼, defined on g by: f ∼ g ⇔ for each n ≤ k there is j ≤ m such that both ‖f(n)‖f ,‖g(n)‖f ∈ b m j . the set gk/∼, is clearly finite. let its cardinality be h, and choose h elements f1, . . . ,fh of g, such that f1 | {0, . . . ,k}, . . . ,fh | {0, . . . ,k}, is a list of representatives, one for each class of the quotient gk/∼. given f ∈g, let i ∈{1, . . . ,h} be such that fi is the representative for which fi | {0, . . . ,k}∼ f | {0, . . . ,k}. then d‖.‖c∗ (fi,f) = ∞∑ n=0 2−nd‖.‖f (fi(n),f(n)) = k∑ n=0 2−nd‖.‖f (fi(n),f(n)) + ∞∑ n=k+1 2−nd‖.‖f (fi(n),f(n)). let n ∈ ω. if fi(n) ≤d‖.‖f f(n), we obtain d‖.‖f (fi(n),f(n)) = 0. otherwise, since ≤d‖.‖f is linear and w is a weighting function on f, we deduce that d‖.‖ f (fi(n),f(n)) = ‖f(n)‖f −‖fi(n)‖f therefore, d‖.‖c∗ (fi,f) ≤ k∑ n=0 2−n b m + 1 + ∞∑ n=k+1 2−n‖f(n)‖f < 2b m + 1 + ε 2 < ε. duality and quasi-normability for complexity spaces 105 we conclude that (g,d‖.‖c∗ ) is precompact. hence, (f,d‖.‖c∗ ) is hereditarily precompact and, thus, totally bounded. � remark 3.23. it is shown in [22] that the dual complexity space (c∗f ,d‖.‖c∗ ) is not precompact and, thus, not totally bounded, in general. hence, the condition of the existence of a lower bound cannot be omitted in the statement of the preceding theorem. for each l ∈ c∗f and each f ⊆c ∗ f , we define fl = {f ∈ f | l is a lower bound for f}. in particular, (c∗f )l = {f ∈c ∗ f | l is a lower bound for f}. let us recall that a subset y of a topological space x is said to be relatively compact if y (in x) is compact. theorem 3.24. let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear. then, for each l ∈ c ∗ f and each f ⊆c ∗ f , fl is relatively compact in the (complete) metric space (c∗f , (d‖.‖c∗ ) s). proof. given l ∈ c∗f and f ⊆c ∗ f , we first prove that l is a lower bound for fl, where fl denotes the closure of fl in (c∗f , (d‖.‖f ) s). assume the contrary. then there is f ∈fl such that d‖.‖f (l(n0),f(n0)) = δ > 0 for some n0 ∈ ω. on the other hand, there is a sequence (fk)k∈ω in fl such that (d‖.‖c∗ ) s(f,fk) → 0. thus, δ ≤ d‖.‖f (l(n0),fk(n0)) + d‖.‖f (fk(n0), f(n0)) = d‖.‖ f (fk(n0),f(n0)). since d‖.‖ f (fk(n0),f(n0)) → 0 as k → +∞, we obtain a contradiction. hence, (fl,d‖.‖c∗ ) is a totally bounded quasi-metric space by theorem 3.22. it then follows from the smyth completeness of (c∗f ,d‖.‖f ) that fl is compact in (c∗f , (d‖.‖c∗ ) s). � corollary 3.25. let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear. then, for each l ∈c ∗ f , ((c ∗ f )l, (d‖.‖c∗ ) s) is a compact metric space. proof. a similar argument to the given in the proof of theorem3.21, shows that (c∗f )l is closed in (c ∗ f , (d‖.‖c∗ ) s). by theorem 3.24, ((c∗f )l, (d‖.‖c∗ ) s) is compact. � the following example shows that the condition that the order ≤d‖.‖f is linear on f cannot be omitted in theorems 3.22 and 3.24 and corollary 3.25. example 3.26. consider the bibanach norm-weightable space (r+,‖.‖ r + ) and the bibanach norm-weightable space (c∗ r +,‖.‖c∗ r+ ) (see theorem 3.15). define l: ω → c∗ r + by (l(n))(m) = 22n if m = n, and (l(n))(m) = 0 otherwise. now consider the sequence (fk)k∈ω such that for each k ∈ ω, fk : ω →c∗r+ is defined by (fk(n))(m) = 22n if m = n = k, and (fk(n))(m) = 0 otherwise. clearly, d‖.‖c∗ (l(n),fk(n)) = 0 for all n,k ∈ ω, because (fk(n))(m) is only different from zero when m = n = k and in such a case one has (fn(n))(n) = 106 s. romaguera and m. schellekens 22n = (l(n))(n). therefore, l is a lower bound for f = {fk | k ∈ ω}. however, d‖.‖c∗ c∗ (fk,fk+1) = 1 for all k ∈ ω, because d‖.‖c∗c∗ (fk,fk+1) = ∞∑ n=0 2−nd‖.‖c∗ (fk(n),fk+1(n)) = ∞∑ n=0 2−n   ∞∑ j=0 2−j[((fk+1(n))(j) − (fk(n))(j)) ∨ 0]   and (fk+1(n)(j)) = 0 except when j = n = k + 1. in this case, one has (fk+1(n))(j) − (fk(n))(j) = 22j, so d‖.‖c∗c∗ (fk,fk+1) = 2−j2−j22j = 1. we conclude that (f,d‖.‖c∗c∗ ) is not totally bounded. we remark that it is possible still to endow the complexity space with a satisfactory structure in this context. to this end we first introduce the notion of a unitary quasi-normed space. definition 3.27. a unitary quasi-normed space is a triple (e,‖.‖ ,?) such that (e,‖.‖) is a quasi-normed space (on r) and ? is an internal commutative (multiplication) law for which there is a unique element 1e ∈ e such that for each x ∈ e\{e} there exists a unique 1 x ∈ e\{e} satisfying x ? 1 x = 1e. if (e,‖.‖ ,?) is a unitary quasi-normed space, we may define a quasi-metric d−1 on e \{e} as follows: d−1(x,y) = ∥∥∥∥1y − 1x ∥∥∥∥ for all x,y ∈ e\{e}. now let f be a (nonempty) subset of e which is closed for the law ? and such that (f,‖.‖f ) is a bibanach norm-weightable space satisfying that for each x ∈ f\{e}, 1 x ∈ f (note that, in fact, 1e ∈ f). then, we construct a set f∞ = (f\{e}) ∪{∞}, where ∞ /∈ e is defined by the conditions: (i) for every x ∈ f, ∞+ x = x +∞ = ∞, (ii) for every x ∈ f, ∞?x = x?∞ = ∞ and (iii) 1 ∞ = e. note that, by (iii), the quasi-metric d−1 |f\{e} can be extended to f∞. this extension will be also denoted by d−1. (moreover, if one defines ∞·r = r ·∞ = ∞ for all r > 0, then (f∞, +, ·) is a cone in the sense of [10].) under the above conditions we define cf∞ = { f : ω → f∞ | ∑∞ n=0 2 −n ∥∥∥ 1f(n)∥∥∥s < +∞} and dc(f,g) = ∞∑ n=0 2−nd−1(f(n),g(n)) for all f,g ∈cf∞. then, it is straightforward to check that dc is a quasi-metric on cf∞. duality and quasi-normability for complexity spaces 107 definition 3.28. the quasi-metric space (cf∞,dc) is called the complexity space (of (f∞,d−1)). remark 3.29. note that (r,‖.‖ ,?) is a unitary quasi-normed space, where (r,‖.‖) is the bibanach space of example 2.1 and ? denotes the usual multiplication on r. moreover the induced quasi-metric d−1 is exactly the quasi-metric u−1 defined in section 1. taking f = r+, we have that f∞ = (0, +∞] and cf∞ = {f : ω → (0, +∞] |∑∞ n=0 2 −n 1 f(n) < +∞}. thus, we obtain the complexity space (with values in (0, +∞]), as discussed in [23]. given a bibanach norm-weightable space (f,‖.‖f ), consider the complexity space (cf∞,dc) and the dual complexity space (c∗f ,dc∗). then, as in [22], we may define an isometry ψ : cf∞ → c∗f by ψ(f) = 1/f for all f ∈ cf∞. combining this fact with the propositions and theorems proved in this section, we obtain, among other, the following results: a) the complexity space (cf∞,dc) is an optimal join semilattice orderconvex quasi-metric space and the point f∞ is its maximum, where f∞(n) = ∞ for all n ∈ ω. b) the complexity space (cf∞,dc) is an upper weightable smyth complete quasi-metric space. c) let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear and let f ⊆cf∞. if (f,dc) has a lower bound, then it is totally bounded. d) let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear. then, for each l ∈ cf∞, ((cf∞)l, (dc)s) is a compact metric space. comment. our assumption regarding complexity lower bounds both for the dual complexity space and the complexity space (see theorem 4 and statement c) above), may seems restrictive at first from a computational point of view. indeed, by the blum speed up theorem ([3] or [9]), there exist problems for which any algorithm computing such a problem can be replaced by a new algorithm which computes the given problem significantly faster. more specifically, an asymptotic gain can be obtained at each time which is logarithmic in the complexity of the program one starts out with. however, such problems may be seen as artificially constructed to prove the theorem according to [9] which continues to state that ”for an important class of problems that can occur in practice an optimal algorithm does exists”, by levin’s theorem, and hence one does obtain a lower bound, in general. as such our assumption is justifiable not only by the concrete examples of complexity lower bounds which one can find in the literature (e.g. [11]), but also finds formal justification by the above cited result. 108 s. romaguera and m. schellekens 4. function spaces and hyperspaces for complexity spaces in this section we investigate completeness of the quasi-metric of uniform convergence and of the hausdorff quasi-pseudometric for dual complexity spaces, in the context of function spaces and hyperspaces, respectively. we will need to consider extended quasi-(pseudo)metrics. they satisfy the usual axioms for a quasi-(pseudo)metric, except that we allow d(x,y) = +∞. let x be a nonempty set and let (y,d) be a quasi-(pseudo)metric space. denote by d the extended quasi-(pseudo)metric defined on the set y x of all functions from x to y by d(f,g) = sup{d(f(x),g(x)) | x ∈ x} for all f,g ∈ y x. d is called the extended quasi-(pseudo)metric of uniform convergence (of (y,d)) (compare [20, p. 88-89]). similarly to [13, proposition 5] we obtain that if x is a nonempty set and (y,d) is a bicomplete quasi-(pseudo)metric space, then d is a bicomplete extended quasi-(pseudo)metric on y x. from this result and theorem 2.4 we deduce the following proposition 4.1. let (e,‖.‖) be a bibanach space. then, for each nonempty set x the extended quasi-metric of uniform convergence of (b∗e,d‖.‖b∗ ) is bicomplete on (b∗e) x. now suppose that (f,‖.‖f ) is a bibanach norm-weightable space. it then follows from propositions 3.11 and 4.1, that for each nonempty set x, the extended quasi-metric of uniform convergence of the dual complexity space (c∗f ,d‖.‖c∗ ) is bicomplete on (c ∗ f ) x. in the light of corollary 3.16, it seems natural to ask if this extended quasi-metric is actually smyth complete. the following example shows that this is not the case. example 4.2. let (r+,‖.‖ r + ) be the bibanach norm-weightable space of example 3.6. denote by d the extended quasi-metric of uniform convergence of (c∗ r +,d‖.‖c∗ ). define a sequence (gk)k∈n of functions from ω to c ∗ r + by gk(m) := { 2mχm if m ≥ k 0 otherwise for all m ∈ ω (here, χm denotes the characteristic function of {m}). note that for each k ∈ n and each m,n ∈ ω, we have (gk+1(m))(n) ≤ (gk(m))(n), so d(gk,gk+1) = sup{d‖.‖c∗ (gk(m),gk+1(m)) | m ∈ ω} = 0 for all k ∈ n. hence, (gk)k∈n is a left k-cauchy sequence with respect to d. since, for each k ∈ n, (gk(k))(k) = 2k and (gk+1(k))(k) = 0, we deduce that (gk)k∈n is not a cauchy sequence in the extended metric ds. we conclude that d is not smyth completable and, thus, it is not smyth complete. duality and quasi-normability for complexity spaces 109 let (x,d) be a quasi-pseudometric space and denote by p0(x) the collection of all nonempty subsets of x. according to [1], the extended hausdorff quasipseudometric of d on p0(x) is defined by dh(a,b) = max{sup a∈a d(a,b), sup b∈b d(a,b)} whenever a,b ∈p0(x). our next example shows that the if (c∗f ,d‖.‖c∗ ) is the dual complexity space (of (f,‖.‖f )), the extended hausdorff quasi-pseudometric of the smyth complete quasi-metric d‖.‖c∗ is not smyth completable, in general. example 4.3. let (r+,‖.‖ r + ) be the bibanach norm-weightable space of example 3.6. as in example 4.2 define a sequence (gk)k∈n of functions from ω to c∗ r + by gk(m) := { 2mχm if m ≥ k 0 otherwise for all m ∈ ω. then (aj)j∈n is a sequence in p0(c∗r+ ), where aj = {gk(m) | k ≥ j, m ∈ ω} for all j ∈ n. fix j ∈ n and let gk(m) ∈ aj. since (gk+1(m))(n) ≤ (gk(m))(n) for all n ∈ ω and gk+1(m) ∈ aj+1, we deduce that d‖.‖c∗ (gk(m),aj+1) = 0. similarly, for each gk(m) ∈ aj+1, d‖.‖c∗ (aj,gk(m)) = 0. hence, ((d‖.‖c∗ )h)(aj,aj+1) = 0 for all j ∈ n, and, thus, (aj)j∈n is a left k-cauchy sequence with respect to the extended hausdorff quasi-pseudometric of d‖.‖c∗ . nevertheless, ((d‖.‖c∗ )h)(aj+1,aj) = 1 for all j ∈ n, because d‖.‖c∗ (aj+1,gj(j)) = 1. so the extended hausdorff quasi-peudometric of d‖.‖c∗ is not smyth completable on p0(c ∗ r + ). however, several interesting kinds of dual complexity (sub)spaces have smyth completable extended hausdorff quasi-pseudometrics (actually totally bounded) as we shall show. indeed, it follows from results of künzi and ryser [14, corollaries 2 and 9], that the extended hausdorff quasi-pseudometric on p0(x) of a totally bounded (resp. totally bounded and bicomplete) quasi-pseudometric d on a set x, is totally bounded (resp. totally bounded and bicomplete). combining these results with theorems 3.22 and 3.24, respectively, we obtain: proposition 4.4. let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear and let f ⊆c ∗ f . if (f,d‖.‖c∗ ) has a lower bound, then the extended hausdorff quasi-pseudometric of d‖.‖c∗ is totally bounded on p0(f). 110 s. romaguera and m. schellekens proposition 4.5. let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear, let l ∈ c ∗ f and f a closed subset of (c ∗ f , (d‖.‖c∗ ) s). then the extended hausdorff quasi-pseudometric of d‖.‖c∗ is totally bounded and bicomplete on p0(fl). hence, the (hyper)space (p0(fl), ((d‖.‖c∗ )h) s) is compact. let us recall that the vietoris topology of a topological space (x,t) is defined as the topology on p0(x) which as a subbase the collection of sets of the form v + = {a ∈p0(x) | a ⊆ v} and w− = {a ∈p0(x) | a∩w 6= ∅}, whenever v and w are open sets in (x,t). let (x,t) be a quasi-pseudometrizable space and let d be any quasi-pseudometric on x compatible with t . it is well known (see [1]) that the topology generated by the extended hausdorff quasi-pseudometric of d is finer than the vietoris topology on the collection of all nonempty compact subsets of (x,t(d)). in [2] it is given an example of a compact quasi-metric space (x,d) for which the topology of the extended hausdorff quasi-pseudometric of d is strictly finer than the vietoris topology of (x,t(d)) on the collection of all nonempty compact subsets of (x,t(d)). however, we can prove that the two topologies coincide when one works on the collection ks0(x) of all nonempty compact subsets of the pseudometric space (x,ds). it is easy to see that the restriction of the extended hausdorff quasi-pseudometric to ks0(x) is actually a quasi-pseudometric. proposition 4.6. let (x,d) be a quasi-pseudometric space. then the vietoris topology of (x,t(d)) coincides with the topology generated by the hausdorff quasi-pseudometric of d on ks0(x). proof. let a ∈ ks0(x) and choose an arbitrary ε > 0. then, there is a finite subset aε of a such that a ⊆∪a∈aε(sd)s(a,ε/4). then, the sets v + |ks0(x)= {b ∈k s 0(x) | b ⊆ ⋃ a∈aε sd(a,ε/4)} and w− |ks0(x)= {b ∈k s 0(x) | b ∩sd(a,ε/4) 6= ∅ for all a ∈ aε} are open neighborhoods of a with respect to the vietoris topology of (x,t(d)) on ks0(x). denote their intersection by g. we shall show that g ⊆ sdh (a,ε). indeed: let b ∈ g. choose any a ∈ a. then, ds(a,aε) < ε/4 for some aε ∈ aε. moreover, there is a b ∈ b such that d(aε,b) < ε/4. therefore, d(a,b) < ε/2. so, supa∈a d(a,b) ≤ ε/2. now, choose any b ∈ b. then, there is an a ∈ aε such that d(a,b) < ε/4. hence, supb∈b d(a,b) ≤ ε/4. consequently, sdh (a,b) ≤ ε/2. we conclude that the vietoris topology of (x,t(d)) coincides with the topology of dh on ks0(x). � as a consequence of the preceding proposition and corollary 3.25, we obtain the following corollary 4.7. let (f,‖.‖f ) be a bibanach norm-weightable space such that ≤d‖.‖f is linear. then, the vietoris topology of the dual complexity space duality and quasi-normability for complexity spaces 111 (c∗f ,d‖.‖c∗ ) coincides with the topology generated by the hausdorff quasi-pseudometric of d‖.‖c∗ on the collection of subsets of p0(c ∗ f ) defined by {(c ∗ f ) l | l ∈ c∗f}. references [1] g. berthiaume, on quasi-uniform hyperspaces, proc. amer. math. soc. 66 (1977), 335– 343. 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[22] s. romaguera and m. schellekens, quasi-metric properties of complexity spaces, topology appl. 98 (1999), 311–322. 112 s. romaguera and m. schellekens [23] m. schellekens, the smyth completion: a common foundation for denotational semantics and complexity analysis, in: proc. mfps 11, electronic notes in theoretical computer science 1 (1995), 211-232. [24] m. schellekens, on upper weightable spaces, in: proc. 11th summer conference on general topology and appl., ann. new york acad. sci. 806 (1996), 348–363. [25] m. schellekens, complexity spaces revisited. extended abstract, in: proc. 8th prague topological symposium, topology atlas, 1996, 337–348. [26] m. schellekens, the correspondence between partial metrics and semivaluations, theoretical computer sci., to appear. [27] m. b. smyth, quasi-uniformities: reconciling domains with metric spaces, in: proc. mfps 3, lncs 298, m. main et al. editors, springer, berlin 1988, 236–253. [28] m. b. smyth, totally bounded spaces and compact ordered spaces as domains of computation, in: topology and category theory in computer science, g.m. reed, a.w. roscoe and r.f. wachter editors, clarendon press, oxford 1991, 207–229. [29] ph. sünderhauf, the smyth-completion of a quasi-uniform space, in: m. droste and y. gurevich editors, semantics of programming languages and model theory, algebra, logic and appl., 5, gordon and breach sci. publ., 1993, 189–212. received april 2001 revised august 2001 s. romaguera departamento de matemática aplicada universidad politécnica de valencia apartado 22012 46022 valencia spain e-mail address : sromague@mat.upv.es m. schellekens department of computation national university of ireland cork, ireland e-mail address : m.schellekens@cs.ucc.ie @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 243–253 quasi-pseudometric properties of the nikodym-saks space jesús ferrer ∗ dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. for a non-negative finite countably additive measure µ defined on the σ-field σ of subsets of ω, it is well known that a certain quotient of σ can be turned into a complete metric space σ(ω), known as the nikodym-saks space, which yields such important results in measure theory and functional analysis as vitali-hahn-saks and nikodym’s theorems. here we study some topological properties of σ(ω) regarded as a quasi-pseudometric space. 2000 ams classification: 54e15, 54e55. keywords: quasi-pseudometric space, nikodym-saks space. 1. introduction. all throughout this paper we shall assume that the measure space (ω, σ,µ) corresponds to a non-negative finite countably additive measure µ defined in the σ-algebra σ of subsets of ω, 0 < µ(ω) < ∞. following the terminology of [5, p.156], by σ(ω) we denote the quotient space obtained after identifying the measurable sets a,b such that their symmetric difference a 5 b has zero measure. for the sake of convenience, we shall not use any special symbol to distinguish between the elements of σ and the equivalence classes in σ(ω). it is shown in [5, p.156], and also in [3, p.86] and [7, p.208], that the function µ(a 5 b) defines a metric in σ(ω) such that it becomes a complete metric space. this property allows one to apply baire category arguments to obtain important results in convergence of measures such as the theorems of vitalihahn-saks and nikodym. ∗supported in part by mcyt and feder project bfm2002-01423 244 j. ferrer besides, the set operations in σ(ω) are well defined and are continuous respect to the metric considered, so that (σ(ω),5,∩) may also be regarded as a topological ring. generalizations of this can be found in [4]. the purpose of this paper is to notice that the metric space σ(ω) admits a quasi-pseudometric structure which determines in the standard way both the topology and the order given by set-inclusion. we shall as well revisit some topological properties of σ(ω), such as completeness, compactness and connectedness from a quasi-pseudometric perspective. 2. nikodym-saks’ complete quasi-pseudometric ordered space. given a,b ∈ σ(ω), we define q(a,b) := µ(b \a). it is immediate to verify that q is a quasi-pseudometric in σ(ω). by q−1 and q∗ we denote the conjugate quasi-pseudometric and the metric associated to q, respectively, that is q−1(a,b) = q(b,a) = µ(a\b), q∗(a,b) = q(a,b) + q−1(a,b) = µ(b \a) + µ(a\b) = µ(a5b). it is also quite simple to see that the set operations ∪ and ∩ are continuous in the quasi-pseudometric space (σ(ω),q). also, the mapping a −→ ω \ a is a quasi-uniform isomorphism from (σ(ω),q) onto (σ(ω),q−1). in the same manner, one may easily see that µ : (σ(ω),q) −→ r is quasi-uniformly upper semicontinuous, i.e., given ε > 0, there is δ > 0 such that, whenever q(a,b) < δ, we have µ(b) −µ(a) < ε. noticing that set-inclusion ⊆ is an ordering compatible with the equivalence relation defined in σ, we may regard (σ(ω),⊆) as an ordered space. again, following the terminology of [6], we have the following result. proposition 2.1. (σ(ω),q∗,⊆) is a metric ordered space determined by the quasi-pseudometric q. proof. it all reduces to see that the graph of the order relation ⊆ coincides with ∩ε>0v −1ε , where v −1ε = {(a,b) ∈ σ(ω) × σ(ω) : q −1(a,b) < ε}. this is simple, since (a,b) ∈ ∩v −1ε>0 if and only if q −1(a,b) = 0, which is equivalent to q(b,a) = µ(a\b) = 0. that is, a\b = ∅ and so a ⊆ b. � notice that the order defined by set-inclusion coincides with the so called ”specialization order” defined by the quasi-pseudometric q, i.e., a ⊆ b ⇔ q(b,a) = 0 ⇔ b ∈{a}, that is, ”it takes no effort to move from b to a, so a must be lower”. we introduce a couple of definitions by means of which we shall show that nikodym-saks’ space is complete from a quasi-pseudometric perspective. nikodym-saks space 245 definition 2.2. in a quasi-pseudometric space (x,q), we say that a subset a is quasi-bounded provided there is an element x0 ∈ a such that the set of reals {q(x,x0) : x ∈ a} is bounded. it is plain that the associated pseudometric space (x,q∗) is bounded if and only if the quasi-pseudometric spaces (x,q) and (x,q−1) are both quasibounded. definition 2.3. by a quasi-pseudometric ordered space we mean a triple (x,q,≤) such that the quasi-pseudometric q determines the topological ordered space (x,q∗,≤) in the sense given in [6]. thus, we say that the quasipseudometric ordered space (x,q,≤) is orderly quasi-complete whenever every quasi-bounded sequence (xn)∞n=1 satisfies the following two conditions: 1) (xn)∞n=1 admits a supremum (least upper bound) and an infimum (greatest lower bound) in (x,≤). 2) for each n, if yn := inf{xj : j ≥ n}, then q(yn,xn) ≤ ∞∑ j=n q(xj+1,xj). following the terminology introduced in [10], if (x,q) is a quasi-pseudometric space, a sequence (xn)∞n=1 in x is said to be right-k-cauchy whenever, given ε > 0, there is k ∈ n such that, for n ≥ m ≥ k, we have q(xn,xm) < ε. we say that (x,q) is right-k-sequentially complete provided every right-k-cauchy sequence converges. proposition 2.4. if (x,q,≤) is orderly quasi-complete, then (x,q) is rightk-sequentially complete. proof. let (xn)∞n=1 be a right-k-cauchy sequence in (x,q). we define inductively an increasing sequence (kj)∞j=1 of positive integers such that, for each j, if n ≥ m ≥ kj, then q(xn,xm) < 2−j. now, since (xkj )∞j=1 is quasi-bounded, if, for each j, yj := inf{xki : i ≥ j}, and y := sup{yj : j ≥ 1} = limjxkj , then, for each j, using condition 2 of the former definition, q(y,xkj ) ≤ q(y,yj) + q(yj,xkj ) = q(yj,xkj ) ≤ ∞∑ i=j q(xki+1,xki) < ∞∑ i=j 2−i = 2−j+1. finally, for ε > 0, let j0 be such that 2−j0+1 < ε/2. then, for n ≥ kj0 , we take j1 ≥ j0 with kj1 ≥ n, and so q(y,xn) ≤ q(y,xkj1 ) + q(xkj1 ,xn) ≤ 2 −j1+1 + 2−j0 < ε. � corollary 2.5. nikodym-saks’ quasi-pseudometric space (σ(ω),q) is right-ksequentially complete. 246 j. ferrer proof. after proposition 2.2, it all reduces to see that (σ(ω),q,⊆) is orderly quasi-complete. for any sequence (an)∞n=1 in σ(ω), it is clear that ∪∞n=1an and ∩∞n=1an are in σ(ω) and they correspond to supn an, infn an, respectively. now, for each n, let bn := ∩∞j=naj, then q(b,an) = µ(an\b) = µ(∪∞j=n(an\aj)) ≤ µ(∪ ∞ j=n(aj\aj+1)) = ∞∑ j=n µ(aj\aj+1) = ∞∑ j=n q(aj+1,aj). � again following [6], a quasi-pseudometric space (x,q) is bicomplete when its associated pseudometric space (x,q∗) is complete. the completeness of nikodym-saks’ space can now be reobtained by means of quasi-pseudometrics. corollary 2.6. nikodym-saks’ quasi-pseudometric space (σ(ω),q) is bicomplete. proof. let (an)∞n=1 be a cauchy sequence in (σ(ω),q ∗). for each j ∈ n, there is kj ∈ n such that, if n,m ≥ kj, then q∗(an,am) < 2−j. so, if n,m ≥ kj, we have q(an,am) < 2 −j, q−1(an,am) < 2 −j, thus obtaining, after what we did previously, that, if b := limjakj and c := limjakj , then (an) ∞ n=1 q-converges to b. now, since (an) ∞ n=1 is q −1-rightk-cauchy, it follows that (ω\an)∞n=1 is q-right-k-cauchy and, given that, for each j, q(ω\an, ω\am) = q−1(an,am) < 2−j, n,m ≥ kj, we have that (ω\an)∞n=1 q-converges to limj(ω\akj ) = ω\c. hence (an)∞n=1 q−1-converges to c. hence, since b ⊆ c, and taking limits in µ(c \b) = q(b,c) ≤ q(b,an) + q(an,c) = q(b,an) + q−1(c,an), it follows that µ(b \ c) = µ(c \ b) = 0. that is, b = c in σ(ω), and so (an)∞n=1 converges in (σ(ω),q ∗). � again after [6, p.84], we recall that a quasi-uniformity u in a space x is convex with respect to the order ≤ whenever, given u ∈ u, there is v ∈ u such that v ⊆ u, and, for each x ∈ x, v (x) = {y ∈ x : (x,y) ∈ v} is convex respect to ≤, i.e., a ≤ c ≤ b, a,b ∈ v (x), imply c ∈ v (x). after proposition 4.19 of [6, p.84], in light of our previous result, we know that (σ(ω),q∗) is a convex metric space in the sense before defined. nevertheless, we cannot conclude, as it happens in many metric convex spaces, that every ball v ∗ε (a) = {x ∈ σ(ω) : q∗(a,x) < ε} has to be a convex set, as our next result proves. proposition 2.7. let ω = [0, 1] and let λ represent the lebesgue measure. then, for each 0 < ε < 1/2, there is a measurable set a such that the ball v ∗ε (a) is not convex with respect to set-inclusion. nikodym-saks space 247 proof. let ε/2 < a < 1 − 3ε 2 . we consider the following measurable sets a = [a,a + 11ε 8 ], x = [a,a + ε 8 ] ∪ [a + � 4 ,a + ε 2 ] ∪ [a + ε,a + 11ε 8 ], y = [a− ε 2 ,a + 3ε 2 ], z = [a− ε 2 ,a + ε 2 ] ∪ [a + ε,a + 11ε 8 ] q∗(a,x) = λ(x \a) + λ(a\x) = λ(a\x) = ε 8 + ε 2 = 5ε 8 < ε, q∗(a,y ) = λ(y \a) + λ(a\y ) = λ(y \a) = ε 2 + ε 8 = 5ε 8 < ε, q∗(a,z) = λ(z \a) + λ(a\z) = ε 2 + ε 2 = ε. hence, we have x ⊆ z ⊆ y , x,y ∈ v ∗ε (a), but z /∈ v ∗ε (a). � the following result will be needed afterwards. let us recall first that a subset f of an ordered set (x,≤) is said to be inductive whenever every totally ordered subset of f has an upper bound in x. proposition 2.8. every non-empty closed subset of (σ(ω),q∗) is inductive with respect to set-inclusion. proof. let f be a non-empty closed set in (σ(ω),q∗). let (fi)i∈i be a totally ordered subset of f. since µ is finite, there exists ρ = supi∈iµ(fi). we now start an inductive process by taking i1 ∈ i such that µ(fi1 ) > ρ− 1 2 . assuming already found fi1 ⊆ fi2 ⊆ ... ⊆ fin in f such that, for j = 1, 2, ...,n, µ(fij ) > ρ− 1 2j , we proceed to find fin+1 with the same properties. if µ(fin) > ρ− 1 2n+1 , then we set fin+1 := fin; if µ(fin) ≤ ρ − 1 2n+1 , we find fin+1 in f such that µ(fin+1 ) > ρ − 1 2n+1 , then fin ⊆ fin+1 , otherwise, since we are dealing with totally ordered elements, we would have fin+1 ⊆ fin, and, µ(fin+1 ) ≤ µ(fin) ≤ ρ − 1 2n+1 , which is a contradiction. we have thus constructed an increasing sequence (fin) ∞ n=1 in f, with µ(fin) > ρ− 1 2n , n ∈ n. we set f := ∪∞n=1fin. then, f ∈ σ(ω), and, for each n, q∗(f,fin) = µ(f\fin) = µ(∪ ∞ j=1(fij\fin)) ≤ µ(∪ n j=1(fij\fin))+µ(∪ ∞ j=n+1(fij\fin)) = µ(∪∞j=n+1(fij \fin)) ≤ µ(∪ ∞ j=n+1(fij \fij−1 )) = ∞∑ j=n+1 µ(fij \fij−1 ) = ∞∑ j=n+1 (µ(fij ) −µ(fij−1 )) ≤ ∞∑ j=n+1 (ρ− (ρ− 1 2j−1 )) = ∞∑ j=n+1 1 2j−1 = 1 2n−1 . hence, (fin) ∞ n=1 converges to f in (σ(ω),q ∗). since f is closed, it follows that f ∈ f. we show finally that f is an upper bound for the chain (fi)i∈i. give i ∈ i, we consider two possibilities: if there is n0 ∈ n such that fi ⊆ fin0 , then it is clear that fi ⊆ fin0 ⊆ f. 248 j. ferrer on the contrary, if fi is not contained in fin, n ∈ n, then again the total ordering guarantees that fin ⊆ fi, n ∈ n, and so f ⊆ fi. then, since ρ ≥ µ(fi) ≥ µ(f) = limnµ(fin) ≥ ρ, we have µ(fi 5f) = µ(fi \f) = µ(fi) −µ(f) = ρ−ρ = 0. that is, fi = f. � 3. connectedness and compactness in nikodym-saks’ space. in this section we study the topological properties of connectedness and compactness in the space (σ(ω),q∗), observing that such properties are directly related with the degree of atomicity of the measure µ. we shall introduce again some notation. for a ∈ σ(ω), by σ(a) we denote the collection of elements of σ(ω) contained in a, we shall also refer to this collection as a lower interval; similarly σ+(a) will stand for all the measurable supersets of a and we will refer to this as an upper interval. let us recall that e ∈ σ(ω) is called an atom when it has positive measure and the lower interval σ(e) is reduced to {∅,e}. when a measure µ does not have any atom then it is said to be non-atomic. it is convenient to recall that a measure can only admit at most a countable amount of disjoint atoms, when ω admits a countable partition formed by atoms, then µ is said to be purely atomic. before characterizing the connectedness of (σ(ω),q∗) in terms of atoms, let us notice that the quasi-pseudometric spaces (σ(ω),q) and (σ(ω),q−1) are always connected: let us suppose that (σ(ω),q) admits two disjoint open sets a, b covering σ(ω). then, one of them, say a, contains ω, so there is δ > 0 such that vδ(ω) ⊆ a. but, vδ(ω) = σ(ω). hence, b = ∅. proposition 3.1. the following are equivalent. (i) no upper interval σ+(a), a 6= ∅, is q−1-open. (ii) µ is non-atomic. (iii) (σ(ω),q∗) is connected. (iv) for each a ∈ σ(ω) and α ∈ [0,µ(a)], there is b ∈ σ(a) such that µ(b) = α. proof. (i) ⇒ (ii). assume that e is an atom. we show that the upper inteval σ+(e) is q−1-open. given a ∈ σ+(e), let δ = µ(e) > 0. if x ∈ v −1δ (a), then q−1(a,x) < δ implies µ(a\x) < δ. but, µ(e \x) = µ(e \x \a) + µ(e ∩a\x) ≤ µ(e \a) + µ(a\x) < δ. hence, µ(e\x) = 0, otherwise, since e is an atom, we would have µ(e\x) = µ(e) = δ. thus, x ∈ σ+(e). (ii) ⇒ (iii). let us assume that (σ(ω),q∗) is disconnected. so, let a, b be two non-empty disjoint closed sets covering σ(ω). we may suppose without restriction that ω ∈ a. applying proposition 4 to the closed set b and after zorn’s lemma, there is a maximal element m in (b,⊆). since b is also open, there is δ > 0 such that v ∗δ (m) ⊆ b. now, since ω \m has non-zero measure nikodym-saks space 249 (otherwise, ω = m would also be in b), the fact that µ is atom-free guarantees, see [1, p.24], that inf{µ(e) : ∅ 6= e ∈ σ(ω \m)} = 0. thus, we may find e ∈ σ(ω \m) such that 0 < µ(e) < δ. let a := m ∪e. then, q∗(m,a) = µ(e) < δ, and so a ∈ b with m ⊆ a, m 6= a, contradicting the maximality of m. (iii) ⇒ (iv). the continuous mapping x −→ x ∩a maps σ(ω) onto σ(a). thus, if σ(ω) is connected, so is σ(a). now, since µ is continuous, it follows that µ(σ(a)) = [0,µ(a)]. (iv) ⇒ (i). let us assume there is e ∈ σ(ω), e 6= ∅, such that the upper interval σ+(e) is q−1-open. clearly, since µ(e) > 0, there is 0 < δ < µ(e) such that v −1δ (e) ⊆ σ+(e). by hypothesis, there is a ∈ σ(e) for which µ(a) = δ/2. let x := e \a. then, µ(e \x) = µ(a) = δ/2 < δ implies that x ∈ v −1δ (e) and consequently e ⊆ x, a contradiction, since µ(e\x) 6= 0. � by recalling that a finite countably additive measure λ in (ω, σ) is µcontinuous whenever limµ(x)→0 λ(x) = 0, we have that in this case the identity mapping is well defined and continuous from (σ(ω),q∗µ) into (σ(ω),q ∗ λ). therefore the following result is straightforward. corollary 3.2. if λ is a finite countably additive µ-continuous measure and µ is non-atomic, then so is λ. we study in the following the compactness of nikodym-saks’ space. as we did in the connectedness part, it is curious to notice that the quasi-pseudometric spaces (σ(ω),q) and (σ(ω),q−1) are always compact; just recall that, for instance, any q-open cover of σ(ω) must have a member containing ω, consequently, this open set has to be σ(ω). we show next that, in some sense, the compactness of (σ(ω),q∗) does not get along with the connectedness. as a matter of fact, we show that compactness is equivalent to µ being purely atomic. we need, in order to do so, to introduce some more notation. given a sequence (an)∞n=1 in σ(ω), for each n, if (i1, i2, ..., in) ∈{0, 1}n, we define the following sets a(i1,i2,...,in) := (∩{aj : 1 ≤ j ≤ n,ij = 1}) ∩ (∩{ω \aj : 1 ≤ j ≤ n,ij = 0}). it is plain that, for each k = 1, 2, ...,n, the collections {a(i1,i2,...in) : ik = 1}, {a(i1,i2,...in) : ik = 0}, are partitions of ak and ω \ak, respectively. lemma 3.3. if µ is non-atomic, then there is a sequence (an)∞n=1 in σ(ω) such that, for each n, µ(a(1,1,...,1)) = µ(a(0,0,...,0)) = ( 1 4 + 1 2n+1 )µ(ω), µ(a(i1,i2,...,in)) = 1 2n+1 µ(ω), (i1, i2, ...in) /∈{(0, 0, ..., 0), (1, 1, ..., 1)}. 250 j. ferrer proof. we give an inductive sketch of proof. the first set a1 appears courtesy of the non-atomicity of the measure µ and proposition 3.1. once already obtained a1,a2, ...,an, again proposition 3.1 guarantees the existence of measurable sets b(i1,i2,...,in) ⊆ a(i1,i2,...,in), (i1, i2, ...in) ∈{0, 1} n, such that µ(b(i1,i2,...,in)) = 1 2n+2 µ(ω), (i1, i2, ..., in) 6= (1, 1, ..., 1), µ(b(1,1,...,1)) = ( 1 4 + 1 2n+2 )µ(ω). defining an+1 := ∪{b(i1,i2,...,in) : (i1, i2, ...in) ∈ {0, 1} n}, the induction process is done. � proposition 3.4. nikodym-saks’ space (σ(ω),q∗) is compact if and only if µ is purely atomic. proof. assuming (σ(ω),q∗) is compact, it all reduces to show that every set a ∈ σ(ω), with µ(a) > 0, contains atoms. if this were not so, then the restricted measure µ|a would be non-atomic in the restricted nikodym-saks’ space σ(a). applying the former lemma, we would find a sequence (an)∞n=1 ⊆ σ(a) satisfying the conditions there stated. it may be easily seen that the distance q∗(an,am) = 14µ(a), n 6= m. hence, such a sequence cannot admit any cauchy subsequence, which contradicts the fact that σ(ω) is compact. conversely, let (en)∞n=1 be an atomic partition of σ(ω). we show that (σ(ω),q∗) is homeomorphic to cantor’s space 2n. consider the one-to-one and onto mapping t : 2n −→ σ(ω) such that t(x) := ∪{en : xn = 1}, x ∈ 2n. given x ∈ 2n, ε > 0, let a = t(x). since µ(a) = ∑ {µ(en) : xn = 1}, µ(ω \a) = ∑ {µ(en) : xn = 0}, there are two finite subsets f,f ′ of n such that x(f) = 1, x(f ′) = 0, and∑ {µ(en) : n ∈ f} > µ(a) − ε 2 , ∑ {µ(en) : n ∈ f ′} > µ(ω \a) − ε 2 . it follows that the set v = {y ∈ 2n : y(f) = 1,y(f ′) = 0} is a neighborhood of x such that µ(a5t(y)) < ε, y ∈ v . finally, to see that t−1 is also continuous, it suffices to notice that, for each p ∈ n, the intervals σ+(ep) and σ(ω \ep) are closed subsets of σ(ω). � 4. nikodym-saks’ space as a topological group. we know that σ(ω) is an abelian nilpotent group and that the symmetric difference 5 is continuous, thus σ(ω) may be regarded also as a topological group. besides, for each a ∈ σ(ω), σ(ω) can be expressed as the topological direct sum of the closed subgroups σ(a) and σ(ω\a), i.e., σ(ω) = σ(a)⊕σ(ω\a). our aim in this section is to show that nikodym-saks’ space can be decomposed, in a unique way, as the topological direct sum of a connected subgroup nikodym-saks space 251 (the component of the zero element ∅) plus a compact totally disconnected subgroup. proposition 4.1. there exists a unique measurable set m such that σ(ω) = σ(m) ⊕ σ(ω \m), with σ(m) connected and σ(ω \m) compact. proof. let a denote the countable collection, possibly empty, of atoms of σ(ω). since the set m := ω \ (∪{e : e ∈ a}) contains no atoms, it follows after proposition 3.1 that the topological group σ(m) is connected. now, assuming m 6= ω (otherwise, σ(ω \ m) = {∅}, clearly compact), we have that the restricted measure µ|ω\m is purely atomic, proposition 3.4 then shows that σ(ω \m) is compact. notice also that in this case, since σ(ω \m) is a copy of a cantor space, it is totally disconnected. � proposition 4.2. the connected subgroup σ(m) before obtained coincides with the connected component of ∅. proof. let us denote by c the connected component of σ(ω) containing ∅. it is well known that c is a closed topological subgroup of σ(ω). hence, since ∅ ∈ σ(m) and σ(m) is connected, it follows that σ(m) ⊆ c. we show the reverse inclusion. let a ∈ c. assume a \ m 6= ∅. hence, after what we have seen before, σ(a \ m) is a totally disconnected subgroup. but, the mapping τ : c −→ σ(a \ m) such that τ(x) = x ∩ (a \ m) is continuous, and so τ(c) is connected in σ(a \ m). thus, τ(c) must be a singleton, but this is not so since ∅,a\m ∈ τ(c). � we finish by studying the properties of compactness and connectedness related with their local properties. corollary 4.3. σ(ω) is compact if and only if it is locally compact. proof. since σ(ω) is hausdorff, we need only show the sufficiency part. so, if σ(ω) is locally compact, we may find δ > 0 such that the closed ball v ∗ δ(∅) = {x ∈ σ(ω) : q∗(∅,x) ≤ δ} is compact. let, as before, m ∈ σ(ω) be such that σ(m) is the connected component of ∅. it all reduces to see that m = ∅. if not, since σ(m) is connected, we can find, after proposition 3.1, a ∈ σ(m) such that 0 < µ(a) ≤ δ, and so σ(a) ⊆ v ∗ δ(∅). hence σ(a) is compact and, after proposition 3.4, a must contain atoms. this would imply that, m containing a would also contain atoms, thus contradicting the definition of m. � corollary 4.4. if σ(ω) is connected, then it is locally connected. proof. let us assume that σ(ω) is connected. we first show that every ball centered at ∅ is connected. this is a simple consequence of the facts v ∗δ (∅) = ∪{σ(x) : x ∈ v ∗ δ (∅)}, ∅ ∈∩{σ(x) : x ∈ v ∗ δ (∅)}, 252 j. ferrer and that, for any x, σ(x) is connected. now, being in a topological group, since every ball centered at a is a translation of the form v ∗δ (a) = a 5v ∗ δ (∅), it follows that v ∗δ (a) is also connected. � as it was pointed out by professor paolo de lucia one can easily find examples of locally connected disconnected (but not totally disconnected) nikodymsaks spaces. example 4.5. consider the measure space (ω, σ,µ) given by (λ represents the lebesgue measure in [0, 1] and we denote by l the class of lebesgue-measurable subsets of [0, 1]): ω = [0, 1] ∪{2}, σ = l ∪{a∪{2} : a ∈ l}, µ|l = λ, µ(a∪{2}) = λ(a) + 1, a ∈ l. clearly, since {2} is a µ-atom, the corresponding nikodym-saks’ space σ(ω) is not connected. notice that it is neither totally disconnected, since σ([0, 1]) is the connected component of ∅. and, after recalling that σ(ω) = σ([0, 1]) ⊕ σ({2}), it can be easily shown that σ(ω) is locally connected. acknowledgements. the author wishes to thank the referee for the interesting suggestions and comments on the paper. also, it means a pleasure for this author to express his gratitude to virtu bertó for her patience and help in the preparation of the talk references [1] de guzmán; m., rubio, b.: integración: teoŕıa y técnicas, alhambra s. a., madrid (1979). [2] diestel, j.; uhl, j.j.: vector measures, math. surveys 15, amer. math. soc., providence (1977). [3] diestel, j.: sequences and series in banach spaces, springer-verlag, new york (1984). [4] drewnowski, l.: topological rings, continuous set functions, integration, bull. acad. polon. scien. 20 , 269-276 (1972). 1981. [5] dunford, n.; schwartz, j.t.: linear operators (i), wiley, new york (1976). [6] fletcher, p.; lindgren, w.: quasi-uniform spaces, marcel dekker, (1982). [7] halmos, p.r.: measure theory, springer-verlag, new york (1974). [8] munroe, m.e.: measure and integration, addison-wesley, london (1968). [9] oxtoby, j.c.: measure and category, springer-verlag, new york (1980). [10] reilly, i.l.; subrahmanyam, p.v.; vamanamurthy, m.k.: cauchy sequences in quasipseudometric spaces, mh. math. 93, 127-140 (1982). [11] williamson, j.h.: integración lebesgue, tecnos , madrid (1973). received november 2001 revised november 2002 nikodym-saks space 253 jesús ferrer departamento de análisis matemático, universidad de valencia, dr. moliner 50, 46100 burjasot (valencia), spain e-mail address : jesus.ferrer@uv.es @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 211–216 unusual and bijectively related manifolds john g. hocking dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. a manifold is “unusual” if it admits of a continuous self-bijection which is not a homeomorphism. the present paper is a survey of work published over yearsaugmented with recent examples and results. 2000 ams classification: 57a05. keywords: continuous bijection, 2-manifold. 1. dissertation. signore e signori, ladies and gentlemen! i am deeply grateful for the opportunity to attend this conference. it is indeed a wonderful occasion, isn’t it? the honor to professor naimpally is richly deserved, of course. he was once my student, i am very proud to say, and i have basked in the glow of his brilliance for well over thirty five years. in a small way this conference to honor him also honors me and i am happy to accept the reflected glory! my topic today is far from the theme of this conference but i must say that professor naimpally approved of this digression from the main flow. i shall speak about a small twig on a branch of geometric topology which has absorbed me for many years. i trust you will not be disappointed. for me, it all began in an undergraduate topology course years and years ago. i had just gotten to the point of defining a ”homeomorphism” as a continuous bijection with a continuous inverse. an eager young lady in the front row interrupted me. ”why do you have to assume the continuity of the inverse? couldn’t that be proved?” of course, i had the standard example to give her: f : [0, 1) → s1 212 john g. hocking given by f(t) = (cos 2πt, sin 2πt). this is surely a continuous bijection with a discontinuous inverse. (i almost said ”so there!” in triumph.) the lady was not really happy. ”wasn’t that kind of cheat? the two spaces are different. sure, your example shows that you must assume continuity of f−1 in the general case. but what if the map is from a space to itself?” it took a moment’s thought to come up with the following example: let x = {(n,y) : n ∈ z, 0 ≤ y < 1}. then a bijection f defined by f(n,y) =   (n,y) if n < 0 (n− 1,y) if n > 1 (0, 1 2 y) if n = 0 (0, 1 2 + 1 2 y) if n = 1 if you do not ”see” this, it simply shrinks two of the ”poles”, picks one up and stacks it on the other. well, that kept the lady quiet and i got on with my lecture. she still had another objection, however. she caught up with me in the hall after class and asked for a connected example. i had it ready for her the next day. you may picture this way: at each negative integer point on the real line erect a smal circle tangent to the line. at each of the other integer points stand a unit interval missing its upper endpoint. the bijection simply applies the first map i gave to the first such interval. if you insist on being a purist, you could then shift left one unit. that finally satisfied the student but it left me feeling a little uneasy. my interest in topology has always been in low-dimensional manifolds. by ”low” i mean just two or three dimensions. anything beyond that is too difficult for me! to make this talk self-contained i define an n-dimensional manifold to be a connected metric space m each point of which has a neighborhood homeomorphic to either irn or irn−1 × [0, 1). points of the first kind are interior points and make up the interior of m, denoted by intm. the second kind are boundary points and constitute ∂m, the boundary of m. while there are only four distinct 1-manifolds, (0, 1), [0, 1), [0, 1], and s1, the 2-manifolds are much more numerous! many subsets of ir2, including ir2 itself, satisfy the definition. the countable infinity of closed orientable surfaces starting with the sphere and going on to the torus and the other spheres with ”handles”. the corresponding non-orientable surfaces beginning with the projective plane and the klein bottle constitute the totality of compact boundary-free 2-manifolds. all of these fall into the category i call ”usual” manifolds, as you shall soon see. unusual and bijectively . . . 213 a minute ago i said i was uneasy about my responses to that nice young lady in my class. i knew she had been recalling the fact that a continuous bijection of the unit interval to itself necessarily has a continuous inverse. i asked myself ”are there manifolds which admit of a continuous bijection with a discontinuous inverse?” of course, such would have to be non-compact and, of course, i found one quite easily. think of it as an infinite 2-dimensional tube centered along the x-axis. on it to the left erect an infinity of handles and to the rigth an infinity of pairs of holes with boundaries and chimneys with their top edges missing. the desired bijection simply bends on of the chimneys over until its ”missing” upper edge coincides with the boundary of a hole. draw a picture if you must. but, notice that this is simply my connected example inflated! as time passed, my colleague, p.h. doyle, and i developed the topic into several papers. i want to mention some of our results and some of our failures as well (it seems to me that we ought to tell of our errors as often as of our successes. the mathematical community would surely profit from such knowledge!). first however, i must say that rajagopalan and wilansky, writing in the journal of the australian mathematical society in 1966, had introduced the term ”non-reversible” to describe a space which admits a continuous bijection which is not a homeomorphism. in our informal talks, however, doyle and i always spoke of ”unusual” spaces and so i now make the following definition 1.1. a (hausdorff) space x is usual if every continuous selfbijection on x is necessarily a homeomorphism: otherwise, x is unusual. incidentally, to save a lot of writing and speaking time, from here on a continuous bijection will be simply a bi-map. theorem 1.2. for every n ≥ 2, there are unusual n-manifolds. proof. look at the example m above and consider m × irk, k ∈ n � any compact manifold is surely usual and, in view of the brouwer invariance of domain theorem, so is every open manifold. you might say that these facts led us to adopt the term ”usual” in the first place. i have to say that with all of the brilliance of pat doyle and my own plodding persistence we were never able to characterize either the usual or the unusual manifolds. a first result in that direction was theorem 1.3. if ∂m is compact, then m is usual. let me describe another 2-manifold for your consideration. in ir2 start with the strip bounded between the x-axis and the line y = 3. at all points (−2n, 2) remove an open square of side one centered at (−2n, 2) with sides parallel to the axes. then remove from the right open rectangles {(x,y) such that 2k < x < 2k + 1, 1 < y ≤ 3}. this leaves a series of ”towers” to the 214 john g. hocking right. finally, remove the top edge of each of these towers. draw this out to see exactly what is entailed. the bijection is rather obvious, just bend a tower over and let its open end join the edge of a tower to its left. you will notice that there are both compact and non-compact boundary components, infinitely many of each type. do not make false conjectures based on this example, as we did at first! by simply taking the product of this manifold with the unit interval we get an unusual manifold with a connected boundary! here is another example to consider: now i ask you to look at ir3 and begin with the set of points {(x,y,z) such that x2 + z2 = 1}. from this tube, then, remove open square ”patches” centered at the points (2 + 4k, 1 2 , 0). from the edge x = −3 of the first hole to the left erect a ”bridge” over to the right edge x = −5 of the hole to the left. continue this bridge building on to infinity. finally, to the right at the right edges of the holes erect upright ”panels” missing their top edges. the bi-map simply bends one of these panels over to meet the edge of the next hole along. again, you might want to draw the picture to see just how easy this is. in this example we see that, in the presence of non-compact boundary components, a compact component can be only partially ”swallowed” by the action of a bi-map. we wasted a few weeks in a futile search before we came up with this one! let me get on to the second of my topics with the following definition 1.4. two (hausdorff) spaces, x and y are bijectively related if there are bi-maps f : x → y and g : y → x. we shall use the symbol fam(x) to denote the equivalence class of all spaces bijectively related to x. for any usual space fam(x) = {x} but this condition in no way characterizes the usual spaces. i will leave to you to prove that the rationals q, with the usual topology, have the property fam(q) = {q}. as you undoubtely expect, there are infinitely many manifolds with nonhomeomorphic bijective relatives. i describe next one of the relatives of the second 2-manifold i described earlier. from the edge of the first ”hole” to the left remove a closed line segment. call the manifold so created n. it is easy to see the bi-map from n to m. for the other direction, simply strech the second tower over the first and identify its missing edge with a part of the boundary beyond the first tower. again, drawing a picture will help considerably in ”seeing” this bi-map. if one draws a simple closed curve j around the hole in n with a closed segment taken away from its edge, one readily proves that m and n are not homeomorphic. also, it is trivial to see that the class fam(m) is infinite. we were never able to answer the following question: if fam(x) is not equal to {x}, is it necessarily infinite? unusual and bijectively . . . 215 it is interesting to note that for the two manifolds m and n above, the products m × [0, 1) and n × [0, 1) are homeomorphic. this led us to another problem we could not solve. assuming that m and n are non-homeomorphic bijectively related manifolds, when does there exists a manifold p such that m ×p and n ×p are homeomorphic? can p be compact? now suppose that f : m → n and g : n → m make m and n bijectively related. if the composite map g ◦f : m → m is itself a homeomorphism, so is f itself. in the example above, we know that f is not a homeomorphism and therefore m is unusual. this give us theorem 1.5. if fam(m) is not equal to {m}, then m is unusual. here are a few more results to give you a flavor of the subject: theorem 1.6. if m and n are bijectively related, then each imbeds in the interior of the other. theorem 1.7. if m is orientable, so is each of its bijective relatives. as an addendum to theorem 1.7 it is interesting to note that there are bimaps from orientable manifolds to non-orientable ones but not conversely. i leave you to find an example of a bi-map from the second manifold i described to a non-orientable one. (this is very easy!) theorem 1.8. if f : m → n is a bi-map and if f(∂m) = ∂n, then f is a homeomorphism. theorem 1.9. if the 2-manifold m has infinitely many compact boundary components, infinitely many handles and infinitely many annular ends, then m is unusual. theorem 1.9 should remind you of my very first example, the genesis of this entire project. now, let me leave you with problem: show that for that first manifold m, fam(m) = {m}. and here is another puzzle for you to chew on: consider the 3-manifold consisting of open lower halph-space in ir3 together with the open annular disks {(x,y, 0) such that 1 4 ≤ (x− 2n)2 + y2 ≤ 1}. prove that this is an unusual manifold. if you need a little help with this, consult the k. whyburn paper listed in the bibliography. now i thank you all again and wish you the very best for the future of which you form such an important part! references [1] p.h. doyle and j.g. hocking, a decomposition theorem for n-dimensional manifolds, proc. amer. math. soc. 13 (1962), 469–471. [2] p.h. doyle and j.g. hocking, continuous bijections on manifolds, j. austral. math. soc. 22 (1976), 257–263. 216 john g. hocking [3] p.h. doyle and j.g. hocking, strongly reversible manifolds, j. austral. math. soc. series a (1983), 172–176. [4] p.h. doyle and j.g. hocking, bijectively related spaces i: manifolds, pac. j. math. 3 no.1 (1984), 23–31. [5] j. eichorn, die kompactifizierung offener mannigfaltigjeiten zu geschossenen i, math. nachr. 85 (1978), 5–30. [6] k. kuratowski, topology vol 2, academic press, (1968). [7] d.h. petty, one-to-one mappings into the plane, fund. math. 67 (1970), 209–218. [8] m. rajagopalan and a. wilansky, reversible topological spaces, j. austral. math. soc. 6 (1966), 129–138. [9] k. whyburn, a non-topological 1 − 1 mapping onto e3, bull. amer. math. soc. 71 (1965), 523–537. received january 2002 revised september 2002 john g. hocking department of mathematics, michigan state university, east lansing, michigan, 48824, usa. e-mail address : hocking@pilot.msu.edu valeroagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 2, 2005 pp. 229-240 on banach fixed point theorems for partial metric spaces oscar valero ∗ abstract. in this paper we prove several generalizations of the banach fixed point theorem for partial metric spaces (in the sense of o’neill) given in [14], obtaining as a particular case of our results the banach fixed point theorem of matthews ([12]), and some well-known classical fixed point theorems when the partial metric is, in fact, a metric. 2000 ams classification: 54h25, 54e50, 54e99, 68q55. keywords: dualistic partial metric, partial metric, complete, quasi-metric, fixed point. 1. introduction and preliminaries in recent years many works on domain theory have been made in order to equip semantics domain with a notion of distance. in particular, matthews ([12]) introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks, showing that the banach contraction mapping theorem can be generalised to the partial metric context for applications in program verification. the existence of several connections between partial metrics and topological aspects of domain theory have been lately pointed by other authors as o’neill ([15]), bukatin and scott ([3], [4]), waszkiewicz ([23], [24]), schellekens ([20], [21]), escardo ([8]), matthews ([12]) and romaguera and schellekens ([19], [18]). throughout this paper the letters r, r+, n will denote the set of real numbers, of nonnegative real numbers and natural numbers, respectively. le us recall that a partial metric on a (nonempty) set x is a function p : x × x → r+ such that for all x,y,z ∈ x : (i) x = y ⇔ p(x,x) = p(x,y) = p(y,y); ∗the author acknowledges the support of the spanish ministry of science and technology, plan nacional i+d+i, and feder, grant bmf2003-02302. 230 o. valero (ii) p(x,x) ≤ p(x,y); (iii) p(x,y) = p(y,x); (iv) p(x,z) ≤ p(x,y) + p(y,z) − p(y,y). a partial metric space is a pair (x,p) such that x is a nonempty set and p is a partial metric on x. as we indicated above, o’neill ([15]) studied several connections between valuations and partial metrics so that he proposed one significant change to matthews’ definition of the partial metrics, such a change consists of extending their range from r+ to r. according to [14], the partial metrics in the o’neill sense will be called dualistic partial metrics and a pair (x,p) such that x is a (nonempty) set and p is a dualistic partial metric on x will be called a dualistic partial metric space. a paradigmatic example of a dualistic partial metric space is the pair (r,p), where p(x,y) = x ∨ y for all x,y ∈ r. each dualistic partial metric p on x generates a t0 topology t (p) on x which has as a base the family of open p-balls {bp(x,ε) : x ∈ x,ε > 0}, where bp(x,ε) = {y ∈ x : p(x,y) < p(x,x) + ε} for all x ∈ x and ε > 0. from this fact it immediately follows that a sequence (xn)n∈n in a dualistic partial metric space (x,p) converges to a point x ∈ x if and only if p(x,x) = limn→∞p(x,xn). following [15] (compare [12]), a sequence (xn)n∈n in a dualistic partial metric space (x,p) is called a cauchy sequence if there exists limn,m→∞p(xn,xm). a dualistic partial metric space (x,p) is said to be complete if every cauchy sequence (xn)n∈n in x converges, with respect to t (p), to a point x ∈ x such that p(x,x) = limn,m→∞p(xn,xm). as usual a function f : r → r is monotone non-decreasing (or monotone or increasing) if x ≥ y implies f(x) ≥ f(y). recently, an extension of the banach contraction mapping theorem have been proved in the dualistic partial metric context (see [14]) in such a way that the matthews contraction mapping theorem can be deduced as a special case of such a result. in this paper we extend the notion of contraction between partial metric spaces to the dualistic partial metric context, and present two generalizations and one “local version” of the banach fixed point theorem given in ([14]), which recuperate the several well-known classical fixed point theorems when the dualistic partial metric converts into a metric. 2. contractions on dualistic partial metric spaces for the following discussion we recall some basics correspondences between dualistic partial metrics and quasi-metric spaces. our basic references for quasi-metric spaces are [9] and [11]. in our context by a quasi-metric on a set x we mean a nonnegative realvalued function d on x × x such that for all x,y,z ∈ x : (i) d(x,y) = d(y,x) = 0 ⇔ x = y (ii) d(x,y) ≤ d(x,z) + d(z,y). fixed point theorems for partial metric spaces 231 a quasi-metric space is a pair (x,d) such that x is a (nonempty) set and d is a quasi-metric on x. each quasi-metric d on x generates a t0-topology t (d) on x which has as a base the family of open d-balls {bd(x,ε) : x ∈ x, ε > 0}, where bd(x,ε) = {y ∈ x : d(x,y) < ε} for all x ∈ x and ε > 0. if d is a quasi-metric on x, then the function ds defined on x × x by ds(x,y) = max{d(x,y),d(y,x)}, is a metric on x. next we prove a local version of the banach fixed point theorem for dualistic partial metric spaces (theorem 2.3 below) obtaining as a particular case of our result the classical local version of banach’s fixed point theorem for metric spaces (see [1] and [7]). the proof of the following auxiliary results can be found in [14] (compare [12], [15] and [13]). lemma 2.1. if (x,p) is a dualistic partial metric space, then the function dp : x × x → r + defined by dp(x,y) = p(x,y) − p(x,x) is a quasi-metric on x such that t (p) = t (dp). as a consequence of lemma 2.1 a mapping between dualistic partial metric spaces (x,p) and (y,q) is continuous if it is continuous between the associated quasi-metic spaces. lemma 2.2. let (x,p) be a dualistic partial metric space. then the following assertions are equivalent: (1) (x,p) is complete (2) the induced metric space (x,(dp) s) is complete. furthermore limn→∞(dp) s(a,xn) = 0 if and only if p(a,a) = limn→∞p(a,xn) = limn,m→∞p(xn,xm). in [14] it was proved the following banach fixed point theorem which generalies the fixed point theorem for partial metrics of matthews given in [12]. theorem 2.3 (banach fixed point theorem). let f be a mapping from a complete dualistic partial metric space (x,p) into itself such that there is a real number c with 0 ≤ c < 1, satisfying (*) |p(f(x),f(y))| ≤ c |p(x,y)| , for all x,y ∈ x. then f has a unique fixed point. corollary 2.4 (matthews). let f be a mapping from a complete partial metric space (x,p) into itself such that there is a real number c with 0 ≤ c < 1, satisfying p(f(x),f(y)) ≤ cp(x,y), for all x,y ∈ x. then f has a unique fixed point. on the other hand, it was showed in [14] that the contractive condition in the statement of theorem 2.3 can not be replaced by the contractive condition of corollary 2.4. 232 o. valero in the sequel a mapping defined from a dualistic partial metric space into itself satisfying the condition (*) will be called a contraction with contraction constant c. the following well-known local version of the banach fixed point theorem (proposition 2.5 below) is a useful result with many applications. in particular it allows to show that the property of having a fixed point is invariant by homotopy for contractions and nonexpansive maps in metric spaces and, as consequence, it is applied to solve practice second order homogeneous dirichlet problems ([1], [7]). next we extend such this result to the dualistic partial metric case. proposition 2.5. let (x,d) be a complete metric space and let x0 ∈ x and r > 0. suppose that f : bd(x0,r) → x is a contraction with contraction constant l such that d(f(x0),x0) < (1−l)r. then f has a unique fixed point in bd(x0,r). let (x,d) be a quasi-metric space. from now on by y d we denote the clousure of y ⊆ x with respect to t (d). remark 2.6. observe that, in general, the set bp(0,ε) = {y ∈ x : p(x,y) ≤ ε + p(x,x)} is not closed with respect to t (p). indeed, consider the dualistic partial metric space (r,p) where the dualistic partial metric p is defined by p(x,y) = x∨y.it is clear that bp(0,1) = (−∞,1] and r\bp(0,1) = (1,+∞).so bp(0,1) is not a closed set in (r,p). proposition 2.7. let (x,p) be a complete dualistic partial metric space, x0 ∈ x and r > 0. suppose that f : bp(x0,r) → x is a contraction with contraction constant c such that |p(f(x0),x0)| < (1 − c)r − 2|p(x0,x0)| − |p(f(x0),f(x0))|. for all x,y ∈ bp(x0,r). then f has a unique fixed point in bp(x0,r). proof. it is clear that there exists r0 with 0 ≤ r0 < r such that |p(f(x0),x0)| ≤ (1 − c)r0 − 2|p(x0,x0)| − |p(f(x0),f(x0))|. next we show that bp(x0,r0) (dp) s = bp(x0,r0). to see this let x ∈ bp(x0,r0) (dp) s and (xn)n∈n ⊂ bp(x0,r0) such that limn→∞(dp) s(x,xn) = 0. then, given ε > 0, there exists n0 ∈ n with p(x0,xn)−p(x0,x0) ≤ r0 and p(xn,x)−p(xn,xn) < ε whenever n ≥ n0.now we only have to show that p(x,x0) − p(x0,x0) ≤ r0. indeed, p(x,x0) − p(x0,x0) ≤ p(x,xn) + p(xn,x0) − p(xn,xn) − p(x0,x0) < ε + r0. consequently p(x,x0)−p(x0,x0) ≤ r0 and bp(x0,r0) (dp) s ⊆ bp(x0,r0). whence bp(x0,r0) (dp) s = bp(x0,r0). fixed point theorems for partial metric spaces 233 we will show that actually f : bp(x0,r0) → bp(x0,r0). to this end note that |p(f(x),f(x0))| − |p(x0,x0)| ≤ |p(f(x),f(x0))| − c|p(x0,x0)| ≤ c · (|p(x,x0)| − |p(x0,x0)|) ≤ c · (p(x,x0) − p(x0,x0)) ≤ cr0 for all x ∈ bp(x0,r0). hence p(f(x),x0) − p(x0,x0) ≤ dp(x0,f(x0)) + dp(f(x0),f(x)) ≤ |p(f(x),f(x0))| + |p(f(x0),x0)| + |p(f(x0),f(x0))| + |p(x0,x0)| ≤ cr0 + (1 − c)r0 = r0, for all x ∈ bp(x0,r0). therefore f(bp(x0,r0)) ⊆ bp(x0,r0). now we can apply theorem 2.3 to deduce that f has a fixed point in bp(x0,r0) ⊂ bp(x0,r). finally we want to show the uniqueness. suppose that there exist x,y ∈ bp(x0,r) such that f(x) = x, f(y) = y and x 6= y. since f is a contraction in bp(x0,r) we have that p(x,y) = p(x,x) = p(y,y) = 0 because of |p(a,b)| = |p(f(a),f(b))| ≤ c|p(a,b)| for all a,b ∈ {x,y}, which implies that x = y.this concludes the proof. � clearly proposition 2.5 is a particular case of proposition 2.7 because of |p(x0,x0)| = |p(f(x0),f(x0))| = 0 when the dualistic partial metric is, in fact, a metric. 3. generalized banach’s fixed point theorems for complete dualistic partial metric spaces in the last years many authors have obtained generalizations of the banach fixed point theorem in complete metric spaces ([2], [5], [6], [10], [16], [17]) recently, and motivated in part by the applications to computer science, some generalizations of a banach fixed point theorem have been obtained for quasimetric and partial metric spaces (see for instance [12], [14], [22]). in this section our interest is focused on giving two banach fixed point type theorems where the contractive condition is weakened. one way of extending the banach theorem arises in a natural way from approximation problems, where the contractive condition depends on a distinguished real function. proposition 3.1 ([7]). let (x,d) be a complete metric space and let d(f(x),f(y)) ≤ φ(d(x,y)) for all x,y ∈ x, where φ : [0,∞) → [0,∞) is any monotone non-decresing function with limn→∞φ n(t) = 0 for any fixed t > 0. then f has a unique fixed point. the following result generalizes the preceding one. 234 o. valero theorem 3.2. let f be a mapping of a complete dualistic partial metric space (x,p) into itself such that |p(f(x),f(y))| ≤ φ(|p(x,y)|) for all x,y ∈ x, where φ : [0,+∞) → [0,+∞) is any monotone non-decreasing function with limn→∞φ n(t) = 0 for each fixed t > 0. then f has a unique fixed point. proof. first note that if t > 0 then φ(t) < t, because if t ≤ φ(t) then φ(t) ≤ φ(φ(t)) and therefore t ≤ φ2(t). thus, it is easy to prove, by induction, that t ≤ φn(t) for n ≥ 1. it follows that t ≤ limn→∞φ n(t) = 0, a contradiction. fix x ∈ x. it is clear that for each n ∈ n we have |p(fn(x),fn(x))| ≤ φn(|p(x,x)|) and ∣ ∣p(fn(x),fn+1(x)) ∣ ∣ ≤ φn(|p(x,f(x))|). let α := max{|p(x,x)|, |p(x,f(x))|}. since, by lemma 2.1, dp(f n(x),fn+1(x)) = p(fn(x),fn+1(x)) − p(fn(x),fn(x)), we deduce that dp(f n(x),fn+1(x)) ≤ |p(fn(x),fn+1(x))| + |p(fn(x),fn(x))| ≤ φn(|p(x,f(x))|) + φn(|p(x,x)|) ≤ 2φn(α + 1). now let n,k ∈ n. then dp(f n(x),fn+k(x)) ≤ dp(f n(x),fn+1(x)) + ... + dp(f n+k−1(x),fn+k(x)) ≤ φn(|p(x,f(x))|) + φn(|p(x,x)|) + ... ... + φn+k−1(|p(x,f(x))|) + φn+k−1(|p(x,x)|). thus dp(f n(x),fn+k(x)) ≤ 2φn(α + 1) + ... + 2φn+k−1(α + 1). similarly we show that dp(f n+1(x),fn(x)) ≤ 2φn(α + 1), and thus dp(f n+k(x),fn(x)) ≤ 2φn+k−1(α + 1) + ... + 2φn(α + 1). therefore (fn(x))n∈n is a cauchy sequence in the metric space (x,(dp) s), which is complete by lemma 2.2. so there is a ∈ x such that limn→∞(dp) s(a,fn(x)) = 0 and p(a,a) = limn→∞p(a,f n(x)) = limn,m→∞p(f n(x),fm(x)). we want to show that a is the unique fixed point of f. first we claim that a is a fixed point of f. clearly we have that limn,m→∞dp(f n(x),fm(x)) = 0. moreover limn→∞p(f n(x),fn(x)) = 0, since |p(fn(x),fn(x))| ≤ φn(|p(x,x)|) ≤ φn(|p(x,x)| + 1) fixed point theorems for partial metric spaces 235 and limn→∞φ n(|p(x,x)|+1) = 0. thus we deduce that limn,m→∞p(f n(x),fm(x)) = 0. consequently by lemma 2.2, p(a,a) = 0. whence |p(f(a),f(a))| ≤ φ(|p(a,a)|) = φ(0) ≤ φ(ε) < ε. it follows that |p(f(a),f(a))| = 0, because the last inequality holds for all ε > 0. on the other hand, since limn→∞p(a,f n(x)) = p(a,a) = 0 we have |p(fn+1(x),f(a))| ≤ φ(|p(fn(a),a)|) ≤ φ(ε) < ε eventually. thus lemma 2.2 shows that f(a) is a limit point of (fn(x))n∈n in (x,(dp) s). then a = f(a). finally we show that a is the unique fixed point of f. to this end let b ∈ x such that f(b) = b and b 6= a. then p(a,a) = p(b,b) = p(f(a),f(b)) = 0, since otherwise we have that the following inequality holds |p(x,y)| = |p(f(x),f(y))| ≤ φ(|p(x,y)|) < |p(x,y)| for any x,y ∈ {a,b}. the proof is concluded. � remark 3.3. observe that theorem 2.3 follows as a special case of the preceding result if we choose φ(t) = ct with 0 ≤ c < 1. we will say that a dualistic partial metric space (x,p) is bounded if there exists k > 0 such that |p(x,y)| < k for every x,y ∈ x. moreover, we define the diameter of a subset y ⊆ x as δp(y ) = sup{|p(x,y)| : x,y ∈ y } if the supremum exists and δp(y ) = +∞ otherwise. note that these notions coincide with the classical notions of bounded metric space and diameter of a metric space, respectively. furthermore δp(y ) ≤ δ(dp)s(y ) + δwp(y ), where δwp(y ) = sup{|p(x,x)| : x ∈ y } if it exists and δwp(y ) = +∞ otherwise. another way of attempting an extension of the banach fixed point theorem does not rely on measuring the difference between p(f(x),f(y)) and p(x,y), but, similarly to the metric case, the required condition relies on the behaviour of the induced quasi-metric dp. the below technical results are useful to prove the desired theorem. lemma 3.4. let (x,p) be a dualistic partial metric space and y ⊆ x. then δ(dp)s(y ) ≤ 4δp(y ). proof. it is easily seen that δwp(y ) ≤ δp(y ). furthermore (dp) s(x,y) ≤ dp(x,y) + dp(y,x) = 2p(x,y) − p(x,x) − p(y,y), whence we have (dp) s(x,y) ≤ 2|p(x,y)| + |p(x,x)| + |p(y,y)|. immediately we deduce that δ(dp)s(y ) ≤ 4δp(y ). � lemma 3.5. let (x,p) be a complete dualistic partial metric space and let ϕ : x → r+ be an arbitrary non-negative function. assume that inf{ϕ(x) + ϕ(y) : |p(x,y)| + |p(x,x)| + |p(y,y)| ≥ a} = µ(a) > 0 for all a > 0. 236 o. valero then each sequence (xn)n∈n for which limn→∞ϕ(xn) = 0 converges with repect to t ((dp) s) to the same point of x. proof. let an = {x ∈ x : ϕ(x) ≤ ϕ(xn)}. clearly an is a non-empty set and any finite family of such subsets has a non-empty intersection. we show that limn→∞δ(dp)s(an) = 0. to this end, let ε > 0. then there exists n0 ∈ n such that ϕ(xn) < 1 2 µ(ε) for all n ≥ n0. thus, ϕ(x)+ϕ(y) < µ(ε) whenever x,y ∈ an and n ≥ n0. hence, by hypothesis |p(x,y)| < ε. therefore δp(an) < ε for all n ≥ n0. consequently limn→∞δp(an) = 0. by lemma 3.4 we obtain that limn→∞δ(dp)s(an) = 0, which gives limn→∞δ(dp)s(an (dp) s ) = 0 because of δ(dp)s(an (dp) s ) = δ(dp)s(an). we conclude from cantor’s intersection theorem that there exists a unique x ∈ ∩+∞n=1an (dp) s . furthermore, limn→∞(dp) s(x,xn) = 0, since xn ∈ an (dp) s for each n ∈ n. whence we follow that p(x,x) = limn→∞ p(x,xn) = limn→∞ p(xn,xn). note that we have actually proved that for any sequence (xn)n∈n such that limn→∞ϕ(xn) = 0 there exists a limit point x ∈ x with respect to t ((dp) s). let (yn)n∈n be any other sequence such that limn→∞ϕ(yn) = 0 and let y ∈ x its limit point. since limn→∞ϕ(xn) = limn→∞ϕ(yn) = 0 we get that, given ε > 0, there exists n1 ∈ n such that ϕ(xn) < 1 2 µ(ε), ϕ(yn) < 1 2 µ(ε) whenever n ≥ n1. in consequence |p(xn,yn)|+|p(xn,xn)|+|p(yn,yn)| < ε whenever n ≥ n1, because otherwise µ(ε) = inf{ϕ(x) + ϕ(y) : |p(x,y)| + |p(x,x)| + |p(y,y)| ≥ ε} ≤ ϕ(xn) + ϕ(yn) < µ(ε), which is a contradiction. we show that x = y. indeed, (dp) s(x,y) ≤ (dp) s(x,xn) + (dp) s(xn,yn) + (dp) s(yn,y) ≤ 2ε + (dp) s(xn,yn) ≤ 2ε + 2|p(xn,yn)| + |p(xn,xn)| + |p(yn,yn)| < 4ε. so x = y, since the preceding inequality is true for every ε > 0. this completes the proof. � theorem 3.6. let (x,p) be a complete dualistic partial metric space (x,p) and let f : x → x be a continuous mapping from (x,(dp) s) to (x,(dp) s) such that the functions ϕ(x) = dp(x,f(x)) and ψ(x) = dp(f(x),x) satisfy the following conditions: (1) inf{ϕ(x) + ϕ(y) + ψ(x) + ψ(y) : |p(x,y)| + |p(x,x)| + |p(y,y)| ≥ a} = µ(a) > 0 for all a > 0 (2) infx∈x(ϕ(x) + ψ(x)) = 0. then f has a unique fixed point. proof. first we construct a sequence (xn)n∈n ⊂ x such that limn→∞ϕ(xn) + ψ(xn) = 0. let ε > 0 be given then there exists xε ∈ x such that ϕ(xε) + fixed point theorems for partial metric spaces 237 ψ(xε) < ε. put xn = x1/n for each n ∈ n. thus the sequence (xn)n∈n satisfies the following: for every ε ≥ 1, ϕ(xn) + ψ(xn) < 1 ≤ ε whenever n > 2. for every ε < 1, there always exists n0 ∈ n such that ϕ(xn)+ψ(xn) < 1 n0 < ε for all n ≥ n0. it follows that limn→∞ϕ(xn) + ψ(xn) = 0 as we claim. since limn→∞ϕ(xn)+ψ(xn) = 0, by lemma 3.5 there exists a unique x ∈ x such that limn→∞(dp) s(x,xn) = 0. then ϕ(x) − ϕ(xn) = p(x,f(x)) − p(x,x) − p(xn,f(xn)) + p(xn,xn) < ε + p(x,f(x)) − p(xn,f(xn)) < 2ε + p(xn,f(x)) − p(xn,f(xn)) ≤ 2ε + p(f(xn),f(x)) − p(f(xn),f(xn)) < 3ε. similarly is showed that ϕ(xn) − ϕ(x) < 3ε eventually. hence, we conclude that ϕ(x) = limn→∞ϕ(xn). on the other hand, limn→∞ϕ(xn) ≤ limn→∞(ϕ(xn) + ψ(xn)) = 0. thus we have ϕ(x) = 0 and p(x,f(x)) = p(x,x). furthermore, ψ(x) − ψ(xn) = p(x,f(x)) − p(f(x),f(x)) − p(xn,f(xn)) + p(f(xn),f(xn)) < ε + p(x,f(x)) − p(xn,f(xn)) < 2ε + p(xn,f(x)) − p(xn,f(xn)) ≤ 2ε + p(f(xn),f(x)) − p(f(xn),f(xn)) < 3ε. again we similarly show that ψ(xn) − ψ(x) < 3ε eventually. so ψ(x) = limn→∞ψ(xn) ≤ limn→∞(ϕ(xn) + ψ(xn)) = 0. whence we deduce that p(x,f(x)) = p(f(x),f(x)) = p(x,x). by condition (i) of the definition of a dualistic partial metric x = f(x), and f has a fixed point. next we show the uniqueness. to this end we consider y ∈ x such that y 6= x and f(y) = y. then we would have α := |p(x,y)|+|p(x,x)|+|p(y,y)| 6= 0, because otherwise p(x,y) = p(x,x) = p(y,y) = 0 and x = y. it follows that 0 < µ(α) = inf{ϕ(x) + ϕ(y) + ψ(x) + ψ(y) : |p(x,y)| + |p(x,x)| + |p(y,y)| ≥ α} ≤ ϕ(x) + ϕ(y) + ψ(x) + ψ(y) = 0, a contradiction. so that x = y. � as a consequence we obtain the following classical result which can be found in [7]. corollary 3.7. let f be a mapping from a complete metric space (x,d) into itself and ϕ : x → r+ the non-negative function defined by ϕ(x) = d(x,f(x)). assume that inf{ϕ(x) + ϕ(y) : d(x,y) ≥ a} = µ(a) > 0 for all a > 0 238 o. valero and that infx∈xd(x,f(x)) = 0. then f has a unique fixed point. remark 3.8. observe that theorem 2.3 follows from theorem 3.6. it is easy to see that p(x,y) − p(f(x),f(y)) ≤ dp(f(x),x) + dp(f(y),y) = ψ(x) + ψ(y), and p(f(x),f(y)) − p(x,y) ≤ dp(x,f(x)) + dp(y,f(y)) = ϕ(x) + ϕ(y). thus |p(x,y) − p(f(x),f(y))| ≤ ϕ(x) + ϕ(y) + ψ(x) + ψ(y). moreover |p(x,x)| − |p(f(x),f(x))| ≤ |p(x,x) − p(f(x),f(x))| ≤ dp(x,f(x)) + dp(f(x),x) = ϕ(x) + ψ(x) and |p(y,y)| − |p(f(y),f(y))| ≤ |p(y,y) − p(f(y),f(y))| ≤ dp(y,f(y)) + dp(f(y),y) = ϕ(y) + ψ(y). now if |p(f(x),f(y))| ≤ c|p(x,y)| for any 0 ≤ c < 1, bearing in mind the preceding inequalities, we follow that (1 − c)(|p(x,y)| + |p(x,x)| + |p(y,y)|) ≤ 2(ϕ(x) + ϕ(y) + ψ(x) + ψ(y)). therefore the condition (1) of theorem 3.6 is satisfied. finally from the contractive condition we have, for each x ∈ x, that limn→∞(dp) s(fn(x),fn+1(x)) = 0 (see proof of theorem 2.3 in [14]). consequently infx∈x(ϕ(x) + ψ(x)) = 0 because infx∈x(ϕ(x) + ψ(x)) ≤ ϕ(f n(x)) + ψ(fn(x)) = dp(f n(x),fn+1(x)) + dp(f n+1(x),fn(x)) ≤ 2(dp) 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[24] p. waszkierwicz, the local triangle axiom in topology and domain theory, appl. gen. topology 4 (2003), 47-70. 240 o. valero received january 2005 accepted april 2005 oscar valero (o.valero@uib.es) departamento de ciencias matemáticas e informática, universidad de las islas baleares, 07122 palma de mallorca, spain protasov.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 191-198 resolvability of ball structures i. v. protasov abstract. a ball structure is a triple b = (x, p, b) where x, p are nonempty sets and, for any x ∈ x, α ∈ p , b(x, α) is a subset of x, which is called a ball of radius α around x. it is supposed that x ∈ b(x, α) for any x ∈ x, α ∈ p . a subset y ⊆ x is called large if x = b(y, α) for some α ∈ p where b(y, α) = ⋃ y∈y b(y, α). the set x is called a support of b, p is called a set of radiuses. given a cardinal κ, b is called κ-resolvable if x can be partitioned to κ large subsets. the cardinal res b = sup {κ : b is κ-resolvable} is called a resolvability of b. we determine the resolvability of the ball structures related to metric spaces, groups and filters. 2000 ams classification: 54a25, 05a18. keywords: ball structures, resolvability, coresolvability. 1. introduction let b1 = (x1,p1,b1) and b2 = (x2,p2,b2) be ball structures, f: x1 −→ x2. we say that f is a ≺-mapping if, for every α ∈ p1, there exists β ∈ p2 such that f(b1(x,α)) ⊆ b2(f(x),β) for every x ∈ x. a bijection f is called an isomorphism if f and f−1 are ≺-mappings. the results from [10], [11], [12] show that the ball structures (with the isomorphisms defined above) are the natural asymptotic counterparts of topological spaces. a good motivation to study ball structures related to metric spaces is in the survey [5]. a topological spaces is called κ-resolvable (κ is a cardinal) if it can be partitioned to κ dense subsets. for resolvability of topological spaces and topological groups see the surveys [3], [4], [9]. let b = (x,p,b) be a ball structure. a subset y ⊆ x is called large if there exists α ∈ p such that x = b(y,α). the large subsets of ball structure can be considered as the duplicates of the dense subspaces of topological space. given a cardinal κ, we say that b is κ-resolvable if x can be partitioned to κ large subsets. the resolvability of b is the cardinal 192 i. v. protasov resb = sup{κ : b is κ − resolvable}. a subset y ⊆ x is called small if x \ b(y,α) is large for every α ∈ p . the small subsets of ball structure can be considered as the duplicates of the nowhere dense subsets of topological space. assume that every singleton of x is small. given a cardinal κ, we say that b is κ-coresolvable if x can be covered by κ small subsets. the coresolvability of b is the cardinal coresb = min {κ : b isκ − coresolvable}. the referee pointed out that the coresolvability can be considered as the asymptotic duplicate of the novak number of topological space x n(x) = min{|u| : u is a cover of x consisting of nowhere dense subsets}. in this paper we determine (or evaluate) the cardinal invariants res b and cores b for a wide spectrum of ball structures b related to metric spaces, groups and filters. we begin with exposition of results (2), continue with proofs (3) and conclude the paper with comments and open problems (4). all ball structures under consideration are supposed to be uniform. a ball structure b = (x,p,b) is called uniform if b is symmetric and multiplicative. we say that b is symmetric if, for every α ∈ p , there exists β ∈ p such that b(x,α) ⊆ b∗(x,β) for every x ∈ x and vice versa, where b∗(x,β) = {y ∈ x : x ∈ b(y,β)}. a ball structure b is called multiplicative if, for any α,β ∈ p , there exists γ(α,β) ∈ p such that b(b(x,α),β) ⊆ b(x,γ(α,β)) for every x ∈ x. note that if b is uniform and y ⊆ x is large, then there is γ ∈ p such that b(x,γ) ⋂ y 6= ∅ for all x ∈ x. for more detailed information concerning the uniform ball structures as the asymptotic counterparts of the uniform topological spaces see [11]. initially, the problem of resolvability of ball structures was motivated by the following question [1]: can every infinite group be partitioned onto two large subsets? for the positive answer to this question see [14] or [15]. 2. results let b = (x,p,b) be a ball structure, κ be a cardinal. we say that a subset y ⊆ x is κ-crowded if there exists α ∈ p such that |b(y,α) ⋂ y | ≥ κ for every y ∈ y . a ball structure b is called κ-crowded if its support x is κ-crowded. the crowdedness of b is the cardinal cr b = sup {κ : b isκ − crowded}. define the preodering ≤ on p by the rule α ≤ β if and only if b(x,α) ⊆ b(x,β) for every x ∈ x. a subset p ′ ⊆ p is called cofinal if, for every α ∈ p , there exists β ∈ p ′ such that α ≤ β. the cofinality cf b is the minimal cardinality of the cofinal subsets of p . resolvability of ball structures 193 proposition 2.1. for every ball structure b = (x,p,b), the following statements hold (i) if b is κ-crowded, then b is κ-resolvable; (ii) cr b ≤ res b ≤ cr b · cf b; (iii) if κ is a finite cardinal and b is κ-resolvable, then b is κ-crowded. by van douwen-illanes’ theorem [7], if a topological space is n-resolvable for every natural number n, then it can be partitioned to countably many dense subsets. the referee pointed out that the generalization of van douwen-illanes’ theorem for the case of countable cofinality lies in [2]. in view of proposition 2.1 (iii), the following statement can be considered as the analogue of this generalization. proposition 2.2. let b = (x,p,b) be a ball structure, < κn >n∈ω be an increasing sequence of cardinals, κ = sup {κn : n ∈ ω}. if b is κn-crowded for every n ∈ ω, then x can be partitioned in κ large subsets. let (x,d) be a metric space. for any x ∈ x, r ∈ r+, put bd(x,r) = {y ∈ x : d(x,y) ≤ r}. the ball structure (x,r+,bd) induced by the metric space (x,d) is denoted by b(x,d). a ball structure b is called metrizable if b is isomorphic to b(x,d) for the appropriate metric space (x,d). by [8], b is metrizable if and only if b is uniform, connected and cf b ≤ ℵ0. a ball structure b = (x,p,b) is called connected if, for any x,y ∈ x, there exists α ∈ p such that x ∈ b(y,α), y ∈ b(x,α). theorem 2.3. for every metric space (x,d), res b(x,d) = cr b(x,d) and x can be partitioned in cr b(x,d) large subsets. let g be an infinite group with the identity e and let κ be an infinite cardinal with κ ≤ |g|. denote by ℑ(g,κ) the family of all subsets of g of cardinality < κ containing e. for all g ∈ g, f ∈ ℑ(g,κ), put bl(g,f) = fg, br(g,f) = gf. denote by bl(g,κ) and br(g,κ) the ball structures (g,ℑ(g,κ),bl) and (g,ℑ(g,κ),br). observe that the mapping g 7−→ g −1 is an isomorphism between bl(g,κ) and br(g,κ). we say that a subset y ⊆ x is left κ-large (left κ-small) if y is large (small) in the ball structure bl(g,κ). in other words, y is called left κ-large if there exists a subset f ⊆ g such that |f | < κ and g = fy . theorem 2.4. let g be an infinite group, κ be an infinite cardinal and κ ≤ |g|. then g can be partitioned in κ left κ-large subsets. let b = (x,p,b) be a ball structure. a subset y ⊆ x is called bounded if there exist x ∈ x, α ∈ p such that y ⊆ b(x,α). we say that b is bounded if x is bounded. assume that b is unbounded and connected. then every bounded subset of x is small. since x = ⋃ {b(x,α) : α ∈ p} for every x ∈ x, we 194 i. v. protasov conclude that cores b ≤ cf b. in particular, cores b(x,d) ≤ ℵ0 for every unbounded metric space (x,d). on the other hand, the family of all small subsets of an arbitrary ball structure is an ideal in the boolean algebra of all subsets of x (see [11]). thus, cores b ≥ ℵ0 for every unbounded ball structure b. hence, cores b(x,d) = ℵ0 for every unbounded metric space (x,d). the following theorem shows that cores b could be much more less than cf b. theorem 2.5. let g be an infinite group, κ be an infinite cardinal and κ ≤ |g|. if κ < cf (|g|), then cores bl(g,κ) = ℵ0. let g be a topological group, c(g) be a family of all compact subsets of g containing the identity of g. a ball structure (g,c(g),bl) is denoted by bl(g,c). clearly, bl(g,c) = bl(g,ℵ0) for every discrete group g. theorem 2.6. let g be a non-compact locally compact group, then cores bl(g,c) = ℵ0. let x be a set and let ϕ be a filter on x such that ⋂ ϕ = ∅. for any x ∈ x, f ∈ ϕ, put b(x,f) = { x\f, if x /∈ f ; {x}, if x ∈ f ; and denote by b(x,ϕ) the ball structure (x,ϕ,b). theorem 2.7. let x be a set, ϕ be a filter on x such that ⋂ ϕ = ∅. then res b = 1, cores b = min {|ψ| : ψ ⊆ ϕ, ⋂ ψ = ∅}. 3. proofs proof of proposition 2.1. (i) choose α ∈ p such that |b(x,α)| ≥ κ for every x ∈ x. by zorn lemma, there exists a subset y ⊆ x such that the family {b(y,α) : y ∈ y } is pairwise disjoint and, for every x ∈ x, there exists y ∈ y such that b(x,α) ⋂ b(y,α) 6= ∅. since |b(y,α)| ≥ κ for every y ∈ y , there exists a family ℑ of κ-many pairwise disjoint subsets of x such that |f ⋂ b(y,α)| = 1 for all y ∈ y and f ∈ ℑ. in view of uniformity of b and by choice of y , every subset f ∈ ℑ is large. hence, b is κ-resolvable. (ii) the left inequality follows from (i). let ℑ be an arbitrary pairwise disjoint family of large subsets of x. pick a cofinal subset p ′ ⊆ p with |p ′| = cf b. for every α ∈ p ′, put ℑ(α) = {f ∈ ℑ : b∗(f,α) = x}, where b∗(f,α) = ⋃ x∈f b∗(x,α). take any x ∈ x and α ∈ p ′. since b(x,α) ⋂ f 6= ∅ for every f ∈ ℑ(α), we have |ℑ(α)| ≤ |b(x,α)|. hence, |ℑ(α)| ≤ cr b and the right inequality holds. (iii) let f1,f2, ...,fm be pairwise disjoint large subsets of x. choose α ∈ p such that b∗(fi,α) = x for every i ∈ {1,2, ...,m}. then b(x,α) ⋂ fi 6= ∅ for all x ∈ x and i ∈ {1,2, ...,m}. it follows that |b(x,α)| ≥ m and b is m-crowded. ✷ resolvability of ball structures 195 proof of proposition 2.2. it suffices to partition x = y ⋃ z so that y is a disjoint union of κ0 large subset and z is κn-crowded for every n ∈ ω. we may suppose that κ0 > 0. choose α ∈ p such that |b(x,α)| ≥ 2κ0 for every x ∈ x. by zorn lemma, there exists a subset a ⊆ x such that {b(a,α) : a ∈ a} is a maximal disjoint family. for every a ∈ a, partition b(a,α) = c(a) ⋃ d(a) so that |c(a)| = κ0, |d(a)| ≥ κ0. put y = ⋃ a∈a c(a), z = x\y and note that y can be partitioned in κ0-many large subsets. fix n ∈ ω and choose β ∈ p such that |b(x,β)| ≥ 2κn for every x ∈ x. since b is multiplicative, there exists γ ∈ p such that b(b(x,β),α) ⊆ b(x,γ) for every x ∈ x. then |b(z,γ) ⋂ z| ≥ κn for every z ∈ z and z is κn-crowded. ✷ proof of theorem 2.3. since cf b(x,d) = ℵ0, the first statement follows from proposition 2.1. the second statement follows from proposition 2.2. ✷ to prove the next three theorems we use the filtrations of groups. let g be an infinite group with the identity e. a filtration of g is a family {gα : α < |g|} of subgroups of g such that (i) g0 = {e}, g = ⋃ {gα : α < |g|}; (ii) gα ⊂ gβ for all α < β < |g|; (iii) ⋃ {gα : α < β} = gβ for every limit ordinal β; (iv) gα < |g| for every α < |g|. using a minimal well-ordering of g it is easy to construct a filtration of g provided that g is not finitely generated. in particular, every uncountable group admits a filtration. for each α < |g|, decompose gα+1 \ gα to right cosets by gα and fix some set xα of representatives so gα+1 \ gα = gαxα. take an arbitrary element g ∈ g, g 6= e and choose the smallest subgroup gα with g ∈ gα. by (iii), α = α1 + 1 for some ordinal α1 < |g|. hence, g ∈ gα1+1 \ gα1 and there exist g1 ∈ gα1, xα1 ∈ xα1 such that g = g1xα1. if g1 6= e, we choose the ordinal α2, the elements g2 ∈ gα2+1 \ gα2 and xα2 ∈ xα2 such that g1 = g2xα2. since the set of ordinals < |g| is well-ordered, after finite number of steps we get the representation g = xαs(g)xαs(g)−1...xα2xα1, αs(g) < ... < α1,xαi ∈ xαi. note that this representation is unique and put γ1(g) = α1, γ2(g) = α2, ...,γs(g)(g) = αs(g), γ(g) = {γ1(g), ...,γs(g)(g)}. for every natural number n, denote dn = {g ∈ g : s(g) = n}. proof of theorem 2.4. first suppose that |g| = κ. if g is countable, then bl(g,κ) is metrizable and we can apply theorem 2.3. assume that g is uncountable and use the above filtration. for every α < |g|, put fα = {g ∈ g : γs(g)(g) = α} and note that {fα : α < g} is a pairwise disjoint family of left κ-large subsets. 196 i. v. protasov if κ < |g|, we choose a subgroup h of g with |h| = κ. by above paragraph, there exists a partition p of h such that each subset p ∈ p is large in bl(h,κ). decompose g to right cosets by h and fix some set x or representatives so g = hx. then {px : p ∈ p} is a pairwise disjoint family of left κ-large subsets of g. ✷ proof of theorem 2.5. if g is countable, then bl(g,κ) is metrizable and we have cores bl(g,κ) = ℵ0. suppose that g is uncountable and use the above filtration. observe that g \ {e} = ⋃ ∞ n=1 dn, fix a natural number n and show that dn is small in bl(g,κ). take an arbitrary subset f ∈ ℑ(g,κ). by assumption, there exists β ∈ p such that f ⊆ gβ so fdn ⊆ gβdn. show that g \ gβdn is left ℵ0-large. choose the elements a1,a2, ...,an+1 of g such that α1 ∈ gβ+1 \ gβ,a2 ∈ gβ+2 \ gβ+1, ...,an+1 ∈ gβ+n+1 \ gβ+n. take an arbitrary element g ∈ gβdn and put g = g0. if β + n ∈ γ(g), put ε0 = 0, otherwise ε0 = 1. note that β + n ∈ γ(a ε0 n+1g0) and put g1 = aε0n+1g0. if β + n − 1 ∈ γ(g1), we put ε1 = 0, otherwise ε1 = 1. note that {β + n − 1,β + n} ⊆ γ(aε1n+1g1) and put g2 = a ε1 n g1. after n + 1 steps we get {β,β + 1, ...,β + n} ⊆ γ(aεn1 a εn−1 2 ...a ε0 n+1g). it follows that (aεn1 a εn−1 2 ...a ε0 n+1g) /∈ gβdn. put a = {e,a1,a2, ...,an+1}, k = an. we have shown that gβdn ⊆ k −1(g\gβdn). hence, g = k −1(g\ gβdn) and g \ gβdn is left ℵ0-large. ✷ proof of theorem 2.6. if g is σ-compact, then cf bl(g,c) = ℵ0 and bl(g,c) is metrizable. hence, cores bl(g,c) = ℵ0. assume that g is not σ-compact. then we can easily construct a filtration {gα : α < |g|} so that every subgroup gα, α > 0 is open. repeat the arguments proving theorem 2.5. ✷ proof of theorem 2.7. two easy observations. a subset y ⊆ x is large if and only if y ∈ ϕ. a subset y ⊆ x is small if and only if x \ y is large. ✷ 4. comments and open problems problem 4.1. let b = (x,p,b) be a ball structure, κ be a cardinal such that b is κ′-crowded for every κ′ < κ. can x be partitioned in κ large subsets? by proposition 2.2, this is so if cfκ = ℵ0. problem 4.2. let g be an infinite group, κ be an infinite cardinal, κ ≤ |g|. can g be κ-partitioned so that each cell of the partition is left and right κ-large? if κ = ℵ0, this is so [14]. let g be an infinite amenable (in particular, abelian) group, µ be a banach measure on g. clearly, µ(a) > 0 for every left ℵ0-large subset a of g. it follows, that res bl(g,ℵ0) = ℵ0. on the other hand, every free group of infinite rank κ can be partitioned in κ left ℵ0-large subsets [11]. resolvability of ball structures 197 problem 4.3. let g be a free abelian group of rank ℵ2. can g be partitioned in ℵ2 ℵ1-large subsets. problem 4.4. let g be an infinite group, κ be an infinite cardinal, κ ≤ |g|. can g be partitioned in ℵ0 left κ-small subsets? by theorem 2.5, this is so if |g| is a regular cardinal. a topological space is called irresolvable if it can not be partitioned in two dense subsets. let us say that a ball structure b is irresolvable if resb = 1. by proposition 2.1, b is irresolvable if and only if crb = 1. a topological space x is called κ-extraresolvable if x admits a family ℑ, |ℑ| = κ of dense subsets such that f1 ⋂ f2 is nowhere dense for all distinct subsets f1,f2 ∈ ℑ. it is important to remark that if 1 < κ < ω then κ-extraresolvability is equivalent to κ-resolvability. the concept of κextraresolvability was introduced by v. i. malykhin [8]. as the referee pointed out, the published paper where this concept appears for the first time in the literature is [6]. let us say that a ball structure b = (x,p,b) is κ-extraresolvable if x admits a family ℑ, |ℑ| = κ of large subsets such that f1 ⋂ f2 is small for all distinct subsets f1,f2 ∈ ℑ. if b is unbounded and κ-crowded, then there exists a family ℑ, |ℑ| = κℵ0 of large subset of x such that f1 ⋂ f2 is finite for all distinct subsets f1,f2 ∈ ℑ, so b is κℵ0-extraresolvable. the extraresolvability of b is the cardinal sup {κ : b is κ-extraresolvable}. problem 4.5. determine (or evaluate) extraresolvability of ball structures of metric spaces and groups. the referee asked ”what about the infinite case in (iii) of proposition 2.1?” let k be an uncountable ordinal and let g be a free group of rank k. by [13], bl(g,ℵ0) can be partitioned in k-many large subsets, but cr bl(g,ℵ0) = ℵ0. on the other hand, the statement (iii) remains true for k = ℵ0. acknowledgements. i would like to thank the referee for the long list of corrections and suggestions to the previous version of the paper. references [1] a. bella and v. i. malykhin, small and other subsets of a group, q and a in general topology, 11 (1999), 183-187. [2] k.p.s. bhaskara rao, on ℵ-resolvability, unpublished manuscript. [3] w. w. comfort, o. masaveau and h. zhou, resolvability in topology and topological groups, annals of new york acad. of sciences, 767 (1995), 17-27. [4] w. w. comfort and s. garcia-ferreira, resolvability: a selective survey and some new results, topology appl. 74 (1996), 149-167. 198 i. v. protasov [5] a. dranishnikov,asymptotic topology, russian math. survey 55 (2000), 71-116. [6] s. garcia-ferreira, v. i. malykhin and a. h. tomita, extraresolvable spaces, topology appl. 101 (2000), 257-271. [7] a. illanes, finite and ω-resolvability, proc. amer. math. soc. 124 (1996), 1243–1246. [8] v. i. malykhin, irresolvability is not descriptive good, unpublished manuscript. [9] i. v. protasov, resolvability of groups (in russian), matem.stud. 9 (1998), 130–148. [10] i. v. protasov, metrizable ball structures, algebra and discrete math. 2002, n 1, 129– 141. [11] i. v. protasov, uniform ball structures, algebra and discrete math., 2003, n 1, 93–102. [12] i. v. protasov, normal ball structures, matem. stud. (to appear). [13] i. v. protasov, combinatorial size of subsets of groups and graphs, algebraic systems and their applications, proc. inst. math. nan ukraine, 2002, 333–345. [14] i. v. protasov, quasiray decompositions of graphs, matem.stud. 17 (2002), 220-222. [15] i. v. protasov and t. banakh,ball structures and colorings of groups and graphs, math. stud. monogr. ser. 11 (2003). received march 2003 accepted february 2004 i. v. protasov (kseniya@profit.net.ua) department of cybernetics, kyiv university, volodimirska 64, kiev 01033, ukraine @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 143–155 on classes of t0 spaces admitting completions eraldo giuli ∗ abstract. for a given class x of t0 spaces the existence of a subclass c, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. a positive example is the class of all t0 spaces, with c the class of sober t0 spaces, and a negative example is the class of tychonoff spaces. we prove that x has the previous property (i.e., admits completions) whenever it is the class of t0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category top of topological spaces. two classes of examples are provided. 2000 ams classification: primary: 54a05, 54b30, 54d25, 54g10, 18a30. secondary: 54d05, 54d10, 54d30. keywords: affine set, t0, sober and injective space, compact space, completion, zariski closure, topological category, coreflective subcategory. 1. introduction let met be the category of metric spaces and non-expansive maps and let cmet be the full subcategory of complete metric spaces. it is well known (see e.g. [14]) that (1) for every metric space x there exist a complete metric space x∗ and a dense embedding γ : x → x∗ such that, for every complete metric space y and non-expansive map f : x → y there exists a (unique) non-expansive map f∗ : x∗ → y for which γ ◦f∗ = f; (2) if f : x → z is a dense embedding into a complete metric space z then z coincides, up to isometries, with x∗. since in met dense non-expansive maps are epimorphisms, (1) says that cmet is epi-embedding reflective in met. it is also well known that the category of compact hausdorff spaces is embedding-epireflective in the category of tychonov spaces, being here x∗ the stone-čech compactification of x. however no property (2) is fulfilled by the latter construction. in this sense we can say ∗dedicated to professors miroslav hušek and gerhard preuss on their sixtieth birthday 144 e. giuli that property (2) distinguishes completions from compactifications (see [6] and [5] for a general treatment of categorical completions). it is also known (see e.g. [6]) that the category sob0 of sober t0 spaces (every irreducible closed set is the closure of a unique point) is epi-embedding reflective in the category top0 of all t0 spaces and that property (2) is fulfilled (i.e., sob0 is firmly epireflective in top0 in the terminology of [6]). on the other hand no non trivial epireflective subcategory of top consisting of hausdorff spaces admits a firm epireflection (cf. [6] example 1.8(2)). the aim of this paper is to give sufficient conditions for a class of t0 objects of a large enough topological category, sset, whose objects are called affine sets, to admit a firm epireflection or, as we shall say, to admit completions. for that we introduce and we study in section 2 the category sset of affine sets and affine maps which properly contains the category top of topological spaces and continuous maps. in section 3, we analyze the extension from top to sset of the ordinary closure, the b-closure and the zariski closure (the last two do not coincide in sset, while they are the same in top). in section 4 we restrict our considerations to affine sets satisfying a mild condition (the corresponding full subcategory is denoted by sset) and there we introduce and study the so called separated ( = t0) affine sets. section 5 contains the main result: a class of separated affine sets admits completions whenever it is the class of all separated affine sets of a subcategory of sset which is stable under affine subsets, disjoint unions and quotients (i.e., of a hereditary coreflective subcategory of sset). two methods to produce hereditary coreflective subcategories of sset and of top are considered. the first goes back to a paper of diers [13] (from which some terminology is derived, e.g., affine set, zariski closure, etc.) and the second goes back to a paper of hušek and the author [16]. finally two proper classes of examples of classes of t0 spaces admitting completions are provided: (a) for each infinite regular cardinal α, the class of all t0 spaces for which every point in the closure of a subset is also in the closure of a smaller subset of cardinality less than α. if α is a successor cardinal we obtain the t0 spaces of tightness less than α. (b) for each infinite cardinal α, those t0 spaces for which the intersection of less than α open sets is open. for categorical terminology see [1] and [20]. for general topology we refer to [14]. all the subcategories considered in the paper are full and isomorphism closed. these are frequently identified with classes of objects defined by a given property. on classes of t0 spaces 145 2. the category sset of affine sets an affine set over the two point set s = {0, 1} is a pair (x,u) , where x is a set and u is a subset of the power set p(x). an affine map from (x,u) to (y,v) is a function f : x → y such that f−1(v ) ∈ u for every v ∈ v. sset will denote the category of affine sets (over s) and affine maps. the functional isomorphic description of sset is as follows: objects are pairs (x,a) where x is a set and a is a subset of the power set sx and the morphisms from (x,a) to (y,b) are functions f : x → y such that β ◦f ∈a whenever β ∈b. both descriptions of sset will be utilized throughout the paper. we will denote by f : sset → set the obvious forgetful functor. proposition 2.1. (sset,f) is a topological category. proof. to show that every f-structured source admits a unique initial lift, let x be a set, {(yi,ui) : i ∈ i} a family of affine sets and {fi : x → yi : i ∈ i} a family of functions. then the subset u = {(fi)−1(v ) : v ∈ui, i ∈ i} is the unique initial structure in x for the given data. � thus an affine map f : (x,u) → (y,v) is initial (with respect to f) if and only if every u ∈ u is of the form f−1v , v ∈ v. as usual an initial monomorphism will be called an embedding. it is clear that every subset m of the underlying set x of an affine set (x,u) carries as initial structure the family v = {u ∩ m : u ∈ u}. in this case we say that (m,v) is an affine subset of (x,u). some consequences (cf. [1, 17] ) of proposition 2.1 are collected in the corollary 2.2. (i) every f -structured sink {gi : f(xi,ui) → y, i ∈ i} admits a unique final lift (i.e., there is a largest affine structure in y for which all the gi are affine maps ); (ii) in sset the epimorphisms are the surjective affine maps and the monomorphisms are the injective affine maps; (iii) in sset the embeddings coincide with the regular monomorphisms ( = equalizers of two affine maps ); (iv) every affine map f admits an essentially unique (surjective, embedding )-factorization. that is: f = mf◦ef for some surjective affine map ef and embedding mf and, for every commutative square f ◦e = m◦g with e surjective and m embedding, there exists a (unique) affine map d such that d◦e = g and m◦d = f; (v) sset is complete. every limit is obtained as an initial lift of the corresponding limit in set. in particular, the product of a family {(xi,ui) : i ∈ i)} of affine sets is the cartesian product x of the family {(xi) : i ∈ i} endowed with the affine structure u = {π−1i (u) : u ∈ ui, i ∈ i} where πi : x → xi are the projections; 146 e. giuli (vi) sset is co-complete. every colimit is obtained as a final lift of the corresponding colimit in set. example 2.3. (a) every closure (hence every topological) space (cf. [11, 9]), is an affine set and a function between closure (resp. topological) spaces is continuous if and only if it is affine. thus both the categories top of topological spaces and cl of closure spaces are fully embedded in sset; (b) affine spaces coincide with so called normal (boolean) chu spaces recently introduced by william pratt as a generalization of nielsen, plotkin and winskel’s notion of event structure for modelling concurrent computation [19]. moreover continuous maps between normal (boolean) chu spaces coincide with the affine maps. thus sset is a full subcategory of the category chus of (boolean) chu spaces and continuous maps. (c) following l. m. brown and m. diker [3] a texture space is a pair (x,u) where u is a subset of the power set p(x) which is a complete, completely distributive lattice with respect to the inclusion, which contains x and ∅, separates the points of x, and for which meet coincides with intersection and finite join with union. thus texture spaces are particular affine spaces. in our context the natural morphisms between texture spaces are the affine maps. for suitable morphisms in the class of texture spaces see [4]. remark 2.4. (1) following the same lines as the functional description of sset, in [15] the category aset, for every set a, was considered. the above category coincides with the full subcategory of the category chua of chu spaces with respect to the set a, consisting of normal chu spaces [19] and, for particular instances of a it contains the category fuz of fuzzy topological spaces (a = [0, 1]) and the category ap of approach spaces (a = [0,∞]) [15]. the categories of the form aset fulfil both the proposition and the corollary above. (2) the category sset is not well-fibred even though every set admits a set of affine structures. in fact the empty set admits two affine structures and every one-point set admits four structures. this defect can be removed by assuming that every affine structure contains the empty and the whole set. we shall consider this full subcategory of sset, denoted by sset, in section 4. 3. kuratowski and zariski closure in this section we extend from top to sset the usual (kuratowski) closure k, the b-closure in the form introduced by baron [2] (to characterize the epimorphisms in the category top0 of t0-spaces), and the b-closure in the form considered by skula [21]. the latter, called zariski closure and denoted by z, in contrast with the situation in top, does not coincide with baron closure in sset. we recall that a closure operator c of a topological category (over set) x is an assignment, to each subset m of (the underlying set of) any object x of x, of a subset cxm of x such that on classes of t0 spaces 147 (c1) m ⊂ cxm; (c2) cxm ⊂ cxn whenever m ⊂ n; (c3) c-continuity. for every f : x → y in x and m subset of x, f(cxm) ⊂ cy (fm). note that by property (c2) (c4) cxm ∪ cxn ⊂ cx(m ∪n). we will drop x in cxm when no confusion is possible. the closure operator c is called (c5) idempotent if cx(cxm) = cxm; (c6) grounded if cx∅ = ∅; (c7) additive if cx(m ∪n) = cxm ∪ cxn. (c8) hereditary if, for every m ⊂ y ⊂ x, cy m = (cxm) ∩ y, where y is endowed with the initial structure induced by the inclusion y ⊂ x. a subset m ⊂ x is called c-closed (respectively c-dense) in x if cxm = m (respectively cxm = x). a morphism f : x → y is called c-dense if f(x) is c-dense in y and it is called c-closed if it sends c-closed subsets into c-closed subsets. an object x is called (1) c-separated if the diagonal ∆x = {(x,x) : x ∈ x} is c-closed in the square x2; (2) absolutely c-closed if it is c-separated and it is c-closed in every cseparated object in which it can be embedded; (3) c-compact if, for every object y , the projection p : x × y → y is c-closed; (4) c-connected if the diagonal ∆x is c-dense in x2. it is well known (e.g. see [14]) that in top, if c is the ordinary closure, then cseparated means hausdorff, absolutely c-closed means h-closed and c-compact means compact in the usual sense (no hausdorff condition is included). the c-connected spaces coincide with the irreducible spaces, that is: any disjoint open sets u,v must satisfy u = ∅ or v = ∅ (cf. [8]). for the general theory of closure operators we refer to [12, 10] and [7]. if (x,u) is an affine set and x ∈ x we will denote by ux the family of all u ∈u such that x ∈ u. let (x,u) be an affine set and let m ⊂ x: (a) the kuratowski closure of m in (x,u) is defined by k(x,u)m = {x ∈ x : (∀u ∈ux)(∃m ∈ m)(m ∈ u)} (b) the baron closure of m in (x,u) (cf. [2]) is defined by b(x,u)m = {x ∈ x : x ∈ k(x,u)(m ∩k(x,u){x})} = {x ∈ x : (∀u ∈ux)(∃m ∈ (u ∩m))(∀v ∈um)(x ∈ v )}; 148 e. giuli (c) the zariski closure of m in (x,u) (cf. [21]) is defined by z(x,u)m = {x ∈ x : (@ u,v ∈u)(u ∩m = v ∩m,x ∈ (u r v )}. proposition 3.1. (i) always b(x,u)m ⊂ z(x,u)m and b(x,u)m ⊂ k(x,u)m. (ii) if ∅ ∈u then z(x,u)m ⊂ k(x,u)m. (iii) if u is a topology then z(x,u)m ⊂ b(x,u)m. proof. (i). if x is not in z(x,u)m and u,v ∈u are such that u ∩m = v ∩m and x ∈ (u r v ), then u ∈ux and for every m ∈ u ∩m, by m ∈ v , we have v ∈um while x is not in v . consequently x does not belong to b(x,u)m. the inclusion b(x,u)m ⊂ k(x,u)m is obvious. (ii). assume ∅ ∈ u and x not in k(x,u)m. then there exists u ∈ ux such that u ∩m = ∅, ∅ = v ∈u, u ∩m = v ∩m and x ∈ (u rv ); consequently x is not in z(x,u)m. (iii). assume x not in b(x,u)m. then there exists u ∈ ux such that, for each m ∈ u ∩ m there is a vm ∈ um with x not in vm. if u is a topology then the set v = ⋃ {vm ∩u : m ∈ (m ∩u)} belongs to u and both conditions u ∩ m = v ∩ m and x ∈ (u r v ) are fulfilled; consequently x is not in z(x,u)m. � theorem 3.2. (i) the kuratowski, baron and zariski closure are idempotent and hereditary closure operators of sset. (ii) the closure operators k and b are grounded in (x,u) if and only if u is a cover of x. if, in addition, the empty set belongs to u, then z is grounded. (iii) the closure operators k, b and z are additive in (x,u) whenever u is stable under intersection of pairs. proof. (i). properties (c1) and (c2) are trivial and k-continuity of every affine map (i.e., property (c3)) directly follows from the defining property of affine map. property (c3) for b: if x ∈ b(x,u)m then x ∈ k(x,u)(m ∩ k(x,u){x}) so that fx ∈ k(y,v)(f(m ∩ k(x,u){x})). now, by k-continuity of f applied twice, k(y,v)(f(m ∩k(x,u){x})) ⊂ k(y,v)(fm ∩f(k(x,u){x})) ⊂ k(y,v)(fm ∩ k(y,v){fx}) consequently fx ∈ b(y,v)(fm). property (c3) for z: assume that y is not z(y,v)(fm) and let u,v ∈v such that u ∩ m = v ∩ m and y ∈ (u r v ). then for every x ∈ f−1y we have x ∈ f−1u while x is not in f−1v and f−1u ∩ m = f−1v ∩ m which means that x is not in z(x,u)m, consequently y is not in f(z(x,u)m). the idempotency of k is clear. let x ∈ b(x,u)(b(x,u)m) which means x ∈ k(x,u)(b(x,u)m∩k(x,u){x}). then for every u ∈ux, u∩b(x,u)m∩k(x,u){x} 6= ∅. let, for a fixed u, y ∈ u ∩ b(x,u)m ∩k(x,u){x}. for every v ∈uy, by y ∈ b(x,u)m, v ∩m∩k(x,u){y} 6= ∅ and by y ∈ k(x,u){x}, k(x,u){y}⊂ k(x,u){x} on classes of t0 spaces 149 consequently v ∩m ∩k(x,u){x} 6= ∅. now u ∈uy consequently x ∈ b(x,u)m. this shows that b is idempotent. for the idempotency of z note that if u,v ∈u satisfy u ∩ m = v ∩ m then also u ∩ z(x,u)m = v ∩ z(x,u)m, so that if x is not in z(x,u)m then there exist u,v ∈ u such that (u ∩ m = v ∩ m, hence) u ∩z(x,u)m = v ∩z(x,u)m and x ∈ (u rv ), consequently x is not in z(x,u)(z(x,u)m). the hereditariness directly follows from the fact that the affine structure in an affine subset y of an affine set (x,u) is {u ∩y : u ∈u}. (ii). k(x,u)∅ = ∅ if and only if u is a cover is clear. then the remaining part of (ii) follows from k(x,u)∅ = b(x,u)∅ ⊂ z(x,u)∅. (iii). this simple example shows that b, z and k are not additive. let x = {1, 2, 3} and let u = {{0, 1},{0, 2},∅}. for each m ⊂ x, z(x,u)m ⊂ k(x,u)m, by ∅ ∈u, b(x,u){i} = k(x,u){i}, for i = 1, 2, and b(x,u){1, 2} = k(x,u){1, 2} = x. hence z(x,u){1, 2} = x. for additivity, one inclusion is (c4), and the other directly follows from the stability under intersection of pairs of u. � the next result follows from the idempotency and hereditariness of our closure operators (see [10, 12]). corollary 3.3. every affine map admits an essentially unique (c-dense, cclosed embedding ) factorization for c = k,b,z. proof. for a given f : x → y denote by ef : x → cy (fx) the codomain restriction of f to cy (fx) and by mf : cy (fx) → y the inclusion map. by c idempotent, cy (fx) is c-closed and, by c hereditary, ef is c-dense. thus mf ◦ef is a (c-dense map, c-closed embedding)-factorization of f. let now f ◦e = m◦g (e : x → y,m : z → t) be a commutative square in sset, with e a c-dense affine map and m a c-closed embedding, and let y ∈ y . since y is in the c-closure of ex in y by assumption, then, by c-continuity of f, fy is in the c-closure of f(ex) = (f ◦e)x. by commutativity, fy is then in the c-closure of (m(gx), hence) mz. now mz is c-closed in t by assumption, so that fy ∈ mz, consequently, by injectivity of m, there is unique z ∈ z such that mz = fy. set dy = z. then d is an affine map since m ◦ d = f and m is initial by assumption. moreover d is the unique affine map satisfying both m◦d = f and d◦e = g since m is injective by assumption. � remark 3.4. it should be noted that in the previous proof we have not used the full power of hereditariness of our closures. indeed what we need is that every subset is c-dense in its closure. that property, called weak hereditariness, is, in the presence of idempotency, weaker than hereditariness (see e.g., [12]). 4. separated (= t0) affine sets in this section we will refer to the functional description of sset 150 e. giuli the extension of the ordinary closure k of top to sset has no interest for the development of such basic topological notions as separation, compactness, absolute closedness and connectedness. indeed, by the form of products in sset as explained in section 2, an affine set is k-separated if and only if it is absolutely closed if and only if it has at most one point. moreover every affine set is k-compact since every projection is k-closed and every affine set is k-connected since the diagonal of every affine set is k-dense in the square of the affine set. the aim of the last two sections is to show that the above topological notions are not trivial in sset if we refer to the zariski (or baron) closure. for that we restrict our considerations to those affine sets whose affine structure contains the two constant functions 0 and 1. the corresponding full subcategory, denoted by sset, which is topological too, has many pleasant properties not shared by sset. among others, (1) it is well-fibred, i.e. every constant function is affine or, equivalently, the affine sets with at most one point have unique affine structure; (2) our closures are grounded there; (3) a nonempty affine set is indiscrete if and only if its structure consists of constant functions. in what follows s will denote the two-point set {0, 1} endowed with the affine structure a = {0, 1, ids} (u = {∅,s,{1}} in the subset description). s will be called the sierpinski affine set. an affine set (x,a) is called separated if a separates the points of x. sset0 will denote the full subcategory of separated affine sets. clearly the sierpinski affine set s is separated. the following result has a trivial proof but plays an important role: lemma 4.1. a function f : x → s is an affine map between (x,a) and s if and only if f ∈a. proposition 4.2. for an affine set (x,a) these are equivalent: (i) (x,a) is separated; (ii) every affine map f : i2 → (x,a) is constant, where i2 is a two-point indiscrete affine set; (iii) ∆x is z-closed in (x,a) × (x,a); (iv) ∆x is b-closed in (x,a) × (x,a); (v) (x,a) is an affine subset of a product of copies of the sierpinski affine set s. proof. (i)⇔(ii). the existence of a non constant affine map from i2 to (x,a) is equivalent to saying that there are distinct points x and y in x such that α(x) = α(y) for every α ∈ a, which is equivalent to saying that (x,a) is not separated. (i)⇒(iii). if (x,y) ∈ x2r∆x then, by assumption, there is α ∈a such that α(x) 6= α(y). then the two affine maps α◦p1,α◦p2 : x ×x → x coincide in on classes of t0 spaces 151 ∆x and do not coincide on (x,y) which says that (x,y) is not in the z-closure of ∆x. consequently ∆x is z-closed. (iii)⇒(iv). it follows from proposition 3.1 (i). (iv)⇒(ii). assume (x,a) is not separated and let x 6= y in x such that, for each α ∈ a, α(x) = α(y). taking into account that the affine structure of x ×x consists of functions of the form α◦pi, i = 1, 2, it is easy to verify that the point (x,y) is in the b-closure of ∆. (i)⇒(v). let sa be the product of a copies of s and let φ : (x,a) → sa be the map whose components are the elements of a. by virtue of lemma 4.1, φ is affine and initial, and it is (injective, hence) an embedding if (and only if) (x,a) is separated. (v)⇒(i). separation is clearly a productive and hereditary property. � remark 4.3. (1) the equivalence (i)⇔(ii) in the above proposition says that the separated affine sets coincide with the so called t0-objects of the well fibred topological category sset, see [18]. in particular sset0 is the largest epireflective non bireflective subcategory of sset. moreover (i)⇔(v) says that sset is simply cogenerated by the sierpinski affine set s. (2) the sset0-reflection r of an affine set (x,a) is the restriction to the image of the affine map φ. indeed let (y,b) be a separated affine set, let ψ : (y,b) → sb be the canonical map, let f : (x,a) → (y,b) be any affine map and let f′ : sa → sb be the affine map associated to f. then, denoting by k : φ(x) → sa the inclusion, we obtain the commutative square (f′ ◦ k) ◦ r = ψ ◦ f. now r is surjective and ψ is an embedding since (y,b) is separated, so that, by corollary 2.2 (iv) there exists a (unique) affine map d : φ(x) → y such that d◦r = f. (3) since the sset0-reflections are initial (see the proof of (i)⇒(v) in proposition 4.2) then the category sset is universal in the sense of marny [18]. corollary 4.4. (i) in sset0 the epimorphisms are precisely the z-dense affine maps and the regular monomorphisms are precisely the z-closed embeddings. (ii) in sset0 every affine map admits an essentially unique factorization by a z-dense (respectively b-dense) affine map followed by a z-closed (respectively, b-closed ) embedding. proof. (i). by lemma 4.1 the morphisms from an affine set (x,a) to the sierpinski affine set are precisely the elements of the structure of a. on the other hand our zariski closure of a subset m is obtained by intersecting all the equalizers, of pairs in a, containing m, so, equivalently intersecting equalizers of pairs of affine maps into s. then, z being the regular closure operator 152 e. giuli induced by s hence, in virtue of (i)⇔(v) for sset0, it gives the epimorphisms and the regular monomorphisms (for that we use weak hereditariness) in sset0 as in (i) (see [10]). (ii). every affine subset of a separated affine set is separated, so the statement follows from corollary 3.3. � 5. completions an affine set (x,a) is called z-injective if it is injective with respect to zdense embeddings. that is: for every z-dense embedding m : (m,c) → (y,b) and affine map f : (m,c) → (x,a) there exists an affine map f′ : (y,b) → (x,a) such that f′ ◦m = f. by a standard argument z-injectivity is preserved by products. since embeddings are initial affine maps, by lemma 4.1 the sierpinski affine set is z-injective. proposition 5.1. for a separated affine set (x,a) these are equivalent: (i) (x,a) is z-injective; (ii) (x,a) is absolutely z-closed; (iii) the canonical map φ : (x,a) → sa is a z-closed embedding. proof. (i)⇒(ii). let (x,a) be separated and z-injective, let k : (x,a) → (y,b) be an embedding with (y,b) separated and m◦e the (z-dense, z-closed embedding)-factorization of k (see corollary 4.4 (ii)). then, since (x,a) is z-injective, there is an affine map g from the codomain of e into (x,a) such that g ◦ e = 1x, which says that e is a section, hence an isomorphism since it is an epimorphism in sset0. (ii)⇒(iii). trivial. (iii)⇒(i). assume that the canonical map φ : (x,a) → sa is a z-closed embedding, let k : (m,c) → (y,b) be a z-dense embedding and f : (m,c) → (x,a) any affine map. since the elements of c are the restrictions to m of those of b, there is an affine map f′′ : sb → sa such that (f′′ ◦ψ) ◦k = φ◦f consequently, by corollary 4.4 (ii), there is an affine map f′ : (y,b) → (x,a) satisfying, in particular, f′ ◦k = f. � we are now ready to prove that sset0 admits completions. theorem 5.2. the full subcategory csset0 of sset0 consisting of all the absolutely z-closed affine sets is firmly epireflective in sset0. proof. the csset0-reflection s of a separated affine set (x,a) is the restriction to the z-closure in sa of the image of the affine map φ. the proof uses the same argument as remark 4.3 (2). the property (c2) directly follows from the fact that the sierpinski affine set is a z-injective cogenerator of sset0. � on classes of t0 spaces 153 recall that a subcategory x of a category y is called coreflective in y if for every object y of y there exist an object ŷ in x and a morphism s : ŷ → y such that, for every x ∈ x and morphism f : x → y there exists a unique f′ : x → ŷ satisfying s◦f′ = f. if y is topological over set (as is our sset), then the coreflection maps s are bimorphisms, so we may assume, in our context, that ̂(x,a) is x endowed with an affine structure finer than or equal to a and that the coreflection maps are identities. if a coreflective subcategory of sset is stable under affine subsets we shall say that it is hereditarily coreflective. let x be a hereditary coreflective subcategory of sset and let us denote by t0x the subcategory of its separated affine sets. we shall show that t0x admits completions. lemma 5.3. if x is a hereditarily coreflective subcategory of sset then: (i) ŝ is separated and z-injective in x. (ii) t0x is cogenerated by ŝ in x. proof. (i). since ŝ is a finer modification of s it is separated. let k : x → y be a z-dense embedding with x,y ∈ x and let f : y → ŝ any affine map. since s is z-injective there exists f′′ : y → s with f′′ ◦k = s◦f (s : ŝ → s). consequently, by the universal property of coreflections, there is f′ : y → s such that s◦f′ = f′′ so that f′ ◦k = f , by injectivity of s. (ii). it is clear that a function from an affine set in x to s is affine if and only if it is so into ŝ. consequently the canonical map φ : (x,a) → ŝa remains an embedding whenever (x,a) is in x and the product ŝa is taken in x. � theorem 5.4. if x is a hereditary coreflective subcategory of sset then the affine sets which are affine subsets of products (taken in x) of copies of ŝ form a firm epireflective subcategory of t0x. proof. the proof follows the lines of proof of theorem 5.2 (and remark 4.3 (2)). � there is a general method to produce hereditary coreflective subcategories of sset: recall that an algebra structure in a set a is a family of functions ω = {ωt : at → a} where t runs in a given class of sets. then for every set x, by point-wise extension, the powerset ax carries an algebra structure. if ω is an algebra structure in the two-point set s we denote by sset(ω) the subcategory of sset consisting of those affine sets (x,a) for which a is a ω-subalgebra of the function algebra sx. it is easy to show that, for every algebra structure ω in s the corresponding subcategory sset(ω) of affine sets over the algebra (s, ω) is hereditarily coreflective in sset. 154 e. giuli in this way, for suitable ω, we obtain among others, the topological categories cs of closure spaces (cf. [9]), top of topological spaces and pros of preordered sets. for cs an internal characterization of the complete ( = absolutely closed) t0 spaces is given in [9]; in top0 they are the sober t0 spaces (see the introduction) and in pros it is easy to see that they are the partially ordered sets. we do not know any example of a hereditary coreflective subcategory of sset which is not of the form sset(ω). there is a second general method to produce hereditary coreflective subcategories, e.g. in top, which is described in [16]. in that paper two proper classes of hereditary coreflective subcategories of top are produced: for every regular cardinal α, alex (α) = {x ∈ top : intersection of less than α open sets in x is open}, tight (α) = {x ∈ top : if x ∈ b̄ ⊂ x then x ∈ ā for some a ⊂ b, |a| < α}. while it is easy to show that every category in the first class is of the form sset(ω) we do not know if (at least) one of the categories of the second class is of the form sset(ω). references [1] j. adamek, h. herrlich and g. strecker, abstract and concrete categories (wiley and sons inc., 1990). [2] s. baron, note on epi in t0, canad. math. bull. 11(1968), 503–504. [3] l. m. brown and m. diker, ditopological texture spaces and intuitionistic sets, fuzzy sets and systems 98 (1998), 217–224. [4] l. m. brown, r. ertürk and ş. dost, ditopological texture spaces and fuzzy topology. i. general concepts. preprint, 2002. [5] g. c. l. brümmer and e. giuli, a categorical concept of completion, comment. math. univ. carolin. 33 (1992), 131–147. [6] g. c. l. brümmer, e. giuli and h. herrlich, epireflections which are completions, cahiers topologie géom. diff. catég. 33 (1992), 71–93. [7] m. m. clementino, e. giuli and w. tholen, topology in a category: compactness, portugal. math. 53 (1996), 397–433. [8] m. m. clementino and w. tholen, separation versus connectedness, topology appl. 75 (1997), 143–179. [9] d. deses, e. giuli and e. lowen-colebunders, on complete objects in the category of t0 closure spaces, applied gen. topology, 4 (2003), 25-34. [10] d. dikranjan and e. giuli, closure operators. i. topology appl. 27 (1987), 129–143. [11] d. dikranjan, e. giuli and a. tozzi, topological categories and closure operators, quaestiones math. 11 (1988), 323–337. [12] d. dikranjan and w. tholen categorical structure of closure operators (kluwer academic publishers, dordrecht, 1995). [13] y. diers, affine algebraic sets relative to an algebraic theory , j. geom. 65 (1999), 54–76. [14] r. engelking, general topology, (heldermann verlag, berlin 1988). [15] e. giuli, zariski closure, completeness and compactness, mathematik-arbeitspapiere (univ. bremen) 54 (2000), 207–216. [16] e. giuli and m. hušek, a counterpart of compactness, boll. un. mat. ital.(7) 11-b (1997), 605–621. on classes of t0 spaces 155 [17] h. herrlich, topological functors, gen. topology appl. 4 (1974), 125–142. [18] th. marny, on epireflective subcategories of topological categories, gen. topology appl. 10 (1979), 175–181. [19] w. pratt, chu spaces and their interpretation as concurrent objects (springer lecture notes in computer science 1000 1995), 392–405. [20] g. preuss, theory of topological structures (d. reidel publishing company, 1988). [21] l. skula, on a reflective subcategory of the category of all topological spaces, trans. amer. math. soc. 142 (1969), 137–141. received march 2002 revised march 2003 eraldo giuli department of mathematics, university of l’aquila, 67100 l’aquila, italy. e-mail address : giuli@univaq.it on classes of t0 spaces admitting completions. by e. giuli applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 65–76 every finite system of t1 uniformities comes from a single distance structure jobst heitzig abstract. using the general notion of distance function introduced in an earlier paper, a construction of the finest distance structure which induces a given quasi-uniformity is given. moreover, when the usual defining condition xuε y :⇔ d(y,x) 6 ε of the basic entourages is generalized to nd(y,x) 6 nε (for a fixed positive integer n), it turns out that if the value-monoid of the distance function is commutative, one gets a countably infinite family of quasi-uniformities on the underlying set. it is then shown that at least every finite system and every descending sequence of t1 quasi-uniformities which fulfil a weak symmetry condition is included in such a family. this is only possible since, in contrast to real metric spaces, the distance function need not be symmetric. 2000 ams classification: primary 54e15; secondary 54e35, 54a10, 54e70. keywords: distance function, free monoid, generalized metric, uniformity. 1. introduction since fréchet’s invention of real metric spaces in [2], many generalizations of this concept have been studied in the literature. much research has been done on generalized metric spaces, in which the distance functions are replaced by certain set systems (cf. [9]). on the contrary, many authors independently suggested more general types of distance functions, the references [8], [12], [7], [5], [6], [11], [10], and [1] are only a small selection. in [3] and [4], a common framework for most if not all of these general concepts of distance functions has been developed to a certain extent. in this paper, the induction of quasi-uniformities on a distance space (x,d,m,p) will be studied. in such a structure, d : x ×x → m is a general distance function on x, that is, it fulfils the zero-distance condition d(x,x) = 0 and the triangle inequality d(x,y) + d(y,z) > d(x,z), and takes its values in a quasi-ordered monoid (q. o. m.) m = (m, +, 0,6). the set p ⊆ m must be a set of positives (or idempotent zero-filter ) for m, that is, a filter of (m,6) with 66 jobst heitzig infimum 0 such that, for every ε ∈ p, there is δ ∈ p with 2δ 6 ε. the triple (d,m,p) is called a distance structure on x. for examples and categorical aspects of distance functions on various mathematical objects, see [3, 4]. using kelley’s metrization lemma, one can easily show that every quasiuniformity is induced by a suitable multi-quasi-pseudo-metric, that is, a “quasipseudo-metric” taking values in a real vector space instead of the non-negative reals. there is no doubt that this fact must have been noticed early. in this article however, we will see that also every finite family of t1 uniformities (and many families of t1 quasi-uniformities) on a fixed set x comes from a single distance structure. in theorem 8, this is proved by constructing the finest such structure. this construction is a combinatorially more complex variant of the construction of a finest distance structure for a given quasi-uniformity, which is given in theorem 2. in contrast to multi-quasi-pseudo-metric spaces, the “topological” information in the resulting spaces will be mostly contained in the set of positives p rather than in the distance function d itself. for example, each t1 quasi-uniformity on some set x can be induced using one and the same distance function. 2. preliminaries in generalization of the usual definition of entourages in a metric space, let un(ε) := { (x,y) ∈ x ×x : nd(y,x) 6 nε } for every ε ∈ p and every positive integer n. as p is a filter, the set en := {un(ε) : ε ∈ p} is a base for a filter un of reflexive relations on x for each n. moreover, when m is commutative, nd(y,x) 6 δ > nd(z,y) implies nd(z,x) 6 n ( d(z,y) + d(y,x) ) 6 2δ, so that, for every ε ∈ p, there is δ ∈ p with un(δ)2 ⊆ un(ε), that is, un is a quasi-uniformity. of course, there are certain relationships between the un, and in many cases most of them coincide. obviously, n = n1 + · · · + nk impliesun1 (ε) ∩·· ·∩unk(ε) ⊆ un(ε). also, nd(x,y) 6 nmd(x,y) + (m− 1)nd(y,x), so that (2m− 1)nδ 6 nε impliesum(δ) ∩u−1n (δ) ⊆ un(ε). for a positive d (that is, when d(x,y) > 0 for all x,y), n 6 m and mδ 6 nε imply um(δ) ⊆ un(ε). (†) on the other hand, a symmetric d (that is, one with d(x,y) = d(y,x)) fulfils 2d(x,y) = d(x,y) + d(y,x) > d(x,x) = 0, so that here the implication (†) holds at least when m−n is even. this proves the following uniformities and distance structures 67 lemma 2.1. (a) n = n1 + · · ·+ nk implies un ⊆ un1 ∨·· ·∨unk , in particular, the map n 7→ un is antitone with respect to divisibility. (b) for all n,m, un ⊆ u−1n ∨ um. (c) for a positive d, all un coincide. (d) for a symmetric d and all k > 1, u2k = u2 ⊆ u1 = u2k−1. note that there are indeed natural distance functions which are neither positive nor symmetric, the most important being perhaps the distance x−1y on groups, introduced by menger [8]: example 2.2. let g := [0, 2π) be the additive group of real numbers modulo 2π, m := (p(g), +,{0},⊆) the power set of g ordered by set inclusion and with the usual element-wise addition, p := { (−δ,δ) : δ ∈ (0, 2π] } . then d(x,y) := {y − x} defines a skew-symmetric distance function (that is, one with d(x,y) + d(y,x) = 0), and u1 is the usual “euclidean” uniformity on g, while un is this uniformity “modulo 2πn ” since xun(−δ,δ) y ⇐⇒ x−y ∈ ⋃ k∈n(−δ + 2kπ n , 2kπ n + δ). likewise, for x := c \ {0}, m′ := m ⊗ [0,∞), p ′ := p × (0,∞), and d′(x,y) := ( d′(arg x, arg y), ∣∣|y|−|x|∣∣), the uniformity un of (d′,m′,p ′) induces the euclidean topology “modulo multiplication with nth roots of unity”. 3. finest distance functions like for other topological structures on a set x, we might compare two distance functions d,d′ resp. distance structures d = (d,m,p) and d′ = (d′,m′,p ′) on x with respect to their fineness. if the implication d(x1,y1) + · · · + d(xn,yn) 6 d(z1,w1) + · · · + d(zm,wm) =⇒ d′(x1,y1) + · · · + d′(xn,yn) 6 d′(z1,w1) + · · · + d′(zm,wm) holds for all xi,yi,zi,wi ∈ x, we say that d is finer than d′. if, additionally, for all ε′ ∈ p ′, there is ε ∈ p such that d(x1,y1) + · · · + d(xn,yn) 6 ε =⇒ d′(x1,y1) + · · · + d′(xn,yn) 6 ε′ for all xi,yi ∈ x, we say that d is finer than d′. for a convenient notation, let me introduce the free monoid f of all words in x that have even length and define d(x1y1 · · ·xnyn) := d(x1,y1) + · · · + d(xn,yn), srd t :⇔ d(s) 6 d(t) (s,t ∈ f). by definition, (f,◦, 0,rd) is a q. o. m., where ◦ is concatenation and 0 is the empty word. given any quasi-order r on f which is compatible to ◦ (that is, whenever (f,◦, 0,r) is a q. o. m.), the following construction leads to a distance function dr if and only if xxr 0 rxx and xz rxyyz for all x,y,z ∈ x. (?) 68 jobst heitzig let (mr,⊆) := θ(f,r) be the lower set completion of (f,r), that is, the system of all lower sets ra := {s : srt for some t ∈ a} of (f,r) with set inclusion as partial order. define an associative operation +r on mr and its neutral element 0r by ra +r rb := r{s◦ t : s ∈ a and t ∈ b} for all a,b ⊆ f and 0r := r{0}. then let dr : { x ×x → mr = (mr, +r, 0r,⊆) (x,y) 7→ r{xy}. it was shown in [3] that drd is equivalent to d, which motivates calling rd the generating quasi-order of d. moreover, when r⊥ is the smallest quasi-order on f which fulfils (?) and is compatible with ◦ then d⊥ := dr⊥ is a finest distance function on x. in this relation, the step from s ∈ f to an upper neighbour w. r. t. r⊥ consists of inserting a pair yy at an arbitrary position in s or removing a pair yy after an even number of letters in s, while the step to a lower neighbour is made by removing a pair yy at an arbitrary position or inserting a pair yy after an even number of letters. 4. induction of a single quasi-uniformity we are now ready for the first main result of this paper: theorem 4.1. every quasi-uniformity v admits a finest distance structure (dv,mv,pv) for which v = u1. proof. let v be some quasi-uniformity on x and v0 := ⋂ v. we will see that the essential information about v is contained in the set of positives pv which we must construct, while the generating quasi-order rdv is fully determined by the very weak condition that xy rdv zz must hold for any triple x,y,z ∈ x which fulfils y v0 x (otherwise dv(x,y) 66 ε for some ε ∈ pv, in contradiction to v0 ⊆ u1(ε)). therefore, let r be the smallest quasi-order on f that is compatible with ◦ and fulfils x′y′r 0 rxx and xz rxyyz for all x,y,z,x′,y′ ∈ x with y′v0 x′. (?′) if we find a suitable s. o. p. p such that (dr,p) induces v then r must obviously be the smallest relation (and thus dr a finest distance function) with this property. now observe that each of the resulting entourages u1(ε) has to include some entourage v1 ∈ v, hence every ε ∈ p must include some set {xy ∈ f : y v1 x} with v1 ∈ v. since 0r = r{xx} is a neutral element, ε must even include the set {xy ∈ f : y v0v1v0 x}⊆ 0r +r {xy ∈ f : y v1 x} +r 0r. the same must be true for any δ ∈ p which fulfils δ +r δ ⊆ ε, so that ε must also include a set {xyx′y′ ∈ f : y v0v2v0 x,y′v0v2v0 x′} ⊆ δ +r δ for some v2 ∈ v. this process of replacing some ε by some 2δ can be continued, and in uniformities and distance structures 69 order to describe it formally, let us define w to be the smallest set of tuples of positive integers that contains the 1-tuple (1) and fulfils (n1, . . . ,ni−1,ni + 1,ni + 1,ni+1, . . . ,nk) ∈ w whenever (n1, . . . ,nk) ∈ w and 1 6 i 6 k. one can think of the elements of w as coding exactly those terms of the form ‘εn1 +· · ·+εnk’ that can be obtained when we start with the term ‘ε1’ and then successively replace an arbitrary summand ‘εn’ by the term ‘εn+1 + εn+1’. accordingly, one shows by induction that for each element ε1 of a set of positives p there is a sequence ε2,ε3, . . . in p such that (n1, . . . ,nk) ∈ w implies εn1 + · · · + εnk 6 ε1. in our situation, this observation implies that for each ε ∈ p there must be a sequence s = (v1,v2, . . . ) in v with the property that ε includes the set as of all words v1w1 · · ·vkwk ∈ f for which there is some (n1, . . . ,nk) ∈ w such that wi v0vniv0 vi for i = 1, . . . ,k. in particular, εs := ras ⊆ rε = ε. it turns out that this is the only restraint on the set of positives pv. more precisely, we will see that the system b := {εs : s is a sequence in v} of lower sets of (f,r) is a base for a set of positives of (mr, +r, 0r,⊆), and that the distance structure (dr,p) induces the quasi-uniformity v. it is then clear that p is the largest set of positives with this property, so that (dv,pv) := (dr,p) is a finest distance structure inducing v. since v is a filter and the map s 7→ εs is isotone in every component of s, b is a filter-base. in order to show that p is a s. o. p., we first observe that (n1, . . . ,nk), (m1, . . . ,ml) ∈ w implies (n1 + 1, . . . ,nk + 1,m1 + 1, . . . ,ml + 1) ∈ w. indeed, after increasing each index by one, the replacements that produce (n1, . . . ,nk) and (m1, . . . ,ml) from the tuple (1) can be combined to a sequence of replacements that produce (n1 + 1, . . . ,nk + 1,m1 + 1, . . . ,ml + 1) from the tuple (2, 2). hence also v1w1 · · ·vkwk,v′1w′1 · · ·v′lw ′ l ∈ ε(v2,v3,v4,... ) implies v1w1 · · ·vkwkv′1w ′ 1 · · ·v ′ lw ′ l ∈ ε(v1,v2,v3,... ) for each sequence (v1,v2, . . . ) in v. secondly, we must prove that ⋂ b = 0r, which is the harder part. let s = x1z1 · · ·xmzm ∈ ⋂ b and v1 ∈ v. i will show that zj v0v1v0 xj holds for all j = 1, . . . ,m. choose a sequence s = (v1,v2, . . . ) in v such that vi+1v0vi+1 ⊆ vi for all i > 1 (such a sequence always exists in a quasi-uniformity). note that (n1, . . . ,nk) ∈ w then implies v0vn1v0vn2v0 · · ·v0vnkv0 ⊆ v0v1v0. now s ∈ ras, that is, there exists a word v1w1 · · ·vkwk and a k-tuple (n1, . . . ,nk) ∈ w such that wi v0vniv0 vi for i = 1, . . . ,k and srv1w1 · · ·vkwk. the latter means that, starting with v1w1 · · ·vkwk, one gets x1z1 · · ·xmzm in finitely many steps in each of which 70 jobst heitzig some pair of letters is inserted or removed corresponding to the condition (?′). now take the k-tuple ψ := (w1 v0vn1v0 v1, . . . ,wk v0vnkv0 vk) of formulae (which express true propositions about the word v1w1 · · ·vkwk) and modify it, analogously to those finitely many steps, in the following way: (i) if (because of xz rxyyz) a pair yy is being removed after an odd number of letters, replace the two consecutive formulae . . .v0 y,y v0 · · · in ψ by one formula . . .v0 · · · (that is, erase the symbols ‘y,y v0’); (ii) if (because of 0 rxx) a pair xx is being removed after an even number of letters, remove the corresponding formula x.. .x from ψ; (iii) if (because of x′y′r 0) a pair x′y′ is inserted, insert the formula y′v0 x′ at the respective position in ψ. by definition of r, all these modifications preserve the truth of all formulae in the tuple, and each formula in the resulting tuple (ψ1, . . . ,ψk) expresses a true proposition of the form ψj = zj v0vnav0vna+1v0 . . .v0vnbv0 xj with 1 6 a,b 6 k. since all vni are reflexive, ψj thus implies zj v0vn1v0vn2v0 . . .v0vnkv0 xj, hence zj v0v1v0 xj. because v1 was chosen arbitrarily, we conclude that zj v0 xj for all j, and therefore x1z1 · · ·xmzm r 0. finally, we have to show that (dr,p) induces the quasi-uniformity v. for v ∈ v, choose v1 ∈ v such that v0v1v0 ⊆ v , then choose a sequence s as in the preceding paragraph. there we have shown that, in particular, dr(x,z) ⊆ ras implies (z,x) ∈ v0v1v0 ⊆ v. on the other hand, for each ε ∈ p there is some sequence s = (v1, . . . ) in v such that εs ⊆ ε, and (z,x) ∈ v1 ⊆ v0v1v0 impliesdr(x,z) ⊆ εs ⊆ ε. � a somewhat astonishing consequence of this construction is that one distance function is compatible to all t1 quasi-uniformities on x: corollary 4.2. the distance function d⊥ is the finest distance function d on x such that for each t1 quasi-uniformity v on x there is a s. o. p. p such that (d⊥,p) induces v (namely p = pv). 5. induction of systems of quasi-uniformities i will now extend this result to certain systems of quasi-uniformities and show that, in particular, every finite system and every descending sequence of t1 uniformities is part of some system (un)n∈ω. some additional notation: intervals of integers will be designated by [a,b]. a pair of letters xy ∈ f is a syllable of a word s ∈ f if and only if it occurs in s after an even number of letters. let s̃ ∈ f be the word s after deletion of all syllables of the form xx (x ∈ x). the length of s̃ in letters is designated by uniformities and distance structures 71 `(s), and sa is the ath letter of s̃ for any position a ∈ [1,`(s)]. the subword of s̃ from position a to b is sa,b. moreover, let λ(x,s) and σ(xy,s) denote the number of occurrences of the letter x resp. the syllable xy in s̃. finally, (xy)r = xy · · ·xy is a word consisting of r equal syllables. the next constructions mainly rely on four lemmata. for the moment, let us fix some words s,t ∈ f with sr⊥ t, where t̃ = (v1w1) r1 · · ·(v%w%)r%, vi 6= wi, and all ri are even. then s̃ can be derived from t̃ by a finite number of successive deletions of pairs of identical letters which are neighbours at the time of deletion. a guiding example: for s = yy xy zz xy uz uz r⊥xy xy zz zuuz uz xxuz = t, the deletion steps could be this: in t̃ = xy xy zuuz uz uz, first delete uu, giving xy xy zz uz uz, then delete zz, giving xy xy uz uz = s̃. we now also fix such a sequence of deletions and let d ⊆ [1,`(t)] be the set of positions in t̃ whose corresponding letters are deleted in one of these steps (in the example: d = [5, 8]). for a ∈ d, let π(a) ∈ [1,`(t)] be that position in t̃ such that ta and tπ(a) build a deleted pair (in the example: π(5) = 8 and π(6) = 7). finally, we write ayb if and only if a and b− 1 are even numbers in d such that a < π(a) = b− 1 (in the example: 6y8). note that because tc and tπ(c) must first become neighbours before they can be deleted, ay· · ·yb implies that (i) [a,b−1] ⊆ d, (ii) π(c) ∈ [a,b−1] for all c ∈ [a,b−1], and thus (iii) λ(x,ta,b−1) is even for all x ∈ x. lemma 5.1. assume ay· · ·yby· · ·yc, ta = tb−1, and tb = tc−1. then (a) ta−1 = tb or tb−1 = tc. (b) if ta−1 6= tb then λ(ta, tc,`(t)) is odd. (c) if tb−1 6= tc then λ(tb, t1,a−1) is odd. proof. let e,f,e′,f′,e′′,f′′ ∈ [1,`(t)] with e < a 6 f < e′ < b 6 f′ < e′′ < c 6 f′′ such that te,f , te′,f′, and te′′,f′′ are three of the defining subwords (viwi)ri of t̃. moreover, let x := ta−1, y := ta = tb−1, z := tb = tc−1, and w := tc, and assume x 6= z. the situation and the parity arguments that will follow are sketched in figure 1. because of x 6= z, we have λ(x,te′,b−1) = 0. moreover, λ(x,tf+1,e′−1) is even (since all ri are even), and λ(x,ta,b−1) is even because of (iii), so that also λ(x,ta,f ) is even and λ(y,ta,f ) is odd (since |[a,f]| is odd). as before, λ(y,tf+1,e′−1) and λ(y,ta,b−1) are even, thus λ(y,te′,b−1) is odd. because all ri are even, λ(y,tb,f′) is also odd. again, λ(y,tf′+1,e′′−1) and λ(y,tb,c−1) are even, hence λ(y,te′′,c−1) is odd. in particular, y ∈{z,w}, that is, y = w (as yz is a syllable of t̃), and λ(y,tc,f′′) is also odd. finally, λ(y,tc,`(t)) is odd because λ(y,tf′′,`(t)) is even. this proves (a) and (b), whereas (c) is strictly analogous to (b). � 72 jobst heitzig figure 1. situation in lemma 5.1 π π� � ? ?· · · � � ? ? t̃ = · · ·(xy · · ·xy · · ·xy) · · · · · · · · · · · · · · ·(yz · · ·y | | (continued on | the next line) |↑ e ↑ a ↑ f ↑ e′ ↑ b− 1 even } λ(x) even even 0 odd even odd } λ(y) even π π� � ? ?· · · � � ? ?| (conti| nued) | | z · · ·yz) · · · · · · · · · · · · · · ·(zw · · ·zw · · ·zw) · · · ↑ b ↑ f′ ↑ e′′ ↑ c ↑ f′′ λ(y) { odd even odd odd even even odd lemma 5.2. (a) assume that a0 yb0 ya1 yb1 · · ·akybkyc with ta0 = · · · = tak = y, and tb0 = · · · = tbk = z. then ta0−1 = z or y = tc. (b) assume that a y · · ·y b with ta = tb−1, and ta−1 6= tb. then both λ(ta, t1,a−1) and λ(ta, tb,`(t)) are odd. proof. (a) define e′′,f′′ as above. similarly, for each i ∈ [0,k], find positions ei,fi,e ′ i,f ′ i ∈ [1,`(t)] with ei < ai 6 fi < e ′ i < bi 6 f ′ i such that tei,fi and te′i,f′i are two of the defining subwords of t̃. assuming ta0−1 = x 6= z, one proves that λ(y,tb0,f′0 ) is odd exactly as before. since, for i ∈ [1,k], all of λ(y,tbi−1,ai−1), λ(y,tai,bi−1), λ(y,tf′i−1+1,ei−1), λ(y,tei,fi), λ(y,tfi+1,e′i−1), and λ(y,te′i,f′i ) are even, and since also λ(y,tbk,c−1) and λ(y,tf′k+1,e′′−1) are even, we conclude that λ(y,te′′,c−1) is odd, hence y = tc. (b) again as in the previous lemma, one proves that, for y := ta, the number λ(y,tb,f′) is odd, so that the first claim follows because λ(y,tf′,`(t)) is even. the second claim is just the dual. � lemma 5.3. assume that se−1se = xz is the syllable of s̃ that remains after all the deletions in a subword ta−1,b of t̃, with a < b, ta−1 = x, and tb = z. then there is y ∈ x such that λ(y,s) > 0, σ(xy,ta−1,b) > 0, and σ(yz,ta−1,b) > 0. proof. although ta and tb−1 may be different, we find k > 2, c1, . . . ,ck ∈ [1,`(t)], and y0,y1, . . . ,yk ∈ x such that a = c1 y· · ·yc2 y· · ·yc3 · · ·ck−1 y· · ·yck 6 b, uniformities and distance structures 73 tci = tci+1−1 = yi for i ∈ [1,k− 1], y0 = x, yk = z, and yi 6= yj for i 6= j (start with a =: c′1 yc ′ 2 y · · ·yc′l := b and y ′ i := tc′i. as long as there are indices j > i > 1 with y′i = y ′ j, remove all the indices i + 1, . . . ,j, so that finally all remaining y′i are different. since y ′ 1 = ta 6= z = y′l, at least k ≥ 2 of the original indices are not removed, including the index 1, and the corresponding c′i build the required positions c1, . . . ,ck). then k = 2 since otherwise lemma 5.1 (a) would imply that either y0 = y2 or y1 = y3. with y1 for y and c2 for b, lemma 5.2 (b) implies that λ(y,t1,a−1) is odd. now, also λ(y,s1,e−1) is odd, because c ∈ [1,a− 1] ∩d implies π(c) ∈ [1,a − 1] (since the letter x at position a − 1 is not deleted). in particular, λ(y,s1,e−1) > 0. � lemma 5.4. assume that k > 2, c0 yc1 · · ·ck−1 yck, ck ∈ d, and π(ck) = c0 − 1, representing a number of deletions of the form � �π � �π · · · · · · · · · · · · � �π� �π ?? tc0 ?? tc1 ? ? tck−1 ?? tck let t′ := tc0−1tc0tc1−1tc1 · · ·tck−1tck be the word consisting only of the “boundary letters”, and i ∈ [0,k]. then σ(tci−1tci, t′) = σ(tcitci−1, t′). proof. put c−1 := ck. obviously, tci−1 = tci−1 for all i ∈ [1,k], and tck = tc0−1. if also tci−1−1 = tci for all i ∈ [0,k] then k must be odd (since tck 6= tc0 ), and σ(tci−1tci, t ′) = σ(tcitci−1, t ′) = k/2. otherwise, there are r > 1 positions i(1) < · · · < i(r) in [0,k] with tci(j)−1−1 6= tci(j) . then i(j + 1) − i(j) is even for all j (otherwise, put a0 := ci(j)−1, b0 := ci(j),. . . , c := ci(j+1)−1 and apply lemma 5.2 (a)). in case that all i(j) are even, we have tck−1 6= tck = tc0−1 = tc1 = tci for all odd i, so that k must be odd. on the other hand, if all i(j) are odd, we have tck = tc0 − 1 6= tc0 = tci for all even i, so that again k must be odd. this shows that t′ is of one of the following two forms: t′ = (yxxy)m0 (yz1z1y)m1 · · ·(yzr−1zr−1y)mr−1 (yxxy)mr or t′ = xy(yxxy)m0 (yz1z1y)m1 · · ·(yzr−1zr−1y)mr−1 (yxxy)mryx, from which the claim follows immediately. � now we are ready for the construction. let pi be the ith odd prime number, and s(a) := {a1 + · · · + ak : k > 1, ai ∈ a} for any set a of integers. in the next theorem, we need the following sets of even numbers: for any positive integer u, let quj = 2pj ∏u i=1 pi for all j ∈ [1,u], qu := {qu1, . . . ,quu}, and quj := qu \ {quj}. it is easy to see that then, for each j ∈ [1,u] and k ∈ s(quj), k −quj /∈ s(quj) (since pj divides k but not quj). 74 jobst heitzig theorem 5.5. (a) let v1, . . . , vu be a finite system of t1 quasi-uniformities such that, for all i,j ∈ [1,u], vj ⊆ v−1j ∨ vi. then there is a finest s. o. p. p such that, for j ∈ [1,u], vj = uquj . (b) let v1 ⊇ v2 . . . be a descending sequence of t1 quasi-uniformities such that, for all j and all u ∈ vj, there are v1 ∈ v1,v2 ∈ v2, . . . with v −1j ∩ ⋃ i 6=j vi ⊆ u. then there is a finest s. o. p. p such that vj = u2j for all j. proof. for part (a), let i := [1,u], while for part (b), let i be the set of natural numbers. in both cases, p is defined quite analogously to the proof of theorem 4.1: its filter-base is now the system b := {εs : s is a sequence in v} of lower sets εs = r⊥as of r⊥, where v := ∏ i vi, and the definition of as changes to this: for s = ((v11,v12, . . . ), (v21,v22, . . . ), . . . ), as is now the set of all words (v1w1)r1 (v2w2)r2 · · ·(v%w%)r% ∈ f for which there is some (n1, . . . ,n%) ∈ w and some tuple of indices (i1, . . . , i%) such that, for all a ∈ [1,ρ], wa vnaia va and either ra = quia (for the proof of (a)) or ra = 2ia (for the proof of (b)). as before, p turns out to be a s. o. p., where the only major change is the proof of ⋂ b = 0r: let s ∈ ⋂ b, σ(xz,s) > 0, and v = (v11,v12, . . . ) ∈ v. choose s so that vk+1,ivk+1,i ⊆ vki for all i ∈ i and all k, and some t ∈ as with sr⊥ t. assume that t̃ = (v1w1)r1 (v2w2)r2 · · ·(vρwρ)rρ. if σ(xz,t) > 0, put yv := x, otherwise choose some yv ∈ x with λ(yv ,s) > 0, σ(xyv , t) > 0, and σ(yv z,t) > 0, according to lemma 5.3. since `(s) is finite and v is filtered, there is some y such that, for all v ∈ v, there is v ′ ∈ v with v ′ 6 v and yv ′ = y, where 6 denotes component-wise set inclusion. consequently, xuv y uv z for all v ∈ v, where uv = ⋃ i v1i. this implies that x,y ∈ ⋂ vi and x,y ∈ ⋂ vi′ for some i, i′ ∈ i, hence x = y = z. since this is a contradiction to x 6= z, we have shown that s̃ is the empty word, that is, s ∈ 0r. finally, let us show that vj = uquj resp. vj = u2j for each j ∈ i. fix some j ∈ i and let v0j ∈ vj. because of the premises, the following choices can now be made. for part (a), choose for all i ∈ i\{j} some v0i ∈ vj and v1i ∈ vi such that (v0i)−1∩v1i ⊆ v0j. then choose v1j ∈ vj such that v1j ⊆ v0i for all of the finitely many i ∈ i \{j}. for part (b), choose instead some (v11,v12, . . . ) ∈ v with v1h = v1j ⊆ v0j for all h 6 j and (v1j)−1 ∩ ⋃ i 6=j v1i ⊆ v0j. after that, choose the remaining components of a sequence s = ( (v11,v12, . . . ), (v21,v22, . . . ), . . . ) in v so that vk+1,ivk+1,i ⊆ vki for all i ∈ i and all k, and assume that rdr⊥(x,y) 6 εs, that is, s := (xz) r r⊥ t ∈ as with (a) r = quj resp. (b) r = 2j. we have to show that z v0j x. uniformities and distance structures 75 by definition of as, we have t̃ = (v1w1)r1 (v2w2)r2 · · ·(vρwρ)rρ, and there is some corresponding tuple (i1, . . . , iρ). since the only letters in s̃ are x and z, there are exactly r occurrences of the syllable xz in t̃ which are not deleted (because otherwise lemma 5.3 would imply the existence of a third letter y in s̃). all other occurrences of xz in t̃ are deleted as part of some set of deletions of the form represented in lemma 5.4, that is, there are c0,. . . ,ck with properties as in lemma 5.4 and with tci−1tci = xz for some i ∈ [0,k]. then the lemma implies that σ(xz,t) = r + σ(zx,t) =: k. for (a): if (vawa)ra = (xz)quj for some a ∈ [1,ρ], then ia = j and (z,x) ∈ vna,ia ⊆ v1j ⊆ v0j. otherwise, we know that k ∈ s(quj), that is, σ(zx,t) = k − quj ∈ s(qu) \ s(quj), so that (vawa)ra = (zx)quj and ia = j for some a ∈ [1,ρ]. also, (vbwb)rb = (xz)qui and ib = i for some b ∈ [1,ρ] and some i ∈ i \{j}, so that (z,x) ∈ (v1j)−1 ∩v1i ⊆ v0j. for (b) instead: if (vawa)ra = (xz)2 i for some a ∈ [1,ρ] and i 6 j, then ia = i and (z,x) ∈ vna,ia ⊆ v1i ⊆ v0j. otherwise, k is a multiple of 2j+1 so that σ(zx,t) = k − 2j is not such a multiple. therefore, (vawa)ra = (zx)2 ia and ia 6 j for some a ∈ [1,ρ]. also, (vbwb)rb = (xz)2 ib and ib 6= j for some b ∈ [1,ρ], so that again (z,x) ∈ (v1ia)−1 ∩v1ib ⊆ v0j. � unfortunately, this proof highly depends on the fact that mr⊥ is not commutative, so that the conjecture that there is also a suitable distance structure with a commutative value monoid is yet unproved. the most familiar example for a descending sequence of uniformities is perhaps the following. let x := cb[0, 1] be the (infinite-dimensional) vector space of bounded, continuous, and real-valued functions on the unit interval [0, 1], and, for positive integers p, let vp be the uniformity on x induced by the usual p-norm. for a second example, take u different primes p1, . . . ,pu and let vi be the pi-adic uniformity on the rationals. as these are transitive uniformities with countable bases, we may use a slightly simpler construction. more precisely, a base for vi is the set of equivalence relations ui,m := {(x,y) : pmi divides ν(|x−y|)}, where m is a positive integer, and ν(z/n) := z whenever z,n have no common divisor (that is, ν(q) is the nominator of q). therefore, it suffices to use only those εs where all tuples in s are equal, that is, vh+1,i = vhi for all i,h. in this case, the resulting s. o. p. p has a countable base b = {εm : m a positive integer}, where εm := ∞⋃ n=0  n · ⋃ j∈[1,u], (x,y)∈uj,m qujd⊥(x,y)   . 76 jobst heitzig as a concluding remark, i note that with similar methods, one can show that, for each pair of comparable t1 uniformities v2 ⊆ v1, there is some symmetric distance structure (d,p) such that ui = vi, which gives a complete characterization of the symmetric t1 case. references [1] m. m. bonsangue, f. van breugel, and j. j. m. m. rutten, generalized metric spaces: completion, topology, and powerdomains via the yoneda embedding, theoret. comput. sci. 193 (1998), no. 1-2, 1–51. [2] maurice fréchet, sur les classes v normales, trans. amer. math. soc. 14 (1913), 320– 325. [3] jobst heitzig, partially ordered monoids and distance functions, diploma thesis, universität hannover, germany, july 1998. [4] , many familiar categories can be interpreted as categories of generalized metric spaces, appl. categ. structures (to appear) (2002). [5] ralph kopperman, all topologies come from generalized metrics, amer. math. monthly 95 (1988), no. 2, 89–97. [6] djuro r. kurepa, general ecart, zb. rad. (1992), no. 6, part 2, 373–379. [7] boyu li, wang shangzi, and maurice pouzet, topologies and ordered semigroups, topology proc. 12 (1987), 309–325. [8] karl menger, untersuchungen über allgemeine metrik, math. annalen 100 (1928), 75– 163. [9] j. nagata, a survey of the theory of generalized metric spaces, (1972), 321–331. [10] maurice pouzet and ivo rosenberg, general metrics and contracting operations, discrete math. 130 (1994), 103–169. [11] hans-christian reichel, distance-functions and g-functions as a unifying concept in the theory of generalized metric spaces, recent developments of general topology and its applications (berlin, 1992), akademie-verlag, berlin, 1992, p. 279–286. [12] b. schweizer and a. sklar, probabilistic metric spaces, north–holland, new york, 1983. received september 2001 revised january 2002 jobst heitzig institut für mathematik universität hannover welfengarten 1 d-30167 hannover germany e-mail address : heitzig@math.uni-hannover.de 15.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 27 31 continuous representability of completepreorders on the space of upper-continuouscapacitiesgianni bosi and romano islerabstract. given a compact metric space (x;d), and itsborel �-algebra �, we discuss the existence of a (semi)continuousutility function u for a complete preorder � on a subset m0(x) ofthe space m(x) of all upper-continuous capacities on �, endowedwith the weak topology.2000 ams classi�cation: 06a05, 91b16.keywords: upper-continuous capacity, compact metric space, weak topology,continuous utility function. 1. introductionin decision theory under uncertainty, it is usual to consider a separable met-ric space (x;d) of possible consequences of a game, endowed with its borel�-algebra � (i.e., the �-algebra generated by the open subsets of x). a playeris required to choose a probability measure from a set p of �-additive probabil-ity measures on the measurable space (x;�), endowed with the induced weaktopology. the preferences of the player among probability measures in p are ex-pressed by a complete preorder (i.e., a re exive, transitive and complete binaryrelation) � on p . this is the usual model for expected utility (see grandmont[5]). in a more general setting, it may be assumed that player's uncertainty isre ected by capacities better than probability measures (see e.g. epstein andwang [4]). in particular, the notion of an upper-continuous capacity general-izes the notion of a �-additive probability measure in the case of additive setfunctions. topological properties of the space m(x) of all upper-continuouscapacities on the measurable space (x;�), endowed with the weak topology,have been studied by zhou [7] in case that (x;d) is a compact (metric) space.in this paper, we use the results proved by zhou [7] in order to discuss theexistence of a continuous or at least upper semicontinuous utility function u 28 g. bosi and r. islerfor a complete preorder � on a subset m 0(x) of the space m(x) of all upper-continuous capacities.2. notation and preliminariesthroughout this paper, we shall always consider a compact metric space(x;d), endowed with its borel �-algebra, denoted by �. the space of all upper-continuous capacities on � will be denoted by m(x) (see zhou [7]). we recallthat a capacity � on � (i.e., a function from � into [0;1] such that �(?) = 0,�(x) = 1, and �(a) � �(b) for all a � b, a;b 2 �) is said to be upper-continuous if limn!1�(an) = �(t1n=1an) for any weakly decreasing sequence ofsets fang with an 2 � for all n. a sequence f�ng � m(x) is said to convergeweakly to � 2 m(x) ifzx fd�n ! zx fd� for all f 2 c(x);with c(x) the space of all continuous real-valued functions on (x;d), andrx fd� the choquet integral of f with respect to �, namelyzx fd� = z 10 �(f � t)dt + z 0�1(�(f � t) � 1)dt:the corresponding topology (i.e., the weak topology on m(x)) will be denotedby �w. the reader could refer to the comprehensive book by denneberg [2] fordetails concerning the basic properties of the choquet integral. more recentresults on the choquet integral with respect to upper-continuous capacities arefound in zhou [7].given a complete preorder (i.e., a re exive, transitive and complete binaryrelation) � on a subset m 0(x) of m(x), we are interested in the existence ofa utility function u for � (i.e., a real-valued function on m 0(x) such that, forevery �;� 2 m 0(x), � � � if and only if u(�) � u(�)) which is continuousor at least upper semicontinuous in the topology induced on m 0(x) by theweak topology �w. we recall that a complete preorder � on a subset m 0(x)of m(x) is said to be upper (lower) semicontinuous if f� 2 m 0(x) : � � �g(f� 2 m 0(x) : � � �g) is a closed set for every � 2 m 0(x). further, a completepreorder � is said to be continuous if it is both upper and lower semicontinuous.3. continuous representationsin the following theorem, we are concerned with the existence of a continuousor at least upper semicontinuous utility function for a complete preorder on anarbitrary set m 0(x) of upper-continuous capacities.theorem 3.1. let (x;d) be a compact metric space. then the following state-ments hold:(i) every upper semicontinuous complete preorder � on every subset m 0(x)of m(x) admits an upper semicontinuous utility function u;(ii) every continuous complete preorder � on every subset m 0(x) of m(x)admits a continuous utility function u. upper-continuous capacities and utility 29proof. from zhou [7, theorem 3], the space (m(x);�w) is a compact metricspace, and therefore it is in particular a separable metric space (see e.g. engelk-ing [3, theorem 4.1.18]). then every subset m 0(x) of m(x) can be metrized asa separable metric space, and therefore as a second countable metric space (seee.g. engelking [3, corollary 4.1.16]). if � is any upper semicontinuous completepreorder on (m 0(x);�wm0(x)), then � admits an upper semicontinuous utilityfunction u by rader's theorem (see rader [6, theorem 1]). if � is any continu-ous complete preorder on (m 0(x);�wm0(x)), then � admits a continuous utilityfunction u by debreu's theorem (see debreu [1, proposition 3]). so the proofis complete. �for an additive capacity � on �, the condition of upper-continuity is equiv-alent to the condition of countable additivity. therefore, since the space �(x)of all countably additive probability measures on (x;�) is contained in m(x),theorem 3.1 generalizes theorem 1 in grandmont [5] in case that a compactmetric space is considered.given a compact metric space (x;d), and a complete preorder � on a sub-set m 0(x) of m(x), containing the set d of all probability measures on themeasurable space (x;�) which are concentrated (i.e., d = fp 2 �(x) : p =px for some x 2 xg with px the probability measure assigning probability 1 tothe borel set fxg), we can consider the complete preorder �x on x whichis induced by the complete preorder � on m 0(x), in the sense that, for everyx;y 2 x, x �x y if and only if px � py. the following corollary to the previoustheorem concerns the representability of �x by means of a continuous or atleast upper semicontinuous utility function u on (x;d).corollary 3.2. let (x;d) be a compact metric space. then the followingstatements hold:(i) for every upper semicontinuous complete preorder � on every subsetm 0(x) of m(x) containing d, the induced complete preorder �x ad-mits an upper semicontinuous utility function u;(ii) for every continuous complete preorder � on every subset m 0(x) ofm(x) containing d, the induced complete preorder �x admits a con-tinuous utility function u.proof. given an upper semicontinuous complete preorder � on any subsetm 0(x) of m(x) containing d, by the previous theorem there exists an up-per semicontinuous utility function u for �. we claim that the real-valuedfunction u on x de�ned by u(x) = u(px) (x 2 x)is an upper semicontinuous utility function for the induced complete preorder�x on x. it is straightforward to show that u is a utility function for �x.indeed, we havex �x y , px � py , u(px) � u(py) , u(x) � u(y) 30 g. bosi and r. islerfor every x;y 2 x. in order to prove that u is upper semicontinuous (i.e., theset fx 2 x : � � u(x)g is closed for every real number �), consider any realnumber �, any point x 2 x, and any sequence fxng � x converging to x suchthat � � u(xn) for every n. since the sequence fpxng � d converges to px 2 d,u is upper semicontinuous, and from the de�nition of u we have � � u(pxn)for every n, it must be � � u(px) = u(x), and therefore the conclusion follows.this consideration �nishes the �rst part of the proof.if � is a complete preorder on any subset m 0(x) of m(x) containing d, bythe previous theorem there exists a continuous utility function u for �. then,by analogous considerations it can be shown that the function u de�ned aboveis a continuous utility function for �x. so the proof is complete. �it is almost immediate to check that the statements named (i) in the previoustheorem and corollary are still valid if we replace the terms \upper semicon-tinuous complete preorder" and \upper semicontinuous utility function" by theterms \lower semicontinuous complete preorder" and respectively \lower semi-continuous utility function". indeed, one can replace functions u and u by �uand respectively �u, and then apply the previous results by considering the dualcomplete preorders �d and �xd de�ned by [� �d � , � � �] and respectively[x �xd y , y �x x].acknowledgements. we are grateful to an anonymous referee for helpfulsuggestions. references[1] g. debreu, continuity properties of paretian utility, international economic review 5(1964), 285{293.[2] d. denneberg, non-additive measure and integral, kluwer, dordrecht, 1994.[3] r. engelking, general topology, polish scienti�c publishers, 1977.[4] l.g. epstein and t. wang, \beliefs about beliefs" without probabilities, econometrica64 (1996), 1343{1373.[5] j.-m. grandmont, continuity properties of a von neumann-morgenstern utility, journalof economic theory 4 (1972), 45{57.[6] t. rader, the existence of a utility function to represent preferences, review of economicstudies 30 (1963), 229{232.[7] l. zhou, integral representation of continuous comonotonically additive functionals,transactions of the american mathematical society 350 (1998), 1811{1822.received october 2000revised version february 2001 gianni bosi, romano islerdipartimento di matematica applicata\bruno de finetti", universit�a di trieste upper-continuous capacities and utility 31piazzale europa 134127 triesteitalye-mail address: giannibo@econ.univ.trieste.ite-mail address: romano.isler@econ.univ.trieste.it @ appl. gen. topol. 15, no. 1 (2014), 55-63doi:10.4995/agt.2014.2221 c© agt, upv, 2014 on the existence of best proximity points for generalized contractions asrifa sultana and v. vetrivel ∗ department of mathematics, indian institute of technology madras, chennai-600036, india. (asrifa.iitg@gmail.com,vetri@iitm.ac.in) abstract in this article we establish the existence of a unique best proximity point for some generalized non-self contractions on a metric space in a simpler way using a geometric result. our results generalize some recent best proximity point theorems and several fixed point theorems proved by various authors. 2010 msc: 54h25; 47h10. keywords: fixed points; generalized contractions; p-property; best proximity point. 1. introduction fixed point theory plays an important role in supplying a uniform treatment for solving equations of the form tx = x where t is a self mapping defined on a subset of a metric space, partially ordered metric space, topological vector space or some suitable space. given two non-empty subsets a and b of a metric space (x,d), consider a non-self mapping t : a → b. the mapping t is said to be a k−contraction if d(tx,ty) ≤ kd(x,y) hold ∀ x,y ∈ a and for some k ∈ [0,1). if t is a self map, that is, if a = b and a is complete, then the famous banach contraction principle implies that t has a unique fixed point in a. as this principle has applications in various fields, many generalizations of this principle have appeared in the literature (see [12, 13] ) by generalizing ∗corresponding author. received 28 october 2013 – accepted 7 march 2014 http://dx.doi.org/10.4995/agt.2014.2221 a. sultana and v. vetrivel the contractive condition used by banach. in 1975, matkowski [2] used the following contractive condition: (1.1) d(tx,ty) ≤ ϕ(d(x,y)), ∀ x,y ∈ a, where ϕ is a function from r+, the set of all nonnegative reals, into r+ such that ϕ is nondecreasing and satisfies limn→∞ϕ n(t) = 0 for any positive t. matkowski [2] proved that t has a unique fixed point if t is a self map and a is complete. on the other hand, rhoades [3] in 2001 gave an existence result of unique fixed point for mappings satisfying the following contractive condition: (1.2) d(tx,ty) ≤ d(x,y) − ψ(d(x,y)), ∀ x,y ∈ a, where ψ : r+ → r+ is a nondecreasing, continuous function with ψ −1(0) = {0} and limt→∞ ψ(t) = ∞ (if a is bounded, then the infinity condition can be omitted). rhoades [3] proved that t has a unique fixed point if t is a self map and a is complete. next, we present a brief discussion about best proximity point. it is clear that t(a)∩a 6= ∅ is a necessary (but not sufficient) condition for the existence of a fixed point for the map t : a → b. if the necessary condition fails, then d(x,tx) > 0, for all x ∈ a. this means that the mapping t : a → b does not have any fixed point, that is, the equation tx = x has no solution. from this point of view, we think of a point x in a which is closest to tx in some sense. best approximation and best proximity point results are being studied in this direction. the well-known best approximation theorem due to ky fan [14] states that if m is a non-empty compact convex subset of a normed linear space e and s : m → e is a continuous function, then there exists a point x ∈ m such that ‖x − sx‖ = d(sx,m) = inf{‖sx− a‖ : a ∈ m}. such an element x ∈ m satisfying ‖x−sx‖ = d(sx,m) is called a best approximant. on the other hand, though a best approximant acts as an approximate solution of the equation sx = x, such element is not an optimal solution in the sense that the distance between x and sx is minimum. naturally for given subsets a and b of a metric space and a mapping t : a → b one can think of finding a point x∗ ∈ a such that d(x∗,tx∗) = min{d(x,tx) : x ∈ a}. as d(x,tx) ≥ dist(a,b) = inf{d(a,b) : a ∈ a, b ∈ b} ∀ x ∈ a, then an optimal solution of min{d(x,tx) : x ∈ a} is one for which the value dist(a,b) is attained. a point x∗ ∈ a is said to be a best proximity point for the function t : a → b if d(x∗,tx∗) = dist(a,b). so a best proximity point of the map t is an approximate solution of the equation tx = x which is optimal in the sense that distance between x and tx is minimum. it is clear that all best proximity point theorems work as a natural generalization of fixed point theorems if t is a self-map. for some interesting best proximity point results one can refer to [6, 7, 9, 10]. some applications of best proximity point results can be found in [15, 16]. recently v. sankar raj [4] obtained the following best proximity point theorem for mappings satisfying (1.2). theorem 1.1 ([4, theorem 3.1]). let a,b be two non-empty closed subsets of a complete metric space (x,d) such that the pair (a,b) has the p-property c© agt, upv, 2014 appl. gen. topol. 15, no. 1 56 on the existence of best proximity points for generalized contractions and a0 6= ∅ and t : a → b be a mapping such that t(a0) ⊆ b0 and it satisfies(1.2). then there exists a unique x∗ in a such that d(x∗,tx∗) = dist(a,b). 1.1. our contribution. in this paper we prove the existence of a unique best proximity point for mappings satisfying the contractive condition (1.1) and for mappings satisfying a condition which is a weaker form of condition (1.2) (where ψ is assumed to be either continuous or nondecreasing and the infinity condition is not needed). our result enables us prove the above theorem 1.1 under weaker assumptions. in addition, our theorem includes the generalization of banach’s contraction principle due to matkowski [2, theorem 1.2] and help us to improve [3, theorem 1] by rhoades. 2. preliminaries in this section we give some definitions and results which are useful and related to context of our results. let a and b be two non-empty subsets of a metric space (x,d). throughout this article we denote by a0 and b0 the following sets: a0 = {x ∈ a : d(x,y) = dist(a,b) for some y ∈ b} b0 = {y ∈ b : d(x,y) = dist(a,b) for some x ∈ a}. for the sufficient conditions for the non-emptiness of a0 and b0, one can refer to [11]. let (a,b) be a pair of two non-empty subsets of a metric space (x,d) with a0 6= ∅. then the pair (a,b) is said to have the p-property [4] if and only if d(x1,y1) = dist(a,b) d(x2,y2) = dist(a,b) } ⇒ d(x1,x2) = d(y1,y2) where x1,x2 ∈ a0 and y1,y2 ∈ b0. it is easy to check that for a non-empty subset a of (x,d), the pair (a,a) has the p-property. example 2.1 ([4]). let a and b be two non-empty closed convex subsets of a real hilbert space h, then the pair (a,b) has the p-property. example 2.2 ([8]). let a and b be two nonempty bounded closed convex subsets of a uniformly convex banach space x, the pair (a,b) has the pproperty. 3. main results we begin this section with the following two auxiliary results. lemma 3.1. let ψ : r+ → r+ be a function such that ψ −1(0) = {0} and ψ is either nondecreasing or continuous. then, for any bounded sequence {tn} of positive reals, ψ(tn) → 0 implies tn → 0. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 57 a. sultana and v. vetrivel proof. let {tn} be a bounded sequence of positive reals such that ψ(tn) → 0. let us assume that ψ is nondecreasing. suppose tn 9 0. then there exists a subsequence {tnk} of {tn} such that tnk ≥ δ for some δ > 0 and ∀ k ∈ n. as ψ is nondecreasing, so ψ(tnk) ≥ ψ(δ) ∀ k, which is a contradiction. let ψ be continuous. suppose that the sequence {tn} is not convergent. then lim tn 6= lim tn. this implies that there exists a subsequence {tnk} of {tn} such that tnk → t0 > 0 which implies that ψ(tnk ) → ψ(t0) > 0, a contradiction. hence tn → lim tn = lim tn = 0. � lemma 3.2. let a and b be two non-empty subsets of a metric space and ψ : r+ → r+ be a function such that ψ −1(0) = {0} and for any bounded sequence {tn} of positive reals, ψ(tn) → 0 implies tn → 0. suppose that t : a → b be a mapping such that d(tx,ty) ≤ d(x,y) − ψ(d(x,y)) ∀ x,y ∈ a. then, for every ǫ > 0, there exist δ > 0 and γ ∈ (0,ǫ) such that for all x,y ∈ a, d(x,y) < ǫ + δ implies d(tx,ty) ≤ γ. proof. suppose that there exists an ǫ0 > 0 such that for every δ > 0 and γ ∈ (0,ǫ0) there exist x,y ∈ a such that d(x,y) < ǫ0 +δ implies d(tx,ty) > γ. let δn = 1 n2 and γn = ǫ0 n2 1 + n2 ∀ n ∈ n, so there exist {xn} and {yn} in a such that (3.1) d(xn,yn) < ǫ0 + 1 n2 and d(txn,tyn) > ǫ0 n2 1 + n2 now, we get ǫ0 n2 1 + n2 < d(txn,tyn) ≤ d(xn,yn) − ψ(d(xn,yn)) < ǫ0 + 1 n2 − ψ(d(xn,yn)) which implies ψ(d(xn,yn)) < ǫ0 1 + n2 + 1 n2 . thus ψ(d(xn,yn)) → 0 as n → ∞ and since {d(xn,yn)} is bounded, by the given hypothesis, d(txn,tyn) → 0 as n → ∞. now by (3.1) limn→∞d(txn,tyn) ≥ ǫ0, which is a contradiction. � now we recall the following result of hegedűs and szilágyi [1, lemma 1]. lemma 3.3. for a given subset d of r2+ = {(x,y) ∈ r 2 : x,y ≥ 0}, the following statements are equivalent: (i) for any ǫ > 0, there exist δ > 0 and γ ∈ (0,ǫ) such that for all (t,u) ∈ d, t < ǫ + δ implies u ≤ γ; (ii) there exists a function ϕ : r+ → r+ where ϕ is continuous and nondecreasing with ϕ(t) < t, ∀ t > 0 and u ≤ ϕ(t) ∀ (t,u) ∈ d. the following theorem is our main result which gives sufficient conditions for the existence of a unique best proximity point for some generalized contractions. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 58 on the existence of best proximity points for generalized contractions theorem 3.4. let a and b be two non-empty closed subsets of a complete metric space (x,d) such that the pair (a,b) has the p-property and a0 6= ∅ and t : a → b be a mapping such that t(a0) ⊆ b0, satisfying any one of the following contractive conditions: (i) d(tx,ty) ≤ φ(d(x,y)), ∀ x,y ∈ a, where φ : r+ → r+ is nondecreasing and satisfies limn→∞φ n(t) = 0 for any t > 0. (ii) d(tx,ty) ≤ d(x,y) − ψ(d(x,y)), ∀ x,y ∈ a, where ψ : r+ → r+ is either nondecreasing or continuous with ψ−1(0) = {0}; then there exists a unique x∗ in a such that d(x∗,tx∗) = dist(a,b). moreover, if x0 ∈ a0 and xn is defined by d(xn,txn−1) = dist(a,b) ∀ n ∈ n, then xn → x ∗ as n → ∞. proof. since a0 is non-empty, let x0 ∈ a0. as t(x0) ∈ t(a0) ⊆ b0, by definition of b0 there exists x1 ∈ a0 such that d(x1,tx0) = dist(a,b). again, as t(x1) ∈ t(a0) ⊆ b0, there exists x2 ∈ a0 such that d(x2,tx1) = dist(a,b). repeating this process, we can obtain a sequence {xn} ⊆ a0 such that (3.2) d(xn+1,txn) = dist(a,b), ∀ n ∈ n. assume that xn+1 6= xn ∀ n, otherwise there is nothing to prove. by the p-property of (a,b) it is clear that (3.3) d(xn,xn+1) = d(txn−1,txn), ∀ n ∈ n. to prove that {xn} is a cauchy sequence, we shall first claim that there exists a function ϕ : r+ → r+ where ϕ is nondecreasing such that ϕ(t) < t, ∀ t > 0, ϕ(0) = 0 and limn→∞ϕ n(t) = 0 for any t > 0 and d(tx,ty) ≤ ϕ(d(x,y)) for any x,y ∈ a. let us assume that t satisfies condition (i). clearly φ(t) < t for t > 0. indeed, if there exists t0 > 0 with φ(t0) ≥ t0, then φ n(t0) ≥ t0 ∀ n ∈ n as φ is increasing, a contradiction. also note that φ(0) = 0. therefore our claim is true by taking ϕ = φ. suppose t satisfies (ii). then by applying lemma 3.1 and lemma 3.2 we see that for any ǫ > 0, there exist δ > 0 and γ ∈ (0,ǫ) such that for all x,y ∈ a, d(x,y) < ǫ + δ implies d(tx,ty) ≤ γ. now applying lemma 3.3 to the set d = {(d(x,y),d(tx,ty)) : x,y ∈ a}, we see that there exists a function ϕ : r+ → r+ such that ϕ is continuous and nondecreasing with ϕ(t) < t, ∀ t > 0 and d(tx,ty) ≤ ϕ(d(x,y)) for any x,y ∈ a. clearly limn→∞ϕ n(t) = 0 for any t > 0. indeed if there exists t0 > 0 such that limn→∞ϕ n(t0) = β 6= 0, then β = limn→∞ϕ(ϕ n−1(t0)) = ϕ(β) < β, a contradiction. therefore our claim is true. now, d(x2,x3) = d(tx1,tx2) ≤ ϕ(d(x1,x2)) ≤ ϕ 2(d(x0,x1)). by induction we get d(xn,xn+1) ≤ ϕ n((d(x0,x1)) ∀ n ∈ n. from the hypothesis it is clear that d(xn,xn+1) → 0 as n → ∞. thus for a given ǫ > 0 there c© agt, upv, 2014 appl. gen. topol. 15, no. 1 59 a. sultana and v. vetrivel exists n ∈ n such that (3.4) d(xn,xn+1) ≤ ǫ − ϕ(ǫ) ∀ n ≥ n. denoting a ball with center x and radius ǫ by b[x,ǫ], we will show the following relations (a) t(b[xn,ǫ] ∩ a) ⊆ b[txn−1,ǫ]; (b) y ∈ b[txn−1,ǫ] with d(x,y) = dist(a,b), x ∈ a0 ⇒ x ∈ b[xn,ǫ] ∩ a. if x ∈ b[xn,ǫ] ∩ a, then d(tx,txn−1) ≤ d(tx,txn) + d(txn,txn−1) ≤ ϕ(d(x,xn )) + d(xn+1,xn) ≤ ϕ(ǫ) + ǫ − ϕ(ǫ) ≤ ǫ and hence (a) follows. let y ∈ b[txn−1,ǫ] with d(x,y) = dist(a,b), x ∈ a0. now by (3.2), d(xn,txn−1) = dist(a,b). therefore by using the p-property of (a,b) we have d(xn,x) = d(txn−1,y) and hence (b) follows. from (3.4), it is clear that xn+1 ∈ b[xn,ǫ] ∩ a and then by (a), we get txn+1 ∈ b[txn−1,ǫ]. from (3.2), d(xn+2,txn+1) = dist(a,b) with xn+2 ∈ a0. therefore (b) implies xn+2 ∈ b[xn,ǫ]∩a. again by (a), we have txn+2 ∈ b[txn−1,ǫ] and from (3.2), d(xn+3,txn+2) = dist(a,b) with xn+3 ∈ a0. again (b) implies that xn+3 ∈ b[xn,ǫ] ∩ a. continuing this process we can conclude that xn+m ∈ b[xn,ǫ] ∩ a, ∀ m ∈ n. hence {xn} is a cauchy sequence. as a is closed, there exists an element x∗ ∈ a such that xn → x ∗ as n → ∞. as ϕ(t) < t for t > 0, we have d(tx,ty) ≤ d(x,y) ∀ x,y ∈ a which implies that t is continuous in a. therefore txn → tx ∗. from the continuity of the distance function we conclude that d(xn,txn) → d(x ∗,tx∗). since d(xn+1,txn) = dist(a,b) ∀ n, we have d(x∗,tx∗) = dist(a,b). if x1 and x2 are two best proximity points of t , by the p-property of (a,b) we have d(x1,x2) = d(tx1,tx2). then, d(x1,x2) = d(tx1,tx2) ≤ ϕ(d(x1,x2)) < d(x1,x2) [ since ϕ(t) < t, ∀ t > 0 ] which implies that x1 = x2. � since, for any nonempty subset a of x, the pair (a,a) has the p-property, we can deduce the following result, as a corollary from the above theorem, by taking a = b. corollary 3.5. let (x,d) be a complete metric space and a be a nonempty closed subset of x. let t : a → a be a self-map satisfying condition (ii). then t has a unique fixed point x in a. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 60 on the existence of best proximity points for generalized contractions remark 3.6. if the mapping satisfies conditions (ii), it follows from theorem 3.4 that the assumptions of theorem 1.1 on the function ψ can be weakened. the following example illustrates that theorem 3.4 generalizes theorem 1.1. example 3.7. let x be the set consists of the interval [0,1] together with the natural numbers 2,3,4, · · · . let d : x × x → r such that d(x,y) = |x − y| if x = y or both x,y ∈ [0,1], = x + y if one of x,y /∈ [0,1]. then (x,d) is a complete metric space (see [5, remarks 3]). let a = [0,1] ∪ {3,5,7, · · ·} and b = [0,1] ∪ {2,4,6, · · ·} be two subsets of x. define the map t : a → b by t(x) = x − 1 2 x2 if x ∈ [0,1], = x − 1 if x = 3,5,7, · · · . now, for x,y ∈ [0,1] with x 6= y, d(tx,ty) = ∣ ∣ ∣ ∣ (x − y)(1 − 1 2 (x + y)) ∣ ∣ ∣ ∣ ≤ d(x,y)(1 − 1 2 d(x,y)), if x ∈ {3,5, · · ·} and y ∈ a with x 6= y, d(tx,ty) = tx + ty ≤ x + y − 1 = d(x,y) − 1. thus, if we consider the map ψ : [0,∞) → [0,∞) by ψ(t) = 1 2 t2 0 ≤ t ≤ 1 = 1 1 < t < ∞, then d(tx,ty) ≤ d(x,y) − ψ(d(x,y)) ∀ x,y ∈ a where ψ is nondecreasing with ψ−1(0) = {0}. it is easy to check that a,b are closed subsets of x and (a,b) has the p-property. also, a0 = b0 = [0,1] and t(a0) ⊆ b0. thus, all the assumptions of theorem 3.4 hold and note that x∗ = 0 is the unique best proximity point. suppose that t satisfies (1.2) for some ϕ where limt→∞ ϕ(t) = ∞ as a is unbounded. consider a sequence {tn}n∈n in r+ where tn = d(0,2n + 1) for n ≥ 1. since tn → ∞ and limt→∞ ϕ(t) = ∞, ϕ(tn) → ∞. now, for n ≥ 1, (2n + 1) − 1 = t(0) + t(2n + 1) = d(t(0),t(2n + 1)) ≤ d(0,2n + 1) − ϕ(d(0,2n + 1)) = (2n + 1) − ϕ(tn). hence, ϕ(tn) ≤ 1 ∀ n ≥ 1, a contradiction. thus theorem 1.1 cannot be used to give the existence of the solution x∗. remark 3.8. as a corollary we get [2, theorem 1.2] due to matkowski (see also [12, p. 15]), from theorem 3.4, by considering a = b when the mapping satisfies (i). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 61 a. sultana and v. vetrivel remark 3.9. as (ii) includes (1.2), corollary 3.5 is a generalized version of [3, theorem 1] due to rhoades and the following example justifies that. example 3.10. let (x,d) be the metric space as in example 3.7 and a = x. define the mapping t : a → a by t(x) = x − 1 2 x2 if x ∈ [0,1], = x − 1 if x = 2,3,4, · · · . similar to example 3.7, it is easy to check that d(tx,ty) ≤ d(x,y)(1 − 1 2 d(x,y)) if x,y ∈ [0,1], ≤ d(x,y) − 1 if x ∈ {2,3, · · ·} and y ∈ a with x 6= y. we see that d(tx,ty) ≤ d(x,y)−ψ(d(x,y)) ∀ x,y ∈ a, where ψ is the function as in example 3.7. thus corollary 3.5 guarantees the existence of unique fixed point of t and note that t(0) = 0. similar to example 3.7, it is easy to verify that t does not satisfy (1.2). thus [3, theorem 1] cannot be applied to get the fixed point. acknowledgements. the authors 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� ( <�&��( !#"$!&%4�+; o "u#�(�&�52, 5 f /0(!3)< � ;�0 ( 3)"�("."./&e5:2�2 ú('�)+*,'�-�.&/+-+0,1�2&*$3$),465�798,*;:<5=-�> @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 99–114 on the use of partial orders in uniform spaces bruce s. burdick abstract. we investigate the use of nets indexed by preorders in uniform spaces. nine different cauchy conditions and four different convergence conditions yield 36 completeness properties, each of which turns out to be equivalent to a known form of completeness. we also use these preordered nets to characterize the functors θ, λ, and ν, which are associated with these completeness properties. in the case of λ we give an example to show that the analogous characterization with predirected nets does not work. 2000 ams classification: 54a20, 54d20, 54e15. keywords: locally fine, complete, supercomplete, cofinally complete, paracompact. 1. introduction we have recently written a paper [3] on completeness properties determined by nets whose directed sets are well ordered; this current paper represents the opposite extreme, namely, the use of preorders as index sets for nets. in the title, we use the term ‘partial order’ after the fashion of kelley, who in chapter 2 of his book [14] does not require the antisymmetry property for orderings. his partial orders need only be transitive and reflexive; and his directed sets have these properties plus upper bounds for finite sets. but over the years, the terminology has changed, and we follow it, using ‘preordered’ and ‘predirected’ for these concepts below. section 2 deals with the completeness properties which arise. section 3 shows how these preordered nets may be used to give new definitions for functors discussed in the literature. these new definitions have some advantages over the definitions previously known. in the case of the functor ν the new definition is internal rather than external. it does not require the use of the completion of the space. in the case of λ the new definition does not make use of ginsberg’s 100 bruce s. burdick and isbell’s quasiuniformities, i.e., filters of coverings which may fail to satisfy the star refinement property. 2. variations on the property of completeness definition 2.1. in a uniform space (x,u) a filter f is weakly cauchy if for every u ∈ u there is a u-small set s ⊆ x which has non-empty intersection with every member of f. a net ξ : d → x is cofinally cauchy if for every u ∈ u there is a cofinal set c ⊆ d such that ξ[c] is u-small. a space is cofinally complete if every cofinally cauchy net clusters, or, equivalently, if every weakly cauchy filter clusters. a filter is stable if for every u ∈ u there is an f ∈f such that for all f ′ ∈f we have f ⊆ u[f ′]. a net ξ : d → x is almost cauchy if for any u ∈u there is a d ∈ d and a set c of cofinal subsets of d such that for each c ∈ c, ξ[c] is u-small, and for each d′ ∈ d, if d′ ≥ d then d′ ∈ ⋃ c. a space is supercomplete if each almost cauchy net clusters, or, equivalently, if every stable filter clusters. recall that, originally, isbell called a space supercomplete if it had a complete hyperspace [12, 13]. that condition is equivalent to the definition given here. this author stated this in [2] before discovering that isbell mentioned this in the last paragraph of [12]. we note that császár gave this as an open problem in [4]. in [12] and [13], isbell uses a notion of nets indexed by preordered sets to characterize supercompleteness. in [2] we obtained a slightly different characterization of supercompleteness using these preordered nets. we wish to extend these results here, but because of the many different cauchy and convergence conditions that are possible we need to develop first a streamlined terminology for all the completeness properties that are generated. to do this we make use of the alexandroff topology. for a preordered set (p,≤), the alexandroff topology [1] is the collection of upper sets, that is, those a for which x ∈ a and x ≤ y ⇒ y ∈ a. this is easily seen to be a topology which is generated by all sets of the form ↑x = {y : x ≤ y} for x ∈ p, and in this guise was called the partial order topology in chapter iii of the book [17]. when we refer below to open, dense, open dense, or somewhere dense (= not nowhere dense) subsets of a preordered set, we will be assuming that these terms refer to such sets as determined by the alexandroff topology. definition 2.2. any function from a preordered set to a uniform space will be called a ponet. we say a ponet s : (p,≤) → (x,u) is open (dense, open dense, somewhere dense) cauchy if for each u ∈ u there is an open (dense, open dense, somewhere dense) set r ⊆ p with s[r] ×s[r] ⊆ u. in addition, a ponet s : (p,≤) → (x,u) satisfies the property open/open dense cauchy if for each u ∈ u the union of some collection of open sets r ⊆ p such that s[r] × s[r] ⊆ u, is open dense, and we may define somewhere dense/open dense, somewhere dense/open, somewhere dense/dense, and dense/open dense in an analogous manner. partial orders in uniform spaces 101 definition 2.3. we say x is an open (dense, open dense, somewhere dense) limit of a ponet s : (p,≤) → (x,u) if for every neighborhood o of x, s−1[o] contains an open (dense, open dense, somewhere dense) set. from these nine cauchy conditions and four convergence conditions we may define 36 completeness properties in the obvious way. for example, the property open/open dense—dense completeness would say that every open/open dense cauchy ponet has a dense limit. these 36 properties are not distinct, however, as we will show below. the next two propositions are the only previous results we know about this type of completeness. proposition 2.4 (isbell [12], equivalence of (c) and (a) in the theorem, 1962). open/open dense—open completeness is equivalent to supercompleteness. proposition 2.5 (burdick [3], equivalence of (c) and (a) in theorem 2, 1991). open/open dense—somewhere dense completeness is equivalent to supercompleteness. in the following table we extend this type of characterization to all 36 of the completeness properties just defined. the nine cauchy conditions correspond to the nine rows and the four convergence conditions correspond to the four columns. we find that these 36 different combinations resolve into just five well known versions of completeness. the 36 completeness properties c = complete sc = supercomplete cc = cofinally complete pf = paracompact and fine i = indiscrete somewhere open dense open dense dense limit limit limit limit open dense c c c ccauchy open/ open dense sc sc i i cauchy dense/ open dense sc i sc i cauchy open cc pf i icauchy dense cc i pf icauchy 102 bruce s. burdick the 36 completeness properties (continued) somewhere open dense open dense dense limit limit limit limit somewhere dense/ sc i i iopen dense cauchy somewhere dense/open cc i i i cauchy somewhere dense/dense cc i i i cauchy somewhere dense cc i i i cauchy we will only prove some of the results contained in the table above—enough so that the key ideas are demonstrated. first of all it is clear that all these properties imply completeness. likewise, dense—somewhere dense completeness clearly implies cofinal completeness. going the other way we have the following two results. proposition 2.6. completeness implies open dense—open dense completeness. proof. suppose that (x,u) is complete and s : p → x is an open dense cauchy ponet. let q be the set of open dense subsets of p and let q = {(d,x) | x ∈ d ∈ q}. define a preorder on q by (d1,x) ≤ (d2,y) if d2 ⊆ d1. then q is predirected. define f : q→ p by f(d,x) = x. then s◦f : q→ x is a cauchy net. let x be a limit of s ◦f. then for every neighborhood o of x, s−1[o] contains an open dense set. so x is an open dense limit of s. � proposition 2.7. a cofinally complete space is somewhere dense—somewhere dense complete. proof. suppose that (x,u) is cofinally complete and s : p → x is a somewhere dense cauchy ponet. let q be the set of open dense subsets of p and let q = {(d,x) | x ∈ d ∈ q}. define a preorder on q by (d1,x) ≤ (d2,y) if d2 ⊆ d1. then q is predirected. define f : q → p by f(d,x) = x. then s ◦ f : q → x is a cofinally cauchy net. let x be a cluster point of s ◦ f. then for every neighborhood o of x, s−1[o] intersects every open dense subset of p so it must be somewhere dense. so x is a somewhere dense limit of s. � we note that there is a symmetry between open and dense in the table. here is one example of this. partial orders in uniform spaces 103 proposition 2.8. a space is open—open complete if and only if it is dense— dense complete. proof. suppose (x,u) is open—open complete. suppose s : p → x is dense cauchy. let q be the collection of dense subsets of p and let q = {(d,x) | x ∈ d ∈ q}. define (d1,x) ≤ (d2,y) if d2 ⊆ d1. define f : q → p by f(d,x) = x. then s ◦ f is open cauchy. let x be an open limit of s ◦ f. then for every neighborhood o of x, s−1[o] is dense. so x is a dense limit of s. suppose (x,u) is dense—dense complete. suppose s : p → x is open cauchy. let q be the collection of dense subsets of p and let q = {(d,x) | x ∈ d ∈ q}. define (d1,x) ≤ (d2,y) if d2 ⊆ d1. define f : q → p by f(d,x) = x. then s ◦ f is dense cauchy. let x be a dense limit of s ◦ f. then for every neighborhood o of x, s−1[o] intersects every member of q so it contain as open set. so x is an open limit of s. � sometimes the method of the last proof doesn’t do the whole job, but the symmetry still holds. proposition 2.9. for a uniform space the following are equivalent: (1) supercompleteness. (2) somewhere dense/open dense—somewhere dense completeness. (3) dense/open dense—somewhere dense completeness. (4) open/open dense—somewhere dense completeness. (5) dense/open dense—dense completeness. (6) open/open dense—open completeness. proof. (1) implies (2). suppose (x,u) is supercomplete. suppose s : p → x is somewhere dense/open dense cauchy. let q be the set of open dense subsets of p and let q = {(d,x) | x ∈ d ∈ q}. define (d1,x) ≤ (d2,y) if d2 ⊆ d1. then q is predirected. define f : q → p by f(d,x) = x. then s ◦ f is an almost cauchy net. let x be a cluster point of s ◦ f. then for every neighborhood o of x, s−1[o] intersects every member of q so it must be somewhere dense. so x is a somewhere dense limit of s. (2) implies (3). trivial. (3) implies (5). a somewhere dense limit of a dense/open dense ponet will always be a dense limit. (2) implies (4). trivial. (4) implies (6). a somewhere dense limit of an open/open dense ponet will always be an open limit. (5) implies (6). suppose (x,u) is dense/open dense—dense complete. suppose s : p → x is open/open dense cauchy. let q be the collection of dense subsets of p and let q = {(d,x) | x ∈ d ∈ q}. define (d1,x) ≤ (d2,y) if d2 ⊆ d1. define f : q→ p by f(d,x) = x. then s ◦f is dense/open dense cauchy. let x be a dense limit of s ◦ f. then for every neighborhood o of 104 bruce s. burdick x, s−1[o] intersects every member of q so it contain an open set. so x is an open limit of s. (6) implies (1). this follows from proposition 2.4. � the method of proof in proposition 2.8 was used here to prove that (5) implies (6), but it doesn’t supply a direct proof that (6) implies (5). if we tried that, at a crucial point we would not be able to say that s ◦ f is open/open dense cauchy. there is an asymmetry here stemming from the fact that while the f−1 image of an open set is dense, the f−1 image of a dense set merely contains an open set. in section 3 we will see a breakdown in the symmetry of the results for this very reason (compare corollary 3.17 with example 3.15). the combination of paracompact and fine is equivalent to saying that every open cover is a uniform cover. this is stronger than cofinal completeness ([5], (a) implies (c) in thereom 1). proposition 2.10. open—open completeness is equivalent to paracompact and fine. proof. suppose that (x,u) is open—open complete and that c is an open cover of x which is not uniform. let p be the collection of all sets a ⊆ x such that no element of c contains a as a subset. let p = {(a,x) | x ∈ a ∈ p}. define a preorder on p by saying that (a,x) ≤ (b,y) if b ⊆ a. define a ponet s : p → x by s(a,x) = x. p is an open cauchy ponet. so let x be an open limit. some o ∈c contains x. but then some a ∈ p would have to be a subset of o, a contradiction. conversely, suppose that (x,u) is paracompact and fine. let s : p → x be an open cauchy ponet with no open limit. then each point in x would have an open neighborhood o such that s−1[o] would not contain an open set. the collection of these o’s is an open cover, therefore a uniform cover. but this contradicts the open cauchy property. � proposition 2.11. open/open dense—dense completeness implies that the uniform space has the indiscrete uniformity. proof. suppose that (x,u) is open/open dense—dense complete. let x be given the trivial order defined by x ≤ y if and only if x = y. then the identity map ι : x → x is an open/open dense cauchy ponet. let x be a dense limit. then every neighborhood of x must contain all of x, making u the indiscrete uniformity. � 3. the functors ν,λ, and θ now we turn to a consideration of certain functors which are associated with completeness properties. the functors ν and λ are due to howes [9] and ginsberg and isbell [8], respectively. the latter functor has been utilized by many authors over the last four decades. our functor θ was announced in the partial orders in uniform spaces 105 book by howes [11] without any details. this is the first time we have used it in a paper. after defining howes’s functor ν, we give several results which illustrate the importance of ν. after that we proceed to a new definition of ν. definition 3.1. for each infinite cardinal κ we say a uniform space (x,u) is κ-bounded if for every u ∈ u there is a subset s of x, of cardinality less than κ, with u[s] = x. for each uniform space (x,u) and each infinite cardinal κ let uκ be the supremum of all the κ-bounded uniformites on x which are coarser than u. for an infinite cardinal κ, a topological space (x,t ) is κ-pseudocompact if every normal cover of x has a subcover of cardinality less than κ. a topological space is [κ,∞)-compact if every open cover has a subcover of cardinality less than κ. the next two definitions and several results following them are due to n. howes. definition 3.2. [9] given a space (x,u) a new space (x,νu) is constructed in the following manner: embed (x,u) in its completion (y,v), let v∗ be the finest uniformity on y generating the same topology as v, and then let νu be the restriction of v∗ to x. it is easy to see that this construction defines a functor (in view of the definitions ν(x,u) = (x,νu) and νf = f) from the category of uniform spaces to itself. definition 3.3. a space is preparacompact [9] if every cofinally cauchy net has a cauchy subnet, and it is almost preparacompact [10] if every almost cauchy net has a cauchy subnet. lemma 3.4 (howes). a space is preparacompact iff its completion is cofinally complete [9]; it is almost preparacompact iff its completion is supercomplete [10]. in [9] and [10], howes used the functor ν to answer a question of tamano as to which spaces had paracompact completions. proposition 3.5 (howes). for a space (x,u) the following are equivalent: (1) (x,u) has a paracompact completion. (2) (x,νu) is preparacompact. (3) (x,νu) is almost preparacompact. this in turn allowed him to characterize the spaces with lindelöf completions in several different ways. proposition 3.6 (howes). for a space (x,u) the following are equivalent: (1) (x,u) has a lindelöf completion. (2) (x,νu) is preparacompact and ℵ1-bounded. (3) (x,νu) is almost preparacompact and ℵ1-bounded. 106 bruce s. burdick the following generalizes these results of howes’s. lemma 3.7. for a space (x,u) and an infinite cardinal κ, the following are equivalent: (1) (x,u) has a κ-pseudocompact completion. (2) (x,νu) is κ-bounded. proposition 3.8. for a space (x,u) and an infinite cardinal κ, the following are equivalent: (1) (x,u) has a paracompact, [κ,∞)-compact completion. (2) (x,νu) is κ-bounded and preparacompact. (3) (x,νu) is κ-bounded and almost preparacompact. these results are mentioned to demonstrate the usefulness of the functor ν. therefore we should ask if there are other ways of defining ν, ways which might facilitate the use of the characterizations above. we give new constructions of ν which involve other uniformities on the given set x but do not require adding any more points to x. we feel that the results above, which give properties of the completion of a space, will be more significant if the construction of ν itself does not make use of the completion. proposition 3.9. for a fixed uniform space (x,u) and a possibly different uniformity u∗ on x, the following are equivalent: (1) any ponet which is open dense cauchy for u is open dense cauchy for u∗. (2) any net which is cauchy for u is cauchy for u∗. (3) any filter which is cauchy for u is cauchy for u∗. (4) u∗ ⊆ νu. proof. (1) implies (2). trivial. (2) implies (3). elementary. (3) implies (4). let the hausdorff completion of (x,u) be i : (x,u) → (y,v) and the hausdorff completion of (x,u∗) be i∗ : (x,u∗) → (z,w). it suffices to show that there is a continuous map f : (y,t (v)) → (z,t (w)) where f ◦ i = i∗. the sets y and z may be regarded as sets of equivalence classes of cauchy filters, and the equivalence relation is such that two filters f and f′ are equivalent if and only if f∩f′ is cauchy. so we may define f by saying that if f is a representative of the equivalence class [f]u ∈ y , then f([f]u) = [f]u∗. our remarks above show that f is well-defined. to show f continuous it suffices to show that if filter f converges to y in (y,v) then f[f] converges to f(y) in (z,w). this is certainly true if every f ∈f intersects i[x], since the trace of f on i[x] would be u-cauchy, therefore i−1[f] would be a representative of the equivalence class y, and so also of f(y), and so f = i[i−1[f]] would converge to f(y). so given f converging to y in (y,v), let g = {v [f] | v ∈ v,f ∈ f}. g still converges to y, so it is a cauchy filter on (y,v) and every member of g partial orders in uniform spaces 107 intersects i[x]. so i−1[g] is cauchy on (x,u). since i−1[g] is a representative of the equivalence class y it is also a representative of the equivalence class f(y). then for any w ∈ w, w [y] will contain g ∩ i[x] for some g ∈ g. suppose g = v [f] for some f ∈f. then every point of f is the limit with respect to (y,v) of a filter on g∩ i[x] and so by the remark in the last paragraph every point of f[f] is the limit with respect to (z,w) of a filter on f[g]. if w has been chosen to be a closed relation then w [f(y)] contains f[f]. this shows that f[f] converges to f(y). (4) implies (1). any ponet which is open dense cauchy for u will have an open dense limit in the completion (x,u) of (x,u), by the table in section 2. this will still be an open dense limit when u is replaced by the fine uniformity for its topology. therefore the ponet will be open dense cauchy for νu and so for u∗. � corollary 3.10. for any uniform space (x,u), νu is equal to (1) the supremum of the uniformities u∗ such that every u-cauchy net is u∗-cauchy, and (2) the supremum of the uniformities u∗ such that every u-open dense cauchy ponet is u∗-open dense cauchy. in each case the supremum is the finest member of the set. corollary 3.11. a uniform space (x,u) has a paracompact completion if and only if every cofinally cauchy net in (x,νu) has a subnet which is cauchy for u. observation 3.12. if a space (x,u) is complete then νu is fine; if the space is paracompact (or even just topologically complete) then the converse is true. another useful functor is the locally fine coreflection, λ. the reader is referred to [8], [11], [12], [13], and [15] for many properties of this functor. we will characterize λ using ponets as we have for ν. definition 3.13. a cover c of x is called uniformly locally uniform if there is a uniform cover c′ such that on each s ∈c′, the trace of c on s is uniform. a space (x,u) is called locally fine if every uniformly locally uniform cover is uniform. given a space (x,u) the uniformity λu is the coarsest one finer than u such that (x,λu) is locally fine. λ is constructed by transfinite recursion (see [8] or [13] ). proposition 3.14. νλ = ν. consequently, for a uniform space (x,u), we have νu finer than λu. proof. it suffices to show that the completion of λ(x,u) is homeomorphic to the completion of (x,u) via a homeomorphism which is the identity on x. this follows from ginsberg’s and isbell’s ([8], theorem 4.4) (λ commutes with completion). � we wish to prove some characterizations of λ similar to those we have done for ν, but first we observe an example that shows that the obvious net property doesn’t hold in this case. 108 bruce s. burdick example 3.15. stable filters for (x,u) need not be stable for λu, and consequently almost cauchy nets for (x,u) need not be almost cauchy for λu, nor do dense/open dense ponets for (x,u) need to be dense/open dense for λu. let u be the usual uniformity on the reals, and let a filter f be generated by the sets f� = ⋃ n∈z[n − �,n + �], for � > 0. then f is u-stable. since u is a complete metric uniformity, λu is fine [8]. but even if u∗ is a uniformity which makes the function f : (r,u∗) → (r,u), where f(x) = x2, uniformly continuous, then f is not u∗-stable. so f is not λu-stable. proposition 3.16. for a fixed uniform space (x,u) and a possibly different uniformity u∗ on x, the following are equivalent: (1) u∗ ⊆ λu. (2) any ponet which is open/open dense cauchy for u is open/open dense cauchy for u∗. (3) any ponet which is open/open dense cauchy for u is open cauchy for u∗. (4) any locally fine uniformity which is finer than u is finer than u∗. proof. (1) implies (2). given a ponet which is open/open dense cauchy for u we can prove by transfinite induction that it is open/open dense cauchy for λu. this is the essence of lemma 40 of chapter vii of [13]. (2) implies (3). trivial. (3) implies (4). given a locally fine uniformity v on x with u ⊆v, suppose that u∗ 6⊆ v. then there is a u∗-uniform cover c which is not v-uniform. let p be the collection of all sets a ⊆ x such that c is not a v-uniform cover of a. let p = {(a,x) | x ∈ a ∈ p}. define a preorder on p by saying that (a,x) ≤ (b,y) if b ⊆ a. define a ponet s : p → x by s(a,x) = x. we show that s is open/open dense cauchy for v. it suffices to show that for any a ∈ p and u ∈v there is a u-small b ∈ p with b ⊆ a. suppose not, i.e., every u-small subset of a is v-uniformly covered by c. consider the cover c′ = {c ∪ (x − a) | c ∈ c}. the u-small subsets of x are all v-uniformly covered by c′, so by local fineness c′ is a v-uniform cover of x. the trace of c′ on a is the same as the trace of c, so c is a v-uniform cover of a, a contradiction. since s is v-open/open dense cauchy it is u-open/open dense cauchy. but it fails to be u∗-open cauchy since no element of p is contained in any element of c, and this contradicts property (3). (4) implies (1). this follows since λu is a locally fine uniformity which is finer than u. � corollary 3.17. for any uniform space (x,u), λu is equal to (1) the supremum of the uniformities u∗ such that every u-open/open dense cauchy ponet is u∗-open/open dense cauchy, and (2) the supremum of the uniformities u∗ such that every u-open/open dense cauchy ponet is u∗-open cauchy. in each case the supremum is the finest member of the set. partial orders in uniform spaces 109 the usual definition of λ gives rise to technical difficulties in that the ginsberg-isbell derivatives used in the transfinite induction need not be covering uniformities, and then it is tricky to show in the end that their union, i.e., the λ uniform covers, is a covering uniformity. if property (1) in corollary 3.17 were taken as the definition of λu instead this wouldn’t be a problem. what’s more it would be easy to show directly that λu preserves the open/open dense ponets of u. the next theorem of isbell’s [12] is our chief reason for interest in the functor λ. isbell’s theorem a space (x,u) is supercomplete iff it is paracompact and λu is fine. the fact that paracompactness plus fineness implies supercompleteness, and that supercomplete implies paracompact, can be proved without use of the functor λ. if we add to these facts the observation that λ preserves the topology of the space (true because it is true for ν) then isbell’s theorem follows from our corollary 3.17 and propostion 2.9. we should not claim too much, however. we have not replaced the traditional development of this subject because we have used previously known properties of λ in the proofs of propositions 3.14 and 3.16. in particular we used the assumption that λ as traditionally defined is in fact a uniformity to prove that (4) implies (1) in proposition 3.16. let us further point out that pelant in [15] has shown that the locally fine uniform spaces are all subfine and so λ coincides with the subfine coreflection described in chapter vii of [13]. this also then provides a relatively straightforward way of defining λ. however, the proof that it coincides with the original definition of λ is quite involved. our corollary 3.17 has the combined advantages of being relatively easy to prove and giving a construction of λu which clearly yields a uniformity. in view of these properties of ν and λ it would be interesting to have a third functor θ with the following properties: (1) cofinal completeness should be equivalent to paracompactness and θ fine. (2) θ should be coarser than λ. (3) θ2 = θ. (4) θ shouldn’t change the underlying set or the generated topology, and should make the uniformity finer than before. (5) θ, like ν and λ, should be a coreflection when considered as a functor from the category of all uniform spaces to its image category. (this actually follows from (3) and (4) and the assumption that θ is a functor). definition 3.18. a cover c of a uniform space (x,u) is uniformly locally finitizible if there is a u ∈ u such that for any x ∈ x, u[x] is covered by some finite subset of c. a uniform space (x,u) is ℵ0-nearly discrete if for any uniformity u∗ on x, if every u∗-uniform cover is u-uniformly locally finitizible then u∗ ⊆ u. for a uniform space (x,u) let θ+u be the supremum of the 110 bruce s. burdick uniformities u∗ on x such that every u∗-uniform cover c is u-uniformly locally finitizible. lemma 3.19. for a uniform space (x,u), θ+u is ℵ0-nearly discrete and any θ+u-uniform cover is u-uniformly locally finitizible. proposition 3.20. for a fixed uniform space (x,u) and a possibly different uniformity u∗ on x, the following are equivalent: (1) u∗ ⊆ θ+u. (2) every cofinally cauchy net for u is cofinally cauchy for u∗. (3) every weakly cauchy filter for u is weakly cauchy for u∗. (4) every somewhere dense cauchy ponet for u is somewhere dense cauchy for u∗. (5) every open cauchy ponet for u is somewhere dense cauchy for u∗. (6) every dense cauchy ponet for u is somewhere dense cauchy for u∗. (7) every ℵ0-nearly discrete uniformity which is finer than u is finer than u∗. proof. (1) implies (2). given a u-cofinally cauchy net s : d → x and a u∗uniform cover c, take a u-uniform cover c′ such that each member of c′ can be covered by finitely many members of c. there is some c ∈ c′ such that s is frequently in c, and then among the finitely many members of c that cover c, s must be frequently in at least one of them. equivalence of (2) and (3). elementary. (2) implies (4). this uses the methods of section 2. (4) implies (5). trivial. equivalence of (5) and (6). this uses the methods of section 2. (5) implies (7). let v be an ℵ0-nearly discrete uniformity on x with u ⊆v. suppose that u∗ 6⊆ v. then there is a u∗-uniform cover c which is not vuniform, so it is not v-uniformly locally finitizible. let p be the collection of all sets a ⊆ x such that c has no finite subset covering a. let p = {(a,x) | x ∈ a ∈ p}. define a preorder on p by saying that (a,x) ≤ (b,y) if b ⊆ a. define a ponet s : p → x by s(a,x) = x. s is clearly v-open cauchy so it is u-open cauchy. to show it is not u∗somewhere dense cauchy we observe that for any a ∈ p and any c ∈ c, a−c ∈ p. (7) implies (1). this follows from lemma 3.19. � corollary 3.21. a uniform space (x,u) which satisfies νu ⊆ θ+u has a paracompact completion if and only if it is preparacompact. we should point out that unlike ν and λ above, θ+ may change the topology of the space since it always contains all the totally bounded uniformities for the discrete topology on the given set of points. definition 3.22. for any uniform space (x,u) define θu = θ+u ∧λu. partial orders in uniform spaces 111 observation 3.23. θ2 = θ since the same is true for θ+ and λ. θ preserves topology because the same is true for λ. the following proposition is an immediate consequence of propositions 3.16 and 3.20. proposition 3.24. for a fixed uniform space (x,u) and a possibly different uniformity u∗ on x, the following are equivalent: (1) u∗ ⊆ θu. (2) every cofinally cauchy net for u is cofinally cauchy for u∗ and every ponet which is open/open dense cauchy for u is open/open dense cauchy for u∗. (3) every somewhere dense cauchy ponet for u is somewhere dense cauchy for u∗ and every ponet which is open/open dense cauchy for u is open/open dense cauchy for u∗. corollary 3.25. for any uniform space (x,u), θu is equal to (1) the supremum of the uniformities u∗ such that every u-open/open dense cauchy ponet is u∗-open/open dense cauchy and every cofinally cauchy net for u is cofinally cauchy for u∗, and (2) the supremum of the uniformities u∗ such that every u-open/open dense cauchy ponet is u∗-open/open dense cauchy and every somewhere dense cauchy ponet for u is somewhere dense cauchy for u∗. in each case the supremum is the finest member of the set. definition 3.26. we will say (see rice’s paper [16] ) a space is uniformly paracompact if every open cover has a uniformly locally finite refinement. proposition 3.27. for a uniform space (x,u) the following are equivalent: (1) (x,u) is cofinally complete. (2) any open cover o of x which is closed under finite unions is uuniform. (3) x is paracompact and all locally finite collections in x are u-uniformly locally finite. (4) (x,u) is uniformly paracompact. proposition 3.27 is a combination of several results in fried [6]. see our paper [3] for more details about the history of this result. in [16], rice gave several different ways of characterizing uniform paracompactness. some of his results are suggestive of isbell’s theorem, but he stops short of defining a functor θ analogous to λ. we will prove a new version of rice’s theorem 3 which uses only functors into the category of uniformities. definition 3.28. let κ be an infinite cardinal. a space (x,u) is locally κ-fine if every u-uniformly locally uκ-uniform cover is a u-uniform cover. froĺık [7] has already called this locally p-fine when κ = ℵ0 and locally e-fine when κ = ℵ1. he states without proof that these properties are coreflective. note that among the uniformities u∗ such that every u∗-uniform cover is a u-uniformly locally uκ-uniform cover, there is a finest one. 112 bruce s. burdick definition 3.29. for a uniform space (x,u) let θ−u be the supremum of all the uniformities u∗ whose uniform covers are all u-uniformly locally uℵ0 uniform. θ− is a functor but it doesn’t always satisfy θ2− = θ−. example 3.30. a space where θ2−u 6= θ−u. let x = ω × ω × 2. let the uniformity u be generated by all equivalences relations satisfying the following properties: (1) for all but finitely many pairs (n,m), (n,m, 0) is related to (n,m, 1). (2) for all but finitely many n, (n,m,i) is related to any (n,m′, i′). the relation r which relates (n,m,i) to (n′,m′, i′) if and only if i = i′ is not a member of θ−u. but the relations rk which have as their equivalence classes the three sets {(n,m,i) | n ≤ k, i = 0}, {(n,m,i) | n ≤ k, i = 1}, and {(n,m,i) | n > k}, are to be found in θ−u, and it is because of them that r is a member of θ2−u. lemma 3.31. for any uniformity u we have θ−u ⊆ θu. proposition 3.32. for any uniform space (x,u) the following are equivalent: (1) (x,u) is cofinally complete. (2) (x,θ−u) is paracompact and fine. (3) (x,θu) is paracompact and fine. proof. (1) implies (2). suppose (x,u) is cofinally complete. given a cover c of x, let c′ be the collection of open sets o such that there is a u ∈u where for any x ∈ x, u[x] is a subset of some member of c. c′ is a cover of x and it is closed under finite unions. so by proposition 3.27 it is uniform. this shows that any open cover of x is uniformly locally uniform. again by proposition 3.27, every open cover of x has a uniformly locally finite refinement. so every open cover of x has a refinement which is uuniformly locally uℵ0 -uniform. therefore the fine uniformity is one of the uniformities which we take the supremum of to get θ−u. since θ− doesn’t change the topology, θ−u must be the fine uniformity. the fact that cofinal completeness implies paracompactness completes the proof. (2) implies (3). θ is trapped between θ− and λ. therefore if θ− is fine so must θ be fine as well. (3) implies (1). if a net is cofinally cauchy for u it is cofinally cauchy for θu by proposition 3.24. then since paracompact and fine implies (x,θu) is cofinally cauchy, the net must have a cluster point. � proposition 3.33. for a fixed uniform space (x,u) and a possibly different uniformity u∗ on x, the following are equivalent: (1) any open cauchy ponet for u is open cauchy for u∗. (2) any dense cauchy ponet for u is dense cauchy for u∗. partial orders in uniform spaces 113 (3) u∗ ⊆u. proof. (1) implies (2). suppose that u∗ 6⊆ u. then there is a cover c which is u∗-uniform but not u-uniform. let p be the set of a ⊆ x such that no c ∈c has a as a subset. proceeding as in several other proofs in this section we can construct a ponet which is u-open cauchy but not u∗-open cauchy. (1) implies (2). use the methods of section 2. (3) implies (1). trivial. � omnibus theorem a uniform space is complete (supercomplete, cofinally complete, paracompact and fine) if and only if the supremum of the uniformities which preserve the open dense (the open/open dense, both the open/open dense and the somewhere dense, the open) cauchy ponets is fine and the underlying topological space is topologically complete (paracompact, paracompact, paracompact ). if this supremum is complete (supercomplete, cofinally complete, paracompact and fine) then so is the original space. references [1] p. alexandroff, diskrete räume, matematicheskij sbornik 2-44 (1937), 501–519. [2] bruce s. burdick, a note on completeness of hyperspaces, in general topology and applications, fifth northeast conference, s. j. andima, et al. (eds.), lecture notes in pure and applied mathematics 134 (marcel dekker, new york, 1991), 19–24. [3] bruce s. burdick, on linear cofinal completeness, topology proceedings 25 (2000), 435–455. [4] á. császár, strongly complete, supercomplete and ultracomplete spaces, in mathematical structures—computational mathematics—mathematical modelling, papers dedicated to professor l. iliev’s 60th anniversary, sofia, 1975, 195–202. [5] h. corson, the determination of paracompactness by uniformities, american journal of mathematics 80 (1958), 185–190. [6] jan fried, on paracompactness in uniform spaces, commentationes mathematicae universitatis carolinae 26 (1985), 373–385. [7] zdenĕk froĺık, locally e-fine measurable spaces, trans. amer. math. soc. 196 (1974), 237–247. [8] seymour ginsberg and j. r. isbell, some operators on uniform spaces, trans. amer. math. soc. 93 (1959), 145–168. [9] n. howes, on completeness, pacific j. math. 38 (1971), 431–440. [10] n. howes, paracompactifications, preparacompactness, and some problems of k. morita and h. tamano, questions and answers in general topology 10 (1992), pp. 191–204. [11] n. howes, modern analysis and topology, (springer-verlag, new york, 1995). [12] j. r. isbell, supercomplete spaces, pacific j. math. 12 (1962), 287–290. [13] j. r. isbell, uniform spaces, (amer. math. soc., providence, 1964). [14] john l. kelley, general topology, (van nostrand, princeton, 1955). [15] jan pelant, locally fine uniformities and normal covers, czechoslovak math. j. 37 (112) (1987), 181–187. [16] michael d. rice, a note on uniform paracompactness, proc. amer. math. soc. 62 (1977), 359–362. [17] mary ellen rudin, lectures on set theoretic topology, (amer. math. soc., providence, 1975). 114 bruce s. burdick received february 2002 revised february 2003 bruce s. burdick department of mathematics, roger williams university, bristol, rhode island 02809, usa. e-mail address : bburdick@rwu.edu on the use of partial orders in uniform spaces. by bruce s. burdick sosagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 137-154 a fuzzification of the category of m-valued l-topological spaces tomasz kubiak and alexander p. šostak abstract. a fuzzy category is a certain superstructure over an ordinary category in which ”potential” objects and ”potential” morphisms could be such to a certain degree. the aim of this paper is to introduce a fuzzy category ft op(l, m) extending the category t op(l, m) of m-valued ltopological spaces which in its turn is an extension of the category t op(l) of l-fuzzy topological spaces in kubiak-šostak’s sense . basic properties of the fuzzy category ft op(l, m) and its objects are studied. 2000 ams classification: primary: 54a40; secondary: 03e72, 18a05. keywords: m-valued l-topology, (l, m)-fuzzy topology, l-fuzzy category, gl-monoid, power-set operators, (l, m)-interior operator, (l, m)-neighborhood system. introducion the concept of an l-fuzzy topological space, that is of a pair (x, t ) where x is a set and t : lx → l is a mapping subjected to certain axioms was introduced (independently) by t.kubiak [5] and a. šostak [9] (actually a prototype of this definition can be traced already in u.höhle’s work [1].) in some cases it seems reasonable to allow different lattices for domain and codomains of t , resp. l and m, thus coming to the concept of an m-valued l-fuzzy topology on x (or an (l, m)-fuzzy topology on x for short), as a mapping t : lx → m subjected to certain axioms. a detailed study of (l, m)-fuzzy topological spaces will be presented in [6], [7]. in a series of papers the second named author considered the concept of a fuzzy category and the problem of fuzzifications of usual categories (see e.g. [11], [12], [13] etc.) actually, a fuzzy category is an ordinary category modified in such a way, that ”potential” objects and ”potential” morphisms are such only to a certain degree, and this degree can be any element of the corresponding 138 t. kubiak and a.šostak lattice. the concept of a fuzzy category lead us to the idea of ”fuzzification” of some known categories that is to construct fuzzy categories on the basis of some standard categories. in particular, in [13] we studied fuzzification of some categories related to topology and algebra. it is the aim of this paper to ”fuzzify” the category t op (l, m) of (l, m)fuzzy topological spaces. as a tool for this fuzzification we use the structure of a gl-monoid on the codomain lattice (that is lattice m in our cotext), and in particular, the corresponding residuation in it. the structure of the paper is as follows. after introducing the fuzzy category ft op (l, m) and other basic definitions in section 2 we discuss the lattice properties of the family of (l, m)-fuzzy topologies on a set x for a fixed level α (section 3). further, in section 4, we proceed to the study of powerset operators in the context of (l, m)-fuzzy topologies, which, appear to be a convenient and powerfull tool for the investigation of such structures. in section 5 we consider basic constructions in the fuzzy category ft op (l, m) of (l, m)-fuzzy topological spaces — namely, products, subspaces, direct sums and quotients. sections 6 and 7 deal with the inner structure of (l, m)-fuzzy topologies. namely, in section 6 we discuss relations between a structure which satisfies the axioms of an (l, m)-fuzzy topology at a level α and the corresponding fuzzy interior operator. further, in section 7 the relations between this fuzzy interior operator and the corresponding neighbourhood system are discussed. 1. preliminaries let l = (l1, ≤l, ∧l, ∨l, ∗l) and m = (m, ≤m, ∧m , ∨m , ∗m ) be glmonoids (cf e.g. [2], [3]). let ⊤l, ⊤m and ⊥l, ⊥m denote the top and the bottom elements of l and m respectively. in what follows we shall usually omit the subscripts l and m since from the context it will be clear in what lattice the operation is applied. it is well known that every gl-monoid l is residuated, i.e. there exists a further binary operation — implication ” ֌ ” connected with ∗ by the galois coonection: α ∗ β ≤ γ ⇐⇒ α ≤ β ֌ γ ∀α, β, γ ∈ l. let x be a set and lx be the family of all l-subsets of x, i.e. mappings a : x → l. then all operations on l in an obvious way can be pointwise extended to lx thus generating the structure of a gl-monoid on lx . in particular, implication a ֌ b ∈ lx for l-sets a, b ∈ lx is defined by (a ֌ b)(x) := a(x) ֌ b(x); the top 1x and the bottom elements 0x in l x are defined respectively as 1x(x) = ⊤l ∀x ∈ x and 0x(x) = ⊥l ∀x ∈ x. to recall the concept of an l-valued or l−fuzzy category [11, 12], consider an ordinary (classical) category c and let ω : ob(c) → l and µ : mor(c) → l be l−fuzzy subclasses of the classes of its objects and morphisms respectively. now, an l−fuzzy category can be defined as a triple (c, ω, µ) satisfying the following axioms ([12], cf also [11] in case ∗ = ∧): a fuzzification of the category of m-valued l-topological spaces 139 10 µ(f) ≤ ω(x) ∧ ω(y ) ∀ x, y ∈ ob(c) and ∀f ∈ mor(x, y ); 20 µ(g ◦ f) ≥ µ(f) ∗ µ(g) whenever the composition g ◦ f is defined; 30 µ(ex ) = ω(x) where ex : x → x is the identity morphism. 2. basic definitions definition 2.1. [m-fuzzy category ft op (l, m).] let c(l,m) be an (ordinary) category whose objects are pairs (x, t ) where x is a set and t : lx → m is a mapping, and whose morphisms f : (x, tx ) → (y, ty ) are arbitrary mappings f : x → y . given a set x and a mapping t : lx → m we define three fuzzy predicates: ω1(t ) = t (1x ) ( or, equivalently ω1(t ) = ⊤ ֌ t (1x )); ω2(t ) = ∧ u⊂lx,|u|<ℵ0 ( ∧ u∈u t (u) ֌ t ( ∧ u∈u u ) ) ; ω3(t ) = ∧ u⊂lx ( ∧ u∈u t (u) ֌ t ( ∨ u∈u u ) ) . let ω(t ) = ω1(t ) ∧ ω2(t ) ∧ ω3(t ). given (x, tx ), (y, ty ) and a mapping f : x → y we set ν(f) = ∧ v∈ly ( ty (v ) ֌ tx (f −1(v )) ) , and µ(f) = ν(f) ∧ ωx (tx ) ∧ ωy (ty ). a mapping f will be called continuous if ν(f) = ⊤. actualy this means that ty (v ) ≤ tx (f −1(v )) for all v ∈ lx. it is easy to note that µ(ex ) = ω(x). further, if f : (x, tx ) → (y, ty ) and g : (y, ty ) → (z, tz ) are mappings, then ν(g ◦ f) = ∧ w∈lz ( tz (w ) ֌ tx (f −1(g−1(w )) ) ) ≥ ≥ ∧ w∈lz ( ( tz(w ) ֌ ty (g −1(w )) ) ∗ ( ty (g −1(w ) ֌ tx (f −1(g−1(w )) ) ) ≥ ∧ w∈lz (tz (w ) ֌ ty (g −1(w )) ∗ ∧ v∈ly ( ty (v ) ֌ tx (f −1(v )) ) = = ν(g) ∗ ν(f), and hence also µ(g ∗ f) ≥ µ(g) ◦ µ(f). thus we arrive at a (m-)fuzzy category ft op (l, m) = (c(l,m), ω, µ). we interpret ω(t ) as the degree to which a mapping t is an (l, m)-fuzzy topology on x. in case ω(t ) ≥ α we say that t is an (l, m)-fuzzy α-topology on x. an (l, m)-fuzzy ⊤-topology is just an (l, m)-fuzzy topology [6], [7] (and an l-fuzzy topology in case m = l, see e.g. [5], [9], [10]). on the other hand any mapping t : lx → m is an (l, m)-fuzzy ⊥-topology on a set x. 140 t. kubiak and a.šostak a pair (x, t ) where t is an (l, m)-fuzzy ⊥-topology will be referred to as an (l, m)-fuzzy ⊥-topological space. remark 2.2. applying ω3 to u = ∅ we get ω3(t ) ≤ ⊤ ֌ t (0x ) = t (0x ). remark 2.3. • ω1(t ) = ⊤ iff t (1x) = ⊤; • ω2(t ) = ⊤ iff ∀ u1, u2 ∈ l x it holds t (u1 ∧ u2) ≥ t (u1) ∧ t (u2); • ω3(t ) = ⊤ iff ∀ u ⊂ l x it holds t ( ∨ u∈u u) ≥ ∧ u∈u t (u). thus the fuzzy predicates ω1, ω2, ω3 are fuzzifications of the corresponding axioms of an (l, m)-fuzzy topology, cf [6], [7]. fuzzy predicate ν can be viewed as a version of fuzzification of the axiom of continuity while µ ”touch it up” in order to take into account the ”defectiveness of topologiness” of t . remark 2.4. [ the case of an idempotent α. ] let α ∈ l be idempotent, i.e. α ∗ α = α, and let fαt op (l, m) denote the subcategory of ft op (l, m) whose objects (x, t ) and morphisms f satisfy conditions ω(t ) ≥ α and µ(f) ≥ α. then fαt op (l) is obviously a usual (crisp) category. in particular, f⊤t op (l, m) = t op (l, m). definition 2.5. given an object (x, t ) of ft op (l, m), we define a mapping σt := σ : l x → m by setting σ(a) = t (a ֌ 0x ) for every a ∈ l x. the mapping σ thus defined is called the degree of closedness in the space (x, t ). proposition 2.6. [ basic properties of σ ] (1) σ1(σ) := σ(0x ) = t (1x ) and hence σ1(σ) = ω1(t ); (2) σ2(σ) := ∧ a⊂lx ,|a|<ℵ0 ( ∧ a∈a σ(a) ֌ σ ( ∨ a∈a a ) ) ≥ ω2(t ) (3) σ3(σ) := ∧ a⊂lx ( ∧ a∈a σ(a) ֌ σ ( ∧ a∈a a ) ) ≥ ω3(t ). proof. σ1(σ) = σ(0x ) = t (0x ֌ 0x ) = t (1x ) = ω1(t ); σ2(σ) : = ∧ a⊂lx |a|<ℵ0 ( ∧ a∈a σ(a) ֌ σ( ∨ a∈a a) ) = = ∧ a⊂lx |a|<ℵ0 ( ∧ a∈a t (a ֌ 0x ) ֌ t ( ( ∨ a∈a a) ֌ 0x ) ) ) = = ∧ a⊂lx |a|<ℵ0 ( ∧ a∈a t (ua) ֌ t ( ∧ a∈a ua) ) ≥ ≥ ∧ u⊂lx |u|<ℵ0 ( ∧ u∈u t (u) ֌ t ( ∧ u∈u u) ) = ω2(t ), a fuzzification of the category of m-valued l-topological spaces 141 where ua := a ֌ 0x. in a similar way, σ3(σ) := ∧ a⊂lx ( ∧ a∈a σ(a) ֌ σ ( ∧ a∈a a ) ) = = ∧ a⊂lx ( ∧ a∈a t (a ֌ 0x ) ֌ t ( ∧ a∈a (a ֌ 0x) ) = = ∧ a⊂lx ( ∧ a∈a t (ua) ֌ t ( ∨ a∈a ua) ) ≥ ≥ ∧ u⊂lx ( ∧ u∈u t (u) ֌ t ( ∨ u∈u u) ) . � reasoning in a similar way it is easy to establish the following proposition 2.7. given a mapping σ : lx → m let m-valued predicates σ1(σ), σ2(σ) and σ3(σ) be defined as in proposition 2.6, and let t := tς be defined by t (a) = σ(a ֌ 0x ). then ω1(t ) = σ1(σ), ω2(t ) ≥ σ2(σ), ω3(t ) ≥ σ3(σ). in case when l is an mv -algebra the l-powerset lx also is an mv -algebra, and hence (a ֌ 0x) ֌ 0x = a for every a ∈ l x . therefore it follows: proposition 2.8. if l is an mv -algebra, then tσt = t and σtς = σ. in particular the structures t and σ mutually define one another. besides, σ1 ( σt ) = ω1(t ), σ2 ( σt ) = ω2(t ), σ3 ( σt ) = ω3(t ). 3. lattice properties of (l, m)-fuzzy α-topologies let α ∈ m be fixed and let tα(x) := tα(l, m, x) be the family of all (l, m)-fuzzy α-topologies on a set x. theorem 3.1. tα(x) is a complete lattice. proof. first, notice that tdis : l x → m defined by tdis(u) = ⊤ for all u ∈ l x (the so called discrete (l, m)-fuzzy topology) is the top element of tα(x) and tind : l x → m defined by tind(0x ) = tind(1x ) = α and tind(u) = ⊥ for u ∈ lx \ {0x, 1x} (the so called indiscrete (l, m)-fuzzy topology) is the bottom element of tα(x). further, let t 0 α(x) ⊂ tα(x) and let t0 : l x → m be defined by the equality t0(u) = ∧ t∈t0α(x) t (u) ∀u ∈ lx. then ω1(t0) = t0(1x ) = ∧ t∈t0α(x) t (1x ) = ∧ t∈t0α(x) t (1x ) ≥ α; 142 t. kubiak and a.šostak ω2(t0) = ∧ u⊂lx |u|<ℵ0 ( ∧ u∈u t0(u) ֌ t0 ( ∧ u∈u u ) ) = = ∧ u⊂lx |u|<ℵ0 ( ∧ u∈u ( ∧ t∈t0α(x) t ) (u) ֌ ∧ t∈t0α(x) ( t ( ∧ u∈u u )) ) ≥ ≥ ∧ t∈t0 α (x) ( ∧ u⊂lx |u|<ℵ0 ( ∧ u∈u t (u) ֌ t ( ∧ u∈u u) ) ) = ∧ t∈t0 α (x) ω2(t ) ≥ α. reasoning in a similar way we get: ω3(t0) = ∧ u⊂lx ( ∧ u∈u t 0(u) ֌ t0( ∨ u∈u u) ) ≥ ≥ ∧ t∈t0 α (x) ( ∧ u∈lx ( ( ∧ u∈u t (u) ) ֌ t ( ∨ u∈u u) ) ) = ∧ t∈t0 α (x) ω3(t ) ≥ α. thus t0 ∈ t 0 α(x) and hence t0 is indeed the minimal element of t 0 α(x) in tα(x). � the previous theorem allows also to write an explicite formula for the supremum of a subset t0α(x) ⊂ tα(x). namely sup t0α(x) = ∧ {t ∈ tα(x) | t ≥ tλ ∀tλ ∈ t 0 α(x)}. remark 3.2. let s : lx → m be a mapping and let the mapping ts : l x → m be defined by ts = ∧ {t : t ∈ tα(x) and t ≥ s}, where as before tα(x) := tα(l, m, x). from theorem 3.1 it follows that ts is an (l, m)-fuzzy α-topology, besides it is the smallest one (≤) of all (l, m)fuzzy α-topologies which are greater or equal than s. in this case s is called a subbase of the (l, m)-fuzzy α-topology ts. proposition 3.3. [ level decomposition of (l, m)-fuzzy -topologies ] let t : lx → m be an (l, m)-fuzzy α-topology and assume that γ ∈ m is such that γ ∗ α = γ. further, let tγ = {u | t (u) ≥ γ}. then tγ is a (chang-goguen) l-topology on x. in particular, if α is idempotent, then tα is a (chang-goguen) l-topology on x. proof. since ω1(t ) = ⊤ it follows that t (1x ) ≥ α ≥ γ, and 1x ∈ tγ. let u1, . . . , un ∈ tγ. then, since ω2(t ) ≥ α, it holds γ ֌ t (u1 ∧ . . . ∧ un) ≥ t (u1) ∧ . . . ∧ t (un) ֌ t (u1 ∧ . . . ∧ un) ≥ α and hence t (u1 ∧ . . . ∧ un) ≥ α ∗ γ = γ. in a similar way, taking into account that ω3(t ) ≥ α, it is easy to verify that if ui ∈ tγ for all i ∈ i, then t ( ∨ i∈i ui ) ≥ α ∗ γ = γ. � a fuzzification of the category of m-valued l-topological spaces 143 theorem 3.4. let s : lx → m be an (l, m)-fuzzy β-topology where α∗β = α then the mapping t : lx → m defined by t (u) = α ֌ s(u) for every u ∈ lx is an (l, m)-fuzzy topology on x. proof. notice first that in this case α = α ∗ β ≤ β, and hence s(1x ) ≥ α. therefore ω1(t ) = t (1x ) = α ֌ s(1x ) = α ֌ α = ⊤. to verify axioms 2 and 3 for t notice first that for every γ ∈ m it holds α ֌ γ ∗ β = α ֌ γ. indeed, α ֌ γ = ∨ {λ | λ ∗ α ≤ γ} ≤ ≤ ∨ {λ | λ ∗ α ∗ β ≤ γ ∗ β} = ∨ {λ | λ ∗ α ≤ γ ∗ β} = α ֌ γ ∗ β. the converse inequality is obvious. we proceed as follows. since ω2(s) ≥ β, we get s( n ∧ i=1 ui) ≥ n ∧ i=1 s(ui) ∗ β and hence α ֌ s ( n ∧ i=1 ui ) ≥ α ֌ n ∧ i=1 s(ui) ∗ β = α ֌ n ∧ i=1 s(ui) = n ∧ i=1 ( α ֌ s(ui) ) ; thus t ( ∧n i=1 ui) ≥ ∧n i=1 t (ui). from ω3(s) ≥ β, reasoning in a similar way as above, we conclude that s ( ∨ i∈i ui ) ≥ ∧ i∈i s(ui) ∗ β, and hence t ( ∨ i∈i ui) ≥ ∧ i∈i t (ui) for any family {ui | i ∈ i } ⊂ l x. � corollary 3.5. if s : lx → m is an (l, m)-fuzzy α-topology and α is idempotent, then the mapping t : lx → m defined by t (u) := α ֌ s(u) for every u ∈ lx is an (l, m)-fuzzy topology. if s : lx → m is an (l, m)-fuzzy topology then for every α the mapping t (u) = α ֌ s(u) is an (l, m)-fuzzy topology. theorem 3.6. let f : (x, tx ) → (y, ty ) be a mapping, ω(tx ) ≥ β, ω(ty ) ≥ α where β ∗ α = α, and let sy : l y → m be a subbase of ty . then the following conditions are equivalent: 10 ty (v ) ֌ tx (f −1(v )) ≥ α ∀v ∈ ly ; 20 sy (v ) ֌ tx (f −1(v )) ≥ α ∀v ∈ ly . in particular, these conditions are equivalent in case when α ≤ β and α is idempotent. 144 t. kubiak and a.šostak proof. since ty (v ) ≥ sy (v ), it holds ty (v ) ֌ tx (f −1(v )) ≤ sy (v ) ֌ tx (f −1(v )) and hence 10 =⇒ 20. conversely, if sy (v ) ֌ tx (f −1(v )) ≥ α for all v ∈ ly , then sy (v ) ≤ α ֌ tx (f −1(v )) ∀v ∈ lx. let now t ′(v ) := tx (f −1(v )). it is easy to verify that t ′ is an (l, m)-fuzzy βtopology since tx is an (l, m)-fuzzy β-topology. further, let t ′′ : ly → m be defined by t ′′(v ) := α ֌ t ′(v ). then by theorem 3.4 t ′′ is an (l, m)-fuzzy topology on y . moreover, sy (v ) ≤ t ′′(v ). thus, since ty is an (l, m)-fuzzy α-topology generated by subbase sy , it follows that sy (v ) ≤ ty (v ) ≤ t ′′(v ), and hence ty (v ) ≤ α ֌ tx (f −1(v )) =⇒ α ≤ ty (v ) ֌ tx (f −1(v )) ∀v ∈ ly . � question 3.7. do the statements of corollary 3.5 and theorem 3.6 hold also in case α = β but without assumption of idempotency of α ? 4. power-set operators and (l, m)-fuzzy α-topologies let x, y be sets, and let f : ly → lx be a mapping preserving arbitrary joins and meets. in particular, f (1y ) = 1x and f (0y ) = 0x. definition 4.1. (cf e.g. [8]) the powerset operator f→ : m(l y ) → m(l x) of a mapping f : ly → lx is defined by the equality f→(ty )(u) = ∨ {ty (v ) : f (v ) = u}, ∀u ∈ l x for every ty : l y → m, definition 4.2. (cf e.g [8]) the powerset operator f← : m(l x) → m(l y ) of a mapping f : ly → lx is defined by the equality f ←(tx )(v ) = tx ( f (v ) ) ∀v ∈ ly for every tx : l x → m, the following two theorems show that the powerset operators f→ and f← do not diminish the topologiness degree of the mappings ty and tx respectively. theorem 4.3. if m is completely distributive, then ω(f→(ty )) ≥ ω(ty ) := α. ( actually, ω1(f →(ty )) ≥ ω1(ty ), ω2(f →(ty )) ≥ ω2(ty ) and ω3(f →(ty )) ≥ ω3(ty ). ) a fuzzification of the category of m-valued l-topological spaces 145 proof. since f→(ty )(1x ) = ∨ {ty (v ) | f (v ) = 1x} ≥ ty (1y ) ≥ α, it follows that ω1(f →(ty )) ≥ α. to verify that ω2(f →(ty )) ≥ α fix some u1, . . . un ∈ l x and let u0 := ∧n i=1 ui. we have to show that ( n ∧ i=1 f→(ty )(ui) ) ֌ f→(ty )(u0) ≥ α. if for some i ∈ {1, . . . , n} there does not exist vi ∈ l y such that f (vi) = ui, then from the definition of f→(ty ) it is clear that f →(ty )(ui) = ⊥ and hence the inequality is obvious. assume therefore that for each i = 1, . . . , n some vi ∈ l y is fixed such that ui = f (vi). then, since ω2(ty ) ≥ α, and since f ( n ∧ i=1 vi) = n ∧ i=1 f (vi) = ∧ ui = u0 it follows that n ∧ i=1 ty (vi) ֌ ty ( n ∧ i=1 vi) ≥ n ∧ i=1 ty (vi) ֌ ∨ v0∈l y f (v0)=u0 ty (v0) ≥ α. this holds for any choice of vi ∈ l y , i ∈ {1, . . . , n}, satisfying f→(vi) = ui, and therefore taking into account that l is infinitely distributive, we conclude that n ∧ i=1 ( f→(ty )(ui) ) ֌ f→(ty )(u0) = n ∧ i=1 ( ∨ f (vi)=ui vi∈ly ty (vi) ) ֌ ∨ f (vi)=ui v0∈ly ty (v0) ) = ( ∨ f (vi)=ui n ∧ i=1 ty (vi) ) ֌ ∨ f (v0)=u0 v0∈ly ty (v0) ≥ α. to verify the third inequality, ω3(f →(ty )) ≥ α, fix a family u = {ui | i ∈ i}, and let u0 := ∨ i∈i ui. we have to show that ∧ i∈i f→(ty )(ui) ֌ f →(ty )( ∨ i∈i ui) ≥ α. let for each i ∈ i an l-set vi ∈ l y be fixed such that f (vi) = ui. (as in the previous situation it is sufficient to assume that such choice of vi ∈ l y for all i ∈ i is possible.) then ∧ i∈i ty (vi) ֌ ty ( ∨ i∈i vi) ≥ α, that is ∧ i∈i ty (vi) ≥ ty ( ∨ i∈i vi) ∗ α. 146 t. kubiak and a.šostak applying complete distributivity we get the following chain of (in)equalities: α ∗ f→(ty )( ∨ i∈i ui) = α ∗ ∨ {ty ( ∨ i∈i vi) : f (∨ivi) = ∨iui} = = ∨ ( α∗{ty ( ∨ i∈i vi) : f (∨ivi) = ∨iui} ) ≥ ∨ { ∧ i∈i ty (vi) : f (∨ivi) = ∨iui} ≥ ≥ ∨ { ∧ i∈i ty (vi) : f (vi) = ui} = ∨ { ∧ i∈i ty (vi) : vi ∈ vi := {v | ui = f (v )}} = = ∨ ϕ∈ ∏ i vi ( ∧ i∈i ty (ϕ(i))) = ∧ i∈i ∨ vi∈vi ty (vi) = ∧ i∈i ∨ f (vi)=ui ty (vi) = ∧ i∈i f→(ty )(ui). and hence we obtain the required inequality: f→(ty )( ∨ i∈i ui) ֌ ∧ i∈i f→(ty )(ui) ≥ α. � theorem 4.4. ω(f←(tx )) ≥ ω(tx ) =: α. ( actually, ω1(f ←(tx )) = ω1(tx ), ω2(f ←(tx )) ≥ ω2(tx ) and ω3(f ←(tx )) ≥ ω3(tx ). ) proof. ω1(f ←(tx )) = f ←(tx )(1y ) = tx (f (1y )) = tx (1x ) = ω1(tx ) ≥ α. to verify condition ω2(ty ) ≥ α, where ty := f ←(tx ), fix {v1, . . . , vn} ⊂ l y , then n ∧ i=1 ty (vi) ֌ ty ( n ∧ i=1 vi ) = n ∧ i=1 f←(tx )(vi) ֌ f ←(tx ) ( n ∧ i=1 vi ) = n ∧ i=1 tx ( f (vi) ) ֌ tx ( n ∧ i=1 f (vi) ) ≥ α. finally, to verify the condition ω3(ty ) ≥ α fix a family v = {vi | i ∈ i} ⊂ l y . then ∧ i∈i ty (vi) ֌ ty ( ∨ i∈i (vi) ) = ∧ i∈i f←(tx )(vi) ֌ f ←(tx )( ∨ i∈i vi) = = ∧ i∈i tx (f (vi)) ֌ tx ( ∨ i∈i f (vi)) ≥ ω3(tx ) ≥ α. � power-set operators f← and f→ can be applied, in particular, for description of final and initial (l, m)-fuzzy α-topologies. here are some details: let f : x → y be a mapping, then by setting f←(v ) := f−1(v ) one defines a mapping f← : ly → lx, which obviously, preserves joins and meets, and so one can apply to it theorems 4.3 and 4.4. namely, one can get the following corollaries from the statements of these theorems and from the definition of power-set operators. a fuzzification of the category of m-valued l-topological spaces 147 corollary 4.5. let ty : l y → m be a mapping where m is completely distributive, and let ω(ty ) ≥ α. then given a mapping f : x → y , it holds ω ( (f←) → (ty ) ) ≥ α. besides, (f←) → (ty ) is the weakest (l, m)-fuzzy α-topology (actually, even the weakest (l, m)-fuzzy ⊥-topology!) on x for which the mapping f : (x, tx ) → (y, ty ) is continuous (i.e. ν(f) = ⊤). corollary 4.6. let tx : l x → m be a mapping and ω(tx ) ≥ α. then given a mapping f : x → y it holds ω ( (f←) ← (tx ) ) ≥ α. besides, (f←) ← (tx ) is the strongest (l, m)-fuzzy α-topology (actually, even the strongest (l, m)fuzzy ⊥-topology!) on y for which f : (x, tx ) → (y, ty ) is continuous (i.e. ν(f) = ⊤). 5. products, subspaces, direct sums and quotients in this section we shall discuss how basic operations for (l, m)-fuzzy αtopological spaces can be defined. 5.1. products. let x = {(xi, ti) : i ∈ i} be a family of (l, m)-fuzzy αtopological spaces, where m is completely distributive, let x = ∏ i∈i xi be the product of the corresponding sets, and let pi : x → xi be the projections. further, let t̂i := (p ← i ) →(ti) : l x → m. then, by corollary 4.5, ω(t̂i) ≥ α. let s := ∨ i∈i t̂i and let tx : l x → m be the (l, m)-fuzzy α-topology generated by the subbase s : lx → m. then, obviously, tx is the weakest (l, m)-fuzzy α-topology for which all projections are continuous (i.e. ν(pi) = ⊤). moreover, the pair (x, tx ) is the product of the family x in the fuzzy category ft op (l, m) in the following sense: given an (l, m)-fuzzy β-topological space (z, tz ) where β∗α = α, and a family of mappings fi : (z, tz ) → (xi, ti), i ∈ i, there exists a unique mapping h : (z, tz) → (x, tx ) such that pi ◦ h = fi for all i ∈ i and ν(h) ≥ α ⇐⇒ ∧ i∈i ν(fi) ≥ α. indeed, let h := △i∈ifi : z → x be the diagonal product of mappings fi, i ∈ i. if ν(h) ≥ α, then for every i ∈ i ν(fi) = ν(pi ◦ h) ≥ ν(pi) ∗ ν(h) ≥ ⊤ ∗ ν(h) ≥ α. conversely, let ν(fi) ≥ α for all i ∈ i. we have to verify that in this case tx (w ) ֌ tz (h −1(w )) ≥ α ∀w ∈ lx. according to theorem 3.6 it is sufficient to verify that s(w ) ֌ tz (h −1(w )) ≥ α ∀w ∈ lx. however, from the definition of s it is clear that s(w ) = ti(vi) if w := ṽi where ṽi = p −1 i (vi) for some vi ∈ l xi and s(w ) = ⊥ otherwise. therefore 148 t. kubiak and a.šostak it is sufficient to verify the above inequality for l-sets of the form ṽi. however, in this case h−1(ṽi) = h −1(p−1i (vi)) = f −1 i (vi), and hence the requested inequality can be rewritten as ti(vi) ֌ tz(f −1 i (vi)) ≥ α which holds according to our assumptions. 5.2. subspaces. let (x, t ) be an (l, m)-fuzzy α-topological space, let x0 ⊂ x and let e : x0 → x be the embedding mapping. further, let t0 := (e←)→(t ). then according to corollary 4.5 ω(t0) ≥ ω(t ) and hence (x0, t0) is an (l, m)-fuzzy α-topological space. from the construction it is clear that ν(e) = ⊤. moreover, it is easy to note that (x0, t0) is a subobject of (x, t ) in the following sense: for every (l, m)-fuzzy ⊥-topological space (z, tz ) and for every mapping f : (z, tz) → (x0, t0) it holds ν(f) = ν(e ◦ f). indeed, let v0 = e −1(v ) for some v ∈ lx. then t0(v0) ֌ tz (f −1(v0)) ≥ t (v ) ֌ tz (f −1(e−1(v ))) = = t (v ) ֌ tz((e ◦ f) −1(v )) ≥ ν(e ◦ f), and hence ν(f) ≥ ν(e ◦ f). the converse inequality is obvious. 5.3. coproducts (direct sums). let x = {(xi, ti) : i ∈ i} be a family of (l, m)-fuzzy α-topological spaces, let x = ⊕i∈ixi be the disjoint union of the corresponding sets, and let ei : xi → x be the inclusion mapping. further, let si := (e ← i ) ←(ti). then by corollary 4.6 ω(si) ≥ α, and hence, according to theorem 3.1 t := ∧ i∈i si is an (l, m)-fuzzy α-topology. besides, it is clear that t is the strongest (l, m)-fuzzy α-topology for which all inclusions ei are continuous, i.e. ν(ei) = ⊤. moreover (x, t ) is the coproduct of the family x in the fuzzy category ft op (l, m) in the following sense: let (z, tz ) be an (l, m)-fuzzy ⊥-topological space and let fi : (xi, ti) → (z, tz ), i ∈ i, be a family of mappings. further, let the mapping f : (x, t ) → (z, tz ) be defined by f(x) = fi(x) iff x ∈ xi. then ν(f) = ∧ i∈i ν(fi). indeed, since fi = f ◦ ei and ν(ei) = ⊤ for all i ∈ i, the inequality ν(f) ≥ ∧ i∈i ν(fi) is obvious. conversely, assume that ∧ i∈i ν(fi) ≥ α. then α ≤ tz(v ) ֌ ti(f −1(v )) = tz(v ) ֌ si(ei(f −1 i (v )) = tz (v ) ֌ si(f −1(v )). a fuzzification of the category of m-valued l-topological spaces 149 now, taking infimum over all i ∈ i, we obtain: tz(v ) ֌ t (f −1(v )) ≥ α. 5.4. quotients. let (x, tx ) be an (l, m)-fuzzy α-topological space and let f : x → y be a surjective mapping. further, let ty = (f ←)←(tx ). then, according to corollary 4.6 ω(ty ) ≥ α and hence (y, ty ) is an (l, m)-fuzzy αtopological space. it is clear that ty is the strongest (l, m)-fuzzy α-topology for which the mapping f is continuous, i.e. ν(f) = ⊤. the pair (y, ty ) can be viewed as the quotient of (x, tx ) under mapping f in the fuzzy category ft op (l, m) in the following sense: let (z, tz ) be an (l, m)-fuzzy α-topological space and let g : (y, ty ) → (z, tz ) be a mapping. then ν(g ◦ f) = ν(g). indeed, the inequality ν(g ◦ f) ≤ ν(g) holds always. to establish the converse inequality let h = g ◦ f and let w ∈ lz. then by surjectivity of the mapping f there exists u ∈ lx such that g−1(w ) = f(u) and, in particular, u = f−1(g−1(w )). hence, by definition of ty we have ty (g −1(w )) = tx (f −1(g−1(w ))) = tx (h −1(w )). it follows from here that tz (w ) ֌ ty (h −1(w )) = tz (w ) ֌ t (g −1(w )) and taking infimum over all w ∈ lz we obtain: ν(g ◦ f) ≥ ν(g). 6. interior operator theorem 6.1. let t : lx → m be a mapping where m is completely distributive and let ω(t ) ≥ α. we define the mapping int := intt : l x × m → lx by setting: int(a, β) = ∨ {u : u ≤ a, t (u) ≥ β} ∀a ∈ lx, ∀β ∈ m. then: (1int) int(1x, β) = 1x ∀β ≤ α; (2int) a ≤ a′, β′ ≤ β =⇒ int(a, β) ≤ int(a′, β′); (3int) ∧ i=1,...,n int(ai, β) ≤ int( ∧ i=1,...,n ai, β ∗ α) ∀β ∈ m; (4int) int(a, ⊥) = a. (5int) int(int(a, β), β ∗ α) ≥ int(a, β) ∀β ∈ m; (6int) if int(a, β) = a0 ∀β ∈ m′, then int(a, ∨m′) = a0. 150 t. kubiak and a.šostak besides, if ω(t ) = ⊤, then int satisfies the following stronger version of the property (5int): (5int0 ) int(int(a, β), β) ≥ int(a, β) conversely, if a mapping int : lx ×m → lx satisfies conditions (1int) (6int) above for a fixed α ∈ m , then the mapping t := tint : l x → m defined by the equality t (a) = ∨ {β ∈ m : int(a, β) = a} is an (l, m)-fuzzy α-topology on x and besides ω3(tint) = 1. (in the sequel mappings int : lx × m → lx satisfying the above properties (1int) (6int) for a fixed α ∈ m will be referred to as an (l, m)-fuzzy αinterior operator.) the (l, m)-fuzzy α-topology and the corresponding (l, m)-fuzzy α-interior operator are related in the following way: tintt ∗ α ≤ t ≤ tintt and inttint ≤ int and inttint (·, β ∗ α) ≥ int(·, β) ∀β ∈ m. in case ω3(t ) = ⊤, the equalities t = tintt and int = inttint hold (cf theorem 8.1.2 in [4]). proof. (1) since ω1(t ) ≥ α, it follows that t (1x ) ≥ α ≥ β and hence int(1x, β) ≥ 1x. (2) obvious. (3) applying infinite distributivity of the lattice m and condition ω2(t ) ≥ α we have ∧ i=1...n int(ai, β) = ∧ i ( ∨ {ui | ui ≤ ai, t (ui) ≥ β} ) ≤ ≤ ∨ { ∧ i ui | ui ≤ ai, t (ui) ≥ β } ≤ ≤ ∨ {v | v ≤ ∧ i ai, t (v ) ≥ β ∗ α} = = int( ∧ i ai, β ∗ α). (4) obvious. (5) int(a, β) = ∨ {u ∈ lx | u ≤ a, t (u) ≥ β}; hence by condition ω3(t ) ≥ α we have t (int(a, β)) ≥ β ∗ α and therefore int(int(a, β), β ∗ α) ≥ int(a, β). (6) int(a, ∨ m′) = ∨ {u ∈ lx | u ≤ a, t (u) ≥ ∨ m′} = ∨ {u ∈ lx | u ≤ a, t (u) ≥ β ∀β ∈ m′} = a0. moreover, if ω3(t ) = ⊤, then t (int(a, β)) ≥ β and hence int(int(a, β), β) ≥ int(a, β). a fuzzification of the category of m-valued l-topological spaces 151 conversely, (1) if int(1x, β) = 1x for all β ≤ α, then tint(1x ) ≥ α, and hence ω1(tint) ≥ α. (2) let u1, . . . , un ∈ l x, and let β0 := tint(u1) ∧ . . . ∧ tint(un). then for every β ≪ β0 int(ui, β) ≥ ui and hence by property (3 int) int( n ∧ i=1 ui, β ∗ α) ≥ n ∧ i=1 int(ui, β) ≥ n ∧ i=1 ui. it follows from here that tint( n ∧ i=1 ui) = ∨ {γ | int( n ∧ i=1 ui, γ) ≥ n ∧ i=1 ui} ≥ β ∗ α for every β ≪ β0 and hence, by complete distributivity of m tint( n ∧ i=1 ui) ≥ β0 ∗ α. therefore n ∧ i=1 tint(ui) ֌ tint( n ∧ i=1 ui) ≥ α, for each finite family {u1, . . . , un} ⊂ l x and hence ω2(t ) ≥ α. (3) let u := {ui | i ∈ i} and let ∧ i∈i tint(ui) =: β0. then for every i ∈ i and for every β ≪ β0 it holds int(ui, β) ≥ ui. applying (2 int) we conclude from here that ∨ i∈i ui ≤ ∨ i∈i int(ui, β) ≤ int( ∨ i∈i ui, β) for every β ≪ β0. and hence tint( ∨ i∈i ui) ≥ β. hence, by complete distributivity of the lattice m we conclude: tint( ∨ i∈i ui) ≥ ∧ i∈i tint(ui). thus ω3(tint) = ⊤ and hence ω(tint) ≥ α to verify the relations between tintt and t , take some u ∈ l x and let t (u) =: β. then intt (u, β) ≥ u, and hence tintt (u) ≥ β, thus the inequality t ≤ tintt is established. conversely, let tintt (u) = ∨ {β | intt (u, β) ≥ u} = β0. then for each β ≪ β0 intt (u, β) = ∨ {v |t (v ) ≥ β} = u, and hence, in view of the property ω3(t ) ≥ α, we conclude that t (intt (u, β)) ≥ β ∗ α. since this holds for every β ≪ β0 and for every u ∈ l x it follows from here that tintt ∗ α ≤ t . in particular, if ω3(t ) = ⊤, then tintt = t . 152 t. kubiak and a.šostak let now a ∈ lx and let int(a, β) =: w. then by the property (5int) of the (l, m)-fuzzy α-interior operator and from the definition of the (l, m)structure tint : l x → m we have tint(w ) ≥ β and hence, taking into account monotonicity and property (4int) of the (l, m)-fuzzy α-interior operator, it follows inttint (a, β ∗ α) ≥ inttint(w, β) ≥ w = int(a, β), i.e. inttint(·, β ∗ α) ≥ int(·, β). in particular, if ω3(t ) = ⊤, then inttint = int. conversely, let inttint (m, β) =: w, then by the definition of inttint , we conclude that ∨ {u|tint(u) ≥ β, u ≤ a} = w. taking into account that, as it was already established above, ω3(tint) = ⊤ it follows that β ≤ tint(w ) = ∨ {β′ |int(w, β′) = w}. properties (6int) and (2int) we conclude that int(a, β) ≥ int(w, β) ≥ w and hence int ≥ inttint, that is inttint(·,β) ≤ int(·, β) ≤ inttint (·, β ∗ α). in particular, if ω3(t ) = ⊤, then int = inttint. � 7. neighborhood systems let int : lx → m be an (l, m)-α-fuzzy interior operator, i.e. int satisifes properies (1int) — (6int). theorem 7.1. let nint := n : x × l x × l → l be defined by the equality n (x, u, β) = int(u, β)(x). then: (1n ) n (x, 1x, β) = ⊤ ∀x ∈ x if β ≤ α; (2n ) u ≤ u′, β′ ≤ β =⇒ n (x, u, β) ≤ n (x, u′, β′); (3n ) ∧n i=1 n (x, ui, β) ≤ n (x, ∧n i=1 ui, β ∗ α); (4n ) n (x, u, 0) = int(u, 0)(x) (= u(x));. (5n ) n (x, u, β) ≤ ∨ {n (x, v, β ∗ α) | v (y) ≤ n (y, u, β) : ∀y ∈ x}; (6n ) if u(x) ≤ n (x, u, β) ∀x ∈ x, ∀β ∈ m′ ⊂ m, then u(x) ≤ n (x, u, ∨m′). conversely, if n : x × lx × l → l satisfies conditions (1n) — (6n) above, then the mapping intn := int : l x × l → lx defined by int(u, β)(x) = n (x, u, β) satisfies axioms (1int) − (6int), i.e. is an (l, m)-fuzzy α-interior operator. moreover, intnint = int and nintn = n . a fuzzification of the category of m-valued l-topological spaces 153 proof. let int : lx × m −→ lx be an (l, m)-fuzzy α-interior operator and let n := nint be defined as above. (1n): for β ≤ α by (1int) it holds n (x, 1x, β) = int(1x, β)(x) = 1x (x) = ⊤. (2n): if u ≤ u′ and β′ ≤ β, then by (2int) it holds n (x, u, β) = int(u, β)(x) ≤ int(u′, β′)(x) = n (x, u′, β′). (3n): applying (3int) we get: ∧ i=1,...,n n (x, ui, β) = ∧ i=1,...,n int(ui, β)(x) ≤ ≤ int( ∧ i=1,...,n ui, β ∗ α)(x) = n (x, ∧ i=1,...,n ui, β ∗ α) (4n) obviously follows from (4int). (5n): applying (5int) and denoting int(u, β ∗ α) = v we get: n (x, u, β) = int(u, β)(x) ≤ int(int(u, β), β ∗ α)(x) = = int(v, β ∗ α)(x) ≤ ∨ {int(w, β ∗ α) | int(w, β ∗ α) ≤ v } = = ∨ n (x, v, β ∗ α) ≤ ≤ ∨ {n (x, w, β ∗ α)|w (y) ≤ n (y, u, β) ∀y ∈ x}. (6n): assume that u(x) ≤ n (x, u, β) for every β ∈ m′ and every x ∈ x. then u(x) = int(u, β)(x) and hence u = int(u, β) for every β ∈ m′. applying property (6int) of the (l, m)-fuzzy α-interior operator we conclude that u = int(u, ∨ m′) and hence u(x) = n (x, u, ∨ m′). conversely, let n : x × lx × m → m satisfy the properties (1int) — (6int) and let int = intn be defined as above. then int is the interior operator. the validity of properties (1int), (2int), (3int), (4int) and (6int) is obvious from the definition of intn and the corresponding properties of n . to show (5int) notice that int(u, β)(x) = n (x, u, β) ≤ ∨ {n (x, w, β ∗ α)|w (y) ≤ n (y, u, β)} = = ∨ {int(w, β ∗ α)(x)|w ∈ lx such that w (y) ≤ int(u, β)(y)} = int(int(u, β), β ∗ α)(x) finally, the equalities intnint = int and nintn = n are obvious from the definitions. � references [1] u.höhle, uppersemicontinuous fuzzy sets and applications, j. math. anal. appl. 78 (1980), 659-673. [2] u.höhle, commutative, residuated l-monoids, in: non-classical logics and their applications to fuzzy subsets: a handbook of the mathematical foundations of fuzzy subsets, e.p. klement and u. höhle eds., kluwer acad. publ., 1994, 53-106. 154 t. kubiak and a.šostak [3] u. höhle, m-valued sets and sheaves over integral commutative cl-monoids, in: applications of category theory to fuzzy sets., s.e. rodabaugh, e.p. klement and u. höhle eds., kluwer acad. publ., 1992, pp. 33-72. [4] u. höhle and a. šostak, axiomatic foundations of fixed-basis fuzzy topology, in: mathematics of fuzzy sets: logics, topology and measure theory,pp. 123-273. u. höhle and s.e. rodabaugh eds., kluwer academic publ., 1999. boston, dodrecht, london. [5] t. kubiak, on fuzzy topologies, ph.d. thesis, adam mickiewicz university, poznan, poland, 1985. [6] t. kubiak and a. šostak, foundations of the theory of (l, m)-fuzzy topologies, part i, to appear. [7] t. kubiak and a. šostak, foundations of the theory of (l, m)-fuzzy topologies, part ii, to appear. [8] s. e. rodabaugh, powerset operator based foundations for point-set lattice theoretic (poslat) fuzzy set theories and topologies, quaest. math. 20 (1997), 463-530. [9] a. šostak, on a fuzzy topological structure, rend. matem. palermo, ser ii, 11 (1985), 89-103. [10] a. šostak, two decades of fuzzy topology, russian mathematical surveys, 44:6 (1989), 125-186. [11] a. šostak, on a concept of a fuzzy category, in: 14th linz seminar on fuzzy set theory: non-classical logics and applications. linz, austria, 1992, pp. 62-66. [12] a. šostak, fuzzy categories versus categories of fuzzily structured sets: elements of the theory of fuzzy categories, in: mathematik-arbeitspapiere, universität bremen, vol 48 (1997), pp. 407-437. [13] a. šostak, fuzzy categories related to algebra and topology, tatra mount. math. publ. 16:1, (1999), 159-186. received march 2003 accepted june 2003 tomasz kubiak (tkubiak@amu.edu.pl) wydzia l matematyki i informatyki, adam mickiewicz university, pl-60-769, poznań, poland alexander šostak (sostaks@com.latnet.lv) departmant of mathematics, university of latvia, lv-1586 riga, latvia @ appl. gen. topol. 16, no. 2(2015), 99-108doi:10.4995/agt.2015.2988 c© agt, upv, 2015 on cyclic relatively nonexpansive mappings in generalized semimetric spaces moosa gabeleh department of mathematics, ayatollah boroujerdi university, boroujerd, iran, school of mathematics, institute for research in fundamental sciences (ipm), p.o. box 19395-5746, tehran, iran. (gab.moo@gmail.com, gabeleh@abru.ac.ir) abstract in this article, we prove a fixed point theorem for cyclic relatively nonexpansive mappings in the setting of generalized semimetric spaces by using a geometric notion of seminormal structure and then we conclude a result in uniformly convex banach spaces. we also discuss on the stability of seminormal structure in generalized semimetric spaces. 2010 msc: 47h10; 46b20. keywords: cyclic relatively nonexpansive mapping; seminormal structure; generalized semimetric space. 1. introduction a closed convex subset e of a banach space x has normal structure in the sense of brodskil and milman ([2]) if for each bounded, closed and convex subset k of e which contains more than one point, there is a point x ∈ k which is not a diametral point of k, that is, sup{‖x − y‖ : y ∈ k} < diam(k). in 1965, kirk proved that if e is a nonempty, weakly compact and convex subset of a banach space x with normal structure and t : e → e is a nonexpansive mapping, that is ‖t x−t y‖ ≤ ‖x−y‖ for all x, y ∈ e, then t has a fixed point ([8]). as well known, every nonempty, bounded, closed and convex subset of a uniformly convex banach space x has normal structure. so, the following fixed point theorem concludes from the kirk’s fixed point theorem. received 21 may 2014 – accepted 5 june 2015 http://dx.doi.org/10.4995/agt.2015.2988 m. gabeleh theorem 1.1. let e be a nonempty, bounded, closed and convex subset of a uniformly convex banach space x. then every nonexpansive mapping t : e → e has a fixed point. now, let (x, d) be a metric space, and let e, f be subsets of x. a mapping t : e ∪ f → e ∪ f is said to be cyclic provided that t (e) ⊆ f and t (f) ⊆ e. the following interesting theorem is an extension of banach contraction principle. theorem 1.2 ([10]). let e and f be nonempty and closed subsets of a complete metric space (x, d). suppose that t is a cyclic mapping such that d(t x, t y) ≤ α d(x, y), for some α ∈ (0, 1) and for all x ∈ e, y ∈ f. then e ∩ f is nonempty and t has a unique fixed point in e ∩ f. if e ∩ f = ∅ then the cyclic mapping t : e ∪ f → e ∪ f cannot have a fixed point, instead it is interesting to study the existence of best proximity points, that is, a point p ∈ e ∪ f such that d(p, t p) = dist(e, f) := inf{d(x, y) : (x, y) ∈ e × f}. existence of best proximity points for cyclic relatively nonexpansive mappings was first studied in [3] (see also [4, 5, 6, 7] for different approaches to the same problem). we recall that the mapping t : e ∪ f → e ∪ f is called cyclic relatively nonexpansive provided that t is cyclic on e ∪ f and d(t x, t y) ≤ d(x, y) for all (x, y) ∈ e × f . next theorem was established in [3]. theorem 1.3 (corollary 2.1 of [3]). let e and f be two nonempty, bounded, closed and convex subsets of a uniformly convex banach space x. suppose t : e ∪ f → e ∪ f is a cyclic relatively nonexpansive mapping. then t has a best proximity point in e ∪ f. we mention that theorem 1.3 is based on the fact that every nonempty, bounded, closed and convex pair of subsets of a uniformly convex banach space x has proximal normal structure (see proposition 2.1 of [3]). in this article, motivated by theorem 1.2, we establish a fixed point theorem for cyclic relatively nonexpansive mappings in generalized semimetric spaces. next we show that if the pair (e, f) considered in theorem 1.3 has an appropriate geometric condition, then e ∩ f must be nonempty and hence, the result follows from theorem 1.1. 2. preliminaries let x be a set and s a linearly ordered set with its order topology having a smallest element, which denoted by 0. a mapping ds : x × x → s is said to be a generalized semimetric provided that for each x, y ∈ x (1) ds(x, y) = 0 ⇔ x = y, (2) ds(x, y) = ds(y, x). c© agt, upv, 2015 appl. gen. topol. 16, no. 2 100 on cyclic relatively nonexpansive mappings in generalized semimetric spaces if s is the set of nonnegative real numbers, then we replace ds with d and we say that d is a semimetric on x. also, if ds is a generalized semimetric on x, then the pair (x, ds) is called generalized semimetric space. an easy example of a continuous semimetric which is not a metric is given by letting x = s = [0, 1] and defining d(x, y) := |x − y|2 for all x, y ∈ x. according to blumenthal ([1]; p.10), ds generates a topology on x as follows: a point p ∈ x is said to be a limit point of a subset e of x if given any α ∈ s with α 6= 0, there exists a point q ∈ e such that ds(p, q) ∈ (0, α) := {β ∈ s : 0 < β < α}. a set e in x is said to be closed if it contains all of its limit points and a set u in x is said to be open if x − u is closed. if ds is a continuous mapping w.r.t. the topology on x induced by ds, then ds is said to be a continuous generalized semimetric. given a generalized semimetric ds, a b-set will be a set like b(x; α) := {u ∈ x : ds(x, u) ≤ α}. we say that a set e ⊆ x is spherically bounded if there exists a b-set which contains e. we also define cov(e) := ⋂ {k : k is a b-set containing e}. definition 2.1. a subset e of a generalized semimetric space (x, ds) is said to be admissible if e = cov(e). the collection of all admissible subsets of a generalized semimetric (x, ds) will be denoted by a(x). we will say that a(x) is compact provided that any descending chain of nonempty members of a(x) has nonempty intersection. the linearly ordered set s is said to have least upper bound property (lubproperty) if each set in s which is bounded above has a smallest upper bound. dually, this implies that s has the greatest lower bound property (glbproperty). we mention that if s is connected relative to its order topology, then s has the lubproperty. let (e, f) be a nonempty pair of subsets of a generalized semimetric (x, ds). we shall adopt the following notations. dist(e, f) := glb {ds(x, y) : (x, y) ∈ e × f}, δx(e) := lub {ds(x, u) : u ∈ e}, ∀x ∈ x, δ(e, f) := lub {δx(f) : x ∈ e}, diam(e) := δ(e, e). e0 := {x ∈ e : ds(x, y) = dist(a, b), for some y ∈ b}, f0 := {y ∈ f : ds(x, y) = dist(a, b), for some x ∈ a}. definition 2.2 ([6]). a pair of sets (e, f) in a generalized semimetric space (x, ds) is said to be a proximal compactness pair provided that every net {(xα, yα)} of e × f satisfying the condition that ds(xα, yα) → dist(e, f), has a convergent subnet in e × f . c© agt, upv, 2015 appl. gen. topol. 16, no. 2 101 m. gabeleh 3. seminormal structure throughout this paper, we shall say that a pair (e, f) of subsets of a generalized semimetric space (x, ds) satisfies a property if both e and f satisfy that property. for example, (e, f) is admissible if and only if both e and f are admissible; (e, f) ⊆ (g, h) ⇔ e ⊆ g, and f ⊆ h. let (e, f) be a nonempty pair of admissible subsets of x. we say that the pair (e, f) satisfies the condition (p) if e contained in a b-set centered at a point of f and the set f contained in a b-set centered at a point of e. also, for the pair (e, f) we define r(e) := {α ∈ s : [ ⋂ y∈f b(y; α)] ∩ e 6= ∅}, r(f) := {β ∈ s : [ ⋂ x∈e b(x; β)] ∩ f 6= ∅}. note that the if the pair (e, f) satisfies the condition (p), then (r(e), r(f)) is a nonempty pair of subsets of s. indeed, if e ⊆ b(v; β) for some v ∈ f and β ∈ s, then ds(x, v) ≤ β for all x ∈ e and so, v ∈ b(x; β) for all x ∈ e. thus v ∈ ⋂ x∈e b(x; β) ∩ f i.e. β ∈ r(f). similarly, we can see that r(e) is nonempty. furthermore, we set r(e) := glb r(e), r(f) := glb r(f) and ρ := lub {r(e), r(f)}, and define cf (e) := {x ∈ e : x ∈ ⋂ y∈f b(y; ρ)}, ce(f) := {y ∈ f : y ∈ ⋂ x∈e b(x; ρ)}. next lemma guarantees that (cf (e), ce(f)) is a nonempty pair. lemma 3.1. let (x, ds) be a generalized semimetric space such that a(x) is compact and s is connected. let (e, f) be a nonempty and admissible pair of subsets of x such that (e, f) satisfies the condition (p). then (cf (e), ce(f)) is a nonempty and admissible pair in x which satisfies the condition (p). proof. let α > ρ and β > ρ be such that the pair (cα(e), cβ(f)) is nonempty, where cα(e) := [ ⋂ y∈f b(y; α)] ∩ e & cβ(f) := [ ⋂ x∈e b(x; β)] ∩ f. we show that cf (e) = ⋂ α≥ρ cα(e) and ce(f) = ⋂ β≥ρ cβ(f). suppose that u ∈ ⋂ α≥ρ cα(e). if u is not member of cf (e), then there exists v ∈ f such that ds(u, v) > ρ. since s is connected, there exists an element γ ∈ s such that ρ < γ < ds(u, v). but this is a contradiction by the fact that u ∈ cγ(e). that c© agt, upv, 2015 appl. gen. topol. 16, no. 2 102 on cyclic relatively nonexpansive mappings in generalized semimetric spaces is, u ∈ cf (e) and so, ⋂ α≥ρ cα(e) ⊆ cf (e). this implies that cf (e) 6= ∅. besides, if u ∈ cf (e), then u ∈ [ ⋂ y∈f b(y; ρ)] ∩ e ⊆ [ ⋂ y∈f b(y; α)] ∩ e = cα(e), ∀α ≥ ρ. hence, u ∈ ⋂ α≥ρ cα(e) which deduces that cf (e) = ⋂ α≥ρ cα(e). similar argument implies that ce(f) = ⋂ β≥ρ cα(f). now, suppose that e ⊆ b(q, γ1) and f ⊆ b(p, γ2) for some (p, q) ∈ e ×f and γ1, γ2 ∈ s. put γ := lub {γ1, γ2}. then for each α ∈ s with α ≥ ρ, we have cα(e) ⊆ b(q, γ) which concludes that cf (e) = ⋂ α≥ρ cα(e) ⊆ b(q, γ). similar argument implies that ce(f) ⊆ b(p, γ). that is, the pair (cf (e), ce(f)) satisfies the condition (p). � let (e, f) be a nonempty and admissible pair of subsets of a generalized semimetric space (x, ds) such that (e, f) satisfies the condition (p). in what follows we set σ(e,f ) := {(g, h) ⊆ (e, f) : g, h ∈ a(x) and (g, h) satisfies the condition (p)}. here, we introduce the following geometric notion on a nonempty and admissible pair in generalized semimetric spaces. definition 3.2. suppose that (e, f) is a nonempty and admissible pair of subsets of a generalized semimetric space (x, ds) such that (e, f) satisfies the condition (p) and a(x) is compact. we say that σ(e,f ) has seminormal structure if for each (g, h) ∈ σ(e,f ), either g ∪ h is singleton or ch(g) g, cg(h) h. we now state the main result of this paper. theorem 3.3. let (x, ds) be a generalized semimetric space, where s is connected w.r.t. its order topology and let a(x) be compact. suppose that (e, f) is a nonempty and admissible pair of subsets of x which satisfies the condition (p) and σ(e,f ) has seminormal structure. if t : e ∪ f → e ∪ f is a cyclic relatively nonexpansive mapping, then e ∩ f is nonempty and t has a fixed point in e ∩ f. proof. put f := {(g, h) : (g, h) ∈ σ(e,f ) and t is cyclic on g ∪ h}. by the fact that a(x) is compact and by using zorn’s lemma, we conclude that f has a minimal element say (k1, k2) ∈ f. since t (k1) ⊆ k2 and k2 ∈ a(x), we deduce that cov(t (k1)) ⊆ k2. then t (cov(t (k1))) ⊆ t (k2) ⊆ cov(t (k2)). similarly, we can see that t (cov(t (k2))) ⊆ cov(t (k1)), that is, t is cyclic on cov(t (k2)) ∪ cov(t (k1)). besides, (cov(t (k2)), cov(t (k1))) satisfies the c© agt, upv, 2015 appl. gen. topol. 16, no. 2 103 m. gabeleh condition (p). indeed, if k1 ⊆ b(q, α) for some q ∈ k2 and α ∈ s, then for each x ∈ k1, we have ds(t x, t q) ≤ ds(x, q) ≤ α, that is, t x ∈ b(t q, α) for each x ∈ k1. so, t (k1) ⊆ b(t q, α). thus cov(t (k1)) ⊆ b(t q, α). similarly, if k2 ⊆ b(p, β) for some p ∈ k1 and β ∈ s, then we can see that cov(t (k2)) ⊆ b(t p, β). hence, (cov(t (k2)), cov(t (k1))) satisfies the condition (p). minimality of (k1, k2) implies that k1 = cov(t (k2)) & k2 = cov(t (k1)). it follows from lemma 3.1 that (ck2(k1), ck1(k2)) is a nonempty member of σ(e,f ). we show that t is cyclic on ck2(k1) ∪ ck1(k2). let x ∈ ck2(k1). then x ∈ [ ⋂ y∈k2 b(y; ρ)] ∩ k1. so, ds(x, y) ≤ ρ for each y ∈ k2. since t is cyclic relatively nonexpansive, ds(t x, t y) ≤ ds(x, y) ≤ ρ, ∀y ∈ k2. thus t (k2) ⊆ b(t x; ρ) which implies that k1 = cov(t (k2)) ⊆ b(t x; ρ). hence, t x ∈ [ ⋂ u∈k1 b(u; ρ)]∩k2 = ck1(k2). that is, t (ck2(k1)) ⊆ ck1(k2). similarly, we can see that t (ck1(k2)) ⊆ ck2(k1). thereby, t is cyclic on ck2(k1) ∪ ck1(k2). so, (ck2(k1), ck1(k2)) ∈ f. again, by the minimality of (k1, k2) we must have ck2(k1) = k1 & ck1(k2) = k2. since σ(e,f ) has the seminormal structure, we deduce k1 = k2 = {p} for some p ∈ x. therefore, p ∈ e ∩ f is a fixed point of t . � remark 3.4. note that in theorem 3.3 we have not the assumption of continuity of ds. we also mention that if the mapping t considered in theorem 3.3 is nonexpansive self-mapping, the the main result of [9] is deduces (see theorem 3 of [9] for more information). definition 3.5. let (e, f) be a nonempty and admissible pair of subsets of a semimetric space (x, d) such that (e, f) satisfies the condition (p). we say that (e, f) has the property uc if for each nonempty pair (g, h) ∈ σ(e,f ) and for any ε > 0, there exists α(ε) > 0 such that for all r > 0 and x1, x2 ∈ g and y ∈ h with d(x1, y) ≤ r, d(x2, y) ≤ r and d(x1, x2) ≥ rε, there exists u ∈ g such that d(u, y) ≤ r(1 − α(ε)) < r. we now prove the following existence theorem. theorem 3.6. let (x, d) be a semimetric space such that d is continuous and a(x) is compact. suppose (e, f) is a nonempty and admissible pair such that e0 6= ∅ and (e, f) satisfies the condition (p). assume that (e, f) is a proximal compactness pair which has the property uc. if t : e ∪f → e ∪f is c© agt, upv, 2015 appl. gen. topol. 16, no. 2 104 on cyclic relatively nonexpansive mappings in generalized semimetric spaces a cyclic relatively nonexpansive mapping, then either e ∩ f is nonempty and t has a fixed point in e ∩ f, or t has a best proximity point in e ∪ f. proof. let f′ := {(g, h) ∈ σ(e,f ) s.t. ∃(x, y) ∈ g × h with d(x, y) = dist(e, f) and t is cyclic on g ∪ h}. since e0 6= ∅, (e, f) ∈ f′. moreover, if (gα, hα) is a descending chain in f′ and put g := ⋂ α gα and we set h := ⋂ α hα, then by the compactness of a(x), (g, h) is a nonempty member of σ(e,f ) and obviously, t is cyclic on g ∪ h. now, suppose for each α there exists (xα, yα) ∈ gα × hα such that d(xα, yα) = dist(e, f). since (e, f) is proximal compactness, {(xα, yα)} has a convergent subnet say {(xαi, yαi)} such that xαi → x ∈ e and yαi → y ∈ f . hence, d(x, y) = lim i d(xαi, yαi) = dist(e, f), that is, there exists an element (x, y) ∈ g × h such that d(x, y) = dist(e, f). so, every increasing chain in f′ is bounded above with respect to revers inclusion relation. using zorn’s lemma, we obtain a minimal element for f′, say (k1, k2). if k1 ∪ k2 is singleton, then t has a fixed point in e ∩ f and we are finished. so, we assume that k1 ∪ k2 is not singleton. similar argument of theorem 3.3 concludes that ck2(k1) = k1 and ck1(k2) = k2. we now consider the following : case 1. if min{diam(k1), diam(k2)} = 0. we may assume that k1 = {p} for some element p ∈ e. let q ∈ k2 be such that d(p, q) = dist(e, f). since t is cyclic relatively nonexpansive mapping, d(t p, p) = d(t p, t q) ≤ d(p, q) = dist(e, f), that is, p is a best proximity point of t and the result follows. case 2. if min{diam(k1), diam(k2)} > 0. put r := δ(k1, k2) and r := min{diam(k1), diam(k2)}. let x1, x2 ∈ k1 be such that d(x1, x2) ≥ 1 2 diam(k1) and let ε > 0 be such that rε ≤ r 2 . now, for each y ∈ k2 we have d(x1, y) ≤ r, d(x2, y) ≤ r and d(x1, x2) ≥ 1 2 r ≥ rε. since (e, f) has the property uc, there exists α(ε) > 0 and u ∈ k1 so that d(u, y) ≤ r(1 − α(ε)), ∀y ∈ k2. then u ∈ [ ⋂ y∈k2 b(y; r(1 − α(ε)))] ∩ k1, that is, [ ⋂ y∈k2 b(y; r(1 − α(ε)))] ∩ k1 6= ∅. similarly, we can see that [ ⋂ x∈k1 b(x; r(1 − α(ε)))] ∩ k2 6= ∅. set r(k1) := inf{s > 0 : [ ⋂ y∈k2 b(y; s)] ∩ k1 6= ∅}, r(k2) := inf{s > 0 : [ ⋂ x∈k1 b(x; s)] ∩ k2 6= ∅}. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 105 m. gabeleh note that for ρ := max{r(k1), r(k2)} we have ρ ≤ r(1 − α(ε)). since ck2(k1) = k1, x ∈ ⋂ y∈k2 b(y; ρ), ∀x ∈ k1, which implies that δx(k2) ≤ ρ for all x ∈ k1. thus r = δ(k1, k2) = sup x∈k1 δx(k2) ≤ ρ ≤ r(1 − α(ε)) < r, which is a contradiction and this completes the proof of theorem. � next corollary is a straightforward consequence of theorem 3.6 in the setting of uniformly convex banach spaces. corollary 3.7 (see [3]). suppose that (e, f) is a nonempty, bounded, closed and convex pair of subsets of a uniformly convex banach space x. let t : e ∪f → e ∪f be a cyclic relatively nonexpansive mapping. then either e ∩f is nonempty and t has a fixed point in e ∩ f or t has a best proximity point in e ∪ f. example 3.8. let x = r and let a := [−1, 1]. define the mapping t : a → a with t (x) =      −x if x ∈ [−1, 0], −x if x ∈ [0, 1] ∩ q, 0 if x ∈ [0, 1] ∩ qc. then t is a self-mapping defined on a nonempty bounded, closed and convex subset of x. note that existence of fixed point of t cannot be deduced from theorem 1.1, because of the fact t is not continuous (and so is not nonexpansive). now, suppose e := [−1, 0] and f := [0, 1] and formulate the mapping t : e ∪ f → e ∪ f as follows: t (x) =      −x if x ∈ e, −x if x ∈ f ∩ q, 0 if x ∈ f ∩ qc. it is easy to see that ‖t x − t y‖ ≤ ‖x − y‖ for all (x, y) ∈ e × f , that is, t is cyclic relatively nonexpansive mapping on the nonempty, bounded, closed and convex pair (e, f). hence, the existence of fixed point for t is concluded from corollary 3.7. 4. stability and seminormal structure we begin our main conclusions of this section with the following notion. definition 4.1. let (x, ds) be a generalized semimetric space and let (e, f) be a nonempty pair of subsets of x. a mapping t : e ∪ f → e ∪ f is said to be cyclic relatively h-nonexpansive for some h ∈ s with h > 0 if ds(t x, t y) ≤ lub {ds(x, y), h}, for all (x, y) ∈ e × f . c© agt, upv, 2015 appl. gen. topol. 16, no. 2 106 on cyclic relatively nonexpansive mappings in generalized semimetric spaces here, we state the following stability result for cyclic relatively h-nonexpansive mappings. theorem 4.2. let (x, ds) be a generalized semimetric space, where s is connected w.r.t. its order topology and let a(x) be compact. suppose that (e, f) is a nonempty and admissible pair of subsets of x which satisfies the condition (p) and σ(e,f ) has seminormal structure. if t : e ∪ f → e ∪ f is a cyclic relatively h-nonexpansive mapping, then there exists an element p ∈ a∪b so that ds(p, t p) ≤ h. proof. similar argument of theorem 3.3 implies that there exists a nonempty and admissible pair of subsets (k1, k2) ⊆ (e, f) which satisfies the condition (p) and by minimality, cov(t (k2)) = k1 and cov(t (k1)) = k2. if k1 ∪ k2 is singleton, the result follows. so, assume that ck2(k1) $ k1 and ck1(k2) $ k2. let u be an arbitrary element of ck2(k1). suppose ρ < h. then ds(u, y) ≤ ρ for all y ∈ k2. since t is cyclic on k1 ∪ k2, we have ds(u, t u) ≤ ρ < h and we are finished. we now suppose that h ≤ ρ. let y ∈ k2. if ds(u, y) ≥ h, then ds(t u, t y) ≤ lub{ds(u, y), h} = ds(u, y) ≤ ρ. besides, if ds(u, y) < h, then ds(t u, t y) ≤ lub{ds(u, y), h} = h ≤ ρ, that is, for each y ∈ k2 we have ds(t u, t y) ≤ ρ which implies that t y ∈ b(t u; ρ) for all y ∈ k2. hence, t (k2) ⊆ b(t u; ρ). so, k1 = cov(t (k2)) ⊆ b(t u; ρ), and then t u ∈ [ ⋂ x∈k1 b(x; ρ)]∩k2. thus t u ∈ ck1(k2). thereby, t (ck2(k1)) ⊆ ck1(k2). by a similar argument we obtain t (ck1(k2)) ⊆ ck2(k1). therefore, t is cyclic on t (ck2(k1)) ∪ ck1(k2). minimality of (k1, k2) deduces that k1 = ck2(k1) and k2 = ck1(k2), which is a contradiction. � acknowledgements. this research was in part supported by a grant from ipm (no. 93470047). c© agt, upv, 2015 appl. gen. topol. 16, no. 2 107 m. gabeleh references [1] l. m. blumenthal, theory and applications of distance geometry, oxford univ. press, london (1953). [2] m. s. brodskii and d. p. milman, on the center of a convex set, dokl. akad. nauk. ussr 59 (1948), 837–840 (in russian). [3] a. a. eldred, w. a. kirk and p. veeramani, proximal normal structure and relatively nonexpansive mappings, studia math. 171 (2005), 283–293. [4] r. esṕınola, a new approach to relatively nonexpansive mappings, proc. amer. math. soc. 136 (2008), 1987–1996. [5] r. esṕınola and m. gabeleh, on the structure of minimal sets of relatively nonexpansive mappings, numer. funct. anal. optim. 34 (2013), 845–860. [6] m. gabeleh, minimal sets of noncyclic relatively nonexpansive mappings in convex metric spaces, fixed point theory, to appear. [7] m. gabeleh and n. shahzad, seminormal structure and fixed points of cyclic relatively nonexpansive mappings , abstract appl. anal. 2014 (2014), article id 123613, 8 pages. [8] w. a. kirk, a fixed point theorem for mappings which do not increase distances, amer. math. monthly 72 (1965), 1004–1006. [9] w. a. kirk and b. g. kang a fixed point theorem revisited, j. korean math. soc. 34 (1997), 285–291. [10] w. a. kirk, p. s. srinivasan amd p. veeramani,fixed points for mappings satisfying cyclic contractive conditions, fixed point theory 4, no. 1 (2003), 79–86. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 108 @ appl. gen. topol. 15, no. 2(2014), 183-202doi:10.4995/agt.2014.2050 c© agt, upv, 2014 function lattices and compactifications tomi alaste university of oulu, department of mathematical sciences, p.o. box 3000, fi-90014 university of oulu, finland (tomi.alaste@gmail.com) abstract let f be a lattice of real-valued functions on a non-empty set x such that f contains the constant functions. using certain filters on x determined by f, we construct a compact hausdorff topological space δx with the property that every bounded member of f extends to δx and these extensions form a dense subspace of c(δx). if a is any c ∗ -subalgebra of ℓ ∞(x) containing the constant functions, then our construction gives a representation of the spectrum of a as a space of filters on x. 2010 msc: 46e05; 54d80; 54d35. keywords: function lattice; f-filter; f-ultrafilter; spectrum. 1. introduction a widely used method to study topological compactifications and semigroup compactifications is to view these compactifications as the spectrums of some c∗-algebras of bounded, complex-valued functions. if x is a tychonoff space, then every topological compactification of x can be realized as the spectrum of some c∗-algebra consisting of continuous functions on x and containing the constant functions. a similar statement holds for any semigroup compactification of a hausdorff semitopological semigroup s (see [5]). some of these compactifications can be considered as spaces of filters. the most familiar example is the stone-čech compactification βx of a discrete topological space x, which may be regarded as the space of all ultrafilters on x (see [6] or [9]). if s is a discrete semigroup, then βs is actually a semigroup compactification of s, and the consideration of βs as the space of all ultrafilters on s is an extremely powerful approach while analyzing algebraic received 17 december 2013 – accepted 7 may 2014 http://dx.doi.org/10.4995/agt.2014.2050 t. alaste properties of βs (see [9]). for a general tychonoff space x, the stone-čech compactification βx of x can also be considered as a space of filters on x, but this time one uses z-ultrafilters on x instead of ultrafilters (see [8] or [15]). (ultrafilters and z-ultrafilters on x coincide if x is discrete.) the uniform compactification (or the samuel compactification, see [10]) of a uniform space (x,u) was represented as the space of all near ultrafilters on x by koçak and strauss in [12]. near ultrafilters on x need not be filters in the ordinary sense of the word, since they need not be closed under finite intersections. recently, a representation of the uniform compactification using filters was given by the author in [1]. in both [12] and [1], the given representation was used to study the luc-compactification of a topological group. the luccompactification of a locally compact topological group g was also studied using filters by budak and pym in [3], where the luc-compactification of g was considered as a suitable quotient space of the stone-čech compactification of βgd. here, gd denotes the group g endowed with the discrete topology. the wap-compactification of a discrete semigroup was studied using filters by berglund and hindman in [4] and a treatment of semigroup compactifications using equivalence classes of z-filters was given by tootkaboni and riazi in [14]. the original aim of this paper is to show that the spectrum of any c∗-algebra f of bounded, complex-valued functions on x, where x is any non-empty set and f contains the constant functions, can be considered as a space of filters on x. since every topological compactification [semigroup compactification] is determined by the spectrum of some c∗-algebra of bounded functions, our development gives a unified treatment of all these compactifications as spaces of filters. as far as we are aware, for many c∗-algebras our approach is actually the first one using filters instead of equivalence classes of filters or quotient spaces of some other compactifications. if x is a discrete topological space, then our approach yields the usual representation of βx as the space of all ultrafilters on x. independently of the c∗-algebra f in question, our approach has a number of similarities with the consideration of βx for a discrete topological space x as the space of all ultrafilters on x. for example, we obtain a bijective correspondence between non-empty, closed subsets of the spectrum of f and f-filters on x. we believe that the method presented in this paper can serve as a valuable tool in the study of both topological compactifications and semigroup compactifications. this method was used by the author in [1] to study the smallest ideal of the luc-compactification of a topological group and in [2] to study the smallest ideal of any semigroup compactification of any semitopological semigroup. for a large part of the theory developed in this paper, it is not necessary that we work with a c∗-algebra of bounded functions. instead, it is the lattice structure of real-valued functions that is important for our development. therefore, we work with a lattice of real-valued functions (which might contain unbounded functions) throughout sections 3-8. in section 3, we introduce the main object of this paper, namely f-filters and f-ultrafilters, and we study some of their basic properties. in section 4, we define a topology on the set c© agt, upv, 2014 appl. gen. topol. 15, no. 2 184 function lattices and compactifications of all f-ultrafilters and we show that the resulting space δx is a compact hausdorff space. furthermore, we show that the f-filters describe the topology of δx in a similar way as filters describe the topology of the stone-čech compactification of a discrete topological space. section 5 contains a study of continuous functions on δx. we show that every bounded member of f extends to δx and that these extensions form a dense subspace of the algebra of all continuous, real-valued functions on δx. it is remarkable that we do not need the stone-weierstrass theorem to prove the density of these extensions. in sections 6-8, our main results concern closed subalgebras of the algebra of all bounded, real-valued functions on x. in section 7, we establish a correspondence between f-filters and closed, proper ideals of f. section 8 contains a treatment of f-filters on a hausdorff topological space x in the case that every member of f is a continuous function on x. in the last section, we turn our attention to c∗-algebras of bounded, complex-valued functions. here, we include a description how the developed theory so far can be used to produce an interpretation of the spectrum of such an algebra [compactification of x] as a space of filters on x. our construction of the space δx as the space of all f-ultrafilters has some similarities with the consideration of the smirnov compactification of a proximity space using maximal round filters (see [13]). if the function lattice f on a non-empty set x separates the points of x, then there is a bijective correspondence between f-ultrafilters on x and maximal round filters on the proximity space (x,p), where p is the proximity on x generated by f. an advantage of our construction is that it applies to any function lattice f on x, and so it applies also to those semigroup compactifications where the evaluation mapping is not necessarily injective. this includes, for example, the bohr compactification of some topological groups. 2. preliminaries throughout the paper, let x be any non-empty set. we denote by f(x) the algebra of all real-valued functions on x. we denote by ℓ∞(x) the subalgebra of f(x) consisting of all bounded members of f(x). recall that the space ℓ∞(x) is equipped with the norm of uniform convergence. a function f ∈ f(x) is positive if and only if f(x) ≥ 0 for every x ∈ x. for all f,g ∈ f(x), the functions (f ∨ g) : x → r and (f ∧ g) : x → r are defined by (f ∨ g)(x) = max{f(x),g(x)} and (f ∧ g)(x) = min{f(x),g(x)} for every x ∈ x, respectively. by a function lattice on x we mean a vector subspace f of f(x) such that f contains the constant functions and f ∨g ∈ f and f ∧ g ∈ f for all f,g ∈ f. note that a vector subspace f of f(x) is a function lattice on x if and only if |f| ∈ f for every f ∈ f. we denote by n the set of all positive integers, that is, n = {1,2,3, . . .}. we denote by p(x) the family of all subsets of x. a filter on x is a non-empty family ϕ of subsets of x with the following properties: c© agt, upv, 2014 appl. gen. topol. 15, no. 2 185 t. alaste (i) if a,b ∈ ϕ, then a ∩ b ∈ ϕ. (ii) if a ∈ ϕ and a ⊆ b ⊆ x, then b ∈ ϕ. (iii) ∅ /∈ ϕ. a filter base on x is a non-empty family b of subsets of x such that ∅ /∈ b and, for all sets a,b ∈ b, there exists some c ∈ b such that c ⊆ a ∩ b. if b is a filter base on x, then the filter ϕ on x generated by b is ϕ = {a ⊆ x : there exists some b ∈ b such that b ⊆ a}. let ϕ be a filter on x. a family b of subsets of x is a filter base for ϕ if and only if b ⊆ ϕ and, for every a ∈ ϕ, there exists some b ∈ b such that b ⊆ a. let (y,τ) be a (not necessarily hausdorff) topological space. for every subset a of y , we denote by int(y,τ)(a) and cl(y,τ)(a) the interior and the closure of a in (y,τ), respectively, or simply by inty (a) and cly (a) if τ is understood. we denote by c(y ) the subalgebra of ℓ∞(x) consisting of all continuous members of ℓ∞(x). if y is locally compact, then the subalgebra c0(x) of c(x) consists of those members of c(x) which vanish at infinity. 3. f-filters throughout this section, let f be a function lattice on x. we introduce the main object of the paper, namely f-filters and f-ultrafilters on x, and we describe some of their basic properties. for all f ∈ f and r > 0, we put z(f) = {x ∈ x : f(x) = 0} and x(f,r) = {x ∈ x : |f(x)| ≤ r}. definition 3.1. an f-family on x is a non-empty family a of non-empty subsets of x such that, for every a ∈ a with a 6= x, there exist some b ∈ a and a function f ∈ f such that f(b) = {0} and f(x \ a) = {1}. an f-filter on x is a filter ϕ on x which is also an f-family on x. since f contains the constant functions, we may assume that the function f ∈ f in the previous definition satisfies f(b) = {1} and f(x \ a) = {0}. also, since f is closed under the lattice operations ∨ and ∧, we may assume, if necessary, that f(x) ⊆ [0,1]. there exists at least one f-filter on x, namely the filter ϕ = {x}. if f contains only the constant functions, then {x} is the only f-filter on x. on the other hand, if f = ℓ∞(x), then every filter ϕ on x is an f-filter on x. let ϕ be an f-filter on x and suppose that a ∈ ϕ satisfies a 6= x. pick some b ∈ ϕ and a function f ∈ f with f(b) = {0} and f(x \a) = {1}. then b ⊆ z(f) ⊆ a. since z(f) ∈ ϕ, the filter ϕ has a filter base consisting of zero sets (determined by f) of x. however, not every zero set of x is contained in any f-filter. for example, let f = c(r). then a = {0} is a zero set of r but there is no f-filter ϕ on r satisfying a ∈ ϕ. we shall apply the following remark frequently without any further notice. remark 3.2. let a be a non-empty family of non-empty subsets of x. suppose that, for every a ∈ a with a 6= x, there exist some b ∈ a, real numbers s and r with s < r, and a function f ∈ f such that f(x) ≤ s for every x ∈ b c© agt, upv, 2014 appl. gen. topol. 15, no. 2 186 function lattices and compactifications and f(x) ≥ r for every x ∈ x \a. then, using the lattice operations, it is easy to see that a is an f-family on x. zorn’s lemma implies that every f-filter on x is contained in some maximal (with respect to inclusion) f-filter on x. definition 3.3. an f-ultrafilter on x is an f-filter on x which is not properly contained in any other f-filter on x. note that if f = ℓ∞(x), then a filter ϕ on x is an f-ultrafilter if and only if ϕ is an ultrafilter on x. also, the following fact about f-ultrafilters is very useful: if p and q are f-ultrafilters on x, then p = q if and only if p ⊆ q. definition 3.4. define f0 = {f ∈ f : x(f,r) 6= ∅ for every r > 0}. for every non-empty subset a of x, define z(a) = {f ∈ f : f(x) = 0 for every x ∈ a}. the next statement follows from remark 3.2. lemma 3.5. the family a = {x(f,r) : f ∈ f′, r > 0} is an f-family on x for every non-empty subset f′ of f0. we will use the following lemma and its corollaries a number of times in this paper. recall that a non-empty family a of subsets of x has the finite intersection property if and only if ⋂n k=1 ak 6= ∅ whenever a1, . . . ,an ∈ a for some n ∈ n. lemma 3.6. if a is an f-family on x such that a has the finite intersection property, then there exists an f-ultrafilter p on x such that a ⊆ p. proof. we sketch the proof briefly. let ϕ be the smallest filter on x containing the family a. let n ∈ n and suppose that a1, . . .an ∈ a satisfy ak 6= x for every k ∈ {1, . . . ,n}. if k ∈ {1, . . . ,n}, then there exist some bk ∈ a and a positive function fk ∈ f with fk(bk) = {0} and fk(x \ ak) = {1}. put b = ⋂n k=1 bk and f = ∑n k=1 fk. since b ∈ ϕ, f ∈ f, f(b) = {0}, and f(x) ≥ 1 for every x ∈ x \ ⋂n k=1 ak, the filter ϕ is an f-filter on x. � the next two corollaries now follow from lemma 3.5. corollary 3.7. let ϕ be an f-filter on x and let f ∈ f. if x(f,r) ∩ b 6= ∅ for every b ∈ ϕ and for every r > 0, then there exists an f-ultrafilter p on x such that ϕ ∪ {x(f,r) : r > 0} ⊆ p. corollary 3.8. let ϕ be an f-filter on x and let a ⊆ x. if a ∩ b 6= ∅ for every b ∈ ϕ, then there exists an f-ultrafilter p on x containing the family ϕ ∪ {x(f,r) : f ∈ z(a), r > 0}. if f = ℓ∞(x), then we may take the members of f in the next theorem to be characteristic functions of subsets of x. then, except for statement (ii), the conclusion of the next theorem is the same as in [9, theorem 3.6]. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 187 t. alaste theorem 3.9. if ϕ ⊆ p(x), then the following statements are equivalent: (i) ϕ is an f-ultrafilter on x. (ii) ϕ is an f-filter on x and, if x(f,r) /∈ ϕ for some f ∈ f and r > 0, then, for every real number t with 0 < t < r, there exists some a ∈ ϕ such that x(f,t) ∩ a = ∅. (iii) ϕ is a maximal f-family on x such that ϕ has the finite intersection property. (iv) ϕ is an f-filter on x and, if ⋃n k=1 ak ∈ ϕ for some n ∈ n and for some a1, . . . ,an ⊆ x, then there exists k ∈ {1, . . . ,n} such that x(f,r) ∈ ϕ for all f ∈ z(ak) and r > 0. (v) ϕ is an f-filter on x and, if a ⊆ x satisfies a 6= ∅ and a 6= x, then, either x(f,r) ∈ ϕ for every f ∈ z(a) and for every r > 0, or x(g,r) ∈ ϕ for every g ∈ z(x \ a) and for every r > 0. proof. (i) ⇒ (ii) this follows from corollary 3.8 with g = (|f| − t) ∨ 0. note that g ∈ z(x(f,t)) and x(g,r − t) ⊆ x(f,r). (ii) ⇒ (iii) this follows from the definition of an f-family. (iii) ⇒ (iv) suppose that (iii) holds. let us first show that ϕ is a filter on x. clearly, x ∈ ϕ, ∅ /∈ ϕ, and b ∈ ϕ whenever a ∈ ϕ and a ⊆ b ⊆ x. so, let a,b ∈ ϕ. pick some c,d ∈ ϕ and functions f,g ∈ f with f(c) = g(d) = {0} and f(x \ a) = g(x \ b) = {1}. since c ∩ d ⊆ x(|f| + |g|,r) for every r > 0, we have x(|f| + |g|,r) ∈ ϕ for every r > 0 by lemma 3.5. since x(|f| + |g|,1/2) ⊆ a ∩ b, we have a ∩ b ∈ ϕ, as required. suppose now that ⋃n k=1 ak ∈ ϕ for some n ∈ n and for some non-empty subsets a1, . . . ,an of x. suppose also that, for every k ∈ {1, . . . ,n}, there exist rk > 0 and a function fk ∈ z(ak) such that x(fk,rk) /∈ ϕ. if k ∈ {1, . . . ,n}, then the family a = ϕ∪ {x(fk, t) : t > 0} is an f-family on x by lemma 3.5. since a contains ϕ properly, there exist some bk ∈ ϕ and tk > 0 such that bk∩x(fk, tk) = ∅. put b = ⋂n k=1 bk. then b ∈ ϕ and b∩[ ⋃n k=1 z(fk)] = ∅, a contradiction. (iv) ⇒ (v) this is obvious. (v) ⇒ (i) suppose that (v) holds. suppose also that there exists an f-filter ψ on x which properly contains ϕ. pick some set a ∈ ψ \ ϕ. pick some b ∈ ψ and a function f ∈ f with f(b) = {0} and f(x \ a) = {1}. pick some c ∈ ψ and a function g ∈ f with g(c) = {1} and g(x \ b) = {0}. since x(f,1/2) ⊆ a, we have x(f,1/2) /∈ ϕ. since f ∈ z(b), we have x(g,1/2) ∈ ϕ by assumption. but now x(g,1/2) ∩ c = ∅, a contradiction. � the two statements given in statement (v) of the previous theorem are not exclusive. indeed, let f = c(r) and a = q. then z(a) = z(r \ a) = {0}, and so x(f,r) = x(g,r) = x for all f ∈ z(a), g ∈ z(r \ a), and r > 0. 4. the topological space δx as in the previous section, we assume that f is a function lattice on x. our next task is to define a topology on the set of all f-ultrafilters on x c© agt, upv, 2014 appl. gen. topol. 15, no. 2 188 function lattices and compactifications and establish some of the properties of the resulting space. in particular, we show that the resulting space is a compact hausdorff space and that f-filters describe its topology. definition 4.1. define δx = {p : p is an f-ultrafilter on x}. for every subset a of x, put â = {p ∈ δx : a ∈ p}. for every f-filter ϕ on x, put ϕ̂ = {p ∈ δx : ϕ ⊆ p}. to be precise, we should include the function lattice f in the notation above, such as δf(x). except in section 6, we consider only one function lattice f in the same context, so we hope that the notation chosen above does not cause any misunderstandings. theorem 4.2. if ϕ and ψ are f-filters on x, then the following statements hold: (i) ϕ̂ = ⋂ a∈ϕ â. (ii) ϕ = ⋂ p∈ϕ̂ p. (iii) ϕ ⊆ ψ if and only if ψ̂ ⊆ ϕ̂. (iv) ϕ = ψ if and only if ϕ̂ = ψ̂. proof. (i) this is obvious. (ii) the inclusion ϕ ⊆ ⋂ p∈ϕ̂ p is obvious, so suppose that a is a subset of x such that a /∈ ϕ. by corollary 3.8, there exists an element p ∈ ϕ̂ such that {x(f,r) : f ∈ z(x \ a), r > 0} ⊆ p. now, it is enough to show that a /∈ p. suppose that a ∈ p. pick some b ∈ p and a function f ∈ f with f(b) = {1} and f(x \ a) = {0}. since f ∈ z(x \ a), we have x(f,1/2) ∈ p. but now b ∩ x(f,1/2) = ∅, a contradiction. (iii) necessity is obvious and sufficiency follows from statement (ii). (iv) this follows from statement (iii). � the family {â : a ⊆ x} is a base for a topology on δx. we define the topology of δx to be the topology which has this family as its base. in particular, {â : a ∈ p} is a neighborhood base of a point p ∈ δx. if y ⊆ δx, then we denote clδx(y ) by y with one exception: if a ⊆ x, then we use clδx(â) instead of the cumbersome notation â. we denote by τ(f) the weakest topology τ on x such that every member of f is continuous with respect to τ. for every subset a of x, we denote int(x,τ(f))(a) by a ◦. for every element x ∈ x, we denote by nf(x) the neighborhood filter of x in (x,τ(f)). we shall apply the following remark frequently without any further notice. remark 4.3. let ϕ be an f-filter on x. suppose that a ∈ ϕ satisfies a 6= x. pick some b ∈ ϕ and a function f ∈ f with f(b) = {0} and f(x \ a) = {1}. then b ⊆ {x ∈ x : |f(x)| < 1} ⊆ a. in conclusion, if c is any subset of x, then c ∈ ϕ if and only if c◦ ∈ ϕ. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 189 t. alaste theorem 4.4. if x ∈ x, then the family ax = {x(f,r) : f ∈ f, f(x) = 0, and r > 0} is a filter base on x. the filter on x generated by ax is nf(x) and it is an f-ultrafilter on x. proof. if f,g ∈ f and r > 0, then x(|f| + |g|,r) ⊆ x(f,r) ∩ x(g,r). this implies that ax is a filter base on x. clearly, ax generates the filter nf(x), and so nf(x) is an f-filter on x by lemma 3.5. then nf(x) is an f-ultrafilter on x by theorem 3.9 (iv). � the following definition is reasonable by the previous theorem. definition 4.5. the function e : x → δx defined by e(x) = nf(x) for every x ∈ x is the canonical mapping. if a ⊆ x and x ∈ x, then e(x) ∈ â if and only if x ∈ a◦. next, let a,b ⊆ x. in general, b̂ ∩ e(a) = ∅ does not imply b ∩ a = ∅. however, this implication holds if b is a τ(f)-open subset of x. we apply this fact repeatedly in what follows. we gather some properties of the space δx in the following lemmas. lemma 4.6. let a ⊆ x and let p ∈ δx. the following statements are equivalent: (i) p ∈ e(a). (ii) a ∩ b 6= ∅ for every b ∈ p. (iii) x(f,r) ∈ p for every f ∈ z(a) and for every r > 0. in particular, p ∈ e(a) for every a ∈ p. proof. (i) ⇒ (ii) if a ∩ b = ∅ for some b ∈ p, then a ∩ b◦ = ∅, and so e(a) ∩ b̂◦ = ∅. since b◦ ∈ p, we have p /∈ e(a). (ii) ⇒ (iii) this follows from corollary 3.8. (iii) ⇒ (i) suppose that p /∈ e(a). then there exists a τ(f)-open subset b of x such that b ∈ p and b̂ ∩ e(a) = ∅, and so b ∩ a = ∅. pick some c ∈ p and a function f ∈ f with f(c) = {1} and f(x \ b) = {0}. then f ∈ z(a). since x(f,1/2) ∩ c = ∅, we have x(f,1/2) /∈ p, and so statement (iii) does not hold. � lemma 4.7. if a,b ⊆ x, then the following statements hold: (i) x̂ \ a = δx \ e(a). (ii) if a is a τ(f)-open subset of x, then e(a) = clδx(â). (iii) â = b̂ if and only if a◦ = b◦. (iv) â = ∅ if and only if a◦ = ∅. (v) â = δx if and only if a = x. proof. (i) suppose first that p ∈ x̂ \ a. since x̂ \ a ∩ e(a) = ∅, we have p /∈ e(a). suppose now that p ∈ δx \ e(a). then there exists a τ(f)-open c© agt, upv, 2014 appl. gen. topol. 15, no. 2 190 function lattices and compactifications subset c of x such that c ∈ p and ĉ ∩ e(a) = ∅. then c ∩ a = ∅, that is, c ⊆ x \ a, and so x \ a ∈ p. (ii) the inclusion clδx(â) ⊆ e(a) holds for any subset a of x and follows from statement (i). suppose now that a is τ(f)-open and that p ∈ e(a). if b ∈ p, then b̂ ∩ e(a) 6= ∅, so b◦ ∩ a 6= ∅, and so b̂ ∩ â 6= ∅. therefore, p ∈ clδx(â). statement (iii) follows from remark 4.3. then statements (iv) and (v) follow from statement (iii). � lemma 4.8. let a,b ⊆ x. then x(f,r) ∩ x(g,r) 6= ∅ for all f ∈ z(a), g ∈ z(b), and r > 0 if and only if e(a) ∩ e(b) 6= ∅. proof. necessity follows from lemma 4.6, so suppose that x(f,r)∩x(g,r) 6= ∅ for all f ∈ z(a), g ∈ z(b), and r > 0. put a = {x(h,r) : h ∈ z(a) ∪ z(b), r > 0}. then a is an f-family on x by lemma 3.5. we claim that a has the finite intersection property. let f1, . . . ,fn ∈ z(a) and let g1, . . . ,gm ∈ z(b) for some n,m ∈ n. then f := ∑n k=1|fk| ∈ z(a) and g := ∑m k=1|gk| ∈ z(b). if r > 0, then x(f,r) ∩ x(g,r) ⊆ ( n⋂ k=1 x(fk,r) ) ∩ ( m⋂ k=1 x(gk,r) ) , thus verifying our claim. by lemma 3.6, there exists an element p ∈ δx such that a ⊆ p. then p ∈ e(a) ∩ e(b) by lemma 4.6, as required. � now, we are ready to prove first of the main theorems of this section. theorem 4.9. the space δx is a compact hausdorff space and e(x) is dense in δx. proof. first, e(x) is dense in δx by lemma 4.7 (iv). to see that δx is hausdorff, let p and q be distinct points of δx. pick some set a ∈ p \ q. pick some b ∈ p and a function f ∈ f with f(b) = {0} and f(x \ a) = {1}. since x(f,1/2) /∈ q, there exists some c ∈ q such that x(f,1/3) ∩ c = ∅ by theorem 3.9 (ii). then b ∩c = ∅, and so b̂ and ĉ are disjoint neighborhoods of p and q, respectively. lemma 4.7 (i) implies that the family b = {e(a) : a ⊆ x} is a base for the closed subsets of δx. suppose that a subset c of b has the finite intersection property. to show that δx is compact, it is enough to show that ⋂ c∈c c 6= ∅. put a′ = {a ⊆ x : e(a) ∈ c} and a = {x(f,r) : a ∈ a′, f ∈ z(a), r > 0}. then a is an f-family on x by lemma 3.5 and a has the finite intersection property by lemma 4.8. by lemma 3.6, there exists an element p ∈ δx such that a ⊆ p. then p ∈ e(a) for every a ∈ a′ by lemma 4.6, and so p ∈ ⋂ c∈c c, thus finishing the proof. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 191 t. alaste we finish this section by showing that f-filters describe the topology of δx. as with the stone-čech compactification of a discrete topological space, we have two natural candidates for the closure of an f-filter in δx, namely ϕ̂ and the following. definition 4.10. define ϕ = ⋂ a∈ϕ e(a) for every f-filter ϕ on x. note that ϕ is a non-empty, closed subset of δx. the next statement follows from lemma 4.6. theorem 4.11. if ϕ is an f-filter on x, then ϕ̂ = ϕ. theorem 4.12. if c is a non-empty, closed subset of δx, then there exists a unique f-filter ϕ on x such that ϕ̂ = c. proof. let c be a non-empty, closed subset of δx. put ϕ = ⋂ p∈c p. clearly, ϕ is a filter on x. let us show that ϕ is an f-family on x, hence, an f-filter on x. suppose that a ∈ ϕ satisfies a 6= x. if p ∈ c, then a ∈ p, and so there exist some bp ∈ p and a function fp ∈ f with f(x) ⊆ [0,1], fp(bp) = {0}, and fp(x \ a) = {1}. now, {b̂p : p ∈ c} is an open cover of c, and so there exist points p1, . . . ,pn ∈ c for some n ∈ n such that c ⊆ ⋃n k=1 b̂pk . put f = ∑n k=1 fpk and b = ⋃n k=1 bpk . then b ∈ ϕ. since f(x) ≤ n − 1 for every x ∈ b and f(x) = n for every x ∈ x \ a, the filter ϕ is an f-family on x. let us verify the equality ϕ̂ = c. the inclusion c ⊆ ϕ̂ is obvious, so suppose that q ∈ δx \c. then there exists a τ(f)-open subset a of x such that a ∈ q and â ∩ c = ∅. for every p ∈ c, pick a τ(f)-open subset bp of x such that bp ∈ p and â ∩ b̂p = ∅. then a ∩ bp = ∅ for every p ∈ c. as above, there exist n ∈ n and points p1, . . . ,pn ∈ c such that b := ⋃n k=1 bpk ∈ ϕ. since a ∩ b = ∅, we have q /∈ ϕ̂, as required. finally, the f-filter ϕ on x satisfying ϕ̂ = c is unique by theorem 4.2 (iv). � 5. continuous functions on δx again, we assume that f is a function lattice on x. this section is devoted to a study of continuous, real-valued functions on the space δx. we show that every bounded member of f extends to δx and that these extensions form a dense subspace of c(δx). we leave the proof of the following lemma to the reader. lemma 5.1. let p ∈ δx, let g ∈ c(δx), and let r > 0. then {x ∈ x : |g(p) − g(e(x))| ≤ r} ∈ p. theorem 5.2. for every bounded function f ∈ f, there exists a unique function f̂ ∈ c(δx) satisfying f = f̂ ◦ e. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 192 function lattices and compactifications proof. let p ∈ δx and put (5.1) c = ⋂ a∈p clr(f(a)). then c is a non-empty subset of r by assumption. choosing any element f̂(p) ∈ c, we obtain a function f̂ : δx → r. next, let us show that if p = e(x) for some x ∈ x, then c = {f(x)}. this will establish the equality f = f̂ ◦ e. clearly, f(x) ∈ c, so let y ∈ r be such that y 6= f(x). pick r > 0 such that y /∈ u := [f(x) − r,f(x) + r]. then f−1(u) ∈ e(x). since y /∈ clr(f(f −1(u))), we have y /∈ c, as required. the density of e(x) in δx implies that the continuous function h on δx satisfying f = h ◦ e is unique. therefore, it is enough to show that f̂ is continuous to finish the proof. to see that f̂ is continuous, let p ∈ δx and put g = f −f̂(p). first, we claim that x(g,r) ∈ p for every r > 0. by corollary 3.7, it is enough to show that x(g,r) ∩ b 6= ∅ for every b ∈ p and for every r > 0. so, let b ∈ p and r > 0 be given. since f̂(p) ∈ clr(f(b)), there exists a point x ∈ b such that |g(x)| = |f(x)− f̂(p)| ≤ r, and so x ∈ x(g,r)∩b, as required. to finish the proof, let r > 0. if q ∈ x̂(g,r), then f̂(q) ∈ clr(f(x(g,r))), so |f̂(q) − f̂(p)| ≤ r, and so f̂ is continuous at p. � although the canonical mapping need not be injective, we call the continuous function f̂ on δx satisfying f = f̂ ◦ e an extension of f to δx. we denote by r∗ the one-point compactification of r, that is, r∗ = r∪{∞}. theorem 5.3. for every function f ∈ f, there exists a unique continuous function f̂ : δx → r∗ satisfying f = f̂ ◦ e. proof. arguing as in the previous proof and using the compactness of r∗, we need only to show that f̂ is continuous at a point p ∈ δx with f̂(p) = ∞. let n ∈ n. by lemma 4.6, it is enough to show that a := {x ∈ x : |f(x)| ≥ n} ∈ p. put b = {x ∈ x : |f(x)| ≥ n + 1} and g = n + 1 − (|f| ∧ (n + 1)). then g ∈ z(b) and x(g,1) ⊆ a. since p ∈ e(b), we have a ∈ p by lemma 4.6, as required. � the points p ∈ δx satisfying f̂(p) = ∞ have a simple characterization. indeed, if f ∈ f is unbounded, then the sets cn = {x ∈ x : |f(x)| ≥ n}, where n ∈ n, determine a filter base b on x. the filter ϕ on x generated by b is an f-filter on x and satisfies ϕ̂ = {p ∈ δx : f̂(p) = ∞}. in the next theorem (and later), we put fb = f ∩ ℓ ∞(x). recall that the space fb is equipped with the norm of uniform convergence. we could deduce the following theorem from the stone-weierstrass theorem. however, we feel that the proof below is worth presenting, since it uses only properties of ffilters instead of the stone-weierstrass theorem. also, in this way we obtain the stone-weierstrass theorem as a corollary in section 8. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 193 t. alaste theorem 5.4. the mapping γ : fb → c(δx) defined by γ(f) = f̂ is a linear isometry and γ(f) is dense in c(δx). if fb is an algebra, then γ is also a homomorphism. proof. using the density of e(x) in δx and the equality f = f̂ ◦ e, it is easy to verify that γ is a linear isometry (homomorphism if fb is an algebra) and we leave the details to the reader. let us show that γ(f) is dense in c(δx). to prove this, it is enough to show that, for every positive function g ∈ c(δx) with ‖g‖ = 1 and for every r > 0, there exists a function f ∈ fb such that ‖f̂ − g‖≤ r. so, let g ∈ c(δx) be positive with ‖g‖ = 1 and let r > 0. pick n ∈ n such that 1/n ≤ r/3. for every k ∈ {1, . . . ,n}, define the following subsets of [0,1], x, and δx, respectively: ik = [k − 1 n , k n ] , ak = {x ∈ x : k − 2 n < g(e(x)) < k + 1 n }, ck = g −1(ik). note that ak ∩ aj = ∅ whenever k,j ∈ {1, . . . ,n} and k + 3 ≤ j. let k ∈ {1, . . . ,n}. if p ∈ ck, then ak ∈ p by lemma 5.1, and so there exist some bp ∈ p and fp ∈ fb with fp(bp) = {k/n}, fp(x \ ak) = {0}, and fp(x) ⊆ [0,k/n]. pick elements p1, . . . ,pm ∈ ck for some m ∈ n such that ck ⊆ ⋃m j=1 b̂pj and put fk = fp1 ∨ . . . ∨ fpm. note that fk(x \ ak) = {0} and fk(x) = k/n for every x ∈ x with e(x) ∈ ck. put f = f1 ∨. . .∨fn. then f ∈ fb and we claim that ‖f̂ −g‖ ≤ r. to verify our claim, it is enough to show that |f(x) − g(e(x))| ≤ r for every x ∈ x. so, let x ∈ x. suppose first that g(e(x)) ≥ (n − 3)/n. then e(x) ∈ ck for some k ∈ n with n − 2 ≤ k ≤ n, so f(x) ≥ (n − 2)/n, and so |f(x) − g(e(x))| ≤ r. suppose now that g(e(x)) < (n − 3)/n. then there exists k ∈ {1, . . . ,n − 3} such that (k − 1)/n ≤ g(e(x)) < k/n. then x ∈ ak and e(x) ∈ ck, and so f(x) ≥ k/n. since ak ∩aj = ∅ for every j ∈ {1, . . . ,n} with j ≥ k+3, we have fj(x) = 0 for every j with k + 3 ≤ j ≤ n, and so f(x) ≤ (k + 2)/n. therefore, |f(x) − g(e(x))| ≤ 3/n ≤ r, thus finishing the proof. � any closed subalgebra of ℓ∞(x) containing the constant functions is a function lattice on x (see [16, p. 291] or [11, p. 265]). therefore, we obtain the following corollary. corollary 5.5. if f is a closed subalgebra of ℓ∞(x) containing the constant functions, then γ : f → c(δx) is an isometric isomorphism. corollary 5.6. if f is a function lattice on x such that f ⊆ ℓ∞(x), then the closure of f in ℓ∞(x) is a closed subalgebra of ℓ∞(x). proof. denote by f′ the closure of f in ℓ∞(x). remark 3.2 implies that a filter ϕ on x is an f-filter if and only if ϕ is an f′-filter, and so the notation δx is unambiguous. corollary 5.5 implies that the mapping γ : f′ → c(δx) is an isometric isomorphism. since c(δx) is an algebra, the statement follows. � next, we show that f-filters describe all dense images of x in compact hausdorff spaces. precise statement and details follow. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 194 function lattices and compactifications theorem 5.7. let y be a compact hausdorff space and let ε : x → y be a function such that ε(x) is dense in y . the following statements hold: (i) the set f = {h◦ε : h ∈ c(y )} is a closed subalgebra of ℓ∞(x) containing the constant functions. (ii) f is isometrically isomorphic with c(y ). (iii) there exists a homeomorphism f : δx → y such that f ◦ e = ε. proof. we prove only statement (iii) and leave the verifications of statements (i) and (ii) to the reader. if p ∈ δx, then c = ⋂ a∈p cly (ε(a)) is a non-empty subset of y , and we claim that c is a singleton. suppose that c contains distinct elements x and y. by urysohn’s lemma, there exists a function h ∈ c(y ) such that h(x) = 0 and h(y) = 1. put f = h ◦ ε and a = {x ∈ x : f̂(p) − 1/3 ≤ f(x) ≤ f̂(p) + 1/3}. then a ∈ p by lemma 5.1, so x,y ∈ cly (ε(a)), and so h(x),h(y) ∈ clr(f(a)). therefore, |h(x)−h(y)| ≤ 2/3, a contradiction. since c is a singleton, we obtain a function f : δx → y . clearly, f ◦e = ε. since e(x) and ε(x) are dense subsets of δx and y , respectively, it is enough to show that f is injective and continuous to finish the proof. to see that f is injective, suppose that p,q ∈ δx satisfy p 6= q. by urysohn’s lemma, there exists a function g ∈ c(δx) such that g(p) = 0 and g(q) = 1. put f = g ◦ e. then f ∈ f by corollary 5.5. put a = {x ∈ x : f(x) ≤ 1/3} and b = {x ∈ x : f(x) ≥ 2/3}. then a ∈ p and b ∈ q by lemma 5.1, and so f(p) ∈ cly (ε(a)) and f(q) ∈ cly (ε(b)). statement (ii) implies that there exists a function h ∈ c(y ) such that f = h◦ ε. then h(f(p)) ∈ clr(f(a)) and h(f(q)) ∈ clr(f(b)), and so f(p) 6= f(q), as required. to show that f is continuous, let p ∈ δx and let u be an open neighborhood of f(p) in y with u 6= y . again, there exists a continuous function h ∈ c(y ) such that h(f(p)) = 0 and h(y \ u) = {1}. put f = h ◦ ε. the continuity of h implies that f̂(p) = 0, and so b = {x ∈ x : −1/2 ≤ f(x) ≤ 1/2} ∈ p by lemma 5.1. if q ∈ b̂, then h(f(q)) ∈ [−1/2,1/2], and so f(q) ∈ u, thus finishing the proof. � 6. some relationships between function lattices throughout this section, we assume that f1 and f2 are function lattices on x contained in ℓ∞(x). we denote by δ1x and δ2x the spaces of f1-ultrafilters on x and f2-ultrafilters on x, respectively. also, we denote by e1 and e2 the canonical mappings from x to δ1x and δ2x, respectively. if a ⊆ x, then the notation â is ambiguous. however, we hope that it is clear from the notation used whether we consider â as a subset of δ1x or δ2x. if f ∈ f1 ∩ f2, then f extends to both δ1x and δ2x. we denote these extension by f δ1 and fδ2, respectively. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 195 t. alaste theorem 6.1. if f1 ⊆ f2, then the following statements are equivalent: (i) f1 is dense in f2. (ii) the set {fδ2 : f ∈ f1} is dense in c(δ2x). (iii) a filter ϕ on x is an f1-filter if and only if ϕ is an f2-filter. (iv) δ1x = δ2x. proof. the equivalence of statements (i) and (ii) and the implication (iv) ⇒ (i) follow from theorem 5.4, and the implication (iii) ⇒ (iv) is obvious. to verify the implication (i) ⇒ (iii), it is enough to show that a non-empty family a of non-empty subsets of x is an f1-family on x if and only if a is an f2-family on x. since f1 ⊆ f2, necessity is obvious, and sufficiency follows from the density of f1 in f2 and remark 3.2. � for the rest of this section, we assume that f1 and f2 are closed subalgebras of ℓ∞(x) containing the constant functions. theorem 6.2. the inclusion f1 ⊆ f2 holds if and only if there exists a continuous, surjective mapping f : δ2x → δ1x such that e1 = f ◦ e2. proof. suppose first that f1 ⊆ f2. let p ∈ δ2x and put c = ⋂ a∈p clδ1x(e1(a)). similar arguments as used in the proof of theorem 5.7 apply to show that c is a singleton, and so we obtain a function f : δ2x → δ1x. clearly, e1 = f ◦ e2. also, arguing as in the last part of the proof of theorem 5.7, we see that f is continuous. therefore, we need only to show that f is surjective. if q ∈ δ1x, then q is an f2-filter on x. pick any p ∈ δ2x with q ⊆ p and let a ∈ q. since δ1x is a regular topological space, there exists a τ(f1)open subset b of x such that b ∈ q and clδ1x(b̂) ⊆ â. then b ∈ p, so f(p) ∈ clδ1x(b̂) by lemma 4.7 (ii), and so a ∈ f(p). therefore, q ⊆ f(p), and so q = f(p), as required. suppose now that there exists a continuous mapping f : δ2x → δ1x with e1 = f ◦ e2. let f ∈ f1. by theorem 5.2, there exists a function g ∈ c(δ1x) such that f = g◦e1. since g◦f ∈ c(δ2x) and f = (g◦f)◦e2, we have f ∈ f2 by corollary 5.5, thus finishing the proof. � for the proof of the next theorem, recall the definition of f̂ from the proof of theorem 5.2. theorem 6.3. suppose that f1 ⊆ f2 and let f : δ2x → δ1x be as in theorem 6.2. if p ∈ δ2x and q ∈ δ1x, then the following statements are equivalent: (i) q ⊆ p. (ii) f(p) = q. (iii) fδ2(p) = fδ1(q) for every f ∈ f1. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 196 function lattices and compactifications proof. (i) ⇒ (ii) this was proved already in the proof of theorem 6.2. (ii) ⇒ (iii) suppose that f(p) = q. let f ∈ f1. since e1 = f ◦ e2, the functions fδ2 and fδ1 ◦ f agree on e2(x), hence, on δ2x. therefore, fδ2(p) = fδ1(q). (iii) ⇒ (i) suppose that q is not contained in p. pick some a ∈ q \ p. pick b ∈ q and a positive function f ∈ f1 with f(b) = {0} and f(x \ a) = {1}. then fδ1(q) = 0. since x(f,1/2) /∈ p, there exists some c ∈ p such that x(f,1/3) ∩ c = ∅ by theorem 3.9 (ii). then fδ2(p) ≥ 1/3, thus finishing the proof. � define two closed equivalence relations ∼ and ≈ on δ2x as follows: p ∼ q if and only if f(p) = f(q), and p ≈ q if and only if fδ2(p) = fδ2(q) for every f ∈ f1. theorem 6.3 shows that these relations are identical. since f is a quotient mapping (see [16, pp. 60-61]), we obtain the following statement. corollary 6.4. if f1 ⊆ f2, then the quotient space δ2x/ ≈ is homeomorphic with δ1x. 7. f-filters and ideals of f throughout this section, we assume that f is a closed subalgebra of ℓ∞(x) containing the constant functions. we establish a correspondence between ffilters on x and closed, proper ideals of f. roughly speaking, we show how the ideals of f can be used to generate f-filters on x. we apply the following convention for the rest of this paper: by an ideal of f, we always mean a closed, proper ideal of f. the next lemma follows from [7, (1.23) proposition]. since the proof of the cited proposition relies on the spectrums of single elements of c∗-algebras, we present the following short proof using only basic properties of banach algebras. lemma 7.1. if f ∈ f \ f0, then 1/f ∈ f. proof. suppose first that f ∈ f \ f0 is positive. pick r > 0 such that r ≤ f(x) for every x ∈ x. then 0 < r ‖f‖ ≤ f(x) ‖f‖ ≤ 1 for every x ∈ x. put g = f/‖f‖. then ‖1 − g‖ < 1 by the inequalities above, and so g is invertible in f (see [7, (1.3) lemma]). therefore, 1/f ∈ f. if f ∈ f \ f0 is any function, then f 2 ∈ f \ f0 is positive. the equality 1/f = f/f2 and the first part of the proof imply that 1/f ∈ f. � corollary 7.2. if i is an ideal of f, then i ⊆ f0. definition 7.3. for every ideal i of f, define (7.1) b(i) = {x(f,r) : f ∈ i, r > 0}. theorem 7.4. if i is an ideal of f, then b(i) is a filter base on x and the filter ϕ on x generated by b(i) is an f-filter. conversely, if ϕ is an f-filter on x, then there exists an ideal i of f such that ϕ is generated by b(i). c© agt, upv, 2014 appl. gen. topol. 15, no. 2 197 t. alaste proof. suppose first that i is an ideal of f. first, x(f,r) 6= ∅ for every f ∈ i and for every r > 0 by corollary 7.2, and so ∅ /∈ b(i). next, let f,g ∈ i and let r > 0. since f2 +g2 ∈ i and x(f2 +g2,r) ⊆ x(f,r) ∩x(g,r), the set b(i) is a filter base on x. since b(i) is an f-family on x by lemma 3.5, the filter ϕ on x generated by b(i) is an f-filter. suppose now that ϕ is an f-filter on x. put i = {f ∈ f : x(f,r) ∈ ϕ for every r > 0}. clearly, 0 ∈ i. let f1,f2 ∈ i, let h ∈ f with h 6= 0, let α ∈ r with α 6= 0, let (gn) be a sequence in i converging to some g ∈ f, and let r > 0. the inclusions x(f1,r/2) ∩ x(f2,r/2) ⊆ x(f1 − f2,r), x(f1,r/‖h‖) ⊆ x(f1h,r), x(f1,r/|α|) ⊆ x(αf1,r), x(gn,r/2) ⊆ x(g,r), where the last one holds if ‖gn − g‖ ≤ r/2, imply that i is an ideal of f. we claim that b(i) is a filter base for ϕ. clearly, b(i) ⊆ ϕ, so suppose that a ∈ ϕ satisfies a 6= x. pick some b ∈ ϕ and a function f ∈ f with f(b) = {0} and f(x \a) = {1}. since b ⊆ x(f,r) for every r > 0 and b ∈ ϕ, we have f ∈ i. since x(f,1/2) ⊆ a, the claim follows. � let ϕ be an f-filter on x. the previous theorem guarantees the existence of an ideal i of f such that ϕ is generated by b(i). the next theorem shows that this ideal i is unique. theorem 7.5. let i be an ideal of f, let ϕ be the f-filter on x generated by b(i), and let f ∈ f. the following statements are equivalent: (i) f ∈ i. (ii) f̂(p) = 0 for every p ∈ ϕ. (iii) x(f,r) ∈ ϕ for every r > 0. proof. (i) ⇒ (ii) suppose that f ∈ i. let p ∈ ϕ and let r > 0. since ϕ is generated by b(i), we have x(f,r) ∈ p by theorem 4.11, and so |f̂(p)| ≤ r by lemma 4.6. therefore, f̂(p) = 0. (ii) ⇒ (iii) this follows from lemma 5.1 and theorem 4.2 (ii). (iii) ⇒ (i) suppose that (iii) holds. it is enough to show that f ∈ clf(i), and so we may assume that f 6= 0. let 0 < r < ‖f‖. then x(f,r) 6= x. since b(i) is a filter base for ϕ, there exist functions h ∈ i and g ∈ f such that g(x(h,1)) = {0} and g(x \ x(f,r)) = {1}. now, 1/(|h| ∨ 1)2 ∈ f by lemma 7.1, so k := h2/(|h| ∨ 1)2 ∈ i, and so fk ∈ i. the inclusion x(h,1) ⊆ x(f,r) implies that f and fk agree on x \ x(f,r). therefore, ‖f − fk‖ = supx∈x(f,r)|f(x)(1 − k(x))| ≤ r, and so f ∈ clf(i), thus finishing the proof. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 198 function lattices and compactifications let ϕ be an f-filter on x. we say that a function f ∈ f tends to zero in the direction of ϕ if and only if, for every r > 0, there exists some a ∈ ϕ with |f(x)| ≤ r for every x ∈ a. the previous theorem, then, says that a subset i of f is an ideal of f if and only if there exists an f-filter ϕ on x such that i consists of those members of f which tend to zero in the direction of ϕ. let i be an ideal of f. the equality x(f,r) = x(|f|,r) for every f ∈ f and for every r > 0 implies that |f| ∈ i for every f ∈ i. for every ideal i of f, we denote by ϕ(i) the f-filter on x generated by b(i). if i and j are ideals of f, then the previous theorem implies that i ⊆ j if and only if ϕ(i) ⊆ ϕ(j). from this we conclude the following: an ideal i of f is a maximal ideal of f if and only if ϕ(i) is an f-ultrafilter on x. we denote by m(f) the set of all maximal ideals of f. if i is an ideal of f, then the hull of i is the set h(i) = {j ∈ m(f) : i ⊆ j}. the kernel k(j ) of a non-empty subset j of m(f) is the set k(j ) = ⋂ j∈j j. note that k(j ) is an ideal of f. the hull-kernel topology on m(f) is defined by declaring a nonempty subset j of m(f) to be closed if and only if j = h(k(j )). in terms of f-filters, this reads as follows: a non-empty subset j of m(f) is closed if and only if there exists an f-filter ϕ on x such that j = {j ∈ m(f) : ϕ ⊆ ϕ(j)}. therefore, the mapping j 7→ ϕ(j) from m(f) to δx is a homeomorphism. the following well-known property of f follows from theorem 4.2 (ii). corollary 7.6. if i is an ideal of f, then k(h(i)) = i. 8. f-filters on topological spaces in the previous sections, we made no assumption about algebraic or topological structure on the set x. in this section, we assume that (x,τ) is a hausdorff topological space and that f is a function lattice on x such that f ⊆ c(x). recall that a◦ denotes the τ(f)-interior of a subset a of x. if a ⊆ x, then e−1(â) = a◦. since f ⊆ c(x), the set a◦ is τ-open in x, and so the canonical mapping e : x → δx is continuous. for every element x ∈ x, we denote by n(x) the neighborhood filter of x in (x,τ). since f ⊆ c(x), we have nf(x) ⊆ n(x) for every x ∈ x. the canonical mapping e : x → δx is an embedding if and only if the equality n(x) = nf(x) holds for every x ∈ x. by remark 3.5, this equality for every x ∈ x is equivalent to statement (ii) below. lemma 8.1. the following statements are equivalent: (i) the canonical mapping e : x → δx is an embedding. (ii) for every x ∈ x and for every neighborhood u ∈ n(x) with u 6= x, there exists a function f ∈ f with f(x) = 1 and f(x \ u) = {0}. the next theorem follows from theorem 5.7. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 199 t. alaste theorem 8.2. if y is a compact hausdorff space and ε : x → y is a continuous mapping such that ε(x) is dense in y , then the following statements hold: (i) the set f = {h◦ε : h ∈ c(y )} is a closed subalgebra of c(x) containing the constant functions. (ii) f is isometrically isomorphic with c(y ). (iii) there exists a homeomorphism f : δx → y such that f ◦ e = ε. statements (ii) and (iii) of the next corollary constitute stone-weierstrass theorem. corollary 8.3. if x is compact, then the following statements are equivalent: (i) the canonical mapping e : x → δx is a homeomorphism. (ii) f separates the points of x. (iii) f is dense in c(x). proof. since x is compact, the canonical mapping e : x → δx is a continuous surjection. therefore, e is a homeomorphism if and only if e is injective, and so (i) and (ii) are equivalent. (i) ⇒ (iii) suppose that e : x → δx is a homeomorphism. then it is easy to verify that the mapping g 7→ g ◦ e from c(δx) to c(x) is an isometric isomorphism. since {f̂ : f ∈ f} is dense in c(δx) by theorem 5.4 and f̂ ◦ e = f for every f ∈ f, the statement follows. (iii) ⇒ (ii) this follows from urysohn’s lemma. � next statement is a consequence of the gelfand-naimark theorem. here, it follows from corollary 8.3. an isometric isomorphism t : c(x) → c(y ) induces a bijection between ideals of c(x) and c(y ). then the maximal ideal spaces m(c(x)) and m(c(y )) are homeomorphic (under their hull-kernel topologies). corollary 8.4. if x and y are compact hausdorff spaces, then x and y are homeomorphic if and only if c(x) and c(y ) are isometrically isomorphic. we finish this section with the following statement concerning locally compact topological spaces. if x is locally compact, then we denote by x∞ the one-point compactification of x. let e1 : x → x∞ denote the natural embedding. then {h ◦ e1 : h ∈ c(x∞)} = c0(x) ⊕ r, where r denotes the constant functions on x. necessity of the following statement follows from corollary 5.5 using the zero extension. sufficiency follows from theorem 6.2 and from the fact that x is embedded and open in x∞. theorem 8.5. suppose that x is non-compact and locally compact and that f is a closed subalgebra of c(x) containing the constant functions. the canonical mapping e : x → δx is an embedding and e(x) is open in δx if and only if c0(x) ⊆ f. remark 8.6. let x and f be as above and suppose that x is embedded in δx. then the family ϕk = {x \ k : k ⊆ x and clx(k) is compact} is an c© agt, upv, 2014 appl. gen. topol. 15, no. 2 200 function lattices and compactifications f-filter on x and δx \ e(x) = ϕ̂k. in particular, if δx = x∞, then ∞ = ϕk as an f-filter. 9. spectrums of unital c∗-subalgebras of ℓ∞(x) in this last section, we change our notation from spaces of real-valued functions to spaces of complex-valued functions. we denote by ℓ∞(x) the c∗algebra of all bounded, complex-valued functions on x. if x is a topological space, then we denote by c(x) the c∗-subalgebra of ℓ∞(x) consisting of continuous members of ℓ∞(x). we explain briefly how the introduced filters can be used to represent the spectrum of any c∗-subalgebra of ℓ∞(x) as a space of filters on x. throughout this section, let f be a c∗-subalgebra of ℓ∞(x) such that f contains the constant functions. we consider the spectrum ∆ of f as the space of all non-zero, multiplicative linear functionals on f, that is, ∆ = {µ ∈ f∗ : µ 6= 0 and µ(fg) = µ(f)µ(g) for all f,g ∈ f}, where f∗ denotes the banach dual of f. the evaluation mapping ε : x → ∆ is defined by [ε(x)](f) = f(x) for every x ∈ x and for every f ∈ f. under the relative weak* topology of f∗, the space ∆ is a compact hausdorff space and ε(x) is a dense subset of ∆. the characteristic property of the space ∆ is the fact that f and c(∆) are isometrically ∗-isomorphic. if µ ∈ ∆, then kerµ = {f ∈ f : µ(f) = 0} is a maximal ideal of f. conversely, if i is a maximal ideal of f, then there exists a unique element µ ∈ ∆ such that i = kerµ (see [7]). the space fr of all real-valued members of f is a closed subalgebra of the space of all bounded, real-valued functions on x, and so fr is a function lattice on x. we define δx to be the space of all fr-ultrafilters on x. if f ∈ f, then theorem 5.2 implies that the real and imaginary parts of f extend to δx, and so there exists a unique function f̂ ∈ c(δx) satisfying f = f̂ ◦ e. then the mapping f 7→ f̂ from f to c(δx) is an isometric ∗-isomorphism by theorem 5.4. small adjustments in section 7 apply to show that, for every f-filter ϕ on x, there exists a unique ideal i of f such that ϕ is generated by b(i). (here, b(i) is defined as in (7.1).) in the second part of the proof of lemma 7.1, we apply the equality 1/f = f/|f|2. here, f(x) = f(x) for every x ∈ x and f(x) denotes the complex-conjugate of f(x). in the last part of the proof of theorem 7.4, we apply the fact that |f|2 + |g|2 ∈ i. in the proof of implication (iii) ⇒ (i) of theorem 7.5, we define k = |h|2/(|h| ∨ 1)2. here, we apply the fact that |f| ∈ f for every f ∈ f (see [11, p. 265]). if i is an ideal of f, then the equalities x(f,r) = x(|f|,r) = x(f,r), which hold for every f ∈ f and for every r > 0, and theorem 7.5 imply that |f| ∈ i and f ∈ i for every f ∈ i. finally, the mapping µ 7→ kerµ from ∆ to the maximal ideal space m(f) of f is a bijection. once m(f) is equipped with the hull-kernel topology, this mapping is a homeomorphism. since the mapping j 7→ ϕ(j) from m(f) to c© agt, upv, 2014 appl. gen. topol. 15, no. 2 201 t. alaste δx, where ϕ(j) is the f-ultrafilter on x generated by b(j), is a homeomorphism, we conclude that the mapping µ 7→ p(µ) from ∆ to δx, where p(µ) is the f-ultrafilter on x generated by b(kerµ), is a homeomorphism. references [1] t. alaste, u-filters and uniform compactification, studia math. 211 (2012), 215–229. [2] t. alaste, semigroup compactifications in terms of filters, (submitted) (arxiv:1302.1742). [3] t. budak and j. pym, local topological structure in the luc-compactification of a locally compact group and its relationship with veech’s theorem, semigroup forum 73 (2006), 159–174. [4] j.f. berglund and n. hindman, filters and the weak almost periodic compactification of a discrete semigroup, trans. amer. math. soc. 284 (1984), 1–38. [5] j. f. berglund, h. d. junghenn and p. milnes, analysis on semigroups, john wiley & sons, inc., new york, 1989. [6] w.w. comfort and s. negrepontis, the theory of ultrafilters, (springer–verlag, new york, 1974). [7] g.b. folland, a course in abstract harmonic analysis, (crc press, boca raton, fl, 1995). [8] l. gillman and m. jerison, rings of continuous functions, (springer–verlag, new york, 1976). [9] n. hindman and d. strauss, algebra in the stone–čech compactification, (walter de gruyter & co., berlin, 1998). [10] j.r. isbell, uniform spaces, (american mathematical society, providence, r.i., 1964). [11] g.j.o. jameson, topology and normed spaces, (chapman and hall, london, 1974). [12] m. koçak and d. strauss, near ultrafilters and compactifications, semigroup forum 55 (1997), 94–109. [13] s. a. naimpally and b.d. warrack, proximity spaces, (cambridge university press, london, 1970). [14] m. a. tootkaboni and a. riazi, ultrafilters on semitopological semigroups. semigroup forum 70 (2005), 317–328. [15] r. c. walker, stone-čech compactification, (springer-verlag, new york, 1974). [16] s. willard, general topology, (addison-wesley, reading, 1970). c© agt, upv, 2014 appl. gen. topol. 15, no. 2 202 @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 281–288 a short note on hit–and–miss hyperspaces rené bartsch and harry poppe dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. compactness here is sometimes viewed splitting into “κ-lindelöfness” and “κcompactness” for cardinals κ. to focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result, [8], of som naimpally, to who this article is hearty dedicated. 2000 ams classification: 54b20, 54d30, 54f99. keywords: hit-and-miss topology, compactness, relative completeness, relative compact unions, upper vietoris topology. 1. introduction. let (x,τ) be a topological space. by p(x), p0(x), cl(x) and k(x) respectively we denote the power set, the power set without the empty set ∅, the family of all closed subsets and the set of all compact subsets of x. for b ∈ p(x) and a ⊆ p(x) we define b−a := {a ∈ a|a ∩ b 6= ∅} (hit–set) and b+a := {a ∈ a|a ∩ b = ∅} (miss–set). specializing a := cl(x), we get the usual symbols b−,b+. by τl,a we denote the topology for a, generated by the subbase of all g−a,g ∈ τ. now consider ∅ 6= α ⊆ p(x); by τα,a we denote the topology for a which is generated from the subbase of all b+a,b ∈ α and g−a,g ∈ τ. of course, for every possible α we have τl,a ⊆ τα,a; for α = cl(x) we get the vietoris topology and for α = k(x) we get the fell topology for a. if α = ∆ ⊆ cl(x), τα,a is called ∆–topology by beer and tamaki [2], and was first introduced by poppe [10]. by f(x) and f0(x) we denote the set of all filters and ultrafilters, respectively, on a set x (a filter is not allowed to contain the empty set ∅); the symbol f(ϕ) (resp. f0(ϕ)) means the set of all filters (resp. ultrafilters) which contain a given filter ϕ; . x is the filter generated by a singleton {x},x ∈ x. the symbol qτ denotes the convergence structure induced by a topology τ, i.e. qτ := {(ϕ,x) ∈ f(x) ×x|ϕ ⊇ . x∩ τ}, so qτ is a relation between filters and points of a set x. if x is a set, τ, a are subsets of p(x), then we call a weakly complementary w.r.t. τ, iff for every subset σ ⊆ τ there exist a subset b ⊆ a, s.t. ⋃ b∈b b = x\ ⋃ s∈σ s. 282 r. bartsch and h. poppe lemma 1.1. let x be a set, τ, a ⊆ p(x) and k ⊆ x. then holds⋃ i∈i gi ⊇ k =⇒ ⋃ i∈i g −a i ⊇ k −a for every collection gi, i ∈ i,gi ∈ τ. if a is weakly complementary w.r.t. τ, then for every collection gi, i ∈ i,gi ∈ τ the implication ⋃ i∈i gi ⊇ k ⇐= ⋃ i∈i g −a i ⊇ k −a holds, too. proof. let ⋃ i∈i gi ⊇ k. a ∈ k −a ⇒ a∩k 6= ∅ ⇒ ∅ 6= a∩ ⋃ i∈i gi ⇒∃i0 ∈ i : a∩gi0 6= ∅ ⇒ a ∈ g −a i0 ⇒ a ∈ ⋃ i∈i g −a i . conversely, let a be weakly complementary w.r.t. τ and ⋃ i∈i g −a i ⊇ k −a . assume⋃ i∈i gi 6⊇ k. then x\ ⋃ i∈i gi ⊇ k\ ⋃ i∈i gi 6= ∅ holds, so there is an a ∈ a,a ⊆ x \ ⋃ i∈i gi with a∩k \ ⋃ i∈i gi 6= ∅. thus a ∈ k −a , implying a ∈ ⋃ i∈i g −a i . this yields ∃i0 ∈ i : a∩gi0 6= ∅ in contradiction to the construction of a. � corollary 1.2. let x be a set, τ, a ⊆ p(x) and k ⊆ x. then holds (1.1) ⋃ i∈i gi ⊇ k ⇐⇒ ⋃ i∈i g −a i ⊇ k −a for every collection gi, i ∈ i,gi ∈ τ if and only if a is weakly complementary w.r.t. τ. proof. we only have to show, that a is weakly complementary w.r.t. τ, if (1.1) holds. assume, a is not weakly complementary w.r.t. τ. then there must be a collection {gi|i ∈ i}⊆ τ, such that ⋃ {a|a ∈ p(x \ ⋃ i∈i gi) ∩ a} 6⊇ x \ ⋃ i∈i gi. now, we chose k := ( x \ ⋃ i∈i gi ) \ ⋃ {a|a ∈ p(x \ ⋃ i∈i gi) ∩a} 6= ∅. then no element of a, which meets k, can be contained in x \ ⋃ i∈i gi, i.e. every element of k−a meets ⋃ i∈i gi, too. so, it must meet a gi0, i0 ∈ i and consequently it is contained in ⋃ i∈i g −a i . but, by construction, the collection {gi|i ∈ i} doesn’t cover k, so (1.1) would fail. � obviously, if for every collection {gi|i ∈ i} ⊆ τ the complement x \ ⋃ i∈i gi itself belongs to a, or if all singletons {x},x ∈ x are elements of a, then a is weakly complementary w.r.t. τ. so, if τ is a topology on x, cl(x) and k(x) are weakly complementary w.r.t. τ. corollary 1.3. let (x,τ) be a topological space, k ⊆ x and ∀i ∈ i : gi ∈ τ. then holds ⋃ i∈i gi ⊇ k ⇐⇒ ⋃ i∈i g−i ⊇ k − we have yet another easy, but useful set-theoretical lemma: lemma 1.4. let x be a set, a ⊆ p(x) and ϕ ∈ f(x). assume, a is closed under finite unions of its elements. then holds ϕ∩ a 6= ∅ ⇐⇒∀ψ ∈ f0(ϕ) : ψ ∩ a 6= ∅ , i.e. a filter contains an a–set, iff each refining ultrafilter contains an a–set. a short note on hit-and-miss hyperspaces 283 proof. suppose ∀ψ ∈ f0(ϕ) : ∃aψ ∈ a : aψ ∈ ψ. now, assume ϕ∩ a = ∅. from this automatically follows x 6∈ a. consider b := {x \a| a ∈ a}. because of the closedness of a under finite unions, b is closed under finite intersection of its elements, and ∅ 6∈ b, because x 6∈ a. for any f ∈ ϕ,b ∈ b we have f ∩ b 6= ∅, because f ∩ b = ∅ would imply f ⊆ x \ b ∈ a and therefore ϕ ∩ a 6= ∅. so, ϕ ∪ b is a subbase of a filter and consequently, there exists an ultrafilter ψ, containing ϕ∪b, therefore containing ϕ and the complement of every a–set in contradiction to ∀ψ ∈ f0(ϕ) : ψ ∩ a 6= ∅. the other direction of the statement of the lemma is obvious. � definition 1.5. let κ be a cardinal. then a topological space (x,τ) is called κ-compact, iff every open cover of x with cardinality at most κ admits a finite subcover. (x,τ) is called κ-lindelöf, iff every open cover of x admits a subcover of cardinality at most κ. a filter is called κ-generated, iff it has a base of cardinality at most κ. a filter ϕ is called κ-completable, iff every subset b ⊆ ϕ with card(b) at most κ fulfills⋂ b∈b b 6= ∅. it is called κ-complete, iff ⋂ b∈b b ∈ ϕ holds under this condition. proposition 1.6. a topological space (x,τ) is κ-compact, if and only if every κ-generated filter on x has a convergent refining ultrafilter. proof. let (x,τ) be κ-compact and ϕ a filter on x with a base b of cardinality at most κ. assume, all refining ultrafilters of ϕ would fail to converge in x. then for each element x ∈ x, all refining ultrafilters of ϕ contain the complement of an open neighbourhood of x. but the set of complements of open neighbourhoods of a point x is closed w.r.t. finite unions, thus by lemma 1.4, ϕ contains the complement of an open neighbourhood of x. so, for each x ∈ x there must exist ox ∈ τ ∩ . x and bx ∈ b, s.t. bx ⊆ x \ox, implying bx ⊆ x \ox and thus x \bx ⊇ ox. now, for each b ∈ b we define ob := x \ b and find, that {ob| b ∈ b} is an open cover of x, because of the preceeding facts. so, there must exist a finite subcover ob1 ∪ ·· · ∪ obn = x, implying ⋃n i=1(x \ bi) = x, just meaning ⋂n i=1 bi = ∅, which is impossible, because all bi belong to the filter ϕ. so, the assumption must be false; there must exist convergent refining ultrafilters of ϕ. otherwise, let all κ-generated filters on x have a convergent refining ultrafilter. assume, there would exist an open cover c := {oi ∈ τ| i ∈ i}, ⋃ i∈i oi = x,card(i) ≤ κ such that all finite subcollections fail to cover x (implying κ to be infinite). but the set of all finite subcollections of the infinite collection c of cardinality at most κ has cardinality at most κ, too. so, b := {x \ ⋃n k=1 oik| n ∈ in,ik ∈ i} is a filterbasis of cardinality at most κ, thus there must exist an ultrafilter ϕ ⊇ b, which converges in x leading to the usual contradiction, because every x ∈ x is contained in an open ox ∈ c and x \ox belongs to b ⊆ ϕ. � analogously we get a characterization of κ-lindelöf-spaces. proposition 1.7. if (x,τ) is κ-lindelöf, then every κ-completable filter on x has a convergent refining ultrafilter. if κ is an infinite cardinal and every κ-complete filter on a topological space (x,τ) has a convergent refining ultrafilter, then (x,τ) is κ-lindelöf. 284 r. bartsch and h. poppe of course, every κ-complete filter is κ-completable, so we may say, that a topological space (x,τ) is κ-lindelöf, if and only if each κ-complete filter on x has a convergent refinement. 2. compactness properties for hyperspaces. lemma 2.1. let κ be a cardinal, (x,τ) a topological space and let a ⊆ p(x) be weakly complementary w.r.t. τ. if a0 := a \{∅} is κ-lindelöf (resp. κ-compact) in τl,a0 , then (x,τ) is κ-lindelöf (resp. κ-compact). proof. if a is weakly complementary w.r.t. τ, then a0 is, too. so, corollary 1.2 is applicable. let {gi|i ∈ i} be an open cover (resp. an open cover with cardinality at most κ) of x. by corollary 1.2, then {g−a0i |i ∈ i} is an open cover of x −a0 = a0 (resp. of card. at most κ), so there exists a subset j ⊆ i of cardinality at most κ (resp. a finite subset j), s.t. ⋃ j∈j g −a0 j ⊇ a0 = x −a0 , implying ⋃ j∈j gj ⊇ x by corollary 1.2. � of course, the assumed topology τl,a0 is not really hit-and-miss, because the miss-sets are missed. but every proper hit-and-miss topology would be stronger and therefore it would enforce the desired properties for (x,τ) as well. lemma 2.2. let (x,τ) be a κ-compact (resp. κ-lindelöf ) topological space and assume cl(x) ⊆ a ⊆ p(x). then a0 := a \{∅} is κ-compact (resp. κ-lindelöf ) in τl,a0 . proof. let ϕ̂ be a κ-generated (resp. κ-complete) filter on a0. then, for an arbitrary h ∈a := {g ∈ xp0(x)| ∀m ∈ p0(x) : g(m) ∈ m} the image h(ϕ̂) is a κ-generated (resp. κ-complete) filter on x and consequently it has a τ-convergent refining ultrafilter ψh. furthermore, there must exist an ultrafilter ψ̂ ⊇ ϕ̂, s.t. h(ψ̂) = ψh. so, the set a := {a ∈ x| ∃f ∈a : (f(ψ̂),a) ∈ qτ} is not empty and consequently the closure a belongs to a0. now, for any o ∈ τ with a ∈ o−a0 (⇔ a ∩ o 6= ∅) we get a ∩ o 6= ∅ (because of the closureproperties). now, the assumption o−a0 6∈ ψ̂ would imply o+a0 ∈ ψ̂, yielding ∀f ∈a : x \o ∈ f(ψ̂), thus ∀f ∈a : ∀b ∈ a∩o : (f(ψ̂),b) 6∈ qτ in contradiction to the construction of a. thus, o ∈ τ,a ∈ o−a0 always imply o−a0 ∈ ψ̂ and consequently ψ̂ τl,a0 -converges to a. � definition 2.3. let (x,τ) be a topological space. a subset a ⊆ x is called weakly relatively complete in x, iff ∀ϕ ∈ f(a) ∩q−1τ (x) : f(ϕ) ∩q −1 τ (a) 6= ∅ , i.e. every filter ϕ on a, which converges in x, has a refinement, converging in a. proposition 2.4. let (x,τ) be a topological space and a ⊆ x. then holds: (a) a is weakly relatively complete in x, iff f0(a)∩q−1τ (x) = f0(a)∩q−1τ (a), i.e. every ultrafilter on a, which converges in x, converges in a. (b) if a is closed in x, then a is weakly relatively complete in x. (c) if a is compact, then a is weakly relatively complete in x. a short note on hit-and-miss hyperspaces 285 (d) if (x,τ) is compact and a is weakly relatively complete in x, then a is compact. (e) if (x,τ) is hausdorff, then every weakly relatively complete subset a ⊆ x is closed in (x,τ). (f) a is compact iff a is weakly relatively complete and relatively compact. (g) if (x,τ) is κ-compact and a is weakly relatively complete in (x,τ), then a is κ-compact. (h) if (x,τ) is κ-lindelöf and a is weakly relatively complete in (x,τ), then a is κ-lindelöf. (i) weak relative completeness is transitive, i.e. for all a ⊆ b ⊆ x with b weakly relatively complete in (x,τ) and a weakly relatively complete in (b,τ|b), the subset a is weakly relatively complete in (x,τ). there is also a useful description by coverings for weak relative completeness. lemma 2.5. let (x,τ) be a topological space and a ⊆ x. then the following are equivalent: (1) a is weakly relatively complete in x. (2) for every open cover a of a and every element x of x, there is an open neighbourhood ux,a of x, s.t. ux,a ∩a is covered by finitely many members of a. (3) for every open cover a of a there exists an open cover a′ ⊇ a of x, such that the intersection of every member of a′ with a can be covered by finitely many members of a, i.e. ∀o ∈ a′ : ∃n ∈ in,p1, ...,pn ∈ a : ⋃n i=1 pi ⊇ o ∩a holds. proof. (1)⇒(2): let a ⊆ τ with ⋃ p∈a p ⊇ a be given. for every x ∈ a we can chose a single member of a as open neighbourhood, whose intersection with a is covered by itself. so, assume (2.2) ∃x ∈ x \a : ∀ux ∈ u(x) ∩ τ : ∀n ∈ in,p1, ...,pn ∈ a : ux ∩a 6⊆ n⋃ i=1 pi then b := {(u∩a)\ ⋃n i=1 pi| u ∈ u(x)∩τ,n ∈ in,pi ∈ a} would be closed under finite intersections and thus there would exist an ultrafilter ϕ on a with ϕ ⊇ b. by construction ϕ → x must hold for this ultrafilter, and now by the weak relative completeness of a it follows ∃a ∈ a : u(a) ⊆ ϕ. but a is an open cover of a, so there is an open set p ∈ a with a ∈ p, implying p ∈ ϕ – in contradiction to the construction of ϕ. thus (2.2) is false and we have ∀x ∈ x \a : ∃ux ∈ u(x) ∩ τ : ∃n ∈ in,p1, ...,pn ∈ a : ux ∩a ⊆ n⋃ i=1 pi (2)⇒(3): note, that (3) is fulfilled with a′ := {ux| x ∈ x \a}∪ a. (3)⇒(1): for a given ultrafilter ϕ on a with ϕ → x ∈ x assume ϕ 6∈ q−1τ (a). then ∀a ∈ a : ∃ua ∈ u(a) ∩ τ : uca = x \ ua ∈ ϕ. with these neighbourhoods define a := {ua| a ∈ a}, which is an open cover of a. by (2) there is an open cover a′ ⊇ a of x such that ∀o ∈ a′ : ∃n ∈ in,p1, ...,pn ∈ a : ⋃n i=1 pi ⊇ o ∩a holds. now, ϕ → x implies ∃o ∈ a′ : o ∈ ϕ (especially a∩o 6= ∅ follows), and then we have ∃n ∈ in,p1, ...,pn ∈ a : o ∩ a ⊆ ⋃n i=1 pi, implying ∃j ∈ {1, ...,n} : pj ∈ ϕ 286 r. bartsch and h. poppe – in contradiction to the construction of a. so, the assumption ϕ 6∈ q−1τ (a) must be false, showing, that every ultrafilter on a, which converges in x, converges in a. � theorem 2.6. let (x,τ) be a topological space, and let α ⊆ p(x) consist of weakly relatively complete subsets of x. then holds for any a with cl(x) ⊆ a ⊆ p(x): (a0,τα) is compact ⇐⇒ (x,τ) is compact. proof. according to lemma 2.1 we need only to show that (a0,τα) is compact, if (x,τ) is compact. so, assuming (x,τ) to be compact, by proposition 2.4 every weakly relatively complete subset of x is compact, and we have α ⊆ k(x). now we will use alexander’s lemma: let u be a cover of a0, consisting of subbase elements k +a0 i ,g −a0 j with ki compact and gj open. a := x \ ( ⋃ {g|g−a0 ∈ u}) is closed. by construction, a 6∈ g−a0 for any g−a0 ∈ u, so for a 6= ∅ there must exist some k +a0 0 ∈ u with a ∈ k +a0 0 , yielding that k0 ⊆ ⋃ {g|g−a0 ∈ u}; k0 compact ⇒ ∃g1, ...,gn ∈ u with k0 ⊆ ⋃n k=1 gk, but then {k +a0 0 }∪{g −a0 1 , ...,g −a0 n } is a cover of a0. if a = ∅, then ⋃ {gi|g −a0 i ∈ u} = x, so from the compactness of x the existence of some g −a0 1 , ...,g −a0 n ∈ u with x = ⋃n k=1 gk follows. by lemma 1.1 then⋃n k=1 g −a0 k = a0 holds. � many known theorems of compactness w.r.t. the fell– or the vietoris–topology follow immediately from the above result. lemma 2.7. let (x,τ) be a topological space, a ⊆ p0(x) with cl(x) ⊆ a and α ⊆ cl(x). if r ⊆ x is relatively compact in x, then p0(r) ∩ a is relatively compact in (a,τα). proof. let b := {o−ai | i ∈ i,oi ∈ τ}∪{c +a j | j ∈ j,cj ∈ α} be an open cover of a by subbase elements of τα. let o := ⋃ i∈i oi. if o = x, then there exists finitely many i1, ..., in ∈ i with ⋃n k=1 oik ⊇ r, because r is relatively compact, and thus ⋃n k=1 o −a ik ⊇ r−a ⊇ p0(r) ∩ a, by lemma 1.1. if o 6= x, then x \o is nonempty and closed, but not covered by the o−ai from b. thus, there must exist a j0 ∈ j with x\o ∈ c+aj0 , implying cj0 ⊆ o. now, we have p0(r)∩a = (p0(r)∩c+aj0 )∪(p0(r)∩c −a j0 ), and, of course, p0(r)∩c+aj0 is covered just by c+aj0 ∈ b. so, we have to find a finite subcover for (p0(r)∩c −a j0 ), if this is not empty. observe, that r∩cj0 is relatively compact in x, because it is a subset of r. furthermore, {oi| i ∈ i}∪{x\cj0} is an open cover of x. thus we find again finitely many i1, .., in ∈ i, s.t. ⋃n k=1 oik ⊇ r∩cj0 (because x\cj0 can be removed from any cover of r∩cj0 without loosing the covering property). therefore⋃n k=1 o −a ik ⊇ (r∩cj0 )−a , by lemma 1.1. but p0(r)∩c −a j0 ⊆ (r∩cj0 )−a holds, because any subset of r, which hits cj0 , automatically hits r∩cj0 . � 3. compact unions. as an interesting application of a simple set-theoretical property, concerning the +-operator, we want to take a brief look at the naturally arising question, whether a union of compact sets itself is compact. michael showed in [6] that a union of a short note on hit-and-miss hyperspaces 287 closed sets is compact, if the unifying family is compact w.r.t. the vietoris-topology. now, the vietoris-topology is induced by the upper-vietoris τ+v (miss sets: a +a with ac ∈ τ) and τl, but τl is not sufficient to enforce compactness of a union of compact sets, as the following example shows: let x := ir, endowed with euclidian topology, m := {[−m,m]| m ∈ in}. then ⋃ m∈m m = ir, is obviously not compact. but every cover of m with elements of the defining subbase for τl must especially cover the element {0} = [0, 0] of m, so it must contain a set o− with 0 ∈ o. now, every element of m contains the point 0, thus m ⊆ o− follows. so, m is compact in τl by alexander’s subbase lemma. and unifying compact sets, τl is not necessary, too, as we will see. proposition 3.1. let x be a set, x ⊆ p(x) and m ⊆ x. then holds⋃ i∈i c +x i ⊇ m =⇒ ⋃ i∈i cci ⊇ ⋃ m∈m m for every collection ci, i ∈ i. proof. for every m ∈ m there must exist an im ∈ i with m ∈ c +x im , because of⋃ i∈i c +x i ⊇ m. thus m ⊆ c c im ⊆ ⋃ i∈i c c i . � in [5] it was shown lemma 3.2. let (x,τ) be a topological space and m ⊆ k(x) compact w.r.t. the upper–vietoris topology. then k := ⋃ m∈m m is compact w.r.t. τ. applying our simple set-theoretical statement, we get a similar result for unions of relatively compact subsets. lemma 3.3. let (x,τ) be a topological space, let x be the family of all relatively compact subsets of x and let m ⊆ x be relatively compact in x w.r.t. the upper vietoris topology. then r := ⋃ m∈m m is relatively compact in (x,τ). proof. let ⋃ i∈i oi ⊇ x with oi ∈ τ,i ∈ i an open cover of x. because of the relative compactness of all p ∈ x, there is a finite subcover oi1 p , ...,oinp p for every p ∈ x, i.e. op := ⋃np k=1 oikp ⊇ m. of course, op ∈ τ and so (op )c is closed w.r.t. τ. furthermore, p ∩ocp = ∅, implying p ∈ (o c p ) +x . thus we have x ⊆ ⋃ p∈x(o c p ) +x , where the (ocp ) +x are open w.r.t. the upper–vietoris topology. because of the relative compactness of x w.r.t. the upper–vietoris topology, there must exist finitely many p1, ...,pn ∈ x with m ⊆ ⋃n j=1(o c pj )+x . now, from proposition 3.1 we get r = ⋃ m∈m m ⊆ ⋃n j=1 opj , where every opj is a finite union of members of the original cover {oi|i ∈ i} by construction. � 288 r. bartsch and h. poppe corollary 3.4. let (x,τ) be a topological space and let m ⊆ p0(x) consist of relatively compact subsets of x. if m is compact w.r.t. the upper–vietoris topology, then r := ⋃ m∈m m is relatively compact in (x,τ). proof. m is compact and therefore relatively compact in every set, which contains m, especially in the family of all relatively compact subsets of x. so, lemma 3.3 applies. � references [1] bartsch, r., dencker, p., poppe, h., ascoli–arzelà–theory based on continuous convergence in an (almost) non–hausdorff setting, in ”categorical topology”; dordrecht (1996). [2] beer, g., tamaki, r.,on hit-and-miss hyperspace topologies, commentat. math. univ. carol. 34 (1993), no.4, 717-728. [3] beer, g., tamaki, r., the infimal value functional and the uniformization of hit–and–miss hyperspace topologies, proc. am. math. soc. 122, no.2 (1994), 601-612. [4] comfort, w.w., negrepontis, s., the theory of ultrafilters, berlin (1974). [5] klein, e., thompson, a.c., theory of correspondences. including applications to mathematical economics. canadian mathematical society series of monographs and advanced texts (1984). [6] michael, e., topologies on spaces of subsets, trans. amer. math. soc. 71 (1951), 152–182. [7] naimpally, s., hyperspaces and function spaces, q & a in general topology 9 (1991), 33-60. [8] naimpally, s., all hypertopologies are hit-and-miss, appl. gen. topol. 3, no.1 (2001), 45-53. [9] poppe, h., eine bemerkung über trennungsaxiome in räumen von abgeschlossenen teilmengen topologischer räume, arch.math. 16 (1965), 197–199. [10] poppe, h., einige bemerkungen über den raum der abgeschlossenen mengen, fund.math. 59 (1966), 159–169. [11] poppe, h., compactness in general function spaces, berlin (1974). received november 2001 revised september 2002 rené bartsch dept. of computer science, rostock university, albert-einstein-straße 21, 18059 rostock, germany e-mail address : rene.bartsch@informatik.uni-rostock.de harry poppe dept. of mathematics, rostock university, universitätsplatz 1, 18055 rostock, germany e-mail address : harry.poppe@mathematik.uni-rostock.de @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 255–261 closure properties of function spaces ljubǐsa d.r. kočinac ∗ dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. in this paper we investigate some closure properties of the space ck(x) of continuous real-valued functions on a tychonoff space x endowed with the compact-open topology. 2000 ams classification: 54a25, 54c35, 54d20, 91a44. keywords: menger property, rothberger property, hurewicz property, reznichenko property, k-cover, countable fan tightness, countable strong fan tightness, t-tightness, groupability, ck(x), selection principles, game theory. 1. introduction. all spaces in this paper are assumed to be tychonoff. our notation and terminology are standard, mainly the same as in [2]. some notions will be defined when we need them. by ck(x) we denote the space of continuous real-valued functions on a space x in the compact-open topology. basic open sets of ck(x) are of the form w(k1, . . . ,kn; v1, . . . ,vn) := {f ∈ c(x) : f(ki) ⊂ vi, i = 1, . . . ,n}, where k1, . . . ,kn are compact subsets of x and v1 . . . ,vn are open in r. for a function f ∈ ck(x), a compact set k ⊂ x and a positive real number ε we let w(f; k; ε) := {g ∈ ck(x) : |g(x) −f(x)| < ε, ∀x ∈ k}. the standard local base of a point f ∈ ck(x) consists of the sets w(f; k; ε), where k is a compact subset of x and ε is a positive real number. the symbol 0 denotes the constantly zero function in ck(x). the space ck(x) is homogeneous so that we may consider the point 0 when studying local properties of ck(x). many results in the literature show that for a tychonoff space x closure properties of the function space cp(x) of continuous real-valued functions on ∗supported by the serbian mstd, grant 1233 256 ljubǐsa d.r. kočinac x endowed with the topology of pointwise convergence can be characterized by covering properties of x. we list here some properties which are expressed in this manner. (a) [arhangel’skii–pytkeev] cp(x) has countable tightness if and only if all finite powers of x have the lindelöf property ([1]). (b) [arhangel’skii] cp(x) has countable fan tightness if and only if all finite powers of x have the menger property ([1]). (c) [sakai] cp(x) has countable strong fan tightness if and only if all finite powers of x have the rothberger property ([13]). (d) [kočinac–scheepers] cp(x) has countable fan tightness and the reznichenko property if and only if all finite powers of x have the hurewicz property ([7]). in this paper we consider these properties in the context of spaces ck(x). to get analogues for the compact-open topology one need modify the role of covers, preferably ω-covers from cp-theory should be replaced by k-covers in ck-theory. recall that an open cover u of x is called a k-cover if for each compact set k ⊂ x there is a u ∈ u such that k ⊂ u. the symbol k denotes the collection of all k-covers of a space. let us mention one result of this sort which should be compared with the arhangel’skii-pytkeev theorem (a) above. in [9] it was remarked (without proof) that the following holds: theorem 1.1. a space ck(x) has countable tightness if and only if for each k-cover u of x there is a countable set v ⊂u which is a k-cover of x. the proof of this result can be found in theorem 3.13 of [11]. 1.1. selection principles and games. in this paper we shall need selection principles of the following two sorts: let s be an infinite set and let a and b both be sets whose members are families of subsets of s. then s1(a,b) denotes the selection principle: for each sequence (an : n ∈ n) of elements of a there is a sequence (bn : n ∈ n) such that for each n bn ∈ an, and {bn : n ∈ n} is an element of b. sfin(a,b) denotes the selection principle: for each sequence (an : n ∈ n) of elements of a there is a sequence (bn : n ∈ n) of finite sets such that for each n bn ⊂ an, and ⋃ n∈n bn is an element of b. for a topological space x, let o denote the collection of open covers of x. then the property s1(o,o) is called the rothberger property [12],[5], and the property sfin(o,o) is known as the menger property [10], [3],[5]. there is a natural game for two players, one and two, denoted gfin(a,b), associated with sfin(a,b). this game is played as follows: there is a round per positive integer. in the n-th round one chooses an an ∈ a, and two closure properties of function spaces 257 responds with a finite set bn ⊂ an. a play a1,b1; · · · ; an,bn; · · · is won by two if ⋃ n∈n bn is an element of b; otherwise, one wins. similarly, one defines the game g1(a,b), associated with s1(a,b). 2. countable (strong) fan tightness of ck(x). for x a space and a point x ∈ x the symbol ωx denotes the set {a ⊂ x \{x} : x ∈ a}. a space x has countable fan tightness [1] if for each x ∈ x and each sequence (an : n ∈ n) of elements of ωx there is a sequence (bn : n ∈ n) of finite sets such that for each n bn ⊂ an and x ∈ ⋃ n∈n bn, i.e. if sfin(ωx, ωx) holds for each x ∈ x. a space x has countable strong fan tightness [13] if for each x ∈ x the selection principle s1(ωx, ωx) holds. in [8], it was shown (compare with the corresponding theorem (b) of arhangel’skii for cp(x)): theorem 2.1. the space ck(x) has countable fan tightness if and only if x has property sfin(k,k). we show theorem 2.2. for a space x the following are equivalent: (a) ck(x) has countable strong fan tightness; (b) x has property s1(k,k). proof. (a) ⇒ (b): let (un : n ∈ n) be a sequence of k-covers of x. for a fixed n ∈ n and a compact subset k of x let un,k := {u ∈un : k ⊂ u}. for each u ∈ un,k let fk,u be a continuous function from x into [0,1] such that fk,u (k) = {0} and fk,u (x \u) = {1}. let for each n, an = {fk,u : k compact in x,u ∈un,k}. then, as it is easily verified, 0 is in the closure of an for each n ∈ n. since ck(x) has countable strong fan tightness there is a sequence (fkn,un : n ∈ n) such that for each n, fkn,un ∈ an and 0 ∈{fkn,un : n ∈ n}. consider the sets un, n ∈ n. we claim that the sequence (un : n ∈ n) witnesses that x has property s1(k,k). let c be a compact subset of x. from 0 ∈{fkn,un : n ∈ n} it follows that there is i ∈ n such that w = w(0; c; 1) contains the function fki,ui. then c ⊂ ui. otherwise, for some x ∈ c one has x /∈ ui so that fki,ui(x) = 1, which contradicts the fact fki,ui ∈ w . (b) ⇒ (a): let (an : n ∈ n) be a sequence of subsets of ck(x) the closures of which contain 0. fix n. for every compact set k ⊂ x the neighborhood w = w(0; k; 1/n) of 0 intersects an so that there exists a function fk,n ∈ an such that |fk,n(x)| < 1/n for each x ∈ k. since fk,n is a continuous function there are neighborhoods ox, x ∈ k, such that for uk,n = ⋃ x∈kox ⊃ k we have fk,n(uk,n) ⊂ (−1/n, 1/n). let un = {uk,n : k a compact subset of x}. for each n ∈ n, un is a k-cover of x. apply that x is an s1(k,k)-set: for each 258 ljubǐsa d.r. kočinac m ∈ n there exists a sequence (uk,n : n ≥ m) such that for each n, uk,n ∈un and {uk,n : n ≥ m} is a k-cover for x. look at the corresponding functions fk,n in an. let us show 0 ∈ {fk,n : n ∈ n}. let w = w(0; c; ε) be a neighborhood of 0 in ck(x) and let m be a natural number such that 1/m < ε. since c is a compact subset of x and x is an s1(k,k)-set, there is j ∈ n, j ≥ m such that one can find a uk,j with c ⊂ uk,j. we have fk,j(c) ⊂ fk,j(uk,j) ⊂ (−1/j, 1/j) ⊂ (−1/m, 1/m) ⊂ (−ε,ε), i.e. fk,j ∈ w . � 3. countable t-tightness of ck(x). the notion of t-tightness was introduced by i. juhász at the iv international conference “topology and its applications”, dubrovnik, september 30 – october 5, 1985 (see [4]). a space x has countable t -tightness if for each uncountable regular cardinal ρ and each increasing sequence (aα : α < ρ) of closed subsets of x the set ∪{aα : α < ρ} is closed. in [14], the t-tightness of cp(x) was characterized by a covering property of x. the next theorem is an analogue of this result in the ck(x) context. theorem 3.1. for a space x the following are equivalent: (a) ck(x) has countable t -tightness; (b) for each regular cardinal ρ and each increasing sequence (uα : α < ρ) of families of open subsets of x such that ⋃ α<ρuα is a k-cover of x there is a β < ρ so that uβ is a k-cover of x. proof. (a) ⇒ (b): let ρ be a regular uncountable cardinal and let (uα : α < ρ) be an increasing sequence of families of open subsets of x such that ⋃ α<ρuα is a k-cover for x. for each α < ρ and each compact set k ⊂ x let uα,k := {u ∈uα : k ⊂ u}. for each u ∈uα,k let fk,u be a continuous function from x into [0,1] such that fk,u (k) = {0} and fk,u (x \u) = {1}. for each α < ρ put aα = {fk,u : u ∈uα,k}. since the t-tightness of ck(x) is countable, the set a = ⋃ α<ρ aα is closed in ck(x). let w(0; k; ε) be a standard basic neighborhood of 0. there is an α < ρ such that for some u ∈uα, k ⊂ u. then u ∈uα,k and thus there is f ∈ aα, hence f ∈ aα ∩w(0; k; ε). therefore, each neighborhood of 0 intersects some of the sets aα, α < ρ, which means that 0 belongs to the closure of the set⋃ α<ρ aα. since this set is actually the set a it follows there exists a β < ρ with 0 ∈ aβ. we claim that the corresponding family uβ is a k-cover of x. let c be a compact subset of x. then the neighborhood w(0; c; 1) of 0 intersects aβ; let fk,u ∈ aβ ∩w(0; c; 1). by the definition of aβ, then from fk,u (x \u) = 1 it follows c ⊂ u ∈uβ. closure properties of function spaces 259 (b) ⇒ (a): let (aα : α < ρ) be an increasing sequence of closed subsets of ck(x), with ρ a regular uncountable cardinal. we shall prove that the set a := ⋃ α<ρ aα is closed. let f ∈ a. for each n ∈ n and each compact set k ⊂ x the neighborhood w(f; k; 1/n) of f intersects a. put un,α = {g←(−1/n, 1/n) : g ∈ aα} and un = ⋃ α<ρ un,α. let us check that for each n ∈ n, un is a k-cover of x. let k be a compact subset of x. the neighborhood w := w(f; k; 1/n) of f intersects a, i.e. there is g ∈ a such that |f(x) − g(x)| < 1/n for all x ∈ k; this means k ⊂ g←(−1/n, 1/n) ∈un. by (b) there is un,βn ⊂un which is a k-cover of x. put β0 = sup{βn : n ∈ n}. since ρ is a regular cardinal, β0 < ρ. it is easy to verify that for each n the set un,β0 is a k-cover of x. let us show that f ∈ aβ0 . take a neighborhood w(f; c; ε) of f and let m be a positive integer such that 1/m < ε. since um,β0 is a k-cover of x one can find g ∈ aβ0 such that c ⊂ g←(−1/m, 1/m). then g ∈ w(f; c; 1/m) ∩ aβ0 ⊂ w(f; c; ε) ∩ aβ0 , i.e. f ∈ aβ0 = aβ0 and thus f ∈ a. so, a is closed. � 4. the reznichenko property of ck(x). in this section we shall need the notion of groupability (see [7]). 1. a k-cover u of a space x is groupable if there is a partition (un : n ∈ n) of u into pairwise disjoint finite sets such that: for each compact subset k of x, for all but finitely many n, there is a u ∈un such that k ⊂ u. 2. an element a of ωx is groupable if there is a partition (an : n ∈ n) of a into pairwise disjoint finite sets such that each neighborhood of x has nonempty intersection with all but finitely many of the an. we use the following notation: • kgp – the collection of all groupable k-covers of a space; • ωgpx – the family of all groupable elements of ωx. in 1996 reznichenko introduced (in a seminar at moscow state university) the following property: each countable element of ωx is a member of ωgpx . this property was further studied in [6] and [7] (see introduction). in [6] it was called the reznichenko property at x. when x has the reznichenko property at each of its points, then x is said to have the reznichenko property. we study now the reznichenko property in spaces ck(x). theorem 4.1. let x be a tychonoff space. if one has no winning strategy in the game g1(k,kgp), then ck(x) has property s1(ω0, ω gp 0 ) (i.e. ck(x) has countable strong fan tightness and the reznichenko property). 260 ljubǐsa d.r. kočinac proof. evidently, from the fact that one has no winning strategy in the game g1(k,kgp), it follows that x satisfies s1(k,kgr) and consequently x is in the class s1(k,k). by theorem 2.2 ck(x) has countable strong fan tightness, and thus countable tightness. therefore, it remains to prove that each countable subset of ω0 is groupable (i.e. that ck(x) has the reznichenko property). let a be a countable subset of ck(x) such that 0 ∈ a. we define the following strategy σ for one in g1(k,kgp). for a compact set k ⊂ x the neighborhood w = w(0; k; 1) of 0 intersects a. let fk ∈ a ∩ w . as fk is continuous, for every x ∈ k there is a neighborhood ox of x such that fk(ox) ⊂ (−1, 1). from the open cover {ox : x ∈ k} of k choose a finite subcover {ox1, . . . ,oxm} and let uk = ox1∪·· ·∪oxm. then fk(uk) ⊂ (−1, 1) and the set u1 = {uk : k a compact subset of x} is a k-cover of x. one’s first move, σ(∅), will be u1. let two’s response be an element uk1 ∈ u1. one considers now the corresponding function fk1 ∈ a (satisfying fk1 (uk1 ) ⊂ (−1, 1)) and looks at the set a1 = a \ {fk1} which obviously satisfies 0 ∈ a1. for every compact subset k of x one chooses a function fk ∈ a ∩ w(0; k; 1/2) and a neighborhood uk of k such that fk(uk) ⊂ (−1/2, 1/2). the set u2 = {uk : k a compact subset of x}\{uk1} is a k-cover of x. one plays σ(uk1 ) = u2. suppose that uk2 ∈ u2 is two’s response. one first considers the function fk2 ∈ a1 with fk2 (uk2 ) ⊂ (−1/2, 1/2) and then looks at the set a2 = a1 \{fk2} the closure of which contains 0, and so on. the strategy σ, by definition, gives sequences (un : n ∈ n), (un : n ∈ n) and (fn : n ∈ n) having the following properties: (i) (un : n ∈ n) is a sequence of k-covers of x and for each n un = σ(u1, . . . ,un−1); (ii) for each n, un ∈un and un /∈{u1, . . . ,un−1)}; (iii) for each n, fn is a member of a\{f1, . . . ,fn−1); (iv) for each n, fn(un) ⊂ (−1/n, 1/n). since σ is not a winning strategy for one, the play u1,u1; . . . ;un,un; . . . is lost by one so that v := {un : n ∈ n} is a groupable k-cover of x. therefore there is an increasing infinite sequence n1 < n2 < · · · < nk < ... such that the sets hk := {ui : nk ≤ i < nk+1}, k = 1, 2, . . . , are pairwise disjoint and for every compact set k ⊂ x there is k0 such that for each k > k0 there is h ∈hk with k ⊂ h. define also mk := {fi : nk ≤ i < nk+1}. then the sets mk are finite pairwise disjoint subsets of a. one can also suppose that a = ⋃ k∈n mk; otherwise we distribute countably many elements of a\ ⋃ k∈n mk among mk’s so that after distribution new sets are still finite and pairwise disjoint. we claim that the sequence (mk : k ∈ n) witnesses that a is groupable (i.e. that ck(x) has the reznichenko property). let w(0; k; ε) be a neighborhood of 0 and let m be the smallest natural number such that 1/m < ε. there is n0 such that for each n > n0 one can choose an element hn ∈hn with k ⊂ hn; choose also a corresponding function fn ∈ mn satisfying fn(hn) ⊂ (−1/n, 1/n). so for each n > max{n0,m} we closure properties of function spaces 261 have fn ∈ mn∩w(0; k; ε), i.e for all but finitely many n w (0; k; ε)∩mn 6= ∅. the theorem is shown. � in a similar way one can prove theorem 4.2. for a space x the statement (a) below implies the statement (b): (a) one has no winning strategy in the game gfin(k,kgp); (b) ck(x) has countable fan tightness and the reznichenko property (i.e. ck(x) has property sfin(ω0, ω gp 0 )). problem 4.3. is the converse in theorem 4.1 and in theorem 4.2 true? references [1] a.v. arhangel’skǐi, topological function spaces (kluwer academic publishers, 1992). 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[14] m. sakai, variations on tightness in function spaces, topology appl. 101 (2000), 273–280. received november 2001 revised november 2002 ljubǐsa d.r. kočinac faculty of sciences, university of nǐs, 18000 nǐs, serbia e-mail address : lkocinac@ptt.yu 19.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 63 75 a contribution to the study of fuzzy metricspacesalmanzor sapenaabstract. we give some examples and properties of fuzzymetric spaces, in the sense of george and veeramani, and charac-terize the t0 topological spaces which admit a compatible unifor-mity that has a countable transitive base, in terms of the fuzzytheory.2000 ams classi�cation: 54a40keywords: fuzzy metric, precompact, strongly zero-dimensional.1. introductionone of the main problems in the theory of fuzzy topological spaces is to obtainan appropriate and consistent notion of a fuzzy metric space. many authorshave investigated this question and several notions of a fuzzy metric space havebeen de�ned and studied. in particular, and modifying the concept of metricfuzziness introduced by kramosil and michalek [9] (which is a generalizationof the concept of probabilistic metric space introduced by k. menger [10] tothe fuzzy setting), george and veeramani [4, 5], have studied a notion of fuzzymetric space. in a previous paper [7], gregori and romaguera proved that theclass of fuzzy metric spaces, in george and veeramani's sense, coincides withthe class of metric spaces. in the light of the results obtained in [7], we thinkthat the george and veeramani's de�nition is an appropriate notion of metricfuzziness in the sense that it provides rich fuzzy topological structures which canbe obtained, in many cases, from classical theorems. on the other hand, metricspaces can be studied from the point of view of fuzzy theory. unfortunately,not much examples of such spaces have been given. in this paper we give newexamples of fuzzy metric spaces and study some properties of these spaces.the structure of the paper is as follows. after preliminaries, in section 3,we construct new fuzzy metrics from a given one, and study some questionsrelative to boundedness. in section 4 we give new examples of fuzzy metrics.in section 5 we study a property of cauchy sequences in standard fuzzy metricspaces, and �nally, in section 6, we de�ne the concept of non-archimedean 64 a. sapenafuzzy metric space and prove that the family of these spaces agrees with theclass of non-archimedean metric spaces, so it provides a characterization of thet0 topological spaces which admit a compatible uniformity, that has a countabletransitive base, in the fuzzy setting.2. preliminariesthroughout this paper the letters n and r will denote the set of all posi-tive integers and real numbers, respectively. our basic reference for generaltopology is [2].according to [11] a binary operation � : [0;1]�[0;1] �! [0;1] is a continuoust-norm if � satis�es the following conditions:(i) � is associative and commutative(ii) � is continuous(iii) a � 1 = a for every a 2 [0;1](iv) a � b � c � d whenever a � c and b � c, for all a;b;c;d 2 [0;1]according to [4],[5], a fuzzy metric space is an ordered triple (x;m;�) suchthat x is a non-empty set, � is a continuous t-norm and m is a fuzzy set ofx � x�]0;+1[ satisfying the following conditions, for all x;y;z 2 x, s;t > 0:(i) m(x;y;t) > 0(ii) m(x;y;t) = 1 if and only if x = y(iii) m(x;y;t) = m(y;x;t)(iv) m(x;y;t) � m(y;z;s) � m(x;z;t + s) (triangular inequality)(v) m(x;y; �) : ]0;+1[�! [0;1] is continuous.if (x;m;�) is a fuzzy metric space, we will say that (m;�), or m (if it is notnecessary to mention �), is a fuzzy metric on x.lemma 2.1. [6] m(x;y; �) is nondecreasing for all x;y 2 x.lemma 2.2. [1] let (x;m;�) be a fuzzy metric space.(i) if m(x;y;t) > 1 � r for x;y 2 x, t > 0, 0 < r < 1, we can �nd a t0,0 < t0 < t such that m(x;r;t0) > 1 � r.(ii) for any r1 > r2, we can �nd a r3 such that r1 � r3 � r2, and for anyr4 we can �nd a r5 such that r5 � r5 � r4, (r1; r2; r3; r4; r5 2]0;1[).let (x;d) be a metric space. de�ne a � b = ab for every a;b 2 [0;1]; and letmd be the function on x � x�]0;+1[ de�ned bymd(x;y;t) = tt + d(x;y)then (x;md;�) is a fuzzy metric space, and md is called the standard fuzzymetric induced by d (see [4]).george and veeramani proved that every fuzzy metric m on x generates ahausdor� topology �m on x which has as a base the family of open sets of theform: fbm(x;r;t) : x 2 x;0 < r < 1; t > 0g a contribution to the study of fuzzy metric spaces 65where bm(x;r;t) = fy 2 x : m(x;y;t) > 1 � rgfor every r 2]0;1[, and t > 0. (we will write b(x;r;t) when confusion is notpossible).de�nition 2.3. a sequence fxng in a fuzzy metric space (x;m;�) is calleda cauchy sequence [5], if for each " > 0, t > 0 there exists n0 2 n such thatm(xn;xm; t) > 1 � ", for all m;n � n0.a subset a of x is said to be f-bounded if there exist t > 0 and r 2]0;1[such that m(x;y;t) > 1 � r for all x;y 2 a.proposition 2.4 ([4]). if (x;d) is a metric space, then:(i) the topology �d on x generated by d coincides with the topology �mdgenerated by the standard fuzzy metric md.(ii) fxng is a d�cauchy sequence (i.e., a cauchy sequence in (x;d)) ifand only if it is a cauchy sequence in (x;md;�).(iii) a � x is bounded in (x;d) if and only if it is f-bounded in (x;md;�).we say that a topological space (x;�) is fuzzy metrizable if there exists afuzzy metric m on x such that � = �m. in [7] it is proved that a topologicalspace is fuzzy metrizable if and only if it is metrizable.unless explicit mention we will suppose r endowed with the usual topology.3. some properties of fuzzy metric spacesfrom now on we will denote by ti (i = 1;2;3) the following continuoust-norms: t1(x;y) = minfx;ygt2(x;y) = xyt3(x;y) = maxf0;x + y � 1gthe following inequalities are satis�ed:t3(x;y) � t2(x;y) � t1(x;y)and t(x;y) � t1(x;y)for each continuous t-norm t .in consequence the following lemma holds.lemma 3.1. let x be a non-empty set. if (m;t) is a fuzzy metric on x andt 0 is a continuous t-norm such that t 0 � t, then (m;t 0) is a fuzzy metric onx.next two properties give methods for constructing f-bounded fuzzy metricsfrom a given fuzzy metric. 66 a. sapenaproposition 3.2. let (x;m;�) be a fuzzy metric space and k 2]0;1[. de�nen(x;y;t) = maxfm(x;y;t);kg; for each x;y 2 x, t > 0:then (n;�) is an f-bounded fuzzy metric on x, which generates the sametopology that m.proof. it is straightforward. �proposition 3.3. let i 2 f1;2;3g and k > 0. suppose that (x;m;ti) is afuzzy metric space, and de�ne:n(x;y;t) = k + m(x;y;t)1 + k for all x;y 2 x;t > 0:then, (n;ti) is an f-bounded fuzzy metric on x, which generates the sametopology that m.proof. we prove this proposition for the case i = 2. for seeing that (n;ti) isa fuzzy metric on x, we only show the triangular inequality.now, it is an easy exercise to verify that the following relationk + a1 + k � k + b1 + k � k + ab1 + kholds, for all a;b 2 [0;1].therefore,k + m(x;y;t)1 + k � k + m(y;z;s)1 + k � k + m(x;y;t) � m(y;z;s)1 + k� k + m(x;z;t + s)1 + kclearly k1+k is a lower bound of n(x;y;t), for all x;y 2 x, t > 0.finally, for t > 0;r 2]0;1[ it is satis�ed thatbm(x;r;t) = bn(x; r1 + k;t)and bn(x;r;t) = bm(x;r(k + 1); t);and so �m = �n.the cases i = 1;3 are left as simple exercises. �problem 3.4. if (m;�) is a fuzzy metric on x and k > 0; then, is�k + m(x;y;t)1 + k ;��a fuzzy metric on x?proposition 3.5. let (m1;�) and (m2;�) be two fuzzy metrics on x. de�ne:m(x;y;t) = m1(x;y;t) � m2(x;y;t)n(x;y;t) = minfm1(x;y;t);m2(x;y;t)gthen: a contribution to the study of fuzzy metric spaces 67(i) (m;�) is a fuzzy metric on x if a � b 6= 0 whenever a;b 6= 0.(ii) (n;�) is a fuzzy metric on x.(iii) the topologies generated by m and n are the same.proof. the proofs of (i) and (ii) are straightforward.(iii) first we will prove that �n < �mlet a 2 �n; then 8x 2 a;9r 2]0;1[ such thatbn(x;r;t) = fy 2 x : n(x;y;t) > 1 � rg � aconsider bm(x;r;t) = fy 2 x : m(x;y;t) > 1 � rg:if z 2 bm(x;r;t); then m(x;z;t) > 1 � r, i.e.,m1(x;z;t) � m2(x;z;t) > 1 � r:notice that m1(x;z;t) � m1(x;z;t) � m2(x;z;t) > 1 � r;and m2(x;z;t) � m1(x;z;t) � m2(x;z;t) > 1 � rso, n(x;z;t) = minfm1(x;z;t);m2(x;z;t)g > 1 � r:then, bm(x;r;t) � bn(x;r;t) � a;thus a 2 �m and hence �n < �m.for seeing that �m < �n, let a 2 �m; then 8x 2 a;9r 2]0;1[ such thatbm(x;r;t) = fy 2 x : m(x;y;t) > 1 � rg � a:let s 2]0;1[ such that (1 � s) � (1 � s) > 1 � r.considerbn(x;s;t) = fy 2 x : n(x;y;t) > 1 � sg= fy 2 x : minfm1(x;y;t);m2(x;y;t)g > 1 � sgif z 2 bn(x;s;t), then m1(x;z;t) > 1 � s and m2(x;z;t) > 1 � s.so, m1(x;z;t) � m2(x;z;t) > (1 � s) � (1 � s)> 1 � r:then bn(x;s;t) � bm(x;r;t) � a, and hence �m < �n. �remark 3.6. if we consider the fuzzy metric (md;�) where d is the usualmetric on r and � is t3, it is easy to verify that m = md � md is not a fuzzymetric on r (compare with 2.10 of [5]).de�nition 3.7. [7] a fuzzy metric space (x;m;�) is called precompact if foreach r 2]0;1[; and t > 0, there exists a �nite subset a of x such that x =sfb(a;r;t) : a 2 ag. in this case, we say that m is a precompact fuzzy metricon x. 68 a. sapenain [7] it is proved that a fuzzy metric space is precompact if and only if everysequence has a cauchy subsequence. using this fact, the proof of the followingproposition is straightforward.proposition 3.8. let (x;d) be a metric space and let md be the standard fuzzymetric deduced from d. then, d is a precompact metric if and only if md is aprecompact fuzzy metric.proposition 3.9. let (x;m;�) be a precompact fuzzy metric space, and sup-pose a � b 6= 0 whenever a;b 6= 0. then, (m;�) is f-bounded.proof. (compare with the end of the proof of [4, theorem 3.9].)let r 2]0;1[ and t > 0. by assumption there is a �nite subset a =fa1; : : : ;ang of x such that x = nsi=1 b(ai;r;t). let� = minfm(ai;aj; t) : i;j = 1; : : : ;ng > 0:let x;y 2 x. then x 2 b(ai;r;t) and y 2 b(aj;r;t) for some i;j 2 f1; : : : ;ng.therefore m(x;ai; t) > 1 � r and m(y;aj; t) > 1 � r. now,m(x;y;3t) � m(x;ai; t) � m(ai;aj; t) � m(aj;y;t)� (1 � r) � � � (1 � r)> 1 � sfor some s 2]0;1[ by the assumption on �, and so m is f-bounded �problem 3.10. is each precompact fuzzy metric space f-bounded?remark 3.11. the converse of the last proposition is false. in fact, the sub-space x of the hilbert metric space (r1;d), formed by the points of unit weight(0; : : : ;0;1;0; : : : ;0), is not precompact and bounded (it has diameter p2), andthen by (iii) of proposition 2.2, (x;md;�) is f-bounded, and by proposition3.8 md is not precompact.4. examples of fuzzy metric spacesin this section we will see examples of fuzzy metrics where the t-norm ist1, and other fuzzy metrics (m;ti);(i = 2;3) which are not fuzzy metrics inconsidering (m;ti�1). before, we need the following lemma.lemma 4.1. let (x;d) be a metric space and s;t > 0. the following inequalityholds, for all n � 1; d(x;z)(t + s)n � max �d(x;y)tn ; d(y;z)sn �proof. we distinguish three cases:(1) d(x;z) � d(x;y)(2) d(x;z) � d(y;z)(3) d(x;z) > d(x;y) and d(x;z) > d(y;z)the inequality chosen is obvious in cases (1) and (2). now, suppose (3) issatis�ed and distinguish two possibilities: a contribution to the study of fuzzy metric spaces 69(3.1) d(x;z) = d(x;y) + d(y;z)(3.2) d(x;z) < d(x;y) + d(y;z)suppose (3.1) is satis�ed. put d(x;y) = �d(x;z) with � 2]0;1[ and henced(y;z) = (1 � �)d(x;z):now, to show the above inequality we have to prove that1(t + s)n � max � �tn ; 1 � �sn � :therefore, consider the functions f(�) = tn� and g(�) = sn1�� which arestrictly decreasing and increasing, respectively. now, the largest value of minntn� ; sn1��ois taken when f(�) = g(�), that is, for � = tntn+sn . then,(t + s)n � tn + sn= f( tntn + sn)� min�tn� ; sn1 � ��and the chosen inequality is stated.the case (3.2) is a consequence of (3.1). �example 4.2. let (x;d) be a metric space, and denote b(x;r) the open ballcentered in x 2 x with radius r > 0.(i) for each n 2 n, (x;m;t1) is a fuzzy metric space where m is given bym(x;y;t) = 1ed(x;y)tn for all x;y 2 x, t > 0;and �m = �(d).(this example when n = 1 has been given in [4].)(ii) for each k;m 2 r+, n � 1, (x;m;t1) is a fuzzy metric space where mis given by m(x;y;t) = ktnktn + md(x;y) for all x;y 2 x;t > 0;and �m = �(d).proof. (i) it is easy to verify that (m;t1) satis�es all conditions of fuzzy metrics;in particular the triangular inequality is a consequence of the previous lemma.now, for x 2 x;r 2]0;1[ and t > 0 we have thatbm(x;r;t) = b(x;�tn ln(1 � r));and b(x;r) = bm(x;1 � 1e rtn ; t);and hence �m = �(d). 70 a. sapena(ii) we will only give a proof of the triangular inequality. indeed, by theprevious lemma1 + md(x;z)k(t + s)n � max �1 + md(x;y)ktn ;1 + md(y;z)ksn �hence k(t + s)nk(t + s)n + md(x;z) � min� ktnktn + md(x;y); ksnksn + md(y;z)� ;and the triangular inequality is stated.now, for x 2 x, t > 0 and r 2]0;1[ we have thatbm(x;r;t) = b(x; ktnrm(1 � r));and b(x;r) = bm(x; mrktn + mr;t);and hence �m = �(d). �remark 4.3. the above expression of m cannot be generalized to n 2 r+(take the usual metric d on r, k = m = 1, n = 1=2). nevertheless it is easy toverify that (m;t2) is a fuzzy metric on x, for n � 0. (compare with 2.9-2.10of [5]).next, we will give fuzzy metrics which cannot be deduced from a metric, inthe sense of last example, since they will not be fuzzy metrics for the t-normt1.example 4.4. let x be the real interval ]0;+1[ and a > 0. it is easy toverify that (x;m;t2) is a fuzzy metric space, where m is de�ned bym(x;y;t) = (�xy�a if x � y�yx�a if y � xfor all x;y 2 x, t > 0.(we notice that this example for x = n and a = 1 was given in [4]).now, for x 2 x, t > 0 and r 2]0;1[, we haveb(x;r;t) = #(1 � r) 1a x; x(1 � r) 1a "and hence b(x;r;t) is an open interval of r, whose diameter converges to zeroas r ! 0. in consequence, �m is the usual topology of r relative to x.finally, (x;m;t1) is not a fuzzy metric space. indeed, for a = 1, if we takex = 1, y = 2 and z = 3, thenm(x;z;t + s) = 13< minf12; 23g= minfm(x;y;t);m(y;z;s)g: a contribution to the study of fuzzy metric spaces 71next, we will give examples of fuzzy metric spaces for the t-norm t3 whichare not for the t-norm t2.example 4.5. let x be the real interval ]1;+1[ and consider the mappingm on x2�]0;+1[ given bym(a;b;t) = 1 � ( 1a ^ b � 1a _ b) for all a;b 2 x;t > 0:we will see that (x;m;t3) is a fuzzy metric space and (x;m;t2) is not.further, the topology �m on x is the usual topology of r relative to x:for seeing that (m;t3) is a fuzzy metric we only prove the triangular in-equality, which becomes (when the left side of the inequality is distinct of zero)(4.1)�1 � � 1a ^ b � 1a _ b�� + �1 � � 1b ^ c � 1b _ c�� 1 � 1 � � 1a ^ c � 1a _ c�for it, �rst, we distinguish 6 cases:(1) suppose a < b < c. in this case, the inequality 4.1 becomes an equality.(2) suppose a < c < b. in this case, the inequality 4.1 becomes:1b + 1b + 1a � 1a + 1c + 1cwhich is true, since 1b < 1c.(3) suppose c < a < b. in this case, the inequality 4.1 becomes:1b + 1b + 1c � 1a + 1c + 1awhich is true, since 1b < 1a.(4) suppose b < a < c. in this case, the inequality 4.1 becomes:1a + 1c + 1a � 1b + 1b + 1cwhich is true, since 1a < 1b.(5) suppose b < c < a. in this case, the inequality 4.1 becomes:1a + 1c + 1c � 1b + 1b + 1awhich is true, since 1c < 1b(6) suppose c < b < a. in this case, the inequality 4.1 becomes an equality.now, if a = b, or a = c, or b = c, the inequality 4.1 is obvious, and thetriangular inequality is stated, so (m;t3) is a fuzzy metric.on the other hand, if we take a = 2, b = 3 and c = 10, thenm(a;b;t) � m(b;c;s) > m(a;c;t + s)and thus, (m;t2) is not a fuzzy metric.finally, if we take x 2 x, r 2]0;1[ with r < 1x, and t > 0, it is easy to verifythat b(x;r;t) = i x1+rx; x1�rxh, then b(x;r;t) is an open interval or r whichdiameter converges to zero as r ! 1, and thus �m is the usual topology of rrelative to x. 72 a. sapenaexample 4.6. let x be the real interval ]2;+1[ and consider the mappingm on x2�]0;+1[ de�ned as followsm(a;b;t) = (1 if a = b1a + 1b if a 6= b, t > 0.it is easy to verify that (x;m;t3) is a fuzzy metric space. on the other handif we take a = 1000, b = 3 and c = 10000, thenm(a;b;t) � m(b;c;s) > m(a;c;t + s)and so (x;m;t2) is not a fuzzy metric space.finally, the topology �m is the discrete topology on x. indeed, for x 2 x,if we take r < 12 � 1x then b(x;r;t) = fxg.next example is based in [8].example 4.7. let fa;bg be a nontrivial partition or the real interval x =]2;+1[. de�ne the mapping m on x2�]0;+1[ as followsm(x;y;t) = (1 � � 1x^y � 1x_y� if x;y 2 a or x;y 2 b1x + 1y elsewhere.then, imitating example 2 of [8], one can prove that (x;m;t3) is a fuzzymetric space, and by example 4.4, clearly (x;m;t2) is not a fuzzy metric space.from examples 4.4 and 4.5 it is deduced that an open base for the neighbor-hood system of a point x 2 x, is i x1+rx; x1�rxh\a if x 2 a, and i x1+rx; x1�rxh\b,with 0 < r < 12 � 1x, if x 2 b.5. some properties of standard fuzzy metricsin this section (x;d) will be a metric space, and md the standard fuzzymetric deduced of d.grosso modo, we can say that all properties of classical metrics can be trans-lated to standard fuzzy metrics. now, an interesting question is to know whichof these properties can be generalized to any fuzzy metric. in this sense wewill see a new property which is satis�ed by standard fuzzy metrics. (noticethat there is no signi�cative di�erence between the standard fuzzy metric mdand the fuzzy metric ktnktn+md(x;y) of example 4.2, unless md is the most simpleexpression depending of t).proposition 5.1. let fxng1n=1 and fyng1n=1 be two cauchy sequences in (x,md, t2) and let t > 0. then, the sequence of real numbers fmd(xn;yn; t)g1n=1converges to some real number in ]0;1[.proof. suppose fxng1n=1 and fyng1n=1 are cauchy sequences in (x;md;t2). by(ii) of proposition 2.2, fxng1n=1 and fyng1n=1 are cauchy sequences in (x;d)and then it is easy to verify that fd(xn;yng1n=1 is a cauchy sequence in r. a contribution to the study of fuzzy metric spaces 73now, let " > 0, t > 0. then, there exists n0 2 n such thatjd(xn;yn) � d(xm;ym)j < "t for all m;n � n0:hence,�� 1md(xn;yn; t) � 1md(xm;ym; t)�� = 1t jd(xn;yn) � d(xm;ym)j< ";for all m;n � n0, and therefore n 1md(xn;yn;t)o is a cauchy sequence in r, whichconverges to some k 2 r, and then the sequence fmd(xn;yn; t)g1n=1 convergesto 1k 2]0;1[, since k 6= 1 and md(xn;yn; t) � 1, for all n 2 n. �corollary 5.2. let fxng1n=1 be a cauchy sequence in the fuzzy metric space(x;md;t2) and a 2 x. then, the sequence of real numbers fmd(a;xn; t)g1n=1converges to some real number in ]0;1[.problem 5.3. let fxng1n=1 be a cauchy sequence in the fuzzy metric space (x,m, �) and let a 2 x, t > 0. does the sequence of real numbers fm(a;xn; t)g1n=1converge to some real number in ]0;1[?the last proposition is not true for any fuzzy metric space as shows thefollowing example.example 5.4. let fa;bg be a partition of the real interval x =]2;+1[, suchthat f2n � 1g1n=2 � a and f2ng1n=1 � b, and consider the fuzzy metric space(x;m;t3) of example 4.6. it is easy to verify that both sequences are cauchyin (x;m;t3). now, if we put an = 2n � 1 and bn = 2n, for n = 2;3; : : : wehave m(an;bn; t) = � 12n � 1 + 12n� �! 0 as n �! 1:6. on non-archimedean fuzzy metricsrecall that a metric d on x is called non-archimedean ifd(x;z) � maxfd(x;y);d(y;z)g; for all x;y;z 2 x:now, we give the following de�nition.de�nition 6.1. a fuzzy metric (m;�) on x is called non-archimedean ifm(x;z;t) � minfm(x;y;t);m(y;z;t)g for all x;y;z 2 x, t > 0:clearly, if m is a non-archimedean fuzzy metric on x, then (m;t1) is afuzzy metric on x:proposition 6.2. let d be a metric on x and md the corresponding stan-dard fuzzy metric. then, d is non-archimedean if and only if md is non-archimedean. 74 a. sapenaproof. suppose d is non-archimedean. then,md(x;z;t) = tt + d(x;z)� tt + maxfd(x;y);d(y;z)g= minfmd(x;y;t);md(y;z;t)g:conversely, if md is non-archimedean then,d(x;z) = t� 1md(x;y;t) � 1�� t� 1minfmd(x;y;t);md(y;z;t)g � 1�= maxfd(x;y);d(y;z)g : �recall that a completely regular space is called strongly zero-dimensional ifeach zero-set is the countable intersection of sets that are open and closed, andthat a t0 topological space (x;�) is strongly zero-dimensional and metrizableif and only if there is a uniformity u compatible with � that has a countabletransitive base ([3, theorem 6.8]).theorem 6.3. a topological space (x;�) is strongly zero-dimensional andmetrizable if and only if (x;�) is non-archimedeanly fuzzy metrizable.proof. suppose (x;�) is strongly zero-dimensional and metrizable. then, from[3, theorem 6.8], (x;�) is non-archimedeanly metrizable and by proposition6.2 it is non-archimedeanly fuzzy metrizable.conversely, suppose m is a compatible non-archimedean fuzzy metric for(x;�). now, for a fuzzy metric space (x;m;�) in [7] it is proved that thefamily fun : n 2 ng whereun = �(x;y) 2 x � x : m(x;y; 1n) > 1 � 1n�is a base for a uniformity u on x which is compatible with �m. now, we willsee that fun : n 2 ng is transitive.indeed, if (x;y);(y;z) 2 un thenm(x;z; 1n) � min�m(x;y; 1n);m(y;z; 1n)�> 1 � 1n;and thus (x;z) 2 un, i.e., un � un � un.now, from the mentioned theorem of [3], the hausdor� topological space(x;�) is a strongly zero-dimensional and metrizable space. � a contribution to the study of fuzzy metric spaces 75references[1] deng zi-de, fuzzy pseudo metric spaces, j. math. anal. appl. 86 (1982), 74{95.[2] r. engelking,general topology, pwn-polish sci. publ, warsaw, 1977.[3] p. fletcher and w. lindgren,quasi uniform spaces, marcel dekker, new york, 1982.[4] a. george and p.v. veeramani, on some results in fuzzy metric spaces, fuzzy sets andsystems 64 (1994), 395{399.[5] a. george and p.v. veeramani, on some results of analysis for fuzzy metric spaces, fuzzysets and systems 90 (1997), 365{368.[6] m. grabiec, fixed points in fuzzy metric spaces, fuzzy sets and systems 27 (1988),385{389.[7] v. gregori and s. romaguera, some properties of fuzzy metric spaces, fuzzy sets andsystems, 115 (2000), 485{489.[8] v. gregori and s. romaguera, on completion of fuzzy metric spaces, preprint.[9] o. kramosil and j. michalek, fuzzy metric and statistical metric spaces, kybernetica, 11(1975), 326{334.[10] b. schweizer and a. sklar, probabilistic metric spaces, elsevier science publishing co.,new york (1983).[11] b. schweizer and a. sklar, statistical metric spaces, paci�c j. math. 10 (1960), 314{334.received october 2000revised version april 2001 almanzor sapena pieradepartamento de matem�atica aplicadaescuela polit�ecnica superior de alcoypza. ferr�andiz-carbonell, 203801 alcoy (alicante), spaine-mail address: alsapie@mat.upv.es @ �� !#"%$'&'� ( )*(,+�-*. � /10" � � � � )2( "�34)*. 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]+�"n1�#��$%��2 2!�!#��,o�#� $�&7!=� "��,o�f ",�� ^'4`_(�t�t�a� ¢ �.�8�a�7boc � ¬=<�d�d!¡y¥ « y ©+ 0) ∨ (∃ϕ ∈a(y) : ϕ(x) > 0)) (4) ∀x,y ∈ x : x 6= y ⇒ (∃γ ∈r : γ(x) 6= γ(y)), (5) ∀f ∈ ap(i2, (x,δ)) : f constant, (6) ∀x,y ∈ x : x 6= y ⇒a(x) 6= a(y). proof. the implication (1)⇒(3) is obvious because it is proved in [8] that for every x ∈ x, {{ψ < ε}|ψ ∈a(x),ε > 0} is a base for the tδ-neighbourhoodsystem at x. to verify the implication (3) ⇒ (2), note that it was proved in [8] that the source (δ(·,a) : (x,δ) −→ p : x 7→ δ(x,a))a∈2x is initial in ap. therefore it suffices to show that it is point-separating in order to conclude that (x,δ) is a subspace of a power of p, so pick x,y ∈ x with x 6= y. assume without loss of generality that ϕ(y) > 0 for some ϕ ∈ a(x). then automatically δ(x,{y}) ≥ ϕ(y) > 0 and δ(y,{y}) = 0. the implication (2)⇒(1) is obvious since the concrete bicoreflector from ap onto top preserves products and subspaces, because ([0,∞],tδp) ∈ |top0| and because the latter is an epireflective subconstruct of top. the implication (1) ⇒(4) is proved using the implication (1) ⇒(3) because for all x ∈ x, δ(·,{x}) ∈ 478 r. lowen and m. sioen r. to prove the converse one, take x,y ∈ x, x 6= y and assume without loss of generality that γ(x) > α > γ(y) for some γ ∈ r,α ∈ r+. because γ : (x,tδ) −→ p is lower semicontinuous, {γ > α} is a tδ-neighbourhoood of x not containing y. furthermore, it is clear that (1) implies (5), whereas the converse implication follows by contraposition, because if x,y ∈ x were distinct points such that all neighbourhoods of x contain y and vice versa, f : i2 −→ (x,δ) defined by f(0) := x,f(1) := y would be a non-constant contraction. the implication (3) ⇒(6) is obvious and we finish with the implication (6) ⇒ (4). take x,y ∈ x with x 6= y. according to (6), we can assume without loss of generality that there exists ϕ ∈ a(x) \ a(y). therefore, it follows from the transition formula (distance −→ approach system) that there exists a ∈ 2x with infz∈a ϕ(z) > δ(y,a), whence automatically δ(x,a) > δ(y,a) and because δ(·,a) ∈r, we are done. � this shows that the t0-property in ap is in fact completely topological and it is proved in the next proposition that, again like in the topological case, the corresponding epireflection arrows are obtained as quotients. proposition 2.3. ap0 is an epireflective subconstruct of ap. for any (x,δ) ∈ |ap|, we define an equivalence relation ∼ on x by x ∼ y ⇔ (∀ϕ ∈r : ϕ(x) = ϕ(y)) ⇔a(x) = a(y). then the ap-quotient of (x,δ) with respect to ∼ gives us is an ap0-epireflection arrow for (x,δ). proof. for every x ∈ x, we write x for the corresponding equivalence class w.r.t. ∼ and we denote the corresponding projection by π : x −→ x/ ∼: x 7→ x. by definition of ∼, it is obvious that for each γ ∈r, the map γ : x/ ∼−→ [0,∞] : x 7→ γ(x) is well-defined. from the descripiton of quotients in ap, it is now clear that the final regular function frame on x/ ∼ with respect to π : (x,r) −→ x/ ∼ is exactly r/ ∼:= {ϕ ∈ [0,∞]x/∼|ϕ◦π ∈r} = {γ|γ ∈r}. then clearly (x/ ∼,r/ ∼) ∈ |ap0| and for every (x′,r′) ∈ |ap0| and f ∈ ap((x,r), (x′,r′)), it is clear that f : (x/ ∼,r/ ∼) −→ (x′,r′) : x 7→ f(x) is a well-defined contraction, being the unique one such that f = f ◦π. � the corresponding epireflector is denoted by t0 : ap −→ ap0. remark 2.4. ap is universal, i.e. it is the bireflective hull of ap0 in ap. moreover, every epireflector from ap onto one of its subconstructs is either a bireflector or the composition of a bireflector, followed by the ap0−epireflector. proof. the universality follows immediately from the result, proved in [8], that for each (x,δ) ∈ |ap|, the source (δ(·,a) : (x,δ) −→ p)a∈2x a note on separation in ap 479 is initial in ap, because p ∈ |ap0|. the second part is proved in [10]. � 2.2. the t1-axiom. remark 2.5. for every (x,δ) ∈ |ap|, the following assertions are equivalent: (1) (x,tδ) ∈ |top1|, (2) ∀x,y ∈ x : x 6= y ⇒ (∃ϕ,ψ ∈r : (ϕ(x) < ϕ(y)) ∧ (ψ(y) < ψ(x)), (3) ∀x,y ∈ x : x 6= y ⇒ ((∃ϕ ∈a(x) : ϕ(y) > 0) ∧ (∃ψ ∈a(y) : ψ(x) > 0)). (4) ∀x,y ∈ x : x 6= y ⇒ ((a(x) 6⊂a(y)) ∧ (a(y) 6⊂a(x))). proof. this is proved in the same way as 2.2 . � comparing this remark with the way t1-objects are defined in top and the characterizations of t0-objects in ap, yields that the following definition is plausible: definition 2.6. we call an approach space t1 if it satisfies the equivalent statements from 2.5. we define ap1 to be the full subconstruct of ap defined by all t1 approach spaces. corollary 2.7. ap1 is an epireflective subconstruct of ap. next we want to give an internal description of the corresponding epireflector from ap onto ap0. it is well-known that a topological space (x,t ) is t1 if and only if it is both t0 and symmetric in the sense of [3], meaning that ∀x,y ∈ x : x ∈ cl({y}) ⇔ y ∈ cl({x}). this, together with remark 2.4 above motivates the following line of working: definition 2.8. an approach space (x,δ) is called r0 if it satisfies the condition ∀x ∈ x : a(x) = ⋂ y:δ(y,{x})=0 a(y). the full subconstruct of ap formed by all r0-objects is denoted by apr0 . constructing the t1−epireflector will carry us outside of ap, into the superconstruct prap of pre-approach spaces and contractions, as introduced in [9]. let us only recall that a pre-approach space is a pair (x,δ) with δ : x × 2x −→ [0,∞] satisfying (d1), (d2) and (d3) (such δ is called a pre-distance) and contractions are defined in the same way as above. just as in the approach case, a pre-approach distance δ can be equivalently characterized by resp. a pre-approach system a, a pre-hull h and a pre-approach limit λ. for details we refer to [9], but we note that, just as for distances, stepping from ap to prap comes down to dropping the triangular axiom for a and λ and the idempotency for h. it was proved in [9] that ap is a concretely 480 r. lowen and m. sioen bireflective subconstruct of prap. take (x,h) ∈ |prap|. if γ ∈ [0,∞]x, define h0(γ) := γ and for each ordinal α ≥ 1, hα(γ) := { h(hα−1(γ)) α not a limit ordinal∧ β<α h β(γ) α limit ordinal. then there exists some ordinal κ such that hκ(γ) = hκ+1(γ) for all γ ∈ [0,∞]x and we put h∗(γ) := hκ(γ) for each γ ∈ [0,∞]x. then it can be proved that h∗ : [0,∞]x −→ [0,∞]x is a hull on x and idx : (x,h) −→ (x,h∗) is the reflection arrow. we will write d : prap −→ ap for the corncrete reflector and to simplify notations, we will write (x, d(δ)) instead of d((x,δ)) for (x,δ) ∈ |prap|, and analoguously for the associated pre-approach systems, pre-hulls and pre-approach limits. first note that obviously, |ap1| = |apr0|∩ |ap0|. we will first show that apr0 is a concretely bireflective subconstruct of ap, yielding at once a description of the concrete bireflector r0 : ap −→ apr0 . if (x,a) ∈ |ap|, we define a relation ∼r0 on x as follows: x ∼r0 y ⇔ δ(x,{y}) = 0. if we put a∗(x) := ⋂ y∼r0x a(y) for all x ∈ x, and a∗ := (a∗(x))x∈x, then (x,a∗) ∈ |prap|. fix (x,a) ∈ |ap|. put a0 := a and for every ordinal α ≥ 1, define bα(x) := { (aα−1)∗(x) α not a limit ordinal,⋂ β<αa β(x) α a limit ordinal , x ∈ x (note that (x,bα := (bα(x))x∈x) ∈ |prap|) and define aα := d(bα), whence (x,aα) ∈ |ap|. proposition 2.9. for every (x,a) ∈ |ap|, there exists an ordinal κ for which aκ = aκ+1. if we denote ar0 := aκ, (x,ar0 ) ∈ |apr0| and idx : (x,a) −→ (x,ar0 ) is the apr0−bireflection arrow. proof. pick (x,a) ∈ |ap|. since for all ordinals β < α, aβ ⊃ aα and hence 0 ≤ δaα ≤ δaβ , it is clear that aκ+1 = aκ, if we take a fixed ordinal κ > card([0,∞]x×2 x ). then define ar0 := aκ. by construction, it is obvious that (x,ar0 ) ∈ |apr0|. now fix (x′,a′) ∈ |apr0| and f ∈ ap((x,a), (x′,a′)). then by definition, f ∈ ap((x,a0), (x′,a′)). now assume that f ∈ ap((x,aα−1), (x′,a′)) for some non-limit ordinal α. in order to verify that f ∈ prap((x,bα), (x′,a′)), pick x ∈ x and ϕ′ ∈ a′(f(x)). we should prove that ϕ′ ◦ f ∈ bα(x), so let y ∈ x such that δaα−1 (y,{x}) = 0. then automatically δ′(f(y),{f(x)}) = 0, so because (x′,a′) ∈ |apr0|, ϕ′ ∈ ⋂ z∈x′:δ′(z,{f(x)})=0 a′(z) ⊂a′(f(y)), a note on separation in ap 481 whence ϕ′◦f ∈aα−1(y). since d is a concrete bireflector, it now immediately follows that f ∈ ap((x,aα), (x′,a′)). next take a limit ordinal α and assume that f ∈ ap((x,aβ), (x′,a′)) for all β < α. it then trivially follows that f ∈ ap((x,aα), (x′,a′)). by transfinite induction, it now follows in particular that f ∈ ap((x,ar0 ), (x′,a′)) so we are done. � for (x,δ) ∈ |ap|, we will also use the notation (x, r0(δ)) for r0((x,δ)) and the same convention applies for the other equivalent axiomizations for approach spaces. proposition 2.10. for every (x,r) ∈ |ap|, the ap1-epireflection is obtained by taking the ap0-epireflection of its apr0−bireflection, i.e. the corresponding ap1-epireflection arrow is given by π : (x,r) −→ (x/ ∼, r0(r)/ ∼), (where ∼ and π are determined by r0(r).) proof. it suffices to verify that (x/ ∼, r0(r)/ ∼) ∈ |ap1|. first note that, with the notations as in 2.3, for all x,y ∈ x δr0(r)/∼(x,{y}) = sup γ∈r0(r),γ(y)=0 γ(x) = sup γ∈r0(r),γ(y)=0 γ(x) = δr0(r)(x,{y}). because (x, r0(r)) ∈ |apr0|, this implies that ∀x,y ∈ x : δr0(r)/∼(x,{y}) = 0 ⇒ δr0(r)/∼(y,{x}) = 0, i.e. that (x/ ∼,tr0(r)/∼) is a symmetric space in the sense of [3] (called an r0space there) and since it also belongs to |top0|, it belongs to |top1| and we are done. � 2.3. the t2-axiom. if x is a set and f ⊂ [0,∞]x, we call 〈f〉 := {γ ∈ [0,∞]x | ∀ε > 0,∀m < ∞ : ∃γεm ∈ f : γ ∧m ≤ γεm + ε}, resp. c(f) := supγ∈f infx∈x γ(x), the saturation, resp. the level of f. if (x,a) ∈ |ap| and x,y ∈ x, then a(x) ∨ a(y) := 〈a(x) ∪ a(y)〉 is the supremum of a(x) and a(y) in the lattice of all saturated ideals in [0,∞]x. remark 2.11. if (x,δ) ∈ |ap|, the following assertions are equivalent: (1) ∀x,y ∈ x : x 6= y ⇒ c(a(x) ∨a(y)) > 0, (2) ∀x,y ∈ x : x 6= y ⇒ (∃ϕ ∈a(x),∃ψ ∈a(y) : infs∈x(ϕ∨ψ)(s) > 0, (3) (x,tδ) ∈ |top2|. proof. obvious since ({{ϕ < ε}|ε > 0,ϕ ∈ a(x)})x∈x is a base for the tδneigbourhood system. � again, a comparison with the classical topological situation and taking into account that the role of neighbourhood filters in topology is played by the so-called approach systems in approach theory, makes it plausible to define hausdorff objects in ap in the following, again topological, way: 482 r. lowen and m. sioen definition 2.12. we call an approach space t2 if it satisfies the equivalent statements from 2.11. we define ap2 to be the full subconstruct of ap defined by all t2 approach spaces. corollary 2.13. ap2 is an epireflective subconstruct of ap. as an answer to a question raised by h. herrlich, an internal description of the epireflector from top onto top2 was given by v. kannan in [5], making use of a transfinite construction. we will now derive an explicit description for the epireflector from ap onto ap2, along the same lines as was done for the t1-case in the section above. first we define a property r for approach spaces, which is inspired by the notion of reciprocity for convergence spaces, as defined in [2]. if (x,a) ∈ |ap|, we define a relation ∼r on x by x ∼r y ⇔ (∃x1 := x,. . . ,xn := y : ∀i ∈{1, . . . ,n− 1} : c(a(xi) ∨a(xi+1)) = 0). definition 2.14. we call (x,δ) ∈ |ap| an rspace if it fulfills the following condition ∀x ∈ x : a(x) = ⋂ y∼rx a(y) and we denote the full subconstruct of ap formed by all r-spaces by apr. first note that |ap2| = |ap0| ∩ |apr|. to begin with, we will prove that apr is a bireflective subconstruct of ap by describing the bireflector r : ap −→ apr. let (x,a) ∈ |ap|. if we put a†(x) := ⋂ y∼rx a(y) for all x ∈ x, and a† := (a†(x))x∈x, then (x,a†) ∈ |prap|. fix (x,a) ∈ |ap|. put a0 := a and for every ordinal α ≥ 1, define bα(x) := { (aα−1)†(x) α not a limit ordinal,⋂ β<αa β(x) α a limit ordinal , x ∈ x (note that (x,bα := (bα(x))x∈x) ∈ |prap| and define aα := d(bα), whence (x,aα) ∈ |ap|. proposition 2.15. for all (x,a) ∈ |ap|, there exists an ordinal κ for which aκ+1 = aκ. if we denote ar := aκ, (x,ar) ∈ |apr| and idx : (x,a) −→ (x,ar) is the apr−bireflection arrow. proof. the proof is exactly the same as that of 2.9 except for verifying that for (x′,a′) ∈ |apr| and f ∈ ap((x,a), (x′,a′)), if we assume that f ∈ ap((x,aα−1), (x′,a′)) for some non-limit ordinal α (*), it follows that f ∈ prap((x,bα), (x′,a′)). therefore, pick x ∈ x and ϕ′ ∈a′(f(x)). assume that y ∈ x and x1 := y, . . . ,xn := x such that c(aα−1(xi) ∨aα−1(xi+1)) = 0 a note on separation in ap 483 for all i ∈ {1, . . . ,n− 1}. then it follows from our assumption (*) that for all i ∈{1, . . . ,n− 1} c(a′(f(xi)) ∨a′(f(xi+1))) = sup γ∈a′(f(xi)) sup µ∈a′(f(xi+1)) inf z′∈x′ (γ ∨µ)(z′) ≤ sup γ∈a′(f(xi)) sup µ∈a′(f(xi+1)) inf z∈x (γ∨µ)(f(z)) ≤ c(aα−1(xi)∨aα−1(xi+1)) = 0, showing that f(y) ∼r f(x). because (x′,a′) ∈ |apr|, this implies that ϕ′ ∈a′(f(y)) whence ϕ′ ◦f ∈aα−1(y) and this completes the proof. � for (x,δ) ∈ |ap|, we will also use the notation (x, r(δ)) for r((x,δ)) and the same convention applies for the other equivalent axiomizations for approach spaces. proposition 2.16. for every (x,r) ∈ |ap|, the ap2-epireflection is obtained by taking the ap0-epireflection of its apr−bireflection, i.e. the corresponding ap2-epireflection arrow is given by π : (x,r) −→ (x/ ∼, r(r)/ ∼), (where ∼ and π are detremined by r(r).) proof. we only need to check that (x/ ∼, r(r)/ ∼) ∈ |ap2|. now assume that x,y ∈ x for which every tr(r)/∼-neighbourhood of x meets every tr(r)/∼neighbourhood of y. note that tr(r) = {{µ > 0} | µ ∈ r(r)} and that with the notation introduced in 2.3 r(r)/ ∼= {γ|γ ∈ r(r)}. this yields that x ∼r y where the ∼r-relation is taken with respect to r(r), and because (x, r(r)) ∈ |apr|, it follows that ar(r)(x) = ar(r)(y), whence x = y. � 2.4. regularity. in [11], three different suggestions for regularity were proposed, and in [5], it was motivated that the strongest one of them is the correct notion of regularity in the construct ap. we simply recall this definition for the sake of completeness. for any set x, let f(x) stand for the set of all filters on x and for each f ∈ f(x) and ε ≥ 0, let f(ε) denote the filter generated by {f (ε) | f ∈f}. definition 2.17. [11], [5] an approach space (x,δ) is called regular if ∀ε ≥ 0,∀f ∈ f(x) : λ(f(ε)) ≤ λ(f) + ε. we write rap for the full subconstruct of ap formed by all regular spaces, and it was proved in [11] that it moreover is concretely bireflective. for some other equivalent characterizations of regularity in terms of distances and approach systems, we refer to [11]. note that, where the lower separation axioms we discussed in ap all turn out to be topological in the sense that an approach space (x,δ) is ti if and only if its topological coreflection in ti in the classical sense (with i ∈{0, 1, 2}), 484 r. lowen and m. sioen the regularity condition stated above is of a purely quantitative nature. this was already noted in [11], where it is shown that δ(x,a) :=   ∞ a = ∅ 0 x ∈ a 1 x 6∈a,a infinite 2 other cases x ∈ r,a ⊂ r defines an approach distance on r such that (r,δ) 6 ∈|rap| but (r,tδ) is a regular topological space. it was also already stated in [11] that for topological spaces, this notion of regularity is equivalent to the classical one. 2.5. complete regularity. we recall the definition of uniform approach spaces from [8]. definition 2.18. an approach space (x,δ) is called uniform if and only if there exists a collection d of ∞p-metrics on x which is closed with respect to taking finite suprema and such that ∀x ∈ x,∀a ∈ 2x : δ(x,a) = sup d∈d δd(x,a). the full subconstruct of ap formed by all uniform approach spaces is denoted by uap and it can be shown (see e.g. [8]) that uap = eap(pmet∞). it was also proved in [8] that for every (x,δ) ∈ |ap|, the corresponding uap−epireflection arrow, which in fact is a concrete bireflection arrow, is given by idx : (x,δ) −→ (x,δu := sup d∈gs(δ) δd), with gs(δ) := {d|d ∞p− metric on x,δd ≤ δ}. the following proposition shows that ‘being uniform’ is precisely the correct quantified generalization of complete regularity to the approach setting and it was proved in [8] that for topological objects, these two notions are equivalent. proposition 2.19. for every (x,δ) ∈ |ap|, the following assertions are equivalent: (1) (x,δ) ∈ |uap|, (2) ∀x ∈ x,∀a ∈ 2x,∀ε > 0,∀ω < ∞ : ∃fωε ∈ ap((x,δ),r) : fωε (x) = 0 and ∀z ∈ a : f ω ε (z) + ε ≥ δ(x,a) ∧ω. (3) ∀x ∈ x,∀a ∈ 2x,∀ε > 0,∀ω < ∞ : ∃fωε ∈ ap((x,δ),r) bounded : fωε (x) = 0 and ∀z ∈ a : f ω ε (z) + ε ≥ δ(x,a) ∧ω. proof. we first show that (1) implies (2). therefore let d be a collection of ∞p-metrics on x which is closed w.r.t. the formation of finite suprema and such that δ = supd∈d δd and fix x ∈ x,a ∈ 2x,ε > 0 and ω < ∞. then we can find dωε ∈d with δdωε (x,a) + ε > δ(x,a) ∧ω a note on separation in ap 485 and because fωε := d ω ε (x, ·) ∈ ap((x,δ),r), we are done. that (2) implies (3) is clear since for every f ∈ ap((x,δ),r) and ω < ∞ , obviously |f| ∧ ω ∈ ap((x,δ),r). finally we show that (3) implies (1). in order to verify that (x,δ) ∈ |uap|, it sufices to show that δ ≤ δu, so fix x ∈ x,a ∈ 2x,ε > 0 and ω < ∞. according to (3), we can find fωε ∈ ap((x,δ),r) bounded such that fωε (x) = 0 and such that f ω ε (a) + ε ≥ δ(x,a) ∧ω for each a ∈ a . if we denote the euclidean metric on r by de, it is clear that de ◦ (fωε ×fωε ) is an ∞p-metric on x, which belongs to gs(δ) because fωε is a contraction. it then is clear that δu(x,a) + ε ≥ inf a∈a |fωε (x) −f ω ε (a)| + ε ≥ δ(x,a) ∧ω, completing the proof. � to see that, like regularity, this is a numerical, non-topological separation axiom, note that d(x,y) := { |x−y|/2 x ≤ y |x−y| x > y x,y ∈ r defines a pseudo-quasi-metric on r, such that (r,td) is a completely regular topological space but (x,δd) 6∈|uap|. references [1] adámek, j. herrlich, h. and strecker, g. abstract and concrete categories, john wiley, new york (1990) [2] bentley, h. l., herrlich h. and lowen-colebunders e. convergence , j. pure appl. alg. 68 (1990), pp. 27-45 [3] herrlich, h. topological structures, math. centre tracts 52 (1974), pp. 59-122 [4] herrlich, h. categorical topology 1971-1981, proc. 5th prague top. symp. 1981, heldermann verlag berlin (1983), pp. 279-383 [5] brock, p and kent, d. on convergence approach spaces, appl. cat. struct. 6 (1998), pp. 117-125 [6] kannan, v. on a problem of herrlich, j. madurai. univ. 6 (1976), pp. 101-104 [7] lowen, r. approach spaces: a common supercategory of top and met, math. nachr. 141 (1989), pp. 183-226 [8] lowen, r. approach spaces: the missing link in the topology-uniformity-metric triad, oxford mathematical monographs, oxford university press (1997) [9] lowen, e and lowen, r. a quasitopos containing conv and met as full subcategories , int. j. math. and math. sci., 11(3) (1988), pp. 417-438 [10] marny, t. on epireflective subcategories of topological categories, gen. top. appl. 10 (1979), pp. 175-181 [11] robeys, k., extensions of products of metric spaces, phd thesis universiteit antwerpen, ruca (1992) received november 2001 revised august 2002 486 r. lowen and m. sioen r. lowen university of antwerp, ruca, department of mathematics and computer science, middelheimlaan 1, b2020 antwerp, belgium e-mail address : rlow@ruca.ua.ac.be m. sioen free university of brussels, department of mathematics, pleinlaan 2, b1000 brussels, belgium e-mail address : msioen@vub.ac.be banackag.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 2548 the topological structure of (homogeneous) spaces and groups with countable cs∗-character taras banakh and lubomyr zdomsky̆ı abstract. in this paper we introduce and study three new cardinal topological invariants called the cs∗-, cs-, and sb-characters. the class of topological spaces with countable cs∗-character is closed under many topological operations and contains all ℵ-spaces and all spaces with point-countable cs∗-network. our principal result states that each non-metrizable sequential topological group with countable cs∗character has countable pseudo-character and contains an open kωsubgroup. this result is specific for topological groups: under martin axiom there exists a sequential topologically homogeneous kω-space x with ℵ0 = cs ∗ χ(x) < ψ(x). 2000 ams classification: 22a05, 54a20, 54a25, 54a35, 54d55, 54e35, 54h11 keywords: sb-network, cs-network, cs∗-network, sequential topological group, kω-group, topologically homogeneous space, small cardinal, cardinal invariant. introduction in this paper we introduce and study three new local cardinal invariants of topological spaces called the sb-character, the cs-character and cs∗-character, and describe the structure of sequential topological groups with countable cs∗character. all these characters are based on the notion of a network at a point x of a topological space x, under which we understand a collection n of subsets of x such that for any neighborhood u ⊂ x of x there is an element n ∈ n with x ∈ n ⊂ u, see [lin]. a subset b of a topological space x is called a sequential barrier at a point x ∈ x if for any sequence (xn)n∈ω ⊂ x convergent to x, there is m ∈ ω such that xn ∈ b for all n ≥ m, see [lin]. it is clear that each neighborhood of a point x ∈ x is a sequential barrier for x while the converse in true for fréchet-urysohn spaces. 26 t.banakh, l.zdomsky̆ı under a sb-network at a point x of a topological space x we shall understand a network at x consisting of sequential barriers at x. in other words, a collection n of subsets of x is a sb-network at x if for any neighborhood u of x there is an element n ⊂ u of n such that for any sequence (xn) ⊂ x convergent to x the set n contains almost all elements of (xn). changing two quantifiers in this definition by their places we get a definition of a cs-network at x. namely, we define a family n of subsets of a topological space x to be a csnetwork (resp. a cs∗-network) at a point x ∈ x if for any neighborhood u ⊂ x of x and any sequence (xn) ⊂ x convergent to x there is an element n ∈ n such that n ⊂ u and n contains almost all (resp. infinitely many) members of the sequence (xn). a family n of subsets of a topological space x is called a cs-network (resp. cs∗-network) if it is a cs-network (resp. cs∗-network) at each point x ∈ x, see [na]. the smallest size |n| of an sb-network (resp. cs-network, cs∗-network) n at a point x ∈ x is called the sb-character (resp. cs-character, cs∗-character) of x at the point x and is denoted by sbχ(x,x) (resp. csχ(x,x), cs ∗ χ(x,x)). the cardinals sbχ(x) = supx∈x sbχ(x,x), csχ(x) = supx∈x csχ(x,x) and cs∗χ(x) = supx∈x cs ∗ χ(x,x) are called the sb-character, cs-character and cs ∗character of the topological space x, respectively. for the empty topological space x = ∅ we put sbχ(x) = csχ(x) = cs ∗ χ(x) = 1. as expected the character χ(x) of a topological space x is the smallest cardinal κ such that each point x ∈ x has a neighborhood base of size ≤ κ. in the sequel we shall say that a topological space x has countable sbcharacter (resp. cs-, cs∗-character) if sbχ(x) ≤ ℵ0 (resp. csχ(x) ≤ ℵ0, cs∗χ(x) ≤ ℵ0). in should be mentioned that under different names, topological spaces with countable sbor cs-character have already occured in topological literature. in particular, a topological space has countable cs-character if and only if it is csf-countable in the sense of [lin]; a (sequential) space x has countable sb-character if and only if it is universally csf-countable in the sense of [lin] (if and only if it is weakly first-countable in the sense of [ar1] if and only if it is 0-metrizable in the sense of nedev [ne]). from now on, all the topological spaces considered in the paper are t1-spaces. at first we consider the interplay between the characters introduced above. proposition 1. let x be a topological space. then (1) cs∗χ(x) ≤ csχ(x) ≤ sbχ(x) ≤ χ(x); (2) χ(x) = sbχ(x) if x is fréchet-urysohn; (3) cs∗χ(x) < ℵ0 iff csχ(x) < ℵ0 iff sbχ(x) < ℵ0 iff cs ∗ χ(x) = 1 iff csχ(x) = 1 iff sbχ(x) = 1 iff each convergent sequence in x is trivial; (4) sbχ(x) ≤ 2 cs∗ χ (x); (5) csχ(x) ≤ cs ∗ χ(x) · sup{ ∣ ∣[κ]≤ω ∣ ∣ : κ < cs∗χ(x)} ≤ ( cs∗χ(x) )ℵ0 where [κ]≤ω = {a ⊂ κ : |a| ≤ ℵ0}. here “iff” is an abbreviation for “if and only if”. the arens’ space s2 and the sequential fan sω give us simple examples distinguishing between on groups with countable cs∗-character 27 some of the characters considered above. we recall that the arens’ space s2 is the set {(0,0),( 1 n ,0),( 1 n , 1 nm ) : n,m ∈ n} ⊂ r2 carrying the strongest topology inducing the original planar topology on the convergent sequences c0 = {(0,0),( 1 n ,0) : n ∈ n} and cn = {( 1 n ,0),( 1 n , 1 nm ) : m ∈ n}, n ∈ n. the quotient space sω = s2/c0 obtained from the arens’ space s2 by identifying the points of the sequence c0 is called the sequential fan, see [lin]. the sequential fan sω is the simplest example of a non-metrizable frécheturysohn space while s2 is the simplest example of a sequential space which is not fréchet-urysohn. we recall that a topological space x is sequential if a subset a ⊂ x if closed if and only if a is sequentially closed in the sense that a contain the limit point of any sequence (an) ⊂ a, convergent in x. a topological space x is fréchet-urysohn if for any cluster point a ∈ x of a subset a ⊂ x there is a sequence (an) ⊂ a, convergent to a. observe that ℵ0 = cs ∗ χ(s2) = csχ(s2) = sbχ(s2) < χ(s2) = d while ℵ0 = cs∗χ(sω) = csχ(sω) < sbχ(sω) = χ(sω) = d. here d is the well-known in set theory small uncountable cardinal equal to the cofinality of the partially ordered set nω endowed with the natural partial order: (xn) ≤ (yn) iff xn ≤ yn for all n, see [va]. besides d, we will need two other small cardinals: b defined as the smallest size of a subset of uncountable cofinality in (nω,≤), and p equal to the smallest size |f| of a family of infinite subsets of ω closed under finite intersections and having no infinite pseudo-intersection in the sense that there is no infinite subset i ⊂ ω such that the complement i \ f is finite for any f ∈ f, see [va], [vd]. it is known that ℵ1 ≤ p ≤ b ≤ d ≤ c where c stands for the size of continuum. martin axiom implies p = b = d = c, [ms]. on the other hand, for any uncountable regular cardinals λ ≤ κ there is a model of zfc with p = b = d = λ and c = κ, see [vd, 5.1]. unlike to the cardinal invariants csχ, sbχ and χ which can be distinguished on simple spaces, the difference between the cardinal invariants csχ and cs ∗ χ is more subtle: they cannot be distinguished in some models of set theory! proposition 2. let x be a topological space. then cs∗χ(x) = csχ(x) provided one of the following conditions is satisfied: (1) cs∗χ(x) < p; (2) κℵ0 ≤ cs∗χ(x) for any cardinal κ < cs ∗ χ(x); (3) p = c and λω ≤ κ for any cardinals λ < κ ≥ c; (4) p = c (this is so under ma) and x is countable; (5) the generalized continuum hypothesis holds. unlike to the usual character, the cs∗-, cs-, and sb-characters behave nicely with respect to many countable topological operations. among such operation there are: the tychonov product, the box-product, producing a sequentially homeomorphic copy, taking image under a sequentially open map, and forming inductive topologies. as usual, under the box-product 2i∈ixi of topological spaces xi, i ∈ i, we understand the cartesian product ∏ i∈i xi endowed with the box-product 28 t.banakh, l.zdomsky̆ı topology generated by the base consisting of products ∏ i∈i ui where each ui is open in xi. in contrast, by ∏ i∈i xi we denote the usual cartesian product of the spaces xi, endowed with the tychonov product topology. we say that a topological space x carries the inductive topology with respect to a cover c of x if a subset f ⊂ x is closed in x if and only if the intersection f ∩ c is closed in c for each element c ∈ c. for a cover c of x let ord(c) = supx∈x ord(c,x) where ord(c,x) = |{c ∈ c : x ∈ c}|. a topological space x carrying the inductive topology with respect to a countable cover by closed metrizable (resp. compact, compact metrizable) subspaces is called an mωspace (resp. a kω-space, mkω-space). a function f : x → y between topological spaces is called sequentially continuous if for any convergent sequence (xn) in x the sequence (f(xn)) is convergent in y to f(limxn); f is called a sequential homeomorphism if f is bijective and both f and f−1 are sequentially continuous. topological spaces x,y are defined to be sequentially homeomorphic if there is a sequential homeomorphism h : x → y . observe that two spaces are sequentially homeomorphic if and only if their sequential coreflexions are homeomorphic. under the sequential coreflexion σx of a topological space x we understand x endowed with the topology consisting of all sequentially open subsets of x (a subset u of x is sequentially open if its complement is sequentially closed in x; equivalently u is a sequential barrier at each point x ∈ u). note that the identity map id : σx → x is continuous while its inverse is sequentially continuous, see [lin]. a map f : x → y is sequentially open if for any point x0 ∈ x and a sequence s ⊂ y convergent to f(x0) there is a sequence t ⊂ x convergent to x0 and such that f(t) ⊂ s. observe that a bijective map f is sequentially open if its inverse f−1 is sequentially continuous. the following technical proposition is an easy consequence of the corresponding definitions. proposition 3. (1) if x is a subspace of a topological space y , then cs∗χ(x) ≤ cs ∗ χ(y ), csχ(x) ≤ csχ(y ) and sbχ(x) ≤ sb(y ). (2) if f : x → y is a surjective continuous sequentially open map between topological spaces, then cs∗χ(y ) ≤ cs ∗ χ(x) and sbχ(y ) ≤ sbχ(x). (3) if f : x → y is a surjective sequentially continuous sequentially open map between topological spaces, then min{cs∗χ(y ),ℵ1} ≤ min{cs ∗ χ(x),ℵ1}, min{csχ(y ),ℵ1} ≤ min{csχ(x),ℵ1}, and min{sbχ(y ),ℵ1} ≤ min{sbχ(x),ℵ1}. (4) if x and y are sequentially homeomorphic topological spaces, then min{cs∗χ(x),ℵ1} = min{csχ(x),ℵ1} = min{csχ(y ),ℵ1} = = min{cs∗χ(y ),ℵ1}, and min{sbχ(y ),ℵ1} = min{sbχ(x),ℵ1}. (5) for any topological space x min{sbχ(x),ℵ1} = min{sbχ(σx),ℵ1} ≤ sbχ(σx) ≥ sbχ(x) and csχ(x) ≤ csχ(σx) ≥ min{csχ(σx),ℵ1} = min{csχ(x),ℵ1} = min{cs ∗ χ(x),ℵ1} = min{cs ∗ χ(σx),ℵ1} ≤ cs ∗ χ(σx) ≥ cs∗χ(x). on groups with countable cs∗-character 29 (6) if x = ∏ i∈i xi is the tychonov product of topological spaces xi, i ∈ i, then cs∗χ(x) ≤ ∑ i∈i cs ∗ χ(xi), csχ(x) ≤ ∑ i∈i csχ(xi) and sbχ(x) ≤ ∑ i∈i sbχ(xi). (7) if x = 2i∈ixi is the box-product of topological spaces xi, i ∈ i, then cs∗χ(x) ≤ ∑ i∈i cs ∗ χ(xi) and csχ(x) ≤ ∑ i∈i csχ(xi). (8) if a topological space x carries the inductive topology with respect to a cover c of x, then cs∗χ(x) ≤ ord(c) · supc∈c cs ∗ χ(c). (9) if a topological space x carries the inductive topology with respect to a point-countable cover c of x, then csχ(x) ≤ supc∈c csχ(c). (10) if a topological space x carries the inductive topology with respect to a point-finite cover c of x, then sbχ(x) ≤ supc∈c sbχ(c). since each first-countable space has countable cs∗-character, it is natural to think of the class of topological spaces with countable cs∗-character as a class of generalized metric spaces. however this class contains very non-metrizable spaces like βn, the stone-čech compactification of the discrete space of positive integers. the reason is that βn contains no non-trivial convergent sequence. to avoid such pathologies we shall restrict ourselves by sequential spaces. observe that a topological space is sequential provided x carries the inductive topology with respect to a cover by sequential subspaces. in particular, each mω-space is sequential and has countable cs∗-character. our principal result states that for topological groups the converse is also true. under an mω-group (resp. mkω-group) we understand a topological group whose underlying topological space is an mω-space (resp. mkω-space). theorem 1. each sequential topological group g with countable cs∗-character is an mω-group. more precisely, either g is metrizable or else g contains an open mkω-subgroup h and is homeomorphic to the product h × d for some discrete space d. for mω-groups the second part of this theorem was proven in [ba1]. theorem 1 has many interesting corollaries. at first we show that for sequential topological groups with countable cs∗character many important cardinal invariants are countable, coincide or take some fixed values. let us remind some definitions, see [en1]. for a topological space x recall that • the pseudocharacter ψ(x) is the smallest cardinal κ such that each one-point set {x} ⊂ x can be written as the intersection {x} = ∩u of some family u of open subsets of x with |u| ≤ κ; • the cellularity c(x) is the smallest cardinal κ such that x contains no family u of size |u| > κ consisting of non-empty pairwise disjoint open subsets; • the lindelöf number l(x) is the smallest cardinal κ such that each open cover of x contains a subcover of size ≤ κ; • the density d(x) is the smallest size of a dense subset of x; 30 t.banakh, l.zdomsky̆ı • the tightness t(x) is the smallest cardinal κ such that for any subset a ⊂ x and a point a ∈ ā from its closure there is a subset b ⊂ a of size |b| ≤ κ with a ∈ b̄; • the extent e(x) is the smallest cardinal κ such that x contains no closed discrete subspace of size > κ; • the compact covering number kc(x) is the smallest size of a cover of x by compact subsets; • the weight w(x) is the smallest size of a base of the topology of x; • the network weight nw(x) is the smallest size |n| of a topological network for x (a family n of subsets of x is a topological network if for any open set u ⊂ x and any point x ∈ u there is n ∈ n with x ∈ n ⊂ u); • the k-network weight knw(x) is the smallest size |n| of a k-network for x (a family n of subsets of x is a k-network if for any open set u ⊂ x and any compact subset k ⊂ u there is a finite subfamily m ⊂ n with k ⊂ ∪m ⊂ u). for each topological space x these cardinal invariants relate as follows: max{c(x), l(x),e(x)} ≤ nw(x) ≤ knw(x) ≤ w(x). for metrizable spaces all of them are equal, see [en1, 4.1.15]. in the class of k-spaces there is another cardinal invariant, the k-ness introduced by e. van douwen, see [vd, §8]. we remind that a topological space x is called a k-space if it carries the inductive topology with respect to the cover of x by all compact subsets. it is clear that each sequential space is a k-space. the k-ness k(x) of a k-space is the smallest size |k| of a cover k of x by compact subsets such that x carries the inductive topology with respect to the cover k. it is interesting to notice that k(nω) = d while k(q) = b, see [vd]. proposition 3(8) implies that cs∗χ(x) ≤ k(x) · ψ(x) ≥ kc(x) for each k-space x. observe also that a topological space x is a kω-space if and only if x is a k-space with k(x) ≤ ℵ0. besides cardinal invariants we shall consider an ordinal invariant, called the sequential order. under the sequential closure a(1) of a subset a of a topological space x we understand the set of all limit point of sequences (an) ⊂ a, convergent in x. given an ordinal α define the α-th sequential closure a(α) of a by transfinite induction: a(α) = ⋃ β<α(a (β))(1). under the sequential order so(x) of a topological space x we understand the smallest ordinal α such that a(α+1) = a(α) for any subset a ⊂ x. observe that a topological space x is fréchet-urysohn if and only if so(x) ≤ 1; x is sequential if and only if clx(a) = a (so(x)) for any subset a ⊂ x. besides purely topological invariants we shall also consider a cardinal invariant, specific for topological groups. for a topological group g let ib(g), the boundedness index of g be the smallest cardinal κ such that for any nonempty open set u ⊂ g there is a subset f ⊂ g of size |f | ≤ κ such that g = f · u. it is known that ib(g) ≤ min{c(g), l(g),e(g)} and w(g) = ib(g) · χ(g) for each topological group, see [tk]. on groups with countable cs∗-character 31 theorem 2. each sequential topological group g with countable cs∗-character has the following properties: ψ(g) ≤ ℵ0, sbχ(g) = χ(g) ∈ {1,ℵ0,d}, ib(g) = c(g) = d(g) = l(g) = e(g) = nw(g) = knw(g), and so(g) ∈ {1,ω1}. we shall derive from theorems 1 and 2 an unexpected metrization theorem for topological groups. but first we need to remind the definitions of some of αi-spaces, i = 1, . . . ,6 introduced by a.v. arkhangelski in [ar2], [ar4]. we also define a wider class of α7-spaces. a topological space x is called • an α1-space if for any sequences sn ⊂ x, n ∈ ω, convergent to a point x ∈ x there is a sequence s ⊂ x convergent to x and such that sn \ s is finite for all n; • an α4-space if for any sequences sn ⊂ x, n ∈ ω, convergent to a point x ∈ x there is a sequence s ⊂ x convergent to x and such that sn ∩ s 6= ∅ for infinitely many sequences sn; • an α7-space if for any sequences sn ⊂ x, n ∈ ω, convergent to a point x ∈ x there is a sequence s ⊂ x convergent to some point y of x and such that sn ∩ s 6= ∅ for infinitely many sequences sn; under a sequence converging to a point x of a topological space x we understand any countable infinite subset s of x such that s \ u if finite for any neighborhood u of x. each α1-space is α4 and each α4-space is α7. quite often α7-spaces are α4, see lemma 7. observe also that each sequentially compact space is α7. it can be shown that a topological space x is an α7-space if and only if it contains no closed copy of the sequential fan sω in its sequential coreflexion σx. if x is an α4-space, then σx contains no topological copy of sω. we remind that a topological group g is weil complete if it is complete in its left (equivalently, right) uniformity. according to [pz, 4.1.6], each kω-group is weil complete. the following metrization theorem can be easily derived from theorems 1, 2 and elementary properties of mkω-groups. theorem 3. a sequential topological group g with countable cs∗-character is metrizable if one of the following conditions is satisfied: (1) so(g) < ω1; (2) sbχ(g) < d; (3) ib(g) < k(g); (4) g is fréchet-urysohn; (5) g is an α7-space; (6) g contains no closed copy of sω or s2; (7) g is not weil complete; (8) g is baire; (9) ib(g) < |g| < 2ℵ0. according to theorem 1, each sequential topological group with countable cs∗-character is an mω-group. the first author has proved in [ba3] that the 32 t.banakh, l.zdomsky̆ı topological structure of a non-metrizable punctiform mω-group is completely determined by its density and the compact scatteredness rank. recall that a topological space x is punctiform if x contains no compact connected subspace containing more than one point, see [en2, 1.4.3]. in particular, each zero-dimensional space is punctiform. next, we remind the definition of the scatteredness height. given a topological space x let x(1) ⊂ x denote the set of all non-isolated points of x. for each ordinal α define the α-th derived set x(α) of x by transfinite induction: x(α) = ⋂ β<α(x(β))(1). under the scatteredness height sch(x) of x we understand the smallest ordinal α such that x(α+1) = x(α). a topological space x is scattered if x(α) = ∅ for some ordinal α. under the compact scatteredness rank of a topological space x we understand the ordinal scr(x) = sup{sch(k) : k is a scattered compact subspace of x}. theorem 4. two non-metrizable sequential punctiform topological groups g, h with countable cs∗-character are homeomorphic if and only if d(g) = d(h) and scr(g) = scr(h). this theorem follows from theorem 1 and main theorem of [ba3] asserting that two non-metrizable punctiform mω-groups g, h are homeomorphic if and only if d(g) = d(h) and scr(g) = scr(h). for countable kω-groups this fact was proven by e.zelenyuk [ze1]. the topological classification of non-metrizable sequential locally convex spaces with countable cs∗-character is even more simple. any such a space is homeomorphic either to r∞ or to r∞ × q where q = [0,1]ω is the hilbert cube and r∞ is a linear space of countable algebraic dimension, carrying the strongest locally convex topology. it is well-known that this topology is inductive with respect to the cover of r∞ by finite-dimensional linear subspaces. the topological characterization of the spaces r∞ and r∞ ×q was given in [sa]. in [ba2] it was shown that each infinite-dimensional locally convex mkω-space is homeomorphic to r∞ or r∞ ×q. this result together with theorem 1 implies the following classification corollary 1. each non-metrizable sequential locally convex space with countable cs∗-character is homeomorphic to r∞ or r∞ × q. as we saw in theorem 2, each sequential topological group with countable cs∗-character has countable pseudocharacter. the proof of this result is based on the fact that compact subsets of sequential topological groups with countable cs∗-character are first countable. this naturally leads to a conjecture that compact spaces with countable cs∗-character are first countable. surprisingly, but this conjecture is false: assuming the continuum hypothesis n. yakovlev [ya] has constructed a scattered sequential compactum which has countable sb-character but fails to be first countable. in [ny2] p.nyikos pointed out that the yakovlev construction still can be carried under the assumption b = c. more precisely, we have on groups with countable cs∗-character 33 proposition 4. under b = c there is a regular locally compact locally countable space y whose one-point compactification αy is sequential and satisfies ℵ0 = sbχ(αy ) < ψ(αy ) = c. we shall use the above proposition to construct examples of topologically homogeneous spaces with countable cs-character and uncountable pseudocharacter. this shows that theorem 2 is specific for topological groups and cannot be generalized to topologically homogeneous spaces. we remind that a topological space x is topologically homogeneous if for any points x,y ∈ x there is a homeomorphism h : x → x with h(x) = y. theorem 5. (1) there is a topologically homogeneous countable regular kω-space x1 with ℵ0 = sbχ(x1) = ψ(x1) < χ(x1) = d and so(x1) = ω; (2) under b = c there is a sequential topologically homogeneous zero-dimensional kω-space x2 with ℵ0 = csχ(x2) < ψ(x2) = c; (3) under b = c there is a sequential topologically homogeneous totally disconnected space x3 with ℵ0 = sbχ(x3) < ψ(x3) = c. we remind that a space x is totally disconnected if for any distinct points x,y ∈ x there is a continuous function f : x → {0,1} such that f(x) 6= f(y), see [en2]. remark 1. the space x1 from theorem 5(1) is the well-known arkhangelskifranklin example [af] (see also [co, 10.1]) of a countable topologically homogeneous kω-space, homeomorphic to no topological group (this also follows from theorem 2). on the other hand, according to [ze2], each topologically homogeneous countable regular space (in particular, x1) is homeomorphic to a quasitopological group, that is a topological space endowed with a separately continuous group operation with continuous inversion. this shows that theorem 2 cannot be generalized onto quasitopological groups (see however [zd] for generalizations of theorems 1 and 2 to some other topologo-algebraic structures). next, we find conditions under which a space with countable cs∗-character is first-countable or has countable sb-character. following [ar3] we define a topological space x to be c-sequential if for each closed subspace y ⊂ x and each non-isolated point y of y there is a sequence (yn) ⊂ y \ {y} convergent to y. it is clear that each sequential space is c-sequential. a point x of a topological space x is called regular gδ if {x} = ∩b for some countable family b of closed neighborhood of x in x, see [lin]. first we characterize spaces with countable sb-character (the first three items of this characterization were proved by lin [lin, 3.13] in terms of (universally) csf-countable spaces). proposition 5. for a hausdorff space x the following conditions are equivalent: (1) x has countable sb-character; 34 t.banakh, l.zdomsky̆ı (2) x is an α1-space with countable cs ∗-character; (3) x is an α4-space with countable cs ∗-character; (4) cs∗χ(x) ≤ ℵ0 and sbχ(x) < p. moreover, if x is c-sequential and each point of x is regular gδ, then the conditions (1)–(4) are equivalent to: (5) cs∗χ(x) ≤ ℵ0 and sbχ(x) < d. next, we give a characterization of first-countable spaces in the same spirit (the equivalences (1) ⇔ (2) ⇔ (5) were proved by lin [lin, 2.8]). proposition 6. for a hausdorff space x with countable cs∗-character the following conditions are equivalent: (1) x is first-countable; (2) x is fréchet-urysohn and has countable sb-character; (3) x is fréchet-urysohn α7-space; (4) χ(x) < p and x has countable tightness. moreover, if each point of x is regular gδ, then the conditions (1)–(4) are equivalent to: (5) x is a sequential space containing no closed copy of s2 or sω; (6) x is a sequential space with χ(x) < d. for fréchet-urysohn (resp. dyadic) compacta the countability of the cs∗character is equivalent to the first countability (resp. the metrizability). we remind that a compact hausdorff space x is called dyadic if x is a continuous image of the cantor discontinuum {0,1}κ for some cardinal κ. proposition 7. (1) a fréchet-urysohn countably compact space is first-countable if and only if it has countable cs∗-character. (2) a dyadic compactum is metrizable if and only if its has countable cs∗character. in light of proposition 7(1) one can suggest that cs∗χ(x) = χ(x) for any compact fréchet-urysohn space x. however that is not true: under ch, csχ(αd) 6= χ(αd) for the one-point compactification αd of a discrete space d of size |d| = ℵ2. surprisingly, but the problem of calculating the cs ∗and cs-characters of the spaces αd is not trivial and the definitive answer is known only under the generalized continuum hypothesis. first we note that the cardinals cs∗χ(αd) and csχ(αd) admit an interesting interpretation which will be used for their calculation. proposition 8. let d be an infinite discrete space. then (1) cs∗χ(αd) = min{w(x) : x is a (regular zero-dimensional) topological space of size |x| = |d| containing no no-trivial convergent sequence}; (2) csχ(αd) = min{w(x) : x is a (regular zero-dimensional) topological space of size |x| = |d| containing no countable non-discrete subspace}. on groups with countable cs∗-character 35 for a cardinal κ we put log κ = min{λ : κ ≤ 2λ} and cof([κ]≤ω) be the smallest size of a collection c ⊂ [κ]≤ω such that each at most countable subset s ⊂ κ lies in some element c ∈ c. observe that cof([κ]≤ω) ≤ κω but sometimes the inequality can be strict: 1 = cof([ℵ0] ≤ω) < ℵ0 and ℵ1 = cof([ℵ1] ≤ω) < ℵℵ01 . in the following proposition we collect all the information on the cardinals cs∗χ(αd) and csχ(αd) we know. proposition 9. let d be an uncountable discrete space. then (1) ℵ1 · log |d| ≤ cs ∗ χ(αd) ≤ csχ(αd) ≤ min{|d|,2 ℵ0 · cof([log |d|]≤ω)} while sbχ(αd) = χ(αd) = |d|; (2) cs∗χ(αd) = csχ(αd) = ℵ1 · log |d| under gch. in spite of numerous efforts some annoying problems concerning cs∗and cs-characters still rest open. problem 1. is there a (necessarily consistent) example of a space x with cs∗χ(x) 6= csχ(x)? in particular, is cs ∗ χ(αc) 6= csχ(αc) in some model of zfc? in light of proposition 8 it is natural to consider the following three cardinal characteristics of the continuum which seem to be new: w1 = min{w(x) : x is a topological space of size |x| = c containing no non-trivial convergent sequence}; w2 = min{w(x) : x is a topological space of size |x| = c containing no non-discrete countable subspace}; w3 = min{w(x) : x is a p-space of size |x| = c}. as expected, a p-space is a t1-space whose any gδ-subset is open. observe that w1 = cs ∗ χ(αc) while w2 = csχ(αc). it is clear that ℵ1 ≤ w1 ≤ w2 ≤ w3 ≤ c and hence the cardinals wi, i = 1,2,3, fall into the category of small uncountable cardinals, see [va]. problem 2. are the cardinals wi, i = 1,2,3, equal to (or can be estimated via) some known small uncountable cardinals considered in set theory? is w1 < w2 < w3 in some model of zfc? our next question concerns the assumption b = c in theorem 5. problem 3. is there a zfc-example of a sequential space x with sbχ(x) < ψ(x) or at least cs∗χ(x) < ψ(x)? propositions 1 and 5 imply that sbχ(x) ∈ {1,ℵ0}∪[d,c] for any c-sequential topological space x with countable cs∗-character. on the other hand, for a sequential topological group g with countable cs∗-character we have a more precise estimate sbχ(g) ∈ {1,ℵ0,d}. problem 4. is sbχ(x) ∈ {1,ℵ0,d} for any sequential space x with countable cs∗-character? as we saw in proposition 7, χ(x) ≤ ℵ0 for any fréchet-urysohn compactum x with csχ(x) ≤ ℵ0. 36 t.banakh, l.zdomsky̆ı problem 5. is sbχ(x) ≤ ℵ0 for any sequential (scattered) compactum x with csχ(x) ≤ ℵ0? now we pass to proofs of our results. on sequence trees in topological groups our basic instrument in proofs of main results is the concept of a sequence tree. as usual, under a tree we understand a partially ordered subset (t,≤) such that for each t ∈ t the set ↓ t = {τ ∈ t : σ ≤ t} is well-ordered by the order ≤. given an element t ∈ t let ↑ t = {τ ∈ t : τ ≥ t} and succ(t) = min(↑ t \ {t}) be the set of successors of t in t . a maximal linearly ordered subset of a tree t is called a branch of t . by maxt we denote the set of maximal elements of a tree t . definition 1. under a sequence tree in a topological space x we understand a tree (t,≤) such that • t ⊂ x; • t has no infinite branch; • for each t /∈ maxt the set min(↑ t \ {t}) of successors of t is countable and converges to t. saying that a subset s of a topological space x converges to a point t ∈ x we mean that for each neighborhood u ⊂ x of t the set s \ u is finite. the following lemma is well-known and can be easily proven by transfinite induction (on the ordinal s(a,a) = min{α : a ∈ a(α)} for a subset a of a sequential space and a point a ∈ ā from its closure) lemma 1. a point a ∈ x of a sequential topological space x belongs to the closure of a subset a ⊂ x if and only if there is a sequence tree t ⊂ x with mint = {a} and maxt ⊂ a. for subsets a,b of a group g let a−1 = {x−1 : x ∈ a} ⊂ g be the inversion of a in g and ab = {xy : x ∈ a, y ∈ b} ⊂ g be the product of a,b in g. the following two lemmas will be used in the proof of theorem 1. lemma 2. a sequential subspace f ⊂ x of a topological group g is first countable if the subspace f−1f ⊂ g has countable sb-character at the unit e of the group g. proof. our proof starts with the observation that it is sufficient to consider the case e ∈ f and prove that f has countable character at e. let {sn : n ∈ ω} be a decreasing sb-network at e in f −1f . first we show that for every n ∈ ω there exists m > n such that s2m ∩ (f −1f) ⊂ sn. otherwise, for every m ∈ ω there would exist xm,ym ∈ sm with xmym ∈ (f−1f) \ sn. taking into account that limm→∞ xm = limm→∞ ym = e, we get limm→∞ xmym = e. since sn is a sequential barrier at e, there is a number m with xmym ∈ sn, which contradicts to the choice of the points xm,ym. now let us show that for all n ∈ ω the set sn ∩ f is a neighborhood of e in f . suppose, conversely, that e ∈ clf (f \ sn0) for some n0 ∈ ω. on groups with countable cs∗-character 37 by lemma 1 there exists a sequence tree t ⊂ f , mint = {e} and maxt ⊂ f \sn0. to get a contradiction we shall construct an infinite branch of t . put x0 = e and let m0 be the smallest integer such that s 2 m0 ∩ f−1f ⊂ sn0. by induction, for every i ≥ 1 find a number mi > mi−1 with s 2 mi ∩f−1f ⊂ smi−1 and a point xi ∈ succ(xi−1) ∩ (xi−1smi). to show that such a choice is always possible, it suffices to verify that xi−1 /∈ maxt . it follows from the inductive construction that xi−1 ∈ f ∩ (sm0 · · ·smi−1) ⊂ f ∩ s 2 m0 ⊂ sn0 and thus xi−1 /∈ maxt because maxt ⊂ f \ sn0. therefore we have constructed an infinite branch {xi : i ∈ ω} of the sequence tree t which is not possible. this contradiction finishes the proof. � lemma 3. a sequential α7-subspace f of a topological group g has countable sb-character provided the subspace f−1f ⊂ g has countable cs-character at the unit e of g. proof. suppose that f ⊂ g is a sequential α7-space with csχ(f −1f,e) ≤ ℵ0. we have to prove that sbχ(f,x) ≤ ℵ0 for any point x ∈ f . replacing f by fx−1, if necessary, we can assume that x = e is the unit of the group g. fix a countable family a of subsets of g closed under group products in g, finite unions and finite intersections, and such that f−1f ∈ a and a|f−1f = {a ∩ (f−1f) : a ∈ a} is a cs-network at e in f−1f . we claim that the collection a|f = {a ∩ f : a ∈ a} is a sb-network at e in f . assuming the converse, we would find an open neighborhood u ⊂ g of e such that for any element a ∈ a with a ∩ f ⊂ u the set a ∩ f fails to be a sequential barrier at e in f . let a′ = {a ∈ a : a ⊂ f ∩ u} = {an : n ∈ ω} and bn = ⋃ k≤n ak. let m−1 = 0 and u−1 ⊂ u be any closed neighborhood of e in g. by induction, for every k ∈ ω find a number mk > mk−1, a closed neighborhood uk ⊂ uk−1 of e in g, and a sequence (xk,i)i∈ω convergent to e so that the following conditions are satisfied: (i) {xk,i : i ∈ ω} ⊂ uk−1 ∩ f \ bmk−1; (ii) the set fk = {xn,i : n ≤ k, i ∈ ω} \ bmk is finite; (iii) uk ∩ (fk ∪ {xi,j : i,j ≤ k}) = ∅ and u 2 k ⊂ uk−1. the last condition implies that u0u1 · · ·uk ⊂ u for every k ≥ 0. consider the subspace x = {xk,i : k,i ∈ ω} of f and observe that it is discrete (in itself). denote by x̄ the closure of x in f and observe that x̄ \x is closed in f . we claim that e is an isolated point of x̄ \ x. assuming the converse and applying lemma 1 we would find a sequence tree t ⊂ x̄ such that min t = {e}, maxt ⊂ x, and succ(e) ⊂ x̄ \ x. by induction, construct a (finite) branch (ti)i≤n+1 of the tree t and a sequence {ci : i ≤ n} of elements of the family a such that t0 = e, |succ(ti) \ tici| < ℵ0 and ci ⊂ ui ∩ (f −1f), ti+1 ∈ succ(ti) ∩ tici, for each i ≤ n. note that the infinite set σ = succ(tn) ∩ tncn ⊂ x converges to the point tn 6= e. on the other hand, σ ⊂ tncn ⊂ tn−1cn−1cn ⊂ · · · ⊂ t0c0 · · ·cn ⊂ u0 · · ·un ⊂ u. it follows from our assumption on a that c0 · · ·cn ∈ a and 38 t.banakh, l.zdomsky̆ı thus (c0 · · ·cn) ∩ f ⊂ bmk for some k. consequently, σ ⊂ x ∩ bmk and σ ⊂ {xj,i : j ≤ k, i ∈ ω} by the item (i) of the construction of x. since e is a unique cluster point of the set {xj,i : j ≤ k, i ∈ ω}, the sequence σ cannot converge to tn 6= e, which is a contradiction. thus e is an isolated point of x̄ \ x and consequently, there is a closed neighborhood w of e in g such that the set v = ({e} ∪x) ∩w is closed in f . for every n ∈ ω consider the sequence sn = w ∩ {xn,i : i ∈ ω} convergent to e. since f is an α7-space, there is a convergent sequence s ⊂ f such that s ∩ sn 6= ∅ for infinitely many sequences sn. taking into account that v is a closed subspace of f with |v ∩ s| = ℵ0, we conclude that the limit point lim s of s belongs to the set v . moreover, we can assume that s ⊂ v . since the space x is discrete, lims ∈ v \ x = {e}. thus the sequence s converges to e. since a′ is a cs-network at e in f , there is a number n ∈ ω such that an contains almost all members of the sequence s. since sm ∩ (sk ∪ an) = ∅ for m > k ≥ n, the sequence s cannot meet infinitely many sequences sm. but this contradicts to the choice of s. � following [vd, §8] by l we denote the countable subspace of the plane r2: l = {(0,0),( 1 n , 1 nm ) : n,m ∈ n} ⊂ r2. the space l is locally compact at each point except for (0,0). moreover, according to lemma 8.3 of [vd], a first countable space x contains a closed topological copy of the space l if and only if x is not locally compact. the following important lemma was proven in [ba1] for normal sequential groups. lemma 4. if a sequential topological group g contains a closed copy of the space l, then g is an α7-space. proof. let h : l → g be a closed embedding and let x0 = h(0,0), xn,m = h( 1 n , 1 nm ) for n,m ∈ n. to show that g is an α7-space, for every n ∈ n fix a sequence (yn,m)m∈n ⊂ g, convergent to the unit e of g. denote by ∗ : g × g → g the group operation on g. it is easy to verify that for every n the subspace dn = {xn,m ∗yn,m : m ∈ n} is closed and discrete in g. hence there exists kn ∈ n such that x0 6= xn,m∗yn,m for all m > kn. consider the subset a = {xn,m ∗ yn,m : n > 0, m > kn} and using the continuity of the group operation, show that x0 6∈ a is a cluster point of a in g. consequently, the set a is not closed and by the sequentiality of g, there is a sequence s ⊂ a convergent to a point a /∈ a. since every space dn is closed and discrete in g, we may replace s by a subsequence, and assume that |s ∩ dn| ≤ 1 for every n ∈ n. consequently, s can be written as s = {xni,mi ∗ yni,mi : i ∈ ω} for some number sequences (mi) and (ni) with ni+1 > ni for all i. it follows that the sequence (xni,mi)i∈ω converges to x0 and consequently, the sequence t = {yni,mi}i∈ω converges to x −1 0 ∗ a. since t ∩ {yni,m}m∈n 6= ∅ for every i, we conclude that g is an α7-space. � on groups with countable cs∗-character 39 lemma 4 allows us to prove the following unexpected lemma 5. a non-metrizable sequential topological group g with countable cscharacter has a countable cs-network at the unit, consisting of closed countably compact subsets of g. proof. given a non-metrizable sequential group g with countable cs-character we can apply lemmas 2–4 to conclude that g contains no closed copy of the space l. fix a countable cs-network n at e, closed under finite intersections and consisting of closed subspaces of g. we claim that the collection c ⊂ n of all countably compact subsets n ∈ n forms a cs-network at e in g. to show this, fix a neighborhood u ⊂ g of e and a sequence (xn) ⊂ g convergent to e. we must find a countably compact set m ∈ n with m ⊂ u, containing almost all points xn. let a = {ak : k ∈ ω} be the collection of all elements n ⊂ u of n containing almost all points xn. now it suffices to find a number n ∈ ω such that the intersection m = ⋂ k≤n ak is countably compact. suppose to the contrary, that for every n ∈ ω the set ⋂ k≤n ak is not countably compact. then there exists a countable closed discrete subspace k0 ⊂ a0 with k0 6∋ e. fix a neighborhood w0 of e with w0 ∩ k0 = ∅. since n is a cs-network at e, there exists k1 ∈ ω such that ak1 ⊂ w0. it follows from our hypothesis that there is a countable closed discrete subspace k1 ⊂ ⋂ k≤k1 ak with k1 ∋ e. proceeding in this fashion we construct by induction an increasing number sequence (kn)n∈ω ⊂ ω, a sequence (kn)n∈ω of countable closed discrete subspaces of g, and a sequence (wn)n∈ω of open neighborhoods of e such that kn ⊂ ⋂ k≤kn ak, wn ∩kn = ∅, and akn+1 ⊂ wn for all n ∈ ω. it follows from the above construction that {e} ∪ ⋃ n∈ω kn is a closed copy of the space l which is impossible. � proofs of main results proof of proposition 1. the first three items can be easily derived from the corresponding definitions. to prove the fourth item observe that for any cs∗-network n at a point x of a topological space x, the family n ′ = {∪f : f ⊂ n} is an sb-network at x. the proof of fifth item is more tricky. fix any cs∗-network n at a point x ∈ x with |n| ≤ cs∗χ(x). let λ = cof(|n|) be the cofinality of the cardinal |n| and write n = ⋃ α<λ nα where nα ⊂ nβ and |nα| < |n| for any ordinals α ≤ β < λ. consider the family m = {∪c : c ∈ [nα] ≤ω, α < λ} and observe that |m| ≤ λ · sup{|[κ]≤ω| : κ < |n|} where [κ]≤ω = {a ⊂ κ : |a| ≤ ℵ0}. it remains to verify that m is a cs-network at x. fix a neighborhood u ⊂ x of x and a sequence s ⊂ x convergent to x. for every α < λ choose a countable subset cα ⊂ nα such that ∪cα ⊂ u and s ∩ (∪cα) = s ∩ (∪{n ∈ nα : n ⊂ u}). it follows that ∪cα ∈ m. let sα = s ∩(∪cα) and observe that sα ⊂ sβ for α ≤ β < λ. to finish the proof it suffices to show that s\sα is finite for some α < λ. then the element ∪cα ⊂ u of m will contain almost all members of the sequence s. 40 t.banakh, l.zdomsky̆ı separately, we shall consider the cases of countable and uncountable λ. if λ is uncountable, then it has uncountable cofinality and consequently, the transfinite sequence (sα)α<λ eventually stabilizes, i.e., there is an ordinal α < λ such that sβ = sα for all β ≥ α. we claim that the set s \ sα is finite. otherwise, s \ sα would be a sequence convergent to x and there would exist an element n ∈ n with n ⊂ u and infinite intersection n ∩(s \sα). find now an ordinal β ≥ α with n ∈ nβ and observe that s ∩ n ⊂ sβ = sα which contradicts to the choice of n. if λ is countable and s \ sα is infinite for any α < λ, then we can find an infinite pseudo-intersection t ⊂ s of the decreasing sequence {s \ sα}α<λ. note that t ∩ sα is finite for every α < λ. since sequence t converges to x, there is an element n ∈ n such that n ⊂ u and n ∩t is infinite. find α < λ with n ∈ nα and observe that n ∩s ⊂ sα. then n ∩t ⊂ n ∩t ∩sα ⊂ t ∩sα is finite, which contradicts to the choice of n. proof of proposition 2. let x be a topological space and fix a point x ∈ x. (1) suppose that cs∗χ(x) < p and fix a cs ∗-network n at the point x such that |n| < p. without loss of generality, we can assume that the family n is closed under finite unions. we claim that n is a cs-network at x. assuming the converse we would find a neighborhood u ⊂ x of x and a sequence s ⊂ x convergent to x such that s \n is infinite for any element n ∈ n with n ⊂ u. since n is closed under finite unions, the family f = {s\n : n ∈ n , n ⊂ u} is closed under finite intersections. since |f| ≤ |n| < p, the family f has an infinite pseudo-intersection t ⊂ s. consequently, t ∩n is finite for any n ∈ n with n ⊂ u. but this contradicts to the facts that t converges to x and n is a cs∗-network at x. the items (2) and (3) follow from propositions 1(5) and 2(1). the item (4) follows from (1,2) and the inequality χ(x) ≤ c holding for any countable topological space x. finally, to derive (5) from (3) use the well-known fact that under gch, λℵ0 ≤ κ for any infinite cardinals λ < κ, see [hj, 9.3.8]. proof of theorem 1. suppose that g is a non-metrizable sequential group with countable cs∗-character. by proposition 2(1), csχ(g) = cs ∗ χ(g) ≤ ℵ0. first we show that each countably compact subspace k of g is first-countable. the space k, being countably compact in the sequential space g, is sequentially compact and so are the sets k−1k and (k−1k)−1(k−1k) in g. the sequential compactness of k−1k implies that it is an α7-space. since csχ((k −1k)−1(k−1k)) ≤ csχ(g) ≤ ℵ0 we may apply lemmas 3 and 2 to conclude that the space k−1k has countable sb-character and k has countable character. next, we show that g contains an open mkω-subgroup. by lemma 5, g has a countable cs-network k consisting of countably compact subsets. since the group product of two countably compact subspaces in g is countably compact, we may assume that k is closed under finite group products in g. we can also on groups with countable cs∗-character 41 assume that k is closed under the inversion, i.e. k−1 ∈ k for any k ∈ k. then h = ∪k is a subgroup of g. it follows that this subgroup is a sequential barrier at each of its points, and thus is open-and-closed in g. we claim that the topology on h is inductive with respect to the cover k. indeed, consider some u ⊂ h such that u ∩k is open in k for every k ∈ k. assuming that u is not open in h and using the sequentiality of h, we would find a point x ∈ u and a sequence (xn)n∈ω ⊂ h \ u convergent to x. it follows that there are elements k1,k2 ∈ k such that x ∈ k1 and k2 contains almost all members of the sequence (x−1xn). then the product k = k1k2 contains almost all xn and the set u ∩ k, being an open neighborhood of x in k, contains almost all members of the sequence (xn), which is a contradiction. as it was proved before each k ∈ k is first-countable, and consequently h has countable pseudocharacter, being the countable union of first countable subspaces. then h admits a continuous metric. since any continuous metric on a countably compact space generates its original topology, every k ∈ k is a metrizable compactum, and consequently h is an mkω-subgroup of g. since h is an open subgroup of g, g is homeomorphic to h × d for some discrete space d. proof of theorem 2. suppose g is a non-metrizable sequential topological group with countable cs∗-character. by theorem 1, g contains an open mkωsubgroup h and is homeomorphic to the product h × d for some discrete space d. this implies that g has point-countable k-network. by a result of shibakov [shi], each sequential topological group with point-countable knetwork and sequential order < ω1 is metrizable. consequently, so(g) = ω1. it is clear that ψ(g) = ψ(h) ≤ ℵ0, χ(g) = χ(h), sbχ(g) = sbχ(h) and ib(g) = c(g) = d(g) = l(g) = e(g) = nw(g) = knw(g) = |d| · ℵ0. to finish the proof it rests to show that sbχ(h) = χ(h) = d. it follows from lemmas 2 and 3 that the group h, being non-metrizable, is not α7 and thus contains a copy of the sequential fan sω. then d = χ(sω) = sbχ(sω) ≤ sbχ(h) ≤ χ(h). to prove that χ(h) ≤ d we shall apply a result of k. sakai [sa] asserting that the space r∞ ×q contains a closed topological copy of each mkω-space and the well-known equality χ(r ∞ × q) = χ(r∞) = d (following from the fact that r∞ carries the box-product topology, see [sch, ch.ii, ex.12]). proof of theorem 5. first we describe two general constructions producing topologically homogeneous sequential spaces. for a locally compact space z let αz = z ∪ {∞} be the one-point extension of z endowed with the topology whose neighborhood base at ∞ consists of the sets αz\k where k is a compact subset of z. thus for a non-compact locally compacts space z the space αz is noting else but the one-point compactification of z. denote by 2ω = {0,1}ω the cantor cube. 42 t.banakh, l.zdomsky̆ı consider the subsets ξ(z) ={(c,(zi)i∈ω) ∈ 2 ω × (αz)ω : zi = ∞ for all but finitely many indices i}; θ(z) ={(c,(zi)i∈ω) ∈ 2 ω × (αz)ω : ∃n ∈ ω such that zi 6= ∞ iff i < n}. observe that θ(z) ⊂ ξ(z). endow the set ξ(z) (resp. θ(z)) with the strongest topology generating the tychonov product topology on each compact subset from the family kξ (resp. kθ), where kξ = {2 ω× ∏ i∈ω ci : ci are compact subsets of αz and almost all ci = {∞}}; kθ = {2 ω × ∏ i∈ω ci : ∃i0 ∈ ω such that ci0 = αz, ci = {∞} for all i > i0 and ci is a compact subsets of z for every i < i0}. lemma 6. suppose z is a zero-dimensional locally metrizable locally compact space. then (1) the spaces ξ(z) and θ(z) are topologically homogeneous; (2) ξ(z) is a regular zero-dimensional kω-space while θ(z) is a totally disconnected k-space; (3) if z is lindelöf, then ξ(z) and θ(z) are zero-dimensional mkωspaces with χ(ξ(z)) = χ(θ(z)) ≤ d; (4) ξ(z) and θ(z) contain copies of the space αz while θ(z) contains a closed copy of z; (5) cs∗χ(ξ(z)) = cs ∗ χ(θ(z)) = cs ∗ χ(αz), csχ(ξ(z)) = csχ(θ(z)) = csχ(αz), sbχ(θ(z)) = sbχ(αz), and ψ(ξ(z)) = ψ(θ(z)) = ψ(αz); (6) the spaces ξ(z) and θ(z) are sequential if and only if αz is sequential; (7) if z is not countably compact, then ξ(z) contains a closed copies of s2 and sω and θ(z) contains a closed copy of s2. proof. (1) first we show that the space ξ(z) is topologically homogeneous. given two points (c,(zi)i∈ω),(c ′,(z′i)i∈ω) of ξ(z) we have to find a homeomorphism h of ξ(z) with h(c,(zi)i∈ω) = (c ′,(z′i)i∈ω). since the cantor cube 2ω is topologically homogeneous, we can assume that c 6= c′. fix any disjoint closed-and-open neighborhoods u,u′ of the points c,c′ in 2ω, respectively. consider the finite sets i = {i ∈ ω : zi 6= ∞} and i ′ = {i ∈ ω : z′i 6= ∞}. using the zero-dimensionality and the local metrizability of z, for each i ∈ i (resp. i ∈ i′) fix an open compact metrizable neighborhood ui (resp. u ′ i) of the point zi (resp. z ′ i) in z. by the classical brouwer theorem [ke, 7.4], the products u × ∏ i∈i ui and u ′ × ∏ i∈i′ u ′ i, being zero-dimensional compact metrizable spaces without isolated points, are homeomorphic to the cantor cube 2ω. now the topological homogeneity of the cantor cube implies the existence of a homeomorpism f : u × ∏ i∈i ui → u ′ × ∏ i∈i′ u ′ i such that f(c,(zi)i∈i) = (c ′,(z′i)i∈i′). let w = {(x,(xi)i∈ω) ∈ ξ(z) : x ∈ u, xi ∈ ui for all i ∈ i} and w ′ = {(x′,(x′i)i∈ω) ∈ ξ(z) : x ′ ∈ u′, x′i ∈ u ′ i for all i ∈ i ′}. on groups with countable cs∗-character 43 it follows that w,w ′ are disjoint open-and-closed subsets of ξ(z). let χ : ω \ i′ → ω \ i be a unique monotone bijection. now consider the homeomorphism f̃ : w → w ′ assigning to a sequence (x,(xi)i∈ω) ∈ w the sequence (x ′,(x′i)i∈ω) ∈ w ′ where (x′,(x′i)i∈i′) = f(x,(xi)i∈i) and x ′ i = xχ(i) for i /∈ i ′. finally, define a homeomorphism h of ξ(z) letting h(x) =      x if x /∈ w ∪ w ′; f̃(x) if x ∈ w ; f̃−1(x) if x ∈ w ′ and observe that h(c,(zi)i∈ω) = (c ′,(z′i)i∈ω) which proves the topological homogeneity of the space ξ(z). replacing ξ(z) by θ(z) in the above proof, we shall get a proof of the topological homogeneity of θ(z). the items (2–4) follow easily from the definitions of the spaces ξ(z) and θ(z), the zero-dimensionality of αz, and known properties of kω-spaces, see [fst] (to find a closed copy of z in θ(z) consider the closed embedding e : z → θ(z), e : z 7→ (z,z0,z,∞,∞, . . .), where z0 is any fixed point of z). to prove (5) apply proposition 3(6,8,9,10). (to calculate the cs∗-, cs-, and sb-characters of θ(z), observe that almost all members of any sequence (an) ⊂ θ(z) convergent to a point a = (c,(zi)) ∈ θ(z) lie in the compactum 2 ω × ∏ i∈ω ci, where ci is a clopen neighborhood of zi if zi 6= ∞, ci = αz if i = min{j ∈ ω : zj = ∞} and ci = {∞} otherwise. by proposition 3(6), the cs∗-, cs-, and sb-characters of this compactum are equal to the corresponding characters of αz.) (6) since the spaces ξ(z) and θ(z) contain a copy of αz, the sequentiality of ξ(z) or θ(z) implies the sequentiality of αz. now suppose conversely that the space αz is sequential. then each compactum k ∈ kξ ∪ kθ is sequential since a finite product of sequential compacta is sequential, see [en1, 3.10.i(b)]. now the spaces ξ(z) and θ(z) are sequential because they carry the inductive topologies with respect to the covers kξ, kθ by sequential compacta. (7) if z is not countably compact, then it contains a countable closed discrete subspace s ⊂ z which can be thought as a sequence convergent to ∞ in αz. it is easy to see that ξ(s) (resp. θ(s)) is a closed subset of ξ(z) (resp. θ(z)). now it is quite easy to find closed copies of s2 and sω in ξ(s) and a closed copy of s2 in θ(s). � with lemma 6 at our disposal, we are able to finish the proof of theorem 5. to construct the examples satisfying the conditions of theorem 5(2,3), assume b = c and use proposition 4 to find a locally compact locally countable space z whose one-point compactification αz is sequential and satisfies ℵ0 = sbχ(αz) < ψ(αz) = c. applying lemma 6 to this space z, we conclude that the topologically homogeneous k-spaces x2 = ξ(z) and x3 = θ(z) give us required examples. 44 t.banakh, l.zdomsky̆ı the example of a countable topologically homogeneous kω-space x1 with sbχ(x1) = ψ(x1) < χ(x1) can be constructed by analogy with the space θ(n) (with that difference that there is no necessity to involve the cantor cube) and is known in topology as the ankhangelski-franklin space, see [af]. we briefly remind its construction. let s0 = {0, 1 n : n ∈ n} be a convergent sequence and consider the countable space x1 = {(xi)i∈ω ∈ s ω 0 : ∃n ∈ ω such that xi 6= 0 iff i < n} endowed with the strongest topology inducing the product topology on each compactum ∏ i∈ω ci for which there is n ∈ ω such that cn = s0, ci = {0} if i > n, and ci = {xi} for some xi ∈ s0 \ {0} if i < n. by analogy with the proof of lemma 6 it can be shown that x1 is a topologically homogeneous kω-space with ℵ0 = sbχ(x1) = ψ(x1) < χ(x1) = d and so(x1) = ω. proof of proposition 5. the equivalences (1) ⇔ (2) ⇔ (3) were proved by lin [lin, 3.13] in terms of (universally) csf-countable spaces. to prove the other equivalences apply lemma 7. a hausdorff topological space x is an α4-space provided one of the following conditions is satisfied: (1) x is a fréchet-urysohn α7-space; (2) x is a fréchet-urysohn countably compact space; (3) sbχ(x) < p; (4) sbχ(x) < d, each point of x is regular gδ, and x is c-sequential. proof. fix any point x ∈ x and a countable family {sn}n∈ω of sequences convergent to x in x. we have to find a sequence s ⊂ x \ {x} convergent to x and meeting infinitely many sequences sn. using the countability of the set ⋃ n∈ω sn find a decreasing sequence (un)n∈ω of closed neighborhoods of x in x such that ( ⋂ n∈ω un ) ∩ ( ⋃ n∈ω sn) = {x}. replacing each sequence sn by its subsequence sn ∩ un, if necessary, we can assume that sn ⊂ un. (1) assume that x is a fréchet-urysohn α7-space. let a = {a ∈ x : a is the limit of a convergent sequence s ⊂ x meeting infinitely many sequences sn}. it follows from our assumption on (sn) and (un) that a ⊂ ⋂ n∈ω un. it suffices to consider the non-trivial case when x /∈ a. in this case x is a cluster point of a (otherwise x would be not α7). since x is fréchet-urysohn, there is a sequence (an) ⊂ a convergent to x. by the definition of a, for every n ∈ ω there is a sequence tn ⊂ x convergent to a and meeting infinitely many sequences sn. without loss of generality, we can assume that tn ⊂ ⋃ i>n si (because a ∈ a \ {x} and thus a /∈ ⋃ n∈ω sn). it is easy to see that x is a cluster point of the set ⋃ n∈ω tn. since x is fréchet-urysohn, there is a sequence t ⊂ ⋃ n∈ω tn convergent to x. now it remains to show that the set t meets infinitely many sequences sn. assuming the converse we would find n ∈ ω such that t ⊂ ⋃ i≤n sn. then t ⊂ ⋃ i≤n tn which is not possible since ⋃ i≤n ti is a compact set failing to contain the point x. on groups with countable cs∗-character 45 (2) if x is fréchet-urysohn and countably compact, then it is sequentially compact and hence α7, which allows us to apply the previous item. (3) assume that sbχ(x) < p and let n be a sb-network at x of size |n| < p. without loss of generality, we can assume that the family n is closed under finite intersections. let s = ⋃ n∈ω sn and fn,n = n ∩ ( ⋃ i≥n si) for n ∈ n and n ∈ ω. it is easy to see that the family f = {fn,n : n ∈ n , n ∈ ω} consists of infinite subsets of s, has size |f| < p, and is closed under finite intersection. now the definition of the small cardinal p implies that this family f has an infinite pseudo-intersection t ⊂ s. then t is a sequence convergent to x and intersecting infinitely many sequences sn. this shows that x is an α4-space. (4) assume that the space x is c-sequential, each point of x is regular gδ, and sbχ(x) < d. in this case we can choose the sequence (un) to satisfy ⋂ n∈ω un = {x}. fix an sb-network n at x with |n| < d. for every n ∈ ω write sn = {xn,i : i ∈ n}. for each sequential barrier n ∈ n find a function fn : ω → n such that xn,i ∈ n for every n ∈ ω and i ≥ fn(n). the family of functions {fn : n ∈ n} has size < d and hence is not cofinal in n ω. consequently, there is a function f : ω → n such that f 6≤ fn for each n ∈ n . now consider the sequence s = {xn,f(n) : n ∈ ω}. we claim that x is a cluster point of s. indeed, given any neighborhood u of x, find a sequential barrier n ∈ n with n ⊂ u. since f 6≤ fn, there is n ∈ ω with f(n) > fn(n). it follows from the choice of the function fn that xn,f(n) ∈ n ⊂ u. since s \ un is finite for every n, {x} = ⋂ n∈ω un is a unique cluster point of s and thus {x} ∪ s is a closed subset of x. now the c-sequentiality of x implies the existence of a sequence t ⊂ s convergent to x. since t meets infinitely many sequences sn, the space x is α4. � proof of proposition 6. suppose a space x has countable cs∗-character. the implications (1) ⇒ (2,3,4,5) are trivial. the equivalence (1) ⇔ (2) follows from proposition 1(2). to show that (3) ⇒ (2), apply lemma 7 and proposition 5(3 ⇒ 1). to prove that (4) ⇒ (2) it suffices to apply proposition 5(4 ⇒ 1) and observe that x is fréchet-urysohn provided χ(x) < p and x has countable tightness. this can be seen as follows. given a subset a ⊂ x and a point a ∈ ā from its closure, use the countable tightness of x to find a countable subset n ⊂ a with a ∈ n̄. fix any neighborhood base b at x of size |b| < p. we can assume that b is closed under finite intersections. by the definition of the small cardinal p, the family {b ∩n : b ∈ b} has infinite pseudo-intersection s ⊂ n. it is clear that s ⊂ a is a sequence convergent to x, which proves that x is fréchet-urysohn. (5) ⇒ (2). assume that x is a sequential space containing no closed copies of sω and s2 and such that each point of x is regular gδ. since x is sequential and contains no closed copy of s2, we may apply lemma 2.5 [lin] to conclude that x is fréchet-urysohn. next, theorem 3.6 of [lin] implies that x is an 46 t.banakh, l.zdomsky̆ı α4-space. finally apply proposition 5 to conclude that x has countable sbcharacter and, being fréchet-urysohn, is first countable. the final implication (6) ⇒ (2) follows from (5) ⇒ (2) and the well-known equality χ(sω) = χ(s2) = d. proof of proposition 7. the first item of this proposition follows from proposition 6(3 ⇒ 1) and the observation that each fréchet-urysohn countable compact space, being sequentially compact, is α7. now suppose that x is a dyadic compact with cs∗χ(x) ≤ ℵ0. if x is not metrizable, then it contains a copy of the one-point compactification αd of an uncountable discrete space d, see [en1, 3.12.12(i)]. then cs ∗ χ(αd) ≤ cs ∗ χ(x) ≤ ℵ0 and by the previous item, the space αd, being fréchet-urysohn and compact, is first-countable, which is a contradiction. proof of proposition 8. let d be a discrete space. (1) let κ = cs∗χ(αd) and λ1 (λ2) is the smallest weight of a (regular zerodimensional) space x of size |x| = |d|, containing no non-trivial convergent sequence. to prove the first item of proposition 8 it suffices to verify that λ2 ≤ κ ≤ λ1. to show that λ2 ≤ κ, fix any cs ∗-network n at the unique non-isolated point ∞ of αd of size |n| ≤ κ. the algebra a of subsets of d generated by the family {d \ n : n ∈ n} is a base of some zero-dimensional topology τ on d with w(d,τ) ≤ κ. we claim that the space d endowed with this topology contains no infinite convergent sequences. to get a contradiction, suppose that s ⊂ d is an infinite sequence convergent to a point a ∈ d \ s. then s converges to ∞ in αd and hence, there is an element n ∈ n such that n ⊂ αd\{a} and n ∩s is infinite. consequently, u = d\n is a neighborhood of a in the topology τ such that s \ u is infinite which contradicts to the fact that s converges to a. now consider the equivalence relation ∼ on d: x ∼ y provided for every u ∈ τ (x ∈ u) ⇔ (y ∈ u). since the space (d,τ) has no infinite convergent sequences, each equivalence class [x]∼ ⊂ d is finite (because it carries the anti-discrete topology). consequently, we can find a subset x ⊂ d of size |x| = |d| such that x 6∼ y for any distinct points x,y ∈ x. clearly that τ induces a zero-dimensional topology on x. it rests to verify that this topology is t1. given any two distinct point x,y ∈ x use x 6∼ y to find an open set u ∈ a such that either x ∈ u and y /∈ u or x /∈ u and y ∈ u. since d \ u ∈ a, in both cases we find an open set w ∈ a such that x ∈ w but y /∈ w . it follows that x is a t1-space containing no non-trivial convergent sequence and thus λ2 ≤ w(x) ≤ |a| ≤ |n| ≤ κ. to show that κ ≤ λ1, fix any topology τ on d such that w(d,τ) ≤ λ1 and the space (d,τ) contains no non-trivial convergent sequences. let b be a base of the topology τ with |b| ≤ λ1, closed under finite unions. we claim that the collection n = {αd \ b : b ∈ b} is a cs∗-network for αd at ∞. fix any neighborhood u ⊂ αd of ∞ and any sequence s ⊂ d convergent to ∞. write {x1, . . . ,xn} = αd \ u and by finite induction, for every i ≤ n find a on groups with countable cs∗-character 47 neighborhood bi ∈ b of xi such that s \ ⋃ j≤i bj is infinite. since b is closed under finite unions, the set n = αd \ (b1 ∪ · · · ∪ bn) belongs to the family n and has the properties: n ⊂ u and n ∩ s is infinite, i.e., n is a cs∗-network at ∞ in αd. thus κ ≤ |n| ≤ |b| ≤ λ1. this finishes the proof of (1). an obvious modification of the above argument gives also a proof of the item (2). proof of proposition 9. let d be an uncountable discrete space. 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[ze2] e. zelenyuk, on group operations on homogeneous spaces, proc. amer. math. soc. 132 (2004), 1219-1222. received october 2002 accepted april 2003 taras banakh and lubomyr zdomsky̆ı (tbanakh@franko.lviv.ua, lubomyr@opari.ltg.lviv.ua) instytut matematyki, akademia świȩtorzyska, kielce, and department of mathematics, ivan franko lviv national university, universytetska 1, lviv, 79000, ukraine @ appl. gen. topol. 15, no. 2(2014), 229-233doi:10.4995/agt.2014.2230 c© agt, upv, 2014 radicals in the class of compact right topological rings mihail ursul and adela tripe a department of mathematics and computer science, png university of technology, private mail bag lae 411, lae, papua new guinea (mihail.ursul@gmail.com) b department of mathematics and computer science, university of oradea, str. universităţii no. 1, oradea, romania (tadela@uoradea.ro) abstract we construct in this article three radicals in the class of compact right topological rings. we also prove that a simple left noetherian compact right topological ring of prime characteristic is finite. 2010 msc: primary 16w80; secondary 54hxx. keywords: right topological ring; radical; left noetherian topological ring. 1. introduction the notion of a compact left (right) topological ring was introduced in [6] and later in [3]. a fundamental question occuring at the study of any class of rings is: what are simple rings which are in this class? it is known, for instance, that a compact ring is simple if and only if it is a matrix ring over a finite division ring [4]. we study in this paper simple compact right topological rings. we prove that every left noetherian compact right topological simple ring with identity of prime characteristic is finite. answering a question posed by professor richard wiegandt, we construct some radicals in the class of compact right topological rings. received 24 march 2014 – accepted 24 july 2014 http://dx.doi.org/10.4995/agt.2014.2230 m. ursul and a. tripe 2. notation and conventions all rings are assumed associative not necessarily with identity. topological groups are assumed hausdorff. the closure of a subset a of a topological space is denoted by a. a j-semismiple ring is a ring whose jacobson radical is zero. 3. a few elementary results differences between elementary properties of topological rings and right topological rings are essential. we will give an example of a right topological ring for which the right annihilator is not closed. we construct also a compact right topological ring with nonclosed center. let r be a compact ring with identity and α : r → r a group endomorphism. define on the group a = r × r with the product topology the multiplication (a, x)(b, y) = (0, aα(b)). then a3 = 0 and a is a compact right topological ring. let p be a prime number and r = fℵ0p with the product topology or r = fp[[t]] be the ring of formal power series over fp. then the additive group of r contains a dense subgroup b of index p. let α : r → r be a group homomorphism for which ker α = b. then annra ⊃ r × b, hence annra is dense. since annra 6= a, it is not closed. theorem 3.1. let (a, t ) be a compact abelian group.then the ring enda of all not necessarily continuous endomorphisms of a with the topology of pointwise convergence is a compact right topological ring. proposition 3.2. for any right topological ring r the closure of its abstract center z = z(r) is a subring. proof. indeed, z · z ⊂ z · z = z · z ⊂ z = z. � example 3.3. a compact topological ring with nonclosed center. the group r admits a compact group topology (see [2]). according to theorem 3.1 endr admits a compact right ring topology. since the center of endr is q, the center is not closed. the subring q of endr is connected. 4. simple compact right topological rings as usual, a ring r is called simple if r2 6= 0 and r has no ideals different from 0 and r. recall that an ideal m of a ring r is called maximal if the factor ring r/m is a simple ring. lemma 4.1. there are no infinite compact right topological simple artinian rings of prime characteristic. proof. assume on the contrary that there exists an infinite compact right topological simple artinian ring r of prime characteristic. there exists an infinite cardinal α such that r (+) ∼=top z (p) α . we identify the additive group r (+) of r with z (p) α . consider the ring s = endc (r (+)) c© agt, upv, 2014 appl. gen. topol. 15, no. 2 230 radicals in the class of compact right topological rings of continuous endomorphisms of r (+) with the compact-open topology. we note that endc (r (+)) has a fundamental system of neighborhoods of 0 consisting of right ideals. indeed, denote for each neighborhood v of 0r by t (v ) the right ideal {ρ ∈ s | ρ (r) ⊂ v }. the family {t (v )}, where v runs all neighborhoods of r, forms a fundamental system of neighborhoods of 0s consisting of right ideals. we shall estimate now the weight of s. fix β < α . put hβ = { ρ ∈ s | ρ (r) ⊂ z (p)β × ∏ γ 6=β {0}γ } . fact 1. hβ is a discrete right ideal of s. indeed, hβ ∩ t ({0} × ∏ γ 6=β z(p)γ}) = 0. fact 2. |hβ| ≤ α. indeed, fix β0, . . . , βn < α, n ∈ ω. put sβ0...βn = {χ ∈ hβ | χ({0}β0 × · · ·× {0}βn × ∏ γ /∈{β0,...,βn} z(p)γ) = 0}. since the values of every π ∈ sβ0...βn are determined by the set z(p)β0 × · · · × z(p)βn × ∏ γ /∈{β0,...,βn} {0}γ, sβ0...βn is finite. this implies that |hβ| ≤ α. fact 3. | ∑ β<α hβ| ≤ α. since |hβ| ≤ α for every β < α, the cardinality of ∑ β<α hβ is ≤ α×α = α. fact 4. ∑ β<α hβ is a dense subgroup of s. denote for each β < α the endomorphism qβ of r(+): qβ : r → r, (xγ) 7→ xβ × ∏ δ 6=β 0δ and by prβ the projection of r on z(p)β = z(p). evidently, qβ ∈ h. we have that qβ0ϕ + · · · + qβnϕ ∈ h for each β0, . . . , βn < α, n ∈ ω, ϕ ∈ s. set v = {0}β0 × · · · × {0}βn × ∏ β 6=β0,...,βn z(p)β. then we have: prβi[(ϕ − σ n j=0qβj ϕ)(x)] = prβi[ϕ(x) − σ n j=0qβj ϕ(x)] = prβi[ϕ(x)] − σ n j=0prβi[qβj ϕ(x)] = prβi[ϕ(x)] − prβi[ϕ(x)] = 0 , for every i ∈ {0, n} and every x ∈ r. we obtain (ϕ − σnj=0qβj ϕ)(r) ⊂ v , i.e., ϕ − σ n j=0qβj ϕ ∈ t (v ), hence ϕ ∈ h + t (v ). we have proved that s = h + t (v ). the topological group s has a fundamental system of neighborhoods of 0 of cardinality ≤ α. it follows that w(s) ≤ α, where w(s) is the weight of s. define χ : r → s, x 7→ rx. then χ is a ring anti-isomorphism of r in s. since s has a fundamental system of neighborhoods of 0 consisting of right ideals, |χ(r)| ≤ α, a contradiction, since |r| = 2α > α. � theorem 4.2. every simple left noetherian compact right topological ring r with identity of prime characteristic is finite. proof. since r is a compact right topological ring, every principal left ideal of r is closed. by compactness, every finitely generated left ideal is closed. since r is left noetherian, every left ideal of r is closed. if f is a filter basis consisting of cosets with respect to left ideals, then by compactness of r its intersection is non-empty. thus, r is a discrete left linearly compact simple c© agt, upv, 2014 appl. gen. topol. 15, no. 2 231 m. ursul and a. tripe ring. it follows that r is a left linearly compact discrete simple ring. then, (see [7], [5]), r is an artinian simple ring. by lemma 4.1, r is finite. � 5. radicals in the class of compact right topological rings lemma 5.1. if r is a compact right topological ring and i an ideal with identity e, then i is a topological direct summand. proof. it is well-known that e will be a central idempotent. we have that re and r(1−e) = {x ∈ r | xe = 0} are closed ideals of r. since r = re⊕r(1−e) is a direct sum, by compactness of r, this sum is topological. � let l be a class of hausdorff right topological rings. we will say that the operator ρ is a radical in the class l if: (i) ρ(r) is a closed ideal of r for each r ∈ l; (ii) ρ(ρ(r)) = ρ(r); (iii) f(ρ(r)) ⊂ ρ(f(r)) for any continuous homomorphism f : r → s, where r, s ∈ l; (iv) ρ(r/ρ(r)) = 0. remark 5.2. our definition differs from the definition of a radical in classes of topological rings introduced in [1]. in [1] it is assumed that the class of topological rings for which is defined a radical is closed under taking of ideals. evidently, this condition is not filled for compact right topological rings. we will construct three radicals in the class of compact right topological rings. first, we note that the connected component of zero is a radical in the class of compact right topological rings. indeed, if r is a compact right topological ring, then its component r0 of zero is equal to ∩n∈nnr, hence r0 is a closed ideal of r. theorem 5.3. let l be the class of all compact right topological rings and ρ(r) = ∩m, where m runs all open maximal ideals of r. then ρ is a radical. proof. since every open ideal is closed, ρ(r) is a closed ideal of r. then ρ(r/ρ(r) = 0 and by kaplansky’s theorem r/ρ(r) has identity. the factor ring r/ρ(r) is a compact j-semisimple ring. denote by h the two-sided ideal generated by ρ(ρ(r)). then ρ(ρ(r)) ⊂ h ⊂ ρ(r). by andrunakievich’s lemma, h3 ⊂ ρ(ρ(r)). since ρ(r)/ρ(ρ(r)) is semiprime, h ⊂ ρ(ρ(r)), hence h = ρ(ρ(r)). we claim that if f : r → s is a continuous homomorphism, then f(ρ(r)) ⊂ ρ(f(r)). indeed, consider f as a surjective homomorphism of r on l = f(r). then ρ(l) = ∩m′ where m′ runs all maximal ideals of l. it follows that f−1(ρ(l)) = ∩f−1(m′) where m′ runs all open maximal ideals of l. if m′ is a maximal open ideal of l, then f−1(m′) will be an open maximal ideal of r. it follows that f−1(ρ(l)) ⊃ ρ(r), hence f(ρ(r)) ⊂ ρ(f(l)). c© agt, upv, 2014 appl. gen. topol. 15, no. 2 232 radicals in the class of compact right topological rings now we will prove that ρ2 = ρ. indeed, consider the factor ring r/ρ(ρ(r)). the ideal ρ(r)/ρ(ρ(r)) has identity, therefore is a topological direct summand. we obtain that r/ρ(ρ(r)) is a topological direct sum of r/ρ(r) and ρ(ρ(r)). it follows that ρ(r/ρ(ρ(r))) = 0, hence ρ(r) ⊂ ρ(ρ(r)) and so ρ(ρ(r)) = ρ(r). � we will obtain another radical φ different from ρ if we will put φ(r) the intersection of all open ideals v of a compact right topological ring r for which r/v is a field. theorem 5.4. for any compact right topological ring r and for any its closed ideal i holds ρ(i) = i ∩ ρ(r). proof. if φ : r → r/ρ(r) is the natural homomorphism, then the restriction φ |i: i → i + ρ(r)/ρ(r) is surjective and i + ρ(r)/ρ(r) is a closed ideal of r/ρ(r). since r/ρ(r) is a product of finite simple rings ([4]), i + ρ(r)/ρ(r) is ρ-semisimple. thus ρ(i) ⊂ ρ(r) ∩ i. denote by h the two-sided ideal of r generated by ρ(i). then ρ(r) ⊂ h ⊂ i. by andrunakievich’s lemma, h3 ⊂ ρ(r). since i/ρ(i) is semiprime, h ⊂ ρ(i), hence h = ρ(i). since i/ρ(i) is a compact semisimple ring, it has an identity [4]. by lemma 5.1 there exists a closed ideal s1 of r/ρ(r) such that r = s1 ⊕ (i/ρ(i)), a topological direct sum. by a standard result, s1 = s/ρ(r), where s is a closed ideal of r. we have that r = s + i and s ∩ i = ρ(i). the factor ring r/s is topologically isomorphic to i/i ∩ s = i/ρ(i), hence is ρ-semisimple. it follows that ρ(r) ⊂ s , hence ρ(r) ∩ i ⊂ s ∩ i = ρ(i). � acknowledgements. the authors are grateful to professor richard wiegandt for his interest in the results of this article. we are also indebted to professor fred van oystaeyen whose remarks stimulated the proof of theorem 4.2. we are also grateful to the refree for her/his important sugestions. references [1] m. cioban, p. soltan and c. gaindric, academician vladimi arnautov 70th anniversary, bul. acad. stiinte repub. mold., 3 (61) (2009), 118–124. [2] p. halmos, comment on the real line, bull. amer. math. soc. 50 (1944), 877–878. [3] n. hindman, j. pym and d. strauss, multiplications in additive compactifications of n and z, topology appl. 131 (2003), 149–176. [4] i. kaplansky, topological rings, amer. j. math. 69, no. 1 (1957), 153–183. [5] h. leptin, linear kompakte moduln und ringe, i, math. z. 62 (1955), 241–267. [6] m. i. ursul, compact left topological rings, scripta scientarum mathematicarum 1 (1997), 257–270. [7] d. zelinsky, raising idempotents, duke math. j. 21 (1954), 315–322. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 233 dowporterstwagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 243-264 spaces whose pseudocompact subspaces are closed subsets alan dow, jack r. porter, r.m. stephenson, jr., and r. grant woods∗ dedicated to professor w. wistar comfort on the occasion of his seventieth birthday abstract. every first countable pseudocompact tychonoff space x has the property that every pseudocompact subspace of x is a closed subset of x (denoted herein by “fcc”). we study the property fcc and several closely related ones, and focus on the behavior of extension and other spaces which have one or more of these properties. characterization, embedding and product theorems are obtained, and some examples are given which provide results such as the following. there exists a separable moore space which has no regular, fcc extension space. there exists a compact hausdorff fréchet space which is not fcc. there exists a compact hausdorff fréchet space x such that x, but not x2, is fcc. 2000 ams classification: primary 54d20; secondary 54d35, 54b10, 54d55 or 54g20 keywords: feebly compact, pseudocompact, fréchet, sequential, product, extension ∗the first author gratefully acknowledges partial research support from the national science foundation, grant no. 2975010131. the third and fourth authors gratefully acknowledge partial research support from the university of kansas and the sabbatical leave programs of their respective institutions, and in the case of the fourth author, from the natural sciences and engineering research council of canada. 244 a. dow, j. r. porter, r. m. stephenson, jr., and r. grant woods 1. introduction and terminology for topological spaces x and y , c(x, y ) will denote the family of continuous functions from x into y , c(x) will denote c(x, r), and c∗(x) will denote the family of bounded functions in c(x). a space x is called pseudocompact provided that c(x) = c∗(x). this definition was first given for tychonoff spaces, i.e., completely regular t1-spaces, by e. hewitt [10]. for terms not defined here, see [5], [6] or [15]. except where noted otherwise, no separation axioms are assumed. some properties of interest that are closely related to pseudocompactness are listed in theorem 1.1. theorem 1.1. let x be a space. then each statement below implies the next one, and all of properties (b1)–(b6) are equivalent. (a) the space x is pseudocompact and completely regular. (b1) every locally finite family of open sets of x is finite. (b2) every pairwise disjoint locally finite family of open sets of x is finite. (b3) every sequence of nonempty open subsets of x has a cluster point in x. (b4) if u = {un : n ∈ n} is a sequence of nonempty open subsets of x such that ui ∩ uj = ∅ whenever i 6= j, then u has a cluster point in x. (b5) every countable open filter base on x has an adherent point. (b6) every countable open cover of x has a finite subcollection whose union is dense in x. (c) x is pseudocompact. we recall that the adherence of a filter base f on a space x is the intersection of the closures of the members of f, and by a cluster point of a sequence {un : n ∈ n} of subsets of a space x is meant a point p ∈ x such that for every neighborhood v of p, v ∩ un 6= ∅ for infinitely many integers n. a sequence denoted {un : n ∈ n} will be referred to as a pairwise disjoint sequence provided that ui ∩ uj = ∅ whenever i 6= j . proofs or references to proofs of the different results in theorem 1.1 can be found in [1], [7], [8], [15] or [26]. these properties have been found useful by a number of authors, especially (b2), which has been referred to in [26] as feebly compact and attributed to s. mardešić and p. papić, and (b1), which was called lightly compact in [1]. i. glicksberg [8] noted that every pseudocompact completely regular space satisfies (b2), and every space satisfying (b2) is pseudocompact. one immediate corollary to theorem 1.1 that will be used below is the following. corollary 1.2 ([8]). let x be a topological space. (a) the union of finitely many feebly compact subspaces of x is feebly compact. (b) if x is feebly compact and u is any open subset of x, then u is a feebly compact subspace of x. spaces whose pseudocompact subspaces are closed subsets 245 (c) if d is a feebly compact subspace of x and d ⊆ g ⊆ d, then g is feebly compact. definition 1.3. we shall call a topological space x feebly compact closed (“fcc”) provided that x is feebly compact and every feebly compact subspace of x is a closed subset of x. definition 1.4. a space x will be called sequentially feebly compact provided that for every sequence {un : n ∈ n} of nonempty open subsets of x there exist a point p ∈ x and a strictly increasing sequence {ni : i ∈ n} in n such that for every neighborhood v of p, v ∩ uni 6= ∅ for all but finitely many i ∈ n. 2. the properties fcc and sequentially feebly compact the property fcc has been studied previously (but not named or labeled) by several authors. it was proved in [23] that every first countable feebly compact hausdorff space, and hence every first countable pseudocompact tychonoff space, is fcc. then a proof was given in [14] that if a feebly compact space x is e1, i.e., if every point x of x is an intersection of countably many closed neighborhoods of x, then x is fcc. the concept has been used in the study of maximal feeble compactness. by a maximal feebly compact space is meant a feebly compact space (x, t ) such that for every feebly compact topology u on x, if t ⊆ u then u = t . using the result of d. cameron [3], that an fcc, submaximal space (i.e., a space in which every dense set is open) is maximal feebly compact, and a result of a.b. raha [17], the authors proved in [16] that a topological space is maximal feebly compact space if and only if it is fcc and submaximal. using the latter, a number of examples of maximal feebly compact spaces are given in [16], e.g., the well-known isbell-mrówka space ψ described in [6, 5i] and in the proof below of theorem 2.12. the property fcc was also considered in the article [9], where the relationship between countably compact regular spaces which are fcc and those which are fréchet was studied. the next lemma provides conditions each of which implies or is implied by, or under suitable restrictions is equivalent to, the property fcc. let us recall that a space x is called semiregular provided that that the regular open sets (i.e., sets having the form int(cl(a)), where a is an open subset of x) form a base for the topology on x. lemma 2.1. let x be a topological space, and consider the conditions below. (f1) every feebly compact subspace of x is a closed subset of x. (f2) for every feebly compact subspace s of x, dense subset d of s, and point p ∈ s\d, there exists a pairwise disjoint sequence k = {kn : n ∈ n} of nonempty open subsets of d such that for every neighborhood v of p in s, v ⊇ kn for all but finitely many n ∈ n. (f3) every feebly compact subspace of x with dense interior is a closed subset of x. 246 a. dow, j. r. porter, r. m. stephenson, jr., and r. grant woods (f4) for every open subset s of x and point p ∈ s\s, there exists a pairwise disjoint sequence k = {kn : n ∈ n} of nonempty open subsets of s such that k has no cluster point in x \{p}, and for every neighborhood v of p in s, v ⊇ kn for all but finitely many n ∈ n. then the following hold. (a) property (f1) implies (f2) and (f3), and if x is a hausdorff space then (f2) implies (f1). (b) property (f4) implies (f3), and if x is feebly compact then (f3) implies (f4). (c) if s is semiregular then in each of the statements (f2) and (f4), the containment “v ⊇ kn” may be replaced by “v ⊇ kn.” (d) if x is fréchet, hausdorff and scattered, then it has property (f1). (e) if x is fréchet and hausdorff and has a dense set of isolated points, then it has property (f3). proof. we prove (b). the proof of (a) is similar. (f4) implies (f3). suppose (f3) is false. then there exist an open subset s of x, a feebly compact subspace f of x with s ⊆ f ⊂ s, and a point p ∈ f \f . it would follow that s = f and thus p ∈ s \ s. by corollary 1.2 (c), the feeble compactness of f , and the relation f ⊆ s \ {p} ⊆ f , the subspace s \ {p} would be feebly compact. by theorem 1.1, every sequence k = {kn : n ∈ n} of nonempty open subsets of x such that each kn ⊂ s would have a cluster point in s \ {p}. therefore, (f4) would not hold. suppose x is feebly compact and (f3) holds. let s and p be as in the hypothesis of (f4). it follows from (f3) and the characterizations in theorem 1.1 that there exists a pairwise disjoint sequence w of nonempty open sets of the space s \ {p} such that w has no cluster point in s \ {p}. define k = {kn : n ∈ n}, where for each n ∈ n, kn = wn ∩ s. since s is dense in s \ {p} and open in x, it follows from the properties of w that k is a pairwise disjoint sequence of nonempty open subsets of x, as well as of s, which has no cluster point in x \ {p}. by the feeble compactness of x, k must have a cluster point, so p is the unique cluster point of k in x. if there were an infinite subset j of n and a neighborhood v of p in s such that kj \ v 6= ∅ for every j ∈ j, then {kj \ v : j ∈ j} would be an infinite locally finite family of open subsets of x, in contradiction of theorem 1.1. statement (c) is obvious. let us prove (d). the proof of (e) is similar. suppose y ⊆ x is feebly compact and p ∈ y . let i be the set of isolated points of the space y . then cly i = y since x is scattered, and thus p ∈ i. as x is fréchet, there is a sequence {xn : n ∈ n} in i which converges to p. then {{xn} : n ∈ n} is a sequence of nonempty open sets of the feebly compact hausdorff space y which has only p as a cluster point. hence p ∈ y . therefore, y is a closed subset of x. � spaces whose pseudocompact subspaces are closed subsets 247 theorem 2.2. let x be a topological space. then the following hold. (a) if x is a feebly compact space which is either (i) e1, or (ii) compact hausdorff and either hereditarily metacompact or hereditarily realcompact, or (iii) fréchet, hausdorff and scattered, then it is fcc. (b) if x is fcc, then it is a feebly compact t1-space and has the properties (f1)–(f4). (c) if x is a countably compact fcc space, then it is (i) (y. tanaka) fréchet and (ii) sequentially compact. (d) if x is feebly compact and either (i) has property (f3) or (ii) is a sequential space, then x is sequentially feebly compact. in particular, fcc implies sequentially feebly compact. (e) if x is feebly compact, fréchet and hausdorff and has a dense set of isolated points, then it has properties (f3)–(f4). (f) if x is sequentially feebly compact, then it is feebly compact. proof. as noted above, part (i) of (a) is obtained in [14]. since by results of e. hewitt [10] and s. watson [28], every realcompact and every metacompact pseudocompact tychonoff space is compact, one obtains (ii) of (a). statement (iii) follows from lemma 2.1 (d). obviously (b) holds. in [9] a proof was given that a statement like (c) (i) holds for regular spaces, and the author of [9] attributed the result to y. tanaka. here is a similar proof that does not require regularity of the space x: suppose a ⊂ x and x ∈ a\a. since a\{x} is not feebly compact and has a as a dense subset, there exists a pairwise disjoint sequence u = {un : n ∈ n} of nonempty open subsets of a which has no cluster point in a \ {x}. choose xn ∈ un for each n ∈ n. then the set c = {xn : n ∈ n} is a discrete subspace of the countably compact t1-space c = c ∪{x}, and consequently, the sequence {xn} in a must converge to x. the statement (c) (ii) follows from the easily verified fact that every countably compact t1 fréchet space is sequentially compact. we prove (d). let u = {un : n ∈ n} be a sequence of nonempty open subsets of the space x. we wish to show that there exist a point p ∈ x and a strictly increasing sequence {ni : i ∈ n} in n such that for every neighborhood v of p, v ∩ uni 6= ∅ for all but finitely many i ∈ n (or equivalently, there exist a point p ∈ x and an infinite subset j of n such that v ∩ uj 6= ∅ for all but finitely many j ∈ j). suppose first that the hypothesis of (d) (i) holds. let us consider two cases. case 1: suppose there are an infinite subset j of n and a point p ∈ x such that p ∈ uj for every j ∈ j. then p and j have the required properties. case 2: suppose that case 1 does not hold. since x is feebly compact, the sequence u has a cluster point p. there exists k ∈ n such that for every integer n > k, the point p /∈ un. define s = ⋃ n≥k+1 un, and for each i ≥ k + 1, let si = ⋃i n=k+1 un. note that p ∈ s \ s and si ⊆ s \ {p} for every i ≥ k + 1. as every feebly compact subspace of x with dense interior is closed, it follows from lemma 2.1 that there exists a pairwise disjoint sequence k = {kn : n ∈ n} of nonempty open subsets of x such that each kn ⊆ s, k 248 a. dow, j. r. porter, r. m. stephenson, jr., and r. grant woods has no cluster point in x \ {p}, and for every neighborhood v of p, v ⊇ kn for all but finitely many n ∈ n. since each si is feebly compact (by corollary 1.2), then for each i ≥ k + 1, si ∩ kn 6= ∅ for at most finitely many n ∈ n. by mathematical induction one can find strictly increasing sequences {mi : i ∈ n} and {ti : i ∈ n} in n such that for each i ∈ n: kmi ∩ uk+ti 6= ∅; and if i > 1, then km ∩ sti−1+k = ∅ for every m ≥ mi. define ni = k +ti for each i ∈ n. then the sequence {ni : i ∈ n} and the point p have the properties required in the definition of sequentially feebly compact. next, we assume the hypothesis of (d) (ii) holds. consider again the two cases named above. the proof in case 1 proceeds as above. assume case 2 holds. then as in case 2 above, there are a cluster point q of u and and k ∈ n so that for every integer n > k, the point q /∈ un. then the set t = ⋃ n≥k+1 un is not a closed set since q ∈ t \ t . because x is a sequential space, it follows that there exists a sequence {xn : n ∈ n} in t which converges to a point p ∈ x \ t . for each integer n ≥ k + 1, note that since p /∈ un then xm ∈ un for at most finitely many m ∈ n. thus, there are strictly increasing sequences {mi : i ∈ n} and {ti : i ∈ n} such that for each i ∈ n, one has xmi ∈ cl(uk+ti ). therefore, the sequence {ni = k + ti : i ∈ n} and point p satisfy the definition of sequentially feebly compact. statement (e) follows from lemma 2.1, and statement (f) follows from the characterizations in theorem 1.1 and the appropriate definitions. � the next result will be used in §5. corollary 2.3. let x be a feebly compact space which has property (f3). suppose u = {un : n ∈ n} is a sequence of nonempty open subsets of x such that um ∩ un = ∅ whenever m 6= n. then there are a point p in x, an infinite subset j of n, and a sequence of nonempty open sets p = {pn : n ∈ j} such that pn ⊆ un for each n ∈ j, and for every neighborhood o of p, o contains pn for all but finitely many n ∈ j. proof. this follows from the proof of case 2 of statement (d) (i) in theorem 2.2. � here are some examples illustrating that these properties are distinct. example 2.4. let x be the one-point compactification of some uncountable discrete space. then x is a scattered, fréchet, compact hausdorff, and hence fcc, space (by theorem 2.2 (a) (iii)) which is not first countable (or e1). example 2.5. there exists a space x which is a countable, compact, maximal feebly compact, and hence fcc, space which is not hausdorff: in [16, 2.12] a proof is given that a certain countable, non-hausdorff, maximal compact space due to v.k. balachandran is also maximal feebly compact. example 2.6. let x be any feebly compact hausdorff space which contains a non-isolated p-point p. then x cannot have property (f3): if u = {un : n ∈ n} were any pairwise disjoint sequence of nonempty open subsets of x \ {p} spaces whose pseudocompact subspaces are closed subsets 249 and one chose, for each n ∈ n, a nonempty open set vn ⊆ un with p /∈ vn, then the sequence {vn : n ∈ n} would have a cluster point in x \ {p}, and hence u would also, i.e., x \ {p} would be feebly compact. the next two spaces are of this type. example 2.7. let x = ω1 + 1, the set of ordinals ≤ ω1, with the order topology. the space x is a sequentially compact, and hence sequentially feebly compact, compact hausdorff space which does not have property (f3). example 2.8. let t be the tychonoff plank, t = (ω1+1)×(ω0+1)\{(ω1, ω0)}. then t is a locally compact hausdorff space that does not have property (f3) and is not sequentially compact [5], [6]. since t has a dense, sequentially compact subspace, namely t \ {(ω1, α) : α < ω0}, it follows easily that t is sequentially feebly compact. example 2.9. let βn be the stone-čech compactification of n, where n has the discrete topology, and let x be any dense, pseudocompact subspace of βn. then x is a feebly compact tychonoff space that is not sequentially feebly compact, and hence is not fcc: it is well-known that no nontrivial sequence in βn is convergent [6], and so for any infinite subset j of n and sequence u = {{j} : j ∈ j}, there would exist no point p ∈ x and infinite subset k of j with every neighborhood of p containing all but finitely many of the sets {{k} : k ∈ k}. example 2.10. let x = n ∪ {−∞, ∞}, where a subset t of x is defined to be open iff t ⊆ n or x \ t is finite. then x is a first countable, scattered, compact t1-space satisfying property (f2), but none of (f1), (f3) and (f4). the properties first countable, e1, fréchet and sequential are well-known to be closely related to one another. we shall give some examples illustrating further similarities and, in some cases, differences between these properties and the properties fcc and sequentially feebly compact. one such is the following familiar space. example 2.11. let (x, s) be [0,1], with its usual topology, and let t be the topology on x generated by s and the family of co-countable subsets of x. then (x, t ) is an fcc space which is e1, but not a sequential space. the latter follows from the fact that no infinite subset of (x, t ) is countably compact. it is also known that for every set t ∈ t , clt t = clst , and consequently every open filter base on (x, t ) has an adherent point. thus (x, t ) has the stated properties. example 2.11 also illustrates the observation that for any fcc space (x, s), if t is any feebly compact topology on x such that s ⊆ t , then (x, t ) is also fcc. more generally, see theorem 3.2 (d) below. a previously defined family of spaces related to fcc spaces was studied in [11], where m. ismail and p. nyikos called a space x c-closed if every countably compact subspace of x is a closed subset of x. they proved that (a) a sequential hausdorff space is c-closed, and (b) a sequentially compact, c-closed 250 a. dow, j. r. porter, r. m. stephenson, jr., and r. grant woods hausdorff space is a sequential space. in their statement (b), if one replaces “sequentially compact, c-closed hausdorff” by “countably compact fcc,” then (as noted above in 2.2 (c)), one can replace their conclusion by “fréchet and sequentially compact.” the next result shows that in (a), even for feebly compact symmetrizable spaces, one cannot replace “c-closed” by “fcc.” (a space (x, t ) is called symmetrizable in the sense of a.v. arhangel′skĭı if there exists a symmetric d on x which induces t , where by a symmetric on x one means a function d : x × x → [0, ∞) which vanishes exactly on the diagonal and satisfies the symmetric property, d(x, y) = d(y, x) for all x, y ∈ x.) before stating the result, let us first recall that an almost disjoint (“ad”) family p on a set x is a collection p ⊆ [x]ω such that p ∩p ′ is finite whenever p , p ′ are distinct members of p. an ad family m on x (such that m ⊆ q ⊆ [x]ω) is called a maximal almost disjoint family (“mad” family) (respectively, maximal almost disjoint subfamily of q) provided that m is properly contained in no ad family on x (respectively, no ad subfamily of q). theorem 2.12. there exists a symmetrizable (therefore, sequential), scattered, c-closed hausdorff space (x, t ) which contains a non-isolated point p such that x \ {p} is first countable, locally compact, feebly compact and zerodimensional, and hence x is sequentially feebly compact and c-closed but not fcc. proof. let ψ be the isbell-mrówka space described in [6, 5i]: let m be an infinite mad family on n and ψ = n∪m, where a subset u of ψ is defined to be open provided that for any set m ∈ m, if m ∈ u then there is a finite subset f of m such that {m} ∪ m \ f ⊆ u. the space ψ is then a first countable pseudocompact locally compact hausdorff space that is not countably compact [6]. list in a 1-1 manner as {mn : n ∈ n} the members of an infinite subset i of m, choose a point p /∈ ψ, and define x = ψ ∪ {p}. next, define d : x × x → [0, ∞) as follows: d(p, mn) = d(mn, p) = 1/n for each mn ∈ i and d(p, y) = d(y, p) = 1 for each y ∈ n ∪ (m \ i); for each n ∈ n and y ∈ ψ \ {n}, d(n, y) = d(y, n) = 1/n whenever n ∈ y ∈ m, and d(n, y) = d(y, n) = 1 whenever either y ∈ m with n /∈ y or y ∈ n \ {n}; and d(x, x) = 0 for all x ∈ x. let t be the topology induced on x by d, i.e., define t to be the collection of all subsets t of x such that for each point t ∈ t there exists ǫ > 0 such that t contains the “ball” {x ∈ x : d(t, x) < ǫ}. it is straightforward to show that d is a symmetric for the space (x, t ), and (x, t ) has the stated properties. furthermore, it is known and not difficult to prove that every symmetrizable space is sequential. � example 2.13. if one lets x be as in the proof above, but weakens the topology t on x by choosing the topology s for which (x, s) is the one-point compactification of ψ, then it was noted in [12] that eric van douwen and peter nyikos had noticed previously the resulting compact hausdorff space (x, s) was sequential but not fréchet. like (x, t ), the space (x, s) is not fcc, and since ψ ∈ s, these spaces do not even have the property (f3). since spaces whose pseudocompact subspaces are closed subsets 251 every symmetrizable compact hausdorff space is known to be metrizable, and every scattered fréchet feebly compact space is fcc, then s 6= t and (x, t ) is not fréchet either. while (x, t ) fails to be countably compact, the space (x, s) is known to be sequentially compact and c-closed. it is natural to ask if the word “scattered” can be removed from the statement in theorem 2.2 (a) (iii). in [9] a very nice proof was given that, assuming [ma], there exists a compact hausdorff fréchet space which is not fcc. the next result, which does not require any special axioms beyond zfc, shows that there is a compact hausdorff fréchet space x which is not fcc. its proof is an elaboration on one due to reznichenko that was outlined in 3.6 of [12]. the authors are grateful to peter nyikos for calling reznichenko’s space to our attention. in addition, we shall show that x can be used to construct a compact hausdorff and fréchet space a(x) which is not fcc, but which has property (f3) and also has a dense set of isolated points. theorem 2.14. there is a compact hausdorff fréchet space which is not fcc. proof. let κ denote the cardinality of continuum. we first define a compact 0-dimensional fréchet topology on t = κ≤ω, i.e., t consists of the functions into κ which have domain either a nonnegative integer or the entire set of nonnegative integers. for any t ∈ κ<ω and α ∈ κ, we will let tα denote the function obtained by extending the domain of t by one and setting the final value to α. for n ∈ ω and t : n → κ, we occasionally denote t by 〈t0, . . . , tn−1〉. recall that t forms a tree when ordered by simple inclusion, i.e., for s, t ∈ t , s ⊆ t if dom(s) ≤ dom(t) and s = t ↾ dom(s). now t is endowed with the following topology. simply for each s ∈ κ<ω, the set [s] = {t ∈ t : s ⊆ t} is defined to be clopen. thus a neighborhood basis for s ∈ κ<ω is the family {[s] \ ⋃ i 0, there exists a δ > 0 such that |t − s| < δ implies ||l̃(t) − l̃(s)|| < ǫ. note that dr(s) divides γ into γ0s and γs1. since c is an open curve, it follows that dr(0) 6= dr(1), and that γ0s and γs1 intersect at only dr(s). by lemma 2.6, dr(s) ∩ l is a single point, so l is divided by dr(s) into two sub-curves, denoted as k1 and k2, that is k1 ⊂ γ0s and k2 ⊂ γs1, as shown in figure 7. figure 7. if |s − τ| < δ, then ||q − q′|| < ǫ case1: the parameter s is such that s 6= 0 and s 6= 1. consider γs1 first. since k2 is oriented, we can let v be the first vertex of k2 that is nearest (in distance along k2) to q. for any 0 < ǫ < ||qv||, let q ′ ∈ qv such that ||qq′|| = ǫ. by lemma 2.7, q′ = l̃(τ) = dr(τ) ∩ l for some τ ∈ (s, 1]. first, note qq′ ∩ intγsτ 6= ∅. to verify this, observe qq′ ⊂ qv ⊂ k2 ⊂ γs1 and γs1 = γsτ ∪ γτ1, so qq′ ⊂ γsτ ∪ γτ1. if qq′ ∩ intγsτ = ∅, then the segment qq′ is contained in dr(s)∪γτ1 which is disconnected. this implies qq′ is disconnected, which is a contradiction. secondly, note that the subset γsτ of a nonsingular pipe section is connected (since c is c1), and qq′ is a line segment jointing the end discs of γsτ , and has intersections with interior of γsτ . this geometry implies that qq′ ⊂ γsτ .(2.12) let δ = τ − s. for an arbitrary t ∈ (s, s + δ) = (s, τ), inclusion 2.12 implies that l̃(t) = dr(t) ∩ qq′. since neither l̃(t) 6= q or l̃(t) 6= q ′, it follows that c© agt, upv, 2014 appl. gen. topol. 15, no. 2 214 computational topology for approximations of knots l̃(t) ∈ int(qq′). so ||l̃(t) − l̃(s)|| < ||qq′|| = ǫ. this shows the right-continuity. we similarly consider the γ0s and obtain the left-continuity. case2: the parameter s is such that s = 0 or s = 1. we similarly obtain the right-continuity if s = 0, or the left-continuity if s = 1. � theorem 2.9. if l and c satisfy conditions 1 and 2, then the map h defined by equation 2.2 is a homeomorphism. proof. by lemma 2.7, l̃(t) is one-to-one and onto. by lemma 2.8, l̃(t) is continuous. since l̃ is defined on a compact domain, it is a homeomorphism. note that c is simple and open, so c(t) is one-to-one, and it is obviously onto. the map c(t) is also continuous and defined on a compact domain, so c(t) a homeomorphism. since h is a composition of c−1 and l̃, h is a homeomorphism. � remark 2.10. a very natural way to define a homeomorphism between simple curves c and l would be by f(p) = l(c−1(p)). an easy method to extend f to a homotopy is the straight-line homotopy. however, we were not able to establish that a straight-line homotopy based upon f would also be an isotopy, where it would be necessary to show that each pair of line segments generated is disjoint. our definition of h in equation 2.2 was strategically chosen so that this isotopy criterion is easily established, since the normal discs are already pairwise disjoint. 3. construction of ambient isotopies note that l and c fit inside a nonsingular pipe section γ of c. for a similar problem, an explicit construction has appeared [17, section 4.4] [9]. the proof of lemma 3.3, below, is a simpler version of a previous proof [9, corollary 4]. the construction here relies upon some basic properties of convex sets, which are repeated here. for clarity, the complete proof of lemma 3.3 is given here. (a) rays outward (b) variant of a push figure 8. convex subset the images in figures 8(a) and 8(b) were created by l. e. miller and are used, here, with permission. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 215 j. li, t. j. peters and k. e. jordan lemma 3.1 ([9, lemma 6]). let a be a compact convex subset of r2 with non-empty interior. for each point p ∈ int(a) and b ∈ ∂a, the ray going from p to b only intersects ∂a at b (see figure 8(a).) lemma 3.2 ([9, lemma 7]). let a be a compact convex subset of r2 with nonempty interior and fix p ∈ int(a). for each boundary point b ∈ ∂a, denote by [p, b] the line segment from p to b. then a = ⋃ b∈∂a [p, b]. lemma 3.3. there is an ambient isotropy between l and c with compact support of γ, leaving ∂γ fixed. proof. we consider each normal disc dr(t) for t ∈ [0, 1]. let p = dr(t)∩c and q = h(p) with h defined by equation 2.2, then define a map fp,q : dr(t) → dr(t) such that it sends each line segment [p, b] for b ∈ ∂dr(t), linearly onto the line segment [q, b] as figure 8(b) shows. the two previous lemmas (lemma 3.1 and lemma 3.2), will yield that fp,q is a homeomorphism, leaving ∂dr(t) fixed [17, lemma 4.4.6]. in order to extend fp,q to an ambient isotopy, define h : dr(t) × [0, 1] → dr(t) [17, corollary 4.4.7] by h(v, s) = { (1 − s)p + sq if v = p fp,(1−s)p+sq(v) if v 6= p, where fp,(1−s)p+sq is a map on dr(t) analogous to fp,q, sending each line segment [p, b] for b ∈ ∂dr(t), linearly onto the line segment [(1 − s)p + sq, b]. it is a routine [17, corollary 4.4.7] to verify that h(v, s) is well defined on the compact set dr(t), continuous, one-to-one and onto, leaving ∂dr(t) fixed. now, we naturally define an ambient isotopy tt : r 2 × [0, 1] → r2 on the plane containing dr(t) by tt(v, s) = { h(v, s) if v ∈ dr(t) v otherwise. we then define t : r3 × [0, 1] → r3 by t (v, s) = { tt(v, s) if v ∈ dr(t) v otherwise. the fact that the normal discs dr(t) are disjoint ensures that t is an ambient isotopy [17, corollary 4.4.8], with compact support of γ, leaving ∂γ fixed. � 4. ambient isotopy for bézier curves now we apply this result to a simple, regular, composite, c1 bézier curve b and the control polygon p. 4.1. ambient isotopy. there exists a nonsingular pipe surface [15] of radius r for b, denoted as sb(r). denote the nonsingular pipe section determined by sb(r) as γb. also, for each sub-control polygon of b, there exists a corresponding nonsingular pipe sections. denote the nonsingular pipe section corresponding to the kth control polygon as γk. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 216 computational topology for approximations of knots theorem 4.1. each sub-control polygon p k of a bézier curve b will eventually satisfy conditions 1 and 2 via subdivision, and consequently, there will be an ambient isotropy between b and p with compact support of γb, leaving ∂γb fixed. proof. by lemma 2.1, conditions 1 and 2 can be achieved by subdivisions. now consider each sub-control polygon p k satisfying conditions 1 and 2, and the corresponding bézier sub-curves bk. use lemma 3.3 to define an ambient isotopy ψk : r 3 × [0, 1] → r3 between bk and p k, for each k ∈ {1, 2, . . . , 2i}. define ψ : r3 × [0, 1] → r3 by the composition ψ = ψ1 ◦ ψ2 ◦ . . . ◦ ψ2i. note that ψk fixes the complement of int(γk), and int(γk) ∩ int(γk′) = ∅ for all k 6= k′. so the composition ψ is well defined. since each ψk is an ambient isotopy, the composition ψ is an ambient isotopy between b and p with compact support of γb, leaving ∂γb fixed. � 4.2. sufficient subdivision iterations. now we consider sufficient numbers of subdivision iterations to achieve the ambient isotopy defined by theorem 4.1, that is, we shall have a control polygon that satisfies conditions 1 and 2. the number of subdivisions for condition 1 is given in the paper [13, lemma 6.3]. to obtain the number of subdivisions for condition 2, we consider the following, for which we let p′(t) = l′(p, i)(t) (the first derivative of the control polygon p), and denote the angle between b′(t) and p′(t) as θ(t), for t ∈ [0, 1]. lemma 4.2 ([13, theorem 6.1]). for any 0 < ν < π 2 , there is an integer n(ν) such that each exterior angle of p will be less than ν after n(ν) subdivisions, where n(ν) = ⌈max{n1, log(f(ν))}⌉,(4.1) n1 = 1 2 log( n∞(n − 1) ‖ △2p ′ ‖ σ ), and f(ν) = 2m (1 − cos(ν))(σ − b′ dist (n1)) . note that for a bézier curve of degree n, there are n − 1 exterior angles for each sub-control polygon p k. to have tκ(p k) < π 2 , it suffices to make each exterior angle smaller than π 2(n−1) . by lemma 4.2, this can be gained by n( π 2(n−1) ) subdivisions. condition 2 is motivated by the weaker condition tκ(p k) < π 2 . we couldn’t derive the same results by using this weaker condition instead, but our condition 2 requires at most one more subdivision, as shown below. lemma 4.3. condition 2 will be fulfilled by at most n( π 2(n−1) )+1 subdivisions. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 217 j. li, t. j. peters and k. e. jordan proof. to prove tκ(p k) + max t∈[0,1] θ(t) < π 2 , it suffices to prove tκ(p k) < π 6 and max t∈[0,1] θ(t) < π 3 . to have tκ(p k) < π 6 , by lemma 4.2, n( π 6(n−1) ) subdivisions will be sufficient. the definition given by equation 4.1 implies that n( π 6(n − 1) ) ≤ n( π 2(n − 1) ) + 1. on the other hand, by [13, section 6.3], for all t ∈ [0, 1], we have 1 − cos(θ(t)) ≤ 2b′dist(i) σ , where b′dist(i) := 1 22i n∞(n − 1)||∆2p ′||. so to have maxt∈[0,1] θ(t) < π 3 , it suffices to set 2b′dist(i) σ < π 3 , which implies i ≥ 1 2 log( n∞(n − 1)||∆2p ′|| σ ) + 1. comparing it with equation 4.1 , we find that it is at most one more than n(ν) for any 0 < ν < π 2 . the conclusion follows. � let n⋆ = max{n( π 2(n − 1) ) + 1, n′(r)},(4.2) where r is the radius of sr(b). theorem 4.4. performing n⋆ or more subdivisions, where n⋆ is given by equation 4.2, will produce an ambient isotopic p for b. proof. according to [13, lemma 6.3], condition 1 is satisfied after n′(r) subdivisions. by lemma 4.3, condition 2 is satisfied after n( π 2(n+1) ) + 1 subdivisions. then theorem 4.1 can be applied to draw the conclusion. � now we compare this result with the existing one [13]. remark 4.5. to obtain ambient isotopy, the previously established result [13] needs max{n( π 2(n−1) ), n′(r)} + 2 subdivision iterations [13, remark 6.1]. in contrast, theorem 4.4 implies max{n( π 2(n−1) ), n′(r)} + 1 will be sufficient. a subdivision doubles the number of line segments. therefore, with only one less subdivision, the work here produces much less line segments, which may be useful especially for applications with very complex shapes. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 218 computational topology for approximations of knots 5. conclusions we give two conditions regarding distance, and total curvature combined with derivative, to guarantee the same knot type. it can be directly applied to bézier curves. this work is alternative to an existence result of requiring the containment of convex hulls of sub-control polygons, and another result using conditions of distance and derivative. the approach here allows fewer subdivision iterations and less line segments by explicitly constructing ambient isotopies. moreover, we showed that it is possible to verify the condition of total curvature only, other than total curvature combined with derivative, with a price of one additional subdivision. testing the global property of total curvature may be easier than testing the local property of derivative in some practical situations. it may find applications in computer graphics, computer animation and scientific visualization. references [1] n. amenta, t. j. peters, and a. c. russell, computational topology: ambient isotopic approximation of 2-manifolds, theoretical computer science 305 (2003), 3–15. [2] l. e. andersson, s. m. dorney, t. j. peters and n. f. stewart, polyhedral perturbations that preserve topological form, cagd 12, no. 8 (1995), 785–799. [3] m. burr, s. w. choi, b. galehouse and c. k. yap, complete subdivision algorithms, ii: isotopic meshing of singular algebraic curves, journal of symbolic computation 47 (2012), 131–152. [4] f. chazal and d. cohen-steiner, a condition for isotopic approximation, graphical models 67, no. 5 (2005), 390–404. [5] w. cho, t. maekawa and n. m. patrikalakis, topologically reliable approximation in terms of homeomorphism of composite bézier curves, cagd 13 (1996), 497–520. [6] e. denne and j. m. sullivan, convergence and isotopy type for graphs of finite total curvature, in: a. i. bobenko, j. m. sullivan, p. schröder, and g. m. ziegler, editors, discrete differential geometry, pages 163–174. birkhäuser basel, 2008. [7] g. e. farin, curves and surfaces for computer-aided geometric design: a practical code, academic press, inc., 1996. [8] m. w. hirsch, differential topology, springer, new york, 1976. [9] k. e. jordan, l. e. miller, t. j. peters and a. c. russell, geometric topology and visualizing 1-manifolds, in: v. pascucci, x. tricoche, h. hagen, and j. tierny, editors, topological methods in data analysis and visualization, pages 1–13. springer ny, 2011. [10] j. li, topological and isotopic equivalence with applications to visualization, phd thesis, university of connecticut, u.s., 2013. [11] j. li and t. j. peters, isotopic convergence theorem, journal of knot theory and its ramifications 22, no. 3 (2013). [12] j. li, t. j. peters, d. marsh and k. e. jordan,computational topology counterexamples with 3d visualization of bézier curves, applied general topology 13, no. 2 (2012), 115–134 [13] j. li, t. j. peters and j. a. roulier, isotopic equivalence from bézier curve subdivision, preprint [14] l. lin and c. yap, adaptive isotopic approximation of nonsingular curves: the parameterizability and nonlocal isotopy approach, discrete & computational geometry 45, no. 4 (2011), 760–795 c© agt, upv, 2014 appl. gen. topol. 15, no. 2 219 j. li, t. j. peters and k. e. jordan [15] t. maekawa, n. m. patrikalakis, t. sakkalis and g. yu, analysis and applications of pipe surfaces, cagd 15, no. 5 (1998), 437–458. [16] d. d. marsh and t. j. peters, knot and bézier curve visualizing tool, http://www.cse.uconn.edu/tpeters/top-viz.html. [17] l. e. miller, discrepancy and isotopy for manifold approximations, phd thesis, university of connecticut, u.s., 2009. [18] j. w. milnor, on the total curvature of knots, annals of mathematics 52 (1950), 248– 257. [19] g. monge, application de l’analyse à la géométrie, bachelier, paris, 1850. [20] e. l. f. moore, t. j. peters and j. a. roulier, preserving computational topology by subdivision of quadratic and cubic bézier curves, computing 79, no. 2–4 (2007), 317–323 [21] g. morin and r. goldman, on the smooth convergence of subdivision and degree elevation for bézier curves, cagd 18 (2001), 657–666. [22] j. munkres, topology, prentice hall, 2nd edition, 1999. [23] d. nairn, j. peters and d. lutterkort, sharp, quantitative bounds on the distance between a polynomial piece and its bézier control polygon, cagd 16 (1999), 613–63. [24] m. reid and b. szendroi, geometry and topology, cambridge university press, 2005. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 220 http://www.cse.uconn.edu/ tpeters/top-viz.html 16.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 33 37 f-points in countably compact spacesa. bella and v. i. malykhin�abstract. answering a question of a. v. arhangel0ski��, weshow that any extremally disconnected subspace of a compactspace with countable tightness is discrete.2000 ams classi�cation: primary 54a25; secondary 54a10keywords: f-point, extremal disconnectedness, compact, countably com-pact, pseudocompact, countable tightness.1. the resultsin [1] arhangel0ski�� noticed that under pfa any extremally disconnectedsubspace of a compact space with countable tightness is discrete. then, heposed the natural question whether this result holds in zfc.the aim of this short note is just to provide the full answer to such question.we actually manage to generalize the result by weakening the compactnessassumption.henceforth, all space are assumed to be tychono� spaces, unless otherwisespeci�ed.recall that, for a given topological space x,the tightness at the point x 2 x,denoted by t(x;x), is the smallest cardinal � such that, whenever x 2 a � x,there exists a set b � a satisfying jbj � � and x 2 b.we say that x 2 x is a f-point if there are no disjoint open f�-sets u andv such that x 2 u \ v .it is evident that x is a f-space if and only if each x 2 x is a f-point in x.if y is a subspace of the space x, we say that y is countably compact inx provided that every in�nite subset of y has an accumulation point in x.of course, a space is countably compact if and only if it is countably compactin itself. if y is countably compact in x, then y is pseudocompact but notnecessarily countably compact.the next lemma is essentially a combination of theorem 1 in [2] and propo-sition 3.1 in [3].�work supported by the research project \analisi reale" of the italian ministero dell'uni-versit�a e della ricerca scienti�ca e tecnologica. 34 a. bella and v. i. malykhinlemma 1.1. let x be a space, y � x and y be a non-isolated point of y . ify is countably compact in x and t(y;x) � @0 then y is in the closure of somecountable discrete subspace of y .proof. let a be a countable subset of y n fyg such that y 2 a and �x afamily u = fun : n 2 !g of closed neighbourhoods of y in x chosen so thata \ tu = ?. let s be the subset of x consisting of all points which are inthe closure of some countable set fzn : n 2 !g where zn 2 un \ a for each n.since y is countably compact in x and y 2 un \ a for each n, it easily followsthat y 2 s. as t(y;x) � @0, we may select a set fsn : n 2 !g � s so thaty 2 fsn : n 2 !g. let fzin : n 2 !g be a sequence witnessing that si 2 s andput b = fzhj : 0 � h � j < !g. b is a subset of a which contains each si in itsclosure and so y 2 b. furthermore, since b \tu = ? and b n un is �nite forevery n, it follows that b is discrete. �lemma 1.2. let x be a f-point of the space x. if x is in the closure of somecountable discrete set n � x, then n [ fxg is homeomorphic to a subspace of�n.proof. by contradiction, let us assume that there are two disjoint subset a;b �n such that x 2 a \ b. as n is discrete, for each y 2 n there is an openneighbourhood v (y) of y such that v (y) \ n = fyg. let fai : i 2 !g andfbi : i 2 !g be enumerations of a and b. let pi be an open f�-set satisfyingai 2 pi � v (ai) n sfv (bj) : j � ig and let qi be an open f�-set satisfyingbi 2 qi � v (bi) n sfv (aj) : j � ig. but now, letting p = sfpi : i 2 !g andq = sfqi : i 2 !g, we get disjoint open f�-sets of x satisfying x 2 p \q { incontrast with the fact that x is a f-point. �a crucial role is played here by the following result, which is a bit more generalversion than the one proved for countably compact spaces in [5], corollary 4.lemma 1.3. let p 2 �n nn. if the subspace n [fpg is countably compact inthe space x, then t(p;x) > @0.proof. let us assume that n [ fpg is countably compact in the space x. ofcourse, we may identify p with the trace in n of the family of all neighbourhoodsof p in x.suppose �rst that p is a p-point. for any p 2 p, choose a partition ofp consisting of two in�nite sets p 0 and p 00 so that p 0 2 p and let x(p) be anaccumulation point of the set p 00 in x. the regularity of x guarantees that p isin the closure of the set a = fx(p) : p 2 pg � x. if x has countable tightnessat p, then we could �nd fpn : n < !g � p such that p 2 fx(pn) : n < !g. but,since p is a p-point, there exists a neighbourhood u of p in x such that u \p 00nis �nite for each n < ! and so p =2 fx(pn) : n < !g { a contradiction.if p is not a p-point, then there is a partition fan : n < !g of n into in�nitesets such that for every n, an 62 p, and if p 2 p then jp \anj = ! for in�nitely f-points in countably compact spaces 35many n. now put t = f [n 0), almost connectedness (g is metrizable modulo the connected component) and various versions of extremality in the sense of comfort and co-authors (s-extremal, if g has no proper dense pseudocompact subgroups, r-extremal, if g admits no proper pseudocompact refinement). we introduce also weakly extremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). it turns out that this “symmetric” version of extremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. we characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. we give also a criterion for r-extremality for connected pseudocompact groups. 2000 ams classification: primary 22b05, 22c05, 40a05; secondary 43a70, 54a20. keywords: pseudocompact group, gδ-dense subgroup, extremal pseudocompact group, dense graph. ∗the first and the third author were partially supported by research grant of the italian miur in the framework of the project “nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani” 2002. the third author was partially supported by research grant of the university of udine in the framework of the project “torsione topologica e applicazioni in algebra, analisi e teoria dei numeri”. 2 d. dikranjan, a. giordano bruno and c. milan 1. introduction the metric spaces and their generalizations are of major interest in general topology. we consider here several weak versions of metrizability that work particularly well for pseudocompact groups. the class of pseudocompact groups, introduced by hewitt [25] was proved to be an important class of topological groups in the last forty years. 1.1. extremal pseudocompact groups. in the framework of pseudocompact groups, extremality is a generalization of metrizability discovered by comfort and co-authors. the starting point was a theorem of comfort and soundararajan [15] who proved that for a compact, metrizable and totally disconnected group there exists no strictly finer pseudocompact group topology. in 1982 comfort and robertson [9] extended this result to all compact, metrizable groups and proved also that a compact abelian group is metrizable if and only if it has no proper dense pseudocompact subgroup (see also corollary 4.14 here for a short simultaneous proof of both theorems). this result motivated the study of the extremal pseudocompact groups, which can be considered as a generalization of the metrizable (compact) ones. in 1988 comfort and robertson [11] defined extremal pseudocompact groups; however, they did not distinguish among different kinds of extremality. this was done explicitly somewhat later (see [1]). a pseudocompact group g is • s-extremal if g has no proper dense pseudocompact subgroups; • r-extremal if there exists no strictly finer pseudocompact group topology on g; • doubly extremal if it is both s-extremal and r-extremal. since the pseudocompact groups of countable pseudocharacter are compact, they cannot contain proper dense subgroups and every continuous bijection between such groups is a homeomorphism; hence it follows that every metrizable pseudocompact group is doubly extremal. note that this result holds for arbitrary pseudocompact groups (not necessarily abelian) and introduces an important class of doubly extremal pseudocompact groups. extremality of zero-dimensional pseudocompact abelian groups was considered by comfort and robertson [11]. it was shown that a zero-dimensional pseudocompact abelian group that is either sor r-extremal is metrizable [11, theorem 7.3]. an important step in the proof was the case of abelian elementary p-groups, for which sand r-extremality coincide and both imply metrizability (cf. [11, theorem 5.19]). hence it is worth asking whether there exist non-metrizable pseudocompact groups that are s-, ror doubly extremal and what is the relation between sand r-extremality. in the following list we collect questions posed in [11], [6], [4], [23] and [1]: (a) is every s-extremal pseudocompact group metrizable? (b) is every r-extremal pseudocompact group metrizable? (c) is every doubly extremal pseudocompact group metrizable? weakly metrizable pseudocompact groups 3 (d) is every s-extremal pseudocompact group also r-extremal? (e) is every r-extremal pseudocompact group also s-extremal? clearly, (a) implies (c) and (d), while (b) implies (c) and (e). moreover, if the conjunction of (d) and (e) is true, then (a), (b) and (c) are equivalent. nevertheless, none of the above questions has a complete answer, even when the attention is restricted to the context of abelian groups, hence it is worth studying extremality in the class of pseudocompact abelian groups. in many particular cases it has been proved that some forms of extremality are equivalent to metrizability (e.g., for totally disconnected pseudocompact abelian groups, countably compact abelian groups, pseudocompact abelian groups of weight at most c, where c denotes the cardinality of the continuum). as far as the relation between sand r-extremality is concerned, in some cases these two notions turn out to be equivalent. 1.2. further levels of extremality. an essential tool in the study of pseudocompact groups is the notion of gδ-density (a subgroup h ≤ g is gδ-dense if non-trivially meets every non-empty gδ-set of g). the first very important theorem on pseudocompact group is due to comfort and ross (cf. theorem 2.1); it shows the relation between a pseudocompact group g, its completion g̃ and gδ-density. in this paper we intend to face the problem of a better description of the relations among the three levels of extremality. to this end we introduce some weaker forms of extremality (in what follows r0(g) will denote the free rank of g). definition 1.1. a pseudocompact abelian group is (a) d-extremal if g/h is divisible for every gδ-dense subgroup h of g; (b) c-extremal if r0(g/h) < c for every gδ-dense subgroup h of g; (c) weakly extremal if it is both d-extremal and c-extremal. while there exist plenty of non-metrizable c-extremal (e.g., torsion) or dextremal (e.g., divisible) pseudocompact abelian groups, we are not aware whether every weakly extremal pseudocompact abelian group is metrizable. this property as well as the one given in the following theorem make the weak extremality the most relevant extremality property: theorem a. (theorem 3.12) if a pseudocompact abelian group g is either s-extremal or r-extremal, then g is weakly extremal. the same conclusion of the above theorem can be deduced from [3, theorem 4.4]; the proof is given in §3.2 and is based on the dense graph theorem (cf. theorem 3.8), proved in 1988 by comfort and robertson [11]. in particular, theorem 3.12 implies that if the problem of extremality for a certain class of pseudocompact abelian groups admits a solution in the case of weak extremality, then this problem is solved also for s-extremality, rextremality and double extremality, which are the main forms of extremality. the notion of weakly extremal group introduces the following relevant question: 4 d. dikranjan, a. giordano bruno and c. milan (f) is every weakly extremal pseudocompact group metrizable? let us note that a positive answer to this question will lead to a simultaneous positive answer to all questions (a)-(e), since r-extremal or s-extremal implies weakly extremal by theorem a. on the other hand, one may anticipate that in all cases where a positive answer to one of questions (a)-(e) is given for both sand r-extremality, it is possible to extend these results to the case of weakly extremal groups. moreover, the notion of weak extremality introduces a new approach to the study of extremal groups and in most cases allows us to extend to r-extremal groups results proved only for s-extremal ones (cf. theorem 4.15). 1.3. singularity and almost connectedness of pseudocompact groups and their connection to extremality. the original aim of this paper (having large intersection with [24]), was to be a comprehensive survey on extremal pseudocompact groups. gradually some original ideas and results appeared. in june 2004, shortly before the submission of this paper, we received a copy of [3], kindly sent by the authors before the publication. although most of the results were announced earlier [22, 23], the proofs became accessible to us only at that point. let us describe in detail the content of the present paper. following [10], we denote by λ(g) the family of all closed (normal) gδsubgroups of a pseudocompact topological group g and by m(g) the minimum cardinality of a dense pseudocompact subgroup of g. then every gδ-set containing 0 contains also a subgroup n ∈ λ(g). in §2 we recall some important facts concerning pseudocompact groups g and the family λ(g), that will be essential in the sequel. most of the proofs are omitted but hints or due references are given in all cases. in §3 we consider some stability properties of the various classes of extremal groups, with particular attention to the behavior of extremal groups under taking closed (pseudocompact) subgroups and quotients with respect to subgroups n ∈ λ(g). §3.2 is opened by the dense graph theorem (theorem 3.8), which is at the basis of the proofs of other results of §§3.2 and 6.1, concerning in particular the relations among various kinds of extremality. comfort, gladdines and van mill showed in [4, corollary 4.6] that s-extremal groups have free rank at most c. the same conclusion was proved for r-extremal groups by comfort and galindo in [3, theorem 5.10 (b)] (announced in [23, theorem 7.3]). these results are simultaneously extended to c-extremal groups in theorem 3.6 (although c-extremal pseudocompact abelian groups g with r0(g) = c need not be metrizable, example 4.4). on the other hand, d-extremal pseudocompact abelian groups need not have bounded free rank (take any large pseudocompact divisible abelian group). in §4 we introduce the concept of singular group generalizing simultaneously metrizability and torsion (cf. theorem 2.14). if m is a positive integer and g is a group, let g[m] = {x ∈ g : mx = 0}. weakly metrizable pseudocompact groups 5 definition 1.2. a topological group g is singular if there exists m ∈ n+ such that g[m] ∈ λ(g). the concept of singular abelian group implicitly appeared already in [20] in the case of compact groups. we prove that for singular pseudocompact abelian groups d-extremality is equivalent to metrizability (cf. theorem 4.6). note that d-extremality and cextremality behave in a completely different way w.r.t. singularity: d-extremality complements singularity to metrizability, whereas c-extremality is a weaker version of singularity (proposition 4.7). on the other hand, theorem 4.13 shows that a c-extremal pseudocompact abelian group g is singular whenever w(g) ≤ c or g is compact. in particular, theorems 4.6 and 4.13 imply theorem 4.15, imposing the restraint w(g) > c for non-metrizable weakly extremal pseudocompact abelian groups g. so this theorem simultaneously strengthens the known results of comfort, gladdines and van mill [4, theorem 4.11] and comfort and galindo [3, theorem 5.10] where the same conclusion is obtained for s-extremal (resp., r-extremal) pseudocompact abelian groups. it is not possible to replace in theorem 4.15 weak extremality neither by c-extremality nor by d-extremality (just take a non-metrizable pseudocompact abelian group of weight ≤ c that is either torsion or divisible). in §5 we define almost connected groups as follows (here c(g) denotes the connected component of g): definition 1.3. a topological group g is almost connected if c(g) ∈ λ(g). if g is an almost connected pseudocompact group, then the quotient g/c(g) is compact. let us recall here, that some authors [5, p.82] call a locally compact group g almost connected if only compactness is required for the quotient g/c(g) (whereas, in our setting this quotient is compact metrizable). throughout this paper almost connected will be understood always in the sense of definition 1.3. since the locally compact pseudocompact groups are compact, there is no interference at all except in the case of compact groups. it follows from the definition that both connected groups and metrizable groups are almost connected. moreover, totally disconnected pseudocompact abelian groups are almost connected if and only if they are metrizable (cf. proposition 5.6). as a matter of fact, almost connectedness and singularity form a well balanced pair of generalizations of metrizability for pseudocompact abelian groups: g is singular if mg is metrizable for some m > 0 (lemma 4.1), whereas g is almost connected if g/mg is metrizable for every m > 0 (theorem 5.8). hence g is metrizable whenever it is simultaneously singular and almost connected. the almost connectedness is stable under taking gδ-dense subgroups and closed gδ-subgroups (cf. theorem 5.8). moreover, if a topological group g admits a dense almost connected pseudocompact subgroup, then g itself is pseudocompact and almost connected (cf. lemma 5.2). 6 d. dikranjan, a. giordano bruno and c. milan for non-metrizable almost connected pseudocompact abelian groups some cardinal invariants are preserved by subgroups that are in λ(g). in particular, the connected component c(g) of a non-metrizable almost connected pseudocompact abelian group satisfies m(g) = m(c(g)) ≤ r0(g) and w(g) ≤ 2 r0(g) (cf. theorem 5.13). the properties of almost connected groups are used in §5 to generalize some results concerning s-extremality of connected pseudocompact abelian groups proved in [6]. the main tool is given by theorem 5.14 which shows that a d-extremal pseudocompact abelian group is almost connected. note that cextremal pseudocompact abelian groups need not be almost connected (take any torsion, non-metrizable pseudocompact abelian group). the proof of the following result, due to comfort and robertson, is not covered by the proofs given here. for the rest this paper is self-contained. theorem 1.1. ([11, corollary 7.5]) let g be a torsion pseudocompact abelian group that is sor r-extremal. then g is metrizable. this theorem can easily be generalized to d-extremal groups: theorem b. (corollary 5.17) let g be a pseudocompact abelian group that is either totally disconnected or singular. then g is d-extremal if and only if it is metrizable. since torsion pseudocompact abelian groups are singular, this is a generalization of theorem 1.1. moreover, since a zero-dimensional topological group is totally disconnected, this corollary generalizes also [11, theorem 7.3], limited to the case of zero-dimensional groups. comfort and van mill [6, theorem 7.1] proved that every connected sextremal group is divisible. an analogous result for r-extremal groups was announced in [1, corollary 5.11] and proved in [3, corollary 5.11]. these results are simultaneously extended to d-extremal groups in corollary 5.22 where it is proved that a d-extremal pseudocompact abelian group is divisible if and only if it is connected. this follows by the more general fact (relying on properties of almost connectedness) that the connected component of an almost connected pseudocompact abelian group is divisible if and only if it is d-extremal (theorem 5.20). the following theorem, due to comfort and galindo (announced in [23, theorem 8.2], [1, theorem 6.1]), establishes some relevant necessary conditions satisfied by a non-metrizable r-extremal (resp. s-extremal) group. it introduces a certain asymmetry between s-extremality and r-extremality for pseudocompact abelian groups. theorem 1.2. ([3, theorem 7.1]) let g be a pseudocompact abelian group. then: (a) if g is s-extremal and non-metrizable, then there exists p ∈ p such that g[p] is not gδ-dense in g̃[p]; (b) if g is r-extremal, then g[p] is dense in g̃[p] for every p ∈ p. weakly metrizable pseudocompact groups 7 as an immediate corollary one can see that for a non-metrizable doubly extremal pseudocompact abelian group g, there exists p ∈ p such that g[p] is dense but not gδ-dense in g̃[p], i.e., g[p] is not pseudocompact ([3, theorem 7.2(ii)]). according to [30]: an abelian group g is almost torsion-free if g[p] is finite for every prime p. this generalization of the notion “torsion-free group” immediately gives: if g is a doubly extremal pseudocompact almost torsion-free abelian group, then g is metrizable. for torsion-free groups this result was obtained in [3]. the problem in attacking question (d) is the current lack of sufficient conditions for r-extremality. theorem 3.8 and its consequences allow us to obtain a sufficient condition for a s-extremal group g to be doubly extremal (hence, can be considered as a partial solution to (d)). theorem c. (theorem 6.10) let g be a s-extremal pseudocompact abelian group. then g is doubly extremal if and only if g[p] = g̃[p] for every p ∈ p. this result strengthens theorem 4.4 (b) of [3] where it is shown that every divisible s-extremal pseudocompact abelian group g such that g[m] = g̃[m] for every m ∈ n is r-extremal. in §6.2 we give a criterion for r-extremality of connected pseudocompact groups making no recourse to “external” issues such as characters or even different topologies on the group: a connected pseudocompact abelian group g is r-extremal if and only if g is weakly extremal and every n ∈ λ(g) with g/n ∼= t is d-extremal (theorem 6.11). according to [4, theorem 4.8], every s-extremal pseudocompact abelian group has cardinality at most c. in §7 we generalize this result as follows: an infinite weakly extremal pseudocompact abelian group g has |g| = c if and only if m(g) = |g| (cf. theorem 7.1 and corollary 7.2). it turns out that c-extremality is stable with respect to taking gδ-subgroups: theorem d. (theorem 4.11) every closed gδ-subgroup of a c-extremal pseudocompact abelian group g is c-extremal. nevertheless, such a stability need not be available for all extremality properties. indeed, it was proved by comfort and galindo [3, theorem 6.1] (announced in [23, lemma 8.1] and [1, theorem 5.16 (b)]) that a pseudocompact abelian group g is metrizable whenever every n ∈ λ(g) is s-extremal. this motivates the necessity to define stronger versions of extremality as follows. definition 1.4. a pseudocompact group g is said to be: (a) strongly extremal if every n ∈ λ(g) is r-extremal; (b) strongly d-extremal if every n ∈ λ(g) is d-extremal. the property in item (a) (without a specific term) was introduced and used in [3]. we will prove in theorem 6.7 that s-extremal pseudocompact abelian groups that are also strongly d-extremal must be metrizable. moreover, in theorem 6.8 we will see that if every n ∈ λ(g) is weakly extremal, then g is 8 d. dikranjan, a. giordano bruno and c. milan strongly extremal (i.e., every c-extremal and strongly d-extremal pseudocompact abelian group is strongly extremal). the next diagram shows the relations among all forms of weak metrizability that appear in the paper: metrizable ↓ strongly extremal ←− doubly extremal ւ ց ւ ց str. d-extr. r-extremal s-extremal ց ւ ց weakly extremal singular ւ ց ւ almost conn. ←− d-extremal c-extremal in the compact case only three distinct classes remain: metrizable groups (as weakly extremal compact groups are metrizable by corollary 4.14), singular groups (as c-extremal compact groups are singular by theorem 4.13) and almost connected groups (as almost connected compact groups are strongly d-extremal according to theorem 5.15). the intersection of that last two is the class of metrizable groups. since every non-metrizable compact connected abelian group is strongly d-extremal, but not singular (hence not c-extremal), strongly d-extremal does not imply c-extremal. 1.4. notation and terminology. the symbols z, p, n and n+ are used for the set of integers, the set of primes, the set of natural numbers and the set of positive integers, respectively. the circle group t is identified with the quotient group r/z of the reals r and carries its usual compact topology. let g be an abelian group. the subgroup of torsion (p-torsion) elements of g is denoted by t(g) (resp., tp(g)). the group g is said to be bounded torsion if there exists n ∈ n+ such that ng = 0. if m is a positive integer, z(m) is the cyclic group of order m. we denote by r0(g) the free rank of g (if g is free abelian this is simply its rank, otherwise this is the maximum rank of a free subgroup of g). for n ∈ n let ϕn : g → g be defined by ϕn(x) = nx for every x ∈ g. then kerϕn = g[n]. if h is a group and h : g → h is a homomorphism, then we denote by γh := {(x,h(x)), x ∈ g} the graph of h. for a topological space x, we denote by w(x) the weight of x (i.e., the minimum cardinality of a base for the topology on x). a space x is said to be zero-dimensional if x has a base consisting of clopen sets. a space x is pseudocompact if every continuous real-valued function on x is bounded, ω-bounded if the closure of every countable subset of x is compact. throughout this paper all topological groups are hausdorff and completeness is intended with respect to the two-sided uniformity, so that every topological group has a completion which we denote by g̃. a group g is precompact if g̃ is compact. with c(g) we indicate the connected component of 0 in g; a group g is totally disconnected if c(g) is trivial. weakly metrizable pseudocompact groups 9 for a topological group g, we will denote by vg(0) the filter of 0-neighborhoods in g, by χ(g) the character of g (that is, the minimal cardinality of a basis of vg(0)) and by ψ(g) the pseudocharacter of g (i.e., the minimum size of a family b of 0-neighborhoods of g such that ⋂ u∈b u = {0}). if m is a subset of a topological group g, then 〈m〉 is the smallest subgroup of g containing m and m is the closure of m. for any abelian group g let hom(g,t) be the group of all homomorphisms from g to the circle group t. when (g,τ) is an abelian topological group, the set of τ-continuous homomorphisms χ : g → t (characters) is a subgroup of hom(g,t) and is denoted by ĝ. for a subset h of g the annihilator of h in ĝ is the subgroup a(h) = {χ ∈ ĝ : χ(h) = {0}} of ĝ. for undefined terms see [19, 26]. 2. background on pseudocompact groups the following theorem guarantees precompactness of the pseudocompact groups and characterizes the pseudocompact groups among the precompact ones. theorem 2.1. ([13, theorems 1.2 and 4.1]) (a) every pseudocompact group is precompact. (b) let g be a precompact group. then the following conditions are equivalent: (b1) g is pseudocompact; (b2) g is gδ-dense in g̃; (b3) g̃ is the stone-cech compactification of g. for the sake of easier reference we isolate the following lemma from theorem 2.1. lemma 2.2. let g be a topological group and let h be a dense, pseudocompact subgroup of g. then g is pseudocompact and h is gδ-dense in g. lemma 2.2 immediately yields that a dense subgroup h of a pseudocompact group g is pseudocompact if and only if h is gδ-dense in g. corollary 2.3. let g be a pseudocompact topological group. (a) if g is metrizable, then g is compact; (b) g is connected if and only if g̃ is connected; (c) g is zero-dimensional if and only if g̃ is zero-dimensional. lemma 2.4. ([11, theorem 3.2]) let g be a pseudocompact group such that {0} is a gδ-set. then g is metrizable and compact. a useful pseudocompact criterion via quotients follows. theorem 2.5. ([11, lemma 6.1]) let g be a precompact group. then g is pseudocompact if and only if g/h is compact metric for every h ∈ λ(g). 10 d. dikranjan, a. giordano bruno and c. milan corollary 2.6. let g be a pseudocompact abelian group. then: (a) n is pseudocompact for every n ∈ λ(g); (b) if w(g) > ω, then w(g) = w(n) for every n ∈ λ(g); (c) if n ∈ λ(g) and h is a closed subgroup of g such that n ⊆ h, then h ∈ λ(g); (d) if n ∈ λ(g) and l ∈ λ(n), then l ∈ λ(g). now we see that gδ-density in g is completely controlled by the subgroups n ∈ λ(g). lemma 2.7. let g be a pseudocompact abelian group and let d be a subgroup of g. then: (a) d is gδ-dense in g if and only if n + d = g for every n ∈ λ(g); (b) if d is gδ-dense in g and n ∈ λ(g), then g/d ∼= n/(d ∩n); (c) if d is gδ-dense in g and n ∈ λ(g), then d∩n is gδ-dense in n. proof. (a) follows from the fact that d is gδ-dense in g if and only if (x + n)∩d 6= ∅ for every x ∈ g and every n ∈ λ(g). (b) by (a) g = n + d, hence g/d ∼= n/(d∩n). (c) follows from (a). � lemma 2.8. let g be a pseudocompact abelian group. then: (a) if n ∈ λ(g), then n ∈ λ(g̃); (b) if g is dense in g1 and n ∈ λ(g), then n g1 ∈ λ(g1). proof. (b) follows from (a) and g̃ = g̃1. � corollary 2.9. let g be a pseudocompact abelian group. if g is dense in g1 and m is a closed subgroup of g1 such that m∩g ∈ λ(g), then m ∈ λ(g1). in particular, m ∈ λ(g̃) for every closed subgroup of g̃ such that m ∩g ∈ λ(g). lemma 2.10. let g be a precompact abelian group, n ∈ λ(g) and n ∈ n+. then nn g̃ = nn g̃ and n ∩nn g̃ = nn g . lemma 2.11. let g be a topological group. let τ and τ′ be pseudocompact group topologies on g such that τ′ ≥ τ. then the following conditions are equivalent: (a) τ = τ′; (b) for every n ∈ λ(g,τ) one has τ|n = τ ′|n ; (c) there exists n ∈ λ(g,τ) such that τ|n = τ ′|n. proof. (a)⇒(b) and (b)⇒(c) are obvious. (c)⇒(a) let τq and τ ′ q be the quotient topologies induced on g/n by τ and τ′ respectively. since τ ≤ τ′, λ(g,τ) ⊆ λ(g,τ′) and so in particular n ∈ λ(g,τ′). this implies that τq and τ ′ q are compact group topologies on g/n. moreover, τq ≤ τ ′ q and consequently τq = τ ′ q. since τ|n = τ ′|n, merzon’s lemma [29] implies that τ = τ′ on g. � weakly metrizable pseudocompact groups 11 remark 2.12. let (g,τ) and h be topological groups and h : (g,τ) → h a homomorphism. consider the map j : g → γh such that j(x) = (x,h(x)) for every x ∈ g. observe that j is a homomorphism such that j(g) = γh, that is j is surjective; moreover j is injective and so j is an isomorphism. if p1 : g×h → h is the projection on the first component, p1 is continuous and also its restriction to γh is continuous; since j is the inverse of p1|γh, it follows that j is open. endow γh with the group topology induced by the product (g,τ) × h. we define the topology τh as the weakest group topology on g such that τh ≥ τ and for which j is continuous. then j : (g,τh) → γh is a homeomorphism. note that the topology τh so defined is the weakest topology on g such that τh ≥ τ and for which h is continuous. indeed, if p2 : g × h → h is the canonical projection on the second component, then its restriction to γh is continuous. the homomorphism h is the composition of j and p2, in the sense that p2|γh ◦j = h : (g,τh) → h and therefore, being j τh-continuous, h has to be τh-continuous too. clearly, if h is τ-continuous then τh = τ. lemma 2.13. let (g,τ) be a topological group and let h : g → t be a homomorphism such that (g,τh) is pseudocompact. then the following conditions are equivalent: (a) h is continuous; (b) every n ∈ λ(g,τ) is such that h|n : n → t is continuous; (c) there exists n ∈ λ(g,τ) such that h|n : n → t is continuous. proof. (a)⇒(b) and (b)⇒(c) are obvious. (c)⇒(a) since h|n : n → t is continuous and τ ≤ τh, it follows that τh|n = τ|n . as τ and τh are pseudocompact by hypothesis, the assertion follows from lemma 2.11. � the next property of pseudocompact groups was announced in [6, remark 2.17] and proved in [21, lemma 2.3]. theorem 2.14. let g be an infinite pseudocompact abelian group. then either r0(g) ≥ c or g is bounded torsion with |g| ≥ c. proof. if g is torsion, then g is bounded by [11, lemma 7.4]. the inequality |g| ≥ c follows from van douwen’s theorem [31]. assume g is not torsion. if g is compact, then the assertion follows from [20, lemma 2.3]. otherwise, pick a non-torsion element x ∈ g and consider the cyclic subgroup c = 〈x〉. since c is countable, there exists a gδ-set o around 0 that meets c in {0}. find n ∈ λ(g) contained in o, hence n ∩c = {0}. then the quotient group g/n is compact and non-torsion. hence the compact case applies. � the following results will be helpful in the sequel to produce gδ-dense subgroups of given pseudocompact groups. in particular, item (a) of the following lemma has been proved by comfort and van mill in [6, lemma 2.13] and it has been announced with its corollary by comfort, gladdines and van mill in [4, lemma 4.1]. 12 d. dikranjan, a. giordano bruno and c. milan lemma 2.15. let g be a pseudocompact group. (a) [6, lemma 2.13] if g = ⋃∞ n=0 an, where an is a subgroup of g for every n ∈ n, then there exist n ∈ n and n ∈ λ(g) such that an ∩n is gδ-dense in n. (b) [4, lemma 4.1 (b)] if n ∈ λ(g) and d is gδ-dense in n, then there exists a subgroup e of g with |e| ≤ c such d+e is a gδ-dense subgroup of g. (c) if g is infinite, then m(n) = m(g) for every n ∈ λ(g). it directly follows from lemma 2.15 that if g is a pseudocompact abelian group such that g = ⋃∞ n=0 an, where an are subgroups of g, then there exist n ∈ n, n ∈ λ(g) and e ≤ g with |e| ≤ c such that (an ∩n) + e is gδ-dense in g. in particular: corollary 2.16. let g be a pseudocompact abelian group such that g =⋃∞ n=0 an, where an ≤ g for every n ∈ n. then there exists a subgroup e of g, with |e| ≤ c, such that an + e is gδ-dense in g. 3. extremal pseudocompact abelian groups 3.1. general properties of extremal groups. lemma 3.1. let g be a s (resp. d,c)-extremal pseudocompact abelian group and let n be a pseudocompact subgroup of g. (a) if n is closed, then g/n is s (resp. d,c)-extremal. (b) if n is an algebraic direct summand of g, then n is s (resp. d,c)extremal. in particular, every divisible pseudocompact subgroup of g is s (resp. d,c)-extremal. proof. (a) let ψ : g → g/n be the canonical homomorphism and let h be a gδ-dense subgroup of g/n. then ψ −1(h) is a gδ-dense subgroup of g (cf. [3, theorem 5.3 (a)]). hence g/ψ−1(h) = {0} (resp. g/ψ−1(h) is divisible, r0(g/ψ −1(h)) < c). since (g/n)/h ∼= g/ψ−1(h) we conclude that (g/n)/h = {0} (resp. (g/n)/h is divisible, r0((g/n)/h) < c). (b) there exists a subgroup l of g such that g = n ⊕ l. let d be a gδ-dense subgroup of n. then d1 = d ⊕l is a gδ-dense subgroup of g. to see this, let m ∈ λ(g). as d is gδ-dense in n, by lemma 2.7 (a) m +d ≥ n and consequently m +d1 ≥ n ⊕l = g. since g is s (resp. d,c)-extremal, one has necessarily g/d1 = {0} (resp. g/d1 is divisible, r0(g/d1) < c). since n/d ∼= g/d1, n is s (resp. d,c)-extremal. � let us recall that stability of sand r-extremality under taking quotients with respect to closed pseudocompact groups was proved by comfort and galindo ([23, lemma 4.5], [3, theorem 5.3]): lemma 3.2 ([23, 3]). let g be a pseudocompact abelian group and let n be a closed pseudocompact subgroup of g. (a) if g is r-extremal (resp., s-extremal), then g/n is r-extremal (resp., s-extremal). weakly metrizable pseudocompact groups 13 (b) if g is doubly extremal, then g/n is doubly extremal. the gδ-subgroups are particularly important since the following stability properties hold. lemma 3.3. let g be a pseudocompact abelian group and let n ∈ λ(g). (a) if n is s (resp. d,c)-extremal, then g is s (resp. d,c)-extremal. (b) [3, theorem 2.1 (b)] if n is r-extremal, then g is r-extremal. (c) if n is doubly extremal, then g is doubly extremal. proof. (a) let h be a gδ-dense subgroup of g. then h ∩ n is gδ-dense in n and so n/h ∩n = {0} (resp. n/h ∩n is divisible, r0(n/h) < c). since g = n + h, the conclusion follows from g/h = (n + h)/h ∼= n/h ∩n. (b) let τ′ be a pseudocompact group topology on g finer than τ. since n ∈ λ(g,τ), then n ∈ λ(g,τ′) and consequently (n,τ′|n) is pseudocompact. then τ|n = τ ′|n, since n is r-extremal by hypothesis. by lemma 2.11 it follows that τ = τ′, hence g is r-extremal. (c) it follows directly from (a) and (b). � corollary 3.4. if g is a r (resp. s,d,c)-extremal pseudocompact abelian group and k is a compact metrizable group, then also g × k is a r (resp. s,d,c)extremal pseudocompact group. remark 3.5. note that strong extremality and strong d-extremality are preserved by taking closed, gδ-subgroups by corollary 2.6(d). in the next theorem we extend to c-extremal groups [4, corollary 4.6] and [23, theorem 7.3] (proved in [23] and [3, theorem 5.10 (b)]), where the same result was announced respectively for sand r-extremal groups. our proof uses ideas from the proof of [4, proposition 4.4] and is very similar to that of [3, theorem 5.10 (b)]. we include it here for the sake of completeness. theorem 3.6. let g be a pseudocompact abelian group. if g is c-extremal, then r0(g) ≤ c. proof. let κ = r0(g). let m be a maximal independent subset of g consisting of non-torsion elements. then |m| = κ and there exists a partition m =⋃∞ n=1 mn such that |mn| = κ for each n. let un = 〈mn〉, vn = u1 ⊕···⊕un and an = {x ∈ g : n!x ∈ vn} for every n ∈ n+. then g = ⋃∞ n=1 an. by corollary 2.16 there exist n ∈ n+ and a subgroup e of g such that ã = an +e is gδ-dense in g and |e| ≤ c. hence |e/(an ∩e)| ≤ c. so the isomorphism ã/an = (an + e)/an ∼= e/(an ∩e) yields |ã/an| ≤ c. since g is c-extremal, r0(g/ã) < c, so r0(g/an) ≤ c by the isomorphism (g/an)/(ã/an) ∼= g/ã. on the other hand, r0(g/an) = κ, as un+1 embeds into g/an (note that an∩un+1 = {0} since every x ∈ an∩un+1 satisfies n!x ∈ vn ∩ un+1 = {0}, so x = 0 as un+1 is a free group), hence κ ≤ c. � 14 d. dikranjan, a. giordano bruno and c. milan 3.2. the dense graph theorem and weakly extremal groups. the dense graph theorem (see theorem 3.8 below) was announced in [11]. it gives a sufficient condition for a group to be neither s-extremal nor r-extremal. a lot of useful results in the study of extremality follow from this theorem. the following lemma has been announced in a similar form in [23] without a proof. lemma 3.7. let g be a topological abelian group and let h be a compact metrizable abelian group with |h| > 1. let h : g → h be a surjective homomorphism. then γh is gδ-dense in g × h if and only if kerh is proper and gδ-dense in g. by means of lemma 3.7 one can prove that for a pseudocompact abelian group g the following statements are equivalent: (a) there exists a pseudocompact abelian group h with |h| > 1 and a homomorphism h : g → h with γh gδ-dense in g×h; (b) there exists a compact metrizable abelian group h with |h| > 1 and a surjective homomorphism h : g → h with γh gδ-dense in g×h; (c) there exists a compact metrizable abelian group h with |h| > 1 and a surjective homomorphism h : g → h with kerh gδ-dense in g. theorem 3.8 (of the dense graph [11, theorem 4.1]). let (g,τ) be a pseudocompact abelian group. suppose that there exist a pseudocompact abelian group h with |h| > 1 and h ∈ hom(g,h) such that γh is a gδ-dense subgroup of g×h. then (a) there exists a pseudocompact group topology τ′ on g such that τ′ > τ and w(g,τ′) = w(g,τ); (b) there exists a proper dense pseudocompact subgroup d of g such that w(d) = w(g,τ). corollary 3.9. let g be a pseudocompact abelian group. assume that there exists a surjective homomorphism h : g → h, where h is a non trivial closed subgroup of t. if γh is gδ-dense in g×h, then g is neither s-extremal nor r-extremal. the next corollary directly follows from lemma 3.7 and corollary 3.9. corollary 3.10. let g be a pseudocompact abelian group. suppose that there exists a surjective homomorphism h : g → h, where h is a closed non-trivial subgroup of t. if kerh is a proper gδ-dense subgroup of g, then g is neither s-extremal nor r-extremal. remark 3.11. note that the hypotheses of theorem 3.8 and corollary 3.10 imply that the group g cannot be metrizable. indeed, if g were metrizable, then kerh = g, that is h ≡ 0 and so the graph γh of h would not be gδ-dense in g×h. thanks to the previous results it is possible to prove theorem a of the introduction showing that both sand r-extremality imply weak extremality (it can be deduced from [3, theorem 4.4]): weakly metrizable pseudocompact groups 15 theorem 3.12. if a pseudocompact abelian group g is either s-extremal or r-extremal, then g is weakly extremal. proof. suppose for a contradiction that g is not weakly extremal. then there exists a gδ-dense subgroup h of g such that either g/h is not divisible or r0(g/h) ≥ c. in both cases, h is a proper subgroup of g. moreover, h is dense and pseudocompact, hence g is not s-extremal. to find a contradiction it remains to prove that g is not even r-extremal. let ψ : g → g/h be the canonical projection. case 1. if g/h is not divisible, then there exists a prime p and a non-trivial homomorphism f : g/h → z(p). composing with the canonical projection ϕ : g → g/h we get a surjective homomorphism h = ϕ◦f : g → z(p). since kerh ⊇ h, it follows that kerh is a proper gδ-dense subgroup of g, then corollary 3.10 applies to conclude that g is not r-extremal. case 2. if r0(g/h) ≥ c, then there exists a surjective homomorphism η : g/h → t (as r0(g/h) ≥ c = r0(t)). define the surjective homomorphism h = η ◦ ψ : g → t and observe that kerh ⊇ h. therefore kerh is a proper gδ-dense subgroup of g. as before it follows from corollary 3.10 that g is not r-extremal. � 4. singular groups 4.1. d-extremality. the next lemma offers an alternative form for singularity of pseudocompact abelian groups (mg is compact metrizable for some m ∈ n+). it is useful when checking stability of this property under taking subgroups and quotients. lemma 4.1. let g be a topological abelian group and m ∈ n+. (a) if mg is metrizable, then g[m] ∈ λ(g). (b) if g is pseudocompact, then g[m] ∈ λ(g) implies that mg is metrizable (hence compact). proof. let ϕm : g → g be the continuous homomorphism defined by ϕm(x) = mx for every x ∈ g. then kerϕm = g[m] and ϕm(g) = mg. let i : g/g[m] → mg be the continuous isomorphism such that i ◦ π = ϕm, where π : g → g/g[m] is the canonical homomorphism. (a) if mg is metrizable, then ψ(mg) = ω and so ψ(g/g[m]) = ω. this implies that g[m] is a gδ-set of g; then g[m] ∈ λ(g). (b) suppose that g[m] ∈ λ(g). then the quotient g/g[m] is metrizable, hence compact by theorem 2.5. by the open mapping theorem the isomorphism i is also open and consequently it is a topological isomorphism. then the group mg is metrizable and so compact. � remark 4.2. it immediately follows from lemma 4.1 that singularity is stable under taking pseudocompact subgroups and quotients w.r.t. closed subgroups. the next lemma characterizes the singular groups in terms of the free rank of their closed gδ-subgroups. 16 d. dikranjan, a. giordano bruno and c. milan lemma 4.3. a pseudocompact abelian group g is not singular if and only if r0(n) ≥ c for every n ∈ λ(g). in such a case, w(g) > ω and r0(n) = r0(g) ≥ c for every n ∈ λ(g). proof. assume that g is not singular and let n ∈ λ(g). suppose for a contradiction that r0(n) < c. since n is pseudocompact, by theorem 2.14 n is bounded torsion, i.e., nn = {0} for some n ∈ n+. in particular, n ⊆ g[n] and therefore corollary 2.6 implies g[n] ∈ λ(g), i.e., g is singular, against the hypothesis. the converse implication is immediate. to prove the second part of the lemma, assume that g is not singular. then obviously w(g) > ω. let n ∈ λ(g). then the quotient g/n is compact and metrizable, hence |g/n| ≤ c. since r0(n) ≥ c, it follows that r0(g) = r0(g/n) ·r0(n) ≤ c ·r0(n) = r0(n), i.e., r0(g) = r0(n). � we start by an example showing that singular pseudocompact abelian groups need not be d-extremal (see theorem 4.6 for a general result). example 4.4. let p be a prime and let h be the subgroup of z(p)c defined by h = ∑ {z(p)i : i ⊆ c, |i| ≤ ω} (i.e., the σ-product of c-many copies of the group z(p)). if we denote by t the circle group, then the group g = t ×h is a singular non-metrizable pseudocompact abelian group with r0(g) = c. thus g is not d-extremal by theorem 4.6. lemma 4.5. let g be a pseudocompact abelian group. if for some n ∈ n+ w(g/ng) > ω, then g is not d-extremal. proof. let h = g/ng. the group h is pseudocompact, non-metrizable and bounded torsion, hence h is not s-extremal by theorem 1.1. let h1 be a proper gδ-dense subgroup of h. then h/h1 is bounded torsion, so h/h1 cannot be divisible. thus h is not d-extremal. the subgroup ng of g is pseudocompact as a continuous image of g, hence also its closure ng is pseudocompact by lemma 2.2. now lemma 3.1 (a) implies that also g is not d-extremal. � it was proved by comfort and robertson that questions (a) and (b) have positive answer in the case of torsion pseudocompact abelian groups [11, corollary 7.5]. this can easily be generalized to d-extremal groups replacing “torsion” by a much weaker condition: theorem 4.6. a singular pseudocompact abelian group is d-extremal if and only if it is metrizable. proof. if g is metrizable then it is weakly extremal, hence d-extremal. suppose that g is not metrizable, i.e., w(g) > ω. by lemma 4.1 there exists m ∈ n+ such that mg is compact and metrizable. let us consider the quotient g/mg, that is pseudocompact. since w(mg) = ω and w(g) = w(g/mg) · w(mg), it follows that w(g/mg) = w(g) > ω. then g is not d-extremal by lemma 4.5. � weakly metrizable pseudocompact groups 17 it follows from theorem 4.6 that for a singular pseudocompact abelian group also weak extremality, s-extremality, r-extremality and double extremality are all equivalent to metrizability. 4.2. c-extremality. we see now that the impact of singularity on c-extremality, compared to that on d-extremality, is quite different. let us start by proving the immediate implication singular ⇒ c-extremal. proposition 4.7. every singular pseudocompact abelian group g is c-extremal. in particular, r0(g) ≤ c. proof. assume that g is singular. then there exists a positive integer m such that mg is metrizable by lemma 4.1. let h be a gδ-dense subgroup of g. we have to see that r0(g/h) < c. since the homomorphism ϕm : g → mg is continuous, the subgroup mh of mg is gδ-dense and so mh = mg as mg is metrizable. therefore mg ≤ h, hence the quotient g/h is bounded torsion. in particular r0(g/h) = 0. the last assertion follows from theorem 3.6. for a direct argument note that metrizability and compactness of mg yield r0(mg) ≤ c and r0(g/mg) = 0. therefore, r0(g) = r0(mg) ≤ c. � to partially invert the implication (see theorem 4.13), we need first some preparation. the construction used in the next lemma follows standard ideas of transfinite induction carried out in similar situations in [6] and [20]. lemma 4.8. let κ be an infinite cardinal, g be an abelian group and {hα : α < κ} be a family of subgroups of g such that r0(hα) ≥ κ for every α < κ. then for every subset x = {xα : α < κ} of g there exists a subgroup l of g such that (a) l∩ (xα + hα) 6= ∅ for every ordinal α < κ; (b) r0(g/l) ≥ κ. proof. our aim will be to build by transfinite recursion two increasing chains {lα : α < κ} and {l ′ α : α < κ} of subgroups of g such that the following conditions are fulfilled for every α < κ: (aα) lα ∩ (xα + hα) 6= ∅; (bα) lα ∩l ′ α = {0}; (cα) l ′ α/ ⋃ β<α l′ β is non-torsion when α > 0; (dα) r0(lα) ≤ |α| and r0(l ′ α) ≤ |α|. if α = 0, take x0 = 0 and l0 = l ′ 0 = {0}. we need the following easy to prove claim for our construction. claim. let g be an abelian group and x ∈ g. let h, l and n be subgroups of g such that l 6= {0}, r0(n) ≥ ω and r0(h) + r0(l) < r0(n). if h ∩l = {0}, then there exists a non-torsion element y ∈ x+n such that (h +〈y〉)∩l = {0} and 〈y〉∩h = {0}. 18 d. dikranjan, a. giordano bruno and c. milan now suppose that α > 0 and lβ and l ′ β are built for all β < α so that the conditions (aβ)-(dβ) are satisfied for all β < α. in order to define lα,l ′ α let m = ⋃ β<α lβ and m ′ = ⋃ β<α l′ β . then r0(m) + r0(m ′) < κ, so by the claim applied with x = 0, n = hα, h = m ′ and l = m, there exists a zα ∈ hα such that m ∩ (m ′ + 〈zα〉) = {0} and 〈zα〉 ∩ m ′ = {0}. put l′α = m ′ + 〈zα〉, hence r0(l ′ α) = r0(m ′) + 1 ≤ |α + 1| and l′α/m ′ is nontorsion. now r0(m) + r0(l ′ α) < κ, so it is possible to apply again the claim with n = hα, x = xα, h = m and l = l ′ α. then there exists yα ∈ xα + hα such that (m + 〈yα〉) ∩ l ′ α = {0}. put lα = m + 〈yα〉 and observe that (aα)-(dα) hold. finally, let l = ⋃ α<κ lα and l ′ = ⋃ α<κ l′α. since (aα) is true for all α, this implies (a). from (bα) and (cα) it follows respectively that the subgroup l′ satisfies l ∩ l′ = {0} and r0(l ′) ≥ κ. let ϕ : g → g/l be the canonical homomorphism. then the restriction ϕ|l′ : l′ → g/l is injective, hence r0(g/l) ≥ r0(ϕ(l ′)) = r0(l ′) = κ. � a subgroup l of a group g is said to be a complement of h if g = h + l. the co-rank of l is r0(g/l). lemma 4.8 will be used in two different ways. in the proof of theorem 4.13 we use it to build a gδ-dense subgroup of a pseudocompact group with large co-rank. in the next corollary we use it to build a complement of a given subgroup with large co-rank. corollary 4.9. let h be a subgroup of an abelian group g, such that r0(h) ≥ [g : h]. then there exists a subgroup l of g such that l + h = g and r0(g/l) ≥ r0(h). proof. let κ = r0(h). then we can write g = ⋃ α<κ (xα+h). then by lemma 4.8 applied to x = {xα : α < κ} and to the family {hα : hα = h, α < κ}, there exists a subgroup l of g with (a) and (b). by (a) l + h ⊇ xα + h for every α, that is l + h ⊇ g, while (b) ensures r0(g/l) ≥ r0(h). � remark 4.10. (a) obviously, the subgroup l in the above corollary has corank precisely r0(h) as g/l ∼= h/h ∩ l holds for every complement l of h. (b) one cannot remove the condition r0(h) ≥ [g : h]. indeed, for every n ∈ n the only complement l of the subgroup h = zn of g = qn is l = g. the next theorem shows that c-extremality is stable under taking closed gδ-subgroups. theorem 4.11. let g be a c-extremal pseudocompact abelian group. then every pseudocompact subgroup of g of index ≤ c is c-extremal. in particular, every closed gδ-subgroup of g is c-extremal. proof. aiming for a contradiction, assume that there exists a pseudocompact subgroup h of g with |g/h| ≤ c such that h is not c-extremal. then there exists a gδ-dense subgroup d of h with r0(h/d) ≥ c, so |g/h| ≤ r0(h/d). set g1 = g/d and let ϕ : g → g1 be the canonical projection. applying to ϕ(h) = h/d corollary 4.9, we find a subgroup l of g1 such that l+ϕ(h) = weakly metrizable pseudocompact groups 19 g1 and r0(g1/l) ≥ r0(ϕ(h)). let h0 = ϕ −1(l). then h + h0 = g. since d is a subgroup of h0 which is gδ-dense in h, the closure of h0 w.r.t. the gδ-topology contains h + h0 = g and so h0 is gδ-dense in g. moreover, r0(g/h0) = r0(g1/l) ≥ r0(ϕ(h)) = r0(h/d) ≥ c (the first equality is due to g/ϕ−1(l) ∼= g1/l). we have produced a gδ-dense subgroup h0 of g with r0(g/h0) ≥ c, a contradiction. now assume that h is a closed gδ-subgroup of g. then g/h is a compact metrizable group, so |g/h| ≤ c. hence the above argument applies to h. � corollary 4.12. let g be a c-extremal pseudocompact abelian group of size c. then every pseudocompact subgroup of g is c-extremal. the next theorem establishes a sufficient condition for a c-extremal pseudocompact abelian group to be singular. theorem 4.13. let g be a c-extremal pseudocompact abelian group. if w(g) ≤ c or g is compact, then g is singular. proof. by theorem 3.6 we have r0(g) ≤ c. if r0(g) < c, then g is bounded torsion by theorem 2.14. hence g is singular. suppose r0(g) = c and w(g) ≤ c. assume for a contradiction that g is not singular. by lemma 4.3 r0(n) = r0(g) = c for every n ∈ λ(g). let {xα + hα : α < c} be a list of all possible cosets of subgroups hα ∈ λ(g) (this is possible as |λ(g)| ≤ w(g)ω = c). since r0(hα) = c for all α < c, by lemma 4.8 there exists a subgroup l of g with (a) and (b). clearly, (a) means that l is a gδ-dense subgroup of g and (b) is r0(g/l) ≥ c. this proves that g is not c-extremal, against the hypothesis. assume that g is compact. according to [18], for every non-singular compact abelian group g there exists a surjective homomorphism g → ∏ i∈i ki, where i is uncountable and each ki is a compact non-torsion abelian group. obviously, such a product cannot be c-extremal, so g is not c-extremal either. � an alternative proof of the first part of the above theorem (w(g) ≤ c) can be derived from [3, lemma 6.7] and lemma 4.3. now we can prove as a corollary of theorem 4.13 a result that generalizes [9, theorem 3.4 and remark 4.3] where it is proved that questions (a) and (b) have positive answer in the case of compact groups. corollary 4.14. a compact abelian group g is metrizable if and only if g is weakly extremal. proof. it suffices to note that if g is weakly extremal, then g is singular by theorem 4.13 and consequently metrizable by theorem 4.6. � using the properties of non singular groups it is possible to prove: theorem 4.15. a pseudocompact abelian group g is metrizable if and only if g is weakly extremal and w(g) ≤ c. 20 d. dikranjan, a. giordano bruno and c. milan proof. if g is metrizable then g is weakly extremal and w(g) ≤ ω. assume that g is weakly extremal and w(g) ≤ c. then g is singular by theorem 4.13, hence theorem 4.6 applies to conclude that g is metrizable. � 5. almost connected groups in this section we study the properties of the connected component and of the subgroups n ∈ λ(g) of an almost connected group g. remark 5.1. let g be an almost connected, pseudocompact abelian group. then: (a) the quotient g/c(g) is metrizable and compact; (b) the connected component c(g) of g is a pseudocompact group (by corollary 2.6 (a)); if g is not metrizable, then w(c(g)) = w(g) (by corollary 2.6 (b)). lemma 5.2. let g be a pseudocompact abelian group. then the following conditions are equivalent: (a) g is almost connected; (b) g̃ is almost connected; (c) if g is dense in a topological group g1, then g1 is almost connected. proof. (a)⇒(c) by lemma 2.2 the group g1 is pseudocompact and so g is gδdense in g1. since g is almost connected, c(g) ∈ λ(g). as c(g) ⊆ c(g1) ∩g and c(g1)∩g is closed in g, it follows from corollary 2.6 (c) that c(g1)∩g ∈ λ(g). now corollary 2.9 applies to conclude that c(g1) ∈ λ(g1), i.e., g1 is almost connected. (c)⇒(b) is obvious. (b)⇒(a) suppose that g̃ is almost connected. then c(g̃) ∈ λ(g̃). since g is gδ-dense in g̃, by lemma 2.7 (c) c(g̃) ∩g is gδ-dense in c(g̃), which is connected. hence c(g̃)∩g is connected too and consequently c(g̃)∩g ⊆ c(g). as c(g̃) ∈ λ(g̃), it follows that c(g̃)∩g ∈ λ(g) and so c(g) ∈ λ(g). � corollary 5.3. if g is an almost connected pseudocompact abelian group and g is dense in a topological group g1, then c(g) = c(g1) ∩ g. in particular, c(g) = c(g̃)∩g. proof. the inclusion c(g) ⊆ c(g1) ∩g holds in general. since g is almost connected, lemma 5.2 yields that also g1 is almost connected, i.e., c(g1) ∈ λ(g1). moreover, as g is gδ-dense in g1, c(g1) ∩ g is gδ-dense in c(g1), that is connected. hence also c(g1) ∩ g is connected and so c(g1) ∩g ⊆ c(g). � now we prove a remarkable property of the almost connected pseudocompact abelian groups. for a topological group g denote by q(g) the quasi component of g (i.e., the intersection of all clopen sets of g [16]). usually, c(g) ≤ q(g), but strict inequality may hold even for pseudocompact abelian groups (see [16] for various levels of the failure of the equality c(g) = q(g)). weakly metrizable pseudocompact groups 21 corollary 5.4. let g be an almost connected pseudocompact abelian group. then q(g) = c(g). proof. it is known that q(g) = c(g̃) ∩g [17]. hence corollary 5.3 yields that c(g) = c(g̃)∩g. then q(g) = c(g). � next we prove that the connected component of a pseudocompact abelian group can be computed in the same way as in the case of compact abelian groups. corollary 5.5. if g is an almost connected pseudocompact abelian group, then c(g) = ⋂∞ n=1 ng g . proof. by corollary 5.3 c(g) = g ∩ c(g̃). since for every compact abelian group k one has c(k) = ⋂∞ n=1 nk [19], lemma 2.10 yields that c(g) = g∩ ∞⋂ n=1 ng̃ = ∞⋂ n=1 (g∩ng̃) = ∞⋂ n=1 ng g . � the next proposition shows that for singular (resp. totally disconnected) pseudocompact abelian groups almost connectedness and metrizability are equivalent. proposition 5.6. let g be a pseudocompact abelian group that is either totally disconnected or singular. then g is almost connected if and only if g is metrizable. proof. if g is metrizable, then g is almost connected. to prove the converse implication, suppose that g is almost connected. if g is totally disconnected, then {0} = c(g) ∈ λ(g). hence g is metrizable by lemma 2.4. if g is singular, then there exists m ∈ n+ such that mg is compact and metrizable by lemma 4.1(b). since c(g) ⊆ mg by corollary 5.5, it follows that w(c(g)) = ω. on the other hand, c(g) ∈ λ(g) by hypothesis, hence w(g) = w(c(g)) = ω, i.e., g is metrizable. � lemma 5.7. [6, theorem 3.4] let g be a connected pseudocompact abelian group. then c(n) ∈ λ(g) for every n ∈ λ(g), i.e., n is almost connected for every n ∈ λ(g). the above lemma can be generalized to almost connected pseudocompact abelian groups as follows. theorem 5.8. let g be a pseudocompact abelian group. then the following conditions are equivalent: (a) g is almost connected; (b) there exists n ∈ λ(g) such that n is almost connected; (c) every n ∈ λ(g) is almost connected; (d) ng ∈ λ(g) for every n ∈ n+; 22 d. dikranjan, a. giordano bruno and c. milan (e) there exists a gδ-dense subgroup h of g such that h is almost connected; (f) every gδ-dense subgroup h of g is almost connected. proof. (a)⇒(c) let n ∈ λ(g) and let c(n) be the connected component of n. then c(n) ⊆ c(g) so in particular c(n) ∈ λ(c(g)) by lemma 5.7 applied to c(g). hence c(n) ∈ λ(g) by corollary 2.6 (d). (c)⇒(b) is obvious. (b)⇒(a) if n ∈ λ(g) is almost connected, then c(n) ∈ λ(g). since c(g) is closed in g and c(n) ⊆ c(g), corollary 2.6 implies c(g) ∈ λ(g). (a)⇒(d) if g is almost connected, then by corollary 5.5 c(g) = ⋂∞ n=1 ng ∈ λ(g). hence c(g) ⊆ ng for every n ∈ n+ and consequently ng ∈ λ(g) by corollary 2.6 (c). (d)⇒(a) if ng ∈ λ(g) for every n ∈ n+, then ⋂∞ n=1 ng ∈ λ(g). the compactness of g̃ implies c(g̃) = ⋂∞ n=1 ng̃. therefore c(g̃) ∩g = ∞⋂ n=1 ng̃∩g = ∞⋂ n=1 ng ∈ λ(g) and so by corollary 2.9 one has that c(g̃) ∈ λ(g̃), i.e., g̃ is almost connected. lemma 5.2 applies to conclude that g is almost connected. (a)⇒(f) assume that g is almost connected and let h be a gδ-dense subgroup of g. since c(g) ∈ λ(g), it follows that c(g) ∩h is gδ-dense in c(g). then c(g)∩h is connected and so c(g)∩h ⊆ c(h). since c(g)∩h ∈ λ(h), corollary 2.6 applies to conclude that c(h) ∈ λ(h). (f)⇒(e) is obvious. (e)⇒(a) suppose that h is an almost connected gδ-dense subgroup of g. then h is pseudocompact and c(h) ∈ λ(h). since c(h) ⊆ c(g) ∩ h and c(g) ∩h is closed in h, it follows from corollary 2.6 that c(g) ∩h ∈ λ(h). then corollary 2.9 implies that c(g) ∈ λ(g), i.e., g is almost connected. � theorem 5.8 can be applied to show that the same conclusion of proposition 5.6 is true if we suppose that there exists n ∈ λ(g) such that n is totally disconnected. corollary 5.9. let g be an almost connected pseudocompact abelian group. if there exists n ∈ λ(g) such that n is totally disconnected, then g is metrizable. proof. since g is almost connected, theorem 5.8 (d) implies that also n ∈ λ(g) is almost connected. on the other hand, n is totally disconnected, hence metrizable by proposition 5.6. then also the group g is metrizable. � in the next corollary we give two opposite properties of the almost connected groups (as far as the similarity with connected groups is concerned). the first one shows that almost connectedness cannot be destroyed by taking direct products with a compact metrizable group, while the second one is a stability property usually possessed by connected groups. weakly metrizable pseudocompact groups 23 corollary 5.10. let g be a pseudocompact abelian group. (a) for every compact metrizable group m, the product g × m is almost connected if and only if g is almost connected. (b) let n be a closed subgroup of g: (b1) if g is almost connected, then also g/n is almost connected; (b2) if both n and g/n are almost connected, then also g is almost connected. the next lemma and its corollary are used in the proof of theorem 5.20 to produce gδ-dense subgroups of connected pseudocompact groups. lemma 5.11. let g be a precompact connected group. then md is dense in g for every m ∈ n+ and for every dense subgroup d of g. in particular, mg is dense in g for every m ∈ n+. proof. since g̃ is connected and compact, g̃ is divisible. then mg̃ = g̃ for every m ∈ n+. since d is dense in g, md is dense in mg. analogously, the density of g in g̃, implies that mg is dense in mg̃ = g̃. then also md is dense in g̃. as md ⊆ g one has that md is dense in g. the last assertion follows from the density of g in g̃. � corollary 5.12. let g be a connected pseudocompact abelian group. then ma is gδ-dense in g for every m ∈ n+ and for every gδ-dense subgroup a of g. theorem 5.13. let g be an almost connected pseudocompact abelian group of uncountable weight α. then: (a) r0(c(g)) = r0(g) ≥ c and α ≤ 2 r0(g); (b) |c(g)| = |g| ≤ 2α ≤ 22 r0(g) ; (c) m(c(g)) = m(g) ≤ r0(g). proof. (a) being almost connected and non-metrizable, the group g is not singular (cf. proposition 5.6), hence r0(c(g)) = r0(g) ≥ c by lemma 4.3. to prove α ≤ 2r0(g) fix a free subgroup f of size r0(g) of g. then w(f) = w(f) ≤ 2r0(g). since g/f is torsion, the pseudocompact group g/f is torsion, hence bounded by theorem 2.14. so there exists n > 0 such that ng ⊆ f . so w(ng) ≤ w(f) ≤ 2r0(g). since g is almost connected, w(ng) = w(g) = α by (d) of theorem 5.8. this proves α ≤ 2r0(g). (b) if |g| = c, then by the first part of (a) c ≤ r0(c(g)) = r0(g) ≤ |g| ≤ c, hence |c(g)| = |g| = c. suppose now that |g| > c. since c(g) ∈ λ(g), the quotient g/c(g) is compact and metrizable (cf. remark 5.1) and so |g/c(g)| ≤ c. now |g| = |g/c(g)| · |c(g)| implies |c(g)| = |g|. (c) the equality m(c(g)) = m(g) follows from (c) of lemma 2.15 as g is almost connected. moreover, w(c(g)) = w(g) > ω and |g/c(g)| ≤ c according by remark 5.1 and r0(c(g)) ≥ c by (a). let m be a maximal independent subset of c(g) and let a = 〈m〉. then the quotient c(g)/a is torsion, so setting an = {x ∈ c(g) : nx ∈ a} for every 24 d. dikranjan, a. giordano bruno and c. milan n ∈ n+, one has c(g) = ⋃∞ n=1 an. by corollary 2.16 there exist n ∈ n+ and e ≤ c(g), with |e| ≤ c, such that an + e is gδ-dense in c(g). then by corollary 5.12 also n(an + e) is gδ-dense in c(g). so for f := ne one has |f | ≤ c and n(an + e) ⊆ nan + ne ⊆ a + f ⊆ c(g). thus a+f is gδ-dense in c(g). by lemma 2.15 (b) there exists a subgroup d of g with |d| ≤ c such that l = a + f + d is gδ-dense in g. since |f + d| ≤ c ≤ |a|, we have |l| ≤ |a| = r0(c(g)) = r0(g). hence m(g) ≤ r0(g). � the next theorem shows the relation between d-extremality and almost connectedness. theorem 5.14. every d-extremal pseudocompact abelian group is almost connected. in particular, weakly extremal pseudocompact abelian groups are almost connected. proof. aiming for a contradiction, assume that g is not almost connected. then by theorem 5.8 (d), there exists n ∈ n+ such that ng 6∈ λ(g), i.e., ng is not a gδ-set. hence w(g/ng) > ω and g is not d-extremal by lemma 4.5, a contradiction. � let us see now that for compact groups the above implication can be inverted. theorem 5.15. for a compact abelian group g the following are equivalent: (a) g is almost connected; (b) g is d-extremal; (c) g is strongly d-extremal. proof. the implication (b) ⇒ (a) follows from theorem 5.14. conversely, if g is almost connected, then c(g) ∈ λ(g) is compact and connected, hence divisible. in particular, c(g) is d-extremal and consequently g is d-extremal too by lemma 3.3 (a). by the equivalence of (a) and (b) and by theorem 5.8, (b) ⇒ (c). � remark 5.16. let us note here that one cannot invert the implication dextremal⇒almost connected even for connected pseudocompact abelian groups. by [21] there exists a dense pseudocompact subgroup g of tc of size c. note that |g| = w(g) = c. since r0(t c) = 2c > c, there exists an infinite cyclic subgroup c of tc such that c ∩g = {0}. then the pseudocompact subgroup g1 = g + c of t c is connected (as tc is connected) and g is gδ-dense in g1. since g1/g ∼= c is not divisible, g1 is not d-extremal. in contrast with remark 5.16 the following corollary shows that for totally disconnected pseudocompact abelian groups d-extremality and almost connectedness are equivalent. let us recall that questions (a) and (b) of the introduction have positive answer in the case of zero-dimensional groups [11, theorem 7.3] and in the case of totally disconnected groups [3]. the following corollary generalizing these results covers theorem b from the introduction. weakly metrizable pseudocompact groups 25 corollary 5.17. let g be a pseudocompact abelian group that is either singular or totally disconnected. then the following conditions are equivalent: (a) g is almost connected; (b) g is d-extremal; (c) g is metrizable. proof. (c)⇒(b) is obvious, (b)⇒(a) follows from theorem 5.14 and (a)⇒(c) follows from proposition 5.6. � in particular one has: corollary 5.18. a torsion pseudocompact abelian group g is weakly extremal iff g is metrizable. corollary 5.19. let g be a pseudocompact abelian group with w(g/c(g)) > ω. then g is not d-extremal. proof. if g were d-extremal, then by theorem 5.14 g would be almost connected, i.e., c(g) ∈ λ(g) and consequently w(g/c(g)) = ω. � the corollary applies also to pseudocompact abelian groups g with w(c(g))< w(g) (since then w(g/c(g)) > ω). theorem 5.20. let g be an almost connected pseudocompact abelian group. then c(g) is divisible if and only if c(g) is d-extremal (so also g is d-extremal). proof. since g is almost connected the subgroup c(g) is pseudocompact by remark 5.1 (b). if c(g) is divisible, then obviously c(g) is d-extremal. assume now that c(g) is not divisible. then there exists n0 ∈ n+ such that n0c(g) is a proper subgroup of c(g). by corollary 5.12 n0c(g) is also gδdense in c(g). since the quotient c(g)/n0c(g) is bounded torsion, it cannot be divisible and so c(g) is not d-extremal. � there is a similar result for strong extremality: for a strongly extremal (strongly d-extremal) pseudocompact abelian group g, c(g) is divisible if and only if c(g) is strongly extremal (strongly d-extremal). remark 5.21. for every s-extremal (resp. weakly extremal, d-extremal) pseudocompact abelian group g the subgroup c(g) is divisible if and only if it is s-extremal (resp. weakly extremal, d-extremal). indeed, by theorem 5.20, if c(g) is s-extremal, then c(g) is divisible. vice versa, assume that c(g) is divisible. note that, c(g) ∈ λ(g) since g is almost connected by theorem 5.14. now lemma 3.1 (b) applies. the case when c(g) is weakly extremal (resp., d-extremal) is analogous. corollary 5.22. a d-extremal pseudocompact abelian group g is divisible if and only if g is connected. proof. if g is divisible and pseudocompact, then g is connected too [33]. if g = c(g) is connected, theorem 5.20 applies. � 26 d. dikranjan, a. giordano bruno and c. milan 6. various characterizations of extremality 6.1. the closure of the graph of a homomorphism. if g and h are abelian topological groups and h : g → h is a homomorphism, the graph γh of h is the subset {(x,h(x)) : x ∈ g} of g × h. then γh is a subgroup of g×h such that (1) if p1 : g×h → g is the canonical projection on the first component, then p1(γh) = g; (2) γh ∩ ({0}×h) = {(0,0)}. consequently g×h = γh ⊕ ({0}×h). let vh be the vertical component of γh in g×h, that is vh = γh∩({0}×h). then γh splits also as γh = γh⊕vh. since vh is a subgroup of {0}×h, it is possible to identify vh with a closed subgroup of h. the fact that vh is a subgroup follows also from the next lemma. lemma 6.1. let g and h be topological abelian groups and h : g → h be a homomorphism. then vh = {t ∈ h : ∃ a net {xα}α∈a ⊆ g such that xα → 0 and h(xα) → t}. proof. if t ∈ h, then (0, t) ∈ γh ∩ ({0}× h) if and only if there exists a net {(xα,h(xα))}α∈a in γh such that 0 = lim xα and t = limh(xα). � remark 6.2. (a) if g and h are abelian topological groups and h : g → h is a homomorphism, then (1) γh is closed in g×h if and only if vh = {0}. (2) γh is dense in g×h if and only if vh = h. hence it follows that if h is compact, h is not continuous and γh is not dense in g×h, then vh is a proper closed subgroup of h. (b) if g is a pseudocompact abelian group, h is a compact subgroup of t and h : g → h is a non-continuous and surjective homomorphism such that γh is not dense in g × h, then by (a) there exists n ∈ n, n > 1, such that vh = z(n). (c) one can easily prove that in the above lemma vh = ⋂ {h(u) : u ∈ vg(0)}. for h = t the latter subgroup was introduced also in [3, notation 3.3]. lemma 6.3. let g be an abelian topological group and let h be a compact abelian group. let h : g → h be a homomorphism. then for every k ∈ n: (a) vkh = kvh; (b) kh is continuous if and only if kvh = 0. proof. (a) first we prove that kvh ⊆ vkh. let t ∈ vh. then by lemma 6.1 there exists a net {xα}α∈a in g such that 0 = lim xα and t = limh(xα). hence kt = limk(h(xα)) = lim(kh)(xα) and consequently kt ∈ vkh by lemma 6.1. to prove the converse inclusion consider t ∈ vkh. in particular, t ∈ h and by lemma 6.1 there exists a net {xα}α∈a in g such that 0 = limxα and t = lim(kh)(xα). thus {h(xα)}α∈a is a net in the compact group h, hence weakly metrizable pseudocompact groups 27 there exists a subnet {h(xαβ )}β∈b of {h(xα)}α∈a which converges to s ∈ h. since {xαβ}β∈b converges to 0, by lemma 6.1 s ∈ vh. moreover it follows that {kh(xαβ )}β∈b converges to ks and so for the uniqueness of limits t = ks ∈ kvh. (b) it follows directly from (a) that γkh = γkh ⊕kvh for every k ∈ n. by the closed graph theorem kh is continuous if and only if γkh is closed, i.e., kvh = 0. � corollary 6.4. let g be a topological abelian group and for n ∈ n+ let h : g → z(n) be a homomorphism. then the following conditions are equivalent: (a) γh is dense in g×z(n); (b) the homomorphism kh : g → z(n) is not continuous for every integer k such that 0 < k < n; (c) kvh 6= {0} for every integer k such that 0 < k < n. proof. (a)⇒(b) if γh is dense in g× z(n) then vh = z(n). applying lemma 6.3 we conclude that the homomorphism kh is not continuous for every k ∈ n+ such that k < n. (b)⇔(c) follows directly from lemma 6.3 (b). (c)⇒(a) by hypothesis vh = z(n), hence remark 6.2 (2) applies. � lemma 6.5. let g be a topological abelian group and let h be a closed subgroup of t. let h : g → h be a surjective homomorphism such that (g,τh) is pseudocompact and vh = z(n) for some n ∈ n+. let n = kernh. then: (a) n = h−1(vh); (b) γh|n is gδ-dense in n ×vh. (c) kerh is a proper gδ-dense subgroup of n. proof. (a) let x ∈ n = kernh. then (nh)(x) = 0, that is n(h(x)) = 0. it follows that h(x) ∈ z(n) = vh. this proves that h(n) ⊆ vh, that is n ⊆ h−1(vh). to prove the opposite inclusion, take y ∈ vh. observe that vh ⊆ h = h(g). then there exists x ∈ g such that h(x) = y. as y ∈ vh and nvh = {0}, in particular ny = 0. consequently 0 = ny = n(h(x)) = (nh)(x) and hence x ∈ n, that is y ∈ h(n). this proves that h−1(vh) ⊆ n. (b) by (a) h|n : n → vh = z(n) is a surjective homomorphism. it follows from lemma 6.3 that the homomorphism kh is not continuous for every k ∈ n such that 0 < k < n. corollary 6.4 implies that γh|n is dense in n × vh. since(g,τh) is pseudocompact by hypothesis and nh is continuous, n ∈ λ(g,τh) and (n,τh|n) = (n,(τ|n )h|n ) is pseudocompact. moreover, (n,(τh|n)h|n ) is homeomorphic to γh|n and then also γh|n is pseudocompact. hence γh|n is gδ-dense in n ×vh. (c) it follows from (b) that γh|n is gδ-dense in n × vh. then by lemma 3.7 kerh|n = kerh is proper and gδ-dense in n. � remark 6.6. let (g,τ) be a topological abelian group and h : g → t be a homomorphism such that h = h(g) is a closed subgroup of t. (a) if vh = z(n) for some n ∈ n+, then lemma 6.3 implies 〈h〉∩ ĝ = 〈nh〉. 28 d. dikranjan, a. giordano bruno and c. milan (b) combining appropriately (a) and the results of this section, it is possible to prove the following theorem, announced in [1, theorem 5.10] and [23, lemma 3.6] and proved in [3, theorem 3.10]. if (g,τ) is pseudocompact and h is not τ-continuous, then: (i) if ĝ∩〈h〉 = {0}, then (g,τh) is pseudocompact if and only if h(g) is closed in t and kerh is gδ-dense in (g,τ); (ii) if ĝ ∩〈h〉 = 〈nh〉 for some n ∈ n+, then (g,τh) is pseudocompact if and only if h(g) is closed in t and kerh is gδ-dense in kernh. since we are not going to use here this property, the interested reader is invited to consult [3] for a detailed proof. 6.2. applications to r-extremality. now we are able to prove several important results on the relations among different kinds of extremality. theorem 6.7. let g be a pseudocompact abelian group. then the following conditions are equivalent: (a) g is metrizable; (b) every n ∈ λ(g) is s-extremal; (c) g is s-extremal and strongly d-extremal. proof. (a)⇒(b) is obvious, (b) ⇒ (a) is proved in [3]. (b)⇒(c) is obvious. (c)⇒(b) let n ∈ λ(g). since g is weakly extremal, g is almost connected by theorem 5.14, hence c(n) ∈ λ(g). therefore, c(n) is a connected, d-extremal pseudocompact abelian group and consequently it is divisible by corollary 5.22. thus c(n) is s-extremal by lemma 3.1 (b). since c(n) ∈ λ(n) by theorem 5.8, n is s-extremal too according to lemma 3.3 (a). � here we see that the stronger version of weak extremality, imposed on all n ∈ λ(g), gives strong extremality. theorem 6.8. a pseudocompact abelian group g is strongly extremal if and only if every n ∈ λ(g) is weakly extremal. proof. if g is strongly extremal, then by theorem 3.12 every n ∈ λ(g) is weakly extremal. to prove the converse implication, it suffices to show that if every n ∈ λ(g) is weakly extremal, then g is r-extremal. indeed, if this is true, then every n ∈ λ(g) is r-extremal (cf. remark 3.5). suppose for a contradiction that g is not r-extremal. then there exists a pseudocompact group topology τ′ on g such that τ′ > τ. since τ and τ′ are, in particular, precompact group topologies, there exists a homomorphism h ∈ (̂g,τ′) \ (̂g,τ), i.e., h : g → t is τ′-continuous but not τ-continuous. let h = h(g). then h is a compact subgroup of t, as the image of the pseudocompact group (g,τ′) under the τ′-continuous homomorphism h. since h is not τ-continuous, γh is not closed in (g,τ) ×h. there are two cases. weakly metrizable pseudocompact groups 29 case 1. suppose that γh is dense in (g,τ) ×h. since h is τ ′-continuous, τh ≤ τ ′ and so (g,τh) is pseudocompact (as (g,τ ′) is pseudocompact by hypothesis). since γh is homeomorphic to (g,τh), also γh is pseudocompact. then γh is gδ-dense in (g,τ) ×h and hence kerh is a gδ-dense subgroup of g by lemma 3.7. moreover, the quotient g/ kerh is algebraically isomorphic to h. being a compact subgroup of t, the group h is all t or it is finite. if h = t then r0(h) = r0(t) = c and so r0(g/ kerh) = r0(h) = c. if h is finite, then also g/ kerh is finite so in particular it cannot be divisible. in both cases g is not weakly extremal, against the hypothesis. case 2. if γh is not dense in (g,τ) × h, then by remark 6.2 (b) there exists n ∈ n+, n > 1, such that vh = z(n). then nh is τ-continuous by lemma 6.3 and so n = kernh ∈ λ(g). as observed in case 1, (g,τh) is pseudocompact, hence kerh is a proper gδ-dense subgroup of n by lemma 6.5. since n/ kerh ∼= z(n) it cannot be divisible, i.e. n is not weakly extremal, a contradiction. � combining with theorem 4.11 we obtain: corollary 6.9. let g be a c-extremal, strongly d-extremal pseudocompact abelian group. then g is strongly extremal. now we can prove theorem c from the introduction. we use essentially some ideas from the proof of theorem 4.4 (b) of [3]. theorem 6.10. let g be a s-extremal pseudocompact abelian group. then g is doubly extremal if and only if g[p] = g̃[p] for every p ∈ p. proof. assume that g[p] = g̃[p] for every p ∈ p and suppose for a contradiction that (g,τ) is not r-extremal. then there exist a pseudocompact group topology τ′ on g such that τ′ > τ and a homomorphism h : g → t which is τ′-continuous but not τ-continuous (cf. the proof of theorem 6.8). take h = h(g); then h is a compact subgroup of t as h is τ′-continuous. since h is not τ-continuous, γh is not closed in (g,τ) ×h. there are two cases: case 1. suppose that γh is dense in (g,τ) ×h. since h is τ ′-continuous, τh ≤ τ ′ and so (g,τh) is pseudocompact. since γh is homeomorphic to (g,τh), also γh is pseudocompact, hence gδ-dense in (g,τ) × h. now theorem 3.8 applies to conclude that g is not s-extremal, a contradiction. case 2. if γh is not dense in (g,τ) × h, then by remark 6.2 (b) there exists n ∈ n+, n > 1, such that vh = z(n). it follows from lemma 6.3 that nh is continuous. let p ∈ p such that p|n. take m = n p ∈ n+; we have that m < n. define h1 := mh. since m < n the homomorphism h1 is not τ-continuous. on the other hand, h1 is τ ′-continuous (as h is τ′-continuous) and ph1 is τ-continuous as ph1 = pmh = nh. moreover vh1 = vmh = mvh = mz(n) = z(p). 30 d. dikranjan, a. giordano bruno and c. milan consider now f = ph1 : (g,τ) → ph ≤ h. since f = ph1 = nh, one has f ∈ (̂g,τ). note that f|g[p] = 0 and consider a continuous extension f̃ of f to g̃ (that is, the continuous homomorphism f̃ : g̃ → t such that f̃|g = f). since f̃|g[p] = f|g[p] = 0 and f̃ is continuous, we have that f̃|g[p] = 0. by hypothesis g[p] = g̃[p], hence f̃| g̃[p] = 0, i.e., f̃ ∈ a(g̃[p]). since a(g̃[p]) = p ̂̃ g, there exists a continuous homomorphism g̃ : g̃ → t such that f̃ = pg̃. if we take g := g̃|g, then f = pg. since f = ph1 by definition, we have p(h1 − g) = 0 and so t := h1 − g is a torsion element of hom(g,t) of order p. clearly, as h1 is not τ-continuous, also t is not τ-continuous. then t : g → z(p) is surjective and γt is not closed. since by remark 6.2 (1) vt is a non trivial subgroup of z(p), it follows that vt = z(p). note that (g,τt) is a pseudocompact group, since t is τ ′continuous and so τ′ ≥ τt. we can now apply lemma 6.5 and conclude that kert is a proper gδ-dense subgroup of kerpt = g. this proves that g is not s-extremal, a contradiction. hence g is r-extremal, and consequently doubly extremal. the converse implication follows from theorem 1.2 (b) since g is s-extremal. � for a compact abelian group g the dimension of g coincides with the free rank r0(ĝ) of the group ĝ of characters of g. in what follows it will be denoted by dimg. definition 6.1. let g be an almost connected pseudocompact abelian group. for n ∈ λ(g) let codimn = dimg/n, and λn(g) = {n ∈ λ(g) : codimn = n}. let us briefly list some properties of this new invariant: (a) λ0(g) consists precisely of those n ∈ λ(g) that contain c(g) (first note that if c(g) ∈ λ(g), then for n ≥ c(g) the quotient g/n is also a quotient of the compact totally disconnected group g/c(g), so g/n is zero-dimensional); (b) if n1,n2 ∈ λn and n1 ≤ n2, then n2/n1 is a compact totally disconnected metrizable group; (c) let λ̃1(g) = {n ∈ λ(g) : g/n ∼= t}. clearly, λ̃1(g) ⊆ λ1(g) as dim t = 1. by (b), if n1,n2 ∈ λ̃1(g) and n1 ≤ n2, then n2/n1 is finite cyclic (as the only compact totally disconnected subgroups of t are the finite ones). the next theorem will make use of the following condition: (d̃) every n ∈ λ̃1(g) is d-extremal. note that (d̃) is obviously weaker than strong d-extremality, but implies dextremality by lemma 3.3 (a). we prove now theorem d which shows that weakly metrizable pseudocompact groups 31 (d̃) in conjunction with weak extremality is equivalent to r-extremality for connected pseudocompact abelian groups. theorem 6.11. let g be a connected pseudocompact abelian group. then g is r-extremal if and only if g is weakly extremal and (d̃) holds. proof. since r-extremality implies weak extremality, it suffices to assume that g is weakly extremal and prove that g is r-extremal if and only if (d̃) holds true. we shall prove that their negations are equivalent. since g is weakly extremal and connected, we conclude by corollary 5.22 that g is divisible. denote by τ the topology of g and assume that (g,τ) is not r-extremal. then there exists a discontinuous character h : (g,τ) → t such that (g,τh) is pseudocompact. since h : (g,τh) → t is continuous and g is divisible, the image h(g) must be a non-trivial divisible compact subgroup of t (by the pseudocompactness of τh). hence h is surjective. according to remark 6.6 (b) one has the two alternatives (i) and (ii). in (i) the subgroup h = kerh of g is gδ-dense and g/h ∼= t has free rank c. this contradicts weak extremality of g. hence we are left with (ii); i.e., there exists n > 0 such that nh is τ-continuous and h is gδ-dense in n = kernh. since obviously n ∈ λ̃1(g) and since n/h is bounded torsion, we conclude that n is not d-extremal, i.e. (d̃) fails. now suppose that some n ∈ λ̃1(g) is not d-extremal. then there exists a gδ-dense subgroup h of n such that n/h is not divisible. it is not restrictive to assume that n/h ∼= z(p) for some prime p. to prove that g is not rextremal we shall find a surjective, discontinuous character h : (g,τ) → t such that kerph = n and kerh = h. then τh will be pseudocompact by remark 6.6 and so (g,τ) will be not r-extremal. to build such an h we fix first a continuous surjective homomorphism l : g → t with ker l = n witnessing n ∈ λ̃1(g). let π : g → g/h be the canonical homomorphism. let n′ = π(n). then n′ ∼= z(p). our first aim is to prove now that g/h ∼= t. for simplicity write x = g/h and recall that x is divisible as a quotient of g. hence the homomorphism ϕp : x → x is surjective and obviously n′ ≤ kerϕp. hence ϕp factorizes as ϕp = γ ◦ λ, where λ : x → x/n ′ is the canonical homomorphism. as x/n′ ∼= g/n ∼= t and γ : x/n′ → x is surjective, we conclude that x is isomorphic to a quotient of t. moreover, the homomorphism γ has kernel k = λ(x[p]), hence pk = 0. since x/n′ ∼= t, this means that k is finite (cyclic), hence x ∼= (x/n′)/k ∼= t. fix an isomorphism ξ : t → x and consider the composition s = λ ◦ ξ : t → t. then kers = z(p). let us split t = z(p∞) ⊕ t and note that s(z(p∞)) = z(p∞). since end(z(p∞)) = zp is a discrete valuation domain, we can write s|z(p∞) = pη1, where η1 is an automorphism of z(p ∞). as t has no elements of order p, the restriction s|t : t → s(t) (1) is an isomorphism and therefore s(t) ∩ z(p∞) = {0}. then we can find an isomorphism η2 : t → s(t) such that s|t = pη2 (note that ϕp : s(t) → s(t) is an isomorphism as s(t) is divisible). now let η : t → t be the isomorphism 32 d. dikranjan, a. giordano bruno and c. milan defined by η = η1 +η2. so, setting h : g → t to be the composition η◦ξ −1◦π, we can claim that ph = l is continuous and kerph = n. since kerh = h, this ends up the proof. � 7. the cardinal invariants of extremal pseudocompact groups comfort and van mill [6, theorem 6.1] proved that |t(g)| ≤ c for connected s-extremal pseudocompact abelian groups (here t(g) denotes the subgroup of torsion elements of g). later comfort, gladdines and van mill [4] proved that r0(g) ≤ c for an s-extremal pseudocompact abelian group g (given as theorem 3.6 here). this allowed them to prove that s-extremal pseudocompact abelian groups have size ≤ c [4, theorem 4.8]. the next theorem extends these results to all weakly extremal pseudocompact abelian groups (as s-extremal groups g obviously satisfy m(g) = |g|). theorem 7.1. let g be an infinite almost connected c-extremal pseudocompact abelian group. then |g| ≤ c if and only if m(g) = |g|. proof. since m(g) ≥ c, obviously |g| ≤ c implies m(g) = |g|. now assume m(g) = |g| holds. by theorem 3.6 r0(g) ≤ c. if r0(g) < c, then g is bounded torsion according to theorem 2.14. in particular, g is singular. by proposition 5.6 g is metrizable, so |g| ≤ c. if r0(g) = c (so in particular g is not torsion), it is sufficient to apply theorem 5.13 (c). � corollary 7.2. let g be a weakly extremal pseudocompact abelian group. then |g| = c if and only if m(g) = |g|. in particular, r0(g) = |g| = c holds true for every non-torsion weakly extremal pseudocompact abelian group g with m(g) = |g|. an elegant application of the inequality |g| ≤ c for s-extremal groups was found by comfort and galindo [3, theorem 6.1] (announced in [23, lemma 8.1] and [1, theorem 5.16 (b)]), namely (b) ⇒ (a) of theorem 6.7 (which shows that for pseudocompact abelian groups the strong s-extremality is equivalent to metrizability). in the sequel we shall apply theorem 7.1 to obtain further connections among the cardinal invariants of the extremal pseudocompact groups. comfort and van mill [6, theorem 5.5] proved that a non-metrizable connected pseudocompact abelian group g with |g| ≥ w(g)ω is not s-extremal. as an immediate consequence of theorem 7.1 we obtain the following stronger result. corollary 7.3. an infinite weakly extremal pseudocompact abelian group g is metrizable if and only if m(g) = |g| ≥ w(g). proof. by theorems 7.1 and 2.14 we have |g| = c. hence g is weakly extremal with w(g) ≤ c. now theorem 4.15 applies to conclude that g is metrizable. � corollary 7.4. let g be a weakly extremal pseudocompact abelian group. then w(g) ≤ 2c. weakly metrizable pseudocompact groups 33 proof. if g is singular, then g is metrizable by proposition 5.6, so w(g) = ω. if g is not singular, then w(g) > ω and r0(g) ≥ c by lemma 4.3. on the other hand, r0(g) = c by theorem 3.6, as g is c-extremal. now theorem 5.13 (a) applies as g is almost connected. � let us recall that according to [17], a group g is hereditarily pseudocompact when every closed subgroup of g is pseudocompact. it was proved in [3] that hereditary pseudocompactness has a strong impact on the s-extremal pseudocompact abelian groups: (a) ([3, theorem 6.9]) every finitely generated subgroup of g is metrizable. (b) ([3, theorem 6.10], under the assumption of lusin’s hypothesis) g itself is metrizable. here we strengthen (a) by replacing “s-extremal” by “weakly extremal and m(g) = |g|” (corollary 7.9). moreover, we show that under this weaker hypothesis one can strengthen further (a) by replacing “finitely generated” by “countable”, if g is connected. this implies that hereditary pseudocompactness coincides with ω-boundedness for weakly extremal pseudocompact groups with m(g) = |g| (corollary 7.11). we were not able to remove lusin’s hypothesis in (b), but in corollary 7.7 we give a stronger version of (b), with “g s-extremal” replaced by “g weakly extremal and m(g) = |g|”. lemma 7.5. let g be a c-extremal, hereditarily pseudocompact abelian group of size c. then there exists m ∈ n+ such that mh is metrizable for every subgroup h of g with w(h) ≤ c. proof. let h be a subgroup of g with w(h) ≤ c. since g is hereditarily pseudocompact, the closure l of h in g is a pseudocompact subgroup of g so l is c-extremal by corollary 4.12. since w(l) = w(h) ≤ c, l is singular by theorem 4.13. then there exists m > 0 such that ml (and so also mh) is metrizable. it remains to see that such an m can be chosen uniformly for all subgroups h ≤ g with w(h) ≤ c. assume the contrary. then for every m > 0 there exists a subgroup hm of g such that w(hm) ≤ c and mhm is not metrizable. consider the subgroup h = ∑∞ m=1 hm of g. to see that w(h) ≤ c it suffices to recall that h is precompact, so w(h) = |ĥ|. if ρm : ĥ → ĥm is the restriction homomorphism, then the diagonal homomorphism ρ : ĥ → ∏∞ m=1 ĥm is injective. since |ĥm| = w(hm) ≤ c, we conclude that |ĥ| ≤ c. by the first part of the argument there exists m0 > 0 such that m0h is metrizable. then m0hm0 is metrizable as well, a contradiction. � lemma 7.6. let g be a connected pseudocompact abelian group of weight > c. then there exists a subgroup h of g of size ≤ ω1 such that mh is not metrizable for any m > 0. proof. following the idea from the proof of [3, theorem 6.10] one can construct by transfinite induction of length ω1 a subgroup h of g of size ≤ ω1 such that r0(ĥ) ≥ ω1. (for every countable subgroup a of g the closure l of a has 34 d. dikranjan, a. giordano bruno and c. milan w(l) ≤ c, so g/l is a non-trivial connected pseudocompact abelian group. hence there exists a non-torsion continuous character ξ : g/l → t, that produces a non-torsion continuous character χ : g → t such that χ|l = 0. the characters χα produced in this way, along with the elements xα ∈ g witnessing χα is non-torsion give rise to the subgroup h = 〈xα : α < ω1〉 of g that has character group ĥ with r0(ĥ) > ω1, witnessed by the independent family {χα|h}.) let m > 0. since m̂h = ĥ/ĥ[m] ∼= mĥ, it follows that w(mh) = |mĥ| > ω, i.e. mh is not metrizable. � corollary 7.7. under the assumption of the lusin’s hypothesis every weakly extremal hereditarily pseudocompact abelian group g with m(g) = |g| is metrizable. proof. it is not restrictive to assume that g is infinite. according to theorems 7.1 and 2.14 |g| = c, hence we can apply lemma 7.5 to produce an m > 0 such that mh is metrizable for every subgroup h of g of weight ≤ c. assume now that g is not metrizable. then w(g) > c by theorem 4.15. since g is almost connected, w(c(g)) = w(g) > c according to remark 5.1. now by lemma 7.6 we can find a subgroup h of c(g) with |h| ≤ ω1 and mh non-metrizable. this means that w(h) > c. since w(h) ≤ 2ω1, we conclude that the lusin’s hypothesis fails. � corollary 7.8. let g be a c-extremal, hereditarily pseudocompact abelian group of size c. then there exists m ∈ n+ such that every countable subgroup of mg is metrizable. in particular, mg is ω-bounded. proof. let m ∈ n+ as in lemma 7.5 and let d be a countable subgroup of mg. for every d ∈ d pick an element gd ∈ g such that d = mgd and let d1 be the subgroup of g generated by the subset x = {gd ∈ g : d ∈ d}. then d1 is countable and w(d1) ≤ c so md1 is metrizable by lemma 7.5. since d ⊆ md1, d is metrizable as well. the last assertion follows from the fact that mg is hereditarily pseudocompact (as continuous image of g), hence the closure of d in mg is a metrizable pseudocompact (hence compact) subgroup of mg containing d. � the next corollary generalizes theorem 6.9 from [3], where s-extremality of g is assumed instead of “c-extremal of size c”. corollary 7.9. let g be a c-extremal, hereditarily pseudocompact abelian group of size c. then every finitely generated subgroup of g is metrizable. proof. let f be a finitely generated subgroup of g. by corollary 7.8 there exists m > 0 such that the (finitely generated) subgroup mf of mg is metrizable. then f is metrizable since the quotient f/mf is finite (as every torsion finitely generated abelian group). � corollary 7.10. let g be a connected, weakly extremal, hereditarily pseudocompact abelian group of size c. then every countable subgroup of g is metrizable. in particular, g is ω-bounded. weakly metrizable pseudocompact groups 35 proof. follows from corollary 7.8 as g is divisible by corollary 5.22. � this corollary implies that for weakly extremal connected pseudocompact group of size c hereditary pseudocompactness coincides with ω-boundedness. in the next corollary we isolate this fact for the smaller class of s-extremal groups. corollary 7.11. a connected s-extremal pseudocompact abelian group is hereditarily pseudocompact if and only if it is ω-bounded. 8. further comments and open questions even if the principal problems (a)-(c) (see introduction) on metrizability of the extremal pseudocompact abelian groups have not been solved, the class of extremal pseudocompact abelian groups has been restricted to groups with specific properties. for the benefit of the reader we list below the most relevant ones: (1) |g| = r0(g) = c for every almost connected c-extremal pseudocompact abelian group g with m(g) = |g| (this includes all s-extremal groups); in case g is also hereditarily pseudocompact, then mg must be ωbounded for some m > 0; (2) if g is weakly extremal and non-metrizable, then c < w(g/n) ≤ 2c for every closed pseudocompact subgroup n 6∈ λ(g) of g; (3) if g is d-extremal, then g is almost connected (so non-totally disconnected if g is not metrizable); moreover, if g is connected, then g is divisible; (4) if g is doubly extremal and non-metrizable, there exists a prime p such that g[p] is dense but not gδ-dense in g̃[p]. the following conjecture, supported by positive evidence in two cases (see theorem 4.13), plays a central role: it obviously implies (f) (and consequently (a)-(e)). main conjecture. every c-extremal pseudocompact abelian group is singular. since weakly extremal groups are c-extremal, the main conjecture implies that every weakly extremal pseudocompact abelian group g is singular. note that this fact is equivalent, by corollary 5.17, to question (f), as weakly extremal singular groups are metrizable according to theorem 4.6. the main conjecture immediately gives “yes” to the following question (suggested by corollary 5.22) since connected groups are almost connected. question 8.1. let g be a connected c-extremal pseudocompact abelian group. must g be divisible (or, equivalently, d-extremal)? note that “yes” to this question implies that the following holds true: for an s-extremal pseudocompact abelian group g, c(g) is s-extremal. the same holds true for r-extremality, for double extremality and for d-extremality. indeed, the subgroup c(g) is c-extremal by theorem 4.11, so would be divisible. 36 d. dikranjan, a. giordano bruno and c. milan now lemma 3.1 (b) implies that c(g) is s-extremal. the same argument holds for d-extremality. along with the main conjecture one can consider also its restricted form: restricted main conjecture. every c-extremal pseudocompact abelian group of size c is singular. by theorem 7.1 the restricted main conjecture implies that every weakly extremal pseudocompact group g with m(g) = |g| is metrizable. so the restricted main conjecture implies a positive answer to questions (a), (c) and (d). note that neither the restricted main conjecture nor (f) follow from a positive answer to all questions (a)–(e), since a singular pseudocompact abelian group of size c need not be weakly extremal (nor metrizable). the following weaker form of question (e) (as s-extremal groups satisfy m(g) = |g| and has size c) is open too: question 8.2. does every r-extremal pseudocompact abelian group g satisfy m(g) = |g|? observe that a positive answer to question 8.2 implies a positive answer to the next question by theorem 7.1: question 8.3. is every infinite r-extremal pseudocompact abelian group of size c? on the other hand, a positive answer to question 8.3, along with the restricted main conjecture, implies a positive answer to (a)-(e). one can expect that connected weakly extremal pseudocompact abelian groups are r-extremal. a positive answer to this conjecture implies positive answer to (d) in the case of connected groups. in terms of theorem 6.10, for the latter implication it suffices to check that g[p] = g̃[p] for every prime p. question 8.4. does weak extremality coincide with the disjunction of s-extremality and r-extremality? if “yes”, then a positive answer to question 8.3, along with the restricted main conjecture, yields a positive answer to (f). note added december 2005. recently comfort and van mill [8] obtained the following impressive result: theorem 8.5. a pseudocompact abelian group is s-extremal or r-extremal if and only if it is metrizable. clearly, this theorem solves the principal open problems related to extremality: (a)–(e) and 8.2, 8.3, 8.4. nevertheless, the restricted main conjecture as well as questions 8.1 and (f) are left open by theorem 8.5. let us see now that an appropriate modification of the argument from [8] can prove our main conjecture, hence all remainig questions formulated in our paper. weakly metrizable pseudocompact groups 37 let g be a c-extremal pseudocompact abelian group. theorem 3.6 yields r0(g) ≤ c. by theorem 2.14 either g is bounded torsion or r0(g) = c. in the former case g is singular. let r0(g) = c and assume that g is not singular. let d = ⊕ s q, with |s| = c, be the divisible hull of the torsion-free quotient g/t(g) and let π : g → d be the composition of the canonical projection g → g/t(g) and the inclusion g/t(g) →֒ d. for a subset a of s let g(a) = π−1( ⊕ a q) and a = {a ⊆ s : g(a) contains a subgroup n ∈ λ(g)}. then a has the countable intersection property and |a| = c for all a ∈ a as r0(n) = c for every n ∈ λ(g) by lemma 4.3. by [8, lemma 3.2] there exists a partition {pn}n∈n of s such that |a ∩ pn| = c for every a ∈ a and for every n ∈ n. define vn = g(p0 ∪ ·· ·∪pn) for every n ∈ n and note that g = ⋃∞ n=0 vn. by lemma 2.15(a) there exist m ∈ n and n ∈ λ(g) such that h = vm ∩ n is gδ-dense in n. by theorem 4.11, to get a contradiction it suffices to show that r0(n/h) = c. let f be a torsion-free subgroup of n such that f ∩h = {0} and maximal with this property. suppose for a contradiction that |f | = r0(n/h) < c. so π(f) ⊆ ⊕ s1 q for some s1 ⊆ s with |s1| < c and w = p0 ∪ ·· · ∪ pm ∪ s1 has |w ∩ pm+1| < c. consequently w 6∈ a and so λ(g) ∋ n 6⊆ g(w). take x ∈ n \g(w). since g/g(w) is torsion-free, 〈x〉∩ g(w) = {0}. but h + f ⊆ g(w) and so 〈x〉∩ (h + f) = {0}, that is (f + 〈x〉) ∩h = {0}, this contradicts the maximality of f . acknowledgements. we profited from intensive exchange of preprints, messages and oral communications on extremal pseudocompact groups with wis comfort, jorge galindo and jan van mill. we are grateful for this generous help. it is a pleasure to thank the referee for her/his useful suggestions that led to an essential improvement of the exposition. references [1] w. w. comfort, tampering with pseudocompact groups, plenary talk at the 2003 summer conference on general topology and its applications (howard university, washington, dc), topology proc. 28 (2) (2004) 401–424. 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[33] h. j. wilcox, pseudocompact groups, pacific j. math. 19 (1966), 365–379. received september 2004 accepted january 2006 dikran dikranjan (dikranja@dimi.uniud.it) dipartimento di matematica e informatica, università di udine, via delle scienze, 206 33100 udine, italy anna giordano bruno (giordano@dimi.uniud.it) dipartimento di matematica e informatica, università di udine, via delle scienze, 206 33100 udine, italy chiara milan (milan@dimi.uniud.it) dipartimento di matematica e informatica, università di udine, via delle scienze, 206 33100 udine, italy @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 1–13 a note on separation and compactness in categories of convergence spaces mehmet baran and muammer kula abstract. in previous papers, various notions of compact, t3, t4, and tychonoff objects in a topological category were introduced and compared. the main objective of this paper is to characterize each of these classes of objects in the categories of filter and local filter convergence spaces as well as to examine how these various generalizations are related. 2000 ams classification: 54b30, 54d10, 54a05, 54a20, 18b99, 18d15 keywords: topological category, filter convergence spaces, tychonoff objects, compact objects, normal objects. 1. introduction the following facts are well known: (1) a topological space x is compact if and only if the projection π2 : x× y → y is closed for each topological space y , (2) a topological space x is hausdorff if and only if the diagonal, ∆, is closed in x ×x, (3) for a topological space x the following are equivalent: (i) x is tychonoff (completely regular t1); (ii) x is homeomorphic to a subspace of a compact hausdorff space; (iii) x is homeomorphic to a subspace of some t4 space. the facts (1) and (2) are used by several authors (see, [7, 14, 22] and [25]) to motivate a closer look at analogous situations in a more general categorical setting. categorical notions of compactness and hausdorffness with respect to a factorization structure were defined in the case of a general category by manes [25] and herrlich, salicrup and strecker [22]. a categorical study of these notions with respect to an appropriate notion of “closedness” based on closure operators (in the sense of [17]) was done in [18] (for the categories of various types of filter convergence spaces) and [14] (for abstract categories). baran in [2] and [4] introduced the notion of “closedness” and “strong closedness” 2 m. baran and m. kula in set-based topological categories and used these notions in [7] to generalize each of the notions of compactness and hausdorffness to arbitrary set-based topological categories. by using (i) and (ii) of (3), in [7] and [14], there are various ways of generalizing the usual tychonoff separation axiom to arbitrary set based topological categories. we further recall from [2] and [8] that for a t1 topological space x, the following are equivalent: (a) x is t3; (b) for every non-void subset f of x, the quotient space x/f (defined in 2.1 below) is t2 if it is t1; (c) for every non-void closed subset f of x, the quotient space x/f is a pret2 space, where a topological space is called pret2 [2](or r1 in [13]) if for any two distinct points, if there is a neighbourhood of one missing the other, then the two points have disjoint neighbourhoods. the equivalence of (b) and (c) follows from the facts that for t1 topological spaces, t2 is equivalent to pret2, and f is closed iff x/f is t1. we note also: (d) a topological space x is t4 iff x is t1 and for every non-void subset f of x, the space x/f is t3 if it is t1. in view of (b) (d), in [2] and [8], there are various ways of generalizing each of the usual t3 and t4 separation axioms to arbitrary set based topological categories. the aim of this paper is to introduce, by using (3), various generalizations of tychonoff objects for an arbitrary set based topological category and compare them with the ones that were given in [7, 9], and [14]. furthermore, each of the classes of t3 and t4-objects, compact and strongly compact objects, and tychonoff objects in the categories of filter and local filter convergence spaces are characterized and relationships among various forms of these tychonoff objects are investigated in these categories. 2. preliminaries let e be a category and set be the category of sets. the functor u : e → set is said to be topological, and e is said to be a topological category over set, if u is concrete (i.e., faithful and amnestic, (i.e., if u(f) = id and f is an isomorphism, then f = id )), has small (i.e., set) fibers, and for which every u-source has an initial lift or, equivalently, for which each u-sink has a final lift [19, 21, 26] or [29]. note that a topological functor u : e → set is said to be normalized if there is only one structure on the empty set and on a point [2] or [26]. let e be a topological category and x ∈ e. then f is called a subspace of x if the inclusion map i : f → x is an initial lift (i.e, an embedding) and we denote this by f ⊂ x. a note on separation and compactness 3 the categorical terminology is that of [20]. let b be a set and p ∈ b. let b ∨ p b be the wedge at p ([2] p. 334), i.e., two disjoint copies of b identified at p, or in other words, the pushout of p : 1 → b along itself (where 1 is a terminal object in set). more precisely, if i1 and i2 : b → b ∨ p b denote the inclusions of b as the first and second factor, respectively, then i1p = i2p is a pushout diagram. a point x in b ∨ p b will be denoted by x1 (x2) if x is in the first (resp. the second) component of b ∨ p b. note that p1 = p2. the skewed p-axis map sp : b ∨ p b → b 2 is given by sp(x1) = (x,x) and sp(x2) = (p,x). the fold map at p, ∇p : b ∨ p b → b is given by ∇p(xi) = x for i = 1, 2 ([2] p. 334 or [4] p. 386). note that the maps sp and ∇p are the unique maps arising from the above pushout diagram for which spi1 = (id,id) : b → b2, spi2 = (p,id) : b → b2, and ∇pij = id,j = 1, 2, respectively, where, id : b → b is the identity map and p : b → b is the constant map at p. the infinite wedge product ∨∞ p b is formed by taking countably many disjoint copies of b and identifying them at the point p. let b∞ = b ×b × . . . be the countable cartesian product of b. define a∞p : ∨∞ p b → b ∞ by a∞p (xi) = (p,p, . . . ,x,p,p, . . .), where xi is in the i-th component of the infinite wedge and x is in the i-th place in (p,p, . . . ,x,p,p, . . .) and 5∞p : ∨∞ p b → b by 5∞p (xi) = x for all i, [2] p. 335 or [4] p. 386. note, also, that the map a∞p is the unique map arising from the multiple pushout of p : 1 → b for which a∞p ij = (p,p,p, . . . ,p,id,p, . . .) : b → b∞, where the identity map, id, is in the j-th place. definition 2.1. (cf. [2] p. 335 or [4] p. 386). let u : e → set be topological and x an object in e with ux = b. let f be a non-empty subset of b. we denote by x/f the final lift of the epi u-sink q : u(x) = b → b/f = (b\f)∪{∗}, where q is the epi map that is the identity on b\f and identifies f with a point ∗ ([2] p. 336). let p be a point in b. (1) x is t1 at p iff the initial lift of the u-source {sp : b ∨ p b → u(x 2) = b2 and ∇p : b ∨ p b → ud(b) = b} is discrete, where d is the discrete functor which is a left adjoint to u. (2) p is closed iff the initial lift of the u-source {a∞p : ∨∞ p b → b ∞ = u(x∞) and 5∞p : ∨∞ p b → ud(b) = b} is discrete. (3) f ⊂ x is strongly closed iff x/f is t1 at ∗ or f = ∅. (4) f ⊂ x is closed iff ∗, the image of f, is closed in x/f or f = ∅. (5) if b = f = ∅, then we define f to be both closed and strongly closed. remark 2.2. (1). in top, the category of topological spaces, the notion of closedness coincides with the usual closedness [2], and f is strongly closed iff f is closed and for each x 6∈ f there exists a neighbourhood of f missing x [2]. if a topological space is t1, then the notions of closedness and strong closedness coincide [2]. 4 m. baran and m. kula (2). in general, for an arbitrary topological category, the notions of closedness and strong closedness are independent of each other [4]. even if x ∈ e is t1, where e is a topological category, then these notions are still independent of each other ([8] p. 64). let a be a set and l a function on a that assigns to each point x of a a set of filters (proper or not, where a filter δ is proper iff δ does not contain the empty set, ∅, i.e., δ 6= [∅]), called the “filters converging to x”. l is called a convergence structure on a (and (a,l) a filter convergence space) iff it satisfies the following two conditions: 1. [x] = [{x}] ∈ l(x) for each x ∈ a (where [f] = {b ⊂ a | f ⊂ b}). 2. β ⊃ α ∈ l(x) implies β ∈ l(x) for any filter β on a. a map f : (a,l) → (b,s) between filter convergence spaces is called continuous iff α ∈ l(x) implies f(α) ∈ s(f(x)) (where f(α) denotes the filter generated by {f(d) | d ∈ α}. the category of filter convergence spaces and continuous maps is denoted by fco (see [15] p.45 or [30] p.354). a filter convergence space (a,l) is said to be a local filter convergence space (in [29], it is called a convergence space) if α ∩ [x] ∈ l(x) whenever α ∈ l(x) ([28] p.1374 or [29] p.142). these spaces are the objects of the full subcategory lfco (in [29] conv) of fco. note that both of these categories are (normalized) topological categories [28], or [29]. more on these categories can be found in [1, 16, 24, 28, 29], and [30]. for filters α and β we denote by α∪β the smallest filter containing both α and β. remark 2.3. an epimorphism f : (a,s) → (b,l) in fco (resp., lfco) is final iff for each b ∈ b, α ∈ l(b) implies that f(β) ⊂ α for some point a ∈ a and filter β ∈ s(a) with f(a) = b ([28] p.1374 or [29] p.143). remark 2.4. a source {fi : (b,l) → (bi,li), i ∈ i} in fco (resp., lfco) is initial iff α ∈ l(a), for a ∈ b, precisely when fi(α) ∈ li(fi(a)) for all i ∈ i ([15] p.46, [28] p.1374 or [29] p.20). we give the following useful lemmas which will be needed later. lemma 2.5. (cf. [3], lemma 3.16). let ∅ 6= f ⊂ b and let q : b → b/f be the epi map that is the identity on b\f and identifies f to the point ∗. (1) for a ∈ b with a 6∈ f, q(α) ⊂ [a] iff α ⊂ [a], (2) q(α) ⊂ [∗] iff α∪ [f] is proper. lemma 2.6. (cf. [10], lemma 3.2). let f : a → b be a map. (1) if α and β are proper filters on a, then f(α) ∪f(β) ⊂ f(α∪β). (2) if δ is a proper filter on b, then δ ⊂ ff−1(δ), where f−1(δ) is the proper filter generated by {f−1(d) | d ∈ δ}. lemma 2.7. (cf. [8], lemma 1.4) let α and β be proper filters on b. then q(α)∪q(β) is proper iff either α∪β is proper or α∪ [f] and β∪ [f] are proper. a note on separation and compactness 5 3. t2-objects recall, in [2] and [6], that there are various ways of generalizing the usual t2 separation axiom to topological categories. moreover, the relationships among various forms of t2-objects are established in [6]. let b be a set and b2 ∨ ∆ b 2 the wedge product of b2, i.e. two disjoint copies of b2 identified along the diagonal, ∆. a point (x,y) in b2 ∨ ∆ b 2 will be denoted by (x,y)1 (resp. (x,y)2)) if (x,y) is in the first (resp., second) component of b2 ∨ ∆ b 2 [10]. recall that the principal axis map a : b2 ∨ ∆ b 2 → b3 is given by a(x,y)1 = (x,y,x) and a(x,y)2 = (x,x,y). the skewed axis map s : b2 ∨ ∆ b 2 → b3 is given by s(x,y)1 = (x,y,y), s(x,y)2 = (x,x,y), and the fold map, 5 : b2 ∨ ∆ b 2 → b2 is given by 5(x,y)i = (x,y) for i = 1,2 [2]. definition 3.1. let u : e → set be topological and x an object in e with ux = b. 1. x is t ′0 iff the initial lift of the u-source {id : b2 ∨ ∆ b2 → u(b2 ∨ ∆ b2)′ = b2 ∨ ∆ b2 and 5 : b2 ∨ ∆ b2 → ud(b2) = b2} is discrete, where (b2 ∨ ∆ b 2)′ is the final lift of the u-sink {i1, i2 : u(x2) = b2 → b2 ∨ ∆ b 2}. here, i1 and i2 are the canonical injections. 2. x is t1 iff the initial lift of the u-source {s : b2 ∨ ∆ b 2 → u(x3) = b3 and 5 : b2 ∨ ∆ b 2 → ud(b2) = b2} is discrete. 3. x is pret ′2 iff the initial lift of the u-source {s : b2 ∨ ∆ b 2 → u(x3) = b3} and the final lift of the u-sink {i1, i2 : u(x2) = b2 → b2 ∨ ∆ b 2} coincide. 4. x is ∆t2 iff the diagonal, ∆, is closed in x2. 5. x is st2 iff ∆ is strongly closed in x2. 6. x is t ′2 iff x is t ′ 0 and pret ′ 2. remark 3.2. (1). note that for the category top of topological spaces, t ′0, t1, pret ′2, and all of the t2’s reduce to the usual t0, t1, pret2 and t2 separation axioms, respectively [2]. (2) if u : e → b, where b is a topos [23], then parts (1) (3), (5), and (6) of definition 3.1 still make sense since each of these notions requires only finite products and finite colimits in their definitions. furthermore, if b has infinite products and infinite wedge products, then definition 3.1 (4) also makes sense. lemma 3.3. let (b,l) be in fco (resp., lfco) and ∅ 6= f ⊂ b. (1) (b,l) is t1 iff for each distinct pair of points x and y in b, [x] 6∈ l(y). (2) all objects (b,l) in fco (resp., lfco) are t ′0. (3) ∅ 6= f ⊂ b is closed iff for any a 6∈ f , if there exists α ∈ l(a) such that α∪ [f] is proper, then [a] 6∈ l(c) for all c ∈ f . 6 m. baran and m. kula (4) ∅ 6= f ⊂ b is strongly closed iff for any a ∈ b with a 6∈ f , [a] 6∈ l(c) for all c ∈ f and α∪ [f] is improper for all α ∈ l(a). (5) (b,l) is ∆t2 iff for all x 6= y in b, [x] 6∈ k(y) iff (b,l) is t1. (6) (b,l) is st2 iff for all x 6= y in b, l(x) ∩l(y) = {[∅]}. (7) (b,l) is pret ′2 (t ′ 2) iff (b,l) is discrete, i.e, for all x in b, l(x) = {[∅], [x]}. proof. (1), (2), and (7) are proved in [5]. the proof of (3)-(6) are given in [4]. � corollary 3.4. let (b,l) be in fco (resp. lfco) and ∅ 6= f ⊂ b. (1) if (b,l) is t1, then b/f is t1 iff f is strongly closed. (2) if (b,l) is t1, then f is always closed. (3) if (b,l) is t1, then f is strongly closed iff ∀x ∈ b if x 6∈ f and α ∈ l(x), then α∪ [f] is improper. (4) if (b,l) is t ′2, then all the subsets of b are both closed and strongly closed. 4. t3-objects we now recall, ([2] and [8]), various generalizations of the usual t3 separation axiom to arbitrary set based topological categories and characterize each of them for the topological categories fco and lfco. definition 4.1. let u : e → set be topological and x an object in e with ux = b. let f be a non-empty subset of b. 1. x is st ′3 iff x is t1 and x/f is pret ′ 2 for all strongly closed f 6= ∅ in u(x). 2. x is t ′3 iff x is t1 and x/f is pret ′ 2 for all closed f 6= ∅ in u(x). 3. x is ∆t3 iff x is t1 and x/f is ∆t2 if it is t1, for all f 6= ∅ in u(x). 4. x is st3 iff x is t1 and x/f is st2 if it is t1, for all f 6= ∅ in u(x). remark 4.2. (1). for the category top of topological spaces, all of the t3’s reduce to the usual t3 separation axiom ([2] and [8]). (2). if u : e → b, where b is a topos [23], then parts (1), (3), and (4) of definition 4.1 still make sense since each of these notions requires only finite products and finite colimits in their definitions. furthermore, if b has infinite products and infinite wedge products, then definition 4.1 (2), also, makes sense. theorem 4.3. let (b,l) be in fco (resp. lfco). (1) (b,l) is ∆t3 iff (b,l) is t1. (2) (b,l) is st3 iff (b,l) is st2. (3) (b,l) is st ′3 iff for all x 6= y in f , [x] 6∈ l(y) and for any x ∈ b and for any proper filter α ∈ l(x), either α = [x] or f ∈ α for all non-empty strongly closed subsets f of b. (4) (b,l) is t ′3 iff for all x 6= y in f , [x] 6∈ l(y) for any x ∈ b and for any proper filter α ∈ l(x) either α = [x] or f ∈ α for any non-empty subset f of b. a note on separation and compactness 7 proof. (1). this follows from definition 4.1 and corollary 3.4. (2). suppose (b,l) is st3. take f ={a}, a one point set. it now follows from lemma 3.3 and corollary 3.4 that (b,l) is st2. conversely, suppose (b,l) is st2. by corollary 3.4, (b,l) is t1. suppose b/f is t1, then by corollary 3.4, f is a strongly closed subset of b. we show that b/f is st2. let x 6= y in b and α ∈ l′(x) ∩ l′(y), where l′ is the quotient structure on b/f induced by the map q : b → b/f that identifies f with a point ∗ and is the identity on b\f. if α is improper, then, by corollary 3.4, we are done. suppose α is proper. since q is the quotient map this implies (see remark 2.3) that ∃β ∈ l(a) and ∃δ ∈ l(b) such that q(β) ⊂ α, q(δ) ⊂ α, and qa = x, qb = y. it follows that q(β) ∪ q(δ) is proper and, by lemma 2.7, either β ∪ δ is proper or β ∪ [f] and δ ∪ [f] are proper. the first case cannot occur since (b,l) is st2. since x 6= y, we may assume a 6∈ f. since f is strongly closed, by corollary 3.4, β ∪ [f] is improper. this shows that the second case also cannot hold. therefore, α must be improper and by corollary 3.4, we have the result. (3). suppose (b,l) is st ′3. since (b,l) is t1, by corollary 3.4, for all x 6= y in b, [x] 6∈ l(y). if α ∈ l(x), where x ∈ b, then q(α) ∈ l′(qx). since b/f is pret ′2, (f is a non-empty strongly closed subset of b) by corollary 3.4, q(α) = [qx] (since α is proper). if x 6∈ f, then, by lemma 2.6, [x] = q−1(x) = q−1q(α) ⊂ α and consequently α = [x]. if x ∈ f, it follows easily that q(α) = [∗] iff f ∈ α. conversely, suppose the conditions hold. by corollary 3.4, clearly, (b,l) is t1. we now show that b/f is pret ′2 for all nonempty strongly closed subsets f of x. if x ∈ b/f and α ∈ l′(x), it follows that there exists β ∈ l(a) such that q(β) ⊂ α and qa = x. if β is improper, then so is α. if β is proper, then by assumption β = [a] or f ∈ β. if the first case holds, then [qa] = q(β) ⊂ α and thus α = [qa]. if the second case holds, then {∗} = q(f) ∈ q(β) ⊂ α and consequently α = [∗]. hence, by lemma 3.3, b/f is pret ′2 and by definition 3.1, (b,l) is st ′ 3. the proof of (4) is similar to the proof of (3), on using definition 3.1, lemma 3.3 and corollary 3.4. � remark 4.4. for the category fco (resp., lfco), we have : (1) by theorem 4.3, st ′3 ⇒ t ′3 ⇒ st3 ⇒ ∆t3, but the converse of each implication is not true in general. (2) by lemma 3.3 and theorem 4.3, st ′3 ⇒ t ′2 ⇒ st3 ≡ st2 ⇒ ∆t3 = ∆t2, but the converse of each implication is not true in general. (3) by corollary 3.4 and theorem 4.3, if (b,l) is st ′3 or t ′ 3, then all subsets of x are both closed and strongly closed. (4) by corollary 3.4 and theorem 4.3, if (b,l) is ∆t3, then f is always closed and f is strongly closed iff ∀x ∈ b if x 6∈ f and α ∈ k(x), then α∪ [f] is improper. 8 m. baran and m. kula 5. t4-objects we now recall various generalizations of the usual t4 separation axiom to arbitrary set based topological categories that are defined in [2] and [8], and characterize each of them for the topological categories fco and lfco. definition 5.1. let u : e → set be topological and x an object in e with ux = b. let f be a non-empty subset of b. 1. x is st ′4 iff x is t1 and x/f is st ′ 3 if it is t1, where f is any nonempty subset of u(x). 2. x is t ′4 iff x is t1 and x/f is t ′ 3 if it is t1, where f is any non-empty subset of u(x). 3. x is ∆t4 iff x is t1 and x/f is ∆t3 if it is t1, for all f 6= ∅ in u(x). 4. x is st4 iff x is t1 and x/f is st3 if it is t1, for all f 6= ∅ in u(x). remark 5.2. (1). for the category top of topological spaces, all of the t4’s reduce to the usual t4 separation axiom by the introduction, [2], and [8]. (2). if u : e → b, where b is a topos [23], then definition 5.1 still makes sense since each of these notions requires only finite products and finite colimits in their definitions. theorem 5.3. let (b,l) be in fco (resp., lfco). (1) (b,l) is ∆t4 iff (b,l) is t1. (2) (b,l) is st4 iff (b,l) is st2. (3) (b,l) is st ′4 (t ′ 4) iff the following two conditions hold: (i) for all x 6= y in b, we have [x] 6∈ l(y). (ii) for any x ∈ b and for any proper filter α ∈ l(x), and for any non-empty disjoint strongly closed (resp., closed ) subsets f and f ′ of b, we have either condition (i ) or (ii ) below: (i) α = [x]; (ii) f ∈ α or f ′ ∈ α. proof. (1). this follows from definition 5.1 and theorem 4.3. (2). the proof has the same form as that of theorem 4.3 (2). one has only to replace the term st3 by st4 and the numbers 3.3, 3.4, 3.4, 3.4, 3.4, 2.3, 3.4, 3.4 respectively by 3.1, 3.3, 4.3, 3.4, 4.3, 2.3, 3.3, 4.3. (3). suppose (b,l) is st ′4. since (b,l) is t1, by corollary 3.4, for all x 6= y in b, [x] 6∈ l(y). if α ∈ l(x), where x ∈ b, then q(α) ∈ l′(qx), where l′ is the quotient structure on b/f induced by the map q of definition 2.1. since b/f is st ′3, (f is a non-empty strongly closed subset of b, i.e., b/f is t1) by corollary 3.4, we have either q(α) = [qx] (since α is proper) or f ′ ∈ q(α), for any non-empty strongly closed subset f ′ of b/f not containing the point ∗ (note that q−1(f ′) = f ′ and f ′ is disjoint from f). suppose that q(α) = [qx]. if x 6∈ f, then, by lemma 2.6, [x] = q−1(x) = q−1q(α) ⊂ α, and consequently α = [x]. if x ∈ f, it follows easily that q(α) = [∗] iff f ∈ α. if f ′ ∈ q(α) for any non-empty strongly closed subset f ′ of b/f not containing the point ∗, then it follows easily that f ′ ∈ α. a note on separation and compactness 9 conversely, suppose the conditions hold. by lemma 3.3, clearly, (b,l) is t1. we now show that b/f is st ′3 for all non-empty strongly closed subsets f of b. if x ∈ b/f and α ∈ l′(x), it follows that there exists β ∈ l(a) such that q(β) ⊂ α and qa = x. if β is improper, then so is α. if β is proper, then by assumption either β = [a] or f ∈ β, or f ′ ∈ β for any strongly closed subset f ′ of b disjoint from f. if the first case holds, then [qa] = q(β) ⊂ α and thus α = [qa]. if the second case holds, then {∗} = q(f) ∈ q(β) ⊂ α, and consequently α = [∗] or f ′ = q(f ′) ∈ q(β) ⊂ α and consequently f ′ ∈ α. hence, by theorem 4.3, b/f is st ′3 and by definition 5.1, (b,l) is st ′ 4. the proof for t ′4 is similar to the proof for st ′ 4. � remark 5.4. for the category fco (resp., lfco), we have : (1). by theorem 4.3, st ′4 ⇒ t ′4 ⇒ st4 ⇒ ∆t4, but the converse of each implication is not true in general. (2). by lemma 3.3, theorem 4.3, and theorem 5.3, st ′4(t ′ 4) ⇒ st ′3(t ′3) ⇒ t ′2 ⇒ st4 = st3 = st2 ⇒ ∆t4 = ∆t3 = ∆t2, but the converse of each implication is not true in general. (3). by remark 4.4 and theorem 5.3, if (b,l) is st ′4 or t ′ 4, then all subsets of x are both closed and strongly closed. (4). by remark 4.4 and theorem 5.3, if (b,l) is ∆t4, then all subsets f of x are closed and f is strongly closed iff ∀x ∈ b, if x 6∈ f and α ∈ l(x), then α∪ [f] is improper. corollary 5.5. let (b,l) be in fco (resp., lfco). if (b,l) is ∆t4,st4, st ′4 or t ′ 4, then any subspace of (b,l) is ∆t4,st4,st ′ 4 or t ′ 4, respectively. proof. this follows from remark 2.4, theorem 5.3, and remark 5.4 (3). � 6. compact objects recall that each of the notions of (strongly) closed morphism and (strongly) compact object in a topological category e over set are introduced in [7]. definition 6.1. let u : e → set be topological, x and y objects in e, and f : x → y a morphism in e. 1. f is said to be closed iff the image of each closed subobject of x is a closed subobject of y . 2. f is said to be strongly closed iff the image of each strongly closed subobject of x is a strongly closed subobject of y . 3. x is compact if and only if the projection π2 : x ×y → y is closed for each object y in e. 4. x is strongly compact if and only if the projection π2 : x ×y → y is strongly closed for each object y in e. remark 6.2. (1). for the category top of topological spaces, the notions of closed morphism and compactness reduce to the usual ones ([12] p. 97 and 10 m. baran and m. kula 103). furthermore, by remark 2.2 and definition 6.1, one can show that the notions of compactness and strong compactness are equivalent. (2). if u : e → b is topological, where b is a topos with infinite products and infinite wedge products, then definition 6.1 still makes sense. (3). since the notions of closedness and strong closedness are, in general, different (see [4] p. 393), it follows that the notions of compactness and strong compactness are different, in general. (4). for an arbitrary topological category, it is not known in general whether the closure used in 2.1 is a closure operator in the sense of dikranjan and giuli [17] or not. however, it is shown, in [10], that the notions of closedness and strong closedness that are defined in 2.1 form appropriate closure operators in the sense of dikranjan and giuli [17] in case the category is one of the categories fco and lfco. the same two facts are proved in [11] for the categories lim (limit spaces) and prtop (pretopological spaces). theorem 6.3. let e be one of the categories fco (resp. lfco). (1) every (b,l) ∈ e is compact. (2) (b,l) ∈ e is strongly compact iff every ultrafilter in b converges. proof. (1). by definition 5.1 (3) we need to show that, for all (a,s) ∈ e, π2 : (b,l)×(a,s) → (a,s) is closed. suppose m ⊂ b×a is closed. suppose that for any a ∈ a there exists c ∈ π2m such that [a] ∈ s(c). it follows that ∃x ∈ b such that (x,c) ∈ m. note that [(x,a)] ∈ l2((x,c)), where l2 is the product structure on b × a, (since [x] ∈ l(x) and [a] ∈ s(c)). since m is closed, (x,a) ∈ m and consequently a = π2(x,a) ∈ π2(m). hence, by lemma 3.3, π2(m) is closed and consequently, (b,l) is compact. (2). suppose every ultrafilter in b converges. we show that (b,l) is strongly compact, i.e., by definition 6.1 (4), we need to show that, for all (a,s) ∈ e, π2 : (b,l) × (a,s) → (a,s) is strongly closed. suppose that m ⊂ b × a is strongly closed. to show that π2m is strongly closed, we assume the contrary and apply lemma 3.3 (4). thus for some point a ∈ a with a 6∈ π2m, we have either [a] ∈ s(c) for some c ∈ π2m or [π2m] ∪α is proper for some α ∈ s(a). if the first case holds, that is for some a ∈ a we have a 6∈ π2m and [a] ∈ s(c) for some c ∈ π2m, then it follows that ∃x ∈ b such that (x,a) 6∈ m. note that [(x,a)] ∈ l2((x,c)), a contradiction, since m is strongly closed. in the second case, suppose that for some a ∈ a with a 6∈ π2m and α ∈ s(a), [π2m] ∪α is proper. let σ = [m] ∪π−12 α. note that σ is proper and π1(σ) is a filter on b. it follows that there exists an ultrafilter β on b with β ⊃ π1(σ). in view of the assumption on (b,l), there exists x ∈ b such that β ∈ l(x). let γ = π−11 β ∪ π −1 2 α . note that γ ∈ l 2(x,a) since π1(γ) = β ∈ l(x) and π2(γ) = α ∈ s(a). since a 6∈ π2m, we have (x,a) 6∈ m. it follows from β ⊃ π1(σ) that [m] ∪γ is proper, a contradiction since m is strongly closed, by lemma 3.3 (4). hence, by lemma 3.3 (4), π2(m) must be strongly closed and consequently, by definition 6.1, (b,l) is strongly compact. a note on separation and compactness 11 conversely, assume that (b,l) is strongly compact and α is a non convergent ultrafilter of b, i.e., for all x ∈ b, α 6∈ l(x). let a be the set obtained by adjoining a new element, say ∞, to b, i.e., a = b∪{∞}. let (a,s), where s is defined by s(x) = {[∅], [x]} for each x 6= ∞ of a, and β ∈ s(∞) iff α = β∪[b], i.e., the trace of β on b coincides with α. note that (a,s) ∈ fco (resp., lfco). let ∆ = {(x,y) ∈ b ×a | x = y}⊂ b ×a. let σ = π−11 [x] ∪π −1 2 α. since π1σ = [x] ∈ l(x) and π2σ = α ∈ s(∞), σ ∈ l2((x,∞)), where l2 is the product structure on b×a. note that σ∪[∆] is improper (let v = a\{x}∈ α and v ∩ ∆ = ∅). since [∞] 6∈ s(c) for all c ∈ b, it follows that [(x,∞)] 6∈ l2(c,c). hence, by lemma 3.3, ∆ is strongly closed in b × a. note that α ∪ [π2(∆)] is proper for α ∈ s(∞), a contradiction since (b,l) is strongly compact. � remark 6.4. results akin to theorem 6.3 have been proved for the categories lim (limit spaces) and prtop (pretopological spaces) in ([11], lemma 4.3). 7. tychonoff objects we now define various forms of tychonoff objects for an arbitrary set-based topological category. furthermore, we characterize each of them for the categories that are mentioned in section 2 and investigate the relationships among them. definition 7.1. let u : e → set be topological and x an object in e. 1. x is ∆t3 12 iff x is a subspace of ∆t4. 2. x is st3 12 iff x is a subspace of st4. 3. x is t ′ 3 12 iff x is a subspace of t ′4. 4. x is st ′ 3 12 iff x is a subspace of st ′4. 5. x is c∆t3 12 iff x is a subspace of a compact ∆t2. 6. x is cst3 12 iff x is a subspace of a compact st2. 7. x is lt3 12 iff x is a subspace of a compact t ′ 2. 8. x is s∆t3 12 iff x is a subspace of a strongly compact ∆t2. 9. x is sst3 12 iff x is a subspace of a strongly compact st2. 10. x is slt3 12 iff x is a subspace of a strongly compact t ′ 2. remark 7.2. (1). for the category top of topological spaces, all ten of the properties defined in definition 7.1 are equivalent and reduce to the usual t3 12 = tychonoff, i.e, completely regular t1, spaces [27], remark 5.2, and remark 6.2. (2). for an arbitrary set-based topological category, properties (3–4) and (5–7) are defined in [8] and [7], respectively. (3). for the categories fco and lfco, it is shown in [10] that the notions of closedness and strong closedness form appropriate closure operators in the sense of [16]. as a consequence, properties (5) and (9) of definition 6.1 reduce to definition 8.1 of [13]. 12 m. baran and m. kula theorem 7.3. let x be in fco (resp., lfco). 1. x is ∆t3 12 (c∆t3 12 ) iff x is t1. 2. x is st3 12 (cst3 12 ) iff x is st2. 3. x is t ′ 3 12 iff x is t ′4. 4. x is st ′ 3 12 iff x is st ′4. 5. x is s∆t3 12 iff x is a subspace of a strongly compact t1. 6. x is sst3 12 iff x is a subspace of a strongly compact st2. 7. x is slt3 12 iff x is a finite discrete space. proof. 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[30] f. schwarz, connections between convergence and nearness, lecture notes in math. no. 719, (springer-verlag, 1978), 345–354. received november 2001 revised december 2002 mehmet baran and muammer kula department of mathematics, erciyes university, kayseri 38039, turkey. e-mail address : baran@erciyes.edu.tr a note on separation and compactness in categories of convergence spaces. by m. baran and m. kula @ appl. gen. topol. 15, no. 1 (2014), 33-42doi:10.4995/agt.2014.2126 c© agt, upv, 2014 on topological groups via a-local functions wadei al-omeri a, mohd. salmi md. noorani a and ahmad. al-omari b a department of mathematics, faculty of science and technology universiti kebangsaan malaysia, 43600 ukm bangi, selangor de, malaysia (wadeimoon1@hotmail.com,msn@ukm.my) b department of mathematics, faculty of science al al-bayat university, p.o.box 130095, mafraq 25113, jordan (omarimutah1@yahoo.com) abstract an ideal on a set x is a nonempty collection of subsets of x which satisfies the following conditions (1)a ∈ i and b ⊂ a implies b ∈ i; (2) a ∈ i and b ∈ i implies a ∪ b ∈ i. given a topological space (x, τ) an ideal i on x and a ⊂ x, ℜa(a) is defined as ∪{u ∈ τ a : u −a ∈ i}, where the family of all a-open sets of x forms a topology [5, 6], denoted by τa. a topology, denoted τa ∗ , finer than τa is generated by the basis β(i, τ) = {v − i : v ∈ τa(x), i ∈ i}, and a topology, denoted 〈ℜa(τ)〉 coarser than τa is generated by the basis ℜa(τ) = {ℜa(u) : u ∈ τ a}. in this paper a bijection f : (x, τ, i) → (x, σ, j ) is called a a∗homeomorphism if f : (x, τa ∗ ) → (y, σa ∗ ) is a homeomorphism, ℜahomeomorphism if f : (x, ℜa(τ)) → (y, ℜa(σ)) is a homeomorphism. properties preserved by a∗-homeomorphism are studied as well as necessary and sufficient conditions for a ℜa-homeomorphism to be a a∗homeomorphism. 2010 msc: 54a05; 54c10. keywords: ℜa-homeomorphism; topological groups; a-local function; ideal spaces; ℜa-operator; a∗-homeomorphism. 1. introduction and preliminaries ideals in topological spaces have been considered since 1930. the subject of ideals in topological spaces has been studied by kuratowski [11] and received 22 march 2013 – accepted 28 january 2014 http://dx.doi.org/10.4995/agt.2014.2126 w. al-omeri, mohd. salmi md. noorani and a. al-omari vaidyanathaswamy [18]. jankovic and hamlett [10] investigated further properties of ideal space. in this paper, we investigate a-local functions and its properties in ideals in topological space [1]. also, the relationships among local functions such as local function [19, 10] and semi-local function [7] are investigated. a subset of a space (x, τ) is said to be regular open (resp. regular closed) [12] if a = int(cl(a)) (resp. a = cl(int(a))). a is called δ-open [20] if for each x ∈ a, there exists a regular open set g such that x ∈ g ⊂ a. the complement of δ-open set is called δ-closed. a point x ∈ x is called a δ-cluster point of a if int(cl(u)) ∩ a 6= φ for each open set v containing x. the set of all δ-cluster points of a is called the δ-closure of a and is denoted by clδ(a) [20]. the δ-interior of a is the union of all regular open sets of x contained in a and its denoted by intδ(a) [20]. a is δ-open if intδ(a) = a. δ-open sets forms a topology τδ. a subset a of a space (x, τ) is said to be a-open (resp. a-closed) [5] if a ⊂ int(cl(intδ(a))) (resp. cl(int(clδ(a))) ⊂ a, or a ⊂ int(cl(intδ(a))) (resp. cl(int(clδ(a))) ⊂ a. the family of a-open sets of x forms a topology, denoted by τa [6]. the intersection of all a-closed sets contained a is called the a-closure of a and is denoted by acl(a). the a-interior of a, denoted by aint(a), is defined by the union of all a-open sets contained in a [5]. an ideal i on a topological space (x, i) is a nonempty collection of subsets of x which satisfies the following conditions: (1) a ∈ i and b ⊂ a implies b ∈ i; (2) a ∈ i and b ∈ i implies a ∪ b ∈ i. applications to various fields were further investigated by jankovic and hamlett [10] dontchev et al. [4]; mukherjee et al. [13]; arenas et al. [3]; navaneethakrishnan et al. [14]; nasef and mahmoud [15], etc. given a topological space (x, i) with an ideal i on x and if ℘(x) is the set of all subsets of x, a set operator (.) ∗ : ℘(x) → ℘(x), called a local function [11, 10] of a with respect to τ and i is defined as follows: for a ⊆ x, a∗(i, τ) = {x ∈ x | u ∩ a /∈ i, for every u ∈ τ(x)} where τ(x) = {u ∈ τ | x ∈ u}. a kuratowski closure operator cl∗(.) for a topology τ∗(τ, i), called the ∗-topology, which is finer than τ is defined by cl∗(a) = a ∪ a∗(τ, i), when there is no chance of confusion. a∗(i) is denoted by a∗ and τ∗ for τ∗(i, τ). x∗ is often a proper subset of x. the hypothesis x = x∗ [7] is equivalent to the hypothesis τ ∩ i = φ. if i is an ideal on x, then (x, τ, i) is called an ideal space. n is the ideal of all nowhere dense subsets in (x, τ). a subset a of an ideal space (x, τ, i) is ⋆-closed [4] (resp. ⋆-dense in itself [7]) if a∗ ⊆ a (resp a ⊆ a∗). a subset a of an ideal space (x, τ, i) is ig−closed [20] if a ∗ ⊆ u whenever a ⊆ u and u is open. for every ideal topological space there exists a topology τ∗(i) finer than τ generated by β(i, τ) = {u − a | u ∈ τ and a ∈ i}, but in general β(i, τ) is not always topology [10]. let (x, i, τ) ba an ideal topological space. we say that the topology τ is compatible with the i, denoted τ ∼ i, if the following holds for c© agt, upv, 2014 appl. gen. topol. 15, no. 1 34 on topological groups via a-local functions every a ⊂ x, if for every x ∈ a there exists a u ∈ τ such that u ∩ a ∈ i, then a ∈ i. given a space (x, τ, i), (y, σ, j ), and a function f : (x, τ, i) → (y, τ, j ), we call f a ∗-homomorphism with respect to τ, i, σ, and j if f : (x, τ∗) → (y, σ∗) is a homomorphism, where a homomorphism is a continuous injective function between two topological spaces, that is invertible with continuous inverse. we first prove some preliminary lemmas which lead to a theorem extending the theorem in [17] and apply the theorem to topological groups. quite recently, in [2], the present authors defined and investigated the notions ℜa : ℘(x) → τ as follows, ℜa(a) = {x ∈ x : there exists ux ∈ τ a containing x such that ux −a ∈ i}, for every a ∈ ℘(x). in [16], newcomb defined a = b[mod i] if (a−b)∪(b −a) ∈ i and observe that = [mod i] is an equivalence relation. in this paper a bijection f : (x, τ, i) → (x, σ, j ) is called a a∗-homeomorphism if f : (x, τa ∗ ) → (y, σa ∗ ) is a homeomorphism, ℜa-homeomorphism if f : (x, ℜa(τ)) → (y, ℜa(σ)) is a homeomorphism. properties preserved by a∗homeomorphism are studied as well as necessary and sufficient conditions for a ℜa-homeomorphism to be a a∗-homeomorphism. 2. a-local function and ℜaoperator let (x, τ, i) an ideal topological space and a a subset of x. then aa ∗ (i, τ) = {x ∈ x : u ∩ a /∈ i, for every u ∈ τa(x)} is called a-local function of a [1] with respect to i and τ, where τa(x) = {u ∈ τa : x ∈ u}. we denote simply aa ∗ for aa ∗ (i, τ). remark 2.1 ([1]). (1) the minimal ideal is considered {∅} in any topological space (x, τ) and the maximal ideal is considered p(x). it can be deduced that aa ∗ ({∅}) = cla(a) 6= cl(a) and a a ∗ (p(x)) = ∅ for every a ⊂ x. (2) if a ∈ i, then aa ∗ = ∅. (3) a * aa ∗ and aa ∗ * a in general. theorem 2.2 ([1]). let (x, τ, i) an ideal in topological space and a, b subsets of x. then for a-local functions the following properties hold: (1) if a ⊂ b, then aa ∗ ⊂ ba ∗ , (2) for another ideal j ⊃ i on x, aa ∗ (j) ⊂ aa ∗ (i), (3) aa ∗ ⊂ acl(a), (4) aa ∗ (i) = acl(aa ∗ ) ⊂ acl(a) (i.e aa ∗ is an a-closed subset of acl(a)), (5) (aa ∗ )a ∗ ⊂ aa ∗ , (6) (a ∪ b)a ∗ = aa ∗ ∪ ba ∗ , (7) aa ∗ − ba ∗ = (a − b)a ∗ − ba ∗ ⊂ (a − b)a ∗ , (8) if u ∈ τa, then u ∩ aa ∗ = u ∩ (u ∩ a)a ∗ ⊂ (u ∩ a)a ∗ , (9) if u ∈ τa, then (a − u)a ∗ = aa ∗ = (a ∪ u)a ∗ , (10) if a ⊆ aa ∗ , then aa ∗ (i) = acl(aa ∗ ) = acl(a). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 35 w. al-omeri, mohd. salmi md. noorani and a. al-omari theorem 2.3 ([1]). let (x, τ, i) an ideal in topological space and a, b subsets of x.then for a-local functions the following properties hold: (1) τa ∩ i = φ; (2) if i ∈ i, then aint(i) = φ; (3) for every g ∈ τa, then g ⊆ ga ∗ ; (4) x = xa ∗ . theorem 2.4 ([1]). let (x, τ, i) be an ideal topological space and a subset of x. then the following are equivalent: (1) i ∼a τ, (2) if a subset a of x has a cover a-open of sets whose intersection with a is in i, then a is in i, in other words aa ∗ = φ, then a ∈ i, (3) for every a ⊂ x, if a ∩ aa ∗ = φ, a ∈ i, (4) for every a ⊂ x, a − aa ∗ ∈ i, (5) for every a ⊂ x, if a contains no nonempty subset b with b ⊂ ba ∗ , then a ∈ i. theorem 2.5 ([1]). let (x, i, τ) be an ideal topological space. then β(i, τ) is a basis for τa ∗ . β(i, τ) = {v − ii : v ∈ τ a(x), ii ∈ i} and β is not, in general, a topology. theorem 2.6 ([2]). let (x, τ, i) be an ideal topological space. then the following properties hold: (1) if a ⊂ x, then ℜa(a) is a-open. (2) if a ⊂ b, then ℜa(a) ⊆ ℜa(b). (3) if a, b ∈ ℘(x), then ℜa(a ∪ b) ⊂ ℜa(a) ∪ ℜa(b). (4) if a, b ∈ ℘(x), then ℜa(a ∩ b) = ℜa(a) ∩ ℜa(b). (5) if u ∈ τa ∗ , then u ⊆ ℜa(u). (6) if a ⊂ x, then ℜa(a) ⊆ ℜa(ℜa(a)). (7) if a ⊂ x, then ℜa(a) = ℜa(ℜa(a)) if and only if (x − a)a ∗ = ((x − a)a ∗ )a ∗ . (8) if a ∈ i, then ℜa(a) = x − x a ∗ . (9) if a ⊂ x, then a ∩ ℜa(a) = int a ∗ (a), where inta ∗ is the interior of τa ∗ . (10) if a ⊂ x, i ∈ i, then ℜa(a − i) = ℜa(a). (11) if a ⊂ x, i ∈ i, then ℜa(a ∪ i) = ℜa(a). (12) if (a − b) ∪ (b − a) ∈ i, then ℜa(a) = ℜa(b). theorem 2.7 ([1]). let (x, τ, i) be an ideal topological space and a subset of x. if τ is a-compatible with i. then the following are equivalent: (1) for every a ⊂ x, if a ∩ aa ∗ = φ implies aa ∗ = φ, (2) for every a ⊂ x, (a − aa ∗ )a ∗ = φ, (3) for every a ⊂ x,(a ∩ aa ∗ )a ∗ = aa ∗ . theorem 2.8 ([2]). let (x, τ, i) be an ideal topological space with τ ∼a i. then ℜa(a) = ∪{ℜa(u) : u ∈ τ a, ℜa(u) − a ∈ i}. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 36 on topological groups via a-local functions proposition 2.9 ([2]). let (x, τ, i) be an ideal topological space with τa ∩i = φ.then the following are equivalent: (1) a ∈ u(x, τ, i), (2) ℜa(a) ∩ aint(a a ∗ ) 6= φ, (3) ℜa(a) ∩ a a ∗ 6= φ, (4) ℜa(a) 6= φ, (5) inta ∗ (a) 6= φ, (6) there exists n ∈ τa − {∅} such that n − a ∈ i and n ∩ a /∈ i. proposition 2.10 ([2]). let (x, τ, i) be an ideal topological space. then τ ∼a i, a ⊆ x. if n is a nonempty a-open subset of aa ∗ ∩ ℜa(a), then n − a ∈ i and n ∩ a /∈ i. theorem 2.11 ([2]). let (x, τ, i) be an ideal topological space. then τ ∼a i if and only if ℜa(a) − a ∈ i for every a ⊆ x. 3. a∗-homeomorphism given an ideal topological space (x, τ, i) a topology τa finer than 〈ℜa(τ)〉 which 〈ℜa(τ)〉 is generated by the basis ℜa(τ) = {ℜa(u) : u ∈ τ a}. definition 3.1 ([5]). a function f : (x, τ) → (x, σ) is called (1) a-continuous if the inverse image of a-open set is a-open. (2) a-open if the image of a-open set is a-open. definition 3.2. let (x, τ, i) and (x, σ, j ) be an ideal topological space. a bijection f : (x, τ, i) → (x, σ, j ) is called (1) a∗-homeomorphism if f : (x, τa ∗ ) → (y, σa ∗ ) is a homeomorphism. (2) ℜa-homeomorphism if f : (x, ℜa(τ)) → (y, ℜa(σ)) is a homeomorphism. theorem 3.3. let (x, τ, i) and (x, σ, j ) be an ideal topological space with f : (x, ℜa(τ)) → (x, σ, j ) an a-open bijective, τ ∼ a i and f(i) ⊆ j . then f(ℜa(a)) ⊆ ℜa(f(a)) for every a ⊆ x. proof. let a ⊆ x and let y ∈ f(ℜa(a)). then f −1(y) ∈ ℜa(a) and there exists u ∈ τa such that f−1(y) ∈ ℜa(u) and ℜa(u) − a ∈ i by theorem 2.8. now f(ℜa(u)) ∈ σ a(y) and f(ℜa(u)) − f(a) = f[ℜa(u) − a] ∈ f(i) ⊆ j . thus y ∈ ℜa(f(a), and the proof is complete. � theorem 3.4. let (x, τ, i) and (x, σ, j ) be an ideal topological space with f : (x, τ) → (x, ℜa(σ)) is a-continuous injection, σ ∼ a j and f−1(j ) ⊆ i. then ℜa(f(a)) ⊆ f(ℜa(a)) for every a ⊆ x. proof. let y ∈ ℜa(f(a)) where a ⊆ x. then by theorem 2.8, there exists u ∈ σa such that y ∈ ℜa(u) and ℜa(u) − f(a) ∈ j . now we have f −1(ℜa(u)) ∈ τa(f−1(y)) with f−1[ℜa(u)−f(a)] ∈ i then f −1[ℜa(u)]−a ∈ i and f −1(y) ∈ ℜa(a) and hence y ∈ f(ℜa(a)), and the proof is complete. � c© agt, upv, 2014 appl. gen. topol. 15, no. 1 37 w. al-omeri, mohd. salmi md. noorani and a. al-omari theorem 3.5. let (x, τ, i) and (x, σ, j ) be a bijective with f(i) = j . then the following properties are equivalent: (1) f is a∗-homeomorphism; (2) f(aa ∗ ) = [f(a)]a ∗ for every a ⊆ x; (3) f(ℜa(a)) = ℜa(f(a)) for every a ⊆ x; proof. (1) ⇒ (2) let a ⊆ x. assume y /∈ f(aa ∗ ). this implies that f−1(y) /∈ aa ∗ , and hence there exists u ∈ τa(f−1(y)) such that u ∩ a ∈ i. consequently f(u) ∈ σa ∗ (y) and f(u) ∩ f(a) ∈ j , then y /∈ [f(a)]a ∗ (j , σa ∗ ) = [f(a)]a ∗ (j , σ). thus [f(a)]a ∗ ⊆ f(aa ∗ ). now assume y /∈ [f(a)]a ∗ . this implies there exists a v ∈ σa ∗ (y) such that v ∩ f(a) ∈ j , then f−1(v ) ∈ τa ∗ (f−1(y)) and f−1(v ) ∩ a ∈ i. thus f−1(y) /∈ aa ∗ (i, τa ∗ ) = aa ∗ (i, τa) and y /∈ f(aa ∗ ). hence f(aa ∗ ) ⊆ [f(a)]a ∗ and f(aa ∗ ) = [f(a)]a ∗ . (2) ⇒ (3) let a ⊆ x. then f(ℜa(a)) = f[x −(x −a) a ∗ ] = y −f(x −a)a ∗ = y − [y − f(a)]a ∗ = ℜa(f(a)). (3) ⇒ (1) let u ∈ τa ∗ . then u ⊆ ℜa(u) by theorem 2.6 and f(u) ⊆ f(ℜa(u)) = ℜa(f(u)). this shows that f(u) ∈ σ a ∗ and hence f : (x, τa ∗ ) → (y, σa ∗ ) is τa ∗ -open. similarly, f−1 : (y, σa ∗ ) → (x, τa ∗ ) is σa ∗ -open and, f is a∗-homeomorphism. � theorem 3.6. let (x, τ, i) be an ideal topological space, then 〈ℜa(τ a ∗ )〉 = 〈ℜa(τ a)〉. proof. note that for every u ∈ τa and for every i ∈ i, we have ℜa(u − i) = ℜa(u). consequently, ℜa(β) = ℜa(τ a) and 〈ℜa(β)〉 = 〈ℜa(τ a)〉, where β is a basis for τa. it follows directly from theorem 11 of [9] that 〈ℜa(β)〉 = 〈ℜa(τ a ∗ )〉, hence the theorem is proved. � theorem 3.7. let f : (x, τ, i) → (y, σ, j ) be a bijection with f(i) = j . then the following are hold: (1) if f is a a∗-homeomorphism, then f is a ℜa-homeomorphism. (2) if τ ∼a i and σ ∼a j and f is a ℜa-homeomorphism, then f is a a∗-homeomorphism. proof. (1) assume f : (x, τa ∗ ) → (y, σa ∗ ) is a a∗-homeomorphism, and let ℜa(u) be a basic open set in 〈ℜa(τ a)〉 with u ∈ τa. then f(ℜa(u)) = ℜa(f(u)) by theorem 3.5. then f(ℜa(u)) ∈ ℜa(σ a ∗ ), but 〈ℜa(τ a ∗ )〉 = 〈ℜa(τ a)〉 by theorem 3.6. thus f : (x, ℜa(τ)) → (y, ℜa(σ)) is a-open. similarly, f−1 : (y, ℜa(σ)) → (x, ℜa(τ)) is a-open and f is ℜa-homeomorphism. (2) assume f is a ℜa-homeomorphism, then f(ℜa(a)) = ℜa(f(a)) for every a ⊆ x by theorems 3.4 and 3.3. thus f is a a∗-homeomorphism by theorem 3.5. � 4. results related to topological groups given a topological group (x, τ, .) and an ideal i on x, denoted (x, τ, i, .) and x ∈ x, we denote by xi = {xi : i ∈ i}. we say that i is left translation c© agt, upv, 2014 appl. gen. topol. 15, no. 1 38 on topological groups via a-local functions invariant if for every x ∈ x we have xi ⊆ i. observe that if i is left translation invariant then xi = i for every x ∈ x. we define i to be right translation invariant if and only if ix = i for every x ∈ x [8]. given a topological group (x, τ, i), i is said to be τa-boundary [2], if τa ∩i = {φ}. note that if i is left or right translation invariant, x /∈ i, and i ∼a i, then i is τa-boundary. definition 4.1 ([2]). let (x, τ, i) be an ideal topological space. a subset a of x is called a baire set with respect to τa and i, denoted a ∈ br(x, τ, i), if there exists a a-open set u such that a = u [mod i]. let u(x, τ, i) be denoted {a ⊆ x : there exists b ∈ br(x, τ, i) − i such that b ⊆ a}. lemma 4.2. let (x, τ) and (x, σ) be two topological spaces and f be a collection of a-open mappings from x to y . let u ∈ τa − {φ} and let a be a non empty subset of u. if f(u) ∈ f(a) = {f(a) : f ∈ f} for every f ∈ f, then f(a) ∈ σa − {φ}. proof. let y ∈ f(a), then there exist f ∈ f such that y ∈ f(a). now, a ⊆ u, then f(a) ⊆ f(u) and y ∈ f(u). then f(u) is a-open in (y, σ) (as f is aopen map). so there exists v ∈ σa(y) such that y ∈ v ⊆ f(u) ⊆ f(a). so f(a) ∈ σa − {φ}. � theorem 4.3. let (x, τ) and (x, σ) be two topological spaces and i be an ideal (x, τ) with τ ∼a i and τa ∩ i = {φ}. moreover, let u ∈ τa − {φ}, a ⊆ x, u ⊆ aa∗ ∩ ℜa(a) and f be a non-empty collection of a-open mappings from x to y . suppose y ∈ f(u) ⇒ u ∩ f−1(y) /∈ i, where f−1(y) = ∪{f−1(y) : f ∈ f}. then f(u ∩ a) ∈ σa − {φ}. proof. since u is a non-empty a-open set contained in aa∗∩ℜa(a) and τ ∼ a i, by proposition 2.10 it follows that u −a ∈ i and u ∩a /∈ i. for any y ∈ f(u), u ∩ f−1(y) /∈ i (by hypothesis) and we have u ∩ f−1(y) = u ∩ f−1(y) ∩ (a ∪ ac) = [u ∩ f−1(y) ∩ a] ∪ [u ∩ f−1(y) ∩ ac] ⊆ [u ∪ f−1(y) ∩ a] ∪ (u − a) (where ac = complement of a). since u ∩ f−1(y) /∈ i and u − a ∈ i, we have u ∩ f−1(y) ∩ a /∈ i. then for any y ∈ f(u), u ∩ f−1(y) ∩ a 6= {φ}. now for a given f ∈ f, k ∈ f(u) ⇒ k ∈ f(u), then there exist x ∈ u ∩ a and x ∈ g−1(k) for some g ∈ f, where k = g(x) ⇒ k ∈ g(u ∩a), and k ∈ f(u ∩a). hence f(u) ⊆ f(u ∩ a), for all f ∈ f. then f(u ∩ a) ∈ σa − {φ} by lemma 4.2. � lemma 4.4. let i be a left (right) translation invariant ideal on a topological group (x, τ, .) and x ∈ x. then for any a ⊆ x the following hold: (1) xℜa(a) = ℜa(xa), and ℜa(a)x = ℜa(ax), (2) xaa ∗ = (xa)a ∗ (resp.aa ∗ x = (ax)a ∗ ). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 39 w. al-omeri, mohd. salmi md. noorani and a. al-omari proof. we assume that i is right translation invariant, the proof is similar for the case when i is left translation invariant would be . (1) we first note that for any two subsets a and b of x, (a− b)x = ax− bx. in fact, y ∈ (a − b)x, then y = tx, for some t ∈ a − b. now t ∈ a then tx ∈ ax. but tx ∈ bx ⇒ tx = bx for some b ∈ b ⇒ t = b ∈ b a contradiction. so y = tx ∈ ax−bx. again, y ∈ ax−bx ⇒ y ∈ ax and y /∈ bx ⇒ y = ax for some a ∈ a and ax /∈ bx ⇒ a /∈ b ⇒ y = ax, where a ∈ a−b ⇒ y ∈ (a−b)x. now, y ∈ ℜa(ax) ⇒ y ∈ ux for some u ∈ τ a with u − a ∈ i. then ux = v ∈ τa and (u − a)x = ux − ax ∈ i where ux ∈ τa. then y ∈ v , where v ∈ τa and v − ax ∈ i ⇒ y ∈ ∪{v ∈ τa : v − ax ∈ i} = ℜa(ax). thus xℜa(a) ⊆ ℜa(ax). conversely, let y ∈ ℜa(ax) = ∪{u ∈ τ a : u − ax ∈ i} ⇒ y ∈ u ∈ τa, where u − ax ∈ i. put v = ux−1. then v ∈ τa. now yx−1 ∈ v and v −a = ux−1 −a = (u −ax)x−1 ∈ i ⇒ yx−1 ∈ ℜa(a) ⇒ y ∈ ℜa(a)x. thus ℜa(ax) ⊆ ℜa(a)x and hence ℜa(a)x = ℜa(ax) (2) in view of (1) ℜa(x −a)x = ℜa((x −a)x), then [x −a a ∗ ]x = x −(ax)a ∗ and x − aa ∗ x = x − (ax)a ∗ thus aa ∗ x = (ax)a ∗ . � lemma 4.5. let i be an ideal space on a topological group (x, τ, .) such that i is left or right translation invariant and τ ∼a i. then i ∩ τa = {φ}. proof. since x /∈ i and τ ∼a i, by theorem 2.4 there exist x ∈ x such that for all u ∈ τa(x), (4.1) u = u ∩ x /∈ i let v ∈ i ∩ τa. if v = {φ} we have nothing to show. suppose v 6= {φ}. without loosing of generality we may assume that i ∈ v (i denoted the identity of x). for y ∈ v then y −1 v ∈ τa and y−1v ∈ y−1i so that y−1v ∈ i where i ∈ y−1v . thus xv ∈ τa and xv ∈ xi and hence xv ∈ i. thus xv ∈ τa ∩ i, where xv is a neighborhood of x, which is contradicting (4.1) and hence i ∩ τa = {φ}. � theorem 4.6. let (x, τ, .) be a topological group and i be an ideal on x such that τ ∼a i. let p ∈ u(x, τ, i) and s ∈ p(x) − i. let u, v ∈ τa such that u ∩ sa ∗ 6= {φ}, v ∩ aint(p a ∗ ) ∩ ℜa(p) 6= {φ}. if a = u ∩ s ∩ s a ∗ and b = v ∩ aint(p a ∗ ) ∩ p ∩ ℜa(p) then the following hold: (1) if i is left translation invariant, then ba−1 is a non-empty a-open set contained in ps−1. (2) if i is right translation invariant, then a−1b is a non-empty a-open set contained in s−1p. proof. (1) since x is a topological group, τ ∼a i and i is right translation invariant, we have by lemma 4.5, i ∩ τa = {φ}. now by theorem 2.2 (u ∩s ∩sa ∗ )a ∗ ⊆ (u ∩s)a ∗ and by theorem 2.7 we get (u ∩s ∩(u ∩s)a ∗ )a ∗ = (u ∩ s)a ∗ . hence (4.2) (u ∩ s ∩ sa ∗ )a ∗ = (u ∩ s)a ∗ c© agt, upv, 2014 appl. gen. topol. 15, no. 1 40 on topological groups via a-local functions thus by theorem 2.2 we have u ∩ sa ∗ = u ∩ (u ∩ s)a ∗ ⊆ (u ∩ s)a ∗ = (u ∩ s ∩ sa ∗ )a ∗ by (*). since u ∩ sa ∗ 6= {φ}, we have a 6= {φ}. again, aa ∗ = (u ∩ s ∩ sa ∗ )a ∗ ⊇ u ∩ sa ∗ ⊇ u ∩ sa ∗ ∩ s = a i.e. a ⊆ aa ∗ . for each a ∈ a, define fa : x → x given by fa(x) = xa −1, and f = {fa : a ∈ a}. since a 6= {φ}, f 6= {φ} and each fa is a homeomorphism. let g = v ∩aint((p) a ∗ )∩ ℜa(p). now it is sufficient to show that g ∩ f −1(y) /∈ i for every y ∈ f(g). because then by theorem 4.3, f(g ∩ p) = f(b) = ba−1 is a non-empty a-open set in x contained in ps−1. let y ∈ f(g). then y = xa−1 for some a ∈ a and x ∈ g ⇒ f−1(y) = xa−1a. thus x ∈ xa−1a ⊆ xa−1aa ∗ (as a ⊆ aa ∗ ) ⊆ (xa−1a)a ∗ (by lemma 4.4) = (f−1(y))a ∗ ⇒ nx ∩ f −1(y) /∈ i for some nx ∈ τ a(x). thus ba−1 is a nonempty a-open subset of ps−1. so in particular, as (2) is similar to (1). � corollary 4.7. let (x, τ, .) be a topological group and i be an ideal on x such that τ ∼a i. let a ∈ u(x, τ, i) and b ∈ p(x) − i. (1) if i is right translation invariant, then [b ∩ ba ∗ ]−1[a ∩ aint(aa ∗ ) ∩ ℜa(a)] is a non-empty a-open set contained in b −1a. (2) if i is left translation invariant, then [a∩aint(aa ∗ )∩ℜa(a)][b∩b a ∗ ]−1 is a non-empty a-open set contained in ab−1. proof. we only show that ba ∗ 6= {φ} and a∩aint(aa ∗ )∩ℜa(a) 6= {φ}, the rest follows from theorem 4.6 by taking u = v = x. in fact, if ba ∗ = {φ}, then b ∩ ba ∗ = {φ} which gives in view of theorem 2.4, b ∈ i, a contradiction. again, a ∈ u(x, τ, i) ⇒ aint(aa ∗ ) ∩ ℜa(a) 6= {φ} (by lemma 4.5 and proposition 2.9) ⇒ aint(aa ∗ )∩ℜa(a) ∈ τ a −{φ}. now, aint(aa ∗ )∩ℜa(a) = [a∩aint(aa ∗ )∩ℜa(a)]∪[a c ∩aint(aa ∗ )∩ℜa(a)] /∈ i (by lemma 4.5). then [ac ∩ aint(aa ∗ ) ∩ ℜa(a)] ⊆ [a c ∩ ℜa(a)] = ℜa(a) − a ∈ i by theorem 2.11. thus a ∩ aint(aa ∗ ) ∩ ℜa(a) /∈ i and hence a ∩ aint(a a ∗ ) ∩ ℜa(a) 6= {φ}. � corollary 4.8. let (x, τ, .) be a topological group and i be an ideal on x such that i ∩ τa = {φ} and a ∈ u(x, τ, i). (1) if i is left translation invariant, then e ∈ aint(a−1a). (2) if i is right translation invariant, then e ∈ aint(aa−1). (3) if i is left as well as right translation invariant, then e ∈ aint(aa−1 ∩ a−1a). proof. it suffices to prove (1) only. we have, a ∈ u(x, τ, i) then there exists b ∈ br(x, τ, i) − i such that b ⊆ a. now for any x ∈ x, ℜa(b)x ∩ ℜa(b) = ℜa(bx) ∩ ℜa(b) = ℜa(bx ∩ b) (by lemma 4.4 and theorem 2.6). thus if ℜa(b)x ∩ ℜa(b) 6= {φ}, then bx ∩ b 6= {φ}. now, if x ∈ [ℜa(b)] −1[ℜa(b)] then x = y−1z for some y, z ∈ ℜa(b), then yx = z = t (say) ⇒ t ∈ ℜa(b)x and t ∈ ℜa(b) ⇒ ℜa(b)x ∩ ℜa(b) 6= {φ} ⇒ x ∈ {x ∈ x : ℜa(b)x ∩ ℜa(b) 6= {φ}} then [ℜa(b)] −1[ℜa(b)] ⊆ {x ∈ x : ℜa(b)x ∩ ℜa(b) 6= {φ}} ⊆ {x ∈ x : bx ∩ b 6= {φ}} ⊆ b−1b ⊆ a−1a. since ℜa(b) 6= {φ} by proposition 2.9 as b ∈ u(x, τ, i) and ℜa(b) is a-open for any b ⊆ x, we have e ∈ [ℜa(b)] −1[ℜa(b) ⊆ aint(a −1a). � c© agt, upv, 2014 appl. gen. topol. 15, no. 1 41 w. al-omeri, mohd. salmi md. noorani and a. al-omari acknowledgements. the authors would like to acknowledge the grant from ministry of high education malaysia ukmtopdown-st-06-frgs0001-2012 for financial support. references [1] w. al-omeri, m. noorani and a. al-omari, a-local function and its properties in ideal topological space,fasciculi mathematici, to appear. [2] w. al-omeri, m. noorani and a. al-omari, on ℜaoperator in ideal topological spaces, submitted. [3] f. g. arenas, j. dontchev and m. l. puertas, idealization of some weak separation axioms, acta math. hungar. 89, no. 1-2 (2000), 47–53. [4] j. dontchev, m. ganster, d. rose, ideal resolvability. topology appl. 93 (1999), 1–16. [5] e. ekici, on a-open sets, a∗-sets and decompositions of continuity and super-continuity, annales univ. sci. budapest. 51 (2008), 39–51. [6] e. ekici, a note on a-open sets and e∗-open sets, filomat 22, no. 1 (2008), 89–96. [7] e. hayashi, topologies denfined by local properties, math. ann. 156 (1964), 205–215. [8] t. r. hamlett and d. rose, remarks on some theorems of banach, mcshane, and pettis, rocky mountain j. math. 22, no. 4 (1992), 1329–1339. [9] t. r. hamlett and d. jankovic, ideals in topological spaces and the set operator ψ, bull. u.m.i. 7 4-b (1990), 863–874. [10] d. jankovic and t. r. hamlett, new topologies from old via ideals, amer. math. monthly, 97 (1990), 295–310. [11] k. kuratowski, topology, vol. i. newyork: academic press (1966). [7] m. khan and t. noiri, semi-local functions in ideal topological spaces. j. adv. res. pure math. 2, no. 1 (2010), 36–42. [12] m. h. stone, application of the theory of boolean rings to general topology, trans. amer. math. soc. 41 (1937), 375–481. [13] m. n. mukherjee, r. bishwambhar and r. sen, on extension of topological spaces in terms of ideals, topology appl. 154 (2007), 3167–3172. [14] m. navaneethakrishnan and j. paulraj joseph, g-closed sets in ideal topological spaces, acta math. hungar. 119, no. 4 (2008), 365–371. [15] a. a. nasef and r. a. mahmoud, some applications via fuzzy ideals, chaos, solitons and fractals 13 (2002), 825–831. [16] r. l. newcomb, topologies which are compact modulo an ideal, ph. d. dissertation, univ. of cal. at santa barbara. (1967). [17] b. j. pettis, remark on a theorem of e. j. mcshane, proc. amer. math. soc. 2(1951), 166–171. [18] r. vaidyanathaswamy, set topology, chelsea publishing company (1960). [19] r. vaidyanathaswamy, the localization theory in set-topology, proc. indian acad. sci., 20 (1945), 51–61 [20] n.v. veličko, h-closed topological spaces, amer. math. soc. trans. 78, no. 2 (1968), 103–118. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 42 rawoagt.dvi @ applied general topology c© universidad politécnica de valencia volume 7, no. 1, 2006 pp. 73-101 on rg-spaces and the regularity degree r. raphael∗ and r. g. woods† abstract. we continue the study of a lattice-ordered ring g(x), associated with the ring c(x). following [10], x is called rg when g(x) = c(xδ). an rg-space must have a dense set of very weak p-points. it must have a dense set of almost-p-points if xδ is lindelöf, or if the continuum hypothesis holds and c(x) has small cardinality. spaces which are rg must have finite krull dimension when taken with respect to the prime z-ideals of c(x). there is a notion of regularity degree defined via the functions in g(x). pseudocompact spaces and metric spaces of finite regularity degree are characterized. 2000 ams classification: primary 54g10; secondary 46e25, 16e50. keywords: p-space, almost-p space, prime z-ideal, rg-space, very weak p-point. 1. introduction and elementary results throughout x will denote a tychonoff space and c(x) will be the ring of continuous real-valued functions on x. the need for examples of epimorphisms of commutative rings led early on to the consideration of rings of the form c(x). later, in connection with the study of the epimorphic hull (cf. [21]), a general lattice-ordered ring called g(x) was defined. this algebra is an epimorphic extension of c(x) and has many interesting properties. there is an associated notion of degree denoted rg(x). it is rarely a ring of continuous functions, but it is when x is an rg-space. there are persistent open questions concerning these objects, the most stubborn being whether rg-spaces must have p-points. these have been previously studied in [21] and [10]. in this article we investigate these topics further. here are the main results. an rg-spaces must have a dense set of what we call “very weak p-points”. it ∗supported by the nserc of canada. †supported by the nserc of canada. 74 r. raphael and r. g. woods must have a dense set of almost-p-points if xδ is lindelöf, or if the continuum hypothesis holds and c(x) has cardinality c. by studying the spectrum of g(x) we show that rg spaces must have finite krull dimension when said dimension is taken with respect to the prime z-ideals. this allow us to characterize rg spaces when x is scattered and perfectly normal. this generalizes previous characterizations (see [10]) in the compact and metric cases. the regularity degree is studied. the main results characterize (pseudo) compact spaces and metric spaces which are of finite regularity degree. for background and notation one should consult [10], [8] and [19]. in particular βx will denote the stone-cech compactification of x, the unique compactification of x in which x is c∗-embedded. some of this work was done while the authors were visiting the mathematics department of the university of auckland. we are grateful to our hosts d. gauld and i. reilly. associated with a function f : x → r is the function f∗ : x → r defined as follows: f∗ vanishes on z(f) and f∗(x) = 1/f(x) for x ∈ coz(f). in the case when f ∈ c(x) the function f∗ is rarely xcontinuous, but it is continuous in the gδ-topology on x. (in ring theory f ∗ is called the quasi-inverse of f in c(xδ))). the subalgebra of c(xδ) generated by c(x) and the quasi-inverses of the functions in c(x) is denoted g(x). significantly, g(x) is itself von neumann regular, which is to say that it is closed under taking quasi-inverses. as well g(x) (like c(x)) is a lattice under point-wise operations. by definition each function h ∈ g(x) can be written in the form ∑i=n i=1 aib ∗ i , ai, bi ∈ c(x). this representation is not unique but the least n for which such a representation for h exists is unique, and is called its regularity degree, rg(h). the regularity degree of x is the supremum of {rg(h)|h ∈ g(x)}. it can be infinite. the following result summarizes some useful technical facts about the behaviour of the quasi-inverse and the regularity degree. proposition 1.1. (i) (bounded functions) each function in g(x) has a representation of the required form using functions from c∗(x). furthermore, the use of bounded functions does not change the value of the regularity degree. (ii) (restrictions) g(x) is well-behaved with respect to restrictions. in particular if s ⊂ x and h ∈ g(x) then h|s ∈ g(s) and rg(h|s) ≤ rg(h). (iii) (compression) if rg(x) = ∞, then this can be witnessed by functions in g(x) whose values lie between 0 and 1. (iv) if f, g ∈ c(x), (fg∗)∗ = f∗g. in particular if rg(h) = 1, then rg(h∗) = 1. proof. claims (ii) and (iv) are easily verified. (i) let h ∈ g(x), say h = ∑1=n 1=1 aib ∗ i , ai, bi ∈ c ∗(x). then replace aib ∗ i with (ai/c)(bi/c) ∗ where c = (1 + a2i ) ∨ (1 + b 2 i ). on rg-spaces and the regularity degree 75 (iii) we claim that for each n ∈ n, there is a function fn ∈ g(x) such that 0 ≤ fn ≤ 1 and rg(fn) > n. suppose if possible that rg(x) = ∞ but that there is a positive integer n that globally bounds the number of terms needed to represent bounded functions in g(x). let f ∈ g(x). since 1 + f2 is not a zero divisor in the regular ring g(x) its inverse 1/(1 + f2) also lies in g(x). then f/(1 + f2) and 1/(1+ f2) can be expressed in terms of n terms. now the formula for the quasi-inverse in [20, remark 3.4] shows that the number of terms (of the form gk∗) needed for the quasi-inverse of a function needing at most n such terms is a function of n alone. thus 1 + f2 can be represented by m terms, where m is an integer that depends only on n. but f is the product of 1 + f2 and f/(1 + f2) so it requires at most mn terms. thus g(x) has bounded functions of arbitrarily high regularity degree. since g(x) is a lattice, it also contains non-negative bounded functions of arbitrarily high regularity degree. since multiplication of a function by a nonzero constant does not change its regularity degree, we can obtain our functions between 0 and 1 as claimed. � 2. the space of prime z-ideals of c(x) versus the stone space of z(g(x)) let pf(x) denote the set of prime z-filters on c(x). clearly these are in 1-to-1 correspondence with the set pz(x) of prime z-ideals on x via the map j : pz(x) → pf(x) given by: j(p) = {z(f) : f ∈ p} (see [8, chapter 2]). in [16, section 1], there is a discussion of the “patch topology” (although this term is not used) on pz(x). montgomery shows that pz(x) with this patch topology is homeomorphic to a compactification of xδ. we will present a proof of this result that is clearer than the proof in [16], and that moreover shows that the patch topology on pz(x) is “naturally” homeomorphic to the space of maximal ideals of g(x), or (equivalently) to the stone space of the boolean algebra of zerosets of members of g(x). it will be technically useful to identify pz(x), equipped with the patch topology, with pf(x), equipped with the topology induced by the bijection j mentioned above. the patch topology. let s be a set of prime ideals of c(x) and let f ∈ c(x). in a slight variation of the notation used in [16, section1], we define: h(f) = {p ∈ s : f ∈ p} and hc(f) = s\h(f). the patch topology on s is defined to be the topology for which {h(f) : f ∈ c(x)} ⋃ {hc(g) : g ∈ c(x)} is a subbase. let s be pz(x). clearly j[h(f)] = {α ∈ pf(x) : z(f) ∈ α} and j[hc(f)] = {α ∈ pf(x) : z(f) /∈ α}. now {α ∈ pf(x) : z(f) ∈ α} is closed under finite intersection (as α is a z -filter and hence closed under finite intersection) and {α ∈ pf(x) : z(f) /∈ α} is closed under finite intersection (as α is a prime z-filter). thus {j[h(f)] ∩ j[hc(g)] : f, g ∈ c(x)} is an open base for the patch 76 r. raphael and r. g. woods topology on pf(x). clearly this family is just {{f ∈ pf(x) : z ∈ f and s /∈ f} : z, s ∈ z(x)}. stone duality and zero-dimensional compactifications. we briefly summarize how to construct zero-dimensional compactifications of zero-dimensional spaces. see [19, 3.2 and 4] for more details. if a is a boolean algebra, denote by s(a) the set of all ultrafilters on a. if a ∈ a, denote by λ(a) the set {α ∈ s(a) : a ∈ α}. then {λ(a) : a ∈ a} is a clopen base for a compact hausdorff zero-dimensional topology on s(a), and thus topologized s(a) is called the stone space of a. suppose that x is a zero-dimensional hausdorff space (i.e. the set b(x) of all clopen sets of x forms an open base of x). suppose that a is a subalgebra of b(x) and that a is a base for the open sets of x. in that case the map i : x → s(a) given by: i(x) = {a ∈ a : x ∈ a} is a homeomorphism from x onto the subspace i[x] of the stone space s(a). if we identify x and i[x], then x is a dense subspace of s(a) so s(a) is a zero-dimensional compactification of x. furthermore the map a → λ(a) = cls(a)a is a boolean algebra isomorphism from a onto b(s(a)). finally, every zero-dimensional compactification of x arises in this way: if k is such a compactification, then define a to be {a∩x : a ∈ b(k)}; then k and s(a) turn out to be naturally equivalent compactifications of x. theorem 2.1. let x be a tychonoff space. then: (i) the set z(g(x)) of zero-sets of functions in g(x) is a boolean algebra a of clopen subsets of xδ that forms an open base for xδ. hence s(a) is a zero-dimensional compactification of xδ. (ii) if α ∈ s(a) define α# to be α ∩ z(x). then the map k : s(a) → pf(x) given by: k(α) = α# is a homeomorphism from s(a) onto pf(x) (with the patch topology) whose restriction to xδ is a bijection onto the fixed z-ultrafilters of c(x). (iii) s(a) is the space of maximal ideals of g(x) (with the hull-kernel topology). proof. (i) we know that z(g(x)) is the collection of finite unions of sets of the form z ∩ c, where z ∈ z(x) and c ∈ coz x (see 5.2 of [21]). this is in fact the boolean subalgebra of the power set of x (where sups are unions, infs are intersections, and boolean-algebraic complements are set-theoretic complements) generated by z(x). let us denote it by a. clearly z(x) ⊆ a so a is a base for the open sets of xδ (as z(x) is). hence as noted above s(a) is a compactification of xδ. (ii) first we must show that the map k is well-defined, i.e. that if α ∈ s(a) then α# ∈ pf(x). it is routine to show that if α ∈ s(a), then α# is a z-filter on x. if z, s ∈ z(x) and z ∪ s ∈ α# then z ∪ s ∈ α and as α is prime (being an ultrafilter on a), either z ∈ α or s ∈ α. thus either z ∈ α# or s ∈ α# and so α# is prime and hence in pf(x). thus the map k is well-defined. on rg-spaces and the regularity degree 77 we show that k is 1-to-1. suppose that α, β ∈ s(a) and that α 6= β. we show that α# 6= β#. without loss of generality suppose that a ∈ α − β. as a is a union of finitely many sets of the form z ∩ c, where z ∈ z(x) and c ∈ coz x, and as α is an ultrafilter and hence prime, there exist z ∈ z(x) and c ∈ coz x such that z∩c ∈ α−β. thus z ∈ α and c ∈ α. but z∩c /∈ β so either z /∈ β or else c /∈ β. if z /∈ β then z ∈ α# − β# and so α# 6= β#. if c /∈ β then as β is an ultrafilter on a, it follows that x − c ∈ β ∩ z(x) = β#. but c ∈ α so x − c /∈ α. thus x − c /∈ α# and again α# 6= β#. thus k is 1-to-1 as claimed. next we show that k maps s(a) onto pf(x). let f ∈ pf(x) and define α(f) = {a ∈ a} such that there exist z ∈ f, s ∈ z(x) − f such that z ∩ (x − s) ⊆ a. we will show that α(f) ∈ s(a) and that k(α(f)) = f. clearly if a ∈ α(f) and a ⊆ b ∈ a then b ∈ α(f). if a, b ∈ α(f) then there exist z, t ∈ f and s, y ∈ z(x) − f such that z ∩ (x − s) ⊆ a and t ∩ (x − y ) ⊆ b. then z ∩ t ∈ f and s ∪ y /∈ f as f is prime. clearly (z ∩ t ) ∩ (x − (s ∪ y )) ⊆ a ∩ b, so a ∩ b ∈ α(f). thus α(f) is a filter on a. next we show that α(f) is an ultrafilter. it suffices to show that if a ∈ a − α(f) then there is an m ∈ α(f) such that m ∩ a = ∅. now a has the form a = ⋃ {z(i)∩c(i) : z(i), x −c(i) ∈ z(x), i = 1 . . . n}. as a ∈ a−α(f) for each i, z(i) ∩ c(i) /∈ α(f) so either z(i) /∈ f or x − c(i) ∈ f. let i = {i : z(i) /∈ f} and j = {i : x − c(i) ∈ f}. then i ∪ j = {1, . . . , n}. let g = ⋃ ((z(i) : i ∈ i) and h = x − ⋃ {c(j) : j ∈ j}. then g ∈ z(x) − f because f is prime, and h ∈ f because f is closed under finite intersection. let m = h∩(x−g). clearly m ∈ α(f). we claim that m ∩a = ∅. it suffices to show that for all i, m ∩z(i)∩c(i) = ∅. for any such i, as i ∪j = {1, . . . , n} either i ∈ i or else i ∈ j. in the former case z(i) ∩ (x − g) = ∅ and in the latter case, c(i) ∩ h = ∅. thus m ∩ a = ∅ and so α(f) is an ultrafilter as claimed, and therefore an element of s(a). next we show that k(α(f)) = f. if z ∈ f it is clear that z ∈ α(f) ∩ z(x) = α# = k(α(f)), so f ⊆ k(α(f)). conversely, suppose that z ∈ α(f)∩z(x) = α# = k(α(f)). thus there exist t ∈ f and s ∈ z(x)−f such that t ∩ (x − s) ⊆ z. thus t ⊆ z ∪ s. as t ∈ f it follows that z ∪ s ∈ f. but f is prime and s /∈ f so z ∈ f. thus k(α(f)) ⊆ f and so k(α(f)) = f. we have established that k is a well-defined bijection from s(a) onto pf(x). it remains to show that k is a homeomorphism. we claim that b = {λ(z ∩ c) : z, x − c ∈ z(x)} is an open base for s(a). to see this, note that each member of a is the union of finitely many members of b, so as λ is a boolean algebra isomorphism (the stone duality ), it preserves finite unions and each λ(a) is the union of finitely many members of b. as each member of b is clopen in s(a), our claim follows. we must show that the map k is both continuous and open. by the above claim, to verify “open” it suffices to show that k[λ(z ∩ c)] is open in pf(x) whenever z ∈ z(x) and c ∈ coz x. but note that k[λ(z ∩ c)] = {α ∩ z(x) : α ∈ λ(z ∩ c)} = {α ∩ z(x) : z ∩ c ∈ α} = {α ∩ z(x) : z ∈ α, x − c /∈ α} = {f ∈ pf(x) : z ∈ f, x − c /∈ f} (because k is onto). 78 r. raphael and r. g. woods but this latter set is open in the patch topology (see the above discussion), so k is an open map. but as k is a bijection, it follows that k−1[{f ∈ pf(x) : z ∈ f andx −c /∈ f}] = λ(z ∩c) and because sets of the form {f ∈ pf(x) : z ∈ f andx −c /∈ f} form an open base for the patch topology on pf(x) and λ(z ∩c) is open in s(a), it follows that k is continuous and a homeomorphism as claimed. finally, if x ∈ xδ then x corresponds to the ultrafilter α(x) = {a ∈ a : x ∈ a} and so k(x) = k(α(x)) = α(x) ∩ z(x) = {z ∈ z(x) : x ∈ z}, which is the fixed z-ultrafilter at x. (iii) this follows from the fact that the correspondence m → {z(f) : f ∈ m} is a bijection from the set m of maximal ideals of the ring g(x) onto s(a), and if m is given the hull-kernel topology then this bijection is a homeomorphism. � the following remark likely appears in [16], but we have not been able to pinpoint it. we retain the same notation as above. proposition 2.2. let i : xδ → x be the identity map on the underlying set of x. then i can be continuously extended to i∧ : s(a) → βx. proof. if α ∈ s(a) let i∧(α) be the unique point in ∩{clβxz : z ∈ α∩z(x) = α#}. (to see that there is indeed such a unique point, note that as α# is a prime z-ideal there is a unique y ∈ βx such that z(oy) ⊆ α# ⊆ z(my) (see [8, 7.15]).) to show that i∧ is continuous, it suffices (because {clβxz : z ∈ z(x)}) is a base for the closed sets of βx) to show that (i∧)−1[clβxz] is closed in s(a) for each z ∈ z(x). but (i∧)−1[clβxz] = {α ∈ s(a) : i ∧(α) ∈ clβxz} = {α ∈ s(a) : z ∈ α} = λ(a) (by stone duality). but λ(a) is a clopen, hence closed, subset of s(a). hence i∧ is continuous. it also maps s(a) onto βx as j maps xδ onto the dense subspace x ⊂ βx. � note that a portion of proposition 2.2 has also appeared (with a different proof) in [4, 3.4, 3.9]. it may be worth noting that if y ∈ βx, then (i∧)−1(y) consists of all prime z-filters f ∈ pf(x) for which z(oy) ⊆ f ⊆ z(my) (we are still identifying s(a) and pf(x) via α → α#). remark 2.3. if x is an rg-space then g(x) = c(xδ), z(g(x)) = z(xδ), and s(z(g(x))) = β(xδ). 3. on regularity degree and prime z-ideal length the following result relies on work in [4] where the pierce sheaf was used to study the regularity degree in general commutative rings. both [4] and [20] are useful references for the algebraic background needed. the following result is the key to theorem 3.4 below. on rg-spaces and the regularity degree 79 lemma 3.1. let x be tychonoff. suppose that c(x) has a strict ascending chain of k + 1 prime z-ideals. then g(x) has a function of regularity degree at least k + 1. it can be constructed directly from the prime z-ideals in the ascending chain. proof. (cf. part (3) of [4, theorem 3.1] and the studies of the universal regular ring by kennison [11], olivier [18], and wiegand [25]). suppose that p0 ⊂ · · · ⊂ pk is a strict chain of prime z-ideals in c(x). for i = 0, . . . , k−1, choose bi ∈ pi+1 − pi and let bk = 1. by the proof of [4, theorem 3.1] the element t = (b0, . . . , bk) ∈ s ′ = qcl((c(x)/p0)) × qcl((c(x)/p1)) × · · · × qcl((c(x)/pk)) cannot be written as the sum of fewer than k+1 terms of the form cid ∗ i , ci, di ∈ c(x). now the fields qcl(c(x)/pi) are stalks (homomorphic images) of the universal regular ring t (c(x)) for c(x) whose spectrum is the set of prime ideals in c(x) under the patch topology. the spectrum of g(x) (cf. 2.1 part (iii)) is the compact space of prime z-ideals of c(x) under the patch topology. the universality of t (c(x)) and the fact that c(x) → g(x) is epic show that g(x) is a homomorphic image of t (c(x))—the spectrum of the former ring is a subspace of the spectrum of the latter. (see also [4, proposition 2.5]). thus because the pi are prime z-ideals, the fields qcl(c(x)/pi) are homomorphic images of g(x). furthermore since g(x) is a regular ring, the finite product s′ is also a homomorphic image of g(x) (use the regularity of g(x) to obtain a function ki ∈ (( ⋂ pj, j 6= i) − pi) and consider the basic idempotent kik ∗ i ). so the element t is the image of say h ∈ g(x). clearly rg(h) ≥ k + 1 because of the constraint on the regularity degree for t in s′. � definition 3.2. by the krull z-dimension of c(x) we will mean the supremum of the lengths of chains of prime z-ideals in c(x). the krull z-dimension of a maximal ideal will mean the supremum of the lengths of chains of prime z-ideals lying in it. remark 3.3. it is interesting to compare the krull z-dimension of fixed and free maximal ideals in c(x) in different cases. when every fixed maximal ideal is a minimal prime, x is necessarily a p-space, and each free maximal ideal is also a minimal prime. but in the examples of [9, 4.4, 7.4] each fixed maximal ideal has krull z-dimension 1, and there is at least one free maximal ideal of krull z-dimension greater than 1. indeed, we will presently see (in the remarks after theorem 4.1) that in some models of ψ, c(ψ) will have infinite krull z-dimension. theorem 3.4. if the krull z-dimension of c(x) is infinite then x is not an rg space and rg(x) = ∞. proof. we will use the fact that if x is rg then specg(x), the space of prime z-ideals of c(x) under the patch topology, is natually homeomorphic to β(xδ) (see remark 2.3 and the discussion that precedes it). let us first recall some properties of β(xδ). 1. there is a boolean isomorphism between the algebra of clopen sets of xδ and the algebra of clopen sets of β(xδ) defined by taking traces in one direction 80 r. raphael and r. g. woods and closures in the other (see the discussion before 2.1). furthermore, if we have a countable family of clopen sets {bn} ⊂ β(xδ) with traces {an} in xδ, then if s = ⋃ bn, with closure s ′ ⊂ β(xδ) then s ′ is clopen in β(xδ) and has trace equal to ⋃ an. (thus β(xδ) is basically disconnected; see [8, 4k(7)]). 2. (all closures are taken in β(xδ)). if h ∈ g(x) , if b = cl[coz(h)] and if a is the trace of b in x, then b = cl(a). in particular, if f ∈ c(x) then coz(f) is clopen in xδ and cl[coz(f)] is the clopen set of β(xδ) that consists of the prime z-ideals of c(x) that do not contain f. 3. let x be an rg space. suppose that b is clopen in β(xδ) with trace a in xδ. let p be a prime z-ideal in b. suppose that h ∈ g(x), a ∈ c(x) and h|a = a|a. then h and a agree at p in the field c(xδ)/m p where mp is the maximal ideal in c(xδ) corresponding to the point p ∈ β(xδ). (reason: h−a has a clopen zero set in xδ , it vanishes on a so it vanishes on its closure b). 4. now assume that c(x) has infinite krull z-dimension. we could conclude that rg(x) is infinite directly from lemma 3.1 but instead our strategy will be to define a family of pairwise disjoint clopen sets of xδ and then use them to define a function on their union that is not in g(x) because it is of ”infinite” degree. the proof will also show that c(xδ) has functions of arbitrarily large regularity degree and thus rg(x) = ∞. since the ring c(x) has infinite krull z-dimension, there are two possibilites. case 1. there exists in c(x) a maximal ideal that contains an infinite ascending sequence {pn} of prime z-ideals. case 2. case 1 fails, so either there is an infinite descending sequence of prime z-ideals, or all ascending chains of prime z-ideals are finite, but their lengths are unbounded globally. one checks easily using [8, 14.8 (a)], if necessary, that in case 2 it is always possible to choose a countable sequence {ck} of finite ascending chains {qk,t} of prime z-ideals with the following properties: (i) distinct chains are disjoint, (ii) chain ck is of length sk, say, and the sk are strictly increasing, (iii) for each k, qi,1 qk,t for all i < k, for all t. we now contradict the rg property in both cases. case 1. use the ascending sequence {pn} to choose for each n, bn ∈ pn+1 − pn. now let b1 = coz(b1) and for n > 1, let bn = coz(bn) − [ ⋃n−1 i=1 coz(bi)]. the {bn} are non-empty clopen and disjoint in xδ. each bn has pn but no other pi its β(xδ)-closure. define h on x by h|bn = bn|bn for each n and h[x − ⋃ bi] = 0. then h ∈ c(xδ) and for each n, h(pn) = bn(pn) by part 3. now, suppose, if possible that h ∈ g(x) say h = ∑i=m i=1 cid ∗ i . take the first m + 2 clopen sets b1, . . . , bm+2 and their closures cl(b1), . . . , cl(bm+2) which are disjoint in β(xδ) [8, 6.5 iii]. let w = {p1, . . . ., pm+2}. (observe the parallel with the situation of lemma 3.1). the set w is closed and discrete in specg(x) and h|w can be written as the sum of m terms. but this contradicts lemma 3.1 since we have a chain of m + 2 primes, and h has been chosen so that h|w plays the role of t in 3.1. case 2. we inductively define a set of elements {bk,t}, t ∈ {1, . . . , sk − 1} so that bk,t ∈ qk,t+1 − qk,t, and bk,t ∈ qi,t for all i < k. start in chain c1 by on rg-spaces and the regularity degree 81 choosing s1 − 1 elements a1,t ∈ q1,i+1 − q1,i. suppose now that the choices have been made for the first k − 1 chains, and consider ck. since qk,t+1 is a prime ideal, condition (iii) shows that there is an element bk,t ∈ qk,t+1 − qk,t, and bk,t ∈ qi,1 for all i < k. as in case 1, we want to get disjoint clopen sets and define a function on their union. consider cl[coz(bk,t)]. it contains the primes {qk,j}, j ≤ t, it excludes the primes qi,t, t < k, and possibly it contains primes from subsequent chains {cr}, r > k. for each k and t ∈ {1, . . . , sk − 1} let bk,t = coz(bk,t) − [ ⋃j=t−1 j=1 coz(bk,j) ∪ ( ⋃ r>k coz(br,t))]. by construction the bk,t are disjoint clopen subsets of xδ and clbk,t contains qk,t and no other prime from the array. now define h on⋃ bk,t to be bk,t on bk,t and let h vanish off ⋃ bk,t. then h ∈ c(xδ) and by property 3, h and bk,t agree on qk,t. now condition (ii) gives us chains of arbitrary length so we argue that h /∈ g(x) as in case 1. � lemma 3.5. suppose that s is a subspace of t that induces an epi c(t ) → c(s). then the preimage of a chain of prime z-ideals in c(s) is a chain of prime z-ideals of c(t ). the lengths of such chains under taking preimages is either maintained or grows—it cannot decline. proof. it is well-known that the preimage of a prime is a prime. it is also well known (cf. [14]) that taking preimages of primes under an epimorphism is 1−1. so it suffices to check that the preimage of a z-ideal is a z-ideal. but this is straightforward: let p be a prime z-ideal in c(s) and let q be its preimage in c(t ). suppose that g ∈ q, f ∈ c(t ) with z(f) = z(g). now f|s and g|s have the same zero set. but g|yn is in the z-ideal p of c(s). so f|s ∈ p , and therefore f ∈ q. � theorem 3.6. a space x cannot be rg and must be of infinite regularity degree if it contains an infinite strictly decreasing sequence of epi-inducing (for example c∗-embedded) subspaces yn such that for each n, yn − yn+1 contains a cozero set of yn that is dense in yn. proof. we will show that c(x) contains chains of prime z-ideals of arbitrary length. then the conclusion follows from theorem 3.4. subspaces that, upon restriction, induce epimorphisms of rings were studied in [1]. it is immediate that the restriction map c(yn) → c(yn+1) is an epi if the map c(x) → c(yn+1) is an epi. suppose that c(yn+1) has a strictly ascending chain of k prime z-ideals p1, . . . , pk. by lemma 3.5 this gives a strict chain of k prime z-ideals in c(yn), say q1, . . . , qk, where qi = {f ∈ c(yn) : f|yn+1 ∈ pi}. now by assumption, yn −yn+1 contains densely coz(h) for some h ∈ c(yn). since h vanishes on yn+1, it lies in q1 because q1 contains the preimage of any function that is zero on yn+1. since coz(h) is dense in yn , h is a non-zero divisor in c(yn). but h ∈ q1, and it is elementary commutative algebra that non zero divisors cannot lie in minimal primes (cf. [13]). so q1 contains properly an additional prime z-ideal, and therefore c(yn) has a chain of at least k + 1 prime z-ideals. 82 r. raphael and r. g. woods now we will see that c(x) has chains of prime z-ideals of arbitrary length. given any natural number n, consider c(yn ) which has a chain of prime zideals of length at least 1. by working back through the chain induced by the inclusions x ⊂ y1 ⊂ y2 ⊂ yn one sees that c(x) has a chain of prime z-ideals of length at least n − 1. � remark 3.7. here are some particular cases of the result or of its method. (1) x cannot be rg if it has a decreasing sequence yn of closed epi-inducing subspaces, with the property that for infinitely many n, yn is weakly lindelöf and yn+1 is nowhere dense in yn. reason: the open set yn − yn+1 is a union of cozero sets so there is a union v of countably many of them that is dense in yn − yn+1. but v is itself a cozero set. (2) suppose x is normal, rg, scattered and of infinite cb-index. then in the derived sequence of isolated points there cannot be infinitely many occurances of countable sets. reason: this is a special case of (i). all of the remainders are closed and hence c-embedded, and at each stage the set of isolated points is dense. in the scattered perfectly normal case we can get a characterization, as follows: theorem 3.8. a scattered perfectly normal space is rg if and only if it is of finite cb-index. proof. by [10, 2.12] finite cb-index implies rg. the converse holds by theorem 3.6 because at each stage the set of isolated points is a dense cozero set. � 4. relationships that exist between cb(x), rg(x), and krull z-dimension theorem 4.1. (i) if x is tychonoff, then rg(x) = rg(υx), (ii) the map λ : ∑ aib ∗ i → ∑ aυi (b υ i ) ∗ is a ring isomorphism from g(x) onto g(υx). in particular, if x is pseudocompact then g(x) and g(βx) are naturally isomorphic. furthermore rg(x) = rg(υx). (iii) rg(x) ≤ rg(βx). proof. (i) and (ii). if f = ∑ aib ∗ i ∈ g(x) let λ(f) = ∑ aυi (b υ i ) ∗. clearly λ(f) ∈ g(υx) and λ(f)|x = f. since x is gδ-dense in υx, xδ is dense in (υx)δ. using this denseness one easily verifies that λ is well-defined and onto. that λ is 1 − 1 and respects ring operations is immediate. clearly rg(λ(f)) = rg(f), so rg(x) = rg(υx). (iii). let f ∈ g(x), with f = ∑i=n i=1 aib ∗ i , ai, bi ∈ c ∗(x). let f =∑i=n i=1 a β i (b β i ) ∗. then f ∈ g(βx). if rg(βx) = k, then rg(f) ≤ k. thus rg(x) ≤ rg(βx). � on rg-spaces and the regularity degree 83 example 4.2. spaces x with rg(x) = 1 but cb(x) > 1 and even infinite. the p-space s of [8, 4n] is scattered and of cb-index 2. there is also a p-space of infinite cb-index as follows : let ωω denote the smallest ordinal of cardinality ℵω. remove the ordinals of countable cofinality (i.e. its non ppoints) and call the resulting subspace x. then x is a scattered p-space of cb-index ω0. example 4.3. spaces with cb(x) = 2 but with rg(x) varying. these will all be achieved using the pseudocompact space ψ of [8] which is never rg because it is separable and has a zero-set which is an uncountable discrete set. the first class furnishes examples of the fact that finite regularity degree does not imply rg (see also example 8.8). the second case shows that one can have pathology even when all zero sets have discrete boundaries. first case (finite regularity degree). take any choice of a maximal almost disjoint family on n yielding a ψ which is almost compact (see [8, 6j]). such mad families exist by [17] and [22]. now βψ is scattered of cb-index 3 so βψ is rg and rg(βψ) ≤ 7. let f ∈ g(ψ). by proposition 1.1 f has a representation as a finite sum using functions from c∗(ψ). each extends to βψ so there exists f ∈ g(βψ) such that f |ψ = f. by [10, 2.12] rg(f) ≤ 7 so we also have rg(f) ≤ 7. second case (infinite regularity degree). we shall use teresawa’s theorem [22, 2.1] choosing [0, 1] as our compact metric space without isolated points, and work with a ψ for which βψ − ψ is homeomorphic to [0, 1]. let n ∈ n. by theorem 10.1 below there exists fn ∈ g[0, 1]such that rg(fn) ≥ n. since [0, 1] is compact it is g-embedded in βψ (cf. [10, 2.1 (c)] ) and there exists fn ∈ g(βψ) such that fn|[0, 1] = fn. thus rg(fn) ≥ n and rg(βψ) = ∞. now by theorem 4.1 that means that rg(ψ) = ∞. remark 4.4. (1) the previous example reveals an interesting phenomenon in the behaviour of rg(x). a space has rg(x) = 1 exactly when it is a p-space (see [10, 1.4]). so if rg(x) > 1, x has a non p-point x and one easily verifies that for every neighbourhood u of x, rg(u) > 1. one might conjecture that in general if rg(x) ≥ n then there must be a point x ∈ x such that rg(u) ≥ n for every neighbourhood u of x. but this is false. in the example above rg(ψ) = ∞, but every point x has a neighbourhood that is a one point compactification of a countable discrete set and hence of regularity degree 2. (2) note as well that the previous example gives a version of ψ for which c(ψ) has infinite krull z-dimension because this holds for c[0, 1]. 84 r. raphael and r. g. woods 5. almost-p-points and spaces x for which |c(x)| = c recall that a point of x is an almost-p-point if it does not belong to a zero-set with empty interior. we denote the set of almost-p-points of x by gx. clearly gx is the intersection of the dense cozero-sets of x. a space x is called an almost-p-space if every non-empty zero-set has a non-empty interior. these are precisely the spaces x for which gx = x. the following facts about gx are useful. the third replies to a question by m. tressl (private communication) about powers of r . proposition 5.1. (i) gx is the intersection of the dense fσ-sets of x. (ii) let u = w ∩ s, where w is open in x and s is dense in x . then gu = u ∩ gx. (iii) if x is realcompact, then gx = g(βx). in particular, if gx = ∅ and x is realcompact, then g(βx) = ∅. proof. (i) let hx denote the intersection of the dense fσ-sets of x. as each cozero-set of x is an fσ-set of x , hx ⊂ gx. conversely, suppose that p /∈ hx. then there is a dense fσ-set f of x that excludes p. by [8, 3.11(b)] there exists a zero-set z of x such that p ∈ z ⊂ x − f . thus x − z is a dense cozero-set of x that excludes p, and p /∈ gx. (ii) first suppose that u is open in x. let p ∈ u. if p /∈ gx, there exists a dense cozero-set c of x that excludes p. then u ∩ c is a dense cozero-set of u that excludes p. thus p /∈ gu, and so gu ⊂ (gx) ∩ u. conversely, suppose that p ∈ u − gu. by (i) there is a dense fσ-set f of u that excludes p. clearly there exists an fσ-set a ⊂ x such that f = u ∩ a. as f is dense in u and u is open in x , it follows that (x − u) ∪ a is a dense fσ-set of x to which p does not belong. thus p /∈ hx, and by (i) p /∈ gx. thus (gx) ∩ u ⊂ gu. next suppose that u is dense in gx. it is routine to show that gu ⊂ u ∩gx. conversely, suppose that p ∈ u ∩ gx and let h be a dense fσ-set of u. then there exists an fσ-set a of x such that h = u ∩ a. as u is dense in x, so is h and hence a. as p ∈ gx, by (i) p ∈ a. thus p ∈ u ∩ a = h. apply (i) again to conclude that p ∈ gu. in the general case, observe that w ∩s is dense in w , so by the results above we have gu = g(w ∩ s) = (w ∩ s) ∩ gw = (w ∩ s) ∩ (w ∩ gx) = u ∩ gx. (iii) that gx ⊂ g(βx) is a special case of (ii). conversely let p ∈ g(βx). as x is realcompact it is the intersection of the (necessarily dense) cozero-sets of βx that contain x (cf. [19, 5.11(c)]). thus p ∈ x. if c is a dense cozeroset of x , then as x is c∗-embedded in βx , there is a cozero-set v of βx (necessarily dense in x) for which c = v ∩ x. thus p ∈ v as p ∈ g(βx). hence p ∈ v ∩ x = c and p ∈ gx. � on rg-spaces and the regularity degree 85 theorem 5.2. let j : xδ → x be the identity map on the underlying set and let jβ : β(xδ) → βx be its stone extension. the following are equivalent: (i) x is an almost-p-space, (ii) jβ is irreducible. proof. (ii) ⇒ (i). let (i) fail. then there exists a z ∈ z(x) such that z 6= ∅ and intxz = ∅. let a = xδ − z. thus a is a proper clopen subset of xδ and therefore clβ(xδ)a is a proper compact subset of β(xδ). but j β[clβ(xδ)a] = clβx(j[a]) = clβxa. but intx(x − a) = ∅ so a is dense in x and therefore clβxa = βx. thus j β maps the proper closed subset clβ(xδ)a onto βx and therefore jβ is not irreducible, and (ii) fails. (i) ⇒ (ii). let (ii) fail. thus there is a proper closed subset k of β(xδ) such that jβ[k] = βx. since β(xδ) is 0-dimensional, there exists a proper clopen subset of β(xδ) that contains k, and this clopen subset will be of the form clβ(xδ)a where a is clopen in xδ and a 6= xδ. since xδ − a 6= ∅ and is open in xδ, there exists an s ∈ z(x) such that ∅ 6= s ⊂ xδ − a. now βx = jβ[clβ(xδ)a] = clβxj[a] = clβxa. therefore a is dense in x, and intxs = ∅. but s 6= ∅, so x is not an almost-p-space. � theorem 5.3. let x be a non-pseudocompact space with no almost-p-points. then x has a non-compact zero-set with empty interior, and hence x has non-remote points. proof. since x is not pseudocompact choose f ∈ c(x) − c∗(x), without loss of generality f ≥ 0. hence there exists a countably infinite subset d = {d(n) : n ∈ n} of x such that f(d(n + 1)) ≥ f(d(n)) + 1 for each n. let z(n) = f−1[f(d(n)) − 0.25, f(d(n)) + 0.25]. then d(n) ∈ z(n) ∈ z(x) and if n 6= k then z(n) ∩ z(k) = ∅. as x has no almost-p-points for each n ∈ n there exists a nowhere dense zero-set s(n) of x such that d(n) ∈ s(n). let t (n) = z(n)∩s(n). then t (n) is a nowhere dense zero-set of x that contains d(n), and clearly f[t (n)] ⊆ [f(d(n))−0.25, f(d(n))+0.25] (since t (n) ⊆ z(n)). let a(n) = x − f−1[(f(d(n)) − 0.3, f(d(n)) + 0.3)]. then a(n) ∈ z(x), a(n)∩t (n) = ∅, and by the choice of the d(n), if n 6= k then a(n)∪a(k) = x. hence for each n there is a gn ∈ c(x) such that a(n) = z(gn), t (n) = g −1 n (1), and 0 ≤ gn ≤ 1. now define g : x → r by: g(x) = ∑ n∈n gn(x) (1) we check whether this definition makes sense (i.e. whether g(x) ∈ r). if m ∈ n and if gm(x) 6= 0 then x /∈ a(m) so as noted above if k 6= m then x ∈ a(k); thus gk(x) = 0 if k 6= m. thus at most one term in the sum in the right hand side of (1) is non-zero, and g is well-defined. now consider the family b = {f−1[(∞, f(d(2)))]} ⋃ {f−1[[f(d(n − 1)), f(d(n + 1))]] : n = 2, 3, 4, . . .} 86 r. raphael and r. g. woods clearly b is a locally finite family of closed subsets of x, and the restriction of g to any one of them equals the restriction of no more than four different gn to that set. thus for each b ∈ b, g|b ∈ c(b) and by [8, 1a(3)], g ∈ c(x). from our construction it is clear that g−1n (1) = ⋃ {t (n) : n ∈ n}. thus d ⊆ g−1(1) and so g−1(1) is a non-compact zero-set of x. now t (n) = g−1(1)∩ f−1[(f(d(n))−0.3, f(d(n))+0.3)] so each t (n) is open in g−1(1). hence for each n ∈ n there exists an open subset w(n) of x such that w(n)∩g−1(1) = t (n). if intxg −1(1) 6= ∅ there exists some k ∈ n such that intxg −1(1) ∩ t (k) 6= ∅. then ∅ 6= intxg −1(1) ∩ w(k) ⊆ t (k), contradicting the construction of t (k). thus g−1(1) is a non-compact zero-set of x with empty interior, as required. as g−1(1) is non-compact there is a point p ∈ clβxg −1(1) − x, and as g−1(1) is nowhere dense, such a p is not a remote point of x. � remark 5.4. interestingly, a pseudocompact space with no almost-p-points and non-measurable cellularity, must also have non-remote points because terada [23] has shown that the points of υx − x cannot be remote. thus any tychonoff space of non-measurable cardinality and with no almost-p-points has non-remote points. lemma 5.5. let v be z-embedded in x, for example a cozero set of x. then |c(v )| ≤ |c(x)|. proof. in [7, 1.4, 1.5] it is shown that for any space x, |z(x)| = |z(x)|ℵ0 and |c(x)| = |z(x)|. now if v is z-embedded, the map z → z ∩ v maps z(x) onto z(v ). thus by the above |c(v )| = |z(v )| ≤ |z(x)| = |c(x)|. � lemma 5.6. let x be an rg-space. if x is the union of ℵ1 nowhere dense zero-sets, then xδ can be partitioned into ℵ1 non-empty clopen subsets. proof. assume that x = ⋃ {z(α) : α < ω1}, where each z(α) is a nowhere dense zero-set of x. inductively define a family {a(α) : α < ω} as follows: a(0) = z(0). now let β < ω1 and assume inductively that we have defined {a(α) : α < β} as follows: (a) {a(α) : α < β} is a pairwise disjoint countable collection of clopen subsets of xδ, (b) ⋃ {z(α) : α < β} = ⋃ {a(α) : α < β}, (c) if α < β then a(α) ⊆ z(α). now define a(β) to be z(β) − ⋃ [a(α) : α < β}. we see that our inductive hypotheses are satisfied for α < β + 1. thus {a(α) : α < ω1} is a partition of x into clopen subsets of xδ, each of which is a nowhere dense subset of x. let s = {α < ω1 : a(α) 6= ∅}. first note that s must be infinite as no topological space is the union of finitely many nowhere dense subsets. if s were countably infinite then x = ⋃ {z(α) : α ∈ s} which would contradict [10, 2.2]. thus |s| = ℵ1, and {a(α) : α ∈ s} is the desired partition of xδ. � theorem 5.7. suppose that x is an rg-space that is the union of ℵ1 nowhere dense zero-sets. then |c(x)| ≥ 2ℵ1. on rg-spaces and the regularity degree 87 proof. by the preceding lemma there is a partition {a(α) : α ∈ s} of x into ℵ1 pairwise disjoint non-empty clopen subsets of xδ, where |s| = ℵ1. if t ⊂ s define gt : xδ → r to be the characteristic function of ⋃ {a(α) : α ∈ t }, which is clopen in xδ. clearly t → gt is a one-to-one map from the power set of s into c(xδ), so |c(xδ)| ≥ 2 |s| = 2ℵ1. � corollary 5.8. if the continuum hypothesis holds then an rg-space x for which |c(x)| = c must have a dense set of almost-ppoints. proof. if gx is not dense in x then there is a cozero-set v of x for which v ∩ gx = ∅. hence gv = ∅ by proposition 5.1 and by the continuum hypothesis v is the union of ℵ1 nowhere dense zero-sets. as |c(v )| ≤ |c(x)| = c < 2 c = 2ℵ1, by theorem 5.7 v cannot be an rg-space. hence by [10, 2.3(f)], x cannot be an rg-space either. � remark 5.9. (1) “dense” is the best that we can do in 5.8, as the one-point compactification of n satisfies the hypotheses of 5.8 but the unique non-isolated point is not an almost-p-point. (2) another way to prove 5.8 is to note that |c(v )| = c (by the argument above) and that v is an rg-space (by [10, 2.3 (f)]) so by 5.7, v must have an almost-p-point, which will be an almost p-point of x as v is open in x. 6. results and examples involving lindelöf spaces we begin with a result that does not require any assumptions on xδ. theorem 6.1. let x = s ∪ t where s and t are complementary dense realcompact z-embedded (for example, lindelöf) suspaces of x. then x is not rg. proof. by [3, 2.6 (b)] s and t are closed and hence clopen in xδ so χs ∈ c(xδ). but since s and t are both dense in x, χs is not x-continuous at any point of x. therefore χs /∈ g(x), as functions in g(x) are continuous on dense open subsets of x ([10, 2.1 (a)] ). thus x is not rg. � the following result in the positive direction is an improvement to [10, 2.3 (d)]. theorem 6.2. suppose that x is rg, and that xδ is normal. let y ⊂ x and let y be z-embedded and realcompact. then y is rg. in particular y is rg if it is lindelöf. proof. by [3, 2.6] yδ is closed in xδ. now apply [10, 2.3 (g)]. � the following result is negative and considers disjoint dense subspaces of a space x. but we again need a constraint on xδ. theorem 6.3. let s and t be disjoint dense realcompact z-embedded (say lindelöf) subspaces of x. if xδ is normal then x is not rg. 88 r. raphael and r. g. woods proof. again s and t are disjoint and closed in xδ and by the normality of xδ there is a function f ∈ c(xδ), 0 ≤ f ≤ 1 such that f[s] = 0 and f[t ] = 1. now a = f−1[0, 1/2] ∈ z(xδ) and hence it is clopen in xδ. again χa[s] = 1, χa[t ] = 0. but s and t are dense in x so χa is not continuous at any point of x. hence as in 6.1 χa /∈ g(x), and x is not rg. � corollary 6.4. [ch] let |x| = c. then if x contains disjoint dense realcompact z-embedded subspaces then x is not rg. proof. by [15, 4] xδ is paracompact and hence normal. now apply the theorem. � we can relax the demand on xδ if we know that sδ is lindelöf. theorem 6.5. let s and t be disjoint dense subspaces of x with sδ lindelöf and t realcompact and z-embedded. then x is not rg. proof. as above, s and t are disjoint closed sets of xδ. now for all x ∈ s, there exists vx ∈ coz(xδ) such that x ∈ vx and vx∩t = ∅ (because the point x does not belong to the closed subset t of xδ). therefore sδ ⊂ { ⋃ vx : x ∈ s}. as s is a lindelöf subset of xδ, we can take a countable subcover and then take its union to get w ∈ coz(xδ) such that s ⊂ w and w ∩ t = ∅. thus w is clopen in xδ and again χw ∈ c(xδ) − g(x) so x is not rg. � one way to ensure that sδ is lindelöf is to know that s is a union of countably many scattered lindelöf spaces (cf. [15, 5.2]). so we have: corollary 6.6. let s and t be disjoint dense lindelöf subspaces of x with s the union of countably many lindelöf scattered subspaces. then x is not rg. remark 6.7. we do not know if this corollary holds if s is the union of uncountably many dense lindelöf subspaces. note that if x satisfies the conditions of the preceding corollary then x is weakly lindelöf and that there are weakly lindelöf spaces whose lindelöf subspaces are nowhere dense—for example see [2, ex. 2.2]. theorem 6.8. if x is an rg-space and if x has a dense subspace a such that aδ is lindelöf then gx is dense in x. proof. case 1. suppose, if possible, that gx = ∅. then x = ⋃ {zα : α ∈ i}, where each zα ∈ z(x) and intx[zα] = ∅. therefore aδ = ⋃ {aδ ∩zα : α ∈ i}, so as aδ is lindelöf there exist {αi.i ∈ n}, such that aδ = ⋃ {aδ∩zαi : i ∈ n}. clearly a = ∪{a ∩ zαi : i ∈ n}. we claim that for each i ∈ n, inta(a ∩ zαi) = ∅. for if not, there exists a v open in x such that ∅ 6= v ∩ a ⊂ a ∩ zαi. as zαi is closed nowhere dense in x, v − zαi is a non-empty open set of x so (v − zαi) ∩ a 6= ∅ as a is dense in x, contradicting the choice of v . thus the claim holds. now by the claim, and the fact that a ∩ zαi ∈ z(a) it follows by [10, 2.2] that a is not an rg-space, and hence by [10, 2.3 (b)] that x is not an rg-space. on rg-spaces and the regularity degree 89 case 2. suppose that gx is not dense in x. then there exists v ∈ coz(x) such that (gx) ∩ v = gv = ∅. now a ∩ v is dense in v because a is dense in x, and (a ∩ v )δ = aδ ∩ vδ. since vδ is clopen in xδ and aδ is lindelöf, (a∩v )δ is a closed subspace of a lindelöf space and hence lindelöf itself. thus by case 1 (with v replacing x and a ∩ v replacing a), we have that v is not rg and hence that x is not rg. � corollary 6.9. if x is rg and has a dense subspace that is a countable union of scattered lindelöf spaces, then gx is dense in x. theorem 6.10. if a space x is the union of countably many scattered lindelöf subspaces, then xδ is lindelöf and |c(xδ)| = |c(x)|. proof. let x = ⋃ {l(n) : n ∈ n}, where each l(n) is a scattered lindelöf space. by [15, 5.2] each (l(n))δ is lindelöf, so as the subspace topology that l(n) inherits from xδ is the same as that of (l(n))δ , it follows that xδ is the union of countably many lindelöf subspaces and hence is lindelöf. let w(y ) and wl(y ) denote respectively the weight and the weak lindelöf number of the space y . in [5] it is proved that for any space y, |c(y )| ≤ w(y )wl(y ). we apply this to xδ. since xδ is lindelöf we have wl(xδ) = ℵ0. furthermore we have w(xδ) ≤ |z(x)| = |c(x)| (the latter equality is part of [7, 1.4]). hence we have |c(xδ)| ≤ |c(x) ℵ0|. but it is well-known that |c(x)|ℵ0| = |c(x)| (see [7]). thus |c(xδ)| ≤ |c(x)|. but the opposite inequality obviously holds as xδ is just x with a stronger topology. � remark 6.11. since the comfort/ hager inequality in the proof of 6.10 uses the weak lindelöf number, one might be tempted to try to strengthen 6.10 by looking for conditions on x that would imply that xδ is weakly lindelöf (rather than lindelöf). however by [21, 5.14] a weakly lindelöf p-space is lindelöf, so no additional generality can be gained this way. 7. some properties incompatible with being rg let i : xδ → x be the identity map on the underlying set of x. theorem 7.1. x is not an rg-space if it satisfies the condition: there exist disjoint subsets c and d of β(xδ) so that (i) iβ(c) and iβ(d) are both dense in βx, (ii) for some h ∈ c(β(xδ)), h[c] = {0} and d ⊂ coz(h). proof. suppose that we have such a function h ∈ c(β(xδ)). we will show that h|x /∈ g(x), a contradiction. throughout the superscript α will denote the stone extension of a bounded function on x and the superscript γ will denote the stone extension of a bounded function on xδ. suppose, if possible, that h|x = ∑n i=1 aib ∗ i , ai, bi ∈ c ∗(x) (using part (i) of 1.1). let b = πbi. we will show that b = 0. clearly b γ = bα ◦ iβ . by computation, (h|x)b2 ∈ c∗(x) and it determines [(h|x)b2]α ∈ c(βx). now the functions [(h|x)b2]α◦iβ , (h|x)b2]γ and h(bγ)2 are all in c(β(xδ)) and they agree on xδ so they are equal. but h vanishes on c and therefore [(h|x)b 2]α 90 r. raphael and r. g. woods vanishes on iβ[c] which is dense in βx. thus [(h|x)b2]α is the zero function and therefore h(bγ)2 is the zero function on β(xδ). but h is non-zero on all elements of d, so bγ vanishes on d which implies that bα vanishes on iβ[d], so again by denseness, bα = 0 and b = 0. the above argument shows the product of all the bi equals 0. in fact, the techniques just used also show that the bi are pairwise orthogonal. for example, if di is the product of the (n − 1) bj obtained by ignoring bi, then in (h|x)d 2 i the term aibi(di) 2 vanishes and since (h|x)d2i ∈ c ∗(x) we get di = 0 by again using the fact that iβ[d] is dense in βx and d ⊂ coz(h). thus all (n − 1)fold products vanish. now continue by finite descent to get that the {bi} are orthogonal. now let t = b21 + · · · + b 2 n, and let s = (b ∗ 1) 2 + · · · + (b∗n) 2. since the {bi} are orthogonal, so are the b∗i and by computation (h|x)t ∈ c ∗(x). again the properties of h show that ht = 0. but by computation, (h|x) = (h|x)ts, so h|x = 0, which implies that h = 0 and this is false since h is non-zero at the elements of d. � remark 7.2. (1) if gx = ∅ and x is rg then there is a compact set in β(xδ) disjoint from x that contains the maximal ideals of c(β(xδ)) that correspond to the minimal prime ideals in c(x). (for each x ∈ x, there is a nonzero divisor fx ∈ c(x) that lies in mx). let lx be the (patch-open) subset of β(xδ) consisting of all prime z-ideals of c(x) that contain fx. no minimal prime lies in any lx and x ∪ lx is open in β(xδ) so one takes its complement. (2) it is easy to derive the non-vanishing of gx in 6.8 from theorem 7.1 using [8, exercise 3b.1] and remark (i). moreover the methods show that if in addition to the hypotheses of 6.8, x is also cozero-complemented and an rg-space then x must have a p-point (if not the minimal primes of c(x) form a compact subset of β(xδ) disjoint from aδ and 7.1 gives a contradiction). (3) it is interesting to compare the condition in the theorem 7.1 with smirnov’s theorem that says that a space is lindelöf if and only if it is normally placed in its stone-cech compactification. if gx = ∅ we can apply 7.1 if the compact subset of β(xδ) from part (i) is a subset of a zero set disjoint from a dense subset of x. one suspects that there are spaces that satisfy the condition in 7.1 but do not satisfy the hypotheses of 6.8. our next result uses p-spaces that are functionally countable. these were characterized by a.w. hager (cf. [15, 3.1,3.2]) and include (properly) all lindelöf p-spaces. theorem 7.3. let x be an rg space for which xδ is functionally countable. then no family of at most ℵ1 nowhere dense zero-sets of x can have x as its union. on rg-spaces and the regularity degree 91 proof. suppose that x is an rg-space that is a union of ℵ1 nowhere dense zero-sets. by lemma 5.6 there is a partition {a(α) : α < ω1} of x into ℵ1 non-empty clopen subsets of xδ. let h = {rα : α < ω1} be a subset of r of cardinality ℵ1 and define f : x → r by f[a(α)] = rα. then f ∈ c(xδ) and |f[xδ]| = ℵ1, so xδ is not functionally countable. � 8. the structure of g(x). g-embedded subspaces proposition 8.1. (g(x) as a functor). let k : x → y be continuous. then under composition k induces a (natural) ring homomorphism g(k) : g(y ) → g(x). also rg(g(k))(f) ≤ rg(f) proof. by a straightforward computation (b ◦ k)∗ = b∗ ◦ k for any b ∈ c(y ). furthermore if f ∈ g(y ), then f ◦ k ∈ g(x) and rg(f ◦ k) ≤ rg(f) as follows. let rg(f) = n, f = ∑i=n i=1 aib ∗ i , ai, bi ∈ c(x). so f ◦ k = ( ∑i=n i=1 aib ∗ i ) ◦ k =∑i=n i=1 (ai ◦ k)(b ∗ i ◦ k) = ∑i=n i=1 (ai ◦ k)(bi ◦ k) ∗. now ai ◦ k, bi ◦ k ∈ c(x) so f ◦ k ∈ g(x) and rg(f ◦ k) ≤ n = rg(f). � example 8.2. note that it is possible that k be a surjection and that rg(f ◦ k) < rg(f). the perfect irreducible surjection βn → n∗ is a simple example. more generally, suppose that x is an almost-p-space with no p-points, and let i : xδ → x be the identity map on the underlying set. we know by 5.2 that k = βi : β(xδ) → βx is a perfect irreducbile continuous surjection, a very well-behaved map. since x has no p-points, rg(x) ≥ 2 so let f ∈ g∗(x) with rg(f) ≥ 2. now fβ ∈ g(βx) and rg(fβ) ≥ 2 since rg(f) = rg(fβ|x) ≤ rg(fβ). but f ◦ i ∈ c∗(xδ) as g ∗(x) ⊂ c∗(xδ). so (f ◦ i) β ∈ c∗((βx)δ) and so rg((f ◦ i)β = 1. but fβ ◦ βi|xδ = (f ◦ i) β|xδ = f ◦ i, so f β ◦ βi = (f ◦ i)β and therefore rg(fβ◦βi) = 1 while rg(fβ) = 2. note that since βxδ is not a p-space, (its regularity degree exceeds 1), yet for all f ∈ g(x), fβ ◦ βi ∈ g(β(xδ)) and rg(fβ ◦ βi) = 1. remark 8.3. if k : x → y is a continuous surjection and s ⊂ y , then one easily has χs ◦ k = χk−1[s]. thus if s ∈ z(y ) is not open, rg(χs) = 2, while rg(χs ◦k) = rg(χk−1[s]). now if s ∈ z(y ) then k −1[s] ∈ z(x) so rg(χk−1[s]) = 1 if and only if k−1[s] is clopen in x. thus if s ∈ z(y ), rg(χs) = rg(χs ◦ k) if and only if either s and k−1[s] are both non-clopen in y and x respectively, or else s and k−1[s] are both clopen respectively in y and x. lemma 8.4. let v be a cozero-set of x and let f : v → r be a function. define f∧ to coincide with f on v and to vanish on x − v . then: (i) (f∧)∗ = (f∗)∧, (ii) if f ∈ g(v ), then f∧ ∈ g(x) and rg(f) = rg(f∧). (in particular v is g-embedded in x). proof. result (i) is readily checked. (ii). first assume that f ∈ c(v ), and let v = coz(g), g ∈ c∗(x). let k = f[1 + |f|]−1. then k ∈ c∗(v ) so by [3, 1.1] there exists h ∈ c(x) such that h[x − v ] = 0 and h|v = kg. now [1+|f|]−1 ∈ c∗(v ) so for the same reason there is a t ∈ c(x) with t[x−v ] = 0 92 r. raphael and r. g. woods and t|v = g[1 + |f|]−1. thus ht∗ ∈ g(x) and rg(ht∗) = 1. one checks that ht∗ = f∧ so rg(f∧) = 1 = rg(f) establishing (b) when f ∈ c(v ). now assume that f ∈ g(v ) and that rg(f) = n. there exist ai, bi ∈ c(v ) so that f = ∑i=n i=1 aib ∗ i . using (i) one easily verifies that f∧ = ∑ a∧i (b ∧ i ) ∗ (∗∗) by part (iv) of 1.1 and the previous paragraph each (b∧i ) ∗ is in g(x) and has regularity degree 1. similarly each a∧i is in g(x) and has regularity degree 1. thus it follows from (∗∗) that f∧ ∈ g(x) and rg(f∧) ≤ n = rg(f). so by part (ii) of proposition 1.1 rg(f∧) = rg(f). � lemma 8.5. let v ∈ coz(x) and v ⊂ t ⊂ x. then if t − v is g-embedded in x, then t is g-embedded in x. proof. let s = t − v . let f ∈ g(t ). now f|s ∈ g(s) so there exists k ∈ g(x) such that k|s = f|s. let v = coz(g). by lemma 8.4 there exists f∧ ∈ g(x) such that f∧|v = f and rg(f∧) = 1. let h = f∧gg∗ + k(1 − gg∗). then h ∈ g(x) and one easily verifies that h|t = f and t is g-embedded in x. � theorem 8.6. let x be a cozero-complemented space. then: (i) if f ∈ g(x) there is a dense cozero-set v of x such that f|v ∈ c(v ). (ii) if x is also normal and the nowhere dense zero-sets of x are rgspaces then g(x) is the set of f ∈ c(xδ) for which there exists a dense cozero-set v of x such that f|v ∈ c(v ). (iii) if x is normal, and there is an integer k such that rg(z) ≤ k for each nowhere dense zero-set z ∈ z(x) then rg(x) ≤ 2k + 1. proof. (i) let f ∈ g(x). by the proof [10, 2.1(a)] there is a finite family {z(i) : i = 1, . . . , n} of zero-sets of x such that f is continuous on ⋂ {(x − z(i)) ∪ intxz(i)}. as x is cozero-complemented, for each i there exists a cozero-set v (i) of x that is dense in intxz(i). (if intxz(i) is empty then so is v (i), but that will cause no problems.) let w(i) = (x − z(i)) ∪ v (i) and let v = ⋂ {w(i)}. then v is a dense cozero-set of x contained in ⋂ {(x − z(i)) ∪ intxz(i)} and f|v ∈ c(w) as required. (ii) suppose that f ∈ c(xδ) and there exists g ∈ c ∗(x) such that coz g is dense in x and f|coz g ∈ c(coz g). by the proof of lemma 8.4 there exists f∧ ∈ g(x) such that f∧|coz g = f|coz g and rg(f∧) = 1. by hypothesis z(g) is an rg-space, and f|z(g) ∈ c((z(g))δ), so f|z(g) ∈ g(z(g)). hence there exist n ∈ n, ai, bi ∈ c(z(g)) such that f|z(g) = ∑n 1 aib ∗ i . as x is normal, z(g) is c-embedded in x and so for each i there exist ji, ki ∈ c(x) such that ji|z(g) = ai and ki|z(g) = bi. let m = j1k1 ∗ · · · + jnkn ∗. then m ∈ g(x) and m|z(g) = f|z(g). we now claim that f = f∧gg∗ + (1 − gg∗)m, (♯) (where f∧ is as defined in 8.4). on rg-spaces and the regularity degree 93 for if x ∈ coz g then (1−gg∗)(x) = 0 and f∧gg∗(x) = f(x)g(x)/g(x) = f(x), so the right hand side of equation ♯ evaluated at x gives f(x) + 0 = f(x). if x ∈ z(g) then g∗(x) = 0 and the right hand side of equation ♯ evaluated at x gives = 0 + (1)m(x) = f(x). our claim is verified and so f ∈ g(x). (iii) let f ∈ g(x). by (i) there exists g ∈ c∗(x) such that z(g) is nowhere dense in x and f|x − z(g) ∈ c(x − z(g)). as f|z ∈ g(z), rg(f|z) ≤ k so there are ai, bi ∈ c(z) so that f|z = ∑k 1 aib ∗ i . as x is normal z is c-embedded in x so there exist a1, . . . , ak, b1, . . . , bk ∈ c(x) such that ai|z = ai and bi|z = bi. by part (ii) of 8.4 (f|x − z(g)) ∧ ∈ g(x) and rg(f|x − z(g))∧ = 1. let m = a1b ∗ 1 + · · · + akb ∗ k; then by part(ii) of 8.4 m ∈ g(x) and rg(m) ≤ k. it is routine to show that f = (f|x − z(g))∧gg∗ + (1 − gg∗)m, from which it follows that rg(f) ≤ 2k + 1. � remark 8.7. (1) example [10, 2.10] (call it x) satisfies all the hypotheses of 8.6(ii) except normality, but it does not satisfy its conclusion. (as usual, let i(x) denote the set of isolated points of x. observe that every nowhere dense set of x is discrete and hence an rg-space, and that if v ∈ coz x then i(x) − v is the complementary cozero-set.) as f|i(x) is continuous for each f ∈ c(xδ), the set occuring in part (b) of the statement of 8.6 is just c(xδ ), but this does not equal g(x) because as noted in [10, 2.10] x is not an rg-space. this shows that the hypothesis of normality cannot be dropped from 8.6(ii). (2) as zero-sets of rg-spaces are rg-spaces, one consequence of 8.6 part (a) is that if x is a normal cozero-complemented rg-space, and if f ∈ c(xδ), then there exists a dense cozero-set v of xsuch that f|v ∈ c(v ). thus if x is a normal basically disconnected rg-space and if f ∈ c(xδ), then there exists a dense cozero-set v of x such that f|v ∈ c(v ). (3) in [10, 5.6] it is shown that nowhere dense z-embedded zero-sets of quasi-p spaces are p-spaces. thus a normal cozero-complemented quasi-p space x will have rg(x) finite (see the argument in the remarks below). in particular: example 8.8. a countable nodec space of finite regularity degree. the space x presented in [9, 5.10] is a countable nowhere locally compact nodec (i.e. closed nowhere dense subspaces are discrete) extremally disconnected quasi-p space without p-points. thus it satisfies the hypotheses of 8.6 and because its nowhere dense zero-sets are rg-spaces, the function m constructed in the proof of 8.6 can be taken to be in c(x) (as the function f|z(g) will be in c(z(g))). from equation (♯) in 8.6 it follows that if f ∈ g∗(x) then rg(f) ≤ 3. as remarked in 8.6 part (ii) above, that means that there is an integer k such that rg(f) ≤ k for all f ∈ g(x). but x is not an rg-space by [10, 2.2]. this is yet another example of a space x with rg(x) finite without x being an rg-space. 94 r. raphael and r. g. woods 9. very weak p-points and rg spaces definition 9.1. a point p ∈ x is called a very weak p-point if p is not a limit point of any countable discrete subset of x. we denote the set of very weak p-points of x by vwp(x). recall that p is a weak p-point of x if p is not a limit point of any countable subset of x. weak p-points and very weak p-points were introduced (with different names) by kunen [12] who showed that the implications p-point ⇒ weak p-point ⇒ very weak p-point are strict. (see also [24]). note that each point of the space of example 8.8 is a very weak p-point, but no point is a weak p-point. the goal of this section is to show that if vwp(x) is not dense in x then x is not an rg space. theorem 9.2. let x be any tychonoff space and let s be any countable scattered subspace of x such that cb(s) is finite. then s is g-embedded in x. proof. we induct on cb(s). if cb(s) = 1 then s is discrete. if s is finite it is c-embedded and we are done. so suppose s = {s(i) : i ∈ n}. as x is tychonoff there is a pairwise disjoint family {v (i) : i ∈ n} of cozero-sets of x such that s(i) ∈ v (i) for each i. let f ∈ c(s). let v = ⋃ {v (i) : i ∈ n} and extend f to f ∈ c(v ) by letting f [v (i)] = {f(s(i))}. as v ∈ coz x by 8.4 (b) f can be extended to a member of g(x). hence s is g-embedded in x if cb(s) = 1. now suppose that t is g-embedded in x if t is any countable scattered subspace of x and if cb(t ) ≤ n. suppose that s is a countable scattered subspace of x and that cb(s) = n + 1. let l = s − i(s). then l is countable, scattered, and cb(l) = n. as i(s) is a countable open subset of s there is a cozero-set coz g of x for which s ∩ coz g = i(s). let f ∈ c(s). by the preceding paragraph there is an f ∈ g(x) such that f |s = f|s. by the induction hypothesis there is a k ∈ g(x) such that k|l = f|l. a straightworward computation shows that fgg∗ + k(1 − gg∗)|s = f. as fgg∗ + k(1 − gg∗) ∈ g(x) we are done. � the following result generalizes [10, 3.3] and also gives a shorter proof. lemma 9.3. let x be a zero-dimensional tychonoff space. let d = {d(i) : i ∈ n} be a countable discrete subset of x. let q be a limit point of d. suppose that f ∈ g(x) and that f satisfies these conditions: (i) if i ∈ n and d(i) ∈ a and a is a clopen subset of x then rg(f|a) ≥ n, (ii) there exists r > 0 such that if i ∈ n then f(d(i)) = r, (iii) there exists s > 0 such that s 6= r and f(q) = s. then rg(f) ≥ n + 1. proof. suppose to the contrary that f = h1k ∗ 1 + · · · + hnk ∗ n where the hi, ki ∈ c(x). let j = {j ∈ {1, . . . , n} : kj(q) 6= 0}. observe that j 6= ∅ or else k∗j (q) = 0 for each j ∈ {1, . . . , n} and so f(q) = 0, contradicting (c). as k ∗ j is continuous on the open set x − z(kj), if w = ⋂ {x − z(kj) : j ∈ j} then on rg-spaces and the regularity degree 95 q ∈ w and if t is defined by t = ∑ {hjk ∗ j : j ∈ j} then t is continuous on w and hence at q. let e be a clopen subset of x contained in w and containing q. let i = {1, . . . , n} − j. then |i| < n and f|e = t|e + ∑ {hjk ∗ j : j ∈ i} (1) claim 1 : f|(d ∩ e) ∪ {q} is not continuous at q. to prove claim 1, note that since e is open and contains q and since q ∈ clxd it follows that q ∈ clx(d ∩e). but f(q) = s and f[d ∩e] = r 6= s so f cannot be continuous at q. claim 2: there exists m ∈ n such that d(m) ∈ d ∩ e and f(d(m)) 6= t(d(m)). to prove claim 2, note that by the definition of j and i that kj(q) = 0 ( and hence k∗j (q) = 0) for every j ∈ i. thus f(q) = t(q). hence if our claim failed then f|(d ∩ e) ∪ {q} = t|(d ∩ e) ∪ {q}, which contradicts claim 1 since t|(d ∩ e) ∪ {q} is continuous at q by our choice of e. so, let m be as in claim 2. by (i) there exists i ∈ i such that k(i)∗(d(m)) 6= 0. as above there is a clopen subset t of e containing d(m) and such that k(i)∗ is continuous on t . by (1) this means that f|t = (t + hik ∗ i )|t + ∑ {hjk ∗ j : j ∈ i − {i}} (2) but |i−{i}| ≤ n−2 and (t+hik ∗ i )|t ∈ c(t ) so rg(f|t ) ≤ n−1, contradicting assumption 1. hence rg(f) ≥ n + 1 as claimed. � theorem 9.4. let x be a countable scattered space for which cb(x) = n. using the notation of [10, 3.2], define fn : x → r by: fn[ii(x)] = {i + 1}, i = 0, . . . , n − 1). then: (i) fn ∈ g(x). (ii) if x ∈ dn−1(x) and a is a clopen set containing x then rg(fn|a) ≥ n. in particular rg(x) ≥ n. proof. induct on n. if n = 1 the result is trivial. suppose that fk satisfies (i) and (ii) for each countable scattered space x for which cb(x) ≤ k. let y be a countable scattered space for which cb(y ) = k + 1. then dk(y ) 6= ∅ and dk+1(y ) = ∅ (see [10, 3.2]). evidently cb(y − dk(y )) = k and fk+1|y − dk(y ) = fk. hence by the induction hypotheses fk+1|y − dk(y ) ∈ g(y −dk(y )). also fk+1|dk(y ) ∈ g(dk(y )) as dk(y ) is discrete. thus by 9.2 there exist functions f, g ∈ g(y ) such that f |y − dk(y ) = fk+1|x − dk(y ) and g|dk(y ) = fk+1|dk(y ). as dk(y ) is closed in the countable space y there is a cozero-set coz g of y for which y − coz g = dk(y ). then fk+1 = fgg∗ + g(1 − gg∗) and so fk+1 ∈ g(y ). this, together with our induction hypotheses, shows that fk+1 satisfies the hypotheses of 9.3. hence by 9.3 rg(fk+1) ≥ k + 1. it remains to show that if p ∈ dk(y ) and if a is any clopen subset of x that contains p, then rg(fk+1|a) ≥ k + 1. but for such an 96 r. raphael and r. g. woods a, cb(a) = k + 1 and fk+1|a has exactly the same properties (re a) as fk+1 has. so the proof that rg(fk+1) ≥ k+1 also shows that rg(fk+1|a) ≥ k+1. � lemma 9.5. let s be a space for which vwp(s) = ∅. then for each n, s has a countable scattered subspace s(n) with cb(s(n)) = n. proof. the argument is by induction. suppose that we have such a space s(k) for the integer k. let i(s(k)) = {dn : n ∈ n}. since the set {dn} is discrete, there exist pairwise disjoint cozero-sets {vn} in x with dn = vn∩s(k). by hypothesis dn is the limit of a countable discrete set t (n) and if we let h(n) = vn ∩ t (n) then dn is a limit point of the countable discrete set h(n). let s(k + 1) = s(k) ∪ ( ⋃ h(n), n ∈ n). one verifies that i(s(k + 1)) = s(k + 1) − s(k) and s(k + 1) is countable scattered of cb-index k + 1. � corollary 9.6. suppose that x is a space whose countable subspaces are scattered, and suppose that vwp(x) is empty. then for each n ∈ n there is a countable scattered subspace s(n) such that cb(s(n)) = n. theorem 9.7. let x be a space with a subspace s such that vwp(s) is empty. then : (i) rg(x) = ∞, (ii) x is not an rg-space. proof. (i) as s has no isolated points it has a countable discrete subset {d(n) : n ∈ n}. as x is tychonoff there exists a pairwise disjoint family {v (n) : n ∈ n} of cozero-sets of x with d(n) ∈ v (n). let n ∈ n. now vwp(v (n) ∩ s) = v (n) ∩ vwp(s) = ∅ so by 9.5 v (n) ∩ s has a countable scattered subspace s(n) for which cb(s(n)) = n. by 9.4 there exists f(n) ∈ g(s(n)) such that rg(f(n)) ≥ n. by 9.2 there exists k(n) ∈ g(x) such that k(n)|s(n) = f(n). thus rg(k(n)) ≥ rg(f(n)). it follows that rg(x) = ∞. (ii) define h : x → r by: h|v (n) = k(n)|v (n) and h[x − ⋃ {v (n) : n ∈ n}] = 0, each set in question being clopen in xδ. since kn|v (n) ∈ g(v (n)) the restriction of h to each set in the partition is continuous so h ∈ c(xδ). but rg(h|v (n)) = rg(kn|v (n)) ≥ n for each n so h /∈ g(x) and x is not an rg-space. � corollary 9.8. if the set of very weak p-points of x is not dense in x then rg(x) = ∞ and x is not an rg-space. proof. if vwp(x) is not dense there exists a cozero-set v of x disjoint from it. it follows that vwp(v ) = ∅. by 9.7 rg(v ) = ∞ and v is not an rg space. since v is g-embedded in x, rg(x) = ∞, and x is not an rg-space since v is not one by [10, 2.3(f)]. � corollary 9.9. if x has a first countable subspace without isolated points then rg(x) = ∞ and x is not an rg-space. proof. if s is first countable without isolated points then vwp(s) = ∅. now apply the theorem. � on rg-spaces and the regularity degree 97 10. regularity degree for compact and related spaces we now apply the above methods to some other spaces. we now use 9.4 to characterize compact (and related) spaces of infinite regularity degree. (our original proof was different and used the fact that a compact space without p-points contains a compact zero set with an infinite boundary). theorem 10.1. let k be a compact space that is not scattered. then for each n, there exists a countable scattered subspace s(n) ⊂ k such that cb(s(n)) = n. consequently rg(k) = ∞ and the krull z-dimension of c(k) is infinite. proof. since k is not scattered it has a compact subspace a without isolated points. as in [15, 3.1], there is a compact subspace l ⊂ a without isolated points that maps continuously onto the cantor set c. if rg(l) = ∞ then rg(k) = ∞ because l is c∗-embedded in k. thus it suffices to assume that there exists a continuous surjection f : k → c. although the following is likely folklore, we will now in detail construct a family f of countably many non-empty clopen subsets of c which, when partially ordered by inclusion, forms a tree of height n with the property (*) for all f ∈ f, |{g ∈ f : g ⊂ f}| = ℵ0. let s = (s1, . . . , sk) ∈ n k be an ordered k-tuple of positive integers. let |s| denote the number of components in s. if s = (s1, . . . , sk) ∈ n k and if j ∈ n we will denote the element (s1, . . . , sk, j) ∈ n k+1 by s + j. if s is an initial sequence of t (i.e. t = (s1, . . . , sn, a, b, . . .)) we will write s ≤ t. (by convention s ≤ t if s = t). now we construct f. let {cs : |s| ∈ n 1} be a pairwise disjoint family of clopen non-empty subsets of c. if s ∈ n1 let {cs+j : j ∈ n}, be a pairwise disjoint family of clopen non-empty subsets of cs. now let k < n and suppose that we have defined {cs : s ∈ n m, 1 ≤ m ≤ k} with the following properties: (1) for all m ∈ {1, . . . , k − 1}, for all s ∈ nm, {cs+j : j ∈ n} is a pairwise disjoint family of clopen subsets of c. (2) if |s| < m ≤ k and j ∈ n, then cs+j ⊆ cs. now define {cs+j : |s| = k, j ∈ n} as follows. the set {cs+j : j ∈ ω} is a pairwise disjoint family of non-empty clopen subsets of cs. then (1) and (2) are satisfied when k is replaced by k + 1 and f = {cs : 1 ≤ s ≤ n} is the required family. we have a continuous surjection f : k → c. then {f−1[f ] : f ∈ f} = {c ′ s : s ∈ ⋃m=n m=1 n m} (where c ′ s = f −1[cs]) satisfies (1) and (2) when cs is replaced by c ′ s and c is replaced by k. now we define s(n) inductively from the “bottom up”. (1) if |s| = n choose d(s) ∈ cs. if |s| = n − 1 , let d(s) be a limit point of {d(s + j), j ∈ n}. (there will be such a limit point as k is compact and {d(s) : |s| = n} is discrete by (1)). 98 r. raphael and r. g. woods if we have chosen {d(s) : s ∈ ⋃n i=m+1 n i} and if s ∈ nm, choose d(s) to be a limit point of {d(s + j) : j ∈ n}. let s(n) = {d(s) : s ∈ ⋃n i=1 n i}. then i(s(n)) = {d(s) : |s| = n}, i(s(n) − i(s(n))) = {d(s) : |s| = n − 1} and so on. we see that cb(s(n)) = n and that s(n) is countable and scattered. the fact that rg(k) is infinite now follows from 9.2 and 9.4. the fact that k is of infinite krull z-dimension also follows from theorem 3.6. � corollary 10.2. let k be a compact space. then the following are equivalent: (i) k is scattered and cb(k) < ∞, (ii) rg(k) < ∞, (iii) k is an rg space. proof. the equivalence (i) ⇔ (ii) is in [10, 3.4] as is the implication (iii) ⇒ (ii). suppose (ii). then k is scattered by the above theorem, and cb(k) is finite by theorem 9.4 so k is rg by [10, 2.12]. thus (ii) ⇒ (iii). � corollary 10.3. any space containing a compact space that is not rg is of infinite regularity degree and of infinite krull z-dimension. in particular this applies to a locally compact space that is not scattered. proof. the first assertion is immediate. for the second, suppose that x is locally compact and not scattered. then its cantor-bendixon kernel a is nonempty, locally compact, and has no isolated points. local compactness shows that a has a compact subset k that has no isolated points. thus k is not rg, and the conclusion follows. � the following is immediate from our theorem and theorem 4.1. corollary 10.4. any pseudocompact space without isolated points is of infinite regularity degree and of infinite krull z-dimension. corollary 10.5. let k be a metric space. then the following are equivalent: (i) k is scattered and cb(k) < ∞, (ii) rg(k) < ∞, (iii) k is an rg space. proof. theorem [10, 3.4] shows that (i) ⇔ (iii) and that (iii) ⇒ (ii). to see that (ii) ⇒ (i) observe that, with (ii), k is scattered by part (i) of theorem 9.7, and cb(k) is finite by theorem [10, 3.3]. � remark 10.6. the example before theorem 9.2 and [10, 2.4] show that we cannot generalize theorem 10.1 to lindelöf spaces. the space ψ is always scattered locally compact, of finite cb-index, and never rg so there is no possibility of a generalizing 10.2 from compact to locally compact spaces. 11. does rg imply finite regularity degree? by [10] and our work above we already have a positive answer to the above question in the pseudocompact, metric, scattered lindelöf, and scattered perfectly normal cases. we now give two other instances where the answer is on rg-spaces and the regularity degree 99 positive. since we don’t know whether perfectly normal rg-spaces are scattered, the first result is pertinent. theorem 11.1. let x be perfectly normal and rg. then rg(x) is finite. proof. let f ∈ g(x). by [10, 2.1 (a)] there is a dense open subset of x on which f is continuous. let v0 be the union of the open subsets of x = x0 on which f is continuous. then v0 is a dense cozero set of x0 whose complement is the nowhere dense zero-set z1 = x0 − v0. since z1 is perfectly normal and also rg (by [10, 2.3(f)]) the process can be repeated— f|z1 is continuous on a dense cozero set v1 of z1 with complement z2 nowhere dense in z1. the process can be continued to get subsets zn with dense cozero sets vn. since the zn are c-embedded in x, by theorem 3.6 the process must stop with an n for which zn+1 = ∅. furthermore by theorem 3.4 the number n is global—it works for all f ∈ g(x). now we need a representation for f. we know that x is the union of the vn and that f is continuous on each of them. suppose that vn = coz(gn). we can assume that each gn ∈ c(x) because each zn is c-embedded in x. one readily verifies that f = fg0g ∗ 0 + ∑j=n j=1 [π k=j−1 k=0 (1 − gkg ∗ k)]gjg ∗ j f. since it uses a number of terms that depends only on n, rg(x) is finite. � proposition 11.2. suppose that x is an rg-space. then x must be of finite regularity degree, if it contains countably many pairwise disjoint subspaces yn each homeomorphic to x with the property that each yn is clopen in xδ. proof. assume if possible that rg(x) = ∞. as each yn is homeomorphic to x, there exists for each n, a function fn ∈ g(yn) such that rg(fn) ≥ n. now define f : xδ → r by f |yn = fn, f |[x − ∪yn] = 0. since each yn is clopen in xδ (as are countable unions of clopen sets), f ∈ c(xδ) by [8, 1a]. since x is rg, we have a representation f = ∑i=k i=1 aib ∗ i , ai, bi ∈ c(x). but if one restricts this representation of f to yn one sees that the constant k globally bounds the regularity degrees of the {fn}, and this is not possible. � corollary 11.3. if x is rg and rg(x) = ∞, then the free union of ω copies of x is not rg. corollary 11.4. suppose that x is an rg-space of countable pseudocharacter. then x must be of finite regularity degree, if it contains countably many disjoint subspaces yn each homeomorphic to x. remark 11.5. it is natural to question the restrictiveness of the hypothesis of proposition 11.2. if such subspaces yn exist then there is a constraint on the p-space xδ—it must have countably many pairwise disjoint clopen subsets each a copy of itself. this demand holds in some but certainly not all p-spaces. for example, under martin’s axiom βn − n will have a dense set of p-points, and if one takes a maximal family of pairwise disjoint clopen subsets {aα} in βn − n, then the set of p-points in their union has this property since it is the free union of c pairwise disjoint clopen copies of itself. there are many 100 r. raphael and r. g. woods similar p-spaces (cf. [6]) where the constraint holds. it clearly fails if xδ is the one-point lindelöfization of an uncountable discrete set. 12. miscellaneous theorem 12.1. (i) tfae for a subspace y of a space x: (a) if f ∈ g(x) then f|y ∈ c(y ) (b) if v ∈ coz x then v ∩ y is clopen in y. (ii) if y satisfies the (equivalent) conditions of (1) and y is z-embedded in x then y is a p-space. proof. to show that (a) implies (b) let v ∈ coz x. then the characteristic function χv of v belongs to g(x). now χv |y = χv ∩y so by (a) χv ∩y ∈ c(y ). this immediately implies (b). to show that (b) implies (a), let f ∈ g(x).then f = g1h ∗ 1 + · · · + gnh ∗ n, gi, hi ∈ c(x). to show that f|y ∈ c(y ) it clearly suffices to show that h ∗ i |y ∈ c(y ) for each i. but h∗i |coz(hi) ∈ c(coz(hi)) so h ∗ i |y ∩coz(hi) ∈ c(y )∩coz(hi). similarly h∗i |z(hi) ∈ c(z(hi)) (as h ∗ i [z(hi)] = 0) so h ∗ i |y ∩ z(hi) ∈ c(y ∩ z(hi)). thus (by (b)) h ∗ i |y is continuous on each of complementary clopen subsets of y . hence h∗i |y ∈ c(y ) and we are done. if y is z-embedded in x then by (b) every cozero-set of y is clopen and so y is a p-space. � 13. open questions (1) if x is rg, is rg(x) finite? 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[25] r. wiegand, modules over universal regular rings, pac. j. math. 39, (1971), 807–819. received july 2004 accepted march 2005 r. raphael (raphael@alcor.concordia.ca) mathematics and statistics, concordia university, montréal, canada, h4b 1r6. r. g. woods (rgwoods@cc.umanitoba.ca) mathematics, university of manitoba, winnipeg, canada, r3t 2n2. @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 327–350 groups with a small set of generators dikran dikranjan, umberto marconi and roberto moresco ∗ dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. following [22] we study the class s of all groups that admit a small set of generators. here we adopt also another notion of smallness (p-small) introduced by prodanov in the case of abelian groups. we push further some results obtained in [22] (by adding some new members of s) and partially resolve an open question posed in [22]. we show that in most cases the groups in s admit a p-small set of generators. 2000 ams classification: 20b30, 20f16, 20k45, 22c05, 54h11. keywords: group, large set, small set, permutation group, linear group, compact group, profinite group. 1. introduction. the question of measuring the size of a set of generators of a group is certainly a relevant one. in the case of topological group one puts topological restrictions on the set of generators to ensure smallness (see [19, 20] for the so called “suitable sets” – “small” sets of generators born in the theory of cohomology of infinite galois groups in the work of tate and douady [8]). in the case of discrete groups the following notion of smallness was proved to be a very useful property in this respect in [22]. a subset b of a group g is large if g = f ·b = b ·f for some finite set f of g. this property has been largely studied in the literature also under different names (big, discretely syndetic, relatively dense). a set s ⊆ g is small if for every finite set f the sets s ·f and f ·s have a large complement in g [2, 3] (clearly, only infinite groups may have small sets). the role of small and large ∗the first author was partially supported by research grant of the italian murst in the framework of the project “nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani” 2000. the second and third author were supported by the grant “progetti di ricerca di ateneo, 2001” of the university of padova. 328 d. dikranjan, u. marconi, r. moresco subsets of groups in number theory, compact representations of groups and dynamics can hardly be overestimated [7, 13, 14]. the question when an infinite group may have a small set of generators was addressed in [22]. let us denote by s the class of groups with small set of generators. it was proved in [22] that s contains all groups that have an infinite abelian normal subgroup (in particular, all groups with infinite center) as well as all solvable groups. call a subset s of an abelian group g small in the sense of prodanov (briefly, p-small) if there exist x1, . . . ,xn . . . in g such that the sets {s + xn}n are pairwise disjoint. it is easy to see that when s − s is not large, then s is p-small. this was the motivation for the introduction of p-small sets in [23]. it was noticed by gusso [15] that p-small sets of the abelian groups are small. their advantage is also that they are much easier to understand and construct. the main contributions of the present paper go in two directions. in §2.1 we define approriate versions of p-smallness in non-abelian groups, as well as other versions of smallness that turn out to be stronger than smallness or p-smallness in many cases. we show that in most of the cases the small generated groups from [22] have a set of generators satisfying also a much stronger property of smallness. on the other hand, we add to the list of groups in s found in [22] some new classes of non-abelian groups with a small set of generators proceding in two ways. in the case of permutation groups and linear groups our arguments are purely algebric. we offer also another approach to the problem that heavily leans on topology. this allows us to produce a wealth of small and p-small sets and to provide many examples of compact-like topological groups that belong to s. finally, we partially answer a question from [22] by showing that s contains all compact groups with eventual exception of the topologically finitely generated pro-p groups and the profinite groups with trivial frattini subgroup. 1.1. notation and terminology. we denote by n and p the sets of natural and prime numbers, respectively; by z the integers, by q the rationals, by r the reals, by t the unit circle group r/z, by z(p) the cyclic group of order p and by zp the p-adic integers (p ∈ p). the cardinality of continuum 2ω will be denoted by c. if s is a set, we denote by p(s) its power set. let g be a group. we denote by 1 the neutral element of g and by z(g) the center of g. for a subset s of g we denote by 〈s〉 the subgroup generated by s. the group g is divisible if for every n ∈ n and g ∈ g there exists x ∈ g with xn = g. the semidirect product of the groups g and k is denoted by gok. abelian groups are mostly written additively. in particular, 0 denotes the neutral element of an additively written abelian group g and for a ⊆ g and n ∈ n we let a(n) = a + · · · + a︸︷︷︸ n when n > 0 and a(n) = {0} for completeness. topological groups are hausdorff. a topological group g is precompact if its completion is compact, pseudocompact if every continuous real-valued function on g is bounded. for a topological group g we denote by c(g) the connected groups with a small set of generators 329 component of the identity and by ψ(g) the pseudocharacter g (the minimum cardinality of a set u of neighborhoods of 1 such that ⋂ u∈u u = {1}). unless explicitly stated, all groups are assumed to be infinite. if x is a topological space and a ⊆ x, the closure of a is denoted by a. for undefined symbols or notions see [7], [9], [12], or [17]. we denote by s(g) and sp (g) the collections of all small and p-small sets of a group g respectively. 2. the various levels of smallness. let us give the following more precise form of largeness. definition 2.1. let g be a group. a subset b of g is left large (resp. right large) if for some finite set f the union f ·b (resp. b ·f) of left (resp. right) translates of b covers g. the following trivial equalities are helpful when passing to complements: {g ∈ g : g ·f 6⊆ a} = (g\a)·f−1, {g ∈ g : g ·f 6⊆ g\a} = a·f−1 (1) clearly, they imply that g\a is right large iff there exists a finite f such that a contains no left translate gf of f. or, g \ a is not right large iff a contains left translates gf of every finite set f. definition 2.2. a subset s of a group g is called: (a) left small if for every finite set f the sets s · f and f · s have a left large complement; right small is defined analogously. (b) weakly left small if for every finite set f the set s ·f has a left large complement; weakly right small is defined analogously. (c) n-small, for a positive n ∈ n, if s is small and the sets (s−1 · s)n−1 and (s ·s−1)n−1 are not large. (d) microscopic if for every n ∈ n the sets (s−1 · s)n and (s · s−1)n are small. a set is small iff it is left small and right small. clearly the 1-small sets are precisely the small ones. the 2-small sets are those small sets s such that s−1 ·s and s ·s−1 are not large. for n > 1 we denote by sn(g) the family of n-small sets of g. in additive notation, s ∈ sn+1(g) for an abelian group g and n > 0 iff (s −s)(n) = s(n) −s(n) is not large, i.e., s(n) ∈ s2(g) (indeed, 2-small implies p-small which in turn implies small by [15]). the equalities (1) give the following useful criterion mentioned in [1] and [22] in the bilateral version of smallness: lemma 2.3. a set s is weakly left small (left small) iff for every finite set f there exists a finite k such that the set s · f contains no right translate kg (and the set f ·s contains no right translate kg). 330 d. dikranjan, u. marconi, r. moresco 2.1. left and right smallness in the sense of prodanov. call a set s of a group g: (a) right small in the sense of prodanov (briefly right p-small) if there exist x1, . . . ,xn . . . in g such that the sets {s ·xn}n are pairwise disjoint (or, equivalently, xn ·x−1m 6∈ s−1 ·s for n 6= m). (b) left small in the sense of prodanov (briefly left p-small) if there exist x1, . . . ,xn . . . in g such that the sets {xn ·s}n are pairwise disjoint (or, equivalently, x−1m ·xn 6∈ s ·s−1 for n 6= m). (c) strongly right p-small if there exist x1, . . . ,xn . . . in g such that the sets {sf ·xn}n are pairwise disjoint for every f ∈ g (or, equivalently, xn · x−1m 6∈ (s−1 · s)f for n 6= m); strongly left p-small is defined analogously. (d) (strongly) p-small if it is (strongly) left and (strongly) right p-small. it is easy to see that s is left p-small (strongly left p-small) if and only if s−1 is right p-small (strongly right p-small). here we give separately the smallnes conditions in terms of the difference sets s ·s−1 and s−1 ·s. claim 2.4. let s be a subset of a group g. then: (al) s is left p -small iff there exists an infinite set x such that (s ·s−1) ∩ (x ·x−1) = {1}; (ar) s is right p -small iff there exists an infinite set x such that (s−1 ·s) ∩ (x−1 ·x) = {1}; (b) s is strongly left p -small iff there exists an infinite set x such that (∀f ∈ g)(s ·s−1)f ∩ (x ·x−1) = {1}. consequently, each one of the properties weakly left (right) small, left (right) small, left (right) p -small, p -small, n-small, microscopic, is invariant under left and right translations. it is clear that strongly p-small implies p-small (by taking f = 1 in the definitions (c) and (d) above). in the non-abelian case, p-small need not imply small (cf. example 2.17). the final part of the claim shows that microscopic implies small. remark 2.5. (1) assume that s is right large. then s ·f = g for some finite f. then for every infinite set x one of the sets sf (f ∈ f) contains infinitely many members x1 of x. in particular, there exist x,y ∈ x with x 6= y and x,y ∈ sf. this gives xy−1 ∈ x ·x−1∩s·s−1. thus x ·x−1 ∩s ·s−1 6= {1} for every infinite x. (more precisely, for every infinite x there exists an infinite x1 ⊆ x such that x1 ·x−11 ⊆ s · s−1). hence s is not left p-small. therefore left p-small sets cannot be right large (but can be left large, cf. example 2.18). the same conclusion may be obtained by using (a) in the next lemma 2.6. groups with a small set of generators 331 (2) if s−1 · s contains a finite index subgroup h, then s is not right psmall. indeed, if x ⊆ g is an infinite set, then some coset ah will contain an infinite subset x1 of x, hence x −1 1 ·x1 ⊆ h ⊆ s −1 ·s, so s is nor right p-small by (ar) of the preceding claim. in the next lemma we give some connection between these notions of smallnes. note that s−1 · s is symmetric, so all three versions of “large” coincide for s−1 ·s. lemma 2.6. let s be a subset of a group g. (a) if s is left (right) p -small, then it is also weakly left (right) small. (b) if s−1 ·s is not large then s is right p -small, if s ·s−1 is not large, then s is left p -small, (c) if s is 2-small then it is p -small. (d) if s is strongly left p -small, then it is left small. (e) if s is strongly p -small, then it is small. (f) if s is microscopic, then it is n-small for every n ∈ n. in particular, microscopic sets are 2-small. if g is abelian, then the first implication can be reversed. proof. (a) there exist x1, . . . ,xn, . . . in g such that x−1m · xn 6∈ s · s−1 for n 6= m. let f be a finite subset of g. to see that s · f has a left large complement choose n such that |f| < n and take k = {x−11 , . . . ,x −1 n }. then s ·f contains no right translates k ·g of k. indeed, assume that k ·g ⊆ s ·f for some g ∈ g. then there exist xi 6= xj in k such that for some f ∈ f one can find s,s1 ∈ s with x−1i g = sf and x −1 j g = s1f. the second equation gives g−1·xj = f−1·s−11 . multiplying this by the first equation we get x −1 i xj = ss −1 1 . hence x−1i xj ∈ s · s −1, that leads to contradiction. analogously one proves that right p-small implies weakly right small. (b) indeed, there exists x1, . . . ,xn . . . in g such that xn 6∈ s−1 ·s · {x1, . . . ,xn−1}, so that xn ·x−1m 6∈ s−1 ·s for m < n. since s−1 ·s is symmetric, we have also xm ·x−1n 6∈ s−1 ·s. thus s is right p-small. one can see analogously, that s is left p-small whenever s ·s−1 is not large. (c) follows from (b). (d) to prove that s is left small take any finite set f. we can assume without loss of generality that 1 ∈ f. we have to find a finite set k such that s ·f contains no right translates k · g and f ·s contains no right translates k · g. the former property was already checked above in (a). to check the latter one note that by assumption there exist x1, . . . ,xn, . . . in g such that x−1m · xn 6∈ sf · (s−1)f for n 6= m and for every f ∈ f. take n > |f| and k = {x−11 , . . . ,x −1 n }. assume that k · g ⊆ f ·s for some g ∈ g. then there exist xi 6= xj in k such that for some f ∈ f one can find t,t1 ∈ s with x−1i g = ft and x −1 j g = ft1. the second equation gives g −1xj = t −1 1 · f −1. 332 d. dikranjan, u. marconi, r. moresco multiplying this with the first equation we get x−1i xj = ftt −1 1 f −1. hence x−1i xj ∈ s f · (s−1)f , that leads to contradiction. (e) follows directly from (d). (f) it suffices to observe that s is small (indeed s−1 ·s contains a translated set of s). for the reverse implication in the abelian case, it suffices to apply (a) and (b) to the set (s −s)(n). � in the next diagram we give all the implications that are always valid between the various symmetric smallness properties we have introduced so far. microscopic . . . n-small . . . 2-small � ��3 q qqs (∗) small p-small q qqk � ��+ (+) strongly p-small the arrow (+) becomes an equivalence for abelian groups, so that the diagram becomes linear with p-small=strongly p-small placed between 2-small and small. the inverse arrow of (∗) fails in two ways: a p-small set need not be small (as mentioned above) and s · s−1, s−1 · s need not be large for a p-small set s (an example for the simultaneous failure of these implications is given in 2.17). the following question remains open even in the abelian case: problem 2.7. do the notions 2-small and strongly p -small coincide? now we produce a series of examples of small sets on z that are not p -small. example 2.8. (a) let pr denote the set of all products of at most r odd primes and let ar = pr ∪−pr. it was proved in [1] that the set pr is small for every r ∈ n, hence ar is small as well. brun proved in 1919 that p9 + p9 contains all sufficiently large even integers. this was improved by rademacher in 1924 who brought r down to 7 and selberg improved it to r = 3 in 1950. according to goldbach’s conjecture p1 + p1 contains all even integers n with |n| > 4. in these terms, the set ar−ar contains the subgroup of all even integers, hence it is surely large for r ≥ 3 (actually, ar is not p-small by remark 2.5) (2)). this shows that if goldbach’s conjecture has positive answer, then the set a1 of all odd primes (positive and negative) is small but not p-small. obviously, to the same result leads the positive answer to the (unsolved) conjugate goldbach’s conjecture (every even integer is a difference of two odd primes). (b) let f(x) = ax2 + bx + c, with a,b,c,∈ z, a 6= 0. then sf = {f(n) : n ∈ z} is small, but not p-small. since these properties are invariant under translation, we can assume without loss of generality that c = 0. we shall see that sf − sf always contains a non-zero subgroup h, so that it is not p-small by remark 2.5. indeed, if b 6= 0, then for every n ∈ z one has 2bn = f(n) − f(−n) ∈ sf − sf , so h = 2bz ⊆ sf − sf . if b = 0, then 4an = f(n + 1)−f(n−1) ∈ sf −sf , so h = 4az ⊆ sf −sf . furthermore sf is small because limn→+∞(f(n + 1) −f(n)) = +∞ [1]. groups with a small set of generators 333 2.2. smallness vs topology, measure, asymptotical density and growth. it was proved in [2, proposition 3.7] that every compact subset of a non-compact topological group is small. in fact, one can prove the following much stronger property. lemma 2.9. a compact set k of a non-compact topological group is always microscopic. proof. indeed, k−1 is compact as well. hence k−1·k and k·k−1 are compact as continuous images under multiplication of the compact spaces k−1×k and k ×k−1. for every n ∈ n and for every finite set f the sets f · (k ·k−1)n and f · (k−1 ·k)n are still compact, hence small by [2, proposition 3.7]. � we shall apply this lemma very often when g is countable, so surely noncompact. also we shall have often k a converging sequence. this is why we make use of the following modified version of a notion introduced by protasov and zelenyuk [29] (the original version was for g abelian and h replaced by g). definition 2.10. a sequence a1,a2, . . . ,an, . . . in an infinite group g is a tsequence if the subgroup h generated by the set {a1,a2, . . . ,an, . . .} admits a hausdorff group topology such that an → 1. since the underlying set of every convergent sequence, along with its limit, is a compact set, we get corollary 2.11. the underlying set of a t -sequence {an}∞n=1 of an infinite group g is a microscopic set. recall, that a banach measure on a group g is a finitely additive invariant measure on g such that every subset of g is measurable and µ(g) = 1 (see [10, 11] for the existence of banach measure on all abelian groups and the existence of groups that admit no banach measure). iv. prodanov noted that banach measures are a very convenient tool when dealing with large sets. indeed, if µ is a right invariant banach measure on the group g, then obviously every right large set has a positive banach measure, while every right p-small set has measure zero (hence a right p-small set is never right large in such a group). therefore, an infinite group g that admits a right invariant banach measure is never a finite union of right p-small sets (see [27] or [7, p. 29] for an elementary proof of this fact in the abelian case that makes no recourse to banach measures). it is proved in [1, theorem 3.4] that every compact group g admits closed small sets a of positive haar measure arbitrarily close to 1 hence small sets need not be neither left nor right p-small. (in this case one can apply also steinhouse-weil theorem to conclude that a−1 · a has non-empty interior, so a−1 ·a is large.) example 2.12. here we give examples of small sets in z with various levels of smallness. 334 d. dikranjan, u. marconi, r. moresco (1) let f(x) ∈ z[x] be polynomial of degree d > 0. then the set sf = {f(n) : n ∈ z} is large if d = 1. it follows from the criterion given in [1] that sf is small if d > 1. according to example 2.8 sf −sf is large if d ≤ 2. as sf−sf ⊇ sg, where g(x) = f(x+1)−f(x) (so deg g = d−1), it is easy to see that in general (sf−sf )(m) = (sf )(m)−(sf )(m) is large if m ≥ 2d−2. hence, sf is never microscopic. the above implications cannot be inverted for arbitrary polynomials f (for f(x) = x2d −x the set sf −sf contains all even integers, hence it is large), but seemingly this can be done for monomials f(x) = axd (e.g., sf is 2-small iff d > 2, etc.). in this way one can construct n-small sets of z that are not n + 1-small for every n ∈ n. (2) for every n > 1 let rn be the rest of n2 modulo 2k, where 2k is the greatest power of 2 with 2k ≤ n. then s = ±{n2 − rn : n > 1} is microscopic by the above corollary (as n2 − rn converges to 0 in the 2-adic topology of z) even if the function defining s grows slower than the polynomial function n 7→ n2. remark 2.13. it will be desireable to understand better the various kinds of small sets of z. here we propose two more points of view. (a) one can connect smallness or largeness of subsets of z with density (for an infinite set a ⊆ z let the density of a be the limit limn |a∩[−n,n]| 2n , whenever it exists). clearly, every large set has positive density (cf. [1, proposition 1.13]). one can build examples of small sets of positive density ([1, proposition 1.14]). on the other hand, by schnilermann’s approach, for every set a with positive density a(n) = z for n sufficiently large. hence a set of positive density cannot be microscopic. we do not know whether such a set can be n-small for some n > 1. (b) another approach to the smallness of a set a = {an} of positive integers is to study the asymptotic behavior of the ratio an+1 an . there exist t-sequences {an} in z such that limn an+1 an = 1 (cf. example 2.12 (2), note that n2 ≥ an = n2 − rn ≥ n2 − n by the choice rn < n, so limn an+1 an = 1 holds true). hence there exists a microscopic set a = {an} in z such that the ratio an+1 an converges to 1. it is known that when limn an+1 an = ∞ or the limit is a transcendental real number, then {an} is a t-sequence [29], while every algebraic number α ≥ 1 admits a sequence {an} with limn an+1 an = α that is not a t-sequence [29]. we do not know whether the condition limn an+1 an = α > 1 may imply that a is microscopic, strongly small or p-small (it yields a is small as an+1 − an → ∞ when α > 1, so that [1, proposition 1.2] applies). if yes, then we obtain new examples of small (microscopic) sets that are not the underlying set of a t-sequence. 2.3. smallness of subgroups and transversals. let us recall that a left transversal of a subgroup h of a group g is a subset t of g such that g = t ·h and (t−1 ·t) ∩h = {1}. (2) groups with a small set of generators 335 lemma 2.14. let h be a subgroup of g and let t be a left transversal of h. then: (a) if h is infinite, then t is right p -small (or, equivalently, t−1 is left p -small); (b) if t is infinite, then h is p -small; moreover, if h is small, then h is microscopic. (c) if (t−1 ·t)∩hf = {1} for every f ∈ g, then h is microscopic if t is infinite (and t is strongly right p -small if h is infinite). proof. (a) if h is infinite, then (2) implies that t is right p-small since every infinite subset {xn} of h will witness right p-smallness. this yields t−1 is left p-small. (b) the part (t−1 ·t) ∩h = {1} of (2), in view of h = h−1 ·h, witnesses h is right p-small. if t1 is a right transversal of h, then analogous argument proves that h is left p-small. since h−1 = h, in both cases h is p-small. clearly, h is not large when t is infinite. since h is a subgroup, this implies that h is microscopic, if h is small. (c) for t 6= t1 in t and every f ∈ g the cosets thf and t1hf are disjoint by hypothesis. then h is strongly left p-small by definition. since h is a subgroup, this implies that h is strongly p-small, hence small (by lemma 2.6). as t is infinite, h must be microscopic, according to item (b). � item (a) cannot be inverted (cf. example 2.18). corollary 2.15. let g be a group and h,d be infinite subgroups of g. (a) if d ∩h = {1}, then both h and d are p -small. (b) if d ∩ hf = {1}, for every f ∈ g then both h and d are strongly p -small and microscopic. corollary 2.16. let h be a normal subgroup of a group g. (a) if h has infinite index, then h is strongly p -small and microscopic. (b) if h is infinite, then any transversal t of h is strongly p -small (hence, small). (c) if h is infinite and has infinite index, then h ∪ t is small for any transversal t of h. proof. (a) to see that h is strongly left p-small fix a left transversal t of h. then it is also a right transversal. for t 6= t1 in t the cosets ht = th and ht1 = t1h are disjoint. moreover, hf = h for every f ∈ f, so that hft and hft1 remain disjoint for every f ∈ f and t 6= t1 in t , i.e., (t−1 ·t)∩h = {1} and (t · t−1) ∩ h = {1} for every f ∈ g. therefore, h is strongly right p-small by lemma 2.14. since h is a subgroup, this implies that h is strongly p-small, hence small. therefore, h is also microscopic. (b) to see that t is strongly left p-small let f be a finite set of g. take any countably infinite subset {xn} of h. then for every f ∈ f the sets tfxn are pairwise disjoint. indeed, if z ∈ tfxn ∩tfxm, then zx−1n = tf and zx−1m = t f 1 336 d. dikranjan, u. marconi, r. moresco for some t,t1 ∈ t . now xmx−1n = (t f 1 ) −1tf = (t−11 t) f . since xmx−1n ∈ h and h is normal, we conclude that t−11 t ∈ h. this yields t = t1 and n = m. (c) follows from (a) and (b). � item (a) implies in particular that h is small – this conclusion was obtained in [2, prop. 1.7]. 2.4. examples distinguishing left/right largness and smallness. it is known [2] that the union of two small sets is again small. we show in the next example (item (b) inspired by [15, example 3.2.7]) that in general the union of two p-small sets need not be neither p-small nor left or right small. example 2.17. let g be the free group of two generators a,b. (a) let s be the set of reduced words that start with a (non-trivial) power of a, and let s′ be the set of reduced words that start with a (nontrivial) power of b. then s and s′, as well as s′′ = s′ ∪{1}, are left p-small, hence weakly left small too. as s and s′′ are complementary, this yields that s and s′′ are left large (as complements of weakly left small sets). therefore, none of them is left small. on the other hand, g = s∪s′′ is large. so in the non-abelian case an infinite group can be the union of two left p-small sets. finally, neither s nor s′′ are right large, consequently, neither s nor s′′ are weakly right small, neither right p-small. hence a weakly left small (actually, left p-small) set need not be weakly right small. (b) let ya,a be the set of all words of g starting and ending by a non-trvial power of a. define analogously yb,b,ya,b and yb,a. clearly y −1a,b = yb,a, (yb,b)a ⊆ yb,a (ya,a)b ⊆ ya,b (ya,b)a ⊆ ya,a and b(ya,b) ⊆ yb,b. (3) all these sets are p-small. indeed, it suffices to check that one of them is p-small and then apply (3) and the fact that the automorphism f : g → g exchanging a and b sends ya,a to yb,b. analogously, one can see that either all four sets are small, or none of them is small. as g is the union of the five sets {1}, ya,a,yb,b,ya,b and yb,a, one of them is not small. since {1} is small it follows from the above argument that none of the sets ya,a,yb,b,ya,b and yb,a is small. the union ya,a ∪ya,b coincides with the set s defined in (a), so it is not even left small. since s−1 = ya,a∪yb,a is not right small, we see that the union of two p-small sets need not be neither left nor right small. moreover, as s is not right p-small, we conclude that the union of two p-small sets need not be p-small. since the set yb,b is weakly left and right small (being p-small), its complement g\yb,b is large. hence the p-small set a = ya,a fails to be small and has a−1a = aa−1 large (this witnesses the strong failure of the reverse implication of the arrow (∗) in the diagram). groups with a small set of generators 337 example 2.18. (see also [27]) a right large set which is right p -small and not left large. let g be the free product of a cyclic group a = 〈a〉 of order two and an infinite cyclic group c = 〈b〉. every element of g\c can be uniquely written as a product w = bn0 ·a · bn1 ·a · bn2 ·a · . . . ·a · bnk−1 ·a · bnk, (3) where k > 0 and n0, . . . ,nk are integers, such that if k > 1, then n1, . . . ,nk−1 are non-zero integers, while the integers n0 and nk may also have value 0. obviously, the elements w ∈ c can be obtained in the form (3) with k = 0. one refers to (1) as to the reduced form of the element w ∈ g. the product w1 ·w2 of two words w1 and w2 can be brought to a reduced form after a finite number of cancelations. let y = {w ∈ g : k > 0 and nk = 0} be the set of all reduced words that end with a and let x = g\y. then y = xa, so that x = y a too. thus, both x and y are right large, since g = x ∪xa = y ∪y a. consequently, neither x nor y are right small. let us see now that neither x nor y are left large. since y = xa, it suffices to see that x is not left large. let us first note that the inverse of an element w as in (3) is given by w−1 = b−nk ·a · b−nk−1 ·a · b−nk−2 ·a · . . . ·a · b−n1 ·a · b−n0. assume that g = ⋃s i=1 gix for some g1, . . . ,gs ∈ g. there exist n 6= m such that bna ∈ gix and bma ∈ gix. then for some x ∈ x one has bna = gix. hence gi = bnax−1. now x = bn0 ·abn1abn2a.. .abnk−1abnk with either x = 1 or nk 6= 0. in both cases the leading term of the reduced form of the word gi is bna.. .. analogous argument shows that the leading term of the reduced form of the word gi is bma.. ., a contradiction. since y is not left large, x is not left small. indeed, if f is any finite set that contains 1 ∈ g, then the complement of f ·x is contained in y , hence cannot be left large. analogous argument shows that y is not left small. therefore we have the following properties: x and y are right large and are neither left large nor left small. let us note that the right large set y is a left transversal of a and right p-small (as all y bn are paiwise disjoint), hence weakly right small too. nevertheless, a is finite (this shows that the implication in lemma 2.14 (a) cannot be inverted). clearly right large sets cannot be even weakly left small. a more careful analysis of example 2.18 shows that the set x has the following curious property: for every w ∈ g there exists a finite subset f of g such that w · x ⊆ x ∪ f. (indeed, let w have the form (1) and assume that nk 6= 0. then f = {w · b−nk,w · b−nk ·a · b−nk−1, . . . ,bn0 ·a} works.) in general, call a subset x of a group g with the above property left absorbing. obviously, every left absorbing set x with infinite complement is not left large. clearly, every cofinite set is both left absorbing and large. so it is reasonable to exclude the cofinite sets to obtain: 338 d. dikranjan, u. marconi, r. moresco lemma 2.19. every non-cofinite left absorbing set in an infinite group g is not left large. proof. it suffices to note that for a left absorbing set x and every finite subset f of g the set f · x can add at most finitely many new points to x, hence cannot cover g when x is not cofinite. � admittedly, the worst failour of coincidence of left and right largeness is presented by the right large sets that are left absorbing. if g admits a left invariant banach measure, then every left large set has positive measure. the free product considered above does not allow a banach measure as it contains a copy of the free group of two generators (see [følner]), but still the set x has a rather strange behaviour: it has “measure 1/2 from right” (in the sense that it contains, roughly, half of the elements of the group), on the other hand it is left absorbing. 2.5. permanence properties of small sets. we collect here some permanence properties of large and small sets related to homomorphisms. according to the next lemma from [16, proposition 1.2] images and counterimages preserve largeness of sets. lemma 2.20. let f : g → g1 be a surjective homomorphism and x be a (left, right) large subset of g. then f(x) is (left, right) large in g1. a subset y of g1 is (left, right) large iff f−1(x) is (left, right) large in g. according to the above lemma the image of a large set cannot be small. on the other hand, trivial examples show that the image of a small set can be large. now we show that inverse images preserve smallness. lemma 2.21. let f : g → g1 be a surjective homomorphism and x be a subset of g1. then x has one of the properties (left, right) small, weakly left (right) small, (left, right) p -small, strongly (left, right) p -small, n-small, microscopic, iff f−1(x) has the same property. proof. assume that s = f−1(x) is left small. it suffices to show that for every finite set f of g the sets f(f) · x and x · f(f) have left large complements in g1. by assumption, f ·s has left large complement. thus f(g\f ·s) is large in g1 by lemma 2.20. since f(g \ f · s) ⊆ g1 \ f(f) · x this proves that g1 \ f(f) · x is left large. analogously we show that x · f(f) has a left large complement in g1. thus x is left small. if x is left small, then g\f ·f−1(x) is left large for every finite f ⊆ g, containing the left large set f−1(g1 \f(f) ·x). analogous proof works for “right small”. the conjunction of these two properties gives the counterpart for “small”. assume that for the infinite set y = {yn} of g1 the sets xyn are pairwise disjoint for yn ∈ g1. let zn ∈ g satisfy f(zn) = yn for every n. then the sets f−1(x)yn are pairwise disjoint. hence f−1(x) is right p-small whenever x is right p-small. similar proof works for “right p-small”. the preservation of remaining versions of smallness may be proved in a similar way. � groups with a small set of generators 339 remark 2.22. gusso [16] showed that “smallness” has the following “transitivity” property: if h is a subgroup of g and s is a small subset of h, then s is small in g. obviously, left and right p-smallness, as well as n-smallness (hence microscopic), have this property. finally, if s ⊆ h is not left (right) large in h, then it is not left (right) large in g. 3. the class s of small generated groups. we start with some direct consequences of corollary 2.16 including for readers convenience also some results from [22, corollary 5]: lemma 3.1. an infinite group has a small set of generators in each of the following cases: (a) if g admits an infinite normal subgroup of infinite index; (b) if g is a semi-direct product of two infinite groups; (c) if g is a restricted direct product of infinitely many non-trivial groups; (d) if g is generated by a finite family of subgroups h1, . . . ,hn ∈s. proof. (a) follows from item (c) of corollary 2.16, (b) follows from (a), (c) follows from (b). for (d) remark 2.22 should be applied. � remark 3.2. if n and h are infinite groups, then the set s = h ∪ n in g = n ×h is small as a union of two small sets, but it is not p-small. indeed, s−1 = s and s2 = g, thus s is neither left nor right p-small (this was proved in the more general case of a normal subgroup n of infinite index of g in [15]). nevetheless, we show in lemma 3.3 (a) that the group g has a p-small set of generators whenever either h or n have this property. let us introduce the classes: (a) sp of all groups that admit a strongly p-small set of generators; (b) sp of all groups that admit a set of generators that is both small and p-small; (c) sn of all groups that admit a n-small set of generators; (d) m of all groups that admit a microscopic set of generators. clearly, sp ⊆ s and sp ⊆ s. as 2-small sets are both small and p-small, we have the following inclusions m⊆ . . . ⊆sn ⊆ . . . ⊆s2 ⊆sp ⊆s ⊇sp . 3.1. some general properties of s, sp , sn, sp and m. lemma 3.3. let g be a group. (a) if g admits a quotient g/n ∈ s (resp. sp , sp , sn, m), then g has the same property; (b) if there exists is a normal subgroup n of g with n ∈s (resp. n ∈sp ), then g has the same property. 340 d. dikranjan, u. marconi, r. moresco proof. (a) by lemma 2.21, g/n ∈ s implies g ∈ s since the inverse image of a small set of generators is a small set of generators. analogous argument shows that if some quotient g/n ∈sp , then g ∈sp , etc. (b) assume that s is a small set of generators of the normal subgroup n of g. this yields that n is infinite. then by lemma 2.16 any transversal t to h is small (see also [22, proposition 3]). then s ∪t is a small set of g. clearly this is a set of generators of g. we prove now that n ∈ sp implies g ∈ sp . let s be a p-small set of generators of n. choose any transversal t0 of n and put t = t0 \n. then the set of generators s1 = s ∪t of g is p-small. indeed, if x ⊆ n witnesses left p-smallness of s, then for n 6= m xns1 ∩xms1 = (xns ∪xnt) ∩ (xms ∪xmt) = (xns ∩xmt) ∪ (xnt ∩xms) ∪ (xnt ∩xmt) = ∅, as xnt ∩xmt = ∅ since t is a transversal of n, xms ∩xnt = ∅ and xns ∩ xmt = ∅ as xms ⊆ n and xns ⊆ n, while xnt ∩n = ∅ and xmt ∩n = ∅. this proves that s1 is left p-small. right p-smallness of s1 can be established in the same way. if in addition we assume that s is small in n, then by remark 2.22 it is also small in g, so that the union s1 is small too. � here we give a topological proof of a result similar to another one (regarding small sets) already known in the case of infinite abstract groups (cf. [2, 3]). theorem 3.4. every non-discrete topological group g admits an infinite microscopic set. proof. if g is uncountable, just take any countable subgroup of g. assume now that g is countable. by a theorem of arhangel′ski (cf. [7, theorem 7.5.3]) the group g admits coarser metrizable group topology τ that in the given case must be non-discrete. let sn → 1 be a non-trivial converging sequence in (g,τ). then the set k = {1}∪{sn : n ∈ n} is compact, so by lemma 2.9 k is microscopic. � there exist infinite groups that admit no hausdorff group topology beyond the discrete one; let us call them markov groups. the question of whether markov groups exist was raised by markov in the early forties, and answered positively by shelah [25]. he constructed (under ch) a markov group g of size ω1 that is also a kurosch group (i.e., all proper subgroups of g are countable; an example of a countable markov group was given in the same year by ol′shankii without any additional set-theoretic assumptions.). call a markov group hereditary markov group, if every infinite subgroup of g is a markov group. it is not known whether such groups exist. clearly, the above theorem works for all abstract groups g that have a countable subgroup admitting a non-discrete hausdorff group topology, i.e., for non-hereditarily-markov groups. we prove a new version of the theorem from [22] that establishes the existence of small sets of generators for some groups. here we claim less (since left p-small sets need not be small), but we gain in generality since no restrictions groups with a small set of generators 341 are needed on the group. in the case of abelian groups the results obviously coincide. in this case we get also a sharper result (cf. corollary 3.6). theorem 3.5. let g be an infinite group. then (a) g has a set of generators x that is a union of a small set and a left (right) p -small set; (b) if g is not hereditarily markov, then x can be chosen to be the union of a microscopic set and a left (right) p -small set. proof. let s be an infinite small subset of g (it exists by [2]). fix a right transversal set tr of 〈s〉 and a left transversal set tl of 〈s〉. by lemma 2.14 tr is left p-small (hence weakly left small by lemma 2.6). analogously, tl is right p-small. clearly, x = s ∪tr as well as s ∪tl generate g. in case g is not hereditarily markov, then g has a countable subgroup h admitting a non-discrete hausdorff group topology. now h has a microscopic set s by theorem 3.4 and s is microscopic also as a subset of g. now the argument goes as before. � theorem 3.4 implies that every infinite abelian group admits an infinite microscopic set. now the compact sets can be obtained also by embeddings in tω. the next theorem offers a more precise result and improves [22, theorem 9]. theorem 3.6. every infinite abelian group has an infinite microscopic set of generators. proof. first we prove the assertion for the infinite abelian groups g which admit a surjective homomorphism onto some prüfer group z(p∞). by lemma 2.21, it suffices to check that z(p∞) ∈m. for this purpose, notice that the sequence s = {1/pn : n ∈ n} converges to 0 in z(p∞) equipped with the topology induced by the circle group t. hence s is a microscopic set in z(p∞) (by corollary 2.11) and s is a set of generators of z(p∞). for the remaining groups, according to a result from [6], a group g that admits no surjective homomorphism g → z(p∞) has finite rank n and for every copy n ∼= zn in g one has g/n ∼= ⊕ p(fp ×bp), where fp is a finite p-group and bp is a bounded p-group. it is clear now that a group g with this property is either finitely generated or admits a surjective homomorphism onto a group of the form h = ⊕∞ n=1 ci, where all groups ci are finite cyclic. to see that h has a microscopic set of generators let ci be a generator of ci and let s = {ci}i∈i. it suffices to observe that ci → 0 in the tychonov topology of h, when each group ci is equipped with the discrete topology. now corollary 2.11 and lemma 2.21 apply. it remains only to note that if g is finitely generated it suffices to consider the case g = z (see (a) in lemma 3.3). now one takes the set s = {2n}∞n=0. this set is microscopic as 2n → 0 in the 2-adic topology of z (cf. corollary 2.11). � recall that the derived series g(n) of a group g is defined by: g(0) = g and g(n+1) is the commutator group of g(n). a group g is solvable if g(n) = {1} 342 d. dikranjan, u. marconi, r. moresco for some integer n, g is perfect if g = g′. the group g is hyperabelian if g contains no perfect subgroups beyond the trivial one. proposition 3.7. (a) if g/g′ is infinite then g has a set of generators that is microscopic and strongly p -small; (b) all free groups belong to sp ∩m. (c) g ∈s2 if g is an infinite solvable group. (d) g ∈s2 when g/g(n) is infinite for some n. (e) g ∈sp if g has an infinite abelian normal subgroup (in particular, if g has infinite center). proof. (a) follows from lemma 3.3 and theorem 3.6 (for abelian groups, microscopic implies strongly p-small). (b) follows from (a) and lemma 3.3 since f/f ′ is infinite for a free group f. (c) pick the smallest k such that the factor g(k)/g(k+1) is infinite (since g is infinite such a k must exist). then by (a) the (infinite) subgroup g(k) has a microscopic and strongly p-small set of generators (recall that microscopic implies 2-small). to see that this yields g ∈ s2 we need the following property. if n is a normal subgroup of a group g with n ∈ s2, then |g/n| < ∞ implies g ∈ s2. indeed, let s be a 2-small set of generators of n. chose a finite transversal f of n. then the set s1 = s ∪ f is 2-small. indeed, it suffices to see that if s is 2-small, i.e., s ∈ s(n) and s−1 · s, s · s−1 are not large in n then also for any x ∈ g the union s1 = s ∪{x} is 2-small in g. now s1 ·s−11 = (s ·s −1) ∪sx−1 ∪xs−1 ∪{1}. now sx−1 ∪xs−1 ∪{1} is a small set, and s ·s−1 is not large (by remark 2.22), therefore their union s1 ·s−11 cannot be large (as the difference between a large set and a small one is a large set [2, theorem 1.4]). (d) the quotient g/g(n) is an infinite solvable group, so (c) and lemma 3.3 apply. (e) follows from (a) of lemma 3.3 and theorem 3.6. � remark 3.8. (i) for the class s (b) and (e) were proved in [22, proposition 6] and [22, prop. 10] respectively. (ii) in the proof of (c) above, it is proved implicitly that the union of a 2small set and a finite set is still 2-small. we do not know if this property holds for microscopic sets as well (this holds true in abelian groups, but the above proof needs a non-abelian version of the property). (iii) a more careful analysis of the proof shows that every solvable group admits a skinny set of generators, i.e., a finite union of strongly left (or right) p-small sets, that are surely small. note that there are small sets that are not skinny, since skinny sets have banach measure zero, while small sets need not have this property. it follows from (e) that aoaut(a) ∈s when a is an infinite abelian group (since a is a normal subgroup of a o aut(a)). groups with a small set of generators 343 3.2. linear groups and permutation groups. according to proposition 3.7 (e) the linear group gln(k) has a small set of generators for every infinite field k and n ≥ 1 (its center is infinite). on the other hand, smaller linear groups (as the one of all upper (resp., lower) triangular matrices in gln(k)) are solvable, hence again have small sets of generators (by proposition 3.7). the following two instances cannot be obtained directly from that proposition. theorem 3.9. the group g = sln(a) has a small set of generators for every infinite euclidean domain a and n > 1. proof. let t+ (t−) denote the subgroup of all upper (resp., lower) triangular matrices in g = sln(k) and let d denote the subgroup of all diagonal matrices in g. then one can easily check that d, t− and t+, along with the finite set {π1, . . . ,πs} of all matrices in g, having a single non-zero entry equal to ±1 on each row and column, generate the group g. now the subgroups t− and t+ are solvable, hence small generated. the same holds for the abelian group d. hence lemma 3.1 applies to conclude g ∈s. � example 3.10. for some euclidean domains a (e.g., a = z) the groups sln(a) are actually finitely generated. this holds true when the additive group (a, +) is torsion-free and finitely generated (so isomorphic to zm for some m ∈ n). indeed, for every i 6= j and for every generator b of the additive group (a, +) consider the matrix αbij = in + beij, where eij is the matrix with only one non-zero entry 1 placed at position ij. then the matrices αbij along with the matrices πi generate sln(a). for a short proof by induction note that starting with an arbitrary matrix ξ ∈ sln(a) after appropriate permutation of the rows and columns (achieved by multiplication by various πi) and multiplication by matrices αbij one can arrange to obtain from ξ a matrix (aij) having the entry a11 6= 0 with minimal δ(a11), where δ denotes the euclidean norm in a, all other entries on the first row or column are zero (as ξ ∈ sln(a), the entry a11 is necessarily invertible). now the inductive hypothesis applies to the minor obtained by removing the first row and the first column. theorem 3.11. let x be an infinite set, let s(x) be the group of all permutations of x and let sω(x) be the subgroup of s(x) of all permutations of finite support. then s(x) ∈ sp and sω(x) has a set of generators that is microscopic and strongly p -small. proof. first we prove that the group g = sω(x) has a microscopic strongly psmall set of generators. obviously the set t of all transpositions of g generates g. we show that this set is microscopic and strongly p-small. indeed, for any subset a ⊆ x denote by ga the subgroup of all permutations with support contained in a. since the set t = t−1 is symmetric and invariant under conjugation, to check that t is strongly p-small it suffices to check that it is right p-small. indeed, s = t · t = t−1 · t = t · t−1 is not large, as every element of s is either a 2-cycle, or a 3-cycle or a product of two disjoint 344 d. dikranjan, u. marconi, r. moresco 2-cycles. hence g 6= s ·gf for any finite f ⊆ x. now we can conclude with remark 2.5 that t is strongly left p-small and strongly right p-small, hence t is small. since an analogous argument shows that sn is not large for every n ∈ n we conclude that t is also microscopic. the normal subgroup sω(x) of s(x) belongs to sp , so lemma 3.3 implies s(x) ∈ sp . for a direct alternative proof of s(x) ∈ s, one can apply lemma 3.1 since g is an infinite normal subgroup of s(x) of infinite index (any permutation of infinite order of x will witness it). � the famous theorem of graham higman, bernhard neumann and hanna neumann [18] says that every countable group is isomorphic to a subgroup of a 2-generated group. in this spirit we have: theorem 3.12. every group is a subgroup of a small generated group and a quotient of an m-generated group. proof. let g be an infinite group. it suffices to embed g in the group s(g) of all permutations of the set g. for the second assertion it suffices to write g as a quotient of a free group and apply lemma 3.1 � it is still an open question whether s coincides with the class g of all groups [22]. it follows from the above theorem that s = g is equivalent to either of the following invariance properties of s: (i) stability under taking subgroups; (ii) stability under taking quotients. another open question from [22] is whether every uncountable group g admits a small set of size |g|. if this is true for all groups of size ω1, then shelah’s group g ([25], let us recall that all proper subgroups of g are countable) admits a small uncountable set s, then s will generate g, so that g will be small generated. in the next section we study the question when g ∈ s for groups g that admit a non-discrete group topology (close to being compact). 3.3. some (compact-like) topological groups have small set of generators. it was shown in [22] that all non-metrizable compact groups belong to s and it was asked whether the class s contains all compact groups. here we obtain further progress in this direction. theorem 3.13. s contains all compact groups that are not totally disconnected. proof. let g be a compact group that is not totally disconnected. then c(g) 6= {1}, hence it is an infinite normal subgroup of g. if it has infinite index, then g ∈ s by lemma 3.1. hence it remains to consider the case when c(g) has finite index. in this case it suffices to prove c(g) ∈ s, this will imply g ∈ s. hence it suffices to prove that every compact connected group belongs to s. let us see first that every connected compact lie group g belongs to s. indeed, let t be a maximal torus of g. then g has a finite number of borel subgroups b1, . . . ,bn containing t and they generate g ([21, §25.2, corollary groups with a small set of generators 345 b]). since the subgroups bi are solvable, we can apply lemma 3.1 and conclude g ∈s. we prove now that every compact connected group g belongs to s. if g is abelian then we apply theorem 3.6. otherwise, the quotient g/z(g) is a product of simple connected lie groups [26], [19, th. 9.24]. in particular, some non-trivial quotient of g is a compact connected lie group, so that we can apply the first part of the argument and lemma 3.3. � following the above proof one can expect that the above theorem extends to all locally compact groups. indeed, it suffices to prove that every connected locally compact group belongs to s. since such groups are projectively lie groups (i.e., inverse limits of lie groups), it suffices to prove that every simple lie group belongs to s. this makes us believe that this extension to the locally compact case is possible. as far as compact groups are concerned, the above theorem reduces the problem to totally disconnected compact groups, i.e., profinite groups. let us recall, that profinite (pro-p) groups are inverse limits of finite (finite p-torsion) groups. a topological group g is said to be topologically finitely generated if it has a dense finitely generated subgroup. we have no proof at hand for the following conjecture (see remark 2.5 for a detailed comment). conjecture 3.14. every topologically finitely generated pro-p group belongs to s for every prime number p. the frattini subgroup φ(g) of a profinite group g is the intersection of all maximal open subgroups of g. it is easy to see that φ(g) is a closed normal subgroup of g. conjecture 3.15. every profinite group with trivial frattini subgroup belongs to s. since non-metrizable profinite groups already belong to s, to verify this conjecture one should consider only metrizable profinite groups g with φ(g) = {1}. then there will be countably many maximal open subgroups mn. for every maximal open subgroup m the largest normal subgroup mg of g contained in m is still open in g. thus we obtain a countable family nn of open normal subgroups, such that each one is an intersection of maximal open subgroups, and ⋂ n nn = {1}. since g is compact, this implies that the open normal subgroups un = ⋂n k=1 nk form a local base at 1. let us see now that a positive answer to these two conjecture would imply that every compact group is small generated. theorem 3.16. if s contains all topologically finitely generated pro-p groups for every prime p and all profinite group with trivial frattini subgroup, then all compact groups belong to s. proof. let g be a compact group. if g is not totally disconnected theorem 3.13 applies. hence from now on we suppose that g is totally disconnected, 346 d. dikranjan, u. marconi, r. moresco i.e., profinite. then its frattini subgroup φ(g) is pronilpotent ([24, corollary 2.8.4]), hence isomorphic to the direct produt ∏ p gp of its p-sylow subgroups gp ([24, proposition 2.3.8]). if the subgroup φ(g) of g has infinite index, then consider the quotient g1 = g/φ(g) that has φ(g1) = {1} (cf. [24, proposition 2.8.2 (a)]), so g1 ∈s by conjecture 3.15 and g ∈s by lemma 3.1. therefore, from now on we assume that the subgroup φ(g) has finite index, i.e., φ(g) is open. hence it suffices to prove that φ(g) ∈s. if at least two of the groups gp are infinite, then we are done by lemma 3.1. if infinitely many groups gp are non-zero then again lemma 3.1 works. so it remains the case when precisely one of the groups gp is infinite and there exists finitely many non-trivial groups gq (with q 6= p) and all of them are finite. in other words, gp is open in g. hence we can assume without loss of generality that g is a pro-p-group. then by [24, lemma 2.8.7 (b)] the quotient group g/φ(g) is an abelian group, hence g/φ(g) ∈s in case it is infinite. therefore, we can assume from now on that g/φ(g) is finite, i.e., φ(g) is open in g. then g is topologically finitely generated by [24, proposition 2.8.10], so that conjecture 3.14 applies now. � according to this theorem, if there exists a compact group without a small set of generators, then there exists also a profinite metrizable group g with this property, that is either a topologically finitely generated pro-p group, or has φ(g) = {1}. remark 3.17. let us discuss here our grounds to believe that conjecture 3.14 is true. when g is a topologically finitely generated pro-p group, the topology of g coincides with its pro-finite topology (that has as typical neighborhoods of 1 all subgroups of finite index of g), since by serre’s theorem [28, §4.3] every finite index subgroup of g is open. moreover, the frattini series {φn(g)}n∈n (defined inductively by φ1(g) = φ(g) and φn+1(g) = φ(φn(g)) for n ≥ 1) forms a fundamental system of neighborhoods of 1 in g for the pro-finite topology of g. taking into account that φ(g) = gpg′, where gp = {xp : x ∈ g} (cf. [24, lemma 2.8.7 (c)]), this describes completely the topological group g in purely algebraic terms. it was proved by e. zel′manov [30] (see also [24, theorem 4.8.5c]) that the torsion finitely generated pro-p groups are finite. so our conjecture holds true for torsion groups. on the other hand, it holds true also for all profinite groups of finite rank (a profinite group g is said to be of finite rank d if every closed subgroup of g has at most d topological generators, [28, chap. 8]). indeed, it is known that every profinite groups of finite rank g admits closed normal subgroups c ≤ n such that n has finite index in g, c is nilpotent and n/c is solvable. now it is clear that g ∈ s when c is infinite (as then c ∈s as a solvable group so that lemma 3.1 applies). when c is finite, then n and n/c are necessarily infinite, so that n/c ∈ s (by proposition 3.7) and consequently g/c ∈ s. now lemma 3.1 applies again. hence the conjecture remains to be proved only for the topologically finitely generated pro-p groups of infinite rank. it should be noted that unlike the finite p-groups, the pro-p groups may have a trivial center even in the case of very nice groups. for example, if k1 groups with a small set of generators 347 denotes the subgroup ( 1 + pzp pzp pzp 1 + pzp ) of the linear group gl2(zp), then the profinite group g = sl2(zp)∩k1 has trivial center. nevertheless, one can prove explicitly that this group belongs to s. indeed, one has to observe that the subgroups t + and t− of upper and lower triangular matrices, respectively, in g and the finite set of all permutation matrices generate g. then one notes also that both groups t + and t− are soluble, so that lemma 3.1 applies. the topologically finitely generated free pro-p groups belong to m ⊆ s as they have infinite abelian quotients. a subset a of a topological group is said to be (totally) bounded if for every non-empty open set u of g there exists a finite set f such that a ⊆ f ·u, a is said to be σ-bounded if a is a countable union of bounded subsets of g. proposition 3.18. if g is a σ-bounded topological group, then g admits a closed normal subgroup n of infinite index with ψ(g/n) ≤ ω. if ψ(g) > ω, then n must be infinite and g is small generated. proof. if g is discrete, then the conclusion is obvious. otherwise, since the union of finitely many bounded sets is again a bounded set, there exists an increasing chain k1 ⊆ k2 ⊆ . . . ⊆ kn ⊆ . . . of bounded sets of g such that g = ⋃∞ n=1 kn. we shall use in the sequel the following property of a bounded set k: for every neighborhood v of 1 there exists a symmetric neighborhood u of 1 such that ux ⊆ v for every x ∈ k. let u0 6= g be a symmetric open neighborhood of 1. define inductively a decreasing chain u0 ⊇ u1 ⊇ . . . ⊇ un ⊇ . . . (1) of symmetric neighborhoods of 1 such that for each n (a) u2n ⊂ un−1 and the inclusion is proper; (b) uxn ⊆ un−1 for every x ∈ kn. note that a) yields un ⊆ un−1. therefore the set n = ⋂ n un is a closed subgroup of g. let x ∈ g. then there exists n ∈ n such that x ∈ kn. since n = ⋂ m≥n um, one can easily see that n x ≤ n. therefore n is a normal subgroup of g. let us show now that the subgroup n has infinite index. indeed, let f : g → g/n be the canonical quotient map. since all inclusions in (a) are chosen to be proper, then also all inclusions f(un) ⊃ f(un+1) are proper as f(un) = f(un+1) yields un ⊆ n · un+1 ⊆ u2n+1, a contradiction. therefore g/n is infinite and obviously ψ(g/n) ≤ ω. finally, n finite would immediately imply ψ(g) ≤ ω. consequently, n is infinite and g is small generated by lemma 3.1 (a). � remark 3.19. for an alternative argument in the case of a non-metrizable locally compact σ-compact group g see [22]. note that every group with those properties satisfies also the hypothesis of our proposition as locally compact non-metrizable groups necessarily have uncountable pseudocharacter. the local compactness was exploited in [22] to get metrizability of the quotient g/n that in general has countable pseudocharacter, as the above proposition shows. 348 d. dikranjan, u. marconi, r. moresco a topological group g is totally minimal if every continous surjective homomorphism with domain g is open. theorem 3.20. a topological group g ∈s in each of the following eight cases: (a) g is σ-bounded with ψ(g) > ω; (b) g is σ-compact with ψ(g) > ω; (c) g is sin-group with ψ(g) > ω; (d) g is precompact with ψ(g) > ω; (e) ([22, theorem 14]) g is compact and non-metrizable; (f) g is pseudocompact and non-compact; (g) g is totally minimal, precompact and non-metrizable; (h) g is not totally disconnected and c(g) has infinite index (in particular, if it is not open in g). proof. (a) by proposition 3.18 we get an infinite normal subgroup n of infinite index. now lemma 3.1 gives g ∈s. (b) every compact set is bounded, thus (a) applies. (c) let us recall that a sin-group has (by definition) a base of invariant neighborhoods of 1 (so that sin stands for small invariant neighborhoods). now the proof of proposition 3.18 can be carried out in this case to get a normal subgroup n of infinite index with ψ(g/n) ≤ ω. now the assumption ψ(g) > ω yields n is infinite. (d) since every precompact group is a sin-group, we can apply (c). (e) since every non-metrizable compact group has uncountable pseudocharacter, we can apply (d). (f) every pseudocompact group is precompact ([5]) and every pseudocompact group of countable pseudocharacter is compact [4]. therefore g is precompact of uncountable pseudocharacter, so that (d) applies. (g) indeed, if g is compact then this follows from (e). now assume that g is not compact. then the completion k of g is a compact non-metrizable group. then k has a closed normal subgroup n of infinite index with ψ(k/n) = ω by proposition 3.18. since k/n is compact, we get χ(k/n) = ψ(k/n) = ω, so that k/n is metrizable. therefore, n is infinite as k is not metrizable. moreover, n∩g is dense in n by the total minimality of g ([7, theorem 4.3.3]), hence n∩g is infinite. moreover, n∩g has infinite index in g, since otherwise it would be an open subgroup of g and consequently n = g∩n would be an open subgroup of k, a contradiction. therefore, g ∈s by lemma 3.1 (a). (h) c(g) is an infinite closed normal subgroup of g. since it has infinite index in g, lemma 3.1 (a) applies. � it follows from (h) that if there exists an infinite topological group failing to have a small set of generators then there exists also such a group g that is either connected or totally diconnected. in the former case the arc-component of g is either trivial or has finite index (note that the the arc-component need not be closed even if g is compact, hence this need not immediately imply, by connectendness of g, that g is also arc-wise connected). groups with a small set of generators 349 4. questions. (i) does s contain all perfect groups? (ii) does s contain all hyperabelian groups of class ω with finite factors (i.e., all groups g such that ⋂∞ n=1 g (n) = {1} and all quotients g(n)/g(n+1) are finite)? (iii) does every countable group have a small set of generators? (iv) for which commutative rings k the group sln(k) has a small set of generators? positive answers to (i) and (ii) yield that s contains all groups. indeed, if the derived series of g stops after a finite number of steps, then g(n) is perfect for some n. if g/g(n) is infinite, then lemma 3.3 and proposition 3.7 apply to give g ∈ s. otherwise, g(n) is infinite and g ∈ s iff g(n) ∈ s (i.e., (i) applies). the alternative is to have all quotients g(n)/g(n+1) non-trivial finite. then to the infinite quotient g/ ⋂ g(n) (ii) applies. note 4.1. added in proof, july 2003: recently, i. protasov and t. banakh resolved the main problem of this paper by establishing that every group has a small set of generators (see theorem 13.1 in the monograph “ball structures and colorings of graphs and groups”. mathematical studies monograph series, 11. vntl publishers, lviv, 1999, 147 pp. isbn 966-7148-99-8). in particular, this answers positively also our conjectures 3.14 and 3.15 and questions (i)–(iv) above. references [1] g. artico, v. malykhin and u. marconi, some large and small sets in topological groups, math. pannon. 12 (2001), no. 2, 157–165. 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[30] e. zel′manov, on periodic compact groups, israel j. math. 77 (1992), no. 1-2, 83–95. received january 2002 revised september 2002 umberto marconi, roberto moresco dipartimento di matematica pura e applicata, università di padova, via belzoni 7, 35131 padova, italy e-mail address : umarconi@math.unipd.it, moresco@math.unipd.it dikran dikranjan dipartimento di matematica e informatica, università di udine, via delle scienze 206, 33100 udine, italy e-mail address : dikranja@dimi.uniud.it applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 77–84 fenestrations induced by perfect tilings f.g. arenas and m.l. puertas ∗ abstract. in this paper we study those regular fenestrations (as defined by kronheimer in [3]) that are obtained from a tiling of a topological space. under weak conditions we obtain that the canonical grid is also the minimal grid associated to each tiling and we prove that it is a t0-alexandroff semirregular trace space. we also present some examples illustrating how the properties of the grid depend on the properties of the tiling and we pose some questions. finally we study the topological properties of the grid depending on the properties of the space and the tiling. 2000 ams classification: primary 54b15; secondary 05b45. keywords: fenestration, tiling; grid, trace spaces, lower semicontinuous decomposition. 1. introduction 1.1. tilings. we quote from [5] the following considerations about tilings. a tiling of a topological space x is a covering of x by sets (called tiles) which are the closures of their pairwise-disjoint interiors. tilings of r2 have received considerable attention (see [2] for a wealth of interesting examples and results as well as an extensive bibliography). on the other hand, the study of tilings of general topological spaces is just beginning (see [1], [4], [5] and [6]). the following definitions will be basic to our discussion. let t = {ti : i ∈ i} be a tiling of a topological space x, we define i(x) = {i ∈ i : x ∈ ti}. we define the set of frontier points of t to be the union of the boundaries of the tiles in t and denote this set by f(t ). the protected points of t constitute the set p(t ) = {x ∈ x : x ∈ ( ⋃ i∈i(x) ti) ◦} and the complement of this set in x is u(t ), the set of unprotected points of t . ∗the authors are supported by grant bfm 2000-1111 from spanish ministry of science and technology. 78 f.g. arenas and m.l. puertas 1.2. fenestrations. according to kronheimer (see [3]) a fenestration e of a topological space x is a family of pairwise disjoint open sets whose union is dense in x. the fenestration is said to be regular if the open sets of the family are regular open sets. if e is a fenestration of x, so is ereg = {u ◦ : u ∈ e}, which will be called the regularization of e. two fenestrations are said to be equivalent if they have the same regularization. the relation between tilings and fenestrations is given in the following definition. definition 1.1. given a tiling t = {ti : i ∈ i} if we set ai = t◦i we clearly obtain a regular fenestration, called e(t ) (the fenestration induced by a tiling). note that not every regular fenestration is induced by a tiling, as the following example shows. example 1.2. let en = {(x,y) ∈ r2 : 1n+1 < y < 1 n } if n ∈ n and e0 = {(x,y) ∈ r2 : y > 1}. e = {en : n ≥ 0} is a regular fenestration of x = {(x,y) : y ≥ 0}; however e = {en : n ≥ 0} is not a tiling of x, since⋃ n≥0 en = {(x,y) : y > 0}. given a fenestration e of a topological space x, a pseudogrid associated to e (see again [3]) is a family 4 of subsets of x such that e ⊂ 4 and 4 is a partition of x. each pseudogrid determines a quotient space and the quotient map is open if and only if the pseudogrid is lower semicontinuous, that is, st(g,4) = ⋃ {a ∈4 : a∩g 6= ∅} is open for every open set g of x. a lower semicontinuous pseudogrid is called a grid. in [3], kronheimer studied under what conditions the grid is minimal. this is an interesting point because then the study of the quotient space associated to each grid becomes easier. to this end given a fenestration e of a space x, there is a canonical way to associate a pseudogrid 4× to e by identifying two points of x if and only if , for every open neighborhood of either, there exists an open neighborhood of the other one which intersects the same collection of elements of e. it is proved in [3] (theorem 6.2.) that 4× is the minimal grid associated to e if and only if 4× is a grid, that is, is a lower semicontinuous decomposition. if it is not a grid, the minimal grid shall be constructed by a different procedure, since minimal grids always exist. moreover, if 4× is primitive (that is, for each of its points, the intersection of all its neighborhoods is a regular open set), kronheimer shows in section 11 of [3] that 4× has properties in the homotopy category which are closely related to those of x. in fact, that is the reason to study the minimal grid associated to any fenestration. the main result of this paper is constructing in a more explicit way the canonical pseudogrid of the fenestration associated to a tiling and obtaining conditions to ensure that it is the minimal grid and is primitive. we also study how the properties of the tiling reflect as properties of the grid. fenestrations 79 2. the minimal grid of a tiling in the case that the fenestration comes from a tiling we can give a more precise description of the canonical pseudogrid. first, we need to restrict our attention to a particular class of tilings, which will be called perfect and that are general enough to be useful. definition 2.1. let t = {ai : i ∈ i}, with ai open in x, be a tiling of a topological space x. given σ ⊂ i, let define aσ = {x ∈ x : i(x) = σ}. we say that the tiling t is complete if aσ ⊂ aτ for every τ ⊂ σ supposed that aσ and aτ are both nonempty, and we say that is perfect if it is complete and u(t ) = ∅. now, our main theorem presents a sufficient condition on a fenestration (induced by a perfect tiling) for its minimal grid to be primitive. the condition is useful in the sense that it is constructive and does not seem excessively restrictive. moreover, the condition is the conjunction of three simpler conditions (being determined by a tiling, the tiling is complete, u(t ) = ∅), the omission of any of which causes the theorem fails, as we will see with some examples. theorem 2.2. let t be as in definition 2.1 a perfect tiling of a topological space x and let e(t ) be the induced regular fenestration. call 4×(t ) the canonical pseudogrid associated to e(t ). then the t0-alexandroff space whose underlying set is 4×(t ) = {σ ⊂ i : aσ 6= ∅} and the minimal neighborhood of every σ ∈4×(t ) is the set v (σ) = {τ ∈4×(t ) : τ ⊂ σ} is called the canonical pseudogrid. moreover, 4×(t ) is semirregular and a lower semicontinuous decomposition; hence 4×(t ) is the minimal grid and is a primitive space. proof. it is clear, using u(t ) = ∅, that x ∼ y if and only if i(x) = i(y), where ∼ is the equivalence relation that leads to the canonical pseudogrid. hence the equivalence classes are just the sets aσ = {x ∈ x : i(x) = σ} for each σ 6= ∅ and consequently the quotient is the set of (nonempty) equivalence classes 4×(t ) = {σ ⊂ i : aσ 6= ∅}. the quotient map π× : x → 4×(t ) is defined as π×(aσ) = σ, so if we define v (σ) = {τ ∈ 4×(t ) : τ ⊂ σ}, we have (π×)−1(v (σ)) = (π×)−1({τ ∈ 4×(t ) : τ ⊂ σ}) = {x ∈ x : i(x) ⊂ σ} = ⋃ i∈σ ai. as every point is protected, given a point x ∈ aσ, {ai : i ∈ σ} is the set of all tiles that contain x, hence x ∈ ( ⋃ i∈σ ai) ◦. hence aσ ⊂ ( ⋃ i∈σ ai) ◦, that is, there is an open set u in x such that (π×)−1(σ) = aσ ⊂ u ⊂ ⋃ i∈σ ai = (π×)−1(v (σ)). applying π× in both sides, we obtain σ ⊂ π×(u) ⊂ v (σ). since we have proved that aτ ⊂ ( ⋃ i∈τ ai) ◦ for each τ ∈4×(t ), we obtain⋃ τ⊂σ aτ ⊂ ( ⋃ i∈σ ai) ◦. in fact the equality holds. to see this, given a point x ∈ ( ⋃ i∈σ ai) ◦, there exists an open neighborhood u of x with u ⊂ ⋃ i∈σ ai. that is, if i /∈ σ, u ∩ai = ∅, then x /∈ ai so if i /∈ σ, we have i /∈ i(x), that is, τ = i(x) ⊂ σ, so finally x ∈ aτ for some τ ⊂ σ and the equality is obtained. hence ⋃ τ⊂σ aτ is open. 80 f.g. arenas and m.l. puertas now, to see that the pseudogrid is lower semicontinuous (a grid) we have to check that for every open set g of x, st(g) = ⋃ σ∈4(g) aσ where 4(g) = {σ ⊂ i : aσ ∩g 6= ∅} is open in x. the condition aσ ⊂ aτ for every τ ⊂ σ given in the hypothesis is clearly equivalent to aτ ∩ g 6= ∅ for every τ ⊂ σ and every open set g of x with aσ ∩g 6= ∅. now, given x ∈ st(g) = ⋃ σ∈4(g) aσ there is σ ∈ 4(g) such that x ∈ aσ; hence we have x ∈ aσ ⊂ ⋃ τ⊂σ aτ ⊂ st(g). since ⋃ τ⊂σ aτ is open, we have that st(g) is open, as desired. since π× is an open mapping and σ ⊂ π×(u) ⊂ v (σ), then we have that v (σ) is a neighborhood of σ. to see that is the least one (and hence the space is alexandroff), suppose there is an open set v ⊂ v (σ), with σ ∈ v 6= v (σ). hence there is τ ⊂ σ with aτ 6= ∅ and τ /∈ v . using that τ ⊂ σ implies aσ ⊂ aτ and that g = (π×)−1(v ) is open in x and aσ ⊂ g we obtain g∩aτ 6= ∅, so g∩aτ 6= ∅, that is (π×)−1(v ) ∩ (π×)−1(τ) 6= ∅, which is a contradiction with τ /∈ v . now, to see that the space is t0 note that, with the given definition, if σ 6= τ clearly v (σ) 6= v (τ). finally, to see that the minimal grid (from theorem 6.2 of [3]) is also semirregular, we have to show that (v (σ))◦ = v (σ) for every σ ∈4×(t ). since τ ∈ v (σ) is equivalent to σ ∩ τ 6= ∅ that is the same that σ ∈ v (τ) (note that τ ∈ v (σ) if and only if v (τ) ∩v (σ) 6= ∅ and that v (τ) ∩v (σ) = v (τ ∩σ)), we have that v (σ) = {τ ∈4×(t ) : τ ∩σ 6= ∅}. now, given a point η ∈ 4×(t ), η ∈ (v (σ))◦ if and only if v (η) ⊂ v (σ), that is, if {i}∈ v (η) ⊂ v (σ) for every i ∈ η, hence {i}∩σ 6= ∅, which means i ∈ σ, for each i ∈ η, so η ⊂ σ. hence η ∈ v (σ), so we obtain (v (σ))◦ = v (σ). then the grid is semirregular. � note that the space can always be defined as we have done but only under the conditions of our theorem we can ensure that it is the canonical pseudogrid associated to the tiling (and we obtain in addition that it is also the minimal grid and that is semirregular and primitive). in the above theorem we use several hypotheses (tiling, complete tiling, tiling without unprotected points) and we obtain several consequences (a canonical construction, t0-alexandroffness, lower semicontinuity, semirregularity) altogether. we can ask exactly what hypothesis gives each selected consequence, since for a particular situation we may need to relax the hypotheses to include more cases. in order to clarify this, we present hereafter some illuminating examples and pose some questions. the first example shows how the description we have given of the canonical pseudogrid depends on the hypothesis. example 2.3. let tn = {(x,y) ∈ r2 : 1n+1 < y < 1 n } if n ∈ n, t0 = {(x,y) ∈ r2 : y > 1} and t−1 = {(x,y) ∈ r2 : y ≤ 0}. t = {tn : n ≥−1} is a complete tiling of x = r2 with u(t ) = {(x, 0) : x ∈ r}. however the canonical fenestrations 81 decomposition associated to t according to 2.2 should be y = {{k},{n,n+1} : k ≥ −1,n ≥ 0} with the correspondent topology. however, this space is not the canonical pseudogrid, proving that the hypothesis u(t ) = ∅ cannot be suppressed. in this case, the canonical decomposition is lower semicontinuous in this example, but we can modify it replacing the tile t−1 = {(x,y) ∈ r2 : y ≤ 0} with the tiles t−2 = {(x,y) ∈ r2 : y ≤ 0,x ≤ 0} and t−3 = {(x,y) ∈ r 2 : y ≤ 0,x ≥ 0}, and the example obtained has the same properties but the canonical decomposition is not lower semicontinuous (take g an open ball of radius 1 of the point (2, 0)). so we pose the following question: problem 2.4. can the hypothesis of 2.2 over the tiling be weakened and still obtain that the canonical pseudogrid associated to a tiling is as described in theorem 2.2? the following example shows how semirregularity can be lost if completeness is suppressed. example 2.5. we quote here an example cited in 4.9 of [3]. let d2 be an open circular disc with center p = {p} in r2. let a1, a2 and a3 be the three open sectors of 120◦ and i, j and k be the three open radii respectively contained in a2 ∩a3, a1 ∩a3 and a1 ∩a2. let i be the midpoint of i and h = {i}. let define x = a1 ∪ a2 ∪ a3 ∪ h ∪ p with the induced topology from d2. t = {a1 ∪ p,a2 ∪ h ∪ p,a3 ∪ h ∪ p} is a tiling of x. the canonical pseudogrid associated to that tiling is according to theorem 2.2 and is lower semicontinuous (is the minimal grid, since is the only one). however it is not semirregular (see 4.9 of [3]). in fact the minimal neighborhood of the point {1, 2, 3} is {{1},{2},{3},{1, 2, 3}}, that is strictly contained in the expected one v ({1, 2, 3}), as defined in 2.2 (v ({1, 2, 3}) also has the element {2, 3}). the reason is that this is not a complete tiling, since τ = {2, 3} ⊂ σ = {1, 2, 3}, aσ = p, aτ = h and p is not contained in h the following two examples show how lower semicontinuity can be lost if completeness is suppressed. example 2.6. let x be ([−1, 1]× [−1, 1])\a where a = [−1 2 , 0[×{0} and let the tiling be {ai : i = 1, 2, 3, 4} where a1 = [0, 1] × [0, 1], a2 = [0, 1] × [−1, 0], a3 = ([−1, 0]× [−1, 0])\a and a4 = ([−1, 0]× [0, 1])\a. clearly τ = {3, 4}⊂ σ = {1, 2, 3, 4}; however aσ is the point (0, 0), that is not in the closure of aτ = [−1,−12 [×{0}. taking g as an open ball centered in (0, 0) with radius less than 1 2 , we obtain an example of an open set whose star with respect to the partition is not open in x. note that this is not a complete tiling since the canonical decomposition is not lower semicontinuous. note that the key property to ensure the completeness of a tiling is not the connectedness of the set of boundary points of the tiling, as one could feel, looking at the former example. the space and tiling obtained in the above 82 f.g. arenas and m.l. puertas example replacing a = [−1 2 , 0[×{0} by a = {1 n : n ∈ n} is complete and the set of boundary points of the tiling is not connected. example 2.7. the following example is quoted from 6.6 of [3]. let consider the tiling {ai : i = 1, 2, 3, 4} of the space s = [−1, 1] × [−1, 1] where a1 = [0, 1]×[0, 1], a2 = [0, 1]×[−1, 0], a3 = [−1, 0]×[−1, 0] and a4 = [−1, 0]×[0, 1]. let x be the möbius band obtained from s by identifying the points (x,−1) with (−x, 1) for each −1 ≤ x ≤ 1 and write ρ for the natural map of s onto x. t = {bi = ρ(ai) : i = 1, 2, 3, 4} is a tiling of x without unprotected points. however it is not perfect, since it is not complete. in fact b{1,2,3,4} has two points: p = ρ((0, 0)) and the point q = ρ((0,−1)) = ρ((0, 1)). now, b1,3 = ρ([−1, 0] ×{−1}) = ρ([0, 1] ×{1}) is not empty; we have that q ∈ b1,3, but p /∈ b1,3, so {1, 3} ⊂ {1, 2, 3, 4} but b{1,2,3,4} 6⊂ b1,3. and we have that 4×(t ) is a semirregular minimal trace space but it is not a lower semicontinuous decomposition (6.6 of [3]). hence, to ensure that the tiling gives a lower semicontinuous decomposition, it is not enough to have a tiling without unprotected points. however, the example 2.5 shows that we can obtain a lower semicontinuous decomposition from a tiling without unprotected points that is not complete. on the other hand the example 2.7 shows that we can obtain a semirregular decomposition from a tiling without unprotected points that is not complete. so we can pose the following question. problem 2.8. can the hypotheses of 2.2 over the tiling be weakened and still obtain that the canonical pseudogrid associated to a tiling is semirregular and lower semicontinuous? 3. topological properties of the minimal grid of a tiling in this final section we ask how topological properties (other than semirregularity and lower semicontinuity) of the minimal grid of a tiling 4×(t ) can be deduced from the topological properties of the space x and properties of the tiling. the following notation is quoted from [3]. for a topological space x, if the set of isolated points xλ = {x ∈ x : {x} is open in x} is dense in x we shall call it the trace of x and x itself a trace space. every pseudogrid is a trace space, whose trace is the image of the fenestration. we summarize the results in the following theorem. theorem 3.1. let t be as in definition 2.1 a perfect tiling of a topological space x and let y = 4×(t ) be the minimal grid associated to e(t ) according to 2.2. (1) if x is compact, connected, locally compact or locally connected, so is y . moreover, w(y ) ≤ w(x), where w(x) is the weight of x. (2) if t is a tiling, then |t | ≤ w(x), and we also have that w(y ) = |y | ≤ 2|t |. (3) s = {{y} : y ∈ y λ} is a tiling of y . fenestrations 83 (4) the image under the quotient map of the set of singular points s(t ) (frontier points such that every neighborhood of which intersects infinitely many tiles in t ) of the tiling is the set of cluster points (see [7]) of the trace (hence a tiling without singular points gives a grid whose trace has no cluster points). if s0(t ) = {x ∈ s(t ) : i(x) is infinite} = ∅, y is a locally finite space. proof. (1) the quotient map is open. (2) first, if t is a tiling, given x ∈ ai, there is b ∈b such that x ∈ b ⊂ ai, since ai is open. since the sets ai are pairwise disjoint, the cardinal of the base b is at least the cardinal of the set of ai’s and this is valid for any base. finally, since y is t0-alexandroff, the cardinal of the space is the same as the weight, and since the points of y are subsets of i, is not greater than 2|t |. (3) clear. (4) since a cluster point of y λ is one whose neighborhoods have infinitely many points of y λ, the first assertion is clear. if s0(t ) = ∅, i(x) is finite for every x ∈ x, hence {σ ⊂ i : aσ 6= ∅} is a set of finite subsets of i, hence v (σ) is finite for every σ ∈4×(t ), hence a locally finite space as defined in section 3 of [3]. � note that the tiling defined in example 2.3 has u(t ) 6= ∅ and s0(t ) = ∅ and the minimal grid (that is not of the form we have obtained in 2.2) is not locally finite. if y is locally connected, we can construct a base formed by open sets that are simultaneously regular open and connected, so we can pose the following question. problem 3.2. find a relation between the connectivity of the tiles and the local connectedness of y . it is clear that 4×(t|a) 6= 4×(t )|π(a) that is, the grid associated to the restriction of the tiling to a is not the restriction of the grid to the image of a, since t|a may not be a tiling under the conditions of 2.2 even if t is (see example 2.5, with x = d2 and a = x). we can also ask if 4×(t1 ×t2) = 4×(t1) ×4×(t2). problem 3.3. find two perfect tilings t1 and t2 of topological spaces x1 and x2 under the conditions of 2.2 such that 4×(t1 ×t2) 6= 4×(t1) ×4×(t2). it is clear that two topologically equivalents tilings (in the sense of section 1 of [6]) give the same induced canonical pseudogrid. now, we say that two perfect tilings are 4×-equivalent if and only if there is an homeomorphism between their induced canonical pseudogrids. problem 3.4. are there two non-topologically equivalent perfect tilings of rn such that they are 4×-equivalent? 84 f.g. arenas and m.l. puertas acknowledgements. the authors are indebted to professor dr. mark j. nielsen, who kindly sent [4], [5], [6] and his ph.d. thesis, and to professor dr. e.h. kronheimer who kindly sent [3]. references [1] f.g. arenas, tilings in topological spaces, int. j. math. math. sci. 22 (1999), no.3, 611–616. [2] b. grunbaum and g.c. shephard, tilings and patterns, freeman, new york, 1986. [3] e.h. kronheimer, the topology of digital images, topology appl. , 46 (1992), 279–303. [4] mark j. nielsen, singular points of a convex tiling, math. ann. 284 (1989), 601–616. [5] mark j. nielsen, singular points of a star-finite tiling, geom. dedic. 33 (1990), 99–109. [6] mark j. nielsen, on two questions concerning tilings, israel j. math. 81 (1993), 129–143. [7] stephen willard, general topology, addison-wesley publ. comp. 1970. received december 2001 f.g. arenas area of geometry and topology faculty of experimental sciences universidad de almeŕıa 04120 almeŕıa spain e-mail address : farenas@ual.es m.l. puertas area of applied mathematics faculty of experimental sciences universidad de almeŕıa 04120 almeŕıa spain e-mail address : mpuertas@ual.es 01.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 1 12 merotopies associated withquasi-uniformities�akos cs�asz�ar�abstract. to an arbitrary quasi-uniformity on the set x,a merotopy on x is assigned. there are results concerning thequestion whether this merotopy is compatible with the topologyinduced by the quasi-uniformity and whether the closure opera-tion induced by the merotopy, admits a compatible uniformity.more precise results are obtained in the case of transitive quasi-uniformities.2000 ams classi�cation: 54e15, 54e17keywords: quasi-uniformity, merotopy, semi-symmetric, transitive1. introductionthe purpose of the present paper is to establish a relation between two well-known kinds of topological structures, namely quasi-uniformities and mero-topies.notation and terminology concerning quasi-uniformities will be used accord-ing to [4]. the concept of a merotopy has been introduced in [8], but we shalluse according to [3] a more advantageous description of them due to [7]. thusa merotopy c on a set x will mean a non-empty collection of covers of x (wedenote by �(x) the collection of all covers of x) with the properties:(1.1) if c 2 c, c0 2 �(x) and c re�nes c0 then c0 2 c,(1.2) c1;c2 2 c implies c1(\)c2 2 cwhere we say that c re�nes c0 (in symbol c < c0) i� c 2 c implies the existenceof c0 2 c0 satisfying c � c0, andc1(\)c2 = fc1 \ c2 : ci 2 cig;�research supported by hungarian foundation for scienti�c research, grant no. t032042. 2 �akos cs�asz�ar(\) is obviously an associative operation. equivalently, (1.2) may be replacedby(1.3) c1;c2 2 c implies the existence of c 2 c satisfying c < ci (i = 1;2).the topological category qunif is composed of the objects of quasi-uniformspaces (x;u) where u is a quasi-uniformity on x, and of the morphisms ofquasi-uniformly continuous maps [4]. the category mer contains the objectsof merotopic spaces (x;c) where c is a merotopy on x and of the morphisms ofmerotopically continuous maps, where f : x ! x0 is said to be merotopicallycontinuous or (c;c0)-continuous, c and c0 being merotopies on x and x0 respec-tively, i� c0 2 c0 implies f�1(c0) 2 c (of course, f�1(c0) = ff�1(c0) : c0 2 c0g).we know ([4]) that each quasi-uniformity u on x induces a topology �(u)on x for which the neighbourhood �lter of x 2 x is given by fu(x) : u 2 ug.similarly, each merotopy c on x induces a closure operation on x (i.e. a mapc : expx ! expx such that c(?) = ?, a � c(a), c(a [ b) = c(a) [ c(b)where exp x is the power set of x) and c = c(c) is de�ned byx 2 c(a) , a 2 sec vc(x)(for b � �(x), where �(x) is the collection of all non-empty subsets of thepower set expx, we writea 2 sec b , a � xa \ b 6= ? for each b 2 b)and the c-neighborhood �lter vc(x) of x 2 x is generated by the �lter basefst(x;c) : c 2 cg. also each topology � on x may be considered as a closurec = c� = cl� special in the sense that c(c(a)) = c(a) for every a � x.2. merotopies associated with quasi-uniformitieslet u be an entourage [4] on x. de�ne cu = fu(x) : x 2 xg. then cu isa cover on x and, both u and u 0 being entourages on x with u � u 0, clearlyu(x) � u 0(x) for x 2 x so that cu < cu0. therefore, if u is a quasi-uniformityon x, then b = fcu : u 2 ug is a base [3] for a merotopy cu. more generally,if b is a base for u and we set b = fcu : u 2 bg then b is still a base forcu. moreover, if (x0;u0) is another quasi-uniform space and f : x ! x0 isquasi-uniformly continuous then f is (cu;cu0)-continuous as well: if u 2 u,u 0 2 u0 and (x;y) 2 u implies (f(x);f(y)) 2 u 0 then f(u(x)) � u 0(f(x)) sothat cu < f�1(cu0).hence we can state:theorem 2.1. if we associate with each quasi-uniformity u on the set x themerotopy cu with base(2.4) b = fcu : u 2 ugwhere(2.5) cu = fu(x) : x 2 xg;then �((x;u)) = (x;cu), �(f) = f for f : x ! x0 de�ne a (covariant)functor � : qunif ! mer. merotopies associated with quasi-uniformities 3it is an interesting question which merotopies can be represented in the formcu with some quasi-uniformity u, or which covers have the form cu for someentourage u. the collection of all covers of the form cu clearly does not coincidewith �(x): if c = cu then there is a surjection f : x ! c such that x 2 f(x) foreach x 2 x, consequently there is a bijection g : x0 ! c for some x0 � x suchthat x 2 g(x) for x 2 x0, or equivalently there is an injection g�1 = h : c ! xsuch that h(c) 2 c for c 2 c, i.e., in the terminology of [8], there is a transversalfor c. now clearly, if t 2 �(x) and h is a transversal for t, then necessarily thefollowing condition must hold:(2.6) t0 � t implies t0j 5 j [ t0jbecause h(t0) � [t0. consequently, if c = cu for some entourage u then (2.6)has to be ful�lled for t = c.according to [6], the condition (2.6) is su�cient for the existence of a trans-versal for t in the case when t and each t 2 t are �nite, or even, according to[5], in the case when t is in�nite but each t 2 t is �nite. however, probablythere are no further results on the su�ciency of (2.6) in the general case (ifsome t 2 t can be in�nite then (2.6) certainly does not guarantee the existenceof a transversal, cf. [9]). so we can formulate:problem 2.2. look for necessary and/or su�cient conditions for a cover c ofx for the existence of an entourage u satisfying c = cu.problem 2.3. look for necessary and/or su�cient conditions for a merotopyc on x for the existence of a quasi-uniformity u satisfying c = cu.if u is a quasi-uniformity on x and we look for the closure c = c(cu) thenit is easy to see:lemma 2.4. c = c(cu) is coarser than c�(u), i.e.c�(u)(a) � c(a) (a � x):proof. clearly vc(x) is generated by the �lter base composed of all sets st(x;cu)where u 2 u, and(2.7) st(x;cu) = [fu(y) : y 2 u(x)g = u(u�1(x)):obviously u(x) � u(u�1(x)). �in general, c 6= c�(u); e.g. if x = r and u is the sorgenfrey quasi-uniformitygenerated by the base fu" : " > 0g where u"(x) = [x;x+") then u"(u�1" (x)) =(x � ";x + ") so that c(cu) is the euclidean topology on r. it is even possiblethat the closure c(cu) it not a topology:example 2.5. let x = fa;b;cg and u be an entourage on x such that u(a) =fag; u(b) = fa;bg; u(c) = fa;cg. clearly u2 = u so that fug is a base for aquasi-uniformity u on x and fcug is a base for the merotopy cu. for c = c(cu),we have c(fbg) = fa;bg and c(fa;bg) = x. 4 �akos cs�asz�arhowever, it is not di�cult to characterize those quasi-uniformities u forwhich c(cu) = c�(u). recall ([4]) that a quasi-uniformity u on x is said tobe point-symmetric i�, for each x 2 x and u 2 u, there is v 2 u such thatv �1(x) � u(x) or, equivalently, i� �(u) is coarser than �(u�1).theorem 2.6. the equality c(cu) = c�(u) holds i� u is point-symmetric.proof. by lemma 2.4, we need, for x 2 x and u 2 u, the existence of w 2 usuch that w(w�1(x)) � u(x). now this condition clearly implies the point-symmetry of u. on the other hand, if, for u 2 u, we choose u0 2 u satisfyingu20 � u, then, given x 2 x, v 2 u such that v �1(x) � u0(x), �nally we setw = v \ u0 2 u, obviously w(w�1)(x) � u0(v �1(x)) � u20 (x) � u(x). �it is easy to �nd examples of point-symmetric quasi-uniformities. in fact,recall (cf. [1]) that a topology c (i.e. a closure c = c� for a topology �) issaid to be s1 i� x 2 g implies c(fxg) � g whenever g is c-open. also recall([4]) that the pervin quasi-uniformity p associated with the topology c (andinducing c) is de�ned by the quasi-uniform subbase fug : g is c-openg whereug(x) = g if x 2 g and ug(x) = x if x 2 x � g. more generally, if b is abase for the topology c then the entourages ub (b 2 b) constitute a subbasefor a transitive quasi-uniformity u(b) compatible with c (see e.g. [2]). if thetopology c is s1, we can also consider the entourages ux;b = ub \ ux�c(fxg)where x 2 b 2 b to obtain a subbase for a transitive quasi-uniformity u1(b)�ner than u(b) and coarser than p, hence still compatible with c.now we can state:proposition 2.7. if c is an s1 topology admitting a base b then every quasi-uniformity u �ner than u1(b) and compatible with c is point-symmetric.proof. given x 2 x and u 2 u, there is a b 2 b such that x 2 b � u(x). bys1, we have c(fxg) � b. let h denote the c-open set h = x � c(fxg). then,for v = ub \ uh 2 u1(b) � u, we have v �1(x) = c(fxg) � b � u(x). �the condition for a quasi-uniformity u of being point-symmetric has anotherimportant consequence for the merotopy cu. recall ([3]) that a merotopy c issaid to be lodato i� c 2 c implies int c 2 c where int c = fint c : c 2 cg andintc = x � c(x � c), c = c(c). now we can state:theorem 2.8. if u is point-symmetric then cu is a lodato merotopy.proof. for c 2 c, choose u 2 u such that cu < c and u0 2 u such thatu20 � u. then, by u0(x) � intu(x), cu0 < int cu < intc and cu0 2 c impliesintc 2 c. �3. semi-symmetric quasi-uniformitiesrecall ([3]) that a semi-uniformity u on a set x is a �lter on x � x havinga base composed of symmetric entourages; it induces a closure c(u) such that,if c = c(u) and x 2 x, then vc(x) = fu(x) : u 2 ug is the neighborhood �lterof x for c. merotopies associated with quasi-uniformities 5now if u is an arbitrary entourage on x then clearly uu�1 (we write abfor a � b if a;b � x � x) is a symmetric entourage on x so that, wheneveru is a quasi-uniformity on x, fuu�1 : u 2 ug is a base for a semi-uniformityu�; by lemma 2.4(3.8) c(u�) = c(cu):we look for those quasi-uniformities u which admit a corresponding semi-uniformity u� that is a uniformity. for this purpose, let us say that u issemi-symmetric i�, given u 2 u, there is v 2 u satisfying v �1v � uu�1;the pair (u;v ) is said to be semi-symmetric in this case and, in particular, theentourage u is said to be semi-symmetric i� (u;u) is semi-symmetric. now itis easy to prove:theorem 3.1. for a quasi-uniformity u, the semi-uniformity u� is a unifor-mity i� u is semi-symmetric.proof. if u� is a uniformity then, for u 2 u, there is v 2 u such thatv v �1v v �1 � uu�1 whence clearly v �1v � uu�1. conversely, if the con-dition in the statement is ful�lled, let u 2 u and u0 2 u be chosen such thatu20 � u, then let v 2 u satisfy v �1v � u0u�10 . now we can suppose v � u0 asv can be replaced by v \u0. then v (v �1v )v �1 � u0u0u�10 u�10 � uu�1. �of course, each uniformity is an example of a semi-symmetric quasi-uniform-ity. but it is easy to �nd non-symmetric examples, too. e.g. if u is the sorgen-frey quasi-uniformity on x = r whose base is composed of the entourages u" =f(x;y) : x 5 y < x + "g (" > 0) then u"u�1" = u�1" u" = f(x;y) : jx � yj < "g.similarly if u is the michael quasi-uniformity on x = r, i.e. the base is com-posed of fu" : " > 0g where u"(x) = (x � ";x + ") if x 2 q and u"(x) = fxg ifx 2 r�q , then u"u�1" (x) = (x�2";x+2"), while clearly u"(x) � (x�";x+")and u�1" (x) � (x � ";x + ") so that u�1" (u"(x)) � u"(u�1" (x)). on theother hand, e.g. example 2.5 is not semi-symmetric: u(u�1(b)) = fa;bg andu�1(u(b)) = x.corollary 3.2. if a quasi-uniformity u is both semi-symmetric and point-symmetric then the topology �(u) is completely regular.proof. by theorem 2.6 c�(u) = c(cu), by (3.8) and theorem 3.1 the latter is atopology induced by a uniformity. �it is easy to see that point-symmetry and semi-symmetry are properties ofa quasi-uniformity independent of each other. in fact, the sorgenfrey quasi-uniformity is semi-symmetric without being point-symmetric, while if c is ans1 topology that is not completely regular then its pervin quasi-uniformity ispoint-symmetric by proposition 2.7 but not semi-symmetric by corollary 3.2.semi-symmetric quasi-uniformities have rather good invariance properties.recall that, if f : x ! y , then the inverse image f�1(u) of a quasi-uniformityu on y is generated by the entourages f̂�1(u) for u 2 u where f̂(x;y) =(f(x);f(y)). 6 �akos cs�asz�arlemma 3.3. if f : x ! y is surjective and u is a semi-symmetric quasi-uniformity on y then f�1(u) is semi-symmetric.proof. if u;v 2 u and v �1v � uu�1, further (f(x);f(y)) 2 v , (f(y);f(z)) 2v �1 then (f(x);f(z)) 2 v �1v � uu�1 so that there is some w 2 y satisfying(f(x);w) 2 u�1, (w;f(z)) 2 u, and choosing u 2 x such that w = f(u),we get (f(x);f(u)) 2 u�1, (f(u);f(z)) 2 u, i.e. (x;u) 2 f̂�1(u�1), (u;z) 2f̂�1(u). �the condition of surjectivity cannot be dropped as semi-symmetry is nothereditary:example 3.4. let x = fa;b;c;dg, u(a) = fag, u(b) = fa;bg , u(c) = fa;cg,u(d) = x. then u2 = u, so that fug is a base for a quasi-uniformity u onx. the semi-symmetry of u is easily checked using the formulas for u(x) andthose u�1(a) = x; u�1(b) = fb;dg; u�1(c) = fc;dg; u�1(d) = fdg. de�nex0 = fa;b;cg; u0 = u \ (x0 � x0). then ujx0 coincides with the quasi-uniformity in example 2.5 which fails to be semi-symmetric.lemma 3.5. if ui is a semi-symmetric quasi-uniformity on xi (i 2 i) andx = qfxi : i 2 ig then u = qui is semi-symmetric on x.proof. let u 2 u be given. we can suppose u = qui where ui 2 ui fori 2 f and a �nite f � i, ui = xi � xi otherwise. choose vi 2 ui such thatv �1i vi � uiu�1i for i 2 f and vi = xi � xi otherwise. for v = qvi, we havev �1v � uu�1. �some partial results concerning heredity may be obtained by introducingthe following de�nition: let us say that u is strongly semi-symmetric i�, givenu 2 u, there is v 2 u such that v �1v � u[u�1; in this case (u;v ) is stronglysemi-symmetric and, in particular, u 2 u is strongly semi-symmetric i� so is(u;u).lemma 3.6. a strongly semi-symmetric quasi-uniformity is semi-symmetricas well.proof. if v �1v � u [ u�1 and (x;y) 2 v �1v then either (x;y) 2 u or(x;y) 2 u�1. in the �rst case, let (x;x) 2 u�1, in the second one let (y;y) 2 u.in both cases, (x;y) 2 uu�1. �e.g. the sorgenfrey quasi-uniformity is strongly semi-symmetric becausef(x;y) : jx � yj < "g = f(x;y) : x 5 y < x + "g [ f(x;y) : x � " < y 5 xg. thesame holds for the michael quasi-uniformity: u"(x) [ u�1" (x) = (x � ";x + ")if x 2 q and = fxg [ ((x � ";x + ") \ q) if x 2 r � q, while u�1� (u�(x)) �(x � 2�;x + 2�) if x 2 q and = fxg [ ((x � �;x + �) \ q) if x 2 r � q.in example 3.4, we �nd a semi-symmetric but not strongly semi-symmetricquasi-uniformity; in fact strong semi-symmetry is hereditary:lemma 3.7. if f : x ! y and u is strongly semi-symmetric on y thenf�1(u) is strongly semi-symmetric on x. merotopies associated with quasi-uniformities 7proof. assume u;v 2 u and v �1v � u [ u�1. if (x;y) 2 f̂�1(v �1) f̂�1(v )then (f(x);f(y)) 2 v �1v � u [ u�1, so (x;y) 2 f̂�1(u) [ f̂�1(u�1). �however, the analogue of lemma 3.5 is not valid for strongly semi-symmetricquasi-uniformities:example 3.8. let x = r2, u be the sorgenfrey quasi-uniformity, and consideru � u. we know that both factors are strongly semi-symmetric. for u =u1 �u1, no v� = u� �u� is suitable: (0; 34�) 2 u�, (34�; 12�) 2 u�1� , (0; 14�) 2 u�,(14�;�12�) 2 u�1� , so ((0;0);(12�;�12�)) 2 v �1� v� but ((0;0);(12�;�12�)) =2 u [u�1 = (u1 � u1) [ (u�11 � u�11 ) because (0; 12�) =2 u�11 and (0;�12�) =2 u1.4. the transitive caseproblems 2.2 and 2.3 have partial solution in the case of transitive entouragesand quasi-uniformities, respectively. in order to see this, consider a systemt 2 �(x) and de�ne an operation � : �(x) ! �(x) by(4.9) �(t) = ft(x) : x 2 xgwhere(4.10) t(x) = \ft 2 t : x 2 tgand we de�ne \? = x. clearly x 2 t(x), hence �(t) is always a cover of x sothat � : �(x) ! �(x).lemma 4.1. the operation � is idempotent.proof. let t 2 �(x) and t0 = �(t). for x;y 2 x and x 2 t(y) we haveft 2 t : y 2 tg � ft 2 t : x 2 tg, consequently t(x) � t(y), so thattft 0 2 t0 : x 2 t 0g = tft(y) 2 t0 : x 2 t(y)g � t(x) while obviouslyt(x) 2 t0; x 2 t(x) imply tft 0 2 t0 : x 2 t 0g � t(x). by this, tft 0 2 t0 : x 2t 0g = t(x) and �(t0) = �(�(t)) = �(t). �let us say that a system t 2 �(x) is point-true i� �(t) = t; hence a point-truesystem is always a cover of x. in other words,lemma 4.2. a system t is point-true i� a) tft 2 t : x 2 tg 2 t if x 2 x andb) if t 2 t, there is x 2 t such that x 2 t 0 2 t implies t � t 0.now let u be a transitive (i.e. such that u2 = u) entourage on x. asx 2 u(y) implies u(x) � u(y) (because (x;z) 2 u and (y;x) 2 u imply(y;z) 2 u), we have u(x) = tfu(y) : x 2 u(y)g, so that:lemma 4.3. if u is a transitive entourage on x then the cover cu is point-true.conversely:lemma 4.4. if c is a point-true cover of x then there is a transitive entourageu on x such that c = cu. 8 �akos cs�asz�arproof. de�ne (x;y) 2 u � x �x i� x 2 c 2 c implies y 2 c. then (x;x) 2 ufor x 2 x and (x;y) 2 u, (y;z) 2 u imply (x;z) 2 u so that u is a transitiveentourage on x. by de�nition, u(x) = tfc 2 c : x 2 cg 2 c by lemma 4.2a), and, if c 2 c, there is by lemma 4.2 b) an x 2 c such that c = u(x).consequently c = fu(x) : x 2 xg. �lemma 4.5. the transitive entourage u in the above lemma is uniquely de-termined by c.proof. let u1 and u2 be transitive entourages on x such that cu1 = cu2. givenx 2 x, there is y 2 x satisfying u1(x) = u2(y). then x 2 u1(x) impliesx 2 u2(y), hence u2(x) � u2(y) = u1(x) and u2(x) � u1(x). thereforeu2 � u1. similarly u1 � u2. �theorem 4.6. there is a bijection from the set of all transitive entourages onx to the set of all point-true covers of x given by the formulas(4.11) u 7! cu;(4.12) c 7! uc;uc(x) = \fc 2 c : x 2 cg(x 2 x):concerning the behaviour of transitive quasi-uniformities, let us �rst remark:lemma 4.7. let ui be transitive entourages on x for i = 1; :::;n and u =tn1 ui. then cu = �((t)n1 cui).proof. let us denote cui = ci, cu = c. then, for x 2 x, we have by (4.12), forthe element of c corresponding to x, u(x) = tn1 ui(x) = tni=1 tfci 2 ci : x 2cig = tfci 2 ci : x 2 ci; i = 1; :::;ng = tfc 2 (t)n1ci : x 2 cg; the latter tis the element of �((t)n1 ci) corresponding to x. �observe that � cannot be omitted because c1(\)c2 may fail to be point-truefor point-true covers ci (i = 1;2).example 4.8. let x = r, c1 = f(2n;2n+2) : n 2 zg[f(2n�2;2n+2) : n 2 zgand c2 = f(2n � 1;2n + 1) : n 2 zg [ f(2n � 1;2n + 3) : n 2 zg. it is easyto check using lemma 4.2 that both c1 and c2 are point-true covers. nowc1(\)c2 = f(n;n +1) : n 2 zg[f(n;n +2) : n 2 zg[f(n;n+3) : n 2 zg[f?gis not point-true since neither (n;n + 3) nor f?g does ful�l lemma 4.2 b).now we can prove:theorem 4.9. if u is a transitive quasi-uniformity then the merotopy c = cuful�ls(4.13) c has a base b composed of point-true coverssuch that(4.14) if ci 2 b for i = 1; : : : ;n then �((t)n1ci) 2 b.conversely if c is a merotopy satisfying (4.13) and (4.14) then there exists atransitive quasi-uniformity u such that c = cu. merotopies associated with quasi-uniformities 9proof. (4.13) is obvious if b = fcu : u 2 u is transitiveg. if ci 2 b (i = 1; :::;n)then there are transitive entourages ui 2 u such that ci = cui. by lemma 4.7,�((t)n1cui) = cu 2 b for u = tn1 ui 2 u and b ful�ls (4.14).conversely, if the merotopy c satis�es (4.13) and (4.14), let b denote thebase for c occurring in (4.13). by lemma 4.4, there are transitive entouragesu such that c = cu for each c 2 b. denote by b the set of all these u. bylemma 4.7 and (4.14), b is a �lter base on x �x and by u2 = u, it is a basefor a transitive quasi-uniformity u. clearly cu = c. �in contrast to lemma 4.5, there is no uniqueness in the above theorem:example 4.10. let x = r, c = f[2n;2n + 2) : n 2 zg and c1 = c [ f[0;1)g,c2 = c [ f[1;2)g. each of the point-true covers c and ci (i = 1;2) de�nemerotopic bases fcg, fcig for the same merotopy c (observe ci < c < ci).however, if we choose transitive entourages ui such that ci = cui (cf. lemma4.4) then fuig is a base for a quasi-uniformity ui and cui = c while u2 " u1(e.g. 1 2 u2(0) � u1(0)), so u1 6= u2.observe that this example shows: if ui (i = 1;2) are transitive entouragesand cu1 < cu2 < cu1 then u1 = u2 need not hold. also fc1;c2g is a base forc but fu1;u2g is not a quasi-uniform base at all as u1 " u2 " u1. certainly,it is a quasi-uniform subbase; however, if u = u1 \ u2, then fug is a basefor a quasi-uniformity u but, since by lemma 4.7 cu = f[2n;2n + 2) : n 2z� f0gg [ f[0;1); [1;2)g, we have c 6= cu as cu < c and c � cu.example 4.10 contains a merotopy and quasi-uniformities inducing very badtopologies. however, it is possible the �nd a better example:example 4.11. let x = r�z = sn2zin where in = (n;n+1). let � denotethe subspace topology on x of the euclidean one on r. denote by b thebase for � composed of all (�)-open sets b contained in some in. consider the(point-true) covers of x cx;b = ffxg;b � fxg;x � fxgg; clearly cx;b = cux;b.denote also c0 = fxg[fi2k�1 : k 2 zg, c00 = fxg[fi2k : k 2 zg. clearly bothc0 and c00 are point-�nite, point-true covers of x. we write c0 = cu0, c00 = cu00with transitive entourages u 0, u 00. let u0 be the transitive quasi-uniformityde�ned by the subbase fux;b : x 2 b 2 bg [ fu 0g, and similarly de�ne u00with the help of the subbase fux;b : x 2 b 2 bg [fu 00g.we have u0 6= u00. in fact, assume the contrary; then u 0 � u = tn1 uxi;bi\u 00for suitable xi 2 bi, 1 5 i 5 n. there is a k 2 z such that i2k�1 is disjoint fromall sets b1; :::;bn so that u 0(x) = i2k�1 for x 2 i2k�1 while u(x) is co�nite asuxi;bi(x) = x � fxig and u 00(x) = x.let us write c0 = cu0, c00 = cu00. for an arbitrary cover c 2 c0, we can �nd,according to lemma 4.7, xi 2 bi 2 b such that �((t)n1cxi;bi(\)u 0) < c. weclaim �((\)n1cxi;bi) < �((\)n1cxi;bi(\)u 0):in fact, if x 2 bi for some i then the member containing x of the left handside is contained either in bi \ i2k�1 = bi for some k or in bi \ x = bi; bothsets belong to the right hand side. if x =2 bi for each i = 1; :::;n, then there 10 �akos cs�asz�aris a k such that i2k is disjoint from all sets bi occurring on the left hand sideand then the member of the left hand side containing some y 2 i2k is the sameas the one containing x; therefore this member is the one containing y of theright hand side. thus the left hand side, belonging to c00, re�nes c and c 2 c00,c0 � c00. a similar argument furnishes c00 � c0 so that �nally c0 = c00 = c.clearly both u0 and u00 induce the (very good) topology �. according toproposition 2.7, they are point-symmetric, so that the merotopy c induces �as well (see theorem 2.6).example 3.4 shows that the invariance properties of semi-symmetry are es-sentially the same in the transitive case as in the general one. however, we canestablish useful criteria guaranteeing the symmetry of a transitive entourage orthe semi-symmetry of a transitive quasi-uniformity.lemma 4.12. if c is a point-true cover of x, u = uc is the correspondingtransitive entourage, then cu�1 = �(cc) where cc = fx � c : c 2 cg.proof. let v = u�1, x 2 x. now y 2 v (x) i� x 2 u(y) = tfc 2 c : y 2 cg i�y 2 c 2 c ) x 2 c i� x =2 c 2 c ) y =2 c i� x 2 x �c; c 2 c ) y 2 x �c i�y 2 tfx�c : c 2 c; x 2 x�cg and the latter t is the element correspondingto x of �(cc). �observe that � cannot be dropped: let x = [0;1] � r, c = f[0;x] : 0 5 x <1g [ f1g; now cc = f(x;1] : 0 5 x < 1g [ [0;1) is not point-true.theorem 4.13. let c be a point-true cover of x and u = uc. u is symmetrici� c is a partition of x.proof. necessity: suppose u(x) \ u(y) 6= ?, say, z 2 u(x) \ u(y). thenu(z) � u(x)\u(y) by the transitivity, x 2 u(z) and y 2 u(z) by the symmetry,and u(x)[u(y) � u(z) by the transitivity again. hence u(x) = u(z) = u(y).su�ciency: if u(x) = c0 then u�1(x) = tfx � c : c 2 c; x =2 cg bylemma 4.12, hence u�1(x) = c0 provided c is a partition. �theorem 4.14. let c = cu for a transitive quasi-uniformity u. the latter issemi-symmetric i� there is a base b for c composed of covers cu with transitiveu 2 u and such that these u constitute a base for u, further, if c 2 b, there isa c0 2 b such that, whenever c0i 2 c0 and c01 \c02 6= ?, there is c 2 c satisfyingc01 [ c02 � c.proof. necessity: let b = fcu : u 2 u is transitive g. given c = cu 2 b,u 2 u transitive, choose a transitive v0 2 u such that v �10 v0 � uu�1 andset v = v0 \ u 2 u. finally let c0 = cv . now if c01 = v (x), c02 = v (y) andc01 \ c02 6= ?, we have some z such that z 2 v (x) \ v (y), hence y 2 v �1(z) �v �1(v (x)) � u(u�1(x)). consequently there is some u satisfying u 2 u�1(x),y 2 u(u), i.e. x;y 2 u(u), therefore c01 [ c02 = v (x) [ v (y) � u(x) [ u(y) �u(u) by the transitivity of u. for c = u(u) 2 c we obtain c01 [ c02 � c.su�ciency: given u 2 u, choose a transitive u0 2 u such that u0 � uand cu0 belongs to the base b in the hypothesis. set c = cu0, then choosec0 2 b satisfying c01 [ c02 � c 2 c whenever c0i 2 c0 and c01 \ c02 6= ?, and merotopies associated with quasi-uniformities 11let c0 = cv for some transitive v 2 u. if x 2 x and y 2 v �1(v (x)), thenv (x); v (y) 2 c0 and v (x)\v (y) 6= ? so that v (x)[v (y) � c = u0(z) � u(z)for a suitable z 2 x. then x;y 2 u(z), hence z 2 u�1(x) and y 2 u(u�1(x)).from v �1(v (x)) � u(u�1(x)) we obtain v �1v � uu�1. �a similar (but simpler) argument furnishes:corollary 4.15. let c = cu for a transitive entourage u. the latter is semi-symmetric i�, whenever ci 2 c and c1 \ c2 6= ?, there exists c 2 c satisfyingc1 [ c2 � c.semi-symmetry and point-symmetry are independent concepts also for tran-sitive quasi-uniformities. in fact, the example given above for a point-symmetricbut not semi-symmetric quasi-uniformity was a pervin quasi-uniformity, hencetransitive. for a semi-symmetric but not point-symmetric, transitive quasi-uniformity, consider:example 4.16. let x = fa;bg, c be the closure associated with the sierpi�nskitopology f?;fag;xg, u the (transitive) pervin quasi-uniformity of c generatedby the base fug where u = ufag and cu = ffag;xg. then u(a) = fag,u(b) = x, u�1(a) = x, u�1(b) = fbg. clearly u�1(u(a)) = u�1(u(b)) =u(u�1(a)) = u(u�1(b)) = x so that u is semi-symmetric, but it is not point-symmetric because u�1(a) " u(a).acknowledgements. the author thanks professor vera t. s�os for helpfulsuggestions. references[1] �a. cs�asz�ar, general topology, akad�emiai kiad�o, budapest, 1978.[2] �a. cs�asz�ar, d-completions of pervin-type quasi-uniformities, acta sci. math. (szeged)57 (1993), 329{335.[3] �a. cs�asz�ar and j. de�ak, simultaneous extensions of proximities, semi-uniformities, con-tiguities and merotopies i, math. pannon. 1/2 (1990), 67{90.[4] p. fletcher and w.f. lindgren, quasi-uniform spaces, marcel dekker, inc., new yorkand basel, 1982.[5] m. hall, jr., distinct representatives of subsets, bull. amer. math. soc. 54 (1948), 922{926.[6] p. hall, on representatives of subsets, j. london math. soc. 10 (1935), 26{30.[7] h. herrlich, topological structures, topological structures (proc. symp. in honour of j.de groot, amsterdam, 1973) math. centre tracts 52 (1974), 59{122.[8] m. kat�etov, on continuity structures and spaces of mappings, comment. math. univ.carolinae 6 (1965), 257{278.[9] l. mirsky, transversal theory, academic press, new york and london, 1971.received january 2000 12 �akos cs�asz�ar�akos cs�asz�ardepartment of analysise�otv�os lor�and universitym�uzeum krt. 6{81088 budapesthungarye-mail address: csaszar@ludens.elte.hu nitkaiagt.dvi @ applied general topology c© universidad politécnica de valencia volume 6, no. 1, 2005 pp. 15-41 the character of free topological groups i peter nickolas and mikhail tkachenko 1 abstract. a systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. in the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. from these results, others follow on the basis of various topological assumptions. amongst these: (i) if x is a compact hausdorff space, then the character of the free abelian topological group on x lies between w(x) and w(x)ℵ0 , where w(x) denotes the weight of x; (ii) if the tychonoff space x is not a p-space, then the character of the free abelian topological group is bounded below by the “small cardinal” d; and (iii) if x is an infinite compact metrizable space, then the character is precisely d. in the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. it is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is ω-narrow. a sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact hausdorff spaces and metrizable spaces. 2000 ams classification: primary 22a05, 54h11, 54a25; secondary 54d30, 54d45. keywords: free (abelian) topological group, uniform space, entourage of the diagonal, character, cardinal invariant, compact, locally compact, pseudocompact, metrizable, kω-space, dominating family 1the first author wishes to thank the second author, and his department, for hospitality extended during the course of this work. the second author was supported by mexican national council of sciences and technology (conacyt), grant no. 400200-5-28411-e. 16 p. nickolas and m. tkachenko 1. introduction in 1948, graev [5] proved that the free topological group f(x) and the free abelian topological group a(x) on a tychonoff space x are metrizable only if x is discrete, in which case the groups are themselves discrete. for our present purposes, we may rephrase graev’s result as saying that when x is a non-discrete tychonoff space, the groups f(x) and a(x) have uncountable character (= minimal size of a base at the identity of the group). it appears that no other estimates of the characters of these groups (except for those valid in the context of general topological groups) have been found to date. in this paper and its sequel [14], we investigate these characters systematically and in some detail. most of our results are in fact for free and free abelian topological groups on uniform spaces, since this gives maximum generality and allows the derivation at will of bounds for the characters of free and free abelian groups over topological spaces. in the abelian case, the free topology has a rather straightforward description in terms of the family of uniformly continuous pseudometrics on the given uniform space, and another in terms of the given uniformity itself, and our initial results express the character of the corresponding group in terms of simple cardinal invariants of the family of pseudometrics (theorem 2.3) or of the uniformity (theorem 2.9). an immediate corollary of these results is that if x is a compact hausdorff space, then the character of a(x) lies between w(x) and w(x)ℵ0 , where w(x) denotes the weight of x. the well known “small cardinal” d (see [3] and our discussion below) plays a role in several other results. for example, we show that if the tychonoff space x is not a p-space, then d is a lower bound for the character of a(x) (corollary 2.16); this applies in particular if x contains an infinite compact subset or a proper dense lindelöf subspace. further, if x is an infinite compact metrizable space, then the character of a(x) is precisely d (corollary 2.22). in the non-abelian case, the situation is intrinsically more complex and description of the free topology in terms of pseudometrics or entourages, for example, is now far from straightforward (see [21] and [16]). our main results on the characters in the non-abelian case make use of a new description of a neighborhood base at the identity in the free topological group on an arbitrary uniform space (theorem 3.6). while it is easy to see that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group (lemma 3.1), the inequality is in general strict, and the two characters may indeed differ arbitrarily largely (see [14]). using our new description of the topology of the free topological group, however, we show that the characters of the free abelian and the free (non-abelian) topological groups are equal whenever the underlying uniform space is ω-narrow (theorem 3.15). the sequel [14] to this paper analyses more closely the cases of the free and free abelian topological groups on compact hausdorff spaces and metrizable spaces. the character of free topological groups i 17 1.1. notation and terminology. we denote by n the set of positive integers and by r the set of real numbers. all topological spaces are hypothesised implicitly to be completely regular, but are not taken to be hausdorff (and therefore tychonoff) without explicit indication. similarly, our uniform spaces are not taken to be hausdorff (or separated) unless this is explicitly signalled. we next establish our conventions for certain cardinal invariants of topological spaces and uniform spaces. in some formulations, such as that of [8], for example, cardinal invariants are taken always to be infinite, that is, to have a minimum value of ℵ0, because this tends to simplify the statements of theorems. for us, however, it is convenient not to follow this convention. thus, if x is a point in a space x, then χ(x, x) denotes the minimum cardinal of a local base at x, and then χ(x), the character of x, is the supremum of the values χ(x, x) for x ∈ x. more generally, if y is a subspace of x, then we write χ(y, x) for the minimum cardinal of a base at y in x. we introduce the ad hoc notation χ∆(x) to denote the least cardinal of a basis at the diagonal ∆ in x × x; that is, χ∆(x) = χ(∆, x × x). for a space x, the weight w(x) of x is defined to be the smallest cardinal of an open base for x, and for a uniform space (x, u), we denote by w(x, u) the least cardinal of a base of (x, u). a space x is a kω-space if there exists a sequence of compact subsets xn ⊆ x for n ∈ n such that x = ⋃∞ n=1 xn and such that x has the weak topology with respect to the family {xn : n ∈ n} (that is, such that u ⊆ x is open in x if and only if u ∩ xn is open in xn for each n ∈ n). in this situation, we call the collection {xn : n ∈ n}, or the corresponding expression x = ⋃∞ n=1 xn for x, a kω-decomposition of x. we may take the sets xn without loss of generality to be non-decreasing. if u is a uniformity on a set x, and if u ∈ u and n ≥ 1, we use nu to denote the n-fold relational composite u ◦ u ◦ · · · ◦ u. if −u denotes the relational inverse of u, then we call u symmetric if u = −u. for x ∈ x and u ∈ u, we denote by b(x, u) the set {y ∈ x : (x, y) ∈ u}. if (x, u) is a uniform space and τ ≥ ℵ0 is a cardinal, then we say that (x, u) is τ-narrow if for each u ∈ u, there is a set {xα : α < τ} ⊆ x such that x = ⋃ α<τ b(xα, u). similarly, we say that a topological group g is τ-narrow if g can be covered by τ many translates of every neighborhood of the identity (the groups with this property was called τ-bounded in [6]). for a non-empty set x, we use fa(x) and aa(x) to denote the abstract free group and abstract free abelian group on x. if n ∈ ω, then fn(x) and an(x) are the subsets of fa(x) and aa(x), respectively, which consist of all elements whose length with respect to the basis x does not exceed n. if x is a space, then f(x) and a(x) stand respectively for the free topological group and free abelian topological group on x. in this case, fn(x) and an(x) refer to the corresponding subspaces of f(x) and a(x), respectively. given a subset y of x, f(y, x) denotes the subgroup of f(x) generated by y , and a(y, x) has a similar meaning. 18 p. nickolas and m. tkachenko if (x, u) is a uniform space, then f(x, u) and a(x, u) stand for the free topological group and free abelian topological group on (x, u), and other related notations follow those already introduced for the free and free abelian topological groups on a topological space. if x and y are sets, we write xy for the set of functions from x to y . if α and β are cardinals, we write cardinal exponentiation in the form βα; thus,∣∣xy ∣∣ = |y ||x|. also, c = 2ℵ0 is the power of the continuum. in any topological group g, we denote by n(e) the family of all open neighborhoods of the neutral element e in g. 1.2. quasi-ordered sets. many of the arguments below make use of the notion of a quasi-ordered set, and related ideas. we establish here the relevant terminology and notation. we say that a pair (p, ≤) is a quasi-ordered set if ≤ is a reflexive transitive relation on the set p . if (p, ≤) has the additional property of antisymmetry, then it is a partially ordered set. a set d ⊆ p is called dominating or cofinal in the quasi-ordered set (p, ≤) if for every p ∈ p there exists q ∈ d such that p ≤ q. similarly, a subset e of p is said to be dense in (p, ≤) if for every p ∈ p there exists q ∈ e with q ≤ p. the minimal cardinality of a dominating family in (p, ≤) is denoted by d(p, ≤), while we use d(p, ≤) for the minimal cardinality of a dense set in (p, ≤). the notions of dominating and dense sets are dual: if a set s is dense in (p, ≤), then it is dominating in (p, ≥) and vice versa. therefore, d(p, ≤) = d(p, ≥) and d(p, ≤) = d(p, ≥). note in any topological group g we have d(n(e), ⊆) = χ(g). if (p, ≤) and (q, ≪) are quasi-ordered sets, then a mapping f : p → q is called order-preserving if x ≤ y implies f(x) ≪ f(y), where x, y ∈ p . similarly, f is order-reversing if x ≤ y implies f(x) ≫ f(y). if the mapping f is a bijection between p and q and if f and f−1 both are order-preserving, then f is called an (order-)isomorphism of (p, ≤) onto (q, ≪). lemma 1.1. let (p, ≤) and (q, ≪) be quasi-ordered sets, and let f : p → q be an order-preserving mapping. if f(p) is a dominating set in q, then d(q) ≤ d(p). proof. let d be an arbitrary dominating set in p . for each q ∈ q, there exists p ∈ p such that q ≤ f(p). also, there exists d ∈ d such that p ≤ d, from which we have q ≤ f(p) ≤ f(d). hence f(d) is a dominating set in q, and so d(q) ≤ |f(d)| ≤ |d|. finally, taking d to be a dominating set in p of minimal cardinality gives d(q) ≤ d(p), as required. � observe that because of the duality noted above between dominating sets and dense sets, there are dualised versions of the lemma: a version for dense sets rather than dominating sets, and others for order-reversing mappings rather than order-preserving ones. we will make frequent use of the lemma and its unstated variants, usually without explicit reference. we will deal later with many different quasi-ordered sets, most, though not all, of which will in fact be partially ordered sets. generally, we will the character of free topological groups i 19 give each ordering distinctive notation by using an appropriate subscript, since some arguments make use of several quasi-orderings simultaneously. if ≪ is an ordering on a set x, then we will usually denote the order defined coordinatewise on a set of functions yx by attaching an asterisk as a superscript, as in ≪∗. an exception is when x = ω, when the ordering defined coordinatewise using ≤ on sets such as ωω and nω will again be denoted just by ≤. consider ωω, the collection of all functions from ω to ω. then following [3], we define two quasi-orders ≤∗ and ≤ on ωω, by specifying that if f, g ∈ ωω, then f ≤∗ g if f(n) ≤ g(n) for all except finitely many n ∈ ω, and f ≤ g if f(n) ≤ g(n) for all n ∈ ω; the quasi-order ≤ is of course in fact a partial order. (the use of the asterisk in ≤∗ is inconsistent with the notational convention just established, but we will not in fact use this ordering again after this paragraph.) the least cardinal of a dominating set in the quasi-ordered set (ωω, ≤∗) is denoted by d, and in the partially ordered set (ωω, ≤) by d1. it is shown in theorem 3.6 of [3] that d = d1. it is also known that ℵ1 ≤ d ≤ c, but that the value of d depends on extra axioms of set theory [3]. below, we will find it most useful to use the characterization of d as the least cardinal of a dominating set with respect to the relation ≤: thus, d = d(ωω, ≤). 2. the character of free abelian topological groups graev’s proof in [5] of the fact that the free topological group on a tychonoff space x is hausdorff proceeds by a construction which extends each continuous pseudometric on x to an invariant pseudometric on the underlying abstract free group fa(x), and an argument which shows that the group topology induced by all the extensions (referred to as graev’s topology) is hausdorff and weaker than the free topology. it is well known that graev’s topology is only equal to the free topology in somewhat pathological cases [9, 19, 22]. in the abelian case, a parallel construction was also outlined by graev, and in this case graev’s topology is always the free topology (see [13, 11, 20]; cf. also 4.4 and 4.8 of [10]). (it seems unlikely that this fact was unknown to graev, but it is not mentioned in his paper.) in what follows, we deal mostly with free and free abelian topological groups on uniform spaces, for convenience and generality, deducing applications to free and free abelian topological groups on topological spaces when appropriate. we therefore need the uniform space analog of the result just noted: that graev’s pseudometrics generate a neighborhood base at the neutral element of the free abelian topological group on a uniform space. for a given pseudometric d on a set x, we denote by d̂ and d̂a graev’s extension [5] of d to the abstract groups fa(x) and aa(x), respectively. note that algebraically the free uniform groups f(x, u) and a(x, u) are fa(x) and aa(x), respectively [12, 15]. if (x, u) is a uniform space and d is a pseudometric on x, then we write vd = {g ∈ a(x, u) : d̂a(g, 0) < 1}, 20 p. nickolas and m. tkachenko where 0 is the neutral element of a(x, u). the following theorem is the uniform space analog of the result of [13, 20]; the arguments in the topological case apply with minimal adjustment in the uniform space case, and we therefore omit the proof. theorem 2.1. let (x, u) be a uniform space. then for every uniformly continuous pseudometric d on (x, u) the set vd is open in a(x, u), and for every neighborhood u of the neutral element in a(x, u) there exists a uniformly continuous pseudometric d on (x, u) such that vd ⊆ u. instead of using uniformly continuous pseudometrics, one can directly use the entourages u ∈ u belonging to a uniform space (x, u) to construct a neighborhood base at the identity of a(x, u). the theorem below gives a simple, explicit description in these terms of the topology of the free abelian topological group on a uniform space, equivalent in a sense to theorem 2.1. this result has certainly been known since the early 1980s, when the first description of the topology of the (non-abelian) free topological group on a uniform space was published [16]. though the paper [20] contains a result equivalent to ours stated in the language of pseudometrics, the first occurrence of essentially our result in the literature appears to have been in [26], though it is formulated there in a context less general than ours. first, we introduce the following notation. if (x, u) is a uniform space and if 〈u0, u1, . . .〉 is a sequence of elements of u, then we denote by n(〈u0, u1, . . .〉) the set of all elements of a(x, u) of the form x0 − y0 + x1 − y1 + · · · + xn − yn, where n ∈ ω and where (xi, yi) ∈ ui for i = 0, 1, . . . , n. theorem 2.2. let (x, u) be a uniform space. then the collection of all sets of the form n(〈u0, u1, . . .〉), as 〈u0, u1, . . .〉 runs over all sequences of elements of u, is an open basis at the neutral element in a(x, u). denote by p(x, u) the family of all uniformly continuous pseudometrics on (x, u) bounded by 1. for d1, d2 ∈ p(x, u), we write d1 ≤ d2 if d1(x, y) ≤ d2(x, y) for all x, y ∈ x. we express our first result on the character of a(x, u)) in terms of the partially ordered set (p(x, u), ≤). theorem 2.3. if (x, u) is a uniform space, then χ(a(x, u)) = d(p(x, u), ≤). proof. by theorem 2.1, there exists a natural correspondence between the family p(x, u) and a base at the neutral element 0 of a(x, u). in fact, the mapping d 7→ vd from (p(x, u), ≤) to the partially ordered set (n(0), ⊇) of open neighborhoods of 0 in a(x, u) ordered by reverse inclusion is orderpreserving and maps p(x, u) to a base at 0 in a(x, u), that is, a dominating set in (n(0), ⊇). this immediately implies (by lemma 1.1) that χ(a(x, u)) ≤ d(p(x, u), ≤). suppose that a subset q ⊆ p(x, u) is such that {vd : d ∈ q} is a base at the neutral element in a(x, u). we claim that for every ̺ ∈ p(x, u) there exists the character of free topological groups i 21 d ∈ q such that ̺ ≤ 2d. indeed, given ̺ ∈ p(x, u), we can find d ∈ q such that vd ⊆ v̺. let x, y ∈ x. if d(x, y) < 1, then x−y ∈ vd ⊆ v̺, whence ̺(x, y) < 1. similarly, if n ∈ n and d(x, y) < 2−n, then d̂a(0, 2 n(x − y)) = 2nd(x, y) < 1 by the linearity of the pseudometric d̂a (see [23]). therefore, 2 n(x − y) ∈ vd ⊆ v̺, whence ̺(x, y) < 2 −n. we have thus proved that d(x, y) < 2−n implies ̺(x, y) < 2−n, for n = 0, 1, . . .. in particular, d(x, y) = 0 implies ̺(x, y) = 0. therefore, the inequality ̺(x, y) ≤ 2d(x, y) holds if d(x, y) = 0. it is also obvious that the same inequality holds if d(x, y) = 1. if 0 < d(x, y) < 1, choose n ∈ ω such that 2−n−1 ≤ d(x, y) < 2−n. then ̺(x, y) < 2−n, so that ̺(x, y) ≤ 2d(x, y), and the inequality again holds. thus, ̺ ≤ 2d, proving our claim. for d ∈ q, define d∗ ∈ p(x, u) by setting d∗ = min{2d, 1}, and consider the family q∗ = {d∗ : d ∈ q} ⊆ p(x, u). by the claim just verified, for all ̺ ∈ p(x, u) there exists d∗ ∈ q∗ such that ̺ ≤ d∗, and so q∗ is a dominating family in p(x, u). therefore, d(p(x, u), ≤) ≤ |q∗| ≤ |q|. let b be any base at 0 in a(x, u). then for every n ∈ b, theorem 2.1 shows that there exists dn ∈ p(x, u) such that vdn ⊆ n. set q = {dn : n ∈ b}. then {vd : d ∈ q} is a base at 0 and satisfies |q| ≤ |b|, so that d(p(x, u), ≤) ≤ |q| ≤ |b|. hence d(p(x, u), ≤) ≤ χ(a(x, u)). combining this with the reverse inequality obtained earlier, we conclude that χ(a(x, u)) = d(p(x, u), ≤). � for a topological space x, denote by px the family of all continuous pseudometrics on x bounded by 1. it is clear that px = p(x, ux), where ux is the fine uniformity of x. since a(x) = a(x, ux), from theorem 2.3 we have: corollary 2.4. the equality χ(a(x)) = d(px, ≤) holds for all tychonoff spaces x. we record in passing a couple of consequences of these last results, not otherwise directly related to our main concerns in this paper. first, let (x, u) be an arbitrary uniform space, and let fg(x, u) denote the abstract group fa(x) equipped with the graev topology, the topology generated by graev’s extensions of all the uniformly continuous pseudometrics on (x, u). then we have: theorem 2.5. if (x, u) is a uniform space, then χ(fg(x, u)) = χ(a(x, u)). proof. by theorem 2.3, χ(a(x, u)) = d(p(x, u), ≤). now if {dα : α ∈ a} is a dominating family of pseudometrics in (p(x, u), ≤), then it is clear that the corresponding family of graev extensions {d̂α : α ∈ a} on the group fa(x) induces an open base at the identity, giving χ(fg(x, u)) ≤ χ(a(x, u)). conversely, the natural continuous homomorphism from fg(x, u)) onto a(x, u) is a quotient, from which the inequality χ(a(x, u)) ≤ χ(fg(x, u)) follows, giving the result. � in particular, if x is an arbitrary topological space and fg(x) denotes the group fa(x) equipped with the graev topology, then we have: 22 p. nickolas and m. tkachenko corollary 2.6. χ(fg(x)) = χ(a(x)). second, as we have noted, it is well known [13, 20] that graev’s topology is the free topology on a free abelian topological group a(x), and theorem 2.1 extended this to the case of a free abelian topological group a(x, u). on the other hand, it has been noted more than once [13, 17] that the fact that a certain family of (uniformly) continuous pseudometrics are sufficient together to define the topology (or uniformity) of x does not imply in general that the corresponding family of graev extensions defines the free topology. this is clear, for example, from the observation of graev [5] noted at the start of the first section: since the free abelian topological group a(x) on a tychonoff space x is metrizable only when x is discrete, the metrizability of x implies the metrizability of a(x) only when x is discrete. we can adapt the proof of theorem 2.3 to obtain a necessary and sufficient condition on a family of uniformly continuous pseudometrics under which their graev extensions do indeed define the free topology. the proof is essentially a recapitulation of the second paragraph of the proof of theorem 2.3, and we omit the details. theorem 2.7. let (x, u) be a uniform space and let q ⊆ p(x, u). then the collection of open sets {g ∈ a(x, u) : d̂a(g, 0) < ǫ}, for d ∈ q and ǫ > 0, is a base at 0 for the topology of the free abelian topological group a(x, u) if and only if for every ̺ ∈ p(x, u) there exists d ∈ q such that ̺ ≤ 2d. let (x, u) be a uniform space. given two sequences s = 〈un : n ∈ ω〉 and t = 〈vn : n ∈ ω〉 in ωu, we write s ≤ t provided that un ⊆ vn for each n ∈ ω. it is easy to see that the correspondence s 7→ n(s), where n(s) is as defined immediately before theorem 2.2, is an order-reversing mapping of (ωu, ≤) to (n(0), ⊇). since by theorem 2.2 the family {n(s) : s ∈ ωu} is a base at 0 in a(x, u), we conclude that χ(a(x, u)) ≤ d(ωu, ≤). in fact, we will show shortly that the latter inequality is equality. lemma 2.8. d(ωu, ≤) ≤ d(p(x, u), ≤) for every uniform space (x, u). proof. for d ∈ p(x, u) and n ∈ ω, put on(d) = {(x, y) ∈ x × x : d(x, y) ≤ 2 −n}. it is clear that the correspondence d 7→ 〈on(d) : n ∈ ω〉 is an order-reversing mapping of (p(x, u), ≤) to (ωu, ≤). consider an arbitrary sequence 〈un : n ∈ ω〉 ∈ ωu. clearly, there exists a sequence 〈vn : n ∈ ω〉 ∈ ωu such that vn is symmetric and 3vn+1 ⊆ vn ⊆ un for each n ∈ ω. by theorem 8.1.10 of [4], we can find d ∈ p(x, u) such that on(d) ⊆ vn for each n ∈ ω, so that the function d 7→ 〈on(d) : n ∈ ω〉 maps p(x, u) to a dense set in ωu. it follows, as required, that d(ωu, ≤) ≤ d(p(x, u), ≤). (we note that [4] uses the standing assumption that all uniform spaces are hausdorff (or separated), but it is well known and easy to see that this assumption is unnecessary for the validity of theorem 8.1.10.) � the character of free topological groups i 23 using the observation made before the lemma together with the lemma itself, we have χ(a(x, u)) ≤ d(ωu, ≤) ≤ d(p(x, u), ≤). but by theorem 2.3, we also have χ(a(x, u)) = d(p(x, u), ≤), so we have proved both of the following results. theorem 2.9. if (x, u) is a uniform space, then χ(a(x, u)) = d(ωu, ≤). theorem 2.10. if (x, u) is a uniform space, then d(ωu, ≤) = d(p(x, u), ≤). it is clear that every uniform space (x, u) satisfies w(x, u) = d(u, ⊆) ≤ d(ωu, ≤) ≤ d(u, ⊆)ℵ0 = w(x, u)ℵ0 . hence theorem 2.9 implies the following bounds for the character of the free abelian topological group on (x, u): corollary 2.11. let (x, u) be a uniform space. then w(x, u) ≤ χ(a(x, u)) ≤ w(x, u)ℵ0 . in the case of a compact hausdorff space x, we have w(x, u) = w(x), so the bounds can be simplified as follows. corollary 2.12. if x is a compact hausdorff space, then w(x) ≤ χ(a(x)) ≤ w(x)ℵ0. in theorem 2.15 and corollary 2.16 below we present a different lower bound for the character of most free abelian topological groups. we shall see later that this bound, the cardinal d, is the exact value of the character of a(x) (and indeed of f(x)) when x is an infinite compact metrizable space. first, we recall two useful notions. if every gδ-set in x is open, then x is said to be a p-space. similarly, given a uniform space (x, u), we say that (x, u) is a uniform p-space if the intersection of countably many elements of u is again an element of u. note that if (x, u) is a uniform p-space, then the underlying topological space x is a p-space. in some of the arguments which follow, it is natural to consider separately the cases when (x, u) is a uniform p-space and when (x, u) is not a uniform pspace. in fact, the character of a(x, u) in the “exceptional” case of a uniform p-space (x, u) can be dealt with simply and conclusively in the following form. theorem 2.13. if (x, u) is a uniform p-space, then χ(a(x, u)) = w(x, u). proof. for any uniform space (x, u), the mapping u 7→ 〈u, u, . . .〉 from (u, ⊆) to (ωu, ≤) is an order-preserving embedding, and if (x, u) is also a uniform p-space, then u is mapped to a dense set in ωu, since for an arbitrary 〈u0, u1, . . .〉 ∈ ωu we have 〈u, u, . . .〉 ≤ 〈u0, u1, . . .〉, where u = ⋂ n∈ω un ∈ u. it follows that d(u, ⊆) = d(ωu, ≤), and then the conclusion that χ(a(x, u)) = w(x, u) follows from theorem 2.9. � we now turn to the “usual” case when (x, u) is not a uniform p-space. 24 p. nickolas and m. tkachenko lemma 2.14. if (x, u) is not a uniform p-space, then d ≤ d(p(x, u), ≤). proof. fix a strictly decreasing sequence 〈un : n ∈ ω〉 ∈ ωu such that u0 = x × x and ⋂ n∈ω un is not in u. for any s = 〈vn : n ∈ ω〉 ∈ ωu, we define a function fs ∈ ωω by fs(n) = max{k ∈ ω : vn ⊆ uk}, for n ∈ ω. since ⋂ n∈ω un /∈ u, we have fs(n) < ∞ for each n ∈ ω, so our definition of fs is valid. moreover, it is easy to see that if 〈vn : n ∈ ω〉 = s ≤ t = 〈wn : n ∈ ω〉 then fs ≥ ft, so that the mapping s 7→ fs from ( ωu, ≤) to (ωω, ≤) is order-reversing. we claim that the set {fs : s ∈ ωu} is dominating in (ωω, ≤). indeed, for any f ∈ ωω, let f̂ be a strictly increasing function in ωω such that f ≤ f̂. now set s = 〈u f̂(n) : n ∈ ω〉 ∈ ωu, and note that s is a strictly decreasing sequence of sets. then fs(n) = max{k ∈ ω : uf̂(n) ⊆ uk} = f̂(n) for all n ∈ ω, so that fs = f̂. therefore, f ≤ fs, proving our claim. this immediately implies that d = d(ωω, ≤) ≤ d(ωu, ≤) = d(p(x, u), ≤), by theorem 2.10, as required. � theorem 2.3 and lemma 2.14 imply the following lower bound, complementing theorem 2.13, for the character of a(x, u). theorem 2.15. if (x, u) is not a uniform p-space, then d ≤ χ(a(x, u)). theorem 2.15 implies several important corollaries. corollary 2.16. if a tychonoff space x is not a p-space, then d ≤ χ(a(x)). corollary 2.17. let (x, u) be a hausdorff uniform space which contains an infinite precompact subset. then d ≤ χ(a(x, u)). proof. suppose that p is an infinite precompact subset of (x, u). let (y, v) be the completion of the space (x, u) and let k be the closure of p in y (we identify x with the corresponding dense subspace of y ). then k is an infinite compact subset of y . the group a(x, u) is topologically isomorphic to a dense subgroup of a(y, v) by nummela’s theorem [15], so that χ(a(x, u)) = χ(a(y, v)). since the infinite compact set k cannot be a p-space, (y, v) fails to be a uniform p-space. therefore, theorem 2.15 implies that d ≤ χ(a(y, v)) = χ(a(x, u)). � corollary 2.18. if a tychonoff space x contains an infinite compact set (in particular, a non-trivial convergent sequence), then d ≤ χ(a(x)). corollary 2.19. if a tychonoff space x contains a proper dense lindelöf subspace, then d ≤ χ(a(x)). proof. suppose that y is a proper dense lindelöf subspace of x. then for every point x ∈ x \ y , there exists a gδ-set px in x containing x such that px ∩ y = ∅. if x were a p-space, the complement x \ y would be open in x, the character of free topological groups i 25 thus contradicting the assumption that y is dense in x. hence the conclusion follows from corollary 2.16. � one cannot omit the word “proper” in corollary 2.19, since the character of the free abelian topological group over the one-point lindelöfication of a discrete space of cardinality ℵ1 is exactly equal to ℵ1 (this follows from [7, lemma 2.9]). lemma 2.20. let (x, u) be a uniform space. if d(u, ⊆) = ℵ0, then d( ωu, ≤) = d. proof. choose a dense set {u0, u1, . . .} in (u, ⊆) such that un strictly contains un+1 for all n ∈ ω. then it is easy to see that the mapping (n0, n1, . . .) 7→ (un0, un1, . . .) from ( ωω, ≤) to (ωu, ≤) is order-reversing and maps ωω onto a dense subset of ωu, which gives d(ωu, ≤) ≤ d(ωω, ≤) = d. conversely, if we map (v0, v1, . . .) ∈ ωu to the sequence (n0, n1, . . .) ∈ ωω defined by setting nk = min{m ∈ ω : um ⊆ vk} for all k ∈ ω, then the mapping is easily seen to be order-reversing and to map ωu onto a dominating subset of ωω, giving d = d(ωω, ≤) ≤ d(ωu, ≤), and the result. � the following result is immediate from lemma 2.20 and theorem 2.9. theorem 2.21. let (x, u) be a uniform space satisfying w(x, u) = ℵ0. then χ(a(x, u)) = d. since a uniform space is pseudometrizable if and only if it has a countable base, we have in particular: corollary 2.22. if x is an infinite compact metrizable space, then χ(a(x)) = d. theorem 2.23. let x be a tychonoff space satisfying χ∆(x) ≤ ℵ0. then either x and a(x) are discrete or χ(a(x)) = d. proof. by theorem 14 in [18], from χ∆(x) ≤ ℵ0 it follows that the set x ′ of all non-isolated points in x is compact and χ(x′, x) ≤ ℵ0. if x ′ = ∅, then both x and a(x) are discrete. suppose, therefore, that x′ 6= ∅. clearly, x admits a perfect mapping onto a space y with a single non-isolated point (map x′ to a point). therefore, both y and x are paracompact, so that every neighborhood of the diagonal in x2 belongs to the fine uniformity u of x. in particular, w(x, u) = χ∆(x) ≤ ℵ0. the result now follows from theorem 2.21. � 3. the character of free topological groups the next lemma establishes a simple relation between the characters of the groups a(x, u) and f(x, u). lemma 3.1. if (x, u) is a uniform space, then χ(a(x, u)) ≤ χ(f(x, u)). 26 p. nickolas and m. tkachenko proof. since a(x, u) is a quotient group of f(x, u) and continuous open homomorphisms do not raise the character, we have χ(a(x, u)) ≤ χ(f(x, u)). � similarly, of course, for a topological space x, we have χ(a(x)) ≤ χ(f(x)). thus, each of the lower bounds we have derived above for χ(a(x, u)) or χ(a(x)) yields automatically a corresponding lower bound for χ(f(x, u)) or χ(f(x)), respectively. corollary 2.19, for example, gives us the bound d ≤ χ(a(x)) ≤ χ(f(x)) for a space x containing a proper dense lindelöf subspace. in fact, however, the conclusion of corollary 2.19 can be strengthened to d ≤ χ(a(x)) = χ(f(x)) (see corollary 3.18 below), but this is not at all straightforward. our aim now is to establish the equality χ(a(x)) = χ(f(x)) for the wide class of ℵ0-narrow spaces, which are also known as pseudo-ω1compact spaces. by definition, a space x is pseudo-ω1-compact if every discrete family of open sets in x is countable. our choice of the new name for this class of spaces is motivated (apart from the aesthetic reason) by the fact that x is pseudo-ω1-compact if and only if the uniform space (x, ux) is ℵ0-narrow, where ux is the fine uniformity of x (see [24, assertion 1.2]). our arguments require some unpleasant work describing a neighborhood base in a free topological group. since the cases of the free topological group on a topological space and a uniform space are very similar, we prefer to present the description in the most general form, for free topological groups on uniform spaces. first we recall some notions related to trees. a partially ordered set (p, ≤) is a tree if the set px = {y ∈ p : y < x} is well ordered by < for each x ∈ p (where we write y < x if and only if y ≤ x and y 6= x). the height of an element x ∈ p , denoted by h(x), is the order type of the set (px, <). for an ordinal α, we call the set p(α) = {x ∈ p : h(x) = α} the αth level of (p, ≤). finally, the height of (p, ≤) is defined to be the smallest ordinal α such that p(α) = ∅. in what follows we shall work only with trees of height ω. clearly, if the height of p is ω, then every x ∈ p has only finitely many predecessors with respect to ≤. definition 3.2. let (x, u) be a uniform space. we say that a tree (p, ≤) of height ω is u-covering if it satisfies the following conditions: (i) each element x ∈ p has the form x = (u0, . . . , un), where u0, . . . , un are non-empty open sets in x and n ∈ ω; (ii) if x = (u0, . . . , un) and y = (v0, . . . , vm) are in p, then x ≤ y if and only if n ≤ m and ui = vi for each i = 0, . . . , n; (iii) the family γp = {v : (v ) ∈ p(0)} is a u-uniform cover of x; (iv) if x = (u0, . . . , un) ∈ p, then the family γp (x) = {v : (u0, . . . , un, v ) ∈ p} is a u-uniform cover of x. the character of free topological groups i 27 denote by t (x, u) the family of all u-covering trees. for (p, ≤) ∈ t (x, u), we define a subset wp of fa(x) as follows: (3.1) wp = ⋃ (u0,...,un)∈p, ε0,...,εn=±1, n∈ω u−ε00 · · · u −εn n u εn n · · · u ε0 0 the next lemma is the first step towards the promised description of a neighborhood base at the identity of free topological groups. lemma 3.3. for every neighborhood o of the identity e in f(x, u), there exists (p, ≤) ∈ t (x, u) such that wp ⊆ o. in addition, if the space (x, u) is τ-narrow for some τ ≥ ℵ0, then one can choose (p, ≤) satisfying |p | ≤ τ. proof. denote by o(e) the family of all open symmetric neighborhoods of e in the group f(x, u). for a given o ∈ o(e), choose w ∈ o(e) such that w 3 ⊆ o. suppose that the uniform space (x, u) is τ-narrow for some τ ≥ ℵ0. then the group f(x, u) is τ-narrow by [6] or [2, lemma 3.2]. for every x ∈ x, put ux = x · w ∩ x ∩ w · x. since the two-sided uniformity of f(x, u) induces on x its original uniformity u [15], we can find a set y ⊆ x with |y | ≤ τ such that x = ⋃ x∈y ux. put p(0) = {(ux) : x ∈ y }. then |p(0)| ≤ τ and γp = {ux : x ∈ y } is a u-uniform cover of x. this defines the initial level of the required tree p . let us describe the second step of the construction. for every x ∈ y , choose vx, wx ∈ o(e) such that x −ε · vx · x ε ⊆ w for ε = ±1 and w 3x ⊆ vx, and put ux,y = y · wx ∩ x ∩ wx · y for each y ∈ x. since the space (x, u) is τ-narrow, we can choose, given any x ∈ y , a set y (x) ⊆ x with |y (x)| ≤ τ satisfying x = ⋃ y∈y (x) ux,y. put γp (x) = {ux,y : y ∈ y (x)}. then γp (x) is a u-uniform cover of x for each x ∈ y , and we define the level p(1) of the tree p by p(1) = {(ux, ux,y) : x ∈ y, y ∈ y (x)}. clearly, |p(1)| ≤ τ. at the third step, for each x ∈ y and for each y ∈ y (x), choose vx,y, wx,y ∈ o(e) such that y−ε · vx,y · y ε ⊆ wx for ε = ±1 and w 3 x,y ⊆ vx,y. for z ∈ x, put ux,y,z = z · wx,y ∩ x ∩ wx,y · z and choose a subset y (x, y) of x such that |y (x, y)| ≤ τ and x = ⋃ z∈y (x,y) ux,y,z. put γp (x, y) = {ux,y,z : z ∈ y (x, y)}. then γp (x, y) is a u-uniform cover of x for each x ∈ y and each y ∈ y (x), and we define the level p(2) of the tree p by p(2) = {(ux, ux,y, ux,y,z) : x ∈ y, y ∈ y (x), z ∈ y (x, y)}. clearly, |p(2)| ≤ |p(1)| · τ ≤ τ. continuing this process, we finally obtain the set p = ⋃ n∈ω p(n), partially ordered according to (ii) of definition 3.2, and such that |p | ≤ τ. one easily verifies that (p, ≤) satisfies (i)–(iv), so that (p, ≤) ∈ t (x, u). it remains to show that wp ⊆ o. let (u0, u1, . . . , un) be an element of p . we have to verify that u−ε00 · · · u −εn n u εn n · · · u ε0 0 ⊆ o for arbitrary ε0, . . . , εn = 28 p. nickolas and m. tkachenko ±1. there exist points x0 ∈ y, x1 ∈ y (x0), . . . , xn ∈ y (x0, . . . , xn−1) such that u0 = ux0, u1 = ux0,x1, . . . , un = ux0,x1,...,xn. by definition, we have u0 ⊆ (x0 · w) ∩ (w · x0), u1 ⊆ (x1 · wx0) ∩ (wx0 · x1), . . . . . . un ⊆ (xn · wx0,...,xn−1) ∩ (wx0,...,xn−1 · xn). we claim that (3.2) u −εk+1 k+1 · · · u −εn n u εn n · · · u εk+1 k+1 ⊆ vx0,...,xk for each k = 0, 1, . . . , n − 1. indeed, if k = n − 1, then u−1n un ⊆ (xnwx0,...,xn−1) −1 · xnwx0,...,xn−1 = w 2 x0,...,xn−1 ⊆ vx0,...,xn−1, and, similarly, un · u −1 n ⊆ wx0,...,xn−1xn · (wx0,...,xn−1xn) −1 = w 2x0,...,xn−1 ⊆ vx0,...,xn−1, giving (3.2) for k = n − 1. suppose that (3.2) holds for some k > 0. if εk = 1, we have uk ⊆ xk · wx0,...,xk−1, whence u−1 k u −εk+1 k+1 · · · u −εn n u εn n · · · u εk+1 k+1 uk ⊆ u−1 k vx0,...,xkuk ⊆ w −1 x0,...,xk−1 · x−1 k · vx0,...,xk · xk · wx0,...,xk−1 ⊆ w 3x0,...,xk−1 ⊆ vx0,...,xk−1. similarly, if εk = −1, then we use the inclusion uk ⊆ wx0,...,xk−1 · xk to deduce that uku −εk+1 k+1 · · · u −εn n u εn n · · · u εk+1 k+1 u −1 k ⊆ vx0,...,xk−1. the inclusion (3.2) now follows. finally, from u0 ⊆ x0 · w , and using (3.2) with k = 0, it follows that u−10 u −ε1 1 · · · u −εn n u εn n · · · u ε1 1 u0 ⊆ w −1 · x−10 · vx0 · x0 · w ⊆ w 3 ⊆ o, and similarly, from u0 ⊆ w · x0 it follows that u0u −ε1 1 · · · u −εn n u εn n · · · u ε1 1 u −1 0 ⊆ o. since wp is the union of the sets of the form u −ε0 0 · · · u −εn n u εn n · · · u ε0 0 , we have proved the inclusion wp ⊆ o. � remark 3.4. given the statement of the lemma just proved, it is worth remarking that the family {wp : p ∈ t (x, u)} does not in general constitute a base at the identity in the group f(x, u), or indeed in any group topology on the group fa(x). in fact, for certain p ∈ t (x, u), one cannot find q ∈ t (x, u) with wq · wq ⊆ wp . we will shortly show how a rather more elaborate family of sets constructed using the sets wp do form an open base in f(x, u). the character of free topological groups i 29 let us first establish some other properties of the sets wp . lemma 3.5. suppose that p, q ∈ t (x, u) and g ∈ fa(x). then one can find r ∈ t (x, u) such that wr ⊆ wp ∩ wq and g −1 · wr · g ⊆ wp . proof. the existence of r ∈ t (x, u) satisfying wr ⊆ wp ∩ wq is immediate. hence it suffices to construct a u-covering tree r such that g−1 · wr · g ⊆ wp . the existence of the required tree r is clear if g is the identity e of fa(x). if g 6= e, then it suffices to consider the case when g ∈ x ∪ x−1 and then apply induction on the length of g in the general case. so we assume that g ∈ x ∪ x−1. since γp = {u : (u) ∈ p(0)} is a cover of x, there exist u0 ∈ γp and ε0 = ±1 such that g ∈ u ε0 0 . put r = {(u1, . . . , un) : (u0, u1, . . . , un) ∈ p, n ≥ 1}. it is easy to see that r ∈ t (x, u), and we claim that g−1 · wr · g ⊆ wp . indeed, if (u1, . . . , un) ∈ r and ε1, . . . , εn = ±1, then g−1 · u−ε11 · · · u −εn n u εn n · · · u ε1 1 · g ⊆ u −ε0 0 u −ε1 1 · · · u −εn n u εn n · · · u ε1 1 u ε0 0 ⊆ wp . this proves the inclusion g−1 · wr · g ⊆ wp . � now we present our description of a neighborhood base at the identity of f(x, u) in terms of u-covering trees. again, we need some definitions. let s = 〈pn : n ∈ n〉 ∈ nt (x, u) be a sequence of u-covering trees. then we define a set os ⊆ fa(x) as follows: (3.3) os = ⋃ n∈n ⋃ π∈sn wpπ(1) · · · wpπ(n), where sn is the group of permutations of the set {1, . . . , n}. theorem 3.6. the family σ = {os : s ∈ nt (x, u)} forms a base at the identity e of the group f(x, u). proof. our argument is close to that of [21, th. 1.1]. it suffices to verify that the following assertions are valid: (a) σ is a base for a group topology t ∗ on fa(x); (b) t ∗ is finer than the topology t of the group f(x, u); (c) the two-sided uniformity v of the group g = (fa(x), t ∗) induces on x a uniformity coarser than u. since f(x, u) carries the finest group topology whose two-sided uniformity induces on x the uniformity u, from (a)–(c) it follows that t ∗ = t . let us start with (a). (a) to verify that σ is a base at e for a group topology on fa(x), it suffices to show that σ has the following four properties: (1) for every u, v ∈ σ there exists w ∈ σ with w ⊆ u ∩ v ; (2) for every u ∈ σ there exists v ∈ σ with v −1 · v ⊆ u; (3) for every u ∈ σ and g ∈ u there exists v ∈ σ with v · g ⊆ u; (4) for every u ∈ σ and g ∈ fa(x) there is v ∈ σ such that g −1 ·v ·g ⊆ u. 30 p. nickolas and m. tkachenko we only check (2), (3) and (4), since (1) is immediate from lemma 3.5. note that by definition, the set wp is symmetric for each p ∈ t (x, u), and so is os for each s ∈ nt (x, u). let u ∈ σ be arbitrary. then u = os for some s ∈ nt (x, u), say s = 〈pn : n ∈ n〉. let us check (2). by lemma 3.5, we can find a sequence t = 〈qn : n ∈ n〉 ∈ nt (x, u) such that wqn ⊆ wp2n−1 ∩ wp2n for each n ∈ n. we claim that o−1t ·ot ⊆ os. thus, we take m, n ∈ n and π ∈ sm and ̺ ∈ sn, and show that wqπ(1) · · · wqπ(m) wq̺(1) · · · wq̺(n) ⊆ os. in fact, however, it suffices to assume that m = n here, since each wqp contains e. thus, let n ∈ n and π, ̺ ∈ sn. define σ ∈ s2n by σ(i) = 2π(i) if 1 ≤ i ≤ n and σ(i) = 2̺(i − n) − 1 if n < i ≤ 2n. then from our definition of σ and t it follows that wqπ(1) · · · wqπ(n)wq̺(1) · · · wq̺(n) ⊆ wpσ(1) · · · wpσ(n) wpσ(n+1) · · · wpσ(2n) ⊆ os. this along with (3.3) implies that o−1t ·ot = ot ·ot ⊆ os, as claimed, and the set v = ot ∈ σ is as required. to verify (3), take an arbitrary g ∈ u = os. then g ∈ wpπ(1) · · · wpπ(k) for some k ∈ n and π ∈ sk. put qn = pn+k for each n ∈ n and consider t = 〈qn : n ∈ n〉. then t ∈ nt (x, u) and the set v = ot satisfies v · g ⊆ u. indeed, for n ∈ n and σ ∈ sn, define ̺ ∈ sn+k by ̺(i) = σ(i) + k if i ≤ n and ̺(i) = π(i − n) if n < i ≤ n + k. then wqσ(1) · · · wqσ(n) · g ⊆ wqσ(1) · · · wqσ(n) wpπ(1) · · · wpπ(k) = wp̺(1) · · · wp̺(n+k) ⊆ os, so that ot · g ⊆ os or, equivalently, v · g ⊆ u. the verification of (4) is similar. let g be an arbitrary element of fa(x). by lemma 3.5, for every n ∈ n there exists qn ∈ t (x, u) such that g −1 ·wqn ·g ⊆ wpn. put t = 〈qn : n ∈ n〉. then t ∈ nt (x, u) and g−1 · ot · g ⊆ os. indeed, if n ∈ n and π ∈ sn, then we have g−1 · wqπ(1) · · · wqπ(n) · g = (g −1 · wqπ(1) · g) · · · (g −1 · wqπ(n) · g) ⊆ wpπ(1) · · · wpπ(n) ⊆ os. this implies that g−1 · ot · g ⊆ os, and so the set v = ot ∈ σ is as required. we conclude, therefore, that σ is a base for a group topology t ∗ on fa(x). this proves (a). (b) let o be an arbitrary neighborhood of e in f(x, u). choose a sequence 〈vn : n ∈ ω〉 of open symmetric neighborhoods of e in f(x, u) such that v0 ⊆ o and v 3 n+1 ⊆ vn for each n ∈ ω. by lemma 3.3, for every n ∈ n there the character of free topological groups i 31 exists pn ∈ t (x, u) such that wpn ⊆ vn. note that if n ∈ n and π ∈ sn, then vπ(1) · · · vπ(n) ⊆ v0 by lemma 1.3 of [21]. this immediately implies that wpπ(1) · · · wpπ(n) ⊆ vπ(1) · · · vπ(n) ⊆ v0, whence it follows that os ⊆ v0 ⊆ o, where s = 〈pn : n ∈ n〉. this proves that the topology t ∗ generated by the family σ is finer than t . (c) let v = os, where s = 〈pn : n ∈ n〉 ∈ nt (x, u) is arbitrary. put w = ⋃ (u)∈p1(0) u × u. then w ∈ u, and from the definition of wp1 it follows that w ⊆ {(x, y) ∈ x × x : x−1y ∈ wp1, xy −1 ∈ wp1 }. since wp1 ⊆ os, we conclude that w ⊆ {(x, y) ∈ x × x : x−1y ∈ os, xy −1 ∈ os}. therefore, the restriction of the two-sided uniformity of the group (fa(x), t ∗) to the set x ⊆ fa(x) is coarser than u. the proof is complete. � suppose that (x, u) is an ℵ0-narrow uniform space. roughly speaking, theorem 3.6 and lemma 3.3 show that one has to use only countably many elements of u to produce a basic neighborhood of the identity in f(x, u). we use this fact as well as the next two lemmas in the proof of theorem 3.10. for any uniform space (x, u), we denote by c(u) the family of all u-uniform covers of x, and for γ, λ ∈ c(u), we write γ ≺ λ provided that γ refines λ. lemma 3.7. if (x, u) is a uniform space, then each of the partially ordered sets (u, ⊆) and (c(u), ≺) admits an order-preserving mapping onto a dense subset of the other, and hence d(u, ⊆) = d(c(u), ≺). proof. for each u ∈ u, we set γu = {b(x, u) : x ∈ x}, where we recall that b(x, u) denotes the set {y ∈ x : (x, y) ∈ u} for each x ∈ x. clearly, γu is a u-uniform cover of x. it is easy to see that the mapping u 7→ γu of (u, ⊆) to (c(u), ≺) is order-preserving, and the set {γu : u ∈ u} is obviously dense in (c(u), ≺). conversely, for every γ ∈ c(u), put wγ = ⋃ {v × v : v ∈ γ}. it is clear that the mapping γ 7→ wγ of (c(u), ≺) to (u, ⊆) is order-preserving. to show that {wγ : γ ∈ c(u)} is dense in (u, ⊆), let u ∈ u and take v ∈ u which is symmetric and satisfies 2v ⊆ u. then wγv = ⋃ x∈x b(x, v ) × b(x, v ), and it is easy to check that wγv ⊆ u, as required. the equality d(u, ⊆) = d(c(u), ≺) now follows from (a version of) lemma 1.1. � 32 p. nickolas and m. tkachenko suppose that s = 〈γn : n ∈ ω〉 and t = 〈λn : n ∈ ω〉 are two sequences of u-uniform covers of x, that is, that s, t ∈ ωc(u). we write s ≺ t if γn ≺ λn for each n ∈ ω. this defines the partially ordered set (ωc(u), ≺), and the next result follows directly from lemma 3.7 and theorem 2.10. corollary 3.8. d(ωc(u), ≺) = d(ωu, ≤) = d(p(x, u), ≤) for every uniform space (x, u). our next step is to show that the difference between the characters of the groups a(x, u) and f(x, u) cannot be too big for any ℵ0-narrow uniform space (x, u) (see theorem 3.10). first, we deal with the special case when the uniform space has a countable base. technically, this is the most difficult part of the work. lemma 3.9. let an ℵ0-narrow uniform space (x, u) satisfy w(x, u) ≤ ℵ0. then χ(f(x, u)) ≤ d. proof. (i) let {un : n ∈ ω} be a countable base for the uniformity u. since (x, u) is ℵ0-narrow, we can choose a sequence γ = 〈γn : n ∈ ω〉 of countable u-uniform covers of x such that γn+1 ≺ γn and ⋃ {v × v : v ∈ γn} ⊆ un for each n ∈ ω. note that the space (x, u) is pseudometrizable (and hence paracompact, using the term without the assumption of separation which most authors include in the definition of paracompactness), and so each cover γn can be additionally chosen to be locally finite. we can further assume that if u ∈ γm and v ∈ γn where m > n, then u does not properly contain v , that is, that u ⊇ v implies u = v . (ii) we define an order ≤t on the collection t (x, u) of u-covering trees by specifying that for p, q ∈ t (x, u), we have p ≤t q if wp ⊇ wq (see equation (3.1)). clearly, ≤t is a quasi-order on t (x, u), though not a partial order. denote by t (γ) ⊆ t (x, u) the set of all u-covering trees p with the property that γp ∈ γ and γp (x) ∈ γ for each x ∈ p (see (iii) and (iv) of definition 3.2). also, set γ∗ = ⋃ n∈ω γn. then our definition of the sequence γ = 〈γn : n ∈ ω〉 implies that every v ∈ γ ∗ is contained only in finitely many distinct elements of γ∗. claim 1. t (γ) is a dominating subset of (t (x, u), ≤t). for subsets u1, . . . , un−1, un of x, we write π̂(u1, . . . , un−1, un) = (u1, . . . , un−1) if n ≥ 2, and we write π(u1, . . . , un−1, un) = un if n ≥ 1. to prove the claim, let p ∈ t (x, u). we construct a tree q as follows. since γp is a u-uniform cover of x, there is δ ∈ γ such that δ ≺ γp . we set q(0) = {x ∈ (γ∗)1 : π(x) ∈ δ}. since δ ≺ γp , we can choose a function p0 : q(0) → p(0) such that π(x) ⊆ π(p0(x)) for all x ∈ q(0). now for each x ∈ q(0), we pick δx ∈ γ such δx ≺ γp (p0(x)). then we put q(1) = {y ∈ the character of free topological groups i 33 (γ∗)2 : x = π̂(y) ∈ q(0), π(y) ∈ δx}. since δx ≺ γp (p0(x)) for each x ∈ q(0), we can choose a function p1 : q(1) → p(1) such that π̂(p1(y)) = p0(π̂(y)) and π(y) ⊆ π(p1(y)) for all y ∈ q(1). now for each y ∈ q(1), we pick δy ∈ γ such δy ≺ γp (p1(y)), and then put q(2) = {z ∈ (γ ∗)3 : y = π̂(z) ∈ q(1), π(z) ∈ δy}. continuing in this way, we finally obtain the tree q = ⋃ n∈ω q(n), ordered according to (ii) of definition 3.2. we clearly have q ∈ t (γ). also, if u = (u0, . . . , un) ∈ q(n), then, by construction, pn(u) ∈ p(n), and if we write pn(u) = (v0, . . . , vn), say, then we have u0 ⊆ v0, . . . , un ⊆ vn, so that each product in the union of the form (3.1) defining wq is contained in some product in the corresponding union defining wp , giving wq ⊆ wp , and hence p ≤t q, as required to prove the claim. (iii) if we define e = {(v0, . . . , vn) : v0, . . . , vn ∈ γ ∗, n ∈ ω}, then it is clear that |e| ≤ ℵ0. let f = eω be the family of all mappings from e to ω. define a partial order ≤e on e as follows: for p = (u0, . . . , uk), q = (v0, . . . , vl) ∈ e, we define p ≤e q if and only if k ≤ l and ui ⊇ vi for i = 0, . . . , k. now we define a “strong” partial order ≤s on f = eω, as follows (the name and notation distinguish the relation from another that we will define later on the same set). for f, g ∈ f, we write f ≤s g if f = g or if p ≤e q in e always implies f(p) ≤ g(q) in ω. it is straightforward to verify that ≤s is indeed a partial order. we also define a partial order ≪s on ω × f coordinate-wise, using the usual order ≤ on ω and the order ≤s on f. we define a function π: ω × f → t (x, u). for a pair (m, f) ∈ ω × f, the u-covering tree p = π(m, f) is defined as follows. define the initial level of p by p(0) = {(u) : u ∈ γm}. suppose that we have defined p(n) ⊆ (γ ∗)n+1 for some n ∈ ω. given an element p = (u0, . . . , un) ∈ p(n), put p+ = {(u0, . . . , un, v ) : v ∈ γf(p)} and then set p(n + 1) = ⋃ {p+ : p ∈ p(n)}. to finish the construction, we put p = ⋃ n∈ω p(n) and define the partial order ≤ on p as in definition 3.2. note that the tree p = π(m, f) is in fact an element of t (γ) ⊆ t (x, u). claim 2. the mapping π is order-preserving as a mapping from (ω×f, ≪s) to (t (x, u), ≤t). thus, we suppose that m, n ∈ ω and m ≤ n and that f, g ∈ f and f ≤s g, and we show that wπ(n,g) ⊆ wπ(m,f). indeed, by (3.1), wπ(n,g) is the union over all k of all the sets of the form v −ε00 · v −ε1 1 · · · v −εk k · v εk k · · · v ε11 · v ε0 0 , where v0 ∈ γn, v1 ∈ γg(q1), v2 ∈ γg(q2), . . . , vk ∈ γg(qk), 34 p. nickolas and m. tkachenko where q1 = (v0), q2 = (v0, v1), . . . , qk = (v0, v1, . . . , vk−1), and where ε0, ε1, . . . , εk = ±1. fix one such set. since γn refines γm, there exists u0 ∈ γm such that v0 ⊆ u0. put p1 = (u0). then from f ≤s g it follows that f(p1) ≤ g(q1). hence, γg(q1) refines γf(p1), so there exists u1 ∈ γf(p1) such that v1 ⊆ u1. from f ≤s g it follows that f(p2) ≤ g(q2), where p2 = (u0, u1). again, γg(q2) refines γf(p2), so there exists u2 ∈ γf(p2) such that v2 ⊆ u2. continuing this way, we finally obtain uk ∈ γf(pk) such that vk ⊆ uk, where pk = (u0, u1, . . . , uk−1). clearly, pk ∈ π(m, f), so that the set u−ε00 · u −ε1 1 · · · u −εk k · uεk k · · · uε11 · u ε0 0 is a summand in the union of the form (3.1) corresponding to wπ(m,f). by construction, we have vi ⊆ ui for each i = 0, . . . , k, whence it follows that v −ε00 · · · v −εk k · v εk k · · · v ε00 ⊆ u −ε0 0 · · · u −εk k · uεk k · · · uε00 . we conclude, therefore, that wπ(n,g) ⊆ wπ(m,f), proving our claim. we claim next that π(ω × f) = t (γ). that π(ω × f) ⊆ t (γ) is clear. for the reverse inclusion, let p ∈ t (γ). now γp ∈ γ, so that γp = γn ∈ γ for some n ∈ n. also, for all x ∈ p , we have γp (x) ∈ γ, so that γp (x) = γnx ∈ γ for some nx ∈ n. define f : e → ω, that is, f ∈ f, by setting f(x) = nx for all x ∈ p and f(x) = 0 for all x /∈ p . then it is easy to see that π(n, f) = p , proving that π(ω × f) ⊇ t (γ), and hence our claim. we now consider the quasi-ordered sets (n(ω × f), ≪∗s) and ( nt (x, u), ≤∗t ), where the orders ≪∗s and ≤ ∗ t are defined coordinate-wise in terms of ≪s and ≤t, respectively. we likewise consider the mapping π∗ : (n(ω × f), ≪∗s) → ( nt (x, u), ≤∗t ) defined coordinate-wise in terms of π. it is immediate from what we have shown above that the mapping π∗ is order-preserving, and maps n(ω × f) to a dominating subset of (nt (x, u), ≤∗t ). thus, we conclude that (3.4) d(nt (x, u), ≤∗t ) ≤ d( n(ω × f), ≪∗s). (iv) we wish to obtain a different expression for the right-hand side of (3.4). to this end, we define a “weak” partial order ≤w on f = eω, as follows. for f, g ∈ f, we write f ≤w g if f(p) ≤ g(p) for each p ∈ e. it is clear that ≤w is a partial order. it is also clear that f ≤s g implies f ≤w g. now we define ≪w on ω × f and then ≪∗w on n(ω × f) in exact analogy to the earlier definitions of ≪s and ≪ ∗ s. claim 3. d(n(ω × f), ≪∗s) = d( n(ω × f), ≪∗w). indeed, it is clear that d(n(ω × f), ≪∗w) ≤ d( n(ω × f), ≪∗s), since every dominating set in (n(ω×f), ≪∗s) remains dominating in ( n(ω×f), ≪∗w). next, we have to verify that d(n(ω × f), ≪∗s) ≤ d( n(ω × f), ≪∗w). for q ∈ e, put e(q) = {p ∈ e : p ≤e q}. the character of free topological groups i 35 from our assumptions about the covers γn, it follows that e(q) is finite for every q ∈ e. in fact, suppose that there exists q = (v0, . . . , vl) ∈ e such that e(q) is infinite. then for some k ≤ l, the set ek(q) = {p ∈ e : p = (u0, . . . , uk), p ≤e q} is infinite. then for some i ≤ k, we can find a sequence of elements pn = (u (n) 0 , . . . , u (n) i , . . . , u (n) k ) ∈ ek(q) for n ∈ n such that the sets u (n) i are distinct for all n and such that u (n) i ⊇ vi for all n. if n0 ∈ n is such that vi ∈ γn0, then u (n) i must properly contain vi for infinitely many n, and it follows that there is n1 ≤ n0 such that infinitely many of the u (n) i are in γn1, contradicting the local finiteness of γn1. this contradiction shows that e(q) is finite, as claimed. it is clear furthermore that each e(q) is also non-empty. now for every f ∈ f and q ∈ e, put f̃(q) = max{f(p) : p ∈ e(q)}, noting that by the argument above the value f̃(q) is defined validly. therefore, we obtain a function f̃ : e → ω, that is, f̃ ∈ f. it is easy to see that the mapping f 7→ f̃ from (f, ≤w) to (f, ≤s) is order-preserving, and from our definition of the order ≤s and the function f̃, it is also easily checked that f ≤s f̃ for each f ∈ f, so that the image of the mapping is a dominating subset of (f, ≤s). for ϕ ∈ n(ω × f), we have ϕ(n) ∈ ω × f for each n ∈ n, and we have ϕ(n) = (π1(ϕ(n)), π2(ϕ(n))), where π1 and π2 are the projections from ω × f into ω and f, respectively. we then define ϕ̃ ∈ n(ω × f) by setting ϕ̃(n) = (π1(ϕ(n)), ˜π2(ϕ(n))) for each n ∈ n. then by the argument above, the mapping ϕ 7→ ϕ̃ from (n(ω × f), ≪∗w) to ( n(ω × f), ≪∗s) is order-preserving and has as its image a dominating set, proving the inequality d(n(ω×f), ≪∗s) ≤ d( n(ω× f), ≪∗w), and hence our claim. (v) we can now complete the proof of the lemma. equation (3.3) (immediately preceding theorem 3.6) defines for us a mapping s 7→ os from nt (x, u) to n(e), the family of open neighborhoods of the identity e in f(x, u). further, it is immediate from the relevant definitions that this mapping is order-reversing from (nt (x, u), ≤∗t ) to (n(e), ⊆), where ≤ ∗ t is defined coordinate-wise in terms of ≤t. moreover, rephrased in this terminology, theorem 3.6 states that the mapping has a dense subset of (n(e), ⊆) as its image. hence we have (3.5) d(n(e), ⊆) ≤ d(nt (x, u), ≤∗t ). 36 p. nickolas and m. tkachenko in addition, we have obvious order-isomorphisms as follows: (n(ω × f), ≪∗w) ∼= (nω, ≤) × (nf, ≤∗w) ∼= (nω, ≤) × (n×eω, ≤) ∼= ( ωω, ≤) × (ωω, ≤) ∼= (ωω, ≤)(3.6) (where ≤∗w is the coordinate-wise extension of ≤w from f to nf). therefore, from (3.5), (3.4), claim 3 and (3.6), we have χ(f(x, u)) = d(n(e), ⊆) ≤ d(nt (x, u), ≤∗t ) ≤ d(n(ω × f), ≪∗s) = d(n(ω × f), ≪∗w) = d(ωω, ≤) = d. this finishes the proof. � the conclusion of the next theorem will be strengthened in theorem 3.15. theorem 3.10. if (x, u) is an ℵ0-narrow uniform space, then χ(f(x, u)) ≤ d · χ(a(x, u)). proof. according to theorem 2.3 and theorem 2.10, we have (3.7) χ(a(x, u)) = d(p(x, u), ≤) = d(ωu, ≤), where p(x, u) is the family of uniformly continuous pseudometrics on (x, u) bounded by 1. therefore, all we have to verify is that (3.8) χ(f(x, u)) ≤ d · d(ωu, ≤). denote by s the family of all sequences v = 〈un : n ∈ ω〉 ∈ ωu such that 3un+1 ⊆ un for each n ∈ ω. it is clear that s is dense in ( ωu, ≤), whence d(s, ≤) = d(ωu, ≤). choose a dense subset d of (s, ≤) of the minimal cardinality. then d is also dense in (ωu, ≤). note that the set of terms of the sequence v is a base for a (non-hausdorff) uniformity ṽ on x for each v ∈ s, and hence lemma 3.9 implies that χ(f(x, ṽ)) ≤ d. so, for every v ∈ d, we can find a base b(v) at the identity in f(x, ṽ) such that |b(v)| ≤ d. it is clear that the family b = ⋃ v∈d b(v) satisfies |b| ≤ d · d( ωu, ≤), and we claim that b is a base at the identity in the group f(x, u). by assumption, the space (x, u) is ℵ0-narrow, so the group f(x, u) is ℵ0narrow according to [2, lemma 3.2] or [6]. hence the topology of f(x, u) is generated by continuous homomorphisms to second countable topological groups (see [6] or [25, lemma 3.7]). in other words, given a neighborhood u of the identity in f(x, u), one can find a continuous homomorphism f : f(x, u) → g to a second countable topological group g and an open neighborhood v of the identity in g such that f−1(v ) ⊆ u. choose a countable base {vn : n ∈ ω} the character of free topological groups i 37 at the identity of g such that v0 = v and v 3 n+1 ⊆ vn for each n ∈ ω. for n = 0, 1, . . ., put (3.9) un = {(x, y) ∈ x × x : f(x) −1 · f(y) ∈ vn, f(x) · f(y) −1 ∈ vn}. evidently 3un+1 ⊆ un for each n ∈ ω, so that v = 〈un : n ∈ ω〉 ∈ s. since d is dense in (d, ≤), we can assume that v ∈ d. our choice of v (see (3.9)) guarantees that the restriction of f to x is a uniformly continuous mapping of (x, ṽ) to (g, ∗v∗), where ∗v∗ is the two-sided uniformity of the group g. hence the homomorphism f : f(x, ṽ) → g remains continuous. take an element w ∈ b(v) such that f(w) ⊆ v . then w ⊆ f−1(v ) ⊆ u, and hence b is a base at the identity in f(x, u). this proves (3.8) and the theorem. � combining corollary 2.17 and theorem 3.10, we obtain the following result, which will be given its final form in theorem 3.15. corollary 3.11. if an ℵ0-narrow hausdorff uniform space (x, u) contains an infinite precompact set, then χ(a(x, u)) = χ(f(x, u)). now we proceed to show, in theorem 3.15, that the existence of an infinite precompact set in (x, u) can be omitted in the assumptions of corollary 3.11. the main additional information we need is given in the following result. lemma 3.12. if (x, u) is an ℵ0-narrow uniform p-space, then the group f(x, u) has a base at the identity consisting of open normal subgroups. in particular, the topology of f(x, u) is generated by graev’s extensions of the uniformly continuous pseudometrics on (x, u). proof. the first assertion follows from the uniform analog of [22, th. 4]. for the second assertion, suppose that (x, u) is a uniform p-space, and let u be an open neighborhood of the identity e in f(x, u). then there exists an open normal subgroup v of f(x, u) with v ⊆ u. consider the open cover γ = {x ∩ xv : x ∈ x} of the space x. it is clear that γ is a u-uniform cover. since v is a subgroup of f(x, u), the family γ is a partition of x, i.e., every two elements of γ are disjoint or coincide. define a pseudometric ̺ on x by setting ̺(x, y) = 0 if x, y ∈ o for some o ∈ γ, and ̺(x, y) = 1 otherwise. clearly, ̺ is uniformly continuous. let ̺̂ be graev’s extension of ̺ to the maximal invariant pseudometric on fa(x) (see [5, section 3]). then the set w̺ = {g ∈ fa(x) : ̺̂(g, e) < 1} is an open neighborhood of e in f(x, u) by the continuity of ̺̂ on f(x, u). it remains to show that w̺ ⊆ v . from the fact that the pseudometric ̺ is {0, 1}-valued, it is immediate from graev’s construction that the extension ̺̂ is integer-valued. we therefore have w̺ = {g ∈ fa(x) : ̺̂(g, e) = 0}. 38 p. nickolas and m. tkachenko (indeed, it follows that w̺ is an open normal subgroup of f(x, u).) further, graev’s construction shows straightforwardly that for g ∈ w̺, there exist (nonreduced) representations g = xε11 · · · x ε2n 2n and e = y ε1 1 · · · y ε2n 2n of g and e, for some n ∈ n, some x1, . . . x2n, y1, . . . y2n ∈ x, and ε1, . . . ε2n = ±1, such that ̺(xi, yi) = 0 for i = 1, . . . , 2n. now we clearly have y εi i y εi+1 i+1 = e for some i, from which it follows that xεii x εi+1 i+1 ∈ v . set g1 = x ε1 1 · · · x εi−1 i−1 and g2 = x εi+2 i+2 · · · x ε2n 2n , and put ĝ = g1g2. note that ĝ ∈ w̺. if we assume inductively that ĝ ∈ v , we also have g2g1 ∈ v by the normality of v , and then we have g−11 gg1 = x εi i x εi+1 i+1 g2g1 ∈ v , from which we have g ∈ v , again by normality. it follows by induction that w̺ ⊆ v , as required. � this allows us to extend theorem 2.13 to the non-abelian case, assuming additionally that our uniform space is ℵ0-narrow. theorem 3.13. if (x, u) is an ℵ0-narrow uniform p-space, then χ(f(x, u)) = w(x, u). proof. by lemma 3.12, the topology of f(x, u) is generated by graev’s extensions of the uniformly continuous pseudometrics on (x, u). it follows, therefore, that χ(f(x, u)) ≤ d(p(x, u), ≤) = w(x, u). however, theorem 2.3 and lemma 3.1 together imply that d(p(x, u), ≤) = χ(a(x, u)) ≤ χ(f(x, u)). combining these inequalities, we obtain the required conclusion. � in contrast to theorem 2.13, the assumption of ℵ0-narrowness cannot be removed in theorem 3.13, as is shown by the following example. example 3.14. for every cardinal τ > ℵ1, there exists a hausdorff uniform p-space (x, u) such that w(x, u) = ℵ1 < τ < χ(f(x, u)). indeed, let x = l⊕d be the topological sum of the one-point lindelöfication l of a discrete space y with |y | = ℵ1 and a discrete space d of cardinality τ > ℵ1. denote by x0 the unique non-isolated point of l (and of x). then a base of open neighborhoods of x0 in l (and in x) consists of the sets l \ c, where c is an arbitrary countable subset of y . since |y | = ℵ1, it is easy to see that χ(x0, l) = χ(x0, x) = ℵ1. let u be the fine uniformity of the space x. then a basic entourage of the diagonal ∆ in x × x has the form uc = {(x, y) ∈ l × l : x, y ∈ l \ c} ∪ ∆, where c ⊆ y is countable. therefore, w(x, u) = ℵ1. let us show that χ(f(x, u)) ≥ τ. for every a ∈ d, put la = a −1x−10 la, and consider the subspace z =⋃ a∈d la of f(x, u) ∼= f(x). apply an argument similar to that in [1, prop. 3.2] to show that z is homeomorphic to the fan v (τ) obtained from the topological sum t of τ copies of l by identifying to a point the set t ′ of all non-isolated points of t . since each of the τ distinct spines of the fan v (τ) is homeomorphic to l, a straightforward diagonal argument implies that τ < χ(e, z) ≤ χ(f(x)). the character of free topological groups i 39 finally, we have the main result of this section. theorem 3.15. the equality χ(a(x, u)) = χ(f(x, u)) holds for every ℵ0narrow uniform space (x, u). proof. if (x, u) is a uniform p-space, then the required equality is given by theorems 2.13 and 3.13, while if (x, u) is not a uniform p-space, then the equality follows from theorems 2.15 and 3.10 and lemma 3.1. � the above theorem has several applications; the following four are immediate. corollary 3.16. χ(a(x)) = χ(f(x)) for every ℵ0-narrow space x. using corollary 2.22, we have in particular: corollary 3.17. if x is an infinite compact metrizable space, then χ(f(x)) = d. corollary 3.18. suppose that a space x contains a dense lindelöf subspace. then χ(a(x))=χ(f(x)). proof. it is easy to see that the space x is ℵ0-narrow. indeed, let y be a dense lindelöf subspace of x. if γ is a discrete family of non-empty open sets in x, then the family µ = {u ∩ y : u ∈ γ} is a discrete family of non-empty open subsets of y . however, every such family of subsets of y is countable, so |γ| = |µ| ≤ ℵ0. now the necessary conclusion follows from corollary 3.16. � clearly, every space of countable cellularity is ℵ0-narrow. therefore, we have the following. corollary 3.19. if a space x has countable cellularity, then χ(a(x)) = χ(f(x)). we believe that corollary 3.21 below compared with theorem 2.3 or corollary 2.4 gives a more comprehensive expression for the character of the groups f(x) and a(x) on a lindelöf space x. it is based on a simple relation between dense subsets of (ωux, ≤) and the character of the diagonal in (x ×ω) 2, where ux is the fine uniformity of the space x and the set ω carries the discrete topology. we recall that if s = {un : n ∈ ω} and t = {vn : n ∈ ω} are elements of ωu, where u is a uniformity on some set, then s ≤ t means that un ⊆ vn for each n ∈ ω. lemma 3.20. if x is a paracompact topological space and ux is the fine uniformity on x, then d(ωux, ≤) = χ∆(x × ω). proof. we can assume that x is not discrete—otherwise, the equality becomes trivial. denote by uy the fine uniformity on y ≡ x ×ω. the paracompactness of x implies that every neighborhood of the diagonal ∆x in x 2 belongs to the fine uniformity ux, and similarly for the diagonal ∆y in y 2. the family of all neighborhoods of ∆y in y 2 contains a base which can be naturally identified 40 p. nickolas and m. tkachenko with the family of all sequences {un : n ∈ ω}, where un ∈ ux for each n ∈ ω. it is now immediate that d(ωux, ≤) = χ∆(x × ω). � finally, theorem 3.15, corollary 2.4, theorem 2.10 (with u = ux) and lemma 3.1 imply the following result. corollary 3.21. let x be a lindelöf space. then χ(f(x)) = χ(a(x)) = χ∆(x × ω). it may be worth remarking that the conclusion of the corollary holds in particular if x is a kω-space or a compact space. in the sequel [14] to this paper, we will investigate the compact case in more depth. references 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[26] k. yamada, characterizations of a metrizable space x such that every an(x) is a k-space, topology appl. 49 (1993), 75–94. received november 2002 accepted june 2004 p. nickolas (peter−nickolas@uow.edu.au) department of mathematics and applied statistics, university of wollongong, nsw 2522, australia m. tkachenko (mich@xanum.uam.mx) departamento de matemáticas, universidad autónoma metropolitana, av. san rafael atlixco 186, col. vicentina, del. iztapalapa, c.p. 09340, méxico, d.f. @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 47–70 the local triangle axiom in topology and domain theory pawe lwaszkiewicz abstract. we introduce a general notion of distance in weakly separated topological spaces. our approach differs from existing ones since we do not assume the reflexivity axiom in general. we demonstrate that our partial semimetric spaces provide a common generalization of semimetrics known from topology and both partial metrics and measurements studied in quantitative domain theory. in the paper, we focus on the local triangle axiom, which is a substitute for the triangle inequality in our distance spaces. we use it to prove a counterpart of the famous archangelskij metrization theorem in the more general context of partial semimetric spaces. finally, we consider the framework of algebraic domains and employ lebesgue measurements to obtain a complete characterization of partial metrizability of the scott topology. 2000 ams classification: 54e99, 54e35, 06a06. keywords: partial semimetric, partial metric, measurement, lebesgue measurement, local triangle axiom, continuous poset, algebraic dcpo. 1. introduction over recent years, a number of attempts have been made to equip semantic domains with a notion of distance between points in order to provide a mechanism for making quantitative statements about programs, such as speed of convergence or complexity of algorithms. this particular branch of research is now known as quantitative domain theory. the work in this area has led to a number of concepts which generalise the classical notion of a metric space to an ordered setting. for example, smyth [21] introduced quasi-metrics to domain theory, which by virtue of being non-symmetric encode topology and order on a domain at the same time. partial metrics were defined by matthews [14] and studied further by o’neill [16, 17]. although they are symmetric, they are still capable of capturing order. this work was further extended by heckmann in [9], leading to a concept of weak partial metrics. 48 p. waszkiewicz more recently, keye martin introduced the idea of a measurement on a domain [13]. at first glance, measurements are quite different from distance functions because they take only one argument. nevertheless, they allow for making precise quantitative statements about domains. the desire to understand the interplay between partial metrics and measurements, discovered by the author in [24], is a major motivation for the present work. research in this direction led to a number of general question about the nature of distance in weakly separated spaces in topology (continuous domains in their scott topology are examples of such spaces). our approach is quite distinct from existing theories of generalized metric spaces in quantitative domain theory, e.g. [18, 5, 6, 23] and topology [8], since we do not in general assume either the reflexivity axiom or the triangle inequality. instead, we introduce partial semimetric spaces, which generalize both partial metrics and semimetrics. the theory of partial semimetrics coincides with the theory of measurements in the framework of continuous domains. in the paper we focus on a certain condition on sequences in partial semimetric spaces substituting the triangle inequality, which we call the local triangle axiom. it was recognised as early as 1910 in the work of fréchet [7] and investigated in [3, 4]. the name local axiom of triangle was used by niemytzki [15] and the property was further analysed in [27]. on continuous domains in their scott topology, stable partial symmetrics (defined in section 5.1) satisfying the local triangle axiom correspond to lebesgue measurements (definition 6.5) introduced in a more general context by martin in [13]. there are two major results of the paper. the first one is a metrization result (theorem 3.2), which is a twin of the famous archangelskij metrization theorem working in partial semimetric spaces. we apply it in section 4 to prove quasi-developability of our distance spaces, improving a similar result by künzi and vajner [10]. in the other part of the paper we consider partial semimetric spaces and the local triangle axiom in the framework of domain theory (section 6) to obtain the second major result of the paper. our result provides a solution to an open problem stated in [9] asking for conditions, which guarantee partial metrizability of continuous domains. we show that an algebraic domain admits a partial metric for its scott topology if and only if it admits a lebesgue measurement (theorem 6.12). the heckmann problem remains open for arbitrary continuous domains. for the reader’s convenience we present a summary of symbols and terms used in the paper in table 1. the local triangle axiom 49 table 1: symbols and terms used in the paper symbol or term meaning definition related results 〈x,ρ〉 distance space def. 2.1. τρ distance topology sect. 2. τopρ dual distance sect. 2. topology (cv) a convergence prop. sect. 2.1. prop. 2.5, 2.10. (symm) symmetry axiom sect. 2.2. (ssd) small self-distance sect. 2.2. axiom (r) reflexivity axiom sect. 2.2. (∆]) sharp triangle sect. 2.2. inequality axiom partial symmetric def. 2.6. partial semimetric def. 2.7. cor. 2.9, 2.12. prop. 2.10, 2.11, 2.14, 2.16. def. 6.2. thm. 6.4. partial metric sect. 2.2 cor. 2.18. local axiom, sect. 2.5, prop. 2.16, 2.17, 2.19, of triangle (l) sect. 6.3. 6.6, 6.7. cor. 2.18. thm. 3.2, 4.1, 6.8, 6.12. induced distance symmetrization sect. 2.4. prop. 2.13. thm. 3.2. dρ of distance ρ and its dual. self-distance µρ sect. 5. sect. 5.1, 6.2. measurement def. 6.1. sect. 6. 2. outline of a general theory of distance first, let us consider the definition of a distance space, its intrinsic topologies and some natural axioms that can be introduced in distance spaces. definition 2.1. a distance on a set x is a map ρ: x ×x → [0,∞). a pair 〈x,ρ〉 is called a distance space. a distance function assigns to each element x of x a filter nx of subsets of x by taking u ∈ nx if and only if there exists ε > 0 such that bρ(x,ε) ⊆ u. the set bρ(x,ε) := {y ∈ x | ρ(x,y) < ρ(x,x) + ε} is called a ball centered at x with radius ε. in the same way one forms a collection nopx by replacing the ball in the definition of nx by a dual ball bopρ (x,ε) := {y ∈ x | ρ(y,x) < ρ(y,y) + ε}. 50 p. waszkiewicz the collection τρ := {u | ∀x ∈ u. u ∈nx} is a topology on x called the distance topology on x. dually, the family of sets τopρ := {v | ∀x ∈ v. v ∈n op x } constitutes a dual (distance) topology on x. in general the collection bx of balls centered at x is not a neighborhood base at x and the balls are not open themselves. the set x together with an operation n : x → p(p(x)), which assigns to each point the filter nx, is an example of a neighbourhood space in the sense of [22]. the distance topology just defined is in fact one of the natural topologies for neighbourhood spaces, studied in more detail by smyth (cf. [22], proposition 2.10). example 2.2. to illustrate the difference between arbitrary distance spaces and metric ones, consider the four-element chain as shown in figure 1. (numbers denote respective self-distances; we set the distances ρ(x,y) and ρ(y,x) between points x,y to be max{ρ(x,x),ρ(y,y)}). the distance is symmetric and satisfies the triangle inequality. note that open sets are upper; in particular, the only open set containing the bottom element (denoted ⊥) is the whole space. indeed, the distance between ⊥ and any other element of the space is 3 and hence any ball containing ⊥ must be the whole space. in contrast, any metric on a finite space induces the discrete topology. note that in this particular example the balls are open themselves. in arbitrary distance spaces it is usually not the case. 3 d 2 d 1 d 0 d figure 1: a non-metric distance. 2.1. basic properties of distance spaces. for a subset a of x and an element x ∈ x we define: ρ(x,a) := inf{ρ(x,a) | a ∈ a}. proposition 2.3. in a distance space 〈x,ρ〉 a subset h ⊆ x is closed iff for all x /∈ h we have ρ(x,h) > ρ(x,x). the local triangle axiom 51 proof. straight from the definition of the distance topology we infer that a subset h of x is closed iff for all x /∈ h there exists ε > 0 such that for all y ∈ h we have ρ(x,y) ≥ ρ(x,x) + ε. it is however equivalent to say that ρ(x,h) ≥ ρ(x,x) + ε > ρ(x,x). � as a corollary we obtain the following interesting characterization of the specialisation preorder vτρ: corollary 2.4. in a distance space 〈x,ρ〉 the following are equivalent: 1. x vτρ y; 2. x = y or ∀ε > 0. ∃z 6= x. (ρ(x,z) < ρ(x,x) + ε and z vτρ y). proof. (⇒) for x,y ∈ x, x v/τρy iff x /∈ cl{y} iff ∃ε > 0. ρ(x, cl{y}) ≥ ρ(x,x) + ε iff ∃ε > 0. ∀z ∈ bρ(x,ε). z /∈ cl{y} iff ∃ε > 0. ∀z ∈ bρ(x,ε). z v/τρy. note that we have used proposition 2.3 in the second equivalence. conversely, assume x 6= y and let u be an open set in x. then x ∈ u implies that there exists ε > 0 such that x ∈ bρ(x,ε) ⊆ u. by assumption, there is z ∈ bρ(x,ε) ⊆ u with z vτρ y. but open sets are upper with respect to the specialisation preorder and so y ∈ u. � proposition 2.5. let 〈x,ρ〉 be a distance space, (xn) be a sequence of elements of x and x ∈ x. then ρ(x,xn) → ρ(x,x) implies xn →τρ x. proof. let u be any open set around x. then there exists ε > 0 such that x ∈ bρ(x,ε) ⊆ u. suppose that ρ(x,xn) → ρ(x,x). then there exists nε ∈ ω such that for all n ≥ nε, |ρ(x,xn) −ρ(x,x)| < ε. it is equivalent to say that either 0 ≤ ρ(x,xn) −ρ(x,x) < ε or 0 ≥ ρ(x,xn) −ρ(x,x) > −ε for all n ≥ nε. in either case, we are able to conclude that xn ∈ bρ(x,ε) ⊆ u for all n ≥ nε. therefore, xn →τρ x. � note that the corresponding equivalence: (cv) ρ(x,xn) → ρ(x,x) iff xn →τρ x may not hold in arbitrary partial symmetric spaces. figure 2 provides a counterexample: we assume that all of the distances between elements, which are not given on the picture are 5. one can check that the bottom element is in every open set and hence every sequence converges to it. on the other hand, for the sequence (xn) we have ρ(⊥,xi) → 5, for i = 1, 2, 3, . . ., while ρ(⊥,⊥) = 3. 52 p. waszkiewicz ⊥ 3 d 2 d 2 d 2 d 2 d · · · 2 2 2 2 33 3 3 h h h h h h h hh j j j jj � � � � � � � �� 1 d 1 d 1 d 1 d · · ·x1 x2 x3 x4 · · · figure 2: (cv) does not hold in general. it is known that (cv) holds if the t2 axiom is assumed [8], lemma 9.3 p.481. 2.2. distance axioms. on a distance space a number of further axioms can be introduced. we consider symmetry of the distance map (symm) ∀x,y ∈ x. ρ(x,y) = ρ(y,x) and the axiom of small self-distances (ssd) ∀x,y ∈ x. ρ(x,x) ≤ ρ(x,y) and ρ(x,x) ≤ ρ(y,x), which obviously reduces to ∀x,y ∈ x. ρ(x,x) ≤ ρ(x,y) in the presence of symmetry. the latter condition is a generalisation of the reflexivity axiom (r) ∀x,y ∈ x. ρ(x,x) = 0 known from the theory of metric spaces. in topology, consult for example [8], a symmetric is a distance map, which satisfies (symm) and (r). on the other hand, a symmetric distance function satisfying the sharp triangle inequality (∆]) ∀x,y,z ∈ x. ρ(x,y) ≤ ρ(x,z) + ρ(z,y) −ρ(z,z) is named a weak partial metric [9]. a partial metric [14] is a weak partial metric with (ssd). lastly, one considers mostly distance topologies, which satisfy the t0 separation axiom. an elegant formulation of the (t0) axiom in terms of the distance mapping will be given later for a wide class of spaces (cf. corollary 2.12). the following definition is a compromise on the existing terminology. definition 2.6. a distance function ρ: x×x → [0,∞) on a set x is a partial symmetric whenever it satisfies (symm), (ssd) and the distance topology τρ is t0. the local triangle axiom 53 2.3. partial semimetric spaces. in this section we introduce a particularly interesting subclass of partial symmetric spaces. both partial metric spaces considered by matthews [14], o’neill [16, 17], schellekens [19, 20], heckmann [9] and waszkiewicz [24] and semimetric spaces in topology are our major examples of partial semimetric spaces. definition 2.7. let 〈x,ρ〉 be a partial symmetric space. the map ρ is a partial semimetric if for any x ∈ x, the collection {int(bρ(x,ε)) | ε > 0} is a base for the filter nx. in other words, we require that the family {bρ(x,ε) | x ∈ x,ε > 0} forms a (not necessarily open) neighborhood base at x with respect to the distance topology τρ on x. to summarize the key differences between “classical” distance maps and ours, we collect their properties in the following table. a ‘+’ indicates that the respective condition is satisfied. all maps in the table satisfy (symm), (ssd) and (t0). distance balls form a neighbourhood base (r) partial symmetric partial semimetric + symmetric + semimetric + + proposition 2.8. let 〈x,ρ〉 be a distance space. the following are equivalent: (1) {bρ(x,ε) | ε > 0} forms a (not necessarily open) neighborhood base at x; (2) ∀x ∈ x. ∀h ⊆ x. ρ(x,h) ≤ ρ(x,x) iff x ∈ cl(h). proof. (⇒) let x ∈ x and ε > 0. x ∈ cl(h) iff ∀ε > 0. bρ(x,ε) ∩h 6= ∅ iff ∀ε > 0. ∃z ∈ h. ρ(x,z) < ρ(x,x) + ε iff ρ(x,h) ≤ ρ(x,x). conversely, since x \bρ(x,ε) = {z | ρ(x,z) ≥ ρ(x,x) + ε}, we have ρ(x,x \bρ(x,ε)) ≥ ρ(x,x) + ε > ρ(x,x). therefore, x /∈ cl(x \bρ(x,ε)) and so x ∈ int(bρ(x,ε)). � as an immediate corollary, we note that a partial semimetric can be introduced by an appropriate closure operator familiar from the theory of metric spaces. this was stated in [8], theorem 9.7, for semimetrics. corollary 2.9. let 〈x,ρ〉 be a symmetric distance space. the following are equivalent: 54 p. waszkiewicz (1) ρ is a partial semimetric. (2) ∀x ∈ x. ∀h ⊆ x. ρ(x,h) = ρ(x,x) iff x ∈ cl(h). one can check that if the distance topology is hausdorff, then x is partially semimetrizable iff it is partially symmetrizable and first countable (see the discussion before definition 9.5 of [8]). in the absence of separation axioms, the property (cv) is necessary for this characterization to hold. proposition 2.10. let 〈x,ρ〉 be a partial symmetric space. then the following are equivalent: (1) the map ρ is a partial semimetric; (2) x is first countable and (cv) holds. proof. if ρ is a partial semimetric, then the collection {int(bρ(x, 1/2n+1)) | n ∈ ω} is a neighborhood base at x ∈ x, which amounts to first countability of x. now, if for some sequence (xn) of elements of x and some x ∈ x, we have xn →τρ x, then for every ε > 0, (xn) is cofinally in int(bρ(x,ε)) and so in bρ(x,ε). therefore, using the (ssd) axiom, ∀ε > 0. ∃n ∈ ω. ∀n ≥ n. |ρ(x,xn) −ρ(x,x)| < ε. that is, ρ(x,xn) → ρ(x,x). conversely, if x is first countable and (cv) holds, then a point x ∈ x is in int(bρ(x,ε)) for any ε > 0. suppose not; then there exists ε0 > 0 and a sequence (xn) with xn →τρ x such that xn /∈ bρ(x,ε0). (choose xn to be in the nth member of a decreasing countable base for x and not in bρ(x,ε0)). so ρ(x,xn) 9 ρ(x,x), which contradicts (cv). now, if x ∈ int(bρ(x,ε)), then the collection {bρ(x,ε) | ε > 0} forms a neighborhood base at x. � in a partial semimetric space the specialisation preorder reflects the distance in a simple way. proposition 2.11. let 〈x,ρ〉 be a partial semimetric space. then for all x,y ∈ x, x vτρ y iff ρ(x,y) = ρ(x,x). proof. x vτρ y iff x ∈ cl({y}) iff x ∈{z | ρ(z,y) = ρ(z,z)} iff ρ(x,y) = ρ(x,x). we use proposition 2.8 (2) and (ssd) in the second equivalence. � as a result we are able to characterize the (t0) axiom in terms known from the theory of partial metric spaces. the local triangle axiom 55 corollary 2.12. the distance topology in a partial semimetric space 〈x,ρ〉 is t0 iff x = y iff ρ(x,y) = ρ(x,x) = ρ(y,y). � 2.4. derived distance functions and their topologies. convention in the rest of this paper we assume that the (t0) axiom holds in all distance spaces that we consider. in a distance space 〈x,ρ〉 one can introduce a number of other maps, which are derived from the distance. whenever the map ρ satisfies (ssd), it is convenient to form the corresponding quasi-distance qρ : x ×x → [0,∞) by qρ(x,y) := ρ(x,y) −ρ(x,x). whenever possible, we will drop the index ρ from qρ for the sake of clarity. the (ssd) axiom assures that the function q is well-defined. the usefulness of the quasi-distance stems from the fact that it satisfies the reflexivity axiom (r) and still induces the same topology as the distance ρ. whenever the map ρ satisfies more axioms, the name for the quasi-distance q will change accordingly. for instance, a quasi-semimetric is a quasi-distance formed from a partial semimetric. the quasi-distance has a dual, namely the map qopρ : x×x → [0,∞) defined by qopρ (x,y) := ρ(x,y) − ρ(y,y). the dual quasi-distance induces the dual distance topology. the symmetrization of the quasi-distance and its dual is the map dρ : x ×x → [0,∞), ∀x,y ∈ x. dρ(x,y) := qρ(x,y) + qopρ (x,y) = 2ρ(x,y) −ρ(x,x) −ρ(y,y). the function dρ is always symmetric and reflexive and hence is a symmetric. it is called the induced symmetric. we do not know whether it is true in general that the induced symmetric derived from a partial semimetric is a semimetric. however, there are two notable cases when the prefix “semi-” is retained: firstly, whenever a partial semimetric topology is hausdorff, secondly, in the presence of the local triangle axiom studied in the next section (the latter claim follows from proposition 2.16 and the observation that a partial semimetric is a semimetric if and only if it satisfies the reflexivity axiom). lastly, for a distance function, the self-distance mapping (called also the weight function) is non-trivial in general and, as we shall see in section 5, exhibits many interesting properties. proposition 2.13. for a distance space 〈x,ρ〉 with (ssd), the induced topology τdρ is the join of the distance topology τρ and its dual τ op ρ . moreover, τdρ is semimetrizable whenever both τρ and τopρ are partially semimetrizable. 56 p. waszkiewicz proof. both statements follow from the fact that bdρ(x,ε) ⊆ b(x,ε) ∩b op ρ (x,ε) ⊆ bdρ(x, 2ε) for any x ∈ x and ε > 0. � proposition 2.14. for a sequence (xn) and an element z ∈ x in a partial semimetric space 〈x,ρ〉, the following are equivalent: (a) lim dρ(xn,z) = 0; (b) lim ρ(xn,z) = ρ(z,z) and lim ρ(xn,z) = lim ρ(xn,xn); (c) (xn) →τρ z and lim ρ(xn,z) = lim ρ(xn,xn). proof. note that all limits are taken with respect to the euclidean topology on the real line. the equivalence of (a) and (b) follows easily since dρ(xn,z) = (ρ(xn,z) −ρ(z,z)) + (ρ(xn,z) −ρ(xn,xn)) and both terms on the right-hand side of the equation are non-negative by (ssd). the equivalence of (b) and (c) is clear by proposition 2.10. � 2.5. the local triangle axiom. in this subsection we study a certain condition on the convergence of sequences in a distance space 〈x,ρ〉, namely: (l) qρ(x,yn) → 0, qρ(yn,zn) → 0 imply qρ(x,zn) → 0, for any two sequences (yn), (zn) and element x from x. the property (l) of a distance space is called here the local triangle axiom. we give several characterization of the local triangle axiom. we show that the condition implies partial semimetrizability of the underlying topology. we discuss its dependence on the other axioms and the similarity to the triangle inequality. for symmetric spaces in topology a similar condition was recognised as early as 1910 in the work of fréchet [7] and investigated in [3, 4]. the local triangle axiom was defined by niemytzki [15] and analysed in [27]. a more modern-style proof of metrizability of a topological space, which admits a symmetric satisfying the local triangle axiom, is given by archangelskij [2] and explained in detail in [8]. the precise formulation of the archangelskij metrization theorem follows: theorem 2.15 (archangelskij). let the set x be a t1 topological space symmetrizable with respect to the symmetric d. if for all x ∈ x and all sequences (yn), (zn) ⊆ x we have d(x,yn) → 0 and d(yn,zn) → 0 imply d(x,zn) → 0, then x is metrizable. � in section 3 we prove a counterpart of archangelskij’s theorem, which works in partial symmetric spaces. here, we demonstrate some useful characterizations of the local triangle axiom and its basic implications. in fact any partial symmetric with property (l) is a partial semimetric. proposition 2.16. let 〈x,ρ〉 be a partial symmetric space. if the condition (l) holds in x, then the map ρ is a partial semimetric. the local triangle axiom 57 proof. for any x ∈ x denote by bx the collection of all ρ-balls centered at x. for all x ∈ x the family bx constitutes a neighbourhood base at x if and only if for all x ∈ x and for all b ∈ bx there exists c ∈ bx such that if y ∈ c, there is some d ∈ by with d ⊆ b. (see for example [26], theorem 4.5 p.33.) equivalently, we have ∀x ∈ x. ∀ε > 0. ∃δ1 > 0. ∀y ∈ x. ∃δ2 > 0. ∀z ∈ x. qρ(x,y) < δ1, qρ(y,z) < δ2 imply qρ(x,z) < ε, or in other words, ∀x ∈ x. ∀ε > 0. ∀(yn), (zm) ⊆ x. limn qρ(x,yn) = 0, limn limm qρ(yn,zm) = 0 imply limm qρ(x,zm) = 0, which is in general a weaker condition than (l). � alternative proof : by corollary 2.9 it is enough to show that for any subset h of x, the set h′ := {x | qρ(x,h) = 0} is closed. suppose not. then qρ(x,h′) = 0 for some x /∈ h′. by definition of the distance between a point and a set, there exist sequences (yn) ⊆ h′ and (zn) ⊆ h such that qρ(x,yn) → 0 and qρ(yn,zn) → 0. by assumption, qρ(x,zn) → 0. hence qρ(x,h) = 0 and consequently, x ∈ h′, a contradiction. � proposition 2.17. for a distance space 〈x,ρ〉, the following are equivalent: (1) ρ satisfies (l); (2) ∀x ∈ x. ∀ε > 0. ∃δ > 0. ∀y,z ∈ x. ρ(x,z) < ρ(x,x) + δ,ρ(z,y) < ρ(z,z) + δ imply ρ(x,y) < ρ(x,x) + ε. proof. (2) holds iff ∀x ∈ x. ∀ε > 0. ¬[∃(yk), (zk) ⊆ x. ρ(x,zk) → ρ(x,x),ρ(zk,yk) → ρ(z,z), ρ(x,yk) ≥ ρ(x,x) + ε], iff ∀x ∈ x. ¬[∃(yk), (zk) ⊆ x. ρ(x,zk) → ρ(x,x),ρ(zk,yk) → ρ(z,z), ρ(x,yk) 9 ρ(x,x)], iff ∀x ∈ x. ∀(yk), (zk) ⊆ x. ρ(x,zk) → ρ(x,x),ρ(zk,yk) → ρ(z,z) imply ρ(x,yk) → ρ(x,x) iff (1) holds. � therefore, we can easily show that every partial metric satisfies the local triangle axiom. from the proof of the following result it is also obvious that the converse claim will not hold in general. 58 p. waszkiewicz corollary 2.18. let 〈x,p〉 be a partial metric space. then the map p satisfies (l). proof. we will prove a more general statement. we claim that a mapping ρ: x ×x → [0,∞) satisfies (∆]) if and only if ∀x,y,z ∈ x. ∀ε1,ε2 > 0. ρ(x,z) < ρ(x,x) + ε1 and ρ(z,y) < ρ(z,z) + ε2 imply ρ(x,y) < ρ(x,x) + ε1 + ε2. the formula above implies (l) (which is easily seen by proposition 2.17 (2)). to prove the claim, let x,y,z ∈ x, ε1,ε2 > 0 and assume the hypothesis of the implication above. then by the sharp triangle inequality we have ρ(x,y) ≤ ρ(x,z) + ρ(z,y) −ρ(z,z) < ρ(x,x) + ε1 + ε2, as required. conversely, take any ε > 0. we have ρ(x,z) < ρ(x,x) + (ρ(x,z)− ρ(x,x) + ε) and ρ(z,y) < ρ(z,z) + (ρ(z,y) − ρ(z,z) + ε). by assumption, ρ(x,y) < ρ(x,x) + (ρ(x,z) − ρ(x,x) + ε) + (ρ(z,y) − ρ(z,z) + ε) = ρ(x,z) + ρ(z,y) − ρ(z,z) + 2ε. hence the sharp triangle inequality follows from the arbitrariness of ε and the claim is now proved. � the following result generalizes a similar result for symmetric spaces obtained by h. w. martin in [12]. proposition 2.19. let 〈x,ρ〉 be a partial symmetric space. then the following are equivalent: (1) the map ρ satisfies (l). (2) if k is compact, h is closed and k ∩h = ∅, then qρ(k,h) > 0. proof. we will show that qρ(k,h) = 0 implies k ∩h 6= ∅. let qρ(k,h) = 0. hence there is a sequence (kn) of elements of k such that qρ(kn,h) < 1/2n+1 for every n ∈ ω. this means that there also exists a sequence (hn) of elements from h such that qρ(kn,hn) → 0. by compactness of k, there exists a convergent subsequence (knm) of (kn) with knm → x ∈ k. proposition 2.16 guarantees that the map ρ is a partial semimetric and hence qρ(x,knm) → 0. then for the corresponding subsequence (hnm) of hn we have qρ(knm,hnm) → 0. by assumption, qρ(x,hnm) → 0 and hence x ∈ cl(h) = h. this means that k ∩h 6= ∅. conversely, for any two sequences (xn), (yn) and an element x of x, suppose that qρ(x,xn) → 0 and qρ(xn,yn) → 0. define a compact subset k of x and a closed subset h of x in the following way: k := {x}∪{xn | n ∈ ω}, h := clτρ{yn | n ∈ ω}. there are two cases to consider: either k ∩ h = ∅ or k ∩ h 6= ∅. in the former case, by assumption (2), we infer that there exists ε > 0 such that qρ(k,h) > ε. therefore, infn{qρ(xn,yn)}≥ qρ(k,h) > ε, the local triangle axiom 59 which is impossible since qρ(xn,yn) → 0. in the latter case we distinguish three simple subcases. if no xn’s belong to k∩h, then x ∈ k∩h and hence (yn) →τρ x, which by proposition 2.10 is equivalent to saying that qρ(x,yn) → 0. if infinitely many xn’s belong to k∩h, then since h is closed, also x ∈ k∩h and then by the same argument as above, qρ(x,yn) → 0. finally, if a finite number of xn’s belong to k∩h, say x1, . . . ,xk−1 ⊆ k∩h, then consider the remaining sequence (xn)n≥k and repeat the proof with k := {x}∪{xn | n ≥ k}. � 3. metrizability of the induced distance space in this section we present the first of the two major results of this paper. we prove a counterpart for the archangelskij metrization theorem (theorem 2.15) working in partial symmetric spaces. lemma 3.1. if 〈x,ρ〉 is a semimetric space that satisfies (l), then the induced distance space 〈x,dρ〉 satisfies (l). proof. suppose that for some sequences (yn), (zn) ⊆ x and an element x ∈ x we have lim dρ(x,yn) = 0 and lim dρ(yn,zn) = 0. then by proposition 2.14.(b), one sees that lim ρ(x,yn) = ρ(x,x), lim ρ(yn,zn) = lim ρ(yn,yn) and ρ(x,x) = lim ρ(zn,zn). the two former equalities imply that lim ρ(x,zn) = ρ(x,x) by (l) for the map ρ. together with the third equality, we get lim dρ(x,zn) = 0 by yet another application of proposition 2.14. � theorem 3.2. for any partial symmetric space 〈x,ρ〉 with (l) the induced distance dρ is metrizable. proof. by lemma 3.1 and proposition 2.16, the induced distance is a semimetric and satisfies (l). we will show that 〈x,dρ〉 is hausdorff. since it is first-countable by proposition 2.10, it will be enough to demonstrate that limits of sequences are unique in x. hence, let (yn) be a sequence of elements of x and suppose that (yn) has two limits x and z in x. then again by proposition 2.10 applied to the mapping dρ we have dρ(x,yn) → 0 and dρ(z,yn) → 0. using (l) we conclude that dρ(x,z) = 0. the last equality is equivalent to ρ(x,z) = ρ(x,x) and ρ(x,z) = ρ(z,z). the characterization of the order in partial semimetric spaces from proposition 2.11 implies that x vτρ z and z vτρ x. hence, x = z using the t0 axiom. taking all of the proved properties together, one can see that 〈x,dρ〉 is a hausdorff semimetric space, which satisfies the local triangle axiom. therefore, archangelskij’s theorem applies and we conclude that 〈x,dρ〉 is metrizable. � 4. quasi-developability of a distance space as an application of the metrizability theorem proved in the last section, we consider the problem of quasi-developability of partial symmetric spaces. let us first introduce the necessary terminology. 60 p. waszkiewicz let 〈x,τ〉 be a topological space, x ∈ x and c be any collection of subsets of x. denote card{c ∈c | x ∈ c} by ord(x,c). a sequence g1,g2,g3, . . . of collections of open subsets of a topological space 〈x,τ〉 is called a quasi-development for x provided that if x ∈ u ∈ τ, then there exists n ∈ ω and g such that x ∈ g ∈gn and st(x,gn) ⊆ u, where st(x,gn) :=⋃ {v ∈gn | x ∈ v}. a topological space 〈x,τ〉 is quasi-developable if it admits a quasi-development. proposition 1 of [10] states that every partial metric space is quasi-developable. since we have proved in corollary 2.18 that every partial metric satisfies (l) and noted that the converse does not hold in general, the following theorem improves künzi and vajner’s result. we adapt the idea of the proof and the notation from [10]. theorem 4.1. let 〈x,ρ〉 be a partial symmetric space with (l). then the topological space 〈x,τρ〉 is quasi-developable. proof. for simplicity, denote the induced semimetric by d, instead of the standard dρ. for each k,n ∈ ω, set ank := {x ∈ x | ρ(x,x) ∈ [(k − 1)2−n,k2−n)}. let ⋃ t∈ω bt be a base for the metrizable induced topology τd such that each collection bt is discrete. for each t, l ∈ ω, define ctl := {x ∈ x | bd(x, 2−l) hits at most one element of bt}. for each k,l,t ∈ ω set rklt := {intbρ(b ∩ctl ∩a(l+1)k, 2−(l+1)) | b ∈bt}. we claim that ⋃ k,l,trklt is a quasi-development for τρ. for an arbitrary x ∈ x and a natural number h, denote by m := m(x,h) the natural number chosen according to proposition 2.17 (that is, for x ∈ x and ε := 2−h we take δ := 2−m). by proposition 2.16, the map ρ is a partial semimetric. therefore, x ∈ intbρ(x, 2−m). using the fact that τρ ⊆ τd, there exists t0 ∈ ω such that x ∈ b0 ⊆ intbρ(x, 2−m) ⊆ bρ(x, 2−m) for some b0 ∈ bt0 . furthermore, since bt0 is discrete, there is l0 ∈ ω with l0 ≥ m such that (†) bd(x, 2−l0 ) ∩b 6= ∅ and b ∈bt0 imply b = b0. finally, there exists k0 ∈ ω such that x ∈ a(l0+1)k0 . note first that x ∈ b0 ∩ct0l0 ∩a(l0+1)k0 . hence x ∈ intbρ(x, 2−(l0+1)) ⊆ intbρ(b0 ∩ct0l0 ∩a(l0+1)k0, 2 −(l0+1)). we claim that bρ(b0 ∩ct0l0 ∩a(l0+1)k0, 2 −(l0+1)) ⊆ bρ(x, 2−h), which will show that ⋃ k,l,trklt is a basis for τρ. let y ∈ bρ(b0 ∩ ct0l0 ∩ a(l0+1)k0, 2 −(l0+1)). then y ∈ bρ(b0, 2−(l0+1)). that is, ρ(z,y) < ρ(z,z) + 2−(l0+1) for some z ∈ b0. hence ρ(z,y) < ρ(z,z) + 2−m. on the other hand, the local triangle axiom 61 since b0 ⊆ bρ(x, 2−m), we have ρ(x,z) < ρ(x,x) + 2−m. from the last two inequalities we conclude that ρ(x,y) < ρ(x,x) + 2−h, again using proposition 2.17. that is, we have shown that y ∈ bρ(x, 2−h). it remains to prove that ord(x,rk0(l0+1)t0 ) = 1. let x ∈ bρ(b0∩ct0(l0+1)∩ a(l0+2)k0, 2 −(l0+2)). then there is y ∈ b0 ∩ ct0(l0+1) ∩ a(l0+2)k0 such that ρ(y,x) < ρ(y,y) + 2−(l0+2). we will demonstrate that ρ(x,y) < ρ(x,x) + 2−(l0+1). if ρ(x,x) > ρ(y,y), then ρ(x,y) = ρ(y,x) < ρ(x,x) + 2−(l0+1), using symmetry of ρ. otherwise, since x,y ∈ a(l0+2)k0 , we have ρ(y,y) < ρ(x,x) + 2 −(l0+2) and so ρ(x,y) = ρ(y,x) < ρ(y,y) + 2−(l0+2) < ρ(x,x) + 2−(l0+1). hence d(x,y) = 2ρ(x,y) − ρ(x,x) − ρ(y,y) < 2−l0 , which means that y ∈ bd(x, 2−l0 ) ∩ b. by (†), we thus have b = b0. we have shown that ord(x,rk0(l0+1)t0 ) = 1. now, it is immediate that the collection ⋃ k,l,trklt is a quasi-development for τρ. � 5. the self-distance mapping in distance spaces the self-distance map µρ : x → [0,∞)op (called also a weight function) associated with a partial semimetric ρ proves to be an important object to study. we start with some basic properties of the mapping. later, in section 5, we will see that in domain theory weight functions correspond to measurements in the sense of martin [13]. in section 6.4 we use measurements to build partial metrics on algebraic domains. note that the codomain of the self-distance map is the set of non-negative real numbers with the opposite of the natural order. in this section we consider the interplay between self-distance maps and the specialisation order in partial semimetric spaces. proposition 5.1. the self-distance map associated with a partial semimetric ρ is monotone and strictly monotone with respect to the specialisation orders of its domain and codomain. proof. for any x,y ∈ x, x vτρ y is equivalent to ρ(x,y) = ρ(x,x) by proposition 2.11. but by (ssd), ρ(y,y) ≤ ρ(x,y) = ρ(x,x). that is, µρy ≤ µρx. hence the map µρ is monotone. here, strict monotonicity is the condition ∀x,y ∈ x. (x vρ y and µρx = µρy) imply x = y and one can note that this condition is an equivalent formulation of the t0 axiom of the space (using corollary 2.12). � for the self-distance map µρ associated with a partial symmetric ρ define (5.1) µρ(x,ε) := {y ∈ x | y vτρ x ∧ µρy < µρx + ε}. we say that µρ(x,ε) is the set of elements of x which are ε-close to x. 62 p. waszkiewicz lemma 5.2. let 〈x,ρ〉 be a partial semimetric space. then ∀ε > 0. ∀x ∈ p. µρ(x,ε) ⊆ bρ(x,ε). proof. suppose z ∈ µρ(x,ε). since z vρ x, we have ρ(x,z) = µρz by semimetrizability. therefore, ρ(x,z) = µρz < µρx + ε. that is, z ∈ bρ(x,ε). � proposition 5.3. let 〈x,ρ〉 be a partial semimetric space, x ∈ x and s any subset of x. if s v x and µρx = inf{µρs | s ∈ s}, then x is the supremum of s. proof. let x ∈ u ∈ τρ and let u be any upper bound of s. since u is open, there exists ε > 0 such that x ∈ bρ(x,ε) ⊆ u. by assumption, there exists s ∈ s with µρs < µρx + ε. since s v x, s ∈ µρ(x,ε). by lemma 5.2, s ∈ bρ(x,ε) and so s ∈ u. but the latter set is upper, and therefore u ∈ u. we have shown that for any u ∈ τρ, x ∈ u implies u ∈ u and hence x vτρ u follows. this means that x = ⊔ s. � 5.1. stability condition for partial semimetrics. it happens that there exists a class of partial semimetric spaces, where the distance topology can be recovered from self-distance maps. continuous domains in their scott topology (see section 6.2) are our major example of such spaces. here, we develop the basics. for more information, consult [13, 25, 24]. definition 5.4. let 〈x,τρ〉 be a partial symmetric space. we say that the map ρ is stable if for all x,y ∈ x we have ρ(x,y) := inf{µρz | z vτρ x,y}. lemma 5.5. let 〈x,ρ〉 be a partial semimetric space. then ∀ε > 0. ∃δ > 0. ∀x ∈ p. µρ(x,δ) ⊆ int(bρ(x,ε)). proof. let x ∈ x and ε > 0. then by definition, there exists δ > 0 such that x ∈ bρ(x,δ) ⊆ int(bρ(x,ε)). by lemma 5.2, µρ(x,δ) ⊆ int(bρ(x,ε)). � lemma 5.6. let 〈x,ρ〉 be a stable partial semimetric space. then for every x ∈ x and ε > 0 we have bρ(x,ε) ⊆↑µρ(x,ε). proof. let x ∈ x, ε > 0 and y ∈ bρ(x,ε). then ρ(x,y) < µρx+ε. by stability, there exists z vρ x,y with µρz < ρ(x,y) + ε and hence z ∈ µρ(x,ε). then, y ∈↑µρ(x,ε) follows. � theorem 5.7. let (x,ρ) be a stable partial semimetric space. then {↑µρ(x,ε) | ε > 0} is a neighborhood base at x in τρ. proof. let x ∈ x and ε > 0. then bρ(x,ε) = ↑µρ(x,ε) by lemma 5.2 and lemma 5.6. � the local triangle axiom 63 6. distance for continuous domains as we have mentioned in the introduction, a major motivation for studying general distance spaces was the desire to understand the concept of distance for continuous domains. in previous sections we outlined a general theory, which, we believe, proves especially useful in domain theory. in this section we demonstrate that for continuous domains the theory of partial semimetric spaces coincides with the theory of measurements introduced by martin [13] and studied further in the author’s phd thesis [25]. furthermore, we introduce so called lebesgue measurements (definition 6.5), which correspond to partial semimetrics, which satisfy condition (l). finally, we prove a characterization of partial metrizability of algebraic domains using lebesgue measurements. first, let us recall some terminology of domain theory. see [1] for more information. let p be a poset. a subset a ⊆ p of p is directed if it is nonempty and any pair of elements of a has an upper bound in a. if a directed set a has a supremum, it is denoted ⊔↑a. a poset p in which every directed set has a supremum is called a dcpo. let x and y be elements of a poset p. we say that x approximates (is waybelow ) y if for all directed subsets a of p, y v ⊔↑a implies x v a for some a ∈ a. we denote this by x � y. if x � x then x is called a compact element. the subset of compact elements of a poset p is denoted k(p). now, ↓↓x is the set of all approximants of x below it. ↑↑x is defined dually. we say that a subset b of a dcpo p is a basis for p if for every element x of p, the set ↓↓x∩b is directed with supremum x. a poset is called continuous if it has a basis. it can be shown that a poset p is continuous iff ↓↓x is directed with supremum x, for all x ∈ p. a poset is called a continuous domain if it is a continuous dcpo. note that k(p) ⊆ b for any basis b of p. if k(p) is itself a basis, the domain p is called algebraic. in a continuous domain, every basis is an example of a so called abstract basis, which is a set b together with a transitive relation ≺ on b, such that (int) m ≺ x implies ∃y ∈ b. m ≺ y ≺ x holds for all elements x and finite subsets m of b. for an abstract basis 〈b,≺〉 and an element x ∈ b set x∗ := {y ∈ b | y ≺ x}. for x ∈ b we also define x∗ := {y ∈ b | x ≺ y}. the collection of all sets of the form x∗ is a basis for a topology on b called the pseudoscott topology [11]. in this paper, an abstract basis such that the relation ≺ is reflexive is named a reflexive abstract basis. for any algebraic domain p, the set k(p) is an example of a reflexive abstract basis. for an abstract basis 〈b,≺〉 let i(b) be the set of all ideals (directed, lower subsets) ordered by inclusion. it is called the (rounded ) ideal completion of b. for any algebraic domain p the rounded ideal completion of k(p) is isomorphic to p, in symbols: i(k(p)) ∼= p. 64 p. waszkiewicz upper sets inaccessible by directed suprema form a topology called the scott topology. the specialisation order of the scott topology on a poset coincides with the underlying order. the collection {↑↑x | x ∈ p} forms a basis for the scott topology on a continuous domain p. the scott topology satisfies only weak separation axioms: it is always t0 on a poset but t1 only if the order is trivial. the scott topology on a poset p will be denoted σ(p) (or σ for short). 6.1. measurements. we say that a monotone mapping µ: p → [0,∞)op induces the scott topology on a poset p if ∀u ∈ σ(p). ∀x ∈ p. ∃ε > 0. µ(x,ε) ⊆ u, where µ(x,ε) := {y ∈ p | y v x and µy < µx + ε}. we denote this by µ−→σ(p). definition 6.1. if p is a continuous poset, µ: p → [0,∞)op a scott-continuous map with µ −→ σ(p), then we will say that µ measures p or that µ is a measurement on p. our definition of a measurement is a special case of the one given by martin. in the language of [13] our maps are measurements, which induce the scott topology everywhere. 6.2. partial semimetrics versus measurements. definition 6.2 (martin). let p be a continuous poset with a measurement µ: p → [0,∞)op. the map pµ : p ×p → [0,∞)op defined by pµ(x,y) := ⊔ {µz | z � x,y} = inf{µz | z � x,y} is the partial semimetric associated with µ (cf. proposition 6.3 below). note that the definition is well-formed if any two elements x,y of p are bounded from below. this condition, however, may be omitted: whenever x,y have no lower bound, we scale the measurement to µ∗ : p → [0, 1)op with µ∗(x) := µx/(1 + µx) for any x ∈ p. such map is again a measurement (cf. lemma 5.3.1 of [13], page 135). now, we define p∗µ to be: ∀x,y ∈ p. p∗µ(x,y) = { inf{µ∗z | z � x,y} if ∃z ∈ p. z v x,y 1 otherwise. proposition 6.3. let µ: p → [0,∞)op be a measurement on a continuous poset p . then: (1) pµ is a scott-continuous map from p ×p to [0,∞)op. (2) pµ(x,x) = µx for all x ∈ p . (3) bpµ(x,ε) = ↑µ(x,ε) for all x ∈ p and ε > 0. that is, pµ is a stable partial semimetric, which induces the scott topology. (4) for a sequence (xn) and any x ∈ p , xn → x in the scott topology on p iff lim pµ(xn,x) = µx. the local triangle axiom 65 proof. for the proof of statements (1)–(3) consult [13]. lastly, (4) follows from proposition 2.10. � we are now ready to discuss the coincidence of the theory of partial semimetric spaces with the theory of measurements in the framework of continuous domains. theorem 6.4. for a continuous poset p the following are equivalent: (1) p admits a stable partial semimetric compatible with the scott topology; (2) p admits a partial semimetric compatible with the scott topology, which is scott-continuous as a map from p ×p to [0,∞)op; (3) p admits a measurement. proof. the implication (1)⇒(2) is trivial. for (2)⇒(3), the self-distance mapping of the partial semimetric has the defining measurement property µ → σ(p) by lemma 5.5. in addition, it is scott-continuous since the partial semimetric is. lastly, (3)⇒(1) is a consequence of proposition 6.3.(3). � 6.3. lebesgue measurements. for the purpose of the next definition we introduce the following notation. (6.2) µ(a,ε) := ⋃ {µ(x,ε) | x ∈ a}, where a is a subset of a continuous domain p, the map µ: p → [0,∞)op is monotone and ε > 0. definition 6.5. let p be a continuous domain. a scott-continuous map µ: p → [0,∞)op is a lebesgue measurement on p if for all scott-compact (we may assume saturated) subsets k ⊆ p and for all scott-open subsets u ⊆ p, k ⊆ u ⇒ ∃ε > 0. µ(k,ε) ⊆ u. one can immediately see from the definitions that lebesgue measurements are measurements. proposition 6.6. let p be a continuous domain equipped with a measurement µ: p → [0,∞)op. the following are equivalent: (1) µ is a lebesgue measurement. (2) if ↑x ⊆ u for some x in p and a scott-open subset u of p , then there exists ε > 0 such that µ(↑x,ε) ⊆ u. proof. for the nontrivial direction, suppose that k ⊆ u, where k is a scottcompact saturated subset of p and u is scott-open in p. then for all k ∈ k choose an element l � k with l ∈ u. the collection {↑↑l | l � k} is an open cover of k. hence a finite subcollection ↑↑l1,↑↑l2, . . . ,↑↑ln covers k already. by assumption, for every li, i = 1, . . . ,n there is εi > 0 with µ(↑li,εi) ⊆ u. set ε := min{εi | i = 1, 2, . . . ,n}. then k ⊆ µ(k,ε) ⊆ u. � next, we demonstrate that whenever µ is a lebesgue measurement on a continuous domain, its induced partial semimetric pµ satisfies condition (l). 66 p. waszkiewicz proposition 6.7. let p be a continuous domain measured by µ. the following are equivalent: (1) µ is a lebesgue measurement; (2) for all x ∈ p and for all sequences (xn), (yn) of p , pµ(x,xn) → µx and pµ(xn,yn) → µxn imply pµ(x,yn) → µx. proof. let x ∈ p and take a scott-open set u with x ∈ u. the assumption pµ(x,xn) → µx is equivalent to saying that (xn) →σ x by proposition 6.3. that is, ∃n1 ∈ ω. ∀n ≥ n1. xn ∈ u. by definition, pµ(xn,yn) = inf{µz | z � xn,yn}, for each n ∈ ω. therefore, there exists a sequence (zn) with zn � xn,yn and lim µzn = lim pµ(xn,yn), for all n ∈ ω. since by assumption, lim pµ(xn,yn) = lim µxn, we have lim µzn = lim µxn. this means ∀ε > 0. ∃n2 ∈ ω. ∀n ≥ n2. µzn −µxn < ε. note that the set k := {xn | n ≥ n1}∪{x} is a compact subset of u, and since µ is a lebesgue measurement, the condition specialises to: ∃λ > 0. ∀xn ∈ k. ∀z ∈ p. [z v xn and µz < µxn + λ] ⇒ z ∈ u. hence zn ∈ u for all n ≥ max{n1,n2(λ)}. since zn � yn for all n ∈ ω, (yn) is cofinally in u. that is, (yn) →σ x, or equivalently, pµ(x,yn) → µx. for the converse, observe that by proposition 2.19 for any scott-open set u ⊆ p and x ∈ u we have qpµ(↑x,p \u) > 0. this is however equivalent to saying that there exists ε > 0 such that bpµ(↑x,ε) := ⋃ xvy bpµ(y,ε) ⊆ u. but the map pµ is stable and hence bpµ(y,ε) = ↑µ(y,ε) by the proof of theorem 5.7. we conclude that µ(↑x,ε) ⊆ u. therefore, the map µ is a lebesgue measurement by proposition 6.6. � the last result can be easily extended and stated in a form analogous to theorem 6.4. theorem 6.8. for a continuous domain p the following are equivalent: (1) p admits a stable partial semimetric with (l) for the scott topology; (2) p admits a partial semimetric with (l) for the scott topology, which is scott-continuous as a mapping from p ×p to [0,∞)op; (3) p admits a lebesgue measurement. proof. it is enough to show (2)⇒(3). let p: p × p → [0,∞) be a partial semimetric for the scott topology, which satisfies (l). then for any scottcompact subset k and for any scott-open subset u of p with k ⊆ u, we have qρ(k,p \ u) > 0, by proposition 2.19. it is equivalent to say that for some ε > 0 we have k ⊆ bp(k,ε) ⊆ u. denote the self-distance map for p by µp. the local triangle axiom 67 now, by lemma 5.2, we have µp(k,ε) ⊆ bp(k,ε) ⊆ u and so the mapping µp is a lebesgue measurement on p. � 6.4. partial metrization of algebraic domains. in this section we apply the knowledge about lebesgue measurements to obtain a complete characterization of partial metrizability of the scott topology on an algebraic domain. we start from a similar result obtained by künzi and vajner in proposition 3 p.73 of [10]. theorem 6.9 (künzi and vajner). a poset x admits a partial metric for its alexandrov topology iff there is a function | · |: x → [0,∞) such that (∗) ∀x ∈ x. ∃ε > 0. ∀y ∈↑x. ∀z ∈↓y \↑x. |z|− |y| ≥ ε. for the following crucial lemma, recall the notation from the beginning of section 6. lemma 6.10. let (b,≺) be a reflexive abstract basis equipped with a mapping µ: b → [0,∞)op, which satisfies (∗). then there exists a partial metric compatible with the scott topology on the rounded ideal completion i(b) of b. proof. note that by the discussion in [10] p.74 we can always assume that µ is bounded by 1. therefore, applying theorem 6.9 we conclude that there exists a partial metric p: b×b → [0,∞) which is bounded by 3 and such that µp = µ. the partial metric p satisfies (+) ∀x ∈ x. ∃εx > 0. ↑x = {y | p(x,y) = p(x,x)} = {y | p(x,y) < p(x,x) + εx}, now, extend the function p to i(b) in the following way: for each i,j ∈i(b) define p̂: i(b) ×i(b) → [0,∞) by p̂(i,j) := inf{p(x,y) | x ∈ i,y ∈ j}. since p is continuous as a map from b equipped with the pseudoscott topology to [0,∞)op in its scott topology, the mapping p̂: i(b) ×i(b) → [0,∞)op is scott continuous. note that for every x,y ∈ b we have p̂(x∗,y∗) = p(x,y). this means that p induces the subspace scott topology on the image of b in i(b) under the canonical embedding. step 1: first we will prove that the map so defined satisfies all the partial metric axioms except (t0). for the sharp triangle inequality, note that for all x ∈ i, y ∈ j and z ∈ k, where i,j,k ∈ i(b), we have p(x,z) ≤ p(x,y) + p(y,z) −p(y,y) and this inequality extends to infima. similarly we prove the (ssd) axiom. symmetry is trivial. step 2: if i ⊆ j in i(b), then {p(x,y) | x ∈ i,y ∈ i}⊆ {p(x,y) | x ∈ i, y ∈ j}. therefore, p̂(i,i) ≥ p̂(i,j). hence p̂(i,i) = p̂(i,j) by (ssd). step 3: to prove that p̂ induces the scott topology on i(b) we will show that ↑↑x∗ = bp̂(x∗,ε) for some x ∈ b and ε > 0. suppose x∗ � k for some k ∈ i(b). then x∗ ⊆ k and so p̂(x∗,k) = p̂(x∗,x∗) by step 2. hence, ↑↑x∗ ⊆ bp̂(x∗,ε) for any ε > 0. conversely, if for some l ∈i(b) we have that 68 p. waszkiewicz p̂(x∗,l) < p̂(x∗,x∗) + εx, then p̂(x∗,w∗) < p̂(x∗,x∗) + εx for some w ∈ l. this is however equivalent to p(x,w) < p(x,x) + εx and so w ∈ bp(x,εx) = ↑x by (+). that is, x ≺ w. this means that x ∈ l and so x∗ � l. we have shown that bp̂(x∗,εx) ⊆↑↑x∗. step 4: by step 3, the order induced by p̂ agrees with subset inclusion, which is the specialisation order for the scott-topology on i(b). hence if p̂(i,j) = p̂(i,i) = p̂(j,j) for some i,j ∈i(b), then i ⊆ j and j ⊆ i. therefore, i = j and so p̂ is a partial metric on i(b). � lemma 6.11. let p be an algebraic domain with a monotone map µ: p → [0,∞)op. then the local triangle axiom for µ implies (∗) for the restriction of µ to k(p). proof. take any x ∈ k(p). the set ↑x is scott open and scott-compact in p. by the local triangle axiom, qµ(↑x,p \↑x) > 0. therefore, there exists an ε > 0 such that qµ(↑x,p \↑x) > ε. this implies in particular that for every y ∈ ↑x and z ∈ ↓y \ ↑x we have pµ(y,z) ≥ µy + ε. hence µz = pµ(y,z) ≥ µy + ε. this shows (∗). � therefore, we have obtained a complete characterization of partial metrizability on algebraic domains: theorem 6.12. let p be an algebraic dcpo. the following are equivalent: (1) p admits a lebesgue measurement µ: p → [0,∞)op. 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[27] w. a. wilson, on semi-metric spaces, trans. amer. math. soc. 53 (1931), 361–373. received december 2001 revised january 2003 70 p. waszkiewicz p. waszkiewicz school of computer science, the university of birmingham, edgbaston, birmingham b15 2tt, uk. e-mail address : p.waszkiewicz@cs.bham.ac.uk the local triangle axiom in topology and domain theory. by p. waszkiewicz @ appl. gen. topol. 15, no. 2(2014), 137-146doi:10.4995/agt.2014.3109 c© agt, upv, 2014 asymptotic structures of cardinals oleksandr petrenko a, igor protasov a and sergii slobodianiuk b a department of cybernetics, kyiv national university, ukraine (opetrenko72@gmail.com, i.v.protasov@gmail.com) b department of mechanics and mathematics, kyiv national university, ukraine (slobodianiuk@yandex.ru) abstract a ballean is a set x endowed with some family f of its subsets, called the balls, in such a way that (x, f) can be considered as an asymptotic counterpart of a uniform topological space. given a cardinal κ, we define f using a natural order structure on κ. we characterize balleans up to coarse equivalence, give the criterions of metrizability and cellularity, calculate the basic cardinal invariant of these balleans. we conclude the paper with discussion of some special ultrafilters on cardinal balleans. 2010 msc: 54a25; 05a18. keywords: cardinal balleans; coarse equivalence; metrizability; cellularity; cardinal invariants; ultrafilter. 1. introduction following [15] we say that a ball structure is a triple b = (x, p, b), where x, p are non-empty sets and, for every x ∈ x and α ∈ p , b(x, α) is a subset of x which is called a ball of radius α around x. it is supposed that x ∈ b(x, α) for all x ∈ x and α ∈ p . the set x is called the support of b, p is called the set of radii. given any x ∈ x, a ⊆ x, α ∈ p , we set b∗(x, α) = {y ∈ x : x ∈ b(y, α)}, b(a, α) = ⋃ a∈a b(a, α). received 15 may 2013 – accepted 15 february 2014 http://dx.doi.org/10.4995/agt.2014.3109 i. v. protasov, o. petrenko and s. slobodianiuk a ball structure b = (x, p, b) is called a ballean if (1) for any α, β ∈ p , there exist α′, β′ such that, for every x ∈ x, b(x, α) ⊆ b∗(x, α′), b∗(x, β) ⊆ b(x, β′); (2) for any α, β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x, α), β) ⊆ b(x, γ); (3) for any x, y ∈ x, there exists α ∈ p such that y ∈ b(x, α). a ballean on x can also be determined in terms of entourages of the diagonal ∆x of x × x, in this case it is called a coarse structure [16]. for balleans as counterparts of uniform topological spaces see [15, chapter 1]. let b = (x, p, b), b′ = (x′, p ′, b′) be balleans. a mapping f : x → x′ is called a ≺-mapping if, for every α ∈ p , there exists α′ ∈ p ′ such that, for every x ∈ x, f(b(x, α)) ⊆ b′(f(x), α′). if there exists a bijection f : x → x′ such that f and f−1 are ≺-mappings, b and b′ are called asymorphic and f is called an asymorphism. for a ballean b = (x, p, b), a subset y ⊆ x is called large if there is α ∈ p such that x = b(y, α). a subset v of x is called bounded if v ⊆ b(x, α) for some x ∈ x and α ∈ p . each non-empty subset y ⊆ x determines a subballean by = (y, p, by ) where by (y, α) = y ∩b(y, α). we say that b and b′ are coarsely equivalent if there exist large subset y ⊆ x and y ′ ⊆ x′ such that the subballeans by and b ′ y ′ are asymorphic. given a cardinal κ > 0 and ordinals x, α ∈ κ, we put −→ b(x, α) = [x, x+α] = {y ∈ κ : x 6 y 6 x+α}, ←− b(x, α) = {y ∈ κ : x ∈ [y, y+α]}, ←→ b (x, α) = −→ b(x, α) ∪ ←− b(x, α), so we have got three ball structures −→ κ = (κ, κ, −→ b),←−κ = (κ, κ, ←− b),←→κ = (κ, κ, ←→ b ). it is easy to see that, for κ > 1, the ball structures −→κ and ←−κ do not satisfy (1), so −→κ and ←−κ are not balleans, but ←→κ is a ballean for each κ > 0. this ballean ←→κ is called a cardinal ballean on κ. if κ is finite then the ballean ←→κ is bounded and any two bounded balleans are coarsely equivalent, so in what follows all cardinal balleans are supposed to be infinite. in section 2, we characterize cardinal balleans up to coarse equivalence. the criterions of metrizability and cellularity are given in section 3. in section 4 we study the basic cardinal invariants of cardinal ballean. in section 5 we consider the action of the combinatorial derivation on cardinals. we conclude the paper with discussion in section 6 of some special ultrafilters on cardinal balleans. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 138 asymptotic structures of cardinals 2. coarse equivalence for a ballean b = (x, p, b), we use a preordering < on p defined by the rule: α < β if and only if b(x, α) ⊆ b(x, β) for each x ∈ x. a cofinality cfb is the minimal cardinality of cofinal subsets of p . a ballean b is called ordinal if there exists a well-ordered by < cofinal subset of p . if b is ordinal then b is asymorphic to b′ = (x, κ, b′) for some cardinal κ. each ballean coarsely equivalent to some ordinal ballean is ordinal. clearly, ←→ κ is ordinal for each κ. we say that a family f of subsets of x is uniformly bounded in b if there exists α ∈ p such that, for every f ∈f, there is x ∈ x such that f ⊆ b(x, α). theorem 2.1. a ballean b(x, κ, b) is coarsely equivalent to ←→κ if there exists x0 ∈ x and γ ∈ κ such that (i) b(x0, α + γ) \b(x0, α) 6= ∅ for each α ∈ κ; (ii) for every β ∈ κ, the family {b(x0, α+β)\b(x0, α) : α ∈ κ} is uniformly bounded in b. proof. we choose a subset a ⊆ κ such that [α, α + γ) ∩ [α′, α′ + γ) = ∅ for all distinct α, α′ ∈ a and κ = ∪α∈a[α, α + γ). for each α ∈ a, we use (i) to pick some element x(α) ∈ b(x0, α + γ) \ b(x0, α) and note that the set l = {x(α) : α ∈ a} is large in b. then we denote by f : a → κ the natural well-ordering of a as a subset of κ and, for each α ∈ a, put h(x(α)) = f(α). clearly, h is a bijection from l to κ. by (ii), h is an asymorphism between l and κ. since l is large in b, we conclude that b and ←→κ are coarsely equivalent. � for distinct cardinals κ and κ′, the balleans ←→κ and ←→ κ′ are not coarsely equivalent because (see theorem 4.1 (i)) each large subset of ←→κ (resp. ←→ κ′ has cardinality κ (resp. κ′), so there are no bijections (in particular, asymorphisms) between large subsets of ←→κ and ←→ κ′ . 3. metrizability and cellularity each metric space (x, d) defines a metric ballean (x, r+, bd), where bd(x, r) = {y ∈ x : d(x, y) 6 r}. by [15, theorem 2.1.1], for a ballean b, the following conditions are equivalent: cfb 6 ℵ0, b is asymorphic to some metric ballean, b is coarsely equivalent to some metric ballean. in this case b is called metrizable. applying this criterion, we get theorem 3.1. for a cardinal κ, the following conditions are equivalent: (i) cfκ = ℵ0; (ii) ←→κ is asymorphic to some metric ballean; (iii) ←→κ is coarsely equivalent to some metric ballean. given an arbitrary ballean b = (x, p, b), x, y ∈ x and α ∈ p , we say that x and y are α-path connected if there exists a finite sequence x0, . . . , xn, c© agt, upv, 2014 appl. gen. topol. 15, no. 2 139 i. v. protasov, o. petrenko and s. slobodianiuk x0 = x, xn = y such that xi+1 ∈ b(xi, α) for each i ∈ {0, . . . , n − 1}. for any x ∈ x and α ∈ p , we set b�(x, α) = {y ∈ x : x, y are α-path connected}. the ballean b� = (x, p, b�) is called a cellularization of b. a ballean b is called cellular if the identity mapping id : x → x is an asymorphism between b and b�. by [15, theorem 3.1.3], b is cellular if and only if asdimb = 0, where asdimb is the asymptotic dimension of b. by [15, theorem 3.1.1], a metric ballean is cellular if and only if b is asymorphic to a ballean of some ultrametric space. theorem 3.2. for a cardinal κ, the ballean ←→κ is cellular if and only if κ > ℵ0. proof. the ballean ←→ ℵ0 is not cellular because b �(0, 1) = ℵ0 so ←→ ℵ0 � is bounded but ←→ ℵ0 is unbounded. assume that κ > ℵ0, fix an arbitrary α ∈ κ and note that b�(x, α) ⊆ ⋃ n∈ℵ0 ←→ b (x, nα) ⊆ ←→ b (x, γ), where γ = supn∈ℵ0 nα. since κ > ℵ0, we have γ ∈ κ. hence, id : κ → κ is an asymorphism between ←→κ and ←→κ �. � 4. cardinal invariants given a ballean b = (x, p, b), a subset a of x is called • large if x = b(a, α) for some α ∈ p ; • small if x \b(a, α) is large for every α ∈ p ; • thick if, for every α ∈ p , there exists a ∈ a such that b(a, α) ⊆ a; • thin if, for every α ∈ p , there exists a bounded subset v of x such that b(a, α) ∩b(a′, α) = ∅ for all distinct a, a′ ∈ a\v . we note that large, small, thick and thin subsets can be considered as asymptotic counterparts of dense, nowhere dense, open and uniformly discrete subsets subsets of a uniform topological space. the following cardinal characteristics of a ballean b were introduced and studied in [1], [10], [11], see also [15, chapter 9]. all these characteristics are invariant under asymorphisms. denb = min{|l|: l is a large subset of x}, resb = sup{|f| : f is a family of pairwise disjoint large subsets of x}, coresb = min{|f| : f is a covering of x by small subsets}, thickb = sup{|f| : f is a family of pairwise disjoint thick subsets of x}, thinb = min{|f| : f is a covering of x by thin subsets}, spreadb = sup{|y |b: y is a thin subset of x}, where |y |b = min{|y \ v | : v is a bounded subset of x}. to prove theorem 4.1, we need some ordinal arithmetics (see [2]). an ordinal γ, γ 6= 0 is called additively indecomposable if one of the following equivalent statements holds c© agt, upv, 2014 appl. gen. topol. 15, no. 2 140 asymptotic structures of cardinals • α + β < γ for any ordinal α, β < γ; • α + γ = γ for every ordinal α < γ; every ordinal α > 0 can be written uniquiely in the cantor normal form α = n1 · γ1 + . . . + nk ·γk, where γ1 > . . . > γk are additively indecomposable, n1, . . . , nk are natural numbers. we put |α| = |{β : β < α}|, ||α|| = n1 + . . . + nk, r(α) = γk. theorem 4.1. for every cardinal κ, the following statements hold (i) den←→κ = spread←→κ = κ; (ii) res←→κ = κ and ←→κ can be partitioned in κ large subsets; (iii) cores←→κ = ℵ0; (iv) thick←→κ = κ and ←→κ can be partitioned in κ thick subsets; (v) thin←→κ = cfκ if κ is a limit cardinal and thin κ+ = κ. proof. (i) for a large subset l of x, we pick α ∈ p such that x = b(l, α), observe that |b(x, α)| 6 2|α| so |x| 6 |l|(2|α|) and |x| = |l|. since the ballean ←→κ is ordinal, by [1, theorem 2.3], spreadb = denb and there is a thin subset y of x such that |y |b = |y |. (ii) for κ = ℵ0, we identify κ \{0} with the set n of natural numbers and partition n = ∪n∈ℵ02 nn, where n is the set of all odd numbers. since each subset 2nn is large in ℵ0, we get a desired statement. assume that κ > ℵ0, we have |l| = κ, so it suffices to show that each subset l(γ) is large in κ. then ←→ b (α, γ)∩ l(γ) 6= ∅ for each α ∈ κ. (iii) the statement is trivial for κ = ℵ0 because each singleton in κ is small. assume that κ > ℵ0 and, for each n ∈ n, put sn = {α ∈ κ : ||α|| = n}. we show that, for all n ∈ n and γ ∈ κ, x \ ←→ b (sn, γ) is large so sn is small. since κ > ℵ0, we have |γ| = κ| and γ is cofinal in κ. thus, we may suppose that γ ∈ γ. we take any x ∈ sn, y ∈ ←→ b (x, γ) and note that if y has a member m ·γ in its cantor normal form then m 6 n + 1. on the other hand, if k ·γ is the last member of x + (n + 2) · γ in its cantor normal form then k > n + 2. therefore ←→ b (x, (n + 2) ·γ)∩ (κ\b(sn, γ)) 6= ∅. it follows that κ\ ←→ b (sn, γ) is large. (iv) since the ballean ←→κ is ordinal, by [11, theorem 3.1], thick←→κ = den←→κ and there is a disjoint family of cardinality den←→κ consisting of thick subsets, so we can apply (ii). (v) assume that γ+ < κ but κ can be partitioned into γ thin subsets κ = ∪β<γtβ. by the definition of thin subsets, for each β < γ, there is x(β) ∈ κ such that |[α, α + γ+] ∩ tβ| 6 1 for each α > x(β). we choose x ∈ κ such that x > x(β) for each β < γ. then |[x, x + γ+] ∩ ∪β<γtβ| 6 γ. since |[x, x+γ+]| = γ+, we have |[x, x+γ+]\∪β<γtβ 6= ∅, contradicting κ = ∪β<γtβ. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 141 i. v. protasov, o. petrenko and s. slobodianiuk if κ is a limit cardinal, by above paragraph, κ can not be partitioned into < cfκ thin subsets. since each bounded subset is thin, we have thin κ = cfκ. to conclude the proof, we partition κ+ into κ thin subsets. let {γα : α < κ+} be the numeration of all additively indecomposable ordinals from [κ, κ+). for α ∈ κ+, we fix some bijection fα : γ → [γα, γα+1) and, for each λ ∈ κ, put tλ = {fα(λ) : α ∈ κ +}. then [κ, κ+) = ∪λ<κtλ. clearly [0, κ) is bounded (and so thin) and if x ∈ tλ and x > γα then ←→ b (x, γα)∩ tλ = {x}, so each subset tλ is thin. � it is worth to be marked that a set γ of all additively indecomposable ordinals is not only small (γ = s1) but also thin in ←→ κ , and so ”very small” in asymptotic sense. on the other hand, for κ > ℵ0, γ is unbounded and closed in the order topology on κ, and so ”very large” in topological sense. let b = (x, p, b) be a ballean and α ∈ p . following [14], we say that a subset y ⊆ x has asymptotically isolated balls if, for every β > α, there is y ∈ y such that b(y, β) \ b(y, α) = ∅. if y has asymptotically isolated balls for some α ∈ p , we say that y has asymptotically isolated balls. a ballean b is called asymptotically scattered if each unbounded subset of x has asymptotically isolated balls. a subset y ⊆ x is called asymptotically scattered if the subballean by is asymptotically scattered. the scattered number of b is the cardinal scatb = {min |f| : f is a covering of x by asymptotically scattered subsets}. question 4.2. for each κ, determine or evaluate scat←→κ . we say that a ballean b is weakly asymptotically scattered if, for every unbounded subset y of x, there is α ∈ p such that, for every β ∈ p , there exists y ∈ y such that (b(y, β)\b(y, α))∩x = ∅. a subset y of x is called weakly asymptotically scattered if the subballean by is weakly asymptotically scattered. for κ > ℵ0, each small subset sn from the proof of theorem 4.1 (iii) is weakly asymptotically scattered, so ←→κ can be partitioned into ℵ0 weakly asymptotically scattered subsets. we note that for n > 2, sn is not asymptotically scattered. let γ be a cardinal. following [13], we say that a ballean b = (x, p, b) is γ-extraresolvable if there exists a family f of large subsets of x such that |f| = γ and f ∩ f ′ is small whenever f, f ′ are distinct members of f. the extraresolvability of b is the cardinal exresb = {sup γ : b is γ-extraesolvable}. by [13, theorem 4], exres ←→ ℵ0 = ℵ0. question 4.3. for each κ, determine or evaluate exres κ. under some additional to zfc assumptions, an existence of ℵ1-kurepa tree (see [6, p. 74]), we prove that exres ←→ ℵ1 > ℵ1. recall that a poset (t, <) is c© agt, upv, 2014 appl. gen. topol. 15, no. 2 142 asymptotic structures of cardinals a tree if for any t ∈ t the set sub(t) = {x ∈ t : x < t} is well-ordered. for any t ∈ t , l(t) denotes an ordinal type of sub(t) and l(t ) = sup{l(t) + 1 : t ∈ t}. a tree t is a κ-tree if l(t ) = κ and |levα(t )| < κ, α < l(t ) where levα(t ) = {t ∈ t : l(t) = α}. a κ-tree t is called κ-kurepa if there exists κ+ maximal chains of cardinality κ in t . now let t be an ℵ1-kurepa tree. fix any bijection fα : levα(t ) → [αω; (α + 1)ω), α < ℵ1, and define f = ⋃ fα, bi = f −1(ci), i < ℵ2, where ci is a maximal chain in t . then every bi is large and, for any i 6= j, |bi ∩ bj| < ℵ1. question 4.4. in zfc, does there exist a cardinal κ such that exres←→κ > κ? 5. the combinatorial derivation given a subset a of κ, we denote ∆(a) = {α ∈ κ : a + α ∩ a is unbounded in κ}, a + α = {β + α : β ∈ a}, and say that ∆ : pκ → pκ is the combinatorial derivation. for a group version of ∆ and motivation of this definition see [12]. theorem 5.1. if a subset a of κ is not small and κ is regular then ∆(a) is large. proof. since a is not small, there exists λ ∈ κ such that ←→ b (a, λ) is thick. we show that κ = ←→ b (∆(a), λ + λ). we take an arbitrary δ ∈ κ. since ←→ b (a, λ) is thick, for each α ∈ κ, we can choose xα ∈ a, xα > α such that [xα, xα + δ + λ + λ] ⊂ ←→ b (a, λ). since xα+δ+λ ∈ ←→ b (a, λ), there exists yα ∈ [δ, δ+λ+λ] such that xα+yα ∈ a. the set x = {xα : α ∈ κ} is cofinal in κ and |[δ, δ + λ + λ]| < κ. since κ is regular, we can choose a cofinal subset x′ ⊆ x and y ∈ [δ, δ + λ + λ] such that yα = y for each xα ∈ y . hence, y ∈ ∆(a) and δ ∈ ←→ b (y, λ + λ) so κ = ←→ b (∆(a), λ + λ). � corollary 5.2. for every finite partition of a regular cardinal κ = a1∪. . .∪an, there exists i ∈{1, . . . , n} such that ∆(ai) is large. proof. it suffices to note that at least one cell of the partition is not small. � by theorem 4.1 (iii), each subset sn = {α ∈ κ : ||α|| = n} of κ is small. it is clear that ∆(sn) ⊆{0}∪s1 ∪ . . .∪sn. hence, ∆(sn) is small and corollary does not hold for countable partition even if κ is arbitrary large. question 5.3. is theorem 5.1 true for all singular cardinals? by the definition of ∆, ∆(a) = 0 for each thin subset a of κ. theorem 5.4. let κ be a regular cardinal, a ⊆ κ and 0 ∈ a. then there exist two thin subsets x, y of κ such that ∆(x ∪ y ) = a. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 143 i. v. protasov, o. petrenko and s. slobodianiuk proof. we enumerate a = {aα : α < |a|}, put aα = 0 for each α ∈ κ, α > |a|, and bα = max{α, aα} for every α ∈ κ. we denote x00 = 0. since κ is regular, for each α ∈ κ, we can choose inductively an increasing sequence (xαβ)β6α in κ such that (1) xα0 > supβ<α xββ + bα + bα; (2) xαβ+1 > xαβ + bα + bα, β < α. after κ steps, we denote x = {xαβ : β 6 α < κ}, y = {xαβ + aβ : β 6 α < κ}. by (1) and (2), x and y are thin and ∆(x ∪ y ) = a. � question 5.5. is theorem 5.4 true for all singular cardinals? for a cardinal κ, we denote by κ# the family of all ultrafilters on κ whose members are unbounded in κ. we say that a subset a of κ is sparse if, for every u ∈ κ, the set {α ∈ κ : a ∈ u + α} is bounded in κ, where u + α is an ultrafilter with the base {u + α : u ∈ u}. clearly, the family of all sparse subsets of κ is an ideal in the boolean algebra pκ. if a is thin (in particular, a is bounded) then |{α ∈ κ : a ∈u + α}| 6 1 so a is sparse. we use sparse subsets to characterize strongly prime ultrafilters in κ#. we endow κ with the discrete topology, denote by βκ the stone-čech compactification of κ and use the universal property of βκ to extend the addition + from κ to βκ in such a way that, for each α ∈ κ, the mapping x 7→ x + α : βκ → βκ is continuous, and, for each u ∈ βκ, the mapping x 7→ u + x : βκ → βκ is continuous (see [4, chapter 4]. to describe a base for the ultrafilter u +v, we take any element v ∈ v and, for every x ∈ v , choose some element ux ∈ u. then ∪x∈v (ux + x) ∈ u + v, and the family of subsets of this form is a base for u +v. we note that κ# is a subsemigroup of βκ and say that an ultrafilter u ∈ κ# is strongly prime if u is not in the closure of κ# + κ#. with above definitions, we get the following characterization. theorem 5.6. an ultrafilter u ∈ κ# is strongly prime if and only if some member u ∈u is sparse in κ. question 5.7. what can be said about inclusions between families of sparse and asymptotically scattered subsets of κ? question 5.8. is a subset a of κ sparse provided that ∆(a) is sparse? 6. around t -points a free ultrafilter u on an infinite cardinal κ is called uniform if |u| = κ for each u ∈ u. we say that a uniform ultrafilter u on κ is a tκ-point if, for each minimal well-ordering < of κ, some member of u is thin in the ballean ((κ, <), (κ, <), ←→ b ). we begin with discussion of tℵ0-points. let g be a transitive group of permutations of κ. we consider a ballean b(g, κ) = (κ, [g]<ω, b), where [g]<ω is the family of all finite subsets of g and b(x, f) = {x}∪ f(x) for each x ∈ x and f ∈ [g]<ω, f(x) = {f(x), f ∈ f}. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 144 asymptotic structures of cardinals a free ultrafilter u on ℵ0 is said to be a t -point if, for every countable transitive group g of permutations of ℵ0, some member of u is thin in b(g,ℵ0). by [8, theorem 1], every p-point and every q-point (in βω) are t -points. under ch, there exist pand q-points but it is unknown [3, question 25] whether in each model of zfc there exists either p-point or q-point. by [5] and [7], this is so if c 6 ℵ2. by [8, proposition 4], under ch, there exist a t -point which is neither q-point nor weak p-point. it is an open question [9, p. 348] whether t -points exist in zfc without additional assumptions. theorem 6.1. every t -point is a tℵ0-point. proof. a ballean b = (x, p, b) is called uniformly locally finite if, for each α ∈ p , there exists a natural number n such that |b(x, α)| 6 n for every x ∈ x. by [9, theorem 9], for x = ℵ0 and a free ultrafilter u on x, the following statements are equivalent (i) u is a t -point; (ii) for every metrizable uniformity locally finite ballean b on x, some member of u is thin in b; (iii) for every sequence (fn)n∈ω of uniformly bounded coverings of x, there exists u ∈u such that, for each n ∈ ω, |f ∩ u| 6 1 for all but finitely many f ∈ fn. a covering f of ω is uniformly bounded if there is m ∈ ℵ0 such that, for each x ∈ x, |st(x,f)| 6 m, st(x,f) = ⋃ {f ∈ f : x ∈ f}. the equivalence (i) ⇔ (ii) implies that every t -point is a tℵ0-point because each ballean ((ℵ0, <), (ℵ0, <), ←→ b ) is metrizable and uniformly locally finite. � question 6.2. does there exist a tℵ0-point but not a t -point? question 6.3. does there exist a tℵ0-point in zfc? let us call a free ultrafilter u on ℵ0 to be s-point if, for every minimal well ordering < of ℵ0, some member of u is small in the ballean ((ℵ0, <), (ℵ0, < ), ←→ b ). question 6.4. does there exist an s-point in zfc? theorem 6.5. in zfc, for every uncountable regular cardinal κ, there exists a tκ-point. proof. we say that a uniform ultrafilter u on κ is a club-ultrafilter if u contains a family of all closed unbounded subsets of κ. by [17], u is a q-point, i.e. u is selective with respect to any partition of κ into the cells of cardinality < κ. now we fix some club-ultrafilter u on κ and prove that u is a tκ-point. let < be a minimal well-ordering of κ defined by some bijection h : κ → (κ, <). we partition inductively (κ, <) = ∪α∈(κ,<)[xα, xα + γα) into consecutive intervals, where γ = {γα : α ∈ (κ, <)} is the set of all additively indecomposable ordinals of (κ, <). by above paragraph, some member u ∈ u meets each subset h−1([xα, xα+γα)) in at most one point. by the choice of intervals [xα, xα+γα), u is thin in the ballean ((κ, <), (κ, <), ←→ b ). � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 145 i. v. protasov, o. petrenko and s. slobodianiuk question 6.6. is theorem 6.5 true for uncountable singular cardinals? references [1] m. filali and i. v. protasov, spread of balleans, appl. gen. topol. 9 (2008), 161–175. [2] t. jech, lectures in set theory, lecture notes in math. 27 (1971). [3] k. p. hart and j. van mill, open problems in βω, in open problems in topology, j. van mill, g.m. reed (editors), elsevier science publishers, north holland, 1990, 98–125. [4] n. hindman and d. strauss, algebra in the stone-čech compactification: theory and applications, walter de gruyter, berlin, new york, 1998. [5] j. ketonen, on the existence of p -points in the stone-čech compactification of integers, fundam. math.62 (1976), 91–94. [6] k. kunen, set theory: an introduction to independence proofs, north-holland, 1980. [7] a. r. d. mathias, o# and p -point problem, lecture notes in mathematics 669 (1978), 375–384. [8] o. petrenko and i. v. protasov, thin ultrafilters, notre dame j. formal logic 53 (2012), 79–88. [9] o.v. petrenko and i. v. protasov, balleans and g-spaces, ukr. math. j. 64 (2012), 344–350. [10] i. v. protasov, resolvability of ball structures, appl. gen. topol. 5 (2004), 191–198. [11] i. v. protasov, cellularity and density of balleans, appl. gen. topol. 8 (2007), 283–291. [12] i. v. protasov, the combinatorial derivation, appl. gen. topol. 14 (2013), 171–178. [13] i. v. protasov, extraresolvability of balleans, comment. math. univ. carolinae 48 (2007), 161–175. [14] i. v. protasov, asymptotically scatterd spaces, preprint (arxiv:1212.0364). [15] i. protasov and m. zarichnyi, general asymptology, math. stud. monogr. ser., vol. 12, vntl publishers, lviv, 2007. [16] j. roe, lectures on coarse geometry, amer. math. soc., providence, r.i, 2003. [17] w. rudin, homogenity problems in the theory of čech compactifications, duke math. j. 23 (1956), 409–419. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 146 () @ appl. gen. topol. 16, no. 1(2015), 45-52doi:10.4995/agt.2015.3006 c© agt, upv, 2015 on faint continuity aisling e. mccluskey a and ivan l. reilly b a school of mathematics, statistics and applied mathematics, national university of ireland, galway, ireland. (aisling.mccluskey@nuigalway.ie) b department of mathematics, university of auckland, new zealand. (i.reilly@auckland.ac.nz) abstract recently the class of strongly faintly α-continuous functions between topological spaces has been defined and studied in some detail. we consider this class of functions from the perspective of change(s) of topology. in particular, we conclude that each member of this class of functions belongs the usual class of continuous functions between topological spaces when the domain and codomain of the function in question have been retopologized appropriately. some consequences of this fact are considered in this paper. 2010 msc: 06a06; 54h10. keywords: change of topology; strongly faint α-continuity; faint αcontinuity. 1. introduction there can be no argument that the concept of continuity is one of the most important ideas in the whole of mathematics. so much so that it has become fashionable to speak of ‘continuous mathematics’ and ‘discrete mathematics’ as a fundamental division of mathematics, rather than the more traditional ‘pure mathematics’ and ‘applied mathematics’. over recent decades, many generalizations and variants of continuous functions between topological spaces have been introduced. in 1982, long and herrington [7] considered the class of faintly continuous functions. three weak forms of faint continuity were introduced by noiri and popa [15]. more recently, nasef and noiri [12] have defined and studied three strong forms of faint continuity under the names of strongly faint semi-continuity, strongly faint precontinuity and strongly faint received 26 may 2014 – accepted 4 december 2014 http://dx.doi.org/10.4995/agt.2015.3006 a. mccluskey and i. reilly β-continuity. very recently, nasef [11] has introduced two more strong forms of faint continuity using the terms strongly faint α-continuity and strongly faint γ-continuity, and has provided arguments for their significance outside of mathematics. nasef [11] takes care to distinguish between strongly faint α-continuity and other variants of continuity, especially in the diagram of remark 4.1 and the set of examples 4.1 to 4.5. our primary purpose is to argue that strongly faint αcontinuity is merely continuity in disguise. indeed, if the domain and codomain spaces of a strongly faintly α-continuous function f are each re-topologized in a suitable fashion (see proposition 3.1), then f is simply a continuous function. this observation puts the notion of strongly faint α-continuity in a more natural context, and it permits alternative proofs of most of the results of nasef [11]. in the language of category theory, we are claiming that a strongly faintly α-continuous function f arises because the wrong source and target have been chosen for the morphism f in the category t op of topological spaces and continuous functions. in section 2, we provide the relevant definitions of the classes of functions we consider in this paper. in particular, we examine the basic properties of αtopologies and θ-topologies. section 3 investigates the class of strongly faintly α-continuous functions introduced by nasef [11], especially from the perspective of change of topology. section 4 provides a decomposition of faint α-continuity. our notation and terminology are standard; see for example dugundji [3]. no separation properties are assumed for topological spaces unless explicitly stated. we denote the interior of a subset a of the topological space (x, τ) by inta, and the closure by cla. 2. definitions and preliminaries definition 2.1. a subset a of a topological space (x, τ) is said to be (i) α-open if a ⊆ int(cl(inta)) (ii) semi-open if a ⊆ cl(inta) (iii) preopen if a ⊆ int(cla) (iv) γ-open if a ⊆ cl(inta) ∪ int(cla) (v) θ-open if for each x ∈ a there exists an open set u such that x ∈ u ⊆ clu ⊆ a. the family of all α-open (respectively semi-open, pre-open, γ-open) subsets of x is denoted by τα (respectively so(x), po(x), γo(x)). the complement of a θ-open (respectively γ-open, α-open) set is said to be θ-closed (respectively γ-closed, α-closed). njastad [14] defined α-open subsets in 1965 and he showed that for any topological space (x, τ), the collection τα of all α-open subsets of (x, τ) is a topology on x larger than τ. in 1963, levine [5] introduced semi-open sets. the term preopen subset was introduced by mashhour, abd el-monsef and eldeeb [10] but the concept had appeared much earlier. for example, carson and michael [2] used the term locally dense for preopen sets in 1964. the family c© agt, upv, 2015 appl. gen. topol. 16, no. 1 46 on faint continuity of γ-open subsets of (x, τ) was studied by andrijević [1] in 1996 under the name of b-open subsets. velicko [17] defined θ-open subsets and showed that the collection of all θ-open subsets of (x, τ) forms a topology τθ on x which is smaller than τ. thus for any topological space (x, τ), we have τθ ⊂ τ ⊂ τ α. there is one very significant difference between the families of α-open subsets and γ-open subsets of a topological space (x, τ). this is that τα is always a topology on x, while γo(x) is not a topology on x in general. consider the following example: example 2.2. let x be the set of real numbers r, and let τ be the usual (euclidean) topology on x. if q is the subset of all rational numbers, then q is γ-open. furthermore, if a = [0, 1), the half-open unit interval, then a is γopen. however b = a∩q is not γ-open, since 0 ∈ b but 0 6∈ cl(intb)∪int(clb). thus γo(x, τ) is not closed under finite intersection, and so γo(x, τ) is not a topology on x. despite the comment of nasef [11, page 540] immediately preceding his theorem 4.4, the collection of all γ-open sets of (y, σ) does not form a topology on y . andrijevic [1] showed no such result. he showed that there is a topology on y generated in a natural way by the family γo(y, σ). in fact, this topology is {v ⊂ y : v ∩ s ∈ γo(y, σ) whenever s ∈ γo(y, σ)}. definition 2.3. a function f : x →y is said to be faintly continuous (respectively faintly semi-continuous, faintly precontinuous, faintly α-continuous) if for each x ∈ x and each θ-open set v containing f(x) there exists an open (respectively semi-open, preopen, α-open) set u containing x such that f(u) ⊆ v . definition 2.4. a function f : x → y is said to be strongly faintly semicontinuous (respectively strongly faintly precontinuous, strongly faintly α-continuous, strongly faintly γ-continuous) if for each x ∈ x and each semi-open (respectively preopen, α-open, γ-open) set v containing f(x), there exists a θ-open set u containing x such that f(u) ⊆ v . 3. change of topology the fundamental defining feature of the class of strongly faintly α-continuous functions between topological spaces is given by the following result, which is an immediate corollary of theorem 3.1 of nasef [11]. proposition 3.1. a function f : (x, τ) → (y, σ) is strongly faintly α-continuous if and only if f : (x, τθ) → (y, σα) is continuous. one immediate conclusion from proposition 3.1 is that each strongly faintly α-continuous function is a morphism in the category t op whose objects are topological spaces and whose morphisms are continuous functions between topological spaces. if f : (x, τ) → (y, σ) is strongly faintly α-continuous, then f is a morphism in t op from (x, τθ) to (y, σα). it is not the case that f lies outside of t op. it is the case that the wrong objects in t op have been chosen for the source and target of the morphism f in the category t op. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 47 a. mccluskey and i. reilly throughout his paper, nasef [11] has presented his results in a way that strongly suggests that there is an exact parallel between strongly faint αcontinuity and strongly faint γ-continuity. example 2.2 shows that there is no analogue of proposition 3.1 for strongly faint γ-continuity. change of topology methods of proof cannot be used for strongly faint γ-continuity, as they are in the rest of this paper for strongly faint α-continuity. the equivalence given in proposition 3.1 shows that strongly faint α-continuity is a µ-continuity property in the sense of gauld, mrsevic, reilly and vamanamurthy [4]. the fact that strongly faint α-continuity is equivalent to continuity under these changes of topology can be exploited to yield alternative elegant proofs of existing results, and to suggest new results. definitions 3.2 and 3.3 of nasef [11] describe variations of the standard hausdorff (or t2) and compactness properties, denoted α-t2 and θ-t2, and α-compact and θ-compact, respectively. the following results are immediate from these definitions. lemma 3.2. (i) (x, τ) is α-t2 if and only if (x, τα) is t2. (ii) (x, τ) is θ-t2 if and only if (x, τθ) is t2. lemma 3.3. (i) (x, τ) is α-compact if and only if (x, τα) is compact. (ii) (x, τ) is θ-compact if and only if (x, τθ) is compact. using the preservation of compactness by continuous functions, we can obtain an elegant proof of a whole space version of theorem 3.8 of nasef [11], as follows: proposition 3.4. if f : (x, τ) → (y, σ) is a strongly faintly α-continuous surjection and (x, τ) is θ-compact, then (y, σ) is α-compact. proof. we have that f : (x, τθ) → (y, σα) is continuous and (y, σα) is the image of the compact space (x, τθ). hence (y, σα) is compact, and therefore (y, σ) is α-compact. � the topic considered in theorem 4.7 of nasef [11] yields to a similar approach. note that (x, τ) is α-connected [θ-connected] if and only if (x, τα) [ (x, τθ)] is connected. as expected, (x, τ) is defined to be θ-connected if x cannot be expressed as the union of two nonempty disjoint θ-open subsets of x. proposition 3.5. if f : (x, τ) → (y, σ) is a strongly faintly α-continuous surjection and (x, τ) is θ-connected, then (y, σ) is α-connected. proof. we have that f : (x, τθ) → (y, σα) is a continuous surjection, and (x, τθ) is connected. thus (y, σα) is connected; that is, (y, σ) is α-connected. � classical topological results can be used with this change of topology technique to obtain new results. we now provide some examples. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 48 on faint continuity proposition 3.6. let f, g : (x, τ) → (y, σ) be strongly faintly α-continuous, and let (y, σ) be α-t2. then the equalizer e = {x ∈ x : f(x) = g(x)} of f and g is θ-closed in (x, τ). proof. we have that f, g : (x, τθ) → (y, σα) are continuous by proposition 3.1, and that (y, σα) is hausdorff by lemma 3.2(i). so the classical result of dugundji [3, page 140, 1.5(1)] implies that e is closed in (x, τθ); that is, e is θ-closed in (x, τ). � proposition 3.7. if f : (x, τ) → (y, σ) is strongly faintly α-continuous and (y, σ) is α-t2, then the graph g(f) of f is closed in (x × y, τθ × σα). proof. we use the fact that (y, σα) is hausdorff and that f : (x, τθ) → (y, σα) is continuous and a classical result ( see for example dugundji [3, page 140, 1.5(3)]) to conclude that g(f) is closed in (x × y, τθ × σα). � our proposition 3.7 is essentially a restatement of theorem 3.9 of nasef [11]. definition 3.8. a function f : (x, τ) → (y, σ) is defined to be: (i) strongly α-irresolute [9] if f−1(v ) is open in (x, τ) for every α-open subset v of (y, σ), (ii) α-irresolute [8] if f−1(v ) is α-open in (x, τ) for every α-open subset v of (y, σ), (iii) α-continuous [16] if f−1(v ) is α-open in (x, τ) for every open subset v of (y, σ), (iv) strongly θ-continuous [14] if for each point x in x and each open set v containing f(x), there is an open set u containing x such that f(clu) ⊂ v , (v) quasi-θ-continuous [14] if for each point x in x and each θ-open set v containing f(x), there is a θ-open set u containing x such that f(u) ⊂ v . each of these properties of functions reduces to continuity provided appropriate changes of topology are made on the domain and/or the codomain. in particular, we have proposition 3.9. let f : (x, τ) → (y, σ) be a function. then (i) f is strongly α-irresolute if and only if f : (x, τ) → (y, σα) is continuous, (ii) f is α-irresolute if and only if f : (x, τα) → (y, σα) is continuous [16], (iii) f is α-continuous if and only if f : (x, τα) → (y, σ) is continuous [16], (iv) f is strongly θ-continuous if and only if f : (x, τθ) → (y, σ) is continuous [6], (v) f is quasi-θ-continuous if and only if f : (x, τθ) → (y, σθ) is continuous, (vi) f is faintly continuous if and only if f : (x, τ) → (y, σθ) is continuous [7], c© agt, upv, 2015 appl. gen. topol. 16, no. 1 49 a. mccluskey and i. reilly (vii) f is faintly α-continuous if and only if f : (x, τα) → (y, σθ) is continuous. we observe that (vi) and (vii) follow from definition 2.3. corollary 3.10. let f : (x, τ) → (y, σ) be a function. then the following are equivalent: (1) f : (x, τ) → (y, σ) is strongly faintly α-continuous. (2) f : (x, τθ) → (y, σα) is continuous. (3) f : (x, τθ) → (y, σ) is strongly α-irresolute. (4) f : (x, τ) → (y, σα) is strongly θ-continuous. change of topology allows us to prove the next set of results simply by observing that the composition of two continuous functions is continuous. there is no need to use first principles to prove such results, as nasef [11, theorem 4.5] has done. note that (2), (3) and (4) of our proposition 3.11 are theorems 4.6 (iii), 4.5 and 4.6 (ii) of nasef [11] respectively, for the α case. proposition 3.11. let f : (x, τ) → (y, σ) and g : (y, σ) → (z, µ) be functions. (1) if f is faintly continuous and g is strongly faintly α-continuous, then g ◦ f is strongly α-irresolute. (2) if f is quasi-θ-continuous and g is strongly faintly α-continuous, then g ◦ f is strongly faintly α-continuous. (3) if f is strongly faintly α-continuous and g is α-irresolute, then g ◦ f is strongly faintly α-continuous. (4) if f is strongly faintly α-continuous and g is α-continuous, then g ◦ f is strongly θ-continuous. (5) if f is strongly θ-continuous and g is strongly α-irresolute, then g ◦ f is strongly faintly α-continuous. (6) if f is α-continuous and g is faintly continuous, then g ◦ f is faintly α-continuous. (7) if f is faintly α-continuous and g is strongly faintly α-continuous, then g ◦ f is α-irresolute. (8) if f is faintly α-continuous and g is strongly θ-continuous, then g ◦ f is α-continuous. (9) if f is is strongly faintly α-continuous and g is faintly α-continuous, then g ◦ f is quasi-θ-continuous. proof. (1) f : (x, τ) → (y, σθ) and g : (y, σθ) → (z, µα) are continuous, so that g ◦ f : (x, τ) → (z, µα) is continuous. (2) f : (x, τθ) → (y, σθ) and g : (y, σθ) → (z, µα) are continuous, so that g ◦ f : (x, τθ) → (z, µα) is continuous. (3) f : (x, τθ) → (y, σα) and g : (y, σα) → (z, µα) are continuous, so that g ◦ f : (x, τθ) → (z, µα) is continuous. (4) f : (x, τθ) → (y, σα) and g : (y, σα) → (z, µ) are continuous, so that g ◦ f : (x, τθ) → (z, µ) is continuous. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 50 on faint continuity (5) f : (x, τθ) → (y, σ) and g : (y, σ) → (z, µα) are continuous, so that g ◦ f : (x, τθ) → (z, µα) is continuous. (6) f : (x, τα) → (y, σ) and g : (y, σ) → (z, µθ) are continuous, so that g ◦ f : (x, τα) → (z, µθ) is continuous. (7) f : (x, τα) → (y, σθ) and g : (y, σθ) → (z, µα) are continuous, so that g ◦ f : (x, τα) → (z, µα) is continuous. (8) f : (x, τα) → (y, σθ) and g : (y, σθ) → (z, µ) are continuous, so that g ◦ f : (x, τα) → (z, µ) is continuous. (9) f : (x, τθ) → (y, σα) and g : (y, σα) → (z, µθ) are continuous, so that g ◦ f : (x, τθ) → (z, µθ) is continuous. � 4. a decomposition of faint α-continuity the following fundamental relationship between classes of generalized open sets in a topological space was first proved in 1985 by reilly and vamanamurthy [16]. lemma 4.1. in any topological space (x, τ), τα = po(x) ∩ so(x). the definition of faint α-continuity, definition 2.3, together with lemma 4.1 immediately implies the following decomposition of faint α-continuity. proposition 4.2. a function f : (x, τ) → (y, σ) is faintly α-continuous if and only if f is faintly precontinuous and faintly semi-continuous. nasef [11] has established by his set of examples 4.1 to 4.5 and the diagram of his remark 4.1 that faint α-continuity is distinct and independent from existing classes of functions. his diagram indicates that faint α-continuity implies each of faint precontinuity and faint semi-continuity. proposition 4.2 reveals that there is a ‘joint’ converse. together, these two notions are equivalent to faint α-continuity. the equivalence does not extend however to a ‘strong’ analogue. while strongly faint precontinuity and strongly faint semi-continuity each clearly imply strongly faint α-continuity (by definition 2.4 and lemma 4.1), nasef and noiri [12, example 3.1] establish that the converse does not hold. references [1] d. andrijević, on b-open sets, mat. vesnik 48 (1996), 59–64. [2] h. corson and e. michael, metrizability of certain countable unions, illinois j. math.8 (1964), 351–360. [3] j. dugundji, topology, allyn and bacon, boston, mass. (1966). [4] d. gauld, m. mrsević, i. l. reilly and m. k. vamanamurthy, continuity properties of functions, coll. math. soc. janos bolyai 41 (1983), 311–322. [5] n. levine, semi open sets and semi-continuity in topological spaces, amer. math. monthly 70 (1963), 36–41. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 51 a. mccluskey and i. reilly [6] p. e. long and l. l. herrington, strongly θ-continuous functions, j. korean math. soc. 18 (1981), 21–28. [7] p. e. long and l. l. herrington, the tθ-topology and faintly continuous functions, kyungpook math. j. 22 (1982), 7–14. [8] s. n. maheshwari and s. s. thakur, on α-irresolute mappings, tamkang j. math. 11 (1980), 209–214. [9] r. a. mahmoud, m. e. abd el-monsef and a. a. nasef, some forms of strongly µcontinuous functions, µ ∈ {α-irresolute, open, closed}, kyungpook math. j. 36 (1996), 143–150. [10] a. s. mashhour, m. e. abd el-monsef and s. n. el-deeb, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. eygpt 53 (1982), 47–53. [11] a. a. nasef, recent progress in the theory of faint continuity, math. comput. modelling 49 (2009), 536–541. [12] a. a. nasef and t. noiri, strong forms of faint continuity, mem. fac. sci. kochi univ. ser. a. math. 19 (1998), 21–28. [13] o. njastad, on some classes of nearly open sets, pacific j. math. 15 (1965), 961– 970. [14] t. noiri, on δ-continuous functions, j. korean math. soc. 16 (1980), 161–166. [15] t. noiri and v. popa, weak forms of faint continuity, bull. math. soc. sci. math. r. s. roumanie 34 (82) (1990), 263– 270. [16] i. l. reilly and m. k. vamanamurthy, on α-continuity in topological spaces, acta math. hungar. 45 (1985), 27–32. [17] n. v. velicko, h-closed topological spaces, amer. math. soc. transl. 78, no. 2 (1968), 103–118. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 52 grabnercwn.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 199-212 relative collectionwise normality elise grabner, gary grabner, kazumi miyazaki and jamal tartir abstract. in this paper we study properties of relative collectionwise normality type based on relative properties of normality type introduced by arhangel’skii and genedi. theorem suppose y is strongly regular in the space x. if y is paracompact in x then y is collectionwise normal in x. example a t2 space x having a subspace which is 1− paracompact in x but not collectionwise normal in x. theorem suppose that y is sregular in the space x. if y is metacompact in x and strongly collectionwise normal in x then y is paracompact in x. 2000 ams classification: primary 54d20, secondary 54a35 keywords: paracompact, collectionwise normal, relative topological properties 1. introduction in this paper properties of relative collectionwise normality type based on relative properties of normality type introduced in [2] and [3] are studied. our study focusses on the following well known theorems and relative properties of paracompactness type introduced in [1] and [4]. theorem 1.1 (bing). every paracompact space is collectionwise normal. theorem 1.2 (michael-nagami). every metacompact collectionwise normal space is paracompact. a theorem concerning the relative properties of a subspace y in a space x becomes a theorem about the corresponding global properties of x by letting y = x. it is not surprising when the proof of a result concerning relative properties is a straight forward modification of the usual proof of the corresponding global result. for example we show that if y is strongly star normal in x then y is strongly collectionwise normal in x, theorem 3.7. the proof is 200 e. grabner, g. grabner, k. miyazaki and j. tartir the natural relative version of the standard proof that t2 paracompact spaces are collectionwise normal using the fully normal characterization of paracompactness. however this is not always true. for example there exist a good number of non-equivalent relative properties of paracompactness type, see [1], [2], [6], [7] and [8]. some of these properties are preserved by closed maps (cp-paracompact in x, [7]) and some are not (paracompact in x from outside, [7]). some imply that the subspace y is paracompact (strongly star normal in x, [4]) while others do not (1paracompact in x, [6]). we give an example of a t2 space having a subspace which is 1− paracompact in x but not collectionwise normal in x, example 5.4. thus to obtain an analog of bing′s theorem for subspaces y paracompact in x it is necessary to assume that y satisfies relative separation properties not implied by the space x being a t2 space and y being paracompact in x. if y is paracompact in x and strongly regular in x then y is collectionwise normal in x, theorem 3.3. we give several relative versions of the michael-nagami theorem. if y is s− regular in x, metacompact in x and strongly collectionwise normal in x then y is paracompact in x, theorem 4.4. if y is closed, s− regular in x, collectionwise normal in x and metacompact then y is paracompact in x, corollary 4.5. throughout this paper all spaces are assumed to be hausdorff. suppose x is a space and y a subspace of x. when a set u is said to be open, we mean open with respect to the topology on x even if u happens to be a subset of y. for a set x, x ∈ x, a subset a of x and a collection u of subsets of x, (u)x = {u ∈ u : x ∈ u}, (u)a = {u ∈ u : a ∩ u 6= φ}, st(x, u) = ∪(u)x and st(a, u) = ∪(u)a. 2. definitions and lemma suppose y is a subset of the space x. the subset y is 1. regular in x , 2. super regular in x, 3. strongly regular in x, 4. sregular in x, 5. normal in x , 6. snormal in x, 7. strongly normal in x provided 1. for each x ∈ y and every subset f of x\{x} closed in x there are disjoint open sets u and v such that x ∈ u and f ∩ y ⊆ v [3]. 2. for each x ∈ y and every subset f of x\{x} closed in x there are disjoint open sets u and v such that x ∈ u and f ⊆ v [3]. 3. for each x ∈ x and every subset f of x\{x} closed in x there are disjoint open sets u and v such that x ∈ u and f ∩ y ⊆ v [3]. 4. y is both super regular and strongly regular in x. 5. for each pair e and f of disjoint closed subsets of x there are disjoint open sets u and v such that e ∩ y ⊆ u and f ∩ y ⊆ v [3]. 6. for each pair, e and f of disjoint closed subsets of x, there are disjoint open subsets of x, u and v such that e ⊆ u and f ∩ y ⊆ v [10]. 7. for each pair e and f of disjoint closed (in y ) subsets of y there are disjoint open sets u and v such that e ⊆ u andf ⊆ v [2]. relative collectionwise normality 201 suppose y is a subset of a space x. if y is super regular or strongly regular in x (s− normal or strongly normal in x) then y is regular (normal) in x. however in general there is no implication between these two stronger conditions. also if y is normal (snormal) in x then y is regular (sregular) in x. if x is a regular (normal) space then every subspace of x is s− regular (snormal but not necessarily strongly normal) in x. the subspace y can be strongly normal in x without being strongly regular in x. suppose y is a subset of a space x. a collection u is said to be locally finite on y provided for every y ∈ y there is an open v containing y such that (u)v is finite. a collection f of closed subsets of x is said to be weakly closure reserving with respect to y provided for all f′ ⊆ (f) y , (∪f′)∩y = (∪f′)∩y, [7]. the following lemmas from [7] are frequently used when working with collections that are locally finite with respect to a subset y of a space x. lemma 2.1. suppose y ⊆ x and u is a collection of open subsets of the space x locally finite on y. then the collection {u : u ∈ u} is weakly closure preserving with respect to y and locally finite on y. lemma 2.2. suppose that y ⊆ x and f is a collection of closed subsets of the space x weakly closure preserving with respect to y. 1. if b ⊆ x is closed then {f ∩ b : f ∈ f} is weakly closure preserving with respect to y. 2. if a ⊆ y then a ⊆ x\∪(f\(f)a). in particular, for all y ∈ y, y /∈ ∪{f ∈ f : y /∈ f}. for a space x and y ⊆ x, a collection a of subsets of the space x is said to be discrete with respect to y provided for all x ∈ y there is an open neighborhood u of x that intersects at most one member of a. we say that y is collectionwise normal in a space x provided for every discrete collection f of closed subsets of x, there is a collection of open subsets of x, u = {u(f) : f ∈ f} discrete with respect to y such that for all f ∈ f, f ∩ y ⊆ u(f) ⊆ x\∪(f\{f}). notice that a collection of subsets of a space x which is discrete with respect to a subspace y of x need not be pairwise disjoint. however in the case of collectionwise normality in x this is not a problem as seen in the following lemma. lemma 2.3. suppose y ⊆ x and u is a collection of open subsets of the space x discrete with respect to y. for each u ∈ u let v (u) = u\∪(u\{u}). then the collection {v (u) : u ∈ u} is a pairwise disjoint collection of open subsets of x discrete with respect to y such that for all u ∈ u, u ∩ y = v (u) ∩ y. theorem 2.4. if y is collectionwise normal in the space x then y is normal in x. we say that a subspace y is strongly collectionwise normal in the space x provided for every collection f of closed subsets of x which is discrete with respect to y there is a collection of open subsets of x, u = {u(f) : f ∈ f} discrete with respect to y such that for all f ∈ f, f ∩ y ⊆ u(f) ⊆ 202 e. grabner, g. grabner, k. miyazaki and j. tartir x\∪(f\{f}). by lemma 2.3 the members of u can be taken to be pairwise disjoint and discrete with respect to y if we choose. notice that if y is a closed subset of x and f is a collection of closed subsets of x which is discrete with respect to y then {f ∩ y : f ∈ f} is a discrete collection of closed subsets of x. theorem 2.5. if y is strongly collectionwise normal in the space x then y is strongly normal in x and a collectionwise normal subspace of x. if y is a closed subset of x then y is strongly collectionwise normal in x if and only if y is collectionwise normal in x. a closed collectionwise normal subspace of a space x need not be collectionwise normal in x, example 5.2. 3. relative paracompact implies relative collectionwise normality the following definitions of the most natural properties of relative paracompactness type are from [2]. the subspace y is said to be 1− paracompact in x provided every open cover of x has an open refinement locally finite on y. the subspace y is paracompact in x provided every open cover of x has an open partial refinement covering y and locally finite on y. in [6] it is observed that if y is strongly regular in x and paracompact in x then y is normal in x. if y is closed and paracompact in x then y is normal in x. however a closed subset of a regular space x can be paracompact in x and not s− normal in x, example 5.3. although it is readily seen that if y is 1− paracompact in x then y is superregular in x it need not be strongly regular in x, example 5.1. the following theorem shows that s− normality in x is a relative property of normality type that relates to 1− paracompactness in x. theorem 3.1. suppose y is strongly regular in the space x. if y is 1− paracompact in x then y is s−normal in x. proof. suppose e and f are disjoint closed subsets of x. since y is strongly regular in x, for every x ∈ e there are disjoint open sets w(x) and g(x) such that x ∈ w(x) and f ∩y ⊆ g(x). let w = {w(x) : x ∈ e}∪{x\e} and v be and open refinement of w locally finite on y. for each v ∈ (v)e let x(v ) ∈ e such that v ⊆ w(x(v )). let u = ∪(v)e and note that since v is a cover of x, e ⊆ u. let o = x\u. suppose x ∈ f ∩y. since v is locally finite on y, let q be an open neighborhood of x meeting only finitely many members of v. let v′ = {v ∈ (v) e : q ∩ v 6= φ} and note that v′ is finite. if v′ = φ then q ∩ u = φ and so x ∈ o. suppose v′ 6= φ, say v′ = {v1, v2, .., vn}. then q ∩ g(x(v1)) ∩ ... ∩ g(x(vn)) is an open neighborhood of x missing u and so again x ∈ o. therefore f ∩ y ⊆ o. � a space x can have a subspace which is 1− paracompact in x but not collectionwise normal in x, example 5.4.this example is not regular and the subspace y is not closed. relative collectionwise normality 203 theorem 3.2. suppose that y is closed and paracompact in the space x. then y is strongly collectionwise normal in x. proof. by theorem 2.5 we need only show that y is collectionwise normal in x. let {fα : α ∈ γ} be a discrete collection of closed subsets of x such that if α, β ∈ γ with α 6= β then fα 6= fβ. for each x ∈ x, let ux be and open neighborhood of x meeting at most one member of f. let v be and open partial refinement of {ux : x ∈ x} covering y locally finite on y. for each α ∈ γ let vα = ∪{v ∈ v : y ∩ v ∩ fα 6= φ}. then {vα : α ∈ γ} is a collection of open subsets of x locally finite on y such that for all α ∈ γ, y ∩ fα ⊆ vα ⊆ x\ ∪ (f\{fα}). since y is closed and paracompact in x it is normal in x. for all α ∈ γ let gα and wα be disjoint open subsets of x such that y ∩ fα ⊆ gα and y ∩ (∪(f\{fα})) ⊆ wα. for all α ∈ γ let hα = gα ∩ vα and uα = hα\∪{hβ : β ∈ γ\{α}}. the collection u = {uα : α ∈ γ} is a pairwise disjoint collection of open subsets of x. since for all α ∈ γ, uα ⊆ vα the collection u is locally finite on y and uα ⊆ x\ ∪ (f\{fα}). thus we need only show that fα ∩ y ⊆ uα. note that for all α ∈ γ, fα ∩ y ⊆ hα and since hα ⊆ vα the collection {hα : α ∈ γ} is also locally finite on y. thus by lemma 2.1 the collection {hα : α ∈ γ} is weakly closure preserving with respect to y and so for all α ∈ γ y ∩ (∪{hβ : β ∈ γ\{α}}) = y ∩ (∪{hβ : β ∈ γ\{α}}). suppose α ∈ γ, x ∈ y ∩ fα and λ ∈ γ\{α}. since λ 6= α and x ∈ y ∩ fα, x ∈ y ∩ (∪(f\{fλ})) ⊆ wλ. since hλ ⊆ gλ and gλ ∩ wλ = φ, x /∈ hλ. hence (y ∩ fα) ∩ (∪{hβ : β ∈ γ\{α}}) = φ and so y ∩ fα ⊆ uα. we now proceed much as in theorem 5.1.17 of [5]. let f = y ∩ (∪f) and k = y \ ∪ u. since f and k are disjoint subsets of y closed in x and y is normal in x there exist disjoint open sets w and w ′such that f ⊆ w, k ⊆ w ′. clearly for all α ∈ γ, y ∩ fα ⊆ w ∩ uα and the collection {w ∩ uα : α ∈ γ} is pairwise disjoint. suppose y ∈ y. if α ∈ γ and y ∈ uα then uα is an open neighborhood of y meeting at most one member of {w ∩ uα : α ∈ γ}, (that member being w ∩ uα) if y /∈ uα for all α ∈ γ then y ∈ k and so w ′ is an open neighborhood of y missing all members of {w ∩ uα : α ∈ γ}. thus the collection {w ∩uα : α ∈ γ} is a pairwise disjoint collection of open sets discrete on y such that for all α ∈ γ, y ∩ fα ⊆ w ∩ uα ⊆ x\ ∪ (f\{fα}). � the following is a natural relative version of bing’s theorem. in light of example 5.4, we need to assume that the subspace y is relatively regular in x. theorem 3.3. suppose y is strongly regular in the space x. if y is paracompact in x then y is collectionwise normal in x. proof. let f = {f α : α ∈ γ} be a discrete collection of closed subsets of x such that if α, β ∈ γ with α 6= β then fα 6= fβ. using the fact that y is strongly regular in x, for each x ∈ ∪{f α : α ∈ γ} let ux be an open neighborhood of x such that |{α ∈ γ : ux ∩ fα 6= φ}| = 1 and |{α ∈ γ : ux ∩ fα ∩ y 6= φ}| = 1. for each x ∈ x \ ∪{f α : α ∈ γ}, let ux be an open neighborhood of x such 204 e. grabner, g. grabner, k. miyazaki and j. tartir that ux ∩ ∪{fα ∩ y : α ∈ γ} = φ. let u = {ux : x ∈ x}. since y is paracompact in x, there is an open partial refinement v of u such that v covers y and v is locally finite on y . note that since v is a partial refinement of u, |{α ∈ λ : v ∩ fα ∩ y 6= φ}| ≤ 1 for all v ∈ v. for each y ∈ y, let vy ∈ v such that y ∈ vy. for each α ∈ γ, let yα = y ∩ fα. for each y ∈ ∪{y α : α ∈ γ}, let wy be an open neighborhood of y such that wy ⊆ vy and |{v ∈ v : wy ∩ v 6= φ}| < ℵ0. also, for each y ∈ ∪{yα : α ∈ γ}, let oy = wy \ ∪{v : v ∈ v, wy ∩ v 6= ∅, and y /∈ v }. for each α ∈ γ, let oα = ∪{oy : y ∈ yα}. clearly, fα ∩ y = yα ⊆ oα ⊆ x\ ∪ (f\{fα}) for all α ∈ γ. it remains to show that {oα : α ∈ γ} is discrete with respect to y . to see this, let z ∈ y, and β, γ ∈ γ with β 6= γ. it suffices to show that either vz ∩ oβ = φ or vz ∩ oγ = φ. by the choice of vz, either vz ∩ yβ = φ or vz ∩ yγ = φ. without loss of generality, suppose that vz ∩ yγ = φ. to see that vz ∩ oγ = φ, let u ∈ yγ. either wu ∩ vz = φ or ou ⊆ wu \ vz. in either case, ou ∩ vz = φ. since u was chosen arbitrarily, vz ∩ oy = φ for all y ∈ yγ. therefore, vz ∩ oγ = φ, as desired. � it is not clear as to how one might modify the definition of collectionwise normality in a space x to obtain a stronger version that would be implied by being 1− paracompact in x but not by being paracompact in x. a space x is said to be discretely expandable if every discrete collection of subsets of x is expandable to a locally finite open collection, [9]. a normal space is collectionwise normal if and only if it is discretely expandable, [9]. for a space x and y ⊆ x,we say that y is (1−) discretely expandable in x provided every discrete collection of closed subsets of x, f there is a collection of open subsets of x, {u(f) : f ∈ f} locally finite on y such that for all f ∈ f, y ∩ f ⊆ u(f) ⊆ x\ ∪ (f\{f}), (f ⊆ u(f) ⊆ x\ ∪ (f\{f})). clearly, if y is (1−) paracompact in x then y is (1−) discretely expandable in x. theorem 3.4. suppose y is s− normal in the space x. if y is 1− discretely expandable in x then y is collectionwise normal in x. proof. let f ={fα : α ∈ γ} be a discrete collection of closed subsets of x such that if α, β ∈ γ with α 6= β then fα 6= fβ. for each x ∈ x let u(x) be an open neighborhood of x meeting at most one member of f. let v ={vα : α ∈ γ} be a collection of open subsets of x locally finite on y such that for all α ∈ γ, fα ⊆ vα ⊆ x\ ∪ (f\{fα}). since y is s− normal in x, for all α ∈ γ there exist open sets wα and mα such that y ∩ fα ⊆ wα ⊆ wα ⊆ vα and y ∩ wα ⊆ mα ⊆ mα ⊆ vα. for all α ∈ γ let gα = wα\∪{mβ ∪ wβ : β ∈ γ\{α}}. note that for all α, β ∈ γ, if α 6= β then gα ∩ gβ = φ. suppose α ∈ γ and x ∈ fα ∩ y. since the collection {mγ ∪ wγ : γ ∈ γ} is locally finite on y, if x ∈ ∪{mβ ∪ wβ : β ∈ γ\{α}} then x ∈ mβ ∪ wβ ∩ y = (mβ ∪wβ)∩y = mβ ∩y for some β ∈ γ\{α}. however mβ ⊆ vβ and vβ ∩fα = φ for all β ∈ γ\{α} a contradiction. hence x /∈ ∪{mβ ∪ wβ : β ∈ γ\{α}} and so fα ∩ y ⊆ gα ⊆ x\ ∪ (f\{fα}) for all α ∈ γ. relative collectionwise normality 205 suppose that x ∈ y. since the collection {wα : α ∈ γ} is locally finite on y, if x ∈ ∪{wα : α ∈ γ} then there is an α ∗ ∈ γ such that x ∈ wα∗ . thus mα∗ is an open neighborhood of x meeting at most one member of {gα : α ∈ γ}, i.e. gα∗. hence the collection {gα : α ∈ γ} is discrete with respect to y. � theorem 3.5. suppose y is closed and s− normal in the space x. if y is discretely expandable in x then y is strongly collectionwise normal in x. proof. proceed as in theorem 3.4 replacing the closed discrete collection {fα : α ∈ γ} with the closed discrete collection {y ∩ fα : α ∈ γ}. � theorem 3.6. suppose y is strongly normal in the space x. if y is discretely expandable in x then y is collectionwise normal in x. proof. let f ={fα : α ∈ γ} be a discrete collection of closed subsets of x such that if α, β ∈ γ with α 6= β then fα 6= fβ. let v ={vα : α ∈ γ} be a collection of open subsets of x locally finite on y such that for all α ∈ γ, fα ∩ y ⊆ vα ⊆ x\ ∪ (f\{fα}). for all α ∈ γ, since y is strongly normal in x and fα ∩ y ⊆ vα, there exist open sets wα and mα such that fα ∩ y ⊆ wα ⊆ vα, wα ∩ y ⊆ mα ⊆ vα and mα ∩ y ⊆ vα. for all α ∈ γ let gα = wα\∪{mβ ∪ wβ : β ∈ γ\{α}}. then as in theorem 3.4 for all α ∈ γ, fα ∩y ⊆ gα ⊆ x\∪(f\{fα}) and the collection {gα : α ∈ γ} is discrete with respect to y. � question 1 suppose y is snormal in the space x and discretely expandable in x. is y collectionwise normal in x? for a normal space x, a subspace z can be collectionwise normal in x without being 1− discretely expandable in x, example 5.2. a subspace y of a normal space x can be 1− paracompact in x but not strongly collectionwise normal in x. in fact a subspace of a compact space x need not be strongly collectionwise normal in x, example 5.5. in [4] a relative property of paracompactness type which does imply strongly collectionwise normality in x is introduced. suppose x is a set, u, v collections of subsets of x and y ∈ x. the collection v is said to star refine u at y provided there is a u ∈ u such that st(y, v) ⊆ u. for a space x, a subspace y is said to be strongly star normal in x provided for every collection u of open subsets of x covering y there is a collection v of open subsets of x covering y which star refines u at every point of ∪v. theorem 3.7. if y is strongly star normal in the space x then y is strongly collectionwise normal in x. proof. (proceed as in theorem 5.1.18 of [5]) let f ={fα : α ∈ γ} be a collection of closed subsets of x which is discrete with respect to y such that if α, β ∈ γ with α 6= β then fα 6= fβ. for each y ∈ y let uy be and open neighborhood of y meeting at most one member of f. let w be a collection of open subsets of x covering y which star refines u = {ux : x ∈ y } at every point of ∪w and v be a collection of open subsets of x which covers 206 e. grabner, g. grabner, k. miyazaki and j. tartir y and star refines w at every point of ∪v. then using the same argument as in lemma 5.1.15 of [5], we see that v is a collection of open subsets of x covering y such that for every v ∈ v there is a u ∈ u with st(v, v) ⊆ u. for each α ∈ γ let vα = ∪{v ∈ v : v ∩ fα 6= φ} and note that for all α ∈ γ fα ∩ y ⊆ vα ⊆ x\ ∪ (f\{fα}) and the collection {vα : α ∈ γ} is discrete with respect to y. � 4. relative versions of the michael-nagami theorem by replacing “locally finite” with “point finite” in the definitions of (1−) paracompactness we obtain relative metacompact analogs [7]. the subspace y of x is strongly metacompact in x provided every open cover of x has an open refinement point finite on y . the subspace y of a space x is metacompact in x provided every open cover of x has an open partial refinement point finite on y . clearly for a space x strongly metacompactness in x is a natural relatively metacompact analog of 1− paracompactness in x and metacompactness in x is the corresponding relative metacompact analog of paracompactness in x. before presenting several relative versions of the michael nagami theorem here are several examples clarifying the limitations of what we can expect. a closed discrete subspace of a normal space x is always strongly metacompact in x and collectionwise normal but need not be paracompact in x, example 5.2. in example 5.6 we give a regular space x having an open subspace y which is strongly collectionwise normal in x and strongly metacompact in x but not 1− paracompact in x. in example 5.7 we give a non regular space x having a closed subspace y which is super regular in x, strongly metacompact in x and 1− discretely expandable in x but not 1− paracompact in x. question 2 suppose y is strongly metacompact in x and 1− discretely expandable in the space x. is y paracompact in x? the proof of theorem 5.3.3 (michael-nagami theorem) of [5] can be readily modified to prove the following relative version. theorem 4.1. suppose that x is a regular space and y ⊆ x. if y is strongly metacompact in x and 1− discretely expandable in x then every open cover of x has an open partial refinement covering y which is the countable union of collections locally finite on y. question 3 suppose that x is a regular space, y ⊆ x and every open cover of x has an open partial refinement covering y which is the countable union of collections locally finite on y. is y paracompact in x? for a closed subspace y of a space x, y is paracompact in x if and only if every open cover of x has an open partial refinement covering y which is the countable union of collections locally finite on y, [8]. also for a closed subset y, y is strongly metacompact in x if and only if y is a metacompact subspace of x, [7]. relative collectionwise normality 207 corollary 4.2. suppose y is closed in the regular space x. if y is 1discretely expandable and metacompact then y is paracompact in x. (is y 1paracompact in x?) question 4 suppose y is (strongly) metacompact in x and collectionwise normal in x. is y paracompact in x? in question 3 if locally finite on y is replaced with discrete with respect to y the answer is yes. lemma 4.3. suppose that y is strongly regular and strongly collectionwise normal in the space x. if every open cover of x has an open partial refinement covering y which is the countable union of collections discrete with respect to y then y is paracompact in x. proof. let u be an open cover of x. for all x ∈ x let wx be an open neighborhood of x such that wx ⊆ u and y ∩ wx ⊆ u for some u ∈ u. let w = {wx : x ∈ x} and v = ∪{vn : n < ω} be an open partial refinement of w covering y such that for all n < ω, the collection vn is discrete with respect to y. for all n < ω, since vn is discrete with respect to y, the collection {v : v ∈ vn} is discrete with respect to y. for each n < ω let gn = {g(v, n) : v ∈ vn} be a collection of open subsets of x discrete with respect to y such that for all v ∈ vn, v ∩y ⊆ g(v, n) and g(v, n) ⊆ u for some u ∈ u. for each n < ω let fn = ∪vn. for each v ∈ vo let h(v, 0) = g(v, 0). for each 0 < n < ω and v ∈ vn let h(v, n) = g(v, n)\ ∪ {fk : k < n}. for each n < ω let hn = {h(v, n) : v ∈ vn} and let h = ∪{hn : n < ω}. we now show that h covers y and is locally finite with respect to y. let y ∈ y. let n = min{k < ω : y ∈ fk}. since y ∈ y ∩ fn and vn is discrete with respect to y, there is a v ∈ vn with y ∈ v ∩ y ⊆ g(v, n) and so y ∈ h(v, n). let m = min{k < ω : y ∈ ∪vk} and v ′ ∈ vm such that y ∈ v ′. since v ′ ⊆ fm, v ′ is an open neighborhood of y missing all members of hk for all m < k < ω. for all k ≤ m, since the collection gk and hence hk is discrete with respect to y, let ok be an open neighborhood of y meeting at most one member of hk. then v ′ ∩ oo ∩ ... ∩ om is an open neighborhood of y meeting only finitely many members of h. � again the proof of theorem 5.3.3 (michael-nagami theorem) of [5] can be readily modified to prove the following relative version. we include a proof here to demonstrate the modifications needed for this theorem and in the proof of theorem 4.1. theorem 4.4. suppose that y is strongly regular in the space x. if y is metacompact in x and strongly collectionwise normal in x then y is paracompact in x. proof. let o be an open cover of x and let u = {uα : α ∈ γ} be an open partial refinement of o covering y point finite on y such that if α, β ∈ γ and α 6= β then uα 6= uβ. let vo = {φ}. suppose k < ω and for all i ≤ k the collection vi has been defined and wi = ⋃ vi such that 208 e. grabner, g. grabner, k. miyazaki and j. tartir 1. vi is an open partial refinement of u discrete with respect to y 2. if x ∈ y such that |{α ∈ γ : x ∈ uα}| ≤ i then x ∈ ∪{wj : j = 0..i}. let tk+1 = {t ⊆ γ : |t | = k + 1} and for all t ∈ tk+1 let ft = (x\ ∪ {wj : j = 0..i}) ∩ (x\ ∪ {uα : α ∈ γ\t }). suppose t ∈ tk+1. if x ∈ y ∩ft then {α ∈ γ : x ∈ uα} ⊆ t and x /∈ ∪{wj : j = 0..k}. hence {α ∈ γ : x ∈ uα} = t and so y ∩ ft ⊆ ∩{uα : α ∈ t }. suppose that x ∈ y. if |{α ∈ γ : x ∈ uα}| ≤ k then ∪{wj : j = 0..k} is an open neighborhood of x missing all members of {ft : t ∈ tk+1}. suppose |{α ∈ γ : x ∈ uα}| ≥ k + 2. let α1, α2, ..., αk+2 be distinct members of {α ∈ γ : x ∈ uα}. then ∩{uαi : i = 1..k + 2} is an open neighborhood of x meeting no member of {ft : t ∈ tk+1}. suppose |{α ∈ γ : x ∈ uα}| = k + 1. let t ′ = {α ∈ γ : x ∈ uα} and note ∩{uα : α ∈ t ′} is a neighborhood of x meeting exactly one member of {ft : t ∈ tk+1}. hence we see that {ft : t ∈ tk+1} is a collection of closed subsets of x which is discrete with respect to y. let {gt : t ∈ tk+1} be a collection of open subsets of x discrete with respect to y such that for all t ∈ tk+1, y ∩ ft ⊆ gt ⊆ x\(∪{ft ′ : t ′ ∈ tk+1\{t }}). also assume that for all t ∈ tk+1, if y ∩ ft = φ then gt = φ. for all t ∈ tk+1, let vt = gt ∩ (∩{uα : α ∈ t }) and note that y ∩ ft ⊆ vt . let vk+1 = {vt : t ∈ tk+1} and wk+1 = ⋃ vk+1. then vk+1 is an open partial refinement of u discrete with respect to y. suppose that x ∈ y such that |{α ∈ γ : x ∈ uα}| ≤ k + 1. then there is a t ∈ tk+1 such that x ∈ x\ ∪ {uα : α ∈ γ\t }. thus x ∈ x\ ∪ {uα : α ∈ γ\t } = ((x\ k ⋃ i=0 wi) ∪ ( k ⋃ i=0 wi)) ∩ (x\ ∪ {uα : α ∈ γ\t }) = [(x\ k ⋃ i=0 wi) ∩ (x\ ∪ {uα : α ∈ γ\t })] ∪ [( k ⋃ i=0 wi) ∩ (x\ ∪ {uα : α ∈ γ\t })] ⊆ ft ∪ k ⋃ i=0 wi. hence for all x ∈ y such that |{α ∈ γ : x ∈ uα}| ≤ k + 1, x ∈ k+1 ⋃ i=0 wi. thus, since u is point finite on y, v = ⋃ n<ω vn is an open partial refinement of u covering y such that for all n < ω the collection vn is discrete with respect to y. by lemma 4.3, y is paracompact in x. � corollary 4.5. suppose y is closed and s− regular in the space x. then y is paracompact in x if and only if y is collectionwise normal in x and metacompact. 5. examples example 5.1. a t2 space x having a subspace y which is 1− paracompact in x but not strongly regular in x. relative collectionwise normality 209 let x = ω ∪ (ω × ω) ∪ {∗}. define a topology on x as follows: 1. points of ω × ω are isolated, 2. for each n < ω, {{n} ∪ ({n} × (k, ω)) : k < ω} is a local base at n, 3. the collection {{∗} ∪ ((k, ω) × ω) : k < ω} is a local base at ∗. then x is t2 and the subspace y = ω is 1− paracompact in x but the closed set y cannot be separated from the point ∗ by open subsets of x. thus y is not strongly− regular in x. example 5.2. bing’s example g. let x be bing’s example g, y the nonisolated points of x and z the isolated points of x. the subset y is a closed discrete subspace of x and therefore is strongly metacompact in x and collectionwise normal. however y is not collectionwise normal in x. the subspace z is an open discrete subspace of x and therefore strongly collectionwise normal in x and paracompact in x but not 1− discretely expandable in x. example 5.3. a regular space x having an open normal subspace which is collectionwise normal in x but which is not 1− discretely expandable in x and a closed subspace z which is paracompact in x but not s− normal in x. the space x is a standard modification of the tychonoff plank. let x = [0, ω1] × [0, ω]\{(ω1, ω)}. define a topology on x as follows: 1. points of ω1 × ω are isolated. 2. for all n < ω let {b(α, n) : α < ω1} be a neighborhood base for the point (ω1, n) where b(α, n) = (α, ω1] × {n} for all α < ω1. 3. for all α < ω1 let {g(α, n) : n < ω} be a neighborhood base for the point (α, ω) where g(α, n) = {ω1} × (n, ω] for all n < ω. clearly x is a regular space. let y = x\({ω1}×ω). since y is an open normal subspace of x it is strongly normal in x. the closed sets ω1 × {ω} and {ω1} × ω cannot be separated by open subsets of x. thus not only is x not normal but y is not s− normal in x. the subset y is collectionwise normal in x since it is an open subset of x and the direct sum of compact subspaces (y = ⊕{{α} × [0, ω] : α < ω1}). however y is not 1− discretely expandable in x. to see this let c = {ω1} × ω and f = {{r} : r ∈ c} and note that f is a discrete collection of closed subsets of x. suppose that for all r ∈ c, u(r) is an open neighborhood of r. for all n < ω let βn < ω1 such that b(βn, n) ⊆ u(ω1, n). let β ∗ = sup{βn : n < ω} and note that β∗ < ω1. choose β ∗ < γ < ω1 and let k < ω. then (γ, m) ∈ g(γ, k) ∩ b(βm, m) ⊆ g(γ, k) ∩ u(ω1, m) for all k < m < ω. hence every neighborhood of the point (γ, ω) meets infinitely many members of {u(r) : r ∈ c}. thus the collection {u(r) : r ∈ c} is not locally finite on y. let z = {ω1} × ω. the closed discrete subspace z is easily seen to be paracompact in x but like y it is not s− normal in x. example 5.4. a t2 lindelöf space x having a subspace which is 1− paracompact in x but not collectionwise normal in x. 210 e. grabner, g. grabner, k. miyazaki and j. tartir let y and z be disjoint subsets of r\q such that for every nonempty open subset u of r |u ∩ y | = ω1 = |u ∩ z|. well order q, y , and z, say q ={qn : n < ω} , y = {yα : α < ω1} and z = {zα : α < ω1}. for any set a ⊆ r let qa = {n < ω : qn ∈ a}, ya = {α < ω1 : yα ∈ a} and za = {α < ω1 : zα ∈ a}. let x = (r × {0, 1}) ∪ (y ∪ z ∪ q) ∪ (ω1 × ω × {0, 1}) and define a topology on x as follows: 1. all points of ω1 × ω × {0, 1} are isolated. 2. for all α < ω1 a basic open neighborhood of yα [zα] is of the form {yα} ∪ ({α} ×q u × {0}) [{zα} ∪ ({α} ×q u × {1})] where u is an open neighborhood of yα [zα] in r. 3. for all n < ω a basic open neighborhood of qn is of the form {qn} ∪ ((α, ω1) × {n} × {0, 1}) where α < ω1. 4. for all x ∈ r a basic open neighborhood of (x, 0) [(x, 1)] is of the form ([x, a) × {0}) ∪ ((x, a) ∩ (y ∪ q)) ∪ (y(x, a) ×q (x, a) × {0}) ∪((α, ω1) ×q (x, a) × {0, 1}) where a ∈ r , x < a and α < ω1. [ ((b, x] × {1}) ∪ ((b, x) ∩ (z ∪ q)) ∪ (z(b, x) ×q (b, x) × {1}) ∪((β, ω1) ×q (b, x) × {0, 1}) where b ∈ r , b < x and β < ω1. ] the space x is t2 lindelöf but not regular. the subspace y ∪ z ∪ q is 1− paracompact in x but not collectionwise normal in x. to see that y ∪z ∪q is not collectionwise normal in x let f = (r×{0})∪y and k = (r×{1})∪z. note f and k are disjoint closed subsets of x. suppose that u and v are disjoint open subsets of x such that f ∩ (y ∪ z ∪ q) = y ⊆ u and k ∩ (y ∪ z ∪ q) = z ⊆ v. then u ∩ v ∩ q 6= φ. example 5.5. a compact space x having a subspace y which is not strongly collectionwise normal in x. let x = (ω1 + 1) × (ω + 1) with the product topology and y = x\{(ω1, ω)} (tychonoff plank). then since x is compact y (and every other subspace of x) is 1− paracompact in x. the collection of closed subsets of x f ={(ω1 + 1) × {ω}} ∪ {{(ω, n)} : n < ω} is discrete with respect to y . using the same argument that the tychonoff plank is not normal using the closed (in y ) sets ω1 × {ω} and {ω1} × ω, one can use f to show that y is not strongly collectionwise normal in x. example 5.6. a regular space having a subspace which is strongly metacompact in x and strongly collectionwise normal in x but not 1− discretely expandable in x. let x = r × r, y = r × {0} and z = x\y. points of z have their usual open neighborhoods. for each x ∈ r a basic neighborhood of (x, 0) will be of the form {x} × (−ǫ, ǫ) where ǫ > 0. clearly x is regular and z is strongly metacompact in x and strongly star normal in x. however the points of the closed discrete subset y cannot be separated by open subsets of x which are discrete with respect to z. relative collectionwise normality 211 example 5.7. a nonregular space having a subspace which is super regular in x, strongly metacompact in x and 1− discretely expandable in x but not 1− paracompact in x. let a = ω1 with the order topology. let b = [0, 1] with points of (0, 1] having usual open neighborhoods in [0, 1] with the order topology and open neighborhoods of 0 are of the form u\{ 1 n : n = 1, 2, ...} where u is a usual open neighborhood of 0 in [0, 1] with the order topology. note that b is t2 but not regular. also { 1 n : n = 1, 2, ...} is closed, 1− discretely expandable in b and super regular in b. the construction of the space x is based on examples in [2] and [7]. let x = a × b with the topology defined as follows: 1. for a ∈ a and y ∈ (0, 1] basic open neighborhoods are of the form {a} × v where v is an open neighborhood of y in b, 2. for a ∈ a basic open neighborhoods of (a, 0) are of the form ∪{{x} × vx : x ∈ u} where u is an open neighborhood of a in a and for all x ∈ u, vx is an open neighborhood of 0 in b. let y = a × { 1 n : n = 1, 2, ...} and note that y is a closed discrete subset of x and therefore strongly metacompact in x. also note that y is super regular in x but not strongly regular in x. it is not difficult to show that y is 1− discretely expandable in x. to see that y ia not 1− paracompact in x, let u = {[0, α] × b : α < ω1}. using the pressing down lemma it is easily seen that u does not have an open refinement that is locally finite on y. references [1] a. arhangel’skii, “from classic topological invariants to relative topological properties”, scientiae math. japonicae 55 no 1 (2002), 153-201. [2] a. arhangel’skii and h genedi, “beginnings of the theory of relative topological properties”, gen. top. spaces and mappings (mgu, moscow, 1989), 3-48 (in russian). [3] a. arhangel’skii and h genedi, “position of subspaces in spaces: relative versions of compactness, lindelöf properties, and separation axioms”, vestnik moskovskogo universiteta, mathematika 44 no. 6 (1989), 67-69. [4] a. arhangel’skii and i. gordienko, “relative symmetrizability and metrizability”, comment. math. univ. carol. 37 no 4 (1996), 757-774. [5] r. engelking, “general topology” (pwn, warsaw, 1977). [6] i. gordienko, “on relative properties of paracompactness and normality type”, moscow univ. nath. bul. 46 no. 1 (1991), 31-32. [7] e. grabner, g. grabner and k. miyazaki, “ properties of relative metacompactness and paracompactness type”, topology proc. 25 (2000), 145-178. [8] k. miyazaki, “on relative paracompactness and characterizations of spaces by relative topological properties”, math. japonica 50 (1999), 17-23. [9] j.c. smith and l.l. krajewski, “expandability and collectionwise normality”, trans. amer. math. soc. 160 (1971), 437-451. [10] y. yasui, “results on relatively countably paracompact spaces”, q and a in gen. top. 17 (1999), 165-174. 212 e. grabner, g. grabner, k. miyazaki and j. tartir received april 2003 accepted july 2003 elise grabner (elise.grabner@sru.edu) dept. math., slippery rock university, slippery rock pa, 16057 usa gary grabner (gary.garbner@sru.edu) dept. math., slippery rock university, slippery rock pa, 16057 usa kazumi miyazaki (bzq22206@nifty.ne.jp) dept. math., osaka elector-communication university, osaka 572-8530, japan jamal tartir (tartir@math.ysu.edu) dept. math. and stat., youngstown state university, youngstown oh, 44555 usa applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 45–53 all hypertopologies are hit-and-miss somshekhar naimpally this paper is dedicated to my guru professor john g. hocking, who introduced me to the excitement of mathematical research abstract. we solve a long standing problem by showing that all known hypertopologies are hit-and-miss. our solution is not merely of theoretical importance. this representation is useful in the study of comparison of the hausdorff-bourbaki or h-b uniform topologies and the wijsman topologies among themselves and with others. up to now some of these comparisons needed intricate manipulations. the h-b uniform topologies were the subject of intense activity in the 1960’s in connection with the isbell-smith problem. we show that they are proximally locally finite topologies from which the solution to the above problem follows easily. it is known that the wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in“nice” metric spaces including the normed linear spaces. with the introduction of a new far-miss topology we show that the wijsman topology is hit-andmiss for all metric spaces. from this follows a natural generalization of the wijsman topology to the hyperspace of any t1 space. several existing results in the literature are easy consequences of our work. 2000 ams classification: 54b20, 54e05, 54e15. keywords: hypertopology, vietoris topology, hausdorff metric, hausdorffbourbaki uniformity, uniformity, proximal topology, hit-and-miss topology, locally finite, wijsman topology, proximal ball topology, ball topology, far-miss topology, bounded vietoris topology, bounded proximal topology, bounded hausdorff topology, attouch-wets topology. 1. introduction during the early part of the last century there were two main studies on the hyperspace of all non-empty closed subsets of a topological space. vietoris introduced a topology consisting of two parts (a) the lower finite hit part and (b) the upper miss part. hausdorff defined a metric on the hyperspace of a 46 s. naimpally metric space and the resulting topology depends on the metric rather than on the topology of the base space. two equivalent metrics on a metrisable space induce equivalent hausdorff metric topologies if and only if they are uniformly equivalent. with the discovery of uniform spaces, the hausdorff metric was generalized to hausdorff-bourbaki or h-b uniformity on the hyperspace of a uniformizable (tychonoff) space. in the 1960’s this led to the celebrated isbellsmith problem to find necessary and sufficient conditions that two uniformities on the base space are h-equivalent, i.e., they induce topologically equivalent h-b uniformities on the hyperspace. ([9, 17, 18, 19]. for an account see [15, section 15, page 87]) the solution presented by ward is quite intricate. in this paper we show that the h-b uniform topology has two parts (a) the lower locally finite hit part and (b) the upper proximal miss part. from this representation and the knowledge that lower and upper parts act separately in any comparison, ward’s theorem is obvious. we then briefly indicate how the recent work on bounded hausdorff metric or attouch-wets topologies can also be represented in a similar vein and derive results effortlessly. in 1966, while working on a problem in convex analysis, wijsman introduced a convergence for the closed subsets of a metric space (x,d) viz. an → a if and only if for each x ∈ x,d(x,an) → d(x,a). the resulting topology was found to be quite useful in applications. moreover, wijsman topologies are the building blocks of many other topologies e.g., (a) the metric proximal topology is the sup of all wijsman topologies induced by uniformly equivalent metrics, and (b) the vietoris topology is the sup of all wijsman topologies induced by topologically equivalent metrics. ([3]) in addition, the wijsman topologies are rather intriguing since there are examples of two uniformly equivalent metrics giving non equivalent wijsman topologies and two non uniformly equivalent metrics, giving equivalent wijsman topologies! attempts to topologize the wijsman topology led to the discovery of (a) the ball topology ([1]) and (b) the proximal ball topology ([4]) which were only partially successful. the lower part of the wijsman topology coincides with the lower vietoris topology. it is the upper one which is difficult to handle. in this paper we show that the upper wijsman topology can be expressed as a far-miss topology and this representation makes it easy to compare wijsman topologies among themselves and with others. we give simple conceptual proofs without any calculations. 2. preliminaries (references for uniformities include [9, 10], and for proximities [15].) suppose (x,u) is a hausdorff uniform space and suppose u induces the efproximity δ which, in turn, induces the topology t on x. in case x is a metrisable space with a metric d, u denotes the metric uniformity and δ the metric proximity. the fine uniformity on (x,t) is denoted by u] and the coarsest (totally all hypertopologies are hit-and-miss 47 bounded) uniformity, compatible with δ, is denoted by u∗. to keep matters simple, we assume that all entourages are open symmetric. we use the symbol δ0 to denote the fine lo-proximity on x given by aδ0b iff cla∩ clb 6= ∅. we denote by δ] the fine ef-proximity on x given by aδ]b iff a and b can be separated by a continuous function on x to [0, 1]. we use the following standard notation : cl(x) = the family of all non-empty closed subsets of x. k(x)= the family of all non-empty compact subsets of x. ∆ denotes a non-empty subfamily of cl(x). without any loss of generality we assume that ∆ is closed under finite unions and contains all singletons. we call ∆ a cobase. ([16]) ( in the literature ∆ is usually assumed to contain merely all singletons. in our view the above assumptions simplify the results, since we get a base for the upper ∆-topologies.) for any set e ⊂ x and e ⊂ t we use the following notation: e− = {a ∈ cl(x) : a∩e 6= ∅} e− = {a ∈ cl(x) : a∩e 6= ∅ for each e ∈ e} e++ = {a ∈ cl(x) : a <<δ e i.e. aδec} e+ = {a ∈ cl(x) : a ⊂ e i.e. a << e w.r.t. δ0} the upper proximal ∆-topology (w.r.t. δ) σ(δ∆+) is generated by the basis {e++ : ec ∈ ∆}. the upper ∆-topology τ(∆+) = σ(δ0∆+). the lower vietoris (or finite) topology τ(v −) has a basis {e− : e ⊂ t is finite}. the lower locally finite topology τ(lf−) has a basis {e− : e ⊂ t is locally finite}. the l-lower locally finite topology τ(l−) has a basis {e− : e ∈ l} where l ⊂{e ⊂ t : e is locally finite} satisfies some simple filter condition, i.e., given e,f ∈ l there is a g ∈ l such that g− ⊂ e− ∩f− (see [11]). the proximal (finite) ∆-topology (w.r.t. δ) σ(δ∆) = σ(δ∆+) ∨ τ(v −). we omit δ if it is obvious from the context and write σ(∆) for σ(δ∆). the ∆-topology τ(∆) = τ(∆+) ∨ τ(v −). the proximal locally finite ∆-topology (w.r.t. δ) σ(lfδ∆) = σ(δ∆+)∨ τ(lf−). the proximal l-locally finite ∆-topology (w.r.t. δ) σ(lδ∆) = σ(δ∆+)∨ τ(l−). we omit the prefix “proximal” and replace σ by τ if δ = δ0. well known special cases are: (a) when ∆ = cl(x), – τ(∆) = τ(v) the vietoris or finite topology ([12]) – σ(δ∆) = σ(δ) the proximal topology ([5]) – τ(lf∆) = τ(lf) the locally finite topology ([2, 11, 14, 20]) – σ(lfδ∆) = σ(lfδ) the proximal locally finite topology ([5]) 48 s. naimpally (b) when ∆ = k(x),τ(∆) = τ(f) = σ(f) the fell topology (see [7]) (c) if (x,d) is a metric space, δ is the metric proximity induced by d and ∆ denotes the family b of finite unions of proper closed balls of non-negative radii, then τ(∆) = τ(b) the ball topology ([1]) σ(∆) = σ(b), the proximal ball topology ([4]) 3. hausdorff-bourbaki uniformity recall that (x,u) is a hausdorff uniform space, the uniformity u induces the ef-proximity δ which, in turn, induces the topology t on x. in this section we show that the topology induced on cl(x) by the hausdorff-bourbaki or h-b-uniformity uh is a hit-and-miss topology. we recall that a typical nbhd. of a ∈ cl(x) in the lower h-b-uniform topology τ(u−h) is {b ∈ cl(x) : a ⊂ u(b)} where u ∈ u. in the case of the upper h-b-uniform topology τ(u+h), a typical nbhd. of a ∈ cl(x) is {b ∈ cl(x) : b ⊂ u(a)} where u ∈ u. it is known and easy to show that σ(δ+) = τ(u+h). the h-b-uniform topology τ(uh) = τ(u + h) ∨ τ(u − h) = σ(δ +) ∨ τ(u−h). lemma 3.1. for each a ∈ cl(x) and each u ∈ u, there is a discrete (and hence a locally finite) family {u(x) : x ∈ q ⊂ a} with the properties: (a) if x,y ∈ q and x 6= y then y 6∈ u(x), and (b) a ⊂ u(q). proof. by zorn’s lemma, there is a set q ⊂ a which is maximal such that if x,y ∈ q and x 6= y then y 6∈ u(x). clearly, (b) is satisfied. � remark 3.2. (a) it is obvious that the uniformity u is totally bounded if and only if for each a ∈ cl(x) and each u ∈ u, maximal u-discrete subsets of a are finite. (b) let lu be the collection of families of open sets of the form {u(x) : x ∈ q ⊂ a}, where a ∈ cl(x), u ∈ u and q is u-discrete. the family lu allows us to define a new lower locally finite “hit” topology. definition 3.3. the lower lu-locally finite topology τ(lu−) on cl(x) consists of {e− : e ∈ lu}. if a ∈ e−, where e ⊂ t is finite, then there is a finite set q ⊂ a and a u ∈ u such that q is u-discrete and a ∈ {u(x) : x ∈ q}− ⊂ e−. so τ(v−) ⊂ τ(lu−). the proximal lu-locally finite topology σ(luδ) = σ(δ+) ∨ τ(lu−). lemma 3.4. τ(u−h) = τ(lu −). proof. suppose a ∈ w = {b ∈ cl(x) : a ⊂ u(b)} ∈ τ(u−h) where u ∈ u. let v ∈ u be such that v 2 ⊂ u. let q be a maximal v -discrete subset of a. then a ∈ s = {v (x)− : x ∈ q} and s ∈ τ(lu−). all hypertopologies are hit-and-miss 49 conversely, suppose a ∈ s′ = {u(x)− : x ∈ q ⊂ a} ∈ τ(lu−). it is easy to see that if a ⊂ u(b), then b ∈ s′ and so a ∈{b ∈ cl(x) : a ⊂ u(b)}⊂ s′. � theorem 3.5. τ(uh) = σ(luδ). therefore, every h-b-uniformity uh on cl(x), associated with a uniformity u on x, induces the proximal lu-locally finite topology on cl(x). we mention two special cases here. remark 3.2 (a) gives us corollary 3.6 ([5]). σ(δ) = τ(u∗h). i.e., the proximal (finite) topology is induced by the h-b uniformity associated with the coarsest totally bounded uniformity compatible with δ. corollary 3.7 ([5]). σ(lfδ]) = τ(u]h). i.e., the proximal locally finite topology w.r.t. the fine ef-proximity δ] is induced by the h-b uniformity associated with the fine uniformity u]. proof. it is sufficient to show that if a ∈ e−, where e is a locally finite family of nonempty open sets, then there is a subset q ⊂ a and a u ∈ u], such that a ∈ w = {u(x)− : x ∈ q} ∈ τ(lu]−) and w ∈ e−. but this is just a standard argument involving continuous functions and locally finite families of open sets. � remark 3.8. the above result includes the following: (a) x is normal if and only if the h-b uniformity associated with the fine uniformity induces the locally finite topology ([14, 20]). (b) in a metrisable space, the supremum of all hausdorff metric topologies, associated with compatible metrics, is the locally finite topology (see [20] for a non-standard treatment and [2] for the usual one). remark 3.9. nachman [13] defined the strong hyperproximity on cl(x) as the ef-proximity induced by u ĥ, where û is the an-uniformity (the union of all uniformities compatible with δ). (see [15]). the following result is now obvious. the topology of the strong hyperproximity on cl(x) is the supremum of proximal locally finite hit-and-miss topologies. we now turn our attention to the celebrated isbell-smith problem mentioned in the introduction. let (x,t) be a tychonoff space with compatible uniformities u,v inducing ef-proximities δ(u),δ(v) and h-b uniformities uh,vh respectively. isbell ([9]) first conjectured that if u 6= v then τ(uh) 6= τ(vh) suggesting that they induce non-equivalent families of nbhds. of the element x ∈ cl(x). smith ([17]) gave a counterexample to show that the latter statement to be false but nevertheless proved several results supporting isbell’s conjecture. ward ([19]), however, disproved the conjecture and proved the following result: 50 s. naimpally theorem 3.10 (see [15], pages 89–90). τ(vh) ⊂ τ(uh) if and only if δ(v) ≤ δ(u), and for every v ∈ v and every v -discrete set a there is u ∈ u such that u(a) ⊂ v (a) for each a ∈ a. proof. the result follows easily from theorem 3.5 and the fact that a coarser ef-proximity induces a coarser upper proximal hypertopology. (cf. [5]) � three other related results also follow easily: theorem 3.11. (a) two uniformities on x that are h-equivalent, are in the same proximity class. (b) two uniformities on x, at least one of which is totally bounded, are not h-equivalent. (c) two different metrisable uniformities on x are not h-equivalent. 4. the wijsman topology an unusual topology for the hyperspace of a metric space (x,d) arose from wijsman’s paper [21]. the wijsman topology is not only useful in applications but also valuable as the building block of many hypertopologies ([3]). it was defined in terms of convergence viz. (4.1) a net an ∈ cl(x) w-converges to a ∈ cl(x) iff for each x ∈ x, d(x,an) → d(x,a), where d(x,a) = inf{d(x,a) : a ∈ a}. attempts to characterize the resulting wijsman topology τ(w) as a hit-andmiss topology led to the discovery of the ball topology τ(b) (see [1] for further references) and the proximal ball topology σ(b) ([4]). in the latter paper it was shown that τ(w) = σ(b) in nice metric spaces including all normed linear spaces. we now show that the wijsman topology is a hit-and-far-miss topology in arbitrary metric spaces by using the following characterization by del pretelignola ([6]): (4.2) an w-converges to a if and only if the following conditions are met: (1) for every non-empty open set e, a ∈ e− implies eventually an ∈ e−; (2) whenever 0 < ε < α and a 6∈ s(x,α)− then eventually an 6∈ s(x,ε)−. it is well known that (1) is equivalent to convergence in the lower vietoris topology τ(v−). and (2) is equivalent to the statement: if a is far from s(x,ε) (in the metric proximity), then eventually an does not intersect s(x,ε). here we find “far” is mixed with “disjoint”. we need a new topology mixing the ball and the proximal ball topologies. suppose b denotes the family of finite unions of all proper closed balls with non-negative radii. definition 4.1. the upper far-miss ball topology τ(fm+) has a local basis at a ∈ cl(x), {e+ : ec ∈ b and a ∈ e++}. all hypertopologies are hit-and-miss 51 the hit-and-far-miss topology τ(fm) = τ(fm+) ∨ τ(v−). it now follows from the remarks made above that the wijsman topology is a hit-and-far-miss topology. theorem 4.2. τ(w) = τ(fm) remark 4.3. it is obvious that the wijsman topology can be generalized to a t1 space with a compatible proximity δ and a cobase ∆ which is a subfamily of cl(x) closed under finite unions. thus we have a ∆ hit-and-far-miss topology defined exactly as in 4.1 with ∆ replacing b. it is a trivial matter to note that the wijsman topology is coarser than both the ball topology and the proximal topology. (cf. [1, page 45]) comparing wijsman, ball and proximal ball topologies among themselves or with one another is now quite easy and we just give a sample below. we note that a set e is said to be weakly totally bounded or w-tb (cf. strictly d-included ) iff for every ε > 0, there exists a b in b such that e << b << s(e,ε). (cf. [1, 4]) theorem 4.4 (cf. [1] page 45). the proximal ball topology equals the wijsman topology if and only if every b ∈ b is w −tb. proof. suppose a0 6= x, is a non-empty closed subset of x and a0 ∈ u++, where uc = b ∈ b. the upper proximal ball topology is coarser than the upper wijsman topology iff there is a v with v c = b′ ∈ b with a0 ∈ v ++ and v + ⊂ u++. rewriting the above in terms of the cobase b and the proximity we get the result. � 5. bounded hypertopologies let (x,d) be a metric space. then using bounded sets in the definitions of some upper, some lower, and some both hypertopologies one can get analogues of those defined in section 2. thus there are the following: the bounded vietoris topology, the bounded hausdorff metric of attouch-wets or aw topology, the bounded proximal topology, etc. (see [1]). many workers in the area do not seem to be aware that bounded sets can be defined in any topological space ([8],[15, page 53]). a family of bounded subsets of a topological space is merely a family of non-empty subsets that are closed under finite unions and hereditary. in the present situation we restrict the concept to closed subsets. thus any cobase ∆, which is closed hereditary, serves as a family of bounded sets. so all the results in a metric space are special cases of their analogues in (proximal) ∆-hypertopologies when the cobase consists of all closed metrically bounded subsets. for this reason, we give a short account here about this topic. (5.1) the bounded proximal topology: this is just the proximal (finite) ∆-topology (w.r.t. δ) σ(δ∆) = σ(δ∆+) ∨ τ(v−) where ∆ is closed hereditary. (5.2) the bounded vietoris topology: this is merely the ∆-topology τ(∆) = τ(∆+) ∨ τ(v−) where ∆ is closed hereditary. 52 s. naimpally (5.3) the bounded h-b topology: suppose (x,u) is a hausdorff uniform space, the uniformity u induces the ef-proximity δ which, in turn, induces the topology t on x. let ∆ be a cobase which is closed hereditary. let lb be the collection of all families of open sets of the form {u(x) : x ∈ q ⊂ a}, where a ∈ ∆, u ∈ u and q is udiscrete. then the bounded h-b topology on cl(x) is τ(lbδ) = τ(lbδ+) ∨ τ(lb−) = σ(δ∆+) ∨ τ(lb−). if x is a metrisable space with a compatible d, u is the metric uniformity generated by d, δ is the metric proximity and ∆ is the family of all non-empty closed metrically bounded subsets of (x,d), then τ(lbδ) is precisely the aw-topology τ(awd) on cl(x). using these representations it is now an easy task to get results comparing them among themselves and with others. we give a couple of samples below. remark 5.1. (a) it is easy to see that σ(δ∆) ⊂ τ(lbδ) and so σ(δ∆) = τ(lbδ) if and only if closed and bounded sets are totally bounded. (cf. [1, theorem 3.1.4]). (b) σ(δ∆) = τ(lbδ+) ∨ τ(v−). (cf. [1, theorem 4.2.1]). acknowledgements. i am grateful to professors giuseppe di maio and enrico meccariello for many discussions and suggestions as well as their hospitality during my visit to dipartimento di matematica, seconda universita degli studi di napoli, caserta, italy in june 2001. thanks are due to the referee for a careful reading of the paper and suggestions. references [1] g. beer, topologies on closed and closed convex sets, kluwer academic pub. (1993). [2] g. beer, c. himmelberg, c. prikry and f. van vleck, the locally finite topology on 2x, proc. amer. math. soc. 101 (1987), 168–171. [3] g. beer, a. lechicki, s. levi and s. naimpally, distance functionals and suprema of hyperspace topologies, ann. mat. pura appl. 162 (1992), 367–381. [4] g. di maio and s. naimpally, comparison of hypertopologies, rend. istit. mat. univ. trieste 22 (1990), 140–161. [5] a. di concilio, s. naimpally and p. sharma, proximal hypertopologies, sixth brazilian topology meeting, campinas, brazil (1988) (unpublished). [6] i. del prete and b. lignola, on convergence of closed-valued multifunctions, boll. un. mat. ita. b 6 (1983), 819–834. [7] j. fell, a hausdorff topology for the closed subsets of a locally compact non-hausdorff space, proc. amer. math. soc. 13 (1962), 472–476. [8] s. t. hu, boundedness in a topological space, j. math. pures appl. 28 (1949), 287–320. [9] j. isbell, uniform spaces, american mathematical society (1964). [10] j. l. kelly, general topology, van nostrand (1955). all hypertopologies are hit-and-miss 53 [11] m. marjanovic, topologies on collections of closed subsets, publ. inst. math. (beograd) 20 (1966), 125–130. [12] e. michael, topologies on spaces of subsets, trans. amer. math. soc. 71 (1951), 152– 182. [13] l. nachman, hyperspaces of proximity spaces, math. scand. 23 (1968), 201–213. [14] s. naimpally and p. sharma, fine uniformity and the locally finite hyperspace topology, proc. amer. math. soc. 103 (1988), 641–646. [15] s. a. naimpally and b. d. warrack, proximity spaces, cambridge tracts in mathematics 59, cambridge university press (1970). [16] h. poppe, eine bemerkung über trennungsaxiome im raum der abgeschlossenen teilmengen eines topologischen raumes, arch. math. 16 (1965), 197–199. [17] d. h. smith, hyperspaces of a uniformizable spaces, proc. camb. phil. soc. 62 (1966), 25–28. [18] a. j. ward, a counter-example in uniformity theory, proc. camb. phil. soc. 62 (1966), 207–208. [19] a. j. ward, on h-equivalence of uniformties: the isbell-smith problem, pacific j. math. 22 (1967), 189–196. [20] f. wattenberg, topologies on the set of closed subsets, pacific j. math. 68 (1977), 537–551. [21] r. wijsman, convergence of sequences of convex sets, cones and functions, ii, trans. amer. math. soc. 123 (1966), 32–45. received july 2001 revised october 2001 som naimpally prof. emeritus of mathematics lakehead university 96 dewson street toronto, ontario m6h 1h3 canada e-mail address : sudha@accglobal.net () @ appl. gen. topol. 16, no. 1(2015), 19-30doi:10.4995/agt.2015.2057 c© agt, upv, 2015 on some topological invariants for morphisms defined in homological spheres nasreddine mohamed benkafadar a and boris danielovitch gel’man b a faculty of sciences, department of mathematics, constantine algeria. (kafadar@gmx.com) b faculty of mathematics, department of functional theory and geometry, voronezh, russia. (gelman@math.vsu.ru) abstract in the paper one defines topological invariants of type degree for morphisms in the category t op(2) of topological pairs of spaces and continuous single valued maps, which admit homological n-spheres as target and arbitrary topological pairs of spaces as source. the different described degrees are acquired by means homological methods, and are a powerful tool in the root theory. several existence theorems are obtained for equations with multivalued transformations. 2010 msc: 55n10; 54h25; 54c60. keywords: homology; homotopy; topological degree; fixed points. 0. introduction the concept of topological degree deg is well know for maps of homological n-spheres and oriented n-dimensional manifolds (see for example, [4] and [2]). recall this concept: if x and y are both homological n-spheres and f : x → y is a continuous single valued map by fixing some generated elements z1 and z2 of homology groups hn(x) and hn(y ) respectively, one obtains an equality fn∗(z1) = k ·z2. this number k is called the degree of f and is denoted by deg f. topological degree theory plays a preponderant role in topology fixed points theory and non linear analysis. the different degrees can be considered as a generalization of the winding number, kronecker’s characteristic and others received 23 december 2013 – accepted 17 october 2014 http://dx.doi.org/10.4995/agt.2015.2057 n. m. benkafadar and b. d. gel’man topological invariants. different generalizations of the topological degree has been studied for multivalued transformations (see for example [7] [10]). 1. homological invariants for some classes of morphisms in the categories t op and t op(2) 1.1. notations and definitions. in the present section one introduces some basic topics which play an important role in the sequel. a pair (x, a) of topological spaces such that a ⊆ x is called a pair of topological spaces, in this context, a topological space x is conceived as the pair (x, ∅). let (x, a) and (s, t ) be some pairs of topological spaces and f ∈ mort op(x, s) such that f(a) ⊆ t, then f is named a continuous single valued map of pairs of topological spaces and denoted f : (x, a) → (s, t ). the collections of pairs of topological spaces and continuous single valued maps of pairs of topological maps with the composition of maps define a category denoted t op(2), it admits as a full subcategory the category t op of topological spaces and continuous single valued maps. two morphisms f, g ∈ mort op(2) ((x, a), (s, t )) are called homotopic if and only if there exists a morphism φ ∈ mort op(2) ((x, a) × [0, 1], (s, t )) such that φ(x, 0) = f(x) and φ(x, 1) = g(x) for every x ∈ x. by h one denotes the covariant functor h of singular homology with coefficients in the abelian ring of integer z, defined from the category t op(2) in the category gd of graded groups and homomorphisms of degree zero, where g the category of abelian groups and homomorphisms of groups is a full subcategory. thus, for a given object (x, a) ∈ obj(t op(2)) and a morphism f ∈ mort op(2) ((x, a), (y, b)) the functor h assigns a graded group {hi(x, a)}i≥0 and a homomorphism of degree zero {hi(f)}i≥0 ∈ morgd({hi(x, a)}i≥0,{hi(y, b)}i≥0. 1.2. degree for a class of morphisms in the category t op. an object y in the category t op will be called a homological n-sphere if the topological space y admits the same homological groups of the n-sphere. consider f ∈ mort op(x, y ) a morphism with source an arbitrary topological space x and target y a homological n-sphere. let hn(f) := fn∗ ∈ morg(hn(x), hn(y )) be the induced homomorphism of f and e be a generator of hn(y ). definition 1.1. the degree of a morphism f ∈ mort op(x, y ) is the integer denoted and defined as dg(f, x, y ) =| k |, where k ∈ z verifies a = k · e and imfn∗ =< a >⊆ hn(sn) =< e > . example 1.2. consider the open subset: u = {(x1, x2) ∈ r2 | (x1 −1)2 + x22 < 1}∪{(x1, x2) ∈ r2 | (x1 + 1)2 + x22 < 1} of r2 and let x = ∂u be the boundary of u. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 20 on some topological invariants for morphisms defined in homological spheres if x0 is a fixed element of r 2\x one can define the morphism f ∈ mort op(x, r2�{0}) given by the rule f(x) = x−x0 for every x ∈ x. then the next equalities are satisfied: dg(f, x, r2�{0}) = { 1, if x0 ∈ u 0, if x0 /∈ u let us give some properties of this topological invariant. one will begin by giving the relation between the winding number deg f of a morphisms f ∈ mort op(sn, sn) defined on the sphere and its topological invariant dg(f, sn, sn). proposition 1.3. let f ∈ mort op(sn, sn) then dg(f, sn, sn) =| deg f | . proof. consider hn(s n) =< e > then imfn∗ =< fn∗(e) >, moreover fn∗(e) = deg f · e, hence dg(f, sn, sn) =| deg f | . � definition 1.4. a morphism f ∈ mort op(2) ((x, a), (y, b)) is called h sectional if f admits a right inverse homotopy. it is not difficult to check the next properties of this degree: proposition 1.5. the topological degree satisfies the following assertions: (1) let f ∈ mort op(x, y ) be a constant map then dg(f, x, y ) = 0; (2) let f ∈ mort op(x, y ) be h sectional morphism then dg(f, x, y ) = 1; (3) let z be is a topological space and (f, g) ∈ mort op(x, z)×mort op(z, y ) then dg(g ◦f, x, y ) is a multiple of dg(g, z, y ); (4) let x0 be a subset of x an object in the category t op and fx0 ∈ mort op(x0, z) be the restriction of a morphism f ∈ mort op(x, y ) then there exists a natural number n ∈ n such that dg(fx0, x0, y ) = n ·dg(f, x, y ). proposition 1.6. let y1 and y2 be both some homological n-spheres in the category t op and (f, g) ∈ mort op(x, y1)×mort op(y1, y2) be a pair of morphisms then dg(g ◦f, x, y2) = dg(f, x, y1) ·dg(g, y1, y2) proof. this is a consequence of the definition 1.1, of the topological degree. � the topological degree is invariant for homotopic morphisms: proposition 1.7. let (f, g) ∈ mort op(x, y ) × mort op(x, y ) then if f and g are homotopic dg(f, x, y ) = dg(g, x, y ). let us consider some aspects of the degree dg(f, x, y ) in that case where the target y = sn. proposition 1.8. let f ∈ mort op(x, sn) such that dg(f, x, sn) 6= 0 then f is an epimorphism in the category t op. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 21 n. m. benkafadar and b. d. gel’man proof. suppose that f ∈ mort op(x, sn) is not an epimorphism so f(x) ⊂ sn. let y ∈ sn�f(x), then one can diagramed: x f→ sn f̃ ց ↑ i sn�{y} where f̃ is the submap of f and i is the canonical injection. one can conclude by remarking that hn(s n�{y}) is a trivial group. � proposition 1.9. let f, g ∈ mort op(x, sn) then if dg(f) 6= dg(g) the morphisms f and g admit at least a coincidence point in x. proof. indeed, if f(x) 6= g(x) for every element x ∈ x then for (x, t) ∈ x×[0, 1] the vector field v(x, t) = (1− t) ·f(x) + t ·(−g)(x) ∈ rn+1 is free of zero. this finding offers the opportunity to get the morphism f ∈ mort op(x × [0, 1], sn) where, f(x, t) = v(x,t ‖v(x,t)‖ for all element (x, t) from the source x × [0, 1]. the morphism f defines a homotopy between f and (−g) . hereafter, from propositions 1.7 and the definition 1.1, one takes dg(f, x, sn) = dg(−g, x, sn) = dg(g, x, sn). � 1.3. degree for a class of morphisms in the category t op(2). an object (y, b) ∈ obj(t op(2)) is called a homological n-sphere if h0(y, b) = hn(y, b) isomorphic to the abelian ring of integers z and hi(y, b) = {0} for all other indices. for more notions one this topics see [9]. for instance, the pairs of spaces (rn, rn�{0}); (bn, sn−1) where bn is the closed ball in rn and sn−1 = ∂bn, are some n-spheres in that category. definition 1.10. the degree of a morphism f ∈ mort op(2) ((x, a), (y, b)) where (y, b) is a homological n-sphere with hn(y, b) =< η > is denoted and defined by dgr(f, (x, a), (y, b)) =| k |, where imfn∗ = and b = k · η. the next properties are obvious. proposition 1.11. the following assertions are satisfied: (1) if a morphism f ∈ mort op(2) ((x, a), (y, b)) is a constant map then dgr(f, (x, a), (y, b)) = 0; (2) if (f, g) ∈ mort op(2) ((x, a), (x′, a′))×mort op(2) ((x′, a′), (y, b)) then there exits an integer k ∈ n such that : dgr(g ◦f, (x, a), (y, b)) = k · dgr(g, (x′, a′), (y, b)); (3) let (x0, a0) ⊆ (x, a) and f0 ∈ mort op(2) ((x0, a0), (y, b)) be the submap of the morphism f ∈ mort op(2) ((x, a), (y, b)) then there exists a natural number k ∈ n such that : dgr(f0, (x0, a0), (y, b)) = k ·dgr(f, (x, a), (y, b)); c© agt, upv, 2015 appl. gen. topol. 16, no. 1 22 on some topological invariants for morphisms defined in homological spheres (4) let (y1, b1) and (y2, b2) be some n-spheres in the category t op(2) and (f, g) ∈ mort op(2) ((x, a), (y1, b1)) × mort op(2)((y1, b1), (y2, b2)) be a pair of morphisms then : dgr(g ◦f, (x, a), (y2, b2)) = dgr(f, (x, a), (y1, b1)) · dgr(g, (y1, b1), (y2, b2)) (5) if f, g ∈ mort op(2) ((x, a), (y, b)) are some homotopic morphisms then dgr(f, (x, a), (y, b)) = dgr(g, (x, a), (y, b)). this homological invariant satisfies some more specific properties. let us describe some of them. proposition 1.12. let z ⊂ int(a) ⊆ a ⊆ a ⊂ x and f ∈ mort op(2) ((x, a), (y, b)) then dgr(f̃, (x�z, a�z), (y, b)) = dgr(f, (x, a), (y, b)) where f̃ ∈ mort op(2) ((x�z, a�z), (y, b)) is the submap of the morphism f on the pair (x�z, a�z). proof. this is a consequence of the following commutative diagram: hn(x, a) ց fn∗ in∗ ↑ hn(y, b) hn(x�z, a�z) ր f̃n∗ where i ∈ mort op(2) ((x�z, a�z), (x, a)) is the natural injection. from excision theorem one infers that in∗ is an isomorphism and concludes the proof. � proposition 1.13. let f ∈ mort op(2) ((x, a), (y, b)) be a morphism such that dgr(f, (x, a), (y, b)) 6= 0 then there exist x ∈ x�a and y ∈ y �b such that f(x) = y. proof. indeed, if f(x) /∈ y �b for every element x ∈ x�a on can get the following commutative diagram: hn(x, a) fn∗→ hn(y, b) f̃n∗ ↓ ր in∗ hn(b, b) where i ∈ mort op(2) ((b, b), (y, b)) is the natural injection and f̃ = f. one concludes by observing that hn(b, b) is a trivial group. � corollary 1.14. let f ∈ mort op(x, rn) be a morphism in the category t op and a be a closed subset of x such that f(a) 6= 0 for every x ∈ a, then if the degree of the morphism f ∈ mort op(2) ((x, a), (rn, rn�{0})) is not zero, there exists x0 ∈ x�a such that f(x0) = 0. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 23 n. m. benkafadar and b. d. gel’man corollary 1.15. let f ∈ mort op(bn, rn) such that f(x) 6= 0 for every element x ∈ ∂bn = sn−1 then if the degree of the morphism f ∈ mort op(2) ((b n, sn−1), (rn, rn�{0})) is not zero there exists at least an element x in the interior of the ball such that f(x) = 0. 2. homological invariant for a class of multivalued transformations let (x, a), (s, t ) ∈ obj(t op(2)) a correspondence f : (x, a) → (s, t ) which assigns for each element x ∈ x a subset f(x) ⊆ s, and f(a) = ∪ a∈a f(a) ⊆ t is named a multivalued transformation, the graph of f denoted γf is the pair (γ x f , γ a f ) ∈ obj(t op(2)), where γxf = {(x, s) ∈ x×s | s ∈ f(x)} and γaf = {(a, t) ∈ a ×t | t ∈ f(a)}. a representation of a multivalued transformation f : (x, a) → (s, t ) is a quintuple q = [(x, a), (s, t ), (m, n), p, q] where (p, q) ∈ mort op(2) ((m, n), (x, a)) ×mort op(2) ((m, n), (s, t )) and q(p−1(x)) = f(x) for every element x ∈ x. in the case when p := tf ∈ mort op(2) ((γ x f , γ a f ), (x, a)) and q := rf ∈ mort op(2)((γxf , γaf ), (s, t )) are the natural projections the quintuple q̃ = [(x, a), (s, t ), (γxf , γ a f ), tf , rf ] is named the canonical representation of f. 2.1. degree for multivalued transformations defined in homological nspheres. let y1 and y2 be both some homological n-spheres and f : y1 → y2 be a multivalued transformation with a representation q = [y1, y2, x, p, q]. definition 2.1. the degree of a multivalued transformation f : y1 → y2 relative to the representation q is denoted and defined by dg(f, q) = dg(p, x, y1)· dg(q, x, y2). the degree dg(f, q̃) of f relative to the canonical representation q̃ will be called the degree of the multivalued transformation f and will denoted by dg(f). let us give some properties of this homological invariant. proposition 2.2. let q = [y1, y2, x, p, q] be a representation of f, then dg(f, q) is a multiple of dg(f). proof. one can consider the following commutative diagram: γf tf ւ ↑ λ ցrf y1 ←− p x → q y2 where λ(x) = (p(x), q(x)) for every element x ∈ x. one concludes by using assertion 3 of proposition 1.5. � c© agt, upv, 2015 appl. gen. topol. 16, no. 1 24 on some topological invariants for morphisms defined in homological spheres proposition 2.3. let f, g : y1 → y2 be two multivalued transformations such that g(x) ⊆ f(x) for every element x ∈ y1 then dg(g) = k · dg(f) for some integer k ∈ n. proof. under the hypothesis, one obtains that γg ⊆ γf . one concludes the proof with the following commutative diagram: tf ւ γf ցrf y1 ↑ i y2 tg տ γg րrg and by referring to the assertion 3 of proposition 1.5. � what happened if one gets a morphism f ∈ mort op(y1, y2) and considers it as a multivalued morphism in the following sense f(x) = {f(x)} for every element x ∈ y1. in such situation, one has which follows: proposition 2.4. let f ∈ mort op(y1, y2) and f : y1 → y2 be the multivalued transformation given by the rule f(x) = {f(x)} := f(x) for every element x ∈ y1, then dg(f) = dg(f, y1, y2). proof. it is a consequence of the following commutative diagram: γf rf =rf→ y2 tf = tf ↓ y1 ր f where the morphism tf ∈ mort op(γf , y1) realizes a homeomorphism. � corollary 2.5. let g : y1 → y2 be a multivalued mapping which admits a selector f ∈ mort op(y1, y2) then dg(f, y1, y2) = k · dg(g) for some natural number k ∈ n. proof. indeed f(x) := {f(x)} ⊆ g(x) for every x ∈ y1 and thus one can conclude by referring to the propositions 2.3 and 2.4. � proposition 2.6. let y1, y2 and y3 be some homological n-spheres, f : y1 → y2 be a multivalued transformation and f ∈ mort op(y2, y3) then dg(f, y1, y2) · dg(f) = k ·dg(f ◦f) for some k ∈ n. proof. of course, the quintuple q = [y1, y2, γf , tf , f◦rf ] is a representation of the multivalued morphism f ◦f : y1 → y2 therefore, from proposition 1.6 one obtains dg(f◦f, q) = dg(f, y1, y2)·dg(f) one concludes thanks to proposition 2.2. � proposition 2.7. let y be a homological n-sphere and f : y → sn be a multivalued transformation such that dg(f) is different from zero then f(y ) = sn. proof. of course, in this case dg(rf , γf , s n) 6= 0 one concludes by referring to the proposition 1.8. � c© agt, upv, 2015 appl. gen. topol. 16, no. 1 25 n. m. benkafadar and b. d. gel’man definition 2.8. two multivalued transformations f0, f1 : y1 → y2 defined on some homological n-spheres y1 and y2 are called homotopic if there exists a quintuple [y1, y2, x × [0, 1], φ, ψ] such that q0 = [y1, y2, x, φ0, ψ0] and q1 = [y1, y2, x, φ1, ψ1] realize some representations of f0 and f1 respectively, where φt : x → sn and ψt : x → sn are defined by the rules φt(x) = φ(x, t), ψt(x) = ψ(x, t) for every element (x, t) ∈ x ×{0, 1}. proposition 2.9. let f0, f1 : y1 → y2 be some multivalued transformations defined on some homological n-spheres y1 and y2 then if f0 and f1 are homotopic there exist some natural numbers k0, k1 ∈ n such that k0 · dg(f0) = k1 ·dg(f1). proof. for this purpose one refers to propositions 1.7 and 2.2. � 2.2. degree for multivalued transformations with images in homological n-spheres in the category t op(2). in this section one displays a homological invariant for multivalued transformations acting between homological n-spheres of the category t op(2) let (y0, b0) and (y1, b1) be some homological n-spheres and f : (y0, b0) → (y1, b1) be a multivalued transformation that admits a quintuple q = [(y0, b0), (y1, b1), (m, n), p, q] as a representation. definition 2.10. the degree of a multivalued transformation f : (y0, b0) → (y1, b1) relative to the representation q is denoted and defined by dgr(f, q) = dgr(p, (m, n), (y0, b0)) · dgr(q, (m, n), (y1, b1)). the degree of f relative to the canonical representation q̃ = [(y0, b0), (y1, b1), (γ y0 f , γb0 f ), tf , rf ] will be denoted by dgr(f) := dgr(f, q̃). in the sequel, one describes some properties of this homological invariant. proposition 2.11. let q = [(y0, b0), (y1, b1), (x, x ′), p, q] be a representation of a multivalued transformation f : (y0, b0) → (y1, b1) then there exists a natural number k ∈ n such that dgr(f, q) = k · dgr(f). proof. of course, one can consider the next commutative diagram: (y0, b0) p←− (x, x′) q→ (y1, b1) t տ f ↓ λ ր rf (γy0 f , γb0 f ) where λ(x) = (p(x), q(x)) for every element x ∈ x. one concludes by using the assertion 2 of the proposition 1.11. � c© agt, upv, 2015 appl. gen. topol. 16, no. 1 26 on some topological invariants for morphisms defined in homological spheres corollary 2.12. let f : (y0, b0) → (y1, b1) be a multivalued transformation q = [y0, y1, x, p, q] be a representation of f : y0 → y1 then the quintuple q = [(y0, b0), (y1, b1), (x, p −1(b0)),p, q] is a representation of f : (y0, b0) → (y1, b1) and there exists a natural number k ∈ n such that dgr(f, q) = k · dgr(f). proof. this is a consequence of the definition 2.10 and the proposition 2.11. � example 2.13. let b1(0) be the unit ball of the complex plane c and s1(0) = ∂b1(0) be the boundary of b1(0) and let f : (b1(0), s1(0)) → (c, c�{0}) be the multivalued transformation defined by the rule f(z) = n √ z. the quintuple q = [(b1(0), s1(0)), (c, c�{0}), (b1(0), s1(0)), p, q] where p(w) = wn and q(w) = w for every element w ∈ b1(0), is a representation of the multivalued mapping f. moreover, dgr(p, (b1(0), s1(0)), (b1(0), s1(0))) = dg(p) = n and dgr(q, (b1(0), s1(0)), (c, c�{0})) = dg(q) = 1 so dgr(f, q) = n. on the other hand, the single valued map λ : (b1(0), s1(0)) → (γb1(0)f , γ s1(0) f ) where λ(w) = (p(w), q(w)) is an isomorphism in the category t op(2) this implies that the induced homomorphism in homology is an isomorphism in the category g of groups and homomorphisms of groups and thus dgr(f) = dgr(f, q) = n. proposition 2.14. let f, g : (y0, b0) → (y1, b1) be both some multivalued transformations such that g(x) ⊆ f(x) for every element x ∈ y0 then dgr(g) = k ·dgr(f) for some natural number k ∈ n. proof. under the hypothesis, one obtains that (γy0 g , γb0 g ) ⊆ (γy0 f , γb0 f ). therefore one infers the assertion from, the following commutative diagram: tf ւ (γy0 f , γb0 f ) ցrf (y0, b0) ↑ i (y1, b1) tg տ (γy0g , γ b0 g ) րrg and by referring to the assertion 2 of the proposition 1.11 � proposition 2.15. let f ∈ mort op(2) ((y0, b0), (y1, b1)) and f : (y0, b0) → (y1, b1) be the multivalued transformation given by the rule f(x) = {f(x)} := f(x) for every element x ∈ y0, then dgr(f) = dgr(f, (y0, b0), (y1, b1)). proof. for this purpose one can consider the next diagram: (y0, b0) tf =tf←− (γy0 f , γb0 f ) rf =rf→ (y1, b1) � corollary 2.16. let g : (y0, b0) → (y1, b1) be a multivalued transformation which admits a selector f ∈ mort op(2)((y0, b0), (y1, b1)) then: dgr(f, (y0, b0), (y1, b1)) = k ·dgr(g) for some natural number k ∈ n. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 27 n. m. benkafadar and b. d. gel’man proof. this is a consequence of the propositions 2.14 and 2.15. � proposition 2.17. let (y0, b0), (y1, b1) and (y2, b2) be some homological n-spheres, f : (y0, b0) → (y1, b1) be a a multivalued transformation and f ∈ mort op(2) ((y1, b1), (y2, b2)) then dgr(f, (y1, b1), (y2, b2)) ·dgr(f) = k · dgr(f ◦f) for some natural number k ∈ n. proof. of course, the quintuple q = [(y0, b0), (y2, b2), (γ y0 f , γb0 f ), tf , f ◦rf ] is a representation of the multivalued transformation f ◦f : (y0, b0) → (y2, b2) therefore, from the assertion 4 of proposition 1.11 one obtains the next equality: dgr(f ◦f, q) = dgr(f, (y1, b1), (y2, b2)) · dgr(f), one concludes due to proposition 2.11 � definition 2.18. let f0, f1 : (x, a) → (s, t ) be some multivalued transformations, f0 and f1 are called homotopic if there exists a quintuple q = [(x, a), (s, t ), (m, n)× [0, 1], φ, ψ] such that the following quintuples: q0 = [(x, a), (s, t ), (m, n), φ0, ψ0] and q1 = [(x, a), (s, t ), (m, n), φ1, ψ1] are some representations of f0 and f1 respectively and where φt : (m, n) → (x, a) and ψt : (m, n) → (x, a) are defined by the rules φt(m) = φ(m, t), ψt(m) = ψ(m, t) for every element (m, t) ∈ m ×{0, 1}. proposition 2.19. let f0, f1 : (y0, b0) → (y1, b1) be some multivalued transformations then if f0 and f1 are homotopic there exist some representations q0 and q1 of f0 and f1 respectively such that dgr(f0, q0) = dgr(f1, q1). proof. it is a consequence of assertion 5 of proposition 1.11. � proposition 2.20. let f : (y0,b0) → (rn, rn�{0}) be a multivalued transformation such that dgr(f) 6= 0 then there exists an element y ∈ y0�b0 such that 0 ∈ f(y). proof. indeed, dgr(f) 6= 0 so dgr(rf , (γy0f , γ b0 f ), (rn, rn�{0})) 6= 0 after which one can conclude due to the proposition 1.13. � let s be the boundary of a closed ball b of rn and f : b → rn be a multivalued transformation. the multivalued vector field induced by f noted by φ is the multivalued transformation given by the rule φ(x) = x − f(x) for every element x ∈ b. it is obvious that if q = [b, rn, γf , p, q] is the canonical representation of f and the multivalued vector field induced by f is such that φ : (b, s) → (rn, rn�{0}) then the quintuple q̂ = c© agt, upv, 2015 appl. gen. topol. 16, no. 1 28 on some topological invariants for morphisms defined in homological spheres [(b, s), (rn, rn�{0}), (γbf , γsf ), p, p−q)] is a representation of the multivalued vector field φ. in the sequel the degrees: dgr(p, (γbf , γ s f ), (r n, rn�{0})) and dgr(p −q, (γbf , γsf ), (rn, rn�{0})) will be denoted by dgr(p) and dgr(p −q) respectively. proposition 2.21. let f : b → rn be a multivalued transformation which is free of fixed point on the boundary s of a closed ball b then if the topological degree dgr(φ) of the multivalued vector field induced by f is not zero i.e. dgr(φ) 6= 0, the multivalued transformation f : b → rn admits a fixed point in the interior of the ball b. proof. this is a consequence of proposition 2.20. � proposition 2.22. let f : b → rn be a multivalued transformation free of fixed point on the boundary s of a ball b and f(s) ⊆ b then the following equivalence is satisfied: dgr(φ, q̂) 6= 0 if and only if dgr(p) 6= 0. proof. consider the morphisms p, p − q ∈ mort op(2) ((γbf , γsf ), (rn, rn\{0})) and let ψ ∈ mort op(γbf × [0, 1]), rn) be a morphism given by the rule: ψ((x, y), λ) = p(x, y) −λ · q(x, y) for every element ((x, y), λ) ∈ γbf × [0, 1]. it follows that the morphism: ψ ∈ mort op(2) ((γ b f , γ s f ) × [0, 1]), (rn, rn\{0})) is a homotopy between the morphisms: p, p−q ∈ mort op(2) ((γ b f , γ s f ), (r n, rn\{0})) therefore from assertions 5 of proposition 1.11 one deduces what follows : dgr(p) = dgr(p −q) and thus one obtains the next equality: dgr(φ, q̂) = (dgr(p)) 2 . hence, dgr(φ, q̂) 6= 0 if and only if dgr(p) 6= 0. � the last proposition 2.22, permits to obtain a generalization of the theorems due to eilenberg-montgomery [3] and kakutani [8]. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 29 n. m. benkafadar and b. d. gel’man theorem 2.23. let f : b → rn be a multivalued transformation which satisfies the following conditions: (1) dgr(p) 6= 0, (2) f(s) ⊆ b. then the multivalued transformation f admits in the ball a fixed point. proof. of course, if f has a fixed point on s then the conclusion of the theorem is satisfied. otherwise, if f is free of fixed point on s then dgr(φ) 6= 0. one concludes the proof from proposition 2.21. � acknowledgements. project supported by cnepru no b00920110092 and pnr / atrst no 8/u250/713. laboratory m.m.e.r.e. u.m.1 constantine and russian fbr grant 11-01 -00382-a. references [1] yu. g. borisovich, b. d. gel’man, a. d. myshkis and v. v. obukhovskii, topological methods in the fixed point theory of multivalued maps, uspehi mat. nauk 35, no. 1 (1980), 59–126 (in russian); english translation: russian math. surveys 35 (1980), 65–143. 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[9] r. m. switzer, algebraic topologyhomotopy and homology, springer verlag, berlinheidelberg-new york, 1975. [10] n. m benkafadar and b. d. gel’man, generalized local degree for multi-valued mappings, international journal of math. game theory and algebra 10, no. 5 (2000), 413–434. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 30 @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 289–299 a better framework for first countable spaces gerhard preuß dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. in the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. several applications of first countability in a broader context than the usual one of topological spaces are studied. 2000 ams classification: 54a05, 54c35, 54d55, 54e15, 18a40, 18d15. keywords: first axiom of countability, second axiom of countability, countably compact, sequentially compact, sequentially complete, continuous convergence, sequentially continuous, semiuniform convergence spaces, convergence spaces, filter spaces, topological spaces, uniform spaces, bicoreflective subconstruct, cartesian closedness. 1. introduction. recently, the author ([4],[5]) has shown that a topologist’s life is easier in convenient topology, where mainly the topological construct suconv of semiuniform convergence spaces is considered. in suconv, topological and uniform concepts are available and fundamental constructions can be easily described. in the realm of topological spaces the first axiom of countability plays an essential role whenever sequences are prefered rather than filters (e.g. in analysis). unfortunately, quotients of first countable topological spaces need not be first countable. this situation can be improved in the framework of semiuniform convergence spaces: first countability is divisible. since first countability is also summable, this implies that the construct fc-suconv of first countable semiuniform convergence spaces is bicoreflective in suconv, i.e. every semiuniform convergence space has an underlying first countable 290 gerhard preuß space. another consequence is the following: fc-suconv is a topological universe (= cartesian closed and extensional topological construct) such that countable products of quotients are quotients. not only first countable (symmetric) topological spaces, but also uniform spaces with a countable base, i.e. pseudometrizable uniform spaces, are first countable (as semiuniform convergence spaces). for uniform spaces, there is no difference between the first and the second axiom of countability, when these axioms are generalized to semiuniform convergence spaces. the applications of first countability, studied here, cover the following themes: 1. the definition of the closure of a subset by means of sequences. 2. the definition of completeness by means of sequences. 3. the equivalence of countable compactness and sequential compactness. 4. the definition of continuous convergence by means of sequences. 5. the definition of continuity by means of sequences. 2. preliminaries. definition 2.1. 1) a semiuniform convergence space is a pair (x, jx), where x is a set and jx a set of filters on x×x such that the following are satisfied: uc1) the filter ẋ× ẋ, generated by {(x,x)}, belongs to jx for each x ∈ x, uc2) g ∈ jx, whenever f ∈ jx and f ⊂ g, uc3) f ∈ jx implies f−1 = {f−1 : f ∈ f} ∈ jx, where f−1 = {(y,x) : (x,y) ∈ f}. if (x, jx) is a semiuniform convergence space, then the elements of jx are called uniform filters. 2) a map f : (x, jx) → (y, jy ) between semiuniform convergence spaces is called uniformly continuous provided that (f ×f)(f) ∈ jy for each f ∈ jx. 3) the construct of semiuniform convergence spaces (and uniformly continuous maps) is denoted by suconv. remark 2.2. suconv is a topological construct, where initial and final structures have an easy description: if x is a set, ((xi, jxi))i∈i a family of semiuniform convergence spaces and (fi : x → xi)i∈i (resp. (fi : xi → x)i∈i) a family of maps, then jx = {f ∈ f(x × x) : (fi × fi)(f) ∈ jxi for each i ∈ i} (resp. jx = {f ∈ f(x ×x): there is some i ∈ i and some fi ∈ jxi with (fi ×fi)(fi) ⊂ f}∪{ẋ× ẋ : x ∈ x}) is the initial (resp. final ) suconv-structure w.r.t. the given data, where f(x ×x) denotes the set of all filters on x ×x. example 2.3. 1) let (x, x) be a (symmetric) topological space. then its corresponding semiuniform convergence space (x, jγqx ) is defined by f ∈ jγqx iff there is some x ∈ x such that f ⊃ ux(x)×ux(x), where ux(x) denotes the neighborhood filter of x in (x, x). a better framework for first countable spaces 291 2) let (x, w) be a uniform space. then its corresponding semiuniform convergence space (x, [w]) is defined by f ∈ [w] iff f ⊃ w. 3) let x = (x, jx) and y = (y, jy ) be semiuniform convergence spaces. then there is a coarsest suconv-structure jx,y on the set [x, y] of all uniformly continuous maps from x into y such that the evaluation map ex,y : x × ([x, y], jx,y ) → y (defined by ex,y (x,f) = f(x)) is uniformly continuous. it is defined by φ ∈ jx,y iff φ(f) ∈ jy for each f ∈ jx (φ(f) denotes the filter generated by {a(f) : a ∈ φ and f ∈ f} with a(f) = {(f(x),g(y)) : (f,g) ∈ a and (x,y) ∈ f}). ([x, y], jx,y ) is called a natural function space. 4) let x = (x, jx) be a semiuniform convergence space. let x* = x ∪ {∞x} with ∞x /∈ x and jx* = {h ∈ f(x* ×x*): the trace of h on x ×x exists and belongs to jx or the trace on x ×x does not exist}. then (x*, jx*) is called a one-point extension of (x, jx). obviously, jx* is the coarsest suconv-structure on x* such that (x, jx) is a subspace of (x*, jx*). remark 2.4. 1) the existence of one-point extensions implies that quotients are hereditary, i.e. if f : x → y is a quotient map in suconv, then for every z ⊂ y , the map f | f−1[z] : f−1[z] → z is again a quotient map. 2) concerning the definitions of (symmetric) kent convergence spaces and filter spaces, and their relations to semiuniform convergence spaces, see [4] (or [5]). 3. first countable spaces. definition 3.1. 1) a semiuniform convergence space (x, jx) is said to fulfill the first axiom of countability or to be first countable iff each uniform filter has a uniform subfilter with a countable base. 2) a filter space (x,γ) is called first countable provided that each f ∈ γ has a subfilter g ∈ γ with a countable base. 3) a kent convergence space (x,q) is first countable iff for each (f,x) ∈ q, there is some (g,x) ∈ q such that g is a subfilter of f with a countable base. proposition 3.2. let (x, jx) be a semiuniform convergence space fulfilling the first axiom of countability. then the underlying filter space (x,γjx ) is first countable. proof. let f ∈ γjx . then f×f ∈ jx and by assumption there is some g ∈ jx with a countable base b = {b1,b2, . . . ,bn . . .} such that g ⊂ f × f. thus, for each n ∈ n = {1, 2, . . .}, there exists some fn ∈ f with fn × fn ⊂ bn . obviously, the finite intersections of elements of {fn : n ∈ n} form a countable base of some subfilter h of f. since g ⊂ h × h , it follows that h × h belongs to jx, i.e. h ∈ γjx . � 292 gerhard preuß proposition 3.3. let (x,γ) be a filter space and (x, jγ) its corresponding semiuniform convergence space. then (x, jγ) is first countable iff (x,γ) is first countable in the sense of 3.1.2). proof. “⇒”. this follows immediately from 3.2, since γ = γjγ . “⇐”. let f ∈ jγ. then there is some g ∈ γ with g×g ⊂ f. furthermore, there is some h ∈ γ with a countable base b such that h ⊂ g. consequently, h × h ⊂ g × g ⊂ f, where h × h ∈ jγ has a countable base, namely {b ×b : b ∈ b}. � proposition 3.4. let (x,γ) be a first countable filter space. then the underlying (symmetric) kent convergence space (x,qγ) is also first countable. proposition 3.5. let (x,q) be a symmetric kent convergence space and (x, jγq ) its corresponding semiuniform convergence space. then (x, jγq ) is first countable iff (x,q) is first countable in the sense of 3.1 3). proof. “⇒”. since γjγq = γq and qγq = q, (x,q) is first countable by means of the above propositions. “⇐”. let f ∈ jγq . then there is some (g,x) ∈ q such that g × g ⊂ f. by assumption, there exists (h,x) ∈ q such that h is a subfilter of g with a countable base b. hence, h × h ⊂ f, where h × h ∈ jγq has the countable base {b ×b : b ∈ b}. � corollary 3.6. the underlying (symmetric) kent convergence space (x,qγjx ) of a first countable semiuniform convergence space (x, jx) is first countable. proof. apply 3.2 and 3.4. � corollary 3.7. let (x, x) be a symmetric topological space (=r0-space) and define qx ⊂ f(x) × x by (f,x) ∈ qx iff f ⊃ ux(x). then the following are equivalent: (1) (x, x) is first countable in the usual sense, i.e. for each x ∈ x, ux(x) has a countable base. (2) (x, x) is first countable as a semiuniform convergence space, i.e. (x, jγqx ) is first countable. (3) (x, x) is first countable as a kent convergence space, i.e. (x,qx) is first countable in the sense of 3.1.3). proof. (1) ⇒ (2). let f ∈ jγqx . then there is some x ∈ x such that ux(x) × ux(x) ⊂ f. since, by assumption, ux(x) has a countable base, the subfilter ux(x) × ux(x) of f has also a countable base and belongs to jγqx . (2) ⇒ (3). apply 3.6. (3) ⇒ (1). let x ∈ x and consider ux(x). then (ux(x),x) ∈ qx and, by assumption, there is a subfilter g of ux(x) with a countable base such that (g,x) ∈ qx. hence, ux(x) ⊂ g ⊂ ux(x), which implies g = ux(x). � proposition 3.8. let (x, w) be a uniform space and (x, [w]) its corresponding semiuniform convergence space. a better framework for first countable spaces 293 then the following are equivalent: (1) (x, w) is first countable as a semiuniform convergence space, i.e. (x,[w]) is first countable. (2) the filter w of entourages has a countable base. proof. since [w] consists of all filters f on x ×x containing w, the proof is obvious. � remark 3.9. it is well-known that a uniform space is pseudometrizable, i.e. its uniformity is induced by a pseudometric, iff its filter of entourages has a countable base. thus, a semiuniform convergence space (x, jx) is pseudometrizable (i.e. there is a pseudometric d on x such that jx = {f ∈ f(x × x) : f ⊃ {vε : ε > 0}} with ∨ε = {(x,y) : d(x,y) < ε}) iff it is uniform (i.e. there is a uniformity w on x such that jx = [w]) and first countable. proposition 3.10. the construct fc-suconv of first countable semiuniform convergence spaces (and uniformly continuous maps) is bicoreflective in suconv, where 1x : (x, (jx)fc) → (x, jx) with (jx)fc = {f ∈ jx: there is some g ∈ jx with g ⊂ f and g has a countable base} is the bicoreflection of (x, jx) ∈| suconv | w.r.t. fc-suconv, and (x, (jx)fc) is called the underlying first countable semiuniform convergence space of (x, jx). proof. obviously, if f : (y, jy ) → (x, jx) is a uniformly continuous map from a first countable semiuniform convergence space (y, jy ) into a semiuniform convergence space (x, jx), then f : (y, jy ) → (x, (jx)fc) is also uniformly continuous. � corollary 3.11. in suconv first countability is summable and divisible. remark 3.12. 1) in classical general topology first countability is not divisible, e.g. the usual topological space rt of real numbers is first countable, whereas the topological quotient space rt/n, obtained by identifying the set n of positive integers to a point is not first countable. if rt is regarded as a semiuniform convergence space and the quotient rt/n is formed in suconv, then rt/n is first countable, but no longer topological. indeed, rt/n is a convergence space (=symmetric kent convergence space), since rt is a convergence space and convergence spaces are closed under formation of quotients in suconv. 2) the convergence spaces are exactly the quotients (in suconv) of the topological semiuniform convergence spaces (= symmetric topological spaces). this has been proved in the language of nearness by w. a. robertson [6] (cf. also [1]). proposition 3.13. 1) fc-suconv is closed under formation of subspaces in suconv. 2) fc-suconv is closed under formation of countable products in suconv. 294 gerhard preuß proof. 1) let (x, jx) ∈|fc-suconv|,a ⊂ x and let i : a → x be the inclusion map. if the initial suconv-structure on a w.r.t. i : a → x is denoted by ja, then let f ∈ ja. by definition, (i × i)(f) ∈ jx, and by assumption, there is some g ∈ jx with a countable base b such that g ⊂ (i × i)(f). hence, the subfilter (i × i)−1(g) of f belongs to ja and has the countable base {b ∩ (a×a) : b ∈ b}. 2) let ((xi, jxi))i∈i be a countable family of first countable semiuniform convergence spaces and let (πi∈ixi, jx) be their product. further, let f ∈ jx. then (pi × pi)(f) ∈ jxi for each i ∈ i, where pi : πi∈ixi → xi denotes the i − th projection. by assumption, there is some subfilter gi ∈ jxi of (pi ×pi)(f) with a countable base bi. hence, πi∈igi ⊂ πi∈i(pi ×pi)(f) ⊂ f where πi∈i(xi ×xi) and (πxi) × (πi∈ixi) are not distinguished. obviously, πi∈igi ∈ jx. the set b consisting of all finite intersections of elements of the countable set b′ = {(pi × pi)−1[bi] : i ∈ i,bi ∈ bi} is a countable base of πi∈igi. � remark 3.14. since the construct tops of symmetric topological spaces (and continuous maps) is closed under formation of products in suconv, arbitrary products of first countable semiuniform convergence spaces need not be first countable, e.g. it is well-known that the topological product space rrt is not first countable. theorem 3.15. 1) fc-suconv is a topological universe, where the natural function spaces (resp. one-point extensions) arise from the natural function spaces (resp. one-point extensions) suconv by forming the underlying first countable spaces. 2) in fc-suconv countable products of quotients are quotients. proof. 1) since fc-suconv is bicoreflective in the topological universe suconv and closed under formation of finite products and subspaces, it is the desired topological universe (cf. [5; 3.1.7 and 3.2.5]). 2) since countable products and quotients are formed in fc-suconv as in suconv, where products of quotients are quotients, in fc-suconv countable products of quotients are quotients. � a much stronger condition than the first axiom of countability is the second axiom of countability which can be defined in the realm of semiuniform convergence spaces as follows: definition 3.16. a semiuniform convergence space (x, jx) is said to fulfill the second axiom of countability or to be second countable if there is a countable set a ⊂ p(x ×x) such that each f ∈ jx has a subfilter g ∈ jx with a base of elements of a. proposition 3.17. every second countable semiuniform convergence space is first countable. proposition 3.18. let (x, w) be a uniform space and (x, [w]) its corresponding semiuniform convergence space. then the following are equivalent: a better framework for first countable spaces 295 (1) (x, [w]) is first countable, (2) (x, [w]) is second countable, (3) w has a countable base. proof. the equivalence of (1) and (3) has been proved under 3.8. since 3.17 is valid, it suffices to prove: (1) ⇒ (2). let f ∈ jx. then f ⊃ w and by (3), w has a countable base a. � proposition 3.19. let (x, x) be a symmetric topological space and (x,jγqx) its corresponding semiuniform convergence space. then (x, jγqx ) is second countable iff (x, x) is second countable in the usual sense. proof. “⇒”. let a ⊂ p(x × x) be countable and let each f ∈ jγqx have a subfilter g ∈ jγqx with a base of elements of a. for each x ∈ x, ux(x) × ux(x) ∈ jγqx , and by assumption there is some filter g ∈ jγqx , i.e. g ⊃ ux(y) × ux(y) for some y ∈ x, such that (∗) ux(y) × ux(y) ⊂ g ⊂ ux(x) × ux(x) where g has a base of elements of a. if p1 : x × x → x denotes the first projection, it follows from (∗) ux(y) ⊂ p1(g) ⊂ ux(x) and, since (x, x) is symmetric, p1(g) = ux(x) = ux(y). thus, ux(x) has a base of elements of the countable set a′ = {p1[a] : a ∈ a}. let a′′ = {(p1[a])0 : a ∈ a}. then a′′ is countable and ux(x) has also a base of elements of a′′. each o ∈ x is a union of elements of a′′, i.e. (x, x) is second countable. “⇐”. let x have a countable base b and let f ∈ jγqx , i.e. there is some x ∈ x such that f ⊃ ux(x) × ux(x). obviously, ux(x) has a base of elements of b, and ux(x) × ux(x) has a base of elements of the countable set a = {b × b : b ∈ b}. consequently, (x, jγqx ) is second countable. � remark 3.20. the second axiom of countability does not have the nice structural behaviour as the first axiom of countability. it is easily checked that second countable semiuniform convergence spaces are hereditary, countably productive, countably summable and divisible, but they do not form a bicoreflective subconstruct of suconv, since arbitrary sums of second countable spaces are not second countable, which is known from second countable topological spaces (note: coproducts [=sums] of topological semiuniform convergence spaces are topological). nevertheless, in the realm of semiuniform convergence spaces, second countability is divisible, a result which is not true in the framework of topological spaces. 296 gerhard preuß 4. applications of first countability. remark 4.1. if (x, jx) is a semiuniform convergence space, then a filter f on x is called a cauchy filter iff f × f ∈ jx (i.e. f ∈ γjx ), and it is said to converge to x (i.e. (f,x) ∈ qγjx ) iff f ∩ ẋ is a cauchy filter. a sequence (xn) in x is a cauchy sequence iff the elementary filter fe of (xn) is a cauchy filter, and it converges to x iff the elementary filter of (xn) converges to x. definition 4.2. for each subset a in a semiuniform convergence space (x, jx), the closure a is defined as follows: a = {x ∈ x : there is some filter f on x converging to x such that a ∈ f}. proposition 4.3. let (x, jx) be a first countable semiuniform convergence space and a a subset in (x, jx). then a = {x ∈ x : there is a sequence in a converging to x in (x, jx)}. proof. let x ∈ a. then there is some filter f on x such that a ∈ f and (f,x) ∈ qγjx , i.e. f converges to x in (x, jx). by 3.6, there is some (g,x) ∈ qγjx such that g ⊂ f has a countable base b = {b1,b2, . . .}, where we may assume without loss of generality that b1 ⊃ b2 ⊃ . . . . for each n ∈ n, choose some xn ∈ bn ∩a. then the elementary filter fe of (xn) contains g , which implies that (xn) converges to x in (x, jx). the inverse implication is evident. � remark 4.4. uniform spaces have been generalized to uniform limit spaces, which are semiuniform convergence spaces fulfilling the following two additional axioms: uc4) if f and g are uniform filters, then f ∩ g is also a uniform filter; uc5) if the composition f ◦ g of two uniform filters f and g exists, it is a uniform filter, which has {f ◦g : f ∈ f,g ∈ g} as a base, where ◦ stands for the composition of relations. the underlying filter spaces of uniform limit spaces are called cauchy spaces. proposition 4.5. a first countable cauchy space (x,γ) is complete iff it is sequentially complete (i.e. each cauchy sequence in (x,γ) converges). proof. “⇒”. obvious. “⇐”. let f ∈ γ. by assumption, there is some g ∈ γ with a countable base b = {b1,b2, . . .} such that g ⊂ f. without loss of generality, let b1 ⊃ b2 ⊃ . . . . for each n ∈ n, choose some xn ∈ bn. since the elementary filter fe of (xn) is finer than g, it is a cauchy filter. furthermore, γ = γj*γ , where jγ* = {g ∈ f(x × x) : there are k1, . . . kn ∈ γ with g ⊃ ∩ni=1ki × ki} is a uniform limit space structure for x. it follows from g × fe ⊃ (g ∩ fe) × (g ∩ fe) ∈ jγ*, g×fe ∈ jγ*, and since by assumption fe converges to some x ∈ x, fe×ẋ ∈ j*γ. thus, g × ẋ = (fe × ẋ) ◦ (g ◦ fe) ∈ j*γ, i.e. g converges to x, which implies that f converges to x. � a better framework for first countable spaces 297 corollary 4.6. a first countable uniform limit space (x, jx) is complete iff each cauchy sequence in (x, jx) converges. proof. by 3.2, the underlying filter space (=cauchy space) of the first countable uniform limit space (x, jx) is first countable. applying 4.5, 4.6 is proved. � definition 4.7. a semiuniform convergence space (x, jx) is called 1) countably compact provided that each filter f on x with a countable base has an adherence point x ∈ x, i.e. there is some filter g on x converging to x such that g ⊃ f, 2) sequentially compact iff each sequence in x has a subsequence converging in (x, jx). proposition 4.8. a semiuniform convergence space (x, jx) is countably compact iff each sequence in x has an accumulation point (in (x,qγjx )). proof. “⇒”. let (xn) be a sequence in x and fe its elementary filter. then fe has a countable base and by assumption there is an adherence point x ∈ x of fe, i.e. x is an accumulation point of (xn). “⇐”. let f be a filter on x with a countable base b = {b1,b2, . . .} such that, without loss of generality, b1 ⊃ b2 ⊃ . . . . for each n ∈ n, choose some xn ∈ bn. the elementary filter fe of (xn) is finer that f and has an adherence point by assumption. hence, f has an adherence point. � proposition 4.9. 1) every sequentially compact semiuniform convergence space is countably compact. 2) every countably compact semiuniform convergence space, which is first countable, is sequentially compact. proof. 1) is obvious. 2) let (xn) be a sequence in x and fe its elementary filter. since (x, jx) is countably compact, fe has an adherence point x ∈ x, i.e. there is some filter f ⊃ fe such that (f,x) ∈ qγjx . by 3.6, there is a subfilter g of f with a countable base b = {b1,b2, . . .} converging to x. without loss of generality, let b1 ⊃ b2 ⊃ . . . . for each m ∈ n, em∩bm 6= ø, where em = {xn : n ≥ m}. choose ym ∈ bm ∩em for each m ∈ n. then (ym)m∈n converges to x, since the elementary filter f′e of (ym) is finer than g. furthermore, there exists a subsequence (xjn) of (xn) such that the elementary filter of (xjn) coincides with the elementary filter of (ym), i.e. (xjn) converges to x. � remark 4.10. continuous convergence, introduced by h. hahn [3] in analysis, is nowadays mainly considered in the realm of limit spaces, where a kent convergence space (x,q) is called a limit space provided that for any x ∈ x and any two filters f and g on x converging to x, the intersection f ∩ g converges to x. let x = (x,q) and y = (y,r) be limit spaces and let [x, y] be the set of all continuous maps between x and y, where a map f : x → y is continuous iff (f(f),f(x)) ∈ r for each (f,x) ∈ q. a filter f on [x, y] converges continuously to f ∈ [x, y] provided that for each x ∈ x and each 298 gerhard preuß filter g on x converging to x in x, the filter ex,y (g × f) converges to f(x) in y, where g × f denotes the product filter and ex,y (x,g) = g(x) for each (x,g) ∈ x×[x, y]). a sequence (fn) in [x, y] is said to converge continuously to f ∈ [x, y] iff the elementary filter of (fn) converges continuously to f. if a kent convergence space x = (x,q) has the property that for each x ∈ x, the neighborhood filter u(x) = ⋂ {f ∈ f(x) : (f,x) ∈ q} converges to x, then it is called a pretopological space. proposition 4.11. let x = (x,q) be a first countable limit space and y = (y,r) a pretopological space. then a sequence (fn) in [x, y] converges continuously to f ∈ [x, y] iff for each x ∈ x and each sequence (xn) in x converging to x, the sequence (fn(xn)) converges to f(x) in y. proof. “⇐” (indirectly). if the elementary filter fe of (fn) does not converge continuously to f ∈ [x, y], there is some (g,x) ∈ q such that ex,y (g × fe) does not converge to f(x), i.e. there is some u ∈ u(f(x)) in y such that for each g ∈ g and each f ∈ fe, there are y ∈ g and g ∈ f with g(y) /∈ u. by assumption, there exists some (h,x) ∈ q such that h has a countable base b = {b1,b2, . . .} and h ⊂ g. without loss of generality, let b1 ⊃ b2 ⊃ . . . . since for each n ∈ n,bn ∈ g and {fm : m ≥ n} ∈ f, one can construct sequences (xn) and m0 < m1 < ... such that xn ∈ bn and fmn(xn) /∈ u. define a sequence (yn) in x by y0 = . . . = ym0 = x0,ym0+1 = . . . = ym1 = x1, . . . and so on. obviously, (yn) converges to x in x, but (fn(yn)) does not converge to f(x), since fmn(ymn) = fmn(xn) /∈ u. “⇒”. this implication is straightforward. � proposition 4.12. let x = (x,q) be a first countable kent convergence space and y = (y,r) a pretopological space. a map f : x → y is continuous, where continuity is defined as under 3.10., iff for each x ∈ x and each sequence (xn) in x converging to x in x, the sequence (f(xn)) converges to f(x) in y proof. “⇒”. this implication is evident. “⇐”. since each filter f on x with a countable base is the intersection of all elementary filters on x containing f (cf. [2; 2.8.6.]) and each map f : x → y preserves intersection of filters, the proof is obvious. � references [1] h.l. bentley, h. herrlich and w.a. robertson, convenient categories for topologists, comm. math. univ. carolinae 17 (1976), 207–227. [2] w. gähler, grundstrukturen der analysis i/ii (birkhäuser, basel, 1977/78). [3] h. hahn, theorie der reellen funktionen (berlin, 1921). [4] g. preuß, semiuniform convergence spaces, math. japonica 41 (1995), 465–491. [5] g. preuß, foundations of topology – an approach to convenient topology (kluwer, dordrecht, 2002). [6] w.a. robertson, convergence as a nearness concept, phd thesis (carlton university, ottawa, 1975). a better framework for first countable spaces 299 received november 2001 revised october 2002 gerhard preuß department of mathematics and computer science, free university berlin, arnimallee 3, 14195 berlin, germany e-mail address : preuss@math.fu-berlin.de 09.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 115 127 fuzzy functions: a fuzzy extension of thecategory set and some related categoriesulrich h�ohle, hans-e. porst, alexander p. �sostakabstract. in research works where fuzzy sets are used, mostlycertain usual functions are taken as morphisms. on the otherhand, the aim of this paper is to fuzzify the concept of a functionitself. namely, a certain class of l-relations f : x � y ! l isdistinguished which could be considered as fuzzy functions froman l-valued set (x;ex) to an l-valued set (y;ey ). we studybasic properties of these functions, consider some properties ofthe corresponding category of l-valued sets and fuzzy functionsas well as brie y describe some categories related to algebra andtopology with fuzzy functions in the role of morphisms.2000 ams classi�cation: 04a72, 18a05, 54a40keywords: l-relation, l-fuzzy function, fuzzy category, fuzzy topology,fuzzy group 1. introductionin research works where fuzzy sets are involved, in particular, in the theoryof fuzzy topological spaces, fuzzy algebra, fuzzy measure theory, etc., mostlycertain usual functions are taken as morphisms: they can be certain mappingsbetween the corresponding sets, or between the fuzzy powersets of these sets,etc. on the other hand, there are only few papers where attempts to fuzzifythe concept of a function itself are undertaken (see e.g. [11, 12], etc). the aimof our work is also to present a possible approach to this problem. namely,a certain class of l-relations (i.e. mappings f : x � y ! l) is distinguishedwhich seem reasonable to be viewed as (l-)fuzzy functions from a set x to a sety . we de�ne composition of fuzzy functions; study images and preimages ofl-sets under fuzzy functions; introduce properties of injectivity and surjectivityfor them; describe products and coproducts in the corresponding category, etc.in the last part of the paper we de�ne some categories related to topology andalgebra where fuzzy functions play the role of morphisms. 116 ulrich h�ohle, hans-e. porst, alexander p. �sostakin conclusion, we would like to mention the following two peculiarities of ourapproach.first, the appropriate context for our work is formed not by usual sets, orby their l-subsets (i.e. mappings f : x ! l), but rather by l-valued sets (i.e.sets, endowed with an l-valued equality e : x � x ! l, see e.g. [6, 7]) andtheir l-subsets. and second, in the result we obtain not a usual category, butthe so called a fuzzy category a concept introduced and studied in [14, 15].2. prerequisiteslet l = (l;�;^;_;�) be an in�nitely distributive gl-monoid (cf. e.g. [6],[7]), i.e. a commutative integral divisible cl-monoid (cf. [1]). it is well knownthat every gl-monoid is residuated, i.e. there exists a further binary operationimplication \7�!" such that� � � � () � � � 7�! 8�;�; 2 l:we set �2 = � � � and further by induction: �n = �n�1 � �. let > and ?denote respectively the top and the bottom elements of l.following u.h�ohle (cf e.g. [7]) by an l-valued set we call a pair (x;e) wherex is a set and e is an l-valued equality, i.e. a mapping e : x � x ! l suchthat(1eq) e(x;y) � e(x;x) ^ e(y;y) 8x;y 2 x;(2eq) e(x;y) = e(y;x) 8x;y 2 x;(3eq) e(x;y) � �e(y;y) 7�! e(y;z)� � e(x;z) 8x;y;z 2 x.an l-valued set (x;e) is called separated if(4eq) e(x;x)we(y;y) � e(x;y) () x = y 8x;y 2 x.an l-valued equality e is called global if(5eq) e(x;x) = > 8x 2 x.further, recall that an l-set, or more precisely, an l-subset of a set xis just a mapping a : x ! l. in case (x;e) is an l-valued set, its l-subset a is called strict, if a(x) � ex(x;x)8x 2 x; a is called extensional ifsupx a(x) � �e(x;x) 7�! e(x;x0)� � a(x0);8x0 2 x.by l � set(l) we denote the category whose objects are triples (x;e;a)where (x;e) is an l-valued set and a is its strict extensional l-subset, andmorphisms from (x;ex;a) to (y;ey ;b) are mappings f : x ! y whichpreserve equalities (i.e. ex(x1;x2) � ey (fx1;fx2)) and "respect l-subsets",i.e. a � b �f. let l�set 0(l) stand for the full subcategory of the categoryl � set(l) determined by global separated l-valued sets.to recall the concept of an l�fuzzy category [14, 15], consider an ordinary(classical) category c and let ! : ob(c) ! l and � : mor(c) ! l be l-fuzzysubclasses of its objects and morphisms respectively. now, an l-fuzzy categorycan be de�ned as a triple (c;!;�) satisfying the following axioms ([15], cf. also[14] in case � = ^):10 �(f) � !(x) ^ !(y ) 8x;y 2 ob(c) and 8f 2 mor(x;y );20 �(g � f) � �(f) � �(g) whenever the composition g � f is de�ned; fuzzy functions: a fuzzy extension of the category set and some related categories 11730 �(ex) = !(x) where ex : x ! x is the identity morphism.our aim is, starting from the category l�set(l), to de�ne a fuzzy categoryl�fset(l) having the same class of objects as l�set(l) but an essentiallywider class of "potential" morphisms.3. fuzzy category l � fset(l).3.1. category l � fset(l). we start with de�ning a usual (i.e. crisp) cat-egory l � fset(l). namely, let l � fset(l) denote the category havingthe same objects as l�set(l) and whose morphisms, called (potential) fuzzyfunctions, from (x;ex;a) to (y;ey ;b) are l-mappings f : x � y ! l suchthat(0�) f(x;y) � ex(x;x) ^ ey (y;y) 8y 2 y;8x 2 x;(1�) supx a(x) � �ex(x;x) 7�! f(x;y)� � b(y) 8y 2 y ;(2�) f(x;y) � �ey (y;y) 7�! ey (y;y0)� � f(x;y0) 8x 2 x;8y;y0 2 y ;(3�) ex(x;x0) � �ex(x;x) 7�! f(x;y)� � f(x0;y) 8x;x0 2 x;y 2 y ;(4�) f(x;y) � �ex(x;x) 7�! f(x;y0)� � ey (y;y0) 8x 2 x;8y;y0 2 y ;in particular, when a = >x and b = >y we write f : (x;ex) ! (y;ey )instead of f : (x;ex>x) ! (y;ey >y ).notice that conditions (0�) (3�) say that f is a certain l-relation, while axiom(4�) speci�es that the l-relation f is a function.remark 3.1. since f(x;y) � ex(x;x), and a � b =) a = b � (b 7�! a) (bydivisibility of l), we havef(x;y) � (ex(x;x) 7�! ex(x;x0))= ex(x;x) � (ex(x;x) 7�! f(x;y)) � (ex(x;x) 7�! ex(x;x0))= ex(x;x0) � (ex(x;x) 7�! f(x;y)):therefore axiom (3�) can be given in the following equivalent form:(30�) f(x;y) � �ex(x;x) 7�! ex(x;x0)� � f(x0;y).remark 3.2. applying (4�) it is easy to establish thatf(x;y1) � f(x;y2) � f(x;y1) � �ex(x;x) 7�! f(x;y2)�� ey (y1;y2)� ey (y1;y1) 7�! ey (y1;y2):remark 3.3. let f : (x;ex) ! (y;ey ) be a fuzzy function, x0 � x y 0 � y ,and let the l-equalities ex0 and ey 0 on x0 and y 0 be de�ned as the restrictionsof the equalities ex and ey respectively. then de�ning a mapping f 0 : x0 �y 0 ! l by the equality f 0(x;y) = f(x;y) 8x 2 x0;8y 2 y 0 a fuzzy functionf 0 : (x0;ex0) ! (y 0;ey 0) is obtained. we refer to it as the restriction of f tothe subspaces (x0;ex0) and (y 0;ey 0).given two fuzzy functions f : (x;ex;a) ! (y;ey ;b) and g : (y;ey ;b) !(z;ez;c) we de�ne their composition g � f : (x;ex;a) ! (z;ez;c) by the 118 ulrich h�ohle, hans-e. porst, alexander p. �sostakformula (g � f)(x;z) = _y2y �f(x;y) � �ey (y;y) 7�! g(y;z)��:since, by divisibility of l, f(x;y) = ey (y;y) � �ey (y;y) 7�! f(x;y)� andg(y;z) = ey (y;y) � �ey (y;y) 7�! g(y;z)�, the composition can be de�nedalso by the formula(g � f)(x;z) = _y2y ��ey (y;y) 7�! f(x;y)� � g(y;z)�:proposition 3.4. g � f : (x;ex;a) ! (z;ez;c) is indeed a fuzzy function.proof. the proof of the validity of (0�) is straightforward.(1�): taking into account divisibility of l, strictness of a and axiom (1�)for f we get:supx �a(x) � �ex(x;x) 7�! (g � f)(x;z)��= supx�ex(x;x) 7�! a(x)� � (g � f)(x;z)= wx;y�ex(x;x) 7�! a(x)� � f(x;y) � �ey (y;y) 7�! g(y;z)�� wy2y b(y) � �ey (y;y) 7�! g(y;z)�� c(z):(2�): by axiom (2�) for g we haveez(z;z) 7�! ez(z;z0) � g(y;z) 7�! g(y;z0) 8y 2 y;8z;z0 2 z:then for �xed x 2 x, y 2 y and z;z0 2 z we havef(x;y) � �ey (y;y) 7�! g(y;z)� � �ez(z;z) 7�! ez(z;z0)�� f(x;y) � �ey (y;y) 7�! g(y;z)� � �g(y;z) 7�! g(y;z0)�� f(x;y) � �ey (y;y) 7�! g(y;z0)�:now taking suprema by y 2 y on the both sides of the inequality we get:(g � f)(x;z) � �ez(z;z) 7�! ez(z;z0)� � (g � f)(x;z0):(3�) (we prove this axiom in the form (30�)): applying (30�) for f we have(g � f)(x;z) � �ex(x;x) 7�! ex(x;x0)�= wy f(x;y) � �ey (y;y) 7�! g(y;z)� � �ex(x;x) 7�! ex(x;x0)�� wy f(x0;y) � �ey (y;y) 7�! g(y;z)�= (g � f)(x0;z)(4�): we have to show that for all x 2 x, z;z0 2 z(g � f)(x;z) � �ex(x;x) 7�! (g � f)(x;z0)� � ez(z;z0):to establish this inequality we have to show that for any y;y0 2 y it holds:�f(x;y) � �ey (y;y) 7�! g(y;z)��ex(x;x) 7�! �f(x;y0) � �ey (y0;y0) 7�! g(y0;z0)�� ez(z;z0): fuzzy functions: a fuzzy extension of the category set and some related categories 119by divisibility of l, axiom (4�) for f and g and axiom (3�) for g, we have:�f(x;y) � �ey (y;y) 7�! g(y;z)��ex(x;x) 7�! �f(x;y0) � �ey (y0;y0) 7�! g(y0;z0)��= f(x;y) � �e(y;y) 7�! g(y;z)��e(x;x) 7�! e(x;x)��e(x;x) 7�! f(x;y0)� � �e(y0;y0) 7�! g(y0;z0)��= �f(x;y) � �ey (y;y) 7�! g(y;z)��e(x;x) 7�! f(x;y0)� � �e(y0;y0) 7�! g(y0;z0)�� f(x;y) � �ey (y;y) 7�! g(y;z)��f(x;y) 7�! ey (y;y0)� � �ey (y0;y0) 7�! g(y0;z0)�� ey (y;y0) � �e(y;y) 7�! g(y;z)� � �e(y0;y0) 7�! g(y0;z0)�� g(y0;z) � �e(y0;y0) 7�! g(y0;z0)�� e(z;z0):by a direct veri�cation it is easy to show that the operation of composition is as-sociative: given fuzzy functions f : (x;ex;a) ! (y;ey ;b), g : (y;ey ;b) !(z;ez;c), and h : (z;ez;c) ! (t;et ;d) it holds (h�g)�f = h�(g�f) :(x;ex;a) ! (t;et ;d). further, the identity morphism is de�ned by the cor-responding l-valued equality: ex : (x;ex;a) ! (x;ex;a). it is easy toverify that it satis�es the conditions (0�) (4�) above and that f � ex = exand ey � f = ey for each fuzzy function f : (x;ex;a) ! (y;ey ;b). thusl � fset(l) is indeed a category. �remark 3.5. in case when the equalities ex and ey on x and y respectively,are global, the condition (0�) becomes redundant and the conditions (1�) (4�)can be reformulated in the following simpler way:(1�) supx a(x) � f(x;y) � b(y) 8y 2 y ;(2�) f(x;y) � ey (y;y0) � f(x;y0) 8x 2 x;8y;y0 2 y ;(3�) ex(x;x0) � f(x;y) � f(x0;y) 8x;x0 2 x;8y 2 y ;(4�) f(x;y) � f(x;y0) � ey (y;y0) 8x 2 x;8y;y0 2 y .3.2. fuzzy category l�fset(l). given a fuzzy function f : (x;ex;a) !(y;ey ;b) let �(f) = infx supy f(x;y):thus we de�ne an l-subclass � of the class of all morphisms of l�fset(l).in case �(f) � � we refer to f as a fuzzy �-function. if f : (x;ex;a) !(y;ey ;b) and g : (y;ey ;b) ! (z;ez;c) are fuzzy functions, then �(g�f) ��(g) � �(f). indeed, let x 2 x and y 2 y be �xed. thensupz f(x;y) � �ey (y;y) 7�! g(y;z)� � f(x;y) � supz g(y;z) � f(x;y) � �(g);and therefore for a �xed x 2 xsupy supz f(x;y) � �ey (y;y) 7�! g(y;z)� � supy f(x;y) � �(g) � �(f) � �(g):since x 2 x is arbitrary, we get �(g � f) � �(g) � �(f). 120 ulrich h�ohle, hans-e. porst, alexander p. �sostakfurther, given an l-valued set (x;e) let!(x;e) := �(e) = infx e(x;x):thus a fuzzy category l �fset(l) = (l � fset(l);!;�) is obtained.remark 3.6. if f 0 : (x0;e0x) ! (y;ey ) is the restriction of f : (x;ex) !(y;ey ) (see remark 3.3 above) and �(f) � �, then �(f 0) � �. however,generally the restriction f 0 : (x0;ex0) ! (y 0;ey 0) of f : (x;ex) ! (y;ey )may fail to satisfy the condition �(f 0) � �.3.3. some (fuzzy) subcategories of the fuzzy category l � fset(l).for a �xed � let l�f�set(l) consist of all objects of l�fset(l) and itsfuzzy �-morphisms. in case � is idempotent, l �f�set(l) is a usual (crisp)category. in particular, it is a crisp category for � = >.if l1;l2;l3 � l, then by l1�fset(l2;l3) we denote the (fuzzy) subcate-gory of l�fset(l), whose objects (x;e;a) satisfy the conditions a(x) � l1and e(x � x) � l2, and whose morphisms satisfy the condition f(x �y ) � l3. by specifying the sets l1, l2 and l3 some known and new (fuzzy)categories related to l-sets can be characterized as (fuzzy) subcategories ofl1 � fset(l2;l3)-type or of l1 � fset 0(l2;l3)-type.4. elementary properties of fuzzy functions. special types offuzzy functions.4.1. images and preimages of l-sets under fuzzy functions. assumethat the gl-monoid (l;^;_;�) is equipped with an additional operation �which is distributive over arbitrary joins and meets and is dominated by �, i.e.(�1 � �1) � (�2 � �2) � (�1 � �1) � (�2 � �2). in particular, ^ can be taken as�. another option: in case when (l;^;_;�) is an mv -algebra, the originalconjunction � can be taken as �. given a fuzzy function f : (x;ex) ! (y;ey )and l-subsets a : x ! l and b : y ! l of x and y respectively, wede�ne the fuzzy set f!(a) : y ! l (the image of a under f) by the equalityf!(a)(y) = wx f(x;y)�a(x) and the fuzzy set f (b) : x ! l (the preimageof b under f) by the equality f (b)(x) = wy f(x;y) � b(y).note that if a 2 lx is extensional, then f!(a) 2 ly is extensional (by(2�)) and if b 2 ly is extensional, then f (b) 2 lx is extensional (by(30�)).proposition 4.1 (basic properties of images and preimages of l-sets underfuzzy functions).(1) f!(wi2i(ai)) = wi2i f!(ai) 8fai : i 2 ig � lx;(2) f!(a1 va2) � f!(a1)vf!(a2) 8a1;a2 2 lx;(3) f (vi2i(bi)) � vi2i f (bi) 8fbi : i 2 ig � ly .(30) in case l is completely distributive(î2i f (bi))5 � f (î2i(bi)) � î2i f (bi) 8fbi : i 2 ig � ly ;in particular, fuzzy functions: a fuzzy extension of the category set and some related categories 121(30̂ ) (vi2i f (bi))3 � f (vi2i(bi)) � vi2i f (bi) 8fbi : i 2 ig �lx, in case � = ^ and(30̂^) vi2i f (bi) = f (vi2i(bi)) 8fbi : i 2 ig � ly , in case � = � =^;(4) f (wi2i(bi)) = wi2i f (bi) 8fbi : i 2 ig � ly ;(50�) in case l is completely distributive and � = �, f (f (b)) � b.proof. (1): �wi f!(ai)�(y) = wi wx�f(x;y) � ai(x)�= wx wi�f(x;y) � ai(x)�= wx(f(x;y) � (wi ai)(x))= f!(_iai)(y):the validity of (2) follows from the monotonicity of f .to prove (3) notice that(vi f (bi))(x) = vi�wy f(x;y) � bi(y)�� wy(vi(f(x;y) � bi(y))� wy(f(x;y) � (vi bi(y)))= f (vi bi)(x):assume now that l is completely distributive. recall that complete distribu-tivity of a lattice l means that the way-below relation � in l is approximative(i.e. � = wf� 2 l : � � �g for every � 2 l) and every element � is asupremum of coprimes way-below � (see e.g. [3]). let( î f (bi))(x) = î _y f(x;y) � bi(y) := �:then 8� � �;8i 2 i;9yi 2 y such that f(x;yi) � bi(yi) � �:in particular, this means that f(x;yi) � � for every i 2 i. we �x some i0 2 iand let yi0 := y0. further, notice that by remark 3.2�2 � f(x;yi) � f(x;y0) � e(yi;yi) 7�! e(yi;y0);and hence for every i 2 i[f(x;yi) � bi(yi)] � �4 � �f(x;yi) � (ey (yi;yi) 7�! ey (yi;y0))��bi(yi) � (ey (yi;yi) 7�! ey (yi;y0))�� f(x;y0) � bi(x;y0):therefore �5 � vi�f(x;yi) � bi(yi)� � �4� vi�f(x;y0) � bi(y0)�= f(x;y0) � vi bi(y0)� f (vi bi)(x)and, since this holds for any � � �, by complete distributivity we obtainf (vi bi)(x) � �5 and hence(î2i f (bi))5 � f (î2i(bi)): 122 ulrich h�ohle, hans-e. porst, alexander p. �sostakin case � = ^ in the above proof it is su�cient to multiply by �2 instead of�4, and therefore the resulting inequality is(î2i f (bi))3 � f (î2i(bi)):finally, in case � = � = ^ by idempotency we get the equalityî2i f (bi) = f (î2i(bi)):the proof of (4) is similar to the proof of (1) and is therefore omitted.to prove (5) assume that f (f (b))(y0) � �, for some y0 2 y;� 2 l, thenfor each � � � there exist x0;y1 2 y such that f(x0;y0)�f(x0;y1)�b(y1) � �.therefore, by extensionality of b:b(y0) � (e(y1;y1) 7�! e(y1;y0)) � b(y1)� f(x0;y0) � f(x0;y1) � b(y1)� �;and hence, since l is completely distributive, it follows thatb(y0) � f (f (b))(y0)and thus b � f (f (b)). �4.2. injectivity, surjectivity and bijectivity of fuzzy functions. a fuzzyfunction f : (x;ex;a) ! (y;ey ;b) is called injective, if(inj) f(x;y) � (ey (y;y) 7�! f(x0;y)) � ex(x;x0) 8x;x0 2 x;8y 2 y .notice that injective fuzzy functions satisfy the following condition(inj#) f(x;y) � f(x0;y) � (ex(x;x) _ ex(x0;x0)) 7�! ex(x;x0) 8x;x0 2x;8y 2 y .indeed, f(x;y) � f(x0;y) � f(x;y) � (e(y;y) 7�! f(x0;y))� e(x;x0)� (e(x;x) 7�! e(x;x0)):notice, that in case when ey is global, then (inj) just means that f(x;y) �f(x0;y) � ex(x;x0).a fuzzy function f : (x;ex;a) ! (y;ey ;b) is called �-surjective if itsatis�es the following two conditions:(sur1�) infy supx f(x;y) � �(sur2) f!(a) � b � �.in case f is injective and �-surjective, it is called �-bijective.remark 4.2. notice that in case a = >x the second condition in the de�nitionof �-surjectivity (for any b 2 ly , in particular, for b = >y ) follows from the�rst one. moreover, in case a = >x, b = >y and if > acts as a unit withrespect to �, the both conditions become equivalent. fuzzy functions: a fuzzy extension of the category set and some related categories 123remark 4.3. let (x;ex);(y;ey ) be l-valued sets and (x0;ex0), (y 0;ey 0)be their subspaces. obviously, the restriction f 0 : (x0;ex0) ! (y 0;ey 0) of aninjection f : (x;ex) ! (y;ey ) is an injection. the restriction f 0 : (x;ex) !(y 0;ey 0) of an �-surjection f : (x;ex) ! (y;ey ) is an �-surjection. onthe other hand, generally the restriction f 0 : (x0;ex0) ! (y 0;ey 0) of an �-surjection f : (x;ex) ! (y;ey ) may fail to be an �-surjection.a fuzzy function f : (x;ex;a) ! (y;ey ;b) de�nes a fuzzy relation f�1 :(y;ey ;b) ! (x;ex;a) by setting f�1(y;x) = f(x;y) 8x 2 x;8y 2 y .proposition 4.4 (basic properties of injections, �-surjections and �-biject-ions).(1) f�1 is a fuzzy function i� f is injective (actually f�1 satis�es (4�) i�f satis�es (inj))(2) f is �-bijective i� f�1 is �-bijective.(3) if l is completely distributive and f satis�es (inj#), then( î f!(ai))5 � f!( î ai) � î f!(ai) 8fai : i 2 ig � lx:in particular,(3^) (vi f!(ai))3 � f!(vi ai) � vi f!(ai) if � = ^ and;(3^̂) f!(vi ai) = vi f!(ai) in case � = ^ = �;(4) if f is >-surjective, then f (f (b)) � b 8b 2 ly ; and hence,in particular, f (f (b)) = b in case � = � and l is completelydistributive.proof. the validity of (1) and (2) is obvious.to show (3) �x y 2 y and let (^if!(ai))(y) � �. then for each coprime� � � and each 2 i one can �nd xi 2 x such that f(xi;y) � ai(xi) � �and hence, in particular, f(xi;y) � �. we �x some i0 and denote xi0 := x0,ai0;= a0. by (inj#) it is easy to conclude that ex(xi;xi) 7�! ex(x0;xi) � �2.now, by extensionality of all ai we get�5 � vi�(f(xi;y) � ai(xi)) � �4�� (f(x0;y) � a0(x0)) ^ �vi6=i0�(f(xi;y) � �2� � (ai(xi) � �2)�� (f(x0;y) � a0(x0))^�vi6=i0 �f(xi;y) � (e(xi;xi) 7�! e(xi;x0))��ai(xi) � (e(xi;xi) 7�! e(xi;x0))�� (f(x0;y) � a0(x0)) ^ �vi6=i0�(f(x0;y)� � ai(x0)�= vi f(x0;y) � ai(x0)= f(x0;y) � vi ai(x0)� f�^iai�(y):since this holds for any � � � and l is completely distributive we get( î f!(ai)))5 � f!( î ai) � î f!(ai): 124 ulrich h�ohle, hans-e. porst, alexander p. �sostakto show (4) let b(y0) � �. thenf (f (b))(y0) = wx�f(x;y0) � �f (b)�(x)= wx wy�f(x;y0) � f(x;y) � b(y)�� wx�f(x;y0) � f(x;y0) � b(y0)�:now, by >-surjectivity of f we complete the proof noticing thatf (f (b))(y0) � > �> � b(y0) � b(y0): �proposition 4.5. let f : x � y ! l be a fuzzy function and �(f) � �.then for each coprime � � � there exists z � y such that the restrictiong := f jx�z: x � z ! l is a �-surjection and �(g) � �.proof. given coprime � � �, let z := fy j 9x 2 x such that (x;y) � �g, andlet g := f : x � z ! l be the restriction of f to x � z.to show that �(g) � � assume that, contrary, infx supy2z f(x;y) = �(g) 6��. then there would exist x0 2 x such that f(x;y) 6� � for each y 2 z.on the other hand, from �(f) � � � �, it follows that for each x 2 x, inparticular, for x0 there exists y0 2 y such that f(x0;y0) � �. besides, byde�nition of z it is clear that y0 2 z. the obtained contradiction implies that�(g) � �.to show that g is �-surjective, assume that infy2z supx2x g(x;y) 6� �.it follows from here that there exists y0 2 z such that supx2x g(x;y0) =supx2x f(x;y0) 6� �. however, this contradicts the de�nition of z. thus the�rst condition of the de�nition of �-surjectivity holds. to conclude the proofit is su�cient to apply remark 4.2. �problem 4.6. is it true (at least in the case of a completely distributive latticel), that given a fuzzy function f : x � y ! l where �(f) � � there existsz � y such that the restriction g := f jx�z: x � z ! l is an �-surjectionand �(g) � �?5. constructions in the fuzzy category l � fset(l)5.1. products. let l�fset}(l) be the subcategory of l�fset(l) havingthe same potential objects as l�fset(l) and only such potential morphismsf : x � y ! l from l � fset(l) which satisfy the following additionalcondition (a certain counterpart of the axiom of strictness and the weaken formof the axiom of preservation of equalities; see e.g. [6]):(}) f(x;y) 6= 0 =) e(x;x) = e(y;y).let y = f(yi;ei;bi) : i 2 ig be a family of l-valued sets, y0 = f(yi)i2i 2�iyi j ei(yi;yi) = ej(yj;yj)8i;j 2 ig, let b0 be the restriction of b = qi2i bito y0, and let e(y;y0) = vi ei(yi;y0i)8y = (yi);y0 = (y0i) 2 y . further, let�i : y0 ! yi be the restriction of the projection pi : qi yi ! yi to y0.the pair (y;e) thus de�ned is the product of the family y in the category fuzzy functions: a fuzzy extension of the category set and some related categories 125l � fset}(l). indeed let fi : (x;ex;a) ! (yi;eyi;bi), i 2 i, be a fam-ily of fuzzy functions in l � fset}(l) and let f := �ifi : (x;ex;a) !(y0;ey ;b), be de�ned by f(x;y) = vi fi(x;yi). then f is a fuzzy function.indeed, the validity of (0�), (1�), (3�) and (4�) is easy to verify directly apply-ing the corresponding axiom for all fi, while the validity of (2�) is guaranteedby the condition (}) for all fi; i 2 i. besides, it is clear that fi = �i � f andthat �(f) = vi �(fi). thus, (y0;ey ;b) is indeed the product of the family(yi;eyi;bi) in l � fset}(l). notice, that the condition } obviously holdsfor the subcategory l � fset 0(l) of l � fset(l). moreover, if all (yi;ei)are taken from l � fset 0(l), then y0 = qi yi.5.2. coproducts. let x = f(xi;ai;ei) : i 2 ig be a family of l-valued sets,let x0 = sxi be the disjoint sum of sets xi and let a0 2 lx be de�ned bya0(x) = ai(x) whenever x 2 xi. further, let qi : xi ! x0 be the inclusionmap. we introduce the l-equality on x0 by setting e(x;x0) = ei(x;x0) if(x;x0) 2 xi � xi for some i 2 i and e(x;x0) = 0 otherwise (cf [6]). then(x0;a0;e) is the coproduct of x in l � fset(l) (and hence also in l �fset}(l)).indeed, let fi : (xi;ai;ei) ! (y;b;ey ), i 2 i, be a family of fuzzy functionsin l � fset(l) and let f := �ifi : �(xi;ai;ei) ! y;b;ey ) be de�ned byf(x;y) = fi(xi;y) whenever x = xi 2 xi. then the direct veri�cation showsthat f is a fuzzy function, fi = f � qi and �(f) = ^i�(fi).theorem 5.1 (factorization of a family of �-morphisms). letfi : (x;e;a) ! (yi;ei;bi)be a family of fuzzy �-functions in l � fset}(l). then for every � � �there exists a fuzzy �-surjective �-function g : (x;e;a) ! (z;ez;c) and afamily of usual functions �i : (z;c;ez) ! (yi;bi;ei) separating points suchthat fi = g � �i for every i 2 i.proof. indeed, let (y;ey ) = qi2i(yi;ei) be the product in l � fset}(l)and let f = 4i2ifi : x � qi2i yi ! l. further, given � � �, let z � y andg : x�z ! l have the same meaning as in proposition 4.1 and let c := g(a).thus, by proposition 4.1 we conclude that g : (x;a;ex) ! (z;c;ez) is a �-surjective fuzzy function and �(g) � �. to complete the proof it is su�cient tonotice that the mappings �i : z ! yi de�ned as the restrictions of projectionspi : y ! yi separate points of z and that fi = �i � g. �6. fuzzy categories related to algebra and topology with fuzzyfunctions as morphisms.on the basis of l�fset(l) some fuzzy categories related to topology andalgebra can be naturally de�ned. here are three examples:de�nition 6.1 (fuzzy category ftop(l)). let (x;ex) be an l-valued setand let �x � lx be the (chang-goguen) l-topology on x, [2], [4], [5]; see also[9]. a fuzzy function f : (x;ex;�x) ! (y;ey ;�y ) is called continuous if 126 ulrich h�ohle, hans-e. porst, alexander p. �sostakf(v ) 2 �x for all v 2 �y . l-topological spaces and continuous fuzzy mappingsbetween them form the fuzzy category ftop(l).de�nition 6.2 (fuzzy category fftop(l)). let (x;ex) be an l-valued setand let tx : lx ! l be the l-fuzzy topology on x, [16], [9]. a fuzzy functionf : (x;ex;tx) ! (y;ey ;ty ) is called continuous if tx(f(v )) � ty (v ) forall v 2 ly . l-fuzzy topological spaces and continuous fuzzy mappings betweenthem form the fuzzy category fftop(l).de�nition 6.3 (a fuzzy category l �fgr(l)). let x be a group and ex bean l-valued equality on x such that ex(x �y;x0 �y0) � ex(x;x0)�ex(y;y0) forall x;x0;y;y0 2 x. further, let gx : x ! l be an (extensional) l-subgroupof x (see e.g. [10], [13]). a fuzzy function f : (x;ex;gx) ! (y;ey ;gy )is called a fuzzy homomorphism if f(x � x0;y � y0) � f(x;y) � f(x0;y0) for allx;x0 2 x, y;y0 2 y . l-subgroups of groups endowed with l-valued equalities,and fuzzy homomorphisms between them form a fuzzy category l � fgr(l).these and some other fuzzy categories with fuzzy functions in the role offuzzy morphisms will be studied elsewhere.acknowledgements. the authors are thankful to tomasz kubiak (universityof poznan, poznan, poland) for reading the manuscript carefully and makingvaluable comments. references[1] g. birkho�, lattice theory, 3rd ed., ams providence, ri, 1967.[2] c.l. chang, fuzzy topological spaces, j. math. anal. appl., 24 (1968), 182-190.[3] g. gierz, k.h. hofmann, k. keimel, j.d. lawson, m. mislove, d.s. scott, \a com-pendium of continuous lattices". springer verlag, berlin, heidelberg, new york, 1980.[4] j.a. goguen, l-fuzzy sets, j. math. anal. appl., 18 (1967) 145-174.[5] j.a. goguen, the fuzzy tychono� theorem, j. math. anal. appl., 43 (1973) 737-742.[6] u.h�ohle, commutative, residuated l-monoids, in: non-classical logics and their ap-plications to fuzzy subsets: a handbook of the mathematical foundations of fuzzysubsets, e.p. klement and u. h�ohle eds., kluwer acad. publ., 1994, 53-106.[7] u.h�ohle, m-valued sets and sheaves over integral commutative cl-monoids, in: applica-tions of category theory to fuzzy sets., s.e. rodabaugh, e.p. klement and u. h�ohleeds., kluwer acad. publ., 1992, pp. 33-72.[8] u.h�ohle, l. stout, foundations of fuzzy sets, fuzzy sets and syst., 40 (1991), 257-296.[9] u. h�ohle, a. �sostak, axiomatics of �xed-basis fuzzy topologies, in: mathematics of fuzzysets: logic, topology and measure theory , u. h�ohle, s.e. rodabaugh eds. handbookseries, vol.3. kluwer academic publisher, dordrecht, boston. -1999. pp. 123 273.[10] c.v. negoita, d.a. ralescu, \application of fuzzy sets to system analysis", john wiley& sons, new york, 1975.[11] a. pultr, fuzzy mappings and fuzzy sets, comm. math. univ. carolinae, 17 (1976),441-459.[12] a. pultr, closed categories of fuzzy sets, vortr�age zur automaten un algorithmentheorie16 (1976), 60-68.[13] a. rosenfeld, fuzzy groups, j.math.anal.appl., 35 (1971), 512-517.[14] a. �sostak, on a concept of a fuzzy category, in: 14th linz seminar on fuzzy set theory:non-classical logics and applications. linz, austria, 1992, pp. 62-66. fuzzy functions: a fuzzy extension of the category set and some related categories 127[15] a. �sostak, fuzzy categories versus categories of fuzzily structured sets: elements of thetheory of fuzzy categories, in: mathematik-arbeitspapiere, universit�at bremen, vol 48(1997), pp. 407-437.[16] a. �sostak, on a fuzzy topological structure, suppl. rend. circ. matem. palermo, ser ii,,11 (1985), 89-103.[17] a. �sostak, two decades of fuzzy topology: basic ideas, notions and results, russian math.surveys, 44: 6 (1989), 125-186.[18] a. �sostak, towards the concept of a fuzzy category, acta univ. latviensis (ser. math.),562, (1991), 85-94. received march 2000 ulrich h�ohlebergische universit�atd-42097, wuppertalgermanye-mail address: ulrich.hoehle@math.uni-wuppertal.de hans-e. porstuniversity of bremend-28334, bremengermanye-mail address: porst@math.uni-bremen.de alexander p. �sostakuniversity of latvialv-1586, rigalatviae-mail address: sostaks@com.latnet.lv () @ appl. gen. topol. 16, no. 1(2015), 31-36doi:10.4995/agt.2015.2305 c© agt, upv, 2015 on fixed-point free selections and multivalued maps on r dennis k. burke a and raushan z. buzyakova b a department of mathematics, miami university, oxford, oh, u.s.a. (burkedk@miamioh.edu) b u.s.a. (raushan buzyakova@yahoo.com) abstract we study fixed-point free multivalued selfmaps on r as well as continuous fixed-point free selection of multivalued selfmaps on r. we justify necessary conditions in some earlier results and motivate some new interesting questions. 2010 msc: 54h25; 58c30; 54b20. keywords: fixed point; hyperspace; multivalued function. 1. introduction one of the classical theorems of de bruijn-erdös [2] states that if f is a fixedpoint free multivalued selfmap on a set x and the sizes of f(x)’s are bounded by a fixed natural number n, then x can be covered by finitely many sets each of which misses its image. a more precise statement is as follows: let p(x) be the set of all non-empty subsets of x and let f : x → p(x) be a map with the property that x 6∈ f(x). if there exists a natural number k such that |f(x)| ≤ k for all x ∈ x, then there exists a finite cover f of x such that f misses ⋃ {f(x) : x ∈ f} for each f ∈ f. the sets fs in the above statement are called colors of f and the family f a coloring of f. in the context of a map (not necessarily continuous) on a topological space, a color of the map is traditionally a closed subset of the space. to be more precise, given a multivalued selfmap f on a topological space x, a closed f ⊂ x is a color of f if f(x) misses f for each x ∈ f . a finite cover of x by colors of f is called a coloring of f. if a coloring of f exists, f is called received 14 april 2014 – accepted 22 september 2014 http://dx.doi.org/10.4995/agt.2015.2305 d. k. burke and r. z. buzyakova colorable. the condition of closed-ness of a color in the topological context is justified by the fact that for a space x from a sufficiently wide class, colorability of a continuous selfmap f on x by closed colors is equivalent to the existence of a continuous fixed-point free extension of f to some compactifcation of x. the notion of colorings for continuous single-valued selfmaps was introduced in [1]. a nice review of results about this topic for single-valued maps is given in [6]. it is proved in [3] that a fixed-point free continuous selfmap on r is colorable if the size of f(x) is at most n, where n is a fixed positive number determined by f. this result is a natural topologization of the mentioned de bruijn-erdös theorem. in the first part of this paper we will consider a few routes to extend this result to certain fixed-point free multivalued selfmaps on r with compact images. for this mission, it is natural to consider the following restrictions on f: (1) there exists m > 0 such that sup{d(x, y) : y ∈ f(x)} < m for all x; and/or (2) there exists m > 0 such that d(x, f(x)) > m for all x. we will show that (1) alone or (2) alone leads to an example. at the same time if a multivalued selfmap on r (not necessarily continuous) satisfies both (1) and (2), then it is colorable (theorem 2.1). it is clear that if f has a fixed point then it is not colorable. if, however, f is multivalued, it may have a fixed-point free continuous selection. it is shown in [4] that a continuous fixed-point free map on rn is always colorable. this observation inspired the second part of the paper, in which we discuss possible conditions that guarantee the existence of continuous fixed-point free selections of multivalued selfmaps on r. by exp r we denote the space of all non-empty closed subsets of r endowed with the vietoris topology. by expk(r) we denote the space of all non-empty compact subsets of r with the topology inherited from exp r. a standard neighborhood in exp r is in form 〈u1, ..., um〉 = {a ∈ exp r : a ⊂ u1∪...∪um and ui∩a 6= ∅ for all i = 1, ..., m}, where u1, ..., um are open sets of r. throughout the paper when we say that f is a multivalued selfmap on r, we mean that f is a map from r to exp r . we reserve the letter ”d” to measure distances in r. in notations and terminology, we will follow [5]. 2. continuous fixed-point free multivalued selfmaps on r in this section, we will consider mulitivalued fixed-point free selfmaps on r with compact images. our goal is to find a large subclass of such maps whose members are colorable. it is natural to assume that a more ”volatile” graph is less likely to be colorable while a fixed-point fee map with a ”more orderly conduct” should naturally be colorable. to continue our discussion, it is appropriate at this point to define how exactly we measure the volatility c© agt, upv, 2015 appl. gen. topol. 16, no. 1 32 on fixed-point free selections and multivalued maps on r of a map f. for this purpose, we introduce the following function-determined constants: sq(f) = inf{d(x, y) : x ∈ r, y ∈ f(x)}, st (f) = sup{d(x, y) : x ∈ r, y ∈ f(x)}. one can view sq(f) as the squeeze measure of f, which is, simply the greatest lower bound of the distances between points and their images. similarly, the constant st (f) can be viewed as a stretch measure. the authors think of a fixed-point free map f as being volatile if its squeeze measure sq(f) is 0 or its stretch measure st (f) is ∞. thus, we may think of f is not volatile if both sq(f) and st (f) are positive real numbers. in this section we will show that any non-volatile fixed-point free (not necessarily continuous) multivalued map on r is colorable. we will also show that for either of the two measures of volatility, there exists an example of a non-colorable continuous fixed-point free multivalued selfmap on r that obeys the given measure and violates the other. we start with our affirmative result. theorem 2.1. let f : r → exp(r) be a map and m, m positive numbers. suppose that m ≤ d(x, y) ≤ m for every x ∈ r and y ∈ f(x). then f is colorable. proof. put i = {[m 2 · n, m 2 · (n + 1)] : n is an integer}. clearly i is a cover of r. let k be an integer such that k · m 2 > m. we define our colors {ai : i = 0, ..., k} as follows. ai is the union of all elements of i that can be obtained by shifting [ m 2 ·i, m 2 ·i+ m 2 ] a multiple of k + 1 units to the right or to the left. that is, ai is the union of the largest subfamily of i that contains [m 2 · i, m 2 · i + m 2 ] and has neighbors at distance exactly m 2 · k from each other. it is clear that each ai is closed and {ai : i = 0, ..., k} is a cover of r. let us show that ai is a color. note that ai is the union of a disjoint collection of segments of length m/2. since d(x, f(x)) ≥ m for each x, we conclude that the image of each segment i of this collection misses i. also, since the distance between any two distinct segments i and j of this collection is m 2 · k > m, the image of i misses j. therefore, the image of ai misses ai. hence, ai is a color. � recall that given a colorable multivalued map f on a topological space, the chromatic number of f is defined as the smallest number of colors necessary to color f. observe that the argument of theorem 2.1 implies that the chromatic numbers does not exceed ⌈2m m ⌉ + 1. we next show that either part (alone) of the inequality in the hypothesis of theorem 2.1 is not sufficient for the desired conclusion. we first describe an example that shows that positiveness of sq alone does not guarantee colorability. example 2.2. there exists a continuous fixed-point free non-colorable map from r to expk(r) such that sq(f) > 0, st (f) = ∞, and the image of each point is a segment. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 33 d. k. burke and r. z. buzyakova construction. for each x ≤ −1, put f(x) = [x + 1/2, 0]. for each x ≥ −1, put f(x) = [x + 1/2, x + 1]. this map is clearly continuous, fixed-point free, sq(f) = 1/2, and st (f) = ∞. let us show that f is not colorable. since n ∈ f(m) for every pair of negative integers n, m with n > m, we conclude that n and m have different colors. therefore, −1, −2, −3, ... must have mutually distinct colors. � next, we will show that finiteness of st alone is not sufficient for colorability. lemma 2.3 and example 2.4 can serve as a manual for constructing examples with the desired properties. we will then demonstrate the manual at work. we start with the following technical statement. lemma 2.3. let a map f : r → exp(r), a ∈ r, and n ∈ ω \ {0, 1} satisfy the following requirement: f(a + m n ) = [a + m + 1 n , a + 1 + m + 1 n ] for each m = 0, ..., n − 1 then f cannot be colored by fewer than n colors. proof. if f is not colorable, then we are done. otherwise, fix a coloring of f. if m > k, then a + m n ≥ a + k+1 n . if m ≤ n − 1 then a + m n ≤ a + 1 + 1 n . therefore, (p) a + m n ∈ f(a + k n ) for k < m ≤ n − 1 property (p) implies that a, a+ 1 n , ..., a+ (n−1)+1 n must have mutually distinct colors. � we are now ready to construct an example showing that finiteness of sq alone is not sufficient for colorability of a fixed-point free continuous map from r to expk(r). example 2.4. there exists a fixed-point free non-colorable continuous map from r into expk(r) such that st (f) is finite, sq(f) = 0, and the image of each point is a unit segment. construction. we state the next claim for further reference and it is a direct consequence of continuity of a composition of continuous functions. claim. let h : r → r and f : r → exp(r) be maps such that f(x) = [h(x), h(x) + 1] for each x ∈ r. if h is continuous, then f is continuous too. fix a sequence of real numbers a2, ..., an, ... such that (a) an+1 − an > 5, for each n for each n = 2, 3... put (b) g(an) = an + 1/n, g(an + 1 n ) = an + 2 n , ..., g(an + n−1 n ) = an + 1 observe that x < g(x) for all x in {an + m/n : m = 0, ..., n − 1}, where n ∈ ω \ {0, 1}. moreover, (c) g(an), g(an + 1 n ), ..., g(an + (n−1) n ) fall in [an, an + 5) ⊂ [an, an+1). c© agt, upv, 2015 appl. gen. topol. 16, no. 1 34 on fixed-point free selections and multivalued maps on r now applying (a), we conclude that x < g(x) for all x in s = {an + m n : n = 2, 3... and m = 0, 1, ..., n − 1}. since s is closed, there exists a continuous extension h : r → r of g such that x < h(x) for all x ∈ r. define f : r → expk(r) by letting f(x) = [h(x), h(x) + 1]. by claim, f is continuous. by (b) and lemma 2.3, f is not colorable. since x < h(x) for all x in the domain, f is fixed-point free. � one can use the argument of the example and the accompanying lemma to create well-defined maps with the desired properties. in particular, the referee of this paper suggested the map f : r → expk(r) defined as follows: f(x) = [x + 1 x , x + 1 x + 1] if x ≥ 1 and f(x) = [x + 1, x + 2] otherwise. following the argument of our example and the lemma, one can see that this map has the desired properties. we have now shown that there are examples of non-colorable continuous maps from r to expk r that have positive squeeze component and ones that have finite stretch component. 3. continuous fixed-point free selections we start this section with a simple observation that motivates this section’s discussion. proposition 3.1. a continuous map f : r → expk(r) has a fixed-point free continuous selection if and only if {x ∈ r : x = max f(x)} or {x ∈ r : x = min f(x)} is empty. proof. to prove necessity, let sf be a continuous fixed-point free selection. then the graph of sf is completely above the diagonal or completely below the diagonal in r2. this means that sf (x) > x for all x or sf (x) < x for all x. assume that sf (x) > x for all x. then x < max f(x) for all x. hence {x ∈ r : x = max f(x)} = ∅. similarly, {x ∈ r : x = min f(x)} = ∅ if sf (x) < x for all x. to prove sufficiency, assume that {x ∈ r : x = max f(x)} = ∅. the map sf defined by letting sf (x) = max f(x) is then fixed-point free and is known to be continuous. � an argument similar to the one in proposition 3.1 implies the following statement. proposition 3.2. let x be a closed subset of r and f : x → expk(x) be continuous. then f has a fixed-point free continuous selection iff the sets {x ∈ r : x = max f(x)} and {x ∈ r : x = min f(x)} are separated by clopen neighborhoods. note that the statement of proposition 3.1 is no longer true if we replace expk(r) by exp(r). this is shown in the next example. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 35 d. k. burke and r. z. buzyakova example 3.3. there exists a continuous map f : r → exp(r) such that the set {x ∈ r : x = max f(x)} is empty but every continuous selection of f has a fixed point. construction. for n ∈ n, let fn = ((−∞, n]) × {n}) ∪ {(x, 2n − x) : n ≤ x ≤ 2n} ∪ [2n, ∞) × {0} and e = ⋃ n∈n fn. define f : r → exp(r) by f(x) = {y : (x, y) ∈ e} = ex (vertical slice at x.) the graph of any continuous selection would have to ”follow” some fn and hence would have n as a fixed point. � our observation prompts the following natural questions. question 3.4. does proposition 3.1 hold of we replace expk(r) by {a ∈ exp(r) : a is connected} ? in general, it would be interesting to indentify conditions under which a continuous multivalued map on rn with compact images has a fixed-point free continuous selection. it would be also interesting to consider the n-dim versions of proposition 3.2 and question 3.4. in addition to continuity we can require our selection to have some other natural properties. question 3.5. suppose that f is a continuous multivalued map on r such that f(x) is a closed non-trivial segment and f has a continuous fixed-point free selection. does f have a differentiable fixed-point free selection? does f have a continuous fixed-point free selection differentiable at at least one point? what if we replace r with rn and ”closed segment” with ”closed ball”? note that ”non-triviality” of images is important since any continuous nondifferentiable (or nowhere differentiable) map with graph strictly above the line y = x would give a trivial example. acknowledgements. the authors would like to thank the referee for many valuable remarks, corrections, and references. references [1] j. m. aarts, r. j. fokkink and j. vremeer, variations on a theorem of lusternik and schnirelmann, topology 35 (1996), 1051–1056. [2] n. g. de bruijn and p. erdös, a color problem for infinite graphs and a problem in the theory of relations, indagationes math. 13 (1951), 371–373. [3] r. buzyakova, on multivalued fixed-point free maps on rn, proc. amer. math. soc. 140 (2012), 2929–2936. [4] r. buzyakova and a. chigogidze, fixed-point free maps of euclidean spaces, fundamenta mathematicae, 212 (2011), 1–16 [5] r. engelking, general topology, pwn, warszawa, 1977. [6] j. van mill, the infinite-dimensional topology of function spaces, elsevier, amsterdam, 2001. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 36 sequeiraagt.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 2, 2004 pp. 265-273 two transfinite chains of separation conditions between t1 and t2 lúıs sequeira abstract. two new families of separation conditions have arisen in the study of the impact that the algebraic properties of topological algebras have on the topologies that may occur on their underlying spaces. we describe the relative strengths of these families of separation conditions for general spaces. keywords: separation axioms 2000 ams classification: 54d10 1. introduction the separation conditions, or axioms, t0, t1 and t2 are very well known, as is the fact that the implications t2 =⇒ t1 =⇒ t0 hold for any topological space. another important separation condition, called sobriety, is known to be stronger than t0, weaker than t2 and independent of t1. john coleman [1], motivated by the study of topologies occurring in some topological algebras , defined new separation conditions called j-step hausdorffness for each j ≥ 1 (hj for short). the relative strengths of the ti conditions and the hj conditions are indicated by t2 ⇐⇒ h1 =⇒ h2 =⇒ h3 =⇒ h4 · · · =⇒ t1 =⇒ t0 where none of the unidirectional arrows are reversible. in [2], keith kearnes and the present author, while extending and clarifying some of the results of [1], introduced symmetrized versions of coleman’s hj conditions, which were labeled shj. in the cited papers, these conditions have been defined for any natural number j, and their occurrence in the underlying spaces of topological algebras with some prescribed algebraic properties was the object of study. 266 lúıs sequeira here, we consider these separation conditions for topological spaces in general, and describe their strengths, relative to each other and to the well known conditions of sobriety and the ti axioms. also, following paolo lipparini’s suggestion, we allow the index j to range over all ordinals, rather than just the natural numbers. 2. preliminaries a topological space x is t0 if whenever a and b are distinct points of x there is a closed subset of x containing one of the points that does not contain the other. x is t1 if for each a ∈ x the singleton set {a} is closed. x is t2, or hausdorff, if for each a ∈ x the intersection of the closures of the neighborhoods of a is {a}. this nonstandard definition of t2 suggests the following generalization: definition 2.1. let a be a topological space. for each a ∈ a and each ordinal α we define a subset ∆aα (a) of a — also denoted by ∆α(a) if there is no cause for ambiguity — recursively by ∆0(a) = a ∆β+1(a) = { b | ∀ open u, v with a ∈ u, b ∈ v, u ∩ v ∩ ∆β(a) 6= ∅ } ∆γ(a) = ⋂ β<γ ∆β(a) (if γ is a limit ordinal) we say that a point a ∈ a is α-step hausdorff if ∆α(a) = {a}. we say that a space is α-step hausdorff, or hα, if each of its points is α-step hausdorff. this definition implies that ∆1(a) is the intersection of the closures of the neighborhoods of a. thus ∆1(a) is a closed subspace of a containing a. each ∆β+1(a) is the intersection of the closures of neighborhoods of a in the subspace ∆β(a) under the relative topology (i. e., ∆β+1(a) coincides with ∆ ∆β(a) 1 (a)). in particular, ∆α(a) is closed in a for all a and α. clearly, a space is h1 if and only if it is hausdorff since both properties say exactly that ∆1(a) = {a} for all a ∈ a. since each ∆α(a) is closed, and since hα asserts that ∆α(a) = { a } for all a ∈ a, it follows that hα =⇒ t1. definition 2.2. for each ordinal α, we let the symbol ∆α also denote the binary relation defined by a ∆α b ⇐⇒ a ∈ ∆α(b) two elements a, b of a topological space are sometimes called unseparable if they cannot be separated by open sets: thus a and b are unseparable if and only if a ∆1 b. we will say a is α-unseparable from b if a ∆α b. one should note, though, that the relation ∆α need not be symmetric, except of course when k = 0 (since ∆0 = a × a) and when k = 1 (since ∆1 is the closure of the diagonal of a × a). chains of separation conditions 267 we will henceforth adopt the following equivalent definition of α-step hausdorffness. definition 2.3. let a be a topological space. for each ordinal α, we will say that a is α-step hausdorff, or hα, if the following condition holds for all a, b ∈ a: a ∆α b =⇒ a = b (hα) in other words, hα is the assertion that ∆α is the equality relation. a new family of separation conditions, related to the hα’s and labeled shα, is defined as follows: definition 2.4 ([2]). let a be a topological space, α an ordinal. a is said to be α-step hausdorff up to symmetry, or shα, if the following condition holds for all a, b ∈ a: a ∆α b ∧ b ∆α a =⇒ a = b (shα) thus shα asserts that ∆α is antisymmetric. the following lemma was present in [2], although only for finite ordinals α. lemma 2.5. for each ordinal α, every shα space is t1. proof. just note that (i) (shα =⇒ t0): if a 6= b either ∆α(a) is a closed set containing a and not b or ∆α(b) is a closed set containing b and not a. (ii) (t0 ∧ ¬t1 =⇒ ¬ shα): a t0 space x that fails to be t1 has a subspace { a, b } with induced topology { ∅, { a }, { a, b } }. for these a and b we have a ∆α b and b ∆α a for all α, thus x fails to satisfy shα for any α. � since each shα condition is formally weaker than the corresponding hα, the relative strengths of these conditions may be described by the following diagram. t2 ⇔ h1 ⇒ h2 ⇒ . . . ⇒ hα ⇒ . . . m ⇓ ⇓ sh1 ⇒ sh2 ⇒ . . . ⇒ shα ⇒ . . . ⇒ t1 ⇒ t0 3. every sh space is t1 and sober in this section we discuss the relation between the separation conditions introduced in the last section and another, well-known, condition called sobriety: definition 3.1. (i) a topological space x is said to be irreducible if it contains no two disjoint nonempty open sets. a subset f of a topological space x is said to be irreducible if it is irreducible as a subspace (i. e., under the induced topology). 268 lúıs sequeira (ii) a topological space x is called sober if every nonempty closed irreducible subset f of x is the closure of a unique point. remark 3.2. (1) the closure of any point is always (closed and) irreducible: if f = cl(a), u, v are open and u ∩ f 6= ∅ 6= v ∩ f , then a ∈ u ∩ v ∩ f . (2) it follows immediately from the definitions that a t1 space is sober if and only if every nonempty irreducible set is a singleton. the following examples attest to the well known fact that sobriety is independent of the t1 axiom: example 3.3. (1) let x be an infinite set endowed with the cofinite topology. then x is t1, but not sober (x itself is closed irreducible). (2) the sierpiński space, ({ 0, 1 }, { ∅, { 0 }, { 0, 1 } }), is not t1, since { 0 } is not closed. it is sober, for the nonempty closed irreducible sets are { 1 } and { 0, 1 } = cl({ 0 }). we wish to describe the relations between sobriety and the hα and shα conditions. we begin with an easy but very useful lemma. lemma 3.4. let f be a nonempty irreducible subset of a topological space x, and let a ∈ f. then f ⊆ ∆α(a), for all α. proof. let f be nonempty irreducible and let a ∈ f . we argue by transfinite induction to show that f ⊆ ∆α(a), for each ordinal α. clearly, f ⊆ ∆0(a) = x. let α > 0 and suppose f ⊆ ∆γ(a), for all γ < α. if α is a limit ordinal, then we clearly have f ⊆ ⋂ γ<α ∆γ(a) = ∆α(a) if α = β + 1 is a successor ordinal, then we have, in particular, f ⊆ ∆β(a). consider any b ∈ f , and let u and v be open sets such that a ∈ u, b ∈ v . clearly, u ∩ f 6= ∅ 6= v ∩ f as a ∈ u ∩ f and b ∈ v ∩ f ; so, by irreducibility of f , we have u ∩ v ∩ f 6= ∅ and, since f ⊆ ∆β(a), u ∩ v ∩ ∆β(a) 6= ∅ thus we see that b ∈ ∆α(a) and, since b was an arbitrary member of f , we have f ⊆ ∆α(a). � theorem 3.5. every shα space (and, a fortiori, every hα space) is t1 and sober. chains of separation conditions 269 proof. let x be an shα space; then, by the results of section 2, x is t1. let f ⊆ x be nonempty and irreducible, and let a, b ∈ f . by lemma 3.4, we have a ∈ f ⊆ ∆α(b), and b ∈ f ⊆ ∆α(a). by the shα property, it follows that a = b. thus f must be a singleton. since every nonempty irreducible set is a singleton, x is sober. � 4. not all t1 and sober spaces are sh in this section, we will provide a counterexample to show that the implication given by theorem 3.5 cannot be reversed. example 4.1. let x = r ∪ { p, q }, where p and q are two distinct points not in r. we topologize x by stipulating that the open sets contained in r are precisely the open sets in the usual euclidean topology of the real numbers and that sets having either p or q as a member are open if and only if they are cofinite. we will show that x is a t1 and sober topological space that is not shα, for any α: x is t1, since the complement of each singleton is clearly open. it is easy to check that ∆1(a) = { a, p, q }, ∀a ∈ r ∆1(p) = ∆1(q) = x from which it follows that, for any α > 1, ∆α(a) = { a }, ∀a ∈ r ∆α(p) = ∆α(q) = x thus x is not shα for any α, as p 6= q but p ∈ ∆α(q) and q ∈ ∆α(p). now we check that x is sober. first, note that if a ∈ f for some real number a, and f ⊆ x is closed and irreducible, then, by lemma 3.4, we have f ⊆ ∆2(a) = { a }, so f = { a } is a singleton. therefore the only possibility for an irreducible set with more than one element is f = { p, q }. but this set is not irreducible: letting u = x \ { p }, v = x \ { q }, we have u, v open and u ∩ f 6= ∅ 6= v ∩ f but u ∩ v ∩ f = ∅ thus every nonempty irreducible set is a singleton, so x is t1 and sober, as claimed. 5. all h and all sh conditions are distinct the purpose of this section is to provide examples of topological spaces that show that the h and sh conditions are all distinct from each other (apart from the noted equivalence h1 ⇔ sh1 ⇔ t2). in order to make the following arguments clearer, we first introduce some terminology and notation. 270 lúıs sequeira definition 5.1. let x be a topological space, let a ∈ x, and let α be an ordinal. we will say that a is strictly hα if ∆α(a) = { a }, but ∆β(a) 6= { a }, for all β < α. we will say that x is strictly hα if x is hα and is not hβ for any β < α. any hausdorff space with more than one point is a strictly h1 space, since h0 only holds in one-point spaces. to establish the desired results, we need to introduce a few constructions of topological spaces. the following is well known. definition 5.2. the sum of a family (xi, τi)i∈i of topological spaces is the space (x, τ) where x is the disjoint union of the xi, and the union of the τi is a basis for τ. a topological space may be strictly hα but fail to contain a strictly hα point. for instance, a sum of strictly hn spaces, for all finite n, is a strictly hω space that has no strictly hω point. this is inconvenient for our purposes, so we will make use of a slightly modified construction. recall that a pointed topological space is a pair (x, ∗) where x is a topological space and ∗ is a point of x. the distinguished point ∗ will be referred to as a base point. in the sequel, we will often need to work at once with several pointed spaces, sharing the same base point. two pointed spaces (x, ∗), (y, ∗), with a common base point ∗, will be called disjoint if x ∩ y = { ∗ }. definition 5.3. a pointed topological space (x, ∗) is strictly hα if x is hα and ∗ is strictly hα. definition 5.4. let ((xi, ∗))i∈i be a family of pointed topological spaces with a common base point, which are pairwise disjoint. the amalgamated sum of the family ((xi, ∗))i∈i is the pointed space (x, ∗), where x = ⋃ i∈i xi and a subset u of x is open if and only if u ∩ xi is xi-open, for all i ∈ i note that the amalgamated sum just described can be viewed as a sum in which the base points are all identified: it is the same as the quotient space obtained by factoring the sum of the spaces by the equivalence relation identifying all base points. definition 5.5 ([1, 2]). let a and b be topological spaces. let ∗ ∈ b be such that { ∗ } is closed in b. we denote by a ∗ b the space with underlying set a . ∪(b\{ ∗ }) in which a subset u ⊆ a ∗ b is open if and only if the following three conditions hold: a) u ∩ a is a-open; b) u ∩ b is b-open; c) if u ∩ a 6= ∅, then (u ∩ b) ∪ { ∗ } is b-open. chains of separation conditions 271 for each subset u of a ∗ b, we will henceforth let ua, ub and u ∗ b denote u ∩ a, u ∩ b and (u ∩ b) ∪ { ∗ }, respectively. the space a ∗ b can be understood as the result of replacing the point ∗ of b with a copy of the space a. thus if u is a neighborhood in a ∗ b of a point a ∈ a, then ua is a neighborhood of a in the space a and ub is a punctured neighborhood of the point ∗ in b. in order to impose an adequate structure on a ∗ b, we will also make the assumption that the singleton { ∗ } is not open in b (and thus any two punctured neighborhoods of ∗ have nonempty intersection). in fact, we require a stronger property to hold. definition 5.6. let (b, ∗) be a pointed topological space, and β > 0 an ordinal. we will say that (b, ∗) is normal strictly hβ if it is strictly hβ and one the following conditions holds: (1) β is a limit ordinal, or (2) β = γ + 1 and { ∗ } is not open in ∆bγ (∗). the following lemma describes how, in a space a ∗ b, the ∆α relations can be computed from the corresponding relations in a and b. the proof of the lemma is not hard, but is somewhat tedious. lemma 5.7. suppose a is a topological space and (b, ∗) is a pointed space which is normal strictly hβ, for some ordinal β. let x denote the space a ∗ b. then we have the following: (i) for each b ∈ b \ { ∗ }, and each ordinal γ, ∆xγ (b) = { ∆bγ (b) if ∗ /∈ ∆ b γ (b) a ∗ ∆ b γ (b) if ∗ ∈ ∆ b γ (b) (ii) for each a ∈ a, and each ordinal γ, ∆xγ (a) = { a ∗ ∆ b γ (∗) if γ < β ∆aδ (a) if γ = β + δ sketch of proof. (i) let b ∈ b \ { ∗ } and let γ be an ordinal. the desired result follows immediately from the two claims below: claim 5.8. ∆xγ (b) ∩ b = ∆ b γ (b) \ { ∗ }. claim 5.9. ∆xγ (b) ∩ a = { a if ∗ ∈ ∆bγ (b) ∅ if ∗ /∈ ∆bγ (b) these two claims may be proved by transfinite induction — the induction step being trivial for limit ordinals and relatively straightforward for successor ordinals. (ii) first, we prove, by transfinite induction, that the desired result holds for all ordinals γ such that 0 ≤ γ ≤ β; in particular, letting γ = β, we get ∆xβ (a) = ∆ a 0 (a) = a. 272 lúıs sequeira it is an easy consequence of the definitions that the equality ∆xβ+δ(a) = ∆ ∆x β (a) δ (a) always holds, for any space. thus the rest of the required result readily obtains. � corollary 5.10. let α and β be ordinals. suppose a is a topological space which is strictly hα and (b, ∗) is a pointed space which is normal strictly hβ. then a ∗ b is strictly hβ+α. furthermore, if, for some point a ∈ a, (a, a) is normal strictly hα, then (a ∗ b, a) is normal strictly hβ+α. theorem 5.11. all hα conditions are distinct. proof. we argue by transfinite induction to show that, for each nonzero ordinal α, there exists a normal strictly hα pointed topological space. let (x1, ∗) denote the space of real numbers, taken with, say, ∗ = 0 as the base point: this is clearly a normal strictly h1 pointed space. let α > 1 and suppose a pointed strictly hβ space (xβ, ∗) has been picked for each ordinal β < α. if α = γ + 1 is a successor ordinal, then we let a = x1, b = xγ and xα = a ∗ b. by corollary 5.10, x is a strictly hα space and we can make it a pointed strictly hα space by choosing ∗ ∈ a as the base point. if α is a limit ordinal, then we let (xα, ∗) be the amalgamated sum of the family ((xβ, ∗))β<α. again, it follows that (xα, ∗) is a normal strictly hα pointed space. � the proof of the preceding theorem provides a recipe for constructing, for each ordinal α, a normal strictly hα pointed space (xα, ∗). using the description of the ∆ relations provided by lemma 5.7, it may be easily shown, by transfinite induction on α, that on these spaces xα all the ∆ relations are symmetric, i. e., for all ordinals α, γ and all elements x, y ∈ xα, we have x ∈ ∆γ(y) ⇐⇒ y ∈ ∆γ(x) (s) clearly, for any ordinal β, a space in which (s) holds will be shβ if and only if it is hβ, so we immediately obtain: theorem 5.12. all shα conditions are distinct. for α > 1, each shα condition is distinct from all the h conditions as well: example 5.13 ([2]). let x = r∪{ p }, where p is a point not in r. topologize x by stipulating that the open subsets of r are the same as under the euclidean topology, and the open sets containing p are the cofinite ones. then it is easily seen that ∆1(p) = x ∆1(a) = { a, p } ∀a ∈ r chains of separation conditions 273 and, for each ordinal α > 1, ∆α(p) = x ∆α(a) = { a } ∀a ∈ r and thus x is sh2, but is not hα, for any α. 6. concluding remarks the following diagram depicts the relative strengths of the separation axioms discussed in this paper. none of the unidirectional arrows may, in general, be reversed. t2 ⇔ h1 ⇒ h2 ⇒ . . . ⇒ hα ⇒ . . . m ⇓ ⇓ sh1 ⇒ sh2 ⇒ . . . ⇒ shα ⇒ . . . ⇒ t1&sober ⇒ t1 ⇓ ⇓ sober ⇒ t0 acknowledgements. the author wishes to thank paolo lipparini for raising the problem of the relation between hk and sobriety and for many useful e-mail communications. references [1] j. p. coleman, separation in topological algebras, algebra universalis 35 (1996), 72-84 [2] k. kearnes and l. sequeira, hausdorff properties of topological algebras, algebra universalis 47 (2002), 343-366 received july 2003 accepted december 2003 l. sequeira (lsequeir@fc.ul.pt) departamento de matemática, faculdade de ciências, universidade de lisboa, 1749-016 lisboa, portugal centro de álgebra, universidade de lisboa, av. prof. gama pinto 2, 1649-003 lisboa, portugal 17.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 39 49 common �xed point theorems for acountable family of fuzzy mappings anna vidal�abstract. in this paper we prove �xed point theorems forcountable families of fuzzy mappings satisfying contractive-typeconditions and a rational inequality in left k-sequentially com-plete quasi-pseudo-metric spaces. these results generalize thecorresponding ones obtained by other authors.2000 ams classi�cation: 54a40.keywords: fuzzy mapping, left k-cauchy sequence, quasi-pseudo-metric.1. introductionheilpern [4] introduced the concept of fuzzy mappings and proved a �xedpoint theorem for fuzzy contraction mappings which is a fuzzy analogue ofnadler's [7] �xed point theorem for multivalued mappings. bose and sahani [1]extended heilpern's �xed point theorem to a pair of fuzzy contraction mappings.park and jeong [8] proved the existence of common �xed points for pairs offuzzy mappings satisfying contractive-type conditions and rational inequality incomplete metric spaces. in [2] the authors extended the theorems of [8] to leftk-sequentially complete quasi-pseudo-metric spaces and in [3] they obtained�xed point theorems for fuzzy mappings in smyth-sequentially complete quasi-metric spaces. this study was motivated by the e�ciency of quasi-pseudo-metric spaces as tools to formulate and solve problems in theoretical computerscience. in this paper we generalize the theorems of [2] and present a partialgeneralization for theorem 3.1 of [1] to countable families of fuzzy mappings inleft k-sequentially complete quasi-pseudo-metric spaces.�while working on this paper the author has been partially supported by the grants fromupv "incentivo a la investigaci�on / 99" and from generalitat valenciana gv00-122-1 40 anna vidal2. preliminariesrecall that (x;d) is a quasi-pseudo-metric space, and d is called a quasi-pseudo-metric if d is a non-negative real valued function on x � x, whichsatis�es d(x;x) = 0 and d(x;z) � d(x;y) + d(y;z) for every x;y;z 2 x. if dis a quasi-pseudo-metric on x, then the function d�1 : x � x ! r, de�nedby d�1(x;y) = d(y;x) for all x;y 2 x, is also a quasi-pseudo-metric on x.only if confusion is possible, we write d-closed or d�1-closed, for example, todistinguish the topological concept in (x;d) or (x;d�1).we will make use of the following notion, which has been studied underdi�erent names by various authors (see e.g. [5], [9]).de�nition 2.1. a sequence (xn) in a quasi-pseudo-metric space (x;d) is calledleft k-cauchy if for each " > 0 there is k 2 n such that d(xr;xs) < " for allr;s 2 n with k � r � s. (x;d) is said to be left k-sequentially complete if eachleft k-cauchy sequence in x converges (with respect to the topology t (d)).a fuzzy set in x is an element of ix where i = [0;1]. the r-level set ofa, denoted by ar, is de�ned by ar = fx 2 x : a(x) � rg if r 2 (0;1], anda0 = cl fx 2 x : a(x) > 0g. for x 2 x we denote by fxg the characteristicfunction of the ordinary subset fxg of x. if a; b 2 ix, as usual in fuzzytheory, we denote a � b when a(x) � b(x), for each x 2 x.let (x;d) be a quasi-pseudo-metric space. we consider the families of [2]w 0(x) = fa 2 ix : a1 is nonempty and d-closedgw�(x) = fa 2 w 0(x) : a1 is d�1-countably compactgand the following concepts for a, b 2 w 0(x):� p(a;b) = inf fd(x;y) : x 2 a1;y 2 b1g = d(a1;b1),� �(a;b) = supfd(x;y) : x 2 a0;y 2 b0g and� d(a;b) = supfh(ar;br) : r 2 ig,where h(ar;br) is the hausdor� distance deduced from the quasi-pseudo-metric d.we will use the following lemmas for a quasi-pseudo-metric space (x;d):lemma 2.2. let x 2 x and a 2 w 0(x). then fxg � a if and only ifp(x;a) = 0:lemma 2.3. p(x;a) � d(x;y) + p(y;a), for any x;y 2 x; a 2 w 0(x).lemma 2.4. if fx0g � a then p(x0;b) � d(a;b) for each a;b 2 w 0(x).lemma 2.5. suppose k 6= ? is countably compact in the quasi-pseudo-metricspace (x;d�1). if z 2 x, then there exists k0 2 k such that d(z;k) = d(z;k0).3. fixed point theoremsfirst we generalize the theorems of [2] to countable families of fuzzy map-pings. from now on (x;d) will be a quasi-pseudo-metric space. common �xed point theorems 41de�nition 3.1. f is said to be a fuzzy mapping if f is a mapping from theset x into w 0(x). we say that z 2 x is a �xed point of f if z 2 f(z)1, i.e.,fzg � f(z).theorem 3.2. let (x;d) be a left k-sequentially complete space and letffi : x ! w�(x)g1i=1 be a countable family of fuzzy mappings. if there existsa constant h, 0 � h < 1, such that for each x;y 2 x,d(fi(x);fi+1(y)) � hmaxf (d ^ d�1)(x;y);p(x;fi(x));p(y;fi+1(y));p(x;fi+1(y))+p(y;fi(x))2 g; i = 1;2;3; : : :d(fi(x);f1(y)) � hmaxf (d ^ d�1)(x;y);p(x;fi(x));p(y;f1(y));p(x;f1(y))+p(y;fi(x))2 g; i = 2;3;4; : : : ;then there exists z 2 x such that fzg � fi(z), i = 1;2;3; ldotsproof. assume � = ph. let x01 2 x and suppose x11 2 (f1(x01))1. by lemma2.5 there exists x12 2 (f2(x11))1 such that d(x11;x12) = d(x11;(f2(x11))1) since(f2(x11))1 is d�1-countably compact. we haved(x11;x12) = d(x11;(f2(x11))1) � d1(x11;f2(x11)) � d(f1(x01);f2(x11))again, we can �nd x21 2 x such that x21 2 (f1(x12))1 and d(x12;x21) �d(f2(x11);f1(x12)). continuing in this manner we produce a sequence�x11;x12;x21;x22;x23;x31;x32;x33;x34; : : : ;xn1;xn2; : : : ;xn(n+1); : : : in x such thatxn1 2 (f1(x(n�1)n))1; d(x(n�1)n;xn1) � d(fn(x(n�1)(n�1));f1(x(n�1)n));xn2 2 (f2(xn1))1; d(xn1;xn2) � d(f1(x(n�1)n);f2(xn1));n = 1;2; : : : andxni 2 (fi(xn(i�1)))1; d(xn(i�1);xni) � d(f(i�1)(xn(i�2));fi(xn(i�1)));i = 3;4; : : : ;(n + 1), n = 2;3; : : :we will prove that (xrs) is a leftk-cauchy sequence. firstlyd(x11;x12) � d(f1(x01);f2(x11))< � maxf(d ^ d�1)(x01;x11);p(x01;f1(x01));p(x11;f2(x11));p(x01;f2(x11)) + p(x11;f1(x01))2 g� � maxf(d ^ d�1)(x01;x11);d(x01;x11);d(x11;x12);d(x01;x12) + d(x11;x11)2 g� � maxfd(x01;x11);d(x11;x12); d(x01;x11) + d(x11;x12)2 g= � maxfd(x01;x11);d(x11;x12)g 42 anna vidalif d(x11;x12) > d(x01;x11), then d(x11;x12) < �d(x11;x12), a contradiction.thus, d(x11;x12) � d(x01;x11), and d(x11;x12) < � d(x01;x11): similarlyd(x12;x21) � d(f2(x11);f1(x12))< � maxf(d ^ d�1)(x11;x12);p(x11;f2(x11));p(x12;f1(x12));p(x11;f1(x12)) + p(x12;f2(x11))2 g� � maxfd(x11;x12);d(x12;x21)gand d(x12;x21) < �d(x11;x12) < �2d(x01;x11);d(x21;x22) � d(f1(x12);f2(x21))< � maxfd(x12;x21);d(x21;x22)gand d(x21;x22) < �d(x12;x21) < �3d(x01;x11);d(x22;x23) � d(f2(x21);f3(x22))< � maxfd(x21;x22);d(x22;x23)gand so d(x22;x23) < �d(x21;x22) < �4d(x01;x11).let y0 = x01. now, we rename the constructed sequence (xrs) as follows:y1 = x11;y2 = x12;y3 = x21;y4 = x22; : : :and so, we obtain the sequence (yn) of points of x such thatyn = xij 2 (fj(yn�1))1 for n = (i+1)i2 + j � 1where i = 1;2; : : : , j = 1; : : : ; i + 1. by the above relations, one can verifythat d(yn;yn+1) < �d(yn�1;yn) < �n d(y0;y1) n = 1;2; ::: and for m > n itis easy to see that d(yn;ym) � �n1��d(y0;y1). then, from [6], (yn) is a leftk-cauchy sequence in x, so there exists z 2 x such that d(z;yn) ! 0 (andd(z;xi(i+1)) ! 0, d(z;xii) ! 0; as i ! 1).next, we show by induction that p(z;fj(z)) = 0, j = 1;2;3; :::by lemmas2.3, 2.4 we have:p(z;f1(z)) � d(z;x12) + p(x12;f1(z))� d(z;x12) + d(f2(x11);f1(z)):similarly p(z;f1(z)) � d(z;x23) + p(x23;f1(z))� d(z;x23) + d(f3(x22);f1(z))p(z;f1(z)) � d(z;x34) + p(x34;f1(z))� d(z;x34) + d(f4(x33);f1(z))and in general, for i = 1;2;3; :::(3.1) p(z;f1(z)) � d(z;xi(i+1)) + d(fi+1(xii);f1(z)) common �xed point theorems 43butd(fi+1(xii);f1(z)) � h maxf(d ^ d�1)(xii;z);p(xii;fi+1(xii));p(z;f1(z));p(xii;f1(z)) + p(z;fi+1(xii))2 g� h maxf(d ^ d�1)(xii;z);d(xii;xi(i+1));d(z;xi(i+1)) + d(fi+1(xii);f1(z));d(xii;xi(i+1)) + d(fi+1(xii);f1(z)) + d(z;xi(i+1)))2 g:(3.2)in the sequel, the expression (2) will be denoted by hmaxfcg. now, thereare four cases:case i: if maxfcg = (d ^ d�1)(xii;z), then the inequality (3.1) becomesp(z;f1(z)) � d(z;xi(i+1))) + h(d ^ d�1)(xii;z)� d(z;xi(i+1)) + hd(z;xii) ! 0; as i ! 1:the other three cases ii-iv coincide with the corresponding ones in [8], andp(z;f1(z)) = 0 in all them. thus, p(z;f1(z)) = 0.suppose p(z;fj(z)) = 0. then, by lemma 2.2 fzg � fj(z) and by lemma 2.4we havep(z;fj+1(z)) � d(fj(z);fj+1(z))� hmax(d ^ d�1)(z;z);p(z;fj(z));p(z;fj+1(z));p(z;fj+1(z)) + p(z;fj(z))2 g= hp(z;fj+1(z))thus (1 � h)p(z;fj+1(z)) � 0;and therefore p(z;fj+1(z)) = 0. hence, bylemma 2.2 it follows that fzg � fj(z), for each j 2 n. �theorem 3.3. let (x;d) be a left k-sequentially complete space and letffi : x ! w�(x)g1i=1 be a countable family of fuzzy mappings. if there existsa constant h 2]0;1[, such that for each x;y 2 xd(fi(x);fi+1(y)) � k [p(x;fi(x)) � p(y;fi+1(y))]1=2; i = 1;2;3; :::d(fi(x);f1(y)) � k [p(x;fi(x)) � p(y;f1(y))]1=2; i = 2;3;4; :::;then there exists z 2 x such that fzg � fi(z), i = 1;2;3; :::proof. let x01 2 x: let (xrs) be the sequence in the proof of theorem 3.2.now, d(x11;x12) � d(f1(x01);f2(x11)) � 1phd(f1(x01);f2(x11))� hph[p(x01;f1(x01)) � p(x11;f2(x11))]1=2� h1=2[d(x01;x11) � d(x11;x12)]1=2 44 anna vidalso, d(x11;x12) � hd(x01;x11). similarlyd(x12;x21) � 1phd(f2(x11);f1(x12))� h1=2[d(x11;x12) � d(x12;x21)]1=2and d(x12;x21) � hd(x11;x12) < h2d(x01;x11);d(x21;x22) � 1phd(f1(x12);f2(x21))� h1=2[d(x12;x21) � d(x21;x22)]1=2and d(x21;x22) � hd(x12;x21) � h3d(x01;x11);d(x22;x23) � 1phd(f2(x21);f3(x22))� h1=2 [d(x21;x22) � d(x22;x23)]1=2and d(x22;x23) � hd(x21;x22) � h4d(x01;x11).let y0 = x01. now, we rename the constructed sequence (xrs) as theorem3.2. by the above relations one can verify that d(yn;yn+1) � hd(yn�1;yn) �hn d(y0;y1); n = 1;2; :::and from [6], (yn) is a left k-cauchy sequence in x.then, there exists z 2 x such that d(z;yn) ! 0.next we will show by induction that p(z;fj(z)) = 0, j = 1;2;3; ::: by lemmas2.3 and 2.4 it follows that for i = 1;2;3; :::p(z;f1(z)) � d(z;xi(i+1)) + p(xi(i+1);f1(z))� d(z;xi(i+1)) + d(fi+1(xii);f1(z))� d(z;xi(i+1)) + h[d(xii;xi(i+1)) � p(z;f1(z))]1=2 ! 0 as i ! 1:then, p(z;f1(z)) = 0: now, suppose p(z;fj(z)) = 0. then, by lemmas 2.2and 2.4 we havep(z;fj+1(z)) � d(fj(z);fj+1(z))� h[p(z;fj(z)) � p(z;fj+1(z))]1=2 = 0:it follows that p(z;fj+1(z)) = 0 and fzg � fj(z), for each j 2 n. �since d(a;b) � �(a;b), 8a;b 2 w 0(x), then we deduce the followingcorollary.corollary 3.4. let (x;d) be a left k-sequentially complete space and letffi : x ! w�(x)g1i=1 be a countable family of fuzzy mappings. if there existsa constant h 2]0;1[, such that for each x;y 2 x�(fi(x);fi+1(y)) � k[p(x;fi(x)) � p(y;fi+1(y))]1=2; i = 1;2;3; : : :�(fi(x);f1(y)) � k[p(x;fi(x)) � p(y;f1(y))]1=2; i = 2;3;4; : : : ;then there exists z 2 x such that fzg � fi(z), i = 1;2;3; : : : common �xed point theorems 45theorem 3.5. let (x;d) be a left k-sequentially complete space and letffi : x ! w�(x)g1i=1 be a countable family of fuzzy mappings. if there existconstants h, k > 0, with h + k < 1, such that for each x;y 2 xd(fi(x);fi+1(y)) � hp(y;fi+1(y))[1+p(x;fi(x))]1+d(x;y) + kd(x;y); i = 1;2;3; : : :d(fi(x);f1(y)) � hp(y;f1(y))[1+p(x;fi(x))]1+d(x;y) + kd(x;y); i = 2;3;4; : : :d(f1(x);fi(y)) � hp(y;fi(y))[1+p(x;f1(x))]1+d(x;y) + kd(x;y); i = 3;4; : : : ;then there exists z 2 x such that fzg � fi(z), i = 1;2;3; : : :.proof. let x01 2 x: let (xrs) be the sequence in the proof of theorem 3.2.now, d(x11;x12) � d(f1(x01);f2(x11)) and using one of the two boundaryconditions for d, it is proved thatd(x11;x12) � k1 � hd(x01;x11) and d(x12;x21) � k1 � hd(x11;x12):similarly we haved(x21;x31) � k1 � hd(x12;x21);d(x32;x31) � k1 � hd(x21;x31); : : : :let y0 = x01. now, we rename the constructed sequence (xrs) as theorem3.2 and we can see thatd(yn;yn+1) � k1 � h d(yn�1;yn) � � k1 � h�n d(y0;y1):furthermore, taking t = k1�h, for m > n the following relation is satis�edd(yn;ym) � tn1 � t d(y0;y1):in consequence (yn) is a left k-cauchy sequence and hence converges to z inx. we will see that p(z;fj(z)) = 0; j = 1;2;3; ::: first,p(z;f1(z)) � d(z;xi(i+1)) + hd(xii;xi(i+1))[1 + p(z;f1(z))]1 + d(z;xii) + kd(z;xii) ! 0;as i ! 1.then we have p(z;f1(z)) = 0: now, suppose p(z;fj(z)) = 0. then bylemmas 2.2 and 2.4 we have p(z;fj+1(z)) � hp(z;fj+1(z)) and it follows thatp(z;fj+1(z)) = 0. hence, by lemma 2.2 it follows that fzg � fj(z), for eachj 2 n. �we consider the following theorem for complete metric spaces.theorem 3.6 (bose and sahani [1]). let (x;d) be a complete linear spaceand let f1 and f2 be fuzzy mappings from x to w(x) satisfying the followingcondition: for any x;y in x,d(f1(x);f2(y)) � a1p(x;f1(x)) + a2p(y;f2(y)) + a3p(y;f1(x))+a4p(x;f2(y)) + a5d(x;y)where a1, a2, a3, a4, a5, are non-negative real numbers, a1+a2+a3+a4+a5 < 1and a1 = a2 or a3 = a4. then there exists z 2 x such that fzg � fi(z), i = 1;2: 46 anna vidalwe will present two similar theorems for a countable family of fuzzy mappingsin a quasi-pseudo-metric space (x;d).theorem 3.7. let (x;d) be a left k-sequentially complete space and letffi : x ! w�(x)g1i=1 be a countable family satisfying the following condition:for any x, y 2 x,d(fi(x);fi+1(y)) � a1p(x;fi(x)) + a2p(y;fi+1(y)) + a3p(y;fi(x))+a4p(x;fi+1(y)) + a5(d ^ d�1)(x;y); i = 1;2; :::d(fi(x);f1(y)) � a1p(x;fi(x)) + a2p(y;f1(y)) + a3p(y;fi(x))+a4p(x;f1(y)) + a5(d ^ d�1)(x;y); i = 2;3; :::where a1, a2, a3, a4, a5, are non-negative real numbers and a1+a2+2a4+a5 < 1.then there exists z 2 x such that fzg � fi(z), i = 1;2;3; :::proof. let x01 2 x: let (xrs) be the sequence in the proof of theorem 3.2.nowd(x11;x12) � d(f1(x01);f2(x11))� a1p(x01;f1(x01)) + a2p(x11;f2(x11)) + a3p(x11;f1(x01))+a4p(x01;f2(x11)) + a5(d ^ d�1)(x01;x11)� a1d(x01;x11) + a2d(x11;x12) + a4(d(x01;x11) + d(x11;x12))+a5d(x01;x11);i.e., d(x11;x12) � a1 + a4 + a51 � a2 � a4 d(x01;x11):let r = a1 + a4 + a51 � a2 � a4 . then 0 < r < 1 and d(x11;x12) � rd(x01;x11): againd(x12;x21) � d(f2(x11);f1(x12))� a1d(x11;x12) + a2d(x12;x21) + a4(d(x11;x12) + d(x12;x21))+a5d(x11;x12);i.e., d(x12;x21) � rd(x11;x12) � r2d(x01;x11):let y0 = x01. now, we rename the constructed sequence (xrs) as theorem3.2. by the above relations one can verify that d(yn;yn+1) � rn d(y0;y1); n =1;2; :::: and there exists z 2 x such that d(z;yn) ! 0.we will show by induction that p(z;fj(z)) = 0, j = 1;2;3; ::: by lemmas 2.3and 2.4 it follows that for i = 1;2;3; :::p(z;f1(z)) � d(z;xi(i+1)) + p(xi(i+1);f1(z))� d(z;xi(i+1)) + d(fi+1(xii);f1(z)) common �xed point theorems 47butd(fi+1(xii);f1(z)) � a1p(xii;fi+1(xii)) + a2p(z;f1(z))+a3p(z;fi+1(xii)) + a4p(xii;f1(z))+a5(d ^ d�1)(xii;z)� a1d(xii;xi(i+1))+a2 �d(z;xi(i+1)) + d(fi+1(xii);f1(z)) +a3d(z;xi(i+1))+a4 �d(xii;xi(i+1)) + d(fi+1(xii);f1(z)) +a5d(z;xii):thus d(fi+1(xii);f1(z)) � a1+a41�a2�a4 d(xii;xi(i+1))+ a2+a31�a2�a4 d(z;xi(i+1))+ a51�a2�a4 d(z;xii):so p(z;f1(z)) � d(z;xi(i+1)) + a1 + a41 � a2 � a4 d(xii;xi(i+1))+ a2 + a31 � a2 � a4 d(z;xi(i+1)) + a51 � a2 � a4 d(z;xii) ! 0as i ! 1:then, p(z;f1(z)) = 0. now, suppose p(z;fj(z)) = 0. then, by lemma 2.2fzg � fj(z) and by lemma 2.4 we havep(z;fj+1(z)) � d(fj(z);fj+1(z))� a1p(z;fj(z)) + a2p(z;fj+1(z))+a3p(z;fj(z)) + a4p(z;fj+1(z)) + (d ^ d�1)(z;z)= (a2 + a4)p(z;fj+1(z)):thus (1 � a2 � a4)p(z;fj+1(z)) � 0, and it follows that p(z;fj+1(z)) = 0. bylemma 2.2 it follows that fzg � fj(z), for each j 2 n. �we notice the above theorem is not a generalization of theorem 3.6. now wepresent a partial generalization of this theorem.theorem 3.8. let (x;d) be a left k-sequentially complete space and letffi : x ! w�(x)g1i=1 be a countable family of fuzzy mappings, satisfying thefollowing condition: for any x, y 2 x,d(fi(x);fi+1(y)) � a1p(x;f1(x)) + a2p(y;f2(y)) + a3p(y;f1(x))+a4p(x;f2(y)) + a5(d ^ d�1)(x;y); i = 1;2; :::d(f1(x);fi(y)) � a1p(x;f1(x)) + a2p(y;fi(y)) + a3p(y;f1(x))+a4p(x;fi(y)) + a5(d ^ d�1)(x;y); i = 3;4;where a1, a2, a3, a4, a5, are non-negative real numbers and a1+a2+2a3+a5 < 1,a1 + a2 + 2a4 + a5 < 1. then there exists z 2 x such that fzg � fi(z),i = 1;2;3; : : : (compare with 3.6). 48 anna vidalproof. let x01 2 x. let (xrs) be the sequence in the proof of theorem 3.2.now d(x11;x12) � d(f1(x01);f2(x11))and, as in the proof of the above theorem, we haved(x11;x12) � a1 + a4 + a51 � a2 � a4 d(x01;x11):again d(x12;x21) � a2 + a3 + a51 � a1 � a3 d(x11;x12):let r = a1 + a4 + a51 � a2 � a4 , and s = a2 + a3 + a51 � a1 � a3 . then 0 < r;s < 1. taket = maxfr;sg < 1. so, we haved(x11;x12) � rd(x01;x11) � td(x01;x11);d(x12;x21) � sd(x11;x12) � td(x11;x12) � t2d(x01;x11):let y0 = x01. now, we rename the constructed sequence (xrs) as theorem 3.2.by the above relations one can verify that d(yn;yn+1) � tn d(y0;y1), n = 1;2; : : :then there exists z 2 x such that d(z;yn) ! 0 and as in the proof of the abovetheorem it can be shown that fzg � fj(z), for each j 2 n. �references[1] r. k. bose and d. sahani, fuzzy mappings and �xed point theorems, fuzzy sets andsystems, 21 (1987), 53{58.[2] v. gregori and s. romaguera, common �xed point theorems for pairs of fuzzy mappings,indian journal of mathematics, 41, n� 1 (1999), 43{54.[3] v. gregori and s. romaguera, fixed point theorems for fuzzy mappings in quasi-metricspaces, fuzzy sets and systems, 115 (2000), 477{483.[4] s. heilpern, fuzzy mappings and �xed point theorem, j. math. anal. appl., 83 (1981),566{569.[5] j. c. kelly, bitopological spaces, proc. london math. soc., 13 (1963), 71{89.[6] h. p. a. k�unzi, m. mrsevic, i. l. reilly, m. k. vamanamurthy, convergence, precom-pactness and symmetry in quasi-uniform spaces, math. japonica, 38 (1993), 239{253.[7] s. b. nadler, multivalued contraction mappings, paci�c j. math., 30 (1969), 475{488.[8] j. y. park and j. u. jeong, fixed point theorems for fuzzy mappings, fuzzy sets andsystems, 87 (1997), 111{116.[9] i. l. reilly, p. v. subrahmanyam and m. k. vamanamurthy, cauchy sequences in quasi-pseudometric spaces, mh. math., 93 (1982), 127{140.received september 2000revised version january 2001 a. vidaldep. de matem�atica aplicadaescuela polit�ecnica superior de gandia common �xed point theorems 49carreyera nazaret-oliva s/n46730-grau de gandia, (valencia), spaine-mail address: avidal@mat.upv.es applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 85–89 topological groups with dense compactly generated subgroups hiroshi fujita and dmitri shakhmatov abstract. a topological group g is: (i) compactly generated if it contains a compact subset algebraically generating g, (ii) σ-compact if g is a union of countably many compact subsets, (iii) ℵ0-bounded if arbitrary neighborhood u of the identity element of g has countably many translates xu that cover g, and (iv) finitely generated modulo open sets if for every non-empty open subset u of g there exists a finite set f such that f ∪ u algebraically generates g. we prove that: (1) a topological group containing a dense compactly generated subgroup is both ℵ0-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both ℵ0-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is σ-compact and finitely generated modulo open sets. 2000 ams classification: 54h11, 22a05. keywords: topological group, compactly generated group, dense subgroup, almost metrizable group, ℵ0-bounded group, paracompact p-space, metric space, σ-compact space, space of countable type. 1. preliminaries all topological groups in this article are assumed to be t1 (and thus tychonoff). for subsets a and b of a group g let ab = {ab : a ∈ a and b ∈ b} and a−1 = {a−1 : a ∈ a}. for a ∈ a and b ∈ b we write ab or ab rather than {a}b or a{b}. if a is a subset of a group g, then the smallest subgroup of g that contains a is denoted by 〈a〉. recall that a topological group g is said to be: (i) compactly generated if g = 〈k〉 for some compact subspace k of g, (ii) sigma-compact provided that there exists a sequence {kn : n ∈ ω} of compact subsets of g such that g = ⋃ {kn : n ∈ ω}, 86 hiroshi fujita and dmitri shakhmatov (iii) ℵ0-bounded if for every neighborhood u of the unit element there exists a countable set s ⊂ g such that us = g ([2]), (iv) totally bounded if for every neighborhood u of the unit element there exists a finite set s ⊂ g such that us = g, (v) finitely generated modulo open sets if for every non-empty open set u ⊆ g, there exists a finite set f ⊆ g such that 〈f ∪u〉 = g ([1]). clearly, compactly generated groups are σ-compact. it is well-known that σ-compact groups, separable groups and their dense subgroups are ℵ0-bounded ([2]). 2. the results the main purpose of this note is to study the following question: when does a topological group contain a dense compactly generated subgroup? our first result provides two necessary conditions: theorem 2.1. if a topological group g contains a dense compactly generated subgroup, then g is both ℵ0-bounded and finitely generated modulo open sets. proof. let g be a topological group and k be its compact subset such that 〈k〉 is dense in g. then g is ℵ0-bounded ([2]), so it remains only to show that g is finitely generated modulo open sets. given a non-empty open set u, the group g is divided into pairwise disjoint left-congruence classes modulo its subgroup 〈u〉. let x be a complete set of representatives of these congruence classes: g = ⋃ x∈x x〈u〉. since each congruence class is an open set, finite number of those classes must cover the compact set k. therefore there is a finite set f ⊂ x such that f〈u〉⊇ k. since 〈k〉 is dense in g, it follows that g = u〈k〉⊆ u〈f〈u〉〉⊆ 〈f ∪u〉⊆ g. � in our future arguments we will make use of the following easy lemma: lemma 2.2. let x be a topological space. let k ⊂ x be a compact set with a neighborhood base {un}n∈ω. suppose that we have compact sets cn ⊂ ⋂ k≤n uk for all n ∈ ω. then the set c = k ∪ ⋃ n∈ω cn is also compact. a topological group g is almost metrizable if there exist a non-empty compact set k ⊂ g and a sequence {un}n∈ω of open subsets of g such that (1) k ⊂ un for all n ∈ ω and (2) if o is an open set containing k, then there is an n ∈ ω such that k ⊂ un ⊂ o. (such a sequence {un}n∈ω is called a neighborhood base of k in g.) both metric groups and locally compact groups are almost metrizable ([3]). our next theorem demonstrates that the necessary conditions for a topological group g to have a dense compactly generated subgroup found in theorem 2.1 are also sufficient in case g is almost metrizable. theorem 2.3. an almost metrizable topological group g contains a dense compactly generated subgroup if and only if it is ℵ0-bounded and finitely generated modulo open sets. dense compactly generated subgroups 87 proof. the “only if” part of our theorem follows from theorem 2.1, so it remains only to prove the “if” part. let k be a compact subset of g with a neighborhood base {un}n∈ω. since g is ℵ0-bounded, for each n ∈ ω there is a countable set sn ⊂ g such that g = snun. the set s = ⋃ n∈ω sn is countable, so we can fix its enumeration s = {sn}n∈ω. let g ∈ g. let v be any neighborhood of the unit element of g. then kv −1 is an open set containing k, and so there is an n ∈ ω such that un ⊆ kv −1. since snun = g, there is an s ∈ sn such that g ∈ sun ⊆ skv −1. let g = skv−1 with k ∈ k and v ∈ v . then gv = sk ∈ gv ∩ sk 6= ∅. since v and g are arbitrary, it follows that sk is dense in g. since g is finitely generated modulo open sets, there are finite sets fn such that g = 〈fn∪un〉 for each n ∈ ω. set e0 = f0∪{s0}. it follows that g = 〈e0∪u0〉. so there is a finite set e1 ⊆ u0 such that f1∪{s1}⊂ 〈e0∪e1〉. from this it follows that 〈e0∪e1∪u1〉 = g. so there is a finite set e2 ⊆ u1 such that f2 ∪{s2}⊂ 〈e0 ∪e1 ∪e2〉. in this way we obtain finite sets en+1 ⊂ un (for n ∈ ω) such that fn+1 ∪{sn+1} ⊆ 〈e0 ∪ e1 ∪ ·· · ∪ en+1〉. by lemma 2.2, the set c = k ∪ ⋃ n∈ω en is compact. the subgroup 〈c〉 is dense, since it contains sk. thus g contains a compactly generated dense subgroup. � since every metrizable group is almost metrizable ([3]), and ℵ0-boundedness is equivalent to separability for metrizable groups, from theorem 2.3 we obtain: corollary 2.4. a metrizable group contains a dense compactly generated subgroup if and only if it is separable and finitely generated modulo open sets. our next result generalizes theorem 4 from [1]. lemma 2.5. if a σ-compact almost metrizable group g contains a dense compactly generated subgroup, then g itself is compactly generated. proof. suppose g = ⋃ n∈ω ln, with ln compact. suppose also that h = 〈l0〉 is dense in g. let k ⊆ g be a compact set with a neighborhood base {un}n∈ω. by regularity of the topology of g and compactness of k, we may assume without loss of generality that each un contains the closure of un+1. by compactness of ln and denseness of h, there is a finite subset fn of h such that ln ⊂ un+1fn. let cn = lnf−1n ∩un+1. then cn is compact, because it is a closed subset of the union of finitely many copies of ln. we also have cn ⊂ un and ln ⊂ cnfn ⊂ 〈cn ∪l0〉. therefore, setting c = l0 ∪k ∪ ⋃ n∈ω cn, we obtain 〈c〉 = g. it follows from lemma 2.2 that c is compact. � combining theorem 2.3 and lemma 2.5, we obtain our next theorem which extends the main result of [1]: theorem 2.6. an almost metrizable topological group is compactly generated if and only if it is σ-compact and finitely generated modulo open sets. theorems 2.3 and 2.6 become especially simple for locally compact groups: theorem 2.7. for a locally compact group g the following conditions are equivalent: 88 hiroshi fujita and dmitri shakhmatov (i) g has a dense compactly generated subgroup, (ii) g is compactly generated, (iii) g is finitely generated modulo open sets. proof. let u be an open neighbourhood of the identity element having compact closure u. (i)→(ii). let k be a compact subset of g such that 〈k〉 is dense in g. then u ∪k is also compact and 〈u ∪k〉 ⊇ u〈k〉 = g because 〈k〉 is dense in g and u is an open neighbourhood of the identity. (ii)→(iii) follows from theorem 2.1. (iii)→(i). assume that g is finitely generated modulo open sets. then there exists a finite set f ⊆ g with 〈f ∪ u〉 = g. now note that g = 〈f ∪ u〉 ⊆ 〈f ∪u〉⊆ g. since 〈f ∪u〉 is compact, g is compactly generated. � since for every non-empty open subset u of a topological group g the set 〈u〉 is an open subgroup of g, it follows that a topological group without proper open subgroups is finitely generated modulo open sets ([1]). therefore, from theorem 2.6 we obtain corollary 2.8. an almost metrizable, σ-compact group without proper open subgroups is compactly generated. corollary 2.9. a metric σ-compact group without proper open subgroups is compactly generated. totally bounded groups are finitely generated modulo open sets, and so we get corollary 2.10. every σ-compact totally bounded almost metrizable group is compactly generated. references [1] h. fujita and d. b. shakhmatov, a characterization of compactly generated metrizable groups, proc. amer. math. soc., to appear. [2] i. guran, topological groups similar to lindelöf groups (in russian), dokl. akad. nauk sssr 256 (1981), no. 6, 1305–1307; english translation in: soviet math. dokl. 23 (1981), no. 1, 173–175. [3] b.a. pasynkov, almost-metrizable topological groups (in russian), dokl. akad. nauk sssr 161 (1965), 281–284. received september 2001 revised march 2002 dense compactly generated subgroups 89 hiroshi fujita department of mathematical sciences faculty of science ehime university matsuyama 790-8577 japan e-mail address : fujita@math.sci.ehime-u.ac.jp dmitri shakhmatov (corresponding author) department of mathematical sciences faculty of science ehime university matsuyama 790-8577 japan e-mail address : dmitri@dpc.ehime-u.ac.jp () @ appl. gen. topol. 16, no. 1(2015), 65-74doi:10.4995/agt.2015.3161 c© agt, upv, 2015 on c-embedded subspaces of the sorgenfrey plane olena karlova chernivtsi national university, department of mathematical analysis, kotsjubyns’koho 2, chernivtsi 58012, ukraine (maslenizza.ua@gmail.com) abstract we prove that every c∗-embedded subset of s2 is a hereditarily baire subspace of r2. we also show that for a subspace e ⊆ {(x, −x) : x ∈ r} of the sorgenfrey plane s2 the following conditions are equivalent: (i) e is c-embedded in s2; (ii) e is c∗-embedded in s2; (iii) e is a countable gδ-subspace of r 2 and (iv) e is a countable functionally closed subspace of s2. 2010 msc: 54c45; 54c20. keywords: c ∗-embedded; c-embedded; the sorgenfrey plane. 1. introduction recall that a subset a of a topological space x is called functionally open (functionally closed) in x if there exists a continuous function f : x → [0, 1] such that a = f−1((0, 1]) (a = f−1(0)). sets a and b are completely separated in x if there exists a continuous function f : x → [0, 1] such that a ⊆ f−1(0) and b ⊆ f−1(1). a subspace e of a topological space x is • c-embedded (c∗-embedded) in x if every (bounded) continuous function f : e → r can be continuously extended on x; • z-embedded in x if every functionally closed set in e is the restriction of a functionally closed set in x to e; • well-embedded in x [7] if e is completely separated from any functionally closed set of x disjoint from e. received 3 july 2014 – accepted 5 december 2014 http://dx.doi.org/10.4995/agt.2015.3161 o. karlova clearly, every c-embedded subspace of x is c∗-embedded in x. the converse in not true. indeed, if e = n and x = βn, then e is c∗-embedded in x (see [4, 3.6.3]), but the function f : e → r, f(x) = x for every x ∈ e, does not extend to a continuous function f : x → r. a space x has the property (c∗ = c) [11] if every closed c∗-embedded subset of x is c-embedded in x. the classical tietze-urysohn extension theorem says that if x is a normal space, then every closed subset of x is c∗embedded and x has the property (c∗ = c). moreover, a space x is normal if and only if every its closed subset is z-embedded (see [9, proposition 3.7]). the following theorem was proved by blair and hager in [2, corollary 3.6]. theorem 1.1. a subset e of a topological space x is c-embedded in x if and only if e is z-embedded and well-embedded in x. a space x is said to be δ-normally separated [10] if every closed subset of x is well-embedded in x. the class of δ-normally separated spaces includes all normal spaces and all countably compact spaces. theorem 1.1 implies the following result. corollary 1.2. every δ-normally separated space has the property (c∗ = c). according to [15] every c∗-embedded subspace of a completely regular first countable space is closed. the following problem is still open: problem 1.3 ([12]). does there exist a first countable completely regular space without property (c∗ = c)? h. ohta in [11] proved that the niemytzki plane has the property (c∗ = c) and asked does the sorgenfrey plane s2 (i.e., the square of the sorgenfrey line s) have the property (c∗ = c)? in the given paper we obtain some necessary conditions on a set e ⊆ s2 to be c∗-embedded. we prove that every c∗-embedded subset of s2 is a hereditarily baire subspace of r2. we also characterize cand c∗-embedded subspaces of the anti-diagonal d = {(x, −x) : x ∈ r} of s2. namely, we prove that for a subspace e ⊆ d of s2 the following conditions are equivalent: (i) e is cembedded in s2; (ii) e is c∗-embedded in s2; (iii) e is a countable gδ-subspace of r2 and (iv) e is a countable functionally closed subspace of s2. 2. every finite power of the sorgenfrey line is a hereditarily α-favorable space recall the definition of the choquet game on a topological space x between two players α and β. player β goes first and chooses a nonempty open subset u0 of x. player α chooses a nonempty open subset v1 of x such that v1 ⊆ u0. following this player β must select another nonempty open subset u1 ⊆ v1 of x and α must select a nonempty open subset v2 ⊆ u1. acting in this way, the players α and β obtain sequences of nonempty open sets (un) ∞ n=0 and (vn) ∞ n=1 such that un−1 ⊆ vn ⊆ un for every n ∈ n. the player α wins if ∞ ⋂ n=1 vn 6= ∅. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 66 on c-embedded subspaces of the sorgenfrey plane otherwise, the player β wins. if there exists a rule (a strategy) such that α wins if he plays according to this rule, then x is called α-favorable. respectively, x is called β-unfavorable if the player β has no winning strategy. clearly, every α-favorable space x is β-unfavorable. moreover, it is known [13] that a topological space x is baire if and only if it is β-unfavorable in the choquet game. if a is a subspace of a topological space x, then a and inta mean the closure and the interior of a in x, respectively. lemma 2.1. let x = n ⋃ k=1 xk, where xk is an α-favorable subspace of x for every k = 1, . . . , n. then x is an α-favorable space. proof. we prove the lemma for n = 2. let g = g1 ∪ g2, where gi = intxi, i = 1, 2. we notice that for every i = 1, 2 the space xi is α-favorable, since it contains dense α-favorable subspace. then gi is α-favorable as an open subspace of the α-favorable space xi. it is easy to see that the union g of two open α-favorable subspaces is an α-favorable space. therefore, x is αfavorable, since g is dense in x. � let p = (x, y) ∈ r2 and ε > 0. we write b[p; ε) = [x, x + ε) × [y, y + ε), b(p; ε) = (x − ε, x + ε) × (y − ε, y + ε). if a ⊆ s2 then the symbol cls2a (clr2a) means the closure of a in the space s2 (r2). we say that a space x is hereditarily α-favorable if every its closed subspace is α-favorable. theorem 2.2. for every n ∈ n the space sn is hereditarily α-favorable. proof. let n = 1 and ∅ 6= f ⊆ s. assume that β chose a nonempty open in f set u0 = [a0, b0)∩f , a0 ∈ f . if u0 has an isolated point x in s, then α chooses v1 = {x} and wins. otherwise, α put v1 = [a0, c0) ∩ f , where c0 ∈ (a0, b0) ∩ f and c0 − a0 < 1. now let u1 = [a1, b1) ∩ f ⊆ v1 be the second turn of β such that a1 ∈ f and the set (a1, b1) ∩ f has no isolated points in s. then there exists c1 ∈ (a1, b1) ∩ f such that c1 − a1 < 1 2 . let v2 = [a1, c1) ∩ f . repeating this process, we obtain sequences (um) ∞ m=0, (vm) ∞ m=1 of open subsets of f and sequences of points (am) ∞ m=0, (bm) ∞ m=0 and (cm) ∞ m=1 such that [am, bm) ⊇ [am, cm) ⊇ [am+1, bm+1), cm − am < 1 m+1 , cm ∈ f , um = [am, bm) ∩ f and vm+1 = [am, cm) ∩ f for every m = 0, 1, . . . . according to the nested interval theorem, the sequence (cm) ∞ m=1 is convergent in s to a point x ∗ ∈ ∞ ⋂ m=0 vm. since f is closed in s, x∗ ∈ f . hence, f ∩ ∞ ⋂ m=0 vm 6= ∅. consequently, f is α-favorable. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 67 o. karlova suppose that the theorem is true for all 1 ≤ k ≤ n and prove it for k = n+1. consider a set ∅ 6= f ⊆ sn+1. let the player β chooses a set u0 = f ∩ n+1 ∏ k=1 [a0,k, b0,k) with a0 = (a0,k) n+1 k=1 ∈ f . denote u+0 = n+1 ∏ k=1 (a0,k, b0,k) and consider the case u+0 ∩ f = ∅. for every k = 1, . . . , n + 1 we set u0,k = {a0,k}× ∏ i6=k [a0,i, b0,i) and f0,k = f ∩u0,k. since u0,k is homeomorphic to s n, by the inductive assumption the space f0,k is α-favorable for every k = 1, . . . , n+1. then f is α-favorable according to lemma 2.1. now let u+0 ∩ f 6= ∅. if there exists an isolated in sn+1 point x ∈ u0, then α put v1 = {x} and wins. assume u0 has no isolated points in s n+1. then there is c0 = (c0,k) n+1 k=1 ∈ u+0 ∩ f such that diam( n+1 ∏ k=1 [a0,k, c0,k)) < 1. we put v1 = f ∩ n+1 ∏ k=1 [a0,k, c0,k). let u1 = f ∩ n+1 ∏ k=1 [a1,k, b1,k) be the second turn of β such that a1 = (a1,k) n+1 k=1 ∈ f and u1 ⊆ v1. again, if u + 1 ∩ f = ∅, where u + 1 = n+1 ∏ k=1 (a1,k, b1,k), then, using the inductive assumption, we obtain that for every k = 1, . . . , n + 1 the space f ∩ ( {a1,k} × ∏ i6=k [a1,i, b1,i) ) is α-favorable. then α has a winning strategy in f by lemma 2.1. if u+1 ∩f 6= ∅ and u1 has no isolated points in s n+1, the player α chooses a point c1 = (c1,k) n+1 k=1 ∈ u+1 ∩f such that diam( n+1 ∏ k=1 [a1,k, c1,k)) < 1/2 and put v2 = f ∩ n+1 ∏ k=1 [a1,k, c1,k). repeating this process, we obtain sequences of points (am) ∞ m=0, (bm) ∞ m=0 and (cm) ∞ m=0, and of sets (um) ∞ m=0 and (vm) ∞ m=1, which satisfy the following properties: 1) um = f ∩ n+1 ∏ k=1 [am,k, bm,k); 2) am ∈ f , cm ∈ u + m ∩ f ; 3) vm+1 = f ∩ n+1 ∏ k=1 [am,k, cm,k); 4) vm+1 ⊆ um ⊆ vm; 5) diam(vm+1) < 1 m+1 for every m = 0, 1, . . . . we observe that the sequence (cm) ∞ m=0 is convergent in rn+1 and x∗ = lim m→∞ cm ∈ ∞ ⋂ m=0 vm = ∞ ⋂ m=0 vm. since cm → x ∗ in sn+1, cm ∈ f and f is closed in sn+1, x∗ ∈ f ∩ ( ∞ ⋂ m=0 vm ) . hence, f is α-favorable. � c© agt, upv, 2015 appl. gen. topol. 16, no. 1 68 on c-embedded subspaces of the sorgenfrey plane 3. every c∗-embedded subspace of s2 is a hereditarily baire subspace of r2. lemma 3.1. a set e ⊆ r2 is functionally closed in s2 if and only if 1) e is gδ in r 2; and 2) if f is r2-closed set disjoint from e, then f and e are completely separated in s2. proof. necessity. let f : s2 → r be a continuous function such that e = f−1(0). according to [1, theorem 2.1], f is a baire-one function on r2. consequently, e is a gδ subset of r 2. condition (2) follows from the fact that every r2-closed set is, evidently, a functionally closed subset of s2. sufficiency. since e is gδ in r 2, there exists a sequence of r2-closed sets fn such that x \ e = ∞ ⋃ n=1 fn. clearly, e ∩fn = ∅. then condition (2) implies that for every n ∈ n there exists a continuous function fn : s 2 → r such that e ⊆ f−1n (0) i fn ⊆ f −1(1). then e = ∞ ⋂ n=1 f−1n (0). hence, e is functionally closed in s2. � lemma 3.2. let x be a metrizable space, a ⊆ x be a set without isolated points and let b ⊆ x be a countable set such that a ∩ b = ∅. then there exists a set c ⊆ a without isolated points such that c ∩ b = ∅. proof. let d be a metric on x, which generates its topological structure. for x0 ∈ x and r > 0 we denote b(x0, r) = {x ∈ x : d(x, x0) < r} and b[x0, r] = {x ∈ x : d(x, x0) ≤ r}. let b = {bn : n ∈ n}. we put a0 = ∅ and construct sequences (an) ∞ n=1 and (vn) ∞ n=1 of nonempty finite sets an ⊆ a and open neighborhoods vn of bn which for every n ∈ n satisfy the following conditions: an−1 ⊆ an;(3.1) ∀x ∈ an ∃y ∈ an \ {x} with d(x, y) ≤ 1 n ;(3.2) d(an, ⋃ 1≤i≤n vi) > 0.(3.3) let a1 = {x1, y1}, where d(x1, y1) ≤ 1 and x1 6= y1. we take ε > 0 such that a1 ∩ b[b1, ε] = ∅ and put v1 = b(b1, ε). assume that we have already defined finite sets a1, . . . ak and neighborhoods v1, . . . , vk of b1, . . . , bk, respectively, which satisfy conditions (3.1)–(3.3) for every n = 1, . . . , k. let ak = {a1, . . . , am}, m ∈ n. taking into account that the set d = a \ ⋃ 1≤i≤k v i has no isolated points, for every i = 1, . . . , m we take ci ∈ d with ci 6= ai and d(ai, ci) ≤ 1 k+1 . put ak+1 = ak ∪ {c1, . . . , cm}. take δ > 0 such that ak+1 ∩ b[bk+1, δ] = ∅. let vk+1 = b(bk+1, δ). repeating this process, we obtain needed sequences (an) ∞ n=1 and (vn) ∞ n=1. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 69 o. karlova it remains to put c = ∞ ⋃ n=1 an. � the following results will be useful. theorem 3.3 ([5]). a subspace e of a topological space x is c∗-embedded in x if and only if every two disjoint functionally closed subsets of e are completely separated in x. theorem 3.4 ([16]). the sorgenfrey plane s2 is strongly zero-dimensional, i.e., for any completely separated sets a and b in s2 there exists a clopen set u ⊆ s2 such that a ⊆ u ⊆ s2 \ b. recall that a space x is hereditarily baire if every its closed subspace is baire. theorem 3.5. let e be a c∗-embedded subspace of s2. then e is a hereditarily baire subspace of r2. proof. assume that e is not r2-hereditarily baire space and take an r2-closed countable subspace e0 without r 2-isolated point (see [3]). notice that e is s2-closed according to [15, corollary 2.3]. therefore, e0 is s 2-closed set. by theorem 2.2 the space e0 is α-favorable, and, consequently, e0 is a baire subspace of s2. let e′0 be a set of all s 2-nonisolated points of e0. since e ′ 0 is the set of the first category in s2-baire space e0, the set g = e0\e ′ 0 is s 2-dense open discrete subspace of e0. we notice that g is r 2-dense subspace of e0. by lemma 3.2 there exists a set c ⊆ g without r2-isolated point such that clr2c ∩ e ′ 0 = ∅. we put f = clr2c ∩ e0. let a and b be any r2-dense in f disjoint sets such that f = a ∪ b. evidently a and b are clopen subsets of f , since f is s2-discrete space. notice that f is z-embedded in e, because f is countable. moreover, f is r2-closed in e. hence, f is s2-functionally closed in e. by theorem 1.1 the set f is c-embedded in c∗-embedded in s2 set e. consequently, f is c∗-embedded in s2. therefore, theorem 3.3 and theorem 3.4 imply that there exist disjoint clopen set u, v ⊆ s2 such that a = u ∩ f and b = v ∩ f . according to lemma 3.1 the sets u and v are gδ in r 2. let d = clr2f . then u ∩ d and v ∩ d are r2-dense in d disjoint gδ-sets, which contradicts to the baireness of d. � 4. every discrete c∗-embedded subspace of s2 is a countable gδ-subspace of r 2 . lemma 4.1. let x be a metrizable separable space and a ⊆ x be an uncountable set. then there exists a set q ⊆ a which is homeomorphic to the set q of all rational numbers. proof. let a0 be the set of all points of a which are not condensation points a (a point a ∈ x is called a condensation point of a in x if every neighborhood of a contains uncountably many elements of a). notice that a0 is countable, c© agt, upv, 2015 appl. gen. topol. 16, no. 1 70 on c-embedded subspaces of the sorgenfrey plane since x has a countable base. put b = a \ a0. then the inequality |a| > ℵ0 implies that every point of b is a condensation point of b. take a countable subset q ⊆ b which is dense in b. clearly, every point of q is not isolated. hence, q is homeomorphic to q by the sierpiński theorem [14]. � lemma 4.2. let e be an r2-hereditarily baire z-embedded subspace of s2. then the set e0 of all isolated points of e is at most countable. proof. assume e0 is uncountable. notice that e0 is an fσ-subset of e, since e0 is an open subset of e and s2 is a perfect space by [6]. then e0 = ∞ ⋃ n=1 en, where every set en is closed in e. take n ∈ n such that en is uncountable. according to lemma 4.1 there exists a set q ⊆ en which is homeomorphic to q. since q is clopen in en and en is a clopen subset of a z-embedded in s 2 set e, there exists a functionally closed subset q1 of s 2 such that q = e ∩q1. by lemma 3.1 the set q1 is a gδ-set in r 2. then q is a gδ-subset of a hereditarily baire space e. hence, q is a baire space, a contradiction. � theorem 4.3. if e is a discrete c∗-embedded subspace of s2, then e is a countable gδ-subspace of r 2. proof. theorem 3.5 and lemma 4.2 imply that e is a countable hereditarily baire subspace of r2. according to [8, proposition 12] the set e is gδ in r2. � the converse implication in theorem 4.3 is not valid as theorem 4.5 shows. lemma 4.4. let a be an s2-closed set, ε > 0 and l(a; ε) = {p ∈ s2 : b[p; ε) ⊆ a}. then l(a; ε) is r2-closed. proof. we take p0 = (x0, y0) ∈ clr2l(a; ε) and show that p0 ∈ l(a; ε). we consider u = intr2b[p0; ε) and prove that u ⊆ a. take p = (x, y) ∈ u and put δ = min{(x − x0)/2, (y − y0)/2, (x0 + ε − x)/2, (y0 + ε − y)/2}. let p1 ∈ b(p0; δ) ∩ l(a; ε). it is easy to see that p ∈ b[p1; ε). then p ∈ a, since p1 ∈ l(a; ε). hence, u ⊆ a. then b[p0; ε) = cls2u ⊆ cls2a = a, which implies that p0 ∈ l(a; ε). therefore, l(a; ε) is closed in r 2. � theorem 4.5. there exists an s2-closed countable discrete gδ-subspace e of r2 which is not c∗-embedded in s2. proof. let c be the standard cantor set on [0, 1] and let (in) ∞ n=1 be a sequence of all complementary intervals in = (an, bn) to c such that diam (in+1) ≤ diam (in) for every n ≥ 1. we put pn = (bn; 1 − an), e = {pn : n ∈ n} and f = {(x, 1 − x) : x ∈ r} ∩ (c × [0, 1]). notice that e is a closed subset of s2, f is functionally closed in s2 and e ∩ f = ∅. let n′ ⊆ n be a set such that {bn : n ∈ n ′} and {bn : n ∈ n\ n ′} are dense subsets of c. to show that e is not c∗-embedded in s2 we verify that disjoint clopen subsets e1 = {pn : n ∈ n ′} and e2 = {pn : n ∈ n \ n ′} c© agt, upv, 2015 appl. gen. topol. 16, no. 1 71 o. karlova of e can not be separated by disjoint clopen subsets in s2. assume the contrary and take disjoint clopen subsets w1 and w2 of s 2 such that wi ∩ e = ei for i = 1, 2. we prove that w1 ∩ f is r 2-dense in f . to obtain a contradiction we take an r2-open set o such that o ∩ f ∩ w1 = ∅. since the set u = s 2 \ w1 is clopen, u = ∞ ⋃ n=1 l(u; 1 n ), where l(u; 1 n ) = {p ∈ s2 : b[p; 1/n) ⊆ u} and the set fn = l(u; 1 n ) is r2-closed by lemma 4.4 for every n ∈ n. since o ∩ f is a baire subspace of r2, there exist n ∈ n and an r2-open in f subset i ⊆ f such that i ∩ o ⊆ fn ∩ f ⊆ s 2 \ e1. taking into account that diam (in) → 0, we choose n1 > n such that bn − an < 1 2n for all n ≥ n1. since the set {an : n ∈ n ′} is dense in c, there exists n2 ∈ n ′ such that n2 > n1 and p = (an2; 1 − an2) ∈ i. clearly, p ∈ f . consequently, b[p; 1 n ) ∩ e1 = ∅. but pn2 ∈ b[p, 1 n ) ∩ e1, a contradiction. similarly we can show that w2 ∩ f is also r 2-dense in f . notice that w1 and w2 are gδ in r 2 by lemma 3.1. hence, w1 ∩ f and w2 ∩ f are disjoint dense gδ-subsets of a baire space f , which implies a contradiction. therefore, e is not c∗-embedded in s2. � 5. a characterization of c-embedded subsets of the anti-diagonal of s2. by d we denote the anti-diagonal {(x, −x) : x ∈ r} of the sorgenfrey plane. notice that d is a closed discrete subspace of s2. theorem 5.1. for a set e ⊆ d the following conditions are equivalent: 1) e is c-embedded in s2; 2) e is c∗-embedded in s2; 3) e is a countable gδ-subspace of r 2; 4) e is a countable functionally closed subspace of s2. proof. the implication (1) ⇒ (2) is obvious. the implication (2) ⇒ (3) follows from theorem 4.3. we prove (3) ⇒ (4). to do this we verify condition (2) from lemma 3.1. let f be an r2-closed set disjoint from e. denote d = f ∩d and u = ⋃ p∈d b[p; 1). we show that u is clopen in s2. clearly, u is open in s2. take a point p0 ∈ cls2u and show that p0 ∈ u. choose a sequence pn ∈ u such that pn → p0 in s 2. for every n there exists qn ∈ d such that pn ∈ b[qn, 1). notice that the sequence (qn) ∞ n=1 is bounded in r 2 and take a convergent in r2 subsequence (qnk) ∞ k=1 of (qn) ∞ n=1. since d is r 2-closed, q0 = lim k→∞ qnk ∈ d. then p0 ∈ clr2b[q0, 1). if p0 ∈ b[q0, 1), then p0 ∈ u. assume p0 6∈ b[q0, 1) and let q0 = (x0, y0). without loss of generality we may suppose that p0 ∈ [x0, x0 + 1] × {y0 + 1}. since pnk → p0 in s 2, qnk ∈ (−∞, x0] × [y0, +∞) for all k ≥ k0 and p0 ∈ [x0, x0 + 1) × {y0 + 1}. then p0 ∈ ⋃∞ k=1 b[qnk , 1) ⊆ u. hence, u is clopen and d = u ∩ d. since d and f \ u are disjoint functionally closed c© agt, upv, 2015 appl. gen. topol. 16, no. 1 72 on c-embedded subspaces of the sorgenfrey plane subsets of s2, there exists a clopen set v such that d ∩ v = ∅ and f \ u ⊆ v . then f ⊆ u ∪ v ⊆ s2 \ e. consequently, f and e are completely separated in s2. therefore, e is functionally closed in s2 by lemma 3.1. (4) ⇒ (1). notice that e satisfy the conditions of theorem 1.1. indeed, e is z-embedded in s2, since |e| ≤ ℵ0. moreover, e is well-embedded in s 2, since e is functionally closed. � remark 5.2. notice that a subset e of r2 is countable gδ if and only if it is scattered in r2. indeed, assume that e is countable gδ-set which contains a set q without isolated points. then q is a gδ-subset of r 2 which is homeomorphic to q, a contradiction. on the other hand, if e is scattered, then lemma 4.1 implies that e is countable. since e is hereditarily baire and countable, e is gδ in r 2. finally, we show that the sorgenfrey plane is not a δ-normally separated space. let e = {(x, −x) : x ∈ q} and f = d \ e. then e is closed and f is functionally closed in s2, since f is the difference of the functionally closed set d and the functionally open set ⋃ p∈e b[p, 1). but e and f can not be separated by disjoint clopen sets in s2, because e is not gδ-subset of d in r 2. references [1] w. bade, two properties of the sorgenfrey plane, pacif. j. math. 51, no. 2 (1974), 349–354. [2] r. blair and a. hager, extensions of zero-sets and of real-valued functions, math. zeit. 136 (1974), 41–52. [3] g. debs, espaces héréditairement de baire, fund. math. 129, no. 3 (1988), 199–206. [4] r. engelking, general topology. revised and completed edition. heldermann verlag, berlin (1989). [5] l. gillman and m. jerison, rings of continuous functions, van nostrand, princeton (1960). [6] r. heath and e. michael, a property of the sorgenfrey line, comp. math. 23, no. 2 (1971), 185–188. [7] t. hoshina and k. yamazaki, weak c-embedding and p -embedding, and product spaces, topology appl. 125 (2002), 233–247. [8] o. kalenda and j. spurný, extending baire-one functions on topological spaces, topology appl. 149 (2005), 195–216. [9] o. karlova, on α-embedded sets and extension of mappings, comment. math. univ. carolin. 54, no. 3 (2013), 377–396. [10] j. mack, countable paracompactness and weak normality properties, trans. amer. math. soc. 148 (1970), 265–272. [11] h. ohta, extension properties and the niemytzki plane, appl. gen. topol. 1, no. 1 (2000), 45–60. [12] h. ohta, k. yamazaki, extension problems of real-valued continuous functions, in: ”open problems in topology ii”, e. pearl (ed.), elsevier, 2007, 35–45. [13] j. saint-raymond, jeux topologiques et espaces de namioka, proc. amer. math. soc. 87, no. 3 (1983), 409–504. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 73 o. karlova [14] w. sierpiński, sur une propriete topologique des ensembles denombrables denses en soi, fund. math. 1 (1920), 11–16. [15] y. tanaka, on closedness of cand c∗-embeddings, pacif. j. math. 68, no. 1 (1977), 283–292. [16] j. terasawa, on the zero-dimensionality of some non-normal product spaces, sci. rep. tokyo kyoiku daigaku sect. a 11 (1972), 167–174. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 74 @ appl. gen. topol. 18, no. 1 (2017), 1-11 doi:10.4995/agt.2017.2250 c© agt, upv, 2017 algebraic and topological structures on rational tangles vida milani a, seyed m. h. mansourbeigi b and hossein finizadeh c a department of mathematics and statistics, utah state university, logan utah, usa (vida milani@yahoo.com) b department of computer science, utah state university, logan utah, usa (smansourbeigi@aggiemail.usu.edu) c department of mathematics, shahid beheshti university, tehran, iran (hfinizadeh@gmail.com) communicated by m. sanchis abstract in this paper we present the construction of a group hopf algebra on the class of rational tangles. a locally finite partial order on this class is introduced and a topology is generated. an interval coalgebra structure associated with the locally finite partial order is specified. irrational and real tangles are introduced and their relation with rational tangles are studied. the existence of the maximal real tangle is described in detail. 2010 msc: 16t05; 11y65; 18b35; 57m27; 57t05. keywords: group hopf algebra; locally finite partial order; tangle; pseudomodule; bi-pseudo-module; pseudo-tensor product; incidence algebra; interval coalgebra; continued fraction; tangle convergent. 1. introduction rational tangles are not only beautiful mathematical objects but also have many applications in other fields such as biology and dna synthesis [5]. the theory of tangles was invented in 1986 by conway in his work [2]. he introduced the notion of rational tangles and with each rational tangle he associated a rational number by the continued fraction method. the associated rational received 30 march 2014 – accepted 22 november 2016 http://dx.doi.org/10.4995/agt.2017.2250 v. milani, s. m. h. mansourbeigi and h. finizadeh number is based on the pattern of tangle twists. according to conway’s theorem [2, 3], two rational tangles are equivalent if and only if they represent the same rational number. kauffman and lambropoulou gave another proof of this theorem in [9]. the key value of tangles is in their hidden algebraic structures. the more we discover these structures, the better we can handle them mathematically and apply in life science. towards the better understanding of tangles as mathematical objects in this paper we concern ourselves to the algebraic structures of tangles. the framework of this paper is as follows: section 2 is devoted to a review of the concept of rational tangles from [6, 7, 8, 9]. in this section we discuss basic definitions and explain the conway’s continued fraction method for assigning a rational number to a rational tangle. we also recall tangle operations. as we will see the class of rational tangles is not invariant under these operations. we modify this in the next section. in section 3 we are concerned in algebraic structures on rational tangles. we introduce a free product operation on rational tangles and show that the class of rational tangles is closed under this operation. the group hopf algebra structure comes up in the next step. we present the group hopf algebra structure of rational tangles explicitly. in section 4 we go towards a topological point of view in tangle study. we introduce a locally partial order to generate a topology on the class of rational tangles. there are tools to study the topology of partially ordered sets and we apply them [14]. the interval coalgebra structure is presented and its relation to the incidence algebra associated with the partial order is given. in section 5 we discuss the notion of irrational and real tangles and their infinite continued fractions. we see that any infinite chain of rational tangles has an upper bound in the set of real tangles and the existence of a maximal real tangle is explained. 2. rational tangles in this section we present a review of the concept of rational tangles from [6, 7, 8, 9]. basic definitions and the conway’s continued fraction method for assigning a rational number to a rational tangle is described in detail. some tangle operations are also recalled. definition 2.1. tangles consist of strings embedded in a 3-dimensional ball. within the ball there are no free ends, the ends of the strings are restricted to move on the surface of the ball while the rest of the tangle remains inside the ball. since a string has two ends, there is an even number of ends on the surface of the ball. inside the ball there may exist closed loops that are linked with the tangle strings and the strings of the tangle may themselves be knotted and linked. all throughout this paper we consider 2-tangles, i.e. 2-string tangles with four ends. we simply refer to them as tangles. definition 2.2. for n ∈ z, the horizontal tangles denoted by [n], are made of n half twists of two horizontal strings and the vertical tangles denoted by c© agt, upv, 2017 appl. gen. topol. 18, no. 1 2 algebraic and topological structures on rational tangles 1 [n] are made of n half twists of two vertical strings while the end points of the strings remain on the boundary of the ball. the directions of the half twists are specified by the ± signs, fig. 1. definition 2.3. the horizontal and vertical sums of two tangles t1 and t2 are denoted respectively by t1 + t2 and t1 ?t2 . they are obtained by connecting the endpoints of t1 to the endpoints of t2 as are shown in fig. 2. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 3 v. milani, s. m. h. mansourbeigi and h. finizadeh rational tangles belong to a subclass of tangles introduced by conway [2]. they are obtained from finite sums of horizontal and vertical tangles. definition 2.4. a tangle is rational if it can be obtained from the zero tangle [0] or the infinity tangle [∞] by twisting the strings while the endpoints of the strings remain moving on the surface of the ball. definition 2.5. two rational tangles t1 and t2 are isotopic if t2 is obtained from t1 by moving the strings of t1 continuously in such a way that no string penetrates either itself or another string. isotopy of rational tangles is an equivalence relation and we use the notation t1 ∼ t2 for equivalent rational tangles t1 and t2. the set of rational tangles is not invariant under the horizontal and vertical sums, as we will see later. in order to modify this we apply the continued fractions representation of conway [2] to encode the twists of rational tangles. in the following we introduce this representation in detail. definition 2.6. a rational tangle is said to be in standard form if it is created by consecutive horizontal sums of the tangles [±1] only on the right (left) and vertical sums of the tangles [±1] only at the bottom (top) starting from [0] or [∞]. so a rational tangle in standard form has an algebraic expression of the type: [an] ? 1 an−1 + [an−2] ? . . . ? 1 a2 + [a1]. for a2,a3, . . . ,an−1 ∈ z− 0. we let a1,an be [0] or [∞]. remark 2.7. in the process of creating a rational tangle, we may start with horizontal or vertical tangles. we always assume that we start twisting from [0] tangle. the shape of a rational tangle in standard form is encoded by associating to it a vector of integers (a1,a2, . . . ,an), where the i-th term represents |ai| half twists in the directions as in fig. 1. when i is odd it is horizontal and when i is even it is vertical. for a rational tangle this vector is unique, up to breaking the last term as (a1,a2, . . . ,an) = (a1,a2, . . . ,an − 1, 1); an > 0 (a1,a2, . . . ,an) = (a1,a2, . . . ,an + 1,−1); an < 0. for this reason the index n can be taken to be odd. remark 2.8. according to [9] every rational tangle is isotopic to a rational tangle in the standard form t = (a1,a2, . . . ,an), where n is odd and all terms are positive or negative (except possibly the first term). furthermore the standard form for a rational tangle is unique. from now on all rational tangles are considered to be in standard form. john conway [2, 3] associated a finite continued fraction to each rational tangle in the following way: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 4 algebraic and topological structures on rational tangles definition 2.9. let t = (a1,a2, . . . ,an) be a rational tangle. let [a1,a2, . . . ,an] := a1 + 1 a2 + . . . + 1 an−1+ 1 an be the continued fraction associated with t. the sum of the continues fraction is called the fraction of the tangle. for more details on continued fractions we refer to [4, 10]. proposition 2.10. two rational tangles are isotopic if and only if they have the same fractions. for the proof see [2]. 3. algebraic structures on rational tangles in this section our concern is in algebraic structures on rational tangles. we introduce a free product operation on rational tangles and show that the class of rational tangles is closed under this operation. we present the hopf algebraic structure of rational tangles. details on hopf algebra structure can be found in [1, 11, 12]. 3.1. free product of rational tangles. as we mentioned before the horizontal and vertical sums of two rational tangles are not necessarily a rational tangle. for example the horizontal sum of two rational tangles t1 = 1 [3] ,t2 = 1 [2] is not a rational tangle. in this section we define the free product of rational tangles and see that the class of rational tangles is closed under free product. definition 3.1. let t1 = (a1,a2, . . . ,an) , t2 = (b1,b2, . . . ,bm) be two rational tangles in the standard form. their free product which is denoted by t1 ~t2 is defined to be the standard form of the tangle (a1,a2, . . . ,an +b1,b2, . . . ,bm). we write t1 ~t2 = standard(a1,a2, . . . ,an + b1,b2, . . . ,bm). proposition 3.2. the class of rational tangles is a non commutative group under the free product. this group is denoted by rt . proof. obviously the class of rational tangles are closed under the free product. moreover the free product is associative and the tangle [0] is the identity element. now to any rational tangle t = (a1,a2, . . . ,an) we associate, as its inverse, the rational tangle t−1 = (−an,−an−1, . . . ,−a1). obviously t~t−1 = t−1~ t = [0]. so the class of rational tangles is a group under the free product. the free product is not commutative (up to isotopy). as we see for t1 = (1, 1, 2) and t2 = (1, 1, 3), we have t1 ~t2 � t2 ~t1 since they have different fractions and by the conway’s theorem they are not isotopic. � now we try to continue with more algebraic structures on rational tangles. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 5 v. milani, s. m. h. mansourbeigi and h. finizadeh 3.2. group hopf algebra structure. definition 3.3. let g be a noncommutative group and (a, +,•) be a (commutative or noncommutative) ring with the multiplicative unit element 1. let ψ : a×g → g be a mapping. the triple (a,g, ψ) is called a noncommutative left pseudo-module over a if • for all g ∈ g : ψ(1,g) = g. • for all g ∈ g and a,b ∈a : ψ(a, ψ(b,g)) = ψ(a• b,g). we can define the noncommutative right pseudo-module over a in the same way. definition 3.4. for t = (a1, . . . ,an) ∈ rt , and r ∈ z, we define the left scalar multiplication by rt := (ra1, . . . ,ran). right scalar multiplication is also defined by tr := (a1r, . . . ,anr). as we see rt = tr. remark 3.5. for r,s ∈ z and for t ∈rt it is not always true that (r + s)t = rt ~ st. for if we take r = −s, then −t = t−1. which is not true since for t = (a1,a2, . . . ,an), −t = (−a1, . . . ,−an); t−1 = (−an, . . . ,−a1). so the above scalar multiplication is not a module action of z on rt . but we have proposition 3.6. let ψ : z ×rt → rt be defined by ψ(r,t) = rt , for all r ∈ z and t ∈ rt . then the triple (z,rt , ψ) is a noncommutative left pseudo-module over z. proof. obviously ψ(1,t) = t , for all t ∈ rt . also for r,s ∈ z and t = (a1,a2, . . . ,an) ∈rt , we have ψ(r, ψ(s,t)) = r(st) = (r(sa1), . . . ,r(san)) = ((rs)a1, . . . , (rs)an) = (rs)(a1, . . . ,an) = ψ(rs,t). � remark 3.7. we can also make rt into a noncommutative right pseudo-module by defining φ : rt ×z →rt ; φ(t,r) = tr for all r ∈ z and t ∈ rt . also for all r,s ∈ z and all t ∈ rt , we have (rt)s = r(ts). we call rt a noncommutative bi-pseudo-module over z. definition 3.8. the pseudo-tensor product of two pseudo bi-modules m and n over the ring of integers z is the quotient of the free abelian group with basis the symbols m⊗n; for m ∈ m and n ∈ n; by the subgroup generated by −(m1 + m2) ⊗n + m1 ⊗n + m2 ⊗n −m⊗ (n1 + n2) + m⊗n1 + m⊗n2 c© agt, upv, 2017 appl. gen. topol. 18, no. 1 6 algebraic and topological structures on rational tangles (m,r) ⊗n−m⊗ (r,n) where m,m1,m2 ∈m,n,n1,n2 ∈n ,r ∈ z. remark 3.9. we can construct the pseudo-tensor product rt ⊗rt over the ring of integers z. we define the following mappings: ∆ :rt −→rt ⊗rt ; ∆(t) = t ⊗t σ :rt −→ z; σ(t) = 1 s :rt −→rt ;s(t) = t−1 µ :rt ×rt −→rt ; µ(t,s) = t ⊗s proposition 3.10. the group of rational tangles rt together with the comultiplication ∆, counit σ and antipodal s as defined above is a group hopf algebra. proof. for all t ∈rt , we have (id⊗ ∆) ◦ ∆(t) = t ⊗ (t ⊗t) = (t ⊗t) ⊗t = (∆ ⊗ id) ◦ ∆(t) (id⊗ σ) ◦ ∆(t) = t ⊗ 1 ∼ 1 ⊗t = (σ ⊗ id) ◦ ∆(t) µ◦ (s⊗ id) ◦ ∆(t) = s(t) ~t = t−1 ~t = [0] = σ(t)1rt µ◦ (id⊗s) ◦ ∆(t) = t ~s(t) = t ~t−1 = [0] = σ(t)1rt where 1rt = [0] is the identity element of the group rt . so the axioms of a group hopf algebra satisfies. � 4. topology generated by partial order on rational tangles in this section we go through topological point of view in tangle study. we introduce a locally finite partial order generating a topology on the class of rational tangles. the interval coalgebra structure is presented and its relation to the incidence algebra associated with the partial order is given. more details on the coalgebra structure can be found in [1, 11, 12]. definition 4.1. let t = (a1,a2, . . . ,an) be a rational tangle. for 1 ≤ k ≤ n, the rational tangle ck = (a1,a2, . . . ,ak) is called the k-th convergent of t. remark 4.2. the following relations are true for the convergents [4, 10]: a2, . . . ,an > 0 =⇒ c1 < c3 < c5 < ... < c6 < c4 < c2 a2, . . . ,an < 0 =⇒ c1 > c3 > c5 > ... > c6 > c4 > c2 definition 4.3. let t1 = (a1,a2, . . . ,an),t2 = (b1,b2, . . . ,bm) be two rational tangles. we write t1 ≤ t2 if n ≤ m and t1 is the n-th convergent of t2. proposition 4.4. the relation ≤ on the class of rational tangles is a locally finite partial order. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 7 v. milani, s. m. h. mansourbeigi and h. finizadeh proof. since each rational tangle is the convergent of itself, so the relation is reflexive. now let t1 = (a1,a2, . . . ,an),t2 = (b1,b2, . . . ,bm) be two rational tangles with t1 ≤ t2,t2 ≤ t1. then n ≤ m,m ≤ n and both t1 and t2 are convergents of each other. so t1 ∼ t2. now let t1 = (a1,a2, . . . ,an),t2 = (b1,b2, . . . ,bm),t3 = (c1,c2, . . . ,cl) be rational tangles with t1 ≤ t2,t2 ≤ t3. then n ≤ m,m ≤ l. also t1 is the n-th convergent of t2 and t2 is the m-th convergent of t3. so n = l and t1 is the n-th convergent of t3. now for each pair of rational tangles t1 ≤ t2 we define the interval [t1,t2] by [t1,t2] = {t : t ∈rt ,t1 ≤ t ≤ t2}. then [t1,t2] has a finite number of elements (up to isotopy) and so the partial order is locally finite. � for each rational tangle t ∈rt , we define λ(t) = {s : s ∈rt ,s ≤ t}. proposition 4.5. the family b = {λ(t)}t∈rt form a basis for a topology on rt . proof. since for t ∈ rt ,t ≤ t , we have t ∈ λ(t). also for the rational tangles t1,t2,t ∈rt , if t ∈ (t1) ∩ λ(t2), then t ∈ λ(t1) =⇒ t ≤ t1 =⇒ λ(t) ⊆ λ(t1) t ∈ λ(t2) =⇒ t ≤ t2 =⇒ λ(t) ⊆ λ(t2) so λ(t) ⊆ λ(t1) ∩ λ(t2). so the family b satisfies the axioms of a basis and generates a topology on rt . � definition 4.6. a subset u ∈rt is open if it is a union of the sets λ(ti) ∈b, where the index i belongs to some index set. so far we have seen that rt is a locally finite partial order set (a locally finite poset). the incidence algebra associated with this poset is defined by the following procedure: let i = {[t,s] : t,s ∈ rt ,t ≤ s}. let c(i) be the set of all functions from i to z. consider the convolution f ? g([t,s]) = ∑ t≤x≤s f([t,x])g([x,s]) c(i) is called the incidence algebra associated with the partial order. details on incidence algebra can be found in [13]. for all f,g ∈ c(i) we define ∆ : i −→i⊗i; ∆([t,s]) = ∑ t≤x≤s [t,x] ⊗ [x,s] σ : i −→ z; σ([t,s]) = 0. proposition 4.7. the set i together with the comultiplication λ and counit σ is a coalgebra. it is called the interval coalgebra. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 8 algebraic and topological structures on rational tangles proof. we check the axioms of the coalgebra. obviously for all [t,s] ∈i, (id⊗ ∆) ◦ ∆([t,s]) = (∆ ⊗ id) ◦ ∆([t,s]) (id⊗ σ) ◦ ∆([t,s]) = (σ ⊗ id) ◦ ∆([t,s]) � remark 4.8. the incidence algebra convolution is related to the comultiplication ∆ by f ? g([t,s]) = µz ◦ (f ⊗g) ◦ ∆([t,s]) where µz is the multiplication in z. 5. irrational tangles to each infinite standard continued fraction, there corresponds a 2-tangle. they are called irrational tangles. in this section we study their properties and their relation to rational tangles. definition 5.1. let {an}n≥1 be a sequence of integers all positive except the first term. the number (a1,a2, . . .) = lim n→∞ (a1, . . . ,an) is called an infinite standard continued fraction. proposition 5.2 ([9]). the above limit exists and is finite. furthermore to each irrational number, there corresponds a unique infinite standard continued fraction. definition 5.3. to each infinite standard continued fraction (a1,a2, . . .) there corresponds an irrational tangle obtained from consecutive ai half twists, starting with a horizontal |a1| half twist in direction according to the sign of a1. the number cn = (a1, . . . ,an) is called the n-th convergent of the irrational tangle. the set of all rational and irrational tangles is called real tangles. definition 5.4. for two irrational tangles t and s we write t ≤ s if every convergent of t is a convergent of s. definition 5.5. two irrational tangles t = (a1,a2, . . .) and s = (b1,b2, . . .) are said to be equivalent if lim n→∞ (a1, . . . ,an) = lim m→∞ (b1, . . . ,bm). in this case we write t ∼ s. proposition 5.6. for any two irrational tangles t and s we have t ≤ s ⇐⇒ t ∼ s ⇐⇒ s ≤ t. moreover for any rational tangle t = (a1, . . . ,an) there exists an irrational tangle s with t ≤ s. proof. the first part is obvious from the definition. for the second part let s = (a1,a2, . . . ,an), where the bar sign is for the repetition of the sequence. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 9 v. milani, s. m. h. mansourbeigi and h. finizadeh from the above proposition we have proposition 5.7. any infinite chain t1 ≤ t2 ≤ t3 ≤ . . . of rational tangles has an upper bound in the set of real tangles. proof. there is a sequence {ni}i≥1 of positive integers and a sequence {an}n≥1 of integers all positive except the first term such that, t1 = (a1, . . . ,an1 ),t2 = (a1, . . . ,an1,an1+1, . . . ,an2 ) , . . . ,tk = (a1,a2, . . . ,an1,an1+1, . . . ,an2, . . . ,ank ), . . . for all k ≥ 1. since all ai s (except possibly a1) are positive, then the irrational tangle s = (a1,a2, . . .) = lim k→∞ (a1, . . . ,ank ) satisfies ti ≤ s, for all i ≥ 1. � corollary 5.8. any chain t1 ≤ t2 ≤ t3 ≤ . . . . of real tangles has an upper bound in the set of real tangles. proof. if all ti s are rational, then from the above proposition, the upper bound exists. now if ti is irrational for some i ≥ 1, then for all j ≥ i we have ti ∼ tj. so the irrational tangle ti is the upper bound. � corollary 5.9. there exists a maximal real tangle. proof. by the zorn’s lemma and the above discussion, the maximal real tangle exists. � references [1] p. cartier, a primer of hopf algebras, preprint ihes (2006). [2] j. h. conway, an enumeration of knots and links and some of their properties, computational problems in abstract algebra (1986), 329–358. [3] j. h. conway and r. k. guy, continued fractions, in: the book of numbers, springer verlag (1996). [4] a. cuyt, a. b. petersen, b. verdonk, h. waadeland and w. b. jones, handbook of continued fractions for special functions, springer (2008). [5] i. k. darcy, modeling protein-dna complexes with tangles, computers and mathematics with applications 55 (2008), 924–937. [6] j. r. goldman and l. h. kauffman, rational tangles, adv. appl. math. 18 (1997), 300–332. [7] l. h. kauffman and s. lambropoulou, classifying and applying rational knots and rational tangles, contemporary math. 304 (2002), 223–259. [8] l. h. kauffman and s. lambropoulou, on the classification of rational knots, enseign. math. 49 (2003), 357–410. [9] l. h. kauffman and s. lambropoulou, on the classification of rational tangles, adv. appl. math. 33 (2004), 199–237. [10] l. lorentzen and h. waadeland, continued fractions, in: convergence theory, vol. 1, world scientific (2008). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 10 algebraic and topological structures on rational tangles [11] yu. i. manin, quantum groups and noncommutative geometry, centre de recherches mathematiques (crm), universite de montreal (1991). [12] j. w. milnor and j. c. moore, on the structure of hopf algebras, annals of math. 81 (1965), 211–264. [13] e. spiegel and c. j. o’donnell, incidence algebras, marcel dekker, new york (1997). [14] m. l. wachs, poset topology: tools and applications, lect. notes ias/parkcity math. inst. (2004). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 11 @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 391–419 hyperconvergences szymon dolecki and frédéric mynard dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. the hyperconvergence (upper kuratowski convergence) is the coarsest convergence on the set of closed subsets of a convergence space that makes the canonical evaluation continuous. sundry reflective and coreflective properties of hyperconvergences are characterized in terms of the underlying convergence. 1. introduction. convergences on hyperspaces (spaces of closed sets) have been attracting attention due to their numerous applications, but also for theoretic reasons. closed sets (of a convergence) can be identified with continuous maps (from that convergence) valued in the sierpiński topology $ = {∅,{1},{0, 1}}. therefore, a hyperspace is an instance of a function space. a convergence on a hyperspace that is of particular interest, is the coarsest among those convergences for which the natural evaluation map (from the product of a convergence space with its hyperspace) valued in the sierpiński space is continuous. we shall call it the hyperconvergence (or the upper kuratowski convergence). it follows from the definition that hyperconvergences constitute a particular instance of continuous convergences ; recall that the continuous convergence (of a convergence) is the coarsest convergence on the set of continuous maps from that convergence (valued in another convergence space) for which the natural evaluation map is continuous. on the other hand, hyperconvergences play a universal role among all continuous convergences (valued in topologies). this is because the sierpiński topology is initially dense in the category of all topologies, which means that every topology is the initial object with respect to maps valued in the sierpiński space [1]. therefore, every continuous convergence valued in a topology is the initial object with respect to maps valued in hyperconvergences. this fact has important consequences for continuous convergences as well as for quotient maps. 392 szymon dolecki and frédéric mynard hyperconvergence can be characterized as the (lower) order convergence on the complete lattice of closed sets; transposed onto the lattice of open sets, it is the scott convergence ([13] in the case of open sets of a topological space). we study hyperspaces with respect to convergences that are not necessarily t1. this is not because of a desire of utmost generality, but because of the very nature of the considered topic. indeed the hyperconvergence (for a nonempty convergence) is never t1, but is always t0. another peculiar property of hyperconvergences is hypercompactness : every filter converges. to express properties of hyperconvergences in terms of the corresponding properties of the underlying convergences, is an essential element of our quest. because hyperconvergences are defined on the spaces of closed sets, hyperconvergence properties typically correspond to hereditary properties (with respect to open sets) of the underlying convergences. for example, the hyperconvergence of a topology is topological if and only if the underlying topology is core-compact [13]. the very fact that the hyperconvergence of a topology need not be topological has been at the origin of convergence theory [5]. in this survey paper we not only present a collection of our recent results related to hyperconvergence [11], [17], [16], [15], [14], but also refine several of them and clarify certain links between them. therefore we have tried to make this paper self-contained providing full proofs in most cases. 2. notation. following engelking, a ⊂ x means that a is a (not necessarily proper) subset of x, and ac = x\a when x has been fixed. we consider now families of subsets of a fixed set. a family b is finer than a family a (in symbols, a≤b) if for every a ∈a there exists b ∈b such that b ⊂ a. families of sets e and h mesh (in symbols, e#h) provided e ∩ h 6= ∅ for each e ∈ e and every h ∈h; we denote by e# the grill of e, that is, the family of all sets that intersect every element of e. the symbol ec stands for the family {ec : e ∈e}. if f is a filter, then we denote by βf the set of all the ultrafilters which are finer than f. if a : 2y → 2y and if a is a family of subsets of y , then we denote by a\a = {aa : a ∈a} the a-regularization of a. if now h is a class of families of sets, then a\h = {a\h : h∈ h}. on the other hand, if v(y) is a family of subsets of x for every y ∈ y , then for a family a of subsets of y , v(a) = ⋃ a∈a ⋂ y∈a v(y) is the contour of v(·) along a. if a = {a} then we abridge v(a) = v({a}). if now x = y and {v(x) : x ∈ x} is given and we define x ∈ ava if and only if a ∈v#(x), then we have (2.1) v(a)#b ⇐⇒a#a\vb. hyperconvergences 393 by oξ(x) we denote the family of all ξ-open sets which contain x. therefore, oξ(a) stands for the family of all ξ-open sets which include a and oξ(a) =⋃ a∈aoξ(a). finally, we denote respectively by f−(y),f−(b) and f−(b) the preimage by f of y,b and b. if h ⊂ x ×y and a ⊂ x then the image of a by h is denoted by ha while h−b stands for the preimage of b ⊂ y by h. 3. hyperspace. the hyperspace associated with a topology (more generally, with a convergence) τ is the set of all τ-closed subsets. it is well known that there exists a one-to-one correspondence between closed subsets and continuous maps which take values in a two-element set endowed with the sierpiński topology $. indeed, if τ is a convergence (see section 5) on a set x and a ⊂ x, then the indicator map (3.1) 〈x,a〉 = { 0, if x ∈ a 1, if x /∈ a is continuous (in the first variable) from τ to $ = {∅,{1},{0, 1}} if and only if a is τ-closed. therefore c(τ, $) stands for the set of all τ-closed sets, equivalently for the set of all maps continuous from τ to $. in particular, the constant maps 〈x,∅〉 = 1 and 〈x,x〉 = 0 correspond to the empty set and the whole of x respectively. we shall sometimes call hyperpoints the elements of a hyperspace, hypersets the subsets of a hyperspace (that is, families of closed sets), and hyperfilters the filters on a hyperspace. the sierpiński topology plays an exceptional role among other topologies: it is initially dense in the category of all topologies. this means that every topology τ is the initial object with respect to maps valued in the sierpiński topological space: (3.2) τ = ∨ f∈c(τ,$) f−$, where f−θ denotes the initial convergence of θ by f. indeed, the statement above is an immediate consequence of the representation of closed sets with the aid of the coupling map (3.1) continuous in the first variable for every fixed a. 4. polarities, galois connections. we will see later that minimality of hyperconvergence can be rephrased in terms of polarity. a map f : x → y (between complete lattices x,y ) is called an inverse polarity whenever f( ∧ a) = ∨ f(a) 394 szymon dolecki and frédéric mynard for every a ⊂ x. each inverse polarity admits its adjoint map f∗(y) = ∧ {x : y ≥ f(x)} which is also a polarity. a couple of maps (4.1) f : x � y : g is called a (−,−)-connection if f and g are order-inverting and if gf and fg are contractive, that is, gf(x) ≤ x and fg(y) ≤ y. it is well known [3] that (4.1) is a (−,−)-connection if and only if f is an inverse polarity and g = f∗. it is often handy to invert order on one or both the lattices x,y . if we invert the order on x only, then we obtain a (+,−)-connection, with f and g order-preserving, with gf expansive (that is, gf(x) ≥ x) and fg contractive. similarly we define (−, +)-connections and (+, +)-connections, and refer to all of them as galois connections. a projector is an order-preserving, contractive and idempotent map, while an order-preserving, expansive, idempotent map is called a coprojector. an important consequence of the definitions above is that if (4.1) is a (−,−)connection, then f = fgf and gfg = g, hence gf and fg are projectors. moreover, the restrictions f : fix(gf) � fix(fg) : g constitute a couple of reciprocal lattice inverse isomorphisms. of course, if (4.1) is a (+,−)-connection, then gf is a coprojector while fg is a projector. finally, let us mention that (4.1) is a (−,−)-connection if and only if f(x) ≥ y is equivalent to x ≤ g(y) for every x and y. 5. convergence. we refer to [11], [17] for details of convergence theory. a convergence ξ on a set x is a relation between filters f on x and elements x of x x ∈ limξf such that limξf ⊂ limξg if f ≤ g and x ∈ limξ{x}↑ for each x ∈ x, where {x}↑ stands for the principal ultrafilter of x; the latter condition is that of strictness. please, have in mind that various authors give various (slightly different) definitions of convergence; a most frequent additional axiom is the one that we use to define prototopologies (see section 11). the adherence of a family h of sets is defined by adhξh = ⋃ e#h limξe. if ξ is a convergence then |ξ| stands for the set on which ξ is defined. a convergence ζ is finer then a convergence ξ (ξ is coarser than ζ) whenever limζf ⊂ limξf for every filter f. if ξ is a convergence on x and τ is a convergence on y, then a map f : x → y is continuous if f(limξf) ⊂ limτf(f). the final convergence (the finest convergence on y for which f is continuous from ξ) is denoted by fξ and the initial convergence (the coarsest convergence on x for which f is continuous to τ) is denoted by f−τ. hyperconvergences 395 a set v is a ξ-vicinity of x if v ∈f for every filter f such that x ∈ limξf; the family vξ(x) of vicinities of x is a filter. a convergence ξ is a pretopology if and only if x ∈ limξvξ(x) for every x. a set o is ξ-open if o ∈ f for every filter f such that limξf ∩ o 6= ∅ (ξ-closed if its complement is ξ-open). in other words, a set is ξ-open if and only if it is a ξ-vicinity of each its elements. a set w is a ξ-neighborhood of x if it includes a ξ-open set that contains x; the set of all ξ-neighborhoods of x is denoted by nξ(x). a convergence ξ is a topology if and only if x ∈ limξnξ(x) for every x. of course, a topological closure anξ = clξ is defined via (2.1). it is the closure for a topology tξ called topologization of ξ. a family g of filters is a base for a convergence ξ whenever x ∈ limξf implies the existence of g ∈ g such that g ≤f and x ∈ limξg. 6. hyperconvergence. the hyperconvergence (also called the upper kuratowski convergence) is the power convergence with respect to the sierpiński topology. in other words, if ξ is a convergence on x, then the hyperconvergence [ξ, $] associated with ξ is the coarsest convergence on c(ξ, $) such that 〈·, ·〉 : ξ × c(ξ, $) → $ is continuous. we say that ξ is a primal convergence of [ξ, $] and that [ξ, $] is the dual convergence of ξ. it follows directly from the definition of power convergence that a0 ∈ lim[ξ,$]g if and only if for every x ∈ |ξ| and each filter f on |ξ| such that x ∈ limξf, (6.1) 〈x,a0〉 ∈ lim$〈f, g〉. here g is a hyperfilter and a0 is a hyperpoint. the filter 〈f, g〉 is generated by 〈f,g〉 = {〈x,g〉 : x ∈ f,g ∈g}. it is possible to characterize the hyperconvergence in terms of the primal convergence. to this end we will need a notion of reduced filter. let us have in mind that if g is a ξ-hyperfilter (that is, a filter on c(ξ, $)) then every g ∈ g is a ξ-hyperset (that is, a family of ξ-closed sets). the reduced filter of a hyperfilter g is defined in [10] by rg ≈{ ⋃ a∈g a : g ∈ g}, where f ≈b means that f is generated by b. notice that the non degeneracy of a hyperfilter g does not imply the non degeneracy of the reduced filter rg; indeed, rg is degenerate if and only if there exists ∅ 6= g ∈ g such that⋃ a∈g a = ∅, that is, whenever g = {∅} which means that g = {∅} ↑ (the principal ultrafilter of the empty set ∅). of course, lim[ξ,$]{∅}↑ contains the hyperpoint ∅ and thus every ξ-closed subset a of |ξ|. on the other hand, adhξr({∅}↑) = adhξ∅↑ = ∅. proposition 6.1. [10] let g be a hyperfilter on c(ξ, $). then (6.2) a0 ∈ lim[ξ,$]g ⇐⇒ adhξrg ⊂ a0. 396 szymon dolecki and frédéric mynard proof. we have already noted that a0 ∈ lim[ξ,$]g is equivalent to (6.1) for every x ∈ |ξ| and each filter f on |ξ| such that x ∈ limξf. because the only $-neighborhood of 0 is the whole of {0, 1}, the formula above is significant only in the case of 〈x,a0〉 = 1, that is, x /∈ a0. therefore (6.1) means that if x /∈ a0 then there exist f ∈f and g ∈ g such that f ∩ ⋃ a∈g a = ∅, that is, f does not mesh with rg. we conclude that if x /∈ a0, then x /∈ adhξrg, and thus the proof is complete. � corollary 6.2. a hyperfilter g converges to a0 in [ξ, $] if and only if for every filter f such that limξf \a0 6= ∅, there is g ∈ g for which ⋂ a∈g a c ∈f. proof. by (6.2) a0 ∈ lim[ξ,$]g if and only if every filter f which meshes with rg does not ξ-converge to an element of ac0. therefore limξf∩ac0 6= ∅ implies that f does not mesh with rg, that is, there exist g ∈ g and f ∈f such that f ∩ ⋃ a∈g a = ∅, in other words, ⋂ a∈g a c ∈f. � a convergence is hypercompact provided that every filter converges. it follows from (6.2) that proposition 6.3. each hyperconvergence is hypercompact. proof. indeed |ξ| ∈ lim[ξ,$]g for every filter g, because adhξrg ⊂ |ξ|. � a convergence is said to be t1 if all its singletons are closed. except for the trivial case (of the discrete topology on a singleton), no hypercompact convergence is t1; in fact, if τ is a hypercompact convergence on y and y0 6= y1, then ∅ 6= limτ{y0,y1}↑ ⊂ limτ{y0}↑ ∩ limτ{y1}↑. hence if τ were t1, then limτ{y0}↑ ∩ limτ{y1}↑ = ∅. proposition 6.4. if |ξ| is non empty, then the hyperconvergence [ξ, $] is not t1. more precisely, if a ∈ c(ξ, $), then (6.3) cl[ξ,$]{a} = {b ∈ c(ξ, $) : a ⊂ b}, so that |ξ| ∈ cl[ξ,$]{∅}. therefore, if x 6= ∅ and ξ is any convergence on x, then the ξ-closed sets ∅ and x are distinct and x ∈ cl[ξ,$]{∅}. proposition 6.5. each hyperconvergence is t0. proof. indeed, if a0,a1 ∈ c(ξ, $) and x ∈ a1 \ a0, then by (6.3) x ∈ a for every a ∈ cl[ξ,$]{a1}, hence a0 /∈ cl[ξ,$]{a1}. � the hyperconvergence of the (unique) convergence on a singleton (this is necessarily the discrete topology ι) is homeomorphic to the sierpiński topology. in fact the (ι-closed) subsets of {x} are ∅ and {x}; for [ι, $], the principal hyperfilter of ∅ converges to ∅ and to {x}, while the principal hyperfilter of {x} converges to {x}; hence the map which associates {x} to 0 and ∅ to 1 is a homeomorphism. hyperconvergences 397 7. point and star topologies. with a view to characterizing reduced filters, we associate with each convergence two special topologies [4], [11]. if ξ is a convergence on x, then clξ•x = clξ{x} defines a (binary) relation on x. for our purposes, it is handy to use a special notation for the inverse relation, namely, clξ∗y = cl − ξ•y = {x : y ∈ clξ{x}}. as usual, the image and the preimage of sets by a relation are clξ•a = ⋃ x∈a clξ{x} and clξ∗b = {x : clξ{x}∩b 6= ∅} = ⋃ y∈b clξ∗y. both the operations clξ• and clξ∗ turn out to be topological closures. in fact, it is straightforward that they are expansive and that clξ•∅ = clξ∗∅ = ∅. the idempotency of clξ• and of clξ∗ follows immediately from that of clξ. as for finite additivity, we get more than needed: as the operations are relations, they are (fully) additive: clξ•( ⋃ i∈i ai) = ⋃ i∈i clξ•ai; clξ∗( ⋃ i∈i ai) = ⋃ i∈i clξ∗ai. the topology ξ• is called the point topology of ξ and the topology ξ∗ is called the star topology of ξ. if ξ is t1, then ξ• and ξ∗ are equal to the discrete topology. of course, ξ• ≥ ξ, but this is not necessarily the case with ξ∗. example 7.1. consider the discrete topology ι on {0, 1}. then the hyperconvergence [ι, $] is a topology and for instance cl[ι,$]•{{0}} = cl[ι,$]{{0}} = {{0},{0, 1}} while cl[ι,$]∗{{0}} = {∅,{0}}. as both the pointand the starclosures are additive, the collection of all ξ•-closed (respectively, ξ∗-closed) sets is also that of open sets of some topology. proposition 7.2. for every convergence ξ, a set is ξ•-closed if and only if it is ξ∗-open. proof. if y /∈ clξ•a = a then x /∈ clξ∗y for every x ∈ a, that is, a∩ clξ∗y = ∅. in other words, clξ∗ac ⊂ ac. � the topology ξ• has been defined with the aid of the ξ-closures of singletons. because there exists a unique filter that contains the singleton {x}, namely the principal ultrafilter {x}↑, clξ•x = clξ{x} = limtξ{x}↑. therefore y ∈ limξ• ∧ x∈clξ∗y {x}↑ = limξ•{clξ∗y}↑ and thus clξ∗y is a neighborhood of y with respect to ξ•. actually, another way of proving that a set is ξ•-open if and only if it is ξ∗-closed is to notice that lemma 7.3. nξ•(x) is the principal filter generated by clξ∗x. proof. observe that w ∈ nξ•(x) whenever x ∈ limξ•{y}↑ = clξ•y implies that y ∈ w , equivalently clξ∗x ⊂ w. this means that clξ∗x is the least ξ•neighborhood of x. � 398 szymon dolecki and frédéric mynard therefore, by (2.1), for two families a and b, (7.1) cl\ξ∗a#b ⇐⇒a#cl \ ξ•b. 8. reduced and erected filters. if ξ is a convergence on x, then for each subset h of x, we define the erected set eξh = {a = clξa : a ⊂ h}, and for a filter h on x, the erected filter of h (8.1) e\ξh = {eξh : h ∈h}. the erected filter of a non degenerate filter is never degenerate. indeed, {∅} belongs to every element of an erected filter. the (possibly degenerate) filter re\ξh is generated by the sets of the form intξ∗h = ⋃ clξa=a⊂h a, where h ∈h, that is, int \ ξ∗h = re \ ξh, and thus re\ξh≥h. a hyperset g is ξ-stable if b = clξb ⊂ g ∈ g implies b ∈ g. notice that g is ξ-stable if and only if gc = oξ(gc). a hyperset g is ξ-saturated if b = clξb ⊂ ⋃ a∈g a implies that b ∈ g. a hyperfilter g on c(ξ, $) is ξstable (respectively, ξ-saturated ) if it admits a filter base consisting of ξ-stable (respectively, ξ-saturated) hypersets. it is straightforward that proposition 8.1. a hyperfilter is an erected filter if and only if it is saturated. proposition 8.2. let f be a ξ-reduced filter and let g be a stable ξ-hyperfilter. then f meshes with rg if and only if the filters g and e\ξf restricted to c(ξ, $)\ {∅} mesh. proof. by definition, a ξ-reduced filter f meshes with rg if and only if ⋃ clξa=a⊂f a ∩ ⋃ b∈g b 6= ∅ for every f ∈ f and g ∈ g. equivalently, there is a non empty ξ-closed set a ∈ eξf ∩g for every f ∈ f and g ∈ g, because g is stable. � if now g is an arbitrary hyperfilter on c(ξ, $), then e\ξrg is coarser than g. indeed, a base of rg consists of ⋃ a∈g a where g ∈ g; therefore, e \ ξrg admits a base of the form {b = clξb ⊂ ⋃ a∈g a} with g ∈ g, and since g ⊂{b = clξb ⊂ ⋃ a∈g a}, e\ξrg ≤ g. it follows that proposition 8.3. the hyperconvergence [ξ, $] has a convergence base consisting of saturated hyperfilters. proof. if a ∈ lim[ξ,$]g, then adhξr(e \ ξrg) = adhξrg ⊂ a, hence a ∈ lim[ξ,$]e \ ξrg; as e\ξrg is erected, it is saturated. � proposition 8.4. [16] a filter is a reduced filter if and only if it is regular for the point topology if and only if it is open for the star topology. hyperconvergences 399 proof. if g ∈ g, and if x ∈ ⋃ a∈g a, then there is a ∈ g and thus clξ{x} ⊂ clξa = a ⊂ ⋃ a∈g a, so that rg is regular for the point topology of ξ. conversely, if clξ•h ⊂ h, then ⋃ clξa=a⊂h a ⊂ ⋃ x∈h clξ{x} = clξ•h ⊂ h, that is, if h is regular for ξ•, then re\ξh = h. by definition, a filter is ξ •-regular if and only if it admits a base of ξ•-closed sets, that is, by proposition 7.2, of ξ∗-open sets; equivalently the filter is ξ∗-open. � in general int\ξ∗h = re \ ξh≥h≥ cl \ ξ•h. by the proposition above, h = rg for some hyperfilter g on c(ξ, $) if and only if int\ξ∗h = h = cl \ ξ•h. consequently, we shall use the term ξ-reduced filter for regular for the point topology of ξ. let us denote by cl\ξ•s the set of filters that are regular for the point topology of ξ. if we observe that g0 ≤ g1 implies r g0 ≤ rg1 and h0 ≤ h1 implies e\ξh0 ≤ e \ ξh1, then we have proved that theorem 8.5. the operations of erection e\ξ and of reduction r constitute a galois (+,−)-connection. hence they form a lattice isomorphism between reduced filters and filters based in saturated families. it follows that for every family {hi : i ∈ i} of filters on |ξ|, (8.2) e\ξ (∨ i∈i hi ) = ∨ i∈i e\ξ hi, and for every family {gi : i ∈ i} of filters on c(ξ, $), (8.3) r( ∧ i∈i gi) = ∧ i∈i rgi. another immediate consequence of general properties of galois connections, is that (8.4) h≤ rg ⇐⇒ e\ξh≤ g. because [ξ, $] has a convergence base consisting of ξ-saturated filters, (8.5) adh[ξ,$]g = adh[ξ,$]e \ ξr g. 9. reflective and coreflective properties of convergences. the category of convergences with continuous maps as morphisms is concrete over the category of sets. this means that every morphism from ξ to τ is uniquely determined by a map from |ξ| to |τ|. a (covariant) functor f is a map on morphisms which preserves the composition; as every object ξ can be identified with the identity morphism ιξ : ξ → ξ, each functor f maps also objects following the rule ιf(ξ) = f(ιξ). a functor f between two concrete categories (over the category of sets) is concrete whenever each morphism and its image by f have the same underlying map. 400 szymon dolecki and frédéric mynard a concrete subcategory of convergences is reflective (respectively, coreflective) if it is stable for initial (respectively, final) convergences; in other words, m is reflective if τ ∈ m implies f−τ ∈ m, and ∨ i∈i τi ∈ m whenever τi ∈ m for each i ∈ i (m is coreflective if ξ ∈ m implies fξ ∈ m, and ∧ i∈i ξi ∈ m whenever ξi ∈ m for each i ∈ i). every concrete functor in the category of convergences is determined by its action on objects [1, remark 5.3]. an order-preserving map m on convergences such that |mξ| = |ξ|, is a concrete functor if and only if m(f−τ) ≥ f−(mτ) (equivalently, f(mξ) ≥ m(fξ)) for every f and every τ on the domain (respectively, each ξ on the range) of f. a concrete functor is called a reflector if it is a projector, a coreflector if it is a coprojector. from our point of view, a property of convergences is tantamount to a subcategory of convergences. we shall consider reflective properties, like topologicity, pretopologicity (corresponding to reflective subcategories) and coreflective properties, like countable character, fréchetness, sequentiality (corresponding to coreflective subcategories). a reflective concrete subcategory can be described as the class of those convergences τ for which (9.1) mτ ≥ τ, where m is a concrete functor; a coreflective concrete subcategory is the class of those convergences τ for which (9.2) τ ≥ mτ, where m is a concrete functor. if j is a class of filters possibly depending on convergence, then a convergence ξ is called adherence-determined by j if limξf = ⋂ j(ξ)3h#f adhξh for every filter f [7]. let (9.3) limajξf = ⋂ j(ξ)3h#f adhξh. we assume that ζ ≤ ξ implies j(ζ) ⊂ j(ξ), j(ajξ) = j(ξ) and that g ∈ j(ξ) implies that (the filter generated by) f−(g) belongs to j(ξ). then aj is a reflector. if j does not depend on convergence, then the first two conditions are automatically fulfilled. a convergence is a pseudotopology, paratopology, pretopology if it is adherence-determined by the class s (of all filters), pω (of countably based filters), p (of principal filters), respectively. these classes of filters are independent of convergence; moreover the preimage by a map of a filter from any of these classes, remains in the class. the corresponding reflectors s,pω,p are called the pseudotopologizer, the paratopologizer and the pretopologizer. a convergence is a topology if it is adherence-determined by the class t of principal filters of closed sets. the corresponding reflector t is the topologizer. hyperconvergences 401 a convergence ξ is called e-based (or, based in e) if x ∈ limξf implies the existence of a filter g ∈ e such that g ≤f and x ∈ limξg [7]. let (9.4) limbeξf = ⋃ e(ξ)3g≤f limξg. we assume that ζ ≤ ξ implies e(ζ) ⊂ e(ξ) and e(beξ) = e(ξ) and that g ∈ e implies that (the filter generated by) f(g) belongs to e. then be is a coreflector. if e does not depend on convergence, then the first two conditions are automatically fulfilled. for instance, a convergence is first-countable (or of countable character ) if and only if it is based in the class pω (of countably based filters); we denote by first = bpω the corresponding coreflector. we shall also consider the categories of convergences τ characterized by the inequality τ ≥ jeτ, where j is a concrete reflector and e is a concrete coreflector. such categories are coreflective, by (9.2). in particular, τ is bisequential if τ ≥ sfirstτ, strongly fréchet if τ ≥ pωfirstτ, fréchet if τ ≥ pfirstτ and sequential if τ ≥ tfirstτ. 10. power convergence. the power convergence (or the continuous convergence) [ξ,σ] of ξ with respect to σ is the coarsest convergence on the set c(ξ,σ) of continuous maps from ξ to σ for which the natural evaluation map is continuous. this is the only convergence structure satisfying the exponential law for arbitrary convergences ξ,τ,σ: (10.1) [ξ × τ,σ] = [τ, [ξ,σ]], where = stands for a homeomorphism via the transposition map tf(y)(x) = f(x,y). of course, hyperconvergences constitute a special case of power convergences, namely with respect to σ = $. recall that for a map f : x → y , the map f∗ : zy → zx is defined by f∗(h) = h ◦ f. it turns out that if f : ξ → τ is a continuous map, then (the restriction of ) f∗ : [τ,σ] → [ξ,σ] is also continuous. in particular, if ξ ≥ τ then c(τ,σ) ⊂ c(ξ,σ) and the injection from [τ,σ] to [ξ,σ] is continuous. moreover, (10.2) [ ∧ i∈i fiτi,σ] = ∨ i∈i (f∗i ) −[τi,σ] for every family of surjective maps {fi : i ∈ i} with a common range. if g : w → z, then the map g∗ : wx → zx is defined by g∗(h) = g ◦ h. let us observe that if g : σ → θ is a continuous map, then for every ξ the map g∗ : [ξ,σ] → [ξ,θ] is also continuous. by (3.2), (10.3) [ξ,σ] = [ξ, ∨ g∈c(σ,$) g−$] = ∨ g∈c(σ,$) g−∗ [ξ, $], so that every continuous convergence with respect to a topology σ is the initial convergence with respect to hyperconvergences. this fact implies that 402 szymon dolecki and frédéric mynard theorem 10.1. if for a given convergence ξ, the hyperconvergence [ξ, $] belongs to a concrete reflective subcategory j of convergences, then [ξ,σ] also belongs to j for every topology σ. a concrete reflective subcategory l of convergences is called cartesian-closed if [ξ,σ] ∈ l whenever ξ,σ ∈ l. the categories of convergences and of pseudotopologies are cartesian-closed, but that of topologies is not. theorem 10.2. if l is a concrete reflective subcategory of convergences and l is the corresponding reflector, then l is cartesian-closed if and only if l commutes with finite products. proof. assume that l commutes with finite products and that ξ,σ ∈ l. by definition, [ξ,σ] is the least convergence (on c(ξ,σ)) for which ξ× [ξ,σ] ≥ e−σ (where e−σ stands for the initial convergence of σ by the natural evaluation). because σ ∈ l and thus e−σ ∈ l, ξ ×l[ξ,σ] ≥ l(ξ × [ξ,σ]) ≥ l(e−σ) = e−σ, hence l[ξ,σ] ≥ [ξ,σ] proving that [ξ,σ] ∈ l. conversely, suppose that l[ξ,σ] ≥ [ξ,σ] for every ξ,σ ∈ l. in order to prove that ξ ×lτ ≥ l(ξ × τ) for all convergences ξ and τ, it is enough to show that c(ξ × τ,σ) ⊂ c(ξ × lτ,σ) for each σ = lσ. if f ∈ c(ξ × τ,σ), then tf ∈ c(τ, [ξ,σ]) hence tf ∈ c(lτ,l[ξ,σ]) because l is a functor. it follows that (tf)∗ ∈ c([l[ξ,σ],σ], [lτ,σ]) ⊂ c([[ξ,σ],σ], [lτ,σ]), because [l[ξ,σ],σ] ≥ [[ξ,σ],σ] by virtue of the assumption. as the injection i|ξ| is continuous from ξ to [[ξ,σ],σ], and (tf)∗ ◦ i|ξ| = f, we conclude that f ∈ c(ξ ×lτ,σ). � let x ⊂ y,θ be a convergence on x and τ be a convergence on y . if the injection ix : x → y is continuous, then we write θ b τ. it follows that i∗x : c(τ,σ) → c(θ,σ) is continuous from [τ,σ] to [θ,σ]; on the other hand, i∗x(f) = f ◦ ix, that is, i ∗ x is the restriction to x of maps on y . in the special case of σ = $, the restriction i∗x(a) is equivalent to the intersection of a τ-closed set a ∈ c(τ, $) with x. indeed, by definition, i∗x(ψa)(x) = ψa(ix(x)), where ψa is the indicator function of a subset a of y , hence i∗x(ψa)(x) = 0 if and only if an element x of x belongs to a. accordingly, for a family a of θ-closed sets, i∗∗x (a) = {a ∈ c(τ, $) : a ∈a}. if ζ ≥ ξ, then c(ξ,σ) ⊂ c(ζ,σ) for every convergence σ, and the injection is continuous from [ξ,σ] to [ζ,σ]. let (10.4) epiσξ = i−|ξ|[[ξ,σ],σ], where i|ξ| is the injection of |ξ| in c(c(ξ,σ),σ) (it is straightforward that i|ξ| is continuous from ξ to [[ξ,σ],σ]). hyperconvergences 403 consider now [·,σ] that associates with every convergence ξ on x, the power convergence [ξ,σ]. on the other hand, [θ,σ] is a convergence on c(θ,σ) for every convergence θ on a subset of |σ|x and each x ∈ x can be identified with a map continuous from [θ,σ] to σ. therefore, the couple of maps [·,σ], i−x[·,σ] (where i−x[θ,σ] is the restriction of [θ,σ] to x) is a (−,−)-connection with respect to the usual order on convergences at one end and to c on the other. it follows from the general properties of galois connections that [epiσξ,σ] = [ξ,σ] and that epiσ is a projector. it is clear from the definition that epiσ is a concrete functor, hence it is a reflector. 11. epitopologies. by taking the supremum on the right-hand side of (10.4) over all topologies σ, we define epiξ = ∨ σ=tσ epiσξ = ∨ σ=tσ i−|ξ|[[ξ,σ],σ], which in view of the preliminaries above is a concrete reflector. it is called the epitopologizer and each convergence τ such that epiτ = τ is called an epitopology (or antoine convergence). it can be deduced from (10.3) that the epitopologizer can be represented by epiξ = i−|ξ|[[ξ, $], $], thus epiξ = epi$ξ. consequently, the epitopologizer is the composition of two branches of the (−,−)-connection [·, $], i−x[·, $] (which is a special case of the connection considered in the previous section). it follows, in particular, that (11.1) [epiξ, $] = [ξ, $] and that epiξ is the coarsest convergence ζ for which [ζ, $] = [ξ, $]. proposition 11.1. the epitopologizer commutes with finite products: (11.2) ξ × epiτ ≥ epi(ξ × τ). notice that (11.2) applied twice yields epiξ × epiτ ≥ epi(ξ × τ). proof. as epi is idempotent, it is enough to prove that ξ×epiτ ≥ epi(ξ×τ). by virtue of the exponential law (10.1) and (11.1), [ξ × epiτ, $] = [ξ, [epiτ, $]] = [ξ, [τ, $]] = [ξ × τ, $]. therefore, ξ × epiτ ≥ epi(ξ × epiτ) = epi(ξ × τ). � hence, in view of theorem 10.2, the category of epitopologies is cartesianclosed. on the other hand, every topology is an epitopology. indeed, denote by ιy the discrete topology on y . because ιc(ξ,$) ≥ [ξ, $], the injection from c([ξ, $], $) to c(ιc(ξ,$), $) is continuous from [[ξ, $], $] to [ιc(ξ,$), $], and since the topologizer t can be characterized via tξ = ∨ f∈c(ξ,$) f−$ = i−|ξ|[ιc(ξ,$), $], 404 szymon dolecki and frédéric mynard it is clear that epiξ ≥ tξ for every convergence ξ, which proves the claim. we say that a convergence is a prototopology if limξf∩ limξg ⊂ limξ(f∧g) for every couple of filters f,g. it is immediate that prototopologies constitute a cartesian-closed reflective subcategory of convergences. theorem 11.2. [4] the category of epitopologies is the least cartesian-closed reflective subcategory of prototopologies that includes all topologies. proof. let l be a cartesian-closed reflective subcategory of prototopologies that contains all topologies. for every prototopology ξ, there exist a family {τi : i ∈ i} of topologies and maps fi : |τi| → |ξ| such that ξ = ∧ i∈i fiτi. by virtue of (10.2), [ξ, $] = ∨ i∈i (f∗i ) −[τi, $]. because l is cartesian-closed and contains all topologies, [τi, $] ∈ l for every i ∈ i, and since l is reflective [ξ, $] ∈ l as the initial object with respect to prototopologies in l. therefore [[ξ, $], $] ∈ l by the cartesian-closedness of l, and thus the initial prototopology epiξ = i−|ξ|[[ξ, $], $] belongs to l, because l is reflective. it follows that every epitopology belongs to l. � we shall now characterize epitopologies internally. we say that ξ is pointdiagonal if (11.3) limξf ⊂ limξnξ•(f) for every filter f. if ξ is a pseudotopology, then (11.3) is equivalent to (11.4) adhξcl \ ξ•h⊂ adhξh for every filter h. indeed, if x ∈ adhξcl \ ξ•h and ξ is point-diagonal, then there is a filter f#cl\ξ•h (equivalently, h#nξ•(f)) and such that x ∈ limξf ⊂ limξnξ•(f), and thus x ∈ adhξh. conversely, if (11.4) holds, ξ is a pseudotopology, and x /∈ limξnξ•(f), then there is a filter h#nξ•(f) (equivalently, f#cl\ξ•h) and such that x /∈ adhξh, hence by (11.4) x /∈ adhξcl \ ξ•h thus x /∈ limξf. it follows from (12.2) (to be proved later) that (11.5) limepiξf = ⋂ cl \ ξ•s3h#f clξ(adhξh). therefore, theorem 11.3. a convergence ξ is an epitopology if and only if it is a pointdiagonal pseudotopology with closed limits. proof. let ξ be an epitopology. then it is a pseudotopology, because limsξf =⋂ h#f adhξh ⊂ limepiξf. the closedness of limits follows from (11.5 ). if x /∈ limξnξ•(f), then by (11.5) there exists a ξ-reduced filter h#nξ•(f) such hyperconvergences 405 that x /∈ clξadhξh . because h#nξ•(f) is equivalent to cl \ ξ•h#f and since h = cl\ξ•h, also h#f, hence x /∈ limξf. conversely, let ξ be a point-diagonal pseudotopology with closed limits, and let x /∈ limξf = limξnξ•(f). as ξ is a pseudotopology, there is an ultrafilter u#nξ•(f) (equivalently, f#cl \ ξ•u) and such that x /∈ adhξu = adhξcl \ ξ•u. because u is an ultrafilter and ξ has closed limits, adhξu = limξu = clξadhξh where h = cl\ξ•u is a reduced filter which meshes with f. in view of (11.5), x /∈ limepiξf. � it follows from (11.5) and (11.4) that (11.6) clξ(adhξ(cl \ ξ•h)) = clξ(adhepiξh). this formula which will be instrumental in several arguments, is a special case of (12.3) that we will prove later. since ξ• is a principally based (called also finitely generated ) topology, and nξ•(x) = clξ∗x, the filter nξ•(f) is generated by {clξ∗f : f ∈ f}, that is, nξ•(f) = cl \ ξ∗f. therefore the condition (11.3) is equivalent to limξf ⊂ limξcl \ ξ∗f, and thus corollary 11.4. [4] a convergence ξ is an epitopology if and only if it is a star-regular pseudotopology with closed limits. proposition 11.5. every hyperconvergence is an epitopology. proof. from i−|ξ|[[ξ, $], $] ≥ [[ξ, $], $] it follows that [i − |ξ|[[ξ, $], $], $] c [[[ξ, $], $], $], hence [ξ, $] = [i−|ξ|[[ξ, $], $], $] ≤ i − c(ξ,$) [[[ξ, $], $], $] = epi[ξ, $]. � 12. duality. our goal is to characterize various properties of hyperconvergences in terms of primal convergences, or rather in terms of equivalence classes of primal convergences which correspond to the same hyperconvergence. as for every convergence ξ the epitopologization epiξ of ξ is the coarsest convergence such that [epiξ, $] = [ξ, $], primal characterizations of sundry properties of hyperconvergences are naturally formulated as properties of epitopologies. if m is a concrete functor such that m ≥ t, then c(mτ,σ) ⊂ c(τ,σ) for every convergence τ and each topology σ. thus i|ξ| is an injection into c(m[ξ, $], $) for every convergence ξ. therefore one can define [17][14] the m-epitopologizer by epimξ = i − |ξ|([m[ξ, $], $]). we say that ξ is an m-epitopology if ξ = epimξ. because $ is an initially dense object of the category of topologies, (12.1) epimξ ≥ i −([m[ξ,σ],σ]) for every topology σ (here i stands for the natural injection of |ξ| to c(c(ξ,σ),σ)). 406 szymon dolecki and frédéric mynard as epim is a composition of two branches of a (−,−)-connection and of m (inserted between them), we refer to this situation as to modified duality. it follows that if m is a coreflector, then epim is a reflector; however, if m is a reflector, then epim need not be a coreflector (in the category of convergences). we can describe explicitly the convergence of filters for epimξ (a filter f converges to x in epimξ if and only if for every ξ-closed set a with x /∈ a and each ξ-hyperfilter g such that rg meshes with f implies that a /∈ limm[ξ,$]g). however significant interpretations of this general description will be given for special cases, where j and e are $-compatible classes of filters, and m is either the reflector aj, or the coreflector be, or the composition ajbe (which is in general neither reflector nor coreflector). in this section we shall merely consider the reflector epibe which will be instrumental in the study of epiajbe that we will undertake in subsequent sections. a class e of filters is said to be $-compatible whenever g ∈ e implies that rg ∈ e and if h ∈ e implies that e\ξh ∈ e. the classes of all filters, of countably based filters, of countably deep filters (a filter f is countably deep if ⋂ a∈a a ∈ f for every countable a⊂f) and of principal filters are all $compatible. according to our notation cl\ξ•e stands for the class of ξ-reduced e-filters. in particular, cl\ξ•s is the class of ξ-reduced filters, cl \ ξ•pω of countably based ξreduced filters, and cl\ξ•p of principal ξ-reduced filters. we notice that be[ξ, $] is based in ξ-saturated filters for every convergence ξ, but ae[ξ, $] is not in general. proposition 12.1. [16] if e is $-compatible, then a filter f converges to x in epibeξ if and only if x ∈ clξ(adhξh) for every filter h ∈ cl \ ξ•e that meshes with f. in other words, (12.2) limepib e ξf = ⋂ cl \ ξ•e3h#f clξ(adhξh), proof. because be[ξ, $] is based in ξ-saturated e-filters, a filter f converges to x in epibeξ if and only if adhξrg \a 6= ∅ for every ξ-closed set a with x /∈ a and each ξ-saturated e-filter g such that rg meshes with f, equivalently, if x ∈ clξ(adhξrg) for every ξ-saturated e-filter g such that rg meshes with f. because of the duality between ξ-saturated and ξ-reduced e-filters for a $compatible class e, this means that x ∈ clξ(adhξh) for every ξ-reduced e-filter h that meshes with f. � since cl\ξ•e ⊂ e and, of course, adhξh ⊂ clξ(adhξh), we infer that aeξ ≥ epibeξ for every convergence ξ. therefore every be-epitopology is adherencedetermined by e-filters. in particular we recover the already established fact that each epitopology is a pseudotopology. another already established fact hyperconvergences 407 that each be-epitopology is an epitopology, is also an immediate consequence of (12.2). let us observe that epibpξ = tξ, that is, epibp is the topologizer. indeed, in this case (12.2) becomes ⋂ clξ•h∈f# clξ(adhξh) which is equal to ⋂ h∈f# clξh. therefore, every topology is a be-epitopology under the provision that e includes p, the class of principal filters. we notice that (12.3) clξ(adhξ(cl \ ξ•h)) = clξ(adhepib e ξh), for every e-filter h. indeed, if x ∈ adhepib e ξh then there exists a filter f that meshes with h and converges to x for epibeξ, hence cl \ ξ•h meshes with f, thus by (12.2), x ∈ clξ(adhξ(cl \ ξ•h)). on the other hand, by (11.4) adhξ(cl \ ξ•h) ⊂ adhepib e ξh because epibeξ is an epitopology. therefore, if b = oξ(b), then (12.4) adhepib e ξh#b ⇐⇒ adhξ(cl \ ξ•h)#b for every e-filter h. 13. compactoidness. it turns out that some properties of hyperconvergences can be rephrased in terms of some compactness-like properties of their underlying convergences. the concept of a set relatively compact with respect to another set has been extended by various authors (see [8] for a historical account) to a filter (and, more generally, family of sets) relatively compact with respect to a set, and in [9], with respect to another family. let f be a class of filters. if a,b are families of subsets of |ξ|, then a is said to be f-compactoid in b (for ξ) if adhξf ∈b# for every f ∈ f such that f#a. a family a is f-compact if it is f-compactoid in itself. in particular, if a ⊂ x, then {a} (equivalently, a = {a}↑) is compactoid in x (for a convergence on x) if and only if a is relatively compact; {a} (equivalently, a = {a}↑) is compact if and only if a is compact. compactoidness is a common generalization of compactness (of sets) and of convergence (of filters). indeed, a filter f is compactoid in {x} for ξ if and only if x ∈ limsξf, where s is the pseudotopologizer. this notion of compactoidness needs to be ultimately extended. let f and g be classes of filters, and let ξ and τ be convergences on a common set. a family a is said to be f g -compactoid in b (for ξ τ ) [8, section 8] if (13.1) ∀f∈f∀g∈g f ≥g, adhτg ∈a# =⇒ adhξf ∈b#. if g = p is the class of principal filters and τ = ι is the discrete topology, then ∀g∈p f ≥g, adhιg ∈a# is equivalent to f#a, and the whole condition (13.1) becomes ∀f∈ff#a =⇒ adhξf ∈b#, that is, to the f-compactoidness of a in b for ξ. 408 szymon dolecki and frédéric mynard the use of two classes of filters enables one to recover such compactness-like notions like lindelöf property, which is in these terms s pω -compactoidness. another important special case of (13.1) is that of cover compactoidness. a family p of subsets of |ξ| is a ξ-cover of a set a if every filter such that limξf ∩a 6= ∅ contains an element of p; a family p is an ideal if q ⊂ p ∈p implies that q ∈ p and if ⋃ r ∈ p for every finite subfamily r of p. it was observed in [8] that a family p is a ξ-cover of a if and only if adhξpc ∩a = ∅ (recall that pc = {pc : p ∈p}). it follows that theorem 13.1. [8] a family a is f g -compactoid in b (for ξ τ ) if and only if for every b ∈ b and each ξ-cover p of b such that pc ∈ f there exist a ∈ a and a refinement r of p which is a τ-cover of a such that rc ∈ g. if ξ = τ in the theorem above, then we say that a is f g -cover-compactoid in b for ξ. for example, a is s pω -cover-compactoid for ξ in b if for every (ideal) ξ-cover p of b ∈b there exist a ∈a and a countable subfamily r of p which is a ξ-cover of a. if ξ is a topology, then ideal can be dropped without altering the meaning [8], hence this means that a is lindelöf in b for ξ. let us recall that cover-compactoidness and compactoidness coincide for topologies, but do not for arbitrary convergences. at first we notice that convergence with respect to the epitopologization is a compactoidness property. more generally, proposition 13.2. if e is $-compatible, then a filter converges to x in epibeξ if and only if it is cl\ξ•e-compactoid in nξ(x) for ξ. proof. on rephrasing proposition 12.1, a filter f converges to x in epibeξ if and only if adhξh∩ o 6= ∅ for every ξ-open set o that contains x and each ξ-reduced filter h∈ e that meshes with f. � in particular, proposition 13.3. a filter converges to x in epiξ if and only if it is cl\ξ•scompactoid in nξ(x) for ξ. in the sequel we shall make repeated use of the following immediate consequence of (12.4): lemma 13.4. [11, lemma 6.3] let e be a $-compatible class of filters. if b is a family of ξ-open sets, then a family is e-compactoid in b for epibeξ if and only if it is cl\ξ•(e)-compactoid in b for ξ. it follows from proposition 13.2 and from lemma 13.4 that corollary 13.5. a filter converges to x in epibeξ if and only if it is ecompactoid in nξ(x) for epiξ. hyperconvergences 409 14. topological core compactness and commutativity with finite products. given a subcategory l of convergences, a convergence ξ is called l-exponential if [ξ,σ] ∈ l for every σ ∈ l. it follows immediately from the definitions that a subcategory is cartesian-closed if and only each of its objects is exponential. by theorem 10.2, a concrete reflective subcategory of convergences is cartesian-closed if and only if the corresponding reflector commutes with finite products. in fact more is true, namely, a convergence ξ is l-exponential if and only if ξ ×lτ ≥ l(ξ × τ) for every convergence τ (where l is the reflector corresponding to l). it is known [18] that a topology is t-exponential if and only if it is core compact, that is, for every x and each neighborhood o of x there is a neighborhood v of x which is compactoid in o [13]. we shall refine this classical result. let j and e be classes of filters. the following functor qj e in the category of convergences was defined in [17]: x ∈ lim q j e ξ f whenever for every o ∈ oξ(x) there exists a filter j ∈ j such that j ≤ f and j is e-compactoid in o for ξ. we say that a convergence ξ is topologically j-core e-compact at x if x ∈ limξf implies that x ∈ limsqj e ξ f. if j = p, the class of principal filters, then we abridge p-core to core. let us observe that (14.1) sqp e = qp e . indeed, if x ∈ limsqp e f then for every ultrafilter u finer than f and each o ∈ oξ(x) there exists fu ∈ u which is e-compactoid in o for ξ. therefore there exists n < ω and u1,u2, . . . ,un such that f = fu1 ∪fu2 ∪ . . .∪fun ∈f and obviously f is e-compactoid in o for ξ. a class g of filters is composable if ha ∈ g(y ) for every h ∈ g(x × y ) and a∈ g(x) and if it includes the class of principal filters. because the star and the point closures can be defined as relations, we conclude that proposition 14.1. every composable class is $-compatible. theorem 14.2. [17, theorem 6.2] if e and j are composable classes of filters, then (14.2) θ × ajτ ≥ t(ξ × τ) for every τ = beτ if and only if (14.3) θ ≥ sqj e epibeξ. proof. assume (14.3) and τ = beτ. since the reflector s commutes with the product, we can assume without loss of generality that θ ≥ qj e epibeξ. consider filters f and g such that x ∈ limθf and y ∈ limajτg and a (ξ × τ)closed set h that meshes with f × g. we will show that (x,y) ∈ h and thus (14.2) holds. by (14.3) for every w ∈ oξ(x) there exists a j-filter 410 szymon dolecki and frédéric mynard dw ≤ f which is e-compactoid in w for epibeξ. it follows that hdw #g and hdw ∈ j because j is composable. hence y ∈ adhτhdw and thus there exists an e-filter lw #hdw such that y ∈ limτlw . therefore dw meshes with the e-filter h−lw (because e is composable). we infer that there is xw ∈ adhepib e ξh −lw ∩w as dw is e-compactoid in w for epibeξ, and thus (xw ,y) ∈ h, because t(epibeξ × τ) = t(ξ × τ). accordingly, x ∈ clξh−y = h−y, that is, (x,y) ∈ h. conversely, if (14.3) does not hold, then there exist x0 and a free ultrafilter u such that x0 ∈ limθu and there is w ∈ oξ(x0) such that for every jfilter j ≤ u, there exists an e-filter ej that meshes with j and such that adhepib e ξej∩w = ∅. denote by j(u) the set of j-filters which are coarser than u. let τ be the convergence on x for which every point is isolated except for x0, and a free filter g converges to x0 if there is j in j(u) for which g ≥ej . this is an e-based convergence and x0 ∈ limajτu thus (x0,x0) ∈ limθ×ajτ (u×u). the filter u×u meshes with the set a = {(x,y) : x ∈ limepib e ξ{y}↑,y 6= x0}. let c be a filter on a such that (x,y) ∈ limξ×τc. if y 6= x0 then (x,y) ∈ a, because y is isolated in τ and limepib e ξ{y}↑ is ξ-closed for every y. if y = x0 then there exist filters j ∈ j(u) and f such that x ∈ limξf and f ×ej ≤c. obviously a meshes with f ×ej , that is, for each f ∈f and every e ∈ ej there exist x ∈ f and y ∈ e with x ∈ limepib e ξ{y}↑ = cl(epib e ξ)∗y. therefore cl \ (epib e ξ)∗ f meshes with ej . because epibeξ is star-regular, x ∈ limepib e ξcl \ (epib e ξ)∗ f, hence x ∈ adhepib e ξej and hence (x,x0) belongs to the (ξ × τ)-closed set wc×{x0}. it follows that (x0,x0) /∈ a∪wc×{x0}⊃ clξ×τa so that (x0,x0) /∈ limt(ξ×τ)(u ×u). � it is obvious that proposition 14.3. (14.2) holds for every τ = beτ if and only if θ × ajbeτ ≥ t(ξ × τ) for every convergence τ. 15. reflective properties of hyperconvergence. it has already been mentioned that a hyperconvergence associated with a topology need not be a pretopology (a fortiori need not be a topology). topologicity (tτ ≥ τ) and pretopologicity (pτ ≥ τ) are examples of reflective properties. we shall provide a general characterization of a reflective property of hyperconvergences in terms of a coreflective property of primal convergences. if j is a property of convergences, then by definition ξ ∈ j∗ whenever [ξ, $] ∈ j. if j is a concrete reflective subcategory of convergences, then by theorem 10.1, ξ ∈ j∗ if and only if [ξ,σ] ∈ j for every topology σ. proposition 15.1. if j is a concrete reflective subcategory, then j∗ is a (concrete) coreflective subcategory. hyperconvergences 411 proof. let fi : |ξi|→ y and ξi ∈ j∗ for every i ∈ i, that is, [ξi, $] ∈ j for each i ∈ i. then by (10.2), [ ∧ i∈i fiξi, $] = ∨ i∈i(f ∗ i ) −[ξi, $], and since by assumption j is a concrete reflective subcategory of convergences, ∨ i∈i(f ∗ i ) −[ξi, $] ∈ j, thus ∧ i∈i fiξi ∈ j∗. � theorem 15.2. [17] if m ≥ t is a concrete functor, then the following statements are equivalent: m[ξ, $] ≥ [ξ, $];(15.1) ξ ≥ epimξ;(15.2) ξ ×mτ ≥ t(ξ × τ) for every τ.(15.3) proof. if m[ξ, $] ≥ [ξ, $], then [[ξ, $], $] ≥ [m[ξ, $], $] so that ξ ≥ epiξ ≥ epimξ. conversely if ξ ≥ epimξ then by (12.1) the injection i : ξ → [m[ξ,σ],σ] is continuous. by (10.1), the evaluation ti : ξ ×m[ξ,σ] → σ is also continuous. but [ξ, $] is the coarsest convergence on c(ξ, $) that makes the evaluation continuous. therefore, m[ξ, $] ≥ [ξ, $]. assume (15.1) and let f ∈ c(ξ × τ,σ). then by (10.1) and because m is a functor, tf ∈ c(τ, [ξ,σ]) = c(mτ,m[ξ,σ]) ⊂ c(mτ, [ξ,σ]) by (15.1). again by (10.1 ), f ∈ c(mτ, [ξ,σ]) = c(ξ ×mτ,σ). therefore (15.3) holds. the hyperconvergence [ξ, $] is the coarsest convergence τ for which the natural evaluation is continuous from ξ×τ to $. therefore if (15.3) holds, then the evaluation is also continuous from ξ×m[ξ, $] to $ and thus m[ξ, $] ≥ [ξ, $]. � it follows from theorems 15.2 and 14.2 and from proposition 14.3 that ξ ≥ epiajbeξ ⇐⇒ ξ ≥ sq j e epibeξ (but this does not mean that epiajbe = sq j e epibe). this property of ξ is equivalent to a (reflective) property of [ξ, $] = [epiξ, $], hence it is a property of epitopologies. we conclude that corollary 15.3. if e and j are composable classes of filters, then ajbe[ξ, $] ≥ [ξ, $] if and only if epiξ is topologically j-core e-compact for epibeξ (equivalently ξ is j-core cl\ξ•e-compact). consider now several special cases of the condition above. as bs is the identity and as is the pseudotopologizer, and since every hyperconvergence is a pseudotopology, the condition as[ξ, $] ≥ [ξ, $] is always fulfilled. on the other hand, by corollary 15.3, if x ∈ limξf then adhepiξ h∩w 6= ∅ for every w ∈ oξ(x) and for each ultrafilter u finer than f and for every filter h that meshes with u; in other words, x ∈ clξ(adhepiξh), which is of course always the case. we apply now corollary 15.3 in the case e = s (the class of all filters) and j = pω (the class of countably based filters). 412 szymon dolecki and frédéric mynard theorem 15.4. the hyperconvergence [ξ, $] is a paratopology if and only if epiξ is topologically pω-core compact. if e = s is the class of all filters and j = p is the class of principal filters, then corollary 15.3 yields theorem 15.5. the hyperconvergence [ξ, $] is a pretopology if and only if epiξ is topologically core compact. thanks to theorem 10.1 we infer that corollary 15.6. if an epitopology ξ is topologically core (respectively, pωcore) compact, then [ξ,σ] is pretopological (respectively, paratopological) for each topology σ. corollary 15.3 cannot be used to characterize the topologicity of hyperconvergences, because t = at, but the class t of principal filters of closed sets is not composable; instead one can apply directly theorem 15.2 to another representation of the topologizer [11]. however, in the next section we will use yet another approach. 16. vicinities and neighborhoods. certain objects related to hyperconvergences, like vicinity or neighborhood, can be characterized in terms of primal convergences. this enables one to characterize, in terms of primal convergences, various reflective and coreflective properties of hyperconvergences that can be formulated with the aid of such objects. some of the reflective properties of hyperconvergences have been characterized already by virtue of theorem 15.2. the situation is different for the coreflective properties of hyperconvergences, because for them there exists no equivalence scheme similar to theorem 15.2. in order to characterize hypervicinities and open hypersets, we introduce a notion of rigid compactoidness. a family h ξ-rigidly meshes with a family a if for every h ∈ h, there is b = clξb ⊂ h such that b ∈ a#. of course, if h ξ-rigidly meshes with a, then it meshes with a. on the other hand, if h is ξ-closed (that is, h = cl\ξh) and meshes with a, then it ξ-rigidly meshes with a. a family a is rigidly e-compactoid in b for ξ if adhξe ∈b# for each e-filter e which ξ-rigidly meshes with a; a family is rigidly e-compact if it is rigidly e-compactoid in itself. every e-compactoid (e-compact) family is rigidly ecompactoid (rigidly e-compact). if ξ is a topology, then the converse also holds, because then for every filter h, one has adhξh = ⋂ h∈h clξh and thus adhξh = adhξcl \ ξh. lemma 16.1. a set f is cl\ξ•s-compactoid in b for ξ if and only if oξ(f) is rigidly compactoid in b for ξ. hyperconvergences 413 proof. we notice that h rigidly meshes with oξ(f) if and only if int \ ξ∗h meshes with f. indeed, the former means that for each h ∈ h there is a subset d = clξd of h such that d ∈ oξ(f)#, equivalently clξd ∩ f 6= ∅, or else int\ξ∗h meshes with f. therefore if oξ(f) is rigidly compactoid in b for ξ and int \ ξ∗h = h = cl \ ξ•h meshes with f, then h rigidly meshes with oξ(f) and thus adhξh∩b 6= ∅. conversely if f is cl\ξ•s-compactoid in b for ξ and h rigidly meshes with oξ(f), then the ξ-reduced filter int\ξ∗h meshes with f and thus adhξ(int \ ξ∗h) ∩b 6= ∅. � it is straightforward that each intersection of stable hyperfilters is stable; in particular, each vicinity filter for a hyperconvergence is stable. on the other hand, an open hyperset (that is, open for a hyperconvergence) is always stable. observe that a is ξ-stable if and only if ac = oξ(ac). let us characterize stable vicinities. theorem 16.2. let e be $-compatible. a hyperset a is a stable vicinity of a0 for be[ξ, $] if and only if ac = oξ(ac) is rigidly e-compactoid in ac0 for ξ. proof. a hyperset a is a vicinity of a0 for be[ξ, $] if and only if a∈ e for every e-hyperfilter e that [ξ, $]-converges to a0. equivalently, if for each ξ-reduced e-filter e with adhξe ⊂ a0 there exists e ∈e such that if b = clξb ⊂ e, then b ∈ a. in other words, for every e ∈ e there exists b = clξb ⊂ e such that b /∈ a#c , then adhξe∩ac0 6= ∅; equivalently, ac is rigidly cl \ ξ•e-compactoid in ac0 for ξ. � corollary 16.3. let e be $-compatible. a hyperset a is open for be[ξ, $] if and only if ac = oξ(ac) and is rigidly e-compact for ξ. this refines a result of [11, corollary 5.3]; if ξ is a topology, in the case where e is the class of all filters this recovers [10, theorem 3.1] of dolecki, greco and lechicki, and for the class e of countably based filters [2, theorem 2.1] of alleche and calbrix. we will use the characterizations above to describe the pretopologicity and the topologicity of hyperconvergences in terms of primal convergences. the theorem below is identical with theorem 15.5; what differs is the proof. first observe that, as every hyperconvergence has a convergence base consisting of saturated filters, every vicinity filter has a filter base consisting of saturated, hence stable, hypersets. theorem 16.4. the hyperconvergence [ξ, $] is a pretopology if and only if epiξ is topologically core compact. proof. by definition, [ξ, $] is a pretopology if and only if a0 ∈ lim[ξ,$]v[ξ,$](a0), where v[ξ,$](a0) denotes the filter of vicinities of a0 for [ξ, $]. in other words, adhξrv[ξ,$](a0) ⊂ a0. equivalently, if limξf ∩ac0 6= ∅, then f does not mesh with rv[ξ,$](a0), in other words, there is f ∈ f and a stable a ∈ v[ξ,$](a0) such that f ∩ ⋃ a∈aa = ∅, that is, f ⊂ ⋂ a∈aa c. by theorem 16.2, this 414 szymon dolecki and frédéric mynard means that for every ξ-open set o and each filter f such that limξf ∩o 6= ∅ there exists a family b = oξ(b) which is rigidly compactoid in o for ξ and such that ⋂ b∈b b ∈ f. it follows that ξ is topologically core cl \ ξ•s-compact. indeed, as b is rigidly compactoid in o for ξ, and b ⊂oξ(f), the latter is also rigidly compactoid in o for ξ, thus by lemma 16.1, f is cl\ξ•s-compactoid in o for ξ. conversely, if ξ is topologically core cl\ξ•s-compact and limξf ∩o 6= ∅, then there exists f which is cl\ξ•s-compactoid in o for ξ, hence by lemma 16.1 b = oξ(f) is rigidly compactoid in o for ξ and of course ⋂ b∈b b ∈ f. finally, since o is ξ-open, by lemma 13.4, it follows that f is compactoid in o for epiξ. � a convergence is a topology if and only if the filter of neighborhoods of every point converges to that point. once again, as [ξ, $] = [epiξ, $], we can formulate any characterization of the pretopologicity of hyperconvergences in terms of the underlying epitopologies. theorem 16.5. the hyperconvergence [ξ, $] is a topology if and only if for every ξ-open set o and each filter f such that limξf ∩ o 6= ∅ there exists a family b = oξ(b) which is rigidly compact for epiξ and such that o ∈ b and⋂ b∈b b ∈f. as every compact family is rigidly compact, the existence of a compact family b in the statement above is a sufficient condition for the topologicity of [ξ, $] (and by theorem 10.1, of [ξ,σ] for every topology σ). in the special case where ξ is a topology, this is a condition of day and kelly [6] (who did not use the name compact family, but gave an equivalent definition) and the characterization of core compactness [13, proposition 4.2] by hofmann and lawson, if we remember that a family b = oξ(b) is compact if and only if it is an open set for the scott topology associated with a topology ξ. therefore it must follow that if the primal convergence is a topology, then the topologicity and the pretopologicity of the hyperconvergence coincide. this is actually the case. we do not provide here the proof, because it uses the technique of multifilters, and would require quite a few preliminaries. theorem 16.6. [11, theorem 5.6] if ξ is a topology, then [ξ, $] is a topology if and only if [ξ, $] is a pretopology. example 16.7. consider the bisequence {x∞}∪{xn : n < ω}∪{xn,k : n,k < ω} endowed with the canonical pretopology π: the filter associated with (xn)n is the coarsest free filter that converges to x∞ and for every n < ω, the filter associated with (xn,k)k is the coarsest free filter that converges to xn. this is a hausdorff locally compact pretopology (hence a point-diagonal pseudotopology with closed limits), so that its hyperconvergence is pretopological in view of theorem 16.4. on the other hand, the topologization of π is hausdorff regular, but not locally compact, thus [tπ, $] is not a pretopology. hyperconvergences 415 17. coreflective properties of hyperconvergence. let us start by considering the case of coreflectors of the form m = be, where be is defined in (9.4). theorem 17.1. let e be a $-compatible class of filters. the hyperconvergence [ξ, $] is e-based if and only if every ξ-open set is s e -compactoid for ξ (equivalently, for epiξ). proof. let [ξ, $] be e-based and let a be an ξ-closed set. to show that ac is s e -compactoid let g to be a filter such that adhξg ⊂ a, that is, a ∈ lim[ξ,$]e \ ξ•g. by the assumption there exists an e-filter e coarser than e\ξ•g for which a ∈ lim[ξ,$]e, hence adhξre ⊂ a. the filter e = re is a ξ-reduced e-filter, because e is $-compatible, and e ≤ re\ξg. for every e ∈ e let ge ∈ g be such that intξ∗ge ⊂ e. then the filter h generated by {ge : e ∈ e} is an e-filter coarser than g and than e and hence adhξh⊂ a. conversely if every ξ-open set is s e -compactoid and a ∈ lim[ξ,$]g, then adhξrg ⊂ a, and thus there exists an e-filter e coarser than rg such that adhξe ⊂ a, because ac is s e -compactoid. hence a ∈ lim[ξ,$]e \ ξe and e \ ξe belongs to e, because e is $-compatible, and e\ξe ≤ e \ ξrg ≤ g. � as a corollary we get (for general convergences, open hereditarily cover lindelöf need not entail hereditarily cover lindelöf ). theorem 17.2. [15] the hyperconvergence [ξ, $] is of countable character if and only if ξ (equivalently, epiξ) is open hereditarily cover lindelöf. observe that a convergence τ is respectively bisequential, strongly fréchet, fréchet and sequential if and only if sτ ≥ sfirstτ,pωτ ≥ pωfirstτ, pτ ≥ pfirstτ and tτ ≥ tfirstτ. in other words, a convergence is fréchet whenever every sequential vicinity of a point is a vicinity. therefore, theorem 17.3. the hyperconvergence [ξ, $] is fréchet if and only if every family b = oξ(b) which is rigidly countably compactoid in a ξ-open set o, is rigidly ξ-compactoid in o. a convergence is sequential whenever every sequentially open set is open. hence theorem 17.4. the hyperconvergence [ξ, $] is sequential if and only if every family b = oξ(b) which is rigidly countably ξ-compact is rigidly ξ-compactoid. recall that a topology τ is tight if x ∈ clτa implies the existence of a countable subset e of a such that x ∈ clτe. as a special case of [15, theorem 3.7], we have theorem 17.5. if ξ is a topology and if t [ξ, $] is tight, then ξ is hereditarily lindelöf. 416 szymon dolecki and frédéric mynard proof. let g be a filter such that adhξg ⊂ a = clξa, that is, a ∈ lim[ξ,$]e \ ξ•g. in particular, a ∈ cl[ξ,$]{clξg : g ∈g}, and since t [ξ, $] is tight, there exists a sequence (gn) such that a ∈ cl[ξ,$]{clξgn : n < ω}, that is, for every ξ-compact family b = oξ(b) such that ac ∈b there exists n < ω such that (clξgn)c ∈b. it follows that adhξ(gn) = ⋂ n<ω clξgn ⊂ a, for if x ∈ ⋂ n<ω clξgn \a, then (clξgn)c /∈ oξ(x) for every n < ω and ac ∈ oξ(x). of course, oξ(x) is tξ-compact, which leads to a contradiction. � corollary 17.6. let ξ be a topology. the following are equivalent: (1) ξ is hereditarily lindelöf; (2) [ξ, $] is first-countable; (3) [ξ, $] is countably-tight; (4) [ξ, $] is sequential; (5) t [ξ, $] is sequential; (6) t [ξ, $] is countably-tight; recall that every perfectly normal lindelöf topology is hereditarily lindelöf [12, exercise 3.8.a]. notice that, unlike in the cited book of engelking, our definition of lindelöf convergence does not involve any separation axiom. on the other hand, the atomic cofinite topology on an uncountable set is (hausdorff) compact, hence lindelöf, but not hereditarily lindelöf (for the atomic cofinite topology on x all elements but one are isolated, and a subset of x is a neighborhood of the non isolated point whenever its complement is finite). indeed, the restriction of this topology to the (open) complement of the non isolated element is the discrete topology on an uncountable set. 18. lattice-theoretic approach to hyperconvergence. if l is a complete lattice with respect to an order ≤, then three convergences arise naturally: lower, upper, and their infimum. a filter f converges to x in the upper convergence (x ∈ lim+f) if lim supf = ∧ f∈f ∨ f ≤ x. this formula obviously defines a convergence, because f ≤g implies lim supf ≥ lim supg, and x = ∨ f3x ∧ f. clearly, lim supf ≤ x ≤ y implies that lim supf ≤ y, that is, every limit is upper stable. it follows that every +closed set is upper stable, hence each +-open set is lower stable. by definition, a subset o of l is +-open if ∧ f∈f ∨ f ≤ x for a filter f and x ∈ o then o ∈ f; in other words, if ∧ a ∈ o then there is a finite subset b of a such that ∧ b ∈ o. open sets for the lower convergence (that is, the upper convergence with respect to the inverse order) were characterized by d. scott [19] with the aid of the relation of being way below: an element x (of a complete lattice) is way below an element y (in symbols, x � y) whenever ∨ a ≤ y implies∨ f ≤ x for some finite subset f of a. accordingly, a set o is open for the lower convergence if x ∈ o and x ≤ y imply y ∈ o, and if ∨ a ∈ o, then there hyperconvergences 417 is a finite subset b of a such that ∨ b ∈ o. in the special case of the lattice of open sets for a topology ξ, this means precisely that b is open for the lower topology if and only if oξ(b) = b is ξ-compact. a complete lattice is called continuous if its lower convergence is topological, that is, if each neighborhood filter converges to its defining point; in other words, if y = ∨ {x : x � y} for every y ∈ l. in [13] hofmann and lawson proved that the lattice of open sets of a topology ξ is continuous if and only if ξ is core compact. this is dual to the result from preceding paragraphs that a topology ξ is core compact if and only if the hyperconvergence [ξ, $] is topological. moreover, hofmann and lawson showed that in the class of so-called sober topologies, core compactness is equivalent to hereditary local compactness. we shall relate these facts in terms of hyperspaces. the hyperspace c(ξ, $) ordered by inclusion is a complete lattice in which the supremum and the infimum are given by∨ a = clξ( ⋃ a∈a a); ∧ a = ⋂ a∈a a. if ξ is a topology, then adhξh = ⋂ h∈h clξh, hence a ∈ lim[ξ,$]f if and only if ∧ f∈f ∨ f ⊂ a, that is, the hyperconvergence (of a topology) is the upper convergence in the lattice of closed sets. an element p of a lattice is prime if a∨ b ≥ p implies that either a ≥ p or b ≥ p. a ξ-closed set is called irreducible if it is prime in the lattice c(ξ, $), that is, if it cannot be written as the union of two proper ξ-closed subsets. notice that clξ{x} is irreducible for every x ∈ |ξ| and for each convergence ξ. the whole space of the cofinite topology on an infinite set is an example of irreducible closed set which is not the closure of a point. a t0 convergence is called sober if every irreducible closed set contains a dense point. each hausdorff topology is sober; the cofinite topology on an infinite set is a t1 non sober topology. being sober is of course a topological property. the lower vietoris (or lower kuratowski ) topology v−(ξ) on c(ξ, $) has the following subbase of open sets: o− = {a ∈ c(ξ, $) : a ∩ o 6= ∅} where o ∈oξ. notice that for a hyperset a, a hyperpoint a0 ∈ clv−(ξ)a if and only if a0 ⊂ clξ( ⋃ a∈aa), that is, whenever a0 ≤ ∨ a in the lattice c(ξ, $). the spectrum σ(ξ) of c(ξ, $) is the set of non empty irreducible ξ-closed sets, endowed with the restriction of the lower vietoris topology. in other words, if we let cl\ξu = {clξ{x} : x ∈ u}, then the topology of σ(ξ) is generated by the hypersets σ(ξ) \ cl\ξa with a ∈ c(ξ, $). in our terms, [13, proposition 2.7] implies that proposition 18.1. the map clξ : |ξ| → σ(ξ) is continuous and open on its image; it is injective if and only if it is an embedding if and only if ξ is t0; it is bijective if and only if it is a homeomorphism if and only if ξ is sober. proof. the first assertion follows from the fact that if o is ξ-open, then clξ{x}∈ o− whenever clξ{x}∩o 6= ∅, that is, whenever x ∈ o. to prove the second statement, notice that ξ is t0 if and only if x0 6= x1 implies clξ{x0} 6= clξ{x1}. 418 szymon dolecki and frédéric mynard the third statement is a consequence of the fact that ξ is sober if and only if the only non empty irreducible ξ-closed sets are the closures of singletons. � we prove here directly the following conclusion of [13, proposition 2.7]. proposition 18.2. the spectrum is sober. proof. let a ⊂ σ(ξ) be a v−(ξ)-closed hyperset (that is, {a ∈ σ(ξ) : a ⊂ clξ( ⋃ b∈ab)}) which is not the closure of a hyperpoint of σ(ξ); in other words, the set clξ( ⋃ b∈ab) is reducible: there exist ξ-closed sets a0,a1 such that clξ( ⋃ b∈ab) = a0 ∪ a1 and a1 \ a0 6= ∅ and a0 \ a1 6= ∅. the hypersets a0 = {a ∈ σ(ξ) : a ⊂ a0} and a1 = {a ∈ σ(ξ) : a ⊂ a1} are v−(ξ)-closed. let a ⊂ a0 ∪ a1, where a is an element of σ(ξ). as a is irreducible, either a∩a0 is a subset of a∩a1 and thus a ⊂ a1 or vice versa and thus a ⊂ a0. this shows that a is reducible and hence σ(ξ) endowed with v−(ξ) is sober. � it follows that every t0 topology ξ can be embedded in a sober topology (the sobrification of ξ). references [1] j. adámek, h. herrlich, and e. strecker. abstract and concrete categories. john wiley and sons, inc., 1990. [2] b. alleche and j. calbrix. on the coincidence of the upper kuratowski topology with the cocompact topology. topology appl., 93:207–218, 1999. [3] g. birkhoff. lattice theory. a.m.s., 1967. [4] g. bourdaud. espaces d’antoine et semi-espaces d’antoine. cahiers de topologie et géométrie différentielle, 16:107–133, 1975. [5] g. choquet. convergences. ann. univ. grenoble, 23:55–112, 1947-48. [6] b. j. day and g. m. kelly. on topological quotient maps preserved by pullbacks or products. proc. camb. phil. soc., 67:553–558, 1970. [7] s. dolecki. convergence-theoretic methods in quotient quest. topology appl., 73:1–21, 1996. [8] s. dolecki. convergence-theoretic characterizations of compactness. topology appl., 125:393-417, 2002. [9] s. dolecki, g. h. greco, and a. lechicki. compactoid and compact filters. pacific j. math., 117:69–98, 1985. [10] s. dolecki, g. h. greco, and a. lechicki. when do the upper kuratowski topology (homeomorphically, scott topology) and the cocompact topology coincide? trans. amer. math. soc., 347:2869–2884, 1995. [11] s. dolecki and f. mynard. convergence-theoretic mechanisms behind product theorems. topology appl., 104:67–99, 2000. [12] r. engelking. topology. heldermann verlag, 1989. [13] k. h. hofmann and j. d. lawson. the spectral theory of distributive continuous lattices. trans. amer. math. soc., 246:285–309, 1978. [14] f. mynard. coreflectively modified duality. rocky mountain mathematical journal., to appear. [15] f. mynard. first-countability, sequentiality and tightness of the upper kuratowski convergence. rocky mountain mathematical journal., 33(4), winter 2003, to appear. [16] f. mynard. strongly sequential spaces. comment. math. univ. carolinae, 41:143–153, 2000. [17] f. mynard. coreflectively modified continuous duality applied to classical product theorems. appl. gen. topology, 2:119–154, 2002. hyperconvergences 419 [18] f. schwarz. powers and exponential objects in initially structured categories and application to categories of limits spaces. quaest. math., 6:227–254, 1983. [19] d. scott. continuous lattices. in f. w. lawvere, editor, toposes, algebraic geometry and logic. springer-verlag, 1972. lecture notes in math. 274. received february 2002 revised september 2002 szymon dolecki département de mathématiques, université de bourgogne, b. p. 47870, 21078 dijon, france e-mail address : dolecki@u-bourgogne.fr frédéric mynard department of mathematics, hume hall 308, university of mississippi, university, ms 38677, usa e-mail address : mynard@olemiss.edu () @ appl. gen. topol. 16, no. 1(2015), 81-87doi:10.4995/agt.2015.3247 c© agt, upv, 2015 a generalized version of the rings ck(x) and c∞(x)– an enquery about when they become noetheri sudip kumar acharyya a, kshitish chandra chattopadhyay b and pritam rooj a,∗ a department of pure mathematics, university of calcutta, 35, ballygunge circular road, kolkata 700019, west bengal, india (sdpacharyya@gmail.com, prooj10@gmail.com) b department of mathematics, university of burdwan, golapbag, burdwan 713104, west bengal, india (kcchattopadhyay2009@gmail.com) abstract suppose f is a totally ordered field equipped with its order topology and x a completely f-regular topological space. suppose p is an ideal of closed sets in x and x is locally-p. let cp(x, f) = {f : x → f | f is continuous on x and its support belongs to p} and cp∞(x, f) = {f ∈ cp(x, f) | ∀ε > 0 in f , clx{x ∈ x : |f(x)| > ε} ∈ p}. then cp(x, f) is a noetherian ring if and only if c p ∞(x, f) is a noetherian ring if and only if x is a finite set. the fact that a locally compact hausdorff space x is finite if and only if the ring ck(x) is noetherian if and only if the ring c∞(x) is noetherian, follows as a particular case on choosing f = r and p = the ideal of all compact sets in x. on the other hand if one takes f = r and p = the ideal of closed relatively pseudocompact subsets of x, then it follows that a locally pseudocompact space x is finite if and only if the ring cψ(x) of all real valued continuous functions on x with pseudocompact support is noetherian if and only if the ring c ψ ∞(x) = {f ∈ c(x) | ∀ε > 0, clx{x ∈ x : |f(x)| > ε} is pseudocompact } is noetherian. finally on choosing f = r and p = the ideal of all closed sets in x, it follows that: x is finite if and only if the ring c(x) is noetherian if and only if the ring c∗(x) is noetherian. 2010 msc: primary 54c40; secondary 46e25. keywords: noetherian ring; artinian ring; totally ordered field; zerodimensional space; pseudocompact support; relatively pseudocompact support. ∗the third author thanks the ugc, new delhi-110002, india, for financial support. received 29 august 2014 – accepted 9 january 2015 http://dx.doi.org/10.4995/agt.2015.3247 j. s. k. acharyya, k. c. chattopadhyay and p. rooj 1. introduction a commutative ring r with or with out identity, is called noetherian/artinian if any ascending/descending chain of ideals i1 ⊆ i2 ⊆ i3 ⊆ · · · /i1 ⊇ i2 ⊇ i3 ⊇ · · · in it is stationary in the sense that there is an m ∈ n with ij = im for all j > m. noetherian rings play an important role in commutative algebra and also in algebraic geometry. a principal result for these rings is that if r is noetherian, then the polynomial ring r[x1, x2, . . . , xn] in finitely many indeterminates is also noetherian. thus quite a large number of “good” rings appear to be noetherian. the theory of rings c(x) of all real valued continuous functions over topological spaces x became an important area of research with the pioneering work of m. h. stone [11], gelfand and kolmogorov [5] and hewitt [6]. two important subrings of the last mentioned ring viz. the ring ck(x) of all real valued continuous functions over x, which have compact support and the ring c∞(x) of all those functions in ck(x) which vanish at infinity have also received the attention of some mathematicians, mentioned may be made of c. w. kohl ([8] and [9]). a large number of well known mathematicians subsequently got attracted to this area, particularly after the classic text book of gillman and jerison [4] came into being in the year 1960. many properties by and large common to most of the “good” rings have also been shared by the rings c(x) and some of these by the rings ck(x) and c∞(x) also. since most of the so-called “good” rings are noetherian one may ask, if so are also the rings c(x), ck(x) and c∞(x). in this article we establish a general result from which it follows that for a tychonoff space x, c(x) is noetherian if and only if x is a finite set and if x is locally compact and hausdorff, then ck(x) is noetherian if and only if c∞(x) is noetherian when and only when x is a finite set. it also follows from the same general result that a locally pseudocompact tychonoff space x is finite if and only if the ring cψ(x) of all functions in c(x) with pseudocompact support is noetherian if and only if the ring cψ ∞ (x) of all functions f in c(x) for which for each ε > 0, the set clx{x ∈ x : |f(x)| > ε} is pseudocompact is noetherian. the last ring may be thought of as the pseudocompact analogue of the ring c∞(x). now we state our principal result. let x be a hausdorff topological space and f , a totally ordered field equipped with the order topology. then c(x, f) = {f : x → f | f is continuous on x} makes a commutative lattice ordered ring with identity if the compositions are defined pointwise on x. we get the familiar ring c(x) on choosing f = r. x is called completely f-regular (cfr in short) if given a closed set k in x and a point x ∈ x−k, there exists an f ∈ c(x, f) such that f(x) = 0 and f(k) = 1. the ring c(x, f) together with a few of its subrings were investigated by acharyya, chattopadhyaya and ghosh [2]. their purpose is to look into a few aspects on the possible interplay between the topological structure on x and the algebraic structure of c(x, f) and their subrings mentioned in the last sentence. it was observed in the same paper [2] that a cfr-space with f not isomorphic to r is zero-dimensional, in particular tychonoff. conversely c© agt, upv, 2015 appl. gen. topol. 16, no. 1 82 noetherianness of a class of function rings each zero-dimensional hausdorff topological space becomes cfr-space for any ordered field f . thus zero-dimensionality of a hausdorff space x is realised as a kind of separation axiom effected by f-valued continuous functions on x. a family p of closed sets in x is called an ideal of closed sets if: (1) a ∈ p, b ∈ p ⇒ a ∪ b ∈ p and (2) a ∈ p and c ⊆ a with c closed in x ⇒ c ∈ p. with any such ideal p, we associate the following two subsets of c(x, f), each of which is a subring of c(x, f) (possibly without identity): cp(x, f) = {f ∈ c(x, f) : clx(x − z(f)) ∈ p} and cp ∞ (x, f) = {f ∈ c(x, f) : ∀ε > 0 in f, {x ∈ x : |f(x)| ≥ ε} ∈ p} = {f ∈ c(x, f) : ∀ε > 0 in f, clx{x ∈ x : |f(x)| > ε} ∈ p} here z(f) = {x ∈ x : f(x) = 0} is the zero set of f. it is easy to check that cp(x, f) ⊆ c p ∞ (x, f). given an ideal p of closed sets in x, x is called locally-p if each point x ∈ x has an open neighbourhood u such that clxu ∈ p. thus locally compact spaces x are locally-p if p is the ideal of compact sets in x. the principal result of this paper is stated as follows: theorem 1.1 (main theorem). given an ideal p of closed sets in x and a totally ordered field f, the following statements are equivalent for a locally-p, cfr-space x: (1) cp(x, f) is a noetherian ring. (2) cp(x, f) is an artinian ring. (3) cp ∞ (x, f) is a noetherian ring. (4) cp ∞ (x, f) is an artinian ring. (5) x is a finite set. 2. two subsidiary results and the proof of the main result lemma 2.1. let {r1, r2, . . . , rn} be a finite family of commutative rings with identity. the ideals of the direct product r1 × r2 × · · · × rn are exactly of the form i1 × i2 × · · · × in, where for k = 1, 2, . . . , n, ik is an ideal of rk. proof. if ik is an ideal of rk for k = 1, 2, . . . , n then it follows trivially that i1 × i2 × · · · × in is an ideal of r1 × r2 × · · · × rn. conversely let i be an ideal of r1 × r2 × · · · × rn. suppose πk : r1 × r2 × · · ·×rn → rk is the k-th projection map for k = 1, 2, . . . , n defined in the usual manner. let ik = {πk(x) : x ∈ i}. then ik is an ideal of rk for k = 1, 2, . . . , n and i ⊆ i1 × i2 × · · · × in. now we choose (x1, x2, . . . , xn) ∈ i1 × i2 × · · · × in. then for k = 1, 2, . . . , n, there exists yk ∈ i such that πk(yk) = xk. this implies that (0, 0, . . . , xk, . . . , 0), with its k-th co-ordinate xk belongs to i, consequently (x1, x2, . . . , xn) ∈ i. hence i = i1 × i2 × · · · × in. � c© agt, upv, 2015 appl. gen. topol. 16, no. 1 83 j. s. k. acharyya, k. c. chattopadhyay and p. rooj lemma 2.2. given a totally ordered field f and a cfr-space x, the following statements are equivalent: (1) x is locally-p. (2) {z(f) : f ∈ cp(x, f)} is a closed base for x. (3) {z(f) : f ∈ cp ∞ (x, f)} is a closed base for x. (a special case of this result with f = r appeared in a paper in 2010 (see [1, theorem 4.3])) proof. the proof is just a simple adaptation of the proof of theorem 4.3 in the paper [1], mentioned above. we have only to take care of the fact that a completely f-regular space is regular. � proof of theorem 1.1. first assume that x is a finite set with ‘n’ elements. since x is hausdorff it becomes a discrete space and therefore p = p(x) ≡ the power set of x. consequently cp(x, f) = c p ∞ (x, f) = c(x, f) = fn ≡ f × f × · · · × f (n times). since {0} and f are the only ideals of the field f , it follows from lemma 2.1 that, there are exactly 2n many ideals of the product ring fn. hence cp(x, f) and c p ∞ (x, f) are both noetherian rings and artinian rings, trivially. conversely let x be an infinite set. as, a commutative ring is noetherian if and only if each ideal in this ring is finitely generated, therefore to show that cp(x, f) is not a noetherian ring, we shall construct an ideal in this ring which is not finitely generated. now like any infinite hausdorff space, x contains a copy of n = {1, 2, 3, . . .}. we take i = {f ∈ cp(x, f) : f(1) = 0 and f(k) = 0 for all but possibly finitely many k’s from n}. it is easy to check that i is an ideal of cp(x, f). we assert that i is not finitely generated. for that purpose we select any finite number of elements f1, f2, . . . , fn from i, n ∈ n. then the set ⋂n i=1 z(fi) contains the point 1 and also all but finitely many points from the set n. so we can choose m 6= 1 from n such that m ∈ ⋂n i=1 z(fi). since m is isolated in the space n, there exists an open neighbourhood u of m in x such that u ∩ n = {m}. as x is locally-p, it follows from lemma 2.2 that there is an f ∈ cp(x, f) such that f(m) 6= 0 and f(x − u) = 0. since x − u contains all the points of n − {m} and m 6= 1, it is clear that f ∈ i. but the choice that f(m) 6= 0, while f1(m) = f2(m) = · · · = fn(m) = 0 tells us that, there does not exists α1, α2, . . . , αn ∈ cp(x, f), for which we can write f = ∑n i=1 αifi. this shows that i is not finitely generated. altogether cp(x, f) is not a noetherian ring. analogous arguments can be made to prove that cp ∞ (x, f) is not a noetherian ring. to complete this theorem we shall show that cp(x, f) (respectively cp ∞ (x, f)) is not an artinian ring (with the same hypothesis that x is an infinite set). at this stage, we may be tempted to argue as follows: since a commutative artinian ring is known to be noetherian, therefore cp(x, f) (respectively cp ∞ (x, f)) is not an artinian ring, because we have just realised that (with x infinite), cp(x, f) is not a noetherian ring (respectively c p ∞ (x, f) is not a noetherian ring). but a word of caution here, the proof of the fact that a c© agt, upv, 2015 appl. gen. topol. 16, no. 1 84 noetherianness of a class of function rings commutative artinian ring r is noetherian crucially uses the tacit assumption that 1 ∈ r (see [3, theorem 3, chapter 16]). but in our situation each of the rings cp(x, f) and c p ∞ (x, f) may well lack identity elements. indeed if x is a non compact locally compact hausdorff space, then none of the rings ck(x) and c∞(x) possesses identity. thus we feel it necessary to prove independently that cp(x, f) (respectively c p ∞ (x, f)) is not an artinian ring. indeed for each k ∈ n, if we set ik = {f ∈ cp(x, f) : f(1) = f(2) = · · · = f(k) = 0}, then on using lemma 2.2, it is not at all hard to check that i1 % i2 % i3 % · · · is a strictly decreasing sequence of ideals in cp(x, f) which never terminates at a finite stage. this proves that cp(x, f) is not an artinian ring. analogously one can see that cp ∞ (x, f) is also not an artinian ring. the theorem is completely proved. � remark 2.3. a careful scrutiny into the proof of the above theorem yields that if x is an infinite locally-p, cfr-space, then none of the rings that lie between cp(x, f) and c p ∞ (x, f) is noetherian (respectively artinian). therefore we can write the following improved version of our main theorem. theorem 2.4. the following statements are equivalent for a locally-p, cfrspace x: (1) there exists a noetherian ring lying between cp(x, f) and c p ∞ (x, f). (2) there exists an artinian ring lying between cp(x, f) and c p ∞ (x, f). (3) x is a finite set. 3. a few interesting special cases of the main theorem the choice f = r and p = the ideal of all closed sets in x together with the fact that c∗(x) is isomorphic to c(βx), where βx is the stone-čech compactification of x yields the following special case: theorem 3.1. the following statements are equivalent for a tychonoff space x: (1) c(x) is a noetherian ring. (2) c(x) is an artinian ring. (3) c∗(x) is a noetherian ring. (4) c∗(x) is an artinian ring. (5) x is a finite set. since a point and a closed subset of x missing that point could always be separated by a function in c∗(x), it follows by making a simple modification of the proof of the converse part of theorem 1.1 that, for an infinite set x, no ring lying between c∗(x) and c(x) can ever be noetherian (respectively artinian). if we choose f = r and p = the ideal of all compact sets in x, then the following special case of our main theorem emerges: c© agt, upv, 2015 appl. gen. topol. 16, no. 1 85 j. s. k. acharyya, k. c. chattopadhyay and p. rooj theorem 3.2. the following statements are equivalent for a locally compact hausdorff space x: (1) ck(x) is a noetherian ring. (2) ck(x) is an artinian ring. (3) c∞(x) is a noetherian ring. (4) c∞(x) is an artinian ring. (5) x is a finite set. it follows from remark 2.3 that for an infinite set x none of the rings lying between ck(x) and c∞(x) is noetherian (respectively artinian). it may be mentioned that in case ck(x) 6= c∞(x), there lies at least 2 ℵ1 many rings between ck(x) and c∞(x) (see [4, 7g(1) and 14.13]). before mentioning the last important special case of the main theorem in this article, we recall that a subset a of x is called relatively pseudocompact if each f ∈ c(x) is bounded on a. we write below the following result, proved in 1971 by mark mandelkar, which we will need for our present purpose. theorem 3.3 (mandelkar’s theorem [10]). a support in x i.e. a subset of the form clx(x − z(f)), f ∈ c(x) is relatively pseudocompact if and only if it is pseudocompact. the closed pseudocompact subsets of a space x may not form an ideal of closed sets. indeed the right edge {ω1} × ω0 of the tychonoff plank t ≡ (ω1+1)×(ω0+1)−{(ω1, ω0)} is a closed subset of t and t is pseudocompact (see [4, 8.20]). nevertheless, the set cψ(x) of all real valued continuous functions on x with pseudocompact support makes a subring, indeed an ideal of c(x). this follows from mandelkar’s theorem and also the fact that the closed relatively pseudocompact subsets of a space x, do constitute an ideal of closed sets in x (see [10, corollary 2]). mandelkar’s theorem further implies that the set cψ ∞ (x) = {f ∈ c(x) : ∀ε > 0, clx{x ∈ x : |f(x)| > ε} is pseudocompact } which we may call the pseudocompact analogue of the ring c∞(x), forms an ideal of c(x) with cψ(x) ⊆ c ψ ∞ (x). since the cozero sets make an open base for the topology of a tychonoff space x, it is also a straight forward consequence of mandelkar’s theorem that x is locally relatively pseudocompact if and only if it is locally pseudocompact, in the sense that each point x of x has an open neighbourhood whose closure is pseudocompact. we now organize the above findings to conclude the following proposition, which we feel an interesting particular case of our main theorem 1.1. theorem 3.4. the statements written below are equivalent for a locally pseudocompact (tychonoff) space x: (1) cψ(x) is a noetherian ring. (2) cψ(x) is an artinian ring. (3) cψ ∞ (x) is a noetherian ring. (4) cψ ∞ (x) is an artinian ring. (5) x is a finite set. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 86 noetherianness of a class of function rings if x is an infinite set, then no ring lying between cψ(x) and c ψ ∞ (x) is noetherian (respectively artinian). this last assertion follows from remark 2.3. remark 3.5 (concluding remark). rings of real valued continuous functions defined over tychonoff spaces and many of their well known subrings are in general far from being noetherian, yet all are important in their own right. acknowledgements. the authors express their gratitude to the referee of this paper for making some valuable comments towards the improvement of the original version of this paper. references [1] s. k. acharyya and s. k. ghosh, functions in c(x) with support lying on a class of subsets of x, topology proc. 35 (2010), 127–148. 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[11] m. h. stone, applications of the theory of boolean rings to general topology, trans. amer. math. soc. 41 (1937), 375–481. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 87 05.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 61 81 useful topologies and separable systemsg. herden, a. pallackabstract. let x be an arbitrary set. a topology t on x issaid to be useful if every continuous linear preorder on x is rep-resentable by a continuous real{valued order preserving function.continuous linear preorders on x are induced by certain familiesof open subsets of x that are called (linear) separable systems onx. therefore, in a �rst step useful topologies on x will be char-acterized by means of (linear) separable systems on x. then, ina second step particular topologies on x are studied that do notallow the construction of (linear) separable systems on x thatcorrespond to non{representable continuous linear preorders. inthis way generalizations of the eilenberg{debreu theorems whichstate that second countable or separable and connected topologieson x are useful and of the theorem of est�evez and herv�es whichstates that a metrizable topology on x is useful, if and only if itis second countable can be proved.2000 ams classi�cation: 54f05, 91b16, 06a05.keywords: completely regular topology, weak topology, normal topology,short topology, countably bounded topology, countably bounded linear pre-order. 1. introductiona topology t on an arbitrary set x is said to be useful, if every continuouslinear (total) preorder on x has a continuous utility representation, i.e. canbe represented by a continuous real{valued order preserving function (utilityfunction) (see [15]). continuity of means that the order topology tinducedby is coarser than t. su�cient conditions for a topology t on x to be usefulare, for instance, given by the classical eilenberg{debreu theorems (edt) and(dt) ([9, 10, 11]). necessary and su�cient conditions for a topology t on xto be useful have been presented by the theorem of est�evez and herv�es (eht)in case that t is a metrizable topology on x ([13], see also [6, 7]) using theconcept of a useful topology t on x these theorems can be restated as follows: 62 g. herden, a. pallack(edt) every connected and separable topology t on x is useful.(dt) every second countable topology t on x is useful.(eht) a metrizable topology t on x is useful, if and only if t is second count-able.the aim of this paper is the characterization of all useful topologies t onx. a theorem which solves this problem, in particular, would generalize theeilenberg-debreu theorems and the theorem of est�evez and herv�es. mean-while banach-spaces or, more generally, convex spaces are frequently studiedin mathematical utility theory. in the in�nite dimensional case these spacesmay fail to be second countable or separable. this means that continuousrepresentation of linear orderings (preference orderings) in these spaces is notguaranteed by the classical eilenberg-debreu theorems. therefore, a charac-terization of useful topologies is of particular interest in mathematical utilitytheory (cf. also remark 6.8).2. a first approachlet throughout this section x be a �xed given set and let t be some topologyon x. for every subset a of x we denote by a its topological closure. themost fundamental result that is known on useful topologies is dt. dt easilyimplies edt (cf. [14, 16]) and also generalizes the su�ciency part of the the-orem of est�evez and herv�es. on the other hand, it is well known that secondcountability, in general, is not necessary for t to be useful (cf., for instance, theniemitzki plane that is extensively discussed in [32]). hence, in order to at leastapproximate second countability we consider linearly ordered subtopologies tlof t. tl is linearly ordered if it is linearly ordered by set inclusion.it is easily to be seen that second countability of t implies second countabilityof all its linearly ordered subtopologies tl. indeed, let tl be a linearly orderedsubtopology of the second countable topology t. then we choose a countablebase b of t and consider the countable subsetbl := �o 2 tl j9b 2 b(b � o ^8o0 2 tl(o0 $ o =) b 6� o0)) [�o 2 tl j9b 2 b(o = [b 6�o02tlo0) [f?;xgof tl in order to immediately verify that bl is a base of tl. let us now assumethat all linearly ordered subtopologies tl of t are second countable and let bea continuous linear (total) preorder on x. then we consider the familyl := fl(x)gx2x := ffy 2 x jy � xggx2xof open decreasing subsets of x. the linearly ordered subtopology tl of t thatis induced by l is second countable which means that there exists a countablesubset lb of l [ f?;xg that is a base of tl. the countability of lb impliesthat the corresponding chain (lb;�) only has countably many jumps. thereader may recall that a jump of (lb;�) is a pair of sets e $ e0 2 lb suchthat there exists no set e00 2 lb such that e $ e00 $ e0. by interposingthe rationals into the jumps of (lb;�) we, thus, obtain some chain (leb;�)that extends (lb;�) and may, without loss of generality, be assumed to be useful topologies and separable systems 63order-isomorphic to the chain ([0;1]q;�) of all rationals in the real interval[0;1]. let g: (leb;�) �! ([0;1]q;�) be some order-isomorphism. then oneveri�es that f: (x;-) �! ([0;1]r;�) de�ned for all points x 2 x by f(x) :=supfg(l(y)) jl(y) 2 lb;y xg is a continuous utility representation of -.clearly, t is not necessarily second countable if all its linearly ordered sub-topologies tl are second countable. in order to obtain a counterexample oneonly has to choose some topology on the natural numbers that is not secondcountable. hence, the above result does not only provide an alternative proofof dt but also generalizes dt.in order to also generalize edt we consider the �rst in�nite ordinal !. thenwe consider the family tc of all linearly ordered subtopologies tl of t that areinduced by some linearly (totally) ordered set (o;�) of open subsets of x thatsatisfy the following conditions:(lo1): 8o0 2 o(o0 � to0$o2o o) or, equivalently, 8o0 2 o 8o 2o(o0 $ o =) o0 � o), and(lo2): ��no 2 oj so3o0$o o0 $ o ^so3o0$o o0 \ x n o 6= ?o�� !.now it follows that in case that t is a separable and connected topology onx every linearly ordered subtopology tl 2 tc of t must be second countable.indeed, let some topology tl 2 tc be arbitrarily chosen. then the separabilityof t implies that no chain (o;�) or (o;�) of open subsets of x which satis-�es the conditions (lo1) and (lo2) and induces tl contains some uncountablewell-ordered subchain, i.e. (o;�) or (o;�) is short which, in particular, meansthat tl is �rst countable (cf. [1]). in addition, the connectedness of t impliesthat none of the sets onso3o0$o o0 such that ? $ so3o0$o o0 � o $ xis empty. let, therefore, s be a countable dense subset of x. then wechoose for every point x 2 s some countable base of tl-neighborhoods of x.the union of the collection of these tl-neighborhoods with the countable setno 2 oj so3o0$o o0 $ o ^so3o0$o o0 \ xno 6= ?o is a countable base oftl. let us now assume that all linearly ordered subtopologies tl 2 tc are secondcountable. then the same arguments that already have been applied in order togeneralize dt allow us to conclude that every continuous linear (total) preorderon x has a continuous utility representation.on the other hand, it also cannot be expected that second countability ofthe linearly ordered subtopologies tl 2 tc of t is necessary in order to guaranteeusefulness of t. indeed, let be some arbitrary continuous linear preorderon x. then the linearly ordered set (l;�) := (fl(x)gx2x ;�) satis�es thefollowing additional condition that strengthens condition (lo2).(lo3) : 8o 2 l( [l3o0$o o0 $ o =) [l3o0$o o0 � o):therefore, it is somewhat surprising that in case that we concentrate onnormal topologies t on x (cf. de�nition 3.3) the conditions (lo1) and (lo2)completed by two straightforward conditions that are necessary in order to also 64 g. herden, a. pallackinclude the case that t is not necessarily connected already characterize usefultopologies. this characterization provides a generalization of edt in the justdescribed way. in particular, it can be shown that our results are generalizationsof the theorem of est�evez and herv�es. the reader may still notice that the afore-discussed generalizations of dt and edt provide a possibility of how to applyour results on useful topologies that will be proved in the following sections.3. r-separable systemsit is well known that continuous linear preorders are closely related to r-separable systems (see [16]). therefore, we shall approach the characterizationof useful topologies in a �rst step with help of r-separable systems.suppose that r is an arbitrary binary relation on some �xed given topologi-cal space (x; t) (brie y we speak of an r-space). then the reader may recall at�rst the following notation: a subset a of x is said to be r-decreasing (or sim-ply decreasing, if the relation is clear from the context), if a 2 a and bra implythat b 2 a. an increasing set is de�ned in an analogous manner. each subsetf of x gives rise to the smallest decreasing (respectively, increasing) subsetd(f) (respectively, i(f)) containing f . if f = fxg for some point x 2 x, thenwe write d(x) (respectively, i(x)) instead of d(fxg) (respectively, of i(fxg)). foreach subset f of x there is a smallest closed decreasing subset d(f) (respec-tively, smallest closed increasing subset i(f)) containing f . if f = fxg forsome point x 2 x, then we write d(x) (respectively, i(x)) instead of d(fxg)(respectively, of i(fxg). notice that for each subset f of x we have f � d(f).in general, this inequality is strict as is seen from the following simple example.let x := f1;2g ; t := f?;f1g ;f2g ;f1;2gg and r := f(i;j) 2 x � x ji � jg.then f2g = f2g $ d(2) = f1;2g = x. with these preliminaries we are fullyprepared for the following de�nition.de�nition 3.1. a family e of open r{decreasing subsets of x is said to be anr-separable system on x, if it satis�es the following conditions:(rs1) there exist sets e1;e2 2 e such that e1 � e2.(rs2) for all sets e1;e2 2 e such that e1 � e2 there exists some set e3 2 esuch that e1 � e3 � e3 � e2.moreover, if r is the equality relation \=" on x, i.e. the discrete order on x,we say that e is a separable system on x.remark 3.2. in mathematical utility theory r-separable systems on x wereconstructed for the �rst time by peleg [30] in order to prove his utility repre-sentation theorem. in peleg`s theorem r is a strict partial order or brie y anorder on x. in 1977 burgess and fitzpatrick [4] studied decreasing scales inx. we recall that a family s := ffrgr2d of open decreasing subsets of x issaid to be a decreasing scale in x, if the following two conditions are satis�ed:(ds1) d is a dense subset of the real interval [0;1] such that 1 2 d andf1 = x.(ds2) for every pair of real numbers r1 < r2 2 d the inclusion fr1 � fr2holds. useful topologies and separable systems 65one immediately veri�es that decreasing scales in x are particular cases ofr-separable systems on x.the concept of an r-separable system on x is closely related to the conceptof a normally preordered space (cf. [29]) or more generally normal r-space.de�nition 3.3. an r-space (x; r; t) is said to be a normal r-space, if forany pair a, b of disjoint closed decreasing (respectively, increasing) subsets ofx there exist disjoint open decreasing, (respectively, increasing) subsets u, vof x such that a � u and b � v .notice that, if r coincides with the equality-relation " = " on x, then(x; r; t) is a normal space.the connections between the concept of an r{separable system and of anormal r{space is described in the following lemma ([16, lemma 2.1]).lemma 3.4. let (x;r;t) be an arbitrary r-space. then in order for (x;r;t)to be a normal r-space it is necessary and su�cient that for every pair c1, c2of disjoint closed subsets of x, c1 being decreasing and c2 increasing, thereexists an r-separable system e on x such that c1 � e and c2 � xne forevery set e 2 e.now we turn our attention to linear r-separable systems. given an arbitraryr-space (x; r; t), an r-separable system e on x is said to be linear if, forevery pair of sets e, e0 2 e such that e 6= e0 at least one of the inclusionse � e0 or e0 � e holds. linear r-separable systems e on x easily can becharacterized ([17, proposition 1.4.1]).proposition 3.5. let e be a family of open decreasing subsets of x, that islinearly ordered by set inclusion, and let b be the family of all sets e 2 e suchthat e $ e and for which there exists some set b 2 e such that e $ b. thenthe following assertions are equivalent:(i) e is a linear r-separable system on x,(ii) 8e 2 b(te$b2e b = te$b2e b),(iii) 8e 2 b((e � te$b2e b) ^ (te$b2e b 2 e =) te$b2e b =te$b2e b)).every r-separable system e on x contains some linear r-separable system.indeed, let q denote the rationals. then this result is an immediate consequenceof the following lemma ([16, lemma 2.2]).lemma 3.6. let e be an r-separable system on x. then there exists a func-tion f : q �! e such that f(p) � f(p) � f(q) for all p < q 2 q.the reader may recall that a real-valued function f on x is said to be in-creasing if, for all pairs (x;y) 2 r, the inequality f(x) � f(y) holds. with helpof this notation we are already able to present the general separation theoremgst of nachbin-urysohn-type, which corresponds to gurt in [16] (see also[29], [34]). 66 g. herden, a. pallacktheorem 3.7. let (x; r; t) be an r-space. then in order for (x; r; t) to bea normal r-space it is necessary and su�cient that for any two disjoint closedsubsets c1; c2 of x such that c1 is decreasing and c2 increasing, there existssome continuous increasing real-valued function f on x such that 0 � f � 1,f(x) = 0 for all x 2 c1 and f(x) = 1 for all x 2 c2.now we present the most important result of this section (see [16, lemma2.3]).theorem 3.8. let (x; r; t) be an r-space. then every linear r-separablesystem e on x induces a linear preorder on x which satis�es the followingproperties:(l1): r �-;,(l2): the order topology tis coarser than te.for later use we recall the de�nition of -. let e be a linear r-separable sys-tem on x and let points x, y 2 x be arbitrarily chosen. set ey := fe 2 ejy 2 egand de�ne by settingx y () ey = ? _ 8e 2 ey8b 2 ey(e 6� b _ x 2 b):it is easily seen that can be divided into the following two less complicatedsubrelations:(i) x � y () there exists some r-separable systemb � e on x such thatx 2 b and y 2 xnb for all sets b 2 b;(ii) x � y () :(x � y) and :(y � x):it seems that theorem 3.8 is closely related to the famous szpilrajn's theo-rem [33] which states that every partially order can be re�ned to a linear order.but is not necessarily a re�nement of a, since we did not require that � iscontained in the set of all pairs (x;y) 2 r such that (y;x) 62 r. for later usewe abbreviate this set by rs. hence, szpilrajn's theorem is not a consequenceof theorem 3.8. as far as the authors know the continuous analogous of szpil-rajn's theorem never has been discussed in the literature. in order to be moreprecise, the reader may recall that a linear preorder on x is continuous if andonly if for each pair of points x � y 2 x there exists some continuous increasingreal-valued function fxy on x such that fxy(x) < fxy(y) (the reader may apply[22, lemma 1] and gst to verify this result). obviously, this characterizationof continuous linear preorders can be generalized to arbitrary preorders. there-fore, a continuous preorder on x is said to satisfy the szpilrajn-property, ifthere exists a continuous linear preorder -� on x such that -� is a re�nementof -. the szpilrajn-property will be discussed in [17, chapter 6].4. real order-embeddingslet throughout this section x be some �xed given set and r some relationon x. we recall some de�nitions. (x; r) is said to be ja�ray-separable ifthere exists a countable subset z of x such that, if x;y 2 x and (x;y) 2 rs,then there exist points z;z0 2 z such that xrxrsz0ry. (x; r) is said to bebirkho�-separable if there exists a countable subset z of x such that, for every useful topologies and separable systems 67pair (x;y) 2 rs \(xnz)�(xnz), there exists some z 2 z such that xrszrsy.the space (x; r) is called debreu-separable if there exists a countable subsetz of x such that for every pair (x;y) 2 rs there exists some z 2 z such thatxrzry, and it is called cantor-separable if there exists a countable subset zof x such that, for every pair (x;y) 2 rs, there exists some z 2 z such thatxrszrsy.now we are fully prepared for presenting the following representation the-orem (see, for example, [3, proposition 1.6.11] and [13, lemma 3.1]).theorem 4.1. let (x; -) be a linearly preordered set. then the followingassertions are equivalent:(i) there exists an order preserving function f : (x; -) �! (r; �),(ii) (x; -) is ja�ray-separable,(iii) (xj�; -j�) is birkho�-separable,(iv) (x; -) is debreu-separable,(v) (x; t-) is separable and (xj�; -j�) has only countably many jumps,(vi) (x; t-) is second countable.the reader may notice that in contrast to proposition 1.6.11 in bridgesand mehta [3] the assertion concerning birkho�-separability has been mod-i�ed somewhat. indeed, the concepts of birkho�-separability and debreu-separability are not equivalent in the context of preorders but only in the contextof orders. this hint is due to mehta [24, november 1999, oral communication].5. the structure of useful topologieslet x be a �xed given set and let t be some topology on x. it is the aimof this section to characterize all useful topologies on x with help of linearseparable systems on x. because of proposition 3.5 the characterization ofuseful topologies with help of linear separable systems is a quite satisfactoryapproximation of the desired results that have been announced in the secondsection. in order to also include the non-connected case we need at �rst thefollowing notation. a topological space (x; t) is said to satisfy the open-closedcountable chain condition (occc), if every family f of non-empty open andclosed subsets f of x that satis�es the following two conditions is countable:(oc1): 8f 2 f(f � f 0 _ f 0 � f),(oc2): 8f 2 f([ff 0 2 f jf 0 $ fg $ f $ \ff 00 2 f jf $ f 00g).let now e be a linear separable system on x. we consider the set z(e)of all pairs b $ e 2 e for which there exists some set c 2 e such thatb $ c � c $ e. then e is said to have a countable re�nement if there existsa countable family o of non{empty open subsets of x such that for every pair(b;e) 2 z(e) there exists some set o 2 o such that o � e \ xnb. e issaid to be second countable, if there exists a countable subset h of e such thatfor every pair of sets (b;e) 2 z(e) there exists some set e+ 2 h such thatb � e+ � e+ � e. in addition, if g(e) is the set of all (open) sets e 2 e suchthat se3b$e b $ e, let gg denote the family of all linear separable systems e 68 g. herden, a. pallackon x for which g(e) is a countable set. with help of this notation the followingproposition characterizes all useful topologies t on x.proposition 5.1. for a topology t on a set x, the following assertions areequivalent:(i) t is useful,(ii) t satis�es occc and every linear separable system e on x has acountable re�nement,(iii) t satis�es occc and every linear separable system e on x is secondcountable,(iv) t satis�es occc and every linearly ordered subtopology tl of t that isinduced by some linear separable system e 2 gg is second countable.proof. (i) =) (ii) at �rst we assume, in contrast, that t does not satisfy occc.then there exists an uncountable family f of non-empty open and closed sub-sets f of x that satis�es the conditions (oc1) and (oc2). since every setf 2 f is open and closed, condition (oc1) implies that the preorder de�nedfor every pair of points x;y 2 x byx y () 8f 2 f(y 2 f =) x 2 f)is linear and continuous. in addition, the uncountability of f allows us toconclude with help of condition (oc2) that has uncountably many jumps.indeed, for every set f 2 f any pair of points x 2 fn [ fb 2 f jb $ fg,y 2 \fc 2 f jf $ cgnf de�nes a jump of -. hence, is not representable.this contradiction implies that t satis�es occc. let now e be a linear separablesystem on x. it remains to show that there exists a countable family o ofopen subsets of x such that for every pair (b;e) 2 z(e) there exists some seto 2 o such that o � e \xnb. as remarked after theorem 3.8 we may de�nea continuous linear preorder on x induced by e such that for every pair(b;e) 2 z(e) and every pair of points x 2 cnb;y 2 enc the strict inequalityx � y holds. this means that we may choose for every pair (b;e) 2 z(e)points x � y 2 x such that ]x;y[� e \ xnb. because of assertion (i), thelinear preorder is representable. this means, in particular, that (xj�; -j�)only has countably many jumps and that tis second countable (cf. theorem4.1, assertions (v) and (vi)). the existence of the desired family o of opensubsets of x, thus, follows immediately, which �nishes the proof of assertion(ii).(ii) =) (i) let be a continuous linear preorder on x. because of the opengap lemma ([9, 10]) it su�ces to prove that is representable. therefore, weconsider the linear separable system l := fl(x)gx2x := ffy 2 x jy � xggx2xon x. since (x; t) satis�es occc it follows that (xj�; -j�) only has countablymany jumps. indeed, let f([xi]; [yi])gi2i be the family of all jumps of (xj�; -j�). then we may choose for every index i 2 i the open and closed subsetfi := fz 2 x jz xig = fz 2 x jz � yig of x. let f be the family of thesesubsets. because (x; -) is a chain, we may conclude that f satis�es condition(oc1). in addition, the de�nition of f implies that f also satis�es condition useful topologies and separable systems 69(oc2). hence, it follows from occc that f is countable, and this means thatthe family f([xi]; [yi])gi2i of all jumps of (xj�; -j�) actually is countable. inorder to now �nish the proof of the representability of it remains to verifythat (x; t-) is separable (cf. theorem 4.1 (v)). l has a countable re�nement.hence, there exists a countable family o of non{empty open subsets of x suchthat for every pair x � y 2 x for which ]x;y[ is neither empty nor contains ajump of (xj�; -j�), there exists some set o 2 o such that o �]x;y[. choosingin every set o 2 o some point x 2 o and considering, in addition, for everyjump ([x]; [y]) of (xj�; -j�) points x 2 [x] and y 2 [y] respectively, we mayconclude that (x; t-) must be separable, and assertion (i) follows.(i)^(ii) =) (iii) let e be a linear separable system on x. it su�ces to showthat e is second countable. let, therefore, o be a countable family of non{empty open subsets of x such that for every pair of sets (b;e) 2 z(e) thereexists some set o 2 o such that o � e \xnb. by eliminating redundant setswe may assume without loss of generality that for every set o 2 o there existsets b � e 2 e such that o � e \ xnb. hence, we may choose for every seto 2 o the non{empty linear separable systems w1 := fb 2 e jonb 6= ?g andw2 := fe 2 e jone = ?g. it follows that there exist countable sets o1 � w1and o2 � w2 such that se2o1 e = se2w1 e and te2o2 e = te2w2 e.indeed, otherwise the construction described after theorem 3.8 implies thatboth continuous linear preorders -1 and -2 on x which are induced by w1and by w2, respectively, are not short and, thus, not representable in contrastto assertion (i). since o is countable we may conclude that b := so2o o1 [so2o o2 is a countable set. the construction of b implies that for every pair ofsets (b;e) 2 z(e) there exists some set e+ 2 b such that b � e+ � e+ � e,as desired.(iii) =) (iv) trivial.(iv) =) (i) let be some continuous linear (total) preorder on x. thenwe consider the linear separable system l := fl(x)gx2x on x. in the proofof the implication (ii) =) (i) it already has been shown that occc impliesthat (xj�; -j�) only has countably many jumps. since l satis�es condition(lo3) it follows that l 2 gg. the reader may recall that condition (lo3)implies condition (lo2). assertion (iv), thus, implies that the linearly orderedsubtopology tl of t that is induced by l is second countable which allows usto conclude with help of the considerations in the second section that hasa continuous utility representation. therefore, the proof of the proposition iscomplete. �clearly, in case that t is connected occc may be omitted. hence, thecharacterization of t to be useful simpli�es somewhat.corollary 5.2. let t be connected. then the following assertions are equiva-lent:(i) t is useful,(ii) every linear separable system e on x has a countable re�nement,(iii) every linear separable system e on x is second countable, 70 g. herden, a. pallack(iv) every linear ordered subtopology tl of t that is induced by some linearseparable system e 2 gg is second countable.6. a different approachlet (x; t) be an arbitrary topological space and let g := feigi2i be a familyof separable systems on x. then g is said to be well-separated, if it satis�esthe following conditions:(ws1): 8i 2 i 8j 2 i 8e 2 ei8b 2 ej(i 6= j =) e \ b = ?).(ws2): 8feigi2i (ei 2 ei =) si2i ei = si2i ei).now the following lemma holds:lemma 6.1. let t be a useful topology on x. then every well-separated familyg := feigi2i of separable systems on x is countable.proof. let g := feigi2i be some well-separated family of separable systems onx. then we may assume without loss of generality that every subset j of i forwhich there exists for every j 2 j some non{empty open and closed set ej 2 ejis countable. indeed, otherwise we consider some well ordering � on j, choosefor every j 2 j some �xed non{empty open and closed set ej 2 ej in orderto consider for every j 2 j the set fj := si�j ei. then the conditions (ws1)and (ws2) imply that f := ffjgj2j is an uncountable family of non{emptyopen and closed subsets of x that satis�es the conditions (oc1) and (oc2)and, thus, contradicts the usefulness of t.let us now assume, in contrast, that i is uncountable. then we consider somewell-ordering � on i, choose the �rst uncountable ordinal !1 and consider somesubfamily fe�g� 0 and someuncountable subset s of xj� such that �(x;y) � � for all points x 6= y 2 s.indeed, otherwise, for every natural number n > 0, every subset zn of xj� suchthat �(x;y) � 1n for all points x 6= y 2 zn is countable. the lemma of zornallows us to choose, for every natural number n > 0, some maximal subset ynof xj� such that �(x;y) � 1n for all points x 6= y 2 yn. then y := sn2nnf0g ynis a countable subset of xj� such that y = xj�, a contradiction. thus, theexistence of s follows. the inclusion t� � t implies that t�j� � tj�. hence, wemay conclude that the family ��y 2 xj� j�(x;y) < �3 x2s is an uncountablefamily of pairwise disjoint non{empty open subsets of xj� that, obviously, islocally �nite. this contradiction �nishes the proof. �now we are ready to summarize our considerations for some interesting re-sults.proposition 6.5. let �(x;c(x)) be induced by some uniformity that has acountable base. then the following assertions are equivalent:(i) t is useful,(ii) �(x; c(x)) is second countable. useful topologies and separable systems 73proof. (i) =) (ii) since �(x;c(x)) is induced by some uniformity which hasa countable base we may conclude that �(x;c(x)) is induced by some pseu-dometric � on x. with help of the inclusion �(x;c(x)) � t and lemma 6.3the desired conclusion now follows from lemma 6.4.(ii) =) (i) let be a continuous linear preorder on x. then t��(x;c(x)) and assertion (i) follows from dt. �corollary 6.6. let (g; �; t) be a �rst countable topological group. then thefollowing assertions are equivalent:(i) t is useful,(ii) t is second countable.proof. every topology t of a �rst countable topological group (g; �; t) is inducedby some uniformity which has a countable base (see, for example, [18]). �corollary 6.7. [13] let t be induced by some metric �. then the followingassertions are equivalent:(i) t is useful,(ii) t is second countable.remark 6.8. corollary 6.7 has an important consequence. it implies, in par-ticular, that for a metric space (x;�) the assumptions of debreu's theorem arenot only su�cient but also necessary for a continuous linear preorder on (x;�)to be representable by a continuous utility function. on the other hand, metricspaces (in particular hilbert spaces or more generally banach spaces) which arenot second countable are meanwhile commonly encountered in economic theory(see [23] or our remark in the introduction). this is the case, for example, if thecommodity space is l1(�), the space of �-essentially bounded �-measurablefunctions on a �-�nite measure space, which arises in the analysis of allocationof resources over time or states of nature ([2]), or ca(k), the space of countablyadditive signed measures on a compact metric space which has been exploitedfor the analysis of commodity di�erentiation ([21] and [19]). linear preordersde�ned on these spaces principally must satisfy more properties than just be-ing continuous in order to have a continuous utility representation. hence, theapproaches of shafer [31], mas-colell [21], monteiro [28], mehta and monteiro[25] and others gain additional importance. the problem which arises is tolook for useful natural additional conditions which a linear preorder on thesespaces should satisfy and which also guarantee its representability by a contin-uous utility function. such a useful condition could be countably boundedness.the reader may recall that a linear preorder on x is countably bounded ifthere exists a countable subset y of x such that for every point x 2 x thereexist points y;y0 2 y such that y x y0. for example, since every convexsubset of the space l1(�) and ca(k) respectively is path connected, it followsfrom monteiro [28] that every continuous countably bounded linear preorderon a convex subset of l1(�) and ca(k) respectively has a continuous utilityrepresentation. another useful condition could be convexity (see [5, theorem3]). 74 g. herden, a. pallackin order to now prove the main result of this section let t be an arbitrarybut �xed given topology on x. in case that t0 is a topology on x that cannotbe excluded to be di�erent from t we denote for every subset a of x by c0(a)the t0-closure of a. in case that we are sure that t0 = t the t-closure of ais abbreviated as usual by a. then a topology t0 on x is said to be well-compatible, if for every t0-open subset o of x and every point x 2 o thet0-closure c0(fxg) of fxg is contained in o. the reader may recall that t0 iswell-compatible, if and only if for each pair of points x;y 2 x the equivalencec0(fxg) = c0(fyg) () c0(fxg) \ c0(fyg) 6= ? holds. now lt is the family of allwell-compatible topologies t0 on x for which there exists some linear separablesystem e 2 gg (cf. section 5) such that e � t0 � te and c0(e0) � e for everypair of sets e0 � e0 � e 2 e. te is the topology on x that is induced by e(cf. section 3). the reader may prove as an easy exercise that c0(e0) � e forevery pair of sets e0 � e0 $ e 2 e, if and only if (c0(e)ne) \ c0(e0) = ? forevery pair of sets e0 � e0 $ e 2 e . in case that t is connected it follows thatlt is the set of all well-compatible topologies t0 on x for which there existssome linear separable system e on x such that e � t0 � te and c0(e0) � e forevery pair of sets e0 � e0 $ e 2 e. let �nally e be some arbitrarily chosenlinear separable system on x. then the reader may verify, in addition, thatte 2 lt, if and only if for every pair of sets e0 � e0 � e 2 e the equatione0 = \�e00 2 e je0 � e00 � e holds and, furthermore, a possible �rst elementor a possible last element of the chain (e;�) is open and closed.now we are fully prepared for proving the main result of this section.proposition 6.9. the following assertions are equivalent:(i) t is useful,(ii) t satis�es occc, every well-separated family g := feigi2i of separablesystems on x is countable and every topology t0 2 lt is pseudometriz-able.(iii) t satis�es occc, �(x;c(x)) satis�es clf and every topology t0 2 ltis pseudometrizable.proof. (i) =) (ii) because of proposition 5.1 and lemma 6.1 it is su�cientto prove that every topology t0 2 lt is pseudometrizable. let, therefore,some topology t0 2 lt be arbitrarily chosen. then there exists some linearseparable system e on x, whose associated set g(e) is countable, such thate � t0 � te and c0(e0) � e for every pair of sets e0 � e0 � e 2 e. letnow t0j� be the quotient topology that is induced by the equivalence relation\x � y () c0(fxg) = c0(fyg)\. the well-compatibility of t0 implies that theequivalence relation � is open, i.e. the canonical projection p : x �! xj�is open. this means, in particular, that t0 is pseudometrizable, if and onlyif t0j� is metrizable. in order to verify that t0j� is metrizable we show at �rstthat t0j� is second countable. then we prove that t0j� is normal which �nallyallows us to conclude with help of the alexandro�-urysohn metrization the-orem that t0j� is metrizable. one more application of the well compatibility useful topologies and separable systems 75of t0 implies that t0j� is second countable, if and only if t0 is second count-able. since g(e) is countable it follows with help of proposition 5.1 (iii)that te is second countable. this means that there exists a countable setc(e) of pairs of sets e0 � e0 � e 2 e [ f?;xg such that the family ofcorresponding open sets e \ xne0 is a base of te. we want to show thatb := �e \ xnc0(e0) je0 � e0 � e 2 c(e) is a base of t0. then the secondcountability of t0 follows. let, therefore, o be some (non-empty) t0-open subsetof x and let x 2 o be some arbitrary point. we must show that there existssome set e+ \ xnc0(e++) 2 b such that x 2 e+ \ xnc0(e++) � o. theinclusion t0 � te implies that there exists some pair of sets e0 � e0 $ e 2 c(e)such that x 2 e \ xne0 � o. we, thus, distinguish between the following twocases:case 1. x 2 enc0(e0). in this case the inclusion e0 � c0(e0) implies thatx 2 e \ xnc0(e0) � e \ xne0 � o and we are done.case 2. x 62 enc0(e0), i.e. x 2 c0(e0)ne0 � c0(e0)ne0. in this situation weshow at �rst that e0ne0 \o 6= ?. in order to verify this inequality the validityof the equation c0(e0ne0) = c0(e0)ne0 is needed. since e0ne0 � c0(e0)ne0 andc0(e0)ne0 is t0-closed the inclusion c0(e0ne0) � c0(e0)ne0 follows immediately.hence, the desired equation will be proved if we are able to show that also theinclusion c0(e0)ne0 � c0(e0ne0) holds. let, therefore, some point y 2 c0(e0)ne0and some t0-neighborhood u of y be arbitrarily chosen. we must show thatu \ e0ne0 6= ?. then y 2 c0(e0ne0) and the inclusion follows. the inclusiont0 � te implies that there exist some pair of sets e�� e�� $ e� 2 c(e) suchthat y 2 e� \ xne�� u. since y 2 e�ne0 the linearity of e implies thate0 � e�. on the other hand, it follows that e�� e0. indeed, otherwise wemay conclude that e0 $ e��, which means that c0(e0) � e��. since y 2 c0(e0)this inclusion contradicts the relation y 2 e \ xne��. a combination of theseconsiderations implies that e0ne0 � e \xne��. hence, u \e0ne0 6= ?, which�nishes the proof of the desired equation. with help of the equation c0(e0ne0) =c0(e0)ne0 the inequality e0ne0\o 6= ? easily can be veri�ed. indeed, otherwisewe may conclude that e0ne0 � xno. since xno is t0-closed it, thus, followsthat c0(e0)ne0 = c0(e0ne0) � xno which contradicts the relation x 2 c0(e0)ne0and x 2 o. now we proceed by choosing some arbitrary point z 2 e0ne0 \ oin order to then consider some pair of sets e++ � e++ $ e+ 2 c(e) suchthat z 2 e+ \ xne++ � o. since z 2 e0ne0 and, thus, z 2 e+ne0 thelinearity of e implies that e0 � e0 � e+ which, in particular, means thatx 2 c0(e0) � e+. because of the relations z 62 e++ and z 2 e0 it follows, onthe other hand, that e++ � e++ $ e0. hence, c0(e++) � e0. we, thus, maysummarize our considerations for the conclusions x 2 e+\xnc0(e++) � o ande+ \ xnc0(e++) 2 b which completes the second case and, therefore, showsthat t0 is second countable. the particular construction of b �nally allows usto apply the arguments of the proof of proposition 1.4.2 in [17] in order toverify that t0 is normal. then the well-compatibility of t0 implies that also t0j� 76 g. herden, a. pallackis normal and the alexandro�-urysohn metrization theorem can be applied.this last conclusion �nishes the proof of assertion (ii).(ii) =) (iii) this implication follows immediately with help of corollary 6.2.(iii) =) (i) let be some arbitrary continuous linear (total) preorderon x. then we consider the linear separable system l := fl(x)gx2x :=ffy 2 x jy � xggx2x on x. since t satis�es occc it follows that only hascountably many jumps which means that g(l) is countable (cf. the corre-sponding argument in the proof of the implication (ii) =) (i) of proposition5.1. clearly, coincides with the linear preorder on x that is induced by l(cf. theorem 3.8). hence, we may apply theorem 3.8 in order to concludethat the order topology tthat is induced by is coarser than tl. for everypoint x 2 x its t--closure c0(fxg) coincides with the equivalence class [x] thatis de�ned by �. hence, it follows that tis well{compatible. since, in addition,l � tand d(x) = fy 2 x jy xg � l(z) = fu 2 x ju � zg for every pair ofpoints y � z 2 x we may conclude that t2 lt. hence, tis pseudometriz-able. the underlying argument which the proof of lemma 6.3 is based uponimplies that t� �(x;c(x)). this means, in particular, that tsatis�es clf.therefore, it follows from lemma 6.4 that tis second countable which impliesthat has a continuous utility representation. this last conclusion settles theimplication (iii) =) (i) and nothing remains to be shown. �corollary 6.10. let t be connected. then the following assertions are equiv-alent:(i) t is useful,(ii) every well-separated family g := feigi2i of separable systems on x iscountable and every topology t0 2 lt is pseudometrizable,(iii) �(x;c(x)) satis�es clf and every topology t0 2 lt is pseudometriz-able.remark 6.11. the condition that every topology t0 2 lt is pseudometrizableseems to be a bit arti�cial. on the other hand, proposition 6.9 means that t isuseful if and only if t satis�es occc, �(x;c(x)) satis�es clf and t allows thede�nition of enough (continuous) pseudometrics on x. hence, proposition 6.9which, in particular, generalizes the nice result of est�evez and herv�es completesproposition 5.1 and may at least serve as basis for �nally obtaining still moresatisfactory results. 7. useful normal topologiesin the second section we already have announced some optimal result on theusefulness of normal topologies. in order to prove this result let t be a �xedgiven normal topology on x. for every subset a of x the interior of a isdenoted by a�. then we choose the family o of all sets o of open subsets oof x that are linearly ordered by set inclusion and satisfy condition (lo1) (cf.section 5).let some set o 2 o be arbitrarily chosen. then the sets z(o), g(o) and ogand the concepts of o to have a countable re�nement or to be second countable useful topologies and separable systems 77are de�ned in the same way as the corresponding sets z(o), g(o) and gg, andthe similar concepts in section 5. in addition, we consider the family f of allsets o 2 o which also satisfy the following condition which completes condition(lo2) in order to also include the case that t is not necessarily connected (cf.section 2).lo2+ : ��8<:o 2 o j [o3o0$o o0 $ o ^ [o3o0$o o0 \ xno 6= ?9=;��++��8<:o 2 o j [o3o0$o o0 � o ^ [o3o0$o o0 = ( [o3o0$o o0)� _ o = o9=;�� ! :the reader may verify that in case that t is connected the conditions (lo2)and (lo2+) coincide.proposition 7.1. let t be a normal topology on x. then the following asser-tions are equivalent:(i) t is useful,(ii) t satis�es occc and every set o 2 o has a countable re�nement,(iii) t satis�es occc and every set o 2 o is second countable,(iv) t satis�es occc and every linearly ordered subtopology tl of t that isinduced by some set o 2 og is second countable,(v) t satis�es occc and every linearly ordered subtopology tl of t that isinduced by some set o 2 f is second countable.proof. (i) =) (ii) let some set o 2 o be arbitrarily chosen. then we considerthe set m(o) of all sets o 2 o for which there exists some maximal seto 3 o0 $ o. since t is a normal topology on x it follows from lemma3.4 and lemma 3.6 with help of condition (lo1) that for every pair of setso 3 o0 $ o 2 m(o) there exists a linear separable system e(o) on x suchthat o0 � e � o for every set e 2 e(o). we, thus, set e(o) := o [(so2m(o) e(o)), and show that e(o) is a linear separable system on x. let,therefore, e0(o) � e(o) be the subset of all sets e 2 e(o) such that e $ e andfor which u(e) := fe0 2 e(o) je $ e0g 6= ?. then we choose some arbitraryset e 2 e0(o) and distinguish between the following two cases:case 1: (u(e);�) does not contain a minimal element. in this case theconstruction of e(o) allows us to conclude with help of condition (lo1) thatte02u(e) e0 = te02u(e) e0.case 2: (u(e);�) contains a minimal element. let e0 be this minimalelement of (u(e);�). then the de�nition of m(o) implies with help of theconstruction of e(o) that e0 is closed.summarizing both cases it follows with help of proposition 3.5 (ii) that e(o),actually, is a linear separable system on x. assertion (ii) now is an immediateconsequence of the corresponding assertion of proposition 5.1. 78 g. herden, a. pallack(ii) =) (i) since every linear separable system e on x satis�es condition(lo1) the desired implication follows with help of the implication (ii) =) (i)in the proof of proposition 5.1.(i) =) (iii) let o 2 o be some arbitrarily chosen set. as in the proofof the implication (i) =) (ii) we consider the set m(o) and construct thelinear separable system e(o) on x. of course, we may assume without loss ofgenerality that for every pair of sets o 3 o0 $ o 2 m(o) such that o0 or ois closed the corresponding linear separable system e(o) on x consists of o0and o. let us abbreviate this assumption by (*). because of proposition 5.1(iii) there exists some countable subset e0 of e0(o) such that for every pair ofsets (e0;e) 2 z(e(o)) there exists some set e+ 2 e0 such that e0 � e+ �e+ � e. now we consider the situation o0 $ o00 � o00 $ o for some pairof sets (o0;o) 2 z(o) and some set o00 2 o. there exists some set e 2 e0such that o0 � e � e � o. if e 62 o, then there exists because of (*) andthe construction of e(o) some pair of sets o 3 o+ $ o++ 2 m(o) suchthat o+ $ e � e $ o++. since o is linearly ordered by set inclusion itfollows with help of condition (lo1) and the chain o0 $ o00 � o00 $ o thato0 $ o+ � o+ � o++ $ o. hence, one immediately veri�es that assertion(iii) will follow with help of assertion (iii) of proposition 5.1, if we are able toshow that the set of all pairs o 3 o+ $ o++ 2 m(o) for which there existssome set e 2 e0no such that o+ � e � e � o++ 2 m(o) is countable.but this is easily seen since the corresponding sets o++ \ xno+ are pairwisedisjoint and e0 is countable.(iii) =) (iv) trivial.(iv) =) (i) in the same way as the implication (ii) =) (i) also this implica-tion follows with help of the proof of the corresponding implication (iv) =) (i)of proposition 5.1.(i) ^ (iv) =) (v) let some set o 2 f be arbitrarily chosen. then weconsider the linear separable system e(o) on x that already has been con-structed in the proof of the implication (i) =) (ii). since the linearly orderedsubtopology tl of t that is induced by o is coarser than the linearly orderedsubtopology t0l of t that is induced by e(o) it su�ces to verify that t0l is sec-ond countable (cf. the argument of the generalization of dt in the secondsection). in order to show that t0l is second countable it is because of assertion(iv) and condition (lo2+) su�cient to prove that the set k(o) of all (open)sets o 2 o such that (so3o0$o o0)� $ so3o0$o o0 $ o $ o is countable.in order to show the countability of k(o) we apply the normality of t in or-der to construct for every (open) set o 2 k(o) some linear separable systeme0(o) on x such that so3o0$o o0 � e0 � e0 � o for every set e0 2 e0(o)(cf. corresponding argument in the proof of the implication (i) =) (ii)). since(so3o0$o o0)� $ so3o0$o o0 $ o $ o we may conclude that e0(o) 6= ?.in the same way as in the corresponding part of the proof of the implication(i) =) (ii) it follows that e0(o) := e(o) [ (so2k(o) e0(o)) also is a linearseparable system on x. let us now assume, in contrast, that k(o) is not useful topologies and separable systems 79countable. then, since e0(o) 6= ? for every (open) set o 2 k(o) we mayconclude that the continuous linear (total) preorder on x that is inducedby e0(o) has uncountably many jumps or contains an uncountable family ofpairwise disjoint open (non-degenerate) intervals. this means, in particular,that has no continuous utility representation, which contradicts assertion (i).hence, assertion (v) follows.(v) =) (iv) since og � f assertion (iv) is an immediate consequence ofassertion (v). �in case that t is connected, proposition 7.1 is the generalization of edt tonormal topologies (cf. the corresponding remark in section 2).corollary 7.2. let t be a normal and connected topology on x. then thefollowing assertions are equivalent:(i) t is useful,(ii) every set o 2 o has a countable re�nement,(iii) every set o 2 o is second countable,(iv) every linearly ordered subtopology tl of t that is induced by some seto 2 og is second countable,(v) every linearly ordered subtopology tl of t that is induced by some seto 2 f is second countable.let, for the moment, a normal topology t on x said to be short, if every seto 2 o that is well-ordered by set inclusion is countable. then the followinginteresting proposition holds which, in particular, shows that condition (lo1)is a generalization of clf. this means that proposition 7.1 and corollary 7.2are generalizations of the theorem of est�evez and herv�es.proposition 7.3. in order for a normal topology t on x to be short it isnecessary that t satis�es clf.proof. let t be short. we assume, in contrast, that t does not satisfy clf.then there exists an uncountable locally �nite family o := foigi2i of pairwisedisjoint (non-empty) open subsets of x. in analogy to the proof of lemma6.1 we may assume that none of the sets oi (i 2 i) contains some non-emptyopen and closed subset. let us abbreviate this assumption by (*). in addition,the proof of lemma 6.1 allows us to assume that i coincides with the �rstuncountable ordinal !1, i.e., o = foigi2i = fo�g�i1 . using the latter, apply the same technique we just employed, this time with (f2, x2) replacing (f1, x1), to find i2 > i1 and g2 ∈ f such that x ∗ 2 = (z ∗ i2 , g2(z ∗ i2 )) ∈ v2 ⊂ u2 with the additional property that g2(z ∗ i2 ) = f2(z) in (case 1). such a process is to be done recursively so that we find a sequence i1 < i2 < · · · in n, a sequence (gi)i∈n in f, bi = {gi} ∈ a, and a subsequence (z ∗ ij )j∈n of (z ∗ i )i∈n such that for each j ∈ n, x∗j = (z ∗ ij , gj(z ∗ ij )) ∈ vj ⊂ uj, and, in (case 1), gj(z ∗ ij ) = fj(z). in the latter case, the sequence (gj(z ∗ ij ))j∈n equals (fj(z))j∈n = (fj(zj))j∈n, so it converges to w ∈ h. in (case 2), x∗j = (z ∗ ij , gj(z ∗ ij )) ∈ vj ⊂ z ×qj. this shows that gj(z ∗ ij ) ∈ qj, and hence the sequence (gj(z ∗ ij ))j∈n converges to w ∈ h. we have proved that under the assumption 2.7(6), a sequence (z∗ij , x ∗ j , bj, gj), j = 1, 2, . . . , can be produced with {z∗ij | j ∈ n} an infinite set as requested above. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 213 i. ivanšić and l. r. rubin we assume therefore that {zi | i ∈ n} is an infinite set. by passing to a subsequence of (zi)i∈n if necessary we may stipulate that for i 6= j, zi 6= zj and that for each i, zi 6= z. by 2.7(3), z is hausdorff, so b = {zi | i ∈ n} ∪ {z} is a convergent sequence in the sense of definition 2.8. define λ : b → h by λ(zi) = fi(zi), and λ(z) = w. then λ is a map. applying 2.7(2) get a map f : z → h such that c = {f} ∈ a along with a subsequence (zij ) of (zi) such that (zij , f(zij )) ∈ uij for all j. since a is a collection of subsets of f, then f ∈ f. � now we can prove our main result about the preservation pseudo-compactness in any finite product of the above types of spaces. theorem 2.13. let n ∈ n and xδ1, . . . , xδn be spaces like those obtained in theorem 2.7. then x = xδ1 × · · · × xδn is pseudo-compact. proof. for simplicity we will prove this in case n = 2; the reader will easily fill in the details needed for the general case. let (fk, zk, hk) correspond to (f, z, h) in lemma 2.12 for k ∈ {1, 2}. suppose that x is not pseudo-compact. then there exist a subset {xi | i ∈ n} of x and a map ω : x → r such that ω(xi) = i for each i. put wi = (i− 1 3 , i+ 1 3 ) and let qi = ω −1(wi). for each i, find open sets u k i of xδk , k ∈ {1, 2}, such that xi ∈ u 1 i × u 2 i ⊂ qi. first apply lemma 2.12 to the nonempty open subsets u1i of xδ1. there exist f1 ∈ f1, z1 ∈ z1, a subsequence (u 1 ij ) of (u1i ), and for all j ∈ n, a point z1j ∈ z1 such that for each j, (z 1 j , f1(z 1 j )) ∈ u 1 ij , (z1j ) converges to z1 ∈ z1, and (f1(z 1 j )) converges to f1(z1) in h1. hence, (z 1 j , f1(z 1 j )) converges to (z1, f1(z1)) in gf1 ⊂ xδ1. by passing to a subsequence we may as well assume that u1ij = u 1 j for each j. now apply the same procedure to the nonempty open sets u2i of xδ2. we get f2 ∈ f2, z2 ∈ z2, a subsequence (u 2 ij ) of (u2i ), and for all j ∈ n, a point z2j ∈ z2 such that for each j, (z 2 j , f2(z 2 j )) ∈ u 2 ij , (z2j ) converges to z2 ∈ z2, and (f2(z 2 j )) converges to f2(z2) in h2. hence, (z 2 j , f2(z 2 j )) converges to (z2, f2(z2)) in gf2 ⊂ xδ2. by passing to subsequences we may assume the following. there are sequences (zki ) in zk, k ∈ {1, 2}, such that (fk(z k i )) converges to fk(zk) in gfk with (r1i , r 2 i ) ∈ u 1 i × u 2 i ⊂ qi for each i where we define r k i = (z k i , fk(z k i )). let dk = {r k i | i ∈ n} ∪ {(zk, fk(zk))}. then each dk ⊂ gfk ⊂ xδk is compact. now, ω(r1i , r 2 i ) ∈ ω(qi) ⊂ wi for each i. so ω({(r 1 i , r 2 i ) | i ∈ n}) ⊂ ω(d1 × d2), a compact subset of r, which is impossible. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 214 finite products of limits of direct systems induced by maps references [1] r. engelking, general topology, pwn–polish scientific publishers, warsaw, 1977. [2] i. ivanšić and l. rubin, pseudo-compactness of direct limits, topology appl. 160 (2013), 360–367. [3] i. ivanšić and l. rubin, the topology of limits of direct systems induced by maps, mediterr. j. math. 11, no. 4 (2014), 1261–1273. [4] r. m. stephenson, jr., pseudocompact spaces, trans. amer. math. soc. 134 (1968), 437–448. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 215 hermanagt.dvi @ applied general topology c© universidad politécnica de valencia volume 8, no. 1, 2007 pp. 93-149 boundaries in digital spaces gabor t. herman ∗ abstract. intuitively, a boundary in an n -dimensional digital space is a connected component of the (n − 1)-dimensional surface of a connected object. in this paper we make these concepts precise, and show that the boundaries so specified have properties that are intuitively desirable. we provide some efficient algorithms for tracking such boundaries. we illustrate that the algorithms can be used, in particular, for computer graphic display of internal structures (such as the skull and the spine) in the human body based on the output of medical imaging devices (such as ct scanners). in the process some interesting mathematical results are proven regarding “digital jordan boundaries,” such as a specification of a local condition that guarantees the global condition of “jordanness.” 2000 ams classification: 54a99, 52c99, 68r99, 68w05 keywords: boundaries, surfaces, tracking, algorithms, digital topology, geometry, digital space. 1. the boundary tracking problem 1.1. a sample application of boundary tracking. many imaging devices will produce estimated values of a physical quantity at certain points (referred to as grid points) in three-dimensional space. for example, a ct (computerized tomography) scanner estimates the x-ray attenuation coefficient inside a human body at points of a three-dimensional rectangular grid. when displaying the results of such an estimation, we usually use a sequence of two-dimensional images, such as that shown in figure 1, in which bone appears light, softer tissues appear grey, and air appears dark. ∗the research of the author is currently supported by nih grant hl070472 and nsf grant dms0306215. 94 gabor t. herman figure 1. a slice of a ct scan of the head of a trauma victim. in figure 1 a cursor is placed at the right edge of a large piece of bone in the upper part of the image. since this ct scan was taken of a trauma victim, an important question to ask is: are all other bones which are normally “connected” to this main piece of bone still connected for this patient? for example, the biggish piece of bone just to the right and below the cursor (it is the mandible) is not connected to any other bone in this slice, but it should be connected to the rest of the skull (at our resolution) somewhere in the whole three-dimensional array. by this we mean that somewhere in the threedimensional space the mandible should be so close to the rest of the skull that, at the resolution of our imaging process, the two would be judged to be “connected.” making precise the sense in which the word “connected” is used here and the analysis of the resulting connectivity properties in threedimensional images is the main topic of the material that follows. we turn aside for a moment to introduce some notation. we use r to denote the set of real numbers and rn to denote the set of n -dimensional (row) vectors, all of whose components are real numbers. if v ∈ rn , vn denotes the nth component of v, for 1 ≤ n ≤ n . the norm of a vector v is denoted by ‖v‖. we use the algebraic notion of a vector in rn and the geometric notion of a point in an n -dimensional euclidean space quite interchangeably, and actually refer to rn as the n -dimensional (euclidean) space. we use z to denote the set of all integers. for any positive real number δ and for any positive integer boundaries in digital spaces 95 n , we define (1.1) δzn = {(δc1, . . . , δcn ) | cn ∈ z, for 1 ≤ n ≤ n} . specifically, we use zn to abbreviate 1zn . let g be any set of points (a grid) in rn . the voronoi neighborhood in g of any element g of g is defined as (1.2) ng (g) = { v ∈ rn | for all h ∈ g, ‖v − g‖ ≤ ‖v − h‖ } . in other words, the voronoi neighborhood of g consists of all those vectors which are not nearer to any other point of g than they are to g. for example, the voronoi neighborhood of (c1, c2) in z 2 is (1.3) { (v1, v2) ∈ r 2 | max (|v1 − c1| , |v2 − c2|) ≤ 1 2 } . this is a closed square with side-length 1 centered at the point (c1, c2). when perceived as a set of points in r2, δz2 is referred to as a square grid. the voronoi neighborhoods in a grid in r2 are referred to as pixels. we now return to the way in which the two-dimensional image of figure 1 was produced. let us assume that the x-ray attenuation coefficients have been estimated at grid points with integer coordinates (c1, c2, c3), 1 ≤ c1 ≤ i, 1 ≤ c2 ≤ j, 1 ≤ c3 ≤ k and that figure 1 is the kth of these k slices. the region over which the images are defined is the union of the pixels associated with those (c1, c2) ∈ z 2 for which 1 ≤ c1 ≤ i and 1 ≤ c2 ≤ j. the grey value assigned to the pixel associated with one of these (c1, c2)s is proportional to the estimated x-ray attenuation coefficient at (c1, c2, k). the overall aim of the type of work on which we are reporting is the display and analysis of selected objects based on the estimated values of a physical quantity at grid points in three(and sometimes higher) dimensional space. examples for bones in the head are shown in figure 2. on the top we show a computer graphic display of the boundary of the connected piece of bone which is indicated by the cursor in figure 1. (the cursor location in the top image in figure 2 corresponds to the cursor location in figure 1.) we now see that the mandible is disconnected from this bone in the whole three-dimensional data set. the boundary of the mandible can be extracted separately, and then the two boundaries can be displayed together in their correct relative locations, as is shown in the bottom image of figure 2. 1.2. a methodology for extracting object boundaries. we now discuss a particular methodology for extracting boundaries of objects based on values assigned to points of the grid z3 in r3. for δ > 0, δz3 is referred to as a cubic grid. the voronoi neighborhoods associated with a grid in r3 are referred to as voxels. the voxel associated with the element (c1, c2, c3) of z3 is the closed unit cube { v ∈ r3 | max (|v1 − c1| , |v2 − c2| , |v3 − c3|) ≤ 1 2 } . the tessellation of r3 into voxels of a cubic grid is sometimes referred to as a cuberille. in figure 3, we illustrate a grid point g of z3 together with all other 96 gabor t. herman figure 2. computer graphic displays of detected boundaries of bones in the head, based on the data set which contains the slice shown in figure 1. grid points which lie in the 2 × 2 × 2 cube whose center is g, as well as the voronoi neighborhood of g. now suppose that we have some methodology to determine which grid points belong to what kind of material. for example, in ct bone attenuates x-rays more than any other type of tissue in the human body and so we may say that if the value assigned to a grid point is above a certain value, then that grid point is in bone. so, as an approximation, the part of space that is occupied by bone may be considered to be the union of the voxels associated with those pixels for which the estimated x-ray attenuation coefficient is above the threshold for bone. this collection of voxels may not necessarily form a connected subset of three-dimensional space: some pieces of bone may be disconnected from the rest and some voxels may be included because noise in the estimation process boundaries in digital spaces 97 figure 3. on the left we show a point g of the cubic grid z3 together with the other 26 grid points which lie in the 2×2×2 cube whose center is g. on the right we show (using heavy lines) the voronoi neighborhood of g (a voxel). caused us to identify a grid point as being in bone, when in fact it is not. let us say that the “object” whose boundary we wish to display is a component of the set of points occupied by the “bone” voxels. even this specification is not precise enough to describe the intuitive notion of “a boundary” of an object. this is because an object of the kind we described in the previous paragraph may have cavities inside it (just look at figure 1; the inside of bones contains lots of less x-ray attenuating tissue and may well be identified as a result of thresholding as “not bone”) and so it may have multiple boundaries (one exterior one and possibly many interior ones). let us say that our task is to identify exactly one of these boundaries. how do we specify which one? one way is to point at a boundary face, that is at a face which separates a bone voxel from a not-bone voxel, and say that we wish to display that boundary of the object which contains that boundary face. (at this point, it is not even clear that this specification is legitimate. is the “boundary containing a boundary face” a well defined concept? we will show that it is; we will set up an environment in which for any boundary face there will be one and only one boundary containing it.) an intuitively useful picture is the following. we consider the cuberille, the tessellation of three-dimensional space into cubic voxels. a finite number of these voxels are occupied by sugar cubes of just the right size. we point at an uncovered face of one of these sugar cubes (i.e., a face such that the voxel on the other side of it is not occupied by a sugar cube) and ask that there be delivered to us the boundary surface which contains that face. the problem is to design an algorithm which is guaranteed to do this for all possible arrangements of the sugar cubes. 98 gabor t. herman at this point a topologist may reasonably ask: are we not reinventing the wheel here? after all, the boundary of a set is a classical topological concept. the problem is that there is no topology (in the classical sense) for the cuberille (and, indeed, for the other digital spaces to be discussed below) that would result in the same sets being “connected” according to the topological definition as those that would be considered connected in the digital environment. this somewhat surprising (and disturbing) fact is stated precisely and proved in section 4.3 of [13]. it is fascinating that, in spite of this, we are able to do many things similar to topics that are normally considered in classical topology: the jordan curve theorem, simple connectedness, etc. however, we are going to do these things not as a consequence of classical topology, but rather as a self-standing theory for digital spaces. the approach presented here is one of a number of different alternatives that can be taken to insert topological ideas into the digital environment. examples of alternatives can be found in [4], [16], [17], [19], [20] and in their references. from an even more general point of view, this paper concentrates on one family of methods for producing computer graphic displays of three-dimensional objects, as illustrated in figure 2. indeed there are many alternatives, for three surveys see [9], [10] and chapter 1 of [25]. broadly speaking, the approaches taken fall into two categories: the so-called “volumetric” and “boundary-based” approaches. what is presented here is an example of the latter, but it is by no means the only one. as discussed above, our approach produces a single boundary of a single component, specified by a user-supplied initial face in the boundary. as opposed to this, the so-called “marching cubes algorithm” [22] automatically detects all surfaces of all components. it is not clear that this really is on advantage: after all, in practical visualization only a few of these surfaces should be displayed (as in figure 2). besides, the mathematical theorems proven below regarding the boundaries produced by our algorithms having well-defined interiors and exteriors allow us to detect automatically all boundaries by repeated applications of the algorithm for the tracking of a single boundary. in fact, as discussed in [21], the set of all surfaces produced by the marching cubes algorithm can also be produced by creating “continuous analogs” of the boundaries produced by the algorithms discussed below, and the total process of tracking all the boundaries (in our sense) and creating their continuous analogs requires less computer time than what is needed by the marching cubes algorithm, provided only that the number of grid points needed for the objects of interest (i.e., i × j × k) is large; and this is even more so if the values originally estimated for these grid points are noisy (i.e., inaccurate). 1.3. flies in flatland. before attacking the three-dimensional problem, we consider its simpler twodimensional analog. following abbott [1], we refer to the two-dimensional plane as flatland and to the computing device which we hope will deliver us the required boundary surfaces as a flat fly. as an example, consider a boundaries in digital spaces 99 configuration of two-dimensional sugar cubes in flatland shown in figure 4 with a flat fly on one of the flat faces (i.e., on an edge) of one of these flat sugar cubes. algorithm 1.1. algorithm for flat flies (1) dirty the face on which you are standing. (2) crawl onto the face which meets the one on which you are standing at the vertex in front of you. (3) see if it is dirty. (a) if it is, fly away. (b) if it is not, start again at instruction (1). this algorithm actually does something that can be made precise. to achieve that precision, we need to introduce some mathematical terminology. 1.4. components determined by binary relations. let f be any set and ρ be a binary relation on f . if (c, d) ∈ ρ, then we say that c is ρ-adjacent to d and that d is ρ-adjacent from c and, in case ρ is a symmetric relation (meaning that (c, d) ∈ ρ if, and only if, (d, c) ∈ ρ), that c and d are ρ-adjacent. in case f is finite, we define for any d in f its indegree (denoted by inp (d)) as the number of elements a of f such that (a, d) ∈ ρ. in the case when f = zn , we will be repeatedly dealing with two symmetric binary relations, αn and ωn , defined as follows. (1.4) (c, d) ∈ αn ⇔ (c 6= d and, for 1 ≤ n ≤ n, |cn − dn| ≤ 1) . (1.5) (c, d) ∈ ωn ⇔ n ∑ n=1 |cn − dn| = 1. for n = 2, (c, d) ∈ α2 means that the associated pixels have a nonempty intersection, but are not identical, and (c, d) ∈ ω2 means that the associated pixels have exactly one edge in common. these are often used notions and go under a number of commonly used names: ω2 is referred to both as the edge-adjacency and as the 4-adjacency, while α2 is referred to both as the edge-or-vertex-adjacency and as the 8-adjacency. for any c and d in a subset a of f , we call a sequence 〈 c(0), . . . , c(k) 〉 of elements of a a ρ-path in a connecting c to d, if c(0) = c, c(k) = d and, for 1 ≤ k ≤ k, c(k−1) is ρ-adjacent to c(k). if such a path exist, we say that c is ρ-connected to d in a. a nonempty subset a of f is said to be a ρ-connected if, for any c and d in a, c is ρ-connected in a to d. if ρ-connectedness in a is symmetric, then it is an equivalence relation. (note that if ρ itself happens to be symmetric, then ρ-connectedness in a is automatically symmetric.) hence, it partitions a into ρ-components (i.e., into nonempty ρ-connected subsets which are not proper subsets of any other ρconnected subset of a). referring to figure 4, we see that there are three ω2-components of the set of grid points whose pixels are painted gray: one has 100 gabor t. herman figure 4. illustration of boundary tracking in two dimensions by the algorithm for flat flies when a flat fly is initially placed on a flat face (i.e., on an uncovered edge of an occupied pixel): the first row is the beginning, the second row is the middle, and the third row is the end of the execution of the algorithm. seven element and the other two have one element each. on the other hand, there are only two α2-components. we denote the complement f − a of a in f by a. in figure 4, a (the set of grid points whose pixels are not painted grey) is an α2-connected subset (and hence it has only one α2-component). however, it is not ω2-connected. in the boundaries in digital spaces 101 special case when a = f , we use the phrases ρ-path and ρ-connected instead of ρ-path in a and ρ-connected in a. for any binary relation ρ on a set f , we define its transitive closure as the binary relation ρ∗ on f by: (c, d) ∈ ρ∗ if, and only if, c is ρ-connected to d. if ρ∗ is symmetric, then we refer to ρ as an adjacency (on f ). in this case f is partitioned into ρ-components. note, in particular, that (for all n ≥ 1) both αn and ωn are adjacencies on z n (and, for these adjacencies, there is only one component in the partition of zn ). 1.5. what does a flat fly do? we first specify the outcome of the algorithm for flat flies in the specific case identified in figure 4. the flat fly is initially put on the face which separates the grid point g (painted grey) from the grid point h (not painted grey). after a while it flies away (see figure 4), leaving behind a bunch of faces that have been dirtied. this set of faces can be described as consisting of exactly those faces which are between a grid point painted grey that is α2-connected in the set of grey grid points to g and a grid point not painted grey that is ω2-connected in the set of not grey grid points to h. this is, in fact, representative of the general behavior of the flat fly. we can say the following. suppose that in the square grid z2 the pixels of a finite number of grid points are filled with flat sugar cubes and that a flat fly is put on top of one of these sugar cubes into a pixel which is not occupied by a sugar cube (see figure 4). let g be the name of the grid point of the sugar cube on top of which the flat fly is placed into the pixel of a grid point named h (see figure 4). claim 1.2. after a finite number of loops through instructions (1), (2), and (3) of the algorithm for flat flies, the flat fly will fly away. at that time, the set of dirty faces is exactly the set of those faces which are between a pixel associated with an element of the α2-component of the set of grid points with sugar cubes that contains g and a pixel associated with an element of the ω2component of the set of grid points without sugar cubes that contains h. we do not give a proof of this claim in this section; such a proof will be a consequence of the general results that we will be deriving in the sections that follow. we will return to proving the claim at the appropriate point later in the paper. however, before going on to other things we state another claim that shows that the set of faces dirtied by the flat fly has some desirable properties. first we need to introduce some additional notation and terminology. we first observe that the face onto which the fly is initially placed can be identified by the pair (g, h) of grid points on either side of it. note that this is an ordered pair, interpreted as “the fly is put into h, with its feet touching g.” if we were to draw the flat fly on (h, g), then its feet would be touching the same edge, but it would be hanging upside down. similarly, every ordered pair of ω2-adjacent grid points can be thought of as such an edge with an orientation across it from the first grid point in the pair toward the second 102 gabor t. herman and, conversely, all such oriented edges can be described by a unique element of ω2. thus we make the fundamental observation that ω2 has a dual purpose: it is an adjacency on z2 and its elements can be used as unique identifiers of elements of the set of all oriented faces between pixels of this square grid. we note immediately that, for arbitrary positive integer n , ωn is an adjacency on the grid zn and its elements can be used as unique identifiers of elements of the set of all oriented (hyper-)faces between points of the grid. we now introduce two important concepts for the grid zn . for any two subsets o and q of zn we define the boundary between them by: (1.6) ϑ (o, q) = {(c, d) | c ∈ o, d ∈ q, (c, d) ∈ ωn } . let a consist of the nonempty finite set of grid points which are filled with sugar cubes and let the flat fly be placed on (g, h), where g ∈ a and h ∈ a. let o be the α2-component of a containing g and q be the ω2-component of a containing h. then claim 1.2 says that the set of faces dirtied by the flat fly is exactly ϑ (o, q). let 〈 c(0), . . . , c(k) 〉 be an ωn -path and s be any subset of ωn . we say that 〈 c(0), . . . , c(k) 〉 crosses s, if there is a k, 1 ≤ k ≤ k, such that either ( c(k−1), c(k) ) ∈ s or ( c(k), c(k−1) ) ∈ s . claim 1.3. let a be any nonempty finite subset of z2. let o be an α2component of a and q be an ω2-component of a, such that ϑ (o, q) is not empty. then there exist two uniquely defined subsets i and e of z2, which have the following properties. (i) o ⊆ i and q ⊆ e. (ii) ϑ (o, q) = ϑ (i, e). (iii) i ∪ e = z2 and i ∩ e = ∅. (∅ denotes the empty set.) (iv) i is α2-connected and e is ω2-connected. (v) every ω2-path connecting an element of i to an element of e crosses ϑ (o, q). the proof of this claim is also delayed till later; in fact it will be proven in a much more general context. here we discuss its relevance with respect to the algorithm for flat flies. since, according to claim 1.2, the output of this algorithm is a boundary ϑ (o, q) satisfying the premises of claim 1.3, the latter claim says that there are two sets of grid points i and e, of which we will think intuitively as the interior and the exterior, which have certain properties reminiscent of the famous jordan curve theorem [23]. according to that theorem, the set of all points in the plane which are not on a simple closed curve can be partitioned into two subsets, one may be called the interior and the other the exterior. both of these are connected sets in the sense that we can get from any point of the interior to any other point by drawing continuously a curve between them which never leaves the interior (and similarly for the exterior), but one cannot draw continuously a curve from a point in the interior to a point boundaries in digital spaces 103 in the exterior which does not contain at least one point of the simple closed curve, which is in fact the boundary between the interior and the exterior. the boundary that is the output of the algorithm for flat flies shares important properties with simple closed curves in r2: its interior and exterior partition the whole space (iii), both are connected in some sense (iv), but one cannot get from the interior to the exterior without crossing the boundary (v) which, in fact, is the boundary between the interior and the exterior (ii). (mathematically speaking there is also a difference: a simple closed curve is a subset of the plane, and the jordan curve theorem says that the complement of the curve, rather than the whole plane, has precisely two components. since our boundary is a subset of ω2, rather than of z 2, it is possible to demand that the interior and exterior partition the whole of z2, if anything, a more attractive-sounding aim than that of the classical theorem.) it is important to recognize that, in spite of this formal similarity to the jordan curve theorem, claim 1.3 is not a result about the continuous plane r2 but a thoroughly digital theorem. notions of “connectedness,” “path,” and “crosses” have all been defined for the square grid z2 and although they are analogous to corresponding notions in r2, the analogy breaks down sometimes. it is the difficulty of applying the continuous notions directly to the digital environment to prove claims regarding algorithms operating in discrete domains which motivates us to develop a geometry directly for digital spaces. for example, it is tempting to reinterpret the boundary output by the algorithm for flat flies as the point set in r2 which is the union of all the points in all the dirtied faces. however, in spite of the discrete jordan curve properties of this boundary (claim 1.3), the jordan curve theorem is not applicable to the corresponding point set in r2, since it does not form a simple closed curve. (it “touches itself” at one vertex and so it is not a simple curve.) having said this, we must admit that the argument is only half convincing. it is all very well that we can give a self-contained theory in digital spaces with nice theorems that resemble those of famous results of continuous mathematics; nevertheless, an interpretation in the underlying continuous space is unavoidably needed for some applications. for example, no physician would think of the boundaries shown in figure 2 as collections of ordered pairs of voxels; rather they would be perceived as continuous biological surfaces. hence, it is not really desirable to have boundaries which touch themselves in the fashion of the boundary consisting of the dirtied faces at the end of figure 4. while we are pondering this, we may as well ask the following question: is it reasonable that two pixels which only share a vertex are considered connected to each other? this came about quite naturally in the way the flat fly crawled about on the sugar cubes in figure 4, but looking at the result the connection is rather suspect. would it not be better to use only ω2 to define connectivity and avoid such flimsy looking connections? once we raise this objection on physical grounds, we also see that the use of α2 is not particularly elegant even from a mathematical point of view. why do we use different types of components for grid points with sugar cubes and for 104 gabor t. herman those without in claim 1.2? why do we use different types of components for a and for a in claim 1.3? since in (v) of claim 1.3 only ω2 is used, would it not be much nicer to replace α2 by ω2 everywhere in claim 1.3? unfortunately, the resulting claim would not be true; one can easily produce counterexamples [11]. thus, we need ω2 (it is used to describe the edges which make up the boundary), and we need a broader adjacency for the validity of the rather desirable claim 1.3. 1.6. on the cuberille. in three-dimensional space we would like to have an algorithm for fat flies (in figure 5 we show one placed on a sugar cube), which would have behavior resembling that of flat flies as expressed in claims 1.2 and 1.3. we first need to decide on the appropriate analog of α2 for the case of a cuberille. one candidate is α3. the geometrical interpretation of α3 is: two points of the cubic grid z3 are α3-adjacent if, and only if, the corresponding voxels have either exactly one face, or one edge, or one vertex in common (see figure 3). the question is: for the purpose of the three-dimensional analog of claim 1.2, do we want to consider two sugar cubes connected if they share only a vertex? even in two dimensions such a connectivity was undesirable, but was forced upon us in order to have claim 1.3. in three dimensions we need ω3 to define the boundary surface, but also something wider to obtain an analog of claim 1.3. the difference is that now we have an alternative, another analog of α2 in which two grid points are adjacent if the corresponding voxels have either exactly one face in common or exactly one edge in common. formally, for any positive integer n , we define the adjacency δn on z n as follows. for any c and d in zn , (1.7) (c, d) ∈ δn ⇔ [ (c, d) ∈ αn and n ∑ n=1 |cn − dn| ≤ 2 ] . clearly, δ2 = α2, and so δ3 and α3 are equally legitimate three-dimensional analogs of α2. however, δ3 is intuitively preferable over α3, inasmuch as it does not allow adjacency by vertices only; see figure 6. (for obvious reasons, ω3 has been referred to both as face-adjacency and as 6-adjacency, δ3 has been referred to both as face-or-edge-adjacency and as 18-adjacency, and α3 has been referred to both as face-or-edge-or-vertex-adjacency and as 26-adjacency.) in view of this, our aim is as follows. problem 1.4. design an algorithm for fat flies that will have the following property. suppose that in the cubic grid z3 (see figure 3) the voxels of a finite number of grid points are filled with sugar cubes and that a fat fly is put on top of one of these sugar cubes into a voxel which is not occupied by a sugar cube (see figure 5). let g be the name of the grid point of the sugar cube on top of which the fat fly is placed into the voxel of a grid point named h. let o be the δ3-component of the set of grid points with sugar cubes which contains g and q be the ω3-component of the set of grid points without sugar cubes which boundaries in digital spaces 105 figure 5. the route of a model 1 fat fly on one of a pair of sugar cubes. figure 6. the grid points c and d on the left are α3-, δ3-, and ω3-adjacent; those in the middle are α3and δ3-adjacent but not ω3-adjacent; those on the right are α3-adjacent but are not δ3-adjacent or ω3-adjacent. contains h. after a finite number of steps of the algorithm for fat flies, the fat fly should fly away and, at that time, the set of faces that have been dirtied should be exactly ϑ (o, q). the output ϑ (o, q) of such an algorithm is an intuitively desirable boundary, due to the truth of the following claim (whose proof is also postponed 106 gabor t. herman until we have reached the appropriate point in the development of our general theory). claim 1.5. let a be any nonempty finite subset of z3. let o be a δ3component of a and q be an ω3-component of a such that ϑ (o, q) is not empty. then there exist two uniquely defined subsets i and e of z3 with the following properties. (i) o ⊆ i and q ⊆ e. (ii) ϑ (o, q) = ϑ (i, e). (iii) i ∪ e = z3 and i ∩ e = ∅. (iv) i is δ3-connected and e is ω3-connected. (v) every ω3-path connecting an element of i to an element of e crosses ϑ (o, q). we see that this claim is a digital three-dimensional version of the jordan curve theorem. apart from it being a mathematically pleasing fact, this observation is of the utmost practical importance. in applications we are interested in finding boundaries of objects for two reasons. one is to create computer graphic displays of them, such as the ones shown in figure 2, and the other is to analyze certain properties of the objects, such as their volumes. from the graphical display point of view, property (v) can be used to prove that when we display the boundary ϑ (o, q) as it appears from any exterior point, then the surface displayed will hide all the points interior to the object. for the purpose of analysis, claim 1.5 works as follows. suppose that a device has given us estimates of some physical quantity at points of the cubic grid z3. suppose further that we can identify from such data that set, a, of grid points which are in cardiac muscle. assuming that a itself is δ3-connected and that the set of grid points in any one of the chambers of the heart is an ω3-component of a, we can estimate the volume of the left ventricle, say, by selecting ω3-adjacent voxels g and h, so that g is in the cardiac muscle and h is in the left ventricle, defining o and q as in problem 1.4, and estimating the volume of the left ventricle of the heart as the combined volume of the voxels associated with the grid points of the uniquely defined exterior e, whose existence is guaranteed by claim 1.5. alternatively, if we wish to find the volume of the whole heart, then assuming that the set of grid points outside the heart form an ω3-component of a, we can select ω3-adjacent voxels g and h, so that g is in the cardiac muscle and h is outside the heart, defining o and q as in problem 1.4, and estimating the volume of the whole heart as the combined volume of the voxels associated with the grid points of the uniquely defined interior i, whose existence is guaranteed by claim 1.5. this also indicates how the achievement of problem 1.4 would allow us to detect individual boundaries of the cardiac muscle. boundaries in digital spaces 107 1.7. algorithms for fat flies. first we observe that just by simply changing the word “vertex” in the algorithm for flat flies into the word “edge” we get a set of instructions that are meaningful in the three-dimensional case. however, the resulting algorithm will not be a solution to problem 1.4. even if there is only one grid point with a sugar cube, this simply modified version will fail. the fat fly will keep moving forward and, having dirtied only four faces of the cube, it will get back to the original face and will fly away. clearly, ϑ (o, q) in this case should consist of all six faces of the sugar cube, and the proposed algorithm resulted in only four of them being dirtied. so consider instead the following. algorithm 1.6. algorithm for fat flies model 1 (1) dirty the face on which you are standing. (2) crawl onto the face which meets the one on which you are standing at the edge in front of you. (3) turn to the right, and dirty the face on which you are standing. (4) crawl onto the face which meets the one on which you are standing at the edge in front of you. (5) see if it is dirty. (a) if it is, fly away. (b) if it is not, turn left and start again at instruction (1). it turns out that this algorithm will behave as required for the case of a single sugar cube. it can be seen in figure 5 that the fat fly under the control of the model 1 algorithm will visit (and hence dirty) every face of a single sugar cube. however, the route of the fat fly as indicated in figure 5 tells us that the model 1 algorithm does not behave as is needed for problem 1.4. since there are edges not traversed by its route, if we attach a second sugar cube to one of these edges, the grid points of the two sugar cubes are δ3-adjacent, and so ϑ (o, q) consists of the twelve faces of the pair of sugar cubes. however, the route of the fat fly does not change; it still dirties only six faces; see figure 5. in fact, we see that this indicates a general principle. if the proposed algorithm is such that the edge toward which the fat fly starts crawling is not influenced by the absence or presence of nearby sugar cubes, then for the algorithm to work, it must traverse each edge when started on a face of a single sugar cube. (otherwise, we can attach a second sugar cube to the untraversed edge, just as indicated in figure 5.) however, if each edge is traversed once during a route on a single sugar cube, then each face must be visited twice. so the simple mechanism of dirtying the faces the first time they are visited and flying away the second time one is visited will not do. this by itself is not a problem; we can have the fly “mark” a face the first time it visits it, “dirty” it when it finds that it has already been marked, and fly away when it finds that the face has already been dirtied. the interesting question is: what should be the rule for deciding the edge towards which the fat fly starts crawling when it is on a particular face? 108 gabor t. herman figure 7. pairs of faces with two types of arrow orientation; only the one on the left is acceptable. we now establish a further principle. having accepted that the route of the fat fly must traverse each edge, let us aim for the shortest route on a single sugar cube which satisfies this condition. if we draw the route of such a fly on the sugar cube as was done for figure 5, then we see that on each face there have to be four arrows (one for each edge of the face), two of which point from the center of the face towards an edge and two which point from the other two edges towards the center of the face. further, there has to be some consistency between these arrows: if an edge has an arrow pointing to it on one face, on the face which meets this edge the arrow has to be pointing away from the edge. finally, we consider the desirable simplicity of the algorithm for fat flies. we impose the condition that the way in which the arrows point on a face depends only on the orientation of that face. (there are six possible orientations, corresponding to the six faces of a single sugar cube.) now consider the arrangements shown in figure 7. we see that the only type of arrangement that satisfies all the conditions just discussed is the one in which the two inward arrows are next two each other (making an angle of 90◦) as is illustrated on the left in the figure. this implies that the route of the fat fly for a single sugar cube must have the general form indicated by figure 8. we say general form, since the considerations above still leave us with some freedom. for example, the horizontal cycle of arrows around the vertical faces could be reversed without violating the conditions described above. the same is true for the two vertical cycles of arrows, leaving us with eight possible legitimate arrangements. however, these are symmetrical versions of each other and, for what we are going to do, there is no harm in restricting our attention to the arrangement shown in figure 8. a legitimate route for the fat fly is defined as one which follows the arrows and traverses each edge once. the following is a fascinating fact. if we restrict (in a reasonable way) the form that an algorithm for fat flies may take, then it can be shown [14] that there is no such algorithm which causes the fat fly to cover a legitimate route if placed on a single sugar cube and which, at the same time, satisfies problem 1.4. in other words, there is no straightforward three-dimensional generalization of the algorithm for flat flies. one way of overcoming this difficulty is to allow fat flies with outlandish capabilities, such as being able to create a clone of themselves [14]. we discuss boundary tracking algorithms based on boundaries in digital spaces 109 figure 8. legitimate routes of the fat fly on a single sugar cube follow the arrows and cross every edge once. the numbers refer to the orientations of the faces and not to the order in which they are visited. such ideas in the following sections. 1.8. a formulation of the boundary tracking problem. we now say good-bye to our flies and prepare ourselves for the more mathematical treatment of the following sections. let g be an arbitrary grid in r3 and let π be the adjacency defined on this grid by (c, d) ∈ π if, and only if, the intersection of the voxels associated with c and d is not a subset of a line in r3. (in case g = z3, this definition implies that π = ω3.) we redefine the concept of a boundary for this more general situation. for any two subsets o and q of g the boundary between them is (1.8) ϑ (o, q) = {(c, d) | c ∈ o, d ∈ q, (c, d) ∈ π} . a (finite binary) picture over a grid g is an assignment of 1 to finitely many points of g and 0 to the rest of the points. (this is a mathematical way of identifying whether or not the voxel associated with the grid point is occupied by a sugar cube.) we refer to the grid points that have 1 assigned to them as 1spels and to the others as 0-spels. (the word spel is an abbreviation of “spatial element.”) we say that (c, d) ∈ π is a bel (short for “boundary element”) if c is a 1-spel and d is a 0-spel. the following is a generalization of problem 1.4 in which the adjacencies δ3 and ω3 on z 3 are replaced by arbitrary adjacencies 110 gabor t. herman κ and λ on the grid g. (the general notion of an adjacency between spels will be rigorously defined in section 3. in section 2 the discussion will be restricted to specific cases for κ and λ.) problem 1.7. design an algorithm for the grid g and adjacencies κ and λ, with the following property. for any picture over g and for any bel (g, h), let o be the κ-component of the set of 1-spels which contains g and q be the λcomponent of the set of 0-spels which contains h. after a finite number of steps the algorithm should terminate and, at that time, its output should be exactly ϑ (o, q). these notions are easily generalized to n -dimensional space. we define the adjacency π on a given grid g by (c, d) ∈ π if, and only if, the intersection of the voronoi neighborhoods of c and d is not a subset of any set formed by all linear combinations of n − 2 vectors in rn . (in case g = zn , this definition implies that π = ωn .) defining boundaries, pictures, 1-spels, 0-spels, and bels exactly as above problem 1.7, the aim itself can be restated without any changes for the n -dimensional case. in what follows we will not fully achieve this very general aim: the algorithms that we will provide in the next section will perform as desired only for a restricted class of the adjacencies κ and λ. however, this class is sufficiently large to justify us claiming to have achieved the n -dimensional generalization of problem 1.4 and some instances of problem 1.7 that are not included in the n -dimensional generalization of problem 1.4. 2. some boundary tracking algorithms 2.1. boundary tracking on the cuberille. in this and the next subsection we concentrate our attention to solving problem 1.4 or, more mathematically, problem 1.7 with g = z3, κ = δ3 and λ = ω3. we first discuss a way of defining the “adjacency of one bel to another” and then use this concept to describe the algorithms. looking at figure 8 we see that the arrangement of arrows on the sugar cube can be interpreted as a digraph with nodes {1, 2, 3, 4, 5, 6}. we now redefine this digraph in a way that will be easy to generalize later on. let, for 1 ≤ n ≤ n , un denote the unit vector in direction n, which is defined as the element of zn for which unn = 1 and all other components are 0. for the case n = 3, consider the digraph with nodes m = { u1, u2, u3, −u1, −u2, −u3 } and with arcs (2.1) ρ = {( u1, u2 ) , ( u2, −u1 ) , ( −u1, −u2 ) , ( −u2, u1 ) , ( u1, u3 ) , ( u3, −u1 ) , ( −u1, −u3 ) , ( −u3, u1 ) , ( u2, u3 ) , ( u3, −u2 ) , ( −u2, −u3 ) , ( −u3, u2 )} (see figure 9). this (m, ρ) is an example of the general concept of a basic digraph, which will be defined in subsection 2.4. note that if (c, d) ∈ ω3, then d − c ∈ m and is ρ-adjacent to exactly two other unit vectors. boundaries in digital spaces 111 figure 9. illustration of a basic digraph (on the right) and its interpretation as a set of directions taken while tracking the boundary between a single voxel and the set of all other voxels in z3 (in the middle, where each face is labelled by the unit vector normal to the face and exterior to the cube). the basic digraph gives rise to a bel-adjacency β, defined as follows. if a = (c, d) is a bel (which means that c is a 1-spel, d is a 0-spel, and (c, d) ∈ ω3), then there are exactly two unit vectors u ∈ m which are ρ-adjacent from d − c and each of these gives rise to exactly one bel bu, which is β-adjacent from (c, d), according to the following definition (see figure 10). if d + u is a 1-spel (case (iii) in figure 10), then bu = (d + u, d). if d + u is a 0-spel and c + u is a 1-spel (case (ii) in figure 10), then bu = (c + u, d + u). if both d + u and c + u are 0-spels (case (i) in figure 10), then bu = (c, c + u). based on these definitions, the following algorithm fulfills the requirements of problem 1.7 with g = z3, κ = δ3 and λ = ω3. in the algorithm l is a queue [24] of bels and s is a set of bels (which is considered the output of the algorithm when it terminates). the algorithm can be considered to be an instance of the well-known breadth-first search algorithm for traversing a strongly-connected component of a digraph from an initial node. algorithm 2.1. inefficient bel-tracking algorithm (1) put (g, h) into l and s. (2) remove a bel a from l. for both bels b which are β-adjacent from a and which are not in s, put b into l and s. (3) check if l is empty. (a) if it is, stop. (b) if it is not, start again at instruction (2). we will not bother proving the correctness (in the sense of fulfilling the requirements of problem 1.7 with g = z3, κ = δ3 and λ = ω3) of this algorithm. this is because we will be able to give a more efficient alternative. the inefficiency of the algorithm above is hidden in the ostensibly innocuous phrase “which are not in s?” how do we check that a bel is not in s? in the early 112 gabor t. herman figure 10. illustration of the definition of bel-adjacency using the basic digraph of figure 9. in each case, the darklyshaded bel is β-adjacent to the lightly-shaded one. note that in addition to the bel that is indicated in this figure, there is a second bel that is β-adjacent to the lightly-shaded one. stages of the algorithm s has only a few elements and so it is easy to carry out such a check. as the algorithm proceeds, s gets larger and larger (quite typically containing well over a million elements in a medical application) and checking for a particular bel belonging to it gets demanding on computer resources (time and/or storage). we next discuss an alternative to the inefficient bel-tracking algorithm which goes a long way towards reducing this computational expense. prior to doing that we point out a common property of these algorithms. at the end of subsection 1.7 we discussed the need for “fat flies with outlandish capabilities, such as being able to create a clone of themselves.” the mathematical equivalent of this is in the phrase “for both bels b which are β-adjacent from a.” we can represent this as a fat fly on bel a cloning itself into two, one clone crawling onto one and the other clone crawling onto the other bel β-adjacent from a. 2.2. artzy’s algorithm. a way of decreasing the computational burden of the inefficient bel-tracking algorithm was proposed by artzy et al. [2]. their trick is to make use of an auxiliary data structure t of once-visited bels and to check for membership in t instead of for membership in s. the important underlying fact is that each bel is β-adjacent from exactly two bels, and so it will be reached exactly twice. hence if a bel is found in t , it has already been reached once before and so it will not be reached again, and can be removed from t . consequently, boundaries in digital spaces 113 the size of t is likely to be a tiny fraction of the size of s and one can expect orders of magnitude of reduction in computational requirements over the inefficient bel-tracking algorithm when dealing with typical problems arising in many application areas, such as medicine. for a detailed discussion of the computational considerations associated with the algorithm of this section, see [6]. algorithm 2.2. artzy’s algorithm (1) put (g, h) into l and s and put two copies of (g, h) into t . (2) remove a bel a from l. for both bels b which are β-adjacent from a, try to find a copy of b in t . (a) if successful, remove this copy of b from t . (b) if not, then put b into l, s and t . (3) check if l is empty. (a) if it is, stop. (b) if it is not, start again at instruction (2). we illustrate the output of artzy’s algorithm in figure 11. ct scans with 0.8mm × 0.8mm pixels were taken at 7 mm interval covering the spine and the ribs of a patient. from these we estimated (by interpolation) values for a cubic grid assuming a unit distance of 0.65 mm and then we assigned 1s and 0s to the grid points using a threshold appropriate for distinguishing bone (1) from not bone (0). we applied artzy’s algorithm to the resulting picture. it produced, upon its termination, an output s containing 1,661,728 bels. this is displayed in figure 11 using a methodology described in [3]. we will not prove the correctness (in the sense of fulfilling the requirements of problem 1.7 with g = z3, κ = δ3 and λ = ω3) of artzy’s algorithm, since in subsection 2.4 we will state an n -dimensional version of it and the proof of correctness of that version will imply the correctness of the three-dimensonal version of this subsection. however, before getting into that generalization we discuss a remarkable adaptation of artzy’s algorithm to an alternative grid in the three-dimensional euclidean space. 2.3. a variant of artzy’s algorithm for the fcc grid. near the end of subsection 1.5 we raised some objections to the use of α2 as one of the adjacencies in claim 1.2, but then discarded them on the basis that using only ω2 would make the claim false. the same consideration lead us to using δ3, in addition to ω3, in problem 1.4 (and consequently to allowing such a flimsy-looking connectivity as is illustrated in figure 5). it turns out, however, that this difficulty is not inherent in the nature of our general problem; rather, it arises due to choosing a cubic (or a square) grid. we now discuss an alternative family of grids in three-dimensional space, one for which objections such as those stated at the end of subsection 1.5 disappear. (there are also other advantages of these grids over the cubic grids, but we will not get into a 114 gabor t. herman figure 11. three-dimensional display of the spine and the ribs of a patient (seen from behind and downward from the cervical vertebrae) approximated as a collection of 1,661,728 faces of cubic voxels. the area of the individual faces is the same as in figure 13. discussion of these and instead refer the interested reader to subsection 2.1 of [13].) for any positive real number φ, we define a face-centered cubic grid (or fcc grid, for short) as the set of points (2.2) fφ = {(φc1, φc2, φc3) | c1, c2, c3 ∈ z and c1 + c2 + c3 ≡ 0 (mod 2)} . each of the associated voxels is a rhombic dodecahedron (i.e., a twelve-faced solid, for which every face is an identical rhombus), as can be seen from figure 12. we will sometimes abbreviate f1 as f . clearly, f is a subset of z3. a remarkable property of f is that for the adjacency π for the grid f , as defined in subsection 1.8, we have that, for any g and h in f , (g, h) ∈ π if, and only if, (g, h) ∈ δ3 (see figure 12). on the other hand, it follows from the definition of f that, for any g and h in f , it cannot possibly be that (g, h) ∈ ω3. since f is a subset of z3, every picture over f gives rise to a derived picture over z3, simply by assigning 0 to those elements of z3 which are not in f . this boundaries in digital spaces 115 figure 12. on the left we show a point g of the fcc grid f = f1 together with the other 12 grid points which lie in the 2 × 2 × 2 cube whose center is g. on the right we show (using heavy lines) one of the faces of the voronoi neighborhood of g (whose shape is a rhombic dodecahedron) and also (using lighter lines) the cubic voxel associated with g when considered a point in z3. note that the edge of the cubic voxel which is specified by (g, h) is a diagonal of that face of the rhombic dodecahedral voxel which is specified by (g, h). is the observation which allows us to adapt artzy’s algorithm to boundary tracking in the fcc grid. the idea [7] is to consider the nature of the beladjacency β in the derived picture. observe figure 10. in case (i), (c, d + u) is a bel of the picture over f . case (ii) cannot possibly arise in a derived picture. in case (iii), (c, d + u) is not a bel of the picture over f , since both c and d + u are 1-spels. note that case (i) can be distinguished from the other cases by the fact that it is the only one in which the bel (e, f ) which is β-adjacent from (c, d) has the property that e = c (and we also happen to have that f = c + u). we make use of this observation by running artzy’s algorithm on the derived picture and producing an output s which contains bels of the picture over f . in order to do this, we need to find a suitable starting point to artzy’s algorithm, since in this case the (g, h) of problem 1.7 is not a bel of the derived picture. however, if we define, for 1 ≤ n ≤ 3, (2.3) h′n = { hn, if ∑n i=1 |gi − hi| ≤ 1, gn, otherwise, then (g, h′) will be a bel of the derived picture in z3. 116 gabor t. herman algorithm 2.3. tracking algorithm for the fcc grid (1) put (g, h′) into l and put two copies of (g, h′) into t . (2) remove a bel (c, d) of the derived picture from l. for both bels (e, f ) of the derived picture that are β-adjacent from (c, d), do the following. if e = c, then put (c, d + f − c) into s. try to find one copy of (e, f ) in t . (a) if successful, remove this copy of (e, f ) from t . (b) if not, then put (e, f ) into l and t . (3) check if l is empty. (a) if it is, stop. (b) if it is not, start again at instruction (2). we illustrate the output of the tracking algorithm for the fcc grid in figure 13. the same ct scans were used as those employed to produce figure 11. from these we estimated (by interpolation) values for an fcc grid assuming a unit distance of 20.25 × 0.65mm and then we assigned 1s and 0s to the grid points using the same threshold that was employed to produce figure 11 for distinguishing bone (1) from not bone (0). we changed the unit of distance so that the faces of the voxels in the two grids will have the same size; we considered this to be reasonable when comparing the boundaries output by two tracking algorithms. consequently, the volumes of the rhombic dodecahedra are more than 3.36 times larger than the volumes of the cubes and the number of fcc grid points needed to cover a particular volume of space is less than a third of the number of cubic grid points needed for the same purpose. (this is one of the advantages of using an fcc grid over using a cubic grid.) we applied the tracking algorithm for the fcc grid to the resulting picture. it produced, upon its termination, an output s containing 1,258,482 bels. this is displayed in figure 13 using the same methodology as was employed to produce figure 11. the appearance of the two computer graphic displays is very similar, but there are some less obvious, but nevertheless significant, differences. first (even though the same set of ct slices, the same threshold, and the same area for the faces were used to produce the boundaries) the number of bels in the approximation to the spine and the ribs based on the cubic grid is approximately a third more than the number of bels in the approximation based on the fcc grid. this is because the larger number of orientations (twelve, as opposed to six) of the rhombic dodecahedral faces allows us to fit the underlying biological surface more tightly. this also has a consequence on computational costs: the time required to track the boundary displayed in figure 13 is less than the time required to track (by essentially the same algorithm) the boundary displayed in figure 11. the time required to produce the computer graphic displays of the already tracked boundary was approximately one and one-half times longer when using the cubic grid than when using the fcc grid. boundaries in digital spaces 117 figure 13. three-dimensional display of the spine and ribs of a patient (seen from behind and downward from the cervical vertebrae) approximated as a collection of 1,258,482 faces of rhombic dodecahedral voxels. the area of the individual faces is the same as in figure 11. 2.4. artzy’s algorithm in n -dimensional space. we now generalize artzy’s algorithm to the n -dimensional euclidean space, with n > 1. the intent is to fulfill the generalization of problem 1.7 which is stated in the last paragraph of subsection 1.8 with g = zn , κ = δn and λ = ωn . to do this we need to generalize the concept of “adjacency of one bel to another,” which is achieved by introducing an appropriate basic digraph using a generalization of (2.1). consider the basic digraph (m, ρ) with the following properties: (i) m = {un | 1 ≤ n ≤ n} ∪ {−un | 1 ≤ n ≤ n}; (ii) ρ = ⋃ 1≤i 1. another spel-adjacency in ( z3, ω3 ) is σ that is defined as follows. in addition to all elements of ω3, σ contains a pair of spels (c, d) if one of the following three conditions is satisfied: (i) c1 = d1 and (c2 − d2) × (c3 − d3) = −1, (ii) c2 = d2 and (c3 − d3) × (c1 − d1) = −1, (iii) c3 = d3 and (c1 − d1) × (c2 − d2) = −1. the following is a generalization of claim 2.7 to arbitrary digital spaces. claim 3.1 (desirability claim). let κ and λ be spel-adjacencies in a digital space (v, π). for any picture over (v, π), let o be a κ-component of the set of 1-spels and q be a λ-component of the set of 0-spels, such that ϑ (o, q) is not empty (and, hence, it is a surface). then there exist two uniquely defined subsets i and e of v with the following properties. (i) o ⊆ i and q ⊆ e. (ii) ϑ (o, q) = ϑ (i, e). (iii) i ∪ e = v and i ∩ e = ∅. (iv) i is κ-connected and e is λ-connected. (v) every π-path connecting an element of i to an element of e crosses ϑ (o, q). boundaries in digital spaces 123 we call an ordered pair (κ, λ) of spel-adjacencies in a digital space (v, π) desirable for (v, π), if the desirability claim as stated above is valid for that particular choice of κ and λ. we see that, in order to prove the desirability claim of subsection 2.6 for the three special cases listed there, what we need to do is to show that: case (i). (γn , ωn ) is desirable for ( zn , ωn ) ; case (ii). (δn , ωn ) is desirable for ( zn , ωn ) ; case (iii). (δ3, δ3) is desirable for (f, δ3). this section is aimed at achieving this, but in a context which provides a general approach to proving the desirability of pairs of spel-adjacencies. 3.2. interiors and exteriors. let (v, π) be a digital space and let s be a surface in it. we define the immediate interior ii (s), the immediate exterior ie (s), and the immediate neighborhood in (s) of s as follows: (3.1) ii (s) = {c | (c, d) ∈ s for some d in v } , ie (s) = {d | (c, d) ∈ s for some c in v } , in (s) = ii (s) ∪ ie (s) . the interior i (s) and the exterior e (s) of s are defined as follows: (3.2) i (s) = {c ∈ v | there exists a π-path connecting c to an element of ii (s) that does not cross s} , e (s) = {c ∈ v | there exists a π-path connecting c to an element of ie (s) that does not cross s} . lemma 3.2. for any surface s in a digital space (v, π), (3.3) i (s) ∪ e (s) = v. proof. let c be an arbitrary element of v and d be an arbitrary element of in (s). (by definition, neither of the sets v and in (s) is empty.) since v is π-connected, there exists a π-path 〈 c(0), . . . , c(k) 〉 from c to d. let k be the smallest nonnegative integer such that c(k) is in in (s). then 〈 c(0), . . . , c(k) 〉 is a π-path connecting c to an element of ii (s) ∪ ie (s) which does not cross s. hence c is in at least one of i (s) and e (s). � a surface s in a digital space (v, π) is said to be near-jordan if every π-path from any element of ii (s) to any element of ie (s) crosses s. we remark that if s is not a near-jordan surface, then the intersection of its interior and its exterior is necessarily nonempty. (this is because there is a path, not crossing s, from its immediate interior to its immediate exterior; all the spels in this path are in both the interior and the exterior of s, as can be seen from (3.2), the symmetry of proto-adjacency and the symmetrical definition of “crosses.”) 124 gabor t. herman lemma 3.3. let s be a surface in a digital space (v, π). then the following three conditions are equivalent. (i) s is near-jordan. (ii) every π-path from any element of i (s) to any element of e (s) crosses s. (iii) i (s) ∩ e (s) = ∅. furthermore, if these conditions are satisfied, then it is also the case that (3.4) s = ϑ (i (s) , e (s)) . proof. let s be any surface in a digital space (v, π). suppose there is a π-path 〈 c(0), . . . , c(k) 〉 not crossing s connecting a c in i (s) to a d in e (s). by (3.2), there is also a π-path 〈 e(0), . . . , e(l) 〉 not crossing s connecting c to an element of ii (s) and a π-path 〈 d(0), . . . , d(m) 〉 not crossing s connecting d to an element of ie (s). then 〈 e(l), . . . , e(0) = c = c(0), . . . , c(k) = d = d(0), . . . , d(m) 〉 is a π-path not crossing s which connects an element of ii (s) to an element of ie (s). therefore, by definition, s is not near-jordan. this argument shows that (i) implies (ii). if i (s) and e (s) have an element c in common, then the π-path 〈c〉 is from an element of i (s) to an element of e (s) and does not cross s. this shows that (ii) implies (iii). that (iii) implies (i) was proven in the paragraph preceding the statement of the lemma. finally, it is the case for any surface s that it is a subset of ϑ (i (s) , e (s)). (this follows trivially from the definitions of immediate interior and exterior and of interior and exterior.) suppose now that (i) (iii) hold and that the surfel (c, d) is in ϑ (i (s) , e (s)). then d belongs to e (s), and so by (iii) d does not belong to i (s); hence (d, c) is not in s. the π-path 〈c, d〉 is from an element of i (s) to an element of e (s) and so, as condition (ii) is satisfied, it crosses s. hence (c, d) is in s. this proves that any of the conditions (i) (iii) implies (3.4). � the following theorem provides two further characterizations of a nearjordan surface. the first of these has a significant property that characterizations (ii) and (iii) in lemma 3.3 do not have: it gives an easy way to construct near-jordan surfaces. it should also help the reader to grasp the essential nature of such surfaces. theorem 3.4. let s be a surface in a digital space (v, π). then the following three conditions are equivalent. (i) s is near-jordan. (ii) there exists a nonempty proper subset o of v such that (3.5) s = ϑ ( o, o ) . (iii) there exists a nonempty proper subset o of v such that i (s) = o and e (s) = o . furthermore, if these conditions are satisfied, then the o of (ii) has to be i (s). boundaries in digital spaces 125 proof. let s be any surface in a digital space (v, π). since, by definition, s is nonempty, both i (s) and e (s) are nonempty. if s is also near-jordan, then it follows from lemma 3.2 and lemma 3.3(iii) that i (s) is a nonempty proper subset of v and e (s) is the complement of i (s) in v . by setting o = i (s), (i) implies (ii) from (3.4). now assume that there is an o that satisfies (ii). to show that for this o we have that i (s) = o and e (s) = o, it suffices (by lemma 3.2) to show that (3.5) implies (3.6) e (s) ∩ o = ∅ and (3.7) i (s) ∩ o = ∅. first we note that (3.5) implies that ii (s) is a subset of o and that ie (s) is a subset of o and also that any π-path from an element of o to an element (necessarily not the same) of o must cross s. now suppose that c is in o. then any π-path from c to an element of ie (s) must cross s, showing that c is not in e (s), which means that (3.6) is true. now suppose that d is in o. then any π-path from d to an element of ii (s) must cross s, showing that d is not in i (s), which means that (3.7) is true. thus, (ii) implies (iii) and the o of (ii) has to be i (s). finally, (iii) of this theorem implies (iii) of lemma 3.3, which is shown there to be equivalent to (i). � 3.3. connectedness in digital spaces. the following is a useful tool for proving connectedness of the interiors and/or the exteriors of surfaces. theorem 3.5. let s be a surface, and let κ and λ be spel-adjacencies in a digital space (v, π). (i) if for some κ-connected subset o of v (3.8) ii (s) ⊆ o ⊆ i (s) , then i (s) is κ-connected. (ii) if for some λ-connected subset q of v (3.9) ie (s) ⊆ q ⊆ e (s) , then e (s) is λ-connected. proof. we prove only (i) since the proof of (ii) is entirely analogous. we assume the truth of the premise of (i). let c1 and c2 be two spels in i (s). we need to prove that they are κ-connected in i (s). by (3.2), for 1 ≤ i ≤ 2, there exists a di in ii (s) such that there is a π-path (hence, by the definition of a speladjacency, a κ-path) not crossing s (and hence entirely in i (s)) connecting ci to di. by (3.8), d1 and d2 are in o and are therefore κ-connected in o and hence (again by (3.8)) in i (s). by combining these three κ-paths in i (s) into a single κ-path (recall that κ is symmetric by the definition of a spel-adjacency), we get the desired result. � 126 gabor t. herman 3.4. boundaries in pictures. let s be a surface and κ and λ be spel-adjacencies in a digital space (v, π). we say that s is a κλ-boundary in the picture (v, π, f ), if there is κ-component o of the set of 1-spels and a λ-component q of the set of 0-spels such that s = ϑ (o, q). theorem 3.6. let (v, π, f ) be a picture and κ and λ be spel-adjacencies in (v, π). if o is a κ-component of the set of 1-spels, q is a λ-component of the set of 0-spels, and s is the κλ-boundary ϑ (o, q), then o ⊆ i (s) and q ⊆ e (s). proof. we prove only that o ⊆ i (s), since the proof that q ⊆ e (s) is strictly analogous. let d ∈ o, c ∈ ii (s) (⊆ o), and 〈 c(0), . . . , c(k) 〉 be a κ-path in o from c to d. we prove by induction that, for 0 ≤ k ≤ k, c(k) ∈ i (s). this suffices to complete our proof. clearly, c(0) ∈ i (s). now assume that the same is true for c(k−1), for some k, 1 ≤ k ≤ k. since ( c(k−1), c(k) ) ∈ κ, there exists a π-path 〈 e(0), . . . , e(t ) 〉 from c(k−1) to c(k), such that, for 1 ≤ t ≤ t , ( e(0), e(t) ) ∈ κ. now we prove by induction that, for 0 ≤ t ≤ t , (3.10) e(t) ∈ i (s) ∪ q. since e(t ) = c(k) is in o and so cannot be in q, this shows that c(k) ∈ i (s) and so completes our proof. if t = 0, then (3.10) is satisfied since e(0) = c(k−1) is in i (s) by the hypothesis for the induction on k. now assume that (3.10) is satisfied for some t < t . in case e(t) ∈ i (s), either e(t+1) ∈ i (s) (in which case we are done) or ( e(t), e(t+1) ) ∈ s (see (3.2)), and so e(t+1) ∈ q (and again we are done). in case e(t) ∈ q, either e(t+1) is a 0-spel and therefore is in q since π ⊆ λ (in which case we are done) or e(t+1) ∈ o, since c(k−1) is in o. in this latter case, ( e(t+1), e(t) ) ∈ s, and so e(t+1) ∈ i (s) (and again we are done). � theorem 3.7. let κ and λ be spel-adjacencies in a digital space (v, π). if s is a κλ-boundary in a picture (v, π, f ), then i (s) is κ-connected and e (s) is λ-connected. proof. let s = ϑ (o, q), for some κ-component o of the set of 1-spels and some λ-component q of the set of 0-spels. then, by theorem 3.6, o ⊆ i (s) and q ⊆ e (s) and so, by theorem 3.5, i (s) is κ-connected and e (s) is λ-connected. � what we have proved so far allows us to establish the following powerful result for proving the desirability of a pair of spel-adjacencies. theorem 3.8. if κ and λ are spel-adjacencies in a digital space (v, π) such that every κλ-boundary in every picture is near-jordan, then (κ, λ) is desirable for (v, π). boundaries in digital spaces 127 proof. let (v, π, f ) be a picture over (v, π) and let o be a κ-component of the set of 1-spels and q be a λ-component of the set of 0-spels, such that s = ϑ (o, q) is not empty (and hence it is a κλ-boundary). by the assumption of the theorem, s is near-jordan. let i = i (s) and e = e (s). in claim 3.1, (i) is satisfied, by theorem 3.6. near-jordanness of s implies the validity of (3.4), which is the same as (ii) in claim 3.1. on the other hand, (iii) in claim 3.1 follows from lemma 3.2 and (iii) of lemma 3.3. theorem 3.7 gives us (iv) of claim 3.1 and (v) follows from (ii) of lemma 3.3. that i (s) is the only choice for i (and hence that e (s) is the only choice for e), satisfying (ii) of claim 3.1 follows from theorem 3.4. � 3.5. isomorphic digital spaces. an isomorphism i from a digital space (v, π) to a digital space (v ′, π′) is a one-to-one function that maps v onto v ′ so that (3.11) (c, d) ∈ π ⇔ (i (c) , i (d)) ∈ π′. theorem 3.9. the digital spaces ( z3, σ ) and (f, δ3) are isomorphic. proof. define, for c ∈ z3, i (c) = (c2 + c3, c3 + c1, c1 + c2). since the sum of its components is even, i (c) ∈ f . to show that i is one-to-one, assume that i (c′) = i (c′′), and let c = c′ − c′′. then i (c) = i (c′) − i (c′′) = 0, which implies that c2 = −c3, c1 = −c3, and −2c3 = c1 + c2 = 0. from this it follows that all three components of c are 0, and hence c′ = c′′. finally, to show that i maps z3 onto f , let d ∈ f , and define c1 = (−d1 + d2 + d3) /2, c2 = (d1 − d2 + d3) /2, and c3 = (d1 + d2 − d3) /2. since the sum of the components of d is even, c1, c2, and c3 are all integers, and, clearly, i (c) = d. this shows that i is a one-to-one function mapping z3 onto f . to show that i is an isomorphism from ( z3, σ ) to (f, δ3), first we observe that (i (c) , i (d)) ∈ δ3 if, and only if, i (c) and i (d) differ by 1 in two coordinates and are the same in the third. a consequence of this observation is that if we let a = i (c) − i (d), then (i (c) , i (d)) ∈ δ3 if, and only if, two of |a1|, |a2|, and |a3| have the value 1 and the third has the value 0. by the definition of i, for a = i (c) − i (d), (3.12) a1 = (c2 − d2) + (c3 − d3) , a2 = (c3 − d3) + (c1 − d1) , a3 = (c1 − d1) + (c2 − d2) . now suppose that (c, d) ∈ σ. there are two different reasons why this might be the case. the first is that (c, d) ∈ ω3. in this case, exactly one of |c1 − d1|, |c2 − d2|, and |c3 − d3| has the value 1 and the other two have the value 0. the other possibility is that there exist two integers k and j (1 ≤ k 6= j ≤ 3) such that (ck − dk) × (cj − dj ) = −1 and cn = dn if n is neither k nor j. looking at the components of a, as expressed in the equation above, we see that either case implies that (i (c) , i (d)) ∈ δ3. conversely, suppose that (i (c) , i (d)) ∈ δ3 and so two of |a1|, |a2|, and |a3| have the value 1 and the third has the value 0. one way that this may 128 gabor t. herman come about is by both the terms in the zero-valued sum being themselves zerovalued. in that case, the other (common) term in the other two sums must have absolute value 1. this implies that (c, d) ∈ ω3. the alternative possibility is that the terms in the zero-valued sum are not zero-valued but are the negatives of each other. then we have the situation that there exist two integers k and j (1 ≤ k 6= j ≤ 3) such that (ck − dk) = − (cj − dj ) 6= 0 and, if n is neither k nor j, |(cn − dn) + (ck − dk)| = 1 and |(cn − dn) − (ck − dk)| = 1. it is easy to see that this can happen only if (ck − dk) × (cj − dj ) = −1 and cn = dn. in either case, (c, d) ∈ σ. � theorem 3.10. let i be an isomorphism from a digital space (v, π) to a digital space (v ′, π′), and let s be a surface in (v, π). if we define (3.13) s′ = {(i (c) , i (d)) | (c, d) ∈ s} , then s is a near-jordan surface in (v, π) if, and only if, s′ is a near-jordan surface in (v ′, π′). proof. first we point out that s′ is indeed a surface in (v ′, π′), since the fact that s is a nonempty subset of π implies, in view of (3.11), that s′ is a nonempty subset of π′. the rest of the theorem is proved by the following sequence of equivalences, each one of which is an immediate consequence of the definitions. s is not a near-jordan surface in (v, π) if, and only if, there exists a π-path 〈 c(0), . . . , c(k) 〉 from an element of ii (s) to an element of ie (s) which does not cross s. this happens if, and only if, there exists a π′-path 〈 i ( c(0) ) , . . . , i ( c(k) )〉 from an element of ii (s′) to an element of ie (s′) that does not cross s′, which means that s′ is not a near-jordan surface in (v ′, π′). � if there is an isomorphism i from the digital space (v, π) to a digital space (v ′, π′) and ρ is an adjacency in (v, π), then we define its isomorphic image (due to i) ρ′ in (v ′, π′) by (3.14) (i (c) , i (d)) ∈ ρ′ ⇔ (c, d) ∈ ρ. it is trivial to check that the ρ′ defined in this fashion is an adjacency in (v ′, π′). since here we will be discussing multiple definitions simultaneously, for any spel-adjacency ρ in a digital space (v, π) and for any subsets o and q of v , we introduce the notation (3.15) ϑρ (o, q) = {(c, d) | c ∈ o, d ∈ q, (c, d) ∈ ρ} . note that ϑπ (o, q) is just the boundary in (v, π) between o and q. theorem 3.11. if i is an isomorphism from the digital space (v, π) to the digital space (v ′, π′), (κ, λ) is a desirable pair of adjacencies for (v, π) and κ′ and λ′ are the respective isomorphic images of κ and λ, then (κ′, λ′) is desirable for (v ′, π′). boundaries in digital spaces 129 proof. by theorem 3.8 we need to prove that any κ′λ′-boundary s′ in any picture (v ′, π′, f ′) over (v ′, π′) is near-jordan. let s′ = ϑπ′ (o ′, q′), where o′ is a κ′-component of the set of 1-spels and q′ is a λ′-component of the set of 0-spels of (v ′, π′, f ′). define the function f on v by f (c) = f ′ (i (c)). then (v, π, f ) is a picture over (v, π), and o = {c | i (c) ∈ o′} and q = {d | i (d) ∈ q′} are clearly a κ-component of the set of 1-spels and a λ-component of the set of 0-spels, respectively, of (v, π, f ). defining s = ϑπ (o, q) we get (3.16) (c, d) ∈ s ⇔ c ∈ o and d ∈ q and (c, d) ∈ π ⇔ i (c) ∈ o′ and i (d) ∈ q′ and (i (c) , i (d)) ∈ π′ ⇔ (i (c) , i (d)) ∈ s′. this implies that s is a surface (since it is nonempty), it is near-jordan (by lemma 3.3) and so s′ is also near-jordan (by theorem 3.10). � 3.6. simple connectedness in digital spaces. let s be a surface in a digital space (v, π). for any practical application, it would be impossible to determine whether s is near-jordan by examining all π-paths from ii (s) to ie (s). we need instead a result which says that s is near-jordan if some local condition is satisfied at every surfel of s. this is the subject matter of this and the next subsection. in classical topology, a simply connected space is (intuitively speaking) a connected space in which every loop can be continuously pulled to a point without leaving the space. here we present one of the corresponding notions for digital spaces (for others, see section 6.1 of [13] and section 3 of [5]). if (3.17) p = 〈 c(1), . . . , c(m), d(0), . . . , d(n), e(1), . . . , e(l) 〉 and (3.18) p ′ = 〈 c(1), . . . , c(m), f (0), . . . , f (k), e(1), . . . , e(l) 〉 are π-paths in v , such that (3.19) f (0) = d(0), f (k) = d(n), and 1 ≤ k + n ≤ 4, then p and p ′ are said to be elementarily equivalent in the digital space (v, π). (note that in this definition, m or l or both in (3.17) may be zero; in other words, the difference between elementarily equivalent π-paths may be at their “head” or at their “tail.”) two π-paths, p and p ′ in v are said to be equivalent in the digital space (v, π), if there is a sequence of π-paths p0, . . . , pl (l ≥ 0) in v , such that p0 = p, pl = p ′ and, for 1 ≤ l ≤ l, pl−1 and pl are elementarily equivalent in the digital space (v, π). we demonstrate the notion of equivalent π-paths in ( z2, ω2 ) in figure 15. that 〈a, b, c, d, e, f, g, h, i, j, k, l, a〉 is elementarily equivalent to 〈a, b, c, d, e, d, g, h, i, j, k, l, a〉 follows by substituting in (3.17) n = 2, d(0) = e, d(1) = f , d(2) = g and in (3.18) k = 2, f (0) = e, f (1) = d, f (2) = g. that 130 gabor t. herman figure 15. demonstration of equivalence of π-paths in ( z2, ω2 ) : 〈a, b, c, d, e, f, g, h, i, j, k, l, a〉 is equivalent to 〈a, b, c, d, g, h, i, j, k, l, a〉, since it is elementarily equivalent to 〈a, b, c, d, e, d, g, h, i, j, k, l, a〉, which is elementarily equivalent to 〈a, b, c, d, g, h, i, j, k, l, a〉. 〈a, b, c, d, e, d, g, h, i, j, k, l, a〉 is elementarily equivalent to 〈a, b, c, d, g, h, i, j, k, l, a〉 follows by substituting in (3.17) n = 2, d(0) = d, d(1) = e, d(2) = d and in (3.18) k = 0, f (0) = d. a loop (of length k) in a digital space (v, π) is a π-path 〈 c(0), . . . , c(k) 〉 such that c(k) = c(0). in particular, for any spel c, 〈c〉 is a loop, and is called a trivial loop. we note that any loop of length 1, 2, or 3 is automatically equivalent to a trivial loop. a digital space is said to be simply connected if every loop in the digital space is equivalent to a trivial loop. theorem 3.12. for any positive integer n , ( zn , ωn ) is simply connected. proof. we show that every loop in ( zn , ωn ) is equivalent to a trivial loop. we do this by induction on the length of the loop. we have already noted that, in general, every loop of length 1 or 2 is equivalent to a trivial loop. suppose that every loop in ( zn , ωn ) of length less than some k > 2 is equivalent to a trivial loop. consider a loop 〈 c(0), . . . , c(k) 〉 in ( zn , ωn ) of length k. we now show that it is equivalent to a loop of length k − 1 or k − 2 and thus, by the induction hypothesis, is equivalent to a trivial loop. since c(1) is ωn -adjacent to c (0), we have c(1) 6= c(0). (to illustrate our argument, consider figure 15. in that figure c(0) = a = (0, 0) and c(1) = b = (0, 1).) then there is a unique j such that c (1) j 6= c (0) j . without loss of generality, assume that c (1) j > c (0) j . (in figure 15, j = 2.) let z = max1≤k≤k { c (k) j } . (in figure 15, z = 3.) let l be the largest integer in the range 0 < l < k such that c (l) j = z. (in figure 15, in the left column l = 5 with c (5) = f = (2, 3) and in the middle column l = 4 with c(4) = e = (1, 3).) clearly, c (l+1) j = z − 1. (in figure 15, in the left column c(l+1) = c(6) = g = (2, 2) and in the middle column c(l+1) = c(5) = d = (1, 2).) let k be the smallest integer such that, for all i in the range 0 < k ≤ i ≤ l < k, we have c (i) j = z. (in figure 15, k = 4 in boundaries in digital spaces 131 both the left and the middle column.) clearly, c (k−1) j = z − 1. (in figure 15, c(k−1) = c(3) = d = (1, 2) in both the left and middle columns.) we will now use induction on l − k. if l−k = 0, then k = l and therefore c(k−1) = c(k+1). (this case is illustrated in figure 15 by the middle column, for which k = l = 4 and c(k−1) = c(k+1) = d.) in this case the loop (3.20) 〈 c(0), . . . , c(k−2), c(k−1), c(k), c(k+1), c(k+2), . . . , c(k) 〉 is elementarily equivalent to the loop (3.21) 〈 c(0), . . . , c(k−2), c(k−1) = c(k+1), c(k+2), . . . , c(k) 〉 which has length k−2 and we are done. (the loop in (3.20) is illustrated by the loop of the middle column of figure 15, while the loop in (3.21) is illustrated by the loop of the right column of figure 15.) suppose now (induction hypothesis) that whenever l−k = h, then 〈 c(0), . . . , c(k) 〉 is equivalent to a loop of length k − 1 or k − 2. we now show that the same conclusion holds if l − k = h + 1. let (3.22) 〈 c(0), c(1), . . . , c(k), . . . , c(l−1), c(l), c(l+1), . . . , c(k) 〉 be a loop. define j, z, l, and k for this loop as above and suppose l − k = h + 1. (this is the case in figure 15 for the loop associated with the left column with l = 5, k = 4 and, hence, h = 0. note also that for this loop j = 2 and z = 3.) let c′(l) be a spel such that, for 1 ≤ n ≤ n , (3.23) c′(l)n = { z − 1, if n = j, c (l−1) n , otherwise. (in the left column of figure 15, c′(l) = c′(5) = (1, 2) = d.) clearly, c′(l) is proto-adjacent to c(l−1). also, c(l+1) differs from c(l) in exactly the jth component (which is z − 1 for the former) and c(l) differs from c(l−1) in exactly one component which is other than the jth; thus c′(l) is proto-adjacent to c(l+1). (in the left column of figure 15, c(l−1) = c(4) = e and c(l+1) = c(6) = g, both of which are proto-adjacent to c′(l) = c′(5) = d.) it follows that (3.24) 〈 c(0), c(1), . . . , c(k), . . . , c(l−1), c′(l), c(l+1), . . . , c(k) 〉 is also a loop and is easily seen to be elementarily equivalent to the loop in (3.22). (the loop in (3.22) is illustrated by the loop of the left column of figure 15, while the loop in (3.24) is illustrated by the loop of the middle column of figure 15.) furthermore, it follows from (3.23) that for the loop in (3.24) the condition of the induction hypothesis holds. (indeed, we have already seen that for the middle column of figure 15 we have l − k = 0 = h.) so, by the induction hypothesis, the loop in (3.24) is equivalent to a loop of length k − 1 or k − 2. from this it follows that the loop in (3.22) is also equivalent to a loop of length k − 1 or k − 2. � 132 gabor t. herman 3.7. locally-jordan surfaces. let s be a surface and p = 〈 c(0), . . . , c(k) 〉 be a π-path in a digital space (v, π). we say that the crossing parity ps p of p through s is even (or zero; i.e., ps p = 0) if the number of π-paths among 〈 c(0), c(1) 〉 , . . . , 〈 c(k−1), c(k) 〉 that cross s is even and we say that it is odd (or one; i.e., ps p = 1) if this number is odd. we use the notation ⊕ for modulo 2 addition of parities (i.e.; 0 ⊕ 0 = 1 ⊕ 1 = 0 and 0 ⊕ 1 = 1 ⊕ 0 = 1). it is easy to see that, for any surface s in a digital space, the crossing parity through s is even for any loop in the digital space whose length is not greater than two. also, cyclic permutation of a loop does not influence its crossing parity through a surface s, since, for 1 ≤ k ≤ k, (3.25) ps 〈 c(0), . . . , c(k−1), c(k), . . . , c(k) 〉 = ps 〈 c(k), . . . , c(k) = c(0), . . . , c(k−1), c(k) 〉 . in addition, reversing a π-path does not change its crossing parity. it is also easy to see that if 〈 c(0), . . . , c(k) 〉 and 〈 d(0), . . . , d(l) 〉 are π-paths such that c(k) = d(0), then (3.26) ps 〈 c(0), . . . , c(k), d(1), . . . , d(l) 〉 = ps 〈 c(0), . . . , c(k) 〉 ⊕ ps 〈 d(0), . . . , d(l) 〉 . theorem 3.13. if s is a near-jordan surface in a digital space, then the crossing parity through s is odd for any π-path p = 〈 c(0), . . . , c(k) 〉 such that ( c(0), c(k) ) ∈ s. proof. first note that (according to lemma 3.3), since s is near-jordan, i (s)∩ e (s) = ∅ and s = ϑ (i (s) , e (s)). we prove by induction that, for any 0 ≤ k ≤ k, (3.27) ps 〈 c(0), . . . , c(k) 〉 = { 0, if c(k) ∈ i (s) , 1, otherwise. since c(k) ∈ e (s), this is sufficient to prove the theorem. clearly, (3.27) is true for k = 0. suppose that (3.27) is true for some k − 1, where 1 ≤ k ≤ k. we prove that it is also true for k. we use the following special case of (3.26): (3.28) ps 〈 c(0), . . . , c(k−1), c(k) 〉 = ps 〈 c(0), . . . , c(k−1) 〉 ⊕ ps 〈 c(k−1), c(k) 〉 . in case c(k−1) ∈ i (s), the first term on the right-hand side of (3.28) is 0, and the second term is 0 if c(k) ∈ i (s) and 1 otherwise. in case c(k−1) /∈ i (s), the first term on the right-hand side of (3.28) is 1, and the second term is 1 if c(k) ∈ i (s) and 0 otherwise. in either case, (3.27) is true for k. � a surface s in a digital space (v, π) is said to be locally-jordan if ps 〈 c(0), . . . , c(k) 〉 is odd for any π-path such that ( c(0), c(k) ) ∈ s and 2 ≤ k ≤ 3. (this restriction on v enforces “locality.” it says that any path from the immediate interior of s to its immediate exterior must cross s exactly once if k = 2 and boundaries in digital spaces 133 must cross s either once or three times if k = 3.) by theorem 3.13, if a surface s in a digital space (v, π) is near-jordan, then it is locally-jordan. lemma 3.14. any loop of length not more than 4 has even crossing parity through any locally-jordan surface in any digital space (v, π). proof. we have already pointed out that the crossing parity through any surface is even for any loop of length not more than two. let s be a locally-jordan surface. consider a loop l = 〈 c(0), . . . , c(k) 〉 with 3 ≤ k ≤ 4. if there does not exist a k, 1 ≤ k ≤ k, such that 〈 c(k−1), c(k) 〉 crosses s, then we are done. otherwise, for such a k, either ( c(k−1), c(k) ) ∈ s or ( c(k), c(k−1) ) ∈ s is true. if ( c(k), c(k−1) ) ∈ s, p = 〈 c(k), . . . , c(k) = c(0), . . . , c(k−1) 〉 is a π-path of length k −1 with 2 ≤ k −1 ≤ 3 and so the fact that s is locally-jordan implies ps p = 1. by application of (3.25) and (3.26), we have (3.29) ps l = ps p ⊕ ps 〈 c(k−1), c(k) 〉 = 1 ⊕ 1 = 0. if ( c(k−1), c(k) ) ∈ s, a similar argument, which also makes use of the fact that reversing a π-path does not change its crossing parity, can be used to derive the same conclusion. � lemma 3.15. let s be a locally-jordan surface in a digital space (v, π). if p and p ′ are equivalent π-paths, then they have the same crossing parity through s. proof. by the definition of equivalent, it is sufficient to prove that if p and p ′ satisfy (3.17), (3.18) and (3.19), then they have the same crossing parity through s. by applying (3.26), we get (3.30) ps p = ps 〈 c(1), . . . , c(m), d(0) 〉 ⊕ ps 〈 d(0), . . . , d(n) 〉 ⊕ ps 〈 d(n), e(1), . . . , e(l) 〉 and (3.31) ps p ′ = ps 〈 c(1), . . . , c(m), f (0) 〉 ⊕ ps 〈 f (0), . . . , f (k) 〉 ⊕ ps 〈 f (k), e(1), . . . , e(l) 〉 . therefore, using (3.19), the invariance of crossing parity under reversal, and (3.26), we get (3.32) ps p ⊕ ps p ′ = ps 〈 d(0), . . . , d(n) 〉 ⊕ ps 〈 f (0), . . . , f (k) 〉 = ps 〈 d(0), . . . , d(n) = f (k), . . . , f (0) = d(0) 〉 = 0. the last equality follows from the previous lemma combined with (3.19). � the results up to now apply to digital spaces which do not have to be simply connected. the next lemma makes essential use of simple connectedness. 134 gabor t. herman lemma 3.16. if s is a locally-jordan surface in a simply connected digital space (v, π), then s is near-jordan if either (and hence both) of the following two equivalent conditions holds. (i) for any c ∈ ii (s) and d ∈ ii (s), there exists a π-path p from c to d such that ps p = 0. (ii) for any c ∈ ie (s) and d ∈ ie (s), there exists a π-path p from c to d such that ps p = 0. proof. evidently the two conditions are equivalent. indeed, if c ∈ ii (s) and d ∈ ii (s), then there exist c′ ∈ ie (s) and d′ ∈ ie (s) such that (c, c′) ∈ s and (d, d′) ∈ s; hence if there exists a π-path of even crossing parity from c′ to d′, then there also exists one from c to d, so that (ii) implies (i). similarly, (i) implies (ii). in what follows, we prove that s is near-jordan if (ii) holds. we do this by supposing that s is not near-jordan and showing that this, together with (ii), leads to a contradiction. first, we show that there exists a π-path p1 = 〈 c(1), . . . , c(k) 〉 such that c(1) ∈ ii (s), c(k) ∈ ie (s) and ps p1 = 0. indeed, since s is supposed to be not near-jordan, there is a π-path from ii (s) to ie (s) that does not cross s. clearly, any such π-path has the required properties. next, we show that there exists a π-path p3 = 〈 e(1), . . . , e(l) 〉 such that ( e(1), e(l) ) ∈ s and ps p3 = 0. let p1 be the π-path of the last paragraph. let c(0) be such that ( c(1), c(0) ) ∈ s. by (ii), there exists a π-path p2 = 〈 c(k) = d(0), . . . , d(l) = c(0) 〉 from c(k) to c(0) such that ps p2 = 0. then p3 = 〈 c(1), . . . , c(k), d(1), . . . , d(l) 〉 is a π-path from c(1) to d(l) such that ( c(1), d(l) ) ∈ s and, by (3.26), ps p3 = ps p1 ⊕ ps p2 = 0. for a π-path p3 satisfying the properties listed at the beginning of the previous paragraph, let e(0) = e(l). by (3.26), p4 = 〈 e(0), e(1), . . . , e(l) 〉 is a loop such that ps p4 = ps 〈 e(0), e(1) 〉 ⊕ ps p3 = 1. since (v, π) is simply connected, p4 is equivalent to a trivial loop, whose crossing parity through s is zero. since s is locally-jordan, according to lemma 3.15, we must also have ps p4 = 0, contradicting the fact that ps p4 = 1. � lemma 3.17. let (v, π, f ) be a picture over the digital space (v, π). (i) let λ be a spel-adjacency in (v, π). let o be a union of π-components of 1-spels and q be a λ-component of 0-spels in (v, π, f ) such that s = ϑ (o, q) is nonempty. for any c and d in q, there exists a π-path p from c to d such that ps p = 0. (ii) let κ be a spel-adjacency in (v, π). let o be a κ-component of 1-spels and q be a union of π-components of 0-spels in (v, π, f ) such that s = ϑ (o, q) is nonempty. for any c and d in o, there exists a π-path p from c to d such that ps p = 0. boundaries in digital spaces 135 proof. we prove only (i), since the proof of (ii) is obviously similar. first, we show that the result is true if (c, d) ∈ λ. in this case, there exists a π-path 〈 c(0), . . . , c(k) 〉 from c to d such that, 0 ≤ k ≤ k, ( c, ck ) ∈ λ. now we prove by induction that, for 0 ≤ k ≤ k, (3.33) ps 〈 c(0), . . . , c(k) 〉 = { 0, if c(k) /∈ o, 1, if c(k) ∈ o. since c(k) is a 0-spel, and so cannot be in o, this inductive proof meets the aim of this paragraph. clearly, (3.33) is true for k = 0, since c(0) is a 0-spel. now suppose that it is true for some k − 1 (1 ≤ k ≤ k). in the proof we repeatedly use (3.28). first we consider the case c(k−1) ∈ o. in this case the first term on the right-hand side of (3.28) has value 1, by the induction hypothesis. if c(k) is a 1-spel, then it must also be in o and so the second term on the right-hand side of (3.28) has value 0. if c(k) is a 0-spel, then it must be in q (since ( c, ck ) ∈ λ and q is a λ-component of 0-spels that contains c) and so the second term on the right-hand side of (3.28) has the value 1. either way, (3.33) is true for k. on the other hand, if c(k−1) /∈ o, the first term on the right-hand side of (3.28) has value 0 by the induction hypothesis. if c(k−1) is a 1-spel, then it cannot be in q and so the second term on the right-hand side of (3.28) has value 0. at the same time c(k) cannot be in o, so (3.33) is true for k. if c(k−1) is a 0-spel, then (as before) it must be in q and so the second term on the right-hand side of (3.28) has value 0 if c(k) /∈ o and value 1 if c(k) ∈ o. thus, again (3.33) is true in either case for k. this completes the induction proof. now consider arbitrary c and d in q. there exists a λ-path 〈 c(0), . . . , c(k) 〉 of 0-spels in q from c to d. by the result in the last paragraph, for 1 ≤ k ≤ k, there exists a π-path pk = 〈 c(k−1) = c (0) k , . . . , c (mk) k = c(k) 〉 such that pspk = 0. therefore, p = 〈 c (0) 1 , . . . , c (m1) 1 , c (1) 2 , . . . , c (m2) 2 , . . . , c (1) k , . . . , c (mk ) k 〉 is a π-path from c to d such that (3.34) ps p = k ∑ k=1 ps 〈 c (0) k , . . . , c (mk) k 〉 = k ∑ k=1 ps pk = 0, as can be shown by repeated application of (3.26). (here ∑ refers to modulo 2 additions.) � theorem 3.18. if κ and λ are spel-adjacencies in a simply connected digital space (v, π), then every locally-jordan κλ-boundary in a picture over (v, π) is near-jordan. proof. let o be a κ-component of 1-spels and q be a λ-component of 0-spels in a picture over (v, π) such that s = ϑ (o, q) is a locally-jordan κλ-boundary. noting that q has to be a union of π-components of 0-spels, we see from lemma 3.17(ii) that for any c and d in o, there exists a π-path p from c to d such that psp = 0. noting that ii (s) ⊆ o, we see that this implies that condition (i) in lemma 3.16 holds and so s is near-jordan. � 136 gabor t. herman 3.8. desirable pairs of spel-adjacencies. lemma 3.19. if s is a κλ-boundary in a picture over a digital space (v, π) and p = 〈a, b, c〉 is a π-path such that (a, c) ∈ s, then psp = 1. proof. suppose that s = ϑ (o, q), where o is a κ-component of 1-spels and q is a λ-component of 0-spels. if b is a 1-spel, then it is in o and (b, c) ∈ s. if b is a 0-spel, then it is in q and (a, b) ∈ s. in either case, ps p = 1. � lemma 3.20. a κλ-boundary s in a picture over a digital space (v, π) is locally-jordan if ps p = 1 for every π-path p = 〈 c(0), c(1), c(2), c(3) 〉 of length 3 such that ( c(0), c(3) ) ∈ s, c(0) 6= c(2), c(1) 6= c(3), c(1) is a 0-spel and c(2) is a 1-spel. proof. let o be the κ-component of 1-spels and q be the λ-component of 0spels such that s = ϑ (o, q). we need to show that ps p = 1 for any π-path p = 〈 c(0), . . . , c(k) 〉 such that ( c(0), c(k) ) ∈ s and k = 2 or 3. by lemma 3.19, we need only show the result for k = 3. if c(0) = c(2) or c(1) = c(3), then it is easy to see that ps p = ps 〈 c(0), c(3) 〉 = 1. in the following, we assume that both c(0) 6= c(2) and c(1) 6= c(3). there are four possibilities: (i) both c(1) and c(2) are 1-spels, (ii) both c(1) and c(2) are 0-spels, (iii) c(1) is a 1-spel and c(2) is a 0-spel, and (iv) c(1) is a 0-spel and c(2) is a 1-spel. in cases (i), (ii) and (iii), it is easily seen that ps p = 1. that this is also true in case (iv) is exactly the condition in our lemma. � theorem 3.21. for n ≥ 1, every γn ωn -boundary in every picture over the digital space ( zn , ωn ) is near-jordan. proof. let s be an arbitrary γn ωn -boundary in a picture over ( zn , ωn ) . let o be the γn -component of 1-spels and q be the ωn -component of 0-spels such that s = ϑ (o, q). since γn and ωn are both spel-adjacencies in the simply connected digital space ( zn , ωn ) (see theorem 3.12), all we have to do, according to theorem 3.18, is to show that s is locally-jordan. we use lemma 3.20. let p = 〈 c(0), c(1), c(2), c(3) 〉 be an ωn -path of length 3 such that ( c(0), c(3) ) ∈ s, c(0) 6= c(2), c(1) 6= c(3), c(1) is a 0-spel and c(2) is a 1-spel. now all we need to show is that psp = 1. the properties that 〈 c(0), c(1), c(2), c(3) 〉 is an ωn -path, ( c(0), c(3) ) ∈ s (and hence ( c(3), c(0) ) ∈ π), c(0) 6= c(2) and c(1) 6= c(3) imply that there exist integers i, j (1 ≤ i 6= j ≤ n ) and u, v (|u| = |v| = 1) such that (3.35) c (1) i = c (0) i + u, c (2) j = c (1) j + v, c (3) i = c (2) i − u, c (0) j = c (3) j − v. if i or j is 1, then c(2) is γn -adjacent to c (0) and, consequently is in o. under these circumstances, it is easy to see that, whether or not c(1) is in q, we have ps p = 1. on the other hand, if i and j are not 1, then we also have that either c(2) ∈ o or c(1) ∈ q (and, consequently, ps p = 1), as we prove now by showing that the alternative leads to a contradiction. boundaries in digital spaces 137 assuming the alternative, we have a loop 〈 c(0), c(1), c(2), c(3), c(0) 〉 in ( zn , ωn ), such that c (k) 1 is the same for 0 ≤ k ≤ 3, and all of the following conditions hold: (i) ( c(0), c(3) ) ∈ s, (ii) there exist integers i, j (1 ≤ i 6= j ≤ n ) and u, v (|u| = |v| = 1) such that (3.35) is satisfied, and (iii) c(1) is a 0-spel not in q and c(2) is a 1-spel not in o. since s is assumed to be finite, we may assume without loss of generality that 〈 c(0), c(1), c(2), c(3), c(0) 〉 is such a loop for which the common value z of c (0) 1 = c (1) 1 = c (2) 1 = c (3) 1 is as great as possible. now consider the four spels c′(0), . . . , c′(3) such that c′(k) is proto-adjacent to c(k) and c ′(k) 1 = c (k) 1 + 1 for 0 ≤ k ≤ 3. clearly, 〈 c′(0), c′(1), c′(2), c′(3), c′(0) 〉 is a loop in ( zn , ωn ) such that c ′(k) 1 is the same (namely, z + 1) for 0 ≤ k ≤ 3. condition (ii) also holds for this new loop. looking at condition (iii), we see that c′(1) must be a 0-spel (otherwise c(2) would be in o) and not in q (otherwise c(1) would be in q). similarly, c′(3) must be a 0-spel and hence must be in q. in view of this, c′(2) must be a 1-spel (otherwise c′(1) would be in q after all) and not in o (otherwise c(2) would be in o). similarly, c′(0) must be a 1-spel and hence must be in o. putting all this together, we see that the loop 〈 c′(0), c′(1), c′(2), c′(3), c′(0) 〉 also satisfies conditions (i) and (iii). this contradicts the maximality of z and thus completes the proof. � combining theorem 3.21 with theorem 3.8 yields the following. corollary 3.22. for n ≥ 1, (γn , ωn ) is desirable for ( zn , ωn ) . theorem 3.23. if (κ, λ) is a desirable pair of adjacencies for a digital space (v, π), then so is (κ′, λ) for any adjacency such that κ ⊆ κ′. proof. by theorem 3.8 all we need to prove that any κ′λ-boundary s in any binary picture (v, π, f ) is near-jordan. let c ∈ ii (s) and d ∈ ie (s). we need to show that every π-path from c to d crosses s. let o be the κ-component and o′ be the κ′-component of the set of 1-spels in (v, π, f ) which contain c and let q be the λ-component of the set of 0-spels in (v, π, f ) which contains d. then s = ϑ (o′, q). since κ ⊆ κ′, we have o ⊆ o′, and ϑ (o, q) ⊆ s. furthermore, ϑ (o, q) is not empty, since the fact that c (which is in o) is in ii (s) implies that there must be an e in ie (s) (and hence in q) such that (c, e) ∈ ϑ (o, q). the fact that (κ, λ) is a desirable pair for the digital space (v, π) implies that the κλ-boundary ϑ (o, q) is nearjordan. by theorem 3.6, we also know that c, which is in o, is in the interior of ϑ (o, q) and d, which is in q, is in the exterior of ϑ (o, q). by lemma 3.3, every π-path from c to d crosses ϑ (o, q) and hence crosses s since s is a superset of ϑ (o, q). � the followings are immediate consequences of the last two theorems and theorem 3.8. corollary 3.24. for n ≥ 1, every δn ωn -boundary in every picture over the digital space ( zn , ωn ) is near-jordan. 138 gabor t. herman corollary 3.25. for n ≥ 1, (δn , ωn ) is a desirable pair for ( zn , ωn ) . theorem 3.26. in the digital space ( z3, ω3 ) , every σσ-boundary in every picture is near-jordan. proof. let s be an arbitrary σσ-boundary in a picture over ( z3, ω3 ) . let o be a σ-component of 1-spels and q be a σ-component of 0-spels such that s = ϑ (o, q). according to theorem 3.18, all we have to do is to show that s is locally-jordan. we use lemma 3.20. let 〈 c(0), c(1), c(2), c(3) 〉 be an ω3-path of length 3 such that ( c(0), c(3) ) ∈ s, c(0) 6= c(2), c(1) 6= c(3), c(1) is a 0-spel and c(2) is a 1-spel. now all we need to show is that ps p = 1. just as in the proof of theorem 3.21, we get that there exists a pair of integers i, j (1 ≤ i 6= j ≤ 3) and u, v (|u| = |v| = 1) such that (3.35) is satisfied. it follows that (3.36) ( c (2) i − c (0) i ) × ( c (2) j − c (0) j ) = u × v = − ( c (3) i − c (1) i ) × ( c (3) j − c (1) j ) and c2k = c 0 k and c 3 k = c 1 k, if k is neither i nor j. now observing the definition of σ in subsection 3.1 we see that exactly one of the diagonals ( c(0), c(2) ) and ( c(1), c(3) ) is in σ. if ( c(0), c(2) ) ∈ σ, then c(2) ∈ o and we see that psp = 1, whether or not c (1) ∈ q. if ( c(1), c(3) ) ∈ σ, then c(1) ∈ q, and again ps p = 1. � corollary 3.27. in the digital space ( z3, ω3 ) , (σ, σ) is a desirable pair. theorem 3.28. let ρ be a spel-adjacency in a digital space (v, π) such that (ρ, ρ) is desirable for (v, π). then (ρ, ρ) is desirable for (v, ρ). proof. by theorem 3.8, all we need to show is that every ρρ-boundary s in every binary picture (v, ρ, f ) is near-jordan. let o be the ρ-component of 1-spels and q be the ρ-component of 0-spels such that s = ϑρ (o, q). consider, t = ϑπ (o, q) = ϑρ (o, q) ∩ π. we want to show first that t is a near-jordan surface in (v, π). for this it is sufficient to show that t is not empty, since in that case the desirability of (ρ, ρ) for (v, π) provides us what we seek. we know that s is not empty (by the definition of a boundary). let (c, d) ∈ s, and hence (c, d) ∈ ρ. by the definition of a spel-adjacency, there exists a π-path 〈 c(0), . . . , c(k) 〉 from c to d such that ( c(0), c(k) ) ∈ ρ, for 1 ≤ k ≤ k. let c(k−1) be the last 1-spel on this path; it exists since c is a 1-spel and d is a 0-spel. then c(k−1) must be in o because c(0) ∈ o is ρ-adjacent to it, and c(k) must be in q because 〈 c(k), . . . , c(k) 〉 is a ρ-path of 0s and c(k) is in q. this shows that ( c(k−1), c(k) ) ∈ t . let 〈 c(0), . . . , c(k) 〉 be an arbitrary ρ-path from the immediate interior of s to the immediate exterior of s. all we need to do now is to show that it crosses s. we adopt the convention that when we talk about the interior of s, we mean its interior in (v, ρ), and when we talk about the interior of t , we mean its boundaries in digital spaces 139 interior in (v, π). similar conventions apply to exteriors, immediate interiors and immediate exteriors. by theorem 3.6, o ⊆ i (t ) and q ⊆ e (t ). by definition, ii (s) ⊆ o and ie (s) ⊆ q. therefore, c(0) ∈ i (t ) and c(k) ∈ e (t ). in view of lemma 3.2, there must exist a k, 1 ≤ k ≤ k, such that c(k−1) ∈ i (t ) and c(k) ∈ e (t ). our proof is complete if we can show that in fact c(k−1) ∈ o and c(k) ∈ q, since this implies that ( c(k−1), c(k) ) ∈ s. we now show that c(k−1) ∈ o. by the definition of a spel-adjacency, there exists a π-path 〈 e(0), . . . , e(l) 〉 from c(k−1) to c(k) such that ( c(k−1), e(l) ) ∈ ρ for 0 ≤ l ≤ l. since this a π-path from the interior to the exterior of the surface t , which is near-jordan in (v, π), there must be an l, 1 ≤ l ≤ l, such that ( e(l−1), e(l) ) ∈ t . if l = 1, then c(k−1) = e(l−1) is in o and we are done. otherwise, if c(k−1) were a 0-spel, then it would be in q (since it is ρ-adjacent to the spel e(l) in q) and hence in e (t ). this would contradict lemma 3.3(iii). therefore, c(k−1) must be a 1-spel. since it is ρ-adjacent to the spel e(l−1) in o, it itself must be in o. to show that c(k) ∈ q, essentially we can repeat the same argument using a π-path from c(k) to c(k−1) such that c(k) is ρ-adjacent to every other point on the path. � from this theorem and the corollary that precedes it we get the following. corollary 3.29. for the digital space ( z3, σ ) , (σ, σ) is a desirable pair. theorem 3.30. for the digital space (f, δ3), (δ3, δ3) is a desirable pair. proof. by theorem 3.9 ( z3, σ ) is isomorphic to (f, δ3), under an isomorphism i defined in the proof of that theorem. since i (σ) = δ3, the previous corollary and theorem 3.11 provide the result we seek. � corollary 3.22 restates the desirability claim 3.1 for case (i) in subsection 3.1, case (ii) is provided by corollary 3.25, while case (iii) is provided by theorem 3.30. 4. correctness of tracking algorithms 4.1. general boundary tracking. let o be the set of all 1-spels and q be the set of all 0-spels in a picture (v, π, f ). recall that we refer to elements of b = ϑ (o, q) as bels (short for boundary elements). since, by the definition of a picture, o is finite, so is b. we now generalize the idea of a bel-adjacency (first introduced in subsection 2.1). we say that the binary relation β on b is a bel-adjacency if β is an adjacency (i.e., β∗ is symmetric). we propose an algorithm for the following task: given a binary picture (v, π, f ), a bel-adjacency β, and a bel o, find the β-component s of b that contains o. 140 gabor t. herman algorithm 4.1. general bel-tracking algorithm (1) put o into l and s and put inβ (o) copies of o into t . (2) remove a bel a from l. for all bels b which are β-adjacent from a, try to find a copy of b in t . (a) if successful, remove this copy of b from t . (b) if not, then put b into l and s and put inβ (b) − 1 copies of b into t . (3) check if l is empty. (a) if it is, stop. (b) if it is not, start again at instruction (2). prior to proving the correctness of this algorithm, two remarks are in order. first, in instruction (2)b, inβ (b) − 1 is guaranteed to be nonnegative. this is because we only get to this point in the algorithm for a b which is β-adjacent from a bel a and, so, inβ (b) ≥ 1. second, the potential efficiency of this algorithm comes from the fact that we check for membership in t (rather than in s). while s keeps getting bigger and bigger as the algorithm is executed, due to instruction (2)b, the same is not true for t : elements from t will be repeatedly removed due to instruction (2)a. hence the size of t is likely to be a small fraction of the size of s after the algorithm has been performing for a while on a large data set. the essence of the proof of correctness is given in the next lemma. to state it easily we use a couple of abbreviations. we let nt (b) abbreviate “the number of copies of the bel b in the list t .” the other definition is more complicated. the value of na (b) is 0 if b /∈ s and is “inβ (b) less the number of bels in s − l which are β-adjacent to the bel b” otherwise. lemma 4.2. both just prior and just after the execution of instruction (2) in the general bel-tracking algorithm it is the case that, for every bel b, (i) nt (b) = na (b), (ii) the bel b has so far been put into l and s either due to instruction (1) or due to instruction (2)b at most once, and (iii) if the bel b is in s, then o is β-connected in b to b. proof. consider the situation just after the execution of instruction (1). we have that nt (o) = inβ (o) = na (o) and the bel o has so far been put into l and s exactly once. for any bel b other than o, nt (b) = 0 = na (b) and b has not so far been put into l and s even once. since only o has been put into s, (iii) is clearly satisfied at this time. now we show that if (i), (ii) and (iii) hold just prior the execution of instruction (2), then they also hold just after its execution. this is sufficient, since the situation cannot change as a result of instruction (3). assume therefore that we are just at the beginning of executing instruction (2). this means that at this time l is not empty and so we remove a bel a from it. this a must have been put into l and s earlier on and, since nothing is ever removed from s, we must have that a ∈ s. we leave it to the reader to supply the easy proof boundaries in digital spaces 141 of the fact that for those bels which are not β-adjacent from a, the inductive step is valid. for the bels b which are β-adjacent from a, we study separately two possibilities. case a: we find a copy of b in t . in this case, by instruction (2)a, we remove this b from t . this reduces nt (b) by 1. since b was in t , it also had to be in s (nothing is ever put into t without being put into s at the same time). therefore the applicable part of the definition of na (b) is that it is inβ (b) less the number of bels in s − l which are β-adjacent to the bel b. the only thing that changes in this definition is that a, which is β-adjacent to b, got removed from l. hence na (b) is also decreased by 1, proving (i) of the lemma for the bel b. since nothing is put into l and s, (ii) and (iii) are also valid at the end of executing instruction (2). case b: there is no copy of b in t ; i.e., nt (b) = 0. first we show that under these circumstances, it cannot be the case that b has been previously put into l and s. this is so, since otherwise prior to the beginning of instruction (2), the applicable part of the definition of na (b) would be “inβ (b) less the number of bels in s − l which are β-adjacent to the bel b.” since at that time the bel a is still in l, the value of na (b) has to be positive, contradicting the truth of (i) in the induction hypothesis. upon executing instruction (2)b, b has been put into l and s (for the first time) and nt (b) = inβ (b) − 1. there is at least one bel, namely a, which is in s − l and is β-adjacent to b. there cannot be another one, since whenever a bel is put into s, it is also put into l at the same time, and so if at a later time it is no longer in l, then it must have been removed from it. at that time, b would have been put into l and s and we would not be in case b. this shows that in this case too, nt (b) = na (b) just after the execution of instruction (2). finally, since a ∈ s just prior to the execution of instruction (2), o is β-connected in b to a, by (iii) of the induction hypothesis. the same must be true for the new bel b in s, since a is β-adjacent to it. � theorem 4.3. if (v, π, f ) is a binary picture, β is a bel-adjacency and o is a bel, then the general bel-tracking algorithm terminates in a finite number of steps and, at that time, s is the β-component of b that contains o. proof. from (iii) of the previous lemma it follows that anything that gets put into s is in the β-component of b which contains o. termination in a finite number of steps now easily follows from (ii) of the previous lemma: since each of the finitely many bels in the β-component of b which contains o is put into l at most once (and nothing else ever gets put into l) and in each execution of instruction (2) of the algorithm a bel is removed from l, sooner or later l has to become empty and the algorithm will stop due to instruction (3). at that time, as all through the execution of the algorithm, o is β-connected in b to every element of s. that the converse is also true (and hence s is the β-component of b which contains o) can be shown as follows. for any bel b of the β-component of b which contains o, there is a β-path 〈 b(0), . . . , b(k) 〉 from o to b. it is a trivial matter to show by induction that, for 1 ≤ k ≤ k, b(k−1) will get put into l and s and, since l gets eventually emptied, b(k−1) must 142 gabor t. herman get removed from l, resulting in b(k) being put into l and s (provided that it is not in t , which would imply that it has been put into l and s in some previous step). � this theorem shows that the general bel-tracking algorithm is powerful stuff. its practical usefulness depends on two properties of the bel-adjacency β. the first is, how easy is it to compute the bels β-adjacent from a given bel? clearly, the efficiency of executing instruction (2) depends on this (as well as on how easy it is to determine for a bel whether or not it is in t ). the other property has to do with the usefulness of the resulting boundaries: are they in fact of the form ϑ (o, q) for some appropriately specified o and q? in the following subsection we will discuss these questions for cases specified in subsection 2.6. 4.2. boundary tracking on hyper-cubes. we now apply the general bel-tracking algorithm to the tracking of boundaries in the spaces ( zn , ωn ) with n ≥ 2. a basic digraph for zn is a pair (m, ρ) for which: (i) m = {un | 1 ≤ n ≤ n} ∪ {−un | 1 ≤ n ≤ n}; (ii) ρ = ⋃ 1≤i 0, then ( c(k), c(k−1) ) ∈ ρ∗ (by the assumption) and ( c(k−1), c(0) ) ∈ ρ∗ (by the induction hypothesis). since ρ∗ is clearly transitive, we have that ( c(k), c(0) ) ∈ ρ∗. this proves (i). we now define the binary relation τ on l by (c, d) ∈ τ if, and only if, c ∈ l, d ∈ l, and (c, d) ∈ ρ. since it is assumed in (ii) that every element of l has exactly one element of l ρ-adjacent from it, the number of elements in τ must be exactly the number of elements in l. this shows that there cannot be an element of l which has more than one element of l ρ-adjacent to it, since otherwise we would have more elements in τ than in l (since it is also assumed that every element of l has at least one element of l ρ-adjacent to it). to complete the proof, let us assume that the premise of (iii) is satisfied and c and d in l are such that c is ρ-adjacent to d. we define an infinite sequence of elements of l as follows: c(0) = d and, for k ≥ 1, c(k) is the unique element of l that is ρ-adjacent from c(k−1). since l is finite, there must be a smallest positive integer l such that c(l) = c(m), for some m < l. if m were positive, then the uniqueness of the element of l that is ρ-adjacent to c(l) = c(m) would imply that c(l−1) = c(m−1), which would contradict the minimality of l. hence, 144 gabor t. herman c(l) = c(0) = d. again by the uniqueness of the element of l which is ρ-adjacent to c(l) = d, we must have that c(l−1) = c, and so 〈 c(0), . . . , c(l−1) 〉 is a ρ-path in l from d to c. � theorem 4.5. for n ≥ 2, let ( m, ⋃ 1≤i 0 and any i1, i2, ..., il. 2.4. sheaf cohomology. we basically define cohomology of sheaves using the derived functors of the global section functor (see [9]), but robin has proven in [7] that sheaf cohomology and c̆ech cohomology coincides on graphs. let consider the open covering x = ∪ v∈v uv, where uv is the largest connected open of x which contains only the node v. for any edge e = |vw|, c© agt, upv, 2017 appl. gen. topol. 18, no. 2 222 sheaf cohomology on network codings: maxflow-mincut theorem the intersection uv ∩ uw := ue is the biggest open contained in e. then c̆0(x;f) = c̆0(u;f) = π v∈v f(uv), c̆ 1(x;f) = c̆1(u;f) = π e∈e f(ue) and the c̆ech complex is reduced to be : 0 −→ c̆0(x;f) δ 0 −→ c̆1(x;f) −→ 0. the sheaf cohomology is the cohomology associated to the c̆ech complex defined as follow: h0(x;f) := ker(δ0), and h1(x;f) := c̆1(x;f)/im(δ0). 2.5. information flow. definition 2.4. the information flow on a network x for a family of transmitted data z = (zs1,zs2, ...,zsk ), where zsi ∈ k nsi , is an assignment ψ(e) ∈ kcap(e) for each edge e, which satisfies the following so-called flow conditions: for e = |vw| and assuming that in(v) = {e1,e2, ...,en}, (i) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = ψ(e), for v /∈ (s ∪ r). (ii) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = ψ(e), for v = si ∈ s \ r. (iii) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = zsi, for v = rj ∈ r \ s and w = si. (iv) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = ψ(e), for v = rj ∈ r \ s and w /∈ s(rj) let σ = {σv} ∈ h 0(x;f), where for each i,σsi = (zsi, σ̃si) ∈ k nsi ⊕ klsi . one defines the map ψ : e −→ ⊕ e∈e kcap(e) as follows: ∀(e = |vw|) ∈ e,ψ(e) := σv|ue = σw|ue. it is clear that the family {ψ(e)} defines an information flow on the network for the data z = (zs1,zs2, ...,zsk ). this construction makes it possible to apply homological algebra tools to network coding problems as we state in the next theorem. theorem 2.5 (information theoretical meaning; [4]). for any network coding sheaf f of a graph x fitted with ψ, h0(x;f) is equivalent to the information flows on the network. namely this theorem tells that h0(x;f) carries all the data that can be transmitted on the network. 2.6. topological cut. α : s → 2r is the function which assigns to each source si the set α(si) of all receiver nodes receiving information from si, definition 2.6 (cut-set and cut value in network theory). (1) a cut-set c on a graph x is a set of edges whose removal breaks the connection between a source and some receivers. namely, if s is a source and d is an open set of a graph x which includes some receivers r1,r2, ...,rj ∈ α(s) but the node s, then the set of incoming edges into d define a cut-set c between s and r1,r2, ...,rj. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 223 m. atontsa and c. tcheka (2) the value of a cut-set c, denoted val(c), is the sum of the capacities of its edges. remark 2.7. the topological cut values of the graph x which satisfy the sheaf f are presented by the cohomology classes of h0(d;f). the following theorem from which we recover the physical definition of cut value shows that h0(d;f) is independent of the sheaf f . proposition 2.8 ([4]). the value of the cut-set is equivalent to topological cut values. more precisely val (c) = dim(h0(d;f)). 3. optimization of the network: maxflow-mincut 3.1. duplication of networks. in this subsection, we construct from a network x, a modified network x which still keep some information from the original one. let γ(v,w) be the reunion of all paths in x connecting v to w, and γ(si) := ∪ r∈α(si) γ(si,r). the modified graph x is obtained by duplicating edges which conduct information from multiple sources. more precisely, x := {(x,i),x ∈ γ(si), i = 1,2, ...,k}, and hence x is the disjoint union of the subgraphs γ(si). heuristically for us, the purpose for this construction is to apply the results of the single source case [4] to each subgraphs γ(si). as an illustration, fig.1. and fig.2 show how we construct the graph x from x. the adding edges e = |rjsi| are not represented on purpose to simplify the construction. fig.1. graph x s1 v1 e4 v e1 s2 e2 w e r1 e5 e6 v2 e3 r2 e8 e7 c© agt, upv, 2017 appl. gen. topol. 18, no. 2 224 sheaf cohomology on network codings: maxflow-mincut theorem (s1,1) (v,1) (w,1) (v1,1) (r1,1) (r2,1) (s2,2) (v,2) (w,2) (r1,2) (v2,2) (r2,2) (e4,1) (e1,1) (e,1) (e5,1) (e6,1) (e7,1) (e6,2) (e7,2) (e8,2) (e,2) (e2,2) (e3,2) fig.2. graph x = γ(s1) ∪ γ(s2) to make the difference with the sheaf f of x, we denote by f : x −→ v ect, the network coding sheaf on x = ∪ i∈{1,...,k} ∪ v∈v∩γ(si) (uv, i). the natural projection j : x −→ x,(x,i) 7→ x, induces an injective map: j∗ : h0(x;f) −→ h0(x;f). therefore dimh0(x;f) ≤ dimh0(x;f), and in terms of information theory, any information flow on the network x can be extended to an information flow on x. 3.2. relative sheaf cohomology. let d ⊂ x be an open of a graph x. that inclusion induces a surjective map p∗ : c∗(x;f) → c∗(d;f). therefore we have the short exact sequence defined as follow: 0 → c∗(x,d;f) i ∗ → c∗(x;f) p ∗ → c∗(d;f) → 0 where c∗(x,d;f) ∼= c ∗(d;f ) c∗(x;f ) the short exact sequence induces the long exact sequence below: 0 → h0(x,d;f) i 0 → h0(x;f) p 0 → h0(d;f) δ 0 → h1(x,d;f) i 1 → ... it has been proven in [4] that if the graph x contains only one source s and the open d includes some receivers, but does not include the source node s, then for any network coding sheaf f,h0(x,d;f) = 0. thus the long exact sequence theorem applied to the above short exact sequence shows that dimh0(x;f) ≤ dimh0(d;f). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 225 m. atontsa and c. tcheka in the sequel we consider that the graph x has multiple sources along with multiple receivers. using the preliminary result on the single source case to the graph x introduced in section 6, we obtain the following inequalities on its subgraphs γ(si) = ∪ r∈α(si) γ(si,r), for all i ∈ {1,2, ...,k}: dimh0i (x;f) ≤ dimh 0(di;f), for any open di of the subgraph γ(si) = ∪ r∈α(si) γ(si,r) which contains some receiver nodes and does not include the source node si, and h ∗ i (x;f) := h0(γ(si);f). if b := ∪ i di, this leads to the following inequality: ∑ i∈{1,2,...,k} dimh0i (x;f) ≤ ∑ i∈{1,2,...,k} dim h0(di; − f) ‖ ‖ dimh0(x;f) ≤ dim h0(b;f) 3.3. maxflow bound. the following theorem generally known as the weak duality generalizes the upper bound theorem proved in [4] for the single-source scenario. theorem a: if d is an open of the graph x which include some receiver nodes, and not the sources, then d defines a cut and we have this inequality: maxflow(x) = max f dimh0(x;f) ≤ mincut(x) = min d dim h0(d;f) to prove this theorem, we will need the next two lemmas. let p : x −→ x the natural surjection, i = ⋃ si,sj∈s p(γ(si)) ∩ p(γ(sj)), and if d denotes a cut on the network x, let j = {e = |vw| ⊂ i,e cutting edge}. lemma 3.1. there exists a finite vector space m of dimension dimm =∑ e=|vw|⊂i cap(e) such that h0(x;f) ∼= h0(x;f) ⊕ m, proof. the surjection p : x −→ x induces the injection: p∗ : h0(x;f) −→ h0(x;f). therefore we have h0(x;f) = p(h0(x;f)) ⊕ m, where dim m ≤ dimh0(x;f). let σ = {σv,i} ∈ h 0(x;f). σ = σ1 + σ2, with σ1 ∈ h 0(x;f) and σ2 ∈ m. σ * p(h 0(x;f)) ⇔ σ2 6= 0. however,σ * p(h0(x;f)) ⇔ ∃(i,j) ∈ {1,2, ...,k}2, i 6= j, and e = |vw| ∈ p(γ(si))∩p(γ(sj)) such that σw,i|(ue,i) 6= σw,j|(ue,j) ⇐⇒ σw,i|(ue,i)−σw,j|(ue,j) 6= 0. i.e (σw,i|(ue,i),σw,j|(ue,j)) ∈ imf, where f : kcap(e) × kcap(e) −→ kcap(e) (x,y) 7−→ x − y . it is clear that σ2 consists of such couple (σw,i|(ue,i),σw,j|(ue,j)) ∈ imf and dim imf = cap(e). hence dimm = ∑ e=|vw|⊂i cap(e) � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 226 sheaf cohomology on network codings: maxflow-mincut theorem lemma 3.2. there exists a finite vector space n of dimension dimn =∑ e=|vw|⊂j cap(e) such that h0(p−1(d);f) ∼= h0(d;f) ⊕ n proof. the surjection p : p−1(d) −→ d implies the injection: p∗ : h0(d;f) −→ h0(p−1(d);f). therefore h0(p−1(d);f) = p∗(h(d;f))⊕n, where dimn ≤ dimh0(p−1(d);f). let p−1(d) = ∪ i∈{1,2,...,k} di. h 0(p−1(d);f) = h0( ∪ i∈{1,2,...,k} di;f) = ⊕ i∈{1,2,...,k} h0(di;f). dimh0(p−1(d);f) = ∑ i dimh0(di;f) = ∑ i val(ci), where ci denotes the cut created by di on the subgraph γ(si). on the other hand, dimh0(d;f) = val(c), where c is the cut created by the open d on the graph x. val(c) = ∑ e is cutting edge cap(e). it is however clear that, ∑ i val(ci) = ∑ e is cutting edge on x by d cap(e) + ∑ e⊂p(γ(si))∩p(γ(sj)),e is cutting edge cap(e) this leads to: dimh0(p−1(d);f) = dimh0(d;f) + ∑ e⊂p(γ(si))∩p(γ(sj)),e is cutting edge cap(e). hence dim n = ∑ e⊂j cap(e). � proof of theorem a. by using the inequality: dimh0(x;f) ≤ dimh0(p−1(d);f) with lemma 3.1. and lemma 3.2., we have this: dimh0(x;f) + dimm ≤ dimh0(d;f) + dimn ⇐⇒ dimh0(x;f) + dimm − dimn ≤ dimh0(d;f) however, it is clear that dim m − dimn ≥ 0, hence dimh0(x;f) ≤ dimh0(d;f), for all open d and network coding sheaf f on the graph x. therefore we have the following result: max f dimh0(x;f) ≤ min d dim h0(d;f) 3.4. mincut bound. as we consider in this paper that the graphs wear the decoding maps, each node is expressed by the incoming edges and one have naturally the isomorphism c0(x;f) ∼= c1(x;f). it turns out using the rank-nullity theorem on the differential δ0, which is linear in our case, that h0(x;f) ∼= h1(x;f). we will use this identification in proving the next result. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 227 m. atontsa and c. tcheka theorem b : if x is a graph with multiple source nodes and some receivers, maxflow(x) := max f dimh0(x;f) ≥ mincut(x) =: min d dim h0(d;f) the proof of this salient theorem uses the following construction on the network. let us consider now a graph x and u an open which contains all the nodes but the source nodes si. we admit to add some virtual nodes on the cutting edges such that they are include in the open set u, and we then obtain the modified graph x1. the following figures are an illustration of that concept where the nodes g and h have been added. fig.4. graph x with an open u s b e1 c e2 j e6 ee4 r1e5 f e3 e7 e8 r2e9 u fig.5. graph x1 s b e1 c e2 j e6 ee4 r1e5 f e3 e7 e8 r2e9 u g h c© agt, upv, 2017 appl. gen. topol. 18, no. 2 228 sheaf cohomology on network codings: maxflow-mincut theorem we consider in particular that the network x1 have the following characteristics: for a cutting edge e = |sw| of x, where we have added a node v, uv will still be the largest connected open of the graph x1 which contains only the node v. moreover we define the local coding maps φvs := φws and φwv := id kcap(e) 1imφws, where 1imφws means the characteristic function on the vectorial space imφws. the network coding sheaf on the network (x1,(φba)e=|ab|∈x1) defined above is denoted f1. heuristically for us, the purpose of this modification is to understand more the properties of the cut created by the open set u without changing the value of the cut-set. to make sure that we have not significantly changed something, we have the following lemma: lemma 3.3. for any graph x fitted with ψ,h0(x1,f1) is equivalent to the information on the graph x. proof. we denote x1 = (v1,e1). let now σ = {σv}v∈v1 ∈ h 0(x1,f1). if e = |sw| ∈ e is a cutting edge by the open set u which have been added a node v as follows: s w e s w e1 v e2 we now have the following proposition which first stresses a link between the two graphs. proposition 3.4. for any open set u satisfying the above conditions, we have the following equality: h1(u,f1) ∼= h 1(x,f). proof. using the above notation, if e = |sw| is a cutting edge and v is the node which is added and include in the cut u, we denote e2 = |vw| and e1 = |sv|. it is clear from the definition that f1(ue2) = f(ue), and a direct computation shows that imφws ∼= imφwv. therefore f1(ue2)/imφwv ∼= f(ue)/imφsw, and we obtain the isomorphism. � proof of theorem b. let u ⊂ x an open set which satisfies the condition stated earlier in proposition 3.4. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 229 m. atontsa and c. tcheka we have the following square: h1(u;f1) (1) −→ ∼= h1(x;f) ∼= ↓(2) ∼= ↓(3) h0(u;f1) (4) −→ ∼= h0(x;f) ‖ h0(u;f), where (1) is the isomorphism of proposition 3.4, (2) and (3) are isomorphisms that results from comments at the beginning of this subsection. then (4) is the unique morphism which makes the diagram to commute. we obtain from the square the equality: dimh0(u;f) = dimh0(x;f) thus, min d dimh0(d;f) ≤ dimh0(u,f) = dimh0(x;f) ≤ max f dimh0(x;f). then, { σs|ue1 = σv|ue1 σv|ue2 = σw|ue2 ⇐⇒ { φvs(σs) = σv φwv(σv) = σw|ue2 (1). but φvs := φws on the network x, and φwv(σv) = φwv(φvs(σs)) := φvs(σs). therefore (1) ⇐⇒ φvs(σs) = σw|ue2 . hence any cocycle σ = {σv}v∈v1 ∈ h 0(x1,f1) is equivalent to the induced element ∼ σ = {σv}v∈v ∈ h 0(x,f). we then have h0(x1,f1) ∼= h 0(x,f). � acknowledgements. the authors would like to thank tchatchiem kamche hermann for useful discussions during this work references [1] j. m. curry, sheaves, cosheaves and applications, arxiv:1303.3255 [math.at] (2013). [2] y. felix, s. halperin and j-c. thomas, rational homotopy theory, volume 205 of graduate texts in mathematics. springer-verlag, new york, 2001. [3] e. gasparim, a first lecture on sheaf cohomology, universidade federal de pernambuco, cidade universitária, recife, pe, brazil, 50670-901. [4] r. ghrist and y. hiraoka, applications of sheaf cohomology and exact sequences to network coding, nolta, 2011. [5] r. ghrist and s. krishnan, a topological max-flow-min-cut theorem, in proc. global sig. inf. proc, aug. 2013. [6] a. hatcher, algebraic topology, cambridge university press, 2002. [7] r. hartshorne, algebraic geometry, springer, 1997. [8] r. koetter and m. médard, an algebraic approach to network coding, ieee trans. on networking 11 (2003), 782–795. [9] l. i. nicolaescu, the derived category of sheaves and the poincaré-verdier duality, april, 2005 (www3.nd.edu/ lnicolae/verdier-ams.pdf). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 230 @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 351–360 developable hyperspaces are metrizable l’ubica holá, jan pelant ∗ and lászló zsilinszky dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. developability of hyperspace topologies (locally finite, (bounded) vietoris, fell, respectively) on the nonempty closed sets is characterized. submetrizability and having a gδ-diagonal in the hyperspace setting is also discussed. 2000 ams classification: 54b20. keywords: developable spaces, vietoris topology, fell topology, locally finite topology, bounded vietoris topology, gδ-diagonal. 1. introduction. let cl(x) (k(x)) denote the hyperspace of nonempty closed (compact) sets of a t2 topological space (x,τ). for notions not defined in the paper see [11], [2] and [15]. historically there have been two hyperspace topologies of particular importance: the vietoris topology τv (see section 1) and the hausdorff metric topology τh, as considered in michael’s fundamental paper on hyperspaces [22]. it is well-known, that τv = τh on k(x) and hence (k(x),τv ) is metrizable iff x is. metrizability of the larger hyperspace (cl(x),τv ) is also characterized, it is equivalent to x being compact metrizable [22]. to investigate generalized metric properties of the vietoris topology, one may start by considering bing’s factorization of metrizability into collectionwise normality (cwn) and mooreness ([11]). there is an abundance of results on cwn and related properties, e.g. (combined results of keesling and velichko from [19], [20], [27]): (cl(x),τv ) is cwn (paracompact, normal, resp.) iff x is compact; however, some stronger (hereditary) properties, such as hereditary normality, stratifiability or monotone normality, coincide with metrizability for (cl(x),τv ) (cf. [6], [12], [13] and [7]). ∗the second author was supported by the grant gacr 201/00/1466. 352 l’ubica holá, jan pelant and lászló zsilinszky as far as the other half of bing’s theorem is concerned, mooreness (developability) has been considered only for k(x); indeed, mizokami has shown that (k(x),τv ) is moore iff x is [23]. it is one of the purposes of this paper to characterize developability (mooreness) of the vietoris topology on cl(x). a good starting point for investigating developability is to look at 1st countability of (cl(x),τv ) first, as was done by holá and levi [16] while extending an older result of choban [8]: theorem 1.1. let x be a t2 space. the following are equivalent: (i) (cl(x),τv ) is 1st countable; (ii) x is perfectly normal, the derived set x′ is countably compact, hereditarily separable and of countable character, and x \x′ is countable. thus, in general, 1st countability of the vietoris topology does not guarantee its metrizability (just consider x = ω, the discrete space of non-negative integers); other cases, e.g. if x is dense-in-itself and metrizable, are treated in [10]. in section 2 we will prove that developability and metrizability of (cl(x),τv ) always coincide. however, see remark 3.4 following theorem 3.3 for a relevant comment. in fact, after obtaining the same relationship for other well-studied hypertopologies, such as the locally finite, fell and bounded vietoris topology, respectively (see section 1 for definitions), as well as for the wijsman topology (which is 1st countable iff it is metrizable [2]), it seems that the coincidence of developability and metrizability in the hyperspace setting could be established for a broad class of hypertopologies. it remains to be seen if it is the case for hit-and-miss and hit-and-far topologies ([2]) or even for the general hyperspace topology studied in [28], incorporating all the above topologies along with some weak hyperspace topologies, including the wijsman topology. since metrizability is to submetrizability as developability is to having a gδdiagonal (see 1.6, 2.2 and 2.5 in [15]), we then turn to studying submetrizability and having a gδ-diagonal in cl(x). it turns out to be a perfect match for the fell and the bounded vietoris topology, respectively, moreover, if x is morita’s m-space, also for the vietoris topology. finally, in section 4, the above generalized metric properties are discussed on k(x) with the vietoris and fell topologies, respectively. 2. preliminaries. to describe the hypertopologies we will work with, we need to introduce some notation: for u ⊂ x put u+ = {a ∈ cl(x) : a ⊂ u} and u− = {a ∈ cl(x) : a∩u 6= ∅}. subbase elements of the vietoris (locally finite) topology τv (τlf ) on cl(x) are of the form u+ with u ∈ τ and ⋂ u∈u u − with u ⊂ τ finite (locally finite). note, that for a metrizable x, the supremum of all hausdorff metric (resp. wijsman) topologies corresponding to topologically equivalent metrics is the locally finite (resp. vietoris) topology ([4], [25], [5]). developable hyperspaces are metrizable 353 another classical hypertopology, the fell topology, has found numerous applications in various fields of mathematics ([21], [1]); it has as a subbase elements of the form u− and v +, where u ∈ τ and v has a compact complement in x. if (x,d) is a metric space the bounded vietoris topology τbvd has as a subbase, elements of the form u− and v +, where u ∈ τ and the complement of v is a closed bounded set in (x,d). proposition 2.1. let x be a t2 space. if (cl(x),τv ) or (cl(x),τlf ) is 1st countable, then x is collectionwise normal. proof. in view of theorem 1.1 (resp. [9]), x is normal and x′ is countably compact; thus, if d is a discrete family of closed subsets of x, only finitely many members d0, . . . ,dn ∈ d intersect x′. since x is normal, there exist pairwise disjoint open sets u0, . . . ,un such that di ⊂ ui for each i ≤ n. denote d0 = {d0, . . . ,dn} and observe that d = ⋃ (d\d0) is closed. consequently, (d\d0) ∪{u0 \d,.. . ,un \d} is a disjoint open expansion of d. � proposition 2.2. let x be a t2 space. if (cl(x),τv ) or (cl(x),τlf ) is developable, then x is metrizable and x′ is compact. proof. by admissibility of the vietoris and the locally finite topology, x is a developable space, which is also collectionwise normal by proposition 2.1; hence, in view of bing’s theorem ([11]), x is metrizable. since x′ is countably compact, in our case, it is compact. � 3. developability in cl(x). theorem 3.1. let x be a t2 space. the following are equivalent: (i) (cl(x),τlf ) is moore; (ii) (cl(x),τlf ) is developable; (iii) (cl(x),τlf ) is metrizable; (iv) x is metrizable and x′ is compact. proof. (ii)⇒(iv) follows from proposition 2.2 and (iv)⇒(iii) from [4], theorem 2.3. � theorem 3.2. (cl(ω),τv ) is not developable. proof. suppose (cl(ω),τv ) is developable. let d = {un : n ∈ ω} be a development of (cl(ω),τv ). without loss of generality we may suppose that un+1 is a refinement of un for every n. for every a ∈ [ω]ω, define l(a) = min{n : st(a,un) ⊂ a+}, and for every a ∈ [ω]ω and f ∈ [a]<ω put π(a,f) = min{l(b) : f ⊂ b,b is a proper infinite subset of a}. for every a ∈ [ω]ω, m = l(a), and every f ∈ [a]<ω choose h(a,f) ∈ [a]<ω such that f is a proper subset of h(a,f) and there is u ∈ um with a+ ∩ ⋂ p∈h(a,f){p} − ⊂ u. 354 l’ubica holá, jan pelant and lászló zsilinszky for every a ∈ [ω]ω and every b,g ∈ [a]<ω such that h(a,b) ⊂ g we have π(a,g) > l(a): otherwise, π(a,g) ≤ l(a) and there is an infinite proper subset b of a with g ⊂ b such that l(b) ≤ l(a). hence, there is u ∈ul(a) such that {a,b} ⊂ u. as un+1 is a refinement of un for every n ∈ ω, we see that for each k < l(a) there is uk ∈ uk with {a,b} ⊂ uk. hence for every k ≤ l(a), st(b,uk) is not subset of b+ (as a\b 6= ∅) and so l(b) > l(a), a contradiction. we use an inductive construction now: • put n0 = π(ω,∅). take a0 ∈ [ω]ω such that l(a0) = n0 and g0 ∈ [ω]<ω such that there is u ∈ un0 with a + 0 ∩p∈g0 {p} − ⊂ u. put f0 = ∅. • let fj+1 = ⋃ i≤j gi and nj+1 = π(aj,fj+1); choose aj+1 ∈ [ω] ω such that fj+1 ⊂ aj+1, where aj+1 is a proper subset aj, l(aj+1) = nj+1 and put gj+1 = h(aj+1,fj+1). clearly, this construction can be repeated ω-many times. finally, put b =⋃ i∈ω gi and take p ∈ ω such that np > l(b). then fp ⊂ b ⊂ ap−1, but l(b) ≥ π(ap−1,fp) = np, a contradiction. � theorem 3.3. let x be a t2 space. the following are equivalent: (i) (cl(x),τv ) is moore; (ii) (cl(x),τv ) is developable; (iii) (cl(x),τv ) is metrizable; (iv) x is compact and metrizable. proof. only (ii)⇒(iv) needs justification: in view of proposition 2.2, it suffices to show that every sequence in x \ x′ has a cluster point in x. otherwise, x \x′ contains a closed copy of ω, thus, (cl(ω),τv ) sits in (cl(x),τv ) and is hence developable, a contradiction with theorem 3.2. � remark 3.4. after l’. holá’s lecture at caserta 2001, prof. arhangel’skǐı was wondering, whether theorem 3.3 could be extended to hyperspaces which are σ-spaces (i.e. spaces with a σ-discrete network) (see [15], also for related notions of (strong) σ-spaces). he was very right. a possible way could be an easy modification of the proof of theorem 3.2 in effect that (cl(ω),τv ) is not a σ-space and an application of the fact that each countably compact σ-space is compact and metrizable [15]. fortunately, popov [26] proved already in 1978 the following theorem 3.5. (cl(ω),τv ) contains the sorgenfrey line s as a subspace. recall that s is not even a σ-space. so it follows from popov’s result that if (cl(x),τv ) is a σ-space then x is countably compact. of course, theorem 3.2 represents a special case, nevertheless we have decided to keep its proof to make the paper more self-contained. as m-spaces are σ-spaces, some other information on the vietoris hyperspaces which are σ-spaces, may be found in developable hyperspaces are metrizable 355 proposition 4.9 below. coming back to the original arhangel’skǐı’s question, we could reformulate theorem 3.3: theorem 3.6. let x be a t2 space. the following are equivalent: (o) (cl(x),τv ) has a σ-discrete network; (i) (cl(x),τv ) is moore; (ii) (cl(x),τv ) is developable; (iii) (cl(x),τv ) is metrizable; (iv) x is compact and metrizable. we proceed with other topologies now: theorem 3.7. let (x,d) be a metric space. then the following are equivalent: (i) (cl(x),τbvd) is moore; (ii) (cl(x),τbvd) is developable; (iii) (cl(x),τbvd) is metrizable; (iv) (x,d) is boundedly compact (i.e. every closed bounded set in (x,d) is compact). proof. since (iii)⇔(iv) is known, only (ii)⇒(iv) needs some comments: let b ∈ cl(x) be bounded in (x,d). developability of (cl(x),τbvd) implies that cl(b) equipped with the relative topology τbvd on cl(b) is also developable. it is easy to verify that the relative topology τbvd on cl(b) coincides with the vietoris topology τv . thus, (cl(b),τv ) is developable and b must be compact by theorem 3.3. � theorem 3.8. let x be a t2 space. the following are equivalent: (i) (cl(x),τf ) is moore; (ii) (cl(x),τf ) is developable; (iii) (cl(x),τf ) is t2 and has a gδ-diagonal; (iv) (cl(x),τf ) is submetrizable; (v) (cl(x),τf ) is metrizable; (vi) x is hemicompact and metrizable. proof. (ii)⇒(iii) developability of (cl(x),τf ) implies that it has a gδ-diagonal; moreover, even 1st countability of (cl(x),τf ) implies, that x is locally compact ([16]), so, (cl(x),τf ) is t2 by a result of fell. (iii)⇒(v) hausdorffness of (cl(x),τf ) implies that (cl(x),τf ) is locally compact. since (cl(x),τf ) has a gδ-diagonal, points of (cl(x),τf ) are gδ; thus, (cl(x),τf ) is 1st countable; hence (see [17]) it is paracompact. in summary, (cl(x),τf ) is paracompact, locally compact with a gδ-diagonal and is therefore metrizable. (v)⇔(vi) is known ([3]) (to prove (vi)⇒(v) realize that every first countable hemicompact hausdorff space is locally compact). the remaining implications are trivial. � 356 l’ubica holá, jan pelant and lászló zsilinszky 4. submetrizability, having a gδ-diagonal and related properties in cl(x). the last theorem of the previous section showed that these properties coincide for (cl(x),τf ). in what follows, we show that similar relationship holds for the (bounded) vietoris topology as well. first, we will see that for the (bounded) vietoris and locally finite topology, respectively, submetrizability and developability are distinct. proposition 4.1. (i) if x is a metrizable space, then (cl(x),τlf ) is submetrizable. (ii) if (x,d) is a separable metric space, then (cl(x),τv ) and (cl(x),τbvd) are submetrizable. proof. (i) if x is a metrizable space, take any compatible metric d on x and consider the hausdorff metric topology τhd. it is known that τhd ⊆ τlf . (ii) since d is a separable metric on x, the wijsman topology τwd is metrizable ([2]). it is known that τwd ⊆ τbvd ⊆ τv . � corollary 4.2. (i) (cl(x),τv ) is submetrizable and not developable, if x is non-compact, separable and metrizable. (ii) (cl(x),τbvd) is submetrizable and not developable, if (x,d) is a separable metric space which is not boundedly compact. (iii) (cl(x),τlf ) is submetrizable and not developable, if x is a non-compact dense-in-itself metrizable space. proposition 4.3. let x be a t2 space. (i) if the points in (cl(x),τv ) are gδ, then x is hereditarily separable and every closed set in x is gδ. (ii) if the points in (cl(x),τf ) are gδ, then x is hereditarily separable, every open set in x is σ-compact and every closed set in x is a gδ-set. (iii) if (x,d) is a metric space, then the points in (cl(x),τbvd) are gδ iff (x,d) is separable. proof. we prove only (i): let a ⊂ x. since a is a gδ-set in (cl(x),τv ), there are τv -open sets gn such that {a} = ⋂ n∈ω gn. without loss of generality we can suppose, that for every n ∈ ω, gn = (gn)+∩∩l≤n(uiln )−, where gn,uiln , l ≤ n are open sets in x. for every n ∈ ω, l ≤ n choose ailn ∈ a∩uiln . it is easy to verify that {ailn : l ≤ n,n ∈ ω} = a. � remark 4.4. let x be a t2 space. (i) if (cl(x),τv ) has a gδ-diagonal, then x is hereditarily separable. (ii) points in (cl(x),τv ) are gδ iff every a ∈ cl(x) is a gδ-set and has a countable pseudobase (in a). proposition 4.5. let (x,d) be a metric space. the following are equivalent: (i) (cl(x),τbvd) is submetrizable; developable hyperspaces are metrizable 357 (ii) (cl(x),τbvd) has a gδ-diagonal; (iii) (x,d) is separable. proof. (iii)⇒(i) see proposition 4.1(ii). (ii)⇒(iii) follows from proposition 4.3(iii). � proposition 4.6. let x be a w∆-space. the following are equivalent: (i) (cl(x),τv ) is submetrizable; (ii) x is a separable metrizable space. proof. (ii)⇒(i) see proposition 4.1(ii). (i)⇒(ii) submetrizability of (cl(x),τv ) implies its hausdorffness, so x is regular by a result of michael [22] and submetrizable, which in turn, being a w∆-space, is an m-space ([15]). however, an m-space with a gδ-diagonal is metrizable ([15]). finally, by remark 4.4(i), x is separable. � proposition 4.7. let x be an m-space. the following are equivalent: (i) (cl(x),τv ) is submetrizable; (ii) (cl(x),τv ) is t2 and has a gδ-diagonal. (iii) x is a separable metrizable space. proof. ((ii)⇒(i) see proposition 4.1(ii). (i)⇒(ii) an m-space with a gδ-diagonal is metrizable ([15]) and by remark 4.4(ii), x is separable. � proposition 4.8. let x be a t2 space. (i) (cl(x),τv ) is a w∆-space (strict p-space,m-space, resp.) ⇒ x is countably compact. (ii) x is countably compact ; (cl(x),τv ) is a w∆-space. proof. (i) by proposition 4.1(ii), (cl(ω),τv ) is submetrizable, so it has a g∗δdiagonal ([15]). by a theorem of hodel ([15]) this implies, that (cl(ω),τv ) is not a w∆-space (neither is a strict p-space or an m-space, since these properties are stronger than w∆). on the other hand, if (cl(x),τv ) is a w∆-space and x is not countably compact, then ω sits in x as a closed subset and hence cl(ω) embeds as a closed subset in (cl(x),τv ). since the w∆-property is closed hereditary, this would imply that (cl(ω),τv ) is a w∆-space, a contradiction. (ii) by example 2 of [24], there is a countably compact space x, such that (f2(x),τv ) (= the space of all sets with at most 2 elements) is not a w∆space. since f2(x) is a closed set in (cl(x),τv ) and the w∆-property is closed hereditary, (cl(x),τv ) is not a w∆-space. � a topological space x is ultracompact iff every net in x with a countable range has a cluster point. this characterization of ultracompactness is due to holá and künzi [18]. proposition 4.9. let x be a linearly ordered topological space. the following are equivalent: (o) (cl(x),τv ) is countably compact; 358 l’ubica holá, jan pelant and lászló zsilinszky (i) (cl(x),τv ) is an m-space; (ii) (cl(x),τv ) is a w∆-space; (iii) x is countably compact. proof. (ii)⇒(iii) see proposition 4.8(i). for the rest of the proof, realize that in linearly ordered topological spaces countable compactness and ultracompactness coincide [14], and x is ultracompact iff (cl(x),τv ) is [18]. further, an ultracompact space is countably compact, which in turn is an m-space. � remark 4.10. it can be inferred from the above proof, that if x is ultracompact, then (cl(x),τv ) is an m-space (w∆-space). so, for every non-compact ultracompact space x, (cl(x),τv ) is a non-compact m-space (w∆-space). proposition 4.8 also offers another proof of developability of (cl(x),τv ): proposition 4.11. let x be a t2 space. the following are equivalent: (i) (cl(x),τv ) is developable; (ii) (cl(x),τv ) is a w∆-space and x has a gδ-diagonal. proof. (i)⇒(ii) is clear. (ii)⇒(i) by proposition 4.8(i), x is countably compact. every countably compact space with a gδ-diagonal is compact and metrizable [15]. now theorem 3.3 applies. � 5. some generalized metric properties in k(x). proposition 5.1. let x be a t2 space. the following are equivalent: (i) (k(x),τv ) is submetrizable; (ii) x is submetrizable. proof. (i)⇒(ii) it is easy to verify that if t is a coarser metrizable topology on (k(x),τv ), then the relative topology t on x is coarser than τ. (ii)⇒(i) let µ ⊆ τ be a metrizable topology. then k(x,τ) (= compact sets in τ) ⊆ k(x,µ). (k(x,µ),µv ) is a metrizable space, where µv is the vietoris topology generated by µ. thus, also µv restricted to k(x,τ) is a metrizable topology on k(x,τ). since µ ⊆ τ, we have that µv restricted on k(x,τ) is coarser than τv . � the situation in (k(x),τf ) is different: proposition 5.2. let x be a t2 space. the following are equivalent: (i) (k(x),τf ) is moore; (ii) (k(x),τf ) is developable; (iii) (k(x),τf ) is t2 and has a gδ-diagonal; (iv) (k(x),τf ) is submetrizable; (v) (k(x),τf ) is metrizable; (vi) x is hemicompact and metrizable. developable hyperspaces are metrizable 359 proof. (iii)⇒(vi) first we show that if (k(x),τf ) is t2, then x is locally compact. suppose that x fails to be locally compact. let x ∈ x be such that for every u ∈b(x) and k ∈ k(x) there is some xu,k ∈ u\k (here b(x) stands for a base of neighborhoods of x). let y ∈ x be a point different from x. it is easy to verify that the net of compact sets {{xu,k,y} : u ∈b(x),k ∈ k(x)} converges both to {x,y} and to {y} in (k(x),τf ), a contradiction. since (k(x),τf ) has a gδ-diagonal, points of (k(x),τf ) are gδ and x has a gδ-diagonal. thus, x is σ-compact, locally compact with a gδ-diagonal, i.e. it must be hemicompact, by [11] 3.8.c (b), and metrizable. (vi)⇒(v) if x is hemicompact and metrizable, then (cl(x),τf ) is metrizable [2]; thus, (k(x),τf ) is also metrizable. (v)⇒(iv), (iv)⇒(iii) and (i)⇒(ii) are trivial. (ii)⇒(iii) it follows from [3], that 1st countability of (k(x),τf ) implies hemicompactness of x and hence its local compactness (since x must be first countable). this in turn is equivalent to hausdorffness of (k(x),τf ). (iii)⇒(i) by the above, (iii) is equivalent to metrizability of (k(x),τf ). (i)⇒(ii) is trivial � references [1] h. attouch, variational convergence for functions and operators, pitman, boston (1984). [2] g. beer, topologies on closed and closed convex sets, kluwer, dordrecht (1993). [3] g. beer, on the fell topology, set-valued anal.1 (1993), 69–80. [4] g. a. beer, c. j. himmelberg, k. prikry and f. s. van vleck, the locally finite topology on 2x, proc. amer. math. soc.101 (1987), 168–172. [5] g. beer, a. lechicki, s. levi and s. naimpally, distance functionals and suprema of hyperspace topologies, ann. mat. pura appl. (4) 162 (1992), 367–381. [6] h. brandsma, monolithic hyperspaces; phd. thesis, vrije universiteit, amsterdam (1998). 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[28] l. zsilinszky, topological games and hyperspace topologies, set-valued anal. 6 (1998), 187– 207 . received january 2002 revised october 2002 l’ubica holá academy of sciences, institute of mathematics, štefánikova 49,81473 bratislava, slovakia e-mail address : hola@mat.savba.sk jan pelant mathematical institute, czech academy of sciences, žitná 25, 11567 praha 1, czech republic e-mail address : pelant@cesnet.cz lászló zsilinszky department of mathematics and computer science, university of north carolina at pembroke, pembroke, nc 28372, usa e-mail address : laszlo@uncp.edu @ appl. gen. topol. 15, no. 1 (2014), 43-54doi:10.4995/agt.2014.2144 © agt, upv, 2014 on some properties of t0−ordered reflection sami lazaar ∗ and abdelwaheb mhemdi department of mathematics, faculty of sciences of tunis. university tunis-el manar. “campus universitaire” 2092 tunis, tunisia. (salazaar72@yahoo.fr, mhemdiabd@gmail.com) abstract in [12], the authors give an explicit construction of the t0−ordered reflection of an ordered topological space (x, τ, ≤) . all ordered topological spaces such that whose t0−ordered reflections are t1−ordered spaces are characterized. in this paper, some properties of the t0−ordered reflection of a given ordered topological space (x, τ, ≤) are studies. the class of morphisms in ordtop orthogonal to all t0−ordered topological space is characterized. 2010 msc: 54f05; 18b30; 54g20; 54c99; 06f30; 18a05. keywords: ordered topological space; t2−ordered; t1−ordered; t0−ordered; ordered reflection; ordered quotient; category and functor. 1. introduction among the oldest separation axioms in topology, there are three famous ones t0, t1 and t2. the t0−, t1− and t2−reflections of a topological space have long been of interest to categorical topologist. the construction of these reflections in the category top of all topological spaces are given in [10]. in [2], the authors introduced some new separation axioms using the ti−reflections ti(x) i ∈ {0, 1, 2} as follow: definition 1.1. let i, j be two integers such that 0 ≤ i < j ≤ 2. a topological space x is said to be a t(i,j) − space if ti(x) is a tj−space. ∗corresponding author. received may 2013 – accepted february 2014 http://dx.doi.org/10.4995/agt.2014.2144 s. lazaar and a. mhemdi precisely, there are two new types of separation axioms namely t(0,1)− and t(0,2)−spaces. the characterization of these spaces are completely stated in [2, theorem 3.5.] and [2, theorem 3.12.]. after this, in [12], h. künzi and t. a. richmond are interested in the corresponding concepts of ti−ordered reflections in the category ordtop with ordered topological spaces (x, τ, ≤) as objects and continuous increasing maps as morphisms (or arrows). the authors show that, given an ordered topological space (x, τ, ≤), the ti−ordered reflection, for i ∈ {0, 1, 2} of (x, τ, ≤) is obtained as a quotient of x ( for more information see [12, theorem 2.5.]). let (x, τ, ≤) be an ordered topological space. for a ⊆ x, the increasing hull of a is i (a) = {y ∈ x : ∃ x ∈ a x ≤ y}. a subset a of x is an increasing set if i (a) = a and we denote by i (a) the closed increasing hull of a, that is, the smallest closed increasing set containing a. decreasing set, decreasing hull d(a) and the closed decreasing hull d(a) are defined dually. a subset a which satisfy a = i(a) ∩ d(a) is called a c − set. we denote by c(a) the smallest c − set containing a. an ordered topological space (x, τ, ≤) is said to be t0−ordered if the following equivalent conditions hold. (1) [i(x) = i(y) and d(x) = d(y)] =⇒ x = y. (2) c(x) = c(y) =⇒ x = y. (3) if x 6= y, there exist a monotone open neighborhood of one of the points which does not contain the other point. let (x, τ, ≤) be an ordered topological space. for x, y ∈ x, define an equivalence relation on x by x ≈ y if and only if [i (x) = i (y) and d (x) = d (y)], which is equivalent to c (x) = c (y) . order the set x/≈ by the finite step order defined by : z0 ≤0 zn ⇐⇒ ∃ z1, ..., zn−1 and ∃ z ′ i, z ∗ i ∈ zi (i = 0, 1, ..., n) with z ′ i ≤ z∗i+1 ∀ i = 0, 1, ..., n − 1 . t. a. richmond and h-p. a. künzi show that ( x/≈, τ/≈, ≤0 ) is the t0−ordered reflection of x. this paper consists of some investigations into the t0−ordered reflection of an ordered topological space (x, τ, ≤) . in the first section we give the characterization of an ordered topological space (x, τ, ≤) such that its t0−ordered reflection is t k1 −ordered and we characterize ordered topological spaces whose t0−ordered reflections are t2−ordered. [2, theorem 3.5] and [2, theorem 3.12] are recovered. the second investigation deals with some categorical properties of the category ordtop0, of t0−ordered topological spaces. more precisely, a characterization of the class of morphisms in ordtop rendered invertible, by the t0−ordered reflection functor, is given. [2, theorem 2.4] is seen to be a particular case of our result. © agt, upv, 2014 appl. gen. topol. 15, no. 1 44 on some properties of t0−ordered reflection 2. separation axioms given an ordered topological space (x, τ, ≤), the construction of its t0−ordered reflection denoted by ( x/≈, τ/≈, ≤0 ) satisfies some categorical properties: for each ordered topological space (y, γ, ⊑) and each continuous increasing map f from (x, τ, ≤) to (y, γ, ⊑) , there exists a unique continuous increasing map ≈ f : ( x/≈, τ/≈, ≤0 ) −→ ( y/≈, γ/≈, ⊑0 ) such that the following diagram commutes: (x, τ, �) f // qx �� (y, γ, ⊑) qy �� ( x/≈, τ/≈, ≤0 ) ≈ f // ( y/≈, γ/≈, ⊑0 ) where qx is the canonical surjection map. from the above properties, it is clear that we have a covariant functor from the category of ordered topological spaces ordtop into the full subcategory ordtop0 of ordtop whose objects are t0−ordered topological spaces. in [12], the authors characterize those ordered topological spaces whose t0−ordered reflections are t1−ordered as follows: theorem 2.1 ([12, theorem 3.2]). the following statements are equivalent: (1) the t0 − ordered reflection x/≈ of x is t1 − ordered. (2) x �0 y in x/≈ implies there exists an open increasing neighborhood of x not containing y and there exists an open decreasing neighborhood of y not containing x. (3) i (x) = ⋂ {n : n is an open increasing neigborhood of x} and d (x) = ⋂ {n : n is an open decreasing neigborhood of x} ∀x ∈ x. on the other hand, recall that an ordered topological space (x, τ, ≤) is said to be a t k1 −ordered space if, for any point x in x, we have c(x) = {x} (for more information see [13]). the following theorem characterizes ordered topological spaces whose t0−ordered reflections are t k1 −ordered. theorem 2.2. let (x, τ, ≤) be an ordered topological space. the following statements are equivalent: (i) the t0 − ordered reflection x/≈ of x is t k1 − ordered; (ii) for each x ∈ x and each monotone closed subset f of x such that f ∩ c(x) 6= ∅, we have x ∈ f ; (iii) for each monotone open subset o of x containing x, we have c(x) ⊆ o. © agt, upv, 2014 appl. gen. topol. 15, no. 1 45 s. lazaar and a. mhemdi proof. • (i) ⇒ (ii) suppose that x/≈ is t k1 − ordered. let f be a closed monotone subset of x such that f ∩ c (x) 6= ∅ and y ∈ f ∩ c (x) . clearly, qx(y) ∈ qx (f)∩c (qx(x)). thus, we can see that qx(x) = qx(y). now, since f is monotone and consequently a saturated subset of x, qx (x) = qx (y) ⊆ f. therefore, x ∈ f. • (ii) ⇒ (i) let y ∈ x be such that qx(y) ∈ c (qx(x)) . clearly c (y) ⊆ c (x) . conversely, since i (y) ∩ c (x) is non empty then by (ii) x ∈ i (y) and by the same way we say that x ∈ d (y) . therefore x ∈ c (y) and c (x) ⊆ c (y) . finally we can see that c (x) = c (y), so we have qx(x) = qx(y). • (ii) ⇒ (iii) let x ∈ x and o be a monotone open subset of x such that x ∈ o. if c (x) * o then by (ii) x /∈ o which is false. • (iii) ⇒ (ii) let f be a closed monotone subset of x such that f ∩ c (x) 6= ∅. if x /∈ f then x ∈ f c, since f c is a monotone open subset of x then by (iii) we obtain c (x) ⊆ f c which is false. � as an immediate consequence of theorem 2.2, we have the following corollary. corollary 2.3 ([2, theorem 3.5]). let (x, τ) be a topological space. then the following statements are equivalent: (i) x is a t(0,1)-space; (ii) for each open subset u of x and each x ∈ u, we have {x} ⊆ u; (iii) for each x ∈ x and each closed subset c of x such that {x} ∩ c 6= ∅, we have x ∈ c. proof. let (x, τ) be a topological space. it is enough to consider the ordered topological space (x, τ, =) in theorem 2.2. � now, let us introduce the following notation and definition: notation 2.4. let (x, τ, ≤) be an ordered topological space and z in x. we denote by: t (z) := {x ∈ x : c (x) = c (z)} . definition 2.5. let (x, τ, ≤) be an ordered topological space. defines on x the finite step preorder �(x,≤) related to ≤, by x �(x,≤) y if there exists z0, ..., zn and ∃ z ′ i, z ∗ i ∈ t (zi) (i = 0, 1, ..., n) such that z0 = x, zn = y and z ′ i ≤ z∗i+1 ∀ i = 0, 1, ..., n − 1. for short, we denote �(x,≤) also by �≤ . © agt, upv, 2014 appl. gen. topol. 15, no. 1 46 on some properties of t0−ordered reflection remarks 2.6. • it is clear that g (≤) ⊆ g (�≤) . • if x is a t0−ordered space, we have ≤=�≤. • for each x, y ∈ x, we have equivalence between x �≤ y and qx (x) ≤0 qx (y) . recall that an ordered topological space (x, τ, ≤) is said to be t2−ordered if there is an increasing neighborhood of x disjoint form some decreasing neighborhood of y whenever x � y, which is equivalent to the order ≤ being closed in (x, τ) × (x, τ) . now, we are in position to give the characterization of ordered topological spaces whose t0−ordered reflections are t2−ordered. theorem 2.7. let (x, τ, ≤) be an ordered topological space. then the following statements are equivalent: (i) the t0 − ordered reflection x/≈ of x is t2 − ordered; (ii) if x �(x,≤) y there exists an increasing neighborhood of x disjoint from some decreasing neighborhood of y; (iii) the graph g (�≤) of �≤ is closed in x × x. proof. • (i) =⇒ (ii) let x, y be two points in x such that x �≤ y. then qx (x) � 0 qx (y) . since x/≈ is t2−ordered, there exists an increasing neighborhood u of qx (x) disjoint from some decreasing neighborhood v of qx (y). now, we can see that q −1 x (u) is an increasing neighborhood of x disjoint from q−1x (v ), which is a decreasing neighborhood of y. • (ii) =⇒ (iii) let x, y ∈ x such that (x, y) /∈ g (�≤) which means that x �≤ y. then, there exists an increasing neighborhood u of x disjoint from some decreasing neighborhood v of y. clearly, we can see that u × v is a neighborhood of (x, y) and we have (u × v ) ∩ g (�≤) = ∅. therefore, g (�≤) is closed in x × x. • (iii) =⇒ (i) for this implication we can see that g (�≤) = q−1x × q −1 x ( g ( ≤0 )) . then , g ( ≤0 ) is closed and thus x/≈ is t2−ordered. � by the same way as in corollary 2.3, the following result holds immediately. corollary 2.8 ([2, theorem 3.12]). let (x, τ) be a topological space. then the following statements are equivalent: (i) x is a t(0,2)-space; (ii) for each x, y ∈ x such that {x} 6= {y}, there are two disjoints open sets u and v in x with x ∈ u and y ∈ v . © agt, upv, 2014 appl. gen. topol. 15, no. 1 47 s. lazaar and a. mhemdi 3. the class of morphisms in ordtop orthogonal to all t0−ordered spaces it is worth noting that reflective subcategories arise throughout mathematics, via several examples such as the free group and free ring functors in algebra, various compactification functors in topology, and completion functors in analysis: cf. [14, p. 90]. recall from [14, p. 89] that a subcategory d of a category c is called reflective (in c) if the inclusion functor i : d −→ c has a left adjoint functor f : c −→ d; i.e., if, for each object a of c, there exist an object f(a) of d and a morphism µa : a −→ f(a) in c such that, for each object x in d and each morphism f : a −→ x in c, there exists a unique morphism f̃ : f(a) −→ x in d such that f̃ ◦ µa = f. the concept of reflections in categories has been investigated by several authors (see for example [3], [4], [5], [6],[9], [11], [15]). this concept serves the purpose of unifying various constructions in mathematics. historically, the concept of reflections in categories seems to have its origin in the universal extension property of the stone-čech compactification of a tychonoff space. a morphism f : a −→ b and an object x in a category c are called orthogonal [7], if the mapping homc(f; x) : homc(b; x) −→ homc(a; x) that takes g to gf is bijective. for a class of morphisms σ (resp., a class of objects d), we denote by σ⊥ the class of objects orthogonal to every f in σ (resp., by d⊥ the class of morphisms orthogonal to all x in d) [7]. the orthogonality class of morphisms d⊥ associated with a reflective subcategory d of a category c satisfies the following identity d⊥⊥ = d [1, proposition 2.6]. thus, it is of interest to give explicitly the class d⊥. note also that, if i : d −→ c is the inclusion functor and f : c −→ d is a left adjoint functor of i, then the class d⊥ is the collection of all morphisms of c rendered invertible by the functor f ( i.e. d⊥ = {f ∈ homc : f (f) is an isomorphism of d})[1, proposition 2.3]. this section is devoted to the study of the orthogonal class ordtop⊥ 0 ; hence we will give a characterization of morphisms rendered invertible by the functor of the t0−ordered reflection. recall that a continuous map q : y → z is said to be a quasihomeomorphism if u → q−1(u) defines a bijection o(z) → o(y ) [8], where o(y ) is the set of all open subsets of the space y. then the following definition is more natural. definition 3.1. let f : (x, τ, ≤) −→ (y, γ, ⊑) be an increasing continuous map between two ordered topological spaces. f is said to be an ordered − quasihomeomorphism if u 7−→ f−1 (u) defines a bijection between the set of saturated open (resp. closed) sets of y and the set of saturated open (resp. closed) sets of x. © agt, upv, 2014 appl. gen. topol. 15, no. 1 48 on some properties of t0−ordered reflection examples 3.2. (1) qx : x −→ x/≈ is an ordered-quasihomeomorphism. (2) let q : (x, τ, ≤) −→ (y, γ, ⊑) be an increasing continuous map between two ordered topological spaces. if q̃ : (x, τ) −→ (y, γ) x 7−→ q (x) is a quasihomeomorphism then q is an ordered-quasihomeomorphism. the converse does not hold as shown in the following example: (3) let x = [0, 3] with the topology induced by the usual topology of r. define on x the order � by g (�) = {(a, b) : a, b ∈ q ∩ x and a ≤ b}∪ {(√ 5, x ) : x ∈ (q ∩ x) ∪ {√ 2, √ 5 }} ∪ {( x, √ 5 ) : x ∈ (q ∩ x) ∪ {√ 2, √ 5 }} ∪ △x. qx is an ordered-quasihomeomorphism which is not a quasihomeomorphism: ]0, 2[ is an open set, since it is not saturate then there is no an open subset v of x/≈ such that ]0, 2[ = q−1 x (v ). let us give an important property of ordered-quasihomeomorphisms. proposition 3.3. if f : x −→ y , g : y −→ z are continuous increasing maps between ordered topological spaces and two of the three maps f, g , g ◦ f are ordered − quasihomeomorphisms, then so is the third one. proof. • suppose that f and g are two ordered-quasihomeomorphisms. for any saturated closed subset u of x, let v be the unique saturated closed subset of y such that u = f−1 (v ) and let w the unique saturated closed subset in z such that v = g−1 (w) . it is clear that w is the unique saturated closed subset of z such that u = (g ◦ f)−1 (w) . we conclude that g ◦ f is an ordered-quasihomeomorphism. • suppose that g and g◦f are ordered-quasihomeomorphisms. let u be a saturated closed subset in x. since g◦f is an ordered-quasihomeomorphism, there exists a unique saturated closed subset w in z such that u = (g ◦ f)−1 (w) = f−1 ( g−1 (w) ) . now, v = g−1 (w) is a saturated closed subset of y satisfying u = f−1 (v ) . let us show that v is the unique saturated closed subset of y such that u = f−1 (v ) . for this, let v ′ be a saturated closed subset in y such that u = f−1 (v ′) . there exists a unique saturated closed subset w ′ in z such that v ′ = g−1 (w ′). so (g ◦ f)−1 (w) = u = f−1 (v ′) = f−1 ( g−1 (w ′) ) = (g ◦ f)−1 (w ′) . finally w = w ′ and consequently v = g−1 (w) = g−1 (w ′) = v ′. • suppose that f and g ◦ f are ordered-quasihomeomorphisms. if v is a saturated closed set in y,f−1 (v ) is a saturated closed set in x. then there exits a unique saturated closed set w in z such that (g ◦ f)−1 (w) = f−1 (v ). it is easy to show that w is the unique © agt, upv, 2014 appl. gen. topol. 15, no. 1 49 s. lazaar and a. mhemdi saturated closed set in z such that v = g−1 (w) . we conclude that f is an ordered-quasihomeomorphism. � now, let’us introduce the following definition: definition 3.4. let f : (x, τ, ≤) −→ (y, γ, ⊑) be an increasing continuous map between two ordered topological spaces. we say that f is strongly − increasing (for short s − increasing) if it satisfies : x �≤ y if and only if f (x) �(y,⊑) f (y) for all x, y ∈ x. examples 3.5. (1) let (x, τ, ≤) be an ordered topological space. then qx is a s-increasing map. (2) an increasing map need not to be s-increasing map. indeed, take (x, τ, ≤) of the example in 3.2 (3) and f the following map. f : [0, 3] −→ [0, 3] x 7−→ 0 clearly for any α ∈ [0, 3] \ ( q ∪ {√ 2, √ 5 }) we have f (α) �≤ f (0) but α �≤ 0. in order to give the main result of this section, we introduce the following definitions. definitions 3.6. let f : (x, τ, ≤) −→ (y, γ, ⊑) be an increasing continuous map. (1) f is said to be t − injective (or t − one − to − one) if, for each x, y in x : if there exists a monotone open subset of x which contains one of this points but not the other, then, the points f (x) , f(y) of y , can be separated by a monotone open subset of y. (2) f is said to be t − surjective (or t − onto) if, for each point y ∈ y, there exists x ∈ x such that we can not separate y and f (x) by a monotone open subset of y. (3) f is said to be t −bijective if it is both t −injective and t −surjective. examples 3.7. (1) every onto continuous increasing map is t-onto. (2) a t-onto map need not be onto as shown the following example : let x = {0, 1, 2} with the topology τx = {∅, x, {0, 2} , {1}} and the order ≤x defined by his graph g (≤x) = {(0, 0) , (0, 1) , (0, 2) , (1, 1) , (1, 2) , (2, 2)} . the map f : (x, τx, ≤x) −→ (x, τx, ≤x) such that f (x) = {0} is t-onto but not onto. (3) a t-one-to-one map need not be one-to-one : qx : (x, τx, ≤x) −→ ( x/≈, τx/≈, ≤0x ) is t-one-to-one but not one-to-one. (4) a one-to-one map need not be t-one-to-one : let τd the discrete topology on x. then the map f : (x, τd, ≤x) −→ (x, τx, ≤x) defined by f (x) = x for all x ∈ x is a one-to-one map but not t-one-to-one. © agt, upv, 2014 appl. gen. topol. 15, no. 1 50 on some properties of t0−ordered reflection before giving the main result of this section we need a lemma: lemma 3.8. let f : (x, τ, ≤) −→ (y, γ, ⊑) be an increasing continuous map. then the following properties hold: (1) f is t-injective if and only if ≈ f is injective. (2) f is t-surjective if and only if ≈ f is surjective. (3) f is t-bijective if and only if ≈ f is bijective. proof. (1) • suppose that ≈ f is injective : let x, y ∈ x. if we can separate x and y by a monotone open subset of x then qx (x) 6= qx (y) . since ≈ f is injective then ≈ f (qx (x)) 6= ≈ f (qx (y)) which means qy (f (x)) 6= qy (f (y)) . therefore, we can separate f (x) and f (y) by a monotone open subset of y . • conversely, suppose that f is t-injective : let x, y ∈ x be such that qx (x) 6= qx (y) which means that we can separate x and y by one monotone open subset of x. since f is t-injective we can separate f (x) and f (y) by one monotone open subset of y which means that qy (f (x)) 6= qy (f (y)) and then ≈ f (qx (x)) 6= ≈ f (qx (y)) . (2) • suppose that ≈ f is surjective : if y ∈ y , since ≈ f is a surjective map there exists x ∈ x such that ≈ f (qx (x)) = qy (y). thus, we have qy (f (x)) = qy (y) and we can not separate f (x) and y by a monotone open subset of y. • conversely, suppose that f is t-onto. if we can’t separate f (x) and y (x ∈ x, y ∈ y ) then we have ≈ f (qx (x)) = qy (y) , and we conclude that ≈ f is an onto map. (3) an immediate consequence of (1) and (2). � now, we are in a position to give the main result of this section. theorem 3.9. let f : (x, τ, ≤) −→ (y, γ, ⊑) be an increasing continuous map between two ordered topological spaces. then the following statements are equivalent: (1) ≈ f is an isomorphism; (2) f satisfies the following properties. (i) f is s-increasing. (ii) f is t-onto. (iii) f is ordered-quasihomeomorphism. © agt, upv, 2014 appl. gen. topol. 15, no. 1 51 s. lazaar and a. mhemdi (x, τ, �) f // qx �� (y, γ, ⊑) qy �� ( x/≈, τ/≈, ≤0 ) ≈ f // ( y/≈, γ/≈, ⊑0 ) proof. (1) ⇒ (2) • by lemma 3.8, f is t-onto . • f is s-increasing. since ≈ f is an isomorphism, then qx (x) ≤0 qx (y) if and only if ≈ f (qx (x)) ⊑0 ≈ f (qx (y)) which means that f (qy (x)) ⊑0 f (qy (y)) . now, by remarks 2.6, we can see that x �≤ y if and only if f (x) �⊑ f (y) . • by proposition 3.3 and example 3.2 it’s clear that f is an orderedquasihomeomorphism. (2) ⇒ (1) • according to lemma 3.8, ≈ f is a surjective map. • ≈ f is injective. by lemma 3.8, it is sufficient to show that f is t-one-to-one. to do this result, let x, y ∈ x and u an open monotone neighborhood of x such that y /∈ u. since f is an ordered-quasihomemorphism, there exists a saturated open subset v of y such that f−1 (v ) = u. let us show that v is monotone. without loss of generality we can suppose u increasing. let a, b ∈ y such that a ∈ v and a ⊑ b. since f is t-onto, there exists α ∈ u and β ∈ x such that t (f (α)) = t (a) and t (f (β)) = t (b) . now, we can see that f (α) �⊑ f (β) and thus α �≤ β. as u is increasing we have β ∈ u. therefore f (β) ∈ v and b ∈ v. • ≈ f −1 is increasing. let y, y′ ∈ y such that qy (y) ⊑0 qy (y′) . since f is t-onto there exist x, x′ ∈ x such that t (f (x)) = t (y) and t (f (x′)) = t (y′) . then, we have qy (f (x)) = qy (y) , we have also qy (y) = qy (f (x)) = ≈ f (qx (x)) so that ≈ f −1 (qy (y)) = qx (x) . by the same way, we have ≈ f −1 (qy (y ′)) = qx (x ′). since qy (y) ⊑0 qy (y′) we have qy (f (x)) ⊑0 qy (f (x ′)) which means that f (x) �⊑ f (x′) . now, according the fact © agt, upv, 2014 appl. gen. topol. 15, no. 1 52 on some properties of t0−ordered reflection that f is s-increasing, we have x �≤ x′ which is equivalent to qx (x) ≤0 qx (x ′) and finally ≈ f −1 (qy (y)) ≤0 ≈ f −1 (qy (y ′)) . • now, let we show that ≈ f is an open map. let u be an open set of x/≈. since q−1 x (u) is an open saturated subset of x and f is an ordered-quasihomeomorphism then there exist a saturated open subset v of y such that f−1 (v ) = q−1x (u) = f−1 ( q−1y ( ≈ f (u) )) . let us show that q−1y ( ≈ f (u) ) = v. let y ∈ v, since f is a t-onto map there exists x ∈ x such that t (f (x)) = t (y) . by saturation of v, f (x) ∈ v and consequently x ∈ f−1 (v ) = q−1x (u) = q −1 x ( ≈ f −1 (≈ f (u) )) = f−1 ( q−1y ( ≈ f (u) )) , and thus, f (x) ∈ q−1 y ( ≈ f (u) ) . now the saturation of q−1 y ( ≈ f (u) ) shows that y ∈ q−1y ( ≈ f (u) ) . we conclude that v ⊆ q−1y ( ≈ f (u) ) . the second inclusion is proved similarly. therefore ≈ f (u) is an open subset of y/≈. finally, ≈ f is a bijective open morphism with its inverse ≈ f −1 is increasing; so that ≈ f is an isomorphism. � we close this paper by giving an immediate consequence of this theorem. corollary 3.10 ([2, theorem 2.4]). let q : (x, τ) −→ (y, γ) be a continuous map. then the following statements are equivalent: (i) q is a topologically onto quasihomeomorphism; (ii) t0(q) is a homeomorphism. acknowledgements. the authors gratefully acknowledge helpful corrections, comments, and suggestions of the anonymous referee which reinforce and clarify the presentation of our paper. © agt, upv, 2014 appl. gen. topol. 15, no. 1 53 s. lazaar and a. mhemdi references [1] a. ayache, o. echi, the envelope of a subcategory in topology and group theory, int. j. math. math. sci. 21 (2005), 3787–3404. 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[15] w. tholen, reflective subcategories, topology appl. 27 (1987), 201–212. © agt, upv, 2014 appl. gen. topol. 15, no. 1 54 13.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 1 7 on topological sequence entropy of circlemapsjos�e s. c�anovas�abstract. we classify completely continuous circle maps fromthe point of view of topological sequence entropy. this improvesa result of roman hric.2000 ams classi�cation: 37b40, 37e10.keywords: topological sequence entropy, circle maps, chaos in the sense ofli{yorke. 1. introductionlet (x;d) be a compact metric space and let f : x ! x be a continuousmap. denote by c(x;x) the set of continuous maps f : x ! x. (x;f) iscalled a discrete dynamical system. the map f is said chaotic in the sense ofli{yorke (or simply chaotic) if there is an uncountable set s � x such thatfor any x;y 2 s, x 6= y, it holds that(1.1) lim infn!1 d(fn(x);fn(y)) = 0;and(1.2) lim supn!1 d(fn(x);fn(y)) > 0:s is said a scrambled set of f (see [10]). when f is chaotic we say that (x;f)is chaotic.the notion of chaos plays an special role in the setting of discrete dynamicalsystems. so, some topological invariants have been porposed to give a chara-terization of chaos. maybe, the most important topological invariant in thissetting is the topological entropy (see [1]). when x = [a;b], a;b 2 r, it iswell{known that positive topological entropy implies that f is chaotic, whilethe converse result is false (see [12]).so, in order to characterize chaotic interval maps we need an extension oftopological entropy called topological sequence entropy (see [7]). given an�this paper has been partially supported by the grant d.g.i.c.y.t. pb98-0374-c03-01. 2 jos�e s. c�anovasincreasing sequence of positive integers a = (ai)1i=0, a number ha(f) can beassociated to each f 2 c(x;x). this number is also a topological invariant.then, de�ning h1(f) = supa ha(f), chaotic interval maps can be characterizedby the following result.theorem 1.1. let f 2 c([a;b]; [a;b]). then(a) f is non{chaotic i� h1(f) = 0.(b) f is chaotic with zero topological entropy i� h1(f) = log 2.(c) f is chaotic with positive topological entropy i� h1(f) = 1.theorem 1.1 establishes a complete classi�cation of maps from the point ofview of topological sequence entropy. the part (a) was proved by franzov�aand sm��tal in [6]. (a) provides that any chaotic map holds h1(f) > 0. in [3]was proved (b) and (c) in case of piecewise monotonic maps. this result wasextended to the general case in [4].following [7], a map f 2 c(x;x) is said null if h1(f) = 0. f is said boundedif h1(f) < 1 and unbounded if h1(f) = 1. in the general case, it is unknownwhen a continuous map is null, bounded or unbounded. it is easy to see thatwhen f is stable in the lyapunov sense (f has equicontinuous powers) the mapis null (see [7]). theorem 1.1 establishes a characterization of null, boundedand unbounded continuous interval maps.the aim of this paper is to prove theorem 1.1 in the setting of continuouscircle maps. this will provide a classi�cation of unbounded, bounded and nullcontinuous circle maps. before starting with this classi�cation, let us point outthat for any f 2 c(s1;s1), hric proved in [8] that it is non{chaotic i� h1(f) =0, which classi�es chaotic circle maps from the point of view of topologicalsequence entropy. 2. preliminarieslet (x;d) be a compact metric space and let f : x ! x be a continuousmap. denote by c(x;x) the set of continuous maps f : x ! x. let f0 be theidentity on x, f1 := f and fn = f � fn�1 for all n � 1. consider an increasingsequence of positive integers a = (ai)1i=1 and let y � x and " > 0. we saythat a subset e � y is (a;";n;y;f){separated if for any x;y 2 e, x 6= y, thereis an i 2 f1;2; :::;ng such that d(fai(x);fai(y)) > ". denote by sn(a;";y;f)the cardinality of any maximal (a;";n;y;f){separated set. de�ne(2.3) s(a;";y;f) := lim supn!1 1n log sn(a;";y;f):it is clear from the de�nition that if y1 � y2 � x, then(2.4) s(a;";y1;f) � s(a;";y2;f):let(2.5) ha(f;y ) := lim"!0 s(a;";y;f):the topological sequence entropy of f respect to the sequence a is de�ned by(2.6) ha(f) := ha(f;x): on t. s. e. of circle maps 3when a = (i)1i=0, we receive the classical de�nition of topological entropy (see[1]).finally, let(2.7) h1(f;y ) := supa ha(f;y )and(2.8) h1(f) := supa ha(f):an x 2 x is periodic if there is an n 2 n such that fn(x) = x. the smallestpositive integer holding this condition is called the period of x. the set ofperiods of f, p(f), is de�ned byp(f) := fn 2 n : 9x 2 x periodic point of periodng:3. results on topological sequence entropyin this section we prove some useful results concerning topological sequenceentropy of continuous maps de�ned on arbitrary compact metric spaces.proposition 3.1. let f 2 c(x;x). for all n 2 n it holds that h1(fn) =h1(f).proof. first, we prove that h1(fn) � h1(f). in order to see this, let a =(ai)1i=1 be an increasing sequence of positive integers and de�ne na = (nai)1i=1.then it is straightforward to see that ha(fn) = hna(f) and henceh1(fn) = supa ha(fn) = supa hna(f)� supb hb(f) = h1(f):now, we prove the converse inequality. let a be an increasing sequence ofpostive integers. by [8], there is another sequence b = b(a) such that ha(f) �hb(fn). then h1(f) = supa ha(f) � supa hb(a)(fn)� supa ha(fn) = h1(fn);which ends the proof. �corollary 3.2. under the conditions of proposition 3.1, the following state-ments hold:(a) f is null i� fn is null for all n 2 n.(b) f is bounded i� fn is bounded for all n 2 n.(c) f is unbounded i� fn is unbounded for all n 2 n.proposition 3.3. let f 2 c(x;x) have positive topological entropy. thenh1(f) = 1. 4 jos�e s. c�anovasproof. since h(f) > 0 it follows by [7] that for any increasing sequence ofpositive integers a = (ai)1i=1, ha(f) = k(a)h(f), where(3.9) k(a) = limk!1 lim supn!1 1ncardfai;ai + 1; :::;ai + k : 1 � i � ng:taking a = (2i)1i=1 it holds that k(a) = 1 and hence ha(f) = 1. �proposition 3.4. let (x;d) and (y;e) be compact metric spaces and let f :x ! x and g : y ! y be continuous maps. let � : x ! y be continuous andsurjective such that � � f = g � �. let a be an increasing sequence of positiveintegers a and let y1 � y . then, for any " > 0 there is a � > 0 such that(3.10) s(a;�;��1(y1);f) � s(a;";y1;g):in particular, h1(f) � h1(g).proof. let e � y1 be a maximal subset (a;n;";y1;g){separated. let f ��1(y1) be a set containing exactly one element from ��1(y) for all y 2 e.we claim that f is an (a;n;�;��1(y1);f){separated subset for some � > 0.assume the contrary. since � is uniformly continuous, there is a � = �(") > 0such that d(x1;x2) < �, x1;x2 2 x, implies e(�(x1);�(x2)) < ". now letx1;x2 2 f be such that(3.11) d(fai(x1);fai(x2)) < �for all i 2 f1;2; :::;ng. let y1;y2 2 e be such that �(xj) = yj for j = 1;2.then, for all i 2 f1;2; :::;ng we have thate(gai (y1);gai(y2)) = e(gai (�(x1));gai(�(y2)))= e(� � fai(x1);� � fai(y2)) � ";which leads us to a contradiction. then sn(a;�;��1(y1);f) � sn(a;";y1;f)and hence s(a;�;��1(y1);f) = lim supn!1 1n log sn(a;�;��1(y1);f)� lim supn!1 1n log sn(a;";y1;g)= s(a;";y1;g);which ends the proof. �under the conditions of proposition 3.4, if � is an homemorphism, then fand g are said to be conjugate. thencorollary 3.5. under the conditions of proposition 3.4, if f and g are conju-gate, then h1(f) = h1(g). on t. s. e. of circle maps 54. main resultsin the sequel we will discuss the space of continuous circle maps denoted byc(s1;s1). let f 2 c(s1;s1) and let l : r ! s1 be de�ned by l(x) = exp(2�ix)for all x 2 r. then, there are a countable number of continuous maps f : r !r such that l � f = f � l. an f holding this condition is called a lifting of f.if ef is another lifting of f, then(4.12) ef � f = k 2 n:by jjj we denote the length of an interval j � r.theorem 4.1. let f 2 c(s1;s1). then(a) f is non{chaotic i� h1(f) = 0:(b) f is chaotic with zero topological entropy i� h1(f) = log 2:(c) f is chaotic with positive topological entropy i� h1(f) = 1:proof. according to chapter 3 from [2], c(s1;s1) can be decomposed into thefollowing classes:(4.13) c1 = ff 2 c(s1;s1) : f has no periodic pointsg;(4.14)c2 = ff 2 c(s1;s1) : p(fn) = f1g or p(fn) = f1;2;22; :::g for some n 2 ng;(4.15) c3 = ff 2 c(s1;s1) : p(fn) = n for some n 2 ng:according to [8], any f 2 c1 is non{chaotic and holds that h1(f) = 0. letf 2 c3. again by [8], it holds that f is chaotic and h(f) > 0. then, byproposition 3.3 we have that h1(f) = 1. so, we must consider only mapsfrom c2.let f 2 c2 and let n 2 n be such that p(fn) = f1g or p(fn) = f1;2;22; :::g.it is well{known that f is chaotic i� fn is chaotic. so, applying proposition 3.1,it is not restrictive to assume that n = 1. since f has a �xed point, by lemma2.5 from [9], there is a lifting f : r ! r and there is a compact interval j, withjjj > 1, such that f(j) = j. for the rest of the proof call l = ljj. first assumethat f is non{chaotic. then by [8] it holds that h1(f) = 0. secondly, assumethat f is chaotic. hence f is also chaotic (see [8]) and has zero topologicalentropy (see [12]). by proposition 3.4, for any " > 0 there is a � = �(") > 0such that(4.16) s(a;";s1;f) � s(a;�;l�1(s1);f) = s(a;�;j;f):on the other hand, by [4], there is a compact interval ji � j, holding thatf2i(ji) = ji such that(4.17) s(a;�;j;f) � s(a;�=6;[2ij=1fj(ji);f) � log 2:by (4.16) and (4.17) we conclude that(4.18) s(a;";s1;f) � log 2:since " was arbitrary chosen, we obtain(4.19) ha(f) � log 2; 6 jos�e s. c�anovasand(4.20) h1(f) = supa ha(f) � log 2:now, we prove the converse inequality. by [12], there is a compact intervalji, with jjij < 1 and f2i(ji) = ji such that f2ijji is chaotic. by [6], there isan increasing sequence of positive integers b such that s(b;";ji;f2i) � log 2for a suitable " > 0. since ljji : ji ! l(ji) is an homemorphism, we can applyproposition 3.4 to ljji = l to obtain a � > 0 such that(4.21) s(a;�;l(ji);f2i) � s(a;";ji;f2i) � log 2:hence(4.22) h1(f) � h2ia(f) = ha(f2i) � s(a;�;l(ji);f2i) � log 2;which concludes the proof. �remark 4.2. when two{dimensional maps are concerned, theorems 1.1 and4.1 are false in general. more precisely, in [11] and [5] a chaotic map f 2c([0;1]2; [0;1]2) with h1(f) = 0 and a non chaotic map g 2 c([0;1]2; [0;1]2)holding h1(g) > 0 have been constructed. it seems that the dimension of thespace x plays a special role in theorems 1.1 and 4.1. we conjecture thattheorem 1.1 remains true for continuous maps de�ned on �nite graphs, that is,in the special setting of one{dimensional dynamics.references[1] r. l. adler, a. g. konheim and m. h. mcandrew, topological entropy, trans. amer. math.soc. 114 (1965), 309{319.[2] ll. alsed�a, j. llibre and m. misiurewicz, combinatorial dynamics and entropy in dimensionone, world scienti�c (1993).[3] j. s. c�anovas, on topological sequence entropy of piecewise monotonic mappings, bull. austral.math. soc. 62 (2000), 21{28.[4] j. s. c�anovas, topological sequence entropy of interval maps, prepublicaci�on 6/00 del departa-mento de matem�atica aplicada y estad��stica. universidad polit�ecnica de cartagena.[5] g. l. forti, l. paganoni and j. sm��tal, strange triangular maps of the square, bull. austral.math. soc. 51 (1995), 395{415.[6] n. franzov�a and j. sm��tal, positive sequence topological entropy characterizes chaotic maps,proc. amer. math. soc. 112 (1991), 1083{1086.[7] t. n. t. goodman, topological sequence entropy, proc. london math. soc. 29 (1974), 331{350.[8] r. hric, topological sequence entropy for maps of the circle, comment. math. univ. carolin. 41(2000), 53{59.[9] m. kuchta, characterization of chaos for continuous maps of the circle, comment. math. univ.carolinae 31 (1990), 383{390.[10] t. y. li and j. a. yorke, period three implies chaos, amer. math. monthly 82 (1975), 985{992.[11] l. paganoni and p. santambrogio, chaos and sequence topological entropy for triangular maps,grazer. math. ber. 339 (1999), 279{290.[12] j. sm��tal, chaotic functions with zero topological entropy, trans. amer. math. soc. 297 (1986),269{282. received july 2000 on t. s. e. of circle maps 7revised version october 2000 jos�e s. c�anovasdep. de matem�atica aplicada y estad��sticauniversidad polit�ecnica de cartagena30203 cartagenaspaine-mail address: jose.canovas@upct.es @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 487–507 some properties of the containing spaces and saturated classes of spaces stavros iliadis dedicated to professor s. naimpally on the occasion of his 70th birthday. abstract. subjects of this paper are: (a) containing spaces constructed in [2] for an indexed collection s of subsets, (b) classes consisting of ordered pairs (q,x), where q is a subset of a space x, which are called classes of subsets, and (c) the notion of universality in such classes. we show that if t is a containing space constructed for an indexed collection s of spaces and for every x ∈ s, qx is a subset of x, then the corresponding containing space t|q constructed for the indexed collection q ≡{qx : x ∈ s} of spaces, under a simple condition, can be considered as a specific subset of t. we prove some “commutative” properties of these specific subsets. for classes of subsets we introduce the notion of a (properly) universal element and define the notion of a (complete) saturated class of subsets. such a class is “saturated” by (properly) universal elements. we prove that the intersection of (complete) saturated classes of subsets is also a (complete) saturated class. we consider the following classes of subsets: (a) ip(cl), (b) ip(op), and (c) ip(n.dense) consisting of all pairs (q,x) such that: (a) q is a closed subset of x, (b) q is an open subset of x, and (c) q is a never dense subset of x, respectively. we prove that the classes ip(cl) and ip(op) are complete saturated and the class ip(n.dense) is saturated. saturated classes of subsets are convenient to use for the construction of new saturated classes by the given ones. 2000 ams classification: 54b99, 54c25 keywords: containing space, universal space, saturated class of spaces, saturated class of subsets. 488 s. iliadis 1. introduction. agreement concerning notations. we denote by τ a fixed infinite cardinal. the set of all finite subsets of τ is denoted by f. by a space we mean a t0-space of weight less than or equal to τ. in this paper we use all notions and notation introduced in [2]. in particular, if an indexed collection of spaces is denoted by the letter “s”, a co-mark of s is denoted by the letter “m”, and an m-admissible family of equivalence relations on s is denoted by the letter “r”, then we always denote by “t” the containing space t(m, r) and by “bt” the standard base for t. if moreover m = {{uxδ : δ ∈ τ} : x ∈ s}, then the elements of bt are denoted by utδ (h), δ ∈ τ and h ∈ c ♦(r). as in [2] we shall be concerned with classes, sets, collections, and families. a class is not necessarily a set. a collection and a family are sets. any equivalence relation on a set s (which is considered as a subset of s×s satisfying the wellknown conditions) is denoted by the symbol ∼ supplied usually with one or more indices. any ordinal α is identified with the set of all ordinals less than α. for every set x we denote by p(x) the set of all subsets of x. for every subset q of a space x we denote by clx(q), intx(q), and bdx(q) the closure, the interior, and the boundary of q in x, respectively. we shall use the symbol “≡” in order to introduce new notations without mention to this fact. this will be done as follows. when we introduce an expression a as a notation of an object (a set, an indexed set, a mapping and so on) writing a ≡ b (or b ≡ a), where b is another new expression, or when we consider a known object with a known expression a as its notation writing a ≡ b (or b ≡ a), where b is a new expression, in both cases we mean that b is considered as another notation of the same object. in the second section we construct some specific subsets of the containing spaces given in [2]. suppose that for every space x of an indexed collection s of spaces a subset qx of x is given. then, the indexed collection q ≡{qx : x ∈ s} is called a restriction of s. such a restriction can be also treat as an indexed collection of spaces. any co-mark m of s defines by a natural manner a co-mark m|q of q and any m-admissible family r of equivalence relations on s defines an admissible family r|q of equivalence relations on q. for “almost all” co-marks m and families r, r|q is m|q-admissible. in this case, in parallels with the containing space t, we can also consider the containing space t(m|q, r|q) ≡ t|q for q corresponding to the co-mark m|q and the family r|q. we show that there exists a natural embedding of t|q into t, which gives us the possibility to consider t|q as a subset of t. in the third section some “commutative” properties of subsets t|q are given. a restriction q ≡{qx : x ∈ s} is called closed (open) if qx is closed (open) in x. also, the notion of a complete restriction of s is given. we show that some properties of the containing spaces and saturated classes of spaces 489 closed and open restrictions are complete. the following restrictions are also considered: cl(q) ≡{clx(qx) : x ∈ s}, int(q) ≡{intx(qx) : x ∈ s}, bd(q) ≡{bdx(qx) : x ∈ s}, and co(q) ≡{x \qx : x ∈ s}. for “almost all” co-marks m and families r of equivalence relations on s the following “commutation” relations are proved: t|cl(q) = clt(t|q), t|int(q) = intt(t|q), t|bd(q) = bdt(t|q), and t|co(q) = t \ t|q. the first relation is true for any restriction q and the others for complete restrictions. classes consisting of ordered pairs (q,x), where q is a subset of a space x, and called classes of subsets, are considered in the fourth section. an element (qt ,t ) of a class ip of subsets is called universal (respectively, properly universal) if for every element (qx,x) of ip there exists a homeomorphism h of x into t such that h(qx) ⊂ qt (respectively, h−1(qt ) = qx). using the above considered properties of subsets t|q of t we define the so-called (complete) saturated classes of subsets. this definition is similar to that of classes of spaces (see [2]). as for the classes of spaces the (complete) saturated classes of subsets not only have (properly) universal elements but they are “saturated” by such elements. we prove that the intersection of (complete) saturated classes is also a (complete) saturated class. we also show that the classes of subsets ip(cl) ≡{(q,x) : q is closed in x} and ip(op) ≡{(q,x) : q is open in x} are complete saturated classes and the class ip(n.dense) ≡{(q,x) : q is never dense in x} is saturated. for classes ip(cl) and ip(op) this follows by the corresponding “commutation” relations. in the fifth section we introduce the notion of a “commutative operator”. the closure, the interior, and the boundary operators are such operators. using 490 s. iliadis these operators we construct new (complete) saturated classes of subsets by the given ones. finally, in the last section we pose some problems. 2. specific subspaces of the space t(m, r). definition 2.1. let {v xδ : δ ∈ τ} be a mark of a space x. then, for every subspace q of x the indexed set {v qδ ≡ q∩v x δ : δ ∈ τ} is a mark of the space q. this mark is called the trace on q of the mark {v xδ : δ ∈ τ} of x. lemma 2.2. let {v xδ : δ ∈ τ} be the mark of a marked space x, q a subspace of x, and {v qδ : δ ∈ τ} the trace on q of the mark {v x δ : δ ∈ τ}. then, dxs (x) = d q s (x) for every x ∈ q and s ∈f \{∅}. therefore, dxs (q) = dqs (q). (we note that the mapping dxs is constructed with respect to the mark {v xδ : δ ∈ τ} of x and the mapping dqs is constructed with respect to the mark {v q δ : δ ∈ τ} of q). proof. let s ∈ f \ {∅}, x ∈ q and dxs (x) = f ∈ 2s. we must prove that dqs (x) = f. by the definition of the mapping d x s we have x ∈ x(s,f) = ∩{x(δ,f(δ)) : δ ∈ s}. therefore, x ∈ q∩ (∩{x(δ,f(δ)) : δ ∈ s}) = ∩{q∩x(δ,f(δ)) : δ ∈ s} = ∩{q(δ,f(δ)) : δ ∈ s} = q(s,f). this means that dqs (x) = f. � definition 2.3. suppose that for every element x of an indexed collection s of spaces a subspace qx of x is given. then, the indexed collection q ≡{qx : x ∈ s} is called a restriction of s. the element qx of q will be also denoted by q(x). some properties of the containing spaces and saturated classes of spaces 491 definition 2.4. let s be an indexed collection of spaces, q ≡{qx : x ∈ s} a restriction of s, m a co-mark of s, and let r ≡ {∼s: s ∈ f} be an indexed family of equivalence relations on s. the trace on q of the co-mark m of s is the co-mark of q denoted by m|q and defined as follows: the mark (m|q)(qx) of an element qx of q is the trace on qx of the mark m(x) of x. the trace on q of an equivalence relation ∼ on s is the equivalence relation on q denoted by ∼|q and defined as follows: two elements qx and qy of q are ∼|q-equivalent if and only if x ∼ y . the indexed family r|q ≡{∼s|q : s ∈f} of equivalence relations on q is called the trace on q of the indexed family r. the trace on q of an element h of c♦(r), denoted by h|q, is the set of all elements qx of q for which x ∈ h. it is easy to see that h|q is an element of c♦(r|q). obviously, if h ∈ c(r), then h|q ∈ c(r|q). by definition, it follows that if ∼0 and ∼1 are two equivalence relations on s and ∼1 is contained in ∼0, then ∼1|q is contained in ∼0|q. therefore, if r0 and r1 are two indexed families of equivalence relations on s and r1 is a final refinement of r0, then r1|q is a final refinement of r0|q. agreement 1. in what follows in this section it is supposed that an indexed collection of spaces denoted by s, a restriction of s denoted by q ≡{qx : x ∈ s}, a co-mark of s denoted by m ≡{{uxδ : δ ∈ τ} : x ∈ s}, and an m-admissible family of equivalence relations on s denoted by r ≡{∼s: s ∈f} are fixed. we note that, in general, the trace on q of the m-standard family of equivalence relations on s is not the m|q-standard family of equivalence relations on q. this justifies the following definition. definition 2.5. the m-admissible family r of equivalence relations on s is said to be (m, q)-admissible if r|q is an m|q-admissible family of equivalence relations on q. lemma 2.6. the family r is (m, q)-admissible if and only if for every element s of f \ {∅} there exists an element t of f \ {∅} such that relation x ∼t y implies relation dxs (qx) = dys (qy ) for every x,y ∈ s. 492 s. iliadis proof. suppose that the condition of the lemma is satisfied. we must prove that the family r is (m, q)-admissible. since r is m-admissible it suffices to prove that the family r|q = {∼s|q : s ∈ f} is a final refinement of the m|q-standard family of equivalence relations on q. denote the last family by rq ≡{∼sq: s ∈f}. let s and t be elements of f \{∅} satisfying the condition of the lemma. we need to prove that the trace on q of the equivalence relation ∼t on s, that is, the equivalence relation ∼t|q on q is contained in the equivalence relation ∼sq on q. let q x,qy ∈ q and qx ∼t|qqy . this means that x ∼t y . by the condition of the lemma, the last relation implies that dxs (q x) = dys (q y ). by lemma 2.2, dq x s (q x) = dq y s (q y ) and by lemma 1.4 of [2], qx ∼sq q y . thus, the equivalence relation ∼t |q is contained in the equivalence relation ∼sq. conversely, suppose that the family r is (m, q)-admissible. let s ∈f\{∅}. since the family r|q is m|q-admissible there exists an element t of f \{∅} such that the equivalence relation ∼t|q on q is contained in the equivalence relation ∼sq on q. let x,y ∈ s and x ∼t y , that is, qx ∼t |qqy . then, qx ∼sq q y . by lemma 1.4 of [2], dq x s (q x) = dq y s (q y ) and by lemma 2.2, dxs (q x) = dys (q y ). � remark 2.7. lemma 2.6 implies the existence of (m, q)-admissible families of equivalence relations on s. for example, such a family is the admissible family r0 ≡ {∼s0: s ∈ f} for which x ∼s0 y if and only if x ∼sm y and dxs (q x) = dys (q y ). (this family is admissible because the set p(2sx) = p(2 s y ) is finite). notation. suppose that r is an (m, q)-admissible family of equivalence relations on s. then, besides of the space t we can also consider the containing space t(m|q, r|q) for the indexed collection q corresponding to the co-mark m|q = {{u qx δ = q∩u x δ : δ ∈ τ} : q x ∈ q} and the m|q-admissible family r|q of equivalence relations on q. this containing space is also denoted by t|q. if h is an element of c♦(r), then we denote by t(h|q) the subset of t consisting of all points a for which there exists an element (x,x) of a such that x ∈ h and x ∈ qx. it is easy to verify that t(h|q) = t|q ∩ t(h). for every δ ∈ τ and e ∈ c♦(r|q) the set of all elements b of t|q for which there exists an element (x,qx) of b such that x ∈ uq x δ and q x ∈ e is denoted by ut|qδ (e). also, we set some properties of the containing spaces and saturated classes of spaces 493 bt|q = {ut|qδ (e) : δ ∈ τ, e ∈ c ♦(r|q)}. lemma 2.8. for every b ∈ t|q there exists a unique element a of t such that for every x ∈ qx the pair (x,qx) belongs to b if and only if the pair (x,x) belongs to a. proof. let b ∈ t|q. consider an element (y,qy ) of b and denote by a the element of t containing the pair (y,y ). now, let (x,qx) ∈ b. then, x ∼s|qy and dq x s (x) = d qy s (y) (2.1) for every s ∈f \{∅}. by lemma 2.2, dxs (x) = d qx s (x) and d y s (y) = d qy s (y). (2.2) therefore, x ∼s y and dxs (x) = dys (y) (2.3) for every s ∈ f \{∅}. this means that the pairs (x,x) and (y,y ) belong to the same element of the set t. thus, (x,x) ∈ a. conversely, let x ∈ qx and (x,x) ∈ a. then, relation (2.3) is true for every s ∈f\{∅}. lemma 2.2 implies relation (2.2). for every s ∈f\{0} relations (2.2) and (2.3) imply relation (2.1), which means that the pairs (x,qx) and (y,qy ) belong to the same element of the set t|q. thus, (x,qx) ∈ b. obviously, the element a ∈ t satisfying the condition of the lemma is uniquely determined. � definition 2.9. let b be an arbitrary element of t|q and a the unique element of t satisfying the condition of lemma 2.8. we define a mapping et|qt of t|q into t setting e t|q t (b) = a. below, we shall prove that this mapping is an embedding, which will be called the natural embedding of the space t|q into the space t. lemma 2.10. let δ ∈ τ, h ∈ c♦(r) and let l be the trace on q of h. then, an element b of t|q belongs to u t|q δ (l) if and only if the element a ≡ e t|q t (b) belongs to utδ (h). proof. for every x ∈ s the indexed set {uq x δ = q x ∩uxδ : δ ∈ τ} 494 s. iliadis is the trace on qx of the mark m(x). let b ∈ t|q and e t|q t (b) = a ∈ t. suppose that b ∈ ut|qδ (l) and let (x,q x) ∈ b. by lemma 2.8, (x,x) ∈ a. we have qx ∈ l and x ∈ uq x δ = q x ∩uxδ and, therefore, x ∈ h and x ∈ uxδ , which means that a ∈ utδ (h). similarly we prove that if a ∈ utδ (h), then b ∈ u t|q δ (l). � proposition 2.11. the mapping et|qt is an embedding of t|q into t. proof. first, we prove that the mapping et|qt is one-to-one. indeed, let b1 and b2 be two distinct elements of the space t|q. since t|q is a t0-space (see proposition 2.9 of [2]) there exists δ ∈ τ and l ∈ c♦(r|q) such that one of the points b1 and b2 belongs to the open set u t|q δ (l) and the other does not belong. let h ∈ c♦(r) such that l is the trace on q of h. by lemma 2.10 one of the points a1 ≡ e t|q t (b1) and a2 ≡ e t|q t (b2) belongs to the set utδ (h) and the other does not belong. this means that a1 6= a2, that is, e t|q t is one-to-one. the continuity of the mappings et|qt and (e t|q t ) −1 follows by lemma 2.10 and by the fact that every element of the base bt|q of the space t|q has the form ut|qδ (l) and every element of the base b t of the space t has the form utδ (h). thus, the mapping e t|q t is an embedding. � agreement 2. in what follows in this paper we identify a point b of t(m|q, r|q) with the point e t|q t (b) ≡ a of t(m, r) and consider the space t(m|q, r|q) as a subspace of the space t(m, r). the definitions of the natural embeddings ext and e t|q t imply the following consequence. corollary 2.12. the following relations is true: t|q = ∪{ext (q x) : x ∈ s}. proposition 2.13. let x ∈ s and let eq x x be the identical embedding of q x into x. then, e t|q t ◦e qx t|q = ext ◦e qx x . (2.4) some properties of the containing spaces and saturated classes of spaces 495 proof. let x ∈ qx. let also (ext ◦e qx x )(x) = e x t (x) = a. then, a is the point of t containing the pair (x,x). on the other hand, by the definition of the mapping eq x t|q , eq x t|q (x) ≡ b is the point of t|q containing the pair (x,qx). by the construction of the embedding et|qt we have e t|q t (b) = a. this proves relation (2.4). � 3. commutative properties of the subspaces t|q. agreement. in the present section it is supposed that s, m, q, and r are the same as in the preceding section and they are fixed. definition 3.1. the restriction q of s is said to be closed (respectively, open) if for every x ∈ s, qx is a closed (respectively, an open) subset of x. the restriction q is said to be an (m, r)-complete restriction if for every point a ∈ t|q and for every element (x,x) of a we have x ∈ qx. notation. below besides of the restriction q we also consider the following restrictions connected with q: cl(q) ≡{clx(qx) : x ∈ s}, bd(q) ≡{bdx(qx) : x ∈ s}, int(q) ≡{intx(qx) : x ∈ s}, and co(q) ≡{x \qx : x ∈ s}. lemma 3.2. suppose that q is a closed restriction of s and r is an (m, q)admissible family of equivalence relations on s. then, the following statements are true: (1) the subset t|q of t is closed. (2) if, moreover, r is (m, co(q))-admissible, then t|co(q) = t \ t|q. (3.1) (3) the restriction q is an (m, r)-complete restriction. proof. let a be a point of t for which there exists an element (x,x) of a such that x /∈ qx. since qx is closed in x there exists δ ∈ τ such that x ∈ uxδ and uxδ ∩q x = ∅. let s = {δ}. since r is (m, q)-admissible by lemma 2.6 there exists an element t of f such that ∼t⊂∼sm and d x s (q x) = dys (q y ) for every y ∈ s for which x ∼t y . let h be the ∼t-equivalence class of x. then, utδ (h) is an open neighbourhood of a in t. we prove that utδ (h) ∩ t|q = ∅. (3.2) indeed, in the opposite case, there exists a point b belonging to the set utδ (h)∩ t|q. let (y,qy ) be an element of b ∈ t|q. by lemma 2.8, b as a point of the 496 s. iliadis space t contains the pair (y,y ). since b ∈ utδ (h) we have y ∈ h and y ∈ u y δ . therefore, x ∼t y . by the choice of t, x ∼sm y and d x s (q x) = dys (q y ). since y ∈ qy there exists a point z ∈ qx such that dxs (z) = dys (y). this equality and the relation y ∈ uyδ imply that z ∈ u x δ . therefore, q x ∩uxδ 6= ∅, which contradicts to the choice of δ. thus, the relation (3.2) is proved. now we prove the statements of the lemma. (1). by corollary 2.12 as the above point a we can take any point of the set t \ t|q. in this case, relation (3.2) implies that the set t|q is closed. (2). as the point a we can take any point of the set t|co(q). in this case, relation (3.2) implies that a /∈ t|q, that is, t|q ∩t|co(q) = ∅. the corollary 2.12 implies that t|q ∪ t|co(q) = t. the last two relations are equivalent to the relation (3.1). (3). if q is not an (m, r)-complete restriction, then there exists a point a of t|q and an element (x,x) of a such that x /∈ qx. then, relation (3.2) is true for some open neighbourhood utδ (h) of a, which is a contradiction. thus, q is an (m, r)-complete restriction. � lemma 3.3. suppose that the restriction q of s is open and the family r of equivalence relations on s is (m, q)-admissible and (m, co(q))-admissible. then: (1) the following relation is true: t|co(q) = t \ t|q. (3.3) (2) the subset t|q of t is open. (3) the restriction q is an (m, r)-complete restriction. proof. (1). relation (3.3) follows immediately by the statement (2) of lemma 3.2 if instead of the restriction q of this lemma we consider the closed restriction co(q). (note that co(co(q)) = q). (2). since the restriction co(q) is closed by lemma 3.2 the subset t|co(q) of t is closed. therefore, by relation (3.3) the subset t|q of t is open. (3). let a ∈ t|q and (x,x) ∈ a. if x /∈ qx, then x ∈ x\qx. by corollary 2.12, a ∈ t|co(q), which contradicts to the relation (3.3). � proposition 3.4. suppose that the family r is (m, q)-admissible and (m, cl(q))-admissible. then, t|cl(q) = clt(t|q). proof. since the restriction cl(q) is closed by lemma 3.2 the subset t|cl(q) of t is closed. therefore, since t|q ⊂ t|cl(q), it suffices to prove that t|cl(q) ⊂ clt(t|q). let a ∈ t|cl(q). then, by corollary 2.12 there exists an element (x,x) of a such that x ∈ clx(qx). suppose that a /∈ clt(t|q). then, there exists a neighbourhood utδ (h) ∈ b t of a such that utδ (h) ∩ t|q = ∅. corollary 2.12 implies that uyδ ∩ q y = ∅ for every y ∈ h. since x ∈ h, uxδ ∩ q x = ∅. this means that x /∈ clx(qx), which is a contradiction. thus, a ∈ clt(t|q) and, therefore, t|cl(q) ⊂ clt(t|q). � some properties of the containing spaces and saturated classes of spaces 497 proposition 3.5. suppose that q is (m, r)-complete (in particular, by lemmas 3.2 and 3.3, q may be a closed or an open restriction of s) and the family r of equivalence relations on s is (m, q)-admissible, (m, int(q))-admissible, and (m, co(int(q)))-admissible. then, t|int(q) = intt(t|q). proof. obviously, the restriction int(q) is open. by assumptions of the lemma and lemma 3.3, the subset t|int(q) is open in t. therefore, since t|int(q) ⊂ t|q, it suffices to prove that intt(t|q) ⊂ t|int(q). let a ∈ intt(t|q). there exists an open neighbourhood utδ (h) ∈ b t of a such that utδ (h) ⊂ intt(t|q). let (x,x) ∈ a. then, x ∈ u x δ and x ∈ h. we prove that uxδ ⊂ q x. indeed, in the opposite case, there exists a point y belonging to the set uxδ \ q x. let b be the point of t containing the pair (y,x). then, b ∈ utδ (h). since q is an (m, r)-complete restriction, (y,x) ∈ b, and y /∈ qx we have b /∈ t|q, which contradicts of the fact that utδ (h) ⊂ intt(t|q) ⊂ t|q. thus, u x δ ⊂ q x, which means that x ∈ intx(qx) and, therefore, a ∈ t|int(q). � proposition 3.6. suppose that q is (m, r)-complete and r is: (a) (m, q)-admissible, (b) (m, cl(q))-admissible, (c) (m, int(q))-admissible, (d) (m, co(int(q)))-admissible, and (e) (m, bd(q))-admissible. then, t|bd(q) = bdt(t|q). (3.4) proof. obviously, bdt(t|q) = clt(t|q) \ intt(t|q). by proposition 3.4, t|cl(q) = clt(t|q) and by proposition 3.5, t|int(q) = intt(t|q). therefore, it suffices to prove that t|bd(q) = t|cl(q) \ t|int(q). let a ∈ t|bd(q). there exists an element (x,x) of a such that x ∈ bdx(qx). then, x ∈ clx(qx) and, therefore, a ∈ t|cl(q). on the other hand, since bd(q) is a closed restriction, by lemma 3.2, bd(q) is an (m, r)complete restriction. this means that for every (y,y ) ∈ a we have y ∈ 498 s. iliadis bdy (qy ), that is, y /∈ inty (qy ). by corollary 2.12, a /∈ t|int(q), that is, a ∈ t|cl(q) \ t|int(q). conversely, let a ∈ t|cl(q) \t|int(q). since cl(q) and int(q) are (m, r)complete restrictions, for every (x,x) ∈ a we have x ∈ clx(qx) \ intx(qx), that is, x ∈ bdx(qx). this means that a ∈ t|bd(q). thus, relation (3.4) is proved. � proposition 3.7. suppose that the restriction q of s is (m, r)-complete and the family r of equivalence relations on s is (m, q)-admissible and (m, co(q))admissible. then, co(q) is also (m, r)-complete restriction and t|co(q) = t \ t|q. (3.5) proof. let a ∈ t|co(q). there exists an element (x,x) of a such that x ∈ x \qx. let (y,y ) ∈ a. if y /∈ y \qy , then a ∈ t|q and since q is a (m, r)complete restriction we have x ∈ qx, which is a contradiction. therefore, y ∈ y \qy , which means that co(q) is a (m, r)-complete restriction. this also means that a /∈ t|q. by the above t|co(q) ⊂ t \ t|q. on the other hand t|co(q) ∪ t|q = t. the last two relations imply (3.5). � definition 3.8. suppose that for every element λ of a set λ, f(λ) ≡{fx(λ) : x ∈ s} is a restriction of s. the union (respectively, the intersection) of the restrictions f(λ) is the restriction {fx : x ∈ s} of s for which fx = ∪{fx(λ) : λ ∈ λ} (respectively, fx = ∩{fx(λ) : λ ∈ λ}) for every x ∈ s. these restrictions are also denoted by ∨{f(λ) : λ ∈ λ} and ∧{f(λ) : λ ∈ λ}, respectively. proposition 3.9. suppose that for every element λ of a set λ of cardinality ≤ τ an (m, r)-complete restriction f(λ) of s is given and let f be either the restriction ∧{f(λ) : λ ∈ λ} or the restriction ∨{f(λ) : λ ∈ λ}. if the family r is (m, f)-admissible and (m, f(λ))-admissible for every λ ∈ λ, then f is an (m, r)-complete restriction. some properties of the containing spaces and saturated classes of spaces 499 proof. suppose that f(λ) = {fx(λ) : x ∈ s}. let f = ∧{f(λ) : λ ∈ λ}≡{fx : x ∈ s}, a ∈ t|f ≡ t(m|f, r|f) and (x,x) ∈ a. there exists a pair (y,y ) ∈ a such that y ∈ fy . therefore, y ∈ fy (λ) for every λ ∈ λ. by corollary 2.12, a ∈ t|f(λ). since by assumption f(λ) is an (m, r)-complete restriction, x ∈ fx(λ) for every λ ∈ λ, which means that x ∈ fx. thus, f is also an (m, r)-complete restriction. the case where f = ∨{f(λ) : λ ∈ λ} is proved similarly. � 4. saturated classes of subsets. definition 4.1. in our considerations by a class of subsets we mean a class ip consisting of ordered pairs (q,x), where q is a subset of a space x. such a class is said to be topological if for every homeomorphism h of a space x onto a space y the condition (q,x) ∈ ip implies that (h(q),y ) ∈ ip. in what follows all considered classes of subsets are assumed to be topological. definition 4.2. let ip be a class of subsets. a restriction q of an indexed collection s of spaces is said to be a ip-restriction if (q(x),x) ∈ ip for every x ∈ s. (we recall that q = {q(x) : x ∈ s}). definition 4.3. a class ip of subsets is said to be saturated if for every indexed collection s of spaces and for every ip-restriction q of s there exists a co-mark m+ of s satisfying the following condition: for every co-extension m of m+ there exists an (m, q)-admissible family r+ of equivalence relations on s such that for every admissible family r of equivalence relations on s, which is a final refinement of r+, and for every elements h and l of c♦(r) for which h ⊂ l, we have (t(hq), t(l)) ∈ ip. the considered co-mark m+ is said to be an initial co-mark of s (corresponding to ip -restriction q) and the family r+ is said to be an initial family of s (corresponding to the co-mark m and ip -restriction q). the proof of the next proposition is similar to the proof of proposition 3.3 of [2]. (about the “intersection of classes” see the note to proposition 3.3 of [2]). proposition 4.4. the intersection of no more than τ many saturated classes of subsets is also a saturated class of subsets. 500 s. iliadis notation. we shall denote by: (a) ip(cl), (b) ip(op), and (c) ip(n.dense) the classes of subsets consisting of all ordered pairs (q,x) such that: (a) q is closed, (b) q is open, and (c) q is nowhere dense in x, respectively. proposition 4.5. the classes ip(cl), ip(op), and ip(n.dence) are saturated classes of subsets. proof. by lemma 3.2 it follows immediately that ip(cl) is a saturated class. (for every ip(cl)-restriction q of an indexed collection s of spaces as an initial co-mark m+ we can take any co-mark of s and for every co-extension m of m+ as an initial family we can take any (m, q)-admissible family of equivalence relations on s). similarly, lemma 3.3 implies that ip(op) is a saturated class. (in this case, for every ip(op)-restriction q of an indexed collection s of spaces, as an initial co-mark m+ we can take any co-mark of s and for every co-extension m of m+ as an initial family we can consider any (m, q)-admissible and (m, co(q))admissible family of equivalence relations on s). now, we prove that ip(n.dense) is a saturated class. consider an indexed collection s of spaces and let q ≡{qx : x ∈ s} be a ip(n.dense)-restriction of s. therefore, qx is a nowhere dense subset of x ∈ s. denote by m+ an arbitrary co-mark of s. we prove that m+ is an initial co-mark of s corresponding to the ip(n.dense)-restriction q. for this purpose we consider an arbitrary co-mark m ≡{{uxδ : δ ∈ τ} : x ∈ s} of s, which is a co-extension of m+, and let r+ be any (m, q)-admissible family of equivalence relations on s. we show that r+ is an initial family corresponding to the co-mark m and the ip(n.dense)-restriction q. indeed, let r ≡{∼s: s ∈f} be an admissible family of equivalence relations on s, which is a final refinement of r+, h, l ∈ c♦(r), and h ⊂ l. then, r is also (m, q)-admissible. in order to prove that r+ is an initial family (and, therefore, m+ is an initial co-mark) we need to prove that the subset t(h|q) of t(l) is nowhere dense. for this purpose it suffices to prove that t|q is a nowhere dense subset of t. let u be an open subset of t. without loss of generality, we can suppose that u has the form utδ (h) for some δ ∈ τ and some ∼ t-equivalence class h, t ∈ f. let x ∈ h. since qx is nowhere dense in x there exists an element some properties of the containing spaces and saturated classes of spaces 501 ε ∈ τ such that uxε ⊂ uxδ and u x ε ∩qx = ∅. let {δ,ε}∪ t = s. by lemma 2.6 there exists an element q of f such that ∼q⊂∼sm and d x s (q x) = dys (q y ) if x ∼q y . let e be the ∼q-equivalence class of x. in order to prove that the subset t|q of t is nowhere dense it suffices to prove that utε (e) ⊂ utδ (h) and u t ε (e) ∩ t|q = ∅. (4.1) let y be an arbitrary element of e. by the choice of q, x ∼sm y . by lemma 1.1 of [2], uyε ⊂ uyδ . therefore, u t ε (e) ⊂ utδ (e) ⊂ u t δ (h). we now prove that the set utε (e) ∩ t|q is empty. indeed, in the opposite case, there exists a point a ∈ t|q belonging to the set utε (e). by corollary 2.12 there exists a pair (y,y ) ∈ a such that y ∈ qy . then, y ∈ uyε and y ∈ e. by the choice of q, x ∼sm y and d x s (q x) = dys (q y ). therefore, there exists a point x ∈ qx such that dxs (x) = dys (y). this means that x ∈ uxε , that is, uxε ∩qx 6= ∅, which is a contradiction proving relation (3.1). thus, the set t|q is a nowhere dense subsets of t and, therefore, ip(n.dense) is a saturated class. � corollary 4.6. if ip is a saturated class of subsets, then the classes ip(cl) ∩ ip , ip(op) ∩ ip , and ip(n.dense) ∩ ip are also saturated classes. definition 4.7. a class ip of subsets is said to be closed (respectively, open) if for every element (q,x) ∈ ip the subset q of x is closed (respectively, open). a restriction q of an indexed collection s of spaces is said to be complete if there exists a co-mark m of s and an (m, q)-admissible family r of equivalence relations on s such that q is an (m, r)-complete restriction. a class ip of subsets is said to be complete if for every indexed collection s of spaces any ip-restriction q of s is complete. lemmas 3.2 and 3.3 imply that any closed or any open restriction of any indexed collection of spaces is complete. therefore, any closed or open class of subsets is complete. in particular, the classes ip(cl) and ip(op) are complete. lemma 4.8. let q be a restriction of a collection s of spaces, m0 a co-mark of s and r0 a family of equivalence relations on s such that q is (m0, r0)complete. then, for every co-extension m of m0 and (m, q)-admissible family r of equivalence relations on s, which is a final refinement of r0, q is an (m, r)-complete restriction. proof. let m be a co-extension of m0 and θ an indicial mapping of this coextension. let also r ≡ {∼s: s ∈ f} be an (m, q)-admissible family of equivalence relations on s, which is a final refinement of r0 ≡ {∼s0: s ∈ f}. consider a point a ∈ t|q and let (x,x) ∈ a. we must prove that x ∈ q(x). there exists an element (y,y ) of a such that y ∈ q(y ). let b be a point of t(m0, r0) such that (y,y ) ∈ b. since y ∈ q(y ), b ∈ t(m0|q, r0|q). we 502 s. iliadis prove that (x,x) ∈ b. indeed, since r is a final refinement of r0 and since x ∼s y for every s ∈f we have x ∼t0 y for every t ∈f. let t be an arbitrary element of f \{∅} and s = θ(t). for every z ∈ s denote by ãzt the t-algebra of z related to the mark m0(z) and by a z s the s-algebra of z related to the mark m(z). also denote by d̃zt the mapping of z into 2t constructed for the algebra ãzt and by d z s the corresponding mapping constructed for the algebra azs . it is easy to verify that if for some point z ∈ z, dzs (z) = f ∈ 2s, then d̃zt (z) = f ◦ θ|t ∈ 2t, where θ|t is the restriction of θ to t ⊂ τ. since (x,x), (y,y ) ∈ a we have dxs (x) = dys (y) for every s ∈ f. by the above, d̃xt (x) = d̃ y t (y) for every t ∈ f. therefore, the pairs (x,x) and (y,y ) belong to the same point of t(m0, r0). since (y,y ) ∈ b we have (x,x) ∈ b. since q is an (m0, r0)-complete restriction and y ∈ q(y ) we have x ∈ q(x), which proves the lemma. � definition 4.9. let ip be a class of subsets. an element (qt ,t ) ∈ ip is said to be universal (respectively, properly universal) in ip if for every element (qz,z) of ip there exists a homeomorphism h of z into t such that h(qz) ⊂ qt (respectively, h−1(qt ) = qz). proposition 4.10. in any non-empty complete saturated class of subsets there exist properly universal elements. proof. let ip be a non-empty complete saturated class of subsets. since ip is a topological class there exists an indexed collection s of spaces and a iprestriction q ≡ {qx : x ∈ s} of s such that for every element (qz,z) of ip there exists an element x of s and a homeomorphism f of z onto x for which f(qz) = qx. since ip is complete, q is a complete restriction of s. therefore, there exists a co-mark m0 of s and an (m0, q)-admissible family r0 of equivalence relations on s such that q is an (m0, r0)-complete restriction. on the other hand, since ip is a saturated class there exists an initial comark m+ of s corresponding to the ip-restriction q. let m be a co-mark of s, which is simultaneously a co-extension of m+ and m0. there exists a family r+ of equivalence relations on s, which is an initial family corresponding to the co-mark m and the ip-restriction q. denote by r an admissible family of equivalence relations on s, which is simultaneously a final refinement of r+ and r0. by lemma 4.8, q is an (m, r)-complete restriction. now, we consider the containing space t and its subset t|q. by construction, the ordered pair (t|q, t) is an element of ip. we prove that this element is properly universal in ip. indeed, let (qz,z) be an element of ip. there exists an element x ∈ s and a homeomorphism f of z onto x such that f(qz) = qx. let ext be the natural embedding of x into t and h = ext ◦f. then, h is a homeomorphism of z into t. we prove that h−1(t|q) = qz. it suffices to prove that some properties of the containing spaces and saturated classes of spaces 503 (ext ) −1(t|q) ⊂ qx. (4.2) let a ∈ t|q and x ∈ (ext ) −1(a). by the definition of the mapping ext , (x,x) ∈ a. since q is an (m, r)-complete restriction we have x ∈ qx proving relation (3.2). this completes the proof of the proposition. � similarly we can prove the following proposition. proposition 4.11. in any non-empty saturated class of subsets there exist universal elements. corollary 4.12. in the classes ip(cl), ip(op), and ip(cl) ∩ ip(n.dense) there exist properly universal elements. 5. commutative operators. definition 5.1. suppose that for every space x a mapping ox of the set p(x) into itself is given. then, the class of all such mappings is said to be an operator. such an operator is said to be topological if for every homeomorphism h of a space x onto a space y we have h(ox(q)) = oy (h(q)) for every q ∈p(x). in what follows, all considered operators are assumed to be topological. notation. the class of all spaces will be denoted by s. let o ≡{ox : x ∈s} be an operator. for every indexed collection s of spaces and for every restriction q ≡{qx : x ∈ s} of s we set o(q) = {ox(qx) : x ∈ s}. obviously, o(q) is a restriction of s. let ip be a class of subsets. denote by o(ip) and o−1(ip) the classes of subsets, which are defined as follows: o(ip) = {(ox(q),x) : (q,x) ∈ ip} and o−1(ip) = {(q,x) : (ox(q),x) ∈ ip}. it is clear that o(ip) and o−1(ip) are topological classes of subsets. below, the following operators will be considered: bd = {bdx : x ∈s}, 504 s. iliadis cl = {clx : x ∈s}, and int = {intx : x ∈s}, where bdx, clx, and intx are the boundary, the closure, and the interior operators in a space x, respectively. definition 5.2. let o ≡ {ox : x ∈ s} be an operator. this operator is said to be commutative with respect to a restriction q of an indexed collection s of spaces if there exists a co-mark m+ of s satisfying the following condition: for every co-extension m of m+ there exists an (m, q)-admissible and (m, o(q))-admissible family r+ of equivalence relations on s such that for every admissible family r of equivalence relations on s, which is a final refinement of r+, and for every elements h and l of c♦(r) for which h ⊂ l, we have ot(l)(t(h|q)) = t(h|o(q)). the considered co-mark m+ is said to be an o-commutative co-mark (corresponding to the restriction q) and the family r+ is said to be an o-commutative family (corresponding to the co-mark m and the restriction q). (we note that any co-extension of m+ is also an o-commutative co-mark and any admissible family of equivalence relation on s, which is a final refinement of r+, is also an o-commutative family). the operator o is said to be commutative with respect to a class ip of subsets if it is commutative with respect to any ip-restriction of any indexed collection of spaces. the operator o is said to be (completely) commutative if it is commutative with respect to any (complete) restriction of any indexed collection of spaces. it is clear that a (completely) commutative operator is commutative with respect to any (complete) class of subsets. lemmas 3.2 and 3.3 and propositions [3.4-3.6] imply the following two consequences. corollary 5.3. the operator cl is commutative. corollary 5.4. the operators bd and int are completely commutative. the proofs of the following two proposition are similar. we prove only the second. proposition 5.5. let ip be a saturated class of subsets and o an operator, which is commutative with respect to ip . then, o(ip) is a saturated class of subsets. some properties of the containing spaces and saturated classes of spaces 505 proposition 5.6. let ip and if be saturated classes of subsets and o an operator, which is commutative with respect to ip . then, ip ∩ o−1(if) is a saturated class of subsets. proof. suppose that o = {ox : x ∈ s}. let s be an indexed collection of spaces and q a (ip ∩o−1(if))-restriction of s. therefore, the restriction q is a ip-restriction, the restriction g ≡ o(q) of s is an if-restriction, and o is commutative with respect to q. since ip and if are saturated classes and o is commutative with respect to ip there exists a co-mark m+ of s, which is simultaneously an initial comark corresponding to the ip-restriction q, an initial co-mark corresponding to the if-restriction g, and an o-commutative co-mark corresponding to the restriction q. we prove that m+ is also an initial co-mark of s corresponding to the (ip ∩ o−1(if))-restriction q. consider a co-extension m of the co-mark m+. there exists a family r+ of equivalence relations on s, which is simultaneously an initial family corresponding to the co-mark m and the ip-restriction q, an initial family corresponding to the co-mark m and the if-restriction g, and an o-commutative family corresponding to the co-mark m and the restriction q. we show that this family is also an initial family corresponding to the co-mark m and the (ip ∩ o−1(if))-restriction q. indeed, let r be an admissible family of equivalence relations on s, which is a final refinement of r+, and h and l two elements of c♦(r) such that h ⊂ l. then, we can consider the space t(l) and its subsets t(h|g) and t(h|q). by construction, (t(h|q), t(l)) ∈ ip and (t(h|g), t(l)) ∈ if. since o is commutative with respect to ip we have t(h|g) = t(h|o(q)) = ot(l)(t(h|q)). this relation shows that (t(h|q), t(l)) ∈ ip ∩ o−1(if), which means that the family r+ is an initial family corresponding to the co-mark m and the (ip ∩ o−1(if))-restriction q and, therefore, the co-mark m+ is an initial comark of s corresponding to the (ip∩o−1(if))-restriction q. thus, ip∩o−1(if) is a saturated class. � corollary 5.7. if if is a saturated class of subsets, then cl(if) and cl−1(if) are also saturated classes of subsets. corollary 5.8. if ip is a complete saturated class of subsets, then bd(ip) and int(ip) are also complete saturated classes of subsets. corollary 5.9. if if is a saturated class of subsets and ip is a complete saturated class of subsets, then ip ∩bd−1(if) and ip ∩int−1(if) are also complete saturated classes of subsets. in the next proposition, which is easy to prove, we use the following definition of a saturated class of spaces. 506 s. iliadis definition 5.10. a class ip of spaces is said to be saturated if for every indexed collection s of elements of ip there exists a co-mark m+ of s satisfying the following condition: for every co-extension m of m+ there exists an m-admissible family r+ of equivalence relations on s such that for every admissible family r of equivalence relations on s, which is a final refinement of r+, and for every element l of c♦(r), the space t(l) belongs to ip. remark 5.11. the above definition of a saturated class of spaces is slightly different from that of [2]. in [2] instead of the space t(l) we consider only the space t. however, for the new notion of a saturated class of spaces all results of [2] concerning saturated classes of spaces are hold. proposition 5.12. the following statements are true: (1) if ip is a (complete) saturated class of subsets and ie is a saturated class of spaces, then the classes {(q,x) ∈ ip : x ∈ ie} and {(q,x) ∈ ip : q ∈ ie} are (complete) saturated classes of subsets. (2) if if is a saturated class of subsets, then the classes {x ∈s : (q,x) ∈ if for some subset q of x} and {q ∈s : (q,x) ∈ if for some space x ∈s} are saturated classes of spaces. (3) if ip and ie are saturated classes of spaces, then the class {(q,x) : q ∈ ip , x ∈ ie, and q ⊂ x} is a saturated class of subsets. 6. concluding remarks and some problems. 1. in [3] (see also [1]) for a given countable ordinal α “very simple examples of borel sets mα and aα (lying in the hilbert cube h) which are exactly of the multiplicative and additive class α respectively” are constructed. these sets satisfy the following property: “if x is a metric space and b ⊂ x is a borel set of the multiplicative (additive) class α in x, then there exists a continuous mapping ϕ of x into h such that ϕ−1(mα) = b (such that ϕ−1(aα) = b).” actually in [3] the class of subsets consisting of all pairs (b,x), where b is a borel set of multiplicative (additive) class α in a metric space x, is considered. the proving property of the elements (mα,h) and (aα,h) shows that these elements in some sense can be considered as a “properly universal element” in this class. by our method we can prove the following result: for a given countable ordinal α the class ipm (respectively, the class ipa) of all pairs (q,x), where q is a borel set of the multiplicative (respectively, some properties of the containing spaces and saturated classes of spaces 507 additive) class α in a separable metrizable space x, is a complete saturated class of subsets. therefore, in these classes there exist properly universal elements. in connection with the above we put the following problem. (1) are the elements (mα,h) and (aα,h) properly universal elements in the classes ipm and ipa, respectively? 2. the problem of the existence of (properly) universal elements for different classes of subsets is arisen. below, we set two questions concerning the class of all subsets, that is, the class of subsets consisting of all pairs (q,x), where q is a subset of a space x, whose answers it seems to be negative. (2) is there a properly universal element in the class of all subsets? (3) is the class of all subsets complete saturated? the second question is equivalent to the following: (3a) is any saturated class of subsets complete saturated? (obviously, the class of all subsets is saturated and, therefore, in this class there are universal elements). another general problem is the following: (4) construct (completely) commutative operators, which are distinct from that of section 5 and its compositions. (the composition of two operators o1 ≡{o1x : x ∈s} and o 2 ≡{o2x : x ∈s} is the operator o ≡{ox ≡ o1x ◦ o 2 x : x ∈s}). references [1] r. engelking, w. holsztyński and r. sikorski, some examples of borel sets, colloq. math. 15 (1966), 271–274. [2] s. d. iliadis, a construction of containing spaces, topology appl. 107 (2000), 97–116. [3] r. sikorski, some examples of borel sets, colloq. math. 5 (1958), 170–171. received january 2002 revised september 2002 stavros iliadis department of mathematics, university of patras, patras, greece. e-mail address : iliadis@math.upatras.gr 20.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 77 98 attractors of reaction-di�usion equations inbanach spacesjos�e valeroabstract. in this paper we prove �rst some abstract theoremson existence of global attractors for di�erential inclusions gener-ated by !-dissipative operators. then these results are appliedto reaction-di�usion equations in which the banach space lp isused as phase space. finally, new results concerning the fractaldimension of the global attractor in the space l2 are obtained.2000 ams classi�cation: 35b40, 35b41, 35k55, 35k57.keywords: attractor, asymptotic behaviour, reaction-di�usion equations,fractal dimension. 1. introductionthe theory of global attractors for partial di�erential equations have beenapplied to a wide range of equations of parabolic and reaction-di�usion types.it is very important in this theory to choose an appropriate phase space forthe system. the existence, uniqueness and properties of solutions of partialdi�erential equations depend strongly on the phase space.the more common phase space used for reaction-di�usion equations is thespace l2 ( ) (see [3], [4], [5], [6], [7], [13], [17], [18], [21], [25], [26], [28], [29],[31], [32], [33]).however, other spaces like l1 ( ) [1], h ( ) ; 0 < � 1 (see [2], [11], [12],[20] and their bibliography) or c ( ) (see [15]-[16], [27]) have been also usedsuccesfully.as far as we know, the space lp ( ), where 2 < p < 1; has not beenconsidered so far.in this paper we shall consider the existence of global attractors for di�eren-tial inclusions of the type � x0(t) 2 a(x(t));u(0) = u0;where a is a multivalued !-dissipative operator. 78 jos�e valerothis general framework allows us to apply the obtained results in order toprove the existence of a global attractor for the following reaction-di�usionequation 8<: @u@t � �u + f(u) 3 !u + h; on ]0;t [ � ;u = 0; on ]0;t [ � �;u(0;x) = u0(x) 2 lp ( ) ;where f : r ! 2r is a multivalued maximal monotone map and 2 � p < 1.therefore, we extend the results of attractors for reaction-di�usion equationsto the case where the phase space lp ( ) is used. moreover, the nonlinearfunction f, which is usually a continuous one, can be in our case multivalued.hence, our results are valid also for di�erential equations with discontinuousnonlinearities. we obtain also a result (see theorem 4.4), which is new in thecase where f 2 c1 and p = 2.in the case p = 2 we obtain some estimates of the fractal dimension of theglobal attractor. such estimates are well known under di�erent conditions onthe function f (see [3]-[5], [7], [18], [25]). in all these papers the function f isat least lipschitz on any bounded set of r. we shall extend these result byconsidering a function f (s) which is lipschitz on a �xed bounded set [�a;a]butcan be even discontinuous for s =2 [�a;a].we shall recall now some de�nitions of the theory of dynamical systems (see[20], [22]-[23], [29] for more details). let y be a complete metric space with themetric denoted by �(�; �), v : r+ � y ! y be a semigroup of operators, i.e.,the following properties holdv (t1;v (t2;x)) = v (t1 + t2;x); for all t1; t2 2 r+;x 2 y;v (0; �) = i:the map x 7�! v (t;x) is supposed to be continuous for each t 2 r+.let us introduce the next notation:b(y ) is the set of all bounded subsets of y ; +� (x) = [t��v (t;x), +� (a) = [x2a +� (x), where t;� 2 r+, x 2 y , a 2 b(y );!(b) = \t�0 +t (b) is the !-limit set.we denote by z the set of stationary points of v , i.e.,z = fu 2 y j v (t;u) = u; for all t 2 r+g:the continuous function l : y ! r is called a lyapunov function on y forv if l(v (t;x)) < l(x) for any t > 0, x 2 y , x =2 z.let us recall the concept of distance between sets. let a;b � y . then thedistance from a to b is determined as follows:d(a;b) = supx2a� infy2b �(x;y)� :de�nition 1.1. the compact set < � y is called a global attractor of thesemigroup v if the following conditions hold: attractors of reaction-di�usion equations 79(1) it attracts each set b 2 b(y ); that is,d(v (t;b);<) ! 0; as t ! +1:(2) it is invariant, i.e., v (t;<) = <, for all t � 0:de�nition 1.2. the semigroup v is called "pointwise dissipative" if there ex-ists a bounded set b1 which attracts each x 2 y .remark 1.3. we note that if for any b 2 b(y ) there exists some t = t(b) 2r+ for which +t (b) 2 b(y ), then pointwise dissipativeness is equivalent to theexistence of a bounded set b0 such that for each x 2 y there exists t (x) forwhich v (t;x) 2 b0. in such a case we can take b1 = +t(b0) (b0).de�nition 1.4. the semigroup v is said to be time-continuous if the mapt 7! v (t;x) is continuous for each x 2 y .theorem 1.5. ([23, p.107-109] and [19, p.4-5])let v (t; �) be compact for somet > 0 and let v be pointwise dissipative. suppose that for any b 2 b(y )there exists t = t(b) 2 r+ such that +t (b) 2 b(y ). then v has the globalattractor <. if v is time-continuous and y is connected, then < is connected.this paper is organized as follows. in the second section we prove the exis-tence of attractors for abstract di�erential inclusions in banach spaces generatedby !-dissipative operators. in the third section we apply the abstract resultsof the preceding section to di�erential inclusions generated by subdi�erentialmaps in hilbert spaces. in the fourth section we prove the existence of globalattractors for reaction-di�usion equations in lp ( ) spaces. finally, in the �fthsection we obtain estimates of the fractal dimension of the global attractor ofreaction-di�usion equations in the hilbert space l2 ( ) :2. existence of attractors of differential inclusions generatedby !-dissipative operatorslet x be a real banach space with its dual denoted by x�. we shall denoteby k�k and k�k� the norms in x and x� respectively. h�; �i will denote pairingbetween the spaces x and x�. consider the next di�erential inclusion(2.1) ( du(t)dt 2 a(u(t)) ; 0 � t < 1;u(0) = u0 2 x;where a : d (a) � x ! 2x is a multivalued nonlinear operator and u : r+ !x.we remind that the dual operator j : x ! 2x� (may be multivalued) isde�ned as follows:j (x) = nj 2 x� j kxk2 = kjk2� = hx; jio :consider the next conditions: 80 jos�e valero(a1) the operator a is !-dissipative, i.e., for all x1;x2 2 d (a) ; y1 2 a(x1),y2 2 a(x2) ; there exists j (xi;yi) 2 j (x1 � x2) such thathy1 � y2; ji � ! kx1 � x2k2 ;where ! � 0:when ! = 0 the operator a is called dissipative. it is easy to checkthat condition (a1) is equivalent to the dissipativeness of the operatora � !i.(a2) d (a) � \0<�<�0im(i � �a), where �0 > 0, �0! < 1:notice that d (a) is a complete metric space endowed with the usual metric�(x;y) = kx � yk ; for all x;y 2 d (a).if conditions a1 and a2 are satis�ed, then there exists a semigroup of oper-ators v : r+ � d (a) ! d (a) corresponding to the operator a (see [8, p.108]or [14, p.63]) such that v is determined by the next formula(2.2) v (t;x) = limn!1 �i � tna��n x; x 2 d (a); t 2 r+:moreover, for any �xed t 2 r+ one has(2.3) kv (t;x) � v (t;y)k � exp(!t)kx � yk ; for all x;y 2 d (a):it follows from inequality (2.3) that the map x 7�! v (t;x) is continuous foreach t 2 r+.a map u : [0;t ] ! x is called a strong solution of inclusion (2.1) on [0;t ]if: (1) u is continuous on [0;t ] and u(0) = u0;(2) u is absolutely continuous on any compact subset of (0;t) ;(3) u is almost everywhere (in short, a.e.) di�erentiable on (0;t) and sat-is�es inclusion (2.1) a.e. on (0;t).remark 2.1. if u(�);u(0) = u0, is a strong solution of (2.1) and (a1), (a2)hold, then u(t) = v (t;u0) [14, theorem 3.1], where v is the semigroup de�nedby (2.2).remark 2.2. suppose that the space x is re exive, (a1), (a2) hold and theoperator a is closed. then the map u(t) = v (t;u0) is a strong solution of (2.1)for any t > 0,u0 2 d (a) [14, p.77].we also must consider the next condition:(a3) im(i � �a) = x,for all � > 0.a dissipative operator which satis�es a3 is called m-dissipative. in such acase a2 is always satis�ed. moreover, every m-dissipative operator is closed [8,p.75].if a = b + !i with b m-dissipative, then a is closed and (a1), (a2) aresatis�ed. indeed,im(i � �a) = im(i � �b � �!i) = im((1 � �!) i � �b) = attractors of reaction-di�usion equations 81= im�i � �1 � �!b� = x;if 1 � �! > 0. hence, \0<�<�0im(i � �a) = x; if �0! < 1:in that case we have the inclusion:(2.4) du(t)dt 2 b (u(t)) + ! u(t) ; u(0) = u0:remark 2.3. it is easy to prove that if u(t) = v (t;x) is a strong solution of(2.1) for each x 2 d (a); then the set of stationary points z can be characterizedas follows: z = fu 2 d (a) j 0 2 a(u)g :it may be proved also that if a = b + !i, where b is m-dissipative, thenthis characterization is also true.first we shall recall the next result, which states the existence of a compactattractor for m-dissipative operators.theorem 2.4. [30, p.609] let a be an m-dissipative operator. suppose that thesemigroup v generated by a is compact for some t0 > 0 and that z is nonemptyand bounded. assume also that the next condition is satis�ed:(2.5) if x 2 d (a) and kv (t;x) � vk = kx � vk for all v 2 z, for all t � 0,then x 2 z.then z is the global attractor of v . moreover, the following conditions hold:(1) z is connected if d (a) is connected;(2) every positive trajectory fx(t) = v (t;x0), t 2 r+, x0 2 d(a)g con-verges to some element z 2 z, i.e., !(x) = z 2 z, for all x 2 d(a).remark 2.5. in fact, m-dissipativeness is too strong. the statement of thetheorem remains valid if (a1) � (a2) hold with ! = 0.remark 2.6. the proof of theorem 2.4 is based on the results from [23]. itis remarked in [19] that in the abstract result of that paper on the connectivityof the global attractor is necessary to supppose the semigroup v to be time-continuous. since the semigroup v is time-continuous, the statement of point(1) remains valid.to obtain suitable conditions on a and also to study the !-dissipative casewe need strong solutions of (2.1).further, we shall prove two results on existence of the global attractor forthe semigroup v:lemma 2.7. suppose that a satis�es (a1) � (a2) and that u(t) = v (t;u0) isa strong solution of (2.1) for any u0 2 d (a). suppose also that there existsu 2 z such that the next condition holds:(2.6)for all x 2 d (a) ; x =2 z; y 2 a(x) ; there exists j 2 j (x � u) for whichhy;ji < 0: 82 jos�e valerothen the function l(x) = kx � uk is a lyapunov function for v on d (a).proof. suppose that x =2 z. since the map x(t) = v (t;x) is continuous on t,there exists an interval [0;s) such that x(t) =2 z for any t 2 [0;s). multiplying(2.1) by j 2 j (x(t) � u) and using (2.6), we have�dx(t)dt ;j� < 0 a.e. t 2 (0;s) :by lemma 1.2 from [8, p.100] we have ddx(t)dt ;je = � ddt kx(t) � uk�kx(t) � uk.hence, � ddt kx(t) � uk�kx(t) � uk < 0 a.e. t 2 (0;s) :by integration we obtainkx(t) � uk < kx(0) � uk ; if t 2 (0;s) .it is easy to see that the last inequality holds for any t > 0. indeed, let x(s) 2 z.then x(t) = x(s) ; for all t � s; andkx(s) � uk = lim�!s kx(�) � uk = inf0<� 0, the map u(t) = v (t;u0) be a strong solution for all u0 2 d (a), z benonempty and bounded and (2.6) hold for some u 2 z. then v has the globalattractor <. it is connected if d (a) is connected.proof. it follows from lemma 2.7 that l(x) = kx � uk is a lyapunov functionfor v on d (a) and also that for some u 2 z one haskv (t;u0) � uk � ku0 � uk ; for all u0 2 b 2 b�d(a)� :then, it is obvious that +0 (b) 2 b�d(a)�.z is nonempty and bounded by assumption. it is well known from [23, the-orems 2.1 and 2.4] that if v is compact for some t > 0, there exists a lyapunovfunction and +0 (x) is a bounded set, then ! (x) is nonempty, compact andattracts x. moreover, ! (x) � z: it follows that z attracts any x 2 d (a), sothat v is pointwise dissipative.we conclude the proof by applying theorem 1.5. �corollary 2.9. under the conditions of theorem 2.8, < = z if we supposethat one of the next conditions holds:(1) (2.6) holds for any u 2 z;(2) a is m-dissipative.moreover, in the second case ! (x) = z 2 z, for all x 2 d (a). attractors of reaction-di�usion equations 83proof. let us prove the �rst statement. let u 2 <. we must prove that in thiscase u 2 z.recall that the function u(�) : r ! x is called a complete trajectory of vif u(t + t1) = v (t;u(t1)), for all t1 2 r, t 2 r+. this function will be called acomplete trajectory of the point x if u(0) = x.from the de�nition, every global attractor is the union of all bounded com-plete trajectories of v (for a proof see [22, p.10]). hence, the point u belongsto some bounded complete trajectory fu(t) ; t 2 rg. without loss of gener-ality we can put u(0) = u. from each sequence fu(tn)g, tn % +1 (thatis, an increasing sequence of times converging to +1) or tn & �1 (that is,a decreasing sequence of times converging to �1), belonging to a completebounded trajectory fu(t); t 2 rg, we can choose a subsequence converging tosome stationary point (see [23, theorem 2.4]). it follows that there exist twosequences fu(tn)g ; tn % +1, fu(tm)g ; tm & �1, such thatlimn!1u(tn) = u1 2 z; limm!1u(tm) = u2 2 z:let us prove by contradiction that u 2 z. suppose the opposite. then by (2.6)and using lemma 2.7 we haveku(tn) � u2k < ku � u2k < ku(tm) � u2k ; for all tn > 0; tm < 0:since limtm!�1 ku(tm) � u2k = 0, we obtain u = u1 = u2 2 z. thus, < � z,from which < = z.let us prove the second statement. since l(x) = kx � uk is a lyapunovfunction for some u 2 z, (2.5) is immediately satis�ed. thus, the secondstatement is a consequence of theorem 2.4. �remark 2.10. if a = b + !i, where b is m-dissipative (or satis�es (a1) and(a2) with ! = 0), (2.6) can be written in the next way: there exists u 2 z suchthat(2.7)for all x 2 d (a) ; x =2 z; y 2 b (x) ; there exists j 2 j (x � u) for whichhy;ji < �! hx;ji :in the preceding theorem we have used conditions providing the existence of acompact attractor. in the next one we shall consider another kind of conditionswithout using the set of stationary points z.theorem 2.11. let a satisfy (a1)-(a2), v be compact for some t0 > 0,u(t) = v (t;u0) be a strong solution for any u0 2 d(a) and the next conditionhold: there exist c > 0; � > 0 such that(2.8) for all u 2 d(a); kuk > c; y 2 a(u); there exists j 2 j(u) for which< y;j >� ��:then v has the global attractor <. it is connected if d (a) is connected.proof. first we shall prove that for all x 2 d(a) there exists t(x), for whichv (t;x) 2 b0, where b0 = nu 2 d (a) j kuk � c + �o ; 84 jos�e valerowith � > 0.let x =2 b0, so that kxk > c + �. suppose that u(t) = v (t;x) =2 b0, for allt � 0. then, using (2.8) and arguing as in the proof of lemma 2.7 we getku(t)k2 � ku(0)k2 � 2�t; for all t � 0:for t great enough u(t) 2 b0. the resulting contradiction proves that u(t) 2 b0for some t.further, let us prove the inclusion +0 (b) 2 b�d(a)�, if b 2 b�d(a)�,b � d(a). for this purpose it su�ces to show that if ku0k � m; u0 2d(a); m > c; then ku(t)k = kv (t;u0)k � m; for all t � 0. indeed, sup-pose thatku(t1)k > m; for some t1 > 0. since the map u(�) is continuouson [0;1) it is clear that there exists t2 < t1 such that ku(t2)k = m andku(t)k > m for t2 < t � t1. but in this case we obtain arguing as before thatku(t1)k2 � ku(t2)k2 � 2� (t2 � t1), which is a contradiction.it remains to consider the case where x 2 d(a)nd(a). let us show �rst thatv (t;x) 2 b0 for some t(x) � 0. let it not be so, i.e., v (t;x) =2 b0 ; for all t �0. since x 2 d(a), we can �nd a sequence fxkg � d(a) such that limk!1 xk =x. let t be �xed. then using (2.3) we obtain that there exists k0 such thatkv (t;xk)k � c + �=2; for all k � k0. since for any m > c and x 2 d(a)we have that kv (�;x)k � m; for all � � 0; if kxk � m, we deduce thatkv (�;xk)k � c + �=2; for all � 2 [0; t]; k � k0. then, using (2.8) again weget kv (t;xk)k2 � kv (�;xk)k2 � 2�(t � �); t > �; for all k � k0:when k ! 1 we obtainkv (t;x)k2 � kv (�;x)k2 � 2�(t � �); t > �:it is easy to see that choosing t great enough we have that v (t;x) 2 b0. hence,we obtain a contradiction.it remains to prove that +0 (b) 2 b�d(a)� ; for all b 2 b�d(a)�. weshall prove that v (t;bm) � bm; for all t � 0, wherebm = nv 2 d (a) j kvk � mo ; m > c:for any b � bm; b � d(a); we have already veri�ed the inclusion v (t;b) �bm; for all t � 0. suppose that there exist x 2 d(a)nd(a); x 2 bm; andt > 0 such that kv (t;x)k > m. choosing fxkg � d(a); kx � xkk !k!1 0; andusing (2.3), we have kv (t;x) � v (t;xk)k !k!1 0. then kv (t;xk)k > m; forall k � k0. the resulting contradiction shows that v (t;bm) � bm and then +0 (b) 2 b�d(a)� ; for all b 2 b�d(a)�.therefore, +0 (b) 2 b�d(a)� ; for all b 2 b�d(a)� ; and, in view ofremark 1.3, the semigroup v is pointwise dissipative . we conclude the proofby using theorem 1.5. �remark 2.12. in view of remark 2.2 if x is re exive and a is closed, thenu(t) = v (t;u0) is a strong solution of (2.1) for any u0 2 d(a). attractors of reaction-di�usion equations 85remark 2.13. it is easy to see that if there exist � > 0; m � 0 such that(2.9) for all u 2 d (a) ; y 2 a(u) ; there exists j 2 j (u) for whichhy;ji � ��kuk2 + m;then (2.8) holds.3. applications to inclusions generated by subdifferential mapslet h be a real hilbert space where is given the scalar product denoted by(�; �). we identify h with its dual h� and then the dual operator j is theidentity map i. we recall that the multivalued operator a : d(a) � h ! 2his called monotone if(y1 � y2;x1 � x2) � 0; for all x1;x2 2 d(a); y1 2 a(x1); y2 2 a(x2):a monotone operator is called maximal monotone if there does not exist anothermonotone operator b such that graph(a) � graph(b), wheregraph(a) = f(x;y) 2 h � h : y 2 a(x)g :remark 3.1. the operator a is maximal monotone if and only if �a is m-dissipative [8, p.71].consider the problem(3.10) � dudt 2 �@'(u(t)) + !u(t) + h;u(0) = u0;where ! � 0; @' : h ! h is the subdi�erential of a convex proper lowersemicontinuous function ' : h ! ]�1;+1] and h 2 h. it is well knownthat d(') = d(@') and also that @' is a maximal monotone operator [8,p.54]. then �@' is m-dissipative in view of remark 3.1 and �@' + h is alsom-dissipative.for any u0 2 d(@') and t > 0 there exists a unique strong solution of(3.10), u(�) 2 c([0;t ];h) (see [9, p.82] or [21, p.1399]). in view of remark 2.1we have u(t) = v (t;u0).let us recall the next well-known criterion of compacity of the semigroup v(see [21, p.1398]). we shall give the proof for the sake of completeness.lemma 3.2. the semigroup v (t; �) is compact (that is, the map v (t; �) iscompact for any t > 0) if the following property is satis�ed:(l) the level sets bc de�ned bybc = fu 2 d(') j kuk � c; '(u) � cgare compact in h for every c 2 r+.moreover, for any b 2 b�d(')� and t > 0 there is r > 0 such thatv (t;b) � br. 86 jos�e valeroproof. we must prove that for every b 2 b(d(')) and any t > 0 the setv (t;b) is precompact in h: let bn = fu 2 h j kuk < ng, n > 0. we take anarbitrary u0 2 bn, u0 2 d (') ; u(t) = v (t;u0). it follows from [9, p.82] thatif u0 2 d (') ; thent dudt 2 + t ddt'(u) = t�h + !u; dudt � a.e. on (0;t);and dudt 2 l2 (0;t ;h). therefore, it follows that '(u(t)) is absolutely continu-ous on [0;t ] [8, p.189]. hence, integrating by parts we havez t0 t dudt 2 dt + t'(u(t)) = z t0 t(h + !u; dudt )dt + z t0 '(u(t))dt;and then t'(u(t)) � 12 z t0 t dudt 2 dt + t'(u(t))(3.11) � 12 z t0 t(khk + ! kuk)2 dt + z t0 '(u(t))dt:on the other hand, without loss of generality we can assume that minf'(u) :u 2 hg = '(x0) = 0. indeed, let x0 2 d(@'), y0 2 @'(x0): if we introduce thenew function e'(u) = '(u) � '(x0) � (y0;u � x0), the inclusiondudt + @'(u) 3 h + !uis equivalent to dudt + @e'(u) 3 h � y0 + !u = eh + !uand minfe'(u) : u 2 hg = e'(x0) = 0. it is clear that e' satis�es (l).hence, since h + !u(t) � du(t)dt 2 @'(u(t)) a.e. on (0;t), we have'(u(t)) � (h + !u(t) � du(t)dt ;u(t) � x0):integrating over (0;t) we get z t0 '(u(t))dt� 12 ku(0) � x0k2 � 12 ku(t) � x0k2 + z t0 (khk + ! ku(t)k)ku(t) � x0kdt� 12 ku(0) � x0k2 + z t0 (khk + ! ku(t)k)ku(t) � x0kdt:let v0 2 d (') be �xed. obviously, there exists a constant c such that kv (t)k �c, for all t 2 [0;t ], where v (t) = v (t;v0). it follows from inequality (2.3) thatku(t) � v (t)k � exp(!t)ku0 � v0k , for 0 � t � t: attractors of reaction-di�usion equations 87therefore, there are constants d1; d2 (depending on t and n, but not onu0 2 bn) for which(3.12) ku(t)k � d1, for all t 2 [0;t ] ;(3.13) z t0 '(u(t))dt � d2 < 1:using (3.12)-(3.13) in relation (3.11) we obtain that for any t > 0 there existsk > 0 such that '(u(t)) � k.consider now that u0 2 d ('), u0 2 bn. we take a sequence fun0g � d (')such that un0 ! u0, as n ! 1. it is clear by inequality (2.3) that un (t) !u(t). since ' is lower semicontinuous, we get'(u(t)) � liminfn!1 '(un (t)) � k:then v (t;bn) � br;where r = maxfk;d1g. it follows from (l) that v (t;bn) is precompact inh: �thus, as a consequence of theorem 2.11 we obtain the following result, whichis a slight improvement of corollary 2.1 from [21].corollary 3.3. let ' satisfy (l) and let the following condition hold: thereexist �;c > 0 such that for all u 2 d(@'); kuk > c; for all y 2 �@'(u) onehas(3.14) (y;u) � �� ! kuk2 � (h;u) :then v has the global attractor <; which satis�es the next property of smooth-ness: there exists c 2 r+ such that < � bc.proof. it is straightforward to prove by using (3.14) that (2.8) holds. hence,by lemma 3.2 all conditions of theorem 2.11 hold. finally, since the attractor< is invariant, we have v (t;<) = <; for all t � 0. taking t > 0 we obtain bylemma 3.2 that < � bc for some c > 0, as claimed. �further, as a consequence of theorem 2.8 and lemma 3.2 we obtain:corollary 3.4. let (l) be satis�ed and let the following conditions hold:(1) there exists u 2 z such that for all x 2 d(@'); x =2 z; for all y 2@'(x), one has(3.15) (y;x � u) > ! (x;x � u) + (h;x � u) :(2) the set of stationary points z; i.e., the set of solutions of the inclusion(3.16) @'(u) 3 !u + h; u 2 h;is bounded.then v has the global attractor <. 88 jos�e valeroremark 3.5. it is possible to transform (3.15) as follows. by the de�nition ofthe subdi�erential map we have(y;x � u) � '(x) � '(u); for all y 2 @'(x):hence, (3.15) will be satis�ed if'(x) � '(u) > !(x;x � u) + (h;x � u); for all x =2 z:as a consequence of corollary 2.9 we have:corollary 3.6. let ! = 0; h = 0; and let (l) be satis�ed. if ' is coercive, i.e.,'(u) ! +1, as kuk ! +1, then < = z 6= ? is the global attractor of v .proof. since ' is coercive, we have [8, p.52](3.17) z = fx j 0 2 @'(x)g = �x 2 d(') j '(x) = miny2d(') '(y)� 6= ?:on the other hand, using again that ' is coercive it is easy to check that z isbounded. moreover, in view of remark 3.5 and since ! and h are equal to zero,in order to prove (3.15) it su�ces to satisfy the next condition: '(x) � '(u) >0; for all x =2 z; and some �xed u 2 z: this condition is satis�ed for any u 2 zin view of (3.17). it remains to apply lemma 3.2 and corollary 2.9. �remark 3.7. from the coerciveness of ' it follows that z is non-empty andbounded. we can also use these two conditions instead of coerciveness. on theother hand, in corollaries 3.4 and 3.6, < � bc for some c > 0.4. applications to reaction-diffusion equationsin the sequel wil be an open bounded subset of rn with su�ciently smoothboundary �. consider the boundary value problem:(4.18) 8><>: @u@t � �u + f(u) 3 !u + h; on ]0;t [ � ;u = 0; on ]0;t [ � �;u(0;x) = u0(x) on ;where ! � 0, f : d(f) � r ! 2r is a maximal monotone multivalued mapsuch that d(f) = r; h 2 lp( ) and 2 � p < 1.we shall use the banach space x = lp ( ) as phase space. de�ne theoperator bp : d (bp) ! 2x;bp (u) = f� 2 lp ( ) : � (x) 2 �u(x) � f (u(x)) ; a.e. on g ;d (bp) = nu 2 w2;p ( ) \ w1;p0 ( ) :9y 2 lp ( ) such that y (x) 2 f (u(x)) a.e. on g :the operator bp is m-dissipative (see [8, p.87]). hence, the operator ap (u)= bp (u) + !u + h, d (ap) = d (bp), satis�es a1{a2.recall that in lp ( ) the dual operator f : lp ( ) ! lq ( ), 1p + 1q = 1; issingle-valued and de�ned by f (u) = jujp�2ukukp�2lp : attractors of reaction-di�usion equations 89let us denote the semigroup generated by (4.18) in the complete metric spaced (bp) � lp ( ) by vp.proposition 4.1. let f satisfy the next condition: there exist m � 0; " > 0such that for all s 2 d(f); y 2 f(s) one has(4.19) y s � ! jsj2 � " jsj2 � m:then condition (2.8) holds.proof. for u 2 d(ap) and y 2 f(u); a.e. on ; y 2 lp( ), we obtain by using(4.19) and integrating over thatz (y(x) � !u(x)) ju(x)jp�2 u(x)kukp�2lp dx � "z ju(x)jpkukp�2lp dx � m z jujp�2kukp�2lp dx:then, using integration by parts, for any � 2 ap (u) we geth�;f (u)i == z �uju(x)jp�2 u(x)kukp�2lp dx �z (y(x) � !u(x) + h(x)) ju(x)jp�2 u(x)kukp�2lp dx� �(p � 1)z jru(x)j2 ju(x)jp�2kukp�2lp dx � "z ju(x)jpkukp�2lp dx+m z jujp�2kukp�2lp dx + z jh(x)j ju(x)jp�1kukp�2lp dx:further, h�older inequality impliesh�;f (u)i � �"kuk2lp + m j j2p kukp�2lpkukp�2lp + khklp kukp�1lpkukp�2lp= �"kuk2lp + m j j2p + khklp kuklp � �"2 kuk2lp + m j j2p + 12" khk2lp ;where j j denotes the lebesgue measure of . fix � > 0. then there existsc > 0 such that if kuklp > c we obtainh�;f (u)i � ��:thus, condition (2.8) holds. �remark 4.2. in the case p = 2 condition (4.19) can be weakened by putting" � �1 instead of ", where �1 is the �rst eigenvalue of �� in h10 ( ) :theorem 4.3. if condition (4.19) is satis�ed and(4.20) 2 � p � 2nn � 2; if n � 3;then the semigroup generated by (4.18) has the global attractor <, which iscompact in lp ( ) and bounded in h10 ( ) : if p = 2, then < is connected. 90 jos�e valeroproof. we have to prove �rst that the semigroup is compact. consider �rst thecase where p = 2. denote in this case the semigroup by v2.note that there exists a proper convex lower semicontinuous map j : r !]�1;+1] such that f = @j, where @j is the subdi�erential of j [8, p.60].let us determine the function ' : h ! ]�1;+1] ; h = l2( ); by'(u) = � 12 r jru(x)j2 dx + r j(u(x))dx; if u 2 d(');+1; otherwise,where d(') = fu 2 h10( ); j(u) 2 l1( )g. it is well known [8, p.88] thaty 2 @'(u) if and only if u 2 d(@'), y(x) 2 ��u(x) + f(u(x)) a.e. on , andy(�) 2 l2( ), where d(@') = fu 2 h2( ) \ h10( ) jthere exists v 2 l2( ) such that v(x) 2 f(x) a.e. on g;d(') = d(@') = l2( ):then b2 = @' and we have problem (3.10).we must prove that '(u) satis�es (l). indeed, since the function j(u) isbounded from below by an a�ne function [8, p.51], for any u 2 bc we havez (� + vu(x)) dx + 12 z jru(x)j2 dx � '(u) � c;where �;v 2 r. since the norms kukh10( ) and krukl2( ) are equivalent, thepreceding inequality implies that bc is bounded in h10( ). finally, we use thefact that the inclusion h10( ) � l2( ) is compact [10, p.169].now lemma 3.2 implies the compacity of v2 (t; �) for any t > 0. moreover,since for any bounded set b � l2 ( ) and t > 0 there exists c such thatv2 (t;b) � bc; the set v2 (t;b) is bounded in h10 ( ).further, let p > 2. it is easy to check using the unicity of solutions thatvp (t;u0) = v2 (t;u0), for any u0 2 d (bp); t � 0. hence, for any bounded setb � d (bp) (in the topology of lp ( )) and t > 0 we have that vp (t;b) =v2 (t;b) is bounded in h10 ( ). since in view of condition (4.20) the injectionh10 ( ) � lp ( ) is compact (see [10, p.169]), vp (t;b) is a precompact set.we note that for any u0 2 d (ap) there exists a unique strong solutionu(�) of inclusion (4.18) (see [8, p.146]), so that since in view of remark 2.1,u(t) = vp (t;u0), the existence of the global attractor follows from proposition4.1 and theorem 2.11.further, since the global attractor is bounded in lp ( ), we get that vp (t;<)is bounded in h10( ): therefore, we obtain that < is bounded in h10( ) by theequality vp (t;<) = <:finally, if p = 2 we have that d (b2) = d(@') = l2 ( ) : then since l2 ( )is connected, the global attractor is connected. �theorem 4.4. let p = 2; h � 0; 0 2 f(0) and let f satisfy the next condition(4.21) ys � (��1 + !) s2; for all s 2 r; y 2 f(s); attractors of reaction-di�usion equations 91where �1 is the �rst eigenvalue of �� in h10( ). moreover, there exists c > 0such that if jsj � c, then(4.22) ys > (��1 + !)s2; for all y 2 f(s):then the semigroup generated by (4.18) has the global connected attractor <,which is compact in l2 ( ) and bounded in h10 ( ).proof. we have seen in the proof of the previous theorem that b2 = @' and(l) holds. we must verify that (3.15)-(3.16) hold. it is clear that the functionv(x) � 0 is a stationary point of v2: let us check (3.15) for this point. it followsfrom (4.21) that for all u 2 d(@'); y 2 f(u) a.e. on , y 2 l2( ); one hasy(x)u(x) � (��1 + !)u2(x); a.e. on ;and so (y;u) = z y(x)u(x)dx � (��1 + !)kuk2l2 :therefore,(��u + y;u) � ! kuk2l2 ; for all u 2 d(@'); y 2 l2( );y 2 f(u) a.e. on :it remains to show that this inequality is strict when u =2 z. suppose that(��u + y;u) = ! kuk2l2 ; y(x) 2 f(u(x)) a.e. on , y 2 l2( ). this impliesthat y(x)u(x) = (��1 + !) u2(x) a.e. on , and u 2 spanfe1;e2; :::;emg, i.e.,it belongs to the space generated by the eigenfunctions corresponding to �1.indeed, if y(x)u(x) > (��1 + !) u2(x) on a set 1 � such that j 1j 6= 0, then(y;u) > (��1 + !)kuk2l2 :hence (��u + y;u) > ! kuk2l2, which is a contradiction. on the other hand,by using the equality y(x)u(x) = (��1 + !) u2(x); we have(��u;u) + (y � !;u) = (��u;u) � �1 kuk2l2 = 0:the last equality can hold only if u 2 spanfe1;e2; :::;emg. let us suppose thatit is not the case. then(��u;u) = 1xi=1 �i iei; 1xi=1 iei! = 1xi=1 �i 2i > �1 1xi=1 2i = �1 kuk2l2 ;and thus we obtain a contradiction. we must check that u 2 z. we take thepartition = 1 [ 2, where u(x) = 0 a.e. on 1, u(x) 6= 0 a.e. on 2, andde�ne the function �(x) = � 0; on 1;y(x); on 2:since 0 2 f(0), we have �(x) 2 f (u(x)) a.e. on . on the other hand �(x) =(��1 + !) u(x) a.e. on . since ��ek = �1ek on ; for all k = 1; :::;m [10,p.192], we get ��u(x) + �(x) = !u(x) a.e. on :thus, u 2 z and condition (3.15) holds. 92 jos�e valeronext, we must prove that z is bounded in l2( ). if u 2 z; then for somey 2 l2( ); y(x) 2 f(u(x)) a.e. on , we have(��u + y;u) = ! kuk2l2 ;and then, as we have already proved,y(x)u(x) = (��1 + !) u2(x) a.e. on :it follows from this equality that ju(x)j < c a.e. on , because by assumptiony(x)u(x) > �(�1 + !)u2(x) if ju(x)j � c. thus, z is bounded in l1( ) andconsequently in l2( ).the properties of the attractor may be proved in the same way as in theorem4.3. �remark 4.5. we note, as a particular case, that if ys > ��1s2, for all s 2rnf0g ; y 2 f(s);then u � 0 is the unique stationary point. it follows fromcorollary 2.9 that < = z = f0g:5. dimension of the global attractor of reaction-diffusionequations in the case p = 2we are now interested in the estimation of the fractal dimension of the globalattractor of (4.18) in the hilbert space l2 ( ) : such estimates are well knownin the case of a di�erentiable function f (see [3]-[5], [25]). for non-di�erentiablemaps a similar result was obtained in [7] supposing that f is lipschitz and in[18] in the case where f 2 w1;1loc (r). in all these papers the function f is atleast lipschitz on any bounded set of r.we shall extend these results by considering a function f (s) which is lips-chitz on a �xed bounded set [�a;a] but can be even discontinuous for s =2 [�a;a].recall that the fractal dimension of a compact seta is de�ned bydf(a) = inffd > 0 j �f(a;d) = 0g;where �f(a;d) = lim�!0 �dn�;and n� is the minimum number of balls of radius r � � which is necessary tocover a.first we have to obtain an estimate of the elements of the global attractorin the norm of the space l1 ( ) : for this goal we need to impose a dissipativecondition which is stronger than (4.19).proposition 5.1. let us assume that there exist " > 0;r > 2;m � 0 such that(5.23) ys � !s2 � " jsjr � m; for any s 2 r;y 2 f (s) :let k � 0, u0 2 d (ak+r) and h 2 l1 ( ). then u(t) = v2 (t;u0) satis�es:(5.24) ku(t)klk+2� j j 1k+2 �2 1k+r ��4m" �1r + �4khkl1" � 1r�1� + �"2 (r � 2) t�� 1r�2� ;for all t > 0. attractors of reaction-di�usion equations 93proof. let k � 0 be arbitrary. we note that since u0 2 d (ak+r) ; we havethat v2 (t;u0) = vk+r (t;u0) ; so that u(�) 2 c ([0;t ] ;lk+r ( )) for any t > 0.denote v (t) = ku(t)kk+2lk+2. due to the regularity of u0 we have that u(�) isa strong solution of (4.18), u 2 w1;1 (0;t ;lk+r ( )) ; for any t > 0; andu(t) 2 w1;k+r0 ( ) \ w2;k+r ( ), for any t 2 [0;t ] (see [8, p.146]). it followsfrom [8, lemma 1.2, p.100] that 1k+2 ddt kukk+2lk+2 = r dudt jujk udx. multiplying(4.18) by jujk u and using the green formula and (5.23) we obtain1k+2 ddt kukk+2lk+2 + (k + 1)r jruj2 jujk dx + "r jujk+r dx� m r jujk dx + r h jujk udx:now the h�older inequalitieskukklk � kukklk+2 j j 2k+2 ; kukk+rlk+2 � kukk+rlk+r j jr�2k+2imply1k + 2 ddt kukk+2lk+2 + " j j2�rk+2 kukk+rlk+2 � m kukklk+2 j j 2k+2 + z h jujk udx:further young inequality with exponent q = k+rk and coe�cientsa = "4 j j� rk+2 ;ca = �"4 j j� rk+2��krgives(5.25)1k + 2 ddt kukk+2lk+2 + 3"4 j j2�rk+2 kukk+rlk+2 � j j�"4��kr m k+rr + z h jujk udx:using the h�older inequalitiesz h jujk udx � khkl1 kukk+1lk+1 ; kukk+1lk+1 � kukk+1lk+2 j j 1k+2we have 1k+2 ddt ku(t)kk+2lk+2 + 3"4 j j2�rk+2 kukk+rlk+2� j j�"4��kr m k+rr + r h jujk udx� j j�"4��kr m k+rr + khkl1 kukk+1lk+1� j j�"4��kr m k+rr + khkl1 kukk+1lk+2 j j 1k+2 :now applying the young inequality with exponent q = k+rk+1 and coe�cientsa = "4 j j1�rk+2 , ca = �"4 j j1�rk+2�k+11�r to the last term of the inequality we get1k+2 ddt kukk+2lk+2 + "2 kukk+rlk+2 j j2�rk+2� j j��"4��kr m k+rr + �"4��k+1r�1 khkk+rr�1l1 � :hence,(5.26) ddtv (t) + (v (t))q � �; 94 jos�e valerowhere � = (k + 2) j j��"4��kr m k+rr + �"4��k+1r�1 khkk+rr�1l1 �, = "2 (k + 2) j j2�rk+2 .q = k+rk+2.further we shall use the following version of the gronwall lemma [29, chapteriii, p.163]:lemma 5.2. let y (t) � 0 be absolutely continuous on (0;1). suppose thatthere exist q > 1; > 0; � � 0 such that(5.27) ddty (t) + yq (t) � �:then(5.28) y (t) � �� 1q + ( (q � 1) t)� 1q�1 ; for all t > 0:thus, lemma 5.2 impliesku(t)klk+2 � � j jk+rk+22 ��"4��k+rr m k+rr + �"4��k+rr�1 khkk+rr�1l1 �� 1k+r+�"2 j j2�rk+2 (r � 2) t�� 1r�2� j j 1k+2 � 12 1k+r ��4m" �1r + �4khkl1" � 1r�1� + �"2 (r � 2) t�� 1r�2� ;for all t > 0. �corollary 5.3. let u0 2 c10 ( ) ; h 2 l1 ( ) and let (5.23) hold. then forany � > 0 we have(5.29) kukl1(�;1;l1( )) � �4m" �1r + �4khkl1" � 1r�1 + 1�(r � 2) � "2� 1r�2 :proof. first we shall show that u0 2 d (ap) for any p � 2. we take an arbitrarysingle-valued function g (s) such that g (s) 2 f (s), for all s 2 r. since f (s)is maximal monotone, g (s) is non-decreasing. any non-decreasing function ismeasurable, so that the composition l (x) = g (u0 (x)) is a measurable selectionof f (u0 (x)). let us check that l (x) 2 l1 ( ). since u0 2 c10 ( ), there existsb > 0 such that ju0 (x)j � b, for all x 2 . hence,g (�b) � g (u0 (x)) � g (b) ; for all x 2 ;so that l (x) 2 l1 ( ). it follows that l (x) 2 lp ( ), for any p � 2; and thatl (x) 2 f (u0 (x)), a.e. on . therefore, u0 2 d (ap), for all p � 2:proposition 5.1 implies that (5.24) is satis�ed for any k � 0. passing to thelimit as k ! 1 we obtainku(t)kl1 � �4m" �1r + �4khkl1" � 1r�1 + 1�(r � 2) t"2� 1r�2 ; for all t > 0; attractors of reaction-di�usion equations 95and then for any � > 0 we getkukl1(�;t;l1( )) � �4m" �1r + �4khkl1" � 1r�1 + 1�(r � 2) � "2� 1r�2 : �corollary 5.4. let u0 2 l2 ( ) ; h 2 l1 ( ) and let (5.23) hold. then in-equality (5.29) holds:proof. let un0 2 c10 ( ) be a sequence such that un0 ! u0 in l2 ( ). then in-equality (2.3) implies that un (t) = v2 (t;un0) converges to u(t) = v2 (t;u0)in c ([0;t ] ;l2 ( )). the sequence un is bounded in l1 (�;t ;l1 ( )) by(5.29). hence, there exists a subsequence converging to u weakly star inl1 (�;t ;l1 ( )). therefore, (5.29) holds. �now we can obtain an estimate of the elements of the global attractor <(which exists in view of theorem 4.3) in the norm of the space l1 ( ) :theorem 5.5. let h 2 l1 ( ) and let (5.23) hold. then for any y 2 < thefollowing estimate holds:(5.30) kykl1( ) � �4m" �1r + �4khkl1" � 1r�1 :proof. let � > 0, y 2 < be arbitrary. we choose � such that�(r � 2) �"2�� 1r�2 < �:since < is invariant, there exists u0 2 < for which y = u(2�) = v2 (2�;u0).corollary 5.4 implieskukl1(�;3�;l1( )) � �4m" �1r + �4khkl1" � 1r�1 + �:since u 2 c (�;3�;l2 ( )) and it is bounded in l1 (�;3�;l1 ( )), we haveu 2 cw (�;3�;lq ( )) ; for any2 � q < 1, where cw denotes the weak topology(see [24, p.275]). hence,kyklq( ) = ku(2�)klq( ) � �4m" �1r + �4khkl1" � 1r�1 + �! j j1q ;so that kykl1( ) = ku(2�)kl1( ) � �4m" �1r + �4khkl1" � 1r�1 + �:since � is arbitrary, (5.30) holds. �theorem 5.6. let h 2 l1 ( ) and let (5.23) hold. suppose that there existsa > 0 such that(5.31) �4m" �1r + �4khkl1" � 1r�1 � a 96 jos�e valeroand in the interval [�a;a] the function f (s) is lipschitz (with lipschitz constant�).then there exists k depending on and n for which(5.32) df (<) � k (! + �)n2 :proof. let f�ng1n=1 be the eigenvalues of �� in h10 ( ) ; pn be the ortho-projector to the subspace generated by the eigenfunctions corresponding to the�rst n eigenvalues and qn = i � pn. it follows from [7, theorem 4] that ifwe �nd t > 0, l 2 [1;+1); � 2 (0; 1p2) such that(5.33) kv2(t;u0) � v2(t;v0)k � l ku0 � v0k ;(5.34) qnv2(t;u0) � qnv2(t;v0) � � ku0 � v0k ;for all u0;v0 2 <, then for any � > 0 such that �p26l�n �p2�� = � < 1 thenext estimate holds(5.35) df(a) � n + �:in view of inequality (2.3) condition (5.33) holds with l (t) = exp(!t) :further we note that from theorem 5.5 and the lipschitz condition of f on[�a;a] it follows that(5.36) kf (u) � f (v)kl2 � � ku � vkl2 , for any u;v 2 <:we take two arbitrary initial conditions u0;v0 2 <. let now w (t) = u(t) �v (t) ; wn (t) = qnw (t) ; where u(t) = v2 (t;u0) ; v (t) = v2 (t;v0). from (4.18)we can easily obtain12 ddt wn (t) 2l2 + rwn 2l2 + �f (u) � f (v) ;wn� = !�w;wn� :using the inequality rwn 2l2 � �n+1 wn (t) 2l2 ; (2.3) and (5.36) we obtainddt wn (t) 2l2 + 2�n+1 wn (t) 2l2 � 2(! + �)exp(2!t)kw (0)k2l2 :multiplying both sides by exp(2�n+1t) and integrating over (0; t) we get wn (t) 2l2� kw (0)k2l2 �exp(�2�n+1t) + !+�!+�n+1 (exp (2!t) � exp (�2�n+1t))�� kw (0)k2l2 �exp(�2�n+1t) + !+�!+�n+1 exp(2!t)�= �2 (t;n)kw (0)k2l2 :repeating exactly the same proof of theorem 7 from [7] we can obtain thatfor t = log(p212)�n+1�! there exists a constant d (depending on ) such that ifn = h(d (! + �))n2 i, where [x] denotes the integer part of x, then � (t;n) < 1p2and (12� (t;n) l (t))n < 1, so that �p26l�n �p2�� < 1 for � = n. it followsfrom (5.35) that df(a) � 2n � k (! + �)n2 ; attractors of reaction-di�usion equations 97with k = 2dn2 : �remark 5.7. we note that if f is locally lipschitz, then it is lipschitz onany interval [�b;b]. we allow the function f (s) to be not lipschitz (and evendiscontinuous) for s =2 [�a;a] : references[1] 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no. 1(2015), 37-44doi:10.4995/agt.2015.2305 c© agt, upv, 2015 nonself kkm maps and corresponding theorems in hadamard manifolds parin chaipunya and poom kumam ∗ department of mathematics, faculty of science, king mongkut’s university of technology thonburi, 126 pracha-uthit road, bang mod, thung khru, bangkok 10140, thailand. (parin.cha@mail.kmutt.ac.th, poom.kum@kmutt.ac.th) abstract in this paper, we consider the kkm maps defined for a nonself map and the correlated intersection theorems in hadamard manifolds. we also study some applications of the intersection results. our outputs improved the results of raj and somasundaram [17, v. sankar raj and s. somasundaram, kkm-type theorems for best proximity points, appl. math. lett., 25(3):496–499, 2012.]. 2010 msc: 53c22; 53c25; 47h04. keywords: kkm maps; hadamard manifolds; generalized equilibrium problems; best proximity points. 1. introduction the kkm theory, as the term coined by park [15], is the study of the equivalent formulations, variants, and extensions to the 1929 geometric result due to knaster, kuratowski, and mazurkiewicz. this result is known nowadays as the kkm lemma, and it provides a firm foundation to many different areas of mathematics, e.g., fixed point theory, minimax theory, game theory, variational inequality, equilibrium theory, and henceforth. this lemma is also known for being equivalent to both the brouwer’s fixed point theorem and the sperner’s lemma (see [16] for further discussions). one of the most important ∗corresponding author received 19 april 2014 – accepted 3 october 2014 http://dx.doi.org/10.4995/agt.2015.2305 p. chaipunya and p. kumam enhancement of the kkm lemma is due to fan [7], whose result is obtained in a topological vector space. in [17], the nonself kkm maps have been introduced and studied under the framework of a normed linear space. as naturally occurs, the best proximity point theorem is deduced in relation to the nonself kkm lemma. on the other hand, colao et al. [6] proved the kkm lemma in a hadamard manifold, as an auxiliary tool for proving several results on the existence of solutions to equilibrium problems. also, the fixed point, variational inequality and nash equilibrium are investigated by the authors. in this paper, we occupy the nonself kkm lemma in hadamard manifolds. the nonself version of the browder’s fixed point theorem as well as the solvability of a generalized equilibrium problem are studied, as applications of our kkm lemma. 2. preliminaries recall first that a hadamard manifold m is a complete simply-connected smooth riemannian manifold whose sectional curvature is non-positive. at each point x ∈ m, we write txm to represent the tangent plane at x, which is at the same time a manifold. with this structure, we can define an exponential map expp : tpm → m by expp(ν) := γν(1), where γν is a geodesic defined by its position p and velocity ν at p. recall that exponential maps are diffeomorphisms. the exponential maps allow us to characterize the minimal geodesic joining a point p to another point q by the function t 7→ expp(texp −1 p (q)), with t ∈ [0,1]. naturally, a subset k ⊂ m is said to be geodesically convex if minimal geodesics correspond to each of its elements are contained in k. for any nonempty subset a ⊂ m, denoted by co(a) the geodesically convex hull of a, i.e., the smallest geodesically convex set containing a. note that the geodesically convex hull of any finite subset is compact. moreover, the geodesic distance d(p,q) between two points p,q ∈ m defined the length of its minimal geodesic induces the original topology of m. a real function f : m → r is said to be geodesically convex if the composition f ◦ γ is convex (in ordinary sense), provided that γ is the minimal geodesic joining two arbitrary points in m. in particular, the geodesic distance is geodesically convex in both of its arguments. referring to [18], hadamard manifolds behave nicely with probability measures defined on them. let p(m) be the collection of probability measures µ on m whose supports are separable and ∫ m d(x,y)dµ(y) < ∞ for every1 x ∈ m. then, to each µ ∈ p(m) and y ∈ m, we associate a point z∗ ∈ m that minimizes the (uniformly) geodesically convex function z 7→ ∫ m [d2(z,x)− d2(y,x)]dµ(x). such point z∗ is independent of y ∈ m, so we prefer writing 1the quantifier ‘for some’ is used in some texts. they are however identical as one can deduce from the triangle inequality. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 38 nonself kkm maps and corresponding theorems in hadamard manifolds b(µ) in place of z∗. moreover, we say that it is the barycenter of µ. if supp(µ) is contained in some closed geodesically convex set k, it is the case that b(µ) ∈ k. we can make p(m) into a metric space by endowing it with the wasserstein metric given by: dw (µ,ν) := inf ∫∫ m×m d(x,y)dλ(x,y), ∀µ,ν ∈ p(m), where the infimum is taken over λ ∈ p(m × m) whose marginals are µ and ν. with respect to this metric, the map µ 7→ b(µ) is nonexpansive. 3. nonself kkm maps the pair (a,b) set up by two given nonempty subsets a and b of a metric space (s,d) is called a proximal pair if to each point (x,y) ∈ a × b, there corresponds a point (x̄, ȳ) ∈ a × b such that d(x, ȳ) = d(x̄,y) = dist(a,b), where dist(a,b) := inf{d(x,y),x ∈ a,y ∈ b}. in addition, if both a and b are convex, we say that (a,b) is a convex proximal pair. in the future contents, we assume that m is a hadamard manifold with the geodesic distance d. given a point x ∈ m and two nonempty subsets a,b ⊂ m, we write d(x,a) := infz∈a d(x,z). definition 3.1. let (a,b) be a proximal pair in a hadamard manifold m. a nonself map t : a ⇒ b is said to be kkm if for each finite subset d := {x1,x2, · · · ,xm} ⊂ a, there is a subset e := {y1,y2, · · · ,ym} ⊂ b such that d(xi,yi) = dist(a,b), ∀i ∈ {1,2, · · · ,m}, and co({yi, i ∈ i}) ⊂ t({xi, i ∈ i}) for every ∅ 6= i ⊂ {1,2, · · · ,m}. theorem 3.2. suppose that (a,b) is a proximal pair in a hadamard manifold m and t : a ⇒ b is a kkm map with nonempty closed values. then, the family {t(x),x ∈ a} has the finite intersection property. proof. assume to the contrary that there is a finite subset d := {x1,x2, · · · ,xm} ⊂ a such that ⋂ x∈d t(x) = ∅. since t is kkm, we can find a subset e := {y1,y2, · · · ,ym} ⊂ b so that co({xj,j ∈ j}) ⊂ t({yj,j ∈ j}), for every ∅ 6= j ⊂ {1,2, · · · ,m}. set k := co(e). define a function λ : k → r by λ(y) := m ∑ i=1 d(y,k ∩ t(xi)), ∀y ∈ k. at each y ∈ k, we have λ(y) > 0 since ⋂ x∈d t(x) = ∅. then, the map y ∈ k 7→ µy := m ∑ i=1 [ d(y,k ∩ t(xi))δyi λ(y) ] , c© agt, upv, 2015 appl. gen. topol. 16, no. 1 39 p. chaipunya and p. kumam where δu is the dirac probability measure corresponding to u ∈ k, is continuous (from k into p(k)). thus, the composition y 7→ µy 7→ b(µy) is continuous from k into itself, and it therefore has a fixed point y0 ∈ k (see [11]). take j := {j ∈ {1,2, · · · ,m},d(y0,k ∩ t(xj)) > 0}. it is immediate that y0 6∈ ⋃ j∈j t(xj). as a matter of fact, we have supp(µy0) ⊂ co({xj,j ∈ j}) which implies that y0 = b(µy0) ∈ co({xj,j ∈ j}) ⊂ ⋃ j∈j t(xj), a contradiction. therefore, the family {t(x),x ∈ a} must possess the finite intersection property. � theorem 3.3. suppose that (a,b) is a proximal pair in a hadamard manifold m and t : a ⇒ b is a kkm map with nonempty closed values. if t(x0) is compact at some x0 ∈ a, then the intersection ⋂ {t(x),x ∈ a} is nonempty. proof. by theorem 3.2, we know that t(x0) ∩ t(x) is nonempty and closed for all x ∈ a. moreover, the family {t(x0) ∩ t(x),x ∈ a} has the f.i.p. the conclusion follows as t(x0) is compact. � remark 3.4. with the same proofs, theorems presented above can also be extended to cat(0) spaces, but with an additional assumption that every continuous map from a compact convex subset of m into itself has a fixed point. in particular, if a and b are identical, we can obtain kkm results as of [6, 12] 4. some applications we observe here some applications of our results in the previous section. before we go into the main subjects, let us observe the following fact about convex hulls of finitely many points between a convex proximal pair. lemma 4.1. let (a,b) be a convex proximal pair of a hadamard manifold m. assume that x1,x2, · · · ,xm ∈ a and y1,y2, · · · ,ym ∈ b are points such that d(xi,yi) = dist(a,b), ∀i ∈ {1,2, · · · ,m}. then, (co({x1,x2, · · · ,xm}),co({y1,y2, · · · ,ym})) is a proximal pair with (4.1) dist(co({x1,x2, · · · ,xm}),co({y1,y2, · · · ,ym})) = dist(a,b). proof. the equity (4.1) is obvious, so let us prove the former part. let us write c1 := {x1,x2, · · · ,ym}, and for j ≥ 2, let cj be the union of minimal geodesics that join pairs of points in cj−1. in the same way, we let d1 := {y1,y2, · · · ,ym}, and for j ≥ 2, let dj be the union of minimal geodesics that join pairs of points in dj−1. one can simply show, by using mathematical induction, that (cj,dj) is a proximal pair for all j ≥ 1. now, apply [14, proposition 2.5.5] to complete the proof. � c© agt, upv, 2015 appl. gen. topol. 16, no. 1 40 nonself kkm maps and corresponding theorems in hadamard manifolds 4.1. generalized equilibrium problems. given a nonempty set q and a bifunction ψ : q × q → r, the equilibrium problem concerns the existence (and the determination) of a point x̄ ∈ q that makes ψ(x̄, ·) into a non-negative function. this equilibrium problem is first considered by fan [7, 8] under euclidean spaces. it is then improved and enriched in [3]. as it unifies many problems in optimization, for examples, minimization problem, variational inequality, minimax inequality, and nash equilibrium problem, the equilibrium theory gained its fame very quickly. consult [10, 13, 2, 5, 9, 1] for richer details. in this section, we shall consider the case where the bifunction ψ is defined on the product p × q, with p,q being nonempty and possibly distinct sets. this leads to a more general aspect of equilibrium problems. theorem 4.2. suppose that (p,q) is a geodesically convex proximal pair in a hadamard manifold m, and ψ : p × q → r is a bifunction. assume that the following conditions hold: (i) ψ(x,y) ≥ 0 provided x ∈ p, y ∈ q, and d(x,y) = dist(p,q), (ii) ∀x ∈ p, the set {y ∈ q,ψ(x,y) < 0} is geodesically convex, (iii) ∀y ∈ q, the function ψ(·,y) is u.s.c., (iv) there exists a nonempty compact set l ⊂ m such that both l ∩ p and l ∩ q are nonempty and ψ(x, ȳ) < 0, ∀x ∈ p \ l, for some point ȳ ∈ l ∩ q. then, there exists a point x̄ ∈ l ∩ p such that ψ(x̄,y) ≥ 0, ∀y ∈ q. proof. define a map g : q ⇒ p by g(y) := {x ∈ p,ψ(x,y) ≥ 0}, ∀y ∈ q. since ψ(·,y) is u.s.c., g(y) is closed for each y ∈ q. from (iv), we have g(ȳ) ⊂ l and so g(ȳ) is compact. we shall prove next that g is a kkm map. suppose that {y1,y2, · · · ,ym} ⊂ q and {x1,x2, · · · ,xm} ⊂ p such that d(xi,yi) = dist(p,q), ∀i ∈ {1,2, · · · ,m}. let us assume to the contrary that there exists a subset ∅ 6= j ⊂ {1,2, · · · ,m} and a point x0 ∈ co({xj,j ∈ j}) such that x0 6∈ g({yj,j ∈ j}). equivalently, ψ(x0,yj) < 0, ∀j ∈ j. by lemma 4.1, we can choose a point y0 ∈ co({yj,j ∈ j}) with d(x0,y0) = dist(p,q). note that yj ∈ {y ∈ q,ψ(x0,y) < 0} for each j ∈ j and {y ∈ q,ψ(x0,y) < 0} is geodesically convex. therefore, we have y0 ∈ co({yj,j ∈ j}) ⊂ {y ∈ q,ψ(x0,y) < 0}, which contradicts the hypothesis (i). hence, g is a kkm map, and the desired result follows immediately from the construction of g. � remark 4.3. if p = q, then the condition (i) reads as follows: (i’) ψ(x,x) ≥ 0, ∀x ∈ p, where it is always assumed in classical equilibrium theory. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 41 p. chaipunya and p. kumam 4.2. best proximity points. suppose that (s,d) is a metric space and a ⊂ s is nonempty. given a map t : a ⇒ s, a point x0 ∈ a is a fixed point of f if x0 ∈ t(x0). in particular, if t is closed valued, a fixed point is expressed metrically by d(x0,t(x0)) = 0. suppose that b ⊂ s is nonempty. then, it may be the case that the map t : a ⇒ b does not have a fixed point. in fact, it is evident that d(x,t(x)) ≥ dist(a,b) for all x ∈ a. in this case, instead of fixed points, we can consider the best proximity point x0 ∈ a, i.e., the point such that d(x0,t(x0)) = dist(a,b). the notion of best proximity point is stronger than the best approximation. in details, if t is single-valued and x0 is a best proximity point of t , then t(x0) is a best approximant to x0 for b. now, we state our nonself version of browder’s fixed point theorem [4] in the setting of hadamard manifolds. theorem 4.4. let (a,b) be a proximal pair of a hadamard manifold m, where a is assumed to be compact and geodesically convex, and t : a ⇒ b be a map such that (i) ∀x ∈ a, t(x) is nonempty and geodesically convex, (ii) t is an open fibre map, i.e., ∀y ∈ b, the inverse image t −1(y) := {x ∈ a,y ∈ t(x)} is open. then, there exists a point x̄ ∈ a such that d(x̄,t(x̄)) = dist(a,b). proof. consider the dual map g : b ⇒ a defined by g(y) := a \ t −1(y), ∀y ∈ b. if g(y0) = ∅ for some y0 ∈ b, i.e., t −1(y0) = a. this means y0 ∈ t(x) for all x ∈ a. now, since (a,b) is a proximal pair, we can find a point x̄ ∈ a such that d(x̄,y0) = dist(a,b). in particular, y0 ∈ t(x̄). thus, we have d(x̄,t(x̄)) ≤ d(x̄,y0) = dist(a,b), yielding the desired result. on the other hand, suppose that g(y) is nonempty for every y ∈ b. moreover, g is closed valued. observe that ⋂ y∈b g(y) = ⋂ y∈b (a \ t −1(y)) = a \ ⋃ y∈b t −1(y). since {t −1(y),y ∈ b} is an open cover of a, we obtain from the above equality that ⋂ y∈b g(y) is empty. we conclude from theorem 3.3 that g is not a kkm map. thus, suppose that {y1,y2, · · · ,ym} ⊂ b and {x1,x2, · · · ,xm} ⊂ a are sets such that d(xi,yi) = dist(a,b), ∀i ∈ {1,2, · · · ,m} and co({x1,x2, · · · ,xm}) is not contained in g({y1,y2, · · · ,ym}). in particular, choose x̄ ∈ co({x1,x2, · · · ,xm}) such that x̄ 6∈ g({y1,y2, · · · ,ym}). hence, x̄ ∈ t −1(yi) for all i ∈ {1,2, · · · ,m}, or equivalently, yi ∈ t(x̄) for all i ∈ {1,2, · · · ,m}. according to lemma 4.1, we can choose a point c© agt, upv, 2015 appl. gen. topol. 16, no. 1 42 nonself kkm maps and corresponding theorems in hadamard manifolds z ∈ co({y1,y2, · · · ,ym}) with d(x̄,z) = dist(a,b). since t(x̄) is convex, we get z ∈ co({y1,y2, · · · ,ym}) ⊂ t(x̄). we have again d(x̄,t(x̄)) ≤ d(x̄,z) = dist(a,b), which leads to the desired result. � in case a and b are identical, we have the following variant of browder’s theorem in the setting of hadamard manifolds. theorem 4.5. let k be a nonempty, compact, and geodesically convex subset of a hadamard manifold m and t : k ⇒ k an open fibre map whose values are nonempty and geodesically convex. then, t has a fixed point. conclusion we have proved the intersection theorem for nonself kkm maps in hadamard manifolds, which extends the existed result of [17]. we have also provide some applications of our intersection result towards the existence of an equilibrium point and a best proximity point. acknowledgements. the first author is supported by the thailand research fund through the royal golden jubilee ph.d. program (grant no. phd/0045/2555) and the king mongkut’s university of technology thonburi under the rgj-ph.d. scholarship. moreover, we would like to gratefully thank the anonymous referees for their suggestions, which improve this paper significantly. references [1] m. bianchi and s. schaible, generalized monotone bifunctions and equilibrium problems, j. optimization theory appl. 90, no. 1 (1996), 31–43. [2] g. bigi, a. capătă and g. kassay, existence results for strong vector equilibrium problems and their applications, optimization 61, no. 4-6 (2012), 567–583. [3] e. blum and w. oettli, from optimization and variational inequalities to equilibrium problems, the mathematics student 63, no. 1-4 (1994), 123–145. [4] f. browder, the fixed point theory of multi-valued mappings in topological vector spaces, mathematische annalen 177, no. 4 (1968), 283–301. [5] m. castellani and m. giuli, on equivalent equilibrium problems, j. optim. theory appl. 147, no. 1 (2010), 157–168. [6] v. colao, g. lópez, g. marino and v. mart́ın-márquez, equilibrium problems in hadamard manifolds, j. math. anal. appl. 388, no. 1 (2012), 61–77. [7] k. fan, a generalization of tychonoff’s fixed point theorem, math. ann. 142 (1961), 305–310. [8] k. fan, a minimax inequality and applications, inequalities, iii (proceedings of the third symposium, university of california, los angeles, ca, 1969; dedicated to the memory of theodore s. motzkin), pages 103–113, 1972. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 43 p. chaipunya and p. kumam [9] a. n. iusem and w. sosa, new existence results for equilibrium problems, nonlinear anal. theory methods appl. 52, no. 2 (2003), 621–635. [10] l.-j. lin and h. i. chen, coincidence theorems for families of multimaps and their applications to equilibrium problems, abstr. appl. anal. 2003, no. 5 (2003), 295–309. [11] s. németh, variational inequalities on hadamard manifolds, nonlinear anal. theory methods appl. 52, no. 5 (2003), 1491–1498. [12] c. p. niculescu and i. rovenţa, fan’s inequality in geodesic spaces, applied mathematics letters 22, no. 10 (2009), 1529–1533. [13] w. oettli, a remark on vector-valued equilibria and generalized monotonicity, acta math. vietnam. 22, no. 1 (1997), 213–221. [14] a. papadopoulos, metric spaces, convexity and nonpositive curvature, irma lectures in mathematics and theoretical physics, european mathematical society, 2005. [15] s. park, some coincidence theorems on acyclic multifunctions and applications to kkm theory, fixed point theory and applications (1992), pp. 248–277. [16] s. park, ninety years of the brouwer fixed point theorem, vietnam j. math. 27 (1997), 187–222. [17] v. sankar raj and s. somasundaram, kkm-type theorems for best proximity points, appl. math. lett. 25, no. 3 (2012), 496–499. [18] k.-t. sturm, probability measures on metric spaces of nonpositive curvature, in: heat kernels and analysis on manifolds, graphs, and metric spaces. lecture notes from a quarter program on heat kernels, random walks, and analysis on manifolds and graphs, april 16–july 13, 2002, paris, france, pp. 357–390. providence, ri: american mathematical society (ams), 2003. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 44 02.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 13 28 an operation on topological spacesa. v. arhangel0ski��abstract. a (binary) product operation on a topologicalspace x is considered. the only restrictions are that some el-ement e of x is a left and a right identity with respect to thismultiplication, and that certain natural continuity requirementsare satis�ed. the operation is called diagonalization (of x). twoproblems are considered: 1. when a topological space x admitssuch an operation, that is, when x is diagonalizable? 2. whatare necessary conditions for diagonalizablity of a space (at a givenpoint)? a progress is made in the article on both questions. inparticular, it is shown that certain deep results about the topo-logical structure of compact topological groups can be extendedto diagonalizable compact spaces. the notion of a moscow spaceis instrumental in our study.2000 ams classi�cation: 54d50, 54d60, 54c35keywords: c-embedding, diagonalizable space, hewitt-nachbin completion,moscow space, pseudocompact space, separability, stone-�cech compacti�ca-tion, tightness 1. diagonalizable spacesin this article we build upon some ideas and techniques from [5], showing thatthey are applicable in a much more general setting. the key new idea is mate-rialized below in a new notion of a diagonalizable space, which turns out to bea very broad generalization of the notion of a semitopological semigroup withidentity. it also generalizes the notion of a mal0tsev space. diagonalizablityis preserved by retracts and by products. thus, a diagonalizable space neednot be homogeneous. moreover, every zero-dimensional �rst countable space isdiagonalizable. however, despite its very general nature, diagonalizablity turnsout to be so strong a property, that we are able to extend some importanttheorems about compact topological groups to compact diagonalizable spaces.these results involve stone-�cech compacti�cations, c-embeddings, and prod-ucts; in particular, they extend the classical results of i. glicksberg [10] ande. van douwen [8] (see also [12]). a central role in what follows also belongs 14 a.v. arhangel0ski��to the notion of a moscow space, which was recently shown to have delicateapplications in topological algebra.a topological space x will be called diagonalizable at a point e 2 x if thereexists a mapping � of the square x � x in x satisfying the following twoconditions:1) �(x;e) = �(e;x), for each x 2 x;2) for each a 2 x, the mappings �a and �a of the space x into itself,de�ned by the formulas �a(x) = �(x;a) and �a(x) = �(a;x) for eachx 2 x, are continuous at x = e.the mapping � in this case is called a diagonalizing mapping (at e), or a diag-onalization of x at e, and the mappings �a and �a are called, respectively, theright action and the left action by a on x, corresponding to the product oper-ation �. if in the de�nition above the mapping � can be chosen to be jointlycontinuous at (e;a) and (a;e) for each a 2 x, we say that x is continuouslydiagonalizable at e. clearly, every space x is diagonalizable at every isolatedpoint of x. if x is (continuously) diagonalizable at every point e 2 x, then xis called (continuously) diagonalizable.a space x with a �xed separately (jointly) continuous mapping � : x�x !x and a �xed point e 2 x will be called a semitopoid (a topoid) with identitye if � is a diagonalization of x at e, that is, if �(x;e) = �(e;x) = x, for eachx 2 x. the next assertion is obvious.proposition 1.1. if a space x is (continuously) diagonalizable at some pointa of x, and x is homogeneous, then x is (continuously) diagonalizable.example 1.2.1) every topological, and even every paratopological, group g is contin-uously diagonalizable: as a continuous diagonalization mapping � atthe neutral element e of g we can take just the product operation:�(x;y) = xy, for each (x;y) in g�g. it remains to refer to homogene-ity of g.therefore, sorgenfrey line is continuously diagonalizable, since it is aparatopological group.2) every semitopological group g, that is, a group g with a topology suchthat the product operation in g is separately continuous (with respectto each argument), is diagonalizable at the neutral element e by theproduct operation. since every semitopological group g is a homoge-neous space, it follows from proposition 1.1 that every semitopologicalgroup is diagonalizable.3) let x be a mal0tsev space, that is, a space with a continuous mappingf of the cube x � x � x in x such that f(x;y;y) = f(y;y;x) = x forall x and y in x (such f is called a continuous antimixer on x). thenx is continuously diagonalizable. indeed, �x any e in x, and de�ne amapping � of x � x in x by the formula:�(x;y) = f(x;e;y): an operation on topological spaces 15clearly, � is continuous, and �(e;x) = f(e;e;x) = x = f(x;e;e) =�(x;e). thus, � is a continuous diagonalization mapping at e, and xis continuously diagonalizable.4) similarly, every space with a separately continuous antimixer is diago-nalizable.5) every retract of a topological group is a mal0'tsev space (see [13]).therefore, every retract of a topological group is continuously diago-nalizable. in particular, every absolute retract is continuously diago-nalizable. hence, every tychono� cube is continuously diagonalizable.notice that a mal0tsev space, unlike a topological group, need not behomogeneous (consider, for example, the closed unit interval). thus,diagonalizable spaces do not have to be homogeneous.several simple statements below demonstrate that the class of di-agonalizable spaces is much larger than the classes of semitopologicalgroups or mal0tsev spaces.proposition 1.3. every linearly ordered topological space x with the smallestelement e is continuously diagonalizable at e.proof. let < be a linear ordering on x, generating the topology of x, suchthat e is the smallest element of x. for arbitrary (x;y) 2 x �x, put �(x;y) =maxfx;yg. clearly, � is a continuous diagonalizing mapping at e. �theorem 1.4. every linearly ordered compact space x is continuously diago-nalizable at least at one point.proof. indeed, every compact space x, the topology of which is generated bya linear ordering <, has the smallest element with respect to < [9]. �corollary 1.5. every homogeneous linearly ordered compact space is continu-ously diagonalizable.proof. this follows from theorem 1.4 and proposition 1.1. �example 1.6. the \double arrow" space is continuously diagonalizable, sinceit is compact, homogeneous, and linearly ordered.the conclusion in theorem 1.4 can be considerably strengthened if we assumethat the topology of x is generated by a well ordering. indeed, we have thefollowingtheorem 1.7. if x is a topological space, the topology of which is generatedby a well ordering <, then x is continuously diagonalizable.proof. assume that e is any point of x. put y = fx 2 x : x � eg andz = fx 2 x : e < xg. then y and z are open and closed subsets of x, thespace y is linearly ordered, and e is the last element of y . from proposition1.3 it follows that y is continuously diagonalizable at e (consider the reverseordering of y ).it remains to apply the next lemma: 16 a.v. arhangel0ski��lemma 1.8. assume that y is an open and closed subspace of x, e 2 y , and yis (continuously) diagonalizable at e. then x is (continuously) diagonalizableat e.proof. put z = x ny . the sets y �y , y �z, z �y , and z �z are pairwisedisjoint and open and closed in x � x. together they cover x � x.let us �x a (continuous) diagonalizing mapping for the space y at e. thenwe de�ne a diagonalizing mapping � for x at e as follows.if (x;y) 2 y � y , we put �(x;y) = (x;y).if (x;y) 2 y � z, we put �(x;y) = y.if (x;y) 2 z � y , we put �(x;y) = x.if (x;y) 2 z � z, we put �(x;y) = e.clearly, � is a (continuous) diagonalizing mapping for x at e. �let us call a space x continuously homogeneous if there exist a point e 2 xand a continuous mapping h of x into the space hp(x) of homeomorphisms ofx onto itself in the topology of pointwise convergence satisfying the followingconditions:(1) hx(e) = x, for each x 2 x, where hx = h(x); and(2) he = h(e) is the identity mapping of x onto itself.such a mapping h will be called a homogeneity cp-structure on x at e.clearly, every continuously homogeneous space is homogeneous.proposition 1.9. every continuously homogeneous space x is diagonalizable.proof. let h : x ! hp(x) be a homogeneity cp-structure on x at a pointe 2 x. put �(x;y) = hx(y), for each (x;y) 2 x � x, where hx = h(x). itis easily veri�ed that the mapping � is separately continuous. we also have:�(e;x) = he(x) = x, since he is the identity mapping, and �(x;e) = hx(e) = x,by the other property of homogeneity cp-structure. thus, x is diagonalizableat e. since x is homogeneous, it follows that x is diagonalizable. �theorem 1.10. every retract of a (continuously) diagonalizable space is (con-tinuously) diagonalizable.proof. assume that x is a (continuously) diagonalizable space, y a subspaceof x, and r a retraction of x onto y . take any point e in y , and �x adiagonalizing mapping � : x � x ! x at e.de�ne a mapping �r of y � y in y by the formula �r = r�(y;z), for ev-ery (y;z) in y � y . clearly, if � is (separately) continuous, then �r is also(separately) continuous.take any y 2 y . then�r(y;e) = r�(y;e) = r(y) = yand �r(e;y) = r�(e;y) = r(y) = y;since y 2 y and r is a retraction of x onto y . therefore, y is (continuously)diagonalizable at e. � an operation on topological spaces 17remark 1.11. notice, that a space x is (continuously) diagonalizable at e 2 xif and only if there exists a (continuous) separately continuous mapping � ofthe product space x � x onto the diagonal �x = f(x;x) : x 2 xg such that�(x;e) = �(e;x) = (x;x), for each x in x. this obvious observation explainsthe name \diagonalizable space".there is another curious result on diagonalizablity involving retractions. ob-serve that for any e 2 x the subspaces feg � x and x � feg are retracts ofx �x (under the obvious projections). now let us ask the following question:when the subspace (x � feg) [ (feg � x) is a retract of x � x? if this is thecase, we will call the space x crosslike at e 2 x. if (x � feg) [ (feg � x) is aretract of x �x under a separately continuous retraction, then x will be saidto be weakly crosslike (at e). for example, the closed unit interval and the realline are crosslike spaces.proposition 1.12. if a space x is weakly crosslike at e 2 x, then x is diag-onalizable at e.proof. fix a separately continuous retraction r of x � x onto the subspace(x � feg) [ (feg � x). for each x 2 x, put f(e;x) = f(x;e) = x. then f isa continuous mapping of (x � feg) [ (feg � x) onto x. clearly, the mapping� = f � r is a diagonalization of x at e. �similarly, the next assertion is proved:proposition 1.13. if a space x is crosslike at e 2 x, then x is continuouslydiagonalizable at e.a space x is called zero-dimensional at a point e 2 x if there exists a baseof x at e consisting of open and closed sets (notation: ind(e;x) = 0).theorem 1.14. if a space x is zero-dimensional at a point e 2 x, and e is ag� in x, then x is crosslike at e, and, hence, continuously diagonalizable at e.proof. we can �x a countable family fvn : n 2 !g of open and closed neighbor-hoods of e in x such that vn+1 � vn, for each n 2 !, and feg = \fvn : n 2 !g.put wn = vn � (x n vn+1), un = (x n vn) � vn, w = [fwn : n 2 !g, andu = [fun : n 2 !g. obviously, the sets u and w are open in x � x, and(feg � (x n feg) � w , (x n feg) � feg) � u.it is easy to check that u and w are disjoint, and that they are closed in(x � x) n f(e;e)g. therefore, the set k = (x � x) n (u [ w [ feg) is openin x � x. for each (x;y) 2 w , put r(x;y) = (e;y). for each (x;y) 2 u, putr(x;y) = (x;e). for each (x;y) 2 (x � x) n (u [ w), put r(x;y) = (e;e).clearly, r is a continuous retraction of x � x onto the cross (x � feg) [(feg � x) at e. hence, x is crosslike at e, and, by proposition 1.13, x iscontinuously diagonalizable at e. �the same idea leads to one more elementary result in the same direction.recall that, for a non-empty space x, the equality ind(x) = 0 signi�es thatfor any two disjoint closed subsets p and f in x there exists an open andclosed subset w such that p � w and f \ w = ? (see [9]). 18 a.v. arhangel0ski��proposition 1.15. let x be a space and e a point in x such that the subspacez = (x � x) n f(e;e)g of x � x satis�es the condition ind(z) = 0. then thespace x is crosslike at e.proof. put p = feg � x and f = x � feg. then p and f are disjoint closedsubsets of z. since ind(z) = 0, there exists an open and closed subset w of zsuch that p � w and f \ w = ?.now we de�ne a mapping r of x �x in (feg �x) [(x �feg) as follows. if(x;y) 2 w , let r(x;y) = (e;y). if (x;y) 2 z n w , let r(x;y) = (x;e). finally,we put r(e;e) = (e;e). clearly, the restriction of r to w is continuous, since itis the restriction of the projection mapping of x �x. similarly, the restrictionof r to z nw is continuous. therefore, r is continuous at all points of z. sincez is open in x � x, to see that the mapping r is continuous, we only have tocheck its continuity at the point (e;e). however, the continuity of r at (e;e) isalso obvious.finally, we observe that r is the identity mapping on (feg �x) [(x �feg).thus, (feg � x) [ (x � feg) is a retract of x � x. �theorem 1.14 shows how much more general is the diagonalizablity assump-tion, than the assumption that the space has a (separately) continuous antim-ixer. indeed, according to [15], every compact space with a separately con-tinuous antimixer is a dugundji compactum, and it is well known that every�rst countable dugundji compactum is metrizable. we also see from theorem1.14 that diagonalizablity of a compact space does not impose any homogeneityrestrictions on the space. in that the diagonalizablity di�ers drastically fromthe assumptions that x is a paratopological group or a semitopological group.example 1.16. let x be a space, e a point of x, and pe(x) the space ofall closed subsets of x containing e, in the vietoris topology. put z = pe(x),and de�ne a mapping � : z � z ! z by the rule: �(a;b) = a [ b, for any(a;b) 2 z � z.it is easily veri�ed that the mapping � is continuous. it is also clear that�(e;a) = a = �(a;e), for each a 2 z, where e = feg. therefore, � is acontinuous diagonalization of the space z = pe(x) at the point e = feg 2pe(x).since there is no reason to believe that the space pe should be diagonalizableat every point, the above conclusion suggests that the space pe normally canbe expected to be not homogeneous and provides some means for proving that.though the proof of the next statement is obvious, the result itself is quiteimportant.proposition 1.17. the product of any family of (continuously) diagonalizablespaces is a continuously diagonalizable space.similar assertion holds for crosslike spaces. in conclusion of this section, wemention a curious corollary of theorem 1.10. an operation on topological spaces 19theorem 1.18. suppose x is a space such that x � y is homeomorphic to a(continuously) diagonalizable space, for some space y . then x is also (contin-uously) diagonalizable.proof. indeed, x is a retract of x �y . it remains to apply theorem 1.10. �2. some necessary conditions for diagonalizablitya space x is called moscow at a point e 2 x if, for every open set u theclosure of which contains e, there exists a g�-subset p of x such that e 2 p � u(see [1, 5]). if x is moscow at every point, we call x a moscow space.a space x is called weakly klebanov at a point e 2 x if for every family ofg�-subsets of x such that the closure of [ contains e there exists a g�-subsetp of x such that e 2 p � [ . we say that x is weakly klebanov if x isweakly klebanov at every point of x. clearly, every weakly klebanov space ismoscow, and every space of countable pseudocharacter is weakly klebanov.the importance of the notion of moscow space comes from the role it playsin connection with c-embeddings; see about that [6] and [4]. besides, a non-trivial result on moscow spaces is the theorem that every dugundji compactumis moscow (see [15]); it follows that every compact (actually, every pseudocom-pact) topological group is a moscow space.the simplest example of a non-moscow space is the one-point (alexandro�)compacti�cation of an uncountable discrete space. note that the tightnessof this space is countable. on the other hand, it was shown in [5] that everytopological (and even semitopological) group of countable tightness is a moscowspace. this again underlines the signi�cance of the concept of a moscow spacefor topological algebra.one of our main results is the next theorem:theorem 2.1. suppose x is a space of countable tightness diagonalizable (ate 2 x). then x is weakly klebanov (at e).proof. let a be a subset of x which is the union of a family of g�-subsets ofx, and e any point in a. we have to show that there exists a g�-subset p suchthat e 2 p � a.since the tightness of x is countable, there exists a countable subset b of asuch that e 2 b. for each b 2 b we �x a g�-set pb such that b 2 pb � a.let us also �x a diagonalizing mapping � of x�x into x at e. for each b 2 bconsider the mapping �b of x into x given by the formula: �b(x) = �(x;b), forevery x 2 x.since � is a diagonalizing mapping at e, �b is continuous at e. therefore,��1(pb) contains a g�-set mb such that e 2 mb, since �b(e) = �(e;b) = b. thenthe set f = \fmb : b 2 bg is also a g�-set in x, and e 2 f .take any point a 2 f . we have �b(a) 2 pb � a, for each b 2 b, sincef � ��1b (pb). thus, �(a;b) = �b(a) 2 a, for each b 2 b. however, e 2 b,and the function �(a;x) is continuous with respect to the second argument atx = e. it follows that �(a;e) 2 a. since � is a diagonalizing mapping at e, wehave �(a;e) = a. therefore, a 2 a, that is, f � a. the proof is complete. � 20 a.v. arhangel0ski��the assumption that the tightness of x is countable can be considerablyweakened but can not be completely removed. the �-tightness of a space x ata point e 2 x is said to be countable [5] if for each open subset u such that eis in the closure of u there exists a countable subset b of u such that e 2 b(notation: t�(e;x) � !). if the �-tightness of x is countable at every pointe 2 x, we say that the �-tightness of x is countable, and write t�(x) � !.introducing a mild, obvious, change in the proof of theorem 2.1, we obtaina proof of the next statement:theorem 2.2. every diagonalizable (at a point e) space x of countable �-tightness is moscow (at e).it is worth noting that for every dyadic compactum x the �-tightness ofx is countable, while if the tightness of a dyadic compactum x is countable,then x is metrizable (see [7]). in particular, the �-tightness of every tychono�cube is countable. this shows that the countability of �-tightness is much,much weaker restriction than the countability of tightness. however, the nextexample shows that we can not completely drop it.example 2.3. the space of ordinals !1 + 1 is continuously diagonalizable, bytheorem 1.7. nevertheless, this space is easily seen to be not moscow [4].of course, this happens because the �-tightness of !1 + 1 is not countable(precisely at the point !1). observe that the space !1 + 1 does not admit aseparately continuous antimixer, since it is compact but not dyadic. observealso that, by theorem 1.17, the space (!1 + 1)� is continuously diagonalizable,for every cardinal number �.example 2.4. let � be an uncountable cardinal number and a� the one-point(alexandro�) compacti�cation of a discrete space of cardinality �. then a� isnot diagonalizable (at the unique non-isolated point of a�). obviously, a� isa fr�echet-urysohn space; hence, the tightness of a� is countable. assume nowthat a� is diagonalizable. then, by theorem 2.1, a� is moscow, a contradic-tion. it follows that a� is not diagonalizable.note that the usual convergent sequence is continuously diagonalizable bytheorem 1.7 or by theorem 1.14. note also, that the space a� is compact,zero-dimensional, hausdor�, and satis�es the �rst axiom of countability at allpoints except one, the non-isolated point. thus, theorem 1.14 can not be muchimproved.it is well known that a compact topological group of countable tightnessis metrizable (see [7]). for diagonalizable compact spaces we have a parallelstatement with a weaker conclusion. recall that a compact space x is said tobe !-monolithic if, for every countable subset a of x, the closure of a in x isa space with a countable base.theorem 2.5. every diagonalizable !-monolithic compact hausdor� space xof countable tightness is �rst countable.proof. take any point x 2 x. since x is compact hausdor�, it is enoughto show that x is a g�-point in x. the space x is fr�echet-urysohn, and x an operation on topological spaces 21is �rst countable at a dense set y of points (since every !-monolithic compacthausdor� space of countable tightness has these properties [3]). therefore,there exists a sequence fyn : n 2 !g of points of y converging to x. on theother hand, x is weakly klebanov, by theorem 2.1.it remains to apply the following obvious lemma:lemma 2.6. suppose x is a weakly klebanov space and fyn : n 2 !g is asequence of g�-points in x converging to x. then x is also a g�-point in x.note that the space a� in example 2.4 is an !-monolithic compact hausdor�space of countable tightness. theorem 2.5 clari�es, why it is not diagonalizable:it is because it is not �rst countable.corollary 2.7. every diagonalizable corson compactum is �rst countable.proof. indeed, every corson compact space is monolithic and fr�echet-urysohn(see [3]). it remains to apply theorem 2.5. �in the next result we assume the continuum hypothesis (ch). it is notclear whether the statement remains true without this assumption.theorem 2.8. (ch) every diagonalizable sequential compact hausdor� spacex is �rst countable.proof. let y be the set of all points of x at which x satis�es the �rst axiom ofcountability. then y is g�-dense in x, by a theorem in [2] (here we use (ch)).on the other hand, from theorem 2.1 it follows that x is weakly klebanov.assume now that x 6= y . then y is not closed in x, since y is dense in x.therefore, since x is sequential, there exists a point x 2 x n y and a sequencefyn : n 2 !g of points of y converging to x. it follows from lemma 2.6 that xis a g�-point in x. since x is compact hausdor�, we conclude that x is �rstcountable at x. this is a contradiction with x =2 y and de�nition of y . �3. diagonalizablity and c-embeddingsin this section, we combine our results on diagonalizablity and a result ofm.g. tkachenko to obtain several new results on c-embeddings and stone-�cechcompacti�cations. for more results on c-embeddings in the context of topo-logical groups see [11]. here is uspenskij's 's modi�cation [15] of tkachenko'sresult from [14]:theorem 3.1 (tkachenko). if x is a moscow tychono� space, then everyg�-dense subspace y of x is c-embedded in x.theorem 3.2. let x be a compact diagonalizable space of countable �-tight-ness. then x is the stone�cech compacti�cation of any g�-dense subspace yof x.proof. indeed, x is a moscow space, by theorem 2.1. therefore, by theorem3.1, y is c-embedded in x. it follows that y is pseudocompact and x =�y . � 22 a.v. arhangel0ski��the next statement is a typical application of theorem 2.2.theorem 3.3. if a tychono� space x is diagonalizable at e 2 x, and the�-tightness of x at e is countable, then either e is a g�-point in x, or thesubspace y = x n feg is c-embedded in x.proof. assume that e is not a g�-point in x. then y is g�-dense in x. bytheorem 2.2, x is moscow at e. since y is g�-dense in x, it follows, by anobvious modi�cation of theorem 3.1 (see [4]), that y is c-embedded in x. �corollary 3.4. assume that a tychono� space x is diagonalizable at e 2 x,the �-tightness of x at e is countable, and the subspace y = x nfeg is hewitt-nachbin complete. then e is a g�-point in x.proof. assume that e is not isolated in x. then, since y = x n feg is hewitt-nachbin complete, y is not c-embedded in x. now it follows from theorem3.3 that e is a g�-point in x. �corollary 3.5. assume that x is a pseudocompact tychono� space diagonal-izable at a point e 2 x such that the �-tightness of x at e is countable. theneither x is �rst countable at e, or the subspace x n feg is pseudocompact.proof. since ever pseudocompact tychono� space is �rst countable at everyg�-point, it follows from theorem 3.3 that the subspace y = x n feg is c-embedded in x. therefore, since x is pseudocompact, the space y must bepseudocompact as well. �here is a result in the same direction, in which the assumption on x doesnot contain explicitly a restriction on the tightness of x.theorem 3.6. assume that x is a pseudocompact tychono� space diagonal-izable at a point e 2 x. assume also that the next condition is satis�ed: (�)for each open subset u of x such that e 2 u n u, the subspace u n feg is notpseudocompact. then x is �rst countable at e.proof. clearly, we can assume that the point e is not isolated in x. thencondition � implies that the subspace x nfeg is not pseudocompact. it followsfrom corollary 3.5 that, to complete the proof, it remains to show that the�-tightness of x at e is countable.take any open set u such that e 2 u nu. by (�), the subspace z = u nfeg isnot pseudocompact. therefore, there exists a discrete family � = fvn : n 2 !gof non-empty open subsets in z. however, the subspace u is pseudocompact,since x is pseudocompact and u is a canonical closed subset of x (see [9]).it follows that the sequence (vn : n 2 !) converges to e. clearly, vn \ u isnon-empty, for each n 2 !. choosing a point xn 2 vn \ u for each n 2 !, weobtain a sequence of points of the set u converging to e. hence, the �-tightnessof x at e is countable. �the condition (�) in theorem 3.6 may look a little arti�cial. however, thereare several natural corollaries of theorem 3.6. recall that a subset a of a spacex is called locally closed if a = b\c, where b is a closed subset of x and c is an operation on topological spaces 23an open subset of x. the next three statements follow directly from theorem3.6.corollary 3.7. assume that x is a pseudocompact tychono� space diagonal-izable at a point e 2 x such that every locally closed pseudocompact subspaceof x is closed in x. then x is �rst countable at e.corollary 3.8. assume that x is a pseudocompact tychono� space diagonaliz-able at a point e 2 x and such that the subspace xnfeg is dieudonn�e complete.then x is �rst countable at e.corollary 3.9. assume that x is a pseudocompact tychono� space diagonal-izable at a point e 2 x and such that the subspace x n feg is metacompact.then x is �rst countable at e.the list of corollaries to theorem 3.6 can be easily expanded.4. diagonalizable separable spacesthe results obtained in the preceding sections are, in particular, applicableto separable spaces. we present several such applications below.theorem 4.1. if a separable space x is diagonalizable at e 2 x, then x ismoscow at e.proof. this statement is a direct corollary of theorem 2.1 and the obvious factthat the �-tightness of every separable space is countable. �a space x is called a g�-extension of a space y if y is a g�-dense subspaceof x. a space x may have many di�erent g�-extensions. for example, everycompacti�cation of a pseudocompact tychono� space x is a g�-extension ofx, and usually there are many such compacti�cations.however, it turns out that few of these extensions should be expected to bediagonalizable. this is demonstrated by the next "uniqueness" result.theorem 4.2. if a tychono� space x is a g�-extension of a separable spacey , and x is diagonalizable and hewitt-nachbin complete, then x is the hewitt-nachbin completion �y of y .proof. the space x is also separable. therefore, by theorem 3.1, x is moscow.since y is g�-dense in x, it follows theorem 3.1 that y is c-embedded inx. therefore, since x is hewitt-nachbin complete, x is the hewitt-nachbincompletion of x. �with the help of theorem 4.2, we could easily construct many further ex-amples of non-diagonalizable separable spaces.the notion of diagonalizablity can be also applied to show that g�-extensionsof spaces, in general, should not be expected to be homogeneous. this is basedon the following key lemma from [6]:lemma 4.3. if a tychono� space x is a homogeneous g�-extension of amoscow space y , then x is also a moscow space and y is c-embedded inx. 24 a.v. arhangel0ski��theorem 4.4. if a tychono� space x is a homogeneous g�-extension of aseparable diagonalizable space y , and x is hewitt-nachbin complete, then x isthe hewitt-nachbin completion �y of y .proof. by theorem 4.1, the space y is moscow. since x is homogeneous andy is g�-dense in x, it follows from lemma 4.3 that x is also moscow and y isc-embedded in x. since x is hewitt-nachbin complete, we can conclude thatx is the hewitt-nachbin completion �y of y . �corollary 4.5. if x is a compact hausdor� homogeneous extension of a sep-arable pseudocompact diagonalizable space y , then x is the stone�cech com-pacti�cation of y .proof. indeed, y is g�-dense in x, since y is pseudocompact, and x is hewitt-nachbin complete, since x is compact hausdor�. it remains to apply theorem4.2. �the next statement is proved by a similar argument.corollary 4.6. if x is a hausdor� compacti�cation of a separable pseudocom-pact space y , and x is diagonalizable, then x is the stone�cech compacti�ca-tion of y .we know that every zero-dimensional hausdor� space of countable pseu-docharacter is diagonalizable. we also established several conditions underwhich diagonalizable spaces are moscow or even have countable pseudocharac-ter. since the class of moscow spaces is an extension of the class of spaces ofcountable pseudocharacter, it is natural to ask if every zero-dimensional moscowspace is diagonalizable. theorem 4.1 is instrumental in �nding a compact coun-terexample.example 4.7. let �! be the stone-�cech compacti�cation of the discrete space!, and e 2 �! n !. let us show that �! is not diagonalizable at e.assume the contrary. then the space z = �! � �! is, obviously, diagonal-izable at the point (e;e). since the space �! is separable, the space z is alsoseparable. now it follows from theorem 4.1 that z is moscow at the point(e;e). however, this is not the case, as it was shown in [5]. thus, not everycompact moscow space of countable �-tightness is diagonalizable.the next two results are related in an obvious way to the classical theoremsin [10] and [12] (see also [6] and [4]).theorem 4.8. assume that y� is a separable tychono� space with a diag-onalizable hewitt-nachbin complete g�-extension x�, for each � 2 a, wherejaj � 2!. then the next formula holds for the hewitt-nachbin extensions �y�:�f�y� : � 2 ag = ��fy� : � 2 ag:proof. by theorem 4.2, �y� = x�, for each � 2 a. therefore, �f�y� : � 2 agis a g�-extension of the space �fy� : � 2 ag. obviously, �f�y� : � 2 ag ishewitt-nachbin complete. applying again theorem 4.2 and proposition 1.17,we conclude that �f�y� : � 2 ag = ��fy� : � 2 ag. � an operation on topological spaces 25corollary 4.9. assume that y� is a separable pseudocompact space with adiagonalizable hausdor� compacti�cation by�, for each � 2 a, where jaj � 2!.then the next formula holds for the stone�cech compacti�cations �y�:�f�y� : � 2 ag = ��fy� : � 2 ag:proof. to deduce this statement from theorem 4.8, it is enough to observe thatevery pseudocompact space is g�-dense in each hausdor� compacti�cation of it,and that the hewitt-nachbin completion of any pseudocompact space coincideswith the stone-�cech compacti�cation of y . �we conclude this section with the next obvious corollary of theorem 4.1.corollary 4.10. if a separable tychono� space x is diagonalizable at a pointe 2 x, and x n feg is hewitt-nachbin complete, then e is a g�-point in x.5. continuously diagonalizable spacesfollowing m.g. tkachenko [14], we say that the o-tightness of a space x at apoint e 2 x is countable (and write ot(e;x) � !) if, for each family of opensubsets of x such that e 2 [ , there exists a countable subfamily � of suchthat e 2 [�. if this is true for every point e in x, we say that the o-tightnessof x is countable.theorem 5.1. if a space x is continuously diagonalizable at a point e 2 x,and the o-tightness of x at e is countable, then x is moscow at e.proof. let u be any open subset of x such that e is in the closure of u.obviously, we may assume that e is not in u.let � be the family of all open subsets w of x such that, for some openneighborhood ow of e (which we now �x), xy 2 u for each x 2 ow and eachy 2 w (that is, �(ow � w) � u). then � is a base of the space u, since theoperation � is jointly continuous at (e;x), for each x 2 x.therefore, e 2 [�. since the o-tightness of x is countable, it follows thatthere exists a countable subfamily � of � such that e is in the closure of [�.put g = [� and p = \fow : w 2 �g. then p is a g�-set in x, since � iscountable, and e 2 p , e 2 g.take any a 2 p . we want to show that a 2 u. we may assume that a is note, since e 2 u. then, for each w 2 �, a 2 ow which implies that aw � u.therefore, ag � u. since ax depends continuously on the second argument xat x = e, and e 2 g, it follows that ae 2 ag � u. finally, since ae = a, weobtain: a 2 u, that is, e 2 p � u, and x is a moscow space. �corollary 5.2. if a space x is continuously diagonalizable at a point e 2 x,and the souslin number of x is countable, then x is a moscow space.proof. it is enough to observe that if the souslin number of x is countable,then the o-tightness of x is also countable [14]. �corollary 5.3. if x is a continuously diagonalizable tychono� space with thecountable souslin number, then every g�-dense subspace y of x is c-embeddedin x. 26 a.v. arhangel0ski��proof. the space x is moscow, by theorem 5.1. it follows from theorem 3.1that every g�-dense subspace y of x is c-embedded in x. the next statementis a typical application of theorem 2.2. �corollary 5.4. if a tychono� space x is continuously diagonalizable at e 2 x,and the o-tightness of x at e is countable, then either e is a g�-point in x, orthe subspace y = x n feg is c-embedded in x.proof. assume that e is not a g�-point in x. then y is g�-dense in x. bytheorem 5.1, x is moscow at e. since y is g�-dense in x, it follows, bytheorem 3.1, that y is c-embedded in x. �corollary 5.5. suppose a tychono� space x is continuously diagonalizable ate 2 x, the o-tightness of x at e is countable, and the subspace y = x n feg ishewitt-nachbin complete. then e is a g�-point in x.proof. assume that e is not isolated in x. then, since y = x n feg is hewitt-nachbin complete, y is not c-embedded in x. now it follows from 5.4 that eis a g�-point in x. �the next result should be compared to 3.5corollary 5.6. assume that x is a pseudocompact tychono� space contin-uously diagonalizable at a point e 2 x such that the o-tightness of x at e iscountable. then either x is �rst countable at e, or the subspace x n feg ispseudocompact.proof. since every pseudocompact tychono� space is �rst countable at everyg�-point, from corollary 5.4 it follows that the subspace y = x n feg is c-embedded in x. therefore, since x is pseudocompact, the space y must bepseudocompact as well. �many results, proved in the previous section for separable diagonalizablespaces, have their counterparts for continuously diagonalizable spaces with thecountable souslin number. their proofs do not di�er much, so we just formulatea few such results below, omitting the proofs.theorem 5.7. assume that a tychono� space x is a g�-extension of a spacey such that the souslin number of y is countable, and assume also that xis continuously diagonalizable and hewitt-nachbin complete. then x is thehewitt-nachbin completion �y of y .theorem 5.8. if a tychono� space x is a homogeneous g�-extension of acontinuously diagonalizable space y with the countable souslin number, and xis hewitt-nachbin complete, then x is the hewitt-nachbin completion �y ofy .corollary 5.9. if x is a compact hausdor� homogeneous extension of a pseu-docompact continuously diagonalizable space y with the countable souslin num-ber, then x is the stone�cech compacti�cation of y . an operation on topological spaces 27corollary 5.10. assume that y� is a separable pseudocompact space with acontinuously diagonalizable hausdor� compacti�cation by�, for each � 2 a.then the next formula holds for the stone�cech compacti�cations �y�:�f�y� : � 2 ag = ��fy� : � 2 ag:in connection with corollaries 5.9 and 5.10, see [8] and [10].references[1] arhangel0ski�� a.v., functional tightness, q-spaces, and �-embeddings, comment. math.univ. carol. 24:1 (1983), 105{120.[2] arhangel0ski�� a.v., on bicompacta hereditarily satisfying souslin's condition. tightnessand free sequences. soviet math. dokl. 12 (1971), 1253{1257.[3] arhangel0ski�� a.v., topological function spaces. kluwer academic publishers, 1992.[4] arhangel0ski�� a.v., topological groups and c-embeddings. submitted, 1999.[5] arhangel0ski�� a.v., on a theorem of w.w. comfort and k.a. ross, comment. math.univ. carolinae 40:1 (1999), 133{151.[6] arhangel0ski�� a.v., moscow spaces, pestov{tkachenko problem, and c-embeddings. toappear in cmuc.[7] arhangel0ski�� a.v. and v.i. ponomarev, fundamentals of general topology in problemsand exercises. (d. reidel publ. co., dordrecht-boston, mass., 1984).[8] van douwen e., homogeneity of �g if g is a topological group. colloquium mathematicum41 (1979), 193{199.[9] engelking r., general topology. pwn, warszawa, 1977.[10] glicksberg i., stone�cech compacti�cations of products, trans. amer. math. soc. 90(1959), 369{382.[11] hernandez s., sanchis m., and m.g. tkachenko, bounded sets in spaces and topologicalgroups. topology and appl. 101:1 (2000), 21{44.[12] hu�sek m., the hewitt realcompacti�cation of a product. comment. math. univ. carol.11 (1970), 393{395.[13] reznichenko e.a., v.v. uspenskij, pseudocompact mal0tsev spaces. topology and appl.86 (1998), 83{104.[14] tkachenko m.g., the notion of o-tightness and c-embedded subspaces of products, topol-ogy and appl. 15 (1983), 93{98.[15] uspenskij v.v., topological groups and dugundji spaces, matem. sb. 180:8 (1989), 1092{1118. received march 2000 arhangel0ski��, a.v.(january 1-june 15):department of mathematicsmorton hall 321ohio university,athens, ohio 45701usae-mail address: arhangel@bing.math.ohiou.edu 28 a.v. arhangel0ski��(june 15{december 30):kutuzovskij prospect, h. 33 apt. 137moscow 121165russiae-mail address: arhala@arhala.mccme.ru @ appl. gen. topol. 15, no. 2(2014), 147-154doi:10.4995/agt.2014.3181 c© agt, upv, 2014 the classical ring of quotients of cc(x) papiya bhattacharjee a, michelle l. knox b and warren wm. mcgovern c a penn state erie, the behrend college, erie, pa 16563 (pxb39@psu.edu) b midwestern state university, wichita falls, tx 76308 (michelle.knox@mwsu.edu) c h. l. wilkes honors college, florida atlantic university, jupiter, fl 33458 (warren.mcgovern@fau.edu) abstract we construct the classical ring of quotients of the algebra of continuous real-valued functions with countable range. our construction is a slight modification of the construction given in [3]. dowker’s example shows that the two constructions can be different. 2010 msc: 54c40, 13b30. keywords: ring of continuous functions; ring of quotients; zero-dimensional space. 1. introduction our aim here is two-fold. we aim to add to the growing knowledge regarding the ring of continuous functions of countable range on the space x, denoted by cc(x), while also supplying a correction to representation of the classical ring of quotients of cc(x), denoted qc(x). in this section we supply the relevant definitions and concepts. in section 2, we construct qc(x) in the same vein as the representation theorems of fine, gillman, and lambek [2]. the third section is devoted to studying a specific space which shows why the construction in section 2 is needed. we also include an example of mysior that is peculiar in its own right. throughout this paper, x will denote a zero-dimensional hausdorff space, that is, a hausdorff space with a base of clopen sets. the ring of all realvalued continuous functions on x is denoted by c(x), and the subring of c(x) consisting of those functions with countable range is denoted by cc(x). received 23 may 2013 – accepted 14 november 2013 http://dx.doi.org/10.4995/agt.2014.3181 p. bhattacharjee, m. l. knox and w. wm. mcgovern we may restrict to the class of zero-dimensional spaces because, as it is argued in [3] and [12], for any space y there is a zero-dimensional hausdorff space x such that cc(y ) and cc(x) are isomorphic as rings. one of the first results concerning cc(x) was by w. rudin [14] who showed that a compact space x satisfies c(x) = cc(x) precisely when x is scattered. in [9] the authors studied general zero-dimensional spaces for which cc(x) = c(x), calling such a space functionally countable. recently, there has been an interest in cc(x) as a ring in its own right. recall that for f ∈ c(x), z(f) denotes its zero-set: z(f) = {x ∈ x : f(x) = 0}. the set-theoretic complement of a zero-set is known as a cozero-set and we label this set by coz(f). we denote the collection of all cozero-sets by coz(x) and use clop(x) to denote the boolean algebra of clopen subsets of x. when we consider cozero-sets arising from functions in cc(x), we get what is denoted in [6] by clop(x)σ: the set of all countable unions of clopen subsets, i.e. the set of all σ-clopen sets. one of the main differences between c(x) and cc(x) is the realization of the maximal ideal spaces. the gelfand-kolmogorov theorem states max(c(x)) is homeomorphic to the stone-čech compactification of x, denoted βx. on the other hand, the maximal ideal space of cc(x) is homeomorphic to the banaschewski compactification of x, denoted β0x. (a proof can be modeled after theorem 5.1 of [6].) the most well-known way of realizing βx is as the collection of all z-ultrafilters of x. a nice way to view β0x is as the stone dual of clop(x). in general, βx and β0x are not homeomorphic. the next theorem characterizing when they are is well-known (see chapter 6.2 of [1]). theorem 1.1. for a zero-dimensional space x the following statements are equivalent. (1) βx = β0x (2) every cozero-set is a σ-clopen set. (3) βx is zero-dimensional. zero-dimensional spaces x for which βx is zero-dimensional are known as strongly zero-dimensional spaces. in this short article we are interested in zerodimensional spaces which are not strongly zero-dimensional. the third section deals with one of the most well-known of such examples. as for references, the text [4] is still pivotal. we also mention [13] and [1] for topological considerations, definitions, and concepts not explicitly discussed here. we end this section with a remark about σ-clopens. remark 1.2. it should be apparent that if u ∈ clop(x)σ, then there is some f ∈ cc(x) such that coz(f) = u. a sketch of this proof is as follows. first write u = ⋃ n kn as a disjoint union of a countable number of clopen sets. next, consider the function (call it f) that maps ki to 1 i and the rest of x to 0. this function is continuous and is an element of cc(x). finally, coz(f) = u. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 148 the classical ring of quotients of cc(x) 2. classical ring of quotients in [12] the authors constructed both the classical ring of quotients and the maximum ring of quotients of cc(x). they modeled their construction after the representation theorems of fine, gillman, and lambek [2]. we recall these after we set up some notation. let g(x) = {u ⊆ x : u is a dense open subset of x} and g0(x) = {o ⊆ x : o is a dense cozero-set of x}. we let q(x) and qc(x) denote the classical rings of quotients of c(x) and cc(x), respectively. we let q(x) and qc(x) denote the maximum rings of quotients of c(x) and cc(x), respectively. theorem 2.1 ([2]). suppose x is a tychonoff space. then q(x) = lim o∈g0(x) c(o) and q(x) = lim u∈g(x) c(u). in the above theorem the use of the limit is to describe that the rings are direct limits of rings of continuous functions. this direct limit can be described as the union of the rings c(u) modulo the equivalence that f1 ∈ c(u1), f2 ∈ c(u2) are equivalent if they agree on u1 ∩ u2. in [12] the authors classified qc(x) and qc(x) for zero-dimensional x as qc(x) = lim o∈g0(x) cc(o) and qc(x) = lim u∈g(x) cc(u). unfortunately, we believe that the classification of qc(x) is incorrect. we now construct the classical ring of quotients of cc(x). let gσ(x) = {k ∈ x : k is a dense σ-clopen set of x}. for the purpose of comparison, we will denote q′(x) = lim o∈g0(x) cc(o). theorem 2.2. let x be a zero-dimensional space. then qc(x) = lim u∈gσ(x) cc(u). proof. given that x ∈ gσ(x) we have an embedding cc(x) ≤ qc(x) ≤ q ′(x). first let f ∈ cc(x) be a non zero-divisor element of cc(x). we claim that coz(f) is a dense subset of x. if not, then x r clx coz(f) is a nonempty open subset and hence there is a nonempty clopen set of x which is disjoint from clx coz(f). the characteristic function on said clopen set is a non-zero function belonging to cc(x) and annihilates f, a contradiction. thus, coz(f) ∈ gσ(x). restricting f to coz(f) produces an element which is invertible in cc(coz(f)) and hence also in qc(x). so every regular element of cc(x) is c© agt, upv, 2014 appl. gen. topol. 15, no. 2 149 p. bhattacharjee, m. l. knox and w. wm. mcgovern invertible in qc(x). it follows (by a straightforward ring theoretic argument) that the classical ring of quotients of cc(x) is embedded inside of qc(x). as for the reverse inclusion, the proof follows mutatis mutandi from the proof of the representation theorem of [2] (theorem 2.6). this was attempted in theorem 2.12 of [12]. the only error made there was that when taking a dense cozeroset u of x there might not be a d ∈ cc(x) such that coz(d) = u. in fact, the only change needed from their proof is the modification we have suggested in using gσ(x) instead of g0(x). � remark 2.3. observe that qc(x) ≤ q ′(x) since gσ(x) ⊆ g0(x). one might question whether both direct limits produce the same rings. in the next section we will exhibit an example of a zero-dimensional space for which qc(x) < q ′(x). of course, if x is strongly zero-dimensional, then g0(x) = gσ(x) and hence qc(x) = q ′(x). remark 3.10 discusses the equality and the question of whether such a space need be strongly zero-dimensional. we end this section with some remarks and results whose proofs are in the same vein as above. corollary 2.4. let f be a proper subfield of r. for a zero-dimensional space x, the classical ring of quotients of c(x, f) is lim u∈gσ(x) c(u, f). in particular, the classical ring of quotients of c(x, q) is q(x, q) = lim u∈gσ(x) c(u, q). 3. a counterexample we let x be the space defined in 4v of [13]. we give a brief sketch of the construction. let w denote the space of countable ordinals equipped with the interval topology. for an ordinal σ, we use w(σ) to denote the set of ordinals smaller than σ. notice that w(ω1) = w. let j = r r q. for x ∈ j, let jx = {x + r : r ∈ q} and j = {jx : x ∈ j}. re-index j by j = {jα : α ∈ w} so that jα ∩ jβ = ∅ whenever α 6= β. for α < ω1, let uα = rr ⋃ {jβ : α < β < ω1}, and let x = ⋃ {{α}×uα : α < ω1}. equip x with the subspace topology from w×r; x is a tychonoff space. the space x is similar to dowker’s example from [1]. x is a classical example of a zero-dimensional space that is not strongly zero-dimensional. another reference for this space is exercise 16m of [4]. for notational purposes, let xσ = {(τ, r) ∈ x : τ < σ}. in other words xσ = x ∩ (w(σ) × r). we denote the x-complement of xσ by x ′ σ and call such a set a cofinal band of x since x′σ = {(τ, r) ∈ x : σ ≤ τ}. since x is not strongly zero-dimensional, we know that not every cozeroset of x is a σ-clopen. we aim to convince the reader of three things. first, that if c is a σ-clopen of x, then either c or x r c is a subset of xσ for some σ ∈ w. second, if c is a σ-clopen, then x r clxc is also a σ-clopen. third, qc(x) < q ′(x). we let π : x → w be the continuous projection map. we recall some subsets of x defined in section 3 of [8]. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 150 the classical ring of quotients of cc(x) definition 3.1. let k be a clopen subset of x such that π(k) is cofinal in w. define the following sets for r ∈ r and ǫ > 0: sǫr = {σ ∈ w : ({σ} × (r − ǫ, r + ǫ)) ∩ x ⊆ k} and tk = {r ∈ r : s ǫ r is cofinal in w for some ǫ > 0}. proposition 3.18 of [8] states tk is an unbounded subset of r. here we show more: that tk is an open subset of r. proposition 3.2. suppose k is a clopen subset of x such that π(k) is cofinal in w. then tk = {r ∈ r : there is a σr ∈ w such that {(τ, r) ∈ x : σr ≤ τ} ⊆ k}. proof. for r ∈ r notice that by the construction of x, if (σ, r) ∈ x, then for all σ ≤ τ ∈ w, (τ, r) ∈ x. if r ∈ tk, then choose σ ∈ w such that (σ, r) ∈ x. set y = {(τ, r) ∈ x : σ ≤ τ} and observe that y is homeomorphic to w. let g = χk ∈ c(x) be the characteristic function on k; k = coz(g). the restriction of g to y is constant on a tail (see chapter 5 [4]), and r ∈ tk, so there is some σr ∈ w such that for all σr ≤ τ ∈ w, (τ, r) ∈ coz(g) = k. next, let r ∈ r have the property that there is σr ∈ w such that {(τ, r) ∈ x : σr ≤ τ} ⊆ k. assume, by way of contradiction, that r /∈ tk, so for each n ∈ n we have that s 1 n r is not cofinal in w. that is, for each n ∈ n there exists σn ∈ w where ({α} × (r − 1 n , r + 1 n )) ∩ x is not a subset of k for all α > σn. let σ = sup{σn : n ∈ n}, then for all α > σ, ({α} × (r − 1 n , r + 1 n )) ∩ x is not a subset of k. however, choosing a β ∈ w for which σ, σr ≤ β,then since k is open there must exist n ∈ n such that ({β} × (r − 1 n , r + 1 n )) ∩ x ⊆ k, a contradiction. therefore r ∈ tk. � proposition 3.3. let k be a clopen subset of x such that π(k) is cofinal in w. the set tk is a nonempty open subset of r. proof. as we did previously, let g = χk ∈ c(x) with coz(g) = k. let t ∈ tk, and suppose, by means of contradiction, that there is no open neighborhood of t contained in tk. then there is (without loss of generality) an increasing sequence of rationals, say {qn}, not belonging to tk which converges to t. note that g is eventually zero on a tail of [w × {qn}] ∩ x for each n ∈ n. thus for each n ∈ n there is a σn ∈ w such that for all σ > σn we have g((σ, qn)) = 0, i.e. (σ, qn) /∈ k for every σ > σn. let ζ = sup{σn : n ∈ n}. then consider an appropriate α > ζ such that for all α ≤ β (β, t) ∈ k. (such an α exists because t ∈ tk.) but then since k is open there is a rational qn < t such that (α, qn) ∈ k, a contradiction. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 151 p. bhattacharjee, m. l. knox and w. wm. mcgovern remark 3.4. we observe that the proof of proposition 3.2 actually shows that if k is clopen subset of x such that π(k) is cofinal in w, then for any r ∈ r the set {σ ∈ w : (σ, r) ∈ k} is either bounded by a countable ordinal or contains a tail of w. in other words, tk ={r ∈ r : there is a cofinal subset of w, say w, such that (τ, r) ∈ k for all τ ∈ w}. this is pivotal in proving our next result. proposition 3.5. let k be a clopen subset of x such that π(k) is cofinal in w. then π(x r k) is not cofinal in w. thus k contains a cofinal band of x. proof. we suppose that both π(k) and π(x r k) are cofinal in w. by the previous remark it follows that tk ∩ txrk = ∅ and tk ∪ txrk = r. by proposition 3.3 both tk and txrk are nonempty open sets. this produces a disconnection of r, the desired contradiction. � proposition 3.6. let k be a σ-clopen subset of x. then x r clxk is also a σ-clopen subset. proof. suppose k is a σ-clopen subset of x. first consider the case where π(k) is not cofinal in w. then k ⊆ xτ for some τ ∈ w. xτ is a separable zero-dimensional metrizable space and hence strongly zero-dimensional. hence xτ r clxk is a σ-clopen subset of xτ. since xτ is a clopen subset of x, it follows that xτ r clxk is a σ-clopen subset of x, thus x r clxk = x ′ τ ∪ (xτ r clxk) is σ-clopen subset of x. next consider the case where π(k) is cofinal in w. then for some τ ∈ w k contains the cofinal band x′τ . since x r clxk is an open subset of xτ . we mentioned in the previous paragraph that xτ is a separable zero-dimensional metrizable space and hence strongly zero-dimensional. it follows that xrclxk is a σ-clopen subset of xτ and thus a σ-clopen subset of x. � corollary 3.7. suppose k is a dense σ-clopen subset of x. then π(k) is cofinal, and thus k contains a cofinal band of x. theorem 3.8. for the space x, qc(x) < q ′(x). proof. let t1 = [w × (−∞, 0)] ∩ x and t2 = [w × (0, ∞)] ∩ x. both t1 and t2 are cozero-sets of x and hence so is t = t1 ∪ t2. moreover, t is a dense subset of x. we note that π(t ) is cofinal in x, but t does not contain a cofinal band. therefore, t is not a σ-clopen subset of x. it follows that t ∈ g0(x) r gσ(x). let f : t → r be defined by f(x) = { 1, if x ∈ t1 0, if x ∈ t2. then f ∈ cc(t ) and so f ∈ q ′(x). we claim that f /∈ qc(x). if it were, then there would exist a dense σ-clopen of x, say v ∈ gσ(x), and g ∈ cc(v ) such c© agt, upv, 2014 appl. gen. topol. 15, no. 2 152 the classical ring of quotients of cc(x) that f and g agree on t ∩ v . but since v is a dense σ-clopen set, v contains a cofinal band of x. therefore, v ∩ t equals t on this band and so g sends t1 to 1 and t2 to 0. but g is defined on the whole band, contradicting continuity at points of the form (τ, 0) for large enough τ. � remark 3.9. proposition 3.6 is interesting on its own. the proposition yields for a zero-dimensional space y , both rings qc(y ) and limu∈gσ(y ) c(u) are von neumann regular rings. the proof would be modeled after the proofs of proposition 1.2 [10] and theorem 1.3 of [7]. simply, you would need that x is σ-clopen complemented. the ring limu∈gσ(x) c(u) has not been studied except in the case that x is strongly zero-dimensional. we conjecture that the ring can be realized as the classical ring of quotients of the alexandroff algebra a(x) (see [6] or [5] for more information). remark 3.10. let x∗ denote the finer topology on r2 defined by mysior [11] using d = q × q. x∗ is another example of a zero-dimensional space that is not strongly zero-dimensional. the countable set d is precisely the set of all isolated points of x∗. thus, d is the smallest dense σ-clopen subset of x∗. it follows that qc(x ∗) = q′(x∗) = c(n) = q(x∗) = q(x∗). therefore, in general it is not the case that the equality qc(x) = q ′(x) forces x to be strongly zero-dimensional. we are unable to characterize in any nice way when qc(x) = q ′(x). acknowledgements. we would like to express our gratitude to the referee for his/her helpful comments. references [1] r. engelking, general topology, sigma ser. pure math. 6 (heldermann verlag, berlin, 1989). [2] n. j. fine, l. gillman and j. lambek, rings of quotients of rings of functions, lecture note series ( mcgill university press, montreal, 1966). [3] m. ghadermazi, o. a. s. karamzadeh and m. namdari, on the functionally countable subalgebra of c(x), rend. sem. mat. univ. padova, to appear. [4] l. gillman and m. jerison, rings of continuous functions, graduate texts in mathematics, 43, (springer verlag, berlin-heidelberg-new york, 1976). [5] a. hager, cozero fields, confer. sem. mat. univ. bari. 175 (1980), 1–23. [6] a. hager, c. kimber and w. wm. mcgovern, unique a-closure for some ℓ-groups of rational valued functions, czech. math. j. 55 (2005), 409–421. [7] m. henriksen and r. g. woods, cozero complemented spaces; when the space of minimal prime ideals of a c(x) is compact, topology appl. 141, no. 1-3 (2004), 147–170. [8] m. l. knox and w. wm. mcgovern, rigid extensions of ℓ-groups of continuous functions, czech. math. journal 58(133) (2008), 993–1014. [9] r. levy and m. d. rice, normal p -spaces and the gδ-topology, colloq. math. 44, no. 2 (1981), 227–240. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 153 p. bhattacharjee, m. l. knox and w. wm. mcgovern [10] r. levy and j. shapiro, rings of quotients of rings of functions, topology appl. 146/147 (2005), 253–265. [11] a. mysior, two easy examples of zero-dimensional spaces, proc. amer. math. soc. 92, no. 4 (1984), 615–617. [12] m. namdari and a. veisi, rings of quotients of the subalgebra of c(x) consisting of functions with countable image, inter. math. forum 7 (2012), 561–571. [13] j. porter and r. g. woods, extensions and absolutes of hausdorff spaces, (springerverlag, new york, 1988). [14] w. rudin, continuous functions on compact spaces without perfect subsets, proc. amer. math. soc. 8 (1957), 39–42. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 154 @ appl. gen. topol. 17, no. 1(2016), 71-81 doi:10.4995/agt.2016.4561 c© agt, upv, 2016 normally preordered spaces and continuous multi-utilities gianni bosi a, alessandro caterino b and rita ceppitelli b a deams, università di trieste, piazzale europa 1, 34127, trieste, italy (gianni.bosi@deams.units.it) b dipartimento di matematica ed informatica, università di perugia, via l. vanvitelli 1, 06123 perugia, italy (alessandro.caterino@unipg.it,rita.ceppitelli@unipg.it) abstract we study regular, normal and perfectly normal preorders by referring to suitable assumptions concerning the preorder and the topology of the space. we also present conditions for the existence of a countable continuous multi-utility representation, hence a richter-peleg multiutility representation, by assuming the existence of a countable net weight. 2010 msc: 54f05; 91b16. keywords: normally preordered space; perfectly normally preordered space; continuous multi-utility representation. 1. introduction the present paper can be viewed, in some sense, as a continuation of the paper [4] by bosi and herden. in that paper, the authors proved that a topological space (x,τ) is normal iff every d-i-closed preorder � on x gives (x,τ,�) the structure of a normally preordered space. moreover, they discussed the existence of continuous multi-utility representations and, in addition, investigated the relation between such a representation and the concept of a normally topological preordered space introduced by nachbin [18]. we recall that a (not necessarily total) preorder � on a topological space (x,τ) admits a (continuous) multi-utility representation if there exists a family f of (continuous) increasing real functions on the preordered topological space received 18 january 2016 – accepted 16 march 2016 http://dx.doi.org/10.4995/agt.2016.4561 g. bosi, a. caterino and r. ceppitelli (x,τ,�) such that, for all x,y ∈ x, x � y is equivalent to f(x) ≤ f(y) for all f ∈f. this kind of representation, whose main feature is to fully characterize the preorder, was first introduced by levin [16], who called functionally closed a preorder admitting a multi-utility representation. the first systematic study of multi-utility representations is due to ok [14], who presented different conditions for the existence of continuous multi-utility representations. minguzzi introduced the concept of a (continuous) richterpeleg multi-utility representation f of a preorder �. this is a particular kind of multi-utility representation where every function f ∈ f is a richter-peleg utility function for � (i.e., every function f ∈f is order-preserving). richterpeleg multi-utilities have been recently studied by alcantud et al. [1], who in particular were concerned with the case of countable richter-peleg multi-utility representations. our attention is primarily focused on regular, perfectly normal and strongly normal preorders. we prove that a topological space (x,τ) is perfectly normal [normal] iff every d-i-closed preorder � on x gives (x,τ,�) the structure of a perfectly normally [strongly normally] preordered space. a similar result does not hold for regular spaces. moreover, we furnish conditions for the existence of a countable continuous multi-utility representation by assuming the existence of a countable net weight, in the spirit of bosi et al. [3]. the concept of a submetrizable space is also profitably used in this direction. 2. notations and preliminaries let � be a preorder, i.e., a reflexive and transitive binary relation on some fixed given set x. the preorder � is said to be total if for any two elements x,y ∈ x either x � y or y � x. define, for every point x ∈ x, the sets d(x) := {y ∈ x | y � x} and i(x) := {z ∈ x | x � z}. a subset d of x is said to be decreasing if d(x) ⊂ d for all x ∈ d. by duality, the concept of an increasing subset i of x is defined. in addition, for every subset a of x we set d(a) := {y ∈ x | ∃x ∈ a | y � x} and i(a) := {z ∈ x | ∃x ∈ a | x � z}, i.e., d(a) is the smallest decreasing and i(a) the smallest increasing subset of x that contains a. ∆x = {(x,x) | x ∈ x} is the diagonal of x. if (x,�) is a preordered set, then a real function u on x is said to be (i) isotone (increasing) if, for every x,y ∈ x, [x � y ⇒ u(x) ≤ u(y)], (ii) order-preserving if it is increasing and, for every x,y ∈ x, [x ≺ y ⇒ u(x) < u(y)]. we denote by τnat the natural topology on the real line r. let τ be a topology on x. if � is a preorder on x, then the triplet (x,τ,�) is referred to as a preordered topological space. for every subset a of x, we denote by cl(a) its topological closure. for every subset a of x we denote, furthermore, by d(a) the smallest closed decreasing subset of x that contains a. analogously, we denote by i(a) the smallest closed increasing subset of x that contains a. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 72 normally preordered spaces and continuous multi-utilities a preorder � on a topological space (x,τ) is said to be (i) closed if � is closed as a subset of x ×x in the product topology τ × τ; (ii) semi-closed if d(x) = d({x}) and i(x) = i({x}) are closed subsets of x for every x ∈ x; (iii) d-i-closed if for every closed subset a of x both sets d(a) and i(a) are closed subsets of x. a preordered topological space (x,τ,�) is said to be (i) regularly preordered if for every point x ∈ x and for every increasing closed set f ⊂ x such that x /∈ f there are two disjoint open subsets u and v , decreasing and increasing respectively, such that x ∈ u and f ⊂ v . a dual property is required if x /∈ f and f is decreasing. (ii) normally preordered if for any two disjoint closed decreasing, respectively increasing, subsets a and b of x, there exist disjoint open decreasing, respectively increasing, subsets u and v of x such that a ⊂ u and b ⊂ v . (iii) strongly normally preordered if for any two closed subsets a and b of x such that not (y � x) for all x ∈ a and all y ∈ b there exist disjoint open decreasing, respectively increasing, subsets u and v of x such that a ⊂ u and b ⊂ v . (iv) perfectly normally preordered if for every a,b ⊂ x closed disjoint subsets of x, decreasing and increasing respectively, there is a continuous isotone function f : (x,τ,�) → ([0, 1],τnat,≤) such that a = f−1(0) and b = f−1(1). finally, we remember the following definition 2.1. a preorder � on a topological space (x,τ) is said to satisfy the continuous multi-utility representation property if a family f of continuous real functions f on (x,τ) can be chosen in such a way that x � y if and only if f(x) ≤ f(y) for every f ∈f. 3. regularly and normally preordered topological spaces bosi and herden [4, theorem 4.3] proved that a space (x,τ) is normal if and only if every d-i-closed preorder � gives (x,τ,�) the structure of a normally preordered space. we now show that a similar result does not hold for regular spaces. we note that a preordered topological space (x,τ,�) is regularly preordered if and only if for every x ∈ x and for every decreasing (increasing) open set c© agt, upv, 2016 appl. gen. topol. 17, no. 1 73 g. bosi, a. caterino and r. ceppitelli a ⊂ x with x ∈ a there are an open decreasing (increasing) set ax and a closed decreasing (increasing) set fx such that x ∈ ax ⊂ fx ⊂ a. proposition 3.1. if (x,τ) is a regular non-normal space, then there is a di-closed preorder defined on x which does not give (x,τ,�) the structure of a regularly preordered space. proof. let (x,τ) be a regular non-normal space and let f,g ⊂ x be two disjoint closed sets which cannot be separated by disjoint open subsets of x. we consider on x the preorder �= ∆x∪(f×f). if c ⊂ x then d(c) = i(c) = c if c ∩ f = ∅, otherwise d(c) = i(c) = c ∪ f. hence, � is d-i-closed. now, choose an element x ∈ f and suppose there are two open disjoint sets u,v ⊂ x, decreasing and increasing respectively, such that x ∈ u and g ⊂ v . then, since every decreasing set which intersects f contains f, we would have f ⊂ u and g ⊂ v , which is a contradiction. � we recall that a preorder � on a topological space (x,τ) is said to be icontinuous if, for every open subset a of x, d(a) and i(a) are both open. it is said to be ic-continuous if it is both i-continuous and d-i-closed (see künzi [15]). proposition 3.2. a space (x,τ) is regular if and only if every ic-continuous preorder gives (x,τ,�) the structure of a regularly preordered space. proof. let (x,τ) be regular and let � be a ic-continuous preorder on x. if x ∈ a, where a is an open decreasing subset of x, then, by regularity of x, there is an open subset o of x such that x ∈ o ⊂ cl(o) ⊂ a. since a is decreasing, we have that x ∈ o ⊂ cl(o) ⊂ d(cl(o)) ⊂ a where d(cl(o)) is closed (and decresasing) because � is d-i-closed. finally, we have that x ∈ d(o) ⊂ d(cl(o)) ⊂ a, where d(o) is open and decreasing. similarly, if x ∈ a and a is an open increasing subset of x, there is an open subset o of x such that x ∈ i(o) ⊂ i(cl(o)) ⊂ a. conversely, since the discrete preorder � is ic-continuous, it easily follows the regularity of (x,τ). � the following proposition furnishes a characterization of perfectly normally preordered spaces in terms of normally preordered spaces. the following definition is needed. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 74 normally preordered spaces and continuous multi-utilities definition 3.3. let (x,τ,�) be a preordered topological space. then (i) a decreasing (increasing) set f ⊂ x is said to be �-fσ if it is a countable union of decreasing (increasing) closed subsets of x; (ii) a decreasing (increasing) set g ⊂ x is said to be �-gδ if it is a countable intersection of decreasing (increasing) open subsets of x. we note that every open decreasing (increasing) subset of x is �-fσ if and only if every closed increasing (decreasing) subset of x is �-gδ. proposition 3.4. let (x,τ,�) be a preordered topological space. then (x,τ,�) is perfectly normally preordered if and only if (x,τ,�) is normally preordered and all closed decreasing and increasing subsets of x are �-gδ. proof. let (x,τ,�) be perfectly normal. then, clearly, it is normally preordered. now, let g be a closed decreasing subset of x and let f : x → [0, 1] be a continuous isotone function such that g = f−1(0). then, one has g = ∞⋂ n=1 f−1 [ 0, 1 n ) where the sets f−1 [ 0, 1 n ) are clearly decreasing open subsets of x. similarly, one can prove that if g is a closed increasing subset of x then it is �-gδ. conversely, suppose that a,b are disjoint closed subsets of x, decreasing and increasing respectively. by our assumptions, we have that x \ a =⋃∞ n=1 an, where every an is a closed increasing subset of x. since x is normally preordered there is a continuous isotone function fn : x → [0, 1] such that fn(a) = 0 and fn(an) = 1. then, f : x → [0, 1] defined by f = ∞∑ n=1 fn 2n : x → [0, 1] is clearly isotone, continuous because the series converges uniformly and a = f−1(0). similarly, there is a continuous isotone function g : x → [0, 1] such that b = g−1(1). then, it is easy to construct a continuous isotone function h : x → [0, 1] such that a = h−1(0) and b = h−1(1) (cf., for instance, minguzzi [17, proposition 5.1]). � we recall that a topological space (x,τ) is perfectly normal if and only if it is a normal topological space and every closed subset of x is a gδ-set. the previous proposition and theorem 4.3 in bosi and herden [4] allow us to prove the following proposition. proposition 3.5. a space (x,τ) is perfectly normal if and only if every di-closed preorder gives (x,τ,�) the structure of perfectly normally preordered space. proof. let (x,τ) be perfectly normal and � be a d-i-closed preorder on x. from bosi and herden [4, theorem 4.3 ], it follows that (x,τ,�) is normally preordered. now, suppose that f is a decreasing open subset of x. from the c© agt, upv, 2016 appl. gen. topol. 17, no. 1 75 g. bosi, a. caterino and r. ceppitelli perfect normality of (x,τ), there is a countable family {cn} of closed subsets of x such that f = +∞⋃ n=1 cn. since f is decreasing and � is a d-i-closed preorder, we also have that f = +∞⋃ n=1 d(cn). it follows that every decreasing open subsets of x is �-fσ. similarly, one proves that every increasing open subsets of x is also �-fσ. by the above proposition 3.5, it follows that (x,τ,�) is perfectly normally preordered. conversely, since the discrete preorder � is a d-i-closed preorder, it easily follows the perfect normality of x. � minguzzi [17, proposition 5.2] proved that every regularly preordered lindelöf space (x,τ,�), where � is semi-closed, is normally preordered. it turns out that this is true even if the preorder is not semi-closed. formally : proposition 3.6. every regularly preordered lindelöf space is normally preordered. we recall that a topological space (x,τ) is said to be hereditarily lindelöf if for every collection {oi}i∈i of open subsets of x there exists a countable subset j of i such that ⋃ i∈i oi = ⋃ j∈j oj. the following result about perfectly normally preordered spaces improves theorem 5.3 in minguzzi [17]. proposition 3.7. every regularly preordered hereditarily lindelöf space is perfectly normally preordered. proof. let (x,τ,�) be a regularly preordered hereditarily lindelöf space. by the previous proposition such a space is normally preordered. it remains to prove that every open decreasing (or increasing) subset of x is �-fσ. let a be an open decreasing (increasing) subset of x and let x ∈ a. since the space is regularly preordered, for every x ∈ a there are an open decreasing (increasing) set ux and a closed decreasing (increasing) set vx such that x ∈ ux ⊂ vx ⊂ a. since the space is hereditarily lindelöf, the open cover {ux : x ∈ a} of a has a countable subcover {uxn : n ∈ n}. clearly a = ⋃ n∈n uxn = ⋃ n∈n vxn and so a is �fσ. � the next proposition follows from theorem 4.3 and from the proof of theorem 4.8 in bosi and herden [4]. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 76 normally preordered spaces and continuous multi-utilities proposition 3.8. a topological space (x,τ) is normal if and only if every d-iclosed preorder � on (x,τ) gives (x,τ,�) the structure of a strongly normally preordered space. we recall that a topological space (x,τ) is said to be limit point compact if every infinite subset l of x has an accumulation point. proposition 3.9. let (x,τ) be a t2 limit point compact topological space. then the following conditions are equivalent (i) (x,τ) is normal (ii) every closed preorder � on x gives (x,τ,�) the structure of a strongly normally preordered space. proof. without supposing any separation axiom on x, as in bosi and herden [4, theorem 4.8], it is possible to prove the implication (i) =⇒ (ii). conversely, if we assume (ii), then the discrete order � on x, which is closed since (x,τ) is t2, gives (x,τ,�) the structure of a strongly normally preordered space, that is to say, (x,τ) is normal. � we observe that there exist non-normal t1 compact spaces (hence limit point compact) that satisfies property (ii) in the above proposition. to this purpose, consider the following example. example 3.10. let x be an infinite set endowed with the cofinite topology τ1, that is τ1 = {a ⊂ x : |x \ a| < ℵ0} ∪ {∅}. it is easy to see that cl(∆x) = x × x. hence the only closed preorder of x is �= x × x. it follows that there is no pair (a,b) of non-empty closed disjoint subsets of x such that ¬(x � y) for all x ∈ a and y ∈ b. so (x,τ1,�) is strongly normally preordered. the compact space (x,τ1) is also an example of a non-normal t1 space, every closed preorder of which has a continuous multi-utility representation. therefore, separation hypotheses stronger than t1 are needed in order to guarantee the validity of the necessary conditions of theorem 4.8 and theorem 4.9 in bosi and herden [4]. another example of a t0 but not t1 space is (y,τ2), where y = {a,b1,b2} and τ2 = {x,∅,{a},{a,b1},{a,b2}}. now we will prove that theorem 4.9 in [4] holds under the hypothesis t3. proposition 3.11. let (x,τ) be a t3-space. if every closed preorder � on (x,τ) has a continuous multi-utility representation then (x,τ) is normal. proof. first we prove that if � is a d-i closed preorder then � is also closed. suppose that ¬ [x � y]. then, x /∈ i(y) and i(y) is a closed subset of x since (x,τ) is t1 and � is d-i closed. so, by regularity of x, there is an open set o in x such that x ∈ o ⊂ cl(o) ⊂ x \ i(y). moreover, since � is d-i-closed, we also get x ∈ o ⊂ cl(o) = d(cl(o)) ⊂ x \ i(y). it follows that u = cl(o) and v = x \ cl(o) are disjoint neighborhoods of x and y, decreasing and increasing respectively. as known, this is equivalent to say that � is closed. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 77 g. bosi, a. caterino and r. ceppitelli by hypothesis, we obtain that every d-i closed preorder has a continuous multiutility representation. finally, from theorem 4.5 [4], that is true for t1 spaces, the thesis follows. � we note that in proposition 3.11 the space x is not assumed to be limit point compact. instead, this hypothesis allows to prove the sufficient condition of theorem 4.9 in [4]: proposition 3.12. let (x,τ) be a limit point compact space. if (x,τ) is a normal space then every closed preorder on (x,τ) has a continuous multi-utility representation. in fact, if (x,τ) is limit point compact then, as showed in the proof of theorem 4.8 in [4], every closed preorder � on x is d-i closed. thus, if � is a closed preorder, (x,τ,�) is normally preordered and, as known, � has a continuous multi-utility representation. propositions 3.11 and 3.12 investigate the relation between the existence of continuous multi-utility representation for closed preorders and the normality of the topological space. 4. countable continuous multi-utilities minguzzi [17] proved that, if (x,τ,�) is a second-countable regularly preordered space, where � is a closed preorder, then there is a countable continuous multi-utility representation for �. actually, using the same proof, the following result holds. proposition 4.1. let (x,τ,�) be a regularly preordered space with a closed preorder. if the product topology τ ×τ on x ×x is hereditarily lindelöf, then � has a countable continuous multi-utility representation. a family n of subsets of a topological space (x,τ) is called a network for x if every open subset of x is a union of elements of n . the network weight (or net weight ) of (x,τ) is defined by nw(x,τ) = min{|n| : n is a network for (x,τ)} + ℵ0. we just mention the fact that the existence of a countable net weight generalizes the concept of second countability of a topology (i.e., the existence of a countable basis). moreover, we remember that if a topological space x has a countable netweight then x ×x is hereditarily lindelöf. corollary 4.2. let (x,τ,�) be a regularly preordered space with a countable net weight and assume that � is a closed preorder. then � has a countable continuous multi-utility representation. the following example shows that, even if (x,τ,�) is a regularly preordered hereditarily lindelöf space and � is a total closed preorder (hence (x,τ,�) is perfectly normally preorded), then, in general, it doesn’t exist a countable continuous multi-utility representation for �. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 78 normally preordered spaces and continuous multi-utilities example 4.3. let j = (−1, 1] ⊂ r endowed with the sorgenfrey topology τ and with the preorder � defined by x � y ⇔| x |>| y | or (| x |=| y | and x < 0) or x = y ∀x,y ∈ (−1, 1]. it is well-known that the space (j,τ) is hereditarily lindelöf. moreover, it is easy to verify that � is semiclosed. in fact, if a > 0, then one has d(−a) = (−1,−a] ∪ (a, 1] , i(−a) = [−a,a] and d(a) = (−1,−a] ∪ [a, 1] , i(a) = (−a,a]. since � is total, then it is also closed. now, we observe that the preorder � defined on j is neither d-i-closed nor i-continuous, and so proposition 3.2 cannot apply to prove that (j,τ,�) is regularly preordered. in fact, if −1 ≤ −a < −b ≤ 0, then i((−a,−b]) = (−a,a) which is open but not closed. moreover, if 0 ≤ a < b ≤ 1, then i((a,b]) = (−1,−a]∪ [a, 1] which is closed but not open. however, it is not difficult to prove directly that (j,τ,�) is regularly preordered. let x ∈ j and suppose x = −a ≤ 0. if a ⊂ j is an open increasing set containing x, then x ∈ i(x) ⊂ a, hence x ∈ [−a,a] ⊂ a. since a is open and increasing, then there is b ∈ j with a < b < 1 such that x ∈ (−b,b) ⊂ a. if c ∈ j, with a < c < b, then x ∈ u = (−c,c] ⊂ a, where u is closed, open and increasing. now, suppose that a ⊂ j is an open decreasing set containing x. then one has x ∈ d(x) ⊂ a and so x ∈ v = (−1,−a] ∪ (a, 1] ⊂ a. clearly, v is closed, open and increasing. now, let x ∈ j and suppose x = a > 0. if a ⊂ j is an open increasing set containing x, then x ∈ i(x) ⊂ a, hence x ∈ u = (−a,a] ⊂ a. as above u is closed, open and increasing. if a ⊂ j is an open decreasing set containing x then x ∈ d(x) ⊂ a and it follows that x ∈ v = (−1,−a] ∪ [a, 1] ⊂ a. since a is open and decreasing, then there is b ∈ j, with 0 < b < a < 1, such that x ∈ (−1,−b) ∪ (b, 1] ⊂ a. finally, if c ∈ j, with b < c < a, then x ∈ u = (−1,−c] ∪ (c, 1] ⊂ a, where u is closed, open and increasing. in conclusion, � has a continuous multi-utility representation, since (j,τ,�) is (perfectly) normally preordered and � is closed. but it does not have a countable one, otherwise (see minguzzi [17, lemma 5.4]) it should also have a countable continuous multi-utility representation consisting of continuous utility functions. but this is not true (see bosi et al. [3]). we recall that a topological space (x,τ) is said to be submetrizable if there is a metric topology τ′ on x which is coarser than τ. moreover, (x,τ) is hemicompact if there is a countable family {kn} of compact subsets of x such that every compact subset of x is contained in some kn. finally, x is a kspace if a subset a ⊂ x is open if and only if a ∩ k is open in k, for every compact subset k of x. in minguzzi [17] it is proved that a kω-space, that is a hemicompact k-space, is normally preordered with respect to every closed preorder. caterino et al. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 79 g. bosi, a. caterino and r. ceppitelli [12, theorem 3.5] showed that every closed preorder defined on a submetrizable kω-space has a continuous utility representation. an important example of a submetrizable kω-space is the space of the tempered distributions [9, 10, 11]. a submetrizable kω-space x is a union of a countable family of compact metrizable subspaces kn. since every kn has a countable basis bn, then b = ∪∞n=1bn is a countable network for x. therefore, we have the following result. corollary 4.4. let (x,τ,�) be a submetrizable kω-space, where � is a closed preorder. then � has a countable continuous multi-utility representation. 5. conclusions in this paper we have presented several results on regularly and normally preordered spaces and we have investigated the existence of continuous multiutility representations for not necessarily total preorders defined on these spaces. different topological conditions are considered to this aim, which can be thought of as interesting in mathematical economics, such as submetrizability or the existence of a countable net weight, as well as suitable “continuity conditions” of the preorder, such as the assumption according to which the preorder is closed. in the last section of this paper, we have proved some sufficient conditions that guarantee the existence of a countable continuous multi-utility representation for every closed preorder. we hope that in the future we shall be able to characterize the preordered spaces in which every closed preorder has a countable continuous multi-utility representation. we note that theorem 3.4 in [5] represents a result of this type with respect to the existence of continuous multi-utility representations of every closed preorder. acknowledgements. this research was carried out within the gruppo nazionale per l’analisi matematica, la probabilità e le loro applicazioni, istituto nazionale di alta matematica (italy). references [1] j. c. r. alcantud, g. bosi and m. zuanon, richter-peleg multi-utility representations of preorders, theory and decision 80 (2016), 443–450. 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[18] l. nachbin, topology and order, van nostrand, princeton, 1965. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 81 almirag.dvi @ applied general topology c© universidad politécnica de valencia volume 5, no. 1, 2004 pp. 4970 a topological approach to best approximation theory samuel g. moreno, jose maŕıa almira, esther m. garćıa–caballero and j. m. quesada∗ abstract. the main goal of this paper is to put some light in several arguments that have been used through the time in many contexts of best approximation theory to produce proximinality results. in all these works, the main idea was to prove that the sets we are considering have certain properties which are very near to the compactness in the usual sense. in the paper we introduce a concept (the wrapping) that allow us to unify all these results in a whole theory, where certain ideas from topology are essential. moreover, we do not only cover many of the known classical results but also prove some new results. hence we prove that exists a strong interaction between general topology and best approximation theory. 2000 ams classification: 41a65, 54-99 keywords: wrapping, best approximation, proximinality, compactness. 1. introduction one of the central problems in best approximation theory can be roughly formulated in the following way. let x be a set, a a nonempty subset of x and x ∈ x. if there exists a real valued function d on x × x that provides a notion of gap between points in x, we want to know about the existence of points a ∈ a such that d(x, a) = inf b∈a d(x, b). the points a ∈ a satisfying the previous relation are called the best approximations to x from a. the subset a is said to be proximinal if for all x ∈ x there exists a best approximation to x from a. ∗research supported by junta de andalućıa, grupos fqm0178 and fqm0268. 50 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada the proximinality of a subset a can be assured if there exists a topology for x large enough to be the functions d(x, ·) : a −→ r sequentially lower semicontinuous and small enough in order to be the domain a countably compact. the main tools used in the known proofs of the results concerning the existence of best approximations can be clearly distinguished in the following cases. (a): if x is a normed space and the gap function is the metric derived from the norm, the functional analysis arguments will give a common thread of compactness. as a sample of the most notorious results of this kind we give the following list. • nonempty closed subsets of finite dimensional subspaces of normed linear spaces are proximinal. • nonempty weakly closed subsets of a reflexive banach space (in particular, any nonempty closed convex subset) are proximinal. • nonempty weakly star closed subsets of the dual space of a normed space are proximinal. (b): if x is not normed, or x is normed but the gap function d is not a metric. in this situation the set x is usually a function space and the techniques used in the proofs will depend on each particular case. the arguments will derive, in general, from measure theory and they will provide the convergence of certain sequences of functions. as an example we mention that the subset of non decreasing functions in the orlicz space lφ(0, 1) is proximinal in the sense that for each function f ∈ lφ(0, 1) there is a non decreasing function g ∈ lφ(0, 1) that minimizes the gap function d(f, ·) = ∫ 1 0 φ(|f − ·|)dµ over the subset of non decreasing functions in lφ(0, 1). another interesting situation is when x is quasi-metric space (see for example [21], where a characterization of best approximation result is proved in this context). the main purpose of this paper is to give a common thread of compactness in the proofs of proximinality. to achieve our aim we introduce and extend some well known facts of best approximation theory to general topological spaces with some structure. our description covers and extends some classical situations in normed spaces (see [6]) and in metric spaces. moreover some recent developments in topological vector spaces are covered through our method. the present paper is organized as follows. in section 2 we state the main definitions. we use the following idea: let (x, ρ) be a metric space, x ∈ x and a a nonempty subset of x. the set of best approximations to x from a is defined by p(x, a) = {a ∈ a : ρ(x, a) = ρ(x, a)}, topological approach to best approximation 51 where ρ(x, a) = inf a∈a ρ(x, a). if b(x, r) denotes the closed ball of center x and radius r, we have p(x, a) = a ∩ b(x, ρ(x, a)) = a ∩   ⋂ r>ρ(x,a) b(x, r)   . this description gives the idea to extend some notions of best approximation theory to more general spaces. in order to estimate the gap between points in some space we will introduce the concept of wrapping as an increasing family of closed sets with nonempty interior whose union is the whole space. section 3 is devoted to the application of the framework introduced in the previous section to metric spaces, topological vector spaces and function spaces. we cover some of the classical situations and some recent descriptions due to different authors. in section 4 we show that our description gives a common thread of compactness for the proofs of proximinality in some of the most interesting and well known examples. finally, we also characterize proximinality in terms of countable compactness. the situation described here in order to introduce the notion of wrapping has been applied by other authors not only in approximation theory but also in selection theory (in michael’s approach, see [15]) and in fixed point theory (see [12]). 2. definitions and preliminaries results let (x, τ) be a hausdorff topological space and let us denote by c(x) the family of closed subsets of x. a wrapping for x is a function ξ : x× [0, ∞) −→ c(x) such that, for each x ∈ x, satisfies: (i) ξ(x, 0) = {x}, (ii) for all r > 0 there is an open set θ(x, r) with x ∈ θ(x, r) ⊂ ξ(x, r), (iii) for all r, s ≥ 0, if r ≤ s then ξ(x, r) ⊆ ξ(x, s), (iv) for all s ≥ 0, ⋃ r>s ξ(x, r) = x. the set ξ(x, r) is called the ξ-ball of center x and radius r. if ξ is a function from x × [0, ∞) into the power set p(x) that fulfills the axioms (iii) and (iv) of the above definition, then it is called a pre–wrapping for x. fixed a wrapping ξ for x, a a nonempty subset of x and x ∈ x, we define: • the radius set of the ξ-balls of center x with nonempty intersection with a (2.1) fξ(x, a) = {r ≥ 0 : a ∩ ξ(x, r) 6= ∅}, • the ξ-distance of x to a (2.2) dξ(x, a) = inf fξ(x, a), 52 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada • the set of best ξ-approximations to x from a (2.3) pξ(x, a) = a ⋂   ⋂ r∈fξ(x,a) ξ(x, r)   . the subset a will be called ξ-proximinal if pξ(x, a) is nonempty for all x ∈ x, and it will be called ξ-chebyshev if there is a function pξ(·, a) : x −→ a such that pξ(x, a) = {pξ(x, a)} for all x ∈ x (i.e. if pξ(x, a) is a singleton for all x ∈ x). by means of the previous definitions we have the following statements: (i) pξ(·, a) is a function from x to p(a), (ii) x ∈ a if and only if pξ(x, a) = {x}, (iii) the set fξ(x, a) is either the interval (dξ(x, a), ∞) or [dξ(x, a), ∞). moreover, in the last case we have pξ(x, a) = a ⋂ ξ(x, dξ(x, a)) 6= ∅, (iv) dξ(x, a) = inf a∈a dξ(x, {a}) = inf a∈a inf{r ≥ 0 : a ∈ ξ(x, r)}, (v) {y ∈ x : dξ(x, {y}) ≤ s} = ⋂ r>s ξ(x, r), for each x ∈ x, (vi) dξ(·, a) is a function from x to [0, ∞) and dξ(·, a)|a = 0. associated to the concept of wrapping, we would like to introduce the following properties: given ξ a wrapping for x, we say that it satisfies the intersection property: if for all x ∈ x and s ≥ 0, ξ(x, s) = ⋂ r>s ξ(x, r). triangular property: if for all x ∈ x, r ≥ 0, y ∈ ξ(x, r) and s ≥ 0, ξ(y, s) ⊂ ξ(x, r + s). a-adherence property: if for all x ∈ x, if a ⋂ ( ⋂ r>0 ξ(x, r) ) 6= ∅ then x ∈ a. if ξ satisfies the intersection property it is clear that (2.4) pξ(x, a) = a ∩ ξ(x, dξ(x, a)) = {a ∈ a : dξ(x, {a}) = dξ(x, a)}, and so pξ(x, a) 6= ∅ if and only if fξ(x, a) = [dξ(x, a), ∞). in case that ξ satisfies the triangular property, and x, y, z ∈ x, we have that for each ε > 0, y ∈ ξ(x, dξ(x, {y}) + ε 2 ) and z ∈ ξ(y, dξ(y, {z}) + ε 2 ), so z ∈ ξ(x, dξ(x, {y}) + dξ(y, {z}) + ε) for all ε > 0. therefore (2.5) dξ(x, {z}) ≤ dξ(x, {y}) + dξ(y, {z}) and hence the function ρ : x × x −→ [0, ∞) defined by ρ(x, y) = dξ(x, {y}) + dξ(y, {x}) topological approach to best approximation 53 is a pseudometric in x. reciprocally, if (2.5) is fulfilled and ξ satisfies the intersection property, it also satisfies the triangular property. without difficulty it can be shown that {x} = ⋂ r>0 ξ(x, r) for all x ∈ x, if and only if for all a ⊂ x the wrapping ξ has the a-adherence property. we can characterize the closed subsets in a by means of the proximinality and adherence properties as follows proposition 2.1. let (x, τ) be a hausdorff regular space. a nonempty subset a of x is closed if and only if there exists a wrapping ξ for x with the aadherence property such that a is ξ-proximinal. note 1. a hausdorff space is regular if for each point x and each closed subset a, if x 6∈ a, then there are disjoint open sets u and v such that x ∈ u and a ⊂ v. proof let a be a closed subset of x. if a = x we can choose the wrapping defined by ξ(x, 0) = {x} and ξ(x, r) = x for r > 0. if a is a proper subset of x and x ∈ x \ a there are disjoint open sets ux, vx such that x ∈ ux and a ⊂ vx; we define the wrapping ξ by: ξ(x, r) =    {x} if r = 0, x \ vx if 0 < r < 1, x if r ≥ 1, if x 6∈ a, and ξ(x, 0) = {x} and ξ(x, r) = x for r > 0 in other case. then, in case that x ∈ a we have pξ(x, a) = {x} and if x 6∈ a it follows that pξ(x, a) = a. consequently a is ξ-proximinal, and it is straightforward to verify that ξ has the a-adherence property. to show the converse suppose that a is not closed and let x ∈ a \ a. for every wrapping ξ, since the open sets θ(x, r) have nonvoid intersection with a, we have that pξ(x, a) = a ⋂ ( ⋂ r>0 ξ(x, r) ) . so if a is ξ-proximinal, the wrapping cannot have the a-adherence property. 2 note 2. the previous result is in the same line as theorem 6 of [11]. let ρ be a metric in a set x. for each x ∈ x, the function ρx : x −→ [0, ∞) defined by ρx(y) = ρ(y, x) gives the distance to x. by definition, the metric topology for x is the smallest one containing the sets s(x, r) := {y ∈ x : ρ(y, x) < r} = ρ−1x ([0, r)), for all x ∈ x and r ≥ 0. hence the metric topology for x is the smallest one that makes the functions ρx continuous at x. if a ⊂ x, the uniform continuity of the function ρ(·, a) follows from the symmetry and the triangle inequality of the metric ρ. with a similar pattern we can state the following result. 54 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada proposition 2.2. let ξ be a wrapping for x and ∅ 6= a ⊂ x. if ξ satisfies the triangular property and dξ(·, {x}) is continuous at x ∈ x, then dξ(·, a) is continuous at x. proof let {xi}i∈i be a net in x such that xi −→ x. given ε > 0, there is a i1 ∈ i such that xi ∈ ξ(x, ε 2 ) for i ≥ i1. for each i ∈ i, let ai ∈ a ∩ ξ(xi, dξ(xi, a) + ε 2 ). since ξ satisfies the triangular property, then ai ∈ ξ(x, dξ(xi, a) + ε) for i ≥ i1, and hence (2.6) dξ(x, a) − dξ(xi, a) ≤ ε. to prove the other inequality, first observe that x ∈ ξ(xi, dξ(xi, {x}) + ε 4 ) for all i ∈ i and let aε ∈ a ∩ ξ(x, dξ(x, a) + ε 4 ). since aε ∈ ξ(xi, dξ(xi, {x}) + dξ(x, a) + ε 2 ), then dξ(xi, a) − dξ(x, a) ≤ dξ(xi, {x}) + ε 2 . finally, notice that there exists i2 ∈ i such that (2.7) dξ(xi, a) − dξ(x, a) ≤ ε, for all i ≥ i2, since dξ(·, {x}) is continuous at x. if i0 is an upper bound of {i1, i2}, it follows from (2.6) and (2.7) that |dξ(xi, a) − dξ(x, a)| ≤ ε, for all i ≥ i0.2 3. metric spaces, topological vector spaces and function spaces. 3.1. metric spaces. let (x, ρ) be a metric space, and let ∅ 6= a ⊂ x. in the sequel, the closed ball of center x and radius r will be denoted by b(x, r). the function ξ : x × [0, ∞) −→ c(x) defined by ξ(x, r) = b(x, r) is the “natural” wrapping for x, and it clearly satisfies the intersection and the triangular properties. we recall that p(x, a) = {a ∈ a : ρ(x, a) = ρ(x, a)} = a ∩ b(x, ρ(x, a)) where ρ(x, a) is the distance of x to a. we can generalize this wrapping by merely introducing a real function f with suitable properties and considering the balls b(x, f(r)). in some sense, the best approximation problem with this new kind of wrapping will be show to be equivalent to the standard problem in the space (x, ρf ), where ρf is a metric derived from ρ and f. more precisely, a function f : [0, ∞) −→ [0, ∞) is said to be of type a (abbr. ta) if (i) f(r) = 0 if and only if r = 0, (ii) f is right continuous (lim r↓s f(r) = f(s)), (iii) f is superadditive (f(r) + f(s) ≤ f(r + s)). topological approach to best approximation 55 let f be a ta function and 0 ≤ r < s. we have f(r) < f(r)+f(s−r) ≤ f(s), thus f is strictly increasing. moreover, nf(1) ≤ f(n) for each nonnegative integer n, so that lim r→∞ f(r) = ∞. let us now give some natural examples of ta functions: the first one is given by f(r) = r + [r] (where [r] denotes the greatest integer smaller or equal to r) is a ta function. on the other hand, if f : [0, ∞) −→ [0, ∞) is a convex function that vanishes only at 0, then it is an strictly increasing continuous ta function: if p ∈ [0, ∞), then (3.8) f(λp) ≤ λf(p) + (1 − λ)f(0) = λf(p), for all λ ∈ [0, 1]. therefore, if 0 < r ≤ s and λ ∈ (0, 1], we have 1 λ f(λ(r + s)) ≤ f(r + s). taking λ = s r + s and using (3.8), we get f(r) + f(s) ≤ r s f(s) + f(s) ≤ f(r + s). let (x, ρ) be a metric space, and let f be a ta function. by the properties of f, it is clear that the function ξf , defined on x × [0, ∞) by ξf (x, r) = b(x, f(r)) is a wrapping for x, that satisfies the intersection and the triangular properties. moreover, we can state the following result. proposition 3.1. let (x, ρ) be a metric space, a a nonempty subset of x and f a ta function. for every x ∈ x, we have pξf (x, a) = a∩b(x, f(dξf (x, a))) where dξf (x, a) = inf{r ≥ 0 : f(r) ≥ ρ(x, a)}. proof the first relation follows from the intersection property of the wrapping ξf . clearly, {r ≥ 0 : f(r) > ρ(x, a)} ⊂ {r ≥ 0 : a∩b(x, f(r)) 6= ∅} ⊂ {r ≥ 0 : f(r) ≥ ρ(x, a)}, so that we have an inverted chain of inequalities of infimums of such sets. now f is right continuous and strictly increasing, so that inf{r ≥ 0 : f(r) > ρ(x, a)} = inf{r ≥ 0 : f(r) ≥ ρ(x, a)} and the second statement of the proposition holds.2 let f be a ta function, and let the right inverse of f be defined by f−1ex (x) = inf{r ≥ 0 : f(r) ≥ x} = inf f −1([x, ∞)), for x ≥ 0. we can prove now the following result. proposition 3.2. let f−1ex be the right inverse of a ta function f. then f−1ex is a function from [0, ∞) onto [0, ∞) which is non decreasing, continuous, subadditive ( f−1ex (x + y) ≤ f −1 ex (x) + f −1 ex (y) ) and vanishes only at 0. 56 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada proof since lim r→∞ f(r) = ∞, then f−1([x, ∞)) is nonempty (and bounded below) for each x ≥ 0. under the hypothesis on f (strictly increasing and right continuous), for each x ≥ 0 there exists s ≥ 0 such that f−1([x, ∞)) = [s, ∞), hence f−1ex (x) = min f −1([x, ∞)). let r ≥ 0 and x = f(r). it is clear that f−1ex (x) = r = f −1(x), hence f−1ex is a surjective extension of the inverse function f−1 : f([0, ∞)) −→ [0, ∞). also, if 0 ≤ x ≤ y, then [y, ∞) ⊂ [x, ∞) and f−1ex (x) = min f −1([x, ∞)) ≤ min f−1([y, ∞)) = f−1ex (y), so that f−1ex is non decreasing. let x > 0 and let a = lim y↓x f−1ex (y) and b = lim y↑x f−1ex (y). by the surjectivity of f−1ex , it follows that a = b. in the same way we get that 0 = f−1ex (0) = lim y↓0 f−1ex (y). hence f −1 ex is continuous in [0, ∞). the condition f−1ex (x) = 0 if and only if x = 0, is a direct consequence of the right continuity of f. finally, let x, y > 0. there exist r, s > 0 such that f(t) < x ≤ f(r) for t ∈ [0, r), and f(t) < y ≤ f(s) for t ∈ [0, s), so that x + y ≤ f(r) + f(s) ≤ f(r + s), and f−1ex (x + y) ≤ f −1 ex (f(r + s)) = r + s = f−1ex (x) + f −1 ex (y). hence f −1 ex is a subadditive function.2 let ρ be a metric for x and let f be a ta function. one can easily verify that the composition ρf = f −1 ex ◦ ρ is a metric in x. in fact, ρ and ρf are equivalent metrics. moreover, if bσ denotes the closed ball in the metric space (x, σ), we have that (3.9) bρ(x, f(r)) = bρf (x, r), which follows from bρ(x, f(r)) = {y ∈ x : ρ(y, x) ≤ f(r)} ⊂ {y ∈ x : f −1 ex (ρ(y, x)) ≤ r} = bρf (x, r), and from bρf (x, r) = {y ∈ x : f −1 ex (ρ(y, x)) ≤ r} ⊂ {y ∈ x : ρ(y, x) ≤ f(r)} = bρ(x, f(r)), (the last inclusion is due to ρ(x, y) ≤ f(f−1ex (ρ(x, y)))). we can also establish the following relation dξf (x, a) = inf{r ≥ 0 : f(r) ≥ ρ(x, a)} = f −1 ex (ρ(x, a)) = f−1ex ( inf a∈a ρ(x, a)) = inf a∈a f−1ex (ρ(x, a)) = inf a∈a ρf (x, a) = ρf (x, a).(3.10) in view of (3.9) and (3.10), proposition 3.1 takes the form topological approach to best approximation 57 proposition 3.3. let (x, ρ) be a metric space, a a nonempty subset of x and f a ta function. for every x ∈ x, we have pξf (x, a) = a ∩ bρf (x, ρf (x, a)). hence the best approximation problem in the metric space (x, ρ) with the wrapping ξf (x, r) = bρ(x, f(r)), is equivalent to the approximation problem in (x, ρf) with the usual wrapping ξ(x, r) = bρf (x, r). 3.2. topological vector spaces. in [17] the following problem has been considered. let x be a separated locally convex space and let f : x −→ r be a continuous convex function satisfying f(0) = 0. if a is a nonempty closed subset of x, the authors define the number fa(x) = inf{f(x − a) : a ∈ a}, and the so-called f–projection set pf,a(x) = {a ∈ a : f(x − a) = fa(x)}. they define properties of a related to the set valued mapping pf,a(·) and explore several relationships between these properties and the continuity of this mapping. taking into account that for each r > 0 the sub-level set cr = {x ∈ x : f(x) ≤ r} is a closed convex absorbing set containing 0 in its interior, is immediate to verify that the function ξ defined by ξ(x, r) = { {x} if r = 0, x − cr if r > 0, is a wrapping for x that satisfies the intersection property. of course we have dξ(x, a) = fa(x) and pξ(x, a) = pf,a(x), hence this minimization problem can be described and studied with the tools we have introduced in this paper. also the following problem can be found in [2]. let c be a closed bounded convex subset of a banach space x which has the origin as an interior point and let fc denote the minkowski functional with respect to c. given a nonempty closed bounded subset a ⊂ x and a point x ∈ x, we consider the minimization problem which consists in proving the existence of a point a0 ∈ a such that fc(a0 − x) = inf{fc(a − x) : a ∈ a}. in this subsection we are going to introduce a wrapping in a topological vector space in order to cover and extend the problem above. let (x, τ) be a (hausdorff) topological vector space, and let c be a convex neighborhood of 0. for all x ∈ x, it is easy to check that (i) for each r > 0, the set x+rc is an convex neighborhood of x contained in the closed convex set x + rc, (ii) if 0 ≤ r ≤ s, then x + rc ⊂ x + sc, 58 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada (iii) for each y ∈ x, there exists t ≥ 0 such that y belongs to x + tc, and so ⋃ r≥0 (x + rc) = x. in the light of the above properties, we can state the following result (we remind the reader that a halfline in x is a set of the form {x + ty : t ≥ 0}, where x, y ∈ x and y 6= 0). proposition 3.4. let (x, τ) be a hausdorff topological vector space with a convex neighborhood of the origin, c, whose closure does not contain halflines. the function ξc : x × [0, ∞) −→ c(x) defined by ξc(x, r) = x + rc is a wrapping for x that satisfies the intersection and the triangular properties. proof we will prove the following two statements: (1): for each x ∈ x, r ≥ 0, and for each y ∈ x + rc and s ≥ 0, y + sc ⊂ x + (r + s)c. (2): for each x ∈ x and s ≥ 0, x + sc = ⋂ r>s (x + rc). as the first step for the proof of (1) we shall prove that for all r, s ≥ 0, (3.11) rc + sc = (r + s)c. let x, y ∈ c and let r, s be positive. by convexity of c, it follows that 1 r + s (rx + sy) = r r + s x + s r + s y ∈ c, which implies that rc + sc ⊂ (r + s)c. the reverse inclusion is trivial. now, let r, s > 0 and let y ∈ x + rc. if z belongs to y + sc, then, using (3.11), we have z ∈ x + rc + sc = x + (r + s)c. this concludes the proof of (1). the proof of (2) goes as follows. since trivially x + sc ⊂ ⋂ r>s (x + rc), we have to prove the reverse inclusion. first consider s > 0, and let y ∈ ⋂ r>s (x + rc). if {rn} is a sequence of numbers greater than s and converging to s, we have that c ∋ 1 rn (y − x) → 1 s (y − x), and hence y ∈ x + sc. this implies ⋂ r>s (x + rc) ⊂ x + sc topological approach to best approximation 59 for each s > 0. finally, let y ∈ ⋂ r>0 (x + rc). by 1 r (y − x) ∈ c for each r > 0, if y 6= x, then the halfline {t(y − x) : t ≥ 0} is contained in c, which is absurd. so ⋂ r>0 (x + rc) ⊂ {x}, and this concludes the proof.2 observe that an open set which does not contain halflines needs not to be bounded. consider, for example, the set c = {{xn} ∈ l1 : |xn| < n for each n}, where l1 denotes the space of sequences {xn} ∞ n=1 of real numbers which are absolutely summable. it is clear that c is a convex set containing 0. moreover, let x = {xn} ∈ c and define εx = min n {n − |xn|}. if y ∈ {z ∈ l1 : ‖z − x‖1 < εx}, then, for each n, |yn| − |xn| ≤ |yn − xn| ≤ ‖y − x‖1 < εx ≤ n − |xn|, and therefore c is open. it is straightforward to verify that the convex set c = {{xn} ∈ l1 : |xn| ≤ n for each n} does not contain halflines and, however, it is unbounded (if xn = {(n − 1)δmn} ∞ m=1, then xn ∈ c and sup n ‖xn‖1 = ∞). let c be a convex subset of a hausdorff topological vector space, with 0 as an interior point and which contains no halflines. the minkowski functional of c, defined by fc(x) = inf{r > 0 : x r ∈ c}, is a non-negative, positive homogeneous, convex, continuous and subadditive function, that vanishes only at 0. moreover, the convex sets c1 = {x ∈ x : fc(x) < 1} and c2 = {x ∈ x : fc(x) ≤ 1} (open and closed, respectively) satisfy c1 ⊂ c ⊂ c ⊂ c2. the minkowski functional provides the following characterization of the set of best ξc-approximations. 60 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada proposition 3.5. let (x, τ) be a hausdorff topological vector space with a convex neighborhood of the origin, c, whose closure does not contain halflines, and let a be a nonempty subset of x. for each x ∈ x, we have pξc (x, a) = a ∩ ( x + dξc (x, a)(c \ c) ) (3.12) = {a ∈ a : f c (a − x) = dξc (x, a)}, where dξc (x, a) = inf a∈a f c (a − x). proof it follows from (2.4) that pξc (x, a) = a ∩ ( x + dξc (x, a)c ) . we first consider the case dξc (x, a) > 0. for a ∈ pξc (x, a), we will show that a 6∈ x+dξc (x, a)c. suppose this is not so. if {tn} is a sequence of positive numbers such that tn → 0, then (1 + tn)a − tnx → a, and hence there exists tm > 0 and c ∈ c such that (1 + tm)a − tmx = x + dξc (x, a)c, so that a = x + dξc (x, a) 1 + tm c ∈ x + dξc (x, a) 1 + tm c. hence we have dξc (x, a) ≤ dξc (x, a) 1 + tm , an absurd. this shows that a ∩ ( x + dξc (x, a)c ) ⊂ a ∩ ( x + dξc (x, a)(c \ c) ) . the reverse inclusion and the case dξc (x, a) = 0 are obvious. thus we have proved that pξc (x, a) = a ∩ ( x + dξc (x, a)(c \ c) ) . the proof ends by noting that dξc (x, {a}) = inf{r ≥ 0 : a ∈ x + rc} = inf{r ≥ 0 : a − x r ∈ c} = f c (a − x). 2 we have also the following characterization of the ξc-distance of x to a. proposition 3.6. if ∅ 6= a ∩ ( x + rc ) ⊂ x + r(c \ c) for some r ≥ 0, then r = dξc (x, a) and consequently pξc (x, a) is nonempty. proof it suffices to consider r > 0. first observe that d = dξc (x, a) ≤ r. supposing d < r, we get ∅ 6= a ∩ ( x + d + r 2 c ) ⊂ a ∩ ( x + rc ) ⊂ x + r(c \ c). thus there exist c1 ∈ c and c2 ∈ c \ c, such that d + r 2 c1 = rc2. topological approach to best approximation 61 let t = d+r 2r ∈ (0, 1). since c1 + 1−t −t c is a neighborhood of c1, there exists c3 ∈ c ⋂ ( c1 + 1−t −t c ) . therefore c2 = tc1 = tc3 + t(c1 − c3) ∈ tc + (1 − t)c ⊂ c. this contradicts that c2 6∈ c, and we conclude d = r. 2 the set of best ξc-approximations to x from a inherits the invariance properties of the space x, and it is easy to verify that for s > 0, t ≥ 0 and y ∈ x, we have pξ(sc) (tx − y, ta − y) = tpξc (x, a) − y. a convex subset u of a topological vector space will be called strictly convex provided that the relations x, y ∈ u \ int(u) and x 6= y imply x+y 2 ∈ int(u). note that if u is a convex neighborhood of the origin, then int(u) = u and u is strictly convex if u is so. it is possible to establish uniqueness as a consequence of strict convexity of the set that generates the wrapping.(for normed linear spaces see lemma 3.2. of [19]) proposition 3.7. let (x, τ) be a hausdorff topological vector space with a convex neighborhood of the origin, c, whose closure does not contain halflines, and let a be a nonempty, convex and ξc-proximinal subset of x. if c is strictly convex, then a is ξc-chebyshev. moreover, if for each convex and ξcproximinal subset a of x we have that a is ξc-chebyshev, then c is strictly convex. proof first consider that c is strictly convex. let a1, a2 ∈ pξc (x, a). in case a1 6= a2, we have d = dξc (x, a) > 0, and so ai−x d ∈ c \ c. by the assumption on c, we have a1+a2 2 − x d ∈ c. since a is convex, this means that a1+a2 2 ∈ a ∩ (x + dc), which contradicts (3.12). this proves the first claim. now suppose that c is not strictly convex. then there exist distinct points a1, a2 ∈ c \ c such that a1+a2 2 ∈ c \ c. let the mapping g from [0, 1] into x be defined by g(t) = (1 − t)a1 + ta2. since g is continuous and [0, 1] is compact, the convex set a = g([0, 1]) is compact. from (3.12) and the fact that f c (· − x) is continuous, it follows that a is ξc-proximinal. moreover, since fc(a) = 1 = dξc (0, a) for each a ∈ a (this is not difficult to check), then a = pξc (0, a). hence a is not ξc-chebyshev. this proves the second claim.2 for a ∈ a, sa will denote the set of all points in x having a as a best ξc-approximation, i.e. sa = {x ∈ x : a ∈ pξc (x, a)} = {x ∈ x : fc(a − x) = dξc (x, a)}. 62 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada clearly sa is nonempty (a ∈ sa). let {xd}d∈d be a net in sa which converges to x. then f c (a−xd) = dξc (xd, a). since fc(a−·) and dξc (·, a) are continuous, we have f c (a − x) = dξc (x, a), hence x ∈ sa and sa is closed. it is interesting to note that if we assume that a is convex we can prove a geometrical condition of the sets sa. this condition has been established, for normed linear spaces, in [19]. proposition 3.8. let (x, τ) be a hausdorff topological vector space with a convex neighborhood of the origin, c, whose closure does not contain halflines, and let a be a nonempty and convex subset of x. then for each a ∈ a, the set sa is a cone with vertex a. proof if x ∈ sa, and we denote d = dξc (x, a), then there exists c1 ∈ c \ c such that a = x + dc1. we shall prove that for each t ≥ 0 (3.13) ∅ 6= a ∩ ( (1 − t)a + tx + tdc ) ⊂ (1 − t)a + tx + td(c \ c). suppose this is not so. then for some b ∈ a and some c2 ∈ c, we would have b = (1−t)a+tx+tdc2 = (1−t)x+(1−t)dc1 +tx+tdc2 = x+d((1−t)c1 +tc2). in case that t ∈ (0, 1), by the fact that (1 − t)c1 + tc2 belongs to c, we have b ∈ x + dc, an absurd. if t ∈ (1, ∞), using that a is convex, we get 1 t b + (1 − 1 t )a = 1 t x + d((1 t − 1)c1 + c2) + (1 − 1 t )x + (1 − 1 t )dc1 = x + dc2 ∈ a, again a contradiction. from (3.13) and proposition 3.6 , we deduce dξc ((1 − t)a + tx, a) = tdξc (x, a), for all non-negative t. therefore f c (a − ((1 − t)a + tx)) = tf c (a − x) = tdξc (x, a) = dξc ((1 − t)a + tx, a). whence (1 − t)a + tx belongs to sa.2 without the hypothesis of convexity on a, we can establish the following corollaries of the above result. corollary 3.9. let (x, τ) be a hausdorff topological vector space with a convex neighborhood of the origin, c, whose closure does not contain halflines, and let ∅ 6= a ⊂ x. if a ∈ a and x ∈ sa, then (1 − t)a + tx ∈ sa for each t ∈ [0, 1]. corollary 3.10. let (x, τ) be a hausdorff topological vector space with a convex neighborhood of the origin, c, whose closure does not contain halflines, and let a be a nonempty and ξc-chebyshev subset of x. then pξc ((1 − t)pξc (x, a) + tx, a) = pξc (x, a), for each x ∈ x and t ∈ [0, 1]. the previous two statements are well known in the context of normed linear spaces (see [24]). topological approach to best approximation 63 3.3. function spaces. we recall that a convex function φ from [0, ∞) into [0, ∞) is said to be ∆2-convex at 0 if there exists k > 0 such that (3.14) φ(2x) ≤ k φ(x), for each x ≥ 0. if φ : [0, ∞) −→ [0, ∞) is a ∆2-convex function at 0, with φ(0) = 0 and φ 6≡ 0 then it follows rather easily that (a) for all x > 0, φ(x) > 0, (b) φ is superadditive , (c) φ is strictly increasing, (d) there exists m ≥ 1 such that φ(x+y) ≤ m(φ(x)+φ(y)) for all x, y ≥ 0. let (ω, a, µ) be a finite measure space, φ : [0, ∞) −→ [0, ∞) a ∆2-convex function at 0 and let lφ(ω) := lφ(ω, a, µ) denote the class of the µ-equivalent measurable functions, f : ω −→ r, such that ∫ ω φ(|f|)dµ < ∞, where we assume that f = g if µ{ω ∈ ω : f(ω) 6= g(ω)} = 0. it is well known that lφ(ω) is a real linear space. for r ≥ 0, f ∈ lφ(ω), we define the sets of functions, θ(f, r) = { g ∈ lφ(ω) : ∫ ω φ(|f − g|)dµ < r } and ξ(f, r) = { g ∈ lφ(ω) : ∫ ω φ(|f − g|)dµ ≤ r } . in order to define a topology in lφ(ω) we consider the family τφ of subsets of lφ(ω), (3.15) τφ = {o ⊂ lφ(ω) : ∀f ∈ o, ∃r > 0 such that θ(f, r) ⊂ o} proposition 3.11. the following conditions hold true (a) (lφ(ω), τφ) is a hausdorff topological space. (b) {θ(f, r) : f ∈ lφ(ω), r > 0} is a basis for τφ. (c) for all f ∈ lφ(ω) and r ≥ 0, ξ(f, r) is τφ–closed. proof (a) by definition (3.15), ∅ and lφ(ω) belong to τφ. moreover, if {oλ}λ∈λ is an arbitrary family of sets in τφ, is immediate that ∪λ∈λoλ ∈ τφ. finally, let oi, i = 1, 2, · · · , n, be a finite family of sets in τφ. for all f ∈ ∩ n i=1oi, there are ri > 0 such that θ(f, ri) ⊂ oi, i = 1, 2, · · · , n. taking r0 = min 1≤i≤n ri, we have θ(f, r0) = n ⋂ i=1 θ(f, ri) ⊂ n ⋂ i=1 oi, and therefore ∩ni=1oi ∈ τφ. this proves that τφ is a topology in lφ(ω). 64 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada now let f, g ∈ lφ(ω), f 6= g and consider α = ∫ ω φ(|f − g|)dµ > 0. in order to prove that (lφ(ω), τφ) is a hausdorff topological space, we will show that (3.16) θ (f, α/(2m)) ∩ θ (g, α/(2m)) = ∅. if (3.16) is false, then there exists h ∈ lφ(ω), such that h ∈ θ (f, α/(2m)) ∩ θ (f, α/(2m)). but this is not possible since ∫ ω φ(|f − g|)dµ ≤ ∫ ω φ(|f − h| + |g − h|)dµ ≤ m ( ∫ ω φ(|f − h|)dµ + ∫ ω φ(|g − h|)dµ ) < α. (b) by definition of τφ it suffices to show that θ(f, r) belongs to τφ. we will prove that for all g ∈ θ(f, r) there exists s > 0 such that θ(g, s) ⊂ θ(f, r). suppose the contrary. then there exists a sequence {gn} in lφ(ω), such that ∫ ω φ(|g − gn)|)dµ < 1 2n and ∫ ω φ(|f − gn|)dµ ≥ r. by the jensen inequality φ ( 1 µ(ω) ∫ ω |g − gn|dµ ) ≤ 1 µ(ω) ∫ ω φ(|g − gn|)dµ, we obtain ∫ ω |g − gn|dµ ≤ µ(ω)φ −1 ( 1 2nµ(ω) ) , and therefore ‖g − gn‖l1(ω) → 0. then there exists a subsequence {gnk} such that gnk → g, µ-almost everywhere in ω. since φ is continuous, then φ(|f − gnk |) → φ(|f − g|), µ-a.e. on the other hand, φ(|f − gnk|) ≤ m (φ(|f − g|) + φ(|g − gnk|)) ≤ m (φ(|f − g|) + h) , where h = ∞ ∑ k=1 φ(|g − gnk|). since, φ(|g − gnk|) ≥ 0 on ω, then ∫ ω hdµ = ∞ ∑ k=1 ∫ ω φ(|g − gnk|)dµ < ∞ ∑ k=1 1 2nk < ∞ and therefore m (φ(|f − g|) + h) ∈ l1(ω). finally, applying the lebesgue dominated convergence theorem, we get ∫ ω φ(|f − g|)dµ = lim k→∞ ∫ ω φ(|f − gnk|)dµ ≥ r, and we obtain a contradiction. (c) if we consider the complement of ξ(f, r), the proof follows the same pattern of (b).2 topological approach to best approximation 65 then we easily deduce the following corollary 3.12. the family {ξ(f, r) : f ∈ lφ(ω), r ≥ 0} is a wrapping for (lφ(ω), τφ). 4. ξ-proximinality in this section we prove a simple yet general proximinality result. it is general because it includes, as special cases, some of the most interesting and well known examples of proximinality. is simple because the proof requires nothing but the definition of countable compactness. we recall that a hausdorff topological space is called countably compact if every countable open covering has a finite subcovering. countable compactness admits a heine-borel type argument: a hausdorff topological space is countably compact if and only if every family of closed subsets having the finite intersection property also has the countable intersection property. as a consequence, each descending sequence of nonempty closed subsets has nonempty intersection. finally, let us remember that a space is countably compact if and only if every sequence of points of the space has an accumulation point. the following result generalizes a result by singer ([24], p. 383). theorem 4.1. let ξ be a pre–wrapping for (x, τ) and let a be a nonempty subset of x. if τ′ is a hausdorff topology in x such that for each x ∈ x there exists a non increasing sequence {εn} of positive numbers tending to 0 such that (i) a ∩ ξ(x, dξ(x, a) + ε1) is τ ′-countably compact, (ii) a ∩ ξ(x, dξ(x, a) + εn) is τ ′-closed for n > 1, then a is ξ-proximinal. proof if for each x ∈ x we define, for n ≥ 1, an = a ∩ ξ(x, dξ(x, a) + εn), then {an+1} is a non increasing sequence of nonempty and τ ′-closed subsets of a1. since a1 is τ ′-countably compact, (4.17) ∅ 6= ⋂ n≥1 an+1 = a ∩ ⋂ {ξ(x, r) : r > dξ(x, a)}. if we assume fξ(x, a) = (dξ(x, a), ∞), then by 4.17 we have that pξ(x, a) is nonempty. the same conclusion follows trivially in case fξ(x, a) = [dξ(x, a), ∞). thus a is ξ-proximinal.2 it may be surprising that some of the most famous existence results in normed linear spaces and also results in function spaces that are not normed can be obtained as consequences of the result above. the following list of examples is intended to be a representative sampling of this fact. a: nonempty closed subsets of finite dimensional subspaces of normed linear spaces are proximinal. 66 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada we consider the “natural” wrapping for the normed linear space, i. e. ξ(x, r) = b(x, r), where b(x, r) stands for the closed ball of center x and radius r. if a is a closed subset of a finite dimensional subspace, then the sets a ∩ b(x, r) are bounded closed subsets contained in a finite dimensional space, hence they are compact. this example is emblematic because it gives an affirmative answer, with an elegant formulation, to the problem that gave rise to best approximation theory, namely, the possibility on finding, within the algebraic polynomials of degree least or equal to a fixed n, the nearest one (in the sense of uniform norm) to some continuous fixed function. this result was proved by f. riesz [20] in 1918, although the corresponding result in the setting of polynomial approximation was proved by chebyshev in 1859 (see [3], [4] and [25]). note 3. for the following example, we recall first that σ(x, x∗) stands for the weak topology on x, i. e., the topology defined by the family of seminorms {px∗ : x ∗ ∈ x∗}, where px∗(x) = |x ∗(x)| for each x ∈ x. on the other hand, σ(x∗, x) stands for the weak star topology on x∗, i. e., the topology defined by the family of seminorms {px : x ∈ x}, where px(x ∗) = |x∗(x)| for each x∗ ∈ x∗. b: every nonempty σ(x, x∗)–closed subset of a reflexive banach space x is proximinal. we consider again the natural wrapping for x, i. e. ξ(x, r) denotes the closed ball of center x and radius r. if x is reflexive (i.e., the natural embedding of x into its double dual, x∗∗, is surjective), then the unit ball b(0, 1) is σ(x, x∗)– compact (in fact, this is a characterization of reflexivity, see [9]). the functions fx and gα (with x ∈ x and α 6= 0 ), defined by fx(y) = x + y and gα(y) = αy for all y ∈ x are homeomorphisms in (x, σ(x, x∗)). then, as the continuous image of a compact set is compact, we have that b(x, r) = x + rb(0, 1) = fx(gr(b(0, 1))) is σ(x, x∗)-compact for all x ∈ x and all r > 0. so if a is a nonempty and σ(x, x∗)–closed subset of a reflexive space, then the sets a ∩ b(x, r) are weakly compact. thus, a is ξ-proximinal. mazur proved (see [14]) that the norm closure of a convex subset equals its weak closure. then the nonempty closed convex subsets of a reflexive banach space are proximinal. this appears firstly in a paper of day (see [5]) in 1941. in fact, he gives this result for banach spaces with a weakly compact unit ball. day used the results of milman and alaoglu and birkhoff which gave the same result for uniformly convex spaces. by other way, smulian (see [23]) gave a characterization of the reflexivity of banach spaces in terms that every descending sequence of nonempty closed, bounded and convex subsets topological approach to best approximation 67 had nonvoid intersection. the previous example can be deduced from this characterization. c: nonempty and σ(x∗, x)-closed subsets of the dual space x∗ of a normed space x are proximinal. for x∗ ∈ x∗ and r ≥ 0, the closed balls (in the usual norm of the dual spaces) of center x∗ and radius r are used to define the wrapping. more precisely, ξ(x∗, r) = b∗(x∗, r) = {y∗ ∈ x∗ : ‖y∗ − x∗‖ ≤ r} where ‖x∗‖ = sup {|x∗(x)| : x ∈ b(0, 1)}. the alaoglu theorem states that b∗(0∗, 1) is σ(x∗, x)-compact. the balls b∗(x∗, r) are then weakly star compact. if a is a nonempty and σ(x∗, x)closed subset of x∗, then the sets a ∩ b∗(x∗, r) are σ(x∗, x)-compact. thus a is ξ-proximinal. the previous example, for the case of linear subspaces, appears firstly in a paper of hirschfeld ([10], 1958) with a wrong proof. later, in 1960, it appeared in a paper of phelps (see [18]) d: the subset of non decreasing functions in l∞(0, 1) is proximinal. let µ be the lebesgue measure on the interval (0, 1). recall that an extended real valued lebesgue measurable function f on (0, 1) is said to be essentially bounded if there exists some real number a ≥ 0 such that µ({x ∈ (0, 1) : |f(x)| > a}) = 0. if f is essentially bounded then the essential supremum of f is defined by ‖f‖∞ = inf{a ≥ 0 : µ({x ∈ (0, 1) : |f(x)| > a}) = 0}. let l∞(0, 1) denote the set of all essentially bounded lebesgue measurable functions on (0, 1), two functions being identified if they differ only on a set of measure zero, and let a ⊂ l∞(0, 1) be the subset of non decreasing functions from (0, 1) into r. under pointwise linear operations, (l∞(0, 1), ‖ · ‖∞) is a real banach space, and each equivalence class in l∞(0, 1) contains a bounded function. it is clear that the function ξ defined by ξ(f, r) = b(f, r) = {g ∈ l∞(0, 1) : ‖g − f‖∞ ≤ r} where f ∈ l∞(0, 1) and r ≥ 0, is a wrapping for (l∞(0, 1), ‖ · ‖∞). using a bounded function as a representative of each equivalence class f ∈ l∞(0, 1) we have, for r > 0, (4.18) b(0, r) ⊂ ∏ x∈(0,1) [−r, r]. with the aid of tychonoff theorem we can state that ∏ x∈(0,1) [−r, r] is compact in the cartesian product topology and without effort we can prove that the nonempty set a∩b(f, ‖f‖∞) is closed in the product topology. the remainder 68 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada of the proof is therefore devoted to showing that the set a∩b(f, ‖f‖∞) is compact in the product topology. but, taking into account the previous comments, this a direct consequence of (4.18) and the fact that b(f, ‖f‖∞) ⊂ b(0, 2‖f‖∞). e: the subset of non decreasing functions in lφ(0, 1) is φ-proximinal, i.e. for each function f ∈ lφ(0, 1) there is a non decreasing function g ∈ lφ(0, 1) such that ∫ 1 0 φ(|f − g|)dµ ≤ ∫ 1 0 φ(|f − h|)dµ for each non decreasing function h ∈ lφ(0, 1). let us consider the wrapping ξ described in subsection 3.3. then, for a fixed f ∈ lφ(0, 1) and for r ≥ 0, we have ξ(f, r) = { g ∈ lφ(0, 1) : ∫ 1 0 φ(|f − g|)dµ ≤ r } . we shall denote, for shortness, d = dξ(f, a) and aε = a ∩ ξ(f, d + ε), where ε is any positive number. we will show that a is ξ-proximinal by proving that for all ε > 0, every sequence in aε has an accumulation point in the topology τφ (hence aε is τφ-countably compact), and that aε is τφ-closed. first consider a sequence {gn} in aε. then using the jensen inequality we get ∫ 1 0 |f − gn|dµ ≤ φ −1 ( ∫ 1 0 φ(|f − gn|)dµ ) , and therefore ‖gn‖l1 ≤ k ‖f‖l1 + φ −1(d + ε). by the previous inequality it is straightforward to verify that the functions gn are uniformly bounded in each closed subinterval [a, b] ⊂ (0, 1). applying the helly theorem we get a subsequence {gnk} such that gnk → g, µ-almost everywhere in (0, 1). since φ is continuous, then φ(|f − gnk|) → φ(|f − g|), µ-a.e. the function g is non decreasing and by fatou lemma we have (4.19) ∫ 1 0 φ(|f − g|)dµ ≤ d + ε. on the other hand, by (4.19) and the fact that φ(|g|) ≤ m (φ(|f − g|) + φ(|f|)), we can assure that g ∈ lφ(0, 1) and therefore g ∈ aε. since |g − gnk| ≤ |f − g| + |f − gnk|, then φ(|g − gnk|) ≤ m (φ(|f − g|) + φ(|f − gnk|)) ∈ l1(0, 1). applying the lebesgue dominated convergence theorem, we have lim k→∞ ∫ 1 0 φ(|g − gnk|)dµ = 0. thus g is an accumulation point (in the τφ topology) of {gn}. the sets aε are τφ-closed since they are τφ-countably compact and τφ is first countable. to close this section, we characterize, for proper closed subsets of regular spaces, the proximinality property in terms of countable compactness. this result is in the same spirit as theorem 5 of [11] topological approach to best approximation 69 proposition 4.2. let (x, τ) be a hausdorff regular space. a proper and closed subset a of x is countably compact if and only if it is ξ-proximinal for any wrapping ξ. proof let us consider x ∈ x \ a. the regularity of x implies that there are open sets ox, oa such that x ∈ ox, a ⊂ oa and ox ∩ oa = ∅. hence x is an interior point of the closed set ua = x \ oa. now suppose that a is not countably compact and let {fn} be a sequence of nonempty and relative closed subsets of a such that fn+1 ⊂ fn and ∩ ∞ n=1fn = ∅. since a is closed, the relative closed subsets fn are closed. for 0 < r ≤ 1, let nr denotes the integer such that 1 2nr < r ≤ 1 2nr−1 . we define the wrapping ξ in x by ξ(x, r) =    {x} if r = 0, ua ∪ fnr if 0 < r ≤ 1, x if r > 1, and, for y 6= x, ξ(y, 0) = {y} and ξ(y, r) = x for r > 0. thus, we have pξ(x, a) = a ∩ ( ∞ ⋂ n=1 (ua ∪ fn) ) = ∅. this implies that a is not proximinal with respect to the wrapping described above.2 references [1] j. m. almira, a. j. lópez-moreno, n. del toro, metrics with good corona properties, questions and answers in general topology 21 (2003), 19-26. [2] f. s. de blasi, j. myjak, on a generalized best approximation problem, j. approx. theory 94 (1998), 54-72. [3] p. l. chebyshev, théorie des mécanismes connus sous le nom de parallélogrammes, mem. acad. sci. petersb. 7 (1854), 539-568. also to be found in oeuvres de p. l. tchebychef, volume 1, 111-143, chelsea, new york, 1961. [4] p. l. chebyshev, sur les questions de minima qui se rattachent ó la représentation approximative des fonctions, mem. acad. sci. petersb. 7 (1859), 199-291. also to be found in oeuvres de p. l. tchebychef, volume 1, 273-378, chelsea, new york, 1961. [5] m. m. day, reflexive banach spaces not isomorphic to uniformly convex spaces, bull. amer. math. soc. 47 (1941), 313-317. [6] f. deutsch, existence of best approximations, j. approximation theory 28 (1980), 132154. [7] j. dujundji, topology, allyn and bacon, 1972. [8] n. dunford, j. t. schwartz, linear operators. part i, john wiley, 1988. [9] w. f. eberlein, weak compactness in banach spaces i, proc. nat. acad. sci. u.s.a. 33 (1947), 51-53. [10] r. a. hirschfeld, on best approximation in normed vector spaces, nieuw archief voor wiskunde 6 (1958), 41-51. [11] v. l. klee, some characterizations of compactness, amer. math. monthly 58 (1951), 389393. 70 s. g. moreno, j. m. almira and e. m. garćıa-caballero and j. m. quesada [12] yu. f. korobeinik, on fixed points of one class of operators, matematychni studii 7 (1997), 187-192. [13] g. köthe, topological vector spaces i, springer–verlag, 1969. [14] s. mazur, über konvexe mengen in linearen normierte räumen, studia math. 4 (1933), 70-84. [15] e. michael, selection theorems with and without dimensional restrictions, in recent developments of general topology and its applications (berlin, 1992) pp. 218-222, akademieverlag, 1992. [16] c. mustata, on the best approximation in metric spaces, rev. anal. numer. theor. approx. 4 (1975), 45-50. [17] d. v. pai, p. govindarajulu, on set–valued f-projections and f-farthest point mappings, j. approx. theory 42 (1984), 4-13. [18] r. r. phelps, uniqueness of hahn-banach extensions and unique best approximation, trans. amer. math. soc. 95 (1960), 238-255. [19] r. r. phelps, convex sets and nearest points, proc. amer. math. soc. 8 (1957), 790-797. [20] f. riesz, über lineare funktionalgleichungen, acta math. 41 (1918), 71–98. [21] s. romagera, m. sanchis, semi-lipschitz functions and best approximation in quasi-metric spaces, j. approx. theory 103 (2000), 292-301. [22] samuel g. moreno, f. zó, mejor aproximación en espacios topológicos, métricos y vectoriales topológicos, magister thesis disertation, universidad nacional de san luis, argentina, 1997. [23] v. l. smulian, about the principle of inclusion in spaces of type b , (in russian) mat. sbornik n. s. 5 (1939), 317-328. traduced survey in math. rev. 1 (1940), 335. [24] i. singer, best approximation in normed linear spaces by elements of linear subspaces, springer–verlag, 1970. [25] a. f. timan, theory of approximation of functions of a real variable, dover publications (1994). received october 2002 accepted february 2003 s. g. moreno, j. m. almira, e. m. garćıa-caballero. (samuel@ujaen.es, jmalmira@ujaen.es, emgarcia@ujaen.es) departamento de matemáticas, universidad de jaén, e.u.p. linares, 23700 linares (jaén), spain. j. m. quesada (jquesada@ujaen.es) departamento de matemáticas, universidad de jaén, e.p.s. jaén. avda. de madrid 35, 23071 jaén, spain. () @ appl. gen. topol. 17, no. 1(2016), 37-49doi:10.4995/agt.2016.4163 c© agt, upv, 2016 a local fixed point theorem for set-valued mappings on partial metric spaces abdessalem benterki lamda-ro laboratory, department of mathematics, university of blida, algeria lmp2m laboratory, university of medea, algeria (benterki.abdessalem@gmail.com) abstract the purpose of this paper is to study the existence and location of fixed points for pseudo-contractive-type set-valued mappings in the setting of partial metric spaces by using bianchini-grundolfi gauge functions. 2010 msc: 47h04; 47h10; 54h25; 54c60. keywords: partial metric space; fixed point; set-valued mapping. 1. introduction in [24], matthews introduced the notion of a partial metric space, which is a generalization of usual metric spaces in which the self-distance for any point need not be equal to zero. the partial metric space has wide applications in many branches of mathematics as well as in the fields of computer domain and semantics. later, many authors studied fixed point theorems for set-valued mapping on partial metric spaces (see, e.g., [1, 4, 5, 11, 22, 23, 32] and references cited therein). a basic result is the nadler fixed point theorem [5, theorem 3.2] for contractive set-valued mappings, using the partial hausdorff metric, which reduces to the banach contraction mapping theorem for single-valued mappings [24, theorem 5.3]. in [10], dontchev and hager presented a fixed point theorem for set-valued mappings on complete metric space speaks about a location of a fixed point with respect to an initial value of the set-valued mapping. let (x, d) be a received 14 october 2015 – accepted 07 february 2016 http://dx.doi.org/10.4995/agt.2016.4163 a. benterki metric space and let a and b be two nonempty closed subsets of x. recall that the generalized hausdorff metric h between a and b is given by h(a, b) = max{e(a, b), e(b, a)} where e(a, b) = sup a∈a d(a, b) = sup a∈a inf b∈b d(a, b) (the excess of a over b). this (extended) metric could take the value +∞; see [7, 18]. we denote by b(x, r) the closed ball of radius r centered at x defined by b(x, r) = {y ∈ x : d(x, y) 6 r}. then the theorem of dontchev and hager [10] reads as follows: theorem 1.1. let (x, d) be a complete metric space, and consider a point x ∈ x, nonnegative scalars r > 0 and λ be such that 0 6 λ < 1, and a setvalued mapping φ from the closed ball b(x, r) to the closed subset of x and the following conditions hold: (i): d(x, φ(x)) < r(1 − λ), (ii): e(φ(x1) ∩ b(x, r), φ(x2)) 6 λd(x1, x2) ∀x1, x2 ∈ b(x, r), then φ has a fixed point in b(x, r), that is, there exists x ∈ b(x, r) such that x ∈ φ(x). if φ is a single-valued mapping, then x is the unique fixed point of φ in b(x, r). this theorem is the main tool to establish the convergence of several iterative methods for variational inclusion problem: find x ∈ x such that (vi) 0 ∈ f(x) + f(x) where f be a single-valued mapping acting between two banach spaces x and y , f is a set-valued mapping from x into the subsets of y . see e.g. [9, 14, 15, 16, 20, 29] for further informations on the applications of this theorem. recall that the variational inclusions (vi) are an abstract model of a wide variety including systems of nonlinear equations (when f = {0}), systems of inequalities (when f is the positive orthant in rm), linear and nonlinear complementary problems, variational inequalities (mixed quasi-variational inequality, hartman-stampacchia variational inequality), first-order necessary conditions for nonlinear programming, etc. in particular, they may characterize optimality or equilibrium (traffic network equilibrium, spatial price equilibrium problems, migration equilibrium problems, environmental network problems, etc.) and then have several applications in engineering and economics (analysis of elastoplastic structures, walrasian equilibrium, nash equilibrium, financial equilibrium problems, etc.) see e.g [12, 13, 17, 21, 30, 31]. in this paper, we extend theorem 1.1 on partial metric spaces by using bianchini-grandolfi gauge functions and give some related corollaries. 2. preliminaries we start by recalling some basic definitions and properties of partial metric spaces. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 38 a local fixed point theorem for set-valued mappings on partial metric spaces definition 2.1. let x be a nonempty set. a function p : x × x → r+ (where r+ denotes the set of all nonnegative real numbers) is said to be a partial metric on x if for any x, y, z ∈ x, the following conditions hold: (p1): p(x, x) = p(y, y) = p(x, y) ⇔ x = y; (t0-separation axiom) (p2): p(x, x) 6 p(x, y); (small self-distance axiom) (p3): p(x, y) = p(y, x); (symmetry) (p4): p(x, y) 6 p(x, z)+p(z, y)−p(z, z). (modified triangular inequality) the pair (x, p) is then called a partial metric space. a basic example of a partial metric space is the pair (r+, p), where p(x, y) = max{x, y} for all x, y ∈ r+. other examples of partial metric spaces may be found in [6, 24]. each partial metric p on x generates a t0 topology τp on x with a base of the family of open p-balls {bp(x, ǫ) : x ∈ x, ǫ > 0}, where bp(x, ǫ) = {y ∈ x : p(x, y) < p(x, x) + ǫ} for all x ∈ x and ǫ > 0. the closed p-ball of radius r centered at x is denoted by bp(x, r) where bp(x, r) = {y ∈ x : p(x, y) 6 p(x, x) + r}. if p is a partial metric on x, then the function ps : x × x → r+ given by ps(x, y) = 2p(x, y) − p(x, x) − p(y, y) is a metric on x. let (x, p) be a partial metric space. then: • a sequence {xn} converges to a point x ∈ x if and only if p(x, x) = lim n→+∞ p(x, xn). this will be denoted by xn → x, as n → +∞. • a sequence {xn} is called a cauchy sequence if there exists (and is finite) lim n,m→+∞ p(xn, xm). • the partial metric space (x, p) is said to be complete if every cauchy sequence {xn} in x converges, with respect to τp, to a point x ∈ x such that p(x, x) = lim n,m→+∞ p(xn, xm). lemma 2.2. let (x, p) be a partial metric space. (a): {xn} is a cauchy sequence in (x, p) if and only if it is a cauchy sequence in the metric space (x, ps). (b): a partial metric space (x, p) is complete if and only if the metric space (x, ps) is complete. furthermore, (2.1) lim n→+∞ ps(xn, x) = 0 ⇔ p(x, x) = lim n→+∞ p(xn, x) = lim n,m→+∞ p(xn, xm) where x is a limit of {xn} in (x, p s). lemma 2.3 ([2]). assume that xn → x as n → +∞ in a partial metric space (x, p) such that p(x, x) = 0. then lim n→+∞ p(xn, y) = p(x, y) for every y ∈ x. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 39 a. benterki let (x, p) be a partial metric space and let cp(x) be the family of all nonempty and closed subsets of the partial metric space (x, p), induced by the partial metric p. for x ∈ x and a, b ∈ cp(x), we define p(x, a) = inf{p(x, a), a ∈ a}, δp(a, b) = sup{p(a, b), a ∈ a}, and hp(a, b) = max{δp(a, b), δp(b, a)}. we adopt the convention that δp(∅, b) = 0. lemma 2.4 ([4]). let (x, p) be a partial metric space and a ⊂ x. then p(a, a) = p(a, a) ⇔ a ∈ a where a denotes the closure of a with respect to the partial metric p. note that a is closed in (x, p) if and only if a = a. it is easy to see that, every closed subset (with respect to τp) of a complete partial metric space is complete. recall some properties of mapping δp : c p(x) × cp(x) → [0, +∞] proposition 2.5. let (x, p) be a partial metric space. for any a, b, c ∈ cp(x), we have the following: (i): δp(a, a) = sup{p(a, a), a ∈ a}; (ii): δp(a, a) 6 δp(a, b); (iii): δp(a, b) = 0 ⇒ a ⊆ b; (iv): δp(a, b) 6 δp(a, c) + δp(c, b) − inf c∈c p(c, c). remark 2.6. the properties mentioned above are satisfied without using the concept of boundedness for a, b and c. see the proof of [5, proposition 2.2] for further details. in the following, j denotes an interval on r+ containing 0, that is an interval of the form [0, a[, [0, a] or [0, +∞[. definition 2.7. let r > 1. a function ϕ : j → j is said to be a gauge function of order r on j if it satisfies the following conditions: (1) ϕ(λt) 6 λrϕ(t) for all λ ∈]0, 1[ and t ∈ j; (2) ϕ(t) < t for all t ∈ j \ {0}. we consider some examples of gauge functions of order r > 1. example 2.8. (1) ϕ(t) = λt (0 < λ < 1) is a gauge function of the first order on j = [0, 1[; (2) ϕ(t) = c tr (c > 0, r > 1) is a gauge function of order r on j = [0, r[, where r = (1/c)1/(r−1); (3) every convex function ϕ on an interval j such that ϕ(0) = 0 and ϕ(t) < t for all t ∈ j \ {0} is a gauge function of the first order on j. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 40 a local fixed point theorem for set-valued mappings on partial metric spaces definition 2.9 (bianchini-grandolfi gauge functions). a nondecreasing function ϕ : j → j is said to be a bianchini-grandolfi gauge function on j if (2.2) s(t) := ∞ ∑ n=0 ϕn(t) is convergent for all t ∈ j where ϕn denotes the n-th iteration of the function ϕ and ϕ0(t) = t i.e. ϕ0(t) = t, ϕ1(t) = ϕ(t), ϕ2(t) = ϕ(ϕ(t)), . . . , ϕn(t) = ϕ(ϕn−1(t)). these functions are known in the literature as (c)-comparison functions in some sources (see e.g. [8, 33]) and as rate of convergence in some other sources (see e.g. [27, 28]). the sum (2.2) is called the corresponding estimate function and noticed that ϕ satisfies the following functional equation (2.3) s(t) = t + s(ϕ(t)) and (with the exception of pathological cases) we have : (2.4) ϕ(t) = s−1(s(t) − t). lemma 2.10 ([25]). every gauge function of order r > 1 on j is a bianchinigrandolfi gauge function on j. 3. the main result before giving our main result, we need the following lemma. lemma 3.1. let (x, p) be a partial metric space. let x ∈ x and a ∈ cp(x). if p(x, a) < µ (µ > 0) then there exists a ∈ a such that p(x, a) < µ. proof. we argue by contradiction. let x ∈ x and a ∈ cp(x) such that p(x, a) < µ. we suppose that p(x, a) > µ for all a ∈ a. then, we have p(x, a) = inf{p(x, a) : a ∈ a} > µ, which is a contradiction. hence, there exists a ∈ a such that p(x, a) < µ. � now, we are ready to state and prove our main result. theorem 3.2. let (x, p) be a complete partial metric space, and consider a point x ∈ x, nonnegative scalar r > 0 and a set-valued mapping φ : bp(x, r) → c p(x). let ϕ : r+ → r+ be an increasing and continuous function such that ϕ is a bianchini-grandolfi gauge function on interval j and lim t↓0 ϕ(t) = 0. if there exists α ∈ j such that the following two conditions hold: (a): p(x, φ(x)) < α where s(α) 6 p(x, x) + r (b): δp(φ(x) ∩ bp(x, r), φ(y)) 6 ϕ (p(x, y)) ∀x, y ∈ bp(x, r), then φ has a fixed point x∗ in bp(x, r). if φ is a single-valued mapping and p(x, x) + 2r ∈ j, then x∗ is the unique fixed point of φ in bp(x, r). c© agt, upv, 2016 appl. gen. topol. 17, no. 1 41 a. benterki proof. if ϕ ≡ 0 then, by proposition 2.5 (iii) and assumption (b), we have for all x1, x2 ∈ bp(x, r) (3.1) { φ(x1) ∩ bp(x, r) ⊆ φ(x2) φ(x2) ∩ bp(x, r) ⊆ φ(x1) ⇒ φ(x1) ∩ bp(x, r) = φ(x2) ∩ bp(x, r) 6= ∅; according to assumption (a), relation (3.1) and lemma 3.1, there exists x ∈ bp(x, r) such that x ∈ φ(x) ∩ bp(x, r) = φ(x) ∩ bp(x, r) which completes the proof. assume now ϕ 6≡ 0, by assumption (a) and lemma 3.1, there exists x1 ∈ φ(x) ∩ bp(x, r) such that (3.2) p(x1, x) < α = ϕ 0(α). denoting x0 = x and, by hypothesis (b), we have p(x1, φ(x1)) 6 δp(φ(x0) ∩ bp(x0, r), φ(x1)) 6 ϕ(p(x0, x1))(3.3) < ϕ(α) = ϕ1(α). this implies that there exists x2 ∈ φ(x1) such that (3.4) p(x2, x1) < ϕ 1(α) and, by the property (p4) of partial metric, we have p(x2, x0) 6 p(x2, x1) + p(x1, x0) − p(x1, x1) < ϕ1(α) + ϕ0(α)(3.5) < s(α) 6 p(x0, x0) + r. hence x2 ∈ φ(x1) ∩ bp(x0, r). proceeding by induction and suppose we have constructed, for k ∈ n (where n denotes the set of nonnegative integers), an element xk+1 such that xk+1 ∈ φ(xk) ∩ bp(x0, r) and (3.6) p(xk+1, xk) < ϕ k(α). by hypothesis (b), we have p(xk+1, φ(xk+1)) 6 δp(φ(xk) ∩ bp(x0, r), φ(xk+1)) 6 ϕ(p(xk+1, xk))(3.7) < ϕ(ϕk(α)) = ϕk+1(α). thus, there exists xk+2 ∈ φ(xk+1) such that (3.8) p(xk+2, xk+1) < ϕ k+1(α). c© agt, upv, 2016 appl. gen. topol. 17, no. 1 42 a local fixed point theorem for set-valued mappings on partial metric spaces moreover, the use of property (p4) of partial metric gives p(xk+2, x0) 6 k+1 ∑ j=0 p(xj+1, xj) − k ∑ j=1 p(xj, xj) < +∞ ∑ j=0 ϕj(α) = s(α) 6 p(x0, x0) + r.(3.9) hence xk+2 ∈ φ(xk+1) ∩ bp(x0, r) and the induction step is complete. on the other hand, we have max{p(xk+1, xk+1), p(xk, xk)} 6 p(xk+1, xk) which implies that (3.10) max{p(xk+1, xk+1), p(xk, xk)} < ϕ k(α). consider now ps(xk+1, xk) = 2p(xk+1, xk) − p(xk+1, xk+1) − p(xk, xk) 6 2p(xk+1, xk)(3.11) < 2ϕk(α). from the conditions on ϕ, it is clear that lim k→+∞ ϕk(t) = 0 for t ∈ j \ {0} and ϕ(t) < t (see[8, 26, 33]). hence, we have lim n→+∞ ps(xn+1, xn) = 0. moreover, for all integers n and m such that n > m, we have ps(xn, xm) 6 n−1 ∑ k=m ps(xk+1, xk) 6 2 n−1 ∑ k=m ϕk(α)(3.12) 6 2s(α). since s(t) is convergent for each t ∈ j, we obtain that {xn} is a cauchy sequence in (x, ps). since (x, p) is complete, by lemma 2.2, (x, ps) is complete and the sequence {xn} is convergent in (x, p s) to x ∈ x. again by lemma 2.2, we have (3.13) p(x, x) = lim n→+∞ p(xn, x) = lim n,m→+∞ p(xn, xm). moreover, since {xn} is a cauchy sequence in the metric space (x, p s), we have lim n,m→+∞ ps(xn, xm) = 0, and, from (3.10), we have lim n→+∞ p(xn, xn) = 0, thus, from definition of ps, we have lim n,m→+∞ p(xn, xm) = 0. therefore, from (3.13), we have (3.14) p(x, x) = lim n→+∞ p(xn, x) = lim n,m→+∞ p(xn, xm) = 0. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 43 a. benterki furthermore, since {xn} is a sequence in the closed p-ball bp(x0, r) which is complete and according to lemma 2.3, using (3.14), we have for y = x0 ∈ x (3.15) p(x, x0) = lim n→+∞ p(xn, x0) 6 p(x0, x0) + r. i.e. x ∈ bp(x0, r). we assert now that x ∈ φ(x). the modified triangle inequality and assumption (b) give p(x, φ(x)) 6 p(x, xn) + p(xn, φ(x)) − p(xn, xn) 6 p(x, xn) + δp(φ(xn−1) ∩ bp(x0, r), φ(x))(3.16) 6 p(x, xn) + ϕ(p(xn−1, x)) taking limit as n → +∞ and using (3.14) and the continuity of ϕ, we obtain p(x, φ(x)) = 0. therefore, from (3.14) (p(x, x) = 0), we obtain p(x, φ(x)) = p(x, x) which from lemma 2.4 implies that x ∈ φ(x) = φ(x). if φ is a single-valued mapping and p(x, x) + 2r ∈ j, we suppose that there exist two fixed points x∗, x∗∗ ∈ bp(x0, r). then, we have p(x∗, x∗∗) 6 p(x∗, x0) + p(x0, x ∗∗) − p(x0, x0) 6 p(x0, x0) + 2r ∈ j and p(x∗, x∗∗) = p(x∗, φ(x∗∗)) 6 δp(φ(x ∗) ∩ bp(x0, r), φ(x ∗∗)) 6 ϕ(p(x∗, x∗∗))(3.17) < p(x∗, x∗∗) which is a contradiction and the proof is completed. � the following example shows the usage of theorem 3.2. example 3.3. let x = r+ = [0, +∞[ be endowed with the partial metric p(x, y) =    0, x = y ∈ [ 0, 361 900 ] ; max{x, y}, otherwise. note that p(x, y) is a metric on [ 0, 361 900 ] and ps(x, y) =    2p(x, y), x, y ∈ [ 0, 361 900 ] ; |x − y|, otherwise. first, observe that, for a > 361 900 , [a, +∞[ is closed with respect to the partial metric p. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 44 a local fixed point theorem for set-valued mappings on partial metric spaces let a > 361 900 , then we have y ∈ [a, +∞[ ⇔ p(y, [a, +∞[) = p(y, y) ⇔ inf z∈[a,+∞[ p(y, z) = p(y, y) ⇔ y > z > a ⇔ y ∈ [a, +∞[ hence, [a, +∞[ is closed for any a > 361 900 . take ϕ(t) =      2 3 t, 0 6 t 6 1 2 ; 5t − 13 6 , t > 1 2 . which is a bianchini-grundolfi gauge function on j = [ 0, 1 2 ] such that s(t) = 3t and lim t↓0 ϕ(t) = 0. we set φ : [0, 1] → cp(x) defined by φ(x) =        {x2}, x ∈ [ 0, 19 30 ] ; [1, +∞[ , x ∈ ] 19 30 , 1 ] . we apply theorem 3.2 with the following specifications x = 1 4 , r = 1, α = 1 3 ∈ j, bp(x, r) = [0, 1]. first, observe that, p ( 1 4 , φ ( 1 4 )) = p ( 1 4 , { 1 16 }) = 1 4 < 1 3 = α and s(α) = 3 ∗ 1 3 = 1 6 p ( 1 4 , 1 4 ) + 1; that is, condition (a) of theorem 3.2 holds. to see that condition (b) of theorem 3.2 holds it is sufficient to consider the following cases: (1) if x = y ∈ [ 0, 19 30 ] then δp(φ(x)∩[0, 1], φ(y)) = δp({x 2}, {x2}) = 0 6      2 3 p(x, x) = ϕ (p(x, y)) , x ∈ [ 0, 1 2 ] ; 5x − 13 6 = ϕ (p(x, y)) , otherwise. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 45 a. benterki (2) if x, y ∈ [ 0, 19 30 ] and x 6= y then δp(φ(x) ∩ [0, 1], φ(y)) = δp({x 2}, {y2}) 6      1 2 max{x, y}, x, y ∈ [ 0, 1 2 ] ; 5 max{x, y} − 9 4 , otherwise. 6      2 3 max{x, y} = ϕ (p(x, y)) , x, y ∈ [ 0, 1 2 ] ; 5 max{x, y} − 13 6 = ϕ (p(x, y)) , otherwise. (3) if x, y ∈ ] 19 30 , 1 ] then δp(φ(x) ∩ [0, 1], φ(y)) = δp({1}, [1, +∞[) = 1 6 5 max{x, y} − 13 6 = ϕ (p(x, y)) (4) if x ∈ [ 0, 19 30 ] and y ∈ ] 19 30 , 1 ] then δp(φ(x) ∩ [0, 1], φ(y)) = δp({x 2}, [1, +∞[) = 1 6 5y − 13 6 = ϕ (p(x, y)) and δp(φ(y) ∩ [0, 1], φ(x)) = δp({1}, {x 2}) = 1 6 5y − 13 6 = ϕ (p(x, y)) hence, all conditions of theorem 3.2 are satisfied and x∗ ∈ {0, 1} ⊂ bp(x, r) are the required points. on the other hand, it is easy to show that theorem 1.1 is not applicable in this case. indeed, for x ∈ [ 0, 19 30 ] and y = 1, we have eps(φ(x) ∩ [0, 1], φ(y)) = eps({x 2}, [1, +∞[) = |1 − x2| > |1 − x| = ps(x, y) hence, no constant λ, 0 6 λ < 1 can be chosen in a way that eps(φ(x)∩ [0, 1], φ(y)) 6 λp s(x, y) for all x, y ∈ [0, 1]. we can obtain the following corollaries from theorem 3.2. corollary 3.4. let (x, p) be a complete partial metric space, and consider a point x ∈ x, nonnegative scalars r > 0 and 0 6 λ < 1 and a set-valued mapping φ : bp(x, r) → c p(x). let the following two conditions hold: (a): p(x, φ(x)) < (p(x, x) + r)(1 − λ), (b): δp(φ(x) ∩ bp(x, r), φ(y)) 6 λp(x, y) ∀x, y ∈ bp(x, r), then φ has a fixed point x∗ in bp(x, r). if φ is a single-valued mapping, then x∗ is the unique fixed point of φ in bp(x, r). c© agt, upv, 2016 appl. gen. topol. 17, no. 1 46 a local fixed point theorem for set-valued mappings on partial metric spaces proof. we apply theorem 3.2 for ϕ(t) = λt which is a bianchini-grandolfi gauge function on j = [0, p(x, x)+2r] and the corresponding estimate function s(t) = t 1 − λ . take α = (p(x, x) + r)(1 − λ) ∈ j. � remark 3.5. corollary 3.4 extends theorem 1.1 on partial metric spaces. the next example demonstrates the usage of theorem 3.2 and corollary 3.4. example 3.6 ([5, example 3.3]). let x = {0, 1, 4} be endowed with the partial metric p(x, y) = 1 4 |x−y|+ 1 2 max{x, y} for all x, y ∈ x. define the mapping φ : x → cp(x) by φ(0) = φ(1) = {0} and φ(4) = {0, 1}. we apply corollary 3.4 for x = 1, r = 2 and λ = 1 2 . first, we have p(1, φ(1)) = p(1, {0}) = 3 4 < (p(1, 1) + 2)(1 − 1 2 ). on the other hand, we get δp(φ(x) ∩ bp(1, 2), φ(y)) 6 1 2 p(x, y) ∀x, y ∈ bp(1, 2) = {0, 1}. thus, all the hypotheses are satisfied and the fixed point of φ is x = 0 ∈ bp(1, 2). corollary 3.7. let (x, p) be a complete partial metric space, and consider a point x ∈ x, nonnegative scalar r > 0 and a set-valued mapping φ : bp(x, r) → c p(x). let ϕ : r+ → r+ be an increasing and continuous function such that ϕ is a bianchini-grandolfi gauge function on interval j and lim t↓0 ϕ(t) = 0. if there exists α ∈ j such that the following two conditions hold: (a): p(x, φ(x)) < α where s(α) 6 p(x, x) + r (b): hp(φ(x), φ(y)) 6 ϕ (p(x, y)) ∀x, y ∈ bp(x, r), then φ has a fixed point x∗ in bp(x, r). if φ is a single-valued mapping and p(x, x) + 2r ∈ j, then x∗ is the unique fixed point of φ in bp(x, r). proof. we must assert that the second condition of theorem 3.2 is satisfied. for any x1, x2 ∈ bp(x, r) we obtain δp(φ(x1) ∩ bp(x, r), φ(x2)) 6 δp(φ(x1), φ(x2)) 6 hp(φ(x1), φ(x2)) 6 ϕ(p(x1, x2)). we complete the proof by applying theorem 3.2. � corollary 3.8. let (x, p) be a complete partial metric space, and consider a point x ∈ x, nonnegative scalars r > 0 and λ be such that 0 6 λ < 1, and a set-valued mapping φ : bp(x, r) → c p(x). let the following two conditions hold: (a): p(x, φ(x)) < (p(x, x) + r) (1 − λ), (b): hp(φ(x1), φ(x2)) 6 λp(x1, x2) ∀x1, x2 ∈ bp(x, r), c© agt, upv, 2016 appl. gen. topol. 17, no. 1 47 a. benterki then φ has a fixed point x∗ in bp(x, r). if φ is a single-valued mapping, then x∗ is the unique fixed point of φ in bp(x, r). remark 3.9. corollary 3.8 extends theorem 9.1 in [3] and lemma 1 in [19] on the partial metric space. the nadler’s fixed point theorem ([5, theorem 3.2]) on partial metric spaces follows readily from corrollary 3.8. observe that no boundedness assumption on the values is required. corollary 3.10. let (x, p) be a complete partial metric space. if φ : x → cp(x) is a set-valued mapping such that hp(φ(x), φ(y)) 6 λp(x, y) for all x, y ∈ x where 0 6 λ < 1. then φ has a fixed point. proof. let x ∈ x. choose r > 0 with p(x, φ(x)) < (p(x, x) + r) (1 − λ). the result now follows from corollary 3.8. � acknowledgements. the author wishes to express his gratitude to the referees for their pertinent comments and their suggestions which improved the contents and presentation of the paper. references [1] m. abbas, b. ali, and c. vetro, a suzuki type fixed point theorem for a generalized multivalued mapping on partial hausdorff metric spaces, topology appl. 160 (2013), 553–563. 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(mhemdiabd@gmail.com) c department of mathematics, faculty of sciences of tunis. university tunis-el manar. “campus universitaire” 2092 tunis, tunisia. (tarek turki@gmail.com) abstract following van douwen, a topological space is said to be nodec if it satisfies one of the following equivalent conditions: (i) every nowhere dense subset of x, is closed; (ii) every nowhere dense subset of x, is closed discrete; (iii) every subset containing a dense open subset is open. this paper deals with a characterization of topological spaces x such that f(x) is a nodec space for some covariant functor f from the category top to itself. t0, ρ and fh functors are completely studied. secondly, we characterize maps f given by a flow (x, f) in the category set such that (x, p(f)) is nodec (resp., t0-nodec), where p(f) is a topology on x whose closed sets are precisely f-invariant sets. 2010 msc: 54b30; 54d10; 54g12; 46m15. keywords: categories; functors; nodec spaces; primal space. 1. introduction recall that a topological space x is called submaximal if every dense subset of x is open. in [8] it was shown that, in a submaximal space without isolated points, every nowhere dense subset (that is the interior of its closure is empty), ∗corresponding author. received 25 june 2014 – accepted 4 december 2014 http://dx.doi.org/10.4995/agt.2015.3141 l. dridi, a. mhemdi and t. turki is closed and discrete. hence, a space satisfying the later property is called nodec. different equivalent conditions for a space to be submaximal are given in [1] and the ones for a space to be nodec in [14], [12] and [4]. theorem 1.1. [4, theorem 2.5] the following statements are equivalent: (1) x is nodec; (2) each nowhere dense subset of x, is closed. (3) for each a ⊆ x, if ◦ a ⊆ a, then a is closed. (4) for each a ⊆ x, if a ⊆ ◦ ◦ a, then a is open. (5) for each a ⊆ x, a\a ⊆ ◦ a. (6) for each a ⊆ x, a = a ∪ ◦ a. (7) for each a ⊆ x, ◦ a = a ∩ ◦ ◦ a. on the other hand, the theory of category and functors play an enigmatic role in topology, specially the notion of reflective subcategory. so it is of importance to recall the standard notion of reflective subcategory a of b that is, a full subcategory such that the embedding a −→ b has a left adjoint f : b −→ a (called reflection). further, recall that for all i = 0, 1, 2, 3, 3.1 2 the subcategory topi of ti-spaces is reflective in top, the category of all topological spaces. t3 1 2 is also called the tychonoff-reflection and will be denoted by ρ. in this paper the functors t0, ρ and fh (the functionally hausdorff reflection) are studied. some authors (see [3],[6], [11]) are interested in separation axioms using the theory of categories and functors as follows. definition 1.2. let c be a category and f, g two (covariant) functors from c to itself. (1) an object x of c is said to be a t(f,g)-object if g(f(x)) is isomorphic with f(x). (2) let p be a topological property on the objects of c. an object x of c is said to be a t(f,p )-object if f(x) satisfies the property p . consequently, some new separation axioms t(0,ρ), t(0,f h), t(ρ,f h) are introduced and characterized. recently, in [2] a characterization of topological spaces x such that their compactification noted k(x) is a nodec space, is given. in [5], l. dridi et al characterized topological spaces x such that f(x) is a submaximal space, for a given covariant functor f. the first section of this paper is devoted to the characterization of t0-nodec space. second section studies the same problem using the functor ρ (resp., fh). © agt, upv, 2015 appl. gen. topol. 16, no. 1 54 f -nodec spaces finally, in the third section we are interested in the relation between nodec spaces and primal spaces. some important results are given. 2. t0-nodec spaces first let us recall the t0-reflection of a topolgical space. let x be a topological space. we define the binary relation ∼ on x by x ∼ y if and only if {x} = {y}. then ∼ is an equivalence relation on x and the resulting quotient space t0(x) := x/ ∼ is the t0-reflection of x. the canonical surjection µx : x −→ t0(x) is a quasihomeomorphism. ( a continuous map q : x −→ y is said to be a quasihomeomorphism if u 7−→ q−1(u) (resp., c 7−→ q−1(c) ) defines a bijection o(y ) −→ o(x) (resp., f(y ) −→ f(x)), where o(x) (resp., f(x)) is the collection of all open sets (resp;, closed sets) of x )[7]. before giving the main result of this section let us introduce some definitions, notations and remarks. notations 2.1. [5, notations 2.2] let x be a topological space, a ∈ x and a ⊆ x. we denote by: (1) d0(a) := {x ∈ x : {x} = {a}} (2) d0(a) = ∪[d0(a); a ∈ a]. remarks 2.2. [5, remarks 2.3] let x be a topological space and a be a subset of x. in [5] the following remarks are given. (i) d0(a) = µ −1 x (µx(a)). (ii) d0(d0(a)) = d0(a). (iii) a ⊆ d0(a) ⊆ a and consequently d0(a) = a. (iv) in particular if a is open (resp., closed ), then d0(a) = a. definition 2.3. let x be a topological space. x is called a t0-nodec space if its t0-reflection is a nodec space. now we are in a position to give the characterization of t0-nodec space. theorem 2.4. let x be a topological space. then the following statements are equivalent: (1) x is a t0-nodec space; (2) for any nowhere dense subset a of x, d0(a) is closed. (3) ∀a ⊆ x; if ◦ a ⊆ d0(a) =⇒ d0(a) = a. (4) ∀a ⊆ x; a\d0(a) ⊆ ◦ a. (5) ∀a ⊆ x; a = d0(a) ∪ ◦ a. proof. (1) =⇒ (2) let a be a nowhere dense subset of x. then µx(a) ⊆ µx(a), since µx is a closed map. suppose that there exist an open set u of t0(x) such that u ⊂ µx(a). so µ −1 x (u) is an open set and µ−1 x (u) ⊆ © agt, upv, 2015 appl. gen. topol. 16, no. 1 55 l. dridi, a. mhemdi and t. turki µ−1 x (µx(a)) ⊆ µ −1 x (µx(a)) = d0(a) = a, which contradict the fact that a is a nowhere dense subset of x. then µx(a) is a nowhere dense subset of t0(x). since the later is nodec, µx(a) is closed. so µ −1 x (µx(a)) = d0(a) is closed. (2) =⇒ (1) let s be a nowhere dense subset of t0(x) and a = µ −1 x (s). then d0(a) = a. suppose that there exists x ∈ ◦ a. thus there exists an open set u of x such that x ∈ u ⊂ a. so µx(u) is an open set containing µx(x), since µx is open. moreover, µx(u) ⊆ µx(a) = µx(µ −1 x (s)) ⊆ µx(µ −1 x (s)) = s, by [5, lemma 2.16]. therefore µx(x) ∈ ◦ s, which contradicts the fact that s is a nowhere dense subset of t0(x). then ◦ a = ∅ and consequently a is a nowhere dense subset of x. thus d0(a) = a is closed, by hypothesis. so s is a closed set of t0(x). (2) =⇒ (4) let a be a subset of x. since x is t0-nodec, µx(a)\µx(a) ⊂ ◦ µx(a), by theorem 1.1. then µ −1 x (µx(a)\µx(a)) = µ −1 x (µx(a))\µ −1 x (µx(a)) = d0(a)\d0(a) = a\d0(a) ⊆ µ −1 x ( ◦ µx(a)) = ◦ µ−1 x (µx(a)) = ◦ d0(a) = ◦ a. (4) =⇒ (5) let a be a subset of x. as ◦ a ⊆ a and d0(a) ⊆ a, it is clear that d0(a) ∪ ◦ a ⊂ a. conversely, a = d0(a) ∪ (a\d0(a)) ⊆ d0(a) ∪ ◦ a, by (4). thus, the equality holds. (5) =⇒ (2) let a be a nowhere dense subset of x. then a = d0(a) ∪ ◦ a = d0(a). (2) =⇒ (3) straightforward. (3) =⇒ (2) if a is a nowhere dense subset of x, then ◦ a = ∅ ⊆ d0(a). thus d0(a) = a, by (3). � remark 2.5. clearly every nodec space is a t0-nodec space. the converse does not hold: indeed, given a set x = {0, 1, 2} equipped with the topology τ = {∅, {2}, x}. we can easily see that t0(x) is a nodec space. however {0} is a nowhere dense subset of x and not closed. 3. ρ-nodec spaces and f h-nodec spaces let x be a topological space, f a subset of x and x ∈ x. x and f are said to be completely separated if there exists a continuous map f : x −→ r such that f(x) = 0 and f(f) = {1}. now, two distinct points x and y in x are called completely separated if x and {y} are completely separated. a space x is said to be completely regular if every closed subset f of x is completely separated from any point x not in f . recall that a topological © agt, upv, 2015 appl. gen. topol. 16, no. 1 56 f -nodec spaces space x is called a t1-space if each singleton of x is closed. a completely regular t1-space is called a tychonoff space [13]. a functionally hausdorff space is a topological space in which any two distinct points of this space are completely separated. remark here that a tychonoff space is a functionally hausdorff space and consequently a hausdorff space (t2-space). now, for a given topological space x, we define the equivalence relation ∼ on x by x ∼ y if and only if f(x) = f(y) for all f ∈ c(x) (where c(x) design the family of all continuous maps from x to r). let us denote by x/ ∼ the set of equivalence classes and let ρx : x −→ x/ ∼ be the canonical surjection map assigning to each point of x its equivalence class. since every f in c(x) is constant on each equivalence class, we can define ρ(f) : x/ ∼−→ r by ρ(f)(ρx(x)) = f(x). one may illustrate this situation by the following commutative diagram. x ▽ ρx // x/ ∼ ρ(f) }}③ ③ ③③ ③ ③③ ③ r �� f ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ now, equip x/ ∼ with the topology whose closed sets are of the form ∩[ρ(fα) −1(fα) : α ∈ i], where fα : x −→ r (resp., fα) is a continuous map (resp., a closed subset of r). it is well known that, with this topology, x/ ∼ is a tychonoff space (see for instance [15]) and its denoted by ρ(x). the construction of ρ(x) satisfies some categorical properties: for each tychonoff space y and each continuous map f : x −→ y , there exists a unique continuous map f̃ : ρ(x) −→ y such that f̃ ◦ ρx = f. we will say that ρ(x) is the ρ-reflection (or tychonoff-reflection) of x. from the above properties, it is clear that ρ is a covariant functor from the category of topological spaces top into the full subcategory tych of top whose objects are tychonoff spaces. on the other hand the quotient space x/ ∼ which is denoted by fh(x) is a functionally hausdorff space. the construction fh(x) satisfies some categorical properties: for each functionally hausdorff space y and each continuous map f : x −→ y , there exists a unique continuous map f̃ : fh(x) −→ y such that f̃◦ρx = f. we will say that fh(x) is the functionally hausdorff-reflection of x (or the fh-reflection of x). consequently, it is clear that fh is a covariant functor from the category of topological spaces top into the full subcategory funhaus of top whose objects are functionally hausdorff spaces. we need to introduce and recall some definitions, notations and results. notations 3.1. [5, notation 3.1] let x be a topological space, a ∈ x and a a subset of x. we denote by: (1) dρ(a) := ∩[f −1(f({a})) : f ∈ c(x)]. © agt, upv, 2015 appl. gen. topol. 16, no. 1 57 l. dridi, a. mhemdi and t. turki (2) dρ(a) := ∪[dρ(a) : a ∈ a]. the following results are given in [5]. proposition 3.2. [5, proposition 3.2] let x be a topological space, a ∈ x and a a subset of x. then: (1) dρ(a) = ρ −1 x (ρx(a)). (2) dρ(a) is a closed subset of x. (3) a ⊆ dρ(a) ⊆ ∩[f −1(f(a)) : f ∈ c(x)]. (4) ∀f ∈ c(x), f(a) = f(dρ(a)). now, we introduce the following definition. definition 3.3. let x be a topological space. x is called a ρ-nodec (resp., fh-nodec ) space if its ρ-reflection (resp., fh-reflection ) is a nodec space. in order to give a characterization of ρ-nodec spaces, we need to recall some elementary proporties which characterize tychonoff spaces in terms of zero-set (resp., cozero-sets). consider a topological space x and a ⊆ x. a is called a zero-set if there exists f ∈ c(x) such that a = f−1({0}). the complement of a zero-set is called a cozero-set. according to [15, proposition 1.7], a space is tychonoff if and only if the family of zero-sets of the space is a base for the closed sets (equivalently, the family of cozero-sets of the space is a base for the open sets). in [5] it is showen that a closed (resp., open) subset of ρ(x) is of the form ∩[ρ(f)−1({0}) : f ∈ h] (resp., ∪[ρ(f)−1(r⋆) : f ∈ h]) , where h is a collection of continuous maps from x to r. recall that a subset v of a topological space x is called a functionally open subset of x (for short f-open ) if and only if dρ(v ) is open in x (see [5]). now in order to characterize ρ-nodec spaces and fh-nodec spaces, we introduce the following definitions: definition 3.4. let x be a topological space and v a set of x. v is called a functionally nowhere dense subset of x (for short f-nowhere dense ) if and only if for any f-open subset w of x, dρ(w) * dρ(v ). definition 3.5. let x be a topological space and a a nonempty subset of x. a is said to be a ρ-nowhere dense subset of x if for any nonzero continuous map f from x to r there exists a continuous map g from x to r satisfying fg 6= 0 and g(a) = {0}. remark 3.6. v is an f-open set of x if and only if ρx(v ) is an open set of fh(x). proposition 3.7. let x be a topological space and a a subset of x. then the following statements are equivalent: (i) a is a ρ-nowhere dense subset of x; (ii) ρx(a) is a nowhere dense subset of ρ(x). © agt, upv, 2015 appl. gen. topol. 16, no. 1 58 f -nodec spaces proof. (i) =⇒ (ii) let a be a ρ-nowhere dense subset of x. according to the universal property of ρ(x), each continuous map g : ρ(x) −→ r may be written as g = ρ(f) with f = g◦ρx. now suppose that there exists a nonzero continuous map f from x to r such that ρ(f)−1(r⋆) ⊂ ρx(a). then there exists a continuous map g such that fg 6= 0 and g(a) = {0}. thus ρx(a) ⊂ ρ(g) −1{0} and consequently ρx(a) ⊂ ρ(g) −1{0}. so ρ(f)−1(r⋆) ⊂ ρx(a) ⊂ ρ(g) −1{0}. therefore ρ(f)−1(r⋆)∩ρ(g)−1(r⋆) = ∅ which contradict the fact that fg 6= 0. (ii) =⇒ (i) let a be a subset of x such that ρx(a) is a nowhere dense subset of ρ(x). then for any nonzero continuous map f from x to r, ρ(f)−1(r⋆) * ρx(a). thus there exists x ∈ x such that ρ(x) ∈ ρ(f) −1(r⋆) and ρ(x) /∈ ρx(a). therefore there exists a nonzero continuous map g such that ρ(x) ∈ ρ(g)−1(r⋆) and ρ(g)−1(r⋆) ∩ ρx(a) = ∅. so g(a) = {0}. hence a is a ρ-nowhere dense subset of x. � now, we are in a position to give the characterization of ρ-nodec spaces. theorem 3.8. let x be a topological space. then the following statements are equivalent: (i) x is a ρ-nodec space; (ii) for any ρ-nowhere dense set a of x, we have dρ(a) is an intersection of zero sets of x. proof. (i) =⇒ (ii) let a be a ρ-nowhere dense subset of x. according to the proposition 3.7 ρx(a) is a nowhere dense subset of ρ(x). since x is a ρ-nodec space, then ρx(a) is a closed set of ρ(x) and thus ρx(a) = ⋂ [ρ(f)−1{0} : f ∈ h] (where h is a subfamily of c(x) ). so that ρ−1 x (ρx(a)) = ⋂ [ρ−1 x (ρ(f)−1{0}) : f ∈ h]. therefore dρ(a) = ⋂ [f−1{0} : f ∈ h] is an intersection of zero sets of x. (ii) =⇒ (i) conversely, let a a subset of x such that ρx(a) is a nowhere dense subset of ρ(x). then, by proposition 3.7, a is a ρ-nowhere dense subset of x and consequently dρ(a) is an intersection of zero sets of x. hence there exists a subfamily {fi : i ∈ i} of c(x) satisfying ρ −1 x (ρx(a)) = ⋂ [f−1i {0} : i ∈ i]. then: ρx(a) = ρx( ⋂ [f−1i {0} : i ∈ i]) = ρx( ⋂ [ρ−1x (ρ(fi) −1{0}) : i ∈ i]) = ρx(ρ −1 x ( ⋂ [ρ(fi) −1{0} : i ∈ i])) = ⋂ [ρ(fi) −1{0} : i ∈ i] finally, ρx(a) is a closed subset of ρ(x). � theorem 3.9. let x be a topological space. then the following statements are equivalent: (i) x is fh-nodec. (ii) for any f-nowhere dense subset a of x, dρ(a) is closed. proof. (i) =⇒ (ii) let a be an f-nowhere dense subset of x. first let us show that ρx (a) is a nowhere dense subset of fh(x). © agt, upv, 2015 appl. gen. topol. 16, no. 1 59 l. dridi, a. mhemdi and t. turki suppose that ◦ ρx (a) 6= ∅. then there exist ρx (u) an open subset of fh(x) such that: ∅ 6= ρx (u) ⊂ ρx (a). thus ρ −1 x (ρx (u)) ⊂ ρ −1 x (ρx (a)) ⊂ ρ−1x (ρx (a)) = dρ (a). so dρ (u) = ρ −1 x (ρx (u)) is open in x that is u is an fopen subset and dρ (u) ⊂ dρ (a), which contradict the fact that a is f-nowhere dense. hence ρx (a) is a nowhere dense subset of fh(x) and consequently a closed set of fh(x). therefore dρ(a) = ρ −1 x (ρx (a)) is closed. (ii) =⇒ (i) let ρx (a) be a nowhere dense subset of fh (x), where a is a subset of x. we prove that a is f-nowhere dense in x. suppose that there exists v an f-open subset of x, such that dρ (v ) ⊂ dρ (a). so ρx (dρ (v )) ⊂ ρx(dρ (a)) ⊂ ρx (dρ (a)). thus ρx (v ) ⊂ ρx (a). v is an f-open set of x, that is dρ (v ) an open subset of x, and thus ρx (v ) is open in fh (x). then ◦ ρx (a) 6= ∅, which contradict the fact that ρx (a) is nowhere dense in fh (x). hence a is f-nowhere dense in x. by (ii), dρ (a) is closed in x and consequently ρx (a) is closed in fh (x). therefore fh (x) is nodec. � 4. alexandroff topology according to kennisson, a flow in a category c is a couple (x, f), where x is an object of c and f : x −→ x is a morphism, called the iterator (see [9] and [10]). recall that the topology p(f) defined on a flow (x, f) of the category set, is a topology such that closed sets are exactly those a which are f-invariant (i.e., f(a) ⊆ a) and consequently open sets are those which are f−1-invariant. it is clearly seen that for any subset a of x, the topological closure a is exactly ∪[fn(a) : n ∈ n]. in particular for any point x ∈ x, {x} = of (x) = {fn(x) : n ∈ n} called the orbit of x by f. one can see easily that the family {vf (x) : x ∈ x} is a basis of open sets of p(f), where vf (x) := {y ∈ x : fn(y) = x, for some integer n}. clearly, p(f) is an alexandroff topology on x. an element x of x is said to be a periodic point if fn(x) = x for some positive integer n. a characterization of maps f such that (x, p(f)) is nodec, is given by the following result. proposition 4.1. let (x, f) be a flow in set. then the following statements are equivalent. (i) (x, p(f)) is a nodec space; (ii) ∀ x ∈ x we have x is either a periodic point or f(x) is a fixed point. proof. (i) =⇒ (ii). let x ∈ x. if x is not a periodic point, then x /∈ {f(x)}. suppose that there exists y ∈ ◦ {f(x)}, then vf (y) ⊆ ◦ {f(x)} ⊆ {f(x)}. hence y = fn(f(x)) = fn+1(x) for some integer n so that x ∈ vf (y) ⊆ {f(x)}, which © agt, upv, 2015 appl. gen. topol. 16, no. 1 60 f -nodec spaces contradict the fact that x is not a periodic point. thus ◦ {f(x)} = ∅. since (x, p(f)) is a nodec space, we get {f(x)} is closed and consequently f(x) is a fixed point. (ii) =⇒ (i). let a be a subset of x such that ◦ a = ∅ and x ∈ a. then we have f−1({x}) * {x} ⊂ a, otherwise {x} is an open set. let y ∈ f−1({x})\{x}, it is clear that y is not a periodic point. indeed, if y is a periodic point, then {y} = {f(y)}. but f(y) ∈ {x}, so we get y ∈ {x}, a contradiction. thus f(y) is a fixed point. since ◦ a = ∅, then we have {x} is not open which means that f−1({x}) \ {x} 6= ∅. let z ∈ f−1({x}) \ {x}, it is clear that z is not a periodic point. hence f(z) = x is a fixed point and consequently {x} is a closed subset. therefore a is a closed set. � example 4.2. consider the map f: n −→ n n 7−→ n + 1 where n is the set of all natural numbers including 0. let p be a positive integer and n ∈ n, then we have fp(n) = n + p. hence n is not a periodic point and f(n) is not a fixed point for every n ∈ n. now, consider the topological space (n, p(f)) and set a = 2n + 1. since every open subset of (n, p(f)) must contain 0, then ◦ a = ∅. however, a is not closed (a = n \ {0}). remark 4.3. let (x, f) be a flow in set, we equip x with the topology p(f). then for every a ∈ x we have d0(a) is closed if and only if a is a periodic point. indeed, suppose that d0(a) is a closed subset of (x, p(f)), that is d0(a) is a f-invariant set. on the other hand, we know that a ∈ d0(a) then f(a) ∈ f(d0(a)) ⊆ d0(a) and consequently {a} = {f(a)}. hence a ∈ {f(a)} which implies that a = fn(f(a)) = fn+1(a) for some n in n. therefore a is a periodic point. for the converse, suppose that a is a periodic point which means that fn(a) = a for some positive integer n. let x ∈ d0(a), then {x} = {a} and so x ∈ {a}. hence d0(a) ⊆ {a}. for the reverse inclusion, let x ∈ {a} then x = fm(a) for some m in n. one can easily see that fpn(a) = a for every p ∈ n, thus we get a = fnm(a) = fnm−m+m(a) = fnm−m(fm(a)) = fnm−m(x). it follows that a ∈ {x} and so {x} = {a}. therefore x ∈ d0(a). finally d0(a) = {a} is closed. it is easily to see that every submaximal space (resp., t0-submaximal space) is a nodec space (resp., t0-nodec space). the converse does not hold. as showen in the following examples. example 4.4. given a set x = {a, b, c} equipped with the trivial topology. clearly x is a nodec space which is not t0 and consequently not submaximal. © agt, upv, 2015 appl. gen. topol. 16, no. 1 61 l. dridi, a. mhemdi and t. turki example 4.5. let l be an infinite set, a, b /∈ l and x = l ∪ {a, b}. we equip x with the topology whose open sets are ∅, x, l, co-finite subset of l and cofinite subset of x containing {a, b}. then {a} = {b} and t0(x) = l∪{µx(a)}. thus t0(x) is nodec. indeed, it is straightforward to see that the nowhere dense sets are finite subsets of t0(x), which are closed. however, x is not t0-submaximal, since for an infinite subset of l such that l \ s is infinite, we have s \ s is not closed. remark 4.6. a primal nodec space is not always a submaximal space as shows the following example. let x = {a, b, c} and f : x −→ x the map defined by:    f(a) = b f(b) = c f(c) = a. it is clear that (x, p(f)) is nodec but not submaximal, since f2 6= f. the following proposition shows that for a given primal space (x, p(f)), there is an equivalence between t0-submaximal and t0-nodec. proposition 4.7. let (x, f) be a flow in set. then the following statements are equivalent. (i) (x, p(f)) is a t0-submaximal space; (ii) (x, p(f)) is a t0-nodec space; (iii) f(x) is a periodic point for every x ∈ x. proof. (i) =⇒ (ii). straightforward. (ii) =⇒ (iii). let x ∈ x. suppose that f(x) is not a periodic point, then x is not a periodic point and x 6= f(x). for the set a = {f(x)}, we have ◦ a = ∅. indeed, if there exists y ∈ ◦ a ⊆ a then y = fn(x) for some positive integer n. hence x ∈ v f (y) ⊆ ◦ a ⊆ a = {f(x)} which implies that f(x) is a periodic point, a contradiction. therefore ◦ a = ∅. by theorem 1.1 we get d0(a) is closed and finally, by remark 4.3, we conclude that f(x) is a periodic point. (iii) =⇒ (i). first, remark that if x is not a periodic point then {x} is open. in fact, if {x} is not open, then there exists y ∈ f−1({x}) \ {x} which implies that x = f(y) is a periodic point. now, let a be a dense subset of x. then every non periodic point of x belongs to a which means that all points of ac are periodic points. since [d0(a)] c ⊆ ac, then all points of [d0(a)] c are also periodic points. let x ∈ [d0(a)] c and y ∈ {x}. so y is a periodic point and {y} = {x}. therefore y ∈ [d0(a)] c and consequently {x} ⊆ [d0(a)] c for each x ∈ [d0(a)] c. since (x, p(f)) is an alexandroff space, then [d0(a)] c is a closed subset of x and [d0(a)] is open. � © agt, upv, 2015 appl. gen. topol. 16, no. 1 62 f -nodec spaces remarks 4.8. let (x, f) be a flow in set. (1) (x, p(f)) is fh-nodec and ρ-nodec, since p(f) is an alexandroff topology and for every a ∈ x, dρ(a) is a closed subset of x. (2) let a be a subset of x and denote b = x \ dρ(a). then dρ(b) = b. indeed suppose that there exists x ∈ dρ(b) ∩ dρ(a). thus there exist a ∈ a and b ∈ b such that f(x) = f(a) = f(b), for all f ∈ c(x). therefore b ∈ dρ(a) which contradict the fact that b = x \ dρ(a). on the other hand, dρ(b) == ∪[dρ(b) : b ∈ b] is a closed set of (x, p(f)), since the latter is alexandroff. hence, dρ(a) is an open subset of x which implies that (x, p(f)) is fh-submaximal and ρ-submaximal. as example of primal space which is t0-nodec but not nodec, we give the following: example 4.9. consider the map f : n −→ n defined by:    f(0) = 0 f(n) = n + 1 if n /∈ 3n f(n) = n − 1 if n ∈ 3n \ {0}. let n ∈ n \ {0}. − if n ≡ 0[3] then n − 1 /∈ 3n. hence f2(n) = f(f(n)) = f(n − 1) = n − 1 + 1 = n, that is n is a periodic point. − if n ≡ 2[3] then n + 1 ≡ 0[3]. hence f2(n) = f(f(n)) = f(n + 1) = n + 1 − 1 = n, that is n is a periodic point. − if n ≡ 1[3] then f(n) = n + 1 ≡ 2[3] that is f(n) is a periodic point. therefore f(n) is a periodic point for every n ∈ n. now, consider the topological space (n, p(f)). we can easily check that for each n ∈ n we have f(n) ≡ 0[3] or f(n) ≡ 2[3]. then {n} is open if and only if n = 0 or n ≡ 1[3]. let a ⊆ n such that ◦ a = ∅, then a is a periodic point for each a ∈ a. therefore d0(a) is a closed subset. therefore, (n, p(f)) is a t0-nodec space which is not a nodec space. acknowledgements. the authors thank the referee for his useful comments and suggestions to improve the presentation and the mathematical content of this paper. © agt, upv, 2015 appl. gen. topol. 16, no. 1 63 l. dridi, a. mhemdi and t. turki references [1] a. v. arhangel’skii and p. j. collins, on submaximal spaces, topology appl. 64 (1995), 219–241. [2] k. belaid and l. dridi, i-spaces, nodec spaces and compactifications, topology appl. 161 (2014), 196–205. [3] k. belaid, o. echi and s. lazaar, t(α,β)-spaces and the wallman compactification, int. j. math. math. sc. 68 (2004), 3717–3735. [4] g. bezhanishvili, l. esakia and d. gabelaia, modal logics of submaximal and nodec spaces, collection of essays dedicated to dick de jongh on occasion of his 65th birthday, j. van benthem, f. veltman, a. troelstra, a. visser, editors. (2004), pp. 1–13. [5] l. dridi, s. lazaar and t. turki, f-door spaces and f-submaximal spaces, appl. gen. topol. 14, no. 1 (2013), 97–113. [6] o. echi and s. lazaar, reflective subcategories, tychonoff spaces, and spectral spaces, top. proc. 34 (2009), 307–319. [7] a. grothendieck and j. dieudonné, eléments de géométrie algébrique i: le langage des schemas, inst. hautes etudes sci. publ. math. no. 4, 1960. [8] e. hewitt, a problem of a set theoretic topology, duke mat. j. 10 (1943), 309–333. 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[15] r. c. walker, the stone-cech compactification, ergebnisse der mathematik und ihrer grenzgebiete, band 83. new york-berlin: springer-verlag, (1974). © agt, upv, 2015 appl. gen. topol. 16, no. 1 64 @ appl. gen. topol. 15, no. 2(2014), 155-166doi:10.4995/agt.2014.3029 c© agt, upv, 2014 rcl-spaces and closedness/completeness of certain function spaces in the topology of uniform convergence j. k. kohli a and d. singh b a department of mathematics, hindu college, university of delhi, delhi, india (jk kohli@yahoo.com) b department of mathematics, sri aurobindo college, university of delhi, delhi, india. (dstopology@rediffmail.com) abstract it is shown that the notion of an rcl-space (demonstratio math. 46(1) (2013), 229-244) fits well as a separation axiom between zero dimensionality and r0-spaces. basic properties of rcl-spaces are studied and their place in the hierarchy of separation axioms that already exist in the literature is elaborated. the category of rcl-spaces and continuous maps constitutes a full isomorphism closed, monoreflective (epireflective) subcategory of top. the function space rcl(x, y) of all rcl-supercontinuous functions from a space x into a uniform space y is shown to be closed in the topology of uniform convergence. this strengthens and extends certain results in the literature (demonstratio math. 45(4) (2012), 947-952). 2010 msc: 54c08; 54c10; 54c35; 54d05; 54d10. keywords: rcl-space; ultra hausdorff space; initial property; monoreflective (epireflective) subcategory; rcl-supercontinuous function; topology of uniform convergence. 1. introduction the notion of an rcl-space evolved naturally in the study of rcl-supercontinuous functions [37]. here we study their basic properties and show that it fits well as a separation axiom between zero dimensionality and r0-spaces. we reflect received 13 july 2013 – accepted 20 may 2014 http://dx.doi.org/10.4995/agt.2014.3029 j. k. kohli and d. singh upon interrelations and interconnections that exist among rcl-spaces and separation axioms which already exist in the lore of mathematical literature and lie between zero dimensionality and r0-spaces. the class of rcl-spaces properly contains each of the classes of zero dimensional spaces and ultra hausdorff spaces [35] and is strictly contained in the class of r0-spaces ([20, 33]) which in its turn properly contains each of the classes of functionally regular spaces ([3, 39]) and functionally hausdorff spaces. the organization of the paper is as follows: section 2 is devoted to preliminaries and basic definitions. in section 3 we elaborate upon the place of rcl-spaces in the hierarchy of separation axioms which lie between zero dimensionality and r0-spaces and already exist in the mathematical literature. section 4 is devoted to study basic properties of rcl-spaces wherein it is shown that (i) the property of being an rcl-spaces is invariant under disjoint topological sums and initial sources so it is hereditary, productive, supinvariant, preimage invariant and projective; (ii) the category of rcl-spaces and continuous maps is a full, isomorphism closed monoreflective (epireflective) subcategory of top; (iii) it is shown that a t0-space is ultra hausdorff if and only if it is an rcl-space. in section 5 we discuss the relation between rcl-supercontinuous functions and rcl-spaces. section 6 is devoted to the study of function spaces wherein it is shown that the function space of all rcl(x, y ) of all rcl-supercontinuous functions from a topological space x into a uniform space y is closed in y x in the topology of uniform convergence and the condition for its completeness is outlined. 2. preliminaries and basic definitions let x be a topological space. a subset a of a space x is called regular gδ-set [23] if a is an intersection of a sequence of closed sets whose interiors contain a, i.e., if a = ∞⋂ n=1 fn = ∞⋂ n=1 f 0 n ,where each fn is a closed subset of x (here f 0 n denotes the interior of fn). the complement of a regular gδ-set is called a regular fσ-set. any union of regular fσ-sets is called dδ-open [17]. the complement of a dδ-open set is referred to as a dδ-closed set. a subset a of a space x is said to be regular open if it is the interior of its closure, i.e., a = a 0 . the complement of a regular open set is referred to as a regular closed set. any union of regular open sets is called δ-open set [40]. the complement of a δ-open set is referred to as a δ-closed set. any intersection of closed gδ-sets is called d-closed set [16]. any intersection of zero sets is called z-closed set ([15, 30]). a collection β of subsets of a space x is called an open complementary system [9] if β consists of open sets such that for every b ∈ β, there exist b1, b2, . . . ∈ β with b = ∪{x \ bi : i ∈ n}. a subset a of a space x is called a strongly open fσ-set [9] if there exists a countable open complementary system β(a) with a ∈ β(a). the complement of a strongly open fσ-set is c© agt, upv, 2014 appl. gen. topol. 15, no. 2 156 rcl-spaces and closedness/completeness of certain function spaces called strongly closed gδ-set. any intersection of strongly closed gδ-sets is called d∗-closed set [31]. definition 2.1. a topological space x is said to be (i) functionally regular ([3, 39]) if for each closed set f in x and each x /∈ f there exists a continuous real-valued function f defined on x such that f(x) /∈ f(f). (ii) ultra hausdorff [35] if every pair of distinct points in x are contained in disjoint clopen sets. (iii) rz-space ([20, 33]) if for each open set u in x and each x ∈ u there exists a z-closed set h containing x such that h ⊂ u; equivalently u is expressible as a union of z-closed sets. (iv) rδ-space [19] if for each open set u in x and each x ∈ u there exists a δ-closed set h containing x such that h ⊂ u; equivalently u is expressible as a union of δ-closed sets. (v) r0-space ([5],[38] 1 [28]) if for each open set u in x and each x ∈ u implies that {x} ⊂ u. (vi) r1-space ([42] 2 [5]) if x /∈ {y} implies that x and y are contained in disjoint open sets. (vii) π2-space [38] 3 (≡ pς-space [41]≡ strongly s-regular space [7]) if every open set in x is expressible as a union of regular closed sets. (viii) π0-space ([38, p 98]) if every nonempty open set in x contains a nonempty closed set. definition 2.2 ([19]). a space x is said to be an (i) rdδ -space if for each open set u in x and each x ∈ u there exists a regular gδ-set h containing x such that h ⊂ u; equivalently u is expressible as a union of regular gδ-sets. (ii) rdδ -space if for each open set u in x and each x ∈ u there exists a dδ-closed set h containing x such that h ⊂ u; equivalently u is expressible as a union of dδ-closed sets. (iii) rd-space if for each open set u in x and each x ∈ u there exists a closed gδ-set h containing x such that h ⊂ u; equivalently u is expressible as a union of closed gδ-sets. (iv) rd-space if for each open set u in x and each x ∈ u there exists a d-closed set h containing x such that h ⊂ u; equivalently u is expressible as a union of d-closed sets. 1vaidyanathswamy calls r0-axiom as π1-axiom in his text book (see [38, p 98]). császár calls an r0-space as s1-space in [4]. 2yang [42] in his studies of paracompactness refers an r1-space as a t2-space. császár calls an r1-space as s2-space in [4]. 3 π2-spaces were defined by vaidyanathswamy [38] (1960) and rediscovered by wong [41] (1981) and ganster [7] (1990) with different terminologies. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 157 j. k. kohli and d. singh definition 2.3 ([20]). a space x is said to be an (i) rd∗-space if for each open set u in x and each x ∈ u there exists a strongly closed gδ-set h containing x such that h ⊂ u; equivalently u is expressible as a union of strongly closed gδ-sets. (ii) rd∗-space if for each open set u in x and each x ∈ u there exists a d∗-closed set h containing x such that h ⊂ u; equivalently u is expressible as a union of d∗-closed sets. definition 2.4. a space x is said to be (i) d-completely regular [9] if it has a base of strongly open fσ-sets. (ii) d-regular [9] if it has a base of open fσ-sets. (iii) weakly regular [9] if it has a base of fσ-neighbourhoods. (iv) dδ-completely regular [18] if it has a base of regular fσ-sets. 3. rcl-spaces and hierarchy of seperation axioms definition 3.1. let x be a topological space. any intersection of clopen sets in x is called cl-closed [32]. an open subset u of x is said to be rcl-open [37] if for each x ∈ u there exists a cl-closed set h containing x such that h ⊂ u; equivalently u is expressible as a union of cl-closed sets. definition 3.2 ([37]). a topological space x is said to be an rcl-space if every open set in x is rcl-open. it is clear from the definitions that every zero dimensional space as well as every ultra hausdorff space is an rcl-space. the space of strong ultrafilter topology [36, example 113, p.133] is a hausdorff extremally disconnected rclspace which is not zero dimensional. the comprehensive diagram (figure 1) well reflects the place of rcl-spaces in the hierarchy of separation axioms related to the theme of the present paper and certain other topological invariants and extends several existing diagrams in the literature (see [9, 18, 19]). however, most of the implications of figure 1 are irreversible (see [9, 18, 19, 20]). we reproduce the diagram (figure 2) from [20] concerning separation axioms between functionally regular space and r0-space, which is complementary to figure 1. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 158 rcl-spaces and closedness/completeness of certain function spaces partition topology zero dimensional space ultra hausdorff developable space pseudo metrizable rcl space perfect space perfectly normal d-completely regular completely regular functionally regular rz space d-regular d -completely regular dr space regular space 2 space ( p -space strongly s-regular space) d r space d r space d r space rd space 1r space weakly regular space r0 space r -space 0 space figure 1. functionally regular rz space * d r space d r space d r space d r space 1 r space z r space * d r space d r space r space 0 r space 0 space figure 2. 4. basic properties of rcl-spaces definition 4.1. let x be a topological space. a point x ∈ x is said to be an rcl-adherent point of a set a ⊂ x if every rcl-open set containing x intersects a. let arcl denote the set of all rcl-adherent points of the set a. then a ⊂ a ⊂ arcl. the set a is rcl-closed if and only if a = arcl. lemma 4.2. the correspondence a → arcl is a kuratowski closure operator. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 159 j. k. kohli and d. singh theorem 4.3. let x be a topological space. consider the following statements: (i) x is an rcl-space (ii) for each x ∈ x and for each open set u containing x, {x}rcl ⊂ u (iii) there exists a subbase s for x such that x ∈ s ∈ s ⇒ {x}rcl ⊂ s (iv) x ∈ {y}rcl ⇒ y ∈ {x}rcl (v) x ∈ {y}rcl ⇒ {x}rcl = {y}rcl then (i) ⇔ (ii) ⇔ (iii) ⇒ (iv) ⇔ (v). proof. (i) ⇒ (ii). let x ∈ x and let u be an open set containing x. since x is an rcl-space, there exists an rcl-closed set a such that x ∈ a ⊂ u. consequently {x}rcl ⊂ u. the assertions (ii) ⇒ (i) and (ii) ⇔ (iii) are trivial. iii) ⇒ (iv). since every subbasic open set containing y contains {y}rcl, every basic open set containing y contains {y}rcl and hence it contains x. so y ∈ {x}rcl. (iv) ⇒ (v). since x ∈ {y}rcl, y ∈ {x}rcl. so x ∈ {y}rcl and y ∈ {x}rcl implies {x}rcl ⊂ {y}rcl and {y}rcl ⊂ {x}rcl, and hence {x}rcl = {y}rcl. the implication (v) ⇒ (iv) is obvious. � theorem 4.4. for a topological space x the following statements are equivalent: (i) {x}rcl 6= {y}rcl implies that x and y are contained in disjoint open sets (ii) x /∈ {y}rcl implies that x and y are contained in disjoint open sets (iii) a is compact set and {x}rcl ∩ a = ∅ implies x and a are contained in disjoint open sets (iv) if a and b are compact sets, and {a}rcl ∩b = ∅ for every a ∈ a, then a and b are contained in disjoint open sets. proof. (i) ⇒ (ii). suppose that x /∈ {y}rcl. then {x}rcl 6= {y}rcl and so by (i) x and y are contained in disjoint open sets. (ii) ⇒ (iii). let a be a compact set and suppose that {x}rcl ∩ a = ∅. so for each a ∈ a, a /∈ {x}rcl by (ii) there exist disjoint open sets ua and va containing a and x, respectively. thus the collection ν = {ua : a ∈ a} is an open cover of the compact set a and so there exists a finite subcollection {ua1, ..., uan} of ν which covers a. let u = ∪ n i=1uai and v = ∩ n i=1vai . then u and v are disjoint open sets containing a and x, respectively. (iii) ⇒ (iv). suppose that a and b are compact and {a}rcl ∩ b = ∅ for every a ∈ a. then by (iii) for each a ∈ a there exist disjoint open sets ua and va containing a and b, respectively. the collection ν = {ua : a ∈ a} is an open cover of the compact set a and so there exists a finite subcollection {ua1, ..., uan} of ν which covers a. let u = ∪ n i=1uai and v = ∩ n i=1vai. then u and v are disjoint open sets containing a and b, respectively. (iv) ⇒ (i). suppose {x}rcl 6= {y}rcl. then either x /∈ {y}rcl or y /∈ {x}rcl. for definiteness assume that y /∈ {x}rcl. then {x}rcl ∩ {y}rcl = ∅ and so by (iv) there exist disjoint open sets u and v containing x and y, respectively. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 160 rcl-spaces and closedness/completeness of certain function spaces theorem 4.5. the disjoint topological sum of any family of rcl-spaces is an rcl-space. theorem 4.6. the property of being an rcl-space is closed under initial sources, i.e., the property of being an rcl-space is an initial property. proof. let {fα : x → yα : α ∈ λ} be a family of functions, where each yα is an rcl-space and let x be equipped with initial topology. let u be any open set in x and let x ∈ u. then there exist α1, ..., αn ∈ λ and open sets vi ∈ yαi(i = 1, ..., n) such that x ∈ f −1 α1 (v1) ∩ ... ∩ f −1 αn (vn) ⊂ u. since each yα is an rcl-space, there exists a cl-closed set aαi in yαi(i = 1, ..., n) such that fαi(x) ∈ aαi ⊂ vi. since each fα is continuous, it follows that each f −1 αi (aαi) is a cl-closed set in x. let a = ∩n i=1f −1 αi (aαi). since any intersection of clclosed sets is a cl-closed, a is a cl-closed set in x and x ∈ a ⊂ u so x is an rcl-space. � as an immediate consequence of theorem 4.6 we have the following. theorem 4.7. the property of being an rcl-space is hereditary, productive, sup-invariant, preimage invariant and projective4. theorem 4.8. the category of rcl-spaces and continuous maps is a full isomorphism closed monoreflective as well as epireflective subcategory of top5. the following result gives a factorization of ultra hausdorff property with rcl-space as an essential ingredient. theorem 4.9. every ultra hausdorff space is an rcl-space. conversely, every t0, rcl-space is an ultra hausdorff space. proof. the first assertion is immediate, because in this case every singleton is cl-closed and so every open set is the union of cl-closed sets. conversely, suppose that x is a t0, rcl-space and let x, y ∈ x, x 6= y. by t0-property of x there exists an open set u containing one of the points x and y but not both. to be precise, assume that x ∈ u. since x is an rcl-space, there exists a cl-closed set a such that x ∈ a ⊂ u. let a = ∩{cα : α ∈ λ}, where each cα is a clopen set. then there exists an α0 ∈ λ such that y /∈ cα0. hence cα0 and x \ cα0 are disjoint clopen sets containing x and y, respectively and so x is an ultra hausdorff space. � 5. rcl-supercontinuous functions and rcl-spaces definition 5.1 ([37]). a function f : x → y from a topological space x into a topological space y is said to be rcl-supercontinuous if for each x ∈ x and for each open set v containing f(x), there exists an rcl open set u containing x such that f(u) ⊂ v . 4a topological property p is said to be projective if whenever a product space has property p every co-ordinate space possesses property p. 5for the definition of categorical terms we refer the reader to herrlich and strecker [11]. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 161 j. k. kohli and d. singh it is immediate from the definition that every continuous function defined on an rcl-space is rcl-supercontinuous. next we quote the following result from [37]. theorem 5.2 ([37, theorem 4.11]). let f : x → ∏ α∈λ xα be defined by f(x) = (fα(x))α∈λ, where fα : x → xα is a function for each α ∈ λ. let∏ α∈λ xα be endowed with the product topology. then f is rcl-supercontinuous if and only if each fα is rcl-supercontinuous. now we give an alternative short proof of the following result from [37]. theorem 5.3 ([37, theorem 4.13]). let f : x → y be a function and g : x → x × y be the graph function defined by g(x) = (x, f(x)) for each x ∈ x. then g is rcl-supercontinuous if and only if f is rcl-supercontinuous and x is an rcl-space. proof. observe that g = 1x ×f, where 1x denotes the identity function defined on x. now by theorem 5.2, g is rcl-supercontinuous if and only if 1x and f both are rcl-supercontinuous. again 1x is rcl-supercontinuous implies that each open set in x is rcl-open and so x is an rcl-space. � theorem 5.4. let f : x → y be an rcl-supercontinuous open bijection. if either of the space x and y is a t0-space, then x and y are homeomorphic ultra hausdorff spaces. proof. by [37, theorem 5.1] x and y are homeomorphic rcl-spaces. the last part of the theorem is immediate in view of the fact that a t0, rcl-space is ultra hausdorff (theorem 4.9). � 6. function spaces it is a well known fact that the function space c(x, y ) of all continuous functions from a topological space x into a uniform space y is not necessarily closed in y x in the topology of pointwise convergence. however, it is closed in y x in the topology of uniform convergence. it is of fundamental importance in topology, analysis and several other branches of mathematics and its applications to know whether a given function space is closed / compact / complete in y x or c(x, y ) in the topology of pointwise convergence / uniform convergence. results of this nature and ascoli type theorems abound in the literature (see [1, 12]). sierpinski [29] showed that the set of all connected (darboux) functions from a topological space x into a uniform space y is not necessarily closed in y x in the topology of uniform convergence. in contrast, naimpally [25] showed that the set of all connectivity functions from a space x into a uniform space y is closed in y x in the topology of uniform convergence. moreover, in [26] naimpally introduced the notion of graph topology γ for a function space and proved that the set of all almost continuous functions in the sense of stalling [34] is not only closed in y x in the graph topology but c© agt, upv, 2014 appl. gen. topol. 15, no. 2 162 rcl-spaces and closedness/completeness of certain function spaces it represents the closure of c(x, y ) in the graph topology. in the same vein, hoyle [10] showed that the set sw(x, y) of all somewhat continuous functions from a space x into a uniform space y is closed in y x in the topology of uniform convergence. furthermore, kohli and aggarwal in [14] proved that the function space sc(x, y ) of quasicontinuous ( ≡ semicontinuous) functions, cα(x, y ) of α-continuous functions, and l(x, y) of cl-supercontinuous functions are closed in y x in the topology of uniform convergence. in this section we strengthen the results of [14] and show that the set rcl(x, y ) ⊃ l(x, y ) of all rcl-supercontinuous functions is closed in y x in the topology of uniform convergence. definition 6.1. a subset a of a topological space x is said to be (i) semi open [22] (≡ quasi open [13]) if there exists an open set u in x such that u ⊂ a ⊂ u (ii) α-open [27] if a ⊂ (a0) 0 (iii) cl-open [32] if for each x ∈ a there exists a clopen set h such that x ∈ h ⊂ a. definition 6.2. a function f : x → y from a topological space x into a topological space y is said to be a (i) connected (darboux) function if f(a) is connected for every connected set a ⊂ x (ii) connectivity function if the graph of every connected subset of x is a connected subset of x × y (iii) semicontinuous [22] (quasicontinuous [13]) if f−1(v ) is semi open in x for every open set v in y (iv) α-continuous [24] if f−1(v ) is α-open in x for every open set v in y (v) somewhat continuous [8] if for each open set v in y such that f−1(v ) 6= ∅, then there exists a nonempty open set u in x such that u ⊂ f−1(v ), i.e. (f−1(v ))0 6= ∅. remark 6.3. somewhat continuous functions have also been referred to as feebly continuous (see [2, 6]) in the literature. however, frolik [6] requires functions to be onto. we now recall the notion of the topology of uniform convergence. let y x = {f : x → y is a function} be the set of all functions from a topological space x into a uniform space (y, ν), where ν is a uniformity on y . let f ⊂ y x. a basis for the uniformity of uniform convergence u for f is the collection {w(v ) : v ∈ ν}, where w(v ) = {(f, g) ∈ f × f : (f(x), g(x)) ∈ v for all x∈ x}. the uniform topology associated with u is called the topology of uniform convergence. for details we refer the reader to [12]. definition 6.4 ([12]). a uniform space (y, ν) is said to be complete if and only if every cauchy net in y converges to a point in y . c© agt, upv, 2014 appl. gen. topol. 15, no. 2 163 j. k. kohli and d. singh theorem 6.5 ([12, p. 194]). a product of uniform spaces is complete if and only if each co-ordinate space is complete. theorem 6.6. let x be a topological space and let (y, ν) be a uniform space. then the set rcl(x, y ) of all rcl-supercontinuous functions from x into y is closed in y x in the topology of uniform convergence. further, if y is a complete uniform space, then so is the function space rcl(x, y ) in the topology of uniform convergence. proof. let f ∈ y x be the limit point of rcl(x, y ) which is not rcl-supercontinuous at x ∈ x. then there exists v ∈ ν such that f−1(v [f(x)]) does not contain any rcl-open set containing x. choose a symmetric member w of ν such that wowow ⊂ v . since f is a limit point of rcl(x, y ), there exists g ∈ rcl(x, y ) such that g(y) ∈ w [f(y)] for all y ∈ x. then g ⊂ wof and g−1 ⊂ f−1ow −1 = f−1ow and hence g−1owog ⊂ f−1owowowof ⊂ f−1ov of. therefore g−1[w(g(x))] ⊂ f−1(v [f(x)]). since f−1(v [f(x)]) does not contain any rcl-open set containing x, neither does g −1[w(g(x))] which contradicts rcl-supercontinuity of g. therefore f ∈ rcl(x, y ). the last assertion is immediate in view of theorem 6.5 and the fact that a closed subspace of complete uniform space is complete. � remark 6.7. in view of the above discussion we extend the following inclusions diagram from [14]. l(x, y ) ⊂ rcl(x, y ) ⊂ c(x, y ) ⊂ cα(x, y ) ⊂ sc(x, y ) ⊂ sw(x, y ) ⊂ y x. since in the topology of uniform convergence each of the above function space is a closed subspace of its succeeding one, the completeness of any one of them implies that of its predecessor. in particular, if y is complete, 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[42] c. t. yang, on paracompact spaces, proc. amer. math. soc. 5, no. 2 (1954), 185–194. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 166 18.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 51 61 new results on topological dynamics ofantitriangular mapsf. balibrea, j. s. c�anovas and a. linero�abstract. we present some results concerning the topologicaldynamics of antitriangular maps, f : x2 ! x2 with the formf(x; y) = (g(y); f(x)); where (x; d) is a compact metric spaceand f; g : x ! x are continuous maps. we make an specialanalysis in the case of x = [0; 1].2000 ams classi�cation: 54h20, 37b20, 37b10, 37e99.keywords: cournot duopoly, antitriangular maps, recurrence, topologicaldynamics. 1. introductionlet (x;d) be a compact metric space and let ' : x ! x be a continuousmap, ' 2 c(x;x): the pair (x;') is called a discrete dynamical system,whose orbits are given by the sequence f'n(x)g1n=0, x 2 x, where 'n = ' �'n�1, n � 1 and '0 = identity. in general, the full knowledge of all orbitsof the system is a di�cult problem and it is only known in some particularcases. nevertheless, good approximations can be given. these approachescan be probabilistic (invariant measures, metric entropy, ...) or topological(periodic structure, topological entropy, ...). in this paper we will follow thislast approach.a point x 2 x is periodic when, for some n > 0; is 'n(x) = x: if n = 1;the periodic point is called �xed point. the order or period of a periodic pointis precisely the smallest of the values m for which 'm(x) = x: we denote byper(') the set of periods that the continuous map ' has.we use a(') to denote one of the following sets: the set of periodic points,p('); the set ap(') of almost periodic points, that is, the points x 2 x suchthat for any neighborhood v = v (x) of x; there is n = n(v ) 2 n such that'kn(x) 2 v; for every k � 0; the set ur(') of uniformly recurrent points, x 2 xsuch that for any neighborhood v = v (x) of x; there is n = n(v ) such that�this paper has been partially supported by the d.g.i.c.y.t. grant pb98{0374{c03{01. 52 f. balibrea, j. s. c�anovas and a. linerofor all q > 0 it holds 'r(x) 2 v for some q � r < q + n; r(') = fx 2 x : x 2!'(x)g is the set of recurrent points, with !'(x) the omega-limit set of the pointx; that is, the points y 2 x such that there is a subsequence fnigi � n with'ni(x) ! y as ni ! 1; c(') = r(') is the centre of '; where a denotes theclosure of a set of a � x; !(') is the (global) !-limit set, !(') = sx2x !'(x); (') is the set of non-wandering points, those points x 2 x such that forany neighborhood u = u(x) of x there exists n = n(u) 2 n in a way that'n(u)\u 6= ?; and �nally the set cr(') of chain-recurrent points, the pointsx 2 x for which given any " > 0; there is fxigni=0 � x such that x0 = x;d(xi+1;'(xi)) < " for i = 0;1; :::;n � 1 and d(x0;'(xn)) < ":it follows from the de�nitions that(1.1) p(') � ap(') � ur(') � r(') � c(')and(1.2) !(') � (') � cr('):in this paper, we are devoted to topological dynamics of antitriangular maps,that is, continuous maps f : x � x ! x � x of the form(1.3) f(x;y) = (g(y);f(x));with (x;y) 2 x � x.antitriangular maps appear in some economical models, particularly with theso{called cournot duopoly (see [12] or [8]). the cournot duopoly consists in aneconomy in which two �rms are competitors in the same sector. this situationis modelled by a map f having the form of (1.3) and such that x = i = [0;1].from (1.3) it is clear that(1.4) f2n(x;y) = ((g � f)n(x);(f � g)n(y))and(1.5) f2n+1(x;y) = (g � (f � g)n(y);f � (g � f)n(x))for any (x;y) 2 x � x and for any n 2 n. so, it is natural to expect thedynamics of f to be strongly connected to the dynamics of g � f and f � g:in [10] a program is developed for triangular maps, that is, continuous mapst : i2 ! i2 of the form t(x;y) = (f(x);g(x;y)). this is made investigatingthe relationship between the sets a(t) � i2 and a(f) � i, wherea(�) 2 fp(�);ap(�);ur(�);r(�);c(�);!(�); (�);cr(�)g:when x = i, a �rst step to follow a similar program for antitriangular mapsis to do the same with the sets a(f) � i2 and a(g � f);a(f � g) � i.for a(�) 2 fp(�);ap(�);c(�);cr(�)g we see that a(f) = a(g � f) �a(f � g)and when a(f) = r(f) the two situations, r(f) = r(g � f) � r(f � g) andr(f) $ r(g�f)�r(f �g) are possible (see [3]). we also see that in the case ofthe set of uniformly recurrent points the same result is true. for a(f) = (f)the situation is more complicated and the case (f) * (g � f) � (f � g)can happen. it remains open what is the situation when a(f) = !(f). we topological dynamics of antitriangular maps 53conjecture that the cases !(f) = !(g�f)�!(f�g) and !(f) $ !(g�f)�!(f�g)could be found.we underline that similarly to the interval case in antitriangular maps on i2it is held c(f) = p(f); which is not true in general in the triangular case (see[10]).similarly to the interval case, we construct examples proving that the follow-ing chain is possible(1.6) p(f) 6= ap(f) 6= ur(f) 6= r(f) 6= c(f) 6= !(f) 6= (f) 6= cr(f):the paper is organized as follows. in the next section, we study the rela-tionship between the sets a(f) and a(g � f) � a(f � g). the last section isconcerned with the introduction of the chain (1.6).2. projection of the topological dynamicsif (x;d) is a compact metric space, we denote the product space by (x�x;�),where �((x1;y1);(x2;y2)) = maxfd(x1;x2);d(y1;y2)gfor all (x1;y1);(x2;y2) 2 x � x. if f;g 2 c(x;x) we de�ne the product mapf � g : x � x ! x � x by (f � g)(x;y) = (f(x);g(y)) for all (x;y) 2 x � x.so, if f(x;y) = (g(y);f(x)) is an antitriangular map, we obtain that f2 =(g � f) � (f � g).in this section we consider an antitriangular map f : x � x ! x � x andwe study if the equality a(f) = a(g � f) � a(f � g) holds, where a(�) denotesone of the subsets p(�), ap(�), ur(�), r(�), c(�), !(�), (�) or cr(�). beforestudying this problem, we need the following result.proposition 2.1. let (x;d) be a compact metric space. let f;g 2 c(x;x).then(a) p(f2) = p(f) and p(f � g) = p(f) � p(g).(b) ap(f2) = ap(f) and ap(f � g) = ap(f) � ap(g).(c) ur(f2) = ur(f) and ur(f � g) � ur(f) � ur(g).(d) r(f2) = r(f) and r(f � g) � r(f) � r(g).(e) c(f2) = c(f) and c(f � g) � c(f) � c(g).(f) !(f2) = !(f) and !(f � g) � !(f) � !(g).(g) (f2) � (f) and (f � g) � (f) � (g).(h) cr(f2) = cr(f) and cr(f � g) = cr(f) � cr(g).additionally, if x = [0;1], then c(f � g) = c(f) � c(g).proof. for the �rst part of (a)-(h) see [4]. the second part of properties (a){(h)follow from de�nitions. for instance, we prove here the equality cr(f � g) =cr(f) � cr(g).let (x0;y0) 2 cr(f �g): given an arbitrary " > 0; we must prove that thereis an "-chain for x and f, and an "-chain for y and g. for " > 0; there is an"-chain for (x0;y0) and f � g;(x0;y0);(x1;y1); :::;(xn;yn);(xn+1;yn+1) = (x0;y0); 54 f. balibrea, j. s. c�anovas and a. linerosuch that d((f � g)(xi;yi);(xi+1;yi+1)) < " for i = 0;1; :::;n: clearlyx0;x1; :::;xn;xn+1 = x0is an "{chain for x0 and f andy0;y1; :::;yn;yn+1 = y0is an "{chain for y0 and g. so cr(f � g) � cr(f) � cr(g).now, let x0 2 cr(f), y0 2 cr(g) and prove that (x0;y0) 2 cr(f � g).fix " > 0 and let x0;x1; :::;xn;xn+1 = x0 be an "{chain for x0 and f; andy0;y1; :::;ym;ym+1 = y0 an "{chain for y0 and g. we can clearly assume thatn = m (repeating the chains if necessary). then,(x0;y0);(x1;y1); :::;(xn;yn); :::;(xn+1;yn+1) = (x0;y0)is an "{chain for (x0;y0) and f � g. hence (x0;y0) 2 cr(f � g).to �nish the proof, assume that x = [0;1] and prove that c(f � g) =c(f) � c(g). since c(f) = p(f) = r(f) (cf. [6]), we obtain thatr(f) � r(g) = p(f) � p(g) = p(f) � p(g) = p(f � g) � c(f � g);and jointly with (e) we conclude the proof. �theorem 2.2. let (x;d) be a compact metric space. consider f;g 2 c(x;x)and let f(x;y) = (g(y);f(x)). then(a) p(f) = p(g � f) � p(f � g).(b) ap(f) = ap(g � f) � ap(f � g).(c) ur(f) � ur(g � f) � ur(f � g).(d) r(f) � r(g � f) � r(f � g).(e) c(f) � c(g � f) � c(f � g).(f) !(f) � !(g � f) � !(f � g):(g) (f2) � (g � f) � (f � g):(h) cr(f) = cr(g � f) � cr(f � g).if in addition x = i, then c(f) = c(g � f) � c(f � g).proof. just notice that f2 = (g � f) � (f � g) and apply proposition 2.1. �now we �x x = i. it was proved in [3] that the inclusion (d) of theorem2.2 can be strict, that is, there is an antitriangular map f holding(2.7) r(f) $ r(g � f) � r(f � g):we are able to give an example showing that(2.8) ur(f) $ ur(g � f) � ur(f � g);that is, the inclusion (c) of theorem 2.2 can be strict. to this end, considerthe trapezoidal tent map f(x) = maxf1 �j2x �1j;�g (� = 0:8249:::) from [11].the idea for constructing the example is the following: any in�nite !{limitset of f is contained in a solenoidal structure. this structure can be labelledby codes which characterizes the elements of in�nite !{limit sets (see below).we take two uniformly recurrent points x0;y0 belonging to the same in�nite topological dynamics of antitriangular maps 55!{limit set and such that x0;y0 are labelled by the same code. we prove that(x0;y0) =2 ur(f) for the map f(x;y) = (y;f(x)).now, we need some de�nitions. for any z � z let z1 = f� = (�i)1i=1 : �i 2z; i 2 ng. for n 2 n let zn = f(�1;�2; :::;�n) : �i 2 z; 1 � i � ng. if � 2 znand # 2 zm, n 2 n, m 2 n [f1g, then � � # 2 zn+m (where n +1 means 1)will denote the sequence � de�ned by �i = �i if 1 � i � n and �i = #i�n for anyi > n. in what follows we denote 0 = (0;0; : : : ;0; : : :) and 1 = (1;1; : : : ;1; : : :),while if � 2 z1 then �jn 2 zn is de�ned by �jn = (�1;�2; : : : ;�n).proposition 2.3. let f be the trapezoidal map de�ned above. consider theantitriangular map f(x;y) = (y;f(x)): thenur(f) $ ur(f) � ur(f):proof. by [11], f has periodic points of periods 2n, n 2 n [f0g. by [9, propo-sition 1], there is a family fk�g�2z1 of pairwise disjoint (possibly degenerate)compact subintervals of [0;1] satisfying the following properties.(p1) the interval k0 contains all absolute maxima of f.(p2) de�ne in z1 the following total ordering: if �;� 2 z1, � 6= � and k isthe �rst integer such that �k 6= �k then � < � if either cardf1 � i k: then f(k�) = k�. also f(k0) � k1:(p4) for any n and � 2 zn, let k� be the least interval including all intervalsk�, � 2 z1, such that �jn = �. then, for any � 2 z1; k� =t1n=1 k�jn.(p5) if !(x;f) is an in�nite !-limit set of f; then !(x;f) � f0;1g1, wheref0;1g1 denotes the set of in�nite sequences (�i)1i=1 with �i 2 f0;1g forall i 2 n.(p2) gives us information on the positions of fk�g�2z1 in [0;1] while (p3)gives us information on the dynamics of the family of intervals fk�g�2z1. onthe other hand, since f has an interval of absolute maxima, (p1) gives us thatk0 is non{degenerate. let [x0;y0] = k0. by (p2) it is straightforward to provethat(2.9) f2n(1+2k)(k0) � k0jn�(1)for all n;k 2 n. then (p3) gives us that(2.10) f22n�1(1+2k)(k0) < k0 < f22n(1+2k)(k0)for n;k 2 n. we claim that x0;y0 2 ur(f). in order to see this �x jk0j >" > 0 small enough and consider the open interval (x0 � ";x0 + "). by (p4)k0j2n�1�(1) � (x0�";x0 +") if 2n�1 � m and k0j2n�(1)\(x0�";x0+") = ? forall n 2 n. by (2.9) and (2.10), f22n�1(1+2k)(x0) 2 k0j2n�1�(1) � (x0 � ";x0 + ") 56 f. balibrea, j. s. c�anovas and a. lineroif 2n � 1 � m for all k 2 n. this gives us x0 2 ur(f) (similarly it can beproved that y0 2 ur(f)). letx0(f) := fn 2 n : f2n(x0) < k0gand y0(f) := fn 2 n : f2n(y0) > k0g:consider the antitriangular map f(x;y) = (y;f(x)). then, it is clear thatf2(x;y) = (f(x);f(y)). by [5, proposition 3.5], (x0;y0) 2 ur(f) if and onlyif x0(f) \ y0(f) is in�nite. however, by (2.10), x0(f) \ y0(f) = ?. therefore(x0;y0) =2 ur(f) while (x0;y0) 2 ur(f)�ur(f). this concludes the proof. �we are not able to say anything about the inclusions (f){(g) of theorem2.2, but we are able to give an example of an antitriangular map f such that (f) * (g � f) � (f � g) (compare with (a){(f)). to this end, we prove thefollowing lemma.lemma 2.4. let f(x;y) = (f(y);f(x)) be an antitriangular map with f 2c(i;i). then x 2 (fn) if and only if (x;x) 2 (fn) for any integer n � 1:proof. first assume that x 2 (fn) and let v � i2 be an open neighborhoodof (x;x). let u � i be an open set such that (x;x) 2 u � u � v . sincex 2 (fn), there is a positive integer m such that (fn)m(u) \ u 6= ?. henceaccording to (1.4) and (1.5) we havefnm(u � u) \ (u � u) 6= ?;which gives us (x;x) 2 (fn). on the other hand, let (x;x) 2 (fn). letu � i be an open set such that x 2 u. since (x;x) 2 (fn), there is anm 2 n such that (fn)m(u � u) \ (u � u) 6= ?. then fnm(u) \ u 6= ? andx 2 (fn). �lemma 2.4 allows us to show that in general(2.11) (f) * (g � f) � (f � g):to see this, let ef be a continuous interval map such that (ef) n (ef2) 6= ?(see [7]) and de�ne ef(x;y) = (ef(y); ef(x)) (in this case g � f = f � g = ef2).let x 2 (ef) n (ef2): by lemma 2.4 we clearly obtain that x 2 (ef) implies(x;x) 2 (f), while (x;x) =2 (ef2)� (ef2) = (g �f)� (f �g). additionally,we obtain that (x;x) 2 (f) n (f2).in [7] it is shown that in the case of continuous interval maps every successionof equalities and strict inclusions is possible in the chain(2.12) (f) % (f2) % (f22) % (f23) % ::: % (f2n) % (f2n+1) % :::according to lemma 2.4, the same happens in the case of antitriangular mapsfor the chain of inclusions (f) % (f2) % (f22) % (f23) % ::: % (f2n) % (f2n+1) % :::it su�ces to take f(x;y) = (f(y);f(x)) where f holds (2.12). topological dynamics of antitriangular maps 57again concerning the non-wandering set, (2.11) gives us that it can happen (g � f) $ �1( (f)) and (f � g) $ �2( (f));where �i represents the canonical projection, i = 1;2. notice that it is straight-forward to see that�1(a(f)) = a(g � f) and �2(a(f)) = a(f � g)for a(�) 2 fp(�);ap(�);ur(�);r(�);c(�);!(�);cr(�)g:finally, we are able to prove that equalities are possible in theorem 2.2 undersome particular assumptions. we will see this in the next section.3. chain of inclusions3.1. general properties about the chain of inclusions. let f : i ! i becontinuous. then [6] provides that c(f) � !(f) and the inclusions from (1.1)and (1.2) can be rewritten as follows:(3.13) p(f) � ap(f) � ur(f) � r(f) � c(f) � !(f) � (f) � cr(f):moreover the above inclusions can be strict.proposition 3.1. there exists a continuous map f0 : i ! i such thatp(f0) 6= ap(f0) 6= ur(f0) 6= r(f0)6= c(f0) 6= !(f0) 6= (f0) 6= cr(f0):proof. in [14, theorem 4.6] we can �nd an interval map ef such thatp(ef) 6= ap(ef) 6= ur(ef) = r(ef) 6= c(ef) 6= !(ef) 6= (ef) 6= cr(ef):then we de�ne a new continuous map byf0(x) = 8<: ef(3x); if x 2 [0; 13];a�ne in [13; 23];f(3x � 2), if x 2 [23;1];where f : i ! i is a continuous map with positive topological entropy (see[1] for de�nition). then ur(f) 6= r(f) (see [14, theorem 4.19]) and by anstandard argument (see e.g. [7]) f0 holds the statement. �here we investigate if (3.13) and proposition 3.1 are true in the setting ofantitriangular maps. from de�nitions it is clear that(3.14) c(f) � (f);but we are unable to say nothing about the inclusion(3.15) c(f) � !(f):for instance, this inclusion does not work for triangular maps, that is, two{dimensional maps with the form t(x;y) = (f(x);g(x;y)) (see [10]).clearly, c(f) = r(f) � !(f). however, it is not known if !(f) is closed.this would give us !(f) = !(f), which would prove (3.15). 58 f. balibrea, j. s. c�anovas and a. lineroit is well known that c(f) = r(f) = p(f) in the case of interval maps(see [6]). however this is false for triangular maps ([10]). now in the case ofantitriangular maps we obtain the following result.theorem 3.2. let f(x;y) = (g(y);f(x)) be an antitriangular map. thenc(f) = p(f).proof. since p(f) � r(f), it is obvious that p(f) � r(f). in order to provethe converse inclusion, we use [6] and theorem 2.2 to writer(f) � r(g � f) � r(f � g) = p(g � f) � p(f � g) = p(f);which ends the proof. �in order to prove more results, we need some additional hypothesis on f. acontinuous interval map f : i ! i is called a piecewise monotone map whenthere are 0 = a1 < a2 < ::: < an = 1 such that fj[ai;ai+1] is either decreasing orincreasing for 1 � i < n. then we can prove the following result.proposition 3.3. let f(x;y) = (g(y);f(x)) be an antitriangular map suchthat f and g are piecewise monotone maps. then!(f) � c(f) = !(f):proof. if f;g are piecewise monotone maps then g�f and f�g are also piecewisemonotone maps. according to [4, proposition 22, chapter iv], p(g � f) =!(g � f) and p(f � g) = !(f � g): by theorem 2.2 and [6]!(f) � !(g � f) � !(f � g) = p(g � f) � p(f � g) == r(g � f) � r(f � g) = r(f) = c(f):on the other hand, since p(f) � !(f); by theorem 3.2 we obtainc(f) = p(f) � !(f) � !(g � f) � !(f � g)= p(g � f) � p(f � g) = p(f) = c(f): �for antitriangular maps it is possible to �nd an interesting periodic structure,similar to the �sarkovski��'s ordering (see [13]). it is known that per(g � f) =per(f � g) and either per(f) = } or per(f) = } [ f2g ; with} = 2 (per(g � f) n f1g) [ fk 2 per(g � f) : k odd, k � 1g ;where 2a = f2a : a 2 ag ; for a � n (see [2]). then we say that f has typeless, equal or bigger than 21 if the related one-dimensional map g � f has thecorresponding type. then the following result makes sense.theorem 3.4. let f(x;y) = (g(y);f(x)) be an antitriangular map such thatper(f) � f2n : n 2 n [ f0gg. assume that p(f) is a closed set. if a(�);b(�) 2fp(�);ap(�);ur(�);r(�);c(�);!(�); (�);cr(�)g it holds(a) a(f) = b(f).(b) a(f) = a(g � f) � a(f � g): moreover (f2) = (f): topological dynamics of antitriangular maps 59proof. if p(f) is closed, according to theorem 2.2 we �nd p(f) = p(g � f) �p(f � g) = p(g�f)�p(f �g) = p(f); so p(g�f) and p(f �g) are closed, hencep(g�f) = a(g�f); p(f �g) = a(f �g); where a(:) represents one of the othersseven sets ([14, theorem 4.11]). thereforecr(f) = cr(g � f) � cr(f � g) = p(g � f) � p(f � g) = p(f):now (1.1), (1.2) and theorem 2.2 end the proof. �3.2. an example of strict inclusions. now we study if proposition 3.1 holdsin the case of antitriangular maps. to this end, �rst we prove the followinglemma.lemma 3.5. let f(x;y) = (y;f(x)) be an antitriangular map de�ned oni2. let x 2 i. then x 2 a(f) if and only if (x;x) 2 a(f) where a(�) 2fp(�);ap(�);ur(�);r(�);c(�);!(�); (�);cr(�)g:proof. the cases p(�); ap(�), c(�), cr(�) hold by theorem 2.2 and ur(�); r(�),!(�) follow easily from de�nitions. so, we prove the case (�). first assumethat x 2 (f) and let v � i2 be an open neighborhood of (x;x). let u � i bean open set such that (x;x) 2 u � u � v . since x 2 (f), there is a positiveinteger n such that fn(u) \ u 6= ?. thenf2n(u � u) \ (u � u) = (fn(u) � fn(u)) \ (u � u) 6= ?;which provides (x;x) 2 (f). second, assume that (x;x) 2 (f) and let u � ibe an open neighborhood of x. since (x;x) 2 (f), there is a positive integerm such that fm(u �u)\(u �u) 6= ?. we have two possibilities: (1) m = 2nfor n 2 n and (2) m = 2n + 1 for n 2 n. if (1) happens, thenf2n(u � u) \ (u � u) = (fn(u) � fn(u)) \ (u � u) 6= ?and fn(u) \ u 6= ?. if (2) happens, thenf2n+1(u � u) \ (u � u) = (fn(u) � fn+1(u)) \ (u � u) 6= ?;and fn(u)\u 6= ? and fn+1(u)\u 6= ?. in both cases x 2 (f), which endsthe proof. �from lemma 3.5 the following result follows.theorem 3.6. there is an antitriangular map f0 such thatp(f0) 6= ap(f0) 6= ur(f0) 6= r(f0)6= c(f0) 6= !(f0) 6= (f0) 6= cr(f0):proof. just de�ne f0(x;y) = (y;f0(x)), f0 given by proposition 3.1 and applylemma 3.5. � 60 f. balibrea, j. s. c�anovas and a. lineroreferences[1] ll. alsed�a, j. llibre and m. misiurewicz, combinatorial dynamics and entropy in dimensionone, advanced series in nonlinear dynamics, 5 world scienti�c publishing co. pte. ltd., sin-gapore (1993).[2] f. balibrea and a. linero, on the periodic structure of the antitriangular maps on the unitsquare, ann. math. sil. 13 (1999), 39-49.[3] f. balibrea and a. linero, some results on topological dynamics of antitriangular maps, actamath. hungar. 88 (1-2) (2000), 169-178.[4] l. block and w. a. coppel, one-dimensional dynamics, lecture notes in math. 1513 (springer-verlag, berlin, heidelberg, 1992).[5] j. s. c�anovas and a. linero, topological dynamics classi�cation of duopoly games, chaos, soli-tons & fractals (to appear).[6] e. m. coven and g. a. hedlund, p = r for maps of the interval, proc. amer. math. soc. 79(1980), 316-318.[7] e. m. coven and z. nitecki, non{wandering sets of the powers of maps of the interval, ergod.th. and dynam. sys. 1 (1981), 9{31.[8] r. a. dana and l. montrucchio, dynamic complexity in duopoly games, j. econ. theory 44(1986), 40{56.[9] v. jim�enez l�opez, an explicit description of all scrambled sets of weakly unimodal functions oftype 21; real analysis exchange 21 (1995/1996), 1{26.[10] s. f. kolyada, on dynamics of triangular maps of the square, ergod. th. and dynam. sys. 12(1992), 749-768.[11] m. misiurewicz and j. sm��tal, smooth chaotic functions with zero topological entropy, ergod.th. and dynam. sys. 8 (1988), 421{424.[12] t. puu, chaos in duopoly pricing, chaos, solitons & fractals 1 (1991), 573{581.[13] a. n. sharkovsky, coexistence of cycles of a continuous map of the line into itself, ukrain. math.j. 16 (1964), 61-71 (in russian); english version in \thirty years after sharkovski��'s theorem:new perspectives", world scienti�c series on nonlinear science, series b 8 (1995), 1-11.[14] a. n. sharkovsky, s. f. kolyada and a. g. sivak, and v.v. fedorenko, dynamics of one-dimensional maps, mathematics and its applications, volume 407 (kluwer academic publish-ers, 1997). received october 2000revised version may 2001 f. balibreadepartamento de matem�aticasuniversidad de murcia30100 campus de espinardo, murcispaine-mail address: balibrea@um.es j. s. c�anovas topological dynamics of antitriangular maps 61departamento de matem�atica aplicada y estad��sticauniversidad polit�ecnica de cartagena30203 cartagena, murciaspaine-mail address: jose.canovas@upct.es a. linerodepartamento de matem�aticasuniversidad de murcia30100 campus de espinardo, murciaspaine-mail address: lineroba@um.es @ appl. gen. topol. 18, no. 1 (2017), 31-44 doi:10.4995/agt.2017.4469 c© agt, upv, 2017 non metrizable topologies on z with countable dual group daniel de la barrera mayoral department of mathematics, ies los rosales, madrid, spain (dbarrera@ucm.es) communicated by e. induráin abstract in this paper we give two families of non-metrizable topologies on the group of the integers having a countable dual group which is isomorphic to a infinite torsion subgroup of the unit circle in the complex plane. both families are related to d-sequences, which are sequences of natural numbers such that each term divides the following. the first family consists of locally quasi-convex group topologies. the second consists of complete topologies which are not locally quasi-convex. in order to study the dual groups for both families we need to make numerical considerations of independent interest. 2010 msc: primary: 22a05; 55m05; secondary: 20k45. keywords: locally quasi-convex topology; d-sequence; continuous character; infinite torsion subgroups of t. 1. introduction and notation the most important class of topological abelian groups are the locally compact groups. in this class, the most interesting theorems can be proved. in this paper we consider a wider class of groups, which is the class of locally quasiconvex groups. clearly, locally compact groups are locally quasi-convex, since they can be considered as a dual group (in fact, a locally compact abelian group is isomorphic to its bidual group) and dual groups are locally quasi-convex. this paper is part of a project about duality, which intends to solve, among others, the mackey problem (first stated in [8]). this problem asks whether received 28 december 2015 – accepted 13 september 2016 http://dx.doi.org/10.4995/agt.2017.4469 d. de la barrera mayoral there exists a maximum element in the family of all locally quasi-convex compatible topologies (see definition 1.12). we present two different kinds of topologies on the group of the integers, having as common feature that topologies in both families have countable dual group, which is isomorphic to an infinite torsion subgroup of the unit circle in the complex plane. on the one hand, the topologies presented in section 3 are metrizable, noncomplete and locally quasi-convex and the results contained are the natural further steps of the ones given in [2]. on the other hand, the topologies presented in section 4 are complete, nonmetrizable but they are not locally quasi-convex. these topologies, introduced by graev, were thoroughly studied by protasov and zelenyuk. precisely, in [16], a neighborhood basis for this topology is given and this allows us to study these topologies on z from a numerical point of view. the fact that these topologies have countable dual group is surprising due to the belief that the more open sets the topology has, the more continuous characters it should have. all the groups considered will be abelian and the notation will be additive. in order to state properly the mackey problem, we need first to recall some notation on duality: definition 1.1. we will consider the unit circle as t = r/z = { x + z : x ∈ ( − 1 2 , 1 2 ]} . we will consider in t the following neighborhood basis: tm := { x + z : x ∈ [ − 1 4m , 1 4m ]} . we set t+ := { x + z : x ∈ [ − 1 4 , 1 4 ]} = t1. definition 1.2. let (g,τ) be a topological group. denote by g∧ the set of continuous homomorphisms (or continuous characters) chom(g,t). this set is a group when we consider the operation (f +g)(x) = f(x)+g(x). in addition, if we consider the compact open topology on g∧ it is a topological group. the topological group g∧ is called the dual group of g. definition 1.3. the natural embedding αg : g → g∧∧, is defined by x 7→ αg(x) : g∧ → t, where χ 7→ χ(x). proposition 1.4. αg is a homomorphism. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 32 non metrizable topologies on z with countable dual group proof. αg(x+y)(χ) = χ(x+y) = χ(x)χ(y) = αg(x)(χ)αg(y)(χ) = (αg(x)αg(y))(χ), for all χ ∈ g∧. thus, αg(x + y) = αg(x)αg(y). � definition 1.5. a topological group (g,τ) is reflexive if αg is a topological isomorphism. definition 1.6. let τ,ν be two group topologies on a group g. we say that τ and ν are compatible if algebraically (g,τ)∧ = (g,ν)∧. now we include the definition of quasi-convex set, which emulates the hahnbanach theorem from functional analysis: definition 1.7 ([17]). let (g,τ) be a topological group and let m ⊂ g be a subset. we say that m is quasi-convex if for every x ∈ g\m, there exists χ ∈ g∧ such that χ(m) ⊂ t+ and χ(x) /∈ t+. we say that τ is locally quasiconvex if there exists a neighborhood basis for τ consisting of quasi-convex subsets. we denote by lqc the class of all locally quasi-convex topological groups. definition 1.8. let τ be a group topology. we define the locally quasi-convex modification of τ as the finest locally quasi-convex group topology (τ)lqc among all those satisfying that they are coarser than τ. for a deeper insight into the locally quasi-convex modification of a topological group, see [10]. lemma 1.9. let (g,τ) be a topological group. then (g,τ)∧ = (g, (τ)lqc) ∧. definition 1.10. let (g,τ) a topological group. the polar of a subset s ⊂ g is defined as s. := {χ ∈ g∧ | χ(s) ⊂ t+}. remark 1.11. it is clear that (g,τ)∧ = ⋃ v∈b v ., where b is a neighborhood basis of 0 for the topology τ. definition 1.12 ([12]). let g be a class of abelian topological groups and let (g,τ) be a topological group in g. let cg(g,τ) denote the family of all g-topologies ν on g compatible with τ. we say that µ ∈ cg(g,τ) is the gmackey topology for (g,g∧) (or the mackey topology for (g,τ) in g) if ν ≤ µ for all ν ∈cg(g,τ). if g is the class of locally quasi-convex groups lqc, we will simply say that the topology is mackey. the problem whether the mackey topology for a topological group (g,τ) exists was first posed in [8]. this problem is called the mackey problem. during recent years several mathematicians have tried to solve the mackey problem: de leo [9], aussenhofer, dikranjan, mart́ın-peinador [4], dı́az-nieto c© agt, upv, 2017 appl. gen. topol. 18, no. 1 33 d. de la barrera mayoral [10], gabriyelyan [11]. a survey including similarities and differences between the mackey topology in spaces and the mackey problem for groups has recently appeared [14]. in [3], the mackey problem for groups of finite exponent is reduced to the problem of finding a top element in the family of compatible linear topologies (that is, topologies having a neighborhood basis consisting in open subgroups). in order to solve the mackey problem (in the negative) we try to find all the locally quasi-convex group topologies which are compatible with a non-discrete linear topology on z and study if the supremum of all these topologies is again compatible. so far, it is not known if the supremum of compatible topologies is again compatible. now we give the basic definitions on d-sequences, which are sequences of natural numbers that encode the relevant information of non-discrete linear topologies on z. definition 1.13. a sequence of natural numbers b = (bn)n∈n0 ∈ nn0 is called a d-sequence if it satisfies: (1) b0 = 1, (2) bn 6= bn+1 for all n ∈ n0, (3) bn divides bn+1 for all n ∈ n0. the following notation will be used in the sequel: • d := {b : b is a d − sequence}. • d∞ := {b ∈d : bn+1 bn →∞}. • d∞(b) := {c : c is a subsequence of b and c ∈d∞}. and for an arbitrary (fixed) natural number ` we define: • d`∞ := {b ∈d : bn+` bn →∞}. • d`∞(b) := {c : c is a subsequence of b and c ∈d`∞}. our interest in d-sequences stems from the fact that several compatible topologies on z can be associated to d-sequences. this family of topologies is interesting due to the fact that we don’t know in general, if the supremum of compatible topologies is again compatible. this is the main difference between the study of the mackey topology in spaces and the mackey problem for groups. definition 1.14. let b be a d-sequence. define (qn)n∈n by qn := bn bn−1 . definition 1.15. let b be a d-sequence. we say that (a) b has bounded ratios if there exists a natural number n, satisfying that qn = bn bn−1 ≤ n. (b) b is basic if qn is a prime number for all n ∈ n. definition 1.16. let b be a d-sequence. we write z(bn) := { k bn + z : k = 0, 1, . . . ,bn − 1 } and z(b∞) := ⋃ n∈n0 z(bn). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 34 non metrizable topologies on z with countable dual group as quotient groups, we can write z(bn) = 〈{ 1 bn }〉 /z and z(b∞) = 〈{ 1 bm : m ∈ n0 }〉 /z. in other words, z(bn) consists of the elements whose order divides bn and z(b∞) is generated by the elements having order bn for some natural number n ∈ n0. proposition 1.17. let b be a d-sequence. suppose that qj+1 6= 2 for infinitely many j. for each integer number l ∈ z, there exists a natural number n = n(l) and unique integers k0, . . . ,kn , such that: (1) l = n∑ j=0 kjbj. (2) ∣∣∣∣∣∣ n∑ j=0 kjbj ∣∣∣∣∣∣ ≤ bn+12 for all n. (3) kj ∈ ( − qj+1 2 , qj+1 2 ] , for 0 ≤ j ≤ n. the proof of proposition 1.17 can be found in [5, proposition 2.2.1] and [6, proposition 1.4] definition 1.18. the family bb := {bnz : n ∈ n0} is a neighborhood basis of 0 for a linear (that is, it has a neighborhood basis consisting in open subgroups) group topology on z, which will be called b-adic topology and is denoted by λb. this topology is precompact. in [2], it is proved that (z,λb)∧ = z(b∞) and the following lemma 1.19. let b be a d-sequence. then bn → 0 in λb. the topologies of uniform convergence on the group z are given by different families in hom(z,t). identifying hom(z,t) with t, we simply have to consider families of subsets in t. in this paper, we concentrate on families of cardinality 1, that is we consider the topology of uniform convergence on {b} for b ⊂ t. we shall write τb instead of τ{b}. the topology τb admits the following neighborhood basis of 0: vb,m := {z ∈ z : χ(z) + z ∈ tm for all χ ∈ b} = ⋂ χ∈b χ−1(tm). a particular case of these topologies is the following. definition 1.20. a d-sequence b in z induces in a natural way a sequence in t. namely, for a d-sequence, b = (bn)n∈n0 , define b := ( 1 bn + z : n ∈ n0 ) ⊂ t. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 35 d. de la barrera mayoral denote by τb the topology of uniform convergence on b. 2. killing λb null sequences. in this section we prove that for a d sequence, b with bounded ratios and any λb convergent sequence, (an), there exists a strictly finer compatible topology (constructed as a topology of uniform convergence on a suitable sequence in d∞(b)) such that the fixed sequence (an) is no longer convergent. this will allow us to find a locally quasi-convex topology which has no convergent sequences, but it is still compatible. theorem 2.1. let b be a basic d-sequence with bounded ratios and let (xn) ⊂ z be a non-quasiconstant (that is, no element in (xn) is repeated infinitely many times) sequence such that xn λb→ 0. then there exists a metrizable locally quasi-convex compatible group topology τ(= τc for some subsequence c of b) on z satisfying: (a) τ is compatible with λb. (b) xn τ9 0. (c) λb < τ. proof. we define τ as the topology of uniform convergence on a subsequence c of b, which we construct by means of the claim: claim 2.2. since b is a d-sequence with bounded ratios, there exists l ∈ n such that qn+1 ≤ l for all n ∈ n. we can construct inductively two sequences (nj), (mj) ⊂ n such that nj+1 −nj ≥ j and xmj bnj+1 + z /∈ tl. proof of the claim: j = 1. choose n1 ∈ n such that there exists xm1 , satisfying bn1 | xm1 but bn1+1 xm1 . by proposition 1.17, we write xm1 = n(xm1 )∑ i=0 kibi. the condition bn1 | xm1 implies ki = 0 if i < n1 and bn1+1 xm1 implies kn1 6= 0. hence, xm1 bn1+1 + z = n(xm1 )∑ i=0 kibi bn1+1 + z = n1∑ i=0 kibi bn1+1 + z = kn1 qn1+1 + z. we have that 1 2 ≥ |kn1| qn1+1 ≥ 1 l . c© agt, upv, 2017 appl. gen. topol. 18, no. 1 36 non metrizable topologies on z with countable dual group therefore, xm1 bn1+1 + z /∈ tl = [ − 1 4l , 1 4l ] + z. j ⇒ j + 1. suppose we have nj,mj satisfying the desired conditions. let nj+1 ≥ nj + j be a natural number such that there exists xmj+1 satisfying bnj+1 | xmj+1 and bnj+1+1 xmj+1 . by proposition 1.17, we write xmj+1 = n(xmj+1 )∑ i=0 kibi. the condition bnj+1 | xmj+1 implies that ki = 0 if i < nj+1 and bnj+1+1 xmj+1 implies knj+1 6= 0. then, xmj+1 bnj+1+1 + z = n(xmj+1 )∑ i=0 kibi bnj+1+1 + z = nj+1∑ i=0 kibi bnj+1+1 + z = knj+1 qnj+1+1 + z. from 1 2 ≥ ∣∣knj+1∣∣ qnj+1+1 ≥ 1 qnj+1+1 ≥ 1 l , we get that xmj+1 bnj+1+1 + z /∈ tl. this ends the proof of the claim. we continue the proof of theorem 2.1. consider now c = ( bnj+1 ) j∈n. let c = { 1 bnj+1 + z : j ∈ n } and let τ = τc be the topology of uniform convergence on c. then: (1) by proposition [2, remark 3.3], τ is metrizable and locally quasiconvex. by [2, proposition 3.7], we have λb < τ. (2) by the claim, we have proved that xmj j→∞ 9 0 in τ. this implies that xn 9 0 in τ. (3) since nj+1 ≥ nj +j, we get that nj+1−nj ≥ j. thus, bnj+1 bnj+1+1 ≤ 1 2j → 0. by [2, theorem 4.4], we have that (z,τ)∧ = z(b∞). � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 37 d. de la barrera mayoral 3. the topology γb. in section 2, we use a subfamily of d∞(b) to eliminate all λb convergent sequences. unfortunately, this subfamily depends on the choices made in theorem 2.1. this is why we introduce a new topology γb, which is the supremmum of the topologies of uniform convergence on sequences in d∞(b). the topology γb is well-defined (in the sense that it does not depend on the choices made in theorem 2.1). definition 3.1. let b be a basic d-sequence with bounded ratios. we define on z the topology γb := sup{τc : c ∈d∞(b)}. now we set some results on γb: remark 3.2. since d∞(b) is non-empty, we have that λb < γb. theorem 3.3. let b a basic d-sequence with bounded ratios. then the topology γb has no nontrivial convergent sequences. hence, it is not metrizable. proof. since every γb-convergent sequence is λb-convergent, we consider only λb-convergent sequences. let (xn) ⊆ z be a sequence such that xn λb→ 0. by theorem 2.1, there exists a subsequence (bnk)k satisfying that τ(bnk ) is a locally quasi-convex topology, (bnk) ∈d∞(b) and xn τbnk9 0. since τbnk ≤ γb, we know that xn γb9 0. hence, the only convergent sequences in γb are the trivial ones. since γb has no nontrivial convergent sequences, it is not metrizable. � corollary 3.4. let p be a prime number and set p = (pn). then γp has no nontrivial convergent sequences. lemma 3.5. let b be a d-sequence. then (z, γb)∧ = z(b∞). for the proof of lemma 3.5 we need the following definition of [5]: definition 3.6 ([5, definition 4.4.4]). let b be a basic d-sequence with bounded ratios. we define on z the topology δb := sup{τc : c ∈d`∞(b)}. proof of lemma 3.5. since λb < γb, it is clear that z(b∞) ≤ (z, γb)∧. by definition, it is clear that γb ≤ δb. hence (z, γb)∧ ≤ (z,δb)∧ = z(b∞). 2 now we set some interesting questions related to the mackey problem and γb: open question 1. is γb = δb? open question 2. is γb (or δb, if different) the mackey topology for (z,λb)? the following lemmas are easy to prove, even for a topological space instead of a topological group. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 38 non metrizable topologies on z with countable dual group lemma 3.7. let k be a countably infinite compact subset of a hausdorff space. then k is first countable, and, hence, every accumulation point in k is the limit point of some sequence. proof. let k = {xn,n ∈ n}. fix a ∈ k. consider for each xn ∈ k a closed neighborhood of a (this is possible because k is regular) say vn ⊂ k\xn. the set {vn,n ∈ n} and its finite intersections constitute a neighborhood basis for a ∈ k. in fact, if w is any open neighborhood of a, we have a family of closed sets given by {k \w,vn,n ∈ n} with empty intersection. since k is compact, the mentioned family cannot have the finite intersection property. thus, there is a finite subfamily {vj,j ∈ f} such that (k \w) ∩i∈f vi = ∅. � lemma 3.8. if a countable topological group g has no nontrivial convergent sequences, then it does not have infinite compact subsets either. proof. suppose, by contradiction, that k ⊂ g is an infinite compact subset. since k is first countable and hausdorff, any accumulation point is the limit of a sequence in k. sincek is infinite and compact, it has an accumulation point, contradicting the fact that there does not exist convergent sequences in g (nor in k). hence any compact subset must be finite. � proposition 3.9. let b be a basic d-sequence with bounded ratios, then the group g = (z, γb) is not reflexive. in fact, the bidual g∧∧ can be identified with z endowed with the discrete topology. proof. the dual group of (z, γb) has supporting set z(b∞). the γb-compact subsets of z are finite by lemma 3.8, therefore, the dual group carries the pointwise convergence topology. thus, g∧ is exactly z(b∞) with the topology induced by the euclidean of t. by [1, 4.5], the group z(b∞) has the same dual group as t, namely, z with the discrete topology. we conclude that the canonical mapping αg is an open non-continuous isomorphism. � corollary 3.10. since (z, γb)∧∧ is discrete, the homomorphism α(z,γb) is not continuous. however, it is an open algebraic isomorphism. proposition 3.11. let b ∈ d`∞ and gγ := (z,τb). then αgγ is a nonsurjective embedding from gγ into g ∧∧ γ . proof. since τb is metrizable, αgγ is continuous. the fact that τb is locally quasi-convex and z(b∞) separates points of z imply that αgγ is injective and open in its image [1, 6.10]. on the other hand, αgγ is not onto, for otherwise gγ would be reflexive. however, a non-discrete countable metrizable group cannot be reflexive. indeed, the dual of a metrizable group is a k-space ([1, 7]) and the dual group of a k-space is complete. hence, the original group must be complete as well. by baire category theorem, the only metrizable complete group topology on a countable group is the discrete one. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 39 d. de la barrera mayoral 4. the complete topology t{bn} in this section we consider some special families of topologies which were introduced by graev and deeply studied in the group of the integers by protasov and zelenyuk ([15, 16]). we need first some definitions about sequences. definition 4.1. let g be a group and g = (gn) ⊂ g a sequence of elements of g. we say that g is a t-sequence if there exists a hausdorff group topology τ such that gn τ→ 0. since the topology must be hausdorff, we can consider that gn 6= gm if n 6= m lemma 4.2. let g ⊂ g be a t -sequence, then there exists the finest group topology t{gn} satisfying that gn t{gn}→ 0. since we can describe t{gn} we are interested in finding a suitable neighborhood basis for this topology. definition 4.3 ([16]). let g be a group and let a = (an) be a t-sequence in g and (ni)i∈n a sequence of natural numbers. we define: • a∗m := {±an|n ≥ m}∪{0g}. • a(k,m) := {g0 + · · · + gk|gi ∈ a∗m i ∈{0, . . . ,k}}. • [n1, . . . ,nk] := {g1 + · · · + gk : gi ∈ a∗ni, i = 1, . . . ,k}. • v(ni) = ∞⋃ k=1 [n1, . . . ,nk]. proposition 4.4. the family {v(ni) : (ni) ∈ n n} is a neighborhood basis of 0g for a group topology t{an} on g, which is the finest among all those satisfying an → 0. the symbol g{an} will stand for the group g endowed with t{an}. next, we include a theorem of great interest from [16]: theorem 4.5 ([16, theorem 2.3.11]). let (gn) be a t -sequence on a group g. then, the topology t{gn} is complete. corollary 4.6. let b be a d-sequence and c ∈d`∞(b). then τc 6= t{bn}. applying the baire category theorem, we can obtain the following corollary: corollary 4.7. let g be a countable group. for any t -sequence g, the topology t{gn} is not metrizable. since we are mainly interested on topologies on the group of integers, we include the following results: proposition 4.8 ([16, theorem 2.2.1]). if limn→∞ an+1 an = ∞ then (an) is a t -sequence in z. this proposition provides a condition to prove that the following sequences are t-sequences: (bn) = (2 n2 ), (ρn) = (p1 · · · · ·pn), where (pn) is the sequence of prime numbers. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 40 non metrizable topologies on z with countable dual group proposition 4.9 ([16, theorem 2.2.3]). if limn→∞ an+1 an = r and r is a transcendental number then (an) is a t -sequence in z. let b be a d-sequence. it is clear (by lemma 1.19) that b is a t-sequence. hence, the topology t{bn} can be constructed on z. by theorem 4.5, it is a complete topology. the following properties of z{an} are interesting by themselves. remark 4.10. let (an) be a t-sequence on z. then: (1) z{an} is sequential, but not frechet-urysohn ([15, theorem 7 and theorem 6]). (2) z{an} is a kω-group ([16, corollary 4.1.5] ). we prove now that, for a d-sequence b with bounded ratios, the locally quasi-convex modification of t{bn} is λb. this fact is also proved in [11, proposition 2.5], but we give here a direct proof. first we study some numerical results in the group t: lemma 4.11. let k ∈ t and m ∈ n satisfy k, 2k,. . . ,mk ∈ t+. if mk ∈ tn, then k ∈ tnm. proof. write k = z + y where z ∈ z and y ∈ ( −1 2 , 1 2 ] . since k ∈ t+, we have |y| ≤ 1 4 . suppose that k /∈ tnm. this means that 14nm < |y| ≤ 1 4 . now, mk = mz + my. since 1 4n < |my| ≤ m 4 and mk ∈ tn, we have that |my| ≥ 1 − 14n. hence, there exists n ≤ k such that 1 4 < |ny| < 3 4 . consequently, nk /∈ t+. � lemma 4.12. let k ∈ t and let m1,m2, . . . ,mn ∈ n\{1}. if k, 2k, 3k,. . . ,m1k, 2m1k,. . . ,m1m2k,. . . ,( n−1∏ i=1 mi ) k, 2 ( n−1∏ i=1 mi ) k,. . . , ( n∏ i=1 mi ) k ∈ t+, then k ∈ t∏n i=1 mi . proof. since (∏n−1 i=1 mi ) k, 2 (∏n−1 i=1 mi ) k,. . . ( ∏n i=1 mi) k ∈ t+ and ( ∏n i=1 mi) k ∈ t1, by lemma 4.11, we have that (∏n−1 i=1 mi ) k ∈ tmn. since (∏n−2 i=1 mi ) k, 2 (∏n−2 i=1 mi ) k,. . . (∏n−1 i=1 mi ) k ∈ t+ and (∏n−1 i=1 mi ) k ∈ tmn, by lemma 4.11, we have that (∏n−2 i=1 mi ) k ∈ tmnmn−1 . repeating the argument n times, we obtain that k ∈ t∏n i=1 mi. � lemma 4.13. let k ∈ t and let (mi) ∈ nn a sequence of natural numbers satisfying mi > 1 for all i ∈ n. if k, 2k, 3k,. . . ,m1k, 2m1k,. . . ,m1m2k,. . . ,( n−1∏ i=1 mi ) k, 2 ( n−1∏ i=1 mi ) k,. . . , ( n∏ i=1 mi ) k, · · · ∈ t+ c© agt, upv, 2017 appl. gen. topol. 18, no. 1 41 d. de la barrera mayoral then k = 0 + z. proof. by lemma 4.12, we have that k ∈ ⋂ n>0 t∏ni=1 mi. hence, k = 0+z. � now we can study the polar of the neighborhoods in the topology t{bn}. proposition 4.14. let b be a d-sequence with bounded ratios and v(ni) a basic neighborhood of t{bn}. then, there exists n ∈ n such that a := {bn, 2bn, . . . ,bn+1, 2bn+1, . . . ,bn+2, 2bn+2, . . .}∈ v(ni). as a consequence v . (ni) are finite subsets of t, for any sequence (ni). proof. since b has bounded ratios, there exists l ∈ n such that qn ≤ l for all n ∈ n. let n1,n2, . . . ,nl be the indexes satisfying that ni ≤ nj if i ∈ {1, 2, . . . ,l} and j /∈ {1, 2, . . . ,l}. that is, the terms n1, . . . ,nl are the l smallest ones in (ni). let n := max{n1,n2, . . . ,nl}. then bn,bn+1, · · · ∈ a∗ni for i ∈ {1, 2, . . . ,l} and a ⊂{bn, 2bn, . . . ,lbn,bn+1, 2bn+1, . . . ,lbn+1, . . .}⊂ [n1,n2, . . . ,nl] ⊂ v(ni). let χ ∈ v . (ni) . since a ⊂ v(ni), it is clear that v . (ni) ⊂ a.. define k := χ(bn ). then χ(a) = {k, 2k,. . .qnk, 2qnk,. . . ,qnqn+1k,. . .}. since χ ∈ a., we have χ(a) ⊂ t+. by lemma 4.13, the fact k = χ(bn ) = 0 + z follows. hence χ ∈ z(bn ) and v .(ni) is finite. � corollary 4.15. let b be a d-sequence with bounded ratios. then z∧{bn} = z(b∞). proof. since z{bn} = ⋃ (ni) v . (ni) , we have that z∧{bn} ≤ z(b ∞). since b → 0 in λb, λb ≤t{bn} and z ∧ {bn} ≥ z(b ∞). � finally, we can prove that for a d-sequence, b, with bounded ratios, the locally quasi-convex modification of t{bn} is λb. theorem 4.16. let b a d-sequence with bounded ratios. then the locally quasi-convex modification of t{bn} is λb. proof. since the polar sets of the basic neighborhoods of t{bn} are finite, the equicontinuous subsets in the dual group are finite as well. hence the locally quasi-convex modification of t{bn} is precompact. since t{bn} and its locally quasi-convex modification are compatible topologies, it follows that( t{bn} ) lqc = λb. � corollary 4.17. let b a d-sequence with bounded ratios. then t{bn} is not locally quasi-convex. proof. if t{bn} were locally quasi-convex, then ( t{bn} ) lqc = t{bn}, but ( t{bn} ) lqc is not complete and t{bn} is complete. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 42 non metrizable topologies on z with countable dual group remark 4.18. although the examples in theorem 4.16 are not locally quasiconvex, there exist examples of graev-type topologies on z which are locally quasi-convex. indeed, in [13] it is proved that there exist graev-type topologies on z which are reflexive (hence locally quasi-convex). remark 4.19. corollary 4.17 gives a family of examples of: • complete topologies whose locally quasi-convex modifications are not complete. • non-metrizable topologies whose locally quasi-convex modifications are metrizable. proposition 4.20. the dual group of z{bn} coincides algebraically and topologically with the dual group of (z,λb); that is, the dual group of z{bn} is z(b ∞) endowed with the discrete topology. proof. from corollary 4.15 the dual group of z{bn} is z ∧ {bn} = z(b ∞). item (2) in remark 4.10 implies that z∧{bn} is metrizable and complete. by baire category theorem and since z∧{bn} is countable, we get that it is discrete. hence, it coincides topologically with (z,λb)∧. � now we answer the question whether z{bn} is map-mackey. proposition 4.21. the topology z{bn} is not map-mackey. proof. if t{bn} were mackey, the condition γb ≤ t{bn} should hold. but, the topology γb has no convergent sequences and bn → 0 in t{bn}. � open question 3. let (gn) a t -sequence in z. if t{gn} is locally quasi-convex, is t{gn} the mackey topology? acknowledgements. the author is thankful to the referee for the kind and deep review. references [1] l. außenhofer, contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, dissertationes math. 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[17] n. ya. vilenkin, the theory of characters of topological abelian groups with boundedness given, izvestiya akad. nauk sssr. ser. mat. 15 (1951), 439–162. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 44 @ appl. gen. topol. 18, no. 1 (2017), 117-129 doi:10.4995/agt.2017.6578 c© agt, upv, 2017 on the generalized asymptotically nonspreading mappings in convex metric spaces withun phuengrattana a,b a department of mathematics, faculty of science and technology, nakhon pathom rajabhat university, nakhon pathom 73000, thailand (withun ph@yahoo.com) b research center for pure and applied mathematics, research and development institute, nakhon pathom rajabhat university, nakhon pathom 73000, thailand. (withun ph@yahoo.com) communicated by s. romaguera abstract in this article, we propose a new class of nonlinear mappings, namely, generalized asymptotically nonspreading mapping, and prove the existence of fixed points for such mapping in convex metric spaces. furthermore, we also obtain the demiclosed principle and a ∆-convergence theorem of mann iteration for generalized asymptotically nonspreading mappings in cat(0) spaces. 2010 msc: 47h09; 47h10. keywords: asymptotically nonspreading mapping; convex metric spaces; cat(0) spaces; demiclosed principle. 1. introduction throughout this paper, we denote the set of positive integers by n. let t be a mapping on a nonempty subset c of a banach space x. we denote by f(t) the set of fixed points of t , i.e., f(t) = {x ∈ c : tx = x}. in 2008, kohsaka and takahashi [14] introduced a nonlinear mapping called nonspreading mapping in a smooth, strictly convex, and reflexive banach space x as follows: let c be a nonempty closed convex subset of x. a mapping received 09 september 2016 – accepted 24 december 2016 http://dx.doi.org/10.4995/agt.2017.6578 w. phuengrattana t : c → c is said to be nonspreading if φ(tx,ty) + φ(ty,tx) ≤ φ(tx,y) + φ(ty,x)(1.1) for all x,y ∈ c, where φ(x,y) = ‖x‖2 − 2〈x,jy〉 + ‖y‖2 for all x,y ∈ x and j is the duality mapping on c. observe that if x is a real hilbert space, then j is the identity mapping and φ(x,y) = ‖x−y‖2 for all x,y ∈ x. so, a nonspreading mapping t in a real hilbert space x is defined as follows: 2‖tx−ty‖2 ≤‖tx−y‖2 + ‖ty −x‖2 for all x,y ∈ c. since then, some fixed point theorems of such mapping has been studied by many researchers; see, for example, [9, 10, 11]. later in 2013, naraghirad [17] introduced a new class of nonspreading-type mappings in a real banach space, called an asymptotically nonspreading mapping, as follows: a mapping t : c → c is called asymptotically nonspreading if ‖tnx−tny‖2 ≤‖x−y‖2 + 2〈x−tnx,j(y −tny)〉(1.2) for all x,y ∈ c and n ∈ n, where j is the normalized duality mapping of c. in the case when x is a real hilbert space, we know that j is the identity mapping. so, an asymptotically nonspreading mapping t in a real hilbert space x is defined as follows: ‖tnx−tny‖2 ≤‖x−y‖2 + 2〈x−tnx,y −tny〉(1.3) for all x,y ∈ c and n ∈ n. in a real hilbert space, it is easy to show that (1.3) is equivalent to 2‖tnx−tny‖2 ≤‖tnx−y‖2 + ‖tny −x‖2 for all x,y ∈ c and n ∈ n. naraghirad [17] proved weak and strong convergence theorems of the iterative sequences generated by an asymptotically nonspreading mapping in a real banach space. motivated by the above works, we define a new class of nonlinear mappings which contains the class of asymptotically nonspreading mappings in convex metric spaces, called a generalized asymptotically nonspreading mapping, and prove some existence theorems for such mapping in convex metric spaces. furthermore, we also obtain the demiclosed principle and a ∆-convergence theorem of mann iteration for generalized asymptotically nonspreading mappings in cat(0) spaces. 2. preliminaries in the sequel, we recall some definitions, notations, and conclusions which will be needed in proving our main results. let (x,d) be a metric space. a mapping w : x ×x × [0, 1] → x is said to be a convex structure [22] on x if for each x,y ∈ x and λ ∈ [0, 1], d(z,w(x,y,λ)) ≤ λd(z,x) + (1 −λ)d(z,y) c© agt, upv, 2017 appl. gen. topol. 18, no. 1 118 on the generalized asymptotically nonspreading mappings n convex metric spaces for all z ∈ x. a metric space (x,d) together with a convex structure w is called a convex metric space which will be denoted by (x,d,w). a nonempty subset c of x is said to be convex if w(x,y,λ) ∈ c for all x,y ∈ c and λ ∈ [0, 1]. it is easy to see that open and closed balls are convex and the intersection of a family of convex subsets of a convex metric space x is also convex, see [22]. clearly, a normed space and each of its convex subsets are convex metric spaces, but the converse does not hold. in 1996, shimizu and takahashi [21] introduced the concept of uniform convexity in convex metric spaces and studied the properties of these spaces. definition 2.1. a convex metric space (x,d,w) is said to be uniformly convex if for any ε > 0, there exists δ(ε) ∈ (0, 1] such that for all r > 0 and x,y,z ∈ x with d(z,x) ≤ r, d(z,y) ≤ r and d(x,y) ≥ rε, imply that d ( z,w ( x,y, 1 2 )) ≤ (1 − δ(ε)) r. obviously, uniformly convex banach spaces are uniformly convex metric spaces. one of the special spaces of uniformly convex metric spaces is a cat(0) space (see more details in [3]). the useful inequality of cat(0) space is (cn) inequality [4], that is, if z,x,y are points in a cat(0) space and if m is the midpoint of the geodesic segment [x,y], then the cat(0) inequality implies (cn) d(z,m)2 ≤ 1 2 d(z,x)2 + 1 2 d(z,y)2 − 1 4 d(x,y)2. by using the (cn) inequality, it is easy to see that cat(0) spaces are uniformly convex. moreover, if x is a cat(0) space and x,y ∈ x, then for any λ ∈ [0, 1], there exists a unique point λx⊕ (1 −λ)y ∈ [x,y] such that d(z,λx⊕ (1 −λ)y) ≤ λd(z,x) + (1 −λ)d(z,y), for any z ∈ x. it follows that cat(0) spaces have a convex structure w(x,y,λ) := λx ⊕ (1 − λ)y. existence theorems and convergence theorems in convex metric spaces and cat(0) spaces have been studied and investigated, see, for examples, [13, 5, 16, 12, 18, 15, 19, 1, 2]. the notion of the asymptotic center can be introduced in the general setting of a cat(0) space x as follows: let {xn} be a bounded sequence in x. for x ∈ x, we define a mapping r (·,{xn}) : x → [0,∞) by r (x,{xn}) = lim sup n→∞ d(x,xn). the asymptotic radius of {xn} is given by r ({xn}) = inf {r (x,{xn}) : x ∈ x} , and the asymptotic center of {xn} is the set a ({xn}) = {x ∈ x : r (x,{xn}) = r ({xn})} . it is known by [7] that in a cat(0) space, the asymptotic center a ({xn}) consists of exactly one point. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 119 w. phuengrattana we now give the definition and collect some basic properties of the ∆convergence which will be used in the sequel. definition 2.2 ([13]). a sequence {xn} in a cat(0) space x is said to ∆converge to x ∈ x if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. in this case, we write ∆-limn→∞ xn = x and call x the ∆-limit of {xn}. we now collect some basic properties of the ∆-convergence which will be used in the sequel. lemma 2.3 ([13]). every bounded sequence in a cat(0) space has a ∆convergent subsequence. lemma 2.4 ([6]). let c be a nonempty closed convex subset of a cat(0) space x. if {xn} is a bounded sequence in c, then the asymptotic center of {xn} is in c. lemma 2.5 ([8]). let {xn} be a sequence in a cat(0) space x with a({xn}) = {x}. if {un} is a subsequence of {xn} with a({un}) = {u} and {d(xn,u)} converges, then x = u. the following lemma is a generalization of the (cn) inequality which can be found in [8]. lemma 2.6. let x be a cat(0) space. then d(z,λx⊕ (1 −λ)y)2 ≤ λd(z,x)2 + (1 −λ)d(z,y)2 −λ(1 −λ)d(x,y)2, for any λ ∈ [0, 1] and x,y,z ∈ x. 3. main results in this section, we study the existence and convergence theorems for a generalized asymptotically nonspreading mapping in both convex metric spaces and cat(0) spaces. we first define a generalized asymptotically nonspreading mapping in convex metric spaces. definition 3.1. let c be a nonempty subset of a convex metric space (x,d,w). a mapping t : c → c is called generalized asymptotically nonspreading if there exist two functions f,g : c → [0,γ], γ < 1 such that (c1) d(tnx,tny)2 ≤ f(x)d(tnx,y)2 + g(x)d(tny,x)2 for all x,y ∈ c and n ∈ n; (c2) 0 < f(x) + g(x) ≤ 1 for all x ∈ c. remark 3.2. the class of generalized asymptotically nonspreading mappings contains the class of asymptotically nonspreading mappings. indeed, we know that if f(x) = g(x) = 1 2 for all x ∈ c, then t is an asymptotically nonspreading mapping. the next example shows that there is a generalized asymptotically nonspreading mapping which is not asymptotically nonspreading. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 120 on the generalized asymptotically nonspreading mappings n convex metric spaces example 3.3. define a mapping t : [0,∞) → [0,∞) by tx = { 0.9, if x ≥ 1, 0, if x ∈ [0, 1). then t is not an asymptotically nonspreading mapping. indeed, if x = 1.2 and y = 0.7, then tx = 0.9, ty = 0, and 2d(tx,ty)2 = 1.62 > 1.48 = 0.04 + 1.44 = d(tx,y)2 + d(ty,x)2. however, t is a generalized asymptotically nonspreading mapping. indeed, let f,g : [0,∞) → [0, 0.9) be defined by f(x) = { 0, if x ≥ 1, 0.81, if x ∈ [0, 1), and g(x) = { 0.81, if x ≥ 1, 0, if x ∈ [0, 1). now, we only need to consider the following two cases: (i) if x ≥ 1 and y ∈ [0, 1), then tx = 0.9,ty = 0,tnx = tny = 0 ∀n ≥ 2, f(x) = 0, and g(x) = 0.81. so, we have d(tx,ty)2 = 0.81 ≤ g(x)x2 = f(x)d(tx,y)2 + g(x)d(ty,x)2. on the other hand, for any n ≥ 2, we have d(tnx,tny)2 = 0 ≤ f(x)d(tnx,y)2 + g(x)d(tny,x)2. (ii) if x ∈ [0, 1) and y ≥ 1, then tx = 0,ty = 0.9,tnx = tny = 0 ∀n ≥ 2, f(x) = 0.81, and g(x) = 0. so, we have d(tx,ty)2 = 0.81 ≤ f(x)y2 = f(x)d(tx,y)2 + g(x)d(ty,x)2. on the other hand, for any n ≥ 2, we have d(tnx,tny)2 = 0 ≤ f(x)d(tnx,y)2 + g(x)d(tny,x)2. therefore, t is a generalized asymptotically nonspreading mapping. 3.1. existence theorems. we now prove existence theorems for generalized asymptotically nonspreading mappings in complete convex metric spaces. theorem 3.4. let c be a nonempty closed convex subset of a complete convex metric space (x,d,w) such that a({xn}) is singleton for all bounded sequence {xn} in c and t : c → c be a generalized asymptotically nonspreading mapping. then the following assertions are equivalent: (i) f(t) is nonempty; (ii) there exists a bounded sequence {xn} in c such that lim inf n→∞ d(xn,txn) = 0. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 121 w. phuengrattana proof. since it is obvious that (i) implies (ii), we show that (ii) implies (i). suppose that there exists a bounded sequence {xn} in c such that lim inf n→∞ d(xn,txn) = 0. consequently, there is a bounded subsequence {xnk} of {xn} such that lim k→∞ d(xnk,txnk ) = 0. suppose a({xnk}) = {z}. since t is a generalized asymptotically nonspreading mapping, we have d(xnk,tz) 2 ≤ (d(xnk,txnk ) + d(tz,txnk )) 2 = d(xnk,txnk ) 2 + d(tz,txnk ) 2 + 2d(xnk,txnk )d(txnk,tz) ≤ d(xnk,txnk ) 2 + f(z)d(tz,xnk ) 2 + g(z)d(txnk,z) 2 + 2d(xnk,txnk )d(txnk,tz) ≤ d(xnk,txnk ) 2 + f(z)d(tz,xnk ) 2 + g(z)(d(txnk,xnk ) + d(xnk,z)) 2 + 2d(xnk,txnk )d(txnk,tz) = (1 + g(z))d(xnk,txnk ) 2 + f(z)d(tz,xnk ) 2 + g(z)d(xnk,z) 2 + 2g(z)d(txnk,xnk )d(xnk,z) + 2d(xnk,txnk )d(txnk,tz). this implies that (1 −f(z))d(xnk,tz) 2 ≤ (1 + g(z))d(xnk,txnk ) 2 + g(z)d(xnk,z) 2 + 2m(1 + g(z))d(xnk,txnk ), where m = supk∈n{d(xnk,z),d(txnk,tz)}. taking lim sup on both sides of the above inequality, we get (1 −f(z)) lim sup k→∞ d(xnk,tz) 2 ≤ g(z) lim sup k→∞ d(xnk,z) 2 ≤ (1 −f(z)) lim sup k→∞ d(xnk,z) 2. so, we have lim sup k→∞ d(xnk,tz) ≤ lim sup k→∞ d(xnk,z). it implies that r(tz,{xnk}) = lim sup k→∞ d(xnk,tz) ≤ lim sup k→∞ d(xnk,z) = r(z,{xnk}). this shows that tz ∈ a({xnk}). by the uniqueness of asymptotic center, we conclude that tz = z. � theorem 3.5. let c be a nonempty closed convex subset of a complete convex metric space (x,d,w) such that a({xn}) is singleton for all bounded sequence {xn} in c and t : c → c be a generalized asymptotically nonspreading mapping. if limn→∞ d(t nx,tn+1x) = 0 for all x ∈ c, then the following assertions are equivalent: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 122 on the generalized asymptotically nonspreading mappings n convex metric spaces (i) f(t) is nonempty; (ii) there exists x ∈ c such that {tnx} is bounded. proof. since it is obvious that (i) implies (ii), we show that (ii) implies (i). suppose that there exists x ∈ c such that {tnx} is bounded. setting yn = tnx for all n ∈ n. then we have lim n→∞ d(tyn,yn) = lim n→∞ d(tn+1x,tnx) = 0. since {yn} is bounded, it implies by theorem 3.4 that f(t) is nonempty. � it follows from the fact that, in a complete uniformly convex metric space, the asymptotic center of a bounded sequence with respect to a closed convex subset is singleton; see [20]. so, we have the following results. theorem 3.6. let c be a nonempty closed convex subset of a complete uniformly convex metric space (x,d,w) and t : c → c be a generalized asymptotically nonspreading mapping. then the following assertions are equivalent: (i) f(t) is nonempty; (ii) there exists a bounded sequence {xn} in c such that lim inf n→∞ d(xn,txn) = 0. theorem 3.7. let c be a nonempty closed convex subset of a complete uniformly convex metric space (x,d,w) and t : c → c be a generalized asymptotically nonspreading mapping. if limn→∞ d(t nx,tn+1x) = 0 for all x ∈ c, then the following assertions are equivalent: (i) f(t) is nonempty; (ii) there exists x ∈ c such that {tnx} is bounded. since every cat(0) space is a uniformly convex metric space, the following results can be obtained from theorems 3.6 and 3.7 immediately. theorem 3.8. let c be a nonempty closed convex subset of a complete cat(0) space (x,d) and t : c → c be a generalized asymptotically nonspreading mapping. then the following assertions are equivalent: (i) f(t) is nonempty; (ii) there exists a bounded sequence {xn} in c such that lim inf n→∞ d(xn,txn) = 0. theorem 3.9. let c be a nonempty closed convex subset of a complete cat(0) space (x,d) and t : c → c be a generalized asymptotically nonspreading mapping. if limn→∞ d(t nx,tn+1x) = 0 for all x ∈ c, then the following assertions are equivalent: (i) f(t) is nonempty; (ii) there exists x ∈ c such that {tnx} is bounded. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 123 w. phuengrattana remark 3.10. theorems 3.4-3.9 improve and extend the main results of naraghirad [17] from an asymptotically nonspreading mapping to a generalized asymptotically nonspreading mapping and from a banach space to a complete convex metric space. 3.2. ∆-convergence theorems. in this section, we study ∆-convergence theorems for a generalized asymptotically nonspreading mapping in complete cat(0) spaces. the following theorem show the demiclosed principle for a generalized asymptotically nonspreading mapping on complete cat(0) spaces. theorem 3.11. let c be a nonempty closed convex subset of a complete cat(0) space (x,d) and t : c → c be a generalized asymptotically nonspreading mapping. if {xn} is a bounded sequence in c that ∆-converges to z and limn→∞ d(xn,txn) = 0, then z ∈ f(t). proof. suppose that {xn} is a bounded sequence in c such that ∆-limn→∞ xn = z and limn→∞ d(xn,txn) = 0. by the definition of t, we have d(xn,tz) 2 ≤ (d(xn,txn) + d(tz,txn)) 2 = d(xn,txn) 2 + 2d(xn,txn)d(tz,txn) + d(tz,txn) 2 ≤ d(xn,txn)2 + 2d(xn,txn)d(tz,txn) + f(z)d(tz,xn)2 + g(z)d(txn,z) 2 ≤ d(xn,txn)2 + 2d(xn,txn)d(tz,txn) + f(z)d(tz,xn)2 + g(z)(d(txn,xn) + d(xn,z)) 2 = (1 + g(z))d(xn,txn) 2 + 2d(xn,txn)d(tz,txn) + f(z)d(tz,xn) 2 + 2g(z)d(txn,xn)d(xn,z) + g(z)d(xn,z) 2. this implies that (1 −f(z))d(xn,tz)2 ≤ (1 + g(z))d(xn,txn)2 + 2m(1 + g(z))d(xn,txn) + g(z)d(xn,z) 2, where m = supn∈n{d(xn,z),d(txn,tz)}. taking lim sup on both sides of the above inequality, we get (1 −f(z)) lim sup n→∞ d(xn,tz) 2 ≤ g(z) lim sup n→∞ d(xn,z) 2 ≤ (1 −f(z)) lim sup n→∞ d(xn,z) 2. thus, we have lim sup n→∞ d(xn,tz) ≤ lim sup n→∞ d(xn,z). by the uniqueness of asymptotic centers, we have tz = z. hence z ∈ f(t). � by using theorem 3.11, we obtain the following ∆-convergence theorem for a generalized asymptotically nonspreading mapping in complete cat(0) spaces. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 124 on the generalized asymptotically nonspreading mappings n convex metric spaces theorem 3.12. let c be a nonempty closed convex subset of a complete cat(0) space (x,d) and t : c → c be a generalized asymptotically nonspreading mapping with f(t) is nonempty. assume that {αn} is a sequence in (0, 1) such that 0 < a ≤ αn ≤ b < 1. let {xn} be a sequence in c generated by xn+1 = (1 −αn)xn ⊕αntnxn, for all n ∈ n.(3.1) then the sequence {xn} ∆-converges to a fixed point of t . proof. let z ∈ f(t). since t is generalized asymptotically nonspreading, we have d(z,tnxn) 2 ≤ f(z)d(z,xn)2 + g(z)d(tnxn,z)2. this implies that (1 −g(z))d(z,tnxn)2 ≤ f(z)d(z,xn)2. since 0 < f(z) + g(z) ≤ 1, we have d(z,tnxn) ≤ d(z,xn).(3.2) in view of lemma 2.6, (3.1), and (3.2), d(xn+1,z) 2 ≤ (1 −αn)d(xn,z)2 + αnd(tnxn,z)2 −αn(1 −αn)d(xn,tnxn)2 ≤ d(xn,z)2 −αn(1 −αn)d(xn,tnxn)2 ≤ d(xn,z)2 −a(1 − b)d(xn,tnxn)2.(3.3) thus, we have d(xn+1,z) ≤ d(xn,z). this implies that limn→∞ d(xn,z) exists for all z ∈ f(t) and hence {xn} is bounded. it follows by (3.3) that a(1 − b)d(xn,tnxn)2 ≤ d(xn,z)2 −d(xn+1,z)2, which yields that lim n→∞ d(xn,t nxn) = 0.(3.4) in view of (3.1), we see that d(xn+1,xn) ≤ αnd(tnxn,xn) ≤ bd(tnxn,xn). then we have lim n→∞ d(xn+1,xn) = 0.(3.5) c© agt, upv, 2017 appl. gen. topol. 18, no. 1 125 w. phuengrattana since t is generalized asymptotically nonspreading, we obtain that d(tn+1xn,t n+1xn+1) 2 ≤ f(xn)d(tn+1xn,xn+1)2 + g(xn)d(tn+1xn+1,xn)2 ≤ f(xn)(d(tn+1xn,tn+1xn+1) + d(tn+1xn+1,xn+1))2 + g(xn)(d(t n+1xn+1,xn+1) + d(xn+1,xn)) 2 = f(xn)d(t n+1xn,t n+1xn+1) 2 + f(xn)d(t n+1xn+1,xn+1) 2 + 2f(xn)d(t n+1xn,t n+1xn+1)d(t n+1xn+1,xn+1) + g(xn)d(t n+1xn+1,xn+1) 2 + g(xn)d(xn+1,xn) 2 + 2g(xn)d(t n+1xn+1,xn+1)d(xn+1,xn) ≤ γd(tn+1xn,tn+1xn+1)2 + γd(tn+1xn+1,xn+1)2 + 2γd(tn+1xn,t n+1xn+1)d(t n+1xn+1,xn+1) + γd(tn+1xn+1,xn+1) 2 + γd(xn+1,xn) 2 + 2γd(tn+1xn+1,xn+1)d(xn+1,xn). thus, we have (1 −γ)d(tn+1xn,tn+1xn+1)2 ≤ 2γd(tn+1xn+1,xn+1)2 + 2γm1d(tn+1xn+1,xn+1) + γd(xn+1,xn) 2 + 2γd(tn+1xn+1,xn+1)d(xn+1,xn), where m1 = supn∈n{d(tn+1xn,tn+1xn+1)}. this implies from (3.4) and (3.5) that lim n→∞ d(tn+1xn,t n+1xn+1) = 0.(3.6) consider d(xn,t n+1xn) ≤ d(xn,xn+1) + d(xn+1,tn+1xn+1) + d(tn+1xn+1,tn+1xn), then, by (3.4), (3.5), and (3.6), we have lim n→∞ d(xn,t n+1xn) = 0.(3.7) since d(tnxn,t n+1xn) ≤ d(tnxn,xn) +d(xn,tn+1xn), it implies by (3.6) and (3.7) that lim n→∞ d(tnxn,t n+1xn) = 0.(3.8) c© agt, upv, 2017 appl. gen. topol. 18, no. 1 126 on the generalized asymptotically nonspreading mappings n convex metric spaces since t is generalized asymptotically nonspreading, we have d(txn,t n+1xn) 2 ≤ f(xn)d(txn,tnxn)2 + g(xn)d(tn+1xn,xn)2 ≤ f(xn)(d(txn,tn+1xn) + d(tn+1xn,tnxn))2 + g(xn)d(t n+1xn,xn) 2 ≤ f(xn)d(txn,tn+1xn)2 + f(xn)d(tn+1xn,tnxn)2 + 2f(xn)d(txn,t n+1xn)d(t n+1xn,t nxn) + g(xn)d(t n+1xn,xn) 2 ≤ γd(txn,tn+1xn)2 + γd(tn+1xn,tnxn)2 + 2γd(txn,t n+1xn)d(t n+1xn,t nxn) + γd(t n+1xn,xn) 2. this implies that (1 −γ)d(txn,tn+1xn)2 ≤ γd(tn+1xn,tnxn)2 + 2γm2d(tn+1xn,tnxn) + γd(tn+1xn,xn) 2, where m2 = supn∈n{d(txn,tn+1xn)}. thus, by (3.7) and (3.8), we have lim n→∞ d(txn,t n+1xn) = 0.(3.9) from d(txn,xn) ≤ d(txn,tn+1xn) + d(tn+1xn,xn), it implies by (3.7) and (3.9) that lim n→∞ d(txn,xn) = 0.(3.10) we now let ω∆(xn) := ⋃ a({un}), where the union is taken over all subsequences {un} of {xn}. we claim that ω∆(xn) ⊂ f(t). let u ∈ ω∆(xn). then there exists a subsequence {un} of {xn} such that a({un}) = {u}. since {un} is bounded, it implies by lemma 2.3 that there exists a subsequence {unk} of {un} such that ∆-limk→∞ unk = y ∈ c. by (3.10) and theorem 3.11, we have y ∈ f(t). then limn→∞ d(xn,y) exists. suppose that u 6= y. by the uniqueness of asymptotic centers, we obtain that lim sup k→∞ d(unk,y) < lim sup k→∞ d(unk,u) ≤ lim sup n→∞ d(un,u) < lim sup n→∞ d(un,y) = lim sup n→∞ d(xn,y) = lim sup k→∞ d(unk,y). this is a contradiction, hence u = y ∈ f(t). this shows that ω∆(xn) ⊂ f(t). next, we show that ω∆(xn) consists of exactly one point. let {un} be a subsequence of {xn} with a({un}) = {u} and let a({xn}) = {z}. since u ∈ ω∆(xn) ⊂ f(t), it implies that limn→∞ d(xn,u) exists. by lemma 2.5, we get z = u. hence, the sequence {xn} ∆-converges to a fixed point z of t. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 127 w. phuengrattana remark 3.13. theorem 3.12 improves and extends theorem 4.1 of naraghirad [17] to a generalized asymptotically nonspreading mapping and to a complete cat(0) space. acknowledgements. the author would like to thank the anonymous reviewers for their helpful comments. references [1] a. abkar and m. eslamian, fixed point and convergence theorems for different classes of generalized nonexpansive mappings in cat(0) spaces, comp. math. appl. 64 (2012), 643–650. 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ø ä ö î ; � = a � å � ú å � ÷@ø�ù�ù�ö�÷� ;ö�ú þ â � / ^ � � ��j ; � = a � m þ õ å ù�÷=ö#\ �� å � ÷=ø�ù�ù�ö�÷� aö�ú # ø ä ö è ÷76]� % \ '�å �6 \ � \7�;�uî ' ök÷=ø�ù�÷�� úwö� �6 è \ / \�^��2þ�� ø! ;ö� �6 è \ h @kj � x @ j}9 @ m \ x \ 9 \ > $'& �)�uîtè ù�ú � ø +6�ö ä öïö�, å�� è �qø å ù 6 2*\%� h @kj � x @ jw9 @ m �?� ÷76q �6 è j+* ��6 m 2*/ % r ^ ` ä j 1 � m þ ;ö æ j c míù�ø '~å æ �� å ö � �6 è r d + ^ ` ä j�1 � m d &èä jfx m�� ` ä j 1 � mkd &èä jfx � ml6^ 4 î�è ÷@ø�ù ä è ú å ÷� å ø�ù '�å �6>+yd r ^ 4 þ 4;6 � �@î + å � ÷=ø�ù�ù�ö�÷� aö�ú þ # x ' ö ä ö è ÷=ø�ù� å ù � ø ��ækè � î + ^ &èä j�x m ' ø � �'ú � ö�÷=ø æ ö è ÷�� ø � ö�ú �� ö� dø # 9x^>\ j%ù�ø! ;ög �6 è \ å �[ä ö�� è4ä�î 6�ö@ù�÷=ö ó è!�� úwø ä 8 m þ õ å ù�÷@ö�+ å � úwö�ù � ö å ùb9x^>\ î å ' ø � �'úr +6�ö@ù # ø�� ø ' �6 è z+ ^ 9x^ \ î�è ÷=ø�ù� ä;è ú å ÷� å ø�ù þ w � ?abd67@�� i��i 4;6�ö è!� øf=�ö`� ä ø��qø �aå å ø�ù ä ö ækè4å ù � = è � å ú # ø ä,ó]è�� ú�ø ä 8 � � è ÷@ö � 9 è ù�ú \ � ù�úwö ä �6�ö è ú�ú å å ø�ù è � è��+��æ � å ø�ù> �6 è � v j}91m >x� v j�\�myx �uþ 4�ø � ö@öl �6 å ��î è!@ ö è0�%�m��è�� ö ( i � yc9 �8� . # 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(elghaoui.mohamed@yahoo.fr) b university of gafsa, faculty of sciences of gafsa, tunisia. (adlenesoo@yahoo.com) abstract in this paper, we give an explicit criterion to decide the density of finitely generated additive subgroups of r n and c n . 2010 msc: 47a06; 47a99. keywords: dense; additive group; rationally independent; kronecker. 1. introduction it is a classical result of kronecker that the additive group z + αz, α ∈ r, is dense in r whenever α is irrational. in higher dimensions, this is generalized as follows: zn + z[θ1, . . . , θn] t is dense in rn if and only if 1, θ1, . . . , θn are rationally independent (see [6] and [5]). for general generated additive groups of the form h = p∑ k=1 zuk, where p ≥ 1 and uk ∈ kn (k = r or c), a criterion for the density was given by waldschmidt ([7], see proposition 2.1 for the real case). however, the use of this theorem in higher dimension or with a large number of generators is more difficult. so the main aim of this paper is to give an explicit arithmetic way for checking the density of any finitely generated additive subgroup of cn and rn, which may be used in a future algorithm. this criterion can be used as a tool to characterize the density of any orbit given by the natural action of any abelian linear or affine group on kn (see for example, [1], [2], [3] and [4]). received 5 september 2014 – accepted 27 july 2015 http://dx.doi.org/10.4995/agt.2015.3257 m. elghaoui and a. ayadi 2. preliminaries first of all, let us introduce the following proposition which characterizes the density of additive subgroups zu1 + · · · + zup of rn. proposition 2.1 ([7], proposition 4.3, chapter ii). let h = zu1 + · · · + zup with uk ∈ rn, k = 1, . . . , p. then h is dense in rn if and only if for every (s1, . . . , sp) ∈ zp\{0}: rank [ u1 . . . . . . up s1 . . . . . . sp ] = n + 1. if p = n + 1 and (u1, . . . , un) is a basis of r n, the additive group h = p∑ i=1 zui, where un+1 = n∑ i=1 θiui is isomorphic (by a linear map) to z n + z[θ1, . . . , θn] t . in this case, proposition 2.1 becomes explicit and have the following form: h is dense in rn if and only if 1, θ1, . . . , θn are rationally independent. now, for the general case, if h is dense in rn then p ≥ n + 1 and the vector space p∑ k=1 ruk is equal to r n (proposition 2.1). the last condition means that a basis of rn can be extracted from the set of vectors uk, k = 1, . . . , p. so let us assume here and after that this basis is (u1, . . . , un) and that p ≥ n + 1. with these assumptions, the rank condition in proposition 2.1 becomes: (2.1) rank   1 0 . . . 0 αn+1,1 . . . αp,1 0 ... ... ... ... ... ... ... ... ... 0 ... ... ... 0 . . . 0 1 αn+1,n . . . αp,n s1 . . . . . . sn sn+1 . . . sp   = n + 1 where the scalars αk,i are the coordinates of uk, k = n + 1, . . . , p in the basis (u1, . . . , un), i.e. uk = n∑ i=1 αk,iui for all k = n + 1, . . . , p c© agt, upv, 2015 appl. gen. topol. 16, no. 2 128 rational criterion for testing the density of additive subgroups simplifying further (2.1) using elementary row operations, we get: rank   1 0 . . . 0 αn+1,1 . . . αp,1 0 ... ... ... ... ... ... ... ... ... 0 ... ... ... 0 . . . 0 1 αn+1,n . . . αp,n 0 . . . . . . 0 sn+1 − n∑ i=1 siαn+1,i . . . sp − n∑ i=1 siαp,i   = n + 1 this condition is fulfilled if the last row is not null, which means that for every (s1, . . . , sp) ∈ zp\{0}, there is at least one integer k0 ∈ {n+1, . . . , p} such that sk0 − n∑ i=1 siαk0,i 6= 0, which gives rise to the following proposition: proposition 2.2. let h = zu1+· · ·+zup, p ≥ n+1 and such that (u1, . . . , un) is a basis of rn with uk = n∑ i=1 αk,iui, for every k = n + 1, . . . , p. then h is dense in rn if and only if for every (s1, . . . , sp) ∈ zp\{0}, there is at least one integer k0 ∈ {n + 1, . . . , p} such that sk0 − n∑ i=1 siαk0,i 6= 0. now, let us suppose that 1, αk,i1, . . . , αk,irk is the longest sequence extracted from the list {1, αk,1, . . . , αk,n} which contains 1 and such that its elements are rationally independent. set ik := {i1, . . . , irk}. • if ik0 = {1, 2, . . . , n} for at least one integer k0 ∈ {n+1, . . . , p} then h is dense in rn. indeed, otherwise, by proposition 2.2, there exists (s1, . . . , sp) ∈ zp\{0} such that for every k = n + 1, . . . , p (2.2) sk − n∑ i=1 siαk,i = 0 as ik0 = {1, 2, . . . , n} then using equation 2.2 with k = k0, we get sk0 = 0 and si = 0 for all i = 1, . . . , n. using again this equation for the other values of k ∈ {n+1, . . . , p}, we get sk = 0. therefore si = 0 for every i = 1, . . . , p, which leads to a contradiction since (s1, . . . , sp) ∈ zp\{0}. • if ik = ∅ all the coordinates of the given vector uk are rational. actually if this condition is fulfilled for every k = n + 1, . . . , p then we have: proposition 2.3. if ik = ∅ for every k = n + 1, . . . , p, then p∑ j=1 zuj is not dense in rn. we need the following lemma: c© agt, upv, 2015 appl. gen. topol. 16, no. 2 129 m. elghaoui and a. ayadi lemma 2.4. let (u1, . . . , un) be a basis of r n, n ≥ 2. then: (i) the group p∑ k=1 zuk is closed in r n, for any 1 ≤ p ≤ n. (ii) the group p∑ k=1 ruk + q∑ k=p+1 zuk is closed in r n, for any 1 ≤ p < q ≤ n. proof. let φ : rp −→ p∑ k=1 ruk the natural isomorphism defined by φ(x1, . . . , xp) = p∑ k=1 xkuk. then φ is a homeomorphism and we have φ(zp) = p∑ k=1 zuk. since z p is closed in rp, so p∑ k=1 zuk is closed in p∑ k=1 ruk and hence in r n. a similar argument can be used to the proof (ii) by considering rp × zq−p which is closed in rq. � proof of proposition 2.3. if ik = ∅ for every k = n + 1, . . . , p, then the coordinates of every vector uk are rational. so there exist qk ∈ n∗ and pk,j ∈ z such that αk,j = pk,j qk . therefore, uk = 1 qk n∑ j=1 pk,juj, for every k = n + 1, . . . , p. hence p∑ j=1 zuj ⊂ 1q n∑ j=1 zuj, where q = qn+1qn+2 . . . qp. by lemma 2.4, 1 q n∑ j=1 zuj is closed in rn, therefore p∑ j=1 zuj is not dense in r n. � • for a fixed k = n + 1, . . . , p, assume that ik 6= ∅ and ik 6= {1, 2, . . . , n}. then rewrite the scalars αk,j for every j /∈ ik as a function of 1 and the scalars {αk,i i ∈ ik}. thus there exist γ(k)j,i1, . . . , γ (k) j,irk , tk,j ∈ q such that αk,j = tk,j + ∑ i∈ik γ (k) j,i αk,i c© agt, upv, 2015 appl. gen. topol. 16, no. 2 130 rational criterion for testing the density of additive subgroups we obtain: uk = n∑ j=1 αk,juj = ∑ j∈ik αk,juj + ∑ j /∈ik ( tk,j + ∑ i∈ik γ (k) j,i αk,i ) uj = ∑ j∈ik αk,juj + ∑ i∈ik αk,i   ∑ j /∈ik γ (k) j,i uj   + ∑ j /∈ik tk,juj = ∑ j∈ik αk,j  uj + ∑ i/∈ik γ (k) i,j ui   + ∑ j /∈ik tk,juj let qk ∈ n∗ and m(k)i,j , pk,j ∈ z such that tk,j = pk,j qk and γ (k) i,j = m (k) i,j qk . therefore, (2.3) qkuk = ∑ j∈ik αk,j  qkuj + ∑ i/∈ik m (k) i,j ui   + ∑ j /∈ik pk,juj notice that the choice of the scalar qk is not unique as it can be replaced by a positive multiple of it. set u′k,j := qkuj + ∑ i/∈ik m (k) i,j ui for every k = n + 1, . . . , p and j ∈ ik. so (2.4) qkuk = ∑ j∈ik αk,ju ′ k,j + ∑ j /∈ik pk,juj for a fixed k, the family of vectors ( u′k,j, j ∈ ik ) and (uj, j /∈ ik) constitute all together a basis of rn since the obtained set is a result of transforming the basis (u1, . . . , un) using elementary operations. 3. the main result: the real case now, assume that ik 6= ∅ for at least one k and that ik 6= {1, . . . , n} for every k = n + 1, . . . , p. definition 3.1. we define mh to be the matrix of the coordinates of the vectors u′k,j, j ∈ ik and k = n + 1, . . . , p. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 131 m. elghaoui and a. ayadi the matrix mh is actually defined up to scaling of its columns which are the vectors {u′k,j, j ∈ ik ; k = n + 1, . . . , p}. indeed, for a fixed k ∈ {n + 1, . . . , p}, the choice of qk in the definition of u ′ k,j is not unique as it can be replaced by any positive multiple of it. however, the rank of the matrix mh does not change with a particular choice of the set of vectors u′k,j. example 3.2. let h = zu1 + · · · + zu7 with u1 = [1, 0, 0]t , u2 = [0, 1, 0]t , u3 = [0, 0, 1] t , u4 = [1, 3 √ 2, 2]t , u5 = [0, √ 2, √ 5]t , u6 = [2 √ 2, √ 3, 1]t , u7 = [3, √ 2, 2 √ 2]t . so n = 3 and p = 7. let k = 4. then 1, 3 √ 2 is the longest sequence which can be extracted from the set {1, 1, 3 √ 2, 2} (of the coordinates of u4 along with 1) such that its elements are rationally independent. since only α4,2 has been selected, so i4 = {2}. once α4,1 and α4,3 are written as a function of 1, 3 √ 2, we get: t4,1 = 1, t4,3 = 2, γ (4) 1,2 = γ (4) 3,2 = 0 using the same procedure for the remaining values of k, we obtain: i5 = {2, 3}, t5,1 = 0, γ(5)1,2 = γ (5) 1,3 = 0 i6 = {1, 2}, t6,3 = 1, γ(6)3,1 = γ (6) 3,2 = 0 i7 = {2}, t7,1 = 3, t7,3 = 0, γ(7)1,2 = 0, γ (7) 3,2 = 2 we choose q4 = q5 = q6 = q7 = 1 so that pk,j = tk,j and m (k) i,j = γ (k) i,j for every i ∈ ik, j /∈ ik and k = 4, 5, 6, 7. the vectors u′k,j, j ∈ ik, k = 4, 5, 6, 7 are: u′4,2 = u2 u′5,2 = u2 u′5,3 = u3 u′6,1 = u1 u′6,2 = u2 u′7,2 = u2 + 2u3 so that mh is given by: mh =   0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 2   theorem 3.3. let h = zu1 + · · · + zup with uk ∈ rn and mh defined as above. then h is dense in rn if and only if rank(mh) = n. we need the following lemmas for the proof of the theorem 3.3: c© agt, upv, 2015 appl. gen. topol. 16, no. 2 132 rational criterion for testing the density of additive subgroups lemma 3.4. let u1, . . . , un+1 ∈ rn be such that (u1, . . . , un) is a basis of rn and un+1 = n∑ i=1 αiui. suppose that 1, αk1, . . . , αkr is the longest sequence extracted from the list {1, α1, . . . , αn} which contains 1 and such that its elements are rationally independent. then there exist q ∈ n∗, mk,1, . . . , mk,r ∈ z such that ∑ j∈i ru′j + ∑ j /∈i zuj ⊂ n+1∑ k=1 zuk ⊂ ∑ j∈i ru′j + 1 q ∑ j /∈i zuj where u′j = quj + ∑ k /∈i mk,juk for every j ∈ i and i = {kj, j = 1, . . . , r}. proof. assume without loss of generality that kj = j, j = 1, . . . , r. in the above discussion, we have introduced the vectors u′k,j when several vectors are added to the basis (u1, . . . , un). but in this case, only one vector has been added (p = n + 1), so we drop the index k from the definition of u′k,j, mk,j, pk,j and ik. thus we have qun+1 = ∑ j∈i αju ′ j + ∑ j /∈i pjuj where u′j = quj + ∑ i/∈i mi,jui. moreover, let h := n+1∑ k=1 zuk and u′n+1 = qun+1 − ∑ j /∈i pjuj = r∑ j=1 αju ′ j. now, consider the vector space e of dimension r equipped with the basis b1 = (u′1, . . . , u ′ r). the vector u ′ n+1 ∈ e and its coordinates with respect to the basis b1 are [α1, . . . , αr]t . moreover, since 1, α1, . . . , αr are rationally independent, so for every (s1, . . . , sr+1) ∈ zr+1\{0}, det   1 0 . . . 0 α1 0 ... ... ... ... ... ... ... 0 ... 0 . . . 0 1 αr s1 . . . . . . sr sr+1   = sr+1 − r∑ i=1 siαi 6= 0. by applying proposition 2.1 to k′ := r∑ j=1 zu′j + zu ′ n+1, we get: k′ = e c© agt, upv, 2015 appl. gen. topol. 16, no. 2 133 m. elghaoui and a. ayadi on the other hand, e ⊕ ( n∑ k=r+1 ruk ) = rn, so r∑ j=1 zu′j + zu ′ n+1 + n∑ k=r+1 zuk = r∑ j=1 zu′j + zu ′ n+1 + n∑ k=r+1 zuk using lemma 2.4, n∑ k=r+1 zuk is closed in r n, thus: r∑ j=1 zu′j + zu ′ n+1 + n∑ k=r+1 zuk = e + n∑ k=r+1 zuk finally, we have for every 1 ≤ j ≤ r:    u′n+1, u ′ j ∈ n+1∑ k=1 zuk, un+1, uj ∈ e + 1q n∑ k=r+1 zuk so k′ ⊂ n+1∑ k=1 zuk. then e + n∑ k=r+1 zuk = k′ + n∑ k=r+1 zuk ⊂ n+1∑ k=1 zuk ⊂ e + 1 q n∑ k=r+1 zuk. the proof is completed. � lemma 3.5. let b = (u1, . . . , un) be a basis of rn, n ≥ 2, and v1, . . . , vq ∈ n∑ i=1 zui with 1 ≤ q < n. then the group q∑ i=1 rvi + n∑ i=1 zui is not dense in r n. proof. without loss of generality, we can assume that the vectors v1, . . . , vq are linearly independent. so they can be completed to the basis b′ = (v1, . . . , vq, vq+1, . . . , vn) using the basis b. we may also assume that vi = ui for every i = q + 1, . . . , n. since vi ∈ n∑ j=1 zuj for every i = 1, . . . , n, it follows that, through a change of basis, we have ui ∈ n∑ j=1 qvj. so there exist p ∈ n∗ and ni,j ∈ z such that ui = n∑ j=1 ni,j p vj c© agt, upv, 2015 appl. gen. topol. 16, no. 2 134 rational criterion for testing the density of additive subgroups hence ui ∈ 1 p n∑ j=1 zvj for every i = 1, . . . , n. therefore, q∑ i=1 rvi + n∑ i=1 zui ⊂ q∑ i=1 rvi + 1 p n∑ i=1 zvi = q∑ i=1 rvi + 1 p n∑ i=q+1 zvi by lemma 2.4, the group q∑ i=1 rvi + 1 p n∑ i=q+1 zvi is closed in r n, so the group q∑ i=1 rvi + n∑ i=1 zui is not dense in r n. � proof of theorem 3.3. let h := p∑ i=1 zui. suppose that h = r n, and define hk := n∑ i=1 zui + zuk, for every k = n + 1, . . . , p. as h ⊂ p∑ k=n+1 hk so, we have (3.1) p∑ k=n+1 hk = r n on the other hand, by lemma 3.4, we have ∑ j∈ik ru′k,j + ∑ j /∈ik zuj ⊂ hk ⊂ ∑ j∈ik ru′j,k + 1 qk ∑ j /∈ik zuj where for every j ∈ ik, u′k,j = qkuj + ∑ i/∈ik m (k) i,j ui, with m (k) i,j ∈ z and qk ∈ n∗. it follows that p∑ k=n+1 hk ⊂ p∑ k=n+1   ∑ j∈ik ru′k,j + ∑ j /∈ik 1 qk zuj   ⊂ p∑ k=n+1   ∑ j∈ik ru′k,j   + p∑ k=n+1   ∑ j /∈ik 1 qk zuj   c© agt, upv, 2015 appl. gen. topol. 16, no. 2 135 m. elghaoui and a. ayadi let q = qn+1 . . . qp, the last formula then simplifies to p∑ k=n+1 hk ⊂ p∑ k=n+1   ∑ j∈ik ru′k,j   + 1 q p∑ k=n+1   ∑ j /∈ik zuj   ⊂ p∑ k=n+1   ∑ j∈ik ru′k,j   + 1 q n∑ j=1 zuj and by equation 3.1, we have (3.2) p∑ k=n+1   ∑ j∈ik ru′k,j   + 1 q n∑ j=1 zuj = r n suppose that p∑ k=n+1   ∑ j∈ik ru′k,j   6= rn, then we can extract a maximal set of independent vectors {v1, . . . , vm} with m < n from the set of vectors {u′k,j, j ∈ ik, k = n + 1, . . . , p}. as u′k,j ∈ 1 q n∑ j=1 zuj so vi ∈ 1 q n∑ j=1 zuj for every i = 1, . . . , m. using lemma 3.5, p∑ k=n+1   ∑ j∈ik ru′k,j   + 1 q n∑ j=1 zuj = m∑ j=1 rvj + 1 q n∑ j=1 zuj is not dense in rn, this leads to a contradiction with equation 3.2. so p∑ k=n+1   ∑ j∈ik ru′k,j   = rn. since p∑ k=n+1   ∑ j∈ik ru′k,j   is the span of the columns of the matrix mh so rank(mh) = n. conversely, suppose rank(mh) = n, i.e. p∑ k=n+1   ∑ j∈ik ru′k,j   = rn. by lemma 3.4, we have: for every k = n + 1, . . . , p ∑ j∈ik ru′k,j ⊂ ∑ j∈ik ru′k,j + ∑ j /∈ik zuj ⊂ hk so p∑ k=n+1   ∑ j∈ik ru′k,j   ⊂ p∑ k=n+1 hk c© agt, upv, 2015 appl. gen. topol. 16, no. 2 136 rational criterion for testing the density of additive subgroups as hk ⊂ h, so p∑ k=n+1 hk ⊂ h. thus r n = p∑ k=n+1   ∑ j∈ik ru′k,j   ⊂ h it follows that h = rn. � example 3.6. let h = zu1 + · · ·+ zu7, where u1 = [1, 0, 0]t , u2 = [0, 1, 0]t , u3 = [0, 0, 1] t , u4 = [1, √ 2, 1]t , u5 = [0, 1, √ 3]t , u6 = [ √ 2, √ 3, 1]t , u7 = [1, √ 2, √ 2]t . so n = 3 and p = 7. the sets ik, k = 4, . . . , 7 are: i4 = {2}, i5 = {3}, i6 = {1, 2}, i7 = {2}. we obtain: u′4,2 = u2 u′5,3 = u3 u′6,1 = u1 u′6,2 = u2 u′7,2 = u2 + u3 so that: mh =   0 0 1 0 0 1 0 0 1 1 0 1 0 0 1   since rank(mh) = 3 then h is dense in r 3. now, let us summarize the approach to follow in order to test the density of a given additive group h = p∑ k=1 zuk of r n: (1) if p ≤ n or p∑ k=1 ruk 6= rn, then h is not dense in rn. (2) otherwise, compute the sets ik, k = n + 1, . . . , p: • if ik = ∅ for every k = n + 1, . . . , p, then h is not dense in rn. • if there is an integer k0 ∈ {n + 1, . . . , p} such that ik0 = {1, . . . , n}, then h is dense in rn. • if ik 6= ∅ for at least one k and ik 6= {1, . . . , n} for every k = n + 1, . . . , p, then compute the vectors u′k,j, j ∈ ik, k = n + 1, . . . , p. (3) determine the matrix mh and its rank. then h is dense in rn iff rank(mh) = n. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 137 m. elghaoui and a. ayadi 4. the complex case: density of additive subgroups of cn let u1, . . . , up ∈ cn, p ≥ 2n + 1. suppose that (u1, . . . , u2n) is a basis of cn over r. denote by ũk = [ℜ(uk), ℑ(uk)]t for every k = 1, . . . , p, where ℜ(w) and ℑ(w) are respectively the real and the imaginary part of a vector w ∈ cn. we let h̃ = zũ1 + · · · + ũp, so h̃ ⊂ r2n. theorem 4.1. let h = zu1 + · · · + zup with uk ∈ cn. then h is dense in cn if and only if rank(m h̃ ) = 2n. proof. the proof results directly from theorem 3.3 and the fact that h = cn if and only if h̃ = r2n. � example 4.2. let h = zu1 + · · · + zu8 with u1 = [1, 0]t , u2 = [0, 1]t , u3 = [i, 0] t ,u4 = [0, i] t , u5 = [1 + 2i, 5 √ 3]t , u6 = [i √ 2, √ 3 + i]t , u7 = [2 √ 3 + i, √ 2 + 2i]t , u8 = [1 + 4i √ 2, 5i √ 3]t . then h̃ = zu1 + · · · + zu8, where ũ1 = [1, 0, 0, 0]t , ũ2 = [0, 1, 0, 0]t , ũ3 = [0, 0, 1, 0] t , ũ4 = [0, 0, 0, 1] t , ũ5 = [1, 5 √ 3, 2, 0]t , ũ6 = [0, √ 3, √ 2, 1]t , ũ7 = [2 √ 3, √ 2, 1, 2]t , ũ8 = [1, 0, 4 √ 2, 5 √ 3]t . the sets ik, k = 5, . . . , 8 are: i5 = {2}, i6 = {2, 3}, i7 = {1, 2} and i8 = {3, 4}. we obtain: ũ′5,2 = ũ2 ũ′6,2 = ũ2 ũ′6,3 = ũ3 ũ′7,1 = ũ1 ũ′7,2 = ũ2 ũ′8,3 = ũ3 ũ′8,4 = ũ4 then m h̃ =   0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1   . since rank(mh̃) = 4 then h̃ is dense and so is h. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 138 rational criterion for testing the density of additive subgroups acknowledgements. this work is supported by the research unit: systèmes dynamiques et combinatoire: 99ur15-15 references [1] a. ayadi and h. marzougui, dense orbits for abelian subgroups of gl(n, c), foliations 2005, world scientific, hackensack, nj, (2006), 47–69. [2] a. ayadi, h. marzougui and e. salhi, hypercyclic abelian subgroups of gl(n, r), j. difference equ. appl. 18 (2012), 721–738. [3] a. ayadi, hypercyclic abelian groups of affine maps on cn, canad. math. bull. 56 (2013), 477–490. [4] n. s. feldman, hypercyclic tuples of operators and somewhere dense orbits, j. math. anal. appl. 346 (2008), 82–98. [5] g. h. hardy and e. m. wright, an introduction to the theory of numbers, fourth edition, clarendon press, oxford, 1960. [6] l. kronecker, nherungsweise ganzzahlige auflsung linearer gleichungen, monatsberichte knigl. preu. akad. wiss. berlin, (1884), 1179–1193 and 1271–1299. [7] m. waldschmidt, topologie des points rationnels, cours de troisième cycle, université p. et m. curie (paris vi), (1994/95). c© agt, upv, 2015 appl. gen. topol. 16, no. 2 139 06.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 83 92 jungck theorem for triangular maps andrelated resultsm. grin�c,� �l. snohayabstract. we prove that a continuous triangular map gof the n-dimensional cube in has only �xed points and no otherperiodic points if and only if g has a common �xed point withevery continuous triangular map f that is nontrivially compat-ible with g. this is an analog of jungck theorem for maps ofa real compact interval. we also discuss possible extensions ofjungck theorem, jachymski theorem and some related results tomore general spaces. in particular, the spaces with the �xed pointproperty and the complete invariance property are considered.2000 ams classi�cation: 54h20, 54h25keywords: compatible maps, complete invariance property, jungck theo-rem, jachymski theorem, �xed point, periodic point, triangular map1. introductioncontinuous selfmaps of a real compact interval for which every periodic pointis a �xed point were studied by many authors (see e.g. [4], [17], [3], [9], [7]).recently some of these results were investigated from the point of view oftriangular maps (cf. [6]). the class of triangular maps for which perg = fixgadmits much stranger behavior than the analogous class of one{dimensionalmaps (see e.g. [12], [5]). among others, the pointwise convergence of thesequence of the iterations does not characterize this class of triangular maps.nevertheless, the concept of compatible mappings introduced in [10] allows usto describe not only the class of selfmaps of the interval (cf. [9]) but, as we aregoing to show, also the class of triangular maps.�m. grin�c died in january 1999ymain results of this paper were presented by the �rst author at the european conferenceon iteration theory ecit'98 (muszyna{z lockie, poland, aug. 30 sept. 5, 1998). the �rstauthor was supported by the state committee for scienti�c research (poland) grant no. 2po3a 033 11 and the second author by the slovak grant agency, grant no. 1/4015/97. 84 m. grin�c, �l. snohathroughout the paper all maps are assumed to be continuous even if we donot state it explicitly. if x and y are topological spaces, c(x;y ) denotes theclass of continuous maps from x to y . let i be a real compact interval, say theunit interval [0;1]. let n be a positive integer. in the n-dimensional cube inwe will use the euclidean metric. by a triangular map we mean a continuousmap g : in �! in of the formg(x1;x2; : : : ;xn) = (g1(x1);g2(x1;x2); : : : ;gn(x1;x2; : : : ;xn));shortly g = (g1;g2; : : : ;gn)4. the map g1 is called the basis map of g. theclass of all triangular maps of in will be denoted by c4(in;in). if n = 1 thenc4(i;i) = c(i;i). by i(v1;v2; : : : ;vk) := (v1;v2; : : : ;vk)�in�k we denote the�bre over the point (v1;v2; : : : ;vk) and the map gv1;v2;:::;vk 2 c4(in�k;in�k)de�ned bygv1;v2;:::;vk(xk+1; : : : ;xn) := ( gk+1(v1;v2; : : : ;vk;xk+1);: : : ;gn(v1;v2; : : : ;vk;xk+1; : : : ;xn) )is the �bre map of g working in the �bre i(v1;v2; : : : ;vk).continuous interest in triangular maps is, among others, caused by the factthat they display a kind of a dualism. on one hand, they are close to one{dimensional maps in the sense that some important dynamical features extendto triangular maps. for instance, sharkovsky's theorem holds for them (see[11]). on the other hand, they already display other important propertieswhich are typical for higher dimensional maps and cannot be found in theone{dimensional maps. for instance, there are triangular maps with positivetopological entropy having only periodic points whose periods are (all) powers oftwo (see [12]). for more information on triangular maps see, e.g., [11], [12], [13],[1], [5]. a triangular map de�nes a discrete dynamical system that is, especiallyin ergodic theory, sometimes called a skew product of (one{dimensional or lessdimensional) dynamical systems. as far as the authors know, in �xed pointtheory triangular maps have not been studied yet.let fixg or perg denote the set of all �xed points or periodic points of g,respectively. recall (cf. [8]) that selfmaps f and g of a set x are compatible ifthey commute on the set of their coincidence points, i.e., on the set coin(f;g) :=fx 2 x : f(x) = g(x)g. if f and g are compatible and coin(f;g) is nonempty,we will say that f and g are nontrivially compatible (cf. [9]).note that throughout the paper we will often prefer another but, as onecan easily show, equivalent de�nition of nontrivial compatibility: f and g arenontrivially compatible if and only if coin(f;g) is nonempty and for everyx 2 coin(f;g), the whole trajectories of x under f and g coincide (i.e., fn(x) =gn(x) for every n 2 n).g. jungck proved the following (cf. [9, theorem 3.6]):theorem 1.1 (jungck theorem). a map g 2 c(i;i) has a common �xed pointwith every map f 2 c(i;i) which is nontrivially compatible with g if and onlyif perg = fixg. jungck theorem for triangular maps and related results 85the main aim of this paper is to show that the analog of this theorem appliesto triangular maps (see theorem 2.7).in section 3 we discuss possible extensions of jungck theorem and some re-lated results for general continuous maps to more general spaces (see theorems3.2, 3.3, and 3.4).before going to our results notice that the fact that jungck theorem holdsfor triangular maps is a new illustration of their dualistic character when theyare compared with selfmaps of i. in fact, in some other aspects of the �xedpoint theory they di�er from interval maps: for instance, in [14] one can �ndan example of a triangular map g in i2 and a compact subinterval j � i suchthat g(j2) � j2 but g has no �xed point in j2.2. jungck theorem for triangular mapslemma 2.1. let g 2 c(i;i). if perg = fixg and x0 2 fixg, then there is amap f 2 c(i;i) and �;� 2 i such that � � x0 � � andcoin(f;g) = f�;x0;�g � g�1(x0):proof. we can additionally suppose that x0 2 inti (if x0 = 0 or x0 = 1 thenthe proof is similar). first we de�ne the function f on the right side of x0.take � := supfx 2 i : g(x) = x0g:then x0 � �. put f(x) = x0 for every x 2 [�;1]. let fj[x0;�] be an arbitrarycontinuous function such that f(x0) = f(�) = x0 and 1 � f(x) > g(x) forall x 2 (x0;�). to see that such a map exists it is enough to realize thatg(x) < 1 for every x 2 [x0;�]. in fact, if g(z) = 1 for some z 2 [x0;�] theng([x0;z]) \g([z;�]) � [x0;z] [ [z;�], i.e., g has a 2-horseshoe. this implies thatg has a periodic point of period greater than 1 (g has even positive topologicalentropy [2, proposition 4.3.2]). this contradicts the assumption on g.similarly we proceed on the left side of x0 by putting � := inffx 2 i : g(x) =x0g, f(x) = x0 for every x 2 [0;�] and taking into account that g(x) > 0 forall x 2 [�;x0]. the obtained �;� and f ful�ll all desired conditions. �for n � m let �m : in �! im be the projection of the space in onto thespace im de�ned by �m(x1; : : : ;xn) = (x1; : : : ;xm). in the notation �m wesuppressed n, since it will always be clear what is the dimension of the domainof �m.the following extension lemma is a generalization of [13, lemma 1]. we willuse in it the notation c4(k;in) for the set of triangular maps from k into inwhich are de�ned analogously as triangular maps from in into in.lemma 2.2. let k � in be a compact set, � = (�1;�2; : : : ;�n)4 2 c4(k;in).then there is a map f = (f1;f2; : : : ;fn)4 2 c4(in;in) such that f jk = �.moreover, we can prescribe any of the maps fm, m = 1;2; : : : ;n, requiringonly that fm 2 c(im;i) be an extension of �m 2 c(�m(k);i).proof. since � 2 c(k;in), for every i the map �i is continuous if we considerit as a map k �! i. note that there is no sense in extending each �i to a 86 m. grin�c, �l. snohamap fi de�ned on the whole in and putting f = (f1;f2; : : : ;fn) because thenf 2 c(in;in), in general, would not be triangular. but the map �i dependsonly on the �rst i variables and so we can consider it also as a map �i(k) �! i.we are going to show that, due to the compactness of k, �i is still continuous,i.e., �i 2 c(�i(k);i). for i = n this is trivial since �n 2 c(k;i). fori = 1;2; : : : ;n � 1 this follows from the repeated use of the followingclaim 2.3. if m � ik is a compact set and = ( 1; 2; : : : ; k)4 2 c4(m;ik)then k 2 c(m;i), �k�1(m) � ik�1 is a compact set and the map � de�nedas � = ( 1; 2; : : : ; k�1)4 belongs to c4(�k�1(m), ik�1).proof. (of the claim). the compactness of �k�1(m) is obvious. we triviallyhave k 2 c(m;i) and so we need only prove that the map � = ( 1; 2; : : : ; k�1)4 : �k�1(m) �! ik�1is continuous. assume, on the contrary, that � is discontinuous at a pointz 2 �k�1(m). then there is a sequence of points zi 2 �k�1(m) for whichlimi!1 zi = z and the sequence ( �(zi)) does not tend to �(z). since ik�1is compact, there is a convergent subsequence of ( �(zi)). without loss ofgenerality we may assume that limi!1 �(zi) = a 6= �(z). take points vi 2 isuch that (zi;vi) 2 m. there is a converging subsequence of (vi). we mayassume that limi!1 vi = v. then (zi;vi) �! (z;v). since m is closed, (z;v) 2m. the point (z;v) belongs to the �bre i( �(z)) and the sequence ( �(zi))does not converge to �(z). so ( (zi;vi)) does not converge to (z;v), and wehave a contradiction with the continuity of . thus claim 2.3 is proved.so we have proved that for m = 1;2; : : : ;n, the map �m : �m(k) �! iis continuous. by tietze extension theorem the functions �m 2 c(�m(k);i),1 � m � n have continuous extensions fm 2 c(im;i), 1 � m � n, respectively.now it su�ces to put f = (f1;f2; : : : ;fn)4 with arbitrary such extensions. �lemma 2.4. let g = (g1;g2; : : : ;gn)4 2 c4(in;in), n � 2 have a common�xed point with every triangular map which is nontrivially compatible with g.then(i) perg1 = fixg1, and(ii) for every a1 2 fixg1, ga1 has a common �xed point with every trian-gular map which is nontrivially compatible with ga1.proof. (i) to shorten the notation we will write y = (x2; : : : ;xn). fix a mapf 2 c(i;i) which is nontrivially compatible with g1. puttingf(x1;y) := (f(x1);g2(x1;x2); : : : ;gn(x1;y)) for every (x;y) 2 in;we see that coin(f;g) = coin(f;g1) � in�1is nonempty and for (x1;y) 2 coin(f;g)f(g(x1;y)) = (f(g1(x1));g2(g1(x1);g2(x1;x2)); : : : ;gn(g1(x1); : : : ))= (g1(f(x1));g2(f(x1);g2(x1;x2)); : : : ;gn(f(x1); : : : ))= g(f(x1;y)): jungck theorem for triangular maps and related results 87thus, by the assumption of the lemma, fixf\fixg 6= ?. let (fx1; : : : , fxn) be acommon �xed point of f and g. then f(fx1) = g1(fx1) = fx1, so fixf \fixg1 6=? and by jungck theorem we get perg1 = fixg1.(ii) let a1 2 fixg1 and let � 2 c4(in�1;in�1) be nontrivially compatiblewith ga1. we will prove that � and ga1 have a common �xed point. takey0 2 coin(ga1;�). by lemma 2.1 there exists a map f1 2 c(i;i) and �;� 2 isuch that � � a1 � � and coin(f1;g1) = f�;a1;�g � g�11 (a1). we are going tode�ne a triangular map f 2 c4(in;in). we start with the triangular selfmapof the compact set f�;a1;�g�in�1 which sends (a1;y) to (a1;�(y)) and (x1;y)with x1 2 f�;�g n fa1g to (a1;y0). using lemma 2.2 we can extend this mapto a triangular map f whose basis map is the above mentioned map f1. it isobvious that f is nontrivially compatible with g. therefore fixf \fixg 6= ?.from the de�nition of f we get that fix � \ fixga1 6= ?. �in the sequel we will use the following simple properties of triangular maps:if f 2 c4(in;in), then�1(fixf) = fixf1 and �1(perf) = perf1.lemma 2.5. if a map g = (g1;g2; : : : ;gn)4 2 c4(in;in), n � 1, has a com-mon �xed point with every triangular map f which is nontrivially compatiblewith g, then perg = fixg.proof. for n = 1 this holds by jungck theorem. so let n � 2. take (a1;a2; : : : ;an) 2 perg. by lemma 2.4 we get a1 2 perg1 = fixg1 which means thatg1(a1) = a1. then (a2; : : : ;an) 2 perga1, wherega1 = (g2(a1; �);g3(a1; �; �); : : : ;gn(a1; �; �; : : : ; �))4:moreover, ga1 has a common �xed point with every triangular map which isnontrivially compatible with ga1.now applying lemma 2.4 to the map ga1 we obtain that a2 2 fix g2(a1; �), sog2(a1;a2) = a2. it means that (a3; : : : ;an) 2 perga1a2. proceeding in this waywe see that also an�1 2 fixgn�1(a1;a2; : : : ;an�2; �) and an 2 perga1a2:::an�1.moreover, ga1a2:::an�1 is an interval selfmap that has a common �xed point withevery continuous map which is nontrivially compatible with ga1a2:::an�1. byjungck theorem an belongs to fixga1a2:::an�1, whence gn(a1;a2; : : : ;an�1;an) =an: thus we have proved that g(a1;a2; : : : ;an) = (a1;a2; : : : ;an). thereforeperg = fixg. �before proving the converse statement �x some notation. when f 2 c4 (in,in) and (x1; : : : ;xn) 2 in, the symbol !f(x1; : : : ;xn) will denote the !-limit setof the point (x1; : : : ;xn) under f , i.e., the set of all limit points of the trajectory(fk(x1; : : : ;xn))1k=1.further recall that if x is a hausdor� topological space and f;g 2 c(x;x)are nontrivially compatible then coin(f;g) is closed and, as one can easily show,for every x 2 coin(f;g) we have !f(x) = !g(x) � coin(f;g).lemma 2.6. assume that g 2 c4(in;in), n � 1. if perg = fixg, theng has a common �xed point with every triangular map f which is nontriviallycompatible with g. 88 m. grin�c, �l. snohaproof. for n = 1 this holds by jungck theorem. assume that the lemma holdsfor some n � 1 and take any g = (g1;g2; : : : ;gn+1)4 2 c4(in+1;in+1) withperg = fixg and any f = (f1;f2; : : : ;fn+1)4 2 c4(in+1;in+1) which isnontrivially compatible with g. to �nish the proof we need to show that gand f have a common �xed point. to this end take a point (a1;a2; : : : ;an+1) 2coin(f;g). since perg = fixg we have also perg1 = fixg1. therefore thesequence (gk1(a1))1k=1 tends to some point v 2 fixg1 (see [4], [17] or [3]). hence!g(a1;a2; : : : ;an+1) = !f(a1;a2; : : : ;an+1) � i(v):the set on the left side of this inclusion is nonempty and is a subset of coin(f;g).so, coin(f;g) contains a point of the form (v;b2; : : : ;bn+1). since v 2 fixg1,this implies that also v 2 fixf1. now consider the maps gv and fv fromc4(in;in). these maps are compatible because f and g are compatible andthe �bre i(v) is mapped into itself both by f and g. moreover, (b2; : : : ;bn+1) 2coin(fv;gv) and so fv and gv are nontrivially compatible. if we �nally takeinto account that perg = fixg and so pergv = fixgv, we can apply theinduction hypothesis to the map gv to get that there is a point(c2; : : : ;cn+1) 2 fixgv \ fixfv:then (v;c2; : : : ;cn+1) 2 fixg \ fixfwhich ends the proof. �from lemma 2.5 and lemma 2.6 we immediately get the following general-ization of jungck theorem:theorem 2.7. [jungck theorem for triangular maps.] for each n � 1, a mapg 2 c4(in;in) has a common �xed point with every map f 2 c4(in;in)which is nontrivially compatible with g if and only if perg = fixg.this result allows us also to extend the list of conditions which characterizethe triangular maps with �xed points as unique periodic points (cf. [6, corollary3.1]):corollary 2.8. let g 2 c4(in;in). then the following conditions are equiv-alent:(i) perg = fixg,(ii) c\fixg 6= ? for any nonempty closed set c � in such that g(c) � c,(iii) g has a common �xed point with every map f 2 c4(in;in) thatcommutes with g on fixf,(iv) g has a common �xed point with every map f 2 c(in;in) that com-mutes with g on fixf,(v) g has a common �xed point with every map f 2 c4(in;in) which isnontrivially compatible with g. jungck theorem for triangular maps and related results 893. on a generalization of results related to jachymski theoremand jungck theoremlooking at conditions in corollary 2.8 it seems to be natural to ask whethersome of the implications do not hold for continuous (not necessarily triangular)selfmaps of more general spaces than the n-dimensional cube. in this sectionwe give answers to some questions of this type.first recall that j. jachymski proved the following (cf. [7, proposition 1]):theorem 3.1. [jachymski theorem.] let a be a nonempty compact and convexsubset of a normed linear space and let g be a continuous selfmap of a. thenthe following conditions are equivalent:(i) c \fixg 6= ? for any nonempty closed set c � a such that g(c) � c,(ii) g has a common �xed point with every map f 2 c(a;a) that commuteswith g on fixf.we are going to give a more general formulation of this result. to thisend recall after l. e. ward (cf. [18]) the following de�nition. a subset f ofa topological space x is a �xed point set of x if there exists a continuousselfmap of x whose set of �xed points is exactly f . the space x has thecomplete invariance property (cip) if each of its nonempty closed subsets is a�xed point set.l. e. ward proved that a convex subset of a normed linear space has the cip(cf. [18, corollary 1.1]; for other examples of topological spaces with the cipsee [15] or [16]). further, by schauder theorem the space a from jachymskitheorem has the �xed point property (fpp). so, the following is a generalizationof jachymski theorem:theorem 3.2. [generalization of jachymski theorem.] let x be a hausdor�topological space with the fpp and the cip and let g be a continuous selfmapof x. then the following conditions are equivalent:(i) c\fixg 6= ? for any nonempty closed set c � x such that g(c) � c,(ii) g has a common �xed point with every map f 2 c(x;x) that commuteswith g on fixf.moreover, (i) =) (ii) does not require the cip and (ii) =) (i) does not requirethat x be hausdor� and have the �xed point property.proof. (i) =) (ii). let a continuous map f : x �! x commute with g onfixf. then g(fixf) � fixf. further, the set fixf is nonempty since x hasthe �xed point property and closed since x is hausdor�. therefore by (i),fixf \ fixg 6= ?.(ii) =) (i). let c be a nonempty closed subset of x such that g(c) � c.by the cip of x, there exists a continuous map f : x �! x with fixf = c.for x 2 c, g(f(x)) = g(x) since f(x) = x, and f(g(x)) = g(x) since g(c) � cand fixf = c. thus f and g commute on fixf. by (ii), fixf \ fixg 6= ?,i.e., c \ fixg 6= ?. �theorem 3.3. let x be a hausdor� topological space with the cip and let gbe a continuous selfmap of x. then the following conditions are equivalent: 90 m. grin�c, �l. snoha(i) c\fixg 6= ? for any nonempty closed set c � x such that g(c) � c,(ii) g has a common �xed point with every map f 2 c(x;x) which isnontrivially compatible with g.moreover, (i) =) (ii) does not require the cip and (ii) =) (i) does not requirethat x be hausdor�.proof. (i) =) (ii). let f 2 c(x;x) be nontrivially compatible with g.the set coin(f;g) is nonempty and, since x is hausdor�, closed. moreover,g(coin(f;g)) � coin(f;g). indeed, if x 2 coin(f;g) then f(x) = g(x) and, bywhat was said in introduction, f2(x) = g2(x). then f(g(x)) = f2(x) = g2(x),so g(x) 2 coin(f;g). by (i), coin(f;g) \ fixg 6= ?. let x0 2 fixg be acoincidence point of f and g. then x0 = g(x0) = f(x0), which means thatfixf \ fixg 6= ?.(ii) =) (i). let c � x be a nonempty closed set such that g(c) � c.by the assumption, there is a map h 2 c(x;x) with fixh = c. put f :=h � g. then c � coin(f;g). take any x 2 coin(f;g). then g(x) 2 c (sinceotherwise we would have f(x) = h(g(x)) 6= g(x)) and so gn(x) 2 c for alln 2 n. from this and the facts that c � coin(f;g) and f(x) = g(x) we getfn(x) = gn(x) for all n 2 n. hence f and g are nontrivially compatible. by(ii), fixf \ fixg 6= ?. let x0 be a common �xed point of f and g. thenx0 = f(x0) = h(g(x0)) = h(x0), so x0 2 c. therefore c \ fixg 6= ?. �if x is a hausdor� space then any periodic orbit of a continuous selfmap ofx, being �nite, is a closed set. thus the condition (i) from theorem 3.3 impliesperg = fixg (the converse is not true: for instance, let x be the unit disc,c its circumference and g an irrational rotation). hence in a hausdor� spacewith the cip the condition (ii) from theorem 3.3 implies perg = fixg. but itturns out that this is true even under weaker assumptions on the space than tobe hausdor� and to have the cip.theorem 3.4. [generalization of a part of jungck theorem).] let x be atopological space with the property that for every nonempty �nite set a � xthere exists a map h 2 c(x;x) such that fixh = a. if a map g 2 c(x;x)has a common �xed point with every map f 2 c(x;x) which is nontriviallycompatible with g, then perg = fixg.proof. suppose on the contrary that g has a periodic orbit a of period greaterthan one. take h 2 c(x;x) with fixh = a and de�ne f := h � g.clearly, f 2 c(x;x) and a � coin(f;g). take any x 2 coin(f;g). theng(x) 2 a since otherwise we would have f(x) = h(g(x)) 6= g(x). hence fn(x) =gn(x) for all n � 1. thus we have proved that f is nontrivially compatible withg. but g and f have no common �xed point since if g(x) = x then x =2 a andso f(x) = h(g(x)) = h(x) 6= x. this contradiction �nishes the proof. �the example below shows that the converse implication is not true.example 3.5. let x be the unit disc and g an irrational rotation of x. thenx has the property from the previous theorem and perg = fixg. nevertheless,there is a map f 2 c(x;x) which is nontrivially compatible with g but has jungck theorem for triangular maps and related results 91no common �xed point with g. to see this, take any h 2 c(x;x) whose �xedpoint set is the circumference of x and put f := h � g. �references[1] alsed�a ll., kolyada s. f. and snoha �l., on topological entropy of triangular maps of thesquare, bull. austral. math. soc. 48 (1993), 55{67.[2] alsed�a ll., llibre j. and misiurewicz m., combinatorial dynamics and entropy in di-mension one, world scienti�c publ., singapore, 1993.[3] chu s. c. and moyer r. d., on continuous functions, commuting functions, and �xedpoints, fund. math. 59 (1966), 90{95.[4] coppel w. a., the solution of equations by iteration, proc. cambridge philos. soc. 51(1955), 41{43.[5] forti g. l., paganoni l. and sm��tal j., strange triangular maps of the square, bull.austral. math. soc. 51 (1995), 395{415.[6] grin�c m., on common �xed points of commuting triangular maps, bull. polish. acad.sci. math. 47 (1999), 61{67.[7] jachymski j. r., equivalent conditions involving common �xed points for maps on theunit interval, proc. amer. math. soc. 124 (1996), 3229{3233.[8] jungck g., common �xed points for commuting and compatible maps on compacta, proc.amer. math. soc. 103 (1988), 977{983.[9] jungck g., common �xed points for compatible maps on the unit interval, proc. amer.math. soc. 115 (1992), 495{499.[10] jungck g., compatible mappings and common �xed points, internat. j. math. math. sci.9 (1986), 771{779.[11] kloeden p. e., on sharkovsky's cycle coexistence ordering, bull. austral. math. soc. 20(1979), 171{177.[12] kolyada s. f., on dynamics of triangular maps of the square, ergodic theory dynam.systems 12 (1992), 749{768.[13] kolyada s. f. and snoha �l., on !-limit sets of triangular maps, real anal. exchange18(1) (1992/93), 115{130.[14] kolyada s. and snoha �l., topological entropy of nonautonomous dynamical systems,random comput. dynam. 4 (1996), 205{233.[15] martin j. r. and weiss w., fixed point sets of metric and nonmetric spaces, trans.amer. math. soc. 284 (1984), 337-353.[16] schirmer h., fixed point sets of continuous sefmaps, fixed point theory proc. (sher-brooke, 1980), lecture notes in math., vol. 886, springer-verlag, berlin, 1981, 417-428.[17] sharkovsky a. n., on cycles and the structure of a continuous mapping, ukrain. mat.zh. 17(3) (1965), 104-111 (russian).[18] ward l. e., jr., fixed point sets, paci�c j. math. 47 (1973), 553-565.received march 2000 m. grin�cinstitute of mathematicssilesian universitybankowa 14, pl-40-007 katowice 92 m. grin�c, �l. snohapoland(m. grin�c died in january 1999)�l. snohadepartment of mathematicsfaculty of natural sciencesmatej bel universitytajovsk�eho 40, sk-974 01 bansk�a bystricaslovakiae-mail address: snoha@fpv.umb.sk @ appl. gen. topol. 16, no. 2(2015), 217-224doi:10.4995/agt.2015.3584 c© agt, upv, 2015 the dynamical look at the subsets of a group igor protasov a and serhii slobodianiuk b a department of cybernetics, kyiv university, volodymyrska 64, kyiv 01033, ukraine (i.v.protasov@gmail.com) b department of mathematics and mechanics, kyiv university, volodymyrska 64, kyiv 01033, ukraine (slobodianiuks@gmail.com) abstract we consider the action of a group g on the family p(g) of all subsets of g by the right shifts a 7→ ag and give the dynamical characterizations of thin, n-thin, sparse and scattered subsets. for n ∈ n, a subset a of a group g is called n-thin if g0a ∩ · · · ∩ gna is finite for all distinct g0, . . . , gn ∈ g. each n-thin subset of a group of cardinality ℵ0 can be partitioned into n 1-thin subsets but there is a 2-thin subset in some abelian group of cardinality ℵ2 which cannot be partitioned into two 1-thin subsets. we eliminate the gap between ℵ0 and ℵ2 proving that each n-thin subset of an abelian group of cardinality ℵ1 can be partitioned into n 1-thin subsets. 2010 msc: 54h20; 05c15. keywords: thin; sparse and scatterad subsets of a group; recurrent point; chromatic number of a graph. 1. introduction let g be a group with the identity e, p(g) denotes the family of all subsets of g, [g]<ω = {f ⊆ g : f is finite}, [g]n = {f ⊆ g : |f | = n}, n ∈ n. we say that a subset a of g is • thin if a ∩ ga is finite for every g ∈ g \ {e}; • n-thin if g0a ∩ · · · ∩ gna is finite for any distinct g0, . . . , gn ∈ g; • sparse if, for every infinite subset x ⊆ g, there exists a finite subset f ⊂ s such that ⋂ g∈f ga is finite; received 18 february 2015 – accepted 21 june 2015 http://dx.doi.org/10.4995/agt.2015.3584 i. protasov and s. slobodianiuk • scattered if, for any subset b ⊆ a, there exists f ∈ [g]<ω such that, for each h ∈ [g]<ω, f ∩ h = ∅, we can find b ∈ b such that hb ∩ b = ∅; • thick if, for any f ∈ [g]<ω, there exists g ∈ g such that fg ⊆ a. in [3] c. chou used thin subsets to prove that there are 22 |g| distinct left invariant banach measures on each infinite amenable groups. clearly, thin subsets are precisely 1-thin subsets, n-thin subsets appeared in [10] in attempt to characterize the ideal in the boolean algebra p(g) generated by thin subsets. sparse subsets appeared in [4] for characterization of strongly prime ultrafilters in the semigroup g∗ of free ultrafilters on g and studied in [9]. scattered subsets were introduced in [1] as asymptotic counterparts of scattered topological spaces. unexplicitely, thick subsets were used in [11] to partition of an infinite totally bounded group g into |g| dense subsets. as to our knowledge, the name ”thick subset” appeared in [2]. for every infinite group g, we have thin ⇒ 2-thin ⇒ · · · ⇒ n-thin ⇒ · · · ⇒ sparse ⇒ scattered and none of these arrows could be reversed. for ”scattered ✟✟⇒ sparse” see remark 3.6. more on these subsets and their applications one can find in the surveys [12], [16]. in this paper, we identify p(g) with {0, 1}g, endow p(g) with the product topology and consider the action of g on p(g) by the right shifts a 7→ ag. after short preliminary section 2, we give the dynamical characterizations to all above defined subsets in section 3. it should be mentioned that the dynamical approach is especially effective for finite partitions of groups [5]. by [10], every n-thin subset of a countable group g can be partitioned into n thin subsets (some cells of the partitions could be empty). answering the question from [10], g. bergman (see [15]) constructed an abelian group g of cardinality ℵ2 and a 2-thin subset of g which cannot be partitioned into two thin subsets. on the other hand [15], every n-thin subset of an abelian group of cardinality ℵm can be partitioned into n m+1 thin subsets but there is a 2-thin subset in some group of cardinality ℵω which cannot be finitely partitioned. in section 4 we eliminate the gap between ℵ0 and ℵ2 proving that every n-thin subsets of an abelian group of cardinality ℵ1 is a union of n thin subsets. 2. some dynamics let g be a group. a topological space x is called a g-space if there is the action x × g → x : (x, g) 7→ xg such that, for each g ∈ g, the mapping x → x : x 7→ xg is continuous. given any x ∈ x and u ⊆ x, we set [u]x = {g ∈ g : xg ∈ u} c© agt, upv, 2015 appl. gen. topol. 16, no. 2 218 the dynamical look at the subsets of a group and denote o(x) = {xg : g ∈ g}, t (x) = clo(x), w(x) = {y ∈ t (x) : [u]x is infinite for each neighbourhood u of y}. we recall also that x ∈ x is a recurrent point if x ∈ w(x). now we consider a group g, identify p(g) with the space {0, 1}g and endow p(g) with the product topology. thus, the subsets u(f, h) = {a ⊆ g : f ⊆ a, h ∩ a = ∅}, where f ∈ [g]<ω, h ∈ [g]<ω, form the base for the open sets on p(g). in what follows, we consider p(g) as a g-space with the action defined by a 7→ ag, ag = {ag : a ∈ a}. we say that a subset a of g is recurrent if a is a recurrent point in (p(g), g). 3. characterizations all groups in this sections are supposed to be infinite. theorem 3.1. for a subset a of a group g, the following statements hold (i) a is finite if and only if w(a) = ∅; (ii) a is thick if and only if g ∈ w(a). proof. (i) it suffices to note that a is finite if and only if, for every x ∈ g, the set {g ∈ g : x ∈ ag} is finite. (ii) suppose that g ∈ w(a) and take an arbitrary finite subset f of g. since u(f, ∅) is a neighborhood of g in p(g), there exists g ∈ g such that ag ∈ u(f, ∅), so fg−1 ⊆ a and a is thick. assume that a is thick and take an arbitrary finite subset f of g. then we choose an injective sequence (gn)n∈ω in g such that fgi ∩ fgj = ∅ for all distinct i, j ∈ ω. for each in n ∈ ω, we take hn ∈ g such that (fg0 ∪ · · · ∪ fgn)hn ⊆ a, so f ⊆ ah −1 n g −1 i , i ∈ {0, . . . , n}. it follows that u(f, ∅) contains infinitely many points of the orbit o(a), so g ∈ w(a). � theorem 3.2. for a subset a of a group g, the following statements hold (i) a is n-thin if and only if |y | ≤ n for every y ∈ w(a); (ii) a is sparse if and only if each subset y ∈ w(a) is finite; (iii) a is scattered if and only if, for every subset b ⊆ a there exists y ∈ [g]<ω in the closure of {bb−1 : b ∈ b}. proof. suppose that a is n-thin but |y | > n for some y ∈ w(a). let {y0, . . . , yn} be distinct elements from y . since y ∈ w(a) and the set u({y0, . . . , yn}, ∅) is a neighborhood of y , the set w = {g ∈ g : {y0, . . . , yn} ⊆ ag} is infinite. we note that w = {g ∈ g : {y0g −1, . . . , yng −1} ⊆ a} = {g ∈ g : g−1 ∈ y−10 a ∩ · · · ∩ y −1 n a}. hence, a is not n-thin. suppose that a is not n-thin. we take g0, . . . , gn ∈ g such that the subset b = g0a ∩ · · · ∩ gna is infinite. if b ∈ b then {g −1 0 b, . . . , g −1 n b} ⊆ a so g−10 , . . . , g −1 n ⊆ ab −1. we take an arbitrary limit point l of the set {ab−1 : b ∈ b}. then l ∈ w(a) but {g−10 , . . . , g −1 n } ⊆ l, so |l| > n. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 219 i. protasov and s. slobodianiuk (ii) suppose that a is sparse but some subset y ∈ w(a) is infinite. we take a countable subset {yn : n ∈ ω} of y and put un = u({y0, . . . , yn}, ∅). then we choose an injective sequence (gn)n∈ω in g such that gn ∈ [un]a for each n ∈ ω. we note that {y0, . . . , yn} ⊆ agn so g −1 n ∈ y0a ∩ · · · ∩ yna. we put x = {y−1n : n ∈ ω}. then ⋂ g∈f ga is infinite for each finite subset f of x. hence, a is not sparse. assume that each subset y ∈ w(a) is finite but a is not sparse. then there exists an injective sequence (gn)n∈ω in g such that g0a ∩ · · · ∩ gna is infinite for each n ∈ ω. we choose an injective sequence (yn)n∈ω in g such that yn ∈ g0a ∩ · · · ∩ gna, so {g −1 0 , . . . , g −1 n } ⊆ ay −1 n for each n ∈ ω. let l be an arbitrary limit point of {ay−1n : n ∈ ω}. then {g −1 n : n ∈ ω} ⊆ l and l ∈ w(a). (iii) suppose that a is scattered and b is a subset of g. we choose corresponding f ∈ [g]<ωand take an arbitrary h ∈ [g] k and xnym ∈ y . let k be a finite subset of g such that e /∈ k. we take n ∈ ω such that k ∩ h ⊆ hn. and xnym ∈ y for some m ∈ ω. by (2), kxnym ∩ a = ∅, in particular, kxnym ∩ y = ∅. case 2. there exists k ∈ ω such that y ⊆ {xnym : n ≤ k, m ≥ n}. we take an arbitrary k ∈ [g]<ω such that k ∩ hk = ∅. then we choose n, m ∈ ω, such that k ∩ h ⊆ hn, n > k and xnym ∈ y . by (1), kxnym ∩ {xnym : n ≤ k, m ≥ n} = ∅, in particular, kxnym ∩ y = ∅. 4. partitions into thin subsets we fix a subset t of a group g and, for every g ∈ g \ {e}, consider the orbital graph γg with the set of vertices t and the set of edges eg = {{s, t} ∈ [t ] 2 : st−1 ∈ {g, g−1}}. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 221 i. protasov and s. slobodianiuk we say that a graph (t, e) is an orbital companion of t if, for each g ∈ g\{e}, the set e contains all but finitely many edges from eg so e \ eg if finite. for n ∈ n, a graph γ is called n-discrete if each connected component of γ has no more than n vertices. given a graph (v, e), we take any subsets e′, e′′, from e such that e = e′ ∪ e′′ and say that (v, e) is a union of the graphs (v, e′) and (v, e′′). in this section we use the following equivalent definition of an n-thin subset t of a group (see theorem 3.3): for every f ∈ [g]<ω, there exists h ∈ [g]<ω such that, for every g ∈ g \ h, we have |fg ∩ t | ≤ n. lemma 4.1. every n-thin subset t of a countable group g has an n discrete orbital companion γ. proof. we write g as the union of an increasing chain {fi : i < ω} of finite subsets such that e ∈ f0, fi ⊂ fi+1 and fi = f −1 i . then we choose a chain {vi : i < ω} of finite subsets of g such that, for any i < ω and g ∈ g \ vi, we have vi ⊂ vi+1 and |f m i g ∩ t | ≤ n. we define the set e of edges of γ by the rule: {s, t} ∈ e ⇔ ∃k : s /∈ vk and st −1 ∈ fk. given any g ∈ g \ {e}, we pick k < ω such that g ∈ fk. if t ∈ t and gt ∈ t then either {t, gt} ⊂ vk or (t, gt) ∈ e. since vk is finite, we conclude that γ is an orbital companion of t . suppose that γ is not n-discrete and choose a subset s ∈ [t ]n+1 such that the induced graph γs is connected. we find the minimal number k such that, for any {s1, s2} ∈ [s] 2 ∩ e, s1s −1 2 ∈ fk and so there are {s, s ′} ∈ [s]2 ∩ e such that s′s−1 ∈ fk \ fk−1. it follows that s ∈ g \ vk and s ⊂ f n k s but |s| > n and we get a contradiction with the choice of vk. � lemma 4.2. let g be an abelian group of cardinality ℵ1 and let t be an n-thin subset of g. then some orbital companion of t is a union of two n-discrete graphs. proof. applying lemma 2 from [15], we represent g as the union of increasing chain {gi : i < ω1} of countable subgroups such that (∗) for any i < ω1 and g ∈ gi+1 \ gi, |gig ∩ t | ≤ n. for every i < ω1, we consider the n-thin subset ti = t ∩ (gi+1 \ gi) of gi+1 and choose n-discrete companion (ti, ei) of ti given by lemma 4.1. we denote by e′i the union of ei and the set of all {t, t ′} ∈ [ti] 2 such that t−1t′ ∈ gi. since g is abelian, we have gig = ggi and by (∗), (ti, e ′ i) is a union of two n-discrete graphs (ti, ei) and (ti, e ′ i \ ei). we put e = ⋃ i<ω1 e′i and note that (t, e) is a union of n-discrete graphs because the subsets {ti : i < ω1} are pairwise disjoint. now it remains to prove that (t, e) is an orbital companion of t . given g ∈ g \ {e}, we choose i < ω1 such that g ∈ gi+1 \ gi. since g is abelian, t ∩t g−1 = t ∩g−1t and then the set {t ∈ t ∩g−1t : (t, gt) /∈ e} is the union c© agt, upv, 2015 appl. gen. topol. 16, no. 2 222 the dynamical look at the subsets of a group of the following sets ⋃ i 1. the confusion in the proof of [19, theorem 3.11] is applying [19, theorem 2.11] to the inequality d(xn, yn) − a 2 d(x, y) � ad(xn, x) + a 2 d(y, yn) where d(xn, yn) − a 2d(x, y) is not positive in general. in fact, the property of lim n→∞ d(xn, yn) is stated in theorem 2.10.(6). finally we show that fixed point results for cyclic maps in [19] may be deduced from certain fixed point results for cyclic maps in the the setting of b-metric spaces. we need not use even the assumption b ∈ a′+ in proofs of corollary 2.12 and corollary 2.13 and it may be replaced by b ∈ a. corollary 2.11 ([19], theorem 4.1). let (x, a, d) be a complete c∗-algebravalued b-metric space with the coefficient a, a and b be non-empty closed subsets of x and t : a ∪ b −→ a ∪ b be a cyclic map, that is t a ⊂ b and t b ⊂ a, such that d(t x, t y) � b∗d(x, y)b c© agt, upv, 2017 appl. gen. topol. 18, no. 2 250 an equivalence of results in c ∗ -algebra valued b-metric and b-metric spaces for all x ∈ a, y ∈ b and some b ∈ a with ‖b‖ < 1 ‖a‖ . then t has a unique fixed point in a ∩ b. proof. let ρ be the b-metric defined as in corollary 2.2. then (x, ρ) is a complete b-metric space. by (2.11) we have 0 � d(t x, t y) � b∗d(x, y)b for all x ∈ a and y ∈ b. then by lemma 1.2 we get, for all x ∈ a and y ∈ b, ρ(t x, t y) = ‖d(t x, t y)‖ ≤ ‖b∗d(x, y)b‖ ≤ ‖b∗‖‖d(x, y)‖‖b‖ = ‖b‖2ρ(x, y). note that ‖b‖2 < 1 since ‖b‖ < 1. by using [23, corallary 3.4] with ϕ(t) = ‖b‖2t for all t ≥ 0 we get that t has unique fixed point in a ∩ b. � corollary 2.12 ([19], theorem 4.5). let (x, a, d) be a complete c∗-algebravalued metric space, a and b be non-empty closed subsets of x and t : a ∪ b −→ a ∪ b be a cyclic map such that d(t x, t y) � b[d(t x, x) + d(t y, y)] for all x, y ∈ x and some b ∈ a′+ with ‖b‖ < 1 2‖a‖ . then t has a unique fixed point in a ∩ b. proof. similar to the argument in the proof of corollary 2.11 and using [23, corallary 3.8] with ϕ(t) = 2‖b‖t for all t ≥ 0. � corollary 2.13 ([19], theorem 4.7). let (x, a, d) be a complete c∗-algebravalued metric space, a and b be non-empty closed subsets of x and t : a ∪ b −→ a ∪ b be a cyclic map such that d(t x, t y) � b[d(t x, y) + d(t y, x)] for all x, y ∈ x and some b ∈ a′+ with ‖b‖ < 1 2‖a‖2 . then t has a unique fixed point in a ∩ b. proof. similar to the argument in the proof of corollary 2.12 and using [23, corallary 3.6] with ϕ(t) = 2‖b‖t for all t ≥ 0. � though many results in c∗-algebra valued metric spaces and c∗-algebra valued b-metric spaces are consequences of certain results in metric spaces or in b-metric spaces, we must say that we do not know whether the caristi’s fixed point theorem in c∗-algebra valued metric spaces [25, theorem 3.5] may be deduced from caristi’s fixed point theorem in metric spaces [8, theorem 1] or not. the difficulty is that from the inequality (3.6) on page 587 of [25] we may not get ρ(x, t x) ≤ ‖φ(x)‖ − ‖φ(t x)‖ for all x ∈ x. so the following question is still open. question 2.14. is it possible to derive caristi’s fixed point theorem in c∗-algebra valued metric spaces [25, theorem 3.5] from caristi’s fixed point theorem in metric spaces [8, theorem 1]? acknowledgements. the authors wish to express their thanks to anonymous reviewers and the editor for several helpful comments concerning the paper. c© agt, upv, 2017 appl. 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-�1�a@-b:<-< c h�q kd((0, 0), (0, 1)), for any k ∈ [0, 1). this proves that t does not satisfy the contractive condition (3.3). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 18 best proximity points of contractive mappings 4. applications let a and b be two non-empty subsets of a metric space (x,d). a mapping f : a → b is called (�,k)-uniformly locally contractive [2] (where k ∈ [0, 1) and � > 0) if d(fx,fy) ≤ kd(x,y) for all x,y ∈ a with d(x,y) < �. an (�,k)uniformly locally contractive mapping need not be a contraction, for example one can refer to [2, 8]. as an application of theorem 3.2, we now establish the following result for uniformly locally contractive mappings. theorem 4.1. let (x,d) be complete metric space, a and b be closed subsets of (x,d) such that a0 6= ∅ and (a,b) satisfies p -property. suppose that t : a → b is an (�,k)-uniformly locally contractive mapping satisfying t(a0) ⊆ b0. then t has a unique best proximity point if the space (a0,d) is �-chainable, that is, given a,b ∈ a0, there exist n ∈ n and a sequence (yi)ni=0 in a0 such that y 0 = a, yn = b and d(yi−1,yi) < � for each i = 1, 2, · · · ,n. proof. consider the graph g where v (g) = x and e(g) as follows: e(g) = {(x,y) ∈ x ×x : d(x,y) < �}. it is clear that e(g) ⊇ ∆ and g has no parallel edges. also, in this case g = g̃. let x,y ∈ a be such that (x,y) ∈ e(g) and for all x1,y1 ∈ a, d(x1,tx) = d(a,b) and d(y1,ty) = d(a,b). since (x,y) ∈ e(g), d(tx,ty) ≤ kd(x,y) where k ∈ [0, 1). hence and by the p-property of (a,b), we have d(x1,y1) < �. therefore t is a g-contraction. since a0 6= ∅ and t(a0) ⊆ b0, there exist x0 and x1 in a0 such that d(x1,tx0) = d(a,b). the �-chainability of (a0,d) implies that there exist a natural number n and a sequence (yi)ni=0 containing points of a0 such that y0 = x0, y n = x1 and d(y i−1,yi) < � for i = 1, · · · ,n. thus (yi)ni=0 ⊆ a0 is a path in g between x0 and x1. if {sn}n∈n is a sequence in a such that sn → s, then there exists m ∈ n such that d(sn,s) < � ∀n ≥ m. hence we can obtain a subsequence {snp}p∈n such that (snp,s) ∈ e(g) ∀p ∈ n. also, it is clear from the �-chainability of (a0,d) that for every x,y ∈ a0, there is a path (qi)li=0 ⊆ a0 in g̃ (i.e., g) between them. thus t has a unique best proximity point by theorem 3.2. � as a corollary to the above theorem, we get the following theorem due to edelstein [2] by considering a = b = x. theorem 4.2 ([2, theorem 5.2]). let (x,d) be a complete metric space. an (�,k)uniformly locally contractive mapping f : x → x has a unique fixed point if (x,d) is �-chainable. in the last part of this section we establish the following result for non-self contractive mapping on a partially ordered metric space. let (x,d) be a metric space endowed with a partial order � and a and b be two non-empty subsets of (x,d). by x�, we denote the following set: x� = {(x,y) ∈ x ×x : x � y or x � y}. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 19 a. sultana and v. vetrivel following [10], we say that a mapping t : a → b is a proximally monotone mapping if for all x1,x2 ∈ a with x1 � x2: d(y1,tx1) = d(a,b) d(y2,tx2) = d(a,b) } ⇒ (y1,y2) ∈ x�, for all y1,y2 ∈ a. theorem 4.3. let (x,d) be complete metric space, a and b be two closed subsets of (x,d) such that (a,b) has the p -property. let t : a → b be a proximally monotone map such that t(a0) ⊆ b0 and d(tx,ty) ≤ kd(x,y) for all x � y and for some k ∈ [0, 1). assume that either t is continuous on a or for any {yn}n∈n in a with yn → y∗ and (yn,yn+1) ∈ x� for n ∈ n, there exists (ynp )p∈n such that (ynp,y∗) ∈ x� for p ∈ n. then t has a best proximity point if there exist x0 and x1 in a0 such that d(x1,tx0) = d(a,b) and (x0,x1) ∈ x�. moreover, the best proximity point of t is unique if for x,y ∈ a0, there exists z ∈ a0 such that (x,z), (y,z) ∈ x�. proof. by considering the graph g where v (g) = x and e(g) := {(x,y) ∈ x ×x : x � y ∨y � x}, the proof follows by theorem 3.2 and remark 3.3. � the above result includes the fixed point results for mappings on a partially ordered metric space due to ran and reurings [12] and j. j. nieto and r. r. lópez [9]. acknowledgements. the authors are grateful to the referees for their valuable comments and suggestions to improve this manuscript. the first author is thankful to university grants commission ( f.2 − 12/2002(sa − i) ) , new delhi, india for the financial support. references [1] t. dinevari and m. frigon, fixed point results for multivalued contractions on a metric space with a graph, j. math. anal. appl. 405 (2013), 507–517. [2] m. edelstein, an extension of banach’s contraction principle, proc. amer. math. soc. 12 (1961), 7–10. [3] k. fan, extensions of two fixed point theorems of f. e. browder, math. z. 122 (1969), 234–240. [4] j. jachymski, the contraction principle for mappings on a metric space with a graph, proc. amer. math. soc. 136 (2008), 1359-1373. [5] w. k. kim and k. h. lee, existence of best proximity pairs and equilibrium pairs, j. math. anal. appl. 316 (2006), 433–446. [6] w. k. kim, s. kum and k. h. lee, on general best proximity pairs and equilibrium pairs in free abstract economies, nonlinear anal. tma 68 (2008), 2216–2227. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 20 best proximity points of contractive mappings [7] w. a. kirk, s. reich and p. veeramani, proximinal retracts and best proximity pair theorems, numer. funct. anal. optim. 24 (2003), 851–862. [8] l. máté, the hutchinson-barnsley theory for certain non-contraction mappings, period. math. hungar. 27 (1993), 21–33. [9] j. j. nieto and r. rodŕıguez-lópez, contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order 22 (2005), 223–239. [10] v. pragadeeswarar and m. marudai, best proximity points: approximation and optimization in partially ordered metric spaces, optim. lett. 7 (2013), 1883–1892. [11] v. sankar raj, best proximity point theorems for non-self mappings, fixed point theory 14 (2013), 447–454. [12] a. c. m. ran and m. c. reurings, a fixed point theorem in partially ordered sets and some applications to matrix equations, proc. amer. math. soc. 132 (2004), 1435–1443. [13] a. sultana and v. vetrivel, fixed points of mizoguchi-takahashi contraction on a metric space with a graph and applications, j. math. anal. appl. 417 (2014), 336–344. [14] a. sultana and v. vetrivel, on the existence of best proximity points for generalized contractions, appl. gen. topol. 15 (2014), 55–63. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 21 @ appl. gen. topol. 18, no. 1 (2017), 173-182 doi:10.4995/agt.2017.6713 c© agt, upv, 2017 contractive definitions and discontinuity at fixed point ravindra k. bishta and r. p. pant b a department of mathematics, national defence academy, khadakwasla, pune, india (ravindra.bisht@yahoo.com) b department of mathematics, kumaun university, nainital, uttarakhand, india. (pant rp@rediffmail.com) communicated by i. altun abstract in this paper, we investigate some contractive definitions which are strong enough to generate a fixed point but do not force the mapping to be continuous at the fixed point. we also obtain a fixed point theorem for generalized nonexpansive mappings in metric spaces by employing meir-keeler type conditions. 2010 msc: primary: 47h09; secondary: 47h10. keywords: fixed point; (�−δ) contractions; power contraction; orbital continuity. 1. introduction the well-known banach-picard-caccioppoli contraction principle states that: theorem 1.1. if a self-mapping t of a complete metric space (x,d) satisfies the condition; d(tx,ty) ≤ ad(x,y), 0 ≤ a < 1, for each x,y ∈ x, then t has a unique fixed point. the picard iteration {xn} defined by xn+1 = txn, (n = 0, 1, 2, ...) converges to x∗ for any initial value x0 ∈ x. it is known that the mapping t of banach-picard-caccioppoli contraction is continuous in the entire domain of x. in an interesting development, kannan [9] proved the following theorem: received 11 october 2016 – accepted 09 january 2017 http://dx.doi.org/10.4995/agt.2017.6713 r. k. bisht and r. p. pant theorem 1.2 ([9]). if a self-mapping t of a complete metric space (x,d) satisfies the condition: d(tx,ty) ≤ b[d(x,tx) + d(y,ty)], 0 ≤ b < 1/2, for each x,y ∈ x, then t has a unique fixed point. the kannan fixed point theorem gave rise to the famous question of continuity of contractive mappings at their fixed points. it may be observed that kannan contractive condition does not require the continuity of the mapping t for the existence of the fixed point. however, a mapping t satisfying kannan contractive condition turns out to be continuous at the fixed point. to see this, suppose that z = tz is a fixed point of t and xn → z. then d(txn,z) = d(txn,tz) ≤ b[d(xn,txn) + d(z,tz)] ≤ b[d(xn,z) + d(z,txn)], that is, (1 − b)d(txn,z) ≤ bd(xn,z). this implies that txn → z = tz and t is continuous at the fixed point z. kannan’s paper generated a far-flung interest in the study of fixed points of generalized contractive mappings and soon these were followed by a flood of papers involving contractive definitions, many of which did not require continuity of the mapping. also, kannan contractive condition contained the geometrically elegant idea of defining generalized contractions (generally referred to as contractive definitions in the literature) by replacing d(x,y) in theorem 1.1 above, by a convex combination of distances between the four points x,y,tx and ty. as a result of this, a large number of contractive definitions were soon introduced and studied by various researchers (for various contractive conditions see [3, 4, 15, 16, 17]). one of the most interesting generalizations of the banach-picard-caccioppoli contraction principle consists of replacing the lipschitz constant k by some real valued function whose functional values are less than 1. in 1969, boyd and wang [2] initiated the work along these lines and proved the following theorem: theorem 1.3 ([2]). let t be a mapping of a complete metric space (x,d) into itself. suppose there exists a function φ, upper semicontinuous from right from r+ into itself such that d(tx,ty) ≤ φ(d(x,y)), for all x,y ∈ x. if φ(t) < t for each t > 0, then t has a unique fixed point. another noteworthy generalizations of both banach-picard-caccioppoli contraction principle and boyd and wang fixed point theorem was obtained by meir and keeler [12] in 1969. they proved the following theorem: theorem 1.4 ([12]). if a self-mapping t of a complete metric space (x,d) satisfies the condition: (i) for a given � > 0 there exists a δ(�) > 0 such that � ≤ d(x,y) < � + δ implies d(tx,ty) < � then t has a unique fixed point. a mapping satisfying boyd and wong or meir-keeler type condition is also continuous in the entire domain of x. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 174 contractive definitions and discontinuity at fixed point the following theorem was established by j. matkowski [11] in 1975 as a generalization of meir and keeler fixed point theorem (see also [6]): theorem 1.5 ([11]). if a self-mapping t of a complete metric space (x,d) satisfy the conditions: (i) d(tx,ty) < d(x,y), for all x,y ∈ x,x 6= y; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < d(x,y) < � + δ implies d(tx,ty) ≤ � then there exists exactly one fixed point of t ; moreover, its domain of attraction coincides with the whole of x. in [8] jachymski listed some meir-keeler type conditions and established relations between them. further he gave some new meir-keeler type conditions ensuring a convergence of the successive approximations (see also [5]). in a survey paper of contractive definitions, rhoades [17] compared 250 contractive definitions and showed that majority of the contractive definitions do not require the mapping to be continuous in the entire domain. however, in all the cases the mapping is continuous at the fixed point. he further demonstrated that the contractive definitions force the mapping to be continuous at the fixed point though continuity was neither assumed nor implied by the contractive definitions. the question whether there exists a contractive definition which is strong enough to generate a fixed point but which does not force the map to be continuous at the fixed point was reiterated by rhoades in [18] as an existing open problem. the question of the existence of contractive mappings which are discontinuous at their fixed points was settled in the affirmative by pant [13]. recently, bisht and pant[1] also gave a contractive definition which does not force the map to be continuity at the fixed point. in this note we provide more solutions to the open question of the existence of contractive definitions which are strong enough to generate a fixed point but which do not force the mapping to be continuous at the fixed point. recall that the set o(x; t) = {tnx : n = 0, 1, 2, ...} is called the orbit of the self-mapping t at the point x ∈ x. definition 1.6. a self-mapping t of a metric space (x,d) is called orbitally continuous at a point z ∈ x if for any sequence {xn} ⊂ o(x; t) (for some x ∈ x) xn → z implies txn → tz as n →∞. it is easy to check that every continuous self-mapping of a metric space is orbitally continuous, but converse need not be true. 2. main results in what follows we shall denote m(x,y) = max{d(x,y),d(x,tx),d(y,ty), [d(x,ty) + d(y,tx)]/2}; n(x,y) = max{d(x,y),d(x,tx),d(y,ty),a[d(x,ty) + d(y,tx)]/2}, 0 ≤ a < 1. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 175 r. k. bisht and r. p. pant theorem 2.1. let (x,d) be a complete metric space. let t be a self-mapping on x such that for any x,y ∈ x; (i) d(tx,ty) ≤ φ(n(x,y)), where φ : r+ → r+ is such that φ(t) < t for each t > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �. suppose t is orbitally continuous. then t has a unique fixed point, say z, and tnx → z for each x ∈ x. moreover, t is continuous at z iff limx→zm(x,z) = 0. proof. let x0 be any point in x. define a sequence {xn} in x given by the rule xn+1 = t nx0 = txn and qn = d(xn,xn+1) for all n ∈ n ⋃ {0}. then by (i) qn = d(xn,xn+1) = d(txn−1,txn) ≤ φ(n(xn−1,xn)) < n(xn−1,xn) = max{qn,qn−1} = qn−1. thus {qn} is a strictly decreasing sequence of positive real numbers and, hence, tends to a limit q ≥ 0. if possible, suppose q > 0. then there exists a positive integer k ∈ n such that n ≥ k implies (2.1) q < qn < q + δ(q). it follows from (ii) and qn < qn−1 that qn ≤ q, for n ≥ k, which contradicts the above inequality. thus we have q = 0. we shall show that {xn} is a cauchy sequence. fix an � > 0. without loss of generality, we may assume that δ(�) < �. since qn → 0, there exists k ∈ n such that qn < 1 2 δ, for n ≥ k. following jachymski [7, 8] we shall use induction to show that, for any n ∈ n, (2.2) d(xk,xk+n) < � + 1 2 δ. inequality (2.2) holds for n = 1. assuming (2.2) is true for some n we shall prove it for n + 1. by the triangle inequlaity, we have (2.3) d(xk,xk+n+1) ≤ d(xk,xk+1) + d(xk+1,xk+n+1). observe that it suffices to show that (2.4) d(xk+1,xk+n+1) ≤ �. to show it we shall prove that m(xk,xk+n) ≤ � + δ, where m(xk,xk+n) =max{d(xk,xk+n),d(xk,txk),d(xk+n,txk+n), [d(xk,txk+n) + d(xk+n,txk)]/2}. (2.5) by the induction hypothesis, we get (2.6) d(xk,xk+n) < � + 1 2 δ,d(xk,xk+1) < 1 2 δ,d(xk+n,xk+n+1) < 1 2 δ. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 176 contractive definitions and discontinuity at fixed point also, 1 2 [d(xk,xk+n+1) + d(xk+1,xk+n)] ≤ 1 2 [d(xk,xk+n) + d(xk+n+1,xk+n) + d(xk,xk+1) + d(xk,xk+n)] < � + δ. thus m(xk,xk+n) < � + δ so by (ii) d(xk+1,xk+n+1) ≤ �, completing the induction. hence (2.2) implies that {xn} is a cauchy sequence. since x is complete, there exists a point z ∈ x such that xn → z as n → ∞. also txn → z. orbital continuity of t implies that limn→∞txn = tz. this yields tz = z, that is, z is a fixed point of t. uniqueness of the fixed point follows from (i). now, let t be continuous at the fixed point z and xn → z. then txn → tz = z. hence lim n m(xn,z) = lim n max{d(xn,z),d(xn,txn),d(z,tz), [d(xn,tz) + d(z,txn)]/2} = 0. on the other hand, if limxn→z m(xn,z) = 0, then d(xn,txn) → 0 as xn → z. this implies that txn → z = tz, i.e., t is continuous at z. this concludes the theorem. � in the next theorem, we replace the orbital continuity of the mapping t by continuity condition on tp, where p is a positive integer ≥ 2. theorem 2.2. let (x,d) be a complete metric space. let t be a self-mapping on x such that tp is continuous and for any x,y ∈ x; (i) d(tx,ty) ≤ φ(n(x,y)), where φ : r+ → r+ is such that φ(t) < t for each t > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �. then t has a unique fixed point, say z, and tnx → z for each x ∈ x. moreover, t is continuous at z iff limx→zm(x,z) = 0. proof. let x0 be any point in x. define a sequence {xn} in x given by the rule xn+1 = t nx0 = txn. then following the proof of above theorem we conclude that {xn} is a cauchy sequence. since x is complete, there exists a point z ∈ x such that xn → z as n → ∞. also txn → z and tpxn → z. by continuity of tp, we have tpxn → tpz. this implies tpz = z. we claim that tz = z. for if tz 6= z, we get d(tz,z) = d(tz,tpz) ≤ φ(n(z,tp−1z)) < n(z,tp−1z) = d(tpz,tp−1z); d(tpz,tp−1z) ≤ φ(n(tp−1z,tp−2z)) < n(tp−1z,tp−2z) = d(tp−1z,tp−2z); . . . d(t 2z,tz) ≤ φ(n(tz,z)) < n(tz,z) = d(tz,z), that is z = tz and z is a fixed point of t. uniqueness of the fixed point follows from (i). � taking m(x,y) = n(x,y) = max{d(x,y),d(x,tx),d(y,ty),a[d(x,ty) + d(y,tx)]/2}, 0 ≤ a < 1 we now state the following theorems: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 177 r. k. bisht and r. p. pant theorem 2.3. let (x,d) be a complete metric space. let t be a self-mapping on x such that for any x,y ∈ x; (i) d(tx,ty) ≤ φ(m(x,y)), where φ : r+ → r+ is such that φ(t) < t for each t > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �. then t has a unique fixed point, say z, and tnx → z for each x ∈ x. moreover, t is continuous at z iff limx→zm(x,z) = 0. proof. it may be completed on the lines of the proof of theorem 2.1 above. � theorem 2.4. let (x,d) be a complete metric space. let t be a self-mapping on x such that tp is continuous and for any x,y ∈ x; (i) d(tx,ty) ≤ φ(m(x,y)), where φ : r+ → r+ is such that φ(t) < t for each t > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �. then t has a unique fixed point, say z, and tnx → z for each x ∈ x. moreover, t is continuous at z iff limx→zm(x,z) = 0. proof. it may be completed on the lines of the proof of theorem 2.2 above. � remark 2.5. the last part of theorems 2.1 and 2.2 can alternatively be stated as: t is discontinuous at z iff limx→zm(x,z) 6= 0. the following example illustrates the above theorems: example 2.6. let x = [0, 2] and d be the usual metric on x. define t : x → x by t(x) = 1 if x ∈ [0, 1], t(x) = 0 if x ∈ (1, 2]. then t satisfies the conditions of theorems 2.1 and 2.2 and has a unique fixed point x = 1 at which t is discontinuous. the mapping t satisfies the contractive condition (i) with φ(t) = 1 for t > 1 and φ(t) = t 2 for t ≤ 1. also, t satisfies condition (ii) with δ(�) = 1 for � ≥ 1 and δ(�) = 1−� for � < 1. it can also be easily seen that limx→1m(x, 1) 6= 0 and t is discontinuous at the fixed point x = 1. however, tp is continuous, since tp(x) = 1 for all x ∈ x(p ≥ 2). theorem 2.7. let (x,d) be a complete metric space. let t be a self-mapping on x such that for any x,y ∈ x; (i) d(tx,ty) ≤ φ(n(x,y)), where φ : r+ → r+ is such that φ(t) < t for each t > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �. then t has a unique fixed point. moreover, t is continuous at z iff limx→zm(x,z) = 0. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 178 contractive definitions and discontinuity at fixed point proof. let x0 be any point in x and let x 6= tx. define a sequence {xn} in x given by the rule xn+1 = t nx0 = txn. then following the proof of theorem 2.1, we conclude that {xn} is a cauchy sequence. since x is complete, there exists a point z ∈ x such that xn → z as n → ∞. also txn → z. we claim that tz = z. for if tz 6= z, we get d(tz,txn) ≤ φ(max{d(z,xn),d(z,tz),d(xn,txn),a[d(z,txn)+d(xn,tz)]/2}). on letting n → ∞ this yields, d(tz,z) ≤ φ(d(tz,z)) < d(tz,z), a contradiction. thus z is a fixed point of t . uniqueness of the fixed point follows from (i). � the following theorem shows that power contraction allows the possibility of discontinuity at the fixed point. in the next theorem we denote: m′(x,y) =max{d(x,y),d(x,tmx),d(y,tmy), [d(x,tmy) + d(y,tmx)]/2}, n′(x,y) =max{d(x,y),d(x,tmx),d(y,tmy),a[d(x,tmy) + d(y,tmx)]/2}, 0 ≤ a < 1 where m ∈ n. theorem 2.8. let (x,d) be a complete metric space. let t be a self-mapping on x such that for any x,y ∈ x: (i) d(tmx,tmy) ≤ φ(n′(x,y)), where φ : r+ → r+ is such that φ(t) < t for each t > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m′(x,y) < � + δ implies d(tmx,tmy) ≤ �. then t has a unique fixed point. proof. by theorem 2.7, tm has a unique fixed point z ∈ x; i.e., tm(z) = z. then t(z) = t(tm(z)) = tm(t(z)) and so t(z) is a fixed point of tm. since the fixed point of tm is unique, tz = z. � taking m′(x,y) = n′(x,y) = max{d(x,y),d(x,tmx),d(y,tmy),a[d(x,tmy)+ d(y,tmx)]/2}, 0 ≤ a < 1 we get the following result: theorem 2.9. let (x,d) be a complete metric space. let t be a self-mapping on x such that for any x,y ∈ x: (i) d(tmx,tmy) ≤ φ(m′(x,y)), where φ : r+ → r+ is such that φ(t) < t for each t > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m′(x,y) < � + δ implies d(tmx,tmy) ≤ �. then t has a unique fixed point. proof. it may be completed following theorem 2.7 above. � remark 2.10. theorems 2.1, 2.2, and 2.3 unify and improve the results due to bisht and pant [1], ćirić [5, 6], jachymski [8], kuczma et al. [10], matkowski [11], and pant [13]. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 179 r. k. bisht and r. p. pant some consequences of the above proved theorems are the following corollaries which also generalize and extend the results of jachymski [8], kuczma et al. [10], matkowski [11], and pant [13]. corollary 2.11. let (x,d) be a complete metric space. let t be a self-mapping on x such that: (i) d(tx,ty) < n(x,y), for any x,y ∈ x with m(x,y) > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �. suppose t is orbitally continuous. then t has a unique fixed point, say z, and tnx → z for each x ∈ x. moreover, t is continuous at z iff limx→zm(x,z) = 0. corollary 2.12. let (x,d) be a complete metric space. let t be a self-mapping on x such that tp is continuous: (i) d(tx,ty) < n(x,y), for any x,y ∈ x with m(x,y) > 0; (ii) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �. then t has a unique fixed point, say z, and tnx → z for each x ∈ x. moreover, t is continuous at z iff limx→zm(x,z) = 0. 3. fixed points of nonexpansive mappings in what follows we shall denote p(x,y) = max{d(x,y),b[d(x,tx) + d(y,ty)]/2,c[d(x,ty) + d(y,tx)]/2}, 0 ≤ b,c < 1. theorem 3.1. let (x,d) be a complete metric space. let t be a self-mapping on x such that for any x,y ∈ x; (i) for a given � > 0 there exists a δ(�) > 0 such that � < m(x,y) < � + δ implies d(tx,ty) ≤ �; (ii) d(tx,ty) ≤ p(x,y). then t has a fixed point, say z, and tnx → z for each x ∈ x. proof. let x0 be any point in x and let x 6= tx. define a sequence {xn} in x given by the rule xn+1 = t nx0 = txn. then following the proof of theorem 2.1 we can easily prove that {xn} is a cauchy sequence. since x is complete, there exists a point z ∈ x such that xn → z as n → ∞. also txn → z. we claim that tz = z. for if tz 6= z, we get d(tz,txn) ≤ max{d(z,xn),b[d(z,tz) + d(xn,txn)]/2,c[d(z,txn) + d(xn,tz)]/2}. on letting n →∞ this yields, d(tz,z) ≤ max{b[d(tz,z)]/2,c[d(tz,z)]/2} < d(tz,z), a contradiction since 0 ≤ b,c < 1. thus z is a fixed point of t . � remark 3.2. theorem 3.1 also remains true if we replace condition (ii) by the following condition: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 180 contractive definitions and discontinuity at fixed point (i). d(tx,ty) ≤ max{d(x,y),d(x,tx),d(y,ty),b[d(x,ty)+d(y,tx)]/2}, 0 ≤ b < 1. the following example illustrates theorem 3.1: example 3.3. let x = [−1, 1] and d be the usual metric on x. define t : x → x by t(x) = −|x|x for each x. then t satisfies all the conditions of theorem 3.1 and has a fixed point x = 0. the mapping t satisfies condition (i) with δ(�) = 1 4 ( √ (2�) − �)) for � < 2 and δ(�) = � for � ≥ 2. however, t does not satisfy the contractive condition d(tx,ty) < max{d(x,y), [d(x,tx) + d(y,ty)]/2, [d(x,ty) + d(y,tx)]/2}. it may be observed that there exist a large number of meir-keeler type nonexpansive conditions which yield more than one fixed point. the following example illustrates this fact: example 3.4. let x = [0, 1] and d be the usual metric on x. define t : x → x by tx = sgn(x) (the signum function), i.e., t0 = 0,tx = 1 if x > 0. then t has two fixed points x = 0 and x = 1. acknowledgements. the authors are thankful to the learned referees for their suggestions which improved the presentation of this paper. references [1] r. k. bisht and r. p. pant, a remark on discontinuity at fixed point, j. math. anal. appl. 445 (2017), 1239–1242. 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[18] b. e. rhoades, contractive definitions and continuity, contemporary mathematics 72 (1988), 233–245. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 182 () @ appl. gen. topol. 17, no. 1(2016), 17-35doi:10.4995/agt.2016.4085 c© agt, upv, 2016 convergence in graded ditopological texture spaces dedicated to the memory of lawrence michael brown ramazan ekmekçi̇a and rıza ertürkb a çanakkale onsekiz mart university, faculty of arts and sciences, department of mathematics, 17100 çanakkale, turkey. (ekmekci@comu.edu.tr) b hacettepe university, faculty of science, department of mathematics, 06800 beytepe, ankara, turkey. (rerturk@hacettepe.edu.tr) abstract graded ditopological texture spaces have been presented and discussed in categorical aspects by lawrence m. brown and alexander šostak in [6]. in this paper, the authors generalize the structure of difilters in ditopological texture spaces defined in [11] to the graded ditopological texture spaces and compare the properties of difilters and graded difilters. 2010 msc: 54a05; 54a20; 06d10. keywords: texture; graded ditopology; graded difilter; fuzzy topology. 1. introduction the concept of fuzzy topological space was defined in 1968 by c.chang as an ordinary subset of the family of all fuzzy subsets of a given set[7]. as a more suitable approach to the idea of fuzzyness, in 1985, šostak and kubiak independently redefined fuzzy topology where a fuzzy subset has a degree of openness rather than being open or not [12, 10] (for historical developments and basic ideas of the theory of fuzzy topology see [13]). in classical topology the notion of open set is usually taken as primitive with that of closed set being auxiliary. however, since closed sets are easily obtained received 19 august 2015 – accepted 09 january 2016 http://dx.doi.org/10.4995/agt.2016.4085 r. ekmekçi and r. ertürk as the complements of open sets they often play an important, sometimes dominating role in topological arguments. a similar situation holds for topologies on lattices where the role of set complement is played by an order reversing involution. it is the case, however, that there may be no order reversing involution available, or that the presence of such an involution is irrelevant to the topic under consideration. to deal with such cases it is natural to consider a topological structure consisting of a priori unrelated families of open sets and of closed sets. this was the approach adapted from the beginning for the topological structures on textures, originally introduced as a point-based representation for fuzzy sets [1, 2]. such topological structures were given the name of a dichotomous topology, or ditopology for short. they consist of a family τ of open sets and a generally unrelated family κ of closed sets. hence, both the open and the closed sets are regarded as primitive concepts for a ditopology. a ditopology (τ, κ) on the discrete texture (x, p(x)) gives rise to a bitopological space (x, τ, κc). this link with bitopological spaces has had a powerful influence on the development of the theory of ditopological texture spaces, but it should be emphasized that a ditopology and a bitopology are conceptually different. indeed, a bitopology consists of two separate topological structures (complete with their open and closed sets) whose interrelations we wish to study, whereas a ditopology represents a single topological structure. ditopological texture spaces were introduced by l. m. brown as a natural extention of the work of the second author on the representation of latticevalued topologies by bitopologies in [9]. the concept of ditopology is more general than general topology, bitopology and fuzzy topology in chang’s sense. an adequate introduction to the theory of texture spaces and ditopological texture spaces may be obtained from [1, 2, 3, 4, 5]. recently, l. m. brown and a. šostak have presented the concept ”graded ditopology” on textures as an extention of the concept of ditopology to the case where openness and closedness are given in terms of a priori unrelated grading functions [6]. the concept of graded ditopology is more general than ditopology and smooth topology. two sorts of neighbohood structure on graded ditopological texture spaces are presented and investigated by the authors in [8]. the aim of this work is to generalize the structure of difilters in ditopological texture spaces defined by s. özçağ, f. yıldız and l. m. brown in [11] to the graded ditopological texture spaces which is introduced by l. m. brown and a. šostak in [6]. furthermore we compare the properties of difilters and graded difilters. the material in this work forms a part of the first named author’s ph.d. thesis, currently being written under the supervision of the second name author dr. rıza ertürk. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 18 convergence in graded ditopological texture spaces 2. preliminaries we recall various concepts and properties from [3, 4, 5] under the following subtitle. ditopological texture spaces: let s be a set. a texturing s on s is a subset of p(s) which is a point separating, complete, completely distributive lattice with respect to inclusion which contains s, ∅ and for which meet ∧ coincides with intersection ⋂ and finite joins ∨ with unions ⋃ . the pair (s, s) is then called a texture or a texture space. in general, a texturing of s need not be closed under set complementation, but there may exist a mapping σ : s → s satisfying σ(σ(a)) = a and a ⊆ b ⇒ σ(b) ⊆ σ(a) for all a, b ∈ s. in this case σ is called a complementation on (s, s) and (s, s, σ) is said to be a complemented texture. for a texture (s, s), most properties are conveniently defined in terms of the p − sets ps = ⋂ {a ∈ s | s ∈ a} and the q − sets qs = ∨ {a ∈ s | s 6∈ a} = ∨ {pu | u ∈ s, s 6∈ pu}. a texture (s, s) is called a plain texture if it satisfies any of the following equivalent conditions: (1) ps * qs for all s ∈ s (2) a = ∨ i∈i ai = ⋃ i∈i ai for all ai ∈ s, i ∈ i recall that m ∈ s is called a molecule in s if m 6= ∅ and m ⊆ a∪b, a, b ∈ s implies m ⊆ a or m ⊆ b. the sets ps, s ∈ s are molecules, and the texture (s, s) is called ”simple” if all molecules of s are in the form {ps | s ∈ s}. for a set a ∈ s, the core of a (denoted by a♭) is defined by a ♭ = ⋂ { ⋃ {ai | i ∈ i} |{ai | i ∈ i} ⊆ s, a = ∨ {ai | i ∈ i} } . theorem 2.1 ([3]). in any texture space (s, s), the following statements hold: (1) s 6∈ a ⇒ a ⊆ qs ⇒ s 6∈ a ♭ for all s ∈ s, a ∈ s. (2) a♭ = {s | a * qs} for all a ∈ s. (3) for aj ∈ s, j ∈ j we have ( ∨ j∈j aj) ♭ = ⋃ j∈j a ♭ j. (4) a is the smallest element of s containing a♭ for all a ∈ s. (5) for a, b ∈ s, if a * b then there exists s ∈ s with a * qs and ps * b. (6) a = ⋂ {qs | ps * a} for all a ∈ s. (7) a = ∨ {ps | a * qs} for all a ∈ s. example 2.2. (1) if p(x) is the powerset of a set x, then (x, p(x)) is the discrete texture on x. for x ∈ x, px = {x} and qx = x \ {x}. the mapping πx : p(x) → p(x), πx(y ) = x \ y for y ⊆ x is a complementation on the texture (x, p(x)). (2) setting i = [0, 1], j = {[0, r), [0, r] |r ∈ i} gives the unit interval texture c© agt, upv, 2016 appl. gen. topol. 17, no. 1 19 r. ekmekçi and r. ertürk (i, j ). for r ∈ i, pr = [0, r] and qr = [0, r). and the mapping ι : j → j , ι[0, r] = [0, 1 − r), ι[0, r) = [0, 1 − r] is a complementation on this texture. (3) the texture (l, l, λ) is defined by l = (0, 1], l = {(0, r] | r ∈ [0, 1]}, λ((0, r]) = (0, 1 − r]. for r ∈ l, pr = (0, r] = qr. (4) s = {∅, {a, b}, {b}, {b, c}, s} is a simple texturing of s = {a, b, c}. pa = {a, b}, pb = {b}, pc = {b, c}. it is not possible to define a complementation on (s, s). (5) if (s, s), (v, v) are textures, the product texturing s ⊗ v of s × v consists of arbitrary intersections of sets of the form (a × v ) ∪ (s × b), a ∈ s, b ∈ v, and (s × v, s ⊗ v) is called the product of (s, s) and (v, v). for s ∈ s, v ∈ v , p(s,v) = ps × pv and q(s,v) = (qs × v ) ∪ (s × qv). a dichotomous topology, or ditopology for short, on a texture (s, s) is a pair (τ, κ) of subsets of s, where the set of open sets τ satisfies (t1) s, ∅ ∈ τ (t2) g1, g2 ∈ τ ⇒ g1 ∩ g2 ∈ τ (t3) gi ∈ τ, i ∈ i ⇒ ∨ i gi ∈ τ and the set of closed sets κ satisfies (ct1) s, ∅ ∈ κ (ct2) k1, k2 ∈ κ ⇒ k1 ∪ k2 ∈ κ (ct3) ki ∈ κ, i ∈ i ⇒ ⋂ i ki ∈ κ. hence a ditopology is essentially a ”topology” for which there is no priori relation between the open and closed sets. let (τ, κ) be a ditopology on (s, s). (1) if s ∈ s♭, a neighborhood of s is a set n ∈ s for which there exists g ∈ τ satisfying ps ⊆ g ⊆ n * qs. (2) if s ∈ s, a coneighborhood of s is a set m ∈ s for which there exists k ∈ κ satisfying ps * m ⊆ k ⊆ qs. if the set of nhds (conhds) of s is denoted by η(s) (µ(s)) respectively, then (η, µ) is called the dinhd system of (τ, κ). theorem 2.3 ([5]). for a ditopology (τ, κ) on (s, s) let the families η(s), s ∈ s♭ and µ(s), s ∈ s be defined as above. (1) for each s ∈ s♭ we have η(s) 6= ∅ and these families satisfy the following conditions: (i) n ∈ η(s) ⇒ n * qs (ii) n ∈ η(s), n ⊆ n′ ∈ s ⇒ n′ ∈ η(s) (iii) n1, n2 ∈ η(s), n1 ∩ n2 * qs ⇒ n1 ∩ n2 ∈ η(s) (iv) (a) n ∈ η(s) ⇒ ∃n⋆ ∈ s, ps ⊆ n ⋆ ⊆ n, so that n⋆ * qt ⇒ n⋆ ∈ η(t), ∀t ∈ s♭ (b) for n ∈ s and n * qs, if there exists n⋆ ∈ s, ps ⊆ n⋆ ⊆ n, which satisfies n⋆ * qt ⇒ n⋆ ∈ η(t), ∀t ∈ s♭, then n ∈ η(s). moreover, the sets g in τ are characterized by the condition that g ∈ η(s) for all s with g * qs. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 20 convergence in graded ditopological texture spaces (2) for each s ∈ s we have µ(s) 6= ∅ and these families satisfy the following conditions: (i) m ∈ µ(s) ⇒ ps * m (ii) m ∈ µ(s), m ⊇ m′ ∈ s ⇒ m′ ∈ µ(s) (iii) m1, m2 ∈ µ(s) ⇒ m1 ∪ m2 ∈ µ(s) (iv) (a) m ∈ µ(s) ⇒ ∃m⋆ ∈ s, m ⊆ m⋆ ⊆ qs, so that pt * m⋆ ⇒ m⋆ ∈ µ(t), ∀t ∈ s (b) for m ∈ s and ps * m, if there exists m⋆ ∈ s, m ⊆ m⋆ ⊆ qs, which satisfies pt * m⋆ ⇒ m⋆ ∈ µ(t), ∀t ∈ s, then m ∈ µ(s). moreover, the sets k in κ are characterized by the condition that k ∈ µ(s) for all s with ps * k. conversely, if η(s), s ∈ s♭ and µ(s), s ∈ s are non-empty families of sets in s which satisfy conditions (1) and (2) above, respectively, then there exists a ditopology (τ, κ) on (s, s) for which η(s) (µ(s)) are the families of nhds (resp., conhds) of the ditopology (τ, κ). difilters on textures: [11] let (s, s) be a texture. (1) f ⊆ s is called a s-filter or a filter on (s, s), if f 6= ∅ and satisfies: (f1) ∅ 6∈ f (f2) f ∈ f, f ⊆ f ′ ∈ s ⇒ f ′ ∈ f, and (f3) f1, f2 ∈ f ⇒ f1 ∩ f2 ∈ f. (2) g ⊆ s is called a s-cofilter or a cofilter on (s, s), if g 6= ∅ and satisfies: (cf1) s 6∈ g (cf2) g ∈ g, g ⊇ g ′ ∈ s ⇒ g′ ∈ g, and (cf3) g1, g2 ∈ g ⇒ g1 ∪ g2 ∈ g. (3) if f is a s-filter and g is a s-cofilter then f × g is called a s-difilter or a difilter on (s, s). a difilter f × g on (s, s) is called regular if it satisfies following equivalent conditions: (1) f ∩ g = ∅. (2) (fi, gi) ∈ f × g, i = 1, 2, ..., n ⇒ ⋂n i=1 fi * ⋃n i=1 gi. (3) a * b for all a ∈ f and b ∈ g. example 2.4. (1) for a plain texture (s, s) and ditopology (τ, κ) on (s, s), then η(s) × µ(s) is a regular s-difilter for all s ∈ s♭ = s on (s, s). (2) let (s, s, τ, κ) be a ditopological texture space. then the families η∗(s) = {a ∈ s | ∃gk ∈ τ : gk * qs, 1 ≤ k ≤ n and g1 ∩ ... ∩ gn ⊆ a}, s ∈ s ♭ µ∗(s) = {a ∈ s | ∃fk ∈ κ : ps * fk, 1 ≤ k ≤ n and a ⊆ f1 ∪ ... ∪ fn}, s ∈ s form a regular difilter η∗(s) × µ∗(s) on (s, s). definition 2.5 ([11]). let (s, s, τ, κ) be a ditopological texture space, f a s-filter and g a s-cofilter. (1) we say f converges to a point s ∈ s♭, and write f → s if η∗(s) ⊆ f; g converges to a point s ∈ s, and write g → s if c© agt, upv, 2016 appl. gen. topol. 17, no. 1 21 r. ekmekçi and r. ertürk µ∗(s) ⊆ g. the difilter f × g is said to be diconvergent if f → s and g → s′ for some s, s′ ∈ s satisfying ps′ * qs. (2) f is called prime if a1, a2 ∈ s, a1 ∪ a2 ∈ f ⇒ a1 ∈ f or a2 ∈ f. g is called prime if b1, b2 ∈ s, b1 ∩ b2 ∈ g ⇒ b1 ∈ g or b2 ∈ g. proposition 2.6 ([11]). the following are equivalent for a regular difilter f×g on (s, s). (1) f × g is maximal. (2) f ∪ g = s. (3) f is a prime s-filter and g = s \ f. (4) g is a prime s-cofilter and f = s \ g. it is obtained in [11] that there exist one to one correspondences among the set of maximal regular difilters on (s, s), the set of prime filters on (s, s) and the set of prime cofilters on (s, s). graded ditopological texture spaces: [6] let (s, s), (v, v) be textures and consider t , k : s → v satisfying (gt1) t (s) = t (∅) = v (gt2) t (a1) ∩ t (a2) ⊆ t (a1 ∩ a2) ∀a1, a2 ∈ s (gt3) ⋂ j∈j t (aj) ⊆ t ( ∨ j∈j aj) ∀aj ∈ s, j ∈ j and (gct1) k(s) = k(∅) = v (gct2) k(a1) ∩ k(a2) ⊆ k(a1 ∪ a2) ∀a1, a2 ∈ s (gct3) ⋂ j∈j k(aj) ⊆ k( ⋂ j∈j aj) ∀aj ∈ s, j ∈ j. then t is called a (v, v)-graded topology, k a (v, v)-graded cotopology and (t , k) a (v, v)-graded ditopology on (s, s). the tuple (s, s, t , k, v, v) is called a graded ditopological texture space. for v ∈ v we define t v = {a ∈ s | pv ⊆ t (a)}, k v = {a ∈ s | pv ⊆ k(a)}. then (t v, kv) is a ditopology on (s, s) for each v ∈ v . that is, if (s, s, t , k, v, v) is any graded ditopological texture space, then there exists a ditopology (t v, kv) on the texture space (s, s) for each v ∈ v if (s, s, σ) is a complemented texture and (t , k) is a (v, v)-graded ditopology on (s, s), then (k ◦ σ, t ◦ σ) is also a (v, v)-graded ditopology on (s, s). (t , k) is called complemented if (t , k) = (k ◦ σ, t ◦ σ). example 2.7. let (s, s, τ, κ) be a ditopological texture space and (v, v) the discrete texture on a singleton. take (v, v) = (1, p(1)) (the notation 1 denotes the set {0}) and define τg : s → p(1) by τg(a) = 1 ⇔ a ∈ τ. then τg is a (v, v)-graded topology on (s, s). likewise, κg defined by κg(a) = 1 ⇔ a ∈ κ is a (v, v)-graded cotopology on (s, s). then (τg, κg) is called the graded ditopology on (s, s) corresponding to ditopology (τ, κ). therefore graded ditopological texture spaces are more general than ditopological texture spaces. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 22 convergence in graded ditopological texture spaces graded dineighborhood systems: [8] let (s, s) and (v, v) be two texture spaces. for any mapping k : s → v, we use the notation vk to denote the family {a ∈ s : k(a) * qv} for all v ∈ v , and thus for each v ∈ v , we have vk ⊆ kv. definition 2.8 ([8]). let (t , k) be a (v, v)-graded ditopology on texture (s, s) and n : s♭ → vs, m : s → vs mappings where n(s) = ns : s → v for each s ∈ s♭ and m(s) = ms : s → v for each s ∈ s. then the mapping ns is called a ”graded neighborhood system of s” if (2.1) vns = {a ∈ s : t (b) * qv and ps ⊆ b ⊆ a * qs for some b ∈ s} for each v ∈ v ♭. the mapping ms is called a ”graded coneighborhood system of s” if (2.2) vms = {a ∈ s : k(b) * qv and ps * a ⊆ b ⊆ qs for some b ∈ s} for each v ∈ v ♭. the mapping n (m) is called a ”graded neighborhood system” (”graded coneighborhood system”) of graded ditopological texture space (s, s, t , k, v, v) if ns (ms) is a graded neighborhood system for each s ∈ s♭ (graded coneighborhood system for each s ∈ s) respectively. (n, m) is called a ”graded dineighborhood system” of graded ditopological texture space (s, s, t , k, v, v) if n is a graded neighborhood system and m is a graded coneighborhood system of graded ditopological texture space (s, s, t , k, v, v). proposition 2.9 ([8]). for the above notations, (n, m) is a graded dinhd system of a graded ditopological texture space (s, s, t , k, v, v) iff (2.3) ns(a) = { sup{t (b) : ps ⊆ b ⊆ a * qs, b ∈ s}, a * qs ∅, a ⊆ qs for each s ∈ s♭, a ∈ s and (2.4) ms(a) = { sup{k(b) : ps * a ⊆ b ⊆ qs, b ∈ s}, ps * a ∅, ps ⊆ a for each s ∈ s, a ∈ s. theorem 2.10 ([8]). let (t , k) be a (v, v)-graded ditopology on a texture (s, s). if (n, m) is the graded dinhd system of the graded ditopological texture space (s, s, t , k, v, v), then the following properties hold for all a, a1, a2 ∈ s: (1) for each s ∈ s♭; (n1) ns(a) 6= ∅ ⇒ a * qs (n2) ns(∅) = ∅ and ns(s) = v (n3) a1 ⊆ a2 ⇒ ns(a1) ⊆ ns(a2) (n4) a1 ∩ a2 * qs ⇒ ns(a1) ∧ ns(a2) ⊆ ns(a1 ∩ a2) (n5) ns(a) ⊆ sup{ ∧ s′∈b♭ ns′(b) : ps ⊆ b ⊆ a * qs, b ∈ s} (2) for each s ∈ s; (m1) ms(a) 6= ∅ ⇒ ps * a (m2) ms(s) = ∅ and ms(∅) = v c© agt, upv, 2016 appl. gen. topol. 17, no. 1 23 r. ekmekçi and r. ertürk (m3) a1 ⊆ a2 ⇒ ms(a2) ⊆ ms(a1) (m4) ms(a1) ∧ ms(a2) ⊆ ms(a1 ∪ a2) (m5) ms(a) ⊆ sup{ ∧ s′∈(s\b) ms′(b) : ps * a ⊆ b ⊆ qs, b ∈ s} 3. graded difilters and convergence definition 3.1. let (s, s) and (v, v) be textures. (1) a mapping f : s → v is called a (v, v)-graded filter on (s, s) if f satisfies: (gf1) f(∅) = ∅ (gf2) a1 ⊆ a2 ⇒ f(a1) ⊆ f(a2) (gf3) f(a1) ∧ f(a2) ⊆ f(a1 ∩ a2) (2) a mapping g : s → v is called a (v, v)-graded cofilter on (s, s) if g satisfies: (gcf1) g(s) = ∅ (gcf2) a1 ⊆ a2 ⇒ g(a2) ⊆ g(a1) (gcf3) g(a1) ∧ g(a2) ⊆ g(a1 ∪ a2) (3) if f is a (v, v)-graded filter and g (v, v)-graded cofilter on (s, s) then the pair (f, g) is called a (v, v)-graded difilter on (s, s). proposition 3.2. the following are equivalent for a (v, v)-graded difilter (f, g) on (s, s). (1) f ∧ g = ∅ i.e. f(a) ∧ g(a) = ∅ for all a ∈ s. (2) ∀n ∈ n, ∧n i=1(f(ai) ∧ g(bi)) 6= ∅ ⇒ ⋂n i=1 ai * ⋃n i=1 bi, for all ai, bi ∈ s (3) f(a) ∧ g(b) 6= ∅ ⇒ a * b, for all a, b ∈ s proof. (1) ⇒ (2) : let n ∈ n, ∧n i=1(f(ai) ∧ g(bi)) 6= ∅ for all ai, bi ∈ s and suppose that ⋂n i=1 ai ⊆ ⋃n i=1 bi. then, from (1) we get ∅ = f( n ⋂ i=1 ai) ∧ g( n ⋂ i=1 ai) ⊇ f( n ⋂ i=1 ai) ∧ g( n ⋃ i=1 bi) ⊇ n ∧ i=1 f(ai) ∧ n ∧ i=1 g(bi) = n ∧ i=1 (f(ai) ∧ g(bi)) which contradicts with ∧n i=1(f(ai) ∧ g(bi)) 6= ∅. (2) ⇒ (3) : clear. (3) ⇒ (1) : if we assume that f ∧ g 6= ∅ then f(a) ∧ g(a) 6= ∅ for some a ∈ s. thus we obtain that a 6⊆ a by (3) but this is a contradiction. so we have f ∧ g = ∅. � definition 3.3. a (v, v)-graded difilter (f, g) on (s, s) is called regular if it satisfies the equivalent conditions in the previous proposition. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 24 convergence in graded ditopological texture spaces example 3.4. (1) let (s, s) be a texture space and f × g a (regular) difilter on it. then the mappings f, g : s → p(1) defined by f(a) = { {0}, a ∈ f ∅, a 6∈ f and g(a) = { {0}, a ∈ g ∅, a 6∈ g for all a ∈ s, form a (regular) (1, p(1))-graded difilter on (s, s). on the other hand, if (f, g) is a (regular) (1, p(1))-graded difilter on (s, s) then the families defined by f = {a ∈ s : f(a) = {0} }, g = {a ∈ s : g(a) = {0} } form a (regular) difilter f × g on (s, s). besides, if (f, g) be a (regular) (v, v)-graded difilter on a texture (s, s) then the families f v = {a ∈ s | pv ⊆ f(a)}, g v = {a ∈ s | pv ⊆ g(a)} form a (regular) difilter fv × gv on (s, s) for each v ∈ v . (2)analogous with the dinhd-difilter situation; if (n, m) is a graded dinhd system of a graded ditopological texture space (s, s, t , k, v, v) then ms is a (v, v)-graded cofilter on (s, s) but in general for s ∈ s♭, ns is not a (v, v)graded filter on (s, s). if (s, s) is plain then ns is a (v, v)-graded filter on (s, s) for each s ∈ s♭ = s. as a result, if (s, s) is plain then (ns, ms) is a (v, v)-graded difilter on (s, s) for each s ∈ s♭ = s. (3) let (s, s, t , k, v, v) be a graded ditopological texture space. if the mappings ∀s ∈ s♭, n∗s : s → v, ∀s ∈ s, m ∗ s : s → v are defined by n ∗ s (a) = { sup{ ⋂n k=1 t (bk) : bk * qs, b1 ∩ ... ∩ bn ⊆ a forbi ∈ s}, a * qs ∅, a ⊆ qs and m∗s (a) = { sup{ ⋂n k=1 k(bk) : ps * bk, a ⊆ b1 ∪ ... ∪ bn forbi ∈ s}, ps * a ∅, ps ⊆ a for all a ∈ s then (n∗s , m ∗ s ) is a regular (v, v)-graded difilter on (s, s) for each s ∈ s♭. proposition 3.5. for the above notations, if the texture (s, s) is plain then ns = n ∗ s and ms = m ∗ s for each s ∈ s ♭ = s. proof. let a ∈ s, s ∈ s. a ⊆ qs implies n ∗ s (a) = ns(a) = ∅ so, let it be a * qs, and suppose that ns(a) * n∗s (a) for some a ∈ s. then there exists v ∈ v such that ns(a) * qv and pv * n∗s (a). considering ns(a) * qv, we have ps ⊆ b ⊆ a * qs and t (b) * qv for some b ∈ s. since (s, s) is plain, we have b * qs. so, t (b) ⊆ n∗s (a) and considering t (b) * qv we get n∗s (a) * qv and pv ⊆ n ∗ s (a) which contradicts with pv * n ∗ s (a). c© agt, upv, 2016 appl. gen. topol. 17, no. 1 25 r. ekmekçi and r. ertürk thus, ns(a) ⊆ n ∗ s (a). now we assume that n ∗ s (a) * ns(a) for some a ∈ s. then there exists v ∈ v such that n∗s (a) * qv and pv * ns(a). considering n∗s (a) * qv, there exist b1, b2, ..., bn ∈ s such that bk * qs for 1 ≤ k ≤ n, b1∩b2∩...∩bn ⊆ a and ⋂n k=1 t (bk) * qv. since ⋂n k=1 t (bk) ⊆ t ( ⋂n k=1 bk) we have t ( ⋂n k=1 bk) * qv and so, pv ⊆ t ( ⋂n k=1 bk). moreover, since bk * qs for 1 ≤ k ≤ n and (s, s) is plain, we get ps ⊆ (b1 ∩b2 ∩...∩bn) ⊆ a * qs. thus, it is obtained that pv ⊆ t ( ⋂n k=1 bk) ⊆ ns(a) which is a contradiction. therefore we get n∗s (a) ⊆ ns(a) and so, ns = n ∗ s . similarly it can be shown that ms = m ∗ s . � definition 3.6. let f be a (v, v)-graded filter and g a (v, v)-graded cofilter on a graded ditopological texture space (s, s, t , k, v, v). we say that f converges to s and write that f → s if n∗s ⊆ f. also we say that g converges to s and write that g → s if m∗s ⊆ g. for s, s′ ∈ s, the graded difilter (f, g) is called diconvergent if ps′ * qs, f → s and g → s′. in this case, s (s′) is called (co-)limit of (f, g). proposition 3.7. if (f, g) is a (v, v)-graded difilter on (s, s, t , k, v, v) then (a) f → s ⇔ ”a * qs ⇒ t (a) ⊆ f(a)” (b) g → s ⇔ ”ps * a ⇒ k(a) ⊆ g(a)” proof. (a) let f → s and a * qs. since f → s, we have n∗s ⊆ f and n∗s (a) ⊆ f(a). considering a * qs we obtain that t (a) ⊆ n ∗ s (a) and so t (a) ⊆ f(a). on the other hand, if we suppose that ”a * qs ⇒ t (a) ⊆ f(a)” then n∗s (a) ⊆ f(a) and so we get f → s. the proof of (b) is similar. � proposition 3.8. let the texture (s, s) be plain. if (f, g) is a graded difilter on (s, s, t , k, v, v) then the following are equivalent: (a) (f, g) is diconvergent. (b) ∃s ∈ s : (ns, ms) ⊆ (f, g) proof. (a) ⇒ (b): if (f, g) is diconvergent then there exists s, s′ ∈ s such that ps′ * qs and (n∗s , m ∗ s′) ⊆ (f, g). since (s, s) is plain, from proposition 3.5 we have (ns, ms′) = (n ∗ s , m ∗ s′). besides, since ps′ * qs, ps * a ⊆ b ⊆ qs ⇒ ps′ * a ⊆ b ⊆ qs′ for all a, b ∈ s and so we have ms ⊆ ms′. thus we get ns = n ∗ s ⊆ f, ms ⊆ ms′ = m ∗ s′ ⊆ g and hence (ns, ms) ⊆ (f, g). (b) ⇒ (a): since (s, s) is plain we get ps * qs and by proposition 3.5 we have (ns, ms) = (n ∗ s , m ∗ s ). therefore (f, g) is diconvergent. � definition 3.9. let (s, s, t , k, v, v) be a graded ditopological texture space, a ∈ s and v ∈ v . the set ⋂ {b ∈ s|a ⊆ b, pv ⊆ k(b)} ∈ s is called v-closure of a and denoted by [a]v. the set ∨ {b ∈ s|b ⊆ a, pv ⊆ t (b)} ∈ s is called v-interior of a and denoted by ]a[v. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 26 convergence in graded ditopological texture spaces note that for each v ∈ v , [a]v (]a[v) is the closure (the interior) of a in the ditopological texture space (s, s, t v, kv). proposition 3.10. let (f, g) be a regular graded difilter on a graded ditopological texture space (s, s, t , k, v, v) and s ∈ s. (a) f → s ⇒ ”∀a ∈ s, v ∈ g(a) ⇒]a[v⊆ qs” (b) g → s ⇒ ”∀a ∈ s, v ∈ f(a) ⇒ ps ⊆ [a] v” proof. (a) let f → s, and suppose that there exists v ∈ g(a) such that ]a[v* qs for a set a ∈ s. then b ⊆ a, pv ⊆ t (b) and b * qs for some b ∈ s. considering f → s and b * qs, from proposition 3.7 we get t (b) ⊆ f(b). so, pv ⊆ t (b) ⊆ f(b) ⊆ f(a) and considering v ∈ g(a) we get f(a) ∩ g(a) 6= ∅ which contradicts with the regularity of (f, g). the proof of (b) is similar. � definition 3.11. let (f, g) be a regular graded difilter on a graded ditopological texture space (s, s, t , k, v, v). (1) s ∈ s is called a cluster point of f if for all a ∈ s, v ∈ f(a) ⇒ ps ⊆ [a]v (2) s ∈ s is called a cluster point of g if for all a ∈ s, v ∈ g(a) ⇒]a[v⊆ qs (3) for s, s′ ∈ s, ps * qs′, if s is a cluster point of f and s′ is a cluster point of g then (f, g) is called diclustering in (s, s, t , k, v, v). corollary 3.12. on a graded ditopological texture space (s, s, t , k, v, v), each diconvergent regular graded difilter is diclustering. proof. if (f, g) is a diconvergent regular graded difilter on (s, s, t , k, v, v) then there exist s, s′ ∈ s such that f → s, g → s′ and ps′ * qs. considering proposition 3.10., s is a cluster point of g and s′ a cluster point of f. since ps′ * qs, (f, g) is diclustering. � definition 3.13. let (f, g) be a (v, v)-graded difilter on (s, s). f is called prime if f(a1 ∪ a2) ⊆ f(a1) ∪ f(a2) and g is called prime if g(a1 ∩ a2) ⊆ g(a1) ∪ g(a2) for all a1, a2 ∈ s. example 3.14. let s = {1, 2}, s = {∅, {1}, {2}, s}, v = {a, b, c} and v = p(v ). in this case, (s, s) and (v, v) are texture spaces. if the mappings f, g : s → v are defined by f(∅) = ∅, f({1}) = {a}, f({2}) = {b}, f(s) = {a, b} g(∅) = {a, b}, g({1}) = {b}, g({2}) = {a}, g(s) = ∅ then (f, g) is a regular (v, v)-graded difilter. moreover, f and g are prime. the structure of graded difilter is more general than the structure of difilter. most of the properties of difilters can be generalized to the graded case and it can be expected that graded difilters satisfy these generalized properties. but this is not possible in each case. for instance, the statements (1) − (4) in proposition 2.6. are equivalent for difilters however the generalizations of c© agt, upv, 2016 appl. gen. topol. 17, no. 1 27 r. ekmekçi and r. ertürk these statements are not always equivalent for graded difilters as in the next example. definition 3.15. let (f, g) be a (v, v)-graded difilter on (s, s). (f, g) is called a maximal (v, v)-graded difilter on (s, s) if whenever (f′, g′) is a (v, v)graded difilter on (s, s) and (f, g) ⊆ (f′, g′) then we have (f, g) = (f′, g′) is hold. example 3.16. let s = {1, 2}, s = {∅, {1}, {2}, s}, v = {a, b, c} and v = {∅, {b}, {c}, {b, c}, v }. in this case, (s, s) and (v, v) are plain texture spaces. if the mappings f, g : s → v are defined by f(∅) = ∅, f({1}) = {b}, f({2}) = {c}, f(s) = v g(∅) = v , g({1}) = {c}, g({2}) = {b}, g(s) = ∅ then (f, g) is a regular (v, v)-graded difilter. moreover, (f, g) is a maximal regular (v, v)-graded difilter but f ∨ g 6= v . (example for (1) ; (2) in proposition 3.17.) proposition 3.17. let (f, g) be a regular (v, v)-graded difilter on (s, s). for the statements (1) (f, g) is a maximal regular (v, v)-graded difilter (2) f ∨ g = v (i.e. ∀a ∈ s, f(a) ∨ g(a) = f(a) ∪ g(a) = v ) (3) f is prime and g = v \ f (4) g is prime and f = v \ g the following implications are hold: (1) ⇐ (2) ⇔ (3) ⇔ (4), (1) ; (2) proof. (1) ⇐ (2): let (f, g) be a regular (v, v)-graded difilter on (s, s) and f ∨ g = v . from regularity of (f, g) we have f ∩ g = ∅. considering f ∨ g = f ∪ g = v we get g = v \ f and f = v \ g. if (f′, g′) is a regular (v, v)-graded difilter on (s, s) and (f, g) ⊆ (f′, g′) then considering f′ ∩ g′ = ∅ we get f(a) = v \ g(a) ⊇ v \ g′(a) ⊇ f′(a) ⊇ f(a) g(a) = v \ f(a) ⊇ v \ f′(a) ⊇ g′(a) ⊇ g(a). thus, (f, g) = (f′, g′) is obtained. so, (f, g) is a maximal regular (v, v)graded difilter on (s, s). (2) ⇔ (3): if f ∨ g = v then from regularity of (f, g), we have f ∩ g = ∅ and so g = v \ f. suppose that f is not prime. then, there exist a1, a2 ∈ s such that f(a1 ∪ a2) * f(a1) ∪ f(a2). thus it is obtained that g(a1) ∩ g(a2) = v \ f(a1) ∩ v \ f(a2) = v \ (f(a1) ∪ f(a2)) * v \ f(a1 ∪ a2) = g(a1 ∪ a2) which contradicts with (gcf3). therefore f is prime. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 28 convergence in graded ditopological texture spaces on the other hand, if g = v \ f then f ∨ g = f ∪ g = v is obtained. (2) ⇔ (4): similar as (2) ⇔ (3). (1) ; (2): example 3.16. � proposition 3.18. a texture space (v, v) is discrete if and only if pv * a ⇒ pv ∩ a = ∅ for all v ∈ v , a ∈ v. proof. let pv * a ⇒ pv ∩ a = ∅ for all v ∈ v , a ∈ v and suppose that (v, v) is not discrete. then there exists v ∈ v such that pv 6= {v}. so, t ∈ pv, t 6= v for some t ∈ v . since textures are point separating lattices, we have v 6∈ pt and pv * pt. considering {v} ⊆ pv ∩ pt 6= ∅, if we take a = pt in the hypothesis then we get a contradiction. thus, (v, v) is discrete. on the other hand, if (v, v) is discrete then pv = {v} and so pv * a ⇒ pv ∩ a = ∅ for all v ∈ v , a ∈ v. � the generalizations of the equivalent statements in proposition 2.6. to the graded case are equivalent if (v, v) is discrete as in the next theorem. hence the concepts studied and results obtained in this paper are more general than those in [11]. theorem 3.19. if (f, g) is a regular (v, v)-graded difilter on (s, s) and (v, v) is discrete then the statements (1) − (4) in proposition 3.17 are equivalent. proof. it is sufficient to show the implication (1) ⇒ (2). let (f, g) be a maximal regular (v, v)-graded difilter on (s, s) and suppose that f ∨ g 6= v . then there exists a0 ∈ s such that f(a0) ∪ g(a0) 6= v and so there exists v ∈ v such that pv = {v} * f(a0) ∪ g(a0). there are two cases: ”∀a, b ∈ s, pv * f(a) ∩ g(b)” or ”∃a, b ∈ s : pv ⊆ f(a) ∩ g(b)”. case 1: ”∀a, b ∈ s, pv * f(a) ∩ g(b)”. in particular, if we take a = s and b = ∅ then pv * f(s) ∩ g(∅). so, there are two subcases: pv * f(s) or pv * g(∅). case 1.1: ”pv * f(s)”. since from (gf2) f(a) ⊆ f(s) for all a ∈ s, we have pv * f(a) for all a ∈ s. if we define a mapping f∗ : s → v by f ∗(b) =    ∅, b = ∅ f(b), a0 * b f(b) ∪ pv, a0 ⊆ b then (f∗, g) is a regular (v, v)-graded difilter on (s, s): (gf1) f∗(∅) = ∅ (gf2) let b1, b2 ∈ s and b1 ⊆ b2. if b1 = ∅ then we have f∗(b1) = ∅ ⊆ f∗(b2). if b1 6= ∅ then b2 6= ∅ and so, (a) if a0 ⊆ b1 then a0 ⊆ b2 and so f ∗(b1) = pv ∪ f(b1) ⊆ pv ∪ f(b2) = f∗(b2). (b) if a0 * b1 then f∗(b1) = f(b1) ⊆ f(b2) ⊆ f∗(b2). (gf3) let b1, b2 ∈ s. if b1 = ∅ or b2 = ∅ then we have f∗(b1) ∧ f∗(b2) = ∅ ⊆ f∗(b1 ∩ b2). if b1, b2 6= ∅ then c© agt, upv, 2016 appl. gen. topol. 17, no. 1 29 r. ekmekçi and r. ertürk (a) if a0 * b1, b2 then a0 * b1 ∩ b2 and f∗(b1) ∧ f∗(b2) = f(b1) ∩ f(b2) ⊆ f(b1 ∩ b2) = f ∗(b1 ∩ b2). (b) if a0 ⊆ b1, b2 then a0 ⊆ b1∩b2 and f ∗(b1)∧f ∗(b2) = (pv ∪f(b1))∩ (pv ∪f(b2)) = pv ∪(f(b1)∩f(b2)) ⊆ pv ∪f(b1 ∩b2) = f ∗(b1 ∩b2). (c) without loss of generality let a0 * b1 and a0 ⊆ b2. then a0 * b1 ∩ b2. considering that pv * f(a) for all a ∈ s and v is discrete we have f(b1) ∩ pv = ∅. thus we get f ∗(b1) ∧ f ∗(b2) = f(b1) ∩ (pv ∪ f(b2)) = (f(b1) ∩ pv) ∪ (f(b1) ∩ f(b2)) = f(b1) ∩ f(b2) ⊆ f(b1 ∩ b2) = f ∗(b1 ∩ b2). let b ∈ s. then since (f, g) is regular; (a) if a0 ⊆ b then, since v is discrete, g(b) ⊆ g(a0) and pv * f(a0) ∪ g(a0) it is obtained that pv * g(b) and so we have pv ∩ g(b) = ∅. therefore we get f∗(b)∩g(b) = (pv ∪f(b))∩g(b) = (pv ∩g(b))∪ (f(b) ∩ g(b)) = ∅, i.e. f∗(b) ∩ g(b) = ∅. (b) if a0 * b then we get f∗(b) ∩ g(b) = f(b) ∩ g(b) = ∅. therefore (f∗, g) is a regular (v, v)-graded difilter on (s, s). however, (f, g) $ (f∗, g) (at least f∗(a0) = pv ∪ f(a0) 6= f(a0)) and this contradicts with the maximality of (f, g). hence, the implication (1) ⇒ (2) is satisfied. case 1.2: ”pv * g(∅)”. since from (gcf2) g(a) ⊆ g(∅) for all a ∈ s, we have pv * g(a) for all a ∈ s. if we define a mapping g∗ : s → v by g ∗(b) =    ∅, b = s g(b), b * a0 g(b) ∪ pv, b ⊆ a0 then (f, g∗) is a regular (v, v)-graded difilter on (s, s) (it can be shown like in case 1.1). however, (f, g) $ (f, g∗) (at least g∗(a0) = pv ∪g(a0) 6= g(a0)) and this contradicts with the maximality of (f, g). hence, the implication (1) ⇒ (2) is satisfied. case 2: ”∃c, d ∈ s, pv ⊆ f(c) ∩ g(d)”. if we suppose that a0 = s then since c ⊆ a0 we have pv ⊆ f(c) ⊆ f(a0) which contradicts with pv * f(a0). similarly if we suppose that a0 = ∅ then since a0 ⊆ d we have pv ⊆ g(d) ⊆ g(a0) which contradicts with pv * g(a0). thus we get that a0 6= ∅, s. now, we show that ∀a, b ∈ s : pv ⊆ f(a) ∩ g(b) ⇒ a0 ∩ a * b or ∀a, b ∈ s : pv ⊆ f(a) ∩ g(b) ⇒ a * a0 ∪ b. contrary, if we assume that there exist a1, b1, a2, b2 ∈ s such that ”pv ⊆ f(a1)∩g(b1), a0∩a1 ⊆ b1” and ”pv ⊆ f(a2)∩g(b2), a2 ⊆ a0∪b2” then we have pv ⊆ f(a1)∩f(a2) ⊆ f(a1 ∩a2) and pv ⊆ g(b1)∩g(b2) ⊆ g(b1 ∪b2) and so f(a1 ∩ a2) ∩ g(b1 ∪ b2) 6= ∅. since (f, g) is regular, from proposition 3.2. (3) we get a1 ∩ a2 * b1 ∪ b2. hence there exists s ∈ a1 ∩ a2 such that c© agt, upv, 2016 appl. gen. topol. 17, no. 1 30 convergence in graded ditopological texture spaces s 6∈ b1 ∪ b2. considering s ∈ a2, s 6∈ b2 and a2 ⊆ a0 ∪ b2 we have s ∈ a0. now, considering s ∈ a0, s ∈ a1 and a0 ∩ a1 ⊆ b1 we have s ∈ b1 which contradicts with s 6∈ b1 ∪ b2. therefore we have again two subcases. case 2.1:”∀a, b ∈ s : pv ⊆ f(a) ∩ g(b) ⇒ a0 ∩ a * b”. if we define a mapping f′ : s → v by f ′(b) =    ∅, b = ∅ f(b), b 6= ∅ and ”pv ⊆ f(a) for all a ∈ s ⇒ a0 ∩ a * b” f(b) ∪ pv, b 6= ∅ and ”∃a ∈ s : pv ⊆ f(a) and a0 ∩ a ⊆ b” then (f′, g) is a regular (v, v)-graded difilter on (s, s): (gf1) f′(∅) = ∅ (gf2) let b1, b2 ∈ s and b1 ⊆ b2. if b1 = ∅ then we have f′(b1) = ∅ ⊆ f′(b2). if b1 6= ∅ then b2 6= ∅ and so, (a) let ”∃a ∈ s : pv ⊆ f(a) and a0 ∩a ⊆ b1”. then a0 ∩a ⊆ b1 ⊆ b2 and so f′(b1) = pv ∪ f(b1) ⊆ pv ∪ f(b2) = f ′(b2). (b) let ”pv ⊆ f(a) ⇒ a0 ∩ a * b1 for all a ∈ s”. then f′(b1) = f(b1) ⊆ f(b2) ⊆ f ′(b2). (gf3) let b1, b2 ∈ s. if b1 = ∅ or b2 = ∅ then we have f′(b1) ∧ f′(b2) = ∅ ⊆ f′(b1 ∩ b2). if b1, b2 6= ∅ then (a) let ”∃a1 ∈ s : pv ⊆ f(a1), a0 ∩ a1 ⊆ b1” and ”∃a2 ∈ s : pv ⊆ f(a2), a0 ∩ a2 ⊆ b2”. then a1 ∩ a2 ∈ s, a0 ∩ (a1 ∩ a2) ⊆ b1 ∩ b2 and so, f′(b1) ∧ f ′(b2) = (pv ∪ f(b1)) ∩ (pv ∪ f(b2)) = pv ∪ (f(b1) ∩ f(b2)) ⊆ pv ∪ f(b1 ∩ b2) = f ′(b1 ∩ b2). (b) let ”pv ⊆ f(a) ⇒ a0 ∩ a * b1” and ”pv ⊆ f(a) ⇒ a0 ∩ a * b2”. then f′(b1) ∧ f ′(b2) = f(b1) ∩ f(b2) ⊆ f(b1 ∩ b2) ⊆ f ′(b1 ∩ b2). (c) without loss of generality let ”∃a1 ∈ s : pv ⊆ f(a1), a0 ∩ a1 ⊆ b1” and ”pv ⊆ f(a) ⇒ a0 ∩ a * b2”. since pv ⊆ f(b2) implies the contradiction a0 ∩b2 * b2 we have pv * f(b2). then pv ∩f(b2) = ∅ because v is discrete. moreover, since ”pv ⊆ f(a) ⇒ a0 ∩ a * b2 we have ”pv ⊆ f(a) ⇒ a0 ∩a * b1 ∩b2 and so f′(b1 ∩b2) = f(b1 ∩b2). thus we get f ′(b1) ∧ f ′(b2) = (pv ∪ f(b1)) ∩ f(b2) = (pv ∩ f(b2)) ∪ (f(b1) ∩ f(b2)) = f(b1) ∩ f(b2) ⊆ f(b1 ∩ b2) = f ′(b1 ∩ b2). let b ∈ s. if b = ∅ then f′(b) ∩ g(b) = ∅. so, assume that b 6= ∅. then since (f, g) is regular; (a) if ”∃a ∈ s : pv ⊆ f(a) and a0 ∩ a ⊆ b” then, because of the implication of case 2.1. we have pv * g(b). since v is discrete we get pv ∩g(b) = ∅. therefore we get f′(b)∩g(b) = (pv ∪f(b))∩g(b) = (pv ∩g(b))∪(f(b)∩g(b)) = ∅∪(f(b)∩g(b)) = f(b)∩g(b) = ∅. that is f′(b) ∩ g(b) = ∅. (b) if ”pv ⊆ f(a) ⇒ a0 ∩ a * b” then we get f′(b) ∩ g(b) = f(b) ∩ g(b) = ∅. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 31 r. ekmekçi and r. ertürk therefore (f′, g) is a regular (v, v)-graded difilter on (s, s). however, since pv ⊆ f(c) and a0∩c ⊆ a0 we have f ′(a0) = pv ∪f(a0) and since pv * f(a0) we get f′ 6= f. thus (f, g) $ (f′, g) and this contradicts with the maximality of (f, g). hence, the implication (1) ⇒ (2) is satisfied. case 2.2:”∀a, b ∈ s : pv ⊆ f(a) ∩ g(b) ⇒ a * a0 ∪ b”. if we define a mapping g′ : s → v by g ′(a) =    ∅, a = s g(a), a 6= s and ”pv ⊆ g(b) for all b ∈ s ⇒ a * a0 ∪ b” g(a) ∪ pv, a 6= s and ”∃b ∈ s : pv ⊆ g(b) and a ⊆ a0 ∪ b” then (f, g′) is a regular (v, v)-graded difilter on (s, s) (it can be shown like in case 2.1). however, (f, g) $ (f, g′) and this contradicts with the maximality of (f, g). hence, the implication (1) ⇒ (2) is satisfied. � corollary 3.20. if (f, g) is a maximal regular (v, v)-graded difilter on (s, s) and (v, v) is discrete then f(s) = g(∅) = v . proof. considering theorem 3.19., since (f, g) is maximal regular we get f ∨ g = v . suppose that f(s) 6= v or g(∅) 6= v . then we have f(s)∪g(s) 6= v or f(∅) ∪ g(∅) 6= v which contradicts with f ∨ g = v . � it is obtained in [11] that if f is a prime filter on a texture (s, s) then s \f is a prime cofilter and f × (s \ f) is a maximal regular difilter on the same texture. however the generalization of this statement is not true for graded difilters. in example 3.14., f is a prime (v, v)-graded filter on (s, s) and the texture (v, v) is discrete but v \f isn’t a prime (v, v)-graded cofilter on (s, s) since f(s) 6= v and (v \ f)(s) 6= ∅. since (v, v) is discrete, by corollary 3.20. we get that there is no maximal regular (v, v)-graded difilter on (s, s) whose first component is f. proposition 3.21. let (f, g) be a regular (v, v)-graded difilter on (s, s). then there exists a maximal regular (v, v)-graded difilter (fm , gm) on (s, s) such that (f, g) ⊆ (fm , gm ). proof. let (fj, gj)j∈j be a chain of regular (v, v)-graded difilters on (s, s) which satisfies (f, g) ⊆ (fj, gj) for all j ∈ j. if the mappings f ′, g′ : s → v are defined by f′ = ∨ j∈j fj and g ′ = ∨ j∈j gj then (f ′, g′) is a regular (v, v)graded difilter on (s, s): (gf1) f′(∅) = ∨ j∈j fj(∅) = ∅ (gf2) let a1, a2 ∈ s and a1 ⊆ a2. since fj(a1) ⊆ fj(a2) for each j ∈ j we get f′(a1) = ∨ j∈j fj(a1) ⊆ ∨ j∈j fj(a2) = f ′(a2). (gf3) let a1, a2 ∈ s. consider the sets ji1 = {j ∈ j | (fi, gi) ⊆ (fj, gj)} and ji2 = {j ∈ j | (fi, gi) % (fj, gj)} for each i ∈ j. since (fj, gj)j∈j is a c© agt, upv, 2016 appl. gen. topol. 17, no. 1 32 convergence in graded ditopological texture spaces chain we have ji1 ∪ ji2 = j for each i ∈ j and so we get f ′(a1) ∧ f ′(a2) = ∨ i∈j fi(a1) ∧ ∨ j∈j fj(a2) = ∨ i∈j (fi(a1) ∧ ∨ j∈j fj(a2)) = ∨ i∈j ( ∨ j∈j (fi(a1) ∧ fj(a2))) = ∨ i∈j ( ∨ j∈j i1 (fi(a1) ∧ fj(a2)) ∨ ∨ j∈j i2 (fi(a1) ∧ fj(a2))) ⊆ ∨ i∈j ( ∨ j∈j i1 (fj(a1) ∧ fj(a2)) ∨ ∨ j∈j i2 (fi(a1) ∧ fi(a2))) ⊆ ∨ i∈j ( ∨ j∈j i1 (fj(a1 ∩ a2)) ∨ ∨ j∈j i2 (fi(a1 ∩ a2))) = ∨ i∈j fi(a1 ∩ a2) = f ′(a1 ∩ a2). thus f′ is a (v, v)-graded filter on (s, s) and similarly it can be shown that g′ is a (v, v)-graded cofilter on (s, s). now, we use the similar method as in (gf3) to show that (f′, g′) is regular. so, consider the sets ji1, ji2 for each i ∈ j as above. let a ∈ s. since (fj, gj) is regular for each j ∈ j we obtain that f ′(a) ∩ g′(a) = ∨ i∈j fi(a) ∩ ∨ j∈j gj(a) = ∨ i∈j (fi(a) ∩ ∨ j∈j gj(a)) = ∨ i∈j ( ∨ j∈j (fi(a) ∩ gj(a))) = ∨ i∈j ( ∨ j∈j i1 (fi(a) ∩ gj(a)) ∨ ∨ j∈j i2 (fi(a) ∩ gj(a))) ⊆ ∨ i∈j ( ∨ j∈j i1 (fj(a) ∩ gj(a)) ∨ ∨ j∈j i2 (fi(a) ∧ gi(a))) = ∅. therefore (f′, g′) is an upper bound for the chain (fj, gj)j∈j in the set z = {(u, r) | (u, r) is a regular (v, v)−graded difilter on (s, s) and (f, g) ⊆ (u, r)}. by zorn’s lemma the set z has a maximal element (fm , gm ). hence (f, g) ⊆ (fm , gm ) and we obtain that (fm , gm) is maximal in the set of all regular (v, v)-graded difilters on (s, s). � proposition 3.22. let (v, v) be a discrete texture space and (f, g) a maximal regular graded difilter on (s, s, t , k, v, v). then (f, g) is diconvergent if and only if it is diclustering. proof. if (f, g) is diconvergent then it is diclustering by corollary 3.12. on the other hand, let (f, g) be diclustering. namely let s be cluster point of f, s′ cluster point of g where ps * qs′. then v ∈ f(a) ⇒ ps ⊆ [a]v and v ∈ g(a) ⇒]a[v⊆ qs′ for all a ∈ s. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 33 r. ekmekçi and r. ertürk let a ∈ s with a * qs′ and v 6∈ f(a). by theorem 3.19. we have f(a) ∪ g(a) = v . so, considering regularity of (f, g) we get v ∈ g(a) and so ]a[v⊆ qs′. hence v 6∈ t (a) because v ∈ t (a) implies a =]a[ v⊆ qs′ which contradicts with a * qs′. thus we have v 6∈ f(a) ⇒ v 6∈ t (a), i.e. t (a) ⊆ f(a). therefore, considering proposition 3.7. we get f → s′. using similar method, it can be obtained that g → s. so (f, g) is diconvergent. � proposition 3.23. let (s, s, t , k, v, v) be a graded ditopological texture space. then, for the staements (a) every regular graded difilter on (s, s, t , k, v, v) is diclustering. (b) every maximal regular graded difilter on (s, s, t , k, v, v) is diconvergent. the implication (b) ⇒ (a) and in case of (v, v) is discrete, (a) ⇒ (b) are hold. proof. (a) ⇒ (b): let (f, g) be a maximal regular graded difilter on (s, s, t , k, v, v). from (a), (f, g) is diclustering. considering that (v, v) is discrete and proposition 3.22. (f, g) is diconvergent. (b) ⇒ (a): let (f, g) be a regular graded difilter on (s, s, t , k, v, v). considering proposition 3.21., there exists a maximal regular graded difilter (fm , gm ) on (s, s, t , k, v, v) with (f, g) ⊆ (fm , gm). from (b) we have fm → s, gm → s′ and ps′ * qs for some s, s′ ∈ s. considering gm → s′ and proposition 3.10. we have v ∈ f(a) ⇒ v ∈ fm (a) ⇒ ps′ ⊆ [a] v and so we get that s′ is a cluster point of f. similarly it can be obtained that s is a cluster point of g. thus (f, g) is diclustering. � 4. conclusion filters and their convergence are convenient tools for topological spaces as filter convergence can describe some of topological concepts. in this work, graded difilters are introduced and their convergence which characterizes interior, closure of sets, etc. is investigated. this new sturucture is helpful to deal with the theory of graded ditopological texture spaces. moreover, the relations between difilters and graded difilters are studied. as expected, graded difilters are based on graded dinhd systems. graded dinhd systems are not graded difilters in general so, the method used in [11] is used to define the convergence of graded difilters. obviously, graded difilters are more general than difilters and naturally some properties of difilters are not valid for graded difilters in general (see prop. 3.17. and theorem 3.19.). clearly, graded difilters and their convergence can be useful to define and investigate the concept of compactness on graded ditopological texture spaces. acknowledgements. the authors would like to thank the referees for their helpful suggestions and comments. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 34 convergence in graded ditopological texture spaces references [1] l. m. brown and m. diker, ditopological texture spaces and intuitionistic sets, fuzzy sets and systems 98 (1998), 217–224. [2] l. m. brown and r. ertürk, fuzzy sets as texture spaces, i. representation theorems, fuzzy sets and systems 110, no. 2 (2000), 227–236. [3] l. m. brown, r. ertürk and ş. dost, ditopological texture spaces and fuzzy topology, i. basic concepts, fuzzy sets and systems147, no. 2 (2004), 171–199. [4] l. m. brown, r. ertürk and ş. dost, ditopological texture spaces and fuzzy topology, ii. topological considerations, fuzzy sets and systems 147, no. 2 (2004), 201–231. [5] l. m. brown, r. ertürk and ş. dost, ditopological texture spaces and fuzzy topology, iii. separation axioms, fuzzy sets and systems 157, no. 14 (2006), 1886–1912. [6] l. m. brown and a. šostak, categories of fuzzy topology in the context of graded ditopologies on textures, iranian journal of fuzzy systems 11, no. 6 (2014), 1–20. [7] c. l. chang, fuzzy topological spaces, j. math. anal. appl. 24 (1968), 182–190. [8] r. ekmekçi and r. ertürk, neighborhood structures of graded ditopological texture spaces, filomat 29, no. 7 (2015), 1445–1459. [9] r. ertürk, separation axioms in fuzzy topology characterized by bitopologies, fuzzy sets and systems 58 (1993), 206–209. [10] t. kubiak, on fuzzy topologies, ph.d. thesis, a. mickiewicz university poznan, poland, 1985. [11] s. özçağ, f. yıldız and l. m. brown, convergence of regular difilters and the completeness of di-uniformities, hacettepe journal of mathematics and statistics 34s (2005), 53–68. [12] a. šostak, on a fuzzy topological structure, rend. circ. matem. palermo, ser. ii, 11 (1985), 89–103. [13] a. šostak, two decades of fuzzy topology: basic ideas, notions and results, russian mathematical surveys 44, no. 6 (1989) 125–186. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 35 @ applied general topology c© universidad politécnica de valencia volume 4, no. 1, 2003 pp. 157–192 di-uniform texture spaces selma özçağ and lawrence m. brown ∗ abstract. textures were introduced by the second author as a point-based setting for the study of fuzzy sets, and have since proved to be an appropriate framework for the development of complement-free mathematical concepts. in this paper the authors lay the foundation for a theory of uniformities in a textural context. analogues are given for both the diagonal and covering approaches to the classical theory of uniform structures, the notion of uniform topology is generalized and an analogue given for the well known result that a topological space is uniformizable if and only if it is completely regular. finally a textural analogue of the classical interplay between uniformities and families of pseudo-metrics is presented. 2000 ams classification: primary: 54e15, 54a05. secondary: 06d10, 03e20, 54a40, 54d10, 54d15, 54e35. keywords: uniformity, texturing, direlation, dicover, difunction, di-uniform texture space, uniform ditopology, uniform bicontinuity, initial di-uniformity, separation, dimetric, complement-free, point-free, fuzzy set. 1. introduction textures were introduced by the second author as a point-based setting for the study of fuzzy sets, and have since proved to be an appropriate setting for the development of complement-free mathematical concepts. in this paper the authors lay the foundation for a theory of uniformities imposed on textures. analogues are given for both the diagonal and covering approaches to the classical theory of uniform structures, the notion of uniform topology is generalized and an analogue given for the well known result that a topological space is uniformizable if and only if it is completely regular. finally the notion of pseudo dimetric is given and a pseudo metrization theorem for di-uniformities and for ditopologies is presented. ∗dedicated to the memory of professor doğan çoker. 158 s. özçağ and l. m. brown let s be a non-empty set. we recall [2] that a texturing on s is a point separating, complete, completely distributive lattice s of subsets of s with respect to inclusion, which contains s, ∅, and for which arbitrary meet ∧ coincides with intersection ⋂ and finite joins ∨ coincide with unions ∪. the pair (s, s) is then called a texture. in general a texturing of s need not be closed under set complementation. the sets ps = ⋂ {a ∈ s | s ∈ a}, qs = ∨ {pu | u ∈ s, s /∈ pu}, s ∈ s, are important in the study of textures, and the following facts concerning these so called p–sets and q–sets will be used extensively below. lemma 1.1. [1] (1) s /∈ a =⇒ a ⊆ qs =⇒ s /∈ a[ for all s ∈ s, a ∈ s. (2) a[ = {s | a 6⊆ qs} for all a ∈ s. (3) for ai ∈ s, i ∈ i we have ( ∨ i∈i ai) [ = ⋃ i∈i a [ i. (4) a is the smallest element of s containing a[ for all a ∈ s. (5) for a, b ∈ s, if a 6⊆ b then there exists s ∈ s with a 6⊆ qs and ps 6⊆ b. (6) a = ⋂ {qs | ps 6⊆ a} for all a ∈ s. (7) a = ∨ {ps | a 6⊆ qs} for all a ∈ s. here a[ is defined by a[ = ⋂{⋃ {ai | i ∈ i} | {ai | i ∈ i}⊆ s, a = ∨ {ai | i ∈ i} } and known as the core of a ∈ s. the above lemma exposes an important formal duality in (s, s), namely that between ⋂ and ∨ , qs and ps, and ps 6⊆ a and a 6⊆ qs. indeed, it is to emphasize this duality that we normally write ps 6⊆ a in preference to s /∈ a. lemma 1.1 (5) is particularly useful in establishing inclusion by reductio ad absurdum, and will be used without comment in the sequel. the simplest example of a texture is (x,p(x)), for which px = {x} and qx = x \ {x}, x ∈ x. a natural texturing of the unit interval i = [0, 1] is defined by i = {[0,r) | r ∈ i}∪{[0,r] | r ∈ i}. for the texture (i, i) we have pr = [0,r] and qr = [0,r), r ∈ i. this texture will prove useful in the later sections. both (x,p(x)) and (i, i) have the property that join coincides with union (equivalently, that ps 6⊆ qs for all s), but certainly this is not the case in general. the definition of a diagonal uniformity on a set s involves binary relations on s, but the standard theory of binary relations and functions is largely inappropriate for general textures (s, s) because of their lack of symmetry. with this in mind, the second author has recently introduced notions of relation and corelation [1] for textures, based on the duality mentioned above. it is shown di-uniform texture spaces 159 in [1] that, working in terms of direlations, which are pairs consisting of a relation and a corelation, a theory is obtained which resembles in many important respects that of classical binary relations and functions. it will be appropriate, therefore, to base our textural analogue of a diagonal uniformity on the concept of direlation and for the convenience of the reader we recall some basic definitions and results from [1]. the reader is referred to [1] for more details, motivation and examples. for textures (s, s), (t, t) we denote by s ⊗ t the product texturing of s × t [4]. thus, s ⊗ t consists of arbitrary intersections of sets of the form (a × t) ∪ (s × b), a ∈ s, b ∈ t. for s ∈ s, ps and qs will always denote the p-sets and q-sets for the texture (s, s), while for t ∈ t , pt and qt will denote the p-sets and q-sets for (t, t). we reserve the notation p(s,t), q(s,t), s ∈ s, t ∈ t , for the p-sets, q-sets in (s ×t, s ⊗ t). on the other hand, p (s,t) and q(s,t) will denote the p-sets and q-sets for the texture (s ×t, p(s) ⊗ t). hence (see [1]) we have p (s,t) = {s}×pt and q(s,t) = [(s\{s})×t ]∪ [s×qt]. likewise, p (t,s) and q(t,s) are the p-sets and q-sets for (t ×s, p(t)⊗s). it is easy to verify that p (s,t) 6⊆ q(s′,t′) ⇐⇒ s = s′ and pt 6⊆ qt′. again, we will use this fact, and its companion p (t,s) 6⊆ q(t′,s′) ⇐⇒ t = t′ and ps 6⊆ qs′, without comment in what follows. now let us recall: definition 1.2. [1] let (s, s), (t, t) be textures. then (1) r ∈p(s) ⊗ t is called a relation on (s, s) to (t, t) if it satisfies r1 r 6⊆ q(s,t),ps′ 6⊆ qs =⇒ r 6⊆ q(s′,t). r2 r 6⊆ q(s,t) =⇒ ∃s′ ∈ s such that ps 6⊆ qs′ and r 6⊆ q(s′,t). (2) r ∈p(s) ⊗ t is called a co-relation on (s, s) to (t, t) if it satisfies cr1 p (s,t) 6⊆ r,ps 6⊆ qs′ =⇒ p (s′,t) 6⊆ r. cr2 p (s,t) 6⊆ r =⇒ ∃s′ ∈ s such that ps′ 6⊆ qs and p (s′,t) 6⊆ r. (3) a pair (r,r), where r is a relation and r a co-relation on (s, s) to (t, t) is called a direlation on (s, s) to (t, t). normally, relations will be denoted by lower case and co-relations by upper case letters, as in the above definition. for direlations (p,p), (q,q) on (s, s) to (t, t) we write (p,p) v (q,q) if and only if p ⊆ q and q ⊆ p. for a general texture (s, s) we define i = is = ∨ {p (s,s) | s ∈ s} and i = is = ⋂ {q(s,s) | s ∈ s}. if we note that i 6⊆ q(s,t) ⇐⇒ ps 6⊆ qt and p (s,t) 6⊆ i ⇐⇒ pt 6⊆ qs then it is trivial to verify that i is a relation and i a co-relation on (s, s) to (s, s). we refer to (i,i) as the identity direlation on (s, s). a direlation (r,r) on (s, s) (that is, on (s, s) to (s, s)) is reflexive if r and r are reflexive, that is if (i,i) v (r,r). 160 s. özçağ and l. m. brown if (r,r) is a direlation on (s, s) to (t, t), the inverse (r,r)← = (r←,r←) of (r,r) is the direlation on (t, t) to (s, s) defined by r← = ⋂ {q(t,s) | r 6⊆ q(s,t)}, r← = ∨ {p (t,s) | p (s,t) 6⊆ r}. a direlation (r,r) on (s, s) is called symmetric if (r,r) = (r,r)←, that is if and only if r = r←. this notion of symmetry is quite different from the classical notion of symmetry for relations. however, as we will see, it will play the same role in the theory of textural uniformities as does classical symmetry in the theory of uniformities. definition 1.3. let (s, s), (t, t) be textures, r a relation and r a co-relation on (s, s) to (t, t). (1) for a ⊆ s the a–section of r is the element r(a) of t defined by r(a) = ⋂ {qt | ∀s, r 6⊆ q(s,t) =⇒ a ⊆ qs}∈ t. (2) for a ⊆ s the a–section of r is the element r(a) of t defined by r(a) = ∨ {pt | ∀s,p (s,t) 6⊆ r =⇒ ps ⊆ a}∈ t. (3) for b ⊆ t the b–presection of r (b–presection of r) is the b–section r←(b) ∈ s of the co-relation r← (respectively, the b–section r←(b) ∈ s of the relation r←) on (t, t) to (s, s). the following lemma gives formulae for directly calculating the presections. lemma 1.4. for a relation r, a co-relation r and b ⊆ t we have: (1) r←(b) = ∨ {ps | ∀t, r 6⊆ q(s,t) =⇒ pt ⊆ b}∈ s. (2) r←(b) = ⋂ {qs | ∀t, p (s,t) 6⊆ r =⇒ b ⊆ qt}∈ s. the following results from [1] will prove useful later on. lemma 1.5. for a direlation (r,r) on (s, s) to (t, t) we have (1) r 6⊆ q(s,t) ⇐⇒ p (t,s) 6⊆ r← and p (s,t) 6⊆ r ⇐⇒ r← 6⊆ q(t,s). (2) r 6⊆ q(s,t) ⇐⇒ r(ps) 6⊆ qt and p (s,t) 6⊆ r ⇐⇒ pt 6⊆ r(qs). proposition 1.6. with the notation as in definition 1.3: (1) for relations r1, r2 with r1 ⊆ r2, co-relations r1, r2 with r1 ⊆ r2, a1, a2 in s with a1 ⊆ a2 and b1, b2 in t with b1 ⊆ b2 we have r1(a1) ⊆ r2(a2), r1(a1) ⊆ r2(a2), r←2 (b1) ⊆ r←1 (b2) and r←2 (b1) ⊆ r←1 (b2). (2) for any relation r we have r(∅) = ∅, a ⊆ r←(r(a)) for a ∈ s and r(r←(b)) ⊆ b for b ∈ t. (3) for any co-relation r we have r(s) = t , r←(r(a)) ⊆ a for a ∈ s and b ⊆ r(r←(b)) for b ∈ t. (4) for the identity direlation (i,i) on (s, s) and a ∈ s we have i(a) = i(a) = a and hence i←(a) = i←(a) = a. di-uniform texture spaces 161 (5) if a relation r (co-relation r) on (s, s) is reflexive then for all a ∈ s we have a ⊆ r(a) (r(a) ⊆ a). (6) for a relation r and co-relation r on (s, s) to (t, t) we have r( ∨ j∈j aj) = ∨ j∈j r(aj) and r( ⋂ j∈j aj) = ⋂ j∈j r(aj) for any aj ∈ s, j ∈ j. (7) for a relation r and co-relation r on (s, s) to (t, t) we have r←( ⋂ j∈j bj) = ⋂ j∈j r←(bj) and r ←( ∨ j∈j bj) = ∨ j∈j r←(bj) for any bj ∈ t, j ∈ j. another important concept for direlations is that of composition. we recall the following: definition 1.7. [1] let (s, s), (t, t), (u, u) be textures. (1) if p is a relation on (s, s) to (t, t) and q a relation on (t, t) to (u, u) then their composition is the relation q ◦ p on (s, s) to (u, u) defined by q ◦p = ∨ {p (s,u) | ∃t ∈ t with p 6⊆ q(s,t) and q 6⊆ q(t,u)}. (2) if p is a co-relation on (s, s) to (t, t) and q a co-relation on (t, t) to (u, u) then their composition is the co-relation q ◦ p on (s, s) to (u, u) defined by q◦p = ⋂ {q(s,u) | ∃t ∈ t with p (s,t) 6⊆ p and p (t,u) 6⊆ q}. (3) with p, q; p, q as above, the composition of the direlations (p,p), (q,q) is the direlation (q,q) ◦ (p,p) = (q ◦p,q◦p). it is shown in [1] that the operation of taking the composition of direlations is associative, and that the identity direlations are identities for this operation. if (r,r) is a direlation on (s, s) then (r,r) ◦ (r,r) = (r ◦r,r◦r) is also a direlation on (s, s), which we denote by (r,r)2. we give the obvious meaning to (r,r)n for any n = 3, 4, . . .. the direlation (r,r) on (s, s) is called transitive if (r,r)2 v (r,r). we will also have occasion to consider the greatest lower bound of direlations. we give the definition for two direlations, but it may be extended in the obvious way to any family of direlations. definition 1.8. [1] let (p,p), (q,q) be direlations on (s, s) to (t, t). then puq = ∨ {p (s,t) | ∃v ∈ s with ps 6⊆ qv and p,q 6⊆ q(v,t)}, p tq = ⋂ {q(s,t) | ∃v ∈ s with pv 6⊆ qs and p (v,t) 6⊆ p,q}, and (p,p) u (q,q) = (puq,p tq). 162 s. özçağ and l. m. brown proposition 1.9. with the notation as in definition 1.8, (1) puq is a relation on (s, s) to (t, t). it is the greatest lower bound of p and q in the set of all relations on (s, s) to (t, t), ordered by inclusion. (2) p t q is a co-relation on (s, s) to (t, t). it is the least upper bound of p and q in the set of all co-relations on (s, s) to (t, t), ordered by inclusion. (3) the direlation (p,p) u (q,q) is the greatest lower bound of (p,p) and (q,q) on the set of all direlations on (s, s) to (t, t), ordered by the relation v. (4) (puq)← = p← tq← and (p tq)← = p← uq←. (5) for a ∈ s, (puq)(a) ⊆ p(a) ∩q(a) and p(a) ∪q(a) ⊆ (p tq)(a). (6) for b ∈ t, p←(b) ∪ q←(b) ⊆ (p u q)←(b) and (p t q)←(b) ⊆ p←(b) ∩q←(b). (7) let (p1,p1), (p2,p2) be direlations on (s, s) to (t, t) and (q1,q1), (q2,q2) direlations on (t, t) to (u, u). then ((q1,q1) u (q2,q2)) ◦ ((p1,p1) u (q2,q2)) v ((q1,q1) ◦ (p1,p1)) u ((q2,q2) ◦ (p2,p2)). the notion of difunction is derived from that of direlation as follows. definition 1.10. [1] let (f,f) be a direlation on (s, s) to (t, t). then (f,f) is called a difunction on (s, s) to (t, t) if it satisfies the following two conditions. df1 for s,s′ ∈ s, ps 6⊆ qs′ =⇒ ∃t ∈ t with f 6⊆ q(s,t) and p (s′,t) 6⊆ f. df2 for t,t′ ∈ t and s ∈ s, f 6⊆ q(s,t) and p (s,t′) 6⊆ f =⇒ pt′ 6⊆ qt. difunctions are preserved under composition. it is easy to see that the identity direlation (is,is) on (s, s) is in fact a difunction on (s, s) to (s, s). in this context we refer to (is,is) as the identity difunction on (s, s). if (f,f) : (s, s) → (t, t) is a difunction, a ∈ s, then f(a) is called the image and f(a) the co-image of a. likewise, for b ∈ t, f←(b) is called the inverse image and f←(b) the inverse co-image of b. it is shown in [1] that f←(b) = f←(b) for all b ∈ t, that is the inverse image and inverse co-image coincide. since a texturing is generally not closed under set complementation, when discussing topological concepts we cannot insist that closed sets should be the complement of open sets. this leads to the notion of a dichotomous topology, or ditopology for short [2]. this is a pair (τ,κ) of subsets of s, where the set of open sets τ satisfies (1) s, ∅ ∈ τ, (2) g1, g2 ∈ τ =⇒ g1 ∩g2 ∈ τ and (3) gi ∈ τ, i ∈ i =⇒ ∨ i gi ∈ τ, and the set of closed sets κ satisfies (1) s, ∅ ∈ κ, (2) k1, k2 ∈ κ =⇒ k1 ∪k2 ∈ κ and (3) ki ∈ κ, i ∈ i =⇒ ⋂ i ki ∈ κ. di-uniform texture spaces 163 the reader is referred to [2, 3, 6, 7] for some results on ditopological texture spaces and their relation with fuzzy topologies. a subset β of τ is called a base of τ if every set in τ can be written as a join of sets in β, while a subset β of κ is a base of κ if every set in κ can be written as an intersection of sets in β. for the unit interval texture (i, i) mentioned above, we may define a natural ditopology (τi,κi) by τi = {[0,s) | s ∈ i}∪{i}, κi = {[0,s] | s ∈ i}∪{∅}. continuity of difunctions is the subject of the following definition. definition 1.11. [6] let (sk, sk,τk,κk), k = 1, 2, be ditopological texture spaces and (f,f) a difunction on (s1, s1) to (s2, s2). then (1) (f,f) is continuous if g ∈ τ2 =⇒ f←(g) ∈ τ1. (2) (f,f) is cocontinuous if k ∈ κ2 =⇒ f←(k) ∈ κ1. (3) (f,f) is bicontinuous if it is continuous and cocontinuous. the reader is referred to [9] for general terms related to lattice theory. this paper comprises part of the first author’s research towards her phd thesis to be submitted to hacettepe university. we pause here to mention our motivation for introducing textures as a substrate for topology. ditopological texture spaces were conceived as a point-set setting for the study of fuzzy topology, and provide a unified setting for the study of topology, bitopology and fuzzy topology. some of the links with hutton spaces, l–fuzzy sets and topologies are expressed in a categorical setting in [6]. here it is the choice of bicontinuous difunctions for the morphisms on the textural side which makes possible a correspondence with the point-free concept of hutton space. despite the close links with fuzzy sets and topologies, the development of the theory of ditopological texture spaces has proceeded largely independently, and has concentrated on the development of concepts which help to compensate for the possible lack of complementation. one such is that of direlation and difunction, another that of dicover ([2,3], see §2 below). both play a crucial role in this paper. if one takes the view that a texturing s can provide a much more economic computational model than p(s), it is important that we do not lose power in other directions. for example i is certainly much simpler than p(i), but if we consider only ordinary open covers (and closed cocovers) it is trivial that for the usual ditopology, every closed subset is compact (and every open set cocompact). however this is non-trivially equivalent [2] to the fact that every open, coclosed dicover has a finite, cofinite subcover and this, via a bitopological argument, can be shown to be equivalent to the compactness of i under its usual topology. hence this compactness property of (i, i) in its dicovering form is as powerful as that of i, and we will see later that with an appropriate di-uniformity (i, i) can again play the same role as i does in the usual theory of uniformities. 164 s. özçağ and l. m. brown duality is an important element in defining such concepts. when applied to ditopologies it often gives rise to pairs of properties, such as compact – cocompact, regular – coregular. in the case of uniform ditopologies, as we will see, it actually links the open and closed sets via symmetry, and this causes the ditopology to be simultaneously completely regular and completely coregular. a form of duality also plays a role in giovanni sambin’s basic picture for formal topology [11]. there are clear parallels here which warrant further study. likewise, links with the theory of locales and with domain theory have yet to be worked out. finally, complement free textural concepts can be expected to find applications in negation free logics, and indeed (ditopological) textures themselves could well prove to be useful models for certain classes of such logics. 2. direlations and dicovers as mentioned in the introduction, the entourages of a diagonal uniformity in the classical sense [13] will be replaced by direlations in the textural setting. a second important formulation of the theory of uniformities is that of the covering uniformity [12], so we will require an appropriate notion of cover in order to obtain an analogous description for textures. in this section we show that the notion of dicover, used in [2] to characterize the important form of compactness mentioned above and in [3] to describe various covering properties of ditopological texture spaces, is associated in a natural way with direlations. hence this notion will form the basis for our description of covering uniformities in the textural sense. let us recall [2,3] that by a dicover of the texture (s, s) we mean a family c = {(ai,bi) | i ∈ i} of elements of s × s which satisfies ⋂ i∈i1 bi ⊆ ∨ i∈i2 ai for all partitions (i1,i2) of i, including the trivial partitions. an important example is the family p = {(ps,qs) | s ∈ s[}, which is shown in [3] to be a dicover for any texture (s, s). if d is a dicover we often write l d m in place of (l,m) ∈ d. we recall the following notions for dicovers given in [3]. (1) c is a refinement of d if given i ∈ i we have l d m so that ai ⊆ l and m ⊆ bi. in this case we write c ≺ d. (2) the star and co-star of c ∈ s with respect to c are respectively the sets st(c,c) = ∨ {ai | i ∈ i, c 6⊆ bi}∈ s, and cst(c,c) = ⋂ {bi | i ∈ i, ai 6⊆ c}∈ s. we say that c is a delta refinement of d, and write c ≺(∆) d, if c∆ = {(st(c,ps), cst(c,qs)) | s ∈ s[}≺ d. we say that c is a star refinement of d, and write c ≺(?) d, if c? = {(st(c,ai), cst(c,bi)) | i ∈ i}≺ d. before describing the link between direlations and dicovers, it will be appropriate for us to define a particular class of dicovers that will arise naturally in this connection. di-uniform texture spaces 165 definition 2.1. a family c ⊆ s×s is called an anchored dicover if it satisfies: (1) p ⊆ c, and (2) given a c b there exists s ∈ s satisfying (a) a 6⊆ qu =⇒ ∃a′ c b′ with a′ 6⊆ qu and ps 6⊆ b′, and (b) pv 6⊆ b =⇒ ∃a′′ c b′′ with pv 6⊆ b′′ and a′′ 6⊆ qs. since p is a dicover, we see by (1) that an anchored dicover is a dicover. it is straightforward to verify that p itself is anchored. the notion of anchored dicover enables us to improve ([3], lemma 4.7 (3)). since these results will be useful later on we present the modified lemma in full. lemma 2.2. let c, d, e be dicovers on (s, s). (1) c ≺(?) d =⇒ c ≺ d. (2) if p ≺ c then c ≺(?) d =⇒ c ≺(∆) d (3) if c is anchored then (i) c ≺(∆) d =⇒ c ≺ d. (ii) c ≺(∆) d ≺(∆) e =⇒ c ≺(?) e. proof. (1) and (2) are proved in [3] so we concentrate on (3). (i). take a c b and s ∈ s as in definition 2.1 (2). it will suffice to show a ⊆ st(c,ps) and cst(c,qs) ⊆ b. if a 6⊆ st(c,ps) then we have u ∈ s with a 6⊆ qu and pu 6⊆ st(c,ps). by (2)(a) there exists a′ c b′ with a′ 6⊆ qu and ps 6⊆ b′. but then a′ ⊆ st(c,ps) so pu 6⊆ a′, which gives the contradiction a′ ⊆ qu. the inclusion cst(c,qs) ⊆ b is proved likewise. (ii). take a c b and for s ∈ s satisfying definition 2.1 (2), choose l d m so that st(d,ps) ⊆ l and m ⊆ cst(d,qs). it will suffice to show st(c,a) ⊆ l and m ⊆ cst(c,b). we prove the first inclusion, the second being dual. suppose st(c,a) 6⊆ l and take w ∈ s with st(c,a) 6⊆ qw and pw 6⊆ l. now we have a1 c b1 satisfying a1 6⊆ qw and a 6⊆ b1. let us choose u ∈ s with a 6⊆ qu and pu 6⊆ b1. by condition (2)(a) we have a′ c b′ with a′ 6⊆ qu and ps 6⊆ b′. choose u d v with st(c,pu) ⊆ u and v ⊆ cst(c,qu). then a1 ⊆ st(c,pu) ⊆ u and v ⊆ cst(c,qu) ⊆ b′. since ps 6⊆ b′ we now have ps 6⊆ v , and so u ⊆ st(d,ps) ⊆ l, whence a1 ⊆ l. a1 6⊆ qw and pw 6⊆ l now give a contradiction, and the proof is complete. � let us now show that we may associate an anchored dicover with each reflexive direlation (d,d) on (s, s). proposition 2.3. let (d,d) be a reflexive direlation on (s, s) and for s ∈ s let d[s] = d(ps) and d[s] = d(qs). then γ(d,d) = {(d[s],d[s]) | s ∈ s[} is an anchored dicover of (s, s). proof. set c = γ(d,d). since (d,d) is reflexive, ps ⊆ d(ps) = d[s] and d[s] = d(qs) ⊆ qs by proposition 1.6 (5)). hence, p ≺ c. let us associate s with d[s] c d[s] and take d[s] 6⊆ qu. now d[s] = d(ps) = d( ∨ {ps′ | ps 6⊆ qs′}) = ∨ {d(ps′) | ps 6⊆ qs′} by proposition 1.6 (6), so there 166 s. özçağ and l. m. brown exists s′ ∈ s with ps 6⊆ qs′ and d[s′] 6⊆ qu. since d[s′] ⊆ qs′ we also have ps 6⊆ d[s′], whence d[s′] c d[s′] satisfies the condition in definition 2.1 (2a). the proof of (2b) is dual to this. � let us denote by rdr the family of reflexive direlations and by adc the family of anchored dicovers on (s, s). the above proposition can now be seen as giving us a mapping γ : rdr → adc. proposition 2.4. for (d,d), (e,e) ∈ rdr we have (d,d)◦(d,d)← v (e,e) =⇒ γ(d,d) ≺(∆) γ(e,e). proof. we establish st(γ(d,d),ps) ⊆ e[s] for s ∈ s[. suppose this is not so. then we have z ∈ s with d[z] 6⊆ e[s] and ps 6⊆ d[z]. take t ∈ s with d[z] 6⊆ qt and pt 6⊆ e[s], and then t′ ∈ s satisfying d[z] 6⊆ qt′ and pt′ 6⊆ qt. from d[z] 6⊆ qt′ we obtain d 6⊆ q(z,t′), while ps 6⊆ d[z] implies p (z,s) 6⊆ d, and hence d← 6⊆ q(s,z). thus p (s,t′) ⊆ d← ◦d ⊆ e, so e 6⊆ q(s,t) which gives the contradiction pt ⊆ e[s]. in just the same way e[s] ⊆ cst(γ(d,d),qs), and the proof is complete. � now let us show that a dicover gives rise to a reflexive, symmetric direlation in a natural way. proposition 2.5. let c = {(aj,bj) | j ∈ j} be a dicover on (s, s) and define δ(c) = (d(c),d(c)) by d(c) = ∨ {p (s,t) | ∃j ∈ j with aj 6⊆ qt and ps 6⊆ bj}, d(c) = ⋂ {q(s,t) | ∃j ∈ j with pt 6⊆ bj and aj 6⊆ qs}. then δ(c) is a reflexive and symmetric direlation on (s, s). proof. write d = d(c), d = d(c) for short. first we verify that d is a relation on (s, s), leaving the proof that d is a co-relation to the reader. take s,t ∈ s with d 6⊆ q(s,t). then we have t′ ∈ s and j ∈ j satisfying p (s,t′) 6⊆ q(s,t), aj 6⊆ qt′ and ps 6⊆ bj. if ps′ 6⊆ qs then ps ⊆ ps′, whence ps′ 6⊆ bj and so p (s′,t′) ⊆ d, which gives d 6⊆ q(s′,t). this establishes r1. on the other hand, since ps 6⊆ bj, we have s′ ∈ s satisfying ps 6⊆ qs′ and ps′ 6⊆ bj. as before, d 6⊆ q(s′,t), which verifies r2. to show d is reflexive, suppose i 6⊆ d and take s,t ∈ s with i 6⊆ q(s,t) and p (s,t) 6⊆ d. then ps 6⊆ qt and for all j ∈ j we have aj ⊆ qt or ps ⊆ bj. put j1 = {j ∈ j | ps ⊆ bj} and let j2 = j \j1. then (j1,j2) is a partition of j, so ps ⊆ ⋂ j∈j1 bj ⊆ ∨ j∈j2 aj ⊆ qt, since c is a dicover. this gives the contradiction ps ⊆ qt, so d is reflexive. the proof that d is reflexive is dual to this. hence, (d,d) is reflexive. to show (d,d) is symmetric it will suffice to verify that d← = d. suppose that d← 6⊆ d and take u,v ∈ s satisfying d← 6⊆ q(u,v) and p (u,v) 6⊆ d. we have u′ ∈ s with pu′ 6⊆ qu and p (u′,v) 6⊆ d by cr2. there exists t ∈ s with p (u′,v) 6⊆ q(u′,t) and j ∈ j for which pt 6⊆ bj and aj 6⊆ qu′, whence p (t,u′) ⊆ d di-uniform texture spaces 167 and so d 6⊆ q(t,u). this is easily seen to be equivalent to p (u,t) 6⊆ d← and so d← ⊆ q(u,t). finally qt ⊆ qv, which gives the contradiction d← ⊆ q(u,v). finally, suppose d 6⊆ d← and take u,v ∈ s with d 6⊆ q(u,v) and p (u,v) 6⊆ d←. as above we have d 6⊆ q(v,u), whence t ∈ s and j ∈ j so that p (v,t) 6⊆ q(v,u), aj 6⊆ qt and pv 6⊆ bj. since qu ⊆ qt we also have aj 6⊆ qu, whence we have the contradiction d ⊆ q(u,v) from the definition of d. � if we denote by dc the set of dicovers and by srdr the set of symmetric reflexive direlations on (s, s), this proposition defines a mapping δ : dc → srdr. proposition 2.6. for c, d ∈ dc, c ≺(?) d =⇒ δ(c) ◦ δ(c) v δ(d). proof. suppose d(c) ◦ d(c) 6⊆ d(d). then we have s,u ∈ s so that p (s,u) 6⊆ d(d) and there exists t ∈ s satisfying d(c) 6⊆ q(s,t) and d(c) 6⊆ q(t,u). now we have t′ ∈ s so that pt′ 6⊆ qt and there exists a1cb1 for which a1 6⊆ qt′, ps 6⊆ b1. also we have u′ ∈ s so that pu′ 6⊆ qu and there exists a2cb2 for which a2 6⊆ qu′, pt 6⊆ b2. since c ≺ (?) d we may choose cde with st(c,a1) ⊆ c and e ⊆ cst(c,b1). hence, since we clearly have a1 6⊆ b2, a2 ⊆ st(c,a1) ⊆ c and e ⊆ cst(c,b1) ⊆ b1. thus c 6⊆ qu′ and ps 6⊆ e, and we obtain the contradiction p (s,u) ⊆ p (s,u′) ⊆ d(d). this establishes d(c) ◦d(c) ⊆ d(d), and the proof of d(d) ⊆ d(c) ◦d(c) is dual to this. � let us now discuss the relation between the mappings γ and δ. theorem 2.7. let (s, s) be a texture. with the notation above, (1) δ(γ(d,d)) = (d,d) ◦ (d,d) for all (d,d) ∈ srdr. (2) γ(δ(c)) = c∆ for all c ∈ dc. proof. (1). take (d,d) ∈ srdr and suppose d(γ(d,d)) 6⊆ d◦d. then we have s,t ∈ s satisfying p (s,t) 6⊆ d◦d for which we have z ∈ s[ satisfying d[z] 6⊆ qt and ps 6⊆ d[z]. however, d[z] 6⊆ qt ⇐⇒ d 6⊆ q(z,t) and ps 6⊆ d[z] ⇐⇒ p (z,s) 6⊆ d = d← ⇐⇒ d 6⊆ q(s,z) by lemma 1.5, since (d,d) is symmetric, and we obtain the contradiction p (s,t) ⊆ d◦d. conversely, suppose d◦d 6⊆ d(γ(d,d)). then we have s,u ∈ s with p (s,u) 6⊆ d(γ(d,d)) for which we have t ∈ s satisfying d 6⊆ q(s,t) and d 6⊆ q(t,u). firstly, d 6⊆ q(t,u) gives us d[t] 6⊆ qu. secondly, d 6⊆ q(s,t) implies t ∈ s[ and also gives p (t,s) 6⊆ d← = d since (d,d) is symmetric. hence we obtain ps 6⊆ d[t]. we deduce that p (s,u) ⊆ d(γ(d,d)), which is a contradiction. this completes the proof that d(γ(d,d)) = d◦d, and the proof of d(γ(d,d)) = d ◦d is dual to this, so δ(γ(d,d)) = (d,d) ◦ (d,d). (2). let c = {(ai,bi) | i ∈ i} and take s ∈ s[. suppose that d(c)[s] 6⊆ st(c,ps). then we have t ∈ s with d(c)[s] 6⊆ qt and pt 6⊆ st(c,ps). now we have w ∈ s with d(c) 6⊆ q(w,t) and ps 6⊆ qw. thus for some t′ ∈ s and i ∈ i, 168 s. özçağ and l. m. brown p (w,t′) 6⊆ q(w,t), ai 6⊆ qt′ and pw 6⊆ bi. we deduce that ps 6⊆ bi, and hence pt ⊆ pt′ ⊆ ai ⊆ st(c,ps), which is a contradiction. conversely, suppose st(c,ps) 6⊆ d(c)[s]. then we have i ∈ i with ai 6⊆ d(c)[s] and ps 6⊆ bi. hence for some t ∈ s we have ai 6⊆ qt and d(c) 6⊆ q(z,t) =⇒ ps ⊆ qz for all z ∈ s. take s′, t′ ∈ s satisfying ps 6⊆ qs′, ps′ 6⊆ bi and ai 6⊆ qt′, pt′ 6⊆ qt. now ai 6⊆ qt′, ps′ 6⊆ bi gives p (s′,t′) ⊆ d(c), so d(c) 6⊆ q(s′,t) and putting z = s′ in the above implication gives the contradiction ps ⊆ qs′. this completes the proof that d(c)[s] = st(c,ps). likewise, d(c)[s] = cst(c,qs), so γ(δ(c)) = c∆. � corollary 2.8. with the notation above: (1) (d,d) v δ(γ(d,d)) for all (d,d) ∈ rsdr. (2) c∆ is anchored for all dicovers c. if c is anchored then c ≺ γ(δ(c)). proof. (1). clear by theorem 2.7 (1) since (d,d) v (d,d)◦(d,d) when (d,d) is reflexive. (2). the first statement is clear from theorem 2.7 (2) and proposition 2.3. for the second we need only note that c ≺ (∆) c∆ and apply lemma 2.2 (1) when c is anchored to give c ≺ c∆. hence c ≺ γ(δ(c)) by theorem 2.7 (2). � let us recall from [3] that the meet of two dicovers c and d is the dicover c ∧ d = {(a∩c,b ∪d) | a c b, c d d}. as might be expected, this notion is closely related to that of the greatest lower bound for direlations. proposition 2.9. let (s, s) be a texture. with the notation above, (1) for (d,d), (e,e) ∈ rdr we have γ((d,d)u(e,e)) ≺ γ(d,d)∧γ(e,e). (2) for c, d ∈ dc we have δ(c ∧ d) v δ(c) uδ(d). proof. (1). since γ((d,d) u (e,e)) = {((d u e)[s], (d t e)[s]) | s ∈ s[}, the result follows trivially from proposition 1.9 (5). (2). if d(c ∧ d) 6⊆ d(c) u d(d) then we have s,t ∈ s satisfying p (s,t) 6⊆ d(c)ud(d) for which we have acb, cde satisfying a∩c 6⊆ qt and ps 6⊆ b∪e. take t′ ∈ s satisfying a∩c 6⊆ qt′ and pt′ 6⊆ qt. now p (s,t′) ⊆ d(c), d(d), so d(c), d(d) 6⊆ p (s,t), which leads to the contradiction p (s,t) ⊆ d(c) ud(d). this verifies d(c∧d) ⊆ d(c)ud(d), and the proof of d(c)td(d) ⊆ d(c∧d) is dual to this, so (2) is proved. � 3. direlational and dicover uniformities we now have the tools necessary to define direlational and dicover uniformities on a texture, and to prove their equivalence. definition 3.1. let (s, s) be a texture and u a family of direlations on (s, s). if u satisfies the conditions (1) (i,i) v (d,d) for all (d,d) ∈ u. that is, u ⊆ rdr. (2) (d,d) ∈ u, (e,e) ∈ dr and (d,d) v (e,e) implies (e,e) ∈ u. di-uniform texture spaces 169 (3) (d,d), (e,e) ∈ u implies (d,d) u (e,e) ∈ u. (4) given (d,d) ∈ u there exists (e,e) ∈ u satisfying (e,e) ◦ (e,e) v (d,d). (5) given (d,d) ∈ u there exists (c,c) ∈ u satisfying (c,c)← v (d,d). then u is called a direlational uniformity on (s, s), and (s, s, u) is known as a direlational uniform texture space. it will be noted that this definition is formally the same as the usual definition of a diagonal uniformity, and the notions of base and subbase may be defined in the obvious way. exactly as for diagonal uniformities we have the following lemma. lemma 3.2. a direlational uniformity u on (s, s) has a base of symmetric direlations. proof. take (d,d) ∈ u. by condition (5) we have (e,e) ∈ u with (e,e)← v (d,d), so (e,e) v (d,d)← and (d,d)← ∈ u by condition (2). but now (f,f) = (d,d)u(d,d)← ∈ u by condition (3), and clearly (f,f) is symmetric and satisfies (f,f) v (d,d). � the following example of a direlational uniformity will prove important later on. example 3.3. let (i, i) be the unit interval texture and for � > 0 define d� = {(r,s) | r,s ∈ i, s < r + �}, d� = {(r,s) | r,s ∈ i, s ≤ r − �}. clearly (d�,d�) is a reflexive, symmetric direlation on (i, i). moreover, (d�,d�)2 v (d2�,d2�), while for � ≤ δ, (d�,d�) v (dδ,dδ) and so (d�,d�)u(dδ,dδ) = (d�,d�). hence ui = {(d,d) | (d,d) ∈ dr and there exists � > 0 with (d�,d�) v (d,d)} is a direlational uniformity on (i, i). we will call ui the usual direlational uniformity on (i, i). definition 3.4. let (s, s, u) be a direlational uniform texture space and c a dicover of s. then c is called uniform if γ(c,c) ≺ c for some (c,c) ∈ u. lemma 3.5. let (s, s, u) be a direlational uniform texture space and υ the family of uniform dicovers. then υ has the following properties: (1) given c ∈ υ there exists d ∈ υ ∩ adc with d ≺ c. (2) c ∈ υ, d ∈ dc and c ≺ d implies d ∈ υ. (3) c, d ∈ υ implies c ∧ d ∈ υ. (4) given c ∈ υ there exists d ∈ υ with d ≺(?) c. proof. (1). by hypothesis there exists (c,c) ∈ u with γ(c,c) ≺ c. but d = γ(c,c) ∈ υ ∩ adc and d ≺ c. (2). immediate. (3). take c, d ∈ υ and (c,c), (d,d) ∈ u with γ(c,c) ≺ c, γ(d,d) ≺ d. then (c,c) u (d,d) ∈ u by definition 3.1 (3), and γ((c,c) u (d,d)) ≺ c ∧ d by proposition 2.9 (1), so c ∧ d ∈ υ. 170 s. özçağ and l. m. brown (4). take c ∈ υ and (c,c) ∈ u with γ(c,c) ≺ c. by definition 3.1 (4) we have (d,d) ∈ u with (d,d) ◦ (d,d) v (c,c), and then by definition 3.1 (5) we have (e,e) ∈ u with (e,e)← v (d,d). if we let (f,f) = (d,d) u (e,e) then (f,f) ∈ u and (f,f) ◦ (f,f)← v (c,c), so γ(f,f) ≺ (∆) γ(c,c) ≺ c by proposition 2.4. in exactly the same way we may find (g,g) ∈ u with γ(g,g) ≺ (∆) γ(f,f). if we let d = γ(g,g) then d ∈ υ is anchored by proposition 2.3, so by lemma 2.2 (3 ii) we have d ≺(?) c. � this leads to the following definition. definition 3.6. let (s, s) be a texture. if υ is a family of dicovers of s satisfying conditions (1)–(4) of lemma 3.5 we say υ is a dicovering uniformity on (s, s), and call (s, s,υ) a dicovering uniform texture space. we can now see lemma 3.5 as associating a dicovering uniformity with a given direlational uniformity. the following theorem expresses the equivalence of these two concepts. theorem 3.7. let (s, s) be a texture. (1) to each direlational uniformity u on (s, s) we may associate a dicovering uniformity υ = γ(u) = {c ∈ dc | ∃(c,c) ∈ u with γ(c,c) ≺ c}. (2) to each dicovering uniformity υ on (s, s) we may associate a direlational uniformity u = ∆(υ) = {(d,d) ∈ rdr | ∃c ∈ υ with δ(c) v (d,d)}. (3) ∆(γ(u)) = u for every direlational uniformity u on (s, s). (4) γ(∆(υ)) = υ for every dicovering uniformity υ on (s, s). proof. (1). this is just lemma 3.5. (2). we need to establish the conditions (1)–(5) of definition 3.1 for u = ∆(υ). conditions (1) and (2) are an immediate consequence of the definition of ∆(υ), and (3) follows trivially from proposition 2.9 (2). take (d,d) ∈ ∆(υ). then we have c ∈ υ satisfying δ(c) v (d,d). now (c,c) = δ(c) ∈ ∆(υ), and since (c,c) is symmetric by proposition 2.5 we have (c,c)← = (c,c) v (d,d), which proves (5). finally we have e ∈ υ satisfying e ≺(?) c, and then (e,e) = δ(e) ∈ ∆(υ) and (e,e)◦(e,e) v (d,d) by proposition 2.6, so (4) is established also. (3). first take (d,d) ∈ ∆(γ(u)). then we have c ∈ γ(u) with δ(c) v (d,d), and then (c,c) ∈ u with γ(c,c) ≺ c. without loss of generality we may take (c,c) ∈ rsdr since the symmetric elements of u form a base, so by corollary 2.8, (c,c) v δ(γ(c,c)) v δ(c) v (d,d), which shows (d,d) ∈ u. conversely, take (d,d) ∈ u and choose (e,e) ∈ u with (e,e) symmetric so that (e,e)◦(e,e) v (d,d). then δ(γ(e,e)) v (d,d) by theorem 2.7 (1), and we have established (d,d) ∈ ∆(γ(u)). (4). first take c ∈ γ(∆(υ)). then we have (c,c) ∈ ∆(υ) with γ(c,c) ≺ c, and then d ∈ υ with δ(d) v (c,c). without loss of generality we may take di-uniform texture spaces 171 d ∈ adc since the anchored elements of υ form a base, so by corollary 2.8 (2). d ≺ γ(δ(d)) ≺ γ(c,c) ≺ c, whence c ∈ υ. conversely, take c ∈ υ and choose e ∈ υ with e ≺ (?) c. without loss of generality we may assume e is anchored, so by lemma 2.2 (2) we have e ≺ (∆) c, whence e∆ ≺ c. now theorem 2.7 (2) gives γ(δ(e)) ≺ c, so c ∈ γ(∆(υ)), as required. � we will use the term di-uniformity to refer to direlational and dicovering uniformities in general. example 3.8. consider the texture (i, i). the dicovering uniformity υi corresponding to the direlational uniformity ui of example 3.3 has a base consisting of the dicovers d�, � > 0, where d� = {([0,r + �), [0,r − �]) | r ∈ i}, and [0,r + �) is understood to be [0, 1] when r + � > 1 and [0,r − �] is ∅ if r − � < 0. 4. the uniform ditopology we begin by associating a ditopology with a direlational uniformity. proposition 4.1. let (s, s, u) be a direlational uniform texture space. then the family (ηu(s),µu(s)), s ∈ s[, defined by ηu(s) = {n ∈ s | n 6⊆ qs, ps 6⊆ qt =⇒ ∃(d,d) ∈ u, d[t] ⊆ n}, µu(s) = {m ∈ s | ps 6⊆ m, pt 6⊆ qs =⇒ ∃(d,d) ∈ u, m ⊆ d[t]}, is the dineighbourhood system for a ditopology on (s, s). proof. we must verify that the family ηu(s), s ∈ s[, satisfies the following conditions [6]: (1) n ∈ ηu(s) =⇒ n 6⊆ qs. (2) n ∈ ηu(s), n ⊆ n′ ∈ s =⇒ n′ ∈ ηu(s). (3) n1,n2 ∈ ηu(s), n1 ∩n2 6⊆ qs =⇒ n1 ∩n2 ∈ ηu(s). (4) (a) n ∈ ηu(s) =⇒ ∃n? ∈ s, ps ⊆ n? ⊆ n so that n? 6⊆ qt =⇒ n? ∈ ηu(t), t ∈ s[. (b) for n ∈ s and n 6⊆ qs, if there exists n? ∈ s,ps ⊆ n? ⊆ n which satisfies n? 6⊆ qt =⇒ n? ∈ ηu(t), t ∈ s[, then n ∈ ηu(s). conditions (1) and (2) are immediate from the definitions, and (3) follows at once from the inclusion (due)(pt) ⊆ d(pt) ∩e(pt) (proposition 1.9 (5)). (4) (a). take n ∈ ηu(s) and define n? = ∨ {pz | pz 6⊆ qt =⇒ ∃(d,d) ∈ u with d[t] ⊆ n}. clearly ps ⊆ n? ⊆ n and if n? 6⊆ qt it is easy to show that n ∈ ηu(t). (4) (b). take n ∈ s with n 6⊆ qs and n? ∈ s with ps ⊆ n? ⊆ n and satisfying n? 6⊆ qt =⇒ n? ∈ ηu(t). to show n ∈ ηu(s) take t ∈ s with 172 s. özçağ and l. m. brown ps 6⊆ qt. since ps ⊆ n? we have n? 6⊆ qt. choose t′ ∈ s with n? 6⊆ qt′ and pt′ 6⊆ qt. by hypothesis n ∈ ηu(t′) so pt′ 6⊆ qt now gives (d,d) ∈ u with d[t] ⊆ n. since n 6⊆ qs this shows that n ∈ ηu(s), as required. in just the same way the sets µu(s) satisfy the dual of conditions (1)–(4) above, and this completes the proof (cf. [6]). � definition 4.2. let (s, s, u) be a direlational uniform texture space and ηu(s), µu(s) defined as above. the ditopology with dineighbourhood system {(ηu(s), µu(s)) | s ∈ s[} is called the uniform ditopology of u and denoted by (τu,κu). lemma 4.3. let (s, s, u) be a direlational uniform texture space with uniform ditopology (τu,κu). (i) g ∈ τu ⇐⇒ (g 6⊆ qs =⇒ ∃(d,d) ∈ u with d[s] ⊆ g). (ii) k ∈ κu ⇐⇒ (ps 6⊆ k =⇒ ∃(d,d) ∈ u with k ⊆ d[s]). proof. we prove (i), leaving (ii) to the reader. it is shown in [6] that the open sets are characterized by the property that g 6⊆ qs =⇒ g ∈ ηu(s). take g ∈ τu and s ∈ s with g 6⊆ qs. now we have s′ ∈ s with g 6⊆ qs′ and ps′ 6⊆ qs. by the above g ∈ ηu(s′) and now ps′ 6⊆ qs implies there exists (d,d) ∈ u with d[s] ⊆ g. conversely suppose g has the property stated in (i). then if g 6⊆ qs we have (d,d) ∈ u with d[s] ⊆ g. now if ps 6⊆ qt we have pt ⊆ ps and so d[t] ⊆ d[s] ⊆ g, which shows that g ∈ ηu(s). thus g ∈ τu. � proposition 4.4. let υ be a dicovering uniformity on (s, s). denote by (τ,κ) the uniform ditopology of the direlational uniformity ∆(υ). then: (i) g ∈ τ ⇐⇒ (g 6⊆ qs =⇒ ∃c ∈ υ with st(c,ps) ⊆ g). (ii) k ∈ κ ⇐⇒ (ps 6⊆ k =⇒ ∃c ∈ υ with k ⊆ cst(c,qs)). proof. (i). take g ∈ τ and g 6⊆ qs. then by lemma 4.3 we have (d,d) ∈ ∆(υ) with d[s] ⊆ g. we may take (e,e) ∈ ∆(υ) with (e,e) ◦ (e,e)← v (d,d) and as in the proof of proposition 2.4 we have st(γ(e,e),ps) ⊆ d[s]. there exists c ∈ υ with δ(c) v (e,e) and without loss of generality we may assume c is anchored. hence by corollary 2.8 (2), c ≺ γ(δ(c)) ≺ γ(e,e), so st(c,ps) ⊆ d[s] ⊆ g. conversely, suppose that given g 6⊆ qs there exists c ∈ υ with st(c,ps) ⊆ g. if we set (d,d) = δ(c) then (d,d) ∈ ∆(υ) and by theorem 2.7 (2) we have d[s] = st(c,ps) ⊆ g. hence g ∈ τ. (ii). the proof is dual to (i), and is omitted. � this justifies the following definition. definition 4.5. let υ be a dicovering uniformity on (s, s). then the ditopology (τυ,κυ) defined by τυ = {g ∈ s | g 6⊆ qs =⇒ ∃c ∈ υ, st(c,ps) ⊆ g}, κυ = {k ∈ s | ps 6⊆ k =⇒ ∃c ∈ υ, k ⊆ cst(c,qs)}, is called the uniform ditopology of υ. di-uniform texture spaces 173 in just the same way the dineighbourhood system (ηυ(s),µυ(s)), s ∈ s[, for (τυ,κυ) is given by ηυ(s) = {n ∈ s | n 6⊆ qs, ps 6⊆ qt =⇒ ∃c ∈ υ, st(c,pt) ⊆ n}, µυ(s) = {m ∈ s | ps 6⊆ m, pt 6⊆ qs =⇒ ∃c ∈ υ, m ⊆ cst(c,qt}. we omit the details. the following lemma enables us to generate open sets and closed sets for the uniform ditopology of a dicovering uniformity. lemma 4.6. let υ be a dicovering uniformity on (s, s) and take l ∈ s. (1) the set g = g(l) = ∨ {pu | ∃d ∈ υ, st(d,pu) ⊆ l} is open for the uniform ditopology. (2) the set k = k(l) = ⋂ {qu | ∃d ∈ υ, l ⊆ cst(d,qu)} is closed for the uniform ditopology. proof. we establish (1), leaving the dual proof of (2) to the reader. take g 6⊆ qs. then we have u ∈ s and d ∈ υ satisfying pu 6⊆ qs and st(d,pu) ⊆ l. take e ∈ υ with e ≺(?) d. by definition 4.5 it will be sufficient to show that st(e,ps) ⊆ g. if this is not so then we have a0 e b0 with ps 6⊆ b0 and a0 6⊆ g so we may take v ∈ s with a0 6⊆ qv and pv 6⊆ g. if we can show that st(e,pv) ⊆ st(d,pu) we will obtain an immediate contradiction to the definition of g, so take a1 e b1 with pv 6⊆ b1, and choose a′0 d b′0 satisfying st(e,a0) ⊆ a′0 and b′0 ⊆ cst(e,b0). since cst(e,b0) ⊆ b0, ps 6⊆ b0 and pu 6⊆ qs we see that pu 6⊆ b′0, whence a1 ⊆ st(e,a0) ⊆ a′0 ⊆ st(d,pu), using the evident fact that a0 6⊆ b1. this establishes the required inclusion and completes the proof. � corresponding results for direlational uniformities may easily be formulated and the details are left to the interested reader. it is well known that a classical uniformity has a base of open members and a base of closed members. we now establish an analogous result for diuniformities. we confine our attention to the dicovering case since there is a well established meaning to the notions of openness and closedness for dicovers [3]. namely, a dicover c of the ditopological texture space (s, s,τ,κ) is open (respectively, closed , co-open, coclosed ) if a c b =⇒ a ∈ τ (a ∈ κ, b ∈ τ, b ∈ κ). first we require the following lemma. lemma 4.7. let υ be a dicovering uniformity on (s, s), c ∈ υ and l ∈ s. consider the uniform ditopology on (s, s). then: (1) l ⊆ ] st(c,l)[ and [cst(c,l)] ⊆ l. (2) [l] ⊆ st(c,l) and cst(c,l) ⊆ ]l[. 174 s. özçağ and l. m. brown proof. 1. if h = h(st(c,l)) is the open set defined in lemma 4.6 (1) it is trivial to verify that l ⊆ h ⊆ st(c,l), whence l ⊆ ] st(c,l)[. the second inclusion follows in the same way from lemma 4.6 (2). 2. if k = k(l) is the closed set defined in lemma 4.6 (2) it is trivial to verify that l ⊆ k ⊆ st(c,l), whence [l] ⊆ st(c,l). the second inclusion follows in the same way from lemma 4.6 (1). � proposition 4.8. a dicovering uniformity has a base of open, coclosed dicovers and a base of closed, co-open dicovers. proof. trivial from lemma 4.7. � definition 4.9. a ditopological texture space (s, s,τ,κ) is called di-uniformizable if there exists a di-uniformity on (s, s) whose uniform ditopology coincides with (τ,κ). we recall the following regularity axioms for ditopological texture spaces. definition 4.10. [3] let (τ,κ) be a ditopology on (s, s). then (τ,κ) is called (1) regular if g ∈ τ, g 6⊆ qs =⇒ ∃h ∈ τ with h 6⊆ qs, [h] ⊆ g. (2) coregular if f ∈ κ, ps 6⊆ f =⇒ ∃k ∈ κ with ps 6⊆ k, f ⊆ ]k[. (3) biregular if it is regular and coregular. using proposition 4.8 it is straightforward to verify that a di-uniformizable ditopology is biregular. however we will shortly prove a more powerful result, and so omit the details. definition 4.11. [7] let (τ,κ) be a ditopology on (s, s). then (τ,κ) is called (1) completely regular if given g ∈ τ, g 6⊆ qs, there exists a bicontinuous difunction (f,f) : (s, s) → (i, i) satisfying ps ⊆ f←(p0) and f←(q1) ⊆ g. (2) completely coregular if given k ∈ τ, ps 6⊆ k, there exists a bicontinuous difunction (f,f) : (s, s) → (i, i) satisfying k ⊆ f←(p0) and f←(q1) ⊆ qs. (3) completely biregular if it is completely regular and completely coregular. we end this section by showing that a di-uniformizable ditopology is completely biregular. since it is easy to see that the complete regularity conditions imply the corresponding regularity conditions it will follow that a diuniformizable ditopology is biregular. we choose to work with direlational uniformities. first we require the following lemma, which is the textural analogue of the metrization lemma ([10], page 185). lemma 4.12. let (s, s) be a texture and rn, n ∈ n, a sequence of reflexive relations satisfying r3n+1 ⊆ rn, n ∈ n. define the function ϕ : s×s → [0, 1] by ϕ(u,v) =   0 if rn 6⊆ q(u,v) ∀n ∈ n, 1 if rn ⊆ q(u,v) ∀n ∈ n, 2−n if ∃n ∈ n, rn 6⊆ q(u,v), rn+1 ⊆ q(u,v). di-uniform texture spaces 175 then there exists a function q : s ×s → [0,∞) satisfying (1) 1 2 ϕ(u,v) ≤ q(u,v) ≤ ϕ(u,v), ∀u,v ∈ s. (2) pu 6⊆ qv =⇒ q(u,v) = 0 ∀u,v ∈ s. (3) q(u,v) ≤ q(u,w) + q(w,v) ∀u,v,w ∈ s. proof. we consider chains u0,u1, . . . ,un of elements of s and write s(u0,u1, . . . ,un) = n−1∑ i=1 ϕ(ui,ui+1), n > 0, s(u0,u0) = ϕ(u0,u0) = 0. consider the function q : s ×s → [0,∞) defined by q(u,v) = inf{s(u0, . . . ,un) | u = u0 and v = un, n ∈ n}. (1) it is clearly sufficient to prove that ϕ(u,v) ≤ 2s(u0, . . . ,un) for any chain u0,u1, . . . ,un with u = u0 and v = un. the proof is by induction on n ∈ n and follows essentially the same steps as the proof of the metrizaton lemma. we therefore omit the details. (2) for each n ∈ n we have is ⊆ rn so pu 6⊆ qv implies is 6⊆ q(u,v), and hence rn 6⊆ q(u,v). by (1) we now have 0 ≤ q(u,v) ≤ ϕ(u,v) = 0, whence q(u,v) = 0. (3) immediate from the definition of q. � lemma 4.13. if we consider a sequence of corelations rn, n ∈ n, satisfying rn ⊆ r3n+1 and define ϕ∗(u,v) =   0 if p (u,v) 6⊆ rn ∀n ∈ n, 1 if p (u,v) ⊆ rn ∀n ∈ n, 2−n if ∃n ∈ n, p (u,v) 6⊆ rn, p (u,v) ⊆ rn+1, we obtain q∗ : s ×s → [0,∞) satisfying (1) 1 2 ϕ∗(u,v) ≤ q∗(u,v) ≤ ϕ∗(u,v), ∀u,v ∈ s. (2) pv 6⊆ qu =⇒ q∗(u,v) = 0 ∀u,v ∈ s. (3) q∗(u,v) ≤ q∗(u,w) + q∗(w,v) ∀u,v,w ∈ s. in case rn = r←n then we clearly have ϕ ∗(u,v) = ϕ(v,u) and q∗(u,v) = q(v,u) for all u,v ∈ s. now we may give: theorem 4.14. a diuniformizable ditopological texture space is completely biregular. proof. let (s, s,τ,κ) be a ditopological texture space and u a compatible direlational uniformity. to show that (τ,κ) is completely regular take g ∈ τ and a ∈ s with g 6⊆ qa. then there exists (r,r) ∈ u with r(pa) ⊆ g. let (r0,r0) = (r,r). by definition 3.1 there exists (r1,r1) ∈ u such that (r1,r1)3 v (r0,r0), (r2,r2) ∈ u with (r2,r2)3 v (r1,r1), and so on. hence we obtain a sequence (rn,rn) of reflexive direlations satisfying (rn+1,rn+1)3 v (rn,rn), and by 176 s. özçağ and l. m. brown lemma 3.2 there is no loss of generality in assuming that the (rn,rn) are symmetric, i.e. rn = r←n for each n ∈ n. let ϕ and q be the functions given in lemma 4.12 for the sequence rn, n ∈ n of reflexive relations and define θ : s → [0, 1] by θ(s) = 2q(a,s) ∧ 1. we take the texture i on i = [0, 1] and verify that the point function θ satisfies the condition pu 6⊆ qv =⇒ pθ(u) 6⊆ qθ(v) of ([1], theorem 3.14). however if pu 6⊆ qv then q(a,v) ≤ q(a,u) + q(u,v) = q(a,u) by lemma 4.12 (2),(3), so θ(v) ≤ θ(u), which is equivalent to pθ(u) 6⊆ qθ(v) in (i, i). it follows that f = ∨ {p (s,t) | ∃v ∈ s with ps 6⊆ qv and t ≤ θ(v)}, f = ⋂ {q(s,t) | ∃v ∈ s with pv 6⊆ qs and θ(v) ≤ t}, define a difunction (f,f) : (s, s) → (i, i). if we take the usual ditopology (τi,κi) on (i, i) then (f,f) is bicontinuous. we prove continuity, leaving the dual proof of cocontinuity to the reader. for s ∈ s suppose f←([0,r)) 6⊆ qs. then we have t ∈ i with p (s,t) 6⊆ f and t < r. from the definition of f we have v ∈ s and t′ ∈ i with p (s,t) 6⊆ q(s,t′), pv 6⊆ qs and θ(v) ≤ t′. from pt 6⊆ qt′ we have t′ ≤ t and so θ(v) < r. clearly θ(v) < 1 and so θ(v) = 2q(a,v) < r, whence there exists n with 2(q(a,v) + 2−n) < r. we verify that rn(ps) ⊆ f←([0,r)). suppose the contrary and take w ∈ s with rn(ps) 6⊆ qw and pw 6⊆ f−1([0,r)). now we have z ∈ s with rn 6⊆ q(z,w) and pv 6⊆ qz, whence rn 6⊆ q(v,w) and so q(v,w) ≤ ϕ(v,w) ≤ 2−n by lemma 4.12. hence we have θ(w) ≤ 2q(a,w) ≤ 2(q(a,v) + q(v,w)) ≤ 2(q(a,v) + 2−n) < r. on the other hand, from pw 6⊆ f←([0,r)) we have w′ ∈ s with pw 6⊆ qw′ for which (4.1) p (w′,u) 6⊆ f =⇒ u ≤ r. choose r′ ∈ i satisfying θ(w) < r′ < r. then pr′ 6⊆ qθ(w) and pw 6⊆ qw′, so by the definition of f we have f ⊆ q(w′,r′), which is equivalent to p (w′,r′) 6⊆ f. applying implication (4.1) with u = r′ now gives the contradiction r ≤ r′, and we have proved f←([0,r)) 6⊆ qs =⇒ rn(ps) ⊆ f←([0,r)). hence f←([0,r)) ∈ τ since (rn,rn) ∈ u and τ = τu. this proves continuity since f←(i) = s ∈ τ. it remains to show that pa ⊆ f←(p0) and f←(q1) ⊆ g. suppose first that pa 6⊆ f←(p0). then we have b ∈ i with f 6⊆ q(a,b) and pb 6⊆ p0, that is b > 0. by the definition of f we have b′ ∈ i with p (a,b′) 6⊆ q(a,b) and v ∈ s with pa 6⊆ qv satisfying b′ ≤ θ(v). hence 0 0. however pa 6⊆ qv implies q(a,v) = 0 by lemma 4.12, which is a contradiction. if now we suppose f←([0, 1)) 6⊆ g then we have s ∈ s satisfying f←([0, 1)) 6⊆ qs and ps 6⊆ g. hence we have t ∈ i with p (s,t) 6⊆ f and [0, 1) 6⊆ qt, that is t < 1. from the definition of f we now have t′ ∈ i with p (s,t) 6⊆ q(s,t′) di-uniform texture spaces 177 and v ∈ s with pv 6⊆ qs and θ(v) ≤ t′. hence θ(v) ≤ t′ ≤ t < 1, whence 2q(a,v) < 1 and so ϕ(a,v) < 1 by lemma 4.12. hence there exists n ∈ n with rn 6⊆ q(a,v) and so r = r0 6⊆ q(a,v). this leads to r(pa) 6⊆ qa and so g 6⊆ qv. on the other hand pv 6⊆ qs and ps 6⊆ g give pv 6⊆ g, and we have the contradiction g ⊆ qv. this completes the proof of complete regularity, and complete coregularity can be proved in a similar way using the conjugate functions ϕ∗ and q∗ of lemma 4.13. � the converse of the above proposition is also true, but we postpone the proof until we have discussed initial di-uniformities in the next section. definition 4.15. a direlational uniformity u satisfying d u = (i,i) is called separated. we recall from [7] the following characteristic property of t0 ditopological spaces: (τ,κ) is t0 if and only if qs 6⊆ qt =⇒ ∃b ∈ τ ∪κ with ps 6⊆ b 6⊆ qt. theorem 4.16. let u be a direlational uniformity. then the uniform ditopology (τu,κu) is t0 if and only if u is separated. proof. =⇒. we know that i ⊆ d {d | (d,d) ∈ u}, so suppose d {d | (d,d) ∈ u} 6⊆ i. then we have s,t ∈ s with p (s,t) 6⊆ i for which we have s′ ∈ s satisfying d 6⊆ q(s′,t) for all (d,d) ∈ u. now p (s,t) 6⊆ i implies pt 6⊆ ps, which with ps 6⊆ qs′ gives qt 6⊆ qs. since (τu,κu) is t0 we have b ∈ τu ∪ κu satisfying pt 6⊆ b 6⊆ qs. there are two cases to consider: (a) b ∈ τu. now b 6⊆ qs′ implies d[s′] ⊆ b for some (d,d) ∈ u. it follows that pt 6⊆ d[s′] = d(ps′). but now d(ps′) ⊆ qt, so by lemma 1.5(2) we have d ⊆ q(s′,t), which is a contradiction. (b) b ∈ κu. noting that u has a base of symmetric direlations, a dual argument again leads to a contradiction. this completes the proof of d {d | (d,d) ∈ u} = i, and ⊔ {d | (d,d) ∈ u} = i is dual. ⇐=. take s,t ∈ s with qs 6⊆ qt. by the definition of qs there exists u ∈ s with ps 6⊆ pu and pu 6⊆ qt. take s′, t′ ∈ s satisfying ps 6⊆ qs′, ps′ 6⊆ pu and pu 6⊆ qt′, pt′ 6⊆ qt. then p (u,s′) 6⊆ i = d {d | (d,d) ∈ u} since ps′ 6⊆ pu, so as pu 6⊆ qt′ there exists (e,e) ∈ u with e ⊆ q(t′,s′), whence e(pt′) ⊆ qs′. now let g = ∨ {pz | z ∈ s, ∃(d,d) ∈ u with d[z] ⊆ qs′}. it may be shown that g ∈ τu (compare lemma 4.6). clearly pt′ ⊆ g, and so g 6⊆ qt. on the other hand if d[z] ⊆ qs′ then pz ⊆ d[z] ⊆ qs′ so g ⊆ qs′ and hence ps 6⊆ g. this verifies that (τu,κu) is t0. � 178 s. özçağ and l. m. brown 5. uniform bicontinuity and initial di-uniformities in order to define uniform bicontinuity it will be necessary to say what we mean by the inverse of a direlation and of a dicover under a difunction. we begin with the following: definition 5.1. let (s, s), (t, t) be textures, (r,r) a direlation on (t, t) and (f,f) a difunction on (s, s) to (t, t). then (f,f)−1(r) = ∨ {p (s1,s2) | ∃ps1 6⊆ qs′1 so that p (s′1,t1) 6⊆ f, f 6⊆ q(s2,t2) =⇒ p (t1,t2) ⊆ r}, (f,f)−1(r) = ⋂ {q(s1,s2) | ∃ps′1 6⊆ qs1 so that f 6⊆ q(s′1,t1),p (s2,t2) 6⊆ f, =⇒ r ⊆ q(t1,t2)}, (f,f)−1(r,r) = ((f,f)−1(r), (f,f)−1(r)). remark 5.2. in definition 5.1, p (t1,t2) ⊆ r may be replaced by r 6⊆ q(t1,t2) and r ⊆ q(t1,t2) by p (t1,t2) 6⊆ r. indeed, if s ′ 1, s2 satisfy the conditions in the definition of (f,f)−1(r), and p (s′1,t1) 6⊆ f, f 6⊆ q(s2,t2), then we may choose t′2 with f 6⊆ q(s2,t′2), p (s2,t′2) 6⊆ q(s2,t2). this gives us p (t1,t′2) ⊆ r, and so r 6⊆ q(t1,t2) since pt′2 6⊆ qt2 . the opposite direction is trivial, and the second property is dual. it is trivial to verify that (f,f)−1(r,r) is indeed a direlation on (s, s), and we omit the proof. let us examine the properties of this inverse mapping. proposition 5.3. let (f,f) be a difunction on (s, s) to (t, t). then (f,f)−1(it ,it ) = (is,is), where (is,is), (it ,it ) are the identity direlations on (s, s), (t, t) respectively. proof. to establish (f,f)−1(it ) = is we first suppose that is 6⊆ (f,f)−1(it ). then is 6⊆ q(s,s′) and p (s,s′) 6⊆ (f,f)−1(it ) for some s,s′ ∈ s. we have ps 6⊆ qs′ since is 6⊆ q(s,s′). by definition 5.1 there exists w1,w2 ∈ t satisfying p (s′,w1) 6⊆ f, f 6⊆ q(s′,w2) and p (w1,w2) 6⊆ it . on the other hand df2 implies pw1 6⊆ qw2 . hence it 6⊆ q(w1,w2) which contradicts p (w1,w2) 6⊆ it . the proof of the reverse inclusion is similar and the proof of the dual equality (f,f)−1(it ) = is is left to the reader. � corollary 5.4. let (f,f) be a difunction on (s, s) to (t, t). if (r,r) is a reflexive direlation on (t, t) then (f,f)−1(r,r) is a reflexive direlation on (s, s). proof. let (r,r) be reflexive. then (it ,it ) v (r,r) =⇒ (is,is) = (f,f)−1(it ,it ) v (f,f)−1(r,r) by the proposition so (f,f)−1(r,r) is reflexive. � di-uniform texture spaces 179 proposition 5.5. let (f,f) be a difunction on (s, s) to (t, t) and (r,r) a direlation on (t, t). then ((f,f)−1(r,r))← = (f,f)−1((r,r)←). proof. it is clearly sufficient to establish ((f,f)−1(r))← = (f,f)−1(r←), since the dual equality ((f,f)−1(r))← = (f,f)−1(r←) then follows by replacing r by r← and taking the inverse of both sides. first suppose that ((f,f)−1(r))← 6⊆ (f,f)−1(r←). then ((f,f)−1(r))← 6⊆ q(s,s′) and p (s,s′) 6⊆ (f,f)−1(r←) for some s,s′ ∈ s. since r← is a corelation, by remark 5.2 we have u,v ∈ s with p (s,s′) 6⊆ q(s,v), pu 6⊆ qs so that (5.2) f 6⊆ q(u,t1), p (v,t2) 6⊆ f =⇒ p (t1,t2) 6⊆ r ← for all t1, t2 ∈ t . on the other hand ((f,f)−1(r))← 6⊆ q(s,s′) gives (f,f)−1(r) ⊆ q(s′,s) and so p (s′,u) 6⊆ (f,f)−1(r). since ps′ 6⊆ qv we have w1,w2 ∈ t for which p (v,w1) 6⊆ f, f 6⊆ q(u,w2) and r ⊆ q(w1,w2) by remark 5.2. putting t1 = w2, t2 = w1 in the implication (5.2) now gives p (w2,w1) 6⊆ r ←, so giving the contradiction r 6⊆ q(w1,w2). hence ((f,f) −1(r))← ⊆ (f,f)−1(r←). the proof of (f,f)−1(r←) ⊆ ((f,f)−1(r))← is similar, and is omitted. � corollary 5.6. let (f,f) be a difunction on (s, s) to (t, t). if (r,r) is a symmetric direlation on (t, t) then (f,f)−1(r,r) is a symmetric direlation on (s, s). proof. immediate. � proposition 5.7. let (f,f) be a difunction on (s, s) to (t, t), (p,p) and (q,q) direlations on (t, t). then (f,f)−1(p,p) ◦ (f,f)−1(q,q) v (f,f)−1((p,p) ◦ (q,q)). proof. suppose that (f,f)−1(p)◦(f,f)−1(q) 6⊆ (f,f)−1(p◦q). by the definition of composition of relations we have s,u,z ∈ s with p (s,u) 6⊆ (f,f)−1(p◦q), (f,f)−1(q) 6⊆ q(s,z) and (f,f)−1(p) 6⊆ q(z,u). by r2 there exists s′ ∈ s with ps 6⊆ qs′ and (f,f)−1(q) 6⊆ q(s′,z). by definition 5.1, p (s,u) 6⊆ (f,f)−1(p◦q) gives w1,w2 ∈ s satisfying p (s′,w1) 6⊆ f, f 6⊆ q(u,w2) and p (w1,w2) 6⊆ p◦q. on the other hand from (f,f)−1(q) 6⊆ q(s′,z) we have z′,s′′ ∈ s with p (s′,z′) 6⊆ q(s′,z), ps′ 6⊆ qs′′ for which (5.3) p (s′′,t1) 6⊆ f, f 6⊆ q(z′,t2) =⇒ q 6⊆ q(t1,t2) for all t1, t2 ∈ t by remark 5.2. likewise, (f,f)−1(p) 6⊆ q(z,u) gives u′,z′′ ∈ s with p (z,u′) 6⊆ q(z,u), pz 6⊆ qz′′ for which (5.4) p (z′′,v1) 6⊆ f, f 6⊆ q(u′,v2) =⇒ p 6⊆ q(v1,v2) for all v1,v2 ∈ t . finally pz′ 6⊆ qz′′ so by df1 we have v ∈ t for which f 6⊆ q(z′,v) and p (z′′,v) 6⊆ f. 180 s. özçağ and l. m. brown by cr1, p (s′,w1) 6⊆ f and ps′ 6⊆ qs′′ gives p (s′′,w1) 6⊆ f so implication (5.3) may be applied with t1 = w1, t2 = v to give q 6⊆ q(w1,v). likewise (5.4) may be applied with v1 = v, v2 = w2 to give p 6⊆ q(v,w2). this gives the contradiction p (w1,w2) ⊆ p◦q and we have shown (f,f) −1(p)◦(f,f)−1(q) ⊆ (f,f)−1(p◦q). the proof of (f,f)−1(p ◦ q) ⊆ (f,f)−1(p) ◦ (f,f)−1(q) is dual to the above, and is omitted. � corollary 5.8. let (f,f) be a difunction on (s, s) to (t, t). if (r,r) is a transitive direlation on (t, t) then (f,f)−1(r,r) is a transitive direlation on (s, s). proof. straightforward. � now let us make the following definition. definition 5.9. let u be a direlational uniformity on (s, s), v a direlational uniformity on (t, t) and (f,f) a difunction from (s, s) to (t, t). if (d,d) ∈ v =⇒ (f,f)−1(d,d) ∈ u the difunction (f,f) is said to be u–v uniformly bicontinuous. example 5.10. let u be a direlational uniformity on (s, s). then the identity difunction (i,i) on (s, s) is u–u–uniformly bicontinuous. to see this it is clearly sufficient to note that (i,i)−1(r,r) = (r,r) for all direlations (r,r) on (s, s). the proof of this equality is straightforward and is left to the interested reader. now let us consider the composition of uniformly bicontinuous difunctions. the following lemma will be useful. lemma 5.11. let (s, s), (t, t) and (w, w) be textures, (f,f) a difunction on (s, s) to (t, t), (g,g) a difunction on (t, t) to (w, w) and (r,r) a direlation on (w, w). then (f,f)−1((g,g)−1(r,r)) = ((g,g) ◦ (f,f))−1(r,r). proof. first suppose that (f,f)−1((g,g)−1(r)) 6⊆ (g◦f,g◦f)−1(r). then we have s,s′,u ∈ s with p (s,u) 6⊆ (g ◦f,g◦f)−1(r), ps 6⊆ qs′, satisfying (5.5) p (s′,t1) 6⊆ f, f 6⊆ q(u,t2) =⇒ (g,g) −1(r) 6⊆ q(t1,t2) for all t1, t2 ∈ t by remark 5.2. now p (s,u) 6⊆ (g ◦f,g◦f)−1(r), ps 6⊆ qs′, gives w1,w2 ∈ w for which p (s′,w1) 6⊆ g◦f, g◦f 6⊆ q(u,w2) and r ⊆ q(w1,w2). now we obtain w′1 ∈ w , v1 ∈ t with p (s′,v1) 6⊆ f, p (v1,w′1) 6⊆ g, and w′2 ∈ w , v2 ∈ t with f 6⊆ q(u,v2) and g 6⊆ q(v2,w′2). hence we may apply (5.5) with t1 = v1, t2 = v2 to give (g,g)−1(r) 6⊆ q(v1,v2). hence we have v ′ 1,v ′ 2 ∈ t satisfying p (v1,v′2) 6⊆ q(v1,v2), pv1 6⊆ qv′1 , for which (5.6) p (v′1,z1) 6⊆ g, g 6⊆ q(v′2,z2) =⇒ r 6⊆ q(z1,z2) di-uniform texture spaces 181 for all z1,z2 ∈ w by remark 5.2. now p (v1,w′1) 6⊆ g, pv1 6⊆ qw′1 and pv1 6⊆ qv′1 gives p (v′1,w1) 6⊆ g. also, g 6⊆ q(v2,w′2), pv′2 6⊆ qv2 and pw′2 6⊆ qw2 gives f 6⊆ q(v′2,w2) . hence we may apply (5.6) with z1 = w1, z2 = w2 to give r 6⊆ q(w1,w2), which is a contradiction. hence (f,f)−1((g,g)−1(r)) ⊆ (g ◦ f,g ◦ f)−1(r), and the proof of the reverse inclusion is similar and is omitted. the proof of the dual equality (f,f)−1((g,g)−1(r)) = (g ◦f,g◦f)−1(r) is left to the interested reader. � the following is now immediate from the definitions: proposition 5.12. uniform bicontinuity is preserved under composition of difunctions. as expected we also have: proposition 5.13. let (τk,κk), k = 1, 2, be the uniform ditopology of the direlational uniformity uk on (sk, sk), and let (f,f) be a difunction on (s1, s1) to (s2, s2). then if (f,f) is u1–u2 uniformly bicontinuous it is (τ1,κ1)–(τ2,κ2) bicontinuous. proof. take g ∈ τ2 and s ∈ s1 with f←(g) 6⊆ qs. then we have t ∈ s2 with p (s,t) 6⊆ f and g 6⊆ qt, whence by lemma 4.3 there exists (d,d) ∈ u2 satisfying d[t] ⊆ g. let (e,e) = (f,f)−1(d,d) ∈ u1. it may be shown that e[s] ⊆ f←(g), whence f←(g) ∈ τ1, again by lemma 4.3. hence (f,f) is τ1–τ2 continuous, and the proof of κ1–κ2 cocontinuity is dual to this. � now let us turn our attention to the notion of initial di-uniformity. theorem 5.14. let (s, s) be a texture, vi, i ∈ i, direlational uniformities on the textures (ti, ti) and (fi,fi) difunctions on (s, s) to (ti, ti), i ∈ i. then the family (fi,fi) −1(di,di), (di,di) ∈ vi, i ∈ i is a subbase for a direlational di-uniformity u on (s, s). proof. let u = {(d,d) ∈ dr | ∃i1, . . . , in ∈ i, (dik,dik) ∈ vik, 1 ≤ k ≤ n with dn k=1(fik,fik) −1(dik,dik) v (d,d)}. we must verify conditions (1)– (5) of definition 3.1. clearly (1) is immediate from proposition 5.3 and (2), (3) are trivial from the definition of u. if (d,d) ∈ u then we have (dik,dik) ∈ vik, 1 ≤ k ≤ n, with dn k=1(fik,fik) −1(dik,dik) v (d,d). for each k we may choose (eik,eik) ∈ vik satisfying (eik,eik) ◦ (eik,eik) v (dik,dik). if we set 182 s. özçağ and l. m. brown (e,e) = dn k=1(fik,fik) −1(eik,eik) then (e,e) ∈ u and (e,e) ◦ (e,e) v nl k=1 ((fik,fik) −1(eik,eik) ◦ (fik,fik) −1(eik,eik)) v nl k=1 (fik,fik) −1((eik,eik) ◦ (eik,eik)) v nl k=1 (fik,fik) −1(dik,dik) v (d,d), by proposition 1.9 (7) and proposition 5.7. hence (4) is satisfied. finally, (5) may be verified in a similar way using proposition 1.9 (4) and proposition 5.5. � definition 5.15. the di-uniformity u on (s, s) defined in theorem 5.14 is called the initial direlational di-uniformity on (s, s) defined by the spaces (ti, ti, vi) and the difunctions (fi,fi), i ∈ i. clearly the initial di-uniformity is the coarsest di-uniformity on (s, s) for which the difunctions (fi,fi) are u–vi uniformly bicontinuous for all i ∈ i. we are now in a position to prove the converse of theorem 4.14, and so complete our characterization of di-uniformizable ditopological texture spaces. theorem 5.16. (s, s,τ,κ) is di-uniformizable if and only if it is completely biregular. proof. it remains to show that if (τ,κ) is completely biregular then there exists a compatible di-uniformity. let u denote the initial direlational uniformity generated by the family of all bicontinuous difunctions from (s, s,τ,κ) to (i, i,τi,κi). we show that (τ,κ) = (τu,κu). first take g ∈ τu and g 6⊆ qs. then there exist z,s′,s′′,s′′′,w ∈ s so that g 6⊆ qz, pz 6⊆ qs′, ps′ 6⊆ qs′′, ps′′ 6⊆ qs′′′, ps′′′ 6⊆ qw and pw 6⊆ qs. choose (d,d) ∈ u with d[z] ⊆ g. now there exist (τ,κ)– (τi,κi) bicontinuous difunctions (f1,f1), . . . , (fn,fn) and � > 0 for which e = (f1,f1) −1(d�) u . . .u (fn,fn)−1(d�) ⊆ d. since ps′′′ 6⊆ qw, by df1 there exists ri ∈ i for each i = 1, . . . ,n, so that fi 6⊆ q(s′′′,ri) and p (w,ri) 6⊆ fi. we show that (a) ⋂n i=1 f ← i ([0,ri + �)) ⊆ e[z] ⊆ d[z] ⊆ g, and (b) ⋂n i=1 f ← i ([0,ri + �)) 6⊆ qs, from which it follows at once that g ∈ τ. suppose that (a) is false. take u,u′,u′′ ∈ s with ⋂n i=1 f ← i ([0,ri+�)) 6⊆ qu′′, pu′′ 6⊆ qu, pu 6⊆ qu′ and pu′ 6⊆ e[z]. now for each i = 1, . . . ,n we have ti ∈ i with p (u′′,ti) 6⊆ fi and [0,ri + �) 6⊆ qti, that is ti < ri + �. take any v1,v2 ∈ i with p (s′′′,v1) 6⊆ fi and fi 6⊆ q(u′′,v2). di-uniform texture spaces 183 by df1 we have pv1 6⊆ qri and pti 6⊆ qv2 , whence ri ≤ v1 and v2 ≤ ti. thus v2 ≤ ti < ri + � ≤ v1 + �, so p (v1,v2) ⊆ d�. since ps′′ 6⊆ qs′′′ and pu′′ 6⊆ qu we deduce that (fi,fi) −1(d�) 6⊆ q(s′′,u) ∀i = 1, . . . ,n, whence p (s′,u) ⊆ (f1,f1)−1(d�) u . . .u (fn,fn)−1(d�). since pu 6⊆ qu′ we now have e 6⊆ q(s′,u′). on the other hand pu′ 6⊆ e[z] gives v ∈ s with pu′ 6⊆ qv for which e 6⊆ q(x,v) =⇒ pz ⊆ qx∀x ∈ s. from the above we have e 6⊆ q(s′,v) so setting x = s′ in the above implication leads to the contradiction pz ⊆ qs′. to prove (b) it will suffice to show pw ⊆ ⋂n i=1 f ← i ([0,ri + �)), so assume the contrary. now we have j, 1 ≤ j ≤ n, and pw 6⊆ qw′ for which p (w′,y) 6⊆ fj =⇒ [0,rj + �) ⊆ qy. however p (w,rj) 6⊆ fj and pw 6⊆ qw′ gives p (w′,rj) 6⊆ fj, and we obtain the contradiction [0,rj + �) ⊆ [0,rj) by taking y = rj in the above implication. conversely, take g ∈ τ and s ∈ s with g 6⊆ qs. since (τ,κ) is completely regular there exists a bicontinuous difunction (f,f) on (s, s) to (i, i) for which f←(p0) 6⊆ qs and f←(q1) ⊆ g. take � > 0 and define e = (f,f)−1(d�). then e ∈ u and we will show that e[s] ⊆ g, whence g ∈ τu. it will be sufficient to show e[s] ⊆ f←(q1), so assume the contrary and take v ∈ s with e[s] 6⊆ qv and pv 6⊆ f←(q1). the latter gives us v′ ∈ s with (5.7) p (v′,w) 6⊆ f =⇒ q1 ⊆ qw =⇒ w = 1, and the former gives z ∈ s with e 6⊆ qv and ps 6⊆ qz. now we have v′′ ∈ s with p (z,v′′) 6⊆ q(z,v) and z′ ∈ s with pz 6⊆ qz′ so that (5.8) p (z′,t1) 6⊆ f, f 6⊆ q(v′,t2) =⇒ p (t1,t2) ⊆ d� =⇒ t2 < t1 + �. clearly pv′′ 6⊆ qv′ so by df1 there exists t′ ∈ i with f 6⊆ q(v′′,t′) and p (v′,t′) 6⊆ f. now (7) with w = t′ gives t′ = 1 and so f 6⊆ q(v′′,1). on the other hand, from f←(p0) 6⊆ qs we have s′ ∈ s with f←(p0) 6⊆ qs′, ps′ 6⊆ qs. hence we have u ∈ s with pu 6⊆ qs′ so that (5.9) f 6⊆ q(u,w) =⇒ pw ⊆ p0 =⇒ w = 0. clearly pu 6⊆ qz so by df1 we have t ∈ i so that f 6⊆ q(u,t), p (z′,t) 6⊆ f. now (9) with w = t gives t = 0, so p (z′,0) 6⊆ f and (8) with t1 = 0, t2 = 1 gives the contradiction 1 < �. we have now established τ = τu, and a dual proof gives κ = κu, so the proof is complete. � before leaving the topics of uniform bicontinuity and initial di-uniformity we must see how these should be defined for dicovering di-uniformities. the following gives a fairly obvious notion of inverse image of a dicover under a difunction. 184 s. özçağ and l. m. brown definition 5.17. let (s, s), (t, t) be textures, (f,f) a difunction on (s, s) to (t, t) and c a dicover of (t, t). then (f,f)−1(c) = {(f←(a),f←(b)) | a c b}. it is a straightforward matter to verify that (f,f)−1(c) is a dicover of (s, s), but the authors do not know if this inverse operation preserves the property of being anchored, or even of being refined by the dicover p. this will cause some technical difficulties, but will not prevent us using this operation to characterize uniform bicontinuity and initial diuniformities in terms of dicovers, as we will see. we begin by relating this inverse image with that given earlier for direlations. proposition 5.18. let (f,f) : (s, s) → (t, t) be a difunction and (d,d) a reflexive direlation on (t, t). then γ((f,f)−1(d,d)) ≺ ((f,f)−1(γ(d,d)))∆. proof. let d = γ(d,d) and c = (f,f)−1(d). if we set c = (f,f)−1(d) and c = (f,f)−1(d) we must verify c[s] ⊆ st(c,ps) and cst(c,qs) ⊆ c[s]. we prove the first inclusion, the second being dual. recall that c[s] = c(ps) and suppose that c(ps) 6⊆ st(c,ps). now we have u ∈ t with c(ps) 6⊆ qu, pu 6⊆ st(c,ps) and hence s′ ∈ s with c 6⊆ q(s′,u), ps 6⊆ qs′. by remark 5.2 we have p (s′,u′) 6⊆ q(s′,u) and ps′ 6⊆ qs′′ for which (5.10) p (s′′,t1) 6⊆ f, f 6⊆ q(u′,t2) =⇒ d 6⊆ q(t1,t2) for all t1, t2 ∈ t . on the other hand, pu 6⊆ st(c,ps) = ∨ {f←(d[z]) | z ∈ t, ps 6⊆ f←(d[z])} = f←( ∨ {d[z] | z ∈ t, ps 6⊆ f←(d[z])}) by proposition 1.6 (7), and so we have w ∈ t satisfying f 6⊆ q(u,w) and pw 6⊆ ∨ {d[z] | z ∈ t, ps 6⊆ f←(d[z])}. since pu′ 6⊆ qu we have f 6⊆ q(u′,w). on the other hand applying df1 to ps′ 6⊆ qs′′ gives v ∈ t satisfying f 6⊆ q(s′,v) and p (s′′,v) 6⊆ f. we may now apply implication (5.10) with t1 = v, t2 = w to give d 6⊆ q(v,w). this is equivalent to d[v] 6⊆ qw, and so pw ⊆ d[v]. to obtain a contradiction it will therefore suffice to show that f←(d[v]) ⊆ qs′, for then ps 6⊆ f←(d[v]) and so pw 6⊆ d[v]. suppose, therefore, that f←(d[v]) 6⊆ qs′. then we have t ∈ t with p (s′,t) 6⊆ f and d[v] 6⊆ qt. using df2 now gives pt 6⊆ qv, whence d[v] ⊆ qv ⊆ qt since d is reflexive. this contradiction completes the proof. � proposition 5.19. let (f,f) : (s, s) → (t, t) be a difunction and (c,c), (d,d) reflexive direlations on (t, t). then (d,d) ◦ (d,d)← v (c,c) =⇒ δ((f,f)−1(γ(d,d))) v (f,f)−1(c,c). proof. assume that (d,d)◦(d,d)← v (c,c), i.e. d◦d← ⊆ c and c ⊆ d◦d←. let d = γ(d,d), e = (f,f)−1(d), and assume that d(e) 6⊆ (f,f)−1(c). now we have s,s′ ∈ s with p (s,s′) 6⊆ (f,f)−1(c) and t ∈ t with ps 6⊆ di-uniform texture spaces 185 f←(d(qt)), f←(d(pt)) 6⊆ qs′. hence we have v ∈ t with f 6⊆ q(s,v), pv 6⊆ d(qt), and v′ ∈ t with p (s′,v′) 6⊆ f, d(pt) 6⊆ qv′. also, by r2 for the relation f, we have u ∈ s with ps 6⊆ qu and f 6⊆ q(u,v). hence, since p (s,s′) 6⊆ (f,f)−1(c), there exists t1, t2 ∈ t such that p (u,t1) 6⊆ f, f 6⊆ q(s′,t2) and p (t1,t2) 6⊆ c. on the other hand, from d(pt) 6⊆ qv′ we have d 6⊆ q(t,v′) and from pv 6⊆ d(qt) we have p (t,v) 6⊆ d, that is d← 6⊆ q(v,t). since (f,f) is a difunction, p (s′,v′) 6⊆ f and f 6⊆ q(s′,t2) imply pv′ 6⊆ qt2 by df2, so d 6⊆ q(t,t2). likewise, pt1 6⊆ qv and so d← 6⊆ q(t1,t) by r1 for the relation d ←. we now obtain p (t1,t2) ⊆ d◦d ← ⊆ c, which is a contradiction. the proof of d(e) ⊆ (f,f)−1(c) is dual to the above, and is omitted. � with the notation of theorem 3.7 we now have: proposition 5.20. let (f,f) : (s, s) → (t, t) be a difunction. if u, v are direlational di-uniformities on (s, s), (t, t), respectively, then (f,f) is u–v uniformly bicontinuous if and only if c ∈ γ(v) =⇒ (f,f)−1(c)∆ ∈ γ(u). proof. suppose (f,f) is u–v uniformly bicontinuous and take c ∈ γ(v). now we have (c,c) ∈ v with γ(c,c) ≺ c, so (d,d) = (f,f)−1(c,c) ∈ u and γ(d,d) ∈ γ(u). however γ(d,d) ≺ (f,f)−1(c)∆ by proposition 5.18, so (f,f)−1(c)∆ ∈ γ(u). conversely suppose c ∈ γ(v) =⇒ (f,f)−1(c)∆ ∈ γ(u) and take (e,e) ∈ v. choose a symmetric (c,c) ∈ v with (c,c) ◦ (c,c) v (e,e) and (d,d) ∈ v with (d,d) ◦ (d,d)← v (c,c). by corollary 5.6, (f,f)−1(c,c) is also symmetric, and it is reflexive by corollary 5.4. hence by theorem 2.7 (1) and proposition 5.7, δ(γ((f,f)−1(c,c))) = (f,f)−1(c,c) ◦ (f,f)−1(c,c) v (f,f)−1((c,c) ◦ (c,c)) v (f,f)−1(e,e). by proposition 5.19, δ((f,f)−1(γ(d,d)) v (f,f)−1(c,c) and so δ((f,f)−1(γ(d,d))∆) = δ(γ(δ((f,f)−1(γ(d,d)))) v δ(γ((f,f)−1(c,c)) v (f,f)−1(e,e) by theorem 2.7 (2). since (d,d) ∈ v, c = γ(d,d) ∈ γ(v) and so, by hypothesis, (f,f)−1(γ(d,d))∆ = (f,f)−1(c)∆ ∈ γ(u). hence δ((f,f)−1(γ(d,d))∆) ∈ ∆(γ(u)) = u by theorem 3.7 (3). it follows from the above inclusion that (f,f)−1 (e,e) ∈ u, and we have shown that (f,f) is u–v uniformly bicontinuous. � this justifies the following definition. 186 s. özçağ and l. m. brown definition 5.21. let υ, ν be dicovering uniformities on (s, s), (t, t) respectively and (f,f) : (s, s) → (t, t) a difunction. then (f,f) is called υ–ν uniformly bicontinuous if c ∈ ν =⇒ (f,f)−1(c)∆ ∈ υ. finally we have the following: proposition 5.22. let (s, s) be a texture and for i ∈ i let (ti, ti, vi) be a direlational di-uniform texture space and (fi,fi) : (s, s) → (ti, ti) a difunction. if u is the initial direlational uniformity on (s, s) for the given system, the family( n∧ k=1 (fik,fik) −1(cik) ∆ )∆ , n ∈ n+, ik ∈ i, cik ∈ γ(vik), 1 ≤ k ≤ n, is a base for the dicovering di-uniformity γ(u). proof. take c ∈ γ(u). then there exists (e,e) ∈ u satisfying γ(e,e) ≺ c, and hence ik ∈ i, (eik,eik) ∈ vik for 1 ≤ k ≤ n with dn k=1(fik,fik) −1(eik,eik) v (e,e). if we choose a symmetric (cik,cik) ∈ vik with (cik,cik) ◦ (cik,cik) v (eik,eik), and then (dik,dik) ∈ vik with (dik,dik) ◦ (dik,dik) ← v (cik,cik), we have δ((fik,fik) −1(γ(dik,dik)) ∆ v (fik,fik) −1(eik,eik), exactly as in the proof of proposition 5.20. in view of proposition 2.9 (2) we deduce δ ( n∧ k=1 (fik,fik) −1(γ(dik,dik)) ∆ ) v nl k=1 δ ( (fik,fik) −1(γ(dik,dik)) ∆ ) v nl k=1 ( (fik,fik) −1(eik,eik) ) v (e,e). applying γ to both sides and using theorem 2.7 (2) now gives( n∧ k=1 (fik,fik) −1(cik) ∆ )∆ ≺ c, where we have set cik = γ(dik,dik) ∈ γ(vik). it remains to show that the dicover on the left belongs to γ(u). now d = γ( dn k=1(fik,fik) −1(dik,dik)) ∈ γ(u) is anchored by proposition 2.3, and d ≺ ∧n k=1 γ((fik,fik) −1(dik,dik) by proposition 2.8 (1), so by lemma 2.2 (i) we have d ≺ ( n∧ k=1 γ((fik,fik) −1(dik,dik)) )∆ ≺ ( n∧ k=1 (fik,fik) −1(cik) ∆ )∆ by proposition 5.18, and this gives the required result. � in view of the above proposition, the following definition is compatible with definition 5.15. definition 5.23. let (s, s) be a texture and for each i ∈ i let (ti, ti,νi) be a dicovering di-uniform texture space and (fi,fi) : (s, s) → (ti, ti) a difunction. di-uniform texture spaces 187 then the covering di-uniformity υ on (s, s) with base{ n∧ k=1 ( (fi,fi) −1(cik) ∆ )∆ | ik ∈ i, cik ∈ νik, 1 ≤ k ≤ n, n ∈ n + } is called the initial covering di-uniformity on (s, s) for the spaces (ti, ti,νi) and difunctions (fi,fi), i ∈ i. it is not known if the above results and definitions can be simplified in general. 6. dimetrics and diuniformities definition 6.1. let (s, s) be a texture, ρ, ρ : s × s → [0,∞) two point functions. then ρ = (ρ,ρ) is called a pseudo dimetric on (s, s) if m1 ρ(s,t) ≤ ρ(s,u) + ρ(u,t) ∀s,u,t ∈ s, m2 ps 6⊆ qt =⇒ ρ(s,t) = 0 ∀s,t ∈ s, dm ρ(s,t) = ρ(t,s) ∀s,t ∈ s, cm1 ρ(s,t) ≤ ρ(s,u) + ρ(u,t) ∀s,u,t ∈ s, cm2 pt 6⊆ qs =⇒ ρ(s,t) = 0 ∀s,t ∈ s. in this case ρ is called the pseudo metric, ρ the pseudo cometric of ρ. if ρ is a pseudo dimetric which satisfies the conditions m3 ps 6⊆ qu, ρ(u,v) = 0, pv 6⊆ qt =⇒ ps 6⊆ qt ∀s,t,u,v ∈ s, cm3 pu 6⊆ qs, ρ(u,v) = 0, pt 6⊆ qv =⇒ pt 6⊆ qs ∀s,t,u,v ∈ s, it is called a dimetric. when giving examples it will clearly suffice to give ρ satisfying the metric conditions, since dm may then be used to define ρ, which will automatically satisfy the cometric conditions. note that for a pseudo dimetric to be a dimetric it is sufficient that ρ(s,t) = 0 =⇒ ps 6⊆ qt, but example (4) below shows this condition is not necessary in general. example 6.2. (1) let (s, s) be any texture and define ρ(s,t) = { 0 if ps 6⊆ qt, 1 otherwise. clearly ρ defines a dimetric ρ, which we will call the discrete dimetric on (s, s). (2) if d is a (pseudo) metric on x in the usual sense, ρ = (d,d) is a (pseudo) dimetric on (x,p(x)). (3) consider the texture (i, i) and set ρ(s,t) = (t−s) ∨ 0. then ρ defines a dimetric ρ on (i, i), which we will call the usual dimetric on (i, i). (4) let (l, l) be the texture l = (0, 1], l = {(0,r] | 0 ≤ r ≤ 1}. again ρ(s,t) = (t−s) ∨ 0 defines a dimetric, called the usual dimetric on (l, l). 188 s. özçağ and l. m. brown note that (2) and (4) may be combined to give a rich supply of pseudo dimetrics on the product of (x,p(x)) and (l, l). since this is the texture corresponding to the lattice of classic fuzzy sets on x [4,6] the connection with fuzzy topology is clear, although we will not pursue this line of enquiry here. as expected, a (pseudo) dimetric ρ gives rise to a ditopology, which we will refer to as the (pseudo) metric ditopology of ρ. proposition 6.3. let ρ be a pseudo dimetric on (s, s) and for s ∈ s[, � > 0 define nρ� (s) = ∨ {pt | ∃u ∈ s with ps 6⊆ qu, ρ(u,t) < �}, mρ� (s) = ⋂ {qt | ∃u ∈ s with pu 6⊆ qs, ρ(u,t) < �}. then βρ = {nρ� (s) | s ∈ s[, � > 0} is a base and γρ = {mρ� (s) | s ∈ s[, � > 0} a cobase for a ditopology (τρ,κρ) on (s, s). proof. by m2 it is clear that ps ⊆ nρ� (s) for all s ∈ s[ and so ∨ βρ = s. now take s1,s2,s ∈ s[, �1,�2 > 0 with nρ�1 (s1) ∩n ρ �2 (s2) 6⊆ qs. choose t ∈ s with nρ�1 (s1) ∩n ρ �2 (s2) 6⊆ qt, pt 6⊆ qs and then for k = 1, 2 take tk ∈ s with ptk 6⊆ qt so that for some psk 6⊆ qs′k, s ′ k ∈ s, we have ρ(s ′ k, tk) < �k. since ρ(tk, t) = 0 by m2 we deduce ρ(s′k, t) < �k for k = 1, 2 by m1, so we may choose � ∈ r satisfying 0 < � < min(�1 −ρ(s′1, t), �2 −ρ(s′2, t)). however it is now straightforward to verify that nρ� (t) ⊆ n ρ �1 (s1) ∩nρ�2 (s2), n ρ � (t) 6⊆ qs whence by ([6], theorem 4.3), βρ is a base for some topology τρ on (s, s). the proof that γρ is a base for some cotopology κρ on (s, s) is dual to this, and is omitted. � clearly the discrete dimetric on (s, s) gives rise to the discrete, codiscrete ditopology. likewise, the metric ditopology of the usual dimetric on (i, i) is the usual ditopology on (i, i), while the same dimetric on (l, l) gives the discrete, codiscrete ditopology. now let us verify that a pseudo dimetric also defines a di-uniformity. theorem 6.4. let ρ be a pseudo dimetric on (s, s). i) for � > 0 let r� = r ρ � = ∨ {p (s,t) | ∃u ∈ s, ps 6⊆ qu and ρ(u,t) < �}, r� = r ρ � = ⋂ {q(s,t) | ∃u ∈ s, pu 6⊆ qs and ρ(u,t) < �}. then the family {(rρ� ,rρ� ) | � > 0} is a base for a direlational uniformity uρ on (s, s). ii) the di-uniformity uρ is separated if and only if ρ is a dimetric. iii) the uniform ditopology of uρ coincides with the pseudo metric ditopology of ρ. di-uniform texture spaces 189 proof. (i) it is trivial to verify that (r�,r�) is a direlation for all � > 0. we must verify the conditions of definition 3.1 for the family uρ = {(r,r) | ∃� > 0, (r�,r�) v (r,r)}. condition (1) is trivial from m1, cm1; and (2) follows from the definition. condition (3) is a consequence of (r�,r�) v (r�1,r�1 ) u (r�2,r�2 ), where � = min(�1,�2), which is trivial since clearly � ≤ δ =⇒ (r�,r�) v (rδ,rδ). to prove (4) we need only show that (r�,r�)2 v (r2�,r2�). if r� ◦ r� 6⊆ r2� there exists s,t ∈ s with p (s,t) 6⊆ r2� so that for some u,v ∈ s we have ps 6⊆ qu, r� 6⊆ q(u,v) and r� 6⊆ q(v,t). by m1, m2 and the definition of r� we obtain ρ(u,v) < �, ρ(v,t) < �, whence ρ(u,t) ≤ ρ(u,v) + ρ(v,t) < 2�. this gives the contradiction p (s,t) ⊆ r2� so r� ◦ r� ⊆ r2�, and the dual result r2� ⊆ r� ◦ r� is proved likewise. finally (5) follows from (r�,r�)← = (r�,r�). to prove this we need only show that r←� = r� for any � > 0. suppose that r� 6⊆ r←� . then we have s,t ∈ s with r� 6⊆ q(s,t) and p (s,t) 6⊆ r←� . since r←� is a corelation, p (s,t) 6⊆ r←� is equivalent to r� 6⊆ q(t,s) and so we have s′ ∈ s satisfying p (t,s′) 6⊆ q(t,s) for which we have t′ ∈ s with pt 6⊆ qt′ and ρ(t′,s′) < �. by m1 we have ρ(t,s′) < �, whence ρ(s′, t) < � by dm. since ps′ 6⊆ qs we obtain r� ⊆ q(s,t), which is a contradiction. this establishes r� ⊆ r←� , and the reverse inclusion is proved in the same way. this completes the proof that uρ is a direlational uniformity on (s, s). (ii) it is sufficient to show that m3 is equivalent to d �>0 r� ⊆ i. suppose that m3 holds but d �>0 r� 6⊆ i. now we have s,t ∈ s with d �>0 r� 6⊆ q(s,t) and p (s,t) 6⊆ i. hence we have t′ ∈ s with p (s,t′) 6⊆ q(s,t) so that for some s′ ∈ s with ps 6⊆ qs′ we have r� 6⊆ q(s′,t′) ∀� > 0. we deduce ρ(s′, t′) = 0 and so ps 6⊆ qt by m3. however now i 6⊆ q(s,t), which contradicts p (s,t) 6⊆ i. the proof that d �>0 r� ⊆ i implies m3 is left to the interested reader. (iii) by lemma 4.3 the set g ∈ s is open for the uniform ditopology if and only if g 6⊆ qs =⇒ ∃� > 0 with r�[s] ⊆ g. since ps ⊆ r�(ps) = r�[s], if we can show that r�[s] is uniformly open it will follow by ([6], theorem 4.2) that the family r�[s], s ∈ s[, � > 0, is a base for τuρ. however, if we take r�[s] 6⊆ qt, we then have t′ ∈ s with p (s,t′) 6⊆ q(s,t), so that ρ(s′, t′) < � for some s′ ∈ s with ps 6⊆ qs′. since pt′ 6⊆ qt we have ρ(t′, t) = 0 and so ρ(s′, t) < �, whence we may choose δ > 0 with ρ(s′, t) + δ < � and it is now easy to show that rδ[t] ⊆ r�[s]. this establishes that r�[s] is uniformly open, as required. finally it is straightforward to verify that r�[s] = n ρ � (s), so the family nρ� (s), s ∈ s[, � > 0, is a base for both τuρ and τρ, whence these topologies coincide. likewise, the cotopologies coincide. � 190 s. özçağ and l. m. brown corollary 6.5. a pseudo metric ditopology is completely biregular. it is t0, and hence bi–t3 12 [7] and in particular bi–t2 [7], if and only if ρ is a dimetric. proof. immediate from theorem 6.4, theorem 4.14 and theorem 4.16. � for the dimetric of example 6.2 (3) we obtain the discrete direlational uniformity with base {(i,i)}. the metric di-uniformity of the usual dimetric on (i, i) is the usual di-uniformity, while the dicovering uniformity corresponding to the usual dimetric on (l, l) has base {((0,s + �], (0,s− �]) | 0 < s < 1} (cf. example 3.8). definition 6.6. a direlational uniformity u on (s, s) is called (pseudo) metrizable if there exists a (pseudo) dimetric ρ with u = uρ. theorem 6.7. a direlational uniformity u is pseudo metrizable if and only if it has a countable base. it is metrizable if and only if it is also separated. proof. if u is pseudo metrizable there is a pseudo dimetric ρ with u = uρ. but now, for example, (rρ 1/n ,r ρ 1/n ), n ≥ 1, is a countable base of uρ, and hence of u. conversely, let u have the countable base (bn,bn), n ≥ 1. take (d1,d1) ∈ u symmetric with (d1,d1) v (b1,b1), and by induction for n > 1 choose a symmetric (dn,dn) ∈ u so that (dn,dn)3 v (dn−1,dn−1) u (bn,bn). then (dn+1,dn+1)3 v (dn,dn) for all n = 1, 2, . . . and (dn,dn) v (dn,dn)3 v (bn,bn), so {(dn,dn) | n = 1, 2, . . .} is also a base of u. now let q, q∗ be as defined in lemma 4.12 and lemma 4.13 for the sequence (dn,dn). clearly ρ = (q,q∗) is a pseudo dimetric, so we may consider the direlational uniformity uρ. however, lemma 4.12 (1) and remark 4.13 (1) immediately give (dn+1,dn+1) v (r ρ 2−n ,r ρ 2−n ) v (dn,dn), whence u = uρ. the final statement is immediate from theorem 4.16 and corollary 6.5. � definition 6.8. a ditopology on (s, s) is called (pseudo) metrizable if it is the metric ditopology of some (pseudo) dimetric on (s, s). theorem 6.9. the ditopology (τ,κ) on (s, s) is pseudo metrizable if and only if there exists a family cn, n = 1, 2, . . . of anchored dicovers of (s, s) satisfying the conditions (1) cn+1 ≺(?) cn for all n ≥ 1. (2) g ∈ τ ⇐⇒ (g 6⊆ qs =⇒ ∃n, st(cn,ps) ⊆ g). (3) f ∈ κ ⇐⇒ (ps 6⊆ f =⇒ ∃n, f ⊆ cst(cn,qs)). proof. suppose first that (τ,κ) = (τρ,κρ) for some pseudo dimetric ρ, and consider the direlational uniformity uρ on (s, s). note that (r ρ 4−n ,r ρ 4−n ), n ≥ 1, is a base of uρ, whence cn = γ(r ρ 4−n ,r ρ 4−n ), n ≥ 1, is a base of anchored dicovers for υρ = γ(uρ). by the hypothesis and theorem 6.4 (iii), (τ,κ) is the uniform ditopology of uρ, while uρ = ∆(υρ) by theorem 3.7 (3). clearly (2) and (3) now follow from proposition 4.4, so it remains to show (1). however di-uniform texture spaces 191 noting that (rρ� ,r ρ � ) is symmetric we may apply proposition 2.4 to (r ρ � ,r ρ � ) 2 v (rρ2�,r ρ 2�) for � = 4 −(n+1) and � = 2 × 4−(n+1) to give cn+1 ≺(∆) γ(r ρ 2×4−(n+1), r ρ 2×4−(n+1) ) ≺(∆) cn, whence cn+1 ≺(?) cn by lemma 2.2 (3 ii), since the dicovers are anchored. conversely, suppose that there exists a sequence of anchored dicovers cn satisfying (1)–(3). then these form a base for a dicovering uniformity υ. moreover, by proposition 4.4, conditions (2) and (3) imply that the uniform topology of υ, and hence of u = ∆(υ), is (τ,κ). clearly δ(cn), n = 1, 2, . . . is a countable base of u, so by theorem 6.7 there is a pseudo dimetric ρ for which the uniform ditopology of uρ = u is the metric ditopology of ρ. hence (τ,κ) = (τρ,κρ), so (τ,κ) is pseudo metrizable. � clearly conditions (2) and (3) may also be given in terms of the dineighbourhood system and theorem 6.9 is then seen as a ditopological analogue of the alexandroff-urysohn metrization theorem [12]. we end by showing that arbitrary di-uniformities may be defined using pseudo dimetrics. definition 6.10. let u be a direlational uniformity on (s, s). then a pseudo dimetric ρ on (s, s) is called uniform for u if (rρ� ,r ρ � ) ∈ u ∀� > 0. for pseudo metrics ρ1, ρ2 on (s, s), ρ1 ∨ρ2 = (ρ1 ∨ρ2, ρ1 ∨ρ2) is a pseudo metric on (s, s), and clearly (rρ1� ,r ρ1 � ) u (rρ2� ,rρ2� ) = (rρ1∨ρ2� ,rρ1∨ρ2� ). hence the family g of pseudo dimetrics on (s, s) uniform for u has the property ρ1,ρ2 ∈ g =⇒ ρ1 ∨ρ2 ∈ g. this leads to the following: theorem 6.11. let g be a non-empty family of pseudo dimetrics on (s, s) which is closed under finite suprema. then ug = {(r,r) | ∃ρ ∈ g, � > 0 with (rρ� ,r ρ � ) v (r,r)} is a direlational uniformity on (s, s). moreover, if u is a direlational uniformity on (s, s) and g the set of pseudo dimetrics uniform for u then ug = u. proof. straightforward. � acknowledgements. the authors would like to thank the referees for their helpful comments references [1] l. m. brown, relations and functions on textures, preprint. [2] l. m. brown and m. diker, ditopological texture spaces and intuitionistic sets, fuzzy sets and systems 98 (1998), 217–224. 192 s. özçağ and l. m. brown [3] l. m. brown and m. diker, paracompactness and full normality in ditopological texture spaces, j. math. anal. appl. 227 (1998), 144–165. [4] l. m. brown and r. ertürk, fuzzy sets as texture spaces, i. representation theorems, fuzzy sets and systems 110 (2) (2000), 227–236. [5] l. m. brown and r. ertürk, fuzzy sets as texture spaces, ii. subtextures and quotient textures, fuzzy sets and systems 110 (2) (2000), 237–245. [6] l. m. brown, r. ertürk and ş. dost, ditopological texture spaces and fuzzy topology, i. general principles, submitted. [7] l. m. brown, r. ertürk and ş. dost, ditopological texture spaces and fuzzy topology, ii. separation axioms, preprint, 2002. [8] m. diker, connectedness in ditopological texture spaces, fuzzy sets and systems 108 (1999), 223–230. [9] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. mislove and d. s. scott, a compendium of continuous lattices (springer–verlag, berlin, 1980). [10] j. l. kelley, general topology (d. van nostrand, princeton, 1955). [11] g. sambin, some points in formal topology, (proc. dagstuhl seminar topology in computer science, june 2000), theoretical computer science, to appear. [12] j. w. tukey, convergence and uniformity in topology, (ann. of math. studies 2, 1940). [13] a. weil, sur les espaces à structure uniforme et sur la topologie générale (actualités sci. ind. 551, paris, 1937). received july 2002 revised december 2002 s. özçağ and l. m. brown hacettepe university, faculty of science, department of mathematics, 06532 beytepe, ankara, turkey. e-mail address : sozcag@hacettepe.edu.tr brown@hacettepe.edu.tr di-uniform texture spaces. by s. özçag and l. m. brown @ appl. gen. topol. 18, no. 1 (2017), 75-90 doi:10.4995/agt.2017.5869 c© agt, upv, 2017 a new cardinal function on topological spaces dewi kartika sari a,b and dongsheng zhao a a mathematics and mathematics education, national institute of education, nanyang technological university, singapore (nie15dewi2662@e.ntu.edu.sg, dongsheng.zhao@nie.edu.sg) b department of mathematics, faculty of mathematics and natural sciences, universitas gadjah mada, indonesia. communicated by j. galindo abstract using neighbourhood assignments, we introduce and study a new cardinal function, namely gci(x), for every topological space x. we shall mainly investigate the spaces x with finite gci(x). some properties of this cardinal in connection with special types of mappings are also proved. 2010 msc: 54a25; 54c08; 54d3. keywords: neighbourhood assignment; gauge compact space; gauge compact index; m-uniformly continuous mapping. 1. introduction a neighbourhood assignment on a topological space is a function that assigns an open neighbourhood to each point in the space. a number of classes of topological spaces can be characterized by means of neighbourhood assignments (for example, compact spaces, d-spaces, connected spaces, metrizable spaces)[14, 3, 5, 9]. neighbourhood assignments have also been employed to characterize special type of mappings, in particular those that are linked to baire class one functions [1][13]. in [2], using neighbourhood assignments, a compactness type of topological property (called gauge compactness), was defined. this property is weaker than compactness in general and equivalent to compactness for tychonoff spaces. in order to refine the classification of gauge compact spaces, here we define a cardinal gci(x) for every topological space x, called the gauge compact index of x. as we shall see, this cardinal received 03 june 2016 – accepted 18 october 2016 http://dx.doi.org/10.4995/agt.2017.5869 d. k. sari and d. zhao indeed reveals some subtle distinctions between certain spaces. an immediate and natural task is to determine the spaces x with gci(x) equal to a given value. the layout of the paper is as follows. in section 2, we define the gauge compact index and prove some general properties. section 3 is mainly about t1 spaces having gauge compact index 2. in section 4, we study spaces having gauge compact index 3. section 5 is devoted to the links between gauge compact indices and special types of mappings between topological spaces. one of the results is that every m-uniformly continuous mapping (in particular, every continuous mapping) from a space with gauge compact index 2 to a t1 space is a constant function. from the results proved in this paper, it appears that having a finite gauge compact index is a property more useful for non-hausdorff spaces than to hausdorff spaces. although in classical topology, the main focus was on hausdorff spaces, the interests in non-hausdorff spaces had been growing for quite some time. this is particularly the case in domain theory which has a strong background in theoretical computer science. the most important topology in domain theory is the scott topology of a poset, which is usually only t0 (it is t1 iff the poset is discrete). see [8] for a systematic treatment of non-hausdorff spaces from the point of view of domain theory. this paper contains some results about the gauge compact indices of scott spaces of algebraic posets. more investigation on gauge compact indices of scott spaces (poset with their scott topology) are expected. 2. the gauge compact index of a space in what follows, in order to be consistent with the definition of gauge compactness, a neighbourhood assignment on a space will be simply called a gauge on the space and the symbol ∆(x) will be used to denote the collection of all gauges on the space x. all topological spaces considered below will be assumed to be non-empty. a space x is called gauge compact if for any δ ∈ ∆(x) there exists a finite subset a ⊆ x such that for every x ∈ x, there exists a ∈ a so that either x ∈ δ(a) or a ∈ δ(x) holds [2]. equivalently, x is gauge compact if and only if for any δ ∈ ∆(x), there is a finite set a ⊆ x such that δ(x)∩a 6= ∅ holds for every x ∈ x − ⋃ a∈a δ(a). let a,b be non-empty subsets of space x and δ ∈ ∆(x). we write a ≺mδ b if for any x ∈ a, there exists y ∈ b such that x ∈ δ(y) or y ∈ δ(x). also {x}≺mδ {y} will be simply written as x ≺ m δ y (thus x ≺ m δ y iff either x ∈ δ(y) or y ∈ δ(x)). so a space x is gauge compact if and only if for any δ ∈ ∆(x), there is a finite set a ⊆ x satisfying x ≺mδ a. definition 2.1. the gauge compact index of a space x, denoted by gci(x), is defined as gci(x) = inf{β : ∀δ ∈ ∆(x),∃a ⊆ x so that |a| < β and x ≺mδ a}, c© agt, upv, 2017 appl. gen. topol. 18, no. 1 76 a new cardinal function on topological spaces where β is a cardinal number and |a| is the cardinality of set a. remark 2.2. (1) a space x is gauge compact if and only if gci(x) ≤ℵ0. (2) since all topological spaces considered here are non-empty, the gauge compact index of a space is at least 2. (3) if τ1 and τ2 are two topologies on x such that τ1 ⊆ τ2, then gci(x,τ1) ≤ gci(x,τ2). (4) if x is a discrete finite space with |x| = n, then gci(x) = n + 1. conversely, if |x| = n and gci(x) = n + 1 with n finite, then x is a discrete space. (5) if there exists δ ∈ ∆(x) such that for any a ⊆ x, x ≺mδ a implies |a| ≥ m, then gci(x) ≥ m + 1. let b be a base of the topology of space x and ∆b(x) be the collection of all gauges δ ∈ ∆(x) satisfying δ(x) ∈ b for every x ∈ x. the following lemma shows that to determine the gauge compact indices we can only consider δ ∈ ∆b(x). lemma 2.3. let b be a base of the topology of space x. then gci(x) = inf{β : ∀λ ∈ ∆b(x),∃a ⊆ x, |a| < β and x ≺mλ a}. proof. let α0 = inf{β : ∀λ ∈ ∆b(x),∃a ⊆ x, |a| < β and x ≺mλ a}. we need to show that gci(x) = α0. for any δ ∈ ∆b(x), δ is also in ∆(x), so there is a ⊆ x satisfying |a| < gci(x) and x ≺mδ a. this implies that α0 ≤ gci(x). now, for any δ ∈ ∆(x), we can construct a λ ∈ ∆b(x) so that λ(x) ⊆ δ(x) for each x ∈ x. then there exists a ⊆ x such that |a| < α0 and x ≺mλ a. but then x ≺mδ a also holds. hence gci(x) ≤ α0. all these show that gci(x) = α0. � example 2.4. let x = (r,τup), where τup is the upper topology on r (u∈ τup iff u = ∅, u = r, or u = (a, +∞) for some a ∈ r). then for any δ ∈ ∆(x) and x ∈ r, 0 ∈ δ(x) if x < 0 and x ∈ δ(0) if 0 ≤ x. hence x ≺mδ {0} holds and so gci(x) = 2. note that x is not compact. we first prove some general results on gauge compact indices. theorem 2.5. the gauge compact index of a hausdorff space x equals a finite integer n if and only if |x| = n− 1. proof. let gci(x) = n. firstly, as every finite hausdorff space is discrete, so by remark 2.2(4), |x| 6< n− 1. now assume that |x| > n−1. take n distinct elements x1,x2, · · · ,xn in x. since x is hausdorff, we can choose disjoint open sets γ(xi) with xi ∈ γ(xi) (i = 1, 2, · · · ,n). define the gauge δ on x as follows: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 77 d. k. sari and d. zhao δ(x) =   γ(xi), if x ∈ γ(xi); x −{x1,x2, · · · ,xn}, if x ∈ x − n⋃ i=1 γ(xi). then x ≺mδ xi if and only if x ∈ γ(xi). also, as γ(xi)’s are n disjoint sets, for any n−1 distinct elements y1,y2, · · · ,yn−1 in x, there must be an i0 such that {y1,y2, · · · ,yn−1}∩γ(xi0 ) = ∅. so, xi0 ≺mδ yi does not hold for each i. hence x ≺ m δ {y1,y2, · · · ,yn−1} fails, which contradicts the assumption that gci(x) = n. it thus follows that |x| = n− 1. conversely, if x is a hausdorff space such that |x| = n − 1, then x is a discrete space. by remark 2.2(4), gci(x) = n. � from theorem 2.5 and remark 2.2(1), we obtain the following. corollary 2.6. (1) if x is hausdorff, then every subspace y ⊆ x with a finite gauge compact index is discrete. (2) every hausdorff space with a finite gauge compact index is finite and therefore compact. (3) if x is a gauge compact hausdorff space and x is an infinite set, then gci(x) = ℵ0. the example below shows that the converse of corollary 2.6(1) is not always true. by proposition 9 of [2], a topological space x is gauge compact if and only if for any net {xn} in x, either the net is gauge clustered or it has a cluster point. here, a net {xn} is called gauge clustered if for any δ ∈ ∆(x), there is a subnet {xnk} such that ⋂ k δ(xnk ) 6= ∅. example 2.7. let x be the set of all real numbers equipped with the topology generated by the euclidean open sets and the co-countable sets. let y = x ×{0, 1} and b = {(u ×{0, 1}) − f : u is open in x and f ⊆ y is finite}. it is easy to see that the space y equipped with the topology generated by the base b is t1. however the two points (0, 0) and (0, 1) do not have disjoint neighbourhoods in y , hence y is not hausdorff. let z be an infinite subset of y . without loss of generality, we assume that the set {(x, 0) : (x, 0) ∈ z} is infinite (otherwise we consider {(x, 1) : (x, 1) ∈ z}), and we further assume that there is a sequence {(xk, 0) : k = 1, 2, · · ·}⊆ z satisfying xk+1 > xk for all k (otherwise we consider a decreasing sequence). for any (x,i) ∈ z, there exists nx such that x 6= xk for all k ≥ nx. then u(x,i) = (((x− 1,x + 1) \{(xk, 0)}k≥nx ) ×{0, 1}) ∩z c© agt, upv, 2017 appl. gen. topol. 18, no. 1 78 a new cardinal function on topological spaces is an open neighbourhood of (x,i) in z. since (xk, 0) 6∈ u(x,i) holds for all large enough k, (x,i) is not a cluster point of the sequence {(xk, 0) : k = 1, 2, · · ·} in z. now, define δ ∈ ∆(z) by δ(z) = { ( xn+xn−1 2 , xn+xn+1 2 ) ×{0, 1}, if z = (xn, 0); z, if z 6= (xn, 0) for any n. since the sets δ(xk, 0) are pairwise disjoint, ⋂ kj δ(xkj, 0) = ∅ for every subnet {(xkj, 0)}. thus {(xk, 0)} is not gauge clustered. by proposition 9 of [2], the subspace z is not gauge compact, thus gci(z) is not finite. hence every subspace z of y with a finite gauge compact index is a finite set and hence discrete because y is a t1 space. proposition 2.8. let x and y be two disjoint spaces whose gauge compact indices are finite. then gci(x ⊕y ) = gci(x) + gci(y ) − 1, where x ⊕y is the sum of x and y . proof. assume that gci(x) = n and gci(y ) = m. let λ ∈ ∆(x ⊕y ). for any x ∈ x and y ∈ y , let λx(x) = λ(x) ∩ x and λy (y) = λ(y) ∩ y . then λx ∈ ∆(x) and λy ∈ ∆(y ). thus, there exist sets {x1,x2, · · · ,xn−1} ⊆ x and {y1,y2, · · · ,ym−1}⊆ y such that x ≺mλx {x1,x2, · · · ,xn−1} and y ≺ m λy {y1,y2, · · · ,ym−1}. let c = {x1,x2, · · · ,xn−1}∪{y1,y2, · · · ,ym−1}. then x ⊕y ≺mλ c. therefore, gci(x ⊕y ) ≤ n + m− 1. next, as gci(x) =n and gci(y ) =m, there exist δx∈∆(x) and δy ∈∆(y ) such that there are no a ⊆ x, b ⊆ y with |a| < n − 1 and |b| < m − 1 satisfying x ≺mδx a and y ≺ m δy b. let δx ⊕ δy be the gauge on x ⊕ y such that δx ⊕ δy |x = δx and δx ⊕ δy |y = δy . for any c ⊆ x ⊕ y , if x ⊕ y ≺mδx⊕δy c, then x ≺ m δx x ∩ c and y ≺mδy y ∩ c. thus by the assumption on δx and δy , we have |x ∩c| ≥ n− 1, |y ∩c| ≥ m− 1. so, |c| ≥ (n− 1) + (m− 1) = (m + n) − 2. by remark 2.2(5), gci(x⊕y ) ≥ (m + n−2) + 1 = m + n−1. it follows that gci(x ⊕y ) = n + m− 1 = gci(x) + gci(y ) − 1. � corollary 2.9. (1) let xi (i = 1, 2, · · · ,m) be pairwise disjoint spaces such that each gci(xi) is finite. then gci( m⊕ i=1 xi) = m∑ i=1 gci(xi) − (m− 1). (2) let x be a topological space with a finite gauge compact index. if y is a clopen proper subspace of x, then gci(y ) < gci(x). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 79 d. k. sari and d. zhao proof. (1) can be proved by repeating the use of proposition 2.8. (2) let y be a proper clopen subspace of x and gci(x) = n be finite. for any δy ∈ ∆(y ), we can extend δy to a δ ∈ ∆(x) by letting δ(x) = δy (x) for x ∈ y and δ(x) = x − y for x ∈ x − y . then there is a subset a of x such that |a| < n and x ≺mδ a. then we have y ≺ m δy y ∩a. since |a∩y | ≤ |a| < n−1, so gci(y ) ≤ n. similarly, gci(x − y ) ≤ n. then, using proposition 2.8, we have gci(x) = gci(y ⊕ (x − y )) = gci(y ) + gci(x − y ) − 1 ≥ gci(y ) + 2 − 1 > gci(y ), as desired. � remark 2.10. note that if at least one of two cardinals α,β is infinite, then α + β = max{α,β}. from this, we deduce that if x and y are disjoint spaces such that at least one of gci(x) and gci(y ) is infinite, then gci(x ⊕y ) = max{gci(x), gci(y )}. the product of two gauge compact spaces need not be gauge compact [2, example 10]. the example below shows that the product space need not be gauge compact even the two factor spaces have finite gauge compact indices. example 2.11. let x = y = (r,τup). by example 2.4, gci(x) = gci(y ) = 2. define the gauge δ on x ×y as follows: δ((x,y)) = (x− 1, +∞) × (y − 1, +∞), (x,y) ∈ x ×y. now, for any finite number of elements (x1,y1), (x2,y2), · · · , (xn,yn) in x ×y with n ≥ 1, let x∗ = min{xi : i = 1, 2, · · · ,n}− 2, y∗ = max{yi : i = 1, 2, · · · ,n} + 2. then for every 1 ≤ i ≤ n, (xi,yi) 6∈ δ((x∗,y∗)) and (x∗,y∗) 6∈ δ((xi,yi)). it follows that for any a ⊆ x × y with |a| < ℵ0, x × y ≺mδ a does not hold. therefore gci(x × y ) 6≤ ℵ0, hence gci(x × y ) ≥ ℵ1. on the other hand, for any gauge η on the product space x ×y , let a = {(a,a) : a is an integer}. then |a| < ℵ1 and x ×y ≺mη a holds. thus gci(x ×y ) = ℵ1. now a natural question is: which spaces x have the property that for any space y with a finite gauge compact index, gci(x × y ) is finite. note that a space x is compact if and only if for any δ ∈ ∆(x), there exists a finite set a ⊆ x such that x = ⋃ x∈a δ(x). proposition 2.12. let x be a space such that gci(x ×y ) is finite for all y with a finite gauge compact index . then gci(x) is finite and x is compact. proof. since the product of x with the trivial space {e} with exactly one element is homeomorphic to x and gci({e}) = 2, by the assumption it follows that gci(x) = gci(x ×{e}) is finite. let y = (n,τ) be the space where n is the set of all natural numbers excluding 0 and τ = {∅,n}∪{↑n : n ∈ n}. then gci(y ) = 2. so gci(x×y ) c© agt, upv, 2017 appl. gen. topol. 18, no. 1 80 a new cardinal function on topological spaces is finite. for any δ ∈ ∆(x), consider the λ ∈ ∆(x ×y ) defined by λ(x,m) = δ(x)×↑m, (x,m) ∈ x ×y. then there exist a finite set a = {(x1,m1), (x2,m2), · · · , (xk,mk)} such that x × y ≺mλ a. we claim that x = ⋃ {δ(xi) : i = 1, 2, · · · ,k}. to see this, assume that x0 6∈ δ(xi) for any i. then (x0,m1 + · · · + mk + 1) 6∈ λ(xi,mi) and (xi,mi) 6∈ λ(x0,m1 + · · · + mk + 1) for every i, which contradicts the assumption on the set a. therefore x = ⋃ {δ(xi) : i = 1, 2, · · · ,k}. hence x is compact. � 3. spaces whose gauge compact indices equal to two we first consider the gauge compact indices of some spaces arising from intrinsic topologies on posets (partially ordered sets). let (p,≤) be a poset. a subset u of p is called an upper set if u =↑u = {y∈p : y ≥ x for some x∈u}. dually u ⊆ p is a lower set, if u =↓u = {y ∈ p : y ≤ x for some x ∈ u}. a topology on a poset p is order compatible if the closure of each point x ∈ p with respect to this topology equals the lower part of x, that is, cl({x}) =↓x = {y ∈ p : y ≤ x}. the finest order compatible topology on p is the alexandroff topology γ(p) which consists of all upper subsets of p . the coarsest order compatible topology on p is the upper interval topology (or weak topology) ν(p), of which {p−↓a : a is a finite subset of p} is a basis (see [12, proposition 1.8]). thus all open sets in an order compatible topology are upper sets. an element a of a poset p is called a linking element if it is comparable with any element in p : for any x ∈ p , either a ≤ x or x ≤ a holds. if a ∈ p is a linking element and τ is an order compatible topology on poset p , then for any δ ∈ ∆(p,τ) we have p ≺mδ {a}, so gci(p,τ) = 2. the following is an example of poset p which does not have a linking element but gci(p,ν(p)) = 2. example 3.1. let p = n ∪{a,b}, where n is the set of all natural numbers excluding 0. define the partial order � on p by (i) 1 � a,b � a. (ii) for m,n ∈ n, m � n iff m ≤ n (in the ordinary sense). figure 1. poset p = n∪{a,b} c© agt, upv, 2017 appl. gen. topol. 18, no. 1 81 d. k. sari and d. zhao for any finite subset a of the poset (p,�), ↓a is a finite set. as {p− ↓a : a ⊆ p is finite} is a base of the upper interval topology ν(p), by lemma 2.3, to determine gci(p,ν(p)), we only need to consider δ ∈ ∆(p,ν(p)) satisfying δ(x) = p−↓ax for some finite subset ax ⊆ p with x 6∈↓ax. then as ↓aa∪↓ab is a finite subset of p , there exists m0 ∈ n − (↓aa∪↓ab), implying m0 ∈ δ(a) and m0 ∈ δ(b). for any other n ∈ n, n ∈ δ(m0) if m0 ≤ n, and m0 ∈ δ(n) if n ≤ m0. therefore p ≺mδ {m0}, implying gci(p,ν(p)) = 2. however, the poset (p,�) clearly does not have a linking element. for the alexandroff topology on a poset, we have a better result. proposition 3.2. for any poset p , gci(p,γ(p)) = 2 if and only if p has a linking element. proof. again, we only need to prove the necessity. assume that gci(p,γ(p)) = 2. define δ(x) =↑x for each x ∈ p . then δ is a gauge on (p,γ(p)). so there is an element e ∈ p such that p ≺mδ {e}, which implies immediately that e is a linking element of p. � another order compatible topology on a poset p is the scott topology which is the most important topological structure in domain theory. a non-empty subset d of a poset p is a directed set if every two elements in d have an upper bound in d. a subset u of a poset p is called a scott open set if (i) u =↑u, and (ii) for any directed set d ⊆ p , ∨ d ∈ u implies d ∩u 6= ∅ whenever ∨ d exists. the family σ(p) of all scott open sets of p is indeed a topology on p [4]. the space σp = (p,σ(p)) is called the scott space of p . for two elements a and b in a poset p , a is way-below b, denoted by a � b, if for any directed subset d of p , if ∨ d exists and b ≤ ∨ d then there exists d ∈ d such that a ≤ d. an element x is compact if x � x. the set of all compact elements of p is denoted by k(p). for any k ∈ k(p), ↑k is scott open. a poset p is called algebraic, if for any a ∈ p , {x ∈ k(p) : x ≤ a} is directed and a = ∨ {x ∈ k(p) : x ≤ a}. for any algebraic poset p , the family {↑u : u ∈ k(p)} is a base of the scott topology on p (see [4, corollary ii-1.15]). lemma 3.3. an element e of an algebraic poset p is a linking element if and only if for any x ∈ k(p), x ≤ e or e ≤ x holds. proof. if e is a linking element of p , then clearly for any x ∈ k(p), we have either x ≤ e or e ≤ x. conversely, assume that e satisfies the condition. let b be any element of p . since p is algebraic, b = ∨ {y ∈ k(p) : y ≤ b}. if there is a y ∈ k(p) such that y ≤ b and y 6≤ e, then e ≤ y, which implies that e ≤ b. otherwise, for every y ∈ k(p) with y ≤ b, we have y ≤ e. so b = ∨ {y ∈ k(p) : y ≤ b} ≤ e. all these show that e is a linking element of p . � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 82 a new cardinal function on topological spaces proposition 3.4. let p be an algebraic poset. then gci(σp) = 2 if and only if p has a linking element. proof. we only need to prove the necessity. note that for every u ∈ k(p), ↑u ∈ σ(p). assume, on the contrary, that p has no linking element. let p −k(p) = {ai : i ∈ i}. then no ai is a linking element. by lemma 3.3, there is ui ∈ k(p) such that ai 6≤ ui 6≤ ai. since ai = ∨ {x ∈ k(p) : x ≤ ai}, there is vi ∈ k(p) such that vi ≤ ai and vi 6≤ ui. now define a gauge δ on p (with respect to the scott topology) as follows: δ(x) = { ↑x, if x ∈ k(p); ↑vi, if x = ai. since gci(σp) = 2, there is x0 ∈ p such that p ≺mδ {x0}. if x0 is a compact element, then δ(x0) =↑x0. for any y ∈ k(p), we must have either x0 ∈ δ(y) or y ∈ δ(x0), implying y ≤ x0 or x0 ≤ y. by lemma 3.3, x0 is a linking element, contradicting our assumption. therefore, x0 = ai for some i ∈ i. but then ui 6∈↑vi = δ(ai) and ai 6∈↑ui = δ(ui), contradicting the assumption on x0. all these show that p must have a linking element. � given a set x, the weakest t1 topology on x is the co-finite topology, denoted by τcof , where u ∈ τcof if and only if either u = ∅ or x−u is a finite set. lemma 3.5. for any infinite set x, gci(x,τcof ) = 3. proof. let δ ∈ ∆(x). take any a ∈ x. if δ(a) = x, then x ≺mδ {a}. if δ(a) 6= x, then x − δ(a) = {a1,a2, · · · ,an} is a finite set and n ≥ 1. since x − ⋃ {δ(ai) : i = 1, 2, · · · ,n} is finite, we have ⋂ {δ(ai) : i = 1, 2, · · · ,n} 6= ∅. choose any element c ∈ ⋂ {δ(ai) : i = 1, 2, · · · ,n}. for each x ∈ x, if x ∈ δ(a) then x ≺mδ a. if x /∈ δ(a), then x = ai for some i, so c ∈ δ(ai) = δ(x) which implies x ≺mδ c. hence x ≺ m δ a where a = {a,c}. thus gci(x,τcof ) ≤ 3. now, we will show that gci(x,τcof ) 6= 2. first note that if two sets have the same cardinality, then their corresponding co-finite topological spaces are homeomorphic. since |x| = |x × {1, 2}|, so the two spaces (x,τcof ) and (x∗,τcof ) are homeomorphic, where x ∗ = x ×{1, 2}. define the gauge δ on x∗ as follows: δ((x, 1)) = x∗ −{(x, 2)}, δ((x, 2)) = x∗ −{(x, 1)}. for any x ∈ x, (x, 1) ≺mδ (x, 2) does not hold. so x ∗ ≺mδ {u} does not hold for any u ∈ x∗. thus, gci(x,τcof ) = gci(x∗,τcof ) 6= 2. hence we conclude that gci(x,τcof ) = 3. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 83 d. k. sari and d. zhao if a t1 space x is finite and gci(x) = 2, then it is hausdorff (every finite t1 space is discrete), thus |x| = 1. lemma 3.6. if x is a t1 space such that gci(x) = 2 and |x| > 1, then x is an infinite set. theorem 3.7. for any t1 space (x,τ), gci(x,τ) = 2 if and only if |x| = 1. proof. trivially, if |x| = 1, then gci(x) = 2. for the sufficiency, if gci(x,τ) = 2 and |x| 6= 1, then by lemma 3.6, x is an infinite set. since (x,τ) is t1, τ is finer than or equal to the co-finite topology τcof on x. hence, by remark 2.2(3), gci(x,τ) ≥ gci(x,τcof ). however, by lemma 3.5, gci(x,τcof ) = 3, thus gci(x,τ) ≥ 3, a contradiction. hence |x| = 1 holds. � given any space x and x ∈ x, if y = cl({x}), it is easy to show that for any δ ∈ ∆(y ), y ≺mδ {x}. therefore, we have the following result. corollary 3.8. a topological space x is t1 if and only if for any subspace y ⊆ x, gci(y ) = 2 implies |y | = 1. 4. spaces whose gauge compactness index is three in this section, we prove more results on spaces whose gauge compact indices are 3. a topological space x is called hyperconnected [11] if no two non-empty open sets are disjoint; equivalently, if x is not the union of two proper closed sets. every hyperconnected space is connected. theorem 4.1. let x be a t1 space with |x| > 2. if gci(x) = 3 then x is hyperconnected. proof. let x be a t1 space with |x| > 2 and gci(x) = 3. if x is not hyperconnected, then there are two non-empty open subsets u and v of x such that u ∩ v = ∅. choose a ∈ u and b ∈ v . if u = {a} and v = {b} then |u| = |v | = 1. moreover, x is t1 so u ∪ v is closed. then x = u ∪ v ∪ (x − (u ∪ v )) is the union of three non-empty disjoint open sets, which then implies gci(x) ≥ 4, by corollary 2.9(1). thus we can assume that |u| > 1 (otherwise |v | > 1). since u, with the subspace topology, is a t1 space, by theorem 3.7, gci(u) ≥ 3. then there exists a gauge β on u such that for every x ∈ u there exists y ∈ u such that x ≺mβ y does not hold. now, define the gauge δ on x as follows: (1) if x ∈ u, let δ(x) = β(x); (2) if x ∈ v , let δ(x) = v ; (3) if x 6∈ (u ∪v ), choose an open set δ(x) such that x ∈ δ(x) and a,b /∈ δ(x). since gci(x) = 3, there exists {x1,x2} ⊆ x such that x ≺mδ {x1,x2} holds. it is easy to see that one of xi’s must be in u and another be in v (if c© agt, upv, 2017 appl. gen. topol. 18, no. 1 84 a new cardinal function on topological spaces a ≺mδ x then x ∈ u, and if b ≺ m δ x then x ∈ v ). assume that x1 ∈ u and x2 ∈ v . by the assumption on β, there exists x ∈ u such that x1 ≺mβ x does not hold, equivalently x ≺mβ x1 does not hold, therefore x ≺ m δ x1 does not hold. clearly x ≺mδ x2 does not hold either. this contradicts that x ≺ m δ {x1,x2}. hence x must be hyperconnected. � remark 4.2. a t1 space with only two elements is not hyperconnected but its gauge compact index is 3. so, the condition |x| > 2 in the above theorem is not removable. definition 4.3. a space x is called an f-space if for any δ ∈ ∆(x) and any proper non-empty closed subset f,⋂ {δ(x) : x ∈ f} 6= ∅. example 4.4. (1) for any infinite set x, (x,τcof ) is an f-space. (2) the set r of real numbers with the co-countable topology is an f-space. (3) let |x| = α be a regular cardinal and τ be the topology on x such that u ∈ τ if and only if u = ∅ or |x −u| < |x|. then (x,τ) is an f-space. (4) the space (p,ν(p)) considered in example 3.1 is an f-space. proposition 4.5. if f : x → y is a surjective continuous mapping and x is an f -space, then y is an f -space. proof. let λ ∈ ∆(y ) and k be a closed non-empty proper subset of y . define the gauge δ on x as follows: δ(x) = f−1(λ(f(x))) for any x ∈ x. the set f = f−1(k) is a closed non-empty proper subset of x. since x is an f-space, so ⋂ x∈f δ(x) 6= ∅. take one element c ∈ ⋂ x∈f δ(x) = ⋂ x∈f f−1(λ(f(x))). now for every y∈ k, since f is surjective, there is x∈ f such that f(x) = y. so, we have f(c) ∈ λ(f(x)) = λ(y). thus, f(c) ∈ ⋂ y∈k λ(y), implying ⋂ y∈k λ(y) 6= ∅. hence y is an f-space. � proposition 4.6. every clopen subspace of an f -space is an f -space. proof. let x be an f-space and a ⊆ x be clopen. given any gauge δ on a and a non-empty proper closed subset k of a, we extend δ to a gauge κ on x by letting κ(x) = { δ(x), if x ∈ a, x −a, if x /∈ a. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 85 d. k. sari and d. zhao since a is clopen, k is closed in x. moreover x is an f-space, so ⋂ x∈k κ(x) 6= ∅. since k ⊆ a, then ⋂ x∈k κ(x) = ⋂ x∈k δ(x) 6= ∅. thus, a is an f-space. � remark 4.7. let x be the set of all real numbers equipped with the cocountable topology. then a = {1, 2, 3, · · ·} is a closed subset of x. let δ(x) = {x} for all x ∈ a. now, k = {2, 4, 6, · · ·} is a proper non-empty closed subset of a and ⋂ x∈k δ(x) = ∅. it follows that a is a closed subspace of the f-space x, and a is not an f-space. hence a closed subspace of an f-space need not be an f-space. theorem 4.8. if x is a t1 f -space with |x| ≥ 2, then gci(x) = 3. proof. firstly, by theorem 3.7, we have that gci(x) 6= 2 because |x| 6= 1. it now remains to show that gci(x) ≤ 3. if x is finite, then x is a discrete space. hence for every x ∈ x, {x} is clopen. let δ be the gauge on x defined by δ(x) = {x}. choose one a ∈ x and consider the proper closed set f = x−{a}. since x is an f-space, ⋂ {δ(x) : x ∈ f} 6= ∅. but this is true only when f has only one element. it thus follows that |x| = 2. thus, gci(x) = 3. now, assume that x is infinite. let δ ∈ ∆(x) and a ∈ x such that δ(a) 6= x (if δ(a) = x, then x ≺mδ {a}). hence, f = x−δ(a) is a proper closed set. by the assumption on x, ⋂ {δ(z) : z ∈ f} 6= ∅. take c ∈ ⋂ {δ(z) : z ∈ f}. for any x ∈ x, if x ∈ f we have c ∈ δ(x); if x /∈ f, x ∈ δ(a). hence x ≺mδ {a,c}. it follows that gci(x) ≤ 3. therefore gci(x) = 3. � remark 4.9. the set of real numbers r with the upper topology is a t0 f-space whose gauge compact index is 2. thus theorem 4.8 need not be true if the space is not t1. we still haven’t been able to obtain a complete characterization of t1 spaces whose gauge compact indices equal 3. we previously conjectured that every such space is an f-space. unfortunately, the example below gives a negative answer. example 4.10. let z be the set of all integers equipped with the topology τ = {∅}∪{a ⊆ z : z\a is finite}∪{b ⊆ 2z : 2z\b is finite}. the topology τ is finer than the co-finite topology, so it is t1. let δ ∈ ∆(z,τ). if δ(1) = x, then z ≺mδ {1}. if δ(1) 6= x, then x−δ(1) = {z1,z2,z3, · · · ,zn} is a finite set and n ≥ 1. since x − n⋃ i=1 δ(zi) is finite,⋂ {δ(x) : x 6∈ δ(1)} 6= ∅. choose one element c ∈ ⋂ {δ(x) : x 6∈ δ(1)}, then z ≺mδ {1,c}. hence gci(z,τ) ≤ 3. but as z is an infinite t1 space, by theorem 3.7, gci(z,τ) 6= 2, so gci(z,τ) = 3. now, consider the gauge λ ∈ ∆(z,τ) given by λ(x) = z −{x− 2,x + 1}, (x ∈ z). the set f = z − 2z is a proper non-empty closed set and ⋃ {z − λ(x) : x ∈ f} = z. hence c© agt, upv, 2017 appl. gen. topol. 18, no. 1 86 a new cardinal function on topological spaces ⋂ {λ(x) : x ∈ f} = ∅. thus (z,τ) is a t1 space with a gauge compact index of 3 but not an f-space. 5. properties of gauge compact indices in respective to mappings in this section we study the relationship between the gauge compact indices of x and y where there is a certain type of surjective mapping f : x → y . as remarked at the beginning of the section 2, all spaces considered in this paper are non-empty. recall from [2] that a mapping f : x → y from a topological space x to a topological space y is said to be m-uniformly continuous, if for every λ ∈ ∆(y ) there exists δ ∈ ∆(x) such that for any x,y ∈ x, x ≺mδ y implies f(x) ≺mλ f(y). every continuous mapping is m-uniformly continuous. the converse is true for all x iff y is an r0 space [2]. theorem 5.1. if there is a surjective m-uniformly continuous mapping f : x → y , then gci(y ) ≤ gci(x). proof. let λ ∈ ∆(y ). since f is m-uniformly continuous, there is a δ ∈ ∆(x) such that x1 ≺mδ x2 implies f(x1) ≺ m λ f(x2). then there exists a subset a ⊆ x with |a| < gci(x) such that x ≺mδ a. furthermore, for every y ∈ y , there exists x ∈ x such that f(x) = y. then x ≺mδ a for some a ∈ a. so y = f(x) ≺mλ f(a). it follows that y ≺ m λ f(a). since |f(a)| ≤ |a| < gci(x), we obtain gci(y ) ≤ gci(x). � a mapping between topological spaces that maps open subsets to open subsets is called an open mapping [14]. open mappings need not be continuous. corollary 5.2. if there is a bijective open mapping from a space x to a space y , then gci(x) ≤ gci(y ). proof. let f be a bijective open mapping from the space x to a space y . then the inverse mapping f−1 : y → x is bijective and continuous. by theorem 5.1, we have gci(x) ≤ gci(y ). � the combination of theorem 5.1 and corollary 5.2 deduces the following. corollary 5.3. if there is a bijective, open and m-uniformly continuous mapping f : x → y , then gci(x) = gci(y ). the example below shows that a bijective, open and m-uniformly continuous mapping need not be a homeomorphism. example 5.4. let p = (r,≤) be the poset of real numbers with the ordinary order of numbers. let x = (p,σ(p)) and y = (p,γ(p)). note that σ(p) = {∅,r}∪{(a, +∞) : a ∈ r} and γ(p) = σ(p) ∪{[a, +∞) : a ∈ r}. for any gauge λ on y and y1,y2 ∈ y , if y1 ≤ y2 then y2 ∈ λ(y1), so y1 ≺mλ y2 holds for any two elements y1,y2 ∈ y . it then follows immediately that every mapping f : x → y is m-uniformly continuous. in particular, the identity mapping c© agt, upv, 2017 appl. gen. topol. 18, no. 1 87 d. k. sari and d. zhao id : x → y , x 7→ x, is m-uniformly continuous. as σ(p) ⊆ γ(p), the identity mapping is open. now u = [0, +∞) ∈ γ(p), but i−1(u) = u /∈ σ(p). so id is not continuous, thus not a homeomorphism. theorem 5.5. if gci(x) = 2, then every m-uniformly continuous mapping from x to a t1 space is a constant function. proof. let gci(x) = 2, f : x → y be m-uniformly continuous, and y be t1. by theorem 5.1, gci(f(x)) ≤ gci(x) = 2, implying gci(f(x)) = 2. the set f(x), with the subspace topology, is a t1 space. by theorem 3.7, |f(x)| = 1. thus, f is a constant function. � one might conjecture that the converse of theorem 5.5 also holds. below, we give a counterexample to this conjecture. in order to simplify the explanation, we first prove a simple lemma. lemma 5.6. let f : x → y be an m-uniformly continuous mapping from a space x to a t1 space y . then for any x1,x2 ∈ x, x1 ∈ cl({x2}) implies f(x1) = f(x2). proof. assume, on the contrary, that x1 ∈ cl({x2}) and f(x1) 6= f(x2). chose a gauge λ ∈ ∆(y ) satisfying λ(f(x1)) = y −{f(x2)} and λ(f(x2)) = y −{f(x1)}. since f is m-uniformly continuous, there is a gauge δ ∈ ∆(x) such that u ≺mδ v implies f(u) ≺mλ f(v). from x1 ∈ cl({x2}) we have x2 ∈ δ(x1), so x1 ≺ m δ x2. therefore f(x1) ≺mλ f(x2). but this is not true by the definition of λ(f(x1)) and λ(f(x2)). this contradiction proves that f(x1) = f(x2) must hold. � example 5.7. let p = {a,b,c,d} be the poset in which b < a,b < c and d < c. figure 2. poset p = {a,b,c,d} for any gauge δ ∈ ∆(p,ν(p)), we have p ≺mδ {a,c}. so, gci(p,ν(p)) ≤ 3. also for the gauge λ given by λ(a) = p−↓c,λ(b) = p−↓d,λ(c) = λ(d) = p−↓a, there is no x ∈ p satisfying p ≺mλ {x}. so, gci(p,ν(p)) 6= 2 and therefore gci(p,ν(p)) = 3. also in the space (p,ν(p)), it holds that cl({a}) = {a,b} and cl({c}) = {b,c,d}. if f : (p,ν(p)) → y is an m-uniformly continuous mapping from (p,ν(p)) to a t1 space y , then by lemma 5.6, we have f(a) = f(b) from cl({a}) = {a,b} and f(b) = f(c) = f(d) from cl({c}) = {b,c,d}. hence f is a constant function. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 88 a new cardinal function on topological spaces 6. remarks on some further work we end the paper with some remarks and possible future work. (1) if x is a hausdorff space such that gci(x×y ) is finite for any y with a finite gauge compact index, then by proposition 2.12, gci(x) is finite. so, x is finite. we still do not know whether this conclusion holds for non-hausdorff spaces, that is whether the following statement is valid: “if x is a t1 space such that gci(x ×y ) is finite for any y with a finite gauge compact index, then x is a finite set.” (2) we have proved that if p is an algebraic poset, then gci(σp) = 2 iff p has a linking element, where σp is the scott space of p . we do not know whether this conclusion is still valid for other classes of posets, such as the continuous directed complete posets (see [4] on continuous posets). (3) quite a number of different other cardinal functions on topological spaces have been introduced and investigated, for example “weight”, “density”, “lindelöf degree”, “extent” (the extent e(x) of x is the supremum of the cardinals of its closed discrete subsets), etc. the two cardinal functions more relevant to gauge compact index are the lindelöf degree and extent. we still do not have any significant result on their connections with gauge compact index. we leave that exploration to our future work. acknowledgements. we must thank the referee for carefully reading the original manuscript and giving us a lot of valuable suggestions that helped us to improve the paper significantly. we would also like to express our appreciation to dr. ho weng kin and dr. lee peng yee for their ideas. the first author wishes to thank lpdp, ministry of finance of republic indonesia, for providing financial support for this research. references [1] a. bouziad, the point of continuity property, neighbourhood assignments and filter convergences, fund. math. 218, no. 3 (2012), 225–242. [2] d. zhao, a compactness type of topological property, questiones mathemeticae 28, (2005), 1–11. [3] e. k. van douwen and w. f. pfeffer, some properties of the sorgenfrey line and related spaces, pacific j. math. 81, no. 2 (1979), 371–377. [4] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. w. mislove and d. s. scott, continuous lattices and domains, vol. 93, cambridge university press, 2003. [5] g. gruenhage, a survey of d-spaces, in: set theory and its applications (l. babinkostova, a. caicedo, s. geschke, m. scheepers, eds.), contemporary mathematics, vol. 533, 2011, pp. 13–28. [6] h. j. k. junnila, neighbornets, pacific j. math. 76, no. 1 (1978), 83–108. [7] j. dugundji, topology, allyn and bacon, inc., boston, 1966. [8] j. goubault-larrecq, non-hausdorff topology and domain theory: selected topics in point-set topology, vol. 22, cambridge university press, 2013. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 89 d. k. sari and d. zhao [9] j. nagata, symmetric neighbourhood assignments and metrizability, questions answers in general topology 22, (2004), 181-184. [10] j. van mill, v. v. tkachuk and r. g. wilson, classes defined by stars and neighbourhood assignments, topology and its applications 154 (2007), 2127–2134. [11] l. a. steen and j. a. seebach, counterexamples in topology, vol.18, dover publications, inc., new york, 1978. [12] p. johnstone, stone spaces, cambridge university press, 1982. [13] p. y. lee, w. k. tang and d. zhao, an equivalent definition of functions of the first baire class, proc. amer. math. soc. 129, (2001), 2273–2275. [14] r. engelking, general topology, sigma series in pure mathematics, vol. 6, heldermann, berlin, 1989. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 90 @ �� !#"%$'&'� ( )*(,+�-*. � /10" � � � � )2( "�34)*. 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0 and r ∈ [0,2r] : ‖x − p‖ ≤ r,‖y − p‖ ≤ r,‖x − y‖ ≥ r ⇒ ∥ ∥ ∥ x + y 2 − p ∥ ∥ ∥ ≤ ( 1 − δ ( r r )) r; (b) strictly convex if for all x,y,p ∈ x and r > 0 : ‖x − p‖ ≤ r,‖y − p‖ ≤ r,x 6= y ⇒ ∥ ∥ ∥ x + y 2 − p ∥ ∥ ∥ < r. definition 2.1 ([7]). let a1,a2, . . . ,ap be nonempty subsets of a metric space x. then t : ∪ p i=1ai → ∪ p i=1ai is called p-cyclic mapping if t(ai) ⊂ ai+1 for i = 1,2, . . . ,p, where ap+i = ai. a point x ∈ ∪ p i=1ai is said to be a best proximity point if d(x,tx) = d(ai,ai+1). definition 2.2 ([1]). let a1,a2, . . . ,ap be nonempty subsets of a metric space x. a p-cyclic map t on ∪ p i=1ai is a p-cyclic contraction mapping if for some k ∈ (0,1), (2.1) d(tx,ty) ≤ kd(x,y) + (1 − k)d(ai,ai+1) for all x ∈ ai,y ∈ ai+1, i = 1,2, . . . ,p. remark 2.3. note that definition 2.2 implies that t satisfies d(tx,ty) ≤ d(x,y), for all x ∈ ai,y ∈ ai+1, moreover, the inequality (2.1) can be written as d(tx,ty) − d(ai,ai+1) ≤ k[d(x,y) − d(ai,ai+1)] for all x ∈ ai,y ∈ ai+1. definition 2.4 ([7]). let a1,a2, . . . ,ap be nonempty subsets of a metric space x. then a p-cyclic mapping t : ∪ p i=1ai → ∪ p i=1ai is called a p-cyclic nonexpansive mapping if d(tx,ty) ≤ d(x,y) for all x ∈ ai,y ∈ ai+1, i = 1,2, . . . ,p. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 120 best proximity points for cyclical contractive mappings the nonexpansive condition ensures the equality of distance between consecutive sets. lemma 2.5 ([7]). let (x,d) be a metric space and let a1,a2, . . . ,ap be nonempty subsets of x. if t : ∪ p i=1ai → ∪ p i=1ai is a p-cyclic nonexpansive mapping then d(ai,ai+1) = d(ai+1,ai+2) = · · · = d(a1,a2), i = 1,2, . . . ,p−1. lemma 2.6 ([1]). let a be a nonempty closed and convex subset and b be a nonempty closed subset of a uniformly convex banach space . let {xn} and {zn} be a sequences in a , and let {yn} be a sequence in b satisfying (i) ‖zn − yn‖ → d(a,b), (ii) for every ǫ > 0, there exists n0 ∈ n, such that for all m > n > n0,‖xm − yn‖ ≤ d(a,b) + ǫ. then, for every ∈> 0, there exists n1 ∈ n, such that for all m > n > n1,‖xm− zn‖ ≤ ǫ. lemma 2.7 ([1]). let a be a nonempty closed convex subset and b be a nonempty closed subset of a uniformly convex banach space. let{xn} and{zn} be a sequences in a and let {yn} be a sequence in b satisfying (i) ‖xn − yn‖ → d(a,b), (ii) ‖zn − yn‖ → d(a,b). then ‖xn − zn‖ converges to zero. theorem 2.8 ([7]). let a1,a2, . . . ,ap be nonempty subsets of a metric space x and let t : ∪ p i=1ai → ∪ p i=1ai be a p-cyclic mapping. if for some x ∈ ai, the sequence {t pnx} ∈ ai contains a convergent subsequence {t pnjx} converging to ξ ∈ ai, then ξ is a best proximity point in ai. definition 2.9. let a1,a2, . . . ,ap be nonempty subsets of a metric space x. a p-cyclic mapping t on ⋃p i=1 ai is said to be a p-cyclic contractive map if d(tx,ty) < d(x,y), for all x ∈ ai,y ∈ ai+1 satisfying d(x,y) > d(ai,ai+1), for all i = 1, . . . ,p. definition 2.10. the nonempty subsets a1,a2, . . . ,ap of a metric space x are said to satisfy cyclical proximal property if there exists xi ∈ ai for all 1 ≤ i ≤ p such that xi = xi+p for all i = 1, . . . ,p whenever ‖xi − xi+1‖ = d(ai,ai+1). 3. main results the following lemma shows that any p-cyclic contractive mapping is also p-cyclic non-expansive. lemma 3.1. let a1,a2, . . . ,ap be nonempty closed and convex subsets of a uniformly convex banach space x. let t : ⋃p i=1 ai → ⋃p i=1 ai be such that (i) t(ai) ⊂ ai+1, i = 1,2, . . . ,p, where ap+i = ai, (ii) ‖tx − ty‖ < ‖x − y‖, for all x ∈ ai, y ∈ ai+1 and ‖x − y‖ 6= d(ai,ai+1). then ‖tx − ty‖ ≤ ‖x − y‖, for all x ∈ ai, y ∈ ai+1. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 121 j. maria felicit and a. anthony eldred proof. it is easy to observe that d(ai,ai+1) = d(ai+1,ai+2), for all i = 1, . . . ,p− 1. we shall prove that ‖tx−ty‖ = d(ai,ai+1), whenever ‖x−y‖ = d(ai,ai+1). assume that ‖x − y‖ = d(ai,ai+1), then it is possible to choose sequences {xn} ∈ ai and {yn} ∈ ai+1 such that ‖xn − yn‖ > d(ai,ai+1) and ‖xn − yn‖ → d(ai,ai+1) with xn 6= x,yn 6= y. since d(ai,ai+1) ≤ ‖txn − ty‖ < ‖xn − y‖, ‖txn − ty‖ → d(ai,ai+1). similar argument asserts that ‖tyn − tx‖ → d(ai,ai+1). since ‖pai+1ty − ty‖ ≤ ‖txn − ty‖, txn → pai+1ty and tyn → pai+2tx. as ‖txn − tyn‖ → d(ai,ai+1), we have ‖pai+1ty−pai+2tx‖ = d(ai,ai+1). by uniqueness of the proximal point, ty = pai+2tx, tx = pai+1ty. hence the lemma. � it is necessary to ensure the non-expansive condition as it may not be explicitly given in the contractive condition for example theorem 3.4, whereas the conditions used in theorem 3.6 directly imply the non-expansive condition. theorem 3.2. let a1,a2, . . . ,ap be nonempty closed and convex subsets of a strictly convex banach space x satisfying cyclical proximal property. further, assume one of the subsets is compact. let t : ⋃p i=1 ai → ⋃p i=1 ai be a pcyclic mapping such that ‖tx − ty‖ < ‖x − y‖ for all x ∈ ai, y ∈ ai+1 and ‖x − y‖ 6= d(ai,ai+1), then for each i, 1 ≤ i ≤ p, there exists a unique best proximity point such that, for any x0 ∈ ai0 (with respect to ai+1), the sequence {xpn} converges to the best proximity point. proof. assume ai is compact. define φ : ai0 → r + by φ(y) = d(y,ty) for all y ∈ ai0. from the lemma 2.7 it is easy to observe that t is continuous on ai0. in general, t m is continuous on any ai, i = 1, . . . ,p, where m is positive integer. so φ is continuous and hence there exists y0 ∈ ai0, such that d(y0,ty0) = φ(y0) = infy∈ai0 d(y,ty). suppose d(y0,ty0) > d(ai,ai+1), then d(t py0,t p+1y0) < d(y0,ty0) which is a contradiction. hence d(y0,ty0) = d(ai,ai+1). assume that x0 ∈ ai0, and {xpn} ∈ ai0, for all n = 1,2, . . . . suppose for some n, xpn = y0, then xpn+1 = txpn = ty0, assume xpn 6= y0 for any n. since ‖t ny0 − t n+1y0‖ = d(ai,ai+1) and t py0 = y0, by cyclical proximal property. d(xpn,pai+1(y0)) = d(t pxpn−p,t p+1y0) ≤ d(xpn−p,ty0) = d(xp(n−1),pai+1(y0)). therefore d(xpn,pai+1(y0)) is a decreasing sequences converging to some r ≥ 0. since ai is compact, it follows that the sequence {xpn} has a subsequence {xpnk} converging to some z ∈ ai. if d(z,pai+1(y0)) ≤ d(ai,ai+1), then there is nothing to prove. assume that d(z,pai+1(y0)) > d(ai,ai+1), then d(z,pai+1(y0)) = lim n→∞ d(xpn,pai+1(y0)) = lim n→∞ d(t pxpn,pai+1(y0)) = lim k→∞ d(t pxpnk,pai+1(y0)) = d(t pz,t p+1y0) (since t p is continuous on ai0) < d(z,ty0) = d(z,pai+1(y0)), c© agt, upv, 2015 appl. gen. topol. 16, no. 2 122 best proximity points for cyclical contractive mappings which is a contradiction. therefore z = y0. since any convergent subsequence of {xpn} converges to y0, {xpn} itself converges to y0 which is the best proximity point. for uniqueness, suppose there exists z ∈ ai with z 6= y0 such that ‖z−tz‖ = d(ai,ai+1), by cyclical proximal property t py0 = y0,t pz = z. if ‖y0 − tz‖ − d(ai,ai+1) > 0 then ‖ty0 − t 2z‖ − d(ai,ai+1) < ‖y0 − tz‖ − d(ai,ai+1) = ‖t py0 − t p+1z‖ − d(ai,ai+1) ≤ ‖ty0 − t 2z‖ − d(ai,ai+1). which is a contradiction. � example 3.3. let a1 = {(0,0,x) ∈ r 3/x ≥ 1}, a2 = {(0,1,x) ∈ r 3/x ≥ 1}, a3 = {(1,1,x) ∈ r 3/x ≥ 1}, and a4 = {(1,0,x) ∈ r 3/x ≥ 1} be subsets in the space r3 with euclidean norm. clearly a1,a2,a3 and a4 satisfy cyclical proximal property. define t on ∪4i=1ai as t(0,0,x) = ( 0,1,x + 1 x ) , for (0,0,x) ∈ a1, t(0,1,x) = ( 1,1,x + 1 x ) , for (0,1,x) ∈ a2 t(1,1,x) = ( 1,0,x + 1 x ) , for (1,1,x) ∈ a3, t(1,0,x) = ( 0,0,x + 1 x ) , for (1,0,x) ∈ a4. for any (0,0,x) ∈ a1, and (0,1,y) ∈ a2. if ‖(0,0,x)−(0,1,y)‖ > d(a1,a2) = 1, then x 6= y. also ‖t(0,0,x) − t(0,1,y)‖ = ‖ ( 0,1,x + 1 x ) − ( 1,1,y + 1 y ) ‖ < (1 + (x − y)2) 1 2 = ‖(0,0,x) − (0,1,y)‖ hence t is a cyclic contractive map. also for any (0,0,x) ∈ a1, ‖(0,0,x) − t(0,0,x)‖ = ‖(0,0,x) − ( 0,1,x + 1 x ) ‖ = ( 1 + (1 x )2) 1 2 > 1 = d(a1,a2). here t does not admit any best proximity point as none of the sets are compact. next we consider two of the famous extensions of banach contraction theorem due to boyd-wong and gregathy. theorem 3.4. let a1,a2, . . . ,ap be nonempty closed subsets of a complete metric space (x,d). let t : ∪ p i=1ai → ∪ p i=1ai be a p-cyclic mapping . suppose d(tx,ty) ≤ ψ(d(x,y) − d(ai,ai+1)) + d(ai,ai+1) for all x ∈ ai, y ∈ ai+1, where ψ : [0,∞) → [0,∞) is upper semi-continuous from the right and satisfies 0 ≤ ψ(t) < t for all t > 0. then c© agt, upv, 2015 appl. gen. topol. 16, no. 2 123 j. maria felicit and a. anthony eldred (i) d(t pnx,t pn+1y) → d(ai,ai+1) as n → ∞ (ii) d(t p(n+1)x,t pn+1y) → d(ai,ai+1) as n → ∞ note: the contractive condition here does not directly guarantee the nonexpansive condition and hence the importance of lemma 3.1. proof. (i) choose x0 ∈ ai, set sn = d(t pnx0,t pn+1x0) − d(ai,ai+1). given ψ(t) < t for all t > 0, from the lemma 3.1, it follows that d(t p(n+1)x0,t p(n+1)+1x0) ≤ d(t pnx0,t pn+1x0). therefore {sn} is a decreasing sequence and hence converges . let r be the limit of sn. then r ≥ 0. suppose r > 0. then d(t p(n+1)x0,t p(n+1)+1x0) − d(ai,ai+1) ≤ d(t p(n+1)−1x0,t p(n+1)x0) ≤ d(t p(n+1)−2x0,t p(n+1)−1x0) ≤ . . . ≤ d(t pn+1x0,t pn+2x0) ≤ ψ(d(t pnx0,t pn+1x0) −d(ai,ai+1)). taking lim sup on both sides, lim supd(t p(n+1)x0,t p(n+1)+1x0) − d(ai,ai+1) ≤ lim supψ(d(t pnx0,t pn+1x0) − d(ai,ai+1)). we obtain r ≤ ψ(r), which is a contradiction. hence d(t pnx0,t pn+1x0) → d(ai,ai+1) as n → ∞. similar argument shows that d(t p(n+1)x,t pn+1y) → d(ai,ai+1) as n → ∞. � theorem 3.5. let a1,a2, . . . ,ap be nonempty closed and convex subsets of a uniformly convex banach space x. let t : ⋃p i=1 ai → ⋃p i=1 ai be a p-cyclic mapping such that d(tx,ty) ≤ ψ(d(x,y) − d(ai,ai+1)) + d(ai,ai+1) for all x ∈ ai, y ∈ ai+1, where ψ : [0,∞) → [0,∞) is upper semi-continuous from the right and satisfies 0 ≤ ψ(t) < t for all t > 0 and ψ(0) = 0. then for each i,1 ≤ i ≤ p,there exists a unique best proximity point such that, for any x0 ∈ ai, {t pnx0} converges to the best proximity point. proof. choose x0 ∈ ai. suppose d(ai,ai+1) = 0, then t has a unique fixed point x ∈ ∩ p i=1ai, see in [8]. assume that d(ai,ai+1) 6= 0, then by theorem 3.4 it follows that ‖t pnx0−t pn+1x0‖ → d(ai,ai+1) and ‖t p(n+1)x0−t pn+1x0‖ → d(ai,ai+1). by lemma 2.7, it follows that ‖t pnx0 − t p(n+1)x0‖ → 0. similarly ‖t pn+1x0 − t p(n+1)+1x0‖ → 0. to complete the proof, we have to show that for every ǫ > 0, there exists n0, such that for all m > n ≥ n0, ‖t pmx0 − t pn+1x0‖ ≤ d(ai,ai+1) + ǫ. suppose not, then there exists ǫ > 0, such that for all k ∈ n there exists mk > nk ≥ k for which ‖t pmkx0 − t pnk+1x0‖ ≥ d(ai,ai+1) + ǫ. this mk can be chosen such that it is the least integer greater than nk to satisfy the above inequality and ‖t p(mk−1)x0 − c© agt, upv, 2015 appl. gen. topol. 16, no. 2 124 best proximity points for cyclical contractive mappings t pnk+1x0‖ < d(ai,ai+1) +ǫ. consequently ‖t pnx0 −t pn+1x0‖ → d(ai,ai+1) and ‖t p(n+1)x0 − t pn+1x0‖ → d(ai,ai+1). by lemma2.7, it follows that ‖t pnx0 − t p(n+1)x0‖ → 0. similarly ‖t pn+1x0 − t p(n+1)+1x0‖ → 0. d(ai,ai+1) + ǫ ≤ ‖t pmkx0 − t pnk+1x0‖ ≤ ‖t pmkx0 − t p(mk−1)x0‖ + ‖t p(mk−1)x0 − t pnk+1x0‖ ≤ ‖t pmkx0 − t p(mk−1)x0‖ + d(ai,ai+1) + ǫ. this implies that limk→∞‖t pmkx0 − t pnk+1x0‖ = d(ai,ai+1) + ǫ. since ‖t p(mk+1)x0 − t p(nk+1)+1x0‖ ≤ ‖t pmk+1x0 − t pnk+2x0‖, ‖t pmkx0 − t pnk+1x0‖ ≤ ‖t pmkx0 − t p(mk+1)x0‖ + ‖t p(mk+1)x0 −t p(nk+1)+1x0‖ + ‖t p(nk+1)+1x0 − t pnk+1x0‖ ≤ ‖t pmkx0 − t p(mk+1)x0‖ +‖t pmk+1x0 − t pnk+2x0‖ +‖t p(nk+1)+1x0 − t pnk+1x0‖ ≤ ‖t pmkx0 − t p(mk+1)x0‖ +ψ(‖t pmkx0 − t pnk+1x0‖ −d(ai,ai+1)) + d(ai,ai+1) +‖t p(nk+1)+1x0 − t pnk+1x0‖, which yields that ‖t pmkx0 − t pnk+1x0‖ − d(ai,ai+1) ≤ ‖t pmkx0 − t p(mk+1)x0‖ + ψ(‖t pmkx0 − t pnk+1x0‖ −d(ai,ai+1)) + ‖t p(nk+1)+1x0 − t pnk+1x0‖. therefore lim supk ‖t pmkx0−t pnk+1x0‖−d(ai,ai+1) ≤ lim supk ψ(‖t pmkx0− t pnk+1x0‖−d(ai,ai+1)), as ‖t pmkx0−t p(mk+1)x0‖ → 0 and ‖t p(nk+1)+1x0 − t pnk+1x0‖ → 0. hence ǫ ≤ ψ(ǫ), a contradiction. by lemma 2.6, {t pnx0} is a cauchy sequence and converges to x ∈ ai. from theorem 2.8, it follows that ‖x − tx‖ = d(ai,ai+1). to see that t px = x, we note that ‖x − t p+1x‖ = lim n→∞ ‖t pnx0 − t p+1x‖ ≤ lim n→∞ ‖t p(n−1)x0 − tx‖ = ‖x − tx‖ = d(ai,ai+1). since ai+1 is convex set and x is uniformly convex banach space, tx = t p+1x. consequently ‖t px − tx‖ = ‖t px − t p+1x‖ ≤ ‖x − tx‖ = d(ai,ai+1). hence t px = x. uniqueness follows as in theorem 3.2. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 125 j. maria felicit and a. anthony eldred the following result on geraghty contractive condition can be proved in a similar fashion. theorem 3.6. let a1,a2, . . . ,ap be nonempty closed and convex subsets of a uniformly convex banach space x and let s = {α : r+ → [0,1) : α(tn) → 1 ⇒ tn → 0}. let t : ⋃p i=1 ai → ⋃p i=1 ai be a p-cyclic mapping such that ‖tx − ty‖ ≤ α(‖x − y‖)(‖x − y‖) + (1 − α(‖x − y‖))d(ai,ai+1) for all x ∈ ai, y ∈ ai+1, where α ∈ s. then for each i,1 ≤ i ≤ p, there exists a unique best proximity point such that, for any x0 ∈ ai, {t pnx0} converges to the best proximity point. acknowledgements. the authors are very grateful to the reviewers for their comments which have been very useful when improving the manuscript. references [1] a. anthony eldred and p. veeramani, existence and convergence of best proximity points, j. math. anal. appl. 323, no.2 (2006), 1001–1006. [2] d. w. boyd and j. s. w. wong, on nonlinear contractions, proc. amer. math. soc. 20 (1969), 458–464. [3] c. vetro, best proximity points: convergence and existence theorem for p-cyclic mappings, nonlinear anal. 73 (2010), 2283–2291. [4] c. di bari, t. suzuki and c. vetro, best proximity points for cyclic meir-keeler contraction, nonlinear anal. 69 (2008) 3790–3794. [5] m. edelstein, on fixed and periodic points under contractive mapping, j. london math. soc. 37 (1962), 74–79. [6] m. geraghty, on contractive mapping, proc. amer. math. soc. 40 (1973), 604–608. [7] s. karpagam and s. agrawal, best proximity point theorems for p-cyclic meir-keeler contractions, fixed point theory appl. 2009 (2009), article id 197308. [8] w. a. kirk, p. s. srinivasan and p. veeramani, fixed points for mapping satisfying cyclical contractive condition, fixed point theory 4 (2003), 79–89. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 126 @ appl. gen. topol. 15, no. 2(2014), 111-119doi:10.4995/agt.2014.2815 c© agt, upv, 2014 new common fixed point theorems for multivalued maps raj kamal a, renu chugh a, shyam lal singh b,∗ and swami nath mishrac a department of mathematics, maharshi dayanand university, rohtak 124001, india. b 21, govind nagar, rishikesh 249201, india. c department of mathematics, walter sisulu university, mthatha, 5117, south africa. abstract common fixed point theorems for a new class of multivalued maps are obtained, which generalize and extend classical fixed point theorems of nadler and reich and some recent suzuki type fixed point theorems. 2010 msc: 54h25; 47h10. keywords: fixed point; banach contraction theorem; hausdorff metric space 1. introduction let (x, d) be a metric space and cl(x) the family of all nonempty closed subsets of x. (cl(x), h) equipped with the generalized hausdorff metric h defined by h(a, b) = max { sup x∈a d(x, b), sup y∈b d(y, a) } , where a, b ∈ cl(x) and d(x, k) = inf z∈k d(x, z), is called the generalized hyperspace of x. ∗corresponding author. received november 2011 – accepted may 2012 http://dx.doi.org/10.4995/agt.2014.2815 r. kamal, r. chugh, s. l. singh and s. n. mishra for any nonempty subsets a, b of x, d(a, b) denotes the gap between the subsets a and b, while ρ(a, b) = sup{d(a, b) : a ∈ a, b ∈ b}, bn(x) = {a : ∅ 6= a ⊆ x and the diameter of a is finite}. as usual, we write d(x, b) (resp. ρ(x, b)) for d(a, b) (resp. ρ(a, b)) when a = {x}. for x, y ∈ x, we follow the following notation, where s and t are maps to be defined specifically in a particular context: m(sx, t y) = { d(x, y), d(x, sx) + d(y, t y) 2 , d(x, t y) + d(y, sx) 2 } . recently suzuki [23] obtained a forceful generalization of the famous banach contraction theorem. subsequently, a number of new fixed point theorems have been established and some applications have been discussed (see, for instance, [1, 5, 6, 7, 8, 9, 10, 13, 16, 20, 21, 22, 24]). the following result is essentially due to kikkawa and suzuki [8] (see also [22]) which generalizes the classical multivalued contraction theorem due to nadler [11] (see also [2, 12, 14, 18]). theorem 1.1. let (x, d) be a complete metric space and let t : x → cl(x). assume there exists r ∈ [0, 1) such that for every x, y ∈ x, d(x, t x) ≤ (1 + r)d(x, y) implies h(t x, t y) ≤ rd(x, y). then there exists z ∈ x such that z ∈ t z. the following generalization of theorem 1.1 is due to singh and mishra [20]. theorem 1.2. let x be a complete metric space and t : x → cl(x). assume there exists r ∈ [0, 1) such that for every x, y ∈ x, d(x, t x) ≤ (1 + r)d(x, y) implies h(t x, t y) ≤ rm(t x, t y). then there exists z ∈ x such that z ∈ t z. the following general common fixed point theorem is due to sastry and naidu [19]. theorem 1.3. let x be a complete metric space and s, t maps from x to itself. assume there exists r ∈ [0, 1) such that for every x, y ∈ x, d(sx, t y) ≤ r max { d(x, y), d(x, sx), d(y, t y), d(x, t y) + d(y, sx) 2 } .(1.1) then s and t have a unique common fixed point. for an excellent discussion on several special cases and variants of theorem 1.3, one may refer to rus [18]. the generality of theorem 1.3 may be appreciated from the fact that the condition (1.1) in theorem 1.3 cannot be replaced by a slightly more general condition: d(sx, t y) ≤ r max{d(x, y), d(x, sx), d(y, t y), d(x, t y), d(y, sx)}.(1.2) c© agt, upv, 2014 appl. gen. topol. 15, no. 2 112 new common fixed point theorems for multivalued maps see [19, ex. 5]. notice that the condition (1.2) with s = t is ćirić’s quasicontraction [4]. we remark that, in rhoades’ comprehensive comparison of contractive conditions [15], the condition (1.2) with s = t is considered the most general contraction for a self-map of a metric space. a particular case of our main result (cf. theorem 2.1) generalizes theorems 1.1 and 1.2. some other special cases are also discussed. 2. main results we shall need the following lemma essentially due to nadler, jr. [11] (see also [2], [3], [16, p. 4], [16, 17], [18, p. 76]). lemma 2.1. if a, b ∈ cl(x) and a ∈ a, then for each ε > 0, there exists b ∈ b such that d(a, b) ≤ h(a, b) + ε. theorem 2.2. let x be a complete metric space and let s and t maps from x to cl(x). assume there exists r ∈ [0, 1) such that for every x, y ∈ x, min{d(x, sx), d(y, t y)} ≤ (1 + r)d(x, y) implies h(sx, t y) ≤ rm(sx, t y). then there exists an element u ∈ x such that u ∈ su ∩ t u. proof. obviously m(sx, t y) = 0 iff x = y is a common fixed point of s and t . so we may assume that m(sx, t y) > 0. let ε > 0 be such that β = r + ε < 1. let u0 ∈ x and u1 ∈ t u0. by lemma 2.1, their exists u2 ∈ su1 such that d(u2, u1) ≤ h(su1, t u0) + m(su1, t u0). similarly, their exists u3 ∈ t u2 such that d(u3, u2) ≤ h(t u2, su1) + εm(t u2, su1). continuing in this manner, we find a sequence {un} in x such that u2n+1 ∈ t u2n, u2n+2 ∈ su2n+1 and d(u2n+1, u2n) ≤ h(t u2n, su2n−1) + m(t u2n, su2n−1), d(u2n+2, u2n+1) ≤ h(su2n+1, t u2n) + εm(su2n+1, t u2n). now, we show that for any n ∈ n, d(u2n+1, u2n) ≤ βd(u2n−1, u2n).(2.1) suppose if d(u2n−1, su2n−1) ≥ d(u2n, t u2n), then min{d(u2n−1, su2n−1)d(u2n, t u2n)} ≤ (1 + r)d(u2n−1, u2n). c© agt, upv, 2014 appl. gen. topol. 15, no. 2 113 r. kamal, r. chugh, s. l. singh and s. n. mishra therefore by the assumption, d(u2n+1, u2n) ≤ h(su2n−1, t u2n) ≤ rm(su2n−1, t u2n) ≤ rm(su2n−1, t u2n) + εm(su2n−1, t u2n) = βm(su2n−1, t u2n) = β max { d(u2n−1, u2n), d(u2n−1, su2n−1) + d(u2n, t u2n) 2 , d(u2n−1, t u2n) + d(u2n, su2n−1) 2 } ≤ β max d(u2n−1, u2n), d(u2n, u2n+1). this yields (2.1). suppose, if d(u2n, t u2n) ≥ d(u2n−1, su2n−1), then min{d(u2n−1, su2n−1), d(u2n, t u2n)} ≤ (1 + r)d(u2n−1, u2n). therefore by the assumption, d(u2n+1, u2n) ≤ h(su2n−1, t u2n) ≤ rm(su2n−1, t u2n) ≤ rm(su2n−1, t u2n) + εm(su2n−1, t u2n) = βm(su2n−1, t u2n) = β max { d(u2n−1, u2n), d(u2n−1, su2n−1) + d(u2n, t u2n) 2 , d(u2n−1, t u2n) + d(u2n, su2n−1) 2 } ≤ β max{d(u2n−1, u2n), d(u2n, u2n+1)}. this prove (2.1). in an analogous manner, we show that d(u2n+2, u2n+1) ≤ βd(u2n+1, u2n).(2.2) we conclude from (2.1) and (2.2) that for any n ∈ n, d(un+1, un) ≤ βd(un, un−1). therefore {un} is a cauchy sequence and has a limit in x. call it u. since un → u, there exists n0 ∈ n (natural numbers) such that d(u, un) ≤ 1 3 d(u, y) for y 6= u and all n ≥ n0. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 114 new common fixed point theorems for multivalued maps then as in [23, p. 1862], (1 + r)−1d(u2n−1, su2n−1) ≤ d(u2n−1, su2n−1) ≤ d(u2n−1, u2n) ≤ d(u2n−1, u) + d(u, u2n) ≤ 2 3 d(y, u) = d(y, u) − 1 3 d(y, u) ≤ d(y, u) − d(u2n−1, u) ≤ d(u2n−1, y). therefore d(u2n−1, su2n−1) ≤ (1 + r)d(u2n−1, y).(2.3) now either d(u2n−1, su2n−1) ≤ d(y, t y) or d(y, t y) ≤ d(u2n−1, su2n−1). in either case, by (2.3) and the assumption, d(u2n, t y) ≤ h(su2n−1, t y) ≤ rm(su2n−1, t y). ≤ r max { d(u2n−1, y), d(u2n−1, su2n−1) + d(y, t y) 2 , d(u2n−1, t y) + d(y, su2n−1) 2 } . making n → ∞, d(u, t y) ≤ r max { d(u, y), d(u, u) + d(y, t y) 2 , d(u, t y) + d(y, u) 2 } ≤ r max { d(u, y), d(u, t y) + d(u, y) 2 } .(2.4) it is clear from (2.4) that d(u, t y) ≤ rd(u, y).(2.5) now we show that h(su, t y) ≤ r max { d(u, y), d(u, su) + d(y, t y) 2 , d(u, t y) + d(y, su) 2 } (2.6) assume that y 6= u. then for every n ∈ n, there exists zn ∈ t y such that d(u, zn) ≤ d(u, t y) + 1 n d(y, u). c© agt, upv, 2014 appl. gen. topol. 15, no. 2 115 r. kamal, r. chugh, s. l. singh and s. n. mishra so we have by (2.5), d(y, t y) ≤ d(y, zn) ≤ d(y, u) + d(u, zn) ≤ d(y, u) + d(u, t y) + 1 n d(y, u) ≤ d(y, u) + rd(u, y) + 1 n d(u, y) = ( 1 + r + 1 n ) d(y, u). hence d(y, t y) ≤ (1 + r)d(y, u).(2.7) now either d(u, su) ≤ d(y, t y) or d(y, t y) ≤ d(u, su). so in either case by (2.7) and the assumption, h(su, t y) ≤ rm(su, t y), which is (2.6). now taking y = u2n in (2.6), we have d(su, u2n+1) ≤ h(su, t u2n) ≤ r max { d(u, u2n), d(u, su) + d(u2n, u2n+1) 2 , d(u, u2n+1) + d(u2n, su) 2 } . passing to the limit this obtains d(su, u) ≤ r 2 d(su, u). so u ∈ su, as su is closed. in an analogous manner, we can show that u ∈ t u. � corollary 2.3. let x be a complete metric space and s, t : x → x. assume there exists r ∈ [0, 1) such that for every x, y ∈ x, min{d(x, sx), d(y, t y)} ≤ (1 + r)d(x, y) implies d(sx, t y) ≤ rm(sx, t y). then s and t have a unique common fixed point. proof. it comes from theorem 2.2 that s and t have a common fixed point. the uniqueness of the common fixed point follows easily. � corollary 2.4. theorem 1.2. corollary 2.5 ([20]). let x be a complete metric space and t : x → x. assume there exists r ∈ [0, 1) such that for every x, y ∈ x, d(x, t x) ≤ (1 + r)d(x, y) implies d(t x, t y) ≤ rm(t x, t y). then t has a unique fixed point. proof. it comes from corollary 2.3 when s = t . � now we give an application of corollary 2.3. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 116 new common fixed point theorems for multivalued maps theorem 2.6. let p, q : x → bn(x). assume there exists r ∈ [0, 1) such that for every x, y ∈ x, min{ρ(x, px), ρ(y, qy)} ≤ (1 + r)d(x, y)(2.8) implies ρ(px, qy) ≤ r max { d(x, y), ρ(x, px) + ρ(y, qy) 2 , d(x, qy) + d(y, px) 2 } (2.9) then there exsits a unique point z ∈ x such that z ∈ pz ∩ qz. proof. choose λ ∈ (0, 1). define single-valued maps s, t : x → x as follows. for each x ∈ x, let sx be a point of px which satisfies d(x, sx) ≥ rλρ(x, px). similarly, for each y ∈ x, let t y be a point of qy such that d(y, t y) ≥ rλρ(y, qy). since sx ∈ px and t y ∈ qy, d(x, sx) ≤ ρ(x, px) and d(y, t y) ≤ ρ(y, qy). so (2.8) gives min{d(x, sx), d(y, t y)} ≤ min{ρ(x, px), ρ(y, qy)} ≤ (1 + r)d(x, y),(2.10) and this implies (2.9). therefore d(sx, t y) ≤ ρ(px, qy) ≤ r.r−λ max { rλd(x, y), rλρ(x, px) + rλρ(y, qy) 2 , rλd(x, qy) + rλd(y, px) 2 } ≤ r1−λ max { d(x, y), d(x, sx) + d(y, t y) 2 , d(x, t y) + d(y, sx) 2 } . so (2.10), viz., min{d(x, sx), d(y, t y)} ≤ (1 + r′)d(x, y) imlpies d(sx, t y) ≤ r′ max { d(x, y), d(x, sx) + d(y, t y) 2 , d(x, t y) + d(y, sx) 2 } , where r′ = r1−λ < 1. hence by corollary 2.3, s and t have a unique point z ∈ x such that sz = t z = z. this implies z ∈ pz ∩ qz. � the following result show that theorem 2.6 is a generalization of the result of singh and mishra [20, theorem 3.6]. corollary 2.7. let p : x → bn(x). assume there exists r ∈ [0, 1) such that ρ(x, px) ≤ (1 + r)d(x, y) c© agt, upv, 2014 appl. gen. topol. 15, no. 2 117 r. kamal, r. chugh, s. l. singh and s. n. mishra implies ρ(px, py) ≤ r max { d(x, y), ρ(x, px) + ρ(y, py) 2 , d(x, py) + d(y, px) 2 } . then there exists a unique point z in x such that z ∈ pz. proof. it comes from theorem 2.6 when q = p . � we remark that corollaries 2.5 and 2.7 generalize fixed point theorems from [11, 14, 18] and others. now we give two examples to show the generality of our results. example 2.8. let x = {(0, 0), (4, 0), (0, 4), (4, 5), (5, 4)} and d be defined by d[(x1, x2), (y1, y2)] = |x1 − y1| + |x2 − y2|. let s and t be such that s(x1, x2) = { (x1, 0) if x1 ≤ x2 (0, x2) if x1 > x2 and t (x1, x2) = { (0, x1) if x1 ≤ x2 (0, x2) if x1 > x2 then maps s and t do not satisfy (1.1) of theorem 1.3 (e.g. (x, y) = ((4, 5), (5, 4))). however, s and t satisfy all the hypotheses of corollary 2.3. example 2.9. let x = {(1, 1), (4, 1), (1, 4), (4, 5), (5, 4)} and d be defined by d[(x1, x2), (y1, y2)] = |x1 − y1| + |x2 − y2| let t be such that t (x1, x2) = { (x1, 1) if x1 ≤ x2 (1, x2) if x1 > x2 then t satisfies all the hypotheses of corollary 2.5, but does not satisfy ciric’s quasi-contraction, viz. (1.2) with s = t (e.g.x = (4, 5), y = (5, 4)). we close this paper with the following. question 2.10. can we replace “h(sx, t y) ≤ rm(sx, t y)” in theorem 2.1 by the following: h(sx, t y) ≤ r max { d(x, y), d(x, sx), d(y, t y), d(x, t y) + d(y, sx) 2 } .(2.11) we remark that (2.11) with s = t is the ciric’s generalized contraction [3] for t : x → cl(x). acknowledgements. the authors thank editor-in-chief professor salvador romaguera for his suggestions in this paper. the third author (sls) acknowledges the support by the ugc, new delhi under emeritus fellowship. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 118 new common fixed point theorems for multivalued maps references [1] a. abkar and m. eslamian, fixed point theorems for suzuki generalized nonexpansive multivalued mappings in banach spaces, fixed point theory appl. 2010 (2010), 10 pp. [2] n. a. assad and w. a. kirk, fixed point theorems for set-valued mappings of contractive type, pacific j. math. 43 (1972), 553–562. 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(thanasismeg13@gmail.com) abstract in this paper we study and discuss the soft set theory giving new definitions, examples, new classes of soft sets, and properties for mappings between different classes of soft sets. furthermore, we investigate the theory of soft topological spaces and we present new definitions, characterizations, and properties concerning the soft closure, the soft interior, the soft boundary, the soft continuity, the soft open and closed maps, and the soft homeomorphism. 2010 msc: 54a05; 06d72. keywords: soft set theory; soft topology. 1. preliminaries for every set x we denote by p(x) the power set of x, that is the set of all subsets of x and by |x| the cardinality of x. also, we denote by ω the first infinite cardinal and by r the set of real numbers. in 1999 d. molodtsov (see [17]) introduced the notion of soft set. later, he applied this theory to several directions (see [18], [19], and [20]). the soft set theory has been applied to many different fields (see, for example, [1], [2], [4], [5], [7], [8], [10], [12], [13], [14], [15], [21], [23], [25]). in 2011 and 2012 few researches (see, for example, [3], [6], [9], [16], [22], [24]) introduced and studied the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. the paper is organized as follows. in section 2 we study and discuss the soft set theory giving new definitions, examples, new classes of soft sets, and properties for mappings between different classes of soft sets. in section 3 we received july 2012 – accepted october 2012 http://dx.doi.org/10.4995/agt.2014.2268 d. n. georgiou and a. c. megaritis investigate the theory of soft topological spaces and we present new definitions, characterizations, and many properties concerning the soft closure, the soft interior, the soft boundary, the soft continuity, the soft open and closed maps, and the soft homeomorphism. 2. soft set theory definition 2.1 (see [17]). let x be an initial universe set and a a set of parameters. a pair (f, a), where f is a map from a to p(x), is called a soft set over x. in what follows by ss(x, a) we denote the family of all soft sets (f, a) over x. definition 2.2 (see [17]). let (f, a), (g, a) ∈ ss(x, a). we say that the pair (f, a) is a soft subset of (g, a) if f(p) ⊆ g(p), for every p ∈ a. symbolically, we write (f, a) ⊑ (g, a). also, we say that the pairs (f, a) and (g, a) are soft equal if (f, a) ⊑ (g, a) and (g, a) ⊑ (f, a). symbolically, we write (f, a) = (g, a). definition 2.3 (see, for example, [17] and [24]). let i be an arbitrary index set and {(fi, a) : i ∈ i} ⊆ ss(x, a). the soft union of these soft sets is the soft set (f, a) ∈ ss(x, a), where the map f : a → p(x) defined as follows: f(p) = ∪{fi(p) : i ∈ i}, for every p ∈ a. symbolically, we write (f, a) = ⊔{(fi, a) : i ∈ i}. example 2.4. let x = r, a = {0, 1}, and i = {1, 2, . . .}. for every i ∈ i we consider the soft set (fi, a), where the map fi : a → p(x) defined as follows: fi(p) = { (0, i), if p = 0, (−i, 0), if p = 1. then, ⊔{(fi, a) : i ∈ i} = (f, a), where the map f : a → p(x) defined as follows: f(p) = { (0, +∞), if p = 0, (−∞, 0), if p = 1. definition 2.5 (see, for example, [17] and [24]). let i be an arbitrary index set and {(fi, a) : i ∈ i} ⊆ ss(x, a). the soft intersection of these soft sets is the soft set (f, a) ∈ ss(x, a), where the map f : a → p(x) defined as follows: f(p) = ∩{fi(p) : i ∈ i}, for every p ∈ a. symbolically, we write (f, a) = ⊓{(fi, a) : i ∈ i}. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 94 soft set theory and topology example 2.6. let x = r, a = {0, 1, 2}, and i = {1, 2, . . .}. for every i ∈ i we consider the soft set (fi, a), where the map fi : a → p(x) defined as follows: fi(p) =      (−1 i , 1 i ), if p = 0, (1 − 1 i , 1 + 1 i ), if p = 1, (2 − 1 i , 2 + 1 i ), if p = 2. then, ⊓{(fi, a) : i ∈ i} = (f, a), where the map f : a → p(x) defined as follows: f(p) =      {0}, if p = 0, {1}, if p = 1, {2}, if p = 2. definition 2.7 (see, for example, [24]). let (f, a) ∈ ss(x, a). the soft complement of (f, a) is the soft set (h, a) ∈ ss(x, a), where the map h : a → p(x) defined as follows: h(p) = x \ f(p), for every p ∈ a. symbolically, we write (h, a) = (f, a)c. example 2.8. let x = r and a = {1, 2, . . .}. we consider the soft set (f, a), where the map f : a → p(x) defined as follows: f(p) = [p, +∞), for every p ∈ a. then, (f, a)c = (h, a), where the map h : a → p(x) defined as follows: h(p) = (−∞, p), for every p ∈ a. definition 2.9 (see [17]). the soft set (f, a) ∈ ss(x, a), where f(p) = ∅, for every p ∈ a is called the a-null soft set of ss(x, a) and denoted by 0a. the soft set (f, a) ∈ ss(x, a), where f(p) = x, for every p ∈ a is called the a-absolute soft set of ss(x, a) and denoted by 1a. the proofs of the following propositions are straightforward verifications of the above definitions. proposition 2.10. let (f, a) ∈ ss(x, a). the following statements are true: (1) (f, a) ⊓ (f, a) = (f, a). (2) (f, a) ⊔ (f, a) = (f, a). (3) (f, a) ⊓ 0a = 0a. (4) (f, a) ⊔ 0a = (f, a). (5) (f, a) ⊓ 1a = (f, a). (6) (f, a) ⊔ 1a = 1a. (7) (f, a) ⊓ (f, a)c = 0a. (8) (f, a) ⊔ (f, a)c = 1a. (9) (0a) c = 1a. (10) (1a) c = 0a. (11) ((f, a)c)c = (f, a). (12) 0a ⊑ (f, a) ⊑ 1a. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 95 d. n. georgiou and a. c. megaritis proposition 2.11. let (f, a), (g, a), (h, a) ∈ ss(x, a). the following statements are true: (1) (f, a) ⊓ ((g, a) ⊓ (h, a)) = ((f, a) ⊓ (g, a)) ⊓ (h, a). (2) (f, a) ⊔ ((g, a) ⊔ (h, a)) = ((f, a) ⊔ (g, a)) ⊔ (h, a). (3) (f, a) ⊓ ((g, a) ⊔ (h, a)) = ((f, a) ⊓ (g, a)) ⊔ ((f, a) ⊓ (h, a)). (4) (f, a) ⊔ ((g, a) ⊓ (h, a)) = ((f, a) ⊔ (g, a)) ⊓ ((f, a) ⊔ (h, a)). proposition 2.12. let i be an arbitrary set and {(fi, a) : i ∈ i} ⊆ ss(x, a). the following statements are true: (1) (fi, a) ⊑ ⊔{(fi, a) : i ∈ i}, for every i ∈ i. (2) ⊓{(fi, a) : i ∈ i} ⊑ (fi, a), for every i ∈ i. (3) (⊔{(fi, a) : i ∈ i}) c = ⊓{(fi, a) c : i ∈ i}. (4) (⊓{(fi, a) : i ∈ i}) c = ⊔{(fi, a) c : i ∈ i}. definition 2.13. let (f, a), (g, a) ∈ ss(x, a). the soft symmetric difference of these soft sets is the soft set (h, a) ∈ ss(x, a), where the map h : a → p(x) defined as follows: h(p) = (f(p)\g(p))∪(g(p)\f(p)), for every p ∈ a. symbolically, we write (h, a) = (f, a) △ (g, a). example 2.14. let x = {1, 2, 3, 4, 5} and a = {0, 1, 2, . . .}. we consider the soft sets (f, a) and (g, a), where the maps f : a → p(x) and g : a → p(x) defined as follows: f(p) = { {1, 2, 3, 4}, if p = 0, ∅, otherwise, g(p) = { {1, 4, 5}, if p = 0, ∅, otherwise. then, (f, a) △ (g, a) = (h, a), where the map h : a → p(x) defined as follows: h(p) = { {2, 3, 5}, if p = 0, ∅, otherwise. the proof of the following proposition is straightforward verification of the definition 2.13. proposition 2.15. let (f, a), (g, a), (h, a) ∈ ss(x, a). the following statements are true: (1) (f, a) △ ((g, a) △ (h, a)) = ((f, a) △ (g, a)) △ (h, a). (2) (f, a) △ (g, a) = (g, a) △ (f, a). (3) (f, a) △ 0a = (f, a). (4) (f, a) △ (f, a) = 0a. (5) (f, a) ⊓ ((g, a) △ (h, a)) = ((f, a) ⊓ (g, a)) △ ((f, a) ⊓ (h, a)). remark 2.16. by proposition 2.15 follows that the pair (ss(x, a), △) is a group of soft sets. the identity element is the soft set 0a and the inverse of the element (f, a) is the soft set (f, a). also, the triad (ss(x, a), △, ⊓) is a ring of soft sets. let x and y be two initial universe sets, px and py two sets of parameters, f : x → y , and e : px → py . in [11] the authors, using f and e, define the c© agt, upv, 2014 appl. gen. topol. 15, no. 1 96 soft set theory and topology notion of a mapping from the family of all soft sets (f, a) over x, where a ⊆ px, to the family of all soft sets (g, b) over y , where b ⊆ py . in [24] the authors gave a mapping from ss(x, a) to ss(y, b) and studied properties of images and inverse images of soft sets. the given below definition is actually the definition of this mapping. definition 2.17. let x and y be two initial universe sets, a and b two sets of parameters, f : x → y , and e : a → b. then, by φfe we denote the map from ss(x, a) to ss(y, b) for which: (1) if (f, a) ∈ ss(x, a), then the image of (f, a) under φfe, denoted by φfe(f, a), is the soft set (g, b) ∈ ss(y, b) such that g(py ) = { ⋃ {f(f(p)) : p ∈ e−1({py })}, if e −1({py }) 6= ∅, ∅, if e−1({py }) = ∅, for every py ∈ b. (2) if (g, b) ∈ ss(y, b), then the inverse image of (g, b) under φfe, denoted by φ−1 fe (g, b), is the soft set (f, a) ∈ ss(x, a) such that f(px) = f −1(g(e(px))), for every px ∈ a. the following propositions are easily proved. proposition 2.18. let (f, a), (f1, a) ∈ ss(x, a), (g, b), (g1, b) ∈ ss(y, b). the following statements are true: (1) if (f, a) ⊑ (f1, a), then φfe(f, a) ⊑ φfe(f1, a). (2) if (g, b) ⊑ (g1, b), then φ −1 fe (g, b) ⊑ φ−1 fe (g1, b). (3) (f, a) ⊑ φ−1 fe (φfe(f, a)). (4) if f is an 1-1 map of x into y and e is an 1-1 map of a into b, then (f, a) = φ−1 fe (φfe(f, a)). (5) φfe(φ −1 fe (g, b)) ⊑ (g, b). (6) if f is a map of x onto y and e is a map of a onto b, then φfe(φ −1 fe (g, b)) = (g, b). (7) φ−1 fe ((g, b)c) = (φ−1 fe (g, b))c. proposition 2.19. let i be an arbitrary set, {(fi, a) : i ∈ i} ⊆ ss(x, a), and {(gi, b) : i ∈ i} ⊆ ss(y, b). the following statements are true: (1) φfe(⊔{(fi, a) : i ∈ i}) = ⊔{φfe(fi, a) : i ∈ i}. (2) φfe(⊓{(fi, a) : i ∈ i}) ⊑ ⊓{φfe(fi, a) : i ∈ i}. (3) φ−1 fe (⊔{(gi, b) : i ∈ i}) = ⊔{φ −1 fe (gi, b) : i ∈ i}. (4) φ−1 fe (⊓{(gi, b) : i ∈ i}) = ⊓{φ −1 fe (gi, b) : i ∈ i}. definition 2.20. define the order of a soft set (f, a) ∈ ss(x, a) as follows: (1) ord(f, a) = n, where n ∈ ω, if and only if the intersection of any n + 2 distinct elements of {f(p) : p ∈ a} is empty and there exist n + 1 distinct elements of {f(p) : p ∈ a}, whose intersection is not empty. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 97 d. n. georgiou and a. c. megaritis (2) ord(f, a) = ∞, if and only if for every n ∈ ω there exist n distinct elements of {f(p) : p ∈ a}, whose intersection is not empty. note. let x be an initial universe set and a a set of parameters. we consider the following subsets of ss(x, a): (1) c(x, a) = {(f, a) ∈ ss(x, a) : ⋃ {f(p) : p ∈ a} = x}. (2) ss(x, a, ν) = {(f, a) ∈ ss(x, a) : |f(p)| = ν, for every p ∈ a}, where ν is an ordinal such that ν ≤ |x|. (3) f(x, a) = {(f, a) ∈ ss(x, a) : |f(p)| < ω, for every p ∈ a}. (4) o(x, a, n) = {(f, a) ∈ ss(x, a) : ord(f, a) = n}, where n ∈ ω∪{∞}. example 2.21. (1) let x be a nonempty set. every cover {ui : i ∈ i} of x, that is ∪{ui : i ∈ i} = x, can be considered as the soft set (f, a) ∈ c(x, a), where a = i and the map f : a → p(x) defined as follows: f(i) = ui, for every i ∈ a. (2) let x be a set with |x| = 5. then, the family of all subsets y of x with |y | = 3 can be considered as the element (f, a) of ss(x, a, 3), where a = {1, 2, . . . , 10} and f is an 1-1 map of a to p(x). proposition 2.22. let (f, a) ∈ ss(x, a) and (g, b) ∈ ss(y, b). the following statements are true: (1) if (f, a) ∈ c(x, a) and the maps f : x → y and e : a → b are onto, then φfe(f, a) ∈ c(y, b). (2) if (f, a) ∈ ss(x, a, ν), the map f : x → y is 1-1, and the map e : a → b is 1-1 and onto, then φfe(f, a) ∈ ss(y, b, ν). (3) if (f, a) ∈ f(x, a), the map f : x → y is 1-1, and the map e : a → b is 1-1 and onto, then φfe(f, a) ∈ f(y, b). (4) if (f, a) ∈ o(x, a, n), the map f : x → y is 1-1, and the map e : a → b is 1-1 and onto, then φfe(f, a) ∈ o(y, b, n). (5) if (g, b) ∈ c(y, b), then φ−1 fe (g, b) ∈ c(x, a). (6) if (g, b) ∈ ss(y, b, ν) and the map f : x → y is 1-1 and onto, then φ−1 fe (g, b) ∈ ss(x, a, ν). (7) if (g, b) ∈ f(y, b) and the map f : x → y is 1-1, then φ−1 fe (g, b) ∈ f(x, a). (8) if (g, b) ∈ o(y, b, n) and the map f : x → y is onto, then φ−1 fe (g, b) ∈ o(x, a, n). proof. suggestively we prove the statements (1), (2), (7), and (8). (1) let (f, a) ∈ c(x, a) and φfe(f, a) = (g, b). then, ⋃ {f(px) : px ∈ a} = x. since the map f : x → y is onto, f(x) = y . therefore, ⋃ {g(py ) : py ∈ b} = ⋃ { ⋃ {f(f(p)) : p ∈ e−1({py })} : py ∈ b} = ⋃ {f(f(px)) : px ∈ a} = f(∪{f(px) : px ∈ a}) = f(x) = y. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 98 soft set theory and topology thus, φfe(f, a) ∈ c(y, b). (2) let (f, a) ∈ ss(x, a, ν) and φfe(f, a) = (g, b). then, |f(p)| = ν, for every p ∈ a. let py ∈ b. since the map e : a → b is 1-1, g(py ) = f(f(p)), where p ∈ e−1({py }). also, since the map f : x → y is 1-1, we have |f(f(p))| = |f(p)|. therefore, |g(py )| = |f(f(p))| = |f(p)| = ν. thus, φfe(f, a) ∈ ss(y, b, ν). (7) let (g, b) ∈ f(y, b) and φ−1 fe (g, b) = (f, a). then, |g(py )| < ω, for every py ∈ b. let px ∈ a. then, f(px) = f −1(g(e(px))). since the map f : x → y is 1-1, we have |f(px)| = |f −1(g(e(px)))| < ω. this means that φ−1 fe (g, b) ∈ f(x, a). (8) let (g, b) ∈ o(y, b, n) and φ−1 fe (g, b) = (f, a). then, the intersection of any n + 2 distinct elements of {g(py ) : py ∈ b} is empty and there exist n + 1 distinct elements of {g(py ) : py ∈ b}, whose intersection is not empty. let p1x, . . . , p n+1 x ∈ a such that g(e(p1x)) ∩ . . . ∩ g(e(p n+1 x )) 6= ∅. then, f(p1x) ∩ . . . ∩ f(p n+1 x ) = f−1(g(e(p1x))) ∩ . . . ∩ f −1(g(e(pn+1 x ))) = f−1(g(e(p1x)) ∩ . . . ∩ g(e(p n+1 x ))). since the map f : x → y is onto, f(p1x) ∩ . . . ∩ f(p n+1 x ) 6= ∅. this means that there exist n+1 distinct elements of {f(px) : px ∈ a}, whose intersection is not empty. now, we prove that the intersection of any n + 2 distinct elements of the set {f(px) : px ∈ a} is empty. let p 1 x, . . . , p n+2 x ∈ a. then, f(p1x) ∩ . . . ∩ f(p n+2 x ) = f−1(g(e(p1x))) ∩ . . . ∩ f −1(g(e(pn+2 x ))) = f−1(g(e(p1x)) ∩ . . . ∩ g(e(p n+2 x ))) = f−1(∅) = ∅. thus, φ−1 fe (g, b) ∈ o(x, a, n). � 3. soft topology definition 3.1 (see, for example, [24]). let x be an initial universe set, a a set of parameters, and τ ⊆ ss(x, a). we say that the family τ defines a soft topology on x if the following axioms are true: (1) 0a, 1a ∈ τ. (2) if (g, a), (h, a) ∈ τ, then (g, a) ⊓ (h, a) ∈ τ. (3) if (gi, a) ∈ τ for every i ∈ i, then ⊔{(gi, a) : i ∈ i} ∈ τ. the triplet (x, τ, a) is called a soft topological space or soft space. the members of τ are called soft open sets in x. also, a soft set (f, a) is called soft closed if the complement (f, a)c belongs to τ. the family of soft closed sets is denoted by τc. remark 3.2. let (x, τ, a) be a soft topological space. then, by proposition 2.10 (11) we have (g, a) ∈ τ if and only if (g, a)c ∈ τc. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 99 d. n. georgiou and a. c. megaritis the proof of the following proposition is straightforward verification of the definition 3.1, proposition 2.10, and proposition 2.12. proposition 3.3. let (x, τ, a) be a soft topological space. the family τc has the following properties: (1) 0a, 1a ∈ τ c. (2) if (q, a), (r, a) ∈ τc, then (q, a) ⊔ (r, a) ∈ τc. (3) if (qi, a) ∈ τ c for every i ∈ i, then ⊓{(qi, a) : i ∈ i} ∈ τ c. example 3.4. (1) let x = {1, 2, . . .}, a = {0, 1}, and τ = {(gn, a) : n = 1, 2, . . .} ∪ {0a, 1a}, where the map gn : a → p(x) defined as follows: gn(p) = { {n, n + 1, . . .}, if p = 0, ∅, if p = 1. the triplet (x, τ, a) is a soft topological space. (2) let (x, t) be a topological space, a a nonempty set, and τ = {(gu, a) : u ∈ t}, where the map gu : a → p(x) defined as follows: gu (p) = u, for every p ∈ a. the triplet (x, τ, a) is a soft topological space. definition 3.5. let (x, τ, a) be a soft topological space, a ∈ a, and x ∈ x. we say that a soft set (f, a) ∈ τ is an a-soft open neighborhood of x in (x, τ, a) if x ∈ f(a). proposition 3.6. let (x, τ, a) be a soft topological space. then, (g, a) ∈ τ if and only if for every a ∈ a and x ∈ g(a) there exists an a-soft open neighborhood (g(a,x), a) of x in (x, τ, a) such that (g(a,x), a) ⊑ (g, a). proof. if (g, a) ∈ τ, then for every a ∈ a and x ∈ g(a) we consider the soft set (g(a,x), a), where g(a,x) = g. obviously, (g(a,x), a) is an a-soft open neighborhood of x. conversely, we suppose that for every a ∈ a and x ∈ g(a) there exists an a-soft open neighborhood (g(a,x), a) of x in (x, τ, a) such that (g(a,x), a) ⊑ (g, a), that is g(a,x)(p) ⊆ g(p), for every p ∈ a. (1) we prove that (g, a) ∈ τ. we set i = {(a, x) : a ∈ a, x ∈ g(a)}. it suffices to prove that (g, a) = ⊔{(g(a,x), a) : (a, x) ∈ i} or equivalently g(p) = ∪{g(a,x)(p) : (a, x) ∈ i}, for every p ∈ a. let p ∈ a. by relation (1) we have g(a,x)(p) ⊆ g(p), for every (a, x) ∈ i. therefore, ∪{g(a,x)(p) : (a, x) ∈ i} ⊆ g(p). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 100 soft set theory and topology we prove that g(p) ⊆ ∪{g(a,x)(p) : (a, x) ∈ i}. let y ∈ g(p). then, by assumption there exists a p-soft open neighborhood (g(p,y), a) of y in (x, τ, a) such that (g(p,y), a) ⊑ (g, a). therefore, y ∈ g(p,y)(p) ⊆ ∪{g(a,x)(p) : (a, x) ∈ i}. thus, (g, a) = ⊔{(g(a,x), a) : (a, x) ∈ i}. � definition 3.7 (see [22]). let (x, τ, a) be a soft topological space. the soft closure cl(f, a) of (f, a) ∈ ss(x, a) is the soft set ⊓{(q, a) ∈ τc : (f, a) ⊑ (q, a)}. definition 3.8. let (x, τ, a) be a soft topological space and a ∈ a. a point x ∈ x is said to be an a-cluster point of (f, a) ∈ ss(x, a) if for every a-soft open neighborhood (g, a) of x we have (f, a) ⊓ (g, a) 6= 0a. the set of all a-cluster points of (f, a) is denoted by cl(f, a). also, the set of all a-cluster points of (f, a)c is denoted by cl((f, a)c). proposition 3.9. let (x, τ, a) be a soft space and (f, a) ∈ ss(x, a). then, cl(f, a) = (rf,a, a), where the map rf,a : a → p(x) defined as follows: rf,a(p) = f(p) ∪ cl(f, p), for every p ∈ a. proof. we need to prove that (a) (f, a) ⊑ (rf,a, a), (b) (rf,a, a) ∈ τ c, and (c) (rf,a, a) ⊑ (q, a), for every (q, a) ∈ τ c such that (f, a) ⊑ (q, a). (a) first we observe that f(p) ⊆ f(p) ∪ cl(f, p) = rf,a(p), for every p ∈ a. thus, (f, a) ⊑ (rf,a, a). (b) we prove that (rf,a, a) ∈ τ c or equivalently (rf,a, a) c ∈ τ. let a ∈ a and x ∈ x \ rf,a(a) = x \ (f(a) ∪ cl(f, a)). by proposition 3.6, it suffices to prove that there exists an a-soft open neighborhood (g(a,x), a) of x such that (g(a,x), a) ⊑ (rf,a, a) c or equivalently g(a,x)(p) ⊆ x \ rf,a(p) = x \ (f(p) ∪ cl(f, p)), for every p ∈ a. since x /∈ cl(f, a), there exists an a-soft open neighborhood (g(a,x), a) of x such that (f, a) ⊓ (g(a,x), a) = 0a. this means that f(p) ∩ g(a,x)(p) = ∅, for every p ∈ a. therefore, g(a,x)(p) ⊆ x \ f(p), for every p ∈ a. we prove that g(a,x)(p) ⊆ x \ cl(f, p), for every p ∈ a. indeed, let y ∈ g(a,x)(p), where p ∈ a. then, the soft set (g(a,x), a) is a p-soft open neighborhood of y such that (f, a)⊓(g(a,x), a) = 0a and, therefore, y ∈ x \cl(f, p). thus, g(a,x)(p) ⊆ (x \ f(p)) ∩ (x \ cl(f, p)) = x \ (f(p) ∪ cl(f, p)), for every p ∈ a. (c) finally, let (q, a) ∈ τc such that (f, a) ⊑ (q, a). (2) we prove that (rf,a, a) ⊑ (q, a). since rf,a(p) = f(p)∪cl(f, p) and f(p) ⊆ q(p), for every p ∈ a, it suffices to prove that cl(f, p) ⊆ q(p) or x \ q(p) ⊆ x \ cl(f, p), for every p ∈ a. indeed, let y ∈ x \ q(p) and y ∈ cl(f, p), where p ∈ a. we observe that the soft set (q, a)c is a p-soft open neighborhood c© agt, upv, 2014 appl. gen. topol. 15, no. 1 101 d. n. georgiou and a. c. megaritis of y such that (f, a) ⊓ (q, a)c 6= 0a which contradicts relation (2). thus, y ∈ x \ cl(f, p). � definition 3.10 (see [24]). let (x, τ, a) be a soft topological space. the soft interior int(f, a) of (f, a) ∈ ss(x, a) is the soft set ⊔{(g, a) ∈ τ : (g, a) ⊑ (f, a)}. definition 3.11. let (x, τ, a) be a soft topological space and a ∈ a. a point x ∈ x is said to be an a-interior point of (f, a) ∈ ss(x, a) if there exists an a-soft open neighborhood (g, a) of x such that (g, a) ⊑ (f, a). the set of all a-interior points of (f, a) is denoted by int(f, a). proposition 3.12. let (x, τ, a) be a soft space and (f, a) ∈ ss(x, a). then, int(f, a) = (rf,a, a), where the map rf,a : a → p(x) defined as follows: rf,a(p) = f(p) ∩ int(f, p), for every p ∈ a. proof. we need to prove that (a) (rf,a, a) ⊑ (f, a), (b) (rf,a, a) ∈ τ, and (c) (g, a) ⊑ (rf,a, a), for every (g, a) ∈ τ such that (g, a) ⊑ (f, a). (a) first we observe that rf,a(p) = f(p)∩int(f, p) ⊆ f(p), for every p ∈ a. thus, (rf,a, a) ⊑ (f, a). (b) we prove that (rf,a, a) ∈ τ. let a ∈ a and x ∈ rf,a(a) = f(a) ∩ int(f, a). by proposition 3.6, it suffices to prove that there exists an a-soft open neighborhood (g(a,x), a) of x such that (g(a,x), a) ⊑ (rf,a, a) or equivalently g(a,x)(p) ⊆ f(p) ∩ int(f, p), for every p ∈ a. since x ∈ int(f, a), there exists an a-soft open neighborhood (g(a,x), a) of x such that (g(a,x), a) ⊑ (f, a). therefore, g(a,x)(p) ⊆ f(p), for every p ∈ a. we prove that g(a,x)(p) ⊆ int(f, p), for every p ∈ a. indeed, let y ∈ g(a,x)(p), where p ∈ a. then, the soft set (g(a,x), a) is a p-soft open neighborhood of y such that (g(a,x), a) ⊑ (f, a) and, therefore, y ∈ int(f, p). thus, g(a,x)(p) ⊆ f(p) ∩ int(f, p), for every p ∈ a. (c) finally, let (g, a) ∈ τ such that (g, a) ⊑ (f, a). (3) we must prove that (g, a) ⊑ (rf,a, a). it suffices to prove that g(p) ⊆ f(p) ∩ int(f, p), for every p ∈ a. indeed, let y ∈ g(p), where p ∈ a. then, (g, a) is a p-soft open neighborhood of y such that (g, a) ⊑ (f, a). therefore, y ∈ int(f, p). also, by relation (3) we have g(p) ⊆ f(p). hence, y ∈ f(p). thus, y ∈ f(p) ∩ int(f, p). � proposition 3.13. let (x, τ, a) be a soft space, a ∈ a, and (f, a) ∈ ss(x, a). then, cl((f, a)c) = x \ int(f, a). proof. we prove that cl((f, a)c) ⊆ x \ int(f, a). let x ∈ cl((f, a)c). then, for every a-soft open neighborhood (g, a) of x we have (f, a)c ⊓ (g, a) 6= 0a. we suppose that x ∈ int(f, a). then, there exists an a-soft open neighborhood c© agt, upv, 2014 appl. gen. topol. 15, no. 1 102 soft set theory and topology (g, a) of x such that (g, a) ⊑ (f, a). therefore, (f, a)c ⊓ (g, a) = 0a, which is a contradiction. thus, x ∈ x \ int(f, a). now, we prove that x \ int(f, a) ⊆ cl((f, a)c). let x ∈ x \ int(f, a) and (g, a) be an a-soft open neighborhood of x. we must prove that (f, a)c ⊓ (g, a) 6= 0a. since x /∈ int(f, a), there exists p ∈ a such that g(p) * f(p). this means that there exists x ∈ x such that x ∈ g(p) and x ∈ x \ f(p). hence, (x \ f(p)) ∩ g(p) 6= ∅ and, therefore, (f, a)c ⊓ (g, a) 6= 0a. thus, x ∈ cl((f, a)c). � definition 3.14. let (x, τ, a) be a soft topological space. the soft boundary bd(f, a) of (f, a) ∈ ss(x, a) is the soft set cl(f, a) ⊓ cl((f, a)c). proposition 3.15. let (x, τ, a) be a soft space and (f, a) ∈ ss(x, a). then, bd(f, a) = (rf,a, a), where the map rf,a : a → p(x) defined as follows: rf,a(p) = (f(p) ∪ cl(f, p)) ∩ ((x \ f(p)) ∪ (x \ int(f, p))), for every p ∈ a. proof. by propositions 3.9, 3.12, and 3.13 for every p ∈ a we have rf,a(p) = (f(p) ∪ cl(f, p)) ∩ ((x \ f(p)) ∪ cl((f, p) c))) = (f(p) ∪ cl(f, p)) ∩ ((x \ f(p)) ∪ (x \ int(f, p))). � definition 3.16. let (x, τ, a) be a soft topological space. a family b ⊆ τ is called a base for (x, τ, a) if for every soft open set (g, a) 6= 0a, there exist (gi, a) ∈ b, i ∈ i, such that (g, a) = ⊔{(gi, a) : i ∈ i}. proposition 3.17. let (x, τ, a) be a soft topological space. then, a family b ⊆ τ is a base for (x, τ, a) if and only if for every a ∈ a, x ∈ x, and every a-soft open neighborhood (g, a) of x there exists an a-soft open neighborhood (g(a,x), a) of x such that (g(a,x), a) ∈ b and (g(a,x), a) ⊑ (g, a). proof. let b be a base for (x, τ, a), a ∈ a, x ∈ x, and (g, a) be an a-soft open neighborhood of x. then, x ∈ g(a). since b is a base for (x, τ, a), there exist (gi, a) ∈ b, i ∈ i, such that (g, a) = ⊔{(gi, a) : i ∈ i}. hence, g(a) = ∪{gi(a) : i ∈ i} and, therefore, x ∈ gi0(a) for some i0 ∈ i. thus, (gi0, a) is an a-soft open neighborhood of x such that (gi0, a) ∈ b and (gi0 , a) ⊑ (g, a). conversely, let b ⊆ τ. suppose that for every a ∈ a, x ∈ x, and every a-soft open neighborhood (g, a) of x there exists an a-soft open neighborhood (g(a,x), a) of x such that (g(a,x), a) ∈ b and (g(a,x), a) ⊑ (g, a). we prove that b is a base for (x, τ, a). indeed, let (g, a) 6= 0a be a soft open set. we consider the set i = {(a, x) : a ∈ a, x ∈ g(a)}. then, as in the proof of proposition 3.6 we have (g, a) = ⊔{(g(a,x), a) : (a, x) ∈ i}. since (g(a,x), a) ∈ b, for every (a, x) ∈ i, the set b is a base for (x, τ, a). � c© agt, upv, 2014 appl. gen. topol. 15, no. 1 103 d. n. georgiou and a. c. megaritis definition 3.18. let (x, τx, a) and (y, τy , b) be two soft topological spaces, x ∈ x, and e : a → b. a map f : x → y is called soft e-continuous at the point x if for every a ∈ a and every e(a)-soft open neighborhood (g, b) of f(x) in (y, τy , b) there exists an a-soft open neighborhood (f, a) of x in (x, τx, a) such that φfe(f, a) ⊑ (g, b). if the map f is soft e-continuous at any point x ∈ x, then we say that the map f is soft e-continuous. proposition 3.19. let (x, τx, a) and (y, τy , b) be two soft topological spaces and e : a → b. then, the following statements are equivalent: (1) the map f : x → y is soft e-continuous. (2) φ−1 fe (g, b) ∈ τx, for every (g, b) ∈ τy . (3) φ−1 fe (q, b) ∈ τcx, for every (q, b) ∈ τ c y . proof. (1) ⇒ (2) let (g, b) ∈ τy . then, φ −1 fe (g, b) is the soft set (f, a) ∈ ss(x, a) such that f(px) = f −1(g(e(px))), for every px ∈ a. let a ∈ a and x ∈ f(a). by proposition 3.6, it suffices to prove that there exists an a-soft open neighborhood (f(a,x), a) of x in (x, τx, a) such that (f(a,x), a) ⊑ (f, a). since x ∈ f(a) = f−1(g(e(a))), we have f(x) ∈ g(e(a)). this means that the soft set (g, b) is an e(a)-soft open neighborhood of f(x) in (y, τy , b). since the map f : x → y is soft e-continuous, there exists an a-soft open neighborhood (f(a,x), a) of x in (x, τx, a) such that φfe(f(a,x), a) ⊑ (g, b). therefore, by proposition 2.18, (f(a,x), a) ⊑ φ −1 fe (φfe(f(a,x), a)) ⊑ φ −1 fe (g, b). thus, φ−1 fe (g, b) ∈ τx. (2) ⇒ (1) let x ∈ x, a ∈ a, and (g, b) be an e(a)-soft open neighborhood of f(x) in (y, τy , b). then, f(x) ∈ g(e(a)) or x ∈ f −1(g(e(a))). therefore, by assumption, φ−1 fe (g, b) is an a-soft open neighborhood of x in (x, τx, a). therefore, by proposition 2.18, φfe(φ −1 fe (g, b)) ⊑ (g, b). thus, the map f : x → y is soft e-continuous at the point x. (2) ⇒ (3) let (q, b) ∈ τcy . then, (q, b) c ∈ τy . by proposition 2.18 we have φ−1 fe ((q, b)c) = (φ−1 fe (q, b))c. since φ−1 fe ((q, b)c) ∈ τx, we have φ−1 fe (q, b) ∈ τcx. (3) ⇒ (2) let (g, b) ∈ τy . then, (g, b) c ∈ τcy . by proposition 2.18 we have φ−1 fe ((g, b)c) = (φ−1 fe (g, b))c. since φ−1 fe ((g, b)c) ∈ τcx, we have φ−1 fe (g, b) ∈ τx. � c© agt, upv, 2014 appl. gen. topol. 15, no. 1 104 soft set theory and topology example 3.20. let x = {x1, x2, x3}, y = {y1, y2, y3}, a = {0, 1}, and b = {0, 1, 2}. we consider the following soft sets (f, a), (g, a), and (h, a) over x defined as follows: f(p) = { {x3}, if p = 0, {x1, x2}, if p = 1, g(p) = { ∅, if p = 0, {x3}, if p = 1, h(p) = { {x3}, if p = 0, x, if p = 1. also, we consider the following soft sets (q, b) and (r, b) over y defined as follows: q(p) =      {y1}, if p = 0, {y3}, if p = 1, ∅, if p = 2, r(p) =      {y1, y2}, if p = 0, {y3}, if p = 1, y, if p = 2. then, the triplets (x, τx, a) and (y, τx, b), where τx = {0a, 1a, (f, a), (g, a), (h, a)} and τy = {0b, 1b, (q, b), (r, b)} are soft topological spaces. let f : x → y be the map such that f(x1) = f(x2) = y1 and f(x3) = y3 and e : a → b the map such that e(0) = 1 and e(1) = 0. then, the map f is soft e-continuous. also, if e′ : a → b is the map such that e′(0) = 1 and e′(1) = 2, then the map f is not soft e′-continuous. proposition 3.21. let (x, τx, a) and (y, τy , b) be two soft topological spaces, by a base for (y, τy , b), and e : a → b. then, the following statements are equivalent: (1) the map f : x → y is soft e-continuous. (2) φ−1 fe (g, b) ∈ τx, for every (g, b) ∈ by . proof. (1) ⇒ (2) follows by proposition 3.19. (2) ⇒ (1) by proposition 3.19 it suffices to prove that φ−1 fe (g, b) ∈ τx, for every (g, b) ∈ τy . let (g, b) ∈ τy . then, there exist (gi, b) ∈ by , i ∈ i, such that (g, b) = ⊔{(gi, b) : i ∈ i}. therefore, by proposition 2.19 we have φ−1 fe (g, b) = φ−1 fe (⊔{(gi, b) : i ∈ i}) = ⊔{φ −1 fe (gi, b) : i ∈ i} ∈ τx. � remark 3.22 (see, for example, [24]). let (x, τ, a) be a soft topological space and (f, a) ∈ ss(x, a). we recall the following properties : (1) (f, a) ∈ τc if and only if cl(f, a) = (f, a). (2) (f, a) ∈ τ if and only if int(f, a) = (f, a). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 105 d. n. georgiou and a. c. megaritis (3) int(f, a) = (cl((f, a)c))c. (4) cl(f, a) = (int((f, a)c))c. (5) if (f, a) ⊑ (g, a), then cl(f, a) ⊑ cl(g, a). proposition 3.23. let (x, τx, a) and (y, τy , b) be two soft topological spaces and e : a → b. then, the following statements are equivalent: (1) the map f : x → y is soft e-continuous. (2) φfe(cl(f, a)) ⊑ cl(φfe(f, a)), for every (f, a) ∈ ss(x, a). (3) cl(φ−1 fe (g, b)) ⊑ φ−1 fe (cl(g, b)), for every (g, b) ∈ ss(y, b). (4) φ−1 fe (int(g, b)) ⊑ int(φ−1 fe (g, b)), for every (g, b) ∈ ss(y, b). proof. (1) ⇒ (2) let (f, a) ∈ ss(x, a). since φfe(f, a) ⊑ cl(φfe(f, a)), by proposition 2.18 we have (f, a) ⊑ φ−1 fe (φfe(f, a)) ⊑ φ −1 fe (cl(φfe(f, a))). therefore, cl(f, a) ⊑ cl(φ−1 fe (cl(φfe(f, a)))). since cl(φfe(f, a)) ∈ τ c y , by proposition 3.19, φ −1 fe (cl(φfe(f, a))) ∈ τ c x and, therefore, cl(φ−1 fe (cl(φfe(f, a)))) = φ −1 fe (cl(φfe(f, a))). hence, cl(f, a) ⊑ φ−1 fe (cl(φfe(f, a))). finally, by proposition 2.18 we have φfe(cl(f, a)) ⊑ φfe(φ −1 fe (cl(φfe(f, a)))) ⊑ cl(φfe(f, a)). (2) ⇒ (3) let (g, b) ∈ ss(y, b). we apply (2) to (f, a) = φ−1 fe (g, b) and we obtain the inclusion φfe(cl(φ −1 fe (g, b))) ⊑ cl(φfe(φ −1 fe (g, b))) ⊑ cl(g, b). therefore, cl(φ−1 fe (g, b)) ⊑ φ−1 fe (φfe(cl(φ −1 fe (g, b)))) ⊑ φ−1 fe (cl(g, b)). (3) ⇒ (4) let (g, b) ∈ ss(y, b). we apply (3) to (g, b)c and we obtain the inclusion cl(φ−1 fe ((g, b)c)) ⊑ φ−1 fe (cl((g, b)c)), which gives (see proposition 2.18) φ−1 fe (int(g, b)) = φ−1 fe ((cl((g, b)c))c) = (φ−1 fe (cl((g, b)c)))c ⊑ (cl(φ−1 fe ((g, b)c)))c = (cl((φ−1 fe (g, b))c))c = int(φ−1 fe (g, b)). (4) ⇒ (1) by proposition 3.19 it suffices to prove that φ−1 fe (g, b) ∈ τx, for every (g, b) ∈ τy . let (g, b) ∈ τy . then, int(g, b) = (g, b). therefore, φ−1 fe (g, b) = φ−1 fe (int(g, b)) ⊑ int(φ−1 fe (g, b)). c© agt, upv, 2014 appl. gen. topol. 15, no. 1 106 soft set theory and topology also, int(φ−1 fe (g, b)) ⊑ φ−1 fe (g, b). thus, int(φ−1 fe (g, b)) = φ−1 fe (g, b), which means that φ−1 fe (g, b) ∈ τx. � definition 3.24. let (x, τx, a) and (y, τy , b) be two soft topological spaces and e : a → b. a map f : x → y is called soft e-open (respectively, soft eclosed) if for every (f, a) ∈ τx (respectively, (f, a) ∈ τ c x) we have φfe(f, a) ∈ τy (respectively, φfe(f, a) ∈ τ c y ). proposition 3.25. let (x, τx, a) and (y, τy , b) be two soft topological spaces and e : a → b. then, the following statements are equivalent: (1) the map f : x → y is soft e-open. (2) φfe(int(f, a)) ⊑ int(φfe(f, a)), for every (f, a) ∈ ss(x, a). proof. (1) ⇒ (2) let (f, a) ∈ ss(x, a). since, int(f, a) ⊑ (f, a), we have φfe(int(f, a)) ⊑ φfe(f, a). since int(f, a) ∈ τx, we have φfe(int(f, a)) ∈ τy . therefore, by the above inclusion we have φfe(int(f, a)) ⊑ ⊔{(g, a) ∈ τy : (g, a) ⊑ φfe(f, a)} = int(φfe(f, a)). (2) ⇒ (1) we prove that φfe(f, a) ∈ τy , for every (f, a) ∈ τx. let (f, a) ∈ τx. then, φfe(f, a) = φfe(int(f, a)) ⊑ int(φfe(f, a)). also, int(φfe(f, a)) ⊑ φfe(f, a). thus, int(φfe(f, a)) = φfe(f, a), which means that φfe(f, a) ∈ τy . � the proof of the following proposition is similar to the proof of proposition 3.25. proposition 3.26. let (x, τx, a) and (y, τy , b) be two soft topological spaces and e : a → b. then, the following statements are equivalent: (1) the map f : x → y is soft e-closed. (2) cl(φfe(f, a)) ⊑ φfe(cl(f, a)), for every (f, a) ∈ ss(x, a). definition 3.27. let (x, τx, a) and (y, τy , b) be two soft topological spaces and e a 1-1 map of a onto b. a soft e-continuous map f of x onto y is called soft e-homeomorphism if the map f is 1-1 and the inverse map f−1 : y → x is soft e−1-continuous. proposition 3.28. let (f, a) ∈ ss(x, a), e a 1-1 map of a onto b, and f a 1-1 map of x onto y . then, (1) φfe(f, a) = φ −1 f−1e−1 (f, a). (2) φfe((f, a) c) = (φfe(f, a)) c. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 107 d. n. georgiou and a. c. megaritis proof. (1) let φfe(f, a) = (g, b), φ −1 f−1e−1 (f, a) = (g′, b), and py ∈ b. we must prove that g(py ) = g ′(py ). let e −1(py ) = px. since the map e : a → b is 1-1, g(py ) = f(f(px)). on the other hand, g′(py ) = (f −1)−1(f(e−1(py ))) = f(f(px)). thus, g(py ) = g ′(py ). (2) let φfe((f, a) c) = (g, b), (φfe(f, a)) c = (g′, b), and py ∈ b. we must prove that g(py ) = g ′(py ). let e −1(py ) = px. since the map e : a → b is 1-1, g(py ) = f(x \ f(px)). since the map f : x → y is 1-1 and onto, f(x \ f(px)) = y \ f(f(px)). therefore, g(py ) = y \ f(f(px)). on the other hand, g′(py ) = y \ f(f(px)). thus, g(py ) = g ′(py ). � proposition 3.29. let (x, τx, a) and (y, τy , b) be two soft topological spaces, e a map of a onto b, and f a 1-1 map of x onto y . then, the following statements are equivalent: (1) the map f is soft e-homeomorphism. (2) the map f is soft e-continuous and soft e-open. (3) the map f is soft e-continuous and soft e-closed. proof. (1) ⇒ (2) we prove that φfe(f, a) ∈ τy , for every (f, a) ∈ τx. let (f, a) ∈ τx. since the map f −1 is soft e−1-continuous and (f, a) ∈ τx, we have φ−1 f−1e−1 (f, a) ∈ τy . by proposition 3.28, φfe(f, a) ∈ τy . (2) ⇒ (3) we prove that φfe(f, a) ∈ τ c y , for every (f, a) ∈ τ c x. let (f, a) ∈ τcx. then, (f, a) c ∈ τx and, therefore, φfe((f, a) c) ∈ τy . by proposition 3.28, (φfe(f, a)) c = φfe((f, a) c), which means that φfe(f, a) ∈ τ c y . (3) ⇒ (1) we prove that the inverse map f−1 : y → x is soft e−1continuous. it suffices to prove that φ−1 f−1e−1 (f, a) ∈ τcy , for every (f, a) ∈ τ c x. let (f, a) ∈ τcx. then, φfe(f, a) ∈ τ c y . by proposition 3.28, φ −1 f−1e−1 (f, a) = φfe(f, a). thus, φ −1 f−1e−1 (f, a) ∈ τcy . � references [1] h. aktaş and n. çağman, soft sets and soft groups, information sciences 177 (2007), 2726–2735. [2] m. i. ali, f. feng, x. liu, w. k. min and m. shabir, on some new operations in soft set theory, comput. math. appl. 57 (2009), 1547–1553. [3] a. aygunoglu and h. aygun, some notes on soft topological spaces, neural. comput. appl. 21 (2011), 113–119. 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[25] y. zou and z. xiao, data analysis approaches of soft sets under incomplete information, knowl. base. syst. 21 (2008), 941–945. c© agt, upv, 2014 appl. gen. topol. 15, no. 1 109 @ applied general topology c© universidad politécnica de valencia volume 4, no. 2, 2003 pp. 421–444 bombay hypertopologies giuseppe di maio, enrico meccariello and somashekhar naimpally dedicated by the first two authors to professor s. naimpally on the occasion of his 70th birthday. abstract. recently it was shown that, in a metric space, the upper wijsman convergence can be topologized with the introduction of a new far-miss topology. the resulting wijsman topology is a mixture of the ball topology and the proximal ball topology. it leads easily to the generalized or g-wijsman topology on the hyperspace of any topological space with a compatible lo-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). further generalization involving a topological space with two compatible lo-proximities and a cobase results in a new hypertopology which we call the bombay topology. the generalized locally finite bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. we show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences. 2000 ams classification: 54b20, 54a10, 54c35, 54d30, 54e05, 54e15. keywords: hyperspace, wijsman topology, ball topology, proximal ball topology, far-miss topology, hit-and-miss topology, bombay topology, vietoris topology, fell topology, proximal locally finite topology, ∆-topology, proximal locally finite ∆-topology. 1. introduction. the main purpose of this work is to give a unified treatment to the problem of comparing various hypertopologies with one another. we accomplish this by representing known hypertopologies as special cases of just one general hypertopology the illocally finite bombay hypertopology. we prove one result, with a simple proof, giving necessary and sufficient conditions for one upper 422 g. di maio, e. meccariello and s. naimpally bombay topology to be coarser than another. we then show how this one result includes, as special cases, many known results scattered throughout the literature. moreover, our approach gives simple transparent proofs in comparison with those in the original articles which involve intricate calculations. hypertopologies, which were born during the early part of the last century, have proven to be useful in continua theory, topologies on spaces of functions, optimization, convex analysis, game theory, differential equations, image analysis and fractal geometry, etc. two early discoveries where: (a) the vietoris topology that can be defined in any topological space, and (b) the hausdorff metric topology which is available only in a metric space. after the discovery of uniform spaces by weil, the second one was generalized to hausdorff-bourbaki uniform topology. in a seminal paper, michael [18] made a detailed study of above topologies in the middle of the last century. then in 1966 wijsman [25], in studying convex analysis, introduced a convergence for nets of closed subsets of a metric space (x,d) viz: an → a iff for each x ∈ x, d(x,an) → d(x,a). the lower part of this convergence is equivalent to convergence in the lower vietoris topology. attempts to topologize the upper part of the convergence were only partially successful until recently. the first attempt resulted in the ball topology ([1]) wherein a typical neighbourhood of a closed set consists of closed subsets that do not intersect a proper closed ball in the metric space. the next attempt led to proximal ball topology ([10]) wherein a typical neighbourhood of a closed set consists of closed sets that are far (w.r.t. the metric proximity) from a proper closed ball. this was more successful since it equalled the wijsman topology in nice metric spaces, which included the normed linear spaces. but the topologization for all metric spaces defied attempts for the past 36 years. a major part of the literature on hyperspaces consists of comparisons of various hypertopologies with one another. in this, the wijsman topologies play an important role as the building blocks of many other topologies ([2]). but they are not easy to handle as the wijsman topology depends strongly on the metric d. for example, even two uniformly equivalent metrics can give rise to unequal wijsman topologies and non-uniformly equivalent metrics can induce equal wijman topologies! moreover, it is not easy to compare objects of different kinds and the absence of topologization of wijsman topology was a major hurdle. so the literature contains long and complicated proofs involving epsilonetics. recently, one of us [20] discovered a simple way of topologizing the upper wijsman convergence. in this approach, a typical neighbourhood of a closed set a in a metric space (x,d) consists of closed sets that do not intersect a proper closed ball b that is far from a in the metric proximity. with this new approach it is possible to give simple conceptual proofs of results involving comparisons of the wijsman topologies among themselves and with others. this new way bombay hypertopologies 423 of looking at upper wijsman topology interprets those results in a simple and direct way. moreover, it leads to a generalization of the wijsman topology to the hyperspace of any topological space satisfying some simple conditions. when it was further scrutinized, it was found that the above generalization of the upper wijsman topology can be further generalized to represent all known upper hypertopologies. in this way we arrived at the upper bombay topology which can be defined in any topological space equipped with two compatible proximities and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). joining the upper bombay topology to the lower one generated by a collection il of locally finite families of open sets give rise to the il-locally finite bombay topology which includes all known hypertopologies as special cases. we thus achieved our goal of unification stated at the beginning. for references on hyperspaces up to 1993, we generally refer to [1], except when a specific reference is needed. for proximities see [13], [22] and [19]. 2. preliminaries. let (x,τ) denote a t1 space. for any e ⊂ x, clxe, inte and ec stand for the closure, interior and complement of e in x, respectively. let γ, γ1 and γ2 be compatible lo-proximities on x. we always assume γ1 ≤ γ2 (i.e. a6γ1b implies a6γ2b (see [22]) where 6γ stands for the negation of γ). we use the symbol γ0 to denote the fine lo-proximity on x given by aγ0b iff clxa∩ clxb 6= ∅. (γ0 is called the wallman proximity). in the case (x,τ) is tychonoff, then: γ] denotes the fine ef-proximity on x given by a6γ]b iff they can be separated by a continuous function f : x → [0, 1]. (γ] is called the functionally indistinguishable proximity on x). if u is a compatible separated uniformity on x, then γ(u) denotes the ef-proximity on x given by aγ(u)b iff u(a) ∩b 6= ∅ for each u ∈u. (γ(u) is called the uniform proximity induced by u). if (x,τ) is a metrizable space with metric d, then γ(d) is the ef-proximity on x given by aγ(d)b iff dd(a,b) = inf{d(a,b) : a ∈ a,b ∈ b} = 0. (γ(d) is called the metric proximity induced by d). cl(x) (resp. k(x)) is the family of all nonempty closed subsets of x (resp. the family of all nonempty closed and compact subsets of x). ∆ is a nonempty subfamily of cl(x) which is closed under finite unions and which contains all singletons. we call ∆ a cobase ([23]). let us note that in the literature ∆ is usually assumed to contain merely all singletons. 424 g. di maio, e. meccariello and s. naimpally in our view the above assumption simplifies the results. in fact, it allows to display transparent statements and makes theory much simpler. for any set e ⊂ x and a subfamily e ⊂ τ we use the following standard notation: e− = {a ∈ cl(x) : a∩e 6= ∅}; e− = {a ∈ cl(x) : a∩e 6= ∅ for each e ∈e}. furthermore, if γ is a compatible proximity on x, we set e++γ = {a ∈ cl(x) : a �γ e, i.e. a6γec}. note that γ1 ≤ γ2 is equivalent to u++γ1 ⊂ u ++ γ2 for every u ∈ τ. we omit γ if it is clear from the context and write e++γ simply as e ++. moreover e+ = {a ∈ cl(x) : a ⊂ e, i.e. a �γ0 e}. now we describe some hypertopologies on cl(x). the upper proximal ∆-topology (w.r.t. γ) σ(γ; ∆)+ is generated by the basis {e++ : ec ∈ ∆}. when γ = γ0 we have the upper ∆-topology τ(∆)+ = σ(γ0; ∆)+. the lower vietoris (or finite) topology τ(v −) has a basis {e− : e ⊂ τ is finite}. the lower locally finite topology τ(lf−) has a basis {e− : e ⊂ τ is locally finite}. the il-lower locally finite topology τ(il−) has a basis {e− : e ⊂ il} provided il ⊂ {e ⊂ τ : e is locally finite} satisfies the following filter condition: (∗) whenever e, f ∈ il, then there exists g ∈ il such that g− ⊂e− ∩f− (see [17]). the proximal (finite) ∆-topology (w.r.t. γ) σ(γ; ∆) = σ(γ; ∆)+∨τ(v −). we omit γ if it is obvious from the context and write σ(∆) for σ(γ; ∆). the ∆-topology τ(∆) = τ(∆)+∨τ(v −) = σ(γ0; ∆)+∨τ(v −) = σ(γ0; ∆). the proximal locally finite ∆-topology (w.r.t. γ) σ(γ; lf, ∆) = σ(γ; ∆)+ ∨ τ(lf−). bombay hypertopologies 425 the proximal il-locally finite ∆-topology (w.r.t. γ) σ(γ; il, ∆) = σ(γ; ∆)+ ∨ τ(il−). we omit the prefix ”proximal” and replace σ by τ if γ = γ0. well known special cases are: (a) when ∆ = cl(x), τ(∆) = τ(v ) the vietoris or finite topology; σ(γ; ∆) = σ(γ) the proximal topology (w.r.t. γ); τ(lf∆) = τ(lf) the locally finite topology; σ(γ; lf, ∆) = σ(γ; lf) the proximal locally finite topology. (b) when ∆ = k(x), τ(∆) = τ(f) the fell topology. (c) let (x,d) be a metric space, γ(d) the metric proximity induced by d and ∆ denote the cobase b generated by all finite unions of closed balls of nonnegative radii. then τ(∆) = τ(b) the ball topology; σ(γ(d); ∆) = σ(γ(d);b) = σ(b) the proximal ball topology. it was shown in [20] that a typical neighbourhood at a ∈ cl(x) in the upper wijsman topology τ(wd)+ is of the form: u+ where uc ∈b and a6γ(d)uc. thus three parameters are involved: (i) γ(d) the metric proximity in a6γ(d)uc, (ii) γ0 the fine lo-proximity in u+, and (iii) the cobase b which contains uc. we note that there are two proximities, namely γ(d), γ0 with γ(d) ≤ γ0 and a cobase. by replacing the two proximal parameters γ(d), γ0 by two loproximities γ1, γ2 with γ1 ≤ γ2 and the cobase b by ∆ we have the following definition: definition 2.1. let (x,τ) be a t1 space with compatible proximities γ1, γ2 with γ1 ≤ γ2 and ∆ a cobase. then a typical neighbourhood of a ∈ cl(x) in the upper bombay topology σ(γ1,γ2; ∆)+ is: u++γ2 where u c ∈ ∆ and a6γ1uc (or equivalently a ∈ u++γ1 ). (note that since γ1 ≤ γ2 a6γ1uc implies a6γ2uc which in turn is equivalent to a ∈ u++γ2 ). γ1, γ2 and ∆ are the three parameters of the upper bombay hypertopology: γ1, γ2 are the proximal parameters and ∆ is the cobase. 426 g. di maio, e. meccariello and s. naimpally furthermore, σ(γ1; ∆)+ (respectively, σ(γ2; ∆)+) represents the first coordinate upper topology (the second coordinate upper topology). (i) the bombay topology σ(γ1,γ2; ∆) is the join of the upper bombay topology σ(γ1,γ2; ∆)+ and the lower vietoris topology τ(v −) i.e. σ(γ1,γ2; ∆) = σ(γ1,γ2; ∆)+ ∨ τ(v −). (ii) the locally finite bombay topology σ(γ1,γ2; lf, ∆) = σ(γ1,γ2; ∆)+ ∨ τ(lf−). let il ⊂{e : e ∈ τ is locally finite} satisfy the following filter condition: (∗) whenever e, f ∈ il, then there is g ∈ il such that g− ⊂e− ∩f−. (iii) the il-locally finite bombay topology σ(γ1,γ2; il, ∆) = σ(γ1,γ2; ∆)+ ∨ τ(il−). remark 2.2. (a) if in (iii) of above definition each member of il is finite, then σ(γ1,γ2; il, ∆) equals σ(γ1,γ2; ∆), since τ(il −) = τ(v −) (see also lemma 3.2 below). (b) again if in (iii) of definition 2.1 we choose γ1 = γ2 = γ, then we have the proximal il-locally finite ∆ topology σ(γ; il, ∆) from which we obtain all classical hypertopologies defined above. (c) let (x,d) be a metric space, ∆ = b the cobase generated by all closed balls, and γ = γ(d) the metric proximity induced by d. then σ(γ,γ0; ∆) = σ(γ,γ0;b) = τ(wd) is the wijsman topology, i.e. the bombay topology is a generalization of the wijsman topology. in addition, suppose (x,τ) is a t1 space, γ a compatible lo-proximity coarser than γ0 and ∆ a cobase, then we refer the upper-bombay topology σ(γ,γ0; ∆)+ (the bombay topology σ(γ,γ0; ∆)) as the upper g-wijsman topology (the gwijsman topology). here g stands for generalized (w.r.t. ∆ and γ since γ0 is kept fixed). (d) let (x,u) be a separated uniform space and γ = γ(u) the compatible ef-proximity induced by u. let il be the collection of all families of open sets of the form {u(x) : x ∈ q ⊂ a}, where a ∈ cl(x), u ∈u and q is u-discrete, i.e. x, y ∈ q, x 6= y implies x 6∈ u(y). then the hausdorff bourbaki or h-b uniform topology associated to u on cl(x) is τ(uh) = τ(uh)+∨τ(u−h) = σ(γ) +∨τ(il−) (see [20]). moreover, if the uniformity u comes from a metric d we get the hausdorff metric topology τ(hd). thus all hausdorff-bourbaki topologies are proximal locally finite. bombay hypertopologies 427 (e) many new hypertopologies can be defined by replacing the lower vietoris topology τ(v −) by the lower il-locally finite topology. thus we have the il-locally finite fell topology, the il-locally finite wijsman topology, the il-locally finite ball topology, the proximal il-locally finite ∆-topology, etc. in addition, by a proper choice of the two proximal coordinates in a bombay topology, one can get infinitely many new hypertopologies. 3. principal results. in this section we plan to find necessary and sufficient conditions for one il-locally finite bombay topology to be coarser than another. it is well known that in the comparison of hit-and-miss hypertopologies, the lower and the upper parts play their roles separately. hence, following lemmas hold. lemma 3.1. let (x,τ) be a t1 space with compatible lo-proximities γ1, γ2, η1, η2 satisfying γ1 ≤ γ2 and η1 ≤ η2. let ∆, λ be cobases and il1, il2 two collections of locally finite families of open sets satisfying condition (∗). the following are equivalent: (a) σ(γ1,γ2; il1, ∆) ≤ σ(η1,η2; il2, λ); (b) τ(il−1 ) ≤ τ(il − 2 ) and σ(γ1,γ2; ∆) + ≤ σ(η1,η2; λ)+. lemma 3.2. let (x,τ) be a t1 space and il, il1, il2 collections of locally finite families of open sets satisfying condition (∗). then: τ(il−1 ) ≤ τ(il − 2 ) if and only if for each e ∈ il1, there exists f ∈ il2 such that f− ⊂e−. thus if all members of il2 are finite, then the same is true of the members of il1 and hence τ(il −) ≤ τ(v −) if and only if all members of il are finite. now, we turn our attention to the upper bombay topologies and consider the principal result of this paper showing that it includes most of the results in the literature involving comparison of various hypertopologies. next definition and lemma play a key role. definition 3.3. let (x,τ) be a t1 space and γ a compatible lo-proximity on x. if b ⊂ x, set γ(b) = {f ⊂ x : fγb} ([24]), i.e. γ(b) is the collection of all subsets f of x near to b w.r.t. γ. lemma 3.4. let (x,τ) be a t1 space, γ and η compatible lo-proximities on x and c and d nonempty closed subsets of x. the following are equivalent: (a) (dc)++η ⊂ (cc)++γ ; (b) c ⊂ d and γ(c) ⊂ η(d). proof. (a) ⇒ (b). assume not, then either i) c 6⊂ d or ii) γ(c) 6⊂ η(d). if i) occurs, then there exists c ∈ c \ d. choose f ∈ (dc)++η and consider 428 g. di maio, e. meccariello and s. naimpally f ′ = f ∪{c}. then f ′ ∈ (dc)++η but f ′ 6∈ (cc)++γ ; a contradiction. if ii) occurs, then there exists an f ⊂ x such that fγc but f 6ηd. since fγc, then f 6= ∅. let e = clxf. then e ∈ cl(x), eγc but e 6ηd. therefore e ∈ (dc)++η but e 6∈ (cc)++γ ; a contradiction. (b) ⇒ (a). assume not, then (dc)++η 6⊂ (cc)++γ . hence there exists f ∈ cl(x) such that f 6ηd but fγc, i.e. f ∈ γ(c) but f 6∈ η(d); a contradiction. � theorem 3.5. (main theorem) let (x,τ) be a t1 space with compatible lo-proximities γ1, γ2, η1, η2 satisfying γ1 ≤ γ2 and η1 ≤ η2 and ∆ and λ cobases. the following are equivalent: (a) σ(γ1,γ2; ∆)+ ≤ σ(η1,η2; λ)+; (b) for each b ∈ ∆ and w ∈ τ, w 6= x, with b �γ1 w , there exists a b′ ∈ λ such that: (i) b ⊂ b′ �η1 w , and (ii) γ2(b) ⊂ η2(b′). proof. σ(γ1,γ2; ∆)+ ≤ σ(η1,η2; λ)+ if and only if for each a ∈ cl(x), a 6= x, a ∈ u++γ2 ∈ σ(γ1,γ2; ∆) + where uc ∈ ∆ and a ∈ u++γ1 there exists a v ∈ τ such that v c ∈ λ, a ∈ v ++η1 and a ∈ v ++ η2 ⊂ u++γ2 . noting that a ∈ v ++ η1 is equivalent to v c �η1 ac and using lemma 3.4 we have (i) and (ii) where w = ac, b = uc and b′ = v c. � corollary 3.6. let (x,τ) be a t1 topological space with compatible lo-proximities γ1, γ2, η1, η2 satisfying γ1 ≤ γ2 and η1 ≤ η2, ∆ and λ cobases and il1 and il2 collections of locally finite families of open sets satisfying condition (∗). the following are equivalent: (a) σ(γ1,γ2; il1, ∆) ≤ σ(η1,η2; il2, λ); (b) il2 refines il1 and whenever b ∈ ∆, w ∈ τ, w 6= x, with b �γ1 w , then there exists a b′ ∈ λ such that: (i) b ⊂ b′ �η1 w , and (ii) γ2(b) ⊂ η2(b′). remark 3.7. (a) for future reference we note that fγb implies that there is a net in x whose range c ⊂ f and cγb (cf. lemma 3.2 in [5]). (b) we note that if η2 ≤ γ2, then (ii) at (b) of the main theorem is automatically satisfied. (c) if γ1 is an ef-proximity, then a ∈ u++γ1 implies the existence of e ∈ cl(x) with e ∈ u++γ1 and a ∈ (inte) ++ γ1 . so setting w ′ = inte, (i) at (b) of the main theorem can be written as b ⊂ b′ �η1 w ′ �γ1 w . (d) the proximal topologies σ(γ1; ∆) and σ(γ2; ∆) are the proximal coordinate topologies of the given (finite) bombay topology σ(γ1,γ2; ∆). bombay hypertopologies 429 let γ and η, with γ ≤ η, be compatible lo-proximities on a given topological space x and ∆ a cobase. in the motivation we saw that the (finite) bombay topology σ(γ,η; ∆) is a generalization of the g-wijsman topology σ(γ,γ0; ∆), which in turn, is a generalization of the wijsman topology σ(γ(d),γ0;b). we conclude this section by deriving some results in the general case and we give appropriate references. we reserve the special cases: 1) (x,τ) is a metrizable space with metric d and the first proximal parameter γ = γ(d) is the ef-proximity associated to d, and 2) (x,τ) is a tychonoff space and the first proximal parameter γ is ef or (x,τ) is a uniformizable space with separated uniformity u and the first proximal parameter γ = γ(u) is the proximity induced by u, to next sections. the results given below are new and follow easily from the main theorem or from the definitions and so we omit the proofs. we start to compare a (finite) bombay topology σ(γ1,γ2; ∆) with its proximal coordinate topologies. theorem 3.8. (cf. [1] pages 45, 53) let (x,τ) be a t1 space with compatible lo-proximities γ1, γ2, satisfying γ1 ≤ γ2 and ∆ a cobase. then: (a) σ(γ1,γ2; ∆) ≤ σ(γ1; ∆); (b) σ(γ1,γ2; ∆) ≤ σ(γ2; ∆). corollary 3.9. let (x,τ) be a t1 space, γ a compatible lo-proximity and ∆ a cobase. then: (a) σ(γ,γ0; ∆) ≤ σ(γ; ∆) (i.e. the g-wijsman topology is coarser than its first proximal coordinate topology); (b) σ(γ,γ0; ∆) ≤ τ(∆) (i.e. the g-wijsman topology is coarser than its second coordinate topology). next theorem and corollary show when a bombay topology σ(γ1,γ2; ∆) and a g-wijsman topology σ(γ,γ0; ∆) are finer than their proximal coordinate topologies. theorem 3.10. (cf. [1] pages 45, 52, 53) let (x,τ) be a t1 space with compatible lo-proximities γ1, γ2 satisfying γ1 ≤ γ2 and ∆ a cobase. the following are equivalent: (a) σ(γ2; ∆) ≤ σ(γ1,γ2; ∆); (b) σ(γ1,γ2; ∆) = σ(γ1; ∆) = σ(γ2; ∆); 430 g. di maio, e. meccariello and s. naimpally (c) for each b ∈ ∆ and w ∈ τ, w 6= x, with b �γ2 w , there exists a b′ ∈ ∆ such that b ⊂ b′ �γ1 w ; (d) whenever a ∈ cl(x), b ∈ ∆ and a6γ2b, then a6γ1b. corollary 3.11. let (x,τ) be a t1 space, γ a compatible lo-proximity on x and ∆ a cobase. the following are equivalent: (a) τ(∆) ≤ σ(γ,γ0; ∆); (b) σ(γ,γ0; ∆) = σ(γ; ∆) = τ(∆); (c) for each b ∈ ∆ and w ∈ τ, w 6= x, with b ⊂ w , there exists a b′ ∈ ∆ such that b ⊂ b′ �γ w ; (d) whenever a ∈ cl(x), b ∈ ∆ and a∩b = ∅, then a6γb. now, we compare two proximal hit-and-miss hypertopologies associated to the same cobase ∆. theorem 3.12. (cf. [1] page 53 and theorem 3.3 in [5]) let (x,τ) be a t1 space with compatible lo-proximities γ1, γ2 satisfying γ1 ≤ γ2 and ∆ a cobase. the following are equivalent: (a) σ(γ1; ∆) ≤ σ(γ2; ∆); (b) for each b ∈ ∆ and w ∈ τ, w 6= x, with b �γ1 w , there exists a b′ ∈ ∆ such that b ⊂ b′ �γ2 w and γ1(b) ⊂ γ2(b′); (c) whenever a ∈ cl(x), b ∈ ∆ and a6γ1b, then there exists a b′ ∈ ∆ such that i) b ⊂ b′, a6γ2b′ and ii) whenever c is the range of a net satisfying c ⊂ (b′)c and cγ1b then cγ2b′. corollary 3.13. let (x,τ) be a t1 space, γ a compatible lo-proximity on x and ∆ a cobase. the following are equivalent: (a) σ(γ; ∆) ≤ τ(∆); (b) for each b ∈ ∆ and w ∈ τ, w 6= x, with b �γ w , there exists a b′ ∈ ∆ such that b ⊂ b′ �γ0 w and γ(b) ⊂ γ0(b′); (c) whenever a ∈ cl(x), b ∈ ∆ and a6γb, then there exists a b′ ∈ ∆ such that i) b ⊂ b′, a∩b′ = ∅ and ii) whenever {xλ : λ ∈ σ} is a net whose range c is contained in (b′)c and cγb, then {xλ : λ ∈ σ} has a cluster point. theorem 3.14. (cf. [1] page 53 and theorem 3.11 in [5]) let (x,τ) be a t1 space with compatible lo-proximities γ1, γ2 satisfying γ1 ≤ γ2 and ∆ a cobase. the following are equivalent: (a) σ(γ2; ∆) ≤ σ(γ1; ∆); (b) σ(γ2; ∆) = σ(γ1; ∆); (c) for each b ∈ ∆ and w ∈ τ, w 6= x, with b �γ2 w , there exists a b′ ∈ ∆ such that b ⊂ b′ �γ1 w ; (d) whenever a ∈ cl(x), b ∈ ∆ and a6γ2b, then a6γ1b. corollary 3.15. let (x,τ) be a t1 space, γ a compatible lo-proximity on x and ∆ a cobase. the following are equivalent: bombay hypertopologies 431 (a) τ(∆) ≤ σ(γ; ∆); (b) τ(∆) = σ(γ; ∆); (c) for each b ∈ ∆ and w ∈ τ, w 6= x, with b ⊂ w , there exists a b′ ∈ ∆ such that b ⊂ b′ �γ w ; (d) whenever a ∈ cl(x), b ∈ ∆ and a∩b = ∅, then a6γb. next theorem compares two bombay topologies which have the same second proximity parameter but different cobases. theorem 3.16. (cf. [1] page 39) let (x,τ) be a t1 space and γ, δ and η compatible lo-proximities on x such that γ ≤ η as well as δ ≤ η. let ∆ and λ be cobases. the following are equivalent: (a) σ(γ,η; ∆) ≤ σ(δ,η; λ); (b) for each b ∈ ∆ and w ∈ τ, w 6= x, with b �γ w , there exists a b′ ∈ λ such that b ⊂ b′ �δ w . corollary 3.17. let (x,τ) be a t1 space, γ and δ compatible lo-proximities on x and ∆ and λ cobases. the following are equivalent: (a) σ(γ,γ0; ∆) ≤ σ(δ,γ0; λ); (b) for each b ∈ ∆ and w ∈ τ, w 6= x, with b �γ w , there exists a b′ ∈ λ such that b ⊂ b′ �δ w . corollary 3.18. let (x,τ) be a t1 space, ∆ and λ cobases. the following are equivalent: (a) τ(∆) ≤ τ(λ); (b) for each b ∈ ∆ and w ∈ τ, w 6= x, with b ⊂ w , there exists a b′ ∈ λ such that b ⊂ b′ ⊂ w . 4. the metric case. a large part of the literature is in the setting of a metric space. let (x,τ) be a metrizable space with metric d, γ = γ(d) the d-metric proximity, b(d) the family of finite unions of all d-closed balls and tb(d) denote the family of all closed d-totally bounded subsets of x (we omit d if it is obvious from the context and write respectively b and tb for b(d) and tb(d)). remark 4.1. let (x,τ) be a metrizable space with metric d and γ = γ(d) the d-metric proximity. let α, δ and ε be positive reals with ε < δ < α and sd(x,ε) and bd(x,ε) be the open and the closed d-balls centered at x of radius ε. we omit the subscript d if it is clear from the context. then: (a) if x is a metrizable space with metric d, then tb = tb(d) is a cobase and it is even hereditarily closed. (b) whenever α, δ and ε are positive reals with ε < δ < α, then s(x,ε) ⊂ b(x,ε) �γ s(x,δ) ⊂ b(x,δ) �γ s(x,α) ⊂ b(x,α). 432 g. di maio, e. meccariello and s. naimpally (c) a set d is said to be strictly d-included in e (d ⊂⊂d e) iff there is a finite set of points {x1, · · · ,xn} of e and positive reals εk < αk, k = 1, · · · ,n, such that (sdi) d ⊂ n⋃ k=1 s(xk,εk) ⊂ n⋃ k=1 s(xk,αk) ⊂e ([1] page 38). so, using (b) it can be shown that open balls can be replaced by closed balls in (sdi) as well as (sdi) is in turn equivalent to d ⊂ n⋃ k=1 s(xk,εk) �γ n⋃ k=1 s(xk,αk) ⊂e. (d) if a is a nonempty subset of x, b′ is a finite unions of balls, w ∈ τ and a ⊂ b′ �γ w , then there exists a finite union of balls b′′ such that a ⊂ b′ ⊂⊂d b′′ �γ w and thus by above a �γ b′′ �γ w . indeed, if b′ �γ w , then d(b′,wc) = inf{d(b,y) : b ∈ b′,y ∈ wc} = ε > 0. since b′ = n⋃ k=1 s(xk,εk), select t = ε 2 and set b′′ = n⋃ k=1 s(xk,εk + t). thus whenever a ∈ cl(x) and w ∈ τ with w 6= x, the following are equivalent: (1) there exists a b′ ∈b such that a ⊂ b′ �γ w ; (2) a ⊂⊂d w ; (3) there exists a b′′ ∈b such that a �γ b′′ �γ w . (e) we note that from remark 3.7 (a) if c and c′ are nonempty subsets of x with c ⊂ c′, then the inclusion γ(c) ⊂ γ0(c′) occurs if and only if whenever there is a sequence of points {xn : n ∈ in} in (c′)c with limn→∞d(xn,c) = 0, then the sequence {xn : n ∈ in} has a cluster point (see [1] page 52). (f) let x be a metrizable space with compatible metrics d and e inducing the metric proximities γ = γ(d) and η = η(e), respectively. let ∆ = b = b(d) and λ = b′ = b(e). the following are equivalent: (1) τ(wd) ≤ τ(we); (2) for each b ∈ b and w ∈ τ with b �γ w there exists a b′ ∈ b′ such that b ⊂ b′ �η w ; (3) for each b ∈ b and w ∈ τ with b �γ w there exists a b′′ ∈ b′ such that b ⊂⊂e b′′ ⊂⊂e w (cf. [3], [15]); (4) each proper open d-sphere is strictly e-included in every its open enlargement (cf. theorem 2.1.10 in [1]). bombay hypertopologies 433 (g) from (d) we have that strict d-inclusion is equivalent to weak total boundedness. we recall that a closed subset e of x is said to be weakly totally bounded or w-tb in a open set w iff there exists a b ∈b such that e �γ b �γ w (see (16.1) in [10]). moreover: (g1) in any infinite dimensional banach space, the closed unit ball is not totally bounded but it is weakly totally bounded in any open ball centered at 0 and radius greater than 1; (g2) let l2 be the hilbert space of square summable sequences, θ the origin and {en : n ∈ in} the standard orthonormal base for l2. let (x,d) be the metric subspace of l2 where x = {θ}∪{en : n ∈ in}. then {e2n+1 : n ∈ in} is not w-tb in x \{e2n : n ∈ in}. theorem 4.2. (cf. 3.9) let (x,d) be a metric space, γ = γ(d) the metric proximity induced by d and b the cobase generated by all closed d-balls. then: (a) τ(wd) ≤ σ(b); (b) τ(wd) ≤ τ(b). theorem 4.3. (cf. 3.11) let (x,d) be a metric space, γ = γ(d) the metric proximity induced by d and b the cobase generated by all closed d-balls. in the following (a), (b), (c) and (d) are equivalent and each implies (e) which is equivalent to (f). (a) τ(b) ≤ τ(wd); (b) τ(wd) = σ(b) = τ(b); (c) for each b ∈b and w ∈ τ, w 6= x, with b ⊂ w implies b ⊂⊂d w ; (d) x is ball atsuji (i.e. disjoint closed sets, one of which is a ball, are far). (e) σ(b) ≤ τ(wd); (f) for each positive real ε and each b ∈ b, b is strictly d-included in its ε-enlargement, i.e. b ⊂⊂d s(b,ε). theorem 4.4. (cf. 3.13) let (x,d) be a metric space, γ = γ(d) the metric proximity induced by d and b the cobase generated by all closed d-balls. the followings are equivalent: (a) σ(γ;b) ≤ τ(b); (b) for each b ∈ b and w ∈ τ, w 6= x, with b �γ w , there exists a b′ ∈b such that b ⊂ b′ ⊂ w and γ(b) ⊂ γ0(b′); (c) for each b ∈ b and w ∈ τ, w 6= x, with b �γ w , there exists a b′ ∈b such that b ⊂ b′ ⊂ w and for each sequence {xn : n ∈ in} in (b′)c with lim n→∞ d(xn,b) = 0 has a cluster point. theorem 4.5. (cf. theorem 3.1 in [8]) let x be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d and ∆ a cobase. the following are equivalent: (a) τ(wd) ≤ τ(∆); 434 g. di maio, e. meccariello and s. naimpally (b) for every x ∈ x and every reals α, β with 0 < α < β and s(x,β) 6= x, there exists a b′ ∈ ∆ such that s(x,α) ⊂ b′ �γ0 s(x,β); (c) τ(wd) ≤ σ(γ; ∆). lemma 4.6. let x be a metrizable space with metric d and a ∈ cl(x). the following are equivalent: (a) a is totally bounded; (b) for each positive real ε and each closed subset e of a, e ⊂⊂d s(e,ε); (c) for each positive real ε and each closed subspace e of a, e is w-tb in s(e,ε) (i.e. there exists b ∈b such that e � b � s(e,ε)). proof. it suffices to show that (c) ⇒ (a), since (a) ⇒ (b) and (b) ⇒ (c) follow from the definition of total boundedness and remark 4.1 (d) respectively. suppose (c) holds but not (a). then, without any loss of generality, we may suppose that a = {an : n ∈ in} is bounded and there is an ε > 0 such that d(ap,aq) > ε for all p 6= q. now we use the ”milky way ” technique. set a1,n = an for all n ∈ in. since {a1,n : n ∈ in} is bounded, it has a subsequence {a2,n : n ∈ in} such that d(a1,1,a2,n) → α1 ≥ ε. inductively, for each r ∈ in, {ar,n : n ∈ in} has a subsequence {ar+1,n : n ∈ in} such that d(ar,r,ar+1,n) → αr ≥ ε. we note that for all infinite subsets d ⊂{as,n : s > r,n ∈ in} d(ar,r,d) = αr. set e = {a2r,2r : r ∈ in} and η = inf{αr : r ∈ in}. clearly, (∗) 0 < ε ≤ η < +∞. we claim that e is not w-tb in s(e,η). for if not, there is a p ∈ in and a t > 0 such that s(a2p,2p; t) � s(e,η) and s(a2p,2p; t) contains an infinite subset f ⊂{a2n,2n : n ∈ in}. from what we have done so far, it follows that t < η and d(a2p,2p,f) = α2p ≤ t. so, α2p ≤ t < η ≤ inf{αr : r ∈ in} which contradicts (∗). � corollary 4.7. let x a metrizable space with metric d and b ∈ cl(x). the following are equivalent: (a) b is compact; (b) for each closed subset e of b and each w ∈ τ with e ⊂ w , e ⊂⊂d w ; (c) for each closed subset e of b and each w ∈ τ with e ⊂ w , e is w-tb in w. proof. (a) ⇒ (b) is clear because since γ is an ef-proximity and b is compact, then b ⊂ w is equivalent to b �γ w . hence apply (d) in remark 4.1. again, (b) ⇔ (c) follows from (d) in remark 4.1. (c) ⇒ (a) it suffices to prove that if every closed subset of b is w-tb in a larger open set, then every sequence {xn : n ∈ in} in b has a convergent subsequence. by above lemma {xn : n ∈ in} has a cauchy subsequence {yk : k ∈ in}. if {yk : k ∈ in} does not converge, then the closed set {y2k+1 : k ∈ in} is not w-tb in the open set x \{y2k : k ∈ in}. � from (d) in remark 4.1, main theorem, lemma 4.6 and corollary 4.7 we have: bombay hypertopologies 435 theorem 4.8. (cf. proposition 4.1 in [8]) let x be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, b = b(d) the family of finite unions of all closed d-balls and ∆ a cobase. the following are equivalent: (a) σ(γ; ∆) ≤ τ(wd); (b) for every b ∈ ∆ \{x} and every positive real ε there exists a b′ ∈ b such that b ⊂ b′ �γ s(b,ε); (c) for every b ∈ ∆ \{x} and every positive real ε, b ⊂⊂d s(b,ε). thus if ∆ is hereditarily closed, then σ(γ; ∆) ≤ τ(wd) if and only if every b ∈ ∆ is totally bounded. corollary 4.9. (cf. theorem (5.5) in [2]) let (x,d) be a metric space and γ the metric proximity induced by d. the following are equivalent: (a) τ(wd) = σ(γ); (b) for each positive real ε and each proper closed set e, e ⊂⊂d s(e,ε); (c) for each positive real ε and each closed subspace b of x, b is w-tb in s(b,ε); (d) x is totally bounded. theorem 4.10. (cf. proposition 4.2 in [8]) let x be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, b = b(d) the family of finite unions of all closed d-balls and ∆ a cobase. the following are equivalent: (a) τ(∆) ≤ τ(wd); (b) for every b ∈ ∆ and every w ∈ τ, w 6= x, with b ⊂ w , there exists a b′ ∈b such that b ⊂ b′ �γ w ; (c) whenever b ∈ ∆ and w ∈ τ, w 6= x, with b ⊂ w , then b ⊂⊂d w . thus if ∆ is hereditarily closed, then τ(∆) ≤ τ(wd) if and only if every b ∈ ∆ is compact. corollary 4.11. (cf. corollary (5.6) in [2]) let (x,d) be a metric space and γ the metric proximity induced by d. the following are equivalent: (a) τ(wd) = τ(v ); (b) whenever e is a proper closed subset of x and w ∈ τ is such that e ⊂ w , then e ⊂⊂d w ; (c) for each closed subspace b of x and each w ∈ τ with b ⊂ w , b is w-tb in w ; (d) x is compact. theorem 4.12. (cf. proposition 3.1 in [7]) let x be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, tb = tb(d) the cobase of all closed d-totally bounded set of x. the following are equivalent: (a) τ(tb) ≤ τ(wd); (b) x is complete. proof. the main theorem shows that the range of any subsequence of a cauchy sequence (which is totally bounded) is far from every disjoint closed set. � 436 g. di maio, e. meccariello and s. naimpally theorem 4.13. let x be a metrizable space and γ = γ(d) the metric proximity induced by d. the following are equivalent: (a) τ(v ) = σ(γ); (b) γ = γ0; (c) x is atsuji. proof. use the main theorem noting that τ(v ) = σ(γ0) = σ(γ) if and only if τ(v )+ = σ(γ0)+ = σ(γ)+. � 5. the tychonoff case. as we noted above, quite a bit of the literature on hyperspaces, especially that concerning the wijsman topology, is based on metric spaces. here we show that many results are valid in (generalized) uniform spaces (see [6], [10], [11] and [5]). in this section (x,τ) denotes a tychonoff space, γ a compatible ef-proximity, whereas γ0 and γ] denote respectively the wallman (the fine lo-) proximity and the functionally indistinguishable (the fine ef-) proximity on x (see preliminaries). it is a well known fact that γ] = γ0 if and only if x is normal (urysohn’s lemma). π(γ) denotes the family of all uniformities u compatible with γ, u? is the coarsest totally bounded member of π(γ) and u] the fine uniformity. π(τ) denotes the family of all uniformities u compatible with τ. it is a well kown fact that π(γ) ⊂ π(τ). without loss of generality we may assume that each u ∈ π(τ) consists of members which are symmetric and closed. for each u ∈ π(τ), γ(u) is the uniform proximity induced by u, whereas b(u) denotes the collection of finite unions of (generalized) closed balls w.r.t. u, where for each x ∈ x and u ∈ u, u[x] is the (generalized) u ball in x centered at x. tb(u) denotes the collection of all closed u-totally bounded subsets of x. the result below cannot be extended to lo-proximity spaces (see lemma 6.1 in [5]). theorem 5.1. (cf. theorem 2.7 in [11], [1] page 50) let (x,τ) be a tychonoff space and γ a compatible ef -proximity on x. then sup{σ(γ,γ0;b(u)) : u ∈ π(γ)} = σ(γ). proof. it suffices to prove σ(γ) ≤ sup{σ(γ,γ0;b(u)) : u ∈ π(γ)}. suppose a ∈ w ++γ ∈ σ(γ), w ∈ τ, w 6= x and a ∈ cl(x). then there are a closed entourage u ∈ u? and a finite subset f of x such that a ⊂ u[f] ⊂ u2[f] �γ w . thus σ(γ) ≤ σ(γ,γ0;b(u?)) and hence the claim holds. � corollary 5.2. (cf. [11]) let (x,τ) be a tychonoff space. then sup{σ(γ,γ0;b(u)) : u ∈ π(τ)} = σ(γ]). bombay hypertopologies 437 corollary 5.3. (cf. [21]) let (x,τ) be a tychonoff space. the following are equivalent: (a) sup{σ(γ,γ0;b(u)) : u ∈ π(τ)} = τ(v ); (b) x is normal; (c) γ] = γ0. corollary 5.4. (cf. [2]) let x be a metrizable space with metric d and γ = γ(d) the metric proximity induced by d. then: (a) sup{τ(we): e varies in the set of all metrics uniformly equivalent to d} = σ(γ); (b) sup{τ(we): e varies in the set of all metrics topologically equivalent to d} = τ(v ). theorem 5.5. (cf. [7] and lemma 6.12 in [5]) let (x,τ) be a tychonoff space, γ a compatible ef -proximity on x, u ∈ π(γ) and tb(u) the family of all totally bounded closed subsets of x w.r.t. u. then σ(γ; tb(u)) ≤ σ(γ,γ0;b(u)). proof. let b ∈ tb(u) and w ∈ τ with b �γ w 6= x. then there are an entourage u ∈ u and a finite set f ⊂ x such that b ⊂ u[f] ⊂ u2[f] ⊂ w . observe that u[f] ∈ b(u) and u[f] ⊂ u2[b] ⊂ w implies u[f] �γ w . the result follows from the main theorem. � let u be a separated uniformity on x and u ∈u. we recall that that u′ ∈u is composably contained in u iff there is a u′′ ∈u such that u′ ◦u′′ ⊂ u (see definition (3.3) in [10]). next result generalizes theorem 3.1 in [8] to tychonoff spaces. theorem 5.6. (cf. theorem 3.1 in [8]) let (x,τ) be a tychonoff space, u ∈ π(τ), the compatible ef -proximity γ = γ(u) and ∆ a cobase. the following are equivalent: (a) σ(γ,γ0;b(u)) ≤ τ(∆); (b) for each x ∈ x, for each u ∈ u with u[x] 6= x and for each u′ ∈ u composably contained in u there is a b′ ∈ ∆ such that u′[x] ⊂ b′ �γ u[x]; (c) σ(γ,γ0;b(u)) ≤ σ(γ; ∆). proof. it suffices to use the main theorem observing that if u′ ∈u is composably contained in u ∈u, then u′[x] �γ u[x] for each x ∈ x. � using the notion of composably contained we have the next definition which extends to uniform setting the concept of strict inclusion. definition 5.7. (cf. [3], [15] and [1] page 38) let (x,τ) a tychonoff space, u ∈ π(τ) and d, e ⊂ x. we say that d is strictly u-included in e (d ⊂⊂u e) iff there exist a finite set f ⊂ d and entourages u, u′ ∈u with u′ composably contained in u such that 438 g. di maio, e. meccariello and s. naimpally (?) d ⊂ u′[f] ⊂ u[f] ⊂ e. remark 5.8. if γ = γ(u), then in above definition condition (?) is equivalent to the following one (?′) d ⊂ u′[f] �γ u[f] ⊂ e. thus whenever γ is a compatible ef-proximity on x and u ∈ π(γ), a set d is stricly u-included in e iff there exist a finite subset f and entourages u, u′ ∈u such that d ⊂ u′[f] �γ u[f] ⊂ e. we also note that if u is the metric uniformity induced by d, then for each u ∈u there exists some positive ε such that u = {(x,y) ∈ x×x : d(x,y) ≤ ε}. thus u[x] = b(x,ε) for each x ∈ x and d ⊂⊂u e is equivalent to d ⊂⊂d e (cf. (c) in remark 4.1). the next result generalizes theorems 4.1 and 4.2 in [8] to the ef-proximities setting. we omit the proof since it is easily derived from definitions, the main theorem and the above remark. theorem 5.9. let (x,τ) be a tychonoff space, γ a compatible ef -proximity on x, u ∈ π(γ) and ∆ a cobase. in the following (a) and (b) are equivalent and each implies (c) which is equivalent to (d). (a) τ(∆) ≤ σ(γ,γ0;b(u)); (b) whenever b ∈ ∆ \{x} and w ∈ τ with b ⊂ w , then b ⊂⊂u w . (c) for every b ∈ ∆ \{x} and u ∈u, b ⊂⊂u u[b]; (d) σ(γ; ∆) ≤ σ(γ,γ0;b(u)). corollary 5.10. (cf. 3.11) let (x,τ) be a tychonoff space, γ a compatible ef -proximity on x and u ∈ π(γ). in the following (a), (b), (c), (d) and (e) are equivalent and each implies (f) which is equivalent to (g). (a) τ(b(u)) ≤ σ(γ,γ0;b(u)); (b) σ(γ,γ0;b(u)) = τ(b(u)) = σ(γ;b(u)); (c) whenever b ∈b(u) and w ∈ τ with b ⊂ w , then b ⊂⊂u w ; (d) for every b ∈ b(u) and w ∈ τ, w 6= x, with b ⊂ w , there exists a b′ ∈b(u) such that b ⊂ b′ �γ w ; (e) x is b(u)-atsuji space w.r.t. γ (i.e. disjoint closed sets, one of which is a member of b(u), are far w.r.t. γ). (f) σ(γ;b(u)) ≤ σ(γ,γ0;b(u)); (g) for every b ∈b(u) and u ∈u, b ⊂⊂u u[b]. we end this section with the following results derived from (d) in remark 2.2 and lemma 3.2. theorem 5.11. (cf. [6]) let (x,τ) be a tychonoff space, γ a compatible ef -proximity on x and u ∈ π(γ). then the proximal topology σ(γ) and the hausdorff-bourbaki topology τ(uh) are equal if and only if the uniformity u equals the coarsest member u? ∈ π(γ), and thus u is totally bounded. bombay hypertopologies 439 proof. the result follows from the fact that u is totally bounded iff maximal u-discrete sets are finite for all u ∈u. � corollary 5.12. (cf. [2]) let (x,τ) be a tychonoff space, γ a compatible ef -proximity on x and u ∈ π(γ). the following are equivalent: (a) σ(γ,γ0;b(u)) = τ(uh); (b) u is totally bounded. 6. bounded hypertopologies. in the literature on hyperspaces of a metric space (x,d), there are several bounded hypertopologies such as (i) the bounded vietoris τ(bv ), (ii) the bounded proximal σ(bγ) w.r.t. γ, (iii) the bounded hausdorff metric topology (or the attouch-wets topology) τ(awd), etc. in this section, with the help of a slight generalization of the concept of abstract boundedness due to hu ([16], also see [12] and [22]) we show that all bounded hypertopologies can be subsumed under the locally finite bombay topologies. definition 6.1. a nonempty collection ∆ ⊂ cl(x) is called a boundedness in a topological space x iff ∆ is closed under finite unions and is closed hereditary. we also assume that ∆ contains the singletons. remark 6.2. (a) the followings are examples of boundedness: (i) b(x) the family of all closed d-bounded subsets of a metric space (x,d). this can be extended easily to a uniform space (x,u) by declaring a closed set a is u-bounded iff there is a finite set {x1, · · · ,xn}⊂ x and entourages uk ∈u\{x×x}, k = 1, · · · ,n, such that a ⊂ n⋃ k=1 uk[xk] or using notation of section 4, there is a b ∈b(u) such that a ⊂ b. let ∆ = b(u) denote the family of all ubounded subsets of x and γ = γ(u) the ef-proximity induced by u. (ii) the family k(x) of all nonempty compact subsets of a hausdorff topological space. (b) let (x,u) be a hausdorff uniform space, ∆ a boundedness and γ = γ(u). then the ∆-bounded hausdorff bourbaki filter (or ∆-attouchwets filter) u∆ is generated by sets of the form: [b,u] = {(a1,a2) ∈ cl(x) ×cl(x) : a1 ∩b ⊂ u[a2] and a2 ∩b ⊂ u[a2]}, where b ∈ ∆ and u ∈u. 440 g. di maio, e. meccariello and s. naimpally the associated topology τ(u∆) is the ∆-bounded hausdorff topology (or the ∆-attouch-wets topology). we note that if x ∈ ∆ we get the hausdorff bourbaki-uniformity. if u is induced by a metric d and ∆ = tb(d), then the τ(u∆) is the bounded hausdorff topology (i.e. the attouch-wets topology) τ(awd). it was shown in [20] that if il = ilu,∆, then the family of sets {u[x] : x ∈q} where q is a maximal discrete subset of b ∈ ∆ for u ∈u describes τ(ilu,∆). thus τ(u∆) = τ(il−u,∆) ∨σ(δ; ∆) +. it is now obvious that ∆-bounded hypertopologies are a part and parcel of ∆-bombay topologies and the usual metric d-boundedness is also included in our definition of ∆, and by above it follows that: theorem 6.3. the ∆-bounded hausdorff topology (i.e. the ∆-attouch-wets topology) τ(u∆) and the bounded hausdorff topology (i.e. the attouch-wets topology) τ(awd) are proximal il-locally finite. moreover, using the main theorem and the decomposition property of the locally finite bombay topology we have: theorem 6.4. let (x,τ) be a t1 topological space with compatible lo-proximities γ1, γ2, η1, η2 satisfyng γ1 ≤ γ2 and η1 ≤ η2, ∆, λ two boundedness and il1 and il2 locally finite collections of open subsets of x. the following are equivalent: (a) σ(γ1,γ2; il1, ∆) ≤ σ(η1,η2; il2, λ); (b) il2 refines il1 and whenever b ∈ ∆, w ∈ τ, w 6= x, with b �γ1 w , then there exists a b′ ∈ λ such that: (i) b ⊂ b′ �η1 w , and (ii) γ2(b) ⊂ η2(b′). let (x,d) be a metric space and as usual γ = γ(d) the metric proximity, u the separated uniformity associated to d, b the cobase generated by all closed d-balls and b(x) the family of all closed d-bounded subsets of x. we have: (see [1] page 115) (1) the bounded proximal topology σ(δ; b(x)) = τ(v −) ∨σ(δ; b(x))+; (2) the dual bounded proximal topology σ(γ; il,b) = τ(il−u,b)∨σ(γ,γ0;b) +, which is clearly a il-locally finite bounded wijsman topology. we point out that from above remark (b) if in (2) if we replace b with b(x), then σ(γ; il,b(x)) = τ(il−u,b(x)) ∨σ(γ,γ0; b(x)) +. bombay hypertopologies 441 thus from theorem 6.4 and above remark the following results are obvious and certainly they have generalizations: theorem 6.5. let x be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, b the cobase generated by all proper closed d balls and b(x) the family of all closed d bounded subsets of x. then: (a) σ(γ,γ0;b) ≤ σ(γ;b) ≤ σ(γ; b(x)) ≤ σ(γ); (b) σ(γ,γ0;b) ≤ σ(γ;b) ≤ σ(γ; b(x)) ≤ τ(awd). theorem 6.6. (cf. [1], page 93) let x be a metrizable space, d and e compatible metrics, γ and η the metric proximities induced by d and e respectively, b(x) the family of all closed d-bounded sets and b′(x) the family of all closed e-bounded sets. the following are equivalent: (a) τ(awd) = τ(awe); (b) δ = δ′, b(x) = b′(x) (i.e. every closed d-bounded subset is ebounded and vice versa) and they have the same uniformly continuous functions on bounded sets. theorem 6.7. let x be a metrizable space with metric d, γ = γ(d). as usual, let b(x) and tb = tb(d) denote respectively the family of all closed d-bounded sets and the family of all closed d-totally bounded sets. the following are equivalent: (a) b(x) ⊂ tb; (b) τ(awd) = τ(wd); (c) τ(awd) = σ(γ; b(x)). 7. uniformizable bombay hypertopologies. let (x,τ) be a t1 space. this section is devoted to the characterization of the uniformizable bombay topologies associated with a cobase ∆ and compatible lo-proximities γ, η satysfying γ < η. we do not consider the case γ = η because from (a) and (b) in remark 2.2 we get the proximal ∆ topologies σ(γ; ∆) (w.r.t. γ) and the uniformizable proximal ∆ topologies σ(γ; ∆) have been treated in [1], [6] and [9] for example. furthermore, we omit the proofs as they are similar to those in [9]. first we need the following definitions. definition 7.1. let (x,τ) be a t1 space, γ a compatible lo-proximity and ∆ ⊂ cl(x). (a) ∆ is called γ-urysohn iff whenever d ∈ ∆ and a ∈ cl(x) are far w.r.t. γ, there exists an s ∈ ∆ such that d �γ s �γ ac (see also [4]) (b) ∆ is called urysohn iff (a) above is true w.r.t. the lo-proximity γ0, i.e. whenever d ∈ ∆ and a ∈ cl(x) are disjoint, there exists an s ∈ ∆ such that d ⊂ ints ⊂ s ⊂ ac. 442 g. di maio, e. meccariello and s. naimpally lemma 7.2. (cf. theorem 1.6 in [9]) let x be a t1 space with compatible lo-proximities γ, η satysfying γ < η and ∆ a cobase. if ∆ is γ-urysohn, then the relation π defined on the power set of x by: (?) a6πb iff cla6γclb and either cla ∈ ∆ and clb �η (cla)c or clb ∈ ∆ and cla �η (clb)c is a compatible ef -proximity on x. moreover, π ≤ γ and ∆ is γ-urysohn iff ∆ is π-urysohn. remark 7.3. (a) observe that even if the starting proximities γ and η are just lo, the new compatible proximity π is always ef. thus the base space x is automatically completely regular. note that we have a procedure that allows us to construct an ef-proximity on a tychonoff space x by using as seeds lo-proximities γ, η with γ < η and a cobase ∆ which is γ-urysohn. (b) let x be an infinite set, τ the cofinite topology, ∆ = cl(x) and γl the coarsest compatible lo-proximity on x defined by a6γlb iff a6γ0b and either a or b is finite. then x is t1 and it is easy to check that ∆ is not γ-urysohn for each compatible lo-proximity γ with γl ≤ γ ≤ γ0. lemma 7.4. (cf. theorem 2.2 in [9]) let x be a t1 space with compatible lo-proximities γ and η satisfying γ < η and ∆ a cobase. if ∆ is γ-urysohn, then the upper bombay topology σ(δ,η; ∆)+ equals the upper proximal ∆-topology σ(π; ∆)+ w.r.t. π, where π is the ef -proximity on x constructed in above lemma 7.2. theorem 7.5. (cf. theorem 2.1 in [9]) let x be a t1 space with compatible lo-proximities γ, η satisfying γ < η and ∆ a cobase hereditarily closed. the following are equivalent: (a) ∆ is γ-urysohn; (b) the bombay topology σ(γ,η; ∆) is tychonoff. let x be a tychonoff space with a compatible uniformity u (as usual we assume that all elements u ∈u are symmetric and closed), b(u) the cobase generated by all (generalized) closed balls w.r.t. u (see preliminaries in section 4) and γ = γ(u) denote the uniform proximity induced by u. then it is easy to show that b(u) is γ-urysohn. it therefore follows that: corollary 7.6. (cf. [14]) let u be a compatible uniformity on x whose elements are symmetric and closed, b(u) the cobase generated by all (generalized) u-balls and γ the uniform proximity induced by u. then the wijsman topology τ(w(u)) is tychonoff. problem 7.7. is it true that in a tychonoff space every ef -proximity can be constructed using two lo-proximities as in lemma 7.2 ? bombay hypertopologies 443 acknowledgements. we thank the referee for a careful reading of the manuscript and valuable suggestions. references [1] g. beer, topologies on closed and closed convex sets, kluwer academic publishers, (kluwer academic publishers, 1993). [2] g. beer, a. lechicki, s. levi and s. naimpally, distance functionals and suprema of hyperspace topologies, annali di matematica pura ed applicata 162 (1992), 367–381. [3] c. costantini, s. levi and j. zieminska, metrics that generate the same hyperspace convergence, set-valued analysis 1 (1993), 141–157. [4] d. di caprio and e. meccariello, notes on separation axioms in hyperspaces, q. & a. in general topology 18 (2000), 65–86. [5] d. di caprio and e. meccariello, g-uniformities, lr-proximities and hypertopologies, acta math. hungarica 88 (1-2) (2000), 73–93. [6] a. di concilio, s. naimpally and p.l. sharma, proximal hypertopologies, proceedings of the vi brasilian topological meeting, campinas, brazil (1988) [unpublished]. [7] g. di maio and ľ. holá, a hypertopology determined by the family of totally bounded sets is the infimum of upper wijsman topologies, q. & a. in general topology, 15 (1997), 51–66. [8] g. di maio and ď. holý, comparison among wijsman topology and other hypertopologies, atti sem. mat. fis. univ. di modena 48 (2000), 121–133. [9] g. di maio, e. meccariello and s. naimpally, uniformizing (proximal) ∆-topologies, topology and its applications, (to appear). [10] g. di maio and s. naimpally, comparison of hypertopologies, rendiconti di trieste 22 (1990), 140–161. [11] g. di maio and s. naimpally, abstract measure of farness and wijsmann convergence, zbornik radova 5 (1991), 109–112. [12] g. di maio and s. naimpally, some notes on hyperspace topologies, ricerche di matematica, (to appear). [13] r. engelking, general topology, revised and completed version, helderman verlag, (helderman, berlin, 1989.) [14] s. francaviglia, a. lechicki and s. levi, quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, j. math. anal. appl. 112 (1985), 347–370. [15] ľ. holá and r. lucchetti, equivalence among hypertopologies, set-valued analysis 3 (1995), 339–350. [16] s.t. hu, boundedness in topological space, j. math. pures. appl. 28 (1949), 287–340. [17] m. marjanovic, topologies on collections of closed subsets, publ. inst. math. (beograd) 20 (1966), 196–130. [18] e. michael, topologies on spaces of subsets, trans. amer. math. soc. 71 (1951), 152–182. [19] c.j. mozzochi, m. gagrat and s. naimpally, symmetric generalized topological structures, exposition press, (hicksville, new york, 1976.) [20] s. naimpally, all hypertopologies are hit-and-miss, applied general topology 3 (2002), 45–53. [21] s. naimpally and p. sharma, fine uniformity and locally finite hyperspace topology on 2x, proc. amer. math. soc. 103 (1988), 641–646. [22] s. naimpally, b. warrack, proximity spaces, cambridge tracts in mathematics 59, (cambridge university press, 1970.) [23] h. poppe, eine bemerkung über trennungsaxiome in raum der abgeschlossenen teilmengen eines topologischen raumes, arch. math. 16 (1965), 197–199. [24] w.j. thron, proximity structures and grills, math. ann. 206 (1973), 35–62. 444 g. di maio, e. meccariello and s. naimpally [25] r. wijsman, convergence of sequences of convex sets, cones, and functions, ii, trans. amer. math. soc. 123 (1966), 32–45. received february 2002 revised february 2003 giuseppe di maio seconda università degli studi di napoli, facoltà di scienze, dipartimento di matematica, via vivaldi 43, 81100 caserta, italia e-mail address : giuseppe.dimaio@unina2.it enrico meccariello università del sannio, facoltà di ingegneria, piazza roma, palazzo b. lucarelli, 82100 benevento, italia e-mail address : meccariello@unisannio.it somashekhar naimpally 96 dewson street, toronto, ontario, m6h 1h3, canada e-mail address : sudha@accglobal.net @ appl. gen. topol. 18, no. 1 (2017), 203-217 doi:10.4995/agt.2017.6889 c© agt, upv, 2017 metric spaces and textures şenol dost hacettepe university, department of mathematics and science education, 06800 beytepe, ankara, turkey. (dost@hacettepe.edu.tr) communicated by s. romaguera abstract textures are point-set setting for fuzzy sets, and they provide a framework for the complement-free mathematical concepts. further dimetric on textures is a generalization of classical metric spaces. the aim of this paper is to give some properties of dimetric texture space by using categorical approach. we prove that the category of classical metric spaces is isomorphic to a full subcategory of dimetric texture spaces, and give a natural transformation from metric topologies to dimetric ditopologies. further, it is presented a relation between dimetric texture spaces and quasi-pseudo metric spaces in the sense of j. f. kelly. 2010 msc: 54e35; 54e40; 18b30; 54e15. keywords: metric space; texture space; uniformity; natural transformation; difunction; isomorphism. 1. introduction texture theory is point-set setting for fuzzy sets and hence, some properties of fuzzy lattices (i.e. hutton algebra) can be discussed based on textures [2, 3, 4, 5]. ditopologies on textures unify the fuzzy topologies and classical topologies without the set complementation [6, 7]. recent works on textures show that they are also useful model for rough set theory [8] and semi-separation axioms [10]. on the other hand, it was given various types of completeness for diuniform texture spaces [13]. as an expanded of classical metric spaces, the dimetric notion on texture spaces was firstly defined in [11]. in this paper, we give the categorical properties of dimetric texture spaces, and present some relation between classical metric spaces and dimetric texture spaces. received 20 november 2016 – accepted 15 february 2017 http://dx.doi.org/10.4995/agt.2017.6889 ş. dost this section is devoted to some fundamental definitions and results of the texture theory from [2, 3, 4, 5, 6]. definition 1.1. let u be a set and u ⊆ p(u). then u is called a texturing of u if (t1) ∅ ∈ u and u ∈ u, (t2) u is a complete and completely distributive lattice such that arbitrary meets coincide with intersections, and finite joins with unions, (t3) u is point-seperating. then the pair (u,u) is called a texture space or texture. for u ∈ u, the p-sets and the q-sets are defined by pu = ⋂ {a ∈ u | u ∈ a}, qu = ∨ {a ∈ u | u /∈ a}, respectively. a texture (u,u) is said to be plain if pu * qu, ∀u ∈ u. a set a ∈ u\{∅} is called a molecule if a ⊆ b∪c, b,c ∈ u implies a ⊆ b or a ⊆ c. the texture (u,u) is called simple if the sets pu, u ∈ u are the only molecules in u. example 1.2. (1) for any set u, (u,p(u)) is the discrete texture with the usual set structure of u. clearly, pu = {u} and qu = u \{u} for all u ∈ u, so (u,p(u)) is both plain and simple. (2) i = {[0, t] | t ∈ [0, 1]}∪{[0, t) | t ∈ [0, 1]} is a texturing on i = [0, 1]. then (i,i) is said to be unit interval texture. for t ∈ i, pt = [0, t] and qt = [0, t). clearly, (i,i) is plain but not simple since the sets qu, 0 < u ≤ 1, are also molecules. (3) for textures (u,u) and (v,v), u⊗v is product texturing of u×v [5]. note that the product texturing u⊗v of u ×v consists of arbitrary intersections of sets of the form (a×v )∪(u×b), a ∈ u and b ∈ v. here, for (u,v) ∈ u×v p(u,v) = pu ×pv and q(u,v) = (qu ×v ) ∪ (u ×qv). ditopology: a pair (τ,κ) of subsets of u is called a ditopology on a texture (u,u) where the open sets family τ and the closed sets family κ satisfy u, ∅ ∈ τ, u, ∅ ∈ κ g1,g2 ∈ τ =⇒ g1 ∩g2 ∈ τ, k1, k2 ∈ κ =⇒ k1 ∪k2 ∈ κ gi ∈ τ,i ∈ i =⇒ ∨ i∈i gi ∈ τ, ki ∈ κ,i ∈ i =⇒ ⋂ i∈i ki ∈ κ. direlation: let (u,u), (v,v) be texture spaces. now we consider the product texture p(u) ⊗ v of the texture spaces (u,p(u)) and (v,v). in this texture, p-sets and the q-sets are denoted by p (u,v) and q(u,v), respectively. clearly, c© agt, upv, 2017 appl. gen. topol. 18, no. 1 204 metric spaces and textures p (u,v) = {u}× pv and q(u,v) = (u \ {u}× v ) ∪ (u × qv) where u ∈ u and v ∈ v . according to: (1) r ∈ p(u) ⊗v is called a relation from (u,u) to (v,v) if it satisfies r1 r * q(u,v),pu′ * qu =⇒ r * q(u′,v). r2 r * q(u,v) =⇒ ∃u′ ∈ u such that pu * qu′ and r * q(u′,v). (2) r ∈ p(u) ⊗v is called a corelation from (u,u) to (v,v) if it satisfies cr1 p (u,v) * r,pu * qu′ =⇒ p (u′,v) * r. cr2 p (u,v) * r =⇒ ∃u′ ∈ u such that pu′ * qu and p (u′,v) * r. (3) if r is a relation and r is a corelation from (u,u) to (v,v) then the pair (r,r) is called a direlation from (u,u) to (v,v). a pair (i,i) is said to be identity direlation on (u,u) where i = ∨ {p (u,u) | u ∈ u} and i = ⋂ {q(u,u) | u * qu}. recall that [5] we write (p,p) v (q,q) if p ⊆ q and q ⊆ p where (p,p) and (q,q) are direlations. let (p,p) and (q,q) be direlations from (u,u)to (v,v). then the greatest lower bound of (p,p) and (q,q) is denoted by (p,p)u(q,q) , and it is defined by (p,p) u (q,q) = (puq,p tq) where puq = ∨ {p (u,v) | ∃z ∈ u with pu * qz, and p,q * q(z,v)}, p tq = ⋂ {q(u,v) | ∃z ∈ u with pz * qu, and p (z,v) * p,q}. inverses of a direlation: if (r,r) is a direlation then the inverse direlation of (r,r)← is a direlation from (v,v) to (u,u), and it is defined by (r,r)← = (r←,r←) where r← = ⋂ {q(v,u) | r * q(u,v)} and r ← = ∨ {p (v,u) | p (u,v) * r} the a-sections and the b-presections under a direlation (r,r) are defined as r→a = ⋂ {qv | ∀u,r * q(u,v) =⇒ a ⊆ qu}, r→a = ∨ {pv | ∀u,p (u,v) * r =⇒ pu ⊆ a}, r←b = ∨ {pu | ∀v,r * q(u,v) =⇒ pv ⊆ b}, r←b = ⋂ {qu | ∀v,p (u,v) * r =⇒ b ⊆ qv}. the composition of direlations: let (p,p) be a direlation from (u,u) to (v,v), and (q,q) be a direlation on (v,v) to (w,w). the composition c© agt, upv, 2017 appl. gen. topol. 18, no. 1 205 ş. dost (q,q) ◦ (p,p) of (p,p) and (q,q) is a direlation from (u,u) to (w,w) and it is defined by (q,q) ◦ (p,p) = (q ◦p,q◦p) where q ◦p = ∨ {p (u,w) | ∃v ∈ v with p * q(u,v) and q * q(v,w)}, q◦p = ⋂ {q(u,w) | ∃v ∈ v with p (u,v) * p and p (v,w) * q}. difunction: a direlation from (u,u) to (v,v) is called a difunction if it satisfies the conditions: (df1) for u,u′ ∈ u, pu * qu′ =⇒ ∃v ∈ v with f * q(u,v) and p (u′,v) * f . (df2) for v,v′ ∈ v and u ∈ u, f * q(u,v) and p (u,v′) * f =⇒ pv′ * qv. obviously, identity direlation (i,i) on (u,u) is a difunction and it is said to be identity difunction. it is well known that [5] the category dftex of textures and difunctions is the main category of texture theory. definition 1.3. let (f,f) : (u,u) → (v,v) be a difunction. if (f,f) satisfies the condition sur. for v,v′ ∈ v , pv * qv′ =⇒ ∃u ∈ u with f * q(u,v′) and p (u,v) * f . then it is called surjective. similarly, (f,f) satisfies the condition inj. for u,u′ ∈ u and v ∈ v , f * q(u,v) and p (u′,v) * f =⇒ pu * qu′ . then it is called injective. if (f,f) is both injective and surjective then it is called bijective. note 1.4. in general, difunctions are not directly related to ordinary (point) functions between the base sets. we note that [5, lemma 3.4] if (u,u), (v,v) are textures and a point function ϕ : u → v satisfies the condition (a) pu * qu′ =⇒ pϕ(u) * qϕ(u′) then the equalities fϕ = ∨ {p (u,v) | ∃z ∈ u satisfying pu * qz and pϕ(z) * qv}, fϕ = ⋂ {q(u,v) | ∃z ∈ u satisfying pz * qu and pv * qϕ(z)}, define a difunction (fϕ,fϕ) on (u,u) to (v,v). for b ∈ v, f←ϕ b = ϕ←b = f←ϕ b, where ϕ ←b = ∨ {pu | pϕ(u′) ⊆ b ∀u′ ∈ u with pu * qu′}. furthermore, the function ϕ = ϕ(f,f) : u → v corresponding as above to the difunction (f,f) : (u,u) → (v,v), with (v,v) plain, has the property (a) and in addition the property: (b) pϕ(u) * b, b ∈ v =⇒ ∃u′ ∈ u with pu * qu′ for which pϕ(u′) * b. conversely, if ϕ : u → v is any function satisfying (a) and (b) then there exists a unique difunction (fϕ,fϕ) : (u,u) → (v,v) satisfying ϕ = ϕ(fϕ,fϕ). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 206 metric spaces and textures on the other hand, if we consider simple textures it is obtained the same class of point functions. the category of textures and point functions which satisfy the conditions (a)(b) between the base sets is denoted by ftex. bicontinuous difunction: a difunction (f,f) : (u,u,τu,κu ) → (v,v,τv ,κv ) is called continuous (cocontinuous) if b ∈ τv (b ∈ κv ) =⇒ f←(b) ∈ τu (f←(b) ∈ κu ). a difunction (f,f) is called bicontinuous if it is both continuous and cocontinuous. the category of ditopological texture spaces and bicontinuous difunctions was denoted by dfditop in [6]. 2. some categories of dimetrics on texture spaces the notion of dimetric on texture space was firstly introduced in [11]. in this section, we will give some properties of dimetric texture spaces, and we present a link between classical metrics and dimetrics with categorical approach. definition 2.1. let (u,u) be a texture, ρ,ρ : u × u → [0,∞) two point function. then ρ = (ρ,ρ) is called a pseudo dimetric on (u,u) if m1 ρ(u,z) ≤ ρ(u,v) + ρ(v,z), m2 pu * qv =⇒ ρ(u,v) = 0, dm ρ(u,v) = ρ(v,u), cm1 ρ(u,z) ≤ ρ(u,v) + ρ(v,z), cm2 pv * qu =⇒ ρ(u,v) = 0. for all u,v,z ∈ u. in this case ρ is called pseudo metric, ρ the pseudo cometric of ρ. if ρ is a pseudo dimetric which satisfies the conditions m3 pu * qv,ρ(v,y) = 0,py * qz =⇒ pu * qz ∀u,v,y,z ∈ u, cm3 pv * qu,ρ(u,y) = 0,pz * qy =⇒ pz * qu ∀u,v,y,z ∈ u it is called a dimetric. if ρ = (ρ,ρ) is (pseudo) dimetric on (u,u) then (u,u,ρ) is called (pseudo) dimetric texture space. let (u,u,ρ) be a (pseudo) dimetric texture space. it was shown in [11, proposition 6.3] that βρ = {nρ� (u) | u ∈ u[,� > 0} is a base and γρ = {mρ� (u) | u ∈ u[,� > 0} a cobase for a ditopology (τρ,κρ) on (u,u) where nρ� (u) = ∨ {pz | ∃v ∈ u, with, pu * qv,ρ(v,z) < �}, mρ� (u) = ⋂ {qz | ∃v ∈ u, with, pv * qu,ρ(v,z) < �}. in this case (u,u,τρ,κρ) is said to be (pseudo) dimetric ditopological texture space. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 207 ş. dost definition 2.2. let (u,u,ρ) be a (pseudo) dimetric texture space. then g ∈ u is called (1) open if for every g * qu, then there exists � > 0 such that nρ� (u) ⊆ g, (2) closed if for every pu * g, then there exists � > 0 such that g ⊆ mρ� (u). we set oρ = {g ∈ u | g is open in (u,u,ρ)} and cρ = {k ∈ u | k is closed in (u,u,ρ)}. proposition 2.3. let (u,u,ρ) be a (pseudo) dimetric texture space. for u ∈ u and � > 0, (i) nρ� (u) is open in (u,u,ρ), (ii) mρ� (u) is closed in (u,u,ρ). proof. we prove (i), and the second result is dual. let nρ� (u) * qv for some v ∈ u. by the definition of nρ� (u), there exists y,z ∈ u such that py * qv and pu * qz, ρ(z,y) < �. we set δ = �−ρ(z,y). now we show that n ρ δ (v) ⊆ n ρ � (u). we suppose n ρ δ (v) * n ρ � (u). then n ρ δ (v) * qr and pr * n ρ � (u) for some r ∈ u. by the first inclusion, there exists m,n ∈ u such that pm * qr, pv * qn and ρ(n,m) < δ. now we observe that ρ(z,y) + ρ(n,m) < � and ρ(z,r) ≤ ρ(z,y) + ρ(y,v) + ρ(v,n) + ρ(n,m) ≤ � by the condition (m2). since pu * qz and ρ(z,r) ≤ �, we have the contradiction pr ⊆ nρ� (u). � definition 2.4. let (uj,uj,ρj), j = 1, 2 be (pseudo) dimetric texture spaces and (f,f) be a difunction from (u1,u1) to (u2,u2). then (f,f) is called (1) ρ1 −ρ2 continuous if p (u,v) * f then n ρ1 δ (u) ⊆ f ←(nρ2� (v)), ∀� > 0 ∃δ > 0, (2) ρ1 −ρ2 cocontinuous if f * q(u,v) then f←(mρ2� (v)) ⊆ m ρ1 δ (u), ∀� > 0 ∃δ > 0, (3) ρ1 −ρ2 bicontinuous if it is continuous and cocontinuous. proposition 2.5. let (f,f) be a difunction from (u1,u1,ρ1) to (u2,u2,ρ2). (i) (f,f) is continuous ⇐⇒ f←(g) ∈ oρ1 , ∀g ∈ oρ2 . (ii) (f,f) is cocontinuous ⇐⇒ f←(k) ∈ cρ1 , ∀k ∈ cρ2 . proof. we prove (i), and the second result is dual. (=⇒:) let (f,f) be a continuous difunction. take g ∈ oρ2 . we show that f←(g) is open in (u1,u1,ρ1). let f ←(g) * qu for some u ∈ u. by the definition of inverse image, there exists v ∈ v such that p (u,v) * f and g * qv. since g is open, we have nρ� (v) ⊆ g for � > 0. by the definition of continuity, there exists δ > 0 such that n ρ1 δ (u) ⊆ f ←(nρ2� (v)). then n ρ1 δ (u) ⊆ f ←(g), and so f←(g) ∈ oρ1 . (⇐=:) let p (u,v) * f. we consider nρ2� (v) for some � > 0. since nρ2� (v) ∈ oρ2 , we have f ←(nρ2� (v)) ∈ oρ1 by assumption. since pv ⊆ nρ2� (v) and pv * f→(qu), nρ2� (v) * f →(qv). hence, we have f ←(nρ2� (v)) * qu. then there exists δ > 0 such that n ρ1 δ (u) ⊆ f ←(nρ2� (v)). � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 208 metric spaces and textures corollary 2.6. let (uj,uj,ρj), j = 1, 2 be (pseudo) dimetric texture spaces and (f,f) be a difunction from (u1,u1) to (u2,u2). (1) (f,f) is ρ1 −ρ2 continuous ⇐⇒ (f,f) is τρ1 − τρ2 continuous. (2) (f,f) is ρ1 −ρ2 cocontinuous ⇐⇒ (f,f) is κρ1 −κρ2 cocontinuous. proof. we prove (1), leaving the dual proof of (2) to the interested reader. (=⇒:) let (f,f) be ρ1 −ρ2 continuous. let g ∈ τρ2 . to prove f←(g) ∈ τρ1 , we take f←(g) * qu for some u ∈ u1. by definition of inverse relation, there exists v ∈ u2 such that p (u,v) * f and g * qv. since g ∈ τρ2 , we have nρ2� (v) ⊆ g for some � > 0. then f← ( nρ2� (v) ) ⊆ f←(g). from the assumption, we have δ > 0 such that n ρ1 δ (u) ⊆ f ← ( nρ2� (v) ) ⊆ f←(g). thus, f←(g) ∈ τρ1 . (⇐=:) let p (u,v) * f. we consider nρ2� (v) for some � > 0. since pv ⊆ nρ2� (v), we have nρ2� (v) * f →(qu). hence, f ← ( nρ2� (v) ) * qu, and since f← ( nρ2� (v) ) is open in (u1,u1,ρ1), we have n ρ1 � (u) ⊆ f← ( n ρ2 δ (v) ) for some δ > 0. thus, (f,f) is ρ1 −ρ2 continuous. � theorem 2.7. (pseudo) dimetric texture spaces and bicontinuous difunctions form a category. proof. since bicontinuity between ditopological texture spaces is preserved under composition of difunction [6], and identity difunction on (s,s,ρ) is ρ − ρ bicontinuous, and the identity difunctions are identities for composition and composition is associative [5, proposition 2.17(3)], (pseudo) di-metric texture spaces and bicontinuous difunctions form a category. � definition 2.8. the category whose objects are (pseudo) di-metrics texture spaces and whose morphisms are bicontinuous difunctions will be denoted by (dfdimp) dfdim. clearly, dfdim is a full subcategory of dfdimp. if we take as objects di-metric on a simple texture we obtain the full subcategory dfsdim and inclusion functor s : dfsdim ↪→ dfdim. also we obtain the full subcategory dfpdim and inclusion functor p : dfpdim ↪→ dfdim by taking as objects di-metrics on a plain texture. in the same way we can use dfpsdim to denote the category whose objects are di-metrics on a plain simple texture, and whose morphisms are bicontinuous difunctions. now, we define g : dfdim → dfditop by g((u,u,ρ) (f,f) −−−→ (v,v,µ)) = (u,u,τρ,κρ) (f,f) −−−→ (v,v,τµ,κµ). obviously, g is a full concrete functor from corollary 2.6. likewise, the same functor may set up from dfdimp to dfditop. we recall [11] that a ditopology on (u,u) is called (pseudo) dimetrizable if it is the (pseudo) dimetric ditopology of some (pseudo) dimetric on (u,u). we c© agt, upv, 2017 appl. gen. topol. 18, no. 1 209 ş. dost denote by dfditopdm the category of dimetrizable ditopological texture space and bicontinuous difunction. clearly it is full subcategory of the category dfditop. proposition 2.9. the categories dfditopdm and dfdim are equivalent. proof. consider the functor g : dfdim → dfditopdm which is defined above. it can be easily seen that g is a full and faithfull functor, since the homset restriction function of g is onto and injective. now we take a dimetrizable ditopological texture space (u,u,τ,κ) such that τ = τρ and κ = κρ, where ρ is a dimetric on (u,u). clearly, the identity difunction (i,i) : (u,u,ρ) → g(u,u,ρ) is an isomorphism in the category dfditopdm. hence, g is isomorphism-closed, and so the proof is completed. � corollary 2.10. the category dfdimp is equivalent to the category of pseudo dimetrizable completely biregular [7] ditopological texture spaces and bicontinuous difunctions. proof. let (u,u,ρ) be a pseudo dimetric space. then the dimetric ditopology (u,u,τρ,κρ) is completely biregular by [11, corollary 6.5]. consequently, the functor g which is the above proposition is given an equivalence between the categories dfdimp and the category of pseudo metrizable completely biregular ditopological texture spaces. � on the other hand, since every pseudo dimetric ditopology is t0 [11, corollary 6.5], so the category dfdimp is equivalent to the category of pseudo metrizable t0 ditopological texture spaces and bicontinuous difunctions. now we give some properties of morphisms in the category dfdim. note that it takes consideration the reference [1] for some concepts of category theory proposition 2.11. let (f,f) be a morphism from (u,u,ρ) to (v,v,µ) in the category dfdim (dfdimp). (1) if (f,f) is a section then it is injective. (2) if (f,f) is injective morphism then it is a monomorphism. (3) if (f,f) is retraction then it is surjective. (4) if (f,f) is surjective morphism then it is an epimorphism. (5) (f,f) is an isomorphism if and only if it is bijective and the inverse difunction (f,f)← is bicontinuous difunction. proof. the proof of (1)−(4) can be obtained easily in the category dftex by [5, proposition 3.14]. we show that the result (5). note that, (f,f) is bijective if and only if it is an isomorphism in dftex. since (f,f) is bijective, its inverse (f,f)← is a morphism in dftex such that (f,f)← ◦ (f,f) = (iu,iu ), (f,f) ◦ (f,f)← = (iv ,iv ). consequently, (f,f) is ρ − µ bicontinuous iff (f,f)← is µ−ρ bicontinuous. � now let (u,d) be a classical (pseudo) metric space. then ρ = (d,d) is a (pseudo) dimetric on the discrete texture space ( u,p(u) ) . as a result, a c© agt, upv, 2017 appl. gen. topol. 18, no. 1 210 metric spaces and textures subset of u is open (closed) in the metric space (u,d) if and only if it is open (closed) in the dimetric texture space (u,p(u),ρ). on the other hand, recall that [5] if (f,f) is a difunction from (u,p(u)) to (v,p(v )), then f and f are point functions from u to v where f = (u ×v )\ f = f′. the category of metric spaces and continuous functions between metric spaces is denoted by met. according to: theorem 2.12. the category met is isomorphic to the full subcategory of dfdim. proof. we consider a full subcategory d-dfdim of dfdim whose objects are dimetric texture spaces on discrete textures and morphisms are bicontinuous difunctions. now we prove that the mapping t :met→ d-dfdim is a functor such that t(u,d) = (u,p(u),ρ) and t(f) = (f,f′) where f is a morphism in met. note that (f,f′) is a bicontinuous difunction in d-dfdim if and only if f is a continuous point function in met. it can be easily seen that if i is identity function on u then (i,i) is identity difunction on (u,p(u)) where i = (u ×u)\ i. since f′◦g′ = (f ◦g)′, we have t(f ◦g) = t(f)◦t(g). hence, t is a functor. furthermore, t is bijective on objects, and the hom-set restriction of t is injective and onto. consequently, t is clearly an isomorphism functor. � by using same arguments, the category pmet of pseudo metric spaces and continuous functions is isomorphic to the full subcategory of dfdimp. now suppose that (u,d) is a classical metric space and (u,td) is the metric topological space. then the pair (td,t c d) is a ditopology on (u,p(u)). on the other hand, we consider the dimetric ditopological texture space (u,p(u),τρ,κρ) where ρ = (d,d). now we consider the functors m : met → dfditop and n : met → dfditop which are defined by m((u,d) f−→ (v,e)) = (u,p(u),td,tcd) (f,f′) −−−→ (v,p(v ),te,tce), n((u,d) f−→ (v,e)) = (u,p(u),τρ,κρ) (f,f′) −−−→ (v,p(v ),τµ,κµ) where ρ = (d,d) and µ = (e,e). according to: proposition 2.13. let τ : m → n be a function such that assigns to each met-object (x,d) a dfditop-morphism τ(x,d) = (i,i) : m(x,d) → n(x,d). then τ is a natural transformation. proof. we prove that naturality condition holds. let f : (u,d) → (v,e) be a met-morphism. from theorem 2.12, (f,f′) : (u,p(u),ρ) → (v,p(v ),µ) is a dfdim-morphism. further, it is a dfditop-morphism by corollary 2.6. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 211 ş. dost (u,p(u),td,t c d) τ(u,d)=(i,i) // (f,f′) �� (u,p(u),τρ,κρ) (f,f′) �� (v,p(v ),te,t c e) τ(v,e)=(i,i) // (v,p(v ),τµ,κµ) on the other hand, the identity difunction τ(u,d) = (i,i) : (u,p(u),td,t c d) → (u,p(u),τρ,κρ) is bicontinuous on (u,p(u)), and so it is a dfditop-morphism. clearly the above diagram is commutative, and the proof is completed. � 3. point functions between dimetric texture spaces as we have noted earlier, however, it is possible to represent difunctions by ordinary point functions in certain situations. the construct fditop, where the objects are ditopological texture spaces and the morphisms bicontinuous point functions satisfying (a) and (b) which is given note 1.4, and we will to define a similar construct of (pseudo) dimetric texture spaces. definition 3.1. let (u,u,ρ) and (v,v,µ) be (pseudo) dimetric texture spaces, and ϕ on u to v a point function satisfy the condition (a). then ϕ is called (1) continuous if ϕ←(pv) * qu implies n ρ δ (u) ⊆ ϕ ←(nµ� (v)), ∀� > 0 ∃δ > 0. (2) cocontinuous if pu * ϕ←(qv) implies ϕ←(mµ� (v)) ⊆ m ρ δ (u), ∀� > 0 ∃δ > 0. (3) bicontinuous if it is continuous and cocontinuous. proposition 3.2. let ϕ be a point function satisfy the condition (a) from (u1,u1,ρ1) to (u2,u2,ρ2). (i) ϕ is continuous ⇐⇒ ϕ←(g) ∈ oρ1 , ∀g ∈ oρ2 . (ii) ϕ is cocontinuous ⇐⇒ ϕ←(k) ∈ cρ1 , ∀k ∈ cρ2 . proof. let ϕ be a point function satisfy the condition (a) and (fϕ,fϕ) be the corresponding difunction. then ϕ←(b) = f←ϕ (b) = f ← ϕ (b) for all b ∈ u2. now we take g ∈ oρ2 . we show that ϕ←(g) is open in (u1,u1,ρ1). let ϕ←(g) * qu for some u ∈ u. by the definition of inverse image, there exists v ∈ v such that p (u,v) * f and g * qv. since g is open, we have nρ2� (v) ⊆ g for � > 0. by the definition of continuity, there exists δ > 0 such that n ρ1 δ (u) ⊆ f←(nρ2� (v)). then n ρ1 δ (u) ⊆ f ←(g), and so f←(g) ∈ oρ1 . (⇐=:) let p (u,v) * f. we consider nρ2� (v) for some � > 0. since nρ2� (v) ∈ oρ2 , we have f ←(nρ2� (v)) ∈ oρ1 by assumption. since pv ⊆ nρ2� (v) and pv * f→(qu), nρ2� (v) * f →(qv). hence, we have f ←(nρ2� (v)) * qu. then there exists δ > 0 such that n ρ1 δ (u) ⊆ f ←(nρ2� (v)). � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 212 metric spaces and textures corollary 3.3. suppose that ϕ : (u1,u1) → (u2,u2) is a point function satisfy the condition (a) and that ρk is a (pseudo) dimetric on (uk,uk), k = 1, 2. then (1) ϕ is bicontinuous if and only if (fϕ,fϕ) is bicontinuous. (2) ϕ is ρ1 − ρ2 bicontinuous if and only if ϕ is (τρ1,κρ1 )–(τρ2,κρ2 ) bicontinuous where (τρj,κρj ), j = 1, 2 is dimetric ditopological texture space. proof. since ϕ←(b) = f←ϕ (b) = f ← ϕ (b) for all b ∈ u2, the proof is automatically obtained by corollary 2.6. � the category whose objects are dimetrics and whose morphisms are bicontinuous point functions satisfying the conditions (a) and (b) will be denoted by fdim. proposition 3.4. let f be a morphism from (u,u,ρ) to (v,v,µ) in the category fdim. (1) if f is a section then it is an fdim-embedding. (2) if f is injective morphism then it is a monomorphism. (3) if f is a retraction then it is a fdim-quotient. (4) if f is a surjective morphism then it is an epimorphism. (5) f is an isomorphism if and only if it is a textural isomorphism and its inverse is bicontinuous. proof. since the category fdim is a construct, the first four results are automatically obtained. recall that f is a textural isomorphism from (u,u) to (v,v) if it is a bijective point function from u to v satisfying a ∈ u =⇒ f(a) ∈ v such that a → f(a) is a bijective from u to v. hence, this is equivalent to requiring that f be bijective with inverse g, and a ∈ u =⇒ f(a) ∈ v and b ∈ v =⇒ g(b) ∈ u. by [5, proposition 3.15], f is textural isomorphism if and only if f is isomorphism in ftex. � we define d : fdim → dfdim by d((u,u,ρ) ϕ−→ (v,v,µ)) = (u,u,ρ) (fϕ,fϕ)−−−−−→ (v,v,µ). theorem 3.5. d : fdim → dfdim defined above is a functor. the restriction dp : fpdim → dfpdim is an isomorphism with inverse vp : dfpdim → fpdim given by vp((u,u,ρ) (f,f) −−−→ (v,v,µ)) = (u,u,ρ) ϕ(f,f )−−−−→ (v,v,µ). likewise we have isomorphism between fsdim and dfsdim. proof. it is easy to show that d(ιu ) = (iu,iu ). now let (u,u), (v,v), (z,z) be textures, ϕ : u → v , ψ : v → z point functions satisfying (a) and (b). we have (fψ◦ϕ,fψ◦ϕ) = (fψ,fψ) ◦ (fϕ,fϕ) by [5, theorem 3.10]. we can also say that a point function is (texturally) bicontinuous if and only if the corresponding c© agt, upv, 2017 appl. gen. topol. 18, no. 1 213 ş. dost difunction is bicontinuous. thus d : fdim → dfdim is a functor. if we restrict to dp : fpdim → dfpdim we again obtain a functor. now let us define vp : dfpdim → fpdim by vp(u,u,ρ) = (u,u,ρ) and vp(f,f) = ϕ(f,f) which is also a functor and the inverse of dp. this means that dp is an isomorphism. the other isomorphisms can be proved similarly. � we recall that a quasi-pseudo metric on a set u in the sense of j. c. kelly [9] is a non-negative real-valued function ρ(, ) on the product u ×u such that (1) ρ(u,u) = 0, (u ∈ u) (2) ρ(u,z) ≤ ρ(u,v) + ρ(v,z), (u,v,z ∈ u) now let ρ(, ) be a quasi-pseudo metric on a set u, and let q(, ) be defined by q(u,v) = ρ(v,u). then it is a trivial matter to verify that q(u,v) is a quasipseudo metric on u. in this case, ρ(, ) and q(, ) are called conjugate, and denote the set u with this structure (u,ρ,q). now let (u1,ρ1,q1) and (u2,ρ2,q2) be quasi-pseudo metric spaces. a function f : u1 → u2 is pairwise continuous if and only if f is ρ1–ρ2 continuous and q1–q2 continuous. so, quasi-pseudo metric spaces and pairwise continuous functions form a category, and we will denote this category pqmet. obviously, met is a full subcategory of pqmet. now let (u,u,ρ) be a dimetric space with (u,u) plain. then u = v =⇒ ρ(u,v) = 0 and ρ(u,v) = 0, by the dimetric condition (m2). so, (u,ρ,ρ) is pseudo-quasi metric space in the usual sense. thus we have a forgetful functor a : fpsdimp → pqmet, if we set a(u,u,ρ) = (u,ρ,ρ) and a(ϕ) = ϕ. likewise, the functor t : met → dfdim becomes a functor t : pqmet → dfdimp on setting t(u,p,q) = (u,p(u), (p,q)) and t(ϕ) = ϕ. now we consider the following diagram. pqmet t uukkk kkk kkk k dfpsdimp vps // fpsdimp a iirrrrrrrrr then: theorem 3.6. a is an adjoint of vps ◦t and t a co-adjoint of a◦vps. proof. take (u,p,q) ∈ ob (pqmet). we show that (ιu, (u,p(u), (p,q))) is an a-universal arrow. it is clearly an a-structured arrow, so take an object (u,u,µ) in fpsdimp and ϕ ∈ pqmet((u,p,q), (u,µ,µ)). then, by [5, theorem 3.12], we know that ϕ ∈ mor fpstex, and that it is the unique such morphism satisfying a(ϕ) ◦ ιu = ϕ, so it remains to verify that ϕ : (u,p(u), (p,q)) → (u,u,µ) is bicontinuous. however, for every open set g in (u,u,µ), we have ϕ←(g) = ϕ−1[g], by [5, lemma 3.9], and ϕ−1[g] is open in (u,p,q) since ϕ is p–µ continuous. likewise, for every closed set k in (u,u,µ) we have ϕ←(k) = u \ϕ−1[u \k] is closed in (u,p,q) since ϕ is q–µ continuous. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 214 metric spaces and textures 4. dimetrics and direlational uniformity in this section, we will give a relation between dimetrics and direlational uniformity by using categorical approach. firstly, we recall some basic definitons and results for direlational uniformity from [11]. let us denote by dr the family of direlations on (u,u). direlational uniformity: let (u,u) be a texture space and d a family of direlations on (u,u). then d is called direlational uniformity on (u,u if it satisfies the following conditions: (1) if (r,r) ∈ d implies (i,i) v (r,r). (2) if (r,r) ∈ d, (e,e) ∈ dr and (r,r) v (e,e) then (e,e) ∈ d. (3) if (r,r), (e,e) ∈ d implies (d,d) u (e,e) ∈ d. (4) if (r,r) ∈ d then there exists (e,e) ∈ d such that (e,e) ◦ (e,e) v (r,r). (5) if (r,r) ∈ d then there exists (c,c) ∈ u such that (c,c)← v (r,r). then the triple (u,u,d) is said to be direlational uniform texture. it will be noted that this definition is formally the same as the the usual definition of a diagonal uniformity, and the notions of base and subbase may be defined in the obvious way. further, if d d = (i,i) then d is said to be separated. inverse of a direlation under a difunction: let (f,f) be a difunction from (u,u) to (v,v) and (r,r) be a direlation on (v,v). then (f,f)−1(r) = ∨ {p (u1,u2) | ∃pu1 * qu′1 so that p (u′1,v1) * f,f * q(u2,v2) =⇒ p (v1,v2) ⊆ r} (f,f)−1(r) = ⋂ {q(u1,u2) | ∃pu′1 * qu1 so thatf * q(u′1,v1),p (u2,v2) * f, =⇒ r ⊆ q(v1,v2)} (f,f)−1(r,r) = ((f,f)−1(r), (f,f)−1(r)). uniformly bicontinuos difunction: let (u,u,d) and (v,v,e) be direlational uniform texture space and (f,f) be a difunction from (u,u) to (v,v). then (f,f) is called d–e uniformly bicontinuous if (r,r) ∈ e =⇒ (f,f)−1(r,r) ∈ d. recall that [12] the category whose objects are direlational uniformities and whose morphisms are uniformly bicontinuous difunctions was denoted by dfdiu. now let us verify that a pseudo dimetric also defines a direlational uniformity. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 215 ş. dost theorem 4.1. let ρ be a pseudo dimetric on (u,u). i) for � > 0 let r� = ∨ {p (u,v) | ∃z ∈ u,pu * qz and ρ(z,v) < �} r� = ⋂ {q(u,v) | ∃z ∈ u,pz * qu and ρ(z,v) < �} then the family {(r�,r�) | � > 0} is a base for a direlational uniformity dρ on (u,u). ii) the uniform ditopology [11] of uρ coincides with the pseudo metric ditopology of ρ. a direlational uniformity d on (u,u) is called (pseudo) dimetrizable if there exists a (pseudo) dimetric ρ with d = dρ. lemma 4.2. let (uj,uj,ρj), j = 1, 2 be (pseudo) dimetrics and (f,f) be a difunction from (u1,u1) to (u2,u2). then (f,f) is ρ1−ρ2 bicontinuous if and only if (f,f) is dρ1 −dρ2 uniformly bicontinuous. proof. let (f,f) be a ρ1 − ρ2 bicontinuous difunction from (u1,u1,ρ1) to (u2,u2,ρ2). from corollary 2.6, (f,f) is also bicontinuous from (u1,u1,τρ1,κρ1 ) to (u2,u2,τρ2,κρ2 ) where (τρj,κρj ) is (pseudo) dimetric ditopology on (uj,uj), j = 1, 2. on the other hand, the uniform ditopology of dρj coincides with the (pseudo) dimetric ditopology of ρj, j = 1, 2. further, (f,f) is also uniformly bicontinuous by [11, proposition 5.13]. � now we define g : dfdim → dfdiu by g((u,u,ρ) (f,f) −−−→ (v,v,µ)) = (u,u,dρ) (f,f) −−−→ (v,v,dµ). obviously, g is a full concrete functor from lemma 4.2. we denote by dfdiudm the category of dimetrizable direlational uniform textures and uniformly bicontinuous difunctions. proposition 4.3. the categories dfdiudm and dfdim are equivalent. proof. it is easy to show that the functor g : dfdim → dfdiudm which is defined above is full and faitfull. now we take an object (u,u,d) in dfdiudm. since it is a metrizable direlational uniform space, then there exists a dimetric ρ on (u,u) such that u = uρ. because of the identity difunction (i,i) : (u,u,ρ) → g(u,u,ρ) is an isomorphism in the category dfdiudm, the functor g is isomorphism-closed. hence, the proof is completed. � recall that [11] a direlational uniformity u is (pseudo) dimetrizable if and only if it has a countable base. if the category of direlational uniformities with countable bases and uniformly bicontinuous difunctions denote by dfdiucb then we have next result automatically from proposition 4.3: corollary 4.4. the categories dfdiucb and dfdim are equivalent. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 216 metric spaces and textures a direlational uniformity d is dimetrizable if and only if it is separated [11]. we denote the category of separated direlational uniformities and uniformly bicontinuous difunctions by dfdius. from proposition 4.3, we have: corollary 4.5. dfdius is equivalent to the category dfdim. acknowledgements. the author would like to thank the referees and editors for their helpful comments that have helped improve the presentation of this paper. references [1] j. adámek, h. herrlich and g. e. strecker, abstract and concrete categories, john wiley & sons, inc., 1990. 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[13] f. yıldız, completeness types for uniformity theory on textures, filomat 29, no. 1 (2015), 159–178. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 217 21.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 101 112 weak completeness of the bourbakiquasi-uniformitym. a. s�anchez-granero�abstract. the concept of semicompleteness (weaker thanhalf-completeness) is de�ned for the bourbaki quasi-uniformityof the hyperspace of a quasi-uniform space. it is proved thatthe bourbaki quasi-uniformity is semicomplete in the space ofnonempty sets of a quasi-uniform space (x;u) if and only if eachstable �lter on (x;u�) has a cluster point in (x;u). as a conse-quence the space of nonempty sets of a quasi-pseudometric spaceis semicomplete if and only if the space itself is half-complete. itis also given a characterization of semicompleteness of the space ofnonempty u�-compact sets of a quasi-uniform space (x;u) whichextends the well known zenor-morita theorem.2000 ams classi�cation: 54e15, 54e35, 54b20.keywords: bourbaki quasi-uniformity, hausdor� quasi-uniformity, half com-pleteness. 1. introductionour basic reference for quasi-uniform spaces is [8].a (base b of a) quasi-uniformity u on a set x is a (base b of a) �lter uof binary relations (called entourages) on x such that (a) each element of ucontains the diagonal �x of x � x and (b) for any u 2 u there is v 2 usatisfying v � v � u.let us recall that if u is a quasi-uniformity on a set x, then u�1 = fu�1 :u 2 ug is also a quasi-uniformity on x called the conjugate of u. the unifor-mity u _ u�1 will be denoted by u�. if u 2 u, the entourage u \ u�1 of u�will be denoted by u�:each quasi-uniformity u on x induces a topology t (u) on x; de�ned asfollows:t (u) = fa � x : for each x 2 a there is u 2 u such that u(x) � ag:�the author acknowledges the support of the spanish ministry of science and technology,under grant bfm2000-1111. 102 m. a. s�anchez-graneroif (x;t ) is a topological space and u is a quasi-uniformity on x such thatt = t (u) we say that u is compatible with t .a quasi-uniform space (x;u) is precompact if for each u 2 u there existsa �nite subset f of x such that x = u(f). (x;u) is u�1-precompact if(x;u�1) is precompact and (x;u) is u�-precompact (totally bounded) if theuniform space (x;u�) is precompact.a sequence (xn)n2n in a quasi-pseudometric space (x;d) is called right k-cauchy [12] if for each " > 0 there is k 2 n such that d(xn;xm) < " for eachn � m � k. (x;d) is said to be right k-sequentially complete if each right k-cauchy sequence converges. a �lter f on a quasi-uniform space (x;u) is calledright k-cauchy [13] if for each u 2 u there is an f 2 f such that u�1(x) 2 ffor each x 2 f . (x;u) is said to be right k-complete if each right k-cauchy�lter converges.obviously a quasi-pseudometric space (x;d) is right k-sequentially completeif the quasi-uniformity ud is right k-complete. it is known that the converseholds for regular spaces [2].a �lter f on a quasi-uniform space (x;u) is called left k-cauchy [13] if foreach u 2 u there is an f 2 f such that u(x) 2 f for each x 2 f . (x;u) issaid to be left k-complete if each left k-cauchy �lter converges.a quasi-uniform space (x;u) is half complete [7], if each cauchy �lter on(x;u�) converges in (x;u).let (x;u) and (y;v) be two quasi-uniform spaces. a mapping f : (x;u) !(y;v) is said to be quasi-uniformly continuous if for each v 2 v there is u 2 usuch that (f(x);f(y)) 2 v whenever (x;y) 2 u.let (x;u) be a quasi-uniform space and let p0(x) be the collection of allnonempty subsets of x. the bourbaki (hausdor�) quasi-uniformity on p0(x)is de�ned by uh = fuh : u 2 ug, where uh is de�ned by uh = f(a;b) 2p0(x) : b � u(a) and a � u�1(b)g for each u 2 u (see [3] and [11]).let (x;u) be a quasi-uniform space. let denote by k0(x) (resp. k�10 (x),k�0(x)) the family of nonempty compact (resp. u�1-compact, u�-compact)subsets of x, by f0(x) the family of nonempty �nite subsets of x, by c0(x)(resp. c�10 (x), c�0(x)) the family of nonempty closed (resp. u�1-closed, u�-closed) subsets of x and by pc0(x) (resp. pc�10 (x), pc�0(x)) the family ofnonempty precompact (resp. u�1-precompact, u�-precompact) subsets of x.we will use the same symbol uh to denote the restriction of uh to any of theprevious subspaces.in this paper the concept of semicompleteness of the bourbaki quasi uni-formity is introduced and used to extend the main theorems concerning com-pleteness in uniform (metric) spaces to the setting of quasi-uniform (quasi-pseudometric) spaces.the well-known zenor-morita theorem states that a uniform space (x;u)is complete if and only if (k0(x);uh) is complete. in [5] it is proved thata compactly symmetric quasi-uniform space (x;u) is complete if and only if(k0(x);uh) is complete, providing a generalization of the zenor-morita theo-rem for compactly symmetric quasi-uniform spaces. here completeness is meant completeness of bourbaki quasi-uniformity 103in the sense used by fletcher and lindgren in their monograph [8]. in section3 it is given a generalization of the zenor-morita theorem for quasi-uniformspaces in terms of semicompleteness.burdick [4, corollary 2], based on former work of isbell [9], answered a ques-tion of cs�asz�ar [6] in the a�rmative by proving the following characterization:the hausdor� uniformity on p0(x) of a uniform space (x;u) is complete ifand only if each stable �lter on (x;u) has a cluster point. in [11] it is proved asatisfactory generalization of this result to the setting of quasi-uniform spaces,since it was proved that (p0(x);uh) is right k-complete if and only if eachstable �lter on the quasi-uniform space (x;u) has a cluster point in (x;u). insection 3 it is given another generalization of isbell-burdick theorem for quasi-uniform spaces. in particular it is proved that (p0(x);uh) is semicomplete ifand only if each stable �lter on (x;u�) has a cluster point in (x;u). moreover,a characterization of half completeness of (p0(x);uh) is obtained in terms ofdoubly stable �lters on (x;u).it is known (see e.g. [4, corollary 6]) that the hausdor� metric of a (bounded)metric space (x;d) is complete if and only if (x;d) is complete. in [11] itis proved a satisfactory generalization of this result to the setting of quasi-pseudometric spaces, since it was proved that (p0(x);dh) is right k-sequentiallycomplete if and only if (x;d) is right k-sequentially complete. in section 3 asimpler proof of this result is given. it is also proved that (p0(x);dh) is semi-complete if and only if (x;d) is half complete.2. preliminary resultslet us denote npc�10 (x) = fa 2 p0(x) : for each u 2 u there exists a�nite subset f of x such that a � u�1(f)g.npc�10 (x) can be used to describe the closure of f0(x) in (p0(x);uh).proposition 2.1. let (x;u) be a quasi-uniform space. then clt (uh)(f0(x))= npc�10 (x).proof. let a 2 clt (uh)(f0(x)), and let u 2 u. then there exists f 2 f0(x)such that f 2 uh(a), and hence a � u�1(f). therefore a 2 npc�10 (x).conversely, let a 2 npc�10 (x) and let u 2 u. then there exists f 2 f0(x)such that a � u�1(f). let f 0 = f \u(a). it is easy to check that f 0 2 uh(a)and hence a 2 clt (uh)(f0(x)). �corollary 2.2. let (x;u) be a quasi-uniform space such that (x;u�1) is pre-compact. then k0(x) is dense in (p0(x);uh).proof. it is clear that clt (uh)(f0(x)) � clt (uh)(k0(x)). since (x;u�1) isprecompact then x 2 npc�10 (x), and hence a 2 npc�10 (x) for each a 2p0(x). by the previous result we conclude that clt (uh)(k0(x)) = p0(x). �proposition 2.3. let (x;u) be a quasi-uniform space. then it holds thatclt ((u�)h)(f0(x)) = pc�0(x) and hence clt ((u�)h)(k�0(x)) = pc�0(x). 104 m. a. s�anchez-graneroproof. let a 2 clt ((u�)h)(f0(x)), and let u 2 u. then there exists f 2f0(x) such that f 2 (u�)h(a), and hence a � u�(f). therefore a 2npc�0(x) = pc�0(x).conversely, let a 2 pc�0(x) and let u 2 u. then there exists f 2 f0(x)such that f � a and a � u�(f). then f 2 (u�)h(a) and hence a 2clt ((u�)h)(f0(x)). �corollary 2.4. let (x;u) be a totally bounded quasi-uniform space. thenk�0(x) is dense in (p0(x);(u�)h) and hence in (p0(x);(uh)�).let us denote c�(f0(x)) = fa 2 p0(x) : there is a (u�)h-cauchy net inf0(x) which t (uh)-converges to ag, c(f0(x)) = fa 2 p0(x) : there is a leftk-cauchy net in (f0(x);uh) which t (uh)-converges to ag and c�1(f0(x)) =fa 2 p0(x) : there is a right k-cauchy net in (f0(x);uh) which t (uh)-converges to ag.the proof of the following result is a slight modi�cation of [10, lemma 1].proposition 2.5. let (x;u) be a quasi-uniform space.(1) pc0(x) � c(f0(x)).(2) pc�10 (x) � c�1(f0(x)).(3) pc�0(x) = c�(f0(x)).proof. let us prove that pc0(x) � c(f0(x)). the proofs of pc�10 (x) �c�1(f0(x)) and pc�0(x) � c�(f0(x)) are analogous to this one.let a 2 pc0(x). let [a] 1, f(s,t) = s ⇒ s = 0 or t = 0; (5) f(s,t) = ln(1 + as)/2, a > e, f(s, 1) = s ⇒ s = 0; (6) f(s,t) = (s + l)(1/(1+t) r) − l, l > 1,r ∈ (0,∞), f(s,t) = s ⇒ t = 0; (7) f(s,t) = s logt+a a, a > 1, f(s,t) = s ⇒ s = 0 or t = 0; (8) f(s,t) = s− ( 1+s 2+s )( t 1+t ), f(s,t) = s ⇒ t = 0; (9) f(s,t) = sβ(s), β : [0,∞) → (0, 1), and is continuous, f(s,t) = s ⇒ s = 0; (10) f(s,t) = s− t k+t ,f(s,t) = s ⇒ t = 0; (11) f(s,t) = s − ϕ(s),f(s,t) = s ⇒ s = 0, here ϕ : [0,∞) → [0,∞) is a continuous function such that ϕ(t) = 0 ⇔ t = 0; (12) f(s,t) = sh(s,t),f(s,t) = s ⇒ s = 0,here h : [0,∞) × [0,∞) → [0,∞)is a continuous function such that h(t,s) < 1 for all t,s > 0; (13) f(s,t) = s− ( 2+t 1+t )t, f(s,t) = s ⇒ t = 0. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 47 v. ozturk and a. h. ansari (14) f(s,t) = n √ ln(1 + sn), f(s,t) = s ⇒ s = 0. (15) f(s,t) = φ(s),f(s,t) = s ⇒ s = 0,here φ : [0,∞) → [0,∞) is a upper semicontinuous function such that φ(0) = 0, and φ(t) < t for t > 0, (16) f(s,t) = s (1+s)r ; r ∈ (0,∞), f(s,t) = s ⇒ s = 0. definition 1.11 ([9]). a function ψ : [0,∞) → [0,∞) is called an altering distance function if the following properties are satisfied: (i) ψ is non-decreasing and continuous, (ii) ψ (t) = 0 if and only if t = 0. see also [2] and [12]. definition 1.12 ([3]). an ultra altering distance function is a continuous, nondecreasing mapping ϕ : [0,∞) → [0,∞) such that ϕ(t) > 0 , t > 0 and ϕ(0) ≥ 0 2. main results through out this section, we assume ψ is altering distance function, ϕ is ultra altering distance function and f is a c-class function. we shall start the following theorem. theorem 2.1. let (x,d) be a b−metric space and f,g,s,t : x → x be mappings with f (x) ⊆ t (x) and g (x) ⊆ s (x) such that (2.1) ψ(d (fx,gy)) ≤ f(ψ(ms (x,y)),ϕ(ms (x,y))), for all x,y ∈ x where, ms (x,y) = max { d (sx,ty) ,d (fx,sx) ,d (gy,ty) , d (fx,ty) + d (sx,gy) 2s } . suppose that one of the pairs (f,s) and (g,t) satisfy the b − (e.a)-property and that one of the subspaces f (x) ,g (x) ,s (x) and t (x) is closed in x. then the pairs (f,s) and (g,t) have a point of coincidence in x. moreover, if the pairs (f,s) and (g,t) are weakly compatible, then f,g,s and t have a unique common fixed point. proof. if the pairs (f,s) satisfies the b − (e.a)-property, then there exists a sequence {xn} in x satisfying limn→∞fxn = limn→∞sxn = q, for some q ∈ x. as f (x) ⊆ t (x) there exists a sequence {yn} in x such that fxn = tyn. hence limn→∞tyn = q. let us show that limn→∞gyn = q. by (2.1), (2.2) ψ (d (fxn,gyn)) ≤ f(ψ (ms (xn,yn)) ,ϕ (ms (xn,yn))) ≤ ψ (ms (xn,yn)) c© agt, upv, 2017 appl. gen. topol. 18, no. 1 48 common fixed point in b-metric spaces where ms (xn,yn) = max { d (sxn,tyn) ,d (fxn,sxn) ,d (tyn,gyn) , d(sxn,gyn)+d(fxn,t yn) 2s } = max { d (sxn,fxn) ,d (fxn,gyn) , d(sxn,gyn)+d(fxn,fxn) 2s } ≤ max { d (sxn,fxn) ,d (fxn,gyn) , s[d(sxn,fxn),d(fxn,gyn)] 2s } . in (2.2), on taking limit, ψ (limn→∞d (q,gyn)) ≤ f(ψ (limn→∞d (q,gyn)) ,ϕ (limn→∞d (q,gyn))). so, ψ (limn→∞d (q,gyn)) = 0, or ,ϕ (limn→∞d (q,gyn)) = 0. thus limn→∞d (q,gyn) = 0. hence limn→∞gyn = q. if t (x) is closed subspace of x, then there exists a r ∈ x, such that tr = q. by (2.1), (2.3) ψ (d (fxn,gr)) ≤ f(ψ (ms (xn,r)) ,ϕ (ms (xn,r))) where ms (xn,r) = max { d (sxn,tr) ,d (fxn,sxn) ,d (tr,gr) , d(fxn,t r)+d(sxn,gr) 2s } = max { d (sxn,q) ,d (fxn,sxn) ,d (q,gr) , d(fxn,q)+d(sxn,gr) 2s } . letting n →∞, limn→∞ms (xn,r) = max { d (q,q) ,d (q,q) ,d (q,gr) , d (q,q) + d (q,gr) 2s } = d (q,gr) . now, (2.3) and definition of ψ and ϕ, as n →∞, ψ(d (q,gr) ≤ f(ψ(d (q,gr)),ϕ(d(q,gr))) which implies ψ(d (q,gr)) = 0 or ϕ(d(q,gr)) = 0 gives gr = q. thus r is a coincidence point of the pair (g,t). as g (x) ⊆ s (x) , there exists a point z ∈ x such that q = sz. we claim that sz = fz. by (2.1), we have (2.4) ψ(d (fz,gr)) ≤ f(ψ(ms (z,r)),ϕ(ms(z,r))) c© agt, upv, 2017 appl. gen. topol. 18, no. 1 49 v. ozturk and a. h. ansari where ms (z,r) = max { d (sz,tr) ,d (fz,sz) ,d (tr,gr) , d (fz,tr) + d (sz,gr) 2s } = max { d (q,q) ,d (fz,q) ,d (q,q) , d (fz,q) + d (q,q) 2s } ≤ max { d (fz,q) , d (fz,q) 2s } = d (fz,q) . thus from (2.4), ψ(d (fz,gr)) = ψ(d (fz,q)) ≤ f(ψ(d (fz,q)),ϕ(d (fz,q))) implies that ψ(d (fz,q)) = 0, or ,ϕ(d (fz,q)) = 0. therefore sz = fz = q. hence z is a coincidence point of the pair (f,s) . thus fz = sz = gr = tr = q. by weak compatibility of the pairs (f,s) and (g,t ), we deduce thatfq = sq and gq = tq. we will show that q is a common fixed point of f,g,s and t . from (2.1) , (2.5) ψ (d (fq,q)) = ψ(d(fq,gr)) ≤ f(ψ (ms (q,r)) ,ϕ (ms (q,r))) where, ms (q,r) = max { d (sq,tr) ,d (fq,sq) ,d (tr,gr) , d (fq,tr) + d (sq,gr) 2s } = max { d (fq,q) ,d (fq,fq) ,d (q,q) , d (fq,q) + d (fq,q) 2s } = d (fq,q) . by (2.5) ψ (d (fq,q)) ≤ f(ψ(d (fq,q)),ϕ (d (fq,q))). so fq = sq = q. similarly, it can be shown gq = tq = q. to prove the uniqueness of the fixed point of f,g,s and t . suppose for contradiction that p is another fixed point of f,g,s and t. by (2.1), we obtain ψ (d (q,p)) = ψ(d (fq,gp)) ≤ f(ψ (ms (q,p)) ,ϕ (ms (q,p))) and ms (q,p) = max { d (sq,tp) ,d (fq,sq) ,d (tp,gp) , d (fq,tp) + d (sq,gp) 2s } = max { d (q,p) ,d (q,q) ,d (p,p) , d (q,p) + d (q,p) 2s } = d (q,p) . hence we have ψ (d (q,p)) ≤ f(ψ (d (q,p)) ,ϕ (d (q,p))), which implies that ψ (d (q,p)) = 0 or ϕ (d (q,p)) = 0. so q = p. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 50 common fixed point in b-metric spaces corollary 2.2. let (x,d) be a b−metric space and f,g,s,t : x → x be mappings with f (x) ⊆ t (x) and g (x) ⊆ s (x) such that d (fx,gy) ≤ f(ms (x,y) ,ϕ(ms (x,y))), for all x,y ∈ x, where ms (x,y) = max { d (sx,ty) ,d (fx,sx) ,d (gy,ty) , d (fx,ty) + d (sx,gy) 2s } . suppose that one of the pairs (f,s) and (g,t) satisfy the b − (e.a)-property and that one of the subspaces f (x) ,g (x) ,s (x) and t (x) is closed in x. then the pairs (f,s) and (g,t) have a point of coincidence in x. moreover, if the pairs (f,s) and (g,t) are weakly compatible, then f,g,s and t have a unique common fixed point. corollary 2.3. let (x,d) be a b−metric space and f,t : x → x be mappings such that ψ(d (fx,fy)) ≤ f(ψ(ms (x,y)),ϕ(ms (x,y))), for all x,y ∈ x, where ms (x,y) = max { d (tx,ty) ,d (fx,tx) ,d (fy,ty) , d (fx,ty) + d (tx,fy) 2s } . suppose that the pair (f,t) satisfies the b−(e.a)-property and t (x) is closed in x. then the pair (f,t) has a common point of coincidence in x. moreover, if the pair (f,t) is weakly compatible, then f and t have a unique common fixed point. example 2.4. let f(s,t) = 99 100 s , x = [0, 1] and define d : x ×x → [0,∞) as follows d (x,y) = { 0,x = y (x + y) 2 ,x 6= y then (x,d) is a b−metric space with constant s = 2. let f,g,s,t : x → x be defined by f (x) = x 4 , g (x) = { 0,x 6= 1 2 1 8 ,x = 1 2 , s (x) = { 2x, 0 ≤ x < 1 2 1 8 , 1 2 ≤ x ≤ 1 and t (x) = { x, 0 ≤ x < 1 2 1 2 , 1 2 ≤ x ≤ 1 . clearly, f (x) is closed and f (x) ⊆ t (x) and g (x) ⊆ s (x). the sequence {xn} , xn = 12 + 1 n , is in x such that limn→∞fxn = limn→∞sxn = 1 8 . so that the pair (f,s) satisfies the b − (e.a)−property. but the pair (f,s) is noncompatible for limn→∞d (fsxn,sfxn) 6= 0. the altering functions ψ,ϕ : [0,∞) → [0,∞) are defined by ψ (t) = √ t . to check the contractive condition (2.1), for all x,y ∈ x, if x = 0 or x = 1 2 , then (2.1) is satisfied. if x ∈ ( 0, 1 2 ) , then c© agt, upv, 2017 appl. gen. topol. 18, no. 1 51 v. ozturk and a. h. ansari ψ (d (fx,gy)) = x 4 ≤ 99 100 9x 4 = 99 100 d (fx,sx) ≤ 99 100 ψ(ms (x,y)). if x ∈ ( 1 2 , 1 ] , then ψ (d (fx,gy)) = x 4 ≤ 99 100 ( x 4 + 1 8 ) = 99 100 d (fx,sx) ≤ 99 100 ψ(ms (x,y)). then (2.1) is satisfied for all x,y ∈ x. the pairs (f,s) and (g,t) are weakly compatible. hence, all of the conditions of theorem 2.1 are satisfied. moreover 0 is the unique common fixed point of f,g,s and t. acknowledgements. the authors would like to thank 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[12] n. saleem, b. ali, m. abbas and z. raza, fixed points of suzuki type generalized multivalued mappings in fuzzy metric spaces with applications, fixed point theory and appl. 2015 (2015), article id 36. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 52 07.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 93 98 separation axioms in topological preorderedspaces and the existence of continuousorder-preserving functionsg. bosi, r. islerabstract. we characterize the existence of a real continu-ous order-preserving function on a topological preordered space,under the hypotheses that the topological space is normal andthe preorder satis�es a strong continuity assumption, called ic-continuity. under the same continuity assumption concerning thepreorder, we present a su�cient condition for the existence of acontinuous order-preserving function in case that the topologicalspace is completely regular.2000 ams classi�cation: 06a06, 54f30keywords: topological preordered space, decreasing scale, order-preservingfunction 1. introductionmccartan [?] introduced a natural continuity hypothesis on a preorder � ona topological space (x; �). such an assumption, which is called ic-continuitythroughout this paper, is stronger than the usual hypothesis according to whichall the lower and upper sections are closed. separation axioms in ordered topo-logical spaces were studied, in connection with suitable continuity assumptions,by burgess and fitzpatrick [?] and later by k�unzi [?]. on the other hand, theexistence of a real continuous order-preserving function on a normally pre-ordered topological space was characterized by mehta [?].in this paper we are concerned with the existence of a real continuous order-preserving function f on a topological preordered space (x; �; �) in case that(x; �) is either a normal or a completely regular space, and the preorder � isic-continuous. such a problem was already faced by bosi and isler [?]. werecall that a full characterization of the existence of a real continuous order-preserving function on a topological preordered space was provided by herden[?], [?] (see also mehta [?] for an excellent review), who introduced the notion 94 g. bosi, r. islerof a separable system. such a concept appears as a generalization of the conceptof a decreasing scale, which was used by burgess and fitzpatrick [?] and seemsmore suitable to our aims.2. definitions and preliminary considerationsgiven a preorder � on an arbitrary set x (i.e., a binary relation on x whichis re exive and transitive), denote by � and � the asymmetric part and re-spectively the symmetric part of �. if (x; �) is a preordered set, and � is atopology on x, then the triplet (x; �; �) will be referred to as a topologicalpreordered space.a subset a of a set x endowed with a preorder � is said to be decreasing(respectively, increasing) if (x 2 a) ^ (y � x) ) y 2 a (respectively, (x 2a) ^ (x � y) ) y 2 a).if a is any subset of a set x endowed with a preorder �, then denote byd(a) (respectively by i(a)) the intersection of all the decreasing (respectively,increasing) subsets of x containing a.given a topological preordered space (x; �; �), we say that � is(i) continuous if d(x) = d(fxg) and i(x) = i(fxg) are closed sets for everyx 2 x,(ii) i-continuous if d(a) and i(a) are open sets for every open subset a ofx,(iii) c-continuous if d(a) and i(a) are closed sets for every closed subset aof x,(iv) ic-continuous if � is both i-continuous and c-continuous.de�nitions (ii) and (iii) were introduced by mccartan [?]. the previousterminology is similar to the terminology adopted by k�unzi [?]. obviously,given a topological preordered space (x;�;�), if (x;�) is a t1 space, and � isc-continuous, then � is continuous. so, if the t1 separation axiom holds, theconcept of c-continuity is stronger than the concept of continuity of a preorderon a topological space.from nachbin [?], a topological preordered space (x;�;�) is said to be nor-mally preordered if, given a closed decreasing set f0 and a closed increasing setf1 with f0 \f1 = ?, there exist an open decreasing set a0 containing f0, andan open increasing set a1 containing f1 such that a0 \ a1 = ?.it is easily seen that a topological preordered space (x;�;�) is normallypreordered if (x;�) is normal and � is ic-continuous. indeed, given a closeddecreasing set f0 and a closed increasing set f1 with f0\f1 = ?, from normalityof (x;�) there exist an open set a00 containing f0, and an open set a01 containingf1 such that a00 \ a01 = ?, and from ic-continuity of the preorder � we havethat a0 = d(a00)ni�a01 n d(a00)� is an open decreasing set containing f0, a1 =i(a01)nd�a00 n i(a01)� is an open increasing set containing f1, and a0\a1 = ?.from herden [?], a topological preordered space (x;�;�) is nachbin sepa-rable if there exists a countable family fan;bngn2n of pairs of closed disjoint separation axioms in topological preordered spaces and... 95subsets of x such that an is decreasing, bn is increasing, and f(x;y) 2 x �x :x � yg � sn2n (an � bn).from burgess and fitzpatrick [?], given a topological preordered space (x,�, �), a family g = fgr : r 2 sg of open decreasing subsets of x is said to bea decreasing scale in (x;�;�) if the following conditions are satis�ed:(i) s is a dense subset of [0; 1] such that 1 2 s and g1 = x, and(ii) for every r1;r2 2 s with r1 < r2, it is gr1 � gr2.observe that any decreasing scale g in a topological preordered space (x;�;�)is a linear separable system in herden's terminology (see herden [?]).if (x;�) is a preordered set, then a real function f on x is said to be(i) increasing if, for every x;y 2 x, [x � y ) f(x) � f(y)],(ii) order-preserving if it is increasing and, for every x;y 2 x, [x � y )f(x) < f(y)].it is well known that, if there exists a continuous order-preserving function fon a topological preordered space (x;�;�), then (x;�;�) is nachbin separable.finally, we recall that, given a topological space (x;�), a family g = fgr :r 2 sg of open subsets of x is said to be a scale in (x;�) if g is a (decreasing)scale in (x;�; =).3. existence of continuous order-preserving functionsour �rst aim is to characterize the existence of a real continuous order-preserving function f on a topological preordered space (x;�;�) with � ic-continuous and (x;�) normal.theorem 3.1. let (x;�;�) be a topological preordered space, with � ic-continuous and (x;�) normal. then the following conditions are equivalent:(i) there exists a real continuous order-preserving function f on the space(x;�;�) with values in [0; 1];(ii) (x;�;�) is nachbin separable;(iii) there exists a countable family fa0n;b0ngn2n of pairs of closed disjointsubsets of x such thatf(x;y) 2 x � x : x � yg � [n2n�a0n � b0n�and, for every n 2 n, if a0n 2 a0n, b0n 2 b0n, then b0n 62 d(a0n).proof. (i) =) (ii) from considerations above, (x;�;�) is normally preordered,and therefore the implication follows from herden [?, corollary 4.2].(ii) =) (iii) just observe that any countable family fa0n;b0ngn2n satisfyingthe condition of nachbin separability also veri�es condition (iii).(iii) =) (i) assume that condition (iii) holds, and let fa0n;b0ngn2n be acountable family of closed disjoint subsets of x with the indicated property.de�ne, for every n 2 n, an = d(a0n), bn = i(b0n). since � is c-continuous, anand bn are closed subsets of x for every n 2 n. further, an and bn are disjointsets for every n 2 n (otherwise, for some n 2 n there exist x 2 x, a0n 2 a0n andb0n 2 b0n such that b0n � x � a0n, and therefore b0n 2 d(a0n)). hence, (x; �; �) is 96 g. bosi, r. islernachbin separable. since � is also i-continuous, it has been already observedthat (x; �; �) is normally preordered. hence, from mehta [?, theorem 1]there exists a continuous order-preserving function f on (x; �; �) with valuesin [0; 1]. �in the sequel, a compact space is a compact-t 2 space, as in engelking [?].corollary 3.2. let (x;�;�) be a topological preordered space, with � ic-continuous and (x;�) compact. then the following conditions are equivalent:(i) there exists a real continuous order-preserving function f on the space(x;�;�) with values in [0; 1];(ii) there exists a countable family fan;bngn2n of pairs of compact dis-joint subsets of x such that an is decreasing, bn is increasing, andf(x;y) 2 x � x : x � yg � [n2n(an � bn) ;(iii) there exists a countable family fa0n;b0ngn2n of pairs of compact dis-joint subsets of x such thatf(x;y) 2 x � x : x � yg � [n2n�a0n � b0n�and, for every n 2 n, if a0n 2 a0n, b0n 2 b0n, then b0n 62 d(a0n).proof. observe that any compact space (x;�) is normal. further, it is wellknown that, given a compact space, any closed subspace is compact, as well asany compact subspace is closed. then the thesis follows from theorem ??. �in the following theorem we provide a su�cient condition for the existenceof a continuous order-preserving function f on a topological preordered space(x;�;�), with (x;�) completely regular, and � ic-continuous.theorem 3.3. let (x;�;�) be a topological preordered space, with � ic-continuous and (x;�) completely regular. there exists a real continuous order-preserving function f on (x;�;�) with values in [0; 1] if the following conditionis veri�ed:(i) there exists a countable family fa0n;b0ngn2n of pairs of disjoint subsetsof x, with a0n compact and decreasing and b0n closed for every n 2 n,such that f(x;y) 2 x � x : x � yg � [n2n�a0n � b0n� :proof. let fa0n;b0ngn2n be a countable family of pairs of subsets of x satisfyingcondition (i) above. from c-continuity of �, i(b0n) is closed and increasing forevery n 2 n. further, it is clear that a0n and i(b0n) are disjoint subsets of xfor every n 2 n. since (x;�) is completely regular, for every n 2 n thereexists a continuous function fn : x ! [0; 1] such that fn(x) = 0 on a0n andfn(x) = 1 on i(b0n) (see e.g. engelking [?, theorem 3.1.7]). hence, for everyn 2 n there exists a scale g0n = fg0nr : r 2 sng such that a0n � g0nr � x ni(b0n)for every r 2 sn n f1g. since � is ic-continuous, gn = fd(g0nr) : r 2 sng is separation axioms in topological preordered spaces and... 97a decreasing scale in (x;�;�) for every n 2 n (see burgess and fitzpatrick[?, lemma 6.1]). de�ne, for every n 2 n, a real function fn : x ! [0; 1] byfn(x) = inffr 2 sn : x 2 d(g0nr)g. since it is fn(x) = inffr 2 sn : x 2 d(g0nr)g,it is easy to show that fn is continuous. further, fn is increasing, since for everyx;y 2 x such that x � y, fr 2 sn : y 2 d(g0nr)g � fr 2 sn : x 2 d(g0nr)g, andtherefore fn(x) � fn(y) from the de�nition of fn. from condition (i), for everyx;y 2 x with x � y, there exists n 2 n such that fn(x) = 0 and fn(y) = 1(see burgess and fitzpatrick [?, theorem 4.1]). de�ne f = pn2n 2�nfn. it isimmediate to observe that f is a real continuous order-preserving function on(x;�;�). �remark 3.4. it is well known that any compact space is completely regular (seee.g. engelking [?, theorem 3.3.1]). so, the situation considered in theorem ??is the most general among those considered in the paper. in the particular casewhen (x;�) is compact, condition (i) of theorem ?? is equivalent to conditions(ii) and (iii) of corollary ??. references[1] g. bosi and r. isler, continuous order-preserving functions on a preordered completelyregular space, paper presented at ii italian-spanish conference on general topology andapplications, trieste (italy), september 8{10, 1999.[2] d.c.j. burgess and m. fitzpatrick, on separation axioms for certain types of topologicalspaces, math. proc. cambridge philosophical soc. 82 (1977), 59{65.[3] r. engelking, general topology (heldermann verlag, berlin, 1989).[4] g. herden, on the existence of utility functions, mathematical social sciences 17 (1989),297{313.[5] g. herden, on the existence of utility function ii, mathematical social sciences 18 (1989),107{111.[6] h.-p. a. k�unzi, completely regular ordered spaces, order 7 (1990), 283{293.[7] s. d. mccartan, bicontinuous preordered topological spaces, paci�c j. of math. 38 (1971),523{529.[8] g. mehta, some general theorems on the existence of order-preserving functions, mathe-matical social sciences 15 (1988), 135{143.[9] g. b. mehta, preference and utility, in: handbook of utility theory, vol. 1, s. barber�a,p. j. hammond and c. seidl eds. (kluwer academic publishers, 1998).[10] l. nachbin, topology and order (d. van nostrand company, 1965).received march 2000 g. bosidipartimento di matematica applicatauniversit�a di triestepiazzale europa 1, 34127 triesteitalye-mail address: giannibo@econ.univ.trieste.it 98 g. bosi, r. islerr. islerdipartimento di matematica applicatauniversit�a di triestepiazzale europa 1, 34127 triesteitaly 14.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 2, no. 1, 2001pp. 9 25 fell type topologies of quasi-pseudo-metricspaces and the kuratowski-painlev�econvergencejes�us rodr��guez-l�opezabstract. we study the double fell topology when thishypertopology is constructed over a quasi-pseudo-metric space.in particular, its relationship with the wijsman hypertopology isstudied. we also propose an extension of the kuratowski-painlev�econvergence in the bitopological setting.2000 ams classi�cation: 54b20, 54e35, 54e55.keywords: quasi-pseudo-metric, double topological space, fell topology, wi-jsman topology, kuratowski-painlev�e convergence.1. introduction and preliminariesrecently, the study of nonsymmetric structures has received a new drive asa consequence of its applications to computer science. this theory began withsmyth (see [20, 21]). he tried to �nd a convenient category for computation andhe proposed the quasi-uniform spaces as the suitable context. continuing thework of smyth, other authors have applied the nonsymmetric topology to thisarea (see [17, 18, 19]). furthermore, some hypertopologies have been success-fully applied to several areas of computer science (see [21, 23]). all these factsmotivate our interest in the nonsymmetric study of several hypertopologies. inthis paper, we continue the work developed by the author in [15].the fell topology was introduced by fell in [8]. in [15] it is introduceda de�nition for the fell hypertopology in the nonsymmetric situation. somesatisfactory results about the relationship of some hypertopologies with thefell topology are obtained in the quasi-uniform setting. we continue this workand obtain extensions of well-known results in the symmetric case about therelationship between the fell and the wijsman hypertopologies. we also studya de�nition for the kuratowski-painlev�e convergence in the bitopological settingand obtain extensions of interesting results as the mrowka's theorem.our basic references for quasi-uniform and quasi-pseudo-metric spaces are[9] and [12]. terms and unde�ned concepts may be found in such references. 10 j. rodr��guez-l�opeza quasi-pseudo-metric on a set x is a nonnegative real valued function don x � x such that for all x;y;z 2 x : (i) d(x;x) = 0 and (ii) d(x;y) �d(x;z) + d(z;y):if, in addition, d satis�es the condition: (iii) d(x;y) = 0 ) x = y; then d issaid to be a quasi-metric on x:a quasi-(pseudo-)metric space is a pair (x;d) such that x is a nonempty setand d is a quasi-(pseudo-)metric on x:if d is a quasi-(pseudo-)metric on x; then the function d�1 de�ned on x �xby d�1(x;y) = d(y;x) for all x;y 2 x, is also a quasi-(pseudo-)metric on x;called the conjugate quasi-pseudo-metric of d, and the function ds de�ned onx�x by ds(x;y) = maxfd(x;y);d�1(x;y)g for all x;y 2 x; is a (pseudo-)metricon x:each quasi-pseudo-metric d on x generates a topology t (d) on x which hasas a base the family of balls of the form bd(x;r) = fy 2 x : d(x;y) < rg; wherex 2 x and r > 0: note that if d is a quasi-metric, then t (d) is a t1 topologyon x: we denote bd(x;") = fy 2 x : d(x;y) � "g.the quasi-pseudo-metric ` on r de�ned by `(x;y) = maxfx � y;0g for allx;y 2 r, is called the lower quasi-pseudo-metric on r. its conjugate quasi-pseudo-metric `�1 is denoted by u and is called the upper quasi-pseudo-metricon r. note that `s = ` _ u is the usual metric on r. a function from atopological space (x;t ) to r is said to be lower semicontinuous (resp. uppersemicontinuous) if it is continuous when we consider the topology generated bythe lower (resp. upper) quasi-pseudo-metric on r.a quasi-uniformity on a set x is a �lter u on x � x which satis�es: (i)� � u for all u 2 u and (ii) given u 2 u there exists v 2 u such thatv 2 � u, where � = f(x;x) : x 2 xg and v 2 = f(x;z) 2 x � x : exists y 2x such that (x;y) 2 v;(y;z) 2 v g. the elements of u are called entourages.the �lter u�1, formed by all sets of the form u�1 = f(x;y) 2 x � x :(y;x) 2 ug where u 2 u, is a quasi-uniformity on x called the conjugatequasi-uniformity of u.if u is a quasi-uniformity on x, then the family fus = u \ u�1 : u 2 ugis a base for a quasi-uniformity us (in fact, it is a uniformity), which is thecoarsest uniformity containing u. this uniformity is called the supremum ofthe quasi-uniformities u and u�1.every quasi-uniformity u generates a topology t (u) on x. a neighborhoodbase for each point x 2 x is given by fu(x) : u 2 ug where u(x) = fy 2 x :(x;y) 2 ug.each quasi-pseudo-metric d on x induces a quasi-uniformity ud on x whichhas as a base the family of entourages of the form f(x;y) 2 x � x : d(x;y) <2�ng; n 2 n. moreover, t (ud) = t (d).in addition of a quasi-uniformity and a quasi-pseudo-metric, we can de�neon a space another structure which makes precise the concept of nearness. thisstructure is a relation � in p0(x). we write a�b for (a;b) 2 � and a�binstead of (a;b) 62 �. fell type topologies and the kuratowski-painlev�e convergence 11de�nition 1.1. let x be a nonempty set. a relation � in p0(x) is a quasi-proximity for x if it satis�es the following conditions:i) x�? and ?�x:ii) c�(a [ b) if and only if c�a or c�b(a [ b)�c if and only if a�c or b�c.iii) fxg�fxg for each x 2 x.iv) if a�b, there exists c 2 p0(x) such that a�c and (xnc)�b.the pair (x;�) is called a quasi-proximity space.obviously, if � is a quasi-proximity on x, then so is the opposite relation��1. this quasi-proximity is called the conjugate quasi-proximity of �. a quasi-proximity � is a proximity if � = ��1.let a and b be subsets of a quasi-proximity space (x;�). if a�b , then ais said to be near b and if a�b, then a is said to be far from b. a set b issaid to be a �-neighborhood of a set a if a�(xnb).every quasi-proximity � on a space x induces in a natural way a topologyon x. if x 2 x, the neighborhoods of x are the �-neighborhoods of x.furthermore, if (x;u) is a quasi-uniform space, then u induces a quasi-proximity �u such that a�ub if and only if (a � b) \ u 6= ? for all u 2 u.a bitopological space (see [10, 13]) is a triple (x;p;q) where x is a set andp and q are topologies on x. a bitopological space is said to be quasi-pseudo-metrizable (resp. quasi-uniformizable) if there exists a quasi-pseudo-metric d(resp. a quasi-uniformity u) on x such that t (d) = p and t (d�1) = q (resp.t (u) = p and t (u�1) = q). in this case we say that d (resp. u) is a quasi-pseudo-metric (resp. quasi-uniformity) compatible with the bitopological space(x;p;q).given a topological space (x;t ) we denote by p0(x) the family of nonemptysubsets of x and by cl0(x) we denote the family of nonempty closed subsetsof x: we also shall use p(x) = p0(x) [ ?. if (x;p;q) is a bitopologicalspace we denote by clp0 (x) (resp. clq0 (x), cls0(x)) the family of nonemptyp-closed (resp. q-closed, p _ q-closed) subsets of x:2. fell type topologies of quasi-pseudo-metric spacesin [15] it can be found a discussion about the de�nition of the fell hyper-topology in the nonsymmetric case. we propose the use of double topologicalspaces rather than bitopological spaces. we recall some de�nitions.de�nition 2.1 ([15]). a double topological space is simply a pair of topologicalspaces ((x;�);(y;�)).in the following, we will also use double space.de�nition 2.2 ([15]). let (x;p;q) be a bitopological space. we de�ne thedouble upper fell topological space as the double topological space ((clq0 (x),f+p ), (clp0 (x), f+q )) where f+p is the topology generated by all sets of theform g+ = fa 2 clq0 (x) : a � gg where g is a p-open set and xng is 12 j. rodr��guez-l�opezp _ q-compact; f+q is de�ned in a similar way by writing p instead of q andq instead of p. the pair (f+p ;f+q ) is called the double upper fell topology.the double lower fell topological space is de�ned as the double topologicalspace ((clq0 (x);f�p ),(clp0 (x);f�q )) where f�p is generated by all sets of theform g� = fa 2 clq0 (x) : a\g 6= ?g where g is p-open; f�q is de�ned in asimilar way by writing p instead of q and q instead of p. the pair (f�p ;f�q )is called the double lower fell topology.the double fell topological space is the double topological space ((clq0 (x),fp ), (clp0 (x), fq)) where fp = f+p _ f�p and fq = f+q _ f�q . the pair(fp ;fq) is called the double fell topology.in this section we study the relationship between the double fell topologyand the double wijsman topology. we also motivate the fact of consideringp _ q-compact sets in the de�nition of the double fell topology (see remark2.10). the results in the symmetric case can be found in [3] and [4].the following is an extension of the wijsman hypertopology de�nition whend is a quasi-pseudo-metric (see [16]).de�nition 2.3. let (x;d) be a quasi-pseudo-metric space. let p = t (d)and q = t (d�1). the double upper wijsman topological space is the doubletopological space ((clq0 (x);t +(wd));(clp0 (x);t +(wd�1))) where t +(wd) isthe weakest topology on clq0 (x) such that for each x 2 x, the functionald(�;x) is lower semicontinuous on clq0 (x). the de�nition for t +(wd�1) issymmetric. the pair (t +(wd);t +(wd�1)) is called the double upper wijsmantopology.the double lower wijsman topological space is the double topological space((clq0 (x);t �(wd)), (clp0 (x);t �(wd�1))) where t �(wd) is the weakest top-ology on clq0 (x) such that for each x 2 x, the functional d(x; �) is uppersemicontinuous on clq0 (x). the de�nition for t �(wd�1) is symmetric. thepair (t �(wd);t �(wd�1)) is called the double lower wijsman topology.the double topological space ((clq0 (x);t (wd));(clp0 (x);t (wd�1))) wheret (wd) = t +(wd) _ t �(wd) and t(wd�1) = t +(wd�1) _ t �(wd�1) is calledthe double wijsman topological space. the pair (t (wd);t (wd�1)) is calledthe double wijsman topology.proposition 2.4. let (x;p;q) be a quasi-pseudo-metrizable bitopological space.then f�p = t �(wd), f�q = t �(wd�1) on p0(x) and f+p � t +(wd) onclq0 (x) and f+q � t +(wd�1) on clp0 (x) where d is a quasi-pseudo-metriccompatible with the bitopological space.proof. it is easy to show that d(x; �)�1(�1;�) = bd(x;�)� so we obtain thatf�p = t �(wd) on p0(x): in a similar way, it can be proved f�q = t �(wd�1)on p0(x):now, we show that f+p � t +(wd) on clq0 (x). let g be a p-open set suchthat xng is p _ q-compact, and a 2 g+. for all x 2 xng, let us consider fell type topologies and the kuratowski-painlev�e convergence 13"x = d(a;x). since a is a q-closed set then "x > 0. choose 0 < �x < "x:thus fbd�1(x;�x) : x 2 xngg is a q-open cover of xng so there existsfx1; : : : ;xng � xng such thatxng � n[i=1 bd�1(xi;�xi):consider the t +(wd)-open set c = \ni=1d(�;xi)�1(�xi;+1). clearly, a 2 c.let us see that c � g+. let b 2 c and suppose that b \ (xng) 6= ?.given b 2 b \ (xng) there exists i 2 f1; : : : ;ng such that d�1(xi;b) < �xi. acontradiction with d(b;xi) > �xi. thus, b 2 c � g+, i.e. g+ 2 t +(wd).similarly, we prove f+q � t +(wd�1) on clp0 (x). �remark 2.5. we give an example showing that the above proposition is nottrue when we de�ne f+p and t +(wd) on clp0 (x) and f+q and t +(wd�1) onclq0 (x).we consider the set n [ f1g with the following quasi-metric:8>>>>>><>>>>>>: d(n;m) = 1 if n 6= md(n;n) = 0 for all n 2 nd(n;1) = 1n for all n 2 nd(1;n) = 1 for all n 2 nd(1;1) = 0 :it is evident that t (d) = p is the discrete topology. therefore n is a t (d)-clopen set, and its complement is obviously a t (ds)-compact set, so we considerthe f+p -open set n+. we shall prove that the set n 2 n+ has not a t +(wd)-neighborhood contained in n+. if n 2 n and n 2 d(�;n)�1(�;+1) where� 2 r we deduce that � < 0 but f1g is a t (d)-closed set which belongs tod(�;n)�1(�;+1) = clp0 (x) so this set is not contained in n+. on the otherhand, if n 2 d(�;1)�1(�;+1) we obtain the same contradiction.it is natural to wonder when the wijsman and fell hypertopologies agree.the following extension of a concept introduced by beer in [1] and reformulatedin [2], gives us the answer.de�nition 2.6. let (x;d) be a quasi-pseudo-metric space. we say that it hasnice closed balls if the proper closed d-balls and the proper closed d�1-balls aret (ds)-compact.now, we can extend a result which can be found in [3].theorem 2.7. let (x;d) be a quasi-pseudo-metric space, p = t (d) and q =t (d�1). then fp = t (wd) on clq0 (x) and fq = t (wd�1) on clp0 (x) ifand only if (x;d) has nice closed balls.proof. let us suppose that there exists a proper closed d�1-ball bd�1(x;�)which is not p _ q-compact. therefore, there is y0 2 x such that d(y0;x) > � 14 j. rodr��guez-l�opezand there exists a sequence fxngn2n � bd�1(x;�) which does not admit a p_q-cluster point in x since bd�1(x;�) is a p-closed set. let an = fxngq [ fy0gqfor all n 2 n. let us prove that the sequence fangn2n fp -converges to fy0gq.let v + \ v �1 \ : : : \ v �n be an fp -open set containing fy0gq. clearly an 2v �1 \ : : : \ v �n . suppose now, to obtain a contradiction, that given k 2 n wecan �nd nk � k such that ank 62 v +. choose ynk 2 ank such that ynk 62 v forall k 2 n. since ank = fxnkgq [ fy0gq and fy0gq 2 v +, it is easy to showthat xnk 62 v . hence, since xnv is a p _ q-compact set, fxnkgk2n admits ap _q-cluster point z. a contradiction, so fangn2n is fp -convergent to fy0gq.let us show now that fd(an;x)gn2n does not converge to d(fy0gq;x) in thelower topology of r, i.e. fangn2n is not t (wd)-convergent to fy0gq. we havethat d(an;x) � d(xn;x) � �. moreover, d(fy0gq;x) > � since if z 2 fy0gqthen d(y0;z) = 0, so � < d(y0;x) � d(y0;z) + d(z;x) = d(z;x). therefored(fy0gq;x) � d(an;x) � d(fy0gq;x) � � > 0:consequently, fp 6= t (wd) on clq0 (x). a contradiction.if there is a proper closed d-ball bd(x;�) which is not t (ds)-compact, we canprove the statement in a similar way.suppose now that (x;d) has nice closed balls. by proposition 2.4, weonly have to show that t +(wd) � f+p on clq0 (x) and t +(wd�1) � f+qon clp0 (x). let x 2 x and � � 0. let us consider the t +(wd)-open setd(�;x)�1(�;+1). we �rst suppose that bd�1(x;�) 6= x. fix � > � such thatbd�1(x;�) is not equal to x. then bd�1(x;�) is p_q-compact. if a 2 clq0 (x)and d(a;x) = �, we can �nd a sequence fangn2n � bd�1(x;�) \ a such thatfd(an;x)gn2n converges to d(a;x). since bd�1(x;�) is p _ q-compact, thereis a p _ q-convergent subsequence fankgk2n of fangn2n. if we denote by a itslimit we obtain that a 2 a and d(a;x) = �. therefore, d(�;x)�1(�;+1) =(xnbd�1(x;�))+ which is a fp -open set.on the other hand, if bd�1(x;�) = x then d(�;x)�1(�;+1) = ? 2 fp .in a similar way it can be proved t +(wd�1) � f+q on clp0 (x): �remark 2.8. we observe that by using the above proof, it can be shown:f+p = t +(wd) on clq0 (x) and f+q = t (w+d�1) on clp0 (x) if and only if(x;d) has nice closed balls. therefore, we deduce that the double fell topologyagrees with the double wijsman topology if and only if the double upper felltopology agrees with the double upper wijsman topology.in the following remark, we give an example where the above theorem doesnot work if we change either the de�nition of the double fell topology or thede�nition of nice closed balls. we will use the following de�nition.de�nition 2.9. let (x;p;q) be a bitopological space. we de�ne the doubleupper vietoris topological space as the double topological space ((clq0 (x);v +p ),(clp0 (x);v +q )) where v +p is the topology generated by all sets of the form fell type topologies and the kuratowski-painlev�e convergence 15g+ = fa 2 clq0 (x) : a � gg where g is a p-open set; v +q is de�ned in asimilar way by writing p instead of q and q instead of p.the double lower vietoris topological space is de�ned as the double topologicalspace ((clq0 (x);v �p );(clp0 (x);v �q )) where v �p is generated by all sets of theform g� = fa 2 clq0 (x) : a \ g 6= ?g where g is p-open; v �q is de�ned ina similar way by writing p instead of q and q instead of p.the double vietoris topological space is de�ned as the double topological space((clq0 (x);vp ), (clp0 (x);vq)) where vp = v +p _ v �p and vq = v +q _ v �q .remark 2.10. now we motivate one fact about the de�nition of the doublefell topology. we think that, maybe, the natural de�nition for fp is to bethe topology generated by the sets of the form g+ and v � where g and vare p-open sets and xng is q-compact. in a similar way, we de�ne the fqhypertopology. we give an example where, with this de�nition, theorem 2.7 isnot true.let d be the quasi-metric on n given byd(n;m) = 8><>: 1m if n < m1 if n > m0 if n = m :we consider the quasi-metric space (n;d). let p = t (d) and q = t (d�1):we claim that fp = vp on p0(x) and fq = vq on p0(x) . since proposition2.4 is also true with this de�nition for the double fell topology, we can deduce,using that t (wd) � vp and t (wd�1) � vq on p0(x) (see [16]), that fp =t (wd) on clq0 (x) and fq = t (wd�1) on clp0 (x) but (x;d) has not niceclosed balls. we only have to prove that v +p � f+p and v +q � f+q .let g 2 p and we consider the v +p -open set g+. since g is a t (d)-openset, it easy to prove that nng is a �nite set, so it is q-compact. therefore,g+ 2 f+p so v +p = f+p on p0(x) .on the other hand, let us suppose that g 2 q and we consider the v +q -openset g+. it is clear that every subset of x is p-compact. hence, f+q = v +q onp0(x) . we observe that this statement is not true if we consider the topologyp _ q, since it is the discrete topology.we consider the closed ball bd(n;1=n) = fn;n + 1; : : :g. it is evident thatthis set is not p _ q-compact.we notice that if we change the de�nition of a quasi-pseudo-metric spacehaving nice closed balls by saying that a quasi-pseudo-metric space has thisproperty if the proper closed d-balls are t (d�1)-compact and the proper closedd�1-balls are t (d)-compact the result is not true either. the preceding exampleshows that. the above ball is not q-compact, since it is an in�nite set and qis the discrete topology.remark 2.11. we claim that if (x;d) is a quasi-metric space having niceclosed balls then t (d) = t (d�1). let us show this. 16 j. rodr��guez-l�opezlet fxngn2n be a t (d�1)-convergent sequence to x. then, if m 2 n thereexists n0 2 n such that d(xn;x) < 1=2m for all n � n0. in addition, wecan �nd a proper closed d�1-ball with center x (otherwise, since (x;t (d�1))is a t1 space, we would have that x = fxg and the result is obvious). letbd�1(x;�) be such a ball. then, xn 2 bd�1(x;�) if n is greater or equal thana certain natural number n1. hence, fxngn2n admits a t (d)-cluster point y,and, furthermore for each m 2 nd(y;x) � d(y;xn) + d(xn;x) < 1mfor a su�cient large n, so x = y and, therefore, t (d) � t (d�1).the other inclusion is similar.in general, the equality t (d) = t (d�1) is not true in a quasi-pseudo-metricspace (x;d) having nice closed balls. let z be the set of integers. the khal-imsky line consists of z with the topology generated by all sets of the formf2n � 1;2n;2n + 1g, n 2 z. it is introduced in image processing in [11]. thenthe quasi-pseudo-metric d de�ned on z by d(2n;2n � 1) = d(2n;2n + 1) =d(n;n) = 0 for all n 2 n and d(x;y) = 1 otherwise, generates the topology ofthe khalimsky line. it is clear that the proper closed d-balls and the properclosed d�1-balls are �nite so they are t (ds)-compact. furthermore, it is obviousthat t (d) 6= t (d�1).when we consider a quasi-metric space we obtain the following result.corollary 2.12. let (x;d) be a quasi-metric space and t (d) = p, t (d�1) =q. the following statements are equivalent.i) fp = t (wd) and fq = t (wd�1) on cls0(x).ii) fp = t (wd) on clp0 (x) and fq = t (wd�1) on clq0 (x).iii) fp = t (wd) on clq0 (x) and fq = t (wd�1) on clp0 (x).iv) (x;d) has nice closed balls.v) p = q and (x;d) has nice closed balls.proof. i) ) ii) and i) ) iii) are obvious. ii) implies iv) can be shown asabove, taking into account that (x;p) and (x;q) are t1 spaces. by the abovetheorem we obtain iii) ) iv). iv) ) v) is the above remark. the implicationv) ) i) is [3, theorem 5.1.10]. �remark 2.13. let us observe that the above corollary is not true when weconsider a quasi-pseudo-metric space. let us show that ii) ) iv) fails. considerthe quasi-pseudo-metric space (r;`) where ` denotes the lower quasi-pseudo-metric. clearly, we have that fp = t (w`) on clp0 (x) and fq = t (wu) onclq0 (x) where p = t (`) and q = t (u). however, (r; `) does not have niceclosed balls, since the closed `-balls and closed u-balls are not bounded.3. other fell type topologiesas we have already observed, we have various possibilities in order to de�nethe fell hypertopology in the nonsymmetric situation. this section is devoted fell type topologies and the kuratowski-painlev�e convergence 17to describe the advantages and disadvantages of our de�nition compared withother ones.we begin giving the de�nition that we think is more natural.de�nition 3.1. let (x;p;q) be a bitopological space. we de�ne the doubleupper �ne fell space as the double space ((clq0 (x);ff+p );(clp0 (x);ff+q ))where ff+p is the topology generated by all sets of the form g+ where g is ap-open set and xng is q-compact; the topology ff+q is de�ned in the corre-sponding natural way.the double �ne fell space is the double space ((clq0 (x);ffp );(clp0 (x);ffq)) where ffp = ff+p _ f�p and ffq = ff+q _ f�q .with this de�nition, not all the results proved in the previous section work(see remark 2.10).another possible de�nition is suggested by burdick's investigations ([5, 6, 7]).he looked for a context in which he considered separately the upper and lowervietoris topologies on a hyperspace and explored the interactions between them.de�nition 3.2. let (x;p;q) be a bitopological space. the double mixed fellspace is the double space ((clq0 (x);mfp );(clp0 (x);mfq)) where mfp =f+p _ f�q and mfq = f+q _ f�p .we call this hypertopology mixed, because we interchange the natural lowerhypertopologies between the two hyperspaces that we construct. we noticethat all results obtained in the previous section are true using this de�nitionwhenever we change the de�nition of the wijsman lower hypertopology. letus observe that our main results only use the upper hypertopologies since thedouble lower wijsman topology always coincides with the double lower felltopology. however, we think that is not a natural de�nition, although it pro-vides a nontrivial topology on the bitopological space (r;t (`);t (u)).furthermore, we can give another de�nition.de�nition 3.3. let (x;p;q) be a bitopological space. the double mixed �nefell space is the double space ((clq0 (x);mffp );(clp0 (x);mffq)) wheremffp = ff+p _ f�q and mffq = ff+q _ f�p .unfortunately, the double mixed �ne fell space has the same problems ofgeneralization as the double �ne fell space. however, it is an appropriate felltype topology to study epiconvergence of lower semicontinuous functions in thedouble setting, which will be discussed elsewhere.4. the kuratowski-painlev�e convergencein this section, we propose a de�nition for the kuratowski-painlev�e conver-gence in the nonsymmetric case and obtain some results about the relationshipsof this type of convergence and some hypertopologies.the kuratowski-painlev�e convergence was introduced to describe the limitof a net in terms of the members of the net itself. we propose the followingde�nitions. 18 j. rodr��guez-l�opezde�nition 4.1. let (x;p;q) be a bitopological space and fa�g�2�a net ofsubsets of x.i) a point x0 belongs to p-lia� (resp. q-lia�) and we say that x0 is ap-limit point (resp. q-limit point) of fa�g�2�if each q-neighborhood(resp. p-neighborhood) of x0 intersects a� for all � in some residualsubset of �.ii) a point x0 belongs to p-lsa� (resp. q-lsa�) and we say that x0is a p-cluster point (resp. q-cluster point) of fa�g�2�if each q-neighborhood (resp. p-neighborhood) of x0 intersects a� for all � insome co�nal subset of �.the proof of the following proposition is straightforward.proposition 4.2. let (x;p;q) be a bitopological space. if fa�g�2�is a net ofsubsets of x then p-lia� (resp. q-lia�) and p-lsa� (resp. q-lsa�) areq-closed sets (resp. p-closed sets).de�nition 4.3. let (x;p;q) be a bitopological space and let fa�g�2�be a netof subsets of x and a 2 p(x).we say that fa�g�2�is p-kuratowski-painlev�e upper convergent (resp. q-kuratowski-painlev�e upper convergent) to a if p-lsa� � a (resp. q-lsa� �a). we write a = k+p -lima� (resp. a = k+q-lima�).we say that fa�g�2�is p-kuratowski-painlev�e lower convergent (resp. q-kuratowski-painlev�e lower convergent) to a if a � p-lia� (resp. a � q-lia�). we write a = k�p -lima� (resp. a = k�q-lima�).we say that fa�g�2�is p-kuratowski-painlev�e convergent (resp. q-kur-atowski-painlev�e convergent) to a if a = p-lia� = p-lsa� (resp. a = q-lia� = q-lsa�). we write a = kp -lima� (resp. a = kq-lima�).with these de�nitions, we can extend a classical result which gives a rela-tionship between the kuratowski-painlev�e convergence and the convergence inthe fell topology. we need the following de�nition.de�nition 4.4 ([15]). a bitopological space (x;p;q) is said to be locallybicompact if every point has a neighborhood base in p and a neighborhood basein q whose elements are p _ q-compact sets.theorem 4.5. let (x;p;q) be a bitopological space, a 2 clq0 (x) (resp.a 2 clp0 (x)) and fa�g�2�a net in clq0 (x) (resp. clp0 (x)).i) a = k�p -lima� (resp. a = k�q-lima�) if and only if a = f�q -lima�(resp. a = f�p -lima�).ii) if a = k+p -lima� (resp. a = k+q-lima�) then a = f+p -lima� (resp.a = f+q -lima�).iii) if (x;p;q) is a locally bicompact quasi-uniformizable bitopological spaceand a = f+p � lima� (resp. a = f+q � lima�) then a = k+p � lima�(resp. a = k+q � lima�). fell type topologies and the kuratowski-painlev�e convergence 19proof. i) this statement is straightforward.ii) let us suppose that a = k+p -lima�. thenp � ls(k \ a�) � p � lsa� � afor all p _q-compact set k. let k0 be a p _q-compact and p-closed set suchthat k \ a� 6= ? for a co�nal subset of �. let us prove that p-lsa� 6= ?.choose k� 2 k \ a� for all � belonging to a co�nal subset �0 of �. we obtainthat fk�g�2�0 admits a p _ q-cluster point k 2 k. it is evident that k 2 p-lsa� \ k � a so a \ k 6= ?. therefore, if a \ k = ? then a� \ k = ?eventually. hence, a = f+p -lima�.the other statement can be proved in a similar way.iii) let us suppose that p-lsa� 6� a. let x 2 (p-lsa�)na. since (x;p;q)is a locally bicompact quasi-uniformizable bitopological space, we can �nd ap-closed and p _q-compact q-neighborhood v of x such that v \a = ? buta� \ v 6= ? frequently. therefore, a 6= f+p -lima� which is a contradiction.consequently, p-lsa� � a.the same reasoning proves the statement for q. �we characterize the kuratowski-painlev�e convergence in terms of sequencesof points.proposition 4.6. let (x;p;q) be a quasi-pseudo-metrizable bitopological spaceand d a quasi-pseudo-metric on x compatible with the bitopological space. a se-quence fangn2n � p(x) is p-kuratowski-painlev�e lower convergent to a set aif and only if each point a 2 a is the limit of some t (d�1)-convergent sequencefangn2n such that an 2 an for all n 2 n.proof. let us suppose that fangn2n is f�q -convergent to a 2 p(x). picka 2 a (if a = ? the result is evident). given k 2 n, we have that a 2bd�1(a;1=k)� so there exists nk 2 n such that an 2 bd�1(a;1=k)� for alln � nk. we can suppose that n1 < n2 < ::: < nk < :::. therefore, we can �ndan 2 an \ bd�1(a;1=k) for all nk+1 � n � nk + 1. if we consider the sequencefangn2n where if n 2 f1; : : : ;n1g we consider a �xed point an 2 an, we havethat this sequence is t (d�1)-convergent to a.conversely, let fangn2n be a sequence and a � x satisfying our assumption.if a 2 a \ g where g is a t (d�1)-open set, there exists a sequence fangn2nt (d�1)-convergent to a verifying that an 2 an for all n 2 n. on the other hand,we can �nd " > 0 and n0 2 n such that bd�1(a;") � g and d�1(a;an) < " forall n � n0. therefore, an 2 g� for all n � n0. �we can also obtain a characterization of the kuratowski-painlev�e upper con-vergence in terms of sequences.proposition 4.7. let (x;p;q) be a quasi-pseudo-metrizable bitopological spaceand d a quasi-pseudo-metric on x compatible with the bitopological space. a se-quence fangn2n � p(x) is p-kuratowski-painlev�e upper convergent to a set aif and only if whenever there exist positive integers n1 < n2 < ::: and ak 2 ankfor all k 2 n such that fakgk2n is t (d�1)-convergent to a then a 2 a. 20 j. rodr��guez-l�opezproof. let us suppose that fangn2n � p(x) is p-kuratowski-painlev�e con-vergent to a. if fangn2n is a sequence as in the statement, it is evident thata 2 p-lsan � a.now, let a 2 p-lsan and fbd�1(a;1=n) : n 2 ng a countable t (d�1)-neighborhood base of a. choose n1 2 n such that bd�1(a;1) \ an1 6= ?.since fn 2 n : bd�1(a;1=2) \ ang is in�nite, we can �nd n2 > n1 verifyingbd�1(a;1=2) \ an2 6= ?. following this procedure, we can construct a strictlyincreasing sequence of positive integers fnkgk2n such that bd�1(a;1=k)\ank 6=? for all k 2 n. if ak 2 bd�1(a;1=k) \ ank for all k 2 n, it is evident that thissequence is t (d�1)-convergent to a, so by assumption a 2 a. �now, we can extend an interesting result due to mrowka (see [14]).theorem 4.8 (mrowka). let (x;p;q) be a bitopological space and let fa�g�2�be a net in p(x). then fa�g�2�has a p-kuratowski-painlev�e convergent sub-net and a q-kuratowski-painlev�e convergent subnet.proof. let b be a base for the topology q. let us consider the space f0;1gwith the discrete topology. for each � 2 �, we de�ne f� : b ! f0;1g as follows:f�(v ) = (1 if a� \ v 6= ?0 if a� \ v = ? :by the tychono�'s theorem, ff�g�2� has a convergent subnet ff�0g�02�0. foreach v 2 b, we obtain that f�0(v ) = 1 eventually if and only if f�0(v ) = 1frequently. therefore, if a� \ v 6= ? frequently then a� \ v 6= ? eventually,so fa�0g�02�0 is p-kuratowski-painlev�e convergent.the reasoning for p is similar. �theorem 4.9. let (x;d) be a quasi-pseudo-metric space. let p = t (d) andq = t (d�1). let us consider fa�g�2�a net in clq0 (x) and fb g 2� a net inclp0 (x).i) if a = t +(wd)-lima� and b = t +(wd�1)-limb then a = k+p -lima� and b = k+q-limb .ii) a = k+p -lima� and b = k+q-limb implies a = t +(wd)-lima� andb = t +(wd�1)-limb if and only if (x;d) has nice closed balls.proof. i) let us suppose that a = t +(wd)-lima� and b = t +(wd�1)-limb .let a 2 p-lsa� and suppose that a 62 a. thus d(a;a) > 0. therefore, given0 < � < d(a;a) since a 2 p-lsa� we obtain that d(a�;a) < � frequently sod(a;a) � d(a�;a) > d(a;a) � � > 0frequently which contradicts that d(a;a) = t (`)-limd(a�;a). the same rea-soning shows that b � q-lsb .ii) this statement is similar to the proof of theorem 2.7. �we recall the following de�nitions (see [15]). fell type topologies and the kuratowski-painlev�e convergence 21de�nition 4.10. let (x;p;q) be a quasi-uniformizable bitopological spaceand u a quasi-uniformity compatible with the bitopological space. the dou-ble upper u-proximal topological space is de�ned as the double topologicalspace ((clq0 (x);t +(�u));(clp0 (x);t +(�u�1))) where t +(�u) is the topologygenerated by all sets of the form g++ = fa 2 clq0 (x) : there exists u 2u such that u(a) � gg where g is a p-open set. the topology t (�u�1) isde�ned in a similar way by writing p instead of q, q instead of p and u�1 in-stead of u. the pair (t +(�u);t +(�u�1)) is called the double upper u-proximaltopology.the double lower u-proximal topological space is de�ned as the double topo-logical space ((clq0 (x);t �(�u));(clp0 (x);t �(�u�1))) where this double topo-logical space coincides with the double lower fell topological space. the pair(t �(�u);t �(�u�1)) is called the double lower u-proximal topology.the double u-proximal topological space is de�ned as the double topologicalspace ((clq0 (x);t (�u));(clp0 (x);t (�u�1))) where t (�u) = t +(�u)_t �(�u)and t (�u�1) = t +(�u�1) _ t �(�u�1). the pair (t (�u);t (�u�1)) is called thedouble u-proximal topology.de�nition 4.11. let (x;p;q) be a quasi-uniformizable bitopological space andu a quasi-uniformity compatible with (x;p;q). we say that (x;p;q) has theproperty pairwise star ifi) given a 2 clq0 (x) and b 2 clp0 (x) with a�ub there exist fx1; : : : ;xng � x and u1; : : : ;un 2 u such that a \ ([ni=1u�1i (xi)p ) = ? andb � [ni=1u�1i (xi).ii) given a 2 clp0 (x) and b 2 clq0 (x) with a�u�1b there exist fx1; : : : ;xng � x and u1; : : : ;un 2 u such that a \ ([ni=1ui(xi)q) = ? andb � [ni=1ui(xi).de�nition 4.12. let (x;u) be a quasi-uniform space. we say that it hasnice closed balls if every proper set of the form u(x)t (u�1) or u�1(x)t (u) ist (us)-compact, where u 2 u and x 2 x:theorem 4.13. let (x;p;q) be a quasi-uniformizable bitopological space andu a quasi-uniformity compatible with the bitopological space. let us considerfa�g�2�a net in clq0 (x) and fb g 2� a net in clp0 (x).i) if a = t +(�u)-lima� and b = t +(�u�1)-limb then a = k+p -lima�and b = k+q-limb .ii) a = k+p -lima� and b = k+q-limb implies a = t +(�u)-lima� andb = t +(�u�1)-limb if and only if (x;p;q) has the property pairwisestar and (x;u) has nice closed balls.proof. i) suppose that a = t +(�u)-lima� and b = t +(�u�1)-limb . ifthere exists a 2 p-lsa�na, we can �nd u 2 u such that u�1(a) \ a =?. it is easy to prove that a 2 (intpv (a))++, where v 2 u and v 2 � u.therefore, a� � (intpv (a))++ for all � in a residual subset of �. furthermore, 22 j. rodr��guez-l�opezv (a) \ v �1(a) = ?. on the other hand, since a 2 p-lsa� we obtain thatv �1(a) \ a� 6= ? for a co�nal subset of � which is not possible. therefore,lsa� � a. b = t +(�u�1)-limb implies b = k+q-limb can be proved in asimilar way.ii) let us suppose that there exist u 2 u and x 2 x such that u�1(x)p is aproper set and is not p _ q-compact. then, we can �nd y0 2 xnu�1(x)p anda net fx�g�2� � u�1(x)p such that it does not admit a p _ q-cluster point.we can easily deduce that the net fa�g�2� is p-kuratowski-painlev�e upperconvergent to fy0gq, where a� = fx�gq [ fy0gq for all � 2 �. on the otherhand, if we consider the t +(�u)-open set (intpv (fy0gq))++ where v;u0 2 u,v 2 � u0 and u0(y0) \ u�1(x)p = ?, we have that x� 62 intpv (fy0gq) for all� 2 � since if there exists z 2 fy0gq verifying (z;x�) 2 v , for some � 2 �,we obtain that (y0;x�) 2 u0 which is not possible. consequently, fa�g�2� isnot t +(�u)-convergent to fy0gq. a contradiction. in a similar way, it can beproved that the proper sets of the form u(x)q are p_q-compact. consequently,(x;u) has nice closed balls.therefore, (x;p;q) is a locally bicompact space. applying theorem 4.5, wededuce that the double fell topology agrees with the double proximal topologywhich implies (see [15]) that the bitopological space has the property pairwisestar.conversely, if (x;p;q) has the property pairwise star and (x;u) has niceclosed balls it can be proved (see [15]) that the double upper fell topologyagrees with the double upper proximal topology, and the statement followsdirectly. �at last, we establish the relationship of the kuratowski-painlev�e convergenceand the convergence in the vietoris hypertopology.theorem 4.14. let (x;p;q) be a quasi-uniformizable bitopological space. letus consider fa�g�2�a net in clq0 (x) and fb g 2� a net in clp0 (x). theni) if a = v +p -lima� and b = v +q -limb then a = k+p -lima� and b =k+q-limb .ii) a = k+p -lima� and b = k+q-limb implies a = v +p -lima� and b =v +q -limb if and only if (x;p _ q) is a compact space.proof. i) the proof is similar to the part i) of the above theorem.ii) let us suppose that (x;p _ q) is not a compact space. therefore, thereexists a net fx�g�2� that does not admit a p _q-cluster point. if we �x y0 2 xit is easy to prove that ffx�gq [ fy0gqg�2� is p-kuratowski-painlev�e upperconvergent to fy0gq. we can also prove that the net ffx�gp [ fy0gpg�2� isq-kuratowski-painlev�e upper convergent to fy0gp .on the other hand, since y0 is not a p _q-cluster point of fx�g�2�, we can �nd fell type topologies and the kuratowski-painlev�e convergence 23u 2 u such that x� 62 us(y0) for all � in whatever co�nal subset �0 of �. wecan choose �0 in such way that we only have to distinguish two possibilities:i) x� 62 u(y0) for all � 2 �0. therefore, it is evident that if we considerthe p-open set intpv (fy0gq) where v 2 u and v 2 � u, we havethat x� 62 v (fy0gq) for all � 2 �0 so ffx�gq [ fy0gqg�2� is not v +p -convergent to fy0gq. a contradiction.ii) x� 62 u�1(y0) for all � 2 �0. reasoning as above we obtain thatffx�gp [ fy0gpg�2� is not v +q -convergent to fy0gp . a contradiction.conversely, since (x;p _ q) is a compact space then f+p = v +p on clq0 (x)and f+q = v +q on clp0 (x) (see [15] ), so the proof is evident. �we can also de�ne the kuratowski-painlev�e convergence in a bitopologicalsense in a di�erent way.de�nition 4.15. let (x;p;q) be a bitopological space and let fa�g�2�be anet of subsets of x and a 2 p(x).we say that fa�g�2�is p-mixed kuratowski-painlev�e convergent (resp. q-mixed kuratowski-painlev�e convergent) to a if a = k�q-lima� and a = k+p -lima� (resp. a = k�p -lima� and a = k+q-lima�). we write a = mkp -lima� (resp. a = mkq-lima�).with this de�nition, we can also wonder under which conditions we cantopologize this convergence. for the other de�nition, the condition was tomake the space locally bicompact. we observe that we use this condition onlyto reconcile the kuratowski-painlev�e upper convergence with the convergencein the upper fell topology. so we have that this is an appropriate concept towork with the double fell topology.proposition 4.16. let (x;p;q) be a locally bicompact bitopological space.then the mixed kuratowski-painlev�e convergence agrees with the convergencein the double fell topology.consequently, the concept of kuratowski-painlev�e convergence is suitable toobtain relationships with the double mixed fell topology and the topologizationof the mixed kuratowski-painlev�e convergence is the double fell topology.it is natural to wonder if we can obtain conditions for the bitopologicalspace (x;p;q) in order to obtain the coincidence of the kuratowski-painlev�econvergence and the other fell topologies de�ned. we give a positive answerto this question. it is natural to look for other de�nitions of local compactnessin bitopological spaces. the next de�nition is due to stoltenberg.de�nition 4.17 ([22]). let (x;p;q) be a bitopological space. we say that p islocally compact with respect to q if for all x 2 x there exists a p-neighborhoodg of x such that the q-closure of g is q-compact.we say that (x;p;q) is pairwise locally compact if p is locally compact withrespect to q and q is locally compact with respect to p. 24 j. rodr��guez-l�opezthis de�nition is not suitable here. if we want that our techniques work withthis de�nition, we have to de�ne another upper fell topology for p consideringthat this topology is generated by the sets of the form g+ where g is p-openand xng is p-compact. the upper fell topology for q would be de�ned in asimilar way. but this topology does not give good results.taking into account this, we propose the following de�nition.de�nition 4.18. let (x;p;q) be a bitopological space. we say that (x;p;q)is bilocally compact if (x;p) and (x;q) are locally compact spaces.it is clear that if (x;p;q) is locally bicompact then it is bilocally compact.with this de�nition we have the following obvious result.proposition 4.19. let (x;p;q) be a bitopological space and suppose that a 2clq0 (x) (resp. a 2 clp0 (x)) and that fa�g�2�is a net in clq0 (x) (resp.clp0 (x)). theni) if a = k+p -lima� (resp. a = k+q-lima�) then a = ff+p -lima� (resp.a = ff+q -lima�).ii) if (x;p;q) is a bilocally compact pairwise hausdor� bitopological spaceand a = ff+p -lima� (resp. a = ff+q -lima�) then a = k+p -lima�(resp. a = k+q-lima�).proof. the proof is similar to theorem 4.5. �acknowledgements. the author is very grateful to professor s. romaguerafor his help, advice and encouragement.references[1] g. beer, on convergence of closed sets in a metric space and distance functions, bull.aust. math. soc. 31 (1985), 421{432.[2] , metric spaces with nice closed balls and distance functions for closed sets, bull.aust. math. soc. 35 (1987), 81{96.[3] , topologies on closed and closed convex sets, vol. 268, kluwer academic pub-lishers, 1993.[4] g. beer, a. lechicki, s. levi and s. naimpally, distance functionals and suprema ofhyperspace topologies, ann. mat. pura ed appl. 162 (1992), 367{381.[5] b. s. burdick, separation properties of the asymmetric hyperspace of a bitopological space,proceedings of the tennessee topology conference, p. r. misra and m. rajagopalan, eds.,world scienti�c, singapore, 1997.[6] , compactness and sobriety in bitopological spaces, topology proc. 22 (1997),43{61.[7] , characterizations of hyperspaces of bitopological spaces, topology proc. 23(1998), 27{43.[8] j. m. g. fell, a hausdor� topology for the closed subsets of a locally compact non-hausdor� space, proc. amer. math. soc. 13 (1962), 472{476.[9] p. fletcher and w. f. lindgren, quasi-uniform spaces, marcel dekker, new york, 1982.[10] j. c. kelly, bitopological spaces, proc. london math. soc. 13 (1963), 71{89.[11] r. kopperman, the khalimsky line as a foundation for digital topology, proceedings in ofnato advanced research workshop "shape in picture", driebergen, the netherlands126 (1994), 3{20. fell type topologies and the kuratowski-painlev�e convergence 25[12] h.-p. a. k�unzi, nonsymmetric topology, bolyai soc. math. stud., topology, szeks�ard,hungary (budapest) 4 (1993), 303{338.[13] e. p. lane, bitopological spaces and quasi-uniform spaces, proc. london math. soc. 17(1967), 241{256.[14] s. mrowka, some comments on the space of subsets, in set-valued mappings, selections,and topological properties of 2x, in proc. conf. suny bu�alo 1969, w. fleischman, ed.,lnm #171, springer-verlag, berlin (1970), 59{63.[15] j. rodr��guez-l�opez, fell type topologies of quasi-uniform spaces, new zealand j. math.,to appear.[16] j. rodr��guez-l�opez and s. romaguera, hypertopologies and quasi-metrics, preprint.[17] s. romaguera and m. schellekens, quasi-metric properties of complexity spaces, topologyappl. 98 (1999), 311{322.[18] , the quasi-metric of complexity convergence, quaestiones math. 23 (2000), 359{374.[19] m. schellekens, the smyth completion: a common foundation for denotational seman-tics and complexity analysis, proc. mfps 11, electronic notes in theoretical computerscience 1 (1995), 211{232.[20] m. b. smyth, quasi-uniformities: reconciling domains with metric spaces, mathematicalfoundations of programming language semantics, 3rd workshop, tulane 1987, lncs298, eds. m. main et al., springer, berlin (1988), 236-253.[21] , totally bounded spaces and compact ordered spaces as domains of computation,topology and category theory in computer science, ed. g.m. reed, a.w. roscoe andr.f. wachter, clarendon press, oxford (1991), 207{229.[22] r. a. stoltenberg, on quasi-metric spaces, duke math. j. 36 (1969), 65{71.[23] ph. s�underhauf, constructing a quasi-uniform function space, topology appl. 67 (1995),1{27. received august 2000revised version november 2000 j. rodr��guez-l�opezescuela polit�ecnica superior de alcoydepartamento de matem�atica aplicadauniversidad polit�ecnica de valenciapza. ferr�andiz-carbonell, 203801 alcoy (alicante)spaine-mail address: jerodlo@mat.upv.es @ appl. gen. topol. 16, no. 2(2015), 225-231doi:10.4995/agt.2015.3830 c© agt, upv, 2015 two general fixed point theorems for a sequence of mappings satisfying implicit relations in gp metric spaces valeriu popa a and alina-mihaela patriciu b a “vasile alecsandri” university of bacău, 600115 bacău, romania (vpopa@ub.ro) b department of mathematics and computer sciences, faculty of sciences and environment, “dunărea de jos” university of galaţi, 800201 galaţi, romania (alina.patriciu@ugal.ro) abstract in this paper, two general fixed point theorem for a sequence of mappings satisfying implicit relations in gp complete metric spaces are proved. 2010 msc: 47h10; 54h25. keywords: gp complete metric space; sequence of mappings; fixed point; implicit relation. 1. introduction and preliminaries in this paper we shall investigate the existence and uniqueness of common fixed point of mappings via implicit relations in the setting of gp metric spaces, inspired from the notion of gp -metric spaces [25],[4],[6],[7] and other papers. we remind that gp metric is inspired from the notions of g metric ([15],[16],[1],[3],[14] and other papers) and partial metric ([13], [1], [2], [8], [9], [10], [11], [12] and other papers). several classical fixed point theorems and common fixed point theorems have been unified considering a general condition by an implicit relation in [17], [18]. some fixed point theorems for mappings satisfying a implicit relation in g metric spaces are established in [19] [22]. recently, fixed point for mappings satisfying implicit relation in partial metric spaces are obtained in received 29 april 2015 – accepted 04 july 2015 http://dx.doi.org/10.4995/agt.2015.3830 v. popa and a.-m. patriciu [5], [9], [10], [24]. quite recently, a fixed point result for mappings satisfying an implicit relation in gp metric spaces is obtained in [23]. we first recall the notion of gp metric. definition 1.1 ([25]). let x be a nonempty set. a function gp : x3 → r+ is called a gp metric on x if the following conditions are satisfied: (gp1) : x = y = z if gp(x, y, z) = gp(x, x, x) = gp(y, y, y) = gp(z, z, z), (gp2) : 0 ≤ gp(x, x, x) ≤ gp(x, x, y) ≤ gp(x, y, z) for all x, y, z ∈ x, with y 6= z, (gp3) : gp(x, y, z) = gp(y, z, x) = ... (symmetry in all three variables), (gp4) : gp(x, y, z) ≤ gp(x, a, a)+gp(a, y, z)−gp(a, a, a) for all x, y, z, a ∈ x. the pair (x, gp) is called a gp metric space. definition 1.2 ([25]). let (x, gp) be a gp metric space and {xn} a sequence in x. a point x ∈ x is said to be the limit of the sequence {xn} or xn → x ({xn} is gp convergent to x) if limn,m→∞ gp(x, xn, xm) = gp(x, x, x). theorem 1.3 ([4]). let (x, gp) be a gp metric space. then, for any {xn} ∈ x and x ∈ x, the following conditions are equivalent: a) {xn} is gp convergent to x, b) gp(xn, xn, x) → gp(x, x, x) as n → ∞, c) gp(xn, x, x) → gp(x, x, x) as n → ∞. definition 1.4 ([25]). let (x, gp) be a gp metric space. 1) a sequence {xn} of x is called a gp cauchy sequence in x if limn,m→∞ gp(xn, xm, xm) exists and is finite. 2) a gp metric space is said to be gp complete if every gp cauchy sequence in x, gp converges to x ∈ x such that limn,m→∞ gp (xn, xm, xm) = gp(x, x, x). lemma 1.5 ([4]). let (x, gp) be a gp metric space. then: 1) if gp(x, y, z) = 0 then x = y = z, 2) if x 6= y then gp(x, x, y) > 0. lemma 1.6. let (x, gp) be a gp metric space and {xn} is a sequence in x which is gp convergent to a point x ∈ x with gp (x, x, x) = 0. then, limn→∞ g (xn, y, z) = g (x, y, z) for all y, z ∈ x. proof. by (gp4) (1.1) gp (x, y, z) ≤ gp (x, xn, xn) + gp (xn, y, z) − gp (xn, xn, xn) ≤ gp (x, xn, xn) + gp (xn, y, z) , which implies gp (x, y, z) − gp (x, xn, xn) ≤ gp (xn, y, z) ≤ gp (xn, x, x) + gp (x, y, z) . by theorem 1.3, gp (xn, x, x) → gp (x, x, x) = 0 c© agt, upv, 2015 appl. gen. topol. 16, no. 2 226 two general fixed point theorems in gp metric spaces and gp (x, xn, xn) → gp (x, x, x) = 0. letting n tends to infinity in (1.1) we obtain lim n→∞ gp (xn, y, z) = gp (x, y, z) . � quite recently, meena and nema [14] initiated the study of fixed points for sequences of mappings in g metric spaces. 2. implicit relations definition 2.1. let fgp be the set of all continuous functions f(t1, ..., t5) : r 5 + → r satisfying the following conditions: (f1) : f is non increasing in variables t2, t3, t4, t5, (f2) : there exists h ∈ [0, 1) such that for all u, v ≥ 0, f(u, v, u, v, u + v) ≤ 0 implies u ≤ hv. in the following examples, the proofs of property (f1) are obviously. example 2.2. f(t1, ..., t5) = t1 − at2 − bt3 − ct4 − dt5, where a, b, c, d ≥ 0 and a + b + c + 2d < 1. (f2) : let u, v ≥ 0 and f(u, v, u, v, u + v) = u − av − bu − cv − d (u + v) ≤ 0, which implies u ≤ hv, where 0 ≤ h = a+c+d 1−(b+d) < 1. example 2.3. f(t1, ..., t5) = t1 − k max{t2, t3, t4, t5}, where k ∈ [ 0, 1 2 ) . (f2) : let u, v ≥ 0 and f(u, v, u, v, u + v) = u − k (u + v) ≤ 0 which implies u ≤ hv, where 0 ≤ h = k 1−k < 1. example 2.4. f(t1, ..., t5) = t1 − k max { t2, t3, t4+t5 2 } , where k ∈ [0, 1). (f2) : let u, v ≥ 0 and f(u, v, u, v, u + v) = u − k max { u, v, u+2v 3 } ≤ 0. if u > v, then u (1 − k) ≤ 0, a contradiction. hence u ≤ v, which implies u ≤ hv, where 0 ≤ h = k < 1. example 2.5. f(t1, ..., t5) = t 2 1 − at2t3 − bt3t4 − ct4t5, where a, b, c ≥ 0 and a + b + 2c < 1. (f2) : let u, v ≥ 0 and f(u, v, u, v, u + v) = u2 − auv − buv − cv (u + v) ≤ 0. if u > v, then u[1 − (a + b + 2c)] ≤ 0, a contradiction. hence u ≤ v, which implies u ≤ hv, where 0 ≤ h = √ a + b + 2c < 1. example 2.6. f(t1, ..., t5) = t1 − at2 − b max{2t3, t4 + t5}, where a, b ≥ 0 and a + 3b < 1. (f2) : let u, v ≥ 0 and f(u, v, u, v, u + v) = u − av − b max{2v, u + 2v} ≤ 0. if u > v, then u[1 − (a + 3b)] ≤ 0, a contradiction. hence u ≤ v, which implies u ≤ hv, where 0 ≤ h = a + 3b < 1. example 2.7. f(t1, ..., t5) = t1 −at2 −b max {t3 + t4, 2t5}, where a, b ≥ 0 and a + 4b < 1. (f2) : let u, v ≥ 0 and f(u, v, u, v, u+v) = u−av−b max{u+v, 2 (u + v)} = u − av − 2b (u + v) ≤ 0. hence u ≤ hv, where 0 ≤ h = a+2b 1−2b < 1. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 227 v. popa and a.-m. patriciu example 2.8. f(t1, ..., t5) = t 2 1 − at22 − bt23 − ct4t5, where a, b, c ≥ 0 and a + b + 2c < 1. (f2) : let u, v ≥ 0 be and f(u, v, u, v, u+v) = u2−av2−bu2−cv (u + v) ≤ 0. if u > v, then u2[1 − (a + b + 2c)] ≤ 0, a contradiction. hence u ≤ v, which implies u ≤ hv, where 0 ≤ h = √ a + b + 2c < 1. example 2.9. f(t1, ..., t5) = t1 − a max{t2, t3} − b max{t4, t5}, where a, b ≥ 0 and a + 2b < 1. (f2) : let u, v ≥ 0 be and f(u, v, u, v, u+v) = u−a max{u, v}−b (u + v) ≤ 0. if u > v, then u[1 − (a + 2b)] ≤ 0, a contradiction. hence u ≤ v, which implies u ≤ hv, where 0 ≤ h = a + 2b < 1. 3. main results theorem 3.1. let (x, gp) be a gp complete metric space and {tn}n∈n : (x, gp) → (x, gp) be a sequence of mappings such that for all x, y, z ∈ x and i, j, k ∈ n: (3.1) f(gp(tix, tjy, tkz), gp(x, y, z), gp(tix, y, tkz), gp(tix, z, tjy), gp(tjy, tkz, x)) ≤ 0 where f ∈ fgp. then, {tn}n∈n has a unique common fixed point. proof. let x0 be any arbitrary point of x. we define a sequence {xn} in s such that xn+1 = tn+1xn, n = 0, 1, 2, ... . by (3.1) we have successively f(gp(tnxn−1, tn+1xn, tn+2xn+1), gp(xn−1, xn, xn+1), gp(tnxn−1, xn, tn+2xn+1), gp(tnxn−1, xn+1, tn+1xn), gp(tn+1xn, tn+2xn+1, xn−1)) ≤ 0 (3.2) f(gp(xn, xn+1, xn+2), gp(xn−1, xn, xn+1), gp(xn, xn, xn+2), gp(xn, xn+1, xn+1), gp(xn+1, xn+2, xn−1)) ≤ 0. by (gp2), gp(xn, xn, xn+2) ≤ gp(xn, xn+1, xn+2) and gp(xn−1, xn, xn) ≤ gp(xn−1, xn, xn+1). by (gp4) and (gp2) gp(xn−1, xn+1, xn+2) ≤ gp(xn−1, xn, xn) + gp(xn, xn+1, xn+2) ≤ gp(xn−1, xn, xn+1) + gp(xn, xn+1, xn+2). by (3.2) and (f1) we obtain f(gp(xn, xn+1, xn+2), gp(xn−1, xn, xn+1), gp(xn, xn+1, xn+2), gp(xn−1, xn, xn+1), gp(xn−1, xn, xn+1) + gp(xn, xn+1, xn+2)) ≤ 0. by (f2) we obtain gp(xn, xn+1, xn+2) ≤ hgp(xn−1, xn, xn+1) c© agt, upv, 2015 appl. gen. topol. 16, no. 2 228 two general fixed point theorems in gp metric spaces which implies (3.3) gp(xn, xn+1, xn+2) ≤ hngp(x0, x1, x2). now for any integers k ≥ m ≥ n ≥ 1 we obtain by (gp4) that gp (xn, xm, xk) ≤ gp (xn, xn+1, xn+2) + gp (xn+1, xn+2, xn+3) + ... + + gp (xk−2, xk−1, xk) ≤ hn ( 1 + h + ... + hk−n ) gp (x0, x1, x2) ≤ h n 1 − h g (x0, x1, x2) → 0 as n → ∞. since by (gp2), gp (xn, xm, xm) ≤ gp (xn, xm, xk) it follows that gp (xn, xm, xm) → 0 as n, m → ∞ and thus, {xn} is a gp cauchy sequence. since (x, gp) is a gp complete metric space, by theorem 1.5, (3.3) and definition 1.4, there exists u ∈ x such that limn,m→∞ gp (xn, xm, xm) = limn→∞ gp (u, xn, xn) = gp (u, u, u) = 0. now we prove that u is a common fixed point of {tn}n∈n. by (3.1) we have successively f(gp(tnxn−1, tju, tku), gp(xn−1, u, u), gp(tnxn−1, u, tku), gp(tn−1xn−1, u, tju), gp(tju, tku, xn−1)) ≤ 0, (3.4) f(gp(xn, tju, tku), gp(xn−1, u, u), gp(xn, u, tku), gp(xn, u, tju), gp(tju, tku, xn−1)) ≤ 0. letting n tends to infinity we obtain f(gp(xn, tju, tku), 0, gp(u, u, tku), gp(u, u, tju), gp(u, tju, tku)) ≤ 0. by (gp2) and (f1) we obtain f(gp(u, tju, tku), gp(u, tju, tku), gp(u, tju, tku), gp(u, tju, tku), gp(u, tju, tku) + gp (u, tju, tku)) ≤ 0. by (f2) it follows that gp(u, tju, tku) ≤ hgp(u, tju, tku) which implies gp(u, tju, tku) = 0. by lemma 1.5 (1), u = tju = tku. thus, u is a common fixed point of {tn}n∈n. suppose that {tn}n∈n has another common fixed point v. then by (3.1) we have successively f(gp(tiu, tju, tkv), gp(u, u, v), gp(tiu, u, tkv), gp(tiu, v, tju), gp(tju, tkv, u)) ≤ 0, f(gp(u, u, v), gp(u, u, v), gp(u, u, v), gp(u, v, v), gp(u, v, v)) ≤ 0. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 229 v. popa and a.-m. patriciu by (f1) we have f(gp(u, u, v), gp(u, u, v), gp(u, u, v), gp(u, u, v), gp(u, v, v) + gp (u, u, v)) ≤ 0. by (f2) we have gp(u, u, v) ≤ kgp(u, v, v), which implies g(u, v, v) = 0. by lemma 1.5 (1), u = v. hence, u is the unique common fixed point. � theorem 3.2. let (x, gp) be a gp complete metric space and {tn}n∈n : (x, gp) → (x, gp) be a sequence of mappings such that for all x, y, z ∈ x and i, j, k ∈ n: (3.5) f(gp(tix, tjy, tkz), gp(x, y, z), gp(tix, y, z), gp(x, tjy, z), gp(x, y, tkz)) ≤ 0 where f ∈ fgp. then, {tn}n∈n has a unique common fixed point. proof. the proof is similar to the proof of theorem 3.1. � acknowledgements. the authors thank the anonimous reviewers for their valuable comments, which improved the initial version of the paper. references [1] t. abdeljawad, e. karapinar and k. tas, existence and uniqueness of common fixed points on partial metric spaces, applied math. lett. 24 (11) (2011), 1894–1899. 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[25] m. r. a. zand and a. n. nezhad, a generalization of partial metric spaces, j. contemporary appl. math. 1, no. 1 (2011), 86–93. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 231 () @ appl. gen. topol. 18, no. 2 (2017), 231-240doi:10.4995/agt.2017.4220 c© agt, upv, 2017 a decomposition of normality via a generalization of κ-normality ananga kumar das and pratibha bhat department of mathematics, shri mata vaishno devi university, katra, jammu and kashmir182320, india (ak.das@smvdu.ac.in, akdasdu@yahoo.co.in and pratibha87bhat@gmail.com) communicated by t. nogura abstract a simultaneous generalization of κ-normality and weak θ-normality is introduced. interrelation of this generalization of normality with existing variants of normality is studied. in the process of investigation a new decomposition of normality is obtained. 2010 msc: primary 54d10; 54d15; secondary 54d20. keywords: regularly open set; regularly closed set; θ-open set; θ-closed set; κ-normal (mildly) normal space; almost normal space; (weakly) (functionally) θ-normal space; weakly κ-normal space; ∆-normal space; strongly seminormal space. 1. introduction and preliminaries several generalized notions of normality such as almost normal, κ-normal, ∆normal, θ-normal, semi-normal, quasi-normal, π-normal, densely normal etc. exist in the literature. recently, interrelation among some of these variants of normality was studied in [4] and factorizations of normality are obtained in [4, 5, 12, 14]. in this paper, we tried to exhibit the interrelations that exist among these generalized notions of normality and introduced a simultaneous generalization of κ-normality and weak θ-normality called weak κ-normality. intrestingly, the class of weakly κ-normal spaces contains the class of almost compact spaces whereas the class of κ-normal spaces does not contain the class of almost compact spaces. this newly introduced notion of weak normality received 29 september 2015 – accepted 10 march 2017 http://dx.doi.org/10.4995/agt.2017.4220 a. k. das and p. bhat is utilized to obtain a factorization of normality. moreover, it is verified that some covering properties which need not imply κ-normality implies weak κnormality. let x be a topological space and let a ⊂ x. throughout the present paper, the closure and interior of a set a will be denoted by a (or cla) and inta (or ao) respectively. a set u ⊂ x is said to be regularly open [15] if u = intu. the complement of a regularly open set is called regularly closed. it is observed that an intersection of two regularly closed sets need not be regularly closed. a finite union of regular open sets is called π-open set and a finite intersection of regular closed sets is called π-closed set. it is obvious that the complement of a π-open set is π-closed and the complement of a π-closed set is π-open, the finite union (intersection) of π-closed sets is π closed, but the infinite union (intersection) of π-closed sets need not be π-closed (see [11]). a point x ∈ x is called a θ-limit point (respectively δ-limit point) [21] of a if every closed (respectively regularly open) neighbourhood of x intersects a. let clθa (respectively clδa) denotes the set of all θ-limit point (respectively δ-limit point) of a. the set a is called θ-closed (respectively δ-closed) if a = clθa (respectively a = clδa). the complement of a θ-closed (respectively δ-closed) set will be referred to as a θ-open (respectively δ-open) set. the family of θ-open sets as well as the family of δ open sets form topologies on x. the topology formed by the set of δ-open sets is the semiregularization topology whose basis is the family of regularly open sets. let y be a subspace of x. a subset a of x is concentrated on y [2] if a is contained in the closure of a ∩ y in x. a subset a of y is said to be strongly concentrated on y [6] if a ⊂ (a ∩ y )o. it is obvious that every strongly concentrated set is concentrated. we say that x is normal on y if every two disjoint closed subsets of x concentrated on y can be separated by disjoint open neighbourhoods in x [2]. similarly, x is said to be weakly normal on y [6] if for every disjoint closed subsets a and b of x strongly concentrated on y , there exist disjoint open sets in x separating a and b respectively. a space x is called densely normal if there exists a dense subspace y of x such that x is normal on y [2]. a topological space x is said to be weakly densely normal [6] if there exist a proper dense subspace y of x such that x is weakly normal on y . it is easy to see that every densely normal space is weakly densely normal and every weakly densely normal space is κ-normal. on the other hand, the converses are not true, as were shown in [10] and [6]. lemma 1.1. a subset a of a topological space x is θ-open if and only if for each x ∈ a, there is an open set u such that x ∈ u ⊂ u ⊂ a. definition 1.2. a topological space x is said to be (i) quasi-normal [23] if any two disjoint π-closed subsets a and b of x there exist two open disjoint subsets u and v of x such that a ⊂ u and b ⊂ v . c© agt, upv, 2017 appl. gen. topol. 18, no. 2 232 a decomposition of normality via a generalization of κ-normality (ii) π-normal [11] if for any two disjoint closed subsets a and b of x one of which is π-closed, there exist two open disjoint subsets u and v of x such that a ⊂ u and b ⊂ v . (iii) ∆-normal [9] if every pair of disjoint closed sets one of which is δ-closed are contained in disjoint open sets. (iv) weakly ∆-normal [9] if every pair of disjoint δ-closed sets are contained in disjoint open sets. (v) weakly functionally ∆-normal (wf ∆-normal) [9] if for every pair of disjoint δ-closed sets a and b there exists a continuous function f : x → [0,1] such that f(a) = 0 and f(b)= 1. (vi) θ-normal [12] if every pair of disjoint closed sets one of which is θ-closed are contained in disjoint open sets; (vii) weakly θ-normal [12] if every pair of disjoint θ-closed sets are contained in disjoint open sets; (viii) functionally θ-normal [12] if for every pair of disjoint closed sets a and b one of which is θ-closed there exists a continuous function f : x →[0,1] such that f(a) = 0 and f(b)=1; (ix) weakly functionally θ-normal (wf θ-normal) [12] if for every pair of disjoint θ-closed sets a and b there exists a continuous function f : x → [0,1] such that f(a) = 0 and f(b)= 1. (x) β-normal [1] if for any two disjoint closed subsets a and b of x, there exist open sets u and v of x such that a ∩ u is dense in a, b ∩ v is dense in b and u ∩ v = φ. (xi) almost β-normal [3] if for every pair of disjoint closed sets a and b, one of which is regularly closed, there exist open sets u and v such that a ∩ u = a, b ∩ v = b and u ∩ v = φ. (xii) θ-regular [12] if for each closed set f and each open set u containing f , there exists a θ-open set v such that f ⊂ v ⊂ u. (xiii) semi-normal [22] if for every closed set f and each open set u containing f, there exists a regular open set v such that f ⊂ v ⊂ u. (xiv) almost normal [18] if every pair of disjoint closed sets one of which is regularly closed are contained in disjoint open sets. (xv) mildly normal [19] ( or κ-normal [20]) if every pair of disjoint regularly closed sets are contained in disjoint open sets. (xvi) ∆-regular [9] if for every closed set f and each open set u containing f, there exists a δ-open set v such that f ⊂ v ⊂ u. 2. weakly κ-normal spaces definition 2.1. a θ-closed set a is said to be a regularly θ-closed set if inta = a. the complement of a regularly θ-closed set will be regularly θ open. clearly every regularly θ-closed set is regularly closed as well as θ-closed but the converse need not be true. example 2.2. let x be the set of positive integers. define a topology on x by taking every odd integer to be open and a set u ⊂ x is open if for every c© agt, upv, 2017 appl. gen. topol. 18, no. 2 233 a. k. das and p. bhat even integer p ∈ u, the predecessor and the successor of p are also in u. here the set {2k, 2k + 1, 2k + 2 : k ∈ z+} is a regularly closed set which is not θ-closed. example 2.3. let x denote the interior of the unit square s in the plane together with the points (0, 0) and (1, 0), i.e. x = so ∪ {(0, 0), (1, 0)}. every point in so has the usual euclidean neighourhoods. the points (0, 0) and (1, 0) have neighbourhoods of the form un and vn respectively, where, un = {(0, 0)} ∪ {(x, y) : 0 < x < 1/2, 0 < y < 1/n} and vn = {(1, 0)} ∪ {(x, y) : 1/2 < x < 1, 0 < y < 1/n}. clearly, the sets {(0, 0)} and {(1, 0)} are θ-closed but not regularly θ-closed. definition 2.4. a topological space x is said to be weakly κ-normal if for every pair of disjoint regularly θ-closed sets a and b there exist disjoint open sets u and v such that a ⊂ u and b ⊂ v . from the definitions it is obvious that every κ-normal space is weakly κnormal and every weakly θ-normal space is weakly κ-normal. the following diagram illustrates the interrelations that exist between weakly κ-normal spaces and variants of normality that exist in literature. but none of the implications below is reversible (see [7], [9], [11], [12], [14], [18] and examples below ). β-normal ))❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚ normal 66♥♥♥♥♥♥♥♥♥♥♥♥ �� ((pp pp pp pp pp pp p // densely normal ))❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚ almost β-normal ∆-normal // �� ,,❨❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨ π-normal ,,❨❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨ $$■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ weak densely normal $$❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ fθ-normal // !!❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ wfθ-normal $$■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ wf∆-normal uu❥❥❥ ❥❥ ❥❥ ❥❥ ❥❥ ❥❥ ❥❥ ❥ almost normal �� dd❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ w∆-normal ))❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ // quasi normal // κ-normal �� θ-normal // wθ-normal // wκ-normal example 2.5. the space defined in example 2.2 is weakly κ-normal but not κ-normal. example 2.6. the example of a tychonoff κ-normal space which is not densely normal was given by just and tartir [10]. since every regular space is θ-regular [12], this space is θ-regular but not normal. thus the space is not weakly θnormal as every θ-regular, weakly θ-normal space is normal [12]. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 234 a decomposition of normality via a generalization of κ-normality theorem 2.7. a topological space x is weakly κ-normal if and only if for every regularly θ-closed set a and a regularly θ-open set u containing a there is an open set v such that a ⊂ v ⊂ v ⊂ u. proof. let x be a weakly κ-normal space and u be a regularly θ-open set containing a regularly θ-closed set a. then a and x − u are disjoint regularly θ-closed sets in x. since x is weakly κ-normal, there are disjoint open sets v and w containing a and x − u, respectively. then a ⊂ v ⊂ x − w ⊂ u. since x − w is closed, a ⊂ v ⊂ v ⊂ u. conversely, let a and b be two disjoint regularly θ-closed sets in x. then u = x −b is a regularly θ-open set containing the regularly θ-closed set a. thus by the hypothesis there exists an open set v such that a ⊂ v ⊂ v ⊂ u. then v and x − v are disjoint open sets containing a and b, respectively. hence x is weakly κ-normal. � theorem 2.8. let x be a finite topological space. for a subset a of x, the following statements are equivalent. (a) a is clopen. (b) a is θ-closed. (c) a is θ-open. proof. the implication (a) =⇒ (b) is obvious. to prove (b) =⇒ (a), let a be a closed subset of x. then (x − a) is θ-open in x. by lemma 1.3.4, for each x ∈ x − a there exists an open set ux containing x such that x ∈ ux ⊂ ux ⊂ x − a. since x is finite, ⋃ x∈x−a ux = x − a, is the union of finitely many closed sets and hence closed. thus a is open. by hypothesis a is θ-closed and hence closed. consequently, a is clopen. the proofs of (a) =⇒ (c) and (c) =⇒ (a) are similar and hence omitted. � from the above result the following observation is obvious. remark 2.9. every finite topological space is weakly κ-normal whereas finite topological spaces need not be κ-normal. theorem 2.10 ([13]). a space x is almost regular if and only if for every open set u in x, intu is θ-open. theorem 2.11. in an almost regular space, the following statements are equivalent (a) x is κ-normal. (b) x is weakly κ-normal. proof. the proof of (a) =⇒ (b) directly follows from definitions. to prove (b) =⇒ (a), let x be an almost regular, weakly κ-normal space. let a and b be two disjoint regularly closed sets in x. by theorem 2.10, a and b are disjoint regularly θ-closed sets in x. thus by weak κ-normality of x, there exist disjoint open sets separating a and b. hence x is κ-normal. � theorem 2.12. in an almost regular space, every π-closed set is θ-closed. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 235 a. k. das and p. bhat proof. let x be an almost regular space and let a ⊂ x be π-closed in x. thus a is finite intersection of π-closed sets in x. since in an almost regular space every regularly closed set is θ-closed [16] and finite intersection of θ-closed sets is θ-closed [21], a is θ-closed. � theorem 2.13. every almost regular, weakly θ-normal space is quasi normal. proof. let x be an almost regular, weakly θ-normal space. let a and b be two disjoint π-closed sets in x. by theorem 2.12, a and b are disjoint θ-closed sets which can be separated by disjoint open set as x is weakly θ-normal. � theorem 2.14. every almost regular, θ-normal space is π-normal. it is well known that every compact hausdorff space is normal. however, in the absence of hausdorffness or regularity a compact space may fail to be normal. thus it is useful to know which topological property weaker than hausdorffness with compactness implies normality. the property of being a t1 space fails to do the job since the cofinite topology on an infinite set is a compact t1 space which is not normal. however, it is well known that every compact r1-space is normal ( see [17]). in [12], it is shown that every compact space in particular every paracompact space in absence of any separation axioms is θ-normal. it is also known that every lindelöf spaces need not be κ-normal. however, by the following theorem of [12] it follows that every lindelöf space is weakly κ-normal. similarly, almost compactness need not implies κ-normality, but by theorem of [12], every almost compact space is weakly κ-normal. theorem 2.15 ([12]). every lindelöf space is weakly θ-normal. corollary 2.16. every lindelöf space is weakly κ-normal. corollary 2.17. every almost regular, lindelöf space is κ-normal. proof. the prove immediately follows from theorem 2.11, since in an almost regular space every weakly κ-normal space is κ-normal. � theorem 2.18 ([12]). every almost compact space is weakly θ-normal. corollary 2.19. every almost compact space is weakly κ-normal. corollary 2.20. every almost regular, almost compact space is κ-normal. proof. the prove immidiately follows from theorem 2.11, since in an almost regular space every weakly κ-normal space is κ-normal. � remark 2.21. corollary 2.17 and corollary 2.20 were independently prooved in [18]. in contrary to the above results the following example establishes that lindelöf spaces need not be κ-normal and almost compactness need not imply κ-normality. example 2.22. let x be the set of positive integers with the topology as defined in example 2.2 and y = {1, 2, 3, ..., 11}. then the subspace topology on y is compact but not κ-normal as disjoint regularly closed sets {2, 3, 4} and {6, 7, 8} can not be separated by disjoint open sets. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 236 a decomposition of normality via a generalization of κ-normality definition 2.23 ([13]). a space x is said to be θ-compact if every open covering of x by θ-open sets has a finite subcollection that covers x. the following result is useful to show that every almost regular, θ-compact space is κ-normal as well as weakly θ-normal. theorem 2.24 ([16]). let a ⊂ x be θ-closed and let x /∈ a. then there exist regular open sets which separate x and a. theorem 2.25. in an almost regular space, every θ-compact space is weakly θ-normal. proof. let x be an almost regular θ-compact space. let a and b be any two disjoint θ-closed subsets of x. by theorem 2.24, for every a ∈ a, there exist disjoint regularly open sets ua and va containing a and b respectively. since x is almost regular, ua and va are disjoint θ-open sets containing a and b. now the collection {ua : a ∈ a} is a θ-open cover of a. then a ⊂ ⋃ a∈a ua = o. since arbitrary union of θ-open sets is θ-open, x − o = d is θ-closed. since a is a θ-closed set disjoint from d, by theorem 2.24, for every d ∈ d, there exist disjoint regularly open sets sd and td containing a and d respectively. again by almost regularity of x, td is a θ-open set which is disjoint from a. now the collection u = {ua : a ∈ a} ∪ {td : d ∈ d} is a θ-open covering of x. by θ-compactness of x, mathcalu has a finite subcollection v which covers x. let the members of v which intersects a be w. each member of w is of the form ua for some a ∈ a as for each d ∈ d, td ∩ a = φ. suppose w = {uai : i = 1, 2, 3, ..., n}. then n⋃ i=1 uai = u and n⋂ i=1 vai = v are disjoint open sets containing a and b respectively. hence x is weakly θ-normal. � corollary 2.26. in an almost regular space, every θ-compact space is weakly κ-normal. proof. the proof immidiately follows from the fact that every θ normal space is weakly κ-normal. � corollary 2.27. in an almost regular space, every θ-compact space is κnormal. proof. the proof immidiately follows from theorem 2.11. � corollary 2.28. in an almost regular space, every almost compact space is weakly κ-normal. proof. the proof is immediate as every almost compact space is θ-compact [13]. � 3. decompositions of normality theorem 3.1. an t1-space is almost normal if and only if it is almost βnormal and weakly κ-normal. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 237 a. k. das and p. bhat proof. necessary part is obvious. conversely, let x be a t1-amlost β normal, weakly κ-normal space. since x is t1-almost β-normal, by theorem 2.9 of [3], x is almost regular. so by theorem 2.11, x is κnormal. hence x is almost normal as every almost β-normal, κ-normal space is almost normal [3]. � corollary 3.2. an t1 space is normal if and only if it is almost β-normal, weakly κ-normal and semi-normal. proof. the prove follows from the result that every almost normal, semi normal space is normal [18]. � definition 3.3. a space x is said to be strongly seminormal if for every closed set a contained in an open set u there exists a regularly θ-open set v such that a ⊂ v ⊂ u. theorem 3.4. every normal space is strongly seminormal. proof. let a be a closed set and u be an open set containing a. let b = x−u. then a and b are disjoint closed sets in x. by urysohn’s lemma there exists a continuous function f : x → [0, 1] such that f(a) = 0 and f(b) = 1. let v = f−1[0, 1/2) and w = f−1(1/2, 1]. then a ⊂ v ⊂ x − w ⊂ u. thus a ⊂ v ⊂ v o ⊂ x − w ⊂ u. we claim that v o is a regularly θ-open set. v o is regularly open, only we have to show that v o is θ-open. let x ∈ v o . then f(x) ∈ [0, 1/2). so there is a closed neighbourhood n of f(x) contained in [0, 1/2). let ux = (f −1(n))o. then x ∈ ux ⊂ f −1(n) ⊂ v o . by lemma 1.1, v o is θ-open. hence x is strongly seminormal. � theorem 3.5. every strongly seminormal space is seminormal. theorem 3.6. every strongly seminormal space is θ-regular. the following implications are obvious but none of these is reversible. normal // strongly semi normal ��uu❦❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦ regular uu❦❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦ �� seminormal ))❙❙ ❙❙ ❙❙ ❙❙ ❙❙ ❙❙ ❙❙ ❙ θ-regular �� semi regular uu❦❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦ ∆-regular example 3.7. let x be the set of positive integers with the topology as defined in example 2.2, then x is seminormal but not strongly seminormal. example 3.8. the space given in [10] by just and tartir is an example of a tychonoff κ-normal space which is not densely normal. since every seminormal κ-normal space is normal [18], thus becoming densely normal, this space is not seminormal but is θ-regular as every regular space is θ-regular. theorem 3.9. a space x is normal if and only if it is strongly seminormal and weakly κ-normal. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 238 a decomposition of normality via a generalization of κ-normality proof. the necessary part i.e., a normal space is strongly seminormal as well as weakly κ-normal directly follows from the definition. conversely, let x be a strongly seminormal and weakly κ-normal space. let a and b be two disjoint closed sets in x. thus a is a closed set contained in an open set u = x − b. since x is strongly seminormal, there exists an regularly θ-open set v such that a ⊂ v ⊂ u. now x − v is a regularly θ-closed set contained in an open set x −a. again by strong seminormality of x, there exists a regularly θ-open set w such that x − v ⊂ w ⊂ x − a. thus x − v and x − w are two disjoint regularly θ-closed sets in x containing b and a respectively. by weak κ-normality of x, there exist two disjoint open sets o and p separating x −w and x − v . hence x is normal. � corollary 3.10. in the class of strongly seminormal spaces, the following statements are equivalent. (a) x is normal. (b) x is ∆-normal. (c) x is wf∆-normal. (d) x is weakly ∆-normal. (e) x is functionally θ-normal. (f) x is θ-normal. (g) x is weakly functionally θ-normal. (h) x is weakly θ-normal. (i) x is π-normal. (j) x is quasi normal. (k) x is almost normal. (l) x is κ-normal. (m) x is weakly κ-normal. remark 3.11. in [9], it is shown that in the class of ∆-regular spaces statements (a)-(d) of corollary 3.10 are equivalent and in the class of θ-regular spaces statements (a)-(h) are equivalent. references [1] a. v. arhangel’skii and l. ludwig, on α-normal and β-normal spaces, comment. math. univ. carolin. 42, no. 3 (2001), 507–519. [2] a.v. arhangel’skii, relative topological properties and relative topological spaces, topology appl. 70 (1996), 87–99. [3] a. k. das, p. bhat and j. k. tartir, on a simultaneous generalization of β-normality and almost β-normality, filomat 31, no. 2 (2017), 425–430. [4] a. k. das and p. bhat, decompositions of normality and interrelation among its variants, math. vesnik 68, 2 (2016), 77-86. [5] a. k. das, p. bhat and r. gupta, factorizations of normality via generalizations of normality, mathematica bohemica 141, no. 4 (2016), 463–473. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 239 a. k. das and p. bhat [6] a. k. das and p. bhat, a class of spaces containing all densely normal spaces, indian j. math. 57, no. 2 (2015), 217–224. [7] a. k. das, a note on spaces between normal and κ-normal spaces, filomat 27, no. 1 (2013), 85–88. [8] a. k. das, simultaneous generalizations of regularity and normality, eur. j. pure appl. math. 4 (2011), 34–41. [9] a. k. das, ∆-normal spaces and decompositions of normality, applied general topology 10, no. 2 (2009), 197–206. [10] w. just and j. tartir, a π-normal, not densely normal tychonof spaces, proc. amer. math. soc. 127, no. 3 (1999), 901–905. [11] l. n. kalantan, π-normal topological spaces, filomat 22, no. 1 (2008), 173–181. [12] j. k. kohli and a. k. das, new normality axioms and decompositions of normality, glasnik mat. 37(57) (2002), 163–173. [13] j. k. kohli and a. k. das, a class of spaces containing all generalized absolutely closed (almost compact) spaces, applied general topology 7, no. 2 (2006), 233–244. [14] j. k. kohli and d. singh, weak normality properties and factorizations of normality, acta math. hungar. 110 (2006), 67–80. [15] c. kuratowski, topologie i, hafner, new york, 1958. [16] p. e. long and l. l. herrington, the tθ topology and faintly continuous functions, kyungpook math. j. 22, no. 1 (1982), 7–14. [17] m. g. murdeshwar, general topology, wiley eastern ltd., 1986. [18] m. k. singal and s. p. arya, on almost normal and almost completely regular spaces, glasnik mat. 5(25) (1970), 141–152. [19] m. k.singal and a. r. singal, mildly normal spaces, kyungpook math j. 13 (1973), 27–31. [20] e. v. stchepin, real valued functions and spaces close to normal, sib. j. math. 13, no. 5 (1972), 1182–1196. [21] n. v. veličko, h-closed topological spaces, amer. math. soc, transl. 78, no. 2, (1968), 103–118. [22] g. vigilino, seminormal and c-compact spaces, duke j. math. 38 (1971), 57–61. [23] v. zaitsev, on certain classes of topological spaces and their bicompactifications, dokl. akad. nauk sssr 178 (1968), 778–779. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 240 @ appl. gen. topol. 16, no. 2(2015), 109-118doi:10.4995/agt.2015.3045 c© agt, upv, 2015 primal spaces and quasihomeomorphisms afef haouati and sami lazaar department of mathematics, faculty of sciences of tunis. university tunis-el manar. “campus universitaire” 2092 tunis, tunisia. (haouati.afef@yahoo.fr, salazaar72@yahoo.fr) abstract in [3], the author has introduced the notion of primal spaces. the present paper is devoted to shedding some light on relations between quasihomeomorphisms and primal spaces. given a quasihomeomorphism q : x → y , where x and y are principal spaces, we are concerned specifically with a main problem: what additional conditions have to be imposed on q in order to render x (resp. y ) primal when y (resp. x) is primal. 2010 msc: 54b30; 54d10; 54f65; 54h20. keywords: quasihomeomorphism; principal space; sober space. 1. introduction first, we recall some notions which were introduced by the grothendieck school (see for example [6] and [7]) such as locally closed sets and quasihomeomorphisms. let x be a topological space and s be a set of x. s is called locally closed if it is an intersection of an open set and a closed set of x. we denote by l(x) the set of all locally closed sets of x. given topological spaces x and y , a continuous map q : x → y is called a quasihomeomorphism if a 7→ q−1(a) defines a bijection from o(y ) (resp., f(y ), resp., l(y )) to o(x) (resp., f(x), resp., l(x)) where o(x) (resp., f(x), resp., l(x)) is the family of all open (resp., closed, resp., locally closed) sets of x. on the other hand, another definition of quasihomeomorphism is given by k.w.yip in [9] as follows. a continuous map q : x → y between topological received 3 june 2014 – accepted 29 may 2015 http://dx.doi.org/10.4995/agt.2015.3045 a. haouati and s. lazaar spaces is said to be a quasihomeomorphism if the following equivalent conditions are satisfied: • for any closed set c of x, q−1(q(c)) = c. • for any closed set f of y , q(q−1(f)) = f . fortunately, the two notions grothendieck’s quasihomeomorphism and yip’s quasihomeomorphism coincide. quasihomeomorphisms are used in algebraic geometry and it has recently been shown that this notion arises naturally in the theory of some foliations associated to closed connected manifolds (one may see [6]). a principal space is a topological space in which any intersection of open sets is open. it is also recognized as alexandroff space. let x be a principal space. then, x provides a quasi-order ≤ (i.e a reflexive, transitive relation) given by x ≤ y if and only if x ∈ {y} which is called the specialization quasi-order (for more informations, one may see [10]). conversely, every quasi-order ≤ on a space x determines a principal topology. indeed, for each x ∈ x we let x ↑ be the upperset of x defined by x ↑ := {y ∈ x : x ≤ y}. then, the family b := {x ↑ : x ∈ x} is a basis of a principal topology on x. note that the closure {x} is exactly the downset ↓ x := {y ∈ x : y ≤ x}. now, let c be a category. then, a flow in c is a couple (x, f) where x is an object of c and f : x → x is an arrow called iterator. if (x, f) and (y, g) are flows in c, then a morphism of flows from (x, f) to (y, g) is an arrow q from x to y such that the following diagram is commutative. x f −→ x q ↓ � ↓ q y g −→ y that is g ◦ q = q ◦ f. for more details, one may see [4] and [5]. let (x, f) be a flow in the category of sets noted set. o.echi has defined the topology p(f) on x with closed sets exactly those a which are f-invariant (a set a of x is called f-invariant if f(a) ⊆ a). clearly p(f) provides a principal topology on x. we can easily see that for any set a of x, the closure a is exactly ∪[fn(a) : n ∈ n] and in particular for any point x ∈ x, {x} = {fn(x), n ∈ n} denoted of (x) and called the orbit of x by f. the family x ↑ = {y ∈ x : fn(y) = x for some n ∈ n} is a basis of open sets of p(f). according to o.echi, a primal space is a topological space (x, τ) such that there is some mapping f : x → x with τ = p(f) (for more informations see [3]). in the first section of this paper, we are interested in some dynamical properties of quasihomeomorphisms between principal spaces. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 110 primal spaces and quasihomeomorphisms the main goal of the second section is to show that given an onto quasihomeomorphism from a primal space x to a principal space y , then y is primal (see theorem 3.2). in the third section, we move our focus to one-to-one quasihomeomorphisms and its effects on primal spaces (see theorem 4.1). finally, some particular cases of quasihomeomorphisms are studied and commented. 2. preliminary results let x be a principal space and ≤ its specialization quasi-order. a point x ∈ x is called minimal if it satisfies the property: for each y ∈ x, y ≤ x ⇒ x ≤ y. let f : x → y be a continuous map between two principal spaces. it follows immediately, from the fact that a map between two principal spaces is continuous if and only if the induced map between the associated quasi-ordered sets is isotone, that for every x, y ∈ x, we have: x ≤ y ⇒ f(x) ≤ f(y). now, the following proposition shows that the converse holds if f is a quasihomeomorphism. proposition 2.1. let f : x −→ y be a quasihomeomorphism where x and y are principal spaces. then, for every x, y ∈ x, we have: x ≤ y ⇐⇒ f(x) ≤ f(y). proof. it is sufficient to show the second implication. for that, let x, y ∈ x such that f(x) ≤ f(y). since f is a quasihomeomorphism, then there exists a unique closed set f of y such that ↓ y = f−1(f). now, the fact that f(y) ∈ f implies ↓ f(y) ⊆ f and consequently f(x) ∈ f , that is x ∈↓ y as desired. � this result leads to the following corollary. corollary 2.2. let q : x −→ y be a quasihomeomorphism where x and y are principal spaces. for every x ∈ x, if q(x) is minimal in y then x is minimal in x. proof. let x be a point in x such that q(x) is a minimal point in y . suppose that there exists x′ ∈ x satisfying x′ ≤ x. then, we have q(x′) ≤ q(x) and thus, by minimality of q(x), q(x) ≤ q(x′). therefore, proposition 2.1 does the job. � question 2.3. let q : x −→ y be a quasihomeomorphism and x a point in x. if x is minimal, then what about q(x) ? the following proposition shows that if in addition q is an onto quasihomeomorphism from x to y , then there is an equivalence between x is minimal in x and q(x) is minimal in y . c© agt, upv, 2015 appl. gen. topol. 16, no. 2 111 a. haouati and s. lazaar proposition 2.4. let q : x → y be an onto quasihomeomorphism where x and y are principal spaces. for any point x ∈ x, the following properties hold: (1) q(↓ x) =↓ q(x) and q(x ↑) = q(x) ↑ (2) x is minimal in x if and only if q(x) is minimal in y . proof. recall that if q : x −→ y is a quasihomeomorphism then q is onto iff q is open iff q is closed (see [2, lemma 1.1]). (1) let x be a point in x. the inclusion q(↓ x) ⊆ ↓ q(x) (resp., q(x ↑) ⊆ q(x) ↑) follows immediately from proposition 2.1. conversely, since q is closed (resp., open) then q(↓ x) is a closed (resp., open) set containing q(x). so that q(↓ x) (resp., q(x ↑)) contains ↓ q(x) (resp., q(x) ↑). (2) according to corollary 2.2, it is enough to show that if x is minimal in x then q(x) is minimal in y . indeed, let z ∈↓ q(x). due to proposition 2.4.(1), we have q(↓ x) =↓ q(x) so that z = q(y) with y ≤ x. since x is minimal, then x ≤ y which leads to q(x) ≤ z as desired. � proposition 2.5. let q : x −→ y be a quasihomeomorphism where x and y are primal spaces. (1) if x is a t0−space, then q is one to one. (2) if y is a t0−space, then q is onto. (3) if x and y are both t0−spaces, then q is a homeomorphism. proof. it follows immediately from [2, lemma 3.7]. (1)(2) let y ∈ y . since q is a quasihomeomorphism then there exists x ∈ x such that q(x) ∈↓ y. using [3, theorem 2.3], the primality of y gives a finite interval [q(x), y]. let q(x0) be the biggest element of [q(x), y] that have an antecedent in x. we claim that q−1(↓ y) = q−1(↓ q(x0)). in fact, since ↓ y is totally ordered, then q(z) ≤ y gives either q(z) ≤ q(x) and so q(z) ≤ q(x0) or q(z) ∈ [q(x), y] which also implies that q(z) ≤ q(x0) because q(x0) is the biggest element in that interval that have an antecedent. so that q−1(↓ y) ⊆ q−1(↓ q(x0)). on the other hand, since q(x0) ≤ y then q −1(↓ q(x0)) ⊆ q −1(↓ y). now, we have q−1(↓ y) = q−1(↓ q(x0)) and q is a quasihomeomorphism. so, {q(x0)} = {y}. finally, since y is t0 then y = q(x0). we conclude that y is onto. (3) combining (1) and (2), we have q is a homeomorphism. � 3. primal spaces and onto quasihomeomorphisms in order to characterize quasihomeomorphisms that conserve the property of primality between principal spaces, we need to recall some notions which were introduced by o.echi in [3]. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 112 primal spaces and quasihomeomorphisms given a quasi-ordered set (x, ≤). • we say that (x, ≤) is causal if for each x, y ∈ x, the interval [x, y] := {z ∈ x : x ≤ z ≤ y} is finite. • (x, ≤) will be called a quasi-forest if the downset of any point is totally quasi-ordered. given a flow (x, f) and x ∈ x. x is said to be a periodic point if fn(x) = x for some n ∈ n. o. echi used the previous concepts to provide in [3] an interesting characterization of primal spaces in order-theoretical terms. before stating one of the main goals of this paper, it is of interest to recall this important result. theorem 3.1 ([3, theorem 2.3]). let x be a principal topological space. then, x is a primal space if and only if the associated quasi-ordered set (x, ≤) is a causal quasi-forest in which each non-minimal point x has singleton interval [x, x]. now, we are in a position to give our main result. theorem 3.2. let (x, p(f)) be a primal space, y a principal space and q : x −→ y a quasihomeomorphism. if q is onto, then y is primal. proof. according to theorem 3.1, we have to show that the associated quasiordered set (y, ≤) is a causal quasi-forest whose non minimal points y have singleton intervals [y, y]. • (y, ≤) is a quasi-forest. let z ∈ y and y1, y2 be two elements in ↓ z. by the surjectivity of q, there exists x ∈ x such that z = q(x). according to proposition 2.4, we have q(↓ x) =↓ q(x). so there exists x1, x2 ∈↓ x such that y1 = q(x1) and y2 = q(x2). the primality of x gives x1 ≤ x2 or x2 ≤ x1 and consequently either q(x1) ≤ q(x2) or q(x2) ≤ q(x1). therefore, ↓ z is totally quasi-ordered. • (y, ≤) is causal. let y1, y2 ∈ y . if y1 ↑ ∩ ↓ y2 is empty then it is finite. otherwise, there exists z an element of this intersection which allows one to claim that y1 ≤ y2. we denote by x1 (resp. x2) an antecedent of y1 (resp. y2). by proposition 2.1, it follows that x1 ≤ x2. now, we show that q(x1 ↑ ∩ ↓ x2) = y1 ↑ ∩ ↓ y2. in fact, the continuity of q gives the first inclusion. conversely, since x1 ↑ ∩ ↓ x2 is a locally closed subset of x and q is a quasihomeomorphism then there exists a locally closed subset l of y satisfying x1 ↑ ∩ ↓ x2 = q −1(l). now, by the surjectivity of q, q(x1 ↑ ∩ ↓ x2) is locally closed in y . let u (resp., f) an open (resp., closed) subset of x such that l = u ∩ f . it is clear that q(x1) ↑⊂ u (resp. ↓ q(x2) ⊂ f) so that q(x1) ↑ ∩ ↓ q(x2) ⊂ q(x1 ↑ ∩ ↓ x2). finally, remark that x1 ↑ ∩ ↓ x2 is finite which implies that y1 ↑ ∩ ↓ y2 is finite. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 113 a. haouati and s. lazaar • every non minimal point y of y has a singleton interval [y, y]. let y ∈ y \ min(y ), using proposition 2.4.(2), there exists x ∈ x \ min(x) such that y = q(x). by the primality of x, we get q(↓ x ∩ x ↑) = q({x}). therefore, ↓ y ∩ y ↑= {y}. � now, in this context, some observations are presented by the following examples. examples 3.3. (1) the following example shows that the surjectivity of q in theorem 3.2 is necessary. indeed, let α, β be two distinct points and set x = {α, β} equipped with the indiscrete topology. on the other hand, let y be an infinite set equipped with the indiscrete topology and define q : x → y by q(α) = q(β) = y with y an arbitrary element of y . we can easily see that x is a primal space and q is a non onto quasihomeomorphism. however, y is a principal space which is not primal since it is not causal ([x, x] = y infinite for every x ∈ y ). (2) the converse of theorem 3.2 does not hold. to see this, consider x = {α, β} and y = {0, 1, 2} both equipped with the indiscrete topology. now, let q : x −→ y defined by q(α) = q(β) = a. therefore, q is a quasihomeomorphism which is not onto despite of the primality of x and y . (3) let x be the set {0, 1, 2} and y the set {a, b} with a 6= b, both equipped with the indiscrete topology τ. set f from x to itself by f(0) = 1, f(1) = 0 and f(2) = 0. clearly the map q from x to y , defined by q(0) = a and q(1) = q(2) = b, is an onto quasihomeomorphism. now, there is a unique map g from y to itself that satisfies (y, τ) = (y, p(f)); it is defined by g(a) = b and g(b) = a. we can see easily that g◦q 6= q◦f and consequently q is not a morphism of flows from (x, f) to (y, g). (4) let (x, f) as in (3) and q : x −→ x the identity map. clearly, there is exactly two maps g1 and g2 from x to itself such that (x, τ) = (x, p(gi)), i ∈ {1, 2}. if we choose g = g1, we have q is a morphism of flows from (x, f) to (x, g) but not if we choose g = g2. before giving an interesting consequence of the previous theorem, we recall the t0−reflection of a topological space. given a topological space x, we define the equivalence relation on x by x ∼ y if and only if {x} = {y}. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 114 primal spaces and quasihomeomorphisms the resulting quotient space x/∼ is a t0−space called the t0−reflection of x and the following properties hold: • the canonical onto map µx : x −→ t0(x) is a quasihomeomorphism ([8]). • x is a principal space if and only if t0(x) is also. corollary 3.4. the t0-reflection of a primal space is primal. one may see by example 3.3.(4) that we have to impose additional conditions in order to render the quasihomeomorphism cited in theorem 3.2 a morphism of flows from (x, f) to (y, g). thus, we provide the following proposition. proposition 3.5. let (x, f) (resp., (y, g)) be a flow in set.we equip (x, f) (resp., (y, g)) with the topology p(f) (resp., p(g)) and let q : x −→ y be a quasihomeomorphism. if y is a t0-space, then q is a morphism of flows from (x, f) to (y, g). proof. according to proposition 2.5, we have q is onto. let x ∈ x and we show that goq(x) = qof(x). we denote y = q(x). the surjectivity of q allows us to denote x′ an antecedent of g(y) with x′ ≤ x since q(x′) ≤ q(x). first case : x ∈ min(x). we have: x′ ≤ x =⇒ x ≤ x′. which leads to f(x) ≤ x′ so that q(f(x)) ≤ q(x′) (1). on the other hand, since x is minimal we have also x ≤ f(x). so x′ ≤ f(x) which gives q(x′) ≤ q(f(x)) (2). now, we have from (1) and (2) {q(x′)} = {q(f(x))}. since y is t0 then q(x ′) = g(q(x)) = q(f(x)) as desired. second case : x /∈ min(x). in this case, we have (↓ x) \ {x} =↓ f(x). so, either x′ = x or x′ ∈↓ f(x). we start with studying the problem when x′ = x. in that case, q(x) = q(x′) = g(q(x)) which means that {q(x)} = {q(x)}. now, f(x) ≤ x ⇒ q(f(x)) ≤ q(x) ⇒ q(f(x)) ∈ {q(x)} which implies that q(f(x)) = q(x) = g(q(x)) as desired. therefore, we give attention now to the case when x′ ∈↓ f(x). indeed, in that case, q(x′) ≤ q(f(x)) (∗). if q(f(x)) = y, then we have q(f(x)) ≤ q(x) and q(x) ≤ q(f(x)). by proposition 2.1, f(x) ≤ x and x ≤ f(x). yet, since x /∈ min(x) then f(x) = x. which means that {x} = {x} and thus x′ = x. this leads to g(q(x)) = q(x′) = q(x) = q(f(x)), as desired. otherwise, if q(f(x)) 6= y, then q(f(x)) ∈ (↓ y) \ {y}. using proposition 2.4.(2), we have y /∈ min(y ), so that (↓ y)\{y} =↓ g(y). thus, q(f(x)) ≤ g(y) (∗∗) now, it follows from (∗) and (∗∗) that {q(f(x))} = {g(y)}. since y is t0 then g(q(x)) = g(y) = q(f(x)). which completes the proof. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 115 a. haouati and s. lazaar � we state a useful remark. remark 3.6. the condition ”y is a t0−space” in proposition 3.5 is a sufficient condition but not a necessary condition. to see this, consider the example 3.3.(4). 4. primal spaces and one-to-one quasihomeomorphisms in this section, our interest is directed towards the characterization of quasihomeomorphisms q : x → y that render x primal when y is. thus, we present the following result. theorem 4.1. let (x, τ) be a principal space, (y, p(g)) a primal space and q : x −→ y a quasihomeomorphism. if q is one-to-one, then x is primal. proof. let ≤ be the specialization quasi-order on x. • (x, ≤) is a quasi-forest. let x ∈ x and x1, x2 ∈↓ x. then, q(x1), q(x2) ∈↓ q(x). since y is primal, we have either q(x1) ≤ q(x2) or q(x2) ≤ q(x1). using proposition 2.1, this leads to x1 ≤ x2 or x2 ≤ x1. • (x, ≤) is causal. let x1, x2 ∈ x. since q is one-to-one then the cardinal of [x1, x2] is equal to the cardinal of its image by q. using proposition 2.1, we have q([x1, x2]) ⊆ [q(x1), q(x2)]. now, since y is primal then [q(x1), q(x2)] is finite and so [x1, x2] is also. • ∀x ∈ x \ min(x), [x, x] = {x}. let x ∈ x \ min(x). by corollary 2.2, q(x) ∈ y \ min(y ). using the primality of y , we have [q(x), q(x)] = {q(x)}. let z ∈ [x, x], then q(z) ∈ [q(x), q(x)] which leads to q(z) = q(x). yet, q is one-to-one. so, z = x. thus, [x, x] = {x}. using theorem 3.1, we conclude that x is a primal space. � now, we give some straightforward remarks. remarks 4.2. (1) the injectivity of the quasihomeomorphism q cited in theorem 4.1 is necessary to conclude that x is primal. to see this, consider x = n, y = {α, β} both equipped with the indiscrete topology and q : x → y such that q(0) = α and q(n∗) = {β}. (2) the converse of theorem 4.1 fails. indeed, consider the example cited in remark 3.3.(2). next, we will localize our interest to the consequences of the previous theorem. recall that a set f of a topological space x is said to be irreducible if for each open sets u and v of x such that f ∩ u 6= ∅ and f ∩ v 6= ∅, we have c© agt, upv, 2015 appl. gen. topol. 16, no. 2 116 primal spaces and quasihomeomorphisms f ∩ u ∩ v 6= ∅ (equivalently, if c1 and c2 are two closed sets of x such that f ⊆ c1 ∪ c2, then f ⊆ c1 or f ⊆ c2). a topological space x is called sober if each nonempty irreducible closed set f of x has a unique generic point(i.e there exists a unique x ∈ x such that f = {x}). let s(x) be the set of all nonempty irreducible closed sets of x. let u be an open set of x; set ũ = {c ∈ s(x) : u ∩ c 6= ∅}. then, the collection {ũ : u is an open set of x} provides a topology on s(x) and the following properties hold: • the map θx : x → s(x) which carries x ∈ x to θx(x) = {x} is a quasihomeomorphism. • s(x) is a sober space. • let f : x −→ y be a continuous map. let s(f) : s(x) −→ s(y ) be the map defined by s(f)(c) = c, for each irreducible closed subset c of x. then s(f) is continuous. • the topological space s(x) is called the sobrification of x, and the assignment s(x) defines a functor from the category of topological spaces to itself. • let f : x −→ y be a continuous map. then, the diagram x f −→ y θx ↓ � ↓ θy s(x) s(f) −→ s(y ) is commutative. in [2], the author has proved that s(f) is a homeomorphism if and only if f is a quasihomeomorphism. now, since µx : x −→ t0(x) is a quasihomeomorphism then s(µx) is a homeomorphism and consequently s(t0(x)) is homeomorphic to s(x). this result allows one to present the following corollary. corollary 4.3. let x be a topological space. if s(x) is a primal space then t0(x) is primal. proof. this follows immediately from theorem 4.1 using the one-to-one quasihomeomorphism θt0(x) : t0(x) → s(t0(x)), x 7→ {x} and considering that s(t0(x)) is homeomorphic to s(x). � now, proposition 3.5 motivates the following question: suppose that q : (x, p(f)) −→ (y, p(g)) is a quasihomeomorphism between two primal spaces. does the condition ”x is a t0−space” allows one to claim that q is morphism of flows from (x, f) to (y, g) ? the following example gives the answer. example 4.4. let x = {α, β} with α 6= β and y = {0, 1, 2, 3}. set f (resp., g) from x (resp., y ) to itself by f(α) = β and f(β) = β (resp., g(0) = 1, g(1) = 2, g(2) = 3 and g(3) = 1). the quasihomeomorphism defined by q(α) = 0 and c© agt, upv, 2015 appl. gen. topol. 16, no. 2 117 a. haouati and s. lazaar q(β) = 2 is not a morphism of flows from (x, f) to (y, g). although x is a t0− space. acknowledgements. the authors gratefully acknowledge the many helpful corrections, comments and suggestions of the anonymous referee. references [1] k. belaid, o. echi and s. lazaar, t(α,β)-spaces and the wallman compactification, int. j. math. math. sc. 68 (2004), 3717–3735. [2] o. echi, quasi-homeomorphisms, goldspectral spaces and jacspectral spaces, boll. unione mat. ital. sez. b artic. ric. mat. (8)6 (2003), 489–507. [3] o. echi, the category of flows of set and top, topology appl. 159 (2012), 2357–2366. [4] j. f. kennisson, the cyclic spectrum of a boolean flow, theory appl. categ. 10 (2002), 392–409. [5] j. f. kennisson, spectra of finitly generated boolean flow, theory appl. categ. 16 (2006), 434–459. [6] a. grothendieck and j. dieudonné, eléments de géométrie algébrique, die grundlehren der mathematischen wissenschaften, vol. 166, springer-verlag, new york, 1971. [7] a. grothendieck and j. dieudonné, eléments de géométrie algébrique. i. le langage des schḿas, inst. hautes études sci. publ. math. no. 4, 1960. [8] m. h. stone, applications of boolean algebra to topology, mat. sb. 1 (1936), 765–772. [9] k. w. yip, quasi-homeomorphisms and lattice-equivalences of topological spaces, j. austral. math. soc. 14 (1972), 41–44. [10] g. gierz, k. h. hofmann, k. keimel, j. d. lawson, m. mislove and d. s. scott, continuous lattices and domains, cambridge univ, press, 2003. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 118 () @ appl. gen. topol. 18, no. 2 (2017), 317-330doi:10.4995/agt.2017.7067 c© agt, upv, 2017 multiple fixed point theorems for contractive and meir-keeler type mappings defined on partially ordered spaces with a distance mitrofan m. choban a and vasile berinde b a department of physics, mathematics and information technologies, tiraspol state university, gh. iablocikin 5., md2069 chişinău, republic of moldova (mmchoban@gmail.com) b department of mathematics and computer science, technical university of cluj-napoca, north university center at baia mare, victoriei 76, 430072 baia mare, romania (vberinde@cunbm.utcluj.ro) communicated by i. altun abstract we introduce and study a general concept of multiple fixed point for mappings defined on partially ordered distance spaces in the presence of a contraction type condition and appropriate monotonicity properties. this notion and the obtained results complement the corresponding ones from [m. choban, v. berinde, a general concept of multiple fixed point for mappings defined on spaces with a distance, carpathian j. math. 33 (2017), no. 3, 275–286] and also simplifies some concepts of multiple fixed point considered by various authors in the last decade or so. 2010 msc: 47h10; 47h09. keywords: distance space; partial order; symmetric space; quasi-metric space; h-distance; contraction condition; multiple fixed point. 1. introduction in a previous paper [32], the authors have introduced and studied a general concept of multidimensional fixed point for mappings defined on a distance space and satisfying a certain contraction condition. received 02 january 2017 – accepted 12 july 2017 http://dx.doi.org/10.4995/agt.2017.7067 m. m. choban and v. berinde several interesting new results that generalise, extend and unify corresponding related results from literature for the case of non ordered distance spaces were obtained. however, the great majority of the multidimensional fixed point theorems existing in literature were established in the setting of a partially ordered metric space or of a partially ordered generalised metric space. therefore, the main aim of this paper is to study the concept of multidimensional fixed point introduced in [32] for the case of mappings defined on partially ordered distance space, thus extending and complementing most of the results established in [32]. we start by presenting a brief survey on the notion of multidimensional fixed point, which naturally emerged from the rich literature produced in the last four decades devoted to coupled fixed points. the concept itself of coupled fixed point has been first introduced and studied by opoitsev, in a series of papers he published in the period 1975-1986, see [61]-[64]. opoitsev has been inspired by some concrete problems arising in the dynamics of collective behaviour in mathematical economics and considered the coupled fixed point problem for mixed monotone nonlinear operators which also satisfy a nonexpansive type condition. in 1987, guo and lakshmikantham [41], apparently not being aware of opoitsev’s previous results [61]-[64], have studied coupled fixed points in connection with coupled quasi-solutions of an initial value problem for ordinary differential equations. amongst the subsequent developments we quote the following works: [40]; [30], containing coupled fixed point results of 1 2 -α-condensing and mixed monotone operators, where α denotes the kuratowski’s measure of non compactness, thus extending some previous results from [41] and [79]; [29], which discusses some existence results and iterative approximation of coupled fixed points for mixed monotone condensing set-valued operators; [28] where the authors obtained coupled fixed point results of 1 2 -α-contractive and generalized condensing mixed monotone operators. more recently, gnana bhaskar and lakshmikantham in [37] established coupled fixed point results for mixed monotone operators in partially ordered metric spaces in the presence of a bancah contraction type condition. essentialy, the results by bhaskar and lakshmikantham in [37] combined, in the context of bivariable mixed monotone mappings, the main fixed point results previously obtained by nieto and rodriguez-lopez [58] and [59], for the case of one variable increasing and decreasing nonlinear operator, respectively. the last two papers are, in turn, a continuation of the hybrid fixed point theorem established in the seminal paper of ran and reurings [66], which has the merit to combine a metrical fixed point theorem (the contraction mapping principle) and an order theoretic fixed point result (tarski’s fixed point theorem). various applications of the theoretical results in coupled fixed point theory were also considered, for the case of: a) uryson integral equations [63]; b) a system of volterra integral equations [30], [28]; c) a class of functional equations arising in dynamic programming [29]; d) initial value problems for first order differential equations with discontinuous right hand side [41]; e) (two point) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 318 multiple fixed point theorems for contractive and meir-keeler type mappings periodic boundary value problems [17], [37], [33], [82]; f) integral equations and systems of integral equations [3], [6], [9], [24], [39], [42], [78], [80], [85]; g) nonlinear elliptic problems and delayed hematopoesis models [84]; h) nonlinear hammerstein integral equations [76]; i) nonlinear matrix and nonlinear quadratic equations [4], [24]; j) initial value problems for ode [8], [75] etc. for a very recent account on the developments of coupled fixed point theory, we also refer to [22]. on the other hand, in 2010, samet and vetro [74] apart of some coupled fixed point results they have established, considered a concept of fixed point of m-order as a natural extension of the notion of coupled fixed point. then, in 2011, mainly inspired by [37], berinde and borcut [18] introduced the concept of triple fixed point and proved existence as well as existence and uniqueness triple fixed point theorems for three-variable mixed monotone mappings, while, in 2012, karapinar and berinde [47], have studied quadruple fixed points of nonlinear contractions in partially ordered metric spaces. after these starting papers, a substantial number of articles were dedicated to the study of triple fixed points, quadruple fixed points, as well as to multiple fixed points (also called fixed point of m-order, or ”a multidimensional fixed point”, or ”an m-tuplet fixed point”, or ”an m-tuple fixed point”), see [1], [2], [7], [48], [49], [50], [53], [60], [67]-[72], [81], [83], [86], which form a very selective list contributions. starting from this background, the main aim of the present paper is to study the concept of multidimensional fixed point introduced in [32] but for mappings defined on partially ordered distance space, in the presence of a contraction type condition and appropriate monotonicity properties, thus extending and complementing the results established in [32]. this approach is based on the idea to reduce the study of multidimensional fixed points and coincidence points to the study of usual one-dimensional fixed points for an associate operator. note that, the first author who reduced the problem of finding a coupled fixed point of mixed monotone operators to the problem of finding a fixed point of an increasing one variable operator was opoitsev, see for example [63]. 2. preliminaries by a space we understand a topological t0-space. we use the terminology from [36, 38, 73, 31]. let x be a non-empty set and d : x × x → r be a mapping such that: (im) d(x, y) ≥ 0, for all x, y ∈ x; (iim) d(x, y) + d(y, x) = 0 if and only if x = y. then d is called a distance on x, while (x, d) is called a distance space. let d be a distance on x and b(x, d, r) = {y ∈ x : d(x, y) < r} be the ball with the center x and radius r > 0. the set u ⊂ x is called d-open if for any x ∈ u there exists r > 0 such that b(x, d, r) ⊂ u. the family t (d) of all d-open subsets is the topology on x generated by d. a distance space is c© agt, upv, 2017 appl. gen. topol. 18, no. 2 319 m. m. choban and v. berinde a sequential space, i.e., a space for which a set b ⊆ x is closed if and only if together with any sequence it contains all its limits [36]. let (x, d) be a distance space, {xn}n∈n be a sequence in x and x ∈ x. we say that the sequence {xn}n∈n is: 1) convergent to x if and only if limn→∞ d(x, xn) = 0. we denote this by xn → x or x = limn→∞ xn (really, we may denote x ∈ limn→∞ xn); 2) convergent if it converges to some point x in x; 3) cauchy or fundamental if limn,m→∞ d(xn, xm) = 0. a distance space (x, d) is called complete if every cauchy sequence in x converges to some point x in x. let x be a non-empty set and d be a distance on x. then: • (x, d) is called a symmetric space and d is called a symmetric on x if (iiim) d(x, y) = d(y, x), for all x, y ∈ x; • (x, d) is called a quasimetric space and d is called a quasimetric on x if (ivm) d(x, z) ≤ d(x, y) + d(y, z), for all x, y, z ∈ x; • (x, d) is called a metric space and d is called a metric if d is a symmetric and a quasimetric, simultaneously. let x be a non-empty set and d(x, y) be a distance on x with the following property: (n) for each point x ∈ x and any ε > 0 there exists δ = δ(x, ε) > 0 such that from d(x, y) ≤ δ and d(y, z) ≤ δ it follows d(x, z) ≤ ε. then (x, d) is called an n-distance space and d is called an n-distance on x. if d is a symmetric, then we say that d is an n-symmetric. spaces with n-distances were studied by niemyzki [56] and by nedev [55]. if d satisfies the condition (f) for any ε > 0 there exists δ = δ(ε) > 0 such that from d(x, y) ≤ δ and d(y, z) ≤ δ it follows d(x, z) ≤ ε, then d is called an f-distance or a fréchet distance and (x, d) is called an f-distance space. any f-distance d is an n-distance, too. if d is a symmetric and an f-distance on a space x, then we say that d is an f-symmetric. remark 2.1. if (x, d) is an f-symmetric space, then any convergent sequence is a cauchy sequence. for n-symmetric spaces and for quasimetric spaces this assertion is not more true. if s > 0 and d(x, y) ≤ s[d(x, z) + d(z, y)] for all points x, y, z ∈ x, then we say that d is an s-distance. any s-distance is an f-distance. a distance space (x, d) is called an h-distance space if, for any two distinct points x, y ∈ x, there exists δ = δ(x, y) > 0 such that b(x, d, δ) ∩ b(y, d, δ) = ∅. we say that (x, d) is a c-distance space or a cauchy distance space if any convergent cauchy sequence has a unique limit point. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 320 multiple fixed point theorems for contractive and meir-keeler type mappings remark 2.2. a distance space (x, d) is an h-distance space if and only if any convergent sequence in x has a unique limit point. hence, any h-distance space is a c-distance space. 3. ordering on cartesian product of distance spaces let (x, d) be a distance space, m ∈ n = {1, 2, ...}. on xm consider the distances dm((x1, ..., xm), (y1, ..., ym)) = sup{d(xi, yi) : i ≤ m} and d̄m((x1, ..., xm), (y1, ..., ym)) = m∑ i=1 d(xi, yi). obviously, (xm, dm) and (xm, d̄m) are distance spaces, too. proposition 3.1 ([32]). let (x, d) be a distance space. then: 1. if d is a symmetric, then (xm, dm) and (xm, d̄m) are symmetric spaces, too. 2. if d is a quasimetric, then (xm, dm) and (xm, d̄m) are quasimetric spaces, too. 3. if d is a metric, then (xm, dm) and (xm, d̄m) are metric spaces, too. 4. if d is an f-distance space, then (xm, dm) and (xm, d̄m) are f-distance spaces, too. 5. if d is an n-distance space, then (xm, dm) and (xm, d̄m) are ndistance spaces, too. 6. if d is an h-distance space, then (xm, dm) and (xm, d̄m) are hdistance spaces, too. 7. if (x, d) is a c-distance space, then (xm, dm) and (xm, d̄m) are cdistance spaces, too. 8. if (x, d) is a complete distance space, then (xm, dm) and (xm, d̄m) are complete distance spaces, too. 9. if d is an s-distance space, then (xm, dm) and (xm, d̄m) are s-distance spaces, too. 10. the spaces (xm, dm) and (xm, d̄m) share the same convergent sequences and the same cauchy sequences. moreover, the distances dm and d̄m are uniformly equivalent, i.e., for each ε > 0, there exists δ = δ(ε) > 0 such that: from dm(x, y) ≤ δ it follows d̄m(x, y) ≤ ε; from d̄m(x, y) ≤ δ it follows dm(x, y) ≤ ε. let � be a (partial) order on a distance space (x, d). a sequence {xn}n∈n is called: • non-decreasing if xn � xn+1 for each n ∈ n; • non-increasing if xn � xn+1 for each n ∈ n; • monotone if it is either non-decreasing or non-increasing. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 321 m. m. choban and v. berinde if (x, d, �) is an ordered distance space and g : x → x is a mapping, then (x, d, �) is said to have the sequential g-monotone property [37, 69] if it verifies: i) if {xn}n∈n is a non-decreasing sequence and lim n→∞ d(x, xn) = 0, then g(xn) � g(x) for all n ∈ n; ii) if {xn}n∈n is a non-increasing sequence and lim n→∞ d(x, xn) = 0, then g(xn) � g(x) for all n ∈ n. an ordered distance space (x, d, �) is called monotonically complete if any monotone cauchy sequence in x converges to some point in x. fix m ∈ n and a subset l ⊆ {1, 2, ..., m}. like in [67, 69, 70], we introduce on x the ordering �l: (x1, ..., xm) �l (y1, ..., ym) iff xi � yi for i ∈ l and yj � xj for j /∈ l. by construction, (xm, dm, �l) and (x m, d̄m, �l) are ordered distance spaces. if l ⊆ {1, 2, ..., m} and m = {1, 2, ..., m} \ l, then x �l y if and only if y �m x for x, y ∈ x m. hence �m is the dual (inverse) order of the order �l. proposition 3.2. let (x, d, �) be a monotonically complete distance space. then (xm, dm, �l) and (x m, d̄m, �l) are ordered monotonically complete distance spaces, too. proof. it is obvious. � 4. multiple fixed point principles for monotone type operators fix m ∈ n. denote by λ = (λ1, ..., λm) a collection of mappings {λi : {1, 2, ..., m} −→ {1, 2, ..., m} : 1 ≤ i ≤ m}. let (x, d) be a distance space and f : xm −→ x be an operator. the operator f and the mappings λ generate the operator λf : xm −→ xm, where λf(x1, ...., xm) = (y1, ..., ym) and yi = f(xλi(1), ..., xλi(m)), for each point (x1, ..., xm) ∈ x m and any index i ∈ {1, 2, ..., m}. a point a = (a1, ..., am) ∈ x m is called a λ-multiple fixed point of the operator f if a = λf(a), i.e., ai = f(aλi(1), ..., aλi(m)) for any i ∈ {1, 2, ..., m}. let (x, d, �) be a (partially) ordered distance space, m ∈ n, l ⊆ {1, 2, ..., m} and f : xm −→ x be an operator. in this context, we consider the following sets of assumptions that include a symmetric type contractive condition, similar to the symmetric contraction introduced and used by the second author in [14]. conditions ω1: 1. (x, �) is a lattice; 2. if x, y, z ∈ x and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x); 3. if x, y ∈ xm, x 6= y and x �l y, then λf(x) �l λf(y) and d m(λf(x), λf(y)) + dm(λf(y), λf(x)) < dm(x, y) + dm(y, x). conditions ω2: 1. (x, �) is a lattice; c© agt, upv, 2017 appl. gen. topol. 18, no. 2 322 multiple fixed point theorems for contractive and meir-keeler type mappings 2. if x, y, z ∈ x and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x); 3. if x, y ∈ xm, x 6= y and x �l y, then λf(y) �l λf(x) and d m(λf(x), λf(y)) + dm(λf(y), λf(x)) < dm(x, y) + dm(y, x). conditions ω3: 1. (x, �) is a lattice; 2. if x, y, z ∈ x and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x); 3. for any i ∈ {1, 2, . . . , m} the mapping λi is a surjection or, more generally, | ∪ {λ−1 i (j) : 1 ≤ j ≤ m}| = m, for each i ∈ {1, 2, . . . , m}; 4. if x, y ∈ xm, x 6= y and x �l y, then λf(x) �l λf(y) and d̄ m(λf(x), λf(y)) + d̄m(λf(y), λf(x)) < d̄m(x, y) + d̄m(y, x). conditions ω4: 1. (x, �) is a lattice; 2. if x, y, z ∈ x and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x); 3. for any i ∈ {1, 2, . . . , m} the mapping λi is a surjection or, more generally, | ∪ {λ−1i (j) : 1 ≤ j ≤ m}| = m, for each i ∈ {1, 2, . . . , m}; 4. if x, y ∈ xm, x 6= y and x �l y, then λf(y) �l λf(x) and d̄ m(λf(x), λf(y)) + d̄m(λf(y), λf(x)) < d̄m(x, y) + d̄m(y, x). now we can state concisely the following general and comprehensive multidimensional fixed point result. theorem 4.1. let a ∈ xm be a multidimensional fixed point of the operator of f : xm −→ x. if any of the conditions ωi, i ∈ {1, 2, 3, 4}, is satisfied, then the operator f has a unique multidimensional fixed point. proof. obviously, (xm, �l) is a lattice, too. let ρ = d m, for i ∈ {1, 2}, and ρ = d̄m, for i ∈ {3, 4}. then for x, y, z ∈ xm and x �l y �l z we have ρ(x, y)+ ρ(y, x) ≤ ρ(x, z)+ ρ(z, x). assume that b ∈ xm is a multidimensional fixed point of the operator f with b 6= a. case 1. the points a and b are comparable. assume that a �l b. then ρ(a, b) + ρ(b, a) = ρ(λf(a), λf(b)) + ρ(λf(b), λf(a)) < ρ(a, b) + ρ(b, a), a contradiction. case 2. the points a and b are not comparable. from any of the conditions ωi it follows that (x, �) is a lattice. hence (xm, �l) is a lattice too, as the cartesian product of lattices. fix c = max{a, b} ∈ xm. we put d = λf(λf(c)). by construction, a �l d and b �l d. hence, c �l d and ρ(a, c) ≤ ρ(a, d), ρ(b, c) ≤ ρ(b, d). therefore ρ(a, c)+ρ(c, a) ≤ ρ(a, d)+ρ(d, a). by virtue of the conditions ωi, we then have ρ(a, d) + ρ(d, a) < ρ(a, c) + ρ(c, a), a contradiction. � now, according to [52], consider the following two classes of meir-keeler type assumptions. conditions mk1: 1. for any two points x, y ∈ x there exist an upper bound and a lower bound; c© agt, upv, 2017 appl. gen. topol. 18, no. 2 323 m. m. choban and v. berinde 2 (meir-keeler monotone contraction condition). there exists a function δ : (0, +∞) −→ (0, +∞) such that from r > 0, x, y ∈ x, d(x, y) < r + δ(r) and x � y it follows that d(x, y) < r; 3. if x, y ∈ xm, x �l y, then λf(x) �l λf(y). conditions mk2: 1. for any two points x, y ∈ x there exist an upper bound and a lower bound; 2 (meir-keeler monotone contraction condition). there exists a function δ : (0, +∞) −→ (0, +∞) such that from r > 0, x, y ∈ x, d(x, y) < r + δ(r) and x � y it follows that d(x, y) < r; 3. if x, y ∈ xm, x �l y, then λf(y) �l λf(x). theorem 4.2. let (x, d) be an h-distance space and let a ∈ xm be a multidimensional fixed point of the operator of f : xm → x. then in any of the conditions mki, i ∈ {1, 2}, the operator f has a unique multidimensional fixed point. proof. obviously, in (xm, �l), for any two points x, y ∈ x m, there exist an upper bound and a lower bound. let ρ = dm. in this case for any two points x, y ∈ xm, from the condition ρ(x, y) < r + δ(r) and x � y, it follows that ρ(x, y) < r. assume that b ∈ xm is a multidimensional fixed point of the operator of f and that b 6= a. case 1. the points a and b are comparable. assume that a �l b and ρ(a, b) = r > 0. since ρ(a, b) < r + δ(r), we have r = ρ(a, b) = ρ(λf(a), λf(b)) < r, a contradiction. case 2. the points a and b are not comparable. we put r = inf{max{ρ(a, c), ρ(b, c)} : c ∈ xm, a �l c, b �l c}. we claim that r = 0. assume that r > 0. then δ(r) > 0 and there exists c such that max{ρ(a, c), ρ(b, c)} < r + δ(r), a �l c, b �l c. we put e = λf(λf(c)). then a �l e and b �l e. since λf(λf(a)) = a and λf(λf(b)) = b, we have max{ρ(a, e), ρ(b, e)} < r, a contradiction. thus r = 0. for each n ∈ n there exists a point cn ∈ x m such that a �l cn, b �l cn and max{ρ(a, cn), ρ(b, cn)} < 2 −n. we can construct a sequence {cn : n ∈ n} for which a = limn→∞ cn and b = limn→∞ cn, a contradiction. � in the particular case m = 2, the following theorems were proved in [21]. their proofs in the general case are similar and we omit them. theorem 4.3. let (x, d, �) be an ordered metric space, m ∈ n, f : xm → x be an operator. suppose that: a) there exists a function δ : (0, +∞) −→ (0, +∞) such that from r > 0, x, y ∈ xm, dm(x, y) < r+δ(r) and x � y it follows that dm(λf(x), λf(y)) < r; b) for any two points x, y ∈ x there exists an upper bound and a lower bound. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 324 multiple fixed point theorems for contractive and meir-keeler type mappings suppose also that one of the following sets of conditions is satisfied: 1. (x, d, �) is monotonically complete; from x, y ∈ xm and x �l y it follows that λf(x) �l λf(y); there exists a ∈ x m such that a �l λf(a). 2. (x, d, �) is complete; from x, y ∈ xm and x �l y it follows that λf(y) �l λf(x); there exists a ∈ x m such that a �l λf(a) or f(a) �l a. then there exists a unique multidimensional fixed point of the operator of f. theorem 4.4. let (x, d, �) be an ordered metric space, m ∈ n, f : xm → x be an operator. suppose that: a) there exists a function δ : (0, +∞) −→ (0, +∞) such that from r > 0, x, y ∈ xm, d̄m(x, y) < r+δ(r) and x � y it follows that d̄m(λf(x), λf(y)) < r; b) for any two points x, y ∈ x, there exist an upper bound and a lower bound; for any i ∈ {1, 2, . . . , m}, the mapping λi is a surjection or, more generally, |∪{λ−1i (j) : 1 ≤ j ≤ m}| = m, for each i ∈ {1, 2, . . . , m}. suppose also that one of the following sets of conditions is satisfied: 1. (x, d, �) is monotonically complete; from x, y ∈ xm and x �l y it follows that λf(x) �l λf(y); there exists a ∈ x m such that a �l λf(a). 2. (x, d, �) is complete; from x, y ∈ xm and x �l y it follows that λf(y) �l λf(x); there exists a ∈ x m such that a �l λf(a) or f(a) �l a. then there exists a unique multidimensional fixed point of the operator of f. 5. some particular cases and a generic application of multiple fixed points if we take concrete values of m ∈ n and consider various particular functions λ = {λi : {1, ..., m} → {1, ..., m} : 1 ≤ i ≤ m} then, most of the concepts of coupled, triple, quadruple,..., multiple fixed points existing in literature are obtained as particular cases of the concept of multiple fixed point considered in [32] and the present paper. for example, if m = 2, λ1(1) = 1, λ1(2) = 2; λ2(1) = 2, λ2(2) = 1, we obtain the concept of coupled fixed point studied in [37] and in various subsequent papers. if m = 3, λ1(1) = 1, λ1(2) = 2, λ1(3) = 3; λ2(1) = 2, λ2(2) = 1, λ2(3) = 2; λ3(1) = 3, λ3(2) = 2, λ3(3) = 1, then the concept of multiple fixed point studied in the present paper reduces to that of triple fixed point, first introduced in [18] and intensively studied in many other research works emerging from it. we note that, as pointed out in [77], the notion of tripled fixed point due to berinde and borcut [18] is different from the one defined by samet and vetro [74] for n = 3, since in the case of ordered metric spaces, in order to keep the c© agt, upv, 2017 appl. gen. topol. 18, no. 2 325 m. m. choban and v. berinde mixed monotone property working, it is necessary to take λ2(3) = 2 and not λ2(3) = 3. it is also important to mention here that some cases of multidimensional coincidence point results (that extend multiple fixed point theorems) are not compatible with the mixed monotone property (see [7]). for other concepts of multiple fixed points considered in literature the condition ”λi is a surjection, for each i ≤ m” is no more valid, see for example [18] and the research papers emerging from it, while the second condition, | ∪ {λ−1i (j) : 1 ≤ j ≤ m}| = m, for each i ≤ m, is satisfied. finally, we point out the fact that our approach in [32] and in this paper is based on the idea to obtain general multiple fixed point theorems by reducing this problem to a unidimensional fixed point problem and by simultaneously working in a more general and very reliable setting, i.e., that of distance space. many other related and relevant results could be obtained in the same way, by reducing the multidimensional fixed point problem to many other independent unidimensional fixed point principles, like the ones established in [5], [11], [12], [13], [15], [16], [19], [21], [23] etc. we end the paper by indicating an interesting generic application of multiple fixed points in game theory. fix an orderable distance space (x, d, �) and a positive integer number m ≥ 2. we put nm = {1, 2, ..., m}. for l ⊆ nm, we introduce on x m the ordering �l: (x1, ..., xm) �l (y1, ..., ym) iff xi � yi for i ∈ l and yj � xj for j /∈ l. denote by λ = (λ1, ..., λm) a collection of mappings {λi : nm −→ nm : i ≤ m}. let f : xm −→ x be an operator. the operator f and the mappings λ generate the operator λf : xm −→ xm, where λf(x1, ...., xm) = (y1, ..., ym) and yi = f(xλi(1), ..., xλi(m)) for each point (x1, ..., xm) ∈ x m and any index i ≤ m. assume now that nm is the set of players and i ∈ nm is the symbol of the ith player. in this case we say that: x is the space of the positions (decisions) of the players; d(x, y) is the measure of the non-convenience of the position x relatively to the position y; ordering ≤ is the relation of domination of positions; a point x = (x1, x2, ..., xm) ∈ x m is a selection of positions, where xi is the position of the player i; the operator f is the operator of correction of the positions. every selection of positions x = (x1, x2, ..., xm) ∈ x m determines the selection of positions y = (y1, y2, ..., ym) = λf(x) ∈ x m. for any player i the number d(xi, yi) is the measure of the non-convenience of the position xi relatively to the position yi for the i th player. one considers that the selection of the positions x = (x1, x2, ..., xm) ∈ x m is optimal if d(xi, yi) is minimal for each i. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 326 multiple fixed point theorems for contractive and meir-keeler type mappings in particular, if λf(x) = x, then the selection of positions x is optimal. one can find distinct concrete examples of the above general model in [54], [51], [61], [62]. acknowledgements. this second author acknowledges the support provided by the deanship of scientific research at king fahd university of petroleum and minerals for funding this work through the projects in151014 and in141047. references [1] r. agarwal, e. karapinar and a.-f. roldán-lópez-de-hierro, some remarks on ‘multidimensional fixed point theorems for isotone mappings in partially ordered metric spaces’, fixed point theory appl. 2014, 2014:245, 13 pp. [2] r. agarwal, e. karapinar and a.-f. roldán-lópez-de-hierro, fixed point theorems in quasi-metric spaces and applications to multidimensional fixed point theorems on gmetric spaces, j. nonlinear convex anal. 16 (2015), no. 9, 1787–1816. 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received 31 may 2017 – accepted 10 september 2017 http://dx.doi.org/10.4995/agt.2018.7721 i. protasov and k. protasova • for any α, β ∈ p , there exists γ ∈ p such that, for every x ∈ x, b(b(x, α), β) ⊆ b(x, γ); • for any x, y ∈ x, there exists α ∈ p such that y ∈ b(x, α). we note that a ballean can be considered as an asymptotic counterpart of a uniform space, and could be defined [7] in terms of entourages of the diagonal ∆x in x ×x. in this case a ballean is called a coarse structure. for categorical look at the balleans and coarse structures as ”two faces of the same coin” see [2]. let b = (x, p, b), b′ = (x′, p ′, b′) be balleans. a mapping f : x → x′ is called coarse if, for every α ∈ p , there exists α′ ∈ p ′ such that, for every x ∈ x, f(b(x, α)) ⊆ b′(f(x), α′). a bijection f : x → x′ is called an asymorphism between b and b ′ if f and f−1 are coarse. in this case b and b ′ are called asymorphic. let b = (x, p, b) be a ballean. each subset y of x defines a subballean by = (y, p, by ), where by (y, α) = y ∩ b(y, α). a subset y of x is called large if x = b(y, α), for α ∈ p . two balleans b and b′ with supports x and x′ are called coarsely equivalent if there exist large subsets y ⊆ x and y ′ ⊆ x′ such that the subballeans by and b ′ y ′ are asymorphic. every infinite group g can be considered as the ballean (g, fg, b), where fg is the family of all finite subsets of g, b(g, f) = fg ⋃ {g}. we note that finitely generated groups are finitary coarsely equivalent if and only if g and h are quasi-isometric [3, chapter 4]. a classification of countable locally finite groups (each finite subset generates finite subgroup) up to asymorphisms is obtained in [4] (see also [5, p. 103]). two countable locally finite groups g1 and g2 are asymorphic if and only if the following conditions hold: (i) for every finite subgroup f ⊂ g1, there exists a finite subgroup h of g2 such that |f | is a divisor of |h|; (ii) for every finite subgroup h of g2, there exists a finite subgroup f of g1 such that |f | is a divisor of |f |. it follows that there are continuum many distinct types of countable locally finite groups and each group is asymorphic to some direct sum of finite cyclic groups. the following coarse classification of countable abelian groups is obtained in [1]. two countable abelian groups are coarsely equivalent if and only if the torsion-free ranks of g and h coincide and g and h are either both finitely generated or infinitely generated. in particular, any two countable torsion abelian groups are coarsely equivalent. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 86 counting coarse subsets of a countable group given a group g, we consider each non-empty subsets as a subballean of g and say that a class of all pairwise coarsely equivalent subsets is a coarse subset of g. for a countable group g, we prove that there as many coarse subsets of g as possible by the cardinal arithmetic. theorem 1.1. for a countable group g, there are 2ω coarse subsets of g. every countable group g contains either countable finitely generated subgroup or countable locally finite subgroup, so we split the proof into corresponding cases. 2. proof: finitely generated case 2.1. we take a finite system s, s = s−1 of generators of g and consider the cayley graph γ with the set of vertices g and the set of edges {{g, h} : gh−1 ∈ s, g 6= h}. we denote by ρ the path metric on γ and choose a geodesic ray v = {vn : n ∈ ω}, v0 is the identity of g, ρ(vn, vm) = |n − m|. then the subballean of g with the support v is asymorphic to the metric ballean (n, n ⋃ {0}, b), where b(x, r) = {y ∈ n : d(x, y) ≤ r}, d(x, y) = |x − y|. thus, it suffices to find a family f, |f| = 2ω of pairwise coarsely non-equivalent subsets of n. 2.2. we choose a sequence (in)n∈ω of intervals of n, in = [an, bn], bn < an+1 such that (1) bn − an > n an. then we take an almost disjoint family a of infinite subsets of ω such that |a| = 2ω. recall that a is almost disjoint if |w ⋂ w ′| < ω for all distinct w, w ′ ∈ a. for each w ∈ a, we denote iw = ⋃ {in : n ∈ w}. to show that f = {iw : w ∈ a} is the desired family of subsets of n, we take distinct w, w ′ ∈ a and assume that iw , iw ′ are coarsely equivalent. then there exist large subsets x, x′ of iw , iw ′ , and an asymorphism f : x −→ x ′. we choose r ∈ n such that iw ⊆ b(x, r), iw ′ ⊆ b(x ′, r) and note that if an interval i of length 2r is contained in iw then i must contain at least one point of x, and the same holds for the pair iw ′ , x ′. since f is an asymorphism, we can take t ∈ n such that, for all x ∈ x, x′ ∈ x′, (2) f(bx(x, 2r + 2) ⊆ bx′(f(x), t); (3) f−1(bx′(x ′, 2r + 2) ⊆ bx(f −1(x′), t). we use (1) to choose m ∈ w\w ′, m > max(w ⋂ w ′) such that (4) bm − am > 2ram; c© agt, upv, 2018 appl. gen. topol. 19, no. 1 87 i. protasov and k. protasova (5) bm − am > 2t. 2.3. we denote z = x ⋂ [am, bm] and enumerate z in increasing order z = {z0, . . . , zk}. then d(zi, zi+1) ≤ 2r + 2 because otherwise the interval [zi + 1, zi+1 − 1] of length 2r has no points of x. if f(z0) < am then, by (2) and (5), f(z) ⊆ [1, am − 1]. on the other hand, k ≥ (bm − am)/2r − 1 and, by (4), (bm − am)/2r > am. hence, k > am − 1 contradicting f(z) ⊆ [1, am − 1] because f is a bijection. if f(z0) > bm then we take s ∈ w ′ such that f(z0) ∈ is. since m > max(w ∧ w ′) and s > m, we have s ∈ w ′ \ w , so we can repeat above argument for f−1 and is in place of f and im with usage (3) instead of (2). 3. proof: locally finite case 3.1. let g be an arbitrary countable group and let x, a be infinite subsets of g. suppose that there exist an infinite subset y of x, a partition a = b ⋃ c and k, l ∈ n, k < l such that (6) there exists h ∈ fg such that, for every y ∈ y , |bx(y, h)| ≥ k; (7) for every f ∈ fg, there exists y ′ ∈ fg such that, for every y ∈ y \y ′, |bx(y, h)| ≥ l; (8) there exists k ∈ fg such that, for every b ∈ b, |ba(b, k)| > l; (9) for every f ∈ fg, there exists c ′ ∈ fg such that, for every c ∈ c\c ′, |ba(c, f)| < k. then x and a are not asymorphic. we suppose the contrary and let f : x −→ a be an asymorphism. we take an infinite subset i of y such that either f(i) ⊂ c or f(i) ⊂ b. assume that f(i) ⊂ c and choose f ∈ fg such that, for every x ∈ x, f(bx(x, h)) ⊆ ba(f(x), f). for this f , we use (9) to choose corresponding c′. we take y ∈ i such that f(y) ∈ c \ c′. by (6), f(b(y, h)) ≥ k. by (9), ba(f(y), f) < k and we get a contradiction because f is a bijection. if f(i) ⊂ b then, by (8), ba(b, k) > l for every b ∈ f(i). since f −1 is coarse, there is f ∈ fg such that, for every a ∈ a f−1(ba(a), k) ⊆ bx(f −1(a), f). for this f , we choose y ′ satisfying (7) and get a contradiction. 3.2. now we assume that g is locally finite and show a plan how to choose the desired family f, |f| = 2ω of pairwise coarsely non-equivalent subsets of g. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 88 counting coarse subsets of a countable group we construct some special sequence (yn)n∈ω of pairwise disjoint subsets of g. then we take a family a of almost disjoint infinite subset of ω, |f| = 2ω, denote (10) xw = ⋃ {yn : n ∈ w}, w ∈ a, and get f as {xw : w ∈ a}. 3.3. we represent g as the union of an increasing chain {fn : n ∈ ω} of finite subgroups such that (11) |fn+1| > |fn| 2. then we choose a double sequence (gnm)n,m∈ω of elements of g such that (12) fnfmgnm ⋂ fifjgij = ∅ for all distinct (n, m), (i, j) from ω ×ω, and put yn = ⋃ {fmgnm : m ∈ ω}. 3.4. we take distinct w, w ′ ∈ a and prove that xw and xw ′ (see (10)) are not coarsely equivalent. we suppose the contrary and choose large asymorphic subsets zw and zw ′ of xw and xw ′. then we take t ∈ ω such that xw ⊆ ftzw , xw ′ ⊆ ftzw ′. if n > t and either fngnm ⊂ xw or fngnm ⊂ xw ′ then (13) |fngnm ⋂ zw | ≥ |fn| |ft| , |fngnm ⋂ zw ′| ≥ |fn| |ft| . to apply 3.1, we choose s ∈ w \ w ′, s > t and denote x = zw , y = ys ⋂ zw , a = zw ′, b = ⋃ {yi : i ∈ w ′, i > s}, c = ⋃ {yi : i ∈ w ′, i < s}, k = |fs| |ft| , l = |fs|. by (13) with s = n, we get (6). by (12) with s = n, we get (7). if n > s then |fn|/|ft| > |fn|/|fs|. by (11), |fn|/|fs| > |fs|, so |fn|/|ft| > |fs| and, by (13), we have (8). if n < s then |fn| < |fs|/|ft| and, by (12), we get (9). 4. comments a subset a of an infinite group g is called • thick if, for every f ∈ fg , there exists g ∈ a such that fg ⊂ a; • small if l \ a is large for every large subset l of g; c© agt, upv, 2018 appl. gen. topol. 19, no. 1 89 i. protasov and k. protasova • thin if, for every f ∈ fg , there exists h ∈ fg such that ba(g, f) = {g} for each g ∈ a \ h. a subset a is thick if and only if l ⋂ a 6= ∅ for every large subset l of g. for a countable group g, in the proof of theorem 1.1, we construct 2ω pairwise coarsely non-equivalent thick subsets of g. every large subset l of g is coarsely equivalent to g, so any two large subsets of g are coarsely equivalent. if g is countable then any two thin subset s, t of g are asymorphic: any bijection f : s −→ t is an asymorphism. every thin subset is small. but a small subset s of g could be asymorphic to g: we take a group g containing a subgroup s isomorphic to g such that the index of s in g is infinite. references [1] t. banakh, j. higes and m. zarichnyi, the coarse classification of countable abelian groups, trans. amer. math. soc. 362 (2010), 4755–4780. [2] d. dikranjan and n. zava, some categorical aspects of coarse spaces and ballean, topology appl. 225 (2017), 164–194. [3] p. de la harpe, topics in geometric group theory, university chicago press, 2000. [4] i. v. protasov, morphisms of ball structures of groups and graphs, ukr. mat. zh. 53 (2002), 847–855. [5] i. protasov and t. banakh, ball structures and colorings of groups and graphs, math. stud. monogr. ser., vol. 11, vntl, lviv, 2003. [6] i. protasov and m. zarichnyi, general asymptology, math. stud. monogr. ser., vol. 12, vntl, lviv, 2007. [7] j. roe, lectures on coarse geometry, amer. math. soc., providence, r.i, 2003. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 90 @ appl. gen. topol. 18, no. 1 (2017), 131-141 doi:10.4995/agt.2017.6644 © agt, upv, 2017 a note on weakly pseudocompact locales themba dube department of mathematical sciences, university of south africa, 0003 pretoria, south africa (dubeta@unisa.ac.za) communicated by s. gaŕıa-ferreira abstract we revisit weak pseudocompactness in pointfree topology, and show that a locale is weakly pseudocompact if and only if it is gδ-dense in some compactification. this localic approach (in contrast with the earlier frame-theoretic one) enables us to show that finite localic products of locales whose non-void gδ-sublocales are spatial inherit weak pseudocompactness from the factors. we also show that if a locale is weakly pseudocompact and its gδ-sublocales are complemented then it is baire. 2010 msc: primary: 06d22; secondary: 54e17. keywords: frame; locale; sublocale; gδ-sublocale; weakly pseudocompact; binary coproduct. 1. introduction and motivation generalizing pseudocompactness in completely regular hausdorff spaces, garćıa-ferreira and garćıa-maynez [11] define a completely regular hausdorff space to be weakly pseudocompact in case it is gδ-dense in some compactification. this generalizes pseudocompactness since pseudocompact spaces (throughout, by “space” we mean a completely regular hausdorff space) are precisely the spaces x that are gδ-dense in βx. the reader will recall that a subspace a of a topological space x is gδ-dense in x if it meets every nonempty gδ-subspace of x. ball and walters-wayland showed in [2] that this concept is expressible frame-theoretically in the following way. a subspace a of x is gδ-dense if and only if the frame homomorphism received 27 september 2016 – accepted 06 january 2017 http://dx.doi.org/10.4995/agt.2017.6644 t. dube ox → oa induced by the subspace inclusion a ↪→ x is coz-codense. it was on the basis of this that in [10] walters-wayland and i defined a completely regular frame l to be weakly pseudocompact if there is a compactification h: m → l of l such that the homomorphism h is coz-codense. assuming the boolean ultrafilter theorem, one quickly observes that this definition is “conservative”, which is to say a space is weakly pseudocompact precisely when the frame of its open sets is weakly pseudocompact. now a natural question to ask is whether weak pseudocompactness in the pointfree setting is equivalent to a condition lifted verbatim from the spatial definition, subject to replacing “subspace” with “sublocale”. that is, is l weakly pseudocompact if and only if it is gδ-dense in some compactification? of course gδ-denseness in locales is to be taken (exactly as in spaces) to mean having non-void intersection with every non-void sublocale which is a countable intersection of open sublocales. before we leap to conclusions and say that should clearly be so in view of the result of ball and walters-wayland mentioned above, we should reflect on the fact that their result is about subspaces, and spaces can have more sublocales than subspaces. it is the main aim of this note to answer the question in the affirmative. this is how we go about doing so. we first show that a sublocale of a compact regular locale is gδ-dense precisely when it has non-void intersection with every non-void zero-set sublocale (lemma 3.2). this then shows that a sublocale is gδ-dense if and only if the associated frame surjection from the ambient locale to the sublocale is coz-codense. thence it is easy to deduce that a locale is weakly pseudocompact if and only if it is gδ-dense in some compactification (proposition 3.4). this done, we then prove the main result in the paper (theorem 3.7). we carry out the proof in frm, and thus deal with coproducts of frames. in remark 3.8 we explain why this result does not follow from the result that the topological product of weakly pseudocompact spaces is weakly pseudocompact. in [11], the authors show that every weakly pseudocompact space is baire. it will be recalled that a space is baire if the intersection of countably many dense open sets is dense. in locales the baire property cannot be defined by decreeing that countable intersections of open dense sublocales be dense, because all intersections of dense sublocales are dense. this led isbell [14] to define baire locales in terms of first and second category sublocales. we will recall the definition at the right place. in proposition 4.3 we show that a weakly pseudocompact locale is baire if its gδ-sublocales are complemented. this rather extraneous condition seems to be necessitated by the fact that the sublocale lattice is not necessarily boolean. 2. preliminaries 2.1. frames, very briefly. we shall use the terms “frame” and “locale” interchangeably. our reference for frames and locales are [16] and [17]. our notation follows these texts, to a large extent. one little deviation is that we write ox for the frame of open sets of a space x. we denote by h∗ the right © agt, upv, 2017 appl. gen. topol. 18, no. 1 132 a note on weakly pseudocompact locales adjoint of a frame homomorphism h. the completely below relation is denoted by ≺≺. all frames in this paper are completely regular. by a point of a frame l we mean an element p such that p < 1 and x∧y ≤ p implies x ≤ p or y ≤ p. we write pt(l) for the set of points of l. a frame is spatial if it as enough points, in the sense that every element is the meet of points above it. compact regular frames have enough points, modulo the boolean ultrafilter theorem, which we will assume throughout. an element of a frame is dense if it has nonzero meet with every nonzero element. this is precisely when its pseudocomplement is 0. an element a of l is a cozero element if it is expressible as a = ∨ an, where an ≺≺ an+1 for every n ∈ n. the set of all cozero elements of l is denoted by coz l. it is a σ-frame, and it generates l if and only if l is completely regular. a frame homomorphism h: l → m is coz-faithful if it is one-one on coz l. this is the case precisely when it is coz-codense, meaning that for any c ∈ coz l, h(c) = 1 implies c = 1. a frame l is normal if whenever a∨ b = 1 in l, there exist u,v ∈ l such that u∧v = 0 and a∨u = 1 = b∨v. the compact completely regular coreflection of l is denoted by βl. let l(r) denote the frame of reals (see [17, chapter xiv]). as in spaces, a frame l is said to be pseudocompact if every frame homomorphism f : l(r) → l is bounded, which is to say f(p,q) = 1 for some p,q ∈ q. there are several characterizations of pseudocompact frames (see [3, 5]). a pertinent one here is that l is pseudocompact if and only if the homomorphism βl → l is cozcodense [19] . 2.2. coproducts of frames. we shall have occasion to deal only with binary coproducts of frames. we write l ⊕ m for the coproduct of l and m. the actual construction of coproducts is described in detail in chapter iv of [17]. we recall only the following. the elements a ⊕ b – which are called basic elements – generate l⊕m by joins. thus, each u ∈ l⊕m is expressible as u = ∨ {a⊕ b | (a,b) ∈ u}. the top element of l⊕m is 1l ⊕ 1m . if 0 6= a⊕ b ≤ u⊕v, then a ≤ u and b ≤ v. the coproduct injections l i // l⊕m m joo map a ∈ l to a⊕1, and b ∈ m to 1⊕b. hence, i∗(a⊕1) = a and j∗(1⊕b) = b, for every a ∈ l and every b ∈ m. if hα : lα → mα is a frame homomorphism for α = 1, 2, then there is an induced frame homomorphism h1⊕h2 : l1⊕l2 → m1 ⊕m2 such that (h1 ⊕h2) (∨ k (ak ⊕ bk) ) = ∨ k ( h1(ak) ⊕h2(bk) ) . © agt, upv, 2017 appl. gen. topol. 18, no. 1 133 t. dube its right adjoint maps as follows (see [6, lemma 2]): (h1 ⊕h2)∗(u) = ∨ {(h1)∗(a) ⊕ (h2)∗(b) | a⊕ b ≤ u}. 2.3. sublocales. we denote the lattice of all sublocales of a locale l by s(l). all joins and meets of sublocales will be computed in this lattice. recall that the meet is simply set-theoretic intersection. the join is not set-theoretic union, in general. we shall not need to know how joins are calculated in s(l). the smallest sublocale of l is denoted by o = {1}, and is called the void sublocale. we say a sublocale s meets a sublocale t if s ∩t 6= o. otherwise, s misses t . the lattice s(l) is a coframe, which is to say for any s ∈ s(l) and any family {tα} of sublocales, the distributive law below holds: s ∨ ∧ α tα = ∧ α (s ∨tα). our notation for open and closed sublocales is that of [17]. thus, for any a ∈ l, o(a) and c(a) denote the open and closed sublocales of l determined by a, respectively. we shall freely use properties of these sublocales, such as o(0) = c(1) = o and o(1) = c(0) = l, and c(a) ⊆ o(b) ⇐⇒ a∨ b = 1. to say a sublocale of l is complemented means that it has a complement in s(l). the sublocales o(a) and c(a) are complements of each other. a complemented sublocale is linear, which means that if s is a complemented sublocale of l, then s ∧ ∨ α tα = ∨ α (s ∧tα), for all families {tα} of sublocales of l. the closure of a sublocale s will be denoted by s. we remind the reader that s = ↑ (∧ s ) . a sublocale is dense if its closure is the whole locale. the smallest dense sublocale of l will be denoted by bl. a sublocale s of l is nowhere dense if s ∧ bl = o. the complement of a complemented nowhere dense sublocale is dense. the closure of a nowhere dense sublocale is nowhere dense (see [18, p. 278]). if h: m → l is a compactification of l, we shall identify l with the sublocale h∗[l] of m, and, by abuse of language, say l is a sublocale of m. 3. a localic approach to weak pseudocompactness extending terminology from spaces, we say a sublocale of a frame l is a gδ-sublocale if it is a countable intersection of open sublocales. isbell [13] calls such sublocales “oδ-sublocales” for reasons he explains. in particular, open sublocales are gδ-sublocales, and every zero-set sublocale, that is, a sublocale © agt, upv, 2017 appl. gen. topol. 18, no. 1 134 a note on weakly pseudocompact locales of the form c(c) for some c ∈ coz l, is a gδ-sublocale [12, remark 2.6]. we should caution the reader that the calculation in [12] is carried out in s(l)op, so the joins there are intersections in s(l). as in spaces, we say a sublocale of l is gδ-dense in l if it meets every non-void gδ-sublocale of l. as mentioned earlier, it is for this reason that the following definition was formulated in [10]. definition 3.1. a frame l is weakly pseudocompact if there is a compactification γ : γl → l of l such that the homomorphism γ is coz-codense. the equivalence of gδ-density of a subspace a ⊆ x to coz-codensity of the homomorphism ox → oa makes it clear that this definition is conservative. less clear is that this frame-theoretic definition agrees with the one that would be adopted verbatim from the spatial one as described in the introduction. to show that they agree, we need a lemma which, although not generalizing the result of ball and walters-wayland, brings to the fore what cannot be deduced from their result. let us expatiate a little on this. the result we are alluding to does not generalize that of ball and walters-wayland in that we do not state it for sublocales of any locale, but rather for sublocales of a compact regular locale. on the other hand, it does not follow from that of ball and walterswayland because even a compact hausdorff space (when viewed as a locale) can have far more sublocales than it has subspaces. recall that if s ∈s(l), then the corresponding frame surjection is νs : l → s defined by νs(a) = ∧ {s ∈ s | s ≥ a}. observe that, for any a ∈ l and s ∈s(l), s ⊆ o(a) ⇐⇒ νs(a) = 1. lemma 3.2. the following are equivalent for a sublocale s of a compact regular locale l. (1) s is gδ-dense in l. (2) s meets every non-void zero-set sublocale of l. (3) νs : l → s is coz-codense. proof. (1) ⇒ (2): this follows from the fact that every zero-set sublocale is a gδ-sublocale. (2) ⇒ (1): we show first that every non-void gδ-sublocale of l is above some non-void zero-set sublocale. so consider a set {an | n ∈ n}⊆ l such that the gδ-sublocale g = ∞⋂ n=1 o(an) is non-void. since l is compact regular, [17, corollary vii 7.3] tells us that g is spatial. since points of a sublocale are precisely the points of the locale belonging to the sublocale, there is a point p ∈ pt(l) such that p ∈ g. consequently, p ∈ o(an) for each n, so that c(p) = {p, 1}⊆ o(an) and hence p∨an = 1. since compact regular locales are normal, for each n there exists cn ∈ coz l such that cn ≤ p and cn ∨ an = 1 (see [1, corollary 8.3.2]). put c = ∨ cn, so that c ∈ coz l, c ≤ p < 1, and © agt, upv, 2017 appl. gen. topol. 18, no. 1 135 t. dube c∨an = 1 for every n. consequently, c(c) ⊆ o(an) for every n, and so o 6= c(c) ⊆ ∞⋂ n=1 o(an). thus, if s meets every non-void zero-set sublocale, then it meets every non-void gδ-sublocale, which shows that (2) implies (1). (2) ⇔ (3): this follows from the fact that, for any c ∈ coz l, s ∩ c(c) = o if and only if s ⊆ o(c) if and only if νs(c) = 1. � remark 3.3. we can actually add another equivalent condition. as in spaces, define the δ-closure of a sublocale s of l to be the sublocale s δ = ⋂ {o(a) | a ∈ coz l and s ⊆ o(a)}. then s is gδ-dense in l if and only if s δ = l. in view of lemma 3.2 we immediately have the following characterization. proposition 3.4. a locale is weakly pseudocompact if and only if it is gδ-dense in some compactification. as an illustration of the usefulness of this characterization, we shall prove that certain binary products of weakly pseudocompact locales are weakly pseudocompact. in spaces, every product of weakly pseudocompact spaces is weakly pseudocompact [11]. the proof relies, among other things, on the fact that the lattice of subspaces of a space is a boolean algebra, so that, for instance, if u is an open subspace of a space x, then for any x ∈ x we have x ∈ u or x ∈ x r u. in locales this latter result of course fails. indeed, if 0 < a ∈ l is not dense, then 0 /∈ o(a) and 0 /∈ c(a), even though c(a) is the complement of o(a). points in a locale behave differently though, as shown in [15, lemma 2.1]. we recall this result, but paraphrase it as follows. lemma 3.5. for any a ∈ l, pt(l) ⊆ o(a) ∪ c(a). let us recall another result; one that describes points in a coproduct in terms of points in the summands. it is taken from [8, lemma 4.2] where it is proved for t1-locales, and hence is valid for completely regular locales. lemma 3.6. pt(l⊕m) = {(p⊕ 1) ∨ (1 ⊕q) | p ∈ pt(l) and q ∈ pt(m)}. in the result that follows we shall place a condition on the locales l and m that all their non-void gδ-sublocales be spatial. this is strictly weaker than requiring all sublocales to be spatial. indeed, if a locale is compact regular, then all its non-void gδ-sublocales are spatial [17, corollary vii 7.3], but of course not all its sublocales need be spatial. as shown in [15], requiring all sublocales of l to be spatial is equivalent to requiring that s(l)op be spatial. theorem 3.7. let l and m be frames all of whose non-void gδ-sublocales are spatial. if l and m are weakly pseudocompact, then l ⊕ m is weakly pseudocompact. © agt, upv, 2017 appl. gen. topol. 18, no. 1 136 a note on weakly pseudocompact locales proof. let γl and γm be compactifications of l and m, respectively, such that l is gδ-dense in γl, and m is gδ-dense in γm. let γl : γl → l and γm : γm → m be the corresponding frame surjections. for brevity, write ρl and ρm for their right adjoints, so that ρl : l → γl is the inclusion map l ↪→ γl, and similarly for ρm . by the result of banaschewski and vermeulen recalled in the preliminaries, for any u ∈ l⊕m, (γl ⊕γm )∗(u) = ∨ {ρl(a) ⊕ρm (b) | a⊕ b ≤ u} = ∨ {a⊕ b | a⊕ b ≤ u} = u, whence (γl⊕γm )∗[l⊕m] = l⊕m, showing that l⊕m is a sublocale of the compact frame γl⊕γm. it is a dense sublocale because l and m are dense sublocales of γl and γm, respectively. thus, γl⊕γm is a compactification of l⊕m. now consider a non-void gδ-sublocale g of γl ⊕ γm. since the elements a⊕b, for a ∈ γl and b ∈ γm, generate γl⊕γm, and γl⊕γm is spatial, we may assume, without loss of generality, that g is of the form g = ∞⋂ n=1 o(an ⊕ bn) 6= o for some sequences (an) and (bn) of non-zero elements of γl and γm, respectively. by [17, corollary vii 7.3], g is spatial, and hence contains a point of γl ⊕ γm. thus, by lemma 3.6, there exist p ∈ pt(γl) and q ∈ pt(γm) such that (†) (p⊕ 1) ∨ (1 ⊕q) ∈ ∞⋂ n=1 o(an ⊕ bn). we claim that p ∈ ⋂ no(an). if not, there in an index m such that p /∈ o(am). then, by lemma 3.5, p ∈ c(am), which implies am ≤ p, whence am ⊕ bm ≤ p⊕ 1 ≤ (p⊕ 1) ∨ (1 ⊕q), and therefore, by (†), (p⊕1)∨(1⊕q) ∈ c(am⊕bm)∩o(am⊕bm); which is false. thus, ⋂ no(an) 6= o. similarly, ⋂ no(bn) 6= o. since l and m are gδ-dense in γl and γm, respectively, l∩ ∞⋂ n=1 o(an) 6= o and m ∩ ∞⋂ n=1 o(bn) 6= o. since l∩o(an) is an open sublocale of l, and since l∩ ∞⋂ n=1 o(an) = ∞⋂ n=1 ( l∩o(an) ) , it follows that l∩ ⋂ no(an) is a (non-void) gδ-sublocale of l. since non-void gδ-sublocales of l are spatial, by hypothesis, there is an s ∈ pt(γl) such that s ∈ l∩ ⋂ no(an). similarly, there is a t ∈ pt(γm) such that t ∈ m ∩ ⋂ no(bn). we claim that the point (s⊕1)∨(1⊕t) of γl⊕γm belongs to (l⊕m)∩g. to show that (s⊕ 1) ∨ (1 ⊕ t) ∈ l⊕m, we need to exercise care because joins in © agt, upv, 2017 appl. gen. topol. 18, no. 1 137 t. dube l⊕m are calculated differently from joins in γl⊕γm. so let us denote binary join in l⊕m by t. now, (s⊕ 1) t (1 ⊕ t) is a point in l⊕m by lemma 3.6, and is therefore a point in γl⊕γm. but (s⊕1)∨(1⊕t) ≤ (s⊕1)t(1⊕t); so the two are equal by maximality of points in regular frames. suppose, by way of contradiction, that (s⊕ 1) ∨ (1 ⊕ t) /∈g. then there is an index k such that (s⊕1)∨(1⊕t) /∈ o(ak⊕bk). then, by lemma 3.5, (s⊕1)∨(1⊕t) ∈ c(ak⊕bk), which implies (ak ⊕ 1) ∧ (1 ⊕ bk) = (ak ⊕ bk) ≤ (s⊕ 1) ∨ (1 ⊕ t), and therefore ak ⊕ 1 ≤ (s⊕ 1) ∨ (1 ⊕ t) or 1 ⊕ bk ≤ (s⊕ 1) ∨ (1 ⊕ t) because (s⊕1)∨(1⊕t) is a point in γl⊕γm. suppose the former, and consider the coproduct injection i: γl → γl⊕γm. then ak = i∗(ak ⊕ 1) ≤ i∗((s⊕ 1) ∨ (1 ⊕ t)) = s; the latter equality holding because s and i∗((s⊕ 1) ∨ (1 ⊕ t)) are points of γl with s = i∗(s⊕ 1) ≤ i∗((s⊕ 1) ∨ (1 ⊕ t)). consequently, s ∈ c(ak), which is false because s ∈ o(ak). a similar contradiction is arrived at if we assume that 1 ⊕ bk ≤ (s ⊕ 1) ∨ (1 ⊕ t). thus, (s⊕ 1) ∨ (1 ⊕ t) ∈g; and so l⊕m is gδ-dense in γl⊕γm, and is therefore weakly pseudocompact. � remark 3.8. this result is of course about the localic product of some weakly pseudocompact spaces; so one may wonder if it does not follow immediately from the spatial result. let us explain via an analogous situation. if x and y are topological spaces for which the (topological) product x × y is pseudocompact, then the frame ox ⊕oy is pseudocompact, even if ox ⊕oy is not isomorphic to o(x × y ). the reason is that o(x × y ) is isomorphic to a dense sublocale of ox ⊕ oy [17, proposition iv 5.4.1], and a locale which has dense pseudocompact sublocale is pseudocompact [9, lemma 4.3]. now, a locale which has a dense weakly pseudocompact sublocale is not necessarily weakly pseudocompact. indeed, let l be a weakly pseudocompact locale which is not pseudocompact. then λl, the lindelöf reflection of l in loc, is not weakly pseudocompact, for if it were then by [10, corollary 2.8] it would be compact, which would make l pseudocompact. so, with reference to the previous theorem, knowing that x ×y is weakly pseudocompact does not tell us anything about the weak pseudocompactness, or otherwise, of ox ⊕oy . 4. weak pseudocompactness and baireness in [14], isbell defines baire locales as follows. a sublocale s of a locale l is of first category if there are countably many nowhere dense sublocales kn, n ∈ n, such that s ≤ ∨ nkn. otherwise, it is of second category. a baire locale is one in which every non-void open sublocale is of second category. observe that since open sublocales are complemented, an open sublocale is of first category © agt, upv, 2017 appl. gen. topol. 18, no. 1 138 a note on weakly pseudocompact locales if and only if it can be written as a join of countably many nowhere dense sublocales. this agrees with the characterization of baire spaces as precisely those spaces in which every countable union of closed sets with empty interior has empty interior. we mentioned in the preliminaries that the closure of a nowhere dense sublocale is nowhere dense [18, p. 278]). plewe’s proof in [18] is in terms of sublocales. for the sake of completeness let us give a proof, but a different one carried out entirely in frm. recall from [7] that a quotient η : l → n of l is called nowhere dense if for every nonzero a ∈ l there exists a nonzero b ≤ a in l such that η(b) = 0. this agrees with the localic notion in the sense that η : l → n is nowhere dense if and only if the sublocale η∗[n] is nowhere dense in l. it is shown in [7, lemma 3.2] that η : l → n is nowhere dense if and only if η∗(0) is a dense element in l. lemma 4.1. the closure of a nowhere dense quotient is nowhere dense. proof. let η : l → n be a nowhere dense quotient of l. the closure of this quotient is ϕ: l →↑η∗(0), where ϕ is the map x 7→ x∨η∗(0). the right adjoint of ϕ is the inclusion map ↑η∗(0) ↪→ l. since the zero of ↑η∗(0) is η∗(0), we have ϕ∗(0↑η∗(0)) = η∗(0), which is a dense element in l. therefore ϕ: l → ↑η∗(0) is nowhere dense. � for use below, we also need to know that if n ⊆ l are sublocales of m with n nowhere dense in l, and l dense in m, then n is nowhere dense in m. to prove this, recall first that if g : a → b is a dense onto frame homomorphism, then g∗(b) is dense in a whenever b is dense in b. for, if g∗(b) ∧a = 0 for any a ∈ a, then 0 = g ( g∗(b) ∧a ) = b∧g(a), implying g(a) = 0 by the density of b, and hence a = 0 by the density of g. lemma 4.2. suppose that in the composite m h−→ l η−→ n of quotient maps h is dense and η is nowhere dense. then ηh: m → n is nowhere dense. proof. since η is nowhere dense, η∗(0) is dense, and hence h∗(η∗(0)) is dense as h is dense. but h∗(η∗(0)) = (ηh)∗(0), so the result follows. � now, in light of lemma 4.1, if u is a first category sublocale of l, then there are countably many closed nowhere dense sublocales kn with u ≤ ∨ nkn. recall that if l is a sublocale of m, then the closed sublocales of l are precisely the sublocales cl(a) = l∩ c(a), for a ∈ m. proposition 4.3. a weakly pseudocompact locale in which gδ-sublocales are complemented is baire. proof. let l be such a locale, and let m be a compactification of l such that l is gδ-dense in m. let u be a first category open sublocale of l. we aim to show that u = o. find countably many elements (an)n∈n in m such that each cl(an) is nowhere dense in l, and u ≤ ∞∨ n=1 cl(an). we claim that, for any n, c(an) is nowhere dense in m. if not, then there exists x ∈ m such that 1 6= © agt, upv, 2017 appl. gen. topol. 18, no. 1 139 t. dube x∗∗ ∈ c(an). since l is dense in m, bm ⊆ l, hence x∗∗ ∈ l∩c(an) = cl(an), contradicting the fact that c(an) ∩ bm = o since cl(an) is nowhere dense in m by lemma 4.2. consequently, o(an) is a dense sublocale of m because complements of complemented nowhere dense sublocales are dense. now pick an open sublocale u of m such that u = l∩u, and observe that u∩ ∞⋂ n=1 o(an) is a gδ-sublocale of m with l∩ ( u∩ ∞⋂ n=1 o(an) ) = (l∩u) ∩ ( l∩ ∞⋂ n=1 o(an) ) = u ∩ ( l∩ ∞⋂ n=1 o(an) ) = u ∧ ∞∧ n=1 ol(an) ≤ ( ∞∨ k=1 cl(ak) ) ∧ ( ∞∧ n=1 ol(an) ) = ∞∨ k=1 ( cl(ak) ∧ ∞∧ n=1 ol(an) ) since ∞∧ n=1 ol(an) is complemented ≤ ∞∨ k=1 (cl(ak) ∧ol(ak)) = o. since l is gδ-dense in m, it follows that u∩ ⋂ no(an) = o. but now ⋂ no(an) is a dense sublocale of m missing the open sublocale u, so we must have u = o, which then implies u = o. therefore l is baire. � remark 4.4. the calculation in the foregoing proof, starting with the equality l∩ ( u∩ ∞⋂ n=1 o(an) ) = (l∩u) ∩ ( l∩ ∞⋂ n=1 o(an) ) , can be used to show that if l is a gδ-dense sublocale of a baire locale, and gδ-sublocales of l are complemented, then l is also baire. acknowledgements. we are grateful to the referee for comments that have helped improve the first version of the paper, especially with regard to presentation. © agt, upv, 2017 appl. gen. topol. 18, no. 1 140 a note on weakly pseudocompact locales references [1] r. n. ball and j. walters-wayland, cand c∗-quotients in pointfree topology, dissertationes mathematicae (rozprawy mat.) vol. 412 (2002), 62 pp. [2] r. n. ball and j. walters-wayland, well-embedding and gδ-density in a pointfree setting, appl. categ. struct. 14 (2006), 351–355. [3] b. banaschewski and c. gilmour, pseudocompactness and the cozero part of a frame, comment. math. univ. carolinae 37 (1996), 579–589. [4] b. banaschewski and c. gilmour, realcompactness and the cozero part of a frame, appl. categ. struct. 9 (2001), 395–417. [5] b. banaschewski, d. holgate and m. sioen, some new characterizations of pointfree pseudocompactness, quaest. math. 36 (2013), 589–599. [6] b. banaschewski and j. vermeulen, on the completeness of localic groups, comment. math. univ. carolinae 40 (1999), 293–307. [7] t. dube, remote points and the like in pointfree topology, acta math. hungar. 123 (2009), 203–222. [8] t. dube and m. m. mugochi, localic remote points revisited, filomat 29 (2015), 111– 120. [9] t. dube, i. naidoo and c. n. ncube, isocompactness in the category of locales, appl. categ. struct. 22 (2014), 727–739. [10] t. dube and j. walters-wayland, weakly pseudocompact frames, appl. categ. struct. 16 (2008), 749–761. [11] s. garćıa-ferreira and a. garćıa-maynez, on weakly-pseudocompact spaces, houston j. math. 20 (1994), 145–159. [12] j. gutiérrez garćıa and j. picado, on the parallel between normality and extremal disconnectivity, j. pure appl. algebra 218 (2014), 784–803. [13] j. r. isbell, first steps in descriptive locale theory, trans. amer. math. soc. 327 (1991), 353–371. [14] j. r. isbell, some problems in descriptive locale theory, can. math. conf. proc. 13 (1992), 243–265. [15] s. b. niefield and k. i. rosenthal, spatial sublocales and essential primes, topology appl. 26 (1987), 263–269. [16] p. t. johnstone, stone spaces, cambridge university press, cambridge, 1982. [17] j. picado and a. pultr, frames and locales: topology without points, frontiers in mathematics, springer, basel, 2012. [18] t. plewe, higher order dissolutions and boolean coreflections of locales, j. pure appl. algebra 154 (2000), 273–293. [19] j. walters-wayland, completeness and nearly fine uniform frames, phd thesis, university catholique de louvain (1995). © agt, upv, 2017 appl. gen. topol. 18, no. 1 141 03.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 29 43 some properties of o-bounded and strictlyo-bounded groupsc. hern�andez, d. robbie, m. tkachenko�abstract. we continue the study of (strictly) o-boundedtopological groups initiated by the �rst listed author and solvetwo problems posed earlier. it is shown here that the product ofa comfort-like topological group by a (strictly) o-bounded groupis (strictly) o-bounded. some non-trivial examples of strictly o-bounded free topological groups are given. we also show thato-boundedness is not productive, and strict o-boundedness can-not be characterized by means of second countable continuoushomomorphic images.2000 ams classi�cation: primary 54h11, 22a05; secondary 22d05, 54c50keywords: o-bounded group, strictly o-bounded group, @0-bounded group,of-undetermined group, comfort-like group1. introductionthe class of �-compact topological groups has many nice properties. forexample, every �-compact group is countably cellular [11] and perfectly �-normal [13, 15]. the subgroups of �-compact groups inherit these properties,but clearly need not be �-compact. the notions of o-boundedness and stricto-boundedness introduced by o. okunev and m. tkachenko respectively, wereconsidered in [9]. the idea was to �nd a wider class of topological groups asclose to the class of �-compact groups as possible which is additionally closedunder taking subgroups. let us recall the corresponding de�nitions.a topological group g is called o-bounded if for every sequence fun : n 2 ngof open neighborhoods of the neutral element in g, there exists a sequenceffn : n 2 ng of �nite subsets of g such that g = sn2n fn � un. it is clearthat all �-compact groups as well as their subgroups are o-bounded. in a sense,o-bounded groups have to be small: the group r! fails to be o-bounded [9,�the research is partially supported by consejo nacional de ciencias y tecnolog��a (cona-cyt) of mexico, grant no. 400200-5-28411e, and grant for special studies program (long)the university of melbourne (science faculty). 30 c. hern�andez, d. robbie, m. tkachenkoexample 2.6]. the class of o-bounded groups has good categorical properties:all subgroups and all continuous homomorphic images of an o-bounded groupare o-bounded [9]. it was not known, however, whether this class was �nitelyproductive [9, problem 5.2]. we show in example 2.12 that there exists asecond countable o-bounded group g whose square is not o-bounded. actually,the group g �rst appeared in [9, example 6.1] in order to distinguish the classesof o-bounded and strictly o-bounded groups. however, the properties of thisgroup were not completely exhausted there. as is shown in [4], the group g isadditionally analytic, that is, g is a continuous image of a separable completemetric space.to de�ne strictly o-bounded groups, we need to describe the of-game (see[9] or [14]). suppose that g is a topological group and that two players, say iand ii, play the following game. player i chooses an open neighborhood u1 ofthe identity in g, and player ii responds choosing a �nite subset f1 of g. inthe second turn, player i chooses another neighborhood u2 of the identity ing and player ii chooses a �nite subset f2 of g. the game continues this wayuntil we have the sequences fun : n 2 ng and ffn : n 2 ng. player ii winsif g = s1n=1 fn � un. otherwise, player i wins. the group g is called strictlyo-bounded if player ii has a winning strategy in the of-game on g. it is easy tosee that �-compact groups are strictly o-bounded and every strictly o-boundedgroup is o-bounded. as we mentioned above, o-bounded groups need not bestrictly o-bounded. in addition, there are lots of strictly o-bounded groupsthat are neither �-compact nor isomorphic to subgroups of �-compact groups[9, example 3.1]. however, an o-bounded continuous homomorphic image of aweil-complete group is �-bounded, hence strictly o-bounded [3]. all this makesthe problem of studying the properties of these two classes of topological groupsfairly interesting.the class of o-bounded groups is not productive in view of example 2.12.however, we have no examples of strictly o-bounded groups g and h such thatthe product g � h is not strictly o-bounded (see problem 4.1). on the otherhand, it was known that a product of an o-bounded group by a �-compact groupwas o-bounded [9, theorem 5.3], and a similar result for strictly o-boundedgroups was recently proved by jian he (see theorem 2.7) who in fact hasproved the result with `�-bounded' instead of `�-compact' and by a methodthat extends the o-bounded result as well. it turns out that there are manytopological groups g (far from being �-compact) with the property that theproduct g �h is (strictly) o-bounded for every (strictly) o-bounded group h.let g be a �-product of countable discrete groups endowed with the @0-boxtopology. we shall call any subgroup of such a group g a comfort-like group.(it was w. comfort who proved that every �-product of countable discretespaces with the @0-box topology inherited from the whole product is lindel�of,see [5]). we prove in section 2 that multiplication by a comfort-like groupg does not destroy (strict) o-boundedness: the product g � h is (strictly) o-bounded for every (strictly) o-bounded group h. it is also shown that thefree topological group f(x) is strictly o-bounded whenever x is the one-point some properties of o-bounded and strictly o-bounded groups 31lindel�o�cation of any uncountable discrete space (theorem 2.8). in fact, theproduct f(x) � h is strictly o-bounded for every strictly o-bounded group h(see theorem 2.11).it is clear that every o-bounded group is @0-bounded in the sense of [6],that is, it can be covered by countably many translates of any neighborhoodof the identity. by theorem 4.1 of [9], if g is @0-bounded and all secondcountable continuous homomorphic images of g are o-bounded, then g itselfis o-bounded. in section 3 we use � to construct an o-bounded group g whosesecond countable continuous homomorphic images are countable (hence strictlyo-bounded), but g itself is not strictly o-bounded. therefore, the class ofstrictly o-bounded groups is considerably more complicated than that of o-bounded groups. in other words, strict o-boundedness is not re ected in theclass of second countable groups.the group g in theorem 3.1 has another interesting feature. let us call atopological group h of-undetermined if neither player i nor player ii has awinning strategy in the of-game in h. it was an open problem whether thereexist of-undetermined groups. it turns out that the group g in example3.1 is of-undetermined. we do not know, however, if such a group can beconstructed in zfc. another problem is considered by t. banakh in [4]: doesthere exist a metrizable of-undetermined group? he shows that such groupsexist under martin's axiom and have necessarily to be second countable.1.1. notation and terminology. we denote by n the positive integers, by zthe additive group of integers, and by r the group of reals. a topological groupg is called @0-bounded [6] if countably many translates of every neighborhoodof the identity in g cover the group g. by a result of [6], g is @0-bounded ifand only if it is topologically isomorphic to a subgroup of a direct product ofsecond countable topological groups. this class of groups is closed under takingdirect products, subgroups and continuous homomorphic images.we say that h is a p-group if the intersection of any countable family of opensets in h is open. every topological group h admits a �ner group topologythat makes it a p-group: a base of such a topology consists of all g�-subsets ofh.if x is a subset of a group g, we use hxi to denote the subgroup of ggenerated by x. finally, the families of all non-empty �nite and countablesubsets of a set a will be denoted by [a] 1 and x1 � � �xn�1 = f �u,where f 2 hki and u 2 uk.if xn 2 d�nk, then it is clear that uxn 2 uk, hence g = f�uxn 2 hki�un. onthe other hand, if xn is in k, then for uk to be a normal subgroup, u0 = x�1n uxnis in uk. therefore, g = fxn � u0 2 hki � uk. this �nishes the proof. �proof of theorem 2.8. for every k 2 [d]�!, let fgkn : n 2 ng be an enu-meration of the group hki. without loss of generality, we may suppose thatplayer i chooses open sets of the form uk (lemma 2.9 applies here). if playeri chooses uk1, player ii chooses f1 = fgk11 g. in general, if player i choosesukn, then player ii chooses fn = fgkij : 1 � i;j � ng. we can also as-sume that k1 � k2 � �� kn � �� . let k = s1i=1 ki. observe thathki = s1n=1hkni = s1n=1 fn. finally, since uk � ukn for each n, lemma 2.10implies thatf(d�) = hki � uk = � 1[n=1 fn� � uk = 1[n=1(fn � uk) � 1[n=1 fn � ukn:then, f(d�) is strictly o-bounded. �in fact, the above theorem admits a stronger form.theorem 2.11. the product f(d�)�h is strictly o-bounded for each strictlyo-bounded group h.proof. we can modify slightly the proof of theorem 2.6 and obtain the proofof our theorem. indeed, the sets uk are now the normal subgroups of f(d�)generated by d� n k, where k 2 [d]�!. these sets were used in the proof oftheorem 2.8 and, as before, form a base for the identity that has the followingproperties:(1) each uk is a normal subgroup of f(d�);(2) the subsets uk are clopen in f(d�);(3) jf(d�)=ukj � @0 for each k 2 [d]�!.we may suppose that player i chooses neighborhoods of the form ui � vi,where ui and vi are neighborhoods of the identity e of f(d�) and eh of hrespectively, and ui = uki, where ki is in [d]�!, i 2 n.as in the proof of theorem 2.8, we choose an enumeration fgkn : n 2 ng ofhki, and put en = fxkij : i;j � ng. at this point, the proof of the theoremcontinues in the same way as the proof of theorem 2.6. �the following example shows that the class of o-bounded groups is not �nitelymultiplicative. this answers the corresponding problem posed in [9] in thenegative. it turns out that the o-bounded group g from [9, example 8] suits.example 2.12. there exists a second countable o-bounded topological groupg such that g � g is not o-bounded. 38 c. hern�andez, d. robbie, m. tkachenkofor every x 2 r!, de�ne suppx = fn 2 n : x(n) 6= 0g. let fnk(x) : k 2 !gbe the enumeration of suppx in the increasing order. denote by x the set ofall x 2 r! such that limk!1 x(nk)nk+1(x) = 0:consider the subgroup g of r! generated by x, i.e., g = hxi. in what followswe use the additive notation for the group operation in r! .we already know that g is o-bounded. we shall prove that g2 is not o-bounded describing a sequence fun : n 2 ng of open neighborhoods of theidentity e 2 g for which no sequence of �nite subsets fen : n 2 ng in gwill make g2 = s1n=1[(en � en) + (un � un)]. for every n 2 n, let un =g \ q1j=1 vn;j, where vn;j = (�1;1) for 0 � j � n and vn;j = r if j > n. now,when considering en +un the only coordinates of the elements en that matterare 0;1; : : : ;n since un is unrestricted on ! nn coordinates. so, we may as wellonly consider en where the elements have 0 at each of the !nn places. moreover,we can assume that en � en+1. let an = maxfjz(i)j : z 2 en; 0 � i � ng.observe that a0 < a1 < � � � . we shall prove that g2 6= s1n=1[(en � en) +(un � un)] for any �nite subsets en � g. that is, there exists at least onepair of elements x, y 2 g such that (x;y) =2 s1n=1[(en � en) + (un � un)]. weconstruct x and y as follows. choose n0 = 0, n1 = 1 and set x(0) = x0 > an1.we now choose any n2 such that x0=n2 < 1=2. now, for all i, 0 an2. then we choose n3 2 ! so that yn1=n3 < 1=3.we set y(j) = 0 if 0 � j < n1 or n1 < j < n3. we continue in this way tode�ne numbers fnk : k 2 !g. we put x(nk) = xnk > ank+1 for k even and suchthat x(nk)=nk+2 < 1=(k + 2). the other values for x(j) so far unde�ned forj < nk+2 are set as 0. similarly, if k is odd, then de�ne y(nk) = ynk > ank+1and nk+2 is de�ned so that y(nk)=nk+2 < 1=(k + 2). it is clear that x,y 2 g.we claim that (x;y) =2 s1n=1[(en � en) + (un � un)]. indeed, suppose thatn 2 n and that nk � n < nk+1. if k is even, then xnk > ank+1 � an, sox =2 en + un. if k is odd, then ynk > ank+1 � an, so y =2 en + un. hence(x;y) =2 s1n=0[(en�en)+(un�un)]. this shows that g2 is not o-bounded. �3. an example of an of-undetermined groupby theorem 4.1 of [9], an @0-bounded group g is o-bounded if and onlyif all second countable continuous homomorphic images of g are o-bounded.here we show that strictly o-bounded groups cannot be characterized this way,thus answering [9, problem 4.2] in the negative. in addition, the group g weconstruct below will be of-undetermined, that is, neither player i nor playerii has a winning strategy in the of-game on g.theorem 3.1. under �, there exists a topological group g with the followingproperties:(a) every countable intersection of open sets in g is open;(b) the image f(g) is countable for every continuous homomorphism f :g ! h to a second countable topological group h; in particular, g iso-bounded; some properties of o-bounded and strictly o-bounded groups 39(c) g is of-undetermined, hence not strictly o-bounded.proof. we shall construct g as a subgroup of the group z!1 endowed withthe @0-box topology, where the group z has the discrete topology. this willguarantee (a). for every � < !1, let �� : z!1 ! z� be the projection andk� be the kernel of ��. then k� is an open subgroup of z!1, and we putn� = g \ k�. clearly, the family fn� : � < !1g forms a decreasing base atthe neutral element of g. the subgroup g of z!1 will also satisfy the followingstrong condition:(b) jgj = @1, but ��(g) is countable for each � < !1.let us show that (b) implies (b). suppose that f : g ! h is a continuoushomomorphism to a second countable topological group h. choose a countablebase fun : n 2 ng at the neutral element of h. for every n 2 n, thereexists an ordinal �n < !1 such that n�n � f�1(un). let � be a countableordinal satisfying �n < � for each n 2 n. then n� � kerf, so by lemma2.1 there exists a homomorphism g : ��(g) ! h such that f = g � ��. sincethe group ��(g) is countable by (b), we have jf(g)j � j��(g)j � !. clearly,every countable group is o-bounded, so theorem 4.1 of [9] implies that g iso-bounded.the di�cult part of our construction is to guarantee (c). this requires somepreliminary work. for a point x 2 z!1, put supp(x) = f� < !1 : x(�) 6= 0gand consider the subgroup � of z!1 de�ned by� = fx 2 z!1 : jsupp(x)j � !g:it is clear that j�j = c = @1. actually, our group g will be constructedas a subgroup of �. since fn� : � < !1g is a base at the neutral elementof g, we can assume without loss of generality that player i always makes hischoice from this family, and this choice, say n�, is de�ned by the correspondingordinal �. therefore, every possible winning strategy for player ii is a function : seq ! [g] 0 then 2n2+1 √ 1 n + 2n 2+1 √ 1 3n 2 ≤ fn,2 (xk, x) ≤ 1 2. if x − xk < 0 then 0 ≤ fn,2 (xk, x) ≤ 2n2+1 √ n−1− 2n 2+1 √ 1 3 2 moreover, these bounds are valid as x ∈ [ a, x1 − b−a3n ] , x ∈ [ xn−1 + b−a 3n , b ] and x ∈ ( xj − b−a3n , xj + b−a 3n ) with j 6= k. lemma 2.4. let pn be a partition of [a, b]. if x ∈ [ xk−1 + b−a 3n , xk − b−a3n ] with k = 1, ..., n − 1, x ∈ [ a, x1 − b−a3n ] , x ∈ [ xn−1 + b−a 3n , b ] , or x ∈ ( xj − b−a3n , xj + b−a 3n ) where j 6= k then for all n ≥ 2 it follows that 1. ∣ ∣ ∣ ∣ 2n2+1 √ 1 n + 2n 2+1 √ 1 3n 2 − 1 ∣ ∣ ∣ ∣ ≤ 1 n √ n 2. ∣ ∣ ∣ ∣ 2n2+1 √ n−1− 2n 2+1 √ 1 3 2 − 0 ∣ ∣ ∣ ∣ ≤ 1 n √ n proofs of lemmas 2.1, 2.2, 2.3 and 2.4 can be obtained by elementary estimates. proposition 2.5. let pn be a partition of [a, b] and en the step function defined by (2.1) en(x) = k1 · χ[a,x1] + n−1 ∑ p=2 kp · χ(xp−1,xp] + kn · χ(xn−1,b] let cn be the radical function associated to en defined by (2.2) cn(x) = k1 + n ∑ j=2 [kj − kj−1] · fn,2 (xj−1, x) then, for all n ≥ 2 it follows that: (1) |cn(x) − en(x)| ≤ 2(mn−mn)n√n , x ∈ [a, b] \ ∪ n−1 k=1 ( xk − b−a3n , xk + b−a 3n ) (2) |cn(x) − [kj(1 − αx) + kj+1αx]| ≤ 2(mn−mn)n√n , x ∈ ( xj − b−a3n , xj + b−a 3n ) , j = 1,..., n − 1 where mn and mn are the maximum and the minimum of the kj and αx ∈ (0, 1) is a number which depends upon x. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 364 uniform reconstruction of continuous functions with rafu method proof. the proof is simlar to the proof given in [7]. part 1. this part is proved considering three possible cases. case 1. suppose that x ∈ [ xj−1 + b−a 3n , xj − b−a3n ] , j = 2, ...,n − 1 then |cn(x) − en(x)| = |cn(x) − kj| = ∣ ∣ ∣ ∣ ∣ cn(x) − ( k1 + j ∑ p=2 [kp − kp−1] ) ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] [1 − fn,2 (xp, x)] + n ∑ p=j+1 [kp − kp−1] [0 − fn,2 (xp, x)] ∣ ∣ ∣ ∣ ∣ ∣ ≤ by lemmas 2.3 and 2.4 ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] · 1 n √ n + n ∑ p=j+1 [kp − kp−1] · − 1 n √ n ∣ ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] · 1 n √ n + n ∑ p=j+1 [kp−1 − kp] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ∣ ≤ 1 n √ n |[kj − k1] + [kj − kn]| ≤ 2 (mn − mn)√ n case 2. suppose that x ∈ [ a, x1 − b−a3n ] . then x − xp < 0, p = 1, ..., n − 1 and proceeding as in case 1 and by using lemmas 2.3 and 2.4, we obtain |cn(x) − en(x)| = |cn(x) − k1| = ∣ ∣ ∣ ∣ ∣ cn(x) − ( k1 + j ∑ p=2 [kp − kp−1] ) ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] [0 − fn,2 (xp, x)] ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] · − 1 n √ n ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp−1 − kp] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ≤ 2 (mn − mn) n √ n case 3. suppose that x ∈ [ xn−1 + b−a 3n , b ] . then x− xp > 0, p = 1, ..., n− 1 and proceeding as in case 1, we can put |cn(x) − en(x)| = |cn(x) − kn| = ∣ ∣ ∣ ∣ ∣ cn(x) − ( k1 + j ∑ p=2 [kp − kp−1] ) ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] [1 − fn,2 (xp, x)] ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ≤ 2 (mn − mn) n √ n taking into account lemmas 2.3 and 2.4. part 2. suppose that x ∈ ( xj − b−a3n , xj + b−a 3n ) , j = 1,..., n − 1, then c© agt, upv, 2017 appl. gen. topol. 18, no. 2 365 e. corbacho [kj(1 − αx) + kj+1αx] − cn(x) = [kj + (kj+1 − kj) αx] − cn(x) = k1 − k1 + j ∑ p=2 [kp − kp−1] [1 − fn,2 (xp−1, x)] + [kj+1 − kj] [αx − fn,2 (xj, x)] + n ∑ p=j+1 [kp+1 − kp] [0 − fn,2 (xp, x)] since for x ∈ ( xj − b−a3n , xj + b−a 3n ) it follows that 0 < fn,2 (xj, x) < 1 we can put αx = fn,2 (xj, x). so that, from lemmas 2.3 and 2.4, taking absolute value and proceeding as in case 1, |cn(x) − [kj(1 − αx) + kj+1αx]| ≤ ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] · 1 n √ n + n ∑ p=j+1 [kp − kp+1] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ∣ = 1 n √ n |[kj − k1] + [kj+1 − kn]| ≤ 2 (mn − mn) n √ n � theorem 2.6. let f be a continuous function defined in [a, b]. then there exists a sequence of radical functions (cn)n defined in [a, b] as in (2.2) such that |cn(x) − f(x)| ≤ 2 (m − m) n √ n + ω ( b − a n ) for all n ≥ 2 and x ∈ [a, b] being m and m the maximum and the minimum of f in [a, b] respectively and ω ( b−a n ) its modulus of continuty. proof. for each n ≥ 2, let pn be a partition of [a, b], let en be the step function defined by en(x) =          f(a) x ∈ [a, x1] f(x2) x ∈ (x1, x2] ... f(b) x ∈ (xn−1, b] and let cn be the corresponding radical function defined from en as (2.2). if x ∈ [a, b] \ ∪n−1 k=1 ( xk − b−a3n , xk + b−a 3n ) then, |cn(x) − f(x)| = |cn(x) − en(x) + en(x) − f(x)| ≤ 2 (m − m) n √ n + |en(x) − f(x)| = 2 (m − m) n √ n + |f(xj) − f(x)| ≤ c© agt, upv, 2017 appl. gen. topol. 18, no. 2 366 uniform reconstruction of continuous functions with rafu method 2 (m − m) n √ n + ω ( b − a n ) taking into account that en(x) = f(xj) for some j and proposition 2.5. if x ∈ ∪n−1 k=1 ( xk − b−a3n , xk + b−a 3n ) , proposition 2.5 applies and we can choose an appropiate index j to obtain |cn(x) − f(x)| ≤ |cn(x) − [f(xj) (1 − αx) + f(xj+1)αx]| + |[f(xj) (1 − αx) + f(xj+1)αx] − f(x)| ≤ 2 (m − m) n √ n + |[f(xj) (1 − αx) + f(xj+1)αx] − [f(x) (1 − αx) + f(x)αx]| ≤ 2 (m − m) n √ n + |f(xj) − f(x)| (1 − αx) + |f(xj+1) − f(x)| (1 − αx) ≤ 2 (m − m) n √ n + ω ( b − a n ) (1 − αx + αx) = 2 (m − m) n √ n + ω ( b − a n ) � remark 2.7. it is well-known (see for instance [6] pp. 147) that if f ∈ c [−1, 1], then there exist an algebraic polynomial pn of degree ≤ n such that for all x ∈ c [−1, 1], |pn(x) − f(x)| ≤ ω ( π n + 1 ) as far as we know, this error estimate is the best possible currently known. by means of theorem 2.6, we have proved an analogous result by using radical continuous functions. in case of the interval [−1, 1], the error bound becomes 2(m−m) n √ n + ω ( 2 n ) . so, depending on the function, this error estimate can be better than error bound in algebraic polynomial approximation. moreover, rafu method provides the explicit form of the function which approximate to the funtion f for each n. however in the case of algebraic polynomials this does not happen. therefore, this is an important contribution of this work. 3. main results 3.1. uniform reconstruction of f from average samples. the following corollary provides a sequence uniformly convergent to the original function f and a uniform error bound. observe that the uniform error bound is the same as theorem 2.6. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 367 e. corbacho corollary 3.1. under the hypothesis of theorem 2.6, if the data ki of the step function (2.1) are substituted by ki = f(xi1)n1+...+f(xip)np n1+...+np , x1q ∈ [a, x1] or xiq ∈ (xi−1, xi], i = 2, ..., n, q = 1, ..., p, n1 + ... + nq 6= 0 then |cn(x) − f(x)| ≤ 2 (m − m) n √ n + ω ( b − a n ) for all x ∈ [a, b], n ≥ 2 and where cn(x) is defined as 2.2 but from the new data ki. proof. in the proof of proposition 2.5 we can put ki = f(xi1)n1+...+f(xip)np n1+...+np , i = 1, ..., n and the same result holds for m and m. moreover, if we define the functions en in proposition 2.5 from ki = f(xi1)n1+...+f(xip)np n1+...+np , i = 1, ..., n and we put that f(x) = f(x)n1+...+f(x)np n1+...+np , then we can easily check that corollary 3.1 is true considering now cn defined as (2.2) but from ki, i = 1, ..., n. � example 3.2. in figure 1 we show the approximation to the piecewise continuous function f(x) defined by 0.5 if x ∈ [0, 0.39), 0.5x−0.185 0.02 if x ∈ [0.39, 0.41), 1 if x ∈ [0.41, 0.69), −0.5x+0.365 0.02 if x ∈ [0.69, 0.71) and 0.5 if x ∈ [0.71, 1] from ki = f(xi1)+...+f(xi15) 15 , i = 1, ..., 200 considering c200(x). (a) approximating function and f (b) approximation error figure 1. uniform reconstruction from average samples. remark 3.3. if ni = 1, we have the usual average values. 3.2. uniform reconstruction of f from approximate values. in [4] j. bustamante, r. c. castillo and a. f. collar solved this problem by means of a regularization method. in [7] we studied this case but here we give a uniform error bound. the reader can compare our error bound with the estimation of the error shown in [4]. when we do not know the values f(xi) but the data f(xi)+ηi, with |ηi| < η for a fixed η > 0 are known, then the following result can be useful to obtain an approximation of the function f. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 368 uniform reconstruction of continuous functions with rafu method corollary 3.4. under the hypothesis of theorem 2.6, if the data ki of the step function (2.1) are changed for ki = f(xi) + ηi, being |ηi| < η, i = 1,...,n then |cn(x) − f(x)| ≤ 2 (m − m + η) n √ n + ω ( b − a n ) + η for all x ∈ [a, b], n ≥ 2 and where cn(x) is defined as (2.2) but from the new data ki. proof. with these data ki, i = 1,..., n we can obtain the error bound 2(m−m+η) n √ n in proposition 2.5. moreover, if we change f(xi) for ki = f(xi)+ηi, i = 1, ..., n in the proof of theorem 2.6 then the new error bound becomes 2(m−m+η) n √ n + ω ( b−a n ) + η. � example 3.5. approximation to the piecewise continuous function function f(x) defined by 4x if x ∈ [0, 0.25), 1 if x ∈ [0.25, 0.5), −0.5x+0.5 0.02 if x ∈ [0.5, 0.75) and 0.5 if x ∈ [0.71, 1] using the data ki = f(xi)+ηi, i = 1, ..., n, with |ηi| ≤ 1100 ( ηi = 1 100 sin4πxi ) and considering c180(x) (figure 2). (a) approximating function and f (b) approximation error figure 2. uniform reconstruction from approximate values. 3.3. uniform reconstruction of f from local average samples. in many applications it is more realistic to assume that the available samples are local average samples near a certain x. we consider the special case in which we know data of the type (3.1) ( χ[−h,h] ⋆ f ) (x) = ∫ +∞ −∞ χ[−h,h](y)f(x − y)dy = ∫ x+h x−h f(z)dz where ⋆ denotes the convolution of the functions χ[−h,h] and f. sometimes we deal with phenomena which involve a function and its integral. for example, in mechanics, the velocity v(t) and the displacement s(t), or the acceleration a(t) and the velocity v(t); in statistics, the probability density function and the cumulative distribution function and in electricity, the current function i(t) and the charge function q(t) are some real examples about this consideration. the tasks are to approximate the function f from integral values as (3.1) and c© agt, upv, 2017 appl. gen. topol. 18, no. 2 369 e. corbacho to give error bounds for this aproximation. there have been only a few research papers to deal with these problems; see for example, h. behforooz [1, 2], e.j.m. delhez [11], f.g. lang and x.p. xu [18] and t. zhanlav and r. mijiddorj [19]. in these papers it was necessary to suppose that the function f had several derivatives and error estimations were not given in some of them. here, with corollary 3.6, rafu method solves easily the problem of the reconstruction of the function from the integral values and provides a uniform error bound for this reconstruction with the only condition that f ∈ c [a, b]. on the other hand, let △ be a subdivision of the interval [a, b] with grids a = x0 < x1 < ... < xn = b whose mesh size is denoted by h = max1≤i≤nhi, hi = xi − xi−1, 1 ≤ i ≤ n and mi(f) = 1hi ∫ xi xi−1 f(x)dx. in practice, due to the measurement error, the exact values mi(f) are unknown but we know approximate average values ui, 1 ≤ i ≤ n such that |ui − mi(f)| < δ where δ is a positive constant describing the level of error of the data. in [16] j. huang and y. chen proposed a regularization method for solving the problem (p): given the approximate values ui, 1 ≤ i ≤ n satisfying the previous condition how does one reconstruct the original function f efficiently? they established the rigorous error estimates in l2 norm for functions f ∈ h1 (a, b) where h1 (a, b) is the usual sobolev space consisting of all l2 (a, b)-integrable functions whose 1-order weak derivative are also l2 (a, b)-integrable. for f ∈ h1 (a, b). they solved this problem in terms of the tikhonov regularization method. in this work, by means of corollaries 3.4 and 3.6, we establish another solution of problem (p) in the uniform norm for all f ∈ c [a, b]. note that h1 (a, b) is continuously embedded in c [a, b]. our solution does not need regularization. see algorithm 4.2 and figure 5. corollary 3.6. with the hypothesis of theorem 2.6, if the data ki of the step function (2.1) are defined by ki = ∫ x̃i+h x̃i−h f(z)dz 2h , with [x̃1 − h, x̃1 + h] ⊆ [a, x1] or [x̃i − h, x̃i + h] ⊆ (xi−1, xi], i = 2, ..., n, then |cn(x) − f(x)| ≤ 2 (m − m) n √ n + ω ( b − a n ) for all x ∈ [a, b], n ≥ 2 and where cn(x) is defined as (2.2) but from the new data ki. proof. we can put that ∫ x̃i+h x̃i−h f(z)dz = f(zi)2h for some value zi ∈ [x̃i − h, x̃i + h] by the integral properties because f is continuous. then, ki = f(zi) for all i and we finish with the same proof of theorem 2.6. � example 3.7. consider the special case given by x̃i = xi−1+xi 2 , i = 1, ..., n and h = b−a n to approximate the continuous function f(x) defined by |sin8πx| if x ∈ [0, 0.5) and x − 0.5 if x ∈ [0.5, 1] from local average samples ki = ∫ x̃i+h x̃i−h f(z)dz 2h , i = 1, ..., 180 with c144(x) (figure 3). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 370 uniform reconstruction of continuous functions with rafu method (a) approximating function and f (b) approximation error figure 3. uniform reconstruction from local average samples. 3.4. uniform reconstruction of f from linear combinations. corollary 3.8. under the hypothesis of theorem 2.6, if the values ki of the step function (2.1) are defined by ki = f(x̃i)−f(x̃i−1) x̃i−x̃i−1 ·(x ′ i − x̃i−1)+f(x̃i−1) with x′1 ∈ [x̃0, x̃1] ⊆ [a, x1] or x′i ∈ [x̃i−1, x̃i] ⊆ (xi−1, xi], i = 2, ..., n, then |cn(x) − f(x)| ≤ 2 (m − m) n √ n + ω ( b − a n ) for all x ∈ [a, b], n ≥ 2 and where cn(x) is defined as (2.2) but from the new data ki. proof. since f ∈ c [a, b], there exists a point x′′i in each interval [x̃i−1, x̃i] such that ki = f(x ′′ i ) for all i = 1, ..., n. then, this proof becomes the proof of theorem 2.6. � example 3.9. consider the special case in which x̃i = xi for all i to approximate the piecewise continuous function f(x) defined by sin4πx if x ∈ [0, 1 20 ) ∪ [1 5 , 3 10 ) ∪ [ 9 20 , 1 2 ), sinπ 5 if x ∈ [ 1 20 , 1 5 ), sin2π 5 if x ∈ [ 3 10 , 9 20 ) and |sin4πx| if x ∈ [1 2 , 1] from the data ki = f(xi)−f(xi−1) xi−xi−1 · (x ′ i − xi−1) + f(xi−1) and the values x′i = xi−i+xi 2 by using c150(x) (figure 4). (a) approximating function and f (b) approximation error figure 4. uniform reconstruction from linear combinations. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 371 e. corbacho 4. algorithms we show three algorithms by using the 4.1.0.0 mathematica program. algorithm 4.1. uniform reconstruction from average samples. f[x−] := as example 3.2; a = 0; b = 1; n = 3000; h = b−a n ; v = n 15 ; t = t able[a + 15 · h · i, {i, 0, v}]; d = t able[f[a + j · h], {j, 0, n − 1}]; for[i = 1, i + +, di = ∑ 15 m=1 dm+15∗(i−1) 15 ]; k = t able[ ∑ 15 m=1 dm+15∗(i−1) 15 , {i, 1, v}];tt = length[t]; kk = length[k]; for[i = 2, i ≤ kk, i + +, mi = (ki−ki−1)· 2n2+1 √ ti−t1 2n2+1 √ ttt−ti+ 2n 2+1 √ ti−t1 ]; for[i = 2, i ≤ kk, i + +, ni = (ki−ki−1)2n2+1√ttt−ti+ 2n2+1√ti−t1 ]; g[x−] = k1 + ∑ kk i=2 ( mi + ni · 2n 2+1 √ abs[x − ti] · sign (x − ti) ) ; plot[{f[x], g[x]}, {x, t1, ttt}] plot[abs[f[x] − g[x]], {x, t1, ttt}] corollary 3.4 can be used together with corollaries 3.1, 3.6 or 3.8. for instance, in algorithm 4.2, we use corollaries 3.4 and 3.6 to reconstruct uniformly an irregular function f from approximate integral values (figure 5). here, random denotes a random number with uniform distribution on [−1, 1] and 0.01 is the considered relative error level of the data. rafu method provides this easy solution to the problem (p) suggested by j. huang and y. chen in [16]. (a) approximating function and f (b) approximation error figure 5. uniform reconstruction from approximate integral values. algorithm 4.2. uniform approximation from approximate integral values. f[x−] := if[0 ≤ x < 0.25, x, if[0.25 ≤ x < 0.5, −x + 0.5, if[0.5 ≤ x < 0.75, x − 0, 5, if[0.75 ≤ x ≤ 1, −x + 1]]]]; a = 0; b = 1; n = 100; h = b−a n ; hh = b−a 2·n ; t = t able[a + j · h, {j, 0, n}]; c© agt, upv, 2017 appl. gen. topol. 18, no. 2 372 uniform reconstruction of continuous functions with rafu method k = t able[ n[integrate[f[x],{x,a+j·h+a+(j+1)·h 2 −hh, a+j·h+a+(j+1)·h 2 +hh}]] 2·hh ·(1 + 0.01 · random[real, {−1, 1}]), {j, 0, n − 1}]; tt = length[t]; kk = length[k]; for[i = 2, i ≤ kk, i + +, mi = (ki−ki−1)· 2n2+1 √ ti−t1 2n2+1 √ ttt−ti+ 2n 2+1 √ ti−t1 ]; for[i = 2, i ≤ kk, i + +, ni = (ki−ki−1)2n2+1√ttt−ti+ 2n2+1√ti−t1 ]; g[x−] = k1 + ∑ kk i=2 ( mi + ni · 2n 2+1 √ abs[x − ti] · sign (x − ti) ) ; plot[{f[x], g[x]}, {x, t1, ttt}] plot[abs[f[x] − g[x]], {x, t1, ttt}] algorithm 4.3. uniform approximation from linear combinations. f[x−] := as example 3.9; a = 0; b = 1; n = 150; h = b−a n ; t = t able[a + j · h, {j, 0, n}]; k = t able[ f[a+(j+1)·h]−f[a+j·h] h · h 2 + f[a + j · h], {j, 0, n − 1}]; tt = length[t]; kk = length[k]; for[i = 2, i ≤ kk, i + +, mi = (ki−ki−1)· 2n2+1 √ ti−t1 2n2+1 √ ttt−ti+ 2n 2+1 √ ti−t1 ]; for[i = 2, i ≤ kk, i + +, ni = (ki−ki−1)2n2+1√ttt−ti+ 2n2+1√ti−t1 ]; g[x−] = k1 + ∑ kk i=2 ( mi + ni · 2n 2+1 √ abs[x − ti] · sign (x − ti) ) ; plot[{f[x], g[x]}, {x, t1, ttt}] plot[abs[f[x] − g[x]], {x, t1, ttt}] 5. uniform reconstruction of f from a non-uniform net from now on, we will consider partitions pn = {x0 = a, x1, ..., xs = b} of [a, b] with non-uniformly spaced data. lemma 5.1. let k be a positive integer. then, for n ≥ 2 it verifies that ∣ ∣ ∣ 2n2+1 √ nk − 1 ∣ ∣ ∣ ≤ 2 k −1 n √ n and ∣ ∣ ∣ 2n2+1 √ 1 nk − 1 ∣ ∣ ∣ ≤ k n √ n proof. by induction on k. case k = 1 can be obtained by elementary estimates. then, we finishes taking into account that ∣ ∣ ∣ 2n2+1 √ n±k − 1 ∣ ∣ ∣ = ∣ ∣ ∣ 2n2+1 √ n±k − 2n 2+1 √ n±1 + 2n2+1 √ n±1 − 1 ∣ ∣ ∣ � lemma 5.2. let pn = {a = x0, x1, ..., xs = b} be a partition of [a, b] with δ (s) = min1≤j≤s |xj − xj−1|. then, for any k = 1, ..., s − 1 and x ∈ [a, b] \ ( xk − δ(s)3 , xk + δ(s) 3 ) it follows that: (1) 2n 2+1 √ δ(s) b−a 1+ 2n 2+1 √ 1 3 2 ≤ fn,2 (xk, x) ≤ 1 if x − xk > 0 (2) 0 ≤ fn,2 (xk, x) ≤ 2n2+1 √ b−a δ(s) − 2n 2+1 √ 1 3 2 if x − xk < 0 c© agt, upv, 2017 appl. gen. topol. 18, no. 2 373 e. corbacho the proof can be obtained by elementary estimates. lemma 5.3. let k ≥ 2 be a positive integer such that 3(b−a) nk ≤ δ (s). then, for all n ≥ 2, it verifies that (1) ∣ ∣ ∣ ∣ 1 − 2n2+1 √ δ(s) b−a 1+ 2n 2+1 √ 1 3 2 ∣ ∣ ∣ ∣ ≤ k n √ n (2) ∣ ∣ ∣ ∣ ∣ 2n2+1 √ b−a δ(s) − 2n 2+1 √ 1 3 2 − 0 ∣ ∣ ∣ ∣ ∣ ≤ 2 k−1 n √ n the proof can be obtained easily from lemmas 2.1 and 5.1. proposition 5.4. let ps = {a = x0, x1, ..., xs = b} be a partition of [a, b] and let es be a step function defined in [a, b] by es(x) = k1 · χ[x0, x1] + s ∑ i=2 ki · χ(xi−1, xi], ki real numbers if 3(b−a) nk ≤ δ (s), being δ (s) = min1≤j≤s |xj − xj−1|and k ≥ 2 a positive integer, then for all n ≥ 2 it follows that: (1) |cn(x) − es(x)| ≤ 2 k(ms−ms) n √ n if x ∈ [a, b] \ ∪s−1j=1 ( xj − δ(s)3 , xj + δ(s) 3 ) (2) |cn(x) − [kj(1 − αx) + kj+1αx]| ≤ 2 k(ms−ms) n √ n if j = 1, ..., s − 1 and x ∈ ( xj − δ(s)3 , xj + δ(s) 3 ) . where ms and ms are the maximum and the minimum of the kj, αx ∈ (0, 1) is a number which depends only on x and (cn)n is the sequence of radical functions associated to es defined as in (2.2). proof. it is analogous to the proof of proposition 2.5 but now we use lemmas 5.1, 5.2 and 5.3. � theorem 5.5. let pn = {a = x0, x1, ..., xsn = b} be a partition of [a, b] with δ (sn) = min1≤j≤sn |xj − xj−1| and ∆ (sn) = max1≤j≤sn |xj − xj−1| such that 3(b−a) nk ≤ δ (sn) ≤ ∆ (sn) ≤ h being h = b−an and k ≥ 2 a positive integer. let f be a continuous function defined in [a, b]. then there exists a sequence (cn)n defined in [a, b] as in (2.2) such that |cn(x) − f(x)| ≤ 2k (m − m) n √ n + ω ( b − a n ) for all n ≥ 2 and x ∈ [a, b], being m, m and ω ( b−a n ) as usual. proof. similiar to the proof of proposition 2.5. here proposition 5.4 applies. � in the same way the results in section 3 have been obtained from theorem 2.6, similar results to section 3 can be derived from theorem 5.5 for the case of non-uniform net and this is another important contribution of this work. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 374 uniform reconstruction of continuous functions with rafu method acknowledgements. the author is grateful to the editor and to the referee for the careful reading of this paper and for their helpful suggestions. references [1] h. behforooz, approximation by integro cubic splines, appl. math. comput. 175 (2006), 8–15. 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[19] t. zhanlav and r. mijiddorj, the local integro cubic splines and their approximation properties, appl. math. comput. 216 (2010), 2215–2219. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 375 () @ appl. gen. topol. 17, no. 2(2016), 105-116doi:10.4995/agt.2016.4495 c© agt, upv, 2016 a construction of a fuzzy topology from a strong fuzzy metric svetlana grecova a, alexander šostak b and ingr̄ıda uļjane b a department of mathematics, university of latvia, riga latvia b department of mathematics, university of latvia, zellu street 25, lv-1002 riga, latvia and institute of mathematics, ul, raina bulv. 29, lv-1459 riga, latvia (sostaks@latnet.lv, ingrida.uljane@lu.lv) abstract after the inception of the concept of a fuzzy metric by i. kramosil and j. michalek, and especially after its revision by a. george and g. veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. in most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. in particular, in the paper (j. j. miñana, a. šostak, fuzzifying topology induced by a strong fuzzy metric, fuzzy sets and systems 300 (2016), 24–39) a fuzzifying topology t : 2x → [0, 1] induced by a fuzzy metric m : x × x × [0, ∞) was constructed. in this paper we extend this construction to get the fuzzy topology t : [0, 1]x → [0, 1] and study some properties of this fuzzy topology. 2010 msc: 54a40. keywords: fuzzy pseudometric; fuzzy metric; fuzzifying topology; fuzzy topology; lower semicontinuous functions; lowen ω-functor. 1. introduction after the concept of a fuzzy metric was defined by i. kramosil and j. michalek in 1975 [16] and later redefined in a slightly revised form in 1994 by a. george and p. veeramani [4], many researches became interested in the topological structure of a fuzzy metric space. in particular different properties received 07 january 2016 – accepted 21 july 2016 http://dx.doi.org/10.4995/agt.2016.4495 s. grecova, a. šostak and i. uļjane of the topologies induced by fuzzy metrics and operations with such topologies were studied by a. george and p. veeramani, v. gregori, s. romaguera, a. sapena, d. mihet, j.j. miñana, s. morillas et al., see e.g. [4], [5], [11], [10], [7], [8], [25], et al. in most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. in particular in the papers [37], [22], [26] fuzzy pseudometrics m : x × x × (0, ∞) were applied to induce fuzzifying topologies t : 2x → [0, 1]. however, as far as we know, there was still no research in the field of “fullbodied” fuzzy topologies t : [0, 1]x → [0, 1] induced by fuzzy metrics. it is the principal aim of this paper to develop such construction. specifically, we consider here an lm-fuzzy topology t m : lx → m generated by a fuzzy metric m, where l and m are complete sublattices of the unit interval [0, 1]. our approach here is based on the construction of a fuzzifying topology presented in [26] which is extended to an lm-fuzzy topology by applying the lowen ω-functor. the structure of the paper is as follows. in the next section we recall basic notions and results used in the sequel. in particular, the concepts of a fuzzy metric, a fuzzy topology (specifically, an lm-fuzzy topology) are recalled here. we recall here also the standard construction of a fuzzy topology from a decreasing family of ordinary topologies and modify it by applying the lowen ω-functor. the next, 3rd section is the main one in this work. here we realize the general construction of a fuzzy topology from a family of topologies in case when these topologies are induced by a strong fuzzy metric and consider some properties of this construction. the main result here is theorem 3.8 showing, in a certain sense, “the good behaviour” of this construction. the next, corollary 3.9 presents the “categorical version” of this theorem. in the last, 4th section some perspectives for the future research are sketched. 2. preliminaries 2.1. fuzzy metrics. basing on the concept of a statistical metric introduced by k. menger [24], see also [29], i. kramosil and j. michalek in [16] defined the notion of a fuzzy metric. a. george and p. veeramani [4] slightly modified the original concept of a fuzzy metric. at present in most cases research involving fuzzy metrics is done in the context of george-veeramani definition. this approach is accepted also in our paper. definition 2.1 ([4]). a fuzzy metric on a set x is a fuzzy set m : x×x×r+ → [0, 1], where r+ = (0, +∞), such that: (1gv) m(x, y, t) > 0 for all x, y ∈ x, and all t ∈ r+; (2gv) m(x, y, t) = 1 if and only if x = y; (3gv) m(x, y, t) = m(y, x, t) for all x, y ∈ x and all t ∈ r+; (4gv) m(x, z, t + s) ≥ m(x, y, t) ∗ m(y, z, s) for all x, y, z ∈ x, t, s ∈ r+; (5gv) m(x, y, −) : r+ → [0, 1] is continuous for all x, y ∈ x. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 106 a construction of a fuzzy topology from a strong fuzzy metric the pair (x, m) is called a fuzzy metric space. if it is important to specify the t-norm in the definition of a fuzzy metric we use notations (m, ∗) and (x, m, ∗) for a fuzzy metric and the fuzzy metric space respectively. in case when axiom (2gv) is replaced by a weaker axiom (2′gv) if x = y, then m(x, y, t) = 1 we get definitions of a fuzzy pseudo-metric, and the corresponding fuzzy pseudometric space. note that axiom (4gv) combined with axiom (2′gv) implies that a fuzzy metric m(x, y, t) is non-decreasing on the third argument. definition 2.2. a fuzzy metric m : x × x × (0, ∞) → [0, 1] is called strong if, in addition to the properties (1gv) (5gv), the following stronger versions of axioms (4gv) and (5gv) are satisfied (4sgv) m(x, z, t) ≥ m(x, y, t) ∗ m(y, z, t) for all x, y, z ∈ x and for all t > 0. (5sgv) m(x, y, −) : r+ → [0, 1] for all x, y ∈ x is continuous and nondecreasing. remark 2.3. in the original definition of a strong fuzzy metric it was defined as a mapping m : x × x × r+ → [0, 1] satisfying axioms (1gv), (2gv), (3gv), (4sgv) and (5gv). however, as it was noticed, such combination of axioms does not imply axiom (4gv) and hence a strong fuzzy metric need not be a fuzzy metric: the corresponding example can be found in [6]. therefore in our definition of a strong fuzzy metric we replace axiom (5gv) by axiom (5sgv) by assuming additionally that m is non-decreasing. this condition, on one hand, can be obtained “gratis” from the system of axioms (1gv), (2gv), (3gv), (4gv) and (5gv) and, on the other hand, it allows to obtain the condition (4gv). thus, under the present definition every strong fuzzy metric is a fuzzy metric. definition 2.4 ([11]). a fuzzy metric m on a set x is said to be stationary, if m does not depend on t, i.e. if for each x, y ∈ x, the function mx,y(t) = m(x, y, t) is constant. in this case we can write m(x, y) instead of m(x, y, t). the next concept implicitly appears in [10]: definition 2.5. given two fuzzy metric spaces (x, m, ∗m) and (y, n, ∗n) a mapping f : x → y is called continuous if for every ε ∈ (0, 1), every x ∈ x and every t ∈ (0, ∞) there exist δ ∈ (0, 1) and s ∈ (0, ∞) such that n(f(x), f(y), t) > 1 − ε whenever m(x, y, s) > 1 − δ. in symbols: ∀ε ∈ (0, 1), ∀x ∈ x, ∀t ∈ (0, ∞) ∃ δ ∈ (0, 1), ∃s ∈ (0, ∞) such that m(x, y, s) > 1 − δ =⇒ n(f(x), f(y), t) > 1 − ε fuzzy metric spaces as objects and continuous mappings between them as morphisms form a category which we denote fuzms. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 107 s. grecova, a. šostak and i. uļjane in a fuzzy metric space (x, m, ∗) a (crisp) topology on x is introduced as follows [4], [5]: given a point x ∈ x, a number ε ∈ (0, 1) and t > 0, we define a ball at a level t with center x and radius ε > 0 as the set bε(x, t) = {y ∈ x | m(x, y, t) > 1 − ε}. obviously t ≤ s =⇒ bε(x, t) ⊆ bε(x, s) and ε ≤ δ =⇒ bε(x, t) ⊆ bδ(x, t). it is shown in [4], see also [5], that the family {bε(x, t) | x ∈ x, t ∈ (0, ∞), ε ∈ (0, 1)} is a base for some topology t m on x. besides one can easily verify the following proposition: proposition 2.6 ([4]). given two fuzzy metric spaces (x, m, ∗m) and (y, n, ∗n) a mapping f : (x, m, ∗m) → (y, n, ∗n) is continuous if and only if the mapping of the induced topological spaces f : (x, t m) → (y, t n) is continuous. hence, by assigning to a fuzzy metric space (x, m, ∗m) the induced topological space (x, t m) and assigning to a continuous mapping f : (x, m, ∗m) → (y, n, ∗n) the mapping f : (x, t m) → (y, t n), we get a functor φ : fuzms → top where top is the category of topological spaces. 2.2. fuzzy topologies. the first approach to the study of topological-type structures in the context of fuzzy sets was undertaken in 1968 by c.l. chang [1]; soon later this approach was essentially developed and extended by j.a. goguen [3]. according to this approach a (chang-goguen) l-fuzzy topology on a set x, where l is a complete infinitely distributive lattice, is a subfamily of the family lx of l-fuzzy subsets of x satisfying certain counterparts of the usual topological axioms. an alternative approach to the fuzzification of the subject of topology was undertaken in 1980 by u. höhle [12]. according to this approach, an (l-)fuzzy topology on a set x is defined as a mapping t : 2x → l satisfying certain functional versions of topological axioms. later, in 1991, the same concept was rediscovered by mingsheng ying [33], by making deep analysis of topological axioms and properties of topological spaces by means of fuzzy logic. mingsheng ying called such structures by fuzzifying topologies and just this term is used now by most authors when speaking about such structures. finally, in 1985 in [17] and [30] (independently) a general view on the concept of a topology in the context of fuzzy sets and fuzzy structures was proposed; later, in [18], [19], this approach led to the concept of an lm-fuzzy topology where l and m are complete infinitely distributive lattices. according to this approach an lm-fuzzy topology is a certain mapping t : lx → m, see definition 2.7. it is the aim of this work to present a construction of an lmfuzzy topology on a fuzzy metric space. besides, since we will deal with fuzzy topological-type structures generated by fuzzy metrics m : x × x × (0, ∞) → [0, 1], we will restrict here with the case when l and m are complete sublattices of the unit interval [0, 1] containing 0 and 1. definition 2.7 ([30, 17, 18]). given a set x, a mapping t : lx → m is called an lm-fuzzy topology on x if it satisfies the following axioms: c© agt, upv, 2016 appl. gen. topol. 17, no. 2 108 a construction of a fuzzy topology from a strong fuzzy metric (1) t (0x) = t (1x) = 1 here given a constant a ∈ [0, 1] by ax we denote the constant function taking value a for all x ∈ x, that is ax : x → {a} ⊆ [0, 1]; (2) t (a ∧ b) ≥ t (a) ∧ t (b) ∀a, b ∈ lx; (3) t ( ∨ i ai) ≥ ∧ i t (ai) ∀{ai : i ∈ i} ⊆ l x. a pair (x, t ) is called an lm-fuzzy topological space. in case when l = [0, 1] and when we do not specify the range m we will call [0, 1], [0, 1]-fuzzy topologies just as fuzzy topologies. remark 2.8. the intuitive meaning of the value t (a) is the degree to which a fuzzy set a ∈ lx is open. remark 2.9. chang-goguen l-fuzzy topological spaces can be characterized now as l2-fuzzy topological spaces where 2 = {0, 1} is the two-element lattice, and l-fuzzifying topological spaces are just 2l-fuzzy topological spaces. definition 2.10. given two lm-fuzzy topological spaces (x, t x) and (y, t y ) a mapping f : (x, t x) → (y, t y ) is called continuous if t x ( f−1(b) ) ≥ t y (b) ∀b ∈ ly . given α ∈ m and a fuzzy topology t : lx → m let tα = {a ∈ l x : t (a) ≥ α}. the following theorem is well-known and easy to prove: theorem 2.11. a mapping f : (x, t x) → (y, t y ) is continuous if and only if the mapping f : (x, t xα ) → (y, t y α ) is continuous for each α ∈ [0, 1], that is f−1(b) ∈ t xα whenever b ∈ t y α . since the composition g◦f : (x, t x) → (z, t z) of two continuous mappings f : (x, t x) → (y, t y ) and g : (y, t y ) → (z, t z) is obviously continuous and since the identity mapping id : (x, t x) → (x, t x) is continuous, we come to the category fuztop(lm) of lm-fuzzy topological spaces as objects and their continuous mappings as morphisms. 2.3. construction of lm-fuzzy topologies from families of crisp topologies. in this section we describe a scheme allowing to construct lm-fuzzy topologies from decreasing families of ordinary topologies. let k be a sup-dense subset of m, that is for every α ∈ m, α 6= 0 there exists a subset kα of k such that α = sup kα. further, let a non-increasing family of topologies {tα : α ∈ k} on a set x be given, that is α < β, α, β ∈ k =⇒ tα ⊇ tβ and t0 = l x whenever 0 ∈ k. further, let ω(tα) be the family of all lower semi-continuous functions a : (x, tα) → [0, 1]. it is well known see [20], [21] (and easy to verify) that ω(tα) satisfies the axioms of a chang-goguen fuzzy topology, (actually even a stratified chang-goguen fuzzy topology) that is (1) ω(tα) contains all constant functions cx : (x, tα) → [0, 1], c ∈ [0, 1], c© agt, upv, 2016 appl. gen. topol. 17, no. 2 109 s. grecova, a. šostak and i. uļjane (2) ω(tα) is closed under finite meets (3) ω(tα) is closed under arbitrary joins. besides t0 = [0, 1] x we shall need also the following well-known fact, see e.g. [20], [21]: proposition 2.12. given two topological spaces (x, t x) and (y, t y ) a mapping f : (x, t x) → (y, t y ) is continuous if and only if the mapping of the corresponding chang-goguen fuzzy topological spaces f : (x, ω(t x)) → (y, ω(t y )) is continuous. let intαa be the interior of a fuzzy set a ∈ [0, 1] x in the chang-goguen fuzzy topology ω(tα). theorem 2.13. by setting t (a) = sup{α : intαa = a} an lm-fuzzy topology t : lx → m is defined. proof. notice first that α ≤ β =⇒ intβa ≤ intαa ≤ a, ∀a ∈ l x and ∀α, β ∈ m, and hence the definition of the mapping t : lx → m is correct. (1) since constant functions cx : (x, tα) → k ⊆ [0, 1], c ∈ k are continuous for every α ∈ k, we conclude that intαcx = cx, and hence t (cx) = 1. (2) let a, b ∈ lx and let t (a) = α, t (b) = β. without loss of generality assume that α ≤ β. then for every ε > 0 such that α − ε ∈ k we have a = intα−εa, and b = intβ−εb = intα−εb. hence a ∧ b = (intα−εa) ∧ (intα−εb) = intα−ε(a ∧ b). thus t (a ∧ b) ≥ α and hence t (a ∧ b) ≥ t (a) ∧ t (b). (3) let {ai : i ∈ i} ⊆ l x be a family of fuzzy subsets of x and let α = ∧ i∈i t (ai). then for every ε > 0 such that α − ε ∈ k and every i ∈ i it holds ai = intα−εai. hence ∨ i∈i ai = ∨ i∈i intα−εai ≤ intα−ε ∨ i∈i ai. since the opposite inequality is obvious, we have ∨ i∈i ai = intα−ε ∨ i∈i ai, and hence t ( ∨ i∈iai) ≥ ∧ i∈it (ai). � remark 2.14. by making restrictions of the construction in an lm-fuzzy topology on the range l of fuzzy sets or on the range m of the fuzzy topology we come to the following special cases: (1) let l = 2 is the two-element lattice, that is 2 = {0, 1} and m = [0, 1]. in this case our construction reduces to the construction described in [26] and gives an 2m-fuzzy topology, that is a fuzzifying topology. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 110 a construction of a fuzzy topology from a strong fuzzy metric (2) let l = [0, 1] and m = 2 be the two element lattice. in this case k = {0, 1} or k = {1} and our construction gives a fuzzy topology t : lx → 2 such that t0 = l x and t1 = ω(t1) is the given topology on the set x. (3) let l = m = 2. then our construction gives t : 2x → 2 such that t0 = 2 x is the discrete topology and t1 = t1 is the given topology. let k be a sup-dense subset of the unit interval and let {t xα : α ∈ k}, {t yα : α ∈ k} be non-increasing families of topologies on the sets x and y respectively. further, let {ω(t xα ) : α ∈ k}, {ω(t y α ) : α ∈ k} be the corresponding chang-goguen fuzzy topologies, and let t x : lx → m and t y : ly → m be the lm-fuzzy topologies constructed from families {ω(t yα ) : α ∈ k} and {ω(t yα ) : α ∈ k}, respectively. theorem 2.15. a function f : (x, t x) → (y, t y ) is continuous if and only if the function f : (x, ω(t xα )) → (y, ω(t y α )) is continuous for every α ∈ k, proof. assume first that f : (x, ω(t xα )) → (y, ω(t y α )) is continuous for every α ∈ k and, given b ∈ ly , let t y (b) = β. we have to show that t x(f−1(b)) ≥ β. in case β = 0 the statement is obvious. therefore we assume that β > 0. then for every ε > 0 such that β −ε ∈ kit holds intβ−εb = b and hence b ∈ ω(t yβ−ε) (here without loss of generality we assume that β −ε ∈ k). from the continuity of all mappings f : (x, ω(t xα )) → (y, ω(t y α )) it follows that f−1(b) ∈ ω(t xβ−ε). hence for every δ > 0 such that β −ε−δ ∈ k it holds f−1(b) = intβ−ε−δf −1(b). from here we easily get the required inequality t x(f−1(b)) ≥ β = t x(b). conversely, assume that f : (x, ω(t xα )) → (y, ω(t y α )) is not continuous for some α. then there exists ε > 0 such that α − ε ∈ k and v ∈ ω(t yα−ε) but f−1(v ) 6∈ ω(t xα−ε). however this means that t x(f−1(v )) ≤ α − ε < t y (v ), and hence the function f : (x, t x) → (y, t y ) is not continuous. � 3. lm-fuzzy topology induced by a strong fuzzy metric 3.1. construction of an lm-fuzzy topology on a strong fuzzy metric space. let (x, m, ∗) be a strong fuzzy metric space. in order to define a relation between properties of a fuzzy metric for a fixed parameter t ∈ r+ and α-levels of the lm-fuzzy topology, that we are going to construct, we take a strictly increasing continuous bijection ϕ : (0, ∞) → (0, 1). we fix α ∈ (0, 1) and consider the family bα = {bm(x, r, t) : x ∈ x, r ∈ (0, 1)}, where t = ϕ −1(α). then, bα is a base of a topology t m α on the set x. indeed, it is easy to verify (see e.g. [10]) that mt : x × x → [0, 1] defined by mt(x, y) = m(x, y, t) for x, y ∈ x, is a stationary fuzzy metric on x which has as a base the family {bmt(x, r) : x ∈ x, r ∈ (0, 1)}. this topology is characterized in the next theorem. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 111 s. grecova, a. šostak and i. uļjane theorem 3.1 ([26]). let α ∈ (0, 1) and let u ∈ 2x. then u ∈ t mα if and only if for each x ∈ u there exists δ ∈ (0, 1) such that bm(x, δ, t) ⊆ u, where t = ϕ−1(α). from this theorem and taking into account that for each x ∈ x, for each δ ∈ (0, 1), and for every t > 0 the inclusion bm(x, δ, s) ⊆ bm(x, δ, t) holds whenever 0 < s < t, we obtain the next corollary. corollary 3.2 ([26]). if u ∈ t mα , then u ∈ t m β whenever β < α, and hence the family {t mα : α ∈ (0, 1)} is non-increasing. now referring to subsection 2.3 from corollary 3.2 we get the following theorem 3.3. let (x, m) be a strong fuzzy metric space. by setting t m(a) = ∨ {α : a ∈ ω(t mα )} for every a ∈ l x we get an lm-fuzzy topology t m : lx → m. in the sequel we refer to the fuzzy topology t m constructed in the previous theorem as a fuzzy topology induced by the fuzzy metric m. 3.2. case of a principal fuzzy metric. definition 3.4. ([8]) (x, m, ∗) is called principal (or just, m is principal) if {b(x, ε, t) : r ∈ (0, 1)} is a local base at x ∈ x, for each x ∈ x and each t > 0. by definition of a principal fuzzy metric and corollary 3.2, it is easy to verify that if (x, m, ∗) is a strong principal fuzzy metric space, then tα = tβ for all α, β ∈ (0, 1). hence also ω(tα) = ω(tβ) for all α, β ∈ (0, 1] and therefore the resulting fuzzy topology is a chang-goguen type topology, or l2-fuzzy topology in our notations. in particular, as one can expect, fuzzy topology generated by the standard fuzzy metric, that is by fuzzy metric md md(x, y, t) = t t + d(x, y) where d : x × x → [0, ∞) is an ordinary metric on the set x, is a changgoguen fuzzy topology. we reformulate this fact as follows: example 3.5. let (x, d) be a metric space and let md be the corresponding standard metric. further, let t md be the fuzzy topology induced by md. then for every u ∈ [0, 1]x t md(u) = { 1 if u is lower semicontinuous 0 otherwise 3.3. continuity of mappings of lm-fuzzy topological spaces versus continuity of mappings of strong fuzzy metric spaces. as different from the concordant situation in case of fuzzy metrics and the induced topologies, (see proposition 2.6), the concept of continuity of mappings of fuzzy metric spaces (definition 2.10) is not coherent with the concept of continuity of the mappings of the induced lm-fuzzy topological spaces (definition 2.10). this fact was known already in case of fuzzyfying topologies induced by fuzzy metrics. [26]. therefore, in order to describe the relations between continuity of c© agt, upv, 2016 appl. gen. topol. 17, no. 2 112 a construction of a fuzzy topology from a strong fuzzy metric mappings between fuzzy metric spaces and the continuity of mappings between the induced fuzzy topological spaces, we need to consider the following stronger version of continuity for fuzzy metric spaces introduced in [7]: definition 3.6. a mapping f : (x, m, ∗m) → (y, n, ∗n) is called strongly continuous at a point x ∈ x if given ε ∈ (0, 1) and t > 0 there exists δ ∈ (0, 1) such that m(x, y, t) > 1 − δ implies n(f(x), f(y), t) > 1 − ε. we say that f : (x, m, ∗m) → (y, n, ∗n) is strongly continuous (on x) if it is strongly continuous at each point x ∈ x. remark 3.7. in paper [7] this property of a mapping f : (x, m, ∗m) → (y, n, ∗n) was called t-continuity. here, we think it is reasonable (following also [26]) to recall this property as strong continuity first because it is well related with the concept of a strong fuzzy metric which is fundamental for this paper, and second, because the prefix t in front of the adjective “continuous” seems to be misleading in this context. theorem 3.8. a mapping f : (x, t m) → (y, t n) of lm-fuzzy topological spaces induced by fuzzy metrics m : x × x × (0, ∞) → [0, 1] and n : y × y × (0, ∞) → [0, 1], respectively, is continuous if and only if the mapping f : (x, m, ∗m) → (y, n, ∗n) is strongly continuous. proof. suppose that a mapping f : (x, t m) → (y, t n) is continuous. then for every α ∈ [0, 1] the mapping f : (x, t mα ) → (y, t n α ) is continuous (theorem 2.11). let x ∈ x and take any bn(f(x), ε, t) ∈ t n α , where α = ϕ(t). since bn(f(x), ε, t) ∈ t n α , where α = ϕ(t) and f is continuous, we have that f−1(bn(f(x), ε, t)) ∈ t m α . therefore, for each x ′ ∈ f−1(bn(f(x), ε, t)) we can find δ ∈ (0, 1) such that bm(x ′, δ, t) ⊆ f−1(bn(f(x), ε, t)). in particular, since x ∈ f−1(bn(f(x), ε, t)) there exists δ ∈ (0, 1) such that bm(x, δ, t) ⊆ f−1(bn(f(x), ε, t). however, this means that if x ′ ∈ bm(x, δ, t), that is if m(x, x′, t) > 1−δ, then x′ ∈ f−1(bn(f(x), ε, t)), that is n(f(x), f(y), t) > 1−ε. therefore, the mapping f : (x, m, ∗m) → (y, n, ∗n) is strongly continuous at a point x, and, since x ∈ x is arbitrary, the mapping f : (x, m, ∗m) → (y, n, ∗n) is strongly continuous. conversely, suppose that a mapping f : (x, m, ∗m) → (y, n, ∗n) is strongly continuous, but f : (x, t m) → (y, t n) is not continuous. then there, applying theorem 2.15 we conclude that there exists α ∈ [0, 1] such that f : (x, ω(t mα )) → (y, (t n α )) is not continuous. then, we can find v ∈ l y such that v ∈ t nα , but f −1(v ) 6∈ t nα . referring to proposition 2.12 without loss of generality we may assume that v ∈ 2x. the inequality f−1(v ) /∈ t mα means that there exists x0 ∈ f −1(v ) such that a 6⊆ f−1(v ) for each a ∈ t mα containing point x0, and, in particular, bm(x0, δ, t) 6⊆ f −1(v ) for each δ ∈ (0, 1), where t = ϕ−1(α). on the other hand, since f(x0) ∈ v ∈ t n α , we can find ε ∈ (0, 1) such that bn(f(x0), ε, t) ⊆ v. therefore, we have found x0 ∈ x and ε ∈ (0, 1) such that for each δ ∈ (0, 1) it holds bm(x0, δ, t) 6⊆ f −1 (bn(f(x0), ε0, t)) where t = ϕ−1(α). however, this means that for each δ ∈ (0, 1) there exists a point c© agt, upv, 2016 appl. gen. topol. 17, no. 2 113 s. grecova, a. šostak and i. uļjane x ∈ x such that m(x0, x, t) > 1−δ, but n(f(x0), f(x), t) ≤ 1−ε and hence the mapping f : (x, m, ∗m) → (y, n, ∗n) is not strongly continuous at the point x0. the obtained contradiction completes the proof. � let fuzsm denote the the subcategory of the category fuzm whose objects are strong metric spaces and whose morphisms are strongly continuous mappings of strong fuzzy metric spaces. from the previous theorem we get the following statement: corollary 3.9. by assigning to every strong fuzzy metric space (x, m, ∗) the lm-fuzzy topological space f(x, m, ∗) = (x, t m) and assigning to every strongly continuous mapping f : (x, m, ∗m) → (y, n, ∗n) the mapping f(f) = f : (x, t m) → (y, t n) we obtain the functor f : fuzsmet → fuztop(lm). 4. conclusion we presented here a method allowing to construct for a given strong fuzzy metric space (x, m, ∗) an lm-fuzzy topological space (x, t m) where l, m are complete sublattices of the unit interval [0, 1]. in case l = {0, 1} this construction comes to the construction of a fuzzifying topology developed in [26]. although we restricted ourselves by the case of strong fuzzy metric spaces, it is clear that all concepts considered here and all results obtained in an obvious way can be reformulated for the case of strong fuzzy pseudometric spaces. at the end of the last section a functor f : fuzsmet → fuztop(l, m) was introduced. an actual problems is to study properties of this functor. in particular, we plan to study preservation of such operations as products, co-products, quotients, etc., by this functor. another challenge is to study categorical properties of the subcategory ffuzsmet in the category fuztop(l, m) as well as categorical properties of the subcategory fuzsmet in the category fuzmet it is important to consider concrete examples of strong fuzzy metrics and the induced fuzzy topologies. as it was said above, in case of a principal fuzzy metric our construction brings forward to a chang-goguen fuzzy topology. in particular, starting with the standard fuzzy metric, we come to a changgoguen fuzzy topology. therefore, to get a general, say lm-fuzzy topology for l = m = [0, 1] we have to start with a strong fuzzy metric which is not principal. some examples of such fuzzy metrics can be found in [6] and [26]. acknowledgements. the authors are thankful to the anonymous referee for making remarks which allowed to improve exposition of the paper. references [1] c. l. chang, fuzzy topological spaces, j. math. anal. appl. 24 (1968), 182–190. 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(abbas.mujahid@gmail.com) abstract in this paper, we introduce a new contraction called dualistic contraction of rational type and used it to obtain some fixed point results in ordered dualistic partial metric spaces. these results generalize various comparable results appeared in the literature. we provide an example to show the usefulness of our results among corresponding fixed point results proved in metric spaces. 2010 msc: 47h09; 47h10; 54h25. keywords: fixed point; dualistic partial metric; dualistic contraction of rational type. 1. introduction matthews [3] introduced the concept of partial metric space as a suitable mathematical tool for program verification and proved an analogue of banach fixed point theorem in complete partial metric spaces. o’neill [7] introduced the notion of dualistic partial metric, which is more general than partial metric and established a robust relationship between dualistic partial metric and quasi metric. in [9], oltra and valero presented a banach fixed point theorem on complete dualistic partial metric spaces and in this way presented a generalization of famous banach fixed point theorem. they also showed that the received 11 june 2016 – accepted 28 august 2016 http://dx.doi.org/10.4995/agt.2016.5920 m. nazam, m. arshad, m. abbas contractive condition in banach fixed point theorem in complete dualistic partial metric spaces cannot be replaced by the contractive condition of banach fixed point theorem for complete partial metric spaces. later on, nazam et al. [4] established a fixed point theorem for geraghty type contractions in ordered dualistic partial metric spaces and applied this result to show the existence of solution of integral equations . harjani et al. [1] extended banach fixed point principle as follows: theorem 1.1 ([1]). let m be complete ordered metric space and t : m → m a continuous and non decreasing mapping satisfying, d(t(j),t(k)) ≤ αd(j,t(j)) ·d(k,t(k)) d(j,k) + βd(j,k), for all comparable j,k ∈ m with j 6= k and 0 < α + β < 1. then t has a unique fixed point m∗ ∈ m. moreover, the picard iterative sequence {tn(j)}n∈n converges to m∗ for every j ∈ m. isik and tukroglu [2] presented an ordered partial metric space version of theorem 1.1, stated below: theorem 1.2 ([2]). let m be complete ordered partial metric space and t : m → m a continuous and non decreasing mapping satisfying, d(t(j),t(k)) ≤ αd(j,t(j)) ·d(k,t(k)) d(j,k) + βd(j,k), for all comparable j,k ∈ m with j 6= k and 0 < α + β < 1. then t has a unique fixed point m∗ ∈ m. moreover, the picard iterative sequence {tn(j)}n∈n converges to m∗ for every j ∈ m. in this paper, we obtain some fixed point theorems for dualistic contractions of rational type. these results extend the comparable results in [2]. we give examples to show that existing results in partial metric space cannot be applied to obtain fixed points of mappings involved herein. 2. preliminaries following mathematical basics will be needed in the sequel. throughout this paper, we denote (0,∞) by r+, [0,∞) by r+0 , (−∞, +∞) by r and set of natural numbers by n. let t be a self mapping on a nonempty set m. an element m∗ ∈ m is called a fixed point of t if it remains invariant under the action of t . if j0 is a given point in m, then a sequence {jn} in m by jn = t(jn−1) = tn(j0), n ∈ n is called sequence is called picard iterative sequence with initial guess j0. o’neill [7] introduced the notion of a dualistic partial metric space as a generalization of partial metric space in order to expand the connections between partial metrics and semantics via valuation spaces. according to o’neill, a dualistic partial metric can be defined as follow. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 200 dualistic contractions of rational type definition 2.1 ([7]). let m be a nonempty set. a function d : m ×m → r is called a dualistic partial metric if for any j,k, l ∈ m, the following conditions hold: (d1) j = k ⇔ d(j,j) = d(k,k) = d(j,k). (d2) d(j,j) ≤ d(j,k). (d3) d(j,k) = d(k,j). (d4) d(j, l) ≤ d(j,k) + d(k,l) −d(k,k). we observe that, as in the metric case, if d is a dualistic partial metric then d(j,k) = 0 implies j = k. in case d(j,k) ∈ r+0 for all j,k ∈ m, then d is a partial metric on m. if (m,d) is a dualistic partial metric space, then the function dd : m ×m → r+0 defined by dd(j,k) = d(j,k) −d(j,j), is a quasi metric on m such that τ(d) = τ(dd). in this case, d s d(j,k) = max{dd(j,k),dd(k,j)} defines a metric on m. remark 2.2. it is obvious that every partial metric is dualistic partial metric but converse is not true. to support this comment, define d∨ : r×r → r by d∨(j,k) = j ∨k = sup{j,k} ∀ j,k ∈ r. it is easy to check that d∨ is a dualistic partial metric. note that d∨ is not a partial metric, because d∨(−1,−2) = −1 /∈ r+0 . nevertheless, the restriction of d∨ to r+0 , d∨|r+0 , is a partial metric. example 2.3. if (m,d) is a metric space and c ∈ r is an arbitrary constant, then d : m ×m → r given by d(j,k) = d(j,k) + c. defines a dualistic partial metric on m. following [7], each dualistic partial metric d on m generates a t0 topology τ(d) on m. the elements of the topology τ(d) are open balls of the form {bd(j,�) : j ∈ m,� > 0} where bd(j,�) = {k ∈ m : d(j,k) < � + d(j,j)}. a sequence {jn}n∈n in (m,d) converges to a point j ∈ m if and only if d(j,j) = limn→∞d(j,jn). definition 2.4 ([7]). let (m,d) be a dualistic partial metric space, then (1) a sequence {jn}n∈n in (m,d) is called a cauchy sequence if limn,m→∞d(jn,jm) exists and is finite. (2) a dualistic partial metric space (m,d) is said to be complete if every cauchy sequence {jn}n∈n in m converges, with respect to τ(d), to a point j ∈ m such that d(j,j) = limn,m→∞d(jn,jm). following lemma will be helpful in the sequel. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 201 m. nazam, m. arshad, m. abbas lemma 2.5 ([7, 9]). (1) a dualistic partial metric (m,d) is complete if and only if the metric space (m,dsd) is complete. (2) a sequence {jn}n∈n in m converges to a point j ∈ m, with respect to τ(dsd) if and only if limn→∞d(j,jn) = d(j,j) = limn→∞d(jn,jm). (3) if limn→∞ jn = υ such that d(υ,υ) = 0 then limn→∞d(jn,k) = d(υ,k) for every k ∈ m. oltra and valero ([6]) extended partial metric space version of the banach contraction principle to dualistic partial metric spaces. theorem 2.6 ([6]). let (m,d) be a complete dualistic partial metric space and t : m → m. if there exists α ∈ [0, 1[ such that |d(t(j),t(k))| ≤ α|d(j,k)|, for any j,k ∈ m. then t has a unique fixed point m∗ ∈ m. moreover, d(m∗,m∗) = 0 and for every j ∈ m, the picard iterative sequence {tn(j)}n∈n converges with respect to τ(dsd) to m ∗. 3. the results in this section, we shall show that, the dualistic contractions of rational type along with certain conditions have unique fixed point. we will support obtained results by some concrete examples. we introduce the following, definition 3.1. let (m,�,d) be an ordered dualistic partial metric space. a self-mapping t defined on m is called dualistic contraction of rational type if for any j,k ∈ m, we have (3.1) |d(t(j),t(k))| ≤ α ∣∣∣∣d(j,t(j)) ·d(k,t(k))d(j,k) ∣∣∣∣ + β|d(j,k)|, for all comparable j,k ∈ m and 0 < α + β < 1. we start with the following result. theorem 3.2. let (m,�,d) be a complete ordered dualistic partial metric space and t : m → m be a continuous and non decreasing dualistic contraction of rational type. then t has a fixed point m∗ in m provided there exists j0 ∈ m such that j0 � t(j0). moreover, d(m∗,m∗) = 0. proof. let j0 be a given point ∈ m and jn = t(jn−1) , n ≥ 1 an iterative sequence starting with j0. if there exists a positive integer r such that jr+1 = jr, then jr is a fixed point of t and d(jr,jr) = 0. suppose that jn 6= jn+1 for any n ∈ n. as j0 � t(j0) = j1 , that is, j0 � j1 which further implies that j1 = t(j0) � t(j1) = j2. continuing this way, we obtain that j0 � j1 � j2 � j3 � ···� jn � jn+1.... c© agt, upv, 2016 appl. gen. topol. 17, no. 2 202 dualistic contractions of rational type since jn � jn+1 by (3.1), we have |d(jn,jn+1)| = |d(t(jn−1),t(jn))| ≤ α ∣∣∣∣d(jn−1,jn) ·d(jn,jn+1)d(jn−1,jn) ∣∣∣∣ + β|d(jn−1,jn)|, ≤ α|d(jn,jn+1)| + β|d(jn−1,jn)|, |d(jn,jn+1)|−α|d(jn,jn+1)| ≤ β|d(jn−1,jn)|, (1 −α)|d(jn,jn+1)| ≤ β|d(jn−1,jn)|, |d(jn,jn+1)| ≤ ( β 1 −α )|d(jn−1,jn)|. if γ = β 1 −α , then 0 < γ < 1 and we have (3.2) |d(jn,jn+1)| ≤ γ|d(jn−1,jn)|. thus (3.3) |d(jn,jn+1)| ≤ γ|d(jn−1,jn)| ≤ γ2|d(jn−2,jn−1)| ≤ ·· · ≤ γn|d(j0,j1)|. as jn � jn, for each n ∈ n, by (3.1) we have |d(jn,jn)| = |d(t(jn−1),t(jn−1))| ≤ α|d(jn−1,jn)|2 |d(jn−1,jn−1)| + β|d(jn−1,jn−1)| ≤ |d(jn−1,jn−1)| { α ∣∣∣∣ d(jn−1,jn)d(jn−1,jn−1) ∣∣∣∣2 + β } ≤ (α + β)|d(jn−1,jn−1)|. indeed ∣∣∣∣ d(jn−1,jn)d(jn−1,jn−1) ∣∣∣∣2 = 1. thus we obtain that (3.4) |d(jn,jn)| ≤ (α + β)n|d(j0,j0)|. now we show that {jn} is a cauchy sequence in (m,dsd). note that, dd(jn,jn+1) = d(jn,jn+1) − d(jn,jn), that is, dd(jn,jn+1) + d(jn,jn) = d(jn,jn+1) ≤ |d(jn,jn+1)|. thus, we have dd(jn,jn+1) + d(jn,jn) ≤ γn|d(j0,j1)|. dd(jn,jn+1) ≤ γn|d(j0,j1)| + |d(jn,jn)| ≤ γn|d(j0,j1)| + (α + β)n|d(j0,j0)| continuing this way, we obtain that dd(jn+k−1,jn+k) ≤ γn+k−1|d(j0,j1)| + (α + β)n+k−1|d(j0,j0)| c© agt, upv, 2016 appl. gen. topol. 17, no. 2 203 m. nazam, m. arshad, m. abbas now dd(jn,jn+k) ≤ dd(jn,jn+1) + dd(jn+1,jn+2) + · · · + dd(jn+k−1,jn+k). ≤ {γn + γn+1 + · · · + γn+k−1}|d(j0,j1)| + {(α + β)n + (α + β)n+1 + · · · + (α + β)n+k−1}|d(j0,j0)|. thus for n + k = m > n (3.5) dd(jn,jm) ≤ γn 1 −γ |d(j0,j1)| + (α + β)n 1 − (α + β) |d(j0,j0)|. similarly, we have (3.6) dd(jm,jn) ≤ γn 1 −γ |d(j1,j0)| + (α + β)n 1 − (α + β) |d(j0,j0)|. on taking limit as n,m →∞, we have lim n,m→∞ dd(jm,jn) = 0 = lim n,m→∞ dd(jn,jm) and hence lim n,m→∞ dsd(jm,jn) = 0, we get that {jn} is a cauchy sequence in (m,dsd). since (m,d) is a complete dualistic partial metric space, so by lemma 2.5 (m,dsd) is also a complete metric space. thus, there exists m∗ in m such that limn→∞d s d(jn,m ∗) = 0, again from lemma 2.5, we get (3.7) lim n→∞ dsd(jn,m ∗) = 0 ⇐⇒ d(m∗,m∗) = lim n→∞ d(jn,m ∗) = lim n,m→∞ d(jm,jn). now limn,m→∞dd(jm,jn) = 0 implies that limn,m→∞[d(jm,jn)−d(jn,jn)] = 0 and hence limn,m→∞d(jn,jm) = limn→∞d(jn,jn). by (3.4), we have limn→∞d(jn,jn) = 0. consequently, limn,m→∞d(jn,jm) = 0. thus (3.8) d(m∗,m∗) = lim n→∞ d(jn,m ∗) = 0. now dd(m ∗,t(m∗)) = d(m∗,t(m∗)) −d(m∗,m∗) = d(m∗,t(m∗)). implies that d(m∗,t(m∗)) ≥ 0. since t is continuous, for a given � > 0, there exists δ > 0 such that t(bd(m ∗,δ)) ⊆ bd(t(m∗),�). since limn→∞d(jn+1,m∗) = d(m∗,m∗) = 0, so there exists r ∈ n such that d(jn,m∗) < d(m∗,m∗) + δ ∀ n ≥ r, therefore {jn} ⊂ bd(m∗,δ) ∀ n ≥ r. this implies that t(jn) ∈ t(bd(m ∗,δ) ⊆ bd(t(m∗),�) and so d(t(jn),t(m∗)) < d(t(m∗),t(m∗)) +� ∀ n ≥ r. now for any � > 0, we know that −� + d(t(m∗),t(m∗)) < d(t(m∗),t(m∗)) ≤ d(jn+1,t(m∗)) which yields that |d(jn+1,t(m∗)) −d(t(m∗),t(m∗))| < � that is d(t(m∗),t(m∗)) = limn→∞d(jn+1,t(m ∗)), finally uniqueness of limit in r implies (3.9) lim n→∞ d(jn+1,t(m ∗)) = d(t(m∗),t(m∗)) = d(m∗,t(m∗)). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 204 dualistic contractions of rational type finally, we have d(t(m∗),m∗) = limn→∞d(t(jn),jn) = limn→∞d(jn+1,jn) = 0. this shows that d(m∗,t(m∗)) = 0. so from (3.8) and (3.9) we deduce that d(m∗,t(m∗)) = d(t(m∗),t(m∗)) = d(m∗,m∗). this leads us to conclude that m∗ = t(m∗) and hence m∗ is a fixed point of t � in order to prove the uniqueness of fixed point of a mapping t in the above theorem, we need an additional assumption. theorem 3.3. let (m,d,�) be complete ordered dualistic partial metric space and t : m → m a mapping which satisfy all conditions of theorem (3.2). then t has a unique fixed point provided that for each fixed point m∗, n∗ of t, there exists ω ∈ m which is comparable to both m∗ and n∗. proof. from theorem (3.2), it follows that the set of fixed points of t is nonempty. to prove the uniqueness: let n∗ be another fixed point of t, that is, n∗ = t(n∗) and d(n∗,n∗) = 0. if m∗ and n∗ are comparable (m∗ � n∗), then we have, |d(m∗,n∗)| = |d(t(m∗),t(n∗))|, ≤ α ∣∣∣∣d(m∗,t(m∗)) ·d(n∗,t(n∗))d(m∗,n∗) ∣∣∣∣ + β|d(m∗,n∗)|. ≤ α ∣∣∣∣d(m∗,m∗) ·d(n∗,n∗)d(m∗,n∗) ∣∣∣∣ + β|d(m∗,n∗)|. that is, (1 −β)|d(m∗,n∗)| ≤ 0 which implies that |d(m∗,n∗)| ≤ 0 and hence d(m∗,n∗) = 0 = d(m∗,m∗) = d(n∗,n∗). the result follows. suppose that m∗ and n∗ are incomparable, there exists ω which is comparable to both m∗, n∗. without any loss of generality, we assume that m∗ � ω, and n∗ � ω. as t is non decreasing, t(m∗) � t(ω) and t(n∗) � t(ω) imply that tn−1(m∗) � tn−1(ω) and tn−1(n∗) � tn−1(ω). thus |d(tn(m∗),tn(ω))| ≤ α ∣∣∣∣d(tn−1(m∗),tn(m∗)) ·d(tn−1(ω),tn(ω))d(tn−1(m∗),tn−1(ω)) ∣∣∣∣ + β|d(tn−1(m∗),tn−1(ω))|. that is, |d(m∗,tn(ω))| ≤ β|d(m∗,tn−1(ω))|. thus, limn→∞d(m∗,tn(ω)) = 0. similarly, we can have limn→∞d(n ∗,tn(ω)) = 0. note that d(n∗,m∗) ≤ d(n∗,tn(ω)) + d(tn(ω),m∗) −d(tn(ω),tn(ω)), ≤ d(n∗,tn(ω)) + d(tn(ω),m∗) −d(tn(ω),m∗) −d(m∗,tn(ω)) + d(m∗,m∗). on taking limit as n →∞ we obtain that d(n∗,m∗) ≤ 0 . now dd(m∗,m∗) = d(n∗,m∗)−d(n∗,n∗) implies that d(n∗,m∗) ≥ 0. hence d(n∗,m∗) = 0 which gives that n∗ = m∗ � c© agt, upv, 2016 appl. gen. topol. 17, no. 2 205 m. nazam, m. arshad, m. abbas example 3.4. let m = r2. define d∨ : m ×m → r by d∨(j,k) = j1 ∨k1, where j = (j1,j2) and k = (k1,k2). note that (m,d∨) is a complete dualistic partial metric space. let t : m → m be given by t(j) = j 2 for all j ∈ m. in m, we define the relation � in the following way: j � k if and only if j1 ≥ k1, where j = (j1,j2) and k = (k1,k2). clearly, � is a partial order on m and t is continuous, non decreasing mapping with respect to �. moreover, t(−1, 0) � (−1, 0). we shall show that for all j,k ∈ m, (3.1) is satisfied. for this, note that |d∨(t(j),t(k))| = ∣∣∣∣d∨ ( j 2 , k 2 )∣∣∣∣ = ∣∣∣∣j12 ∣∣∣∣ for all j1 ≥ k1, |d∨(j,t(j))| = ∣∣∣∣d∨ ( j, j 2 )∣∣∣∣ = { ∣∣j1 2 ∣∣ if j1 ≤ 0 |j1| if j1 ≥ 0, |d∨(k,t(k))| = ∣∣∣∣d∨ ( k, k 2 )∣∣∣∣ = { ∣∣k1 2 ∣∣ if k1 ≤ 0 |k1| if k1 ≥ 0, and |d∨(j,k)| = |j1| for all j1 ≥ k1. now for α = 1 3 , β = 1 2 . if j1 ≤ 0, k1 ≤ 0, then |d∨(t(j),t(k))| ≤ α ∣∣∣∣d∨(j,t(j)) ·d∨(k,t(k))d∨(j,k) ∣∣∣∣ + β|d∨(j,k)| for all j � k holds if and only if 6|j1| ≤ |k1| + 6|j1|. for if j1 ≥ 0, k1 ≥ 0, then contractive condition |d∨(t(j),t(k))| ≤ α ∣∣∣∣d∨(j,t(j)) ·d∨(k,t(k))d∨(j,k) ∣∣∣∣ + β|d∨(j,k)| for all j � k holds if and only if j1 ≤ 23k1 + j1. finally, if j1 ≥ 0, k1 ≤ 0, then |d∨(t(j),t(k))| ≤ α ∣∣∣∣d∨(j,t(j)) ·d∨(k,t(k))d∨(j,k) ∣∣∣∣ + β|d∨(j,k)| ∀ j � k holds if and only if 3j1 ≤ |k1| + 3j1. thus, all the conditions of theorem 3.2 are satisfied. moreover, (0, 0) is a fixed point of t . remark 3.5. as every dualistic partial metric d is an extension of partial metric p, therefore, theorem 3.2 is an extension of theorem 1.2. there arises the following natural question: whether the contractive condition in the statement of theorem 3.2 can be replaced by the contractive condition in theorem 1.2. the following example provides a negative answer to the above question. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 206 dualistic contractions of rational type example 3.6. define the mapping t0 : r+ → r by t0(j) = { 0 if j > 1 −5 if j = 1 . clearly, for any j,k ∈ r, following contractive condition is satisfied d∨(t0(j),t0(k)) ≤ αd∨(j,t0(j)) ·d∨(k,t0(k)) d∨(j,k) + βd∨(j,k) where d∨ is a complete dualistic partial metric on r. here, t has no fixed point. thus a fixed point free mapping satisfies the contractive condition of theorem 1.2. on the other hand, for all 0 < α + β < 1, we have 5 = |d∨(−5,−5)| = |d∨(t0(1),t0(1))| > α ∣∣∣∣d∨(1,t0(1)) ·d(1,t0(1))d∨(1, 1) ∣∣∣∣+β|d∨(1, 1)|. thus contractive condition of theorem 3.2 does not hold. theorem 3.2 remains true if we replace the continuity hypothesis by the following property: (h): if {jn} is a non decreasing sequence in m such that jn → υ, then (3.10) jn � υ for all n ∈ n. this statement is given as follows: theorem 3.7. let (m,�,d) be a complete ordered dualistic partial metric space and if, (i) t : m → m be a non decreasing dualistic contraction of rational type. (ii) there exists j0 ∈ m such that j0 � t(j0). (ii) (h) holds. then t has a fixed point m∗ in m. moreover, d(m∗,m∗) = 0. proof. following the proof of theorem 3.2, we know that {jn} is non decreasing sequence in m such that jn → m∗. by (h), we have jn � m∗. as t is non decreasing, we have t(jn) � t(m∗), that is, jn+1 � t(m∗). also, j0 � j1 � t(m∗) and jn � m∗ , n ≥ 1 imply that (3.11) m∗ � t(m∗). from the proof of theorem 3.2, we deduce that {tn(m∗)} is non decreasing sequence. suppose that limn→+∞t n(m∗) = µ for some µ ∈ m. now j0 � m∗ gives tn(j0) � tn(m∗), that is, jn � tn(m∗) for all n ≥ 1. thus we have jn � m∗ � t(m∗) � tn(m∗) n ≥ 1. by (3.1), we have |d(jn+1,tn+1(m∗))| = |d(t(jn),t(tn(m∗)))|, ≤ α ∣∣∣∣d(jn,jn+1) ·d(tn(m∗),tn+1(m∗))d(jn,tn(m∗)) ∣∣∣∣ + β|d(jn,tn(m∗))|. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 207 m. nazam, m. arshad, m. abbas on taking limit as n approaches to plus infinity, we obtain that |d(m∗,µ)| ≤ β|d(m∗,µ)|. which implies that m∗ = µ. thus limn→+∞t n(m∗) = µ implies that limn→+∞t n(m∗) = m∗. hence (3.12) t(m∗) � m∗ from (3.11) and (3.12), it follows that m∗ = t(m∗). � some deductions are given below. corollary 3.8. let (m,�,d) be a complete ordered dualistic partial metric space and t : m → m be a non decreasing mapping such that, (1) |d(t(j),t(k))| ≤ α ∣∣∣∣d(j,t(j)) ·d(k,t(k))d(j,k) ∣∣∣∣ , where 0 < α < 1. (2) there exists j0 ∈ m such that j0 � t(j0). (3) either t is continuous or (h) holds. then t has a fixed point m∗ in m. moreover, d(m∗,m∗) = 0. proof. set β = 0 in theorem 3.2. � the next deduction generalizes theorem 2.6 presented by valero in [6] corollary 3.9. let (m,�,d) be a complete ordered dualistic partial metric space and t : m → m be a non decreasing mapping such that, (1) |d(t(j),t(k))| ≤ β|d(j,k)|, where 0 < β < 1. (2) there exists j0 ∈ m such that j0 � t(j0). (3) either t is continuous or (h) holds. then t has a fixed point m∗ in m. moreover, d(m∗,m∗) = 0. proof. set α = 0 in theorem 3.2. � remark 3.10. (1) if we set d(j,k) ∈ r+0 for all j,k ∈ m in corollary 3.8 and in corollary 3.9, we obtain results in partial metric spaces. (2) if we set d(j,k) ∈ r+0 for all j,k ∈ m and d(j,j) = 0 for all j ∈ m in corollary 3.8 and in corollary 3.9, we obtain results in metric spaces. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 208 dualistic contractions of rational type acknowledgements. the authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper. references [1] j. harjani, b. lópez and k. sadarangani, a fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, abstr. appl. anal. (2010), art. id 190701. [2] h.isik and d.tukroglu, some fixed point theorems in ordered partial metric spaces, journal of inequalities and special functions 4 (2013), 13–18. [3] s. g. matthews, partial metric topology, in proceedings of the 11th summer conference on general topology and applications, 728 (1995), 183-197, the new york academy of sciences. 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[9] o. valero, on banach fixed point theorems for partial metric spaces, applied general topology 6, no. 2 (2005), 229–240. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 209 () @ appl. gen. topol. 17, no. 2(2016), 185-198doi:10.4995/agt.2016.5660 c© agt, upv, 2016 best proximity point for z-contraction and suzuki type z-contraction mappings with an application to fractional calculus somayya komal a, poom kumam a,b,∗ and dhananjay gopal b,c a department of mathematics, faculty of science, king mongkut’s university of technology thonburi (kmutt), 126 pracha uthit rd., bang mod, thung khru, bangkok 10140, thailand. (komal.musab@gmail.com, poom.kum@mail.kmutt.ac.th) b theoretical and computational science center (tacs), science laboratory building, faculty of science, king mongkut’s university of technology thonburi (kmutt), 126 pracha uthit rd., bang mod, thung khru, bangkok 10140, thailand. (poom.kum@mail.kmutt.ac.th) c department of applied mathematics and humanities, sv national institute of technology, surat, gujarat, india. (gopal.dhananjay@rediffmail.com) abstract in this article, we introduced the best proximity point theorems for zcontraction and suzuki type z-contraction in the setting of complete metric spaces. also by the help of weak p-property and p-property, we proved existence and uniqueness of best proximity point. there is a simple example to show the validity of our results. our results extended and unify many existing results in the literature. moreover, an application to fractional order functional differential equation is discussed. 2010 msc: 54h25; 34a08. keywords: best proximity point; weak p-property; suzuki type zcontraction; functional differential equation. ∗corresponding author: poom.kumam@mail.kmutt.ac.th and poom.kum@kmutt.ac.th received 26 april 2016 – accepted 23 june 2016 http://dx.doi.org/10.4995/agt.2016.5660 s. komal, p. kumam and d. gopal 1. introduction when we study about fixed points of different mappings satisfying certain conditions, then it is observed that this theory has enormous applications in various branches of mathematics and mathematical sciences and hence become the source of inspiration for many researchers and mathematicians working in the metric fixed point theory (see for instant [5, 16, 12, 26]). when a self mapping in a metric space has no fixed points, then it could be interesting to study the existence and uniqueness of some points that minimize the distance between the point and its corresponding image. these points are known as best proximity points. best proximity points theorems for several types of non-self mappings have been derived in [1], [2], [3], [6], [7], [8], [10], [9] and [24]. the best proximity points were introduced by [13] and modified by sadiq basha in [7]. the results about best proximity point theory have been found very briefly in the work of [6] to [9]. now, after the new generalization of banach contraction principle given by khoj. et al. in [15] by defining a notion of z-contraction, after that kumam et. al. in [16] introduced suzuki type z-contraction and unified many fixed point results. some recent contribution in this field can be found in ([18, 17, 4, 20, 21, 22, 23]). because of its importance in nonlinear analysis, we extend these generalizations and contractions to find out the unique best proximity point in metric spaces and introduced these notions for non self mappings in the light of yaq. et al. [25] by using some suitable properties. some examples and an application to fractional order functional differential equation is given to illustrate the usability of new theory. 2. preliminaries in this section, we collect some notations and notions which will be used throughout the rest of this work. let a and b be two nonempty subsets of a metric space (x, d). we will use the following notations: d(a, b) := inf{d(a, b) : a ∈ a, b ∈ b}; a0 := {a ∈ a : d(a, b) = d(a, b) for some b ∈ b}; b0 := {b ∈ b : d(a, b) = d(a, b) for some a ∈ a}. definition 2.1. an element x∗ ∈ a is said to be a best proximity point of the non-self-mapping t : a → b if it satisfies the condition that d(x∗, t x∗) = d(a, b). remark 2.2. it can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping. definition 2.3 ([15]). let ζ : [0, ∞) × [0, ∞) → r be a mapping, then ζ is called a simulation function if it satisfies the following conditions: (1) ζ(0, 0) = 0; (2) ζ(t, s) < s − t for t, s > 0; c© agt, upv, 2016 appl. gen. topol. 17, no. 2 186 best proximity point for z-contraction and suzuki type z-contraction mappings (3) if {tn}, {sn} are sequences in (0, ∞) such that lim n→∞ tn = lim n→∞ sn > 0, then lim sup n→∞ ζ(tn, sn) < 0. we denote the set of all simulation functions by z. definition 2.4 ([15]). let (x, d) be a metric space, f : x → x is a mapping and ζ ∈ z. then f is called a z-contraction with respect to ζ if the following condition holds: (2.1) ζ(d(fx, fy), d(x, y)) ≥ 0 where x, y ∈ x, with x 6= y. definition 2.5 ([16]). let (x, d) be a metric space, f : x → x is a mapping and ζ ∈ z. then f is called a suzuki type z-contraction with respect to ζ if the following condition holds: (2.2) 1 2 d(x, fx) < d(x, y) ⇒ ζ(d(fx, fy), d(x, y)) ≥ 0 where x, y ∈ x, with x 6= y. definition 2.6 ([19]). let (a, b) be a pair of nonempty subsets of a metric space (x, d) with a0 6= φ. then the pair (a, b) is said to have the p-property if and only if d(x1, y1) = d(a, b) and d(x2, y2) = d(a, b) ⇒ d(x1, x2) = d(y1, y2), where x1, x2 ∈ a0 and y1, y2 ∈ b0. definition 2.7 ([27]). let (a,b) be a pair of nonempty subsets of a metric space (x,d) with a0 6= ∅. then the pair (a,b) is said to have weak p-property if and only if for any x1, x2 ∈ a0 and y1, y2 ∈ b0 d(x1, y1) = d(a, b) d(x2, y2) = d(a, b) } ⇒ d(x1, x2) ≤ d(y1, y2). theorem 2.8 ([16]). let (x, d) be a complete metric space. define a mapping f : x → x satisfying the following conditions: (1) f is suzuki type z-contraction with respect to ζ; (2) for every bounded picard sequence there exists a natural number k such that 1 2 d(xmk , xmk+1) < d(xmk , xnk ) for mk > nk ≥ k. then there exists unique fixed point in x and the picard iteration sequence {xn} defined by xn = fxn−1, n = 1, 2, ... converges to a fixed point of f, remark 2.9 ([15]). every z-contraction is contractive and hence banach contraction. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 187 s. komal, p. kumam and d. gopal theorem 2.10 ([5]). let (x, d) be a complete metric space. then every contraction mapping has a unique fixed point. it is known as banach contraction principle. 3. main results in this section, we will introduced the notion of generalized contraction principle for non self mappings by combining suzuki and zcontraction mappings and will find the unique best proximity point. definition 3.1. let (x, d) be a metric space, f : a → b is a mapping and ζ ∈ z. then f is called a z-contraction with respect to ζ if the following condition holds: (3.1) ζ(d(fx, fy), d(x, y)) ≥ 0 where a, b ⊆ x and x, y ∈ a, with x 6= y. definition 3.2. let (x, d) be a metric space, f : a → b is a mapping and ζ ∈ z. then f is called a suzuki type z-contraction with respect to ζ if the following condition holds: (3.2) 1 2 d(x, fx) < d(x, y) ⇒ ζ(d(fx, fy), d(x, y)) ≥ 0 where a, b ⊆ x and x, y ∈ a, with x 6= y. remark 3.3. since the definition of simulation function implies that ζ(t, s) < 0 for all t ≥ s > 0. therefore f is suzuki type z contraction with respect to ζ, then 1 2 d(x, fx) < d(x, y) ⇒ d(fx, fy) < d(x, y) for any distinct x, y ∈ a. remark 3.4. every suzuki type z-contraction is also a z-contraction. now, we are in a position to prove best proximity point theorems for z and suzuki type z-contractions in metric spaces. theorem 3.5. let (a, b) be the pair of nonempty closed subsets of a complete metric space (x, d) such that a0 is nonempty. define a mapping f : a → b satisfying the following conditions: (1) f is z-contraction with f(a0) ⊆ b0; (2) the pair (a, b) has weak p-property. then there exists unique best proximity point in a and the iteration sequence {x2n} defined by x2n+1 = fx2n, d(x2n+2, x2n+1) = d(a, b), n = 0, 1, 2, ... converges, to x∗, for every x0 ∈ a0. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 188 best proximity point for z-contraction and suzuki type z-contraction mappings proof. first of all, we have to show that b0 is closed. for this, let us take {yn} ⊆ b0 a sequence such that yn → t ∈ b. since the pair (a, b) has weak p-property, it follows from the weak p-property that d(yn, ym) → 0 ⇒ d(xn, xm) → 0, as m, n → ∞, and xn, xm ∈ a0 and d(xn, yn) = d(xm, ym) = d(a, b). thus {xn} is a cauchy sequence and converges strongly to a point s ∈ a. by the continuity of the metric d, we have d(s, t) = d(a, b), that is t ∈ b0 and hence b0 is closed. let a0 be the closure of a0; now we have to prove that f(a0) ⊆ b0. if we take x ∈ a0 \ a0, then there exists a sequence {xn} ⊆ a0 such that xn → x. by the continuity of f and the closeness of b0, we get as fx = limn→∞ fxn ∈ b0. that is, f(a0) ⊆ b0. since f is z-contraction,which implies that 0 ≤ ζ(d(fx1, fx2), d(x1, x2)) < d(x1, x2) − d(fx1, fx2), implies that (3.3) d(fx1, fx2) < d(x1, x2). define an operator pa0 : f(a0) → a0, by pa0y = {x ∈ a0 : d(x, y) = d(a, b)}. since the pair (a, b) has weak p-property and using (5), we have d(pa0fx1, pa0fx2) ≤ d(fx1, fx2) < d(x1, x2) for any x1, x2 ∈ ao. hence ζ(d(pa0 fx1, pa0fx2), d(x1, x2)) ≥ 0. so, pa0f : a0 → a0 is a z-contraction from complete metric subspace a0 into itself. since by using remark (2.1), every z-contraction is a contraction and hence a banach contraction. thus by using theorem (2.2), pa0f has unique fixed point, that is pa0fx ∗ = x∗ ∈ a0, which implies that d(x∗, fx∗) = d(a, b). therefore, x∗ is unique in a0 such that d(x ∗, fx∗) = d(a, b). it is easily seen that x∗ is unique one in a such that d(x∗, fx∗) = d(a, b). the picard iterative sequence xn+1 = pa0fxn, n = 0, 1, 2, ... converges, for every x0 ∈ a0, to x ∗. the iteration sequence {x2n}, for n = 0, 1, 2, ... defined by, x2n+1 = fx2n, d(x2n+2, x2n+1) = d(a, b), n = 0, 1, 2, ... is exactly a subsequence of {xn}, so that it converges to x ∗, for every x0 ∈ a0. � theorem 3.6. let (a, b) be the pair of nonempty closed subsets of a complete metric space (x, d) such that a0 is nonempty. define a mapping f : a → b satisfying the following conditions: c© agt, upv, 2016 appl. gen. topol. 17, no. 2 189 s. komal, p. kumam and d. gopal (1) f is suzuki type z-contraction with f(a0) ⊆ b0; (2) the pair (a, b) has the weak p-property. then there exists unique x∗ in a such that d(x∗, fx∗) = d(a, b) and the iteration sequence {x2n} defined by x2n+1 = fx2n, d(x2n+2, x2n+1) = d(a, b), n = 0, 1, 2, ... converges, for every x0 ∈ a0 to x ∗. proof. first of all, we have to show that b0 is closed. for this, let us take {yn} ⊆ b0 a sequence such that yn → g ∈ b. since the pair (a, b) has weak p-property, it follows from weak p-property that d(yn, ym) → 0 ⇒ d(xn, xm) → 0, as m, n → ∞, and xn, xm ∈ a0 and d(xn, yn) = d(xm, ym) = d(a, b). thus {xn} is a cauchy sequence and converges strongly to a point f ∈ a. by the continuity of the metric d, we have d(f, g) = d(a, b), that is g ∈ b0 and hence b0 is closed. let a0 be the closure of a0; now we have to prove that f(a0) ⊆ b0. if we take x ∈ a0 \ a0, then there exists a sequence {xn} ⊆ a0 such that xn → x. by the continuity of f and the closeness of b0, we get as fx = limn→∞ fxn ∈ b0. that is, f(a0) ⊆ b0. define an operator pa0 : f(a0) → a0, by pa0y = {x ∈ a0 : d(x, y) = d(a, b)}. since f is suzuki type z-contraction, such that for 1 2 d(x1, fx1) < d(x1, y1), we have ζ(d(fx1, fy1), d(x1, y1)) ≥ 0. now, we claim that pa0f is suzuki type z-contraction. for this, we have to prove that 1 2 d(x1, pa0fx1) < d(x1, y1), for all x, y ∈ a. since f is suzuki type z-contraction, that is d(fx, fy) < d(x, y). by using p-property, p a0 y = {x ∈ a0 : d(x, y) = d(a, b)} and triangular inequality, we obtain 1 2 d(x1, pa0fx1) ≤ 1 2 [d(x1, y1) + d(y1, pa0fx1)] = 1 2 [d(x1, y1) + d(y1, x1)] = d(x1, y1) = d(fx1, fy1) < d(x1, y1) hence, (3.4) 1 2 d(x1, pa0fx1) < d(x1, y1). for any x1, y1 ∈ a0. which shows that ζ(d(pa0 fx1, pa0fy1), d(x1, y1)) ≥ 0, where x1, y1 ∈ a0. thus, pa0f : a0 → a0 is a suzuki type z-contraction from c© agt, upv, 2016 appl. gen. topol. 17, no. 2 190 best proximity point for z-contraction and suzuki type z-contraction mappings complete metric subspace a0 into itself. consequently, one may write by using the fact that pa0f is a suzuki type z-contraction and remark (3.1) as ⇒ d(pa0fx1, pa0fy1) < d(x1, y1). then by using theorem (2.1), pa0f has unique fixed point, that is pa0fx ∗ = x∗ ∈ a0, which implies that d(x∗, fx∗) = d(a, b). therefore, x∗ is unique in a0 such that d(x ∗, fx∗) = d(a, b). it is easily seen that x∗ is unique one in a such that d(x∗, fx∗) = d(a, b). the picard iterative sequence xn+1 = pa0fxn, n = 0, 1, 2, ... converges, for every x0 ∈ a0, to x ∗. the iteration sequence {x2n}, for n = 0, 1, 2, ... defined by, x2n+1 = fx2n, d(x2n+2, x2n+1) = d(a, b), n = 0, 1, 2, ... is exactly a subsequence of {xn}, so that it converges to x ∗, for every x0 ∈ a0. � corollary 3.7. let (x, d) be a complete metric space. define a mapping f : x → x satisfying the following conditions: (1) f is z-contraction. then there exists unique fixed point in x and the iteration sequence {x2n} defined by x2n+1 = fx2n, d(x2n+2, x2n+1) = d(a, b), n = 0, 1, 2, ... converges to x∗, for every x0 ∈ a0. proof. taking self mapping a = b = x in theorem (3.1), then we get desired result. � remark 3.8. by taking self mapping in theorem (3.2), we obtain theorem (2.1). there is an example to justify our results and remarks. example 3.9. consider x = r2, with the usual metric d. define the sets a = {(x, 1) : x ≥ 0} and b = {(x, 0) : x ≥ 0}. let a0 = a and b0 = b and clearly, the pair (a, b) has the p-property, also satisfies weak p-property. also define f : a → b as: f(x, 1) = ( x2 x + 1 , 0), c© agt, upv, 2016 appl. gen. topol. 17, no. 2 191 s. komal, p. kumam and d. gopal we take a0 = a 6= ∅, b0 = b, f(a0) ⊆ b0. then, d(f(x1, 1), f(x2, 1)) = | x21 x1 + 1 − x22 x2 + 1 | = |x21(x2 + 1) − x 2 2(x1 + 1) (x1 + 1)(x2 + 1) = |(x1x2 + x1 + x2)(x2 − x1)| |(x1 + 1)(x2 + 1)| = x1x2 + x1 + x2 (x1 + 1)(x2 + 1) |x1 − x2| = x1x2 + x1 + x2 x1x2 + x1 + x2 + 1 |x1 − x2| < |x1 − x2| = d((x1, 1), (x2, 1)). i.e. d(f(x1, 1), f(x2, 1)) < d((x1, 1), (x2, 1)), which implies that ζ(d(f(x1, 1), f(x2, 1)), d((x1, 1), (x2, 1))) ≥ 0, i.e. f is ζcontraction. thus, all the conditions of the theorem (3.1) are satisfied, and the conclusion of that theorem is also correct, that is, f has a unique best proximity point z∗ = (0, 1) ∈ a0 such that d(z ∗, fz∗) = d((0, 1), (0, 0)) = d(a, b) = 1. on the other hand, it is clear that the iteration sequence {z2k}, k = 0, 1, 2, ... defined by z2k+1 = f{z2k}, d(z2k+2, z2k+1) = d(a, b) = 1, k = 0, 1, 2, ..., converges for every z0 ∈ a0, to z ∗, since z2(k+1) = (x2(k+1), 1) = ( x22k x2k + 1 , 1) → (0, 1). in fact, from x2(k+1) = x 2 2k x2k+1 , we know that x2k+1 ≤ x2k, so there exists a number x∗ such that x2k → x ∗. furthermore, x∗ = (x∗)2 x∗+1 and hence x∗ = 0. example 3.10. consider x = r2, with the usual metric d. define the sets a = {(x, 1) : x ≥ 0} and b = {(x, 0) : x ≥ 0}. let a0 = a and b0 = b and clearly, the pair (a, b) has the p-property, also satisfies weak p-property. also define f : a → b as: f(x, 1) = ( x2 x + 1 , 0), c© agt, upv, 2016 appl. gen. topol. 17, no. 2 192 best proximity point for z-contraction and suzuki type z-contraction mappings we take a0 = a 6= ∅, b0 = b, f(a0) ⊆ b0. then, 1 2 d((x1, 1), f(x1, 1)) = 1 2 d((x1, 1), ( x21 x1 + 1 , 0)) = 1 2 |1 + (x1 − x21 x1 + 1 )| = 1 2 |1 + 1 1 + x1 | = 1 2 |1 + 1 1+x1 | |x1 − x2| |x1 − x2| = 1 2 |x1 + 2| |(x1 − x2)(x1 + 1)| |x1 − x2| < |x1 − x2| = d((x1, 1), (x2, 1)). thus, d((x1, 1), f(x1, 1)) < d((x1, 1), (x2, 1)), which implies that ζ(d(f(x1, 1), f(x2, 1)), d((x1, 1), (x2, 1))) ≥ 0, and f is suzuki type z-contraction with respect to ζ. thus, all the conditions of the theorem (3.2) are satisfied, and the conclusion of that theorem is also correct, that is, f has a unique best proximity point z∗ = (0, 1) ∈ a0 such that d(z ∗, fz∗) = d((0, 1), (0, 0)) = d(a, b) = 1 on the other hand, it is clear that the iteration sequence {z2k}, k = 0, 1, 2, ... defined by z2k+1 = f{z2k} d(z2k+2, z2k+1) = d(a, b) = 1, k = 0, 1, 2, ..., converges for every z0 ∈ a0, to z ∗, since z2(k+1) = (x2(k+1), 1) = ( x22k x2k + 1 , 1) → (0, 1). in fact, from x2(k+1) = x 2 2k x2k+1 , we know that x2k+1 ≤ x2k, so there exists a number x∗ such that x2k → x ∗. furthermore, x∗ = (x∗)2 x∗+1 and hence x∗ = 0. example 3.11. if we change the defined mapping on same conditions of above example and on little change on given sets like for a = {(1, y) : y ≥ 0} and b = {(0, y) : y ≥ 0} and a0 = a and b0 = b. define f : a → b as: f(1, y) = (0, y2 y + 1 ), as given in [25], then also with this defined mapping there exists a best proximity point for both z and suzuki type z-contractions, also after such change in the conditions, examples (3.1) and (3.2), theorems (3.1) and (3.2) verified, and that best proximity point is (1, 0) for both, that is, d(x, fx) = d((1, 0), (0, 0)) = d(a, b) = 1. if there are two best proximity points for same sets, (1, 0) and (0, 1), then their uniqueness can be proved easily as d((1, 0), (0, 0)) = d(a, b) and d((0, 1), (0, 0)) = d(a, b) = 1, then one can write as: d((1, 0), (0, 0)) = d((0, 1), (0, 0)) = d(a, b), this implies that (1, 0) = (0, 1). hence, existence and uniqueness of best proximity point in the metric space has proved. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 193 s. komal, p. kumam and d. gopal 4. application in this section, we present an application of our fixed point results derived in previous section to establish the existence of solution of fractional order functional differential equation. consider the following initial value problem (ivp for short) of the form (4.1) dαy(t) = f(t, yt), for each t ∈ j = [0, b], 0 < α < 1, (4.2) y(t) = φ(t), t ∈ (−∞, 0] where dα is the standard riemann-liouville fractional derivative, f : j × b → r, φ ∈ b, φ(0) = 0 and b is called a phase space or state space satisfying some fundamental axioms (h-1, h-2, h-3) given below which were introduced by hale and kato in [14]. for any function y defined on (−∞, b] and any t ∈ j , we denote by yt the element of b defined by yt(θ) = y(t + θ), θ ∈ (−∞, 0]. here yt(·) represents the history of the state from −∞ up to present time t. by c(j, r) we denote the banach space of all continuous functions from j into r with the norm ||y||∞ := sup{|y(t)| : t ∈ j} where | · | denotes a suitable complete norm on r. (h-1) if y : (−∞, b] → r, and y0 ∈ b, then for every t ∈ [0, b] the following conditions hold: (i) yt is in b, (ii) ||yt||b ≤ k(t) sup{|y(s)| : 0 ≤ s ≤ t} + m(t)||y0||b, (iii) |y(t)| ≤ h||yt||b, where h ≥ 0 is a constant, k : [0, b] → [0, ∞) is continuous, m : [0, ∞) → [0, ∞) is locally bounded and h, k, m are independent of y(·). (h-2) for the function y(·) in (h-1), yt is a b-valued continuous function on [0, b]. (h-3) the space b is complete. by a solution of problem (4.1)-(4.2), we mean a space ω = {y : (−∞, b] → r : y|(−∞,0] ∈ b and y|[0,b] is continuous}. thus a function y ∈ ω is said to be a solution of (4.1)-(4.2) if y satisfies the equation dαy(t) = f(t, yt) on j, and the condition y(t) = φ(t) on (−∞, 0]. the following lemma is crucial to prove our existence theorem for the problem (4.1)-(4.2). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 194 best proximity point for z-contraction and suzuki type z-contraction mappings lemma 4.1 (see [11]). let 0 < α < 1 and let h : (0, b] → r be continuous and lim t→0+ h(t) = h(0+) ∈ r. then y is a solution of the fractional integral equation y(t) = 1 γ(α) ∫ t 0 (t − s)α−1h(s)ds, if and only if y is a solution of the initial value problem for the fractional differential equation dαy(t) = h(t), t ∈ (0, b], y(0) = 0. now we are ready to prove following existence theorem. theorem 4.2. let f : j × b → r. assume (h) there exists q > 0 such that |f(t, u) − f(t, v)| ≤ q||u − v||b, for t ∈ j and every u, v ∈ b. if b α kbq γ(α+1) = λ < 1 where kb = sup{|k(t)| : t ∈ [0, b]}, then there exists a unique solution for the ivp (4.1)-(4.2) on the interval (−∞, b]. proof. to prove the existence of solution for the ivp (4.1)-(4.2), we transform it into a fixed point problem. for this, consider the operator n : ω → ω defined by n(y)(t) = { φ(t) t ∈ (−∞, 0], 1 γ(α) ∫ t 0 (t − s)α−1f(s, ys)ds t ∈ [0, b]. let x(·) : (−∞, b] → r be the function defined by x(t) = { φ(t) t ∈ (−∞, 0], 0 t ∈ [0, b]. then x0 = φ. for each z ∈ c([0, b], r) with z(0) = 0, we denote by z̄ the function defined by z̄(t) = { 0 if t ∈ (−∞, 0], z(t) if t ∈ [0, b]. if y(·) satisfies the integral equation y(t) = 1 γ(α) ∫ t 0 (t − s)α−1f(s, ys)ds, we can decompose y(·) as y(t) = z̄(t)+x(t), 0 ≤ t ≤ b, which implies yt = z̄t+xt, for every 0 ≤ t ≤ b, and the function z(·) satisfies z(t) = 1 γ(α) ∫ t 0 (t − s)α−1f(s, z̄s + xs)ds set c0 = {z ∈ c([0, b], r) : z0 = 0}, c© agt, upv, 2016 appl. gen. topol. 17, no. 2 195 s. komal, p. kumam and d. gopal and let || · ||b be the seminorm in c0 defined by ||z||b = ||z0||b + sup{|z(t)|; 0 ≤ t ≤ b} = sup{|z(t)|; 0 ≤ t ≤ b}, z ∈ c0. c0 is a banach space with norm ||·||b. let the operator p : c0 → c0 be defined by (4.3) (pz)(t) = 1 γ(α) ∫ t 0 (t − s)α−1f(s, z̄s + xs)ds, t ∈ [0, b]. that the operator n has a fixed point is equivalent to p has a fixed point, and so we turn to proving that p has a fixed point. indeed, consider z, z∗ ∈ c0. then we have for each t ∈ [0, b] |p(z)(t) − p(z∗)(t)| ≤ 1 γ(α) ∫ t 0 (t − s)α−1|f(s, z̄s + xs) − f(s, z̄ ∗ s + xs)| ds ≤ 1 γ(α) ∫ t 0 (t − s)α−1q||z̄s − z̄ ∗ s ||b ds ≤ 1 γ(α) ∫ t 0 (t − s)α−1qkb sup s∈[o,t] ||z(s) − z∗(s)|| ds ≤ kb γ(α) ∫ t 0 (t − s)α−1q ds ||z − z∗||b. therefore ||p(z) − p(z∗)||b ≤ qbαkb γ(α + 1) ||z − z∗||b, i.e. d(p(z), p(z∗)) ≤ λd(z, z∗). now we observe that the function ζ : [0, ∞) × [0, ∞) → r defined by ζ(t, s) = λs−t for all t, s ∈ [0, ∞), is in z and so we deduce that the operator p satisfies all the hypothesis of corollary (3.7). thus p has unique fixed point. � acknowledgements. the authors thank editor-in-chief and referee(s) for their valuable comments and suggestions, which were very useful to improve the paper significantly. the authors would like to thank the petchra pra jom klao ph.d. research scholarship for financial support. also, somayya komal was supported by the petchra pra jom klao doctoral scholarship academic for ph.d. program at kmutt. this work was completed while the third author (dr. gopal) was visiting theoretical and computational science center (tacs), science laboratory building, faculty of science, king mongkut’s university of technology thonburi (kmutt), bangkok, thailand. he thanks professor poom kumam and the university for their hospitality and support. moreover, this project was supported by the theoretical and computational science (tacs) center under computational and applied science for smart innovation research cluster (classic), faculty of science, kmutt. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 196 best proximity point for z-contraction and suzuki type z-contraction mappings references [1] m. abbas, a. hussain and p. kumam, a coincidence best proximity point problem in g-metric spaces, abst. and appl. anal. 2015 (2015), article id 243753, 12 pages. 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[27] j. zhang, y. su and q. chang, a note on ’a best proximity point theorem for geraghtycontractions’, fixed point theory appl. 2013 (2013), 99. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 198 @ appl. gen. topol. 18, no. 1 (2017), 53-59 doi:10.4995/agt.2017.4676 c© agt, upv, 2017 on quasi-orbital space hattab hawete laboratoire des systèmes dynamiques et combinatoires, faculté des sciences de sfax, tunisia. umm al qura university makkah ksa. (hattab.hawete@yahoo.fr) communicated by f. balibrea abstract let g be a subgroup of the group homeo(e) of homeomorphisms of a hausdorff topological space e. the class of an orbit o of g is the union of all orbits having the same closure as o. we denote by e/g̃ the space of classes of orbits called quasi-orbit space. a space x is called a quasi-orbital space if it is homeomorphic to e/g̃ where e is a compact hausdorff space. in this paper, we show that every infinite second countable quasi-compact t0-space is the quotient of a quasiorbital space. 2010 msc: 54f65; 54h20. keywords: homeomorphism; group; quasi-orbit space; quasi-orbital space. 1. introduction the standard setting for topological dynamics is a group of homeomorphisms g on a compact hausdorff space e [6]. this group induces an open equivalence relation defined by the family of orbits (gx = {gx : g ∈ g},x ∈ e). we denote by e/g the orbit space equipped with the quotient topology. the study of this space is difficult: just consider the example of a group generated by an irrational rotation on the circle; indeed the orbit space does not verify the weaker separation axioms, as the t0 separation axiom. for this reason [8, 1, 2, 7] consider an intermediary quotient, called the quasi-orbit space. the class of the orbit gx is g̃x = ⋃ o=gx o. the family (g̃x,x ∈ e) determines an open equivalent relation on e [8]. let e/g̃ the space of classes of received 09 february 2016 – accepted 21 july 2016 http://dx.doi.org/10.4995/agt.2017.4676 h. hawete orbits equipped with the quotient topology. the space of classes of orbits is called the quasi-orbit space. the space e/g̃ is a t0-space and its the universal t0-space associated to the orbit space e/g as in bourbaki [3, exercice 27 page i-104]. let p : e → e/g̃ be the canonical projection. the map p is open. the map ϕ : e/g → e/g̃ which associates to each orbit its class is an onto quasihomeomorphism1. thus e/g̃ is a good representative of e/g. according to [8, 1], the space e/g̃ keeps information on the initial dynamical system. a space x is a quasi-orbital space if it is homeomorphic to a quasi-orbit e/g̃ where e is a compact hausdorff space and g is a subgroup of homeomorphisms of e. in [1], the authors asked the following problem: under which conditions a t0-space is quasi-orbital? in [2] the authors showed that a finite t0-space is quasi-orbital. note that, according to [1, example 3.4], if x is a non quasicompact space then e is not in general compact. in this paper we study this problem for an infinite t0-space. our main result is the following: theorem 1.1. every second countable quasi-compact t0-space is the quotient of a quasi-orbital space. if e is a locally compact second countable topological space and g is a subgroup of homeomorphisms of e then, according to [8, 7], e/g̃ satisfies the following properties: (1) e/g̃ is sober2; (2) if g has a minimal set then, e/g̃ is quasi-compact. in this paper, we show that if e is a locally compact topological space and g is a subgroup of homeomorphisms of e then, if e/g̃ is quasi-compact then it is quasi-orbital. the paper consists of three sections. after introduction we will show some properties of the quasi-orbital space. in section 3 we prove the main theorem. 2. quasi-orbital spaces in this section we study some properties of the quasi-orbital spaces. proposition 2.1. a closed subspace of a quasi-orbital space is quasi-orbital. proof. let y be a closed subset of a quasi-orbital space x. there exist a compact and haudorff space e and a subgroup g of homeo(e) such that x is homeomorphic to to the quasi-orbit space e/g̃; let ϕ such homeomorphism. s = p−1(ϕ(y )) is an invariant compact subset of e. we denote by h = g/s 1a continuous map f : x → y between two topological spaces is called a quasihomeomorphism if the map which assigns to each open set v ⊂ y the open set f−1(v ) is a bijective map. 2a space x is sober if every irreducible, nonempty, closed subset m of x has a unique generic point m, i.e. m = {m}. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 54 on quasi-orbital space the induced subgroup of g on s. since s is an invariant subset of e, we have for each x ∈ s, h(x) = g(x). we will show that s/h̃ is homeomorphic to ϕ(y ) and so to y . let f : s/h̃ → ϕ(y ) which maps any class of an orbit hx to the class of the orbit gx. we will prove now that the bijective map f is a homeomorphism. let v be an open subset of ϕ(x), that means that v = u ∩ϕ(x) where u is an open subset of e/g̃. so we have p−1(v ) = p−1(u) ∩p−1(ϕ(x)) = p−1(u) ∩s since p−1(u) is an open subset of e, p−1(v ) is an open subset of s. thus v is an open subset of s/h̃ and so f is a continuous map. let p1 : s → s/h̃ be the canonical projection and let v be an open subset of s/h̃, that means that p−11 (v ) is an open subset of s and so there exists an open subset u of e such that p−11 (v ) = u ∩s. we have v = p(p−11 (v )) = p(u ∩s) since s is invariant, we deduce that v = p(u) ∩p(s) = p(u) ∩ϕ(x) the fact that p is an open map implies that v is an open subset of ϕ(x). therefore f is an open map. thus f is a homeomorphism and so y is a quasi-orbital space. � example 2.2. this example shows that proposition 2.1 minus the hypothesis that y is closed is false. let f be an increasing homeomorphism of [0, 1] without fixed point in ]0, 1[ such that f(0) = 0, f(1) = 1 and f( 1 2 ) = 3 4 . let (an) be an increasing sequence such that a0 = 1 2 and converges to 5 8 . let (bn) be a decreasing sequence such that b0 = 3 4 and converges to 5 8 . let g be a homeomorphism of [0, 1] such that its support is ⋃ n≥0 f n([an,bn]) and g(fn(an+1)) = f n(an+1). let g be the group of homeomorphisms of [0, 1] generated by f and g. let x = [0, 1]/g̃ be the quasi-orbital space. the subspace y = x − p( 5 8 ) is not closed. on the other hand y can not be a quasi-orbital space because it is irreducible without generic point [8, lemma 2.2]. proposition 2.3. let x be a quasi-orbital space and r be an equivalence relation on x which have a closed continuous cross-section s3. then x/r is quasi-orbital. proof. since s is closed, s(x/r) is a closed subset of x and so, according to proposition 2.1, s(x/r) is quasi-orbital. since s is closed and continuous, it will be an embedding and so x/r is homeomorphic to s(x/r) which implies that x/r is quasi-orbital. � 3according to [13], if x/r is a t1-space and zero-dimensional, then there exists a continuous cross-section for r. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 55 h. hawete remark 2.4. if an open equivalence relation r has a closed and continuous cross-section, then x/r is a t0-space. indeed, let a and b two elements of x/r such that {a} = {b}. since s is continuous and closed, s({a}) = s({a}) = {s(a)} and s({b}) = s({b}) = {s(b)} and so {s(a)} = {s(b)}. the fact that x is a t0-space implies that s(a) = s(b) and so a = b (s is injective). therefore x/r is a t0-space. proposition 2.5. let (xi, i ∈ i) be a family of quasi-orbital spaces. then the product ∏ i∈i xi is quasi-orbital. proof. for every i ∈ i, xi is quasi-orbital, then there exist a compact space ei and a subgroup gi of homeo(ei) such that xi is homeomorphic to the quasi-orbits space ei/g̃i. let e = ∏ i∈i ei be the product space and g = ∏ i∈i gi be the product group. by applying [3, proposition 7 tg i.27], we have, for each x = (xi, i ∈ i), g(x) = ∏ i∈i gi(xi) and so g̃ = ∏̃ i∈i gi = ∏ i∈i g̃i. by applying [3, corollaire p.tg i.34] it follows that ∏ i∈i xi is homeomorphic to e/g̃. since e is compact, ∏ i∈i xi is quasi-orbital. � proposition 2.6. if e is a locally compact space and g is a subgroup of homeomorphisms of e, then if e/g̃ is quasi-compact then it is a quasi-orbital space. proof. since e/g̃ is a quasi-compact space, according to [7, proposition 2.1], g has a minimal set m. the fact that e − m is an open set of a locally compact set implies that e −m is a locally compact space [3, proposition 13 tg i.66]. we denote by h = g/e −m the induced subgroup of g on e −m. since e − m is invariant, we have for each x ∈ e − m, h(x) = g(x). let ê = (e − m) ∪{ω} be the one point compactification of e − m. we can suppose that h is a group of homeomorphisms of ê by putting h(ω) = {ω}. it is easy to see that the bijection f : ê/h̃ → e/g̃ which maps any class of an orbit hx to the class of the orbit gx for all x ∈ e −m and f(ω) = p(m) is a homeomorphism. thus e/g̃ is homeomorphic to ê/h̃. � 3. proof of main theorem recall that, a topological space x is a k-space (compactly generated) if the following holds: a subset a ⊂ x is closed in x if and only if a∩k is closed in k for every compact subset k ⊂ x [10]. it is easy to see that the family of closed compact sets determines the topology of a k-space. any locally compact space is a k-space and any first countable topological space (in particular a metric space) is a k-space. according to [4, p. 248], x is a k-space if and only if it is a quotient space of a locally compact space z. the space z is a disjoint sum of c© agt, upv, 2017 appl. gen. topol. 18, no. 1 56 on quasi-orbital space all compact subsets (ki, i ∈ i) of x: z = ∐ i∈i ki = {(x,i) : i ∈ i and x ∈ ki}. the equivalence relation r on z is defined by: (x,i)r(y,j) if x = y. note that z is equipped with the disjoint sum topology defined by: u is an open set of z if ϕ−1j (u) is an open set of kj where the map ϕj : kj → z is defined by ϕj(x) = (x,j). recall that, for all j, the map ϕj is continuous closed and open and f : z → y is continuous if and only if f ◦ϕj is continuous. remark 3.1. the set s = {0, 1} equipped with the topology {∅,s,{1}} is called the sierpinski space; it is a connected t0-space but it is not a t1-space. if g1 is a finitely generated abelian subgroup of diff ∞ + (s 1) of finite rank k ≥ 2 having only a one fixed point e ∈ s1, then all other orbits are everywhere dense (n. kopell, g. reeb [11], [12]). thus the quasi-orbits space s1/g̃1 is homeomorphic to the sierpinski space s. proof (main theorem). since x is a t0-space, by applying [5, theorem 2.3.26 p.84], there exists an embedding ψ : x → ∏ i∈i si (where si is the sierpinski space {0, 1}). we can suppose that i ⊂ n; indeed, x is second countable. we know that for each i ∈ i there is a homeomorphism fi : si → s1i /g̃i where s 1 i is the unit circle s1 and gi is the group g1 defined in remark 3.1. the product map ∏ i∈i fi : ∏ i∈i si → ∏ i∈i s 1 i /g̃i is also a homeomorphism. according to [3, corollaire p.tg i.34], ∏ i∈i s 1 i /g̃i is homeomorphic to ∏ i∈i s 1 i / ∏ i∈i g̃i. the space ti = ∏ i∈i s 1 i is a compact second countable metric space. we put gi = ∏ i∈i gi. the group g i is abelian. then we conclude that there exists an embedding ϕ : x → ti/g̃i . let p : ti → ti/g̃i be the canonical projection. we denote by e = p−1(ϕ(x)) and we denote by g = gi/e the induced subgroup of gi on e. since e is a saturated subset of ti . we have for each x ∈ e, g(x) = gi (x). we will show that e/g̃ is homeomorphic to ϕ(x) and so to x. let f : e/g̃ → ϕ(x) ⊂ ti/g̃i which maps any class of an orbit g(x) to the class of the orbit gi (x). we will prove now that this bijective map f is a homeomorphism: let v be an open subset of ϕ(x), that means that v = u ∩ϕ(x) where u is an open subset of ti/g̃i . so we have p−1(v ) = p−1(u) ∩p−1(ϕ(x)) = p−1(u) ∩e since p−1(u) is an open subset of ti , p−1(v ) is an open subset of e. thus v is an open subset of e/g̃ and so f is a continuous map. let p1 : e → e/g̃ be the canonical projection and let v be an open subset of e/g̃, that means that p−11 (v ) is an open subset of e and so there exists an open subset u of ti such that p−11 (v ) = u ∩e. we have v = p(p−11 (v )) = p(u ∩e) since e is saturated, we deduce that v = p(u) ∩p(e) = p(u) ∩ϕ(x) c© agt, upv, 2017 appl. gen. topol. 18, no. 1 57 h. hawete the fact that p is an open map implies that v is an open subset of ϕ(x). therefore f is an open map. we conclude that f is a homeomorphism. since e is a metric space, it is first countable and so e is a k-space. thus e is the quotient of a locally compact metric space f by the relation r. note that f is the disjoint union of all compact subsets of e. let q : f → f/r = e be the canonical projection. let g be an element of g. we define on f the map g : f → f by g(x,i) = (g(x),j) where g(ki) is the compact kj. it is easy to see that g is a well defined bijection. let u be an open set of f , then u = ∐ i∈i ui ∩ki where ui is an open set of e. g−1(u) = ∐ i∈i g−1(ui) ∩g−1(ki) and g(u) = ∐ i∈i g(ui) ∩g(ki) and since g is a homeomorphism g(ui) and g −1(ui) are open sets of e and g is a permutation of the set of all compact subsets. then g−1(u) = ∐ i∈i g−1(ui) ∩ki and g(u) = ∐ i∈i g(ui) ∩ki are open sets of f. therefore g is a homeomorphism of f. the set g = {g : g ∈ g} is a subgroup of homeomorphisms of f . since e/g̃ is quasi-compact, we show now that g has a minimal set. we start by showing that e/g̃ contains a point a such that {a} is closed. since e/g̃ is quasi-compact, by zorn’s lemma, it contains a minimal set m. therefore for all z ∈ m we have {z} = m. from the fact that e/g̃ is a t0-space, it follows that m is a single point set {a} (indeed {a} = {b} ⇒ a = b). let x be an element of e such that p(x) = a. the fact that {a} is closed implies that p−1({a}) = g̃x is a closed invariant set of e such that if y ∈ g̃x then gy = gx and so g̃x is a minimal set of g. q−1(g̃x) is a closed subset of f . if there exist (x,i) ∈ q−1(g̃x) and g ∈ g such that g(x,i) = (g(x),j) is not in q−1(g̃x), then q(g(x),j)) is not in g̃x and so g(x) is not in g̃x which contradicts the fact that g̃x is an invariant set. we conclude that q−1(g̃x) is a minimal set of g. the fact that f is locally compact, according to [7, proposition 2.1], implies that f/g̃ is quasi-compact. then, by applying proposition 2.6, we have f/g̃ is a quasi-orbital space ê/h̃. let h be the homeomorphism of ê/h̃ and f/g̃. let p2 : f → f/g̃ and p3 : e → e/g̃ be the canonical projections. let q̃ : f/g̃ → e/g̃ be the map defined by q̃ ◦p2 = p3 ◦ q. q̃ is a continuous and onto map. the map q̂ = ϕ−1 ◦f ◦ q̃ ◦h is a continuous and onto map of ê/h̃ to x which implies that x is a quotient of a quasi-orbital space. 2 c© agt, upv, 2017 appl. gen. topol. 18, no. 1 58 on quasi-orbital space references [1] c. bonatti, h. hattab and e. salhi, quasi-orbits spaces associated to t0-spaces, fund. math. 211 (2011), 267–291. [2] c. bonatti, h. hattab, e. salhi and g. vago, hasse diagrams and orbit class spaces, topology appl. 158 (2011), 729–740. [3] n. bourbaki, topologie générale chapitre 1 à 4, masson, 1990. [4] j. dugundji, topology, allyn and bacon, inc., boston (1966). [5] r. engelking, general topology, 2nd ed., helderman verlag, berlin, 1989. [6] w. h. gottschalk and g. a. hedlund, topological dynamics, ams colloquium publications, vol. 36, 1955. [7] h. hattab, characterization of quasi-orbit spaces, qualitative theory of dynamical systems (2012). [8] h. hattab and e. salhi, groups of homeomorphisms and spectral topology, topology proc. 28, no. 2 (2004), 503–526. [9] j. g. hocking and g. s. young, topology, (1969). [10] j. l. kelley, general topology, van nostrand, new work (1955). [11] n. kopell, commuting diffeomorphisms, proc. sympos. pure math. 14 (1970), 165–184. [12] g. reeb, sur les structures feuilletées de codimension un et sur un théorème de a. denjoy, ann. inst. fourier 11 (1961), 185–200. [13] d. e. miller, a selector for equivalence relations with gδ orbits, proc. amer. math. soc. 72, no. 2 (1978), 365–369. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 59 () @ appl. gen. topol. 16, no. 1(2015), 75-80doi:10.4995/agt.2015.3244 c© agt, upv, 2015 when is a space menger at infinity? leandro f. aurichi a and angelo bella b a instituto de ciências matemáticas e de computação, universidade de são paulo, são carlos, sp, brazil (aurichi@icmc.usp.br) b dipartimento di matematica, città universitaria, catania, italy (bella@dmi.unict.it) abstract we try to characterize those tychonoff spaces x such that βx \ x has the menger property. 2010 msc: 54f65; 54d40; 54d20. keywords: menger; remainder. 1. introduction a space x is menger (or has the menger property) if for any sequence of open coverings {un : n < ω} one may pick finite sets vn ⊆ un in such a way that⋃ {vn : n < ω} is a covering. this equivals to say that x satisfies the selection principle sfin(o, o). it is easy to see the following chain of implications: σ-compact −→ menger −→ lindelöf an important result of hurewicz [4] states that a space x is menger if and only if player 1 does not have a winning strategy in the associated game gfin(o, o) played on x. this highlights the game-theoretic nature of the menger property, see [7] for more. henriksen and isbell ([3]) proposed the following: definition 1.1. a tychonoff space x is lindelöf at infinity if βx \ x is lindelöf. they discovered a very elegant duality in the following: received 26 august 2014 – accepted 2 january 2015 http://dx.doi.org/10.4995/agt.2015.3244 l. f. aurichi and a. bella proposition 1.2 ([3]). a tychonoff space is lindelöf at infinity if and only if it is of countable type. a space x is of countable type provided that every compact set can be included in a compact set of countable character in x. a much easier and well-known fact is: proposition 1.3. a tychonoff space is čech-complete if and only if it is σcompact at infinity. these two propositions suggest the following: question 1.4. when is a tychonoff space menger at infinity? before beginning our discussion here, it is useful to note these well known facts: proposition 1.5. the menger property is invariant by perfect maps. corollary 1.6. x is menger at infinity if, and only if, for any y compactification of x, y \ x is menger. fremlin and miller [6] proved the existence of a menger subspace x of the unit interval [0, 1] which is not σ-compact. the space x can be taken nowhere locally compact and so y = [0, 1] \ x is dense in [0, 1]. since the menger property is invariant under perfect mappings, we see that βy \ y is still menger. therefore, a space can be menger at infinity and not σ-compact at infinity. another example of this kind, stronger but not second countable, is example 3.1 in the last section. on the other hand, the irrational line shows that a space can be lindelöf at infinity and not menger at infinity. consequently, the property m characterizing a space to be menger at infinity strictly lies between countable type and čech-complete. of course, taking into account the formal definition of the menger property, we cannot expect to have an answer to question 1.4 as elegant as henriksenisbell’s result. 2. a characterization definition 2.1. let k ⊂ x. we say that a family f is a closed net at k if each f ∈ f is a closed set such that k ⊂ f and for every open a such that k ⊂ a, there is an f ∈ f such that f ⊂ a. lemma 2.2. let x be a t1 space. if (fn)n∈ω is a closed net at k, for k ⊂ x compact, then k = ⋂ n∈ω fn. proof. simply note that for each x /∈ k, there is an open set v such that k ⊂ v and x /∈ v . � lemma 2.3. let y be a regular space and let x be a dense subspace of y . let k ⊂ x be a compact subset. if (fn)n∈ω is a closed net at k in x, then (fn y )n∈ω is a closed net at k in y . c© agt, upv, 2015 appl. gen. topol. 16, no. 1 76 when is a space menger at infinity? proof. in the following, all the closures are taken in y . let a be an open set in y such that k ⊂ a. by the compactness of k and the regularity of y , there is an open set b such that k ⊂ b ⊂ b ⊂ a. thus, there is an n ∈ ω such that k ⊂ fn ⊂ b ∩ x. note that k ⊂ f n ⊂ b ⊂ a. � lemma 2.4. let x be a compact hausdorff space. if k = ⋂ n∈ω fn, where (fn)n∈ω is a decreasing sequence of closed sets, then (fn)n∈ω is a closed net at k. proof. if not, then there is an open set v such that k ⊂ v and, for every n ∈ ω, fn \ v 6= ∅. by compactness, there is an x ∈ ⋂ n∈ω fn \ v = k \ v . contradiction with the fact that k ⊂ v . � theorem 2.5. let x be a tychonoff space. x is menger at infinity if, and only if, x is of countable type and for every sequence (kn)n∈ω of compact subsets of x, if (f np )p∈ω is a decreasing closed net at kn for each n, then there is an f : ω −→ ω such that k = ⋂ n∈ω f n f(n) is compact and ( ⋂ k≤n f k f(k) )n∈ω is a closed net for k. proof. in the following, every closure is taken in βx. suppose that x is menger at infinity. by lemma 1.2 x is of countable type. let (f np )p,n∈ω be as in the statement. note that, by lemma 2.3 and lemma 2.2, ⋂ p∈ω f np = ⋂ p∈ω f np for each n ∈ ω. thus, for each n ∈ ω, (v n p )p∈ω, where v np = βx \ f n p , is an increasing covering for βx \ x. since βx \ x is menger, there is an f : ω −→ ω such that βx \ x ⊂ ⋃ n∈ω v n f(n) . note that k = ⋂ n∈ω f n f(n) is compact and it is a subset of x. by lemma 2.4, ( ⋂ k≤n f k f(k) )n∈ω is a closed net at k in βx, therefore, ( ⋂ k≤n f k f(k) )n∈ω is a closed net at k in x. conversely, for each n ∈ ω, let wn be an open covering for βx \ x. we may suppose that each w ∈ wn is open in βx. by regularity, we can take a refinement vn of wn such that, for every x ∈ βx \ x, there is a v ∈ vn such that x ∈ v ⊂ v ⊂ wv for some wv ∈ wn. since x is of countable type, by lemma 1.2 we may suppose that each vn is countable. fix an enumeration for each vn = (v n k )k∈ω. define a n k = βx \ ( ⋃ j≤k v nj ). note that each kn = ⋂ k∈ω an k is compact and a subset of x. by lemma 2.4, (an k )k∈ω is a closed net at kn. thus, (a n k ∩ x)k∈ω is a closed net at kn in x. therefore, there is f : ω −→ ω such that k = ⋂ n∈ω (an f(n) ∩ x) is compact and ( ⋂ k≤f(n) a k f(k) ∩ x)n∈ω is a closed net at k. so, by lemma 2.3, k = ⋂ n∈ω (an f(n) ∩ x). since ⋂ n∈ω (an f(n) ∩ x) = ⋂ n∈ω an f(n) and by the fact that k ⊂ x, it follows that βx \x ⊂ ⋃ n∈ω βx \an f(n) ⊂ ⋃ n∈ω int( ⋃ j≤f(n) v n j ) ⊂⋃ n∈ω ⋃ j≤f(n) wv nj . therefore, letting un = {wv n j : j ≤ f(n)} ⊂ wn, we see that the collection ⋃ n∈ω un covers βx \ x, and we are done. � property m given in the above theorem does not look very nice and we wonder whether there is a simpler way to describe it, at least in some special cases. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 77 l. f. aurichi and a. bella recall that a metrizable space is always of countable type. moreover, a metrizable space is complete if and only if it is σ-compact at infinity. therefore, we could hope for a “nicer” m in this case. question 2.6. what kind of weak completeness characterizes those metrizable spaces which are menger at infinity? proposition 2.7. let x be a tychonoff space. if x is menger at infinity then for every sequence (kn)n∈ω of compact sets, there is a sequence (qn)n∈ω of compact sets such that: (1) each kn ⊂ qn; (2) each qn has a countable base at x; (3) for every sequence (bn k )n,k∈ω such that, for every n ∈ ω, (b n k )k∈ω is a decreasing base at kn, then there is a function f : ω −→ ω such that k = ⋂ n∈ω bn f(n) is compact and ( ⋂ k≤n bk f(k) )n∈ω is a closed net at k. proof. suppose x is menger at infinity. let (kn)n∈ω be a sequence of compact sets. since x is menger at infinity, x is lindelöf at infinity. thus, by proposition 1.2, for each kn, there is a compact qn ⊃ kn such that qn has a countable base. now, let (bn k )k,n be as in 3. since each qn is compact and x is regular, each (bn k )k∈ω is a decreasing closed net at qn. thus, by theorem 2.5, there is an f : ω −→ ω as we need. � to some extent, the menger property is closer to σcompactness rather than to lindelöfness. since a čechcomplete space has the baire property, we may ask: question 2.8. is it true that a space menger at infinity has the baire property? we thank m. sakai for calling our attention to the above question. he also noticed a partial answer to it: theorem 2.9 (sakai). let x be a first countable tychonoff space. if x is menger at infinity, then x is hereditarily baire. proof. according to a result of debs [2], a regular first countable space is hereditarily baire if and only if it contains no closed copy of the space of rationals q. to finish, it suffices to observe that q is not menger at infinity. � we end this section presenting a selection principle that at first glance could be related with the menger at infinity property. definition 2.10. we say that a family u of open sets of x is an almost covering for x if x \ ⋃ u is compact. we call a the family of all almost coverings for x. note that the property “being menger at infinity” looks like something as sfin(a, a), but for a narrow class of a. we will see that the “narrow” part is important. proposition 2.11. if x satisfies sfin(a, a), then x is menger. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 78 when is a space menger at infinity? proof. let (un)n∈ω be a sequence of coverings of x. by definition, for each n ∈ ω, there is a finite un ⊂ un, such that k = x \ ⋃ n∈ω ⋃ un is compact. therefore, there is a finite w ⊂ un such that k ⊂ ⋃ w . thus, x = w ∪⋃ n∈ω ⋃ un. � example 2.12. the space of the irrationals is an example of a space that is menger at infinity but does not satisfy sfin(a, a) (by the proposition 2.11). example 2.13. the one-point lindelöfication of a discrete space of cardinality ℵ1 is an example of a menger space which does not satisfy sfin(a, a). example 2.14. ω is an example of a space that satisfies sfin(a, a), but it is not compact. proof. let (vn)n∈ω be a sequence of almost coverings for ω. therefore, for each n, fn = ω \ ⋃ vn is finite. for each n, let vn ⊂ vn be a finite subset such that fn+1 \fn ⊂ ⋃ vn and min(ω \ ⋃ k 0 if and only if x = y; (fm3) m(x,y,t) = m(y,x,t) for all x,y ∈ x and t > 0; (fm4) m(x,y, ·) : [0,∞) → [0,1] is left continuous for all x,y ∈ x; (fm5) m(x,z,t+s) ≥ m(x,y,t)∗m(y,z,s) for all x,y,z ∈ x and t,s > 0. from the above definition, if (fm5) is replaced by the following: (na) m(x,z,max{t,s}) ≥ m(x,y,t) ∗ m(y,z,s) for all x,y ∈ x and t,x > 0, then (x,m,∗) is called a non-archimedean fuzzy space. obviously, every nonarchimedean fuzzy metric space is itself a fuzzy metric space. definition 2.3 ([16]). a weak non-archimedean fuzzy metric space is a triple (x,m,∗), where x is a nonempty set, ∗ is a continuous t-norm and m is a fuzzy set on x2 × [0,∞), satisfying (fm1)-(fm4) and (wna) m(x,z,t) ≥ max{m(x,y,t)∗m(x,z,t/2),m(x,y,t/2)∗m(y,z,t/2)} for all x,y,z ∈ x and t > 0. remark 2.4. (1) every non-archimedean fuzzy metric spaces is itself a weak non-archimedean fuzzy metric space. (2) a weak non-archimedean fuzzy metric space is not necessarily a fuzzy metric space. example 2.5 ([16]). let x = [0,∞) and a ∗ b = ab for all a,b ∈ [0,1]. define a mapping m : x2 × [0,∞) → [0,1] by: m(x,y,0) = 0, m(x,x,t) = 1 for all t > 0, m(x,y,t) = t for x 6= y and 0 < t ≤ 1, m(x,y,t) = t/2 for x 6= y and 1 < t ≤ 2, m(x,y,t) = 1 for x 6= y and t > 2. then (x,m,∗) is a weak non-archimedean fuzzy metric space, but it is not a fuzzy metric space. definition 2.6 ([16]). let (x,m,∗) be a weak non-archimedean fuzzy metric space. we define an open ball in x by b(x,r,t) = {y ∈ x : m(x,y,t) > 1 − r} for any x ∈ x, r ∈ (0,1) and t > 0. proposition 2.7 ([16]). let (x,m,∗) be a weak non-archimedean fuzzy metric space. then we have the following: (1) every open ball is an open set; (2) the family τ = {a ⊂ x : x ∈ a iff there exist t > 0 and r ∈ (0,1) with b(x,r,t) ⊂ a} is a topology on x; (3) every weak non archimedean fuzzy metric space (x,m,∗) is a hausdorff space. proposition 2.8 ([16]). let (x,m,∗) be a weak non-archimedean fuzzy metric space. a sequence {xn} in a weak non-archimedean fuzzy metric space (x,m,∗) is convergent to x ∈ x if and only if limn→∞ m(xn,x,t) = 1 for all t > 0. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 3 s. suantai, y. j. cho and j. tiammee proposition 2.9 ([16]). let (x,m,∗) be a weak non-archimedean fuzzy metric space and (xn) ⊂ x be a sequence convergent to x ∈ x. then lim n→∞ m(y,xn, t) = m(y,x,t) for all y ∈ x and t > 0. proposition 2.10. let (x,m,∗) be a weak non-archimedean fuzzy metric space. suppose that {xn} and {yn} are two sequences in x such that limn→∞ xn = x and limn→∞ yn = y. then lim n→∞ m(xn,yn, t) = m(x,y,t) for all t > 0. proof. let {xn} and {yn} be two sequences in x such that limn→∞ xn = x and limn→∞ yn = y. then, by the condition (wna) and proposition 2.9, we have m(y,xn, t) ≥ m(y,yn, t/2) ∗ m(yn,xn, t) and m(xn,yn, t) ≥ m(xn,x,t/2) ∗ m(x,yn, t). it follows that m(x,y,t) ≤ lim inf n→∞ m(xn,yn, t) ≤ lim sup n→∞ m(xn,yn, t) ≤ m(x,y,t). therefore, limn→∞ m(xn,yn, t) = m(x,y,t) for all t > 0. this completes the proof. � definition 2.11 ([16]). let (x,m,∗) be a weak non-archimedean fuzzy metric space. a sequence {xn} in x is called a cauchy sequence if, for each ǫ ∈ (0,1) and t > 0, there exists n ∈ n such that m(xn,xm, t) > 1 − ǫ for all m,n ≥ n. a weak non-archimedean fuzzy metric space (x,m,∗) is said to be complete if every cauchy sequence is convergent. definition 2.12 ([10]). an element x ∈ x is called a common fixed point of the mappings s,t,a,b : x → x if x = sx = tx = ax = bx. definition 2.13 ([16]). the self-mappings s and t of a weak non-archimedean fuzzy metric space (x,m,∗) are said to be compatible if lim n→∞ m(stxn,tsxn, t) = 1 for all t > 0 whenever {xn} is a sequence in x such that lim n→∞ sxn = lim n→∞ txn = u for some u ∈ x. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 4 common fixed points for generalized ψ-contractions definition 2.14 ([8]). the self-mappings s and t of a nonempty set x are said to be weak compatible if stz = tsz whenever sz = tz for some z ∈ x. 3. common fixed points for compatible mappings in this section, we introduce the notion of ψ-contractions of four self-mappings and prove some common fixed points theorems for a ψ-contraction in complete non-archimedean fuzzy metric spaces. let ψ : [0,1] → [0,1] be a function such that (a) ψ is nondecreasing and left continuous; (b) ψ(t) > t for all t ∈ (0,1). we denote ψ := {ψ : [0,1] → [0,1] : ψ satisfies (a) − (b)}. lemma 3.1 ([8]). if ψ ∈ ψ, then (1) limn→∞ ψ n(t) = 1 for all t ∈ (0,1); (2) ψ(1) = 1. definition 3.2. let x be a nonempty set and m be a fuzzy set on x2×[0,∞). let a,b,s,t : x → x be four mappings. the four couple (a,b;s,t) is called ψ-contractive mappings if there exists ψ ∈ ψ such that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,by,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{m(sx,ty,t),m(ax,sx,t),m(by,ty,t)}. let a,b,s,t : x → x be four mappings such that a(x) ⊂ t(x) and b(x) ⊂ s(x). suppose that there exist x0,x1 ∈ x such that ax0 = tx1 and m(x0,x1, t) > 0. let x0 ∈ x. since a(x) ⊂ t(x) and b(x) ⊂ s(x), there exist x0,x1 ∈ x such that ax0 = tx1 = y0. also, since a(x) ⊂ t(x) and b(x) ⊂ s(x), there exists x2 ∈ x such that bx1 = sx2 = y1. inductively, we can define a sequence {yn} in x such that ax2n = tx2n+1 = y2n, bx2n+1 = sx2n+2 = y2n+1(ω) for each n ≥ 0. lemma 3.3. let (x,m,∗) be a weak non-archimedean fuzzy metric space and a,b,s,t be four self-mappings of x satisfying the following conditions: (1) a(x) ⊂ t(x) and b(x) ⊂ s(x); (2) the pairs a,s and b,t are compatible; (3) one of s,t,a and b is continuous; (4) the four couple (a,b;s,t) is ψ-contractive mappings. suppose that there exist x0,x1 ∈ x such that ax0 = tx1 and m(x0,x1, t) > 0. then the sequence {yn} in x generated by (ω) with initial point x0,x1 is a cauchy sequence. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 5 s. suantai, y. j. cho and j. tiammee proof. first, from the sequence {yn} in x generated by (ω), we show that lim n→∞ m(yn,yn+1, t) = 1 for all t > 0. assume that m(yn,yn+1, t) < 1 for all n ≥ 1. since m(ax0,bx1, t) = m(y0,y1, t) > 0, we obtain m(y2,y1, t) = m(ax2,bx1, t) ≥ ψ(m(x2,x1, t)) = ψ(min{m(sx2,tx1, t),m(ax2,sx2, t),m(bx1,tx1, t)}) = ψ(min{m(y1,y0, t),m(y2,y1, t),m(y1,y0, t)}) = ψ(min{m(y1,y0, t),m(y2,y1, t)}). suppose that m(y1,y0, t) > m(y2,y1, t) then m(y2,y1, t) ≥ ψ(m(y2,y1, t)) > m(y2,y1, t), which is a contradiction. therefore m(y1,y0, t) ≤ m(y2,y1, t), which implies that m(y2,y1, t) ≥ ψ(m(y1,y0, t)) > 0. again, we consider m(y2,y3, t) = m(ax2,bx3, t) ≥ ψ(m(x2,x3, t)) = ψ(min{m(sx2,tx3, t),m(ax2,sx2, t),m(bx3,tx3, t)}) = ψ(min{m(y1,y2, t),m(y2,y1, t),m(y3,y2, t)}) = ψ(min m(y2,y1, t),m(y3,y2, t)}) suppose that m(y2,y1, t) > m(y3,y2, t) then m(y2,y3, t) ≥ ψ(m(y2,y3, t)) > m(y2,y3, t), which is a contradiction. therefore m(y2,y1, t) ≤ m(y2,y3, t), which implies that m(y2,y3, t) ≥ ψ(m(y2,y1, t)) ≥ ψ2(m(y0,y1, t)) > 0. therefore, for any n ∈ n we have m(yn+1,yn, t) ≥ ψn(m(y0,y1, t)) > 0. by lemma 3.1, as n → ∞, we obtain lim n→∞ m(yn+1,yn, t) = 1. next, we show that the sequence {yn} is a cauchy sequence. if {yn} is not cauchy, then there exist ǫ ∈ (0, 1 2 ) and t > 0 such that, for each k ≥ 1, there exist m(k),n(k) ∈ n such that m(k) > n(k) ≥ k and m(ym(k),yn(k), t) ≤ 1−2ǫ. by (wna), we have 1 − 2ǫ ≥ m(ym(k),yn(k), t) ≥ m(ym(k),yn(k)+1, t) ∗ m(yn(k)+1,yn(k), t/2), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 6 common fixed points for generalized ψ-contractions which implies, by k → ∞, that 1 − 2ǫ ≥ lim sup k→∞ m(ym(k),yn(k)+1, t) ∗ m(yn(k)+1,yn(k), t/2) = lim sup k→∞ m(ym(k),yn(k)+1, t). similarly, we obtain 1 − 2ǫ ≥ lim sup k→∞ m(ym(k)+1,yn(k), t), 1 − 2ǫ ≥ lim sup k→∞ m(ym(k)+1,yn(k)+1, t). so, we can assume that m(k) are odd numbers, n(k) are even numbers and m(xm(k),xn(k), t) ≤ 1 − ǫ for all k ≥ 1. define q(k) = min{m(k) : m(ym(k),yn(k), t) ≤ 1−ǫ, m(k) is odd number}. by (wna), we have 1 − ǫ ≥ m(yq(k),yn(k), t) ≥ m(yq(k),yq(k)−2, t/2) ∗ m(yq(k)−2,yn(k), t) ≥ m(yq(k),yq(k)−1, t/4) ∗ m(yq(k)−1,yq(k)−2, t/2) ∗ (1 − ǫ). letting k → ∞, we obtain limk→∞ m(yq(k),yn(k), t) = 1 − ǫ. by (wna) and the condition (4), we have m(yq(k),yn(k), t) ≥ m(yq(k),yn(k)+1, t) ∗ m(yn(k)+1,yn(k), t/2) ≥ m(yq(k)+1,yn(k)+1, t) ∗ m(yq(k)+1,yq(k), t) ∗ m(yn(k)+1,yn(k), t/2) = m(sxq(k)+1,txn(k)+1, t) ∗ m(yq(k)+1,yq(k), t) ∗ m(yn(k)+1,yn(k), t/2) ≥ ψ(m(xq(k)+1,xn(k)+1, t)) ∗ m(yq(k)+1,yq(k), t) ∗ m(yn(k)+1,yn(k), t/2),(*) where m(xq(k)+1,xn(k)+1, t) = min{m(sxq(k)+1,txn(k)+1, t),m(axq(k)+1,sxq(k)+1, t), m(bxn(k)+1,txn(k)+1, t)} = min{m(yq(k),yn(k), t),m(yq(k)+1,yq(k), t),m(yn(k)+1,yn(k), t)}. since lim k→∞ m(xq(k)+1,xn(k)+1, t) = min{ lim k→∞ m(yq(k),yn(k), t), lim k→∞ m(yq(k)+1,yq(k), t), lim k→∞ m(yn(k)+1,yn(k), t)} = min{1 − ǫ,1,1} = 1 − ǫ by taking k → ∞ in (*), we obtain 1 − ǫ ≥ ψ(1 − ǫ) ∗ 1 ∗ 1 > 1 − ǫ, which is a contradiction. therefore, {yn} is a cauchy sequence. this completes the proof. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 7 s. suantai, y. j. cho and j. tiammee now, we are ready to state and prove our main results. theorem 3.4. let (x,m,∗) be a complete weak non-archimedean fuzzy metrci space and a,b,s,t be the self-mappings of x satisfying the following conditions: (1) a(x) ⊂ t(x) and b(x) ⊂ s(x); (2) the pairs a,s and b,t are compatible; (3) one of a,b,s and t is continuous; (4) the four couple (a,b;s,t) is a ψ-contractive mapping. suppose that there exist x0,x1 ∈ x such that ax0 = tx1 and m(x0,x1, t) > 0. assume that, for any x,y ∈ x with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. then a,b,s and t have a unique common fixed point. proof. by lemma 3.3, the sequence {yn} is a cauchy sequence. since x is complete, there exists z ∈ x such that limn→∞ yn = z. from ax2n = tx2n+1 = y2n and bx2n+1 = sx2n+2 = y2n+1 for all n ≥ 1, we obtain lim n→∞ ax2n = lim n→∞ tx2n+1 = lim n→∞ bx2n+1 = lim n→∞ sx2n+2 = z.(3.1) for the proof, we divide 4 cases for the continuity of a,b,s and t . case 1. suppose that s is continuous. then we have lim n→∞ sax2n = lim n→∞ s2x2n = sz.(3.2) since a and s are compatible mappings, limn→∞ m(sax2n,asx2n, t) = 1. thus, from m(asx2n,sz,t) ≥ m(asx2n,sax2n, t) ∗ m(sax2n,sz,t/2) as n → ∞, we obtain limn→∞ m(asx2n,sz,t) = 1, i.e. lim n→∞ asx2n = sz.(3.3) first, wet prove that z is a common fixed point of a and s. if sz 6= z, then there exists t > 0 such that 0 < m(sz,z,t) < 1. by the condition (4), we have m(asx2n,bx2n+1, t) ≥ ψ(m(sx2n,x2n+1, t)), where m(sx2n,x2n+1, t) = min{m(s2x2n,tx2n+1, t),m(asx2n,s2x2n, t),m(tx2n+1,bx2n+1, t)}. lettinge n → ∞, by (3.1), (3.2) and (3.3), we obtain m(sz,z,t) ≥ ψ(min{m(sz,z,t),m(sz,sz,t),m(z,z,t)}) = ψ(min{m(sz,z,t),1,1}) = ψ(m(sz,z,t)) > m(sz,z,t), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 8 common fixed points for generalized ψ-contractions which is a contradiction. therefore, sz = z. if az 6= z, then there exists t > 0 such that 0 < m(az,z,t) < 1. by the condition (4), we have m(az,bx2n+1, t) ≥ ψ(m(z,x2n+1, t)), where m(z,x2n+1, t) = min{m(sz,tx2n+1, t),m(az,sz,t),m(tx2n+1,bx2n+1, t)}. letting n → ∞, by (3.1), (3.2) and (3.3), we obtain m(az,z,t) ≥ ψ(min{m(sz,z,t),m(az,sz,t),m(z,z,t)}) = ψ(min{m(z,z,t),m(az,z,t),m(z,z,t)}) = ψ(min{1,m(az,z,t),1}) = ψ(m(az,z,t)) > m(az,z,t), which is a contradiction and hence az = z. therefore, z is a common fixed point of a and s. since a(x) ⊂ bt(x), there exists z∗ ∈ x such that z = az = tz∗. if z 6= tz∗, then there exists t > 0 such that 0 < m(z,bz∗, t) < 1. by the condition (4) and lemma 3.1, we have m(z,bz∗, t) = m(az,bz∗, t) ≥ ψ(m(z,z∗)) = ψ(m(z,bz∗, t)) > m(z,bz∗, t), which is a contradiction. then z = bz∗. since b and t are compatible, we obtain m(tz,bz,t) = m(tbz∗,btz∗t) = 1 for any t > 0, which implies tz = bz. next, if z 6= bz, then there exists t > 0 such that 0 < m(z,bz,t) < 1. by the condition (4), we have m(z,bz,t) = m(az,bz,t) ≥ ψ(m(z,z)) = ψ(m(z,bz,t)) > m(z,bz,t), which is a contradiction. hence z = tz and so z = tz = bz = sz = az. case 2. suppose that t is continuous. in the same way as in case 1, we can obtain the result. case 3. suppose that a is continuous. then lim n→∞ asx2n = lim n→∞ a2x2n = az.(3.4) since s and a are compatible mappings, limn→∞ m(asx2n,sax2n, t) = 1. by (wna), we have m(sax2n,az,t) ≥ m(sax2n,asx2n, t) ∗ m(asx2n,az,t/2). letting n → ∞, we obtain limn→∞ m(sax2n,az,t) = 1, i.e., lim n→∞ sax2n = az.(3.5) if az 6= z, then there exists t > 0 such that 0 < m(az,z,t) < 1. by the condition (4), we have m(a2x2n,bx2n+1, t) ≥ ψ(m(ax2n,x2n+1, t)), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 9 s. suantai, y. j. cho and j. tiammee where m(ax2n,x2n+1, t) = min{m(sax2n,tx2n+1, t),m(a2x2n,sax2n, t),m(tx2n+1,bx2n+1, t)}. letting n → ∞, by (3.1), (3.4) and (3.5), we obtain m(az,z,t) ≥ ψ(min{m(az,z,t),m(az,az,t),m(z,z,t)}) = ψ(min{m(az,z,t),1,1}) = ψ(m(az,z,t)) > m(az,z,t), which is a contradiction and hence az = z. since a(x) ⊂ t(x), there exists z∗ ∈ x such that z = az = tz∗. if z 6= bz∗, then there exists t > 0 such that 0 < m(z,bz∗, t) < 1. by the condition (4), we have m(z,bz∗, t) = m(az,bz∗, t) ≥ ψ(m(z,z∗, t)) = ψ(m(z,bz∗, t)) > m(z,bz∗, t), which is a contradiction. then z = bz∗. since b and t are compatible, we obtain m(tz,bz,t) = m(tbz∗,btz∗t) = 1 for any t > 0, which implies tz = bz. next, if z 6= bz, then there exists t > 0 such that 0 < m(z,bz,t) < 1. by the condition (4) and lemma 3.1, we have m(z,bz,t) = m(az,bz,t) ≥ ψ(m(z,z,t)) = ψ(m(z,bz,t)) > m(z,bz,t), which is a contradiction and hence z = bz. since b(x) ⊂ s(x), there exists z∗∗ ∈ x such that z = bz = sz∗∗. if z 6= az∗∗, then there exists t > 0 such that 0 < m(z,az∗∗, t) < 1. by the condition (4), we have m(az∗∗,z,t) = m(az∗∗,bz,t) ≥ ψ(m(z∗∗,z,t)) ≥ ψ(m(az∗∗,z,t)) > m(az∗∗,z,t), which is a contradiction. then z = az∗∗. since s and a are compatible mappings, we have m(sz,az,t) = m(saz∗∗,asz∗∗, t) = 1 for any t > 0 and so sz = az. therefore, sz = tz = az = bz = z and so z is a common fixed point of a,b,s and t . case 4. suppose that b is continuous. in the same way as in case 1, we can obtain the result. now, we prove the uniqueness of the common fixed point of a,b,s and t . assume that x,y ∈ x are two common fixed points of a,b,s and t . if x 6= y, then there exists t > 0 such that 0 < m(x,y,t) < 1. by the condition (4), we c© agt, upv, 2019 appl. gen. topol. 20, no. 1 10 common fixed points for generalized ψ-contractions have m(x,y,t) = m(ax,by,t) ≥ ψ(m(x,y,t)) = ψ(min{m(sx,ty,t),m(ax,sx,t),m(by,ty,t)}) = ψ(min{m(x,y,t),m(x,x,t),m(y,y,t)}) = ψ(min{m(x,y,t),1,1}) = ψ(m(x,y,t)) > m(x,y,t), which is a contradiction. therefore, x = y. this completes the proof. � if we put s = t = ix (the identity mapping on x) in theorem 3.4, then we obtain the result of vetro [16] as follows: corollary 3.5 ([16]). let (x,m,∗) be a complete weak non-archimedean fuzzy metric space and a,b : x → x be two mappings. assume that there exists ψ ∈ ψ such that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,by,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{m(x,y,t),m(ax,x,t),m(y,by,t)}. suppose that, for any x,y ∈ x with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. if there exists x0 ∈ x such that m(x0,sx0, t) > 0 for all t > 0, then a and b have a unique common fixed point. if we put a = b and s = t = ix in theorem 3.4, then we obtain the following: corollary 3.6. let (x,m,∗) be a complete weak non-archimedean fuzzy metric space and a : x → x be a mapping. assume that there exists ψ ∈ ψ such that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,ay,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{m(x,y,t),m(ax,x,t),m(y,ay,t)}. suppose that, for any x,y ∈ x, with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. if there exists x0 ∈ x such that m(x0,ax0, t) > 0 for all t > 0, then s has a unique fixed point. as a consequence of corollary 3.6, by remark 2.4, we obtain the result of mihet [12] as follows: corollary 3.7 ([12]). let (x,m,∗) be a complete non-archimedean fuzzy metric space and a : x → x be a mapping. assume that there exists ψ ∈ ψ such c© agt, upv, 2019 appl. gen. topol. 20, no. 1 11 s. suantai, y. j. cho and j. tiammee that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,ay) ≥ ψ(m(x,y,t)), suppose that, for any x,y ∈ x with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. if there exists x0 ∈ x such that m(x0,ax0, t) > 0 for all t > 0, then a has a unique fixed point. we now give an example to illustrate theorem 3.4. example 3.8. let (x,m,∗) be a complete weak non-archimedean fuzzy metric space, where x = [0,∞) with the t-norm defined by a ∗ b = ab for any a,b ∈ [0,1] and the fuzzy set m given by: m(x,y,0) = 0, m(x,x,t) = 1 for all t > 0, m(x,y,t) = 0 for x 6= y and 0 < t ≤ 1, m(x,y,t) = t2/4 for x 6= y and 1 < t ≤ 2, m(x,y,t) = 1 for x 6= y and t > 2. define a function ψ : [0,1] → [0,1] by ψ(t) = √ t for all t ∈ [0,1]. then we see that ψ ∈ ψ. define four mappings a,b,s,t : x → x by ax = x, bx = √ x, sx = 2x, tx = 4 √ x. then we have the following: (1) ax = bx = sx = tx; (2) a,b,s and t are all continuous mappings; (3) the pair a,s and b,t are compatible; (4) the four couple (a,b;s,t) is a ψ-contractive mapping; (5) if we choose x0 = x1 = 0, then a0 = t0 and m(x0,x1, t) = m(0,0, t) > 0; (6) if we choose t = 3 2 , then, for any x,y ∈ x with x 6= y, we have 0 < m(x,y, 3 2 ) = 9 16 < 1. therefore, all the conditions of theorem 3.4 are satisfied. also, we see that a0 = b0 = s0 = t0 and so 0 is a unique common fixed point of a,b,s and t . 4. common fixed points for mappings with the common limit in this section, we prove some common fixed points for mappings with the common limit with respect to the value of the given mappings in weak nonarchimedean fuzzy metric spaces. definition 4.1. two pairs (a,s) and (b,t) of self-mappings of a weak nonarchimedean fuzzy metric space (x,m,∗) are said to have the common limit with respect to the value of the mapping s (resp., t) if there exist two sequence {xn} and {yn} of x such that lim n→∞ axn = lim n→∞ sxn = lim n→∞ byn = lim n→∞ tyn = sz (resp., tz) for some z ∈ x. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 12 common fixed points for generalized ψ-contractions theorem 4.2. let (x,m,∗) be a weak non-archimedean fuzzy metric space and a,b,s,t : x → x be four mappings. suppose that the following conditions holds: (1) a(x) ⊂ t(x) or b(x) ⊂ s(x); (2) the pairs a,s and b,t are also weakly compatible; (3) the pairs (a,s) and (b,t) have the common limit with respect to the value of the mapping s ( or t ); (4) there exists ψ ∈ ψ such that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,by,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{m(sx,ty,t),m(ax,sx,t),m(by,ty,t), m(by,sx,t),m(ax,ty,t)}. (5) for any x,y ∈ x with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. then a,b,s and t have a unique common fixed point. proof. since the pairs (a,s) and (b,t) have the common limit with respect to the value of the mapping s, there exists two sequences {xn} and {yn} in x such that lim n→∞ axn = lim n→∞ sxn = lim n→∞ byn = lim n→∞ tyn = sz for some z ∈ x. first, we show that az = sz. suppose that az 6= sz. then, by the condition (5), there exists t > 0 such that 0 < m(az,sz,t) < 1. by using the condition (4), we obtain m(az,byn, t) ≥ ψ(m(z,yn, t)),(4.1) where m(z,yn, t) = min{m(sz,tyn, t),m(az,sz,t),m(byn,tyn, t), m(byn,sz,t),m(az,tyn, t)}. by taking the limit as n → ∞ in (4.1), we have m(az,sz,t) ≥ ψ(m(az,sz,t)) > m(az,sz,t), which is a contradiction and so az = sz. since a(x) ⊂ t(x), there exists v ∈ x such that az = tv. next, we show that bv = tv. suppose that bv 6= tv. then, by the condition (5), there exists t > 0 such that 0 < m(bv,tv,t) < 1. by using the condition (4), we obtain m(tv,bv,t) = m(az,bv,t) ≥ ψ(m(z,v,t)),(4.2) c© agt, upv, 2019 appl. gen. topol. 20, no. 1 13 s. suantai, y. j. cho and j. tiammee where m(z,v,t) = min{m(sz,tv,t),m(az,sz,t),m(bv,tv,t), m(bv,sz,t),m(az,tv,t)}. = min{m(tv,tv,t),m(az,sz,t),m(bv,tv,t), m(bv,tv,t),m(tv,tv,t)} = min{1,m(bv,tv)}. hence, in (4.2), we obtain m(tv,bv,t) ≥ ψ(m(tv,bv,t)) > m(tv,bv,t), which is a contradiction and so tv = bv. therefore, we have u = az = sz = bv = tv. since the pairs a,s and b,t are weakly compatible, az = sz and bv = tv, we have au = aaz = asz = saz = su, bu = btv = tbv = tu.(4.3) next, we show that au = u. suppose that au 6= u. then there exists t > 0 such that 0 < m(au,u,t) < 1. by the condition (4), we obtain m(au,u,t) = m(au,bv,t) ≥ ψ(m(u,v,t)),(4.4) where m(u,v,t) = min{m(su,tv,t),m(au,su,t),m(bv,tv,t) m(bv,su,t),m(au,tv,t)} = min{m(au,bv,t),m(au,au,t),m(bv,bv,t) m(bv,au,t),m(au,bv,t)} = min{1,m(au,u,t)}. hence, in (4.4), we obtain m(au,u,t) ≥ ψ(m(au,u,t)) > m(au,u,t), which is a contradiction and so au = u. next, we show that bu = u. suppose bu 6= u. then there exists t > 0 such that 0 < m(bu,u,t) < 1. by the condition (4), we obtain m(u,bu,t) = m(au,bu,t) ≥ ψ(m(u,u,t)),(4.5) where m(u,u,t) = min{m(su,tu,t),m(au,su,t),m(bu,tu,t) m(bu,su,t),m(au,tu,t)} = min{m(au,bu,t),m(au,au,t),m(bu,bu,t) m(bu,au,t),m(au,bu,t)} = min{1,m(au,bu,t)}. hence, in (4.5), we obtain m(bu,u,t) ≥ ψ(m(bu,u,t)) > m(bu,u,t), c© agt, upv, 2019 appl. gen. topol. 20, no. 1 14 common fixed points for generalized ψ-contractions which is a contradiction and so bu = u. therefore, we have u = au = bu = su = tu, that is, u is a common fixed point of a,b,s and t . the uniqueness of the common fixed point follows the proof of theorem 3.4. this completes the proof. � remark 4.3. we don’t need the completeness of a weak non-archimedean fuzzy metric space (x,m,∗) in the proof of theorem 4.2. as a consequence of theorem 4.2, by putting s = t = ix, we obtain the following: corollary 4.4. let (x,m,∗) be a weak non-archimedean fuzzy metric space and a,b : x → x be two mappings. suppose that the following conditions holds: (1) the pairs (a,ix) and (b,ix) have the common limit with respect to the value of the mapping ix; (2) there exists ψ ∈ ψ such that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,by,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{m(x,y,t),m(ax,x,t),m(by,y,t), m(by,x,t),m(ax,y,t)}. (3) for any x,y ∈ x with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. then a and b have a unique common fixed point. by putting a = b and s = t in theorem 4.2, we obtain the following: corollary 4.5. let (x,m,∗) be a weak non-archimedean fuzzy metric space and a,s : x → x be a mapping. suppose that the following conditions holds: (1) a(x) ⊂ s(x); (2) a pair a,s is weakly compatible; (3) there exists a sequence {xn} in x such that lim n→∞ axn = lim n→∞ sxn = sz for some z ∈ x; (4) there exists ψ ∈ ψ such that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,ay,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{m(sx,sy,t),m(ax,sx,t),m(ay,sy,t), m(ay,sx,t),m(ax,sy,t)}. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 15 s. suantai, y. j. cho and j. tiammee (5) for any x,y ∈ x with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. then a and s have a unique common fixed point. by putting a = b and s = t = ix in theorem 4.2, we obtain the following: corollary 4.6. let (x,m,∗) be a weak non-archimedean fuzzy metric space and a,s : x → x be two mappings. suppose that the following conditions holds: (1) there exists a sequence {xn} in x such that lim n→∞ axn = lim n→∞ xn = z for some z ∈ x; (2) there exists ψ ∈ ψ such that, for all x,y ∈ x and t ∈ (0,∞) with m(x,y,t) > 0, the following condition holds: m(ax,ay,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{m(x,y,t),m(ax,x,t),m(ay,y,t), m(ay,x,t),m(ax,y,t)}. (3) for any x,y ∈ x with x 6= y, there exists t > 0 such that 0 < m(x,y,t) < 1. then a has a unique fixed point. now, we give an example to illustrate theorem 4.2. example 4.7. let (x,m,∗) be a complete weak non-archimedean fuzzy metric space, where x = [0,30) with the t-norm defined by a ∗ b = ab for any a,b ∈ [0,1] and the fuzzy set m given by: m(x,y,0) = 0, m(x,x,t) = 1 for all t > 0, m(x,y,t) = 0 for x 6= y and 0 < t ≤ 1, m(x,y,t) = t2/4 for x 6= y and 1 < t ≤ 2, m(x,y,t) = 1 for x 6= y and t > 2. then, for any x,y ∈ x with x 6= y, 0 < m(x,t, 3 2 ) = 9 16 < 1. define a function ψ : [0,1] → [0,1] by ψ(t) = √ t for all t ∈ [0,1]. then we see that ψ ∈ ψ. define four mappings a,b,s,t : x → x by ax = { 1, if x ∈ {1} ∪ (5,30), x + 7, if x ∈ (1,5], bx = { 1, if x ∈ {1} ∪ (5,30), x + 6, if x ∈ (1,5], sx =      1, if x = 1, 7, if x ∈ (1,5], x+1 6 if x ∈ (5,30) tx =      1, if x = 1, 9, if x ∈ (1,5], x − 4, if x ∈ (5,30). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 16 common fixed points for generalized ψ-contractions if we choose two sequences {xn} and {yn} defined by xn = yn = 5 + 1n for each n ≥ 1, then lim n→∞ axn = lim n→∞ sxn = lim n→∞ byn = lim n→∞ tyn = s1 = 1 ∈ x. this implies that the pairs (a,s) and (b,t) have the common limit with respect to the value of the mapping s. we see that a(x) = {1} ∪ (8,12], b(x) = {1} ∪ (7,11], s(x) = [1,5) ∪ {7}, t(x) = [1,26) and so a(x) ⊂ t(x), but b(x) 6⊂ s(x). it is easy to show that the mappings a,b,s,t satisfy the condition (4) in theorem 4.2. therefore, all the conditions of theorem 4.2 are satisfied and, also, we see that 1 is a unique common fixed point of a,b,s and t . remark 4.8. example 3.8 and example 4.7 show how significant of our main results (themrem 3.4 and theorem 4.2). these two theorems can guarantee existence of a common fixed point of four nonlinear mappings satisfying new type of contractive conditions while the previous known results cannot be applied. acknowledgements. s. suantai was partially supported by chiang mai university. yeol je cho was supported by basic science research program through national research foundation of korea (nrf) funded by the ministry of science, ict and future planning (2014r1a2a2a01002100). j. tiammee would like to thank the thailand research fund and office of the higher education commission under grant no. mrg6180050 for the financial support and chiang mai rajabhat university. references [1] y. j. cho, fixed points in fuzzy metric spaces, j. fuzzy math. 5 (1997), 949–962. [2] a. george and p. veeramani, on some results in fuzzy metric spaces, fuzzy sets syst. 64 (1994), 395–399. [3] a. george and p. veeramani, on some results of analysis for fuzzy metric spaces, fuzzy sets and systems. 90 (1997), 365–368. [4] m. grabiec, fixed points in fuzzy metric spaces, fuzzy sets and systems 27 (1989), 385–389. [5] v. gregori and a. sapena, on fixed-point theorems in fuzzy metric spaces, fuzzy sets and systems 125 (2002), 245–252. [6] m. jain, k. tas, s. kumar and n. gupta, coupled fixed point theorems for a pair of weakly compatible maps along with clrg-property in fuzzy metric spaces, j. appl. math. 2012 (2012) art. id 961210, 13 pp. [7] g. jungck, compatible mappings and common fixed points, internat. j. math. math. sci. 9 (1986), 771–779. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 17 s. suantai, y. j. cho and j. tiammee [8] g. jungck, common fixed points for noncontinuous nonself maps on nonmetric spaces, far east j. math. sci. 4 (1996), 19–215. [9] i. kramosil and j. michalek, fuzzy metric and statistical metric spaces, kybernetica 11 (1975), 336–344. [10] s. manro and c. vetro, common fixed point theorems in fuzzy metric spaces employing clrg and jclrst properties, ser. math. inform. 29 (2014), 77–90. [11] j. mart́ınez-moreno, a. roldán, c. roldán and y. j. cho, multi-dimensional coincidence point theorems for weakly compatible mappings with the clrg-property in (fuzzy) metric spaces, fixed point theory appl. 2015, 2015:53. [12] d. mihet, fuzzy ψ-contractive mappings in non-archimedean fuzzy metric spaces, fuzzy sets and systems 159 (2008), 739–744. [13] a.-f. roldán-lópez-de-hierro and w. sintunavarat, common fixed point theorems in fuzzy metric spaces using the clrg-property, fuzzy sets and systems 282 (2016), 131– 142. [14] a. sapena, a contribution to the study of fuzzy metric spaces, applied general topology 2, no. 1 (2001), 63–75. [15] b. schweizer and a. sklar, statistical metric space, pacific j. math. 10 (1960), 314–334. [16] c. vetro, fixed points in weak non-archimedean fuzzy metric spaces, fuzzy sets syst. 162 (2011), 84–90. [17] l. a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 18 22.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 113 118 flows equivalencesgabriel soler l�opez�abstract. given a di�erential equation on an open set o ofan n-manifold we can associate to it a pseudoow, that is, a owwhose trajectories may not be de�ned in the entire real line. inthis paper we prove that this pseudoow is always equivalent toa ow with its trajectories de�ned in all r. this result extends asimilar result of vinograd stated in the n-dimensional euclideanspace.2000 ams classi�cation: primary 37e35, 37c15. secondary 58c25.keywords: flow, pseudoow, equivalence di�eomorphism.1. introduction.it is well known that given a crow (r � 1) on an n-manifold m, :r � m ! m, we can associate to it a cr�1-autonomous di�erential equationy0 = f(y). where f maps m onto its tangent bundle, tm, in the followingway f(y) = @ dt (0;y).the converse does not work in general because the solutions of a di�erentialequation could not be de�ned in the entire real line. for example, if we takethe autonomous di�erential equation (x0;y0) = (1;1 + tan2(x)), the solutionsare de�ned for each initial condition (x0;y0) in an interval of length �. thenwe can not associate to this autonomous di�erential equation a ow. however,if the manifold is compact, the converse does work [1, theorem 4, x1.9] and [5,p.11].let us introduce some terminology. given two ows and � on an n-manifold m we say that they are cr-equivalent if there exists a cr-di�eo-morphism h : m ! m such that h conserves the orbits of �. that is, thesubsets h(�(r;p)) and (r;h(p)) of m are equal for any p 2 m. moreover theorientations of the curves h(p)(t) = (t;h(p)) and h��p(t) = h��(t;p) coincidefor any p 2 m, that is, there exists a continuous increasing map ip : r ! rfor which h � �p(t) = h(p)(ip(t)). when we use the norm of a vector x 2 rnwe are always using the norm kxk = kxk1 = maxi2f1;2;:::;ngfjx1j; jx2j; : : : ; jxnjg.�this paper has been partially supported by the d.g.i.c.y.t. grant pb98-0374-c03-01. 114 gabriel soler l�opezby r+ we denote the set of positive real numbers. as usual, given o � m,bd(o) denote the topological boundary of the set o.from now on, when speaking of c0-di�erential equations they are supposedto be continuous and locally lipschitz. let r 2 n[f0g and take a cr-di�erentialequation y0 = f(y) on an open set o � m, f : o ! to. the classical theoryof di�erential systems assures that there exists a cr-map : d ! o calledpseudoow, where d is an open subset of r �o and for each p 2 o the curve p(t) = (t;p) is the solution of the equation y0 = f(y) with initial conditiony(0) = p. analogously to the de�nition of equivalence between ows we say thattwo pseudoows � : d � r�o ! o and : e � r�o ! o are cr-equivalentif there exists a cr-di�eomorphism h : o ! o such that h conserves the orbitsof � and the orientations of the curves h(p)(t) and h � �p(t) coincide for allp 2 o. with this terminology we will say that two autonomous di�erentialequations are cr-equivalent if their associated pseudoows are cr-equivalent,moreover the di�eomorphism h will be called equivalence di�eomorphism.the basic question in which we are interested is to prove that for any cr-autonomous di�erential system in an open set o, we can �nd a cr-equivalentautonomous di�erential equation such that the associated pseudoow is in facta ow, that is, de�ned in all r � o. this question was solved by vinograd [4,pp. 19-21] when the phase space is rn.theorem 1.1 (vinograd). let o be an open set of rn and let f : o ! rnbe a cr-map (r � 0). then there exists a cr-map g : o � rn ! rn suchthat the equations y0 = f(y) and y0 = g(y) are cr-equivalent and the associatedpseudoow to g is a ow. moreover, the equivalence di�eomorphism is theidentity map. (when r = 0 we consider f and g to be locally lipschitz)the aim of this paper is to prove the following theorem that generalizes theprevious one:theorem 1.2 (main result). let m be an n-manifold, o an open set of mand f : o ! to a cr-map (r � 0) . then there exists a cr-map g : o ! tosuch that the equations y0 = f(y) and y0 = g(y) are cr-equivalent and theassociated pseudoow to g is a ow. (when r = 0 we consider f and g to belocally lipschitz)section 2 is devoted to state some classical results that we need in the proofof our result. we also construct a positive c1-function that vanish only inthe boundary of o. this function will be essential in the proof of the maintheorem in section 3. 2. preliminary resultsin the sequel we are going to use the whitney theorem that provides a c1n-manifold m embedded in r2n+1 (see [2, x1.3]). another whitney theoremabout function extensions is stated and used in the proof of lemma 2.4 toconstruct a scalar c1-function f : rn ! r vanishing only in the boundary ofan open set o and being strictly positive in o. flows equivalences 115theorem 2.1 (whitney). let m be an n-manifold of class cr, r � 1. thenthere exists a cr-embedding f : mn ! r2n+1 such that f(m) is a closedc1-submanifold of r2n+1.we will use another less known whitney's theorem. its proof can be foundcombining [7, p. 177,th. 4] and [8]. we introduce some necessary terminologyfor its statement: if � 2 (f0g [ n)n, y 2 rn and f is a map de�ned on anopen subset of rn, we denote �! = �1!�2! : : :�n!, j�j = �1 + �2 + � � � + �n, y� =y�11 y�22 : : :y�nn and d�f(y) = @�1+�2+��+�n@x�11 @x�22 :::@x�nn f(y). as usual we mean d0f = f.theorem 2.2 (whitney). let c � rn be a closed set (as a subset of rn).then the following statements hold.(1) let f0 : c ! rm be a bounded lipschitz map. then there is a boundedlipschitz map f : rn ! rm such that f(x) = f0(x) for any x 2 c.(2) let 1 � k � 1 and let f� : c ! rm be arbitrary maps for any� 2 (f0g [ n)n with 0 � j�j � k. let f ;r : c � c ! rm be de�ned byf ;r(x;y) = f (y) � p0�j�j�r f +�(x)(y�x)��!kx � ykrif x 6= y and f ;r(x;x) = 0otherwise, for any 2 (f0g [ n)n and 0 � r < 1 with j j + r � k.suppose that all maps f ;r are continuous. then there is a ck mapf : rn ! rm such that d�f = f� for any � 2 (f0g [ n)n, 0 � j�j � k.the following result is an easy consequence of the previous theorem:corollary 2.3. let c � rn be a closed set decomposed into disjoint sets aand b, c = a [ b. given two real numbers a and b de�ne f� : c ! rm asfollows: f�(x) = a for any x 2 a and f�(x) = b for any x 2 b. then there isa c1-map f : rn ! r such that f(x) = f�(x) for any x 2 c and any partialderivate of f is equal to 0 in c.proof. take for each � 2 (f0g[n)n, f� : c ! r with f� � 0 for any 0 < j�j <1 and f0 = f�. it is clear that the functions f� satisfy the conditions of part2 of theorem 2.2. then there exists a c1-function f : rn ! r that extendsf0 and whose derivates are 0 in c. �we also need some previous lemmas:lemma 2.4. let o � rn be a nonclosed set. there exists a c1-map f : rn ![0;1[ such that f(x) = 0 for any x 2 bd(o) and f(x) 2]0;1[ for any x 2 o.proof. we are going to construct the c1-map as the sum of a function series.thus we are going to construct c1-functions fi : rn ! [0;1[ for every i 2 n.de�ne cj = ? for j 2 znn, c1 = fx 2 o : 1 < d(x;bd(o)g and for j 2 nnf1gconsider cj = fx 2 o : 1j < d(x;bd(o)) < 1j�1g (eventually cj = ? for j 116 gabriel soler l�opezsmall). let a = ci and b = [j>i+1cj s[j 0, s is strictly increasing and there exists itsinverse t : s(i) ! (a;b). de�ne z : s(i) ! o as z(s) = y(t(s)) and notice thatz0(s) = y0(t(s)) 1s0(t(s)) = f(y(t(s))) (y(t(s)))�(y(t(s))) = g(z(s)) = g(z(s))and z(c) = y(t(c)) = y(c). thus the orbits of and � coincide and also theirorientations because s is strictly increasing. �references[1] a. andronov and e. leontovich, qualitative theory of second-order dynamic systems,(john wiley and sons, new york, 1973).[2] m.w. hirsch, di�erential topology. (springer verlag, new york, 1976).[3] v. jim�enez l�opez, ecuaciones diferenciales, (universidad de murcia, murcia, 2000).[4] v. nemytskii and v. stepanov, qualitative theory of di�erential equations. (princetonuniversity press, princeton, 1960).[5] jacob palis and wellington de melo, geometric theory of dynamical systems. (springer-verlag, new-york, 1982).[6] jorge sotomayor, li�c~oes de equa�c~oes diferenciais ordin�arias. (instituto de matem�aticapura e aplicada rio de janeiro, rio de janeiro, 1979).[7] e. m. stein, singular integrals and di�erentiability properties of functions. (princetonuniv. press, princeton, 1970).[8] h. whitney, analytic extensions of di�erentiable functions de�ned in closed sets, trans.amer. math. soc. 36 (1934), 63-89. 118 gabriel soler l�opez received february 2001revised version april 2001 gabriel soler l�opezdepartamento de matem�atica aplicada y estad��sticauniversidad polit�ecnica de cartagenapaseo alfonso xiii 5230203-cartagenaspaine-mail address: gabriel.soler@upct.es () @ appl. gen. topol. 16, no. 1(2015), 1-13doi:10.4995/agt.2015.3439 c© agt, upv, 2015 baire property in product spaces constancio hernández, leonardo rodŕıguez medina and mikhail tkachenko∗ departamento de matemáticas, universidad autónoma metropolitana, av. san rafael atlixco 186, col. vicentina, iztapalapa, c.p. 09340, méxico d.f., mexico. (chg@xanum.uam.mx, leonardo.rodriguez@unam.mx, mich@xanum.uam.mx) abstract we show that if a product space π has countable cellularity, then a dense subspace x of π is baire provided that all projections of x to countable subproducts of π are baire. it follows that if xi is a dense baire subspace of a product of spaces having countable π-weight, for each i ∈ i, then the product space ∏ i∈i xi is baire. it is also shown that the product of precompact baire paratopological groups is again a precompact baire paratopological group. finally, we focus attention on the so-called strongly baire spaces and prove that some baire spaces are in fact strongly baire. 2010 msc: 54h11; 54e52. keywords: baire space; strongly baire space; skeletal mapping; banachmazur-choquet game; paratopological group; semitopological group. 1. introduction baire spaces, i.e. topological spaces which satisfy the conclusion of the baire category theorem, constitute an important class in several branches of mathematics. all čech-complete spaces, the pseudocompact spaces or, more generally, the regular feebly compact spaces are baire. in a sense, the baire property is one of the weakest forms of topological completeness. ∗the research is partially supported by consejo nacional de ciencias y tecnoloǵıa (conacyt), grant cb-2012-01-178103 received 10 august 2013 – accepted 10 august 2014 http://dx.doi.org/10.4995/agt.2015.3439 c. hernández, l. rodŕıguez medina and m. tkachenko the class of baire spaces is closed under taking open subspaces as well as gδdense subspaces. also, images of baire spaces under open continuous mappings and, more generally, under d-open mappings (see [23]) are baire. the problem of whether a product of a family of baire spaces is baire is an old one and is also well known that the answer to the problem is negative even for the product of two metric baire spaces [9, 11] or two linear normed spaces [16, 26]. however, there are several cases when products (finite, countable or arbitrary) of baire spaces are again baire. in the following list we collect some results of this kind. let x and y be hausdorff spaces with the baire property. then x × y is a baire space provided that the second factor satisfies one of the following conditions (all the terms will be explained in section 4): a) y is quasi-regular and has a countable π-base (oxtoby [18] and froĺık [12]); b) y is pseudo-complete or countably complete (aarts and lutzer [1], froĺık [12]); c) y is α-favorable in the choquet game g(y ) (white [29]); d) y is metric and hereditarily baire (moors [17]). examples of spaces as in a) are baire separable metrizable spaces, as in b) are feebly compact spaces, as in c) are čech-complete or countably complete spaces and as in d) are completely metrizable spaces. our aim is to present some other productive classes of baire spaces. the first of them is the class of dense baire subspaces of products of spaces having countable π-weight. the corresponding fact is established in corollary 2.10 which extends a theorem of oxtoby [18] and also implies a result of tkachuk [25, theorem 4.8] on the productivity of the baire property in the spaces of the form cp(x). a second productive class consists of the precompact paratopological groups with the baire property (see theorem 3.3). it is worth noting that the productivity of the class of precompact topological groups with the baire property was proved earlier in [7]. in section 4 we establish that under additional conditions, baire spaces turn out to be strongly baire. along with the main theorem of [15] our results imply that every regular baire semitopological group g is a topological group provided that g is either a lindelöf σ-space or a w∆-space (see corollaries 4.4 and 4.6). 2. products with the baire property there exist several countably productive classes of spaces that fail to be productive. this is the case of (complete) metric spaces or quasi-regular spaces with a countable π-base. it is also well known, however, that an arbitrary product of complete metric spaces has the baire property (bourbaki [5]), as does the product of quasi-regular baire spaces of countable π-weight (oxtoby [18]). later on, chaber and pol [8] proved that a similar result holds for products of hereditarily baire metric spaces. in corollary 2.10 below we extend the result of oxtoby to dense baire subspaces of products of spaces with countable c© agt, upv, 2015 appl. gen. topol. 16, no. 1 2 baire property in product spaces π-weight. in fact, we prove a more general fact in theorem 2.9 for spaces with a ‘good’ lattice of skeletal mappings and then apply it to dense subspaces of product spaces. let us recall that a continuous mapping f : x → y is said to be skeletal if the preimage f−1(a) of every nowhere dense subset a of y is nowhere dense in x. it is clear that f is skeletal iff f−1(u) is dense in x for every dense open subset u of y iff the closure of f(v ) has a non-empty interior in y for every non-empty open set v in x. skeletal mappings need not be surjective (consider the identity embedding of the rational numbers q into the real line r, where both spaces carry the usual interval topologies). the following fact clarifies how close a skeletal mapping has to be to surjective ones (see [27, lemma 2.2]). needless to say, the mappings in the next two lemmas are not assumed to be surjective. lemma 2.1. if f : x → y is a skeletal mapping, then the set inty f(v ) is dense in f(v ), for each open set v in x. in particular, inty f(x) is dense in f(x). lemma 2.2. suppose that f : x → y and g : y → z are continuous mappings. if both f and h = g ◦ f are skeletal, then so is g. proof. suppose for a contradiction that g fails to be skeletal. then y contains a non-empty open set v such that the image g(v ) is nowhere dense in z. since h is skeletal, the set h−1(g(v )) is nowhere dense in x. it follows from v ⊂ g−1g(v ) and h−1(g(v )) = f−1g−1(g(v )) that the non-empty open set f−1(v ) is nowhere dense in x. this contradiction completes the proof. � we know that open surjective mappings preserve the baire property [12]. the same conclusion is valid for the wider class of skeletal mappings [13, theorem 1]. for completeness sake we present a short proof of this fact. lemma 2.3. let f : x → y be a continuous onto mapping. if f is skeletal and x is baire, then y is also baire. proof. take a countable family {un : n ∈ ω} of open dense subsets of y . since f is skeletal and continuous, vn = f −1(un) is a dense open subset of x, for each n ∈ ω. the space x being baire, the intersection s = ⋂ n∈ω vn is dense in x. it follows from our definition of the sets vn’s that s = f −1f(s), so f(s) = ⋂ n∈ω un is dense in y . hence y is baire. � suppose that f : x → y and g : x → z are continuous onto mappings. we write f ≺ g if there exists a continuous mapping h: y → z satisfying g = h◦f. let x be a space and m a family of continuous onto mappings of x elsewhere. we say that m is an ω-directed lattice for x if m generates the topology of x and for every countable subfamily c of m, there exists f ∈ m such that f ≺ g for each g ∈ c (i.e. every countable subfamily of m has a lower bound in m with respect to the partial order ≺). c© agt, upv, 2015 appl. gen. topol. 16, no. 1 3 c. hernández, l. rodŕıguez medina and m. tkachenko proposition 2.4. let x be a space with an ω-directed lattice m of skeletal mappings and suppose that x has countable cellularity. then x is baire if and only if f(x) is baire, for each f ∈ m. proof. by lemma 2.3, we have to prove the sufficiency only. suppose that x is not baire. it suffices to find an element f ∈ m such that f(x) fails to be baire. notice first that the restriction of a skeletal mapping to an open subspace of x is again skeletal. therefore, assuming that x is of the first category, we will find f ∈ m such that f(x) is also of the first category. this requires the following simple fact. claim. if a is a nowhere dense subset of x, then there exists f ∈ m such that f(a) is nowhere dense in f(x). indeed, denote by b the family of sets of the form g−1(u), where g ∈ m and u is a non-empty open set in g(x). since m generates the topology of x, the family b is a base for x. let γa be a maximal disjoint subfamily of b such that every element of γa is disjoint from a. by the maximality of γa, the open set ⋃ γa is dense in x \ a and, hence, in x. since the cellularity of x is countable, we see that |γa| ≤ ω. therefore, for every v ∈ γa, we can find gv ∈ m and an open set uv ⊂ gv (x) such that v = g −1 v (uv ). since γa is countable, there is an element f ∈ m satisfying f ≺ gv for each v ∈ γa. it is clear that v = f−1f(v ) and f(v ) is open in f(x), for each v ∈ γa. hence f( ⋃ γa) is a dense open set in f(x). since a ∩ ⋃ γa = ∅, we infer that the sets f(a) and f( ⋃ γa) are disjoint, which in turn implies that f(a) is nowhere dense in f(x). this proves our claim. finally, suppose that x = ⋃ n∈ω an, where each an is nowhere dense in x. by our claim, for every n ∈ ω, there exists fn ∈ m such that fn(an) is nowhere dense in fn(x). take f ∈ m such that f ≺ fn for each n ∈ ω. it follows from our choice of f that for every n ∈ ω, there exists a continuous mapping gn : f(x) → fn(x) satisfying fn = gn ◦ f. by lemma 2.2, the mappings gn’s are skeletal. therefore, f(an) is nowhere dense in f(x), for each n ∈ ω—otherwise some of the sets fn(an) = gn(f(an)) would have a non-empty interior. this implies that the image f(x) is of the first category and, hence, completes the proof. � corollary 2.5. let {zi : i ∈ i} be a family of spaces such that the product z = ∏ i∈i zi has countable cellularity. then a dense subspace x of z is baire if and only if πj(x) is baire for each countable set j ⊂ i, where πj is the projection of z to ∏ i∈j zi. proof. for every j ⊂ i, the projection πj is an open surjective mapping. in particular, πj is skeletal. since x is dense in z, the restriction of πj to x, say, pj is again skeletal. notice that the family {pj : j ⊂ i, |j| ≤ ω} is an ω-directed lattice for x. therefore, the required conclusion follows from proposition 2.4. � c© agt, upv, 2015 appl. gen. topol. 16, no. 1 4 baire property in product spaces lemma 2.6. suppose that fi : xi → yi is a skeletal onto mapping, for each i ∈ i. then the product mapping ∏ i∈i fi : ∏ i∈i xi → ∏ i∈i yi is also skeletal. proof. first we prove the lemma in the special case of two mappings, say, f1 and f2. let g = f1 × f2 and take a non-empty open set u in x1 × x2. we can find non-empty open sets u1 and u2 in x1 and x2, respectively, such that u1 × u2 ⊂ u. since f1 and f2 are skeletal, the set fi(ui) contains a non-empty open set vi, for i = 1, 2. it is clear that g(u) ⊃ g(u1 × u2) = f1(u1) × f2(u2). hence the set g(u) contains the non-empty open set v1 × v2 in y1 × y2. this proves that g = f1 × f2 is skeletal. applying induction, we see that the claim of the lemma is valid for finite products of skeletal mappings (notice that the surjectivity assumption has not been used so far). finally, we prove the lemma in the general case. let g = ∏ i∈i fi. it suffices to verify that g(u) has a non-empty interior, for every non-empty canonical open set u in x = ∏ i∈i xi. therefore, let u = n ⋂ i=1 π −1 ik (uik ), where i1, . . . , in are pairwise distinct elements of the index set i, πik is the projection of x onto xik , and uik is a non-empty open set in xik, where 1 ≤ k ≤ n. since the mappings fi’s are surjective, it follows from the definition of g that g(u) = n ⋂ k=1 p −1 ik (dk), where pik is the projection of y = ∏ i∈i yi onto the factor yik and dk = fik(uik ), k = 1, . . . , n. denote by vk the interior of the set dk in yik , 1 ≤ k ≤ n. since the mappings fik ’s is skeletal, the sets v1, . . . vn are non-empty. hence the set g(u) contains the non-empty open set ⋂n k=1 p −1 ik (vk) in y . so g is skeletal. � proposition 2.7. let {xi : i ∈ i} be a family of spaces such that every xi has an ω-directed lattice of skeletal mappings onto separable (first countable, metrizable) spaces. then the product space x = ∏ i∈i xi also has an ω-directed lattice of skeletal mappings onto separable (first countable, metrizable) spaces. proof. for every i ∈ i, let mi be an ω-directed lattice of skeletal mappings onto separable (first countable, or metrizable) spaces. for every countable set j ⊂ i, put mj = { ∏ i∈j fi : fi ∈ mi for each i ∈ j } . by lemma 2.6, the family mj consists of skeletal surjective mappings. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 5 c. hernández, l. rodŕıguez medina and m. tkachenko for every j ⊂ i, denote by πj the natural projection of x onto the subproduct xj = ∏ i∈j xi. clearly the projections πj’s are open, continuous, and surjective. hence the family m = {f ◦ πj : j ⊂ i, |j| ≤ ω, f ∈ mj} consists of skeletal mappings of x onto separable (first countable, metrizable) spaces. since each mi is an ω-directed lattice, it follows from our definitions that so is m. � suppose that x is a space with an ω-directed lattice m of mappings of x elsewhere. then m is said to be a σ-lattice for x if for every sequence {fn : n ∈ ω} in m, the diagonal product ∆n∈ωfn of the mappings fn’s is also an element of m. a typical example of a σ-lattice for a subspace x′ of a product x = ∏ i∈i xi comes if we consider consider the family of restrictions to x′ of projections of x to countable subproducts. proposition 2.8. if a space x has a σ-lattice of skeletal mappings onto spaces of countable cellularity, then x also has countable cellularity. proof. let m be a σ-lattice of skeletal mappings of x onto spaces of countable cellularity. since m generates the topology of x, the family b = {f−1(u) : f ∈ m, u is open in f(x), and u 6= ∅} forms a base for x. consider a disjoint family γ of non-empty open sets in x. we can assume without loss of generality that γ ⊂ b and that γ is maximal with respect to the inclusion relation. hence ⋃ γ is dense in x. take an arbitrary element v0 ∈ γ and put γ0 = {v0}. since v0 ∈ γ ⊂ b, we can find f0 ∈ m and a non-empty open set u0 in f(x) such that v0 = f −1 0 (u0). let c0 = {f0}. suppose that for some n ∈ ω, we have defined subfamilies γ0, . . . , γn of γ and elements f0, . . . , fn of m satisfying the following conditions for all k, m with 0 ≤ k < m ≤ n: (i) |γm| ≤ ω; (ii) fk ≺ fm; (iii) every v ∈ γm has the form f −1 m (u), for a non-empty open set u in fm(x); (iv) the set fk( ⋃ γk+1) is dense in fk(x). for every v ∈ γ, put uv = int fn(v ). by lemma 2.1, uv is dense in fn(v ), for each v ∈ γn. since ⋃ γ is dense in x and fn is continuous, the union of the family {vu : v ∈ γ} is dense in xn = fn(x). by our assumptions, the space xn has countable cellularity. hence there exists a countable subfamily γn+1 of γ such that ⋃ v ∈γn+1 vu is dense in xn. for every v ∈ γn+1, take an element fv ∈ m and a non-empty open set uv in fv (x) such that v = f −1 v (uv ). since the family {fv : v ∈ γn+1} is countable, there exists fn+1 ∈ m such that fn+1 ≺ fn and fn+1 ≺ fv for each v ∈ γn+1. hence the families c© agt, upv, 2015 appl. gen. topol. 16, no. 1 6 baire property in product spaces γ0, . . . , γn, γn+1 and elements f0, . . . , fn, fn+1 of m satisfy (i)–(iv) at the step n + 1. it follows from (i) that γ∗ = ⋃ n∈ω γn is a countable subfamily of γ. since m is a σ-lattice for x and fn+1 ≺ fn ∈ m for each n ∈ ω, we can assume that the diagonal product of the family {fn : n ∈ ω}, say, f∗ is an element of m. then (iii) implies that v = f∗(v ) is open in x∗ = f∗(x) and v = f −1 ∗ f∗(v ), for each v ∈ γ∗. it also follows from our choice of f∗ ∈ m and condition (iv) that the open set f∗( ⋃ γ∗) is dense in x∗. since f∗ is skeletal, the set ⋃ γ∗ = f −1 ∗ f∗( ⋃ γ∗) is dense in x. however, the family γ is disjoint and γ∗ ⊂ γ, whence it follows that γ∗ = γ. we conclude that |γ| ≤ ω and, hence, the cellularity of x is countable. � combining propositions 2.4 and 2.8, we obtain the following result: theorem 2.9. if a space x has a σ-lattice of skeletal mappings onto baire spaces of countable cellularity, then x is baire as well. for a tychonoff space x, let cp(x) be the space of continuous real-valued functions on x endowed with the pointwise convergence topology. it was shown by tkachuk in [25, theorem 4.8] that the product of an arbitrary family of spaces of the form cp(x) is baire provided that each factor is baire. since cp(x) is dense in r x, where r is the real line, tkachuk’s theorem follows from the next result: corollary 2.10. let {xi : i ∈ i} be a family of baire spaces, where each xi is a dense subspace of a product of regular spaces of countable π-weight. then the product space x = ∏ i∈i xi is also baire. proof. it is clear that x is a dense subspace of a product of spaces with countable π-bases, say, y = ∏ j∈j yj. taking the restriction to x of the projections of y to countable subproducts, we obtain a σ-lattice of skeletal mappings of x onto spaces with countable π-bases (notice that a dense subspace of a space with a countable π-base also has a countable π-base). it is also clear that every space with a countable π-base has countable cellularity. by oxtoby’s theorem in [18], every (countable) subproduct of y is baire. hence x has a σ-lattice of skeletal mappings onto baire spaces of countable cellularity. the required conclusion now follows from theorem 2.9. � 3. products of paratopological groups in independent papers, oxtoby [18] and froĺık [12] introduced a number of classes of baire spaces, including as examples the class of čech-complete spaces and the class of regular feebly compact spaces, i.e. those where locally finite families of open subsets are finite. we also recall that a space x is countably complete (pseudo-complete) if there exists a sequence of (pseudo-) bases {bn} for x such that for each decreasing sequence of open sets {un} in x, where each un is contained in some element of bn, we have ⋂ un 6= ∅. it was proved in [18, 12] that if {xα}α∈a is a family of spaces from any of the aforementioned c© agt, upv, 2015 appl. gen. topol. 16, no. 1 7 c. hernández, l. rodŕıguez medina and m. tkachenko classes, then the product x = ∏ α∈a xα also belongs to the same class, so x is a baire space. on the other hand, according to the well-known theorem of comfort and ross [10], the product of an arbitrary family of pseudocompact topological groups is pseudocompact. hence such products have the baire property. it is also known that pseudocompact topological groups are precompact [10, theorem 1.1]. it was recently proved in [7] that the class of baire precompact topological groups is closed under taking arbitrary products. now we extend the latter result to the class of baire precompact paratopological groups, i.e. the groups with jointly continuous multiplication. suppose that (g, τ) is a paratopological group. let us denote by τ∗ the finest topological group topology coarser than τ. also, let g∗ be the corresponding topological group (g, τ∗). we need the following fact proved in [24, theorem 10]: lemma 3.1. let {gi}i∈i be a family of paratopological groups. then g∗ ∼= ∏ i∈i (gi)∗. in general, given two topologies τ1 and τ2 on a set x with τ2 ⊂ τ1, the baire property of x1 = (x, τ1) does not imply that x2 = (x, τ ′) is baire nor vice versa. in the next lemma we find an additional condition on τ1 and τ2 which is responsible for the preservation of the baire property in both directions. lemma 3.2. let τ1 and τ2 be two topologies on a set x such that τ2 ⊂ τ1. if τ2 is a π-base for x1, then the families of nowhere dense sets in x1 and x2 coincide. in particular, x2 is a baire space if and only if so is x1. proof. given a nowhere dense set f ⊂ x1 and a non-empty open set o2 ⊂ x2, there exists a non-empty open set o1 ⊂ x1 such that o1 ⊂ o2 and o1∩f1 = ∅. let u2 ⊂ x2 be a non-empty open set contained in o2. then u2 ⊂ o2 and u2 ∩ f1 = ∅, so f1 is nowhere dense in x2. conversely, if f2 is nowhere dense in x2, then the inclusion τ2 ⊂ τ1 implies that f2 is nowhere dense in x. now, if a non-empty open set u1 ⊂ x1 is of the first category in x1, then a non-empty open set u2 ⊂ x2 contained in u1 is also of the first category in x2. � remark. it follows from lemma 3.2 that the closure in x2 of every nowhere dense subset of x1 is nowhere dense in x2. we recall that a paratopological group g is called saturated if u−1 has a non-empty interior in g, for every neighborhood u of the neutral element in g. the sorgenfrey line is an example of a saturated paratopological group which is not a topological group. it is easy to see that an arbitrary product of saturated paratopological groups is again saturated. theorem 3.3. let {gi}i∈i be a family of baire precompact paratopological groups. then g = ∏ i∈i gi is a baire precompact paratopological group. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 8 baire property in product spaces proof. since each gi is baire and precompact, it is saturated by [20, proposition 3.1] or [2, theorem 2.5]. hence the topology of (gi)∗ forms a π-base for gi (see [4, theorem 5]). apply lemma 3.1 to conclude that g∗ ∼= ∏ i∈i (gi)∗, where each (gi)∗ is a baire precompact topological group by lemma 3.2. hence, according to [4, theorem 3.4], g∗ is a baire precompact topological group. since g is precompact and the topology of g∗ is a π-base for g, lemma 3.2 implies that g is baire. � the comfort-ross theorem on products of pseudocompact topological groups was extended by ravsky [21] to products of feebly compact paratopological groups. furthermore, ravsky noted that every feebly compact paratopological group satisfying the t3 separation axiom was a topological group, thus weakening the regularity assumption in a theorem of arhangel’skii and reznichenko [3, theorem 1.7]. on the other hand, sanchis and tkachenko [22] constructed the first examples of hausdorff feebly compact paratopological groups with the baire property that were not topological groups. this makes it tempting to prove or refute the following conjecture. conjecture 3.4. let {gi}i∈i be a family of baire feebly compact paratopological groups. then g = ∏ i∈i gi is a baire feebly compact paratopological group. 4. topological games and the baire property in this section we apply the topological game methods to handle a number of complete-type topological spaces close to the baire ones. the so-called banachmazur-choquet game or choquet game, for brevity, g(x) is played on a space (x, τ) by players α and β. both players alternatively choose elements from τ∗ = τ \ {∅} as follows. i) player β moves first choosing a set b1 ∈ τ ∗. ii) player α then responds by choosing a set a1 ∈ τ ∗ contained in b1. iii) at the n-th step player β chooses a set bn ∈ τ ∗ contained in an−1. iv) player α responds by choosing a set an ∈ τ ∗ contained in bn. the sequence b1 ⊃ a1 ⊃ b2 ⊃ a2 ⊃ . . . obtained according to i)–iv), also denoted by {(an, bn)}n, is called a play. player α wins the play {(an, bn)}n if ⋂ n∈n an 6= ∅, otherwise β wins the play. by a strategy for a player we mean a rule that specifies each move of the player in every possible situation. more precisely, a strategy t = {tn : n ∈ n} for β consists of a sequence of τ∗-valued mappings such that 1. t1 : {∅} → τ ∗, b1 = t1(∅); 2. tn : {(a1, . . . , an−1) ∈ (τ ∗)n−1 : aj ⊂ tj(a1, . . . , aj−1), 1 ≤ j ≤ n − 1} → τ∗, bn = tn(a1, . . . , an−1). such a sequence of subsets {an}n is called a t-play. a strategy t is called a winning strategy for player β if β wins each t-play. strategies and winning c© agt, upv, 2015 appl. gen. topol. 16, no. 1 9 c. hernández, l. rodŕıguez medina and m. tkachenko strategies for player α are defined similarly. the importance of this type of games is based, in particular, on the following characterization of baire spaces. theorem 4.1 (banach-mazur-choquet). a space x is baire if and only if player β does not have a winning strategy in the choquet game g(x). in particular, a space with a winning strategy for player α, or α-favorable space, is a baire space. for example, a countably complete space is α-favorable. the class of α-favorable spaces is different from the class of β-unfavorable spaces or baire spaces. in fact, this class is closed under arbitrary products, as it was proved by white in [29]. using topological games, one can define new classes of topological spaces. given a dense subset d ⊂ x, the game gs(d) played on x by two players, α and β, has the same playing rules as the choquet game, with the only difference in the winner decision rule. we declare that the player α wins a play {(an, bn)}n provided that (1) ⋂ an 6= ∅; (2) every sequence {an}n with an ∈ an ∩ d, for each n ∈ n, has an accumulation point in x. following kenderov, kortezov, and moors [15], we say that a regular space x is strongly baire if there exists a dense subset d of x such that player β does not have a winning strategy in the gs(d)-game played on x. in [15], the authors extended the results from [6] on the continuity of the group operations in čech-complete semitopological groups to the wider class of strongly baire spaces. the following theorem is the main result established in [15]. theorem 4.2 (kenderov-kortezov-moors). let g be a regular semitopological group. if g is strongly baire, then g is a topological group. since there exist regular baire paratopological groups that are not topological groups (the sorgenfrey line is an example), theorem 4.2 distinguishes the classes of baire and strongly baire paratopological groups. further, as in the case of baire spaces, it is easy to see that the class of strongly baire spaces is closed under taking open subsets, dense gδ-subsets, and images of continuous open mappings. moreover, a space having a dense strongly baire subspace is itself strongly baire. we will show that some generalized metric baire spaces are, in fact, strongly baire. let us recall that a hausdorff space x is called a lindelöf σ-space if it has a cover c by compact sets and a countable family f of closed subsets such that for each c ∈ c and each open set u containing c, there exists f ∈ f such that c ⊂ f ⊂ u. notice that for a fixed x ∈ x, if {fn}n is an enumeration of f(x) = {f ∈ f : x ∈ f}, then every sequence {xn}n such that xn ∈ fn for each n ∈ n, has an accumulation point in x. lemma 4.3. every regular, baire, lindelöf σ-space x is strongly baire. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 10 baire property in product spaces proof. let c be a cover of x by compact sets and f = {fn : n ∈ n} a countable family of closed subsets of x witnessing that x is a lindelöf σspace. let also t = {tn}n be a strategy for player β in the game gs(x). we will prove that there exists a t-sequence {an}n such that player α wins the t-play {an}n. for this purpose, we define inductively a new strategy t ′ = {t′n} for player β as follows. in the first play, we use t′1(∅) = t1(∅). suppose that we have already defined functions t′1, . . . , t ′ n−1 in such a way that each partial t′-sequence (a1, . . . , aj−1) is also a partial t-sequence (j ≤ n − 1). then we define a function t′n by the rule t′n(a1, . . . , an−1) = { tn(a1, . . . , an−1) ∩ int fn, if this set is non-empty; tn(a1, . . . , an−1) \ fn, otherwise. since x is baire, there exists a t′-sequence {an}n such that the intersection ⋂ n∈n an is not empty. then {an}n is also a t-sequence and we claim that this t-sequence witnesses that player α wins in gs(x). indeed, let {an}n be a sequence such that an ∈ an, for each n ∈ n. pick a point x ∈ ⋂ n∈n an and a sequence {nk}k in n such that f(x) = {fnk }k. then, by the definition, ank ∈ ank ⊂ b ′ nk ⊂ intfnk , for each k ∈ n. so the subsequence {ank}k of {an}n has an accumulation point in x. � combining theorem 4.2 and lemma 4.3, we obtain the following fact: corollary 4.4. let g be a regular baire semitopological group. if g is a lindelöf σ-space, then g is a topological group. according to hodel [14], a space x is called a w∆-space if x has a sequence {gn}n of open covers such that for every x ∈ x, if {xn}n is a sequence with xn ∈ st(x, gn) for each n ∈ n, then {xn}n has an accumulation point in x. lemma 4.5. every regular baire w∆-space x is strongly baire. proof. let {gn}n be a sequence of open covers of x witnessing that x is a w∆space. suppose that t = {tn}n is a strategy for player β in the game gs(x). again, we define a new strategy t′ = {t′n}n, just as we did it in the proof of lemma 4.3, using the slightly different rule t′n(a1, . . . , an−1) = tn(a1, . . . , an−1) ∩ gn, where gn ∈ gn is any element intersecting tn(a1, . . . , an−1). according to this definition, the baire property of x implies the existence of a t′-sequence {an}n such that ⋂ n∈n an 6= ∅. now we choose a sequence {an}n, where an ∈ an for each n ∈ n. pick a point x ∈ ⋂ an. note that {x, an} ⊂ an ⊂ b ′ n ⊂ gn ⊂ st(x, gn), for each n. hence {an}n has an accumulation point in x and {an}n is a t-sequence for player α in gs(x). so α wins the t-play {an}n in the game gs(x). � corollary 4.6. let g be a regular baire semitopological group. if g is w∆space, then g is a topological group. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 11 c. hernández, l. rodŕıguez medina and m. tkachenko since every moore space is a w∆-space [14], the fact below is immediate from corollary 4.6. corollary 4.7 (piotrowski, [19]). let g be a baire semitopological group. if g is a moore space, then g is a topological group. as metrizable spaces are moore spaces, every metrizable baire space is automatically a strongly baire one. it was proved by van douwen in [28] that every baire σ-space contains a dense metrizable gδ-subset. this implies that all baire σ-spaces are also strongly baire. 5. open problems in view of the importance of strongly baire spaces we formulate several problems about them, whose solutions can help for better understanding the properties of this class. problem 5.1. let x and y be strongly baire (metric) spaces or topological groups. (a) when is x × y strongly baire? (b) is the product x × y strongly baire provided it is a baire space? (c) what if y is additionally hereditarily baire? in all known examples showing that the baire property is not productive, the corresponding spaces are far from separable. this fact motivates the problem below. problem 5.2. do there exist separable (regular, tychonoff) baire spaces x and y such that the product x × y fails to be baire? references [1] j. m. aarts and d. j. lutzer, pseudo-completeness and the product of baire spaces, pacific j. math. 48 (1973), 1–10. [2] o. t. alas and m. sanchis, countably compact paratopological groups, semigroup forum 74, no. 3 (2007), 423–438. [3] a. v. arhangel’skii and e. a. reznichenko, paratopological and semitopological groups versus topological groups, topology appl. 151 (2005), 107–119. 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[9] p. e. cohen, products of baire spaces, proc. amer. math. soc. 55 (1976), 119–124. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 12 baire property in product spaces [10] w. w. comfort and k. a. ross, pseudocompactness and uniform continuity in topological groups, pacific j. math. 16 (1996), 483–496. [11] w. fleissner and k. kunen, barely baire spaces, fund. math. 101, no. 3 (1978), 229– 240. [12] z. froĺık, baire spaces and some generalizations of complete metric spaces, czechoslovak math. j. 11, no. 86 (1961), 237–248. [13] z. froĺık, concerning the invariance of baire spaces under mappings, czechoslovak math. j. 11, no. 3 (1961), 381–385. [14] r. e. hodel, moore spaces and w∆-spaces, pacific j. math. 38 (1971), 641–652. [15] p. s. kenderov, i. s. kortezov and w. b. moors, topological games and topological groups, topology appl. 109, no. 2 (2001), 157–165. 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[29] h. e. white, jr., topological spaces that are α-favorable for a player with perfect information, proc. amer. math. soc. 50 (1975), 477–482. c© agt, upv, 2015 appl. gen. topol. 16, no. 1 13 @ appl. gen. topol. 15, no. 2(2014), 121-136doi:10.4995/agt.2014.3156 c© agt, upv, 2014 convergence s-compactifications bernd losert and gary richardson department of mathematics, university of central florida, orlando, fl 32816, usa (berndlosert@knights.ucf.edu, gary.richardson@ucf.edu) abstract properties of continuous actions on convergence spaces are investigated. the primary focus is the characterization as to when a continuous action on a convergence space can be continuously extended to an action on a compactification of the convergence space. the largest and smallest such compactifications are studied. 2010 msc: primary 54a20; secondary 54d35. keywords: convergence space; convergence semigroup; continuous action; s-compactification. 1. introduction and preliminaries the study of the notion of a topological transformation group or g-space dates back to the work of gottshalk and hedlund [4] who generalized several classical dynamical results from the theory of differential equations and other fields of mathematics. the problem of characterizing when a g-space has a compactification where the action can be continuously extended to the compactification, called a g-compactification, seems to have originated with de vries [3]. the term “s-compactification” is used in the semigroup context. our work is devoted to the study of s-compactifications where the underlying spaces are convergence spaces rather than topological spaces. it is known that the category of convergence spaces has nicer categorical properties. for example, quotient maps are productive in the category of convergence spaces but not in the category of topological spaces. some results on s-spaces in the context of convergence spaces can be found in [1, 2]. a good reference on convergence spaces and categorical terminology is the book by preuss [11]. received 10 february 2013 – accepted 30 january 2014 http://dx.doi.org/10.4995/agt.2014.3156 b. losert and g. richardson let x be a set, p(x) the power set of x, and f(x) the set of all filters on x. for x ∈ x, ẋ denotes the fixed ultrafilter on x generated by {{x}}. define the following partial order on f(x): given f, g ∈ f(x), f ≥ g (read “f is finer than g”) if and only if g ⊆ f. given f, g ∈ f(x), the least upper bound f ∨ g of f and g exists provided that f ∩ g 6= ∅ for each f ∈ f and g ∈ g and it is the smallest filter containing both f and g. definition 1.1. a pair (x, q) is called a convergence space whenever x is a set and q : f(x) → p(x) obeys: (cs1) x ∈ q(ẋ) (cs2) g ≥ f implies q(f) ⊆ q(g) (cs3) x ∈ q(f) implies x ∈ q(f ∩ ẋ) a function q obeying (cs1) through (cs3) is called a convergence structure on x. the notation x ∈ q(f) is read as “f q-converges to x” or as “f converges to x” and is usually written as f q → x or f → x. most of the time we will not need to make explicit reference to the convergence structure so we will normally write (x, q) as x and f q → x as f → x. a function f : x → y between two convergence spaces is continuous provided that f→f → x whenever f → x. here, f→f denotes the filter on y generated by {f(f): f ∈ f}. given g ∈ f(y ), we use f←g to denote the filter on x generated by {f−1(g): g ∈ g} whenever the latter does not contain ∅. given two convergence structures p and q on a set x, we say that q is finer than p, denoted q ≥ p, whenever the identity mapping idx : (x, q) → (x, p) is continuous. let conv denote the category of convergence spaces and continuous maps and let x be an object in conv. the closure and interior of a subset a of x are defined as follows: cl a = {x ∈ x : f → x for some f and a ∈ f} int a = {x ∈ a: f → x implies that a ∈ f} the operators cl and int are in general not idempotent. the neighborhood filter of x is defined by u(x) = {v ⊆ x : x ∈ int(v )} and a ⊆ x is open whenever int(a) = a. the convergence structure q on x is called a pretopology on x if u(x) → x for each x ∈ x. a pretopology on x is said to be a topology whenever u(x) has a base of open sets for each x ∈ x. we say x is hausdorff provided that each filter converges to at most one point, and regular if cl f → x whenever f → x, where cl f denotes the filter on x generated by {cl f : f ∈ f}. a hausdorff regular convergence space is called t3. a point x ∈ x is an adherent point of f ∈ f(x) whenever there exists a g ≥ f such that g → x; adh f denotes the set of all adherent points of f. we say x is compact provided that adh f 6= ∅ for each f ∈ f(x) or equivalently if each ultrafilter on x converges. a convergence space y is said to be a compactification of x in conv if y is compact hausdorff and c© agt, upv, 2014 appl. gen. topol. 15, no. 2 122 convergence s-compactifications if there is a dense embedding of x into y . observe that compactifications are required to be hausdorff. a compactification y is called regular whenever y is regular. if y and z are two compactifications of x in conv, define y ≥ z to mean that there exists a continuous map h: y → z such that h ◦ f = g, where f is the dense embedding of x into y and g is the dense embedding of x into z. note that ≥ is a partial order on the set of compactifications of x if we agree not to distinguish between isomorphic objects in conv. definition 1.2. the triple (s, ·, p) is said to be a convergence monoid if it satisfies: (cm1) (s, ·) is a commutative monoid with identity e. (cm2) (s, p) is a convergence space. (cm3) the binary operation (x, y) 7→ x · y of s is continuous. let cm denote the category of convergence monoids and continuous homomorphisms. to simplify notation, we will write s for the object (s, ·, p) in cm. let x be a convergence space, let s be a convergence monoid and let λ: x × s → x. consider the following conditions on λ: (a1) λ(x, e) = x for all x ∈ x. (a2) λ(λ(x, s), t) = λ(x, s · t) for all x ∈ x and all s, t ∈ s. (a3) λ is continuous. if λ satisfies (a1) and (a2), then λ is an action of s on x, and if in addition it satisfies (a3), we say that λ is a continuous action of s on x. definition 1.3. let ca be the category whose objects consist of all triples (x, s, λ), where x is a convergence space, s is a convergence monoid and λ is a continuous action of s on x, and whose morphisms are pairs (f, k) of functions of the form (x, s, λ) → (y, t, µ) such that: (c1) f : x → y is a morphism in conv, (c2) k : s → t is a morphism in cm and (c3) µ ◦ (f × k) = f ◦ λ. definition 1.4. a compactification (regular compactification) of an object (x, s, λ) in ca is an object (y, s, µ) in ca such that: (com1) y is a compact hausdorff (compact t3) convergence space, (com2) x is densely embedded in y , where the dense embedding f is such that (com3) (f, ids) is a morphism in ca. remark 1.5. throughout the remainder of this work, s will always denote a convergence monoid, x a convergence space and λ a continuous action of s on x. we will always write p for the convergence structure on s and q for the convergence structure on x. an object in ca of the form (y, s, µ) is called an s-space and for notational convenience will be denoted as (y, µ) or y . a morphism (f, ids) between two s-spaces in ca will be written more simply as f. also, any compactification of x in ca will be called an s-compactification of x. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 123 b. losert and g. richardson assume that y and z are two s-compactifications of x in ca with dense embeddings f : x → y and g : x → z. define y ≥ z to mean that there exists a morphism h: y → z in ca such that h ◦ f = g. we say y and z are equivalent s-compactifications of x if y ≥ z and z ≥ y . verification of the following lemma is straightforward and omitted here. lemma 1.6. (i) the relation ≥ between s-compactifications of x defined above is a partial order on the set of all s-compactifications of x if we agree not to distinguish between equivalent s-compactifications. (ii) suppose that y and z are s-compactifications of x satisfying y ≥ z. then the following diagram commutes: y z x h f g and (a) h(y − f(x)) = z − g(x) and (b) f(x) is open in y if and only if g(x) is open in z. definition 1.7. (i) we call x adherence-restrictive if for each f ∈ f(x) and each convergent filter g ∈ f(s), adh f = ∅ implies adh λ→(f × g) = ∅. (ii) let y be an s-compactification of x, let f : x → y be the dense embedding and let µ be the action of s on y . we say that y is remainder-invariant provided that µ((y − f(x)) × s) ⊆ y − f(x). 2. s-compactifications recall from section 1 that a compactification must be hausdorff. unlike the topological context, a one-point compactification in conv is not necessarily unique up to homeomorphism. the following one-point compactification defined below is used. pick an ω 6∈ x, let x∗ = x ∪ {ω} and let j : x → x∗ be the natural injection. define convergence in x∗ as follows: h ∈ f(x∗) converges to j(x) ⇔ h ≥ j → f for some f ∈ f(x) that converges to x h ∈ f(x∗) converges to ω ⇔ h ≥ j → f ∩ ω̇ for some f ∈ f(x) with adh f = ∅ one can check that x∗ with the above convergence is a compactification of x in conv provided that x is non-compact and hausdorff. moreover, y ≥ x∗ in conv for any other one-point compactification y of x. let η denote the set of all ultrafilters on x which fail to converge. theorem 2.1. suppose x is not compact and let y be an s-compactification of x. then x is adherence-restrictive if and only if y is remainder-invariant. proof. a contrapositive argument is used in each direction. let f : x → y be the dense embedding, let µ be the action of s on y and suppose that y fails to be remainder-invariant. then µ(y, s) = f(x) for some y ∈ y −f(x), s ∈ s and c© agt, upv, 2014 appl. gen. topol. 15, no. 2 124 convergence s-compactifications x ∈ x. let f ∈ η such that f→f → y. then adh f = ∅ and f→(λ→(f × ṡ)) = (f ◦ λ) → (f × ṡ) = (µ ◦ (f × ids)) → (f × ṡ) = µ→(f→f × ṡ) → µ(y, s) = f(x). since f is an embedding, λ→(f×ṡ) → x and thus adh(λ→(f×ṡ)) 6= ∅, proving that x is not adherence-restrictive. conversely, suppose that x is not adherence-restrictive. then there exists f ∈ f(x) with adh f = ∅ and x ∈ adh(λ→(f×g)) for some g p → s and x ∈ x. let h be an ultrafilter on x such that h → x and h ≥ λ→(f × g). let k be an ultrafilter such that k ≥ f×g and h = λ→k. note that the projection kx of k onto x is finer than f. since y is compact and adh f = ∅, f→kx → y for some y ∈ y − f(x), hence f→h = f→(λ→k) ≥ (f ◦ λ) → (kx × g) = (µ ◦ (f × ids)) → (kx × g) = µ →(f→kx × g) → µ(y, s). however, h → x implies that f→h → f(x), and thus µ(y, s) = f(x). hence y fails to be remainder-invariant. � theorem 2.2. if x is non-compact and hausdorff, then it has a one-point remainder-invariant s-compactification if and only if x is adherence-restrictive. proof. assume that x is adherence-restrictive and let x∗ be the compactification of x in conv discussed above. define λ∗ : x∗ × s → x∗ by λ∗(y, s) = { j(λ(y, s)), if y ∈ j(x) ω, if y = ω. some simple algebra reveals that λ∗ is an action of s on x∗. it is shown that λ∗ is continuous. suppose that h ∈ f(x∗) converges to j(x) and g ∈ f(x) converges to s. then there exists f ∈ f(x) that converges to x such that h ≥ j→f. it follows from the definition of λ∗ and the continuity of λ that λ∗ → (j→f × g) = (j ◦ λ) → (f × g) → j(λ(x, s)). next, suppose that h ∈ f(x∗) converges to ω and g ∈ f(x) converges to s. then there exists f ∈ f(x) such that adh f = ∅ and h ≥ j→f ∩ ω̇. since x is adherence-restrictive, adh(λ→(f × g)) = ∅. it follows that λ→ ∗ (h × g) ≥ λ→ ∗ ((j→f ∩ ω̇) × g) = λ→ ∗ (j→f × g) ∩ λ→ ∗ (ω̇ × g) = j→(λ→(f × g)) ∩ ω̇ → ω. this proves that λ∗ is a continuous action of s on x∗. moreover, λ∗ ◦ (j × ids) = j ◦ λ and thus x∗ (with λ∗ as the action) is an s-compactification of x. the compactification is also remainder-invariant by construction. the converse follows from theorem 2.1. � theorem 2.3. assume that x is non-compact and hausdorff. then it has a smallest remainder-invariant s-compactification if and only if x is adherence-restrictive and the image of x is open in each of its remainder-invariant s-compactifications. moreover, if there is a smallest remainder-invariant scompactification, it is equivalent to the one-point s-compactification x∗. proof. suppose that x has a smallest remainder-invariant s-compactifiction. according to theorem 2.1, x is adherence-restrictive, and thus by theorem 2.2, x∗ is a remainder-invariant s-compactification of x. applying lemma c© agt, upv, 2014 appl. gen. topol. 15, no. 2 125 b. losert and g. richardson 1.6 (ii), it follows that x∗ is equivalent to the smallest remainder-invariant scompactification. moreover, part (b) of lemma 1.6 (ii) implies that the image of x is open in each of its remainder-invariant s-compactifications. conversely, suppose that x is adherence-restrictive and that the image of x is open in each of its remainder-invariant s-compactifications. according to theorem 2.2, x∗ is a one-point s-compactification of x and j(x) is open in x∗. let y be any remainder-invariant s-compactication of x having dense embedding f and action µ. define h: y → x∗ by h(f(x)) = j(x) and h(y) = ω for each x ∈ x and y ∈ y − f(x). we show that h is continuous: if a filter h on y converges to f(x), then f(x) ∈ h since f(x) is open in y , and thus f := f←h → x and h→h = (h ◦ f) → f = j→f → j(x) = h(f(x)). next, assume that h → y ∈ y −f(x). if y −f(x) ∈ h, then h→h = ω̇ → ω = h(y). otherwise, f = f←h exists, adh f = ∅ and h→h ≥ j→f ∩ ω̇ → ω. next, we show λ∗ ◦ (h × ids) = h ◦ µ: if x ∈ x and s ∈ s, then (λ∗ ◦ (h × ids))(f(x), s) = λ∗(h(f(x)), s) = j(λ(x, s)) = (h ◦ f)(λ(x, s)) = (h ◦ (f ◦ λ))(x, s) = h(µ ◦ (f × ids))(x, s) = (h ◦ µ)(f(x), s), and thus (λ∗ ◦ (h × ids))(f(x), s) = (h ◦ µ)(f(x), s). finally, if y ∈ y − f(x), then (λ∗(h ◦ ids))(y, s) = λ∗(h(y), s) = λ∗(ω, s) = ω = (h ◦ µ)(y, s). the last equality is valid since y is remainder-invariant. therefore, λ∗ ◦ (h × ids) = h ◦ µ, and thus h is a morphism in ca. this concludes the proof that x∗ is the smallest remainder-invariant s-compactification of x. � we say that s is a convergence group whenever s is a group and the function that maps elements of s to their inverses is continuous. in what follows, we will write s−1 for the inverse of s ∈ s. also, given a filter g on s, we will write g−1 for the filter on s generated by {g−1 : g ∈ g}, where g−1 denotes the set of inverses of the elements of g. the next result should be contrasted with theorem 2.1. theorem 2.4. if s is a convergence group, then x is adherence-restrictive. proof. suppose x fails to be adherence-restrictive. then there exists f ∈ f(x), g ∈ f(s), s ∈ s and x ∈ x such that adh f = ∅ and g → s and x ∈ adh(λ→(f × g)). let h be an ultrafilter such that h ≥ λ→(f × g) and h → x. choose an ultrafilter l ≥ f × g for which h = λ→l. since s is a convergence group, g−1 → s−1, and thus λ→(h × g−1) → λ(x, s−1). it is shown that λ→(λ→l × g−1) ∨ lx exists, lx being the projection of l on x. let l ∈ l, let lx denote the projection of l on x and let g ∈ g. choose any f ∈ f. since l ≥ f × g, l ∩ (f × g) 6= ∅. let (y, t) ∈ l ∩ (f × g). then (λ(y, t), t−1) ∈ λ(l) × g−1 and thus y ∈ λ(λ(l) × g−1) ∩ lx, which means λ→(λ→l × g−1) ∨ lx exists. since λ →(h × g−1) → λ(x, s−1), we have that f ≤ lx ≤ λ →(λ→l × g−1) ∨ lx = λ →(h→ × g−1) ∨ lx → λ(x, s −1), which contradicts adh f = ∅. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 126 convergence s-compactifications the next result appears as theorem 3.2 [9]. theorem 2.5 ([9]). if x is non-compact and hausdorff, then the following are equivalent: (i) x has a smallest compactification in conv. (ii) the image of x is open in each of its compactifications. (iii) the set η of non-convergent ultrafilters on x is finite. (iv) x has a largest compactification in conv. example 2.6. suppose s = {e} is the trivial group, x hausdorff and η is infinite. define λ: x × s → x by λ(x, s) = x. then λ is a continuous action and thus x is an s-space. let y be any compactification of x with dense embedding f. define µ: y × s → s by µ(y, s) = y. then µ is a continuous action of s on y and µ ◦ (f × ids) = f ◦ λ. it follows that y is an s-compactification of x. in this case, there is a bijection between the compactifications of x and the s-compactifications of x. since η is infinite, it follows from theorem 2.5 that there fails to exists either a smallest or a largest s-compactification of x. example 2.7. let x := [0, 1) and s := (0, 1] be equipped with the usual topologies and let the operation on s be multiplication. define λ: x × s → x by λ(x, s) = xs. then λ is a continuous action and thus x is an s-space. let x∗ = [0, 1] have the usual topology (note that ω = 1 in this case). define µ: x∗ × s → x∗ by µ(y, s) = ys. then µ is a continuous action of s on x∗ and the s-space (x∗, µ) is an s-compactification of x. since µ(ω, s) = s 6= ω whenever s 6= 1, (x∗, µ) is not remainder-invariant. since the action λ∗ of s on x∗ fails to be continuous at each (ω, s), s 6= 1, the s-space (x∗, λ∗) is not an s-compactification of x. ergo, x is not adherence-restrictive. 3. regular s-compactifications recall that u(x) denotes the neighborhood filter of x ∈ x, and that the convergence structure q on x is a pretopology provided that u(x) q → x for each x ∈ x. if q is not a pretopology, define f πq → x if and only if f ≥ u(x). then πq is a pretopology on x and πx := (x, πq) is called the pretopological modification of x. theorem 3.1 ([6]). the convergence space x has a regular compactification in conv if and only if (i) x and πx agree on convergence of ultrafilters, and (ii) πx is a completely regular topological space. we say x is completely regular provided that it is t3 and agrees on ultrafilter convergence with a completely regular topological space. with this definition, we can restate theorem 3.1 above as: x has a regular compactification in conv if and only if x is a completely regular convergence space. recall that an object y in ca is a regular s-compactification of x if y is a regular compactification of x and if µ◦(f ×ids) = f ◦λ, where f : x → y is c© agt, upv, 2014 appl. gen. topol. 15, no. 2 127 b. losert and g. richardson the dense embedding. a convergence space is locally compact provided that each convergent filter contains a compact subset. according to proposition 2.3 [8], a convergence space is locally compact if and only if each convergent ultrafilter contains a compact subset. theorem 3.2. if x is adherence-restrictive and completely regular, then the statements (i) – (iii) are equivalent and (iii) implies (iv), where (i) x has a one-point regular remainder-invariant s-compactification. (ii) x is locally compact. (iii) x has a regular remainder-invariant s-compactification and the image of x is open in each such compactification. (iv) x has a smallest regular remainder-invariant s-compactification. proof. (i) implies (ii): let y be a one-point regular s-compactification of x and let f : x → y be the dense embedding. according to theorem 3.1, f : πx → πy is a dense embedding in the category top of topological spaces and thus πy is a t3-compactification of πx in top, which means πx is locally compact. again, by theorem 3.1, x and πx agree on ultrafilter convergence so x is also locally compact. (ii) implies (i): let x∗ be the one-point s-compactification defined in section 2 with dense embedding j : x → x∗. it must be shown that x∗ is regular. assume that h is a filter on x∗ that converges to j(x). then h ≥ j →f for some f ∈ f(x) that converges to x. since x is completely regular and locally compact, cl f → x and f contains a compact subset a of x. observe that ω 6∈ cl j(a): otherwise, if there is an ultrafilter k on x∗ that contains j(a) and converges to ω, then j←k is a filter on x that contains a and has empty adherence, contradicting that a is compact. it follows that x∗ − j(a) ∈ k for each ultrafilter k on x∗ that converges to ω. hence cl j(a) = j(cl a) = j(a) and thus cl h ≥ cl j→f = j→(cl f) → j(x). moreover, if h → ω, then h ≥ j→f ∩ ω̇ for some f ∈ f(x) such that adh(f) = ∅. since x is completely regular, it has the same ultrafilter convergence as πx, hence adh(cl f) = ∅ and thus cl h ≥ cl(j→f ∩ ω̇) = j→(cl f) ∩ ω̇ → ω. this proves that x∗ is regular and thus x∗ is a one-point regular remainder-invariant s-compactification of x. (ii) implies (iii): since (ii) implies (i), x has a regular remainder-invariant scompactification. let y be any regular remainder-invariant s-compactification with dense embedding f : x → y . we now show that f(x) is open in y . since f : πx → πy is a dense embedding in top, πy is a hausdorff compactification of πx in top. since x and πx agree on ultrafilter convergence, πx and x are locally compact. thus, f(x) is open in πy and hence open in y . (iii) implies (ii): let y be a regular s-compactification of x with dense embedding f : x → y such that f(x) is open in y . since πy is a compactification of πx in top, πx is locally compact and thus x is locally compact. (iii) implies (iv): since (iii) implies (i), x∗ is a regular remainder-invariant s-compactification. suppose that y is any regular remainder-invariant s-compactification of x with dense embedding f : x → y . by hypothesis, f(x) is c© agt, upv, 2014 appl. gen. topol. 15, no. 2 128 convergence s-compactifications open in y , so the proof of theorem 2.3 shows that y ≥ x∗, hence x∗ is the smallest regular remainder-invariant s-compactification of x. � theorem 3.3. if x has a regular s-compactification, then it has a largest regular s-compactification, which is remainder invariant whenever x is adherencerestrictive. proof. let (yα) be the family of all regular s-compactifications of x. for each α, let fα : x → yα be the dense embedding and let µα be the action of s on yα. let y = ∏ yα be the product space in conv and define µ: y × s → y so that µ(y, s) = (µα(yα), s), where y = (yα). it is straightforward to check that µ is an action of s on y . let πα : y → yα denote the projection onto yα. since the diagram below is commutative, y × s y yα × s yα µ πα×ids πα µα it follows that µ is continuous. this proves that y is an s-space as well as a compact t3 object in conv. define f : x → y by f(x) = (fα(x)). since fα = πα ◦ f for each α, f is continuous. moreover, if f ∈ f(x) and f→f → f(x), then f→α f = π → α (f →f) → fα(x). since fα is an embedding, f → x and consequently f is an embedding. since µ ◦ (f × ids) = f ◦ λ, f is an embedding in ca. note that the following diagram is commutative in ca: y yα x πα f fα finally, let z = cl f(x) and let δ denote the restriction of µ on z × s. observe that z is an s-space since δ(z×s) = δ(cl f(x)×s) ⊆ cl δ(f(x)×s) = cl µ(f(x)×s) = cl(µ◦(f ×ids))(x ×s) = cl(f ◦λ)(x ×s) = cl f(λ(x ×s)) ⊆ cl f(x) = z. it follows that z is the largest regular s-compactification of x. in the event that z is adherence-restrictive, it follows from theorem 2.1 that z is remainder-invariant. � consideration is now given to the question as to when an s-space has a regular s-compactification. for this discussion, it is convenient to introduce the notion of a cauchy space. definition 3.4. the pair (x, c ) is called a cauchy space and c a cauchy structure whenever c ⊆ f(x) obeys: (ca1) ẋ ∈ c for each x ∈ x. (ca2) f ≥ g ∈ c implies f ∈ c . (ca3) f ∩ g ∈ c whenever f, g ∈ c and f ∨ g exists. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 129 b. losert and g. richardson the elements of a cauchy structure are called cauchy filters. a function f between cauchy spaces is called cauchy continuous if f→f is a cauchy filter whenever f is a cauchy filter. we use chy to denote the category of cauchy spaces and cauchy continuous functions. the first study of completions resembling a cauchy space defined above seems to be due to keller [5]. the interested reader is referred to lowen [10], preuss [11] and reed [12] for a thorough treatment of cauchy spaces. if for all f, g ∈ f(x) and all x ∈ x, f → x and g → x imply f ∩ g → x, then x is called a limit space. we use lim to denote the full subcategory of conv whose objects are limit spaces. if c ⊆ f(x) is a cauchy structure, then qc : f(x) → p(x) defined by f qc → x if and only if f ∩ ẋ ∈ c is a convergence structure making (x, qc ) a limit space. if f : (x, c ) → (y, d) is cauchy continuous, then f : (x, qc ) → (y, qd ) is continuous, but the converse is in general false. if x is a hausdorff limit space, then c = {f ∈ f(x): f converges} is a cauchy structure and qc = q. a compactification y of x in conv with dense embedding f is said to be strict provided that whenever h ∈ f(y ) converges to some y ∈ y , there exists an f ∈ f(x) such that f→f → y and h ≥ cl f→f. all our previous definitions (s-compactifications, etc.) apply to the objects in lim. theorem 3.5. suppose both x and s are hausdorff limit spaces. let y be any strict regular compactification of x in lim with dense embedding f, let c = {f ∈ f(x): f→f converges} and let s = {g ∈ f(s): g converges}. then there exists a continuous action µ of s on y making y into a regular s-compactification of x if and only if λ: (x, c ) × (s, s ) → (x, c ) is cauchy continuous. proof. suppose there exists a continuous action µ of s on y making y into a regular s-compactification of x. then c is a cauchy structure. we now show that λ: (x, c ) × (s, s ) → (x, c ) is cauchy continuous. let f ∈ c and g ∈ s . then f→f → y and g → s for some y ∈ y and s ∈ s. it follows that f→(λ→(f × g)) = (µ ◦ (f × ids)) → (f × g) = µ→(f→f × g) → µ(y, s), which means λ→(f × g) ∈ c , which means λ is cauchy continuous. conversely, suppose that λ is cauchy continuous. define µ: y × s → y by µ(y, s) = { f(λ(x, s)), if y = f(x) for some x ∈ x, lim (f ◦ λ) → (f × g), where f→f → y ∈ y − f(x) and g → s. note that the above is well-defined. indeed, if f1 and f2 are filters on x such that f→f1 and f →f2 converge to y and g1 and g2 are filters on s that converge to s, then f→(f1∩f2) → y and g1∩g2 → s since y and s are limit spaces, hence f1 ∩ f2 ∈ c . moreover, since λ is cauchy continuous, λ →(f1 × g1) ∩ λ →(f2 × g2) ≥ λ →((f1 ∩ f2) × (g1 ∩ g2)) ∈ c . thus, f →(λ→(f1 × g1) ∩ λ →(f2 × g2)) converges and lim (f ◦ λ) → (f1 × g1) = lim (f ◦ λ) → (f2 × g2) in y . we now show that µ is an action. first, we have that µ(f(x), e) = f(λ(x, e)) = f(x) and µ(y, e) = lim (f ◦ λ) → (f × ė) = lim f→(λ→(f × ė)) = y, where f→f → y ∈ y − f(x). second, if x ∈ x and s, t ∈ s, then µ(µ(f(x), s), t) = c© agt, upv, 2014 appl. gen. topol. 15, no. 2 130 convergence s-compactifications µ(f(λ(x, s)), t) = f(λ(λ(x, s), t)) = f(λ(x, s · t)) = µ(f(x), s · t). third, suppose y ∈ y − f(x) and f→f → y and let s, t ∈ s. if µ(y, s) = f(x) for some x ∈ x, then since f is an embedding and f(x) = µ(y, s) = lim (f ◦ λ) → (f × ṡ) = lim f→(λ→(f × ṡ)), we have that λ→(f × ṡ) → x and so µ(y, s · t) = lim (f ◦ λ) → (f× ˙s · t) = lim f→(λ→(λ→(f×ṡ)×ṫ)) = f(λ(x, t)) = µ(f(x), t) = µ(µ(y, s), t). otherwise, if µ(y, s) ∈ y − f(x), then since f→(λ→(f × ṡ)) → µ(y, s) we have that µ(µ(y, s), t) = lim (f ◦ λ) → (λ→(f×ṡ)×ṫ) = lim (f ◦ λ) → (f× ˙s · t). we now show that µ is continuous. first, we prove that if a ⊆ x and g ⊆ s, then µ(cl f(a) × g) ⊆ cl(f ◦ λ)(a × g): let (y, s) ∈ cl f(a) × g. if y = f(x), then x ∈ cl a, and since (f ◦ λ)(cl a × g) ⊆ cl(f ◦ λ)(a × g), µ(f(x), s) = f(λ(x, s)) ∈ cl(f ◦ λ)(a × g). otherwise, if y ∈ y − f(x) and f→f → y with a ∈ f, then a × g ∈ f × ṡ, (f ◦ λ) → (f × ṡ) → µ(y, s), hence µ(y, s) ∈ cl(f ◦ λ)(a × g), hence µ(cl f(a) × g) ⊆ cl(f ◦ λ(a × g)) as claimed. now suppose h is a filter on y and that g is a filter on s converging to s. if h → f(x), then since y is a strict regular compactification of x in lim, there exists an f ∈ f(x) such that f→f → y and h ≥ cl f→f, hence λ→(f×g) → λ(x, s) and µ→(h×g) ≥ µ→(cl f→f×g) ≥ cl (f ◦ λ) → (f×g) → f(λ(x, s)) = µ(f(x), s). if h → y ∈ y − f(x), then since y is strict, there is an f ∈ f(x) that converges to x such that h ≥ cl f→f, hence µ→(h × g) ≥ µ→(cl f→f × g) ≥ cl (f ◦ λ) → (f × g), hence by the regularity of y and the fact that (f ◦ λ) → (f × g) → µ(y, s) we have that µ→(h × g) → µ(y, s). this proves that µ is continuous, and since µ ◦ (f × ids) = f ◦ λ, y is a regular s-compactification of x. � recall from theorem 3.1 that a t3 convergence space has a regular compactification in conv if and only if it is compeletely regular. this result is also valid in the subcategory lim. the following additional result is proved in theorem 2 [6]. theorem 3.6. if x is a completely regular convergence (limit) space, then it has a largest regular compactification in conv (respectively, lim). the largest regular compactification is strict. the largest regular compactification guaranteed by theorem 3.6 is called the stone-čech compactification for convergence (limit) spaces. combining the two previous theorems yields the next result. theorem 3.7. suppose x is a completely regular limit space with stone-čech compactification βx in lim and dense embedding f : x → βx and suppose s is a hausdorff limit space. let s = {g ∈ f(s): g converges} and c = {f ∈ f(x): f→f converges}. then λ can be extended to a continuous action of s on βx making it into a regular s-compactification of x if and only if λ is cauchy continuous. moreover, βx is the largest regular s-compactification of x whenever λ is cauchy continuous. proof. let µ be the action of s on βx. according to theorem 3.6, βx is strict, so by theorem 3.5, βx is a regular s-compactification of x if and only c© agt, upv, 2014 appl. gen. topol. 15, no. 2 131 b. losert and g. richardson if λ is cauchy continuous. suppose λ is cauchy continuous. then f is a dense embedding in ca and µ ◦ (f × ids) = f ◦ λ. it remains to show that βx is the largest regular s-compactification of x. let y be any regular scompactification of x with dense embedding g and action δ. then δ ◦ (g × ids) = g ◦ λ. since βx is the stone-čech compactification of x in lim, there exists a continuous function h: βx → y such that h ◦ f = g. we show that δ◦(h×ids) = h◦µ on f(x): if x ∈ x and s ∈ s, then (δ◦(h×ids))(f(x), s) = δ((h ◦ f)(x), s) = δ(g(x), s) = (δ ◦ (g × ids))(x, s) = (g ◦ λ)(x, s) = (h ◦ f ◦ λ)(x, s) = (h◦ (f ◦ λ))(x, s) = (h◦ (µ◦ (f × ids)))(x, s) = (h◦ µ)(f(x), s). since f(x) is dense in βx, δ ◦ (h × ids) = h ◦ µ on βx and thus h is a morphism in ca. this means βx ≥ y , hence βx is the largest regular s-compactification of x. � 4. generalized quotient spaces let gq denote the full subcategory of ca whose objects (y, t, µ) satisfy: (i) t is a commutative. (ii) µ(·, t) is injective for each fixed t ∈ t . suppose x is an object in gq. define a relation ∽ on x × s so that (x, s) ∽ (y, t) if and only if λ(x, t) = (y, s). one can readily check that ∽ is an equivalence relation. we use 〈x, s〉 for the equivalence class containing (x, s) and θ : x × s → (x × s)/ ∽ for the quotient map. we call (x × s)/ ∽ the generalized quotient of x and we denote it by b(x). define λ: (x × s) × s → (x × s) by λ((x, s), t) = (λ(x, t), s) and define λb : b(x) × s → b(x) by λb(〈x, s〉 , t) = 〈λ(x, t), s〉. it is shown in theorem 2.4 (b) of [2] that λ and λb are continuous actions, which means that x × s and b(x) are s-spaces. in fact, the following diagram is commutative: (x × s) × s x × s b(x) × s b(x) λ (θ,ids) θ λb theorem 4.1. suppose x is an object in gq. if s is compact and x is not compact and hausdorff, then b(x∗) is a one-point s-compactification of b(x), where x∗ is the one-point s-compactification of x given in thereom 2.2. proof. first we show that λ∗ b : b(x)×s → b(x) defined by λ∗ b (〈j(x), s〉 , t) = 〈(j ◦ λ)(x, t), s〉 and λ∗ b (〈ω, s〉 , t) = 〈λ∗(ω, t), s〉 = 〈ω, s〉 is an action. note that λ∗ b (〈j(x), s〉 , e) = 〈j ◦ λ(x, e), s〉 = 〈j(x), s〉 and λ∗ b (〈ω, s〉 , e) = 〈ω, s〉. also, λ∗ b (λ∗ b (〈j(x), t〉 , s), u) = λ∗ b (〈(j ◦ λ)(x, t), s〉 , u) = 〈(j ◦ λ)(λ(x, t), u), s〉 = 〈(j ◦ λ)(x, t · u), s〉 = λ∗ b (〈j(x), s〉 , tu) c© agt, upv, 2014 appl. gen. topol. 15, no. 2 132 convergence s-compactifications and λ∗ b (λ∗ b (〈ω, s〉 , t), u) = λ∗ b (〈λ∗(ω, t), s〉 , u) = λ∗ b (〈ω, s〉 , u) = 〈λ∗(ω, u), s〉 = 〈ω, s〉 = λ∗ b (〈ω, s〉 , t · u) this proves that λb is an action. define λ∗ : (x∗×s)×s → x∗×s so that λ ∗((j(x), s), t) = (λ∗(j(x), t), s) = ((j ◦ λ)(x, t), s) and λ∗((ω, s), t) = (λ∗(ω, t), s) = (ω, s). note that λ ∗ is an action of s on x∗ and it is continuous since it is the composition: ((z, s), t) 7→ ((z, t), s) 7→ (λ∗(z, t), s) it follows that x∗ × s is an object in gq. consider the following commutative diagram: (x∗ × s) × s x∗ × s b(x∗) × s b(x∗) λ∗ (θ∗,ids) θ λ ∗ b where θ∗ : x∗ × s → b(x∗) is the quotient map in conv. since (θ ∗, ids) is a quotient map in conv, the diagram above shows that λ∗ b is continuous, which means that b(x∗) is an s-space. since 〈x, s〉 ∈ b(x) if and only if 〈j(x), s〉 ∈ b(x∗) and since 〈ω, s〉 = {(ω, t): t ∈ s}, it follows that γ : b(x) → b(x∗) defined by γ(〈x, s〉) = 〈j(x), s〉 is an injection and b(x∗) − γ(b(x)) is a singleton set containing 〈ω, s〉. we now proceed to show that γ is a dense embedding. first, observe that the diagram below is commutative: x × s b(x) x∗ × s b(x∗) θ (j,ids) γ θ ∗ moreover, since θ is a quotient map and γ ◦ θ = θ∗ ◦ (j, ids) is continuous in conv, it follows that γ is continuous. next, assume that h ∈ f(b(x)) such that γ→h → 〈j(x), s〉. then there exists (j(x′), s′) ∽ (j(x), s) and k → (j(x′), s′) such that θ∗→k = γ→h. the product convergence structure on x∗×s implies that for some f ∈ f(x) and some g ∈ f(s), f → x ′, g → s′ and k ≥ j→f ×g, hence (γ ◦ θ) → (f ×g) = (θ∗ ◦ (j × ids)) → (f ×g) = θ∗ → (j→f × g) ≤ θ∗→k = γ→h, hence l := θ→(f × g) → 〈x′, s′〉 and γ→l ≤ γ→h, hence l ≤ h since γ is an injection. it follows that h → 〈x′, s′〉 = 〈x, s〉, proving that γ is an embedding. for any fixed s ∈ s, b(x∗) − γ(b(x)) = {〈ω, s〉}. since x is not compact, there exists an ultrafilter f ∈ f(x) that fails to converge, hence j→f → ω, j→f×ṡ → (ω, s) and θ∗→(j→f×ṡ) → 〈ω, s〉. it follows that γ→(θ→(f×g)) = c© agt, upv, 2014 appl. gen. topol. 15, no. 2 133 b. losert and g. richardson (θ∗ ◦ (j, ids)) → (f × g) = θ∗→(j→f × g) → 〈ω, s〉, and since θ→(f × g) ∈ f(b(x)), we conclude that γ is a dense embedding. moreover, since x∗ and s are compact, b(x∗) is a one-point s-compactification of b(x) in conv. we now verify that γ is a morphism in ca: since (λ∗ b ◦(γ ×ids))(〈x, s〉 , t) = λ∗ b (〈j(x), s〉 , t) = 〈(j ◦ λ)(x, t), s〉 = γ(〈λ(x, t), s〉) = (γ ◦ λb)(〈x, s〉 , t), we have that λ∗ b ◦ (γ × ids) = γ ◦ λb. in conclusion, b(x∗) is a one-point scompactification of b(x). � a regular s-compactification of a generalized quotient s-space is given below. the following definition is needed: a continuous surjection f : x → y between convergence spaces is said to be proper if for each ultrafilter f on x, f→f → y implies f → x for some x ∈ f−1({y}). it is shown in proposition 3.2 in [7] that proper maps preserve closures. observe that if f : x → y is a continuous surjection and x is compact and y is hausdorff, then f is a proper map. theorem 4.2. suppose x is an adherence-restrictive object in gq and let y be a strict regular s-compactification of x with dense embedding f and action µ. assume that s is compact and regular and define h: b(x) → b(y ) by h(〈x, s〉) = 〈f(x), s〉. then b(y ) is a regular s-compactification of b(x) with dense embedding h. proof. consider the following commutative diagram: x × s b(x) y × s b(y ) θ (f,ids) h θy note that h is continuous since θ is a quotient map in conv and h◦θ = θy ◦(f× ids) is continuous. we now prove that h is an injection: suppose that 〈x, s〉 6= 〈z, t〉. then λ(x, t) 6= λ(z, s), h(〈x, s〉) = 〈f(x), s〉 and h(〈z, t〉) = 〈f(z), t〉. however, f is injective, so µ(f(x), t) = f(λ(x, t)) 6= f(λ(z, s)) = µ(f(z), s). it follows that h(〈x, s〉) = 〈f(x), s〉 6= 〈f(z), t〉 = h(〈z, t〉), proving that h is an injection as claimed. next, suppose that h is a filter on b(x) such that h→h → h(〈x, s〉) = 〈f(x), s〉. we show that h → 〈x, s〉: the quotient convergence structure on b(y ) implies that there is a filter k on y that converges to y and a filter g on s that converges to t such that h→h ≥ θ→y (k × g) and 〈y, t〉 = 〈f(x), s〉. since y is a strict regular compactification of x in conv, there exists an f ∈ f(x) such that f→f → y and k ≥ cl f→f. it follows that h→h ≥ θ→y (k× g) ≥ θ→y (cl f →f × g), hence h ≥ h←(θ→y (cl f →f × g)). we now show that h←(θ→y (cl f →f × g)) ≥ θ→(cl f × g) by verifying that h−1(θy (cl f(f) × g)) ⊆ θ(cl f × g) for arbitrary f ∈ f and g ∈ g. let 〈v, a〉 ∈ h−1(θy (cl f(f) × g)). then h(〈v, a〉) = 〈f(v), a〉 ∈ θy (cl f(f) × g) implies that (w, b) ∈ cl f(f) × g for some 〈w, b〉 = 〈f(v), a〉, hence µ(w, a) = µ(f(v), b) = f(λ(v, b)) ∈ f(x). however, by theorem 2.1, y is remainder-invariant and thus w = f(z) ∈ c© agt, upv, 2014 appl. gen. topol. 15, no. 2 134 convergence s-compactifications cl f(f) for some z ∈ x. since f is an embedding, it follows that z ∈ cl f . moreover, µ(w, a) = µ(f(z), a) = f(λ(z, a)) = f(λ(v, b)) and so λ(z, a) = λ(v, b) because f is an injection. hence 〈v, a〉 = 〈z, b〉 ∈ θ(z, b) ∈ θ(cl f × g) shows that h−1(θy (cl f(f)×g)) ⊆ θ(cl f ×g), and thus h ←(θ→y (cl f →f×g)) ≥ θ→(cl f × g) as claimed. recall k is a filter on y that converges to y and that g is a filter on s that converges to t where 〈y, t〉 = 〈f(x), s〉. again, since y is remainder-invariant, µ(y, s) = µ(f(x), t) = f(λ(x, t)) ∈ f(x) and so y = f(x′) for some x′ ∈ x. thus, µ(y, s) = µ(f(x′), s) = f(λ(x′, s)) = f(λ(x, t)) and so λ(x, t) = λ(x′, s), which means 〈x, s〉 = 〈x′, t〉. since f→f → y = f(x′), f → x′ and x is regular, we have that cl f → x′, hence h ≥ θ→(cl f × g) → θ(x′, t) = 〈x′, t〉 = 〈x, s〉, hence h is an embedding. since s is compact, it follows that y × s is compact, proving that b(y ) is compact. let 〈y, s〉 ∈ b(y ). then there exists f ∈ f(x) such that f→f → y, hence (h ◦ θ) → (f × ṡ) = (θy ◦ (f × ids)) → (f × ṡ) = θ→y (f →f × ṡ) → θy (y, s) = 〈(y, s)〉, which proves that h is a dense embedding. since y is hausdorff, it follows from theorem 4.1 in [1] that b(y ) is also hausdorff. in conclusion, b(y ) is a compactification of b(x) in conv, and by employing theorem 2.4 (b) in [2], b(y ) is an s-compactification of b(x). we now show that b(y ) is regular: suppose that h is a filter on b(y ) that converges to 〈y, s〉. then there exists a filter k on y that converges to some y′ and a filter g on s that converges to s′ such that h ≥ θ→y (k × g) and 〈y, s〉 = 〈y′, s′〉. since y and s are compact and b(y ) is hausdorff, it follows from proposition 3.2 in [7] that θy is a proper map and thus closure preserving. this means that cl h ≥ cl θ→y (k × g) = θ → y (cl k × cl g) → θy (y ′, s′) = 〈y, s〉 since y and s are regular. therefore, cl h → 〈y, s〉, proving that b(y ) is a regular. � employing theorems 3.2, 4.1, and 4.2 gives the following result. corollary 4.3. suppose x is an object in gq that is adherence-restrictive, locally compact and completely regular and suppose s is compact and completely regular. then b(x∗) is a one-point regular s-compactification of b(x), where x∗ is the one-point regular s-compactification of x. let us now consider the topological case. let τq be the finest topology on x coarser than the convergence structure q of x and let τx := (x, τq) denote the topological modification of x. recall that θ : x × s → b(x) is a quotient map in conv. according to theorem 4.2 in [8], if either x or s is a locally compact hausdorff topological space, then τ(x × s) = τx × τs and thus θ is a quotient map in top. in this case, τb(x) is the generalized quotient of τx (acted upon by τs) in top. the final result pertains to generalized quotient spaces in top and the proof follows from the preceeding remarks along with theorem 4.2. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 135 b. losert and g. richardson corollary 4.4. suppose x is an adherence-restrictive object in gq and that s is a compact hausdorff topological space. let y be a topological s-compactification of x. then τb(x) and τb(y ) are generalized quotients in top and τb(y ) is a topological s-compactification of τb(x). 5. conclusion in general, an s-space has neither a smallest nor a largest s-compactification. conditions are given for the existence of a smallest (regular) s-compactification in theorem 2.3 (respectively, theorem 3.2). whether or not (iv) of theorem 3.2 is equivalent to the other three statements listed in the theorem is unknown to the authors. each s-space having a regular s-compactification has a largest such. theorem 3.5 and theorem 3.7 address the question as to when a regular compactification in lim can be used to form a regular s-compactification in ca. it is shown in theorem 4.2 that the process of forming a regular s-compactification remains invariant under the operation of taking “generalized quotients.” references [1] h. boustique, p. mikusinski and g. richardson, convergence semigroup actions: generalized quotients, appl. gen. topol. 10 (2009), 173–186. [2] h. boustique, p. mikusinski and g. richardson, convergence semigroup categories, appl. gen. topol. 11, no. 2 (2010), 67–88. [3] j. de vries, on the existence of g-compactifications bull. acad. polon. sci. ser. sci. math. astronom. phys. 26, no. 3 (1978), 275–280. [4] w. h. gottschalk and g. a. hedlund, topological dynamics, volume 36. american mathematical society, 1955. [5] h. h. keller, die limes-uniformisierbarkeit der limesräume, mathematische annalen 176, no. 4 (1968), 334–341. [6] d. c. kent and g. d. richardson, regular compactifications of convergence spaces, proc. amer. math. soc. 31 (1972), 571–573. [7] d. c. kent and g. d. richardson, open and proper maps between convergence spaces, czechoslovak mathematical journal 23, no. 1 (1973), 15–23. [8] d. c. kent and g. d. richardson, locally compact convergence spaces, the michigan mathematical journal 22, no. 4 (1975), 353–360. [9] d. c. kent and g. d. richardson, compactifications of convergence spaces internat. j. math. math. sci. 2, no. 3 (1979), 345–368. [10] e. lowen-colebunders, function classes of cauchy continuous maps, m. dekker, 1989. [11] g. preuss, foundations of topology: an approach to convenient topology, kluwer academic publishers, dordrecht, 2002. [12] e. e. reed, completions of uniform convergence spaces mathematische annalen 194, no. 2 (1971), 83–108. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 136 () @ appl. gen. topol. 17, no. 1(2016), 51-55doi:10.4995/agt.2016.4376 c© agt, upv, 2016 on topological groups with remainder of character k maddalena bonanzinga and maria vittoria cuzzupé dipartimento di scienze matematiche e informatiche, scienze fisiche e scienze della terra, universitá di messina, italia (mbonanzinga@unime.it, mcuzzupe@unime.it) abstract in [a.v. arhangel’skii and j. van mill, on topological groups with a first-countable remainder, topology proc. 42 (2013), 157-163] it is proved that the character of a non-locally compact topological group with a first countable remainder doesn’t exceed ω1 and a non-locally compact topological group of character ω1 having a compactification whose remainder is first countable is given. we generalize these results in the general case of an arbitrary infinite cardinal κ. 2010 msc: 54h11; 54a25; 54b05. keywords: character; compactification; π-base; remainder; topological group. 1. introduction in [3] the authors answer in the negative to the following problem: problem 1.1 ([3, problem 1.1]). suppose that g is a non-locally compact topological group with a first countable remainder. is g metrizable? also, the following necessary condition for a non-locally compact topological group to have a first countable remainder is established: theorem 1.2 ([3, theorem 2.1]). suppose that g is a non-locally compact topological group with a first countable remainder. then the character of the space g doesn’t exceed ω1. as a consequence of the previous result the following holds: received 23 november 2015 – accepted 04 january 2016 http://dx.doi.org/10.4995/agt.2016.4376 m. bonanzinga and m. v. cuzzupé theorem 1.3 ([3, theorem 2.4]). if g is a non-locally compact topological group with a first countable remainder, then |g| ≤ 2ω1. in [3] it is proved that theorem 1.2 is the best possible giving the following: example 1.4 ([3, section 3]). a non locally compact topological group g of character ω1 which has a compactification bg such that bg\g is first countable. in this paper we show that the methods used by arhangel’skii and van mill permit to generalize theorem 1.2 and example 1.4 in the general case of an arbitrary infinite cardinal κ. by a space we understand a tychonoff topological space. by a remainder of a space we mean the subspace bx \ x of a hausdorff compactification bx of x. we follow the terminology and notation in [4]. 2. generalizations of arhangel’skii and van mill’s results in the general case of an arbitrary infinite cardinal κ we show that arhangel’skii and van mill’s proof of [3, theorem 2.1], works in the general case of an arbitrary infinite cardinal κ. theorem 2.1. let κ be an infinite cardinal and let g be a non-locally compact topological group. assume that g has a compactification such that its remainder bg \ g has character κ. then the character of the space g doesn’t exceed κ+. to prove theorem 2.1, we need the following propositions 2.2 and 2.3. in particular, proposition 2.2 is known (see for example [1], also note that the concept of free sequence was introduced in [2]). we include the proof of proposition 2.2 for completness of the exposition. proposition 2.2. suppose that y is a space with tightness t(y ) = κ satisfying the following condition: (s) for any subset a of y such that |a| ≤ κ+, the closure of a in y is compact. then y is compact. proof. striving for a contradiction, assume that y is not compact and let x be a compactification of y . pick an arbitrary point x ∈ x \y . then: fact 1: every nonempty gκ-subset p of x that contains x meets y . indeed, let p = ⋂ {vα : α < κ}, where each vα is open. for each α take an open set uα in x such that x ∈ uα ⊆ vα. put {uα : α < κ} = u. we may assume without any loss of generality that u is closed under finite intersections. for any u ∈u pick a point yu ∈ u∩y and let a = {yu : u ∈u}. by condition (s), the set s = a y is compact. as the family f = {u ∩ s : u ∈ u} has the finite intersection property, we must have ⋂ f 6= ∅. since ⋂ f ⊆ p ∩ y , we are done. using fact 1, we define for every ξ < κ+ a point yξ ∈ y and a closed gκsubset pξ of x containing x, as follows. let y0 be any element of y , and put c© agt, upv, 2016 appl. gen. topol. 17, no. 1 52 on topological groups with remainder of character k p0 = x. now assume that ξ < κ +, and that the points yβ ∈ y and the closed gκ-subsets pβ of x have been defined for every β < ξ. denote by fξ the closure of the set {yβ : β < ξ} in x. then fξ ⊆ y and x /∈ fξ. since fξ is closed in x and x is tychonoff, it follows that there exists a closed gδ-subset v of x in x such that x ∈ v and v ∩fξ = ∅. put pξ = v ∩ ⋂ β<ξ pβ. clearly, x ∈ pξ, and pξ is a closed gκ-subset of x. by fact 1, we have pξ ∩ y 6= ∅. this completes the transfinite construction. obviously, the following statements hold for any ξ < κ+ (fact 4 follows directly from facts 2 and 3). fact 2: {yβ : β < ξ}∩ pξ = ∅. fact 3: {yβ : ξ ≤ β < κ+}⊆ pξ. fact 4: {yβ : β < ξ}∩{yβ : ξ ≤ β < κ+} = ∅. fact 4 implies that η = {yξ : ξ < κ +} is a free sequence in x. its closure is compact and is contained in y . hence this contradicts the fact that the tightness of y is at most κ (juhász [5, 3.12]). � following the argument from [3, proposition 2.3], and using proposition 2.2 instead of [3, proposition 2.2] we obtain the following result. proposition 2.3. suppose that x is a nowhere locally compact space with remainder y such that χ(y ) = κ, where κ is an infinite cardinal. then the π-character of the space x doesn’t exceed κ+ at some point of x. proof of theorem 2.1. it follows from proposition 2.3 that there exists a πbase p of g at the neutral element e of g such that |p| ≤ κ+. then, clearly, the family µ = {uu−1 : u ∈p} is a base of g at e such that |µ| ≤ κ+. � theorem 2.4. if g is a non-locally compact topological group with remainder y such that χ(y ) = κ, then |g| ≤ 2κ + . proof. let bg be a compactification of the space g such that the remainder y = bg \ g has character κ. by theorem 2.1, the character of the space g doesn’t exceed κ+. since χ(y ) = κ and y and g are both dense in bg, we conclude that χ(bg) ≤ κ+. since bg is compact, it follows that |bg| ≤ 2κ + . hence, |g| ≤ 2κ + . � following the method in [3, section 3] we construct the following example. example 2.5. a non-locally compact topological group g of character κ+ which has a compactification bg such that bg\g has character κ. let x be a space with a dense subset d and consider the subspace x(d) = (x ×{0})∪ (d ×{1}) of the alexandroff duplicate of x. observe that x(d) is compact if x is compact. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 53 m. bonanzinga and m. v. cuzzupé the idea used by authors in [3, section 3] is to replace every isolated point of the form (d, 1) in x(d) by a copy of a fixed non-empty space y . also they note that if both x and y are compact, then so is x(d, y ) and that the function π : x(d, y ) → x ×{0} that collapses each set of the form {d}× y ×{1} to (d, 0) is a retraction. let κ ≥ ω and let k = 2κ(2κ), i.e., the alexandroff duplicate of the cantor cube 2κ. following the idea used in [3, section 3] and using this building block repeatedly, we will construct an inverse sequence of compact spaces xα, α < κ+. in particular following step by step [3, section 3] and defining x0 = 2 κ instead of 2ω, we construct all xα, where α < ω1 and xω1 = lim←− {xα, π α β}. let πω1α : xω1 → xα denotes the projection for all α < ω1. also the points p ∈ xω1, for which π ω1 α (p) is isolated for every successor ordinal number α < ω1, form a dense subspace h in xω1. now put xω1+1 = xω1(h), and let π ω1+1 ω1 be the standard retraction. we continue as before, replacing each isolated point by a copy of k, etc. let xω1+ω1 be the inverse limit of spaces xω1+β, β < ω1. continuing in this way for all α < ω2, we get an inverse sequence {xα, π α β} of compact spaces having character equal to κ. let xω2 = lim←− {xα, π α β} with retractions π ω2 α : xω2 → xα for all α < ω2. continuing in this way for all α < κ+, we get an inverse sequence {xα, π α β} of compact spaces having character equal to κ. let xκ+ = lim←− {xα, π α β} with retractions πκ + α : xκ+ → xα for all α < κ +. the following fact holds. if p ∈ xκ+ and there exists a successor ordinal number α < κ + such that πκ + α (p) is not isolated, then (πκ + α ) −1({πκ + α (p)}) = {p}. hence, xκ+ has character equal to κ at p. the points p ∈ xκ+, for which π κ + α (p) is isolated for every successor ordinal number α < κ+, form a dense subspace g in xκ+. the space g is easily seen to be homeomorphic to the space (2κ)κ + with the gκ-topology (where the topology on 2κ is the standard product topology). the reason that we get the gκ-topology is clear: because if p ∈ g, then, for every α < κ +, we have that πκ + α+1(p) is isolated. hence, g is a topological group, and so we are done. 3. suggestions for further research it seems natural to pose the following question: question 3.1. is it possible to generalize theorem 2.1 and example 2.5 in some “generalized” class of topological groups? in particular, is this possible in the class of paratopological groups? c© agt, upv, 2016 appl. gen. topol. 17, no. 1 54 on topological groups with remainder of character k acknowledgements. the authors express gratitude to jan van mill for useful communications. references [1] a. v. arhangel’skii, construction and classification of topological spaces and cardinal invariants, uspehi mat. nauk. 33, no. 6 (1978), 29–84. [2] a. v. arhangel’skii, on the cardinality of bicompacta satisfying the first axiom of countability, doklady acad. nauk sssr 187 (1969), 967–970. [3] a. v. arhangel’skii and j. van mill, on topological groups with a first-countable remainder, topology proc. 42 (2013), 157–163. [4] r. engelking, general topology, heldermann verlag, berlin, second ed., 1989. [5] i. juhász, cardinal functions in topology–ten years later, mathematical centre tract, vol. 123, mathematisch centrum, amsterdam, 1980. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 55 () @ appl. gen. topol. 19, no. 1 (2018), 1-7doi:10.4995/agt.2018.7012 c© agt, upv, 2018 partially topological group action m. a. al shumrani department of mathematics, king abdulaziz university, p.o.box: 80203 jeddah 21589, saudi arabia (maalshmrani1@kau.edu.sa) communicated by s. garćıa-ferreira abstract the concept of partially topological group was recently introduced in [3]. in this article, we define partially topological group action on partially topological space and we generalize some fundamental results from topological group action theory. 2010msc:primary: 54a25; secondary: 54b05. keywords: partially topological space; partially topological group; group action. 1. partially topological spaces in this section, we recall definition of the categorygtspt of partially topological spaces and strictly continuous mappings which was defined in [4]. definition 1.1. let x be any set, τx be a topology on x. a family of open families covx ⊆ p(τx) will be called a partial topology if the following conditions are satisfied: (i) if u ⊆ τx and u is finite, then u ∈covx; (ii) if u ∈covx and v ∈ τx, then {u∩v :u ∈u}∈covx; (iii) if u ∈ covx and, for each u ∈ u, we have v(u) ∈ covx such that ⋃ v(u)=u, then ⋃ u∈u v(u)∈covx; (iv) ifu ⊆ τx andv ∈covx are such that ⋃ v = ⋃ u and, for eachv ∈v there existsu ∈u such that v ⊆u, then u ∈covx. received 09 december 2016 – accepted 18 september 2017 http://dx.doi.org/10.4995/agt.2018.7012 m. a. al shumrani elementsof τx are calledopen sets, andelements ofcovx are calledadmissible families.we say that (x,covx) is apartially topological generalized topological space or simply partially topological space. for simplicity, from now on, we shall denote a partially topological space (x,covx) byx. letx andy bepartially topological spaces and let f :x →y bea function. then f is called strictly continuous if f−1(u)∈covx for anyu ∈covy . a bijection f :x →y is called a strictly homeomorphism if both f and f−1 are strictly continuous functions. ifwe have a strictly homeomorphismbetween x and y we say that they are strictly homeomorphic and we denote that byx ∼=y . remark 1.2. theabovenotion of partial topology is a special case of the notion of generalized topology in the sense of h. delfs and m. knebusch considered in [2, 4, 5, 6, 7], when the family opx of open sets of the generalized topology forms a topology. definition 1.3. let (x,covx) be a partially topological space and let y be a subset ofx. then the partial topology covy =(〈covx ∩2y 〉y )pt, that is: the smallestpartial topologycontainingcovx∩2y , is calledasubspace partial topology ony , and (y,covy ) is a subspace of (x,covx). (it is also the smallest generalized topology containing covx ∩2y .) fact 1.4. let ϕ : x → x′ be a mapping between partially topological spaces and let y be a subspace of x. then the following are equivalent: a) ϕ is strictly continuous, b) the restriction mapping ϕ|y :y →x ′ is strictly continuous. definition 1.5. let (x,covx) and (y,covy ) be two partially topological spaces. the product partial topology on x × y is the partial topology covx×y =(〈covx ×2covy 〉x×y )pt in the notation of definition 4.6 of [7]; in otherwords: the smallestpartial topology inx×y that containscovx×2covy . recall that amapping f :x →y is said to be an openmapping if for every open setu of x, the set f(u) is open in y . it is said to be a closed mapping if for every closed set a of x, the set f(a) is closed in y . also, recall that a surjective mapping f : x → y is said to be a quotient mapping provided a subsetu of y is open in y if and only if f−1(u) is open inx. 2. partially topological groups in this section, we recall the definition of partially topological group. this notion was recently introduced in [3]. definition 2.1. apartially topological groupg is an ordered pair ((g,∗), covg) such that (g,∗) is a group, while covg is a generalized topology ong such that ⋃ covg is a t1 topology on g and the multiplication mapping of (g×g,covg×g) into (g,covg), which sends ordered pair (x,y) ∈ g×g c© agt, upv, 2018 appl. gen. topol. 19, no. 1 2 partially topological group action to x ∗ y, is strictly continuous and the inverse mapping from (g,covg) into (g,covg),which sends eachx∈g tox −1, is strictly continuous.for simplicity, from now on, we shall denote a partially topological group ((g,∗),covg) by g. definition 2.2. any subgrouph of a partially topological groupg is a partially topological group and it is called apartially topological subgroup of g. definition 2.3. letϕ :g→g′ be a function. thenϕ is called amorphism ofpartially topological groups ifϕ is both strictly continuousandgrouphomomorphism.moreover,ϕ is an isomorphism if it is strictly homeomorphism and group isomorphism. if we have an isomorphism between two partially topological groupsg and g′, then we say that they are isomorphic and we denote that byg∼=g′. remark 2.4. obviously composition of twomorphisms of partially topological groups is amorphism. in addition, the identitymapping is an isomorphism. so partially topological groups and their morphisms form a categoryptgr. 3. partially topological group action on partially topological space in this section, we introduce partially topological group action on partially topological space andwe extend some fundamental results in [1] of action of a topological group on a topological space to this new concept. definition 3.1. ifg is a partially topological groupwith identity e andx is a partially topological space, then anaction ofgonx is amappingg×x →x, with the image of (g,x) being denoted by g(x), such that (gh)(x) = g(h(x)) and e(x)=x for all g,h∈g and x∈x. if this mapping is strictly continuous, then the action is said to be strictly continuous. the space x, with a given strictly continuous action of g on x, is called partially g-space. for a point x∈x, the setg(x)= {gx : g∈g} is called the orbit of x. definition 3.2. let g be a partially topological group and x a partially topological space. letg act onx. for a point x ofx, the set gx = {g∈g : gx=x} (or gx = {g∈g :xg=x}) is called the stabilizer of x. fact 3.3. the stabilizer gx of any point x∈x is a subgroup of g. definition 3.4. let g be a partially topological group and x a partially topological space. letg act onx. for a point x ofx, we define amapping µx :g→x by µx(g)= gx (or µx(g)=xg). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 3 m. a. al shumrani note that µx is strictly continuous by strictly continuity of the action. the action is called transitive if for each x∈x,gx =x. then obviously we have the following fact. fact 3.5. µx is surjective iff g acts transitively on x. proposition3.6. every strictly continuous action θ :g×x →x of a partially topological group g on a partially topological space x is an open mapping. proof. it suffices to prove that the images under θ of the elements of some base for g×x are open in x. leto=u ×v ⊂g×x, whereu and v are open sets ing andx, respectively. then θ(o) = ⋃ g∈g θg(v ) is open inx since every θg is a strictly homeomorphism of x onto itself. since the open sets u ×v form a base forg×x, the mapping θ is open. � proposition 3.7. the strictly continuity of an action θ : g×x → x of a partially topological groupgwith identity e on a partially topological spacex is equivalent to the strictly continuity of θ at the points of the set {e}×x ⊂g×x. proof. let g ∈ g and x ∈ x be arbitrary and u be a neighborhood of gx in x. since θh is a homeomorphism of x for each h ∈ g, the set v = θg−1(u) is a neighborhood of x in x. by the strictly continuity of θ at (e,x), we can find a neighborhood o of e in g and a neighborhood w of x in x such that hy ∈ v for all h ∈ o and y ∈ w . clearly, if h ∈ o and y ∈ w , then (gh)(y) = g(hy) ∈ gv = θg(v ) = u. thus, ky ∈ u, for all k ∈ go and all y ∈ w , where o′ = go is a neighborhood of g in g. hence, the action θ is strictly continuous. � next we present two examples of strictly continuous actions of partially topological groups. example 3.8. any partially topological groupg acts on itself by left translations, that is, θ(x,y)=xy for allx,y∈g. the strictly continuity of this action follows from the strictly continuity of the multiplication ing. example 3.9. letg be a partially topological group,h a closed subgroup of g, and let g/h be the corresponding left coset space. the action φ of g on g/h, defined by the rule φ(g,xh) = gxh, is strictly continuous. indeed, let y0 ∈g/h, and fix an open neighborhoodo of y0 ing/h. choosex0 ∈g such that π(x0) = y0, where π : g → g/h is the quotient mapping. there exist open neighborhoodsu and v of the identity e ing such that π(ux0)⊂o and v 2 ⊂u. clearly, w = π(vx0) is open in g/h and y0 ∈w. by the choice of u and v , if g ∈ v and y ∈ w , then φ(g,y) ∈ o. indeed, let x1 ∈ vx0 with π(x1) = y. then y = x1h and φ(g,y) = gx1h ∈ vvx0h ⊂ π(ux0) ⊂ o. therefore,φ is continuous at (e,y0)∈g×g/h. hence,φ is strictly continuous by proposition 3.7. suppose that a partially topological groupg acts strictly continuously on a partially topological spacex and thatx/g is the corresponding orbit set. let c© agt, upv, 2018 appl. gen. topol. 19, no. 1 4 partially topological group action x/g have the partially quotient topology generated by the orbital projection π : x → x/g (a subset u ⊂ x/g is open in x/g if and only if π−1(u) is open in x). the partially topological space x/g is called the orbit space or the orbit space of the partillayg-spacex. the following result shows that the orbital projection is always an open mapping. proposition 3.10. if θ : g×x → x is a strictly continuous action of a partially topological groupg on a partially topological spacex, then the orbital projection π :x →x/g is an open mapping. proof. for any open set u ⊂ x, consider the set π−1π(u) = gu. every left translation θg is a strictly homeomorphism of x onto itself, so the set gu = ⋃ g∈g θg(u) is open in x. since π is a quotient mapping, π(u) is open in x/g. hence, π is an openmapping. � theorem 3.11. suppose a compact partially topological group h acts strictly continuously on a hausdorff partially space x, then the orbital projection π : x →x/h is both open and perfect mapping. proof. first note that π is open by proposition 3.10. next we show that π is perfect. let y ∈x/h, choose x∈x such that π(x) = y. note that π−1(y) = hx is the orbit of x inx. since themapping ofh ontohx assigning to every g ∈h the point gx∈x is strictly continuous, the image hx of the compact grouph is also compact. hence, all fibers of π are compact. we show that the mapping π is closed. let y ∈ x/h and x ∈ x such that π(x) = y. let o be an open set in x containing π−1(y) = hx. since the action of h on x is strict continuous, we can find, for every g ∈ h, open neighborhoods g ∈ ug and x ∈ vg in h and x, respectively, such that ugvg ⊂o. by the compactness ofh and of the orbithx, there exists a finite set f ⊂h such that h = ⋃ g∈f ug and hx⊂ ⋃ g∈f gvg. then v = ⋂ g∈f vg is an open neighborhood of x in x, and we claim that hv ⊂ o. indeed, if h ∈ h and z ∈ v , then h ∈ ug, for some g ∈ f , so that hz ∈ ugv ⊂ ugvg ⊂ o. thus, w = π(v ) is an open neighborhood of y in x/h, and we have that π−1π(v )=hv ⊂o. hence, π is closed. � definition 3.12. letx and y be partiallyg-spaces with strictly continuous actions θx :g×x →x and θy :g×y →y.a strictly continuous mapping f : x → y is called partially g-equivariant if θy (g,f(x)) = f(θx(g,x)), that is, gf(x) = f(gx), for all g ∈ g and all x ∈ x. clearly, f is partially g-equivariant if and only if the following diagram c© agt, upv, 2018 appl. gen. topol. 19, no. 1 5 m. a. al shumrani g×x θx −−−−→ x   y f   y f g×y θy −−−−→ y commutes, where f = idg × f is the product of the identity mapping idg of g and the mapping f. example 3.13. let h be a closed subgroup of a partially topological group g, and y = g/h be the left coset space. denote by θg the action of g on itself by left translations, and by θy the natural strictly continuous action of g on y . then the quotient mapping π :g→g/h defined by π(x) = xh for each x∈g is equivariant. indeed, the equality g(π(x)) = gxh = π(gx) holds for all g,x∈g. equivalently, the following diagram g×g θg −−−−→ g   y π   y π g×y θy −−−−→ y commutes, where π= idg×π. let η = {xi : i ∈ i} be a family of partially g-spaces. then the product spacex = ∏ i∈i xi, ifx ishausdorff, is apartiallyg-space.todefineanaction of g on x, take any g ∈g and any x= (xi)i∈i ∈x, and put gx= (gxi)i∈i. thus,g acts onx coordinatewise. the following result shows the strictly continuity of this action. proposition3.14. the coordinatewise action ofg on the productx = ∏ i∈i xi of partially g-spaces is strictly continuous, that is, x is a partially g-space, if x is hausdorff. proof. byproposition 3.7, it suffices to verify the continuity of the action ofg onx at the neutral element e∈g. let x=(xi)i∈i ∈x be an arbitrary point ando⊂x aneighborhoodof gx inx. since canonical open sets formabase of x, we can assume that o= ∏ i∈i oi, where eachoi is an open neighborhood of xi in xi and the set f = {i ∈ i : oi 6= xi} is finite. since all factors are partiallyg-spaces, we can choose, for every i∈f , open neighborhoods e∈ui and xi ∈ vi in g and xi, respectively, such that uivi ⊂ oi. put u = ⋃ i∈f vi andw = ∏ i∈i wi, wherewi =vi if i∈f andwi =xi otherwise. therefore, it follows from the definition of the sets u and w that uw ⊂ o. hence, the action ofg onx is strictly continuous. � theorem 3.15. letg be a partially topological group andx a partially topological space. let g act on x. suppose that both g and x/g are connected, then x is connected. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 6 partially topological group action proof. suppose that x is the union of two disjoint nonempty open subsets u and v . now π(u) and π(v ) are open in x/g. since x/g is connected, π(u) and π(v ) cannot be disjoint. if π(x)∈π(u)∪π(v ), then bothu∪o(x) and v ∪o(x) are nonempty, whereo(x) is the orbit of x. it means o(x) is a disjoint union of two nonempty open sets. but o(x) is the image of g under the strictly continuous function f :g→x defined by f(g)= g(x). therefore, o(x) is connected which is a contradiction. hence,x is connected. � theorem 3.16. if x is a compact partially topological group and g a closed subgroup acting on x by left translation, then x/g is regular. proof. sinceg is closed subgroupandthe left translationmappinglx :x →x is strictly homeomorphism then π−1π(x) = xg=lx(g) is closed. thus every point π(x) ofx/g is closed, and it follows thatx/g is t1 space. nowwe show that for a closed subsetf ofx/g and a point p /∈f there are open sets u,v satisfying p ∈ u,f ⊂ v,u ∩v = ∅. since x acts transitively on x/g, we may assume that p is an element of the class eg = g of the identity element e. since f is closed, there exists an open set u0 such that f ∩u0 = ∅ and p ∈ u0. from the strictly continuity of group action of x, there is an open set w such that e ∈ w and w−1w ⊂ π−1(u0). the set wπ−1(f)= ⋃ x∈π−1(f) wx is open. sinceπ is an openmapping, bothu =π(w) and v =π(wπ−1(f)) are open sets and p∈u and f ⊂v . next we show thatu ∩v = ∅. suppose that there exists y∈u∩v . then there existx1,x2 ∈w andx∈π −1(f) such that y=π(x2)=π(x1x).thus,we haveg∈g suchthatx2g=x1x, fromwhichwededuce thatπ(xg −1)∈f∩u0 = ∅ from xg−1 =x1 −1x2 ∈w −1w ⊂π−1(u0). therefore,u∩v = ∅. � references [1] a. arhangel’skii and m. tkachenko, topological groups and related structures, world scientific, 2008. [2] h. delfs and m. knebusch, locally semialgebraic spaces, lecture notes in math. 1173, springer, berlin-heidelberg, 1985. [3] c.ozel, a. piękosz,m.a.al shumrani ande.wajch, partially paratopological groups, topology appl. 228 (2017), 68–78. [4] a. piękosz, on generalized topological spaces i, ann. polon. math. 107, no. 3 (2013), 217–241. [5] a. piękosz, on generalized topological spaces ii, ann. polon. math. 108, no. 2 (2013), 185–214. [6] a. piękosz, o-minimal homotopy and generalized (co)homology, rocky mountain j. math. 43, no. 2 (2013), 573–617. [7] a. piękosz and e. wajch, compactness and compactifications in generalized topology, topology appl. 194 (2015), 241–268. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 7 () @ appl. gen. topol. 19, no. 1 (2018), 163-172doi:10.4995/agt.2018.7918 c© agt, upv, 2018 fixed point theorems for nonlinear contractions with applications to iterated function systems rajendra pant department of mathematics, visvesvaraya national institute of technology, nagpur 440010, india (pant.rajendra@gmail.com) communicated by s. romaguera abstract we introduce a new type of nonlinear contraction and present some fixed point results without using continuity or semi-continuity. our result complement, extend and generalize a number of fixed point theorems including the well-known boyd and wong theorem [on nonlinear contractions, proc. amer. math. soc. 20(1969), 458–464]. also we discuss an application to iterated function systems. 2010 msc: primary 47h10; 54h25. keywords: suzuki type contraction; self-similarity; iterated function systems; fractals. 1. introduction and preliminaries in 1981, hutchinson [12] introduced the concept of iterated function system (ifs) and self-similarity. a set is said to be self-similar if it is made up of a finite transformed copies of itself. self-similar sets are special class of fractals and there are no objects in nature which have exact structures of self-similar sets. these sets are perhaps the simplest and the most basic structures in the theory of fractals. in recent years this area received great attention of many mathematicians, scientists and a huge developments took place (cf. [1, 2, 3, 4, 5, 7, 10, 11, 14, 18, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33]). the purpose of this paper is to introduce a new class of nonlinear contractions and present some fixed point results for this class of mapping without received 01 august 2017 – accepted 18 december 2017 http://dx.doi.org/10.4995/agt.2018.7918 r. pant using any kind of continuity. our result complement, extend and generalize a number of fixed point theorems in the literature. we also discuss an application of our results to iterated function systems. the following result which generalizes the classical banach contraction principle (bcp) is due to boyd and wong [6]: theorem 1.1. let (x, d) be a complete metric space and t : x → x a selfmapping such that for all x, y ∈ x d(t x, t y) ≤ ϕ(d(x, y)), where ϕ : [0, ∞) → [0, ∞) is upper semicontinuous from the right on [0, ∞), and satisfies ϕ(t) < t for all t > 0. then t has a unique fixed point in x. jachymski [13] established equivalence between various ϕ-contractive type conditions (see also [22]). definition 1.2 ([13]). let (x, d) be a metric space and ϕ : [0, ∞) → [0, ∞) a function such that ϕ(t) < t for t > 0. a self-mapping t : x → x is said to be ϕ-contractive if d(t x, t y) ≤ ϕ(d(x, y)) for all x, y ∈ x. theorem 1.3 ([13]). let (x, d) be a metric space and t : x → x a selfmapping. the following statements are equivalent: (i) there exists an increasing and right continuous function ϕ : [0, ∞) → [0, ∞) such that t is ϕ-contractive; (ii) there exists a continuous function ϕ : [0, ∞) → [0, ∞) with ϕ(t) > 0 for t > 0, such that d(t x, t y) ≤ d(x, y) − ϕ(d(x, y)) for all x, y ∈ x; (iii) there exists an upper semicontinuous function ϕ : [0, ∞) → [0, ∞) such that t is ϕ-contractive; (iv) there exists a function ϕ : [0, ∞) → [0, ∞) with lim sup s→t ϕ(s) < t for all t > 0 such that t is ϕ-contractive; (v) there exists a strictly increasing function ϕ : [0, ∞) → [0, ∞) such that lim n→∞ ϕn(t) = 0 for all t ∈ [0, ∞) and t is ϕ-contractive; (vi) there exists a strictly increasing and continuous ϕ : [0, ∞) → [0, ∞) such that t is ϕ-contractive; on the other hand, suzuki [30] obtained the following forceful generalization of the bcp: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 164 nonlinear contractions and iterated function systems theorem 1.4. let (x, d) be a complete metric space and t : x → x a selfmapping. define a non decreasing function θ : [0, 1) → (1/2, 1] such that θ(r) =      1, if 0 ≤ r ≤ ( √ 5 − 1)/2, (1 − r)r−2, if ( √ 5 − 1)/2 ≤ r ≤ 2−1/2, (1 + r)−1, if 2−1/2 ≤ r < 1. assume that there exists r ∈ [0, 1) such that for all x, y ∈ x (1.1) θ(r)d(x, t x) ≤ d(x, y) implies d(t x, t y) ≤ rd(x, y). then t has a unique fixed point in x. the above theorem has been extended and generalized by many authors in various ways (cf. [8, 9, 15, 16, 17, 19, 20, 26] and elsewhere). 2. suzuki type generalized ϕ-contractive mappings in this section, we present a generalization of theorem 1.1 which also extends theorem 1.4. we begin with the following definition: definition 2.1. let (x, d) be a metric space. a self-mapping t : x → x will be called a suzuki type generalized ϕ-contractive if for all x, y ∈ x, (2.1) 1 2 d(x, t x) ≤ d(x, y) implies d(t x, t y) ≤ ϕ(m(x, y)), where m(x, y) = max{d(x, y), d(x, t x), d(y, t y)} and ϕ : [0, ∞) → [0, ∞) is a function such that ϕ(t) < t for all t > 0 and lim sup s→t+ ϕ(s) < t for all t > 0. the following theorem is the main result of this section. theorem 2.2. let (x, d) be a complete metric space and t : x → x a suzuki type generalized ϕ-contractive mapping. then t has a unique fixed point in x. proof. pick x0 ∈ x arbitrary and define a sequence {xn} by xn = t nx = t xn−1 for all n ∈ n. since 1 2 d(xn, xn+1) ≤ d(xn, xn+1) by (2.1), we have d(xn+1, xn+2) = d(t xn, t xn+1) ≤ ϕ(m(xn, xn+1)) = ϕ(max{d(xn, xn+1), d(xn, xn+1), d(xn+1, xn+2)})(2.2) = ϕ(d(xn, xn+1)) < d(xn, xn+1). for all n ∈ n. set an = d(xn, xn+1) then an ≥ 0. if there exists some n0 ∈ n such that an0 = d(xn0 , xn0+1) = 0, then since 1 2 d(xn0, xn0+1) ≤ d(xn0 , xn0+1) by (2.1), we have an0+1 = d(xn0+1, xn0+2) ≤ ϕ(m(xn0, xn0+1)). but ϕ(0) = 0 so, an0+1 = 0 and 0 ≤ an0+1 = d(t xn0, t xn0+1) ≤ ϕ(m(xn0 , xn0+1)). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 165 r. pant hence the sequence {an} monotone decreasing and bounded below and an = 0 for all n ≥ n0. therefore lim n→∞ d(t nx, t n+1x) = 0 for each x ∈ x. if an > 0 for every n ∈ n then since ϕ(t) < t for t > 0 by (2.2), we get 0 < an+1 = d(xn+1, xn+2) = d(t xn, t xn+1) ≤ ϕ(m(xn, xn+1)) < d(xn, xn+1), we obtain 0 < an+2 ≤ ϕ(an+1) < an+1 ≤ ϕ(an) < an hence {an} and {ϕ(an)} are strictly decreasing sequences, which are bounded below. so, lim n→∞ an and lim n→∞ ϕ(an) exist. suppose 0 < a = lim n→∞ an and an = a + εn (εn > 0). if lim sup s→t+ ϕ(s) < t for all t > 0, then ∀ {tn}, tn ↓ a+ (as n → ∞); lim sup ϕ(tn) tn→a+ < a. hence 0 < a = lim n→∞ an+1 ≤ lim n→∞ ϕ(an) ≤ lim n→∞ sup ϕ(s) s∈(a,an+1) = lim εn+1→0+ sup ϕ(s) s∈(a,a+εn+1) ≤ lim ε→0+ sup ϕ(s) s∈(a,a+ε) < a, a contradiction and lim n→∞ an = lim n→∞ d(t nx, t n+1x) = 0 for each x ∈ x. now, we show that the sequence {xn} is a cauchy. suppose {xn} is not cauchy. then there exists an ε > 0 and integers mk, nk ∈ n such that mk > nk > k and d(xnk , xmk ) ≥ ε and d(xnk , xmk−1) < ε. hence for each k ∈ n, we have ε ≤ d(xnk , xmk ) ≤ d(xnk , xmk−1) + d(xmk−1, xmk) < ε + d(xmk−1, xmk ). since lim k→∞ d(xmk−1, xmk) = 0, we get lim k→∞ d(xnk , xmk ) = ε. note that lim n→∞ an = lim n→∞ d(xn, xn+1) = 0. so, there exists some k ∈ n such that 1 2 d(xnk , xnk+1) ≤ d(xnk , xmk) for mk > nk ≥ k. now by (2.1), we have d(t xnk , t xmk) ≤ ϕ(m(xnk , xmk )). by the triangle inequality d(xnk , xmk) ≤ d(xnk , xnk+1) + d(xnk+1, xmk+1) + d(xmk , xmk+1) = d(xnk , xnk+1) + d(t xnk , t xmk) + d(xmk , xmk+1) ≤ ank + ϕ(m(xnk , xmk)) + amk = ank + ϕ(max{d(xnk , xmk ), ank, amk }) + amk c© agt, upv, 2018 appl. gen. topol. 19, no. 1 166 nonlinear contractions and iterated function systems letting k → ∞ and using ϕ(t) < t and lim sup s→t+ ϕ(s) < t for all t > 0, we obtain ε = lim k→∞ d(xnk , xmk ) ≤ lim k→∞ ϕ(d(xnk , xmk )) ≤ lim ε1→0+ sup ϕ(s) s∈(ε,ε+ε1) < ε, a contradiction. hence {xn} is a cauchy sequence. since x is complete, {xn} has a limit in x. call it z. now for all n ∈ n, we show that (2.3) either 1 2 d(xn, xn+1) ≤ d(xn, z) or 1 2 d(xn+1, xn+2) ≤ d(xn+1, z). arguing by contradiction, we suppose that for some n ∈ n d(xn, z) < 1 2 d(xn, xn+1) and d(xn+1, z) < 1 2 d(xn+1, xn+2). by the triangle inequality and (2.2) d(xn, xn+1) ≤ d(xn, z) + d(xn+1, z) < 1 2 d(xn, xn+1) + 1 2 d(xn+1, xn+2) < 1 2 [d(xn, xn+1) + d(xn, xn+1)] = d(xn, xn+1), a contradiction. thus for all n ∈ n (2.3) holds. in the first case, since 1 2 d(xn, xn+1) = 1 2 d(xn, t xn) ≤ d(xn, z), by (2.1), we have d(xn+1, t z) = d(t xn, t z) ≤ ϕ(d(xn, z)). letting n → ∞ and using ϕ(t) < t and lim sup s→t+ ϕ(s) < t for all t > 0, we obtain d(z, t z) = lim n→∞ d(xn+1, t z) ≤ lim n→∞ ϕ(d(xn+1, t z)) < d(z, t z), a contradiction unless t z = z. similarly, in the other case we can deduce that t z = z. uniqueness of fixed point follows easily. � corollary 2.3. let (x, d) be a complete metric space, ϕ : [0, ∞) → [0, ∞) an increasing and right continuous function such that ϕ(t) < t for all t > 0 and t : x → x a suzuki type generalized ϕ-contractive mapping. then t has a unique fixed point in x. proof. it may be completed by following the proof of theorem 2.2. � corollary 2.4. let (x, d) be a complete metric space and t : x → x a self-mapping such that for all x, y ∈ x 1 2 d(x, t x) implies d(t x, t y) ≤ ϕ(m(x, y)), where ϕ is as in theorem 1.1. then t has a unique fixed point in x. proof. it comes from theorem 2.2 when ϕ : [0, ∞) → [0, ∞) is upper semicontinuous from the right on [0, ∞), and ϕ(t) < t for all t > 0. � corollary 2.5. theorem 1.1. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 167 r. pant example 2.6. let x = {0, 2} ∪ {1, 3, 5, ...} be endowed with the usual metric d. then (x, d) is a complete metric space. define t : x → x and ϕ : [0, ∞) → [0, ∞) by t x =      0, if x = 5, 2, if x = 7, 1, otherwise. and ϕ(t) =      t2 2 , if t ≤ 1, t − 1 3 , if t > 1. for x = 5 and y = 7, we have d(t x, t y) = d(0, 2) = 2 > 5 3 = ϕ(2) = ϕ(d(5, 7)) = ϕ(d(x, y)). therefore t does not satisfy theorem 1.1. on the other hand 1 2 d(5, t 5) = 1 2 d(5, 0) = 5 2 > 2 = d(5, 7) and 1 2 d(7, t 7) = 1 2 d(7, 2) = 5 2 > 2 = d(5, 7). therefore theorem 2.2 is applicable and z = 1 is the unique fixed point of t. further, we get the same conclusion when ϕ(t) =    t2 2 , if t ≤ 1, t − 1 4 , if t > 1. we note that in this case ϕ is not upper semicontinuous on [0, ∞). 3. applications to fractal spaces let (x, d) be an metric space and c(x), the collection of all nonempty compact subsets of x. define (a) d(a, b) := inf{d(a, b) : a ∈ a, b ∈ b} (the distance between two sets). (b) δ(a, b) := sup{d(x, b) : x ∈ a}. the hausdorff metric induced by d is defined by h(a, b) = max { sup x∈a d(x, b), sup y∈b d(y, a) } = max{δ(a, b), δ(b, a)} for all a, b ∈ c(x), where d(x, b) = inf y∈b d(x, y). hutchinson [12] and barnsley [1] initiated an ingenius way to define and construct fractals as compact invariant subsets of an abstract complete metric space with respect to the union of contractions fi, i = 1, 2, 3, . . .n. hutchinson showed that the operator f(a) = f1(a) ∪ f2(a) ∪ ... ∪ fn(a), a ⊂ x, is a contraction with respect to the hausdorff distance. thus, the contraction mapping principle can be applied to the iteration of hutchinson operator f. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 168 nonlinear contractions and iterated function systems consequently, whatever the initial image is chosen to start the iteration under the ifs, for example a0, the generated sequence ak+1 = f(ak), k = 0, 1, ... will tend towards a distinguished image, the attractor a∞ of the ifs. moreover, this image is invariant, i.e., f(a∞) = a∞. now onwards ϕ : [0, ∞) → [0, ∞) a non-decreasing continuous function. the following lemma which is modeled on the pattern of [22, lem. 3.2] is a crucial result of this section. lemma 3.1. let (x, d) be a metric space and t : x → x a continuous suzuki type generalized ϕ-contractive mapping. then 1 2 h(a, t (a)) ≤ h(a, b) implies h(ft (a), ft (b)) ≤ ϕ(mt (a, b)) for all a, b ∈ c(x), where mt (a, b) = max{h(a, b), h(a, t (a)), h(b, t (b))}. that is, ft : c(x) → c(x) is also a suzuki type generalized ϕ-contractive (with the same ϕ), where ∀ d ∈ c(x), ft (d) := t (d). proof. following the proof of lemma 3.2 [22], for all x ∈ a and y ∈ b, we have sup x∈a inf y∈b ϕ(d(x, y)) ≤ sup x∈a ϕ( inf y∈b d(x, y)) ≤ ϕ(mt (a, b)); and sup y∈b inf x∈a ϕ(d(x, y)) ≤ sup y∈b ϕ( inf x∈a d(x, y)) ≤ ϕ(mt (a, b)). further, for all x ∈ a, y ∈ b 1 2 d(x, t x) ≤ d(x, y) implies 1 2 δ(a, t (a)) ≤ 1 2 h(a, t (a)) ≤ h(a, b) and 1 2 d(y, t y) ≤ d(x, y) implies 1 2 δ(b, t (b)) ≤ 1 2 h(b, t (b)) ≤ h(a, b). now δ(ft (a), ft (b)) = max t x∈f (a) min t y∈f (b) d(t x, t y) = max x∈a min y∈b d(t x, t y). since t is suzuki type generalized ϕ-contractive mapping, 1 2 δ(a, t (a)) ≤ 1 2 h(a, t (a)) implies δ(ft (a), ft (b)) ≤ sup x∈a inf y∈b ϕ(d(x, y)) ≤ ϕ(mt (a, b)). similarly, 1 2 δ(b, t (b)) ≤ 1 2 h(b, t (b)) implies δ(ft (b), ft (a)) ≤ sup y∈b inf x∈a ϕ(d(x, y)) ≤ ϕ(mt (a, b)). since h(a, b) = h(b, a) (symmetric), and h(ft (a), ft (b)) = max{δ(ft (a), ft (b)), δ(ft (b), ft (a))}, c© agt, upv, 2018 appl. gen. topol. 19, no. 1 169 r. pant we conclude that 1 2 h(a, t (a)) ≤ h(a, b) implies h(ft (a), ft (b)) ≤ ϕ(mt (a, b)), for all a, b ∈ c(x). therefore ft is suzuki type generalized ϕ-contractive mapping. � lemma 3.2 ([1]). let (x, d) be a complete metric space. then (c(x), h) is a complete metric space. lemma 3.3. let (x, d) be a metric space and tn : c(x) → c(x) (n = 1, 2, 3, ..., p) continuous suzuki type generalized ϕ-contractive mappings, i.e., for all a, b ∈ c(x). 1 2 h(a, tn(a)) ≤ h(a, b) implies h(tn(a), tn(b)) ≤ ϕn(mtn(a, b)). define t : c(x) → c(x) by t (a) = t1(a) ∪ t2(a) ∪ ... ∪ tp(a) = p ⋃ n=1 tn(a) for each a ∈ c(x). then t also satisfies 1 2 h(a, t (a)) ≤ h(a, b) implies h(t (a), t (b)) ≤ η(mt (a, b)). for all a, b ∈ c(x), where η = max{ϕn : n = 1, 2, 3..., p}. proof. we shall prove this by induction. for n = 1, the statement is obviously true. for n = 2, we have h(t (a), t (b)) = h(t1(a) ∪ t2(a), t1(b) ∪ t2(b)) ≤ max{h(t1(a), t1(b)), h(t2(a), t2(b))}. since each t1 and t2 are suzuki type generalized ϕ-contractive, that is 1 2 h(a, t1(a)) ≤ h(a, b) implies h(t1(a), t1(b)) ≤ ϕ1(mt1(a, b)) 1 2 h(a, t2(a)) ≤ h(a, b) implies h(t2(a), t2(b)) ≤ ϕ2(mt2(a, b)), we get h(t (a), t (b)) ≤ max{ϕ1(mt1(a, b)), ϕ2(mt2(a, b))} = η(max{h(a, b), h(a, t1(a) ∪ t2(a)), h(b, t1(b) ∪ t2(b))}) = η(max{h(a, b), h(a, t (a)), h(b, t (b))}) = η(mt (a, b)), where η = max{ϕ1, ϕ2}. � as a consequence of theorem 2.2, and lemmas 3.1 and 3.3, we get following result in fractal spaces. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 170 nonlinear contractions and iterated function systems theorem 3.4. let (x, d) be a complete metric space and tn : c(x) → c(x) continuous suzuki type generalized ϕ-contractive mappings. then the transformation t : c(x) → c(x) defined by t (a) = p ⋃ n=1 tn(a) for each a ∈ c(x) satisfies the following condition 1 2 h(a, t (a)) ≤ h(a, b) implies h(t (a), t (b)) ≤ η(m(a, b)). for all a, b ∈ c(x), where η = max{ϕn : n = 1, 2, 3..., p}. moreover, (a): t has a unique fixed point a in c(x); 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(sa.ahmdi@gmail.com, sa.ahmadi@math.usb.ac.ir) abstract in this paper we associate a pseudo-metric to a dynamical system on a compact metric space. we show that this pseudo-metric is identically zero if and only if the system is chain transitive. if we associate this pseudo-metric to the identity map, then we can present a characterization for connected and totally disconnected metric spaces. 2010 msc: 337b35; 54h20. keywords: chain recurrent; chain transitive; chain component; inverse limit space. 1. introduction one of the main problems in dynamical systems is the description of the orbit structure of a system from a topological point of view [1, 3, 4]. recurrence behavior is one of the most important concepts in topological dynamics. various notions of recurrence have been considered in dynamics such as recurrent points, chain recurrent points and non-wandering points [7]. in this paper (x, d) is a compact metric space and f : x → x is a continuous map. an ǫ-pseudo-orbit (or ǫ-chain) of f from x to y is a sequence {xi} n i=0 with x0 = x, xn = y and d(f(xk), xk+1) < ǫ for k = 0, 1, ..., n − 1. a point x in x is called chain recurrent if there is an ǫ-chain from x to itself. we can define an equivalence relation on the set of chain recurrent points in such received 3 october 2013 – accepted 20 may 2014 http://dx.doi.org/10.4995/agt.2014.3050 s. a. ahmadi a way that two points x and y are said to be equivalent if for every ǫ > 0 there exist an ǫ-chain from x to y and an ǫ-chain from y to x. the equivalence classes of this relation are called chain components. these are compact invariant sets and cannot be decomposed into two disjoint compact invariant sets, hence serve as building blocks of the dynamics. the topology of chain recurrent set and chain components have been always in particular interest [2, 5, 6, 8]. we use the symbol oδ(f, x, y) for the set of δ-pseudo-orbits {xi} n i=0 of f with x0 = x and xn = y. for given points x, y ∈ x we write x ǫ y if oǫ(f, x, y) 6= ∅ and we write x y if oǫ(f, x, y) 6= ∅ for each ǫ > 0. we write x ! y if x y and y x. the set {x ∈ x : x ! x} is called the chain recurrent set of f and is denoted by cr(f). if we define a relation r on x ×x with x r y ⇔ x ! y, then r is an equivalence relation on cr(f). a dynamical system f is called chain recurrent if cr(f) = x. a dynamical system f is called chain transitive if for each x, y ∈ x we deduce x ! y. we say that a dynamical system f has the pseudo-orbit tracing property (potp) on x if for each ǫ > 0 there is δ > 0 so that for a given sequence ξ = {xk}n∈z with d(f(xk), xk+1) < δ for k ∈ n there exists a point x ∈ x such that d(fk(x), xk) < ǫ for k ∈ n (in this case we say that p ∈ x ǫ-shadowed ξ). let x0 be a nonempty set. then a map d0 : x0 × x0 → r is called pseudometric if for all x, y ∈ x0 the following hold (1) d0(x, x) = 0; (2) d0(x, y) = d(y, x) ≥ 0; (3) d0(x, y) ≤ d0(x, z) + d0(z, y). the pair (x0, d0) is called a pseudo-metric space. let (x0, d0) be a pseudometric space. then, the open balls in x0 together with the empty set form a basis for a topology on x0. this topology is first countable and in it closed balls are closed. moreover, this topology is a hausdorff topology if and only if x0 is a metric space. now we are going to present a pseudo-metric on cr(f). definition 1.1. let x, y ∈ cr(f), we define df,ǫ(x, y) = inf{ k∑ i=0 d(pi, qi) : p0 = x, qk = y, k ∈ n} where the infimum is taken over all choices of pi and qi so that qi ǫ ! pi+1 for all i = 0, 1, ..., k − 1. x = p0 p1 p2 p3 ... pk q0 ǫ ;{ ;{ ;{ ;{ ;{ q1 ǫ >~>~ >~ >~ q2 ǫ >~>~ >~ >~ q3 ǫ >~ >~ >~ >~ >~ ... qk = y c© agt, upv, 2014 appl. gen. topol. 15, no. 2 168 on the topology of the chain recurrent set of a dynamical system we also define df (x, y) = inf{ k∑ i=0 d(pi, qi) : p0 = x, qk = y, k ∈ n} where the infimum is taken over all choices of pi and qi so that qi ! pi+1 for all i = 0, 1, ..., k − 1. x = p0 p1 p2 p3 ... pk q0 ;{ ;{ ;{ ;{ ;{ q1 >~ >~ >~ >~ q2 >~ >~ >~ >~ q3 >~ >~ >~ >~ >~ ... qk = y the straightforward calculations imply that for ǫ1 ≤ ǫ2 we deduce df,ǫ2(x, y) ≤ df,ǫ1(x, y) ≤ df (x, y) ≤ d(x, y) and (cr(f), df ) is a pseudo-metric space. if we define bfr (x) = {y ∈ x; df(x, y) < r}, then the collection τf = {b f r (x) : x ∈ x, r > 0} ∪ {∅} is a basis of a topology on cr(f) which is finer than τd. so (cr(f), df ) is a compact space. obviously ! is an equivalence relation on cr(f). let c̃r(f) = cr(f)/r, and π : cr(f) → c̃r(f) be the quotient map, i.e. π(x) = {y ∈ cr(f) : x ! y}. then we can define a metric d̃f (π(x), π(y)) = df (x, y) for x, y ∈ cr(f) on c̃r(f). with this metric π is a distance preserving map. the topology induced by d̃f is denoted by τ̃f . the induced map f̃ : c̃r(f) → c̃r(f) with f̃(π(x)) = π(f(x)) is the identity map. in this paper we are going to prove the following theorems. theorem 1.2. let f : x → x be a chain recurrent continuous map. then f is chain transitive if and only if df (x, y) = 0 for all x, y ∈ x. theorem 1.3. let f : x → x be a chain recurrent continuous map. then df (x, y) = d(x, y) for all x, y ∈ x if and only if f is the identity map and x is totally disconnected. theorem 1.4. let (x, d) be a compact metric space. then the following conditions are mutually equivalent: (1) x is connected; (2) the identity map ι : x → x is chain transitive; (3) for each x, y ∈ x, dι(x, y) = 0. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 169 s. a. ahmadi 2. proof of theorems for the proof of theorem 1.2 we first prove the following lemma. lemma 2.1. let x, y in cr(f) and ǫ > 0 be given. then df,ǫ(x, y) = 0 if and only if x ǫ ! y. proof. clearly if x ǫ ! y then df,ǫ(x, y) = 0. now let df,ǫ(x, y) = 0. we choose 0 < δ < ǫ so that the inequality d(t, s) < δ implies d(f(t), f(s)) < ǫ/2. thus there exist sequences {pi} k i=0 and {qi} k i=0 with p0 = x and qk = y so that qi ǫ ! pi+1 for i = 0, 1, ..., k − 1 and k∑ i=0 d(pi, qi) < δ. so d(f(pi), f(qi)) < ǫ/2 for i = 0, 1, ..., k. if {qi,n} li n=0 ∈ oǫ/2(qi, pi+1), then d(qi,1, f(qi)) < ǫ/2. hence d(qi,1, f(pi)) < ǫ. let {yi} m i=0 ∈ oǫ/2(y, y). then d(y1, f(pk)) ≤ d(y1, f(y)) + d(f(y), f(pk)) < ǫ. therefore the sequence {p0, q0,1, q0,2, ..., q0,l0, q1,1, q1,2, ... ..., qk−1,1, qk−1,2, ..., qk−1,lk−1 = pk, y1, ..., ym} is belong to oǫ(f, x, y). so x ǫ y. since df,ǫ(y, x) = 0, then y ǫ x. � corollary 2.2. let x, y ∈ cr(f). then df (x, y) = 0 if and only if x ! y. corollary 2.3. let f : x → x be a chain recurrent continuous map. then f is chain transitive if and only if df (x, y) = 0 for all x, y ∈ x. corollary 2.4. if x ! x′ and y ! y′ for x, x′, y, y′ ∈ x, then df (x, y) = df (x ′, y′) corollary 2.5. if x ∈ cr(f) then df ≡ 0 on o(f, x) × o(f, x), where o(f, x) = {fn(x); n ∈ n} ∪ {x}. proof. it is enough to show that x ! f(x). given ǫ > 0 clearly {x, f(x)} ∈ oǫ(f, x, f(x)), i.e. x ǫ f(x). we can choose 0 < δ < ǫ/2 so that d(x, y) < δ implies to d(f(x), f(y)) < ǫ/2. now let {x0, ..., xn} ∈ oδ(f, x, x), then d(x1, f(x)) < δ implies that d(f(x1), f 2(x)) < ǫ/2. so d(x2, f 2(x)) < ǫ. thus {f(x), x2, ..., xn} ∈ oǫ(f, f(x), x). therefore x ǫ ! f(x). � corollary 2.6. the map f : (cr(f), df ) → (cr(f), df ) is an isometry. proof of theorem 1.3. first suppose that for each x, y ∈ x, df (x, y) = d(x, y). hence d(x, f(x)) = df (x, f(x)) = 0 for each x ∈ x. let α be a connected component of x and α contains x. given ǫ > 0 we consider the sets πǫ(x) = {y ∈ α : x ǫ ! y} and π(x) = {y ∈ α : x ! y}. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 170 on the topology of the chain recurrent set of a dynamical system if y ∈ πǫ(x) then bǫ(y) ⊆ πǫ(x). so πǫ(x) is an open set. now let y ∈ πǫ(x) then there is a sequence {yn} ⊆ πǫ(x) such that yn → y. so y ∈ bǫ(yn) for some n ∈ n. thus y ∈ πǫ(x). hence πǫ(x) is both open and closed. since πǫ(x) 6= ∅ then πǫ(x) = α. therefore α = ∩ǫ>0πǫ(x) = π(x) = {x}. now let x be totally disconnected and let there exist x, y ∈ x so that dι(x, y) 6= d(x, y) where ι : x → x is the identity map. then there exist points p0, ..., pn, q0, ..., qn ∈ x so that qi ! pi+1 for all i = 0, 1, ..., k − 1, p0 = x, qk = y and∑k i=0 d(pi, qi) < d(x, y). hence there is at least one index i such that qi 6= pi+1. so if α is a connected component contains qi, as the same as the first part we deduce π(qi) = α. hence qi, pi+1 ∈ α which contradicts the totally disconnectedness of x. ✷ let x be a compact metric space. topological dimension of the space x is said to be less than n if for all ǫ > 0 there exists a cover α of x by open sets with diameter less than ǫ such that each point belongs to at most n + 1 sets of α. we know that x is 0-dimensional if and only if it is totally disconnected [1]. corollary 2.7. let ι : x → x be the identity map. then dι(x, y) = d(x, y) for all x, y ∈ x if and only if x has dimension zero. proof of theorem 1.4. clearly 2 is equivalent to 3. suppose that x is connected and x ∈ x. it is enough to show that for each ǫ > 0 the set πǫ = {y ∈ x : ǫ x ! y} is both open and closed. if y ∈ πǫ(x) then bǫ(y) ⊆ πǫ(x). so πǫ(x) is open. if {yn} is a sequence with yn → y then y ∈ bǫ(yn) for some n ∈ n. hence y ∈ πǫ(x). if x is not connected then there is a nonempty proper subset a of x such that it is both open and closed. therefore a and ac are disjoint nonempty compact subsets of x. so ǫ = d(a, ac)/2 > 0. by assumption there is an ǫ-pseudo orbit x0, x1, ..., xn so that x0 = x and xn = y. thus there is an index 0 ≤ i ≤ n − 1 such that xi ∈ a and xi+1 ∈ a c, which is a contradiction. ✷ proposition 2.8. if f : (cr(f), df ) → (cr(f), df ) has the potp with respect to df , then (cr(f), df ) is complete. proof. given ǫ > 0, by assumption there exists δ > 0 so that any δ-pseudoorbit in cr(f) can be ǫ-shadowed with a point in cr(f). let {xn} be a cauchy sequence. so there is n ∈ n such that df (f(xn), xn+1) = df (xn, xn+1) < δ for n ≥ n. thus there exists x ∈ cr(f) so that df (xn, x) = df (xn, f n(x)) < ǫ for n ≥ n. � let f : x → x be a continuous surjection. then if xf = lim ← (x, f) = {(xi) : xi ∈ x and f(xi+1) = xi, i ≥ 0} and d̄((xi), (yi)) = ∞∑ i=0 d(xi, yi) 2i c© agt, upv, 2014 appl. gen. topol. 15, no. 2 171 s. a. ahmadi then (xf , d̄) is a metric space called inverse limit space. the homeomorphism σ : xf → xf with σ((xi) ∞ i=0) = (f(xi)) ∞ i=0 is called the shift map. we know that cr(σ) = lim ← (cr(f), f) [1]. proposition 2.9. let f be a continuous surjection on a compact metric space x to itself and cr(σ) be the chain recurrent set for the shift map σ : xf → xf . then we deduce 2df(x0, y0) ≤ d̄σ((xi), (yi)) proof. suppose that (xi), (yi) ∈ cr(σ) and (p j i ), (q j i ) ∈ cr(σ), j = 0, 1, ..., k so that (p0i ) = (xi), (q k i ) = (yi) and (q j i ) ! (p j+1 i ) for j = 0, 1, ..., k − 1. we show that for each m ≥ 0 and j = 0, 1, ...k − 1, qjm ! p j+1 m . fixed m, j ≥ 0, for given ǫ > 0 oǫ/2m(σ, (q j i ), (p j+1 i )) 6= ∅. let {(r 0 i ), (r 1 i ), ..., (r n i )} ∈ oǫ/2m(σ, (q j i ), (p j+1 i )). then d(f(rlm), r l+1 m ) 2m = d(rlm−1, r l+1 m ) 2m ≤ d̄(σ(rli), (r l+1 i )) < ǫ/2 m for l = 0, 1, ..., n − 1. thus {r0m, r 1 m, ..., r n m} ∈ oǫ(f, q j m, p j+1 m ) so q j m ǫ ! pj+1m . since ǫ > 0 is arbitrary then qjm ! p j+1 m . hence df (xi, yi) ≤ ∑k j=0 d(p j i , q j i ). therefore ∞∑ i=0 df (xi, yi) 2i ≤ k∑ j=0 ∞∑ i=0 d(p j i , q j i ) 2i = k∑ j=0 d̄((p j i ), (q j i )). so ∞∑ i=0 df (xi, yi) 2i ≤ d̄σ((xi), (yi)). corollary 2.5 implies 2df (x0, y0) = ∞∑ i=0 df (xi, yi) 2i ≤ d̄σ((xi), (yi)). � corollary 2.10. let x be a compact metric space and f be a chain recurrent continuous surjection from x to itself. then if the shift map σ : xf → xf is chain transitive then f : x → x is so. theorem 2.11. the topology τ̃f coincide with quotient topology on c̃r(f) proof. every continuous bijection from a compact topological space to a hausdorff space is a homeomorphism. since x is hausdorff, any two elements π(x), π(y) ∈ c̃r(f) as compact subsets posses disjoint saturated neighborhood, so c̃r(f) is a hausdorff space with the quotient topology. also (c̃r(f), τ̃f ) is compact. thus the identity map is a homeomorphism. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 172 on the topology of the chain recurrent set of a dynamical system recall that the hausdorff metric on the compact subsets a, b of x is defined as follows dh(a, b) = max{max a∈a d(a, b), max b∈b d(a, b)}. let τh be the topology induced by hausdorff metric dh on c̃r(f). then we deduce the following proposition. proposition 2.12. the topology τh is finer than τ̃f . proof. suppose that π(x), π(y) ∈ c̃r(f). we can choose y′ ∈ π(y) so that d(x, y′) = d(x, π(y)). thus d̃f (π(x), π(y)) = df (x, y ′) ≤ d(x, y′) ≤ dh(π(x), π(y)). therefore τ̃f ⊂ τh. � the next example shows that d̃f and dh are not equal in general. example 2.13. let f : [0, 1] → [0, 1] be a strictly increasing continuous map so that • f(x) = x for each x ∈ [2−(2i+1), 2−(2i)], i = 0, 1, ...; • f(x) > x for each x ∈ [2−(2i+2), 2−(2i+1)], i = 0, 1, ...; • f(0) = 0. then we deduce cr(f) = {0} ∪ ∞⋃ i=0 [2−(2i+1), 2−2i] and c̃r(f) = {[2−(2i+1), 2−2i]; i = 0, 1, ...} ∪ {0}. if x ∈ [2−(2i+1), 2−2i] and y ∈ [2−(2j+1), 2−2j] for somej < i, then df (x, y) = 2 −(2j+1) − i∑ k=j+1 2−(2k+1). thus we deduce d̃f (π(0), π(1)) = 1/3, but dh(π(0), π(1)) = 1. proposition 2.14. let (x, d) and (y, d′) be two compact metric spaces and f : x → x and g : y → y be continuous maps. then if f and g are topologically conjugate then (c̃r(f), d̃f ) and (c̃r(g), d̃g) are isometric. proof. suppose that h : x → y is a homeomorphism so that h ◦ f = g ◦ h. given ǫ > 0 there is δ > 0 so that for every x, y ∈ x, the inequality d(x, y) < δ implies to d′(h(x), h(y)) < ǫ and the inequality d′(x, y) < δ implies to d(h−1(x), h−1(y)) < ǫ. if {xi} n i=0 ∈ oδ(f, p, q) for some p, q ∈ x, then {h(xi)} n i=0 ∈ oǫ(g, h(p), h(q)). this implies that if p ! q then h(p) ! h(q). hence for every x, y ∈ cr(f) we deduce df (x, y) ≥ d ′ g(h(x), h(y)). if c© agt, upv, 2014 appl. gen. topol. 15, no. 2 173 s. a. ahmadi {xi} n i=0 ∈ oδ(g, h(p), h(q)), then {h −1(xi)} n i=0 ∈ oδ(f, p, q). thus df (x, y) ≤ d′g(h(x), h(y)). so d̃f (π(x), π(y)) = df (x, y) = d ′ g(h(x), h(y)) = d̃′g(π(h(x)), π(h(y))) = d̃ ′ g(h̃(π(x), h̃(π(y)). therefore h̃ : c̃r(f) → c̃r(g) is an isometry. � 3. conclusions in this paper we introduce a pseudo-metric df associated to the dynamical system f. we show that the topology induced by df has a significant relation to some dynamical properties of f, such as transitivity and shadowing. by considering the identity map we obtained some equivalence conditions for connectedness of space. investigate the relation between this topology and topological entropy of f will be a topic for future research. acknowledgements. the author is grateful to prof. m.r. molaei for his instructive advice. references [1] n. aoki and k. hiraide, topological theory of dynamical systems, recent advances. north-holland math. library 52. (north-holland, amsterdam 1994) [2] k. athanassopoulos, one-dimensional chain recurrent sets of flows in the 2-sphere, math. z. 223(1996), 643–649. [3] f. balibrea, j. s. cánovas and a. linero, new results on topological dynamics of antitriangular maps, appl. gen. topol. 2 (2001), 51–61. [4] c. fujita and h. kato, almost periodic points and minimal sets in topological spaces, appl. gen. topol. 10 (2009), 239–244. [5] d. richeson and j. wiseman, chain recurrence rates and topological entropy, topology appl. 156 (2008), 251–261. [6] k. sakai, c1-stably shadowable chain components, ergodic theory dyn. syst. 28 (2008), 987–1029. [7] t. shimomura, on a structure of discrete dynamical systems from the view point of chain components and some applications, japan. j. math. (ns) 15 (1989), 99–126. [8] x. wen, s. gan and l. wen, c1-stably shadowable chain components are hyperbolic, j. differ. equations 246 (2009), 340–357. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 174 () @ appl. gen. topol. 17, no. 2(2016), 129-137doi:10.4995/agt.2016.4593 c© agt, upv, 2016 homeomorphisms on compact metric spaces with finite derived length v. kannan a and sharan gopal b a school of mathematics and statistics, university of hyderabad, hyderabad, india. (vksm@uohyd.ernet.in) b department of mathematics, bits-pilani, hyderabad campus, hyderabad, india. (sharanraghu@gmail.com) abstract the sets of periodic points of self homeomorphisms on an ordinal of finite derived length are characterised, thus characterising the same for homeomorphisms on compact metric spaces with finite derived length. a partition of ordinal is introduced to study this problem which is also used to solve two more problems: one about an equivalence relation and the other about a group action, both on an ordinal of finite derived length. 2010 msc: 06a99; 54h20. keywords: ordinal; homeomorphism; periodic point. 1. introduction periodicity is one of the important properties that are well studied in dynamical systems. the study of sets of periodic points for a class of dynamical systems has been an interesting one in the literature (see [1], [3]). some recent results characterise these sets for toral automorphisms and solenoids (see [12], [11]). in this paper, we do this for all self-homeomorphisms on compact metric spaces with finite derived length. it can be proved that a compact metric space with finite derived length is countable and a countable compact metric space is homeomorphic to a countable ordinal (see [6]). so the sets of periodic points for self homeomorphisms on an ordinal with finite derived length are characterised here. received 27 january 2016 – accepted 12 july 2016 http://dx.doi.org/10.4995/agt.2016.4593 v. kannan and s. gopal another main result is on the group actions. given a topological space x, the study of action of group of homeomorphisms on it is well studied in the literature (see [5]). here we consider the actions of a particular kind of subgroups of this group of homeomorphisms on metric spaces. it is proved that the separable metric spaces with finite derived length have finitely many orbits under the action of all these subgroups. an account of some preliminaries and notations that will be used in the paper is given in this and the next paragraph. by a dynamical system (x,f), we mean a topological space x together with a continuous self map f on x. the set {fl(x) : l ∈ n0} is called the orbit of x ∈ x. a subset a ⊂ x is said to be forward f−invariant if f(a) ⊂ a and backward f−invariant if f−1(a) ⊂ a. a is said to be f−invariant (or just invariant, when the context is clear) if a is both forward f−invariant and backward f−invariant. if a and b are two ordered sets, then we define a relation, saying that a ∼ b if there is an order preserving bijection from a to b. this is an equivalence relation and each equivalence class (in fact a representative of this class) is called an ordinal number. hereafter, by an ordinal number, we actually refer to a representative of the corresponding class. this makes it easy to convey the idea of order among the ordinals. if α and β are two ordinal numbers such that there is an order preserving bijection from α to a subset of β, then define α ≤ β. this relation “≤” is an order on the set of ordinal numbers according to which, this is well ordered. note that any non-negative integer can be regarded as an ordinal and such ordinals are called as finite ordinals and the least infinite ordinal is denoted by ω (which is equivalent to the set of all non-negative integers). on an ordered set x, with more than one element, let b be the collection of all sets of the following types: (i) all open intervals (a,b) in x, (ii) all intervals of the form [a0,b), where a0 is the smallest element (if any) of x and (iii) all intervals of the form (a,b0], where b0 is the largest element (if any) of x. the collection b is a basis for a topology on x, which is called the order topology. in this paper, we consider the order topology on ordinal spaces. the set of limit points of an ordinal α is precisely the set of limit ordinals less than α. zero and the successor ordinals less than α are isolated points in α. it can be seen that any ordinal of finite derived length can be written as ωn · mn + ω n−1 · mn−1 + ... + ω · m1 + m0, mi ∈ n0. hereafter, we use the notation o(n) = ωn ·mn +ω n−1 ·mn−1 + ...+ω ·m1 +m0. the notions of limit ordinal and arithmetic of ordinals can be seen in [10]. we also use the notation |w | to denote the cardinality of a set w . 2. partition of o(n) here, a partition is induced on the ordinal o(n). the definition of this partition involves only elementary operations: intersection, complementation and formation of derived set. this partition is done with respect to a given subset of o(n) i.e., every subset s ⊂ o(n) gives rise to a partition ps of o (n). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 130 homeomorphisms on compact metric spaces this is finer than the partition of o(n) in to the different levels of limit points. to define the partition ps, the following notations are introduced. 1. let x be a topological space. for a subset z ⊂ x, d(z) denotes the set of limit points of z in x and for every n ∈ n, define inductively dn+1(z) = d(dn(z)). then, for k ∈ n, denote by xk, the set d k(x) \ dk+1(x) and let x0 denote the set of isolated points of x. xk is called the set of k th level limit points of x in x for every k ∈ n0. 2. define a sequence j (l), l ∈ n0 inductively as follows : j (0) = {a,b}, j (l+1) = (p(j (l)) \ {∅}) × j (0) (p(x) stands for the set of all subsets of x). remark 2.1. (1) here, a and b can be any two distinct symbols. what is actually needed to define the partition is a set containing two elements, which we denote by j (0). (2) the letters p and p are used in three different contexts. ps denotes the partition associated with the set s, p(x) stands for the collection of all subsets of x and p(h) will be used to denote the set of periodic points of h. the partition ps: let s ⊂ o (n), n ∈ n. for every v ∈ ⋃n l=0 j (l), we associate a subset sv of o (n) in the following way. note that these sv ’s are defined recursively. define sa = (o (n) \ s) ∩ (o(n))0 and sb = s ∩ (o (n))0. if v ∈ j (k+1) for some 0 ≤ k ≤ n − 1 and say v = (w,i) (i ∈ {a,b}) for some w ⊂ j (k), then define sv = [( ⋂ a∈w sa) \ ( ⋃ a∈j (k)\w sa)] ∩ s i ∩ (o(n))k+1, where si = { o(n) \ s, i = a s, i = b . remark 2.2. in the above partition, o(n) is partitioned in to (o(n))k’s and each (o(n))k is partitioned into s ∩ (o (n))k and (o (n) \ s) ∩ (o(n))k. this partition is further refined in such a way that the common limit points of some and only those partition classes in (o(n))k−1 constitute the partition classes of (o (n))k; these are denoted by sv , v ∈ ⋃n l=0 j (l). in other words, for every partition class sv ⊂ (o (n))k, there exist partition classes sv1, sv2, . . . svm (m depends on v ) in (o(n))k−1 such that sv is precisely the set ∩ m i=1d(svi). thus every subset s ⊂ o(n) gives rise to a partition ps of o (n). this is finer than the partition of o(n) in to the different levels of limit points. ps helps to prove the following four different results. • a subset s ⊂ o(n) arises as the set of periodic points of a homeomorphism on o(n) if and only if every finite partition class in ps is contained in s. • using this partition ps, it is proved that in a separable metric space x with finite derived length, there are finitely many orbits under the c© agt, upv, 2016 appl. gen. topol. 17, no. 2 131 v. kannan and s. gopal action of the group gs = {h : x → x : h is a homeomorphism such that h(s) = s}, for any s ⊂ x. • ps is the smallest partition of ω n such that (i)s is a union of partition classes and (ii)if a ⊂ ωn is a union of partition classes, then d(a) is also a union of partition classes. ps is the smallest partition in the sense that any finer refinement of ps will not satisfy the above conditions (i) and (ii). • starting from a subset s ⊂ ωn, go on forming many other sets using the operation of derived set, complement and union i.e, s, ωn\s, s, ωn \ s and so on. in other words, if n sets are already formed, the (n + 1)th set can be the derived set of any of the n sets, or the complement of any of the n sets, or the union of any two of these n sets. among the distinct subsets that can be formed in this way, the minimal ones are precisely the partition classes in ps i.e., in the collection of sets formed as above, there is no set which is strictly contained in any of the partition classes of the partition ps. the first two results form the main part of this paper and the last two follow from these. the last result is similar to, and is a natural sequel to the closurecomplementation problem described in [7] and further investigated in [4]. 3. sets of periodic points 3.1. invariance of partition classes. proposition 3.1. if x is a topological space with finite derived length n and h is a homeomorphism on x, then xk is h−invariant for every 0 ≤ k ≤ n. proof. since, being an isolated point is a topological property, x0 is h−invariant for any homeomorphism h on x. for any 1 ≤ m < n, by the definition of xm+1, there is no sequence in ∪ n i=m+1xi that converges to a point in xm+1; because, the limit of such a sequence would be a limit point of a level larger than m + 1. thus the points of xm+1 consists of the isolated points of x \ ∪ m i=0xi i.e., xm+1 = (x \ ∪ m i=0xi)0. in particular, x1 = (x \ x0)0. since x0 is h−invariant, h is a self homeomorphism on x \ x0. thus by the above argument, it follows that x1 is h−invariant. now let 1 ≤ k < n. suppose that xl is h−invariant for every 0 ≤ l ≤ k. then, h is a self homeomorphism on x \ ⋃k i=0 xi and since xk+1 = (x \ ∪ k i=0xi)0, xk+1 is h−invariant. hence xk is h−invariant for every 0 ≤ k ≤ n. � theorem 3.2. if s ⊂ o(n) is h−invariant for some homeomorphism h on o(n), then sv is h−invariant ∀v ∈ ⋃n k=0 j (k). proof. from the above proposition, it is clear that sa and sb are h−invariant. we now prove that if sw is h−invariant ∀w ∈ j (k) for some 0 ≤ k ≤ n − 1, then sv is h−invariant ∀v ∈ j (k+1). consider a partition class sv , where v ∈ j (k+1). then v = (w,i) for some w ⊂ j (k) and i ∈ {a,b}. by definition, sv = ( ⋂ a∈w sa \ ⋃ a∈j (k)\w sa) ∩ c© agt, upv, 2016 appl. gen. topol. 17, no. 2 132 homeomorphisms on compact metric spaces si ∩ (o(n))k+1. since h is a homeomorphism, observe that for any a ∈ j (k), h(sa) = h(sa) = sa. then h( ⋂ a∈w sa) = ⋂ a∈w sa and h( ⋃ a∈j (k)\w sa) = ⋃ a∈j (k)\w sa. hence sv is h−invariant. � 3.2. periodic points. lemma 3.3. let n ∈ n and s ⊂ ωn+1. if 1 ≤ k ≤ n and (ωn+1 \ s) ∩ sa is either empty or infinite for every a ∈ j (k−1) ∪ j (k) then any bijection f on (ωn+1)k such that (1) f has no orbit of even length (2) sv is f−invariant ∀v ∈ j (k) and (3) p(f) = s ∩ (ωn+1)k can be extended to a homeomorphism h on (ωn+1)k−1 ∪ (ω n+1)k such that p(h) = s ∩ ((ωn+1)k−1 ∪ (ω n+1)k). proof. f being a bijection on (ωn+1)k with no orbits of even length, defines three types of orbits in (ωn+1)k : (1) an infinite orbit ({xl : l ∈ z and f(xi) = xi+1}), (2) a periodic orbit with odd length greater than 1 ({xl : −m ≤ l ≤ m for some fixed m ∈ n; f(xi) = xi+1 ∀i < m and f(xm) = x−m}) and (3) a singleton orbit ({x : f(x) = x}) i.e., a fixed point. since p(f) = s ∩ (ωn+1)k, (ω n+1)k \ s is a union of infinite orbits and s ∩ (ωn+1)k is a union of finite orbits. we now define a homeomorphism h on (ωn+1)k−1 ∪ (ω n+1)k with p(h) = s ∩ ((ωn+1)k−1 ∪ (ω n+1)k) in such a way that h = f on (ω n+1)k and moreover the orbit of each point in (ωn+1)k−1 falls in to one of the three types. to cover the general case, we assume that each point in (ωn+1)k is in both s as well as (ωn+1)k−1 \ s; and the same method works for other cases also. we first define maps on some clopen neighbourhoods of points of (ωn+1)k. this is done in one of the three different ways depending on the type of orbit in which the point lies. type i : {xi : i ∈ z and f(xi) = xi+1} here, xi ∈ sv ∀i ∈ z for some v ∈ j (k). let bi and b ′ i be two deleted clopen neighborhoods of xi (i.e., xi /∈ bi ∪ b ′ i and bi ∪ {xi}, b ′ i ∪ {xi} are clopen neighborhoods of xi) such that bi ⊂ s and b ′ i ⊂ (ω n+1 \ s). choose bi’s and b′i’s in such a way that bi ∩ sw and b ′ i ∩ sw are either empty or infinite ∀w ∈ j (k−1). since all the xi’s are in the same sv , we can assume that for any w ∈ j (k−1), |bi ∩ sw | = |bj ∩sw | and |b ′ i ∩ sw | = |b ′ j ∩ sw |, ∀i,j ∈ z. say bi ∩ (ω n+1)k−1 = {xij : j ∈ n} and b ′ i ∩ (ω n+1)k−1 = {x ′ ij : j ∈ n}. define a bijection hi on ( ⋃ i∈z(bi ∪ b ′ i)) ∩ (ω n+1)k−1 as hi(xij) =    x(i+1)j if − j ≤ i < j x(−j)j if i = j xij if |i| > j and hi(x ′ ij) = x ′ (i+1)j . extend hi to ( ⋃ i∈z bi) ⋃ ( ⋃ i∈z b ′ i) ∪ {xi : i ∈ z} by defining hi(xi) = f(xi). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 133 v. kannan and s. gopal type ii : {xi : −m ≤ i ≤ m for some fixed m ∈ n and f(xi) = xi+1 ∀i < m and f(xm) = x−m} let bi and b ′ i be two deleted clopen neighborhoods of xi such that bi ⊂ s and b′i ⊂ (ω n+1 \s). choose bi’s and b ′ i’s in such a way that bi∩sw and b ′ i∩sw are either empty or infinite ∀w ∈ j (k−1). here again, xi ∈ sv ∀ − m ≤ i ≤ m for some v ∈ j (k). also, |bi ∩ sw | = |bj ∩ sw | and |b ′ i ∩ sw | = |b ′ j ∩ sw |, ∀−m ≤ i,j ≤ m and for any w ∈ j (k−1). say bi ∩(ω n+1)k−1 = {xij : j ∈ n} and b′i ∩ (ω n+1)k−1 = {x ′ ij : j ∈ n}. define a bijection hii on [ ⋃ −m≤i≤m(bi ∪b ′ i)]∩(ω n+1)k−1 as hii(xij) = xψ(i)j and hii(x ′ ij) = x ′ ψ(i)φ(j) , where ψ is a bijection on {−m,−m + 1, ..,0,1, ...,m} defined as ψ(i) = { i + 1 ∀i < m −m if i = m and φ : n → n is defined as φ(2i − 1) = 2i + 1 ∀i ∈ n, φ(2i) = 2i − 2 ∀i ≥ 2 and φ(2) = 1. extend hii to ( ⋃ −m≤i≤m bi) ⋃ ( ⋃ −m≤i≤m b ′ i)∪{xi : −m ≤ i ≤ m} by defining hii(xi) = f(xi). type iii : {x : f(x) = x} (i.e., x is a fixed point). let b and b′ be two deleted clopen neighborhoods of x such that b ⊂ s and b′ ⊂ (ωn+1 \ s). say b = {xi : i ∈ n} and b ′ = {x′i : i ∈ n}. define hiii on (b ∪b′)∩(ωn+1)k−1 as hiii(xi) = xi and hiii(x ′ i) = φ(x ′ i), where φ is defined as above. extend hiii to b ∪ b ′ ∪ {x} by defining hiii(x) = x. the neighborhoods considered above will form a partition of a clopen set, say x ⊂ (ωn+1)k−1 and we can assume that for every w ∈ j (k−1), sw ⊂ x or sw ⊂ (ω n+1)k−1 \ x. thus (ω n+1)k−1 \ x is a union of su’s for some u ∈ j (k−1) such that d(su) = ∅. thus any bijection on (ω n+1)k−1 \ x will be a homeomorphism on it. so, we define a bijection on (ωn+1)k−1 \ x so that su is a single infinite orbit (type i) if su ⊂ (ω n \ s) or otherwise each point of su is fixed. since the neighborhoods defined above for the three types form a partition of x,, the domains of hi, hii and hiii are mutually disjoint and further each of them is a clopen set. hence hi, hii and hiii can be pasted to get a homeomorphism on x and by pasting this homeomorphism and the bijection on (ωn+1)k−1 \x, we get a homeomorphism h on (ω n+1)k∪(ω n+1)k−1 such that p(h) = s ∩ ((ωn+1)k−1 ∪ (ω n+1)k). � remark 3.4. similar to the bijection f, the extended homeomorphism h satisfies the following : (1) h has no orbit of even length (2) sv is h−invariant ∀v ∈ j (k) ∪ j (k−1) and (3) p(h) = s ∩ ((ωn+1)k−1 ∪ (ω n+1)k). theorem 3.5. let s ⊂ ωn for some n ∈ n. the following are equivalent: (1) s = p(h) for some self-homeomorphism h on ωn. (2) sv ∩ (ω n \ s) is either empty or infinite for every v ∈ ⋃n−1 k=0 j (k). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 134 homeomorphisms on compact metric spaces proof. if s = p(h), then s is h−invariant and thus by theorem 3.2, sv is h−invariant for every v . also, either sv ⊂ s or sv ⊂ (ω n \ s). since a finite non-empty invariant set certainly contains a periodic point, it follows that sv ∩ (ω n \ s) is either empty or infinite for every v . for the converse, let s ⊂ ωn such that sv ∩ (ω n \ s) is either empty or infinite for every v ∈ ⋃n−1 k=0 j (k). suppose v1, v2, ...,vm ∈ j (n−1) such that (ωn)n−1\s = ⋃m j=1 svj and svj 6= ∅ ∀j ∈ {1,2, ...,m}. say svj = {xjl : l ∈ z}. define f : (ωn)n−1 → (ω n)n−1 as f(x) = { x if x ∈ s xj(l+1) if x = xjl ∈ (ω n)n−1 \ s . if m does not exist i.e., if there is no v ∈ j (n−1) such that sv ⊂ (ω n \ s), then define f(x) = x ∀x ∈ (ωn)n−1. then f is a bijection on (ω n)n−1. if n = 1, then f is a homeomorphism on ω such that p(f) = s. otherwise, using the above lemma, this f can be extended to a homeomorphism hn−2 on (ωn)n−1 ∪ (ω n)n−2 such that p(hn−2) = s ∩ ((ω n)n−1 ∪ (ω n)n−2). hn−2 defines a bijection on (ω n)n−2 which can be further extended to a homeomorphism hn−3 on (ω n)n−2 ∪ (ω n)n−3 such that p(h) = s ∩ ((ω n)n−2 ∪ (ωn)n−3). by pasting hn−2 and hn−3, we get a homeomorphism on (ω n)n−1 ∪ (ωn)n−2 ∪(ω n)n−3 with s∩((ω n)n−1 ∪(ω n)n−2 ∪(ω n)n−3) as the set of periodic points. continuing this way, we get a homeomorphism h on ωn with p(h) = s. � now, we have the main theorem: theorem 3.6. the following are equivalent for a subset s ⊂ o(n): (1) s = p(h) for some self-homeomorphism h on o(n). (2) sv ∩ (o (n) \ s) is either empty or infinite for every v ∈ ⋃n k=0 j (k). proof. the first part of the proof is same as that in the above proof. for the converse, recall that o(n) = ωn · mn + ω n−1 · mn−1 + ... + ω · m1 + m0 where mi ∈ n0. here, the highest level of limit points is n and since (o (n))n is a finite set, (o(n))n ⊂ s. the rest of the proof follows by taking f to be the identity map on (o(n))n in lemma 3.3 and also using the ideas of theorem 3.5. � 4. an equivalence class and a group action definition 4.1. let z be a topological space and s ⊂ z. let x, y ∈ z. x is said to be topologically same as y in z with respect to s if there exists a homeomorphism h on z such that h(s) = s and h(x) = y. given s ⊂ z, this definition induces an equivalence relation rs on z as : x rs y if x is topologically same as y with respect to s. it is easy to see that this is an equivalence relation on z and the following theorem describes the equivalence classes of o(n). theorem 4.2. the family {sv : v ∈ ⋃n l=0 j (l), sv 6= ∅} is the set of equivalence classes of o(n) with respect to rs. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 135 v. kannan and s. gopal proof. from the proof of theorem 3.2, it follows that sv is invariant for every v ∈ ⋃n l=0 j (l), under any homeomorphism h on o(n) such that h(s) = s. so if v 6= v ′, then x ∈ sv and y ∈ sv ′ are not related. now, let x,y ∈ sv for v ∈ j (k) for some k ∈ {0,1, ...,n}. define a bijection h on (on)k such that h(z) =    z if z /∈ {x,y} y if z = x x if z = y . using the ideas of the proof of lemma 3.3, this map can be extended to a homeomorphism h on ⋃k i=0(o (n))i such that h(s ∩ ( ⋃k i=0(o (n))i)) = s ∩ ( ⋃k i=0(o (n))i), which can be further extended to o(n) by defining h(z) = z, ∀z ∈ o(n) \ ⋃k i=0(o (n))i. hence the family {sv : v ∈ ⋃n k=0 j (k), sv 6= ∅} is the set of equivalence classes. � if a group g is acting on a set x and x ∈ x, then the set gx = {gx : g ∈ g} is called the g−orbit of x. when a finite group acts on a finite set, one natural problem is to count the number of orbits. burnside’s lemma (see [2]) and polya’s theorem (see [9]) are in this direction. occasionally, there are some infinite groups acting on infinite sets, but having only finitely many orbits. the problem of the present section has such background. to every topological space x, we can associate a natural group, namely the group of self homeomorphisms on it. here, we consider a subgroup of this group. given a subset s of a topological space x, let gs = {f : x → x : f is a homeomorphism on x such that f(s) = s} , i.e., the collection of selfhomeomorphisms on x under which s is invariant. it can be easily seen that gs is a group. the following theorem gives a neat description of the gs−orbits in o (n) in terms of sv ’s. its proof follows from theorem 4. theorem 4.3. the family {sv : v ∈ ⋃n k=0 j (k), sv 6= ∅} is the collection of gs−orbits in o (n) for any subset s ⊂ o(n). it follows from the above theorem that a compact metric space x with finite derived length has finitely many gs−orbits for any subset s ⊂ x. in fact, any separable metric space with finite derived length has this property. this is proved in the following theorem. theorem 4.4. if x is a separable metric space with finite derived length, then x has finitely many gs-orbits for every subset s ⊂ x. proof. let x be a separable metric space of finite derived length. if x is compact, then the result follows from theorem 4.3. now, suppose x is not compact. since x is separable, it is known that x is homeomorphic to a subspace of ωn for some n (see [8]). so, it is enough c© agt, upv, 2016 appl. gen. topol. 17, no. 2 136 homeomorphisms on compact metric spaces to consider the number of gs−orbits in x, where s ⊂ x ⊂ ω n. let h = {h : ωn → ωn : h(x) = x and h(s) = s}. then h is isomorphic to a subgroup of gs. thus the number of gs−orbits in x cannot exceed the number of h−orbits in ωn. it follows from theorem 4.3 and the proof of theorem 4.2, that the non-empty members of the family {s ∩ xv : v ∈ ⋃n k=0 j (k)} ∪ {(ωn \ s) ∩ xv : v ∈ ⋃n k=0 j (k)} are precisely the collection of h−orbits in ωn. so, the number of h−orbits is atmost twice the number of xv ’s, which are finite in number. thus there are finitely many h−orbits in ωn and hence finitely many gs−orbits in x. � 5. conclusion we conclude this article by posing a question. the problem of characterising the sets of periodic points of homeomorphisms on ωn leads to a very natural and interesting question of characterising the same sets for continuous maps on ωn. in another direction, the same problem can be extended to the problem of characterising the sets of periodic points of homeomorphisms on ordinals of infinite derived length. it is hoped that the ideas in this paper will be useful in discussing the later question. acknowledgements. we thank the referee for his suggestions. references [1] i. n. baker, fixpoints of polynomials and rational functions, j. london math. soc. 39 (1964), 615–622. [2] p. b. bhattacharya, s. k. jain and s. r. nagpaul, basic abstract algebra, second edition, cambridge university press, 1995. [3] j. p. delahaye, the set of periodic points, amer. math. monthly 88 (1981), 646–651. [4] b. j. gardener and m. jackson, the kuratowski closure-complementation theorem, new zealand journal of mathematics, 38 (2008), 9–44. [5] k. h. hofmann, introduction to topological groups, an introductory course (2005). [6] v. kannan, a note on countable compact spaces, publicationes mathematicae debrecen 21 (1974), 113–114. [7] j. l.kelley, general topology, graduate texts in mathematics-27, springer, 1975. [8] s. mazurkiewicz and w.sierpinski, contribution a la topologie des ensembles denombrables, fund. math 1 (1920), 17–27. [9] g. polya and r. c. read, combinatorial enumeration of groups, graphs, and chemical compounds, springer-verlag, new york, 1987. [10] s. m. srivastava, a course on borel sets, graduate texts in mathematics-180, springer, 1998. [11] s. gopal and c. r. e. raja, periodic points of solenoidal automorphisms, topology proceedings 50 (2017), 49–57. [12] i. subramania pillai, k. ali akbar, v. kannan and b. sankararao, sets of all periodic points of a toral automorphism, j. math. anal. appl. 366 (2010), 367–371. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 137 () @ appl. gen. topol. 19, no. 2 (2018), 261-268doi:10.4995/agt.2018.9058 c© agt, upv, 2018 on the essentiality and primeness of λ-super socle of c(x) s. mehran a, m. namdari b and s. soltanpour c a shoushtar branch, islamic azad university, shoushtar, iran. (s.mehran@iau-shoushtar.ac.ir) b department of mathematics, shahid chamran university of ahvaz, ahvaz, iran. (namdari@ipm.ir) c petroleum university of technology, iran. (s.soltanpour@put.ac.ir) communicated by o. okunev dedicated to professor o.a.s. karamzadeh on the occasion of his retirement and to appreciate his peerless activities in mathematics (especially, popularization of mathematics) for nearly half a century in iran abstract spaces x for which the annihilator of sλ(x), the λ-super socle of c(x) (i.e., the set of elements of c(x) that cardinality of their cozerosets are less than λ, where λ is a regular cardinal number such that λ ≤ |x|) is generated by an idempotent are characterized. this enables us to find a topological property equivalent to essentiality of sλ(x). it is proved that every prime ideal in c(x) containing sλ(x) is essential and it is an intersection of free prime ideals. primeness of sλ(x) is characterized via a fixed maximal ideal of c(x). 2010 msc: primary: 54c30; 54c40; 54c05; 54g12; secondary: 13c11; 16h20. keywords: λ-super socle of c(x); λ-isolated point; λ-disjoint spaces. 1. introduction unless otherwise mentioned all topological spaces are infinite tychonoff and we will employ the definitions and notations used in [11] and [7]. c(x) is the ring of all continuous real valued functions on x. the socle of c(x), denoted by cf (x), is the sum of all minimal ideals of c(x) which plays an important role in the structure theory of noncommutative noetherian rings, see [12], but received 07 december 2017 – accepted 19 june 2018 http://dx.doi.org/10.4995/agt.2018.9058 s. mehran, m. namdari and s. soltanpour o.a.s. karamzadeh initiated the research regarding the socle of c(x) (see [16]), which is the intersection of all essential ideals in c(x) (recall that, an ideal is essential if it intersects every nonzero ideal nontrivially), see[12] and [16]. also the minimal ideals and the socle of c(x) are characterized via their corresponding z-filters; see [16]. in [10] and [15], the socle of cc(x) (the functionally countable subalgebra of c(x)), and lc(x) (the locally functionally countable subalgebra of c(x)), are investigated. the concept of the super socle is introduced in [8], denoted by scf (x), which is the set of all elements f in c(x) such that coz(f) is countable. clearly, scf (x) is a z-ideal containing cf (x). recently, the concept of scf (x) has been generalized to the λ-super socle of c(x), sλ(x), where sλ(x) = {f ∈ c(x) : |x \ z(f)| < λ}, in which λ is a regular cardinal number with λ ≤ |x|, is introduced and studied in [17]. it is manifest that cf (x) = sℵ0(x) and scf (x) = sℵ1(x). it turns out, in this regard, the ideal cf (x) plays an important role in both concepts. as we know the prime ideals are very important in the context of c(x). it turns out that every prime ideal in c(x) is either an essential ideal or a maximal one, therefore the study of essential ideals in c(x) is worthwhile. it is easy to see that for any ideal i in any commutative ring r, the ideal i + ann(i), where ann(i) = {x ∈ x : xi = (0)} is the annihilator of i, is an essential ideal in r. hence an ideal i in a reduced ring is an essential ideal if and only if ann(i) = (0) (note: it suffices to recall that r is reduced if and only if z(r) = {x ∈ r : ann(x) is essential in r} = (0)). in [16, proposition 2.1], it is proved that cf (x) is an essential ideal in c(x) if and only if the set of all isolated points of x is dense in x. we note that in this case the socle is the smallest essential ideal in c(x). also the ideal scf (x) (the super socle of c(x)) is an essential ideal in c(x) if and only if the set of countably isolated points of x is dense in x, see [8, corollary 3.2]. similarly, in what follows, we aim to relate the density of the set of λ-isolated points to an algebraic property of c(x). in [3, proposition 2.5], it is shown that the socle of c(x), i.e., cf (x) is never a prime ideal in c(x), but in [8], it is seen that scf (x) can be a prime ideal (or even a maximal ideal) which this may be considered as an advantage of scf (x) over cf (x). in this article we will see that sλ(x) can be a prime ideal, as well. in section 2, some concepts and preliminary results which are used in the subsequent sections are given. in section 3, we deal with the essentiality of sλ(x) and also with the essential ideals containing sλ(x). in this section, we characterize spaces x for which the annihilator of sλ(x) is generated by an idempotent. consequently, this enables us to find an algebraic property equivalent to the density of the set of λ-isolated points in a space x. in contrast to the fact that cf (x) is never a prime ideal in c(x), in section 4, we characterize spaces x for which sλ(x) is a prime ideal (even maximal ideal). in the final section, for a class of topological spaces, including maximal λ-compact ones, we prove that the λ-super socle of c(x) is the intersection of the essential ideals ox containing sλ(x), where x runs through the set of c© agt, upv, 2018 appl. gen. topol. 19, no. 2 262 on the essentiality and primeness of λ-super socle of c(x) non-λ-isolated points in x. also we show that the z-filter corresponding to the λ-super socle of c(x) is the intersection of all essential z-filters containing sλ(x). 2. preliminaries first we cite the following results and definitions which are in [14] and [17]. definition 2.1. an element x ∈ x is called a λ-isolated point if x has a neighborhood with cardinality less than λ. the set of all λ-isolated points of x is denoted by iλ(x). if every point of x is λ-isolated, then x is called a λ-discrete space, i.e., iλ(x) = x. definition 2.2. a topological space x is said to be λ-compact whenever each open cover of x can be reduced to an open cover of x whose cardinality is less than λ, where λ is the least infinite cardinal number with this property. definition 2.3. x is a pλ-space if every intersection of a family of cardinality less than λ of open sets (i.e., gλ-set) is open. we begin with the following well-known result for sλ(x), see [17, lemma 2.6]. theorem 2.4. ⋂ z[sλ(x)] is equal to the set of non-λ-isolated points, i.e., ⋂ z[sλ(x)] = x \ iλ(x). in particular, if x ∈ x is a λ-isolated point, then there exists f ∈ sλ(x), such that f(x) = 1. corollary 2.5. for any space x the following statements hold. (1) an element x ∈ x is a λ-isolated point if and only if mx + sλ(x) = c(x). (2) x is a λ-discrete space if and only if for all x ∈ x, mx + sλ(x) = c(x). (3) the ideal sλ(x) is a free ideal in c(x) if and only if for all x ∈ x, mx + sλ(x) = c(x). (4) an element x ∈ x is non-λ-isolated point if and only if sλ(x) ⊆ mx. (5) if |x| ≥ λ and |iλ(x)| < λ, then sλ(x) = ⋂ x∈x\iλ(x) mx. 3. on the essentiality of sλ(x) in c(x) we begin with the following theorem, which is, in fact, our main result in this section. theorem 3.1. ann(sλ(x)) = (e), where e is an idempotent in c(x) if and only if x = a∪b, where a and b are two disjoint open subsets of x such that the set of λ-isolated points of x is a dense subset of a and b has no λ-isolated points of x . c© agt, upv, 2018 appl. gen. topol. 19, no. 2 263 s. mehran, m. namdari and s. soltanpour proof. let us first get rid of the case that ann(sλ(x)) = (1). clearly, this case holds if and only if sλ(x) = (0), or equivalently if and if x has no λ-isolated point, since 1.g = 0, for each g ∈ sλ(x), i.e., sλ(x) = (0). conversely, if sλ(x) = (0), then ann(sλ(x)) = c(x) = (1). so put x = a ∪ b, where a = φ and b = x, see theorem 2.4. now let ann(sλ(x)) = (e), where e is an idempotent in c(x) and h = iλ(x) be the set λ-isolated points of x. we claim cl(h) = z(e). in view to theorem 2.4, for each x ∈ h, there exists f ∈ sλ(x) such that f(x) = 1. but by assumption, ef = 0, implies e(x) = 0, i.e., h ⊆ z(e) and consequently cl(h) ⊆ z(e). now let x ∈ z(e) \ cl(h) and seek a contradiction. by complete regularity of x, there exists g ∈ c(x), such that g(x) = 1 and g(cl(h)) = (0). on the other hand for each y ∈ x \ h and every f ∈ sλ(x), we have f(y) = 0, see theorem 2.4, this implies that gf = 0, for every f ∈ sλ(x), which in turn implies g ∈ ann(sλ(x)) = (e). since x ∈ z(e) and g = he, g(x) = h(x).e(x) = 0, which is a contradiction. consequently, cl(h) = z(e) and so cl(h) is clopen. now put a = cl(h) and x \ cl(h) = b, thus we are done. conversely, let x = a ∪ b such that a and b are two disjoint open subsets of x, where a and b have the assumed properties. we may define e(x) = { 0 , x ∈ a 1 , x ∈ b it is clear e ∈ c(x) and e2 = e. we claim ann(sλ(x)) = (e). if f ∈ sλ(x) then |x \ z(f)| < λ and this implies x \ z(f) ⊆ a = z(e), i.e., fe = 0 or e ∈ ann(sλ(x)). it reminds to be shown that if f ∈ ann(sλ(x)), then f ∈ (e). first, we prove that if f ∈ ann(sλ(x)), then z(e) ⊆ z(f). to see this, put h = iλ(x), since for each x ∈ h, we infer that there exists g ∈ sλ(x) such that g(x) = 1. hence (fg)(x) = 0 implies that f(x) = 0, for every x ∈ h. so f(cl(h)) = 0 (note, f(cl(h)) ⊆ clf(h) ). so cl(h) = a = z(e) ⊆ z(f), and since z(e) is clopen, z(e) ⊆ int z(f) and by [11, problem 1d], f is a multiple of e, thus f ∈ (e) and we are done. � as previously mentioned, the set of isolated points in a space x is dense if and only if the socle of c(x) is essential. similarly, in [8, corollary 3.2], it has shown that the ideal scf (x) is an essential ideal if and only if the set of countably isolated points of x is dense in x. but in the following corollary, we generalize this result for λ-super socle. corollary 3.2. the ideal sλ(x) is an essential ideal in c(x) if and only if the set of λ-isolated points of x is dense in x. proof. let sλ(x) be essential ideal, as the previous result ann(sλ(x)) = (0), see[1, proposition 3.1]. therefore by the comment preceding theorem 3.1, e = 0 and a = z(e) = x, i.e., iλ(x) is dense in x. conversely, let cl(iλ(x)) = x, since int( ⋂ z[sλ(x)]) = int((iλ(x)) c) = (cl(iλ(x)) c = φ, we infer that sλ(x) is essential in c(x), see[1, proposition 3.1]. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 264 on the essentiality and primeness of λ-super socle of c(x) clearly, every essential ideal in any commutative ring r contains the socle of r. now the following definition is in order. definition 3.3. an essential ideal in c(x) containing sλ(x) is called a λessential ideal where λ is a cardinal number greater than or equal ℵ0. it is well known that the intersection of the essential ideals in a commutative ring r is equal to the socle of r. more generally, any ideal containing the socle of r is also an intersection of essential ideals, see [13, 3n]. it is obvious that sλ(x) is the intersection of the λ-essential ideals of c(x). proposition 3.4. let x be a λ-discrete space, then the set of λ-essential ideals and the set of free ideals containing sλ(x) coincide. in particular, sλ(x) is the intersection of free ideals containing it. proof. let x be a λ-discrete space and e be a free ideal containing sλ(x), it is well known that every free ideal in c(x) is an essential ideal, see [2, proposition 2.1] and the comment preceding it, hence e is a λ-essential ideal which implies that the set of λ-essential ideals and the set of free ideals containing sλ(x) coincide. � it is clear that every maximal ideal containing the socle of any commutative ring is essential, see [16]. so each maximal ideal m containing sλ(x) is λessential, since cf (x) ⊆ sλ(x). we also recall that every prime ideal in c(x) is either essential or it is a maximal ideal which is generated by idempotent and it is a minimal prime too, see [4]. in view of these facts and using the above proposition and the fact that sλ(x) is a z-ideal (hence it is an intersection of prime ideals), we immediately have the following proposition. proposition 3.5. every prime ideal p in c(x) containing sλ(x) (or even cf (x)) is an essential ideal. in particular if x is a λ-discrete space, then sλ(x) is an intersection of free prime ideals. 4. on the primeness of sλ(x) in c(x) our main aim in this section is to investigate the primeness of the λ-super socle. first, we give an example to show that sλ(x) can be a prime ideal (even a maximal ideal), which is the difference between sλ(x) and cf (x). example 4.1. let x = y ∪ {x} be one point λ-compactification of a discrete space y , see [17, definition 2.11]. we claim that c(x) = r + sλ(x), i.e., sλ(x) is a real maximal ideal. let f ∈ c(x), then we consider two cases. let us first take x ∈ z(f), since x is a pλ-space, z(f) is open and so |x\z(f)| < λ implies f ∈ sλ(x) ⊆ r + sλ(x). now, we suppose x /∈ z(f), so there exists 0 6= r ∈ r such that f(x) = r. put g = f − r, hence x ∈ z(g) and therefor g ∈ sλ(x). we are done. using corollary 2.5, it is evident that if x ∈ x is the only non-λ-isolated point of x, then mx is the unique fixed maximal ideal in c(x) such that sλ(x) ⊆ mx. it is well-known that every prime ideal in c(x) is contained in c© agt, upv, 2018 appl. gen. topol. 19, no. 2 265 s. mehran, m. namdari and s. soltanpour a unique maximal ideal, see [11, theorem 2.11]. now let sλ(x) be a prime ideal in c(x), then sλ(x) is contained in the unique maximal ideal mx, such that x is the only non-λ-isolated point. so the space x has only one non-λisolated point. consequently, if x has more than one non-λ-isolated point then sλ(x) can not be a prime ideal in c(x), see 2.5. now we have the following results. proposition 4.2. if x is a topological space with more than one non-λ-isolated point in x, i.e., |x \ iλ(x)| > 1, then sλ(x) is not a prime ideal in c(x). theorem 4.3. let x be a pλ-space, then the following statements are equivalent. (1) sλ(x) = mx, for som x ∈ x. (2) x is a λ-compact space containing only one non-λ-isolated point. proof. ((1) ⇒ (2)) evidently, x ∈ x is the only non-λ-isolated point in x, see corollary 2.5 and proposition 4.2. now we show that x is a λ-compact space. put x = ⋃ i∈i gi, such that gi is an open set in x, for each i ∈ i and |i| ≥ λ. since x ∈ ⋃ i∈i gi, there exists k ∈ i, such that x ∈ gk. but by complete regularity of x, there exists f ∈ c(x) such that x ∈ int(z(f)) ⊆ gk. since x is a pλ-space, x ∈ z(f) and therefore f ∈ mx = sλ(x). thus |x \ gk| ≤ |x \ z(f)| = |coz(f)| < λ, i.e., x = ( ⋃ j∈j gj) ⋃ gk, where j ⊆ i and |j| < λ. now, it is sufficient to show that λ is the least infinite cardinal number with this property. to see this we show that there exists an open cover of x with cardinality β < λ which is not reducible to a subcover with cardinality less than β. by [17, lemma 2.13], there exists a closed subspace f ⊂ x, such that |f | = β and x ∈ f . now, by complete regularity of x, for each s ∈ f and y ∈ f \ {s}, there exists fy ∈ c(x), such that fy(s) = 0 and fy(y) = 1. therefore s ∈ ⋂ y∈f \{s} z(fy) = gs and since x is a pλ-space, gs is an open set of x. so x = (x \ f) ∪ {gs}s∈f is an open cover of x. it goes without saying that gs ∩ f = {s} and therefore the above cover cannot reduce to an open cover of x with cardinality less than β. consequently, x is a λ-compact space. ((2) ⇒ (1)) it is sufficient to show that mx ⊆ sλ(x), where x is the only non-λ-isolated point of x. let f ∈ mx, i.e., x ∈ z(f). since each point of x except x is a λ-isolated point we infer that for every y ∈ x \ z(f), there exists a neighborhood of y in x, say gy, with cardinality less than λ. hence (x \ z(f)) ⊆ ⋃ i∈i gyi, where |i| < λ and yi is a λ-isolated point, for each i ∈ i. thus | ⋃ i∈i gyi| < λ implies that |x \ z(f)| < λ and we are done. � we note that if x has at most one non-λ-isolated point, then by criterion for recognizing the essential ideals in c(x), see [1, theorem 3.1], sλ(x) is essential in c(x) and by proposition 4.2, it is an essential prime ideal of c(x). if x is the one point λ-compactification of a discrete space, then sλ(x) is an essential maximal ideal, see theorem 4.3. the above discussion refers to the following proposition which is proved in [1, proposition 4.1]. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 266 on the essentiality and primeness of λ-super socle of c(x) proposition 4.4. if x is an infinite space, there is an essential ideal in c(x) which is not a prime ideal. the following theorem is the counterpart of the above proposition. theorem 4.5. let x be a topological space with |x| ≥ λ such that |x \ iλ(x)| > 1, then there exists a λ-essential ideal in c(x) which is not a prime ideal. proof. by assumption, there exist two distinct non-λ-isolated points, say x and y. now, define e = {f ∈ c(x) : {x, y} ⊆ z(f)}, then ⋂ z[e] = {x, y} and therefore by the criterion for recognizing the essential ideals, e is essential. since x, y ∈ ⋂ z[sλ(x)], by theorem 2.4 we infer that sλ(x) ⊆ e. it is evident that e is not a prime ideal, see [11, theorem 2.11] and we are done. � acknowledgements. the authors would like to thank professor o.a.s. karamzadeh for introducing the topics of this article and for his helpful discussion. the authors are also indebted to the well-informed, meticulous referee for his/her carefully reading the article and giving valuable and constructive comments. references [1] f. azarpanah, essential ideals in c(x), period. math. hungar. 31 (1995), 105–112. 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[17] s. mehran and m. namdari, the λsuper socle of the ring of continuous functions, categories and general algebraic structures with applications 6, no. 1 (2017), 37–50. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 268 () @ appl. gen. topol. 17, no. 2(2016), 117-122doi:10.4995/agt.2016.4521 c© agt, upv, 2016 results about the alexandroff duplicate space khulod almontashery and lutfi kalantan king abdulaziz university, department of mathematics, p.o.box 80203, jeddah 21589, saudi arabia (khuloodalim@hotmail.com, lk274387@hotmail.com and lkalantan@kau.edu.sa) abstract in this paper, we present some new results about the alexandroff duplicate space. we prove that if a space x has the property p , then its alexandroff duplicate space a(x) may not have p , where p is one of the following properties: extremally disconnected, weakly extremally disconnected, quasi-normal, pseudocompact. we prove that if x is αnormal, epinormal, or has property wd, then so is a(x). we prove almost normality is preserved by a(x) under special conditions. 2010 msc: 54f65; 54d15; 54g20. keywords: alexandroff duplicate; normal; almost normal; mildly normal; quasi-normal; pseudocompact; rroperty wd; α-normal; epinormal. there are various methods of generating a new topological space from a given one. in 1929, alexandroff introduced his method by constructing the double circumference space [1]. in 1968, r. engelking generalized this construction to an arbitrary space as follows [6]: let x be any topological space. let x′ = x × {1}. note that x ∩ x′ = ∅. let a(x) = x ∪ x′. for simplicity, for an element x ∈ x, we will denote the element 〈x, 1〉 in x′ by x′ and for a subset b ⊆ x, let b′ = {x′ : x ∈ b} = b × {1} ⊆ x′. for each x′ ∈ x′, let b(x′) = {{x′}}. for each x ∈ x, let b(x) = {u ∪ (u′ \ {x′}) : u is open in x with x ∈ u }. then b = {b(x) : x ∈ x} ∪ {b(x′) : x′ ∈ x′} will generate a unique topology on a(x) such that b is its neighborhood system. a(x) with this topology is called the alexandroff duplicate of x. now, if p is a topological property and x has p , then a(x) may or may not have p . throughout this paper, we denote an ordered pair by 〈x, y〉, the set of positive integers by n and the set of real numbers by r. for a subset a of a space x, received 09 january 2016 – accepted 03 august 2016 http://dx.doi.org/10.4995/agt.2016.4521 k. almontashery and l. kalantan inta and a denote the interior and the closure of a, respectively. an ordinal γ is the set of all ordinal α such that α < γ. the first infinite ordinal is ω and the first uncountable ordinal is ω1. a topological space x is called α-normal [3] if for any two disjoint closed subsets a and b of x, there exist two open disjoint subsets u and v of x such that a ∩ u dense in a and b ∩ v dense in b. theorem 0.1. if x is α-normal, then so is its alexandroff duplicate a(x). proof. let e and f be any two disjoint closed sets in a(x). write e = e1∪e2, where e1 = e ∩ x, e2 = e ∩ x ′ and f = f1 ∪ f2, where f1 = f ∩ x, f2 = f ∩ x ′. so, we have e1 and f1 are two disjoint closed sets in x. by α-normality of x, there exist two disjoint open sets u and v of x such that e1 ∩u is dense in e1 and f1 ∩v is dense in f1. let w1 = (u ∪u ′ ∪e2)\f and w2 = (v ∪v ′ ∪f2)\e. then w1 and w2 are disjoint open sets in a(x). now, we prove w1 ∩ e is dense in e. note that w1 ∩ e = (w1 ∩ e1) ∪ (w1 ∩ e2) = (u ∩e1)∪e2, hence w1 ∩ e = (u ∩ e1) ∪ e2 = (u ∩ e1)∪e2 ⊃ e1 ∪e2 ⊃ e. therefore, w1 ∩ e is dense in e. similarly, w2 ∩ f is dense in f . therefore, a(x) is α-normal. � a space x is called extremally disconnected [5] if it is t1 and the closure of any open set is open. extremally disconnectedness is not preserved by the alexandroff duplicate space and here is a counterexample. example 0.2. consider the stone-čech compactification space βω which is compact hausdorff, hence tychonoff. it is well-known that βω is extremally disconnected. clearly ω is open in a(βω) and ωa(βω) = βω which is not open in a(βω). a space x is called weakly extremally disconnected [8] if the closure of any open set is open. weakly extremally disconnected is not preserved and the above example is a counterexample. the following question is interesting and still open: “ does there exist a tychonoff non-discrete space x such that a(x) is extremally disconnected?”. a subset b of a space x is called a closed domain [5] if b = intb. a finite intersection of closed domains is called π-closed [9]. a topological space x is called mildly normal [7] if for any two disjoint closed domains a and b of x, there exist two open disjoint subsets u and v of x such that a ⊆ u and b ⊆ v . a topological space x is called quasi-normal [9] if for any two disjoint π-closed subsets a and b of x, there exist two open disjoint subsets u and v of x such that a ⊆ u and b ⊆ v . it is clear from the definitions that every quasi-normal space is mildly normal. mild normality is not preserved by the alexandroff duplicate space [7]. quasi-normality is not preserved by the alexandroff duplicate space and here is a counterexample. we denote the set of all limit points of a set b by bd and call it the derived set of b. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 118 results about the alexandroff duplicate space example 0.3. consider rω1, which is tychonoff separable non-normal space ([5], 2.3.15). it is well-known that every closed domain in rω1 depends on a countable set [12]. it follows that every π-closed set in rω1 depends on a countable set. now, if a and b are disjoint π-closed sets in rω1, then there is a countable s ⊂ ω1 such that a = πs(a) × r ω1\s and b = πs(b) × r ω1\s, where πs is the projection function πs : r ω1 −→ rs. it follows that πs(a) and πs(b) are disjoint closed sets in r s. since s is countable, rs is metrizable. so, there exist two open disjoint sets u1, v1 ⊂ r s such that πs(a) ⊆ u1 and πs(b) ⊆ v1. then a ⊆ u = u1 × r ω1\s and b ⊆ v = v1 × r ω1\s where u and v are open in rω1 and disjoint. therefore, rω1 is quasi-normal. we show that the alexandroff duplicate space a(rω1) is not quasi-normal by showing that it is not mildly normal. let e = {〈nξ : ξ < ω1〉 ∈ n ω1 : ∀ m ∈ n \ {1}(|{ξ < ω1 : nξ = m}| ≤ 1) }. f = {〈nξ : ξ < ω1〉 ∈ n ω1 : ∀ m ∈ n \ {2}(|{ξ < ω1 : nξ = m}| ≤ 1) }. e and f are disjoint closed subsets in nω1, hence closed in rω1. they cannot be separated by disjoint open sets, see [14], and they are perfect, i.e., e = ed and f = f d. by a theorem from [7] which says: “if a and b are disjoint subsets of a space x such that ad and bd cannot be separated, then a(x) is not mildly normal.”, we conclude that a(rω1) is not mildly normal. a space x is pseudocompact [5] if x is tychonoff and any continuous realvalued function defined on x is bounded. equivalently, a tychonoff space x is pseudocompact if and only if any locally finite family consisting of non-empty open subsets is finite [5]. we will conclude that pseudocompactness is not preserved by the alexandroff duplicate space by the following theorem. theorem 0.4. let x be a tychonoff space. the alexandroff duplicate a(x) is pseudocompact if and only if x is countably compact. proof. if x is not countably compact, then there exists a countably infinite closed discrete subset d of x. then d × {1} = d′ is closed and open set in a(x) which is also countably infinite, hence a(x) is not pseudocompact. now, assume that x is countably compact. since for a set b ⊆ x and a point a ∈ x, we have that a is a limit point for b′ if and only if a is a limit point for b, we do have that a(x) is also countably compact and hence pseudocompact. � so, any pseudocompact space which is not countably compact will be an example of a pseudocompact space whose alexandroff duplicate space a(x) is not pseudocompact. a mrówka space ψ(a) , where a ⊂ [ω]ω is maximal, is such a space, see [4] and [10]. arhangel’skii introduced the notions of epinormality and c-normality in 2012, when he was visiting the department of mathematics, king abdulaziz university, jeddah, saudi arabia. a topological space x is called c-normal [2] if there exist a normal space y and a bijective function f : x −→ y such c© agt, upv, 2016 appl. gen. topol. 17, no. 2 119 k. almontashery and l. kalantan that the restriction f|c : c −→ f(c) is a homeomorphism for each compact subspace c ⊆ x. it was proved in [2] that if x is c-normal, then so is its alexandroff duplicate. a topological space ( x , τ ) is called epinormal [2] if there is a coarser topology τ ′ on x such that ( x , τ ′ ) is t4. theorem 0.5. if x is epinormal, then so is its alexandroff duplicate a(x). proof. let x be any space which is epinormal, let τ be a topology on x, since x is epinormal, then there is a coarser topology τ∗ on x such that ( x , τ∗ ) is t4. since t4 is preserved by the alexandroff duplicate space, then a(x , τ ∗ ) is also t4 and it is coarser than a(x , τ) by the topology of the alexandroff duplicate. hence, a(x) is epinormal. � a space x is said to satisfy property wd [11] if for every infinite closed discrete subspace c of x, there exists a discrete family {un : n ∈ ω} of open subsets of x such that each un meets c at exactly one point. theorem 0.6. if x satisfies property wd, then so does its alexandroff duplicate a(x). proof. let x be any space which satisfies property wd and consider its alexandroff duplicate a(x). to show that a(x) has property wd, let c ⊆ a(x) be any infinite closed discrete subspace of a(x). write c = (c ∩ x) ∪ (c ∩ x′). for each x ∈ c ∩ x, fix an open set ux in x such that vx = ux ∪ (u ′ x \ {x ′}) open in a(x) and (*) vx ∩ c = {x} case 1: c ∩ x is finite. this implies that c ∩ x′ is infinite. let {x′n : n ∈ ω} ⊆ c ∩ x′ such that x′i 6= x ′ j, for all i, j ∈ ω with i 6= j. now, consider the family {{x′n}: n ∈ ω}, then it consists of open sets and each {x ′ n} meets c at exactly one point. now, we will show that {{x′n}: n ∈ ω} is a discrete family in a(x). it is obvious that {x′n : n ∈ ω} is discrete and it is closed because if x ∈ a(x) \ c, then there is open set ux containing x such that ux ∩ c = ∅, and if x ∈ c ∩ x hence, by (*) there is an open set vx in a(x) containing x such that vx ∩ c = {x}. thus, {{x ′ n}: n ∈ ω} is discrete family. therefore, in this case, a(x) satisfies property wd. case 2: c ∩ x is infinite. then c ∩ x is an infinite closed discrete subspace of x. since x satisfies the property wd, then there exists a discrete family {vn : n ∈ ω} of open subsets of x such that each vn meets c ∩ x at exactly one point {xn}. hence, {vn ∪(v ′ n \ {x ′ n}): n ∈ ω} is discrete in a(x) and then meets c in exactly one point {xn}. therefore, also in this case, a(x) satisfies the property wd. � a space x is called almost normal [8] if for any two disjoint closed subsets a and b of x one of which is a closed domain there exist two disjoint open sets u and v of x such that a ⊆ u and b ⊆ v . c© agt, upv, 2016 appl. gen. topol. 17, no. 2 120 results about the alexandroff duplicate space proposition 0.7. let x be a topological space. the following are equivalent: (1) x is the only non-empty closed domain in x. (2) each non-empty open subset is dense in x. (3) the interior of each non-empty proper closed subset of x is empty. proof. (1) =⇒ (2) let u be any non-empty open set. suppose u is not dense in x, then u is a non-empty proper closed domain in x, which contradicts the hypothesis, hence u is dense in x. (2) =⇒ (3) suppose that e is a non-empty proper closed subset of x such that inte 6= ∅, then x = inte ⊆ e = e which is a contradiction. (3) =⇒ (1) suppose that there exists a closed domain b such that ∅ 6= b 6= x, then ∅ 6= intb 6= x , which contradicts the hypothesis, thus x is the only nonempty closed domain in x. � it is clear that any space that satisfies the conditions of proposition 7 will be almost normal. corollary 0.8. if x satisfies the conditions of proposition 7, then its alexandroff duplicate a(x) is almost normal. proof. let e and f be any non-empty disjoint closed subsets of a(x) such that e is a closed domain. let w be an open set in a(x) such that w = e. if w ∩ x 6= ∅, then w ∩ x is dense in x by proposition 7, so x ⊂ w = e. it follows that f ⊂ x′, so e is closed and open, hence there are two disjoint open sets u = a(x) \ f and v = f in a(x) containing e and f respectively. if w ∩ x = ∅, then e ⊂ x′, so e is closed and open, thus there are two disjoint open sets u = e and v = a(x) \ e in a(x) containing e and f respectively. � corollary 0.9. the alexandroff duplicate of the countable complement space [13], the finite complement space [13], and the particular point space [13] are all almost normal. the following problems are still open: (1) if x is almost normal, is then its alexandroff duplicate a(x) almost normal? (2) if x is β-normal [3], is then its alexandroff duplicate a(x) β-normal? acknowledgements. the authors would like to thank the referee for valuable suggestions and comments. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 121 k. almontashery and l. kalantan references [1] p. s. alexandroff and p. s. and urysohn, mémoire sur les espaces topologiques compacts, verh. akad. wetensch. amsterdam, 14 (1929). [2] s. alzahrani and l. kalantan, c-normal topological property, filomat, to appear. [3] a. v. arhangelskii and l. ludwig, on α-normal and β-normal spaces, comment. math. univ. carolinae. 42, no. 3(2001), 507–519. [4] e. k. van douwen, the integers and topology, in: k. kunen, j. e. vaughan (eds.), handbook of set-theoretic topology, north-holland,amsterdam, 1984, pp. 111–167. [5] r. engelking, general topology, vol. 6, berlin: heldermann (sigma series in pure mathematics), poland, 1989. [6] r. engelking, on the double circumference of alexandroff, bull. acad. pol. sci. ser. astron. math. phys. 16, no 8 (1968), 629–634. [7] l. kalantan, results about κ-normality, topology and its applications 125, no. 1 (2002), 47–62. [8] l. kalantan and f. allahabi, on almost normality, demonstratio mathematica xli, no. 4 (2008), 961–968. [9] l. kalantan, π-normal topological spaces, filomat 22, no. 1 (2008), 173–181. [10] s. mrówka, on completely regular spaces, fundamenta mathematicae 41 (1954), 105– 106. [11] p. nyikos, axioms, theorems, and problems related to the jones lemma, general topology and modern analysis (proc. conf., univ. california, riverside, calif., 1980), pp. 441–449, academic press, new york-london, 1981. [12] k. a. ross and a. h. stone, product of separable spaces, amer. math. month. 71(1964), 398–403. [13] l. a. steen and j. a. seebach, counterexamples in topology, dover publications, inc., new york, 1995. [14] a. h. stone, paracompactness and product spaces, bull. amer. soc. 54 (1948), 977–982. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 122 () @ appl. gen. topol. 19, no. 1 (2018), 91-99doi:10.4995/agt.2018.7731 c© agt, upv, 2018 controlled shadowing property alireza zamani bahabadi department of pure mathematics, ferdowsi university of mashhad, mashhad, iran (zamany@um.ac.ir) communicated by f. balibrea abstract in this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing and specification properties. the relation between the controlled shadowing and chaos in sense of li-yorke is studied. at the end we give some examples to investigate the controlled shadowing property. 2010 msc: 37b20; 37c50; 54h20. keywords: controlled shadowing property; chaos; topologically ergodic; specification property; topologically mixing. 1. introduction the concept of shadowing is investigated by many authors (see e.g. [4, 5, 11, 13, 14, 15]). topological mixing and the specification properties are important notions in dynamical systems which has been related to chaos in global sense [10]. to investigate the topological mixing, specification and chaos, the shadowing property along is not useful. so some authors used the another definition of the shadowing property such as average shadowing, ergodic shadowing and d-shadowing to investigate specification property and chaos [6, 7, 12]. in their definitions, there are some large mistakes in jumping of pseudo orbits, but they still obtained some regularity of this mistakes. in this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing, specification property and chaos. first, motivated by [8, 9], we show that any surjective map with controlled shadowing property is chain transitive and received 03 june 2017 – accepted 17 january 2018 http://dx.doi.org/10.4995/agt.2018.7731 a. zamani bahabadi prove that any stable map with controlled shadowing property is topological ergodic which is stronger than transitivity. as well we will relate the controlled shadowing property with the specification property. in fact we will prove that for any surjective map with shadowing property on a compact metric space, the controlled shadowing and the specification properties are equivalent. the relation between the controlled shadowing and chaos in sense of li-yorke is the next result of this paper. at the end we give some examples to investigate the controlled shadowing property. 2. preliminaries let (x, d) be a metric space and let f : x → x be a continuous map. a sequence {xn} ∞ n=0 is called an orbit of f, denoted by o(x, f), if for each n ∈ n, xn+1 = f(xn) and we call it a δ-pseudo-orbit of f if d(f(xi), xi+1) < δ, for all i ∈ n. a continuous map f is said to have the shadowing property if for each ε > 0 there exists δ > 0 such that every δ-pseudo-orbit {xi} ∞ i=0 is ε-shadowed by the orbit of some point y ∈ x, i.e d(fn(y), xn) < ε, for all n ∈ n. a map f is called chain transitive if for any x, y ∈ x and for every ǫ > 0, there exists an ǫ-pseudo orbit (ǫ-chain) from x to y. a point x ∈ x is stable point if for any ǫ > 0 there is a δ > 0 such that if d(x, y) < δ, then d(fi(x), fi(y)) < ǫ for every i ∈ n. a surjective continuous map f is stable map if any point of x is stable point. let u and v be two nonempty open subsets of x and consider n(u, v ) = {n ∈ n; fn(u) ∩ v 6= ∅}. a map f is called transitive if for each nonempty open subsets u, v of x, n(u, v ) 6= ∅. f is called topologically ergodic if for every nonempty open subsets u, v of x, n(u, v ) has positive upper density, that is d ( n(u, v ) ) = lim sup n→∞ 1 n card { n(u, v ) ∩ {0, · · · , n − 1} } > 0, where card(a) denotes the number of members of the finite set a. f is called topologically mixing if f × f is transitive. we say that a sequence {xn} ∞ n=0 is a controlled-δ-pseudo orbit if lim sup n→∞ 1 n card { i ∈ {0, · · · , n − 1}; d(f(xi), xi+1) ≥ δ } < δ. we say that a controlled-δ-pseudo-orbit {xn} ∞ n=0 is control-ε-shadowed by y ∈ x if lim sup n→∞ 1 n card { i ∈ {0, · · · , n − 1}; d(fi(y), xi) ≥ ε } < ε. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 92 controlled shadowing property we say that f has controlled shadowing property, if for every ǫ > 0 there is a δ > 0 such that any controlled-δ-pseudo orbit is control-ε-shadowed by some point of x. 3. main results in this section we obtain chain transitivity, topological ergodicity, topological mixing, specification property and chaos by using the controlled shadowing property. theorem 3.1. let f : x → x be a surjective continuous map with the controlled shadowing property. then f is chain transitive. proof. let x, y ∈ x and 0 < ǫ < 1 2 be given. we show that there is an ǫ-chain from x to y. suppose that δ ≥ 0 is as in the definition of controlled shadowing for ǫ > 0. choose a positive integer n such that 1 n < δ. since f is surjective hence there is a sequence {y−3n, y−3n−1, · · · , y0 = y} such that f(yi) = yi−1 for −3n ≤ i < 0. consider controlled-δ-pseudo orbit as follows: {x, f(x), · · · , f3n (x), y−3n , y−3n−1, · · · , y, x, · · · , f 3n (x), y−3n , · · · , y, · · · } = {xi} ∞ i=0 . therefore we have lim n→∞ 1 n card { i ∈ {0, · · · , n} : d ( f(xi), xi+1 ) ≥ δ } = lim 1 6kn card { i ∈ {0, · · · , 6kn} : d ( f(xi), xi+1 ) ≥ δ } ≤ 3k 6kn = 1 2n ≤ 1 n < δ. consequently {xi} ∞ i=0 can be control-ǫ-shadowed by some point z ∈ x. this means lim n→∞ 1 n card { i ∈ {0, · · · , n} : d(fi(z), xi) ≥ ǫ } < ǫ. now, we claim that there are two infinite sequences of positive integers {i1 < i2 < · · · } and {l1 < l2 < · · · } such that for any j > 0, we have xij ∈ {x, f(x), · · · , f 3n(x)} and d(fij (z), xij ) < ǫ, and xlj ∈ {y−3n, y−3n−1, · · · , y} and d(f lj (z), xlj ) < ǫ. the reason is that if do not exist such sequences, then we must have lim k→∞ 1 6kn card { i ∈ {0, · · · , 6kn} : d(fi(z), xi) ≥ ǫ } ≥ lim k→∞ 3kn 6kn = 1 2 > ǫ, which is a contradiction. so we can find i0 > l0 > 1 and 0 ≤ k0, m0 ≤ 3n such that d ( fi0(z), fk0(x) ) < ǫ and d ( fl0(z), ym0 ) < ǫ. therefore { x, f(x), · · · , fk0−1(x), fi0(z), · · · , fl0−1(z), ym0, · · · , y } is an ǫ-chain from x to y. this shows that f is chain transitive. � theorem 3.2. let f : x → x be a stable map with the controlled shadowing property. then f is topologically ergodic. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 93 a. zamani bahabadi proof. let u and v be two open subsets of x and x ∈ u, y ∈ v . consider ǫ > 0 such that nǫ(x) ⊂ u, nǫ(y) ⊂ v . since f is stable, there is a δ > 0 such that if d(u, v) < δ then d ( fi(u), fi(v) ) < ǫ for i > 0. suppose that δ1 ≥ 0 is as in the definition of controlled shadowing for δ > 0. choose a positive integer n such that 1 n < δ1 and consider controlled-δ1-pseudo orbit as follows: {x, f(x), · · · , f3n (x), y−3n , y−3n−1, · · · , y, x, · · · , f 3n (x), y−3n , · · · , y, · · · } = {xi} ∞ i=0 hence we have lim n→∞ 1 n card { i ∈ {0, · · · , n} : d ( f(xi), xi+1 ) ≥ δ1 } = lim 1 6kn card { i ∈ {0, · · · , 6kn} : d ( f(xi), xi+1 ) ≥ δ1 } ≤ 3k 6kn = 1 2n ≤ 1 n < δ1. so {xi} ∞ i=0 can be control-δ−shadowed by some point z ∈ x. this means lim n→∞ 1 n card { i ∈ {0, · · · , n} : d(fi(z), xi) ≥ δ } < δ. consider lx = { i : xi ∈ {x, f(x), · · · , f 3n(x) and d(fi(z), xi) < δ } , ly = { i : xi ∈ {y−3n, y−3n−1, · · · , y and d(f i(z), xi) < δ } . we claim that lx and ly have positive upper density. that is d(lx) > 0 and d(ly) > 0. we prove for lx and the proof for ly is similar. suppose d(lx) = 0, limn→∞ 1 n card(lx ∩ {0, · · · , n}) = 0. let l′x = { i : xi ∈ {x, f(x), · · · , f 3n(x) and d(fi(z), xi) ≥ δ } . we have d(lx ∪ l ′ x) = d(lx) + d(l ′ x) and d(lx ∪ l ′ x) ≥ 1 2 . since d(lx) = 0 so lim n→∞ 1 n card ( l′x ∩ {0, · · · , n} ) ≥ 1 2 . but lim n→∞ 1 n card { i ∈ {0, · · · , n} : d(fi(z), xi) ≥ δ } ≥ lim n→∞ 1 n card ( l′x ∩ {0, · · · , n} ) ≥ 1 2 . this is a contradiction. so lx and ly have positive upper density. we can find i0 > 3n and 0 ≤ k0 ≤ 3n such that d ( fi0(z), fk0(x) ) < δ, so d ( fi0−k0(z), x ) < ǫ. now for any j ∈ ly with j ≥ i0 − k0 + 3n, d ( fj(z), fm0(y) ) < δ for some 0 ≤ m0 ≤ 3n. so d ( fj−m0(z), y) < ǫ. put nj = (j − m0) − (i0 − k0) > 0 and therefore fnj ( nǫ(x) ) ∩ nǫ(y) 6= ∅. so f nj (u) ∩ v 6= ∅. hence for n large we have 1 n card { n(u, v ) ∩ {0, · · · , n} } ≥ 1 n card { ly ∩ {0, · · · , n} } , c© agt, upv, 2018 appl. gen. topol. 19, no. 1 94 controlled shadowing property so lim sup n→∞ 1 n card { n(u, v ) ∩ {0, · · · , n} } > 0. this shows that f is topologically ergodic. � remark that in the proof of the above theorems we need that for y ∈ x, f−1(y) is not empty. so we need surjectivity in f. theorem 3.3. if f has the controlled shadowing property, then the same holds for fk, for any positive integer k. proof. let ε > 0 be given and δ > 0 be as in the definition of controlled shadowing for f. suppose that {xn} ∞ n=0 is a controlled-δ-pseudo orbit for f k. consider {yi} ∞ i=0 = {x0, f(x0), · · · , f k−1(x0), x1, f(x1), · · · , f k−1(x1), x2, · · · }. we can see that {yn} is a controlled-δ-pseudo orbit for f. so there is z ∈ x such that lim sup n→∞ 1 n card { i ∈ {0, · · · , n}; d(fi(z), yi) ≥ ε } < ε. therefore lim sup n→∞ 1 n card { i ∈ {0, · · · , n}; d(fki(z), xi) ≥ ε } < ε. so fk has controlled shadowing property. � we say that f is totally transitive if all its iterates fn are transitive; f is topological mixing if for any nonempty open sets u and v in x, there is an n > 0 such that fn(u) ∩ v 6= ∅ for all n ≥ n. the specification property was introduced by bowen in [2]. we say that f has periodic specification property if for any ε > 0, there is an integer k > 0 such that for any integer n ≥ 2, any set {y1, · · · , yn} of n poins of x, and any sequence 0 = a1 ≤ b1 < a2 ≤ b2 < · · · < an ≤ bn with ai+1 − bi ≥ k for i = 1, · · · , n − 1, there is a point x ∈ x such that for each 1 ≤ m ≤ n and i with am ≤ i ≤ bm, the following conditions hold: d ( fi(x), fi(ym) ) < ε,(3.1) fl(x) = x where l = k + bn.(3.2) if we omit the condition (3.2), then f has the specification property. theorem 3.4. let x be a compact metric space. if f : x → x is a continuous surjective map with the controlled shadowing and the shadowing properties, then f is topological mixing and f has the specification property. proof. by theorems 3.1 and 3.3, since f has the controlled shadowing and the shadowing properties, then f is totally transitive. so by theorem 1 in [12], the proof is complete. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 95 a. zamani bahabadi theorem 3.5. let f : x → x be a surjective continuous map on the compact metric space x. if f has the specification property and the shadowing property, then f has the controlled shadowing property. proof. let ε > 0 be given and k as in the definition of specification for ε > 0. choose m such that k m < ε. suppose that 0 < δ < 1 m is as in the definition of shadowing. take any controlled δ-pseudo-orbit {xn} ∞ n=0. by the proof of lemma 12 in [12], we can find infinite sequences of integers {an} ∞ n=0 and {bn} ∞ n=0 such that 0 = a0 ≤ b0 < a1 ≤ b1 < · · · and an+1 − bn = k for n = 0, 1, 2, · · · . therefore there is a point z ∈ x such that for any an ≤ j ≤ bn we have d(fj(z), xj) < ε. so we can see card{0 ≤ j < n; d(fj(z), xj) ≥ ε} ≤ k card{0 ≤ j ≤ n; d(f(xj), xj+1) > δ}. hence we have lim sup n→∞ 1 n card{0 ≤ j < n; d(fj(z), xj) ≥ ε} ≤ k lim sup n→∞ 1 n card{0 ≤ j ≤ n; d(f(xj), xj+1) ≥ δ} ≤ kδ < k m < ε. so f has controlled shadowing property. � in the following f is a surjective continuous map on the compact metric space x. corollary 3.6. if f has the shadowing property, then the following conditions are equivalent: 1) f has controlled shadowing property, 2) f has specification property. proof. by theorems 3.4 and 3.5 the proof is complete. � if f is topological mixing, then by [3], it has the specification property. so we have the following corollary. corollary 3.7. if f has the shadowing and topological mixing properties, then f has the controlled shadowing property. theorem 3.8. if f has the controlled shadowing property, then the same holds for f × f. proof. suppose ε > 0 is given and δ > 0 is as in the definition of controlled shadowing property for ε 2 . let {(xn, yn)} ∞ n=0 be a controlled-δ-pseudo orbit for f × f. we define metric ρ on x × x as follows ρ ( (x, y), (x′, y′) ) = max{d(x, x′), d(y, y′)}. so we can see {xn} ∞ n=0 and {yn} ∞ n=0 are two controlled-δ-pseudo orbits for f, then they can be controlε 2 -shadowed by some points z1 and z2 respectively. so c© agt, upv, 2018 appl. gen. topol. 19, no. 1 96 controlled shadowing property there is n ∈ n such that for any n ≥ n, 1 n card { i ∈ {0, · · · , n}; d(fi(z1), xi) ≥ ε 2 } < ε 2 , 1 n card { i ∈ {0, · · · , n}; d(fi(z2), yi) ≥ ε 2 } < ε 2 . let card { i ∈ {0, · · · , n}; d(fi(z1), xi) ≥ ε 2 } = kn, card { i ∈ {0, · · · , n}; d(fi(z2), yi) ≥ ε 2 } = ln. so card { i ∈ {0, · · · , n}; ρ ( (f × f)i(z1, z2), (xi, yi) ) ≥ ε } = card { i ∈ {0, · · · , n}; d(fi(z1), xi) ≥ ε or d(f i(z2), yi) ≥ ε } ≤ kn + ln. therefore for n ≥ n we have 1 n card { i ∈ {0, · · · , n}; ρ ( (f × f)i(z1, z2), (xi, yi) ) ≥ ε } ≤ kn + ln n < ε 2 + ε 2 = ε. so f × f has the controlled shadowing property. � theorem 3.9. if f has the controlled shadowing and the shadowing properties, then f is chaotic in the sense of li-yorke. proof. since f has controlled shadowing property by theorem 3.8, f × f has controlled shadowing property. so by theorem 3.1, f × f is chain transitive. so the controlled shadowing property implies the chain transitivity of f × f. also this is well known that the shadowing property of f implies the shadowing property of f ×f. hence by the shadowing property and the chain transitivity, f × f is transitive. hence f is topologically weakly mixing. but any weakly mixing map is chaotic in the sense of li-yorke (see [10]). therefore f is chaotic in the sense of li-yorke. � a continuum is a nondegenerate compact connected metric space. a continuous map f from a compact metric space x to itself is said to be pchaotic if f has the shadowing property and the periodic points of f are dense in x. corollary 3.10. every p-chaotic map from a continuum to itself has the controlled shadowing property. proof. by corollary 3.3 in [1], every p-chaotic map from a continuum to itself is mixing. so by corollary 3.7, f has the controlled shadowing property. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 97 a. zamani bahabadi example 3.11. let f : [0, 1] → [0, 1] be the tent map which is defined by f(x) = { 2x 0 ≤ x ≤ 1 2 −2x + 2 1 2 ≤ x ≤ 1 . by [1], f is p-chaotic. by corollary 3.10 f has the controlled shadowing property. at the end we explain in the following examples that the shadowing property does not necessarily imply the controlled shadowing property. the first example is in a compact metric space and the second example is in the non-compact metric space. example 3.12. let f : [0, 1] → [0, 1] f(x) = { 2x if x ≤ 1 2 1 if 1 2 ≤ x ≤ 1 . f has the shadowing property but does not have the specification property (see example 2.8 in [1]). therefore f has not the controlled shadowing property. example 3.13. let f : r → r, f(x) = 4x. one can see that f has the shadowing property but is not transitive. indeed for every ǫ > 0 consider δ = ǫ 8 . if {xi} ∞ i=0 is a δ-pseudo orbit of f, then we can see that ⋂ f−i(b ǫ 2 (xi)) 6= ∅. this show that f has the shadowing property. if u = (1, 2), v = (0, 1), then fn(u) ∩ v = ∅ for every n ∈ n. therefore f is not transitive and so f is not chain transitive (chain transitivity and shadowing property imply transitivity). hence by theorem 3.1, f has not the controlled shadowing property. 4. conclusion in this paper we have shown that any surjective map with controlled shadowing property is chain transitive and every stable map with controlled shadowing property is topological ergodic which is stronger than transitivity. as well we proved that for any surjective map with shadowing property on a compact metric space, the controlled shadowing and the specification properties are equivalent. the relation between the controlled shadowing and chaos in sense of li-yorke has been the next result of this paper. finally, we gave an example having the controlled shadowing property. at the end by examples we showed that the shadowing property does not necessarily imply the controlled shadowing property. acknowledgements. the author would like to thank the respectful referee for his/her comments on the manuscript. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 98 controlled shadowing property references [1] t. arai and n. chinen, p-chaos implies distributional chaos and chaos in the sense of devaney with positive topological entropy, topology appl. 154 (2007), 1254–1262. [2] r. bowen, entropy for group endomorphisms and homogeneous spaces, trans. amer. math. soc. 153 (1971), 401–414. [3] j. buzzi, specification on the interval, trans. amer. math. soc. 349 (1997), 2737–2754. [4] b. a. coomes, h. koak and k. j. palmer, periodic shadowing, chaotic numerics (geelong,1993), in: contemp. math., vol. 172, amer. math. soc., providence, ri, 1994, pp. 115–130. [5] r. m. corless and s. yu. pilyugin, approximate and real trajectories for generic dynamical systems, j. math. anal. appl. 189 (1995), 409–423. [6] d. ahmadi dastjerdi and m. hosseini, sub-shadowings, nonlinear anal. 72 (2010), 3759–3766. [7] a. fakhari and f. helen ghane, on shadowing: ordinary and ergodic, j. math. anal. appl. 364 (2010), 151–155. [8] r. gu, the asymptotic average shadowing property and transitivity, nonlinear anal. 67, no. 6 (2007), 1680–1689. [9] r. gu, the average-shadowing property and topological ergodicity, j. comput. appl. math. 206, no. 2 (2007), 796–800. [10] w. huang and x. ye, devaney’s chaos or 2-scattering implies li-yorke’s chaos, topology appl. 117, no. 3 (2002), 259–272. [11] p. e. kloeden and j. ombach, hyperbolic homeomorphisms are bishadowing, ann. polon. math. 65 (1997), 171–177. [12] d. kwietniak and p. oprocha, a note on the average shadowing property for expansive maps, topology. appl. 159 (2012), 19–27. [13] m. mazur, tolerance stability conjecture revisited, topology appl. 131 (2003), 33–38. [14] s. yu. pilyugin and o. b.plamenevskaya, shadowing is generic, topology appl. 97 (1999), 253–266. [15] p. walters, on the pseudo-orbit tracing property and its relationship to stability, in: lecture notes in math., vol. 668, springer, berlin, 1978, pp. 224–231. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 99 @ appl. gen. topol. 18, no. 1 (2017), 183-202 doi:10.4995/agt.2017.6716 c© agt, upv, 2017 transitions between 4-intersection values of planar regions kathleen bell and tom richmond department of mathematics, western kentucky university, 1906 college heights blvd., bowling green, ky 42101 usa (kathleen.bell085@topper.wku.edu, tom.richmond@wku.edu) communicated by s. romaguera abstract if a(t) and b(t) are subsets of the euclidean plane which are continuously morphing, we investigate the question of whether they may morph directly from being disjoint to overlapping so that the boundary and interior of a(t) both intersect the boundary and interior of b(t) without first passing through a state in which only their boundaries intersect. more generally, we consider which 4-intersection values— binary 4-tuples specifying whether the boundary and interior of a(t) intersect the boundary and interior of b(t)—are adjacent to which in the sense that one may morph into the other without passing through a third value. the answers depend on what forms the regions a(t) and b(t) are allowed to assume and on the definition of continuous morphing of the sets. 2010 msc: 54c60; 26e25. keywords: upper semicontinuous; lower semicontinuous; vietoris topology; spatial region; 4-intersection value. 1. introduction given two sets a and b in the euclidean plane, the 4-intersection value associated with a and b is the binary 4-tuple ( χ(∂a∩∂b), χ(a◦ ∩b◦), χ(∂a∩b◦), χ(a◦ ∩∂b) ) where c◦ and ∂c denote the interior and boundary, respectively, of c, χ(c) = 0 if c = ∅, and χ(c) = 1 if c 6= ∅. the 4-intersection values are used received 13 october 2016 – accepted 25 january 2017 http://dx.doi.org/10.4995/agt.2017.6716 k. bell and t. richmond in geographic information systems to quantify the nature of the intersection of two regions a and b in the plane. for example, regions a and b may represent the habitats of a predator and its prey, the extent of a nature preserve and the moist regions of a desert, or the area protected by a military base and the area covered by cellular telephone service. such regions are not static. as they dynamically change, their 4-intersection values, or simply values, may also change. we say two values v1 and v2 are adjacent if there exist dynamically changing sets a(t) and b(t) which pass from value v1 to value v2 without passing though any other intermediate values. our goal is to determine which values are adjacent. the answer depends heavily on what restrictions are imposed. a common restriction in geographic applications is to assume the regions are spatial regions, that is, proper nonempty subsets of the plane which are regular closed and have connected interior. in particular, note that spatial regions must have positive area. spatial regions behave relatively nicely, but they limit the situations which may be modeled. the habitats of species or cellular coverage areas may be disconnected regions. as an elliptical puddle of water dries up, it may shrink to the major axis (which is not regular closed) before disappearing entirely. to allow the modeling of such situations, we will not restrict our attention to spatial regions. throughout, we will consider functions a,b : r →p(r2) which give a subset of the euclidean plane at each time t. we must stipulate how regions a(t) and b(t) are allowed to change. an elliptical puddle of water may dry up uniformly with the entire moist region disappearing at an instant. while this may be continuous in some measure (namely, the moisture content), the area is not continuously changing: it jumps from positive area to zero area discontinuously. discontinuous areas may be useful for some models. for regions determined by electronic transmission coverage, such as wifi accessibility, turning on a new transmitter will instantly and discontinuously increase the coverage area. the 4-intersection values were introduced in [6], where they were applied to spatial regions. the work of [15] connects these concepts to relational algebras. 4-intersection values were applied to regions homeomorphic to 2-dimensional disks in [7] and to regions with holes in [5]. by comparing the exteriors (i.e., complements) of two regions along with the interiors and boundaries, there are 32 = 9 possible matchings which could be empty or not, giving rise to the 9-intersection value model. this was introduced for spatial regions and has been studied for particular shapes in [12] and [13]. a variation of the 9-intersection model replacing the exterior with another set is studied in [2]. our work is closest to that of [4], where the authors consider the adjacency graph for the 9-intersection values when restricted to specific transformations of spatial regions, such as scalings, translations, and rotations. the restriction to spatial regions already limits the number of attainable intersection values to 8 (using 4-intersection or 9-intersection). for example, if a and b are spatial regions with ∂a∩b◦ 6= ∅, then a◦∩b◦ 6= ∅ since neighborhoods of every boundary point of a include interior points of a (see [7]). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 184 transitions between 4-intersection values of planar regions without restricting to spatial regions, all 24 = 16 possible 4-intersection values are attainable, giving ( 16 2 ) = 120 possible adjacencies to consider. there are 29 = 512 possible 9-intersection values, giving ( 512 2 ) = 130, 816 possible adjacencies to consider. as this number would be unwieldy, we focus on the 4-intersection values. the techniques would be similar for 9-intersection values. we impose weaker restrictions on the allowed transformations than considered in [4]. intersections of dynamically moving directed lines and regions have also been considered in [16] and [11]. besides the 4and 9-intersection models, geographers use a point-free approach called regional connection calculus (rcc). in [9], it is shown that rcc is equivalent to considering regular closed, nonempty sets in a regular connected space. see [14] for further discussion of the interplay between these and other approaches. a standard way to create a closed set in the plane which is not regular closed is to add a whisker, that is, a line segment protruding from the set. the closed ball centered at (h,k) of radius r is the set b̄((h,k),r) = {(x,y) ∈ r2 : (x−h)2 + (y −k)2 ≤ r}. 2. the 4-intersection values if a and b are not required to be spatial regions, all 16 possible 4-intersection values may be realized. table 1 systematically lists and labels the 16 possible intersection values, giving one possible realization of each. the table shows that with the exception of value 5, each may be realized with compact sets a and b. the values which may be realized with spatial regions also include the name of the intersection value when applied to spatial regions. if the 4-intersection value of a(t) and b(t) changes from v1 to v2 without assuming any other values, either there is a first instant of v2 occurring or a last instant of v1 occurring. we say the v1 transforms to v2 instantly, at the instant t = a, if a(t) and b(t) have the value v1 for t < a or t > a and have the value v2 at t = a. we say that v1 transforms to v2 directly if there exists a such that a(t) and b(t) have the value v1 for t = a and have the value v2 for t < a or for t > a. thus, v1 transforms to v2 at an instant if and only if v2 transforms to v1 directly. if v1 transforms to v2 instantly or directly, we say that v1 and v2 are adjacent. 3. continuous area in this section, we investigate possible adjacencies assuming that regions a(t) and b(t) are closed at all times and the functions a(t),b(t), and ab(t) giving the area of a◦(t),b◦(t), and a◦(t) ∩b◦(t), respectively, are continuous extended real-valued functions. continuity of area seems to be a natural requirement for morphing regions, although a simple example will show that this alone will not be adequate in many situations. if a = b((0, 0), 2)∪b((5, 0), 1) for t ≤ 0 and a = b((0, 0), 1)∪ b((5, 0), 2) for t > 0 and b = [−3, 3]× [−3, 3] for t ≤ 0 and b = [3, 9]× [−2, 4] c© agt, upv, 2017 appl. gen. topol. 18, no. 1 185 k. bell and t. richmond (∂a∩∂b,a◦ ∩b◦,∂a∩b◦,a◦ ∩∂b) value 1 value 2 value 3 value 4 (0, 0, 0, 0) (0, 0, 0, 1) (0, 0, 1, 0) (0, 0, 1, 1) �� a b �� a b b �� a a b �� a a b b disjoint value 5 value 6 value 7 value 8 (0, 1, 0, 0) (0, 1, 0, 1) (0, 1, 1, 0) (0, 1, 1, 1) a = b = r2 �� a b b �� ia a b �� ia a b b contains inside value 9 value 10 value 11 value 12 (1, 0, 0, 0) (1, 0, 0, 1) (1, 0, 1, 0) (1, 0, 1, 1) �� a b �� a b �� a b �� a b �� meets value 13 value 14 value 15 value 16 (1, 1, 0, 0) (1, 1, 0, 1) (1, 1, 1, 0) (1, 1, 1, 1)� � � �a,b� �� a b �� a b b �� la a b �� a b equals covers covered by overlaps table 1. the 16 values of 4-intersection values for t > 0, then the areas of a(t), b(t), and a(t) ∩b(t) are constant and thus continuous, though this hardly seems to be a good description of continuous morphing. still, we investigate the possible adjacencies under the weak assumption of continuous areas. we start with spatial regions in example 3.1 and gradually weaken the assumptions on the spaces throughout the section. for spatial regions a and b, disjoint is adjacent to overlaps. one may initially be tempted to believe that as disjoint regions a and b morph continuously from value 1 (0, 0, 0, 0) = disjoint to value 16 (1, 1, 1, 1) = overlaps, they should pass through value 9 (1, 0, 0, 0) = meets. we present two examples which show that this transformation need not pass through value 9. example 3.1 (disjoint is adjacent to overlaps). (a) consider a static set a = {(x,y) ∈ r2 : x > 0,y ≥ 1/x} and a dynamic set b(t) = {(x,y) ∈ r2 : y ≤ t}. now a and b are disjoint for t ≤ 0 and overlap for t > 0, showing that c© agt, upv, 2017 appl. gen. topol. 18, no. 1 186 transitions between 4-intersection values of planar regions disjoint transforms directly to overlap, and overlap transforms instantly to disjoint. this example depends on the choice of spatial regions which are not compact. (b) to see that it is possible with compact spatial regions, let a = [−1, 1]× [1/2, 2] and b = ([−1, 1] × [−1, 0]) ∪ (cl(−r,r) × [0, 1]). when r ≤ 0, a and b are disjoint rectangles. for r > 0, a and b overlap. for the next few results, we will assume that all regions have finite area. this will be satisfied if the regions are compact. in particular, the finite area assumption rules out value 5 in which a = b = r2. proposition 3.2. suppose a(t) and b(t) are closed with connected interiors and positive finite areas and the areas a(t), b(t), and ab(t) of a(t),b(t), and a(t) ∩ b(t), respectively, are continuous. then the possible adjacencies are those given in the graph of figure 1. figure 1. adjacency graph for closed regions with connected interior and finite nonzero area, with a(t),b(t), and ab(t) continuous. proof. it is easy to see that the adjacencies shown are possible, with the surprising exception of disjoint begin adjacent to overlaps. example 3.1(b) shows this adjacency under the hypotheses given. it remains to show that disjoint and meet are not adjacent to any of covers, equal, covered by, contains, or inside, and that neither of covers or contains is adjacent to either covered by or inside. under the assumptions, covers, contains, covered by, inside, and equals imply that the interior of one set is contained in the interior of the other. for example, if a covers b, and b◦ 6⊆ a◦, then a◦ and x−a provide a separation of b◦, contrary to b◦ being connected (cf. [6, prop. 5.5]). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 187 k. bell and t. richmond suppose that disjoint is adjacent to covers, with a and b satisfying the disjoint condition for t < 0 and covers for t > 0. now ab(t) = 0 for t < 0, so by continuity, ab(0) = 0. since b◦ ⊆ a◦ for t > 0, f(t) = b(t) −ab(t) = 0 for t > 0, so by continuity, f(0) = 0. now b(0) = f(0) + ab(0) = 0, contrary to b(t) > 0. the same argument shows that disjoint and meet are not adjacent to any of covers, equal, covered by, contains, or inside are similar. to see that neither of covers or contains is adjacent to either covered by or inside, we will show that covers is not adjacent to covered by. the same argument works for the other proofs. suppose a and b satisfy covers for t < 0 and covered by for t > 0, and do not assume a third value at t = 0. now b(t) < a(t) for t < 0 and b(t) > a(t) for t > 0. the continuity conditions imply a(0) = b(0), which is not possible if a and b satisfy covers or covered by. � we note that if a(t) and b(t) are spatial regions, then they are closed sets with connected interiors and positive areas. the proof was clearly based on a and b having positive area, which we will now drop. this permits many more adjacencies. proposition 3.3. suppose a(t) and b(t) are closed with connected interiors and finite (possibly zero) areas and a(t), b(t), and ab(t) are continuous. then all values are adjacent except those shown in the non-adjacency graph of figure 2. figure 2. non-adjacency graphs for closed regions with connected interior and finite area, with a(t),b(t), and ab(t) continuous. proof. the pairs covers and covered by, covers and inside, and covered by and contains are not adjacent (even under weaker hypotheses, omitting the connected interior condition) by theorem 3.10 below. under the hypotheses of the proposition, suppose a(t) and b(t) satisfy contains for t < 0 and inside for t > 0, and do not assume a third value at t = 0. then b◦(t) ⊆ a◦(t) and a(t) ≥ b(t) for t < 0 while a◦(t) ⊆ b◦(t) and a(t) ≤ b(t) for t > 0. by continuity, a(0) = b(0), so a◦(0) and b◦(0) have the same positive area and either b◦(0) ⊆ a◦(0) or a◦(0) ⊆ b◦(0). it follows that a◦(0) = b◦(0) 6= ∅, and thus (since they do not have value 5 of a = b = r2) there exists x ∈ ∂a◦(0) = ∂b◦(0). now ∂(c◦) ⊆ ∂c (see proposition 3.5 below), so x ∈ ∂a(0) ∩ ∂b(0) and thus a(t) and b(t) assume a 4-intersection value of c© agt, upv, 2017 appl. gen. topol. 18, no. 1 188 transitions between 4-intersection values of planar regions form (1,y,z,w) at t = 0 and in particular, is neither contains nor inside at that instant. to complete the proof that the non-adjacency graph in figure 2 is complete, we present examples confirming adjacencies between all remaining states which were not shown in figure 1. disjoint and covers are adjacent. let a = {(0, 0)}∪(cl(−r,r)×[−1, 1]) and b = {(0, 1)}∪(cl(−r/2,r/2)×[−1, 1]). for r ≤ 0, a = {(0, 0)} and b = {(0, 1)} are disjoint. for r > 0, a = [−r,r]×[−1, 1] covers b = [−r/2,r/2]×[−1, 1]. interchanging a and b shows that disjoint and covered by are adjacent. changing the singleton of b so that b = {(0, 0)} for r ≤ 0 shows meets and covers are adjacent, and then interchanging a and b shows meets and covered by are adjacent. disjoint and equals are adjacent. with a as above, let b = {(0, 1)} ∪ (cl(−r,r) × [0, 1]). for r ≤ 0, a = {(0, 0)} and b = {(0, 1)} are disjoint. for r > 0, a = b = [−r,r] × [−1, 1]. changing the singleton of b so that b = {(0, 0)} for r ≤ 0 shows meets and equals are adjacent. disjoint and contains are adjacent. with a as above, let b = {(0, 1/2)}∪ (cl(−r/2,r/2) × [0, 1/2]). for r ≤ 0, a = {(0, 0)} and b = {(0, 1/2)} are disjoint. for r > 0, a = [−r,r] × [−1, 1] contains b = [−r/2,r/2] × [0, 1/2]. interchanging a and b show that disjoint and inside are adjacent. changing the singleton of b so that b = {0, 0) for r ≤ 0 shows meets and contains are adjacent, and then interchanging a and b shows meets and inside are adjacent. overlap and contains are adjacent. let a = [−2, 2] × [−2, 2] and b = ([−1, 1] × [−1, 1]) ∪ (cl(−r,r) × [0, 4]). a contains b for r ≤ 0 and a and b overlap for r > 0. interchanging a and b shows overlaps and inside are adjacent. � next, we present several elementary results needed to prove the nonadjacencies of theorem 3.10. recall that a topological space x is locally connected if for every x ∈ x and every neighborhood u of x, there exists a connected neighborhood v of x with v ⊆ u. in particular, r2 is locally connected. proposition 3.4. suppose b is a subset of a locally connected topological space x. for x ∈ b, let bx be the connected component of b which contains x. then⋃ x∈b ∂bx ⊆ ∂b = ∂( ⋃ x∈b bx), and equality holds if and only if b is closed. proof. suppose z ∈ ∂bx for some x ∈ b. then every neighborhood of z intersects bx ⊆ b, and every neighborhood of z intersects x −bx. if there is a connected neighborhood v of z which does not intersect x−b, then v ⊆ b. now bx ∪v is the union of two connected sets with a point z in common, so c = bx ∪ v is a connected set. furthermore, c is strictly larger than bx, contrary to the fact that bx was the largest connected subset of b containing c© agt, upv, 2017 appl. gen. topol. 18, no. 1 189 k. bell and t. richmond x. thus, every neighborhood of z intersects b and x − b, so z ∈ ∂b. this proves the inclusion. suppose b is closed and z ∈ ∂b. then z ∈ clb ∩ cl(x − b). now z ∈ b since b = clb, so z ∈ bz ⊆ ⋃ x∈b bx. also, since bz ⊆ b, z ∈ cl(x −b) ⊆ cl(x−bz). thus, z ∈ clbz∩cl(x−bz) = ∂bz. this shows that the inclusion is equality if b is closed. to see that equality holds only if b is closed, suppose b is not closed. then there exists a boundary point a of b which is not in b. given x ∈ b, a ∈ cl(x −b) ⊆ cl(x −bx), so a ∈ ∂bx if and only if a ∈ cl(bx) = bx ⊆ b, which does not occur since a 6∈ b. thus, for any point a ∈ ∂b −b, we have a 6∈ ⋃ x∈b bx, so equality fails. � we observe that if b has a finite number of connected components, then as a finite union of closed sets, b is closed, and thus equality would hold in proposition 3.4. proposition 3.5. for any set b in a topological space x, ∂(b◦) ⊆ ∂b, and equality holds if and only if b ⊆ cl(b◦). in particular, equality holds if b is open or is regular closed. proof. if x ∈ ∂(b◦) = cl(b◦) ∩ cl(x − b◦), then x ∈ cl(b◦) ⊆ cl(b), and x ∈ cl(x −b◦) = x −b◦. thus, no neighborhood of x is contained in b, so every neighborhood of x intersects x −b, so x ∈ cl(x −b) ∩ cl(b) = ∂b. it remains to show that ∂b ⊆ ∂(b◦) if and only if b ⊆ cl(b◦). suppose b ⊆ cl(b◦) and x ∈ ∂b = clb ∩ cl(x −b). now x ∈ clb ⊆ clcl(b◦) = cl(b◦) and x ∈ cl(x−b) ⊆ cl(x−b◦) (since b◦ ⊆ b), so x ∈ cl(b◦)∩cl(x−b◦) = ∂(b◦). conversely, suppose b 6⊆ cl(b◦) and choose an x ∈ b−cl(b◦). now x 6∈ cl(b◦) implies x 6∈ ∂(b◦) = cl(b◦) ∩ cl(x − b◦). now x 6∈ cl(b◦) implies x 6∈ b◦, so every neighborhood of x intersects x −b, and thus x ∈ cl(x −b). since x ∈ b ⊆ cl(b), we have x ∈ cl(b) ∩ cl(x −b) = ∂b. thus, ∂b 6⊆ ∂(b◦). � for examples where ∂b 6⊆ ∂(b◦), take b = q in x = r or b = b̄((0, 0), 1)∪ ({0}× [0, 3]) in r2. from proposition 3.5, we deduce the following. proposition 3.6. if a and b are subsets of a topological space with a◦ ⊆ b◦ and ∂a∩∂b = ∅, then ∂(a◦) ⊆ b◦. for the next proposition, we will use the following lemma. lemma 3.7 ([1, lemma 6.16]). suppose a and b are subsets of a topological space, ∂a∩b = ∅, and b is connected. then either b ⊆ a◦ or b∩cl(a) = ∅. part (a) of the next result follows from proposition 3.4 and lemma 3.7, and part (b) follows from part (a), and propositions 3.5 and 3.6. proposition 3.8. suppose a and b are subsets of a locally connected topological space x with ∂a∩b◦ = ∅, ∂a∩∂b = ∅, and a◦ ∩b◦ 6= ∅. let ax be the connected component of a which contains x, with bx defined analogously. then (a) for any x ∈ a◦ ∩b◦, bx ⊆ a◦x, and c© agt, upv, 2017 appl. gen. topol. 18, no. 1 190 transitions between 4-intersection values of planar regions (b) if ∂bx 6= ∅ and ∂bx ∩a◦x = ∅, then b◦x ∩a◦ = ∅. corollary 3.9. if a and b are subsets of r2 with 4-intersection value 5 (0, 1, 0, 0), then a = b = r2. theorem 3.10. suppose a(t) and b(t) are closed subsets of r2 and the area of the components ax(t) and bx(t) containing x as well as the areas of the intersections of the components ax(t) ∩ bx(t) are continuous extended realvalued functions of time. then the transitions from value 6 (0, 1, 0, 1) to value 7 (0, 1, 1, 0), value 6 (0, 1, 0, 1) to value 15 (1, 1, 1, 0), and value 7 (0, 1, 1, 0) to value 14 (1, 1, 0, 1) are not possible without passing through intermediate values. proof. suppose value 6 transforms to value 7, with value 6 occurring for time t < a and value 7 occurring for time t > a. note that a◦ ∩ b◦ remains nonempty at all times, and for every x ∈ a◦∩b◦, the components ax and bx of a and b, respectively, containing x must either have an intersection value of value 6 or of value 7, by lemma 3.7. for t < a, we have ∂bx ∩ a◦x 6= ∅ for those components ax,bx of x ∈ a◦ ∩ b◦ which have value 6. when these ∂bx ∩ a◦x become empty at or after t = a, b◦x ∩ a◦x becomes empty by proposition 3.8(b). thus, the area of a◦ ∩b◦ in components of value 6 goes to zero at or after t = a. since this area changes continuously, there is a first instant of no area, so this area in components of value 6 is zero at time t = a. to maintain a◦∩b◦ 6= ∅ at time t = a, there must be nonzero area in a◦∩b◦ in components of value 7 at time t = a. however, the same argument applied to the area in components of value 7 as time decreases to t = a, shows that there is no last instant of area in components of value 7. thus, if such area is nonzero at t = a, it was nonzero for some values t < a, when there was also nonzero area in components of value 6, and the presence of nonzero area in components of values 6 and 7 simultaneously for t ≤ a gives an intermediate value 8 (0, 1, 1, 1). the proof that value 6 cannot transform to value 15 is similar, and its dual shows that value 7 cannot transform to value 14. � note that in theorem 3.10 we have weakened the assumptions so that a(t) and b(t) need only be closed sets with continuous extended real-valued areas and areas of intersections. in this generality, we may now again consider the exceptional value 5, which by corollary 3.9, is only realized as a = b = r2. proposition 3.11. if a(t) and b(t) are closed subsets of the plane with a(t), b(t) and ab(t) continuous extended real-valued functions, then value 5 is not adjacent to values 1–4 nor values 9–12, and is adjacent to values 6, 7, 8, and 13–16. proof. values 1–4 and 9–12 have a◦∩b◦ = ∅. value 5 has the area of a◦∩b◦ being infinite. this jump from 0 area to infinite area is not permitted by the assumption of continuously changing area. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 191 k. bell and t. richmond value 5 adjacencies are most easily seen moving to value 5. value 6 is realized by taking b to be the closed ball b̄((0, 0),r) = {(x,y) ∈ r2 : x2 +y2 ≤ r} and a = b̄((0, 0),r + 1). as r converges to infinity, a and b converge to r2, giving value 5. value 7 is dual to value 6. value 8 is realized by a = b̄((0, 0),r + 1)∪b̄((2r, 0), 1) and b = b̄((0, 0),r)∪b̄((2r, 0), 2), value 13 by a = b = b̄((0, 0),r), value 14 by a = [−r,r+1]2 and b = [−r,r]2, value 15 by interchanging a and b in value 14, and value 16 by a = (−∞,r] × [r,∞) and b = [−r,∞) × (−∞,r]. in each case, these values converges to value 5, a = b = r2, as r goes to infinity. � theorem 3.10 and propostion 3.11 listed the non-adjacencies for closed sets under continuity of area conditions, shown in figure 3. indeed, these are the only non-adjacencies under these assumptions. 9 10 11 12 1 2 3 4 5 6 7 15 14 a a h hh� � � �� a a h hh figure 3. non-adjacency graph for closed regions with a(t),b(t), and ab(t) continuous extended real-valued functions. the following constructions may be used to show adjacencies. construction 1: create a whisker instantly. delete an interior instantly. consider the set c = [−r,r] × [0, 1] as r changes continuously with time. for r < 0, c = ∅, for r = 0, c = {0}× [0, 1] is a whisker, and for r > 0, c is a regular closed set with nonempty interior. as r increases to zero, the whisker appears instantly when r = 0. as r decreases to zero, the area disappears at the instant r = 0. note that the area of c changes continuously with r, and thus with time. construction 2: delete a whisker at an instant. consider the set d = cl(−r,r) × {0} as r changes continuously with time. if r ≤ 0, d = ∅. if r > 0, d = [−r,r] ×{0}, a whisker. letting r decrease to zero, the whisker will disappear instantly when r = 0. note that the length of the whisker d changes continuously with r, and thus with time. construction 3: create interior and boundary directly and simultaneously, without first creating a boundary. consider the set e = cl(−r,r) × [0, 1] as r changes continuously with time. if r ≤ 0, e = ∅. if r > 0, e = [−r,r] × [0, 1], a regular closed set with nonempty interior. letting r decrease to zero, the interior and boundary both disappear simultaneously at the instant when r = 0. reversing this operation, the area and boundary appear simultaneously and directly for r > 0. note that the area of e changes continuously with r and c© agt, upv, 2017 appl. gen. topol. 18, no. 1 192 transitions between 4-intersection values of planar regions thus with time. we observe that since area is a continuous function, it can vanish at an instant, but it cannot appear at an instant—only directly. constructions 1 and 2 allow us to create whiskers at an instant or delete whiskers at an instant, facilitating many transitions between 4-intersection values. if area needs to be created to cause a◦ to intersect b◦ or ∂b, then construction 3 allows us to create area directly, and whiskers may be added or deleted directly by constructions 1 or 2 in reverse. while construction 3 creates area directly and continuously, we cannot allow arbitrary use of it. for example, for x ∈ r, let ex = cl(x − r,x + r) × [0, 1]. as r continuously increases, the area of ex continuously increases, becoming nonnegative when r > 0. however, if we let e = ⋃ {ex : x ∈ q ∩ [0, 1]}, the area of e is not continuous. for r ≤ 0, the area is zero, and for every r > 0, the area is greater than 1. the problem here is that we are adding area based on a dense set q ∩ [0, 1]. using any nowhere dense index set would produce a continuous area function, and in particular, using any finite number of sets created by construction 3 will produce a continuous area function. 3.1. realizations of adjacencies. with the aid of these three constructions forwards and in reverse, it is not surprising that the remaining constructions are achievable. specific examples showing the adjacencies are given below. by construction n← we mean construction n in reverse. value 1 is adjacent to values 2, 3, 4, 9, 10, 11, and 12 by adding a whisker to a, b, or both simultaneously using construction 1. it is adjacent to values 6, 7, 8, 13, 14, 15, and 16 by adding a new component of a, b, or both simultaneously using construction 3. values 1–4 are not adjacent to value 5 for reasons given in the discussion of value 5. value 2 is adjacent to value 3 by instantly adding a whisker of a in the interior of b and instantly deleting the whiskers of b in the interior of a simultaneously, using constructions 1 and 2. it is adjacent to value 4 by adding a whisker to a in an existing second component of b◦ by construction 1. fattening the whisker of b in a◦ to have positive area by construction 1← shows adjacency to value 6. deleting the whiskers of b in components of a◦ and simultaneously adding a component of b with boundary and interior in a component of a◦ gives value 7, using constructions 1← and 3. value 8 is obtained by creating new components of a with area and boundary in b◦ and of b in a◦ directly and simultaneously using construction 3 twice. values 9 and 10 or obtained by deleting the whiskers of b in a◦ instantly and simultaneously adding a whisker to a component of a which intersects the boundary or boundary and interior of a component of b, using constructions 1 and 2. value 11 requires no construction: simply slide the whisker of b in a◦ to intersect ∂a. value 12 is obtained by adding new whiskers to a and b simultaneously using construction 1. for value 13, use construction 3 to directly create a new region with area and boundary, which will become a new component of both a and b c© agt, upv, 2017 appl. gen. topol. 18, no. 1 193 k. bell and t. richmond while simultaneously deleting the whisker of b in a◦ using construction 1←. using construction 3 to directly add a component of b with positive area in a which shares a boundary with a gives value 14. value 15 is obtained by deleting the whisker of b in a◦ and directly adding a new component of b with positive area in a by which shares a boundary with a, using constructions 1← and 3. for value 16, slight modifications of example 3.1 or the argument above proposition 3.2 by adding a whisker of b in a gives the needed transition. value 3 is symmetric to value 2 by interchanging a and b. thus, it is adjacent to every other value except value 5. value 4 is adjacent to values 6, 7, and 8 by fattening a whisker (or two) to have positive area, using construction 1←. it is adjacent to values 9, 10, and 11 by simply sliding a one-point interior whisker until it intersects the boundary, or a linear whisker until it intersects a parallel linear boundary. value 12 is achieved by staring with a component of a◦ containing a linear whisker of b and a component of b◦ containing a linear whisker of b, and sliding the whiskers so that one endpoint of each simultaneously touches the boundary of its enclosing set. for value 13, directly delete all whiskers of a in b◦ and all whiskers of b in a◦ by construction 1← and simultaneously create a region with positive area and boundary by construction 3 which will be a new component of both a and b. for value 14, directly delete whiskers of a in b◦ by construction 1← and simultaneously create a component of b with positive area and boundary by construction 3 which is in a and intersects ∂a. value 15 is symmetric to value 14 by interchanging a and b, and value 16 is achieved by performing the transition to values 14 and 15 simultaneously in some components of a and b. value 5 adjacencies were given in proposition 3.11. value 6 is not adjacent to values 7 and 15 by theorem 3.10. value 6 transitions to value 8 by an application of construction 3, and to value 9 by an application of construction 1← simultaneous with the intersection of boundaries as components of a and b slide together. or, with a = [0, 3]2 and br = [1, 2] × [ 1r+1, 1 r ], letting r approach infinity gives the transition from value 9 to value 6. values 10, 11, and 12 are adjacent to value 6 by simultaneous application of construction 1← to delete that part of b in a◦ and construction 1 to add whiskers. values 13 and 14 are adjacent to value 6 using spatial regions such as two nested squares, with the inner one expanding to equal or sliding to intersect the outer. value 6 is realized by a = [−2, 2]2 and b = [−1, 1]2 ∪ [3, 5]2; adding a whisker [4, 6]×{4} to a by construction 1 gives value 16. value 7 is symmetric to value 6. value 8 transforms to value 9 (value 12) by two copies of the transition from value 6 to value 9 (value 10), with a and b interchanged in the second copy. with a = [0, 3]2∪[5, 6]×[ 1 r+1 , 1 r ] and br = [1, 2]×[1− 1r+1, 1+ 1 r ]∪[4, 7]×[0, 3], as r goes to infinity, a and b go from value 8 to value 10. value 11 is symmetric to value 10. with a = b̄((0, 0), 1)∪b̄((5, 5),r) and b = b̄((0, 0),r)∪b̄((5, 5), 1), a and b go from value 8 to value 13 as r increases to 1. with this a and b for c© agt, upv, 2017 appl. gen. topol. 18, no. 1 194 transitions between 4-intersection values of planar regions r = 1/2, adding a whisker [0, 5]×{0} to a or b by construction 1 gives value 16. value 8 is realized by a = [0, 3]2 ∪([5, 6]×{y}) and b = [1, 2]2 ∪([4, 7]× [0, 3]) for y ∈ (0, 1] and transitions to value 14 when y decreases to 0. value 15 is symmetric to value 14. value 9 is realized by a = [0, 3]2∪([3, 4]×{1} and b = ([4, 5]×[0, 3])∪([3, 4]× {2}). extending one or both of the whiskers into the other set transitions to values 10, 11, and 12. with a = b = [0, 1] × [0,r], we have value 9 for r = 0 and value 13 for r > 0. with a = [−2, 2] × [0, 1] and b = [−1, 1] × [0,r], we have value 9 for r = 0 and value 14 for r > 0; value 15 is symmetric. value 16 is easily obtained by translating the sets. value 10 transform to values 11 and 12 by the deletion and addition of a whisker using constructions 1 and 2, and to values 13, 14, and 15 by deleting a whisker directly (construction 1←) and adding a region with positive area as a new component of both a and b, a component of b in a, or of a in b, directly by construction 3. with a = [−2, 2]2 and b = [1, 3] × [1,r], we have value 10 for r = 0 and value 16 for r > 0. value 11 is symmetric to value 10. value 12 transforms to values 13, 14, and 15 just as value 10 does, with the simultaneous deletion of a whisker. value 13 is realized by a = b = b̄((0, 0),r). by shrinking the radius on one of the sets, we transform to values 14 and 15, and by shifting the center of one, to from 16. value 14 is realized by a = [0, 3]2 ∪ ([4, 6] × [0, 3]) and b = ([1, 2] ×{1}) ∪ ∪([4, 6] × [0, 3]) ∪ ([7, 10] × [0, 3]). by deleting the whisker [1, 2] ×{1} of b in a◦ and adding a whisker [8, 9] ×{1} to a in b◦ by constructions 1 and 2, we transform to value 15. value 16 is realizable by spatial regions by translating one region. value 15 is symmetric to value 14. 4. upper and lower semicontinuity as seen in the previous sections, continuity of the areas of a, b, and a∩b is not always a good model for continuous morphing. another way to model continuous deformation involves upper and lower semicontinuity. definition 4.1. a function b : r →p(r2) is upper semicontinuous (or u.s.c. or upper vietoris continuous) at t if for every open set m ⊆ r2 with b(t) ⊆ m, there exists a neighborhood n of t with b(t′) ⊆ m for all t′ ∈ n. a function b : r → p(r2) is lower semicontinuous (or l.s.c. or lower vietoris continuous) at t if for every open set m ⊆ r2 with b(t)∩m 6= ∅, there exists a neighborhood n of t with b(t′) ∩ m 6= ∅ for all t′ ∈ n. a function which is both u.s.c. and l.s.c. at t is vietoris continuous at t, or continuous with respect to the vietoris topology at t. see [10, 8]. we note that the well-known hausdorff distance between nonempty compact sets in a metric space x generates a topology on the collection k0 of nonempty c© agt, upv, 2017 appl. gen. topol. 18, no. 1 195 k. bell and t. richmond compact subsets of x known as the hausdorff topology. the hausdorff topology agrees with the vietoris topology on k0 (corollary 4.2.3 of [10]). upper semicontinuity prevents b from expanding beyond a neighborhood of it quickly. construction 3 of section 3 was not u.s.c. however, neither u.s.c., l.s.c, nor u.s.c. and l.s.c. together imply continuity of area. for example, if b(t) = [−2, 2]2 for t ≥ 0 and b(t) = [−1, 1]2 for t < 0, then b(t) is u.s.c at every point, but the area is discontinuous at t = 0, and b(t) is not l.s.c. at t = 0. with a(t) = [−2, 2]2 for t > 0 and a(t) = [−1, 1]2 for t ≤ 0, a is l.s.c. everywhere but not u.s.c. at t = 0 where the area jumps discontinuously. example 4.2(a) shows that u.s.c. and l.s.c. together do not imply continuity of area. the non-compact sets of example 3.1 are both changing upperand lowersemicontinuously, so the additional assumption of u.s.c and l.s.c is not sufficient to prevent closed sets from morphing from disjoint to overlapping directly without first passing through the “meets” value. the next example shows that such a transition is still possible for compact sets. example 4.2 (comb spaces). the following values are adjacent using closedvalued u.s.c. and l.s.c. functions a(t) and b(t). (a) disjoint is adjacent to overlaps. define a(t) as follows: a(t) = [0, 1] ×{0} for t ≥ 1 a(2−n) = a(1) ∪{2−nm}× [0, 1] : m = 0, 1, . . . , 2n} for n ∈ n a(t) = a(1) ∪{2−nm : m = 0, 1, . . . , 2n}× [0, 2n+1(t− 2n+1)] for t ∈ [2−n−1, 2−n] a(t) = [0, 1]2 for t ≤ 0. thus, at t = 2−n, a(t) is a comb with base [0, 1]×{0} and teeth of height 1 at each x = m 2n (m ∈ {0, 1, 2, . . . , 2n}). in the time interval between t = 2−n and t = 2−n−1, the new teeth grow continuously from the base at the midpoints between existing teeth until they reach height 1. it is easy to check that a(t) is both u.s.c. and l.s.c, but has a discontinuous jump in area at t = 0. furthermore, a(t) is a compact set at every t ∈ r. let b(t) be the reflection of a(t) over the line y = 7 8 translated to the left by s(t) where s(t) is piecewise linear, s(t) = 0 for t ≤ 0, s(t) = 1 2 for t ≥ 1, and s(2−n) = 2−n−1 = half the distance between existing teeth at time t = 2−n. then a(t) and b(t) are u.s.c and l.s.c. compact-valued functions with a(t) ∩b(t) = ∅ for t > 0, but a(0) ∩b(0) = [0, 1] × [ 3 4 , 1]. in particular, a(t) and b(t) transform from disjoint to overlaps without passing through meets. (b) disjoint is adjacent to value 2 (0, 0, 0, 1). let a(t) be as above, and let b′(t) = {1 2 −s(t)}× [ 3 4 , 7 8 ] be the segment of the center tooth of the comb b(t) with 3 4 ≤ y ≤ 7 8 for t < 1, and b′(t) = {1 2 }× [ 3 4 , 7 8 ] for t ≥ 1. example 4.2(b) seems to show that a whisker of b may instantly appear in the interior of a without introducing any other intersections among boundaries c© agt, upv, 2017 appl. gen. topol. 18, no. 1 196 transitions between 4-intersection values of planar regions and interiors, even though a and b were disjoint and separated by open sets before that instant. this may seem to violate u.s.c. of b. however, the crux of our example is that the whisker of b is not created at that instant, but rather the interior of a engulfs the whisker at that instant. the comb spaces in the example above are compact at each value of t, but are not always regular closed. this can be readily remedied by fattening each segment of the comb slightly. alternately, our next example gives a variation of the comb space obtained by making the teeth triangular spikes and shows that even if the spaces are always regular closed (or indeed, spatial regions), disjoint is adjacent to overlaps under the u.s.c. and l.s.c. continuity conditions. example 4.3. by the spike centered at x = a of width w and height h, we mean the closed triangular region s(a,w,h) having vertices (a− w 2 , 0), (a,h), and (a + w 2 , 0). define a(t) as follows. for t = 1 − 2−n (n ∈ n), a(t) = [0, 2] × [−1 2 , 0] ∪{s(2−nm, 2−(n+1), 1) : m ∈ {1, 2, . . . , 2n}}. thus, a(t) consists of a rectangular base together with 2n spikes of width 2−(n+1) and height 1. the combined area of the base and spikes is 1 + 1 4 . for t ∈ (1 − 2−n, 1 − 2−(n+1)), we will shrink the widths of the existing spikes by half linearly with time (so their total area decreases from 1 4 to 1 8 ) and create new spikes midway between existing spikes whose areas increase linearly with time from 0 to 1 8 as their heights increase from 0 to 1. specifically, for t ∈ (1 − 2−n, 1 − 2−(n+1)), let t′ = t − (1 − 2−n), so t′ ∈ (0, 2−(n+1)). let h(t′) = 2n+1t′ be the linear function with h(0) = 0 and h(2−(n+1)) = 1. define a(t′) = [0, 2]×[−1 2 , 0]∪{s(2−nm, h(t ′) 2 2−(n+1), 1) : m ∈{1, 2, . . . , 2n},m even}∪{s(2−nm, t ′ 2 ,h(t′) : m ∈ {1, 2, . . . , 2n},m odd}. now for all t ∈ (1 − 2−n, 1 − 2−(n+1)), the area of a(t) is 1 + 1 4 . for t ≤ 1 2 , put a(t) = a( 1 2 ). for t ≥ 1, put a(t) = ([0, 2] × [−1 2 , 0]) ∪ ([0, 1] × [0, 1]). now a(t) is u.s.c. and l.s.c., and a(t) is regular closed and compact (indeed, is a compact spatial region) at every value of t. but, the area of a(t) jumps discontinuously at t = 1. with b(t) defined in terms of a(t) precisely as in the last paragraph of example 4.2, the comments still apply, and a(t) and b(t) transform directly from disjoint to overlaps. below we use a spiral construction for a similar space-filling example. example 4.4 (spirals). throughout, adjacencies refer to those achieved by u.s.c. and l.s.c. functions. these examples are also compact-valued. (a) disjoint and equals are adjacent. for t ∈ [0, 1), let a(t) = {(r,θ) : r = (1 − t)θ,θ ∈ [0, 1 1−t]} and b(t) = {(r,θ) : r = (1 − t)(θ + π),θ ∈ [0, 1 1−t −π]}. for t < 0, put a(t) = a(0) and b(t) = b(0). for t ≥ 1, put a(t) = b(t) = {(r,θ) : r ≤ 1}. now for t ∈ [0, 1), a(t) and b(t) are disjoint archimedean spirals with an increasing number of coils winding tighter around the origin and staying inside the unit circle. now a(t) and b(t) are seen to be u.s.c. and l.s.c. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 197 k. bell and t. richmond compact-valued functions with discontinuous area (at t = 1). furthermore, a(t) and b(t) are disjoint for t < 1 and are equal for t ≥ 1, showing that disjoint and equals are adjacent. (b) disjoint and contains are adjacent. for t ≥ .9, let a(t) and b(t) be as above and put b′(t) = b(t) ∩{(r,θ) : r ≤ 1 2 } for t ≥ .9. (note that we take t ≥ .9 only to assure that b(t) 6= ∅.) now the unit disk a(1) contains the disk b′(1) of radius 1 2 , and a(t) and b′(t) are the desired functions. (c) disjoint and covers are adjacent. for t ≥ .9, let a(t) and b(t) be as above and put b′(t) = b(t) ∩{(r,θ) : r ≥ 1 2 } for t ≥ .9. now the unit circle a(1) covers the annulus b′(1). (d) disjoint and value 2 (0, 0, 0, 1) are adjacent. for t ≥ .9, let a(t) and b(t) be as above. now b(t) restricted to the closed first quadrant q1 contains many components. let b′′(t) be the component of b(t) ∩ q1 closest to the origin, and b′′(0) = {(0, 0)}. now as time increases to 1, the spirals wind tighter and b′′(t) converges to the origin in a u.s.c., l.s.c. manner, as needed. (e) disjoint and value 5 are adjacent. recall that value 5 only occurs if a = b = r2. for t > 0, let a(t) = {(r,θ) : r = tθ,θ ≥ 0} and b(t) = {(r,θ) : r = t(θ + π),θ ≥ 0}. for t ≤ 0, let a(t) = b(t) = r2. now for t > 0, a(t) and b(t) are disjoint non-compact closed archimedean spirals whose coils are becoming more tightly coiled as t decreases to 0. these functions have the required properties to prove the claim. (f) disjoint and overlap are adjacent. this may be achieved by two disjoint copies of sets as in (c), with the second copy translated to remain disjoint and with the labels for a and b interchanged on that copy. while the sets a(t) and b(t) of example 4.4 are not regular closed sets for t < 1, , it is easy to see that these spiral curves may be fattened slightly to obtain regular closed sets illustrating the desired properties. formally, the sets a(t) for t < 1 may be replaced by their �(t)-fattening a(t)�(t) ⋃ {b(x,�(t)) : x ∈ a(t)} and similarly b(t) by b(t)�(t), for a function �(t) decreasing to zero quickly enough to insure that a(t)�(t) and b(t)�(t) remain disjoint. indeed, such a modification of example 4.4(f) shows that disjoint and overlaps are adjacent for compact, regular closed, nonempty, connected spaces even if a◦(t) and b◦(t) are u.s.c. and l.s.c. transitioning from one value to another requires the introduction or deletion of intersections between boundaries and interiors. we summarize some possible transitions below. recall that transitioning from value vi to value vj at the instant t = 0 means that vi exists for t < 0 (or t > 0) and vj exists at t = 0. note that the conditions on deleting intersections at an instant are more restrictive and have implications on other intersection values. proposition 4.5. assuming a(t) and b(t) are u.s.c. and l.s.c. closed-valued functions from r to p(r2), it is possible to introduce the following combinations of new intersections at an instant: (a) ∂a ∩ ∂b, (b) a◦ ∩ b◦ and ∂a ∩ b◦, (c) ∂a ∩ ∂b and a◦ ∩ b◦, (d) ∂a ∩ b◦, and (e) ∂a ∩ ∂b and a◦ ∩ b◦ and ∂a∩b◦. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 198 transitions between 4-intersection values of planar regions the following intersections may be deleted at an instant: (f ) a◦∩b◦, leaving ∂a ∩ ∂b, (g) ∂a ∩ b◦, leaving ∂a ∩ ∂b, (h) a◦ ∩ b◦, leaving a◦ ∩ ∂b, (i) ∂a ∩ b◦ and a◦ ∩ b◦, introducing ∂a ∩ ∂b, and (j) ∂a ∩ b◦, introducing ∂a∩∂b. proof. (a) let a(t) = [t, 1 + t] × [0, 1] for t > 0, a(t) = [0, 1]2 for t ≤ 0, and b(t) = [−1, 0] × [0, 1]. (b), (c), (d), and (e) are shown by parts (b), (a), (d) and (c) of example 4.4, respectively. (f) with a(t) as in part (a), let b(t) = [1, 2] × [−2, 2]. (g) let a(t) = [t, 1 + t] ×{0} for t > 0, a(t) = [0, 1] ×{0} for t ≤ 0, and b(t) = [1, 2]× [−2, 2]. (h) let a(t) = [−2, 2]2, b(t) = [−1, 1]× [0, t] for t > 0 and b(t) = [−1, 1] ×{0} for t ≤ 0. (i) let a(t) = [−2, 2] × [0, 2], b(t) = [−1, 1]× [t/2, t] for t ∈ (0, 1], and b(t) = [−1, 1]×{0} for t ≤ 0. (j) let a(t) = [−2, 2]×[0, 2], b(t) = [−1, 1]×{t} for t ∈ (0, 1], and b(t) = [−1, 1]×{0} for t ≤ 0. � proposition 4.5(a) says that it is possible to transition from value (0,y,z,w) to (1,y,z,w) at an instant. proposition 4.5(b) shows that it is possible to transition from (x, 0, 0,w) to (x, 1, 1,w), or indeed if nonzero intersection values exist in other static components of a and b, it is possible to transform from (x,y,z,w) to (x, 1, 1,w) at an instant. the dual of (b) obtained by interchanging the labels of a and b shows (x,y,z,w) to (x, 1, 1,y). similarly, (b) through (e) show transformations at an instant which toggle zeros to ones. parts (f) through (j) describe some possible transitions toggling a one to a zero, but these have restrictions on other intersection values. part (f) shows that (1, 1,z,w) transitions to (1, 0,z,w) at an instant; (g) and its dual show (1,y, 1,w) is adjacent to (1,y, 0,w) and (1,y,z, 1) is adjacent to (1,y,z, 0); (j) shows that (0,y, 1,w) is adjacent to (1,y, 0,w). corollary 4.6. using u.s.c. and l.s.c functions, disjoint is adjacent to each of the other 15 values. proof. disjoint is the value (0, 0, 0, 0). transitions to other values only add intersections. using the parts of example 4.4 in the proper combinations allows adding all possible intersections. specifically, (a), (d) and the dual of (d) show, respectively, that zeros in the first, third, or fourth positions may be toggled to ones, while (c), (b), and the dual of (b) show that a zero in the second position may be toggled to one together with, respectively, zeros in the first, third, or fourth positions. combining these, the only toggling from zeros to ones not accounted for is from (0, 0, 0, 0) to (0, 1, 0, 0). this is shown in example 4.4(e). � if value vi transitions to vj at an instant, then reversing time, the functions remain u.s.c. and l.s.c. and show that vj transitions to vi directly. as noted in the proof of corollary 4.6, all combinations of toggling from zeros to ones are possible at an instant. thus, transitions between values which only require deletions are all possible directly, except possibly transitions from the exceptional value 5 a = b = r2. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 199 k. bell and t. richmond some transitions, such as from covers to covered by (value 14 (1, 1, 0, 1) to value 15 (1, 1, 1, 0)), require simultaneous creation and deletion of certain intersection values. that is, both a zero and a one must be toggled. some of these will be possible at an instant or directly using the results of proposition 4.5, but some are not. the next result shows some transitions are not possible at an instant. proposition 4.7. if a(t) and b(t) are closed-valued, u.s.c., and a(t)∩b(t) 6= ∅ for t ∈ (−�, 0) for some � > 0, then a(0) ∩b(0) 6= ∅. thus, none of the 15 other values can transition to disjoint at an instant. furthermore, if a(t) and b(t) are regular closed for all t and a(t)∩b(t) 6= ∅ for t ∈ (−�, 0) for some � > 0, then they may not transition to any value (0, 0,z,w) at the instant t = 0. proof. suppose to the contrary that for every � > 0 there exists a t ∈ (−�, 0) with a(t) ∩ b(t) 6= ∅, and a(0) ∩ b(0) = ∅. since r2 is normal, there exist disjoint open sets ga and gb with a ⊆ ga, b ⊆ gb. by u.s.c., a(t) ⊆ ga and b(t) ⊆ gb for all t in (−�, 0), contradicting our the assumption. furthermore, if the sets are regular closed, the only permissible value (0, 0,z,w) is (0, 0, 0, 0) (when a(0) and b(0) are disjoint) since for regular closed sets, the boundary of one intersecting the interior implies the interiors intersect. � recall that under the continuity of area restrictions of theorem 3.10, values 6 and 15 were not adjacent. they are adjacent using u.s.c. and l.s.c. functions. indeed, let a1(t) and b1(t) be as in example 4.4(b) for t ≥ .9, a2(t) = [11, 12] × [(1 − t)/2, 1 − t] for t ∈ (.9, 1), a2(t) = [11, 12] ×{0} for t ≥ 1, and b2(t) = [10, 13] × [0, 1] for t ≥ .9. now a(t) = a1(t) ∪ a2(t) and b(t) = b1(t) ∪b2(t) show that value 6 transforms to value 15 at the instant t = 1. 5. continuous area, u.s.c, and l.s.c. we have seen that the values disjoint and overlaps are, somewhat surprisingly, adjacent under our previous assumptions of (a) continuity of areas of a(t),b(t), and the intersections of their components, or (b) u.s.c. and l.s.c. if we assume both sets of assumptions, then disjoints is not adjacent to overlaps. theorem 5.1. suppose a(t) and b(t) are u.s.c. functions with values being closed sets with finite areas, and the area of a(t)∩b(t) is a continuous function. then (0, 0, 0, 0) is not adjacent to (x, 1,z,w). proof. the basic idea is that u.s.c. prevents a◦ from hopping inside b◦ to introduce nonempty intersection of the interiors, and the continuity of the area prevents b◦ from engulfing a◦. indeed, proposition 4.7 shows that (x, 1,z,w) cannot transform to disjoint at an instant. if disjoint transformed to (x, 1,z,w) at an instant, then there would exist a(t), b(t) with a(t)∩b(t) = ∅ for t < 0 and a◦(0) ∩ b◦(0) 6= ∅. then if a(t) is the area of a(t) ∩ b(t), we have (−∞, 0) ⊆ a−1({0}) but 0 6∈ a−1({0}), so the inverse image of the closed set {0} is not closed and thus a(t) is not continuous. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 200 transitions between 4-intersection values of planar regions in conclusion, which 4-intersection values are adjacent to which depends heavily on the assumptions. in applications, there are many examples of dynamic regions which need not be connected or regular closed, but we have seen that allowing this generality, all transitions between 4-intersection values are possible except those given in theorem 3.10 and proposition 3.11 (see figure 3). restricting the regions to be spatial regions allows fewer adjacencies, but also fewer applications. the permissible adjacencies depend also on the types of dynamic morphing allowed. continuity of area was a weak assumption and vietoris continuity provides some better results, but still allowed the adjacency of disjoint and overlaps. assuming continuity of area together with u.s.c. and l.s.c. provided one setting where disjoints was not adjacent to overlaps. references [1] c. adams and r. franzosa, introduction to topology: pure and applied, pearson prentice hall, upper saddle river, nj, 2008. 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[11] y. kurata and m. egenhofer, the 9+-intersection for topological relations between a directed line segment and a region, in: proceedings of the 1st workshop on behavioral monitoring and interpretation. tzi-bericht, technologie-zentrum informatik, universität bremen, germany, vol. 42, 2007, pp. 62–76. [12] d. mark and m. egenhofer, an evaluation of the 9-intersection for region-line relations, san jose, ca: gis/lis ’92, (1992) 513–521. [13] k. nedas, m. egenhofer and d. wilmsen, metric details of topological line-line relations, international journal for geographical information science 21, no. 1 (2007), 21–48. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 201 k. bell and t. richmond [14] a. j. roy and j. g. stell, indeterminate regions, internat. j. approximate reasoning 27 (2001), 205–234. [15] t. smith and k. park, algebraic approach to spatial reasoning, international journal for geographical information systems 6, no. 3 (1992), 177–192. [16] j. wu, c. claramunt and m. deng, modelling movement patterns using topological relations between a directed line and a region, iwgs ’14 proceedings of the 5th acm sgispatial international workshop on geostreaming. new york: acm, 2014, 43–52. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 202 @ appl. gen. topol. 18, no. 1 (2017), 143-152 doi:10.4995/agt.2017.6663 c© agt, upv, 2017 digital shy maps laurence boxer department of computer and information sciences, niagara university, new york 14109, usa; and department of computer science and engineering, state university of new york at buffalo (boxer@niagara.edu) communicated by s. romaguera abstract we study properties of shy maps in digital topology. 2010 msc: 54c99; 05c99. keywords: digital image; continuous multivalued function; shy map; isomorphism; cartesian product; wedge. 1. introduction continuous functions between digital images were introduced in [15] and have been explored in many subsequent papers. a shy map is a continuous function with certain additional restrictions (see definition 2.5). shy maps were studied in [4, 6, 9]. in the current paper, we develop additional fundamental properties of shy maps. in section 3, we examine relationships between shy maps and other types of functions between digital images, such as constant functions and isomorphisms. theorem 3.2 characterizes shy maps in terms of properties of their multivalued inverse functions. this suggests studying relations between other types of continuous surjections f : (x,κ) → (y,λ) between digital images and their multivalued inverse functions f−1 : y ( x. in section 4, we show that composition, cartesian products using the normal product adjacency, and wedges preserve shyness. in section 5, we develop several special properties of shy maps into (z,c1). received 28 september 2016 – accepted 24 december 2016 http://dx.doi.org/10.4995/agt.2017.6663 l. boxer 2. preliminaries let n be the set of natural numbers, and let z be the set of integers. a digital image is a pair (x,κ) where x ⊂ zn for some positive integer n, and κ is an adjacency relation for zn. thus, a digital image is a graph in which x is the set of vertices and edges correspond to κ-adjacent points of x. much of the exposition in this section is quoted or paraphrased from the papers referenced. 2.1. digitally continuous functions. we will assume familiarity with the topological theory of digital images. see, e.g., [2] for the standard definitions. all digital images x are assumed to carry their own adjacency relations (which may differ from one image to another). we write the image as (x,κ), where κ represents the adjacency relation, when we wish to emphasize or clarify the adjacency relation. among the commonly used adjacencies are the cu-adjacencies. let x,y ∈ zn, x 6= y. let u be an integer, 1 ≤ u ≤ n. we say x and y are cu-adjacent if • there are at most u indices i for which |xi −yi| = 1. • for all indices j such that |xj −yj| 6= 1 we have xj = yj. we often label a cu-adjacency by the number of points adjacent to a given point in zn using this adjacency. e.g., • in z1, c1-adjacency is 2-adjacency. • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency. • in z3, c1-adjacency is 6-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. a subset y of a digital image (x,κ) is κ-connected [15], or connected when κ is understood, if for every pair of points a,b ∈ y there exists a sequence p = {yi}mi=0 ⊂ y such that a = y0, b = ym, and yi and yi+1 are κ-adjacent for 0 ≤ i < m. the set p is called a path from a to b. the following generalizes a definition of [15]. definition 2.1 ([3]). let (x,κ) and (y,λ) be digital images. a function f : x → y is (κ,λ)-continuous if for every κ-connected a ⊂ x we have that f(a) is a λ-connected subset of y . � when the adjacency relations are understood, we will simply say that f is continuous. given positive integers u,v such that u ≤ v a digital interval is a set of the form [u,v]z = {z ∈ z |u ≤ z ≤ v} treated as a digital image with the c1-adjacency. the term path from a to b is also used for a continuous function p : ([0,m]z,c1) → (y,κ) such that f(0) = a and f(m) = b. context generally clarifies which meaning of path is appropriate. continuity can be reformulated in terms of adjacency of points: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 144 digital shy maps theorem 2.2 ([15, 3]). a function f : x → y between digital images is continuous if and only if, for any adjacent points x,x′ ∈ x, the points f(x) and f(x′) are equal or adjacent. � theorem 2.3 ([2, 3]). let f : (a,κ) → (b,λ) and g : (b,λ) → (c,µ) be continuous functions between digital images. then g ◦ f : (a,κ) → (c,µ) is continuous. � definition 2.4. a function f : x → y is an isomorphism [5] (called a homeomorphism in [2]) if f is a continuous bijection and f−1 is continuous. � definition 2.5 ([4]). let f : (x,κ) → (y,λ) be a continuous surjection of digital images. we say f is shy if • for every y ∈ y , f−1(y) is κ-connected; and • for every λ-adjacent y0,y1 ∈ y , f−1({y0,y1}) is κ-connected. � it is known [4] that shy maps induce surjections of fundamental groups. we also have the following. theorem 2.6 ([6, 9]). let f : x → y be a continuous surjection of digital images. then f is shy if and only if for every connected subset y ′ of y , f−1(y ′) is connected. 2.2. normal product adjacency. the normal product adjacency has been used in many papers for cartesian products of graphs. definition 2.7 ([1]). let (x,κ) and (y,λ) be digital images. the normal product adjacency k∗(κ,λ) for x × y is defined as follows. two members (x0,y0) and (x1,y1) of x × y are k∗(κ,λ)-adjacent if and only if one of the following is valid. • x0 = x1, and y0 and y1 are λ-adjacent; or • x0 and x1 are κ-adjacent, and y0 = y1; or • x0 and x1 are κ-adjacent, and y0 and y1 are λ-adjacent. � we will use the following properties. proposition 2.8 ([13]). the projection maps p1 : (x ×y,k∗(κ,λ)) → (x,κ) and p2 : (x × y,k∗(κ,λ)) → (y,λ) defined by p1(x,y) = x, p2(x,y) = y are (k∗(κ,λ),κ)-continuous and (k∗(κ,λ),λ)-continuous, respectively. � proposition 2.9 ([8]). in zm+n, k∗(cm,cn) = cm+n; i.e., given points x,x′ ∈ zm and y,y′ ∈ zn, (x,y) and (x′,y′) are k∗(cm,cn)-adjacent in zm+n if and only if they are cm+n-adjacent. � 2.3. digital multivalued functions. to ameliorate limitations and anomalies that appear in the study of continuous functions between digital images, several authors have considered multivalued functions with various forms of continuity. functions with weak continuity and strong continuity were introduced in [16] and studied further in [9]. connectivity preserving multivalued functions were introduced in [14] and studied further in [9]. continuous multivalued functions were introduced in [10, 11] and studied further in [12, 6, 9]. we use the following. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 145 l. boxer definition 2.10 ([14]). a multivalued function f : x ( y is connectivity preserving if for every connected subset a of x, f(a) is a connected subset of y . � definition 2.11. let a and b be subsets of a digital image (x,κ). we say a and b are κ-adjacent, or adjacent for short, if there exist a ∈ a and b ∈ b such that either a = b or a and b are κ-adjacent. � definition 2.12 ([16]). a multivalued function f : x ( y has weak continuity if for every pair x,y of adjacent points in x, the sets f(x) and f(y) are adjacent in y . � proposition 2.13 ([9]). let f : x ( y be a multivalued function between digital images. then f is connectivity preserving if and only if f has weak continuity and for all x ∈ x, f(x) is a connected subset of y . � theorem 2.14 ([9]). a continuous surjection f : x → y of digital images is shy if and only if f−1 : y ( x is a connectivity preserving multivalued function. � 3. shy, constant, and isomorphism functions proposition 3.1. a constant map between connected digital images is shy. i.e., if x is a connected digital image and y ∈ zn, then the function f : x → {y} is shy. proof. this is obvious. � previous results give the following characterizations of shy maps. theorem 3.2. let f : x → y be a continuous surjection between digital images. the following are equivalent. (1) f is a shy map. (2) for every connected y ′ ⊂ y , f−1(y ′) is a connected subset of x. (3) f−1 : y ( x is a connectivity preserving multi-valued function. (4) f−1 : y ( x is a multi-valued function with weak continuity such that for all y ∈ y , f−1(y) is a connected subset of x. proof. the equivalence of the first three statements follows from theorem 2.6 and theorem 2.14. the equivalence of the third and fourth statements follows from proposition 2.13. � theorem 3.3. let f : (x,κ) → (y,λ) be a continuous surjection. • if f is an isomorphism then f is shy. • if f is shy and one-to-one, then f is an isomorphism. proof. since a single point is connected, the first assertion follows easily from definitions 2.4 and 2.5. suppose f is shy and one-to-one. shyness implies that f is a surjection, hence a bijection; and, from definitions 2.1 and 2.5, that f−1 is continuous. therefore, f is an isomorphism. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 146 digital shy maps 4. operations that preserve shyness we show that composition preserves shyness. theorem 4.1. let f : a → b and g : b → c be shy maps. then g◦f : a → c is shy. proof. let c′ be a connected subset of c. by theorem 2.6, g−1(c′) is a connected subset of b. therefore, by theorem 2.6, (g ◦f)−1(c′) = f−1(g−1(c′)) is a connected subset of a. the assertion follows from theorem 2.6. � the following generalizes proposition 4.1 of [8]. proposition 4.2. let f : (a,α) → (c,γ) and g : (b,β) → (d,δ). then f and g are continuous if and only if the function f × g : (a × b,k∗(α,β)) → (c ×d,k∗(γ,δ)) defined by (f ×g)(a,b) = (f(a),g(b)) is continuous. proof. suppose f and g are continuous. let (a,b) and (a′,b′) be k∗(α,β)adjacent. then a and a′ are equal or α-adjacent, so f(a) and f(a′) are equal or γ-adjacent; and b and b′ are equal or β-adjacent, so g(b) and g(b′) are equal or δ-adjacent. it follows from definition 2.7 that (f×g)(a,b) and (f×g)(a′,b′) are equal or k∗(γ,δ)-adjacent. therefore, f ×g is continuous. conversely, suppose f ×g is continuous. from proposition 2.8, we know the projection maps p1 : (x × y,k∗(κ,λ)) → (x,κ) and p2 : (x × y,k∗(κ,λ)) → (y,λ) defined by p1(x,y) = x, p2(x,y) = y, are continuous. it follows from theorem 2.3 that f = p1 ◦ (f ×g) and g = p2 ◦ (f ×g) are continuous. � proposition 4.3. let f : a → c and g : b → d be functions. then the function f × g : a × b → c × d defined by (f × g)(a,b) = (f(a),g(b)) is a surjection if and only if f and g are surjections. proof. let (c,d) ∈ c×d. if f and g are surjections, there are a ∈ a and b ∈ b such that f(a) = c and g(b) = d. therefore, (f ×g)(a,b) = (c,d). thus, f ×g is a surjection. conversely, if f ×g is a surjection, it follows easily that f and g are surjections. � cartesian products preserve shyness with respect to the normal product adjacency, as shown in the following. theorem 4.4. let f : (a,α) → (c,γ) and g : (b,β) → (d,δ) be continuous surjections. then f and g are shy maps if and only if the function f × g : (a×b,k∗(α,β)) → (c ×d,k∗(γ,δ)) is a shy map. proof. suppose f and g are shy. then they are surjections. by propositions 4.2 and 4.3, f ×g is a continuous surjection. let (c,d) ∈ c × d and let (a,b), (a′,b′) ∈ (f × g)−1(c,d). since f−1(c) is connected, there is a path p in f−1(c) from a to a′. therefore, p ×{b} is a path in f−1(c) ×{b} ⊂ (f × g)−1(c,d) from (a,b) to (a′,b). since g−1(d) is connected, there is a path q in g−1(d) from b to b′. therefore, {a′}× q is a path in {a′} × g−1(d) ⊂ (f × g)−1(c,d) from (a′,b) to (a′,b′). thus, c© agt, upv, 2017 appl. gen. topol. 18, no. 1 147 l. boxer (p×{b})∪({a′}×q) is a path in (f×g)−1(c,d) from (a,b) to (a′,b′). therefore, (f ×g)−1(c,d) is connected. let (c,d) and (c′,d′) be k∗(γ,δ)-adjacent in c×d. then c and c′ are equal or γ-adjacent, and d and d′ are equal or δ-adjacent. let {(a,b), (a′,b′)} ⊂ (f×g)−1({(c,d), (c′,d′)}. since f is shy, f−1({c,c′}) is connected, so there is a path p in f−1({c,c′}) from a to a′. similarly, there is a path q in g−1({d,d′}) from b′ to b. thus, (p ×{b}) ∪ ({a′}×q) is a path in (f−1({c,c′}) ×{b}) ∪ ({a′}×g−1({d,d′}) ⊂ (f ×g)−1({(c,d), (c′,d′)}) from (a,b) to (a′,b′). therefore, (f ×g)−1({(c,d), (c′,d′)}) is connected. it follows from definition 2.5 that f ×g is a shy map. conversely, suppose f × g is a shy map. it follows from propositions 4.2 and 4.3 that f and g are continuous surjections. let u be a connected subset of c. then for d ∈ d, u ×{d} is a connected subset of (c ×d,k∗(γ,δ)). the shyness of f ×g implies f−1(u) ×g−1({d}) = (f ×g)−1(u ×{d}) is connected in (a×b,k∗(α,β)). from proposition 2.8, it follows that f−1(u) = p1(f −1(u) ×g−1({d})) is connected in a. thus, f is shy. a similar argument shows g is shy. � corollary 4.5. let a ⊂ zm, b ⊂ zn, c ⊂ zu, d ⊂ zv. suppose f : (a,cm) → (c,cu) and g : (b,cn) → (d,cv) are functions. then the function f ×g : (a×b,cm+n) → (c ×d,cu+v) is a shy map if and only if f and g are shy maps. proof. by proposition 2.9, in zm+n, k∗(cm,cn) = cm+n and in zu+v, k∗(cu,cv) = cu+v. the assertion follows from theorem 4.4. � the digital image (c,κ) is the wedge of its subsets x and y , denoted c = x ∧ y , if c = x ∪ y and x ∩ y = {x0} for some point x0, such that if x ∈ x, y ∈ y , and x and y are κ-adjacent, then x0 ∈ {x,y}. let f : a → c and g : b → d be functions between digital images, with a ∩ b = {x0}, c ∩d = {y0}, and f(x0) = y0 = g(x0). we define f ∧g : a∧b → c ∧d by (f ∧g)(x) = { f(x) if x ∈ a; g(x) if x ∈ b. since f(x0) = y0 = g(x0), f ∧g is well defined. we have the following. theorem 4.6. let f : a → c and g : b → d be functions between digital images, with a∩b = {x0}, c ∩d = {y0}, and f(x0) = y0 = g(x0). then f and g are shy maps if and only if f ∧g is a shy map. proof. if f and g are shy, it is easy to see that f∧g is continuous and surjective. let u be a connected subset of c ∧ d. let v0,v1 ∈ f−1(u). the connectedness of u implies there is a path p in u from f(v0) to f(v1), i.e., a continuous p : [0,m]z → u such that p(0) = f(v0) and p(m) = f(v1), where p = p([0,m]z). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 148 digital shy maps • if u ⊂ c, then (f ∧g)−1(p) = f−1(p) is a connected subset of a. • if u ⊂ d, then (f ∧g)−1(p) = g−1(p) is a connected subset of b. • otherwise, there are integers 0 = i1 < i2 < · · · < in = m such that the sets p([ij, ij+1]z) alternate between containment in c and containment in d, i.e., without loss of generality, p([ij, ij+1]z) ⊂ c ∩u for even j, and p([ij, ij+1]z) ⊂ d ∩u for odd j. therefore, (f ∧g)−1(p) = ⋃ even j f−1(p([ij, ij+1]z)) ∪ ⋃ odd j g−1(p([ij, ij+1]z)) is a union of connected sets, each containing x0. hence (f∧g)−1(p) is a connected subset of (f∧g)−1(u) containing {v0,v1}. thus, (f∧g)−1(u) is connected. in all cases, (f ∧ g)−1(u) is connected. we conclude from theorem 2.6 that f ∧g is a shy map. conversely, suppose f ∧g is a shy map. then both f and g are continuous surjections. if u is a connected subset of c and v is a connected subset of d, we have by theorem 2.6 that f−1(u) = (f ∧g)−1(u) is a connected subset of a and g−1(v ) = (f ∧g)−1(v ) is a connected subset of b. from theorem 2.6, it follows that f and g are shy maps. � 5. shy maps into z theorem 5.1. let x and y be connected subsets of (z,c1), and let f : x → y be a continuous surjection. then f is shy if and only if f is either monotone non-decreasing or monotone non-increasing. proof. suppose f is shy. if f is not monotone, then either (5.1) for some a,b,c ∈ x with a f(c), or (5.2) for some a,b,c ∈ x with a f(b) and f(b) < f(c). in case (5.1), the continuity of f implies there exist s,t ∈ x such that a ≤ s < b < t ≤ c and f(s) = f(t) = f(b) − 1. thus, s,t ∈ f−1(f(b) − 1) and b 6∈ f−1(f(b) − 1), so f−1(f(b) − 1) is not c1-connected, a contradiction of the shyness of f. case (5.2) generates a similar contradiction. thus, we obtain a contradiction by assuming that f is not monotone. suppose f is monotone. we may assume without loss of generality that f is non-decreasing. let y ′ be a connected subset of y and let x0,x1 ∈ f−1(y ′). we need to show there is a connected subset y ′′ of f−1(y ′) such that {x0,x1}⊂ y ′′. • if x0 = x1 we can take y ′′ = {x0,x1}. • otherwise, without loss of generality, x0 < x1. since f is continuous and non-decreasing, f([x0,x1]z) = [f(x0),f(x1)]z is a connected set containing {f(x0),f(x1)}⊂ y ′, so we can take y ′′ = [x0,x1]z. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 149 l. boxer thus, [x0,x1]z ⊂ f−1([f(x0),f(x1)]z) ⊂ f−1(y ′), so [x0,x1]z is a connected subset of f −1(y ′) containing {x0,x1}. since x0 and x1 were arbitrarily chosen, we must have that f −1(y ′) is c1-connected. therefore, f−1 is a connectivity preserving multivalued function. it follows from theorem 2.6 that f is shy. � a digital simple closed curve is a set s = {xi}mi=0 ⊂ (z n,κ), for some m,n ∈ n with m ≥ 4 and some adjacency κ, such that i 6= j implies xi 6= xj, and xi and xj are κ-adjacent if and only if j ∈{(i− 1) mod m, (i + 1) mod m}. theorem 5.2. let s be a digital simple closed curve. let f : s → y ⊂ (z,c1) be a shy map. then either y = {z} or y = {z,z + 1} for some z ∈ z. proof. if f is not a constant function, i.e., if y 6= {z}, then z0 = min{f(x) |x ∈ s} < max{f(x) |x ∈ s} = z1. let xi ∈ f−1(zi), i ∈ {0, 1}. there are two distinct digital arcs, a and b, connecting x0 and x1 in s. if z1 −z0 > 1, then the continuity of f|a and f|b implies there are points a ∈ a and b ∈ b such that f(a) = f(b) = z0 + 1. since a and b are in distinct components of s \ (f−1({z0,z1}), f−1(z0 + 1) is disconnected. this is contrary to the assumption that f is shy. the assertion follows. � theorem 5.3. let (x,κ) be a connected digital image and let r ∈ x be such that x \ {r} is κ-disconnected. let f : (x,κ) → y ⊂ (z,c1) be a shy map. then there are at most 2 components of x \{r} on which f is not equal to the constant function with value f(r). proof. suppose a is a component of x \{r} on which f is not constant. since x is connected and x \{r} is not, by continuity of f, there exists a ∈ a such that |f(a) −f(r)| = 1. suppose b is another component of x \{r} on which f is not constant. similarly, there exists b ∈ b such that |f(b) −f(r)| = 1. we must have f(a) 6= f(b), since f−1(f(a)) is connected and every path in x from a to b contains r. therefore, we may assume f(a) = f(r) − 1, f(b) = f(r) + 1. suppose c is a component of x\{r} that is distinct from a and b. suppose there exists c ∈ c such that f(c) 6= f(r). if f(c) < f(r) then, by continuity of f, there exists c′ ∈ c such that f(c′) = f(r)−1 = f(a). but this is impossible, since f−1(f(a)) is connected and every path in x from a to c′ contains r. similarly, if we assume f(c) > f(r) we get a contradiction. the assertion follows. � example 5.4. let t be a tree. let r be the root vertex of t . let {vi}mi=0 be the set of vertices adjacent to r. let ti be the subtree of t with vertices r,vi, and the descendants of vi in t (see figure 1). let f : t → y ⊂ (z,c1) be a shy function. then f is constant on all but at most 2 of the ti. proof. the assertion follows from theorem 5.3. � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 150 digital shy maps figure 1. a tree t to illustrate example 5.4. the vertex sets of t0, t1, and t2 are, respectively, {r,v0}, {r,v1,p0,p2,p3}, and {r,v2,p1,p4,p5,p6}. a shy map from t to a subset of (z,c1) is non-constant on at most 2 of t0, t1, and t2. 6. further remarks we have made several contributions to our knowledge of digital shy maps. in section 3, we studied the relations between shy maps and both constant functions and isomorphisms. in section 4, we showed that shyness is preserved by compositions, certain cartesian products, and wedges. in section 5, we demonstrated several restrictions on shy maps onto subsets of (z,c1). additional results concerning shy maps, obtained after the initial submission of this paper, appear in [7]. this research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. acknowledgements. the corrections and suggestions of the anonymous reviewers are gratefully acknowledged. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 151 l. boxer references [1] c. berge, graphs and hypergraphs, 2nd edition, north-holland, amsterdam, 1976. [2] l. boxer, digitally continuous functions, pattern recognition letters 15 (1994), 833– 839. [3] l. boxer, a classical construction for the digital fundamental group, pattern recognition letters 10 (1999), 51–62. [4] l. boxer, properties of digital homotopy, journal of mathematical imaging and vision 22 (2005),19–26. [5] l. boxer, digital products, wedges, and covering spaces, journal of mathematical imaging and vision 25 (2006), 159–171. [6] l. boxer, remarks on digitally continuous multivalued functions, journal of advances in mathematics 9, no. 1 (2014), 1755–1762. [7] l. boxer, generalized normal product adjacency in digital topology, submitted (available at https://arxiv.org/abs/1608.03204). [8] l. boxer and i. karaca, fundamental groups for digital products, advances and applications in mathematical sciences 11, no. 4 (2012), 161–180. 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[15] a. rosenfeld, ‘continuous’ functions on digital images, pattern recognition letters 4 (1987), 177–184. [16] r. tsaur and m. smyth, continuous multifunctions in discrete spaces with applications to fixed point theory, in: digital and image geometry, bertrand, g., imiya, a., klette, r. (eds.), lecture notes in computer science, vol. 2243, pp. 151–162, springer, berlin (2001). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 152 () @ appl. gen. topol. 19, no. 1 (2018), 173-187doi:10.4995/agt.2018.7997 c© agt, upv, 2018 some aspects of isbell-convex quasi-metric spaces olivier olela otafudu school of mathematics, university of the witwatersrand johannesburg 2050, south africa (olivier.olelaotafudu@wits.ac.za) communicated by h.-p. a. künzi abstract we introduce and investigate the concept of geodesic bicombing in t0-quasi-metric spaces. we prove that any isbell-convex t0-quasimetric space admits a geodesic bicombing which satisfies the equivariance property. additionally, we show that many results on geodesic bicombing hold in quasi-metric settings, provided that nonsymmetry in quasi-metric spaces holds. 2010 msc: 54e50; 30l05. keywords: isbell-convexity; geodesic bicombing; injectivity. 1. introduction let (x,d) be a metric space. a map σ : x × x × [0,1] → x is said to be a geodesic bicombing if for every (x,y) ∈ x × x, (1.1) σ(x,y,0) = x, σ(x,y,1) = y and (1.2) d(σ(x,y,t),σ(x,y,t′) = |t − t′|d(x,y) whenever t,t′ ∈ [0,1] (see [4, 10]). furthermore, a geodesic bicombing σ on a metric space (x,d) is conical if d(σ(x,y,t),σ(x′,y′, t)) ≤ (1 − t)d(x,x′) + td(y,y′) whenever t,t′ ∈ [0,1] and x,y,x′,y′ ∈ x. received 11 september 2017 – accepted 14 november 2017 http://dx.doi.org/10.4995/agt.2018.7997 o. olela otafudu in [4] descombes and lang discussed and compared three different convexity notions (convex and consistent, convex and conical) for geodesic bicombings. they proved that busemann spaces, and in particular cat(0) spaces admit geodesic bicomings which are convex and consistent. additional to this, they proved that every gromov hyperbolic group acts geometrically on a proper finite-dimensional metric space with convex and consistent geodesic bicombing. isbell [6] and dress [5] developed independently the concept of injective hull in the category of metric spaces with nonexpansive maps as morphisms. isbell proved that the injective hull (unique up to isometry) of a metric space is hyperconvex by appealing to zorn’s lemma. later on lang [10] presented a new proof which is more constructive of the same result. furthermore, lang proved that every metric space which is injective, admits a conical geodesic bicombing. the injective hull construction for metric spaces has been generalized in the category of t0-quasi-metric spaces with nonexpansive maps as morphisms (see [7]). furthermore, an explicit description of the algebraic and vector lattice operations on the isbell-convex hull of an asymmetrically normed linear vector space is proved in [3]. naturally this led to the speculation that the isbellconvex hull of an isbell-convex t0-quasi-metric space admits a conical geodesic bicombing. the aim of this article is to give a careful and complete proof of the aforementioned speculation. we also discuss the continuity of a geodesic bicombing on a t0-quasi-metric space. furthermore, we prove that for a conical geodesic bicombing σ on a t0-quasi-metric space (x,q), if a set a is bounded σ-convex on the set p0(x) of nonempty subsets of (x,q), then its double closure clτ(q)a∩clτ(q−1)a is also bounded σ-convex on p0(x). let us mention that a conical geodesic bicombing on a t0-quasi-metric space enjoys some property with the takahashi convex structure on the same t0-quasi-metric space. for details on takahashi convex structures on a t0-quasi-metric space, we refer the reader to [9]. 2. preliminaries we start by recalling some useful concepts that we are going to use in the sequel. definition 2.1. let x be a nonempty set and q : x × x → [0,∞) be a map. then q is a quasi-pseudometric on x if (a) q(x,x) = 0 whenever x ∈ x, and (b) q(x,z) ≤ q(x,y) + q(y,z) whenever x,y,z ∈ x. if q is a quasi-pseudometric on a set x, then the pair (x,q) is called a a quasipseudometric space. moreover, we say that q is a t0-quasi-metric provided that it satisfies the additional condition that for any x,y ∈ x, q(x,y) = 0 = q(y,x) implies that x = y. the set x together with a t0-quasi-metric on x is called a quasi-metric space. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 174 some aspects of isbell-convex quasi-metric spaces remark 2.2. note that if q is a quasi-metric on x, then q−1 : x × x → [0,∞) defined by q−1(x,y) = q(y,x) whenever x,y ∈ x is also a quasipseudometric on x, called the conjugate quasi-pseudometric of q. as usual, a quasi-pseudometric d on x such that q = q−1 is called a pseudometric on x. furthermore, the map qs = max{q,q−1} is a pseudometric on x. if q is a t0-quasi-metric on x, then q s is a metric on x. let (x,q) be a quasi-pseudometric space and for each x ∈ x and r ∈ [0,∞), let cq(x,r) = {y ∈ x : q(x,y) ≤ r} be the τ(q −1)-closed ball of centre x and radius r. furthermore, the open ball with centre x and radius r is represented by bq(x,r) = {y ∈ x : q(x,y) < r}. example 2.3. let (x,q) be a t0-quasi-metric space. for any x,y ∈ x with x 6= y and q(x,y) + q(y,x) 6= 0, the function uq(x,y),q(y,x) : r × r → r defined by uq(x,y),q(y,x)(λ,λ ′ ) = { (λ − λ ′ )q(x,y) if λ ≥ λ ′ (λ ′ − λ)q(y,x) if λ < λ ′ is a t0-quasi-metric. take any t0-quasi-metric space (x,q). if q(x,y) = 1 and q(y,x) = 0 whenever x,y ∈ x, then the t0-quasi-metric uq(x,y),q(y,x) is the standard t0-quasimetric u on r, where u(x,y) = max{0,x − y} = x−̇y whenever x,y ∈ r. consider a t0-quasi-metric space (x,q). let p0(x) be the set of all nonempty subsets of x. we recall that for any given p ∈ p0(x), q(p,x) = inf{q(p,x) : p ∈ p} and q(x,p) = inf{q(x,p) : p ∈ p} for all x ∈ x. for any p,q ∈ p0(x), the so-called hausdorff (-bourbaki) quasi-pseudometric qh on p0(x) is defined by qh(p,q) = sup x∈q q(p,x) ∨ sup x∈p q(x,q). it is well-known that qh is an extended quasi-pseudometric (qh may attain the value ∞, then the triangle inequality is interpreted in the obvious way). moreover, qh is a t0-quasi-metric if we restrict the set p0(x) to the nonempty subsets of p of x which satisfy p = clτ(q)p ∩ clτ(q−1)p (see [2, 8]). definition 2.4. ([9, definition 7]) let (x,q) be a t0-quasi-metric space. for any subset p of x, we call clτ(q)p ∩clτ(q−1)p the double closure of p . moreover if p = clτ(q)p ∩ clτ(q−1)p , we say that p is doubly closed. definition 2.5. ([7, definition 2]) a quasi-pseudometric space (x,q) is called isbell-convex (or q-hyperconvex) provided that for any family (xi)i∈i of points in x and families (ri)i∈i and (si)i∈iof nonnegative real numbers satisfying q(xi,xj) ≤ ri + sj whenever i,j ∈ i, the following condition hold: ⋂ i∈i [cq(xi,ri) ∩ cq−1(xi,si)] 6= ∅. for more details about the theory of isbell-convex t0-quasi-metric spaces, we refer the reader to [3, 7, 11]. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 175 o. olela otafudu 3. geodesic bicombing we start this section with the following observation. remark 3.1. we observed that the condition (1.2) is unsuitable for a t0-quasimetric space (x,d) that is not a metric: indeed if d is a t0-quasi-metric space with properties (1.1) and (1.2), then it satisfies |0 − 1|d(x,y) = d(σ(x,y,0),σ(x,y,1)) = d(σ(y,x,1),σ(y,x,0)) = |1 − 0|d(y,x) whenever x,y ∈ x and thus d would be a metric. therefore, for a t0-quasimetric space (x,d) we propose the condition (1.2) differently in definition 3.2. let i = [0,1] be the set of real unit interval. using a different terminology, it was essentially observed by remark 3.1 that in a t0-quasi-metric space the concept of geodesic bicombing can be modified in the following way: definition 3.2. let (x,q) be a t0-quasi-metric space. a geodesic bicombing σ on (x,q) is a map σ : x × x × i → x such that for each (x,y) ∈ x × x, σ(x,y,0) = x, σ(x,y,1) = y and (3.1) q(σ(x,y,λ),σ(x,y,λ′)) = (λ′ − λ)q(x,y) if λ′ ≥ λ and (3.2) q(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)q(y,x)) if λ′ < λ whenever λ,λ′ ∈ i. definition 3.3. let σ be a geodesic bicombing on a t0-quasi-metric space (x,q). we say that σ satisfies the equivariance property if σ(x,y,λ) = σ(y,x,1 − λ) whenever x,y ∈ x and λ ∈ [0,1]. lemma 3.4. if σ is a geodesic bicombing on a t0-quasi-metric space (x,q), then σ is a geodesic bicombing on the conjugate t0-quasi-metric space (x,q −1). furthermore, σ is a geodesic bicombing on the metric space (x,qs) proof. suppose that σ is a geodesic bicombing on (x,q). let x,y ∈ x and λ,λ′ ∈ i. obviously, σ(x,y,0) = x, σ(x,y,1) = y. if λ′ ≥ λ, then q−1(σ(x,y,λ),σ(x,y,λ′)) = q(σ(x,y,λ′),σ(x,y,λ)) = (λ′ − λ)q−1(x,y). if λ > λ′, we have q−1(σ(x,y,λ),σ(x,y,λ′) = q(σ(x,y,λ′),σ(x,y,λ)) = (λ − λ)q−1(y,x). so σ is a geodesic bicombing on (x,q−1). if λ′ ≥ λ for λ,λ′ ∈ i, then we have qs(σ(x,y,λ),σ(x,y,λ′)) = max{q(σ(x,y,λ),σ(x,y,λ′)),q−1(σ(x,y,λ),σ(x,y,λ′))} = max{(λ′−λ)q(x,y),(λ′−λ)q−1(x,y)} = (λ′−λ)qs(x,y). similarly if λ > λ′, qs(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)qs(x,y). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 176 some aspects of isbell-convex quasi-metric spaces hence qs(σ(x,y,λ),σ(x,y,λ′)) = |λ − λ′|qs(x,y) whenever λ,λ′ ∈ i. � example 3.5. if we equip (r,u) with σ(x,y,λ) = (1 − λ)x + λy whenever x,y ∈ r and λ ∈ i, then σ is a geodesic bicombing on (r,u) and σ is called the standard geodesic bicombing on (r,u). indeed, if λ′ ≥ λ, we have u(σ(x,y,λ),σ(x,y,λ′)) = u((1 − λ)x + λy,(1 − λ′)x + λ′y) = max{0,(1 − λ)x + λy − [(1 − λ′)x + λ′y]}. so u(σ(x,y,λ),σ(x,y,λ′)) = max{0,(λ′ − λ)x − (λ′ − λ)y} = (λ′ − λ)u(x,y). by similar arguments if λ > λ′, then u(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)u(y,x). lemma 3.6. let σ be a geodesic bicombing on a t0-quasi-metric space (x,q). then the map σ−1 defined by σ−1(x,y,λ) = σ(y,x,1 − λ) whenever x,y ∈ x and λ ∈ i, is a geodesic bicombing on (x,q). the geodesic bicombing σ−1 is called reversible geodesic bicombing of σ (see [4, p.2]). proof. let x,y ∈ x and λ,λ′ ∈ i. we have σ−1(x,y,0) = σ(y,x,1) = x and σ−1(x,y,1) = σ(y,x,0) = y. if λ′ ≥ λ, then (1 − λ) > (1 − λ′). it follows that q(σ−1(x,y,λ),σ−1(x,y,λ′)) = q(σ(y,x,1 −λ),σ(y,x,1 −λ′)) = (λ′ −λ)q(y,x). if λ > λ′, then one sees that q(σ−1(x,y,λ),σ−1(x,y,λ′)) = (λ − λ′)q(x,y). � example 3.7. let c be a convex subset of a real linear space x equipped with the asymmetric norm ‖.|. then σ(x,y,λ) = (1 − λ)x + λy whenever x,y ∈ c and λ ∈ i, is a geodesic bicombing on (c,d), where d(x,y) = ‖x− y| whenever x,y ∈ c. lemma 3.8. let (x,q) be a t0-quasi-metric space with a geodesic bicombing σ. then we have σ(x,x,λ) = x whenever x ∈ x and λ ∈ i. proof. let x ∈ x and λ ∈ i. then q(x,σ(x,x,λ)) ≤ q(x,σ(x,x,0))+q(σ(x,x,0),σ(x,x,λ)) = q(x,x)+λq(x,x) = 0. thus q(x,σ(x,x,λ)) = 0. furthermore, q(q(σ(x,x,λ),x) ≤ q(σ(x,x,λ),σ(x,x,0))+q(σ(x,x,0),x) = λq(x,x)+q(x,x) = 0. hence q(σ(x,x,λ),x) = 0 = q(x,σ(x,x,λ)). we have σ(x,x,λ) = x by t0property of (x,q). � note that a geodesic bicombing need not to be unique. to obtain the following embedding we assume that the geodesic bicombing is unique. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 177 o. olela otafudu lemma 3.9 (compare [9, proposition 4]). suppose that σ is the unique geodesic bicombing on the t0-quasi-metric space (x,q). if for every x,y ∈ x with x 6= y, the map ψ : (i,uq(x,y),q(y,x)) → (x,q) defined by ψ(λ) = σ(x,y,λ) whenever λ ∈ i is an isometric embedding. (here uq(x,y),q(y,x) is the restriction of the t0-quasi-metric uq(x,y),q(y,x) given in example 2.3 to i). proof. for λ,λ′ ∈ i, we show that q(σ(x,y,λ),σ(x,y,λ′)) = uq(x,y),q(y,x)(λ,λ ′). if λ′ ≥ λ, then q(σ(x,y,λ),σ(x,y,λ′)) = (λ′−λ)q(x,y) = uq(x,y),q(y,x)(λ,λ ′). if λ > λ′, then q(σ(x,y,λ),σ(x,y,λ′)) = (λ−λ′)q(y,x) = uq(x,y),q(y,x)(λ,λ ′). � lemma 3.10. let σ be a geodesic bicombing on a t0-quasi-metric space (x,q). then whenever x,y,u ∈ x and λ ∈ i, σ satisfies the following inequalities: (3.3) q(u,σ(x,y,λ)) ≤ q(u,x) + λq(x,y) (3.4) q(u,σ(x,y,λ)) ≤ q(u,y) + (1 − λ)q(u,x) (3.5) q(σ(x,y,λ),u) ≤ λq(y,x) + q(x,u) (3.6) q(σ(x,y,λ),u) ≤ (1 − λ)q(x,y) + q(u,y). proof. we prove only (3.3) and (3.6), then (3.4) and (3.5) follow analogously. let x,y,x′,y′ ∈ x and λ ∈ i. then q(u,σ(x,y,λ)) ≤ q(u,σ(x,y,0)) + q(σ(x,y,0),σ(x,y,λ)) by triangle inequality. since σ(x,y,0) = x and from the equality (3.1) (λ ≥ 0), it follows that q(u,σ(x,y,λ)) ≤ q(u,x) + λq(x,y). furthermore, q(σ(x,y,λ),u) ≤ q(σ(x,y,λ),σ(x,y,1)) + q(σ(x,y,1),u), then q(σ(x,y,λ),u) ≤ (1 − λ)q(x,y) + q(y,u) by σ(x,y,1) = y and from the equality (3.1) (1 ≥ λ). � proposition 3.11 (compare [9, proposition 2]). let σ be a geodesic bicombing on a t0-quasi-metric space (x,q). then for each x ∈ x and λ ∈ i, σ is continuous at (x,x,λ), where x carries the topology τ(q) (or τ(q−1)). proof. consider the convergent sequence ((xn,yn,λn)) in x × x × i. suppose that ((xn,yn,λn)) converges to (x,x,λ) with respect to the topology induced by q on x. the topology on i does not really matter. we have to prove that the sequence (σ(xn,yn,λn)) converges to (σ(x,x,λ)). from lemma 3.8, we c© agt, upv, 2018 appl. gen. topol. 19, no. 1 178 some aspects of isbell-convex quasi-metric spaces know that σ(x,x,λ) = x whenever λ ∈ i. by inequality (3.4) of lemma 3.10, whenever n ∈ n we have q(x,σ(xn,yn,λn)) ≤ q(x,yn) + (1 − λn)q(x,xn). therefore, the sequence (σ(xn,yn,λn)) converges to (σ(x,x,λ)). one obtains the similar result by using the topology induced by q−1 on x and inequality (3.3). � 4. conical geodesic bicombing definition 4.1. let σ be a geodesic bicombing on a t0-quasi-metric space (x,q). then σ is said to be conical if (4.1) q(σ(x,y,λ),σ(x′,y′,λ)) ≤ (1 − λ)q(x,x′) + λq(y,y′) whenever x,y,x′,y′ ∈ x and λ ∈ i. furthermore, the geodesic bicombing σ is called convex if the function λ 7→ q(σ(x,y,λ),σ(x′,y′,λ)) is convex on i whenever x,y,x′,y′ ∈ x and λ ∈ i. the following ideas are not new and were inspired from [9]. let σ be a conical geodesic bicombing on a t0-quasi-metric space (x,q). a subset c of x is called σ-convex provided that σ(c,c′,λ) ∈ c whenever c,c′ ∈ c and λ ∈ i. observe that x is σ-convex subset of itself. moreover, each σ-convex subset c of x carries a natural conical geodesic bicombing, which is the restriction of σ to c × c × i. proposition 4.2. let σ be a conical geodesic bicombing on a t0-quasi-metric space (x,q). then whenever x ∈ x and r > 0, the closed balls cq(x,r) and cq−1(x,s) and the open balls bq(x,r) and bq−1(x,s) are σ-convex subsets of x. proof. we only prove that cq(x,r) is σ-convex, the proofs related to the balls cq−1(x,s), bq(x,r) and bq−1(x,s) follow analogously. suppose that x ∈ x, r > 0 and λ ∈ i. let y,z ∈ cq(x,r). then q(x,σ(y,z,λ)) = q(σ(x,x,λ),σ(y,z,λ)) by lemma 3.8. furthermore, q(x,σ(y,z,λ)) ≤ (1 − λ)q(x,y) + λq(x,z) ≤ (1 − λ)r + λr = r since σ is conical, q(x,y) ≤ r and q(x,z) ≤ r. thus σ(y,z,λ) ∈ cq(x,r). � obviously, one can prove that the intersection of any family of σ-convex subsets of (x,q) is σ-convex too. let cb0(x) be the subcollection of bounded σ-convex elements of p0(x). in this case qh is a quasi-pseudometric since qh(a,b) < ∞. for more details about how qh(a,b) < ∞, we refer the reader to [9, p.13]. let σ be a conical geodesic bicombing on a t0-quasi-metric space (x,q). for any a,b ∈ cb0(x) and λ ∈ i, set σ(a,b,λ) := {σ(a,b,λ) : a ∈ a,b ∈ b}. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 179 o. olela otafudu then we have that σ(a,b,λ) is nonempty and bounded. let a,a′ ∈ a and b,b′ ∈ b. then we have that q(σ(a,b,λ),σ(a′,b′,λ)) ≤ (1−λ)q(a,a′)+λq(b,b′) ≤ (1−λ)diam(a)+λdiam(b) since σ is conical. one can add some conditions on σ in order the set σ(a,b,λ) preserves σ-convexity. we leave the proof of the following lemma to the reader. lemma 4.3 (compare [9, lemma 3]). let σ be a conical geodesic bicombing on a t0-quasi-metric space (x,q). if a ∈ cb0(x), then its double closure clτ(q)a ∩ clτ(q−1)a is contained in cb0(x). 5. injective spaces we start by recalling the construction of isbell-convex hull (injective hull) of a t0-quasi-metric space (x,q) and we refer the reader to [7] for more details. let fp(x,q) be the set of all pairs of functions f = (f1,f2), where fi : x → [0,∞)(i = 1,2). a function pair (f1,f2) is said to be ample on (x,q) if q(x,y) ≤ f2(x) + f1(y) whenever x,y ∈ x. the set of all function pairs which are ample on (x,q) will be denoted by a(x,q). for a ∈ x, it is easy to see that the function pair fa(x) = (q(a,x),q(x,a)) is element of a(x,q). for each (f1,f2),(g1,g2) ∈ a(x,q), the map d defined by d((f1,f2),(g1,g2)) = sup x∈x (f1(x)−̇g1(x)) ∨ sup x∈x (g2(x)−̇f2(x)) is an extended t0-quasi-metric on a(x,q). let (f1,f2) ∈ a(x,q). we say that (f1,f2) is extremal (or minimal) on (x,q) if take (g1,g2) ∈ a(x,q) such that g1(x) ≤ f1(x) and g2(x) ≤ f2(x) for x ∈ x, then f1(x) = g1(x) and f2(x) = g2(x). the isbell-convex hull or the injective hull of (x,q) is the set e(x,q) of all extremal function pairs on (x,d). it is well-known that if a function pair (f1,f2) is extremal, then f1 : (x,q −1) → (r,u) and f2 : (x,q) → (r,u) are nonexpansive map, that is f1(x) − f1(y) ≤ q(y,x) and f2(x) − f2(y) ≤ q(x,y) whenever x,y ∈ x. furthermore, f1(x) = sup y∈x (q(y,x)−̇f2(y)) and f2(x) = sup y∈x (q(x,y)−̇f1(y)) whenever x ∈ x. for a ∈ x, it is easy to see that fa(x) = (q(a,x),q(x,a)) ∈ e(x,q). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 180 some aspects of isbell-convex quasi-metric spaces let fpnexp(x,r) be the set of all function pairs whose the first component is nonexpansive on (x,q−1) and the second component is nonexpansive on (x,q). observe that if (f1,f2) ∈ e(x,q), then (f1,f2) ∈ fpnexp(x,r). we now consider the set a1(x,q) := a(x,q) ∩ fpnexp(x,r). lemma 5.1. let (x,q) be a t0-quasi-metric space. if we equip fpnexp(x,r) with the restriction of the extended t0-quasi-metric d, then d((f1,f2),(g1,g2)) < ∞. (moreover d is a t0-quasi-metric). proof. let (f1,f2) ∈ fpnexp(x,r). then q(x,y) ≤ f2(x) + f1(y) whenever x,y ∈ x and so sup x∈x (q(x,y)−̇f2(x)) ≤ f1(y). moreover, we have sup x∈x (f1(x)−̇q(y,x)) ≤ f1(y) whenever x,y ∈ x, since f1 is nonexpansive on (x,q −1). thus d((f1,f2),((fy)1,(fy)2)) ≤ f1(y) whenever y ∈ x. by similar arguments one shows that d(((fy)1,(fy)2),(f1,f2)) ≤ f2(y) whenever y ∈ x. therefore, for y ∈ x we have d((f1,f2),(g1,g2)) ≤ f1(y) + g2(y) < ∞ whenever (f1,f2),(g1,g2) ∈ fpnexp(x,r). � the following useful result is due to [1]. its proof is based on zorn’s lemma but a different proof of proposition 5.2 can be given without appealing to zorn’s lemma. proposition 5.2. let (x,q) be a t0-quasi-metric space. there exists a retraction map p : a(x,q) → e(x,q), i.e., a map that satisfies the conditions (a) d(p((f1,f2)),p((g1,g2))) ≤ d((f1,f2),(g1,g2)) whenever (f1,f2),(g1,g2) ∈ a(x,q). (b) p((f1,f2)) ≤ (f1,f2) whenever (f1,f2) ∈ a(x,q).(in particular p((f1,f2)) = (f1,f2) whenever (f1,f2) ∈ e(x,q)). remark 5.3. from proposition 5.2, it follows that if (f1,f2),(g1,g2) ∈ a(x,q), then d(p((f1,f2)),p((g1,g2))) can be ∞. but if (f1,f2),(g1,g2) ∈ fpnexp(x,r), then d(p((f1,f2)),p((g1,g2))) is finite. therefore, the restriction of the map p : a(x,q) → e(x,q) to fpnexp(x,r) is a nonexpansive retraction. proposition 5.4. let (x,q) be a t0-quasi-metric space. then the t0-quasimetric spaces (fpnexp(x,r),d) and (e(x,q),d) are injective. proof. let ∅ 6= a ⊆ b ⊆ x. consider a map f : a → fpnexp(x,r) defined by f(a) = fa = (q(a,.),q(.,a)) whenever a ∈ a. obviously fa is a function pair which is ample, where q(a,.) is nonexpansive on (x,q−1) and q(.,a) is nonexpansive on (x,q). let b ∈ b, we set fb = ((fb)1,(fb)2) where, (fb)1(x) := inf a∈a {(fa)1(x) + q(b,a)} c© agt, upv, 2018 appl. gen. topol. 19, no. 1 181 o. olela otafudu and (fb)2(x) := inf a∈a {(fa)2(x) + q(a,b)} whenever x ∈ x. we have to show that fb ∈ fpnexp(x,r). for each x,y ∈ x, we have (fb)1(x) − (fb)1(y) = inf a∈a {(fa)1(x) + q(b,a)} − inf a′∈a {(fa′)1(y) + q(b,a ′)} ≤ (fa)1(x) + q(b,a) − (fa)1(y) − q(b,a) with a = a ′ ≤ q(a,y) + q(y,x) − q(a,y) = q(y,x). similarly, (fb)2(x) − (fb)2(y) ≤ q(x,y) whenever x,y ∈ x. so (fb)1 and (fb)2 are nonexpansive. to show that the function pair fb is ample, let x,y ∈ x. then (fb)2(x) + (fb)1(y) ≥ inf a,a′∈a {(fa)2(x) + q(a,b) + (fa′)1(x) + q(b,a ′)} ≥ inf a,a′∈a {(fa)2(x) + (fa′)1(x) + q(a,a ′)} ≥ inf a∈a {(fa)2(x) + q(a,y)} = inf a∈a {q(x,a) + q(a,y)} ≥ q(x,y). therefore, fb ∈ fpnexp(x,r). let b,b′ ∈ b and x ∈ x. we show that d(fb,fb′) ≤ q(b,b ′). indeed, (fb′)2(x) − q(b,b ′) = inf a∈a {(fa)2(x) + q(a,b ′)} − q(b,b′) = inf a∈a {(fa)2(x) + q(a,b ′) − q(b,b′)} ≤ inf a∈a {(fa)2(x) + q(a,b)} ≤ (fb)2(x). hence (5.1) sup x∈x ( (fb′)2(x)−̇(fb)2(x) ) ≤ q(b,b′) whenever b,b′ ∈ b and x ∈ x. similarly, we have (5.2) sup x∈x ( (fb)1(x)−̇(fb′)1(x) ) ≤ q(b,b′) for b,b′ ∈ b and x ∈ x. combining (5.1) and (5.2) we have d(fb,fb′) ≤ q(b,b ′) for b,b′ ∈ b. we now show that fb = fb whenever b ∈ a. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 182 some aspects of isbell-convex quasi-metric spaces if b ∈ a, then (5.3) (fb)1(x) ≤ q(b,x) = (fb)1(x) whenever x ∈ x since (fb)1(x) ≤ q(b,x) + q(b,b) and b ∈ a. moreover, (fb)1(x) = q(b,x) ≤ q(b,a) + q(a,x) = (fa)1(x) + q(b,a) for all x ∈ x. thus (5.4) (fb)1(x) ≤ inf a∈a {(fa)1(x) + q(b,a)} = (fb)1(x) for all x ∈ x and b ∈ b. hence, (fb)1(x) = (fb)1(x) for all x ∈ x and b ∈ a from (5.3) and (5.4). analogously, one shows that (fb)2(x) = (fb)2(x) whenever x ∈ x and b ∈ a. therefore the map f : b → fpnexp(x,r) defined by f(b) = fb whenever b ∈ b, extends f . so (fpnexp(x,r),d) is injective. the injectivity of (e(x,q),d) follows from remark 5.3 since (e(x,q),d) is nonexpansive retract of (fpnexp(x,r),d). ✷ the following observations are not new since they have been discussed in [10] from the metric point of view. remark 5.5. let (x,q) be a t0-quasi-metric space. (a) if l : (e(x,q),d) → (e(x,q),d) is a nonexpansive map that fixes ex(x) pointwise, then l is an identity on e(x,q). indeed, if f = (f1,f2) ∈ e(x,q) such that l(f) = (g1,g2) for some g = (g1,g2) ∈ e(x,q), then, g1(x) = d(g,fx) = d(l(f),l(fx)) ≤ d(f,fx) = f1(x) for x ∈ x. similarly, g2(x) ≤ f2(x) whenever x ∈ x. by minimality of (f1,f2), we have g1(x) = f1(x) and g2(x) = f2(x) whenever x ∈ x. hence f = (f1,f2) = (g1,g2) = g. therefore, l(f) = f whenever f ∈ e(x,q). (b) since (e(x,q),d) is injective and ex is quasi-essential by [11, remark 16], (e(x,q),ex) is an injective hull of (x,q). (c) if (y,i) is another injective hull of (x,q), then there exists an isometric embedding of (x,q), and then there exists a unique isometry i : (e(x,q),d) → (y,i) such that i ◦ ex = i by [11, proposition 10]. in [1], agyingi et al. proved that if (y,qy ) is a t0-quasi-metric space and x is a subspace of (y,qy ), then there exists an isometric embedding τ : (e(x,q),d) → (y,qy ) such that τ(f)|x = f whenever f ∈ e(x,q). proposition 5.6. let (x,q) be a t0-quasi-metric space. if l : (x,q) → (x,q) is an isometry, then there exists a unique isometry l̄ : (e(x,q),d) → (e(x,q),d) such that l̄◦ex = ex ◦l. furthermore, l̄(f) = (f1◦l −1,f2◦l −1) whenever f ∈ e(x,q). proof. suppose l : (x,q) → (x,q) is an isometry. then ex ◦ l : (x,q) → (e(x,q),d) is quasi-essential and since (e(x,q),d) is injective, it follows that (e(x,q),ex ◦ l) is injective hull of (x,q) by remark 5.5 (b). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 183 o. olela otafudu moreover, by remark 5.5 (c), there exists a unique isometry such that l̄ ◦ ex = ex ◦ l. if f = (f1,f2) ∈ e(x,q) and x ∈ x, then (l̄(f))1(x) = d(l̄(f),fx) = d(l̄(f), l̄(l̄ −1(fx))) = d(f,l̄ −1(fx)). since ex ◦ l −1 = l̄−1 ◦ ex, fl−1(x) = (ex ◦ l −1)(x) = (l̄−1 ◦ ex)(x) = l̄ −1(fx), whenever x ∈ x. hence (l̄(f))1(x) = d(f,l̄ −1(fx)) = d(f,fl−1(x)) = f1(l −1(x)) = (f1 ◦ l −1)(x). by similar arguments we have (l̄(f))2(x) = (f1 ◦ l −1)(x) whenever x ∈ x. � proposition 5.7. let (x,q) be a t0-quasi-metric space. if l : (x,q) → (x,q) is an isometry, then the function pair ψ(f) = l̄(f) is ample whenever f = (f1,f2) ∈ a(x,q). furthermore, we have l̄(p(f)) = p(l̄(f)) whenever f = (f1,f2) ∈ a(x,q), where p is the map in proposition 5.2 and l̄ is the unique isometry map in proposition 5.6. proof. let f = (f1,f2) ∈ a(x,q). then for any x,y ∈ x, we have (l̄(f))2(x) + (l̄(f))1(y) = (f2 ◦ l −1)(x) + (f1 ◦ l −1)(y) = f2(l −1(x)) + f1(l −1(y)) ≥ q(l−1(x),l−1(y)) = q(x,y). let y ∈ x. consider f∗1 (y) = sup x′∈x {q(x′,y)−̇f2(x ′)}, f∗2 (y) = sup x′∈x {q(y,x′)−̇f1(x ′)} and the operator q(f) = (1 2 (f1 + f ∗ 1 ), 1 2 (f2 + f ∗ 2 )) defined in the proof (given in [1]) of proposition 5.2. then (f∗1 ◦ l −1)(y) = f∗1 (l −1(y)) = sup x′∈x {q(x′,l−1(y))−̇f2(x ′)} = sup l−1(l(x′))∈x {q(l−1(l(x′)),l−1(y))−̇f2(l −1(l(x′)))} = f1(l −1)∗(y) = (f1 ◦ l −1)∗(y). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 184 some aspects of isbell-convex quasi-metric spaces thus, we have that (f∗1 ◦ l −1)(y) = (f1 ◦ l −1)∗(y) whenever y ∈ x. hence, whenever x ∈ x we have (q(f)1 ◦ l −1)(x) = ( 1 2 (f1 + f ∗ 1 ) ◦ l −1 ) (x) = 1 2 ( f1(l −1) + f∗1 (l −1) ) (x) = 1 2 ( f1 ◦ l −1 + f∗1 ◦ l −1 ) (x) = q(f ◦ l−1)1(x). similarly, we can show that (q(f)2 ◦ l −1)(x) = q(f ◦ l−1)2(x) whenever x ∈ x. therefore, l̄(p(f))1 = p(f)1 ◦ l −1 = p(f ◦ l−1)1 = p(l̄(f))1 and l̄(p(f))2 = p(f)2 ◦ l −1 = p(f ◦ l−1)2 = p(l̄(f))2. � proposition 5.8. every isbell-convex t0-quasi-metric space admits a conical geodesic bicombing which satisfies the equivariance property. proof. suppose that (x,q) is an isbell-convex t0-quasi-metric space. let x,y ∈ x and λ ∈ [0,1], we define a function pair ϕλxy = (ϕ λ xy,1,ϕ λ xy,2) by ϕ λ xy,1(z) = (1 − λ)(fx)1(z) + λ(fy)1(z) and ϕλxy,2(z) = (1 − λ)(fx)2(z) + λ(fy)2(z) whenever z ∈ x. we will prove that ϕλxy ∈ a1(x,q). we first show that ϕλxy is ample. let z,z ′ ∈ x, then ϕλxy,2(z) + ϕ λ xy,1(z ′) = (1 − λ)q(z,x) + λq(z,y) + (1 − λ)q(x,z′) + λq(y,z′) = (1 − λ)[q(z,x) + q(x,z′)] + λ[q(z,y) + q(y,z′)] ≥ (1 − λ)q(z,z′) + λq(z,z′) = q(z,z′). we now show that ϕλxy,2 is a nonexpansive map on (x,q) and the proof of the fact that ϕλxy,1 is a nonexpansive map on (x,q −1) follow by similar arguments. let z,z′ ∈ x, then ϕλxy,2(z) − ϕ λ xy,2(z ′) = [(1 − λ)q(z,x) + λq(z,y)] − [(1 − λ)q(z′,x) + λq(z′,y)] = (1 − λ)[q(z,x) − q(z′,x)] + λ[q(z,y) − q(z′,y)] ≤ q(z,z′). thus ϕλxy ∈ a1(x,q). since (x,q) an isbell-convex t0-quasi-metric space, (x,q) is injective. then the map ex : (x,q) → e(x,q) defined by ex(x) = fx whenever x ∈ x, is an isometry. we consider the retraction map p : a(x,q) → e(x,q) in proposition 5.2. for any x,y ∈ x and λ ∈ [0,1], we set σ(x,y,λ) := (e−1 x ◦ p) ◦ ϕλxy. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 185 o. olela otafudu now we have to show that σ is a conical geodesic bicombing on x. on sees that σ is well defined. observe that if λ = 0, then ϕλxy = ((fx)1,(fx)2). moreover, if λ = 1, then ϕλxy = ((fy)1,(fy)2). it follows that σ(x,y,0) = (e−1 x ◦ p) ◦ (((fx)1,(fx)2)) = e −1 x (ex(x)) = x and σ(x,y,1) = (e−1 x ◦ p) ◦ (((fy)1,(fy)2)) = e −1 x (ex(y)) = y, since ((fx)1,(fx)2),((fy)1,(fy)2) ∈ e(x,q). let x,y ∈ x and λ,λ ′ ∈ [0,1]. then q(σ(x,y,λ),σ(x,y,λ′)) = d(ex(σ(x,y,λ)),ex (σ(x,y,λ ′))) = d[ex(e −1 x (p(ϕλxy))),ex(e −1 x (p(ϕλ ′ xy)))] = d(p(ϕλxy),p(ϕ λ ′ xy)). hence q(σ(x,y,λ),σ(x,y,λ′)) ≤ d(ϕλxy,ϕ λ ′ xy) since p is a retraction. furthermore, d(ϕλxy,ϕ λ ′ xy) = sup z∈x [(1 − λ)q(x,z) + λq(y,z)−̇(1 − λ′)q(x,z) + λ′q(y,z)]. if λ′ ≥ λ, then by triangle inequality we have d(ϕλxy,ϕ λ ′ xy) ≤ (λ ′ − λ)q(x,y). if λ′ < λ, then by triangle inequality we have d(ϕλxy,ϕ λ ′ xy) ≤ (λ − λ ′)q(y,x). it follows that if λ′ ≥ λ, then (5.5) q(σ(x,y,λ),σ(x,y,λ′)) ≤ (λ′ − λ)q(x,y) and if λ′ < λ, then (5.6) q(σ(x,y,λ),σ(x,y,λ′)) ≤ (λ − λ′)q(y,x). observe that for any x,y ∈ x and 0 ≤ λ ≤ λ′ ≤ 1, since σ(x,y,0) = x and σ(x,y,1) = y we have the following equality from the inequality (5.5) q(σ(x,y,λ),σ(x,y,λ′)) = (λ′ − λ)q(x,y). similarly, we obtain from inequality (5.6) q(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)q(y,x). therefore, σ is a geodesic bicombing on x. it remains to show that σ satisfies the property (4.1) to be conical. let x,y,x′,y′ ∈ x and λ ∈ [0,1]. then d(ϕλxy,ϕ λ x′y′) = sup z∈x [(1 − λ)q(x,z) + λq(y,z)−̇(1 − λ)q(x′,z) + λq(y′,z)] ≤ (1 − λ)q(x,x′) + λq(y,y′). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 186 some aspects of isbell-convex quasi-metric spaces hence q(σ(x,y,λ),σ(x′,y′,λ) ≤ d(ϕλxy,ϕ λ x′y′) ≤ (1 − λ)q(x,x ′) + λq(y,y′). thus σ is a conical geodesic bicombing on x. the equivariance follows from the observations below: for z ∈ x, we have ϕ 1−λ yx,1(z) = λ(fy)1(z) + (1 − λ)(fx)1(z) = ϕ λ xy,1(z) and ϕ 1−λ yx,2(z) = λ(fy)2(z) + (1 − λ)(fx)2(z) = ϕ λ xy,2(z) whenever x,y ∈ x and λ ∈ [0,1]. � acknowledgements. the author would like to thank the south african national research foundation (nrf) and the faculty of science research committee (frc) of university of the witwatersrand for partial financial support. references [1] c. a. agyingi, p. haihambo and h.-p. a. künzi, tight extensions of t0-quasi-metric spaces, logic, computation, hierarchies, ontos math. log., 4, de gruyter, berlin, 2014, pp 9–22. [2] g. berthiaume, on quasi-uniformities in hyperspaces, proc. amer. math. soc. 66 (1977), 335–343. [3] j. conradie, h.-p. künzi and o. olela otafudu, the vector lattice structure on the isbell-convex hull of an asymmetrically normed real vector space, topology appl. 231 (2017), 92–112. [4] d. descombes and u. lang, convex geodesic bicombings and hyperbolicity, geom. dedicata 177 (2015), 367–384. [5] a. w. m. dress, trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, adv. math. 53 (1984), 321–402. [6] j. r. isbell, six theorems about injective metric spaces, comment. math. helv. 39 (1964), 65–76. [7] e. kemajou, h.-p.a. künzi and o. olela otafudu, the isbell-hull of a di-space, topology appl. 159 (2012), 2463–2475. [8] h.-p. a. künzi and c. ryser, the bourbaki quasi-uniformity, topology proc. 20 (1995), 161–183. [9] h.-p. a. künzi and f. yildiz, convexity structures in t0-quasi-metric spaces, topology appl. 200 (2016), 2–18. [10] u. lang. injective hulls of certain discrete metric spaces and groups. j. topol. anal. 5 (2013) 297-331. [11] o. olela otafudu and z. mushaandja, versatile asymmetrical tight extensions, topol. algebra appl. 5 (2017), 6–12 c© agt, upv, 2018 appl. gen. topol. 19, no. 1 187 () @ appl. gen. topol. 19, no. 1 (2018), 129-144doi:10.4995/agt.2018.7849 c© agt, upv, 2018 characterization of quantale-valued metric spaces and quantale-valued partial metric spaces by convergence gunther jäger a and t. m. g. ahsanullah b a school of mechanical engineering, university of applied sciences stralsund, 18435 stralsund, germany (gunther.jaeger@hochschule-stralsund.de) b department of mathematics, college of science, king saud university, riyadh 11451, saudi arabia. (tmga1@ksu.edu.sa) communicated by v. gregori abstract we identify two categories of quantale-valued convergence tower spaces that are isomorphic to the categories of quantale-valued metric spaces and quantale-valued partial metric spaces, respectively. as an application we state a quantale-valued metrization theorem for quantale-valued convergence tower groups. 2010 msc: 54a20; 54a40; 54e35; 54e70. keywords: l-metric space; l-partial metric space; l-convergence tower space; l-convergence tower group; metrization. 1. introduction there are different generalizations of metric spaces. one of them solves the problem of assigning a precise value to the distance between two points by allowing instead the assignment of a probability distribution for each pair of points, the value of which at u ∈ [0,∞] giving the probabilty that the distance between the points is less than u. a thorough treatment of these probabilistic metric spaces can be found in [28]. from a different perspective, metric spaces are viewed as categories in [19] and later, in [9] it has been shown that not only received 05 july 2017 – accepted 10 october 2017 http://dx.doi.org/10.4995/agt.2018.7849 g. jäger and t. m. g. ahsanullah classical metric spaces but also probabilistic metric spaces are special instances of this approach. the main idea here is to replace the interval [0,∞] as the codomain of a metric by a quantale. for this reason, we speak of quantalevalued metric spaces. another generalization of metric spaces deals with the problem, that self-distances are not always zero. relaxing this requirement and at the same time enforcing the transivity axiom leads to the theory of partial metric spaces [17, 21]. this concept was independently, and in much wider generality, introduced under the name m-valued set by höhle [11], where the relationship with the general view point of [19] becomes obvious, as also m-valued sets take their values in a quantale. all these generalizations allow the introduction of underlying topological spaces and, in consequence, of a concept of convergence. in this paper, we look at convergence for quantale-valued metric spaces from a different perspective. rather than describing a concept of convergence underlying a quantale-valued metric space, we are looking for a concept of convergence that characterizes such spaces. the key point is here to allow different grades of convergence, where these grades are interpreted as values in the quantale. in this sense, a filter in a quantale-valued (partial) metric space converges to a point with a certain grade. we obtain in this way a family of convergence structures on a set indexed by the quantale. for the unit interval as ”index set” spaces with such towers of convergence structures were first studied by richardson and kent under the name probabilistic convergence spaces [26]. in a more general setting, quantale-valued convergence towers are considered in [15]. in this paper, we identify a set of axioms, such that the quantale-valued (partial) convergence tower spaces satisfying these axioms can be identified with quantale-valued (partial) metric spaces. the paper is organized as follows. in the second section, we collect the necessary concepts and notions from lattice theory and fix the notation. in the third section, we study quantale-valued metric spaces and quantale-valued convergence tower spaces and their relationship. the fourth section then states an axiom which ensures the isomorphy of quantale-valued metric spaces and a category of quantale-valued convergence tower spaces satisfying this axiom. in a similar fashion, in section 5, it is shown that quantale-valued partial convergence tower spaces satisfying certain axioms can be used to characterize quantale-valued partial metric spaces. finally, in section 6, we apply our results and state a quantale-valued metrization theorem for quantale-valued convergence tower groups. 2. preliminaries let l be a complete lattice. we assume that l is non-trivial in the sense that ⊤ 6= ⊥ for the top element ⊤ and the bottom element ⊥. in any complete lattice l we can define the well-below relation α ✁ β if for all subsets d ⊆ l such that β ≤ ∨ d there is δ ∈ d such that α ≤ δ. then α ≤ β whenever α✁β and α ✁ ∨ j∈j βj iff α ✁ βi for some i ∈ j. a complete lattice is completely c© agt, upv, 2018 appl. gen. topol. 19, no. 1 130 characterization of l-(partial) metric spaces by convergence distributive (sometimes called constructively completely distributive) if and only if we have α = ∨ {β : β ✁ α} for any α ∈ l, [25]. (for a more accessible proof of the equivalence of this condition with the classical concept of complete distributivity in the presence of the axiom of choice see e.g. theorem 7.2.3 in [1].) in a completely distributive lattice l, from α ✁ β = ∨ {γ ∈ l : γ ✁ β} we infer the existence of γ ∈ l such that α✁γ ✁β, i.e. l satisfies the so-called interpolation property. for more results on lattices we refer to [10]. the triple l = (l,≤,∗), where (l,≤) is a complete lattice, is called a quantale [27] if (l,∗) is a semigroup, and ∗ is distributive over arbitrary joins, i.e. ( ∨ i∈j αi ) ∗ β = ∨ i∈j (αi ∗ β) and β ∗ ( ∨ i∈j αi ) = ∨ i∈j (β ∗ αi). a quantale l = (l,≤,∗) is called commutative if (l,∗) is a commutative semigroup and it is called integral if the top element of l acts as the unit, i.e. if α ∗ ⊤ = ⊤ ∗ α = α for all α ∈ l. a quantale l = (l,≤,∗) is called an mvalgebra [13], if for all α,β ∈ l we have (α → β) → β = α∨β. in a quantale we can define an implication operator by α → β = ∨ {γ ∈ l : α ∗ γ ≤ β}. then δ ≤ α → β if and only if δ ∗ α ≤ β. we consider in this paper only commutative and integral quantales l = (l,≤,∗) with completely distributive lattices l. example 2.1. a triangular norm or t-norm is a binary operation ∗ on the unit interval [0,1] which is associative, commutative, non-decreasing in each argument and which has 1 as the unit. the triple l = ([0,1],≤,∗) can be considered as a quantale if the t-norm is left-continuous. the three most commonly used (left-continuous) t-norms are: • the minimum t-norm: α ∗ β = α ∧ β, • the product t-norm: α ∗ β = α · β, • the lukasiewicz t-norm: α ∗ β = (α + β − 1) ∨ 0. for the minimum t-norm we obtain α → β = { ⊤ if α ≤ β β if α > β . for the product t-norm we have α → β = β α ∧ 1 and for the lukasiewicz t-norm we have α → β = (1 − α + β) ∧ 1. example 2.2. [19] the interval [0,∞] with the opposite order and addition as the quantale operation α ∗ β = α + β (extended by α + ∞ = ∞ + a = ∞ for all α,β ∈ [0,∞]) is a quantale l = ([0,∞],≥,+), see e.g. [9]. we have here α → β = (β − α) ∨ 0. example 2.3. a function ϕ : [0,∞] −→ [0,1], which is non-decreasing, leftcontinuous on (0,∞) – in the sense that for all x ∈ (0,∞) we have ϕ(x) = supz 0 and ϕ ∈ ∆+ the (ϕ,ǫ)-neighbourhood of x ∈ x by nϕ,ǫx = {y ∈ x : d(x,y)(u + ǫ) + ǫ ≥ ϕ(u) ∀u ∈ [0, 1 ǫ )}. and define the ϕ-neighbourhood filter of x ∈ x, nϕx as the filter generated by the sets nϕ,ǫx , ǫ > 0. if we define x ∈ c̃dϕ(f) ⇐⇒ f ≥ n ϕ x then we obtain a left-continuous and pretopological l-convergence tower space (x,c̃d). in order to show that this l-convergence tower space coincides with the l-convergence tower space (x,cd), we need the following results from [29]. for ϕ ∈ ∆+ and 0 ≤ ǫ ≤ 1 we define ϕǫ ∈ ∆+ by ϕǫ(u) =    0 if u = 0 (ϕ(u + ǫ) + ǫ) ∧ 1 if 0 1 ǫ . clearly then ϕ ≤ ϕǫ and tardiff [29] shows that y ∈ nϕ,ǫx if and only if d(x,y)ǫ ≥ ϕ and ϕ ≥ ψ if and only if for all ǫ > 0 we have ϕǫ ≥ ψ. the last assertion implies that for ϕ ∈ ∆+ we have ϕ = ∧ ǫ>0 ϕ ǫ. we will need the following results. lemma a. let ϕj ∈ ∆ + for all j ∈ j and let 0 ≤ ǫ ≤ 1. then ( ∨ j∈j ϕj) ǫ = ∨ j∈j(ϕ ǫ j) and ( ∧ j∈j ϕj) ǫ = ∧ j∈j(ϕ ǫ j). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 135 g. jäger and t. m. g. ahsanullah proof. we only show the second assertion, the first one being similar. for u = 0 or u > 1 ǫ the assertion is obvious. let 0 < u ≤ 1 ǫ . then we have ( ∧ j∈j ϕj) ǫ(u) =  ( ∧ j∈j ϕj)(u + ǫ) + ǫ   ∧ 1 = ( ( sup v 0. so we conclude that for all u ∈ u we have nψ,ǫx ∩ u 6= ∅, and hence, u being an ultrafilter, n ψ,ǫ x ∈ u for all ǫ > 0, i.e. nψx ≤ u. therefore x ∈ c̃ d ψ(u) for all ψ ✁ ϕ and from the left-continuity then also x ∈ c̃dϕ(u). this is true for all ultrafilters u ≥ f and hence, by pretopologicalness, x ∈ c̃dϕ(f). conversely, let f ≥ nϕx. then, for ǫ > 0, we have ∨ f∈f ∧ y∈f d(x,y)ǫ ≥ ∧ y∈n ϕ,ǫ x d(x,y)ǫ ≥ ϕ. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 136 characterization of l-(partial) metric spaces by convergence from lemma a we conclude ϕ ≤ ∧ ǫ>0 ∨ f∈f ∧ y∈f d(x,y)ǫ = ∧ ǫ>0 ( ∨ f∈f ∧ y∈f d(x,y))ǫ = ∨ f∈f ∧ y∈f d(x,y) and we have x ∈ cdϕ(f). 4. the isomorphy of l-met and l-met-cts we introduce the following axiom for an l-convergence tower space. we say that (x,c) ∈ |l-cts| satisfies the axiom (lm) if for all u ∈ u(x) and all α ∈ l we have (lm) x ∈ cα(u) ⇐⇒ ∀u ∈ u,β ✁ α∃y ∈ u s.t. x ∈ cβ([y]). this axiom was introduced in [7] for probabilistic convergence spaces in the sense of richardson and kent [26]. theorem 4.1. let (x,d) ∈ |l-met|. then (x,cd) satisfies (lm). proof. let u ∈ u(x) and let α ∈ l. let first x ∈ cdα(u) and let u ∈ u and β ✁ α. then there is fβ ∈ u such that for all y ∈ fβ we have d(x,y) ≥ β. choose y ∈ u ∩ fβ. then ∨ f∈[y] ∧ z∈f d(x,z) ≥ ∧ z∈u∩fβ d(x,z) ≥ β, i.e. x ∈ cβ([y]). conversely, let for all u ∈ u, β✁α there is y = yβ ∈ u such that x ∈ c d β([y]), i.e. such that ∨ f∈[y] ∧ z∈f d(x,z) ≥ β. let further f ∈ u x = ∧ x∈cd β (f) f. then, for u ∈ u in particular f ∈ [y], i.e. y ∈ f ∩ u. hence u ∨ ux exists and because u is an ultrafilter, we get u ≥ ∧ x∈cd β (f) f. as (x,c d) is pretopological, we conclude cdβ(u) ⊇ ⋂ x∈cd β (f) c d β(f) and we have x ∈ c d β(u). this is true for any β ✁ α and by left-continuity we obtain x ∈ cdα(u). � proposition 4.2. let (x,c) ∈ |l-premet-cts| satisfy the axiom (lm). then c (dc) α (f) = cα(f). proof. let u ∈ u(x) be an ultrafilter and let x ∈ cα(u). by the axiom (lm) we obtain, for β ✁ α that nxβ = {y ∈ x : x ∈ cβ([y])} satisfies n x β ∩ u 6= ∅ for all u ∈ u and hence nxβ ∈ u. furthermore, for x ∈ cβ([y]) we have dc(x,y) ≥ β. hence ∨ u∈u ∧ y∈u dc(x,y) ≥ ∧ y∈nx β dc(x,y) = ∧ x∈cβ([y]) dc(x,y) ≥ β. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 137 g. jäger and t. m. g. ahsanullah this is true for all β✁α and hence also ∧ y∈nx β dc(x,y) = ∧ x∈cβ([y]) dc(x,y) ≥ α, which is equivalent to x ∈ c (dc) α (u). hence we have shown cα(u) ⊆ c (dc) α (u) for all u ∈ u(x) and because both (x,c) and (x,c(d c)) are pretopological, we have for f ∈ f(x) that cα(f) ⊆ c (dc) α (f). the converse implication is always true and so we have equality. � if we denote the subcategory of l-premet-cts with objects the l-metric spaces that satisfy the axiom (lm) by l-met-cts, then we obtain the following main result. theorem 4.3. the categories l-met-cts and l-met are isomorphic. remark 4.4. for l = (∆+,≤,∗) with a continuous triangle function ∗, we introduced in [14] for (x,d) ∈ |l-cts| a different axiom (pm): for all u ∈ u(x), all ϕ ∈ ∆+ and all x ∈ x we have x ∈ cϕ(u) ⇐⇒ ∀u ∈ u,ǫ > 0 ∃y ∈ u s.t. ∨ x∈cψ([y]) ψ(u+ǫ)+ǫ ≥ ϕ(u)∀u ∈ [0, 1 ǫ ). with the notation of this paper and of remark 3.10 then d(c̃ d) = d and if (x,c) is ∗-transitive, left-continuous and pretopological, then c̃ (dc) ϕ (f) = cϕ(f) for all ϕ ∈ ∆+ and all f ∈ f(x). it follows from this, that for an l-convergence tower space (x,c) that is ∗-transitive, left-continuous and pretopological, the axioms (pm) and (lm) are equivalent. in fact, if (pm) is true, then c̃ (dc) ϕ = cϕ and hence, using remark 3.10, then also c (dc) ϕ = cϕ and as d c is an l-metric on x we know that (x,c) = (x,c(d c)) satisfies (lm). a similar argument shows that (lm) implies (pm). 5. l-partial metric spaces as l-convergence tower spaces an l-partial metric space [17, 22] is a pair (x,p) of a set x and a mapping p : x × x −→ l with (lpm1) p(x,y) ≤ p(x,x) for all x,y ∈ x; (lpm2) p(x,y) = p(y,x) for all x,y ∈ x; (lpm3) p(x,y) ∗ (p(y,y) → p(y,z)) ≤ p(x,z). morphisms are defined as in l-met and the category of l-partial metric spaces is denoted by l-pmet. for l = ([0,∞],≥,+), an l-partial metric space is a partial metric space [22]. these spaces were introduced motivated by problems in computer science, where the self-distances d(x,x) are not always zero, [21]. independently and in much wider generality, l-partial metric spaces were introduced and studied under the name m-valued sets in [11, 12]. for l = (∆+,≤,∗), l-partial metric spaces are called probabilistic partial metric spaces in [31] and fuzzy partial metric spaces in [30]. we note that p(y,z) ≤ p(y,y) → p(y,z) and hence (lpm3) implies the transitivity axiom (lm2). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 138 characterization of l-(partial) metric spaces by convergence in the sequel, we need to adapt the definition of l-convergence tower spaces. we relax the axiom (lc1) and replace it by (wlc1) x ∈ cα([x]) whenever cα([x]) 6= ∅. an l-partial convergence tower space is a pair (x,c) which satisfies the axioms (wlc1), (lc2), (lc3) and (lc4). with morphisms as defined before, we denote the category of l-partial convergence tower spaces by l-pcts. we will use the same functors as defined above to embed the category of l-partial metric spaces into the category of l-partial convergence tower spaces. only few adaptations are necessary, so that we simply repeat the results and only prove the modifications. proposition 5.1. let (x,p) ∈ |l-pmet|. define x ∈ cpα(f) ⇐⇒ ∨ f∈f ∧ y∈f p(x,y) ≥ α. then (x,cp) ∈ |l-pcts|. proof. we only need to prove (wlc1). as before, we can show that y ∈ cpα([x]) if and only if p(x,y) ≥ α. if y ∈ cpα([x]), then p(x,y) ≥ α and then by (lpm1) we have p(x,x) ≥ α, i.e. x ∈ cpα([x]). � we call an l-(partial) convergence tower space (x,c) strongly ∗-transitive if (lst) x ∈ cα∗(e(y)→β)([z]) whenever x ∈ cα([y]) and y ∈ cβ([z]), where e(y) = ∨ y∈cγ([y]) γ. it is called symmetric if (ls) x ∈ cα([y]) whenever y ∈ cα([x]). lemma 5.2. if the l-(partial) convergence tower space (x,c) is strongly ∗transitive, then it is transitive. proof. this follows from α ∗ β ≤ α ∗ (e(y) → β) and the axiom (lc3). � proposition 5.3. let (x,p) ∈ |l-pmet|. then (x,cp) is strongly ∗-transitive, left-continuous, symmetric and pretopological. proof. we need to check the strong ∗-transitivity (lst) and the symmetry (ls). for (lst) we first note that e(y) = ∨ y∈c p β ([y]) β = ∨ β≤p(y,y) β = p(y,y). let x ∈ cpα([y]) and y ∈ c p β ([z]). then α ≤ p(x,y) and β ≤ p(y,z) and hence α ∗ (e(y) → β) ≤ p(x,y) ∗ (p(y,y) → p(y,z)) ≤ p(x,z) and hence x ∈ pα∗(e(y)→β)([z]). for (ls), let x ∈ c p α([y]). then p(x,y) = p(y,x) ≥ α and hence y ∈ cpα([x]). � proposition 5.4. let f : (x,d) −→ (x′,d′) be an l-pmet-morphism. then f : (x,cd) −→ (x′,cd ′ ) is continuous. hence we have a functor from l-pmet into the category of strongly ∗transitive, left-continuous, symmetric and pretopological l-partial convergence tower spaces, l-prepmet-pcts. again, this functor is injective on objects. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 139 g. jäger and t. m. g. ahsanullah in the sequel, we have to restrict the lattice context quite strongly. we say that the quantale l = (l,≤,∗) satisfies the axiom (dm2) if for all non-empty index sets j we have (dm2) α → ∨ j∈j βj = ∨ j∈j(α → βj) for all α,βj ∈ l(j ∈ j). examples for quantales that satisfy (dm2) are complete mv-algebras and also l = ([0,∞],≥,+). in general, l = (∆+,≤,∗) does not satisfy (dm2). we show this with the following example. example 5.5. we consider the triangle function induced by the product tnorm defined by ϕ ∗ ψ(u) = ϕ(u) · ψ(u) for all u ∈ [0,∞]. we define for each natural number n ∈ in the distance distribution function ϕn ∈ ∆ + by ϕn(u) = n(u−1) for 1 ≤ u ≤ 1+ 1 n . then ∨ n∈in ϕn = ε1 and hence ε1 → ∨ n∈in ϕn = ε0. on the other hand it is not difficult to show that ε1 → ϕn = ϕn and hence ∨ n∈in (ε1 → ϕn) = ε1. proposition 5.6. let the quantale l = (l,≤,∗) satisfy the axiom (dm2). let (x,c) ∈ |l-prepmet-pcts| and define pc(x,y) = ∨ x∈cα([y]) α. then (x,pc) ∈ |l-pmet|. proof. (lpm1) we have, using (wlc1), pc(x,y) = ∨ y∈cα([x]) α = { ⊥ if cα([x]) = ∅ ≤ ∨ x∈cα([x]) α if cα([x]) 6= ∅ } ≤ pc(x,x). (lpm2) follows from the symmetry (ls). we show (lpm3). first we note that e(y) = ∨ y∈cβ([y]) β = pc(y,y). let now x ∈ cα([y]) and y ∈ cβ([z]). with the axiom (lst) then x ∈ cα∗(e(y)→β)([z]) and hence α ∗ (e(y) → β) ≤ p c(x,z). we conclude with (dm2) and by the distributivity of the quantale operation over joins ∨ x∈cα([y]) ∨ y∈cβ([z]) (α∗(e(y) → β)) = ( ∨ x∈cα([y]) α)∗  e(y) → ∨ y∈cβ([z]) β   ≤ pc(x,z), which is nothing else than pc(x,y) ∗ (pc(y,y) → pc(y,z)) ≤ pc(x,z). � proposition 5.7. let the quantale l satisfy the axiom (dm2). let f : (x,c) −→ (x′,c′) be continuous. then f : (x,pc) −→ (x′,pc ′ ) is an l-pmet-morphism. proposition 5.8. let the quantale l satisfy the axiom (dm2). let (x,p) ∈ |l-pmet|. then p(c p) = p. proposition 5.9. let the quantale l satisfy the axiom (dm2). let (x,c) ∈ |l-pmet-pcts|. then c (pc) α (f) ⊆ cα(f). theorem 5.10. let the quantale l satisfy the axiom (dm2). then the category l-pmet can be coreflectively embedded into the category l-prepmet-pcts. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 140 characterization of l-(partial) metric spaces by convergence we extend the axiom (lm) to l-partial convergence tower spaces. (lm) ∀u ∈ u(x),α ∈ l we have x ∈ cα(u) ⇐⇒ ∀u ∈ u,β ✁ α∃y ∈ u s.t. x ∈ cβ([y]). the proofs of the following results do not make use of the axiom (lc1) and hence they carry over to l-partial metric spaces without any alterations. theorem 5.11. let (x,p) ∈ |l-pmet|. then (x,cp) satisfies (lm). proposition 5.12. let the quantale l satisfy the axiom (dm2) and let (x,c) ∈ |l-prepmet-pcts| satisfy the axiom (lm). then c (pc) α (f) = cα(f). if we denote the subcategory of l-prepmet-pcts with objects the l-partial metric spaces that satisfy the axiom (lm) by l-pmet-pcts, then we obtain the following main result. theorem 5.13. let the quantale l satisfy the axiom (dm2). then the categories l-pmet-pcts and l-pmet are isomorphic. 6. l-metrization of l-convergence tower groups let (x, ·) be a group with neutral element e. for filters f,g ∈ f(x), the filter f ⊙ g is generated by the sets f ⊙ g = {xy : x ∈ f,y ∈ g} for f ∈ f and g ∈ g and the filter f−1 is generated by the sets f−1 = {x−1 : x ∈ f} for f ∈ f. definition 6.1 (see [4]). a triple (x, ·,c), where (x, ·) is a group and (x,c) is an l-convergence tower space, is called an l-convergence tower group if for all x,y ∈ x and all f,g ∈ f(x) (lctgm) xy ∈ cα∗β(f ⊙ g) whenever x ∈ cα(f) and y ∈ cβ(g); (lctgi) x−1 ∈ cα(f −1) whenever x ∈ cα(f). a mapping f : x −→ x′, where (x, ·,c) and (x′, ·′,c′) are l-convergence tower groups, is called an l-ctg-morphism, if f is a homomorphism between the groups (x, ·),(x′, ·′) and a morphism in l-cts. the category of l-convergence tower groups and l-ctg-morphisms is denoted by l-ctg. for l = {0,1}, we obtain classical convergence groups [18, 6], for l = ([0,1],≤,∗) we obtain the probabilistic convergence groups in the sense of [16] and for l = (∆+,≤,∗) we obtain the probabilistic convergence groups of [4]. for l = ([0,∞],≥,+) we obtain limit tower groups [3]. an l-convergence tower group is a stratified {0,1}{0,1}l-convergence tower group in the definition of [5]. lemma 6.2. let (x, ·,c) ∈ |l-ctg| and let α ∈ l, x ∈ x and f ∈ f(x). then x ∈ cα(f) if and only if e ∈ cα([x −1] ⊙ f). proof. if x ∈ cα(f) then by (lc1) and (lctgm) we conclude e = x −1x ∈ c⊤∗α([x −1] ⊙ f) = cα([x −1] ⊙ f). conversely, if e ∈ cα([x −1] ⊙ f), then x = xe ∈ c⊤∗α([x] ⊙ [x −1] ⊙ f) = cα(f). � lemma 6.3. let (x, ·,c) ∈ |l-ctg|. then (x,c) is ∗-transitive. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 141 g. jäger and t. m. g. ahsanullah proof. let x ∈ cα([y]) and y ∈ cβ([z]). then e ∈ cα([x −1] ⊙ [y]) and e ∈ cβ([y −1] ⊙ [z]). by (lctgm) then e = ee ∈ cα∗β([x −1] ⊙ [y] ⊙ [y−1] ⊙ [z]) = cα∗β([x −1] ⊙ [z]) and hence x ∈ cα∗β([z]). � definition 6.4. a triple (x, ·,d) is called an l-metric group if d is invariant, i.e. if d(x,y) = d(xz,yz) = d(zx,zy) for all x,y,z ∈ x. a group homomorphism f : (x, ·) −→ (x′, ·′) between the l-metric groups (x, ·,d),(x′, ·′,d′) is called an l-metg-morphism if it is an l-metric morphism between (x,d) and (x′,d′). the category of l-metric groups is denoted by l-metg. this definition is motivated by the following result, where we use, for an l-metric d : x × x −→ l on x, the product l-metric on x × x defined by d ⊛ d : (x × x) × (x × x) −→ l, d ⊛ d((x,y),(x′,y′)) = d(x,x′) ∗ d(y,y′). lemma 6.5. let (x, ·) be a group and let d : x × x −→ l be an l-metric which is symmetric, i.e. for which d(x,y) = d(y,x) for all x,y ∈ x holds. then the l-metric d is invariant if and only if the mappings m : x ×x −→ x, m(x,y) = xy and i : x −→ x, i(x) = x−1 are l-metric morphisms. proof. let first d be an invariant metric on x. then using the transitivity, we obtain d ⊛ d((x,y),(x′,y′)) = d(x,x′) ∗ d(y,y′) = d(xy,x′y) ∗ d(x′y,x′y′) ≤ d(xy,x′y′) = d(m(x,y),m(x′,y′)), i.e. multiplication is an l-metric morphism. furthermore, using the symmetry of d, we obtain d(x,y) = d(y−1xx−1,y−1yx−1) = d(y−1,x−1) = d(x−1,y−1), i.e. inversion is an l-metric morphism. for the converse, we note that, multiplication being an l-metric morphism, we have for all x,x′,y,y′ ∈ x, d(x,y) ∗ d(x′,y′) = d ⊛ d((x,x′),(y,y′)) ≤ d(xy,x′y′). in particular, we have d(x,y) = d(x,y) ∗ d(z,z) ≤ d(xz,yz) and similarly d(xz,yz) = d(xz,yz) ∗ d(z−1,z−1) ≤ d(xzz−1,yzz−1) = d(x,y). similarly we can show that d(x,y) = d(zx,zy) and hence d is invariant. � we call an l-convergence tower group (x,c) l-metrizable if there is a symmetric and invariant l-metric d on x such that c = cd. theorem 6.6. an l-convergence tower group (x, ·,c) is l-metrizable if and only if it is left-continuous, pretopological, symmetric and satisfies the axiom (lm). proof. we have seen above that if there is an l-metric d such that c = cd, then (x,c) is left-continuous, pretopological and satisfies the axiom (lm). symmetry of (x,cd) follows easily from the symmetry of d. conversely, let (x,c) be left-continuous, pretopological, symmetric and satisfy the axiom (lm). then dc(x,y) = ∨ x∈cα([y]) α is a symmetric l-metric on x that satisfies cd c = c. we only need to show that dc is invariant. to this end, we note that by (lctgm) and (lc1) we have for x,y,z ∈ x that x ∈ cα([y]) if and only if xz ∈ cα([yz]). hence dc(xz,yz) = ∨ xz∈cα([yz]) α = ∨ x∈cα([y]) α = dc(x,y). similarly, we see that dc(zx,zy) = dc(x,y) and hence dc is invariant. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 142 characterization of l-(partial) metric spaces by convergence references [1] s. abramsky and a. jung, domain theory, in: s. abramsky, d.m. gabby, t. s. e. maibaum (eds.), handbook of logic and computer science, vol. 3, claredon press, oxford 1994. 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[29] r. m. tardiff, topologies for probabilistic metric spaces, pacific j. math. 65 (1976), 233–251. [30] j. wu and y. yue, formal balls in fuzzy partial metric spaces, iranian j. fuzzy systems 14 (2017), 155–164. [31] y. yue, separated ∆+-valued equivalences as probabilistic partial metric spaces, journal of intelligent & fuzzy systems 28 (2015), 2715–2724. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 144 @ appl. gen. topol. 17, no. 1(2016), 7-16 doi:10.4995/agt.2016.3919 c© agt, upv, 2016 a uniform approach to normality for topological spaces ankit gupta a and ratna dev sarma b a department of mathematics, university of delhi, delhi 110007, india. (ankitsince1988@yahoo.co.in) b department of mathematics, rajdhani college, university of delhi, delhi 110015, india. (ratna sarma@yahoo.com) abstract (λ,µ)-regularity and (λ,µ)-normality are defined for generalized topological spaces. several variants of normality existing in the literature turn out to be particular cases of (λ,µ)-normality. uryshon’s lemma and tietze extension theorem are discussed in the light of (λ,µ)normality. 2010 msc: 54a05; 54d10. keywords: generalized topology; normality; regularity. 1. introduction a large amount of research in topology is devoted to the study of classes of subsets of topological spaces, which posses properties similar to those of open sets. in the literature, several such classes are available which include amongst others semi-open sets [10], α-open sets [12], β-open sets [1], pre-open sets [11], etc. some other such classes are a-sets [15], b-sets [16], c-sets [7], etc. since these classes have some features common in them, it is quite natural to enquire if these classes can be obtained by using one common definition? á. császàr has successfully provided an answer in this regard. the main tool he has used is, the class of mappings γ : p(x) → p(x) from the power set x into x itself possessing the property of monotonicity (that is, for a ⊆ b implies γ(a) ⊆ γ(b)). in a topological space (x,τ), the operators such as int, cl, int cl, cl int, int cl int, cl int cl etc. are found to belong to this class of mappings. accordingly, the weaker form of open sets including semi-open sets, received 26 may 2015 – accepted 25 march 2016 http://dx.doi.org/10.4995/agt.2016.3919 a. gupta and r. d. sarma pre-open sets, α-open sets, β-open sets are nothing but γ-open sets for different γ’s. all these families form “generalized topologies” on x. in [4], császàr has formulated separation axioms for such spaces. accordingly, separation axioms using semi-open sets[5], β-open sets[13], etc. become particular cases in [4]. in the same spirit, we introduce and investigate a generalized form of normality called (λ,µ)-normality for generalized topologies in this paper. however, unlike in [4], we use two gt’s simultaneously in our definition. this gives us a more general definition of normality, yet it covers almost all the relevant variants of normality existing in the literature. for example, if x has a topology, then by taking λ = µ = int, we get normality for x; λ = int, µ = cl∗θ give θ-normality; λ = int, µ = cl∗δ give ∆-normality for x. if (x,τ1,τ2) is a bitopological space, then λ = intτ1 , µ = intτ2 gives rise to pairwise normality of (x,τ1,τ2). thus our study provides a uniform approach towards various notions of normality existing in the literature. we have shown that the two most important results on normalitythe urysohn’s lemma and tietze extension theorem are valid for (λ,µ)-normality, although in a milder form. we have also defined and studied (λ,µ)-regularity in the process and provided its characterization. 2. preliminaries á. császàr has defined a generalized topological space [3] in the following way: definition 2.1. a collection g of subsets of x is called a generalized topology (in brief gt ) [3] on x if (i) ∅ ∈g; (ii) gi ∈g for i ∈ i 6= ∅, implies g = ⋃ i∈i gi ∈g. the same has been defined and studied as semi topological spaces by peleg[14]. for a topological space (x,τ), each family of semi-open sets, α-open sets, preopen sets and β-open sets etc. form a generalized topology on x. á. császàr [2] has used a map γ : p(x) −→ p(x) where p(x) is the power set of x, as his main tool for developing a generalized form of topological spaces. the map γ possesses the property of monotonicity, which says that, if a ⊆ b then γ(a) ⊆ γ(b). the collection of all such mappings on x is denoted by γ(x), or simply by γ. definition 2.2 ([2]). consider a non empty set x and a map γ ∈ γ(x). we say that a subset a of x is γ-open if a ⊆ γ(a). for a topological space (x,τ), an open set (resp. semi-open, α-open, β-open, pre-open) is γ-open for γ = int (resp. cl int, int cl int, cl int cl, int cl). also for each γ ∈ γ(x), it may be verified that the γ-open sets form a generalized topology on x. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 8 a unified approach to normality definition 2.3 ([2]). let a be a subset of x and γ be a monotonic mapping on x. then the union of all γ-open sets contained in a is called the γ-interior of a, and is denoted by iγ(a). proposition 2.4 ([2]). a subset a of x is γ-open if and only if a = iγ(a) if and only if a is iγ-open. definition 2.5 ([2]). a subset a of x is called γ-closed if x�a is γ-open. definition 2.6 ([2]). the intersection of all γ-closed sets containing a is called γ-closure of a and is denoted by cγ(a). it can be shown that cγ(a) is the smallest γ-closed set containing a. another operator called γ∗ is defined, with the help of γ in the following way: definition 2.7 ([2]). for any a ⊆ x and γ ∈ γ(x), we define γ∗(a) = x�(γ(x�a)) proposition 2.8 ([2]). if γ ∈ γ, then γ∗ ∈ γ. proposition 2.9 ([2]). a subset a of x is γ∗-closed if and only if γ(a) ⊆ a. definition 2.10 ([19]). let x be a topological space and let a ⊆ x. a point x ∈ x is in θ-closure of a if every closed neighbourhood of x intersects a. the θ-closure of a is denoted by clθ(a). the set a is called θ-closed if a = clθa. the complement of a θ-closed set is called θ-open set. definition 2.11 ([19]). let x be a topological space and let a ⊆ x. a point x ∈ x is in δ-closure of a if every regular open neighbourhood of x intersects a. the δ-closure of a is denoted by clδ(a). the set a is called δ-closed if a = clδ(a). the complement of a δ-closed set is called δ-open set. definition 2.12. a topological space x is said to be (1) [8] θ-normal if every pair of disjoint closed sets one of which is θ-closed are contained in disjoint open sets; (2) [8] weakly θ-normal if every pair of disjoint θ-closed sets are contained in disjoint open sets; (3) [6] ∆-normal if every pair of disjoint closed sets one of which is δ-closed are contained in disjoint open sets; (4) [6] weakly ∆-normal if every pair of disjoint δ-closed sets are contained in disjoint open sets. definition 2.13 ([9]). a bitopological space (x,τ1,τ2) is said to be pairwise normal if given a τ1-closed set a and a τ2-closed set b with a∩b = ∅, there exist τ2-open set o2 and τ1-open set o1 such that a ⊆ o2, b ⊆ o1, and o1 ∩o2 = ∅. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 9 a. gupta and r. d. sarma 3. (λ,µ)-regularity and (λ,µ)-normality for defining (λ,µ)-regularity and (λ,µ)-normality, no topology is required on x. it is because, for a non-empty set x, γ(x) is also non-empty. however, we may call x a space, once we define some topological property on x, such as (λ,µ)-regularity, (λ,µ)-normality etc. definition 3.1. let x be a non-empty set and λ,µ ∈ γ(x). then x is said to be λ-regular with respect to µ if for each point x ∈ x and each λ-closed set p such that x /∈ p , there exist a λ-open set u and a µ-open set v such that x ∈ u, p ⊆ v and u ∩v = ∅. x is said to be (λ,µ)-regular if x is λ-regular with respect to µ and vice versa. definition 3.2. a non-empty set x is called (λ,µ)-normal if for a given λclosed set a and a µ-closed set b with a∩b = ∅, there exist a µ-open set u and a λ-open set v such that a ⊆ u, b ⊆ v and u ∩v = ∅. below we provide characterizations for (λ,µ)-regularity and (λ,µ)-normality. theorem 3.3. let x be a non-empty set. then x is (λ,µ)-regular if and only if (i) for a given x ∈ x and λ-open neighbourhood u of x, there exists a µclosed neighbourhood v of x (that is, x ∈ v0 ⊆ v , for some v0 ⊆ x, where v0 is λ-open and v is µ-closed) such that x ∈ v ⊆ u. and (ii) for a given y ∈ x and µ-neighbourhood p of y, there exists a λ-closed neighbourhood q of y such that y ∈ q ⊆ p . proof. let x ∈ x and u be a λ-open neighbourhood of x. therefore x /∈ x�u, a λ-closed set. thus, there exists a disjoint pair of λ-open set o and µ-open set w such that x ∈ o, x�u ⊆ w and o ∩ w = ∅. that is, x�w ⊆ u. hence x ∈ o ⊆ x�w ⊆ u, that is, x ∈ v ⊆ u, where v = x�w , a µ-closed set. similarly, if x is µ-regular with respect to λ, then for a given point x ∈ x and a µ-open neighbourhood u of x, there exists a λ-closed neighbourhood v of x such that x ∈ v ⊆ u. conversely, let x ∈ x and f be a λ-closed set such that x /∈ f. then x ∈ x�f and x�f is λ-open. hence by (i), there exists a λ-open set v0 and a µ-closed set v such that x ∈ v0 ⊆ v ⊆ x�f. then by (ii), there exists a λ-open set q0 and a µ-closed set q such that x ∈ q0 ⊆ q ⊆ v0. thus we have, x ∈ q0, f ⊆ p0, where p0 = x�v , such that q0 is λ-open, p0 is µ-open and p0 ∩q0 = ∅. hence x is (λ,µ)-regular. � theorem 3.4. let x be a non-empty set. then x is (λ,µ)-normal if and only if for a given µ-closed set c and a λ-open set d such that c ⊆ d, there are a λ-open set g and a µ-closed set f such that c ⊆ g ⊆ f ⊆ d. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 10 a unified approach to normality proof. let c and d be the µ-closed and λ-open sets respectively such that c ⊆ d. then x�d is a λ-closed set such that c ∩ (x�d) = ∅. then, from the (λ,µ)-normality, there exist a λ-open set g and a µ-open set v such that c ⊆ g, x�d ⊆ v and g ∩ v = ∅. therefore x�v ⊆ d and hence c ⊆ g ⊆ x�v ⊆ d, where x�v is a µ-closed set. hence c ⊆ g ⊆ f ⊆ d, where x�v = f(say). conversely, consider c as d are λ-closed and µ-closed sets respectively such that c ∩ d = ∅. then x�c is λ-open set containing d. then by the given hypothesis, there exist a λ-open set g and a µ-closed set f such that d ⊆ g ⊆ f ⊆ x�c. thus, we have d ⊆ g, c ⊆ v and g∩v = ∅, where v = x�f , µ-open set. hence x is (λ,µ)-normal. � in our next section, we provide generalized versions of uryshon’s lemma and tietze extension theorem, which holds for (λ,µ)-normality. 4. uryshon’s lemma and tietze extension theorem for (λ,µ)-normality definition 4.1 ([18]). let (x,λ) be a generalized topological space and r be the real line with the usual topology. a mapping f : x → r is said to be generalized upper semi-continuous or g.u.s.c. in brief (resp. generalized lower semi-continuous or g.l.s.c. in brief) if for each a ∈ r, the set {x ∈ x : f(x) < a} (resp. {x ∈ x : f(x) > a}) is λ-open. unlike in topology, a mapping which is both generalized upper semi-continuous and generalized lower semi-continuous, may fail to be generalized continuous in generalized topology. example 4.2. let x = [0, 1], λ consist of the unions of the members of the type [0,a), (b, 1], a,b ∈ [0, 1]. let y = x, under the usual subspace topology of r. then the identity mapping i : x → y is g.u.s.c. and g.l.s.c. but not generalized continuous as f−1(a,b) /∈ λ. theorem 4.3. let x be a (λ,µ)-normal space. then for any disjoint pair of λ-closed set h and µ-closed set f , there exists a real valued function g on x such that (i) g(x) = 0 for x ∈ f , g(x) = 1 for x ∈ h, 0 ≤ g(x) ≤ 1, for all x ∈ x; (ii) g is λ-upper semi-continuous and µ-lower semi-continuous. proof. let x be a (λ,µ)-normal space and g and h be two disjoint subsets of x such that g is µ-closed and h is λ-closed. let us consider, g0 = g and k1 = x�h. then g0 is µ-closed and k1 is λ-open set such that g0 ⊆ k1. since x is (λ,µ)-normal, therefore there exist a λ-open set k1/2 and a µ-closed set g1/2 such that g0 ⊆ k1/2 ⊆ g1/2 ⊆ k1. again applying the hypothesis to each pair of sets (g0 and k1/2) and (g1/2 and k1), we obtain λ-open sets k1/4,k3/4 and µ-closed sets g1/4,g3/4 such that c© agt, upv, 2016 appl. gen. topol. 17, no. 1 11 a. gupta and r. d. sarma g0 ⊆ k1/4 ⊆ g1/4 ⊆ k1/2 ⊆ g1/2 ⊆ k3/4 ⊆ g3/4 ⊆ k1. continuing this process, we obtain two families {gi} and {ki}, where i = p/2q, where {p = 1, 2, . . . , 2q − 1,q = 1, 2, . . .}. if i is any other dyadic rational number other than p/2q, then let ki = ∅, whenever i ≤ 0 and ki = x, for i > 1. similarly, gi = ∅ for i < 0 and gi = x for i ≥ 1. thus, for every r ≤ s ≤ t, we have kr ⊆ ks ⊆ gs ⊆ gt and for s < t, we have gs ⊆ kt. now, we define a function g : x → [0, 1] such that g(x) = inf{t | x ∈ kt} clearly, g(x) ∈ [0, 1]. if x ∈ g, then x ∈ ki for all i, therefore g(x) = 0, when x ∈ h = x�k, then x /∈ ki for all i ∈ [0, 1], hence g(x) = 1. now, we have to show that g is λ-upper semi-continuous and µ-lower semicontinuous. first we show that (i) if x ∈ gp then g(x) ≤ p (ii) if x /∈ kp, then g(x) ≥ p. let x ∈ gp, then x ∈ ks for every s > p. therefore g(x) ≤ p. similarly, if x /∈ kp, then x /∈ ks for any s < p, hence g(x) ≥ p. thus we can say that whenever g(x) > p, we have x /∈ gp and g(x) < p, we have x ∈ kp. now, we consider, x ∈ g−1([0,a)), then g(x) ∈ [0,a), that is, there exists t < a such that g(x) < t and hence x ∈ kt, therefore g−1([0,a)) ⊆ ⋃ t t and hence x /∈ gt, that is, x ∈ x�gt, therefore g−1((a, 1]) ⊆ ⋃ t>a (x�gt). conversely, let x ∈ ⋃ t>a (x�gt), then x ∈ x�gi for some i > a, thus g(x) ≥ i > a and x ∈ g−1((a, 1]). therefore g−1((a, 1]) = ⋃ t>a (x�gt), a µ-open set. hence g is µ-lower semi-continuous function. � our next theorem resembles with the classical tietze extension theorem. but before that, we quote a result which will be used in our main theorem. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 12 a unified approach to normality theorem 4.4 ([3]). let (x,λ) be a generalized topological space and (y,λy) be a generalized topological subspace of x. then a subset a of y is λ-closed in y if and only if it is the intersection of y with a λ-closed set in x. now we come to our main proposed result. theorem 4.5. let x be a (λ,µ)-normal space. let a ⊆ x be a λ-closed as well as µ-closed set and f be a real valued function defined on a which is λ-upper semi-continuous as well as µ-lower semi-continuous function. then there exists an extension f of f to the whole of x such that f is λ-upper semi-continuous and µ-lower semi-continuous in x. proof. let x be a (λ,µ)-normal space and a be a λ-closed and µ-closed subset of x. suppose f is a real valued function on a which is λ-upper semi-continuous and µ-lower semi-continuous. let n be a positive integer, then for each integer k, let unk = {x : f(x) ≥ k/n} and l n k = {x : f(x) ≤ (k − 1)/n} then, for every integer k, unk and l n k are λ-closed and µ-closed subsets of a respectively. therefore unk and l n k are λ-closed and µ-closed subsets of x also and unk ∩l n k = ∅. by theorem 4.3, for each k = 0, 1, 2, . . ., there is a function uk defined on x, which is λ-upper semi-continuous and µ-lower semi-continuous, such that uk(x) = 0, for x ∈ lnk uk(x) = 1/n, for x ∈ u n k and 0 ≤uk(x) ≤ 1/n for all x ∈ x also, for each k = 0,−1,−2, . . ., there is a function vk defined on x which is λ-upper semi-continuous and µ-lower semi-continuous function, such that vk(x) = −1/n, for x ∈ lnk vk(x) = 0, for x ∈ u n k and −1/n ≤vk(x) ≤ 0 for all x ∈ x now, we know that lnk ⊆ l n k+1 and u n k+1 ⊆ u n k . thus for k = 0,−1,−2, . . ., if f(x) ≥ 0, then x ∈ unk and hence vk(x) = 0. similarly, for k = 1, 2, . . ., if f(x) ≤ 0, then x ∈ lnk and hence uk(x) = 0. now, we define a real valued function fn on x as follows: fn(x) = ∞∑ k=1 uk(x) + 0∑ k=−∞ vk(x) for x ∈ x since ui and vi are λ-upper semi continuous and µ-lower semi-continuous function therefore fn is also a λ-upper semi continuous and µ-lower semi-continuous function. now, consider cnk = u n k−1 ∩l n k+1 for k ∈ z. thus c n k ⊆ u n k−1 and c n k ⊆ l n k+1, that is, cnk = {x : k−1 n ≤ f(x) ≤ k n }. therefore a = ⋃ {cnk ,k ∈ z}. now, let k ≥ 1, and x ∈ cnk , that is, 0 ≤ f(x) ≤ k/n. since x ∈ l n k+1, therefore uj(x) = 0 for j ≥ k + 1 and x ∈ unk−1, therefore uj(x) = 1/n for 1 ≤ j ≤ k−1. also 0 ≤uk(x) ≤ 1/n. thus c© agt, upv, 2016 appl. gen. topol. 17, no. 1 13 a. gupta and r. d. sarma fn(x) = u1(x) + u2(x) + . . . + uk−1(x) + uk(x) + uk+1(x) + . . . fn(x) = k−1 n + uk(x) therefore |f(x) −fn(x)| ≤ |k/n− (k − 1)/n−uk(x)| ≤ 1/n + |uk(x)| ≤ 2/n, for x ∈ cnk . now, let k ≤ 0, and x ∈ cnk , that is, −(k + 1)/n ≤ f(x) ≤ 0. since x ∈ l n k+1, therefore vj(x) = −1/n for (k − 1) ≤ j ≤ 0 and x ∈ unk−1, therefore vj(x) = 0 for j ≤ (k + 1). thus fn(x) = v0(x) + v−1(x) + v−2(x) + . . .vk−1(x) + vk(x) + vk+1(x) + . . . fn(x) = k n + vk(x), for a non positive integer k therefore |f(x) −fn(x)| ≤ 2/n hence |f(x) −fn(x)| ≤ 2/n for all x ∈ a. we recall that g-nets defined in [17] behave almost the same way in generalized topology as the nets do in topology and a sequence is just a particular case of g-nets in generalized topology. due to this fact, fn|a for n = 1, 2, . . . converge uniformly to f on a. also (fn|a) forms a cauchy sequence with respect to the uniform norm on a. as a result, as in[9], f has an extension f to x. it may be easily shown that f is λ-upper semi-continuous and µ-lower semi-continuous function. � 5. conclusion under different set of conditions, we get different variants of normality. if we take (i) λ = µ = interior operator of a topology on x, then the λ-closed sets and µ-closed sets are nothing but the closed sets of x. therefore (λ,µ)normality just becomes normality. (ii) λ = intτ1 and µ = intτ2 , two different interior operators over two different topologies τ1 and τ2, then (λ,µ)-normality becomes pairwise normality [9] of (x,τ1,τ2) (iii) λ =interior operator and µ = cl∗θ operator, then (λ,µ)-normality becomes θ-normality[8]. this is due to fact that clθ ∈ γ, that is, clθ operator is monotonic. first we verify that for a,b ⊆ x such that a ⊆ b, we have clθ(a) ⊆ clθ(b). let x ∈ clθ(a), then every closed neighbourhood of x intersects a. since a ⊆ b, therefore every closed neighbourhood of x intersects b also. hence x ∈ clθ(b). thus clθ(a) ⊆ clθ(b). therefore clθ ∈ γ. hence by proposition 2.8, cl∗θ ∈ γ. now, let a be µ-closed, that is, cl∗θ-closed set. then by proposition 2.9 clθ(a) = (cl ∗ θ) ∗(a) and by proposition 2.8 clθ(a) = (cl ∗ θ) ∗(a) ⊆ a, that is, a is θ-closed. since every θ-open set is open therefore in the light of (λ,µ)-normality, we have disjoint pair of sets in which one is λ-closed, that is, closed set and the other is µ-closed, that is, θ-closed set separated c© agt, upv, 2016 appl. gen. topol. 17, no. 1 14 a unified approach to normality by disjoint µ-open, that is, θ-open set and hence open set and λ-open set, that is, open set respectively. (iv) λ =interior operator and µ = cl∗δ operator, then (λ,µ)-normality becomes ∆-normality [6]. because clδ ∈ γ, that is, clδ operator is monotonic. as, consider a,b ⊆ x such that a ⊆ b. then clδ(a) ⊆ clδ(b). since x ∈ clδ(a), then every regular open neighbourhood of x intersects a. since a ⊆ b, therefore every regular open neighbourhood of x intersects b also. hence x ∈ clδ(b). thus clδ(a) ⊆ clδ(b). therefore clδ ∈ γ. from the proposition 2.8 cl∗δ ∈ γ also. now, a set a is µ-closed set, that is, cl∗δ-closed implies that clδ(a) = (cl∗δ) ∗(a) ⊆ a, that is, a is δ-closed. as every δ-open set is open therefore in the light of (λ,µ)-normality, we have disjoint pair of sets in which one is λ-closed, that is, closed set and the other is µ-closed, that is, δ-closed sets separated by disjoint µ-open, that is, δ-open set and hence open set and λ-open set, that is, open set respectively. 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[19] n. v. veličko, h-closed topological spaces, amer. math. soc.transl. 78, no. 2 (1968), 103–118. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 16 () @ appl. gen. topol. 19, no. 2 (2018), 189-201doi:10.4995/agt.2018.7409 c© agt, upv, 2018 fixed points and coupled fixed points in partially ordered ν-generalized metric spaces mortaza abtahi a, zoran kadelburg b and stojan radenović c a school of mathematics and computer sciences, damghan university, p.o.b. 36715-364, damghan, iran (abtahi@du.ac.ir, mortaza.abtahi@gmail.com) b university of belgrade, faculty of mathematics, studentski trg 16, 11000 beograd, serbia (kadelbur@matf.bg.ac.rs) c university of belgrade, faculty of mechanical engineering, kraljice marije 16, 11120 beograd, serbia (radens@beotel.rs) communicated by m. abbas abstract new fixed point and coupled fixed point theorems in partially ordered ν-generalized metric spaces are presented. since the product of two ν-generalized metric spaces is not in general a ν-generalized metric space, a different approach is needed than in the case of standard metric spaces. 2010 msc: 47h10; 54h25. keywords: meir-keeler contractions; ćirić-matkowski contractions; proinov-type contractions; ν-generalized metric space; coupled fixed point theorems. 1. introduction and preliminaries throughout the paper, the sets of integers, nonnegative integers, and positive integers are denoted, respectively, by z, z+, and n; the sets of real numbers and nonnegative real numbers are denoted, respectively, by r and r+. received 15 march 2017 – accepted 13 april 2018 http://dx.doi.org/10.4995/agt.2018.7409 m. abtahi, z. kadelburg and s. radenović 1.1. ν-generalized metric spaces. there are lots of works done on fixed point theory by weakening the requirements of the banach contraction principle. one direction of these generalizations was introduced by branciari in [6], where the triangle inequality was replaced by a so-called polygonal inequality. in what follows, we briefly recall concepts of ν-generalized metric spaces. see also [3, 8, 11, 19, 20]. definition 1.1 (branciari [6]). let e be a nonempty set and ν ∈ n. a mapping dν : e × e → r + is called a ν-generalized metric and the pair (e, dν) is called a ν-generalized metric space if the following hold: (1) dν(x, y) = 0 if and only if x = y; (2) dν(x, y) = dν(y, x), for all x, y ∈ e; (3) dν(x, y) ≤ dν(x, z1) + dν(z1, z2) + · · · + dν(zν, y), for each set {x, z1, . . . , zν, y} of ν + 2 distinct elements of e. clearly, (e, dν) is a metric space if ν = 1, i.e., it is a 1-generalized metric space. it is shown in [19], that the topology of a ν-generalized metric space may be non-compatible. definition 1.2 ([3]). let (e, dν) be a ν-generalized metric space. given k ∈ n, a sequence {xn} in e is said to be k-cauchy if lim n→∞ sup{dν(xn, xn+1+mk) : m ∈ z +} = 0. the sequence {xn} is called cauchy if it is 1-cauchy. cauchy sequences in ν-generalized metric spaces were investigated in [3, 6, 20]. proposition 1.3 ([3, 20]). let {xn} be a ν-cauchy sequence with distinct terms in (e, dν). if ν is odd, or if ν is even and dν(xn, xn+2) → 0 as n → ∞, then {xn} is cauchy. as shown in [16] and [18], a sequence in a 2-generalized metric space may converge to more than one point and a convergent sequence may not be a cauchy sequence. it is said [3, 20] that a sequence {xn} in e converges to x in the strong sense if {xn} is cauchy and {xn} converges to x. [18, example 1.1] shows that there exist convergent sequences that do not converge in the strong sense. the completeness of ν-generalized metric spaces is investigated in [3]. proposition 1.4 ([20]). let {xn} and {yn} be sequences in (e, dν) that converge to x and y in the strong sense, respectively. then dν(x, y) = limn→∞ dν(xn, yn). branciari proved in [6] a generalization of the banach contraction principle. since, as was already said, a ν-generalized metric space does not necessarily have a compatible topology, his proof needed some corrections, see [9, 16, 18, 19, 21]. proofs of kannan’s [10] and ćirić’s [7] fixed point theorems in νgeneralized metric spaces appear in [20]. the analogue of proinov’s result from [15], as an ultimate generalization of the banach contraction principle in the setting of ν-generalized metric spaces, was proved in [2]. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 190 coupled fixed points in ν-generalized metric spaces 1.2. partially ordered spaces and coupled fixed points. nieto and rodŕıguez-lópez initiated in [14] the use of another enrichment of metric space structure by using additional partial order. a lot of researchers obtained several results in such structures. among them, bhaskar and lakshmikantham started in [5] investigation of so-called coupled fixed points. they proved the existence of coupled fixed points for contractive mappings in partially ordered complete metric spaces. these and similar results were later obtained by different methods; see, e.g. [4, 12, 13, 17]. assume that (e, �) is a partially ordered set and that f : e × e → e is a mapping. the notions of a coupled fixed point of f and the (strict) mixed monotone property has become standard, so we omit them here. given a pair (x, y) of elements in e, the picard iterates {f n(x, y)} and {f n(y, x)} are defined, inductively, as follows. let f 0(x, y) = x, f 0(y, x) = y, and then, for n ∈ z+, f n+1(x, y) = f ( f n(x, y), f n(y, x) ) , f n+1(y, x) = f ( f n(y, x), f n(x, y) ) . (1.1) in 2012, berinde and păcurar [4] presented more general coupled fixed point theorems in partially ordered metric spaces (e, �, d). theorem 1.5 ([4]). let (e, �, d) be a complete partially ordered metric space, and f : e ×e → e be a generalized symmetric meir-keeler type mapping, i.e., for every ǫ > 0, there exists δ > 0 such that, for x � u and y � v, ǫ ≤ d(x, u) + d(y, v) <ǫ + δ =⇒ d(f(x, y), f(u, v)) + d(f(y, x), f(v, u)) < ǫ. suppose that (i) the mapping f is continuous and has the mixed strict monotone property, (ii) there exist x0, y0 ∈ e such that (1.2) x0 � f(x0, y0), y0 � f(y0, x0), or x0 � f(x0, y0), y0 � f(y0, x0). then f has a coupled fixed point. 1.3. fixed points of monotone contractions. fixed point theorems of ćirić-matkowski and proinov types for monotone contractions in partially ordered ν-generalized metric spaces can be deduced from a sequence of lemmas and propositions, similarly as it has been done in the setting of (ν-generalized) metric spaces in [1] and [2]. hence, we just state the respective formulations, omitting the proofs. theorem 1.6. let (e, �, dν) be a complete partially ordered ν-generalized metric space and t : e → e be a monotone contraction of ćirić-matkowski type, i.e., (1) the mapping t is nondecreasing, (2) dν(t x, t y) < dν(x, y), for x ≺ y (that is x � y and x 6= y), c© agt, upv, 2018 appl. gen. topol. 19, no. 2 191 m. abtahi, z. kadelburg and s. radenović (3) for every ǫ > 0, there exists δ > 0 such that ( x � y, ǫ < dν(x, y) < ǫ + δ ) =⇒ dν(t x, t y) ≤ ǫ. then t has a fixed point provided there exists x0 ∈ e such that x0 � t x0. moreover, for any x ∈ e with x � t x, the sequence {t nx} converges to a fixed point of t in the strong sense. we also have a proinov type fixed point theorem. theorem 1.7. suppose that (e, �, dν) is a complete partially ordered ν-generalized metric space, and that t : e → e is sequentially continuous and asymptotically regular, i.e., lim n→∞ ( dν(t nx, t n+1x) + dν(t nx, t n+2x) ) = 0, x ∈ e. for γ > 0, define m(x, y) = dν(x, y) + γ ( dν(x, t x) + dν(y, t y) ) . suppose that dν(t x, t y) < m(x, y), x, y ∈ e, x ≺ y, and that, for any ǫ > 0, there exist δ > 0 and n ∈ z+ such that, for all x, y ∈ e, (1.3) x � y, m(t nx, t ny) < δ + ǫ =⇒ dν(t n+1x, t n+1y) ≤ ǫ. then t has a fixed point provided there exists x0 ∈ e such that x0 � t x0. moreover, for any x ∈ e with x � t x, the sequence {t nx} converges to a fixed point of t in the strong sense. remark 1.8. similarly as in various other known fixed point results in ordered spaces, the presented conditions are not sufficient to conclude that the fixed point is unique. an additional assumption is needed, and this can be either that arbitrary two elements of the fixed point set are comparable, or that there exists a third element, comparable with both of them. we do not go into details here, leaving them for the case of coupled fixed points in the next section. 1.4. outline. the next (main) section is devoted to coupled fixed points, and is divided into three parts. in subsection 2.1, we let e be a partially ordered metric space, and investigate the existence of coupled fixed points for different types of symmetric contractions on e. our technique in this section involves considering induced metric and order on the set e = e × e and reducing a symmetric contraction f : e×e → e to a monotone contraction t : e → e and then applying results obtained in section 1.3 to t . this technique appears in several papers. however, we will show that this method is not applicable in the case of partially ordered ν-generalized metric spaces (see further example 2.2). hence, in subsection 2.2, we shall take a different approach to obtain coupled fixed point results in such spaces. finally, in subsection 2.3, we present a brief discussion of the uniqueness of coupled fixed points. we conclude by an illustrative example in the last subsection. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 192 coupled fixed points in ν-generalized metric spaces 2. coupled fixed points of symmetric contractions in this section, we present coupled fixed point theorems for symmetric contractions on (ν-generalized) metric spaces. we start with the following definition of symmetric contractions of ćirić-matkowski type. definition 2.1. let (e, �, dν) be a partially ordered ν-generalized metric space. a mapping f : e × e → e is called a symmetric contraction of ćirićmatkowski type if (1) for x � u and y � v, with (x, y) 6= (u, v), (2.1) dν(f(x, y), f(u, v)) + dν(f(y, x), f(v, u)) < dν(x, u) + dν(y, v), (2) for every ǫ > 0, there exists δ > 0 such that, for x � u and y � v, ǫ < dν(x, u) + dν(y, v) < ǫ + δ =⇒ dν(f(x, y), f(u, v)) + dν(f(y, x), f(v, u)) ≤ ǫ. (2.2) to avoid repetitive writings and simplify calculations, the following convention seems to be convenient. convention. let (e, �, dν) be a partially ordered (ν-generalized) metric space. set e = e × e and, for all elements x = (x, y) and u = (u, v) of e, define x ⊑ u if and only if x � u and v � y. clearly, (e, ⊑) is a partially ordered set. define ρν : e × e → r + by (2.3) ρν(x, u) = dν(x, u) + dν(y, v), x = (x, y), u = (u, v). obviously, if (e, �, dν) is a (complete) metric space then (e, ⊑, ρν) is a (complete) metric space. in general, however, as the following example shows, if e is a ν-generalized metric space (ν ≥ 2) then (e, ρν) may fail to be a ν-generalized metric space. example 2.2 ([8, example 4.2]). let e = {a, b, c} and define dν : e×e → r + by dν(a, b) = 4, dν(a, c) = dν(b, c) = 1, and dν(x, x) = 0, dν(x, y) = dν(y, x) for all x, y ∈ e. since four distinct points in e do not exist, the rectangular inequality is trivially satisfied. hence, (e, dν) is a 2-generalized metric space, which is obviously not a metric space. now, consider the mappings ρ+, ρmax : e × e → r + defined by ρ+(x, u) = dν(x, u) + dν(y, v), ρmax(x, u) = max{dν(x, u), dν(y, v)}, where x = (x, y) and u = (u, v). then, for the quadrilateral {(a, b), (b, c), (a, c), (c, c)} in e, we have ρ+ ( (a, b), (b, c) ) = 5 > 1 + 1 + 1 = ρ+ ( (a, b), (a, c) ) + ρ+ ( (a, c), (c, c) ) + ρ+ ( (c, c), (b, c) ) , c© agt, upv, 2018 appl. gen. topol. 19, no. 2 193 m. abtahi, z. kadelburg and s. radenović and ρmax ( (a, b), (b, c) ) = 4 > 1 + 1 + 1 = ρmax ( (a, b), (a, c) ) + ρmax ( (a, c), (c, c) ) + ρmax ( (c, c), (b, c) ) . hence, in both cases, rectangular inequality is not satisfied, and so (e, ρ+) and (e, ρmax) are not 2-generalized metric spaces. the following notion of regularity for mappings f : e × e → e is needed in this section. definition 2.3. given x, y ∈ e, the mapping f : e × e → e is called asymptotically regular at x = (x, y) if the picard iterates xn = f n(x, y) and yn = f n(y, x), defined by (1.1), satisfy the following condition (2.4) ρν(xn, xn+1) + ρν(xn, xn+2) → 0, xn = (xn, yn). note that, if (e, dν) is a metric space, the summand ρν(xn, xn+2) in (2.4) is redundant. in the case of metric spaces, coupled fixed point results are usually obtained by considering the induced space (e, ⊑, ρν) and reducing a symmetric contraction f : e × e → e to a monotone contraction t : e → e. this strategy, as example 2.2 shows, does not work in the case of ν-generalized metric spaces. hence, we shall take a different approach in this case. 2.1. coupled fixed points in partially ordered metric spaces. in this section, we assume that (e, �, d) is a partially ordered metric space. given a mapping f : e × e → e, define t : e → e by (2.5) t x = (f(x, y), f(y, x)), x = (x, y). the following properties are straightforward. (i) if f is continuous then t is continuous. (ii) if f is asymptotically regular in the sense of definition 2.3, then t is asymptotically regular in the sense of theorem 1.7. (iii) if f has the mixed monotone property, then t is nondecreasing on (e, ⊑). (iv) if f is a symmetric contraction of ćirić-matkowski type, then t is a monotone contraction of ćirić-matkowski type, in the sense of theorem 1.6. (v) f has a (unique) coupled fixed point if and only if t has a (unique) fixed point. these properties along with the results in section 1.3 yield the following coupled fixed point results. theorem 2.4. let (e, �, d) be a complete partially ordered metric space. suppose that f : e × e → e has the following properties. (i) f is continuous and has the mixed strict monotone property, c© agt, upv, 2018 appl. gen. topol. 19, no. 2 194 coupled fixed points in ν-generalized metric spaces (ii) f is a symmetric contraction of ćirić-matkowski type, (iii) there exist x0, y0 ∈ e such that x0 � f(x0, y0) and y0 � f(y0, x0). then f has a coupled fixed point. proof. since f is continuous, t is continuous. since f is a symmetric contraction of ćirić-matkowski type and has the mixed strict monotone property, t is a monotone contraction of ćirić-matkowski type. since x0 � f(x0, y0) and y0 � f(y0, x0), we get x0 ⊑ t x0 with x0 = (x0, y0). all conditions of theorem 1.6 are satisfied. hence t has a fixed point, which in turn implies that f has a coupled fixed point. � finally, we have the following proinov type coupled fixed point theorem. theorem 2.5. suppose that (e, �, d) is a complete partially ordered metric space, and that f : e ×e → e satisfies conditions (i)-(ii) of theorem 1.5. for γ > 0, define m : e × e → r+ by (here x = (x, y), u = (u, v)) m(x, u) = d(x, u) + d(y, v) + γ ( d(x, f(x, y)) + d(y, f(y, x)) ) + γ ( d(u, f(u, v)) + d(v, f(v, u)) ) . (2.6) suppose that (2.7) d ( f(x, y), f(u, v) ) + d ( f(y, x), f(v, u) ) < m(x, u), x ⊑ u, x 6= u, and that, for any ǫ > 0, there exists δ > 0 such that, for x ⊑ u, (2.8) m(x, u) < δ + ǫ =⇒ d ( f(x, y), f(u, v) ) + d ( f(y, x), f(v, u) ) ≤ ǫ. if f is asymptotically regular, then f has a coupled fixed point. proof. it is easily seen that m(x, u) = ρν(x, u) + γ(ρν(x, t x) + ρν(u, t u)). the assumptions of the theorem imply that ρν(t x, t u) < m(x, u), x ⊑ u, x 6= u. and that, for any ǫ > 0, there exists δ > 0 such that, for x ⊑ u, (2.9) m(x, u) < δ + ǫ =⇒ ρν(t x, t u) ≤ ǫ. hence t is a monotone contraction of ćirić-matkowski type on (e, ρν) that satisfies all conditions of theorem 1.7. hence t has a fixed point which, in turn, implies that that f has a coupled fixed point. � 2.2. coupled fixed points in partially ordered ν-generalized metric spaces. in this subsection, we assume that (e, �, dν) is a partially ordered νgeneralized metric space. as example 2.2 shows, the induced space (e, ⊑, ρν) may not be a partially ordered ν-generalized metric space. hence, we take a different approach to get coupled fixed point theorems. when we call a mapping f : e × e → e continuous (since, in general, we do not have a topological structure in e), we mean that f(xn, yn) → f(x, y) in e whenever {xn} and {yn} are sequences in e such that xn → x and yn → y. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 195 m. abtahi, z. kadelburg and s. radenović lemma 2.6. let {xn} and {yn} be two sequences in a ν-generalized metric space (e, dν) and let xn = (xn, yn). the following statements are equivalent. (i) both {xn} and {yn} are ν-cauchy sequences. (ii) lim n→∞ sup{ρν(xn, xn+1+mν) : m ∈ z +} = 0. proof. this follows easily from definition 1.2 and the following simple inequalities. max{sup{dν(xn, xn+1+mν) : m ∈ z +}, sup{dν(yn, yn+1+mν) : m ∈ z +}} ≤ sup{dν(xn, xn+1+mν) + dν(yn, yn+1+mν) : m ∈ z +} = sup{ρν(xn, xn+1+mν)m ∈ z +} ≤ sup{dν(xn, xn+1+mν) : m ∈ z +} + sup{dν(yn, yn+1+mν) : m ∈ z +}. � lemma 2.7. suppose that {xn} and {yn} are sequences in a ν-generalized metric space (e, dν) satisfying (2.4), each of which consists of mutually distinct elements. suppose that, for every ǫ > 0 and for any two subsequences {xpi} and {xqi}, if lim sup i→∞ ρν(xpi, xqi ) ≤ ǫ then, for some n, ρν(xpi+1, xqi+1) ≤ ǫ (i ≥ n). then both {xn} and {yn} are cauchy sequences. proof. first, we show that both {xn} and {yn} are ν-cauchy. towards a contradiction, assume that, for example, {xn} is not ν-cauchy. then condition (2.6) of lemma 2.6 fails to hold. therefore, for some ǫ > 0, we have (2.10) ∀k ≥ 1, ∃n ≥ k, sup{ρν(xn, xn+1+mν) : m ∈ z +} > ǫ. condition ρν(xn, xn+1) → 0 implies the existence of a sequence {ki} of positive integers such that ki−1 < ki and (2.11) ρν(xn, xn+1) < ǫ/i (n ≥ ki). for each ki, by (2.10), there exist ni ≥ ki + 1 and mi ≥ 0 such that ρν(xni, xni+1+miν) > ǫ. by (2.11), ρν(xni , xni+1) < ǫ. hence, we must have mi ≥ 1. let mi be the smallest number with this property so that ρν(xni, xni+1+miν−ν) ≤ ǫ. take pi = ni −1 and qi = ni +miν. then, for every i ≥ 1, we get qi > pi ≥ ki, and ρν(xpi+1, xqi+1) > ǫ,(2.12) ρν(xpi+1, xqi+1−ν) ≤ ǫ.(2.13) since both {xn} and {yn} consist of mutually different elements, property (3) in definition 1.1 shows that, for every i ≥ 1, dν(xpi , xqi) ≤ dν(xpi , xpi+1) + dν(xpi+1, xqi+1−ν) + dν(xqi+1−ν, xqi+2−ν) + · · · + dν(xqi−1, xqi ), dν(ypi, yqi) ≤ dν(ypi, ypi+1) + dν(ypi+1, yqi+1−ν) + dν(yqi+1−ν, yqi+2−ν) + · · · + dν(yqi−1, yqi). c© agt, upv, 2018 appl. gen. topol. 19, no. 2 196 coupled fixed points in ν-generalized metric spaces the above two inequalities along with (2.11) and (2.13) imply that ρν(xpi, xqi) ≤ 2νǫ/i + ǫ, for all i ≥ 1, from which we get lim sup i→∞ ρν(xpi , xqi) ≤ ǫ. this along with (2.12) violate our assumption. hence both {xn} and {yn} are ν-cauchy. finally, the assumption ρν(xn, xn+2) → 0 as n → ∞ implies that dν(xn, xn+2) → 0 and dν(yn, yn+2) → 0 as n → ∞. proposition 1.3 now shows that both {xn} and {yn} are cauchy sequences. � the following is a ćirić-matkowski type coupled fixed point theorem in partially ordered ν-generalized metric spaces. theorem 2.8. let (e, �, dν) be a complete partially ordered ν-generalized metric space. if f : e ×e → e is a symmetric contraction of ćirić-matkowski type satisfying conditions (i)-(ii) of theorem 1.5, then f has a coupled fixed point. proof. suppose that (1.2) holds for x0 = (x0, y0) and let xn = (xn, yn) be the picard iterates of f at x0 defined by (1.1). note that xp ⊑ xq if p ≤ q. in fact, xn = t nx0, n ≥ 1, where t is defined by (2.5). an argument similar to that of [2, theorem 3.4] shows that ρν(xn, xn+1) + ρν(xn, xn+2) → 0. now, let ǫ > 0 and assume that lim sup i→∞ ρν(xpi, xqi ) ≤ ǫ. since f is a symmetric contraction of ćirić-matkowski type, by (2.2), there exists δ > 0 such that p ≤ q, ǫ < ρν(xp, xq) < δ + ǫ =⇒ ρν(xp+1, xq+1) ≤ ǫ. by [1, lemma 3.1], there is n ∈ n, such that ρν(xpi+1, xqi+1) ≤ ǫ (i ≥ n). now, lemma 2.7 shows that the sequences {xn} and {yn} are both cauchy sequences. since e is complete, there exist x and y in e such that xn → x and yn → y. since f is continuous, we conclude that f(x, y) = x and f(y, x) = y, so that (x, y) is a coupled fixed point. � the following is a proinov type coupled fixed point result in the setting of partially ordered ν-generalized metric spaces. theorem 2.9. let (e, �, dν) be a complete partially ordered ν-generalized metric space. given f : e × e → e, define m : e × e → r+ by (2.6), and assume that (2.8) hold. if f is asymptotically regular and satisfies conditions (i)-(ii) of theorem 1.5, then f has a coupled fixed point. proof. suppose that (1.2) holds for x0 = (x0, y0) and let xn = (xn, yn) be the picard iterates of f at x0 defined by (1.1). note that xp ⊑ xq if p ≤ q. since f is asymptotically regular, we have (2.14) ρν(xn, xn+1) + ρν(xn, xn+2) → 0. let {xpi} and {xqi} be two subsequences of {xn}. then m(xpi, xqi) = ρν(xpi , xqi) + γ ( ρν(xpi, xpi+1) + ρν(xqi, xqi+1) ) . c© agt, upv, 2018 appl. gen. topol. 19, no. 2 197 m. abtahi, z. kadelburg and s. radenović since ρν(xn, xn+1) → 0, we get (2.15) lim sup i→∞ m(xpi, xqi) = lim sup i→∞ ρν(xpi, xqi). note that by (2.14) we get dν(xn, xn+1) → 0 and dν(yn, yn+1) → 0. also, by the condition (1.2) and the strict mixed monotone property of f , we have that the sequences {xn} and {yn} consist of mutually distinct terms. now, let ǫ > 0 and assume that lim sup i→∞ ρν(xpi, xqi ) ≤ ǫ. the equality in (2.15) implies that lim sup i→∞ m(xpi, xqi ) ≤ ǫ. by (2.8), there exists δ > 0 such that, for p ≤ q, m(xp, xq) < δ + ǫ =⇒ ρν(xp+1, xq+1) ≤ ǫ. by [1, lemma 3.1], there is n ∈ n, such that ρν(xpi+1, xqi+1) ≤ ǫ (i ≥ n). all conditions of lemma 2.7 are fulfilled and so the sequences {xn} and {yn} are both cauchy sequences. since e is complete, there exist x and y in e such that xn → x and yn → y. since f is continuous, we conclude that f(x, y) = x and f(y, x) = y, so that (x, y) is a coupled fixed point. � 2.3. uniqueness. in order to obtain the uniqueness of coupled fixed point in the previous results, we need some additional assumption. we formulate it just in the case of theorem 2.8. proposition 2.10. let (e, �, dν) and f be as in theorem 2.8 and let cfix(f) be the set of coupled fixed points of f. then any two comparable elements of cfix(f) (in the sense of order ⊑) are equal. in particular, if all the elements of cfix(f) are comparable, then this set reduces to a singleton. proof. suppose, to the contrary, that there exist two distinct coupled fixed points (x, y) and (u, v) of f which are comparable, e.g., (x, y) ⊑ (u, v) and (x, y) 6= (u, v). then by (2.1) we get that dν(x, y) + dν(u, v) < dν(x, y) + dν(u, v), a contradiction. � 2.4. illustrative examples. the following is a very easy example illustrating a possible use of theorem 2.8. example 2.11. let (e, dν) be the space defined in example 2.2. introduce an order � on e by a � a, b � b, b � a and c � c. consider a mapping f : e × e → e given by f(x, x) = a, for all x ∈ e, f(a, b) = f(b, a) = a, and f(x, y) = c otherwise. it is easy to see that all conditions of theorem 2.8 are satisfied. in particular, the only nontrivial case when conditions (2.1) and (2.2) have to be checked (i.e., when x � u, y � v and (x, y) 6= (u, v)) is when (x, y) = (a, a) and (u, v) = (b, a). it is easily seen that both of them are then satisfied. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 198 coupled fixed points in ν-generalized metric spaces example 2.12. consider the following 2-generalized metric space, which is a slight modification of the space considered in [18, example 1.1]. let e = a∪b, with a = {0, 2}, b = (0, 1], and define dν : e × e → [0, +∞) by dν(x, y) =          0, if x = y 1, if x 6= y and ({x, y} ⊆ a or {x, y} ⊆ b) y, if x ∈ a, y ∈ b x, if x ∈ b, y ∈ a. take the usual order ≤ on e. then (e, ≤, dν) is a complete, partially ordered, 2-generalized metric space which is not a metric space. note that if we define ρν on e = e ×e by (2.3), then (e, ρν) is not a 2-generalized metric space. indeed, if we take the points (0, 0), (b1, 0), (b1, b2), (2, 2) from e (here, 0 < b1, b2 ≤ 1), we have ρν((0, 0), (2, 2)) = dν(0, 2) + dν(0, 2) = 1 + 1 = 2, ρν((0, 0), (b1, 0)) = dν(0, b1) + dν(0, 0) = b1, ρν((b1, 0), (b1, b2)) = dν(b1, b1) + dν(0, b2) = b2, ρν((b1, b2), (2, 2)) = dν(b1, 2) + dν(b2, 2) = b1 + b2. hence, if 2b1 + 2b2 < 2, we have ρν((0, 0), (2, 2)) > ρν((0, 0), (b1, 0)) + ρν((b1, 0), (b1, b2)) + ρν((b1, b2), (2, 2)). consider now the mapping f : e × e → e given by f(x, y) =    x − y 2 , if x ≥ y 0, if x < y. the conditions (i)-(ii) of theorem 1.5 are easy to check (for example, the second one is satisfied for x0 = 2 and y0 = 0). in order to check the condition (2.8), consider the mapping m given by m(x, u) = dν(x, u) + dν(y, v) + dν(x, f(x, y)) + dν(y, f(y, x)) + dν(u, f(u, v)) + dν(v, f(v, u)), for x = (x, y), u = (u, v) (i.e., take γ = 1). for u ⊑ x (i.e., u ≤ x, y ≤ v) and, for example, 1 ≥ x > u > v > y > 0 (other possible cases can be treated in a similar way), we have m(x, u) = dν(x, u) + dν(y, v) + dν ( x, x − y 2 ) + dν(y, 0) + dν ( u, u − v 2 ) + dν(v, 0) = 1 + 1 + 1 + y + 1 + v = 4 + y + v, hence m(x, u) < δ + ǫ implies that ǫ > 4 + y + v − δ > 1 (if δ < 3). on the other hand dν(f(x, y), f(u, v))+dν(f(y, x), f(v, u)) = dν (x − y 2 , u − v 2 ) +dν(0, 0) = 1 < ǫ, and the condition (2.8) is satisfied. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 199 m. abtahi, z. kadelburg and s. radenović thus, all the conditions of theorem 2.9 are fulfilled and we conclude that f has a (unique) coupled fixed point (which is (0, 0)). acknowledgements. the second author is thankful to the ministry of education, science and technological development of serbia, grant no. 174002. references [1] m. abtahi, fixed point theorems for meir-keeler type contractions in metric spaces, fixed point theory 17, no. 2 (2016), 225–236. 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[21] m. turinici, functional contractions in local branciari metric spaces, romai j. 8 (2012),189–199. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 201 @ appl. gen. topol. 16, no. 2(2015), 141-165doi:10.4995/agt.2015.3312 c© agt, upv, 2015 some classes of minimally almost periodic topological groups w. w. comfort a and franklin r. gould b,1 a department of mathematics and computer science, wesleyan university, wesleyan station, middletown, ct 06459, usa (wcomfort@wesleyan.edu) b department of mathematical sciences, central connecticut state university, new britain, 06050, usa (gouldfrr@ccsu.edu) to the memory of ivan prodanov on the occasion of the 30th anniversary of his death abstract a hausdorff topological group g = (g, t ) has the small subgroup generating property (briefly: has the ssgp property, or is an ssgp group) if for each neighborhood u of 1g there is a family h of subgroups of g such that ⋃ h ⊆ u and 〈 ⋃ h〉 is dense in g. the class of ssgp groups is defined and investigated with respect to the properties usually studied by topologists (products, quotients, passage to dense subgroups, and the like), and with respect to the familiar class of minimally almost periodic groups (the m.a.p. groups). additional classes ssgp(n) for n < ω (with ssgp(1) = ssgp) are defined and investigated, and the class-theoretic inclusions ssgp(n) ⊆ ssgp(n + 1) ⊆ m.a.p. are established and shown proper. in passing the authors also establish the presence of ssgp(1) or ssgp(2) in many of the early examples in the literature of abelian m.a.p. groups. 2010 msc: primary 54h11; secondary 22a05. keywords: ssgp group, m.a.p. group; f.p.c. group. 1this paper derives from and extends selected portions of the doctoral dissertation [19], written at wesleyan university (middletown, connecticut, usa) by the second-listed coauthor under the guidance of the first-listed co-author. received 10 october 2014 – accepted 12 july 2015 http://dx.doi.org/10.4995/agt.2015.3312 w. w. comfort and f. r. gould 1. introduction 1.1. conventions. (a) as usual, a topological group is a pair (g, t ) with g a group and with t a topology on g for which the maps g × g → g and g → g given by (x, y) → xy and x → x−1 are continuous. (b) the topological spaces we hypothesize, in particular our hypothesized topological groups, are assumed to be completely regular and hausdorff (i.e., to be tychonoff spaces). when a topology is defined or constructed on a set or a group, the tychonoff property will be verified explicitly (if it is not obvious). in this context we recall ([24](8.4)) that in order that a topological group be a tychonoff space, it suffices that it satisfy the hausdorff separation property. (c) for x a space and x ∈ x we write nx(x) := {u ⊆ x : u is a neighborhood of x}. when ambiguity is unlikely we write n(x) in place of nx(x). (d) the identity of a group g is denoted 1 or 1g; if g is known or assumed to be abelian and additive notation is in play, the identity may be denoted 0 or 0g. (e) when g is a group and κ ≥ ω, we use the notations ⊕ κ g and g(κ) interchangeably: ⊕ κ g = g(κ) := {x ∈ gκ : |{η < κ : xη 6= 1g}| < ω}. when g is a topological group, ⊕ κ g has the topology inherited from gκ. the minimally almost periodic groups (briefly: the m.a.p. groups) to which our title refers are by definition those topological groups g for which every continuous homomorphism φ : g → k with k a compact group satisfies φ[g] = {1k}. it follows from the gel ′fand-răıkov theorem [17] (see [24](§22) for a detailed development and proof) that every compact group k is algebraically and topologically a subgroup of a group of the form πi∈i ui with each ui a (finite-dimensional) unitary group [24](22.14). therefore, to check that a topological group g is m.a.p. it suffices to show that each continuous homomorphism φ : g → u(n) with u(n) the n-dimensional unitary group satisfies φ[g] = {1un}. similarly, since every compact abelian group k is algebraically and topologically isomorphic to a subgroup of a group of the form ti [24](22.17), to check that an abelian topological group g is m.a.p., it suffices to show that each continuous homomorphism φ : g → t satisfies φ[g] = {1t}. sometimes for convenience we denote by m.a.p. the (proper) class of m.a.p. groups, and if g is a m.a.p. group we write g ∈ m.a.p.. similar conventions apply to the classes ssgp(n) (0 ≤ n < ω) defined in definition 3.3. algebraic characterizations of those abelian groups which admit an m.a.p. group topology have been achieved only recently. for a brief historical account of the literature touching this issue, see discussion 2.1(h),(i) below. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 142 some classes of minimally almost periodic topological groups 2. m.a.p. groups: a brief historical survey 2.1. discussion. with no pretense to completeness, we here discuss briefly some of the literature relating to the development of the class of m.a.p. groups. (a) in effect, the class m.a.p. was introduced in 1930 by von neumann [27], who then together with wigner [28] proved that (even in its discrete topology) the matrix group sl(2, c) is an m.a.p. group. (b) in the period 1940–1952, several workers showed that certain real topological linear spaces are m.a.p. groups; several examples, with detailed verification, are given by hewitt and ross [24](23.32). (c) in what follows we will quote at length from the 1980 paper of prodanov [33], which showed by “elementary means” that the group ⊕ ω z admits an m.a.p. topology. (d) ajtai, havas and komlós [1] proved that each group g of the form z, z(p∞), or ⊕ n z(pn) (with all pn ∈ p either identical or distinct) admits a m.a.p. group topology. (e) nienhuys [29] showed that the additive group r admits an m.a.p. topology t contained in its usual topology. since q remains dense in (r, t ) and the m.a.p. property is inherited by dense subgroups, this allowed remus [35] to infer that q admits an m.a.p. topology, hence (since the weak sum of m.a.p. groups is again a m.a.p. group) that every infinite divisible abelian group admits a m.a.p. topology. in the same paper [35], remus showed that every free abelian group admits an m.a.p. topology. (f) protasov [34] and remus [35] asked whether every infinite abelian group admits an m.a.p. group topology; the question was deftly settled in the negative by remus [36] with the straightforward observation that for distinct p, q ∈ p, every group topology on the infinite group g := z(p) × (z(q))κ (with κ ≥ ω) has the property that the homomorphism x → qx maps g continuously onto the compact group z(p). (see [3](3.j), [5](4.6) for additional discussion.) (g) in view of the cited examples z(p) × (z(q))κ of remus [36], it was natural for comfort [3](3.j.1) to ask: (1) does every abelian group which is not torsion of bounded order admit an m.a.p. topology? (2) what about the countable case? (h) gabriyelyan [14], [16] answered question (g)(2) affirmatively, showing that indeed the witnessing topology may be chosen complete in the sense that every cauchy net converges. gabriyelyan [15] showed further that an abelian torsion group of bounded order admits an m.a.p. topology if and only if each of its leading ulm-kaplansky invariants is infinite. (the reader unfamiliar with the ulm-kaplansky invariants might consult [13](§77); those cardinals also play a significant role in [4] in a setting closely related to the present paper.) c© agt, upv, 2015 appl. gen. topol. 16, no. 2 143 w. w. comfort and f. r. gould (i) dikranjan and shakhmatov [8] gave a definitive positive answer to question (g)(1) in its fullest generality, indeed they gave several equivalent conditions on a (not necessarily abelian) group g which are necessary and sufficient that g admit an m.a.p. topology. among those conditions, evidently satisfied by each abelian group not torsion of bounded order, are these: (1) g is connected in its zariski topology; (2) m ∈ z ⇒ mg = {0} or |mg| ≥ ω; (3) the group fin(g) is trivial, i.e., fin(g) = {0}. (the group fin(g), whose study was initiated in [7](4.4) and continued in [4](§2), may be defined by the relation fin(g) = 〈 ⋃ {mg : m ∈ z, |mg| < ω}〉 ) . detailed subsequent analysis of the theorems and techniques of [8] have allowed those authors to answer the following two questions in the negative; these questions were posed in [19] and in a privately circulated pre-publication copy of the present manuscript. (1) let g be a group with a normal subgroup k for which k and g/k admit topologies u and v respectively such that (k, u) ∈ m.a.p and (g/k, w) ∈ m.a.p. is there then necessarily a group topology t on g such that (k is closed in (g, t ) and) (k, u) = (k, t |k) and (g/k, u) = (g/k, tq) with tq the quotient topology? (2) let g be a group with a normal subgroup k such that both k and g/k admit m.a.p. topologies. must g admit a m.a.p. topology? 3. ssgp groups: some generalities definition 3.1. let g = (g, t ) be a topological group and let a ⊆ g. then a topologically generates g if 〈a〉 is dense in g. definition 3.2. a hausdorff topological group g = (g, t ) has the small subgroup generating property if for every u ∈ n(1g) there is a family h of subgroups of g such that ⋃ h ⊆ u and ⋃ h topologically generates g. a (hausdorff) topological group with the small subgroup generating property is said to have the ssgp property, or to be an ssgp group, or simply to be ssgp. now for 0 ≤ n < ω the classes ssgp(n) are defined as follows. definition 3.3. let g = (g, t ) be a hausdorff topological group. then (a) g ∈ ssgp(0) if g is the trivial group. (b) g ∈ ssgp(n + 1) for n ≥ 0 if for every u ∈ n(1g) there is a family h of subgroups of g such that (1) ⋃ h ⊆ u, (2) h := 〈 ⋃ h〉 is normal in g, and (3) g/h ∈ ssgp(n). c© agt, upv, 2015 appl. gen. topol. 16, no. 2 144 some classes of minimally almost periodic topological groups remarks 3.4. (a) for 0 ≤ n < ω the class-theoretic inclusion ssgp(n) ⊆ ssgp(n + 1) holds, hence ssgp(n) ⊆ ssgp(m) when n < m < ω. to see this, note that when g ∈ ssgp(n) and u ∈ n(1g) then we have, taking h := {{1g}}, that h := 〈 ⋃ h〉 = {1g} and g/h ≃ g ∈ ssgp(n), so indeed g ∈ ssgp(n + 1). (b) clearly the class ssgp of definition 3.2 coincides with the class ssgp(1) of definition 3.3. a topological group g is said to be precompact if g is a (dense) topological subgroup of a compact group. it is a theorem of weil [40] that a topological group g is precompact if and only if g is totally bounded in the sense that for each u ∈ n(1g) there is finite f ⊆ g such that g = fu. it is obvious that a precompact group g with |g| > 1 is not m.a.p. indeed if g is dense in the compact group g then the continuous function id : g →֒ g does not satisfy id[g] = {1 g }. theorem 3.5. the class-theoretic inclusion ssgp(n) ⊆ m.a.p. holds for each n < ω. proof. the proof is by induction on n. clearly if g ∈ ssgp(0) and φ ∈ hom(g, u(m)) then φ[g] = {1u(m)}, so g ∈ m.a.p. suppose now that ssgp(n) ⊆ m.a.p., let g be a topological group such that g ∈ ssgp(n+1), and let φ : g → u(m) be a continuous homomorphism. choose v ∈ n(1u(m)) so that v contains no subgroups of u(m) other than {1u(m)}. then u := φ −1[v ] ∈ n(1g), and φ maps every subgroup of u to 1u(m). let h be a family of subgroups of g such that ⋃ h ⊆ u and h := 〈h〉 is normal in g, with g/h ∈ ssgp(n). since a homomorphism maps subgroups to subgroups we have φ[h] = {1u(m)}. it follows that φ defines a continuous homomorphism φ̃ : g/h → u(m) (given by φ̃(xh) := φ(x)). by the induction hypothesis, φ̃ is the trivial homomorphism, so φ is trivial as well; the relation g ∈ m.a.p. follows. � now in 3.6–3.18 we clarify what we do and do not know about the classes of groups mentioned in theorem 3.5. (the reader familiar with the literature may recall that a group as hypothesized in lemma 3.6 is referred to frequently as a group with no small subgroups.) lemma 3.6. let g be a nontrivial (hausdorff) topological group for which some u ∈ n(1g) contains no subgroup other than {1g}. then there is no n < ω such that g ∈ ssgp(n). proof. clearly g /∈ ssgp(0). suppose there is a minimal n > 0 such that g ∈ ssgp(n), and let u ∈ n(1g) be as hypothesized. then the only choice for h (as required in definition 3.3) is h := {{1g}}, yielding h = 〈∪h〉 = {1g}. thus, g/h = g ∈ ssgp(n − 1), which contradicts the assumption that n is minimal. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 145 w. w. comfort and f. r. gould evidently lemma 3.6 furnishes a plethora of topological groups which belong to none of the classes ssgp(n). the interested reader will readily augment the following incomplete list. corollary 3.7. let g be a nontrivial topological group which is either discrete or a lie group. then there is no n < ω such that g ∈ ssgp(n). for another statement in parallel with corollary 3.7, see theorem 3.16 below. definition 3.8. let g be a group and let 1g /∈ c ⊆ g. then c cogenerates g if every subgroup h of g such that |h| > 1 satisfies h ∩ c 6= ∅. theorem 3.9. let g be a nontrivial finitely cogenerated topological group. then there is no n < ω such that g ∈ ssgp(n). proof. let c be a finite set of cogenerators for g, and choose u ∈ n(1g) such that u ⋂ c = ∅. then u contains no subgroup other than {1g}, and the statement follows from lemma 3.6. � we have noted for every n < ω the class-theoretic inclusion ssgp(n) ⊆ m.a.p. on the other hand, there are many examples of g ∈ m.a.p. such that g ∈ ssgp(n) for no n < ω. but more is true: there are groups which admit an m.a.p. topology which do not for any n < ω admit an ssgp(n) topology. indeed from corollary 3.14 and theorem 3.9 respectively we see that the groups g = z and g = z(p∞) admit no ssgp(n) topology; while ajtai, havas, and komlós [1], and later zelenyuk and protasov [41], have shown the existence of m.a.p. topologies for z and for z(p∞). in theorem 3.13 we show that in the context of abelian groups, theorem 3.9 can be strengthened. we use the following basic facts from the theory of abelian groups. lemma 3.10. a finitely cogenerated group is the direct sum of finite cyclic p-groups and groups of the form z(p∞), hence is torsion ([12](3.1 and 25.1)). lemma 3.11. a finitely generated abelian group is the direct sum of cyclic groups and cyclic torsion groups ([12](15.5)). lemma 3.12. if g is a finitely generated abelian group and h is a torsionfree subgroup then there is a decomposition g = k ⊕ t where t is the torsion subgroup of g, k is torsionfree, and h ⊆ k ([12](chapter iii)). theorem 3.13. a nontrivial abelian group which is the direct sum of a finitely generated group and a finitely cogenerated group does not admit an ssgp(n) topology for any n < ω. proof. we proceed by induction on the torsionfree rank, r0(g). suppose first that r0(g) = 0. then g is finitely co-generated and does not admit an ssgp(n) topology by theorem 3.9. now suppose that the theorem has been proved up to rank r−1 and we have r0(g) = r ≥ 1 and g = f ⊕t , with f finitely generated and t finitely co-generated. using lemmas 3.10 and 3.11, we rewrite g in the c© agt, upv, 2015 appl. gen. topol. 16, no. 2 146 some classes of minimally almost periodic topological groups form g = f ′ ⊕ t ′ where t ′ is the (finitely cogenerated) torsion subgroup and f ′ is free. then r0(f ′) = r0(g) = r. let a ∈ f ′ be an element of infinite order and choose u ∈ n(0) so that a /∈ u and so that u ⋂ c = ∅ where c is a finite set of cogenerators of t ′ (with 0 /∈ c). if all subgroups contained in u are torsion, then each such subgroup is a subgroup of t ′ and is therefore the zero subgroup, since it misses c. in that case, by lemma 3.6 there is no n < ω such that g ∈ ssgp(n). alternatively, if u contains a cyclic subgroup h of infinite order, we have r0(h) > 0. furthermore, since h ⊆ u, we have h ⋂ t ′ = {0}. it follows from lemma 3.12 that there is a decomposition g = f ′′ ⊕ t ′ which is isomorphic to the original decomposition and is such that h ⊆ f ′′. since a quotient of a finitely generated group is also finitely generated, it follows that f ′′/h is finitely generated. then we have g/h = (f ′′/h) ⊕ t ′. since r0(g) = r0(h) + r0(g/h) (see for example [12](§16, ex. 3(d))), we also have r0(g/h) < r. further, the group g/h is nontrivial since h ⊆ u and a /∈ u. it follows from the induction assumption that g/h does not admit an ssgp(n) topology, and so by theorem 3.15(b) (below), neither does g. � corollary 3.14. the group z does not admit an ssgp(n) topology for any n < ω. the following theorem lists several inheritance properties for groups in the classes ssgp(n). theorem 3.15. (a) if k is a closed normal subgroup of g, with k ∈ ssgp(n) and g/k ∈ ssgp(m) then g ∈ ssgp(m + n). (b) if g ∈ ssgp(n) and π : g ։ b is a continuous homomorphism from g onto b, then b ∈ ssgp(n). in particular, if k is a closed normal subgroup of g ∈ ssgp(n) then g/k ∈ ssgp(n). (c) if k is a dense subgroup of g and k ∈ ssgp(n) then g ∈ ssgp(n). (d) if gi ∈ ssgp(n) for each i ∈ i then ⊕ i∈i gi ∈ ssgp(n) and ∏ i∈i gi ∈ ssgp(n). proof. we proceed in each case by induction on n. each statement is trivial when n = 0. we address (a), (b), (c) and (d) in order, assuming in each case for 1 ≤ n < ω that the statement holds for n − 1. (a) let u ∈ n(1g), so that u ∩ k ∈ n(1k). then there is a family h of subgroups of k such that ⋃ h ⊆ u ∩ k and k/h ∈ ssgp(n − 1) where h := 〈∪h〉 k = 〈∪h〉 g . since g/k is topologically isomorphic with (g/h)/(k/h), we have (g/h)/(k/h) ∈ ssgp(m) along with k/h ∈ ssgp(n − 1). then by the induction hypothesis, g/h ∈ ssgp(m + n − 1). since ⋃ h ⊆ (u) with u arbitrary, we have g ∈ ssgp(m + n), as required. (b) given g ∈ ssgp(n) and continuous π : g ։ b, let u ∈ n(1b). then π−1[u] ∈ n(1g) and there is a family h of subgroups of g such that⋃ h ⊆ (π−1[u]) and g/〈∪h〉 ∈ ssgp(n−1). let h̃ be the family of subgroups of b given by h̃ := {π[l] : l ∈ h}. then ⋃ h̃ ⊆ (u). set h := 〈∪h〉 and set c© agt, upv, 2015 appl. gen. topol. 16, no. 2 147 w. w. comfort and f. r. gould h̃ := 〈∪h̃〉. then h̃ is normal in b since by assumption h is normal in g. by invoking the induction hypothesis we will show that b/h̃ ∈ ssgp(n − 1) and thus that b ∈ ssgp(n). note that h ⊆ π−1[h̃] since 〈∪h〉 ⊆ π−1[〈∪h̃〉]; further, π−1[h̃] is closed. we have then that g/h ∈ ssgp(n−1) so by induction, (g/h)/ ( π−1[h̃]/h ) ∈ ssgp(n − 1) and this is topologically isomorphic with g/π−1[h̃] by the second topological isomorphism theorem. now, we claim that the algebraic isomorphism π̃ : g/π−1[h̃] → b/h̃ induced by π is continuous (though it may not be open). clearly, π maps cosets of π−1[h̃] to cosets of h̃. if ṽ is an open union of cosets of h̃, then π−1[ṽ ] is an open union of cosets of π−1[h̃] and the claim follows. again, by the induction hypothesis, since π̃ is continuous and surjective, we now conclude that b/h̃ ∈ ssgp(n − 1) and thus b ∈ ssgp(n), as required. (c) given g and k as hypothesized, let u ∈ n(1g). since u ∩ k ∈ n(1k), there is a family h of subgroups of k such that ⋃ h ⊆ u ∩ k and k/h ∈ ssgp(n−1), where h := 〈 ⋃ h〉 k . note that h = h g ∩k. let φ : kh/h → k/(k ∩ h) be the natural isomorphism from the first (algebraic) isomorphism theorem for groups. the corresponding theorem for topological groups says that φ is an open map, i.e., φ−1 is a continuous map. then from part (a) of this theorem, kh/h ∈ ssgp(n − 1). now kh/h is dense in g/h, because the subset of g that projects onto the closure of kh/h must be closed and must contain kh. then g/h ∈ ssgp(n − 1) by the induction hypothesis. since h g = 〈 ⋃ h〉 g we have g ∈ ssgp(n), as required. (d) since ⊕ i∈i gi is dense in πi∈i gi, it suffices by part (c) to treat the case g := ⊕ i∈i gi. let u ∈ ng(1g), say u = ⊕ i∈i ui where ui ∈ n(1gi) and ui = gi for i > nu. since each gi ∈ ssgp(n), we have, for each i ∈ i, a family hi of subgroups of gi such that ⋃ hi ⊆ ui and gi/hi ∈ ssgp(n − 1) where hi := 〈∪ hi〉. now consider the family of subgroups of g given by h := {l ⊆ g : l = ⊕ i∈i li with li ∈ hi}. then ⋃ h ⊆ u, 〈∪h〉 is identical to ⊕ i∈i 〈∪hi〉, and h := 〈∪h〉 is identical to ⊕ i∈i hi. we also have that g/h is topologically isomorphic with ⊕ i∈i gi/hi (cf. [24](6.9)). from the induction hypothesis we have g/h ∈ ssgp(n − 1), so g ∈ ssgp(n), as required. � we give a noteworthy consequence of theorem 3.15(b). theorem 3.16. let g be a topological group which contains a proper open normal subgroup. then there is no n < ω such that g ∈ ssgp(n). proof. if g is a counterexample with proper open normal subgroup u, then by theorem 3.15(b) we have g/u ∈ ssgp(n) with g/u discrete, contrary to corollary 3.7. � remark 3.17. certain other tempting statements of inheritance or permanence type, parallel in spirit to those considered in theorem 3.15, do not hold in general. we give some examples. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 148 some classes of minimally almost periodic topological groups (a) we show below, using a construction of hartman and mycielski [22] and of dierolf and warken [6], that a closed subgroup of an ssgp group may lack the ssgp(n) property for every n < ω. indeed, every topological group can be embedded as a closed subgroup of an ssgp group (theorem 3.20). (b) the conclusion of part (a) of theorem 3.15 can fail when s < m + n replaces m + n in its statement. for example, the construction used in lemma 4.5 shows that a topological group g /∈ ssgp(n) may have a closed normal subgroup k ∈ ssgp(1) with also g/k ∈ ssgp(n), so s = m + n is minimal when m = 1. we did not pursue the issue of minimality of m + n in theorem 3.15(a) for arbitrary m, n > 1. (c) the converse to theorem 3.15(c) can fail. in [19] a certain monothetic m.a.p. group constructed by glasner [18] is shown to have ssgp, but we noted above in corollary 3.14 that z admits an ssgp(n) topology for no n < ω. in contrast to that phenomenon, it should be mentioned that (as has been noted by many authors) in the context of m.a.p. groups, a dense subgroup h of a topological group g satisfies h ∈ m.a.p. if and only if g ∈ m.a.p. thus in particular in the case of glasner’s monothetic group, necessarily the dense subgroup z inherits an m.a.p. topology. we now restrict our discussion to abelian groups and to the class ssgp= ssgp(1), and examine which specific abelian groups do and do not admit an ssgp topology. we have already noted (theorem 3.13) that the product of a finitely cogenerated abelian group with a finitely generated abelian group does not admit an ssgp topology even though it may admit an m.a.p. topology. we now give additional examples of abelian groups which admit not only an m.a.p. topology but also an ssgp topology. theorem 3.18. the following abelian groups admit an ssgp topology. (a) q, and those subgroups of q in which some primes are excluded from denominators, as long as an infinite number of primes and their powers are allowed; (b) q/z and q′/z where q′ is a subgroup of q as in described in (a); (c) direct sums of the form ⊕ i<ω zpi where the primes pi all coincide or all differ; (d) z(ω) (the direct sum); (e) zω (the full product); (f) g(α) for |g| > 1 and α ≥ ω; (g) f α for 1 < |f | < ω and α ≥ ω; (h) arbitrary sums and products of groups which admit an ssgp topology. item (h) is a special case of theorem 3.15(d), and item (e) is demonstrated in the second author’s paper [20]. the “coincide” case of item (c) follows from item (f), the “differ” case is established below in theorem 4.9. theorem 3.22((c) and (d)) below demonstrates the validity of item (f) for ω ≤ α ≤ c. this together c© agt, upv, 2015 appl. gen. topol. 16, no. 2 149 w. w. comfort and f. r. gould with (h) gives (d) and (f) in full generality. item (g) then follows from the relation f α ≃ ⊕ 2α f ([12](8.4, 8.5)). the remaining items are demonstrated in [19]. we note that from items (a) and (h) of theorem 3.18 and the familiar algebraic structure theorem r = ⊕ c q ([12](p.1̇05), [24](a.14)) it follows that r admits an ssgp topology. there are many examples of nontrivial ssgp(1) groups (that is, of ssgp groups). it has been shown by hartman and mycielski [22] that every topological group g embeds as a closed subgroup into a connected, arcwise connected group g∗; two decades later dierolf and warken [6], working independently and without reference to [22], found essentially the same embedding g ⊆ g∗ and showed that g∗ ∈ m.a.p.. indeed the arguments of [6] show with minimal additional effort that g∗ ∈ ssgp (of course with property ssgp not yet having been defined or named). we now describe the construction and we supply briefly the necessary details. definition 3.19. let g be a hausdorff topological group. then algebraically g∗ is the group of step functions f : [0, 1) → g with finitely many steps, each of the form [a, b) with 0 ≤ a < b ≤ 1. the group operation is pointwise multiplication in g. the topology t on g∗ is the topology generated by (basic) neighborhoods of the identity function 1g∗ ∈ g ∗ of the form n(u, ǫ) := {f ∈ g∗ : λ({x ∈ [0, 1) : f(x) /∈ u}) < ǫ}, where ǫ > 0, u ∈ ng(1g), and λ denotes the usual lebesgue measure on [0, 1). theorem 3.20. let g be a topological group. then (a) g is closed in g∗ = (g∗, t ); (b) g∗ is arcwise connected; and (c) g∗ ∈ ssgp. proof. note first that the association of each x ∈ g with the function x∗ ∈ g∗ (the function given by x∗(r) := x for all r ∈ [0, 1)) realizes g algebraically as a subgroup of g∗. furthermore the map x → x∗ is a homeomorphism onto its range, since for ǫ < 1, u ∈ n(1g) and x ∈ g one has x ∈ u ⇔ x∗ ∈ n(u, ǫ). (a) let f0 ∈ g ∗ and f0 /∈ g. there are distinct (disjoint) subintervals of [0, 1) on which f0 assumes distinct values g0, g1 ∈ g respectively. by the hausdorff property there is u ∈ n(1g) such that g1u ∩ g2u = ∅. choose ǫ smaller than the measure of either of the two indicated intervals. then f0n(u, ǫ) is a neighborhood of f0 such that f0n(u, ǫ)∩ g = ∅. therefore, g is closed in g∗. (b) let f ∈ g∗ and for each t ∈ [0, 1) define ft : [0, 1) → g by ft(x) = f(x) for 0 ≤ x < t and ft(x) = 1g for t ≤ x < 1; and define f1 := f. then t 7→ ft is a continuous map from [0, 1] to g ∗ such that f0 = 1g∗ and f1 = f. to show that the map is continuous, let ftn(u, ǫ) be a c© agt, upv, 2015 appl. gen. topol. 16, no. 2 150 some classes of minimally almost periodic topological groups basic neighborhood of ft and let s ∈ (t − ǫ/4, t + ǫ/4) ∩ [0, 1]. then fs ∈ ftn(u, ǫ), since λ({x ∈ [0, 1) : fs(x) − ft(x) ∈ u}) < ǫ. we conclude that g∗ is arcwise connected. (c) let n(u, ǫ) ∈ n(1g∗), and for each interval i = [t0, t1) ⊆ [0, 1) with t1 − t0 < ǫ let f(i) := {f ∈ g∗ : f is constant on i, f ≡ 1g on [0, 1)\i}. then f(i) is a subgroup of g∗ and f(i) ⊆ n(u, ǫ), and with hǫ := {f(i)} we have that each f ∈ g∗ is the product of finitely many elements from ⋃ hǫ—i.e., f ∈ 〈 ⋃ hǫ〉 ⊆ 〈 ⋃ hǫ〉. it follows that g ∗ ∈ ssgp, as asserted. � for later use we identify certain subgroups g∗a of g ∗ that retain properties (a) and (c) (but not (b)) of theorem 3.20. when g and a are countable the group g∗a also is countable. an equivalent definition and some related consequences can also be found in [7] and in [19] (definition 2.3.1). here is the relevant definition. definition 3.21. let g be a topological group and let a ⊆ [0, 1) where a is dense in [0, 1) and 0 ∈ a. then g∗a = (g ∗ a, t ) is the subgroup of (g ∗, t ) obtained by restriction of step functions on [0, 1) to those steps [a, b) such that a, b ∈ a ∪ {1}, a < b. theorem 3.22. let g be a topological group. then (a) g is closed in g∗a = (g ∗ a, t ); (b) g∗a is dense in g ∗; (c) g∗a ∈ ssgp; and (d) if g is abelian, then the groups g∗a, g (α) (with α = |a|) are isomorphic as groups. proof. with the obvious required change, the proofs of (a) and (c) coincide with the corresponding proofs in theorem 3.20. (b) let f ∈ g∗ have n steps (n < ω) and let f · n(u, ǫ) ∈ ng∗(f). then there is f̃ ∈ f ·n(u, ǫ)∩g∗a such that f̃ has step end-points in a∪{1}, each within ǫ/n of the corresponding end-point for f. (d) we give an explicit isomorphism. g(α) can be expressed as the set of functions φ : a → g with finite support and pointwise addition. each such function is the sum of finitely many elements of the form φa,g with a ∈ a, g ∈ g, φa,g(a) = g and φa,g(x) = 0 for x 6= a. now we define corresponding functions fa,g ∈ g ∗ a. let f0,g(x) = g for all x ∈ [0, 1), and for a > 0 let fa,g be the two-step function defined by fa,g(x) = g for 0 ≤ x < a and fa,g(x) = 0 for a ≤ x < 1. then the map φa,g 7→ fa,g extends linearly to an isomorphism from g(α) onto g∗a. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 151 w. w. comfort and f. r. gould � remark 3.23. note that theorem 3.15(c) cannot be used to prove (c) from (b) in theorem 3.22. note also that the isomorphism given in the proof of (d) provides a way of imposing an ssgp topology on g(α) for ω ≤ α ≤ c and g an abelian group. when g is nonabelian the corresponding mapping still exists but it need not be an isomorphism. in that case, each element of g∗a is a product of two-step functions, but the order in which they are multiplied can affect the product. some other ssgp groups arise as a consequence of the following fact. theorem 3.24. let g = (g, t ) be a (possibly nonabelian) torsion group of bounded order such that (g, t ) has no proper open subgroup. then g ∈ ssgp. proof. there is an integer m which bounds the order of each x ∈ g, and then n := m! satisfies xn = 1g for each x ∈ g. we must show: each u ∈ n(1g) contains a family h of subgroups such that 〈 ⋃ h〉 is dense in g. given such u, let v ∈ n(1g) satisfy v n ⊆ u. for each x ∈ v we have xk ∈ u for 0 ≤ k ≤ n, hence x ∈ v ⇒ 〈x〉 ⊆ u. thus with h := {〈x〉 : x ∈ v } we have: h is a family of subgroups of u (that is, of subsets of u which are subgroups of g). then v ⊆ ⋃ h, so g = 〈v 〉 ⊆ 〈 ⋃ h〉—the first equality because 〈v 〉 is an open subgroup of g. � in corollaries 3.25 and 3.28 we record two consequences of theorem 3.24. corollary 3.25. if (g, t ) is a (possibly nonabelian) connected torsion group of bounded order, then (g, t ) ∈ ssgp. proof. a connected group has no proper open subgroup, so theorem 3.24 applies. � lemma 3.26. let g ∈ m.a.p. and g abelian. then g does not contain a proper open subgroup. proof. suppose that h is a proper open subgroup of g. since g/h is a nontrivial abelian discrete (and therefore locally compact) group, there is a nontrivial (continuous) homomorphism φ : g/h → t. then the composition of φ with the projection map from g to g/h is a nontrivial continuous homomorphism from g to a compact group, contradicting the m.a.p. property of g. � remark 3.27. we are grateful to dikran dikranjan for the helpful reminder that lemma 3.26 fails when the “abelian” hypothesis is omitted. examples to this effect abound, samples including: (a) the infinite algebraically simple groups whose only group topology is the discrete topology, as concocted by shelah [38] under [ch], and by hesse [23] and ol′shanskĭı [30] (and later by several others) in [zfc]; and (b) such matrix groups as sl(2, c), shown by von neumann [27] to be m.a.p. even in the discrete topology (the later treatments [28], [24](22.22(h)) and [2](9.11) of this specific group follow closely those of [27]). in connection with this comment, note however that theorem 3.16 above is an appropriate nonabelian analogue of lemma 3.26 for ssgp(n) groups. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 152 some classes of minimally almost periodic topological groups corollary 3.28. for an abelian torsion group g of bounded order, these conditions are equivalent for each group topology t on g. (a) (g, t ) ∈ ssgp; (b) (g, t ) ∈ m.a.p.; and (c) (g, t ) has no proper open subgroup. proof. the implications (a) ⇒ (b), (b) ⇒ (c), and (c) ⇒ (a) are given respectively by theorem 3.5, lemma 3.26, and theorem 3.24. � remark 3.29. it is worthwhile to note that connected torsion groups of bounded order, as hypothesized in theorem 3.25, do exist. here we give two quite different proofs that for 0 < n < ω there is a nontrivial connected torsion group of exponent n. of these (a), as remarked by the referee, uses the construction given in theorem 3.20; while (b), drawing freely on the expositions [37] and [2](2.3–2.4), derives from the “free topological group” constructions first given by markov [25], [26] and graev [21]. (a) let g be a group of exponent n (for example, g = z(n)) and define g∗ as in definition 3.19. since algebraically g∗ ⊆ g[0,1), also g∗ has exponent n; and g∗ is connected by theorem 3.20(b). (b) let x be a tychonoff space and let g := {σni=1 kixi : ki ∈ z, n < ω, xi ∈ x} be the free abelian topological group on the alphabet x with 0g = 0, and for continuous f : x → h with h a topological abelian group define f : g → h by f(σni=1 kixi) = σ n i=1 kif(xi) ∈ h. it is easily checked, as in the sources cited, that (a) in the (smallest) topology t making each such f continuous, (g, t ) is a (hausdorff) topological group; (b) the map x → 1 · x from x to g maps x homeomorphically onto a closed topological subgroup of g; and (c) g is connected if (and only if) x is connected. now take x compact connected and fix n such that 0 < n < ω. it suffices to show that (1) nx is a proper closed subset of g, and (2) every proper closed subset f ⊆ x generates a proper closed subgroup 〈f〉 of (g, t ); for then the group g/〈nx〉 will be as desired, since a ∈ g ⇒ na ∈ 〈nx〉. (1) nx is compact in g, hence closed. define f0 : x → r by f0 ≡ 1; then f0 ≡ n on nx, while for x ∈ x we have f0((n + 1)x) = n + 1, so (n + 1)x /∈ nx. (2) given x ∈ x\f choose continuous f1 : x → r such that f1(x) = 1, f1 ≡ 0 on f . then f1 ≡ 0 on 〈f〉 and f1(x) = 1, so x = 1 · x /∈ 〈f〉; so 〈f〉 is proper in g. if a = σni=1 kixi ∈ g\〈f〉 there is i0 such that xi0 /∈ f , and with continuous f2 : x → r such that f2(xi0) = 1, f2(xi) = 0 for i 6= i0 and f2 ≡ 0 on f we have f2(a) = ki0 and f2 ≡ 0 on 〈f〉. then u := f2 −1 (ki0 −1/3, ki0 +1/3) ∈ ng(a) and u ∩〈f〉 = ∅; so 〈f〉 is closed in g. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 153 w. w. comfort and f. r. gould 4. ssgp(n) groups: some specifics the question naturally arises whether for n < ω the class-theoretic inclusion ssgp(n) ⊆ ssgp(n + 1) is proper. that issue is addressed below in theorem 4.6. en route to that we describe one of the earliest examples of an abelian m.a.p. group, constructed by prodanov [33]; his proof illustrates use of the ssgp(2) property to prove m.a.p. (of course, the classes ssgp(n) had not been formally defined in 1980.) we credit the referee with posing a useful question which allowed us eventually to prove that prodanov’s group (g, t ) satisfies not only (g, t ) ∈ ssgp(2) (theorem 4.2) but even (g, t ) ∈ ssgp(1) (theorem 4.3); this corrects a misstatement given in [19] and in an early version of this paper. algebraically, prodanov’s group g is the group g := z(ω) = ⊕ ω z. we begin our verification that (g, t ) is as desired by quoting directly from prodanov [33]. for this, denote by {em : 1 ≤ m < ω} the canonical basis for g and use induction to define a sequence of finite subsets of z(ω): “let a1 = {e1−e2, e2}, and suppose that the sets a1, a2, . . . , am−1 (m = 2, 3, . . .) are already defined. by αm we denote an integer so large that the s-th co-ordinates of all elements of a1 ∪ a2 ∪ . . . ∪ am−1 are zero for s ≥ αm. now we define am to consist of all differences (1) ei+kαm − ei+(k+1)αm (1 ≤ i ≤ m, 0 ≤ k ≤ 2 m−1 − 1) and of the elements (2) ei+2m−1αm (1 ≤ i ≤ m) . thus the sequence {am} ∞ m=1 is defined. now for arbitrary n ≥ 1 we define (3) un := (n+1)!z (ω) ±an ±2an+1 ±. . .±2 lan+l ± . . . .” (by the notation of (3) prodanov means that un consists of those elements of z(ω) which can be represented as a finite sum consisting of an element divisible by (n + 1)! plus at most one element of an with arbitrary sign, plus at most two elements of an+1 with arbitrary signs, plus at most four elements of an+2 with arbitrary signs, and so on.) “it follows directly from that definition that the sets un are symmetric with respect to 0, and that un+1 + un+1 ⊂ un (n = 1, 2, . . .). therefore they form a fundamental system of neighborhoods of 0 for a group topology t on z(ω).” since we need it later, we give a careful proof of an additional fact outlined only briefly by prodanov [33]. theorem 4.1. the group z(ω) with the group topology t defined above is hausdorff. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 154 some classes of minimally almost periodic topological groups proof. it suffices to show ⋂ n<ω un = {0}. let 0 6= g = ∑ i aiei where the ei form the canonical basis. then there is a least integer r such that ai = 0 for i ≥ r + 1, and there is a least integer s such that (s + 1)! does not divide g. let n := max(r, s). we claim that if n > 0 then g /∈ un. suppose otherwise. since (n + 1)! does not divide g, there is some p ≤ n such that (n + 1)! does not divide ap. thus any representation of g in the form (3) must include, for at least one m > n, one or more terms of the form ±(ep − ep+αm), all with the same sign. this means that the components ±ep+αm must be cancelled by the same components from additional terms of the form ±(ep+αm − ep+2αm). this chain of implications continues until k reaches its maximum value, 2m−1 − 1, with the inclusion of terms ±(ep+(2m−1−1)αm − ep+2m−1αm). finally, the components ±ep+2m−1αm must be cancelled by terms of type (2) with i = p and the same value of m. this means that we have necessarily included at least 2m−1 + 1 elements from am in our expansion of g, contradicting the requirement that no more than 2m−n elements of am be included as summands for such representations of g ∈ un. � theorem 4.2. prodanov’s group (g, t ) satisfies (g, t ) ∈ ssgp(2). proof. to see that (z(ω), t ) ∈ ssgp(2), we show first that each un generates z(ω). it suffices to show that each ei ∈ 〈un〉 (1 ≤ i < ω). choosing m such that m ≥ i and m ≥ n, we have from (1) and (2) above that ei = [σ 2m−1−1 k=0 (ei+kαm − ei+(k+1)αm)] + ei+2m−1αm ∈ 〈am〉 ⊆ 〈um〉 ⊆ 〈un〉. thus (z(ω), t ) has no proper open subgroups, so with hn := (n + 1)!z (ω) and gn := z (ω)/hn we have that hn ⊆ un, gn is of bounded order, and also gn has no proper open subgroups. then gn ∈ ssgp by theorem 3.24, so (z(ω), t ) ∈ ssgp(2). � prodanov proves explicitly that (g, t ) ∈ m.a.p.. theorem 4.2 exploits his argument insofar as it applies to our more demanding context. then of course, the following theorem 4.3 strengthens theorem 4.2. theorem 4.3. prodanov’s group (g, t ) satisfies (g, t ) ∈ ssgp(1). proof. since we have already shown that (g, t ) is hausdorff, it remains only to show that every un ∈ n0g contains a family h of subgroups such that g = 〈 ⋃ h〉. in fact, we show that h may be chosen so that 〈 ⋃ h〉 = g. with n given, choose k such that 2k ≥ (n + 1)! − 1 and let x ∈ am with m := n + k. we claim that 〈x〉 ⊆ un. since un contains elements formed from sums which include up to 2k elements from an+k, each y ∈ 〈x〉 can be written in the form y = rx + sx with r, s ∈ z, where rx ∈ (n + 1)!z(ω) and sx ∈ 2m−nam. since ei ∈ 〈am〉 for all i and m (by the proof of theorem 4.2), we have that ei ∈ 〈 ⋃ h〉 for every i < ω, where h includes all subgroups of the form 〈x〉 with x ∈ am and 2 m−n ≥ (n + 1)! − 1. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 155 w. w. comfort and f. r. gould now as promised we construct a family of topological groups demonstrating that the class-theoretic inclusion ssgp(n) ⊆ ssgp(n + 1) is proper for each n. for n = 0 this is clear since we already have many examples of nontrivial groups in the class ssgp(1). the strategy is to find particular groups gn ∈ ssgp(n) and h ∈ ssgp(1) and then construct a topology for h ⊕ gn in such a way that (a) h with its original topology is a closed normal subgroup, (b) (h ⊕ gn)/h is topologically isomorphic with gn, (c) h ⊕ gn /∈ ssgp(n), and (d) h ⊕ gn ∈ ssgp(n + 1). such a construction in the case n = 1 was given in the second author’s dissertation [19]; our task here is to generalize that construction to arbitrary n. the properties of gn that will be required in the induction step are that gn = (gn, tn) is abelian, countable and torsionfree, with the group topology tn defined by a metric. for convenience, we assume that the maximum distance is 1. these properties are satisfied in the case n = 0 by the trivial group, but it is more illuminating to begin the induction with g1 rather than g0. let h be the topological group z∗a as in definition 3.21, where z has the discrete topology and a := {x ∈ [0, 1) : x = t 2m for m, t < ω and 0 ≤ t < 2m}. set g1 = h. it is clear that g1 is abelian, countable and torsionfree, and is not the trivial group. g1 also has ssgp(1) (theorem 3.22). the topology on h = g1 = z ∗ a, defined as in definitions 3.19 and 3.21 can be seen as a metric topology given by the norm ‖h‖ := λ(supp(h)) for h ∈ h, where supp(h) is the support of h as a function on [0, 1). (here the “norm” designation follows historical precedent; we use it both out of respect and for convenience, but we do not require that ‖ng‖ = |n| · ‖g‖.) fix n > 1 and suppose there is a countable, torsionfree abelian group gn−1 with a metric ρ that defines a group topology on gn−1 such that gn−1 ∈ ssgp(n − 1) and gn−1 /∈ ssgp(n − 2). now, define (algebraically) gn := h ⊕gn−1; we give gn a metric topology tn which is different from the product topology, using a technique taken from m. ajtai, i. havas, and j. komlós [1]. we create a metric group topology on gn starting with a function ν : s → r +, where s is a specified generating set for gn with 0 /∈ s. s will typically be highly redundant as a generating set. we refer to ν together with the generating set s as a “provisional norm” (in terms of which a norm on gn will be defined). for x ∈ gn we write x = (h, g) with h ∈ h and g ∈ gn−1. we designate a double sequence of generating functions em,t ∈ h for m, t < ω and t < 2m: em,t(x) = 1 for x ∈ [ t 2m , t+1 2m ) and em,t(x) = 0 otherwise. we note that ‖p · em,t‖ = 1 2m for all p ∈ z, p 6= 0. we also name a basic set of neighborhoods of 0 ∈ gn−1 and we label all the non-zero elements in each neighborhood: um := {g ∈ gn−1 : ‖g‖ ≤ 1 2m } for m < ω c© agt, upv, 2015 appl. gen. topol. 16, no. 2 156 some classes of minimally almost periodic topological groups um\{0} = {gm,t : t < ω}. in addition, let r(m, t) : ω × ω ։ ω be an arbitrary bijection. we set s to be the collection of elements of the following two types: (1) (p · em,t , 0) for p ∈ z\{0} and m, t < ω with t < 2 m (2) (fr , gm,t) for m < ω, 0 ≤ t < ω and r = r(m, t), where fr = 2r−1−1∑ i=0 er,2i . the set of functions fr are linearly independent, so the set of elements of type (2) is also linearly independent. we use that fact along with the fact that each fr has support of measure 1 2 . now, we make the provisional norm assignments, (1) ν( (p · em,t , 0) ) = ‖p · em,t‖h = 1 2m (2) ν( (fr , gm,t) ) = ‖gm,t‖gn−1 ≤ 1 2m with r = r(m, t). notice that (1) gives the same provisional norm to every nonzero element in a subgroup of gn, whereas the assignments hiven by (2) are for a linearly independent set of elements of gn. now we define a seminorm ‖ · ‖ on gn in terms of the provisional norm ν. definition 4.4. for g ∈ gn, ‖g‖ := inf( { n∑ i=1 |αi|ν(si) : g = n∑ i=1 αisi, si ∈ s, αi ∈ z, n < ω } ⋃ {1}). this defines a seminorm because s generates gn and because the use of the infimum in the definition guarantees that the triangle inequality will be satisfied. therefore, the neighborhoods of 0 defined by this seminorm will generate a (possibly non-hausdorff) group topology on gn. now in lemma 4.5 we use the notation and definition just introduced. lemma 4.5. (a) gn is a torsionfree, countable abelian group; (b) the seminorm on gn is a norm (resulting in a hausdorff metric); (c) gn ∈ ssgp(n); and (d) gn /∈ ssgp(n − 1). proof. (a) is clear. (b) to show that ‖·‖ is a norm on gn, we need to show that for 0 6= x ∈ gn we have ‖x‖ > 0. let x = (h, g). if g 6= 0 then an expansion of (h, g) by elements of s must include at least one element of type (2). for those elements, we have ν( (fr, g) ) = ‖g‖. because the metric on gn−1 satisfies the triangle inequality, any expansion of (h, g) by elements of s must yield a value of ‖g‖ or greater for the expression within the curly brackets in definition 4.4. we conclude that ‖(h, g)‖gn ≥ ‖g‖gn−1. on the other hand, if g = 0 then there is an expression for (h, 0) in terms of elements of s of type (p · em,t, 0) such c© agt, upv, 2015 appl. gen. topol. 16, no. 2 157 w. w. comfort and f. r. gould that ∑n i=1 |αi|ν(si) = ‖h‖ = λ(supp(h)) and this value is minimal. if, instead, the expansion includes elements of type s = (fr , gm,t) then there is a minimal ν-value such a term can have. this is because there is a minimal size, 1 2m , for an interval on which h is constant. an expansion of (h, 0) by elements of s that includes an element s = (fr , gm,t) such that r(m, t) > m would also have to include 2r−1 elements of s of the form (er,i, 0), each with coefficient −1. the contribution of these terms to the sum ∑n i=1 |αi|ν(si) is greater than or equal to 1 2 . such an expansion cannot affect the infimum. so if ‖(h, 0)‖ 6= ‖h‖h then ‖(h, 0)‖ ≥ min({‖gm,t‖gn : r(m, t) < m}). we conclude that ‖(h, g)‖ is bounded away from 0 except when (h, g) = (0, 0). (c) we show next that gn ∈ ssgp(n). we claim first that the subgroup h×{0} of gn is an ssgp group in the topology inherited from gn. this is clear because from the provisional norm assignment, ν( (p · em,t , 0) ) = ‖p · em,t‖h, it follows that ‖(h, 0)‖ ≤ ‖h‖h for each h ∈ h, so any ǫ-neighborhood of (0, 0) contains a family h of subgroups such that 〈 ⋃ h〉 = h×{0} = {(h, 0) : h ∈ h}. we will show that the quotient topology for gn/(h × {0}) coincides with the original topology for gn−1 (which also implies that h × {0} is closed in gn). for each g ∈ gn−1 there is an h ∈ h such that ‖(h, g)‖ = ‖g‖gn−1, namely h = fr where g = gm,t and r = r(m, t). (there are, in general, many such pairs m, t and corresponding fr.) on the other hand, as we showed above, ‖(h, g)‖ ≥ ‖g‖gn−1 for each h ∈ h. we conclude that g is in the εneighborhood of 0 ∈ gn−1 if and only if there is h ∈ h such that (h, g) is in the ε-neighborhood of (0, 0) ∈ gn. in other words, the neighborhoods of 0 in gn−1 coincide with the projections onto gn/(h, ×{0}) of the neighborhoods of (0, 0) in gn. thus the topologies of gn/(h × {0}) and gn−1 coincide. (note, however, that the subgroup topology on gn−1 does not coincide with its original topology.) since by assumption gn−1 ∈ ssgp(n − 1), we have indeed that gn ∈ ssgp(n). (d) it remains to show gn /∈ ssgp(n − 1). suppose the contrary. then every ǫ-neighborhood uǫ of (0, 0) ∈ gn (say with ǫ < 1 4 ) contains a family kǫ of subgroups such that gn/ 〈 ⋃ kǫ〉 ∈ ssgp(n − 2). let g ∈ kǫ and (h, g) ∈ g with g 6= 0. for |n| < ω we must have ‖(nh, ng)‖ < ǫ, so each (nh, ng) has an expansion (nh, ng) = ∑mn i=1 αn,i(hi, 0) + ∑ln j=1 βn,i(h ′ i, gi) such that ∑mn i=1 η(αn,i) ν((hi, 0)) + ∑ln j=1 |βn,i| ν((h ′ i, gi)) < ǫ where each (hi, 0), (h ′ i, gi) ∈ s and where η : z → {0, 1} is defined by η(k) := 1 when k 6= 0 and η(k) := 0 when k = 0. we consider two cases. case 1. in the expansion above, each coefficient βn,i has the form βn,i = nβ1,i. then clearly for sufficiently large |n| we have ‖(nh, ng)‖ > 1 4 > ǫ. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 158 some classes of minimally almost periodic topological groups case 2. there is n < ω for which, for some k, the expansion for (nh, ng) is such that βn,k 6= nβ1,k. for the h-component of the given expansion of (nh, ng), we can write nh = ∑m i=1 αn,ihi + ∑l i=1 βn,ih ′ i where m = max(m1, mn) and l = max(l1, ln). multiplying the specified expansion of (h, g) by the number n, we also have nh = ∑m i=1 nα1,ihi + ∑l i=1 nβ1,ih ′ i. equating the two expansions and rearranging, we can write ∑l j=1(βn,j − nβ1,j)h ′ j = ∑m i=1(nα1,i − αn,i)hi where, for the specified index k, we have (βn,k − nβ1,k)h ′ j 6= 0. since the h′j are linearly independent, each h ′ j that has a nonzero coefficient in the expression above must be balanced by terms on the right. this implies that∑m i=1 η(nα1,i−αn,i) ≥ 1 2 , which in turn means that either ∑m i=1 η(α1,i) ≥ 1 4 or ∑m i=1 η(αn,i) ≥ 1 4 . we conclude that ‖(h, g)‖ ≥ 1 4 > ǫ or ‖(nh, ng)‖ ≥ 1 4 > ǫ, contradicting g ∈ kǫ. we conclude that for ǫ < 1 4 we must have ⋃ kǫ ⊆ h, so that 〈∪kǫ〉 is a closed subgroup of h. in such cases we have gn/〈∪kǫ〉 /∈ ssgp(n − 2) because, by theorem 3.15(b), gn/〈∪kǫ〉 ∈ ssgp(n − 2) would imply that (gn/〈∪kǫ〉)/(h/〈∪kǫ〉) ∈ ssgp(n− 2) or, equivalently, that gn/h ≃ gn−1 ∈ ssgp(n − 2), contrary to the induction assumption for gn−1. � we emphasize the essential content of lemma 4.5. theorem 4.6. for 1 ≤ n < ω there is an abelian topological group g such that g ∈ ssgp(n) and g /∈ ssgp(n − 1). in the paragraph following theorem 3.9 we noted the existence of abelian topological groups g ∈ m.a.p. such that g ∈ ssgp(n) for no n < ω. (z and z(p∞), appropriately topologized, are examples.) in the following corollary we note the availability of other examples to the same effect. corollary 4.7. there is a group g of the form g = πk<ω gk such that (a) for each k < ω there is nk < ω such that gk ∈ ssgp(nk); (b) there is no n such that g ∈ ssgp(n). proof. using theorem 4.6, for k < ω choose gk ∈ ssgp(k + 1)\ssgp(k), and set g := πk<ω gk. then (a) holds with nk = k + 1. if there is n < ω such that g ∈ ssgp(n) then since the projection πn : g ։ gn is continuous we would from theorem 3.15(b) have the contradiction gn ∈ ssgp(n); thus (b) holds. � remark 4.8. any group g satisfying the conditions of corollary 4.7 is necessarily an m.a.p. group. this is the case since each gk ∈ ssgp(nk) ⊆ m.a.p. and since the m.a.p. property is preserved by products. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 159 w. w. comfort and f. r. gould while theorem 3.13 furnishes an ample supply of well-behaved abelian groups which admit no ssgp(n) topology, we have found that an ssgp topology can be constructed for many of the standard building blocks of infinite abelian groups. we now verify theorem 3.18(c), that is, we give a construction of an ssgp topology for groups of the form g := ⊕pi zpi (with (pi) a sequence of distinct primes); this illustrates the method used throughout the second-listed co-author’s thesis [19]. using additive notation, write 0 = 0g and let {ei : i = 1, 2, ...} be the canonical basis for g, so that p1e1 = p2e2 = . . . = piei . . . = 0. we define a provisional norm ν, just as in the description preceding lemma 4.5. this will generate a norm || · || via definition 4.4 in such a way that in the generated topology every neighborhood of 0 contains sufficiently many subgroups to generate a dense subgroup of g. suppose we can show that g is hausdorff and that each u ∈ n(0) contains a family of subgroups h such that g/h is torsion of bounded order, where h := 〈∪h〉. then if also g/h has no proper open subgroup, we have from theorem 3.24 that g/h ∈ ssgp, so that g ∈ ssgp(2). our plan is to choose a norm so that g/h, and thus g/h, is finite. then if g contains no proper open subgroup, it is necessarily the case that h = g. we will, then, define a norm ‖ · ‖ so that (1) every neighborhood of 0 contains a set of subgroups of g whose union generates a subgroup h such that g/h is finite; (2) g has no proper open subgroups, or equivalently, every neighborhood of 0 generates g; and (3) g is hausdorff. to that end, define ν(men) = 1 n for every m < ω such that m 6≡ 0 mod pn. we then obtain (1) because the neighborhood of 0 defined by ||g|| < 1 n contains subgroups which generate h := p1p2...pn−1g. thus, g/h is finite, as desired. to satisfy (2) define ên := ∑n i=1 ei for n < ω, and define ν(ên) := 1 n for each n < ω. what then remains (the most difficult piece) is to show that g with this topology is hausdorff. we will then have the following result. theorem 4.9. let g = ⊕ i<ω zpi where p1 < p2 < p3 < ... are primes. let s = {men : n < ω, 0 < m < pn} ⋃ {ên : n ∈ n}, with en, ên defined as above. let ν(men) = 1 n for 0 < m < pn, and let ν(ên) = 1 n . then the norm defined by ‖g‖ = inf { n∑ i=1 |αi|ν(si) : g = α1s1 + ... + αnsn, si ∈ s, αi ∈ z, n < ω } generates an ssgp topology on g. proof. as noted above, our construction for the norm ‖ · ‖ guarantees that every ǫ-neighborhood u of 0 generates g and also contains subgroups whose union generates an h such that g/h is finite. then, as also noted, if g is hausdorff we are done. suppose 0 6= g ∈ g and n is the largest nonzero coordinate index for g. we show that ‖g‖ ≥ 1 n . for convenience we extend the domain of ν to all formal c© agt, upv, 2015 appl. gen. topol. 16, no. 2 160 some classes of minimally almost periodic topological groups finite sums of elements from s with coefficients from z: for ϕ = ∑n i=m (aiei + biêi), let ν(ϕ) = ∑n i=m (ηi + |bi|) 1 i where each ηi is either 0 or 1, according as to whether or not ai ≡ 0 mod pi . in addition, we will assume (1) g = val(ϕ), so the formal sum ϕ evaluates to g ∈ g; (2) each ei and each êi appears at most once in any formal sum; and (3) 0 ≤ ai < pi for each i, item (2) being justified by the fact that we are ultimately interested in the norm, which minimizes ν(ϕ). let f(m, n) be the set of such formal sums where m is the smallest coordinate index for a nonzero coefficient am or bm and where n is the largest such index. (here for bi, “nonzero” indicates that bi is not a multiple of p1p2...pi.) we want to show that ν(ϕ) ≥ 1 n where g = val(ϕ), where ϕ = ∑n i=m (aiei + biêi) and where either am or bm is nonzero and either an or bn is nonzero. in other words, ϕ ∈ f(m, n). clearly ν(ϕ) ≥ 1 n when m ≤ n. suppose that n = m = n + 1. then, since the n + 1 component of g is 0 we have that an+1 + bn+1 ≡ 0 mod pn+1. both coefficients are 0 only if g = 0, so either both are nonzero or an+1 = 0 and bn+1 = mpn+1 for some m 6= 0. in the first case we have ν(ϕ) ≥ 2 n+1 > 1 n and in the second case we have ν(ϕ) ≥ pn+1 n+1 > 1 ≥ 1 n . suppose instead that m = n > n+1. in this case, we know that the (n −1) component of g is 0. then, since g 6= 0 can be written as ϕ = anen +bn ên, we have bn = mpn−1 for some m 6= 0. but then we have ν(ϕ) ≥ pn−1 n ≥ 1 ≥ 1 n . finally, we fix m and use induction on n. assume that we have already shown that ν(ϕ) ≥ 1 n when ϕ ∈ f(m, q) for n < m ≤ q ≤ n −1, and suppose that ϕ ∈ f(m, n). we treat three cases separately. (a) case 1. |bn−1 + bn| ≥ pn−1. then ν(ϕ) ≥ |bn−1| n − 1 + |bn| n ≥ pn−1 n ≥ 1 ≥ 1 n . (b) case 2. bn−1 + bn = 0. recalling that all coordinates of g vanish after the nth, we note that (an + bn)en = 0, and so bn−1ên−1 + bn ên + anen = 0. this means that we can delete these terms from ϕ without affecting its value, and with that done, our induction assumption can be applied. (c) case 3. |bn−1 + bn| < pn−1 and bn−1 + bn 6= 0. again, from (an + bn)en = 0 we obtain the equality bn−1ên−1 + bn ên + anen = (bn−1 + bn)ên−1. let ϕ′ be the formal sum obtained from ϕ by replacing the three terms on the left with the one on the right, and compare the provisional c© agt, upv, 2015 appl. gen. topol. 16, no. 2 161 w. w. comfort and f. r. gould norms: ν(ϕ′) − ν(ϕ) = |bn−1 + bn| n − 1 − ( |bn−1| n − 1 + |bn| n + 1 n ) ≤ |bn| n(n − 1) − 1 n . we see that this difference is negative or zero as long as |bn| ≤ n − 1. then, since ϕ′ ∈ f(m, n − 1), our induction assumption applies. if, on the contrary, |bn| ≥ n, we already have ν(ϕ) ≥ 1 ≥ 1 n . we conclude in each case that ν(ϕ) ≥ 1 n , so g is hausdorff, as desired. � 5. concluding comments here we discuss briefly some other classes of groups which are closely related to the classes ssgp(n) and the class m.a.p. remark 5.1. in the dissertation [19], the second-listed co-author found it convenient to introduce the class of weak ssgp groups (briefly, the wssgp groups), that is, those topological groups g = (g, t ) which contain no proper open subgroup and have the property that for every u ∈ n(1g) there is a family h of subgroups of g such that ⋃ h ⊆ (u), h = 〈∪h〉 is normal in g, and g/h is torsion of bounded order. subsequent analysis (as in theorem 3.24 above) along with the definitions of the classes ssgp(n) has revealed the class-theoretic inclusions ssgp(1) ⊆ wssgp ⊆ ssgp(2). a consequence of theorem 3.24 is that the markov-graev-remus examples (as in remark 3.29) are not just wssgp but are, in fact, ssgp. the same is true of prodonov’s group g: g ∈ ssgp = ssgp(1) (theorem 4.3). from these facts we conclude that the class of wssgp groups contributes little additional useful information to the present inquiry, and we have chosen to suppress its systematic discussion in this paper. in theorems 3.9 and 3.13 we have identified several classes of groups which do not admit an ssgp topology. that suggests the following natural question. question 5.2. what are the (abelian) groups which admit an ssgp topology? our work also leaves open this intriguing question: question 5.3. does every abelian group which for some n > 1 admits an ssgp(n) topology also admit an ssgp topology? there is another important and much-studied class of groups related to the class of m.a.p. groups, namely the class of groups whose every continuous action on a compact space has a fixed point, the so-called fixed point on compacta groups (hereafter, the f.p.c. groups); for basic facts, some recent developments and a bit of history, see for example [18], [32] and [11]. (the reader will recall that a continuous action of a topological group g on a space x is a continuous map φ : g × x ։ x such that (a) φ(g, ·) : x ։ x is a bijection for each g ∈ g with φ(eg, ·) = idx and (b) φ(g, φ(h, x)) = φ(gh, x) for all g, h ∈ g and x ∈ x. a fixed point for the configuration (g, x, φ) is a point x ∈ x such that φ(g, x) = x for all g ∈ g.) it is easy to see that every f.p.c. group is a m.a.p. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 162 some classes of minimally almost periodic topological groups group (we write simply f.p.c. ⊆ m.a.p.): given a nontrivial homomorphism h from a (non-m.a.p.) group g into a compact group k, the continuous map φ : g × k ։ k given by φ(g, x) := h(g) · x ∈ k is a g-action on k with no fixed point. the class-theoretic inclusion f.p.c. ⊆ m.a.p. is proper since, as we remarked in discussion 2.1(a), the group sl(2, r) is an m.a.p. group even in its discrete topology, while veech [39](2.2.1) has shown that every locally compact group, m.a.p. or not, has a continuous action on a compact space such that each non-identity element of the group moves every element of the compact space. (this is called a “free action”.) whether or not every abelian f.p.c. group is a m.a.p. group, however, is a difficult long-standing open question in abelian topological group theory raised in 1998 by glasner [18]: question 5.4. do the f.p.c. abelian groups constitute a proper subclass of the m.a.p. abelian groups? even the characterization of abelian m.a.p. groups and abelian f.p.c. groups by different “big set” conditions (see [31] and [10]) did not settle question 5.4. unfortunately, and contrary to our hopes, our own work with the ssgp property also has so far not shed light on this question. it is known [18], however, that there are f.p.c. topologies for z, so the class-theoretic inclusion f.p.c. ⊆ ssgp fails. we have not successfully addressed the issue of the reversed inclusion, so we list it as another question to be resolved: question 5.5. do the ssgp groups constitute a subclass of the f.p.c groups? what about the abelian case? note added in proof shortly after this paper had been completed in final form and accepted for publication, we received a preprint of [9] from its authors. they build substantially on our results, reformulating and possibly generalizing ssgp(n) with the use of some pleasing algebraic characterizations, and extending the concept to families ssgp(α) for ordinals α. in the process they have provided a positive solution to our question 5.3, and they have made considerable progress in answering question 5.2 for abelian groups. acknowledgements. we gratefully acknowledge helpful comments received from dieter remus, dikran dikranjan, and saak gabriyelyan. each of them improved the exposition in pre-publication versions of this manuscript, and enhanced our historical commentary with additional bibliographic references. we are grateful also for an unusually thoughtful and detailed referee’s report, which helped us (a) to correct an error and several minor expository ambiguities in our early draft and (b) to reorganize and rewrite some of the proofs, most notably the proof of theorem 4.6. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 163 w. w. comfort and f. r. gould references [1] m. ajtai, i. havas and j. komlós, every group admits a bad topology, studies in pure mathematics. to the memory of p. turán (paul erdős, ed.), birkhäuser verlag, basel and akademiai kiado, budapest, 1983, 21–34. 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[41] e. g. zelenyuk and i. v. protasov, topologies on abelian groups, math. ussr izvestiya 37 no. 2 (1991), 445–460. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 165 () @ appl. gen. topol. 19, no. 2 (2018), 217-222doi:10.4995/agt.2018.7952 c© agt, upv, 2018 topological characterization of gelfand and zero dimensional semiring jorge vielma and luz marchan ∗ escuela superior politécnica del litoral. espol, fcnm, campus gustavo galindo km. 30.5 v́ıa perimetral, p.o.box 09-01-5863. guayaquil, ecuador (jevielma@espol.edu.ec, lmarchan@espol.edu.ec) communicated by j. galindo abstract let r be a conmutative semiring with 0 and 1, and let spec(r) be the set of all proper prime ideals of r. spec(r) can be endowed with two topologies, the zariski topology and the d-topology. let maxr denote the set of all maximal prime ideals of r. we prove that the two topologies coincide on spec(r) and on maxr if and only if r is zero dimensional and gelfand semiring, respectively. 2010 msc: 54a10; 54f65; 13c05; 16y60. keywords: zariski topology; d-topology; conmutative semiring; gelfand semiring; zero dimensional semiring. 1. basic facts recall that a semiring (conmutative with non zero identity) is an algebra (r, +, ·, 0, 1), where r is a set with 0, 1 ∈ s, and + and · are binary operations on r called sum and multiplication, respectively, which satisfy the following: (1) (r, +, 0) and (r, ·, 1) are conmutative monoid with 1 6= 0. (2) a · (b + c) = a · b + a · c for every a, b, c ∈ r. (3) a · 0 = 0 for every a ∈ r. ∗the authors are supported by the research project espol fcnm-09-2017. received 22 august 2017 – accepted 09 february 2018 http://dx.doi.org/10.4995/agt.2018.7952 j. vielma and l.marchan a subset i of r will be called an ideal of r if a, b ∈ i and r ∈ r implies a + b ∈ i and ra ∈ i. a prime ideal of r is a proper ideal p of r in which x ∈ p or y ∈ p whenever xy ∈ p . the nilradical of r, denoted by n(r), is the intersection of all the prime ideals de r. max(r) and min(r) denote the set of all maximal and minimal prime ideals of r, respectively. r is said to be gelfand if every prime ideal is contained in at most one maximal ideal. r is said to be zero dimensional if every prime ideal of r is maximal. for x ∈ r, let (0 : x) = {y ∈ r : xy = 0}. an ideal i of r is called a ó-ideal if for every x ∈ r, i(0 : x) = r, that is to say there exist x1 ∈ i and y ∈ (0 : x) such that 1 = x1y. a element x ∈ r is called a complemented element in r if there is y ∈ r such that xy = 0 and x + y = 1, y is called the complement of x. for any semiring r, spec(r) denotes the set of all proper prime ideals of r. this set can be given the zariski topology τz as follows: for every proper set i of r, let (i)0 = {p ∈ spec(r) : i ⊆ p} and let d(i) = spec(r) \ (i)0 = {p ∈ spec(r) : i * p}. if i is the ideal generated by a ∈ s, we write i = (a). note that (a)0 = {p ∈ spec(r) : a ∈ p} and d(a) = {p ∈ spec(r) : a /∈ p}. the sets d(a) ⊆ spec(r) with a ∈ r, constitute a basis for τz, and the sets (i)0 with i ideal of r are the closed sets for τz. let (x, τ) a topological space, τ∗ denote the family of τ-closed subset of x. τ is said to be alexandroff if it is closed under arbitrary intersections. by identifying a set with its characteristic function, we can view τ as a subset of 2x with the product topology, then its closure τ is also a topology, even more, τ = { a ⊆ x : a = ⋂ θ∈l θ, l ⊆ τ } and it is the smallest alexandroff topology containing τz (see [5]). note that a ∈ τ∗ if and only if ac is τ-open, let say ac = ⋂ θ∈l θ for some l ⊆ τ, then x ∈ a ⇒ x /∈ ac ⇒ x /∈ θ, for some θ ∈ l ⇒ x ∈ θc, for some θ ∈ l ⇒ {x} ⊆ θc = θc ⇒ {x} ⊆ ( ⋂ θ∈τ θ )c = a and a is just the union of the τ-closure of each of its points. a subset a of a topological space (x, τ) is τ-saturated if {a} ⊆ a for all a ∈ a, that is to say, if a ∈ τ∗. in particular, a ⊆ spec(r) is τz-saturated if and only if for each p ∈ a, (p)0 ⊆ a. the set of all τz-open and τz-saturated subsets of spec(r) defines a topology on spec(r) called the d-topology, this is to say that the d-topology is just τz ∩ τz ∗. remember that a topology τ on x is said t0 if for each pair of distinct elements x and y in x, exist a open set containing either x or y, and τ is t1 if c© agt, upv, 2018 appl. gen. topol. 19, no. 2 218 topological characterization of gelfand and zero dimensional semiring for each pair of distinct elements x and y in x, exist a open set containing x and not y and an open set containing y and not x. the following results, given in [2] and [5], characterize the topologies t0 and t1 of the following manner: c© agt, upv, 2018 appl. gen. topol. 19, no. 2 219 j. vielma and l.marchan theorem 1.1. let τ be a topology on x then, (i) τ is t0 if and only if τ is t0. (ii) τ is t0 if and only if τ ∨ τ ∗ = ℘(x) (iii) τ is t1 if and only if τ = ℘(x). in [1], al-ezeh endowed spec(l), where l is a distributive lattice with 0 and 1, with two topologies, the τz-topology and the d-topology, and he proved that this two topologies coincide on spec(l) and max(l) iff l is a boolean and normal lattice, respectively. in [3], rafi and rao introduced the concept of d-topology on spec(r), where r is a almost distributive lattice (adl), and characterized those adls for which topologies coincide on spec(r) and min(r). in this paper, the concept of d-topology is introduced on spec(r), where r is a semiring, we do a similar study, as a consequence, we obtain a result given in [1] for distributive lattices. 2. main results we begin by establishing some relationships between the τz-open sets and d-open and between the τz-clopen and the d-clopen. remark 2.1. if i is a ó-ideal of a semiring r, then d(i) is d-open. in effect, let p ∈ d(i), we will show that (p)0 ⊆ d(i). let q ∈ (p)0, since p ∈ d(i), i * p , hence exists x ∈ i such that x /∈ p . since i is a ó-ideal, there exist x1 ∈ r and y ∈ (0 : x) such that x1 + y = 1, note that y ∈ p ⊆ q (because xy = 0 ∈ p and x /∈ p) so x1 /∈ q (otherwise 1 = x1 + y ∈ q) implying i * q, in consequence, q ∈ d(i). the reciprocal of the previous remarks it is not true, as shown in the following example. example 2.2. let a a non-empty subset of a set x, and let l = {∅, a, ac, x}, (l, ∪, ∩) is a semiring where the sum and multiplication are the union and intersection, respectively, and the identities of the sum and multiplication are the empty set and the whole set x, even more (l, ∪, ∩) is a distributive lattice. the ideals of l are {∅}, 〈a〉 = {∅, a}, 〈ac〉 = {∅, ac}, 〈x〉 = l. spec(l) = {〈a〉, 〈ac〉} and d(〈a〉) = {〈ac〉}, clearly d(〈a〉) is τ-saturated, but 〈a〉 it is not a ó-ideal, since (∅ : ac) = {∅, a} and 〈a〉 ∪ (∅ : ac) = {∅, a} 6= l. proposition 2.3. let r be a semiring with trivial nilradical and let i an ideal of r. then d(i) is d-clopen if and only if d(i) = d(x) for some complemented element x in r. proof. assume that d(i) is clopen. then spec(r) \ d(i) is also an open set, so there exists an ideal j of r such that d(j) = spec(r) \ d(i). now d(i) ∩ d(j) = d(ij) = ∅, this implies ij ⊆ p for all p ∈ spec(r), this is, ij ⊆ n(r) = {0}. also now, spec(r) = d(i) ∪ d(j) = d(i + j), this implies i + j = r, thus exist x ∈ i and y ∈ j such that x + y = 1. since ij = {0}, we have xy = 0, then x is complemented. we see that i = (x), let z ∈ i, z = z1 = z(x + y) = zx + zy = zx ∈ (x) since zy ∈ ij = {0}. thus i c© agt, upv, 2018 appl. gen. topol. 19, no. 2 220 topological characterization of gelfand and zero dimensional semiring is a principal ideal generated by x, in consequence d(i) = d(x). conversely, assume that x is a complemented element in r, then there exists an element y ∈ r such that xy = 0 and x+y = 1. now d(x)∩d(y) = d(xy) = d(0) = ∅ and d(x)∪d(y) = d(x+y) = d(1) = spec(r). therefore d(x) is clopen. � now we characterize those semiring for which the zariski topology and the d-topology coincide on spec(r). theorem 2.4. let r be a semiring. then the zariski topology and the dtopology coincide on spec(r) if and only if r is zero dimensional. proof. note that w is d-open if and only if w ∈ τz ∩ τz ∗, thus the zariski topology and the d-topology coincide on spec(r) if and only if τz = τz ∩ τz ∗, but τz = τz ∩ τz ∗ ⇔ τz ⊆ τz ∗ ⇔ τz ⊆ τz ∗ ⇔ τz = τz ∗ ⇔ τz ∨ τz ∗ = τz ⇔ τz = ℘(spec(s)) (τz is t0 and by theorem 1.1 part (i)) ⇔ τz is t1 ( by theorem 1.1 part (iii)) now τz is t1 if and only if every p ∈ spec(r) is closed impliying {p} = (p)0, or equivalently, every prime ideal of r is maximal, this is, r is zero dimensional. � theorem 2.5 ([4]). let l be a distributive lattice. then l is a boolean algebra if and only if every prime ideal of l is a maximal ideal. as a consequence of the theorem 2.4 we obtain the following result given in [1] for distributive lattices. corollary 2.6. let l be a distributive lattice with 0 and 1. then the zariski topology and the d-topology coincide on spec(r) if and only if l is a boolean lattice (a lattice every element of which has a complement). proof. immediately of theorem 2.4 and theorem 2.5, since every lattice is a semiring. � theorem 2.7. r is a gelfand semiring if and only if τz and τz ∩ τz ∗ agree on max(r). proof. suppose r is a gelfand semiring. we want to prove that τz and τz ∩τz ∗ agree on max(r). since τz ∩ τz ∗ ⊆ τz, it remains to show that τz ⊆ τz ∩ τz ∗ on max(r). so we want to show that each d(x) in τz restricted to max(r) is an open set in τz ∩ τz ∗ restricted to max(r). for each x ∈ r, let d∗ x = d(x) ∩ max(r) and let w = {p ∈ spec(r) : x /∈ mp }, c© agt, upv, 2018 appl. gen. topol. 19, no. 2 221 j. vielma and l.marchan where mp is the unique maximal ideal of r containing p . let us prove then that d∗ x is a τz ∩ τz ∗-open subset in max(r). now, for each p ∈ w we have that (p)0 ⊆ w , then w ∈ τ ∗ z . since d∗ x = w ∩ max(r) then, d∗ x is a τz ∩ τz ∗-open set in max(r). therefore the conclusion follows. conversely, suppose τz and τz ∩τz ∗ agree on max(r) and take p ∈ spec(r) which is contained in two different maximal ideals m and n. take, without lost of generality, x0 ∈ m such that x0 /∈ n. so n ∈ d(x0) ∩ max(r) = w ∩ max(r) for some τz ∩ τz ∗ open set w . now since p ⊆ n and n ∈ w it follows that (p)0 ⊆ w . therefore (p)0 ∩ max(r) ⊆ w ∩ max(r) impliying that x0 /∈ m, which is a contradiction. � question 2.8. under what conditions on r, the τz-topology and the d-topology coincide on min(r)? references [1] h. al-ezeh, topological characterization of certain classes of lattices, rend. sem. univ. padova 83 (1990), 13–18. 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[5] c. uzcátegui and j. vielma, alexandroff topologies viewed as closed sets in the cantor cube, divulg. mat. 13, no. 1 (2005), 45–53. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 222 08.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 99 114 preservation of completeness undermappings in asymmetric topologyhans-peter a. k�unzi�abstract. the preservation of various completeness proper-ties in the quasi-metric (and quasi-uniform) setting under open,closed and uniformly open mappings is investigated. in partic-ular, it is noted that between quasi-uniform spaces the propertythat each costable �lter has a cluster point is preserved underuniformly open continuous surjections. furthermore in the realmof quasi-uniform spaces conditions under which almost uniformlyopen mappings are uniformly open are given which generalize cor-responding classical results for uniform spaces. as a by-productit is shown that a quasi-metrizable moore space admits a left k-complete quasi-metric if and only if it is a complete aronszajnspace.2000 ams classi�cation: 54c10, 54e15, 54e35, 54e50keywords: uniformly open mapping, almost uniformly open mapping, openmapping theorem, quasi-metrizable, left k-complete, open mapping, closedmapping, supercomplete, aronszajn space1. introductionin section 2 the preservation of topological completeness properties related toleft k-completeness under open continuous mappings between quasi-metrizablespaces is studied. section 3 contains similar investigations for uniformly openmappings between quasi-metric and quasi-uniform spaces. furthermore section4 deals with the classical problem of determining conditions under which almostuniformly open mappings are uniformly open, again for the case of mappingsbetween quasi-metric and quasi-uniform spaces. those two sections show thatin order to obtain satisfactory results, for our purposes it is appropriate to�this paper was written while the author was supported by the swiss national sciencefoundation under grant 2000-056811.99. he also acknowledges support during a visit to theuniversity of oxford by the stiftung zur f�orderung der wissenschaftlichen forschung an deruniversit�at bern. 100 hans-peter a. k�unziwork under some conditions of supercompleteness. section 5 �nally recordsseveral results on the preservation of these completeness properties under closedcontinuous mappings between quasi-metrizable spaces.2. preservation of completeness properties under open mappingslooking for an adequate version of completeness for the present investiga-tions, we found that conditions from the area of left k-completeness were espe-cially useful. therefore in the following we shall concentrate on such propertiesand notions. to �x our notation and terminology let us recall the following ba-sic concepts and conventions. by n we shall denote the positive integers. letx be a set. as usual, a function d : x � x ! [0;1) that satis�es d(x;y) = 0if and only if x = y; and d(x;z) � d(x;y) + d(y;z) whenever x;y;z 2 x iscalled a quasi-metric on x: the induced topology �(d) is the topology gener-ated by the base consisting of the balls b2�n(x) = fy 2 x : d(x;y) < 2�ngwhere x 2 x and n 2 n: a sequence (xn)n2n in a quasi-metric space (x;d)is called left k-cauchy provided that for each k 2 n there is nk 2 n suchthat d(xn;xm) < 2�k whenever n;m 2 n and nk � n � m: a quasi-metricspace (x;d) is left k-complete provided that each left k-cauchy sequence con-verges in (x;�(d)) (compare [32]). for further concepts from the theory ofquasi-uniform spaces we refer the reader to [9]. (note however that we shalluse us instead of u� to denote the coarsest uniformity �ner than some givenquasi-uniformity u:)it is well known that hausdor� showed that each open continuous imageof a completely metrizable space is completely metrizable provided that it ismetrizable. a modern proof of this fact is now usually based on the result thata paracompact open continuous (hausdor�) image of a �cech complete spaceis �cech complete (compare [4, pp. 114{116]). on the other hand we do notknow whether each open continuous quasi-metrizable image of a left k-completequasi-metric space admits a left k-complete quasi-metric. (recall that kofner[20] showed that quasi-metrizability need not be preserved under open compactcontinuous mappings.) for the further discussion of that problem it is usefulto be aware of the characterization of r0-spaces possessing a �-base given bywicke and worrell in [38, theorems 3.2 and 3.3]: a topological r0-space xhas a �-base if and only if there exists a sequence (gn)n2n of bases for x suchthat every decreasing representative (gn)n2n of (gn)n2n with nonempty termsconverges to some x 2 x and also to every element of \n2ngn: (in the followingwe shall call such a sequence of bases a �-base sequence.) it is straightforward toverify that a quasi-metric space (x;d) is left k-complete if and only if (gn)n2n,where gn = fb2�k(x) : x 2 x;k � n;k 2 ng whenever n 2 n; is a �-basesequence (compare [33, theorem 1]). since each open continuous r0-image ofa space possessing a �-base has a �-base [37, theorem 1], the question ariseswhether each quasi-metrizable space with a �-base admits a left k-completequasi-metric. we observe that it was shown in [24, propositions 10 and 11]that a (tychono�) �cech complete or scattered quasi-metrizable space admits aleft k-complete quasi-metric. we also note that for regular spaces, because a preservation of completeness under mappings in asymmetric topology 101regular t0-space is a complete aronszajn space if and only if it has a �-base (see[37, p. 256]), our problem was already formulated by romaguera (question3 of [33]) when he asked whether each complete aronszajn quasi-metrizablespace admits a left k-complete quasi-metric. while the latter question remainsopen, in this section we shall show that romaguera's problem has a positiveanswer in the class of quasi-metrizable moore spaces. our method of proofmay be of independent interest, since as a by-product of our slightly moregeneral argument we obtain a new proof of kofner's classical result [19] thateach -space with an ortho-pair-base is quasi-metrizable which seems easierto comprehend than the original one. our proof will make use of some ideascontained in junnila's thesis [11] (see also [12]) and in [14]. in particular, thefollowing concept will be used: a neighbornet u of a topological space x iscalled unsymmetric provided that x;y 2 x;x 2 u(y) and y 2 u(x) imply thatu(x) = u(y):the de�nition of an ortho-pair-base (for t1-spaces) is due to kofner [19]. acollection b of pairs (g;g0) (with g � g0) of open sets in a topological spacex is called a pair-base for x provided that whenever h is open and x 2 h thenthere is (g;g0) 2 b such that x 2 g � g0 � h: the concept of a local pair-baseat some x 2 x will now be self-explanatory. a pair-base p of a topologicalspace x is called an ortho-pair-base provided that for each subcollection p0 ofp and each x 2 \fg : (g;g0) 2 p0g such that x 62 int \ fg0 : (g;g0) 2 p0g;the collection p0 is a local pair-base at x.proposition 2.1. let x be a topological space that possesses an ortho-pair-base. then for every unsymmetric neighbornet s of x there exists a neighbornetv of x such that v 2 � s.proof. suppose that g is an ortho-pair-base for x that is well-ordered by �.for each x 2 x choose the �rst element (g;g0) 2 g with respect to � suchthat x 2 g � g0 � s(x) and call it (gx;g0x); furthermore set v (x) = tfg0y :x 2 gyg \ gx: first we want to show that v (x) is a neighborhood at x :otherwise x cannot have a smallest neighborhood and fg0y : x 2 gyg is aneighborhood base at x; since g is an ortho-pair-base. therefore there is gysuch that x 2 gy � g0y � gx: then fx;yg � gx � g0x � s(x) and fx;yg �gy � g0y � s(y): thus s(x) = s(y) by unsymmetry of s: furthermore we have(gx;g0x) < (gy;g0y) or (gx;g0x) > (gy;g0y) which contradicts the de�nition of(gy;g0y) resp. (gx;g0x): we conclude that v is a neighbornet of x: supposethat x 2 x and y 2 v (x): then y 2 gx: thus v (y) � g0x � s(x): we haveshown that v 2 � s. �recall that for a neighbornet v of a topological space x the neighbornet v +is de�ned as follows: v +(x) = tfv (g) : g is a neighborhood at xg wheneverx 2 x (see [19]). note that v + � v 2:proposition 2.2. let x be a topological t1-space with an ortho-pair-base.then for each neighbornet u of x, u+ contains an unsymmetric neighbornets of x. 102 hans-peter a. k�unziproof. suppose that g0 = f(g�;g0�) : � < �g [ f(fxg;fxg) : x is isolated inxg is an ortho-pair-base for x, where we can assume that each g0� is not asingleton. set h0 = x and de�ne inductively, given an ordinal �; h�+1 =intfx 2 h� : sfg0 : x 2 g;(g;g0) 2 g�g * u(x)g and g�+1 = f(g;g0) 2 g� :g0 � h�+1g n f(g;g0) 2 g� : g0� � g0g where the second set of that de�nitionis assumed to be empty if g0� is unde�ned; furthermore for a limit ordinal �set g� = t�<� g� and h� = t�<� h�:clearly, for each �; h�+1 is open and the trans�nite sequence (h�)� is de-creasing. note that certainly h�+1 = ?: so we assume that the induction stopsat the �rst ordinal � such that h� = ?: observe also that sfg0 : (g;g0) 2g�g � h� for each limit ordinal �: we want to show next by induction on� that for each x 2 h� we have that g� contains a neighborhood pair-baseat x : if this condition is satis�ed for some �; then clearly it is also ful�lledfor � + 1; since no g0� is a singleton. for a limit ordinal � we argue as fol-lows: suppose that x 2 t�<� h�: since x 2 h�+1 whenever � < �; we havesfg0 : x 2 g;(g;g0) 2 g�g * u(x) whenever � < �: consequently for each� < � there exists (e�;e0�) 2 g� such that x 2 e� and e0� * u(x): thusx 2 intt�<� e0� by de�nition of the characteristic property of the ortho-pair-base g0: let (l;l0) 2 g0 be such that x 2 l and l0 � intt�<� e0�: supposethat (l;l0) 62 g�: then there is some minimal � < � such that (l;l0) 62 g�:note that � necessarily is a successor ordinal and so (l;l0) 2 g��1: thenl0 * h� or g0��1 � l0: therefore e0� * h� or g0��1 � e0�: thus (e�;e0�) 62 g�|a contradiction. we conclude that f(l;l0) : x 2 l;(l;l0) 2 g�g is a neigh-borhood pair-base at x: in particular, since (l;l0) 2 \�<�g�; we deduce thatx 2 l0 � \�<�h� = h�: so h� is also open if � is a limit ordinal.let x 2 x and let �x be the �rst ordinal � such that x 62 h�: note that�x necessarily is a successor ordinal and so x 2 h�x�1: let (gx;g0x) be the�rst element of g�x�1 with respect to the well-ordering of g0 such that x 2 gx:set s(x) = gx: then the neighbornet s = sx2x(fxg � s(x)) is unsymmetric:if x;y 2 x and fx;yg � s(x) \ s(y); then �x = �y: otherwise suppose forinstance that �x < �y: then x 62 h�y�1; but x 2 gy where (gy;g0y) 2 g�y�1and hence x 2 g0y � h�y�1 |a contradiction. therefore we conclude that�x = �y and so s(x) = s(y):let x 2 x and let g be an arbitrary neighborhood of x: by de�nition ofh�x; there is y 2 s(x)\g\h�x�1 such that sfg0 : y 2 g;(g;g0) 2 g�x�1g �u(y) � u(g): thus s(x) = gx � g0x � u(g): hence we have shown thats � u+: �remark 2.3. combining the two preceding propositions we obtain the resultdue to kofner that in a t1-space x with an ortho-pair-base for each neighbornetu of x there is a neighbornet v of x such that v 2 � u+. as kofner observedin [19, proposition 3], the latter result implies that for each neighbornet u of x,u+ is a normal neighbornet (compare [19, theorem 1]) so that in particular each -space with an ortho-pair-base is quasi-metrizable [19, theorem 2]. (recallthat a t1-space x is a -space provided that it possesses a sequence (vn)n2n of preservation of completeness under mappings in asymmetric topology 103neighbornets such that fv 2n (x) : n 2 ng is a neighborhood base at x wheneverx 2 x.) obviously it also follows from these results that in a topological spacewith an ortho-pair-base for each unsymmetric neighbornet u there exists anunsymmetric neighbornet v such that v 2 � u. kofner noted in [19, p. 1440]that each developable -space possesses an ortho-pair-base. next we want toapply the preceding results to our discussion of the �-base property.proposition 2.4. let x be a t1-space possessing a �-base and having theproperty that for each unsymmetric neighbornet u there exists an unsymmetricneighbornet v such that v 2 � u. then x admits a left k-complete quasi-metric.proof. since x has a base of countable order [37] and thus a primitive base [39,theorem 4.1], x possesses a sequence (hn)n2n of unsymmetric neighbornetssuch that fhn(x) : n 2 ng is a neighborhood base at x whenever x 2 x (see[8, p. 147]). let (bn)n2n be a �-base sequence of x: we can suppose thateach base bn is well-ordered by �n. inductively we shall de�ne unsymmetricneighbornets vn and bn such that v 2n+1 � hn \ bn and bn � vn whenevern 2 n: set v1(x) = x whenever x 2 x: suppose now that, for some n 2 n;the unsymmetric neighbornet vn is de�ned. then for each x 2 x we �nd the�rst element b 2 bn such that x 2 b � vn(x) and set bn(x) = b: similarlyas above, note �rst that the neighbornet bn is unsymmetric: if x;y 2 x andx;y 2 bn(x) \ bn(y); then vn(x) = vn(y) by unsymmetry of vn: by de�nitionof bn it follows that bn(x) = bn(y):by our assumption on unsymmetric neighbornets of x we can �nd an un-symmetric neighbornet vn+1 of x such that v 2n+1 � hn \ bn; since hn \ bnis unsymmetric. the induction having carried out, we note that b2n+1 � bnand bn+1 � hn whenever n 2 n: then fbn : n 2 !g is a base for a com-patible quasi-metrizable quasi-uniformity v on x: let d be a quasi-metric onx inducing v and let (xn)n2n be a left k-cauchy sequence in (x;d): thereis a strictly increasing sequence (nk)k2n in n such that for each k 2 n;(xn;xm) 2 bk whenever nk � n � m and n;m 2 n: for each k 2 n n f1g wehave xnk+1 2 bk(xnk) and thus bk(xnk+1) � bk�1(xnk): since bk(xnk+1) 2 bkwhenever k 2 n we conclude by the �-base property that fbk(xnk+1) : k 2 ngand thus (xn)n2n converges to some x 2 x (compare [34, lemma 1]). we haveshown that (x;d) is left k-complete. �corollary 2.5. each t1-space with an ortho-pair-base that also possesses a�-base admits a left k-complete quasi-metric.in particular we conclude that a moore space admits a left k-complete quasi-metric if and only if it is a complete aronszajn quasi-metrizable space (compare[33, theorem 1]). moore spaces that are complete aronszajn spaces are alsocalled semicomplete moore spaces in the literature [30]. the tychono� exampledue to [4, example 2.9] shows that a quasi-metrizable semicomplete moorespace need not be �cech complete. quasi-metrizability of this space is clear, sinceit is a metacompact moore space (see [9, theorem 7.26]). moreover it has a �-base, because it is locally completely metrizable (compare [4, proposition 2.2]). 104 hans-peter a. k�unziobserve that this example answers negatively another question of romaguera[33, question 2], since each sequentially complete quasi-metric tychono� spaceis �cech complete [22, proposition 4]. let us �nally state explicitly the twoquestions discussed in this section.problem 2.6. does each quasi-metrizable space with a �-base admit a left k-complete quasi-metric?problem 2.7. suppose that x admits a left k-complete quasi-metric and f :x ! y is an open continuous surjection onto a quasi-metric space y: does yadmit a left k-complete quasi-metric? (as mentioned above, these conditionsimply that y possesses a �-base [37, theorem 1]. observe also that theorem 8of [37] asserts that a regular t0-space has a �-base if and only if it is an opencontinuous image of a completely metrizable space.)3. preservation of completeness properties under uniformly openmappingsin this section we shall show that as in the classical, symmetric case betterresults than in section 2 can be achieved if we assume that the mappings areuniformly open with respect to some given (quasi-)uniform structures on thespaces under consideration. let (x;u) and (y;v) be quasi-uniform spaces. a(multi-valued) mapping f : x ! y is called uniformly open provided that foreach u 2 u there is v 2 v such that v (f(x)) � f(u(x)) whenever x 2 x(compare [5]). it is known that in the area of uniform (hausdor�) spaceseach open continuous mapping with compact domain is uniformly open [7,proposition 2.2]. in fact the following more general result holds.proposition 3.1. let (x;u) be a compact uniform space and let the mappingf : (x;u) ! (y;v) be open and continuous where (y;v) is a quasi-uniformspace. then f is uniformly open.proof. let u 2 u: there is p 2 u such that p2 � u: since f is open, foreach a 2 x we �nd wa 2 v such that w2a (f(a)) � f(p(a)): by continuity off and since u is a uniformity, we can consider the open cover fint(p�1(a) \f�1wa(f(a))) : a 2 xg of x: since x is compact, there is a �nite subset fof x such that sa2f int(p�1(a) \ f�1wa(f(a))) = x: set w = ta2f wa andnote that w 2 v: consider x 2 x: there is b 2 f such that x 2 p�1(b) \f�1wb(f(b)): therefore f(x) 2 wb(f(b)) and w(f(x)) � w2b (f(b)) � fp(b) �fp2(x) � fu(x): we have shown that f is uniformly open. �applying the preceding result to the identity mapping on a compact haus-dor� space, we draw the following conclusion.corollary 3.2. [9, proposition 1.47] the uniformity is the coarsest quasi-uniformity on a compact hausdor� space.the identity mapping on a topological space x admitting two quasi-uniform-ities u and v such that u is not contained in v also shows that the conclusionof proposition 3.1 can only hold under some strong conditions. preservation of completeness under mappings in asymmetric topology 105the following classical result from kelley's book [15, p. 203] is well known:let f be a uniformly open continuous mapping from a complete pseudo-metriz-able space into a uniform hausdor� space. then the range of the mapping fis complete. on the other hand, it is known that if g is a topological groupwhose left uniformity is complete and n is a closed normal subgroup, thenthe left uniformity of the quotient group g=n need not be complete, althoughthe quotient mapping is continuous and uniformly open (compare [31, p. 195]and [27]). such examples show that completeness of the domain space is notsu�cient to generalize the afore-mentioned result from kelley's book to uniformspaces.in order to extend our investigations on quasi-metric spaces from section 2to general quasi-uniform spaces, we recall that a �lter f on a quasi-uniformspace (x;u) is called left k-cauchy provided that for each u 2 u there isf 2 f such that u(x) 2 f whenever x 2 f . a quasi-uniform space (x;u)is called left k-complete provided that each left k-cauchy �lter converges(compare [34]). the negative uniform result mentioned above however suggeststhat in an arbitrary quasi-uniform space (x;u) we should consider a propertystronger than left k-completeness, for instance, that each costable �lter has a�(u)-cluster point, where a �lter f on a quasi-uniform space (x;u) is calledcostable provided that for each u 2 u we have tf2f u�1(f) 2 f: costable�lters characterize hereditary precompactness in the sense that a quasi-uniformspace (x;u) is hereditarily precompact if and only if each �lter on (x;u) iscostable (see e.g. [13, proposition 2.5]). an ultra�lter on a quasi-uniformspace is costable if and only if it is a left k-cauchy �lter [34, proposition 1].costable �lters were called cs�asz�ar �lters by p�erez-pe~nalver and romaguerain [29]. they said that a quasi-uniform space (x;u) is cs�asz�ar complete pro-vided that each costable �lter of (x;u) has a �(us)-cluster point. the latterconditions strengthens the well-known property of smyth completeness, whichmeans that each left k-cauchy �lter has a �(us)-cluster point (equivalently, a�(us)-limit point). p�erez-pe~nalver and romaguera also remarked that for anytopological space x the well-monotone quasi-uniformity wx has the propertythat each costable �lter on (x;wx) has a cluster point [29, proposition 2].it was noted (compare [26, p. 169], [32]) that for a quasi-pseudometric space(x;d), each costable �lter of the induced quasi-uniform space (x;ud) clustersif and only if each left k-cauchy sequence (resp. each left k-cauchy �lter)converges. so for quasi-pseudometric spaces the property considered in thefollowing is indeed equivalent to left k-completeness. for uniform spaces theproperty under consideration is equivalent to supercompleteness. a uniformspace x is called supercomplete if each stable �lter has a cluster point [2, 10].for instance that condition is satis�ed by a complete bilateral uniformity ofa topological group of pointwise countable type [35]. it is well known thatsupercompleteness characterizes completeness of the hausdor� uniformity onthe hyperspace of nonempty closed subsets (equivalently, nonempty subsets)of a uniform space. on the other hand, for a quasi-uniform space (x;u) thecondition that each costable �lter clusters in (x;u) is only necessary, but not 106 hans-peter a. k�unzisu�cient that its hausdor� quasi-uniformity (on the collection of nonemptysets) is left k-complete (see [25]).proposition 3.3. let f be a uniformly open continuous mapping from a quasi-uniform space (x;u) in which each costable �lter f has a cluster point ontoa quasi-uniform space (y;v): then each costable �lter on (y;v) has a clusterpoint.proof. let f be a costable �lter on (y;v) and �x u 2 u: since f is uniformlyopen, there is v 2 v such that v (f(x)) � f(u(x)) whenever x 2 x: because the�lter f is costable in (y;v); there is f0 2 f such that f0 � tf2f v �1(f): wewant to show that f�1(f0) � tf2f u�1(f�1(f)) : let f 2 f and a 2 f�1(f0):hence f(a) = f0 for some f0 2 f0: thus f0 2 v �1(e) for some e 2 f: thene 2 v (f0) � f(u(a)): therefore e = f(c) for some c 2 u(a): it follows thata 2 u�1(c) and c 2 f�1(f): we have shown that a 2 u�1(f�1(f)): weconclude that f�1(f0) � tf2f u�1(f�1(f)) and f�1f = �lff�1(f) : f 2fg is costable in (x;u): suppose now that x is a cluster point of f�1f: bycontinuity of f; f(x) is a cluster point of f: �corollary 3.4. a uniform space that is the image of a supercomplete uniformspace under a uniformly open continuous mapping is supercomplete.since in a quasi-uniform space each left k-cauchy �lter is costable and con-verges to its cluster points (see [34]), the next result is a consequence of the pre-ceding proposition and the observation about quasi-pseudometric spaces men-tioned above.corollary 3.5. let (x;u) and (y;v) be quasi-uniform spaces and f : (x;u) !(y;v) be a uniformly open continuous surjection. if u is quasi-pseudometrizableand left k-complete, then v is left k-complete.because uniformly continuous mappings between quasi-uniform spaces arecontinuous with respect to the associated supremum uniformities, the followingcorollary is also readily veri�ed.corollary 3.6. let (x;u) and (y;v) be quasi-uniform spaces and f : (x;u) !(y;v) a uniformly open uniformly continuous surjection. if u is cs�asz�ar com-plete, then v is cs�asz�ar complete.a �lter f on a quasi-uniform space (x;u) is called a weakly cauchy �lteror corson �lter provided that tf2f u�1(f) 6= ? whenever u 2 u (see e.g.[29]). obviously, each costable �lter is weakly cauchy. the property (compare[9, proposition 5.32]) that each weakly cauchy �lter has a cluster point is oftencalled co�nal completeness and even in metric spaces is strictly stronger thancompleteness (see e.g. [2, example 1]). it is known that each uniformly locallycompact and each paracompact �ne uniform space is co�nally complete (e.g.[2, corollaries 4 and 5]). in [36] it is shown that a (tychono�) topologicalgroup is locally compact if and only if it is of pointwise countable type and itsleft uniformity is co�nally complete. the following strengthening of cs�asz�arcompleteness was considered in [29]. a quasi-uniform space (x;u) is called preservation of completeness under mappings in asymmetric topology 107corson complete provided that each weakly cauchy �lter has a �(us)-clusterpoint. as we show next, these two completeness properties are preserved underuniformly open uniformly continuous surjections, too.proposition 3.7. let (x;u) and (y;v) be quasi-uniform spaces and f : (x;u)! (y;v) a uniformly open continuous surjection. if x is co�nally complete,then y is co�nally complete.proof. it will su�ce to show that f�1f is a weakly cauchy �lter on (x;u)provided that f is uniformly open and f is a weakly cauchy �lter on (y;v):so suppose that f is a weakly cauchy �lter on (y;v): let u 2 u: by uniformopenness of f, there is v 2 v such that v (f(x)) � f(u(x)) whenever x 2 x:by our assumption, there is y 2 y such that v (y) \ f 6= ? whenever f 2 f:let x 2 x be such that y = f(x): then fu(x)\f 6= ? whenever f 2 f: thusu(x) \ f�1f 6= ? whenever f 2 f: therefore f�1f = �lff�1f : f 2 fg is aweakly cauchy �lter. �corollary 3.8. let f : (x;u) ! (y;v) be a uniformly open uniformly con-tinuous mapping from a corson complete quasi-uniform space (x;u) onto aquasi-uniform space (y;v): then (y;v) is corson complete.our �nal proposition in this section applies corollary 3.5 to the questionconsidered in section 2.proposition 3.9. suppose that f : x ! y is an open continuous mapping froma topological space x onto a t1-space y: if x admits a left k-complete quasi-metric d such that all �bers of f are precompact in (x;d�1) then y admits a leftk-complete quasi-metric. in particular, a t1-image of a completely metrizablespace under an open compact mapping admits a left k-complete quasi-metric.proof. we shall work with the quasi-metric quasi-uniformity ud = �lfb2�n :n 2 ng on x: for each y 2 y and n 2 n set vn(y) = tx2f�1fyg f(b2�n(x))whenever y 2 y: then fvn : n 2 ng is a base for a quasi-metrizable quasi-uniformity v on y; because v 2n+1 � vn whenever n 2 n and tn2n vn = f(y;y) :y 2 y g: since the �bers are precompact in (x;d�1); we see that v is compatible:by our assumption for each y 2 y and u 2 ud there exists a �nite subsetf � f�1fyg such that f�1fyg � sx2f u�1(x) and thus for each x0 2 f�1fygthere is x 2 f such that x0 2 u�1(x) and so f(u(x)) � f(u2(x0)): thereforetx2f f(u(x)) � tx02f�1fyg f(u2(x0)): since tx2f f(u(x)) is a neighborhoodof y and f is continuous, we deduce that v is compatible. since f : (x;ud) !(y;v) is uniformly open by de�nition of v, we conclude that v is left k-completeby corollary 3.5. �4. almost uniformly open mappingsin this article a (multi-valued) mapping f : x ! y between quasi-uniformspaces (x;u) and (y;v) is called almost uniformly open provided that for eachu 2 u there is v 2 v such that v (f(x)) � cl�(v�1)f(u(x)): note that thisde�nition yields the usual concept of almost uniform openness for mappings 108 hans-peter a. k�unzibetween uniform and metric spaces. extending classical work on metric spaces(see [15, p. 202]) dektjarev [6] proved the following result for supercompleteuniform spaces: let f be an almost uniformly open multi-valued mapping withclosed graph from the supercomplete uniform space x into an arbitrary uniformspace y: then, for any entourages u and v in x and any point x0 2 x; theinclusion fu(x0) � fv u(x0) is valid.in this section we want to address the problem under which conditions an al-most uniformly open mapping between quasi-uniform spaces is uniformly open.to this end we �rst recall that a quasi-uniform space (x;u�1) is called rightk-complete provided that each left k-cauchy �lter on (x;u) converges with re-spect to the topology �(u�1) (compare [34]). in the following we shall considera stronger condition and further variant of the uniform property of supercom-pleteness, namely the condition that each costable �lter on the quasi-uniformspace (x;u) has a �(u�1)-cluster point. the latter condition was already stud-ied to some extent by k�unzi and ryser [26, proposition 6] where it was shownto be equivalent to right k-completeness of the hausdor� quasi-uniformitytransmitted by u�1 onto the collection of nonempty subsets of x: we also re-call that a quasi-metric space (x;d�1) is called right k-sequentially completeprovided that each left k-cauchy sequence of (x;d) converges in (x;�(d�1)):it is known that right k-sequential completeness (for non-r1-spaces) can bestrictly weaker than right k-completeness of the induced quasi-uniformity inquasi-metric spaces [1, remark 2]. this complication suggests that we should�rst establish a version of dektjarev's result for quasi-metric spaces and only af-terwards consider the more general quasi-uniform case. we remark that khanhhas already obtained a quantitative version of our next proposition in [16, the-orem 2]. on the other hand, cao and reilly [3, lemma 5.3] gave some bitopo-logical version of that result. related to khanh's studies further investigationsin quasi-uniform spaces were conducted by chou and penot [5].proposition 4.1. (compare [16]) each almost uniformly open mapping f :x ! y from a quasi-metric space (x;d) into a quasi-metric space (y;d0)such that the graph of f is �(d�1) � �((d0)�1)-closed and (x;d�1) is right k-sequentially complete is uniformly open.proof. let u resp. v be the quasi-metric quasi-uniformities on (x;d) resp.(y;d0) generated by the standard bases fu� : � > 0g resp. fv� : � > 0g: byour assumption on f for each u 2 u there is v 2 v such that v (f(x)) �cl�(v�1)fu(x) whenever x 2 x: hence it su�ces to show that cl�(v�1)fu�(x) �fu�+�(x) whenever �;� > 0 and x 2 x:fix �;� > 0: for each n 2 n; set �n = �2n and choose �n � 1n suchthat v�n(f(x)) � cl�(v�1)fu�n(x) whenever x 2 x: fix x 2 x and let y 2cl�(v�1)fu�(x): find x1 2 u�(x) such that (f(x1);y) 2 v�1: inductively we de�nea sequence (xn)n2n in x such that (f(xn);y) 2 v�n and (xn;xn+1) 2 u�n when-ever n 2 n : suppose that xn was chosen for some n 2 n such that the inductionhypothesis is satis�ed. therefore we have y 2 v�n(f(xn)) � cl�(v�1)f(u�n(xn)):hence we �nd xn+1 2 u�n(xn) such that f(xn+1) 2 v �1�n+1(y): this completes the preservation of completeness under mappings in asymmetric topology 109induction. it follows that (xn)n2n is a left k-cauchy sequence in (x;d): by ourassumption on completeness of x there is x0 2 x such that (xn)n2n convergesto x0 in (x;�(u�1)): we conclude that d(x;x0) < � + �, because d(xn;x0) ! 0and thus d(x1;x0) � � by the triangle inequality. consequently x0 2 u�+�(x):since the graph of f is �(d�1) � �((d0)�1)-closed and d0(f(xn);y) ! 0; we seethat y = f(x0): thus cl�(v�1)fu�(x) � fu�+�(x): we have shown that f isuniformly open. �we shall now give a version of dektjarev's argument for quasi-uniform spaces.proposition 4.2. let (x;u) be a quasi-uniform space such that each costable�lter on (x;u) has a �(u�1)-cluster point and let f be an almost uniformlyopen multi-valued mapping from (x;u) into an arbitrary quasi-uniform space(y;v). suppose that the graph of f is �(u�1) � �(v�1)-closed. then for anyentourages u und w in u and any point x0 2 x; we have cl�(v�1) fu(x0) �fwu(x0), in particular f is uniformly open.proof. suppose that fui : i 2 ig is a base for u and fvi : i 2 ig is a base for v.with every entourage p of u, we associate a sequence of entourages (pn)n2nsuch that p21 � p and p2n+1 � pn whenever n 2 n: by our assumption onf , we can suppose that for each u 2 u there is uf 2 v such that uff(z) �cl�(v�1)fu(z) whenever z 2 x: fix now u;w 2 u: without loss of generalitywe assume that (ui)n � wn+1 whenever i 2 i and n 2 n: let x0 2 x andy 2 cl�(v�1)fu(x0): furthermore let d be the collection of nonempty �nitesubsets of i partially ordered by inclusion and for any � 2 d denote the numberof elements of � by j�j: we shall construct for each � 2 d a nonempty setb� � w1u(x0) such that b� � (ti2�(ui)j�j)�1(b�) whenever � 2 d and� � �:the sets b� are constructed by induction on j�j: for each i 2 i, set bi =fx 2 u(x0) : v �1i (y) \ (((ui)1)f )�1(y) \ f(x) 6= ?g. furthermore for each� 2 d with j�j � 2 set b� = fx 2 s��(ti2�(ui)j�j)(b�) : (\i2�v �1i (y)) \((ti2�(ui)j�j)f)�1(y) \ f(x) 6= ?g. we shall verify next that the sets b�(� 2 d) satisfy the stated conditions: since y 2 cl�(v�1) fu(x0), there is anet (z�)�2e in fu(x0) converging to y in (y;v�1). for each � 2 e chooseu� 2 u(x0) such that (u�;z�) 2 f . we conclude that for each i 2 i; u� 2bi eventually, and, thus, also, for each � 2 d, we have u� 2 b� eventually.hence each b� (� 2 d) is nonempty. for all i 2 i; the inclusion bi � u(x0)holds by de�nition. let j�j = k � 2. inductively we can assume that for all� for which j�j < k, the inclusion b� � wj�jwj�j�1 : : :w2u(x0) is satis�ed.(in particular, we have b� � u(x0) for j�j = 1:) then, by de�nition, b� �s��(ti2�(ui)j�j)(b�) � s�� wj�j+1(b�) � s�� wj�j+1wj�j : : :w2u(x0) =wj�j : : : w2u(x0). hence b� � w1u(x0) whenever � 2 d. consider now�;� 2 d such that � � � and x 2 b�. from ((\i2�(ui)j�j)f )�1(y) \f(x) 6= ?,that is y 2 (ti2�(ui)j�j)f (f(x)) � cl�(v�1)f(\i2�(ui)j�j)(x), we see that thereexists x0 2 (\i2�(ui)j�j)(x) such that (\i2�v �1i (y)) \ ((ti2�(ui)j�j)f )�1(y) \f(x0) 6= ?: therefore x0 2 b� by de�nition of b�. furthermore we deduce that 110 hans-peter a. k�unzix 2 (\i2�(ui)j�j)�1(x0), that is b� � (ti2�(ui)j�j)�1(b�). this concludes theveri�cation of the stated conditions.for each � 2 d set c� = s�� b�: clearly fc� : � 2 dg is a �lterbase onx: we shall show that the generated �lter f is costable in (x;u): let h 2 uand � 2 d: there is i 2 i such that ui � h: consider x 2 cfig: consequentlyx 2 b� for some � 2 d such that i 2 �: note that � = � [ � 2 d: thenx 2 b� � (tj2�(uj)j�j)�1(b�) � u�1i (b�) � h�1(b�) � h�1(c�): hence wehave shown that cfig � t�2d h�1(c�): thus f is costable in (x;u). observenext that the set of cluster points of f in (x;u�1) belongs to wu(x0); sinceeach c� � w1u(x0) (� 2 d):by our assumption there exists a �(u�1)-cluster point x of f: considerarbitrary i;k 2 i: choose � 2 d such that fi;kg � � and u�1i (x) \ b� 6= ?:find x0 2 u�1i (x) \ b�: then v �1k (y) \ f(x0) 6= ? by de�nition of b�: weconclude that (u�1i (x) � v �1k (y)) \ f 6= ?: thus (x;y) 2 f by closedness of fwith respect to the topology �(u�1)��(v�1): we have shown that y 2 f(x) �fwu(x0): it follows that f is uniformly open. �corollary 4.3. (compare [28]) an almost uniformly open mapping with a closedgraph from a supercomplete uniform space into an arbitrary uniform space isuniformly open. in particular, an almost uniformly open continuous mappingfrom a supercomplete uniform space into a uniform hausdor� space is uniformlyopen.corollary 4.4. let (x;u) be a cs�asz�ar complete quasi-uniform space andf : (x;u) ! (y;v) an almost uniformly open uniformly continuous mappingonto a quasi-uniform t1-space (y;v): then f is uniformly open and (y;v) iscs�asz�ar complete.proof. only the �nal paragraph of the proof of proposition 4.2 has to be modi-�ed. this time f has a �(us)-cluster point x in x: let k 2 i: by continuity off there is i 2 i such that f(ui(x)) � vk(f(x)): find � 2 d such that fi;kg � �and there is x0 2 ui(x) \ u�1i (x) \ b�: thus f(x0) 2 vk(f(x)); furthermoref(x0) 2 v �1k (y) by de�nition of b�: consequently (f(x);y) 2 \v and thusy = f(x) 2 fwu(x0): we conclude that f is uniformly open. the secondassertion is a consequence of corollary 3.6. �5. preservation of completeness under closed mappingswe �nish this article with three results on closed continuous mappings be-tween quasi-metrizable spaces. let us recall that kofner [18] has shown thateach �rst-countable closed continuous image of a quasi-metrizable space is quasi-metrizable. his techniques can be modi�ed to yield the following two results.proposition 5.1. the image of a left k-complete quasi-metric space under aperfect continuous mapping admits a left k-complete quasi-metric.proof. let f : x ! y be a perfect continuous mapping from a left k-completequasi-metric space (x;d) onto a topological space y: for each y 2 y and n 2 n preservation of completeness under mappings in asymmetric topology 111set vn(y) = fy0 2 y : f�1fy0g � b2�n(f�1fyg)g: then clearly, by the assump-tion made on f; fvn : n 2 ng is a base of a compatible quasi-metrizable quasi-uniformity v on y (see [17, theorem 2]). let e be a quasi-metric on y inducingv: furthermore let (yn)n2n be a left k-cauchy sequence in (y;e): there is astrictly increasing sequence (nk)k2n in n such that (ynk;yp) 2 vk wheneverp 2 n and p � nk: hence f�1fynk+1g � b2�k(f�1fynkg) whenever k 2 n:by compactness of the �bers of f; we �nd �nite subsets fnk of f�1fynkg suchthat f�1fynkg � b2�k(fnk) and therefore fnk+1 � f�1fynk+1g � b2�(k�1)(fnk)whenever k 2 n: by k�onig's lemma [21] applied to the sequence of �nite sets(fnk)k2n we see that there exists a sequence (y0nk)k2n such that y0nk 2 fnkand d(y0nk;y0nk+1) < 2�(k�1) whenever k 2 n: thus by left k-completenessof (x;d) we can �nd x 2 x such that the left k-cauchy sequence (y0nk)k2nconverges to x: therefore by continuity of f; the sequence (ynk)k2n and henceby [34, lemma 1] the sequence (yn)n2n converges to f(x): hence (y;e) is leftk-complete. we conclude that the topological property of admitting a leftk-complete quasi-metric is preserved under perfect continuous surjections. �proposition 5.2. a �rst-countable image y of a right k-sequentially completequasi-metric space (x;d) under a closed continuous mapping f admits a rightk-sequentially complete quasi-metric.proof. for any y 2 y , let fvn(y) : n 2 ng be a decreasing basic sequence ofopen neighborhoods at y: set wn(y) = fz 2 y : f�1fzg � b2�n(f�1fyg) \f�1vn(y)g whenever y 2 y and n 2 n: furthermore set cwn = sfwkp � : : : �wk1 : 2�k1 + : : : + 2�kp � 2�n and k1; : : : ;kp;p 2 ng whenever n 2 n: notethat cw2n+1 � cwn whenever n 2 n: kofner's argument [18, p. 334] showsthat the quasi-metrizable quasi-uniformity w generated by fcwn : n 2 ng iscompatible on y: note that if a;b 2 y , s 2 n and a 2 ws(b) we can �nd for anya0 2 f�1fag some b0 2 f�1fbg such that a0 2 b2�s(b0): let e be a quasi-metricon y inducing w: it su�ces to show that e is right k-sequentially complete.let (yn)n2n be a left k-cauchy sequence in (y;e�1): for each k 2 n there isa strictly increasing sequence (nk)k2n in n such that (yl;ynk) 2 cwk wheneverl 2 n and l � nk: in particular (ynk+1;ynk) 2 cwk whenever k 2 n: it followsthat for each k 2 n there are p 2 n, s1; : : : ;sp 2 n and a1; : : : ;ap�1 2 y suchthat 2�s1 + : : : + 2�sp � 2�k and (ynk+1;a1) 2 ws1; : : : ;(ap�1;ynk) 2 wsp: (inparticular, (ynk+1;ynk) 2 ws1 if p = 1:) choose y0n1 2 x such that yn1 = f(y0n1):inductively over k 2 n we can �nd points a0p�1; : : : ;a01 and y0nk+1 2 x such thatf(y0nk+1) = ynk+1; for each i = 1; : : : ;p � 1 we have f(a0i) = ai and (y0nk+1;a01) 2b2�s1 ; : : : ;(a0p�1;y0nk) 2 b2�sp: thus for each k 2 n; (y0nk+1;y0nk) 2 b2�(k�1): weconclude that (y0nk)k2n is a left k-cauchy sequence in (x;d�1) and convergesto some x in (x;d): then the sequence (ynk)k2n converges to f(x) by continuityof f: since (yn)n2n is a left k-cauchy sequence in (y;e�1); it also convergesto f(x) in (y;e) (see [34, lemma 1]). we deduce that y admits a right k-sequentially complete quasi-metric. � 112 hans-peter a. k�unziproblem 5.3. does a �rst-countable image of a left k-complete quasi-metricspace under a closed continuous mapping admit a left k-complete quasi-metric?finally we would like to mention that it is well known that under appro-priate hypotheses preimages of quasi-uniform spaces which possess some kindof completeness property also satisfy that type of completeness condition (seee.g. [26, proposition 7]). we �nish this article with another such result. (itis known on the other hand that the property of quasi-metrizability behavesrather badly under preimages (compare [23]).)proposition 5.4. let f : x ! y be a closed continuous mapping from aquasi-metric space (x;d) such that all �bers are left k-complete onto a leftk-complete quasi-metric space (y;d0): then x admits a left k-complete quasi-metric. (the analogous result also holds for right k-sequential completenessinstead of left k-completeness.)proof. for each n 2 n set vn = f(x;y) 2 x � x : d0(f(x);f(y)) < 2�nand d(x;y) < 2�ng: let e be a quasi-metric on x inducing the (compati-ble) quasi-uniformity generated by fvn : n 2 ng: furthermore let (xn)n2nbe a left k-cauchy sequence in (x;e): note �rst that the left k-cauchy se-quence (f(xn))n2n converges to some y 2 y: by our assumption on the �bers,(xn)n2n has a cluster point and thus, by [34, lemma 1], converges providedthat (f(xn))n2n has a constant subsequence. so let us assume that this is notthe case. in particular we can suppose that f(xn) 6= y for n larger than somen0 2 n: by closedness of f, we deduce that y 2 f(cl�(d)fxn : n > n0;n 2 ng):choose x 2 cl�(d)fxn : n > n0;n 2 ng such that f(x) = y: then evidently x isa cluster point and thus by [34, lemma 1] a limit point of the sequence (xn)n2n:we conclude that (x;e) is left k-complete. a similar argument establishes thesecond assertion. �references[1] e. alemany and s. romaguera, on right k-sequentially complete quasi-metric spaces, actamath. hung. 75 (1997), 267{278.[2] b.s. burdick, a note on completeness of hyperspaces, general topology and applications: fifthnortheast conference at the college of staten island, 1989, ed. by s.j. andima et al., dekker,new york, 1991, 19{24.[3] j. cao and i.l. reilly, on pairwise almost continuous multifunctions and closed graphs, indianj. math. 38 (1996), 1{17.[4] j. chaber, m.m. �coban and k. nagami, on monotonic generalizations of moore spaces, �cechcomplete spaces and p-spaces, fund. math. 84 (1974), 107{119.[5] c.c. chou and j.-p. penot, in�nite products of relations, set-valued series and uniform opennessof multifunctions, set-valued anal. 3 (1995), 11{21.[6] i.m. dektjarev, a closed graph theorem for ultracomplete spaces, sov. math. dokl. 5 (1964),1005{1007.[7] v.v. fedorchuk and h.-p.a. k�unzi, uniformly open mappings and uniform embeddings of func-tion spaces, topology appl. 61 (1995), 61{84.[8] p. fletcher and w.f. lindgren, �-spaces, gen. top. appl. 9 (1978), 139{153.[9] p. fletcher and w.f. lindgren, quasi-uniform spaces, dekker, new york, 1982.[10] j.r. isbell, supercomplete spaces, paci�c j. math. 12 (1962), 287{290. preservation of completeness under mappings in asymmetric topology 113[11] h.j.k. junnila, covering properties and quasi-uniformities of topological spaces, ph. d. thesis,virginia polytechnic institute and state university, blacksburg, va., 1978.[12] h.j.k. junnila, neighbornets, paci�c j. math. 76 (1978), 83{108.[13] h.j.k. junnila and h.-p.a. k�unzi, stability in quasi-uniform spaces and the inverse problem,topology appl. 49 (1993), 175{189.[14] h.j.k. junnila and h.-p.a. k�unzi, ortho-bases and monotonic properties, proc. amer. math.soc. 119 (1993), 1335{1345.[15] j.l. kelley, general topology, d. van nostrand, new york, 1955.[16] p.q. khanh, an induction theorem and general open mapping theorems, j. math. anal. appl.118 (1986), 519{534.[17] j. kofner, quasi-metrizable spaces, paci�c j. math. 88 (1980), 81{89.[18] j. kofner, closed mappings and quasi-metrics, proc. amer. math. soc. 80 (1980), 333{336.[19] j. kofner, transitivity and ortho-bases, can. j. math. 33 (1981), 1439{1447.[20] j. kofner, open compact mappings, moore spaces and orthocompactness, rocky mountain j.math. 12 (1982), 107{112.[21] d. k�onig, sur les correspondances multivoques des ensembles, fund. math. 8 (1926), 114{134.[22] h.-p.a. k�unzi, complete quasi-pseudo-metric spaces, acta math. hung. 59 (1992), 121{146.[23] h.-p.a. k�unzi, perfect preimages having g�-diagonals of quasi-metrizable spaces, indian j.math. 41 (1999), 33{37.[24] h.-p.a. k�unzi and s. romaguera, some remarks on doitchinov completeness, topology appl.74 (1996), 61{72.[25] h.-p.a. k�unzi and s. romaguera, left k-completeness of the hausdor� quasi-uniformity, ros-tock. math. kolloq. 51 (1997), 167{176.[26] h.-p.a. k�unzi and c. ryser, the bourbaki quasi-uniformity, topology proc. 20 (1995), 161{183.[27] m. leischner, an incomplete quotient of a complete topological group, arch. math. (basel) 56(1991), 497{500.[28] v.l. levin, open mapping theorem for uniform spaces, am. math. soc., transl., ii. ser. 71(1968), 61{66; translation from izv. vyssh. uchebn. zaved., mat. 2(45) (1965), 86{90.[29] m.j. p�erez-pe~nalver and s. romaguera, weakly cauchy �lters and quasi-uniform completeness,acta math. hung. 82 (1999), 217{228.[30] t.m. phillips, completeness in aronszajn spaces, stud. topol., proc. conf. charlotte, n.c.,1974, 457{465 (1975).[31] w. roelcke and s. dierolf, uniform structures on topological groups and their quotients,mcgraw-hill, new york, 1981.[32] s. romaguera, left k-completeness in quasi-metric spaces, math. nachr. 157 (1992), 15{23.[33] s. romaguera, on complete aronszajn quasi-metric spaces and subcompactness, in papers ongeneral topology and applications, eighth summer conf. at queens college, 1992, ann. newyork acad. sci. 728 (1994), 114{121.[34] s. romaguera, on hereditary precompactness and completeness in quasi-uniform spaces, actamath. hung. 73 (1996), 159{178.[35] s. romaguera and m. sanchis, completeness of hyperspaces on topological groups, j. pure appl.algebra (to appear).[36] s. romaguera and m. sanchis, locally compact topological groups and co�nal completeness, j.london math. soc. (to appear).[37] h.h. wicke and j.m. worrell, jr., open continuous mappings of spaces having bases of countableorder, duke math. j. 34 (1967), 255{271; errata 813{814.[38] h.h. wicke and j.m. worrell, jr., topological completeness of �rst countable hausdor� spacesi, fund. math. 75 (1972), 209{222.[39] h.h. wicke and j.m. worrell, jr., a characterization of spaces having bases of countable orderin terms of primitive bases, can. j. math. 27 (1975), 1100{1109.received march 2000 114 hans-peter a. k�unzihans-peter a. k�unzidepartment of mathematics and applied mathematicsuniversity of cape townrondebosch 7701south africa current address:institut de math�ematiquesuniversit�e de fribourgchemin du mus�ee 23ch-1700 fribourgsuisseanddepartment of mathematicsuniversity of bernesidlerstr. 5ch-3012 berneswitzerlande-mail address: hans-peter.kuenzi@math-stat.unibe.ch () @ appl. gen. topol. 17, no. 2(2016), 83-91doi:10.4995/agt.2016.4116 c© agt, upv, 2016 on monotonic fixed-point free bijections on subgroups of r raushan z. buzyakova new york, usa (raushan buzyakova@yahoo.com) abstract we show that for any continuous monotonic fixed-point free automorphism f on a σ-compact subgroup g ⊂ r there exists a binary operation +f such that 〈g, +f 〉 is a topological group topologically isomorphic to 〈g, +〉 and f is a shift with respect to +f . we then show that monotonicity cannot be replaced by the property of being periodic-point free. we explore a few routes leading to generalizations and counterexamples. 2010 msc: 06f15; 54h11; 26a48. keywords: ordered group; topological group; homeomorphism; shift; monotonic function; fixed point; periodic point. 1. introduction in this paper we present a few results in the direction of the following general problem. problem. let g be a topological group and let f be a continuous automorphism on g. is it possible to restructure the algebra of g without changing the topology so that f is a shift, or taking the inverse, or possibly some other function nicely defined in terms of the new binary operation? we show that any fixed-point free monotonic bijection on a σ-compact subgroup g ⊂ r is a shift with respect to some group structure on g topologically isomorphic to g. in particular, any continuous fixed-point free bijection on the reals r is a shift with respect to some group operation on r compatible with the euclidean topology of r. this result can be used in particular to received 12 september 2015 – accepted 01 june 2016 http://dx.doi.org/10.4995/agt.2016.4116 r. z. buzyakova show that any fixed-point free continuous bijection f on r can be colored in three colors. in other words, there exists a cover {fi : i = 1, 2, 3} such that f(fi)∩fi = ∅ for each i = 1, 2, 3. this colorability fact and an ǫ− δ argument for shifts was communicated to the author by carlos nicolas. our theorem shows that an argument for shifts covers the bijection case as any bijective fixed-point free map is a shift with respect to some topology-compatible group operation on r. for a reader interested in coloring, a possible tricolor for the shift f(x) = x + 3 is a = ⋃ n∈z[6n, 6n + 2], b = ⋃ n∈z[6n + 2, 6n + 4], and c = ⋃ n∈z[6n + 4, 6n + 6]. we then discuss failures and successes of certain natural generalization routes. in particular, we show that in our main theorem monotonicity cannot be replaced by the property of being periodic-point free. we use standard notations and terminology. for topological basic facts and terminology one can consult [2]. since we do not use any intricate algebraic facts, any abstract algebra textbook is a sufficient reference. we consider only continuous maps. all group shifts under discussion are shifts by a non-neutral element. 2. study by r we denote the set of reals endowed with the euclidean topology. if a different binary operation or topology is used, it will be specified. let us agree that given a bijection f and a positive integer n, by fn we denote the composition of n copies of f and by f−n we denote the composition of n copies of f−1. the expression f0 is the identity map. we will use the following folklore facts: facts: (1) if 〈g, +〉 is a group and f : g → g is a bijection, then 〈g, ⊕f 〉 is a group, where f(x) ⊕f f(y) = f(x + y). (2) the groups in fact 1 are isomorphic by virtue of f. (3) if 〈g, tg, +〉 is a topological group and f : 〈g, tg〉 → 〈g, tg〉 is a homeomorphism, then 〈g, tg, +〉 and 〈g, tg, ⊕f 〉 are topologically isomorphic by virtue of f. theorem 2.1. let f : r → r be a continuous fixed-point free bijection. then there exists a binary operation +f on r such that 〈r, +f 〉 is a topological group topologically isomorphic to r and f is a shift with respect to +f . proof. since f is fixed-point free, we conclude that either f(x) > x for all x or f(x) < x for all x. we will carry out our argument assuming the former but will make necessary comments for the latter case. to define our new binary operation, we will first put each x ∈ r into correspondence with xf ∈ r. definition of xf . we define xf for each x ∈ r in three steps a follows: (1) put 0f = 0 and nf = f n(0) for each n ∈ z \ {0}. (2) let h : [0, 1) → [0, f(0)) be an order preserving bijection (hence homeomorphism). for each x ∈ [0, 1), put xf = h(x). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 84 on monotonic fixed-point free bijections on subgroups of r remark. for ”f(x) < x”-case, h is onto (f(0), 0] and order-reversing. note that h(0) = 0. thus, this definition agrees with the first step. (3) fix any x ∈ r. then there exist a unique integer n and a unique x′ in [0, 1) such that x = x′ + n. put xf = f n(h(x′)). note, this definition agrees with steps (1) and (2). indeed, if x ∈ (0, 1), then x′ = x and n = 0. then xf = f 0(h(x′)) = h(x), which agrees with step (2). if x is an integer, then x′ = 0 and x = 0 + n. then nf = xf = f n(h(0)) = fn(0), which agrees with step (1). for further reference, we denote by g the correspondence x 7→ xf . claim 1. the correspondence g is an order-preserving bijection on r and, hence, a homeomorphism. for ”f(x) < x”-case, g is order-reversing. let us first show that g is surjective. fix y ∈ r. since f is a fixed-point free continuous bijection, r = ⋃ n∈ω([f −(n+1)(0), f−n(0)] ∪ [fn(0), fn+1(0)]). therefore, there exists x ∈ [0, f(0)) such that y = fn(x) for some integer n. by part (2) of xf -definition, x = h(z ′) for some z′ ∈ [0, 1). put z = z′ + n. then, by part (3), g(z) = zf = f n(h(z′)) = fn(x) = y. let us show that g is order-preserving. fix a, b ∈ r such that a < b. let a = a′ + na and b = b ′ + nb, where a ′, b′ ∈ [0, 1) and na, nb ∈ z. we assume that na, nb are non-negative. other cases are treated similarly. case (na < nb): by part (2) of the definition of xf , we have h(a ′), h(b′) ∈ [0, f(0)). since f is a fixed-point free homeomorphism, it is orderpreserving. therefore, fna(0) ≤ fna(h(a′)) ≤ fna+1(0). therefore, af ∈ [fna(0), fna+1(0)]. similarly, bf ∈ [fnb(0), fnb+1(0)]. since nb > na, we conclude that af ≤ bf. since f and h are one-to-one, we conclude that af < bf . case ¬(na < nb): re-write b − a > 0 as (b′ − a′) + (nb − na) > 0. since |b′ − a′| < 1, we conclude that nb − na ≥ 0. by the case’s assumption, n = na = nb. hence, a ′ < b′. by part (2) of the definition, h(a′) < h(b′). since f is order-preserving, af = f n(h(a′)) < fn(h(b′)) = bf . the claim is proved. claim 1 and fact 3 imply that 〈r, ⊕g〉 is a topological group topologically isomorphic to r by virtue of g. the next claim completes the proof of the theorem claim 2. f is an ⊕g-shift and f(xf ) = xf ⊕g 1f for all xf ∈ r. fix xf ∈ r. let x = x′ + n, where x′ ∈ [0, 1) and n is an integer. then xf = f n(h(x′)). therefore, f(xf ) = f n+1(h(x′)). put pf = f n+1(h(x′)). then p = x′ + (n + 1) = (x′ + n) + 1. by the definition of ⊕g, we have g(p) = g(x′ + n) ⊕g g(1), that is, pf = xf ⊕g 1f . since pf = f(xf ), the claim is proved. to stress the dependence of ⊕g on f we put +f = ⊕g, which completes our proof. � c© agt, upv, 2016 appl. gen. topol. 17, no. 2 85 r. z. buzyakova our discussion prompts a question of whether theorem 2.1 can be generalized to any subgroup g of r and any continuous periodic-point free bijection f on g. we will show later that a generalization of such a magnitude is not possible. however, certain relaxations on hypotheses can be made. we will next show that the conclusion of theorem 2.1 holds if we replace r by any σ-compact subgroup of r and f by any fixed-point free monotonic bijection. we believe that ”σ-compact subgroup” can be replaced by ”any subgroup”. we will identify the single statement in our argument that requires additional work for a desired generalization. to make argument clearer, let us handle a few cases informally. if g is a discrete subgroup of r, then it is order-isomorphic to z. therefore, any monotonic bijection of g is necessarily a shift. if g has a non-trivial connected component, then g = r and theorem 2.1 applies. to handle the case when g is zero-dimensional and dense in r let us recall a few classical facts. it is due to sierpienski [4] (see also [5, 1.9.6]) that any countable metric space with no isolated points is homeomorphic to the space of rationals q. it is due to alexandroff and urysohn [1] that any σ-compact zero-dimensional metric space which is nowhere countable is homeomorphic to the product q × c of the rationals q and the cantor set c. the immediate applications of these characterizations are the following useful fact: fact: let x and y be homeomorphic to q (or both homeomorphic to q × c). let a, b ∈ x be distinct and let c, d ∈ y be distinct. then there exists a homeomorphism h : x → y such that h(a) = c and h(b) = d. this fact, zero-dimensionality, and homogeneity of g imply the following statement. lemma 2.2. let g be a zero-dimensional dense σ-compact subgroup of r and u a non-empty open subset of g. let a, b ∈ u be distinct and c, d ∈ g satisfy c < d. then there exists a homeomorphism h : u → [c, d]∩g such that h(a) = c and h(b) = d. proof. since g is a group, it is either countable or nowhere countable. therefore, g is homeomorphic to q or q×c. since both cases are handled similarly, we assume that the latter is the case. then any non-empty open subset of g as well as any infinite closed interval of g are σ-compact and nowhere countable. therefore, both u and [c, d] ∩ g are homeomorphic to q × c. next apply fact. � the argument of our next result follows that of theorem 2.1. to avoid unnecessary repetition, we will reference the already presented argument in a few places. even though we could have put both theorems under one umbrella, for readability purpose, the author decided to present them separately. theorem 2.3. let g be a σ-compact subgroup of r and let f : g → g be a fixed-point free monotonic bijection. then there exists a binary operation +f on g such that 〈g, +f 〉 is a topological group topologically isomorphic to g and f is a shift with respect to +f . c© agt, upv, 2016 appl. gen. topol. 17, no. 2 86 on monotonic fixed-point free bijections on subgroups of r proof. we now may assume that g is a non-discrete zero-dimensional subgroup of r. since g is closed under addition and contains a nontrivial sequence converging to 0, we conclude that g is dense in r. since f is a monotonic bijection and g is dense in r, there exists a continuous bijective extension f̄ : r → r of f. let f be the set of all fixed points of f̄. let j = {jn : n = 0, 1, ...} consist of all maximal convex sets of r \ f . note that if f is empty, then j = {j0 = r}. put i = {in = jn ∩ g : n = 0, 1, ...}. claim 1. if i ∈ i, then there exists a convex clopen o ⊂ i in g such that fn(o) ∩ fm(o) = ∅ whenever n 6= m and i = ⋃ n∈z f n(o). to prove the claim, fix any point p in j \ g, where j ∈ j such that i = j ∩ g. such a point exists due to zero-dimensionality of g. due to absence of fixed points and monotonicity of f̄ on j, we conclude that {fn(p) : n ∈ z} is unbounded in j on either side. let ip be the closed interval in r with the endpoints p and f(p). we then have o = ip ∩ g is as desired. the claim is proved. for each in ∈ i, fix on that satisfies the conclusion of the claim. next for each x ∈ g we will define xf as follows. definition of xf . define xf in three steps. (1) fix p∗ ∈ i0 and c ∈ g such that c > 0. put 0f = p∗ and (nc)f = fn(p∗) for each n ∈ z. (2) let ip∗ be the interval of g with endpoints p ∗ and f(p∗). let a and b be convex clopen subsets of ip∗ such that p ∗ ∈ a, f(p∗) ∈ b, and a ∪ b = ip∗. by lemma 2.2, there exists a homeomorphism h of [0, c] ∩ g with a ⊕ o1 ⊕ ... ⊕ on ⊕ ... ⊕ b such that h(0) = p∗ and h(c) = f(p∗). for each x ∈ [0, c] ∩ g, put xf = h(x). remark. note that our use of lemma 2.2 is the only part in which the author could not carry out the argument for an arbitrary zerodimensional subgroup of r. note that our definition agrees with step 1 for p∗ and f(p∗). (3) for each x ∈ g, there exist a unique x′ ∈ [0, c) ∩ g and a unique n ∈ z such that x = x′ + nc. put xf = f n(h(x′)). due to uniqueness of x′ and n, xf is well-defined. note that this definition agrees with our definitions at steps 1 and 2. denote by g the correspondence x 7→ xf . claim 2. g is surjective. to prove the claim, fix y ∈ g. then y ∈ in ∈ i for some n. by the choice of on, there exist m ∈ z and z ∈ on such that y ∈ fm(z). if n = 0, we may assume that z 6= f(p∗). by the definition of h, there exists x′ ∈ [0, c) such that h(x′) = z. put x = x′ + mc. then g(x) = fm(h(x′)) = fm(z) = y. the claim is proved. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 87 r. z. buzyakova claim 3. g is a one-to-one. to show that g is one-to-one, fix distinct a, b ∈ g and let a = a′ + nc and b = b′ + mc, where a′, b′ ∈ [0, c) and n, m ∈ z. then g(a) = fn(h(a′)) and g(b) = fm(h(b′)). if n = m, then g(a) 6= g(b) because both h and fn are one-to-one. assume now that n 6= m. let h(a′) ∈ oi and h(b′) ∈ oj. assume that i = j. then g(a) ∈ fn(oi) and g(b) ∈ fm(oi). by the choice of oi, we have fn(oi) ∩ fm(oi) = ∅. if i 6= j, then g(a) ∈ ii and g(b) ∈ ij. by the definition of i, ii ∩ ij = ∅, which proves the claim. claim 4. g is a homeomorphism. observe that if x ∈ [cn, c(n + 1)] ∩ g, then g(x) = fn(h(x − nc)). thus, g is a homeomorphism on [cn, c(n + 1)] ∩ g. since {[cn, c(n + 1)] ∩ g}n forms a locally finite closed cover of g and g is closed on each element of the cover, we conclude that g is a closed map. by claims 2 and 3, g is a homeomorphism on g. the claim is proved. denote by +f the operation ⊕g defined in facts 1-3. by fact 3, 〈g, +f 〉 is a topological group topologically isomorphic to g. following the argument of theorem 2.1, f(xf ) = xf +f cf for all xf ∈ g. � it is natural to wonder if monotonicity of f in theorem 2.3 can be replaced by a periodic-point free homeomorphism. since the latter can be of wild nature, a counterexample is expected. before we present our construction, we prove the following lemma. lemma 2.4. suppose f : q → q is not an identity map and p ∈ q satisfy the following property: (∗) ∀n > 0∃m > 0 such that fm+1((p−1/n, p+1/n)q) meets f−m((p−1/n, p+1/n)q) if 〈q, ⊕〉 is a topological group topologically isomorphic to q, then f is not a ⊕-shift. proof. let h : q → 〈q, ⊕〉 be a topological isomorphism . let ≺ be the order on 〈q, ⊕〉 defined by a ≺ b if and only if h−1(a) < h−1(b). clearly, 〈q, ⊕, ≺〉 is an ordered topological group. let f be a ⊕-shift. since f is not the identity map, there exists c ∈ q \ {h(0)} such that f(x) = x ⊕ c. without loss of generality, assume that h(0) ≺ c. fix a, b ∈ q such that a ≺ p ≺ b ≺ (a ⊕ c) ≺ (p ⊕ c). clearly fm+1((a, b)≺) misses f −m((a, b)≺) whenever m > 0. since the topology on 〈q, ⊕〉 is euclidean, there exists n such that (p− 1/n, p+ 1/n)q ⊂ (a, b)≺. therefore, fm+1((p − 1/n, p + 1/n)) misses f−m((p − 1/n, p + 1/n)) for any m > 0. � note that a non-identity homeomorphism with periodic points cannot be a shift in any group structure isomorphic to q. therefore, the importance of monotonicity in theorem 2.3 should be demonstrated by a non-monotonic homeomorphism with no periodic points. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 88 on monotonic fixed-point free bijections on subgroups of r example 2.5. there exists a periodic-point free homeomorphism f : q → q such that f is not a shift in any topological group structure on q topologically isomorphic to q. construction. all intervals under consideration are in q. therefore, instead of (a, b)q we write (a, b). we will construct a homeomorphism f : q → q that satisfies property (*) of lemma 2.4 for p = 0. let ǫ be any irrational number strictly between 0 and 1 2 √ 2 . we put ǫ = 1√ 7 for better visualization. step 1. let g1 and g−1 be any two order-preserving homeomorphisms with the following ranges and domains: g1 : [ 0 − 1 21 √ 2 , 0 + 1 21 √ 2 ] → [1 − ǫ, 1 + ǫ] g−1 : [−1 − ǫ, −1 + ǫ] → [ 0 − 1 21 √ 2 , 0 + 1 21 √ 2 ] such maps exist since the endpoints in all intervals are irrational, and therefore, not in g. step n > 1. let gn and g−n be any two order-preserving homeomorphisms with the following ranges and domains: gn : gn−1 ◦ ... ◦ g1 ([ 0 − 1 2n √ 2 , 0 + 1 2n √ 2 ]) → [n − ǫ, n + ǫ] g−n : [−n − ǫ, −n + ǫ] → g−1−(n−1) ◦ ... ◦ g −1 −1 ([ 0 − 1 2n √ 2 , 0 + 1 2n √ 2 ]) note that due to irrationality of √ 2 and √ 7, ranges and domains of these homeomorphisms are clopen intervals in g. for better visualization, let us write out the domains of the defined function with ǫ = 1√ 7 . we have dom(g1) = [ − 1 2 √ 2 , 1 2 √ 2 ] . if n ≤ −1, then dom(gn) = [ n − 1√ 7 , n + 1√ 7 ] . if n ≥ 2, then dom(gn) ⊂ [ (n − 1) − 1√ 7 , (n − 1) + 1√ 7 ] . due to smallness of 1√ 7 , these sets are mutually disjoint. let us summarize this observation for further reference. claim. if n, m ∈ z \ {0} are distinct, then the domains of gn and gm are disjoint. next we select special clopen intervals as follows. definition of an, bn for n > 0: fix any non-empty clopen intervals an ⊂ (n − ǫ, n + ǫ) and bn ⊂ (−n − ǫ, −n + ǫ) with the following properties: p1: an misses the domain of gn+1. such a set exists because the domain of gn+1 is a proper clopen interval of (n − ǫ, n + ǫ). p2: bn misses the range of g−(n+1). such a set exists because the range of g−(n+1) is a proper clopen interval of (−n − ǫ, −n + ǫ). p3: gn ◦ ... ◦ g1 ◦ g−1 ◦ ... ◦ g−n(bn) misses an. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 89 r. z. buzyakova for each n, fix a homeomorphism hn from an onto bn. define g as follows: g(x) = { gn(x) x is in the domain of gn for n ∈ z \ {0} hn(x) x ∈ an to show that g is a function on d = ( ⋃ n≥1 an ) ∪( ⋃ {dom(gn) : n ∈ z \ {0}}), we need to show that any x ∈ d belongs to exactly one element of {an : n = 1, 2, ...} ∪ {dom(gn) : n ∈ z \ {0}}. by claim, we may assume that x ∈ an. by property p1, x is not in the domain of gn+1. since an ⊂ [n − ǫ, n + ǫ], x is not in the domain of any gm with m 6= n + 1. by claim and selection of an’s, we conclude that x 6∈ am for n 6= m. let us show that g is a homeomorphism between its domain and range. by property p2, g is one-to-one. the domain of g is the union of the clopen discrete family {dom(gn) : n ∈ z \ {0}} ∪ {an : n ∈ z+}. since g is one-to-one and is a homeomorphism on each member of the family, g is a homeomorphism. by property p3, g has no periodic points. the complements of the domain and range of g are clopen proper subsets of q that are unbounded on both sides. therefore, g has a homeomorphic extension f : q → q that has no periodic points. it remains to show that f and 0 satisfy the hypothesis of lemma 2.4. fix any n > 0. we have fn (( 0 − 1 2n √ 2 , 0 + 1 2n √ 2 )) = (n − ǫ, n + ǫ). since an ⊂ (n − ǫ, n + ǫ), we conclude that bn ⊂ fn+1 (( 0 − 1 2n √ 2 , 0 + 1 2n √ 2 )) . on the other and, bn ⊂ (−n, −ǫ, −n + ǫ) = f−n (( 0 − 1 2n √ 2 , 0 + 1 2n √ 2 )) . therefore, m = n is as desired. � since zero-dimensional spaces may admit very wildly mannered automorphisms, one may ask if our theorem for r can be extended to rn. alas, an example is in order. example 2.6. there exists a periodic-point free homeomorphism f : r3 → r3 such that f is not a shift in any group structure of r3 topologically isomorphic to r3. construction. let h : r3 → r3 be the rotation of the space by √ 2 degrees about the z-axis in the positive direction. put s = {〈x, y, z〉 ∈ r3 : x2 + y2 ≤ 1}. define g : s → s by letting g(x, y, z) = 〈x, y, z +1−x2 −y2〉. in words, g slides vertically every point by the distance equal to the distance from the point to the wall of the cylinder. thus, the points on the boundary of the cylinder are not moved. clearly, both h and g are homeomorphisms. define f : r3 → r3 as follows: f(x, y, z) = { h(x, y, z) 〈x, y, z〉 6∈ s g ◦ h(x, y, z) 〈x, y, z〉 ∈ s since g is the identity on the boundary of the cylinder, we conclude that g ◦h is equal to h on the boundary of s. therefore, f is a homeomorphism. points on c© agt, upv, 2016 appl. gen. topol. 17, no. 2 90 on monotonic fixed-point free bijections on subgroups of r the z-axis are slided up by 1 unit. points off the z-axis undergo a rotation by√ 2 degrees. thus, f has no periodic points. since the points on the boundary of the cylinder are only rotated, the set {fn(1, 0, 0) : n ∈ ω} is a an infinite subset of the unit disc in the xy-plane centered at the origin. therefore, the set has a cluster point. however,{〈x, y, z〉 + n〈a, b, c〉 : n ∈ ω} is a closed discrete subset of r3 for any 〈x, y, z〉, 〈a, b, c〉 ∈ r3. therefore, f cannot be a shift in any group structure on r3 topologically isomorphic to r3. � we would like to finish this study with a few remarks of categorical nature. assume that f : r → r is a map that is a shift by a non-neutral element with respect to some group operation +f on r that is compatible with the euclidean topology. then, f is a homeomorphism and fixed-point free. therefore, we have a characterization of all maps on r that are shifts by non-neutral elements after some refitting of algebraic structure of r. it is therefore justifiable to view fixed-point free homeomorphisms on r as generalized shifts. similarly, we can define generalized polynomial (trigonometric, etc.) functions as maps in form f(x) = an ⋆ x n ⊕ ... ⊕ a0, for some addition ⊕ and multiplication ⋆ compatible with the topology of r. therefore, it would be interesting to consider the following general problem. problem. characterize generalized polynomials, trigonometric functions, and generalized versions of other standard calculus functions. at last, recall that we were unsuccessful in generalizing theorem 2.1 to a desired extent. therefore, the question is in order. question. let g be a topological subgroup of r and let f : g → g be a fixedpoint free monotonic homeomorphism. does there exist a binary operation ⊕ on g such that 〈g, ⊕〉 is a topological group topologically isomorphic to g and f is a ⊕-shift? acknowledgements. the author would like to thank the referee for valuable remarks and corrections. references [1] p. alexandroff and p. urysohn, uber nuldimensionale punktmengen, math ann. 98 (1928), 89–106. [2] r. engelking, general topology, pwn, warszawa, 1977. [3] c. nicolas, private communication, 2009 [4] w. sierpinski, sur une propriete topologique des ensembles denombrablesdense en soi, fund. math. 1 (1920), 11–16. [5] j. van mill, the infinite-dimensional topology of function spaces, elsevier, 2001. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 91 @ appl. gen. topol. 18, no. 1 (2017), 61-74 doi:10.4995/agt.2017.5818 c© agt, upv, 2017 on quasi-uniform box products olivier olela otafudu and hope sabao school of mathematical sciences, north-west university, south africa (olivier.olelaotafudu@nwu.ac.za, hope@aims.edu.gh) communicated by h.-p. a. künzi abstract we revisit the computation of entourage sections of the constant uniformity of the product of countably many copies the alexandroff one-point compactification called the fort space. furthermore, we define the concept of a quasi-uniformity on a product of countably many copies of a quasi-uniform space, where the symmetrised uniformity of our quasiuniformity coincides with the constant uniformity. we use the concept of cauchy filter pairs on a quasi-uniform space to discuss the completeness of its quasi-uniform box product. 2010 msc: 54b10; 54e35; 54e15. keywords: uniform box product; quasi-uniform box product; quasiuniformity; d-completeness. 1. introduction the theory of uniform box products was conveyed for the first time in 2001 by scott williams during the ninth prague international topological symposium (toposym). he proved, for instance, that the box product of equal factors has a compatible complete uniformity whenever its factor does and he showed that the box product of realcompact spaces is realcompact whenever the index set has no subset of measurable cardinality. some progress has been made on the concept of uniform box products. for instance in [1] and [3], bell defined a uniformity on the product of countably many copies of a uniform space which she called the constant uniformity base. it turns out that the topology induced by this uniformity is coarser than the box received 21 may 2016 – accepted 05 october 2016 http://dx.doi.org/10.4995/agt.2017.5818 o. olela otafudu and h. sabao product but finer than the tychonov product. her new product was motivated by the idea of the supremum metric on countably many copies of (compact) metric spaces. moreover, she gave an answer to the question of scott williams which asks whether the uniform box product of compact (uniform) spaces is normal. furthermore, bell introduced some new ideas on the problem “is the uniform box product of countably many compact spaces collectionwise normal?” that enabled her to prove that the uniform box product of countably many copies of the one-point compactification of a discrete space of cardinality ℵ1 is normal, countably paracompact, and collectionwise hausdorff in the uniform box topology. in additional to bell’s work, in [8] hankins modified bell’s proof of the collectionwise hausdorff property and thereafter, he answered the question“is the uniform box product of denumerably many compact spaces paracompact?” in this note, we study the concept of a quasi-uniform box product of countably many copies of a quasi-uniform space. we show, for instance, that the quasi-uniformity on a box product of countably many copies of a quasi-uniform space (x,u) is included in the constant uniformity base on the box product of countably many copies of the symmetrized uniform space (x,us) of (x,u). moreover, we look at the quasi-uniform box product of countably many copies of the one-point compactification of a countable discrete space. we revisit the computation of entourage sections of the constant uniform box product of countably many copies of the fort space due to bell [3]. furthermore, we study c-completeness and d-completeness in the quasi-uniform box product and in particular, we show that if the factor space of the quasi-uniform box product is quiet, then c-completeness implies d-completeness in quasi-uniform box products. 2. preliminaries this section recalls and reviews some well-known results on computation of entourage sections of the constant uniformity of the product of countably many copies of the one-point compactifaction known as the fort space. for more information on uniform box products we refer the reader to [1, 3, 13]. let v be an uncountable discrete space, and x = v ∪{∞} be its fort space, that is, its alexandroff one-point compactification. then (x,τ) is a topological space where a ∈ τ if a ⊆ v or if x \a is a finite set and ∞ ∈ a. then the fort space x can be equipped with the uniformity base d = {df : f ⊆ v,f is finite} on x compatible with the topology τ where df = 4∪(x\f)2. if x ∈ f, then df (x) = {x} and if x /∈ f , then df (x) = x \f ⊆ v . consider the uniform box product (∏ α∈n x,d ) of the above fort space where d = {df : df ∈d} and df = { (x,y) ∈ ∏ α∈n x × ∏ α∈n x : for all α ∈ n (x(α),y(α)) ∈ df } . c© agt, upv, 2017 appl. gen. topol. 18, no. 1 62 on quasi-uniform box products it was stated in [1, remark, p. 2164] that for each x = (x(α))α∈n ∈ ∏ α∈n x, then (2.1) df (x) = ∏ x(α)∈f {x(α)}× ∏ x(α)/∈f (x \f). the above formula has the following explanations: for x,y ∈ ∏ α∈n x, we have that y ∈ df (x) ⊆ ∏ α∈n x if and only if (x,y) ∈ df . furthermore, (x,y) ∈ df if and only if (x(α),y(α)) ∈ df whenever α ∈ n if and only if y(α) ∈ df (x(α)) whenever α ∈ n. if x(α) ∈ f whenever α ∈ n, then (2.2) y = (y(α))α∈n ∈ df (x) = ∏ x(α)∈f {x(α)} by proposition [1, proposition 3.1]. if x(α) /∈ f whenever α ∈ n, then (2.3) y = (y(α))α∈n ∈ df (x) = ∏ x(α)/∈f (x \f). remark 2.1. we point out that the equality in equation (2.1) can be understood to mean “homeomorphic to” under the natural homeomorphism which possibly rearranges the order of factors. example 2.2. if we equip the fort space x with the pervin quasi-uniformity p with the subbase s = {sa : a ⊆ v finite}, where sa = [a×a]∪[(x\a)×x]. then s−1a = [a×a] ∪ [x × (x \a)] with a ∈ τ. indeed, (a,b) ∈ s−1a if and only if (b,a) ∈ sa = [a × a] ∪ [(x \ a) × x]. furthermore, sa ∩s−1a = ( [a×a] ∪ [(x \a) ×x]) ∩ ([a×a] ∪ [x × (x \a)] ) , and thus sa ∩s−1a = [a×a] ∪ [(x \a) × (x \a)]. we observe that sa ∩ s−1a ⊇ da = 4∪ (x \ a) 2, and this latter set is an element of the open subbase of the uniformity d on x if a is a finite subset of v . for more details on the uniform subbase on x, we refer the reader to [1]. it turns out that if a ∈ a, then sa(a) = {b ∈ x : (a,b) ∈ sa} = a. if a /∈ a, then sa(a) = {b ∈ x : (a,b) ∈ sa} = {b ∈ x} = x. similarly if a ∈ a, then s−1a (a) = {b ∈ x : (b,a) ∈ sa} = {b ∈ x : b ∈ a or b ∈ x \a} = x c© agt, upv, 2017 appl. gen. topol. 18, no. 1 63 o. olela otafudu and h. sabao and if a ∈ a, then s−1a (a) = {b ∈ x : (b,a) ∈ sa} = {b ∈ x : b ∈ x \a} = x \a. remark 2.3. it follows that if p is the pervin quasi-uniformity of the fort space, then p−1 is the conjugate pervin quasi-uniformity of p on x with base s−1 where s−1 = {s−1a : a ⊆ v finite}. moreover, the uniformity p s = p ∨p−1 is finer than the uniformity subbase d = {da : a ⊆ v finite} on x (see [1, proposition 4.2]) compatible with the topology on x, where, for any finite subset a of v , da = 4∪ (x \f) × (x \f). example 2.4. if we equip the fort space x = v ∪ {∞} with the quasiuniformity wf which has subbase {wf : f ⊆ v, f is finite}, where wf = 4∪ [x × (x \f)], we have that (2.4) wf ∩w−1f = (4∪ [x×(x\f)])∩(4∪ [(x\f)×x]) = 4∪(x\f) 2. it follows that if x ∈ f, then wf (x) = {y ∈ x : (x,y) ∈ wf} = {y ∈ x : x = y or (x ∈ x and y ∈ x \f)} = {x}∪ (x \f) and w−1f (x) = {y ∈ x : (y,x) ∈ wf} = {y ∈ x : x = y or (y ∈ x and x /∈ f)} = {x}. if x /∈ f, then we have wf (x) = {y ∈ x : x = y or (x ∈ x and y ∈ x\f)} = {x}∪(x\f) = x\f and w−1f (x) = x. moreover, it follows that wf (x) ∩w−1f (x) = [{x}∪ (x \f)] ∩{x} = {x} whenever x ∈ f. whenever x /∈ f , we have wf (x) ∩w−1f (x) = x \f ∩x = x \f. remark 2.5. for the fort space x = v ∪{∞}, we observe from (1) that the coarsest uniformity wsf finer than wf coincides with the uniformity subbase d = {df : f ⊆ v finite} on x where df = 4∪(x\f)2 = wsf . furthermore, if f is a finite subset of v , we have df (x) = wf (x)∩w−1f (x) = w s f whenever x ∈ f and whenever x /∈ f . c© agt, upv, 2017 appl. gen. topol. 18, no. 1 64 on quasi-uniform box products 3. the box product of a quasi-uniform space in this section we define a quasi-uniformity whose symmetrized quasi-uniformity (uniformity) generates the tychonov product topology on the product set∏ α∈n x of countably many copies of a quasi-uniform space (x,u). we also look at a quasi-uniformity whose uniformity generates the box topology on the product set ∏ α∈n x. in [12], stoltenberg defined the product topology on the cartesian product∏ i∈i xi of a family (xi,ui)i∈i of quasi-uniform spaces as the topology induced by ∏ i∈i ui, the smallest quasi-uniformity on ∏ i∈i xi such that each projection map πi : ∏ i∈i xi → xi is quasi-uniformly continuous. furthermore, the sets of the form {((xi)i∈i, (yi)i∈i) : (xi,yi) ∈ ui} whenever ui ∈ ui and i ∈ i are sub-base for the quasi-uniformity ∏ i∈i ui. the quasi-uniformity ∏ i∈i ui is called the product quasi-uniformity on ∏ i∈i xi. we are going to omit proof of the following lemma since it is straightforward. lemma 3.1. let (x,u) be a quasi-uniform space and ∏ α∈n x be the product set of countably many copies of x. then ǔα = {ǔα : u ∈ u and α ∈ n} is a filter base generating a quasi-uniformity on ∏ α∈n x, where ǔα = { (x,y) ∈ ∏ β∈n x × ∏ β∈n x : (x(α),y(α)) ∈ u } whenever α ∈ n and u ∈u. the following has been observed by bell [3] for uniform spaces. remark 3.2. note that g ∈ τ(ǔα) if and only if for any x = (xα)α∈n ∈ g there exists ǔα ∈ ǔα such that ǔα(x) ⊆ g whenever u ∈ u and α ∈ n. thus for any x,y ∈ g, we have (xα,yα) ∈ u whenever u ∈u and α ∈ n. hence g is an open set with respect to the topology induced by the product quasi-uniformity on ∏ α∈n x. observe that the uniformity (ǔα) s coincides with the uniformity base on ∏ α∈n x and the topology τ((ǔα) s) induced by the uniformity (ǔα)s is the tychonov product topology on ∏ α∈n x. lemma 3.3. let (x,u) be a quasi-uniform space and ∏ α∈n x be the product set of countably many copies of x. then ûψ = {ûψ : ψ : n →u is a function} is a fiter base generating a quasi-uniformity on ∏ α∈n x where ûψ = { (x,y) ∈ ∏ α∈n x × ∏ α∈n x : whenever α ∈ n, (x(α),y(α)) ∈ ψ(α) } whenever u ∈u and ψ : n →u is a function. remark 3.4. if (x,u) is a uniform space, then the quasi-uniformity ûψ is exactly the uniformity ď in [3, definition 4.2]. therefore, for any quasi-uniform space (x,u), the topology τ(ûψ) s induced by the uniformity base ûsψ is the box topology on ∏ α∈n x. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 65 o. olela otafudu and h. sabao 4. quasi-uniform box products in this section we present the quasi-uniform box product of countably many copies of a quasi-uniform space. the theory of uniform box product of countably many copies of a uniform space was developed by bell [1]. she proved that the uniform box product has a topology that sits between the tychonov product and the box product topology. theorem 4.1. let (x,u) be a quasi-uniform space and ∏ α∈n x be the product set of countably many copies of x. then u = {u : u ∈ u} is a filter base generating a quasi-uniformity on ∏ α∈n x where u = { (x,y) ∈ ∏ α∈n x × ∏ α∈n x : (x(α),y(α)) ∈ u whenever α ∈ n } whenever u ∈u. proof. for u ∈ u and x ∈ ∏ α∈n x, we have (x,x) ∈ u since for any α ∈ n, (x(α),x(α)) ∈ u. observe that for any u,v ∈ u with u ⊆ v , it follows that u ⊆ v . thus {u : u ∈u} is a filter base on ∏ α∈n x × ∏ α∈n x. let u,v ∈ u be such that v 2 ⊆ u. suppose that (x,y) ∈ v 2 . then there exists z ∈ ∏ α∈n x such that (x,z) ∈ v and (z,y) ∈ v . hence (x(α),z(α)) ∈ v and (z(α),y(α)) ∈ v whenever α ∈ n. moreover, (x(α),y(α)) ∈ v 2 ⊆ u whenever α ∈ n. thus (x,y) ∈ u. therefore, u = {u : u ∈u} is a quasi-uniformity on ∏ α∈nx. � note that if for any given u ∈u, the function ψ in lemma 3.3 is a constant function ψ(α) = u whenever α ∈ n, then the quasi-uniformity ûψ in lemma 3.3 coincides with the quasi-uniformity u in theorem 4.1. this is going to motivate the following definition. we point out that this remark was observed by bell (see [3, p. 15]) for uniform box products. definition 4.2. let (x,u) be a quasi-uniform space. then the quasi-uniformity u is called the constant quasi-uniformity on the product ∏ α∈n x and the pair(∏ α∈nx,u ) is called the quasi-uniform box product. remark 4.3. if a quasi-uniform space (x,u) is such u = u−1, then u = u −1 = u s . therefore, the quasi-uniform box product (∏ α∈nx,u ) is exactly the constant uniform box product (see [3, theorem 4.3]). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 66 on quasi-uniform box products remark 4.4. for any quasi-uniform box product (∏ α∈nx,u ) of a quasiuniform space (x,u). we have u ∩v = u ∩v whenever u,v ∈u. indeed (x,y) ∈ u ∩v if and only if (x(α),y(α)) ∈ u∩v for all α ∈ n if and only if (x(α),y(α)) ∈ u and (x(α),y(α)) ∈ v for all α ∈ n. this is equivalent to (x,y) ∈ u and (x,y) ∈ v . lemma 4.5. for any quasi-uniform box product (∏ α∈nx,u ) of a quasiuniform space (x,u), the following are true. (1) u −1 = u−1 whenever u ∈u. (2) u−1 ∩u = us = u s whenever u ∈u. proof. we prove (1). then (2) will follow from (1) and remark 4.4. let u ∈ u. then (x,y) ∈ u −1 if and only if (y,x) ∈ u if and only if (y(α),x(α)) ∈ u whenever α ∈ n if and only if (x(α),y(α)) ∈ u−1 whenever α ∈ n if and only if (x,y) ∈ u−1 � remark 4.6. if (x,u) is a quasi-uniform space and (∏ α∈nx,u ) is its quasiuniform box product, then the quasi-uniform space (∏ α∈nx,u−1 ) is again a quasi-uniform box of (x,u), where u−1 = {u−1 : u ∈ u} is also a quasiuniform base on ∏ α∈nx. moreover, u−1 ∨u = u s is a uniformity base on∏ α∈nx and the pair (∏ α∈nx,u s ) is a uniform box product of the uniform space (x,us) which corresponds to the uniform box product in the sense of bell (see [1, definition 3.2]). proposition 4.7. if (x,u) is a quasi-uniform space and (∏ α∈nx,u ) is its quasi-uniform box product, then u(x) = ∏ α∈n (u(x(α))) whenever u ∈u and x ∈ ∏ α∈n x. proof. consider u ∈ u and let y ∈ u(x). then (x,y) ∈ u if and only if (x(α),y(α)) ∈ u whenever α ∈ n if and only if y(α) ∈ u(x(α)) whenever α ∈ n if and only if y ∈ ∏ α∈n(u(x(α))). � c© agt, upv, 2017 appl. gen. topol. 18, no. 1 67 o. olela otafudu and h. sabao corollary 4.8. if (x,u) is a quasi-uniform space and (∏ α∈nx,u ) is its quasi-uniform box product, then u −1 (x) = ∏ α∈n (u−1(x(α))) = u−1(x) whenever u ∈u and x ∈ ∏ α∈n x. furthermore, u(x)∩u −1 (x) = ∏ α∈n (u(x(α)))∩ ∏ α∈n (u−1(x(α))) ⊇ ∏ α∈n (u(x(α))∩u−1(x(α))) whenever u ∈u and x ∈ ∏ α∈n x. example 4.9. let x be the fort space in example 2.2 that we equip with its pervin quasi-uniformity p. consider the quasi-uniform box product (∏ α∈n x,s ) of x. for any finite subset a ⊆ v , if x ∈ ∏ α∈n x, then sa(x) = { y ∈ ∏ α∈n x : (x(α),y(α)) ∈ sa whenever α ∈ n } = { y ∈ ∏ α∈n x : y(α),x(α) ∈ a or x(α) /∈ a and y(α) ∈ x whenever α ∈ n } . hence sa(x) = ∏ x(α)∈a a× ∏ x(α)/∈a x moreover, if x ∈ ∏ α∈n x, then sa −1(x) = { y ∈ ∏ α∈n x : (y(α),x(α)) ∈ sa whenever α ∈ n } = { y ∈ ∏ α∈n x : y(α),x(α) ∈ a or y(α) ∈ x\a and x(α) ∈ x whenever α ∈ n } . hence sa −1(x) = ∏ x(α)∈a a× ∏ x(α)/∈a (x \a). therefore sa(x) ∩sa−1(x) = ∏ x(α)∈a a× ∏ x(α)/∈a (x \a). observe that sa(x) ∩sa−1(x) ⊇ da(x) = ∏ x(α)∈a {x(α)}× ∏ x(α)/∈a (x \a), the basic closed and open set of bell (see [1, p. 2164]). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 68 on quasi-uniform box products example 4.10. consider the quasi-uniformity wf on the fort space, as given in example 2.4. let (∏ α∈n x,wf ) be the quasi-uniform box product of x. then whenever x ∈ ∏ α∈n x, we have wf (x) = ∏ x(α)∈f ( {x(α)}∪x \f ) × ∏ x(α)/∈f (x \f) and w−1f (x) = ∏ x(α)∈f {x(α)}× ∏ x(α)/∈f x. 5. properties of filter pairs in this section, we discuss some properties of filters on quasi-uniform box products. in particular, we prove some properties of filters on a quasi-uniform space that are preserved by filters on their quasi-uniform box products. we begin by considering one way of defining a filter on a quasi-uniform box product given any filter on its factor space. proposition 5.1. let (x,u) be a quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. if f is a filter on (x,u), then f defined by f = {∏ α∈n fα : fα ∈f and fα = x for all but finitely many α ∈ n } is a filter base on (∏ α∈n x,u ) . in a similar way, we can define a filter on the factor space from any filter on the quasi-uniform box product in the following way: proposition 5.2. let (x,u) be a quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. if f is a filter on (∏ α∈n x,u ) , then f, defined by f = { f : ∏ α∈n f ∈f } , is a filter on (x,u). suppose (x,u) is a quasi-uniform space and f and g are filters on x. then following [10], we say (f,g) is cauchy filter pair provided that for each u ∈u there is f ∈ f and g ∈ g such that f × g ⊆ u. a cauchy filter pair on a quasi-uniform space (x,u) is called constant provided that f = g. lemma 5.3. let (x,u) be a quasi-uniform space. if (f,g) is a cauchy filter pair on (∏ α∈n x,u ) , then the filter pair (f,g), where f = { f : ∏ α∈n f ∈f } and g = { g : ∏ α∈n g ∈g } , is a cauchy filter pair on (x,u). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 69 o. olela otafudu and h. sabao proof. consider the cauchy filter pair (f,g) on (∏ α∈n x,u ) . one sees that f and g are filters on x from proposition 5.2. we need to show that for any u ∈u, there are f ∈f and g ∈g such that f ×g ⊆ u. since (f,g) is a cauchy filter pair on (∏ α∈n x,u ) , it follows that for any u ∈ u, there exists f ∈ f and g ∈ g such that f × g ⊆ u. we choose f ∈ f and g ∈ g such that ∏ α∈n f ⊆ f and ∏ α∈n g ⊆ g. let (x(α),y(α)) ∈ f ×g for all α ∈ n. then (x(α))α∈n ∈ f and (y(α))α∈n ∈ g. thus ((x(α))α∈n, (y(α))α∈n) ∈ f ×g ⊆ u. this implies that ((x(α))α∈n, (y(α))α∈n) ∈ u. hence for all α ∈ n, (x(α),y(α)) ∈ u. therefore, f ×g ⊆ u. � a filter g on a quasi-uniform space (x,u) is said to be a d-cauchy filter if there is a filter f on x such that (f,g) is a cauchy filter pair. we call f a cofilter of g. lemma 5.4. let (x,u) be a quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. if g is a d-cauchy filter on (x,u), then the filter g, defined by g = { ∏ α∈n g : g ∈g}, is a d-cauchy filter on (∏ α∈n x,u ) . proof. suppose g = { ∏ α∈n g : g ∈g} where g is a d-cauchy filter on (x,u). then there exists a filter f on x such that (f,g) is a cauchy filter pair on (x,u). define f by f = { ∏ α∈n f : f ∈ f}. then we need to show that (f,g) is a cauchy filter pair on (∏ α∈n x,u ) . suppose f ∈ f and g ∈ g. let (x(α)α∈n, (y(α))α∈n) ∈ f ×g. then for all α ∈ n, (x(α),y(α)) ∈ f ×g, where f = ∏ α∈n f and g = ∏ α∈n g. since (f,g) is a cauchy filter pair, (x(α),y(α)) ∈ u for all α ∈ n. this implies that (x(α)α∈n, (y(α))α∈n) ∈ u and so f ×g ⊆ u. � 6. c-completeness and d-completness in quasi-uniform box products in this section, we present some notions of completeness in quasi-uniform spaces that are preserved by their quasi-uniform box products. in particular, we present the notion of c-completeness in the quasi-uniform box product of a quasi-uniform space and show the relationship between d-completeness and c-completeness in the quasi-uniform box product of a quiet quasi-uniform space. also, since the notion of pair completeness coincides with bicompleteness in quasi-uniform spaces, we show that the quasi-uniform box product of a d-complete quiet quasi-uniform space is bicomplete by showing that it is pair complete. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 70 on quasi-uniform box products we first consider the notion of quietness in the quasi-uniform box product of a quasi-uniform space. following [5], we say a quasi-uniform space (x,u) is quiet provided that for each u ∈ u, there is an entourage v ∈ u such that if f and g are filters on x and x and y are points of x such that v (x) ∈g and v −1(y) ∈f and (f,g) is a cauchy filter pair on (x,u), then (x,y) ∈ u. if v satisfies the above conditions, we say that v is quiet for u. theorem 6.1. let (x,u) be a quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. if (x,u) quiet, then (∏ α∈n x,u ) is quiet. proof. let u ∈u. suppose there exists v ∈u such that (f, g) is a cauchy filter pair on ( ∏ α∈n x,u) and (x(α))α∈n, (y(α))α∈n ∈ ∏ α∈n x satisfy v ((x(α))α∈n) ∈ g and v −1((y(α))α∈n) ∈ f. then (f,g), where f = {f : ∏ α∈n f ∈ f} and g = {g : ∏ α∈n g ∈g}, is a cauchy filter pair on (x,u) and v (x(α)) ∈g and v −1(y(α)) ∈ f whenever α ∈ n. since (x,u) is quiet, then (x(α),y(α)) ∈ u whenever α ∈ n. therefore, ((x(α))α∈n, (y(α))α∈n) ∈ u. � we now look at c-completeness and d-completeness in quasi-uniform box products. a quasi-uniform space (x,u) is called c-complete provided that each cauchy filter pair (f,g) converges. a quasi-uniform space (x,u) is dcomplete if each d-cauchy filter converges, that is, each second filter of the cauchy filter pair (f,g) converges with respect to τ(u). theorem 6.2. let (x,u) be a quiet quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. if (x,u) is c-complete, then (∏ α∈n x,u ) is c-complete. proof. suppose (f,g) is a cauchy filter pair on (∏ α∈n x,u ) . this implies that (f,g), where f = {f : ∏ α∈n f ∈ f} and g = {g : ∏ α∈n g ∈ g}, is a cauchy filter pair on (x,u). since (x,u) is c-complete, then f converges to x0 ∈ x with respect to τ(u−1). also, g converges to x0 (a constant sequence (x0,x0, · · ·)) with respect to τ(u). then for each u ∈u, there is f ∈f, such that f ⊆ u−1(x0). therefore,∏ α∈n f ⊆ u −1 ((x0)α∈n) = ∏ α∈n u−1(x0)α∈n ∈f. this implies f converges to (x0)α∈n with respect to τ(u −1 ). also, for each u ∈u, there is g ∈g, such that g ⊆ u(x0). therefore,∏ α∈n g ⊆ u((x0)α∈n) = ∏ α∈n u(x0)α∈n ∈g. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 71 o. olela otafudu and h. sabao this implies g converges to (x0)α∈n (a constant sequence (x0,x0, · · ·)) with respect to τ(u −1 ). therefore, ( ∏ α∈n x,u) is c-complete. � theorem 6.3. let (x,u) be a quiet quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. if (x,u) is d-complete, then (∏ α∈n x,u ) is d-complete. proof. suppose g is a d-cauchy filter on (∏ α∈n x,u ) . then there exists a filter f such that (f,g) is a cauchy filter pair on (∏ α∈n x,u ) . thus (f,g), where f = {f : ∏ α∈n f ∈ f} and g = {g : ∏ α∈n g ∈ g}, is a cauchy filter pair on (x,u). since (x,u) is d-complete, then g converges to x0 with respect to τ(u). then for each u ∈u, there is g ∈g, such that g ⊆ u(x0). therefore,∏ α∈n g ⊆ u((x0)α∈n) = ∏ α∈n u(x0) ∈g. this implies g converges to (x0)α∈n (a constant sequence (x0,x0, · · ·)) with respect to τ(u). therefore, (∏ α∈n x,u ) is d-complete. � remark 6.4. let (x,u) be a quasi-uniform space. it is not difficult to prove that (∏ α∈n x,u ) is bicomplete whenever (x,u) is bicomplete. we have seen from our previous results that if (f,g) is a cauchy filter pair on(∏ α∈n x,u ) , then f converges with respect to τ(u−1) and g converges with respect to τ(u). furthermore, one can use the argument that if (x,u) is bicomplete, then (x,us) is complete, therefore (∏ α∈n x,u s ) is complete as a uniform box product of (x,us). hence (∏ α∈n x,u ) is bicomplete. we now show the relationship between c-completeness and d-completeness in the quasi-uniform box product of a uniformly regular quasi-uniform space. following [10], we say quasi-uniform space (x,u) is uniformly regular if for any u ∈u, there is v ∈u such that clτ(u)v (x) ⊆ u(x) whenever x ∈ x. lemma 6.5. let (x,u) be a quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. if (x,u) is uniformly regular, then (∏ α∈n x,u ) is uniformly regular. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 72 on quasi-uniform box products proof. suppose that (x,u) is uniformly regular. then for any u ∈ u, there exists v ∈u such that (6.1) clτ(u)v (t) ⊆ u(t) whenever t ∈ x. we need to prove clτ(u)v (x) ⊆ u(x) whenever x ∈ ∏ α∈n x. let y ∈ clτ(u)v (x). then there exists w ∈ u such that w((y(α))α∈n) ∩ v ((x(α))α∈n) 6= ∅. this implies w(y(α)) ∩ u(x(α)) 6= ∅ whenever α ∈ n. hence y(α) ∈ clτ(u)v (x(α)) whenever α ∈ n. furthermore by (6.1), it follows that y(α) ∈ u(x(α)) whenever α ∈ n. therefore, (y(α))α∈n ∈ u((x(α))α∈n) and this implies that clτ(u)v (x) ⊆ u(x). � corollary 6.6. let (x,u) be a d-complete uniformly regular quiet quasiuniform space and (∏ α∈n x,u ) be its quasi-uniform box product. then(∏ α∈n x,u −1 ) is d-complete. proof. since (x,u) is d-complete and uniformly regular, then by theorem 6.3 and lemma 6.5, (∏ α∈n x,u ) is d-complete and uniformly regular. therefore, by [6, lemma 2.1], (∏ α∈n x,u −1 ) is d-complete. � we recall that a quasi-uniform space is said to be pair complete provided that whenever (f,g) is a cauchy filter pair, there exists a point p ∈ x such that the filter g −→ τ(u) p and f −→ τ(u−1) p (see [6]). corollary 6.7. let (x,u) be a d-complete uniformly regular quasi-uniform space and (∏ α∈n x,u ) be its quasi-uniform box product. then (∏ α∈n x,u ) is pair complete. proof. from corollary 6.6, we see that (∏ α∈n x,u ) is d-complete and uniformly regular. then by [6, proposition 2.2], (∏ α∈n x,u ) is pair complete. � corollary 6.8. let (x,u) be a d-complete quiet quasi-uniform space and(∏ α∈n x,u ) be its quasi-uniform box product. then (∏ α∈n x,u ) is ccomplete. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 73 o. olela otafudu and h. sabao proof. since (x,u) is quiet and d-complete, then by theorems 6.1 and 6.3,(∏ α∈n x,u ) is quiet and d-complete. therefore, by proposition [11, proposition 3.3.2], (∏ α∈n x,u ) is c-complete. � acknowledgements. the authors would like to thank the referee for several suggestions that have clearly improved the presentation of this paper. references [1] j. r. bell, the uniform box product, proc. amer. math. soc. 142 (2014), 2161–2171. [2] j. r. bell, an infinite game with topological consequences, topology appl. 175 (2014), 1–14. [3] j. r. bell, the uniform box product problem, phd thesis, suny at buffalo, 2010. [4] d. doitchinov, on completeness of quasi-uniform spaces, c.r. acad. bulgare sci. 41, no. 4 (1988), 5–8. [5] p. fletcher and w. hunsaker, a note on totally bounded quasi-uniformities, serdica math. j. 24 (1998), 95–98. [6] p. fletcher and w. hunsaker, completeness using pairs of filters, topology appl. 44 (1992), 149–155. [7] g. gruenhage, infinite games and generalizations of first-countable spaces, general topology and appl. 6 (1976), 339–352. [8] j. hankins, the uniform box product of some spaces with one non-isolated point, phd thesis, university of south carolina, 2012. [9] h.-p. künzi and c. makitu kivuvu, a double completion for an arbitrary t0-quasimetric space, j. log. algebr. program. 76 (2008), 251–269. [10] h.-p. a. künzi, an introduction to quasi-uniform spaces. contemp. math. 486 (2009), 239–304. [11] c. m. kivuvu, on doitchinov’s quietness for arbitrary quasi-uniform spaces, phd thesis, university of cape town, 2010. [12] r. stoltenberg, some properties of quas-uniform spaces, proc. london math. soc. 17 (1967), 226–240 [13] s. w. williams, box products, handbook of set-theoretic topology, 69–200, northholland, amsterdam, 1984. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 74 () @ appl. gen. topol. 17, no. 1(2016), 1-5doi:10.4995/agt.2016.4616 c© agt, upv, 2016 the equivalence of two definitions of sequential pseudocompactness paolo lipparini a dipartimento di matematica, viale della ricerca scient̀ıfica, ii università gelmina di roma (tor vergata), i-00133 rome, italy (lipparin@axp.mat.uniroma2.it) abstract we show that two possible definitions of sequential pseudocompactness are equivalent, and point out some consequences. 2010 msc: primary 54d20; secondary 54b10. keywords: sequentially pseudocompact; feebly compact; topological space. 1. the equivalence according to artico, marconi, pelant, rotter and tkachenko [1, definition 1.8], a tychonoff topological space x is sequentially pseudocompact if the following condition holds. (1) for any family (on)n∈ω of pairwise disjoint nonempty open sets of x, there are an infinite set j ⊆ ω and a point x ∈ x such that every neighborhood of x intersects all but finitely many elements of (on)n∈j. notice that in [1] x is assumed to be a tychonoff space, but the above definition makes sense for an arbitrary topological space. according to dow, porter, stephenson, and woods [2, definition 1.4], a topological space is sequentially feebly compact if the following condition holds. (2) for any sequence (on)n∈ω of nonempty open subsets of x, there are an infinite set j ⊆ ω and a point x ∈ x such that every neighborhood of x intersects all but finitely many elements of (on)n∈j. (the difference is that in condition (1) the on’s are assumed to be pairwise disjoint, while they are arbitrary in condition (2)) received 10 october 2013 – accepted 11 september 2015 http://dx.doi.org/10.4995/agt.2016.4616 p. lipparini the above two notions have been rather thoroughly studied by the mentioned authors. in this note we show their equivalence. putting together the results from [1] and [2] shows that the class of sequentially pseudocompact tychonoff topological spaces is closed under (possibly infinite) products and contains significant classes of pseudocompact spaces. unless otherwise specified, we shall assume no separation axiom. theorem 1.1. for every topological space x, conditions (1) and (2) above are equivalent. proof. condition (2) trivially implies condition (1). for the converse, suppose that x satisfies condition (1), and let (on)n∈ω be a sequence of nonempty open sets of x. suppose by contradiction that (*) for every infinite set j ⊆ ω and every point x ∈ x there is some neighborhood u(j,x) of x such that n(j,x) = {n ∈ j | u(j,x)∩on = ∅} is infinite. without loss of generality, we can assume that u(j,x) is open. we shall construct by simultaneous induction a sequence (mi)i∈ω of distinct natural numbers, a sequence (ji)i∈ω of infinite subsets of ω, and a sequence of pairwise disjoint nonempty open sets (ui)i∈ω such that (a) ui ⊆ omi for every i ∈ ω, (b) ui ∩ on = ∅, for every i ∈ ω, and n ∈ ji, and (c) ji ⊇ jh, whenever i ≤ h ∈ ω. put m0 = 0 and pick x0 ∈ o0 (this is possible, since o0 in nonempty). apply (*) with j = ω and x = x0, and let u0 = u(ω,x0) ∩ o0 ⊆ o0 = om0 and j0 = n(ω,x0). u0 is nonempty, since x0 ∈ u(ω,x0)∩o0. by (*), j0 is infinite, and clause (b) is satisfied for i = 0. the basis of the induction is completed. suppose now that 0 6= i ∈ ω, and that we have constructed finite sequences (mk)k β. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 4 the equivalence of two definitions of sequential pseudocompactness if we modify the above definition by further requesting that the (oβ)’s are pairwise disjoint, we say that x is d-sequentially α-feebly compact. clearly, for every α, sequential α-feeble compactness implies d-sequential α-feeble compactness, and, for α = ω, both notions are equivalent (and equivalent to sequential feeble compactness), by theorem 1.1. acknowledgements. we wish to express our gratitude to x. caicedo and s. garćıa-ferreira for stimulating discussions and correspondence. we thank our students from tor vergata university for stimulating questions. references [1] g. artico, u. marconi, j. pelant, l. rotter and m. tkachenko, selections and suborderability, fund. math. 175 (2002), 1-33. [2] a. dow, j. r. porter, r. m. stephenson, jr. and r. g. woods, spaces whose pseudocompact subspaces are closed subsets, appl. gen. topol. 5 (2004), 243-264. [3] p. lipparini, products of sequentially pseudocompact spaces, arxiv:1201.4832 (2012). c© agt, upv, 2016 appl. gen. topol. 17, no. 1 5 @ appl. gen. topol. 18, no. 1 (2017), 91-105 doi:10.4995/agt.2017.6322 c© agt, upv, 2017 fixed point theorems for simulation functions in b-metric spaces via the wt-distance chirasak mongkolkehaa, yeol je chob,∗ and poom kumamc,d,∗ a department of mathematics statistics and computer sciences, faculty of liberal arts and science, kasetsart university, kamphaeng-saen campus, nakhonpathom 73140, thailand (faascsm@ku.ac.th) b department of mathematics education and the rins, gyeongsang national university, chinju 660-701, korea, center for general education, china medical university taichung, 40402, taiwan (yjcho@gnu.ac.kr) c kmuttfixed point research laboratory, department of mathematics, room scl 802 fixed point laboratory, science laboratory building, faculty of science, king mongkut university of technology thonburi (kmutt), 126 pracha-uthit road, bang mod, thrung khru, bangkok 10140, thailand. (poom.kum@kmutt.ac.th) d kmutt-fixed point theory and applications research group (kmutt-fpta), theoretical and computational science center (tacs), science laboratory building, faculty of science, king mongkut university of technology thonburi (kmutt), 126 pracha-uthit road, bang mod, thrung khru, bangkok 10140, thailand communicated by s. romaguera abstract the purpose of this article is to prove some fixed point theorems for simulation functions in complete b−metric spaces with partially ordered by using wt-distance which introduced by hussain et al. [12]. also, we give some examples to illustrate our main results. 2010 msc: 47h09; 47h10; 54h25. keywords: fixed point; simulation function; b-metric space; wt-distance; w-distance; generalized distance. received 02 july 2016 – accepted 06 december 2016 http://dx.doi.org/10.4995/agt.2017.6322 c. mongkolkeha, y. j. cho and p. kumam 1. introduction since banach’s fixed point theorem (or banach’s contraction principle) proved by banach [4] in 1922, many authors have extended, improved and generalized in several ways. in 2015, khojasteh et al. [15] introduced the notion of a simulation function to generalize banach’s contraction principle. recently, roldán-lópez-dehierroet et al. [18] modified the notion of a simulation function and showed the existence and uniqueness of coincidence points of two nonlinear mappings using the concept of a simulation function. on the other hand, in 1989, bakhtin [3] (see also czerwik [8]) introduced the concept of a b-metric space (or a space of metric type) and proved some fixed point theorems for some contractive mappings in b-metric spaces which are generalizations of banach’s contraction principle in metric spaces. in 1996, kada et al. [14] introduced some generalized metric, which is called the w-distance and gave some examples of w-distance and, using the w-distance, they also improved caristi’s fixed point theorem, ekeland’s variational principle and the nonconvex minimization theorem of takahashi [20]. later, shioji et al. [19] studied the relationship between weakly contractive mappings and weakly kannan mappings under the conditions, the w-distance and the symmetric wdistance. in 2012, imdad and rouzkard [13] proved some fixed point theorems in a complete metric space equipped with a partial ordering via the w-distance. recently, hussain et al. [12] introduced the concept of the wt-distance in generalized b-metric spaces, which is a generalization of the w-distance, and also proved some fixed point theorems in a partially ordered b-metric space by using the wt-distance. also, abdou et al. [1] proved some common fixed point theorems in menger probabilistic metric type spaces by using the wt-distance. in this paper, we consider some simulation functions to show the existence of fixed points of some nonlinear mappings in complete b-metric spaces via the wt-distance. furthermore, we also give some examples to illustrate the main results. our result improve, extend and generalize several results given by some authors in literatures. 2. preliminaries and generalized distances now, we give some definitions and their examples definition 2.1. let (x,≤) be a partially ordered set.the elements x,y ∈ x are said to be comparable with respect to the order ≤ if either x ≤ y or y ≤ x. let us denote x≤ by the subset of x ×x defined by x≤ = {(x,y) ∈ x ×x : x ≤ y or y ≤ x}. definition 2.2. let (x,≤) be a partially ordered set and f : x → x be a self-mapping of x. we say that (1) f is inverse increasing if, for all x,y ∈ x, f(x) ≤ f(y) implies x ≤ y; (2) f is nondecreasing if, for all x,y ∈ x, x ≤ y implies f(x) ≤ f(y). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 92 fixed point theorems for simulation functions in b-metric spaces via the wt-distance definition 2.3. let (x,≤) be a partially ordered set and t : x → x be a self-mapping of x. then (1) f(t) = {x ∈ x : t(x) = x}, i.e., f(t) denotes the set of all fixed points of t ; (2) t is called a picard operator (briefly, po) if there exists x∗ ∈ x such that f(t) = {x∗} and {tn(x)} converges to x∗ for all x ∈ x; (3) t is said to be orbitally u-continuous for any u ⊂ x × x if, for any x ∈ x, tni (x) → a ∈ x as i → ∞ and (tni (x),a) ∈ u for any i ∈ n imply that tni+1(x) → ta ∈ x as i →∞; (4) t is said to be orbitally continuous on x if x ∈ x and tni (x) → a ∈ x as i →∞ imply that tni+1(x) → t(a) ∈ x as i →∞. definition 2.4. let (x,d) be a metric space. a function p : x ×x → [0,∞) is said to be the w-distance on x if the following are satisfied: (1) p(x,z) ≤ p(x,y) + p(y,z) for all x,y,z ∈ x; (2) for any x ∈ x, p(x, ·) : x → [0,∞) is lower semi-continuous (i.e., if x ∈ x and yn → y ∈ x, then p(x,y) ≤ lim infn→∞p(x,yn); (3) for any ε > 0, there exists δ > 0 such that p(z,x) ≤ δ and p(z,y) ≤ δ imply d(x,y) ≤ ε. let x be a metric space with a metric d. a w-distance p on x is said to be symmetric if p(x,y) = p(y,x) for all x,y ∈ x. obviously, every metric is the w-distance, but not conversely. next, we recall some examples in [21] to show that the w-distance is a generalized metric. example 2.5. let (x,d) be a metric space. a function p : x ×x → [0,∞) defined by p(x,y) = c for all x,y ∈ x is a w-distance on x, where c is a positive real number. but p is not a metric since p(x,x) = c 6= 0 for any x ∈ x. example 2.6. let (x,‖·‖) be a normed linear space. a function p : x×x → [0,∞) defined by p(x,y) = ‖x‖ + ‖y‖ for all x,y ∈ x is a w-distance on x. example 2.7. let f be a bounded and closed subset of a metric spaces x. assume that f contain at least two points and c is a constant with c ≥ δ(f), where δ(f) is the diameter of f. then a function p : x ×x → [0,∞) defined by p(x,y) = { d(x,y), if x,y ∈ f, c, if x /∈ f or y /∈ f, is a w-distance on x. definition 2.8. let x be a nonempty set and s ≥ 1 be a given real number. a functional d : x ×x → [0,∞) is called a b-metric if, for all x,y,z ∈ x, the following conditions are satisfied: (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x); (3) d(x,z) ≤ s[d(x,y) + d(y,z)]. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 93 c. mongkolkeha, y. j. cho and p. kumam a pair (x,d) is called a b-metric space with coefficient s. in definition 2.8, every metric space is a b-metric space with s = 1 and hence the class of b-metric spaces is larger than the class of metric spaces. some examples of b-metric spaces are given by berinde [5], czerwik [9], heinonen [11] and, further, some examples to show that every b-metric space is a real generalization of metric spaces are as follows: example 2.9. the set r of real numbers together with the functional d : r×r → [0,∞) defined by d(x,y) := |x−y|2 for all x,y ∈ r is a b-metric space with coefficient s = 2. however, we know that d is not a metric on x since the ordinary triangle inequality is not satisfied. indeed, d(3, 5) > d(3, 4) + d(4, 5). in 2014, hussain et al. [12] introduced the concept of the wt-distance as follow: definition 2.10. let (x,d) be a b-metric space with constant k ≥ 1. a function p : x × x → [0,∞) is called the wt-distance on x if the following are satisfied: (1) p(x,z) ≤ k(p(x,y) + p(y,z)) for all x,y,z ∈ x; (2) for any x ∈ x, p(x, ·) : x → [0,∞) is k-lower semi-continuous (i.e., if x ∈ x and yn → y ∈ x, then p(x,y) ≤ lim infn→∞kp(x,yn); (3) for any ε > 0, there exists δ > 0 such that p(z,x) ≤ δ and p(z,y) ≤ δ imply d(x,y) ≤ ε. example 2.11 ([12]). let (x,d) be a b-metric space. then the metric d is a wt-distance on x. example 2.12 ([12]). let x = r and d1 = (x−y)2. a function p : x×x → [0,∞) defined by p(x,y) = ‖x‖2 +‖y‖2 for all x,y ∈ x is a wt-distance on x. example 2.13 ([12]). let x = r and d1 = (x−y)2. a function p : x×x → [0,∞) defined by p(x,y) = ‖y‖2 for all x,y ∈ x is a wt-distance on x. the following two lemmas are crucial for our resuts. lemma 2.14 ([12]). let (x,d) be a b-metric space with constant k ≥ 1 and p be a wt-distance on x. let {xn}, {yn} be two sequences in x and {αn}, {βn} two sequences in [0,∞) converging to zero. then the following conditions hold: for all x,y,z ∈ x, (1) if p(xn,y) ≤ αn and p(xn,z) ≤ βn for all n ∈ n, then y = z. in particular, if p(x,y) = 0 and p(x,z) = 0, then y = z; (2) if p(xn,yn) ≤ αn and p(xn,z) ≤ βn for all n ∈ n, then {yn} converges to z; c© agt, upv, 2017 appl. gen. topol. 18, no. 1 94 fixed point theorems for simulation functions in b-metric spaces via the wt-distance (3) if p(xn,xm) ≤ αn for all n,m ∈ n with m > n, then {xn} is a cauchy sequence; (4) p(y,xn) ≤ αn for all n ∈ n, then {xn} is a cauchy sequence. 3. the classes of simulation functions in 2015, khojasteh et al. [15] introduced the notion of a simulation function which generalizes the banach contraction as follow: definition 3.1 ([15]). a simulation function is a mapping ζ : [0,∞)×[0,∞) → r satisfying the following conditions: (ζ1) ζ(0, 0) = 0; (ζ2) ζ(t,s) < s− t for all s,t > 0; (ζ3) if {tn} and {sn} are two sequences in (0,∞) such that limn→∞ tn = limn→∞sn > 0, then lim sup n→∞ ζ(tn,sn) < 0. now, we recall some examples of the simulation function given by khojasteh et al. [15]. example 3.2. let ζi : [0,∞) × [0,∞) → r for i = 1, 2, 3 be defined by (1) ζ1(t,s) = ψ(s) −φ(t) for all t,s ∈ [0,∞), where φ,ψ : [0,∞) → [0,∞) are two continuous functions such that ψ(t) = φ(t) = 0 if and only if t = 0 and ψ(t) < t ≤ φ(t) for all t > 0; (2) ζ2(t,s) = s− f(t,s) g(t,s) t for all t,s ∈ [0,∞), where f,g : [0,∞)× [0,∞) → (0,∞) are two continuous functions with respect to each variable such that f(t,s) > g(t,s) for all t,s > 0. (3) ζ3(t,s) = s−ϕ(s) − t for all t,s ∈ [0,∞), where ϕ : [0,∞) → [0,∞) is a continuous function such that ϕ(t) = 0 if and only if t = 0 then ζi for i = 1, 2, 3 are a simulation function. recently, roldán-lópez-de-hierro et al. [18] modified the notion of a simulation function as follow: definition 3.3 ([18]). a simulation function is a mapping â ζ : [0,∞) × [0,∞) → r satisfying the following conditions: (ζ1) ζ(0, 0) = 0; (ζ2) ζ(t,s) < s− t for all s,t > 0; (ζ3) if {tn} and {sn} are two sequences in (0,∞) such that limn→∞ tn = limn→∞sn > 0 and tn < sn for all n ∈ n, then lim sup n→∞ ζ(tn,sn) < 0. note that the classes of all simulation functions ζ : [0,∞) × [0,∞) → r denote by z and every simulation function in the original sense of khojasteh et al. [15] is also a simulation function in the sense of roldán-lópez-de-hierroet et al. [18], but the converse is not true as in the following example. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 95 c. mongkolkeha, y. j. cho and p. kumam example 3.4 ([18]). let k ∈ r be such that k < 1 and let ζ ∈ z be the function defined by ζ(t,s) = { 2s− 2t, if s < t, ks− t, otherwise. then ζ is a simulation function in the sense of definition 3.3, but ζ does not satisfy the condition (ζ3) of definition 3.1. definition 3.5. let (x,d) is a complete metric space. a mapping t : x → x is called z-contraction if there exists ζ ∈z such that (3.1) ζ(d(tx,ty),d(x,y)) ≥ 0 for all x,y ∈ x. remark 3.6. if we take ζ(t,s) = λs − t for all s,t ≥ 0, where λ ∈ [0, 1) in definition 3.5, then the z-contraction become to the banach contraction. 4. fixed point theorems for simulation functions in this section, we consider the concept of a simulation function and show the existence of a fixed point for such mapping in complete b-metric spaces via the wt-distance. first, we improve the notion of a simulation function for our considerations as follow: definition 4.1. let k be a given real number such that k ≥ 1. a simulation function is a mapping ζ : [0,∞)×[0,∞) → r satisfying the following conditions: (ζ1) ζ(0, 0) = 0; (ζ2) ζ(kt,s) < s−kt for all s,t > 0; (ζ3) if {tn} and {sn} are two sequences in (0,∞) such that lim supn→∞ktn = lim supn→∞sn > 0 and tn < sn for all n ∈ n, then lim sup n→∞ ζ(ktn,sn) < 0. example 4.2. let λ,k ∈ r be such that λ < 1 and k ≥ 1. define the mapping â ζ : [0,∞) × [0,∞) → r by ζ(kt,s) =   s−kt, if s < t, λs−kt ks + 1 , otherwise. clearly, ζ verifies (ζ1), and ζ satisfies (ζ2). indeed, s,t > 0,   0 < s < t ⇒ ζ(kt,s) = s−kt, 0 < t < s, ⇒ ζ(kt,s) = λs−kt ks + 1 < s−kt ks + 1 < s−kt. next, we will show that ζ satisfies (ζ3). if {tn} and {sn} are sequences in (0,∞) such that lim supn→∞ktn = lim supn→∞sn > 0 and tn < sn for all n ∈ n. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 96 fixed point theorems for simulation functions in b-metric spaces via the wt-distance then lim sup n→∞ ζ(ktn,sn) = lim sup n→∞ ( λsn −ktn ksn + 1 ) < lim sup n→∞ ( sn −ktn ktn + 1 ) < lim sup n→∞ ( sn −ktn ktn ) < lim sup n→∞ ( sn ktn − ktn ktn ) ≤ lim sup n→∞ ( sn ktn ) − lim inf n→∞ (1) ≤ 1 − 1 = 0. then ζ is a simulation function in the sense of definition 4.1, but ζ does not satisfy the condition (ζ3) of definition 3.1. indeed, if we take k = 1, tn = 2 √ 2 and sn = 2 √ 2 − 1 n , for all n ∈ n. then, sn < tn lim sup n→∞ ζ(tn,sn) = lim sup n→∞ ( 2 √ 2 − 1 n − 2 √ 2 ) = lim sup n→∞ ( − 1 n ) = 0. theorem 4.3. let (x,≤) be a partially ordered set, (x,d) be a complete b−metric space with constant k ≥ 1 and p be a wt-distance on x. suppose that t : x → x is a nondecreasing mapping satisfying the following conditions: (i) there exists ζ ∈z such that (4.1) ζ(kp(tx,t2x),p(x,tx)) ≥ 0 for all (x,tx) ∈ x≤; (ii) for all x ∈ x with (x,tx) ∈ x≤, inf{p(x,y) + p(x,tx)} > 0 for all y ∈ x with y 6= ty; (iii) there exists x0 ∈ x such that (x0,tx0) ∈ x≤. then t has a fixed point in x. moreover, if tx = x, then p(x,x) = 0. proof. if tx0 = x0, then we are done. suppose that the conclusion is not true. then there exists x0 ∈ x such that (x0,tx0) ∈ x≤. since t is nondecreasing, we have (tx0,t 2x0) ∈ x≤. continuing this process, we obtain (tnx0,tmx0) ∈ x≤ for all n,m ∈ n. now, we claim that (4.2) lim n→∞ p(tnx0,t n+1x0) = 0. by the assumption (i) and the property of ζ, we observe that (4.3) 0 ≤ ζ(kp(tnx0,tn+1x0),p(tn−1x0,tnx0)) ≤ p(tn−1x0,tnx0) −kp(tnx0,tn+1x0) for all n ∈ n. since k ≥ 1 and using (4.3), we get (4.4) p(tnx0,t n+1x0) ≤ kp(tnx0,tn+1x0) ≤ p(tn−1x0,tnx0). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 97 c. mongkolkeha, y. j. cho and p. kumam this mean that the sequence {p(tnx0,tn+1x0)} is a decreasing sequence of nonnegative real numbers and so it is convergent to some r ≥ 0. suppose that r > 0. case i. if k > 1, letting n → ∞ in (4.4), we get r ≤ kr ≤ r which is a contradiction. case ii. if k = 1, putting tn = p(t n+1x0,t n+2x0) and sn = p(t nx0,t n+1x0), the sequences {ktn} and {sn} have the same positive limit. also, the sequences {ktn} and {sn} have the same positive limit superior and verify that tn < sn for all n ∈ n. by the condition (ζ3) of definition 4.1 we have lim sup n→∞ ζ(kp(tn+1x0,t n+2x0),p(t nx0,t n+1x0)) = lim sup n→∞ ζ(ktn,sn) < 0, which is a contradiction. therefore r = 0, that is, the claim (4.3) holds. next, we show that (4.5) lim m,n→∞ p(tnx0,t mx0) = 0. suppose that this is not true. then we can find ε0 > 0 with the sequences {mk}, {nk} such that, for any mk > nk such that (4.6) p(tnkx0,t mkx0) > ε0 for all k ∈{1, 2, 3, · · ·}. we can assume that mk is a minimum index such that (4.6) holds. then we also have (4.7) p(tnkx0,t mk−1x0) ≤ ε0. hence we have ε0 < p(t nkx0,t mkx0) ≤ k[p(tnkx0,tmk−1x0) + p(tmk−1x0,tmkx0)] < kε0 + kp(t mk−1x0,t mkx0). taking limit superior as k → ∞ in the above inequality and using (4.2), we have (4.8) ε0 < lim sup k→∞ p(tnkx0,t mkx0) ≤ kε0. now, we claim that lim sup n→∞ p(tnk+1x0,t mk+1x0) < ε0. if lim sup k→∞ p(tnk+1x0,t mk+1x0) ≥ ε0, then there exists {kr} and δ > 0 such that (4.9) lim sup r→∞ p(tnkr +1x0,t mkr +1x0) = δ ≥ ε0. by the assumption (i) and the property of ζ, we have (4.10) 0 ≤ ζ(kp(tnkr +1x0,tmkr +1x0),p(tnkr x0,tmkr x0)) ≤ p(tnkr x0,tmkr x0) −kp(tnkr +1x0,tmkr +1x0). hence, (4.11) kp(tnkr +1x0,t mkr +1x0) ≤ p(tnkr x0,tmkr x0), c© agt, upv, 2017 appl. gen. topol. 18, no. 1 98 fixed point theorems for simulation functions in b-metric spaces via the wt-distance it follows from (4.8), (4.9) and (4.11), we get that kδ = lim sup r→∞ kp(tnkr +1x0,t mkr +1x0) ≤ lim sup r→∞ p(tnkr x0,t mkr x0) ≤ kε0 ≤ kδ. therefore the sequence {ktkr := kp(tnkr +1x0,tmkr +1x0)} and {skr := p(tnkr x0,tmkr x0)} have the same positive limit superior and verify that tkr < skr for all r ∈ n. by the property (ζ3), we conclude that 0 ≤ lim sup r→∞ ζ(kp(tnkr +1x0,t mkr +1x0),p(t nkr x0,t mkr x0)) = lim sup r→∞ ζ(ktkr,skr ) < 0, which is a contradiction and hence (4.5) hold. it follows from lemma 2.14 (iii) that {tnx0} is a cauchy sequence. since x is a complete b−metric space, the sequence {tnx0} converges to some element z ∈ x. from the fact that limm,n→∞p(t nx0,t mx0) = 0, for each ε > 0, there exists nε ∈ n such that n > nε implies p(tnεx0,t nx0) < ε. since p(x, ·) is k-lower semi-continuous and the sequence {tnx0} converges to z, we have (4.12) p(tnεx0,z) ≤ lim inf n→∞ kp(tnεx0,t nx0) ≤ kε. setting ε = 1 k2 and nε = nk, by (4.12), we have (4.13) lim k→∞ p(tnkx0,z) = 0. now, we prove that z is a fixed point of t. suppose that tz 6= z. since (tnkx0,t nk+1x0) ∈ x≤ for each n ∈ n, using the assumption (ii), (4.2) and (4.13), we have 0 < inf{p(tnkx0,z) + p(tnkx0,tnk+1x0)}→ 0 as n →∞, which is a contradiction. therefore, tz = z. if tx = x, we distinguish two cases. case i if k = 1, then 0 ≤ ζ(p(tx,t2x),p(x,tx)) = ζ(p(x,x),p(x,x)) ≤ p(x,x) −p(x,x) = 0. hence ζ(p(tx,t2x),p(x,tx)) = 0 and so, by (ζ1), we obtain p(x,x) = 0. case ii if k > 1, then 0 ≤ ζ(kp(tx,t2x),p(x,tx)) = ζ(kp(x,x),p(x,x)) ≤ p(x,x) −kp(x,x) = (1 −k)p(x,x), it follow that p(x,x) ≤ 0 and thus we must have p(x,x) = 0. this completes the proof. � now, we give an example to illustrate theorem 4.3. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 99 c. mongkolkeha, y. j. cho and p. kumam example 4.4. let x = [0, 1] and d(x,y) = (x−y)2 with the wt-distance p on x defined by p(x,y) = |y|2. we consider the following set: x≤ = { (x,y) ∈ x ×x : x = y or x,y ∈{0}∪{ 1 2n : n ≥ 1} } with the usual ordering. let t : x → x be a mapping defined by t(x) =   12n+1 , if x = 1 2n , n ≥ 1, 0, otherwise. for all x ∈ x. obviously, t is nondecreasing. also, t satisfies the condition (ii). indeed, for any n ∈ n, we have 1 2n 6= t( 1 2n ). moreover, for each n ∈ n, we have inf { p ( 1 2m , 1 2n ) + p ( 1 2m , 1 2m − 1 22m+1 ) : m ∈ n } = 1 22n > 0. let ζ : [0,∞) × [0,∞) → r define by ζ(t,s) = s−kt 1 + ks for all s,t ∈ [0,∞). similarly, in example 4.2, the function define as above is simulation function in the sense of definition 4.1. now, we show that t satisfies the condition (i). let given x = 1 2n with ( 1 2n ,t( 1 2n )) ∈ x≤. then we have ζ(2p(tx,t2x),p(x,tx)) = ζ(2p( 1 2n+1 , 1 2n+2 ),p( 1 2n , 1 2n+1 )) = ζ(2 1 22n+4 , 1 22n+2 ) = 1 22n+2 − 2 · 1 22n+4 1 + 2 · 1 22n+2 = 22n+3 − 22n+2 (22n+2)(22n+3) · 22n+1 22n+1 + 1 = 22n+2(2 − 1) (22n+4)(22n+1 + 1) = 22n+2 (22n+4)(22n+1 + 1) > 0. therefore, all the hypothesis of theorem 4.3 are satisfied and, further, x = 0 is a fixed point of t . c© agt, upv, 2017 appl. gen. topol. 18, no. 1 100 fixed point theorems for simulation functions in b-metric spaces via the wt-distance corollary 4.5. let (x,≤) be a partially ordered set and (x,d) be a complete metric type space with constant k ≥ 1 and p be a wt-distance on x. suppose that t : x → x is a nondecreasing mapping satisfying the following conditions: (i) there exists α ∈ [0, 1 k ) such that p(tx,t2x) ≤ αp(x,tx) for all x ≤ tx; (ii) for all x ∈ x with x ≤ tx, inf{p(x,y) + p(x,tx)} > 0 for all y ∈ x with y 6= ty; (iii) there exists x0 ∈ x such that x0 ≤ tx0. then t has a fixed point in x. theorem 4.6. let (x,≤) be a partially ordered set and (x,d) be a complete b-metric space with constant k ≥ 1 and p be a wt-distance on x. suppose that t : x → x is a nondecreasing mapping and there exists ζ ∈z such that ζ(kp(tx,t2x),p(x,tx)) ≥ 0 for all (x,tx) ∈ x≤. assume that one of the following conditions holds: (i) for all x ∈ x with (x,tx) ∈ x≤, inf{p(x,y) + p(x,tx)} > 0 for all y ∈ x with y 6= ty; (ii) if both {xn} and {txn} converge to z, then z = tz; (iii) t is continuous on x. if there exists x0 ∈ x such that (x0,tx0) ∈ x≤, then t has a fixed point in x. moreover, if tx = x, then p(x,x) = 0. proof. in the case of t satisfying the condition (i), the conclusion was proved in theorem 4.3. let us prove that (ii) =⇒ (i). suppose that the condition (ii) holds. let y ∈ x with y 6= ty such that inf{p(x,y) + p(x,tx) : (x,tx) ∈ x≤} = 0. then we can find a sequence {zn} such that (zn,tzn) ∈ x≤ and inf{p(zn,y) + p(zn,tzn)} = 0. so we have lim n→∞ p(zn,y) = lim n→∞ p(zn,tzn) = 0. again, by lemma 2.14, we have limn→∞tzn = y. moreover, limn→∞t 2zn = y. in fact, since (4.14) 0 ≤ ζ(kp(tzn,t2zn),p(zn,tzn)) ≤ p(zn,tzn) −kp(tzn,t2zn), it follow from (4.14) and k ≥ 1, we get that lim n→∞ p(tzn,t 2zn) ≤ lim n→∞ kp(tzn,t 2zn) ≤ lim n→∞ p(zn,tzn) = 0. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 101 c. mongkolkeha, y. j. cho and p. kumam letting xn = tzn, the sequences {xn} and {txn} converge to y. hence, by the assumption (ii), y = ty and so (ii) =⇒ (i). obviously, (iii) =⇒ (ii). this completes the proof. � now, we prove new theorems by replacing some conditions in theorem 4.3 with other conditions. theorem 4.7. let (x,≤) be a partially ordered set and (x,d) be a complete b-metric space with constant k ≥ 1 and p be a wt-distance on x. suppose that t : x → x is a nondecreasing satisfying the following conditions: (i) there exists ζ ∈z such that ζ(kp(tx,t2x),p(x,tx)) ≥ 0 for all (x,tx) ∈ x≤; (ii) there exists x0 ∈ x such that (x0,tx0) ∈ x≤, (iii) either t is orbitally continuous at x0 or (iv) t is orbitally x≤-continuous and there exists a subsequence {tnkx0} of {tnx0} converges to some element x? ∈ x such that (tnkx0,x?) ∈ x≤ for any k ∈ n. then t has a fixed point in x. moreover if tx = x, then p(x,x) = 0. proof. if tx0 = x0, then we are done. suppose that the conclusion is not true. then there exists x0 ∈ x such that (x0,tx0) ∈ x≤. since t is monotone, we have (tx0,t 2x0) ∈ x≤. continuing this process, we have a sequence {tnx0} such that (tnx0,t mx0) ∈ x≤ for any n,m ∈ n. as in the same argument in theorem 4.3, we can see that (4.15) lim n→∞ p(tnx0,t n+1x0) = 0. moreover, (4.16) lim m,n→∞ p(tnx0,t mx0) = 0. and {tnx0} is a cauchy sequence converges to some element z ∈ x. next, we prove that z is a fixed point of t. if the condition (iii) holds, then tn+1x0 → tz. by p(x, ·) is k-lower semi-continuous and (4.16), we have (4.17) p(t nx0,z) ≤ lim inf m→∞ kp(tnx0,t mx0) ≤ α ′ n (say) and (4.18) p(t nx0,tz) ≤ lim inf m→∞ kp(tnx0,t m+1x0) ≤ β ′ n, (say) where the sequences {α ′ n := αn k } and {β ′ n := βn k } which converges to 0. by lemma 2.14 (i), we conclude that z = tz. suppose that the condition (iv) hold. from the fact that {tnkx0} → z as k → ∞, (tnkx0,z) ∈ x≤ and t is orbitally x≤-continuous, it follows that {tnk+1x0}→ tz as k →∞. similarly, since p(x, ·) is k-lower semi-continuous c© agt, upv, 2017 appl. gen. topol. 18, no. 1 102 fixed point theorems for simulation functions in b-metric spaces via the wt-distance as above, we conclude that z = tz and the remaining part of the proof follow from the proof of theorem 4.3. � corollary 4.8. let (x,≤) be a partially ordered set and (x,d) be a complete metric space and p be a w-distance on x. suppose that t : x → x is a nondecreasing satisfying the following conditions: (i) there exists ζ ∈z such that ζ(p(tx,t2x),p(x,tx)) ≥ 0 for all (x,tx) ∈ x≤; (ii) there exists x0 ∈ x such that (x0,tx0) ∈ x≤, (iii) either t is orbitally continuous at x0 or (iv) t is orbitally x≤-continuous and there exists a subsequence {tnkx0} of {tnx0} converges to some element x? ∈ x such that (tnkx0,x?) ∈ x≤ for any k ∈ n. then t has a fixed point in x. moreover if tx = x, then p(x,x) = 0. corollary 4.9. let (x,≤) be a partially ordered set and (x,d) be a complete b-metric space with constant k ≥ 1 and p be a wt-distance on x. suppose that t : x → x is a nondecreasing satisfying the following conditions: (i) there exists λ ∈ [0, 1 k ) such that p(tx,t2x) ≤ λp(x,tx) for all (x,tx) ∈ x≤; (ii) there exists x0 ∈ x such that (x0,tx0) ∈ x≤, (iii) either t is orbitally continuous at x0 or (iv) t is orbitally x≤-continuous and there exists a subsequence {tnkx0} of {tnx0} converges to some element x? ∈ x such that (tnkx0,x?) ∈ x≤ for any k ∈ n. then t has a fixed point in x. moreover, if tx = x, then p(x,x) = 0. example 4.10. let x = [0, 1] and d(x,y) = (x−y)2 with the wt-distance p on x defined by p(x,y) = |y|2. we consider the following set: x≤ = { (x,y) ∈ x ×x : x = y or x,y ∈{0}∪{ 1 n : n ≥ 1} } , where ≤ is the usual ordering. let t : x → x be a mapping define by t(x) =   x2, if x = 1 n , n ≥ 2, x 2 , otherwise. then t is a nondecreasing mapping. also, x = 0 is an element in x such that 0 ≤ t(0) = 0 and so (0,t(0)) ∈ x≤. hence t satisfies the condition (ii). next, we show that t satisfies the condition (i) of theorem 4.7 with the simulation function in given in example 4.4. if x 6= 1 n for all n ≥ 2, then c© agt, upv, 2017 appl. gen. topol. 18, no. 1 103 c. mongkolkeha, y. j. cho and p. kumam (x,t(x)) ∈ x≤ and it is easy to see that t satisfies the condition (i). if x = 1 n for all n ≥ 2, then ( 1 n ,t 1 n ) ∈ x≤. further, we have ζ(2p(tx,t2x),p(x,tx)) = ζ ( 2p ( 1 n2 , 1 n4 ) ,p ( 1 n , 1 n2 )) = ζ ( 2 ( 1 n4 )2 , ( 1 n2 )2) = ( 1 n2 )2 − 2 ( 1 n4 )2 1 + 2 · ( 1 n2 )2 = n8 − 2n4 n12 · n4 n4 + 2 = n8 − 2n4 n8(n4 + 2) = n4 − 2 n4(n4 + 2) > 0. hence t satisfies the condition (i). furthermore, for each x ∈ x, tni (x) → 0 ∈ x as i → ∞, and also tni+1(x) → t(0) ∈ x as i → ∞. hence all the conditions of theorem 4.7 are satisfied. furthermore, x = 0 is fixed points of t. acknowledgements. this project was supported by the theoretical and computational science (tacs) center under computational and applied science for smart innovation research cluster (classic), faculty of science, kmutt. the first author was supported by thailand research fund (grant no. trg5880221) and kasetsart university research and development institute (kurdi). also, yeol je cho was supported by basic science research program through the national research foundation of korea (nrf) funded by the ministry of science, ict and future planning (2014r1a2a2a01002100). the authors are also grateful to the referee by several useful suggestions that have improved the first version of the paper. references [1] a. n. abdou, y. j. cho and r. saadati, distance type and common fixed point theorems in menger probabilistic metric type spaces, appl. math. comput. 265 (2015), 1145–1154. 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[9] s. czerwik, nonlinear set-valued contraction mappings in b-metric spaces, atti sem. mat. fis. univ. modena 46 (1998), 263–276. [10] m. geraghty, on contractive mappings, proc. amer. math. soc. 40 (1973), 604–608. [11] j. heinonen, lectures on analysis on metric spaces, springer, berlin, 2001. [12] n. hussain, r. saadati and r. p. agrawal, on the topology and wt-distance on metric type spaces, fixed point theory appl. (2014), 2014:88. [13] m. imdad and f. rouzkard, fixed point theorems in ordered metric spaces via wdistances, fixed point theory appl. (2012), 2012:222. [14] o. kada, t. suzuki and w. takahashi, nonconvex minimization theorems and fixed point theorems in complete metric spaces, math. japon. 44 (1996), 381–391. [15] f. khojasteh, s. shukla and s. radenovi ć, a new approach to the study of fixed point theorems via simulation functions, filomat 96 (2015), 1189–1194. [16] b. e. rhoades, a comparison of various definitions of contractive mappings, trans. amer. math. soc. 226 (1977), 257–90. [17] b. e. rhoades, some theorems on weakly contractive maps, nonlinear anal. 47 (2001), 2683–2693. [18] a. roldán-lopez-de-hierro, e. karapinar , c. roldán-lopez-de-hierro and j. martinezmorenoa, coincidence point theorems on metric spaces via simulation function, j. comput. appl. math. 275 (2015), 345–355. [19] n. shioji, t. suzuki and w. takahashi contractive mappings, kanan mapping and metric completeness, proc. amer. math. soc. 126 (1998), 3117–3124. [20] w. takahashi, existence theorems generalizing fixed point theorems for multivalued mappings, in fixed point theory and applications, marseille, 1989, pitman res. notes math. ser. 252: longman sci. tech., harlow, 1991, pp. 39–406. [21] w. takahashi, nonlinear functional analysis–fixed point theory and its applications, yokohama publishers, yokahama, japan, 2000. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 105 04.dvi @ applied general topologyuniversidad polit�ecnica de valenciavolume 1, no. 1, 2000pp. 45 60 extension properties and the niemytzkiplaneharuto ohtaabstract. the �rst part of the paper is a brief survey onrecent topics concerning the relationship between c�-embeddingand c-embedding for closed subsets. the second part studiesextension properties of the niemytzki plane np . a zero-set, z-,c�-, c-, and p-embedded subsets of np are determined. finally,we prove that every c�-embedded subset of np is a p-embeddedzero-set, which answers a problem raised in the �rst part.2000 ams classi�cation: 54c45, 54g20keywords: c�-embedded, c-embedded, p-embedded, property (u�), uni-formly locally �nite, niemytzki plane, tychono� plank1. introductionall spaces are assumed to be completely regular t1-spaces. a subset y of aspace x is said to be c-embedded in x if every real-valued continuous functionon y can be continuously extended over x, and y is said to be c�-embedded inx if every bounded real-valued continuous function on y can be continuouslyextended over x. obviously, every c-embedded subset is c�-embedded, butthe converse is not true, in general. in the �rst part of the paper, formed bysections 2, 3 and 4, we discuss several problems concerning the relationshipbetween c�-embedding and c-embedding for closed subsets. for example, thefollowing problem is still open as far as the author knows:problem 1.1. does there exist a �rst countable space having a closed c�-em-bedded subset which is not c-embedded?since a space which answers the above problem positively cannot be normal,the following problem naturally arises:problem 1.2. let x be one of the following spaces: the niemytzki plane (i.e.,the space np de�ned in section 4 below); the sorgenfrey plane ( [3, example2.3.12]); michael's product space ([3, example 5.1.32]). then, does the spacex have a closed c�-embedded subset which is not c-embedded? 46 haruto ohtain the second part, formed by sections 5, 6 and 7, we answer problem 1.2for the niemytzki plane np negatively by determining a zero-set, z-, c�-, c-and p-embedded subsets of np . the problem, however, remains open for thesorgenfrey plane and michael's product space.throughout the paper, let r denote the real line with the euclidean topology,q the subspace of rational numbers and n the subspace of positive integers.the cardinality of a set a is denoted by jaj. as usual, a cardinal is the initialordinal and an ordinal is identi�ed with the space of all smaller ordinals withthe order topology. let ! denote the �rst in�nite ordinal and !1 the �rstuncountable ordinal. all unde�ned terms will be found in [3].2. c�-embedding versus c-embeddingit is an interesting problem to �nd a closed c�-embedded subset which is notc-embedded. we begin by showing typical examples of such subsets. first, letus consider the subspace � = �r n (�n n n) of �n. the subset n is closed c�-embedded but not c-embedded in �, because � is pseudocompact (cf. [4, 6p,p.97]). more generally, noble proved in [16] that every space y can be embeddedin a pseudocompact space py as a closed c�-embedded subspace. thus, everynon-pseudocompact space y embeds in py as a closed c�-embedded subsetwhich is not c-embedded. shakhmatov [20] constructed a pseudocompact spacex with a much stronger property that every countable subset of x is closedand c�-embedded.now, we give another examples which does not rely on pseudocompactness.for every space x there exist an extremally disconnected space e(x), calledthe absolute of x, and a perfect onto map ex : e(x) ! x (cf. [3, 6.3.20 (b)]).we now call a space x weakly normal if every two disjoint closed sets in x, oneof which is countable discrete, have disjoint neighborhoods.lemma 2.1. let x be a space which is not weakly normal. then e(x) con-tains a closed c�-embedded subset which is not c-embedded.proof. by the assumption, x has a closed set a and a countable discrete closedset b = fpn : n 2 ng such that a \ b = ? but they have no disjoint neigh-borhoods. we show that the closed set f = e�1x [b] in e(x) is c�-embeddedbut not c-embedded. since b is countable discrete closed in x, we can �nda disjoint family u = fun : n 2 ng of open-closed sets in e(x) such thate�1x (pn) � un � e(x) n e�1x [a] for each n 2 n. let u = sfun : n 2 ng;then u is a cozero-set in e(x). since f and e(x) nu cannot be separated bydisjoint open sets, it follows from theorem 3.1 below that f is not c-embeddedin e(x). on the other hand, f is c-embedded in u, because each e�1x (pn) iscompact and u is disjoint, and further, u is c�-embedded in e(x) by [4, 1h6,p.23]. consequently, a is c�-embedded in e(x). �corollary 2.2. let x be one of the following spaces: the niemytzki plane np ;the sorgenfrey plane s2; michael's product space rq � p ; the tychono� plankt (see example 3.3 below). then e(x) contains a closed c�-embedded subsetwhich is not c-embedded. extension properties and the niemytzki plane 47proof. it is well known (and easily shown) that the spaces np , s2 and t arenot weakly normal. now, we show that michael's product space rq � p isnot weakly normal. the space rq is obtained from r by making each pointof p = r n q isolated. enumerate q as fxn : n 2 ng and choose yn 2 pwith jxn � ynj < 1=n for each n 2 n. let a = fhxn;yni : n 2 ng andb = fhx;xi : x 2 pg. then a and b have no disjoint neighborhoods inrq � p . since a is discrete closed in rq � p , rq � p is not weakly normal.hence, the corollary follows from lemma 2.1. �as another application, we have the following example concerning problem1.1.example 2.3. there exists a space x in which every point is a g� and thereexists a closed c�-embedded subset which is not c-embedded. in fact, letr be a maximal almost disjoint family of in�nite subsets of n. pick a pointpa 2 cl�n a n a for each a 2 r and let r = fpa : a 2 rg. then the subspacex = n [r of �n is extremally disconnected (i.e., e(x) = x) and r is discreteclosed in x. let e be a countable in�nite subset of r. then e and rne haveno disjoint neighborhoods in x by the maximality of r. hence, by the proofof lemma 2.1, e is closed c�-embedded in x but not c-embedded. �we change the topology of the space x = n [r in example 2.3 by declaringthe sets fpag [ (a n f1;2; � � � ;ng), n 2 n, to be basic neighborhoods of pafor each a 2 r. the resulting space is �rst countable and is usually called a -space (see [4, 5i, p.79]). a positive answer to the following problem answersproblem 1.1 positively.problem 2.4. does there exist a -space having a closed c�-embedded subsetwhich is not c-embedded?for an in�nite cardinal , a subset y of a space x is said to be p -embeddedin x if for every banach space b with the weight w(y ) � �, every continuousmap f : y ! b can be continuously extended over x. a subset y of x is saidto be p-embedded in x if y is p -embedded in x for every . it is known thaty is p -embedded in x if and only if for every locally �nite cozero-set coveru of y with juj � , there exists a locally �nite cozero-set cover v of x suchthat fv \ y : v 2 vg re�nes u. in particular, y is c-embedded in x if andonly if y is p!-embedded in x. for further information about p -embedding,the reader is referred to [1]. the following problem concerning the relationshipbetween c-embedding and p-embedding is also open:problem 2.5. does there exist an example in zfc of a space x, with jxj =!1, having a closed c-embedded subset which is not p-embedded?problem 2.6. does there exist an example in zfc of a �rst countable spacehaving a closed c-embedded subset which is not p-embedded?it is known that under certain set-theoretic assumption such as ma+:ch,there exists a �rst countable, normal space x which is not collectionwise normal(see [21]). since a space is collectionwise normal if and only if every closed subset 48 haruto ohtais p-embedded, such a space x has a closed c-embedded subset which is notp-embedded (cf. remark 6.4 in section 6 below).3. spaces in which every closed c�-embedded set is c-embeddedwe say that a space x has the property (c� = q) if every closed c�-embedded subset of x is q-embedded in x, where q 2 fc;p ;pg. a subset yof a space x is said to be z-embedded in x if every zero-set in y is the restrictionof a zero-set in x to y (cf. [2]). every c�-embedded subset is z-embedded.two subsets a and b are said to be completely separated in x if there exists areal-valued continuous function f on x such that f[a] = f0g and f[b] = f1g.the following theorem was proved by blair and hager in [2, corollary 3.6.b].theorem 3.1. [blair-hager] a subset y of a space x is c-embedded in x ifand only if y is z-embedded in x and y is completely separated from everyzero-set in x disjoint from y .recall from [11] that a space x is �-normally separated if every two disjointclosed sets, one of which is a zero-set, are completely separated in x. allnormal spaces and all countably compact spaces are �-normally separated. bytheorem 3.1, we have the following corollary:corollary 3.2. every �-normally separated space has the property (c� = c).the converse of corollary 3.2 does not hold as the next example shows:example 3.3. the tychono� plank t = ((!1+1)�(!+1))nfh!1;!ig is not �-normally separated but every closed c�-embedded subset of t is p-embedded,i.e., t has the property (c� = p). to prove these facts, let a = f!1g � !and b = !1 � f!g; then a is closed in t and b is a zero-set in t . since aand b cannot be completely separated in t , t is not �-normally separated.next, let f be a closed c�-embedded subset of t . we have to show that fis p-embedded in t . since there is no uncountable discrete closed set in t ,every locally �nite cozero-set cover of f is countable. hence, it su�ces to showthat f is c-embedded in t . since f is closed in t , either f includes a closedunbounded subset of b or f \fh�;mi : � < � < !1;n < m � !g = ? for some� < !1 and some n < !. in the former case, every zero-set in t disjoint fromf must be compact. in the latter case, a \ f is �nite since f is c-embedded,which implies that f is compact. in both cases, f is completely separatedform a zero-set disjoint from it. hence, it follows from theorem 2.1 that f isc-embedded.the following example shows that the product of a space with the property(c� = p) and a compact space need not have the property (c� = c).example 3.4. let t be the tychono� plank. as we showed in example 3.3,t has the property (c� = p). we show that t � �e(t) fails to have theproperty (c� = c), where e(t) is the absolute of t . let et : e(t) ! t bethe perfect onto map. then the subspace g = fhet (x);xi : x 2 e(t)g is closedin t � �e(t), because et is perfect. since t is not weakly normal, it follows extension properties and the niemytzki plane 49from lemma 2.1 that e(t) does not have the property (c� = c), and hence, galso fails to have the property (c� = c), because g is homeomorphic to e(t).hence, if we prove that g is c�-embedded in t � �e(t), then it would followthat t � �e(t) does not have the property (c� = c). for this end, let f bea bounded real-valued continuous function on g and de�ne g : e(t) ! r byg(x) = f(het (x);xi) for x 2 e(t). since g is bounded continuous, g extendsto a continuous function h on �e(t). then h � � is a continuous extension off over t � �e(t), where � : t � �e(t) ! �e(t) is the projection. hence, gis c�-embedded in t � �e(t). �problem 3.5. does there exist a space x with the property (c� = c) and ametric space m such that x � m fails to have the property (c� = c)?the positive answer to problem 1.2 for michael's product space answersproblem 3.5 positively. we conclude this section by giving a class of spaceshaving the property (c� = p ). recall from [10, 14] that a family f of subsetsof a space x is uniformly locally �nite in x if there exists a locally �nite cozero-set cover u of x such that every u 2 u intersects only �nitely many membersof f. let be an in�nite cardinal. a subset y of a space x is said to beu -embedded in x if every uniformly locally �nite family f of subsets in ywith jfj � is uniformly locally �nite in x (cf. [7]). the following theoremwas proved in [15] (see also [7, proposition 1.6]).theorem 3.6. [morita-hoshina] for every in�nite cardinal , a subset y ofa space x is p -embedded in x if and only if y is both z-embedded and u -embedded in x.recall from [7] that a space x has the property (u ) (resp. property (u )�)if every locally �nite (resp. discrete) family f of subsets of x with jfj � isuniformly locally �nite in x. all -collectionwise normal and countably para-compact spaces have the property (u ), and all -collectionwise normal spaceshave the property (u )�. hoshina [7] proved that a space x has the property(u )� if and only if every closed subset of x is u -embedded. combining thiswith theorem 3.6, we have the following corollary:corollary 3.7. for every in�nite cardinal , every space having the property(u )� has the property (c� = p ).it will be worth noting that every -collectionwise normal dowker space (see[17]) has the property (u )� for every but does not have the property (u!).4. productsit is quite interesting to consider the relationship between c�and c-em-beddings in the realm of product spaces. in spite of extensive studies, thefollowing problem is still unanswered.problem 4.1. let a be a closed c-embedded subset of a space x, y a space,and assume that a�y is c�-embedded in x �y . then, is a�y c-embeddedin x � y ? 50 haruto ohtain this section, we summarize partial answers to problem 4.1 and also discussthe following problem:problem 4.2. let x and y be spaces with the property (c� = c). under whatconditions on x and y does x � y have the property (c� = c)?first, we consider product spaces with a compact factor. morita-hoshina[15] proved the following theorem which answers problem 4.1 positively wheny is a compact space.theorem 4.3. [morita-hoshina] let a be a subset of a space x, y an in�nitecompact space, and assume that a�y is c�-embedded in x �y . then a�yis pw(y )-embedded in x � y , where w(y ) is the weight of y .from now on, let denote an in�nite cardinal. the next theorem is ananswer to problem 4.2.theorem 4.4. if a space x has the property (u ), then x�y has the property(c� = p ) for every compact space y .proof. if x has the property (u ) and y is a compact space, then it is easilyproved that x � y has the property (u ). hence, x � y has the property(c� = p ) by corollary 3.7. �example 3.4 shows that `property (u )' in theorem 4.4 cannot be weakenedto `property (c� = p )'. the following problem remains open:problem 4.5. if x�y has the property (c� = p ) for every compact space y ,then does x have the property (u )? more specially, does theorem 4.4 remaintrue if `property (u )' is weakened to `property (u )�'?a space is called �-locally compact if it is the union of countably many closedlocally compact subspaces. concerning products with a �-locally compact,paracompact factor, the following theorem was proved by yamazaki in [23] and[25]:theorem 4.6. [yamazaki] let a be a c-embedded subset of a space x, y a�-locally compact, paracompact space, and assume that a � y is c�-embeddedin x�y . then a�y is c-embedded in x�y . moreover, if a is p -embeddedin x in addition, then a � y is also p -embedded in x � y .problem 4.7. does theorem 4.4 remain true if `compact' is weakened to `�-locally compact, paracompact'?next, we consider products with a metric factor. the di�culty of this caseis in the fact that a � y need not be u!-embedded in x � y even if a is p-embedded in x (consider michael's product space). nevertheless, the followingtheorems 4.8 and 4.9 were proved by gutev-ohta [6]:theorem 4.8. [gutev-ohta] let a be a subset of a space x, y a non-discretemetric space, and assume that a � y is c�-embedded in x � y . then a � yis c-embedded in x � y . extension properties and the niemytzki plane 51theorem 4.9. [gutev-ohta] let a be a p -embedded subset of a space x andy a metric space. then the following conditions are equivalent:(1) a � y is p -embedded in x � y ;(2) a � y is c�-embedded in x � y ;(3) a � y is u!-embedded in x � y .corollary 4.10. let a be a p -embedded subset of a space x, y the product ofa �-locally compact, paracompact space k with a metric space m, and assumethat a � y is c�-embedded in x � y . then a � y is p -embedded in x � y .proof. since (a � k) � m is c�-embedded in (x � k) � m, a � k is c�-embedded in x � k. hence, a � k is p -embedded in x � k by theorem4.6. finally, it follows from theorem 4.9 that (a�k)�m is p -embedded in(x � k) � m. �problem 4.11. does theorem 4.8 remain true if `metric space' is weakenedto `paracompact m-space' or `la�snev space'?problem 4.12. let a be a p -embedded subset of a space x and y a paracom-pact m-space. then, does the condition (2) in theorem 4.9 imply the condition(1)?problem 4.13. let a be a p -embedded subset of a space x and let y be oneof the following spaces (i){(iii): (i) a la�snev space; (ii) a strati�able space; (iii)a paracompact �-space. then, are the conditions (1), (2), (3) in theorem 4.9equivalent?for the de�nitions of the spaces (i), (ii) and (iii) in problem 4.13, we referthe reader to [5]. problems 4.12 and 4.13 were raised in [6].now, we try to extend theorems 4.3 and 4.8 to products with a factor spacein wider class of spaces. for this end, we write y 2 �(q) if for every space xand every closed subset a of x, if a�y is c�-embedded in x �y , then a�yis q-embedded in x �y , where q 2 fc;p g. by theorem 4.3, y 2 �(pw(y ))for every in�nite compact space y , and by theorem 4.8, y 2 �(c) for everynon-discrete metric space y . the following results show that the classes �(p )and �(c) are much wider than we expected.theorem 4.14. let y be a space with y 2 �(p ). then y � z 2 �(p ) forevery space z.proof. let x be a space with a closed subset a such that a � (y � z) isc�-embedded in x � (y � z). then, it is obvious that (a � z) � y is c�-embedded in (x � z) � y . since y 2 �(p ), (a � z) � y is p -embedded in(x � z) �y , which means that a � (y � z) is p -embedded in x � (y � z).hence, y � z 2 �(p ). �corollary 4.15. for every space y , y � (! + 1) 2 �(c).proof. since ! + 1 2 �(c) by theorem 4.3 (or theorem 4.8), this followsimmediately from theorem 4.14. � 52 haruto ohtathe next theorem and its corollary were proved by hoshina and yamazakiin [9].theorem 4.16. [hoshina-yamazaki] let y be a space which is homeomorphicto y � y and contains an in�nite compact subset k. then y 2 �(pw(k)).corollary 4.17. [hoshina-yamazaki] for every space y with jy j � 2, y 2�(p ).finally, we consider some miscellaneous products. the following theoremwas proved by yamazaki in [24] and [25]. by a p-space, we mean a p-space inthe sense of morita [13]. for the de�nition of a �-space, see [5].theorem 4.18. [yamazaki] let a be a closed subset of a normal p-space x,y a paracompact �-space, and assume that a � y is c�-embedded in x � y .then a � y is c-embedded in x � y . moreover, if a is p -embedded in x inaddition, then a � y is p -embedded in x � y .since a p-space is countably paracompact, all normal p-spaces have theproperty (u!) and all -collectionwise normal p-spaces have the property (u ).hence, the following problem naturally arises after theorem 4.18.problem 4.19. let x be a normal p-space and y a paracompact �-space.then, does x �y have the property (c� = c)? moreover, if x is -collection-wise normal in addition, then does x � y have the property (c� = p )?recently, a partial answer to problem 4.19 was given by yajima [22].theorem 4.20. [yajima] let x be a collectionwise normal p-space and ya paracompact �-space. then every closed c-embedded subset of x � y isp-embedded in x � y .5. zero-sets in the niemytzki planein the remainder of this paper, we consider extension properties of theniemytzki plane np , and in the �nal section, we answer problem 1.2 for npnegatively. the niemytzki plane np is the closed upper half-plane r�[0; +1)with the topology de�ned as follows: for each p = hx;yi 2 np and " > 0, lets"(p) = (fq 2 np : d(hx;"i;q) < "g [ fpg for y = 0;fq 2 np : d(p;q) < "g for y > 0;where d is the euclidean metric on the plane. the topology of np is generatedby the family fs"(p) : p 2 np;" > 0g. let l = fhx;0i : x 2 rg � np .from now on, we always consider a subset of r to be a subspace of r, andconsider a subset of np to be a subspace of np unless otherwise stated. forexample, an interval i is a subspace of r but i � f0g is a subspace of np .when a � x � np , we say that a is "-open in x if a is open with respectto the relative topology on x induced from the euclidean topology. the words"-closed and "-continuous are used similarly.in this section, we determine a zero-set in np . we �rst state the main resultsin this section, then proceed to the proofs. extension properties and the niemytzki plane 53theorem 5.1. let f be a closed subset of np. then f is a zero-set in npif and only if the set fx 2 r : hx;0i 2 fg is a g�-set in r.corollary 5.2. if s is a subset of np with s\l = ?, then clnp s is a zero-setin np. in particular, every closed subset s of np with s\l = ? is a zero-setin np.proof. this follows from theorem 5.1 above and lemma 5.11 below. �the next corollary follows from corollary 5.2, since f = clnp (f n l) forevery regular-closed set f in np .corollary 5.3. every regular-closed set in np is a zero-set.theorem 5.1 also shows that every zero-set in np is a g�-set with respectto the euclidean topology. on the other hand, every "-closed set in the closedupper half-plane is a zero-set in np . hence, we have the following corollary.corollary 5.4. for a subset s of np, s is a baire set in np if and only ifs is a borel set with respect to the euclidean topology.the �nal theorem of this section describes a zero-set in a subspace of np .theorem 5.5. let y be a subspace of np and y0 = cly (y n l). let f be aclosed subset of y . then f is a zero-set in y if and only if a is a g�-set inb, where a = fx 2 r : hx;0i 2 f \ y0g and b = fx 2 r : hx;0i 2 y0g.before proving theorems 5.1 and 5.5, let us observe some examples of non-trivial zero-sets in np .example 5.6. (1) the �rst one is a zero-set e in np such that e\l = ? butthe set fx 2 r : hx;0i 2 cl" eg is the cantor set k, where cl" e is the closure ofe with respect to the euclidean topology. let i be the set of all componentsof [0;1] n k. for each open interval i = (a;b) 2 i, de�neei = fhx;yi : a < x < b; y = minf1 �p1 � (x � a)2;1 �p1 � (x � b)2gg:then ei is a closed set in np such that cl" ei n ei = fha;0i;hb;0ig. de�nee = sfei : i 2 ig. then e is a closed set in np such that e \ l = ? andk = fx 2 r : hx;0i 2 cl" eg, as required. by corollary 5.2, e is a zero-set innp .(2) the second one is a zero-set f of np such that f = clnp (f nl) and fx 2r : hx;0i 2 fg = rnq . since q�f0g is countable and discrete closed in np , wecan �nd a disjoint family s = fs"(x)(hx;0i) : x 2 qg of basic open sets in np .de�ne f = np nsfs : s 2 sg. then, fx 2 r : hx;0i 2 fg = rnq clearly. toshow that f = clnp(f n l), consider a point q = hx;0i 2 (r n q) � f0g. then,s"(q) \ (f n l) 6= ? for each " > 0, because s is disjoint and the open intervalfy 2 r : hx;yi 2 s"(q) n fqgg cannot be covered by disjoint open intervals jwith inf j > 0. hence, q 2 clnp (f n l), which implies that f = clnp (f n l).finally, f is a zero-set in np by corollary 5.2. � 54 haruto ohtato prove theorems 5.1 and 5.5, we need some de�nitions and lemmas. letr] = r [ f�1;+1g and consider �1 < x < +1 for each x 2 r. for eacha 2 r], we de�ne a function ha : r ! [0;1] as follows: for a 2 r, de�neha(x) = (1 �p1 � (x � a)2 if jx � aj � 1;1 otherwise,and de�ne h+1(x) = h�1(x) = 1 for x 2 r. by an open interval in r, wemean a set of the form (a;b) = fx 2 r : a < x < bg for a;b 2 r] with a < b.for an open interval j = (a;b) in r, we de�neuj = fhx;yi : a < x < b;0 � y < minfha(x);hb(x)gg:lemma 5.7. for every open interval j = (a;b) in r, the following are valid:(1) j � f0g � uj,(2) j � f0g is a zero-set in np and uj is a cozero-set in np.proof. (1) is obvious. to prove (2), let h = j � [0;+1). since uj is "-open inh, there is an "-continuous function f : h ! [0;1] such that f�1(0) = j �f0gand f�1(1) = h n uj. we extend f to the function f� : np ! [0;1] by lettingf�jh = f and f�(p) = 1 for each p 2 np n h. then f� is continuous on npby the de�nition of uj. since j � f0g = f�1� (0) and uj = f�1� [[0;1)], we have(2). �lemma 5.8. if j is a family of disjoint open intervals in r, then the familyu = fuj : j 2 jg is discrete in np.proof. let p = hx;yi 2 np . if y = 0, then s1(p) meets at most one member ofu. if y > 0, then sy=2(p) meets at most one member of u. �let f be a family of subsets of a space x. it is known [14, 18] that f isuniformly locally �nite in x if and only if there exist a locally �nite familyfg(f) : f 2 fg of cozero-sets in x and a family fz(f) : f 2 fg of zero-sets in x such that f � z(f) � u(f) for each f 2 f. now, we say thatf is uniformly discrete in x if there exist a discrete family fu(f) : f 2 fgof cozero-sets in x and a family fz(f) : f 2 fg of zero-sets in x such thatf � z(f) � u(f) for each f 2 f.lemma 5.9. [15, lemma 2.3] the union of a uniformly locally �nite family ofzero-sets in a space x is a zero-set in x.lemma 5.10. if a is a g�-set in r, then a � f0g is a zero-set in np.proof. there exist open sets gn, n 2 n, in r such that a = tn2n gn. for eachn 2 n, gn is the union of a family fji : i 2 mg of disjoint open intervals inr. by lemmas 5.7 and 5.8, fji � f0g : i 2 mg is a uniformly discrete familyof zero-sets in np . hence, gn � f0g is a zero-set in np by lemma 5.9. sincethe intersection of countably many zero-sets is a zero-set, a � f0g is a zero-setin np . �lemma 5.11. if s is a subset of np with s \ l = ?, then the set a = fx 2r : hx;0i 2 clnp sg is a g�-set in r. extension properties and the niemytzki plane 55proof. for each x 2 rna, there exists n(x) 2 n such that s1=n(x)(hx;0i)\s =?. for each n 2 n, let bn = fx 2 r : n(x) = ng. then it is easily proved thata \ clr bn = ?. since r n a = sn2n clr bn, a is a g�-set in r. �lemma 5.12. let y be a subspace of np such that y = cly (y nl) and let fbe a zero-set in y . then a is a g�-set in b, where a = fx 2 r : hx;0i 2 fgand b = fx 2 r : hx;0i 2 y g.proof. since f is a zero-set in y , there exist open sets gn, n 2 n, in y suchthat f = tn2n cly gn. let h = tn2n clnp (gn n l); then f = h \ y by thecondition of y . moreover, the set c = fx 2 r : hx;0i 2 hg is a g�-set in r bylemma 5.11. since a = b \ c, a is a g�-set in b. �lemma 5.13. let e and f be closed sets in np such that l � e and e\f =?. then there exists an open set u in np such that e � u � clnp u � npnf.proof. for each p 2 e, there is n(p) 2 n such that s1=n(p)(p)\f = ?. for eachn 2 n, let en = fp 2 e : n(p) = ng and un = sfs1=2n(p) : p 2 eng. then unis an open set in np with en � un. we show that clnp un \ f = ? for eachn 2 n. suppose on the contrary that there is a point q = hx;yi 2 clnp un \ ffor some n 2 n. then y > 0, because f \l = ?. thus, we can �nd � > 0 suchthat for every x 2 r, if q 62 s1=n(hx;0i), then s�(q) \ s1=2n(hx;0i) = ?. if weput " = minf�;1=2ng, then8p 2 np (q 62 s1=n(p) ) s"(q) \ s1=2n(p) = ?):now, since q 2 clnp un, s"(q) \ s1=2n(p) 6= ? for some p 2 en. by (1), thisimplies that q 2 s1=n(p), which contradicts the fact that s1=n(p) \ f = ?.hence, clnp un \f = ? for every n 2 n, and obviously, e � sn2n un. on theother hand, since f is lindel�of, there exists a countable family fvn : n 2 ng ofopen sets in np such that f � sn2n vn and clnp vn \ e = ? for each n 2 n.finally, the set u = sn2n(un nti�n clnp vi) is a required open set in np . �we are now ready to prove theorems 5.1 and 5.5.proof. (of theorem 5.1) let a = fx 2 r : hx;0i 2 fg. if f is a zero-set in np ,then a is a g�-set in r by lemma 5.12. conversely, assume that a is a g�-setin r, i.e., there exist open sets gn, n 2 n, in r with a = tn2n gn. for eachn 2 n, let kn = (rngn)�f0g. then both a�f0g and kn are zero-sets in npby lemma 5.10. hence, there exists a continuous function fn : np ! [0;1]such that fn[a � f0g] = f0g and fn[kn] = f1g. let hn = f \ f�1[[1=2;1]].then hn is a closed set in np with hn \ l = ?. by using lemma 5.13 andthe technique used in the proof of urysohn's lemma, we can de�ne anothercontinuous function gn : np ! [0;1] such that gn[l] = f0g and gn[hn] = f1g.de�ne zn = f�1n [[0;1=2]] [ g�1n [f1g]. then zn is a zero-set in np such thatf � zn and zn \ kn = ?. on the other hand, f [ l is a zero-set in np ,because it is an "-closed set. since f = (f [ l) \ tn2n zn, f is a zero-set innp . �proof. (of theorem 5.5) if f is a zero-set in y , then f \ y0 is a zero-set in y0.since y0 = cly (y0 nl), it follows from lemma 5.12 that a is a g�-set in b. to 56 haruto ohtaprove the converse, assume that a is a g�-set in b. since y n y0 is a discrete,open and closed subset of y , f ny0 is a zero-set in y . hence, it su�ces to showthat f \ y0 is a zero-set in y . to show this, let z1 = clnp (f n l) \ y0 andz2 = (f \ y0) \ l. then f \ y0 = z1 [ z2. by corollary 5.2, z1 is a zero-setin y0. on the other hand, by the assumption, there exists a g�-set c in r suchthat a = b \ c. since z2 = (c � f0g) \ y0, z2 is a zero-set in y0 by lemma5.10. consequently, f \ y0 is a zero-set in y0, and hence, in y , because y0 isopen and closed in y . �6. z-embedded subsets in npa subset a of r is called a q-set if every subset of a is a g�-set in a. allcountable sets are q-sets and the existence of an uncountable q-set is knownto be independent of the usual axioms of set theory (cf. [12]). it is quite easyto determine a z-embedded set y in np such that y � l. indeed, the �rsttheorem immediately follows from theorem 5.1:theorem 6.1. for a subset a of r, a�f0g is z-embedded in np if and onlyif a is a q-set in r.next, we consider a z-embedded subset in np which is not necessarily asubset of l.lemma 6.2. let y be a subset of np such that y = cly (y n l). then y isz-embedded in np.proof. let f be a zero-set in y . let a = fx 2 r : hx;0i 2 fg and b = fx 2r : hx;0i 2 y g. then by lemma 5.12, a is a g�-set in b, i.e., there is a g�-setc in r with a = b \c. let z = (c �f0g)[clnp (f nl). then z is a zero-setin np , because both c � f0g and clnp (f n l) are zero-sets in np by lemma5.10 and corollary 5.2, respectively. since f = z \ y , y is z-embedded innp . �theorem 6.3. let y be a subspace of np and y0 = cly (y n l). then y isz-embedded in np if and only if a is a q-set in r and is a g�-set in b, wherea = fx 2 r : hx;0i 2 y n y0g and b = fx 2 r : hx;0i 2 y g.proof. first, assume that y is z-embedded in np . then y ny0 is z-embeddedin np , because y0 is open and closed in y . hence, it follows from theorem 6.1that a is a q-set. moreover, since y is z-embedded in np , there is a zero-setf in np such that y n y0 = f \ y . by theorem 5.1, the set c = fx 2 r :hx;0i 2 fg is a g�-set in r. since a = b \ c, a is a g�-set in b. next,we prove the converse. by the assumption, there is a g�-set d in r such thata = b \ d. let z1 = d � f0g and z2 = clnp (y n l). then both z1 and z2are zero-sets in np by lemma 5.10 and corollary 5.2, respectively, and theysatisfy that y n y0 � z1, y0 � z2, z1 \ y0 = ? and z2 \ (y n y0) = ?. hence,it su�ces to show that both y n y0 and y0 are z-embedded in np . since a isa q-set, y n y0 is z-embedded in np by theorem 6.1, and y0 is z-embeddedin np by lemma 6.2. � extension properties and the niemytzki plane 57remark 6.4. it is known that if a � r is a q-set, then the subspace y =(a�f0g) [(np nl) of np is normal (cf. [21, example f]). hence, the closedset a�f0g is then c-embedded in y . however, this does not mean that a�f0gis c-embedded in np even if a is countable. in fact, it is known ([8, example3.14]) that q �f0g is not c�-embedded in np ; this also follows from theorem7.1 below. 7. p-, cand c�-embedded subsets in nprecall from [6] that a subset y of a space x is cu-embedded in x if for everypair of a zero-set e in y and a zero-set f in x with e \f = ?, e and f \ yare completely separated in x. the extension properties we have consideredare related as the following diagram, where the arrow `a ! b' means thatevery a-embedded subset is b-embedded:p �! c �! c? �! z# #u! �! cumoreover, we say that a subset y � x is uniformly discrete in x if the familyffxg : x 2 y g is uniformly discrete in x, in other words, there exists a discretefamily fu(x) : x 2 y g of cozero-sets in x such that x 2 u(x) for each x 2 y .as is easily shown, every uniformly discrete set in x is p-embedded in x.finally, we brie y review scattered sets in r. let a � r. for every ordinal �,we de�ne the set a(�) inductively as follows: a(0) = a; if � = � + 1, then a(�)is the derived set of a(�); and if � is a limit, then a(�) = tfa(�) : � < �g.a subset a of r is called scattered if a(�) = ? for some �, and then we write�(a) = minf� : a(�) = ?g. it is known that �(a) < !1 for every scattered seta in r.theorem 7.1. for a subset a of r, the following conditions are equivalent:(1) a is a scattered set in r;(2) a � f0g is uniformly discrete in np;(3) a � f0g is p-embedded in np;(4) a � f0g is cu-embedded in np.proof. (1) ) (2): we prove this implication by induction on �(a). if �(a) = 0,it is obviously true since a = ?. now, let � > 0 and assume that the implicationholds for every subset a0 � r with �(a0) < �. let a � r be a scattered setwith �(a) = �. in case � = � + 1, (a n a(�)) � f0g is uniformly discrete innp by inductive hypothesis, because �(a n a(�)) < �. since a(�) is discrete,there is a family fix : x 2 a(�)g of disjoint open intervals in r such that x 2 ixfor each x 2 a(�). hence, it follows from lemmas 5.7 and 5.8 that a(�) � f0gis also uniformly discrete in np . since the union of �nitely many uniformlydiscrete subsets is uniformly discrete, a � f0g is uniformly discrete in np .in case � is a limit, then u = fa n a(�) : � < �g is an open cover of a.since every scattered set in r is zero-dimensional, there exists a disjoint openre�nement v of u. by considering order components of each member of v, we 58 haruto ohtacan �nd a family j = fjn : n 2 mg of disjoint open intervals in r such thatj covers a and fjn \ a : n 2 mg re�nes v. by lemmas 5.7 and 5.8 again,the family fjn � f0g : n 2 mg is uniformly discrete in np , and hence, so isf(jn \a) �f0g : n 2 mg . moreover, each (jn \a)�f0g is uniformly discretein np by inductive hypothesis. since the union of a uniformly discrete familyof uniformly discrete subsets is also uniformly discrete, a � f0g is uniformlydiscrete in np .(2) ) (3) ) (4): obvious.(4) ) (1): suppose that a is not scattered; then a includes a perfect subsetb which is closed in a. let k = clr b and take a countable dense subset b0 ofb such that the set b1 = b n b0 is also dense in b, i.e., k = clr b0 = clr b1.let e = (k n b0) � f0g; then e is a zero-set in np by lemma 5.10. now,b0�f0g is a zero-set in a�f0g, because a�f0g is discrete. on the other hand,b1 �f0g = e \ (a � f0g). since a� f0g is cu-embedded in np , there existsa continuous function f : np ! [0;1] such that f[bi � f0g] = i for i = 0;1.let ci = fx 2 r : f(hx;0i) = ig for each i = 0;1. then c0 and c1 are disjointg�-sets in r by theorem 5.1. hence, we can write k n ci = sj2n di;j, whereeach di;j is "-closed in k, for each i = 0;1. since b � c0[c1 and both b0 andb1 are dense in k, di;j is nowhere dense in k for all i and j. this contradictsthe completeness of k. � �lemma 7.2. every cu-embedded subset y in a �rst countable space x isclosed.proof. if y is not closed in x, then there exists a sequence fpn : n 2 ng � ywhich converges to a point p 2 x n y . we may assume that pm 6= pn if m 6= n.let e = fp2n : n 2 ng and f = fp2n�1 : n 2 ng [ fpg. it is easily provedthat f is a compact g�-set in x, and hence, a zero-set in x, because x iscompletely regular. on the other hand, since e [ fpg is also a zero-set in x,e is a zero-set in y . since y is cu-embedded in x, e and f n fpg must becompletely separated in x, which is impossible. �lemma 7.3. every scattered subset a of r is a g�-set in r.proof. this is well-known and also follows from our results. in fact, by theorem7.1, a�f0g is uniformly discrete in np , which implies that a�f0g is a zero-setin np by lemma 5.9. hence, a is a g�-set in r by theorem 5.1. �by lemma 7.2, we can restrict our attention to closed subsets of np . the fol-lowing theorem shows that every cu-embedded subset of np is p-embedded,which answers problem 1.2 for the niemytzki plane negatively.theorem 7.4. let y be a closed subspace of np and let y0 = cly (y n l).then the following conditions are equivalent:(1) the set a = fx 2 r : hx;0i 2 y n y0g is a scattered set in r;(2) y n y0 is uniformly discrete in np;(3) y is p-embedded in np;(4) y is cu-embedded in np. extension properties and the niemytzki plane 59proof. (1) , (2): this equivalence follows from theorem 7.1.(1) ) (3): suppose that (1) is true. then a is a g�-set in r by lemma7.3. hence, the set a � f0g(= y n y0) is a zero-set in np by lemma 5.10.on the other hand, by the de�nition of y0, it follows from corollary 5.2 thaty0 is a zero-set. consequently, y0 and y n y0 are completely separated in np .hence, it su�ces to show that both y0 and y n y0 are p-embedded in np .by theorem 6.3 and corollary 5.2, y0 is a z-embedded zero-set in np , whichimplies that y0 is c-embedded in np by theorem 3.1. since y0 is separable,y0 has no uncountable locally �nite cozero-set cover. hence, y0 is p-embeddedin np . on the other hand, y n y0 is p-embedded in np by theorem 7.1.(3) ) (4): obvious.(4) ) (1): if y is cu-embedded in np , then the set a � f0g(= y n y0) isalso cu-embedded in np , because y ny0 is open and closed in y . hence, thisimplication follows from theorem 7.1. �by theorem 7.4, both of the zero-sets e and f de�ned in example 5.6 arep-embedded in np .corollary 7.5. every cu-embedded subset in np is a p-embedded zero-set.proof. let y be a cu-embedded set in np and let y0 = cly (y n l). bytheorem 7.4, y is p-embedded in np . moreover, as i showed in the proof oftheorem 7.4 (1) ) (3), both y0 and y n y0 are zero-sets in np . hence, y is azero-set in np . �recall from [19] that a subset a of a space x is �-embedded in x if a � yis c�-embedded in x � y for every space y . the following problem is open:problem 7.6. is every p-embedded subset in np �-embedded in np?acknowledgment. the author would like to thank ken-iti tamano for hishelpful suggestions. in particular, theorem 5.1 was born from the discussionwith him. references[1] r. a. al�o and h. l. shapiro, normal topological spaces (cambridge univ. press, cam-bridge, 1974).[2] l. blair and a. w. hager, extensions of zero-sets and of real-valued functions, math. z.136 (1974), 41{52.[3] r. engelking, general topology (heldermann verlag, berlin, 1989).[4] l. gillman and m. jerison, rings of continuous functions (van nostrand, new york,1960).[5] g. gruenhage, generalized metric spaces, in: handbook of set-theoretical topology, k.kunen and j. e. vaughan, eds. 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(north-holland, amsterdam, 1984), 685{732.[22] y. yajima, special re�nements and their applications on products, preprint.[23] k. yamazaki, c�-embedding and c-embedding on product spaces, tsukuba j. math. 21(1997), 515{527.[24] k. yamazaki, c�-embedding and por m-embedding on product spaces, preprint.[25] k. yamazaki, e-mail, january 16, 1998. received march 2000 h. ohtafaculty of educationshizuoka universityohya, shizuoka, 422-8529, japane-mail address: echohtaipc.shizuoka.ac.jp @ appl. gen. topol. 17, no. 2(2016), 159-172 doi:10.4995/agt.2016.4704 c© agt, upv, 2016 digital fixed points, approximate fixed points, and universal functions laurence boxer a, ozgur ege b, ismet karaca c, jonathan lopez d, and joel louwsma e a department of computer and information sciences, niagara university, ny 14109, usa; and department of computer science and engineering, suny at buffalo, buffalo, ny, usa. (boxer@niagara.edu) b department of mathematics, celal bayar university, muradiye, manisa 45140, turkey. (ozgur.ege@cbu.edu.tr) c department of mathematics, ege university, bornova, izmir 35100, turkey. (ismet.karaca@ege.edu.tr) d department of mathematics, canisius college, buffalo, ny 14208, usa. (lopez11@canisius.edu) e department of mathematics, niagara university, ny 14109, usa. (jlouwsma@niagara.edu) abstract a. rosenfeld [23] introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. in the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. among these are several results concerning the relationship between universal functions and the approximate fixed point property (afpp). 2010 msc: primary 55m20; secondary 55n35. keywords: digital image; digitally continuous; digital topology; fixed point. 1. introduction in digital topology, we study geometric and topological properties of digital images via tools adapted from geometric and algebraic topology. prominent among these tools is a digital version of continuous functions. in the current received 16 february 2016 – accepted 17 july 2016 http://dx.doi.org/10.4995/agt.2016.4704 l. boxer, o. ege, i. karaca, j. lopez and j. louwsma paper, we study fixed points and approximate fixed points of digitally continuous functions. we present a number of original results and some corrections of previously published assertions. the paper is organized as follows. section 2 reviews background material. in section 3, we show that a digital image x has the fixed point property (fpp) if and only if x has a single point. in section 4 we introduce approximate fixed points and the approximate fixed point property (afpp). we give examples of digital images that have, and that don’t have, this property. in section 5 we study universal functions on digital images and their relation to the afpp. in section 6 we correct errors that appeared in earlier papers. concluding remarks appear in section 7. 2. preliminaries 2.1. general properties. a fixed point of a function f : x → x is a point x ∈ x such that f(x) = x. for a finite set x, we denote by |x| the number of distinct members of x. let n be the set of natural numbers and let z denote the set of integers. then zn is the set of lattice points in euclidean n−dimensional space. a digital image is a pair (x,κ), where ∅ 6= x ⊂ zn for some positive integer n and κ is an adjacency relation on x. technically, then, a digital image (x,κ) is an undirected graph whose vertex set is the set of members of x and whose edge set is the set of unordered pairs {x0,x1} ⊂ x such that x0 6= x1 and x0 and x1 are κ-adjacent. adjacency relations commonly used for digital images include the following [22]. two points p and q in z2 are 8 −adjacent if they are distinct and differ by at most 1 in each coordinate; p and q in z2 are 4 − adjacent if they are 8-adjacent and differ in exactly one coordinate. two points p and q in z3 are 26−adjacent if they are distinct and differ by at most 1 in each coordinate; they are 18−adjacent if they are 26-adjacent and differ in at most two coordinates; they are 6−adjacent if they are 18-adjacent and differ in exactly one coordinate. for k ∈ {4, 8, 6, 18, 26}, a k − neighbor of a lattice point p is a point that is k−adjacent to p. the adjacencies discussed above are generalized as follows. let u,n be positive integers, 1 ≤ u ≤ n. distinct points p,q ∈ zn are called cu-adjacent if there are at most u distinct coordinates j for which |pj − qj| = 1, and for all other coordinates j, pj = qj. the notation cu represents the number of points q ∈ zn that are adjacent to a given point p ∈ zn in this sense. thus the values mentioned above: if n = 1 we have c1 = 2; if n = 2 we have c1 = 4 and c2 = 8; if n = 3 we have c1 = 6, c2 = 18, and c3 = 26. yet more general adjacency relations are discussed in [19]. let κ be an adjacency relation defined on zn. a digital image x ⊂ zn is κ − connected [19] if and only if for every pair of points {x,y} ⊂ x, x 6= y, there exists a set {x0,x1, . . . ,xc} ⊂ x such that x = x0, xc = y, and xi and xi+1 are κ−neighbors, i ∈{0, 1, . . . ,c− 1}. a κ-component of x is a maximal κ-connected subset of x. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 160 digital fixed and approximate fixed points often, we must assume some adjacency relation for the white pixels in zn, i.e., the pixels of zn\x (the pixels that belong to x are sometimes referred to as black pixels). in this paper, we are not concerned with adjacencies between white pixels. definition 2.1 ([3]). let a,b ∈ z, a < b. a digital interval is a set of the form [a,b]z = {z ∈ z | a ≤ z ≤ b} in which 2−adjacency is assumed. definition 2.2 ([4]; see also [23]). let x ⊂ zn0 , y ⊂ zn1 . let f : x → y be a function. let κi be an adjacency relation defined on z ni , i ∈ {0, 1}. we say f is (κ0,κ1)−continuous if for every κ0−connected subset a of x, f(a) is a κ1−connected subset of y . see also [11, 12], where similar notions are referred to as immersions, gradually varied operators, and gradually varied mappings. if a and b are members of a digital image (x,κ), we write a ↔κ b, or a ↔ b when κ is understood, to indicate that either a = b or a and b are κ-adjacent. we say a function satisfying definition 2.2 is digitally continuous. this definition implies the following. proposition 2.3 ([4]; see also [23]). let x and y be digital images. then the function f : x → y is (κ0,κ1)-continuous if and only if for every {x0,x1}⊂ x such that x0 and x1 are κ0−adjacent, f(x0) ↔κ1 f(x1). for example, if κ is an adjacency relation on a digital image y , then f : [a,b]z → y is (2,κ)−continuous if and only if for every {c,c + 1} ⊂ [a,b]z, f(c) ↔κ f(c + 1). we have the following. proposition 2.4 ([4]). composition preserves digital continuity, i.e., if f : x → y and g : y → z are, respectively, (κ0,κ1)−continuous and (κ1,κ2)−continuous functions, then the composite function g ◦f : x → z is (κ0,κ2)−continuous. we say digital images (x,κ) and (y,λ) are (κ,λ) − isomorphic (called (κ,λ) − homeomorphic in [3, 5]) if there is a bijection h : x → y that is (κ,λ)-continuous, such that the function h−1 : y → x is (λ,κ)-continuous. classical notions of topology [2] yielded the concept of digital retraction in [3]. let (x,κ) be a digital image and let a be a nonempty subset of x. a retraction of x onto a is a (κ,κ)-continuous function r : x → a such that r(a) = a for all a ∈ a. a digital simple closed curve is a digital image x = {xi}m−1i=0 , with m ≥ 4, such that the points of x are labeled circularly, i.e., xi and xj are adjacent if and only if j = (i− 1) (mod m) or j = (i + 1) (mod m). 2.2. digital homotopy. a homotopy between continuous functions may be thought of as a continuous deformation of one of the functions into the other over a time period. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 161 l. boxer, o. ege, i. karaca, j. lopez and j. louwsma definition 2.5 ([4]; see also [21]). let x and y be digital images. let f,g : x → y be (κ,κ′)-continuous functions. suppose there is a positive integer m and a function f : x × [0,m]z → y such that • for all x ∈ x, f(x, 0) = f(x) and f(x,m) = g(x); • for all x ∈ x, the induced function fx : [0,m]z → y defined by fx(t) = f(x,t) for all t ∈ [0,m]z is (2,κ′)−continuous. • for all t ∈ [0,m]z, the induced function ft : x → y defined by ft(x) = f(x,t) for all x ∈ x is (κ,κ′)−continuous. then f is a digital (κ,κ′)−homotopy between f and g, and f and g are digitally (κ,κ′)−homotopic in y . when the adjacency relations κ and κ′ are understood in context, we say f and g are digitally homotopic to abbreviate “digitally (κ,κ′)−homotopic in y .” definition 2.6. a digital image (x,κ) is κ-contractible [21, 3] if its identity map is (κ,κ)-homotopic to a constant function p for some p ∈ x. when κ is understood, we speak of contractibility for short. 2.3. digital simplicial homology. our presentation of digital simplicial homology is taken from that of [16]. a set of m + 1 distinct mutually adjacent points is an m-simplex. definition 2.7. if αq is the number of (κ,q)-simplices in x and m = max{q ∈ n∗ |αq 6= 0}, then m is the dimension of (x,κ), denoted dim(x,κ) or dim(x), and the euler characteristic of (x,κ), χ(x,κ), is defined by χ(x,κ) = m∑ q=0 (−1)qαq. 2 for q ∈ n∗, the group of q-chains of (x,κ), denoted cκq (x), is the free abelian group with basis being the set of q-simplices of x. let δq : c κ q (x) → cκq−1(x) defined by δq(< p0,p1, . . . ,pq >) = { ∑q i=0(−1) i < p0,p1, . . . , p̂i, . . . ,pq > if 0 ≤ q ≤ dim(x); 0 if q > dim(x), where p̂i means that pi is omitted from the vertices of the simplex considered. then δq is a homomorphism, and we have δq−1 ◦ δq = 0 [1]. this gives rise to the following groups [9]. • zκq (x) = kerδq, the group of digital simplicial q-cycles of x. • bκq (x) = imδq+1, the group of digital simplicial q-boundaries of x. • the quotient group hκq (x) = zκq (x) /bκq (x), the q-th digital simplicial homology group of x. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 162 digital fixed and approximate fixed points we have the following. theorem 2.8 ([9]). let (x,κ) be a directed digital simplicial complex of dimension m. • hq(x) is a finitely generated abelian group for every q ≥ 0. • hq(x) is a trivial group for all q > m. • hm(x) is a free abelian group, possibly {0}. 3. fixed point property we say a digital image (x,κ) has the fixed point property (fpp) if every (κ,κ)-continuous function f : x → x has a fixed point. some properties of digital images with the fpp were studied in [14]. however, the following shows that for digital images with cu-adjacencies, the fpp is not very interesting. theorem 3.1. let (x,κ) be a digital image. then (x,κ) has the fpp if and only if |x| = 1. proof. clearly, if |x| = 1 then (x,κ) has the fpp. now suppose |x| > 1. if (x,κ) has more than 1 κ-component, then there is a (κ,κ) continuous map f : x → x such that for all x ∈ x, x and f(x) are in different κ-components of x. such a map does not have a fixed point. therefore, we may assume x is κ-connected. since |x| > 1, there are distinct κ-adjacent points x0,x1 ∈ x. consider the map f : x → x given by f(x) = { x0 if x 6= x0; x1 if x = x0. consider a pair y0,y1 of κ-adjacent members of x. • if one of these points, say, y0, coincides with x0, we have f(y0) = f(x0) = x1 and, since y1 6= x0, f(y1) = x0, so f(y0) and f(y1) are κ-adjacent. • if both y0 and y1 are distinct from x0, then f(y0) = x1 = f(y1) therefore, f is (κ,κ)-continuous. clearly, f does not have a fixed point. therefore, (x,κ) does not have the fpp. � 4. approximate fixed points given a digital image (x,κ) and a (κ,κ)-continuous function f : x → x, we say p ∈ x is an approximate fixed point of f if either f(p) = p, or p and f(p) are κ-adjacent. we say a digital image (x,κ) has the approximate fixed point property (afpp) if every (κ,κ)-continuous function f : x → x has an approximate fixed point. theorem 4.1 ([23]). let i = πni=1 [ai,bi]z. then (i,cn) has the afpp. theorems 3.1 and 4.1 show that it is worthwhile to consider the afpp, rather than the fpp, for digital images. we have the following. theorem 4.2. suppose (x,κ) has the afpp and there is a (κ,λ)-isomorphism h : x → y . then (y,λ) has the afpp. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 163 l. boxer, o. ege, i. karaca, j. lopez and j. louwsma proof. let f : y → y be (λ,λ)-continuous. by proposition 2.4, the function g = h−1 ◦f ◦h : x → x is (κ,κ) continuous, so our hypothesis implies there exists p ∈ x such that p ↔κ g(p). then h(p) ↔λ h◦g(p) = h◦h−1 ◦f ◦h(p) = f(h(p)), so h(p) is an approximate fixed point of f. � proposition 4.3. a digital simple closed curve of 4 or more points does not have the afpp. proof. let s = ({si}m−1i=0 ,κ) with si and sj κ-adjacent if and only if i = (j + 1) modm or i = (j − 1) modm. then the function f : s → s defined by f(si) = s(i+2) modm is (κ,κ)-continuous, and, for each i, si and f(si) are neither equal nor κ-adjacent. � next, we show retractions preserve the afpp. theorem 4.4. let (x,κ) be a digital image, and let y ⊂ x be a (κ,κ)-retract of x. if (x,κ) has the afpp, then (y,κ) has the afpp. proof. let r : x → y be a (κ,κ) retraction. let f : y → y be a (κ,κ)continuous function. let i : y → x be the inclusion map. by proposition 2.4, g = i ◦ f ◦ r : x → x is (κ,κ)-continuous. therefore, g has an approximate fixed point x0 ∈ x. let x1 = g(x0) ∈ y . by choice of x0, it follows that x0 ↔ x1. then x1 = g(x0) ↔ g(x1) = i◦f ◦r(x1) = i◦f(x1) = f(x1). thus x1 is an approximate fixed point of f. � following a classical construction of topology, the wedge of two digital images (a,κ) and (b,λ), denoted a ∧ b, is defined [17] as the union of the digital images (a′,µ) and (b′,µ), where • a′ ∩b′ has a single point, p; • if a ∈ a′ and b ∈ b′ are µ-adjacent, then either a = p or b = p; • (a′,µ) and (a,κ) are isomorphic; and • (b′,µ) and (b,λ) are isomorphic. in practice, we often have κ = λ = µ, a = a′, b = b′. we have the following. theorem 4.5. let a and b be digital images. then (a∧b,κ) has the afpp if and only if both (a,κ) and (b,κ) have the afpp. proof. let a∩b = {p}. let pa,pb : a∧b → a∧b be the functions pa(x) = { x if x ∈ a; p if x ∈ b. pb(x) = { p if x ∈ a; x if x ∈ b. it is easily seen that both of these functions are well defined and (κ,κ)-continuous. also, let ia : a → a∧b and ib : b → a∧b be the inclusion functions, which are clearly (κ,κ)-continuous. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 164 digital fixed and approximate fixed points suppose (a,κ) and (b,κ) have the afpp. let f : a∧b → a∧b be (κ,κ)continuous. we must show that there exists a point of a ∧ b that is equal or κ-adjacent to its image under f. if f(p) = p, then we have realized that goal. otherwise, without loss of generality, f(p) ∈ a\{p}. by proposition 2.4, h = pa ◦ f ◦ ia : a → a is (κ,κ)-continuous. since a has the afpp, there exists a ∈ a such that (4.1) h(a) ↔κ a. if f(a) ∈ b, then p = pa ◦f(a) = pa ◦f ◦ ia(a) = h(a). it follows from statement (4.1) that (4.2) p ↔κ a. if f(a) 6= p then f(a) ∈ b \ {p} and f(p) ∈ a \ {p}, so f(a) and f(p) are distinct, non-adjacent points. this is a contradiction of statement (4.2), since f is continuous. therefore, we have f(a) ∈ a. then f(a) = pa ◦f(a) = pa ◦f ◦ ia(a) = h(a). it follows from statement (4.1) that f(a) ↔κ a. since f was arbitrarily selected, it follows that (a∧b,κ) has the afpp. conversely, suppose (a∧b,κ) has the afpp. since the maps pa and pb are (κ,κ)-retractions of (a ∧ b,κ) onto (a,κ) and (b,κ), respectively, it follows from theorem 4.4 that (a,κ) and (b,κ) have the afpp. � 5. universal functions and the afpp in this section, we define the notion of a universal function and study its relation to the afpp. definition 5.1. let (x,κ) and (y,λ) be digital images. a (κ,λ)-continuous function f : x → y is universal for (x,y ) if given a (κ,λ)-continuous function g : x → y , there exists x ∈ x such that f(x) ↔λ g(x). the notion of a dominating set in graph theory corresponds to the notion of a dense set in a topological space. definition 5.2 ([10]). let (x,κ) be a nonempty digital image. let y be a nonempty subset of x. we say y is κ-dominating in x if for every x ∈ x there exists y ∈ y such that x ↔κ y. theorem 5.3. let (x,κ) and (y,λ) be digital images. let f : x → y be a universal function for (x,y ). then f(x) is λ-dominating in y . proof. let y ∈ y and consider the constant function cy : x → y defined by cy(x) = y for all x ∈ x. this function is clearly (κ,λ) continuous. since f is universal, there exists xy ∈ x such that f(xy) is either equal to or λ-adjacent to y. since y was arbitrarily chosen, the assertion follows. � c© agt, upv, 2016 appl. gen. topol. 17, no. 2 165 l. boxer, o. ege, i. karaca, j. lopez and j. louwsma proposition 5.4. let x be a κ-connected digital image of m points. let (y,λ) be a digital interval or a digital simple closed curve of n points, with n > m+ 2. then there is no universal function from x to y . proof. let f : x → y be a (κ,λ) continuous function. then f(x) is a λconnected subset of y , and |f(x)| ≤ m < n. we show that y \f(x) has a component with at least 2 points, one of which is not λ-adjacent to any member of f(x). • if y is a digital interval [a,a + n−1]z, then, since f(x) is a connected subset of y , f(x) = [u,v]z. consider the following possibilities. – v ≤ a + n − 3. then the endpoint a + n − 1 of y \ f(x) is not adjacent to any point of f(x). – v > a + n− 3. therefore, v ≥ a + n− 2. then u = v −|f(x)| + 1 ≥ a + n− 2 −|f(x)| + 1 ≥ a + n− (m + 2) + 1 > a + 1. i.e., u ≥ a + 2, so the point a of y \f(x) is not adjacent to any point of f(x). • if y is a digital simple closed curve, we may assume y = {yj}n−1j=0 , where ya and yb are adjacent if and only if a = (b + 1) mod n or a = (b− 1) mod n. since f(x) is connected, we may assume without loss of generality that f(x) = {yj}rj=0 where 0 ≤ r < m < n−2. then yr+2 is a point of y \f(x) that is not adjacent to any point of f(x). thus, f(x) is not λ-dominating in y . the assertion follows from theorem 5.3. � proposition 5.5. let (x,κ) be a digital image. then (x,κ) has the afpp if and only if the identity function 1x is universal for (x,x). proof. the function 1x is universal if and only if for every (κ,κ)-continuous f : x → x, there exists x ∈ x such that f(x) ↔κ 1x(x) = x, which is true if and only if (x,κ) has the afpp. � theorem 5.6. let (x,κ) and (y,λ) be digital images and let u ⊂ x. if the restriction function f|u : (u,κ) → (y,λ) is a universal function for (u,y ), then f is a universal function for (x,y ). proof. let h : x → y be (κ,λ)-continuous. since f|u is universal, there exists u ∈ u ⊂ x such that h(u) = h|u (u) ↔λ f|u (u) = f(u). hence f is universal for (x,y ). � theorem 5.7. let (w,κ), (x,λ), and (y,µ) be digital images. let f : w → x be (κ,λ)-continuous and let g : x → y be (λ,µ)-continuous. if g◦f is universal, then g is also universal. proof. let h : x → y be (λ,µ)-continuous. since g◦f is universal, there exists w ∈ w such that (g ◦ f)(w) ↔µ (h ◦ f)(w). i.e., for x = f(w) ∈ x we have g(x) ↔µ h(x). since h was arbitrarily chosen, the assertion follows. � c© agt, upv, 2016 appl. gen. topol. 17, no. 2 166 digital fixed and approximate fixed points theorem 5.8. if g : (u,µ) → (x,κ) and h : (y,λ) → (v,ν) are digital isomorphisms and f : x → y is (κ,λ)-continuous, then the following are equivalent. (1) f is a universal function for (x,y ). (2) f ◦g is universal. (3) h◦f is universal. proof. (1 implies 2): let k : u → y be (µ,λ)-continuous. since f is universal, there exists x ∈ x such that (k ◦ g−1)(x) ↔ f(x). by substituting x = g(g−1(x)), we have k(g−1(x)) ↔ (f◦g)(g−1(x)). since k was arbitrarily chosen and g−1(x) ∈ u, it follows that f ◦g is universal. (2 implies 1): this follows from theorem 5.7. (1 implies 3): let m : x → v be (κ,ν)-continuous. since f is universal, there exists x ∈ x such that (h−1 ◦ m)(x) ↔ f(x). then m(x) = h((h−1 ◦ m)(x)) ↔ν (h◦f)(x). since m was arbitrarily chosen, it follows that h◦f is universal. (3 implies 1): suppose h ◦ f is universal. then given a (κ,λ)-continuous r : x → y , there exists x ∈ x such that h ◦ f(x) ↔ν h ◦ r(x). therefore, f(x) = (h−1 ◦ h ◦ f)(x) ↔λ (h−1 ◦ h ◦ r)(x) = r(x). since r was arbitrarily chosen, it follows that f is universal. � corollary 5.9. let f : (x,κ) → (y,λ) be a digital isomorphism. then f is universal for (x,y ) if and only if (x,κ) has the afpp. proof. the function f is universal, by theorem 5.8, if and only if f◦f−1 = 1x is universal, which, by proposition 5.5, is true if and only if (x,κ) has the afpp. � it may be useful to remind the reader for the following theorem that points that are cn-adjacent in z n may differ in every coordinate. concerning products, we have the following. theorem 5.10. let (xi,cni ) ⊂ zni , i = 1, 2, . . . ,m. let s = ∑m i=1 ni. consider the digital image x = πmi=1 xi ⊂ z s. if (x,cs) has the afpp then each (xi,cni ) has the afpp. proof. suppose (x,cs) has the afpp. let fi : xi → xi be (cni,cni )-continuous. then the function f : x → x defined by f(x1,x2, . . . ,xm) = (f1(x1),f2(x2), . . . ,fm(xm)) is (cs,cs)-continuous. by proposition 5.5, 1x is universal for (x,x). therefore, there is a point x∗ = (x1,∗,x2,∗, . . . ,xm,∗) ∈ x with xi,∗ ∈ xi such that x∗ ↔cs f(x∗). therefore, xi,∗ ↔cni fi(xi,∗) for all i. since fi was arbitrarily chosen, it follows that (xi,cni ) has the afpp. � c© agt, upv, 2016 appl. gen. topol. 17, no. 2 167 l. boxer, o. ege, i. karaca, j. lopez and j. louwsma 6. corrections of published assertions in this section, we correct some assertions that appear in [14, 16]. we show below that the function f : [0, 1]z → [0, 1]z defined by f(x) = 1−x (i.e., f(0) = 1, f(1) = 0) provides a counterexample to several of the assertions of [14]. clearly this function is (2, 2)-continuous and does not have a fixed point. we will need the following. definition 6.1 ([15]). let (x,κ) be a digital image whose digital homology groups are finitely generated and vanish above some dimension n. let f : x → x be a (κ,κ)-continuous map. the lefschetz number of f, denoted λ(f), is defined as λ(f) = n∑ i=0 (−1)i tr(fi,∗), where fi,∗ : h κ i (x) → h κ i (x) is the map induced by f on the i th homology group of (x,κ) and tr(fi,∗) is the trace of fi,∗. in studying digital maps from a sphere to itself, there is a question of how to represent a euclidean sphere digitally. • as in [16], we will represent s1 by the set s1 = [−1, 1]2z\{(0, 0)}⊂ z 2 and c1-adjacency with points {xj}7j=0 labeled circularly. • more generally, as in [16], we will represent sn by the set sn = [−1, 1]n+1z \{0n+1}⊂ z n+1 and c1-adjacency, where 0n+1 is the origin in zn+1. definition 6.2 ([16, 24]). suppose a continuous function f : (sn,κ) → (sn,κ) induces a homomorphism on the n-th homology group, f∗ : h κ n(sn) → hκn(sn), such that f∗([x]) = m[x] for some fixed m ∈ z, where [x] is a generator of hκn(sn). the value of m is the degree of f. theorem 6.3 ([8]). let s be a digital simple closed curve. for an isomorphism of s and a continuous non-surjective self-map of s to be homotopic, we must have |s| = 4. we state the following corrections. • incorrect assertion stated as theorem 3.3 of [14]: if (x,κ) is a finite digital image and f : x → x is a (κ,κ)-continuous function with λ(f) 6= 0, then f has a fixed point. in fact, the function f defined above is a counterexample to this assertion, since it is easily seen that λ(f) 6= 0. • incorrect assertion stated as theorem 3.4 of [14]: every (c1,c1)continuous function f : [0, 1]z → [0, 1]z has a fixed point. in fact, [23] shows that this assertion is false, and the function f defined above is a counterexample. • incorrect assertion stated as theorem 3.5 of [14]: let x = {(0, 0), (1, 0), (0, 1), (1, 1)}⊂ z2. then every (c1,c1)-continuous function f : x → x has a fixed point. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 168 digital fixed and approximate fixed points in fact, we can use the function f above to obtain a counterexample. let g : x → x be defined by g(x,y) = (x,f(y)). then g is (c1,c1)-continuous and has no fixed point. alternately, it follows from theorem 3.1 that the assertion is incorrect. • incorrect assertion stated as theorem 3.8 of [14]: let (x,κ) be a κcontractible digital image. then every (κ,κ)-continuous map f : x → x has a fixed point. in fact, since [0, 1]z is c1-contractible, the function f above provides a counterexample to this assertion. alternately, it follows from theorem 3.1 that the assertion is incorrect. • incorrect assertion stated as example 3.9 of [14]: let x = {(0, 0), (0, 1), (1, 1)}⊂ z2. then (x,c2) has the fpp. in fact, the map f : x → x defined by f(0, 0) = (0, 1), f(0, 1) = (1, 1), f(1, 1) = (0, 0) is (c2,c2)-continuous and has no fixed points. alternately, it follows from theorem 3.1 that the assertion is incorrect. • incorrect assertion stated as corollary 3.10 of [14]: any digital image with the same digital homology groups as a single point image has the fpp. to show this assertion is incorrect, observe that if x = {(0, 0), (0, 1), (1, 1)}⊂ z2 and y is a digital image of one point in z2, then [1, 9] h8q (x) = h 8 q (y ) = { z if q = 0; 0 if q 6= 0. it follows from theorem 3.1 that the assertion is incorrect. • incorrect assertions stated as example 3.17 and corollary 3.18 of [14]: the digital images (mss′6, 6) and (p 2, 6), each with more than one point, have the fpp. it follows from theorem 3.1 that these assertions are incorrect. • incorrect assertion stated as theorem 3.5 of [16]: if (x,κ) is a finite digital image and f : x → x is a (κ,κ)-continuous function with λ(f) 6= 0, then any map homotopic to f has a fixed point. in fact, we observed above that the function f, which is homotopic to itself and has λ(f) 6= 0, does not have a fixed point. • incorrect assertion stated as theorem 3.7 of [16]: if (x,κ) is a digital image such that χ(x,κ) 6= 0, then any map homotopic to the identity has a fixed point. in fact, we can take x = [0, 1]z, for which χ(x,c1) = (−1)1(2) + (−1)2(1) 6= 0, and the function f discussed above is homotopic to 1x and does not have a fixed point. • incorrect assertion stated as theorem 3.11 of [16]: let (sn,c1) ⊂ zn+1 be a digital n-sphere as described above, where n ∈{1, 2}. if f : sn → sn is a continuous map of degree m 6= 1, then f has a fixed point. in fact, we have the following. elementary calculations show that hc11 (s1) ≈ z; also, h c1 1 (s2) ≈ z 23 [13]. for n ∈ {1, 2}, as in the proof of theorem 3.1, we can choose distinct and adjacent x0 and x1 c© agt, upv, 2016 appl. gen. topol. 17, no. 2 169 l. boxer, o. ege, i. karaca, j. lopez and j. louwsma in sn and let f : sn → sn be given by f(x) = x0 for x 6= x0 and f(x0) = x1. clearly, f is continuous and does not have a fixed point. since f∗ : h1(sn) → h1(sn) is 0, the degree of f is 0. • proposition 3.12 of [16] depends on an unstated assumption that (recall definition 2.7) ακq (x) is finite for all q, a condition that is satisfied if and only if x is finite; after all, one can study infinite digital images (x,κ), as in [7], for which, e.g., ακ0 (x) = ∞. e.g., we could take x = z; according to definition 2.7, χ(z,c1) is undefined, since α c1 1 (z) = αc10 (z) = ∞. therefore, the proposition should be stated as follows. let (x,κ) be a finite digital image and suppose f : (x,κ) → (x,κ) is continuous. if f∗ : h κ ∗ (x) → hκ∗ (x) is defined by f∗(z) = kz where k ∈ z, i.e., if there exists k ∈ z such that in every dimension i we have f inducing the homomorphism fi∗ : h κ i (x) → h κ i (x) defined by fi∗(z) = kz, then λ(f) = k χ(x). • a theoretically minor, but possibly confusing, error in theorem 3.14 of [16]: in discussing an antipodal map f : x → x, one needs the property that for every x ∈ x we have −x ∈ x; this property does not characterize the version of s2 used in theorem 3.14 of [16]. in the following, we use s2 = [−1, 1]3z \{(0, 0, 0)}, as described above. theorem 3.14 of [16] asserts that if αi : (si,c1) → (si,c1) is the antipodal map between two digital i-spheres si ⊂ zi+1, for i ∈{1, 2}, then αi has degree (−1)i+1. in fact, we show that this assertion is correct for i = 1, although an argument different from that of [16] must be given, as the argument of [16] makes use of theorem 3.4 of [16] (= theorem 3.3 of [14]), which, as noted above, is incorrect. for i = 2, we show the assertion is not well defined. – for i = 1, we have the following. let the points {ej}7j=0 of s1 be circularly ordered. for notational convenience, let e8 = e0, and, more generally, index arithmetic is assumed to be modulo 8. the generators of the 1-chains of s1 are the members of {< ejej+1 >}7j=0. we have 0 = δ( 7∑ j=0 uj < ejej+1 >) = 7∑ j=0 uj(ej −ej+1) = 8∑ j=1 (uj −uj−1)ej implies u0 = u1 = · · · = u7. therefore, z1(s1) is generated by ∑7 j=0 < ejej+1 >. since, clearly, b1(s1) = {0}, we have h1(s1) = z1(s1)/b1(s1) is isomorphic to z. therefore, the homomorphism (α1)∗ : h1(s1) → h1(s1) induced by α1 must satisfy (α1)∗(x) = kx for some integer k. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 170 digital fixed and approximate fixed points indeed, since the antipode of ej is ej+4, we have (α1)∗( 7∑ j=0 < ejej+1 >) = 7∑ j=0 < ej+4ej+5 >= 7∑ j=0 < ejej+1 >, so k = 1 = (−1)1+1, as asserted. – for i=2, we observe that, using the c1 adjacency, there is no triple of distinct, mutually adjacent points in s2. therefore, h2(s2) = {0}. therefore, the degree of α2 is not well defined since for any integer k we have (α2)∗(x) = kx for all x = 0 ∈ h2(s2). • incorrect assertion stated as theorem 3.15 of [16]: let s1 be a digital simple closed curve in z2. if h : (s1,c1) → (s1,c1) is a continuous function that is homotopic to a constant function in s1, then h has a fixed point. in fact, we can take s1 as above with its points ordered circularly, s1 = {xj}7j=0 where distinct points xu,xv are adjacent if and only if u + 1 = v mod 8 or u − 1 = v mod 8. then, as in the proof of theorem 3.1, the function h : s1 → s1 given by h(x) = { x0 if x 6= x0; x1 if x = x0, is continuous and homotopic to the constant function x0 in s1 but has no fixed point. • correct (for |s1| > 4) assertion incorrectly “proven” as corollary 3.16 of [16]: let s1 be as above. let h : (s1,c1) → (s1,c1) be given by h(xi) = x(i+1) mod m, where m = |s1|. then h is not homotopic in s1 to a constant map. the argument given for this assertion in [16] depends on theorem 3.15 of [16], which, we have shown above, is incorrect. however, since h is easily seen to be an isomorphism, by theorem 6.3, the current assertion is true if and only if |s1| > 4. 7. summary we have shown that only single-point digital images have the fixed point property. however, digital n-cubes have the approximate fixed point property with respect to the cn-adjacency [23]. we have shown that the approximate fixed point property is preserved by digital isomorphism and by digital retraction, and we have a result concerning preservation of the afpp by cartesian products. we have studied relations between universal functions and the afpp. we have corrected several errors that appeared in previous papers. acknowledgements. the remarks of an anonymous reviewer were very helpful and are gratefully acknowledged. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 171 l. boxer, o. ege, i. karaca, j. lopez and j. louwsma references [1] h. arslan, i. karaca, and a. ŏztel, homology groups of n-dimensional digital images, xxi turkish national mathematics symposium (2008), b, 1–13. 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[23] a. rosenfeld, ‘continuous’ functions on digital pictures, pattern recognition letters 4 (1986), 177–184. [24] e. h. spanier, algebraic topology, mcgraw-hill, new york, 1966. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 172 () @ appl. gen. topol. 17, no. 2(2016), 123-127doi:10.4995/agt.2016.4555 c© agt, upv, 2016 a note on uniform entropy for maps having topological specification property sejal shah a, ruchi das b and tarun das b a department of mathematics, faculty of science, the maharaja sayajirao university of baroda, vadodara 390002, gujarat, india. (sks1010@gmail.com) b department of mathematics, faculty of mathematical sciences, university of delhi, delhi 110007, india. (rdasmsu@gmail.com, tarukd@gmail.com) abstract we prove that if a uniformly continuous self-map f of a uniform space has topological specification property then the map f has positive uniform entropy, which extends the similar known result for homeomorphisms on compact metric spaces having specification property. an example is also provided to justify that the converse is not true. 2010 msc: 37b40; 37b20. keywords: topological specification property; uniform entropy; uniform spaces. 1. introduction in [1], authors have defined and studied the notion of topological entropy of a continuous self-map f of a compact topological space as an analogue of the measure theoretic entropy. the relation between topological entropy and measure theoretic entropy is established by the variational principle, which asserts that h(t ) = sup{hµ(t )|µ ∈ pt (x)}, i.e., topological entropy equals the supremum of the measure theoretic entropies hµ(t ), where µ ranges over all t -invariant borel probability measures on x [10]. a dynamical system is called deterministic if its topological entropy vanishes [9]. one may argue that the future of a deterministic dynamical system can be predicted if its past is known [16]. in a similar way positive entropy may received 18 january 2016 – accepted 21 july 2016 http://dx.doi.org/10.4995/agt.2016.4555 s. shah, r. das and t. das be related to randomness and chaos. topological entropy also plays an important role as an invariant for classification of continuous maps up to conjugacy. a remarkable contribution of bowen is the definition of entropy for uniformly continuous self-maps on metric spaces [4]. authors in ([11], [12], [14]) have extended bowen’s definition of entropy to uniformly continuous self-maps on uniform spaces. the uniform entropy considered in this paper is the one introduced in [12]. for details on uniform entropy on a uniform space one can also refer [8]. the survey [8] is also devoted to the study of entropy for topological groups wherein the uniform entropy is used to define topological entropy for continuous endomorphisms of locally compact groups. for a review of entropy in various fields of mathematics and science one can refer [2]. recently in [7], authors have extended smale’s spectral decomposition theorem for anosov diffeomorphisms of compact manifolds to general topological spaces wherein they have extended the notions of expansivity and shadowing for general topological spaces. sensitivity and devaney chaos are also studied for continuous group actions on uniform spaces in [6]. let x be a non-empty set and let △x = {(x, x)|x ∈ x}, called as the diagonal of x × x. a subset m of x × x is said to be symmetric if m = mt , where mt = {(y, x)|(x, y) ∈ m}. the composite u ◦v of two subsets u and v of x × x is defined to be the set {(x, y) ∈ x × x| there exists z ∈ x satisfying (x, z) ∈ u and (z, y) ∈ v }. definition 1.1 ([13]). let x be a non-empty set. a uniform structure on x is a non-empty set u of subsets of x × x satisfying the following conditions: i: if u ∈ u then △x ⊂ u, ii: if u ∈ u and u ⊂ v ⊂ x × x then v ∈ u, iii: if u ∈ u and v ∈ u then u ∩ v ∈ u, iv: if u ∈ u then ut ∈ u, v: if u ∈ u then there exists v ∈ u such that v ◦ v ⊂ u. the elements of u are then called the entourages of the uniform structure and the pair (x, u) is called a uniform space. we work here with uniform space (x, u). note that the fact that points x and y are close (in terms of distance) in a metric space x is equivalent to the fact that point (x, y) is close to the diagonal △x of x × x in a uniform space (x, u). if (x, u) is a uniform space, then there is an induced topology on x characterized by the fact that the neighborhoods of an arbitrary point x ∈ x consists of the sets u[x], where u varies over all entourages of x. the set u[x] = {y ∈ x|(x, y) ∈ u} is called the cross section of u at x ∈ x. we denote the subsets of x × x by u and the subsets of x by u̇. the specification property for homeomorphisms on a compact metric space has turned out to be an important notion in the study of dynamical systems. it was first introduced by bowen to give the distribution of periodic points for axiom a diffeomorphisms [5]. informally the specification property means that it is possible to shadow two distinct pieces of orbits which are sufficiently apart in time by a single orbit. let f be a self homeomorphism on a compact metric c© agt, upv, 2016 appl. gen. topol. 17, no. 2 124 topological specification property and uniform entropy space (x, d). the map f is said to satisfy specification property if for every ǫ > 0 there exists a positive integer m(ǫ) such that for any finite sequence x1, x2, ..., xk in x, any integers a1 ≤ b1 < a2 ≤ b2 < ... < ak ≤ bk with aj − bj−1 ≥ m (2 ≤ j ≤ k) and p > m + (bk − a1), there exists x ∈ x such that fp(x) = x and d(fi(x), fi(xj)) < ǫ (aj ≤ i ≤ bj, 1 ≤ j ≤ k). in [3], it has been proved that on a compact metric space x if f : x → x is a homeomorphism satisfying specification property then the topological entropy of f is positive. in section 2, we extend this result to uniformly continuous self-maps defined on uniform spaces. in particular, we prove that a uniformly continuous map defined on a uniform space having topological specification property has positive uniform entropy. 2. main result in [15], we have introduced the notion of topological specification property for a homeomorphism on a uniform space however the notion of topological specification property can be defined for continuous maps as follows: definition 2.1. a continuous self-map f is said to have topological specification property if for every symmetric neighborhood u of the diagonal △x there exists a positive integer m such that for any finite sequence of points x1, x2, ..., xk in x, any integers 0 = a1 ≤ b1 < a2 ≤ b2 < ... < ak ≤ bk with aj − bj−1 ≥ m (2 ≤ j ≤ k) and any p > m + (bk − a1), there exists x ∈ x such that fp(x) = x and f i(x, xj) ∈ u, aj ≤ i ≤ bj, 1 ≤ j ≤ k. remark 2.2. if (x, d) is a compact metric space, then for any neighborhood u of △x, we can find ǫ > 0 such that uǫ = d −1(0, ǫ) ⊂ u. on the other hand, every uǫ is a neighborhood of △x. thus for compact metric spaces, definitions of topological specification property and of specification property coincide. let f : x → x be a uniformly continuous map, us denote the set of all symmetric elements of u, uo denote the set of open elements of u, n be a positive integer and let u ∈ us. a subset ė of x is said to be (n, u)-separated with respect to f if for each pair of distinct points x, y in ė there exists j such that 0 ≤ j < n and f j(x, y) 6∈ u, where f = f × f. for a subset ż of x, a subset ȧ of x is said to be an (n, u)-spanning set for ż with respect to f if for each x ∈ ż there exists y ∈ ȧ such that for all j with 0 ≤ j < n, we have f j(x, y) ∈ u. let k(x) denote the set of all compact subsets of x. for k̇ ∈ k(x), let sn(u, k̇, f) denote the maximal cardinality of (n, u)-separated sets contained in k̇ and let rn(u, k̇, f) denote the minimal cardinality of (n, u)spanning sets for k̇. note that a maximal (n, u)-separated subset of k̇ is an (n, u)-spanning set for k̇. define rf (u, k̇) = lim sup n→∞ 1 n logrn(u, k̇, f) sf (u, k̇) = lim sup n→∞ 1 n logsn(u, k̇, f) and c© agt, upv, 2016 appl. gen. topol. 17, no. 2 125 s. shah, r. das and t. das h(f, k̇, u) = lim{rf (u, k̇)|u ∈ u s} = lim{rf (u, k̇)|u ∈ u o} = lim{sf (u, k̇)|u ∈ u s} = lim{sf (u, k̇)|u ∈ u o} . for the proofs of above equalities one can refer lemma 1 in [12]. definition 2.3 ([12]). the number h(f, u) defined by sup{h(f, k̇, u)|k̇ ∈ k(x)} is called the uniform entropy of f with respect to the uniformity u. remark 2.4. for a uniformly continuous self-map of a complete metric space, the uniform entropy is equal to the bowen’s entropy [14]. theorem 2.5. let (x, u) be a uniform space and let f : x → x be a uniformly continuous map having topological specification property then h(f, u) is positive. proof. let x, y ∈ x, x 6= y, u be a symmetric neighborhood of △x such that (x, y) 6∈ u2 = u ◦ u and m be a number as in the definition of topological specification property. consider two distinct (n + 1)-tuples, (z0, z1, z2, ..., zn) and (z ′ 0, z ′ 1, z ′ 2, ..., z ′ n) with z0 = x, z ′ 0 = y, zi, z ′ i ∈ {x, y}(1 ≤ i ≤ n) and integers a0 = b0 = 0, a1 = b1 = m, a2 = b2 = 2m, ..., an = bn = nm. since f has topological specification property, there exist z, z′ ∈ x satisfying the definition of topological specification property. note that z 6= z′. for if z = z′ then f i(z, zi) ∈ u, aj ≤ i ≤ bj and 0 ≤ i, j ≤ n and f i(z, z′i) ∈ u, aj ≤ i ≤ bj and 0 ≤ i, j ≤ n. therefore f im (z, zi) ∈ u, 0 ≤ i ≤ n and f im (z, z′i) ∈ u, 0 ≤ i ≤ n. for i = 0, (z, z0) ∈ u and (z, z ′ 0) ∈ u implying that (x, y) ∈ u 2, which contradicts our choice of u. using similar arguments, one can prove that for distinct (n + 1)-tuples (z0, z1, z2, ..., zn) with zi ∈ {x, y}(0 ≤ i ≤ n) associated z are different. thus there are at least 2n+1 points which are (nm, u)-separated which implies h(f, u) = sup{h(f, k̇, u)|k̇ ∈ k(x)} ≥ lim u∈us sf (u, k̇) = lim n→∞ sup 1 n logsn(u, k̇, f) = lim n→∞ sup 1 nm log2n+1 = log2 m > 0. � the following example justifies that the converse of the above result is not true. example 2.6. let f : r → r be defined by f(x) = 2x, for all x ∈ r, where the set of real numbers r has the usual uniformity u. note that the bowen’s entropy of f is log2 [11]. by remark 2.4, h(f, u) = log2. it is known that a c© agt, upv, 2016 appl. gen. topol. 17, no. 2 126 topological specification property and uniform entropy map f having topological specification property has the dense set of periodic points [15]. since f(x) = 2x does not have dense set of periodic points therefore it does not have the topological specification property. acknowledgements. the second author is supported by ugc major research project f.n. 42-25/2013(sr) references [1] r. adler, a. konheim and m. mcandrew, topological entropy, trans. amer. math. soc. 114 (1965), 309–319 . [2] j. amigó, k. keller and v. unakafova, on entropy, entropy-like quantities, and applications, discrete contin. dyn. syst. ser. b 20 (2015), 3301–3343. [3] n. aoki, topological dynamics, topics in general topology. amsterdam, north-holland publishing co. 41 (1989), 625–740. [4] r. bowen, entropy for group endomorphisms and homogeneous spaces, trans. amer. math. soc. 153 (1971), 401–414. [5] r. bowen, periodic points and measures for axiom a diffeomorphisms, trans. amer. math. soc. 154 (1971) 377–397. [6] t. ceccherini-silberstein and m. coornaert, sensitivity and devaney’s chaos in uniform spaces, j. dyn. control syst. 19 (2013), 349–357. [7] t. das, k. lee, d. richeson and j. wiseman, spectral decomposition for topologically anosov homeomorphisms on noncompact and non-metrizable spaces, topology appl. 160 (2013), 149–158. [8] d. dikranjan, m. sanchis and s. virili, new and old facts about entropy in uniform spaces and topological groups, topology appl. 159 (2012), 1916–1942. [9] h. furstenberg, disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, math. systems theory 1 (1967), 1–49. [10] t. n. t. goodman, relating topological entropy and measure entropy, bull. london math. soc. 3 (1971), 176–180. [11] j. hofer, topological entropy for noncompact spaces, michigan math. j. 21 (1974), 235–242. [12] b. hood, topological entropy and uniform spaces, j. london math. soc. 8 (1974), 633– 641. [13] j. kelley, general topology, d. van nostrand company (1955). [14] t. kimura, completion theorem for uniform entropy, comment. math. univ. carolin. 39 (1998), 389–399. [15] s. shah, r. das and t. das, specification property for topological spaces, j. dyn. control syst. 22 (2016), 615–622. [16] b. weiss, single orbit dynamics, cbms regional conference series in mathematics, american mathematical society, providence, ri 95 (2000). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 127 () @ appl. gen. topol. 19, no. 2 (2018), 269-280doi:10.4995/agt.2018.9737 c© agt, upv, 2018 more on the cardinality of a topological space m. bonanzinga a, n. carlson b, m. v. cuzzupè a and d. stavrova c a dipartimento di scienze matematiche e informatiche, scienze fisiche e scienze della terra, university of messina, italy (mbonanzinga@unime.it, mcuzzupe@unime.it) b department of mathematics, california lutheran university, usa. (ncarlson@callutheran.edu) c department of mathematics, sofia university, sofia, bulgaria. (stavrova@fmi.uni-sofia.bg) communicated by d. georgiou abstract in this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. many authors have been working in that field. to mention a few, let us refer to results by arhangel’skii, alas, hajnaljuhász, bell-gisburg-woods, dissanayake-willard, schröder and to the excellent survey by hodel “arhangel’skĭı’s solution to alexandroff’s problem: a survey”. here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. we also provide partial answer to arhangel’skii’s question concerning whether the continuum is an upper bound for the cardinality of a hausdorff lindelöf space having countable pseudo-character (i.e., points are gδ). shelah in 1978 was the first to give a consistent negative answer to arhangel’skii’s question; in 1993 gorelic established an improved result; and further results were obtained by tall in 1995. the question of whether or not there is a consistent bound on the cardinality of hausdorff lindelöf spaces with countable pseudo-character is still open. in this paper we introduce the hausdorff point separating weight hpw(x), and prove that (1) |x| ≤ hpsw(x)alc(x)ψ(x), for hausdorff spaces and (2) |x| ≤ hpsw(x)wlc(x)ψ(x), where x is a hausdorff space with a π-base consisting of compact sets with non-empty interior. in 1993 schröder proved an analogue of hajnal and juhasz inequality |x| ≤ 2c(x)χ(x) for hausdorff spaces, for urysohn spaces by considering weaker invariant urysohn cellularity uc(x) instead of cellularity c(x). we introduce the n-urysohn cellularity n-uc(x) (where n ≥ 2) and prove that the previous inequality is true in the class of n-urysohn spaces replacing uc(x) by the weaker n-uc(x). we also show that |x| ≤ 2uc(x)πχ(x) if x is a power homogeneous urysohn space. 2010 msc: 54d25; 54d10; 54d15; 54d30. received 24 february 2018 – accepted 19 june 2018 http://dx.doi.org/10.4995/agt.2018.9737 m. bonanzinga, n. carlson, m. v. cuzzupè and d. stavrova keywords: n-hausdorff space; n-urysohn space; homogeneous spaces; cardinal invariants. 1. introduction we will follow the terminology and notation in [16]. firstly, we will discuss the classical hajnal and juhasz’s inequality |x| ≤ 2c(x)χ(x) proven for hausdorff spaces [13]. an improvement of this inequality is obtained by bonanzinga in [6] for the general class of n-hausdorff spaces. a space x is defined to be n-hausdorff (where n ≥ 2) if h(x) = n where h(x) is the hausdorff number of x, i.e. the smallest cardinal τ such that for every subset a ⊂ x, |a| ≥ τ, there exist neighborhoods ua,a ∈ a, such that ⋂ a∈a ua = ∅. for every n-hausdorff space x, the n-hausdorff pseudocharacter of x, denoted n-hψ(x), is defined as the smallest κ such that for each point x there is a collection {v (α,x) : α < κ} of open neighborhoods of x such that if x1,x2, ..,xn are distinct points from x, then there exist α1,α2, ...,αn < κ such that ⋂n i=1 v (αi,xi) = ∅. it was then proved that |x| ≤ 2 c(x)χ(x) holds if replacing the character with the hausdorff pseudo-character, and that for every 3hausdorff space the inequality |x| ≤ 2c(x)3-hψ(x) holds. in [12] gotchev proved that the latter inequality is true for every space x having finite hausdorff number. in [16] schröder investigated the inequality of hajnal and juhasz for urysohn spaces replacing cellularity c(x) with the weaker invariant urysohn cellularity uc(x) (as uc(x) ≤ c(x)). in section 2 we prove that schröder’s inequality |x| ≤ 2uc(x)χ(x), for a urysohn space x, can be restated for n-urysohn spaces provided the urysohn cellularity is replaced by the nurysohn cellularity (theorem 2.11 below). an analogue of the hajnal-juhasz inequality in the setting of homogeneous spaces was established in [8] where carlson and ridderbos use the erdös-rado theorem to show that if x is a power homogeneous hausdorff space then |x| ≤ 2c(x)πχ(x). in section 2 we prove that this result can be modified in the setting of urysohn spaces to give the homogeneous analogue of schröder’s result. in particular, we prove that if x is urysohn power homogeneous space then |x| ≤ 2uc(x)πχ(x). in section 3 we give a partial solution to arhangel’skii’s problem [3] concerning whether the continuum is an upper bound for the cardinality of a hausdorff lindelöf space having countable pseudo-character. in [9] charlesworth proved that |x| ≤ psw(x)l(x)ψ(x) for every t1 space x, where psw(x) is the mininum infinite cardinal κ such that x has an open cover s (called separating open cover) having the property that for each distinct x and y in x there is an s ∈ s such that x ∈ s and y /∈ s and such that each point of x is in at most κ elements of s. charlesworth’s result is one of the few that provided partial answer to both of the above arhangel’skii’s problem and another one formulated in the same paper: “is continuum an upper bound for t1 lindelöf space having countable character?”. shelah, in an unpublished c© agt, upv, 2018 appl. gen. topol. 19, no. 2 270 more on the cardinality of a topological space paper in 1978, was the first to provide a consistent negative answer to the question of arhangel’skii’s (whether or not 2ℵ0 is an upper bound for the cardinality of a hausdorff lindelöf space of countable pseudo-characher) by constructing a model of zfc + ch in which there is a lindelöf regular space of countable pseudo-character with cardinality c+ = ℵ2. shelah’s paper was eventually published in 1996 [17]. then, gorelic [11] proved that is consistent with ch that 2ω1 is arbitrarily large and there is a lindelöf, 0-dimensional hausdorff space x of countable pseudo-character with |x| = 2ω1, and thus improving shelah’s result. the question of whether or not there is a consistent bound on the cardinality of hausdorff lindelöf spaces with countable pseudo-character is still open. we introduce an analogue of psw(x) in the class of hausorff spaces, denoted hpsw(x), and prove that |x| ≤ hpsw(x)alc(x)ψ(x) for a hausdorff space x thus giving a partial answer to arhangel’skii’s problem in zfc by even replacing l(x) with the weaker invariant alc(x). this is also a partial answer to a question in [10] if in the main result which states that for hausdorff spaces x, |x| ≤ 2alc(x)χ(x), χ(x) can be replaced by ψ(x). we also prove that |x| ≤ hpsw(x)wlc(x)ψ(x), for a hausdorff space x with a π-base consisting of compact sets with non-empty interior. this result is closely related to results in [5], [4] and [1]. 2. a generalization of schröder’s inequality in [16], schröder gives the following definition: definition 2.1 ([16]). let x be a topological space. a collection v of open subsets of x is called urysohn-cellular, if o1,o2 in v and o1 6= o2 implies o1 ∩ o2 = ∅. the urysohn-cellularity of x, uc(x), is defined by uc(x) = sup{|v| : v is urysohn-cellular } + ℵ0. recall that a topological space x is said to be quasiregular provided for every open set v , there is a nonempty open set u such that the closure of u is contained in v . we observe the following properties. lemma 2.2. if x is a quasiregular space, then for every cellular family u such that |u| = κ there exists a urysohn cellular family u′ such that |u′| = κ. proof. let x be a quasiregular space and u be a cellular family with |u| = κ. for every u ∈ u there exists an open set vu ⊂ u such that vu ⊂ u. clearly, if u1 and u2 are distinct elements of u such that u1 ∩ u2 = ∅, we have vu1 ∩ vu2 = ∅. hence u ′ = {vu : u ∈ u} is a urysohn cellular family for x such that |u′| = κ. � property 2.3. if x is a quasiregular space, c(x) = uc(x). proof. clearly, uc(x) ≤ c(x). let uc(x) = κ and suppose that c(x) > κ. then by lemma 2.2 there exists a urysohn cellular family u such that |u| > κ; a contradiction. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 271 m. bonanzinga, n. carlson, m. v. cuzzupè and d. stavrova recall the following: theorem 2.4 ([12, corollary 3.2]). let x be a space with h(x) finite. then |x| ≤ 2c(x)χ(x). the previous result together with property 2.3 gives the following: corollary 2.5. if x is a quasiregular n-hausdorff (where n ≥ 2) space, |x| ≤ 2uc(x)χ(x). recall that in [7] the authors define a space x to be n-urysohn (where n ≥ 2) if u(x) = n where u(x) is the urysohn number of x, i.e. the smallest cardinal τ such that for every subset a ⊂ x, |a| ≥ τ, there exist neighborhoods ua,a ∈ a, such that ⋂ a∈a ua = ∅. we introduce the following: definition 2.6. let x be a topological space. a collection v of open subsets of x is called n-urysohn-cellular, where n ≥ 2, if o1,o2, ...,on in v and o1 6= o2 6= ... 6= on implies o1 ∩ o2 ∩ ... ∩ on = ∅. the n-urysohn-cellularity of x, n-uc(x), is defined by n-uc(x) = sup{|v| : v is n-urysohn-cellular } + ℵ0. clearly, if v is a urysohn cellular collection of open subsets, then v is nurysohn cellular for every n ≥ 2. also if uc(x) ≤ κ, then n-uc(x) ≤ κ for every n ≥ 2. question 2.7. is there a space x such that (n+1)-uc(x) = κ and n-uc(x) 6= κ? recall that the θ-closure of a set a in the space x is the set clθ(a) = {x ∈ x : for every neighborhood u ∋ x,u ∩ a 6= ∅} [19]. proposition 2.8. let {aα}α∈a be a collection of subsets of x, then ⋃ α∈a clθ(aα) ⊆ clθ( ⋃ α∈a aα). proof. if x ∈ ⋃ α∈a clθ(aα), then there exists α ∈ a such that x ∈ clθ(aα). therefore for every neighborhood ux we have ux ∩ aα 6= ∅. then ux ∩ ( ⋃ α∈a aα) 6= ∅. this implies x ∈ clθ( ⋃ α∈a aα). � the next lemma represents a modification of lemma 7 in [16]: lemma 2.9. let x be a topological space and µ = n-uc(x). let (uα)α∈a be a collection of open sets. then there are b1,b2, ...,bn−1 ⊆ a with |bi| ≤ µ∀i = 1,2, ...,n − 1 and ⋃ α∈a uα ⊆ clθ( ⋃ α∈b1∪b2∪...∪bn−1 uα). proof. let v = {v ⊂ x : v is open and ∃α ∈ a such that v ⊆ uα}. by zorn’s lemma, take a maximal n-urysohn-cellular family w ⊆ v and |w| ≤ µ. for every w ∈ w take β = β(w) such that uβ(w) ∈ {uα : α ∈ a} and w ⊆ uβ(w). we may assume β ∈ b = b1 ⊔ b2 ⊔ ... ⊔ bn−1,bi ⊆ a and |bi| ≤ µ,∀i = 1,2, ...,n − 1. we want to prove that ⋃ α∈a uα ⊆ clθ(( ⋃ α∈b1 uα) ∪ ... ∪ ( ⋃ α∈bn−1 uα)). c© agt, upv, 2018 appl. gen. topol. 19, no. 2 272 more on the cardinality of a topological space assume the contrary, then there exists x ∈ ⋃ α∈a uα and x /∈ clθ(( ⋃ α∈b1 uα)∪ ... ∪ ( ⋃ α∈bn−1 uα)). then we can find α0 ∈ a and a neighborhood ux of x such that x ∈ uα0 and ux ∩ (( ⋃ α∈b1 uα) ∪ ... ∪ ( ⋃ α∈bn−1 uα)) = ∅. then (uα0 ∩ ux) ⊆ ux and (uα0 ∩ ux) ∪ w is a n-urysohn cellular family containing w; this contradicts the maximality of w. � corollary 2.10 ([16]). let x be a topological space and µ = uc(x). let (uα)α∈a be a collection of open sets. then there is b ⊆ a with |b| ≤ µ and ⋃ α∈a uα ⊆ clθ ⋃ α∈b uα. theorem 2.11. let x be a n-urysohn space. then |x| ≤ 2n−uc(x)χ(x). proof. set µ = n-uc(x)χ(x). for every x ∈ x let b(x) denote an open neighbourhood base of x with |b(x)| ≤ µ. construct an increasing sequence {cα : α < µ +} of subsets of x and a sequence {vα : α < µ +} of open collections of open subsets of x such that: (1) |cα| ≤ 2 µ for all α < µ+. (2) vα = ⋃ {b(c) : c ∈ ⋃ τ<α cτ},α < µ +. 3 if {gγ1,γ2,...,γn−1 : (γ1,γ2, ...,γn−1) ⊆ µ} is a collection of subsets of x and each gγ1,γ2,...,γn−1 is the union of closures of ≤ µ many elements of vα and ⋃ {γ1,γ2,...,γn−1}⊆µ clθgγ1,γ2,...,γn−1 6= x then cα \ ⋃ {γ1,γ2,...,γn−1}⊆µ clθgγ1,γ2,...,γn−1 6= ∅. the construction is by transfinite induction. let x0 be a point of x and put c0 = {x0}. let 0 < α < µ + and assume that cβ has been constructed for each β < α. note that vα is defined by (2) and vα ≤ 2 µ. for each collection {gγ1,γ2,...,γn−1 : (γ1,γ2, ...,γn−1) ⊆ µ} of subsets of x where each gγ1,γ2,...,γn−1 is the union of closures of ≤ µ many elements of vα and ⋃ {γ1,γ2,...,γn−1}⊆µ clθgγ1,γ2,...,γn−1 6= x, choose a point of x \ ⋃ {γ1,γ2,...,γn−1}⊆µ clθgγ1,γ2,...,γn−1. let hα be the set of points chosen in this way, (clearly, |hα| ≤ 2 µ) and let cα = hα ∪ ( ⋃ β<α cβ). it is clear that the family {cα : 0 < α < µ +} constructed in this way satisfies condition (1),(2) and (3). let c = ⋃ α<µ+ cα. we shall show that c = x. assume there is y ∈ x \c. for every bγ1,bγ2, ...,bγn−1 ∈ b(y), with |bγi| > 1∀i = 1,2, ...,n − 1 and γ1,γ2, ...,γn−1 ⊆ µ define fγ1,γ2,...,γn−1 = {vc : c ∈ c,vc ∈ b(c),vc ∩ bγ1 ∩ bγ2 ∩ ... ∩ bγn−1 = ∅}. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 273 m. bonanzinga, n. carlson, m. v. cuzzupè and d. stavrova since x is n-urysohn, we have c ⊆ ⋃ {γ1,γ2,...,γn−1}⊆µ ⋃ fγ1,γ2,...,γn−1. by lemma 2.9 we find for every {γ1,γ2, ...γn−1} ⊆ µ subcollections gγ1,gγ2, ...,gγn−1 ⊆ f{γ1,γ2,...,γn−1}, |gγi| ≤ µ∀i = 1,2, ...,n − 1 such that ⋃ f{γ1,γ2,...,γn−1} ⊆ clθ n−1 ⋃ i=1 ( ⋃ gγi). note y /∈ clθ ⋃n−1 i=1 ( ⋃ gγi). indeed, since ( n−1 ⋃ i=1 ( ⋃ gγi)) ∩ bγ1 ∩ bγ2 ∩ ... ∩ bγn−1 = ∅ and then ( n−1 ⋃ i=1 ( ⋃ gγi)) ∩ (bγ1 ∩ bγ2 ∩ ... ∩ bγn−1) = ∅. find α < µ+ such that ⋃ {γ1,γ2,...,γn−1}⊆µ (gγ1 ∪ gγ2 ∪ ... ∪ gγn−1) ⊆ vα. then y /∈ ⋃ {γ1,γ2,...,γn−1}⊆µ clθ n−1 ⋃ i=1 ( ⋃ gγi) but cα ⊆ c ⊆ ⋃ {γ1,γ2,...,γn−1}⊆µ ⋃ fγ1,γ2,...,γn−1 ⊆ ⋃ {γ1,γ2,...,γn−1}⊆µ clθ n−1 ⋃ i=1 ( ⋃ gγi). put gγ1,γ2,...,γn−1 = ⋃n−1 i=1 ( ⋃ gγi). this contradicts 3. � corollary 2.12 ([16]). let x be a urysohn space. then |x| ≤ 2uc(x)χ(x). we end this section with a new cardinality bound for power homogeneous urysohn spaces involving the urysohn cellularity uc(x). recall that a space x is homogeneous if for every x,y ∈ x there exists a homeomorphism h : x → x such that h(x) = y. x is power homogeneous if there exists a cardinal κ for which xκ is homogeneous. it is well established that cardinality bounds on a topological space can be improved if the space possesses homogeneouslike properties. for example, while |x| ≤ 2c(x)χ(x) holds for any hausdorff space x, carlson and ridderbos [8] have shown that if x is additionally power homogeneous then |x| ≤ 2c(x)πχ(x), where πχ(x) denote the π-character of the space x. by modifying this result, we show below that an analogous result holds for urysohn power homogeneous spaces when uc(x) is used in place of c(x). it is first helpful to establish this result in the homogeneous setting. to prove the following result we will use the well-known erdös-rado theorem c© agt, upv, 2018 appl. gen. topol. 19, no. 2 274 more on the cardinality of a topological space which states that if f : [x]2 → κ is a function and |x| > 2κ, then there is some subset y of x with |y | ≥ κ+ such that f(y) = f(z) for all y,z ∈ [y ]2, where [y ]2 denotes the family of all subsets of y of cardinality = 2. theorem 2.13. if x is a homogeneous hausdorff space that is urysohn or quasiregular then |x| ≤ 2uc(x)πχ(x). proof. if x is quasiregular, then, by property 2.3 (following lemma 2 of this section), uc(x) = c(x) and the result follows from proposition 2.1 in [8]. so we assume x is urysohn. let κ = uc(x)πχ(x), fix p ∈ x, and let b be a local π-base at p such that |b| ≤ κ. as x is homogeneous, for all x ∈ x there exists a homeomorphism hx : x → x such that hx(p) = x. as x is urysohn, for all x 6= y ∈ x there exist open sets u and v such that x ∈ u, y ∈ v and u ∩v = ∅. then p ∈ h−1x [u]∩h −1 y [v ], an open set. as b is a local π-base at p, there exists b(x,y) ∈ b such that b(x,y) ⊆ h−1x [u]∩h −1 y [v ]. thus, hx[b(x,y)] ⊆ u, hy[b(x,y)] ⊆ v , and hx[b(x,y)] ∩ hy[b(x,y)] = ∅. the existence of b(x,y) for each x 6= y ∈ x defines a function b : [x]2 → b. suppose by way of contradiction that |x| > 2κ. as |b| ≤ κ, we can apply the erdös-rado theorem to the function b. thus, there exists a subset y of x with |y | = κ+ and a ∈ b such that b(x,y) = a for all distinct x,y ∈ y . observe that for every x 6= y ∈ y , we have hx[a] ∩ hy[a] = hx[b(x,y)] ∩ hy[b(x,y)] = ∅. this shows {hx[a] : x ∈ y } is a urysohn cellular family. however, |{hx[a] : x ∈ y }| = |y | = κ + > uc(x), which is a contradiction. thus, |x| ≤ 2κ = 2uc(x)πχ(x). � to establish the more general theorem in the power homogeneous case, we adapt the proof of theorem 2.3 in [8]. importantly, we adopt the following notation: if x is a power homogeneous space, let µ be a cardinal such that xµ is homogeneous. fix a projection π : xµ → x and a point p in the diagonal ∆(x,µ). let κ be a cardinal such that πχ(x) ≤ κ and fix a local π-base u at π(p) in x such that |u| ≤ κ. we may assume without loss of generality that κ ≤ µ. for any b ⊆ a ⊆ µ, let πa→b be the projection of x a to xb, and for a ⊆ µ, define u(a) by u(a) = { π−1 a→b [ ∏ b∈b ub ] : b ∈ [a]<ω, and ub ∈ u for all b ∈ b } . observe that the family a is a local π-base at pa in x a. the following is lemma 2.2 in [8]. this lemma establishes that if xµ is homogeneous that not only are there homeomorphisms hx : x µ → xµ such that hx(p) = x for all x ∈ x, but that we can guarantee these homeomorphisms have special properties. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 275 m. bonanzinga, n. carlson, m. v. cuzzupè and d. stavrova lemma 2.14. for every x ∈ ∆(x,µ) there is a homeomorphism hx : x µ → xµ such that hx(p) = x and the following conditions are satisfied; (1) for all z ∈ xµ, if zκ = pκ then π(hx(z)) = π(x), (2) for all u ∈ u(κ), there is a point q(u) ∈ π−1κ [u] and a basic open neighbourhood ux of hx(q(u))κ in x κ such that; (a) q(u)α = pα for all α ∈ µ \ κ, (b) π−1κ [ux] ⊆ hx[π −1 κ [u]]. we now prove a generalization of theorem 2.13. theorem 2.15. if x is a power homogeneous hausdorff space that is urysohn or quasiregular then |x| ≤ 2uc(x)πχ(x). proof. if x is quasiregular then again uc(x) = c(x) and the proof follows directly from theorem 2.3 in [8]. so as in the last proof we assume x is urysohn. let κ = uc(x)πχ(x). for every x ∈ ∆(x,µ), fix a homeomorphism hx : xµ → xµ as in lemma 2.14 above. now, for x ∈ ∆(x,µ) and u ∈ u(κ), the set ux obtained from lemma 2.14 is a basic open set in x κ. we may thus fix a collection {ux,α : α < κ} of open sets in x such that ux = ⋂ α<κ π−1α [ux,α]. for every α ∈ κ we can fix a local π-base {v (x,u,α,β) : β < κ} of the point hx(q(u))α in x. in claim 1 in the proof of theorem 2.3 in [8], it is shown that whenever x 6= y ∈ ∆(x,µ) there exists u ∈ u(κ) and α,β < κ such that v (x,u,α,β) ⊆ ux,α\uy,α. we make a related claim in our proof. we omit the proof as it is very similar. claim. whenever x 6= y ∈ ∆(x,µ), there is u ∈ u(κ) and α,β < κ such that v (x,u,α,β) ⊆ ux,α and v (x,u,α,β) ∩ uy,α = ∅. continuing with our main proof, by way of contradiction, assume that |x| > 2κ. define a a map g : [x]2 → u(κ) × κ × κ as follows. for {x,y} ∈ [x]2 and x 6= y, we apply the claim and let g({x,y}) = 〈u,α,β〉 be such that the conclusion of the claim is satisfied. here we have identified ∆(x,µ) with x. as |u(κ)×κ×κ| = κ and |x| > 2κ, we can apply the erdös-rado theorem to find y ⊂ x and 〈u,α,β〉 ∈ u(κ)×κ×κ such that |y | = κ+ and for all {x,y} ∈ [y ]2, g({x,y}) = 〈u,α,β〉. thus for all y ∈ y we have v (y,u,α,β) ⊆ uy,α. let c = {v (x,u,α,β) : x ∈ y } and note that c is a collection of open subsets of x. if x 6= y ∈ y then v (x,u,α,β) ∩ uy,α = ∅ and v (y,u,α,β) ⊆ uy,α, and therefore v (x,u,α,β) and v (y,u,α,β) are disjoint. this means c is a urysohn cellular family. however, |c|= |y | = κ+ > uc(x), which is a contradiction. thus |x| ≤ 2κ. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 276 more on the cardinality of a topological space this above result shows that schröder’s cardinality bound 2uc(x)χ(x) for urysohn spaces can be improved in the power homogeneous setting. question 2.16. if x is power homogeneous and n-urysohn, is |x| ≤ 2n−uc(x)πχ(x)? 3. the hausdorff point separating weight recall the following properties which represent weaker forms of the lindelöf degree l(x): • the almost lindelöf degree of x with respect to closed sets, denoted alc(x), is the smallest infinite cardinal κ such that for every closed subset h of x and every collection v of open sets in x that covers h, there is a subcollection v′ of v such that |v′| ≤ κ and {v : v ∈ v′} covers h; • the weak lindelöf degree of x with respect to closed sets, denoted wlc(x), is the smallest infinite cardinal κ such that for every closed subset h of x and every collection v of open sets in x that covers h, there is a subcollection v′ of v such that |v′| ≤ κ and h ⊆ ⋃ v′. the following holds: wlc(x) ≤ alc(x) ≤ l(x) recall the following definition: definition 3.1 ([9]). a point separating open cover s for a space x is an open cover of x having the property that for each distinct points x and y in x there is s in s such that x is in s but y is not in s. the point separating weight of a space x is the cardinal psw(x) = min{τ : x has a point separating cover s such that each point of x is contained in at most τ elements of s} + ℵ0 definition 3.2. a hausdorff point separating open cover s for a space x is an open cover of x having the property that for each distinct points x and y in x there is s in s such that x is in s but y is not in s. the hausdorff point separating weight of a hausdorff space x is the cardinal hpsw(x) = min{τ : x has a hausdorff point separating cover s such that each point of x is contained in at most τ elements of s} + ℵ0 recall the following: theorem 3.3 ([9, theorem 2.1]). if x is t1, then nw(x) ≤ psw(x) l(x). following the proof of theorem 2.1 in [9], we prove the following: theorem 3.4. if x is a hausdorff space, then nw(x) ≤ hpsw(x)alc(x). c© agt, upv, 2018 appl. gen. topol. 19, no. 2 277 m. bonanzinga, n. carlson, m. v. cuzzupè and d. stavrova proof. let alc(x) = κ and let s be a hausdorff point separating open cover for x such that for each x ∈ x we have |sx| ≤ λ, where sx denotes the collection of members of s containing x. we first show that d(x) ≤ λκ. for each α < κ+ construct a subset dα of x such that: (1) |dα| ≤ λ κ. (2) if u is a subcollection of ⋃ {sx : x ∈ ⋃ β<α dβ} such that |u| ≤ κ and x \ ⋃ u 6= ∅, then dα \ ⋃ u 6= ∅. such a dα can be constructed since the number of possible u’s at the αth stage of construction is ≤ (λκ · κ · λ)κ = λκ. let d = ⋃ α<κ+ dα. clearly |d| ≤ λ κ. furthermore d is a dense subset of x. indeed, if there is a point p ∈ x \ d, since hpsw(x) ≤ λ, for every x ∈ d there exists an open set vx ∈ sx such that x ∈ vx and p /∈ vx. since x ∈ d, we have vx ∩ d 6= ∅. fix y ∈ vx ∩ d. then vx ∈ ⋃ {sy : y ∈ d}. put w = {vx : x ∈ d} ⊆ ⋃ {sy : y ∈ d}. clearly, w is an open cover of d. since alc(x) ≤ κ, we can select a subcollection w′ ⊆ w with |w′| ≤ κ such that d ⊆ ⋃ {v : v ∈ w′} and p /∈ ⋃ {v : v ∈ w′}; this contradicts 2. since d(x) ≤ λκ we have that |s| ≤ λκ. let n = {x \ s : s is the union of at most κ members of s}. then |n| ≤ λκ and n is a network for x. � theorem 3.5. if x is hausdorff space, then |x| ≤ hpsw(x)alc(x)ψ(x). proof. it is known that if x is a t1 space, |x| ≤ nw(x) ψ(x). then by theorem 3.4, we have |x| ≤ hpsw(x)alc(x)ψ(x). � corollary 3.6. if x is a hausdorff space with l(x) = ω,ψ(x) = ω and hpsw(x) ≤ c, then |x| ≤ c. the previous corollary gives a partial solution to arhangel’skii’s problem [2, problem 5.2] concerning whether the continuum is an upper bound for the cardinality of a hausdorff lindelöf space having countable pseudo-character. remark 3.7. using remark 2.5 in [9] we note that countable pseudo-character is essential in corollary 3.6: if x is the product of 2ω copies of the two point discrete space, then x is hausdorff, lindelöf and ψ(x) > ω but |x| > 2ω. the following theorem, under additional hypothesis, gives a result similar to theorem 3.4 in which the weakly lindelöf degree with respect to closed sets takes the place of the almost lindelöf degree with respect to closed sets. theorem 3.8. if x is hausorff space with a π-base consisting of compact sets with non-empty interior, then nw(x) ≤ hpsw(x)wlc(x). proof. let wlc(x) = κ and let s be a hausdorff point separating open cover for x such that for each x ∈ x we have |sx| ≤ λ, where sx denotes the collection of members of s containing x. without loss of generality, we can suppose that the family sx is closed under finite intersection. we first show that d(x) ≤ λκ. for each α < κ+ construct a subset dα of x such that: c© agt, upv, 2018 appl. gen. topol. 19, no. 2 278 more on the cardinality of a topological space (1) dα ≤ λ κ. (2) if u is a subcollection of ⋃ {sx : x ∈ ⋃ β<α dβ} such that |u| ≤ κ and x \ ⋃ u 6= ∅, then dα \ ⋃ u 6= ∅. such a dα can be constructed since the number of possible u’s at the αth stage of construction is (≤ λκ · κ · λ)κ = λκ. let d = ⋃ α<κ+ dα. clearly |d| ≤ λκ. furthermore d is a dense subset of x. indeed if d 6= x, x \ d is a non-empty open set. since x has a π-base consisting of compact sets with non-empty interior, we can find a non empty open subset w ⊆ x such that w is compact and w ⊂ x \ d, hence w ∩ d = ∅. fix x ∈ d. for every p ∈ w there exists an open subset vp ∈ sx such that p /∈ vp. then, we can find a family {vp : p ∈ w} of open subsets of x such that ⋂ {vp : p ∈ w} ∩ w = ∅. so, for the compactness of w the family {vp ∩ w : p ∈ w} can not have the finite intersection property. so put fx = vp1 ∩ ... ∩ vpk , where p1, ...,pκ ∈ w are such that fx ∩ w = ∅. put gx = vp1 ∩ ... ∩ vpk . since sx is closed under finite intersection, gx ∈ sx and gx ∩ w = ∅. since gx ∈ sx then gx ∈ sy for some y ∈ d. clearly, v = {gx : x ∈ d} is an open cover of d. using wlc(x) ≤ κ we can select a subcollection v ′ ⊆ v, |v′| ≤ κ such that d ⊆ ⋃ {v : v ∈ v′}. for every u ∈ ⋃ v′, u ∩ w = ∅, hence ⋃ v′ ∩ w = ∅. since w is a nonempty open set, ⋃ v′ ∩ w = ∅ and then x \ ⋃ v′ 6= ∅. this contradicts 2. since d(x) ≤ λκ we have that |s| ≤ λκ. let n = {x \ s|s is the union of at most κ members of s}. then |n| ≤ λκ and n is a network for x. � then we have the following result: theorem 3.9. if x is a hausdorff space with a π-base consisting of compact sets with non-empty interior, then |x| ≤ hpsw(x)wlc(x)ψ(x). acknowledgements. the authors are strongly indebted to the referee for the very careful reading of the paper. references [1] o. t. alas, more topological cardinal inequalities, colloq. math. 65, no. 2 (1993), 165–168. 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'`$z\ 9�^ [�9 k!u#t�l�tml8&>"%+]=;" v � tv&%=�'*&,","(+�9d[�[�eon=g=a�e=\!p+^�q4_�wy\#r�bscyz�w�btg4] @ appl. gen. topol. 17, no. 1(2016), 57-70 doi:10.4995/agt.2016.4401 c© agt, upv, 2016 two classes of metric spaces m. isabel garrido a,∗ and ana s. meroño b,∗ a instituto de matemática interdisciplinar (imi), departamento de geometŕıa y topoloǵıa, universidad complutense de madrid, 28040 madrid, spain (maigarri@mat.ucm.es) b departamento de análisis matemático, universidad complutense de madrid, 28040-madrid, spain. (anasoledadmerono@ucm.es) abstract the class of metric spaces (x, d) known as small-determined spaces, introduced by garrido and jaramillo, are properly defined by means of some type of real-valued lipschitz functions on x. on the other hand, b-simple metric spaces introduced by hejcman are defined in terms of some kind of bornologies of bounded subsets of x. in this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties. 2010 msc: primary 54e35; secondary 46a17; 54e40. keywords: metric spaces; real-valued uniformly continuous functions; realvalued lipschitz functions; bornologies; bourbaki-boundedness; countable uniform partitions; small-determined spaces; b-simple spaces. 1. introduction and preliminaries we are concerned here with two classes of metric spaces, namely smalldetermined spaces and b-simple spaces, which appear in separate frameworks into the general context of metric spaces. more precisely, one of them is related with the approximation and the extension of real-valued uniformly continuous functions (see [2]), whereas the other one is closer to boundedness and uniform ∗partially supported by mineco project mtm2012-34341 (spain). received 30 november 2015 – accepted 10 february 2016 http://dx.doi.org/10.4995/agt.2016.4401 m. i. garrido and a. s. meroño bornologies (see [6]). more recently, in a paper devoted to the study of some special realcompactifications of metric spaces ([4]), we have noticed that these worlds apparently far away can be connected. our main purpose in this note is to present a common framework where these classes of metric spaces can be studied. this setting will be simply of those “spaces in which all uniform partitions are countable”. spaces having this property are for instance every connected metric space, or more generally every uniformly connected, and also every separable metric space. we will see that on such spaces we can define a countable family of metrics uniformly equivalent to the original one that will be the key to obtaining the main results. 1.1. small-determined spaces. the class of small determined metric spaces, introduced in [2], is properly defined in terms of the so-called lipschitz in the small functions. recall that a function between metric spaces f : (x,d) → (y,d ′) is said to be lipschitz in the small if there exist δ > 0 and k > 0 such that d ′(f(x),f(y)) ≤ k ·d(x,y) whenever d(x,y) < δ. definition 1.1 ([2]). a metric space (x,d) is said to be small-determined whenever every real-valued lipschitz in the small function is lipschitz. if we denote by lipd(x) the set of all real-valued lipschitz functions and by lsd(x) the set of all real-valued lipschitz in the small functions on the metric space (x,d) then, clearly lipd(x) ⊂ lsd(x). to see that the reverse inclusion is not true, it is enough to consider the space of natural numbers n endowed with the usual metric and the function f : n → r defined by f(n) = n2, since f is lipschitz in the small but not lipschitz. hence n is not a small-determined space. nevertheless, as we can see in [2], small-determined metric spaces form a huge class of metric spaces containing for instance all the normed spaces and more generally all the length spaces. moreover these spaces have good properties of approximation and extension of real-valued functions. more precisely, they can be characterized by the fact that every real-valued uniformly continuous function can be uniformly approximated by lipschitz functions, and also characterized in terms of the notion of u-embedding. recall that a subspace x of a metric space (y,d) is said to be u-embedded in y if every real-valued uniformly continuous function defined on x admits a uniformly continuous extension to the whole space y . next result, taken from [2], summarizes all the above comments. theorem 1.2 ([2]). for a metric space (x,d) the following statements are equivalent: (1) x is small-determined. (2) every real uniformly continuous function can be uniformly approximated by lipschitz functions. (3) x is u-embedded in every metric space bi-lipschitz containing it. (4) x is u-embedded in every metric space isometrically containing it. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 58 two classes of metric spaces (5) x is u-embedded in some normed space isometrically containing it. (6) x is u-embedded in some normed space bi-lipschitz containing it. metric spaces fulfilling above condition (4), were studied by ramer in [9] and by levi and rice in [8]. their results as well as the characterizations of small-determined contained in [2] are somehow similar. and, in fact, they can be considered as results given in an external way, since they need to embed the initial metric space into a family of different spaces constructed for each ε > 0. here, we will see that an internal characterization can be done (theorem 3.1). 1.2. b-simple spaces. in order to introduce the second class of metric spaces we are interested in, we need to recall the notion of bornology on a set. so, a family b of subsets of a non-empty set x is said to be a bornology in x when it satisfies the following conditions: • for every x ∈ x the set {x}∈ b. • if b ∈ b and a ⊂ b then a ∈ b. • if a,b ∈ b then a∪b ∈ b. clearly the smallest bornology is the family pfinite(x) of finite subsets of x, while the biggest one is just p(x) the power set of x. one of the most interesting examples of bornology is obtained when we consider a metric space (x,d) and b = bd(x) is the family of all d-bounded subsets. we say that b ⊂ x is d-bounded when it has finite d-diameter, i.e., diamd(b) = sup{d(x,y) : x,y ∈ b} < ∞. another interesting bornology in a metric space is the formed by the socalled bourbaki-bounded subsets. a subset b in the metric space (x,d) is called bourbaki-bounded in x if for every ε > 0 there exist some points x1, ...,xn and some m ∈ n such that, b ⊂ bmd (x1,ε) ∪·· ·∪b m d (xn,ε) where bmd (x,ε) denotes the set of points y ∈ x that can be joined to x by means of an ε-chain in x of length m. recall that an ε-chain in x of length m from x to y, is any collection of points u0, . . . ,um ∈ x such that, u0 = x, um = y and d(ui,ui−1) < ε, for i = 1, . . . ,m. note that if in the above definition, we have always m = 1 for every ε > 0 then we get the notion of totally boundedness, since b1d(x,ε) is just bd(x,ε) the open ball centered in x and radius ε. bourbaki-bounded subsets in metric spaces were firstly considered by atsuji under the name of finitely-chainable subsets ([1]). in the general context of uniform spaces, they were introduced by hejcman in [5] where they are called uniformly bounded subsets. it is easy to check that bourbaki-boundedness is preserved by uniformly continuous functions, and therefore uniformly equivalent metrics define the same bourbaki-bounded subsets. moreover, if we denote by bbd(x) the family of bourbaki-bounded subsets in (x,d), then it is clear that it is a bornology in x such that bbd(x) ⊂ bd(x). while the reverse inclusion is not true in c© agt, upv, 2016 appl. gen. topol. 17, no. 1 59 m. i. garrido and a. s. meroño general. for instance if we consider in r the 0−1 discrete metric d, then every subset is d-bounded but only the finite ones are bourbaki-bounded. one of the most interesting characterization of bourbaki-boundedness was given by hejcman in [5]. namely, bourbaki-bounded subsets in the metric space (x,d) are those subsets being ρ-bounded for all the metrics ρ uniformly equivalent to d. that is, bbd(x) = ⋂{ bρ(x) : ρ u ' d } . so, our second class of metrics spaces defined by hejcman in [6] can be now introduced. definition 1.3 ([6]). a metric space (x,d) is said to be b-simple when there exists some metric ρ, uniformly equivalent to d, such that bbd(x) = bρ(x). that is, b-simple metric spaces are those for which bourbaki-bounded subsets can be determined (“recognized” as hejcman says in [6]) by just only one uniformly equivalent metric. note that, for these spaces their bourbakibounded subsets are precisely the bounded subsets for a uniformly equivalent metric, in other words the bourbaki-bounded bornology is uniformly metrizable. we refer to [3] where the general problem of uniformly metrizability of bornologies is studied. examples of b-simple metric spaces are again all normed spaces, all length spaces, and also every countable uniformly discrete spaces as n. nevertheless, any uncountable uniformly discrete space is not b-simple. indeed, note that for these spaces the bourbaki-bounded subsets are only the finite ones, and then the whole space can not be a countable union of bourbaki-bounded subsets. since any metric space is clearly a countable union of bounded sets, this means that bourbaki-boundedness and boundedness are different for these spaces. in order to give more information about b-simple spaces we present in the next result two characterizations taken from [6] and [3], respectively. recall that for a subset a of the metric space (x,d) and δ > 0, we define its δenlargement as aδd = ⋃ x∈a bd(x,δ). we say that a bornology b in (x,d) is stable under uniform enlargement if there exists some δ > 0 such that aδd ∈ b, for every a ∈ b. theorem 1.4. for a metric space (x,d) the following statements are equivalent: (1) x is b-simple. (2) x is a countable union of bourbaki-bounded sets, and bbd(x) is stable under uniform enlargement. (3) x is a countable union of bourbaki-bounded sets, and for some δ > 0, bmd (x,δ) ∈ bbd(x), for all x ∈ x and m ∈ n. in relation with b-simple spaces, we will see that they can be characterized with only a countable family of uniformly equivalent metrics (theorem 3.5). in particular, by means of these metrics we will give an useful characterization c© agt, upv, 2016 appl. gen. topol. 17, no. 1 60 two classes of metric spaces of bourbaki-boundedness (proposition 3.3) that will allow us to get complete answers (theorem 3.4) to some open questions posed by hejcman in [6]. we will finish this paper exhibiting an intermediate class of spaces that appear in the study of realcompactifications of metric spaces (see [4]). we are going to see that these spaces fit in a natural way in this context since they can be also characterized in terms of the above mentioned metrics (theorem 4.3). 2. a common framework let (x,d) be a metric space and ε > 0. as we have said before an ε-chain in x joining the points x and y is any collection of points u0, . . . ,um ∈ x such that, u0 = x, um = y and d(ui,ui−1) < ε, for i = 1, . . . ,m. so, we can define an equivalence relation on x as follows: x ε ' y if and only if there exists an ε-chain in x joining x and y. every equivalence class is called an ε-chainable component. note that different ε-chainable components are ε − d-apart, and hence they forms a uniform clopen partition of x. next result is the first one showing that our metric spaces have some common property. namely, they have at most countable many ε-chainable components, for every ε > 0. proposition 2.1. let (x,d) a metric space. then, (i) if x is small-determined then it has finitely many ε-chainable components, for every ε > 0. (ii) if x is b-simple then it has countably many ε-chainable components, for every ε > 0. proof. (i) suppose that for some ε > 0, x has not finitely many ε-chainable components, then we can write x = ⋃∞ k=1 ak as an infinite union of subsets that are ε−d-apart. now, if we choose xk ∈ ak, and define f(x) = k ·d(x1,xk) for x ∈ ak, then f is lipschitz in the small but not lipschitz. (ii) now if x is b-simple, let ρ be a metric uniformly equivalent to d such that bρ(x) = bbd(x). then x can be written as a countable union of bourbaki-bounded subsets, namely x = ⋃ k∈n bρ(x0,k) where bρ(x0,k) denotes the open ρ-ball centered in x0 ∈ x and radius k ∈ n. therefore, for every ε > 0, bρ(x0,k) meets at most finitely many ε-chainable components of (x,d), then it is clear that x has at most countably many of these ε-chainable components. � remark 2.2. note that the condition of the metric space (x,d) to have finitely or countably many ε-chainable components, for every ε > 0, is equivalent to say that every uniform partition of x is finite or countable, respectively. along the paper we will indistinctly use both equivalent conditions. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 61 m. i. garrido and a. s. meroño 2.1. a countable family of uniformly equivalent metrics. now, for a metric space (x,d) having at most countable many ε-chainable components, for every ε > 0, we are going to construct a countable family {ρn}n of metrics uniformly equivalent to d. indeed, let (x,d) a metric space with this property and let n ∈ n. take c1,c2, . . . ,ci, . . . , the 1/n-components in x, and choose representative points x1 ∈ c1,x2 ∈ c2, . . . ,xi ∈ ci, . . . . next, on each 1/ncomponent consider the metric defined by dn(x,y) = inf m∑ k=1 d(uk−1,uk) where the infimum is taken over all the 1/n-chains, x = u0,u1, ...,um = y, joining x and y. note that we only need to consider those chains such that, if m ≥ 2 then d(uk−1,uk) +d(uk,uk+1) ≥ 1/n since otherwise, due to the triangle inequality d(uk−1,uk+1) ≤ d(uk−1,uk) + d(uk,uk+1) < 1/n, the point uk can be removed from the initial chain. we will call these chains irreducible chains. it is easy to check that dn is in fact a metric on each 1/n-chainable component in a separately way, but we can extend it to the whole space x. for that, we define ρn as follows: definition 2.3. according to the above notation, we define for every n ∈ n, ρn(x,y) =   dn(x,y) if x,y ∈ ci dn(x,xi) + d(xi,x1) + i + dn(y,xj) + d(xj,x1) + j if x ∈ ci, y ∈ cj, i 6= j. proposition 2.4. let (x,d) be a metric space for which every uniform partition is countable. then for every n ∈ n, the function ρn : x ×x → r above defined is a metric on x. proof. it is clear that only it is necessary to check the triangle inequality for ρn and for points x,y,z which not all of them are in the same 1/n-chainable component. thus we have the next three cases: (a) if x,y ∈ ci and z ∈ cj, with i 6= j, then ρn(x,y) = dn(x,y) ≤ dn(x,xi) + dn(xi,y) ≤ ρn(x,z) + ρn(z,y). (b) if x,z ∈ ci and y ∈ cj, with i 6= j, we have ρn(x,y) = dn(x,xi) + d(xi,x1) + i + dn(y,xj) + d(xj,x1) + j ≤ ≤ dn(x,z) + dn(z,xi) + d(xi,x1) + i + dn(y,xj) + d(xj,x1) + j = = ρn(x,z) + ρn(z,y). (c) if x ∈ ci, y ∈ cj and z ∈ ck, with i 6= j 6= k 6= i, then ρn(x,y) = dn(x,xi)+d(xi,x1)+i+dn(y,xj)+d(xj,x1)+j ≤ ρn(x,z)+ρn(z,y). � c© agt, upv, 2016 appl. gen. topol. 17, no. 1 62 two classes of metric spaces note that, from the triangular inequality of the metric d, we deduce that d(x,y) ≤ ρn(x,y), for all x,y ∈ x. thus, the identity map id : (x,ρn) → (x,d) is in fact lipschitz. on the other hand, ρn(x,y) = d(x,y) whenever d(x,y) < 1/n, i.e., both metrics are what is called uniformly locally identical, for every n ∈ n (notion defined by janos and williamson in [7]). therefore the identity map id : (x,d) → (x,ρn) is now lipschitz in the small. that is, d and ρn are not only uniformly equivalent metrics but they are lipschitz in the small equivalent. it must be pointed here that if we change the representative points in each 1/n-chainable component, and we define the corresponding metric ωn : x × x → r similarly to ρn but with the new representative points, then we still have that d(x,y) = ωn(x,y) whenever d(x,y) < 1/n. so that, the three metrics ρn, ωn and d are uniformly locally identical. moreover, the election of these points will be irrelevant as we can see along the paper. in the next two results we are going to give a characterization of bounded sets for the metric ρn, n ∈ n, as well as the relationships between all of the bornologies of ρn-bounded sets. proposition 2.5. let (x,d) be a metric space for which every uniform partition is countable and let n ∈ n. then b ⊂ x is ρn-bounded if and only if for some xi1, . . . ,xik in x and m ∈ n, we have b ⊂ k⋃ j=1 bmd (xij, 1/n). proof. suppose that b ⊂ x is ρn-bounded, then for some x0 ∈ x and r > 0, we have that b ⊂ bρn (x0,r). now, from the definition of the metric ρn, the set b only meets a finite number of 1/n-chainable components, say ci1, . . . ,cik . let xi1, . . . ,xik be the corresponding representative points of these components, and take k > ρn(x0,xij ), j = 1, . . . ,k. now, if x ∈ b ∩cij we have that dn(x,xij ) = ρn(x,xij ) ≤ ρn(x,x0) + ρn(x0,xij ) < r + k. then there exists an irreducible 1/n-chain in cij joining x and xij , x = u0,u1, ...,um = xij such that ∑m l=1 d(ul−1,ul) < r+k. since the chain is irreducible, if m ≥ 2, every two consecutive sums satisfies d(ul−1,ul)+d(ul,ul+1) ≥ 1/n, and then (1/n)(m − 1)/2 ≤ r + k. in particular, the length of every irreducible chain must be less than m, where m is a natural number with m > 2n(r + k) + 1. we finish, since we have that, b = k⋃ j=1 ( b ∩cij ) ⊂ k⋃ j=1 bmd (xij, 1/n). conversely, we just need to see that every set of the form ⋃k j=1 b m d (xij, 1/n) is ρn-bounded. and for that it is enough to check that b m d (xij, 1/n) is ρnbounded, for every j = 1, . . . ,k. indeed, let x ∈ bmd (xij, 1/n) then ρn(x,xij ) = c© agt, upv, 2016 appl. gen. topol. 17, no. 1 63 m. i. garrido and a. s. meroño dn(x,xij ) ≤ m/n, that is x ∈ bρn (xij,m/n). hence we finish since bmd (xij, 1/n) ⊂ bρn (xij,m/n). � proposition 2.6. for a metric space (x,d) having countable uniform partitions, we have the following chain of inclusions: bd(x) ⊃ bρ1 (x) ⊃ bρ2 (x) ⊃ ···⊃ bρn (x) ⊃ bρn+1 (x) ⊃ ···⊃ bbd(x). proof. note that the first inclusion is clear since d ≤ ρn, for all n ∈ n, as we have pointed. regarding to the last inclusion, recall that the bourbakibounded subsets are the same with uniformly equivalent metrics, and also it is true that every bourbaki-bounded set for a given metric is a bounded set with this metric. and finally the remaining inclusions follow from the above proposition 2.5. indeed, for every n ∈ n, if b ∈ bρn+1 (x) then, for some xi1, . . . ,xik in x and m ∈ n, we have b ⊂ k⋃ j=1 bmd ( xij, 1/(n + 1) ) ⊂ k⋃ j=1 bmd (xij, 1/n). hence b ∈ bρn (x), as we wanted. � now we are ready to say which is the common framework where we are going to study our metric spaces. namely, it will be the frame of “metric spaces for which every uniform partition is countable together with the associated family of metrics {ρn}n”. 3. main results as we have seen, small-determined and b-simple metric spaces are into the above defined framework and we are going to see how the metrics {ρn} are good to describe them. theorem 3.1. a metric space (x,d) is small-determined if and only if, every uniform partition of x is finite and d is lipschitz equivalent to ρn, for every n ∈ n. proof. suppose x is small-determined. then, from proposition 2.1 and remark 2.2, it is only necessary to check the lipschitz equivalence between d and ρn, for every n ∈ n. indeed, as we have said above the identity map id : (x,ρn) → (x,d) is always lipschitz, while the other identity map id : (x,d) → (x,ρn) is lipschitz in the small. so, we finish taking into account that every lipschitz in the small function defined on a small-determined space is also lipschitz (see [2]). conversely, in order to see that (x,d) is small-determined, let f ∈ lsd(x). then there exist k ≥ 0 and n0 ∈ n such that |f(x) − f(y)| ≤ k · d(x,y) whenever d(x,y) < 1/n0. we are going to see that f : (x,ρn0 ) → r is lipschitz. indeed, since x has finitely many 1/n0-chainable components, let c© agt, upv, 2016 appl. gen. topol. 17, no. 1 64 two classes of metric spaces {xi1, ...,xik} be the finite set of representative points in the corresponding components ci1, ...,cik . if x,y ∈ cij for j = 1, ...,k, then it is routine to prove (by using 1/n0-chains) that |f(x) −f(y)| ≤ k ·dn0 (x,y) = k ·ρn0 (x,y). on the other hand, if x ∈ cij and y ∈ cil , with j 6= l, take m = max{k, |f(xij )− f(xil )|,j, l = 1, ...,k} then |f(x) −f(y)| ≤ |f(x) −f(xij )| + |f(xij ) −f(xil )| + |f(xil ) −f(y)| ≤ ≤ k ·dn0 (x,xij ) + m + k ·dn0 (xil,y) ≤ m ·ρn0 (x,y). finally since ρn0 and d are lipschitz equivalent we follow that f ∈ lipd(x). � last result can be considered as an internal characterization of the smalldetermined metric spaces. and we could derive, as a consequence of it, certain characterizations of them given in [2] as well as the corresponding results obtained in [8] and [9] in the context of extension of uniformly continuous functions. as an example we are going to see how obtaining easily the following result from [2]. recall that a metric space is uniformly connected when it can not be the union of two sets at positive distance. for instance the space of rational numbers q with the usual metric is clearly uniformly connected. corollary 3.2 ([2]). let (x,d) be a uniformly connected metric space. then x is small-determined if and only if the metrics d and dn are lipschitz equivalent, for every n ∈ n. proof. the proof follows at once from theorem 3.1. indeed, if x is uniformly connected then, for every n ∈ n, x has only one 1/n-chainable component and therefore ρn = dn. � next, before to see the relationship between b-simple metric spaces and the metrics {ρn}n, we are going to give an useful characterization of bourbakiboundedness in this context. proposition 3.3. let (x,d) be a metric space for which every uniform partition is countable. for a subset b ⊂ x the following are equivalent: (1) b is bourbaki-bounded in x. (2) b is ρn-bounded, for every n ∈ n. proof. from proposition 2.6, it is clear that (1) implies (2). conversely, suppose b is bounded for all the metrics ρn. in order to see that b is bourbakibounded let ε > 0 and take n ∈ n with 1/n < ε. now, from proposition 2.5, there are xi1, . . . ,xik in x and m ∈ n, such that b ⊂ k⋃ j=1 bmd (xij, 1/n) ⊂ k⋃ j=1 bmd (xij,ε). and then, b is bourbaki-bounded, as we wanted. � c© agt, upv, 2016 appl. gen. topol. 17, no. 1 65 m. i. garrido and a. s. meroño note that above result tell us that only countably many uniformly equivalent metrics are needed to recognize bourbaki-boundedness in metric spaces for which every uniform partition is countable. at this point we would like to recall that hejcman in [6] wondered which spaces would have precisely this property. moreover he also wonders if it would be reasonable to consider metrics in x not necessarily uniformly equivalent to the initial one in order to recognize bourbaki-boundedness. next result gives a complete answer to these open questions. theorem 3.4. let (x,d) be a metric space. then following assertions are equivalent: (1) every uniform partition x is countable. (2) for the family of metrics {ρn}n∈n, above defined, we have bbd(x) =⋂ n bρn (x). (3) there is a family {%n}n∈n of metrics uniformly equivalent to d such that bbd(x) = ⋂ n b%n (x). (4) there is a family {%n}n∈n of metrics in x such that bbd(x) =⋂ n b%n (x). proof. that (1) implies (2) is just proposition 3.3. that (2) implies (3) and (3) implies (4) are obvious. finally, suppose that (4) holds, but there is an uncountable uniform partition {ci : i ∈ i} of x. since the partition is uniform, there exist ε > 0 such that ci and cj are ε−d-apart, for i 6= j. now choose some point xi ∈ ci, for i ∈ i, and let a = {xi : i ∈ i}. note that none infinite subset of a can be bourbaki-bounded. now we are going to construct an infinite subset b ⊂ a being %n-bounded, for every n ∈ n, and that will contradict (4). indeed, let x0 a point in x. now, since x = ⋃∞ k=1 b%1 (x0,k), there is some open %1-ball b1 containing an uncountable subset a1 ⊂ a. analogously, there is an uncountable subset a2 ⊂ a1 contained in some open %2-ball b2. with an inductive process, we get a countable family of uncountable sets a1 ⊃ a2 ⊃ ···⊃ an ⊃ . . . , such that every an ⊂ bn for some open %n-ball bn. finally, let b = {x1,x2, . . . ,xn, . . .} be an infinite subset taking different points xn ∈ an. then b is an infinite subset of a that is %1-bounded since it is contained in b1, b is %2-bounded since b\{x1} is contained in a2 and also in b2, and so on. that is, for every n ∈ n, we have that b is %n-bounded since b \{x1, . . . ,xn−1} is contained in bn. � now turning to b-simple spaces we are going to characterize them by means of the metrics {ρn}n. theorem 3.5. a metric space (x,d) is b-simple if and only if every uniform partition is countable, and there is n0 ∈ n such that bbd(x) = bρn (x), for n ≥ n0. proof. one implication is clear from the definition of b-simple metric space. conversely, if x is b-simple then every uniform partition of x is countable (proposition 2.1 and remark 2.2). c© agt, upv, 2016 appl. gen. topol. 17, no. 1 66 two classes of metric spaces now let ρ a metric uniformly equivalent to d such that bbd(x) = bρ(x). now, from the uniform equivalence between d and ρ, there exists n0 ∈ n such that d(x,y) < 1/n0 implies ρ(x,y) < 1. that is, bd(x, 1/n) ⊂ bρ(x, 1), for every x ∈ x and n ≥ n0. and this implies that bmd (x, 1/n) ⊂ bρ(x,m), for every x ∈ x,n ≥ n0 and m ∈ n. we are going to see that for every n ≥ n0 we have that bbd(x) = bρn (x). indeed, one inclusion is clear since bbd(x) = bbρn (x) ⊂ bρn (x). on the other hand, let b ∈ bρn (x). now, from proposition 2.5, there are xi1, . . . ,xik in x and m ∈ n, such that b ⊂ k⋃ j=1 bmd (xij, 1/n) ⊂ k⋃ j=1 bρ(xij,m). then b ∈ bρ(x) = bbd(x) as we wanted. � a natural question here is if it is possible to change the above condition “for every n ≥ n0” by the condition “for every n ∈ n”. next example gives a negative answer to this question. example 3.6. let n be the set of natural numbers, let d the 0 − 1 discrete metric, and n0 > 1. consider the metric space (n, (1/n0) ·d), then it is clear that it is a countable uniformly discrete space and therefore b-simple. on the other hand, it is easy to check (see proposition 2.5) that for the associated metrics {ρn}n we have that bρn (n) = { p(n) for n = 1, 2, . . . , (n0 − 1) pfinite(n) for n ≥ n0. and then bb(1/n0)·d(n) = bρn (n) if and only if n ≥ n0. we finish this section with an example, taken precisely from [6], giving a positive answer to another question posed there by hejcman himself about the existence of non b-simple spaces with countably many uniformly equivalent metrics determining their bourbaki-bounded subsets. example 3.7. let x = rn the product metric space of countably many copies of the real line with the usual metric. we endowed x with the product metric d∞. since x is connected then any uniform partition has only one element, and then the bourbaki-bounded subsets can be determined by a countable family of uniform metrics (theorem 3.4). on the other hand, (x,d∞) is not b-simple since it can not be a countable union of bourbaki-bounded sets. otherwise, suppose x = ⋃ bn, where bn ∈ bbd∞ (x). since, for every n, the projection map πn : x → r is uniformly continuous, then πn(bn) is a bourbaki-bounded subset of r, and then it must be bounded with the usual metric. now take xn ∈ r \ πn(bn) and let x = (xn)n. then clearly x ∈ x \ ⋃ bn, which is a contradiction. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 67 m. i. garrido and a. s. meroño 4. an intermediate class of metric spaces in this brief section we are going to present a different class of metric spaces that is also in our general framework. first of all, we are going to see the relationship between small-determined and b-simple spaces. proposition 4.1. every small-determined metric space is b-simple. proof. the proof can be obtained easily using properly theorem 3.1 and proposition 3.3. indeed, if (x,d) is small-determined then d and ρn are lipschitz equivalent and then both metrics have the same bounded sets, for every n ∈ n. since bd(x) = bρn (x), for every n, it follows that bbd(x) = ⋂ bρn (x) = bd(x) and then x is clearly b-simple. � note that in the above proof we have seen indirectly that if (x,d) is smalldetermined then it is not only b-simple but it satisfies an strong property, namely bbd(x) = bd(x). in fact, for a metric space (x,d), we have the following chain of implications: x is small−determined ⇒ { x has the property bbd(x) = bd(x) } ⇒ x is b −simple. to see that the above reverse implications are not true the next easy examples can be considered. examples 4.2. let n be the set of natural numbers with the usual metric. then it satisfies bbd(n) = bd(n) but it is not small-determined. on the other hand, the real line r with the metric d̂ = min{1,d} where d denotes the usual metric, is b-simple but bb d̂ (r) 6= b d̂ (r). last examples show that the three classes of metric spaces are different. we wonder if this intermediate property corresponds to some already known class of metric spaces. we are very interested in these spaces since, as we have seen in [4], they are precisely those spaces for which the so-called samuel realcompactification and lipschitz realcompactification are equivalent, and also the spaces where every real-valued uniformly continuous function is bounded on bounded sets. we finish this paper giving a new characterization of this intermediate property by means of the metrics ρn. recall that two metrics are called boundedly equivalent if they are the same bounded subsets. theorem 4.3. for the metric space (x,d) the following assertions are equivalent. (1) x has the property bbd(x) = bd(x). (2) every uniform partition x is countable and d and ρn are boundedly equivalent, for every n ∈ n. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 68 two classes of metric spaces proof. obviously, condition (1) implies that x is b-simple, and therefore every uniform partition of x is countable. moreover, from proposition 2.6, if the first and last link of the chain coincide, then all the links are equal. that means that bd(x) = bρn (x), for every n ∈ n. that (2) implies (1) is an easy consequence of proposition 3.3. indeed, if d and ρn are boundedly equivalent, for every n ∈ n, then bd(x) = bρn (x) for every n ∈ n. finally, we have bbd(x) = ⋂ n bρn (x) = bd(x), as we wanted. � 5. conclusions and future research in this paper we have found a general uniform property that, in the frame of metric spaces, allows us to collect together different classes of spaces that were apparently far away. we refer to this mild property as to “having countably many elements in every uniform partition”. so, spaces coming from the bornological setting, as the b-simple spaces, as well as from the general study of lipschitz functions, as the small-determined spaces, lie in this common context. as we have seen here, the key to obtain the main results is the fact that we can define countably many metrics uniformly equivalent to the initial one. these metrics give a way to describe bornological properties (bounded sets, bourbaki-bounded sets) as well as some lipschitz functions (lipschitz in the small functions). moreover, along the present study a new kind of metric spaces appears in a natural way. namely those spaces for which the bourbaki-bounded subsets are exactly the bounded subsets. we do not know if these intermediate spaces correspond to some already studied class, and in fact we propose as future research to find more properties of them. we are very interested in these spaces since we already know that they can be characterized as those for which the samuel realcompactification and the lipschitz realcompactification are the same, as we have seen in our recent work [4], devoted precisely to the study of new realcompactifications for metric spaces. references [1] m. atsuji, uniform continuity of continuous functions of metric spaces, pacific j. math. 8 (1958), 11–16. [2] m. i. garrido and j. a. jaramillo, lipschitz-type functions on metric spaces, j. math. anal. appl. 340 (2008), 282–290. [3] m. i. garrido and a. s. meroño, uniformly metrizable bornologies, j. convex anal. 20 (2013), 285–299. [4] m. i. garrido and a. s. meroño, the samuel realcompactification of a metric space, submitted. [5] j. hejcman, boundedness in uniform spaces and topological groups, czechoslovak math. j. 9 (1959), 544–563. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 69 m. i. garrido and a. s. meroño [6] j. hejcman, on simple recognizing of bounded sets, comment. math. univ. carol. 38 (1997), 149–156. [7] l. janos and r. williamson, constructing metric with the heine-borel property, proc. amer. math. soc. 100 (1987), 568–573. [8] r. levy and m. d. rice, techniques and examples in u-emebdding, topology appl. 22 (1986), 157–174. [9] r. ramer, the extensions of uniformly continuous functions i, ii, indag. math. 31 (1969), 410–429. c© agt, upv, 2016 appl. gen. topol. 17, no. 1 70 () @ appl. gen. topol. 19, no. 2 (2018), 203-216doi:10.4995/agt.2018.7667 c© agt, upv, 2018 on rings of real valued clopen continuous functions susan afrooz a, fariborz azarpanah b and masoomeh etebar b a khoramshahr university of marine science and technology, khoramshahr, iran (afrooz@yahoo.com) b department of mathematics, shahid chamran university of ahvaz, ahvaz, iran (azarpanah@ipm.ir, m.etebar@scu.ac.ir) communicated by d. georgiou abstract among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper. we investigate and study the ring cs(x) of all real valued clopen continuous functions on a topological space x. it is shown that every f ∈ cs(x) is constant on each quasi-component in x and using this fact we show that cs(x) ∼= c(y ), where y is a zero-dimensional s-quotient space of x. whenever x is locally connected, we observe that cs(x) ∼= c(y ), where y is a discrete space. maximal ideals of cs(x) are characterized in terms of quasi-components in x and it turns out that x is mildly compact (every clopen cover has a finite subcover) if and only if every maximal ideal of cs(x) is fixed. it is shown that the socle of cs(x) is an essential ideal if and only if the union of all open quasi-components in x is s-dense. finally the counterparts of some familiar spaces, such as ps-spaces, almost ps-spaces, s-basically and s-extremally disconnected spaces are defined and some algebraic characterizations of them are given via the ring cs(x). 2010 msc: 54c40. keywords: clopen continuous (cl-supercontinuous); zero-dimensional; psspace; almost ps-space; baer ring; p.p. ring; quasi-component; socle; mildly compact; s-basically and s-extremally disconnected space. received 11 may 2017 – accepted 01 july 2017 http://dx.doi.org/10.4995/agt.2018.7667 s. afrooz, f. azarpanah and m. etebar 1. introduction if x and y are topological spaces, then a function f : x → y is said to be clopen continuous [9] or cl-supercontinuous [10] if for every x ∈ x and for each open set v in y containing f(x), there exists a clopen (closed and open) set u in x containing x such that f(u) ⊆ v . since this is a strong form of continuity, let us rename “clopen continuous” as strongly continuous and for brevity write s-continuous. if a is a subset of x, then s-interior of a denotes the set of all x ∈ a such that a contains a clopen set containing x and we denote it by intsa. a subset g of x is said to be s-open if g = intsg. in fact a set is s-open if and only if it is a union of clopen sets. the set of all x ∈ x such that every clopen set containing x intersects a is called s-closure of a and it is denoted by clsa. similarly a set h is called s-closed if h = clsh and a set is s-closed if and only if it is an intersection of clopen sets. a bijection function θ : x → y is called s-homeomorphism [10, under the name of cl-homeomorphism] if both θ and θ−1 are s-continuous. if such function from x onto y exists, we say that x and y are s-homeomorphic and we write x ∼=s y . we denote by c(x) the ring of all real-valued continuous functions on a space x and by cs(x) the set of all real valued s-continuous functions on x. it is easy to see that cs(x) is a ring and in fact it is a subring of c(x). for each f ∈ c(x), the zero-set of f, denoted by z(f), is the set of zeros of f and x \ z(f) is the cozero-set of f. the set of all zero-sets in x is denoted by z(x) and we also denote by zs(x) the set of all zero-sets z(f), where f ∈ cs(x). clearly zs(x) ⊆ z(x). if z ∈ zs(x), then it is the inverse image of the closed subset {0} of r under an element of cs(x) and this implies that every zero-set in zs(x) is s-closed. hence every cozero-set x \ z(f), where f ∈ cs(x), is s-open. the converse is not necessarily true. for instance let s be an uncountable space in which all points are isolated except for a distinguish point s. neighborhoods of s are considered to be those sets containing s with countable complement, see problem 4n in [6]. since {s} = ⋂ s6=a∈s(s \ {a}), the singleton {s} is s-closed but it is not a zero-set. it is well-known that a space x is a completely regular hausdorff space if and only if z(x) is a base for closed subsets of x, if and only if the set of all cozero-sets is a base for open subsets of x, see theorem 3.2 in [6]. whenever x is zero-dimensional (i.e., a t1 space with a base consisting of clopen sets), then clearly c(x) and cs(x) coincide, see also remark 2.3 in [10]. if x is a completely regular hausdorff space, the converse is also true, we cite this fact as a lemma for later use. lemma 1.1. whenever x is zero-dimensional, then c(x) = cs(x) and if x is a completely regular hausdorff space, the converse is also true. proof. the first implication is obvious, see also remark 2.3 in [10]. for the converse, as we have already mentioned the collection c = {x \ z(f) : f ∈ c(x)} is a base for open sets in x. now let f ∈ c(x) and x ∈ x \z(f). since f(x) 6= 0 and f ∈ c(x) = cs(x), there exists a clopen set u in x containing x such that f(y) 6= 0 , for each y ∈ u. hence u ⊆ x \ z(f) which means that x has a base with clopen sets. since x is t1, it is zero-dimensional. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 204 on rings of real valued clopen continuous functions we recall that a completely regular hausdorff space x is a p-space if every gδ-set or equivalently every zero-set in x is open and it is an almost p-space if every non-empty gδ-set or equivalently every nonempty zero-set in x has a non-empty interior. hence every p-space is an almost p-space but the converse fails, for instance, the one-point compactification of an uncountable discrete space is an almost p-space which is not a p-space, see example 2 in [8] and problem 4k(1) in [6]. basically (extremally) disconnected spaces are those spaces in which every cozero-set (open set) has an open closure. clearly every extremally disconnected space is basically disconnected but not conversely, see problem 4n in [6]. several algebraic and topological characterizations of aforementioned spaces are given in [3], [4], [6] and [8]. an ideal i in a commutative ring is called a z-ideal if ma ⊆ i for each a ∈ i, where ma is the intersection of all maximal ideals in the ring containing a. it is easy to see that an ideal i in c(x) (cs(x)) is a z-ideal if and only if whenever f ∈ i, g ∈ c(x) (g ∈ cs(x)) and z(f) ⊆ z(g), then g ∈ i, see problem 4a in [6]. a non-zero ideal in a ring is said to be essential if it intersects every non-zero ideal non-trivially. intersection of all essential ideals in a ring is called the socle of the ring. a topological characterization of essential ideals in c(x) and the socle of c(x) are given in [2] and [7] respectively. an ideal i in c(x) or cs(x) is said to be fixed if ⋂ f∈i z(f) 6= ∅, otherwise it is called free. the space βx is the stone-c̆ech compactification of x and for any p ∈ βx, mp (resp., op) is the set of all f ∈ c(x) for which p ∈ clβxz(f) (resp., p ∈ intβxclβxz(f)). the component cx of a point x in a topological space x is the union of all connected subspaces of x which contain x. the quasi-component qx of x is the intersection of all clopen subsets of x which contain x. clearly cx ⊆ qx for each x ∈ x and the inclusion may be strict, see example 6.1.24 in [5]. it is wellknown that for any two distinct points x and y in a space x, either qx = qy or qx ∩ qy = ∅, see [5]. components and quasi-components in a space x are closed and whenever x is locally connected, then they are also open, see corollary 27.10 in [12]. the converse of this fact is not true in general, see the example which is given preceding lemma 1.1. whenever the components in a space x are the points, then x is called totally disconnected. equivalently, x is totally disconnected if and only if the only non-empty connected subsets of x are the singleton sets. 2. some properties of clopen continuous functions behaviour of s-continuous functions on quasi-components is investigated in this section. the results of this section play an important role in the next sections. proposition 2.1. let x and y be topological spaces and f : x → y be a s-continuous function. then the following statements hold. (1) if x ∈ x, y ∈ y and f(x) = y, then f(qx) ⊆ qy. (2) if y is a t1-space, then f is constant on each quasi-component in x. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 205 s. afrooz, f. azarpanah and m. etebar (3) whenever f is one-to-one and y is a t1-space, then x is totally disconnected. (4) if f is s-homeomorphism and f(x) = y, then f(qx) = qy. moreover, if x or y is t1, then both x and y are totally disconnected. proof. (1) let z ∈ qx but f(z) /∈ qy. then there is a clopen set u in y such that f(z) ∈ u and u ∩ qy = ∅ which implies y /∈ u. since f is s-continuous, there exists a clopen set g in x containing z such that f(g) ⊆ u. but qx ⊆ g implies that f(qx) ⊆ u and this yields that f(x) = y ∈ u, a contradiction. (2) let y ∈ qx and f(x) 6= f(y). then there exists an open set h in y such that f(x) ∈ h but f(y) /∈ h. since f is s-continuous, there exists a clopen set g in x containing x such that f(g) ⊆ h. but any clopen set containing x contains y as well, for y ∈ qx, so f(y) ∈ h, a contradiction. (3) using part (2), f is constant on qx for every x ∈ x. but f is one-to-one, so qx is singleton for each x ∈ x. this implies that every quasi-component in x, and thus every component of x is singleton, i.e., x is totally disconnected. (4) it is evident by parts (1) and (3). � corollary 2.2. for every f ∈ cs(x), the following statements hold. (1) f is constant on each quasi-component in x. (2) the zero-set z(f) is a union of quasi-components in x. in fact z(f) =⋃ x∈z(f) qx. a space x is called ultra hausdorff [11] if for each pair of distinct points in x, there exists a clopen set in x containing one but not the other. two disjoint subsets of a space are called s-completely separated if there is a function f ∈ cs(x) which separates them. similar to the proof of theorem 1.15 in [6], it is easy to see that two disjoint sets are s-completely separated if and only if they are contained in two disjoint members of zs(x). by the following proposition, ultra hausdorff spaces are characterized by quasi-components in the spaces and every s-closed set is s-completely separated from every quasi-component disjoint from it. proposition 2.3. let x be a topological space. then the following statements hold. (1) if a is a s-closed subset of x and x ∈ x\a, then there exists g ∈ cs(x) such that g(a) = {1} and g(qx) = {0}. (2) x is ultra hausdorff if and only if every quasi-component in x is singleton. proof. (1) since x \ a is s-open, there exists a clopen set u containing x such that u ∩ a = ∅. now define an idempotent e with z(e) = u, i.e., e(u) = {0} and e(x \ u) = {1}. clearly e ∈ cs(x), e(qx) = 0, since qx ⊆ u and e(a) = 1. (2) whenever x is ultra hausdorff and x, y ∈ x are distinct, then there exists a clopen set u containing x but not y. this implies that y /∈ qx, i.e. qx c© agt, upv, 2018 appl. gen. topol. 19, no. 2 206 on rings of real valued clopen continuous functions is singleton. conversely, let x, y ∈ x be distinct points. since qx = {x}, there is a clopen set u such that x ∈ u and y /∈ u, i.e., x is ultra hausdorff. � 3. cs(x) is a c(y ) as an equivalent definition, a space x is zero-dimensional if and only if it is t1 and for each point x ∈ x and each closed subset a of x not containing x, there exists a clopen set g in x containing x such that g∩a = ∅. clearly every zero-dimensional space is tychonoff. whenever we consider the collection of all clopen subsets of (x, τ) as a base for a topology τ∗ on x, then cs(x, τ) = c(x, τ∗), by theorem 5.1 in [10]. but the space (x, τ∗) is not necessarily t1 and so it may not be zero-dimensional. in the following theorem we show that cs(x) is in fact a c(y ) for a zero-dimensional space y which is also a s-quotient space of x. theorem 3.1. for each topological space x, there exists a zero-dimensional space xz such that cs(x) ∼= c(xz). proof. for each x ∈ x, let qx be the quasi-component of x and consider the decomposition xz = {qx : x ∈ x}. take a topology τ on xz so that g ∈ τ if and only if ⋃ qx∈g qx is s-open in x. to see that τ is a topology, clearly x = ⋃ qx∈xz qx and ∅ = ⋃ qx∈∅ qx imply that xz and ∅ are open. whenever h and k are open sets in xz, then ⋃ qx∈h∩k qx = ( ⋃ qx∈h qx) ⋂ ( ⋃ qx∈k qx) imply that h ∩ k is open in xz. it is also easy to see that ⋃ α hα is open in xz for each open set hα in xz. the space xz is hausdorff, in fact if qx and qy are two distinct points in xz, where x, y ∈ x, then x /∈ qy and since qy is s-closed, there is an idempotent e ∈ cs(x) such that e(qy) = 0 and e(qx) = 1, by part (1) of proposition 2.3. if we set h = {qz : z ∈ z(e)}, then z(e) = ⋃ qz∈h qz implies that h is a clopen subset of xz. moreover qy ∈ h but qx /∈ h, i.e., xz is ultra hausdorff and hence it is hausdorff as well. to show that xz is zero-dimensional, let h be a closed set in xz and qy /∈ h. hence t = ⋃ qx∈h qx is a s-closed subset of x and y /∈ t . therefore by proposition 2.3, there exists a clopen set u in x containing t but not containing y. now ⋃ z∈u qz = u implies that k = {qz : z ∈ u} is a clopen subset of xz. clearly h ⊆ k and qy /∈ k, hence xz is zero-dimensional. now it remains to show that cs(x) ∼= c(xz). to this end, define ϕ : cs(x) → c(xz) by ϕ(f) = fz for each f ∈ cs(x), where fz is defined by fz(qx) = f(x), for each x ∈ x. by corollary 2.2, clearly ϕ is well-defined. moreover fz ∈ c(xz) for each f ∈ cs(x). in fact if fz(qx) = f(x) = c, then for each ε > 0, there exists a clopen set g in x containing x such that f(g) ⊆ (c−ε, c+ε), for f ∈ cs(x). now we take the open set h = {qz : z ∈ g} in xz containing qx. hence fz(h) = f(g) ⊆ (c − ε, c + ε) implies that fz ∈ c(xz). whenever ϕ(f) = ϕ(g), f, g ∈ cs(x), then fz = gz implies that f(x) = fz(qx) = gz(qx) = g(x), for all x ∈ x. hence f = g, i.e., ϕ is oneto-one. ϕ is also homomorphism, for ϕ(f + g) = (f + g)z and (f + g)z(qx) = (f+g)(x) = f(x)+g(x) = fz(qx)+gz(qx), for each qx ∈ xz. this implies that c© agt, upv, 2018 appl. gen. topol. 19, no. 2 207 s. afrooz, f. azarpanah and m. etebar ϕ(f +g) = ϕ(f)+ϕ(g), for all f, g ∈ cs(x) and hence ϕ is homomorphism. to complete the proof, we must show that ϕ is onto. to this end, let g ∈ c(xz). the function f : x → r defined by f(x) = g(qx), for all x ∈ x is s-continuous. in fact, if x ∈ x, f(x) = g(qx) = c and ε > 0 is given, then there is an open set h in xz containing qx such that g(h) ⊆ (c − ε, c + ε). now it is enough to take the s-open subset g = ⋃ qz∈h qz of x. clearly x ∈ g and f(g) ⊆ (c − ε, c + ε) which implies that f ∈ cs(x). by definitions of f and ϕ, it is clear that ϕ(f) = g and we have thus shown that ϕ is onto. � corollary 3.2. if every quasi-component in x is open, in particular, if x is locally connected, then cs(x) ∼= c(y ) for a discrete space y . proof. whenever each quasi-component of x is open, then each set {qx} is open in the space xz, defined in the proof of theorem 3.1. therefore each qx is an isolated point in xz, so y = xz is discrete and cs(x) ∼= c(y ). � the space xz defined in the proof of theorem 3.1 is a s-continuous image of x. in fact, if we regard the natural function τ : x → xz, with τ(x) = qx for each x ∈ x, then τ is continuous and fz ◦τ = f or equivalently, ϕ(f)◦τ = f. in order to prove that τ is s-continuous at x ∈ x, let h be a clopen subset of xz containing qx (in fact h is an element of a base of the zero-dimensional space xz). now it is enough to take u = ⋃ q∈h q which is s-open in x containing x and clearly τ(u) = h, i.e., τ is continuous at x. in this case we may say naturally, like the notion of quotient space, that xz is a s-quotient of x and the induced map τ is a s-quotient map. the equality fz ◦ τ = f is evident. we conclude this section by the following proposition. proposition 3.3. for two spaces x and y , if x ∼=s y , then xz ∼= yz. proof. let σ : x → y be a s-homeomorphism. by proposition 2.1, if σ(x) = y, then σ(qx) = qy. in fact every quasi-component in y is exactly the image of a unique quasi-component in x under σ. define φ : xz → yz by φ(q) = σ(q) for each quasi-component q in x. clearly φ is a bijection map. given a quasicomponent qx in x, we show that φ is continuous at qx. let h be an open subset of yz containing φ(qx) = σ(qx) = qy and take v = ⋃ q∈h q. hence by definition of open sets in yz, v is s-open in y containing σ(qx) = qy. since σ is onto, there exists an element of qx, say x without loss of generality, such that σ(x) = y. therefore there exists a clopen set u in x containing x (and hence containing qx) such that σ(u) ⊆ v . now we set g = {qz : z ∈ u}. clearly g is open in xz containing qx and φ(g) ⊆ h. similarly, φ −1 is also continuous and we are done. � the converse of the proposition 3.3 is not true. if we take the discrete space x = {a, b} and the space y = (0, 1) ∪ (1, 2) as a subspace of r, then clearly x ∼= xz and yz is a discrete space containing two elements (0, 1) and (1, 2). hence xz ∼= yz, but x and y are not s-homeomorphic. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 208 on rings of real valued clopen continuous functions 4. maximal ideals of cs(x) in this section, for each space x, we consider xz and the isomorphism ϕ : cs(x) → c(xz) as defined in the proof of theorem 3.1. first we show that ϕ takes fixed (free) ideals to fixed (free) ideals and using this, we transfer some well-known facts in the context of c(x) to that of cs(x). lemma 4.1. an ideal i in cs(x) is fixed if and only if ϕ(i) is a fixed ideal in c(xz). in particular, ϕ takes fixed maximal ideals to fixed maximal ideals. proof. if g ∈ ϕ(i), then there exists f ∈ i such that g = ϕ(f) = fz. now f(x) = 0 if and only if g(qx) = fz(qx) = f(x) = 0. therefore, x ∈ ⋂ f∈i z(f) if and only if qx ∈ ⋂ g∈ϕ(i) z(g) = ⋂ f∈i z(fz). � theorem 4.2. for a topological space x, the fixed maximal ideals in cs(x) are precisely the sets mqx = {f ∈ cs(x) : qx ⊆ z(f)} x ∈ x. the ideals mqx are distinct for distinct qx. for each x ∈ x, cs(x)/mqx is isomorphic with the real field r. proof. using lemma 4.1, fixed maximal ideals of cs(x) are precisely of the form ϕ−1(my), where y ∈ xz (i.e., y = qx for some x ∈ x). now f ∈ ϕ −1(my) if and only if fz ∈ my = mqx, or equivalently qx ⊆ z(f), by theorem 3.1. hence ϕ−1(my) = mqx. whenever qp 6= qq for p, q ∈ x, using proposition 2.3, there exists g ∈ cs(x) such that g(qp) = 0 and g(qq) = 1 and this means that g ∈ mqp \ mqq . finally σ : cs(x) → r with σ(f) = f(x) (note that f(y) = f(x), for all y ∈ qx by corollary 2.2), for all f ∈ cs(x) is a homomorphism with kernel mqx, so cs(x)/mqx ∼= r. � a space is said to be mildly compact [11] if every clopen cover of x has a finite subcover. clearly every compact space is mildly compact but not conversely. for instance, consider the space x = (0, 1) ∪ (1, 2) as a subspace of r. by the following proposition, for a space x, the compactness of xz is equivalent to mildly compactness of x. proposition 4.3. for a space x, the following statements are equivalent. (1) x is mildly compact. (2) xz is compact. (3) every ideal of cs(x) is fixed. (4) every maximal ideal of cs(x) is fixed. proof. a collection {hα : α ∈ s} is an open cover of xz if and only if the collection {gα : α ∈ s}, where gα = ⋃ q∈hα q, is a s-open cover of x. this implies the equivalence of parts (1) and (2). the equivalence of third and fourth parts with part (1) is an immediate consequence of lemma 4.1 and theorem 4.11 in [6]. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 209 s. afrooz, f. azarpanah and m. etebar using theorem 3.1, and in view of the fact that there is a correspondence between elements of the space βxz and the set of all maximal ideals of c(xz) by theorem 7.3 in [6], all maximal ideals of cs(x), fixed or free, will be characterize by the following theorem for each space x. theorem 4.4. for every space x, the maximal ideals of cs(x) are precisely of the form mp = {f ∈ cs(x) : p ∈ clβxz z(fz)} p ∈ βxz. remark 4.5. as in c(x), for each maximal ideal m of cs(x), we define om = {f ∈ cs(x) : fg = 0 for some g /∈ m}, see 7.12(b) in [6] and the argument preceding theorem 2.12 in [1]. whenever m is fixed, then m = mqx for some x ∈ x by theorem 4.2, and therefore we have om = oqx = {f ∈ cs(x) : qx ⊆ intsz(f)}. in fact, if f ∈ om , then fg = 0 for some g /∈ mqx. hence qx ⊆ x \ z(g) ⊆ z(f). but x \ z(g) is s-open, hence qx ⊆ intsz(f), so f ∈ oqx. whenever f ∈ oqx, then qx ⊆ intsz(f). since intsz(f) is s-open, there exists a clopen set u containing qx contained in intsz(f). now if we take an idempotent e such that z(e) = u, then qx ⊆ z(e) ⊆ intsz(f). therefore 1 − e /∈ mqx and (1 − e)f = 0 which implies that f ∈ oqx = om . by theorem 2.4 in [4], x is zero-dimensional if and only if for each x ∈ x, the ideal ox is generated by a set of idempotents. hence for each y ∈ xz (y = qx for some x ∈ x), the ideal oy is generated by a set of idempotents. now in view of theorem 2.4 in [4] and using the isomorphism ϕ defined in the proof of theorem 3.1, the following corollary is evident. corollary 4.6. for each x ∈ x, the ideal oqx in cs(x) is generated by a set of idempotents. remark 4.7. by theorem 4.9 in [6], two compact spaces x and y are homeomorphic if and only if c(x) and c(y ) are isomorphic. it is clear that if x ∼=s y , then cs(x) ∼= cs(y ). in fact, if σ : y → x is a s-homeomorphism, then f → f ◦ σ is a s-isomorphism between cs(x) and cs(y ). but in contrast to theorem 4.9 in [6], we observe that cs(x) ∼= cs(y ) does not necessarily imply x ∼=s y even if x and y are mildly compact. to see this, consider spaces x = { 1 n : n ∈ n} ∪ {0} and y = ( ⋃∞ n=1( 1 n+1 , 1 n )) ∪ {0} as subspaces of r. clearly, x and y are mildly compact. also cs(x) ∼= cs(y ), in fact, every g ∈ cs(y ) is constant on each interval ( 1 n+1 , 1 n ), by proposition 2.1, say g(( 1 n+1 , 1 n )) = {an}. now if we define fg : x → r by fg( 1 n ) = an and fg(0) = g(0), then fg ∈ cs(x) and θ : cs(y ) → cs(x), θ(g) = fg for each g ∈ cs(y ) is an isomorphism. since there is no bijection function between x and y , these two spaces are not s-homeomorphic. proposition 4.8. if xz ∼= yz, then cs(x) ∼= cs(y ) and whenever x and y are mildly compact, then the converse is also true. proof. xz ∼= yz implies that c(xz) ∼= c(yz) and hence by theorem 3.1, cs(x) ∼= cs(y ). for the converse, cs(x) ∼= cs(y ) implies c(xz) ∼= c(yz) c© agt, upv, 2018 appl. gen. topol. 19, no. 2 210 on rings of real valued clopen continuous functions by theorem 3.1. now using proposition 4.3, xz and yz are compact, hence xz ∼= yz, by theorem 4.9 in [6]. � 5. some relations between algebraic properties of cs(x) and topological properties of x we call a space x a ps-space if every zero-set in zs(x) is open. clearly, every p-space is a ps-space but not conversely. for instance, x = (0, 1)∪(1, 2) as a subspace of r is not a p-space whereas it is a ps-space, for zs(x) = {∅, x, (0, 1), (1, 2)}, by corollary 2.2. every ps-space is not necessarily a completely regular space. it is enough to consider a non-completely regular space with two components. whenever f ∈ cs(x), then z(fz) = {qx : f(x) = 0} and z(f) = ⋃ qx∈z(fz) qx. these imply, by definition of open sets in xz, that z(f) is s-open in x if and only if z(fz) is open in xz. on the oder hand, since cs(x) ∼= c(xz), by theorem 3.1, the ring c(xz) is regular if and only if cs(x) is a regular ring. in view of these points, the following result is an immediate consequence of problem 4j in [6]. proposition 5.1. a space x is a ps-space if and only if cs(x) is a regular ring. the counterparts of the other conditions of problem 4j in [6] can be obtained more or less for regularity of cs(x). for example, cs(x) is regular if and only if mqx = oqx, for each x ∈ x if and only if every ideal in cs(x) is a z-ideal and so on. note that for each f, g ∈ cs(x), it is easy to see that z(f) ⊆ z(g) if and only if z(fz) ⊆ z(gz) and this implies that an ideal i in cs(x) is a z-ideal if and only if ϕ(i) is a z-ideal in c(xz). we already observed that every ps-space is not necessarily a p-space. by the following result, this happens if and only if x is zero-dimensional. proposition 5.2. a space x is a p-space if and only if x is a zero-dimensional ps-space. proof. clearly, every p-space is basically disconnected, hence using problem 16o in [6], every p-space is zero-dimensional. every p-space is a ps-space as well. conversely, since x is zero-dimensional, c(x) = cs(x) by lemma 1.1 and since x is a ps-space, cs(x) is a regular ring, by proposition 5.1. this implies that c(x) is also a regular ring. now using problem 4j in [6], x is a p-space. � we call a space x an almost ps-space if every non-empty zero-set in zs(x) has a non-empty s-interior. however the notion of almost ps-space is the counterpart of that of almost p-space but the class of almost p-spaces and the class of almost ps-spaces are dissimilar. the following example shows that these classes are not comparable and non of them is larger than the other. example 5.3. whenever every quasi-component in a space x is open, in particular, if x is locally connected, then x is a ps-space. in fact if z(f) 6= ∅, c© agt, upv, 2018 appl. gen. topol. 19, no. 2 211 s. afrooz, f. azarpanah and m. etebar f ∈ cs(x), then z(f) is a union of quasi-components in x, by corollary 2.2. since each quasi-component in x is clopen, z(f) is s-open. this implies that y = (0, 1) ∪ (1, 2) as a subspace of r is a ps-space. hence y is an almost ps-space but clearly, it is not an almost p-space. also every almost p-space need not be an almost ps-space. to see this let x be a (completely regular hausdorff) connected almost p-space, see proposition 2.3 in [8] for existence of such a space. take a point σ ∈ x and let y = x ∪ n with the topology as follows: elements of n are considered to be isolated points, neighborhoods of all points of x, except σ will be the same as in the space x while each neighborhood of σ in y will be of the form g ∪ a, where g is an open set in x containing σ and a is a subset of n such that n \ a is finite. if we define f : y → r with f(n) = 1 n and f(x) = {0}, then f ∈ cs(y ). since x is connected and x = ⋂∞ n=1(y \ {n}), it is a quasi-component in y . also it does not contain any clopen subset of y (note that x itself is not open in y , for σ is the cluster point of the subset n of y and hence n is not closed in y ). hence intsz(f) = ∅, i.e., y is not an almost ps-space. it remains to show that y is an almost p-space. let f ∈ c(y ). if z(f)∩n 6= ∅, then clearly inty z(f) 6= ∅. now suppose that z(f) ∩ n = ∅. whenever σ /∈ z(f), then inty z(f) = intxz(f|x) 6= ∅, for x is an almost p-space. finally, suppose that σ ∈ z(f), then z(f) 6= {σ}, for otherwise intxz(f|x) = intx{σ} 6= ∅ implies that σ is an isolated point of x which contradicts the connectedness of x. therefore there exists x 6= σ such that x ∈ z(f). since y is completely regular hausdorff, define h ∈ c(y ) so that h(x) = 0 and h(σ) = 1. now take g = f2 + h2, then σ /∈ z(g) ⊆ x implies that ∅ 6= intxz(g) = inty z(g) ⊆ inty z(f). hence y is an almost p-space. for the proof of the following proposition, we need the following lemma. lemma 5.4. let f, g ∈ cs(x). (1) if z(g) ⊆ intsz(f), then f is a multiple of g. (2) zs(x) is closed under countable intersection. proof. (1) let xz, ϕ and fz for each f ∈ cs(x) be as in the proof of theorem 3.1. let qx ∈ z(gz), where x ∈ x. hence x ∈ z(g) and so x ∈ intsz(f), by our hypothesis. since intsz(f) is s-open, there exists a clopen set u such that x ∈ u ⊆ z(f). now h = {qy : y ∈ u} is clopen in xz and qx ∈ h ⊆ z(fz). this implies that z(gz) ⊆ intxz z(fz) and using problem 1d in [6], there is kz ∈ c(xz), where k ∈ cs(x) such that fz = kzgz. now it is clear that f = kg. (2) it is easy to see that whenever {sn} is a sequence in cs(x) converges uniformly to a function f, then f ∈ cs(x). now, as in 1.14(a) in [6], if for each n ∈ n, we consider zn = z(fn), where fn ∈ cs(x) and |fn| ≤ 1 (note that if f ∈ cs(x), then f 1+|f| ∈ cs(x), z( f 1+|f| ) = z(f) and | f 1+|f| | ≤ 1), then the sequence sn = ∑n i=1 fi/2 i converges uniformly to a function f ∈ cs(x). clearly ⋂∞ n=1 z(fn) = z(f). � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 212 on rings of real valued clopen continuous functions proposition 5.5. for a topological space x, the following statements are equivalent. (1) x is an almost ps-space. (2) for each non-unit f ∈ cs(x), f = ef for some idempotent e 6= 1 in cs(x). (3) for each non-unit f ∈ cs(x), fe = 0 for some idempotent e 6= 0 in cs(x). (4) every non-empty countable intersection of s-open sets has a non-empty s-interior. proof. (1)⇒(2) if f ∈ cs(x) is not unit, then intsz(f) 6= ∅. since intsz(f) is s-open, there exists a non-empty clopen set u contained in z(f). take the idempotent e with z(e) = u. clearly e 6= 1, for u 6= ∅. since z(e) ⊆ intsz(f), f is a multiple of e, by lemma 5.4. hence f = eg for some g ∈ cs(x). but f = g on x \ z(e), so f = ef. (2)⇒(3) f = ef implies f(1 − e) = 0, where 1 − e is a non-zero idempotent. (3)⇒(4) let a = ⋂∞ n=1 an 6= ∅, where each an is s-open. let x ∈ a. hence there is an idempotent en ∈ cs(x) such that x ∈ z(en) ⊆ an, for each n ∈ n. now by lemma 5.4, ⋂∞ n=1 z(en) is a zero-set, say z(g), where g ∈ cs(x). since g is non-unit (x ∈ z(g)), there exists an idempotent 0 6= e ∈ cs(x) such that ge = 0, by our hypothesis. therefore ∅ 6= z(1 − e) ⊆ z(g) ⊆ a which means that a has a non-empty s-interior. (4)⇒(1) since every zero-set in zs(x) is a countable intersection of s-open sets, the proof is evident. we note that whenever f ∈ cs(x), then z(f) =⋂∞ n=1 f −1((− 1 n , 1 n )) and each f−1((− 1 n , 1 n )) is s-open, by theorem 2.2 in [10]. � we call a space x s-basically (s-extremally) disconnected if for every zeroset z(f) ∈ zs(x) (s-closed subset h of x), intsz(f) (intsh) is s-closed. equivalently, x is a s-basically (s-extremally) disconnected space if and only if for each x \ z(f), f ∈ cs(x) (s-open subset g of x), cls(x \ z(f)) (clsg) is s-open. we show the counterparts of theorems 3.3 and 3.5 in [4] that the s-basically (s-extremally) disconnectedness of x is equivalent to saying that cs(x) is a p.p. ring (baer ring). recall that a ring r is said to be p.p. ring (baer ring) if for each a ∈ r ( s ⊆ r), ann(a) (anns) is generated by an idempotent, where ann(a) := {r ∈ r : ar = 0} (anns := {r ∈ r : rs = 0, ∀s ∈ s}). first we need the following lemma. lemma 5.6. let x be a topological space and xz be the space mentioned in the proof of theorem 3.1. (1) if f ∈ cs(x), then ⋂ g∈ann(f) z(g) = cls(x \ z(f)). (2) x is s-extremally (s-basically) disconnected if and only if xz is extremally (basically) disconnected. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 213 s. afrooz, f. azarpanah and m. etebar proof. (1) since ⋂ g∈ann(f) z(g) = ⋂ x\z(f)⊆z(g) z(g), we have x \ z(f) ⊆ ⋂ g∈ann(f) z(g). but ⋂ g∈ann(f) z(g) is s-closed, hence cls(x \ z(f)) ⊆ ⋂ g∈ann(f) z(g). conversely, let x ∈ ⋂ g∈ann(f) z(g) but x /∈ cls(x \ z(f)). hence there exists an idempotent e ∈ cs(x) such that e(x) = 1 and e(cls(x \ z(f))) = 0, by proposition 2.3. this implies that e ∈ ann(f), but e(x) = 1 yields that x /∈ ⋂ g∈ann(f) z(g), a contradiction. (2) let x be s-extremally disconnected and h be an open set in xz. then g = ⋃ qx∈h qx = {x ∈ x : qx ∈ h} is a s-open set in x. hence clsg is s-open, by our hypothesis. now let qx ∈ clxz h. then every clopen set in xz containing qx intersects h and this implies that every clopen set in x containing x intersects g as well. therefore x ∈ clsg. but clsg is s-open, then there exists a clopen set u in x containing x contained in clsg. now v = {qy : y ∈ u} is a clopen set in xz containing qx contained in clxz h, i.e., clxz h is open in xz and hence xz is extremally disconnected. the proof of the converse is similar. in case of s-basically disconnectedness, the proof goes along the lines of the above arguments, so it is left to the reader. � proposition 5.7. let x be a topological space. (1) cs(x) is a p.p. ring if and only if x is a s-basically disconnected space. (2) cs(x) is a baer ring if and only if x is a s-extremally disconnected space. proof. (1) we may apply each part of lemma 5.6, we prefer to use part (1). if cs(x) is a p.p. ring, then for each f ∈ cs(x), ann(f) = (e) for some idempotent e. now by lemma 5.6, z(e) = ⋂ g∈ann(f) z(g) = cls(x \ z(f)) which implies that cls(x \ z(f)) is clopen and hence it is s-open. therefore x is a s-basically disconnected space. conversely, suppose that x is s-basically disconnected. hence ⋂ g∈ann(f) z(g) = cls(x \ z(f)) is s-open and hence it is clopen. now take an idempotent e with z(e) = cls(x \ z(f)). since x \ z(f) ⊆ cls(x \ z(f)) = z(e), we have ef = 0, i.e., e ∈ ann(f). on the other hand if g ∈ ann(f), then z(e) = cls(x \ z(f)) ⊆ z(g) implies that z(e) ⊆ intsz(g) and by lemma 5.4, g ∈ (e), i.e., ann(f) ⊆ (e). (2) if cs(x) is a baer ring, then c(xz) is also a baer ring, for cs(x) ∼= c(xz), by theorem 3.1. now by theorem 2.5 in [4], xz is extremally disconnected. thus using lemma 5.6, x is s-extremally disconnected. the proof of the converse is similar. � it is manifest that every basically (extremally) disconnected space is a sbasically (s-extremally) disconnected space. the converse is not true in general. for example let x = (0, 1) ∪ (1, 2) be as a subspace of r. in fact x is a psspace which is not basically disconnected and it is not extremally disconnected c© agt, upv, 2018 appl. gen. topol. 19, no. 2 214 on rings of real valued clopen continuous functions as well. it is not hard to see that every s-basically disconnected almost ps-space is a ps-space. the following result states that the zero-dimensionality and sbasically (s-extremally) disconnectedness is equivalent to basically (extremally) disconnectedness. proposition 5.8. a space is basically (extremally) disconnected if and only if it is s-basically (s-extremally) disconnected zero-dimensional. proof. by problem 16o in [6], every basically (extremally) disconnected space is zero-dimensional and since every basically (extremally) disconnected space is also s-basically (s-extremally) disconnected, the left-to-right implication is immediate. for the converse, whenever x is zero-dimensional, then by lemma 1.1, c(x) = cs(x). now if x is s-basically (s-extremally) disconnected, then by proposition 5.7, c(x) = cs(x) is a p.p. (baer) ring. now by theorems 3.3 and 3.5 in [4], x is basically (extremally) disconnected. � the socle cf (x) of c(x) which is the intersection of all essential ideals in c(x) is the set of all functions which vanish everywhere except on a finite number of isolated points of x, see proposition 3.3 in [7]. corollary 2.3 in [2] and proposition 2.1 in [7] show that the socle of c(x) is essential if and only if the set of isolated points of x is dense in x. proposition 5.9. let x be a topological space and xz be the space defined in the proof of theorem 3.1. (1) the socle ss(x) of cs(x) is free if and only if every quasi-component in x is open, if and only if xz is discrete. (2) the socle of cs(x) is essential if and only if the union of open quasicomponents in x is s-dense in x (a subset d of x is called s-dense in x if every non-empty s-open subset of x intersects d). proof. we remind the reader that a subset h of xz is open if and only if⋃ qx∈h qx is s-open in x. this implies that for each x ∈ x, the quasicomponent qx is an isolated point of xz if and only if qx is s-open in x and hence it should be clopen. since cf (xz) is the set of all functions in c(xz) which vanish everywhere except on a finite set of isolated points of xz, ss(x) will be the set of all functions in cs(x) which vanish everywhere except on a finite union of open quasi-components in x. therefore, ⋂ f∈ss(x) z(f) is the union of all non-open quasi-components in x. now it is clear that ss(x) is free if and only if every quasi-components in x is open and this is equivalent to saying that xz is discrete. for the proof of part (2), it is easy to see that the density of isolated points of xz is equivalent to the density of the union of open quasi-components in x. since cs(x) ∼= c(xz), the socle of c(xz) is essential if and only if the socle of cs(x) is. now using corollary 2.3 in [2], we are done. � it is known that every extremally disconnected p-space of non-measurable cardinal is discrete, see problem 12h in [6]. we conclude the paper by the counterpart of this fact. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 215 s. afrooz, f. azarpanah and m. etebar proposition 5.10. every quasi-component in a s-extremally disconnected psspace of non-measurable cardinal is open. proof. let x be s-extremally disconnected ps-space of non-measurable cardinal. then cs(x) is a baer ring by proposition 5.7. but using proposition 3.1, cs(x) ∼= c(xz), hence c(xz) is also a baer ring. therefore xz is extremally disconnected, by theorem 3.5 in [4]. on the other hand, since x is ps-space, cs(x) will be regular by proposition 5.1, whence c(xz) is also regular and hence xz is a p-space by problem 4j in [6]. finally, |xz| ≤ |x| implies that the cardinal of xz is non-measurable, see part (i) in the proof of theorem 12.5 in [6]. now xz is extremally disconnected p-space of non-measurable cardinal which means that xz is discrete, by problem 12h in [6]. therefore each qx is an isolated point in xz and hence each qx should be open in x. � acknowledgements. the authors would like to thank the referee for a careful reading of this article. references [1] s. afrooz, f. azarpanah and o. a. s. karamzadeh, goldie dimension of rings of fractions of c(x), quaest. math. 38, no. 1 (2015), 139–154. [2] f. azarpanah, intersection of essential ideals in c(x), proc. amer. math. soc. 125, no. 7 (1997), 2149–2154. [3] f. azarpanah, on almost p -spaces, far east j. math. sci. special volume (2000), 121132. [4] f. azarpanah and o. a. s. karamzadeh, algebraic characterizations of some disconnected spaces, italian j. pure appl. math. 12 (2002), 155–168. [5] r. engelking, general topology, sigma ser. pure math., vol. 6, heldermann verlag, berlin, 1989. [6] l. gillman and m. jerison, rings of continuous functions, springer-verlag, 1976. [7] o. a. s. karamzadeh and m. rostami, on the intrinsic topology and some related ideals of c(x), proc. amer. math. soc. 93 (1985), 179–184. [8] r. levy, almost p -spaces, can. j. math. xxix, no. 2 (1977), 284–288. [9] i. l. reilly and m. k. vamanamurthy, on supercontinuous mappings, indian j. pure appl. math. 14, no. 6 (1983), 767-772. [10] d. singh, cl-supercontinuous functions, applied gen. topol. 8, no. 2 (2007), 293–300. [11] r. staum, the algebra of bounded continuous functions into non-archimedean field, pacific j. math. 50, no. 1 (1974), 169–185. [12] s. willard, general topology, addison-wesley publishing company, inc., 1970. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 216 @ appl. gen. topol. 15, no. 2(2014), 235-248doi:10.4995/agt.2014.3157 c© agt, upv, 2014 subgroups of paratopological groups and feebly compact groups manuel fernández a and mikhail tkachenko1 b a academia de matemáticas, universidad autónoma de la ciudad de méxico, prolongación san isidro 151, col. san lorenzo tezonco, del. iztapalapa, c.p. 09790, méxico, d.f. (mafevil5@gmail.com, manuel.fernandezvillanueva@uacm.edu.mx) b departamento de matemáticas, universidad autónoma metropolitana, av. san rafael atlixco 186, col. vicentina, del. iztapalapa, c.p. 09340, méxico, d.f. (mich@xanum.uam.mx) abstract it is shown that if all countable subgroups of a semitopological group g are precompact, then g is also precompact and that the closure of an arbitrary subgroup of g is again a subgroup. we present a general method of refining the topology of a given commutative paratopological group g such that the group g with the finer topology, say, σ is again a paratopological group containing a subgroup whose closure in (g, σ) is not a subgroup. it is also proved that a feebly compact paratopological group h is perfectly κ-normal and that every gδ-dense subspace of h is feebly compact. 2010 msc: 22a30; 54h11 (primary); 54b05 (secondary). keywords: feebly compact; precompact; paratopological group; subsemigroup; topologically periodic. 1. introduction our aim is to study closures of subgroups of paratopological groups. we will say that a group g with a topology is an sp-group (abbreviation for subgroup preserving) if the closure of every subgroup of g is again a subgroup of g. every 1this author was supported by conacyt of mexico, grant cb-2012-01 178103. received 15 may 2014 – accepted 29 june 2014 http://dx.doi.org/10.4995/agt.2014.3157 m. fernández and and m. tkachenko topological group is clearly an sp-group. a similar statement fails in the class of paratopological groups, see [2, example 1.4.17]. nevertheless, there exists a wide class of paratopological groups containing the sorgenfrey line which is closed under taking quotient groups, subgroups, and arbitrary products and which contains only sp-groups [13]. the main result of [13] was extended in [6] for the class of almost topological groups (the corresponding definition is given section 6). in section 3 of the article we show that under certain conditions, the topology of a commutative paratopological group can be refined in such a way that the group with the new topology is again a paratopological group which fails to be an sp-group (see propositions 3.7 and 3.8, and corollary 3.9). in fact, we refine the topology of a given paratopological group by declaring open a ‘single’ subsemigroup of the group. in section 4 we consider compact subsets of paratopological groups and the action of internal automorphisms on open neighborhoods of the identity. it is well known that for a compact (even precompact) subset b of a topological group g, and an arbitrary neighborhood u of the identity e in g, one can find a neighborhood v of e such that bv b−1 ⊆ u, for each b ∈ b. in example 4.1 we show that this fact is not valid in the class of hausdorff paratopological groups. a feebly compact space is a topological space in which every locally finite family of open sets is finite. in tychonoff spaces the concepts of pseudocompactness and feeble compactness coincide. it is known that every pseudocompact (hence tychonoff) paratopological group is a topological group [10, theorem 2.6]. this result remains valid for regular feebly compact paratopological groups [1, theorem 1.7]. however, a hausdorff feebly compact paratopological group can fail to be a topological group [9, example 3]. furthermore, under martin’s axiom, there exists a hausdorff countably compact paratopological group with discontinuous inversion [9, example 2]. thus we focus our attention on the study of feebly compact paratopological groups in section 5. we prove that every feebly compact paratopological group is perfectly κ-normal and that any gδ-dense subspace of a feebly compact paratopological group is feebly compact. in section 6 we prove that an almost topological group g is precompact (or baire) if and only if the underlying topological group g has the same property, and that the indices of narrowness of g and g coincide. 2. preliminaries, notation and definitions let g be an abstract group with a topology. the group g is left topological (right topological) provided that the left (right) translations are continuous in g. a group g that is both left and right topological is a semitopological group. if multiplication on g is continuous, we say that g is a paratopological group. if, in addition, the inversion is continuous in g, then g is a topological group. a left (right) topological group g is left (right) precompact if for every open c© agt, upv, 2014 appl. gen. topol. 15, no. 2 236 subgroups of paratopological groups and feebly compact groups neighborhood u of the identity of g there is a finite subset f of g such that fu = g (uf = g). a semitopological group g is precompact provided that it is both left and right precompact. we denote by 〈h〉 the subgroup generated by a subset h of a group g. if x ∈ g, we write 〈x〉 instead of 〈{x}〉. a nonempty subset s of the group g is called a subsemigroup of g if xy ∈ s, for all x, y ∈ s. note that s does not necessarily contain the neutral element of g. a semitopological (paratopological) group g is called saturated provided that u−1 has nonempty interior, for every neighborhood u of the identity in g. every precompact paratopological group is saturated [8, proposition 3.1]. we also say that a semitopological group g is topologically periodic if for every neighborhood u of the neutral element e in g and every element x ∈ g, there exists a positive integer n such that xn ∈ u. clearly every semitopological torsion group is topologically periodic. in this article, a regular space will be a topological space x such that for every x ∈ x and every open neighborhood u of x there is an open neighborhood v of x such that v ⊆ u, or equivalently, for every closed subset f of x and x /∈ f , there are open disjoint sets u and v in x such that x ∈ u, and f ⊆ v . a space will be called t3 provided that it is regular and t1. let x be a topological space and u ⊆ x. we say that u is a regular open subset of x provided that u = int u. the family of regular open sets in x is a base for a topology ρ on x weaker than the topology of x. we denote the topological space (x, ρ) by rx; in general, the space rx is called the semiregularization of x, for rx is a semiregular space. it turns out that for any paratopological group g, the space rg is a regular paratopological group (see [7, proposition 1.5]). since we are interested primarily in paratopological groups, we will refer to rg as the regularization of g. the results of the rest of this section are proved by ravsky in [9]. lemma 2.1. let x be a topological space. then x is feebly compact if and only if rx is feebly compact. let g be a paratopological group and n the family of all open neighborhoods of the neutral element e of g. then the set h = ⋂ u∈n (u ∩ u−1) is an invariant subgroup of g, i.e. xhx−1 = h for each x ∈ g. denote by t0g the quotient paratopological group g/h and let π : g → g/h be the quotient homomorphism. it is easy to verify that u = π−1π(u), for each open set u ⊆ g and that the group t0g is a t0-space (see [9, section 5]). lemma 2.2. a paratopological group g is feebly compact if and only if t0g is feebly compact. lemma 2.3. for every feebly compact paratopological group g, the group t0(rg) is a pseudocompact topological group. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 237 m. fernández and and m. tkachenko 3. closures of subgroups here we find conditions under which a semitopological group turns out to be an sp-group or when a paratopological group g admits a finer paratopological group topology, say, σ such that (g, σ) fails to be an sp-group. proposition 3.1. let g be a left (right) topological group such that every countable subgroup of g is left (right) precompact. then g is left (right) precompact. proof. suppose, on the contrary, that g is not left precompact. choose an open neighborhood u of the identity in g such that fu 6= g, for every finite subset f of g. we will define an increasing countable family {cn}n∈ω of countable subgroups of g as follows. let c0 be any countable subgroup of g. once we have defined the subgroups c0, . . . , cn such that c0 ⊆ · · · ⊆ cn, for any finite set f ⊆ cn, we choose xf ∈ g\fu. we define cn+1 = 〈cn ∪ {xf : f ⊆ cn, |f | < ω}〉. let c = ⋃ n∈ω cn. since every cn is countable, the group c is countable. by hypothesis c is left precompact, thus there is a finite set f ⊆ c such that f (u ∩ c) = c. on the other hand, there exists k ∈ ω such that f ⊆ ck. then xf /∈ fu ⊇ c, a contradiction with the definition of c. thus g is left precompact. � corollary 3.2. let g be a semitopological group such that every countable subgroup of g is precompact. then g is also precompact. it is worth mentioning that subgroups of a precompact paratopological group can fail to be precompact. one of the standard examples of this phenomenon is as follows. let ts be the circle group endowed with the sorgenfrey topology, i.e. a local base at the neutral element 1 of ts is formed by the sets un = {e πix : 0 ≤ x < 1/n}, with n ∈ n+. then ts is a commutative zero-dimensional (hence tychonoff) paratopological group. the paratopological group t2s is also precompact and it contains the closed discrete uncountable subgroup ∆2 = {(x, x−1) : x ∈ ts}. hence the subgroup ∆2 is not precompact. furthermore, every discrete abelian group can be embedded as a subgroup into a precompact hausdorff paratopological group [4, corollary 5]. for an element x of a paratopological group g, we denote by sx the subsemigroup {xn : n ∈ ω} of g. note that sx contains the neutral element of g. proposition 3.3. suppose that a semitopological group g is not an sp-group. then there exists an element x ∈ g of infinite order such that the subsemigroup sx is open in the cyclic group 〈x〉. proof. let h be a subgroup of g such that h fails to be a subgroup of g. as g is a semitopological group, h is a subsemigroup of g [2, proposition 1.4.10]. since h is not a subgroup of g, there exists an element y ∈ h such that y−1 /∈ h. it is clear that the subsemigroup sy is contained in h. since y−1 /∈ h, the element y is of infinite order. moreover, y−n /∈ h, for every c© agt, upv, 2014 appl. gen. topol. 15, no. 2 238 subgroups of paratopological groups and feebly compact groups n ∈ n. if we denote the element y−1 by x, then s = ( g\h ) ∩ 〈x〉 is open in 〈x〉 and sx = x −1s is open in 〈x〉 as well. � corollary 3.4. every topologically periodic semitopological group is an spgroup. proof. suppose that a semitopological group g fails to be an sp-group. then, by proposition 3.3, g contains an element x of infinite order such that the subsemigroup sx = {x n : n ∈ ω} is open in the cyclic group 〈x〉. choose an open set u in g such that u ∩ 〈x〉 = sx. then u contains the neutral element of g and yn /∈ u for each positive integer n, where y = x−1. hence the group g is not topologically periodic. � the following lemma is known in the folklore. we present its proof to ease the reader’s job. lemma 3.5. if all cyclic subgroups of a semitopological group g are left precompact, then g is topologically periodic. proof. if g is not topologically periodic, we can find an open neighborhood u of the neutral element e in g and an element x ∈ g distinct from e such that xn /∈ u, for each positive integer n. in particular, the cyclic group 〈x〉 is infinite and the subsemigroup sy = {y n : n ∈ ω}, where y = x−1, is open in 〈y〉 = 〈x〉. then fsy 6= 〈y〉 for every nonempty finite subset f of 〈y〉. indeed, if k = min{n ∈ z : yn ∈ f}, then yk−1 /∈ fsy. thus the subgroup 〈y〉 of g is not left precompact. � combining corollary 3.4 and lemma 3.5, we obtain the following fact: corollary 3.6. let g be a semitopological group such that every cyclic subgroup of g is left precompact. then g is an sp-group. it is worth mentioning that our corollary 3.4 follows from a more general result established in [14, theorem 3.3]: the closure of every subsemigroup of a topologically periodic semitopological group is a subgroup. the corresponding argument in [14] is, however, quite different. in the sequel we present several results on refinements of topologies of paratopological (sp-)groups turning them into non-sp-groups. a non-empty subset t of an abelian group g is called independent if the equality n1x1+· · ·+nkxk = 0g with n1, . . . , nk integers and distinct x1, . . . , xk ∈ t implies n1x1 = · · · = nkxk = 0g. thus a set t ⊆ g of elements of infinite order is independent if and only if the former equality implies that n1 = · · · = nk = 0. given a paratopological group g with open neighborhood base n at the identity e, and a subsemigroup s of g containing e, we denote by gs the group g with topology τs whose local base at e is the family b = {u ∩ s : u ∈ n} ∪ n . that is, gs = (g, τs). it is easy to verify that gs is also a paratopological group. by the definition, the topology of gs refines the topology of g. note that if g is first countable, so is gs. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 239 m. fernández and and m. tkachenko proposition 3.7. let (g, τ) be a first countable, abelian paratopological group. if g contains an infinite independent subset k of elements of infinite order which accumulates at the identity e of g, then gs is not an sp-group for some countable subsemigroup s of g with e ∈ s. proof. choose any element x ∈ k. let {un}n∈ω be a local base at e. since e is an accumulation point of the set k, we can choose tn ∈ (k \ {x}) ∩ un, for every n ∈ ω, with tn 6= tm if n 6= m. let s be the subsemigroup of g generated by the subset t = {e} ∪ {tn : n ∈ ω}. consider the group gs. since k is an independent subset of g, the equality 〈x〉 ∩ 〈t 〉 = {e} holds. let hn = x + tn, for every n ∈ ω, and h = 〈{hn : n ∈ ω}〉. by the definition of h, x ∈ h, here the closure is taken in the group gs. for let un∩s be a basic open neighborhood of e in gs, then hn = x+tn ∈ (x + (un ∩ s))∩h. now we prove that −x /∈ h. suppose, on the contrary, that −x ∈ h. since s is open in gs, (−x + s)∩h 6= ∅. then, since h is a subgroup, −x+ l1tn1 + · · ·+ litni = k1hm1 + · · · + kjhmj for some positive integers l1, . . . , li, some nonzero integers k1, . . . , kj, and n1, . . . , ni ∈ ω, and distinct m1, . . . , mj ∈ ω. from here it follows that (−1 − k1 − · · · − kj) x = k1tm1 + . . . kjtmj − l1tn1 − · · · − litni. since 〈x〉 ∩ 〈t 〉 = {e}, we have on one hand that −1 − k1 − · · · − kj = 0, hence k1 + · · · + kj = −1. on the other hand, the independence of t \ {e} implies that i = j and, after reindexing the set {m1, . . . , mi}, nr = mr, for every r = 1, . . . i. thus kr = lr, for each r = 1, . . . , i. then every kr is positive, which is a contradiction. thus h is not a subgroup of gs. � in the next proposition we present sufficient conditions on a subsemigroup s of a commutative, first countable paratopological group g, in order that gs won’t be an sp-group. if t is an independent set of elements of infinite order in a commutative group g with identity 0, every non-zero element t ∈ 〈t 〉 can be written in a unique way as t = k1t1 + · · · + kntn, with k1, . . . , kn non-zero integers and distinct t1, . . . , tn ∈ t . we put expt (t) = k1 + · · · + kn. proposition 3.8. let (g, τ) be a commutative, first countable paratopological group with identity e and s = s ∪ {e} a subsemigroup of g. suppose that (1) there is a countable infinite independent set k ⊆ s of elements of infinite order such that e is an accumulation point of k; (2) there exists x∗ ∈ g of infinite order such that 〈x∗〉 ∩ 〈s〉 = {e} ; (3) if t = 〈k〉, and t ∈ t ∩ s, then expt (t) ≥ 0. then the first countable paratopological group gs = (g, τs) is not an sp-group. proof. let {vn}n∈ω be a local base at e in (g, τ). for every n ∈ ω, choose an element vn ∈ vn ∩ k in such a way that vn 6= vm if n 6= m. consider the subgroup h of g generated by the set {x∗ + vn : n ∈ ω}. since the family {vn}n∈ω is a local base at e, we have that x ∗ ∈ h, here the closure is taken in gs. we claim that −x ∗ /∈ h. suppose, on the contrary, that −x∗ ∈ h. since s is an open neighborhood of e in gs, we have that (−x ∗ + s) ∩ h 6= ∅. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 240 subgroups of paratopological groups and feebly compact groups then −x∗ + s = k1 (x ∗ + vn1) + · · · + ki (x ∗ + vni), for some s ∈ s, k1, . . . , ki integers, and distinct n1, . . . , ni ∈ ω. from here, (k1 + · · · + ki + 1) x ∗ = s − k1vn1 − · · · − kivni ∈ 〈x ∗〉 ∩ 〈s〉, which implies by (2) that k1 + · · · + ki + 1 = 0. but then s = k1vn1 + · · · + kivni ∈ s ∩ t , and expt (s) < 0, which contradicts (3). hence, −x∗ /∈ h, and h is not a subgroup of gs. � corollary 3.9. there exists a first countable topology containing the usual topology of r that makes the additive group r a paratopological group but not an sp-group. proof. the additive group r with its usual topology is first countable. we choose a countable neighborhood base {un : n ∈ ω} at 0 in r such that un+1 ⊂ un, for every n ∈ ω. we can define by induction an infinite set k = {xn : n ∈ ω} ⊂ r as follows. let x0 ∈ (r \ q)∩u0. once we have defined independent elements x0, . . . , xn, with xi ∈ ui for every i = 1, . . . , n, we put an = qx0 + · · · + qxn. since an is countable, we can choose an irrational number xn+1 ∈ (r \ an) ∩ un+1. let us verify that the set k is independent. suppose not, then we can find a linear combination k1xi1 + · · · + knxin = 0 of elements of k, with non-zero integers k1, . . . , kn and i1 < · · · < in, where n > 1. then xin = − k1 kn xi1 − · · · − kn−1 kn xin−1 ∈ qxi1 + · · · + qxin−1 ⊆ ain−1, a contradiction. thus k is an independent set of elements of infinite order. clearly k accumulates at 0. let s be the subsemigroup of r generated by k \ {x0} and x ∗ = x0. by proposition 3.8, the paratopological group rs is first countable and fails to be an sp-group. by its definition, the topology of rs is first countable and refines the usual topology of r. � the character of the group (g, τ) in propositions 3.7 and 3.8 is countable. with some adjustments, these results can be extended to groups (g, τ) of an arbitrary character λ ≥ ω. we just need to require that |k ∩ u| ≥ λ for each u ∈ n(e), where n(e) is a local base at the identity e for (g, τ). 4. precompact subsets and internal automorphisms it is known that, given a precompact subset b of a topological group g and any neighborhood u of the identity e of g, there is a neighborhood v of e such that b−1v b ⊆ u, for every b ∈ b. in particular, if b is compact and u is open, then the set ⋂ b∈b bub−1 is again an open neighborhood of e. this property does not hold in the class of paratopological groups, as example 4.1 below shows. following [5], we say that a paratopological group g is ♭-separated provided that g admits a continuous one-to-one homomorphism onto a hausdorff topological group or, equivalently, if g admits a weaker hausdorff topological group topology. example 4.1. there exist a hausdorff saturated paratopological group h, a compact subset b of h, and a neighborhood u of the identity e of h such that no neighborhood v of e satisfies b−1v b ⊆ u, for every b ∈ b. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 241 m. fernández and and m. tkachenko proof. let g = f (ω) be the direct sum of ω copies of the free group f on two generators, x and y, and s be the minimal subsemigroup of f containing the set {1, x, y}, where 1 is the identity of f . we continue as in [5, proposition 6]. for every n ∈ ω, let un be the set of elements z of g such that z(k) = 1, for all k ≤ n, and z(k) ∈ s, for every k > n. the family n = {un : n ∈ ω} satisfies the pontryagin conditions for a neighborhood base at the identity eg of g of a paratopological group topology τ on g (see [11, proposition 2.1]). it is easy to verify that the topology τ on g is hausdorff, for let z 6= eg be an element of g and let k = min{n ∈ ω : z(n) 6= 1}. the sets zuk and uk are disjoint open neighborhoods of z and eg, respectively. thus g is a hausdorff and [5, proposition 6] implies that g is a ♭-separated paratopological group. for every n ∈ ω, let bn the element of g defined by bn(n) = x, and bn(k) = 1 if k 6= n. let b′ = {bn : n ∈ ω}, and b = b ′ ∪ {eg}. every element of n contains all but a finite number of elements of b, thus b is a compact subset of g. let n ∈ ω, and consider the element z ∈ un defined as follows: z(n + 1) = y and z(k) = 1 for every k 6= n + 1. we have that t = b−1n+1zbn+1 /∈ u0, for t(n + 1) = x−1yx /∈ s. the element n ∈ ω is arbitrary, so there is no neighborhood v of eg in g such that b −1v b ⊆ u0 for each b ∈ b. since g is a ♭-separated paratopological group, [4, corollary 3] implies that the group g can be embedded as a subgroup into a hausdorff saturated paratopological group h. suppose that the conclusion of the example does not hold. choose an open neighborhood u of the identity e in h such that u ∩ g = u0. then for some open neighborhood v of e, b −1v b ⊆ u, for each b ∈ b. let v0 = v ∩ g. clearly b and v0 are subsets of g and it follows that b−1v0 b ⊆ u ∩ g = u0, for each b ∈ b, a contradiction. this proves that there is no open neighborhood v of e in h such that b−1v b ⊆ u for each b ∈ b. � it is not clear whether one can refine example 4.1 by choosing the group h precompact. notice that every precompact paratopological group is saturated [8]. 5. feebly compact paratopological groups here we prove that a feebly compact paratopological group h is perfectly κ-normal and that every gδ-dense subspace of h is feebly compact. we start with a simple lemma in which rx denotes the semiregularization of a given space x (see section 2). lemma 5.1. let x be a space and ix : x → rx the identity mapping of x onto the semiregularization of x. if d is a dense subspace of a space x, then rd is naturally homeomorphic to the subspace ix(d) of rx. proof. it follows from the definition of the operation of semiregularization that the families of regular open sets in x and rx coincide. since d is dense in x, we obtain the required conclusion. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 242 subgroups of paratopological groups and feebly compact groups corollary 5.2. a dense subset d of a space x is feebly compact as a subspace of x if and only if d is feebly compact as a subspace of rx. lemma 5.3. let f : x → y be an open continuous mapping of the space x onto a feebly compact space y , and suppose that f−1f(u) = u for every open set u in x. then x is feebly compact. proof. let u be a locally finite family of open sets in x. we claim that u′ = {f(u) : u ∈ u} is a locally finite family of open sets in y . since f is open, u′ is a family of open subsets of y . let y ∈ y . choose x ∈ f−1(y) and an open neighborhood v of x in x meeting only a finite number of elements of u. suppose that f(v ) meets f(u) for some u ∈ u. then v ∩ u = f−1f(v ) ∩ f−1f(u) 6= ∅. thus the open neighborhood f(v ) of y meets only a finite number of elements of u′. hence u′ is locally finite, and since y is feebly compact, u′ is finite. from here, using that for every open set u of x the equality f−1f(u) = u holds, we conclude that u is finite. thus x is feebly compact. � lemma 5.4. let f : x → y be an open continuous mapping of a space x onto a feebly compact space y . suppose that f−1f(u) = u for every open set u of x, and that d is a dense subspace of x. then the mapping g = f↾d : d → f(d) is open and g−1g(v ) = v for every open set v in d. proof. clearly g is continuous. let v = u ∩ d be an open set in d, with u open in x. then g(v ) = f(u ∩ d) ⊆ f(u) ∩ f(d). let y ∈ f(u) ∩ f(d). choose x ∈ f−1(y) ∩ d. since f−1f(u) = u, we have that f−1(y) ⊆ u. thus x ∈ u ∩ d, and y ∈ f(u ∩ d) = g(v ). we conclude that g(v ) = f(u) ∩ f(d) is open in f(d). now we prove that g−1g(v ) = v for every open set v in d. let u be an open set in x such that v = u ∩d. then g−1g(v ) = f−1f(v )∩d ⊆ f−1f(u)∩d = u ∩ d = v . since v ⊆ g−1g(v ), we conclude that g−1g(v ) = v . � corollary 5.5. a dense subset d of a paratopological group g is feebly compact provided that the subspace π(d) of t0g is feebly compact, where π : g → t0g is the quotient homomorphism. proof. the homomorphism π is open and u = π−1π(u), for every open set u in g. hence the required conclusion follows from lemmas 5.3 and 5.4. � a subset d of a space x is gδ-dense in x if it meets every nonempty gδ-set in x. proposition 5.6. let g be a feebly compact paratopological group and d a gδ-dense subset of g. then d is feebly compact. proof. we know that d is feebly compact if and only if rd is feebly compact. by lemma 5.1, rd is homeomorphic to d considered as a subspace of rg. then, by corollary 5.5, d is feebly compact if π(d) is a feebly compact subspace of t0(rg). by lemma 2.3, t0(rg) is a pseudocompact topological c© agt, upv, 2014 appl. gen. topol. 15, no. 2 243 m. fernández and and m. tkachenko group, and clearly π(d) is a gδ-dense subset of t0(rg). by corollary 6.6.3 in [2], π(d) is feebly compact. we conclude that d is feebly compact. � corollary 5.7. every gδ-dense subgroup of a feebly compact paratopological group is feebly compact. a set a ⊆ x is a zero-set in a space x if there exists a continuous function f : x → r such that a = f−1(0). a subset c of a space x is regular closed in x provided that c = int c. definition 5.8. a space x is perfectly κ-normal if every regular closed subset of x is a zero-set in x. a subspace a of a space x is z-embedded in x if every zero-set in a is the intersection with a of a zero-set in x, i.e. b is a zero-set of a if and only if b = a ∩ c, with c a zero-set in x. in [3], blair proves that a space x is perfectly κ-normal if and only if every dense subset of x is z-embedded in x. the next lemma follows the definition of semiregularization. lemma 5.9. every regular closed subset of a space x is regular closed in rx. lemma 5.10. let c be a regular closed set in a paratopological group g. then π(c) is regular closed in t0g, where π : g → t0g is the quotient homomorphism. proof. let c be a regular closed set in g. we prove first that c = π−1π(c). it suffices to show that ch ⊆ c, where h is the kernel of π. let x = ch ∈ ch, with c ∈ c and h ∈ h, and u ∈ n(e). then xu ∈ n(x). we have that xu = chu ∈ n(c). since c = int c, we have that xu ∩ int c = chu ∩ int c 6= ∅. thus x ∈ int c = c. therefore c = π−1π(c). using the above property of π as well as the assumption that π is open and continuous, we conclude that π(c) is a regular closed set in t0g. � lemma 5.11. every pseudocompact topological group is a perfectly κ-normal space. proof. it follows from [2, corollary 5.3.29] that every compact topological group is a perfectly κ-normal space. the răıkov completion of a pseudocompact topological group g, say, ̺g is a compact topological group which is thus perfectly κ-normal. hence so is g as a dense subspace of ̺g. � theorem 5.12. every feebly compact paratopological group is perfectly κnormal. proof. let g be a feebly compact paratopological group and c a regular closed set in g. by lemmas 2.3, 5.9, 5.10, and 5.11, π(c) is a regular closed set in the perfectly κ-normal topological group t0g, where π : g → t0g is the quotient mapping. thus π(c) is a zero-set in t0g. it follows that c = π −1π(c) is a zero-set in g. we conclude that g is perfectly κ-normal. � c© agt, upv, 2014 appl. gen. topol. 15, no. 2 244 subgroups of paratopological groups and feebly compact groups proposition 5.13. suppose that every cyclic subgroup of a hausdorff feebly compact paratopological group g is precompact. then g is a topological group. proof. let us show that g is topologically periodic. suppose that x ∈ g and u is an open neighborhood of the identity e in g. if x has finite order, there is nothing to verify. we can assume therefore that the cyclic subgroup h = 〈x〉 of g is infinite. since h is precompact, there is a finite subset f of h such that h = f · v , where v = u ∩ h. since f is finite, there exists a finite set c ⊂ z such that f = {xn : n ∈ c}. take k ∈ n+ such that k > n for each n ∈ c. by the equality h = f · v , there exist n ∈ c and m ∈ z such that xk = xnxm. then k = n + m and from k > n it follows that m = k − n > 0. hence xm ∈ v ⊆ u and the group g is topologically periodic. finally, by proposition 5 in [9], every feebly compact topologically periodic paratopological group is a topological group. � 6. precompact almost topological groups the following notion was introduced by the first listed author in [6]. definition 6.1. an almost topological group is a paratopological group (g, τ) that satisfies the following conditions: (a) the group g admits topological group topology σ weaker than τ. (b) there exists a local base b at the identity e of the paratopological group (g, τ) such that the set ũ = u \{e} is open in g = (g, σ), for every u ∈ b. if g and g are as in the above definition, we say that g is the underlying topological group of g. proposition 6.2. let g be an almost topological group with underlying topological group g. then g is precompact if and only if g is precompact. proof. suppose that g is precompact. since the topology of g is finer than the topology of g, the group g is precompact. suppose that g is precompact. we assume that g is non-discrete, otherwise there is nothing to prove. let b be a local base at the identity e of g as in part (b) of definition 6.1 and take u ∈ b. choose an arbitrary element x ∈ ũ = u \ {e} and put v = x−1ũ. the set v is an open neighborhood of e in the precompact group g. let k be a finite subset of g such that kv = g and v k = g. for the finite set f = k ∪ kx−1, we have that g = xg = xv k = ũk ⊆ uf , and g = kv = kx−1xv = kx−1ũ ⊆ fu. thus g = uf and g = fu. we conclude that g is precompact. � it is worth mentioning that ‘precompact’ cannot be replaced with ‘pseudocompact’ or ‘feebly compact’ in proposition 6.2. indeed, let ts be the circle group endowed with the sorgenfrey topology (see the comment after corollary 3.2). it is clear that the underlying topological group of ts is the circle group t with the usual compact topology. the group ts is hausdorff and zero-dimensional, hence tychonoff. however, it is not pseudocompact. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 245 m. fernández and and m. tkachenko let us say that the left index of narrowness of a paratopological group g is less than or equal to an infinite cardinal τ or, in symbols, inl(g) ≤ τ, if g can be covered by at most τ translates of any neighborhood of the identity in g. the least cardinal τ ≥ ω such that the group g satisfies inl(g) ≤ τ is called the left index of narrowness of g. one defines the right index of narrowness of a paratopological group in a similar way. the left and right indices of narrowness of g are denoted respectively by inl(g) and inr(g). then the index of narrowness of g is defined as in(g) = inl(g) · inr(g). in general, inl(g) and inr(g) can be different. modifying slightly the argument in proof of proposition 6.2, we obtain the following: proposition 6.3. every almost topological group g satisfies inl(g) = inl(g) and inr(g) = inr(g), where g is the underlying topological group of g. it is easy to see that inl(h) = inr(h) for every topological group h, since inversion in topological groups is continuous. hence the next fact is immediate from proposition 6.3. corollary 6.4. every almost topological group g satisfies inl(g) = in(g) = inr(g). it turns out that the baire property behaves similarly to precompactness in almost topological groups: proposition 6.5. an almost paratopological group g is baire iff the underlying topological group g is baire. proof. we can assume without loss of generality that g is not discrete. the basic fact we are going to use is that the groups g and g have the same nowhere dense subsets. indeed, take a nowhere dense set a in g and an arbitrary nonempty open set u in g. then u is open in g, so there exists a nonempty open set v in g such that v ⊆ u \ a. pick an element x ∈ v . as the group g is almost topological, we can find an open neighborhood w of the identity e in g such that w̃ = w \ {e} is open in g and xw ⊆ v . hence o = xw̃ is a nonempty open set in g satisfying o ⊆ u \ a. hence a is nowhere dense in g. conversely, suppose that b is a nowhere dense subset of g and consider a nonempty open set u in g. arguing as above, we take an element x ∈ u and an open neighborhood v of e in g such that xv ⊆ u and the set ṽ = v \{e} is open in g. then xṽ is a nonempty open set in g, so there exists a nonempty open set w in g such that w ⊆ xṽ \b. then w is open in g and w ⊆ u \b, whence it follows that b is nowhere dense in g. finally, since the families of nowhere dense sets in g and g coincide, we conclude that g is baire iff so is g. � proposition 6.5 can be given an alternative proof as follows. it is known that every almost topological group is saturated [6, proposition 2.6]. given a saturated paratopological group (g, τ), denote by σ the finest topological group topology on g weaker than τ. according to [5, theorem 5], the family σ \ {∅} c© agt, upv, 2014 appl. gen. topol. 15, no. 2 246 subgroups of paratopological groups and feebly compact groups is a π-base for (g, τ). it follows from [6, proposition 2.5] that the groups (g, σ) and g are topologically isomorphic. hence the family of nonempty open sets in g forms a π-base for g as well. we conclude that the families of nowhere dense sets in g and g coincide and g is baire iff so is g. the fact that the nonempty open sets in g form a π-base for g can be applied to deduce the coincidence of several cardinal characteristics of an almost topological group g and its “twin” g. for example, one easily verifies that c(g) = c(g), d(g) = d(g), and πw(g) = πw(g), where the symbols ‘c’, ‘d’, and ‘πw’ stand for the cellularity, density, and π-weight, respectively. 7. some questions it is an interesting task to find out the permanence properties of the class of paratopological (or semitopological) sp-groups. for example, the quotient group of an sp-group is again an sp-group (see [13, proposition 3.3]). it is also clear that an arbitrary subgroup of an sp-group is an sp-group. we do not know, however, if the class of sp-groups is finitely productive: question 7.1. is the product of two paratopological (semitopological) spgroups an sp-group? if the answer to question 7.1 were in the affirmative, then the class of spgroups would be productive, i.e. arbitrary products of sp-groups would again be sp-groups. this fact can be easily verified using the argument from [13, theorem 3.2]. the relations between the properties of an almost topological group g and the underlying topological group g are not completely clear. we present here only one question in this respect (for the concept of r-factorizability, see [2, chapter 8] and [12, 15]). question 7.2. suppose that h is an r-factorizable almost topological group. is the underlying topological group h of h r-factorizable? it is easy to see that the inverse implication is false. indeed, the additive topological group r with its usual topology is r-factorizable, while the sorgenfrey line s is not. clearly the underlying topological group of s is r. question 7.3. suppose that h is a hausdorff precompact paratopological group. is every regular closed subset of h a gδ-set? a topological space x is an efimov space if for every family γ of gδ-sets in x, the closure of ⋃ γ is again a gδ-set in x [2, section 1.6, p. 52]. we also recall that a paratopological group g is 2-pseudocompact if the set ⋂ n∈ω u−1n is not empty, for every decreasing sequence {un} of nonempty open sets in g [9]. question 7.4. is every 2-pseudocompact (or feebly compact) paratopological group an efimov space? c© agt, upv, 2014 appl. gen. topol. 15, no. 2 247 m. fernández and and m. tkachenko references [1] a. v. arhangel’skii and e. a. reznichenko, paratopological and semitopological groups versus topological groups, topology appl. 151 (2005), 107–119. [2] a. v. arhangel’skii and m. g. tkachenko, topological groups and related structures, atlantis studies in mathematics, vol. 1, atlantis press and world scientific, paris– amsterdam, 2008. [3] r. l. blair, spaces in which special sets are z-embedded, canad. j. math. 28, no. 4 (1976), 673–690. [4] t. banakh and o. ravsky, on subgroups of saturated or totally bounded paratopological groups, algebra discrete math. 2003, no. 4 (2003), 1–20. [5] t. banakh and o. ravsky, oscillator topologies on a paratopological group and related number invariants, algebraic structures and their applications, kyiv: inst. mat. nanu (2002), 140–152. [6] m. fernández, on some classes of paratopological groups, topology proc. 40 (2012), 63–72. [7] o. ravsky, paratopological groups, ii, matematychni studii, 17 (2002) 93–101. [8] o. ravsky, the topological and algebraical properties of paratopological groups, ph.d. thesis, lviv university, 2003 (in ukrainian). [9] o. ravsky, pseudocompact paratopological groups, arxiv:1003.5343 [math. gn], september 2013. [10] e. a. reznichenko, extensions of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, topolology appl. 59 (1994), 233–244. [11] s. romaguera, m. sanchis and m. tkachenko, free paratopological groups, topology proc. 27, no. 2 (2003), 613–640. [12] m. g. tkachenko, paratopological and semitopological groups vs topological groups, ch. 20 in: recent progress in general topology iii (k.p. hart, j. van mill, p. simon, eds.), atlantis press, 2014; pp. 825–882. [13] m. g. tkachenko, g. delgadillo piñón and e. rodŕıguez cervera, a property of powers of the sorgenfrey line, q &a in general topology 27, no. 1 (2009), 45–49. [14] m. g. tkachenko, a. h. tomita, cellularity in subgroups of paratopological groups, preprint. [15] l.-h. xie, s. lin and m. tkachenko, factorization properties of paratopological groups, topology appl. 160 (2013), 1902–1917. c© agt, upv, 2014 appl. gen. topol. 15, no. 2 248 @ appl. gen. topol. 20, no. 1 (2019), 33-41doi:10.4995/agt.2019.9135 c© agt, upv, 2019 when is x × y homeomorphic to x ×l y ? raushan buzyakova miami, florida, usa (raushan buzyakova@yahoo.com) communicated by o. okunev abstract we identify a class of linearly ordered topological spaces x that may satisfy the property that x × x is homeomorphic to x ×l x or can be embedded into a linearly ordered space with the stated property. we justify the conjectures by partial results. 2010 msc: 06b30; 54f05; 06a05; 54a10. keywords: linearly ordered topological space; lexicographical product; homeomorphism, ordinal. 1. questions in this paper we provide a discussion that justifies our interest in the question of the title. we also identify more specific questions that may lead to affirmative resolutions. we back up our curiosity by some partial results and examples. the main result of this work is theorem 2.7. to proceed further let us agree on some terminology. a linear order will also be called an order. an order < on x is compatible with the topology of x, if the topology induced by < is equal to the topology of x. a linearly ordered topological space (abbreviated as lots) is a pair 〈x, <〉 of a topological space x and a topology-compatible order < on x. a topological space x is orderable if its topology can be induced by some order on x. when we consider the lexicographical product x ×l y of two lots x and y , we first take the lexicographical products of the ordered sets x and y and then induce the topology as determined by the lexicographical order on x ×l y . for the purpose of readability we will assume an informal style when describing some folklore-type structures or arguments. received 01 january 2018 – accepted 15 october 2018 http://dx.doi.org/10.4995/agt.2019.9135 r. buzyakova the operations of cartesian product and lexicographical product produce (more often than not) completely different structures. the former results in a visually ”more voluminous” structure, while the latter keeps ”visual linearity” but introduces ”stretches”. in rare cases, however, both operations produce the same results from a topological point of view. for example, q × q is homeomorphic to q ×l q. also, s × s is homeomorphic to s ×l s, where s = {±1/n : n = 1, 2, ...}. note that s is homeomorphic to the space n of natural numbers. however, n × n is not homeomorphic to n ×l n. indeed, the former is discrete while the latter has non-isolated points such as 〈2, 1〉, 〈3, 1〉, etc. following this discussion, it is not hard to see that given any discrete space d, it is possible to find a topology-compatible order ≺ on d such that d∗ = 〈d, ≺〉 is discrete and d∗ × d∗ is homeomorphic to d∗ ×l d ∗. our discussion prompts the following general problem. problem 1.1. what conditions on x guarantee that there exists a topologycompatible order ≺ on x such that x×x is homeomorphic to 〈x, ≺〉×l〈x, ≺〉? note that homogeneity is not a necessary condition as follows from the following folklore fact. example 1.2 (folklore). (ω+1)×l(ω+1) is homeomorphic to (ω+1)×(ω+1). proof. first observe that y = [(ω + 1) ×l (ω + 1)] \ {〈ω, n〉 : n = 1, 2, ..} is homeomorphic to (ω + 1) ×l (ω + 1). we will, therefore, provide a homeomorphism between x = (ω + 1) × (ω + 1) and y . we define our homeomorphism in three stages as follows: stage 1: for every n ∈ ω, fix a bijection fn between {〈n, k〉 ∈ x : k = n, ..., ω} ⊂ (ω + 1) × (ω + 1) and {〈2n, m〉 ∈ y : m ∈ ω + 1} ⊂ (ω + 1) ×l (ω + 1). such a homeomorphism exists since both subspaces are homeomorphic to ω + 1. stage 2: for every n ∈ ω, fix a bijection gn between {〈k, n〉 ∈ x : k = n + 1, ..., ω} and {〈2n + 1, m〉 ∈ y : m ∈ ω + 1}. stage 3: define the promised homomorphism from x to y as follows: f(x) =    fn(x) x ∈ {〈n, k〉 ∈ x : k = n, ..., ω} gn(x) x ∈ {〈k, n〉 ∈ x : k = n + 1, ..., ω} 〈ω, 0〉 x = 〈ω, ω〉 visually, f maps the n-th vertical at or above the diagonal in (ω +1)× (ω + 1) onto the (2n)-th copy of (ω + 1) in (ω + 1) ×l (ω + 1). also f maps the n-th horizontal under the diagonal in (ω + 1) × (ω + 1) onto the (2n + 1)-st copy of (ω + 1) in (ω + 1) ×l (ω + 1). finally, f maps the upper right corner point of the cartesian product to 〈ω, 0〉 of the lexicographical product, which is the only point that is the limit of a sequence of non-isolated points. clearly, f is a bijection. let us show that f and f−1 are continuous. since the domains and images of fn and gn are clopen in the respective superspaces, it c© agt, upv, 2019 appl. gen. topol. 20, no. 1 34 when is x × y homeomorphic to x ×l y ? remains to show that f is continuous at 〈ω, ω〉 and f−1 is continuous at 〈ω, 0〉. for this let un = [n, ω] × [n, ω]. then f(u) = {〈(a, b) ∈ y : a ≥ 2n}, which is an open neighborhood of 〈ω, 0〉 in y . we have {un}n is a basis at 〈ω, ω〉 in x and {f(un)}n is a basis at 〈ω, 0〉 in y . since y is bijective, f −1 is continuous at 〈ω, 0〉. we proved that (ω + 1) × (ω + 1) is homeomorphic to y , and therefore, to (ω + 1) ×l (ω + 1). � even though (ω + 1) is not homogeneous, it is homogeneous at all nonisolated points (since there is only such point). but even this property is not necessary for the two types of products to be homeomorphic. a similar argument can be used to verify the presence of the studied phenomenon in the following example. example 1.3. x ×l x is homeomorphic to x × x, where x = (ω ×l ω) + 1 . the limit points in this example have different natures. the leftmost point cannot be carried by a homeomorphism to any internal limit point. we omit the proof of the statement of example 1.3 since we will prove a more general one later (lemma 2.6). following example 1.2 and the fact that any discrete space has the property under discussion, one may wonder if any linearly ordered space with a single non-isolated point has the property. the following example shows that the answer is negative and opens another direction for our study. example 1.4. let x = (ω + 1) ⊕ d, where d is an ω1-sized discrete space. then the following hold: (1) x is orderable. (2) x × x is not homeomorphic to 〈x, ≺〉 ×l 〈x, ≺〉 for any topologycompatible order ≺ on x. proof. to see why x is orderable, first observe that we can think of x as the subspace of ω1 that contains only all isolated ordinals of ω1 and the ordinal ω. to order x, simply reverse the order of every sequence in form {α+1, α+2, ..., }, for each limit ordinal greater than ω. to prove part (2), fix an arbitrary topology-compatible order ≺ on x. the space x≺ = 〈x, ≺〉 has at least one of extreme points or neither. let us consider all possibilities. case (x≺ has neither minimum nor maximum): then {x}×lx≺ is clopen in x≺ ×l x≺ for each x. therefore, x≺ ×l x≺ is the free sum of ω1 many topological copies of x. hence, x≺ ×l x≺ is not homeomorphic to x × x. case (x≺ has minimum but not maximum): assume first that x≺ has a strictly increasing sequence {an}n converging to ω. then any neighborhood of 〈ω, min x≺〉 contains {an}×lx≺. therefore, any neighborhood of 〈ω, min x≺〉 has size ω1, while no point in x × x has such populous base neighborhoods. we now assume that x≺ has no strictly increasing sequences converging to ω. this and the absence of a maximum imply that x≺ ×x≺ c© agt, upv, 2019 appl. gen. topol. 20, no. 1 35 r. buzyakova does not have a topological copy of ω ×l ω + 1. however, x × x does, which is 〈ω, ω〉. in other words, the second derived set of the lexicographical product is empty but (x × x)′′ = {〈ω, ω〉}. case (〈x, ≺〉 has maximum but not minimum): similar to case 2. case (〈x, ≺〉 has both maximum and minimum): similar to the first part of case 2. since we have exhausted all cases, the proof is complete. � it is known (see, for example [2]) that given a subspace x of an ordinal, the square of x is homeomorphic to a subspace of a linearly ordered space if and only if x has no stationary subsets and is character homogeneous at all non-isolated points. this statement and the preceding discussion lead to the following question. question 1.5. let x be a subset of an ordinal, character homogeneous at non-isolated points, and have no stationary subsets. can x be embedded in a linearly ordered space l for which l × l and l ×l l are homeomorphic? note that even though spaces in examples 1.2 and 1.3 are not homogeneous, each point has a basis of mutually homeomorphic neighborhoods. this observation prompts the following question. question 1.6. let x be a subspace of an ordinal and every point of x has a basis of mutually homeomorphic neighborhoods. can x be embedded in a linearly ordered space l for which l × l and l ×l l are homeomorphic? in the next section we will justify the discussed questions by proving a statement that generalizes example 1.2. namely, we will show that if x is a subspace of an ordinal and is homogeneous on its derived set x′, then x is embeddable in a linearly ordered space l that has homeomorphic cartesian and lexicographical products (theorem 2.7). to prove this we will first identify a special class of spaces for which the two types of products are homeomorphic (lemma 2.6). the structure of these spaces is similar to that of the space in example 1.3. we, therefore, generalize example 1.3 too. in notations and terminology we will follow [3]. if x is a linearly-ordered set, by [a, b]x we denote the closed interval in x. if it is clear that the interval is considered in x but not in some larger ordered set, we simply write [a, b]. the same concerns other types of intervals. by x′ we will denote the set of all non-isolated points of x, that is, the derived set of x. we also say that x is homogeneous on its subset a if for every x, y ∈ a there exists a homeomorphism f : x → x such that f(x) = y and f(y) = x. 2. partial results in what follows, by l we denote the class of all subspaces of ordinals that are homogeneous on their derived sets. to prove our main statement (theorem 2.7), we start with two technical lemmas about the key properties of the members of l that will be used in further arguments. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 36 when is x × y homeomorphic to x ×l y ? lemma 2.1. let x ∈ l. then, for any x ∈ x′ there exists αx < x such that x is the single non-isolated point of [αx, x]x . proof. by homogeneity of x on x′ it suffices to show that the conclusion holds for some element of x′. we may assume that x′ is not empty. then z = min x′ is defined. then αz = min x is as desired for z. � lemma 2.2. let x ∈ l. then, x can be written as (⊕x∈x′ix) ⊕ d so that the following hold: (1) d is clopen and discrete, (2) ix and iy are homeomorphic for any x, y ∈ x ′, (3) x is the only non-isolated point of ix for each x ∈ x ′. proof. for any x ∈ x′, let αx be as in lemma 2.1. we can find βx between αx and x such that ix = (βx, x]x has the same cardinality as any smaller neighborhood of x. then d = x \ ∪{ix : x ∈ x ′} is a clopen discrete subset of x and x = (⊕x∈x′ix) ⊕ d is a desired representation. � to prove our target statement, first for each infinite cardinal γ, we identify a linearly ordered topological space 〈lγ, ≺〉 for which 〈lγ, ≺〉 × 〈lγ, ≺〉 is homeomorphic to 〈lγ, ≺〉 ×l 〈lγ, ≺〉. next, we will direct our efforts on the task of embedding the members of l into such spaces. construction of 〈lγ, ≺〉 for an infinite cardinal γ. definition of lγ. denote by λγ the ordinal (γ ×l γ) + 1. define lγ as the subspace of λγ that consists of all points α that fall into one of the following three categories: (1) α = max λγ (2) [α0, α] is order-isomorphic to γ + 1 for some α0 < α. (3) α is isolated. remark. to help visualize lγ, put i = {α < γ : α is isolated} ∪ {γ}. then lγ can be thought of as a γ-long sequence of γ-many clopen copies of i converging to max λγ. definition of ≺. if γ = ω, then lγ = λγ and we let ≺ be equal to the existing ordering <. for γ > ω, we will define ≺ using a folklore ordering procedure. we first define the order formally and then follow up with a simple demonstration. for each α ∈ λγ \ lγ, put rα = {α + 1, α + 2, ...}. by the definition of lγ and the fact that γ > ω, we conclude that rα is a closed subset of lγ . define ≺α on rα as follows: ...α + 5 ≺ α + 3 ≺ α + 1 < α + 2 ≺ α + 4.... define ≺ as follows: (1) x ≺ y if x, y ∈ rα and x ≺α y. (2) x ≺ y if {x, y} is not a subset of rα for any α and x < y. construction of 〈lγ, ≺〉 is complete. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 37 r. buzyakova to convince a reader that the above definition is legal without going into painful details, let us demonstrate a folklore construction of a topology-compatible order for the space x = {−1/n : n = 1, 2, 3, ...} ∪ {5 − 1/n : n = 1, 2, 3, ...}. the space x is not a linearly ordered space but there are many simple topologycompatible orders on x. the one that mimics the above construction is defined as follows. first, reverse the order on {5 − 1/n : n = 1, 2, 3...}. the resulting set becomes order isomorphic to {±1/n : n = 1, 2, 3, ..} and is homeomorphic to x. this short construction is formalized in the above definition in which we top every ”missing limit point” by the reversed sequence ”converging to the next missing limit point”. note that in our definition of ≺ for lγ we do not change the order position of limit points of lγ, which means that the new order coincides with the natural order when one of the compared elements is in l′γ. in a sense, the new order ≺ on lγ is almost indistinguishable from the standard order < if ”observed from far away”. also note that if x is a subspace of an ordinal that is homogeneous on the derived set, then by lemma 2.2, x can be embedded into lγ for some γ. let us record these observations for future reference. lemma 2.3. the following hold: (1) every x ∈ l embeds in lγ for some cardinal γ. (2) if x ∈ l′γ , y ∈ lγ, and x < y, then x ≺ y. (3) if x ∈ l′γ , y ∈ lγ, and y < x, then y ≺ x. we will often use the facts in this summary lemma without explicit referencing. our next goal is to show that lexicographical and cartesian product operations produce topologically equivalent results when applied to an lγ. we start by considering the two operations on smaller pieces of lγ’s. in the following three statements the arguments will be very similar to each other. for clarity, we will also use similar wording. lemma 2.4. let γ be an infinite cardinal. then [0, γ]lγ × [0, γ]lγ is homeomorphic to lγ. proof. to prove the statement we will visualize [0, γ]lγ and lγ as described in the remark after the definition of lγ. namely, [0, γ]lγ = i = {α < γ : α is isolated} ∪ {γ} and lγ is a γ-long sequence of γ-many clopen copies of i converging to ∞ = max lγ. we can write then lγ = (⊕{iα = i : α < γ, α is isolated}) ∪ {∞}, where every neighborhood of ∞ contains all iα’s starting from some moment. having these visuals in mind we will construct a desired homeomorphism in three stages as follows: stage 1: partition the set of isolated ordinals of γ into pairs {{aα, bα} : α < γ, α is isolated} so that bα = aα + 1 and indexing agrees with the natural well-ordering < of the partitioned set. stage 2: since γ is an infinite cardinal, [α, γ]lγ is homeomorphic to i for any α < γ. therefore, for each isolated α < γ we can fix homeomorphisms gα : {α} × [α, γ]lγ → ibα and hα : (α, γ]lγ × {α} → iaα. that c© agt, upv, 2019 appl. gen. topol. 20, no. 1 38 when is x × y homeomorphic to x ×l y ? is, gα maps the α’s vertical of [0, γ]lγ ×[0, γ]lγ at or above the diagonal onto bα’s copy of i in lγ and hα maps the α’s horizontal strictly below the diagonal onto aα’s copy of i. stage 3: define a homomorphism f from [0, γ]lγ × [0, γ]lγ to lγ as follows: f(p) =    gα(p) if p ∈ {α} × [α, γ]lγ hα(p) if p ∈ (α, γ]lγ × {α} ∞ if p = 〈γ, γ〉 the argument similar to that in example 1.2 shows that f is a homeomorphism. � lemma 2.5. let γ be an infinite cardinal. then [0, γ]lγ ×lγ is homeomorphic to [0, γ]〈lγ,≺〉 ×l 〈lγ, ≺〉. proof. denote the spaces in the statement by x and y , respectively. since y is homeomorphic to z = y \ ({γ} ×l [1, γ]〈lγ,≺〉), it suffices to construct an isomorphism from x to z, which we will do next. when treating [0, γ]〈lγ,≺〉 and 〈lγ, ≺〉 as topological spaces with regard to order, we will visualize them as described in lemma 2.4. for convenience, let us copy our notation from lemma 2.4 next: [0, γ]lγ = i = {α < γ : α is isolated} ∪ {γ} lγ = (⊕{iα = i : α < γ, α is isolated}) ∪ {∞}, where ∞ is the maximum element of lγ in either of the two orders. we are now ready to construct a desired homeomorphism in three stages as follows: stage 1: partition the set of isolated ordinals of γ into pairs {{aα, bα} : α < γ, α is isolated} so that bα = aα + 1 and indexing agrees with the natural well-ordering < of the partitioned set. stage 2: by lemma 2.4, for each isolated ordinal α < γ there exists a homeomorphism hα of [α, γ]lγ × iα onto {aα} ×l 〈lγ, ≺〉. since γ is an infinite cardinal, vα = lγ \ ⋃ β≤α iβ is homeomorphic to lγ. hence, we can find a homeomorphism gα from {α} × vα onto {bα} ×l 〈lγ, ≺〉. stage 3: define a homomorphism f from x to z as follows: f(p) =    gα(p) if p ∈ {α} × vα hα(p) if p ∈ [α, γ]lγ × iα 〈γ, ∞〉 if p = 〈γ, ∞〉 in words, f maps most of the α’s horizontal strip corresponding to iα onto the aα’s copy of 〈lγ, ≺〉, most of the α’s vertical onto bα’s copy of 〈lγ, ≺〉, and the corner point of the cartesian product to the maximum of z. the argument similar to that in example 1.2 shows that f is a homeomorphism. � we are now ready to prove a generalization of the statement of example 1.3. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 39 r. buzyakova lemma 2.6. for every infinite cardinal γ, the space lγ ×lγ is homeomorphic to 〈lγ, ≺〉 ×l 〈lγ, ≺〉. proof. denote by x and y the two spaces in the statement. as in lemma 2.5, it suffices to construct a homeomorphism from x to z = y \({∞}×l[1, ∞]〈lγ,≺〉) as in the previous two lemmas, we visualize [0, γ]lγ and lγ as follows: [0, γ]lγ = i = {α < γ : α is isolated} ∪ {γ} lγ = (⊕{iα = i : α < γ, α is isolated}) ∪ {∞}, where ∞ is the maximum element of lγ in either of the two orders. we will closely follow our constructions in the previous two lemmas and construct the promised homeomorphism in three stages as follows: stage 1: partition the set of isolated ordinals of γ into pairs {{aα, bα} : α < γ, α is isolated} so that bα = aα + 1 and indexing agrees with the natural well-ordering < of the partitioned set. stage 2: by lemma 2.5, for each isolated α < γ, we can fix two homeomorphisms: hα : lγ × iα → 〈iaα, ≺〉 ×l 〈lγ, ≺〉 gα : iα ×  lγ \ ⋃ β≤α iβ   → 〈ibα, ≺〉 ×l 〈lγ, ≺〉 stage 3: define a homomorphism f from x to z as follows: f(p) =      gα(p) if p ∈ iα × ( lγ \ ⋃ β≤α iβ ) hα(p) if p ∈ lγ × iα 〈∞, ∞〉 if p = 〈∞, ∞〉 in words, f maps most of the α’s horizontal strip onto the aα’s copy of i ×l lγ, most of the α’s vertical strip onto the bα’s copy of i ×l lγ, and the corner point of the cartesian product to the maximum of z. an argument similar to one of example 1.2 shows that f is a homeomorphism. � lemmas 2.6 and 2.3 imply the following main statement of our discussion. theorem 2.7. let x be a subspace of an ordinal that is homogeneous on the derived set. then x can be embedded into a lot z such that z ×l z is homeomorphic to z × z. in search for candidates with the discussed phenomenon, it is clear that we should immediately eliminate any ordered spaces with stationary subsets. indeed, the square of such a space is not orderable as follows from a standard generalization of katetov’s example [5]. therefore, by the characterization of hereditary paracompactness for go-spaces due to engelking and lutzer ([1] or [4]), we should consider only hereditary paracompact ordered spaces. it is clear that if x has no stationary subset, then x2 does not have such either. thus, we c© agt, upv, 2019 appl. gen. topol. 20, no. 1 40 when is x × y homeomorphic to x ×l y ? need to concentrate on spaces with orderable hereditary paracompact squares. while the engelking-lutzer characterization is incredibly handy for testing an ordered space for hereditary paracompactness, the author is not aware of any criterion for the square of a lots to be hereditary paracompact. is there such a criterion? if not, let us find one! references [1] h. bennet and d. lutzer, linearly ordered and generalized ordered spaces, encyclopedia of general topology, elsevier, 2004. [2] r. buzyakova, ordering a square, topology appl. 191 (2015), 76–81. [3] r. engelking, general topology, pwn, warszawa, 1977. [4] d. lutzer, ordered topological spaces, surveys in general topology, edited by g. m. reed., academic press, new york (1980), 247–296. [5] m. katetov, complete normality of cartesian products, fund. math. 36 (1948), 271–274. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 41 () @ appl. gen. topol. 18, no. 2 (2017), 277-287doi:10.4995/agt.2017.6776 c© agt, upv, 2017 existence of common fixed points of improved f-contraction on partial metric spaces muhammad nazam a, muhammad arshad a and mujahid abbas b a department of mathematics and statistics, international islamic university, islamabad, pakistan (nazim.phdma47@iiu.edu.pk, marshadzia@iiu.edu.pk) b department of mathematics and applied mathematics, university of pretoria, lynnwood road, pretoria 0002, south africa. (abbas.mujahid@gmail.com) communicated by s. romaguera abstract following the approach of f-contraction introduced by wardowski [13], in this paper, we introduce improved f-contraction of rational type in the framework of partial metric spaces and used it to obtain a common fixed point theorem for a pair of self mappings. we show, through example, that improved f-contraction is more general than fcontraction and guarantees fixed points in those cases where f-contraction fails to provide. moreover, we apply this fixed point result to show the existence of common solution of the system of integral equations. 2010 msc: 47h10; 47h04; 54h25. keywords: fixed point; improved f-contraction; integral equations; partial metric. 1. introduction matthews [11] introduced the concept of partial metric spaces and proved an analogue of banach’s fixed point theorem in partial metric spaces. in fact, a partial metric space is a generalization of metric space in which the self distances p(r1, r1) of elements of a space may not be zero and follows the inequality p(r1, r1) ≤ p(r1, r2). after this remarkable contribution, many authors took interest in partial metric spaces and its topological properties and presented received 29 october 2016 – accepted 18 march 2017 http://dx.doi.org/10.4995/agt.2017.6776 m. nazam, m. arshad and m. abbas several well known fixed point results in the framework of partial metric spaces (see [1, 2, 3, 4, 12] and references therein). in 1922, banach presented a landmark fixed point result (banach contraction principle). this result proved a gateway for the fixed point researchers and opened a new door in metric fixed point theory. a number of efforts have been made to enrich and generalize banach contraction principle (see [6, 7] and references therein). following banach, in 2012, wardowski [13] presented a new contraction (known as f contraction). since 2012, a number of fixed point results have been established by using f-contraction (see [5, 9, 10] and references therein). recently, vetro et al.[8] proved some fixed point theorems for hardy-rogers-type self-mappings in complete metric spaces and complete ordered metric spaces for fcontractions. in this article, following wardowski [13] and vetro et al.[8], we prove a common fixed point theorem for a pair of self mappings satisfying improved fcontraction of rational type in complete partial metric spaces. an example is constructed to illustrate this result. we apply the mentioned theorem to show the existence of solution of system of volterra type integral equations. 2. preliminaries throughout this paper, we denote (0, ∞) by r+, [0, ∞) by r+0 , (−∞, +∞) by r and set of natural numbers by n. following concepts and results will be required for the proofs of main results. definition 2.1 ([13]). a mapping t : m → m, is said to be f-contraction if it satisfies following condition (2.1) (d(t (r1), t (r2)) > 0 ⇒ τ + f(d(t (r1), t (r2)) ≤ f(d(r1, r2))), for all r1, r2 ∈ m and some τ > 0. where f : r + → r is a function satisfying following properties. (f1) : f is strictly increasing. (f2) : for each sequence {rn} of positive numbers limn→∞ rn = 0 if and only if limn→∞ f(rn) = −∞. (f3) : there exists θ ∈ (0, 1) such that limα→0+(α) θf(α) = 0. wardowski [13] established the following result using f-contraction. theorem 2.2 ([13]). let (m, d) be a complete metric space and t : m → m be a f-contraction. then t has a unique fixed point υ ∈ m and for every r0 ∈ m the sequence {t n(r0)} for all n ∈ n is convergent to υ. we denote by ∆f , the set of all functions satisfying the conditions (f1)− (f3). example 2.3 ([13]). let f : r+ → r be given by the formula f(α) = ln(α). it is clear that f satisfies (f1)−(f3) for any κ ∈ (0, 1). each mapping t : m → m satisfying (2.1) is a f-contraction such that d(t (r1), t (r2)) ≤ e −τ d(r1, r2), for all r1, r2 ∈ m, t (r1) 6= t (r2). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 278 improved f -contraction on partial metric spaces obviously, for all r1, r2 ∈ m such that t (r1) = t (r2), the inequality d(t (r1), t (r2)) ≤ e −τ d(r1, r2) holds, that is t is a banach contraction. remark 2.4. from (f1) and (2.1) it is easy to conclude that every f-contraction is necessarily continuous. definition 2.5 ([11]). let m be a nonempty set and if the function p : m × m → r+0 satisfies following properties, (p1) r1 = r2 ⇔ p (r1, r1) = p (r1, r2) = p (r2, r2) , (p2) p (r1, r1) ≤ p (r1, r2) , (p3) p (r1, r2) = p (r2, r1) , (p4) p (r1, r3) ≤ p (r1, r2) + p (r2, r3) − p (r2, r2) . for all r1, r2, r3 ∈ m. then p is called a partial metric on m and the pair (m, p) is known as partial metric space. in [11], matthews proved that every partial metric p on m induces a metric dp : m × m → r + 0 defined by (2.2) dp (r1, r2) = 2p (r1, r2) − p (r1, r1) − p (r2, r2) ; for all r1, r2 ∈ m. notice that a metric on a set m is a partial metric p such that p(r, r) = 0 for all r ∈ m and p(r1, r2) = 0 implies r1 = r2 ( using (p1) and (p2)). matthews [11] established that each partial metric p on m generates a t0 topology τ(p) on m. the base of topology τ(p) is the family of open p-balls {bp (r, ǫ) : r ∈ m, ǫ > 0}, where bp (r, ǫ) = {r1 ∈ m : p (r, r1) 0. a sequence {rn}n∈n in (m, p) converges to a point r ∈ m if and only if p(r, r) = limn→∞ p(r, rn). definition 2.6 ([11]). let (m, p) be a partial metric space. (1) a sequence {rn}n∈n in (m, p) is called a cauchy sequence if limn,m→∞ p(rn, rm) exists and is finite. (2) a partial metric space (m, p) is said to be complete if every cauchy sequence {rn}n∈n in m converges, with respect to τ(p), to a point r ∈ x such that p(r, r) = limn,m→∞ p(rn, rm). the following lemma will be helpful in the sequel. lemma 2.7 ([11]). (1) a sequence rn is a cauchy sequence in a partial metric space (m, p) if and only if it is a cauchy sequence in metric space (m, dp) (2) a partial metric space (m, p) is complete if and only if the metric space (m, dp) is complete. (3) a sequence {rn}n∈n in m converges to a point r ∈ m, with respect to τ(dp) if and only if limn→∞ p(r, rn) = p(r, r) = limn,m→∞ p(rn, rm). (4) if limn→∞ rn = υ such that p(υ, υ) = 0 then limn→∞ p(rn, r) = p(υ, r) for every r ∈ m. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 279 m. nazam, m. arshad and m. abbas in the following example, we shall show that there are mappings which are not f-contractions in metric spaces, nevertheless, such mappings follow the conditions of f-contraction in partial metric spaces. example 2.8. let m = [0, 1] and define partial metric by p(r1, r2) = max {r1, r2} for all r1, r2 ∈ m. the metric d induced by partial metric p is given by d(r1, r2) = |r1 − r2| for all r1, r2 ∈ m. define the mappings f : r + → r by f(r) = ln(r) and t by t (r) =      r 5 if r ∈ [0, 1); 0 if r = 1 then t is not a fcontraction in a metric space (m, d). indeed, for r1 = 1 and r2 = 5 6 , d(t (r1), t (r2)) > 0 and we have τ + f (d(t (r1), t (r2))) ≤ f (d(r1, r2)) , τ + f ( d(t (1), t ( 5 6 )) ) ≤ f ( d(1, 5 6 ) ) , τ + f ( d(0, 1 6 ) ) ≤ f ( 1 6 ) , 1 6 < 1 6 , which is a contradiction for all possible values of τ. now if we work in partial metric space (m, p), we get a positive answer that is τ + f (p(t (r1), t (r2))) ≤ f (p(r1, r2)) implies τ + f ( 1 6 ) ≤ f (1) , which is true. similarly, for all other points in m our claim proves true. 3. main result we begin with following definitions. definition 3.1. let (m, p) be a partial metric space. the mapping t : m → m is called a improved f-contraction of rational type, if for all m1, m2 ∈ m, we have (3.1) τ + f(p(t (m1), t (m2))) ≤ f (n(m1, m2)) , for some f ∈ ∆f , τ > 0 and n(m1, m2) = max        p(m1, m2), p(m1, t (m1))p(m2, t (m2)) 1 + p(m1, m2) , p(m1, t (m1))p(m2, t (m2)) 1 + p(t (m1), t (m2))        . c© agt, upv, 2017 appl. gen. topol. 18, no. 2 280 improved f -contraction on partial metric spaces definition 3.2. let (m, p) be a partial metric space. the mappings s, t : m → m are called a pair of improved f-contraction of rational type, if for all m1, m2 ∈ m , we have (3.2) (p(s(m1), t (m2)) > 0 implies τ + f(p(s(m1), t (m2))) ≤ f (m(m1, m2))) , for some f ∈ ∆f , τ > 0 and m(m1, m2) = max        p(m1, m2), p(m1, s(m1))p(m2, t (m2)) 1 + p(m1, m2) , p(m1, s(m1))p(m2, t (m2)) 1 + p(s(m1), t (m2))        . the following theorem is one of our main results. theorem 3.3. let (m, p) be a complete partial metric space and s, t : m → m be a pair of mappings such that (1) s or t is a continuous mapping, (2) (s, t ) is a pair of improved f-contraction of rational type. then there exists a common fixed point υ of the pair (s, t ) in m such that p(υ, υ) = 0. proof. we begin with the following observation: m(m1, m2) = 0 if and only if m1 = m2 is a common fixed point of (s, t ). indeed, if m1 = m2 is a common fixed point of (s, t ), then t (m2) = t (m1) = m1 = m2 = s(m2) = s(m1) and m(m1, m2) = max        p(m1, m1), p(m1, s(m1))p(m2, t (m2)) 1 + p(m1, m2) , p(m1, s(m1))p(m2, t (m2)) 1 + p(s(m1), t (m2))        , = p(m1, m1). from contractive condition (3.2), we get τ + f(p(m1, m1)) = τ + f(p(s(m1), t (m2))) ≤ f (p(m1, m1)) . this is only possible if p(m1, m1) = 0, which entails m(m1, m1) = 0. conversely, if m(m1, m2) = 0, it is easy to check that m1 = m2 is a fixed point of s and t . in order to find common fixed points of s and t for the situation when m(r1, r2) > 0 for all r1, r2 ∈ m with r1 6= r2, we construct an iterative sequence {rn} of points in m such a way that, r2i+1 = s(r2i) and r2i+2 = t (r2i+1) where i = 0, 1, 2, . . . . assume that p(s(r2i), t (r2i+1)) > 0, then from contractive condition (3.2), we get f (p(r2i+1, r2i+2)) = f (p(s(r2i), t (r2i+1))) ≤ f (m(r2i, r2i+1)) − τ, c© agt, upv, 2017 appl. gen. topol. 18, no. 2 281 m. nazam, m. arshad and m. abbas for all i ∈ n ∪ {0}, where m(r2i, r2i+1) = max        p(r2i, r2i+1), p(r2i, s(r2i))p(r2i+1, t (r2i+1)) 1 + p(r2i, r2i+1) , p(r2i, s(r2i))p(r2i+1, t (r2i+1)) 1 + p(s(r2i), t (r2i+1))        = max        p(r2i, r2i+1), p(r2i, r2i+1)p(r2i+1, r2i+2) 1 + p(r2i, r2i+1) , p(r2i, r2i+1)p(r2i+1, r2i+2) 1 + p(r2i+1, r2i+2)        ≤ max {p(r2i, r2i+1), p(r2i+1, r2i+2)} . for if m(r2i, r2i+1) ≤ p(r2i+1, r2i+2), then f (p(r2i+1, r2i+2)) ≤ f (p(r2i+1, r2i+2)) − τ, which is a contradiction due to f1. therefore, f (p(r2i+1, r2i+2)) ≤ f (p(r2i, r2i+1)) − τ, for all i ∈ n ∪ {0}. hence, (3.3) f (p(rn+1, rn+2)) ≤ f (p(rn, rn+1)) − τ, for all n ∈ n ∪ {0}. following (3.3), we obtain f (p(rn, rn+1)) ≤ f (p(rn−2, rn−1)) − 2τ. repeating these steps we get, (3.4) f (p(rn, rn+1)) ≤ f (p(r0, r1)) − nτ. from (3.4), we obtain limn→∞ f (p(rn, rn+1)) = −∞. since f ∈ ∆f , (3.5) lim n→∞ p(rn, rn+1) = 0. from the property (f3) of f-contraction, there exists κ ∈ (0, 1) such that (3.6) lim n→∞ ((p(rn, rn+1)) κ f (p(rn, rn+1))) = 0. following (3.4), for all n ∈ n, we obtain (3.7) (p(rn, rn+1)) κ (f (p(rn, rn+1)) − f (p(r0, x1))) ≤ − (p(rn, rn+1)) κ nτ ≤ 0. considering (3.5), (3.6) and letting n → ∞ in (3.7), we have (3.8) lim n→∞ (n (p(rn, rn+1)) κ ) = 0. since (3.8) holds, there exists n1 ∈ n, such that n (p(rn, rn+1)) κ ≤ 1 for all n ≥ n1 or, (3.9) p(rn, rn+1) ≤ 1 n 1 κ for all n ≥ n1. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 282 improved f -contraction on partial metric spaces using (3.9), we get for m > n ≥ n1, p(rn, rm) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) − m−1 ∑ j=n+1 p(rj, rj) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) = m−1 ∑ i=n p(ri, ri+1) ≤ ∞ ∑ i=n p(ri, ri+1) ≤ ∞ ∑ i=n 1 i 1 k . the convergence of the series ∑ ∞ i=n 1 i 1 κ entails limn,m→∞ p(rn, rm) = 0. hence {rn} is a cauchy sequence in (m, p). due to lemma 2.7, {rn} is a cauchy sequence in (m, dp). since (m, p) is a complete partial metric space, so (m, dp) is a complete metric space and as a result there exists υ ∈ m such that limn→∞ dp(rn, υ) = 0, moreover, by lemma 2.7 (3.10) lim n→∞ p(υ, rn) = p(υ, υ) = lim n,m→∞ p(rn, rm). since limn,m→∞ p(rn, rm) = 0, from (3.10) we deduce that (3.11) p(υ, υ) = 0 = lim n→∞ p(υ, rn). now from (3.11) it follows that r2n+1 → υ and r2n+2 → υ as n → ∞ with respect to τ(p). the continuity of t implies υ = lim n→∞ rn = lim n→∞ r2n+1 = lim n→∞ r2n+2 = lim n→∞ t (r2n+1) = t ( lim n→∞ r2n+1) = t (υ), and from contractive (3.2), we have τ + f(p(υ, s(υ))) = τ + f(p(s(υ), t (υ))) ≤ f(m(υ, υ)) = f(p(υ, υ)). this implies that p(υ, s(υ)) = 0 and due to (p1), (p2) we conclude that υ = s(υ). thus we have s(υ) = t (υ) = υ. hence (s, t ) has a common fixed point υ. now we show that υ is the unique common fixed point of s and t . assume the contrary, that is, there exists ω ∈ m such that υ 6= ω and ω = t (ω). from the contractive condition (3.2), we have (3.12) τ + f(p(s(υ), t (ω))) ≤ f (m(υ, ω)) , c© agt, upv, 2017 appl. gen. topol. 18, no. 2 283 m. nazam, m. arshad and m. abbas where m(υ, ω) = max        p(υ, ω), p(υ, s(υ))p(ω, t (ω)) 1 + p(υ, y) , p(υ, s(υ))p(ω, t (ω)) 1 + p(s(υ), t (ω))        . from (3.12), we have (3.13) τ + f(p(υ, ω)) ≤ f (p(υ, ω)) , the inequality (3.13), leads to p(υ, ω) < p(υ, ω), which is a contradiction. hence, υ = ω and υ is a unique common fixed point of a pair (s, t ). � the following example illustrates theorem 3.3 and shows that condition (3.2) is more general than contractivity condition given by wardowski ([13]). example 3.4. let m = [0, 1] and define p(r1, r2) = max {r1, r2}, then (m, p) is a complete partial metric space. moreover, define d (r1, r2) = |r1 − r2|, so, (m, d) is a complete metric space. define the mappings s, t : m → m as follows: t (r) =      r 5 if r ∈ [0, 1); 0 if r = 1 and s(r) = 3r 7 for all r ∈ m clearly, s, t are self mappings. define the function f : r+ → r by f(r) = ln(r), for all r ∈ r+ > 0. let r1, r2 ∈ m such that p(s(r1), t (r2)) > 0 and suppose that r1 ≤ r2. then m(r1, r2) = max { r2, r1r2 1 + r1 , r1r2 1 + max { 3r1 7 , r2 5 } } . since r1 1+r1 < 1 and r1 1+max{ 3r17 , r2 5 } < 1, we have that m(r1, r2) = r2. in a similar way, if r1 ≥ r2, we obtain that m(r1, r2) = r1, i.e., m(r1, r2) = p(r1, r2). let τ ≤ ln(7 3 ). then τ + (p(s(r1), t (r2))) = τ + ln ( max { 3r1 7 , r2 5 }) ≤ ln( 7 3 ) + ln ( max { 3p(r1, r2) 7 , p(r1, r2) 5 }) = ln( 7 3 ) + ln ( 3p(r1, r2) 7 ) = ln (p(r1, r2)) = f (m(r1, r2)) . thus, the contractive condition (3.2) is satisfied for all r1, r2 ∈ m. hence, all the hypotheses of the theorem 3.3 are satisfied, note that (s, t ) have a unique c© agt, upv, 2017 appl. gen. topol. 18, no. 2 284 improved f -contraction on partial metric spaces common fixed point r = 0. as we have seen in example 2.8, t is not a fcontraction in (m, d) and consequently we can not apply theorem 2.2. corollary 3.5. let (m, p) be a complete partial metric space and t : m → m be a mapping such that (1) t is a continuous mapping, (2) t is an improved f-contraction of rational type. then t has a unique fixed point υ in m such that p(υ, υ) = 0. proof. setting s = t in theorem 3.3, we obtain required result. � remark 3.6. if we set n(r1, r2) = max {p(r1, r2), p(r1, t (r1)), p(r2, t (r2))} in inequality (3.1), corollary 3.5 remains true. similarly, by setting m(r1, r2) = max {p(r1, r2), p(r1, t (r1)), p(r2, s(r2))} in inequality (3.2), theorem 3.3 remains true. 4. application of theorem 3.3 in this part of paper, we shall apply theorem 3.3 to show the existence of common solution of the system of volterra type integral equations. such system is given by the following equations. u(t) = f(t) + t ∫ 0 kn(t, s, u(s))ds,(4.1) w(t) = f(t) + t ∫ 0 jn(t, s, w(s))ds,(4.2) for all t ∈ [0, a], and a > 0. we shall show, by using theorem 3.3, that the solution of integral equations (4.1) and (4.2) exists. let c([0, a], r) be the space of all continuous functions defined on [0, a]. for u ∈ c([0, a], r), define sup norm as: ‖u‖τ = sup t∈[0,a] {u(t)e−τt}, where τ > 0. let c([0, a], r) be endowed with the partial metric (4.3) pτ(u, v) = dτ (u, v) + cn = sup t∈[0,a] ‖ |u(t) − v(t)| e−τt‖τ + cn for all u, v ∈ c([0, a], r) and {cn} is a sequence of positive real numbers such that limn→∞ cn = 0. obviously, c([0, a], r, ‖ · ‖τ) is a banach space. now we prove the following theorem to ensure the existence of solution of system of integral equations. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 285 m. nazam, m. arshad and m. abbas theorem 4.1. assume the following conditions are satisfied. (i) kn, jn : [0, a] × [0, a] × r → r and f, g : [0, a] → r are continuous. (ii) define su(t) = f(t) + t ∫ 0 kn(t, s, u(s))ds, t u(t) = f(t) + t ∫ 0 jn(t, s, u(s))ds, and when n → ∞ there exists τ ≥ 1 such that |kn(t, s, u) − jn(t, s, v)| ≤ τe −τ [m(u, v)] for all t, s ∈ [0, a] and u, v ∈ c([0, a], r), where m(u, v) = max        p(u(t), v(t)), p(u(t), su(t))p(v(t), t v(t)) 1 + p(u(t), v(t)) , p(u(t), su(t))p(v(t), t v(t)) 1 + p(su(t), t v(t))        . then the system of integral equations given in (4.1) and (4.2) has a solution. proof. following assumption (ii), we have p(su(t), t v(t)) = d(su(t), t v(t)) + cn = t ∫ 0 |kn(t, s, u(s) − jn(t, s, v(s)))| ds + cn ≤ t ∫ 0 τe −τ ([m(u, v)]e−τs)eτsds (by taking limit n → ∞) ≤ t ∫ 0 τe−τ ‖m(u, v)‖τe τsds ≤ τe−τ ‖m(u, v)‖τ t ∫ 0 e τs ds ≤ τe−2τ ‖m(u, v)‖τ 1 τ e τt ≤ e−τ‖m(u, v)‖τe τt . this implies p(su(t), t v(t))e−τt ≤ e−τ‖m(u, v)‖τ, that is ‖p(su(t), t v(t))‖ τ ≤ e−τ‖m(u, v)‖τ, c© agt, upv, 2017 appl. gen. topol. 18, no. 2 286 improved f -contraction on partial metric spaces and consequently, τ + ln ‖p(su(t), t v(t))‖τ ≤ ln ‖m(u, v)‖τ . so conditions of theorem 3.3 are satisfied. hence the system of integral equations given in (4.1) and (4.2) have a unique common solution. � acknowledgements. the authors sincerely thank the learned referees for a careful reading and thoughtful comments. the present version of the paper owes much to the precise and kind remarks of anonymous referees. references [1] m. abbas, t. nazir and s. romaguera, fixed point results for generalized cyclic contraction mappings in partial metric spaces, racsam 106 (2012), 287–297. [2] t. abdeljawad, e. karapinar and k. tas, existence and uniqueness of a common fixed point on partial metric spaces, appl. math. lett. 24 (2011), 1900–1904. [3] i. altun, f. sola and h. simsek, generalized contractions on partial metric spaces, topology appl. 157 (2010), 2778–2785. [4] i. altun and s. romaguera. characterizations of partial metric completeness in terms of weakly contractive mappings having fixed point, applicable analysis and discrete mathematics 6, no. 2 (2012), 247–256. [5] m. arshad, a. hussain and m. nazam, some fixed point results for multivalued fcontraction on closed ball, func. anal.-tma 2 (2016), 69–80 [6] s. chandok, some fixed point theorems for (α, β)-admissible geraghty type contractive mappings and related results, math. sci. 9 (2015), 127–135. [7] s. h. cho, s. bae and e. karapinar fixed point theorems for α-geraghty contraction type maps in metric spaces, fixed point theory appl. 2013, 2013:329. [8] m. cosentino and p. vetro, fixed point results for f-contractive mappings of hardyrogers-type, filomat 28, no. 4 (2014), 715–722. [9] a. hussain, m. nazam and m. arshad, connection of ciric type f-contraction involving fixed point on closed ball, gu j. sci. 30, no. 1 (2017), 283–291. [10] a. hussain, h. f. ahmad, m. nazam and m. arshad, new type of multivalued fcontraction involving fixed points on closed ball, j. math. computer sci. 10 (2017), 246–254. [11] s. g. matthews, partial metric topology, in: proceedings of the 11th summer conference on general topology and applications, vol. 728, pp.183-197, the new york academy of sciences, august, 1995. [12] m. nazam, m. arshad and c. park fixed point theorems for improved α-geraghty contractions in partial metric spaces, j. nonlinear sci. appl. 9 (2016), 4436–4449. [13] d. wardowski, fixed point theory of a new type of contractive mappings in complete metric spaces, fixed point theory appl. (2012) article id 94. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 287 () @ appl. gen. topol. 19, no. 1 (2018), 145-153doi:10.4995/agt.2018.7883 c© agt, upv, 2018 k-semistratifiable spaces and expansions of set-valued mappings peng-fei yan, xing-yu hu and li-hong xie∗ school of mathematics and computational science, wuyi university, jiangmen 529020, p. r. china (ypengfei@sina.com, 121602580@qq.com, yunli198282@126.com) communicated by a. tamariz-mascarúa abstract in this paper, the concept of k-upper semi-continuous set-valued mappings is introduced. using this concept, we give characterizations of k-semistratifiable and k-mcm spaces, which answers a question posed by xie and yan [9]. 2010 msc: 54c65; 54c60. keywords: locally bounded set-valued mappings; k-mcm spaces; ksemistratifiable spaces; lower semi-continuous (l.s.c.); k-upper semi-continuous (k-u.s.c.). 1. introduction before stating the paper, we give some definitions and notations. for a mapping φ : x → 2y and w ⊆ y , the symbols φ−1[w ] and φ♯[w ] stand for {x ∈ x : φ(x) ⋂ w 6= ∅} and {x ∈ x : φ(x) ⊆ w}, respectively. a set-valued mapping φ : x → 2y is lower semi-continuous (l.s.c) if φ−1[w ] is open in x for every open subset w of y . also, a set-valued mapping φ : x → 2y is upper semi-continuous (u.s.c) if φ♯[w ] is open in x for every open subset w of y . for mappings φ, φ ′ : x → 2y , we express by φ ⊆ φ ′ if φ(x) ⊆ φ ′ (x) for each x ∈ x. an operator φ assigning to each set-valued ∗supported by nsfc(nos. 11601393; 11526158), the phd start-up fund of natural science foundation of guangdong province (no. 2014a0303101872) received 17 july 2017 – accepted 05 february 2018 http://dx.doi.org/10.4995/agt.2018.7883 p.-f. yan, x.-y. hu and l.-h. xie mapping φ : x → 2y , φ(φ) : x → 2y , φ is called as a preserved order operator if φ(φ) ⊆ φ(φ′) whenever φ ⊆ φ′. for a space y , define f(y ) = {f ⊆ y : f is a nonempty closed set in y }. for a metric space (y, ρ), a subset b of y is called bounded if the diameter of b (with respect to ρ) is finite, and we define b(y ) = {f ⊆ y : f 6= ∅, f is closed and bounded in y }. a sequence {bn}n∈n of closed subsets of a space y is called a strictly increasing closed cover [10] if ⋃ n∈n bn = y and bn ( bn+1 for each n ∈ n. for a space y having a strictly increasing closed cover {bn}, a subset b of y is said to be bounded [10] (with respect to {bn}) if b ⊆ bn for some n ∈ n. define b(y ; {bn}) = {f ⊆ y : f 6= ∅, f is closed and bounded in y }. for a space y with a strictly increasing closed cover {bn}, a mapping φ : x → b(y ; {bn}) is called locally bounded at x if there exist a bounded set v of (y ; {bn}) and a neighborhood o of x such that o ⊆ φ ♯[v ]; if φ is locally bounded at each x ∈ x, then φ is called locally bounded [10] on x. let (y, ρ) be a metric space. for a mapping φ : x → f(y ), define uφ = {x ∈ x : φ is locally bounded at x with respect to ρ}. similarly, let y has a strictly increasing closed cover {bn}. we also define uφ = {x ∈ x : φ is locally bounded at x with respect to {bn})} for a mapping φ : x → f(y ). clearly, uφ is an open set in x. the insertions of functions are one of the most interesting problems in general topology and have been applied to characterize some classical cover properties. for example, j. mack characterized in [5] countably paracompact spaces with locally bounded real-valued functions as follows: theorem 1.1 (j. mack [5]). a space x is countably paracompact if and only if for each locally bounded function h : x → r there exists a locally bounded l.s.c. function g : x → r such that |h| ≤ g. c. good, r. knight and i. stares [3] and c. pan [6] introduced a monotone version of countably paracompact spaces, called monotonically countably paracompact spaces (mcp) and monotonically cp-spaces, respectively, and it was proved in [3, proposition 14] that both these notions are equivalent. also, c. good, r. knight and i. stares [3] characterized monotonically countably paracompact spaces by the insertions of semi-continuous functions. inspired by those results, k. yamazaki [10] characterized mcp spaces by expansions of locally bounded set-valued mappings as follows: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 146 k-semistratifiable spaces and expansions of set-valued mappings theorem 1.2 (k. yamazaki [10]). for a space x, the following statements are equivalent: (1) x is mcp; (2) for every space y having a strictly increasing closed cover {bn}, there exists a preserved order operator φ assigning to each locally bounded mapping ϕ : x → b(y ; {bn}), a locally bounded l.s.c. mapping φ(ϕ) : x → b(y ; {bn}) with ϕ ⊆ φ(ϕ); (3) for every metric space y , there exists a preserved order operator φ assigning to each locally bounded set-valued mapping ϕ : x → b(y ), a locally bounded l.s.c. set-valued mapping φ(ϕ) : x → b(y ) such that ϕ ⊆ φ(ϕ); (4) there exists a preserved order operator φ assigning to each locally bounded mapping ϕ : x → b(r), a locally bounded l.s.c. mapping φ(ϕ) : x → b(r) such that ϕ ⊆ φ(ϕ); (5) there exists a space y having a strictly increasing closed cover {bn}, there exists a preserved order operator φ assigning to each each locally bounded mapping ϕ : x → b(y ; {bn}), a locally bounded l.s.c. mapping φ(ϕ) : x → b(y ; {bn}) such that ϕ ⊆ φ(ϕ). recently, xie and yan [9] gave the following characterizations of stratifiable and semistratifiable spaces by expansions of set-valued mappings along same lines, and asked whether there are similar characterizations for k-mcm and k-semistratifiable spaces. theorem 1.3 (xie and yan [9]). for a space x, the following statements are equivalent: (1) x is stratifiable(resp. semi-stratifiable); (2) for every space y having a strictly increasing closed cover {bn}, there exists a preserved order operator φ assigning to each set-valued mapping ϕ : x → f(y ), an l.s.c. set-valued mapping φ(ϕ) : x → f(y ) such that φ(ϕ) is locally bounded(resp. bounded) at each x ∈ uϕ and that ϕ ⊆ φ(ϕ); (3) for every metric space y , there exists a preserved order operator φ assigning to each set-valued mapping ϕ : x → f(y ), an l.s.c. setvalued mapping φ(ϕ) : x → f(y ) such that φ(ϕ) is locally bounded (resp. bounded) at each x ∈ uϕ and that ϕ ⊆ φ(ϕ); (4) there exists a preserved order operator φ assigning to each set-valued mapping ϕ : x → f(r), an l.s.c. set-valued mapping φ(ϕ) : x → f(r) such that φ(ϕ) is locally bounded (resp. bounded) at each x ∈ uϕ and that ϕ ⊆ φ(ϕ); (5) there exist a space y having a strictly increasing closed cover {bn} and a preserved order operator φ assigning to each set-valued mapping ϕ : x → f(y ), an l.s.c. set-valued mapping φ(ϕ) : x → f(y ) such that φ(ϕ) is locally bounded (resp. bounded) at each x ∈ uϕ and that ϕ ⊆ φ(ϕ). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 147 p.-f. yan, x.-y. hu and l.-h. xie recently, xie and yan posed the following question: question 1.4 ([9, question 3.3]). are there monotone set-valued expansions for k-stratifiable spaces and k-mcm along the same lines? the purposes of this paper is to attempt to answer this question by the concept of k-u.s.c set-valued mappings. throughout this paper, all spaces are assumed to be regular, and all undefined topological concepts are taken in the sense given engelking [2]. 2. main results in this section we shall give characterization of k-mcm and k-semi stratifiable spaces. the following concept plays an important role in this paper. definition 2.1. for a space y with a strictly increasing closed cover {bn}, a mapping φ : x → b(y ; {bn}) is called k-upper semi-continuous (k-u.s.c.) if for every compact subset k of x, φ(k) is bounded. obviously, for every space y with a strictly increasing closed cover {bn} satisfying bn ⊂ int bn+1 and mapping φ : x → b(y ; {bn}): φ is u.s.c ⇒ φ is locally bounded ⇒ φ is k-u.s.c.. firstly, we shall give the characterization of k-mcm by expansion of setvalued mappings. peng and lin gave the kβ characterization as following. they renamed the kβ as k-mcm in [7]. proposition 2.2 ([7]). for a space x, the following statements are equivalent: (1) x is k-mcm; (2) there is an operator u assigning to a decreasing sequence of closed sets (fj)j∈n with ⋂ j∈n fj = ∅, a decreasing sequence of open sets (u(n, (fj)))n∈n such that (i) fn ⊆ u(n, (fj)) for each n ∈ n; (ii) for any compact subset k in x, there is n0 ∈ n such that u(n0, (fj)) ⋂ k = ∅; (iii) given two decreasing sequences of closed sets (fj)j∈n and (ej)j∈n such that fn ⊆ en for each n ∈ n and that ⋂ j∈n fj = ⋂ j∈n ej = ∅, then u(n, (fj)) ⊆ u(n, (ej)), for each n ∈ n. theorem 2.3. for a space x, the following statements are equivalent: (1) x is k-mcm; (2) for every space y having a strictly increasing closed cover {bn}, there exists a preserved order operator φ assigning to each locally bounded set-valued mapping ϕ : x → f(y ), an l.s.c. and k-u.s.c. set-valued mapping φ(ϕ) : x → f(y ) such that ϕ ⊆ φ(ϕ); (3) for every metric space y , there exists a preserved order operator φ assigning to each locally bounded set-valued mapping ϕ : x → f(y ), an l.s.c and k-u.s.c set-valued mapping φ(ϕ) : x → f(y ) such that ϕ ⊆ φ(ϕ); c© agt, upv, 2018 appl. gen. topol. 19, no. 1 148 k-semistratifiable spaces and expansions of set-valued mappings (4) there exists a preserved order operator φ assigning to each locally bounded set-valued mapping ϕ : x → f(r), an l.s.c. and k-u.s.c. set-valued mapping φ(ϕ) : x → f(r) such that ϕ ⊆ φ(ϕ); (5) there exists a space y having a strictly increasing closed cover {bn}, there exists a preserved order operator φ assigning to each locally bounded set-valued mapping ϕ : x → f(y ), an l.s.c. and k-u.s.c. set-valued mapping φ(ϕ) : x → f(y ) such that ϕ ⊆ φ(ϕ). proof. the implications of (2)⇒(3)⇒(4)⇒ (5) are trivial. (1)⇒ (2). assume that x is a k-mcm space. then there exists an operator u satisfying (i), (ii) and (iii) in proposition 2.2. let y be a space having a strictly increasing closed cover {bn}. for each locally bounded set-valued mapping ϕ : x → f(y ) and each n ∈ n, define fn,ϕ = {x ∈ x : ϕ(x) * bn}. then we have that ⋂ n∈n fn,ϕ = ∅. indeed, since ϕ is locally bounded, for each x ∈ x there exist an open neighborhood v of x and some i ∈ n such that ϕ(y) ⊆ bi for each y ∈ v , which implies that v ∩ fi,ϕ = ∅. it implies that x /∈ fi,ϕ and ⋂ n∈n fn,ϕ = ∅. define φ(ϕ) : x → f(y ) as follows: φ(ϕ)(x) = b1 whenever x ∈ x − u(1, (fn,ϕ)), φ(ϕ)(x) = bi+1 whenever x ∈ u(i, (fn,ϕ)) − u(i + 1, (fn,ϕ)). then, φ(ϕ) is lower semi-continuous. to see this, let w be an open subset of y and put k = min {i ∈ n : w ∩ bi 6= ∅}. then, one can easily check that (φ(ϕ))−1[w ] = u(k − 1, (fn,ϕ)) (we set u(0, (fn,ϕ)) = x). this implies that φ(ϕ) is lower semi-continuous. let k be a compact subset of x, then there exists k ∈ n such that k ⋂ u(k+ 1, (fn,ϕ)) = ∅. it implies that φ(ϕ)(k) ⊂ bk+1. hence φ(ϕ) is k-upper semicontinuous. to show that ϕ ⊆ φ(ϕ). for each x ∈ x, there exists some i ∈ n such that x ∈ u(i − 1, (fn,ϕ)) \ u(i, (fn,ϕ))(we set u(0, (fn,ϕ)) = x). since x /∈ u(i, (fn,ϕ)), we have x /∈ fi,ϕ. hence, ϕ(x) ⊆ bi = φ(ϕ)(x). this completes the proof of ϕ ⊆ φ(ϕ). finally, to show that φ is order-preserving, let ϕ, ϕ′ : x → f(y ) be setvalued mappings such that ϕ ⊆ ϕ′. then, fi,ϕ ⊆ fi,ϕ′ for each i ∈ n, and therefore, by (iii) of proposition 2.2, we have u(i, (fn,ϕ)) ⊆ u(i, (fn,ϕ′)) for each i ∈ n. for each x ∈ x. then, φ(ϕ′)(x) = bk′ for some k ′ ∈ n. this implies that x ∈ u(k′ − 1, (fn,ϕ′)) \ u(k ′, (fn,ϕ′)). similarly, φ(ϕ)(x) = bk for some k ∈ n and x ∈ u(k − 1, (fn,ϕ)) \ u(k, (fn,ϕ)). clearly, k ≤ k ′. hence, φ(ϕ)(x) = bk ⊆ bk′ = φ(ϕ ′)(x). this completes the proof of φ(ϕ) ⊆ φ(ϕ′) whenever ϕ ⊆ ϕ′. (5) ⇒ (1). let y be a space having a strictly increasing closed cover {bn} possessing the property in (5). let (fj)j∈n be a sequence of decreasing closed subsets of x with ⋂ j∈n fj = ∅. define a set-valued mapping ϕ(fj) : x → f(y ) as follows: ϕ(fj)(x) = b0 whenever x ∈ x − f1, ϕ(fj)(x) = bi+1 whenever x ∈ fi − fi+1. then, ϕ(fj) is locally bounded. by the assumptions, there exists a preserved operator φ assigning to each ϕ(fj ), an l.s.c. and ku.s.c set-valued mapping φ(ϕ(fj)) : x → f(y ) such that ϕ(fj) ⊆ φ(ϕ(fj )). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 149 p.-f. yan, x.-y. hu and l.-h. xie for every n ∈ n, define u(n, (fj)) = x − (φ(ϕ(fj ))) ♯[bn] it suffices to show the operator u satisfies (i), (ii) and (iii) of proposition 2.2 since ϕ(fj) ⊆ φ(ϕ(fj )), for each n ∈ n we have fn ⊆ x \ (ϕ(fj)) ♯[bn] ⊆ x \ (φ(ϕ(fj ))) ♯[bn] = u(n, (fj)). in addition, φ(ϕ(fj)) is lower semi-continuous, so u(n, (fj)) is an open set of x for each n ∈ n. this shows that the condition (i) is satisfied. for each x ∈ x, φ(ϕ(fj))(x) is bounded, so there exists some n0 ∈ n such that x ∈ (φ(ϕ(fj ))) ♯[bn0]. it implies that x /∈ u(n0, (fj)). hence,⋂ n∈n u(n, (fj)) = ∅. let k be a compact subset of x, then φ(ϕ(fj ))(k) is bounded. there exists some k0 ∈ n such that k ⊂ (φ(ϕ(fj))) ♯[bk0]. it implies that k ⋂ u(k0, (fj)) = ∅. finally, we show the operator satisfies (iii). let (fj)j∈n and (f ′ j)j∈n be sequences of decreasing closed subsets of x such that fj ⊆ f ′ j for each j ∈ n. then one can easily show that ϕ(fj) ⊆ ϕ(f ′j ), hence by the assumption, we have φ(ϕ(fj )) ⊆ φ(ϕ(f ′j )). therefore, u(n, (fj)) = x \ (φ(ϕ(fj))) ♯[bn] ⊆ x \ (φ(ϕ(f ′ j ))) ♯[bn] = u(n, (f ′ j)) holds for each n ∈ n. thus, x is a k-mcm space. � next, we consider the k-semi-stratifiable space. definition 2.4. a space x is said to be semi-stratifiable [1], if there is an operator u assigning to each closed set f , a sequence of open sets u(f) = (u(n, f))n∈n such that (1) f ⊆ u(n, f) for each n ∈ n; (2) if d ⊆ f , then u(n, d) ⊆ u(n, f) for each n ∈ n; (3) ⋂ n∈n u(n, f) = f . x is said to be k-semi-stratifiable [4], if, in addition, (3′) obtained from (3) by requiring (3) a further condition ‘if a compact set k such that k ⋂ f = ∅, there is some n0 ∈ n such that k ⋂ u(n0, f) = ∅’. the following result was proved in [8]. for the completeness, we give its proof. proposition 2.5. for any topological space x, the following statements are equivalent: (1) space x is k-semistratifiable; (2) there is an operator u assigning to a decreasing sequence of closed sets (fj)j∈n, a decreasing sequence of open sets (u(n, (fj)))n∈n such that (i) fn ⊆ u(n, (fj)) for each n ∈ n; c© agt, upv, 2018 appl. gen. topol. 19, no. 1 150 k-semistratifiable spaces and expansions of set-valued mappings (ii) for any compact subset k in x, if ⋂ n∈n fn ∩ k = ∅, there is n0 ∈ n such that u(n0, (fj)) ∩ k = ∅; (iii) given two decreasing sequences of closed sets (fj)j∈n and (ej)j∈n such that fn ⊆ en for each n ∈ n, then u(n, (fj)) ⊆ u(n, (ej)) for each n ∈ n. proof. (1) ⇒ (2) let u0 be an operator having the properties: (1), (2) and (3 ′ ) in definition 2.4. given any decreasing sequences of closed sets (fj)j∈n, we can define an operator u by u((fj)) = (u(n, (fj)))n∈n, where u(n, (fj)) = u0(n, fn) for each n ∈ n. we shall prove that the operator u has the properties (i)-(iii) in (2). because of u0 having properties (i) and (ii) in definition 2.4, one can easily verify that u has the properties (i) and (iii) in (2). we show that the property (ii) in (2) holds for u. take any decreasing sequences of closed sets (fn)n∈n and any compact subset k in x such that ⋂ n∈n fn ∩k = ∅. then, there exists n0 ∈ n such that fn0 ∩ k = ∅. since x is k-semi-stratifiable, there is i ∈ n such that u0(i, fn0)∩k = ∅. if i < n0, we have u(n0, (fn))∩k = u0(n0, fn0)∩k = ∅; if i ≥ n0, we also have u(i, (fn))∩k = u0(i, fi)∩k = ∅. hence the operator u holds for (ii). (2) ⇒ (1) let u0 be an operator having the properties (i)-(iii) in (2). given any closed set f in x by letting fn = f for each n ∈ n, we can define an operator u by u(j, f) = u0(j, (fn)) where (u0(j, (fn)))j∈ω = u0((fn)). one can easily verify that the operator u has the properties in definition 2.4. � theorem 2.6. for a space x, the following statements are equivalent: (1) x is k-semistratifiable; (2) for every space y having a strictly increasing closed cover {bn}, there exists a preserved order operator φ assigning to each set-valued mapping ϕ : x → f(y ), an l.s.c. set-valued mapping φ(ϕ) : x → f(y ) such that φ(ϕ)|uϕ is k-u.s.c. and ϕ ⊆ φ(ϕ) ; (3) for every metric space y , there exists a preserved order operator φ assigning to each set-valued set-valued mapping ϕ : x → f(y ), an l.s.c set-valued set-valued mapping φ(ϕ) : x → f(y ) such that φ(ϕ)|uϕ is k-u.s.c. and ϕ ⊆ φ(ϕ); (4) there exists an order-preserving operator φ assigning to each set-valued set-valued mapping ϕ : x → f(r), an l.s.c. set-valued mapping φ(ϕ) : x → f(r) such that φ(ϕ)|uϕ is k-u.s.c and ϕ ⊆ φ(ϕ); (5) there exists a space y having a strictly increasing closed cover {bn}, there exists a preserved order operator φ assigning to each set-valued set-valued mapping ϕ : x → f(y ), an l.s.c set-valued mapping φ(ϕ) : x → f(y ) such that φ(ϕ)|uϕ is k-u.s.c. and ϕ ⊆ φ(ϕ). c© agt, upv, 2018 appl. gen. topol. 19, no. 1 151 p.-f. yan, x.-y. hu and l.-h. xie proof. the implications of (2)⇒(3)⇒(4)⇒ (5) are trivial. (1) ⇒ (2). assume that x is a k-semistratifiable space. then there exists an operator u satisfying (i), (ii) and (iii) in proposition 2.5. let y be a space having a strictly increasing closed cover {bn}. for each set-valued mapping ϕ : x → f(y ) and each n ∈ n, define fn,ϕ = {x ∈ x : ϕ(x) /∈ bn}. then we have uϕ = x\ ⋂ n∈n fn,ϕ. indeed, for each x ∈ uϕ, then there exists an open neighborhood v of x and some i ∈ n such that ϕ(y) ⊆ bi for each y ∈ v , which implies that v ⋂ fi,ϕ = ∅. it implies that uϕ ⊆ x − ⋂ n∈n fn,ϕ. on the other hand, take any y ∈ x − ⋂ n∈n fn,ϕ. then there is fj,ϕ such that y /∈ fj,ϕ, and therefore, there exists an open neighborhood v of y such that v ∩ {x ∈ x : ϕ(x) * bj} = ∅. it implies that y ∈ v ⊆ uϕ. define φ(ϕ) : x → f(y ) as follows: φ(ϕ)(x) = b0 whenever x ∈ x − u(0, (fn,ϕ)), φ(ϕ)(x) = bi+1 whenever x ∈ u(i, (fn,ϕ))−u(i+1, (fn,ϕ)), φ(ϕ)(x) = y if x ∈ x − uϕ. then, φ(ϕ) is lower semi-continuous and ϕ ⊆ φ(ϕ). we only need to show that φ(ϕ)|uϕ is k-u.s.c. let k be a compact subset of uϕ. by proposition 2.5, there exists k ∈ n such that k ⋂ u(k + 1, (fn,ϕ)) = ∅. it implies that φ(ϕ)(k) ⊆ bk+1. (5) ⇒ (1). let y be a space having a strictly increasing closed cover {bn} possessing the property in (5). let (fj)j∈n be a sequence of decreasing closed subsets of x. define a set-valued mapping ϕ(fj ) : x → f(y ) as follows: ϕ(fj)(x) = b1 whenever x ∈ x − f1, ϕ(fj)(x) = bi+1 whenever x ∈ fi − fi+1, ϕ(fj)(x) = y if x ∈ x − ⋂ i∈n fi. by the assumptions, there exists a preserved operator φ assigning to each ϕ(fj), an l.s.c set-valued mapping φ(ϕ(fj)) : x → f(y ) such that φ(ϕ)|uϕ(fj ) is k-u.s.c. and ϕ(fj ) ⊆ φ(ϕ(fj )). for every n ∈ n, define u(n, (fj)) = x − (φ(ϕ(fj ))) ♯[bn]. it suffices to show the operator u satisfies (i), (ii) and (iii) of proposition 2.5. the proof that the operator u satisfies (i) and (iii) of proposition 2.5 is as same as theorem 2.3, so we only shows that the operator u satisfies (ii) of proposition 2.5. let k be a compact subset of x satisfying k∩( ⋂ n∈n fn) = ∅, then k ⊆ uϕ. there exists k ∈ n such that φ(ϕ(fj))(k) ⊆ bk. hence k ∩ u(k, (fj)) = ∅. thus, x is a k-semistratifiable space. � acknowledgements. we wish to thank the referee for the detailed list of corrections, suggestions to the paper, and all her/his efforts in order to improve the paper. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 152 k-semistratifiable spaces and expansions of set-valued mappings references [1] g. d. creede, semi-stratifiable, in: proc arizona state univ topological conf. (1967, 1969), 318–323. [2] r. engelking, general topology, revised and completed edition, heldermann verlag, 1989. [3] c. good, r. knight and i. stares, monotone countable paracompactness, topology appl. 101 (2000), 281–298. [4] d. l. lutzer, semistratifiable and stratifiable, general topology appl. 1 (1971), 43–48. [5] j. mack, on a class of countably paracompact spaces, proc. amer. math. soc. 16 (1965), 467–472. [6] c. pan, monotonically cp spaces, questions ans. gen. topol. 15 (1997) 25–32. [7] l. x. peng and s. lin, on monotone spaces and metrization theorems, acta. math. sinica 46 (2003), 1225–1232 (in chinese). [8] l.-h. xie, the insertions of semicontinuous functions and strafiable spaces, master’s thesis, jiangmen: wuyi university, 2010. [9] l.-h. xie and p.-f. yan, expansions of set-valued mappings on stratifiable spaces, houston j. math. 43 (2017), 611–624. [10] k. yamazaki, locally bounded set-valued mappings and monotone countable paracompactness, topology appl. 154 (2007), 2817–2825. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 153 () @ appl. gen. topol. 18, no. 2 (2017), 301-316doi:10.4995/agt.2017.7048 c© agt, upv, 2017 quasi-uniform convergence topologies on function spaces revisited wafa khalaf alqurashi a and liaqat ali khan b,∗ a department of mathematics, faculty of science (girls section), king abdulaziz university, p.o. box 80203, jeddah-21589, saudi arabia. (wafa-math@hotmail.com) b department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah-21589, saudi arabia. (lkhan@kau.edu.sa) communicated by h.-p. a. künzi abstract let x and y be topological spaces and f(x, y ) the set of all functions from x into y . we study various quasi-uniform convergence topologies ua (a ⊆ p(x)) on f(x, y ) and their comparison in the setting of y a quasi-uniform space. further, we study ua-closedness and right kcompleteness properties of certain subspaces of generalized continuous functions in f(x, y ) in the case of y a locally symmetric quasi-uniform space or a locally uniform space. 2010 msc: 54c35; 54e15; 54c08. keywords: quasi-uniform space; topology of quasi-uniform convergence on a family of sets; locally uniform spaces, right k-completeness; quasi-continuous functions; somewhat continuous functions. 1. introduction let x and y be two topological spaces, f(x, y ) the set of all functions from x into y and c(x, y ) the set of all continuous functions in f(x, y ). in the case of y = (y, u), a uniform space, various uniform convergence topologies (such as ux, uk, up) on f(x, y ) and c(x, y ) were systematically studied by kelley ([17], chapter 7). it is shown there that: (i) up ≤ uk ≤ ux; (ii) c(x, y ) ∗corresponding author received 24 december 2016 – accepted 18 february 2017 http://dx.doi.org/10.4995/agt.2017.7048 w. k. alqurashi and l. a. khan is ux-closed in f(x, y ); (iii) if y is complete, then f(x, y ) is ux-complete, hence c(x, y ) is also ux-complete. since every topological space is quasi-uniformizable ([8, 39]; [11], p. 27; [7], p. 34), we may assume that y = (y, u) with u a quasi-uniformity. main advantage of this assumption is that one can introduce various notions of cauchy nets and completeness. in this setting, some quasi-uniform convergence topologies ua (a ⊆ p(x)) on f(x, y ) were first discussed by naimpally [31]. in recent years, this topic has been further investigated by papadopoulos [37, 38], cao [6] and kunzi and romaguera [25, 26], among others. there is also a parallel notion of ”set-open topologies” sa (a ⊆ p(x)) on f(x, y ) which were introduced by fox [12] and further developed by arens [2], arens-dugundji [3], and more recently in the papers [4, 9, 23, 33, 34, 35, 36]. these sa topologies are, in general, different from their corresponding uniform convergence topologies ua (a ⊆ p(x)) even in the case of y a metric space, but the two notions coincide in some other particular cases. regarding completeness in quasi-uniform spaces, the formulation of the notion of ”cauchy net” or ”cauchy filter” in such spaces has been fairly difficult, and has been approached by several authors (see, e.g., [1, 8, 10, 41, 42, 43, 44, 45]). we shall find it convenient to restrict ourselves to the notions of a ”right k-cauchy net” and ”right k-complete space” on function spaces, as in [26]. in this paper, we consider various quasi-uniform convergence topologies on f(x, y ) and study their comparison and equivalences. further, we extend some results of above authors on closedness and completeness to more general classes of functions (not necessarily continuous). these include the subspaces of quasi-continuous, somewhat continuous and bounded functions [16, 22, 37, 40]. here, we shall need to assume that y is a locally symmetric quasi-uniform space or a locally uniform space (as appropriate), both notions being equivalent to y a regular topological space [31, 46]. we have included multiple references for certain concepts for the convenience of readers to access the literature. some open problems are also stated. 2. preliminaries definition 2.1 ([17], p. 175-176). let y be a non-empty set. for any u, v ⊆ y × y, we define u−1 = {(y, x) : (x, y) ∈ u} u ◦ v = {(x, y) ∈ y × y : ∃ z ∈ y such that (x, z) ∈ u and (z, y) ∈ v }. if u = v , we shall write u◦u = u2. if u = u−1, then u is called symmetric. the subset ∆(y ) = {(y, y) : y ∈ y } of y × y is called the diagonal on y . if △(y ) ⊆ u, then clearly u ⊆ u ◦ u = u2 ⊆ u3 ⊆ ..... for any x ∈ y , a ⊆ y and u ⊆ y × y , let u[x] = {y ∈ y : (x, y) ∈ u} and u[a] = ∪x∈au[x]. definition 2.2. a family u of subsets of y × y is called a quasi-uniformity on y [7, 11, 24, 29] if it satisfies the following conditions: (qu1) △(y ) ⊆ u for all u ∈ u. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 302 quasi-uniform convergence topologies on function spaces revisited (qu2) if u ∈ u and u ⊆ v , then v ∈ u. (qu3) if u, v ∈ u, then u ∩ v ∈ u. (qu4) if u ∈ u, there is some v ∈ u such that v 2 ⊆ u. in this case, the pair (y, u) is called a quasi-uniform space. if, in addition, u satisfies the symmetry condition: (u5) u ∈ u implies u −1 ∈ u, then u is called a uniformity on y and the pair (y, u) is called a uniform space. the pair (y, u) is called a semi-uniform space [46] if u satisfies (qu1)(qu3) and (u5). definition 2.3. (i) a quasi-uniform space (y, u) is called locally symmetric if, for each y ∈ y and each u ∈ u, there is a symmetric v ∈ u such that v 2[y] ⊆ u[y] [11]. (ii) a semi-uniform space (y, u) is called locally uniform [46] if, for each y ∈ y and each u ∈ u, there is a v ∈ u such that v 2[y] ⊆ u[y]. definition 2.4 ([11], p. 2-3; [46], p. 436). let (y, u) be a quasi-uniform space or a locally uniform space. then the collection t (u) = {g ⊆ y : for each y ∈ g, there is u ∈ u such that u[y] ⊆ g} is a topology, called the topology induced by u on y . equivalently, for each y ∈ y , the collection by = {u[y] : u ∈ u} forms a local base at y for the topology t (u). if (y, τ) is a topological space, then a quasi-uniformity u on y is said to be compatible with (y, τ) provided τ = t (u). it is well-known that a topological space (y, τ) is completely regular iff there exists a compatible uniformity u on y . csaszar [8] showed that every topological space has a compatible quasi-uniformity. in [39], pervin greatly simplified csaszar’s proof by giving a direct method of constructing a compatible quasi-uniformity for an arbitrary topological space. for more information, see ([29], p. 14-16; [7], p. 34). definition 2.5. a net {yα : α ∈ d} in a topological space (y, τ) is said to be τ-convergent to y ∈ y if, for each τ-open neighborhood g of y in y , there exists an α0 ∈ d such that yα ∈ g for all α ≥ α0 ([17], p. 65-66). in particular, a net {yα : α ∈ d} in a quasi-uniform or locally uniform space (y, u) is said to be t (u)-convergent to y ∈ y if, for each u ∈ u, there exists an α0 ∈ d such that yα ∈ u[y] for all α ≥ α0. definition 2.6 ([41, 26, 24, 7]). let (y, u) be a quasi-uniform space. a net {yα : α ∈ d} in y is called a right k-cauchy net provided that, for each u ∈ u, there exists some α0 ∈ d such that (yα, yβ) ∈ u for all α, β ∈ d with α ≥ β ≥ α0. (y, u) is called right k-complete if each right k-cauchy net is t (u)-convergent in y (cf. [26], lemma 1, p. 289). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 303 w. k. alqurashi and l. a. khan definition 2.7. let (y, u) be a quasi-uniform space or a locally uniform space, and let s ⊆ y . then: (i) s is called precompact [11, 29] if, given any u ∈ u, there exists a finite set f ⊆ y such that s ⊆ u[f ]. (ii) s is called totally bounded [11, 29] if, given any u ∈ u, there exists a finite cover {g1, g2, ...., gn} of s such that ∪ n i=1(gi × gi) ⊆ u. (iii) s is bounded [30] if given any u ∈ u, there exists an m ∈ n and a finite set f ⊆ y , such that s ⊆ um[f ] = ∪y∈f u m[y]. note. by ([29], p. 49; [30], p. 368), for any s ⊆ (y, u), a quasi-uniform space, s is totally bounded ⇒ s is precompact ⇒ s is bounded, but the converses need not be true ([11], p. 152; [29], p. 49). in fact, by [27], even a compact quasi-uniform space is not necessarily totally bounded. however, if (y, u) is a uniform spaces, s is precompact iff s is totally bounded ([11], p. 52; [29], p. 49). if (x, τ) is a topological space and a ⊆ x, the closure of a is denoted by a τ or τ-cl(a) (or simply a or cl(a)); the interior of a is denoted by τ −int(a) (or simply int(a)). we shall denote the power set of x by p(x). 3. quasi-uniform convergence topologies on f(x, y ) let x be a topological space and (y, u) a quasi-uniform space, and let a = a(x) be a certain collection of subsets of x which covers x. for any a ∈ a(x) and u ∈ u, let ma,u = {(f, g) ∈ f(x, y ) × f(x, y ) : (f(x), g(x)) ∈ u for all x ∈ a}. then the collection {ma,u : a ∈ a(x) and u ∈ u} forms a subbase for a quasi-uniformity, called the quasi-uniformity of quasi-uniform convergence on the sets in a(x) induced by u. the resultant topology on f(x, y ) is called the topology of quasi-uniform convergence on the sets in a(x) and is denoted by ua [25, 26]. (i) if a = {x}, ua is called the quasi-uniform convergence topology on f(x, y ) and is denoted by ux. (ii) if a = k(x)={a ⊆ x : a is compact}, ua is called the quasiuniform compact convergence topology on f(x, y ) and is denoted by uk (iii) if a = σk(x)={a ⊆ x : a is σ-compact}, ua is called the quasiuniform σ-compact convergence topology on f(x, y ) and is denoted by uσ. (iv) if a = σ0(x)={a ⊆ x : a is countable}, ua is called the quasiuniform countable convergence topology on f(x, y ) and is denoted by uσ0. (v) if a = k0(x)={a ⊆ x : a is finite}, ua is called the quasi-uniform pointwise convergence topology on f(x, y ) and is denoted by up. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 304 quasi-uniform convergence topologies on function spaces revisited since each of the collection a in (i)-(v) is closed under finite unions, the collection {ma,u : a ∈ a(x) and u ∈ u} actually forms a base for the topology ua (cf. [28], p. 7). lemma 3.1. let x be a topological space and (y, u) a quasi-uniform space, and let a, b ⊆ x and u, v ∈ u be such that ma,u ⊆ mb,v . (i) if b 6= ∅, then u ⊆ v . (ii) if v 6= y × y, then b ⊆ a. proof. (i) suppose b 6= ∅, but u * v , and let (a, b) ∈ u with (a, b) /∈ v . consider the constant functions f, g : x → y defined by f(x) = a (x ∈ x), g(x) = b (x ∈ x). then (f, g) ∈ ma,u, but (f, g) /∈ mb,v . indeed, if a = ∅, then (f(a), g(a)) ∈ ∅ ⊆ u; if a 6= ∅ and x ∈ a, then (f(x), g(x)) = (a, b) ∈ u. hence (f, g) ∈ ma,u. on the other hand, since b 6= ∅, for any x ∈ b, (f(x), g(x)) = (a, b) /∈ v , (f, g) /∈ mb,v . this contradicts ma,u ⊆ mb,v . (ii) suppose b * a, and let x0 ∈ b\a. since v 6= y × y , choose c, d ∈ y such that (c, d) /∈ v . fix (p, q) ∈ u. define f, g : x → y by f(x) = p if x ∈ a, f(x) = c if x ∈ x\a; g(x) = q if x ∈ a, g(x) = d if x ∈ x\a. then (f, g) ∈ ma,u, but (f, g) /∈ mb,v . indeed, if a = ∅, then (f(a), g(a)) ∈ ∅ ⊆ u; if a 6= ∅ and x ∈ a, then (f(x), g(x)) = (p, q) ∈ u. hence (f, g) ∈ ma,u. on the other hand, since x0 ∈ b and (f(x0), g(x0)) = (c, d) /∈ v , (f, g) /∈ mb,v . this contradicts ma,u ⊆ mb,v . therefore b ⊆ a. � theorem 3.2. let x be a hausdorff topological space and (y, u) a quasiuniform space. let up, uσ0, uσ, uk and ux be the topologies on f(x, y ) as defined above. then (a) up ≤ uk ≤ uσ ≤ ux and up ≤ uσ0 ≤ uσ. (b) uk = ux iff x is compact. (c) up = uk iff every compact subset of x is finite. in particular, if x is discrete, then up = uk. (d) uσ = ux iff x = a for some σ-compact subset a of x. (e) uk = uσ iff every σ-compact subset of x is relatively compact. (f) uσ0 = ux iff x is separable. (g) uσ0 ≤ uk iff every countable subset of x is relatively compact. (h) uσ0, uσ and ux have the same bounded sets in f(x, y ). proof. (a) clearly, k0(x) ⊆ k(x) ⊆ σk(x), and so up ≤ uk ≤ uσ ≤ ux on f(x, y ). further, k0(x) ⊆ σ0(x) ⊆ σk(x), and so up ≤ uσ0 ≤ uσ ≤ ux on f(x, y ). (b) suppose ux ≤ uk, and let u ∈ u, with u 6= y × y . then there exist a compact subset k of x and a v ∈ u such that mk,v ⊆ mx,u. by lemma 3.1(ii), x ⊆ k = k (since x is hausdorff). thus x is compact. conversely, suppose x is compact. to show ux ≤ uk, take c© agt, upv, 2017 appl. gen. topol. 18, no. 2 305 w. k. alqurashi and l. a. khan arbitrary mx,v ∈ ux. taking k = x, which is compact, mk,v ∈ uk and mx,v ⊆ mk,v . hence mx,v ∈ uk, and so ux ≤ uk. (c) suppose uk ≤ up, and let k ⊆ x be a compact set and u ∈ u, with u 6= y × y . then there exist a finite subset a of x and a v ∈ u such that ma,v ⊆ mk,u . by lemma 3.1(ii), k ⊆ a = a; hence k is finite. conversely, suppose that every compact subset of x is finite. to show uk ≤ up, take arbitrary mk,u ∈ uk with k ⊆ x a compact set. then k is finite. taking a = k, ma,v ∈ up and ma,u ⊆ mk,u. hence mk,u ∈ up, and so uk ≤ up . in particular, if x is discrete, then every compact subset of x is finite and hence up = uk. (d) suppose that ux ≤ uσ, and let u ∈ u, with u 6= y × y . then there exist a σ-compact set a ⊆ x and a v ∈ u such that ma,v ⊆ mx,u. by lemma 3.1(ii), x = a, as required. conversely, suppose x = a for some σ-compact subset a of x. to show ux ≤ uσ, take arbitrary mx,u ∈ ux. clearly, ma,u ∈ uσ and mx,u ⊆ ma,u ⊆ ma,u. hence mx,u ∈ uσ, and so ux ≤ uσ. (e) suppose that uσ ≤ uk and let a be any σ-compact subset of x. if u ∈ u, with u 6= y × y , then there exist a compact set b ⊆ x and a v ∈ u such that mb,v ⊆ ma,u. by lemma 3.1(ii), a ⊆ b, which implies that a is also compact. conversely, suppose that every σcompact subset of x is relatively compact. take arbitrary ma,u ∈ uσ with a a σ-compact subset of x. since a is compact, m a,u ∈ uk and clearly, m a,u ⊆ ma,u. hence ma,u ∈ uk, and so uσ ≤ uk . (f) suppose ux ≤ uσ0. then, for any u ∈ u, with u 6= y ×y , mx,u ∈ ux and hence mx,u ∈ uσ0 . so there exist a countable set a ⊆ x and a v ∈ u such that ma,v ⊆ mx,u. by lemma 3.1(ii), x = a and so x is separable. conversely, suppose x is separable, and let a ⊆ x be countable set such that a = x. take arbitrary mx,u ∈ ux. clearly, ma,u ∈ uσ0 and mx,u ⊆ ma,u ⊆ ma,u. hence mx,u ∈ uσ0, and so ux ≤ uσ0. (g) suppose that uσ0 ≤ uk and let a be any countable subset of x. if u ∈ u, with u 6= y × y , then there exist a compact set b ⊆ x and a v ∈ u such that mb,v ⊆ ma,u. by lemma 3.1(ii), a ⊆ b, which implies that a is also compact. conversely, suppose that every countable subset of x is relatively compact. to show uσ0 ≤ uk, take arbitrary ma,u ∈ uσ0 with a a countable subset of x. since a is compact, m a,u ∈ uk and clearly, ma,u ⊆ ma,u. hence ma,u ∈ uk, and so uσ0 ≤ uk. (h) since uσ0 ≤ ux, every ux-bounded subset of f(x, y ) is easily seen to be uσ0-bounded. in fact, let s ⊆ f(x, y ) be ux-bounded set. then for arbitrary ma,u ∈ uσ0 with a a countable subset of x and u ∈ u, there exists an m ∈ n and a finite set j ⊆ f(x, y ) such c© agt, upv, 2017 appl. gen. topol. 18, no. 2 306 quasi-uniform convergence topologies on function spaces revisited that s ⊆ (mx,u) m[j]. then s ⊆ (ma,u) m[j], showing that s is uσ0bounded. on the other hand, suppose that there exists a uσ0-bounded set t ⊆ f(x, y ) which is not ux-bounded. then there exist a v ∈ u such that t " (mx,v )n[k] = mx,v n[k] for all n ∈ n and all finite sets k ⊆ f(x, y ). choose sequences {hn} ⊆ t, {xn} ⊆ x such that (f(xn), hn(xn)) /∈ v n for all n ∈ n and all f ∈ f(x, y ). let a = {xn}. then ma,v ∈ uσ0, and hn /∈ (ma,v ) n[f] for any n ∈ n and f ∈ f(x, y ); hence t " (ma,v )n[k] for any n ∈ n and finite set k ⊆ f(x, y ). therefore t is not uσ0-bounded, a contradiction. � now, let x be a completely regular hausdorff space and y = (e, τ) a hausdorff topological vector space (tvs, in short) over k(= r or c) with a base we(0) of balanced τ−neighborhoods of 0 in e ([18], theorem 5.1), and let cb(x, e) denote the vector space of all continuous bounded functions from x into e. in this setting, the collection v = {vg : g ∈ we(0)} is a uniformity on e, where vh = {(x, y) ∈ e × e : x − y ∈ h}. for any a ∈ a(x) and h ∈ we(0), let m∗a,vh = {(f, g) ∈ cb(x, y ) × cb(x, y ) : (f(x), g(x)) ∈ vh for all x ∈ a}. then the collection {m∗a,vh (0) : a ∈ a(x) and h ∈ we(0)} forms a base of neighbourhood of 0 in cb(x, e) for a linear topology, denoted by ta. indeed, this follows from ([18], corollary 8.2) and the fact that m∗a,vh (0) = {g ∈ cb(x, y ) : (0, g) ∈ m ∗ a,vh } = {g ∈ cb(x, y ) : g(x) ∈ h for all x ∈ a} : = ncb(a, h). the quasi-uniform topologies up, uσ0, uσ, uk and ux on cb(x, y ) become the linear topologies, denoted by tp, tσ0, tσ, tk and tu in the terminology of [21]. consequently, we can deduce the following from above two results: corollary 3.3 ([21], lemma 3.2). let x be a completely regular hausdorff space and (e, τ) a hausdorff tvs. suppose that a, b ⊆ x and that g, h ∈ we(0) are such that ncb(a, g) ⊆ ncb(b, h). then: (i) if b 6= ∅, then g ⊆ h. (ii) if w 6= e, then b ⊆ a. � corollary 3.4 ([14, 15]; [21], theorem 3.3). let x be a completely regular hausdorff space and (e, τ) a hausdorff tvs. then: (i) tσ = tu iff x = a for some σ-compact subset a of x. (ii) tk = tσ iff every σ-compact subset of x is relatively compact. (iii) tσ0 = tx iff x is separable. (iv) tσ0 ≤ tk iff every countable subset of x is relatively compact. (v) tσ0, tσ and tu have the same bounded sets in cb(x, y ). � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 307 w. k. alqurashi and l. a. khan finally, in this section, we give a brief account of set-open topologies sa (a ⊆ p(x)) on f(x, y ) and their comparison with the corresponding quasiuniform convergence topologies ua (a ⊆ p(x)) let x and y be topological spaces and let a ⊆ p(x). for any a ∈ a and any open set h ⊆ y , let n(a, h) = {f ∈ f(x, y ) : f(a) ⊆ h}. then the collection {n(a, h) : a ∈ a, open sets h ⊆ y } form a subbase for a topology on f(x, y ), called the set-open (or a-open ) topology generated by a and denoted by sa. in particular, if a = {x} (resp. k(x), σk(x), σ0(x), f(x)), then sa is called the uniform (resp. compact-open, σ-compact-open, countable-open, point-open) topology and denoted by su (resp. sk, sσ, sσ0, sp). the relation between the set-open topology and the topology of uniform convergence on a family a ⊆ k(x) was investigated by kelley ([17], p. 230) and mccoy and ntantu ([28], p. 9) in the case of y a uniform space (see also [23]). these sa topologies are, in general, different from their corresponding uniform convergence topologies ua even in the case of y a metric space. more recently, there has been a renewed interest on the problem for coincidence of these two notions and some interesting partial answers have been obtained in [33, 34, 4, 35, 36]. 4. closedness and completeness in function spaces the results of this section are motivated by those given in [17, 31, 26] regarding the closedness and completeness of c(x, y ) and ca(x, y ) in (f(x, y ), ux). it is well-known (e.g., [17, 32]) that c(x, y ) is ux-closed in f(x, y ) but not necessarily up-closed. later, some authors obtained variants of these results for spaces of quasi-continuous, somewhat continuous and bounded functions in the case of y a uniform space [16, 22, 37, 40]. in this section, we examine their ua-closedness and right k-completeness in the setting of y a locally symmetric quasi-uniform or locally uniform spaces. let x be a topological space and (y, u) a quasi-uniform space. let {fα : α ∈ d} be a net in f(x, y ) and a ⊆ x. we recall from [25, 26] that: (i) {fα} is said to be right k-cauchy in (f(x, y ), ua) if, for any u ∈ u, there exists an index α0 ∈ d such that (fα, fβ) ∈ ma,u for all α ≥ β ≥ α0. (ii) {fα} is said to be ua-convergent to f ∈ f(x, y ) if, for any u ∈ u, there exists an index α0 ∈ d such that (f, fα) ∈ ma,u for all α ≥ α0. in this case, we shall write fα ua −→ f. (iii) (f(x, y ), ua) is called right k-complete if each right k-cauchy net in (f(x, y ), ua) is ua-convergent to some function in f(x, y ). lemma 4.1. let x be a topological space and (y, u) a quasi-uniform space, and let a ⊆ x. let {fα : α ∈ d} be a net in f(x, y ) such that (a) {fα : α ∈ d} is a right k-cauchy net in (f(x, y ), ua) (b) fα(x) → f(x) for each x ∈ a (i.e. fα up −→ f on a). then fα ua −→ f. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 308 quasi-uniform convergence topologies on function spaces revisited proof. let u ∈ u. choose v ∈ u such that v 2 ⊆ u. since {fα} is right-kcauchy, there exists α0 ∈ d such that (fα, fβ) ∈ ma,u for all α ≥ β ≥ α0. we claim that (f, fγ) ∈ ma,u for all γ ≥ α0. fix any γ ≥ α0 and x0 ∈ a. since {fα(x0) : α ∈ d} −→ f(x0), also its subnet {fα(x0) : α ≥ γ}−→ f(x0). so there is α(x0) ∈ d with α(x0) ≥ γ such that (f(x0), fα(x0)(x0)) ∈ v . since α(x0) ≥ γ ≥ α0, (f(x0), fα(x0)(x0)) ∈ v , (fα(x0)(x0), fγ(x0)) ∈ v ; hence (f(x0), fγ(x0)) ∈ v ◦ v ⊆ u. � the following result is due to kunzi and romaguera ([26], proposition 1). using lemma 4.1, we can present a somewhat shorter proof of this result for reader’s benefit. theorem 4.2. let x be a topological space and y = (y, u) a right k-complete quasi-uniform space, and let a ⊆ p(x) which covers x. then (f(x, y ), ua ) is right k-complete. proof. let {fα : α ∈ d} be a right k-cauchy net in (f(x, e), ua), and let u ∈ u and x ∈ x be fixed. since a covers x, x ∈ ax for ax ∈ a. there exists α0 ∈ d such that for each α ≥ β ≥ α0, (fα, fβ) ∈ max,u for all α ≥ β ≥ α0. in particular, (fα(x), fβ(x)) ∈ u for all α ≥ β ≥ α0, and so {fα(x) : α ∈ d} is a right k-cauchy net in y. since (y, u) a right k-complete, {fα(x) : α ∈ d} is t (u)-convergent to a point f(x) ∈ y. hence we have an f ∈ f(x, y ) such that fα up −→ f. consequently, by lemma 4.1, fα ua −→ f. thus (f(x, y ), ua ) is complete. � we next obtain variants of some results in [26] for spaces of quasi-continuous, somewhat continuous and bounded functions in the setting of locally symmetric quasi-uniform and locally uniform spaces. a subset a of topological space x is called semi-open (or quasi-open) if there exists an open set g such that g ⊆ a ⊆ cl(g); equivalently, a is semi-open iff a ⊂ cl(int(a)). a function f : x → y is said to be quasicontinuous [22, 40] if f−1(h) is semi-open in x for each open set h in y , or equivalently, if, for each point x ∈ x and for each open set h ⊆ y containing f(x), there exists a semi-open set a ⊆ x such that x ∈ a and f(a) ⊆ h. let q(x, y ) denote the set of all quasi-continuous functions from x into y . theorem 4.3. let x be a topological space and (y, u) a locally symmetric quasi-uniform space, and let a ⊆ x. let {fα : α ∈ d} be a net in q(x, y ) which is ua-convergent to f. then f ∈ q(x, y ). proof. let x0 ∈ x and suppose h is any t (u)-open set containing f(x0) in y . we need to show that there exists a semi-open set g containing x0 in x such that f(g ∩ a) ⊆ h. by definition of t (u), there exists a u ∈ u such that u[f(x0)] ⊆ h. by local symmetry, choose a symmetric v ∈ u such that c© agt, upv, 2017 appl. gen. topol. 18, no. 2 309 w. k. alqurashi and l. a. khan v 2[f(x0)] ⊆ u[f(x0)]. next choose a closed w ∈ u such that w 2 ⊆ v since fα ua −→ f, there exists α0 ∈ d such that (f, fα) ∈ ma,w for all α ≥ α0. in particular, we can write (f(z), fα0(z)) ∈ w ⊆ w 2 ⊆ v for all z ∈ x. since fα0 is quasi-continuous at x0, there exists a semi-open set g containing x0 in x such that fα0(z) ⊆ v [f(x0)] for all z ∈ g ∩ a. finally, let z ∈ g ∩ a. then, using symmetry of v , we obtain f(z) ∈ v −1[fα0(z)] = v [fα0(z)] ⊆ v [v [f(x0)]] ⊆ u[f(x0)] ⊆ h. therefore, f(g ∩ a) ⊆ h; hence f ∈ q(x, y ). � theorem 4.4. let x be a topological space and (y, u) a locally symmetric quasi-uniform space. then: (a) q(x, y ) is ux-closed in f(x, y ). (b) if y is right k-complete, then (q(x, y ), ux) is right k-complete. proof. (a) this follows from theorem 4.3. (b) suppose y is right k-complete. let {fα : α ∈ d} be a right k-cauchy net in (q(x, y ), ux). let u ∈ u and let x0 ∈ x be fixed. since {fα : α ∈ d} be a right k-cauchy net, there exists α0 ∈ d such that (fα, fβ) ∈ mx,u for all α ≥ β ≥ α0. in particular (fα(x0), fβ(x0)) ∈ u for all α ≥ β ≥ α0. and so {fα(x0) : α ∈ d} is a right k-cauchy net in y . since y is right k-complete, {fα(x0)} is t (u)-convergent to a point f(x0) ∈ y . hence we have a function f ∈ f(x, y ) such that fα up −→ f. consequently, by lemma 4.1, fα ux −→ f, and, by part (a), f ∈ q(x, y ). thus (q(x, y ), ux) is kcomplete. � corollary 4.5 ([22], theorem 3.1; [40], theorem 2.2). let x be a topological space and (y, u) a uniform space. then: (a) q(x, y ) is ux-closed in f(x, y ). (b) if y is complete, then (q(x, y ), ux) is complete. a function f : x → y is said to be somewhat continuous [13] if for each open set v in y such that f−1(v ) 6= ∅, there exists a nonempty open set u in x such that u ⊂ f−1(v ); or equivalently, if, for any m ⊆ x, m is dense in x implies f(m) is dense in f(x) ([13], p. 6) . let sw(x, y ) denote the set of all somewhat continuous functions from x into y . theorem 4.6. let x be a topological space and (y, u) a locally symmetric quasi-uniform space. let {fα : α ∈ d} be a net in sw(x, y ) which is uxconvergent to f. then f ∈ sw(x, y ). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 310 quasi-uniform convergence topologies on function spaces revisited proof. let m be a dense subset of x. we need to show that f(m) is dense in f(x). let y0 ∈ f(x), and let h be an open neighborhood of y0 in f(x). (we need to show that h ∩ f(m) 6= ∅.) there exists a g ∈ t (u) such that h = g ∩ f(x). choose x0 ∈ x such that f(x0) = y0. since g ∈ t (u), there exists u ∈ u such that u[y0] ⊆ g. by local symmetry, choose a symmetric v ∈ u such that v 2[y0] ⊆ u[y0]. since (y, u) is a quasi-uniform space, choose a closed w ∈ u such that w 2 ⊆ v . since fα ux −→ f, there exists α0 ∈ d such that fα ∈ mx,w [f] for all γ ≥ α0. in particular, (f(z), fα0(z)) ∈ w ⊆ w 2 ⊆ v for all z ∈ x. since fα0 is somewhat continuous, fα0(m) is dense in fα0(x). since w [fα0(x0)] is a neighbohood of fα0(x0) in the t (u)-topology, there exists some m ∈ m such that fα0(m) ∈ w [fα0(x0)]. then (f(x0), fα0(x0)) ∈ w and (fα0(x0), fα0(m)) ∈ w ; hence (f(x0), fα0(m)) ∈ w ◦ w ⊆ v . since m ∈ m ⊆ x, (f(m), fα0(m)) ∈ v . hence (f(x0), f(m)) ∈ v ◦ v −1 = v 2. consequently, f(m) ∈ v 2[f(x0)] ⊆ u[f(x0)] = u[y0] ⊆ g ⊆ h. thus f(m) ∈ h, and so h ∩f(m) 6= ∅. hence f(m) is dense in f(x), showing that f ∈ sw(x, y ). � theorem 4.7. let x be a topological space and (y, u) a locally symmetric quasi-uniform space. then: (a) sw(x, y ) is ux-closed in f(x, y ). (b) if y is right k-complete, then (sw(x, y ), ux) is right k-complete. proof. (a) this follows from theorem 4.6. (b) suppose y is right k-complete. let {fα : α ∈ d} be a right k-cauchy net in (sw(x, y ), ux) . let u ∈ u and let x0 ∈ x be fixed. there exists α0 ∈ d such that (fα, fβ) ∈ mx,u for all α ≥ β ≥ α0. in particular (fα(x0), fβ(x0)) ∈ u for all α ≥ β ≥ α0, and so {fα(x0) : α ∈ d} is a right k-cauchy net in y . since y is right kcomplete, {fα(x0)} is t (u)-convergent to a point f(x0) ∈ y . hence we have a function f ∈ f(x, y ) such that fα up −→ f. consequently, by lemma 4.1, fα ux −→ f, and, by part (a), f ∈ sw(x, y ). thus (sw(x, y ), ux) is right k-complete. � corollary 4.8 ([16], theorem 1, p. 32). let x be a topological space and (y, u) a uniform space. then: (a) sw(x, y ) is ux-closed in f(x, y ). (b) if y is complete, then (sw(x, y ), ux) is complete. we next present analogues of some results from [37] for functions having range as precompact or bounded sets. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 311 w. k. alqurashi and l. a. khan theorem 4.9. let x be a non-empty set and (y, u) a locally uniform space. let a ⊆ p(x) and ba(x, y ) ⊆ f(x, y ) the set of all functions which are bounded on each member of a. let {fα : α ∈ d} be a net in ba(x, y ) which is ua-convergent to f. then f ∈ ba(x, y ). (hence ba(x, y ) is ua-closed in f(x, y ).) proof. let u ∈ u be symmetric and a ∈ a(x). choose an α0 ∈ d such that (f, fα0) ∈ ma,u for all α ≥ α0. in particular, this implies that for each x ∈ a, f(x) ∈ u−1[fα0(x))] = u[fα0(x))] ⊆ u[fα0(a))]; hence f(a) ⊆ u[fα0(a)]. but fα0 ∈ ba(x, y ), so there exists an integer m ≥ 1 and a finite set f ⊆ y , such that fα0(a) ⊆ u m[f ]. thus f(a) ⊆ u[fα0(a)] ⊆ u[u m[f ] = um+1[f ], which means that f ∈ b(x, y ). � problem 4.10. the authors do not know whether or not the above result can be established for (y, u) a locally symmetric quasi-uniform space. theorem 4.11. let x be a non-empty set and (y, u) a uniform space, and let a ⊆ p(x) and pca(x, y ) ⊆ f(x, y ) be the set of all functions which have precompact range on each member of a. let {fα : α ∈ d} be a net in pca(x, y ) which is ua-convergent to f. then f ∈ pca(x, y ). (hence pca(x, y ) is ua-closed in f(x, y ) and is ua-complete if (y, u) is complete.) proof. let u ∈ u and a ∈ a(x). choose symmetric w ∈ u with w ◦w ⊆ u. there exists a α0 ∈ d such that for each α ≥ α0, fα ∈ ma,w [f]. in particular, (f, fα0) ∈ ma,w , which implies that f(a) ⊆ w −1[fα0(a)] = w [fα0(a)]. but fα0 ∈ pca(x, y ), so there exists a finite set f ⊆ y , such that fα0(a) ⊆ w [f ]. thus f(a) ⊆ (w ◦ w)[f ] ⊆ u[f ], which means that f ∈ pca(x, y ). � problem 4.12. the authors do not know whether or not the above result can be established for (y, u) a locally symmetric quasi-uniform space or a locally uniform space. next we consider the notion of functions f ∈ f(x, y ) which are ”small off compact set”. first, let y = e, a tvs over k (=r or c) with we(0) a base of balanced neighborhoods of 0 in e. recall that: a function f : x → e is called small off compact set (or vanish at infinity) [5, 18, 20] if, for each g ∈ we(0), there exists a compact set k ⊆ x such that f(x) ∈ g for all x ∈ x\k. note that if f ∈ f(x, e) is small off compact set, then given any g ∈ we(0), there exists a compact set k ⊆ x such that f(x) − f(y) ∈ g for all x, y ∈ x\k. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 312 quasi-uniform convergence topologies on function spaces revisited in fact, choose a balanced (or symmetric) h ∈ we(0) such that h + h ⊆ g. since f ∈ f0(x, e), there exists a compact set k ⊆ x such that f(x) ∈ h for all x ∈ x\k. then, for any x, y ∈ x\k, f(x) − f(y) ∈ h − h = h + h ⊆ g. motivated by this observation, we can introduce the notion of ”small off compact set” in the setting of quasi-uniform spaces, as follows. let x be a topological space and (y, u) a quasi-uniform space. (i) a function f : x → y is said to be small off compact set if, for each u ∈ u, there exists a compact set k ⊆ x such that (f(x), f(y)) ∈ u for all x, y ∈ x\k. (ii) f : x → y is said to have compact support if there exists a compact set k ⊆ x such that (f(x), f(y)) ∈ u for all x, y ∈ x\k and all u ∈ u. note that a quasi-uniform space (y, u) is t1 iff ∩{u : u ∈ u} = ∆(y ) holds ([11], p. 6; [29], p. 36). hence, in this case, the condition in (ii) is equivalent to: f(x) = f(y) for all x, y ∈ x\k. let f0(x, y ) (resp. f00(x, y )) denote the subset of f(x, y ) consisting of those functions which are small of compact set (resp. have compact support), and let c0(x, y ) = f0(x, y )∩c(x, y ) and c00(x, y ) = f00(x, y )∩c(x, y ). clearly, f00(x, y ) ⊆ f0(x, y ) and c00(x, y ) ⊆ c0(x, y ). lemma 4.13. c0(x, y ) ⊆ b(x, y ). proof. let f ∈ c0(x, y ), and let u ∈ u. there exists a compact set k ⊆ x such that (f(x), f(y)) ∈ u for all x, y ∈ x\k. in particular, for a fixed x0 ∈ x\k, f(y) ∈ u[f(x0)] for all y ∈ x\k. since f(k) is compact and hence bounded in y ([30], p. 368), there exists an integer m and a finite set f ⊆ y , such that f(k) ⊆ um[f ]. taking f1 = f ∪{f(x0)}, we have f(x) ⊆ u m[f1], which means that f ∈ b(x, y ). � theorem 4.14. let x be a topological space and (y, u) a uniform space. then both f0(x, y ) and c0(x, y ) are ux-closed in f(x, y ). proof. let f ∈ f(x, y ) with f ∈ ux − cl(f0(x, y )). let u ∈ u. choose a symmetric v ∈ u such that v 3 ⊆ u. there exists g ∈ f0(x, y ) such that (f(x), g(x)) ∈ v for all x ∈ x. there exists a compact set k ⊆ x such that (g(x), g(y)) ∈ v for all x, y ∈ x\k. then, for any x, y ∈ x\k, (f(x), g(x)) ∈ v, (g(x), g(y)) ∈ v, (g(y), f(y)) ∈ v −1. hence (f(x), f(y) ∈ v ◦ v ◦ v −1 = v ◦ v ◦ v ⊆ u for all x, y ∈ x\k. therefore f ∈ f0(x, y ), and so f0(x, y ) is ux-closed in f(x, y ). by ([17], theorem 7.9), c(x, y ) is ux-closed in f(x, y ). thus c0(x, y ) is also uxclosed in f(x, y ). � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 313 w. k. alqurashi and l. a. khan remark 4.15. if x is not locally compact, then c0(x, y ) may consist of only constant functions. for example, if x = q (rationals), then c0(x, r) = {0} ([20], p. 12). problem 4.16. if x is locally compact and y = e, a topological vector space, then it is well-known that c00(x, e) is uk-dense in c(x, e) and also uxdense in c0(x, e) ([5], p. 96-98; [18], p. 81; [19], theorem 3.2; [20], theorem 1.1.10). we leave an open problem that whether or not these denseness results hold for y a uniform space. acknowledgements. the authors wish to thank professors h. p. a. künzi, r. a. mccoy and a. v. osipov for communicating to us useful information of some concepts used in this paper, and the referee for suggesting a number of improvements to the original version of the paper. references [1] a. andrikopoulos, completeness in quasi-uniform spaces, acta math. hungar. 105 (2004), 151–173. 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[46] j. williams, locally uniform spaces, trans. amer. math. soc. 168 (1972), 435–469. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 316 @ appl. gen. topol. 20, no. 1 (2019), 19-31doi:10.4995/agt.2019.8843 c© agt, upv, 2019 a note on measure and expansiveness on uniform spaces pramod das and tarun das department of mathematics, faculty of mathematical sciences, university of delhi, delhi-110007, india (pramod.math.ju@gmail.com,tarukd@gmail.com) communicated by f. balibrea abstract we prove that the set of points doubly asymptotic to a point has measure zero with respect to any expansive outer regular measure for a bi-measurable map on a separable uniform space. consequently, we give a class of measures which cannot be expansive for denjoy homeomorphisms on s1. we then investigate the existence of expansive measures for maps with various dynamical notions. we further show that measure expansive (strong measure expansive) homeomorphisms with shadowing have periodic (strong periodic) shadowing. we relate local weak specification and periodic shadowing for strong measure expansive systems. 2010 msc: 54h20; 54e15. keywords: expansiveness; measure expansiveness; expansive measures; equicontinuity; shadowing; specification. 1. introduction the fact that symbolic flows are expansive has been recognized for several years and has provided the impetus for the study of expansive homeomorphisms in a more general settings beginning with the utz’s paper [17] in the middle of the twentieth century. evidently, mathematicians used the size of some particular subsets of the phase space to unleash many interesting phenomena of such homeomorphisms. for instance, j. f. jacobsen [10] proved that there exists no expansive self-homeomorphism of a closed 2-cell by using the fact that the set received 29 october 2017 – accepted 13 january 2019 http://dx.doi.org/10.4995/agt.2019.8843 p. das and t. das of points doubly asymptotic to any fixed point of such a homeomorphism of a compact metric space is at most countable. in recent years, a. arbieto proved [1] that the set of sinks of any homeomorphism with canonical coordinates has measure zero with respect to any positively expansive measure by using the fact that a set on which a measurable map of a separable metric space is lyapunov stable has measure zero with respect to any positively expansive measure. further, he proved that the set of heteroclinic points of a homeomorphism on a compact metric space has measure zero with respect to any expansive measure with the help of the fact that the set of points with converging semiorbits under a homeomorphism of a compact metric space has measure zero with respect to any expansive measure. many of these results have been generalized [15] for expansive measures for maps on non-compact, non-metrizable spaces. treading on the same path, in section 3, we conclude some results analogous to that of expansive self-homeomorphisms on compact metric spaces by using the fact that the set of points positively asymptotic to one point and negatively asymptotic to another point has measure zero with respect to any expansive outer regular measure for a bi-measurable map on a separable uniform space. for instance, we give a class of measures which cannot be expansive for denjoy homeomorphisms on s1. we also prove that every stable class has measure zero with respect to any positively expansive outer regular measure of a measurable map on a separable uniform space. then, we prove that the set of equicontinuous homeomorphisms and the set of homeomorphisms admitting expansive measure on a lindelöf uniform space are disjoint. we prove that sink does not exist for a bi-measurable map having canonical coordinates and strictly positive positively expansive measure. finally, in section 4, we prove that measure expansive (strong measure expansive) homeomorphisms with shadowing have periodic (strong periodic) shadowing. we further relate local weak specification and periodic shadowing for strong measure expansive systems. 2. preliminaries we denote the set of non-negative integers by n and the set of all integers by z. a point x ∈ x is called atom for a measure µ if µ({x}) > 0. a measure µ on x is said to be non-atomic if it has no atom. let us denote the set of all borel measures, the set of all non-atomic borel measures and the set of all strictly positive non-atomic borel measures by m(x), nam(x) and spnam(x) respectively. in 2013, a. arbieto defined [1] the sets φδ(x) = {y ∈ x | d(f i(x), fi(y)) ≤ δ for all i ∈ n} and γδ(x) = {y ∈ x | d(f i(x), fi(y)) ≤ δ for all i ∈ z} for a measurable and bi-measurable map respectively, on a metric space (x, d) to introduce the following concepts. definition 2.1. an expansive (positively expansive) measure for a bi-measurable (measurable) map f : x → x on a metric space (x, d) is a borel measure µ on x for which there is δ > 0 such that µ(γδ(x)) (µ(φδ(x))) = 0 for all x ∈ x. in both the cases, such δ is called an expansive constant for µ. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 20 expansive measures on uniform spaces observe that a positively expansive measure for a bi-measurable map is also an expansive measure. in 2015, c. a. morales generalized [15] these concepts to uniform spaces with the sole difference being the role of spherical neighborhoods now played by uniform neighborhoods. to understand his generalization we need the following terminology. let x be a non-empty set. then the diagonal of x × x is given by ∆(x) = {(x, x)|x ∈ x}. for a subset r of x × x, we define r−1 = {(y, x)|(x, y) ∈ r}. we say that r is symmetric if r = r−1. for two subsets u and v of x ×x, we define their composition as u ◦ v = {(x, y) ∈ x × x| there is z ∈ x satisfying (x, z) ∈ u and (z, y) ∈ v }. in this paper, we assume that the phase space of a dynamical system is a uniform space (x, u), where u is a collection of subsets of x × x satisfying the following properties ([11],[12]): (1) every d ∈ u contains ∆(x). (2) if d ∈ u and e ⊃ d, then e ∈ u. (3) if d, d′ ∈ u, then d ∩ d′ ∈ u. (4) if d ∈ u, then d−1 ∈ u. (5) for every d ∈ u there is symmetric d′ ∈ u such that d′ ◦ d′ ⊂ d. the members of u are called entourages of x. if (x, u) is a uniform space, then we can generate a topology on x by characterizing that a subset y ⊂ x is open if and only if there exists u ∈ u such that for each x ∈ y the cross section u[x] = {y ∈ x | (x, y) ∈ u} is contained in y . morales introduced [15] φd(x) = {y ∈ x | (f i(x), fi(y)) ∈ d for all i ∈ n} and γd(x) = {y ∈ x | (f i(x), fi(y)) ∈ d for all i ∈ z } for some d ∈ u to generalize the respective concepts of expansive and positively expansive measures. definition 2.2. an expansive (positively expansive) measure for bi-measurable (measurable) map f : x → x on a uniform space is a borel measure µ on x for which there exists d ∈ u such that µ(γd(x))(µ(φd(x))) = 0 for all x ∈ x. in both the cases, such d ∈ u is referred to as expansive entourage for µ. remark 2.3. one can verify that remark 3.2 [8] holds for all dynamical notions in the present paper. definition 2.4 ([15]). we say that a bi-measurable (measurable) map f : x → x on a uniform space x is expansive (positively expansive) if there exists d ∈ u such that for any distinct x, y ∈ x there exists n ∈ z (n ∈ n) such that (fn(x), fn(y)) /∈ d. remark 2.5. (i) any non-atomic borel measure on a uniform space x is expansive (positively expansive) measure for any expansive (positively expansive) map on x. recall from [13] that if x has no perfect subset and every borel probability measure on x is net additive, then a borel probability measure is atomic. so, there exists no expansive (positively expansive) map c© agt, upv, 2019 appl. gen. topol. 20, no. 1 21 p. das and t. das with expansive (positively expansive) probability measures on a lindelöf uniform space without any perfect subset. for instance, (a) consider x = r and u = {ut | 0 < t < ∞}, where ut = {(x, y) ∈ r 2 | x2+y2 < t}∪{(x, x) | x ∈ r}. then, f : r → r given by f(x) = 2x is expansive homeomorphism. (b) consider x = {1/n, 1−1/n | n ∈ n}. then f : x → x given by f(0) = 0, f(1) = 1 and f(x) = x+ for all x 6= 0, 1 where x+ denotes the point immediate right to x, is an expansive homeomorphism. but both of the homeomorphism have no expansive measure. thus, expansiveness of a homeomorphism neither imply nor is implied by the existence of an expansive measure for the map. (ii) if f : x → x is an expansive homeomorphism with expansive entourage e, then x must be t1 space. indeed, if x 6= y in x, then there exists n ∈ z such that y /∈ f−n(e[fn(x)]). on the other hand, proposition 2.18 [15] shows that there are homeomorphisms on non-t1 uniform spaces admitting expansive measure. (iii) in [15], authors proved that any expansive probability measure for a bimeasurable map on a lindelöf uniform space is non-atomic. thus, the measure of a countable set under such a measure is equal to zero. (iv) let f : x → x be a measurable map and µ an ergodic invariant probability measure for f. then µ is positively expansive with expansive entourage d if and only if µ(φd(x)) = 0 for all x ∈ e with µ(e) > 0. indeed, if ed = {x ∈ x | µ(φd(x)) = 0}, then by lemma 2.10 [15] it is enough to show that µ(ed) = 1. for fix x ∈ x, we have that φd(x) ⊂ f −1(φd(f(x))). then by the fact that µ is invariant under f we have, µ(φd(x)) ≤ µ(φd(f(x))) which implies that f−1(ed) ⊂ ed and hence µ(f −1(ed) = µ(ed). since µ is ergodic, we have µ(ed) ∈ {0, 1}. but since e ⊂ ed we must have µ(ed) = 1. example 2.6. let z be the sorgenfrey line, i.e. r equipped with the topology based on the intervals of the form [x, y), where (x, y) ∈ r × q with x < y. (i) the map f : z → z given by f(x) = x for x ∈ q and f(x) = 2x for x ∈ r \ q is not expansive but the lebesgue measure is an expansive measure for f. (ii) the map f : z → z given by f(x) = 0 for x ∈ q and f(x) = 2x for x ∈ r \ q is not positively expansive but the lebesgue measure is a positively expansive measure for f. 3. expansive measures in [10], jacobsen and utz stated that the set of points doubly asymptotic to a fixed point of an expansive homeomorphism on a compact metric space is at most countable. in [18], r. williams proved a stronger result which states that the set of points doubly asymptotic to a given point of an expansive homeomorphism on a compact metric space is at most countable. the purpose of this section is to extend this result to expansive measures for bi-measurable c© agt, upv, 2019 appl. gen. topol. 20, no. 1 22 expansive measures on uniform spaces maps on separable uniform space. we prove a stronger result and conclude our result as a direct consequence. definition 3.1. let f : x → x be a bi-measurable map on a uniform space x and let x ∈ x be given. a point y ∈ x is said to be positively asymptotic to x if for every e ∈ u there is a positive integer n such that (fn(x), fn(y)) ∈ e for all n ≥ n. on the other hand, a point y ∈ x is said to be negatively asymptotic to x if for every e ∈ u there is a positive integer m such that (fn(x), fn(y)) ∈ e for all n ≤ −m. a point y ∈ x is said to be doubly asymptotic to x if for every e ∈ u there is a positive integer n such that (fn(x), fn(y)) ∈ e for all | n |≥ n. recall that a borel measure µ on a topological space is outer regular if for every measurable subset a and any ǫ > 0 there is an open subset o ⊃ a such that µ(o \ a) < ǫ. the following lemma was proved in [15]. for the sake of completeness we present it here. lemma 3.2. let µ be a borel measure on a topological space. then for every measurable lindelöf subset k with µ(k) > 0 there are z ∈ k and an open neighborhood u of z such that µ(k ∩ w) > 0 for every open neighborhood w ⊂ u of z. proof. otherwise, for every z ∈ k there is an open neighborhood uz ⊂ u satisfying µ(k ∩ uz) = 0. since k is lindelöf, the open cover {k ∩ uz : z ∈ k} of k admits a countable sub-cover, i.e. there is a sequence {zl}l∈n in k satisfying k = ⋃ l∈n(k ∩ uzl). so, µ(k) = ∑ l∈n µ(k ∩ uzl) = 0, a contradiction. � theorem 3.3. the set of points positively asymptotic to p and negatively asymptotic to q for given p, q ∈ x for a bi-measurable map f : x → x on a separable uniform space has measure zero with respect to any expansive outer regular measure µ of f. proof. let p, q ∈ x be given. let d be an expansive entourage for µ and d′ ∈ u be symmetric such that d′ ◦d′ ⊂ d. let a be the set of all points positively asymptotic and negatively asymptotic to p and q respectively. if an = {x ∈ x | (fn(x), fn(p)) ∈ d′ for all n ≥ n and (fn(x), fn(q)) ∈ d′ for all n ≤ −n}. it is easy to verify that a ⊂ ⋃ n≥0 an and each an is measurable. we show that µ( ⋃ n≥1 an) = 0. if possible, suppose µ( ⋃ n≥1 an) > 0. so, there is m ≥ 1 such that µ(am ) > 0. since x is separable uniform space, it is second countable and since µ is outer regular, lusin theorem [9] implies that for every ǫ > 0 there is a measurable subset cǫ with µ(x \cǫ) < ǫ such that f i |cǫ is continuous for all integer i with | i |≤ m. taking ǫ = µ(am )/2 we obtain a measurable subset c = cµ(am )/2 such that fi |c continuous for all integer i with | i |≤ m and µ(am ∩ c) > 0. since k = am ∩c is lindelöf, we can apply lemma 3.2 to obtain z ∈ am ∩c and d0 ∈ u satisfying µ(am ∩ c ∩ d1[z]) > 0 for all entourage d1 ⊂ d0...(*). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 23 p. das and t. das fix such a z and d0. since z ∈ c and f i |c is continuous for all | i |≤ m, we can fix d∗ ∈ u with d∗ ⊂ d0 such that f i(w) ∈ d[fi(z)] whenever | i |≤ m and w ∈ c ∩ d∗[z]. now, take w ∈ am ∩ c ∩ d ∗[z]. since w ∈ c ∩ d∗[z], we already have fi(w) ∈ d[fi(z)] for all | i |≤ m. since z ∈ am ∩c, we have w, z ∈ am . hence, (fi(w), fi(p)) ∈ d′ and (fi(z), fi(q)) ∈ d′ for all i ≥ m and (fi(w), fi(p)) ∈ d′ and (fi(z), fi(q)) ∈ d′ for all i ≤ −m. since d′ is symmetric and d′ ◦d′ ⊂ d, we get (fi(z), fi(w)) ∈ d for all | i |≥ m, i.e., fi(w) ∈ d[fi(z)] for all | i |≥ m. thus fi(w) ∈ d[fi(z)] for all i ∈ z which implies am ∩c ∩d ∗[z] ⊂ φd(z). since d is an expansive entourage for µ, we have µ(am ∩ c ∩ d ∗[z]) = 0. on the other hand, since d∗ ⊂ d0, we can take d1 = d ∗ in (*) to obtain µ(am ∩ c ∩ d ∗[z]) > 0, a contradiction. this completes the proof. � lemma 1 [2] shows that if f : x → x is an expansive homeomorphism on a compact metric space, then a = ⋃ n≥0 an but it is not true in general for bi-measurable map admitting an expansive measure. example 3.4. consider f : r → r given by f(x) = x for all x ∈ q and f(x) = 2x for all x ∈ r \ q. the lebesgue measure is an expansive measure for f with any δ > 0 as expansive constant. let p, q ∈ q with d(p, q) < δ and choose x = (p + q)/2. then, x ∈ an for all n ≥ 0 but x /∈ a. we have the following corollary to theorem 3.3 and the well-known fact that every borel probability measure on a metric space is outer regular. corollary 3.5. the set of points positively asymptotic and negatively asymptotic to two given points under a homeomorphism f : x → x of a separable metric space has measure zero with respect to any expansive probability measure. theorem 3.6 ([14]). (i) there exists no self-homeomorphism on [0, 1] admitting an expansive measure. (ii) a self-homeomorphism of s1 admits expansive measure if and only if it is denjoy. the following corollary significantly cuts down the collection of expansive measures for denjoy homeomorphisms on s1. corollary 3.7. an outer regular measure µ ∈ spnam(x) cannot be expansive for a denjoy homeomorphism on s1. proof. suppose by contradiction that an outer regular measure µ ∈ spnam(x) is expansive for a denjoy homeomorphism f on s1 with an expansive constant δ. as is well known, f exhibits unique minimal set m which is infinite, totally disconnected and has no isolated point. since s1 \ m is an open subset of s1, it can be written as disjoint union of countably many open arcs, say {ij} so c© agt, upv, 2019 appl. gen. topol. 20, no. 1 24 expansive measures on uniform spaces that diam(ij) → 0 as j → ∞ and for fixed j, f n(ij) 6= ij for all n 6= 0 because of theorem 3.6(i). thus, we have diam(fn(ij)) → 0 as | n |→ ∞. for a, b ∈ ij, let fn0(ab) be the longest length in {fn(ab) | n ∈ z}. take a, b ∈ ij such that the length between fn0(a) and fn0(b) is less than δ. then, diam(fn(ab)) < δ for all n ∈ z. since µ(ab) > 0, it follows that δ is not expansive constant for µ, a contradiction. � as in [7], let the stable set of x ∈ x is w s(x) = {y ∈ x | ∀b ∈ u, ∃n ∈ n such that (fi(x), fi(y)) ∈ b for all i ≥ n} and in the invertible case, the unstable set of x ∈ x is w u(x) ={y ∈ x | ∀b ∈ u, ∃n ∈ n such that (fi(x), fi(y)) ∈ b for all i ≤ −n}. if x ∈ x, then for e ∈ u, we define the local stable set w s(x, e) = {y ∈ x | (fn(x), fn(y)) ∈ e, ∀n ∈ n} and in the invertible case, the local unstable set w u(x, e) = {y ∈ x | (f−n(x), f−n(y)) ∈ e, ∀n ∈ n}. definition 3.8. a point x ∈ x is called heteroclinic point for a map f : x → x if x ∈ w s(o(p)) ∩ w u(o(q)) for some periodic points p, q ∈ x. corollary 3.9. the set of heteroclinic points of a bi-measurable map f : x → x of separable uniform space, with at most countably many periodic points has measure zero with respect to any expansive outer regular measure µ of f. proof. from theorem 3.3, it follows that w s(x) ∩ w u(y) for any x, y ∈ x has measure zero and then the result follows from the fact that the countable union of measure zero set has measure zero. � next example shows that the set of heteroclinic points of a bi-measurable map with more than countably many periodic points and an expansive measure on a separable uniform space may have measure zero. example 3.10. let c be the cantor set in [−1, 0] and x = c ∪(0, ∞) and let f : x → x be given by f(x) = x for all x ∈ c and f(x) = 2x for all x ∈ (0, ∞). the lebesgue measure is expansive measure for f. theorem 3.11. the set of points doubly asymptotic to a given point for a bi-measurable map f : x → x on a separable uniform space has measure zero with respect to any expansive outer regular measure µ of f. proof. it follows in the same fashion if we take p = q in theorem 3.3. � note that any non-atomic borel probability measure is an expansive measure for an expansive homeomorphism of a compact metric space. so, theorem 3.11 implies that the set of points doubly asymptotic to a given point under an expansive homeomorphism has measure zero with respect to any non-atomic borel c© agt, upv, 2019 appl. gen. topol. 20, no. 1 25 p. das and t. das probability measure. from this and well-known measure theoretical results [16], we obtain that the set of points doubly asymptotic to a given point under an expansive homeomorphism is at most countable, which is the fact used by r. williams [18] to prove that there exists no expansive self-homeomorphisms on a closed 2-cell. corollary 3.12. let x be a separable uniform space and f : x → x be a bi-measurable map with an expansive measure µ ∈ spnam(x). then, for each x ∈ x and each neighbourhood n of x there exists y ∈ n such that x and y are not doubly asymptotic. one can prove the following theorem in same fashion as in the proof of theorem 3.3. theorem 3.13. every stable class of a measurable map f on a separable uniform space has measure zero with respect to any positively expansive outer regular measure µ of f. corollary 3.14. let x be a separable uniform space and µ ∈ spnam(x) be a positively expansive measure of a measurable map f on x. then, for each x ∈ x and each neighbourhood n of x there exists y ∈ x such that x and y are not asymptotic. definition 3.15. let x be a uniform space. a homeomorphism (continuous map) f : x → x is called equicontinuous (positively equicontinuous) if for each x ∈ x and every e ∈ u there exists dx ∈ u (depends on x) such that (x, y) ∈ dx implies (f n(x), fn(y)) ∈ e for all n ∈ z (n ∈ n). remark 3.16. observe that corollary 3.11 and corollary 3.14 are obvious generalizations of theorem 3 [2] of b. f. bryant. from these corollaries it is clear that an equicontinuous (positively equicontinuous) map on a separable uniform space cannot have any expansive (positively expansive) measure. we now give a direct proof of this result for maps on lindelöf uniform spaces. theorem 3.17. let f : x → x be equicontinuous (positively equicontinuous) map on a lindelöf uniform space x. then, f cannot have any expansive (positively expansive) measure. proof. suppose by contradiction that f has an expansive measure µ with expansive entourage d. then, for each x ∈ x there exists dx ∈ u such that (x, y) ∈ dx implies (f n(x), fn(y)) ∈ d for all n ∈ z. thus, dx[x] ⊂ γd(x). since d is expansive entourage for µ, µ(dx[x]) ≤ µ(γd(x)) = 0 for all x ∈ x. now, {dx[x] | x ∈ x} is an open cover for x and since x is lindelöf there is {xi}i∈n such that {dxi[xi] | i ∈ n} is an open covering for x. therefore, µ(x) ≤ σi∈nµ(dxi[xi]) which implies µ(x) = 0, which is not the case. the proof for positively equicontinuous map is similar. � example 3.18. the map f : (0, 1) → (0, 1) given by f(x) = xn for some n ∈ n+, the rotation of the unit circle, a contraction [15] on any lindelöf c© agt, upv, 2019 appl. gen. topol. 20, no. 1 26 expansive measures on uniform spaces uniform space are equicontinuous and therefore, they cannot have expansive measure. thus, the two classes of homeomorphisms important in topological dynamics namely equicontinuous homeomorphisms and homeomorphisms admitting an expansive measure are disjoint. on the other hand, the identity map on a discrete uniform space is both expansive and equicontinuous. definition 3.19. let f : x → x be a map on a uniform space x and let d, e ∈ u be given. then, a sequence {xi}i∈n is said to be d-pseudo orbit if (f(xi), xi+1) ∈ d for all i ∈ n. a sequence {xi}i∈n is said to be e-shadowed by some point x ∈ x if (fi(x), xi) ∈ e for all i ∈ n. (i) f is said to have shadowing if for every e ∈ u there exists d ∈ u such that every d-pseudo orbit is e-shadowed by some point x ∈ x. (ii) f is said to have h-shadowing if for every e ∈ u there exists d ∈ u such that if {x0, x1, ..., xm} satisfy (f(xi), xi+1) ∈ d for 0 ≤ i ≤ (m − 1), then there exists x ∈ x such that (fi(x), xi) ∈ e for 0 ≤ i ≤ (m − 1) and fm(x) = xm. corollary 3.20. let x be a uniform space and let f : x → x be a bimeasurable map which satisfies either one of the following. (i) f has h-shadowing. (ii) f is positively expansive and has shadowing. then f−1 cannot have positively expansive measure. proof. (i) let e ∈ u be given and let d ∈ u be given for e by the h-shadowing of f. let x ∈ x be fixed and y be such that (x, y) ∈ d. for any fixed natural number m > 0 the finite sequence {f−m(x), f−m+1(x), ..., f−1(x), y} is a dchain. hence, by h-shadowing there is z ∈ x such that (fi(z), f−m+i(x)) ∈ e for 0 ≤ i ≤ (m − 1) and fm(z) = y which implies ((f−1)m(y), (f−1)m(x)) = (f−m(x), z) ∈ e. this shows that f−1 is positively equicontinuous and hence, by theorem 3.17 it cannot have positively expansive measure. (ii) let e ∈ u be expansive entourage for f and d ∈ u with d ⊂ e be given for e by the shadowing of f. fix a finite d-chain {x0, x1, x2, x3, ..., xm} and extend to a d-pseudo orbit {x0, x1, x2, x3, ..., xm, f(xm), f 2(xm), f 3(xm), ...}. if z ∈ x is a point which e-shadows the above d-pseudo orbit then (fi(z), xi) ∈ e for i < m and (fj+m(z), fj(xm)) ∈ e for all j ≥ 0 which implies (f i(z), xi) ∈ e for i < m and (fj(fm(z)), fj(xm)) ∈ e for all j ≥ 0. hence, (f i(z), xi) ∈ e for i < m and fm(z) = xm. this shows that f has h-shadowing. then by (i), f−1 cannot have positively expansive measure. � definition 3.21. let f : x → x be a bijective map on a uniform space x. then, a point x ∈ x is called sink [15] (source) for f if w u(x, d) = {x} (w s(x, d) = {x}) for some entourage d. we say that f has canonical coordinates if for every entourage e there is an entourage d such that (x, y) ∈ d implies w s(x, e) ∩ w u(y, e) 6= φ. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 27 p. das and t. das proposition 3.22. sink does not exist for a bi-measurable map f : x → x with canonical coordinates admitting a positively expansive measure µ ∈ spnam(x). proof. let x ∈ x be a sink and let e ∈ u be an expansive entourage for the positively expansive measure µ. then by lemma 4.6 [15], there exists d ∈ u such that d[x] ⊂ w s(x, e). but w s(x, e) has measure zero and hence, d[x] must have measure zero. this is a contradiction to the fact that the measure is strictly positive. thus, there does not exist any sink for such f. � 4. measure expansive systems in this section, we study measure expansive and strong measure expansive homeomorphisms on uniform spaces. for the metric definitions of the following notions reader may see in [6]. definition 4.1. let x be a uniform space and f : x → x be a homeomorphism. let d, e ∈ u be a given entourage. then, (i) a sequence {xi}i∈z is said to be d-pseudo orbit if (f(xi), xi+1) ∈ d for all i ∈ z. a d-pseudo orbit is said to be periodic with period n ≥ 1 if xi+n = xi for all i ∈ z. a sequence {xi}i∈z is said to be e-shadowed by some point x ∈ x if (fi(x), xi) ∈ e for all i ∈ z. (ii) f is said to have shadowing if for every e ∈ u there exists d ∈ u such that every d-pseudo orbit is e-shadowed by some point in x. (iii) f is said to have periodic shadowing if for every e ∈ u there exists d ∈ u such that every periodic d-pseudo orbit is e-shadowed by some periodic point in x. (iv) f is said to have strong periodic shadowing if for every e ∈ u there exists d ∈ u such that every periodic d-pseudo orbit with period n is e-shadowed by some periodic point with period n. (v) f is said to have local weak specification if for every e ∈ u there exists m ≥ 1 and d ∈ u such that if {x0, x1, x2, ..., xk} satisfy (f n(xi), xi+1) ∈ d for some n ≥ m, then there exists x ∈ x such that (fj+in(x), fj(xi)) ∈ e for 0 ≤ i ≤ (k − 1) and 0 ≤ j < n. definition 4.2. let x be a uniform space and f : x → x be a homeomorphism. then, (i) f is said to be measure expansive if there exists d ∈ u such that for every invariant non-atomic probability measure µ on x, we have µ(γd(x)) = 0 for all x ∈ x. such d is called measure expansive entourage. (ii) f is said to be strong measure expansive if there exists d ∈ u such that for every invariant probability measure µ on x, we have µ(γd(x)) = µ({x}) for all x ∈ x. such d is called strong measure expansive entourage. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 28 expansive measures on uniform spaces theorem 4.3. let x be a locally compact, para-compact, hausdorff uniform space and let f : x → x be an uniform equivalence. then, f has local weak specification if and only if it has shadowing. proof. suppose that f has local weak specification. let e ∈ u be symmetric and let m ≥ 1, d ∈ u be given by the local weak specification of f. by paracompactness, we can assume that e be such that e[k] is compact for compact subset k ⊂ x. let {xi}i∈z be a sequence such that (f n(xi), xi+1) ∈ d for some n ≥ m and for all i ∈ z. then for each k ≥ 1, there exists yk such that (fj+in(yk), f j(xi)) ∈ e for 0 ≤| i |≤ (k −1) and 0 ≤ j ≤ (n−1). thus for each k ≥ 1, yk ∈ e[x0]. by local compactness, there exists a compact neighborhood k of x0 such that yk ∈ e[k], which is compact. therefore, {yk}k∈n has a convergent subsequence converging to some point, say y in x. by continuity of f, we have (fj+in(y), fj(xi)) ∈ e for all i ∈ z and 0 ≤ j ≤ (n − 1). this means that fn has shadowing for each n ≥ m and similarly as in proposition 3.4(a) [8], we conclude that f has shadowing. on the other hand, by definition shadowing implies local weak specification with n = 1. � theorem 4.4. let x be a first countable, hausdorff uniform space and f : x → x a measure expansive homeomorphism. if f has shadowing, then it has periodic shadowing. proof. let e′′ ∈ u be a measure expansive entourage for f. let e ∈ u be closed, symmetric such that e2 ⊂ e′′. further, let e′ ∈ u be symmetric such that e′2 ⊂ e. let d be given for e′ by the shadowing of f. if {xi}i∈z is a periodic d-pseudo orbit with period n, then there exists x ∈ x such that (fi(x), xi) ∈ e ′ for all i ∈ z. if x is periodic, then nothing to show. suppose that x is not periodic. let us fix 0 ≤ l < n and k ∈ z. since xi+l+kn = xi+l, (f i(fl(x)), xi+l) ∈ e ′ and (xi+l, f i(fkn (fl(x))) ∈ e′ for all i ∈ z we have (fi(fl(x)), fi(fkn(fl(x)))) ∈ e′2 ⊂ e for all i ∈ z. thus, fkn(fl(x)) ⊂ γe(f l(x)) for every k ∈ z. thus, the closure of the orbit of x is contained in ⋃n−1 l=0 γe(f l(x)). if µ is a weak accumulation point of the uniform distribution supported on the finite orbit {x, f(x), f2(x), ..., fn−1(x)}, then it is invariant and µ( ⋃n−1 l=0 γe(f l(x))) = 1. hence, µ(γe(f l(x))) > 0 for some 0 ≤ l < n. since f is measure expansive, µ must be atomic. thus there exists z ∈ ⋃n−1 l=0 γe(f l(x)) such that µ({z}) > 0. since µ is invariant probability measure, z must be periodic. let z ∈ γe(f l(x)) for some 0 ≤ l < n. then, (fi(x), fi(f−l(z))) = (fi(x), fi−l(z)) ∈ e and also, (fi(x), xi) ∈ e ′ ⊂ e for all i ∈ z. therefore, (fi(f−l(z)), xi) ∈ e 2 for all i ∈ z. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 29 p. das and t. das this completes the proof. � example 4.5. (i) similarly as in theorem a [5], we can produce n-expansive homeomorphisms with shadowing. such homeomorphisms are clearly measure expansive and therefore, has periodic shadowing. (ii) let f be an expansive homeomorphism with shadowing. then, one can verify that the n-expansive homeomorphisms constructed in proposition 2.18 [15] have shadowing. since such homeomorphisms are measure expansive, they have periodic shadowing. corollary 4.6. let x be a first countable, locally compact, para-compact, hausdorff uniform space and f : x → x a strong measure expansive uniform equivalence with per(f) 6= φ. if f has local weak specification then it has periodic shadowing. proof. it follows from theorem 4.3 and theorem 4.4. � lemma 4.7. let x be a uniform space and f : x → x a strong measure expansive homeomorphism such that per(f) 6= φ. then, f |p er(f) is expansive. proof. let d ∈ u be a strong measure expansive entourage for f. if per(f) is a single point, then we have nothing to show. therefore, assume that per(f) contains more than one point. if possible, suppose there exists x 6= y in per(f) such that y ∈ γd(x). let µ be an invariant probability measure such that µ(x) > 0 and µ(y) > 0. thus, µ(γd(x)) ≥ µ(x)+µ(y) > µ(x), a contradiction. so, f |p er(f) is expansive with expansive entourage d. � lemma 4.8. let x be a uniform space and f : x → x a homeomorphism such that per(f) = φ. then, f is measure expansive if and only if it is strong measure expansive. hint: an invariant atomic probability measure for f must be supported on periodic points. theorem 4.9. let x be a first countable, hausdorff uniform space and f : x → x a strong measure expansive homeomorphism with per(f) 6= φ. if f has shadowing, then it has strong periodic shadowing. proof. by theorem 4.4, f has periodic shadowing. let e′ ∈ u be a strong measure expansive entourage for f. let e ∈ u be symmetric such that e2 ⊂ e′ and let d ∈ u be given for e by shadowing of f. let {xi}i∈z be a d-pseudo orbit with period n ≥ 1. then, by periodic shadowing there exists x ∈ per(f) such that (fi(x), xi) ∈ e for all i ∈ z. since xi+n = xi for all i ∈ z, we have (xi, f i(fn(x))) = (xi+n , f i+n(x)) ∈ e for all i ∈ z. therefore, (fi(x), fi(fn(x))) ∈ e2 ⊂ e′ for all i ∈ z. since c© agt, upv, 2019 appl. gen. topol. 20, no. 1 30 expansive measures on uniform spaces by lemma 4.7, f/per(f) is expansive with expansive entourage e′, we must have fn(x) = x. this completes the proof. � example 4.10. let z be the sorgenfrey line. then, the map f : z → z given by f(x) = 2x is strong measure expansive homeomorphism with shadowing. thus, f has strong periodic shadowing. acknowledgements. the first author is supported by the department of science and technology, government of india, under inspire fellowship (registration noif150210) program since march 2015. references [1] a. arbieto and c. a. morales, some properties of positive entropy maps, ergodic theory dynam. systems 34 (2014), 765–776. 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[10] j. f. jacobsen and w. r. utz, the non-existence of expansive homeomorphisms on a closed 2-cell, pacific j. math. 10 (1960), 1319–1321. [11] i. m. james, uniform and topological spaces, springer-verlag (1994). [12] j. kelley, general topology, van nostrand company (1955). [13] j. d. knowles, on the existence of non-atomic measures, mathematika 14 (1967), 62–67. [14] c. a. morales, measure-expansive systems, preprint, impa, d083 (2011). [15] c. a. morales and v. sirvent, expansivity for measures on uniform spaces, trans. amer. math. soc. 368 (2016), 5399–5414. [16] k. r. parthasarathy, r. r. ranga and s. r. s. varadhan, on the category of indecomposable distributions on topological groups, trans. amer. math. soc. 102 (1962), 200–217. [17] w. r. utz, unstable homeomorphisms, proc. amer. math. soc. 1 (1950), 769–774. [18] r. williams, some theorems on expansive homeomorphisms, amer. math. month. 8 (1966), 854–856. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 31 @ appl. gen. topol. 20, no. 1 (2019), 75-79doi:10.4995/agt.2019.9817 c© agt, upv, 2019 on monotonous separately continuous functions yaroslav i. grushka department of nonlinear analysis, institute of mathematics nas of ukraine, kyiv (grushka@imath.kiev.ua) communicated by o. valero abstract let t = (t, ≤) and t1 = (t1, ≤1) be linearly ordered sets and x be a topological space. the main result of the paper is the following: if function f(t, x) : t × x → t1 is continuous in each variable (“t” and “x”) separately and function f x (t) = f(t, x) is monotonous on t for every x ∈ x, then f is continuous mapping from t × x to t1, where t and t1 are considered as topological spaces under the order topology and t × x is considered as topological space under the tychonoff topology on the cartesian product of topological spaces t and x. 2010 msc: 54c05. keywords: separately continuous mappings; linearly ordered topological spaces; young’s theorem. 1. introduction in 1910 w.h. young had proved the following theorem. theorem a (see [9]). let f : r2 → r be separately continuous. if f(·, y) is also monotonous for every y, then f is continuous. in 1969 this theorem was generalized for the case of separately continuous function f : rd → r (d ≥ 2): theorem b (see [5]). let f : rd+1 → r (d ∈ n) be continuous in each variable separately. suppose f (t1, . . . , td, τ) is monotonous in each ti separately (1 ≤ i ≤ d). then f is continuous on rd+1. note that theorems a and b were also mentioned in the overview [2]. in the papers [6,7] authors investigated functions of kind f : t × x → r, where received 12 march 2018 – accepted 10 september 2018 http://dx.doi.org/10.4995/agt.2019.9817 ya. i. grushka (t, ≤) is linearly ordered set equipped by the order topology, (x, τx) is any topological space and the function f is monotonous relatively to the first variable as well continuous (or quasi-continuous) relatively to the second variable. in particular in [7] it was proven that each separately quasi-continuous and monotonous relatively to the first variable function f : r × x → r is quasicontinuous relatively to the set of variables. the last result may be considered as the abstract analog of young’s theorem (theorem a) for separately quasicontinuous functions. however, we do not know any direct generalization of theorem a (for separately continuous and monotonous relatively to the first variable function) in abstract topological spaces at the present time. in the present paper we prove the generalization of theorems a and b for the case of (separately continuous and monotonous relatively to the first variable) function f : t × x → t1, where (t, ≤), (t1, ≤1) are linearly ordered sets equipped by the order topology and x is any topological space. 2. preliminaries let t = (t, ≤) be any linearly (ie totally) ordered set (in the sense of [1]). then we can define the strict linear order relation on t such that for any t, τ ∈ t the correlation t < τ holds if and only if t ≤ τ and t 6= τ. together with the linearly ordered set t we introduce the linearly ordered set t±∞ = (t ∪ {−∞, +∞} , ≤) , where the order relation is extended on the set t ∪ {−∞, +∞} by means of the following clear conventions: (a): −∞ < +∞; (b): (∀t ∈ t) (−∞ < t < +∞). recall [1] that every such linearly ordered set t = (t, ≤) can be equipped by the natural “internal” order topology tpi [t], generated by the base consisting of the open sets of kind: (τ1, τ2) = {t ∈ t | τ1 < t < τ2} ,(2.1) where τ1, τ2 ∈ t ∪ {−∞, +∞} , τ1 < τ2. let (x, τx), (y, τy ) and (z, τz) be topological spaces. by c(x, y ) we denote the collection of all continuous mappings from x to y . for a mapping f : x × y → z and a point (x, y) ∈ x × y we write f x(y) := fy(x) := f(x, y). recall [3] that the mapping f : x × y → z is refereed to as separately continuous if and only if f x ∈ c(y, z) and fy ∈ c(x, z) for every point (x, y) ∈ x × y (see also [6–8]). the set of all separately continuous mappings f : x × y → z is denoted by cc (x × y, z) [3,6–8]. let t = (t, ≤) and t1 = (t1, ≤1) be linearly ordered sets. we say that a function f : t → t1 is non-decreasing (non-increasing) on t if and only if for every t, τ ∈ t the inequality t ≤ τ leads to the inequality f(t) ≤1 f(τ) c© agt, upv, 2019 appl. gen. topol. 20, no. 1 76 on monotonous separately continuous functions (f(τ) ≤1 f(t)) correspondingly. function f : t → t1, which is non-decreasing or non-increasing on t is called by monotonous. 3. main results let (x1, τx1), . . . , (xd, τxd) (d ∈ n) be topological spaces. further we consider x1 × · · · × xd as a topological space under the tychonoff topology τx1×···×xd on the cartesian product of topological spaces x1, . . . ,xd. recall [4, chapter 3] that topology τx1×···×xd is generated by the base of kind: { u1 × · · · × ud | (∀j ∈ {1, . . . , d}) ( uj ∈ τxj )} . theorem 3.1. let t = (t, ≤) and t1 = (t1, ≤1) be linearly ordered sets and (x, τx) be a topological space. if f ∈ cc (t × x, t1) and function fx(t) = f(t, x) is monotonous on t for every x ∈ x, then f is continuous mapping from the topological space (t × x, τ t×x) to the topological space (t1, tpi [t1]). proof. consider any ordered pair (t0, x0) ∈ t × x. take any open set v ⊆ t1 such that f (t0, x0) ∈ v . since the sets of kind (2.1) form the base of topology tpi [t1], there exist τ1, τ2 ∈ t1 ∪ {−∞, +∞} such that τ1 <1 f (t0, x0) <1 τ2 and (τ1, τ2) ⊆ v , where <1 is the strict linear order, generated by (non-strict) order ≤1 (on t1 ∪ {−∞, +∞}). the function f is separately continuous. so, since the sets of kind (2.1) form the base of topology tpi [t] , there exist t1, t2 ∈ t ∪ {−∞, +∞} such that t1 < t0 < t2 and(3.1) f [(t1, t2) × {x0}] ⊆ (τ1, τ2) .(3.2) further we need the some additional denotations. • in the case, where (t1, t0) 6= ∅ we choose any element α1 ∈ t such that t1 < α1 < t0 and denote α̃1 := α1. in the opposite case we denote α1 := t0, α̃1 := t1. • in the case (t0, t2) 6= ∅ we choose any element α2 ∈ t such that t0 < α2 < t2 and denote α̃2 := α2. in the opposite case we denote α2 := t0, α̃2 := t2. it is not hard to verify, that in the all cases the following conditions are performed: α1, α2 ∈ t, α̃1, α̃2 ∈ t ∪ {−∞, +∞} ; α1 ≤ α2; α̃1 < α̃2; [α1, α2] ⊆ (t1, t2) , where [α1, α2] = {t ∈ t | α1 ≤ t ≤ α2} ;(3.3) t0 ∈ (α̃1, α̃2) ⊆ [α1, α2] .(3.4) according to (3.3), α1, α2 ∈ (t1, t2). hence, according to (3.2), interval (τ1, τ2) is an open neighborhood of the both points f (α1, x0) and f (α2, x0). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 77 ya. i. grushka since the function f is separately continuous on t × x, then there exist an open neighborhood u ∈ τx of the point x0 (in the space x) such that: f [{α1} × u] ⊆ (τ1, τ2) ;(3.5) f [{α2} × u] ⊆ (τ1, τ2) .(3.6) the set (α̃1, α̃2)×u is an open neighborhood of the point (t0, x0) in the topology τ t×x of the space t × x. now our aim is to prove that (3.7) ∀ (t, x) ∈ (α̃1, α̃2) × u (f (t, x) ∈ (τ1, τ2) ⊆ v ) . so, chose any point (t, x) ∈ (α̃1, α̃2) × u. according to the condition (3.4), we have (t, x) ∈ [α1, α2] × u, that is α1 ≤ t ≤ α2 and x ∈ u. in accordance with (3.5), (3.6), we have f (α1, x) ∈ (τ1, τ2) and f (α2, x) ∈ (τ1, τ2). hence, since the function fx(·) = f(·, x) is monotonous on t and α1 ≤ t ≤ α2, we deduce f (t, x) ∈ (τ1, τ2) ⊆ v . thus, the correlation (3.7) is proven. hence, the function f is continuous in (every) point (t0, x0) ∈ t × x. � theorem a is a consequence of theorem 3.1 in the case t = x = r, where r is considered together with the usual linear order on the field of real numbers and usual topology. corollary 3.2. let t0 = (t0, ≤0), t1 = (t1, ≤1), . . . , td = (td, ≤d) (d ∈ n) be linearly ordered sets, and (x, τx) be a topological space. if the function f : t1 × · · · × td × x → t0 is continuous in each variable separately and f (t1, . . . , td, τ) is monotonous in each ti separately (1 ≤ i ≤ d) then f is a continuous mapping from the topological space (t1 × · · · × td × x, τ t1×···×td×x) to the topological space (t0, tpi [t0]). proof. we will prove this corollary by induction. for d = 1 the corollary is true by theorem 3.1. assume, that the corollary is true for the number d − 1, where d ∈ n, d ≥ 2. suppose, that function f : t1 × · · · × td × x → t0 is satisfying the conditions of the corollary. then we may consider this function as a mapping from t1 ×x(d) to t0, where x(d) = t2 ×· · ·×td ×x. according to inductive hypothesis, function f (t1, ·) is continuous on x(d) for every fixed t1 ∈ t1. so f is a separately continuous mapping from t1 × x(d) to t0. moreover, f is monotonous relatively to the first variable (by conditions of the corollary). hence, by theorem 3.1, f is continuous on t1 × x(d). � theorem b is a consequence of corollary 3.2 in the case t0 = t1 = · · · = td = x = r, where r is considered together with the usual linear order on the field of real numbers and usual topology. in the case t0 = r, tj = (aj, bj), x = (ad+1, bd+1) where aj, bj ∈ r and aj < bj (j ∈ {1, . . . , d + 1}) and intervals (aj, bj) are considered together with the usual linear order and topology, induced from the field of real numbers, we obtain the following corollary. corollary 3.3. if the function f : (a1, b1) × · · · × (ad, bd) × (ad+1, bd+1) → r (d ∈ n) is continuous in each variable separately and f (t1, . . . , td, τ) is monotonous in each ti separately (1 ≤ i ≤ d) then f is a continuous mapping from (a1, b1) × · · · × (ad+1, bd+1) to r. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 78 on monotonous separately continuous functions remark 3.4. in fact in the paper [5] the more general result was formulated, in comparison with theorem b. namely the author of [5] had considered the real valued function f (t1, . . . , td, τ) defined on an open set g ⊆ r d+1, d ∈ n such that f is continuous in each variable separately and monotonous in each ti separately (1 ≤ i ≤ d). but this result of [5] can be delivered from corollary 3.3, because for each point t = (t1, . . . , td, τ) ∈ g in the open set g there exists the set of intervals (a1, b1) , . . . , (ad+1, bd+1) such that t ∈ (a1, b1)× · · · × (ad+1, bd+1) ⊆ g. references [1] g. birkhoff, lattice theory, third edition. american mathematical society colloquium publications, vol. xxv, american mathematical society, providence, r.i., new york, 1967. [2] k. c. ciesielski and d. miller, a continuous tale on continuous and separately continuous functions, real analysis exchange 41, no. 1 (2016), 19–54. [3] o. karlova, v. mykhaylyuk and o. sobchuk, diagonals of separately continuous functions and their analogs, topology appl. 160, no. 1 (2013), 1–8. [4] j. l. kelley, general topology, university series in higher mathematics, van nostrand, 1955. [5] r. l. krusee and j. j. deely, joint continuity of monotonic functions, the american mathematical monthly 76, no. 1 (1969), 74–76. [6] v. mykhajlyuk, the baire classification of separately continuous and monotone functions, scientific herald of yuriy fedkovych chernivtsi national university 349 (2007), 95–97 (ukrainian). [7] v. nesterenko, joint properties of functions which monotony with respect to the first variable, mathematical bulletin of taras shevchenko scientific society 6 (2009), 195–201 (ukrainian). [8] h. voloshyn, v. maslyuchenko and o. maslyuchenko, on layer-wise uniform approximation of separately continuous functioins by polynomials, mathematical bulletin of taras shevchenko scientific society 10 (2013), 135–158 (ukrainian). [9] w. young, a note on monotone functions, the quarterly journal of pure and applied mathematics (oxford ser.) 41 (1910), 79–87. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 79 @ appl. gen. topol. 18, no. 1 (2017), 153-171 doi:10.4995/agt.2017.6694 c© agt, upv, 2017 fixed points of α-θ-geraghty type and θ-geraghty graphic type contractions wudthichai onsod a, poom kumam a,b and yeol je cho c,d a kmuttfixed point research laboratory, department of mathematics, room scl 802 fixed point laboratory, science laboratory building, faculty of science, king mongkut’s university of technology thonburi (kmutt), 126 pracha-uthit road, bang mod, thrung khru, bangkok 10140, thailand (wudthichai.ons@mail.kmutt.ac.th, poom.kum@kmutt.ac.th) b kmutt-fixed point theory and applications research group (kmutt-fpta), theoretical and computational science center (tacs), science laboratory building, faculty of science, king mongkut’s university of technology thonburi (kmutt), 126 pracha-uthit road, bang mod, thrung khru, bangkok 10140, thailand (poom.kum@kmutt.ac.th) c department of mathematics education and the rins, gyeongsang national university, jinju 660-701, korea (yjcho@gnu.ac.kr) d center for general education, china medical university, taichung, 40402, taiwan (yjcho@gnu.ac.kr) communicated by s. romaguera abstract in this paper, by using the concept of the α-garaghty contraction, we introduce the new notion of the α-θ-garaghty type contraction and prove some fixed point results for this contraction in partial metric spaces. also, we give some examples and applications to illustrate the main results. 2010 msc: 47h09; 47h10; 54h25; 37c25. keywords: α-θ-garaghty type contraction; θ-geraghty graphic type contractions; partial order; partial metric spaces; fixed points; common fixed points. 1. introduction in 1922, banach [4] proved a theorem, which is called banach’s fixed point theorem, to show the existence of a solution for an integral equation. in fact, received 04 october 2016 – accepted 24 december 2016 http://dx.doi.org/10.4995/agt.2017.6694 w. onsod, p. kumam and y. j. cho banach’s fixed point theorem plays an important role in several branches of mathematics and applied sciences because of its importance and usefulness to show the existence and uniqueness of solutions of many kinds of nonlinear problems. especially, in 1973, geraghty [9] generalized banach’s fixed point theorem as follows: theorem g. let (x,d) be a metric space and t : x → x be a mapping. suppose that there exists β ∈ f such that, for all x,y ∈ x, d(tx,ty) ≤ β(d(x,y))d(x,y), where f denotes the family of all functions β : [0,∞) → [0, 1) which satisfies the following condition: lim n→∞ β(tn) = 1 =⇒ lim n→∞ tn = 0. then t has a unique fixed point z ∈ x and {tnx} converges to the point z for each x ∈ x. since geraghty’s fixed point theorem, some authors have studied this theorem in several ways (see [11, 23, 21, 8, 25, 7]). on the other hand, in 2012 and 2013, samet et al. [27] and hussain et al. [13] introduced the concept of α-admissible mappings in metric spaces and proved some fixed point theorems for these mappings. subsequently, in 2013, abdeljawad [1] introduced a pair of α-admissible mappings satisfying new sufficient contractive conditions, which are different from those in [27, 13], and obtained fixed point and common fixed point theorems. afterward, some authors have obtained fixed point theorems for some kinds of α-admissible mappings (see [27, 8, 13, 24, 2, 3, 10]). on the other hand, in 2014, jleli et al. [17] introduced a class θ of all the functions satisfying the following conditions: (θ1) θ is nondecreasing; (θ2) for any sequence {tn} in (0,∞), limn→∞θ(tn) = 1 if and only if limn→∞ tn = 0; (θ3) there exist r ∈ (0, 1) and l ∈ (0,∞] such that limt→0+ θ(t)−1 tr = l; (θ4) θ is continuous. also, they generalized banach’s fixed point theorem in generalized metric spaces (see branciari [6], sometime, a generalized metric space is called a branciari metric space) as follows: theorem js. let (x,d) be a complete generalized metric space and t : x → x be a mapping. suppose that there exist θ ∈ θ and k ∈ (0, 1) such that d(tx,ty) 6= 0 =⇒ θ(d(tx,ty)) ≤ [θ(d(x,y))]k for all x,y ∈ x. then t has a unique fixed point in x. also, in 2014, jleli et al. [16] established a new fixed point theorem, which is an extension of their recent result, theorem js. recently, in 2016, liu et c© agt, upv, 2017 appl. gen. topol. 18, no. 1 154 fixed points of α-θ-geraghty type contractions al. [20] introduced the notion of a θ-type contraction and a θ-type suzuki contraction and established some new fixed point theorems for such kinds of contractions in complete metric spaces. motivated by the above results, in this paper, we introduce the notion of an α-θ-geraghty type contraction and prove some common fixed point theorems for this contraction in complete partial metric spaces. moreover, we give some examples and applications to illustrate our main results. 2. preliminaries in this section, we give some definitions, examples and fundamental results. definition 2.1 ([22]). let x be a nonempty set and p : x × x → r+ be a mapping satisfying following conditions: for all x,y,z ∈ x, (pm1) x = y ⇐⇒ p(x,x) = p(x,y) = p(y,y); (pm2) p(x,x) ≤ p(x,y); (pm3) p(x,y) = p(y,x); (pm4) p(x,y) ≤ p(x,z) + p(z,y) −p(z,z). then p is called a partial metric on x and the pair (x,p) is called a partial metric space. in 1995, matthews [22] proved that every partial metric p on x induces a metric dp : x ×x → r+ defined by dp(x,y) = 2p(x,y) −p(x,x) −p(y,y) for all x,y ∈ x. notice that a metric on a set x is a partial metric d such that d(x,x) = 0 for all x ∈ x. definition 2.2 ([22]). let (x,p) be a partial metric space. (1) a sequence {xn}n∈n in (x,p) is said to be convergent to a point x ∈ x if p(x,x) = limn→∞p(x,xn). (2) a sequence {xn}n∈n in (x,p) is called a cauchy sequence in x if limn,m→∞p(xn,xm) exists and is finite. (3) a partial metric space (x,p) is said to be complete if every cauchy sequence {xn} in x converges, with respect to τ(p), to a point x ∈ x such that p(x,x) = limn,m→∞p(xn,xm). definition 2.3 ([27]). let s : x → x and α : x × x → [0,∞) be two mappings. s is said to be α-admissible if α(x,y) ≥ 1 =⇒ α(sx,sy) ≥ 1 for all x,y ∈ x. example 2.4 ([19]). consider x = [0,∞) and define two mappings s : x → x, α : x ×x → [0,∞) by sx = 2x for all x,y ∈ x and α(x,y) = { ey/x, if x ≥ y, x 6= 0, 0, if x < y. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 155 w. onsod, p. kumam and y. j. cho then s is α-admissible. definition 2.5 ([1]). let s,t : x → x and α : x × x → [0,∞) be two mappings. the pair (s,t) is said to be α-admissible if α(x,y) ≥ 1 =⇒ α(sx,ty) ≥ 1, α(tx,sy) ≥ 1 for all x,y ∈ x. example 2.6. let x = [0,∞) and define the mappings s,t : x → x and α : x ×x → [0,∞) by sx = 2x, tx = x2 for all x,y ∈ x and α(x,y) = { exy, if x,y ≥ 0, 0, otherwise. then the pair (s,t) is α-admissible. definition 2.7 ([12]). let s : x → x and α : x × x → [0,∞) be two mappings. s is called a triangular α-admissible mapping if (t1) α(x,y) ≥ 1 implies α(sx,sy) ≥ 1 for all x,y ∈ x; (t2) α(x,z) ≥ 1 and α(z,y) ≥ 1 imply α(x,y) ≥ 1 for all x,y,z ∈ x. example 2.8 ([12]). let x = r and define the mappings s : x → x and α : x×x → [0,∞) by sx = 3 √ x and α(x,y) = ex−y for all x,y ∈ x. then s is a triangular α-admissible mapping. indeed, if α(x,y) = ex−y ≥ 1, then x ≥ y, which implies sx ≥ sy, that is, α(sx,sy) = esx−sy ≥ 1. also, if α(x,z) ≥ 1 and α(z,y) ≥ 1, then x − z ≥ 0 and z − y ≥ 0, that is, x − y ≥ 0 and so α(x,y) = ex−y ≥ 1. definition 2.9. [1] let s,t : x × x and α : x × x → [0,∞) be three mappings. the pair (s,t) is said to be triangular α-admissible if (t1) α(x,y) ≥ 1 implies α(sx,ty) ≥ 1 and α(tx,sy) ≥ 1 for all x,y ∈ x; (t2) α(x,z) ≥ 1 and α(z,y) ≥ 1 imply α(x,y) ≥ 1 for all x,y,z ∈ x. example 2.10. let x = r and define the mappings s,t : x → x and α : x ×x → [0,∞) by sx = √ x, tx = x2 and α(x,y) = exy for all x,y ∈ x. then the pair (s,t) is triangular α-admissible. definition 2.11 ([26]). let s : x → x and α,η : x ×x → [0,∞) be three mappings. s is called an α-admissible mapping with respect to η if α(x,y) ≥ η(x,y) =⇒ α(sx,sy) ≥ η(sx,xy) for all x,y ∈ x. note that, if we take η(x,y) = 1, then definition 2.11 reduces to definition 2.7 (see [27]). also, if we take α(x,y) = 1, then we say that s is an ηsubadmissible mapping. example 2.12. let x = [0,∞) and s : x → x be a mapping defined by sx = x 2 for all x ∈ x. also, define the mappings α,η : x × x → [0,∞) by α(x,y) = 3 and η(x,y) = 1 for all x,y ∈ x. then s is α-admissible mapping with respect to η. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 156 fixed points of α-θ-geraghty type contractions lemma 2.13 ([22]). (1) a partial metric space (x,p) is complete if and only if the metric space (x,dp) is complete. (2) a sequence {xn} in x converges to a point x ∈ x with respect to τ(dp) if and only if lim n→∞ p(x,xn) = p(x,x) = lim n,m→∞ p(xn,xm). (3) if limn→∞xn = v such that p(v,v) = 0, then limn→∞p(xn,y) = p(v,y) for all y ∈ x. lemma 2.14 ([18, 7]). let (x,d) be a metric space and s : x → x be a triangular α-admissible mapping. assume that there exists x0 ∈ x such that α(x0,sx0) ≥ 1. define a sequence {xn} by xn+1 = sxn for each n ≥ 0. then we have α(xn,xm) ≥ 1 for all m,n ≥ 0 with n < m. lemma 2.15 ([1]). let (x,d) be a metric space and s,t : x → x be triangular α-admissible mappings. assume that there exists x0 ∈ x such that α(x0,sx0) ≥ 1. define a sequence {xn} in x by x2n+1 = sx2n and x2n+2 = tx2n+1 for each n ≥ 0. then we have α(xn,xm) ≥ 1 for all m,n ≥ 0 with n < m. in the sequel, we denote by θ̃ the set of all the functions θ : (0,∞) → (1,∞) satisfying the following conditions: (θ1)′ θ is non-decreasing and continuous; (θ2)′ inft∈(0,∞) θ(t) = 1. example 2.16. it is obvious that the following functions belong to θ̃: (1) θ1(t) := e e − 1 tp for all p > 0; (2) θ2(t) := 1 + t for all t > 0; (3) θ3(t) := e √ t for all t > 0; (4) θ4(t) := 2 − 2π arctan( 1 tα ) for all 0 < α < 1 and t > 0. 3. main results in this section, we prove some fixed point theorems for α-θ-geraghty type contractions in complete partial metric spaces. first, we begin with the following definition: definition 3.1. let (x,p) be a partial metric space and s,t : x → x, α : x ×x → [0,∞) be three mappings. (1) the pair (s,t) is called the modified α-θ-geraghty type contraction if there exist θ ∈ θ̃, k ∈ (0, 1) and β ∈ f such that (3.1) α(x,y)θ(p(sx,ty)) ≤ [θ(β(m(x,y))m(x,y))]k c© agt, upv, 2017 appl. gen. topol. 18, no. 1 157 w. onsod, p. kumam and y. j. cho for all x,y ∈ x, where m(x,y) = max{p(x,y),p(x,sx),p(y,ty)} . (2) if s = t in (1), then t is called a generalized α-θ-geraghty type contraction if there exist θ ∈ θ̃, k ∈ (0, 1) and β ∈ f such that (3.2) α(x,y)θ(p(tx,ty)) ≤ [θ(β(n(x,y))n(x,y))]k, for all x,y ∈ x, where n(x,y) = max{p(x,y),p(x,tx),p(y,ty)} . the following theorem is our main result in this paper: theorem 3.2. let (x,p) be a complete partial metric space and α : x×x → [0,∞) be a mapping. suppose that s,t : x ×x are two continuous mappings satisfying the following conditions: (i) the pair (s,t) is the modified α-θ-geraghty type contraction; (ii) the pair (s,t) is triangular α-admissible; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. then s and t have a unique common fixed point z ∈ x. proof. first, we prove that m(x,y) = 0 if and only if x = y is a common fixed point of the mappings s and t. in fact, if x = y is a common fixed point of (s,t), then ty = tx = x = y = sy = sx and m(x,y) = max{p(x,x),p(x,x),p(x,x)} = p(x,x). from the condition (3.1), it follows that θ(p(x,x)) = θ(p(sx,ty)) ≤ α(x,y)θ(p(sx,ty)) ≤ [θ(β(m(x,y))m(x,y))]k. it is only possible if p(x,x) = 0, which implies that m(x,y) = 0. conversely, if m(x,y) = 0, then, using (pm1) and (pm2), it is easy to prove that x = y is a fixed point of s and t. on the other hand, if m(x,y) > 0, we construct an iterative sequence {xn} in x such that x2n+1 = sx2n, x2n+2 = tx2n+1 for each n ≥ 0. we observe that, if xn = xn+1, then xn is a common fixed point of the mappings s and t . so, assume that xn 6= xn+1 for each n ≥ 0. since α(x0,x1) ≥ 1 and (s,t) is triangular α-admissible, using lemma 2.15, we obtain (3.3) α(xn,xn+1) ≥ 1 for each n ≥ 0. thus we have (3.4) θ(p(x2n+1,x2n+2)) = θ(p(sx2n,tx2n+1)) ≤ α(x2n,x2n+1)θ(p(sx2n,tx2n+1)) ≤ [θ(β(m(x2n,x2n+1))m(x2n,x2n+1))]k c© agt, upv, 2017 appl. gen. topol. 18, no. 1 158 fixed points of α-θ-geraghty type contractions for each n ≥ 0. now, also, we have m(x2n,x2n+1) = max{p(x2n,x2n+1),p(x2n,sx2n),p(x2n+1,tx2n+1)} = max{p(x2n,x2n+1),p(x2n,x2n+1),p(x2n+1,x2n+2)} = max{p(x2n,x2n+1),p(x2n+1,x2n+2)} for each n ≥ 0. if m(x2n,x2n+1) = p(x2n+1,x2n+2) for each n ≥ 0, then it follows from (3.4) that θ(p(x2n+1,x2n+2)) ≤ [θ(β(p(x2n+1,x2n+2))p(x2n+1,x2n+2))]k, which implies that ln[θ(p(x2n+1,x2n+2))] ≤ k ln[θ(β(p(x2n+1,x2n+2))p(x2n+1,x2n+2))]. this is a contradiction to k ∈ (0, 1). thus we have m(x2n,x2n+1) = p(x2n,x2n+1) for each n ≥ 0 and so it follows from (3.4) that (3.5) θ(p(x2n+1,x2n+2)) ≤ [θ(β(p(x2n,x2n+1))p(x2n,x2n+1))]k < [θ(p(x2n,x2n+1))] k < θ(p(x2n,x2n+1)) and so (3.6) θ(p(x2n+1,x2n+2)) < θ(p(x2n,x2n+1)). this implies that (3.7) θ(p(xn+1,xn+2)) < θ(p(xn,xn+1)) for each n ≥ 0. taking n →∞ in (3.7), we have (3.8) θ(p(xn,xn+1)) → 1. thus, from (θ2), it follows that (3.9) lim n→∞ p(xn,xn+1) = 0. now, we show that {xn} is a cauchy sequence in x. suppose that {xn} is not a cauchy sequence in x, that is, there exists ε > 0, we can find the sequences {xmk} and {xnk} such that, for all k ≥ 1, if mk > nk > k, then p(xmk,xnk) ≥ ε, p(xmk,xnk−1 ) < ε. so, we have ε ≤ p(xmk,xnk) ≤ p(xmk,xnk−1 ) + p(xnk−1,xnk) −p(xnk−1,xnk−1 ) ≤ p(xmk,xnk−1 ) + p(xnk−1,xnk) < ε + p(xnk−1,xnk), that is, ε < ε + p(xnk−1,xnk). c© agt, upv, 2017 appl. gen. topol. 18, no. 1 159 w. onsod, p. kumam and y. j. cho thus, from (3.9) and the above inequality, it follows that (3.10) lim k→∞ p(xmk,xnk) = ε. by the triangle inequality, we have p(xmk,xnk) ≤ p(xmk,xmk+1 ) + p(xmk+1,xnk) −p(xmk+1,xmk+1 ) ≤ p(xmk,xmk+1 ) + p(xmk+1,xnk) ≤ p(xmk,xmk+1 ) + p(xmk+1,xnk+1 ) + p(xnk+1,xnk) −p(xnk+1,xnk+1 ) ≤ p(xmk,xmk+1 ) + p(xmk+1,xnk+1 ) + p(xnk+1,xnk) and p(xmk+1,xnk+1 ) ≤ p(xmk+1,xmk) + p(xmk,xnk+1 ) −p(xmk,xmk) ≤ p(xmk+1,xmk) + p(xmk,xnk+1 ) ≤ p(xmk+1,xmk) + p(xmk,xnk) + p(xnk,xnk+1 ) −p(xnk,xnk) ≤ p(xmk+1,xmk) + p(xmk,xnk) + p(xnk,xnk+1 ). taking k →∞, it follows from (3.9) and (3.10) that lim k→∞ p(xmk+1,xnk+1 ) = ε. by lemma 2.15, since α(xnk,xmk+1 ) ≥ 1, we obtain θ(p(xnk+1,xmk+2 )) = θ(p(sxnk,txmk+1 )) ≤ α(xnk,xmk+1 )θ(p(sxnk,txmk+1 )) ≤ [θ(β(m(xnk,xmk+1 ))m(xnk,xmk+1 ))] k < [θ(m(xnk,xmk+1 ))] k < θ(m(xnk,xmk+1 )). by using (3.8) and taking k →∞, we conclude that lim k→∞ θ(p(xnk,xmk+1 )) = 1 and so limk→∞p(xnk,xmk+1 ) = 0 < ε, which is a contradiction. therefore, we have lim n,m→∞ p(xn,xm) = 0, which implies that {xn} is a cauchy sequence in (x,p). since x is complete, there exists z ∈ x such that xn → z as n →∞ and so x2n+1 → z and x2n+2 → z. since s and t are continuous, we have tx2n+1 → tz and sx2n+2 → sz. hence, from the definition of the sequence {xn}, we have z = sz. similarly, we have z = tz, that is, sz = tz = z. therefore, z is a common fixed point of s and t . now, we show that z is the unique common fixed point of the mappings s and t . assume the contrary, that is, there exists w ∈ x such that z 6= w and w = tw. from (3.1), we have θ(p(z,w)) ≤ [θ(β(m(z,w))m(z,w))]k < [θ(m(z,w))]k < θ(m(z,w)), c© agt, upv, 2017 appl. gen. topol. 18, no. 1 160 fixed points of α-θ-geraghty type contractions that is, p(z,w) < m(z,w). but, we have m(z,w) = max{p(z,w),p(z,sz),p(w,tw)} = p(z,w). this means that p(z,w) < p(z,w), which is a contradiction and so p(z,w) = 0. therefore, z is a unique common fixed point of s and t . this completes the proof. � in theorem 3.2, it is possible to remove the continuity of the mappings s and t by replacing the following condition: (a) if {xn} is a sequence in x such that α(xn,xn+1) ≥ 1 for all n ≥ 0 and xn → z ∈ x as n → ∞, then there exists a subsequence {xnk} of {xn} such that α(xnk,z) ≥ 1 for all k ≥ 0. theorem 3.3. let (x,p) be a complete partial metric space, α : x × x → [0,∞) be a function. suppose that s,t : x × x are two mappings satisfying the following conditions: (i) the pair (s,t) is the modified α-θ-geraghty type contraction; (ii) the pair (s,t) is triangular α-admissible; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ 1; (iv) (a) holds. then s and t have a unique common fixed point z ∈ x. proof. following the proof lines of theorem 3.2, we know that x2n+1 → z and x2n+2 → z as n →∞. now, we show that z is a common fixed point of s and t. due to the condition (iv), there exists a subsequence {xnk} of {xn} such that α(x2nk,z) ≥ 1 for all k ≥ 1. using (3.1), we have θ(p(x2nk+1,tz)) = θ(p(sx2nk,tz)) ≤ α(x2nk,z)θ(p(sx2nk,tz)) ≤ [θ(β(m(x2nk,z))m(x2nk,z))] k and so (3.11) θ(p(x2nk+1,tz)) ≤ [θ(β(m(x2nk,z))m(x2nk,z))] k, where m(x2nk,z) = max{p(x2nk,z),p(x2nk,sx2nk),p(z,tz)}. taking k →∞, we have (3.12) lim k→∞ m(x2nk,z) = max{p(z,sz),p(z,tz)}. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 161 w. onsod, p. kumam and y. j. cho case i. suppose that limk→∞m(x2nk,z) = p(z,tz) and p(z,tz) > 0. from (3.12), for sufficiently large k, we have m(x2nk,z) > 0, which implies that β(m(x2nk,z)) < m(x2nk,z) and so [θ(β(m(x2nk,z))m(x2nk,z))] k < [θ(m(x2nk,z))] k < θ(m(x2nk,z)). then we have θ(p(x2nk+1,tz)) < θ(m(x2nk,z)), which implies that p(x2nk+1,tz) < m(x2nk,z). taking k →∞ in the above inequality, we obtain p(z,tz) < p(z,tz), which is a contradiction. so we obtain that p(z,tz) = 0. by (pm1) and (pm2), we have z = tz. case ii. suppose that limk→∞m(x2nk,z) = p(z,sz). similarly, from case i, we obtain z = sz. thus, from two cases, we have z = tz = sz. therefore, z is a common fixed point of s and t . � if s = t and m(x,y) = max{p(x,y),p(x,sx),p(y,sy)} in theorem 3.2 and 3.3, then we have the following corollaries: corollary 3.4. let (x,p) be a complete partial metric space and α : x×x → [0,∞) be a function. suppose that s : x×x is a continuous mapping satisfying the following conditions: (i) s is a generalized α-θ-geraghty type contraction; (ii) s is triangular α-admissible; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. then s has a unique fixed point z ∈ x. corollary 3.5. let (x,p) be a complete partial metric space and α : x×x → [0,∞) be a function. suppose that s : x × x is a mapping satisfying the following conditions: (i) s is a generalized α-θ-geraghty type contraction; (ii) s is triangular α-admissible; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ 1; (iv) (a) holds. then s has a unique fixed point z ∈ x. if m(x,y) = max{p(x,y),p(x,sx),p(y,sy)} and p(x,x) = 0 for all x ∈ x in theorem 3.2 and 3.3, then we have the following corollaries: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 162 fixed points of α-θ-geraghty type contractions corollary 3.6. let (x,p) be a complete metric space and α : x×x → [0,∞) be a function. suppose that s : x ×x is a continuous mapping satisfying the following conditions: (i) s is a generalized α-θ-geraghty type contraction; (ii) s is triangular α-admissible; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. then s has a unique fixed point z ∈ x. corollary 3.7. let (x,p) be a complete metric space and α : x×x → [0,∞) be a function. suppose that s : x × x is a mapping satisfying the following conditions: (i) s is a generalized α-θ-geraghty type contraction; (ii) s is triangular α-admissible; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ 1; (iv) (a) holds. then s has a unique fixed point z ∈ x. definition 3.8. let (x,p) be a partial metric space, s,t : x → x be two mappings and α,η : x ×x → [0,∞) be two functions. (1) the pair (s,t) is called the modified (α,η)-θ-geraghty type contraction if there exist θ ∈ θ̃, k ∈ (0, 1) and β ∈ f such that (3.13) α(x,y) ≥ η(x,y) =⇒ θ(p(sx,ty)) ≤ [θ(β(m(x,y))m(x,y))]k for all x,y ∈ x, where m(x,y) = max{p(x,y),p(x,sx),p(y,ty)} . (2) if s = t in (1), then s is called a generalized (α,η)-θ-geraghty type contraction if there exist θ ∈ θ̃, k ∈ (0, 1) and β ∈ f such that (3.14) α(x,y) ≥ η(x,y) =⇒ θ(p(sx,sy)) ≤ [θ(β(n(x,y))n(x,y))]k for all x,y ∈ x, where n(x,y) = max{p(x,y),p(x,sx),p(y,sy)} . theorem 3.9. let (x,p) be a complete partial metric space and α,η : x×x → [0,∞) be two functions. suppose that s,t : x×x are two continuous mappings satisfying the following conditions: (i) the pair (s,t) is the improved (α,η)-θ-geraghty type contraction; (ii) the pair (s,t) is triangular α-admissible with respect to η; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0). then s and t have a unique common fixed point z ∈ x. proof. let x1 ∈ x be such that x1 = sx0 and x2 = tx1. then, iteratively, we can construct a sequence {xn} in x such that (3.15) x2n+1 = sx2n, x2n+2 = tx2n+1 c© agt, upv, 2017 appl. gen. topol. 18, no. 1 163 w. onsod, p. kumam and y. j. cho for each n ≥ 0. by the conditions (ii) and (iii), we have α(sx0,tx1) ≥ η(sx0,tx1) and so α(x1,x2) ≥ η(x1,x2), which implies that α(sx1,tx2) ≥ η(sx1,tx2). by induction, we have α(xn,xn+1) ≥ η(xn,xn+1) for all n ≥ 0 and so, by (i), we have (3.16) θ(p(x2n+1,x2n+2)) = θ(p(sx2n,tx2n+1)) ≤ α(x2n,x2n+1)θ(p(sx2n,tx2n+1)) ≤ [θ(β(m(x2n,x2n+1))m(x2n,x2n+1))]k for all n ≥ 0. now, we have m(x2n,x2n+1) = max{p(x2n,x2n+1),p(x2n,sx2n),p(x2n+1,tx2n+1)} = max{p(x2n,x2n+1),p(x2n,x2n+1),p(x2n+1,x2n+2)} = max{p(x2n,x2n+1),p(x2n+1,x2n+2)}. if m(x2n,x2n+1) = p(x2n+1,x2n+2) for all n ≥ 0, then, from (3.16), we have θ(p(x2n+1,x2n+2)) ≤ [θ(β(p(x2n+1,x2n+2))p(x2n+1,x2n+2))]k, which implies that ln[θ(p(x2n+1,x2n+2))] ≤ k ln[θ(β(p(x2n+1,x2n+2))p(x2n+1,x2n+2))]. this is a contradiction to k ∈ (0, 1). so, we have m(x2n,x2n+1) = p(x2n,x2n+1) for all n ≥ 0. thus it follows from (3.4) that (3.17) θ(p(x2n+1,x2n+2)) ≤ [θ(β(p(x2n,x2n+1))p(x2n,x2n+1))]k < [θ(p(x2n,x2n+1))] k < θ(p(x2n,x2n+1)) and so (3.18) θ(p(x2n+1,x2n+2)) < θ(p(x2n,x2n+1)). this implies that (3.19) θ(p(xn+1,xn+2)) < θ(p(xn,xn+1)) for all n ≥ 0. taking n →∞ in (3.19), we have (3.20) θ(p(xn,xn+1)) → 1 and so, from (θ2), (3.21) lim n→∞ p(xn,xn+1) = 0. therefore, as in the proof lines of theorem 3.2, we can get the conclusion. this completes the proof. � it is possible to remove the continuity of the mappings s and t in theorem 3.9 by replacing the following condition: c© agt, upv, 2017 appl. gen. topol. 18, no. 1 164 fixed points of α-θ-geraghty type contractions (b) if {xn} is a sequence in x such that α(xn,xn+1) ≥ η(xn,xn+1) for all n ≥ 0 and xn → z ∈ x as n → ∞, then there exists a subsequence {xnk} of {xn} such that α(xnk,z) ≥ η(xnk,z) for all k ≥ 0. theorem 3.10. let (x,p) be a complete partial metric space and α,η : x × x → [0,∞) be two functions. suppose that s,t : x × x are two mappings satisfying the following conditions: (i) the pair (s,t) is the modified (α,η)-θ-geraghty type contraction; (ii) the pair (s,t) is triangular α-admissible with respect to η; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0); (iv) (b) holds. then (s,t) has a unique common fixed point z ∈ x. proof. following the proof lines of theorem 3.3 and 3.9, we can get the conclusion. � if s = t and m(x,y) = max{p(x,y),p(x,sx),p(y,sy)} in theorem 3.9 and 3.10, then we have the following corollaries: corollary 3.11. let (x,p) be a complete partial metric space and α,η : x × x → [0,∞) be two functions. suppose that s : x×x is a continuous mapping satisfying the following conditions: (i) s is a generalized (α,η)-θ-geraghty type contraction; (ii) s is triangular α-admissible with respect to η; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0). then s has a unique fixed point z ∈ x. corollary 3.12. let (x,p) be a complete partial metric space and α,η : x × x → [0,∞) be two functions. suppose that s : x ×x is a mapping satisfying the following conditions: (i) s is a generalized (α,η)-θ-geraghty type contraction; (ii) s is triangular α-admissible with respect to η; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0); (iv) (b) holds. then s has a unique fixed point z ∈ x. if m(x,y) = max{p(x,y),p(x,sx),p(y,sy)} and p(x,x) = 0 for all x ∈ x in theorem 3.9 and 3.10, then we have the following corollaries: corollary 3.13. let (x,p) be a complete metric space and α,η : x × x → [0,∞) be two functions. suppose that s : x × x is a continuous mapping satisfying the following conditions: (i) s is a generalized (α,η)-θ-geraghty type contraction; (ii) s is triangular α-admissible with respect to η; c© agt, upv, 2017 appl. gen. topol. 18, no. 1 165 w. onsod, p. kumam and y. j. cho (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0). then s has a unique fixed point z ∈ x. corollary 3.14. let (x,p) be a complete metric space and α,η : x × x → [0,∞) be two functions. suppose that s : x × x is a mapping satisfying the following conditions: (i) s is a generalized (α,η)-θ-geraghty type contraction; (ii) s is triangular α-admissible with respect to η; (iii) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0); (iv) (b) holds. then s has a unique fixed point z ∈ x. now, we give an example to illustrate theorem 3.2 as follows: example 3.15. let x = {1, 2, 3} and define a mapping p : x ×x → [0,∞) by p(1, 2) = p(2, 1) = 3 7 , p(2, 3) = p(3, 2) = 4 7 , p(1, 3) = p(3, 1) = 5 7 , p(1, 1) = 1 10 , p(2, 2) = 2 10 , p(3, 3) = 3 10 . define a function θ : (0,∞) → (1,∞) by θ(x) = 1 + x for all x ∈ x. it is easy to check that p is a partial metric. define a function α : x ×x → [0,∞) by α(x,y) = { 1, if x,y ∈ x, 0, otherwise, define two mappings s,t : x → x by s(x) = 1, t(1) = t(3) = 1, t(2) = 3 for all x ∈ x and define a function β : [0,∞) → [0, 1) by β(m(x,y)) = 9 10 for all x,y ∈ x. since α(x,y) = 1 and α(sx,ty) = 1 for all x,y ∈ x, the pair (s,t) is α-admissible. now, we show that the condition (3.1) holds. if x = 2 and y = 3, then α(2, 3) = 1 and m(2, 3) = max{p(2, 3),p(2,s(2)),p(3,t(3))} = max{p(2, 3),p(2, 1)),p(3, 1)} = max { 4 7 , 3 7 , 5 7 } = 5 7 c© agt, upv, 2017 appl. gen. topol. 18, no. 1 166 fixed points of α-θ-geraghty type contractions and so α(2, 3)θ(p(s(2),t(3)) = 1 ·θ(p(1, 1)) = θ ( 1 10 ) = 1 + 1 10 = 11 10 . now, if we choose k = 1 2 ∈ (0, 1), then we have [θ(β(m(2, 3))m(2, 3)]k = [ θ ( 9 10 · 5 7 )]1/2 = [ θ ( 9 14 )]1/2 = ( 1 + 9 14 )1/2 = ( 23 14 )1/2 . therefore, we have 11 10 = α(2, 3)θ(p(s(2),t(3)) ≤ [θ(β(m(2, 3))m(2, 3)]k = ( 23 14 )1/2 . similarly, for other cases, it is easy to check that the condition (3.1) holds. therefore, all the conditions (i)-(iii) of theorem 3.2) are satisfied. further, s and t have a unique common fixed point and 1 is a unique common fixed point of s and t. 4. applications following the results of jachymski [15], let (x,p) be a partial metric space and ∆ denotes the diagonal of the cartesian product x×x. consider a directed graph g such that v (g) the set of vertices coincides with x and e(g) the set of edges contains all loops. suppose that g has no parallel edges. then we can analyze g with the pair (v (g),e(g)). if x and y are vertices in g, then a path in g from x to y of length l is a sequence {xn}li=0 of (l + 1) vertices such that x0 = x,xl = y and (xi−1,xi) ∈ e(g) for each i = 1, 2, . . . , l. a graph g is said to be connected if there exists a path between any two vertices. definition 4.1 ([15]). a mapping t : x → x is called the banach gcontraction or, simply, g-contraction if t preserves edge of g, i.e., for all x,y ∈ x, (4.1) (x,y) ∈ e(g) =⇒ (tx,ty) ∈ e(g) and t decreases weights of edges of g in the following way: there exists α ∈ (0, 1) such that, for all x,y ∈ x, (4.2) (x,y) ∈ e(g) =⇒ d(tx,ty) ≤ αd(x,y). definition 4.2 ([15]). a mapping t : x×x is said to be g-continuous if, for any x ∈ x and a sequence {xn} with xn → x as n → ∞, (xn,xn+1) ∈ e(g) for all n ∈ n implies txn → tx as n →∞. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 167 w. onsod, p. kumam and y. j. cho definition 4.3. let (x,p) be a partial metric space endowed with a graph g and t : x → x be a self-mapping. t is called the θ-geraghty graphic type contraction if there exist θ ∈ θ̃, k ∈ (0, 1) and β ∈ f such that (4.3) θ(pg(tx,ty)) ≤ [θ(β(m(x,y))m(x,y)]k for all x,y ∈ x, where, m(x,y) = max{pg(x,y),pg(x,tx),pg(y,ty)}. from theorem 3.2, we have the following: theorem 4.4. let (x,p) be a complete partial metric space endowed with a graph g. t : x → x is self-mapping satisfying the following conditions: (i) (x,y) ∈ e(g) =⇒ (tx,ty) ∈ e(g) for all (x,y) ∈ x; (ii) there exists x0 ∈ x such that (x0,tx0) ∈ e(g); (iii) t is g-continuous on (x,p); (iv) t is θ-geraghty graphic type contraction. then t has a unique fixed point z ∈ x. proof. define a function α : x ×x → [0,∞) by α(x,y) = { 1, if (x,y) ∈ e(g), 0, otherwise for all x,y ∈ x. now, we prove that t is α-admissible. let x,y ∈ x such that α(x,y) ≥ 1. then, by the definition of α and the condition (i), we have (x,y) ∈ e(g) and (tx,tx) ∈ e(g). so, we have α(tx,ty) ≥ 1. therefore, t is α-admissible. from the condition (ii), there exists x0 ∈ x such that (x0,tx0) ∈ e(g), that is, α(x0,tx0) ≥ 1 and, from the condition (iv), t is θ-geraghty graphic type contraction. since α(x,y) ≥ 1, we have α(x,y)θ(pg(tx,ty)) ≤ [θ(β(m(x,y))m(x,y)]k. thus all the conditions of theorem 3.2 are satisfied and so t has a unique fixed point in x. this completes the proof. � now, we give an example to illustrate theorem 4.4 as follows: example 4.5. let x = {1, 2, 3} be endowed with the function p : x ×x → [0,∞) defined by p(1, 2) = p(2, 1) = 3 7 , p(2, 3) = p(3, 2) = 4 7 , p(1, 3) = p(3, 1) = 1 7 , p(1, 1) = 1 20 , p(2, 2) = 2 20 , p(3, 3) = 3 20 . define a function θ : (0,∞) → (1,∞) by θ(x) = 1 + x for all x ∈ x. it is easy to check that p is a partial metric. define a mapping t : x → x by t(1) = t(3) = 1, t(2) = 3 and define a function β : [0,∞) → [0, 1) by β(m(x,y)) = 9 10 for all x,y ∈ x. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 168 fixed points of α-θ-geraghty type contractions let g be a direct graph such that v (g) = x and e(g) = {(x,y) : x,y ∈ {1, 2, 3}}. it is easy to show that t preserves edges in g and t is g-continuous. also, there exists x0 = 1 ∈ x such that (1,t1) = (1, 1) ∈ e(g). with out loss of generality, let x,y ∈ x such that x 6= y. now, we show that the condition (4.3) holds. consider the following cases: case i. if x = 1 and y = 2, then we have θ(p(t(1),t(2)) ≤ [θ(β(m(1, 2))m(1, 2)]1/2 θ(p(1, 3)) ≤ [ θ ( 9 10 · 4 7 )]1/2 θ ( 1 7 ) ≤ [ θ ( 18 35 )]1/2 8 7 ≤ ( 53 35 )1/2 . case ii. if x = 2 and y = 3, then we have θ(p(t(2),t(3)) ≤ [θ(β(m(2, 3))m(2, 3)]1/2 θ(p(3, 1)) ≤ [ θ ( 9 10 · 4 7 )]1/2 θ ( 1 7 ) ≤ [ θ ( 18 35 )]1/2 8 7 ≤ ( 53 35 )1/2 . case iii. if x = 3 and y = 1, then we have θ(p(t(3),t(1)) ≤ [θ(β(m(3, 1))m(3, 1)]1/2 θ(p(1, 1)) ≤ [ θ ( 9 10 · 1 7 )]1/2 θ ( 1 20 ) ≤ [ θ ( 9 70 )]1/2 21 20 ≤ ( 79 70 )1/2 . the following figure represents the graph with all the possible cases. therefore, all the conditions of theorem 4.4 are satisfied and z = 1 is a fixed point of t . c© agt, upv, 2017 appl. gen. topol. 18, no. 1 169 w. onsod, p. kumam and y. j. cho 1 2 3 p(1, 2) = 3/7 p(2, 3) = 4/7 p(3, 1) = 1/7 figure : graph g defined in example 4.5 acknowledgements. the first author would like to thank the research professional development project under the science achievement scholarship of thailand (sast) for the master’s degree program at kmutt. this project was supported by the theoretical and computational science (tacs) center under computational and applied science for smart innovation research cluster (classic), faculty of science, kmutt. references [1] t. abdeljawad, meir-keeler α-contractive fixed and common fixed point theorems, fixed point theory appl. 19 (2013). 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[27] b. samet, c. vetro and p. vetro, fixed point theorems for α-ψ-contractive type mappings, nonlinear anal. 75 (2012), 2154–2165. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 171 () @ appl. gen. topol. 17, no. 2(2016), 93-104doi:10.4995/agt.2016.4154 c© agt, upv, 2016 induced dynamics on the hyperspaces puneet sharma department of mathematics, indian institute of technology jodhpur, india (puneet.iitd@yahoo.com) abstract in this paper, we study the dynamics induced by finite commutative relation. we prove that the dynamics generated by such a non-trivial collection cannot be transitive/super-transitive and hence cannot exhibit higher degrees of mixing. as a consequence we establish that the dynamics induced by such a collection on the hyperspace endowed with any admissible hit and miss topology cannot be transitive and hence cannot exhibit any form of mixing. we also prove that if the system is generated by such a commutative collection, under suitable conditions the induced system cannot have dense set of periodic points. in the end we give example to show that the induced dynamics in this case may or may not be sensitive. 2010 msc: 37b20; 37b99; 54c60; 54h20. keywords: hyperspace; combined dynamics; relations; induced map; transitivity; super-transitivity; dense periodicity. 1. introduction 1.1. motivation: dynamical systems were introduced to investigate different physical and natural phenomenon occurring in nature. using the theory of dynamical systems, mathematical models for various physical/natural phenomenon were developed and long term behavior of the natural phenomenon/systems were investigated. while logistic maps were used to develop the population model for any species, lorentz system of differential equations was used for developing mathematical models for weather predictions. since then, dynamical systems (both discrete and continuous) have found applications in various branches of science and engineering and various phenomenon occurring in a variety of disciplines have been investigated. in some of the recent received 09 october 2015 – accepted 21 july 2016 http://dx.doi.org/10.4995/agt.2016.4154 p. sharma studies, it has been observed that many systems observed in different branches of science and engineering can be investigated using set-valued dynamics (c.f. [6, 7, 15, 17]). while [15] used set-valued dynamics to study handwheel force feedback for lanekeeping assistance, [17] used set-valued dynamics to investigate the collective dynamics of an electron and a nuclei. these examples suggest that the dynamics of different systems evolving in various disciplines of science and engineering can be modeled using set-valued dynamics. thus, it is important to study the set-valued dynamics induced by a continuous self map which inturn can help characterizing the dynamics of a general dynamical system. many researchers have addressed the problem and many of the questions in this direction have been answered [1, 8, 11, 13, 14, 16]. in the process, the dynamical behavior of a system and its corresponding set-valued counterpart has been investigated and several interesting results have been obtained. in [1, 16], authors proved that while weakly mixing and topological mixing on the two spaces are equivalent, transitivity on the base space need not imply transitivity on the hyperspace. interesting results relating the topological entropy of the two spaces have been obtained [8]. in [13], sharma and nagar investigated some of the natural questions arising from this setting. they investigated the influence of each of the individual units and the role of the underlying hyperspace topology in determining the dynamics induced by a continuous self map on a general topological space. in the process they investigated properties like dense periodicity, transitivity, weakly mixing and topological mixing. they also investigated notions like sensitive dependence on initial conditions, topological entropy, li-yorke chaos, existence of li-yorke pairs, existence of horseshoe and the corresponding results were established. investigating the inverse of the problem stated, they also investigated the behavior of an individual component of the system, given the dynamical behavior of the induced system on the hyperspace [13, 14]. generalizing the stated problem, nagar and sharma [11] investigated the dynamics induced by a finite collection of continuous self maps. they derived necessary and sufficient conditions for the induced collective dynamics to exhibit various dynamical notions. they introduced the notion of supertransitivity, super-weakly mixing and super-topological mixing for investigating the dynamics induced on the hyperspace. they proved that super-transitivity of a relation is necessary to induce transitivity on the hyperspace. they proved that for any finite relation f on the space x, the induced map on the hyperspace k(x) is weakly mixing (topologically mixing) if and only if the relation f is super-weakly mixing (super-topologically mixing). however, the existence of such systems was left open and some natural questions were raised. when does a system induced by a non-trivial family (family of more than one map) exhibit transitivity? when does the dynamics induced by such a family exhibit stronger forms of mixing? can the dynamics induced by such a system exhibit dense set of periodic points? in this paper, we try to answer some of the questions raised in [11]. we now give some of the preliminaries needed to establish our results. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 94 induced dynamics on the hyperspaces 1.2. dynamics of a relation. let (x, d) be a compact metric space and let f = {f1, f2, . . . , fk} be a finite collection of continuous self maps on x. the pair (x, f) generates a multi-valued dynamical system via the rule f(x) = {f1(x), f2(x), . . . , fk(x)}. for convenience, we denote such systems by (x, f). such a system generalizes the concept of the dynamical system generated by a single map f. the objective of a study of dynamical system is to study the orbit {f n(x) : n ∈ n} of an arbitrary point x, where f n(x) = {fi1(x) ◦ fi2(x) ◦ . . . ◦ fin(x) : 1 ≤ i1, i2, . . . , in ≤ k} is the n-fold composition of f . we now define some of the basic dynamical notions for such a system. a point x is called periodic for if there exists n ∈ n such that x ∈ f n(x). the least such n is known as the period of the point x. the relation f is transitive if for any pair of non-empty open sets u, v in x, there exists n ∈ n such that f n(u) ∩ v 6= φ. the relation f is super-transitive if for any pair of non-empty open sets u, v in x, there exists x ∈ u, n ∈ n such that f n(x) ⊂ v . the relation f is said to be weakly mixing if for every two pair of nonempty open sets u1, u2 and v1, v2, there exists a natural number n such that f n(ui) ⋂ vi 6= φ, i = 1, 2. the relation f is said to be super-weakly mixing if for every two pair of non-empty open sets u1, u2 and v1, v2, there exists xi ∈ ui and a natural number n such that f n(xi) ⊆ vi, i = 1, 2. the relation f is said to be topologically mixing if for every pair of non-empty open sets u, v there exists a natural number k such that f n(u) ⋂ v 6= φ for all n ≥ k. the relation f is said to be super-topologically mixing if for every pair of non-empty open sets u, v there exists k ∈ n such that for each natural number n ≥ k, there exists xn ∈ u such that f n(xn) ⊆ v . a relation f is sensitive if there exists a δ > 0 such that for each x ∈ x and each ǫ > 0, there exists n ∈ n and y ∈ x such that d(x, y) < ǫ but dh(f n(x), f n(y)) > δ. it may be noted that as f n(x) and f n(y) are subsets (and need not be elements) of x, metric d cannot be used to measure the distance between them. as f n(x), f n(y) are elements in the hyperspace and the metric dh is a natural extension of the metric d, dh is used to compute the distance between any f n(x) and f n(y) (refer to sec. 1.3 for the details). incase the relation f is map, the above definitions coincide with the known dynamical notions of a system. see [3, 4, 5, 11] for details. 1.3. some hyperspace topologies. let (x, τ) be a hausdorff topological space and let ψ be a subfamily of all non-empty closed subsets of x. let ψ be endowed with topology ∆, where the topology ∆ is generated using the topology τ of x. then the pair (ψ, ∆) is called the hyperspace associated with (x, τ). a hyperspace topology is called admissible if the map x → {x} is continuous. in this paper, we are interested in only admissible hyperspaces (ψ, ∆) associated with (x, τ). we now give some of the notations and terminologies used in the article. cl(x) = {a ⊂ x : a is non-empty and closed} k(x) = {a ∈ cl(x) : a is compact} f(x) = {a ∈ cl(x) : a is finite} c© agt, upv, 2016 appl. gen. topol. 17, no. 2 95 p. sharma fn(x) = {a ∈ cl(x) : |a| = n, where |a| denotes number of elements in a} e− = {a ∈ ψ : a ∩ e 6= φ} e+ = {a ∈ ψ : a ⊂ e} e++ = {a ∈ ψ : ∃ ǫ > 0 such that sǫ(a) ∩ e}, where sǫ(a) = ⋃ a∈a s(a, ǫ), where sǫ(x) = {y ∈ x : d(x, y) < ǫ} we now give some of the standard hyperspace topologies. vietoris topology: for any n ∈ n and any finite collection of non-empty open sets {u1, u2, . . . , un}, define < u1, u2, . . . , un >= {a ∈ ψ : a ⊂ n⋃ i=1 ui, a ⋂ ui 6= φ ∀i} varying n ∈ n and ui over the collection of all non-empty open sets of x generates a basis for a topology on the hyperspace known as the vietoris topology. hausdorff metric topology: let (x, d) be a metric space. for any a, b ∈ ψ, define dh(a, b) = inf{ǫ > 0 : a ⊆ sǫ(b) and b ⊆ sǫ(a)}. then dh defines a metric on ψ and the topology generated is known as the hausdorff metric topology. it may be noted that dh({x}, {y}) = d(x, y) and hence the metric dh preserves the metric d on x. it is known that the hausdorff metric topology and the vietoris topology coincide, incase x is a compact metric space. see [2, 12] for details. hit and miss topology: let φ ⊆ cl(x) be a subfamily of all non-empty closed subsets of x. the hit and miss topology generated by the family φ is the topology generated by sets of the form u− where u is open in x, and (ec)+ with e ∈ φ, where ec denotes the complement of e. as a terminology, u is called the hit set and any member e of φ is referred as the miss set. hit and far-miss topology: let (x, d) be a metric space and let φ be a given collection of closed subsets of x. the hit and far miss topology generated by the collection φ is the topology generated by the sets of the form u− where u is open in x and (ec)++ with e ∈ φ. here a sub-basic open set in the hyperspace hits an open set u ⊂ x or far misses the complement of a member of φ and hence forms a hit and far miss topology. it is known that any topology on the hyperspace is of hit and miss or hit and far-miss type [12]. lower and upper vietoris topology: consider the collection of sets of the form u− in the hyperspace, where u is non-empty open set in x. the smallest topology on the hyperspace in which all the sets of the form u− considered are open is known as the lower vietoris topology. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 96 induced dynamics on the hyperspaces consider the collection of sets of the form u+ in the hyperspace, where u is non-empty open set in x. the smallest topology on the hyperspace in which all the sets of the form u+ considered are open is known as the upper vietoris topology. it can be seen that the vietoris topology equals the join of upper vietoris and lower vietoris topology, and is an example of a hit and miss topology. a detailed survey on the hyperspace topologies may be found in [2, 9, 10, 12]. 1.4. dynamics induced by a relation. let (x, f) be a dynamical system generated by a finite family of continuous self maps on x, say {f1, f2, . . . , fk}. for any ψ ⊂ cl(x), the collection ψ is said to be admissible with respect to f if f(a) (= k⋃ i=1 fi(a)) ∈ ψ for all a ∈ ψ. it may be noted that any collection ψ admissible to f generates a map f on ψ via the rule f(a) = f(a). consequently, endowing ψ with any suitable hyperspace topology (such that the map f is continuous) generates a dynamical system on the hyperspace. it is interesting to investigate the relation between the dynamical behavior of (x, f) and the induced system (ψ, f). a special case when the family f is a singleton has been investigated by several authors and a lot of work in this direction has already been done[1, 8, 11, 13, 14, 16]. it was proved that while weakly mixing and topological mixing on the two spaces are equivalent, transitivity on the base space need not imply transitivity on the hyperspace[1, 16]. while [8] investigated the topological entropy of induced system of all nonempty compact subsets of x, [13] investigated the problem for a general hyperspace endowed with a general hyperspace topology. they discussed various dynamical notions like dense periodicity, transitivity, weakly mixing, topological mixing and topological entropy. they also investigated metric related dynamical notions like equicontinuity, sensitivity, strong sensitivity and li-yorke chaoticity[13, 14]. authors extended their studies to the general case when the dynamics on x is generated by a finite family and derived the necessary and sufficient conditions for the induced system to exhibit various dynamical notions[11]. in the process, they discussed properties like dense periodicity, transitivity, weakly mixing and topological mixing. they introduced the notion of super-weakly mixing and super-topological mixing for relations to study the induced maps on the hyperspace. authors proved that for any finite relation f on the space x, the induced map on the hyperspace k(x) is weakly mixing (resp. topologically mixing) if and only if the relation f is super-weakly mixing (resp. super-topologically mixing). however, the existence of any such collection of maps was not confirmed and the problem was left open. for the sake of completion, we mention some of their results below. theorem 1.1 ([11]). let β be any base for the topology on x and ∆ be the topology on ψ ⊆ cl(x) such that u+ is non-empty and u+ ∈ ∆ for every c© agt, upv, 2016 appl. gen. topol. 17, no. 2 97 p. sharma u ∈ β. then, the system (ψ, f) is transitive implies that the relation (x, f) is super-transitive. theorem 1.2 ([11]). let f(x) ⊆ ψ. if f is super-weakly mixing, then f is weakly mixing. the converse holds if there exists a base β for topology on x such that u+ ∈ ∆ for every u ∈ β. theorem 1.3 ([11]). let f(x) ⊆ ψ. if f is super-topologically mixing, then f is topologically mixing. the converse holds if there exists a base β for topology on x such that u+ ∈ ∆ for every u ∈ β. in this paper, we answer some of the questions raised in [11] when x is a compact metric space. we prove that the dynamics generated by any such commutative family cannot be transitive/super-transitive and hence cannot generate any of the complex mixing notions on the hyperspace. we also prove that dynamics induced by such a collection on the hyperspace cannot exhibit dense set of periodic points. in the end we give example to prove that the dynamics generated by such family may be sensitive. 2. main results lemma 2.1. let (x, f) be a dynamical system generated by a finite commutative family of continuous self maps on x. then, f is super-transitive if and only if for each non-empty open set u and any point x ∈ x, there exists u ∈ u and a sequence (ni) in n such that f ni(u) → {x} in (k(x), dh). proof. let f = {f1, f2, . . . , fk} and let d be a compatible metric on x. let u be a non-empty open subset of x and let v1 = s1(x). as f is supertransitive, there exists x1 ∈ u and n1 ∈ n such that f n1(x1) ⊂ v1. as f n1(x) = {fi1 ◦fi2 ◦. . .◦fin1 : 1 ≤ i1, i2, . . . , in1 ≤ k} and each fi1 ◦fi2 ◦. . .◦fin1 is continuous (and are finitely many maps), there exists a neighborhood u1 of x1 such that u1 ⊂ u and f n1(u1) ⊂ v1. let v2 = s 1 2 (x). as f is super-transitive (applying transitivity to u1 and v2), there exists x2 ∈ u1 and n2 ∈ n such that f n2(x2) ⊂ v2. as f n2(x) = {fi1 ◦ fi2 ◦ . . . ◦ fin2 : 1 ≤ i1, i2, . . . , in2 ≤ k} and each fi1 ◦ fi2 ◦ . . . ◦ fin2 is continuous (and number of maps are finite), there exists a neighborhood u2 of x2 such that u2 ⊂ u1 and f n2(u2) ⊂ v2. inductively, let ur, vr of non-empty open sets in x such that ur ⊂ ur−1, vr = s 1 r (x) and f nr (ur) ⊂ vr. let vr+1 = s 1 r+1 (x). as f is super-transitive (applying super-transitivity to the pair (ur, vr+1)), there exists xr+1 ∈ ur and nr+1 ∈ n such that f nr+1(xr+1) ⊂ vr+1. one again, as f nr+1(x) = {fi1 ◦fi2 ◦. . .◦finr+1 : 1 ≤ i1, i2, . . . , inr+1 ≤ k} and each fi1 ◦fi2 ◦. . .◦finr+1 is continuous (and number of maps are finite), there exists a neighborhood ur+1 of xr+1 such that ur+1 ⊂ ur and f nr+1(ur+1) ⊂ vr+1. consequently, we obtain a nested sequence of open sets (ur) contained in u such that dh(f nr (u), x) < 1 r for any u ∈ ur. as ui is a decreasing sequence of non-empty compact subsets of x, a = ∩ui ⊂ u is non-empty. let u ∈ a, then c© agt, upv, 2016 appl. gen. topol. 17, no. 2 98 induced dynamics on the hyperspaces u ∈ ur and hence f nr (u) ⊂ vr for all r. consequently, dh(f nr (u), x) < 1 r for all r and hence f ni(u) → {x}. conversely, let u, v be a pair of non-empty open sets ion x. for any {v} ∈< v >, there exists u ∈ u and a sequence (ni) in n such that f ni(u) → {v}. consequently there exists r ∈ n such that f nk (u) ⊂ v ∀ k ≥ r and hence f is super-transitive. � remark 2.2. it may be noted that transitivity of a system generated by single map f is equivalent to the existence of a dense orbit. consequently, the above result is trivially true when the family f is a singleton. thus the result above is a generalization of the known result to the case when the system is generated using more than one map. it may be noted that f n is a union of repeated application of the maps {f1, f2, . . . , fk} (n times) in all possible orders and continuity of each component map of f n guarantees extension of a behavior at a point to a similar behavior in the neighborhood of the point. further, as the number of maps constituting f n(x) is finite at each iteration, a neighborhood exhibiting similar behavior for all components of f n is ensured and hence the existence of ui is guaranteed. it is worth mentioning that the commutativity of the family f is not used for establishing the result. hence the result is true even when the generating family f is non-commutative. proposition 2.3. let (x, f) be a dynamical system generated by a finite commutative family of continuous self maps on x. if f is super-transitive, then f is a singleton. proof. let u be an non-empty open set in x and ǫ > 0 be a real number. let v be a non-empty open set and let v ∈ v . by lemma, there exists u ∈ u and a sequence (ni) such that f ni(u) → v. as each fi is continuous, there exists δ > 0 such that d(x, y) < δ implies d(fi(x), fi(y)) < ǫ 2 for all i = 1, 2, . . . , n. as f ni(u) → v, for any two distinct elements f, g of f , fni(u) → v and gni(u) → v. further, as d(f ni(u)) → 0 (where d(a) denotes the diameter of the set a), there exists r ≥ 1 such that the relation d(fni(u), gni(u)) < δ and d(f ◦ gni−1(u), gni(u)) < δ is true for all i ≥ r. consequently, d(fni+1(u), f ◦ gni(u)) < ǫ 2 and d(f ◦ gni(u), gni+1(u)) < ǫ 2 for all i ≥ r. using triangle inequality we get d(fni+1(u), gni+1(u)) < ǫ for all i ≥ r. consequently, (fni+1(u)), (gni+1(u)) are parallel sequences and hence have the same limit(say y). also fni(u) → v and gni(u) → v implies fni+1(u) → f(v) and gni+1(u) → g(v). consequently, f(v) = g(v). as the argument holds for any open set v , the points at which f and g coincide is dense in x. hence f = g. � remark 2.4. the above proof establishes that the system induced by more than one map cannot be super-transitive. the proof establishes that under stated conditions, if (fni(x)) and (gni(x)) are parallel then (fni+1(x)) and (gni+1(x)) c© agt, upv, 2016 appl. gen. topol. 17, no. 2 99 p. sharma are also parallel and hence have the same limit. further as fni(x), gni(x) converge to v, f(v) and g(v) are unique limit points of (fni+1(x)) and (gni+1(x)) respectively. consequently f and g coincide on a dense set and hence are equal. remark 2.5. the above result proves that if the dynamics on x is generated by more than one map then the system cannot be super-transitive and hence cannot exhibit any stronger forms of mixing. further as super-transitivity of the system (x, f) is necessary for any induced admissible system (ψ, f) to be transitive [11], the system induced on the hyperspace by a family of more than one map cannot be transitive and hence cannot exhibit stronger notions of mixing. in light of the remark stated, we obtain the following corollary. corollary 2.6. let (x, f) be a dynamical system generated by a finite commutative family of continuous self maps on x. let (ψ, ∆) be the associated hyperspace and let f be the corresponding induced map. if the family f contains more than one map then f cannot be transitive. we now show that the dynamics induced by a commutative family on the hyperspace cannot exhibit dense set of periodic points. proposition 2.7. let (x, f) be a dynamical system generated by a finite commutative family of continuous self maps on x and let (k(x), f ) be the induced system endowed with the vietoris topology on the hyperspace. if (k(x), f) exhibits dense set of periodic points then f is a singleton. proof. let if possible, the induced map f exhibit dense set of periodic points. let x ∈ x be arbitrary and let ǫ > 0 be given. for any two members f and g of f , as f and g are continuous on a compact set, they are uniformly continuous, i.e. there exists δ > 0 such that whenever d(x, y) < δ, we have d(f(x), f(y)) < ǫ 3 and d(g(x), g(y)) < ǫ 3 . let u = s δ 2 (x). as is open in the hyperspace and f has dense set of periodic points, there exists a ∈ and n ∈ n such that f n (a) = a. let a ∈ a. as f n (a) = a ⊂ u we have d(x, fn(a)) < δ 2 . consequently, we have d(f(x), fn+1(a)) < ǫ 3 and d(g(x), g ◦fn(a)) < ǫ 3 . also f n (a) = a ⊂ u, implies d(fn(a), g ◦ fn−1(a)) < δ and hence d(fn+1(a), g ◦ fn(a)) < ǫ 3 (as f is commutative). using triangle inequality we have, d(f(x), g(x)) ≤ d(f(x), fn+1(a)) + d(fn+1(a), g ◦ fn(a)) + d(g ◦ fn(a), g(x)) < ǫ. as ǫ > 0 was arbitrary, d(f(x), g(x)) = 0 which implies f(x) = g(x). as the proof holds for any x ∈ x, any two members of the family f coincide and hence f is a singleton. � remark 2.8. the result establishes that if the dynamics on the space x is generated by more than one map then the induced dynamics on the hyperspace cannot exhibit dense set of periodic points. the proof uses the openness of the sets of the form , where u is non-empty open in x and does utilize the complete structure of the vietoris topology. further, the proof does not c© agt, upv, 2016 appl. gen. topol. 17, no. 2 100 induced dynamics on the hyperspaces utilize the structure of the hyperspace k(x) and holds good for any general admissible hyperspace (ψ, ∆). hence the induced map cannot have dense set of periodic points in this case and the result is true for the induced system when the admissible hyperspace is endowed with topology finer than the upper vietoris topology. in light of the remark stated, we get the following corollary. corollary 2.9. let (x, f) be a dynamical system generated by a finite commutative family of continuous self maps on x and let (ψ, f) be the induced system endowed with any hyperspace topology finer than the upper vietoris topology. if the family f contains more than one map then f cannot exhibit dense set of periodic points. remark 2.10. the above corollary establishes the generalization of the proved result to the system induced on a more general admissible hyperspace. further, it is worth mentioning that if any of the members of f exhibit dense set of periodic points, then the system (x, f) exhibits dense set of periodic points as fn(x) ∈ f n(x) for any member f of f . hence if the dynamics on x is induced by more than one function, the system may exhibit dense set of periodic points but the induced system can not exhibit dense set of periodic points which is contrary to the case when f is a singleton. in the light of the remark stated, we get the following corollary. corollary 2.11. let (x, f) be a dynamical system and let (ψ, f) be the corresponding induced system. then, (x, f) has dense set of periodic points ; (ψ, f) has dense set of periodic points. remark 2.12. the above corollary shows that there can exist dynamical systems (x, f) with the dense set of periodic points such that the induced system does not have dense set of periodic points. such a scenario happens when the dynamics on x is induced by more than one map. such a behavior for the induced system is due to the fact that although the dynamics on the space x is generated by a finite commutative relation f , the dynamics on the hyperspace is generated by a function and hence conventional methodology for investigating the dynamics on the hyperspace are to be used. in particular, x is periodic for f if there exists n ∈ n such that x ∈ f n(x) but a is periodic for f if there exists n ∈ n such that a = f n (a) (= f n(a)). consequently, there is a significant difference in the basic dynamical notions of the system which induces such a contrasting behavior on the hyperspace. we now give an example in support of our argument. example 2.13. let x = s1 be the unit circle and let f1 and f2 be the rational rotations on unit circle defined as f1(θ) = θ + p and f2(θ) = θ + q respectively. let f = {f1, f2} be the commutative relation and let (x, f) be the corresponding dynamical system generated. as fi is rotation on the unit circle by a rational multiple of 2π, fi exhibits dense set of periodic points. further as fni (x) ∈ f n(x) for all i and n, f exhibits dense set of periodic points. however, as each fi is a rotation on the unit circle (by different angles), for any set a 6= x f n(a) = a never holds good and hence the hyperspace has a unique c© agt, upv, 2016 appl. gen. topol. 17, no. 2 101 p. sharma periodic point x. consequently, the example shows that there exists dynamical systems (x, f) with the dense set of periodic points such that the induced system does not have dense set of periodic points. remark 2.14. the above example proves that the induced dynamics cannot exhibit dense set of periodic points when induced by a commutative family of more than one map. on similar lines, considering fi (i = 1, 2) to be irrational rotations on the unit circle gives an example of a relation where each component is transitive but the relation is not super-transitive. the relation is not supertransitive as f1 and f2 are isometries and hence for each natural number 2n (or 2n + 1) fn1 ◦ f n 2 and f1 ◦ f 2n−1 2 (or f n 1 ◦ f n 2 and f1 ◦ f 2n 2 ) cannot be pushed inside an open set of arbitrary arc length. consequently, the dynamics induced on the hyperspace by the family considered is not super-transitive. remark 2.15. the above results establish that when the dynamics on x is generated by a commutative family of more than one function, the dynamics on the hyperspace cannot be super-transitive and cannot have dense set of periodic points. the commutativity of the family f plays an important role in proving the result and hence cannot be dropped. in absence of commutativity, the order of f and g in any member of f n(x) cannot be altered and the proof of the result does not hold good. however under non-commutativity, g ◦ fn(x) and f ◦g◦fn−1(x) (and others where fi’s are applied same number of times but in different order) do not coincide and hence f n(x) contains more elements. consequently, it is expected that as super-transitivity (or dense periodicity) does not hold under commutativity, it will not hold in the non-commutative case (as f n(x) contains more elements). however, any proof for the belief is not available and hence is left open. we now give an example to show, that the induced dynamics, when induced by more than one function, might be sensitive. example 2.16. let σ2 be the sequence space of all bi-finite sequences of two symbols 0 and 1. for any two sequences x = (xi) and y = (yi), define d(x, y) = ∞∑ i=−∞ |xi−yi| 2|i| it is easily seen that the metric d generates the product topology on ∑ 2 . let σ : σ2 → σ2 be defined as σ(. . . x−2x−1.x0x1 . . .) = . . . x−2x−1x0.x1x2x3 . . . the map σ is known as the shift map and is continuous with respect to the metric d defined. let f = {σ, σ2} and let f be the corresponding induced map on the hyperspace. we claim that the induced map is sensitive on k(x). equivalently, it is sufficient to show that the map is sensitive on a dense subset of k(x). let a = {x1, x2, . . . , xk} be a finite set where each xi = (. . . x −2 i x −1 i .x 0 i x 1 i . . . x n i . . .) is an element in σ and let ǫ > 0 be given. let r ∈ n such that 1 2r < ǫ. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 102 induced dynamics on the hyperspaces let yi = (. . . x −2 i x −1 i .x 0 i . . . x r+1 i 000 . . .) and zi = (. . . x −2 i x −1 i x 0 i . . . x r+1 i 111 . . .). then b = {y1, y2, . . . , yk} and c = {z1, z2, . . . , zk} are elements in s(a, ǫ) such that f r+2(yi) = (. . . x −2 i x −1 i . . . x r i .0000 . . .) and f r+2(zi) = (. . . x −2 i x −1 i . . . x r i .11111 . . .). consequently, dh(f r+2 (b), f r+2 (c)) ≥ 1 and hence f is sensitive at a. as the proof holds for any finite subset s of sigma2, f is sensitive at finite subsets of σ. consequently, f is sensitive on σ. 3. conclusion the paper discusses the dynamics of the induced function on the hyperspace, when the function is induced by a non-trivial family of commuting continuous self maps on x. it is observed that the dynamics is contrary to the case when the map on the hyperspace is induced using a single function. while the map induced by single function can exhibit complex dynamical behavior (for example weakly mixing or topological mixing), the dynamics induced by a collection of two or more commuting maps cannot even be transitive and hence cannot exhibit any of the higher notions of mixing. further, it is established that the dynamics induced by such a family cannot have dense set of periodic points. this once again is a contrary to the case when the map is induced by a single function, as the map induced in that case always has dense set of periodic points, if the original system has dense set of periodic points. we also give an example to show that the dynamics induced by a commutative family may be sensitive. it is worth mentioning that although the problem has been solved when the dynamics on the hyperspace is induced by a commutative family, the non-commutative case is still open for investigation. as f n(x) contains more points in the non-commutative case, similar results are expected to hold. however as the proof derived does not work for non-commutative case, we leave it open for further investigation. references [1] j. banks, chaos for induced hyperspace maps, chaos, solitons and fractals 25 (2005), 681–685. [2] g. beer, topologies on closed and closed convex sets, kluwer academic publishers, dordrecht/boston/london (1993). [3] l. block and w. coppel, dynamics in one dimension, springer-verlag, berlin hiedelberg (1992). [4] m. brin and g. stuck, introduction to dynamical systems, cambridge unversity press (2002). [5] r. l. devaney, introduction to chaotic dynamical systems, addisson wesley (1986). [6] s. dirren and h. davies, combined dynamics of boundary and interior perturbations in the eady setting, journal of the atmospheric sciences 61 (13) (2004), 1549–1565. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 103 p. sharma [7] r. klaus and p. rohde, fuzzy chaos : reduced chaos in the combined dynamics of several independently chaotic populations, the american naturalist 158, no. 5 (2001), 553–556. [8] d. kwietniak and p. oprocha, topological entropy and chaos for maps induced on hyperspaces, chaos solitions fractals 33 (2007), 76–çô86. [9] e. micheal, topologies on spaces of subsets, trans. amer. math. soc. 71(1951), 152–82. [10] g. di maio and s. naimpally, some notes on hyperspace topologies. ricerche mat. 51, no. 1 (2002), 49–60. [11] a. nagar and p. sharma, combined dynamics on hyperspaces, topology proceedings 38 (2011), 399–410. [12] s. naimpally, all hypertopologies are hit-and-miss, appl. gen. topol. 3, no. 1 (2002), 45–53. [13] p. sharma and a. nagar, topological dynamics on hyperspaces, appl. gen. topol. 11, no. 1 (2010), 1–19. [14] p. sharma and a. nagar, inducing sensitivity on hyperspaces, topology and its applications 157 (2010), 2052–2058. [15] j. p. switkes, e. j. rossettter, i. a. co and j. c. gerdes, handwheel force feedback for lanekeeping assistance: combined dynamics and stability, journal of dynamic systems, measurement and control 128, no. 3 (2006), 532–542. [16] h. roman-flores, a note on transitivity in set valued discrete systems, chaos, solitons and fractals, 17 (2003), 99–104. [17] y. zhao , s. yokojima and g. chen, reduced density matrix and combined dynamics of electron and nuclei, journal of chemical physics 113, no. 10 (2000), 4016–4027. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 104 @ appl. gen. topol. 17, no. 2(2016), 173-183 doi:10.4995/agt.2016.5180 c© agt, upv, 2016 global optimization using α-ordered proximal contractions in metric spaces with partial orders somayya komal a and poom kumam a,b,∗ a department of mathematics, faculty of science, king mongkut’s university of technology thonburi (kmutt),126 pracha uthit rd., bang mod, thung khru, bangkok 10140, thailand (somayya.komal@mail.kmutt.ac.th, poom.kumam@mail.kmutt.ac.th) b theoretical and computational science (tacs) center, science laboratory building, faculty of science king mongkut’s university of technology thonburi (kmutt), 126 pracha uthit rd., bang mod, thung khru, bangkok 10140, thailand (poom.kumam@mail.kmutt.ac.th) abstract the purpose of this article is to establish the global optimization with partial orders for the pair of non-self mappings, by introducing a new type of contractions like α-ordered contraction and α-ordered proximal contraction in the frame work of complete metric spaces. also to calculate some fixed point theorems with the help of these generalized contractions. in addition, to establish an example which shows the validity of our main result. these results extend and unify many existing results in the literature. 2010 msc: 58c30; 47h10. keywords: common best proximity point; global optimal approximate solution; proximally increasing mappings; α-ordered contractions; α-ordered proximal contraction; α-ordered proximal cyclic contraction. 1. introduction it is obvious that best proximity point serves as an optimal approximate solution to the equation zx = x, where z is a non-self mapping from any two non-empty subsets of a metric space, a normed linear space or any other ∗corresponding author: poom.kumam@mail.kmutt.ac.th and poom.kum@kmutt.ac.th received 25 march 2016 – accepted 19 july 2016 http://dx.doi.org/10.4995/agt.2016.5180 s. komal and p. kumam topological space. also it is very interesting point that best proximity point theorems actually generalize the fixed point theorems in natural fashion by taking self mapping instead of non-self mapping in best proximity point theorem then we can get fixed point. since d(x,zx) ≥ d(a,b), for any x ∈ a, we obtain the global minimum of the mapping x 7→ d(x,zx) as a best proximity point. for more details on this approach, we refer the reader to [2], [3], [4], [5], [6], [10], [7], [13], [11], [12], [14], [16], [1] and [15]. the basic purpose of this article is to establish some generalized notions and to derive new theorem of global optimization with partial orders in metric spaces. we have defined in this work an α-ordered contraction to find common best proximity points. the motivation of this paper is [9], we generalized that contraction of [9]. also presented an example to verify the results. 2. preliminaries in this section let us take that a and b are non-void subsets of a metric space (x,d). we recall some definitions and notations in this section which will be used throughout this work. definition 2.1 ([8]). let x be a metric space, a and b two nonempty subsets of x. define d(a,b) = inf{d(a,b) : a ∈ a,b ∈ b}, a0 = {a ∈ a : there exists some b ∈ b such that d(a,b) = d(a,b)}, b0 = {b ∈ b : there exists some a ∈ a such that d(a,b) = d(a,b)}. definition 2.2 ([8]). given non-self mappings s : a → b and t : a → b, an element x∗ is called common best proximity point of the mappings if this condition satisfied: d(x∗,sx∗) = d(x∗,tx∗) = d(a,b). we noticed here that common best proximity point is that element at which both functions s and t attain their global minimum, since d(x,sx) ≥ d(a,b) and d(x,tx) ≥ d(a,b) for all x. definition 2.3 ([9]). a mapping s : a → b is said to be an ordered contraction if there exists a non-negative real number γ < 1 such that x1 � x2 ⇒ d(sx1,sx2) ≤ γd(x1,x2), for all x1,x2 ∈ a. definition 2.4 ([9]). a mapping s : a → b is said to be an ordered proximal contraction if there exists γ < 1 such that x1 � x2, d(u1,sx1) = d(a,b) and d(u2,sx2) = d(a,b), implies that d(u1,u2) ≤ γd(x1,x2), for all u1,u2,x1,x2 ∈ a. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 174 global optimization using α-ordered proximal contractions in metric spaces with partial orders definition 2.5 ([9]). given non-self mappings s : a → b and t : b → a, the pair (s,t) is said to form an ordered proximal cyclic contraction if there exists a non-negative real number k < 1 such that x � y, d(u,sx) = d(a,b) and d(v,ty) = d(a,b), implies that d(u,v) ≤ kd(x,y) + (1 −k)d(a,b), for all u,x ∈ a and v,y ∈ b. definition 2.6 ([9]). given non-self mappings s : a → b and t : b → a, the pair (s,t) is said to be proximally increasing if x � y, d(u,sx) = d(a,b) and d(v,ty) = d(a,b), implies that u ≤ v, for all u,x ∈ a and v,y ∈ b. definition 2.7 ([9]). given non-self mapping s : a → b is said to be proximally increasing if it satisfies the condition: x � y, d(u,sx) = d(a,b) and d(v,sy) = d(a,b), implies that u ≤ v, for all u,v,x,y ∈ a. definition 2.8 ([9]). given non-self mapping s : a → b is said to be increasing if it satisfies the condition: x � y, implies that sx ≤ sy, for all x,y ∈ a. similarly, iteratively snx ≤ sny, for n ∈ n. 3. main results now, we are in position to define some notions and to prove some results. definition 3.1. a mapping s : a → b is said to be an α-ordered contraction if there exists β ∈f and α : x ×x → r+ be a function such that x1 � x2 ⇒ α(x1,x2)d(sx1,sx2) ≤ β(d(x1,x2))d(x1,x2), for all x1,x2 ∈ a. we denote by f the class of all functions β : [0,∞) → [0, 1) satisfying β(tn) → 1, implies tn → 0 as n →∞. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 175 s. komal and p. kumam definition 3.2. a mapping s : a → b is said to be an α-ordered proximal contraction if there exists β ∈f and α : x ×x → r+ such that x1 � x2, d(u1,sx1) = d(a,b) and d(u2,sx2) = d(a,b), implies that α(x1,x2)d(u1,u2) ≤ βd(x1,x2), for all u1,u2,x1,x2 ∈ a. definition 3.3. given non-self mappings s : a → b and t : b → a, the pair (s,t) is said to form an α-ordered proximal cyclic contraction if there exists a non-negative real number k < 1 such that x � y, d(u,sx) = d(a,b) and d(v,ty) = d(a,b), implies that α(x,y)d(u,v) ≤ kd(x,y) + (1 − k)d(a,b), for all u,x ∈ a and v,y ∈ b. theorem 3.4. let x be a non-empty set such that (x,�) is a partially ordered set and (x,d) is a complete metric space, α : x ×x → r+ be a function and let a,b be nonempty closed subsets of (x,d) such that a0 and b0 are nonvoid. let s : a → b, t : b → a and g : a∪b → a∪b satisfy the following conditions: (1) s and t are α-ordered proximal contractions, proximally increasing; (2) g is surjective isometry, its inverse is an increasing mapping; (3) the pair (s,t) is proximally increasing, α-ordered proximal cyclic contraction; (4) s(a0) ⊆ b0, t(b0) ⊆ a0; (5) a0 ⊆ g(a0) and b0 ⊆ g(b0); (6) s and t are α-proximal admissible maps; (7) α(x0,x1) ≥ 1 for x0,x1 ∈ x; (8) there exist elements x0 and x1 in a0 and y0,y1 ∈ b0 such that d(gx1,sx0) = d(a,b), and d(gy1,ty0) = d(a,b). x0 � x1, y0 � y1, x0 � y0. (9) if {xn} is an increasing sequence of elements in a converging to x, then xn � x, for all n. also, if {yn} is an increasing sequence of elements in b converging to y, then yn � y for all n. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 176 global optimization using α-ordered proximal contractions in metric spaces with partial orders then there exists a point x ∈ a and a point y ∈ b such that d(gx,sx) = d(gy,ty) = d(x,y) = d(a,b). moreover, the sequence {xn} in a0, defined by d(gxn+1,sxn) = d(a,b) (n ≥ 1), converges to the element x, and the sequence {yn} in b0, defined by d(gyn+1,tyn) = d(a,b) (n ≥ 1), converges to the element y. proof. since α(x0,x1) ≥ 1 for x0,x1 ∈ x, and for x1 ∈ a0, s(a0) ⊆ b0 there exists x2 ∈ a0 such that d(x2,sx1) = d(a,b), for x2 ∈ a0, s(a0) ⊆ b0 there exists x3 ∈ a0 such that d(x3,sx2) = d(a,b). since s is α-proximal admissible mapping, then from d(x2,sx1) = d(a,b) d(x3,sx2) = d(a,b), implies that α(x2,x3) ≥ 1. proceeding in the same manner, we have α(xn,xn+1) ≥ 1, for n ∈ n. the hypothesis (8) implies the existence of elements x0 and x1 in a0 such that d(gx1,sx0) = d(a,b) and x0 � x1. in view of the fact that s(a0) ⊆ b0, also it is given that a0 ⊆ g(a0), there exists an element x2 ∈ a0 such that d(gx2,sx1) = d(a,b). since s is proximally increasing, gx1 � gx2. as the inverse of mapping g is increasing, so x1 � x2. again, since s(a0) ⊆ b0 and a0 ⊆ g(a0), there exists an element x3 ∈ a0 such that d(gx3,sx2) = d(a,b). continuing in a similar fashion, one can find an element xn in a0 such that d(gxn,sxn−1) = d(a,b) and xn−1 � xn. in light of the fact that g is an isometry and that s is α-ordered proximal contraction, we obtain d(xn,xn+1) = d(gxn,gxn+1) ≤ α(xn−1,xn)d(gxn,gxn+1) ≤ β(d(xn−1,xn))d(xn−1,xn). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 177 s. komal and p. kumam this shows that {d(xn+1,xn)} is a decreasing sequence and bounded below. hence there exists r ≥ 0 such that limn→∞d(xn+1,xn) = r. suppose that r > 0. observed that d(xn+1,xn) d(xn,xn−1) ≤ β(d(xn−1,xn)). taking limit as n →∞, we get lim n→∞ β(d(xn,xn−1)) = 1. since β ∈ f, so that r = 0, which is a contradiction to our supposition and hence (3.1) lim n→∞ d(xn,xn−1) = 0. now, we claim that {xn} is a cauchy sequence. suppose that {xn} is not cauchy sequence. then there exists � > 0 and subsequences {xmk},{xnk} of {xn} such that for any positive integers nk > mk ≥ k rk := d(xmk,xnk ) ≥ �, d(xmk,xnk−1) < �, for any k ∈{1, 2, 3, ...}. for each n ≥ 1, let αn := d(xn+1,xn). then, we have � ≤ rk = d(xmk,xnk ) ≤ d(xmk,xnk−1) + d(xnk−1,xnk ) < � + γnk−1.(3.2) taking limit as k →∞, we get � ≤ lim k→∞ rk < � + lim k→∞ γnk−1. (3.3) it follows that � ≤ lim k→∞ rk < � + 0 (3.4) lim k→∞ d(xmk,xnk ) = �. notice also that � ≤ rk = d(xmk,xnk ) ≤ d(xmk,xmk+1) + d(xnk+1,xnk ) + d(xmk+1,xnk+1) = γmk + γnk + d(xmk+1,xnk+1) = γmk + γnk + d(gxmk+1,gxnk+1) ≤ γmk + γnk + α(xmk,xnk )d(gxmk+1,gxnk+1) ≤ γmk + γnk + β(d(xmk,xnk ))d(xmk,xnk ), c© agt, upv, 2016 appl. gen. topol. 17, no. 2 178 global optimization using α-ordered proximal contractions in metric spaces with partial orders implies that d(xmk,xnk ) −γmk −γnk d(xmk,xnk ) ≤ β(d(xmk,xnk )). taking limit as k →∞, we obtain lim k→∞ β(d(xmk,xnk )) = 1, since β ∈f, so lim k→∞ d(xmk,xnk ) = 0. hence � = 0, which is a contradiction. so {xn} is a cauchy sequence and converges to some element x ∈ a. so, we have xn � x for any n. similarly, in view of the fact that t(b0) ⊆ a0 and b0 ⊆ g(b0), it is ascertained that there is a sequence {yn} of elements in b0 such that (gyn+1,tyn) = d(a,b). since t is proximally increasing and the inverse of g is an increasing mapping, yn � yn+1. since g is an isometry and t is an α-ordered proximal contraction, it follows that d(yn,yn+1) = d(gyn,gyn+1) ≤ α(yn−1,yn)d(gyn,gyn+1) ≤ β(d(yn−1,yn))d(yn−1,yn). similarly, there exists a cauchy sequence {yn} such that it converges to some element y ∈ b. therefore, it follows that yn � y for all n. further, since the pair (s,t) is proximally increasing and the inverse of g is an increasing mapping, we have xn � yn, for all n. since the pair (s,t) forms an α-ordered proximal cyclic contraction and g is an isometry, it follows that d(xn+1,yn+1) = d(gxn+1,gyn+1) ≤ α(xn,yn)d(gxn+1,gyn+1) ≤ kd(xn,yn) + (1 −k)d(a,b). letting n →∞, it follows that d(x,y) = kd(x,y) + (1 −k)d(a,b) (3.5) ⇒ d(x,y) = d(a,b). thus x ∈ a0 and y ∈ b0. since s(a0) ⊆ b0 and t(b0) ⊆ a0, there exists u ∈ a and v ∈ b such that (3.6) d(u,sx) = d(a,b) d(v,ty) = d(a,b). } since s is α-ordered proximal contraction, we get from d(u,sx) = d(a,b) and d(gxn+1,sxn) = d(a,b) as (3.7) d(u,gxn+1) ≤ α(xn,x)d(u,gxn+1) ≤ β(d(x,xn))d(x,xn). c© agt, upv, 2016 appl. gen. topol. 17, no. 2 179 s. komal and p. kumam letting n →∞ in the above inequality, we have d(u,gx) = 0 and so u = gx. it follows that {gxn} converges to u. further, as g is an isometry, the sequence {gxn} converges to gx as well. thus, we write as d(gx,sx) = d(u,sx) = d(a,b). in the same manner, we have v = gy and so it can be prove that d(gy,ty) = d(v,ty) = d(a,b). � example 3.5. consider x = r2 be an euclidean metric space with partially ordered set x. let us define the sets a = {1}× [0,∞) and b = {2}× [0,∞). take a0 = a and b0 = b. obviously, d(a,b) = 1. let g : a∪b → a∪b be an identity mapping, the mapping g is surjective isometry, its inverse is an increasing mapping, a0 ⊆ g(a0) and b0 ⊆ g(b0). let us define s : a → b and t : b → a as: s(1,x) = (2, x x + 1 ), and t(2,x) = (1, x x + 1 ). where (1,x) ∈ a, (2,x) ∈ b and x ∈ [0,∞). let α : r2 ×r2 → [0,∞) defined as: α(x,y) = { 1 if x=1 or x=2 and y ∈ [0,∞), 0 elsewhere. clearly, s and t are proximally increasing and α-ordered proximal contractions with these assumptions such that s(a0) ⊆ b0 and t(b0) ⊆ a0. the pair (s,t) is proximally increasing, α-ordered proximal cyclic contraction. thus, all other assumptions of the theorem (3.1) are also satisfied. finally, very easily one can say that the element (1, 0) in a and the element (2, 0) in b satisfy the conclusion of the preceding result. if g is the identity mapping in the theorem 3.4, then we obtain the following: corollary 3.6. let x be a non-empty set such that (x,�) is a partially ordered set and (x,d) is a complete metric space, α : x ×x → r+ be a function and let a,b be nonempty closed subsets of (x,d) such that a0 and b0 are non-void. let s : a → b, t : b → a satisfy the following conditions: (1) s and t are α-ordered proximal contractions, proximally increasing; (2) the pair (s,t) is proximally increasing, α-ordered proximal cyclic contraction; (3) s(a0) ⊆ b0, t(b0) ⊆ a0; (4) s and t are α-proximal admissible maps; (5) α(x0,x1) ≥ 1 for x0,x1 ∈ x; c© agt, upv, 2016 appl. gen. topol. 17, no. 2 180 global optimization using α-ordered proximal contractions in metric spaces with partial orders (6) there exist elements x0 and x1 in a0 and y0,y1 ∈ b0 such that d(gx1,sx0) = d(a,b), and d(gy1,ty0) = d(a,b). x0 � x1, y0 � y1, x0 � y0. (7) if {xn} is an increasing sequence of elements in a converging to x, then xn � x, for all n. also, if {yn} is an increasing sequence of elements in b converging to y, then yn � y for all n. then there exists a point x ∈ a and a point y ∈ b such that d(x,sx) = d(y,ty) = d(x,y) = d(a,b). moreover, the sequence {xn} in a0, defined by d(xn+1,sxn) = d(a,b) (n ≥ 1), converges to the element x, and the sequence {yn} in b0, defined by d(yn+1,tyn) = d(a,b) (n ≥ 1), converges to the element y. if α(x0,x1) = 1 and β(t) = k, where k ∈ [0, 1) in the corollary (3.1), then we obtain the following corollary of [9]. corollary 3.7. let x be a non-empty set such that (x,�) is a partially ordered set and (x,d) is a complete metric space, let a,b be nonempty closed subsets of (x,d) such that a0 and b0 are non-void. let s : a → b, t : b → a satisfy the following conditions: (1) s and t are ordered proximal contractions, proximally increasing; (2) the pair (s,t) is proximally increasing, ordered proximal cyclic contraction; (3) s(a0) ⊆ b0, t(b0) ⊆ a0; (4) there exist elements x0 and x1 in a0 and y0,y1 ∈ b0 such that d(gx1,sx0) = d(a,b), and d(gy1,ty0) = d(a,b). x0 � x1, y0 � y1, x0 � y0. (5) if {xn} is an increasing sequence of elements in a converging to x, then xn � x, for all n. also, if {yn} is an increasing sequence of elements in b converging to y, then yn � y for all n. then there exists a point x ∈ a and a point y ∈ b such that d(x,sx) = d(y,ty) = d(x,y) = d(a,b). moreover, the sequence {xn} in a0, defined by d(xn+1,sxn) = d(a,b) (n ≥ 1), c© agt, upv, 2016 appl. gen. topol. 17, no. 2 181 s. komal and p. kumam converges to the element x, and the sequence {yn} in b0, defined by d(yn+1,tyn) = d(a,b) (n ≥ 1), converges to the element y. by taking α(x0,x1) = 1 and β(t) = k, where k ∈ [0, 1) in the theorem (3.1), we get the main result of [9] as: corollary 3.8. let x be a non-empty set such that (x,�) is a partially ordered set and (x,d) is a complete metric space and let a,b be nonempty closed subsets of (x,d) such that a0 and b0 are non-void. let s : a → b, t : b → a and g : a∪b → a∪b satisfy the following conditions: (1) s and t are ordered proximal contractions, proximally increasing; (2) g is surjective isometry, its inverse is an increasing mapping; (3) the pair (s,t) is proximally increasing, ordered proximal cyclic contraction; (4) s(a0) ⊆ b0, t(b0) ⊆ a0; (5) a0 ⊆ g(a0) and b0 ⊆ g(b0); (6) there exist elements x0 and x1 in a0 and y0,y1 ∈ b0 such that d(gx1,sx0) = d(a,b), and d(gy1,ty0) = d(a,b). x0 � x1, y0 � y1, x0 � y0. (7) if {xn} is an increasing sequence of elements in a converging to x, then xn � x, for all n. also, if {yn} is an increasing sequence of elements in b converging to y, then yn � y for all n. then there exists a point x ∈ a and a point y ∈ b such that d(gx,sx) = d(gy,ty) = d(x,y) = d(a,b). moreover, the sequence {xn} in a0, defined by d(gxn+1,sxn) = d(a,b) (n ≥ 1), converges to the element x, and the sequence {yn} in b0, defined by d(gyn+1,tyn) = d(a,b) (n ≥ 1), converges to the element y. if we take a = b = x, and α(x0,x1) = 1 in our main result (3.3), we get the following fixed point corollary, which is also the result of [9]. corollary 3.9. let x be a non-empty set such that (x,�) is a partially ordered set and (x,d) is a complete metric space. let s : x → x satisfy the following conditions: (1) s is increasing, ordered contraction; (2) there exist elements x0 in a such that x0 � sx0; (3) if {xn} is an increasing sequence of elements in a converging to x, then xn � x, for all n. then s has a fixed point. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 182 global optimization using α-ordered proximal contractions in metric spaces with partial orders acknowledgements. somayya komal was supported by the petchra pra jom klao doctoral scholarship academic for ph.d. program at kmutt. this project was supported by the theoretical and computational science (tacs) center under computational and applied science for smart innovation research cluster (classic), faculty of science, kmutt. references [1] a. akbar and m. gabeleh, global optimal solutions of noncyclic mappings in metric spaces, j. optim. theory appl. 153 (2012), 298–305. 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[10] s. s. basha, n. shahzad and r. jeyaraj, best proximity points: approximation and optimization, optim. lett. 7 (2011), 145–155. [11] s. s. basha, n. shahzad and r. jeyaraj, optimal approximate solutions of fixed point equations, abstract appl. anal. 2011 (2011), article id 174560, 9 pages. [12] s. s. basha, n. shahzad and r. jeyaraj, common best proximity points: global optimization of multi-objective functions, appl. math. lett. 24 (2011), 883–886. [13] s. s. basha and p. veeramani, best proximity pair theorems for multifunctions with open fibres, j. approx. theory 103 (2000), 119–129. [14] a. a. eldered and p. veeramani, proximinal normal structure and relatively nonexpansive mappings, studia math. 171 (2005), 283–293. [15] j. m. felicit and a. a. eldred, best proximity points for cyclical contractive mappings, applied gen. topol. 16 (2015), 119–126. [16] c. vetro, best proximity points: convergence and existence theorems for p-cyclic mappings, nonlinear anal. 73 (2010), 2283–2291. c© agt, upv, 2016 appl. gen. topol. 17, no. 2 183 @ appl. gen. topol. 18, no. 1 (2017), 107-115 doi:10.4995/agt.2017.6376 c© agt, upv, 2017 on cardinalities and compact closures mike krebs department of mathematics, california state university los angeles, 5151 state university drive, los angeles, california 90032 (mkrebs@calstatela.edu) communicated by s. garćıa-ferreira abstract we show that there exists a hausdorff topology on the set r of real numbers such that a subset a of r has compact closure if and only if a is countable. more generally, given any set x and any infinite set s, we prove that there exists a hausdorff topology on x such that a subset a of x has compact closure if and only if the cardinality of a is less than or equal to that of s. when we attempt to replace “less than or equal to” in the preceding statement with “strictly less than,” the situation is more delicate; we show that the theorem extends to this case when s has regular cardinality but can fail when it does not. this counterexample shows that not every bornology is a bornology of compact closure. these results lie in the intersection of analysis, general topology, and set theory. 2010 msc: 54a99. keywords: topology; hausdorff; cardinality; compact closure. 1. introduction a bornology on a set x is a covering b of x such that (i) if a,b ∈b, then a∪b ∈b, and (ii) if b ∈b and a ⊂ b, then b ∈b. bornologies are objects of much study in analysis—see, for example, [2]. the prototypical example of a bornology is the collection of all bounded sets in a metric space. another standard example, for a hausdorff space x, is the collection of all subsets of x with compact closure. the latter construction is rather general, and one may well wonder: given a bornology on x, does there necessarily exist a topology received 15 july 2016 – accepted 18 january 2017 http://dx.doi.org/10.4995/agt.2017.6376 m. krebs on x with respect to which b is precisely the bornology of sets with compact closure? in this paper, we answer that question in the negative by constructing a set y for which the bornology of subsets of y with cardinality strictly less than that of y cannot be a bornology of compact closure—see example 3.10. this example leads to the following general question. given two sets x and s, take the bornology b of subsets of x with cardinality less than or equal to that of s. is b a bornology of compact closure? one can also ask this question for the bornology of subsets of x with cardinality strictly less than that of s. for the first question, we show that the answer is always yes. for the second question, our counterexample y mentioned above shows that the answer is not always yes; however, we prove that it is whenever s has regular cardinality. considerations in general topology may lead one to ask the same questions without reference to bornologies. we now re-introduce this topic from this new point of view. when a set x is endowed with the discrete topology, a subset a of x is compact if and only if a is finite. one may wonder next, does there necessarily exist a topology on x such that a subset a of x is compact if and only if a is countable? one quickly realizes that unless x is finite, no such topology can be hausdorff. for if so, then let a = {an | n ∈ n} be a countably infinite subset of x with an 6= am whenever n 6= m. (here n denotes the set of natural numbers.) note that each set ak := {an | n ≥ k} is countable, hence compact, hence closed because the topology is hausdorff. but then {ak} is a nested collection of nonempty closed subsets of the compact set a, yet it has empty intersection, which is a contradiction. hausdorff being a typical property to impose a topological space, we therefore modify the question slightly: does there exist a hausdorff topology on x in which a set has compact closure if and only if it is countable? in particular, what about the case x = r, where r is the set of real numbers? if we assume both the continuum hypothesis (ch) and the axiom of choice (ac), then the answer to this last question is an immediate yes, for the following reason. recall that ch states that no uncountable set has cardinality strictly less than that of r. let ω be the least uncountable ordinal, that is, an uncountable well-ordered set such that every subset of the form {y ∈ ω | y ≤ x} for x ∈ ω is countable. it follows from ch that the cardinality of r equals that of ω. we may then identify r with ω and give it the topology induced by the order on ω. a straightforward exercise shows that with this topology, the closure of a set a in r is compact if and only if a is countable. although this logic no longer holds when we do not assume ch, it suggests an approach. begin by taking a well-ordering of r. recursively define a topology on r by constructing a neighborhood basis at each point, assuming one has been constructed at each previous point. we carefully select this neighborhood basis so that the sets with compact closure are precisely the countable sets. indeed, we may generalize this reasoning considerably. the details are carried out in section 3, where we prove our two main theorems. the first (theorem 3.1) states that given any set x and any infinite set s, there exists c© agt, upv, 2017 appl. gen. topol. 18, no. 1 108 on cardinalities and compact closures a hausdorff topology on x such that the sets with compact closure are precisely those whose cardinality is less than or equal to that of s. we obtain the first sentence in the abstract by taking x = r and s = n. in section 2, we discuss some set-theoretic preliminaries, including the definitions of “regular” and “singular” cardinals. the second main theorem (theorem 3.9) states that when the cardinality of s is regular, the phrase “less than or equal to” in theorem 3.1 can be replaced by the phrase “strictly less than.” we conclude with an example to show that the regularity condition in theorem 3.9 cannot be eliminated. 2. background from set theory throughout this paper, we work within the zermelo-fraenkel axiom system (zf). recall that a linear ordering on a set x is said to be a well-ordering if every nonempty subset of x has a smallest element. theorem 2.1 (well-ordering principle). every set admits a well-ordering. it is well-known that the well-ordering principle is equivalent to the axiom of choice (ac). the first step in our proofs will be to well-order the set x, so the proofs depend on the well-ordering principle, and hence ac, right from the git-go. it is also well-known that a countable union of countable sets is countable. more generally, a union over a set no bigger than x of sets no bigger than x is no bigger than x. more precisely, we have the following theorem, where the notation |b| ≤ |c| means that the cardinality of b is less than or equal to that of c. (likewise, we will later use the notation |b| < |c| to indicate that the cardinality of b is strictly less than that of c.) theorem 2.2. let x be an infinite set, and let i be a set with |i| ≤ |x|. for each i ∈ i, let ai be a set with |ai| ≤ |x|. then | ⋃ i∈i ai| ≤ |x|. theorem 2.2 is proved in [1]. the proof depends on ac. our third and final use of ac comes as we define the terms regular and singular for cardinalities. roughly speaking, we say that the cardinality of a set is regular if the set cannot be written as a smaller union of smaller sets, and that it is singular otherwise. we now make this concept more precise. definition 2.3. let x be a set. we say that x has singular cardinality if there exists a set i with |i| < |x| such that for each i ∈ i there exists a set ai with |ai| < |x|, and that x = ⋃ i∈i ai. we say that x has regular cardinality if x does not have singular cardinality. we say that this definition relies on ac because although it is not the standard definition, it is equivalent to the standard definition under the assumption of ac. we refer to [1] for details. there is no purpose to definition 2.3 unless both regular and singular cardinals exist. as the name suggests, regular cardinals are not hard to find. for c© agt, upv, 2017 appl. gen. topol. 18, no. 1 109 m. krebs instance, ℵ0 := |n| is regular, because n does not equal a finite union of finite sets. producing a singular cardinal requires a deliberate construction, such as the following. example 2.4. let x1 = n. for each n ∈ n, define xn+1 to be the power set of xn, i.e., the set of all subsets of xn. by cantor’s theorem, |xn| < |xn+1|. hence y := ⋃ n∈n xn has singular cardinality. 3. main theorems throughout this section, fix a nonempty set x and an infinite set s. if a is a subset of a topological space, then we denote its closure by a. our first objective in this section is to prove the following theorem. theorem 3.1. there exists a hausdorff topology on x so that if a ⊂ x, then a is compact if and only if |a| ≤ |s|. choose a well-ordering ≤ on x. assume that with respect to this ordering, x has a maximal element m. (if not, then create a new ordering by reversing all inequalities involving the minimal element.) to prove theorem 3.1, we begin by defining a topology. the definition is recursive and depends on knowing that what has been defined so far already forms a topology, a fact that in turn requires proof. so we must simultaneously make a recursive definition and an inductive proof. for y ∈ x, we define the closed ray (−∞,y] := {x ∈ x | x ≤ y} and the open ray (−∞,y) := {x ∈ x | x < y}. observe that x = (−∞,m]. lemma/definition 3.2. for any given x ∈ x, define nx, bx, tx, and wx according to (1)–(6) below with y = x, assuming that (1)–(6) are true for all y < x. (1) we define ny to be the collection of all sets of the form (−∞,y] \ k such that k is ty-closed in (−∞,y) and such that if c is a ty-closed subset of k with |c| ≤ |s|, then c is ty-compact. here ty is defined as in (3). (2) we define by := ⋃ z |s|. let m be the smallest element of x such that |(−∞,m]∩a| > |s|. let ` be the least upper bound of a∩(−∞,m). (the fact that x is well-ordered guarantees that m and ` exist.) let q = {a ∈ a | a < m}. let c = ⋃ a∈q((−∞,a] ∩ a). by theorem 2.2, |c| ≤ |s|. observe that (−∞,m] \ (−∞,`] is an open set disjoint from a; therefore it is disjoint from a. so (−∞,m]∩a contains c and at most two other points, namely ` and m. becasue s is infinite, therefore |(−∞,m] ∩a| ≤ |s|, a contradiction. � lemma 3.8. if a ⊂ x and |a| > |s|, then a is not compact. proof. temporarily assume that a is compact. observe that |a| > |s|. let m be the smallest element of x such that |(−∞,m]∩a| > |s|. by lemma/definition 3.2, we have that (−∞,m] is a closed subspace of x = (−∞,m], so (−∞,m]∩a is wm-compact. let k = (−∞,m) ∩ a. note that |k| > |s|, by definition of m. let c be any tm-closed subset of k such that |c| ≤ |s|. we will show that c is tm-compact. by definition of wm, this will show that k is a wm-closed subset of the wm-compact set (−∞,m] ∩a and therefore that k is wm-compact. let ` be the least upper bound of c in x. we must have that ` < m, for otherwise, by theorem 2.2, we would have that |k| = ∣∣∣∣∣⋃ c∈c ((−∞,c] ∩a) ∣∣∣∣∣ ≤ |s|. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 113 m. krebs so c ⊂ (−∞,`]. it follows then from lemma/definition 3.2 that because c is tm-closed, therefore c is w`-closed, and therefore c is wm-closed. but c is a subset of the wm-compact set a, so c is wm-compact, therefore tm-compact. note that {(−∞,k] : k ∈ k} is a wm-open cover of k. hence k ⊂ (−∞,k1] ∪·· ·∪ (−∞,kn] for some k1, . . . ,kn. so k ⊂ n⋃ j=1 ((−∞,kj] ∩a). but each kj < m, so |(−∞,kj] ∩ a| ≤ |s|, by definition of m. but then |k| ≤ |s|, because s is infinite. this contradicts the fact that |k| > |s|. � theorem 3.1 follows at once from lemmas 3.5, 3.6, 3.7, and 3.8. theorem 3.9. let x be a set, and let s be an infinite set with regular cardinality. then there exists a hausdorff topology on x so that if a ⊂ x, then a is compact if and only if |a| < |s|. proof. the proof is identical to that of theorem 3.1 with two small modifications. one must replace every instance of “≤ |s|” with “< |s|.” also, one must use definition 2.3 in place of theorem 2.2 whenever the latter is invoked. � the following example illustrates how theorem 3.9 can fail when s has singular cardinality. example 3.10. consider the sets defined in example 2.4. we will show that there does not exist a topology on y such that a ⊂ y has compact closure if and only if |a| < |y |. suppose otherwise. the fact that y does not have strictly smaller cardinality than itself implies that y = y is not compact. let {uα} be an open cover of y with no finite subcover. we know that |xn| < |y | for all n, so xn is compact. cover xn with finitely many sets uαn,1, . . . ,uαn,jn from the collection {uα}, and let vn = ⋃jn `=1 uαn,` . by our assumption on {uα}, there exists a point bn ∈ y such that bn /∈ wn, where wn = ⋃n k=1 vk. note that the sets wn form an increasing chain of open sets. also note that xn ⊂ xn ⊂ vn ⊂ wn. let s = {bn | n ∈ n}. then |s| < |y |. we will show that s is not compact, thereby producing the desired contradiction. for each ` ∈ n, let s` = {bn | n ≥ `}. then {s`} is a collection of closed subsets of s with the finite intersection property. it suffices to show that ⋂∞ `=1 s` = ∅. let y ∈ y . then y ∈ xm for some m. hence y is in the open set wm. observe that wm ∩ sm = ∅, because of how we chose the points bn. therefore y /∈ sm. therefore {s`} has empty intersection, and so s cannot be compact. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 114 on cardinalities and compact closures acknowledgements. the author would like to acknowledge gerald beer for discussions of bornologies that prompted the questions answered in this paper, and would like to thank the anonymous referee for several useful suggestions. references [1] a. dasgupta, set theory, birkhäuser/springer, new york, 2014. [2] h. hogbe-nlend, bornologies and functional analysis, north-holland publishing co., amsterdam-new york-oxford, 1977. c© agt, upv, 2017 appl. gen. topol. 18, no. 1 115 @ appl. gen. topol. 16, no. 2(2015), 167-181doi:10.4995/agt.2015.3323 c© agt, upv, 2015 lebesgue quasi-uniformity on textures selma özçağ department of mathematics, hacettepe university, ankara, turkey. (sozcag@hacettepe.edu.tr) abstract this paper considers the lebesgue property on quasi di-uniform textures. it is well known that the quasi-uniform space with a compact topology has the lebesgue property. this result is extended to direlational quasi-uniformities and dual dicovering quasi-uniformities. additionally we discuss the completeness of lebesgue di-uniformities. 2010 msc: 54e15; 54a05; 06d10; 03e20. keywords: texture; di-uniformity; quasi-uniformity; lebesgue quasiuniformity. 1. introduction a texturing on a set s is a point-separating, complete, completely distributive lattice s of subsets of s with respect to inclusion, which contains s and ∅, and for which arbitrary meet ∧ coincides with intersection ⋂ and finite joins ∨ with unions ⋃ . the pair (s, s) is called a texture. this definition was first introduced by l. m. brown to represent hutton algebras and lattices of l fuzzy sets in a point based setting [4]. however the development of the theory has proceeded largely independently and the work on di-uniformities has shown that it has much closer links with topological ideas than might be expected. di-uniformity on a texture was first defined in [13] by giving descriptions in terms of direlations, dicovers and dimetrics and the concepts of completeness and total boundedness were introduced in [14]. the effect of a complementation and the relation with quasi-uniformity and uniformity were discussed in [15]. in this context the work [15] pointed out that di-uniformities provide a more unified setting for the study of quasiuniformity and uniformity than does the classical approach. received 19 october 2014 – accepted 4 april 2015 http://dx.doi.org/10.4995/agt.2015.3323 s. özçağ as is well known a quasi-uniformity is obtained by omitting the symmetry condition in the definition of a uniformity. we recall the notion of direlational uniform texture space as follows. definition 1.1 ([13]). let (s, s) be a texture and u a family of direlations on (s, s). if u satisfies the conditions, (1) (i, i) ⊑ (d, d) for all (d, d) ∈ u. that is, u ⊆ rdr. (2) (d, d) ∈ u, (e, e) ∈ dr and (d, d) ⊑ (e, e) implies (e, e) ∈ u. (3) (d, d), (e, e) ∈ u implies (d, d) ⊓ (e, e) ∈ u. (4) given (d, d) ∈ u there exists (e, e) ∈ u satisfying (e, e) ◦ (e, e) ⊑ (d, d). (5) given (d, d) ∈ u there exists (c, c) ∈ u satisfying (c, c)← ⊑ (d, d). then u is called a direlational uniformity on (s, s), and (s, s, u) is known as a direlational uniform texture. this definition is formally same as the usual definition of diagonal uniformity. it should be noted, that the symmetry condition (5) which guarantees a base of symmetric direlations for the direlational uniformity is quite different from the notion of symmetry for relations. in [15] an important result was obtained that a direlational uniformity on the discrete texture (x, p(x)) corresponds not to uniformity but to quasi uniformity. when the symmetry condition (5) is removed we obtain a direlational quasi-uniform texture space (s, s, uq) [17]. another representation for di-uniformities is in terms of dicovers. we recall from [2] that by a difamily we mean a set c = {(aj, bj) | j ∈ j} of elements of s×s and c is called a dicover of (s, s) if ⋂ j∈j1 bj ⊆ ∨ j∈j2 aj for all partitions (j1, j2) of j. a dicover corresponds to a dual cover in the sense of [1] and this notion is related to the notion of pairs of covers with a common index used by gantner and steinlage [8] to characterize quasi uniformities. as in the classical case dicovers generate symmetric direlations and are not appropriate to characterize quasi di-uniformities. hence in [17] the authors used a new notion called dual dicover to introduce dual dicovering quasi-uniformity. below we recall these definitions. dual dicover([17]) a dual difamily cd = {(( c 1,1 j , c 1,2 j ) , ( c 2,1 j , c 2,2 j )) | j ∈ j } of elements of (s × s) × (s × s) is called a dual dicover of (s, s) if {( c 1,1 j ∩ c 2,1 j , c 1,2 j ∪ c 2,2 j ) | j ∈ j } is a dicover of (s, s). definition 1.2 ([17]). let (s, s) be a texture. if υq is a family of dual dicovers satisfying the conditions (1) given cd ∈ υ q there exists an anchored dual dicover dd ∈ υ q with dd ≺ cd, (2) cd ∈ υ q, cd ≺ dd implies dd ∈ υ q, (3) cd, dd ∈ υ q implies cd ∧ dd ∈ υ q, (4) given cd ∈ υ q there exists dd ∈ υ q with dd ≺(⋆) cd. we say υq is a dual dicovering quasi-uniformity on (s, s). c© agt, upv, 2015 appl. gen. topol. 16, no. 2 168 lebesgue quasi-uniformity on textures in [17] besides these definitions there is another approach by using quasipseudometrics. since this work will be based on the direlational and dual dicovering representations we will omit it. this paper is a continuation of the work [16] where lebesgue and co-lebesgue di-uniformities were first introduced and the relationship between lebesgue quasi-uniformity on x and the corresponding lebesgue di-uniformity on discrete texture (x, p(x)) was investigated. moreover, the notions of lebesgue quasi di-uniformity and dual dicovering lebesgue quasi-uniformity were introduced and discussed some of their properties. in this work our source of inspiration is [11] where the notion of pair lebesgue quasi-uniformity was first introduced by j. marin and s. romaguera and we confine our attention to dual dicovering bi-lebesgue quasi di-uniformities. the aim of this work is to continue to develop the notion of lebesgue property on quasi di-uniform textures and investigate dicompleteness of lebesgue di-uniform textures. after a brief introduction, in section 2 we introduce the notion of a bilebesgue quasi di-uniformity and show that on plain textures each quasi diuniformity with a dicompact topology is a bi-lebesgue quasi di-uniformity. we obtained the analogous result for the dual dicovering bi-lebesgue quasi di-uniform spaces which will be defined in definition 2.11. in section 3 we first consider the dual covering lebesgue quasi-uniformy in the sense of brown [1] and discuss the completeness of lebesgue di-uniformities on the discrete texture. general references on ditopological texture spaces include [1, 2, 3, 4, 5, 6] and constant reference will be made to [13, 14, 15, 16, 17] for definitions and results relating to di-uniformities. our standart references for quasi uniformity are [7, 8, 9]. for the conveince of the reader we recall some more special definitions. let (s, s) be a texture. for s ∈ s the sets ps = ⋂ {a ∈ s | s ∈ a} and qs = ∨ {a ∈ s | s /∈ a} are called respectively, the p-sets and q-sets of (s, s). for a ∈ s the core a♭ of a is given by a♭ = {s ∈ s | a * qs}. the set a ♭ does not necessarily belong to s. in general, a texturing of s need not be closed under set complementation, but sometimes we have a notion of complementation. complementation: [2] a mapping σ : s → s satisfying σ(σ(a)) = a, ∀ a ∈ s and a ⊆ b =⇒ σ(b) ⊆ σ(a), ∀ a, b ∈ s is called a complementation on (s, s) and (s, s, σ) is then said to be a complemented texture. examples : 1. for any set x, (x, p(x), πx), πx(y ) = x \ y for y ⊆ x, is the complemented discrete texture representing the usual set structure of x. clearly, px = {x} and qx = x \ {x} for all x ∈ x. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 169 s. özçağ 2. for i = [0, 1] define i = {[0, t] | t ∈ [0, 1]} ∪ {[0, t) | t ∈ [0, 1]}, ι([0, t]) = [0, 1 − t) and ι([0, t)) = [0, 1 − t], t ∈ [0, 1]. then (i, i, ι) is a complemented texture, which we will refer to as the unit interval texture. here pt = [0, t] and qt = [0, t) for all t ∈ i. ditopology: a dichotomous topology on (s, s) or ditopology for short, is a pair (τ, κ) of subsets of s, where the set of open sets τ satisfies 1. s, ∅ ∈ τ, 2. g1, g2 ∈ τ =⇒ g1 ∩ g2 ∈ τ and 3. gi ∈ τ, i ∈ i =⇒ ∨ i gi ∈ τ, and the set of closed sets κ satisfies 1. s, ∅ ∈ κ, 2. k1, k2 ∈ κ =⇒ k1 ∪ k2 ∈ κ and 3. ki ∈ κ, i ∈ i =⇒ ⋂ ki ∈ κ. for a ∈ s the sets [a] = ⋂ {k ∈ κ | a ⊆ k} and ]a[= ∨ {g ∈ τ | g ⊆ a} are called the closure and interior of a. a plain texture is one for which the texturing is closed under arbitrary unions or equivalently join coincides with union in s. there is a considerable simplification in plain textures. we have ps * qs for each s ∈ s. hence, for a ∈ s, s ∈ a, ps ⊆ a and a * qs are equivalent to each other. one of the most useful notions in the theory of di-uniformities is that of direlation. direlations: [5] let (s, s), (t, t) be textures. we use p (s,t), q(s,t) to denote the p-sets and q-sets for the product texture (s × t, p(s) ⊗ t). then: (1) r ∈ p(s) ⊗ t is called a relation from (s, s) to (t, t) if it satisfies r1 r * q(s,t), ps′ * qs =⇒ r * q(s′,t). r2 r * q(s,t) =⇒ ∃ s ′ ∈ s such that ps * qs′ and r * q(s′,t). (2) r ∈ p(s) ⊗ t is called a corelation from (s, s) to (t, t) if it satisfies cr1 p (s,t) * r, ps * qs′ =⇒ p (s′,t) * r. cr2 p (s,t) * r =⇒ ∃ s ′ ∈ s such that ps′ * qs and p (s′,t) * r. a pair (r, r) consisting of a relation r and corelation r is called a direlation. now let (r, r) be a direlation from (s, s) to (t, t). the inverses of r and r are given by r← = ⋂ {q(t,s) | r * q(s,t)}, r ← = ∨ {p (t,s) | p (s,t) * r}. where r← is a relation and r← a corelation. the direlation (r, r)← = (r←, r←) from (t, t) to (s, s) is called the inverse of (r, r). for a ⊆ s the a–section of a relation r and a–section of a corelation r is defined by r→a = ⋂ {qt | ∀ s, r * q(s,t) =⇒ a ⊆ qs}, r→a = ∨ {pt | ∀ s, p (s,t) * r =⇒ ps ⊆ a} c© agt, upv, 2015 appl. gen. topol. 16, no. 2 170 lebesgue quasi-uniformity on textures compositions of direlations: let (s, s), (t, t), (u, u) be textures. (1) if p is a relation on (s, s) to (t, t) and q a relation on (t, t) to (u, u) then their composition is the relation q ◦ p on (s, s) to (u, u) defined by q ◦ p = ∨ {p (s,u) | ∃ t ∈ t with p * q(s,t) and q * q(t,u)}. (2) if p is a co-relation on (s, s) to (t, t) and q a co-relation on (t, t) to (u, u) then their composition is the co-relation q ◦ p on (s, s) to (u, u) defined by q ◦ p = ⋂ {q(s,u) | ∃ t ∈ t with p (s,t) * p and p (t,u) * q}. (3) with p, q; p , q as above, the composition of the direlations (p, p), (q, q) is the direlation (q, q) ◦ (p, p) = (q ◦ p, q ◦ p). 2. lebesgue quasi di-uniform spaces in this section we consider the lebesgue property on quasi di-uniform textures. we introduce lebesgue and co-lebesgue direlational quasi-uniformities. we also define bi-lebesgue quasi di-uniformity, dual dicovering bi-lebesgue quasi di-uniformity and give an analog of the well known result that each quasiuniformity compatible with a compact space is a lebesgue quasi-uniformity. the definition of a direlational uniformity u on a texture (s, s) has been introduced in definition 1.1. we obtain a direlational quasi-uniformity on (s, s) by removing the symmetry condition from the definition of the direlational uniformity. now we begin by recalling the following definition. definition 2.1 ([17, definition 2.1]). let (s, s) be a texture and uq a family of direlations on (s, s). if uq satisfies the conditions (1) (i, i) ⊑ (d, d) for all (d, d) ∈ uq, (2) (d, d) ∈ uq, (e, e) ∈ dr and (d, d) ⊑ (e, e) implies (e, e) ∈ uq, (3) (d, d), (e, e) ∈ uq implies (d, d) ⊓ (e, e) ∈ uq, (4) given (d, d) ∈ uq there exists (e, e) ∈ uq satisfying (e, e) ◦ (e, e) ⊑ (d, d), then uq will be called a direlational quasi-uniformity on (s, s) and (s, s, uq) a direlational quasi-uniform texture space. as in the classical case for the direlational quasi-uniformity uq on (s, s) (uq)← = {(d, d)← : (d, d) ∈ uq} is also a direlational quasi-uniformity on (s, s) and (s, s, (uq)←) is called the conjugate of (s, s, uq) (see, [17]). a direlational quasi-uniformity uq on (s, s) induces a uniform ditopology (τuq , κuq ) as follows, in exactly the same way that a direlational uniformity does [13, lemma 4.3]. (i) g ∈ τuq ⇐⇒ (g * qs =⇒ ∃ (d, d) ∈ u q with d[s] ⊆ g), c© agt, upv, 2015 appl. gen. topol. 16, no. 2 171 s. özçağ (ii) k ∈ κuq ⇐⇒ (ps * k =⇒ ∃ (d, d) ∈ u q with k ⊆ d[s]). here d[s] = d→ps and d[s] = d →qs. when we speak of the ditopology of (s, s, uq) we will always mean the uniform ditopology. in order to consider lebesgue direlational quasi-uniformity it is necessary to recall [6] the open cover and closed cocover for the textures. let (τ, κ) be a ditopology on the texture (s, s) and let a ∈ s. the family {gi | i ∈ i} is said to be an open cover [6] of a if gi ∈ τ for all i ∈ i and a ⊆ ∨ i∈i gi. dually we may speak of a closed cocover of a, namely a family {fi | i ∈ i} with fi ∈ κ for all i ∈ i satisfying ⋂ i∈i fi ⊆ a. for the cocovers we need a notion of dual refinement. definition 2.2 ([16]). let k1, k2 be cocovers. then k1 will be called a dual refinement of k2, and write k1 ⊳ k2 if for a given k2 ∈ k2 there exists k1 ∈ k1 such that k1 ⊆ k2. now we may give: definition 2.3. a direlational quasi-uniformity uq on (s, s) is called (1) lebesgue direlational quasi-uniformity provided that for each cover c of s which is open for the uniform ditopology there is a direlation (r, r) ∈ uq such that {r[s] | s ∈ s♭} is a refinement of c. (2) co-lebesgue direlational quasi-uniformity provided that for each cocover k of ∅ which is closed for the uniform ditopology there is a direlation (r, r) ∈ uq such that k is a dual refinement of {r[s] | s ∈ s♭}. in [16, proposition 2.6] it is proved that each direlational uniformity compatible with a compact (cocompact) ditopological texture space is a lebesgue (co-lebesgue) direlational uniformity on (s, s). we now have the analogous result for the direlational quasi-uniformities. let us recall the following definition. definition 2.4 ([6]). let (τ, κ) be a ditopology on the texture (s, s) and a ∈ s. (1) a is called compact if whenever {gi | i ∈ i} is an open cover of a then there is a finite subset j of i with a ⊆ ⋃ j∈j gj. in particular the ditopological texture space (s, s, τ, κ) is called compact if s is compact. (2) a is called cocompact if whenever {fi | i ∈ i} is a closed cocover of a then there is a finite subset j of i with ⋂ j∈j fj ⊆ a. in particular the ditopological texture space (s, s, τ, κ) is called cocompact if ∅ is cocompact. we recall from [6] that a ditopological texture space (s, s, τ, κ) is stable (costable) if every f ∈ κ\{s} (g ∈ τ\{∅}) is compact (cocompact). the ditopological texture space is called dicompact if it is compact, cocompact, stable and costable. we may now give the promised result. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 172 lebesgue quasi-uniformity on textures proposition 2.5. let (s, s, τ, κ) be a ditopological texture space. (1) if uq is a direlational quasi-uniformity compatible with a compact ditopology, then uq is a lebesgue direlational quasi-uniformity on (s, s). (2) if uq is a direlational quasi-uniformity compatible with a cocompact ditopology, then uq is a co-lebesgue direlational quasi-uniformity on (s, s). proof. the proof follows from the same lines as in the proof of [16, proposition 2.6]. � the notion of lebesgue quasi di-uniformity was introduced in [16] and the anchored property for the dicover was omitted in that definition. nevertheless we now begin by introducing a new notion called bi-lebegue quasi di-uniform space. first let us recall the notion of an anchored dicover which plays an important role in the development of dicovering uniformities and bi-lebesgue quasi diuniformities. definition 2.6 ([13]). a family c ⊆ s × s is called an anchored dicover if it satisfies: (1) p ≺ c, and (2) given a c b there exists s ∈ s satisfying (a) a * qu =⇒ ∃ a ′ c b′ with a′ * qu and ps * b ′, and (b) pv * b =⇒ ∃ a ′′ c b′′ with pv * b ′′ and a′′ * qs. a dicover c = {(aj, bj) | j ∈ j} is finite if the set {aj | j ∈ j} is finite and cofinite if the set {bj | j ∈ j} is finite. if c is defined on a ditopological texture space (s, s, τ, κ), it is said to be open coclosed if aj ∈ τ and bj ∈ κ and closed co-open if aj ∈ κ and bj ∈ τ for all j ∈ j. c is a refinement of d if given j ∈ j we have l d m so that aj ⊆ l and m ⊆ bj. in this case we write c ≺ d. if (d, d) is a reflexive direlation on (s, s) then γ(d, d) = {(d[s], d[s]) | s ∈ s♭} is an anchored dicover of (s, s). definition 2.7. let uq be a quasi di-uniformity on a ditopological texture space (s, s, τ, κ). then uq is a bi-lebesque quasi di-uniformity provided that for each open coclosed anchored dicover c of (s, s) there is a direlation (r, r) ∈ uq such that the dicover γ(r, r) = {(r[s], r[s]) | s ∈ s♭} refines c and (s, s, uq) is called a bi-lebesgue quasi di-uniform texture space. the following important theorem, which is proved in [3] gives a characterization of dicompactness in textures. theorem 2.8. the following are equivalent for (s, s, τ, κ). (1) (s, s, τ, κ) is dicompact. (2) every closed co-open difamily with the finite exclusion property is bound. (3) every open coclosed dicover has a finite and cofinite subdicover. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 173 s. özçağ now we may give the following theorem in the case of plain textures. theorem 2.9. let (s, s, uq) be a plain direlational quasi-uniform texture space such that (τuq , κuq ) is dicompact. then (s, s, u q) is a bi-lebesgue quasi diuniform space. proof. let c = {(aj, bj) | j ∈ j} be an open coclosed anchored dicover of (s, s). for each s ∈ s♭ and j(s) ∈ j we have aj(s)c bj(s) with aj(s) ∈ τuq and bj(s) ∈ κuq . since (s, s) is plain and c is an anchored dicover, for s ′ ∈ s♭ we have aj(s) * qs′ and there exists (es, es) ∈ u q with es[s ′] ⊆ aj(s). now take bj(s) ∈ κuq and ps′ * bj(s) then we have bj(s) ⊆ es[s ′]. since uq is a direlational quasi-uniformity there exists (ds, ds) ∈ u q satisfying (ds, ds) ◦ (ds, ds) ⊑ (es, es). hence {( ] ds[s ′] [ , [ ds[s ′] ] ) | s′ ∈ s♭} is an open coclosed anchored dicover of (s, s) with ps′ ⊆ ] ds[s ′] [ and [ ds[s ′] ] ⊆ qs′. since (s, s) is dicompact, the open coclosed dicover {( ] ds[s ′] [ , [ ds[s ′] ] ) | s′ ∈ s♭} has a finite cofinite subdicover {( ] dsk [s ′ k] [ , [ dsk[s ′ k] ] ) | s ′ ∈ s♭} for k = 1, ..., n by theorem 2.8. if we set (d, d) = dn k=1(dsk , dsk), then (d, d) ∈ u q. since (s, s) is plain, we have dsk [s ′ k] * qs′ and ps′ * dsk[s ′ k] for s ′ ∈ s♭ and 1 ≤ k ≤ n. we shall show that γ(d, d) = {(d[s′], d[s′]) | s′ ∈ s♭} ≺ c. for the given s′ ∈ s♭ there is k ∈ {1, 2, 3, ..., n} such that esk[s ′ k] ⊆ aj(sk) and bj(sk) ⊆ esk [s ′ k]. we need to prove d[s′] ⊆ esk [s ′ k] ⊆ aj(sk) and bj(sk) ⊆ esk [s ′ k] ⊆ d[s ′]. now let us prove bj(sk) ⊆ esk [s ′ k] ⊆ d[s ′]. first suppose that esk [s ′ k] * d[s′]. then there exists z ∈ s♭ with esk [s ′ k] * qz and pz * d[s ′]. because of d = ⊔n k=1 dsk we have dsk ⊆ d so dsk [s ′] ⊆ d[s′] and we have pz * dsk [s ′]. by the definition of composition of co-relations [13, definition 1.7] we have esk ⊆ dsk ◦ dsk ⊆ q(s′ k ,z). from ps′ * dsk[s ′ k] = d → sk (qs′ k ) we obtain p (s′ k ,s′) * dsk and due to pz * dsk[s ′] we have p (s′,z) * dsk by [13, lemma 1.5]. on the other hand e→sk (qs′k) = esk [s ′ k] * qz gives z ′ ∈ s with pz′ * qz and for t ∈ s♭ (2.1) p (t,z′) * esk =⇒ pt ⊆ qs′k . by [13, definition 1.3.(2)]. from esk ⊆ q(s′ k ,z) and pz′ * qz we have p (s′k,z′) * esk and since esk is a co-relation we have s ′′ k ∈ s with ps′′k * qs′k and p (s′′k ,z′) * esk by cr2. now we may apply the implication (1) with t = s ′′ k to give the contradiction ps′′ k ⊆ qs′ k . (2) the proof of d[s′] ⊆ esk[s ′ k] ⊆ aj(sk) is dual to the above and is omitted. � we will use the term quasi di-uniformity [17] to refer to direlational quasiuniformities and dual dicovering quasi-uniformities in general. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 174 lebesgue quasi-uniformity on textures let us recall from [17, definition 3.8] that a dual difamily cd is called an anchored dual dicover if (i) pd = {((ps, qs), (ps, qs))} | s ∈ s ♭} ≺ cd and (ii) cd ≺ c ∆ d . if (r, r) is a reflexive direlation on (s, s) then the family γ q(r, r) = {(γ(r, r), γ(r, r)←) | s ∈ s♭} is an anchored dual dicover where γ(r, r)← = {(r←[s], r←[s]) | s ∈ s♭}. moreover a dual dicover cd satisfying ( c 1,1 j , c 1,2 j ) ∈ (τuq , κuq ) and ( c 2,1 j , c 2,2 j ) ∈ (τ(uq)←, κ(uq)←) is called open co-closed. we recall the definition of a refinement for the dual dicovers. definition 2.10 ([17]). let cd = {((c 1,1 j , c 1,2 j ), (c 2,1 j , c 2,2 j )) | j ∈ j} and dd be dual dicovers. then cd is a refinement of dd, written cd ≺ dd, if given j ∈ j we have ((d1,1, d1,2), (d2,1, d2,2)) ∈ dd so that (c 1,1 j , c 1,2 j ) ⊑ (d 1,1, d1,2) and (c 2,1 j , c 2,2 j ) ⊑ (d 2,1, d2,2) ⇐⇒ c 1,1 j ⊆ d 1,1 ; d1,2 ⊆ c 1,2 j and c 2,1 j ⊆ d 2,1 ; d2,2 ⊆ c 2,2 j now let us make the following definition: definition 2.11. let (s, s, uq) be a quasi di-uniform space. uq is called a dual dicovering bi-lebesgue quasi di-uniformity if for each open coclosed anchored dual dicover cd of (s, s, u q) there is a direlation (r, r) ∈ uq such that γq(r, r) refines cd. we also recall from [17, definition 2.7] that the direlational uniformity with subbase uq ∪(uq)← is called the direlational uniformity associated with uq and is denoted by u∗ = uq ∨ (uq)←. j. marin and s. romaguera [11] obtained a result states that if (x, u) is a quasi uniform space such that (x, τ(u∗)) is compact then (x, u) is a pair lebesgue quasi-uniform space. we end this section by obtaining a similar result to the classical case. theorem 2.12. let (s, s, uq) be a plain quasi di-uniform texture space such that (τu∗, κu∗) is dicompact. then (s, s, u q) is a dual dicovering bi-lebesque quasi di-uniform space. proof. let cd = {(( c 1,1 j , c 1,2 j ) , ( c 2,1 j , c 2,2 j )) | j ∈ j } be an open coclosed anchored dual dicover. for each s ∈ s♭ and j(s) ∈ j we have (c 1,1 j(s) , c 1,2 j(s) ) cd (c 2,1 j(s) , c 2,2 j(s) ) with (c 1,1 j(s) , c 1,2 j(s) ) ∈ (τuq × κuq ) and (c 2,1 j(s) , c 2,2 j(s) ) ∈ (τ(uq)← × κ(uq)←) since (s, s) is plain and cd is an anchored dual dicover we have c 1,1 j(s) * qs′ for s′ ∈ s♭ and there exists (ds, ds) ∈ u q with ds[s ′] ⊆ c 1,1 j(s) . in this case there exist (rs, rs) ∈ u q with (rs, rs) 2 ⊑ (ds, ds) and rs 2[s′] ⊆ ds[s ′] ⊆ c 1,1 j(s) since uq is a direlational quasi-uniformity. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 175 s. özçağ now take c 2,1 j(s) ∈ τ(uq)← . since cd is an anchored dual dicover and (s, s) is plain, we have c 2,1 j(s) * qs′ for s ′ ∈ s♭. hence, there exist (ds, ds) ← ∈ (uq) ← and (rs, rs) ← ∈ (uq) ← such that (rs ←)2[s′] ⊆ ds ←[s′] ⊆ c 2,1 j(s) . dually for c 1,2 j(s) ∈ κuq and c 2,2 j(s) ∈ κ(uq)← with ps′ * c 1,2 j(s) and ps′ * c 2,2 j(s) we have c 1,2 j(s) ⊆ ds[s ′] ⊆ rs 2[s′] and c 2,2 j(s) ⊆ ds ←[s′] ⊆ (rs ←)2[s′]. thus {((rs[s ′], rs[s ′]), (rs ←[s′], rs ←[s′])) | s′ ∈ s♭} is an anchored dual dicover by [17, proposition 3.10]. then {( ] rs[s ′] ∩ rs ←[s′] [ ), ( [ rs[s ′] ∪ rs ←[s′] ] ) | s′ ∈ s♭} is an open coclosed anchored dicover of s satisfying ps′ ⊆ ] rs[s ′]∩rs ←[s′] [ and [ rs[s ′]∪rs ←[s′] ] ⊆ qs′ since (s, s) is dicompact the open coclosed dicover {( ] rs[s ′] ∩ rs ←[s′] [ ), ( [ rs[s ′] ∪ rs ←[s′] ] ) | s′ ∈ s♭} has a finite cofinite subdicover {( ] rsk [s ′ k] ∩ rsk ←[s′k] [ ), ( [ rsk [s ′ k] ∪ rsk ←[s′k] ] ) | s′ ∈ s♭} for k = 1, ..., n by theorem 2.8. now we set (r, r) = dn k=1(rsk , rsk ) and note that (r, r) ∈ u q. since (s, s) is plain we have rsk [s ′ k] ∩ r ← sk [s′k] * qs′ and ps′ * rsk [s ′ k] ∪ r ← sk [s′k] for s ′ ∈ s♭ and 1 ≤ k ≤ n. we will complete the proof by showing γq(r, r) ≺ cd. for the given s ′ ∈ s♭ there is k ∈ {1, 2, 3, ..., n} such that dsk [s ′ k] ⊆ c 1,1 j(sk) , d←sk [s ′ k] ⊆ c 2,1 j(sk) , c 1,2 j(sk) ⊆ dsk[s ′ k] and c 2,2 j(sk) ⊆ d←sk [s ′ k]. now let us prove that r[s ′] ⊆ dsk [s ′ k] ⊆ c 1,1 j(sk) . first suppose that r[s′] * dsk [s ′ k]. then there exists z ∈ s ♭ with r[s′] * qz and pz * dsk [s ′ k]. since rsk [s ′ k] ⋂ r←sk[s ′ k] * qs′ we have rsk [s ′ k] * qs′ and r←sk[s ′ k] * qs′. then we have rsk * q ′ (s′ k ,s) and also by r = ⋂n k=1 rsk for each k = 1, ..., n we have r ⊆ rsk , whence r[s ′] ⊆ rsk [s ′] and we have rsk [s ′] * qz which gives rsk * q(s′,z). hence we obtain p (s′k,z) ⊆ r 2 sk ⊆ dsk . on the other hand pz * dsk [s ′ k] = d → sk ps′ k gives pz * qz′ for z ′ ∈ s♭ and (2.2) dsk * q(v,z′) =⇒ ps′k ⊆ qv, for v ∈ s ♭ from p (s′ k ,z) ⊆ dsk and pz * qz′ we have dsk * q(s′ k ,z′), and since r is a relation we have s′′k ∈ s ♭ with ps′ k * qs′′ k such that dsk * q(s′′ k ,z′) by r2. applying the implication (2) with v = s′′k we deduce ps′k ⊆ qs ′′ k , which is a contradiction. this verifies r[s′] ⊆ dsk [s ′ k] ⊆ c 1,1 j(s) . now it is easy to prove that r←[s′] ⊆ d←sk [s ′ k] ⊆ c 2,1 j(sk) . the other two inclusions can be shown dually and hence the proof is omitted. this completes the proof of γq(r, r) ≺ cd, thus (s, s, u q) is a dual dicovering bi-lebesque quasi di-uniform space. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 176 lebesgue quasi-uniformity on textures 3. completeness of lebesgue di-uniform spaces we conclude this paper by discussing the completeness of lebesgue diuniformities. the subjects of completeness and total boundedness in di-uniform spaces are discussed in [14]. in the classical theory it is known from [7, 11] that every lebesgue uniformity is complete and every lebesgue quasi-uniformity is convergence complete. at the beginning of this work, we expect to find an analogue result for the lebesgue di-uniformities. however there are considerable difficulties such as the convergence of the cauchy di-filter to obtain a similar result for the general textures. since there is a close relationship between quasi-uniformities and diuniformities on discrete textures so we turn our attention to the completeness of lebesgue di-uniformities which correspond to lebesgue quasi-uniformity on the discrete textures. quasi-uniform spaces can be defined in various equivalent ways; by relations that satisfy all the axioms of a uniformity except symmetry; by quasipseudometrics and by (pair, dual) covers. gartner and steinlage [8] presented a description of quasi-uniformities in terms of pairs of covers and marin and romaguera [11] used a similar notion called open pairs, that is {(gα, hα) | α ∈ a} such that gα is τq-open and hα is τq−1-open and for each x ∈ x there is α ∈ a with x ∈ gα ∩hα where τq is the topology generated by q and τq−1 that generated by q−1. brown [1] independently developed a theory of quasi-uniformities by using a new concept of dual cover and showed the equivalence with the notion of open pairs mentioned in [12]. in this context the notion of q-completeness was considered for the completeness of quasi-uniform spaces. now we find it convenient to use the representation in terms of dual covers in this section. throughout this section we are interested in the concept of completeness namely q-completeness for quasi-uniformities which is based on the use of dual covers. since dual covers are not well known concepts, we now recall from [1, 15] some definitions and properties. however the dual covering quasi-uniformity and the equivalence with the diagonal quasi-uniformity were studied widely in [15]. let x be a set. a family u = {(aj, bj) | j ∈ j} of subsets of x is called a dual cover of x if ⋃ {(aj ⋂ bj) | j ∈ j} = x. if u and v are dual covers of x we say u refines v and write u ≺ v if whenever aub there exists cv d satisfying a ⊆ c and b ⊆ d. given a binary point relation d ∈ x we may associate with d the dual family called dual cover γ∗(d) = {(d[x], d−1[x]) | x ∈ x} where, as usual d[x] = {y ∈ x | (x, y) ∈ d} and d−1[x] = {y ∈ x | (y, x) ∈ d}. now let us turn our attention to the completeness of quasi-uniformity and lebesgue quasi-uniformity. in the literature, several authors defined various kinds of completeness on quasi-uniform spaces. we will mention particulary two of these definitions. according to [7] a quasi uniform space (x, q) is called c© agt, upv, 2015 appl. gen. topol. 16, no. 2 177 s. özçağ bicomplete if (x, q∗) is a complete uniform space where q∗ = q ∨ q−1. since lebesgue uniformity is complete, marin and romaguera [11] obtained a result which states that each pair lebesgue quasi-uniformity is bicomplete. on the other hand a quasi-uniform space (x, q) is convergence complete [7] provided that each cauchy filter is τq-convergent and since every lebesgue quasi-uniformity is convergence complete, marin and romaguera also proved that each pair lebesgue quasi-uniformity is convergence complete. starting from this point, we focus our attention on another type of completeness namely q-completeness in the brown’s sense (see, [1]). we recall that a bifilter b on the set x is defined as a product of two filters bu and bv on x denoted by b = bu × bv. any bifilter b is ı-regular if f ∩ g 6= ∅ whenever (f, g) ∈ b. if (x, u, v) is a bitopological space and x ∈ x then b(x) = {(h(x), k(x)) | h(x) is a u-nhd. and k(x) is a v-nhd of x} is an ı-regular bifilter which we will call the nhd. bifilter of x. the bifilter b converges to x if b(x) ⊆ b. if q is a dual covering quasi-uniformity compatible with (x, u, v) then the bifilter b will be called q-cauchy if u ∩ b 6= ∅ for all u ∈ q. definition 3.1 (see [1]). a quasi-uniform space (x, q) is called q-complete if every ı-regular q-cauchy bifilter is convergent in the bitopological space (x, τq, τq−1). now we shall work with dual covers in the sense of brown [1] instead of pair open cover and because of the equivalence of these two concepts we expect to have similar results as given in the paper of marin and romaguera [11]. we first give the definition of a notion dual covering lebesgue quasi-uniformity which was defined by marin and romaguera under the name of pair lebesgue quasi-uniformity. definition 3.2. let q be a quasi-uniformity on x. we say that q is a dual covering lebesgue quasi-uniformity if for each open dual cover u = {(aj, bj) | j ∈ j} of (x, q) there is d ∈ q such that the dual cover {(d[x], d−1[x]) | x ∈ x} refines {(aj, bj) | j ∈ j} (i.e. for each x ∈ x there is j ∈ j such that d[x] ⊆ aj and d−1[x] ⊆ bj). thus, we say that (x, q) is a dual covering lebesgue quasiuniform space. here we recall from [16] that a quasi-uniformity q on a set x is a lebesgue quasi-uniformity provided that for each τq-open cover g of x there is d ∈ q such that the cover {d[x] | x ∈ x} refines g. proposition 3.3. let (x, q) be a dual covering lebesgue quasi-uniform space. then q and q−1 are lebesgue quasi-uniformities. proof. let q be a dual covering lebesgue quasi-uniformity on x. we shall show that q is a lebesgue quasi-uniformity. let {aj : j ∈ j} be a τq-open cover of x. for each j ∈ j let bj = x. then {(aj, x) : j ∈ j} is an open dual cover of (x, q). so there is d ∈ q such that the open dual cover {(d[x], d−1[x]) | x ∈ x} refines {(aj, x) : j ∈ j} which gives the required result. similarly we see that q−1 is a lebesgue quasi-uniformity. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 178 lebesgue quasi-uniformity on textures theorem 3.4. every lebesgue quasi-uniformity is q-complete. proof. let b = bu × bv be a regular cauchy bifilter that does not converge to x. then for each x ∈ x we have m(x) a τq-nhd. of x and n(x) a τq−1 nhd. of x such that (m(x), n(x)) /∈ b. also there exists d ∈ q such that {d[x] : x ∈ x} refines m(x). since b = bu × bv is a cauchy bifilter, d[x] ∈ bu for some x ∈ x, which is a contradiction. � the following result is clear from the above discussion so we omit the proof. theorem 3.5. let q be a dual covering lebesgue quasi-uniformity on x. then both (x, q) and (x, q−1) are q-complete quasi-uniform spaces. in the remainder of this section we consider the completeness of lebesgue di-uniformities on discrete texture (x, p(x)). we will investigate how the relation between quasi-uniformity and di-uniformity effects the completeness of lebesgue di-uniformities. the reader is referred to [15] for more backround material, for the benefit of the reader however we will briefly recall the necessary definitions and results. in [15] it is shown that di-uniformities on the discrete texture correspond to quasi uniformities on x. moreover a direlational uniformity on (x, p(x)) corresponds to a uniformity if and only if it is complemented. let d ⊆ x×x be a point relation then u(d) = (d, d←) is a direlation on (x, p(x)) and if q is a quasi-uniformity on x, the family u(q) = {(e, e) | ∃d ∈ q and u(d) ⊑ (e, e)} is a direlational uniformity on the discrete texture (x, p(x)). proposition 3.6 ([15]). let q be a quasi-uniformity on x and q−1 its conjugate. then the direlational uniformity on (x, p(x), πx) corresponding to q −1 is the complement of the direlational uniformity corresponding to q. that is, u(q−1) = u(q)′. theorem 3.7 ([15]). let q be a quasi-uniformity on x. then q is a uniformity if and only if the corresponding di-uniformity u(q) on (x, p(x), πx) is complemented. we can now tie the completeness of a quasi-uniformity in with the dicompleteness of a di-uniformity on (x, p(x)). proposition 3.8. the quasi-uniform space (x, q) is q-complete if and only if the di-uniform discrete texture space (x, p(x), u(q)) is dicomplete. proof. it is similar to the proof of [19, proposition 2.16]. � now we have the following theorems. theorem 3.9 ([16, theorem 2.3]). let q be a lebesgue quasi-uniformity on x. then the corresponding di-uniformity u(q) on (x, p(x), πx) is a lebesgue direlational uniformity. conversely if u is a lebesgue direlational uniformity on (x, p(x), πx) then u−1(u) is a lebesgue quasi-uniformity on x. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 179 s. özçağ theorem 3.10 ([16, theorem 2.5]). let q be lebesgue quasi uniformity on x. then the complement of the direlational uniformity corresponding to q, that is u(q)′, is a co-lebesgue direlational uniformity on (x, p(x), πx). conversely, if u is the co-lebesgue direlational uniformity corresponding to q−1, then u−1(u′) is a lebesgue quasi uniformity on x. we are now in a position to give the promised result. theorem 3.11. let q be a lebesgue quasi-uniformity on x. then the corresponding lebesgue di-uniformity u(q) on (x, p(x), πx) is dicomplete. proof. let q be a lebesgue quasi-uniformity on x. we know from theorem 3.9 that the corresponding di-uniformity u(q) on (x, p(x)) is lebesgue. since every lebesgue quasi-uniformity is q-complete by theorem 3.4, the corresponding lebesgue di-uniformity u(q) is dicomplete by proposition 3.8. � theorem 3.12. let q be a lebesgue quasi-uniformity on x. if q is a uniformity, then the complement of the corresponding lebesgue di-uniformity u(q) on (x, p(x), πx) is dicomplete. proof. if q is a uniformity then q = q−1 and u(q−1) = u(q)′ = u(q) by proposition 3.6 and theorem 3.7. then by theorem 3.11 the co-lebesgue di-uniformity u(q)′ is dicomplete. � remark 3.13. since every dual covering lebesgue quasi-uniformity is a lebesgue quasi-uniformity it is clear that the last two results hold for the dual covering lebesgue quasi-uniformities. 4. conclusion remarks in [10] hutton gave the definition of uniformities and quasi uniformities on a hutton algebra lx using functions on lx. a similar representation was obtained in [18] for di-uniformities and quasi di-uniformities called difunctional uniformity and difunctional quasi-uniformity [18, definition 2.7]. for the further studies it would be interesting to investigate the lebesgue property on the difunctional uniformities and quasi-uniformities. acknowledgements. the author would like to thank the referees for their helpful suggestions and comments. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 180 lebesgue quasi-uniformity on textures references [1] l. m. brown, dual covering theory, confluence structures and the lattice of bicontinuous functions, ph.d. thesis, glasgow university, 1981. [2] l. m. brown and m. diker, paracompactness and full normality in ditopological texture spaces, journal of mathematical analysis and applications 227 (1998), 144–165. [3] l. m. brown and m. diker, ditopological texture spaces and intuitionistic sets, fuzzy sets and systems 98 (1998), 217–224. [4] l. m. brown, r. ertürk, fuzzy sets as texture spaces i. representations theorems, fuzzy sets and systems 110, no. 2 (2000), 227–236. [5] l. m. brown, r. ertürk and ş. dost, ditopological texture spaces and fuzzy topology, i. basic concepts, fuzzy sets and systems 147, no. 2 (2004), 171–199. [6] l. m. brown and m. m. gohar, compactness in ditopological texture spaces, hacettepe journal of mathematics and statistics 38, no. 1 (2009), 21–43. [7] p. fletcher and w. f. lindgren, quasi-uniform spaces, marcel dekker, new york and basel, 1982. [8] t. e. gantner and r. g. steinlage, characterizations of quasi-uniformities, journal of london mathematical sociey 11, no. 5 (1972), 48–52. [9] h. p. a. kunzi, an introduction to quasi-uniform spaces, in: beyond topology, contemporary mathematics, (f. mynard and e. pearl eds.), american mathematical society 468 (2009), 239–304. [10] b. hutton, uniformities on fuzzy topological spaces, journal of mathematical analysis and applications 58 (1977), 559–571. [11] j. marin and s. romaguera, on quasi uniformly continuous functions and lebesgue spaces, publicationes mathematicae debrecen 48 (1996), 347–355. [12] j. marin and s. romaguera on the bitopological extension of the bing metrization theorem, journal of australian mathematics society 44 (1988), 233–241. [13] s. özçağ and l. m. brown, di-uniform texture spaces, applied general topology 4, no. 1 (2003), 157–192. [14] s. özçağ, f. yıldız and l. m. brown, convergence of regular difilters and the completeness of di-uniformities, hacettepe journal of mathematics and statistics, 34 (2005), 53–68. [15] s. özçağ and l. m. brown, a textural view of the distinction between uniformities and quasi-uniformities, topology and its applications 153 (2006), 3294–3307. [16] s. özçağ, lebesgue and co-lebesgue di-uniform texture spaces, topology and its applications 156 (2009), 3021–3028. [17] s. özçağ, the concept of quasi-uniformity in texture space and its representations, questiones mathematicae 33 (2010), 457–476. [18] s. özçağ l. m. brown and b. krsteska, di-uniformities and hutton uniformities, fuzzy sets and systems 195 (2012), 58–74. [19] f. yıldız and l. m. brown, dicompleteness and real dicompactness of ditopological texture space, topology and its applications 158 (2011), 1976–1989. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 181 () @ appl. gen. topol. 18, no. 2 (2017), 377-390doi:10.4995/agt.2017.7452 c© agt, upv, 2017 some fixed point theorems on non-convex sets m. radhakrishnana, s. rajeshb and sushama agrawala a ramanujan institute for advanced study in mathematics, university of madras, chennai 600 005, india. (radhariasm@gmail.com,sushamamdu@gmail.com) b department of mathematics, indian institute of technology, tirupati 517 506, india. (srajeshiitmdt@gmail.com) communicated by e. a. sánchez-pérez abstract in this paper, we prove that if k is a nonempty weakly compact set in a banach space x, t : k → k is a nonexpansive map satisfying x+t x 2 ∈ k for all x ∈ k and if x is 3−uniformly convex or x has the opial property, then t has a fixed point in k. 2010 msc: 47h09; 47h10. keywords: fixed points; nonexpansive mappings; t −regular sets; k−uniform convex banach spaces; opial property. 1. introduction let k be a nonempty subset of a banach space x. a mapping t : k → k is said to be nonexpansive if ‖t x − t y‖ ≤ ‖x − y‖ for all x, y ∈ k. the following theorem was proved independently by browder [2] and göhde [8] in the setting of uniformly convex banach spaces. theorem 1.1 ([2]). let k be a nonempty weakly compact convex subset of a uniformly convex banach space x and t : k → k be a nonexpansive map. then t has a fixed point in k. using the notion of normal structure, kirk [10] proved the following theorem which is more general than theorem 1.1. received 28 march 2017 – accepted 08 june 2017 http://dx.doi.org/10.4995/agt.2017.7452 m. radhakrishnan, s. rajesh and sushama agrawal theorem 1.2 ([10]). let k be a nonempty weakly compact convex subset having normal structure in a banach space x and t : k → k be a nonexpansive map. then t has a fixed point in k. the convexity assumption cannot be dispense in the above theorems as can be seen from the following simple example. let k = [−2, −1]∪ [1, 2] ⊆ r and t is a self map on k defined by t x = −x for all x ∈ k. then t is nonexpansive, but t has no fixed points in k. this implies that nonexpansive map on a non-convex set in a banach space need not have a fixed point. motivated by theorem 1.1 and theorem 1.2, veeramani [20] introduced the notion of t −regular set as follows: let t be a self map on a nonempty subset k of a banach space x. then k is said to be a t −regular set if x+t x 2 ∈ k for all x ∈ k. clearly, if k is a convex set and t : k → k, then k is t −regular. but a t −regular set need not be a convex set(see example 3.2). further, veeramani [20] proved the following fixed point theorem. theorem 1.3 ([20]). let k be a nonempty weakly compact subset of a uniformly convex banach space x and t : k → k be a nonexpansive map. further, assume that k is t −regular. then t has a fixed point in k. khan and hussain [9] used the notion of t −regular sets to prove the existence of fixed points for nonexpansive mappings in the setting of metrizable topological vector space. also, goebel and schöneberg [6] proved the existence of fixed point for a nonexpansive map on certain nonconvex sets in a hilbert space. sullivan [18] introduced the concept of k−uniform convexity, k−uc in short, where k is any positive integer and proved that every k−uniformly convex banach space has normal structure. note that for k = 1, it is uniformly convex. sullivan [18] observed that every k−uc banach space is a (k + 1)−uc. but the converse is not true. for example, the banach space lp,1(n) [1] for 1 < p < ∞ is 2−uc but not 1−uc where lp,1(n) is the lp(n) space with suitable renorm. motivated by theorem 1.2, theorem 1.3 and the fact that k−uc banach spaces have normal structure [18], we raise the following question: does a nonexpansive map t on a nonempty weakly compact set k in a k−uc banach space have a fixed point if x+t x 2 ∈ k for all x ∈ k? in this paper, we give an affirmative answer to the above question, if x is a 3−uc banach space. for the proof of this result, lemma 3.3 and lemma 3.4 (the geometric inequality on k−uc banach space) are crucial. in another direction, opial [16] introduced a class of spaces for which the asymptotic center of a weakly convergent sequence coincides with the weak limit point of the sequence. gossez and lami dozo [7] have observed that all such spaces have normal structure. hence, in view of kirk’s theorem, every nonempty weakly compact convex set in a banach space which satisfy c© agt, upv, 2017 appl. gen. topol. 18, no. 2 378 some fixed point theorems on non-convex sets opial’s condition has fixed point property for a nonexpansive mapping. recently, suzuki [19] introduced a new class of mappings which also includes nonexpansive maps and proved that every nonempty weakly compact convex set in a banach space which satisfy opial’s condition also has fixed point property for all such maps. in this paper, we prove that if k is a nonempty weakly compact set in a banach space x having the opial property, t : k → k is a nonexpansive map and if k is t −regular set, then t has a fixed point point in k. moreover, the krasnoseleskii’s [12] iterated sequence {xn} where xn+1 = xn+t xn 2 for all n ∈ n and x1 ∈ k weakly converges to a fixed point. 2. preliminaries now, we give some basic definitions and results which are used in this paper. let x be a banach space. for a nonempty subset a of x, let co(a) = { n ∑ i=1 λixi : xi ∈ a, λi ≥ 0, for i = 1, 2, . . . , n and n ∑ i=1 λi = 1, n ∈ n } aff(a) = { n ∑ i=1 λixi : xi ∈ a, λi ∈ r, for i = 1, 2, . . . , n and n ∑ i=1 λi = 1, n ∈ n } the sets co(a) and aff(a) are called the convex hull and the affine hull of a respectively. a set a is affine if a = aff(a). every affine set is a translation of a subspace and the subspace is uniquely defined by the affine set. the dimension of an affine set is the dimension of the corresponding subspace. further, the dimension of a convex set a is defined as the dimension of the smallest affine set which contains a. this shows that the dimension of co(a) is the dimension of aff(a). sliverman [17] introduced the notion of volume of k + 1 vectors, denoted by v (x1, x2, . . . , xk+1), as follows: given x1, x2, . . . , xk+1 ∈ x, v (x1, x2, . . . , xk+1) = 1 k! sup          ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ f1(x2 − x1) . . . f1(xk+1 − x1) f2(x2 − x1) . . . f2(xk+1 − x1) . .. . .. . .. fk(x2 − x1) . . . fk(xk+1 − x1) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ : f1, . . . , fk ∈ bx∗          by the consequences of hahn-banach theorem, v (x1, x2) = ‖x1 − x2‖ for any x1, x2 ∈ x. note that v (x1, x2, . . . , xk+1) = 0 iff the dimension of the convex hull of {x1, x2, . . . , xk+1} does not exceed k − 1. using the notion of volume of k+1 vectors, sullivan [18] defined the concept of k−uniform convexity. we put µ (k) x = sup{v (x1, . . . , xk+1) : x1, . . . , xk+1 ∈ bx}. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 379 m. radhakrishnan, s. rajesh and sushama agrawal definition 2.1 ([18]). the modulus of k−convexity is defined as δ (k) x (ǫ) = inf { 1 − 1 k + 1 ∥ ∥ ∥ ∥ ∥ k+1 ∑ i=1 xi ∥ ∥ ∥ ∥ ∥ : x1, . . . , xk+1 ∈ bx and v (x1, . . . , xk+1) ≥ ǫ } where ǫ ∈ [0, µ (k) x ). a banach space x is said to be k−uniformly convex if δ (k) x (ǫ) > 0 for every 0 < ǫ < µ (k) x . note that all banach spaces of dimension less than k + 1 are k−uc. for more information on k−uc, one can refer to [11, 14, 15]. lim [13] proved the continuity of modulus δ (k) x of k−convexity using the following inequality. theorem 2.2 ([13]). let x be a banach space and k be any positive integer. for every 0 < ǫ1 < c < ǫ2 < µ (k) x , δ (k) x (c) − δ (k) x (ǫ1) c − ǫ1 ≤ 1 k(ǫ 1/k 2 − ǫ 1/k 1 )ǫ 1−1/k 1 corollary 2.3 ([13]). let x be a banach space. then δ (k) x (ǫ) is continuous on [0, µ (k) x ). definition 2.4 ([16]). a banach space x is said to have the opial property if {xn} is a weakly convergent sequence in x with limit z, then lim inf n→∞ ‖xn − z‖ < lim inf n→∞ ‖xn − y‖ for all y ∈ x with y 6= z. it is known that [5] hilbert spaces, finite dimensional banach spaces and lp(n) (1 < p < ∞) have the opial property. edelstein [3] introduced the notion of asymptotic center as follows: definition 2.5 ([3]). let k be a nonempty subset of a banach space x and {xn} be a bounded sequence in x. for each x ∈ x, define r(x) = lim sup n→∞ ‖x − xn‖. the number r = inf x∈k r(x) and the set a(k, {xn}) = {x ∈ k : r(x) = r} are called the asymptotic radius and asymptotic center of {xn} with respect to k respectively. we use the next lemma in the sequel, which is proved by goebel and kirk [4]. lemma 2.6 ([4]). let {zn} and {wn} be bounded sequences in a banach space x and let λ ∈ (0, 1). suppose that zn+1 = λwn + (1 − λ)zn and ‖wn+1 − wn‖ ≤ ‖zn+1 − zn‖ for all n ∈ n. then lim n→∞ ‖wn − zn‖ = 0. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 380 some fixed point theorems on non-convex sets 3. main results 3.1. 3−uc banach spaces. in this section, we first give the convergence theorem for a nonexpansive map t defined on a compact t −regular set in a banach space x. also, we prove the existence of fixed points for a nonexpansive map t defined on a weakly compact t −regular set in a 3−uc banach space x. theorem 3.1. let k be a nonempty compact subset of a banach space x and t : k → k be a nonexpansive map. further, assume that k is t −regular. define a sequence {xn} in k by xn+1 = xn+t xn 2 for n ∈ n and x1 ∈ k. then t has a fixed point in k and {xn} strongly converges to a fixed point of t. proof. since xn+1 = xn+t xn 2 for n ∈ n, by lemma 2.6, we have lim n→∞ ‖xn − t xn‖ = 0. since k is compact and {xn} ⊆ k, there exists a subsequence {xnk} of {xn} and z ∈ k such that {xnk } converges to z. now, by the continuity of t , {t xnk} converges to t z. but, note that lim k→∞ ‖xnk − t xnk‖ = 0. hence {xnk} also converges to t z. this implies that t z = z. also, note that {‖xn − z‖} is a decreasing sequence. for, ‖xn+1 − z‖ ≤ 1 2 ‖xn − z‖ + 1 2 ‖t xn − z‖ ≤ ‖xn − z‖, for all n ∈ n therefore {xn} converges to z, as {xnk} converges to z in norm. � example 3.2. let k = {(x, 0, 1 2n ), (0, y, 1 2n ), (x, x, 1 2n ), (x, 0, 0), (0, y, 0), (x, x, 0) : 0 ≤ x, y ≤ 1 and n ∈ n} be a subset of (r3, ‖.‖2). define a map t : k → k by t (x, y, z) = (y, x, 0) for all (x, y, z) ∈ k. it is easy to see that k is t −regular. also, note that t is nonexpansive. for, let x = (x1, y1, z1), y = (x2, y2, z2) ∈ k. then ‖t x − t y‖2 = ‖(y1 − y2, x1 − x2, 0)‖2 ≤ ‖(x1 − x2, y1 − y2, z1 − z2)‖2 = ‖x − y‖2 by theorem 3.1, t has a fixed point in k, since k is compact and t −regular. also, note that fix(t ) = {(x, x, 0) : 0 ≤ x ≤ 1}. lemma 3.3. let k be a nonempty weakly compact subset of a banach space x and t : k → k be a nonexpansive map. further, assume that k is t −regular. define a sequence {xn} in k by xn+1 = xn+t xn 2 for n ∈ n and x1 ∈ k. then the asymptotic center a(k, {xn}) of {xn} with respect to k is also a nonempty weakly compact t −regular subset of k. moreover, if k is a minimal weakly compact t −regular set, then a(k, {xn}) = k. proof. since r(x) = lim sup n→∞ ‖x−xn‖ is a weakly lower semicontinuous function on x and k is weakly compact, a(k, {xn}) = {x ∈ k : r(x) = inf y∈k r(y) = r} is nonempty. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 381 m. radhakrishnan, s. rajesh and sushama agrawal also {x ∈ x : r(x) ≤ inf y∈k r(y)} is a weakly closed set, this implies that a(k, {xn}) = {x ∈ x : r(x) ≤ inf y∈k r(y)} ∩ k is a weakly closed set. moreover, since t is nonexpansive and lim n→∞ ‖xn − t xn‖ = 0, a(k, {xn}) is t −invariant. now, it is claimed that a(k, {xn}) is a t −regular set. let x ∈ a(k, {xn}). then t x ∈ a(k, {xn}) and ∥ ∥ ∥ ∥ x + t x 2 − xn ∥ ∥ ∥ ∥ ≤ 1 2 ‖x − xn‖ + 1 2 ‖t x − xn‖. this implies that lim sup n→∞ ∥ ∥ ∥ ∥ x + t x 2 − xn ∥ ∥ ∥ ∥ = r. therefore x+t x 2 ∈ a(k, {xn}). hence a(k, {xn}) is a nonempty weakly compact t −regular subset of k. suppose that k is a nonempty minimal weakly compact t −regular set. then a(k, {xn}) = k, as a(k, {xn}) ⊆ k is also a nonempty weakly compact t −regular set. � lemma 3.4. let x be a k−uc banach space, for some k ∈ n and x1, x2, . . . , xk+1 ∈ bx such that v (x1, x2, . . . , xk+1) = ǫ > 0. then ‖t1x1 + t2x2 + · · · + tk+1xk+1‖ ≤ 1 − (k + 1) min{t1, t2, . . . , tk+1}δ (k) x (ǫ), where k+1 ∑ i=1 ti = 1, ti ≥ 0 for i = 1, 2, . . . , k + 1. proof. without loss of generality, we can assume that t1 = min{t1, t2, . . . , tk+1}. ‖t1x1 + t2x2 + · · · + tk+1xk+1‖ = ‖t1(x1 + · · · + xk+1) + (t2 − t1)x2 + (t3 − t1)x3 + · · · + (tk+1 − t1)xk+1‖ ≤ (k + 1)t1 ∥ ∥ ∥ ∥ x1 + x2 + · · · + xk+1 k + 1 ∥ ∥ ∥ ∥ + (t2 − t1)‖x2‖ +(t3 − t1)‖x3‖ + · · · + (tk+1 − t1)‖xk+1‖ ≤ (k + 1)t1(1 − δ (k) x (ǫ)) + t2 + t3 + · · · + tk+1 − kt1 = (k + 1)t1 − (k + 1)t1δ (k) x (ǫ) + 1 − (k + 1)t1 = 1 − (k + 1)t1δ (k) x (ǫ) hence ‖t1x1+t2x2+· · ·+tk+1xk+1‖ ≤ 1−(k+1) min{t1, t2, . . . , tk+1}δ (k) x (ǫ). � remark 3.5. now from lemma 3.4, we have: (1) if k = 2 and t1 = t2 = 1 4 , then ∥ ∥ ∥ x1 4 + x2 4 + x3 2 ∥ ∥ ∥ ≤ 1 − 3 4 δ (2) x (ǫ). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 382 some fixed point theorems on non-convex sets (2) if k = 3 and t1 = t2 = 1 8 , t3 = 1 4 then ∥ ∥ ∥ x1 8 + x2 8 + x3 4 + x4 2 ∥ ∥ ∥ ≤ 1 − 1 2 δ (3) x (ǫ). (3) if k = 3 and t1 + t2 + t3 = 1 2 , then ∥ ∥ ∥ ∥ t1x1 + t2x2 + t3x3 + 1 2 x4 ∥ ∥ ∥ ∥ ≤ 1 − 4 min{t1, t2, t3}δ (3) x (ǫ). we obtain the intuitive and geometric idea for the proof of our main result theorem 3.7 from the proof technique of the following theorem. theorem 3.6. let k be a nonempty weakly compact subset of a 2−uniformly convex banach space x and t : k → k be a nonexpansive map. further, assume that k is t −regular. then t has a fixed point in k. proof. define f = {f ⊆ k : f is nonempty weakly compact t −regular set} . it is easy to see that the set inclusion ⊆, defines a partial order relation on f. by zorn’s lemma, we get a minimal element in f. without loss of generality, we can assume that k is minimal in f. let x1 ∈ k and define xk+1 = xk+t xk 2 ∈ k, for k ∈ n. by lemma 3.3, we have a(k, {xk}) = k i.e., r(x) = lim sup k→∞ ‖x − xk‖ = r, for all x ∈ k. note that r = 0 if and only if k is singleton. for, if r = 0, then lim sup k→∞ ‖x − xk‖ = 0, for all x ∈ k. this gives {xk} converges to every point in k. hence k is singleton. conversely, suppose that k is singleton. then it is easy to see that r = 0, as {xk} ⊆ k. we claim that r = 0. suppose that r > 0. this implies that x 6= t x, for all x ∈ k. it is claimed that t xn ∈ aff{x1, t x1} for all n ∈ n. suppose that there exists n ∈ n such that t xn /∈ aff{x1, t x1}. without loss of generality, we can assume that t x2 /∈ aff{x1, t x1}. this gives {x1, t x1, t x2} is affinely independent and dim(co{x1, t x1, t x2} = 2. hence v (x1, t x1, t x2) = ǫ for some ǫ > 0. since x is 2−uc and δ (2) x is continuous, we have lim ρ→0 (r + ρ) ( 1 − 3 4 δ (2) x ( ǫ (r + ρ)2 )) = r ( 1 − 3 4 δ (2) x ( ǫ r2 ) ) < r this implies that there is a ρ0 > 0 such that (r + ρ0) ( 1 − 3 4 δ (2) x ( ǫ (r + ρ0)2 )) < r. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 383 m. radhakrishnan, s. rajesh and sushama agrawal since a(k, {xk}) = k and for this ρ0 > 0, there exists n ∈ n such that for k ≥ n, we have ‖x1 − xk‖ ≤ r + ρ0 ‖t x1 − xk‖ ≤ r + ρ0 ‖t x2 − xk‖ ≤ r + ρ0 as x is 2−uc, we have ∥ ∥ ∥ ∥ x1 + t x1 + t x2 3 − xk ∥ ∥ ∥ ∥ ≤ (r + ρ0) ( 1 − δ (2) x ( ǫ (r + ρ0)2 )) , for k ≥ n. note that x3 = x1 4 + t x1 4 + t x2 2 ∈ co{x1, t x1, t x2} and by lemma 3.4, we get ‖x3 − xk‖ = ∥ ∥ ∥ ∥ x1 4 + t x1 4 + t x2 2 − xk ∥ ∥ ∥ ∥ ≤ (r + ρ0) ( 1 − 3 4 δ (2) x ( ǫ (r + ρ0)2 )) , for k ≥ n. this implies that r(x3) = lim sup k→∞ ‖x3 − xk‖ ≤ (r + ρ0) ( 1 − 3 4 δ (2) x ( ǫ (r + ρ0)2 )) < r. this gives a contradiction to a(k, {xk}) = k. therefore t xn ∈ aff{x1, t x1}, for all n ∈ n. this implies that {xn} ⊆ aff{x1, t x1}. since {xn} is a bounded sequence and dim(aff{x1, t x1}) = 1, so it has a convergent subsequence say {xnj } of {xn} and z ∈ k such that xnj → z as j → ∞. since lim j→∞ ‖xnj − t xnj ‖ = 0 and t is nonexpansive, t z = z. hence r = 0. this implies that k is singleton and t has a fixed point in k. � next we prove the main result of this paper. theorem 3.7. let k be a nonempty weakly compact subset of a 3−uniformly convex banach space x and t : k → k be a nonexpansive map. further, assume that k is t −regular. then t has a fixed point in k. proof. note that by using zorn’s lemma, we get a nonempty minimal weakly compact t −regular subset of k. without loss of generality, we can assume that k is a nonempty minimal weakly compact t −regular set. let x1 ∈ k and define xk+1 = xk+t xk 2 ∈ k, for k ∈ n. by lemma 3.3, we have a(k, {xk}) = k i.e., r(x) = lim sup k→∞ ‖x − xk‖ = r, for all x ∈ k. we claim that r = 0. suppose that r > 0. this implies that x 6= t x, for all x ∈ k. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 384 some fixed point theorems on non-convex sets suppose that for every n ∈ n, t xn ∈ aff{x1, t x1}. then {xn} is a bounded sequence in aff{x1, t x1}, as k is bounded. hence {xn} has a convergent subsequence. this implies that t has a fixed point in k. suppose that there exists n ∈ n such that t xn 6∈ aff{x1, t x1}. without loss of generality, we can assume that t x2 6∈ aff{x1, t x1}. it is claimed that t xn ∈ aff{x1, t x1, t x2}, for all n ∈ n. we use mathematical induction to prove our claim. case 1. it is claimed that t x3 ∈ aff{x1, t x1, t x2}. suppose that t x3 6∈ aff{x1, t x1, t x2}. this gives {x1, t x1, t x2, t x3} is affinely independent and dim(co{x1, t x1, t x2, t x3}) = 3. hence v (x1, t x1, t x2, t x3) = ǫ, for some ǫ > 0. since x is 3−uc and δ (3) x is continuous, there is a ρ0 > 0 such that (r + ρ0) ( 1 − 1 2 δ (3) x ( ǫ (r + ρ0)3 )) < r. since a(k, {xk}) = k, there exists n ∈ n such that for k ≥ n, we have ‖x1 − xk‖ ≤ r + ρ0 ‖t x1 − xk‖ ≤ r + ρ0 ‖t x2 − xk‖ ≤ r + ρ0 ‖t x3 − xk‖ ≤ r + ρ0 as x is 3−uc, we have for k ≥ n ∥ ∥ ∥ ∥ x1 + t x1 + t x2 + t x3 4 − xk ∥ ∥ ∥ ∥ ≤ (r + ρ0) ( 1 − δ (3) x ( ǫ (r + ρ0)3 )) . note that x4 = x3+t x3 2 = x2+t x2 4 +t x3 2 = x1 8 +t x1 8 +t x2 4 +t x3 2 ∈ co{x1, t x1, t x2, t x3}. now, by lemma 3.4, we get ‖x4 − xk‖ = ∥ ∥ ∥ ∥ x1 8 + t x1 8 + t x2 4 + t x3 2 − xk ∥ ∥ ∥ ∥ ≤ (r + ρ0) ( 1 − 1 2 δ (3) x ( ǫ (r + ρ0)3 )) , for k ≥ n. this implies that r(x4) = lim sup k→∞ ‖x4 − xk‖ ≤ (r + ρ0) ( 1 − 1 2 δ (3) x ( ǫ (r + ρ0)3 )) < r. this gives a contradiction to a(k, {xk}) = k. hence t x3 ∈ aff{x1, t x1, t x2}. case 2. it is claimed that t x4 ∈ aff{x1, t x1, t x2}. suppose that t x4 /∈ aff{x1, t x1, t x2}. this gives {x1, t x1, t x2, t x4} is affinely independent and dim(co{x1, t x1, t x2, t x4}) = 3. since t x3 ∈ aff{x1, t x1, t x2}, we have the following cases: c© agt, upv, 2017 appl. gen. topol. 18, no. 2 385 m. radhakrishnan, s. rajesh and sushama agrawal (a). t x3 ∈ aff{x2, t x2} (b). t x3 6∈ aff{x2, t x2}. subcase 2(a). suppose that t x3 ∈ aff{x2, t x2}. then t x3 = (1 − µ3)x2 + µ3t x2, for some µ3 ∈ r. by the nonexpansiveness of t, we have 1 2 ‖t x2 − x2‖ = ‖x3 − x2‖ ≥ ‖t x3 − t x2‖ = |1 − µ3|‖t x2 − x2‖. this gives 1 2 ≤ µ3 ≤ 3 2 . note that µ3 6= 1 2 . for, if µ3 = 1 2 , then t x3 = x3. now x4 = x3 + t x3 2 = 1 2 ( x2 + t x2 2 + t x3 ) = x2 4 + 1 4 ( t x3 − (1 − µ3)x2 µ3 ) + t x3 2 = ( 2µ3 − 1 4µ3 ) x2 + ( 2µ3 + 1 4µ3 ) t x3 = ( 2µ3 − 1 8µ3 ) x1 + ( 2µ3 − 1 8µ3 ) t x1 + ( 2µ3 + 1 4µ3 ) t x3 = t1x1 + t1t x1 + (1 − 2t1)t x3 where t1 = 2µ3 − 1 8µ3 . since µ3 > 1 2 , we have t1 > 0 and 1 − 2t1 > 0. this gives x4 lies in the interior of co{x1, t x1, t x3}. since {x1, t x1, t x2, t x4} is affinely independent and t x3 ∈ aff{x2, t x2}, we have {x1, t x1, t x3, t x4} is affinely independent and dim(co{x1, t x1, t x3, t x4}) = 3. hence v (x1, t x1, t x3, t x4) = ǫ for some ǫ > 0. since δ (3) x is continuous and x is 3−uc, there is a ρ0 > 0 such that (r + ρ0) ( 1 − 2 min{t1, 1 − 2t1}δ (3) x ( ǫ (r + ρ0)3 )) < r as a(k, {xk}) = k, there exist n ∈ n such that for k ≥ n, we have ∥ ∥ ∥ ∥ x1 + t x1 + t x3 + t x4 4 − xk ∥ ∥ ∥ ∥ ≤ (r + ρ0) ( 1 − δ (3) x ( ǫ (r + ρ0)3 )) . note that x5 = x4+t x4 2 = 1 2 (t1x1 + t1t x1 + (1 − 2t1)t x3 + t x4) . this implies that x5 lies in the interior of co{x1, t x1, t x3, t x4}. now, by lemma 3.4, for k ≥ n we have ‖x5 − xk‖ = ∥ ∥ ∥ ∥ 1 2 (t1x1 + t1t x1 + (1 − 2t1)t x3 + t x4) − xk ∥ ∥ ∥ ∥ ≤ (r + ρ0) ( 1 − 2 min{t1, 1 − 2t1}δ (3) x ( ǫ (r + ρ0)3 )) . this implies that r(x5) = lim sup k→∞ ‖x5 − xk‖ ≤ (r + ρ0) ( 1 − 2 min{t1, 1 − 2t1}δ (3) x ( ǫ (r + ρ0)3 )) < r. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 386 some fixed point theorems on non-convex sets this gives a contradiction to a(k, {xk}) = k. hence t x4 ∈ aff{x1, t x1, t x2}. subcase 2(b). suppose that t x3 /∈ aff{x2, t x2}. then {x2, t x2, t x3} is affinely independent and dim(co{x2, t x2, t x3}) = 2. since t x3 ∈ aff{x1, t x1, t x2} and t x3 6∈ aff{x2, t x2}, we have t x3 = ax1 + bt x1 + (1 − (a + b))t x2, for a, b ∈ r with a 6= b. since {x1, t x1, t x2, t x4} is affinely independent and t x3 = ax1 + bt x1 + (1 − (a + b))t x2, we have {x2, t x2, t x3, t x4} is affinely independent and dim(co{x2, t x2, t x3, t x4}) = 3. this implies that v (x2, t x2, t x3, t x4) = ǫ, for some ǫ > 0. therefore by case 1, we get r(x5) < r. this gives a contradiction to a(k, {xk}) = k. hence t x4 ∈ aff{x1, t x1, t x2}. case 3. now, we assume that t xn ∈ aff{x1, t x1, t x2}, for 1 ≤ n ≤ m − 1. to prove that t xm ∈ aff{x1, t x1, t x2}. suppose not. then {x1, t x1, t x2, t xm} is affinely independent. since t xk ∈ aff{x1, t x1, t x2} for 3 ≤ k ≤ m − 1, we have the following cases: (a). t xk ∈ aff{x2, t x2} for k = 3, 4, . . . , m − 1 (b). t xk 6∈ aff{x2, t x2} for some k ∈ {3, 4, . . . , m − 1}. subcase 3(a). suppose that t xk ∈ aff{x2, t x2} for 3 ≤ k ≤ m − 1. then xk ∈ aff{x2, t x2} for 3 ≤ k ≤ m, as xk = xk−1+t xk−1 2 . let xk = (1 − λk)x2 + λkt x2 for some λk ∈ r, 2 ≤ k ≤ m and t xk = (1 − µk)x2 + µkt x2 for some µk ∈ r, 2 ≤ k ≤ m − 1. note that λk+1 = λk+µk 2 , for 2 ≤ k ≤ m − 1, as xk+1 = xk+t xk 2 . hence λ3 = 1 2 , as λ2 = 0, µ2 = 1. since we work with the aff{x2, t x2}, we can identify the aff{x2, t x2} with the real line r by assuming x2 = 0 and t x2 = 1. in this way, we get that xk = λk and t xk = µk for 2 ≤ k ≤ m − 1. as t xk 6= xk, we have λk 6= µk and λk 6= λk+1 for 2 ≤ k ≤ m − 1. note that, from case 2(a), we have λ3 < µ3. this implies that λ3 < λ4 < µ3, as λk+1 = λk+µk 2 . it is claimed that λk < λk+1 and λk < µk, for 4 ≤ k ≤ m − 1. since t is nonexpansive, we have |µ4 − µ3|‖x2 − t x2‖ = ‖t x3 − t x4‖ ≤ ‖x3 − x4‖ = (λ4 − λ3)‖x2 − t x2‖. this implies that −λ4 + λ3 ≤ µ4 − µ3 ≤ λ4 − λ3. now, since λ4 = λ3+µ3 2 , we have λ4 < µ4. this gives λ4 < λ5 < µ4. continuing in this way, we get λk < λk+1 < µk for 3 ≤ k ≤ m − 1. hence 0 = λ2 < λ3 < λ4 < · · · < λm−1 < λm < µm−1. this implies that λk lies in the interior of co{λ2, µm−1} for 3 ≤ k ≤ m. hence xk lies in the interior of co{x2, t xm−1} for 3 ≤ k ≤ m. this implies that xm lies in the interior of co{x1, t x1, t xm−1}, as x2 = x1+t x1 2 . now, since aff{x1, t x1, t x2} =aff{x1, t x1, t xm−1} and t xm 6∈ aff{x1, t x1, t x2}, we have {x1, t x1, t xm−1, t xm} is affinely independent and dim(co{x1, t x1, t xm−1, t xm}) = 3. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 387 m. radhakrishnan, s. rajesh and sushama agrawal hence xm+1 lies in the interior of co{x1, t x1, t xm−1, t xm}, as xm+1 = xm+t xm 2 . now, by using the arguments as in case 2(a), it is easy to see that r(xm+1) = lim sup k→∞ ‖xm+1 − xk‖ < r. this gives a contradiction to a(k, {xk}) = k. hence t xm ∈ aff{x1, t x1, t x2}. subcase 3(b). suppose that there exists k ∈ n such that 3 ≤ k ≤ m − 1 and t xk 6∈ aff{x2, t x2}. let k0 be the least integer satisfying t xk0 6∈ aff{x2, t x2}. this implies t x3, t x4, . . . , t xk0−1 ∈ aff{x2, t x2}. then {xk0−1, t xk0−1, t xk0} is affinely independent and aff{xk0−1, t xk0−1, t xk0} = aff{x1, t x1, t x2}. now, we consider the set {xk0−1, t xk0−1, t xk0}. suppose that t xk ∈ aff{xk0, t xk0} for k0 + 1 ≤ k ≤ m − 1. then using the arguments as in case 3(a), it is easy to see that xm+1 lies in the interior of co{xk0−1, t xk0−1, t xm−1, t xm} and {xk0−1, t xk0−1, t xm−1, t xm} is affinely independent. now, it is apparent that r(xm+1) < r, as x is 3−uc. this gives a contradiction to a(k, {xk}) = k. hence t xm ∈ aff{x1, t x1, t x2}. suppose that there exists k ∈ n such that k0 + 1 ≤ k ≤ m − 1 and t xk 6∈ aff{xk0, t xk0}. let k1 be the least integer satisfying t xk1 6∈ aff{xk0, t xk0}. this implies that t xk0+1, t xk0+2, . . . , t xk1−1 ∈ aff{xk0, t xk0}. then {xk1−1, t xk1−1, t xk1} is affinely independent and aff{xk1−1, t xk1−1, t xk1} = aff{xk0−1, t xk0−1, t xk0}. now, we consider the set {xk1−1, t xk1−1, t xk1}. continuing in this way, we can find n0 is the largest integer such that k1 ≤ n0 ≤ m − 1 and t xn0 6∈ aff{xn0−1, t xn0−1}. this implies that t xn ∈ aff{xn0, t xn0} for n0 ≤ n ≤ m − 1. then using the arguments as in case 3(a), it is easy to see that xm+1 lies in the interior of co{xn0−1, t xn0−1, t xm−1, t xm} and {xn0−1, t xn0−1, t xm−1, t xm} is affinely independent. now, it is apparent that r(xm+1) < r, as x is 3−uc. this gives a contradiction to a(k, {xk}) = k. hence t xm ∈ aff{x1, t x1, t x2}. hence, by mathematical indution t xn ∈ aff{x1, t x1, t x2}, for all n ∈ n. this implies that {xn} ⊆ aff{x1, t x1, t x2}. since {xn} is a bounded sequence and dim(aff{x1, t x1, t x2}) = 2, so it has a convergent subsequence i.e., there exists a subsequence {xnj } of {xn} and z ∈ k such that xnj → z as j → ∞. since lim j→∞ ‖xnj − t xnj ‖ = 0 and t is nonexpansive, we have t z = z. hence r = 0. this implies that k is singleton and t has a fixed point in k. � remark 3.8. in the light of theorem 3.6 and theorem 3.7, it is natural to expect that if k is a nonempty weakly compact subset of a k−uc banach space x, for k > 3 and if t : k → k is a nonexpansive map satisfying x+t x 2 ∈ k for all x ∈ k, then t has a fixed point in k. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 388 some fixed point theorems on non-convex sets 3.2. banach space with opial property. theorem 3.9. let k be a nonempty weakly compact subset of a banach space x having the opial property and t : k → k be a nonexpansive map. further, assume that k is t −regular. define a sequence {xn} in k by xn+1 = xn+t xn 2 for n ∈ n and x1 ∈ k. then t has a fixed point in k and {xn} converges weakly to a fixed point of t. proof. by lemma 2.6, we have lim n→∞ ‖xn − t xn‖ = 0. since k is weakly compact, there exists a subsequence {xnk} of {xn} and z ∈ k such that {xnk } converges weakly to z. also, we have ‖xnk − t z‖ ≤ ‖xnk − t xnk‖ + ‖t xnk − t z‖, for all k ∈ n. hence lim inf k→∞ ‖xnk − t z‖ ≤ lim inf k→∞ ‖xnk − z‖. since x has the opial property, we obtain t z = z. also note that, {‖xn − z‖} is a decreasing sequence. it is claimed that {xn} converges weakly to z. suppose that {xn} does not converge weakly to z. then there exists a subsequence {xn̂j } of {xn} which does not converge weakly to z. since k is weakly compact and {xn̂j } ⊆ k, there exists a subsequence of {xn̂j } whose weak limit is w ∈ k and z 6= w. without loss of generality, we can assume that {xn̂j } converges weakly to w. it is easy to see that t w = w, as lim j→∞ ‖xn̂j −t xn̂j ‖ = 0. also, it is apparent that {‖xn − w‖} is a decreasing sequence, as t w = w. since x has the opial property, {xn̂j } converges weakly to w and {xnk } converges weakly to z, we have lim n→∞ ‖xn − z‖ = lim k→∞ ‖xnk − z‖ < lim k→∞ ‖xnk − w‖ = lim n→∞ ‖xn − w‖ = lim j→∞ ‖xn̂j − w‖ < lim j→∞ ‖xn̂j − z‖ = lim n→∞ ‖xn − z‖. this is a contradiction. hence {xn} weakly converges to z. � acknowledgements. the authors thank the anonymous reviewer for the comments and suggestions. also, the authors thank prof. p. veeramani, department of mathematics, iit madras (india) for the fruitful discussions regarding the subject matter of this paper. the first author thank the university grants commission (india), for providing financial support to carry out this research work in the form of project fellow through ramanujan institute for advanced study in mathematics, university of madras, chennai. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 389 m. radhakrishnan, s. rajesh and sushama agrawal references [1] j. m. ayerbe toledano, t. 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[20] p. veeramani, on some fixed point theorems on uniformly convex banach spaces, j. math. anal. appl. 167 (1992), 160–166. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 390 () @ appl. gen. topol. 18, no. 2 (2017), 391-400doi:10.4995/agt.2017.7673 c© agt, upv, 2017 generalized c-distance on cone b-metric spaces endowed with a graph and fixed point results kamal fallahi a,∗, mujahid abbas b and ghasem soleimani rad a,c,∗ a department of mathematics, payame noor university, p.o. box 19395-3697, tehran, iran. (fallahi1361@gmail.com, gh.soleimani2008@gmail.com) b department of mathematics, government college university katchery road, lahore 54000, pakistan & department of mathematics, king abdulaziz university, p. o. box 80203, jeddah 21589, saudi arabia. (abbas.mujahid@gmail.com) c young researchers and elite club, central tehran branch, islamic azad university, tehran, iran. (gha.soleimani.sci@iauctb.ac.ir) abstract the aim of this paper is to present fixed point results of contractive mappings in the framework of cone b-metric spaces endowed with a graph and associated with a generalized c-distance. some corollaries and an example are presented to support the main result proved herein. our results unify, extend and generalize various comparable results in the literature. 2010 msc: 46a19; 47h10; 05c20. keywords: cone b-metric space; generalized c-distance; fixed point; orbitally g-continuous mapping. 1. introduction the interplay between the notion of a nearness among abstract objects of a set and fixed point theory is very strong and fruitful. this gives rise to an interesting branch of nonlinear functional analysis called metric fixed point theory. this theory is studied in the framework of a set equipped with some notion of ∗corresponding author. received 12 may 2017 – accepted 15 july 2017 http://dx.doi.org/10.4995/agt.2017.7673 k. fallahi, m. abbas and g. soleimani rad a distance along with appropriate mappings satisfying certain contraction conditions and has many applications in economics, computer science and other related disciplines. the concept of a b-metric is one of the important measure of nearness defined by bakhtin [2] and boriceanu [5]. the reader interested in fixed point results in setup of b-metric spaces is referred to ([1, 6, 12, 14]). huang and zhang [15] defined the concept of cone metric space by replacing the range of a distance function with an ordered normed space equipped with an order structure induced by a cone and proved some fixed point results for contraction type mappings on such spaces ([26]). after that, the concept of bmetric spaces to cone b-metric spaces or cone metric type spaces are introduced ([11, 18]). kada et al. [20] introduced the concept of w-distance on metric spaces and solved non-convex minimization problems. cho et al. [7] defined the notion of a c-distance which is the cone version of a w-distance. recently, hussain et al. [17] defined the concept of wt-distance on b-metric spaces and proved some fixed point theorems under a wt-distance in partially ordered b-metric spaces (also, see [21]). bao et al. [3] defined generalized c-distance in cone b-metric spaces and obtained some fixed point results in ordered cone b-metric spaces. the aim of this paper is to prove the existence and uniqueness of fixed points for contractive mappings defined on cone b-metric spaces endowed with a graph and associated with a generalized c-distance. our results generalize and extend various results in the existing literature. it is worth mentioning that we have employed the weaker version of continuity of the mapping called orbitally g-continuity. 2. preliminaries let e be a real banach space. a subset p of e is called a cone if and only if: (i) p is nonempty, closed and p 6= {θ} (where θ is the zero element of e); (ii) a,b ∈ r,a,b ≥ 0 and x,y ∈ p implies that ax + by ∈ p ; (iii) p ∩ (−p) = {θ}. partial ordering on e is defined with help of a cone p as follows: x � y if and only if y − x ∈ p . we shall write x ≺ y to indicate that x � y but x 6= y and x ≪ y stands for y − x ∈ intp , where intp denotes the interior of p . unless or otherwise stated, it is assumed that e is a banach space, p is a cone in e with intp 6= ∅ and � is partial ordering on e induced by p . a cone p is normal or semi monotone if inf{‖ x + y ‖: x,y ∈ p and ‖ x ‖=‖ y ‖= 1} > 0 or equivalently, if there is a number k > 0 such that for all x,y ∈ p , θ � x � y implies that ‖x‖ ≤ k ‖y‖. the least positive number satisfying the above inequality is called a normal constant of p . if x = (x1, ...,xn) t ,y = (y1, ...,yn) t ∈ rn, then x � y means that xi ≤ yi, i = 1, ...,n. in this case, the c© agt, upv, 2017 appl. gen. topol. 18, no. 2 392 generalized c-distance in cone b-metric spaces endowed with a graph set p = {x = (x1, ...,xn) t ∈ rn : xi ≥ 0 for i = 1,2, ...,n} is a normal cone with k = 1. lemma 2.1. let u,c ∈ e and {xn} a sequence in e. then we have the following properties: (p1) if u � λu where u ∈ p and 0 < λ < 1, then u = θ; (p2) if c ∈ intp, θ � xn and xn → θ, then there exists n0 such that for all n > n0 we have xn ≪ c. definition 2.2 ([11, 18]). let x be a nonempty set and s ≥ 1 a given real number. a mapping d : x × x → e is said to be a cone b-metric on x if for any x,y,z ∈ x, the following conditions hold: (d1) θ � d(x,y) and d(x,y) = θ if and only if x = y; (d2) d(x,y) = d(y,x); (d3) d(x,z) � s(d(x,y) + d(y,z)). the pair (x,d) is called a cone b-metric space. obviously, for s = 1, the cone b-metric space is a cone metric space. moreover, if x is any nonempty set, e = r and p = [0,∞), then cone b-metric on x is a b-metric on x. definition 2.3 ([3]). let (x,d) be a cone b-metric space and s ≥ 1 a given real number. a mapping q : x × x → e is said to be a generalized c-distance on x if for any x,y,z ∈ x, the following properties are satisfied: (q1) θ � q(x,y), (q2) q(x,z) � s[q(x,y) + q(y,z)], (q3) if for all n ≥ 1, q(x,yn) � u for some u = ux, then q(x,y) � su, where {yn} is a sequence in x which converges to y ∈ x; (q4) for any c ∈ intp , there exists e ∈ e with θ ≪ e such that q(z,x) ≪ e and q(z,y) ≪ e imply that d(x,y) ≪ c. if (x,d) is a b-metric space, e = r and p = [0,∞). then, wt-distance [17] on a b-metric space x is a generalized c-distance. but the converse does not hold. furthermore, if s = 1, the generalized c-distance is a c-distance defined in [7]. also, if in the above definition, we take s = 1, e = r and p = [0,∞), then we obtain the definition of w-distance [20]. note that, if q is a generalized c-distance, then q(x,y) = θ is not necessarily equivalent to x = y. moreover, q(x,y) = q(y,x) does not necessarily hold for all x,y ∈ x. lemma 2.4. let (x,d) be a cone b-metric space and q a generalized c-distance on x. let {xn} and {yn} be sequences in x, {un} and {vn} two sequences in p converging to θ. for any x,y,z ∈ x, the following conditions hold: (qp1) if for all n ∈ n, q(xn,y) � un and q(xn,z) � vn, then y = z. in particular, if q(x,y) = θ and q(x,z) = θ, then y = z; (qp2) if for all n ∈ n, q(xn,yn) � un and q(xn,z) � vn, then {yn} converges to z; c© agt, upv, 2017 appl. gen. topol. 18, no. 2 393 k. fallahi, m. abbas and g. soleimani rad (qp3) if for m,n ∈ n, with m > n, we have q(xn,xm) � un , then {xn} is a cauchy sequence in x; (qp4) if for all n ∈ n, q(y,xn) � un then {xn} is a cauchy sequence in x. proof. following arguments similar to those given in [7], the lemma follows. � on the other hand, the interplay between the order among abstract objects of underlying mathematical structure and fixed point theory is very strong and fruitful. this gives rise to an interesting branch of nonlinear functional analysis called order oriented fixed point theory. this theory is studied in the framework of a partially ordered sets along with appropriate mappings satisfying certain order conditions and has many applications in economics, computer science and other related disciplines. existence of fixed points in partially ordered metric spaces was first investigated in 2004 by ran and reurings [25], and then by nieto and lopez [23]. jachymski [19] introduced a new approach in metric fixed point theory by replacing order structure with a graph structure on a metric space. in this way, the results obtained in ordered metric spaces are generalized. employing the notion of orbits, nicolae et al. [22] obtained some fixed point results for a new type of contraction mappings and for g-asymptotic contraction mapping in metric spaces endowed with a graph. bojor [4] defined the notion of greich type mappings and obtained a fixed point theorem for such mappings in metric spaces endowed with a graph. cholamjiak [8] proved fixed point theorems for a banach type contractive mapping on a complete tvs-cone metric spaces endowed with a graph. also, hussain et al. [16] proved new fixed point results for graphic weak ψ-contractive mappings. the following definitions and notations will be needed in the sequel. let (x,d) be a cone b-metric space and ∆ denotes the diagonal of x×x. let g be a directed graph such that set v (g) of its vertices is x and e(g) be the set of edges of a graph g which contains all loops; that is, (x,x) ∈ ∆ ⊂ e(g) for all x ∈ x. assume further that graph g has no parallel edges. thus one can identify the graph g with the ordered pair (v (g),e(g)). if x,y ∈ x, then a finite sequence {xi} n i=0 consisting of n + 1 vertices is called a path in g from x to y whenever x0 = x, xn = y and (xi−1,xi) is an edge of g for i = 1, . . . ,n. the graph g is called connected if there exists a path in g between any two vertices of g. the symbols g−1 and g̃ denote the graph which is obtained from g by reversing the directions of its edges and an undirected graph obtained from g by ignoring the directions of the edges, respectively. in other words, v (g−1) = v (g̃) = x, e(g−1) = { (x,y) : (y,x) ∈ e(g) } and e(g̃) = e(g) ∪ e(g−1). we denote by fix(t) the set of all fixed points of a self mapping t on x and xt the set of all points x ∈ x such that (x,tx) is an edge of a graph g. in other words, xt = {x ∈ x : (x,tx) ∈ e(g)}. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 394 generalized c-distance in cone b-metric spaces endowed with a graph following is the analogue of the concept of picard operators [24] in cone b-metric spaces. definition 2.5. let (x,d) be a cone b-metric space. a self mapping t on x is called a picard operator if t has a unique fixed point x∗ in x and t nx → x∗ for any x ∈ x. consistent with jachymski [19, definition 2.4], we introduce the concept of orbitally g-continuous for self mapping t on a cone b-metric space (see also [9]). definition 2.6. a mapping t : x → x is called orbitally g-continuous on x if for any x,y ∈ x and a sequence {bn} of positive integers with (t bnx,t bn+1x) ∈ e(g) for all n ≥ 1, t bnx → y implies that t(t bnx) → ty. note that a continuous mapping on a cone b-metric space is orbitally gcontinuous for all graphs g but the converse is not true in general. 3. main results the following is the main result of this paper. theorem 3.1. let (x,d) be a complete cone b-metric space associated with the generalized c-distance q and endowed with the graph g and s ≥ 1 be a given real number. also, let t : x → x be a orbitally g-continuous mapping such that the following conditions holds. (t1) t preserves the edges of g; that is, (x,y) ∈ e(g) implies that (tx,ty) ∈ e(g) for all x,y ∈ x; (t2) there exist nonnegative constants α,β,γ such that q(tx,ty) � αq(x,y) + βq(x,tx) + γq(y,ty) for all x,y ∈ x with (x,y) ∈ e(g), where s(α + β) + γ < 1. then t has a fixed point if and only if xt 6= ∅. moreover, for any x∗ in x with tx∗ = x∗, we have q(x∗,x∗) = θ. also, if the subgraph of g with the vertex set fix(t) is connected, then the restriction of t to xt is a picard operator. proof. as fix(t) ⊆ xt , if t has a fixed point then xt is nonempty. let x0 be a given point in xt . define a sequence {xn} in x by xn = txn−1 = t nx0. clearly, (xn,xn+1) ∈ e(g) for all n ∈ n. thus q(xn,xn+1) = q(txn−1,txn) � αq(xn−1,xn) + βq(xn−1,txn−1) + γq(xn,txn) � ( α + β ) q(xn−1,xn) + γq(xn,xn+1), which implies that q(xn,xn+1) � α + β 1 − γ q(xn−1,xn) for all n ∈ n. continuing this way, we have q(xn,xn+1) � λ nq(x0,x1) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 395 k. fallahi, m. abbas and g. soleimani rad for all n ∈ n, where 0 ≤ λ = α+β 1−γ < 1 s . let m > n. it follows from (q2) and 0 ≤ sλ < 1 that q(xn,xm) � s[q(xn,xn+1) + q(xn+1,xm)] � sq(xn,xn+1) + s[sq(xn+1,xn+2) + q(xn+2,xm)] ... � sq(xn,xn+1) + s 2q(xn+1,xn+2) + · · · + s m−nq(xm−1,xm)] � (sλn + s2λn+1 · · · + sm−nλm−1)q(x0,x1) � sλn 1 − sλ q(x0,x1) → θ when n → ∞. by (iii) of lemma 2.4, {xn} is a cauchy sequence in x. next we assume that there exists a point x∗ ∈ x such that xn = t nx0 → x∗ as n → ∞. as x0 ∈ xt , (t nx0,t n+1x0) ∈ e(g) for all n ≥ 0. from orbital g-continuity of t , we get t n+1x0 → tx∗ and hence tx∗ = x∗. now if tx∗ = x∗ for any x∗ ∈ x. then, from (t2) we have q(x∗,x∗) = q(tx∗,tx∗) � αq(x∗,x∗) + βq(x∗,tx∗) + γq(x∗,tx∗) = (α + β + γ)q(x∗,x∗). since 0 ≤ α + β + γ < s(α + β) + γ < 1, therefore q(x∗,x∗) = θ. now, if the subgraph of g with the vertex set fix(t) is connected and x∗∗ ∈ x is a fixed point of t . then there exists a path {xi} n i=0 in g from x∗ to x∗∗ such that x1, . . . ,xn−1 ∈ fix(t); that is, x0 = x∗, xn = x∗∗ and (xi,xi+1) ∈ e(g) for i = 0, . . . ,n −1. by (t2) and q(xi+1,xi+1) = q(xi,xi) = θ, we have q(xi,xi+1) = q(txi,txi+1) � αq(xi,xi+1) + βq(xi,txi) + γq(xi+1,txi+1) = αq(xi,xi+1) + βq(xi,xi) + γq(xi+1,xi+1) = αq(xi,xi+1). it follows from (p1) that q(xi,xi+1) = θ. since q(xi,xi) = θ and q(xi,xi+1) = θ, by definition 2.3 we have d(xi,xi+1) = 0; that is, xi = xi−1. consequently, x∗ = x0 = x1 = · · · = xn−1 = xn = x∗∗ and hence the fixed point of t is unique and the restriction of t to xt is a picard operator. � example 3.2. let x = [0,1], e = c1 r [0,1] with the norm ‖ϕ‖ = ‖ϕ‖∞ + ‖ϕ′‖∞, and p = {ϕ ∈ e : ϕ(t) ≥ 0 on [0,1]} a non-normal cone . define the mapping d : x × x → y by d(x,y) = |x − y|2 · ϕ(t), where ϕ(t) = 2t ∈ p ⊂ e with t ∈ [0,1]. then (x,d) is a cone b-metric space with constant s = 2. let the mapping q : x × x → e be given by q(x,y)(t) = y2 · 2t c© agt, upv, 2017 appl. gen. topol. 18, no. 2 396 generalized c-distance in cone b-metric spaces endowed with a graph where t ∈ [0,1]. then q is a generalized c-distance. define t : x → x by t(x) = { 1 2 if x = 1, x 2 4 if x 6= 1. clearly, t is not continuous at x = 1. now assume that x is endowed with a graph g = (v (g),e(g)), where v (g) = x and e(g) = {(x,x) : x ∈ x}. note that for any x,y ∈ x with (x,y) ∈ e(g), we have x = y. if x,y ∈ x and {bn} is a sequence of positive integers with (t bnx,t bn+1x) ∈ e(g) for all n ≥ 1 such that t bnx → y, then {t bnx} is necessarily a constant sequence. thus, for some y in x, we have t bnx = y for all n ≥ 1 and hence t(t bnx) → ty. take α = 1 4 , β = 1 5 and γ = 0. then 1) s(α + β) + γ = 2(1 4 + 1 5 ) = 9 10 < 1; 2) let x ∈ x with (x,x) ∈ e(g). if x 6= 1, then q(tx,tx)(t) = ( x2 4 )2 · 2t = x4 16 · 2t ≤ αq(x,x)(t) + βq(x,tx)(t) + γq(x,tx)(t). if x = 1, then we have q(t1,t1)(t) = ( 1 2 )2 · 2t = 1 4 · 2t ≤ αq(1,1)(t) + βq(1,t1)(t) + γq(1,t1)(t). 3) also (0,t0) = (0,0) ∈ e(g), so xt 6= ∅. thus, all the conditions of theorem 3.1 are satisfied. moreover, x∗ = 0 is a fixed point of t has a fixed point and q(0,0) = 0. remark 3.3. (i) since we need not to the continuity of mapping, the method of mentioned theorem generalize, extend and unify all of research papers on fixed point theorems in cone b-metric spaces associated with a generalized c-distance and cone metric spaces associated with a c-distances such as: cho et al. [7], bao et al. [3] and hussain et al. [17] (and also, all references contained in them about w-distance and c-distance). (ii) in 2012, ćirić et al. [10] show that the method of du [13] for contraction mappings in cone metric spaces cannot be applied for contraction mappings in cone metric spaces with a associated c-distance. also, their notes are hold for generalized c-distance in cone b-metric spaces. thus, our results are new and cannot to derived from the version of wt-distance in b-metric spaces. if a cone b-metric space x is endowed with the complete graph g0 whose vertex set coincides with x; that is, v (g0) = x and e(g0) = x × x and we set g = g0 in theorem 3.1, then the set xt coincides with the whole set x, where t is a self mapping on x. thus, we have the following corollary. corollary 3.4. let (x,d) be a complete cone b-metric space with constant s ≥ 1 associated with the generalized c-distance q and t : x → x a orbitally c© agt, upv, 2017 appl. gen. topol. 18, no. 2 397 k. fallahi, m. abbas and g. soleimani rad continuous mapping. if there exist nonnegative constants α,β,γ such that q(tx,ty) � αq(x,y) + βq(x,tx) + γq(y,ty) for all x,y ∈ x, where s(α + β) + γ < 1. then t is a picard operator. suppose that (x,⊑) is a partially ordered set (poset). let g1 be the graph such v (g1) = x and e(g1) = { (x,y) ∈ x × x : x ⊑ y } . since ⊑ is reflexive, it follows that e(g1) contain all the loops. if we take g = g1 in theorem 3.1, then we obtain the following corollary. corollary 3.5. let (x,d,⊑) be a partially ordered complete cone b-metric space with constant s ≥ 1 associated with the generalized c-distance q and endowed with the graph g1. suppose that t : x → x is a nondecreasing orbitally g1-continuous mapping. if there exist nonnegative constants α,β,γ such that q(tx,ty) � αq(x,y) + βq(x,tx) + γq(y,ty) for all x,y ∈ x with x ⊑ y, where s(α + β) + γ < 1. then t has a fixed point in x if and only if there exists x0 ∈ x such that x0 ⊑ tx0. moreover, if tx∗ = x∗ for any x∗ ∈ x, then q(x∗,x∗) = θ. also, if the subgraph of g1 with the vertex set fix(t) is connected, then the restriction of t to the set of all points in x ∈ x satisfying x ⊑ tx is a picard operator. let x be a poset endowed with the graph g2 given by v (g2) = x and e(g2) = { (x,y) ∈ x × x : x ⊑ y ∨ y ⊑ x } . that is, an ordered pair (x,y) ∈ x × x is an edge of g2 if and only if x and y are comparable elements of (x,⊑). if we set g = g2 in theorem 3.1, then we obtain the following corollary. corollary 3.6. let (x,d,⊑) be a partially ordered complete cone b-metric space with s ≥ 1 associated with the generalized c-distance q and endowed with the graph g2. suppose that t : x → x is a nondecreasing orbitally g2continuous mapping which maps comparable elements of x onto comparable elements. if there exist nonnegative constants α,β,γ such that q(tx,ty) � αq(x,y) + βq(x,tx) + γq(y,ty) for all x,y ∈ x, where x and y are comparable and s(α+β)+γ < 1. then t has a fixed point in x if and only if there exists x0 ∈ x such that x0 and tx0 are comparable. moreover, tx∗ = x∗ for any x∗ in x implies that q(x∗,x∗) = θ. also, if every two elements of fix(t) are comparable, then the restriction of t to the set of all x ∈ x such x and tx are comparable is a picard operator. let e ∈ int p with θ ≪ e be a fixed. recall that two elements x,y ∈ x are said to be e-closed if d(x,y) ≺ e. define the e-graph g3 by v (g3) = x and e(g3) = { (x,y) ∈ x × x : d(x,y) ≺ e } . note that e(g3) contains all loops. finally, if we set g = g3 in theorem 3.1, then we obtain the following result. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 398 generalized c-distance in cone b-metric spaces endowed with a graph corollary 3.7. let (x,d) be a complete cone b-metric space with s ≥ 1 associated with the generalized c-distance q endowed with the graph g3. suppose that t : x → x is an orbitally g3-continuous mapping which maps e-close elements of x onto e-close elements. if there exist nonnegative constants α,β,γ such that q(tx,ty) � αq(x,y) + βq(x,tx) + γq(y,ty) for all x,y ∈ x, where x and y are e-close elements and s(α+β)+γ < 1. then t has a fixed point in x if and only if there exists x0 ∈ x such that x0 and tx0 are e-close. moreover, if tx∗ = x∗ for any x∗ ∈ x, then q(x∗,x∗) = θ. also, if every two elements of fix(t) are ε-close, then the restriction of t to the set of all x ∈ x such x and tx are e-close is a picard operator. note that our results can be easily proved for α,β,γ : x → [0,1) be functions with suitable conditions instead to be constants and for other well-known contractive conditions. also, as a new work, it will be interesting to study common fixed point results for two or more than two mappings with respect to the generalized c-distance on cone b-metric spaces endowed with the graph g by considering functions α,β,γ : x → [0,1) with suitable conditions. acknowledgements. the authors are grateful to the editorial team and referees for their accurate reading and their helpful suggestions. also, the first and the third authors are thankful to the department of mathematics of payame noor university for financial 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[26] p. p. zabrejko, k-metric and k-normed linear spaces: survey, collect. math. 48 (1997), 825–859. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 400 () @ appl. gen. topol. 19, no. 2 (2018), 223-237doi:10.4995/agt.2018.7955 c© agt, upv, 2018 completely simple endomorphism rings of modules victor bovdi a, mohamed salim a and mihail ursul b a department of mathematical sciences, uae university, united arab emirates (vbovdi@gmail.com, msalim@uaeu.ac.ae) b department of mathematics and computer science, university of technology, lae, papua new guinea (mihail.ursul@gmail.com) communicated by f. lin abstract it is proved that if ap is a countable elementary abelian p-group, then: (i) the ring end (ap) does not admit a nondiscrete locally compact ring topology. (ii) under (ch) the simple ring end (ap)/i, where i is the ideal of end (ap) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) the finite topology on end (ap) is the only second metrizable ring topology on it. moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained. 2010 msc: 16w80; 16n20; 16s50; 16n40. keywords: topological ring; endomorphism ring; bohr topology; finite topology; locally compact ring. 1. introduction the notion of associative simple ring can be extended for associative topological rings in several ways: (i) simple abstract ring endowed with a nondiscrete ring topology (for instance, the classification of nondiscrete locally compact division rings, see [25, chapter iv] and [4, 15, 16]; we refer to some historical notes about locally compact division rings to [29]); received 24 august 2017 – accepted 09 july 2018 http://dx.doi.org/10.4995/agt.2018.7955 v. bovdi, m. salim and mihail ursul (ii) topological ring without nontrivial closed ideals (see [22, 31]). (iii) topological ring r with the property that if f : r → s is a continuous homomorphism in a topological ring s, then either f = 0 or f is a topological embedding of r into s (see [24]). in all cases it is assumed that multiplication is not trivial. i. kaplansky has mentioned (see [20], p. 56) that the classification of locally compact simple rings in positive characteristic p is difficult. he proved that every simple nondiscrete locally compact simple torsion-free ring is a matrix ring over a locally compact division ring. however in [26] (see also [30]) has been constructed a nondiscrete locally compact simple ring of positive characteristic which is not a matrix ring over a division ring. thereby the program of classification of nondiscrete locally compact simple rings was finished. nevertheless it is interesting to look for new examples of locally compact simple rings. if ap is a countable elementary abelian p-group and i is the ideal of the ring end (ap) consisting of endomorphisms with finite images, then the factor ring end (ap)/i is a simple von neumann regular ring. we prove that under (ch) this ring does not admit a nondiscrete locally compact ring topology. s. ulam (see [23, problem 96, p. 181]) posed the following problem: ”can the group s∞ of all permutations of integers so metrized that the group operation (composition of permutations) is a continuous function and the set s∞ becomes, under this metric, a compact space? (locally compact?)”. e.d. gaughan (see [10]) has solved this problem in the negative. we study in §3 an analogous problem for the endomorphism ring of a countable elementary abelian p-group, namely: ”does the endomorphism ring end (ap) of a countable elementary p-group ap admit a nondiscrete locally compact ring topology?”. similarly to the ulam’s problem we obtain a negative answer to this question. moreover, we prove that tfin is the only ring topology t on end (ap) such that (end (ap),t ) is complete and second metrizable. we classify in §4 the completely simple rings (end (m),tfin) of vector spaces m over division rings. corollary 4.4 gives a characterization of semisimple left linearly compact minimal rings. it should be mentioned that corollary 4.4 is related to a result from [3] stating that any semisimple ring admits at most one linearly compact topology. furthermore, we obtain in §5 a description of completely simple rings of the form (end (mr),tfin) of modules m over a commutative ring r. we extend the result of [28] to topological rings (end (mr),tfin). 2. notation, conventions and preliminary results rings are assumed to be associative, not necessarily with identity. topological spaces are assumed to be completely regular. the weight (see [8], p.12) of the space x is denoted by w(x). the pseudocharacter of a point x ∈ x (see [8], p.135) is the smallest cardinal of the form |u|, where u is a family c© agt, upv, 2018 appl. gen. topol. 19, no. 2 224 completely simple endomorphism rings of modules of open subsets of x such that ∩u = {x}. the closure of a subset a of the topological space x is denoted by a and the interior by int(a) (see [8], p.14). a topological space x is called a baire space (see [8], p.198) if for each sequence {x1,x2, . . .} of open dense subsets of x the intersection ∩ ∞ i=1gi is a dense set. an abelian group a is called elementary abelian p-group (p prime) if pa = 0 for all a ∈ a. such group is a direct sum of copies of the cyclic group z(p). the subring of a ring r generated by a subset s, is denoted by 〈s〉. a ring r is called locally finite if every its finite subset is contained in a finite subring. a topological ring (r,t ) is called metrizable if its underlying additive group satisfies the first axiom of countability. a ring r with 1 is called dedekindfinite if each equality xy = 1 implies yx = 1. it is well-known that every finite ring with identity is dedekind-finite. since every compact ring with identity is a subdirect product of finite rings, it follows that every compact ring with identity is dedekind-finite. if a ⊆ r, then annl(a) := {x ∈ r | xa = 0}. if x,y are the subsets of r, then x · y := {xy | x ∈ x,y ∈ y }. a topological ring r is called compactly generated (see [27, chapter ii]) if there exists a compact subset k such that r = 〈k〉. if (r,t ) is a topological ring and i is an ideal of r, then the quotient topology of the factor ring r/i is denoted by t /i. if k is a subset of an abelian group a, then set t(k) = {α ∈ end (a) | α(k) = 0}. when k runs over all finite subsets of a, the family {t(k)} defines a ring topology tfin on end (a). this topology is called the finite topology. lemma 2.1. for any abelian group a the ring (end (a),tfin) is complete. proof. see [27, theorem 19.2]. � lemma 2.2 (cauchy’s criterion). in a hausdorff complete commutative group g, in order that a family (xα)α∈ω should be summable it is necessary and sufficient that, for each neighborhood v of zero in g, there is a finite subset ω0 of ω such that σα∈kxα ∈ v for all finite subsets k of ω which do not meet ω. proof. see [5], p.263. � lemma 2.3. if (xα)α∈ω is a summable subset in (end (a),tfin) then every subset ∆ of ω the family (xβ)β∈∆ is summable. proof. let v be a neighborhood of zero of (end (a),tfin). we can consider without loss of generality that v is a left ideal of end (a). there exists a finite subset ω0 of ω such that σα∈kxα ∈ v for every finite subset k of ω for which k ∩ ω0 = ∅. let f be a finite subset of ∆ such that f ∩ (ω0 ∩ ∆) = ∅. if α ∈ f , then α /∈ ω0, hence σα∈f xα ∈ v . by cauchy’s criterion the family (xβ)β∈∆ is summable. � a topological ring (r,t ) is called minimal (see, for instance, [7]) if there is no ring topology u such that u ≤ t and u 6= t . a topological ring (r,t ) is called simple if r is simple as a ring without topology. a topological ring (r,t ) is called weakly simple if r2 6= 0 and every its closed ideal is either 0 c© agt, upv, 2018 appl. gen. topol. 19, no. 2 225 v. bovdi, m. salim and mihail ursul or r. a topological ring (r,t ) is called completely simple (see [24]) if r2 6= 0 and for every continuous homomorphism f : (r,t ) → (s,u) in a topological ring (s,u) either ker(f) = r or f is a homeomorphism of (r,t ) on im(f). equivalently, r2 6= 0 and (r,t ) is weakly simple and minimal. let m be a unitary right r-module over a commutative ring r with 1. the module m is called divisible if mr = m for every 0 6= r ∈ r. a right r-module m is called faithful if mr = 0 implies r = 0 (r ∈ r). a right r-module m is called torsion-free if mr = 0 implies that either m = 0 or r = 0, where m ∈ m and r ∈ r. recall that a submodule n of an r-module m is called fully invariant α(n) ⊆ n for every endomorphism α of mr. we use in the sequel the notion and results from the books [8, 27]. remark 2.4. if r is a von neumann regular ring, then r2 = r. lemma 2.5. an ideal i of a von neumann regular ring is von neumann regular. proof. let i ∈ i. thus there exists x ∈ r such that ixi = i. it follows that ixixi = i and xix ∈ i. � corollary 2.6. if i an ideal of a von neumann regular ring r, then any ideal h of i is an ideal of r, too. proof. rh = rh2 ⊆ ih ⊆ h. similarly, hr ⊆ h. � if ap is a p-elementary countable group, then i = {α ∈ end (ap) | |im(α)| < ℵ0}. fix a linear basis {vi | i ∈ n} of ap over the field fp. using this fixed basis, we define the map ei : a → a such that ei(vj) = δijvj, (i,j ∈ n) where δij is the kronecker delta. lemma 2.7. we have for end (ap): (i) i is a von neumann regular ring. (ii) i is a simple ring. (iii) the factor ring end(ap)/i is simple von neumann regular. (iv) i is a locally finite ring. proof. (i): the ring end (ap) is regular (see [21, theorem 4.27, p. 63]), so i is von neumann regular by lemma 2.5. (ii), (iii): the ideal i is the only nontrivial ideal of the ring end (ap) (see [17, §17, theorem 1, p. 93]). this means that end (a)/i is simple. it is regular by the part (i). (iv) since i is simple (see [17, §12, proposition 1]), it suffices to show that i contains a nonzero locally finite right ideal. let us show that the left ideal ie1 of i is locally finite as a ring (equivalently, as a fp-algebra). we have 0 6= e1 ∈ ie1. if h is the left annihilator of ie1, then, c© agt, upv, 2018 appl. gen. topol. 19, no. 2 226 completely simple endomorphism rings of modules obviously, h is a locally finite ring, hence it is locally finite as a fp-algebra. we claim that ie1/h is finite. define βn ∈ h (n ≥ 2) in the following way βn(vi) = { vn, for i = 1; 0, for i 6= 1. let us prove that ie1 = fpe1 + σ ∞ n=2fpβn. if α ∈ i, then α(v1) = r1v1 + · · · + rnvn, where ri ∈ fp and n ∈ n, so αe1(v1) = r1e1(v1) + r2β2(v1) + · · · + rnβn(v1) = (r1e1 + r2β2 + · · · + rnβn)(v1); αe1(vj) = (r1e1 + r2β2 + · · · + rnβn)(vj) (j 6= 1). this yields αe1 = r1e1 + r2β2 + · · · + rnβn and so ie1 = fpe1 + σ ∞ n=2fpβn. in particular, ie1 = fpe1 + h, and so h has a finite index in ie1. clearly, ie1 is a locally finite fp-algebra (see [17, proposition 1, p. 241]) and i is a locally finite fp-algebra (see [17, proposition 2, p. 242]). � the next result can be deduced from [27, lemma 36.11]. lemma 2.8. let a be a locally compact, compactly generated, and totally disconnected ring. if a contains a dense locally finite subring b, then a is compact. proof. let a = 〈v 〉, where v is a compact symmetric neighborhood of zero. since v is compact, the subset v +v +v ·v also is compact. since b is dense, a = b+v . by compactness of v +v +v ·v there exists a finite subset h ⊆ b such that v +v +v ·v ⊆ h +v . since b is a locally finite ring, we can assume without loss of generality that h is a subring. let h \{0} = {h1, . . . ,hk}. the subset h + h1v + · · · + hkv + v is an open subgroup of r(+). indeed, this subset is symmetric and (h + h1v + · · · + hkv + v ) + (h + h1v + · · · + hkv + v ) ⊆ h + h1(v + v ) + · · · + hk(v + v ) + v + v ⊆ h + h1v + · · · + hkv + v . we prove by induction on m that v [m] ⊆ h + h1v + · · · + hkv + v, (m ∈ n) where v [1] = v and v [m] = v [m−1] · v for all m. the inclusion is obvious for m = 1. assume that the assertion has been proved for m ≥ 1. clearly, v [m+1] = v [m] · v ⊆ h · v + h1(v · v ) + · · · + hk(v · v ) + v · v ⊆ h1v + · · · + hkv + h1(h + v ) + · · · + hk(h + v ) + h + v ⊆ h + h1v + · · · + hkv + v . c© agt, upv, 2018 appl. gen. topol. 19, no. 2 227 v. bovdi, m. salim and mihail ursul consequently, a = h + h1v + · · · + hkv + v , therefore a is compact. � an element x of a topological ring is called discrete if there exists a neighborhood v of zero such that xv = 0 (i.e., the right annihilator of x is open). lemma 2.9. the set of all discrete elements of a topological ring is an ideal. a simple ring with identity does not contain nonzero discrete elements. 3. locally compact ring topologies on end (a) of a countable elementary abelian p-group a theorem 3.1. let r be a simple, nondiscrete and locally compact ring of char(r) = p > 0 and 1 ∈ r. if v is a compact open subring of r and {eα | α ∈ ω} is a set of orthogonal idempotents in r, then |ω| ≤ w(v ). proof. the ring r does not contain nonzero discrete elements by lemma 2.9. since r is locally compact and char(r) = p, it is totally disconnected. additionally, r has a fundamental system of neighborhoods of zero consisting of compact open subrings by [19, lemma 9]. if v is a compact open subring of r, then by continuity of the ring operations for each α ∈ ω there exists an open ideal vα of v such that eαvα ⊆ v . clearly, there exists yα ∈ vα for which eαyα 6= 0 since r has no nonzero discrete elements. we claim that hold the following two properties: (i) eαyα 6∈ {eβyβ | β 6= α} for each α ∈ ω; (ii) the set x = {eαyα | α ∈ ω} is a discrete subspace of v . indeed, if were eαyα ∈ {eβyβ | β 6= α} for some α ∈ ω, then eαyα = eαeαyα ∈ eα{eβyβ | β 6= α} ⊆ {eαeβyβ | β 6= α} = {0}, so eαyα = 0, a contradiction. the part (i) is proved. (ii) now, for each α ∈ ω we have v \ {eβyβ | β 6= α} is open and, consequently, (v \ {eβyβ | β 6= α}) ∩ x = {eαyα}, by (i). therefore the point eαyα(α ∈ ω) of x is isolated. in other words, the subspace x of v is discrete. since x is discrete, |ω| = |x| = w(x) ≤ w(v ) (see [1, exercises 98-99, p. 72]). � theorem 3.2. let ap be a countable elementary abelian p-group. then the ring i = {α ∈ end (ap) | |im(α)| < ℵ0} does not admit a nondiscrete ring topology u such that (i,u) is a baire space. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 228 completely simple endomorphism rings of modules proof. put sn = {α ∈ i | α(a) ⊆ fpv1 + · · · + fpvn}, where n ∈ n. clearly, i = ∪n∈nsn and sn = {α ∈ i | eiα = 0 for i > n} = annr ( {ek | k > n} ) . this yields that the subset sn is closed due the continuity of the ring operations. since i is a baire space, there exists n ∈ n such that int(sn) 6= ∅, hence sn is an open subgroup. set β ∈ i such that β(vi) = { vn+i, for i = 1, . . . ,n; 0 , for i > n. let w ⊆ sn be a neighborhood of zero of (i,u) such that βw ⊆ sn. if w ∈ w \ {0}, then there exist a ∈ a and r1, . . . ,rn ∈ fp such that 0 6= w(a) = n ∑ i=1 rivi and β(w(a)) = n ∑ i=1 rivn+i. there exists j ∈ 1, . . . ,n such that rj 6= 0. then en+jβw(a) = rjvn+j 6= 0, hence en+jβw 6= 0 and so βw 6∈ sn, a contradiction. � corollary 3.3. under the notation of theorem 3.2 the ring i does not admit a nondiscrete locally compact ring topology. proof. this follows from the fact that each locally compact space is a baire space (see [6, theorem 1, p. 117]). � our main result is the following. theorem 3.4. the endomorphism ring end (ap) of a countable elementary abelian p-group ap does not admit a nondiscrete locally compact ring topology. proof. we use the notation and results from section 2. denote r = end (ap). assume on the contrary that there exists on r a nondiscrete locally compact ring topology t . fact 1. the ring (r,t ) has a fundamental system of neighborhoods of zero consisting of compact open subrings. since the additive group of the ring r has exponent p, it is totally disconnected (this follows from [12, theorem 9.14, p. 95]). by i. kaplansky’s result (see [19, lemma 9]), the ring (r,t ) has a fundamental system of neighborhoods of zero consisting of compact open subrings. fact 2. the group rei is countable for each i ∈ n. we claim that rei is infinite. indeed, for each j ∈ n put βj ∈ r such that βj(vk) = { vj, for k = i; 0, for k 6= i. if j 6= s, then βjei(vi) = βj(vi) = vj and βsei(vi) = βs(vi) = vs, hence βjei 6= βsei, so rei is infinite. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 229 v. bovdi, m. salim and mihail ursul the ring rei is countable. indeed, consider the mapping ψ : rei → a fpvi p , where ψ(αei)(rvi) = α(rvi) for all r ∈ fp. if αei 6= βei (α,β ∈ r), then there exists an element x = ∑ j rjvj ∈ ap such that αei(x) 6= βei(x), hence, α(rivi) 6= β(rivi). thus ψ(αei)(rivi) = α(rivi) 6= β(rivi) = ψ(βei)(rivi). the latter means that ψ is an injective mapping of rei into a fpvi. since afpvi is countable, rei is countable, too. fact 3. i is a closed ideal of r. we claim that i is not dense in the topological ring (r,t ). assume the contrary. since i is locally finite and is a maximal ideal, (r,t ) is topologically locally finite by lemma 2.8. the ring r contains two elements x,y such that xy = 1 and yx 6= 1. the subring 〈x,y〉 is compact, hence dedekind-finite, a contradiction. we obtained that (r/i,t /i) is a nondiscrete metrizable locally compact ring. fact 4. i is a discrete ideal of r. this follows from theorem 3.2. fact 5. rei is a discrete left ideal of r for every i ∈ n. indeed, rei ⊆ i and i is discrete by fact 4 for every i ∈ n. fact 6. annl(ei) is open in r for every i ∈ n. indeed, the group homomorphism q : r → rei,r 7→ rei, is continuous. since rei is discrete q −1(0) = annl(ei) is open. fact 7. ∩iannl(ei) = 0. obvious. fact 8. t ≥ tfin. we notice that annl(ei) = t({vi}) for every i ∈ n. for, if αei = 0, then α(vi) = αei(vi) = 0, i.e., α ∈ t({vi}). conversely, if α ∈ t({vi}), then αei(vi) = α(vi) = 0. if j 6= i then αei(vj) = 0. therefore αei = 0. moreover t({v1, . . . ,vn}) = ∩ n i=1t({vi}) = ∩ n i=1annl(ei) ∈ t (∀n ∈ n). since the family {t({v1, . . . ,vn})} forms a fundamental system of neighborhoods of zero of (r,tfin), we get that tfin ≤ t . fact 9. the ring (r,t ) is metrizable. since ∩i∈nannl(ei) = 0, the pseudocharacter of (r,t ) is ℵ0. if v is a compact open subring of (r,t ) (see fact 1), then the pseudocharacter of v also is ℵ0. however in every compact space the pseudocharacter of a point coincides with its character. therefore (r,t ) is metrizable. fact 10. (r/i,t /i) has an open compact subring. indeed, it is well-known (see [19]) that every totally disconnected ring has a fundamental system of neighborhood of zero consisting of compact open subrings. henceforth v is a fixed open compact subring of (r/i,t /i). fact 11. r/i contains a family of orthogonal idempotents of cardinality 2ℵ0. indeed, the family {ei}i∈n of idempotents of the ring (r,tfin) is summable and 1a = σn∈nen, where 1a is the identity of r. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 230 completely simple endomorphism rings of modules the first ordinal number of cardinality c of continuum is denoted by ω(c). let {n(α) | α < ω(c)} be a family of infinite almost disjoint subsets of n (see [8, example 3.6.18, p. 175–176]). put fn(α) = σi∈n(α)ei for each α < ω(c). the element fn(α) exists by lemma 2.3. then: (i) fn(α) /∈ i for every α < ω(c); (ii) fn(α)fn(β) ∈ i for each α,β < ω(c) and α 6= β. if gα = fn(α) + i for each α < ω(c), then {gα | α < ω(c)} is the required system of orthogonal idempotents. the subring v is metrizable (by fact 9). since v is compact and r/i is a simple von neumann regular ring by lemma 2.7 and w(v ) ≤ ℵ0, we obtain a contradiction to theorem 3.1. � theorem 3.5. (ch) under the notation of theorem 3.4, the ring r/i does not admit a nondiscrete locally compact ring topology. proof. assume on the contrary that the factor ring r/i admits a nondiscrete locally compact ring topology t , so (r/i,t ) contains an open compact subring v . since the cardinality of r/i is continuum and v is infinite, the power of v is continuum. since we have assumed (ch), the subring v is metrizable, hence second metrizable (see [14, 18]). however we have proved in theorem 3.4 that the ring r/i contains a family of orthogonal idempotents of cardinality c, a contradiction with theorem 3.1. � theorem 3.6. the finite topology tfin is the only second metrizable ring topology t on r for which (r,tfin) is complete. proof. let k = 〈f〉, where f is a finite subset of a. clearly, there exists a subgroup a′ of a such that a = k ⊕a′. choose ef ∈ r such that ef ↾k= idk and ef (a ′) = 0. clearly, t(k) = r(1 − ef ) and αk = 0 if and only if α ∈ r(1 − ef ), so the family {r(1 − ef )}, where f runs over all finite subset of a, forms a fundamental system of neighborhoods of zero for (r,tfin). there exists an injective map of ref to hom(k,a), so the left ideal ref is countable, due to countability hom(k,a). since e2f = ef , the peirce decomposition r = ref ⊕ r(1 − ef ) of r with respect to the idempotent ef is a decomposition of the topological group (r,+,t ). it follows that ref is discrete, hence r(1−ef ) is open (in the topology t ). hence t ≥ tfin, so t = tfin (see [9, theorem 30] or [11]). � 4. completely simple topological endomorphism rings of vector spaces theorem 4.1. let af be a right vector space over a division ring f and s = end (af ). the following conditions are equivalent: (i) (s,tfin) is a completely simple topological ring. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 231 v. bovdi, m. salim and mihail ursul (ii) dim(af ) = ∞ or dim(af ) < ∞ and f does not admit a nondiscrete ring topology. proof. (i)⇒ (ii): if af is finite-dimensional, then s is discrete and isomorphic to the matrix ring m(n,f), where n is the dimension of af . then, obviously, f does not admit a nondiscrete ring topology. (ii)⇒ (i): if dim(af ) = n < ∞, then s ∼= m(n,f). since f does not admit nondiscrete ring topologies, the same holds for m(n,f). let af be infinite dimensional. fix a basis {xα}α<τ over f , where τ is an infinite ordinal number. it is well-known that the topological ring (s,tfin) is weakly simple (see [22, satz 12, p. 258]) and the family {t(xα)}α<τ is a prebase at zero for the finite topology tfin of s. assume on the contrary that there exists a hausdorff ring topology t , coarser that tfin and different from it. let eα ∈ s such that e 2 α = eα and eα(xβ) = δαβxα for each α < τ, where δαβ is the kronecker delta. fact 1. t(xα) = annl(eα) for each α < τ. indeed, if p ∈ t(xα), then peα(xα) = p(xα) = 0. if β 6= α, then eα(xβ) = 0, hence peα = 0, i.e. p ∈ annl(eα). conversely, if peα = 0, then we have p(xα) = peα(xα) = 0, i.e. p ∈ t(xα). fact 2. there exists α0 < τ for which seα0 is nondiscrete in (s,t ). assume on the contrary that for every α < τ there exists a neighborhood vα of zero of (s,t ) such that seα ∩ vα = 0. if uα is a neighborhood of zero of (s,t ) such that uαeα ⊆ vα, then uαeα = 0, hence annl(eα) = t(xα) is open in (s,t ). hence tfin ≤ t and t = tfin, a contradiction. fact 3. (seα0 ∩ v )xα0 * ⊕β∈kxβf for any neighborhood v of of zero of (s,t ) and any finite subset k of the set [0,τ) of all ordinal numbers less than τ. assume on the contrary that there exists a finite subset k of [0,τ) and a neighborhood v of zero of (s,t ) such that (4.1) (seα0 ∩ v )xα0 ⊆ ⊕β∈kxβf. fix γ ∈ [0,τ) \ k. for each β ∈ k define qβ ∈ s such that qβ(xβ) = xγ and q(xδ) = 0 for δ 6= β. let v0 be a neighborhood of zero of (s,t ) such that v0 ⊆ v and qβv0 ⊆ v for all β ∈ k. there exists 0 6= h ∈ seα0 ∩ v0 by fact 2 and hxα0 6= 0 by fact 1. since seα0 ∩ v0 ⊆ seα0 ∩ v , we obtain that hxα0 = σβ∈kxβfβ, (fβ ∈ f) by (4.1). there exists β0 ∈ k such that fβ0 6= 0 (because hxα0 6= 0), so qβ0h = qβ0(σβ∈kxβfβ) = rβ0xγ 6∈ ⊕β∈kxβf, a contradiction. therefore fact 3 is proved. now let v be a neighborhood of zero of (s,t ). pick up a neighborhood v0 of zero of (s,t ) such that v0 · v0 ⊆ v . since t ≤ tfin, there exists a finite subset k of [0,τ) such that t({xβ | β ∈ k}) ⊆ v0. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 232 completely simple endomorphism rings of modules we have (seα0 ∩ v0)xα0 * ⊕β∈kxβf by fact 3. it follows that there exists q ∈ seα0 ∩ v0 such that q(xα0) 6∈ ⊕β∈kxβf. clearly, q(xα0 ) ∈ af , so it can be written as q(xα0 ) = ∑ α<τ xαfα, where fα ∈ f and there exists β0 6∈ k such that fβ0 6= 0. consider the element s ∈ s such that s(xβ0) = xα0f −1 β0 and s(xλ) = 0 for λ 6= β0. evidently, s ∈ t(k), hence sq ∈ t(k) · v0 ⊆ v0 · v0 ⊆ v. moreover, sq(xα0) = s(xβ0fβ0 + · · · ) = xα0. since q ∈ seα0, we obtain that sq(xβ) = 0 for β 6= α0. consequently, eα0 = sq ∈ v for every neighborhood v of zero of (s,t ), a contradiction. � remark 4.2. the question of existence of a uncountable division ring which does not admit a nondiscrete hausdorff ring topology is open. several results on this topic can be found in chapter 5 of [2]. theorem 4.3. let ∏ α∈ω rα be a family of compact rings with identity. then the product ( ∏ α∈ω rα, ∏ α∈ω tα) is a minimal ring if and only if every (rα,tα) is a minimal topological ring. (here ∏ α∈ω tα is the product topology on the ring ∏ α∈ω rα.) proof. ⇒: assume on the contrary that there exists β ∈ ω and a ring topology t ′ on rβ such that t ′ ≤ tβ and t ′ 6= tβ. consider the product topology u on ∏ α∈ω rα, where rα is endowed with tα when α 6= β and rβ is endowed with t ′. obviously, u ≤ ∏ α∈ω tα and u 6= ∏ α∈ω tα, a contradiction. ⇐: denote by πα(α ∈ ω) the projection of ∏ α∈ω rα on rα. by definition of the product topology, ∏ α∈ω tα is the coarsest topology on ∏ α∈ω rα for which the projections πα(α ∈ ω) are continuous. let u be a ring topology on ∏ α∈ω rα, u ≤ ∏ α∈ω tα and β ∈ ω. since u ↾rβ× ∏ γ 6=β {0γ}≤ ( ∏ α∈ω tα ) ↾rβ× ∏ γ 6=β {0γ}, it follows that u ↾rβ× ∏ γ 6=β {0γ }= ( ∏ α∈ω tα) ↾rβ× ∏ γ 6=β {0γ} by minimality of (rβ,tβ). then the family {v × ∏ γ 6=β{0γ}} when v runs all neighborhoods of zero of (rβ,tβ) is a fundamental system of neighborhoods of zero of ( rβ × ∏ γ 6=β {0γ}, u ↾rβ× ∏ γ 6=β {0γ} ) . since rβ × ∏ γ 6=β{0γ} is an ideal with identity of ∏ α∈ω rα, the topological ring ( ∏ α∈ω rα,u) is a direct sum of ideals rβ× ∏ γ 6=β{0γ} and {0β}× ∏ γ 6=β rγ. let v be a neighborhood of zero of (rβ,tβ). then v × ∏ γ 6=β rγ be a neighborhood of zero of ( ∏ α∈ω rα,u) and πβ(v × ∏ γ 6=β rγ) = v . we have proved that πβ is a continuous function from ( ∏ α∈ω rα,u) to (rβ,tβ). it follows that ∏ α∈ω tα ≤ u and so u = ∏ α∈ω tα. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 233 v. bovdi, m. salim and mihail ursul corollary 4.4. a left linearly compact semisimple ring is minimal if and only if has no direct summands of the form m(n,∆), where ∆ is a division ring which does not admit a nondiscrete hausdorff ring topology. proof. this follows from theorems 4.1, 4.3 and the theorem of leptin (see [22, theorem 13, p. 258]) about the structure of left linearly compact semisimple rings. � corollary 4.5. a semisimple linearly compact ring (r,t ) having no ideals isomorphic to matrix rings over infinite division rings is minimal. 5. completely simple endomorphism rings of modules the endomorphism ring of a right r-module m is denoted by end (mr). lemma 5.1. let m be a divisible, torsion-free module over a commutative domain r and k the field of fractions of r. the additive group of m has a structure of a vector k-space such that r-endomorphisms of m are exactly the k-linear transformations. proof. we define a structure of a right vector k-space as follows: if a b ∈ k and m ∈ m, then there exists a unique x ∈ m such that ma = xb; set m ◦ a b = x. moreover, if a b = c d and 0 6= m ∈ m, then m ◦ a b = m ◦ c d . indeed, if m ◦ a b = x and m ◦ c d = y, then mad = xbd and mbc = ybd which means that xbd = ybd, hence x = y. let α ∈ end (mr), a b ∈ k, m ∈ m. by definition, am = b(a b ◦ m), hence, aα(m) = bα(a b ◦ m), which means that α(a b ◦ m) = a b ◦ α(m), so α is a k-linear transformation. note that, if a ∈ r and m ∈ m, then m ◦ a 1 = ma. conversely, if α is a k-linear transformation, a ∈ r, m ∈ m, then α(a 1 ◦ m) = a 1 ◦ αm, i.e. α(am) = aα(m). we have proved that every k-linear transformation is an right r-module homomorphism. � remark 5.2. the center z(r) of a weakly simple ring r is a domain. remark 5.3. for every right r-module m the underlying group m(+) is a discrete left topological (end (mr),tfin)-module. indeed, t(m)(m) = 0 for every m ∈ m. moreover, end (mr){0} = {0}, so m is a discrete left topological (end (mr),tfin)-module. theorem 5.4. let mr be a module over a commutative ring r. if the topological ring (end (mr),tfin) is weakly simple, then: (i) p = {r ∈ r | mr = 0} is a prime ideal of r. (ii) m is a vector space over the field k of fractions of r/p and the rendomorphisms of m are exactly the k-linear transformations. conversely, if mr is an r-module and are satisfied (i) and (ii), then the ring (end (mr),tfin) is a weakly simple topological ring. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 234 completely simple endomorphism rings of modules proof. ⇒: if (end (mr),tfin) is weakly simple, then the mapping: (5.1) αr : m → m, m 7→ mr (r ∈ r) is an r-module homomorphism and αr ∈ z(= the center of end (mr)). first we show that the part (i) holds. indeed, if a,b ∈ r and ab = 0, then αaαb = 0 (see (5.1)). thus (end (mr)αa) · (end (mr)αb) = 0, so (end (mr)αa) · (end (mr)αb) = 0. since end (mr) is weakly simple, one of them, say end (mr)αa, is zero. this implies that αa = 0, hence a ∈ p . (ii) the structure of r/p-module on m is defined as follows: if r ∈ r and m ∈ m, then put m(r + p) = mr. note that m is a torsion-free right r/p-module. assume that m(r+p) = 0, where 0 6= r + p ∈ r/p and 0 6= m ∈ m. then mr = 0 = αr(m) (see (5.1)). thus end (mr)αr(m) = 0. it follows that ( end (mr)αr ) (m) = 0 by remark 5.3. since end (mr) is weakly simple end (mr)αr = end (mr). we obtained that end (mr)(m) = 0, so m = 0, a contradiction. under this convention r-submodules are exactly r/p-submodules and rendomorphisms are exactly r/p-endomorphisms. the module m is a divisible r/p-module. indeed, if 0 6= r+p ∈ r/p , then 0 6= m(r + p) = mr. suppose that mr 6= m. consider i = {α ∈ end (mr) | α(m) ⊆ mr}. since mr is a fully invariant submodule, i is a two-sided ideal of the ring (end (mr),tfin). the ideal i is closed. indeed, let α ∈ i. if m ∈ m, then there exists β ∈ i such that α − β ∈ t(m). clearly, α(m) = β(m) ∈ mr and so α ∈ i. we have proved that i is closed. since 1m /∈ i, i = 0. it follows that αr = 0 (see (5.1)), a contradiction. the module m has a structure of a right k-vector space and end (mr) is exactly the ring of endomorphisms of m by lemma 5.1. the converse follows from theorem 4.1. � a characterization of completely simple topological ring end (mr) is given by the following. theorem 5.5. let mr be a module over a commutative ring r. the topological ring (end (mr),tfin) is completely simple if and only are satisfied the conditions (i) and (ii) of theorem 5.4 and either (i) m is finite or (ii) m is infinite and the dimension of m over the field k is infinite. proof. ⇒: according to theorem 5.4, the ideal p is prime and the topology of end (mr) coincide with the finite topology of end (mk), where k is the field of fractions of r/p . if m is finite, we have the part (i). assume that c© agt, upv, 2018 appl. gen. topol. 19, no. 2 235 v. bovdi, m. salim and mihail ursul m is infinite. if r/p is finite, then the dimension of m over k is infinite. suppose that r/p is infinite and dimk(m) = n < ℵ0. then m is isomorphic to m(n,k). since k is an infinite field, it admits a nondiscrete ring topology (see [13]) and we obtain a contradiction because end (mr) is a discrete ring. consequently dimk(m) is infinite. ⇐ this follows from theorems 4.1 and 5.4. � corollary 5.6. the topological ring (end (a),tfin) of an abelian group a is completely simple if and only one of the following conditions holds: (i) a is a elementary abelian p-group. 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and department of computer science and engineering, suny at buffalo (boxer@niagara.edu) communicated by s. romaguera abstract we study properties of cartesian products of digital images, using a variety of adjacencies that have appeared in the literature. 2010 msc: 54c99; 05c99. keywords: digital topology; digital image; retraction; approximate fixed point property; continuous multivalued function; shy map. 1. introduction we study various adjacency relations for cartesian products of multiple digital images. we are particularly interested in “product properties” properties that are preserved by taking cartesian products and “factor properties” for which possession by a cartesian product of digital images implies possession of the property by the factors. many of the properties examined in this paper were considered in [9] for adjacencies based on the normal product adjacency. we consider other adjacencies in this paper, including the tensor product adjacency, the cartesian product adjacency, and the composition or lexicographic adjacency. received 23 january 2017 – accepted 05 october 2017 http://dx.doi.org/10.4995/agt.2018.7146 l. boxer 2. preliminaries much of the material that appears in this section is quoted or paraphrased from [9, 12], and other papers cited in this section. we use n, z, and r to represent the sets of natural numbers, integers, and real numbers, respectively, a digital image is a graph. usually, we consider the vertex set of a digital image to be a subset of zn for some n ∈ n. further, we often, although not always, restrict our study of digital images to finite graphs. we will assume familiarity with the topological theory of digital images. see, e.g., [3] for many of the standard definitions. all digital images x are assumed to carry their own adjacency relations (which may differ from one image to another). when we wish to emphasize the particular adjacency relation we write the image as (x, κ), where κ represents the adjacency relation. 2.1. common adjacencies. to denote that x and y are κ-adjacent points of some digital image, we use the notation x ↔κ y, or x ↔ y when κ can be understood. the cu-adjacencies are commonly used. let x, y ∈ z n, x 6= y. let u be an integer, 1 ≤ u ≤ n. we say x and y are cu-adjacent, x ↔cu y, if • there are at most u indices i for which |xi − yi| = 1, and • for all indices j such that |xj − yj| 6= 1 we have xj = yj. a cu-adjacency is often denoted by the number of points adjacent to a given point in zn using this adjacency. e.g., • in z1, c1-adjacency is 2-adjacency. • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency. • in z3, c1-adjacency is 6-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. for cartesian products of digital images, the normal product adjacency (see definitions 2.1 and 2.2) has been used in papers including [22, 6, 11, 9] (errors in [22] are corrected in [6]). the tensor product adjacency (see definition 2.3), cartesian product adjacency (see definition 2.4), and the lexicographic adjacency (see definition 2.6) have not to our knowledge been studied in digital topology, so their respective roles in digital topology remain to be determined. given digital images or graphs (x, κ) and (y, λ), the normal product adjacency np(κ, λ), also called the strong product adjacency (denoted κ∗(κ, λ) in [11]) generated by κ and λ on the cartesian product x × y is defined as follows. definition 2.1 ([1, 28]). let x, x′ ∈ x, y, y′ ∈ y . then (x, y) and (x′, y′) are np(κ, λ)-adjacent in x × y if and only if • x = x′ and y ↔λ y ′; or • x ↔κ x ′ and y = y′; or • x ↔κ x ′ and y ↔λ y ′. as a generalization of definition 2.1, we have the following. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 22 product adjacencies in digital topology definition 2.2 ([9]). let u and v be positive integers, 1 ≤ u ≤ v. let {(xi, κi)} v i=1 be digital images. let npu(κ1, . . . , κv) be the adjacency defined on the cartesian product πvi=1xi as follows. for xi, x ′ i ∈ xi, p = (x1, . . . , xv) and q = (x′1, . . . , x ′ v) are npu(κ1, . . . , κv)-adjacent if and only if • for at least 1 and at most u indices i, xi ↔κi x ′ i, and • for all other indices i, xi = x ′ i. definition 2.3 ([20]). the tensor product adjacency on the cartesian product πvi=1xi of (xi, κi), denoted t (κ1, . . . , κv), is as follows. given xi, x ′ i ∈ xi, we have (x1, . . . , xv) and (x ′ 1, . . . , x ′ v) are t (κ1, . . . , κv)-adjacent in π v i=1xi if and only if for all i, xi ↔κi x ′ i. figure 1. a digital simple closed curve and its cartesian product with [0, 1]z. (a) shows the simple closed curve msc8 ⊂ (z 2, c2) [21]. (b) shows the set msc8 × [0, 1]z ⊂ z 3 with either the c2 × c1or the np1(c2, c1)-adjacency. (c) shows the set msc8 × [0, 1]z ⊂ z 3 with the t (c2, c1)adjacency, where adjacencies are shown by the solid lines. if the points of msc8 are circularly labeled p0, . . . , p5, then the t (c2, c1)-neighbors of (pi, t) are (p(i−1) mod 6, 1 − t) and (p(i+1) mod 6, 1 − t), t ∈ {0, 1}. definition 2.4 ([26]). the cartesian product adjacency on the cartesian product πvi=1xi of (xi, κi), denoted × v i=1κi or κ1 × . . . × κv, is as follows. given xi, x ′ i ∈ xi, we have (x1, . . . , xv) and (x ′ 1, . . . , x ′ v) are × v i=1κi-adjacent in πvi=1xi if and only if for some i, xi ↔κi x ′ i, and for all indices j 6= i, xj = x ′ j. the following has an elementary proof. proposition 2.5. for πvi=1(xi, κi), × v i=1κi = np1(κ1, . . . , κv). definition 2.6 ([19]). let (xi, κi) be digital images, 1 ≤ i ≤ v. let xi, x ′ i ∈ xi. let p = (x1, . . . , xv), p ′ = (x′1, . . . , x ′ v). we say p and p ′ are adjacent in the composition or lexicographic adjacency on πvi=1xi if x1 ↔κ1 x ′ 1, or if for some c© agt, upv, 2018 appl. gen. topol. 19, no. 1 23 l. boxer index j, 1 ≤ j < v, we have (x1, . . . , xj) = (x ′ 1, . . . , x ′ j) and xj+1 ↔κj+1 x ′ j+1. the adjacency is denoted l(κ1, . . . , κv). figure 2. an illustration of lexicographic adjacency. this is [0, 1]z × {−2, 0, 2}, with both factors regarded as subsets of (z, c1), and the l(c1, c1) adjacency. remark 2.7. notice that for p and p′ to be l(κ1, . . . , κv)-adjacent with xk and x′k κk-adjacent, for indices m > k we do not require that xm and x ′ m be either equal or adjacent. see, e.g., figure 2, where (0, 0) and (1, 2) are l(c1, c1)-adjacent. this is unlike other adjacencies discussed above. 2.2. connectedness. a subset y of a digital image (x, κ) is κ-connected [25], or connected when κ is understood, if for every pair of points a, b ∈ y there exists a sequence {yi} m i=0 ⊂ y such that a = y0, b = ym, and yi ↔κ yi+1 for 0 ≤ i < m. for two subsets a, b ⊂ x, we will say that a and b are adjacent when there exist points a ∈ a and b ∈ b such that a and b are equal or adjacent. thus sets with nonempty intersection are automatically adjacent, while disjoint sets may or may not be adjacent. it is easy to see that a finite union of connected adjacent sets is connected. 2.3. continuous functions. the following generalizes a definition of [25]. definition 2.8 ([4]). let (x, κ) and (y, λ) be digital images. a function f : x → y is (κ, λ)-continuous if for every κ-connected a of x we have that f(a) is a λ-connected subset of y . when the adjacency relations are understood, we will simply say that f is continuous. continuity can be reformulated in terms of adjacency of points: theorem 2.9 ([25, 4]). a function f : x → y is continuous if and only if, for any adjacent points x, x′ ∈ x, the points f(x) and f(x′) are equal or adjacent. note that similar notions appear in [14, 15] under the names immersion, gradually varied operator, and gradually varied mapping. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 24 product adjacencies in digital topology theorem 2.10 ([3, 4]). if f : (a, κ) → (b, λ) and g : (b, λ) → (c, µ) are continuous, then g ◦ f : (a, κ) → (c, µ) is continuous. example 2.11 ([25]). a constant function between digital images is continuous. example 2.12. the identity function 1x : (x, κ) → (x, κ) is continuous. definition 2.13. let (x, κ) be a digital image in zn. let x, y ∈ x. a κ-path of length m from x to y is a set {xi} m i=0 ⊂ x such that x = x0, xm = y, and xi−1 and xi are equal or κ-adjacent for 1 ≤ i ≤ m. if x = y, we say {x} is a path of length 0 from x to x. notice that for a path from x to y as described above, the function f : [0, m]z → x defined by f(i) = xi is (c1, κ)-continuous. such a function is also called a κ-path of length m from x to y. 2.4. digital homotopy. a homotopy between continuous functions may be thought of as a continuous deformation of one of the functions into the other over a finite time period. definition 2.14 ([4]; see also [23]). let (x, κ) and (y, κ′) be digital images. let f, g : x → y be (κ, κ′)-continuous functions. suppose there is a positive integer m and a function f : x × [0, m]z → y such that • for all x ∈ x, f(x, 0) = f(x) and f(x, m) = g(x); • for all x ∈ x, the induced function fx : [0, m]z → y defined by fx(t) = f(x, t) for all t ∈ [0, m]z is (2, κ′)−continuous. that is, fx(t) is a path in y . • for all t ∈ [0, m]z, the induced function ft : x → y defined by ft(x) = f(x, t) for all x ∈ x is (κ, κ′)−continuous. then f is a digital (κ, κ′)−homotopy between f and g, and f and g are digitally (κ, κ′)−homotopic in y . if for some x0 ∈ x we have f(x0, t) = f(x0, 0) for all t ∈ [0, m]z, we say f holds x0 fixed, and f is a pointed homotopy. we denote a pair of homotopic functions as described above by f ≃κ,κ′ g. when the adjacency relations κ and κ′ are understood in context, we say f and g are digitally homotopic (or just homotopic) to abbreviate “digitally (κ, κ′)−homotopic in y ,” and write f ≃ g. proposition 2.15 ([23, 4]). digital homotopy is an equivalence relation among digitally continuous functions f : x → y . definition 2.16 ([5]). let f : x → y be a (κ, κ′)-continuous function and let g : y → x be a (κ′, κ)-continuous function such that f ◦ g ≃κ′,κ′ 1x and g ◦ f ≃κ,κ 1y . c© agt, upv, 2018 appl. gen. topol. 19, no. 1 25 l. boxer then we say x and y have the same (κ, κ′)-homotopy type and that x and y are (κ, κ′)-homotopy equivalent, denoted x ≃κ,κ′ y or as x ≃ y when κ and κ′ are understood. if for some x0 ∈ x and y0 ∈ y we have f(x0) = y0, g(y0) = x0, and there exists a homotopy between f ◦ g and 1x that holds x0 fixed, and a homotopy between g ◦ f and 1y that holds y0 fixed, we say (x, x0, κ) and (y, y0, κ ′) are pointed homotopy equivalent and that (x, x0) and (y, y0) have the same pointed homotopy type, denoted (x, x0) ≃κ,κ′ (y, y0) or as (x, x0) ≃ (y, y0) when κ and κ ′ are understood. it is easily seen, from proposition 2.15, that having the same homotopy type (respectively, the same pointed homotopy type) is an equivalence relation among digital images (respectively, among pointed digital images). 2.5. continuous and connectivity preserving multivalued functions. given sets x and y , a multivalued function f : x → y assigns a subset of y to each point of x. we will write f : x ⊸ y . for a ⊂ x and a multivalued function f : x ⊸ y , let f(a) = ⋃ x∈a f(x). definition 2.17 ([24]). a multivalued function f : x ⊸ y is connectivity preserving if f(a) ⊂ y is connected whenever a ⊂ x is connected. as is the case with definition 2.8, we can reformulate connectivity preservation in terms of adjacencies. theorem 2.18 ([12]). a multivalued function f : x ⊸ y is connectivity preserving if and only if the following are satisfied: • for every x ∈ x, f(x) is a connected subset of y . • for any adjacent points x, x′ ∈ x, the sets f(x) and f(x′) are adjacent. definition 2.17 is related to a definition of multivalued continuity for subsets of zn given and explored by escribano, giraldo, and sastre in [16, 17] based on subdivisions. (these papers make a small error with respect to compositions, that is corrected in [18].) their definitions are as follows: definition 2.19. for any positive integer r, the r-th subdivision of zn is z n r = {(z1/r, . . . , zn/r) | zi ∈ z}. an adjacency relation κ on zn naturally induces an adjacency relation (which we also call κ) on znr as follows: (z1/r, . . . , zn/r), (z ′ 1/r, . . . , z ′ n/r) are adjacent in znr if and only if (z1, . . . , zn) and (z ′ 1, . . . , z ′ n) are adjacent in z n. given a digital image (x, κ) ⊂ (zn, κ), the r-th subdivision of x is s(x, r) = {(x1, . . . , xn) ∈ z n r | (⌊x1⌋, . . . , ⌊xn⌋) ∈ x}. let er : s(x, r) → x be the natural map sending (x1, . . . , xn) ∈ s(x, r) to (⌊x1⌋, . . . , ⌊xn⌋). definition 2.20. for a digital image (x, κ) ⊂ (zn, κ), a function f : s(x, r) → y induces a multivalued function f : x ⊸ y if x ∈ x implies f(x) = ⋃ x′∈e −1 r (x) {f(x′)}. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 26 product adjacencies in digital topology definition 2.21. a multivalued function f : x ⊸ y is called continuous when there is some r such that f is induced by some single valued continuous function f : s(x, r) → y . figure 3. [12] two images x and y with their second subdivisions. (subdivisions are drawn at half-scale.) note [12] that the subdivision construction (and thus the notion of continuity) depends on the particular embedding of x as a subset of zn. in particular we may have x, y ⊂ zn with x isomorphic to y but s(x, r) not isomorphic to s(y, r). e.g., in figure 3, when we use 8-adjacency for all images, x and y are isomorphic, each being a set of two adjacent points, but s(x, 2) and s(y, 2) are not isomorphic since s(x, 2) can be disconnected by removing a single point, while this is impossible in s(y, 2). the definition of connectivity preservation makes no reference to x as being embedded inside of any particular integer lattice zn. proposition 2.22 ([16, 17]). let f : x ⊸ y be a continuous multivalued function between digital images. then • for all x ∈ x, f(x) is connected; and • for all connected subsets a of x, f(a) is connected. theorem 2.23 ([12]). for (x, κ) ⊂ (zn, κ), if f : x ⊸ y is a continuous multivalued function, then f is connectivity preserving. the subdivision machinery often makes it difficult to prove that a given multivalued function is continuous. by contrast, many maps can easily be shown to be connectivity preserving. 2.6. other notions of multivalued continuity. other notions of continuity have been given for multivalued functions between graphs (equivalently, between digital images). we have the following. definition 2.24 ([27]). let f : x ⊸ y be a multivalued function between digital images. • f has weak continuity if for each pair of adjacent x, y ∈ x, f(x) and f(y) are adjacent subsets of y . c© agt, upv, 2018 appl. gen. topol. 19, no. 1 27 l. boxer • f has strong continuity if for each pair of adjacent x, y ∈ x, every point of f(x) is adjacent or equal to some point of f(y) and every point of f(y) is adjacent or equal to some point of f(x). proposition 2.25 ([12]). let f : x ⊸ y be a multivalued function between digital images. then f is connectivity preserving if and only if f has weak continuity and for all x ∈ x, f(x) is connected. example 2.26 ([12]). if f : [0, 1]z ⊸ [0, 2]z is defined by f(0) = {0, 2}, f(1) = {1}, then f has both weak and strong continuity. thus a multivalued function between digital images that has weak or strong continuity need not have connected point-images. by theorem 2.18 and proposition 2.22 it follows that neither having weak continuity nor having strong continuity implies that a multivalued function is connectivity preserving or continuous. example 2.27 ([12]). let f : [0, 1]z ⊸ [0, 2]z be defined by f(0) = {0, 1}, f(1) = {2}. then f is continuous and has weak continuity but does not have strong continuity. proposition 2.28 ([12]). let f : x ⊸ y be a multivalued function between digital images. if f has strong continuity and for each x ∈ x, f(x) is connected, then f is connectivity preserving. the following shows that not requiring the image of a point f(p) to be connected can yield topologically unsatisfying consequences for weak and strong continuity. example 2.29 ([12]). let x and y be nonempty digital images. let the multivalued function f : x ⊸ y be defined by f(x) = y for all x ∈ x. • f has both weak and strong continuity. • f is connectivity preserving if and only if y is connected. as a specific example [12] consider x = {0} ⊂ z and y = {0, 2}, all with c1 adjacency. then the function f : x ⊸ y with f(0) = y has both weak and strong continuity, even though it maps a connected image surjectively onto a disconnected image. 2.7. shy maps and their inverses. definition 2.30 ([5]). let f : x → y be a continuous surjection of digital images. we say f is shy if • for each y ∈ y , f−1(y) is connected, and • for every y0, y1 ∈ y such that y0 and y1 are adjacent, f −1({y0, y1}) is connected. shy maps induce surjections on fundamental groups [5]. some relationships between shy maps f and their inverses f−1 as multivalued functions were studied in [7, 12, 8]. shyness as a factor or product property for the normal product adjacency was studied in [9]. we have the following. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 28 product adjacencies in digital topology theorem 2.31 ([12, 8]). let f : x → y be a continuous surjection between digital images. then the following are equivalent. • f is a shy map. • for every connected y0 ⊂ y , f −1(y0) is a connected subset of x. • f−1 : y ⊸ x is a connectivity preserving multi-valued function. • f−1 : y ⊸ x is a multi-valued function with weak continuity such that for all y ∈ y , f−1(y) is a connected subset of x. 2.8. other tools. other terminology we use includes the following. given a digital image (x, κ) ⊂ zn and x ∈ x, the set of points adjacent to x ∈ zn and the neighborhood of x in zn are, respectively, nκ(x) = {y ∈ z n | y is κ-adjacent to x}, n∗κ(x) = nκ(x) ∪ {x}. 3. maps on products in this section, we consider various product adjacencies with respect to continuity of functions. 3.1. general properties. definition 3.1. let κ1 and κ2 be adjacency relations on a set x. we say κ1 dominates κ2, κ1 ≥d κ2, or κ2 is dominated by κ1, κ2 ≤d κ1, if for x, x ′ ∈ x, if x and x′ are κ1-adjacent then x and x ′ are κ2-adjacent. example 3.2. we have the following comparisons of adjacencies. • for x ⊂ zn and 1 ≤ u ≤ v ≤ n, cu ≥d cv. • for πvi=1(xi, κi) and 1 ≤ u ≤ v ≤ n, npu(κ1, . . . κv) ≥d npv(κ1, . . . κv). • for πvi=1(xi, κi), t (κ1, . . . κv) ≥d npv(κ1, . . . κv). • for πvi=1(xi, κi), we have: – npu(κ1, . . . , κv) ≥d l(κ1, . . . , κv) for 1 ≤ u ≤ v; – t (κ1, . . . , κv) ≥d l(κ1, . . . , κv); – ×vi=1κi ≥d l(κ1, . . . , κv). proof. these follow immediately from the definitions of these adjacencies. � the next example shows that there are adjacencies that can be applied to the same set x such that neither dominates the other. example 3.3. in x = z6 = z3 × z3, neither of t (c2, c2) nor t (c1, c3) dominates the other. proof. consider the points p = (0, 0, 0, 0, 0, 0) and q = (1, 1, 0, 1, 1, 0). we have p ↔t (c2,c2) q but p and q are not t (c1, c3)-adjacent. therefore t (c2, c2) does not dominate t (c1, c3). now consider r = (1, 0, 0, 1, 1, 1). we have p ↔t (c1,c3) r but p and r are not t (c2, c2)-adjacent. therefore t (c1, c3) does not dominate t (c2, c2). � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 29 l. boxer domination, and being dominated, are transitive relations among the adjacencies of a graph. i.e., we have the following. proposition 3.4. given adjacencies κ, λ, µ for a graph, if κ ≤d λ and λ ≤d µ, then κ ≤d µ. proof. elementary, and left to the reader. � proposition 3.5. let f : x → y be a function. • let λ1 and λ2 be adjacency relations on y . if f is (κ, λ1) continuous and λ1 ≥d λ2, then f is (κ, λ2) continuous. • let κ1 and κ2 be adjacency relations on x. if f is (κ1, λ)-continuous and κ1 ≤d κ2, then f is (κ2, λ)-continuous. proof. the assertions follows from the definitions of continuity and the ≥d relation. � given functions fi : (xi, κi) → (yi, λi), 1 < i ≤ v, the function πvi=1fi : π v i=1xi → π v i=1yi is defined by (πvi=1fi)(x1, . . . , xv) = (f1(x1), . . . , fv(xv)), where xi ∈ xi. 3.2. normal product. here, we recall continuity properties of the normal product adjacency. theorem 3.6 ([9]). let fi : (xi, κi) → (yi, λi), 1 < i ≤ v. then the product map f = πvi=1fi : (π v i=1xi, npv(κ1, . . . , κv)) → (π v i=1yi, npv(λ1, . . . , λv)) is continuous if and only if each fi is continuous. theorem 3.7 ([9]). let x = πvi=1xi. let fi : (xi, κi) → (yi λi), 1 ≤ i ≤ v. • for 1 ≤ u ≤ v, if the product map f = πvi=1fi : (x, npu(κ1, . . . , κv)) → (πvi=1yi, npu(λ1, . . . , κv)) is an isomorphism, then for 1 ≤ i ≤ v, fi is an isomorphism. • if fi is an isomorphism for all i, then the product map f = π v i=1fi : (x, npv(κ1, . . . , κv)) → (π v i=1yi, npv(λ1, . . . , κv)) is an isomorphism. theorem 3.8 ([22, 9]). the projection maps pi : (π v j=1xj, npu(κ1, . . . , κv)) → (xi, κi) defined by pi(x1, . . . , xv) = xi for xi ∈ (xi, κi), are all continuous, for 1 ≤ u ≤ v. 3.3. tensor product. for the tensor product adjacency, we have the following. proposition 3.9. suppose x = πvi=1xi has a pair of t (κ1, . . . , κv)-adjacent points. then • each xi has 2 κi-adjacent points; and c© agt, upv, 2018 appl. gen. topol. 19, no. 1 30 product adjacencies in digital topology • if f : (x, t (κ1, . . . , κv)) → (π w j=1yj, t (λ1, . . . , λw)) is continuous and not constant on some component of x, then for every j, yj has 2 λjadjacent points. proof. let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) be t (κ1, . . . , κv)-adjacent in x. then for each i, xi and x ′ i are κi-adjacent in xi, which establishes the first assertion. further, if f is as hypothesized, the continuity of f implies there are t (κ1, . . . , κv)-adjacent p, p ′ such that f(p) = (y1, . . . , yw) and f(p ′) = (y′1, . . . , y ′ w) are unequal, hence t (λ1, . . . , λw)-adjacent. therefore, for all j, yj and y′j are λj-adjacent. � it is easy to construct examples showing that the assertions obtained from proposition 3.9 by substituting the normal product adjacency npv for t are false. theorem 3.10. let x = πvi=1xi, y = π v i=1yi. if the product map f = πvi=1fi : (x, t (κ1, . . . , κv)) → (y, t (λ1, . . . , λv)) is continuous, then for each i, fi : (xi, κi) → (yi, λi) is continuous. proof. if xi, x ′ i are κi-adjacent in xi, then p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) are t (κ1, . . . , κv)-adjacent in x. thus f(p) and f(p ′) are equal or t (λ1, . . . , λv)adjacent in y . this implies fi(xi) and fi(x ′ i) are equal or λi-adjacent in yi. thus fi is continuous. � however, the converse to theorem 3.10 is not generally true, as shown in the following. example 3.11. let f : [0, 1]z → [0, 1]z be the identity function. let g : [0, 1]z → [0, 1]z be the constant function g(x) = 0. then, using examples 2.12 and 2.11, f and g are each (c1, c1)-continuous, but f × g : [0, 1]z × [0, 1]z → [0, 1]z × [0, 1]z is not (t (c1, c1), t (c1, c1))-continuous. proof. this follows from the observations that (0, 0) and (1, 1) are t (c1, c1)adjacent, but (f × g)(0, 0) = (0, 0) and (f × g)(1, 1) = (1, 0) are neither equal nor t (c1, c1)-adjacent. � a partial converse to theorem 3.10 is obtained by using the following notion. definition 3.12. a continuous function f : (x, κ) → (y, λ) is locally one-toone if f|n∗ κ (x,1) is one-to-one for all x ∈ x. note any function between digital images that is one-to-one must be locally one-to-one. theorem 3.13. suppose fi : (xi, κi) → (yi, λi) is continuous and locally oneto-one for 1 ≤ i ≤ v. then the product function f = πvi=1fi : π v i=1xi → π v i=1yi is (t (κ1, . . . , κv), t (λ1, . . . , λv))-continuous and locally one-to-one. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 31 l. boxer proof. suppose fi : (xi, κi) → (yi, λi) is continuous and locally one-to-one for 1 ≤ i ≤ v. let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) be t (κ1, . . . , κv)adjacent, where xi and x ′ i are κi-adjacent in xi. since fi is continuous and locally one-to-one, we must have that fi(xi) and fi(x ′ i) are λi-adjacent in yi. thus, f(p) and f(p′) are t (λ1, . . . , λv)-adjacent, so f is continuous and locally one-to-one. � theorem 3.14. let x = πvi=1xi, y = π v i=1yi. then the product map f = πvi=1fi : (x, t (κ1, . . . , κv)) → (y, t (λ1, . . . , λv)) is an isomorphism if and only if each fi is an isomorphism. proof. if f is an isomorphism, each fi must be one-to-one and onto. therefore, f−1i : yi → xi is a single-valued function. by theorem 3.10, each fi is continuous. since f −1 = πvi=1f −1 i , it follows from theorem 3.10 that each f−1i is continuous. hence fi is an isomorphism. conversely, if each fi is an isomorphism, then f is one-to-one and onto, so f−1 = πvi=1f −1 i is a single-valued function. by theorem 3.13, f is continuous. similarly, f−1 is continuous. therefore, f is an isomorphism. � theorem 3.15. the projection maps pi : (π v i=1xi, t (κ1, . . . , κv)) → (xi, κi) defined by pi(x1, . . . , xv) = xi for xi ∈ xi are all continuous. proof. let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) be t (κ1, . . . , κv)-adjacent in πvi=1xi, where xi, x ′ i ∈ xi. then for all indices i, xi = pi(p) and x ′ i = pi(p ′) are κi-adjacent. thus, pi is continuous. � a seeming oddity is that a common method of injection that is often continuous, is not continuous when the tensor product adjacency is used, as shown in the following. proposition 3.16. let (x, κ) and (y, λ) be digital images. let y ∈ y . if x has a pair of κ-adjacent points, then the function f : x → (x × y, t (κ, λ)) defined by f(x) = (x, y) is not continuous. proof. this is because given κ-adjacent x, x′ ∈ x, f(x) = (x, y) and f(x′) = (x′, y) are not t (κ, λ)-adjacent. � 3.4. cartesian product. theorem 3.17. let fi : (xi, κi) → (yi, λi) be functions between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. then the product function f = πvi=1fi : x → y is (× v i=1κi, × v i=1λi)-continuous if and only if each fi is continuous. proof. suppose f is continuous. let xi ↔κi x ′ i in xi. let p = (x1, . . . , xv), p′ = (x1, . . . , xi−1, x ′ i, xi+1, . . . , xv). then p ↔×vi=1κi p ′, so either f(p) = f(p′) or f(p) ↔×v i=1 λi f(p ′). the former case implies fi(xi) = fi(x ′ i) and the latter case implies fi(xi) ↔λi fi(x ′ i). hence fi is continuous. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 32 product adjacencies in digital topology suppose each fi is continuous. let p and p ′ be ×vi=1κi-adjacent points of x. then there is only one index k in which p and p′ differ, i.e., for some xi ∈ xi and x ′ k ∈ xk, p = (x1, . . . , xv), p ′ = (x1, . . . , xk−1, x ′ k , xk+1, . . . , xv), and xk ↔κk x ′ k. then f(p) and f(p ′) have the same ith coordinate for i 6= k, and have kth coordinates of fk(xk) and fk(x ′ k), respectively. continuity of fk implies either fk(xk) = fk(x ′ k) or fk(xk) ↔κk fk(x ′ k). therefore, f is continuous. � theorem 3.18. the projection maps pi : (π v i=1xi, × v i=1κi) → (xi, κi) defined by pi(x1, . . . , xv) = xi for xi ∈ xi are all continuous. proof. this follows from proposition 2.5 and theorem 3.8. � by contrast with proposition 3.16, we have the following. proposition 3.19. let (xi, κi) be digital images, 1 ≤ i ≤ v. let xi ∈ xi. the functions ii : xi → (π v i=1xi, × v i=1κi) defined by ii(x) =    (x, x2, . . . , xv) for i = 1; (x1, . . . , xi−1, x, xi+1 . . . , xv) for 1 < i < v; (x1, . . . , xv−1, x) for i = v, are continuous. proof. this follows immediately from definition 2.4. � theorem 3.20. let x = πvi=1xi, y = π v i=1yi. then the product map f = πvi=1fi : (x, × v i=1κi) → (y, × v i=1λi) is an isomorphism if and only if each fi is an isomorphism,. proof. suppose f is an isomorphism. then it follows from proposition 2.5 and theorem 3.7 that fi is an isomorphism. suppose each fi is an isomorphism. then f must be one-to-one and onto, and by theorem 3.17, f is continuous. similarly, f−1 = πvi=1f −1 i is continuous. therefore, f is an isomorphism. � 3.5. lexicographic adjacency. theorem 3.21. suppose fi : (xi, κi) → (yi, λi) is a function between digital images, 1 ≤ i ≤ v. let f = πvi=1fi : π v i=1xi → π v i=1yi be the product function. • if f is (l(κ1, . . . , κv), l(λ1, . . . , λv))-continuous, then each fi is (κi, λi)continuous. further, if f is locally one-to-one, then each fi is locally one-to-one. • if each fi is a continuous function that is locally one-to-one, then f is (l(κ1, . . . , κv), l(λ1, . . . , λv))-continuous. proof. suppose f is (l(κ1, . . . , κv), l(λ1, . . . , λv))-continuous. let xi, x ′ i ∈ xi such that xi ↔κi x ′ i. let p0 = (x1, x2, . . . , xv) and let pi =    (x′1, x2, . . . , xv) for i = 1; (x1, . . . , xi−1, x ′ i, xi+1, . . . , xv) for 1 < i < v; (x1, . . . , xv−1, x ′ v) for i = v. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 33 l. boxer notice (3.1) p0 and pi differ only at index i and p0 ↔l(κ1,...,κv) pi for 1 ≤ i ≤ v. therefore, f(p0) and f(pi) are l(λ1, . . . , λv)-adjacent or equal. it follows from statement (3.1) that fi(xi) and fi(x ′ i) are λi-adjacent or equal. since {xi, x ′ i} is an arbitrary set of κi-adjacent members of xi, fi is (κi, λi)-continuous. further if f is locally one-to-one, then from statement (3.1), fi(xi) and fi(x ′ i) are not equal, so fi is locally one-to-one. suppose each fi is continuous and locally one-to-one. let p, p ′ ∈ x = πvi=1xi, where p = (x1, . . . , xv), p ′ = (x′1, . . . , x ′ v), for xi, x ′ i ∈ xi. assume p ↔l(κ1,...,κv) p ′. let k be the smallest index such that xk ↔κk x ′ k. since fk is locally one-to-one, (3.2) fk(xk) ↔λk fk(x ′ k). • if k = 1, it follows from definition 2.6 that f(p) ↔l(λ1,...,λv) f(p ′). • otherwise, i < k implies xi = x ′ i, hence fi(xi) = fi(x ′ i). together with statement (3.2), this implies f(p) ↔l(λ1,...,λv) f(p ′). then f is (l(κ1, . . . , κv), l(λ1, . . . , λv))-continuous, since p and p ′ were arbitrarily chosen. � the following example illustrates the importance of the locally one-to-one hypothesis in theorem 3.21. example 3.22. let xi = [0, i]z for i ∈ {1, 2}. let f : x1 → x2 be the constant function with value 0. then f and 1x2 are (c1, c1) continuous. however, f × 1x2 : x1 × x2 → x 2 2 is not (l(c1, c1), l(c1, c1))-continuous. proof. consider the points p = (0, 0) and p′ = (1, 2). these points are l(c1, c1)adjacent in x1 × x2. however, (f × 1x2)(p) = (0, 0) and (f × 1x2)(p ′) = (0, 2) are neither equal nor l(c1, c1)-adjacent in x 2 2. � theorem 3.23. suppose fi : (xi, κi) → (yi, λi) is a function between digital images, 1 ≤ i ≤ v. let f = πvi=1fi : π v i=1xi → π v i=1yi be the product function. then f is an (l(κ1, . . . , κv), l(λ1, . . . , λv))-isomorphism if and only if each fi is a (κi, λi)-isomorphism. proof. this follows easily from theorem 3.21. � proposition 3.24. the projection map p1 : (π v i=1xi, l(κ1, . . . , κv)) → (x1, κ1) is continuous. proof. let p ↔l(κ1,...,κv) p ′ in πvi=1xi. then p = (x1, . . . , xv), p ′ = (x′1, . . . , x ′ v) for some xi, x ′ i ∈ xi, where either x1 = x ′ 1 or x1 ↔κ1 x ′ 1. since p1(p) = x1 and p1(p ′) = x′1, it follows that p1 is continuous. � by contrast, we have the following. example 3.25. let v > 1. the projection maps pi : ([0, 2] v z , l(c1, . . . , c1)) → ([0, 2]z, c1) are not continuous for 1 1 implies pi(x) = 0 and pi(y) = 2, which are not c1-adjacent in [0, 2]z. the assertion follows. � 3.6. more on isomorphisms. we have the following. theorem 3.26. let σ : {i}vi=1 → {i} v i=1 be a permutation. let fi : (xi, κi) → (yσ(i), λσ(i)) be an isomorphism of digital images, 1 ≤ i ≤ v. let 1 ≤ u ≤ v. let (κ, λ) be any of (npu(κ1, . . . , κv), npu(λσ(1), . . . , λσ(v))), (t (κ1, . . . , κv), t (λσ(1), . . . , λσ(v))), or (×vi=1κi, × v i=1λσ(i)). let x = πvi=1xi, y = π v i=1yσ(i). then the function f : x → y defined by f(x1, . . . , xv) = (f1(x1), . . . , (fv(xv)) is an isomorphism. proof. it is easy to see that f is one-to-one and onto. continuity of f and of f−1 follows easily from the definitions of the adjacencies under discussion. thus, f is an isomorphism. � the following example shows that the lexicographic adjacency does not yield a conclusion analogous to that of theorem 3.26. example 3.27. let x1 = {0, 1} ⊂ (z, c1). let x2 = {0, 2} ⊂ (z, c1). then x = (x1 × x2, l(c1, c1)) and y = (x2 × x1, l(c1, c1)) are not isomorphic. proof. observe that x is connected, since the 4 points of x form a path in the sequence (0, 0), (1, 0), (0, 2), (1, 2) (see figure 2). however, y is not connected, as there is no path in y from (0, 0) to (2, 0). the assertion follows. � 4. connectedness in this section, we compare product adjacencies with respect to the property of connectedness. theorem 4.1 ([9]). let (xi, κi) be digital images, i ∈ {1, 2, . . . , v}. then (xi, κi) is connected for all i if and only (π v i=1xi, npv(κ1, . . . , κv)) is connected. theorem 4.2. let (xi, κi) be digital images, i ∈ {1, 2, . . . , v}. if π v i=1xi is t (κ1, . . . , κv)-connected, then xi is κi-connected for all i. proof. these assertions follow from definition 2.8 and theorem 3.15. � however, the converse to theorem 4.2 is not generally true, as shown by the following. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 35 l. boxer example 4.3. let x = {0} ⊂ z, y = [0, 1]z ⊂ z. then x and y are each c1-connected. however: • x × y = {(0, 0), (0, 1)} is not t (c1, c1)-connected. • y × y has two t (c1, c1)-components, {(0, 0), (1, 1)} and {(1, 0), (0, 1)}. see also figure 1(c), which illustrates that msc8 × [0, 1]z is not t (c2, c1)connected, although msc8 is c2-connected and [0, 1]z is c1-connected. for the cartesian product adjacency, we have the following. theorem 4.4. let (xi, κi) be digital images, i ∈ {1, 2, . . . , v}. then π v i=1xi is ×vi=1κi-connected if and only if xi is κi-connected for all i. proof. suppose x = πvi=1xi is × v i=1κi-connected. it follows from proposition 3.18 that each xi is κi-connected. suppose each xi is κi-connected. let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) be points of x such that xi, x ′ i ∈ xi. there are κi-paths pi in xi from xi to x′i. if the functions ii are as in proposition 3.19, then it is easily seen that ⋃v i=1 ii(pi) is a × v i=1κi-path in x from p to p ′. since p and p′ were arbitrarily chosen, it follows that x is ×vi=1κi-connected. � proposition 4.5. let (x, κ) and (y, λ) be digital images, such that |x| > 1. then (x × y, l(κ, λ)) is connected if and only if (x, κ) is connected. proof. suppose (x, κ) is connected. let p = (x, y), p′ = (x′, y′), with x, x′ ∈ x, y, y′ ∈ y . • if x = x′ then, since |x| > 1 and x is connected, there exists x0 ∈ x such that x ↔κ x0. therefore, p ↔l(κ,λ) (x0, y) ↔l(κ,λ) (x, y ′) = p′. • suppose x 6= x′. since x is connected, there is a path in x, p = {xi} n i=0, such that x = x0 ↔κ x1 ↔κ . . . ↔κ xn−1 ↔κ xn = x ′. therefore, p = (x0, y) ↔l(κ,λ) (x1, y ′) ↔l(κ,λ) (x2, y ′) ↔l(κ,λ) . . . ↔l(κ,λ) (xn, y ′) = p′. therefore, (x × y, l(κ, λ)) is connected. suppose (x, κ) is not connected. then there exist x, x′ ∈ x such that x and x′ are in distinct components of x. let y, y′ ∈ y . by definition 2.6, there is no path in (x × y, l(κ, λ)) from (x, y) to (x′, y′). therefore, (x × y, l(κ, λ)) is not connected. � an argument similar to that used for the proof of proposition 4.5 yields the following. theorem 4.6. let (xi, κi) be digital images, 1 ≤ i ≤ v. suppose k is the smallest index for which |xk| > 1. then (π v i=1xi, l(κ1, . . . , κv)) is connected if and only if (xk, κk) is connected. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 36 product adjacencies in digital topology 5. homotopy 5.1. tensor product. in [9], it is shown that many homotopy properties are preserved by cartesian products with the npv adjacency. we show that we cannot make analogous claims for the tensor product adjacency. example 5.1. there are digital images (xi, κi) and (yi, λi) and continuous functions fi, gi : xi → yi, i ∈ {1, 2}, such that fi ≃ gi but f1 × f2 6≃t (κ1,κ2),t (λ1,λ2) g1 × g2. proof. we can use example 4.3. e.g., if x1 = x2 = y1 = y2 = [0, 1]z, f1 = f2 : x1 → y1 is the identity function, and g1 = g2 : x2 → y2 is the constant function taking the value 0, we have f1 ≃c1,c1 g1 and f2 ≃c1,c1 g2. as we saw in example 4.3, [0, 1]2 z is not t (c1, c1)-connected, so its identity function f1 × f2 is not homotopic to the constant function g1 × g2. � example 5.2. there are digital images (xi, κi) and (yi, λi) for i ∈ {1, 2}, such that xi and yi have the same homotopy type, but (x1 × x2, t (κ1, λ1)) and (y1 × y2, t (κ2, λ2)) do not have the same homotopy type. proof. we saw in example 4.3 that [0, 1]2 z is not t (c1, c1)-connected; however, it is trivial that {0}2 = {(0, 0)} is t (c1, c1)-connected. therefore, we can take x1 = x2 = [0, 1]z ⊂ (z, c1), y1 = y2 = {0} ⊂ (z, c1). � 5.2. cartesian product adjacency. theorem 5.3. let fi, gi : (xi, κi) → (yi, λi) be continuous functions between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, f = π v i=1fi : x → y , g = πvi=1gi : x → y . then f ≃×vi=1κi,× v i=1 λi g if and only if for all i, fi ≃κi,λi gi. further, f and g are pointed homotopic if and only if for each i, fi and gi are pointed homotopic. proof. suppose f ≃×v i=1 κi,× v i=1 λi g. then there is a homotopy h : πvi=1xi × [0, m]z → π v i=1xi such that h(p, 0) = f(p) and h(p, m) = g(p) for all p ∈ x. let xi ∈ xi and let hi : xi × [0, m]z → yi be defined by hi(x, t) = pi(h(ii(x), t)), where ii is the continuous injection of proposition 3.19 corresponding to the point (x1, . . . , xv) ∈ x and pi is the continuous projection map of theorem 3.18. then hi(x, 0) = pi(f(ii(x)) = fi(x) and hi(x, m) = pi(g(ii(x)) = gi(x). since the composition of continuous functions is continuous (theorem 2.10), it follows that hi is a homotopy from fi to gi. further, if h holds some point p0 of x fixed, then we can take p0 = (x1, . . . , xv) to be the point of x used in proposition 3.19, and we can conclude that hi holds pi(p) = xi fixed. suppose for all i, fi ≃κi,λi gi. let hi : xi × [0, mi]z → yi be a (κi, λi)homotopy from fi to gi. we execute these homotopies “one coordinate at a c© agt, upv, 2018 appl. gen. topol. 19, no. 1 37 l. boxer time,” as follows. for x = (x1, . . . , xv) ∈ x such that xi ∈ xi, let mi = ∑i k=1 mi for all i and let h : x×[0, mv]z → y be defined by h(x1, . . . , xv, t) = • (h1(x1, t), f2(x2), . . . , fv(xv)) if 0 ≤ t ≤ m1; • (g1(x1) . . . , gj−1(xj−1), hj(xj, t−mj−1), fj+1(xj+1), . . . , fv(xv)) if mj−1 ≤ t ≤ mj; • (g1(x1) . . . , gv−1(xv−1), hv(xv, t − mv−1)) if mj−1 ≤ t ≤ mj. it is easily seen that h is well defined and is a homotopy from f to g. further, if hi holds xi fixed, then h holds x fixed. � corollary 5.4. let (xi, κi) and (yi, λi) be digital images, 1 ≤ i ≤ v. then x = πvi=1xi and y = π v i=1yi are (× v i=1κi, × v i=1λi)-(pointed) homotopy equivalent if and only if for each i, (xi, κi) and (yi, λi) are (pointed) homotopy equivalent. proof. this follows from theorem 5.3. � 5.3. lexicographic adjacency. theorem 5.5. let (xi, κi) be digital images for 1 ≤ i ≤ v. let x = π v i=1xi. if there is a smallest index k such that |xk| > 1, then (x, l(κ1, . . . , κv)) and (xk, κk) have the same pointed homotopy type. proof. for each i 6= k, let xi ∈ xi. let ik : xk → x be the injection of proposition 3.19. by choice of k, ik is (κk, l(κ1, . . . , κv))-continuous. also by choice of k, the projection map pk : (x, l(κ1, . . . , κv)) → (xk, κk) is continuous. notice pk ◦ ik = 1xk . also, the function h : x × [0, 1]z → x defined for p = (y1, . . . , yv) ∈ x with yi ∈ xi by h(p, t) =        p if t = 0; (y1, x2, . . . , xv) if t = 1 and k = 1; (x1, . . . , xk−1, yk, xk+1, . . . , xv) if t = 1 and 1 < k < v; (x1, . . . , xv−1, yv) if t = 1 and k = v, is easily seen from the choice of k to be a homotopy from 1x to ik ◦ pk that holds fixed the point (x1, . . . , xv). the assertion follows. � corollary 5.6. let (x, κ) and (y, λ) be digital images of different pointed homotopy types. if |x| > 1 and |y | > 1, then (x × y, l(κ, λ)) and (y × x, l(λ, κ)) have different pointed homotopy types. proof. this follows immediately from theorem 5.5. � corollary 5.7. let (xi, κi) and (yi, λi) be digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. suppose there exist a smallest index j such that |xj| > 1, and a smallest index k such that |yk| > 1. if (xj, κj) and (yk, κk) have the same (pointed) homotopy type, then (x, l(κ1, . . . , κv)) and (y, l(λ1, . . . , λv)) have the same (pointed) homotopy type. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 38 product adjacencies in digital topology proof. by theorem 5.5, (x, l(κ1, . . . , κv)) and (xj, κj) have the same pointed homotopy type, and (yk, λk) and (y, l(λ1, . . . , λv)) have the same pointed homotopy type. since we also have assumed (xj, κj) and (yk, λk) have the same (pointed) homotopy type, the assertion follows from the transitivity of (pointed) homotopy type. � 6. retractions definition 6.1 ([2, 3]). let y ⊂ (x, κ). a (κ, κ)-continuous function r : x → y is a retraction, and a is a retract of x, if r(y) = y for all y ∈ y . theorem 6.2 ([12]). let ai ⊂ (xi, κi), i ∈ {1, . . . , v}. then ai is a retract of xi for all i if and only if π v i=1ai is a retract of (π v i=1xi, npv(κ1, . . . , κv)). 6.1. tensor product adjacency. the following example shows that one of the assertions obtained by using the tensor product adjacency rather than npv in theorem 6.2 is not generally valid. example 6.3. let x = {(0, 0), (1, 0), (1, 1)} ⊂ (z2, c2). observe that x ′ = {(0, 0), (1, 0)} is a c2-retract of x, and {0} is a c1-retract of [0, 1]z. however, x′ × {0} is not a t (c2, c1)-retract of x × [0, 1]z. proof. note x × [0, 1]z is t (c2, c1)-connected, since (0, 0, 0), (1, 0, 1), (1, 1, 0), (0, 0, 1), (1, 0, 0), (1, 1, 1) is a listing of its points in a t (c2, c1)-path; but x ′ × {0} = {(0, 0, 0), (1, 0, 0)} is not t (c2, c1)-connected. the assertion follows. � the question of whether πvi=1ai being a retract of (π v i=1xi, t (κ1, . . . , κv)) implies ai is a κi-retract of xi, for all i, is unknown at the current writing. 6.2. cartesian product adjacency. for the cartesian product adjacency, we have the following analog of theorem 6.2. theorem 6.4. suppose ai ⊂ (xi, κi). let x = π v i=1xi, a = π v i=1ai. then there is a retraction ri : xi → ai, 1 ≤ i ≤ v if and only if there is a retraction r : (x, ×vi=1κi) → (a, × v i=1κi). proof. suppose there is a retraction ri : xi → ai, 1 ≤ i ≤ v. let r = π v i=1ri : x → a. clearly r(x) ∈ a for all x ∈ x, and r(a) = a for all a ∈ a. by theorem 3.17, r is continuous. therefore, r is a retraction. conversely, suppose there exists a retraction r : (x, ×vi=1κi) → (a, × v i=1κi). let ri = pi ◦ r ◦ ii : (xi, κi) → (ai, κi), where ii is the injection of proposition 3.19 and the xi of proposition 3.19 satisfies xi ∈ ai. since composition preserves continuity, theorem 3.18 and proposition 3.19 imply ri is continuous. further, for ai ∈ ai we clearly have ri(ai) = ai. thus, ri is a retraction. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 39 l. boxer 6.3. lexicographic adjacency. for the lexicographic adjacency, we do not have an analog of theorem 6.2, as shown by the following example. example 6.5. {0} is a c1-retract of [0, 1]z and [1, 4]z is a c1-retract of [0, 5]z. however, a = {0} × [1, 4]z is not an l(c1, c1)-retract of x = [0, 1]z × [0, 5]z. proof. we give a proof by contradiction. suppose there is an l(c1, c1)-retraction r : [0, 1]z × [0, 5]z → {0} × [1, 4]z. notice p = (0, 1) ↔l(c1,c1) (1, 5) = p ′. since r(p) = p, the continuity of r requires that r(p′) = p or r(p′) ↔l(c1,c1) p, hence r(p′) ∈ {p, (0, 2)}. but also p′ ↔l(c1,c1) (0, 4) = q, and since r(q) = q, the continuity of r similarly requires that r(p′) ↔l(c1,c1) {q, (0, 3)}. therefore, r(p′) ∈ {p, (0, 2)} ∩ {q, (0, 3)} = ∅. since this is impossible, no such retraction r can exist. � 7. approximate fixed point property some material in this section is quoted or paraphrased from [9, 10]. in both topology and digital topology, • a fixed point of a continuous function f : x → x is a point x ∈ x satisfying f(x) = x; • if every continuous f : x → x has a fixed point, then x has the fixed point property (fpp). however, a digital image x has the fpp if and only if x has a single point [10]. therefore, it turns out that the approximate fixed point property is more interesting for digital images. definition 7.1 ([10]). a digital image (x, κ) has the approximate fixed point property (afpp) if every continuous f : x → x has an approximate fixed point, i.e., a point x ∈ x such that f(x) is equal or κ-adjacent to x. the following is a minor generalization of theorem 5.10 of [10]. theorem 7.2 ([9]). let (xi, κi) be digital images, 1 ≤ i ≤ v. then for any u ∈ z such that 1 ≤ u ≤ v, if (πvi=1xi, npu(κ1, . . . , κv)) has the afpp then (xi, κi) has the afpp for all i. determining whether analogs of theorem 7.2 for the tensor product adjacency, or for the cartesian product adjacency, are generally true, appear to be difficult problems. the following examples show that the analogs of converses to theorem 7.2 for the tensor product adjacency and for the cartesian product adjacency are not generally true. example 7.3. although ([0, 1]z, c1) has the afpp [25], ([0, 1] 2 z , t (c1, c1)) does not have the afpp. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 40 product adjacencies in digital topology proof. consider the function f : [0, 1]2 z → [0, 1]2 z defined by f(a, b) = (1 − a, b), i.e., f(0, 0) = (1, 0), f(0, 1) = (1, 1), f(1, 0) = (0, 0), f(1, 1) = (0, 1). one can easily check that f is continuous and has no approximate fixed point when the t (c1, c1) adjacency is used. � example 7.4. although ([0, 1]z, c1) has the afpp, ([0, 1] 2 z , c1 × c1) does not have the afpp. proof. consider the function f : [0, 1]2 z → [0, 1]2 z defined by f(a, b) = (1−a, 1− b), i.e., f(0, 0) = (1, 1), f(0, 1) = (1, 0), f(1, 0) = (0, 1), f(1, 1) = (0, 0). one can easily check that f is continuous and has no approximate fixed point when the c1 × c1 adjacency is used. � we have the following. theorem 7.5. let (xi, κi) be digital images, 1 ≤ i ≤ v. suppose there is a smallest index k such that xk is κk-connected and |xk| > 1. if the product (πvi=1xi, l(κ1, . . . , κv)) has the afpp property, then (xk, κk) has the afpp property. proof. let x = πvi=1xi. suppose the product (x, l(κ1, . . . , κv)) has the afpp property. let g : xk → xk be κ-continuous. let xi ∈ xi. notice this means xi = {xi} for i < k. let x = πvi=1xi. let g : x → x be defined by g(y1, . . . , yv) =    (g(y1), x2, . . . , xv) if k = 1; (x1, . . . , xk−1, g(yk), xk+1, . . . , xv) if 1 < k < v; (x1, . . . , xv−1, g(yv)) if k = v. since g is κk-continuous, our choice of k implies g is l(κ1, . . . , κv)-continuous. by hypothesis, there is a p = (y′1, . . . , y ′ v) ∈ x with y ′ i ∈ xi such that g(p) = p or g(p) ↔ p. therefore, either g(y′k) = pk(g(p)) = pk(p) = y ′ k or g(y ′ k) ↔κk y ′ k. thus, y′k is an approximate fixed point for g. � 8. multivalued functions we study various product adjacencies with respect to properties of multivalued functions. the following has an elementary proof. proposition 8.1. let f : (x, κ) → (y, λ) be a single-valued function between digital images. then the following are equivalent. • f is continuous. • as a multivalued function, f has weak continuity. • as a multivalued function, f has strong continuity. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 41 l. boxer for multivalued functions fi : xi ⊸ yi, 1 ≤ i ≤ v, define the product multivalued function πvi=1fi : π v i=1xi ⊸ π v i=1yi by (πvi=1fi)(x1, . . . , xv) = π v i=1fi(xi). 8.1. weak continuity. for npv, we have the following results. theorem 8.2 ([9]). let fi : (xi, κi) ⊸ (yi, λi) be multivalued functions for 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, and f = π v i=1fi : (x, npv(κ1, . . . , κv)) ⊸ (y, npv(λ1, . . . , λv)). then f has weak continuity if and only if each fi has weak continuity. for the tensor product, we have the following. theorem 8.3. for each index i such that 1 ≤ i ≤ v, let fi : (xi, κi) ⊸ (yi, λi) be a multivalued map between digital images. let x = πvi=1xi, y = π v i=1yi. if the product multivalued map f = πvi=1fi : (x, t (κ1, . . . , κv)) ⊸ (y, t (λ1, . . . , λv)) has weak continuity, then for each i, fi has weak continuity. proof. for all indices i, let xi ↔κi x ′ i in xi. then, in x, we have p = (x1, . . . , xv) ↔t (κ1,...,κv) p ′ = (x′1, . . . , x ′ v). the weak continuity of f implies f(p) and f(p′) are adjacent subsets of (y, t (λ1, . . . , λv)). therefore, there exist y ∈ f(p) and y′ ∈ f(p′) such that y = y′ or y ↔t (λ1,...,λv) y ′. now, y = (y1, . . . , yv) where yi ∈ fi(xi), and y ′ = (y′1, . . . , y ′ v) where y ′ i ∈ fi(x ′ i). if y = y ′ then we have yi = y ′ i for all indices i. if y ↔t (λ1,...,λv) y ′ then we have yi ↔λi y ′ i for all indices i. in either case, we have for all i that fi(xi) and fi(x ′ i) are adjacent subsets of yi. it follows that each fi has weak continuity. � the converse of theorem 8.3 is not generally true, as shown by the following. example 8.4. let f and g be the single-valued functions of example 3.11. by proposition 8.1, f and g have weak continuity. however, example 3.11 shows that f × g is not (t (c1, c1), t (c1, c1))-continuous, so by proposition 8.1, f × g does not have (t (c1, c1), t (c1, c1))-weak continuity. for the cartesian product adjacency, we have the following. theorem 8.5. let fi : (xi, κi) ⊸ (yi, λi) be multivalued maps between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. then the product multivalued map f = πvi=1fi : (x, × v i=1κi) ⊸ (y, × v i=1λi) has weak continuity if and only if for each i, fi has weak continuity. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 42 product adjacencies in digital topology proof. suppose f has weak continuity. let xi ↔κi x ′ i in xi. let x = (x1, . . . , xv) ∈ x, x′ = (x1, . . . , xj−1, x ′ j, xj+1, . . . , xv) ∈ x for some index j. we have x ↔×v i=1 κi x ′. therefore, there exist y = (y1, . . . , yv) ∈ f(x) = π v i=1fi(xi), y′ = (y′1, . . . , y ′ v) ∈ f(x ′) = π j−1 i=1 fi(xi) × fj(xj) × π v i=j+1fi(xi) such that y ↔×v i=1 λi y ′. therefore, we have yj ∈ fj(xj), y ′ j ∈ fj(x ′ j), and yj = y ′ j or yj ↔λj y ′ j. thus, fj has weak continuity. suppose each fi has weak continuity. let p ↔×v i=1 κi p ′ in x, where p = (x1, . . . , xv), p ′ = (x′1, . . . , x ′ v), xi, x ′ i ∈ xi, and, from the definition of the ×vi=1κi adjacency, there is one index j such that xj ↔κj x ′ j and for all indices i 6= j, xi = x ′ i and therefore fi(xi) = fi(x ′ i). since fj has weak continuity, there exist yj ∈ fj(xj) and y ′ j ∈ fj(x ′ j) such that yj = y ′ j or yj ↔λj y ′ j. for i 6= j we can take yi ∈ fi(xi). then y = (y1, . . . , yv) and y′ = (y1, . . . , yj−1, y ′ j, yj+1, . . . , yv) are equal or × v i=1λi-adjacent, and we have y ∈ f(p), y′ ∈ f(p′). therefore, f has weak continuity. � for the lexicographic adjacency, example 8.10 below shows there is no general product property for weak continuity, and example 8.11 below shows there is not a general factor property for weak continuity. 8.2. strong continuity. theorem 8.6 ([9]). let fi : (xi, κi) ⊸ (yi, λi) be multivalued functions for 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, f = π v i=1fi : (x, npv(κ1, . . . , κv)) ⊸ (y, npv(λ1, . . . , λv)). then f has strong continuity if and only if each fi has strong continuity. for the tensor product adjacency, we have the following. theorem 8.7. let fi : (xi, κi) ⊸ (yi, λi) be multivalued maps between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. if the product multivalued map f = πvi=1fi : (x, t (κ1, . . . , κv)) ⊸ (y, t (λ1, . . . , λv)) has strong continuity, then for each i, fi has strong continuity. proof. let xi ↔κi x ′ i in xi. let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v). note p ↔t (κ1,...,κv) p ′ in x. since f has strong continuity, for every q = (y1, . . . , yv) ∈ f(p) = π v i=1fi(xi) where yi ∈ fi(xi), there exists q ′ = (y′1, . . . , y ′ v) ∈ f(p′) = πvi=1fi(x ′ i) where y ′ i ∈ fi(x ′ i) such that either q = q ′ or q ↔t (λ1,...,λv) q′; and therefore yi = y ′ i for all i or yi ↔λi y ′ i for all i. also, for every r′ = (r′1, . . . , r ′ v) ∈ f(p ′) where r′i ∈ fi(x ′ i), there exists r = (r1, . . . , rv) ∈ f(p) where ri ∈ fi(xi) such that either r = r ′ or r ↔t (λ1,...,λv) r ′; and therefore ri = r ′ i for all i or ri ↔λi r ′ i for all i. thus fi has (κi, λi)-strong continuity. � the converse of theorem 8.7 is not generally true, as shown by the following. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 43 l. boxer example 8.8. let f1 : ([0, 1]z, c1) ⊸ ([0, 1]z, c1) be defined by f1(x) = {0}. let f2 : ([0, 1]z, c1) ⊸ ([0, 1]z, c1) be defined by f2(x) = {x}. then f1 and f2 both have strong continuity. however, f1×f2 does not have (t (c1, c1), t (c1, c1))strong continuity. proof. it is easily seen that f1 and f2 both have strong continuity. however, in example 8.4, we showed that f1 × f2 does not have (t (c1, c1), t (c1, c1))weak continuity. therefore, f1 × f2 does not have (t (c1, c1), t (c1, c1))-strong continuity. � theorem 8.9. let fi : (xi, κi) ⊸ (yi, λi) be multivalued maps between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. then the product multivalued map f = πvi=1fi : (x, × v i=1κi) ⊸ (y, × v i=1λi) has strong continuity if and only if for each i, fi has strong continuity. proof. suppose f has strong continuity. let xi ↔κi x ′ i in xi. then p = (x1, . . . , xv) ↔×v i=1 κi (x1, . . . , xj−1, x ′ j, xj+1, . . . , xv) = p ′ in x, for some index j. since f has strong continuity, we must have that for every q = (q1, . . . , qv) ∈ f(p) there exists q ′ = (q′1, . . . , q ′ v) ∈ f(p ′) such that q = q′ or q ↔×v i=1 λi q ′, so qi = q ′ i or qi ↔λi q ′ i; and for every r ′ = (r′1, . . . , r ′ v) ∈ f(p′) there exists r = (r1, . . . , rv) ∈ f(p) such that r = r ′ or r ↔×v i=1 λi r ′, so ri = r ′ i or ri ↔λi r ′ i. therefore, fi has strong continuity. suppose for each i, fi has strong continuity. let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) with xi, x ′ i ∈ xi be such that p ↔×vi=1κi p ′. then for some index j, xj ↔κj x ′ j and for all indices i 6= j, xi = x ′ i. therefore, i 6= j implies there exists qi ∈ fi(xi) = fi(x ′ i); and since fj has strong continuity, for every qj ∈ fj(xj) there exists q ′ j ∈ fj(x ′ j) such that qj = q ′ j or qj ↔λj q ′ j. let q′ = (q1, . . . , qj−1, q ′ j, qj+1, . . . , qv). then q = (q1, . . . , qv) = q ′ or q ↔×v i=1 λi q ′ with q ∈ f(p), q′ ∈ f(p′). similarly, for every r′ ∈ f(p′) there exists r ∈ f(p) such that r = r′ or r ↔×v i=1 λi r ′. thus, f has strong continuity. � for the lexicographic adjacency, the following shows there is not a general product property for weak or strong continuity. example 8.10. let f1 : ([0, 1]z, c1) ⊸ ([0, 1]z, c1) be the multivalued function f1(x) = {0}. let f2 : ({0, 2}, c1) ⊸ ({0, 2}, c1) be the function f2(x) = {x}. then f1 and f2 have weak continuity and strong continuity, but f1 × f2 lacks both (l(c1, c1), l(c1, c1))-weak continuity and (l(c1, c1), l(c1, c1))-strong continuity. proof. it is easy to see that f1 and f2 have weak continuity and strong continuity, and that p = (0, 0) ↔l(c1,c1) (1, 2) = p ′. however (f1 × f2)(p) = {(0, 0)} and (f1 × f2)(p ′) = {(0, 2)}, are not l(c1, c1)-adjacent, so f1 ×f2 lacks (l(c1, c1), l(c1, c1))-weak continuity and therefore lacks (l(c1, c1), l(c1, c1))-strong continuity. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 44 product adjacencies in digital topology for the lexicographic adjacency, the following shows there is not a general factor property for weak or strong continuity. example 8.11. let f1 : ([0, 1]z, c1) ⊸ ([0, 1]z, c1) be the multivalued function f1(x) = [0, 1]z. let f2 : ([0, 1]z, c1) ⊸ ({0, 2}, c1) be the multivalued function f2(x) = {2x}. then f1 × f2 : [0, 1] 2 z ⊸ [0, 1]z × {0, 2} has (l(c1, c1), l(c1, c1))weak and (l(c1, c1), l(c1, c1))-strong continuity, although f2 lacks both weak and strong continuity. proof. it is easy to see that f2 lacks weak and strong continuity. since (f1 × f2)(0, 0) = (f1 × f2)(1, 0) = {(0, 0), (1, 0)}, (f1 × f2)(0, 1) = (f1 × f2)(1, 1) = {(0, 2), (1, 2)}, it follows easily that f1 × f2 has both (l(c1, c1), l(c1, c1))-weak continuity and (l(c1, c1), l(c1, c1))-strong continuity. � 8.3. continuous multifunctions. lemma 8.12 ([9]). let x ⊂ zm, y ⊂ zn. let f : (x, ca) ⊸ (y, cb) be a continuous multivalued function. let f : (s(x, r), ca) → (y, cb) be a continuous function that induces f. let s ∈ n. then there is a continuous function fs : (s(x, rs), ca) → (y, cb) that induces f. for the npv adjacency, we have the following. theorem 8.13 ([9]). given multivalued functions fi : (xi, cai) ⊸ (yi, cbi), 1 ≤ i ≤ v, each fi is continuous if and only if the product multivalued function πvi=1fi : (π v i=1xi, npv(ca1, . . . , cav)) ⊸ (π v i=1yi, npv(cb1, . . . , cbv )) is continuous. for the tensor product, since a single-valued function can be considered as multivalued, example 3.11 shows there is no general product rule for the continuity of multivalued functions. however, we have the following. theorem 8.14. let fi : (xi, cai) ⊸ (yi, cbi) be a continuous multivalued function between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, f = πvi=1fi : x ⊸ y . if for some positive integer r and for all i there is a continuous locally one-to-one function fi : (s(xi, r), cai) → (yi, cbi) that generates fi, then f is (t (ca1, . . . , cav ), t (cb1, . . . , cbv ))-continuous and is generated by a function that is locally one-to-one. proof. let f = πvi=1fi : π v i=1s(xi, r) → y . it follows from theorem 3.13 that f is (t (ca1, . . . , cav), t (cb1, . . . , cbv ))-continuous. further, given q ∈ f(p) where p = (x1, . . . , xv) for xi ∈ xi and q = (y1, . . . , yv) where yi ∈ fi(xi), there exists x′i ∈ s({xi}, r) ⊂ s(xi, r) such that fi(x ′ i) = yi. therefore, f(x ′ 1, . . . , x ′ v) = q. for w ↔t (ca1 ,...,cav ) w ′ in s(x, r) = πvi=1s(xi, r), we have w = (w1, . . . , wv) and w′ = (w′1, . . . , w ′ v), where wi, w ′ i ∈ s(xi, r) and wi ↔cai w ′ i. since fi is locally one-to-one and continuous, we have fi(wi) ↔cbi fi(w ′ i). it follows that c© agt, upv, 2018 appl. gen. topol. 19, no. 1 45 l. boxer f(w1, . . . , wv) ↔t (cb1 ,...,cbv ) f(w ′ 1, . . . , w ′ v). this allows us to conclude that f is (t (ca1, . . . , cav), t (cb1, . . . , cbv ))-continuous. thus, f generates f . let p′ = (x′1, . . . , x ′ v) ↔t (κ1,...,κv) p in x, where x ′ i ∈ xi. since fi is locally one-to-one, fi(xi) ↔λi fi(x ′ i) for all i. therefore, f(p) ↔t (λ1,...,λv) f(p ′), so f is locally one-to-one. � deciding whether the converse of theorem 8.14 is true appears to be a difficult problem. for the cartesian product adjacency, we have the following. theorem 8.15. let fi : (xi, κi) ⊸ (yi, λi) be a multivalued function between digital images, where κi = cai, λi = cbi, 1 ≤ i ≤ v. let x = π v i=1xi, y = πvi=1yi, f = π v i=1fi : x ⊸ y . if each fi is continuous, then f is (×vi=1κi, × v i=1λi)-continuous. proof. suppose each fi is continuous. by lemma 8.12, there exists r ∈ n and generating functions fi : s(xi, r) → yi of fi. we wish to show that f = πvi=1fi generates f . suppose p ↔×vi=1κi p ′ in s(x, r). then p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) where xi, x ′ i ∈ s(xi, r) and xi = x ′ i for all but one index j, with xj ↔κj x ′ j. since each fi is (κi, λi)continuous, we have fj(xj) = fj(x ′ j) or fj(xj) ↔λj fj(x ′ j) and for all indices i 6= j we have fi(xi) = fi(x ′ i). thus we have f(p) = f(p ′) or f(p) ↔×v i=1 λi f(p′). thus, f is (×vi=1κi, × v i=1λi)-continuous. let y = (y1, . . . , yv) ∈ f(x), where yi ∈ yi. then there exists xi ∈ s(xi, r) such that fi(xi) = yi. for p = (x1, . . . , xv), we have f(p) = (y1, . . . , yv). thus, f generates f , so f is continuous. � deciding whether the converse of theorem 8.15 is true appears to be a difficult problem. for the lexicographic adjacency, there is no general product rule for the continuity of multivalued functions, as shown in example 3.22 (since a single-valued function can be regarded as multivalued). however, we have the following. theorem 8.16. let fi : (xi, κi) ⊸ (yi, λi) be a continuous multivalued function between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, f = πvi=1fi : x ⊸ y . if each fi is generated by a function fi : (s(xi, r), κi) → yi that is locally one-to-one, then f is (l(κ1, . . . , κv), l(λ1, . . . , λv))-continuous. proof. by theorem 3.21, the single-valued function f = πvi=1fi : π v i=1s(xi, r) → y is (l(κ1, . . . , κv), l(λ1, . . . , λv))-continuous. further, given y = (y1, . . . , yv) ∈ f(x) with yi ∈ yi, there exist x ′ i ∈ s({xi}, r) ⊂ s(xi, r) such that fi(x ′ i) = yi. therefore, y = f(x′1, . . . , x ′ v) ∈ f(x1, . . . , xv). therefore, f generates f , and the assertion follows. � the paper [16] has several results concerning the following notions. definition 8.17 ([16]). let (x, κ) ⊂ zn be a digital image and y ⊂ x. we say that y is a κ-retract of x if there exists a κ-continuous multivalued function f : x ⊸ y (a multivalued κ-retraction) such that f(y) = {y} if y ∈ y . c© agt, upv, 2018 appl. gen. topol. 19, no. 1 46 product adjacencies in digital topology we generalize theorem 6.2 as follows. theorem 8.18 ([9]). for 1 ≤ i ≤ v, let ai ⊂ (xi, κi) ⊂ z ni. suppose fi : xi ⊸ ai is a continuous multivalued function for all i. then fi is a multivalued κi-retraction for all i if and only if f = π v i=1fi : π v i=1xi ⊸ π v i=1ai is a multivalued npv(κ1, . . . , κv)-retraction. for the cartesian product adjacency, we have the following. theorem 8.19. let ri : xi ⊸ ai be multivalued κi-retractions, 1 ≤ i ≤ v. let x = πvi=1xi, a = π v i=1ai, r = π v i=1ri : x ⊸ a. then r is a × v i=1κimultivalued retraction. proof. since ri is a multivalued retraction, we must have that ri(xi) = ai and ri(ai) = {ai} for all ai ∈ ai. therefore, r(x) = a and r(a) = {a} for all a ∈ a. by theorem 8.15, r is continuous, and therefore is a multivalued retraction. � 8.4. connectivity preserving multifunctions. theorem 8.20 ([9]). let fi : (xi, κi) ⊸ (yi, λi) be a multivalued function between digital images, 1 ≤ i ≤ v. then the product map πvi=1fi : (π v i=1xi, npv(κ1, . . . , κv)) ⊸ (π v i=1yi, npv(λ1, . . . , λv)) is a connectivity preserving multifunction if and only if each fi is a connectivity preserving multifunction. the tensor product adjacency does not yield a similar result, as shown in the following. example 8.21. consider {0} ⊂ z, [0, 1]z ⊂ z. the multivalued function f : ({0}, c1) ⊸ ([0, 1]z, c1) defined by f(0) = [0, 1]z is connectivity preserving. however, f × f : {0}2 = {(0, 0)} ⊸ [0, 1]2 z is not (t (c1, c1), t (c1, c1))connectivity preserving. proof. this follows from the observations that {(0, 0)} has a single point, hence must be t (c1, c1)-connected; but, by example 4.3, (f × f)(0, 0) = [0, 1] 2 z is not t (c1, c1)-connected. � however, we have the following. theorem 8.22. let fi : (xi, κi) ⊸ (yi, λi) be multivalued functions, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. suppose f = π v i=1fi : x ⊸ y is (t (κ1, . . . , κv), t (λ1, . . . , λv))-connectivity preserving. then each fi is connectivity preserving. proof. let p = (x1, . . . , xv) ∈ x, where xi ∈ xi. by assumption, f(p) = πvi=1fi(xi) is t (λ1, . . . , λv)-connected. from theorem 4.2, it follows that fi(xi) is λi-connected. suppose x′i ↔κi xi in xi. then p ′ = (x′1, . . . , x ′ v) ↔t (κ1,...,κv) p. since f is connectivity preserving, f(p′) and f(p) are t (λ1, . . . , λv)-adjacent subsets of c© agt, upv, 2018 appl. gen. topol. 19, no. 1 47 l. boxer y . this implies there exist q′ = (y′1, . . . , y ′ v) ∈ f(p ′), q = (y1, . . . , yv) ∈ f(p) such that q′ ↔t (κ1,...,κv) q or q ′ = q. therefore, for each index i, y′i ↔λi yi or y′i = yi. since y ′ i ∈ fi(x ′ i) and yi ∈ fi(xi), we have that fi(x ′ i) and fi(xi) are λi-adjacent subsets of yi. from theorem 2.18, fi is connectivity preserving. � for the cartesian product adjacency, we have the following. theorem 8.23. let (xi, κi) and (yi, λi) be digital images, for 1 ≤ i ≤ v. let fi : xi ⊸ yi be a multivalued function. let f = π v i=1fi : x = π v i=1xi ⊸ y = πvi=1yi be the product function. then f is (× v i=1κi, × v i=1λi)-connectivity preserving if and only if each fi is connectivity preserving. proof. suppose f is connectivity preserving. let p = (x1, . . . , xv) ∈ x, where xi ∈ xi. then f(p) = π v i=1fi(xi) is × v i=1λi-connected. by theorem 3.18, fi(xi) = pi(f(p)) is λi-connected. for any given index k, let xk ↔κk x ′ k in xk. for all indices i 6= k, let xi ∈ xi. then p = (x1, . . . , xv) and p ′ = (x1, . . . , xk−1, x ′ k , xk+1, . . . , xv) are ×vi=1κi-adjacent. since f is connectivity preserving, f(p) and f(p ′) are ×vi=1λiadjacent subsets of y . therefore, theorem 3.18 implies fk(xk) = pk(f(p)) and fk(x ′ k ) = pk(f(p ′)) are λk-adjacent subsets of yk. it follows from theorem 2.18 that fk is connectivity preserving. since k was an arbitrarily selected index, fi is connectivity preserving for all i. now suppose each fi is connectivity preserving. let p = (x1, . . . , xv) ∈ x where xi ∈ xi. then f(p) = π v i=1fi(xi) is, by theorem 4.4, × v i=1λi-connected. suppose p ↔×v i=1 λi p ′ in x. then for some index k, xk ↔κi x ′ k in xk and for i 6= k there exist xi ∈ xi such that p = (x1, . . . , xv), p ′ = (x1, . . . , xk−1, x ′ k, xk+1, . . . , xv). since fk is connectivity preserving, there exist yk ∈ fk(xk) and y ′ k ∈ fk(x ′ k ) such that yk ↔λk y ′ k or yk = y ′ k. for i 6= k, let yi ∈ fi(xi). then q = (y1, . . . , yv) ∈ f(p) and q ′ = (y1, . . . , yk−1, y ′ k , yk+1, . . . , yv) ∈ f(p ′) are ×vi=1λiadjacent or equal. therefore, f(p) and f(q) are ×vi=1λi-adjacent subsets of y . it follows from theorem 2.18 that f is connectivity preserving. � for lexicographic adjacency, • example 3.22 shows that there is no product property for connectivity preservation; and • there is no factor property for connectivity preservation, as the following example shows. example 8.24. let f1 : ({0}, c1) ⊸ ([0, 1]z, c1) be the multivalued function f1(0) = [0, 1]z. let f2 : ({0}, c1) ⊸ ({0, 2}, c1) be the multivalued function f2(0) = {0, 2}. then f = f1 × f2 : {0} 2 = {(0, 0)} ⊸ [0, 1]z × {0, 2} is (l(c1, c1), l(c1, c1))-connectivity preserving, but f2 is not (c1, c1)-connectivity preserving. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 48 product adjacencies in digital topology proof. this follows from the observations that the single point (0, 0) is connected, and f(0, 0) = [0, 1]z × {0, 2} is l(c1, c1)-connected. � 9. shy maps we have the following. theorem 9.1. let f : (x, κ) → (y, λ) be a shy map of digital images. then f is an isomorphism if and only if f is locally one-to-one. proof. it is obvious that if f is an isomorphism, then f is locally one-to-one. to show the converse, we argue as follows. since f is shy, we know f is a continuous surjection. to show f is one-to-one, suppose there exist x, x′ ∈ x such that y = f(x) = f(x′) ∈ y . since f is shy, f−1(y) is κ-connected. therefore, if x 6= x′ then there is a path of distinct points p = {xi} m i=1 ⊂ f −1(y) such that x = x1, xi ↔ xi+1 for 1 ≤ i < m, and xm = x ′. but since f is locally one-to-one, f|n∗κ(x) is oneto-one, so f(x2) 6= f(x), contrary to the assumption p ⊂ f −1(y). therefore, we must have x = x′, so f is one-to-one. since f is one-to-one, f−1 is one-to-one. since f is shy, given y ↔ y′ in y , f−1({y, y′}) is connected. thus, f−1 is continuous. this completes the proof that f is an isomorphism. � the following generalizes a result of [8]. theorem 9.2 ([9]). let fi : (xi, κi) → (yi, λi) be a continuous surjection between digital images, 1 ≤ i ≤ v. then the product map πvi=1fi : (π v i=1xi, npv(κ1, . . . , κv)) → (π v i=1yi, npv(λ1, . . . , λv)) is shy if and only if each fi is a shy map. for the tensor product, we have the following. theorem 9.3. let fi : (xi, κi) → (yi, λi) be a surjection between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. if the product function f = πvi=1fi : (x, t (κ1, . . . , κv)) → (y, t (λ1, . . . , λv)) is shy, then fi is shy for each i. proof. since f is shy, it is continuous, so by theorem 3.10, each fi is continuous. clearly, each fi is a surjection. let yi ∈ yi. let y = (y1, . . . , yv) ∈ y . since f is shy, f −1(y) = πvi=1f −1 i (yi) is t (κ1, . . . , κv)-connected. by theorem 4.2, fi(yi) is κi-connected. let y′i ↔λi yi in yi. then y ′ = (y′1, . . . , y ′ v) ↔t (λ1,...,λv) y. since f is shy, f−1({y, y′}) = f−1({y}) ∪ f−1({y′}) = πvi=1f −1 i (yi) ∪ π v i=1f −1 i (y ′ i) is t (κ1, . . . , κv)-connected. by theorem 3.15, pi(f −1({y, y′})) = f−1i (yi) ∪ f −1 i (y ′ i) is κi-connected. from definition 2.30, we conclude that fi is a shy map. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 49 l. boxer the converse to theorem 9.3 is not generally true, as shown by the following. example 9.4. let f1 : ([0, 1]z, c1) → ({0}, c1) be the function f1(x) = 0. let f2 : ([0, 1]z, c1) → ([0, 1]z, c1) be the function f2(x) = x. then f1 and f2 are shy, but f1 × f2 : ([0, 1] 2 z , t (c1, c1)) → ({0} × [0, 1]z, t (c1, c1)) is not shy. proof. that f1 and f2 are shy is easily seen. further, f1 × f2 is a surjection. notice that (0, 0) ↔t (c1,c1) (1, 1), but (f1 × f2)(0, 0) = (0, 0) and (f1 × f2)(1, 1) = (0, 1) are neither equal nor t (c1, c1)-adjacent. therefore, f1×f2 is not (t (c1, c1), t (c1, c1))-continuous, hence is not (t (c1, c1), t (c1, c1))shy. � for the cartesian product adjacency, we have the following. theorem 9.5. let fi : (xi, κi) → (yi, λi) be a surjection between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. then the product function f = πvi=1fi : (x, × v i=1κi) → (y, × v i=1λi) is shy if and only if fi is shy for each i. proof. suppose f is shy. then clearly each fi is a surjection, and by theorem 3.17, fi is continuous. let yi ∈ yi. let y = (y1, . . . , yv) ∈ y . since f is shy, f −1(y) = πvi=1f −1 i (yi) is ×vi=1κi-connected. by theorem 3.18, the projection map pi is continuous, so pi(f −1(y)) = f−1 i (yi) is κi-connected. let y′ ∈ y be such that y′ ↔×v i=1 λi y. then y ′ must be among the points qi = (y1, . . . , yi−1, y ′ i, yi+1, . . . , yv), where y ′ i ∈ yi satisfies y ′ i ↔λi yi. since f is shy, f−1({y, qi}) = f −1(y) ∪ f−1(qi) is × v i=1κi-connected. since pi is continuous, pi(f −1({y, qi})) = pi(f −1(y) ∪ f−1(qi)) = f −1 i (yi) ∪ f −1 i (y ′ i) = f −1 i ({yi, y ′ i}) is κi-connected. this completes the proof that each fi is shy. suppose each fi is shy. then clearly f is a surjection, and by theorem 3.17, f is continuous. let yi ∈ yi. let y = (y1, . . . , yv) ∈ y . since fi is shy, f −1 i (yi) is κiconnected. by theorem 4.4, (9.1) f−1(y) = πvi=1f −1 i (yi) is × v i=1 κi-connected. let y′ ∈ y be such that y′ ↔×v i=1 λi y. then for some index i, y ′ = (y1, . . . , yi−1, y ′ i, yi+1, . . . , yv), where y ′ i ∈ yi satisfies y ′ i ↔λi yi. similarly, (9.2) f−1(y′) is ×vi=1 κi-connected. since fi is shy, f −1 i ({yi, y ′ i}) is connected, so there exist xi ∈ f −1 i (yi), x ′ i ∈ f−1i (y ′ i) such that xi ↔κi x ′ i or xi = x ′ i. for indices j 6= i, let xj ∈ f −1 j (yj). then w = (x1, . . . , xv) and w ′ = (x1, . . . , xi−1, x ′ i, xi+1, . . . , xv) satisfy (9.3) w ∈ f−1(y), w′ ∈ f−1(y′), and w ↔×v i=1 κi w ′ or w = w′. from statements (9.1), (9.2), and (9.3), we conclude that f−1({y, y′}) is ×vi=1κiconnected. therefore, f is shy. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 50 product adjacencies in digital topology for the lexicographic adjacency, we have the following. theorem 9.6. let fi : (xi, κi) → (yi, λi) be functions between digital images, 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, f = π v i=1fi : (x, l(κ1, . . . , κv)) → (y, l(λ1, . . . , λv)). if each fi is shy, then f is shy. proof. let y = (y1, . . . , yv) ∈ y , where yi ∈ yi. then f −1(y) = πvi=1f −1 i (yi). since each fi is shy, f −1 i (yi) is κi-connected. by theorem 4.6, f −1(y) is l(κ1, . . . , κv)-connected. let p = (y′1, . . . , y ′ v) ↔l(λ1,...,λv) y in y . then for some smallest index k, y′k ↔λk yk and if k > 1 then yi = y ′ i for i < k. since fk is shy, f −1 k ({yk, y ′ k}) is κk-connected. further, if k > 1 then f −1 i ({yi, y ′ i}) = f −1 i (yi) is connected, since fi is shy. now, (9.4) f−1(p) = πikf −1 i (yi), (9.5) f−1(p′) = πikf −1 i (y ′ i) by the shyness of the fi and theorem 4.6, each of f −1(p) and f−1(p′) is l(κ1, . . . , κv)-connected. further, since yi = y ′ i for i < k and, by shyness of fk, (9.6) f−1 k ({yk, y ′ k}) is κk-connected, from statements (9.4), (9.5), and (9.6) we can conclude that f−1(p) and f−1(p′) are l(κ1, . . . , κv)-adjacent sets. therefore, f −1({p, p′}) = f−1(p) ∪ f−1(p′) is l(κ1, . . . , κv)-connected. therefore, f is shy. � the following shows that the converse of theorem 9.6 is not generally true. example 9.7. let f1 : ([0, 1]z, c1) → {0} ⊂ (z, c1) be the function f1(x) = 0. let f2 : ({0, 2}, c1) → {0} ⊂ (z, c1) be the function f2(x) = 0. then f1 × f2 : ([0, 1]z × {0, 2}, l(c1, c1)) → ({(0, 0)}, l(c1, c1)) is shy, but f2 is not shy. proof. since f−12 (0) is not connected, f2 is not shy. however, [0, 1]z × {0, 2} is l(c1, c1)-connected, as discussed in example 3.27, so, from definition 2.30, f1 × f2 is shy. � 10. further remarks we have studied the tensor product, cartesian product, and lexicographic adjacencies for finite cartesian products of digital images. we have obtained many results for “product” and “factor” properties that parallel results obtained for extensions of the normal product adjacency in [9]. however, there are many properties known [9] for the normal product adjacency whose analogs for the adjacencies studied here are either false or we c© agt, upv, 2018 appl. gen. topol. 19, no. 1 51 l. boxer were not able to derive. by comparing the results of [9] with those of the current paper, it appears that the normal product adjacency is the adjacency that yields the most satisfying results for cartesian products of digital images. acknowledgements. 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[28] j. h. van lint and r. m. wilson, a course in combinatorics, cambridge university press, cambridge, 1992. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 53 @ appl. gen. topol. 16, no. 2(2015), 183-207doi:10.4995/agt.2015.3445 c© agt, upv, 2015 on the locally functionally countable sub algebra of c(x) o. a. s. karamzadeh, m. namdari and s. soltanpour department of mathematics, shahid chamran university of ahvaz, ahvaz, iran (karamzadeh@ipm.ir, namdari@ipm.ir, s-soltanpour@phdstu.scu.ac.ir) abstract let cc(x) = {f ∈ c(x) : |f(x)| ≤ ℵ0}, c f (x) = {f ∈ c(x) : |f(x)| < ∞}, and lc(x) = {f ∈ c(x) : cf = x}, where cf is the union of all open subsets u ⊆ x such that |f(u)| ≤ ℵ0, and cf (x) be the socle of c(x) (i.e., the sum of minimal ideals of c(x)). it is shown that if x is a locally compact space, then lc(x) = c(x) if and only if x is locally scattered. we observe that lc(x) enjoys most of the important properties which are shared by c(x) and cc(x). spaces x such that lc(x) is regular (von neumann) are characterized. similarly to c(x) and cc(x), it is shown that lc(x) is a regular ring if and only if it is ℵ0-selfinjective. we also determine spaces x such that soc ( lc(x) ) = cf (x) (resp., soc ( lc(x) ) = soc ( cc(x) ) ). it is proved that if cf (x) is a maximal ideal in lc(x), then cc(x) = c f (x) = lc(x) ∼= n ∏ i=1 ri, where ri = r for each i, and x has a unique infinite clopen connected subset. the converse of the latter result is also given. the spaces x for which cf (x) is a prime ideal in lc(x) are characterized and consequently for these spaces, we infer that lc(x) can not be isomorphic to any c(y ). 2010 msc: primary: 54c30; 54c40; 54c05; 54g12; secondary: 13c11; 16h20. keywords: functionally countable space; socle, zero-dimensional space; scattered space; locally scattered space, ℵ0-selfinjective. received 9 december 2014 – accepted 22 april 2015 http://dx.doi.org/10.4995/agt.2015.3445 o. a. s. karamzadeh, m. namdari and s. soltanpour 1. introduction c(x) denotes the ring of all real valued continuous functions on a topological space x. in [10] and [11], cc(x), the subalgebra of c(x), consisting of functions with countable image are introduced and studied. it turns out that cc(x), although not isomorphic to any c(y ) in general, enjoys most of the important properties of c(x). this subalgebra has recently received some attention, see [10], [23], [24], [4], and [11]. since cc(x) is the largest subring of c(x) whose elements have countable image, this motivates us to consider a natural subring of c(x), namely lc(x), which lies between cc(x) and c(x). our aim in this article, similarly to the main objective of working in the context of c(x), is to investigate the relations between topological properties of x and the algebraic properties of lc(x). in particular, we are interested in finding topological spaces x for which lc(x) = c(x). an outline of this paper is as follows: in section 2, we show that if x is a locally compact space, then lc(x) = c(x) if and only if x is locally scattered, which is somewhat similar to a classical result due to rudin in [27], and pelczynski and semadeni in [25] (of course, by no means as significant). this classical result says that a compact space x is scattered if and only if c(x) = cc(x). let us for the sake of the brevity, call the latter classical result, rps-theorem. if x is an almost discrete space or a p-space, then l1(x) = lf (x) = lc(x) = c(x), where lf (x) and l1(x) are the locally functionally finite (resp., constant) subalgebra of c(x), see definition 2.7. in section 3, we introduce zl-ideals in lc(x) and trivially observe that most of the facts related to z-ideals are extendable to zl-ideals. in section 4, topological spaces in which points and closed sets are separated by elements of lc(x), are called locally countable completely regular space (briefly, lc-completely regular). clearly, every zero-dimensional space is lc-completely regular (note, in the zero-dimensional case, points and closed sets are separated even by the elements of cc(x), which is a subring of lc(x)), see [10, proposition 4.4]. spaces x, for which lc(x) is regular, are called locally countably p-space (briefly, lcp-space) and are characterized both algebraically and topologically in this section. it is shown that p-spaces and lcp-spaces coincide when x is lccompletely regular. finally, in this section similar to c(x) and cc(x), we prove that lc(x) is a regular ring if and only if it is ℵ0-selfinjective. the socle of c(x) (i.e., cf (x)) which is in fact a direct sum of minimal ideals of c(x) is characterized topologically in [20, proposition 3.3], and it turns out that cf (x) is a useful object in the context of c(x), see [20], [1], [2], [8], [3], and [6]. the socle of cc(x), denoted by soc ( cc(x) ) , is studied in [11, proposition 5.3], and spaces x for which soc ( cc(x) ) = cf (x) are determined in [11, theorem 5.6]. motivated by the latter facts, we characterize the socle of lc(x) both topologically and algebraically, in section 5. spaces x for which soc ( lc(x) ) = soc ( cc(x) ) and soc ( lc(x) ) = cf (x) are also characterized. in [8, proposition 1.2], [3, remark 2.4], it is shown that cf (x) can not be a prime ideal in c(x), where x is any space. but, in [11, proposition 6.2], spaces c© agt, upv, 2015 appl. gen. topol. 16, no. 2 184 on the locally functionally countable sub algebra of c(x) x such that cf (x) is prime in cc(x) are characterized. the latter characterization is similarly extended to lc(x). consequently, this implies that lc(x) is not isomorphic to any c(y ) in general. all topological spaces that appear in this article are assumed to be infinite completely regular hausdorff, unless otherwise mentioned. for undefined terms and notations the reader is referred to [13], [7]. 2. the subalgebra lc(x) of c(x) definition 2.1. let f ∈ c(x) and cf be the union of all open sets u ⊆ x, such that f(u) is countable. we define lc(x) to be the set of all f ∈ c(x) such that cf is dense in x, i.e., cf = ⋃ {u| u is open in x and |f(u)| ≤ ℵ0} lc(x) = {f ∈ c(x) : cf = x} we shall briefly and easily notice that, lc(x) is a subalgebra as well as a sublattice of c(x) containing cc(x), and we call it the locally functionally countable subalgebra of c(x). it is manifest that cf (x) ⊆ cf (x) ⊆ cc(x) ⊆ lc(x) ⊆ c(x), where cf (x) = {f ∈ c(x) : |f(x)| < ∞}, see [10]. the following example shows that the equality between any two of these objects may not necessarily hold. example 2.2. let the basic neighborhood of x be the set {x}, for each point x ≥ √ 2 and for the rest of the real numbers (i.e., x < √ 2) the basic neighborhoods be the usual open intervals containing x. this is a topology t on r and in this case we put x = r. clearly, x is a completely regular hausdorff space which is finer than the usual topology of r. the function f : x → r, where f(x) = 1 for x ≥ √ 2, and f(x) = 0 otherwise, is continuous and x\z(f) is infinite, hence f ∈ cf (x)\cf (x), see [20, proposition 3.3]. we define g : x → r, such that g(x) = x for x ∈ [ √ 2, ∞) ∩ q and g(x) = 0 for x ∈ ([ √ 2, ∞) ∩ qc) ∪ (−∞, √ 2), hence g ∈ cc(x)\cf (x). also we observe that for the function h : x → r, where h(x) = x for x ≥ √ 2, and h(x) = √ 2 otherwise, we have h ∈ lc(x)\cc(x). the identity function i : x → r is continuous and ci = [ √ 2, ∞), see definition 2.1. hence i ∈ c(x)\lc(x). we note that cf = x if and only if for every open subset g ⊆ x, there exists an open subset u ⊆ x such that |f(u)| ≤ ℵ0 and u ∩ g 6= ∅ or equivalently if and only if for each open subset g ⊆ x, there exists a nonempty open subset v ⊆ g with |f(v )| ≤ ℵ0. lemma 2.3. for the space x the following statements hold. (1) if f, g ∈ c(x), then cf+g ⊇ cf ∩ cg. (2) if f, g ∈ c(x), then cfg ⊇ cf ∩ cg. (3) if f ∈ c(x), then c|f| = cf . (4) if f ∈ c(x), then c 1 f = cf . (5) if f, g ∈ lc(x), then cf ∩ cg = x. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 185 o. a. s. karamzadeh, m. namdari and s. soltanpour proof. let cf = ⋃ u⊆x |f(u)|≤ℵ0 u and cg = ⋃ v ⊆x |g(v )|≤ℵ0 v , where u and v are open subsets of x, then cf ∩ cg = ⋃ u,v ⊆x |f(u)|,|g(v )|≤ℵ0 (u ∩ v ) hence (1), (2), (3), (4) are evident. for part (5) we recall that if y is a dense subset of x and g is an open subset of x, then g ∩ y = g. since cf , cg are open and dense in x we infer that cf ∩ cg = cf = cg = x. � the following examples show that the equalities in (1), (2) of the previous lemma do not necessarily hold, in general. example 2.4. (1) let i : r → r be the identity function and f : r → r with f(x) = −x, then ci = cf = ∅, but ci+f = r. hence ci+f ) ci ∩ cf . (2) let i : r\{0} → r be the identity function and f : r\{0} → r with f(x) = 1/x, then ci = cf = ∅, but cif = r\{0}. hence cif ) ci ∩ cf . the following fact shows that lc(x) is indeed a subalgebra of c(x) such that whenever z(f) = ∅ where f ∈ lc(x), then f is a unit in lc(x). we remind the reader that the latter fact is not true for c∗(x). corollary 2.5. for the space x the following statements hold. (1) if f, g ∈ lc(x), then f + g ∈ lc(x) and fg ∈ lc(x). (2) f ∈ lc(x) if and only if |f| ∈ lc(x). (3) let f be a unit element in c(x), then f ∈ lc(x) if and only if 1f ∈ lc(x). corollary 2.6. lc(x) is a sublattice of c(x). definition 2.7. let f ∈ c(x) and cff be the union of all open sets u ⊆ x such that f(u) is finite. we define lf (x) to be the set of all f ∈ c(x) such that cff is dense in x, and call it locally functionally finite subalgebra of c(x), i.e., cff = ⋃ {u| u is open in x and |f(u)| < ∞} lf (x) = {f ∈ c(x) : cff = x} in particular, let f ∈ c(x) and ccf be the union of all open sets u ⊆ x such that f(u) is constant. we define l1(x) to be the set of all f ∈ c(x) such that ccf is dense in x, and we call it locally functionally constant subalgebra of c(x), i.e., ccf = ⋃ {u| u is open in x and |f(u)| = 1} l1(x) = {f ∈ c(x) : ccf = x} clearly, lf (x) and l1(x) are subalgebras of lc(x). in [26] and [15], e0(x) is defined, and by the above notation we have e0(x) = l1(x). it is evident that cf (x) ⊆ lf (x). c© agt, upv, 2015 appl. gen. topol. 16, no. 2 186 on the locally functionally countable sub algebra of c(x) remark 2.8. we note that lemma 2.3, corollary 2.5, and corollary 2.6 are also valid for lf (x) and l1(x). remark 2.9. it is manifest that cc(x) = r, where x = [0, 1]. but the cantor function f is a monotonic nonconstant continuous function, and ccf = [0, 1]\c = [0, 1], where c is the cantor set, see [9], [5]. therefore the cantor function f belongs to l1([0, 1]), and r ( l1([0, 1]), hence r ( lc([0, 1]). we emphasize that cc(x) = r, but r ( lc(x), and this can be considered as an advantage of lc(x) over cc(x), in this case. remark 2.10. in [15], a first countable compact space x (resp., in [26], a nonfirst countable compact space x) is constructed such that l1(x) = r. we are interested in characterizing topological spaces x for which lc(x) = c(x). in the following proposition we have a simple result, which is similar to rps-theorem. let us recall that in a commutative ring r by an annihilator ideal i, we mean i = ann(s) = {r ∈ r : rs = 0}, where s 6= {0} is a nonempty subset of r. proposition 2.11. if x is an almost discrete space (i.e., i(x), the set of isolated points of x, is dense in x), then l1(x) = lf (x) = lc(x) = c(x). in particular, if every nonzero annihilator ideal of c(x), where x is any space, contains a nonzero minimal ideal, then the latter equalities hold. proof. if f ∈ c(x), then ccf ⊇ ⋃ x∈i(x){x} = i(x). hence ccf = x, i.e., f ∈ l1(x). finally, we first recall that c(x) contains many nonzero zerodivisors (note, for each 0 6= f ∈ c(x), (f − |f|)(f + |f|) = 0. hence nontrivial annihilator ideals in c(x) always exist. consequently, by our assumption the socle of c(x) is not zero, i.e., cf (x) 6= 0. we now claim that ann ( cf (x) ) = 0. to see this, if i = ann ( cf (x) ) 6= 0, then i must contain a nonzero minimal ideal, hence i ∩cf (x) 6= 0. but, (i ∩cf (x))2 = 0 and since c(x) is reduced, we infer that i ∩cf (x) = 0, which is absurd. this means that we have already shown that ann ( cf (x) ) = 0, which by [20, proposition 2.1] is equivalent to the density of i(x) in x, hence we are done by the first part. � before, presenting the next fact, we evidently note that every scattered space is an almost discrete space, for if x ∈ x and ux is a neighborhood of x, then ux has an isolated point x0. since ux is open, x0 is an isolated point of x, too. hence x0 ∈ ux ∩ i(x) 6= ∅, therefore i(x) = x. proposition 2.12. if x is a scattered space, then l1(x) = lf (x) = lc(x) = c(x). in particular, if x is a compact scattered space, then the latter rings coincide with cc(x). proof. by the above comment and rps-theorem we are done. � the following example shows that the converse of the above corollary is not valid. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 187 o. a. s. karamzadeh, m. namdari and s. soltanpour example 2.13. let for each point x ∈ q, the basic neighborhood of x be the singleton {x}, and for each x ∈ qc, the basic neighborhood of x be the usual open interval containing x. this constitutes a topology on x = r, and it is clearly, a hausdorff normal space which is almost discrete, since i(x) = q. hence lc(x) = c(x), but x is not scattered. in view of rps-theorem we may naturally define a compact space x to be scattered if given any f ∈ c(x) and any x ∈ x, there exists a compact neighborhood vf of x such that |f(v ◦f )| ≤ ℵ0. motivated by this we give the following definition. definition 2.14. a space x is called locally scattered if given any f ∈ c(x) and a nonempty open set g, there exists a compact subset vf of x in g, with ∅ 6= v ◦f ⊆ g and |f(v ◦f )| ≤ ℵ0. the space βx where x is discrete is locally scattered. clearly, every scattered space is a locally scattered space, but the converse is not true. for example, βn is a locally scattered space which is not scattered, for βn\n has no isolated point (note, each clopen subset of βn\n has the same cardinality as βn\n, see [13, 6s(4)]). lemma 2.15. let x be a locally scattered space. then every open c-embedded subset of x (e.g., any clopen subset) is also locally scattered. proof. let y be an open c-embedded subset of x, and g be an open subset in y , and f ∈ c(y ). since y is c-embedded in x, we infer that there exists g ∈ c(x) such that g|y = f. clearly, g is open in x and by our assumption, there exists a compact subset vg in g such that ∅ 6= v ◦g ⊆ g ⊆ y , |g(v ◦g )| ≤ ℵ0. thus vg is compact in y in g with |f(v ◦g )| = |g(v ◦g )| ≤ ℵ0, i.e., y is locally scattered. � let us recall that a hausdorff space x is locally compact if and only if each point in x has a compact neighborhood. clearly, every compact hausdorff space is locally compact. the following result is somewhat similar to rpstheorem. theorem 2.16. let x be a compact space. then lc(x) = c(x) if and only if x is locally scattered. in particular, if x is a discrete space and y is a non-scattered clopen subset of βx (e.g., x = n and y = βn), then lc(y ) = c(y ) = c ∗(y ) 6= cc(y ). proof. first, we assume that x is compact and lc(x) = c(x). now, for each f ∈ c(x) we have cf = x. hence for any nonempty open subset g in x there exists an open subset uf in x such that |f(uf )| ≤ ℵ0, uf ∩ g 6= ∅. since the open subsets of a locally compact space are locally compact, we infer that uf ∩ g is locally compact. consequently, any neighborhood of a point x ∈ uf ∩ g contains a compact neighborhood, vf say, of x. hence x ∈ v ◦f ⊆ vf ⊆ uf ∩ g ⊆ x and |f(v ◦f )| ≤ |f(uf )| ≤ ℵ0, which means that x is locally scattered and we are done. the converse is evident by definition 2.14, c© agt, upv, 2015 appl. gen. topol. 16, no. 2 188 on the locally functionally countable sub algebra of c(x) and the definition of lc(x). for the last part, we notice that y as a closed subset of βx is compact and by lemma 2.15, it is locally scattered. now by the first part and the compactness of y , we have lc(y ) = c(y ) = c ∗(y ). but, in view of rps-theorem and the fact that y is not scattered, we infer that cc(y ) 6= c(y ), and we are done. � the previous proof immediately yields the following fact, too. corollary 2.17. let x be a locally compact space. then lc(x) = c(x) if and only if x is locally scattered. we recall that if the set of open neighborhoods of a point p in x is closed under countable intersection, then p is called a p-point. the set of all p-points of x is denoted by px and x is called a p-space if px = x. an interesting result due to a. w. hager asserts that a p-space x is functionally countable (i.e., c(x) = cc(x)) if and only if it is pseudo-ℵ1-compact (i.e., each locally finite family of open sets is countable), see [21, proposition 3.2]. this result is extended to cc(x) = c f (x) in [11, proposition 4.2]. the following is also a counterpart of the latter result. proposition 2.18. if px = x (in particular, if x is a p-space), then l1(x) = lf (x) = lc(x) = c(x). proof. for each f ∈ c(x) and x ∈ px there exists an open neighborhood ux of x such that f is constant on ux, see [13, 4l(3)]. therefore c c f ⊇ ⋃ |f(ux)|=1 ux ⊇ px, hence f ∈ l1(x). � we note that βn is not a p-space while l1(βn) = lf (βn) = lc(βn) = c(βn). by [13, 6v(6)], βn\n has a dense set of p-points, hence l1(βn\n) = lf (βn\n) = lc(βn\n) = c(βn\n). remark 2.19. let x be a p-space without isolated points, see [13, 13 p], then x is not almost discrete. but by proposition 2.18, l1(x) = lf (x) = lc(x) = c(x), see also proposition 2.11. let us borrow the following definition from [16]. definition 2.20. a topological space x is called locally functionally countable if every point x ∈ x is countably p-point, in the sense that there exists an open neighborhood ux of x such that c(ux) = cc(ux). the following result implies that if a space x is second countable or a compact space, then x is locally functionally countable if and only if it is functionally countable (i.e., c(x) = cc(x)). proposition 2.21. let x be a lindelöf space. then x is locally functionally countable if and only if it is functionally countable. proof. it is evident that every functionally countable space is locally functionally countable (note, for each x ∈ x take ux = x). conversely, let x be locally functionally countable, then for each f ∈ c(x), f(x) = f( ⋃ x∈x ux), where c© agt, upv, 2015 appl. gen. topol. 16, no. 2 189 o. a. s. karamzadeh, m. namdari and s. soltanpour ux is an open neighborhood of x with c(ux) = cc(ux). since x is lindelöf and c(x) ⊆ c(ux), for each x ∈ x, we infer that f(x) = f( ⋃∞ i=1 uxi) =⋃∞ i=1 f(uxi) is countable, and we are done. � the next result shows that for every locally functionally countable space x, c(x) coincides with lc(x). but the converse is not true in general, see example 2.13 (note, r with the topology in this example is not locally functionally countable, for no irrational number is a countably p-point). proposition 2.22. if x is a locally functionally countable space, then lc(x) = c(x). proof. we must show that for each f ∈ c(x), cf = x. let g ⊆ x be an open set in x and x ∈ g. since x is locally functionally countable, there exists an open neighborhood ux of x such that c(ux) = cc(ux). clearly |f(ux)| = |(f|ux)(ux)| ≤ ℵ0. now, x ∈ ux ⋂ g 6= ∅ and ux ⊆ cf imply that cf ⋂ g 6= ∅, hence cf = x. � it is clear that if y is a subset of x such that for each f ∈ c(x), f|y is constant, then y must be a singleton. for otherwise, if y1, y2 ∈ y and y1 6= y2, then by complete regularity of x there exists f ∈ c(x) such that f(y1) 6= f(y2), which is absurd. hence the following definition, which is also needed, is now in order. definition 2.23. if y is a subset of a space x, then the set of all f ∈ c(x) such that f|y is constant is a subalgebra of c(x), denoted by c1(y ). naturally, we say that y is constant with respect to a subring a of c(x) if a ⊆ c1(y ). we note that for every topological space x, c1(x) = c(x) if and only if x is singleton. if y is a proper closed subset of x, then r ( c1(y ). the following proposition is evident. proposition 2.24. let x be a topological space and y be a connected subset of x, then cc(x) ⊆ c1(y ). in particular, if x\y is countable, then a ⊆ c1(y ) if and only if a ⊆ cc(x). we conclude this section with the following fact whose proof is evident by the complete regularity of x. corollary 2.25. for any subspace y of x, r ⊆ c1(y ) ⊆ c(x). moreover, c1(y ) = r if and only if y is dense in x. proof. for the last part we note that if x /∈ y , then there exists f ∈ c(x) with f(x) = 0 and f(y ) = 1, i.e., c1(y ) 6= r. this implies that y = x in case c1(y ) = r. conversely, let y = x and take f ∈ c(x) such that f ∈ c1(y ), then f(y ) = c, where c ∈ r. consequently, f = c in c(x), for y is dense in x, hence we are done. � c© agt, upv, 2015 appl. gen. topol. 16, no. 2 190 on the locally functionally countable sub algebra of c(x) 3. zl-ideals we remind the reader that many facts in the context of c(x) can be extended naturally to lc(x), similarly to cc(x), see [10]. the proofs of most of the results in this section follow mutatis mutandis from the proofs of their corresponding results in [10]. therefore, we state them without proofs, for the record, but give pertinent references for their corresponding proofs (note, the reason that we emphasize on the recording of these facts here is because we do believe that lc(x) and cc(x), are eligible to play appropriate roles as companions of c(x), in the future studies in the context of c(x), see for example, the comment in the first two lines of the introduction in [4]. definition 3.1. a space x is said to be locally countably pseudocompact (briefly, lc-pseudocompact) if l∗c(x) = lc(x), where l ∗ c(x) = lc(x)∩c∗(x). the next three results are the counterparts of [13, theorem 1.7, corollary 1.8, and theorem 1.9]. proposition 3.2. every homomorphism ϕ : lc(x) → lc(y ) takes l∗c(x) into l∗c(y ). corollary 3.3. if y is not a lc-pseudocompact space, then lc(y ) can not be a homomorphic image of any l∗c(x). corollary 3.4. let ϕ be a homomorphism from lc(x) into lc(y ) whose image contains l∗c(y ), then ϕ(l ∗ c(x)) = l ∗ c(y ). if f ∈ lc(x) and f > 0, then there exists g ∈ lc(x) with f = g2. we also note that whenever f ∈ lc(x) and fr ∈ c(x) where r ∈ r, then fr ∈ lc(x). we recall that all positive units in lc(x) have the same number of square roots, see [13, 1b(1)]. the following proposition and its corollary are the counterparts of [13, 1d(1)] and [10, lemma 2.4] for lc(x). since the latter facts play a basic role in the context of c(x), we present sketch of proofs for these counterparts. proposition 3.5. if f, g ∈ lc(x) and z(f) is a neighborhood of z(g), then f = gh for some h ∈ lc(x). proof. we have zl(g) ⊆ intzl(f). put h(x) = { 0 , x ∈ zl(f) f(x) g(x) , x /∈ intzl(f) therefore h ∈ c(x), and ch ⊇ cf ∩ c1/g = cf ∩ cg = x. hence h ∈ lc(x) and f = gh. � corollary 3.6. if f, g ∈ lc(x), and |f| ≤ |g|r, r > 1, then f = gh for some h ∈ lc(x). in particular, if |f| ≤ |g|, then whenever fr is defined for r > 1, fr is a multiple of g. proof. let h(x) = { 0 , x ∈ zl(g) f(x) g(x) , x /∈ zl(g) c© agt, upv, 2015 appl. gen. topol. 16, no. 2 191 o. a. s. karamzadeh, m. namdari and s. soltanpour then h ∈ c(x), ch ⊇ cf ∩ cg = x. hence h ∈ lc(x) and f = gh. � proposition 3.7. if f ∈ lc(x), then there exists a positive unit u ∈ lc(x) with (−1 ∨ f) ∧ 1 = uf. proof. put u(x) = { 1 , − 1 ≤ f(x) ≤ 1 1 |f(x)| , 1 ≤ |f(x)| clearly cu = cf , hence u ∈ lc(x) and (−1 ∨ f) ∧ 1 = uf. so if f ∈ lc(x), then f and (−1 ∨ f) ∧ 1 belongs to an ideal of lc(x) � remark 3.8. the previous results are also true if we replace lc(x) by either lf (x) or l1(x). convention. let us put zl(x) = {z(f) : f ∈ lc(x)}, zf (x) = {z(f) : f ∈ lf (x)}, and z1(x) = {z(f) : f ∈ l1(x)}, where x is a topological space. definition 3.9. two subsets a and b of a topological space x are said to be locally countably separated (briefly, lc-separated) in x if there is an element f ∈ lc(x) such that f(a) = 1, f(b) = 0. the following result is the counterpart of [13, theorem 1.15], [10, theorem 2.8] . theorem 3.10. two subsets a, b of a space x are lc-separated if and only if they are contained in disjoint members of zl(x). moreover, lc-separated sets have disjoint zero-set neighborhoods in zl(x). clearly, if a < b and f ∈ lc(x) such that f(x) ≤ a, ∀x ∈ a, and f(x) ≥ b, ∀x ∈ b, where a, b are subsets of x, then a, b are lc-separated in x. corollary 3.11. if a, b are lc-separated in x, then there are zero-sets z1, z2 in zl(x) with a ⊆ x \ z1 ⊆ z2 ⊆ x \ b. definition 3.12. ∅ 6= f ⊆ zl(x) is called a zl-filter on x if f satisfies the following conditions. (1) ∅ /∈ f . (2) z1, z2 ∈ f , then z1 ∩ z2 ∈ f . (3) z ∈ f , z′ ∈ zl(x) with z′ ⊇ z, then z′ ∈ f . prime zl-filter and zl-ultrafilter are defined similarly to their counterparts in [13]. if i is an ideal of lc(x), then zl[i] = {z(f) : f ∈ i} is a zl-filter on x. conversely, if f is a zl-filter on x, then z −1[f ] = {f ∈ lc(x) : z(f) ∈ f} is an ideal in lc(x). moreover, every zl-filter f is of the form f = zl[i] for some ideal i in lc(x) and for any ideal j in lc(x), z −1[zl[j]] is an ideal in lc(x) containing j. in example 2.13, we consider the identity function i : (r, t ) → r, clearly i ∈ lc(r) = c(r). now, put i = (i), then zl(i) = {0}. clearly, f(x) = x1/3 ∈ lc(r), f ∈ z−1[zl[i]]\i. hence the following definition is in order. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 192 on the locally functionally countable sub algebra of c(x) definition 3.13. an ideal i in lc(x) is called a zl-ideal if whenever z(f) ∈ zl[i] and f ∈ lc(x), then f ∈ i. similarly, zf -ideal and z1-ideal are defined, see the previous convention. clearly, every zl-ideal is an intersection of prime ideals in lc(x). similarly, every zf -ideal and z1-ideal is an intersection of prime ideals in lf (x) and l1(x). we emphasize again that the proofs of the following results are the same as the proofs of their counterparts in c(x) and cc(x), see [10], and [13]. one can easily observe that these results up to proposition 3.24 and including it are also valid for lc(x) and lf (x). the next theorem is the counterpart of [13, theorem 2.9], [10, theorem 2.13]. theorem 3.14. let p be any zl-ideal in lc(x). then the following statements are equivalent. (1) p is a prime ideal in lc(x). (2) p contains a prime ideal in lc(x). (3) for all f, g ∈ lc(x), if fg = 0, then f ∈ p or g ∈ p. (4) for each f ∈ lc(x), there exists a zero-set in zl[p ] on which f does not change sign. corollary 3.15. every prime ideal in lc(x) is contained in a unique maximal ideal in lc(x). clearly if p is a prime ideal in lc(x), then zl[p ] is a prime zl-filter, and if f is a prime zl-filter, then z −1 l [f ] is a prime zl-ideal. it is evident that every prime zl-filter is contained in a unique zl-ultrafilter. the following lemma is the counterpart of [10, lemma 3.1], also see [28]. lemma 3.16. let f, g, l ∈ lc(x), z(f) ⊇ z(g) ∩ z(l) and define h(x) = { 0 , x ∈ z(g) ∩ z(l) fg2 g2+l2 , x /∈ z(g) ∩ z(l) , k(x) = { 0 , x ∈ z(g) ∩ z(l) fl2 g2+l2 , x /∈ z(g) ∩ z(l) then we have the following conditions. (1) |k| ∨ |h| ≤ |f|. (2) f = h + k. (3) fl2 = k(g2 + l2), fg2 = h(g2 + l2). (4) h, k ∈ lc(x). (5) ch ⊇ cf ∩ cg ∩ cl and ck ⊇ cf ∩ cg ∩ cl. the following results are the counterparts of [10, corollary 3.2 to corollary 3.8]. lemma 3.17. let a, b be two zl-ideals in lc(x). then either a+b = lc(x) or a + b is a zl-ideal. corollary 3.18. let f = {ai}i∈i be a collection of zl-ideals in lc(x). then either ∑ i∈i ai = lc(x) or ∑ i∈i ai is a zl-ideal. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 193 o. a. s. karamzadeh, m. namdari and s. soltanpour proposition 3.19. every minimal prime ideal in lc(x) is a zl-ideal. corollary 3.20. let f = {pi}i∈i be a collection of minimal prime ideals in lc(x). then either ∑ i∈i pi = lc(x) or p = ∑ i∈i pi is a prime ideal in lc(x). proposition 3.21. a prime ideal p in lc(x) is absolutely convex. proposition 3.22. the sum of a collection of semiprime ideals in lc(x) is a semiprime ideal or is the entire ring lc(x). proposition 3.23. let p be a prime ideal in lc(x). then the ring lc(x)/p is totally ordered and its prime ideals are comparable. the next corollary is much stronger than corollary 3.20 whose proof is similar to [10, corollary 3.9]. proposition 3.24. let {pi}i∈i be a collection of semiprime ideals in lc(x) such that at least one of pi’s is prime, then ∑ i∈i pi is a prime ideal or all of lc(x). all the previous results beginning with theorem 3.14, are also valid for l1(x). the following theorem is the counter part of [10, theorem 3.10], see also the comment preceding [10, theorem 3.10]. theorem 3.25. let i be an ideal in lc(x). then i and √ i have the same largest zl-ideal. 4. locally countable completely regular spaces definition 4.1. a hausdorff space x is called locally countable completely regular (briefly, lc-completely regular) if whenever f ⊆ x is a closed set and x ∈ x\f , then there exists f ∈ lc(x) with f(f) = 0 and f(x) = 1. we should remind the reader that, in this section, whenever the proof of a result is very similar to the proof of its counterpart in the literature, the proof is avoided. the proof of the following result is evident. proposition 4.2. a hausdorff space x is lc-completely regular if and only if whenever f ⊆ x is closed and x ∈ x \ f, then x and f have two disjoint zero-set neighborhoods in zl(x). consequently, there exist g, h ∈ lc(x) with x ∈ x \ z(h) ⊆ z(g) ⊆ x \ f. clearly x is a lc-completely regular space if and only if f = {z(f) : f ∈ lc(x)} is a base for the closed sets in x or equivalently if and only if b = {int(z(f)) : f ∈ lc(x)} is a base for the open sets in x. the next proposition is the counterpart of [13, 3.11(a)], [10, proposition 4.3]. proposition 4.3. let x be a lc-completely regular space and a, b be two disjoint closed sets in x such that a is compact, then there is f ∈ lc(x) with f(a) = 0 and f(b) = 1. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 194 on the locally functionally countable sub algebra of c(x) proposition 4.4. let x be a compact space. then x is lc-completely regular if and only if lc(x) separates points in x. since cc(x) is a subring of lc(x), the next result is evident, see [10, proposition 4.4] proposition 4.5. if x is a zero-dimensional space, then x is a lc-completely regular space. the following fact is also similar to [13, theorem 3.6], and [10, corollary 4.5]. proposition 4.6. let x be a hausdorff space. then x is a lc-completely regular space if and only if its topology coincides with the weak topology induced by lc(x). we recall that x is a p-space (resp., cp-space) if and only if c(x) (resp., cc(x)) is a regular ring, see [13, 4j] and [10]. in [10], it is shown that if c(x) is regular, then so too is cc(x). if x is zero-dimensional, then the regularity of c(x) and cc(x) coincide. we have already observed, see proposition 2.18, that if x is a p-space, then l1(x) = lf (x) = lc(x) = c(x). the next definition is now in order. definition 4.7. a space x is called a locally countably p-space (briefly, lcp-space) if lc(x) is regular. by the above comment we have the following result. proposition 4.8. every p-space is lcp-space. proposition 4.9. if a is any regular subring of c(x) such that cc(x) ⊆ a ⊆ c(x), then cc(x) is regular. in particular, if lc(x) is regular, then cc(x) is regular, too. proof. let a be a regular ring, we must show that for each f ∈ cc(x), there exists g ∈ cc(x) such that f = f2g. since a is regular, there is h ∈ a with f = f2h. consequently, f = f2g, where g = h2f. it is also evident that z(f) ⊆ z(g) and g(x) = 1 f(x) , whenever x /∈ z(f). hence |g(x)| = |f(x)| , i.e., g ∈ cc(x), and we are done. � corollary 4.10. let x be a zero-dimensional space. then x is p-space if and only if any of the rings cc(x), lc(x) is regular. remark 4.11. it is wroth mentioning that if x is a zero-dimensional space, then the regularity of c(x), lc(x), cc(x), and c(x, k) (where c(x, k), is a subring of c(x) whose elements take values in k, a subfield of r) coincide, see the above proposition and [10, remark 7.5]. the following theorem is the counterpart of [10, theorem 5.5] and its proof is also the same as the proof of its counterpart. we present a proof for the sake of completeness. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 195 o. a. s. karamzadeh, m. namdari and s. soltanpour theorem 4.12. a space x is a lcp-space if and only if every zero-set in zl(x) is open. moreover, in this case whenever {fi}∞i=1 is a countable set in lc(x), then ∞⋂ i=1 zl(fi) is an open zero-set in zl(x). proof. let x be a lcp-space and f ∈ lc(x), hence f = f2g for some g ∈ lc(x). it is evident that e = fg is an idempotent in lc(x) and z(f) = z(e) = x\z(1 − e) is clopen. conversely, let z(f) be open for each f ∈ lc(x), we are to show that lc(x) is regular. since z(f) = z(f 2) for all f ∈ lc(x), we infer that f = f2g for some g ∈ lc(x), by proposition 3.5. hence lc(x) is regular. finally, let i be an ideal in lc(x), generated by {fi}∞i=1, i.e., i = ∑∞ i=1 filc(x). if i = c(x), then we are done in this case, for ∞⋂ i=1 zl(fi) = ∅. hence we assume that i 6= c(x). since lc(x) is regular, we infer that i = ∑∞ i=1 ⊕eilc(x), where each ei, i ∈ i is an idempotent in lc(x), and for each i 6= j, eiej = 0, see [10, theorem 5.5], or [17, lemma 2], [8, proposition 1.4]. if x ∈ x, ej(x) 6= 0, then for each i 6= j, ei(x) = 0, and ⋂ z[i] = ∞⋂ i=1 z(fi) = ∞⋂ i=1 z(ei) now, we may define g = ∑∞ i=1 ei pi(1+ei) , where p ≥ 2 is a real number. clearly g ∈ c(x), and z(g) = ⋂∞ i=1 z(ei). on the other hand for each x ∈ x, there exists at most a unique i ≥ 1 such that ei(x) 6= 0. therefore g(x) = ei(x) pi(1+ei(x)) = 1 2pi . hence g(x) ⊆ {0, 1 2p , 1 2p2 , . . .} i.e., g ∈ cc(x), therefore g ∈ lc(x). � remark 4.13. in view of the previous proof we may record an interesting fact, which follows. let x be a lcp-space and {fi}i∈i be an infinite countable set of elements in c(x), then ⋂ i∈i z(fi) = z(g), where g ∈ cc(x) ⊆ lc(x) can be chosen with the property that g(x) is an infinite subset of an arbitrary subfield of r. it is well known that x is a p-space if and only if every gδ-set is open, see [13, 4j(3)]. the following theorem is the counterpart of this result, see also [10, corollary 5.7]. corollary 4.14. let x be a lc-completely regular lcp-space. then every gδ-set a containing a compact set s contains a zero-set in zl(x) containing s. in particular, every lc-completely regular lcp-space is a p-space. if mlp = mp ∩lc(x) and olp = op ∩lc(x), where p ∈ x and op is the ideal of c(x) consisting of all f in c(x) for which z(f) is a neighborhood of p. it goes without saying that mlp is a maximal ideal in lc(x) and o l p is a zl-ideal in lc(x). the following theorem is the counterpart of [13, 4j], [10, theorem 5.8]. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 196 on the locally functionally countable sub algebra of c(x) theorem 4.15. let x be a topological space. then the following statements are equivalent. (1) x is a lcp-space. (2) lc(x) is a regular ring. (3) each ideal in lc(x) is a zl-ideal. (4) each prime ideal in lc(x) is a maximal ideal. (5) for each p ∈ x, mlp = olp. (6) every zero-set in zl(x) is open. (7) each ideal in lc(x) is an intersection of maximal ideals. (8) for all f, g ∈ lc(x), (f, g) = (f2 + g2). (9) for every f ∈ lc(x), zl(f) (x \ zl(f)) is c-embedded. (10) if {fi : i ∈ n} ⊆ lc(x), then ⋂∞ i=1 zl(fi) is an open zero-set in zl(x). the following results are the counterparts of [11, proposition 2.5] and [11, corollary 2.6]. proposition 4.16. lc(x) is regular if and only if every pseudoprime ideal in lc(x) is prime. corollary 4.17. let x be a lc-completely regular. then every pseudoprime ideal in c(x) is prime if and only every pseudoprime ideal in lc(x) is prime. the following theorem is similar to [13, theorem 4.11], [11, theorem 3.8]. theorem 4.18. let x be a lc-completely regular space, then the following statements are equivalent. (1) x is compact. (2) every ideal of lc(x) is fixed. (3) every maximal ideal of lc(x) is fixed. (4) every prime ideal of lc(x) is fixed. if x is any topological space and x ∈ x, mlx = mx ∩ lc(x), then as we pointed out earlier mlx is a maximal ideal of lc(x) and in fact lc(x) mlx ∼= r. consequently, the jacobson radical of lc(x) is zero. definition 4.19. a maximal ideal m in lc(x) is called a real maximal ideal of lc(x) if lc(x) m ∼= r. a topological space x is called locally countably realcompact space (briefly, lc-realcompact) if every real maximal ideal m of lc(x) is of the form m = m l x for some x ∈ x. the following results are the counterparts of [13, 10.5(c)] and [11, theorem 3.11]. theorem 4.20. x is a lc-realcompact space if and only if each nonzero homomorphism from lc(x) into r is a valuation map. if x is a compact zero-dimensional space, the corresponding x → mlx is one-one from x onto the set of maximal ideals of lc(x), say max(lc(x)), and hence the space x is homeomorphic to max(lc(x)) with the stone topology c© agt, upv, 2015 appl. gen. topol. 16, no. 2 197 o. a. s. karamzadeh, m. namdari and s. soltanpour (note, the proof is similar to [13, 4.9(a)], see also the comment above [10, theorem 3.9] and [19]). the proof of the following result which is similar to its counterpart in [13, theorem 8.3], is omitted. proposition 4.21. two zero-dimensional lc-realcompact spaces x and y are homeomorphic if and only if lc(x) ∼= lc(y ). we recall that if x, y are compact zero-dimensional spaces, then c(x) ∼= c(y ) if and only if cc(x) ∼= cc(y ). in what follows we show that this result also holds if we replace cc(x) by lc(x), but the proof is not as evident. theorem 4.22. let x and y be two lc-completely regular compact spaces (e.g., zero-dimensional compact spaces). then x and y are homeomorphic if and only if lc(x) ∼= lc(y ). in particular, if x, y are compact zerodimensional spaces, then lc(x) ∼= lc(y ) if and only if cc(x) ∼= cc(y ) if and only if cf (x) ∼= cf (y ) if and only if c(x) ∼= c(y ). proof. clearly if lc(x) ∼= lc(y ), then max(lc(x)) and max(lc(y )) are homeomorphic (with the stone topology), i.e., x, y are homeomorphic, see the comment preceding proposition 4.21. conversely, let ϕ : x → y be a homeomorphism from x onto y . if f ∈ lc(y ), then we claim that foϕ ∈ lc(x). to see this, since f ∈ lc(y ), we infer that y = cf = ⋃ i∈i vi, where for each i ∈ i, vi is open in y and |f(vi)| ≤ ℵ0. let us put ui = ϕ−1(vi), where i ∈ i. clearly ui is open in x and |foϕ(ui)| = |foϕ(ϕ−1(vi))| = |f(vi)| ≤ ℵ0, hence cfoϕ ⊇ ⋃ i∈i ui. since ϕ is open (note, ϕ −1 is continuous), we infer that x = ϕ−1(y ) = ϕ−1( ⋃ i∈i vi) ⊆ ϕ−1( ⋃ i∈i vi) = ⋃ i∈i ϕ−1(vi) = ⋃ i∈i ui therefore cfoϕ = x, i.e., foϕ ∈ lc(x). now we define σ : lc(y ) → lc(x) with σ(f) = foϕ. it is evident that σ is an isomorphism from lc(y ) onto lc(x). the last part is evident. � remark 4.23. the above result shows that if x, y are compact zero-dimensional spaces, such that c(x) ∼= c(y ), then lc(x) ∼= lc(y ). in the comment following [11, corollary 9.5], it is observed that whenever x, y are two arbitrary spaces (not necessary compact zero-dimensional) and c(x) ∼= c(y ), then cc(x) ∼= cc(y ) and cf (x) ∼= cf (y ) (i.e., cc(x) and cf (x) are algebraic objects). this naturally raises the question that whether lc(x) is also an algebraic object, too (i.e., if c(x) ∼= c(y ), then is lc(x) ∼= lc(y ))? clearly, if x, y are strongly zero-dimensional spaces with c(x) ∼= c(y ), then lc(βx) ∼= lc(βy ). let us recall that a commutative ring r is selfinjective (resp., ℵ0-selfinjective), if every homomorphism f : i → r, where i is an ideal (resp., countably generated ideal) in r, can be extended to f̂ : r → r. we recall that a subset s of a commutative ring r is said to be orthogonal, provided xy = 0 for all x, y ∈ s with x 6= y. in the following result we show that [10, theorem 6.10] is also true for lc(x). in contrast to the proofs of some of the previous results, c© agt, upv, 2015 appl. gen. topol. 16, no. 2 198 on the locally functionally countable sub algebra of c(x) we should emphasize that the next proof can not be easily obtained from the proof of its counterpart (i.e., [10, theorem 6.10]). it is well known that the ℵ0-selfinjectivity of a ring is not a consequence of its regularity, in general, see [14, examples 14.7, 14.9]. but, the following worthwhile fact shows that lc(x) as well as c(x) and cc(x) have this rare property, see [8], [10]. we should remind the reader that cf (x) does not satisfy this property in general, see [10, remark 6.11, example 7.1] (note, cf (x) is always regular, see [11, the comment preceding proposition 4.2]. theorem 4.24. let x be a topological space. then lc(x) is regular if and only if lc(x) is ℵ0-selfinjective. proof. if lc(x) is ℵ0-selfinjective, then lc(x) is regular by [18, proposition 1.2], or [10, lemmas 6.7, 6.8, remark 6.9]. conversely, by [18, lemma 1.9] and [10, lemma 6.8, remark 6.9], it suffices to show that if s is an orthogonal subset in lc(x), then there exists f ∈ lc(x) such that for each g ∈ s, fg = g2. let s = {fi}∞i=1, where fi 6= 0, for each i ∈ i. since lc(x) is regular, ⋂∞ i=1 z(fi) = z(h) is an open zero-set in lc(x), by theorem 4.12. put gi = x\z(fi), for each i ≥ 1. since fifj = 0, hence gi ∩ gj = ∅, for each i 6= j, and gi’s are clopen for each i ≥ 1. let us put g = ⋃∞ i=1 gi, hence x = ⋃∞ i=1 gi ∪ (x\g). we may define f : x → r by f(x) = { fi(x) , x ∈ gi 0 , x /∈ g i.e., f|gi = fi for all i ≥ 1 and f(x) = 0 for all x ∈ x\g. hence f is continuous by [13, 1a(2)] and we must show that f ∈ lc(x). let v ⊆ x be an arbitrary open set, then we are to show that there exists an open set u in x such that |f(u)| ≤ ℵ0 and u ∩ v 6= ∅. now we consider two cases. first let v ⊆ x\g, then f(v ) = 0, hence v ⊆ cf . otherwise v ∩ g 6= ∅, hence there exists a nonempty open subset gi such that v ∩ gi 6= ∅. since fi ∈ lc(x) i.e., cfi = x, hence there exists an open set h ⊆ cfi such that |fi(h)| ≤ ℵ0 and ∅ 6= h ∩ (v ∩ gi) = u. now clearly, |f(u)| = |fi(u)| ≤ |fi(h)| ≤ ℵ0 i.e., we are done. finally, we claim that ffi = f 2 i , for each fi ∈ s and this complete the proof, by [18, lemma 1.9]. to this end, we note that if f(x) = 0, then x /∈ g, hence x /∈ gi for all i ≥ 1, i.e., x ∈ z(fi), for all i ≥ 1. thus ffi = f2i , on z(f) for each fi ∈ s. since f(x) = fi(x), for each x ∈ gi = x\z(fi) and z(f) ⊆ z(fi) for each i ≥ 1, we infer that ffi = f2i , for each fi ∈ s, hence we are done. � remark 4.25. let x be an uncountable discrete space, then c(x) = lc(x) is selfinjective but cc(x) is not selfinjective, see [10, example 7.1, remark 7.5]. more generally, if c(x) is ℵ0-selfinjective, then by [8, theorem 1], x is a p-space. hence in view of proposition 2.18, we have l1(x) = lf (x) = lc(x) = c(x). moreover in view of theorem 4.22 and remark 4.11, we note that the ℵ0-selfinjectivity of c(x), lc(x), cc(x), and c(x, k) coincide if x is a zero-dimensional space. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 199 o. a. s. karamzadeh, m. namdari and s. soltanpour 5. the socle of lc(x) we recall that the socle of any commutative ring r, soc(r), is the sum of its nonzero minimal ideals (in fact, it can be written as the direct sum of some of its nonzero minimal ideals). we recall that cf (x) = {f ∈ c(x) : |x\z(f)| < ∞}, the socle of c(x), is a z-ideal. clearly, cf (x) ⊆ lf (x) ⊆ lc(x). we show that cf (x) ⊆ soc ( lc(x) ) , i.e., cf (x) is a sum of minimal ideals of lc(x). in [20, proposition 3.1], it is shown that if i is a minimal ideal of c(x), then i = ec(x), where e is an idempotent such that e(x) = 1 and e(x\{x}) = 0, where x is an isolated point of x. clearly c(x) = ec(x) ⊕ (1 − e)c(x), ec(x) = elc(x) = ecc(x), and lc(x) = elc(x) ⊕ (1 − e)lc(x). we also note that (1 − e)lc(x) = (1 − e)c(x) ∩ lc(x) is a maximal ideal in lc(x), therefore elc(x) = ec(x) is a minimal ideal in lc(x). hence every minimal ideal of c(x) is a minimal ideal in lc(x), too. therefore cf (x) is an ideal in lc(x), and cf (x) ⊆ soc ( lc(x) ) . we should also emphasize that since soc ( lc(x) ) is a semisimple lc(x)-module, hence cf (x) is a direct summand of soc ( lc(x) ) as a lc(x)-module. in the proof of theorem 5.4, we shall briefly observe that soc ( lc(x) ) ⊆ soc ( cc(x) ) . let us also recall that cf (x) is the subring of c(x) whose elements have finite image. hence, we have cf (x) ⊆ cf (x) ⊆ cc(x) ⊆ lc(x) ⊆ c(x) and soc ( cc(x) ) = soc ( cf (x) ) , see [10], [11]. the following lemma which is similar to [11, lemma 5.1], somehow determines the minimal ideals of lc(x). let us first remind the reader that if i is a nonzero minimal ideal in a reduced commutative ring r, then i = er, where e ∈ r is an idempotent (note, i = (a) = (a2), for every 0 6= a ∈ i, and a = a2r, for some r ∈ r, now put e = ar). in [11, lemma 5.1], it is shown that if 0 6= e is an idempotent, then ecc(x) is a minimal ideal in cc(x) if and only if z(1 − e) is connected. in the next lemma the minimal ideals in lc(x) are characterized, too. lemma 5.1. let i be a nonzero minimal ideal in lc(x), then i = elc(x) where e is an idempotent in lc(x) such that z(1−e) is connected. conversely, if i = elc(x) where e 6= 0 is an idempotent in lc(x) such that z(1 − e) is a constant subset of x with respect to lc(x), then i is a minimal ideal in lc(x). proof. let i be a nonzero minimal ideal in lc(x). since lc(x) is reduced, i = elc(x), where e is an idempotent in lc(x). if z(1 − e) is not connected, there exists a nonempty clopen subset a ( z(1 − e) (note, a is clopen in x, too). now define the idempotent e1 ∈ lc(x) such that a = z(1 − e1). clearly z(1 − e1) ( z(1 − e). consequently, e1 = ee1 but e 6= e1e. hence e1lc(x) ( elc(x) = i and this contradicts the minimality of i. conversely, let i = elc(x), where e ∈ lc(x) such that y = z(1 − e) ⊆ x is a constant subset of x with respect to lc(x) (i.e., lc(x) ⊆ c1(y ), see definition 3.16). we are to show that i is minimal in lc(x). it suffices to show that (1−e)lc(x) is a maximal ideal in lc(x). now we define ϕ : lc(x) → r by ϕ(f) = f(y ). c© agt, upv, 2015 appl. gen. topol. 16, no. 2 200 on the locally functionally countable sub algebra of c(x) clearly, kerϕ = (1 − e)lc(x) and lc(x)(1−e)lc(x) ∼= r, hence (1 − e)lc(x) is maximal in lc(x). � in [11, proposition 5.3], the socle of cc(x) is characterized and in [11, remark 5.2, and the introduction of section 5], it’s observed that cf (x) ⊆ soc ( cc(x) ) = soc ( cf (x) ) . next, we topologically characterize the socle of lc(x). the next proof is similar to the proof of [11, proposition 5.3], but it’s given for the sake of the reader. proposition 5.2. let f ∈ lc(x) be a nonunit element. if f ∈ soc ( lc(x) ) , then x\z(f) ⊆ ⋃n i=1 ai, where n ∈ n and {a1, a2, . . . , an} is a set of mutually disjoint clopen connected subsets of x. conversely, if x\z(f) ⊆ ⋃n i=1 ai, where n ∈ n and {a1, a2, . . . an} is a set of mutually disjoint clopen constant subsets of x with respect to lc(x), then f ∈ soc ( lc(x) ) . in particular, soc ( lc(x) ) is a zl-ideal in lc(x). proof. we put soc ( lc(x) ) = ∑ i∈i ⊕eilc(x), where each ei is an idempotent in lc(x), and eilc(x) is a nonzero minimal ideal in lc(x). let f = ei1f1 + ei2f2 + . . .+ einfn be an element in soc ( lc(x) ) , where fk ∈ lc(x) and ik ∈ i, k = 1, 2, . . . , n. we put aik = z(1 − eik), for each ik ∈ i, k = 1, 2, . . . , n. clearly, aik , k = 1, 2, . . . , n, are clopen and connected, by lemma 5.1. since the idempotent elements {ei : i ∈ i} are mutually orthogonal, we infer that {ai : i ∈ i, ai = z(1 − ei)} is a set of mutually disjoint clopen connected subsets of x. if x /∈ ⋃n i=1 ai, then eik (x) = 0, k = 1, 2, . . . , n, hence x ∈ z(f). therefore x\z(f) ⊆ ⋃n i=1 ai. conversely, let x\z(f) ⊆ ⋃n i=1 ai, where {ai : i ∈ i} is a set of mutually disjoint clopen constant subsets in x with respect to lc(x), we show that f ∈ soc ( lc(x) ) . since each ai is a clopen set, there exists an idempotent ei, such that ai = z(1 − ei), where i = 1, 2, . . . , n. we also note that each ai is constant with respect to lc(x), hence there is a set of idempotents in lc(x), {e1, . . . , en} say, which are mutually orthogonal and each eilc(x) is a minimal ideal in lc(x), by lemma 5.1. clearly, f = e1f + e2f + . . . + enf ∈ lc(x) which belongs to soc ( lc(x) ) = ∑ i∈i ⊕eilc(x). � remark 5.3. one can easily observe that if in the previous two results we trade off lc(x) with any r-subalgebra of lc(x), a say, which contains cc(x), then the two results are also valid for a. the next result determines spaces x such that the socles of lc(x), cc(x) and hence of cf (x) coincide. theorem 5.4. soc ( lc(x) ) = soc ( cc(x) ) if and only if the clopen connected subsets of x coincide with the clopen constant subsets of x with respect to lc(x). proof. soc ( lc(x) ) ⊆ soc ( cc(x) ) , for if i is a minimal ideal in soc ( lc(x) ) , then i = elc(x) where e is an idempotent such that z(1 − e) is connected, by lemma 5.1. hence i is a minimal ideal in cc(x), by [11, lemma 5.1]. now, let i be a nonzero minimal ideal in soc ( cc(x) ) , so i = ecc(x), where e 6= 0, 1 is c© agt, upv, 2015 appl. gen. topol. 16, no. 2 201 o. a. s. karamzadeh, m. namdari and s. soltanpour an idempotent and z(1 − e) is a clopen connected subset in x, by [11, lemma 5.1]. hence by our hypothesis z(1 − e) is constant with respect to lc(x). therefore i = elc(x) = ecc(x) is a minimal ideal of lc(x), by lemma 5.1, hence it is in soc ( lc(x) ) . conversely, let soc ( lc(x) ) = soc ( cc(x) ) , and ∅ 6= y ⊆ x be a clopen constant subspace of x with respect to lc(x) we are to show that y is connected. clearly, there exists e ∈ lc(x) such that e(y ) = 1, e(x\y ) = 0. but y = z(1 − e), hence by lemma 5.1, we infer that e ∈ soc ( lc(x) ) = soc ( cc(x) ) . consequently, y = z(1 − e) must be connected, by [11, proposition 5.3]. � we remind the reader in [11, theorem 6.6], it is proved that soc ( cc(x) ) = cf (x) if and only if each clopen connected subsets of x consists of a single isolated point. motivated by this fact and theorem 5.4, we present the next result. theorem 5.5. if every proper nonempty clopen connected subset of x is singleton, (e.g., any totally disconnected space), then soc ( lc(x) ) = cf (x). conversely, if soc ( lc(x) ) = cf (x), then every proper nonempty clopen constant subspace of x with respect to lc(x) is singleton. proof. let every proper nonempty clopen connected subset of x be singleton, we are to show that soc ( lc(x) ) = cf (x). it is evident that cf (x) ⊆ soc ( lc(x) ) . let i be a nonzero minimal ideal in soc ( lc(x) ) , so by lemma 5.1, i = elc(x), where e 6= 0, 1 is an idempotent and z(1 − e) is a clopen connected subset in x. hence by our hypothesis z(1 − e) is singleton. therefore i = elc(x) = ec(x) is a minimal ideal in cf (x), by [20, proposition 3.3]. conversely, let cf (x) = soc ( lc(x) ) , and ∅ 6= y ⊆ x be a clopen constant subspace of x with respect to lc(x). there exists e ∈ lc(x) such that e(y ) = 1, e(x\y ) = 0. clearly, by lemma 5.1, e ∈ soc ( lc(x) ) = cf (x), hence ec(x) is a minimal ideal in cf (x), therefore y = z(1 − e) is singleton. � the following remark is now immediate. remark 5.6. cf (x) = soc ( lc(x) ) = soc ( cc(x) ) if and only if each clopen connected subset of x consists of a single isolated point. consequently, if x is zero-dimensional or totally disconnected, we have cf (x) = soc ( lc(x) ) = soc ( cc(x) ) . let us recall that an ideal in a commutative ring r is essential if it intersects every nonzero ideal of r nontrivially. it is well known and easy to show that a nonzero ideal i in a reduced ring r (i.e., no nonzero element in r is nilpotent) is essential if and only if ann(i) = 0, see [3, background and preliminary results]. the proof of the following corollary is similar to [11, corollary 5.4], but we include the proof for the sake of the reader. corollary 5.7. let x be a lc-completely regular space, and soc ( lc(x) ) =∑ i∈i ⊕eilc(x), where eilc(x) is a nonzero minimal ideal of lc(x), and ei c© agt, upv, 2015 appl. gen. topol. 16, no. 2 202 on the locally functionally countable sub algebra of c(x) is an idempotent for each i ∈ i. put y = ⋃ i∈i z(1 − ei), then soc ( lc(x) ) is essential in lc(x) if and only if y is dense in x. proof. let y = ⋃ i∈i z(1−ei) be dense in x, we are to show that soc ( lc(x) ) is essential in lc(x). since lc(x) is reduced, in order to prove that soc ( lc(x) ) is essential in lc(x) it suffices to show that ann(soc ( lc(x) ) ) = (0). we note that f ∈ ann(soc ( lc(x) ) ) if and only if fei = 0 for each i ∈ i. now, if fei = 0, then f(z(1 − ei)) = 0, hence f(y ) = {0}. since y is dense in x we infer that f = 0, and we are done. conversely, let soc ( lc(x) ) be essential in lc(x), hence ann(soc ( lc(x) ) ) = (0) in lc(x). let us now take x ∈ x\y and obtain a contradiction. by lc-complete regularity of x, there exists 0 6= f ∈ lc(x) with f(y ) = f(y ) = 0. therefore f(z(1 − ei)) = 0, hence fei = 0 for all i ∈ i. thus 0 6= f ∈ ann(soc ( lc(x) ) ) = (0), which is a contradiction. � we recall that cf (x) is never a prime ideal of c(x), see [8, proposition 1.2], or [3, remark 2.4]. the following result characterizes spaces x such that cf (x) 6= 0 is a prime ideal in lc(x) (note, cf (x) 6= 0 if and only if x has isolated points). proposition 5.8. let |i(x)| < ∞, where i(x) is the set of isolated points in x. if 0 6= cf (x) is a prime ideal in lc(x), then x\i(x) is connected in x. conversely, if x\i(x) is constant with respect to lc(x), then 0 6= cf (x) is prime in lc(x). proof. let y = x\i(x) = a ∪ b, where a, b are two nonempty infinite disjoint clopen subsets of y and seek a contradiction. since y is clopen in x we infer that a, b are also clopen in x. clearly, x = i(x) ∪ a ∪ b. now define f, g ∈ lc(x) such that f(a∪i(x)) = 1, f(b) = 0 and g(a∪i(x)) = 0, g(b) = 1. clearly fg = 0 ∈ cf (x), but by [20, proposition 3.3], we infer that f, g /∈ cf (x), which is a contradiction. conversely, let y = x\i(x) be constant with respect to lc(x) and take f, g ∈ lc(x) such that fg ∈ cf (x). clearly x = y ∪ i(x), so x\z(fg) ⊆ i(x) and fg(y ) = 0. since f and g are constant on y , we infer that either f(y ) = 0 or g(y ) = 0, i.e., x\z(f) ⊆ i(x) or x\z(g) ⊆ i(x), therefore f ∈ cf (x) or g ∈ cf (x), by [20, proposition 3.3], and we are done. � in the following corollary, we consider spaces x, such that cf (x) is not a prime ideal in lc(x). corollary 5.9. if i(x) is an infinite set or y = x\i(x) is disconnected, then cf (x) is never a prime ideal in lc(x). proof. let i(x) be an infinite set and take a = {xn : n ∈ n}, b = {yn : n ∈ n} to be two disjoint countably infinite subsets of i(x). we now define f(x) = { 1 n , x = xn ∈ a 0 , x /∈ a and g(x) = { 1 n , x = yn ∈ b 0 , x /∈ b . let ǫ > 0 be given, then there exists k ∈ n such that 1 n < ǫ, for all n ≥ k. now, for the clopen subsets g = x\{x1, x2, . . . , xk}, h = x\{y1, y2, . . . , yk} and c© agt, upv, 2015 appl. gen. topol. 16, no. 2 203 o. a. s. karamzadeh, m. namdari and s. soltanpour for each x ∈ g, y ∈ h, we have |f(x)| < ǫ, |g(y)| < ǫ, hence f, g ∈ c(x). clearly, f, g ∈ cc(x). therefore f, g ∈ lc(x) and 0 = fg ∈ cf (x), but f, g /∈ cf (x), by [20, proposition 3.3]. consequently, in this case cf (x) is not prime in cc(x), a fortiori, in lc(x)). finally let |i(x)| < ∞ and x\i(x) be disconnected, hence by proposition 5.8, we are done. � in the next result, which is our main theorem in this section, we consider the maximality of cf (x) in lc(x). first, let us recall that if ϕ : c(x) → c(y ) is a ring homomorphism with ϕ(1) = 1, then ϕ(cc(x)) ⊆ cc(y ). this is an easy consequence of the fact that whenever f ∈ cc(x), then im(ϕ(f)) ⊆ im(f) (note, let r ∈ im(ϕ(f)), then ϕ(f) − r is non-unit, but ϕ(f − r) = ϕ(f) − r, hence f − r is non-unit too, i.e., z(f − r) 6= ∅, and we are done), see also[11, the comment following corollary 3.5]. theorem 5.10. let cf (x) be a maximal ideal in lc(x). then c f (x) = cc(x) = lc(x) and lc(x) is isomorphic to a finite direct product of fields, each of which, is isomorphic to r and x has a unique infinite clopen connected subset. conversely, let x have a unique infinite clopen connected subset, and assume that every element of lc(x) is constant on it, and lc(x) ∼= n∏ i=1 fi, where each fi is a field. then c f (x) = cc(x) = lc(x), and cf (x) is maximal in lc(x). proof. let cf (x) be a maximal ideal in lc(x). let us first take care of the case, when cf (x) = 0. clearly in this case lc(x) = r (note, in this case x is connected and cc(x) = c f (x) = r), and we are done. hence, we may assume that that cf (x) 6= 0. in view of the previous corollary we infer that i(x), the set of isolated points of x must be finite. let us assume that |i(x)| = n, where n is a positive integer (note, cf (x) 6= 0 if and only if i(x) 6= ∅, see [20, proposition 3.3]). hence cf (x) is a finitely generated ideal in c(x) (note, by [20, proposition 3.1], there is a one-one correspondence between i(x) and the set of nonzero minimal ideals in c(x)). consequently, cf (x) = n∑ i=1 ⊕eic(x), where each ei is an idempotent and eic(x) = eilc(x) is a minimal ideal in c(x) as well as in lc(x), see the comment preceding lemma 5.1. clearly, cf (x) = ecf (x) = ec(x) = elc(x), where e = e1 + e2 + · · · + en (note, eiej = 0 for i 6= j). since cf (x) is maximal in lc(x), we infer that e 6= 1, which implies that (1−e)lc(x) is a nonzero minimal ideal in lc(x). inasmuch as cf (x) is maximal in lc(x) and cf (x) ⊆ soc ( lc(x) ) , we infer that either cf (x) = soc ( lc(x) ) or lc(x) = soc ( lc(x) ) . we claim that cf (x) = soc ( lc(x) ) leads us to a contradiction. to see this, we note that (1−e)lc(x) is a nonzero minimal ideal in lc(x). hence if the latter equality holds, we infer that (1 − e)lc(x) must be in cf (x). but cf (x) ∩ (1 − e)lc(x) = 0, which is absurd. consequently, we must have lc(x) = soc ( lc(x) ) = cf (x) ⊕ (1 − e)lc(x). now, for each ei we can easily show that eilc(x) ∼= r ∼= (1−e)lc(x). to see this, let x ∈ z(1 − ei) and define ϕ : lc(x) → r by ϕ(f) = f(x) for c© agt, upv, 2015 appl. gen. topol. 16, no. 2 204 on the locally functionally countable sub algebra of c(x) all f ∈ lc(x). hence, (1 − ei)lc(x) ⊆ kerϕ. since (1 − ei)lc(x) is maximal in lc(x), we infer that (1 − ei)lc(x) = kerϕ, hence eilc(x) ∼= lc(x)(1−ei)lc(x) ∼= r. similarly (1 − e)lc(x) ∼= lc(x)elc(x) ∼= r. consequently, we have already shown that lc(x) ∼= n+1∏ i=1 ri, where ri = r. in view of lemma 5.1 , and by the fact that (1 − e)lc(x) is minimal in lc(x), we infer that z(e) is connected. consequently, by the comment preceding lemma 5.1, (1 − e)cc(x) is a minimal ideal in cc(x). hence cc(x) = cf (x) ⊕ (1 − e)cc(x), which is equal to soc ( cc(x) ) = soc ( cf (x) ) ⊆ cf (x), see the comment preceding proposition 5.2. thus cf (x) = cc(x). but, cc(x) = cf (x) ⊕ (1 − e)cc(x) is the direct sum of n + 1 minimal ideals in cc(x), hence by the above proof for lc(x), we can also show that cc(x) ∼= n+1∏ i=1 ri, where ri = r. let us consider the natural isomorphism ϕ : cc(x) → lc(x) ⊆ c(x). now in view of the comment preceding the theorem we have lc(x) ⊆ cc(x), hence lc(x) = cc(x) = c f (x). finally, in view of [20, proposition 3.3], it is clear that the connected clopen set z(e) is infinite (in fact z(e) = x \ i(x)). it is also manifest that every non-singleton connected subset of x must be a subset of z(e), hence z(e) is the only clopen connected subset of x which is infinite, and we are done. conversely, since lc(x) ∼= n∏ i=1 fi, where each fi is a field, we infer that lc(x) = n∑ i=1 ⊕uilc(x) = soc ( lc(x) ) , where each uilc(x) is a nonzero minimal ideal in lc(x), and each ui is idempotent with 1 = u1 +u2 · · · un. now let 1 6= u ∈ lc(x) be an idempotent such that z(1−u) is the unique infinite clopen subset of x, on which, every element of lc(x) is constant. consequently, ulc(x) is a minimal ideal in lc(x), by lemma 5.1. multiplying, 1 = u1 + u2 · · · un by u, we get u = uu1 + uu2 + · · · uun. clearly, u 6= 0, hence uui 6= 0 for some i. we now claim that there is a unique i, with 1 ≤ i ≤ n such that uui 6= 0. to see this, let uui 6= 0 6= uuj for some i 6= j and obtain a contradiction. but uui 6= 0 implies that ulc(x)uilc(x) 6= 0, hence ulc(x)uilc(x) = ulc(x) = uilc(x) and similarly ulc(x) = ujlc(x), which is a contradiction. consequently, we may assume that uui = 0 for 1 ≤ i ≤ n − 1 and uun 6= 0. this means that ulc(x) = unlc(x). in view of [20, proposition 3.3], and the fact that z(1 − u) is infinite, we infer that u /∈ cf (x), i.e, un /∈ cf (x). by lemma 5.1, and the fact that each uilc(x) for 1 ≤ i ≤ n − 1 is minimal, we infer that each z(1 − ui) is connected, which by our assumption is not an infinite set, hence it must be a singleton. consequently, in view of [20, proposition 3.3], ui ∈ cf (x) for 1 ≤ i ≤ n − 1. inasmuch as lc(x) = n∑ i=1 ⊕uilc(x) = soc ( lc(x) ) , we infer that cf (x) = n−1∑ i=1 ⊕uilc(x) ⊕ unlc(x) ⋂ cf (x). since unlc(x) is minimal, we infer that either unlc(x) ⋂ cf (x) = 0 or unlc(x) ⋂ cf (x) = unlc(x). the latter c© agt, upv, 2015 appl. gen. topol. 16, no. 2 205 o. a. s. karamzadeh, m. namdari and s. soltanpour equality is impossible, for by what we have already observed above un /∈ cf (x). consequently, cf (x) = n−1∑ i=1 ⊕uilc(x) and lc(x) = cf (x)⊕unlc(x), hence cf (x) is maximal in lc(x). now by the proof of the first part we also have cf (x) = cc(x) = lc(x), hence we are done. � the following theorem shows that for the spaces x in which there exist certain constant subsets with respect to lc(x), lc(x) can not be isomorphic to any c(y ). theorem 5.11. let |i(x)| < ∞ and x\i(x) be constant with respect to lc(x) (note, in this case lc(x) = c f (x)). then there is no space y with lc(x) ∼= c(y ). proof. let |i(x)| < ∞ and x\i(x) be constant with respect to lc(x), then cf (x) is a prime ideal in lc(x) by proposition 5.8. if there exists a space y such that lc(x) ∼= c(y ), then soc ( lc(x) ) ∼= cf (y ). now, since soc ( lc(x) ) is a zl-ideal containing a prime ideal cf (x), soc ( lc(x) ) is a prime ideal in lc(x), by theorem 3.14. hence cf (y ) is a prime ideal in c(y ), which is a contradiction, see the comment preceding proposition 5.11. � remark 5.12. if we replace lc(x) by lf (x) or by l1(x) in this section, then some of the results of this section remain valid for these two rings, too. remark 5.13. let x = w ∪{x1, x2, . . . , xn}, where w is constant with respect to l1(x) (e.g., if we take w as in remark 2.10) and x1, x2, . . . , xn are the only isolated points of x (note, w is connected and has no isolated point) i.e., |i(x)| < ∞ and x\i(x) = w is a constant subset of x with respect to l1(x). hence, by theorem 5.11, remark 5.12, l1(x) can not be isomorphic to any c(y ), in general. but in some special cases, namely, l1(w) and l1(x) we have l1(w) = r and l1(x) ∼= ∏n i=1 ri, where ri = r, for i = 1, 2, . . . , n. that is to say l1(w) = c(y ), where y is a singleton, and l1(x) = c(z), where |z| < ∞. but, we should remind the reader that we are interested in infinite spaces. remark 5.14. let k be a 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[28] d. rudd, on two sum theorems for ideals of c(x), michigan math. j. 17 (1970), 139– 141. c© agt, upv, 2015 appl. gen. topol. 16, no. 2 207 () @ appl. gen. topol. 19, no. 1 (2018), 155-161doi:10.4995/agt.2018.7888 c© agt, upv, 2018 few remarks on maximal pseudocompactness angelo bella department of mathematics and computer science, university of catania, cittá universitaria viale a. doria 6, 95125, catania, italy (bella@dmi.unict.it) communicated by d. georgiou abstract a pseudocompact space is maximal pseudocompact if every strictly finer topology is no longer pseudocompact. the main result here is a counterexample which answers a question raised by alas, sanchis and wilson. 2010 msc: 54d55; 54d99. keywords: pseudocompact; maximal pseudocompact; hereditarily maximal pseudocompact; accessible set. 1. introduction for undefined notions we refer to [5] and [3]. given a space x, we denote by τ(x) its topology. a tychonoff space is pseudocompact if every real valued continuous function defined on it is bounded. equivalently, a tychonoff space x is pseudocompact if and only if every locally finite family of open sets is finite [5]. in a serie of papers ofelia alas, richard wilson and their coauthors have investigated the notion of maximal pseudocompactness (see [1], [2] and [8]). this notion is justified because a pseudocompact space can have a strictly finer tychonoff topology which is still pseudocompact: consider for instance the compact space ω1+1 with the order topology. indeed, let x be the space obtained by isolating the point ω1, i.e. x = ω1 ⊕{ω1}. as x is the topological sum of two countably compact tychonoff spaces, it is pseudocompact and it clearly has a topology strictly finer than ω1 +1. a tychonoff space (x, τ) is maximal pseudocompact received 19 july 2017 – accepted 07 october 2017 http://dx.doi.org/10.4995/agt.2018.7888 if (x, τ) is pseudocompact but (x, σ) is not pseudocompact for any tychonoff topology σ strictly finer than τ. 2. results an easy but useful fact is in the following: lemma 2.1. let (x, τ) be a t1 space and p ∈ x be a point of countable character. if σ is a tychonoff topology on x such that σ ⊇ τ and (x, σ) is pseudocompact, then σ coincides with τ at p (i.e. p has the same system of neighbourhoods in both topologies). proof. fix a decreasing local base of open sets {un : n < ω} at p in τ. if σ differs from τ at p, then there exists a closed neigbourhood v of p in σ which is not a neighbourhood of p in τ. but then, {un \ v : n < ω} would be a locally finite family of non-empty open sets in (x, σ). this family is infinite because (x, τ) is t1 and we reach a contradiction. � therefore, a first countable pseudocompact space is always maximal pseudocompact. we begin by formulating a better sufficient condition. let us say that a collection s hits a set a if s ∩ a 6= ∅ for each s ∈ s. a set s ⊆ x is co-pseudocompact [resp. co-singleton] if x \ s is pseudocompact [resp. |x \ s| = 1]. proposition 2.2. let (x, τ) be a pseudocompact space and assume that for every co-pseudocompact set a ⊆ x and every point p ∈ a there exists a sequence of open sets in x which hits a and converges to p. then x is maximal pseudocompact. proof. assume by contradiction that there is a topology σ strictly finer than τ such that (x, σ) is still pseudocompact. if σ 6= τ at a point p, we may fix a regular closed neighbourhood v of p in σ which is not a neighbourhood of pin τ. the set x \ v is co-pseudocompact and p ∈ x \ v τ . therefore, there exists a sequence {un : n < ω} ⊆ τ converging to p and satisfying un ∩ (x \ v ) 6= ∅ for every n. but then, {un \ v : n < ω} would be a locally finite infinite family of open sets in (x, σ), in contrast with the pseudocompactness of σ. � the next observation shows that maximal pseudocompact spaces are “very close to” first countable. proposition 2.3. if x is maximal pseudocompact, then each p ∈ x is the limit of a convergent sequence of non-empty open sets. proof. proposition 3.1 in [2] states that a maximal pseudocompact space has countable π-character, but the proof of this result actually establishes the stronger statement that every point is the limit of a convergent sequence of non-empty open sets. indeed, let p be a non-isolated point of x. the maximal pseudocompactness of x implies that x \ {p} is not pseudocompact and so there is an infinite family of disjoint open sets {un : n < ω} ⊆ x \ {p} which c© agt, upv, 2018 appl. gen. topol. 19, no. 1 156 few remarks on maximal pseudocompactness is discrete in x \ {p}. we claim that the sequence {un : n < ω} converges to p. if not, there would be an infinite set s ⊆ ω and a closed neighbourood v of p such that un \ v 6= ∅ for each n ∈ s. but then, the infinite family of open sets {un \ v : n ∈ s} would be discrete in x, in contrast with the pseudocompactness of x. � the above proposition shows that maximal pseudocompactness imposes strong conditions to the topology. another non-trivial consequence is described in the following: corollary 2.4. if x is maximal pseudocompat, then |x| ≤ 2c(x). proof. by proposition 3.1 in [2] and šapirovskĭı’s formula w(x) ≤ πχ(x)c(x), there exists a dense set d such that |d| ≤ 2c(x). but, by proposition 2.3 each point of x is the limit of a sequence contained in d and so |x| ≤ |d|ω ≤ 2c(x). � thus, there are plenty of compact spaces which are not maximal pseudocompact. in addition, by corollary 3.5 in [2] every compactification of a noncompact pseudocompact space is not maximal pseudocompact. propositions 2.2 and 2.3 seem to suggest that in the class of pseudocompact spaces maximal pseudocompactness (briefly mp) is a convergent-like property. indeed, if p+ := “for every co-pseudocompact set a ⊆ x and every point p ∈ a there exists a sequence of open sets in x which hits a and converges to p” and p− := “every point is the limit of a converging sequence of non-empty open sets”, then pseudocompact + p+ =⇒ mp =⇒ p− the one-point compactification of an uncountable discrete space is a compact space satisfying p+ which is not first countable. we believe there should exist a maximal pseudocompact space which does not satisfy p+, but at moment we do not have such a space. on the other direction, let x = a ∪ ω be a ψ-space over a mad family a on ω and let x ∪ {∞} be its one-point compactification. fix a0 ∈ a and let z be the quotient space of x ∪{∞} obtained by identifying a0 and ∞ to a point p. z is a compact space which satisfies p −. this is evident for each point of z \ {p}. for p observe that {{n} : n ∈ a0} is a sequence of open sets in z converging to p. but, z is not maximal pseudocompact, because the function f : x → z, defined by letting f(a0) = p and f(x) = x for every x ∈ x \ {a0}, is a continuous bijection which is not open. we conclude that the unknown property p which characterizes maximal pseudocompactness within the class of pseudocompact space lies in between p+ and p− and differs from the latter. since p− is just p+ restricted to co-singleton sets (a subclass of co-pseudocompact sets), property p should involve an appropriate subclass of copseudocompact sets. question 2.5. what is the convergent property p such that a pseudocompact space x is maximal pseudocompact if and only if x satisfies p? c© agt, upv, 2018 appl. gen. topol. 19, no. 1 157 as pointed out in [1], a relevant role in studying maximal pseudocompactness is played by the notion of accessibility from a dense subset. given a space x and a dense set d ⊆ x, we say that x is strongly accessible from d if for any x ∈ x \ d and any a ⊆ d such that x ∈ a, there exists a countable sequence s ⊆ a converging to x. proposition 2.6 ([1, theorem 2.4]). let x be a pseudocompact space and d a dense set of isolated points. if x is strongly accessible from d, then x is maximal pseudocompact. therefore, any compactification γ(n) of the set of integers n with the discrete topology such that γ(n) is strongly accessible from n is maximal pseudocompact [1]. in some case, γ(n) \ n can be homeomorphic to ω1 + 1, thus showing for instance that a compact maximal pseudocompact space need not be fréchet. the first construction of this kind, discovered in the attempt to find a compact radial separable non fréchet space, is the space δ(n) given in [7] by assuming the continuum hypothesis. a similar example, obtained under the weaker assumption d = ω1, is given in [6]. but perhaps, the easiest way to obtain it is by using a tower. recall that the cardinal t is the smallest size of a tower, i.e. a well-ordered by reverse almost inclusion family of subsets of n without any infinite pseudointersection (see [3] for more). fix a family a = {aα : α ∈ ω1} of subsets of n, well-ordered by ⊂ ∗. furthermore, put a−1 = ∅ and aω1 = n. we define a topology on the set γ(n) = n∪ω1 +1 by declaring each point of n isolated and by taking as a local base at each α ∈ ω1 +1 the sets ]β, α]∪aα \ (aβ ∪f), where f is a finite subset of n and −1 ≤ β < α. to be more formally correct, we should replace in the previous definition ω1 + 1 for instance with {xα : α ∈ ω1 + 1}. however, we believe our semplified notation does not cause any trouble to the reader. the space γ(n) is compact hausdorff and first countable at each α < ω1. proposition 2.7. the space γ(n) may fail to be maximal pseudocompact if and only if t = ω1. proof. we begin by showing that t > ω1 implies the maximal pseudocompactness of γ(n). by proposition 2.6, it suffices to check that γ(n) is strongly accessible from n. as the only point of uncountable character in γ(n) is ω1, we only need to consider the case of a set a ⊆ n such that ω1 ∈ a. this clearly implies |(n \ aα) ∩ a| = ω for each α. since t > ω1, the family {(n \ aα) ∩ a : α < ω1} is not a tower and hence we may take an infinite set s ⊆ a satisfying s ⊆∗ n \ aα for every α < ω1. this actually means that s converges to ω1 and we are done. to complete the proof, we now verify that t = ω1 implies that γ(n) may fail to be maximal pseudocompact. the point is that, by assuming t = ω1, we may choose the family a in such a way that {n \ aα : α < ω1} is a tower. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 158 few remarks on maximal pseudocompactness consequently, if we take an infinite set s ⊆ n, then there exists some α < ω1 such that |s ∩ aα| = ω. if α is the least ordinal with this property, then the set s ∩ aα is actually a sequence converging to α. therefore, no subsequence of n can converge to ω1. in particular, the point ω1 cannot be the limit of a sequence of non-empty open subsets of γ(n) and so by proposition 2.3 the space γ(n) is not maximal pseudocompact. � the above discussion suggests the following: question 2.8 (zfc). is there a compactification γ(n) of n which is maximal pseudocompact but not fréchet? any radial compactification of n is obviously maximal pseudocompact, but dow [4] has shown that there are models where every compact separable radial space is fréchet. in [1], question 2.17 asks whether proposition 2.6 is reversible, namely: “suppose that x is a maximal pseudocompact space with a dense set of isolated points d. is x strongly accessible from d?” mimiking the construction given in [1], example 2.14, we can give a consistent negative answer to the above question. example 2.9. [t > ω1] or [d = ω1] a maximal pseudocompact space which is not accessible from a dense set of isolated points. proof. take the space γ(n) = n∪ω1 +1 described above under the assumption t > ω1 or d = ω1and let y = (ω + 1) × ω1. let x be the quotient space of y ⊕ γ(n), obtained by identifying the set {ω} × ω1 ⊆ y with the copy of ω1 in γ(n) (i.e. (ω, α) ≡ α for each α ∈ ω1). recall that if q : y ⊕ γ(n) → x is the quotient map, then v ∈ τ(x) if and only if q−1(v ) ∈ τ(y ⊕ γ(n)) if and only if q−1(v ) ∩ y ∈ τ(y ) and q−1(v ) ∩ γ(n) ∈ τ(γ(n)). we claim that x is maximal pseudocompact. indeed, let σ be a pseudocompact topology finer than the topology τ on x. since γ(n) is compact, q ↾ γ(n) is an embedding and so we may identify γ(n) with q(γ(n)) ⊆ x. we must have ω1 ∈ n σ , otherwise a sequence s ⊆ n, converging in τ to ω1, would provide a discrete infinite family of open singletons in σ. since by lemma 2.1 σ coincides with τ at each α ∈ ω1, we see that n σ = γ(n). thus γ(n) is regular closed in σ and hence pseudocompact in σ. since γ(n) is maximal pseudocompact, we conclude that σ coincides with τ on γ(n), i.e. σ ↾ γ(n) = τ ↾ γ(n). since x is first countable at each point of q(y ), again by lemma 2.1, σ coincides with τ on q(y ), so we also have σ ↾ q(y ) = τ ↾ q(y ). now, take any v ∈ σ. from v ∩ q(y ) ∈ σ ↾ q(y ) = τ ↾ q(y ), it follows q−1(v ) ∩ y ∈ τ(y ). in a similar manner, from v ∩ γ(n) ∈ σ ↾ γ(n) = τ ↾ γ(n) = τ ↾ q(γ(n)), it follows q−1(v ) ∩ γ(n) ∈ τ(γ(n)). this suffices to conclude that v ∈ τ and hence σ = τ(x). x has a dense set of isolated points, namely d = (ω × {0, α + 1 : α < ω1})∪n. but, x is not strongly accessible from d, because ω1 is in the closure of ω × {α + 1 : α < ω1}, but no subsequence of it can converge to ω1. this last thing depends on the countable compactness of y . � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 159 in [1] a space x was called hereditarily maximal pseudocompact (briefly hmp) if each closed subspace of x is maximal pseudocompact. since pseudocompactness is preserved by passing to regular closed subspaces, the following way to define the hereditary version of maximal pseudocompactness certainly makes sense. a space is weakly hereditarely maximal pseudocompact (briefly whmp) if every regular closed subspace is maximal pseudocompact. example 2.14 in [1] as well as the space x in example 2.9 above provide maximal pseudocompact spaces which are not whmp. regarding example 2.9, observe that the set q(ω × ω1) is open in x and q(ω × ω1) is homeomorphic to y ∪{ω1}. but the latter is a regular closed subspace of x which is not maximal pseudocompact. a non-trivial difference between hmp and whmp emerges from the following: proposition 2.10. if the pseudocompact space x is strongly accessible from a dense set d of isolated points, then x is whmp. proof. let y be a regular closed subset of x. since y = u for some open set u, y is strongly accessible from the dense subset of isolated points u ∩ d and we are done. � any version of the space γ(n), mentioned above, is therefore whmp but not hmp. in connection with question 2.17 in [1], consider the following: proposition 2.11. let x be a whmp space. if d is a dense set of isolated points, then x is strongly accessible from d. proof. take a point x ∈ x \ d and a set a ⊆ d such that x ∈ a. since a is open, we see that the subspace a is maximal pseudocompact. therefore, by proposition 2.3 (or lemma 2.8 in [1]), there is a sequence in a converging to x. � propositions 2.10 and 2.11 show that proposition 2.6 is reversible precisely for whmp spaces in the class of pseudocompact spaces. acknowledgements the author thanks richard wilson for the useful discussion concerning the presentation of example 2.9. a great thank also to the referee for the useful comments and the careful reading. this research was partially supported by a grant of the group gnsaga of indam. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 160 few remarks on maximal pseudocompactness references [1] o. t. alas, m. sanchis and r. g. wilson, maximal pseudocompact and maximal rclosed spaces, houston j. math. 38 (2012), 1355–1367. [2] o. t. alas, w. w. tkachuk and r. g. wilson, maximal pseudocompact spaces and the preiss-simon property, central eur. j. math. 12 (2014), 500–509. [3] e.k. van douwen, the integers and topology, in: handbook of set-theoretic topology (k. kunen and j. e. vaughan editors), elsevier science publishers b.v., amsterdam, (1984), 111–160. [4] a. dow, on compact separable radial spaces, canad. math. bull. 40 (1997), 422–432. [5] r. engelking, general topology, heldermann verlag, berlin (1989). [6] p. j. nyikos, convergence in topology, in: recent progress in general topology. elsevier science publishers b. v. (amsterdam). m. husek and j. van mill ed. (1992), 537–570. [7] k. m. devi, p. r. meyer and m. rajagopalan, when does countable compactness imply sequential compactness?, general topology appl. 6 (1976), 279–289. [8] v. v. tkachuk and r. g. wilson, maximal countably compact spaces and embeddings in mp spaces, acta math. hungar. 145, no. 1 (2015), 191–204. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 161 @ appl. gen. topol. 20, no. 1 (2019), 43-55doi:10.4995/agt.2019.9168 c© agt, upv, 2019 approximation of common fixed points in 2-banach spaces with applications d. ramesh kumar a and m. pitchaimani b a department of mathematics, school of advanced sciences, vellore institute of technology, vellore 632014, india. (rameshkumard14@gmail.com) b ramanujan institute for advanced study in mathematics, university of madras, chennai 600005, india. (mpitchaimani@yahoo.com) communicated by e. a. sánchez-pérez abstract the purpose of this paper is to establish the existence and uniqueness of common fixed points of a family of self-mappings satisfying generalized rational contractive condition in 2-banach spaces. an example is included to justify our results. we approximate the common fixed point by mann and picard type iteration schemes. further, an application to well-posedness of the common fixed point problem is given. the presented results generalize many known results on 2-banach spaces. 2010 msc: 47h10; 54h25. keywords: common fixed point; mann iteration; picard iteration; wellposedness; 2-banach space. 1. introduction the banach contraction principle [5] is one of the pivotal results in nonlinear analysis which guarantees the existence and uniqueness of fixed points of mappings. there is a great number of extensions of the banach contraction principle using different forms of contractive conditions in various generalized metric spaces, we refer the reader to [1, 2, 6, 17, 13, 14, 20, 23, 24, 25, 27, 28, 29, 30]. some of such generalizations are obtained through rational contractive conditions. das and gupta [8] studied the extention of banach contraction in term received 09 january 2018 – accepted 26 september 2018 http://dx.doi.org/10.4995/agt.2019.9168 d. ramesh kumar, m. pitchaimani of rational inequality and derived a fixed point result for a self mapping which runs as follows: theorem 1.1. let f be a mapping of a complete metric space x into itself such that (i) d(f(x), f(y)) ≤ α d(y, f(y))[1 + d(x, f(x))] 1 + d(x, y) + βd(x, y), for all x, y ∈ x, α > 0, β > 0, α + β < 1 and (ii) for some x0 ∈ x, the sequence of iterates {f n(x0)} has a subsequence {fnk (x0)} with ξ = lim n→∞ fnk (x0). then ξ is a unique fixed point of f. meanwhile pachpatte [19] extended it to a pair of self mappings. recently, azam et al. [4] obtained fixed point theorems for a pair of contractive mappings using an analogical rational inequality of [8] in a complex-valued metric space setting. also, in [18], nashine et al. studied some common fixed point theorems for a pair of mappings under certain rational contractions in complex valued metric spaces. in [33], shahkoohi and razani established the existence of fixed point of a self mapping satisfying rational geraghty contractive mappings in partially ordered b-metric spaces. for related study [3, 32]. in the 1960’s, gähler [10, 11, 12] generalized the idea of metric space and introduced a new theory of 2-metric space. on the other hand, white [35] started the investigation on the concept of 2-banach spaces. since then, many authors have focused on these spaces and presented papers that dealt with fixed point theory for single-valued and multi-valued operators in 2-banach spaces (see [7, 15]). recently, pitchaimani and ramesh kumar [21] obtained common fixed points under implicit relation in 2-banach spaces and proved some common and coincidence fixed point theorems for asymptotically regular mappings in [22]. inspired by the concept of 2-banach spaces and notion of rational type conditions, an attempt has been made in this paper to prove the existence and uniqueness of common fixed point of a family of self mappings satisfying the generalized rational contractive condition in 2-banach spaces with an example which illustrates our result. further, the approximation of the common fixed point by means of mann and picard iteration method is given. finally, we prove the well-posedness of the common fixed point problem. 2. preliminaries in this section, we recall the notions which will be required in the sequel. throughout this paper, n denotes the set of all positive integers and r denotes the set of all real numbers. definition 2.1. let x be a real linear space and ‖·, ·‖ be a non-negative real valued function defined on x × x satisfying the following conditions: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 44 approximation of common fixed points in 2-banach spaces (i) ‖x, y‖ = 0 if and only if x and y are linearly dependent; (ii) ‖x, y‖ = ‖y, x‖, for all x, y ∈ x; (iii) ‖x, ay‖ = |a|‖x, y‖, for all x, y ∈ x and a ∈ r; (iv) ‖x, y + z‖ ≤ ‖x, y‖ + ‖x, z‖, for all x, y, z ∈ x; then ‖·, ·‖ is called a 2 norm and the pair (x, ‖·, ·‖) is called a linear 2-normed space. some of the basic properties of 2-norms are that they are non-negative satisfying ‖x, y + ax‖ = ‖x, y‖, for all x, y ∈ x and a ∈ r. definition 2.2. a sequence {xn} in a linear 2-normed space (x, ‖·, ·‖) is called a cauchy sequence if lim m,n→∞ ‖xm − xn, y‖ = 0 for all y ∈ x. definition 2.3. a sequence {xn} in a linear 2-normed space (x, ‖·, ·‖) is said to converge to a point x ∈ x if lim n→∞ ‖xn − x, y‖ = 0 for all y ∈ x. definition 2.4. a linear 2-normed space (x, ‖·, ·‖) in which every cauchy sequence is convergent is called a 2-banach space. example 2.5. let x = r3 and a 2-norm ‖·, ·‖ be defined as follows: ‖x, y‖ = |x × y| = det   ~i ~j ~k x1 x2 x3 y1 y2 y3   where x = (x1, x2, x3), y = (y1, y2, y3) ∈ x and~i,~j,~k are the unit vectors along the axes. note that (x, ‖·, ·‖) is a 2-banach space. definition 2.6. a sequence {xn} in a 2-banach space x is said to be asymptotically t regular if lim n→∞ ‖xn − t xn, y‖ = 0 for all y ∈ x. definition 2.7. let x be a nonempty set and s, t : x → x be self mappings. then (i) an element x ∈ x is said to be a fixed point of t if x = t x. (ii) if x = sx = t x then x is called a common fixed point of s and t . let c be a nonempty convex subset of a 2-banach space (x, ‖·, ·‖) and t : x → x be a mapping then (i) the sequence {xn} defined by x0 ∈ c, xn+1 = (1 − βn)xn + βnt (xn), ∀n ≥ 0, where {βn} satisfies 0 < βn ≤ 1, ∀n and ∞ ∑ n=0 βn = ∞, is called the mann iteration scheme. (ii) the sequence {xn} defined by x0 ∈ c, xn+1 = t (xn), ∀n ≥ 0, is called the picard iteration scheme which is particular case of the mann iteration scheme. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 45 d. ramesh kumar, m. pitchaimani 3. main results in this section, we prove the existence and uniqueness of common fixed point of a family of self mappings. theorem 3.1. let (x, ‖·, ·‖) be a 2-banach space and s, t : x → x be two self mappings such that ‖sx − t y, u‖ ≤ a ‖y − t y, u‖ [ 1 + ‖x − sx, u‖ ] 1 + ‖x − y, u‖ + b ‖x − sx, u‖ [ 1 + ‖y − t y, u‖ ] 1 + ‖x − y, u‖ +c ‖x − t y, u‖ [ 1 + ‖y − sx, u‖ ] 1 + ‖x − y, u‖ + d ‖y − sx, u‖ [ 1 + ‖x − t y, u‖ ] 1 + ‖x − y, u‖ +e‖x − y, u‖,(3.1) for all x, y, u ∈ x, x 6= y, where a, b, c, d, e are non negative constants such that a + b + c + d + e < 1. then s and t have a unique common fixed point in x. proof. for x0 ∈ x, define a sequence as follows: x2n+1 = sx2n, x2n+2 = t x2n+1, n = 0, 1, 2, . . . now for all u ∈ x, using (3.1), we get ‖x2n+1 − x2n, u‖ = ‖sx2n − t x2n−1, u‖ ≤ a ‖x2n−1 − x2n, u‖[1 + ‖x2n − x2n+1, u‖] 1 + ‖x2n − x2n−1, u‖ +b ‖x2n − x2n+1, u‖[1 + ‖x2n − x2n−1, u‖] 1 + ‖x2n − x2n−1, u‖ +c ‖x2n − x2n, u‖[1 + ‖x2n−1 − x2n+1, u‖] 1 + ‖x2n − x2n−1, u‖ +d ‖x2n−1 − x2n+1, u‖[1 + ‖x2n − x2n, u‖] 1 + ‖x2n − x2n−1, u‖ +e‖x2n − x2n−1, u‖, which gives ‖x2n+1 − x2n, u‖ ≤ kn‖x2n − x2n−1, u‖, where kn = (a + d + e) + e‖x2n − x2n−1, u‖ (1 − b − d) + (1 − a − b)‖x2n − x2n−1, u‖ . note that kn < 1, as a+b+c+d+e < 1. repeating the same argument several times, we obtain ‖x2n+1 − x2n, u‖ ≤ (kn) 2n‖x1 − x0, u‖. now, for n > m, we have ‖xn − xm, u‖ ≤ ‖xn − xn−1, u‖ + ‖xn−1 − xn−2, u‖ + · · · · · · + ‖xm+1 − xm, u‖ ≤ ((kn) n−1 + (kn) n−2 + · · · · · · + (kn) m)‖x1 − x0, u‖ ≤ (kn) m 1 − kn ‖x1 − x0, u‖. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 46 approximation of common fixed points in 2-banach spaces thus ‖xn −xm, u‖ → 0 as m, n → ∞, since (kn) m 1 − kn → 0 as m → ∞. this shows that {xn} is a cauchy sequence in x. hence, there exists a point w ∈ x such that xn → w as n → ∞. further, we get ‖w − t w, u‖ ≤ ‖w − x2n+1, u‖ + ‖x2n+1 − t w, u‖ = ‖w − x2n+1, u‖ + ‖sx2n − t w, u‖ ≤ ‖w − x2n+1, u‖ + a ‖w − t w, u‖[1 + ‖x2n − x2n+1, u‖] 1 + ‖x2n − w, u‖ +b ‖x2n − x2n+1, u‖[1 + ‖w − t w, u‖] 1 + ‖x2n − w, u‖ +c ‖x2n − t w, u‖[1 + ‖w − x2n+1‖] 1 + ‖x2n − w, u‖ +d ‖w − x2n+1, u‖[1 + ‖x2n − t w, u‖] 1 + ‖x2n − w, u‖ +e‖x2n − w, u‖. letting n → ∞, we have ‖w − t w, u‖ ≤ (a + c)‖w − t w, u‖, which implies that t w = w, as (a + c) < 1 for all u ∈ x. similarly, we get w = sw. thus w is a common fixed point of s and t. for the uniqueness, assume that v ∈ x is another common fixed point of s and t , that is, v = sv = t v. then ‖w − v, u‖ = ‖sw − t v, u‖ ≤ a ‖v − t v, u‖[1 + ‖w − sw, u‖] 1 + ‖w − v, u‖ +b ‖w − sw, u‖[1 + ‖v − t v, u‖] 1 + ‖w − v, u‖ +c ‖w − t v, u‖[1 + ‖v − sw, u‖] 1 + ‖w − v, u‖ +d ‖v − sw, u‖[1 + ‖w − t v, u‖] 1 + ‖w − v, u‖ +e‖w − v, u‖, which yields ‖w − v, u‖ + ‖w − v, u‖2 ≤ (c + d + e)‖w − v, u‖ + (c + d + e)‖w − v, u‖2, which is a contradiction, since c + d + e < 1. hence w is the unique common fixed point of s and t . � taking s = t in theorem 3.1, we obtain the following result. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 47 d. ramesh kumar, m. pitchaimani corollary 3.2. let (x, ‖·, ·‖) be a 2-banach space and t : x → x be a self mapping such that ‖t x − t y, u‖ ≤ a ‖y − t y, u‖ [ 1 + ‖x − t x, u‖ ] 1 + ‖x − y, u‖ + b ‖x − t x, u‖ [ 1 + ‖y − t y, u‖ ] 1 + ‖x − y, u‖ +c ‖x − t y, u‖ [ 1 + ‖y − t x, u‖ ] 1 + ‖x − y, u‖ + d ‖y − t x, u‖ [ 1 + ‖x − t y, u‖ ] 1 + ‖x − y, u‖ +e‖x − y, u‖,(3.2) for all x, y, u ∈ x, x 6= y, where a, b, c, d, e are non negative constants and a + b + c + d + e < 1. then t have a unique fixed point in x. proof. the proof follows from theorem 3.1. � remark 3.3. in the case b = c = d = 0, corollary 3.2 reduces to theorem 1.1. now we extend the theorem 3.1 to the case of pair of mappings sp and t q where p and q are some positive integers. theorem 3.4. let (x, ‖·, ·‖) be a 2-banach space and s, t : x → x be two self mappings such that ‖spx − t qy, u‖ ≤ a ‖y − t qy, u‖ [ 1 + ‖x − spx, u‖ ] 1 + ‖x − y, u‖ + b ‖x − spx, u‖ [ 1 + ‖y − t qy, u‖ ] 1 + ‖x − y, u‖ +c ‖x − t qy, u‖ [ 1 + ‖y − spx, u‖ ] 1 + ‖x − y, u‖ + d ‖y − spx, u‖ [ 1 + ‖x − t qy, u‖ ] 1 + ‖x − y, u‖ +e‖x − y, u‖,(3.3) for all x, y, u ∈ x, x 6= y, where p and q are some positive integers and a, b, c, d, e are non negative constants with a + b + c + d + e < 1. then s and t have a unique common fixed point in x. proof. note that sp and t q satisfy the conditions of theorem 3.1, so sp and t q have a unique common fixed point. let w be the common fixed point. then spw = w ⇒ s(spw) = sw, sp(sw) = sw. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 48 approximation of common fixed points in 2-banach spaces if sw = x0 then s p(x0) = x0. so, sw is a fixed point of s p. similarly, t q(t w) = t w. now, we have ‖w − t w, u‖ = ‖spw − t q(t w), u‖ ≤ a ‖t w − t q(t w), u‖[1 + ‖w − spw, u‖] 1 + ‖w − t w, u‖ +b ‖w − spw, u‖[1 + ‖t w − t q(t w), u‖] 1 + ‖w − t w, u‖ +c ‖w − t q(t w), u‖[1 + ‖t w − spw, u‖] 1 + ‖w − t w, u‖ +d ‖t w − spw, u‖[1 + ‖w − t q(t w), u‖] 1 + ‖w − t w, u‖ +e‖w − t w, u‖, which implies that ‖w −t w, u‖+‖w −t w, u‖2 ≤ (c+d+e)‖w −t w, u‖+(c+d+e)‖w −tw, u‖2, which is a contradiction, as we are having c + d + e < 1. thus, w = t w for all u ∈ x. with the similar arguments, we obtain w = sw. finally, in order to prove the uniqueness of v, let w 6= v be another common fixed point of s and t . then clearly w is also a common fixed point of sp and t q which implies w = v. therefore, s and t have a unique common fixed point. � hence we have proved that if x0 is a unique common fixed point of s p and t q, where p, q are some positive integers then x0 is a unique common fixed point of s and t . next we extend theorem 3.1 to a case of family of mappings satisfying the condition (3.1). theorem 3.5. let (x, ‖·, ·‖) be a 2-banach space and {fα} be a family of self mappings on x such that ‖fαx − fβy, u‖ ≤ a ‖y − fβy, u‖ [ 1 + ‖x − fαx, u‖ ] 1 + ‖x − y, u‖ + b ‖x − fαx, u‖ [ 1 + ‖y − fβy, u‖ ] 1 + ‖x − y, u‖ +c ‖x − fβy, u‖ [ 1 + ‖y − fαx, u‖ ] 1 + ‖x − y, u‖ + d ‖y − fαx, u‖ [ 1 + ‖x − fβy, u‖ ] 1 + ‖x − y, u‖ +e‖x − y, u‖, for all α, β ∈ λ with α 6= β and x, y, u ∈ x with x 6= y, where a, b, c, d, e are non negative constants such that a + b + c + d + e < 1. then there exists a unique w ∈ x satisfying fαw = w for all α ∈ λ. proof. let us take fα and fβ in place of s and t respectively in theorem 3.1, an application of which gives a unique w ∈ x to satisfy fαw = fβw = w. for any other member fγ, uniqueness of w gives fγw = w and this completes the proof. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 49 d. ramesh kumar, m. pitchaimani remark 3.6. theorem 3.5 generalizes and improves the results of [4, 8, 18, 19, 32] in 2-banach space setting. example 3.7. let x = r2 and a 2-norm ‖·, ·‖ be defined by ‖x, y‖ = |x1y2 − x2y1|. note that (x, ‖·, ·‖) is a 2-banach space. let s, t : x → x be two self mappings defined as follows: s(x, y) = ( x + y 2 , x + y 2 ) and t (u, v) = ( u 2 , u 2 ) . for a, b, c, d, e ∈ [1 2 , 1), it can be easily seen that (3.1) is satisfied. hence, by theorem 3.1, s and t have a unique common fixed point in x. here, ( 1 2 , 1 2 ) is the unique common fixed point of s and t . 4. approximation by mann and picard type iteration in this section, we approximate the common fixed point of s and t by a mann and picard type iteration schemes. theorem 4.1. let c be a nonempty closed convex subset of a 2-banach space x and s, t : c → c be two self mappings such that (1−k)s(c)+kt (c) ⊂ s(c) for 0 < k ≤ 1 and s(c) is closed. suppose that s and t satisfy all the conditions of theorem 3.1, then s and t have a unique common fixed point. moreover, if, for arbitrary y0 ∈ c, the sequence {yn} defined by (4.1) s(yn+1) = (1 − βn)s(yn) + βnt (yn), ∀n ≥ 0, where {βn} satisfies 0 < βn ≤ 1, ∀n and ∞ ∑ n=0 βn = ∞, is asymptotically t regular, then it converges to the unique common fixed point of s and t , with a rate estimated by ‖(s(yn+1) − w, u‖ ≤ λ n+1l, where λ ∈ [0, 1) and l ≥ 0 are some constants. proof. from theorem 3.1, s and t have a unique common fixed point w ∈ x. let y0 ∈ c and the sequence {yn} be defined by (4.1). then, for all u ∈ x and n ≥ 0, we have the following ‖s(yn+1) − w, u‖ = ‖(1 − βn)s(yn) + βnt (yn) − w, u‖ ≤ (1 − βn)‖s(yn) − w, u‖ + βn‖t (yn) − w, u‖(4.2) c© agt, upv, 2019 appl. gen. topol. 20, no. 1 50 approximation of common fixed points in 2-banach spaces from (3.1), we get ‖t (yn) − w, u‖ = ‖t (yn) − sw, u‖ ≤ a ‖w − sw, u‖[1 + ‖yn − t yn, u‖] 1 + ‖yn − w, u‖ +b ‖yn − t yn, u‖[1 + ‖w − sw, u‖] 1 + ‖yn − w, u‖ +c ‖yn − sw, u‖[1 + ‖w − t yn, u‖] 1 + ‖yn − w, u‖ +d ‖w − t yn, u‖[1 + ‖yn − sw, u‖] 1 + ‖yn − w, u‖ +e‖yn − w, u‖. since {yn} is asymptotically t -regular and sw = t w = w, letting n → ∞, we have ‖t yn−w, u‖+‖t yn−w, u‖ 2 ≤ (c+d+e)‖t yn−w, u‖+(c+d+e)‖t yn−w, u‖ 2, which is not possible as c + d + e < 1. hence lim n→∞ ‖t (yn) − w, u‖ = 0. now it follows from (4.2) that ‖s(yn+1) − w, u‖ ≤ (1 − βn)‖s(yn) − w, u‖. continuing in this way, we have (4.3) ‖(s(yn+1) − w, u‖ ≤ λ n+1l, where λ = (1 − βn) ∈ [0, 1) and l = ‖sy0 − w, u‖ ≥ 0 are some constants. since λ ∈ [0, 1), from (4.3) we get lim n→∞ ‖(s(yn+1) − w, u‖ → 0. this completes the proof. � corollary 4.2. let c be a nonempty closed convex subset of a 2-banach space x and t : c → c be a mapping satisfying all the conditions of corollary 3.2. then t has a unique fixed point w ∈ x. in addition, if for arbitrary x0 ∈ c, the sequence {xn} defined by (4.4) xn+1 = (1 − βn)xn + βnt (xn, . . . , xn), ∀n ≥ 0, where {βn} satisfies 0 < βn ≤ 1, ∀n and ∞ ∑ n=0 βn = ∞, is asymptotically t regular, then it converges to the unique fixed point of t , with a rate estimated by ‖xn − w, u‖ ≤ λ nl, where λ ∈ [0, 1) and l ≥ 0 are some constants. remark 4.3. from corollary 4.2, we obtain an approximation of fixed point of a self mapping by the mann iteration scheme in 2-banach spaces. note that the result holds even if {xn} is asymptotically s-regular. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 51 d. ramesh kumar, m. pitchaimani corollary 4.4. let (x, ‖·, ·‖) be a 2-banach space and s, t : x → x be two mappings such that t (x) ⊂ s(x) and s(x) is a closed subset of x. suppose all the conditions of theorem 3.1 are satisfied, then s and t have a unique common fixed point w ∈ x. further, if for arbitrary x0 ∈ c, the sequence {yn} defined by yn = s(xn) = t (xn−1), ∀n ∈ n is asymptotically t -regular, then it converges to the unique common fixed point of s and t , with a rate estimated by ‖yn − w, u‖ ≤ λ n l, where λ ∈ [0, 1) and l ≥ 0 are some constants. corollary 4.5. let (x, ‖·, ·‖) be a 2-banach space and t : x → x be a mapping such that all the conditions of corollary 3.2 are satisfied. then t has a unique fixed point x. in addition, if for arbitrary x0 ∈ c, the sequence {zn} defined by zn = t (xn−1), ∀n ∈ n, is asymptotically t -regular, then it converges to the unique fixed point of t , with a rate estimated by ‖zn − w, u‖ ≤ λ nl, where λ ∈ [0, 1) and l ≥ 0 are some constants. remark 4.6. from corollary 4.5, we obtain an approximation of fixed point of self mapping by the picard iteration scheme in 2-banach space. 5. applications the notion of well-posedness of a fixed point problem was introduced in [9] and has generated much interest to several mathematicians, for example [16, 26, 31]. in this section, we study well-posedness of the common fixed point obtainined in theorem 3.1. definition 5.1. let (x, ‖·, ·‖) be a 2-banach space and t be a self mapping on x. then the fixed point problem of t is said to be well-posed if (i) t has a unique fixed point x0 ∈ x (ii) for any sequence {xn} ⊂ x and lim n→∞ ‖xn − t xn, u‖ = 0, we have lim n→∞ ‖xn − x0, u‖ = 0. let cfp(s, t, x) denote a common fixed point problem of self mappings t and f on x and cf(s, t ) denote the set of all common fixed points of t and f. definition 5.2. cfp(s, t, x) is called wellposed if cf(s, t ) is singleton and for any sequence {xn} in x with x̂ ∈ cf(s, t ) and lim n→∞ ‖xn − sxn, u‖ = lim n→∞ ‖xn − t xn, u‖ = 0 implies x̂ = lim n→∞ xn. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 52 approximation of common fixed points in 2-banach spaces theorem 5.3. let (x, ‖·, ·‖) be a 2-banach space, s and t be two self mappings on x as in theorem 3.1. then the common fixed point problem of s and t is well posed. proof. from theorem 3.1, the mappings s and t have a unique common fixed point, say w ∈ x. let {xn} be a sequence in x and lim n→∞ ‖sxn − xn, u‖ = lim n→∞ ‖t xn − xn, u‖ = 0. without loss of generality, assume that w 6= xn for any non-negative integer n. using (3.1) and sw = t w = w, we get ‖w − xn, u‖ ≤ ‖t v − t xn, u‖ + ‖t xn − xn, u‖ = ‖t xn − xn, u‖ + ‖sw − t xn, u‖ ≤ ‖t xn − xn, u‖ + a ‖xn − t xn, u‖[1 + ‖w − sw, u‖] 1 + ‖w − xn, u‖ +b ‖w − sw, u‖[1 + ‖xn − t xn, u‖] 1 + ‖w − xn, u‖ +c ‖w − t xn, u‖[1 + ‖xn − sw, u‖] 1 + ‖w − xn, u‖ +d ‖xn − sw, u‖[1 + ‖w − t xn, u‖] 1 + ‖w − xn, u‖ +e‖w − xn, u‖. taking limit n → ∞, we get ‖w − xn, u‖ + ‖w − xn, u‖ 2 ≤ (c + d + e)‖w − xn, u‖ + (c + d + e)‖w − xn, u‖ 2, which gives the contradiction as c + d + e < 1. this completes the proof. � corollary 5.4. let (x, ‖·, ·‖) be a 2-banach space and t be a self mapping on x as in corollary 3.2. then the fixed point problem of t is well posed. remark 5.5. notice that well-posedness of the common fixed points obtained in theorems 3.4 and 3.5 can easily be viewed. acknowledgements. the authors thank the reviewers for valuable comments. the first author d. ramesh kumar would like to thank the university grants commission, new delhi, india for providing the financial support in preparation of this manuscript. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 53 d. ramesh kumar, m. pitchaimani references [1] m. abbas, b. e. rhoades and t. nazir, common fixed points for four maps in cone metric spaces, 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[35] a. white, 2-banach spaces, math. nachr. 42 (1969), 43–60. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 55 @ appl. gen. topol. 19, no. 2 (2018), 281-289 doi:10.4995/agt.2018.9943 c© agt, upv, 2018 a note about various types of sensitivity in general semiflows alica miller department of mathematics, university of louisville, usa (alica.miller@louisville.edu) communicated by f. balibrea abstract we discuss the implications between various types of sensitivity in general semiflows (sensitivity, syndetic sensitivity, thick sensitivity, thick syndetic sensitivity, multisensitivity, periodic sensitivity, thick periodic sensitivity), including the weak mixing as a very strong type of sensitivity and the strong mixing as the strongest of all type of sensitivity. 2010 msc: primary 54h15; 20m20; secondary 54h20. keywords: sensitivity; strong mixing; weak mixing; strong sensitivity; multisensitivity; syndetic sensitivity; thick sensitivity; thick syndetic sensitivity; periodic sensitivity; thick periodic sensitivity. 1. introduction in this note we discuss the implications between various types of sensitivity in general semiflows (sensitivity, syndetic sensitivity, thick sensitivity, thick syndetic sensitivity, multisensitivity, periodic sensitivity, thick periodic sensitivity), including the weak mixing as a very strong type of sensitivity and the strong mixing as the strongest of all type of sensitivity. under general semiflow we assume a semiflow (t, x) where t is a commutative noncompact hausdorff acting topological monoid (with additive operation) and x is a metric space with at least two points. (we do not assume neither the compactness of the phase space x, nor that the transition maps x 7→ tx have dense images.) so from now on we assume that every monoid t (including groups) and every phase space x are as described above. received 10 april 2018 – accepted 01 september 2018 http://dx.doi.org/10.4995/agt.2018.9943 a.miller all notions that we mention, but not define, are standard and can be, for example found in [1, 4, 6]. stronger forms of sensitivity were introduced in [8], and discussed, for example, in [3, 5, 8, 9]. a nonempty open subset is called a nopen, or a nopen subset. if (x, d) is a metric space, x ∈ x and r > 0, the open ball with center x and radius r is denoted by b(x, r). it consists of all points y ∈ x such that d(x, y) < r. the closed ball with center x and radius r is denoted by b−(x, r). it consists of all points y ∈ x such that d(x, y) ≤ r. it is a closed subset of x. we say that a subset a of t is syndetic if there is a compact k ⊆ t such that for every t ∈ t , (t + k) ∩a 6= ∅. we say that a subset b of t is thick if for every compact k ⊆ t there is a t ∈ t such that t + k ⊆ b. we say that a subset c of t is thickly syndetic if for every compact k ⊆ t there is a syndetic subset s ⊆ t such that s + k ⊆ c. we say that a subset d of t is periodic if it contains a translate t + s of a closed syndetic submonoid of t . we say that a subset e of t is thickly periodic if for every compact k ⊆ t there is a periodic subset p ⊆ t such that p + k ⊆ e. definition 1.1. we say that a monoid t satisfies the syndetic property, shortly sp property, or that t is an sp monoid, if no syndetic subset of t is compact. we say that a monoid t satisfies the dual syndetic property, shortly dsp property, or that t is a dsp monoid, if for every compact subset k of t , the set t \ k is a syndetic subset of t . the condition (sp) can be equivalently formulated in the following way: (sp’) for any two compact subsets k and k′ of t there is an element t ∈ t such that (t + k) ∩k′ = ∅. another equivalent way is the following one: (sp”) for every compact subset k of t , the set t \k is a thick subset of t . let us show that the conditions (sp) and (sp’) are equivalent. suppose (sp) holds. let k, k′ be two compact subsets of t . by (sp) none of them is sysndetic, hence there is an element t ∈ t such that (t + k) ∩ k′ = ∅. conversely, suppose that (sp’) holds. let s be a syndetic subset of t and let k be a corresponding compact for s. suppose s is compact. then there is a t ∈ t such that (t + k) ∩ s = ∅ (it exists by (sp’)). this contradicts to the syndeticity of s. it is easy to see that the conditions (sp’) and (sp”) are equivalent. the condition (dsp) can be equivalently formulated in the following way: (dsp’) for any compact subset k of t there exists a compact subset k′ of t such that no translate t + k′, t ∈ t , is contained in k. another equivalent formulation is the following one. (dsp”) for any compact subset k of t there exists a compact subset k′ of t such that no translate k + k′, k ∈ k, is contained in k. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 282 various types of sensitivity in general semiflows note that the condition (dsp’) is just a reformulation of the condition (dsp). let us show that (dsp’) is equivalent with (dsp”). clearly, (dsp’) implies (dsp”). we will show the contrapositive of the converse, i.e., that the negation (∼dsp’) of (dsp) implies the negation (∼dsp”) of (dsp”). assume that there is a compact k ⊆ t such that for every compact k′ ⊆ t there is a t ∈ t with t + k′ ⊆ k. for this k and any compact k′ ⊆ t there is a t∗ ∈ t such that (t∗ + (k ∪ {0})) ⊆ k. hence t∗ ∈ k. thus t∗ + k′ ⊆ k with t∗ ∈ k, i.e., (∼dsp”) holds. the statement is proved. the condition (sp) was introduced in our paper [7], where we discussed chaos-related properties on the product of semiflows. the property (dsp) is for the first time considered in this paper. example 1.2. (1) every topological group is sp. suppose to the contrary, i.e., that there are two compact subsets k, k′ of the topological group t such that for every t ∈ t we have (t + k)∩ k′ 6= ∅. then every t ∈ t is of the form t = k′−k for some k ∈ k and k′ ∈ k′. since k′ − k is compact and t is noncompact, if we select t /∈ k′ −k, we get a contradiction. the groups r and z are dsp. (2) every directional monoid is both sp and dsp. (a topological monoid t is said to be directional if for every compact subset k of t there is a t ∈ t such that (t + t ) ∩k = ∅. this notion was introduced in our paper [7].) in particular, nn0 and r n + are both sp and dsp. (here n0 denotes the additive monoid of nonnegative integers, while r+ denotes the additive monoid of nonnegative real numbers, both sets with the topology induced from r.) (3) the monoid t = [0, 1) with the topology induced from r and the operation x + y = max{x, y} is both sp and dsp. (4) the monoid t = {0}∪ (1/2, 1] with the topology induced from r and the operation x + y = max{x, y} is neither sp nor dsp. indeed, to show that t is not sp we can consider the compacts k = [2/3, 1] and k′ = [5/8, 5/6]. to show that t is not dsp we consider the compact subset k = {1}. proposition 1.3. the condition (sp) for topological monoids is stronger than the condition (dsp). proof. suppose to the contrary, i.e., that there is a topological monoid t in which (sp) and (∼dsp”) hold. let k ⊆ t be a compact such that for every compact k′ ⊆ t there is a k ∈ k with k + k′ ⊆ k. by (sp’), there is a t∗ ∈ t such that (t∗ + k) ∩ k = ∅. however, for k′ = {t∗} there is a k ∈ k such that k + t∗ ∈ k. this contradicts to (t∗ + k) ∩k = ∅. � definition 1.4. a semiflow (t, x) is: (a) strongly mixing (strm) if for any two nopens u, v in x the set d(u, v ) = {t ∈ t | tu ∩v 6= ∅} contains t \k for some compact k ⊆ t ; c© agt, upv, 2018 appl. gen. topol. 19, no. 2 283 a.miller (b) weakly mixing (wm) if for any nopens u1, v1, u2, v2 in x, d(u1, v1)∩ d(u2, v2) 6= ∅; (c) sensitive (s) if there is a sensitivity constant c > 0 such that for any nopen u ⊆ x, d(u, c) = {t ∈ t | (∃x, y ∈ u) d(tx, ty) > c} 6= ∅; (d) strongly sensitive (strs) if there is a sensitivity constant c > 0 such that for every nopen u in x the set d(u, c) contains t \ k for some compact k ⊆ t ; (e) multisensitive (muls) if there is a sensitivity constant c > 0 such that for any integer n ≥ 1 and any nopens u1, u2, . . . , un in x, d(u1, c) ∩ ·· ·∩d(un, c) 6= ∅; (f) strongly multisensitive (strmuls) if there is a sensitivity constant c > 0 such that for any integer n ≥ 1 and any nopens u1, u2, . . . , un in x, d(u1, c) ∩·· ·∩d(un, c) contains t \k for some compact k ⊆ t ; (g) thickly sensitive (ts) if there is a sensitivity constant c > 0 such that for every nopen u in x the set d(u, c) is a thick subset of t ; (h) syndetically sensitive (synds) if there is a sensitivity constant c > 0 such that for every nopen u in x the set d(u, c) is a syndetic subset of t ; (i) thickly syndetically sensitive (tsynds) if there is a sensitivity constant c > 0 such that for every nopen u in x the set d(u, c) is a thickly syndetic subset of t ; (j) periodically sensitive (pers) if there is a sensitivity constant c > 0 such that for every nopen u in x the set d(u, c) is a periodic subset of t ; (k) thickly periodically sensitive (tpers) if there is a sensitivity constant c > 0 such that for every nopen u in x the set d(u, c) is a thickly periodic subset of t . the next lemma is well-known, see, for example, [1]. lemma 1.5. let (t, x) be a weakly mixing semiflow and u1, . . . , un, v1, . . . , vn nonempty open subsets of x (n ≥ 1). then there is a t ∈ t such that tui ∩vi 6= ∅ for all i = 1, . . . , n. 2. relations between various types of sensitivity in general semiflows in this section we will justify the implication diagram below. in the diagram a crossed implication arrow between two conditions means that that implication does not hold, i.e., that there is a counterexample for that implication. if the condition sp or dsp is given by the arrow, that means that the implication holds when that condition is assumed. proposition 2.1. every strongly mixing semiflow is strongly sensitive. proof. let a, b be two points of x with d(a, b) = ∆ > 0 and let ba = b(a, ∆/4), bb = b(b, ∆/4). let c = ∆/4. then for any a ′ ∈ ba and any b′ ∈ bb, d(a′, b′) > c. let u be a nopen in x. since (t, x) is strongly mixing, there is a compact k ⊆ t such that for every t ∈ t \ k there are x, y ∈ u with c© agt, upv, 2018 appl. gen. topol. 19, no. 2 284 various types of sensitivity in general semiflows tx ∈ ba, ty ∈ bb. hence d(tx, ty) > c. thus d(u, c) ⊇ t \ k, so that (t, x) is strongly sensitive. � s muls ts synds tsynds pers tpers wm strmuls ≡ strs strm x x x x ? sp spx ? x x x ? dsp proposition 2.2. if t is a dsp monoid, then every strongly mixing semiflow is syndetically sensitive. proof. let t be a dsp monoid and (t, x) a strongly mixing semiflow. let p, q be two distinct points of x, and let d = d(p, q) and c = d/3. we claim that the constant c can serve as a sensitivity constant such that for every nopen u of x the set d(u, c) is a syndetic subset of t . indeed, let o1 = b(p, c) and o2 = b(q, c). then for any two points x1 ∈ o1 and x2 ∈ o2 we have d(x1, x2) > c. fix a nopen u ⊆ x. since (t, x) is strongly mixing, there is a compact k1 ⊆ t such that for every t ∈ t \ k1 there is a point x ∈ u with tx ∈ o1. also there is a compact k2 ⊆ t such that for every t ∈ t \k2 there is a point y ∈ u with ty ∈ o2. hence for every t ∈ t \(k1 ∪k2) there is a pair of points (x, y) from u such that tx ∈ o1 and ty ∈ o2, so that d(tx, ty) > c. since t is dsp, t \ (k1 ∪k2) is syndetic and so the proposition is proved. � proposition 2.3. a semiflow is strongly sensitive if and only if it is strongly multisensitive. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 285 a.miller proof. suppose a semiflow (t, x) is strongly sensitive. let n ≥ 1 and let the ui, i = 1, . . . , n be nopens in x. for each i ∈{1, . . . , n} there is a compact ki such that for every t ∈ t \ ki there are xi, yi ∈ ui with d(txi, tyi) > ci. let k = k1 ∪·· ·∪kn and let c = min{c1, . . . , cn}. then for every t ∈ t \k and every i ∈ {1, . . . , n} there are xi, yi ∈ ui with d(txi, tyi) > c. hence (t, x) is strongly munltisensitive. the other direction is clear. � proposition 2.4. there is a strongly multisensitive semiflow which is not weakly mixing. proof. let x = [0,∞) with the metric d(x, y) = |ex − ey|. let t = [0,∞) act on x by t.x = t + x. then (t, x) is strongly multisensitive. indeed, for any n pairs (x1, y1), (x2, y2), . . . , (xn, yn) of elements of x (for any n ∈ n) the distances d(xi + t, yi + t), i = 1, 2, . . . , n, will be bigger than any c > 0 for all t ∈ r+ from some point on as the function f(x) = ex tends to infinity as x tends to infinity. however (t, x) it is not weakly mixing since for u = (a, b) and v = (c, d) with d < a there is no t ∈ t with tu ∩v 6= ∅. � proposition 2.5. every strongly sensitive semiflow with an sp acting monoid is thickly sensitive. proof. let (t, x) be strongly sensitive and let c > 0 be its sensitivity constant. let u ⊆ x be nopen. then there is a compact k ⊆ t such that for every t ∈ t \k, t ∈ d(u, c). let k′ be a compact in t . we need to show that there is a t ∈ t such that t + k′ ⊆ d(u, c). it is enough to show that there is a t ∈ t such that t + k′ ⊆ t \k. otherwise, for every t ∈ t , (t + k′)∩k 6= ∅, contradicting the assumption that t is sp. � proposition 2.6. there is a weakly mixing semiflow which is not thickly sensitive. proof. let t be a one-dimensional torus r/z, i.e., t = [0, 1) with the metric d(x, y) = min{|x − y|, 1 −|x − y|}. define a continuous function f : t → t by f(x) = 2x (mod 1) for every x ∈ t. a point x ∈ t in the cascade (t, f) is said to be eventually fixed if there is an n ≥ 0 such that fn(x) = 0. the set of all eventually fixed points is x = {k/2n | k, n ≥ 0 integers, k < 2n}, which is a dense subset of t. note that f(x) ⊆ x, so that we can consider the restricted semiflow (x, f). each point in this semiflow has a finite orbit whose last term is 0. the point 0 is the only fixed point. as shown in [6], (x, f) is weakly mixing. let now t1 = {0, 1} be a discrete monoid with the operation 0+0 = 0, 1+0 = 0 + 1 = 1 + 1 = 1. let t = n0 × t1 = {(n, t) | n ∈ n0, t ∈ t1} be the product monoid of the discrete monoids n0 and t1 (with componentwise addition). define a monoid action of t on x by (n, t) . x = { fn(x) if t = 0, 0 if t = 1. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 286 various types of sensitivity in general semiflows it is easy to verify that this is indeed a monoid action. all transition maps are clearly continuous, so we have a topological semi-flow (t, x). this semiflow is weakly mixing since (x, f) is weakly mixing. it was shown in [6] that (t, x) is not thickly sensitive. � example 2.7. an example of a weakly mixing semiflow which is not strongly sensitive is the semiflow from the proof of proposition 2.6. indeed, by propositions 2.6 and 2.5, it is enough to see that the monoid t in the proof of proposition 2.6 is sp, i.e., that for any two compact subsets k and k′ of t there is an element (n, t) ∈ t such that ((n, t) + k) ∩ k′ = ∅. since t is discrete, compacts are finite, so that for any sufficiently big n ∈ n0 the element (n, 0) will work. proposition 2.8. every weakly mixing semiflow is multi-sensitive. proof. we will follow [6]. if the diameter of x is infinite let d be any positive real number, otherwise let diam(x) = 12d > 0. then for any ball b(x, 4d), x ∈ x, we have x \b(x, 4d) 6= ∅. we will show that (t, x) is multi-sensitive with sensitivity constant c = d. let m ≥ 1 be an integer and let u1, u2, . . . , um be nonempty open subsets of x. let xi ∈ ui (i = 1, 2, . . . , m). for each i = 1, 2, . . . , m let bi = b(xi, ri), where ri < d is such that bi ⊆ ui. let also c−i = b −(xi, 2d). then each vi = x \ c−i is a nonempty open subset of x. note that for any a ∈ bi and b ∈ vi, d(a, b) > d. by lemma 1.5 there is a t ∈ t such that at the same time tbi ∩ bi 6= ∅ and tbi ∩ vi 6= ∅ for i = 1, 2, . . . , m. let yi, zi ∈ bi ⊆ ui (i = 1, 2, . . . , m) be such that tyi ∈ bi and tzi ∈ vi for i = 1, 2, . . . , m. then d(tyi, tzi) > d (i = 1, 2, . . . , m). � corollary 2.9. there is a multisensitive semiflow which is not thickly sensitive. proof. otherwise using proposition 2.8 we would be able to conclude that every weakly mixing semiflow is thickly sensitive, which would contradict to propositon 2.6. � proposition 2.10. every strongly sensitive semiflow whose acting monoid is sp is thickly syndetically sensitive. proof. let (t, x) be strongly sensitive with sensitivity constant c and let u be a nopen subset of x. since (t, x) is strongly sensitive, there is a compact k ⊆ t such that d(u, c) ⊇ t\k. hence, since t is sp and since, by proposition 1.3, sp implies dsp, d(u, c) is thick and syndetic, or, equivalently, thickly syndetic. � proposition 2.11. there is a syndetically sensitive semiflow which is not thickly sensitive, nor thickly syndetically sensitive. proof. an example is given in [5, example 10]. � proposition 2.12. there is thickly syndetically sensitive semiflow which is not strongly sensitive. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 287 a.miller proof. an example is given in [5, example 11]. it is a uniformly rigid weakly mixing minimal semiflow. as stated in [5], the existence of such semiflows follows from the paper [2], with a general acting monoid t in place of n0. � proposition 2.13. every strongly mixing semiflow is weakly mixing. every strongly multisensitive semiflow is multi sensitive. every thickly syndetically sensitive semiflow is thickly sensitive. every thickly syndetically sensitive semiflow is syndetically sensitive. every thickly periodically sensitive semiflow is periodically sensitive. every thickly periodically sensitive semiflow is thickly syndetically sensitive. every periodically sensitive semiflow is syndetically sensitive. every multisensitive (resp. thickly sensitive; syndetically sensitive) semiflow is sensitive. proof. the first statement is well-known and easy to see. the remaining ones follow from the definitions. � 3. concluding remarks we analyzed a variety of “sensitivity-properties”, starting with the strong mixing as the strongest one and ending with the sensitivity as the weakest one. we organized them into an implication diagram and proved that some of those implications are true, some are not true, and some are left as open questions. in the process we introduced the properties (sp) and (dsp) of topological monoids. here are the remaining questions. question 1. is (sp) a strictly stronger property than (dsp), i.e., is there a topological monoid which is dsp but not sp? question 2. find examples showing that in the implications strs ⇒ ts and strs ⇒ tsynds the condition (sp) is indeed needed, and that in the implication sm ⇒ synds the condition (dsp) is indeed needed. question 3. investigate if, in general, (strs) implies (tpers), (ts) implies (muls), and (synds) implies (ts). acknowledgements. i would like to thank the referee for the careful reading and good questions, which helped to significantly improve the presentation of this article. the author was partially supported by the national science foundation grant dms-1405815. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 288 various types of sensitivity in general semiflows references [1] e. glasner, ergodic theory via joinings, mathematical surveys and monographs, american mathematical society, 2003. [2] e. glasner and d. maon, rigidity in topological dynamics, ergod. th. & dynam. sys. 9 (1989),309–320. [3] l. he, x. yan and l. wang, weak-mixing implies sensitive dependence, j. math. anal. appl. 299 (2004), 300–304. [4] e. kontorovich, m. megrelishvili, a note on sensitivity of semigroup actions, semigroup forum 76 (2008), 133–141. [5] h. liu, l. liao and l. wang, thickly syndetical sensitivity of topological dynamical system, discrete dyn. nature soc. 2014, article id 583431. [6] a. miller, weak mixing in general semiflows implies multi-sensitivity, but not thick sensitivity, j. nonlinear sci. appl., to appear. [7] a. miller and c. money, chaos-related properties on the product of semiflows, turkish j. math. 41 (2017), 1323–1336. [8] t. s. moothathu, stronger forms of sensitivity for dynamical systems, nonlinaerity 20 (2007), 2115–2126. [9] t. wang, j. yin and q. yan, the sufficient conditions for dynamical systems of semigroup actions to have some stronger forms of sensitivities, j. nonlinear sci. appl. 9 (2016), 989–997. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 289 () @ appl. gen. topol. 19, no. 2 (2018), 245-252doi:10.4995/agt.2018.8981 c© agt, upv, 2018 the hull orthogonal of the unit interval [0,1] sami lazaar a and saber nacib b a faculty of sciences of tunis, university of tunis el manar, tunisia. (salazaar72@yahoo.fr) b faculty of sciences of gafsa, university of gafsa, tunisia. (sabernacib83@hotmail.com) communicated by s. romaguera abstract in this paper, the full subcategory hcomp of top whose objects are hausdorff compact spaces is identified as the orthogonal hull of the unit interval i = [0, 1]. the family of continuous maps rendered invertible by the reflector β ◦ ρ is deduced. 2010 msc: 54d30;18b30; 54d60. keywords: completely regular spaces; categories; stone-čech compactification. 1. introduction in the literature, various approaches to the stone-čech compactification βx of a topological space x are given, using constructions based on products of the interval unit i, ultrafilters, and c⋆-algebras, respectively ([5], [7], [12] and [10]). more than a compactification, the embedding of x into βx defines hcomp as a reflective subcategory in the category tych of tychonoff spaces. thus hcomp is a reflective subcategory of top with reflector β ◦ ρ, where ρ is the tychonoff reflector. the year 1937 was an important one in establishing nice connections between topology and algebra. m. h. stone and e. čech published papers giving several fundamental properties of the compactification βx, which had been introduced by tychonoff. for instance, stone showed that any tychonoff space x is c⋆embedded in βx, and this can be interpreted algebraically as showing that the rings c⋆(x) and c⋆(βx) are isomorphic. received 22 november 2017 – accepted 06 may 2018 http://dx.doi.org/10.4995/agt.2018.8981 s. lazaar and s. nacib recall that if d is a reflective subcategory in a category c, with reflector f, then d⊥ = {f ∈ homc : f(f) is an isomorphism} and d ⊥⊥ = d (for more information see [1], [2] and [4]). in our case, we have hcomp⊥ = {f ∈ homtop : β ◦ ρ(f) is an isomorphism} and hcomp ⊥⊥ = hcomp. so on the one hand, if we consider the category sob of sober spaces, it is not difficult to show that sob⊥ = {δ}⊥, where δ is the sierpiński space, and thus sob = {δ}⊥⊥ which gives a characterization of sober spaces using only the space δ. on the other hand, in [6], a. haouati and s. lazaar showed that the reflective subcategory hewitt of top, whose objects are real-compact spaces, is the orthogonal hull of the real line r. analogous to sob⊥ = {δ}⊥ and hewitt⊥ = {r}⊥, we show in this paper that hcomp⊥ = {i}⊥ where i is the unit interval, and consequently the family of continuous maps rendered invertible by β ◦ ρ are those maps which are orthogonal to i. 2. some preliminary results let c be a category. an arrow f in c from a to b is said to be orthogonal to an object x in c if and only if for any arrow g from a to x, there exists a unique arrow g̃ from b to x satisfying g̃ ◦ f = g. the orthogonal σ⊥ of a class of morphisms σ is the class of objects orthogonal to every morphism in σ [4]. the orthogonal of a class of objects is defined analogously. recall that a topological space is called completely regular (or tychonoff) if it is t1 and every closed subset f of the space is completely separated from any point x not in f . an other important characterization of completely regular spaces is given by the following theorem. theorem 2.1 ([12, proposition 1.7]). a space is completely regular if and only if the family of zero-sets of the space is a base for the closed sets (or equivalently, the family of cozero-sets is a base for the open sets). notations 1. lat x be a topological space. we denote by: • c(x) the family off all continuous maps from x to r. • c⋆(x) the family off all bounded continuous maps from x to r. • c⋆i(x) the family off all continuous maps from x to i. • c[0,+∞[(x) the family off all positive continuous maps from x to r. remark 2.2. let f be a continuous map from a topological space x to r. consider the map fi from x to i defined by fi := inf{|f|, 1}. clearly f(x) = 0 if and only if fi(x) = 0 if and only if |f|(x) = 0. therefore by theorem 2.1 a topological space x is completely regular if and only if the family {h−1(]0, 1]) : h ∈ c⋆i(x)} (resp., {h −1(]0, +∞[) : h ∈ c[0,+∞[(x)} is a base for the open sets of x. the following result is an easy observation from [2]. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 246 the hull orthogonal of the unit interval [0,1] proposition 2.3. let c be a category and d a reflective subcategory of c, with reflector f. an arrow in c is orthogonal to an object in d if and only if its f-identification is also. remark 2.4. in our case when we consider the reflective subcategory tych of top, with reflector ρ, where ρ is the tychonoff reflector, a continuous map f between topological spaces is orthogonal to the unit interval i = [0, 1] equipped with its usual topology if and only if its tychonoff reflection ρ(f) is orthogonal to i. let us now give some properties of a continuous map orthogonal to i (resp., [0, +∞[). proposition 2.5. let f be a continuous map from a functionally hausdorff space to a topological space y . if f is orthogonal to i, then f is one-to-one. proof. let x and y be two points in x such that f(x) = f(y). if x and y are distinct then there exists g ∈ c(x) such that g(x) = 0 and g(y) = 1 and thus gi defined in remark 2.2 satisfies also gi(x) = 0 and gi(y) = 1 . the mapping g̃i in c ⋆ i(y ) obtained by orthogonality of f to i gives a contradiction. � remark 2.6. by the same way as in proposition 2.5, we can see easily that if we consider a continuous map from a functionally hausdorff space to a topological space y which is orthogonal to [0, +∞[, then it is one to one. indeed, it is enough to replace gi in proposition 2.5 by |g|. proposition 2.7. let f be a continuous map from a topological space x to a completely regular space y . if f is orthogonal to i, then, f is a dense mapping. proof. assume that f(x) 6= y and let y be in y and not in f(x). since y is completely regular, there exists a mapping h in c(y ) such that h(y) = 0 and h(f(x)) = {1}. then the mapping hi from c ⋆ i(x) satisfies hi(y) = 0 and hi(f(x)) = inf{|f|, 1}(f(x)) = inf{1, 1}(f(x)) = {1}. now if we denote by 1y the constant map equal to 1 from y to i, we get: ∀x ∈ x, (1y ◦ f)(x) = (hi ◦ f)(x) = 1. so, 1y ◦ f = hi ◦ f. this leads to a contradiction because f is orthogonal to i and the continuous maps 1y and hi are not equal. � remark 2.8. by the same way as in proposition 2.7, we can see easily that if we consider a continuous map from a topological space x to a completely regular space y which is orthogonal to [0, +∞[, then it is a dense mapping. indeed, it is enough to replace hi in proposition 2.7 by |h|. proposition 2.9. let f be a continuous map from a completely regular space x to a topological space y . if f is orthogonal to i, then f(x) and x are homeomorphic. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 247 s. lazaar and s. nacib proof. let f1 be the restriction of f to f(x). using proposition 2.5, f1 is a continuous bijective map, so it is sufficient to show that it is an open map. indeed, let g−1(]0, 1]) be an element of the base of open sets, cited in remark 2.2, where g ∈ c⋆i(x). since f is orthogonal to i, the unique map g̃ ∈ c ⋆ i(y ) such that g̃ ◦ f = g satisfies f1(g −1(]0, 1])) = g̃−1(]0, 1]) ∩ f(x), which is open in f(x). � remark 2.10. by the same way as in proposition 2.9, any continuous map f from a completely regular space x to a topological space y which is orthogonal to [0, +∞[, then f(x) and x are homeomorphic. to conclude the three previous results, we can cite the following result. proposition 2.11. every map f : x −→ y in the category tych which is orthogonal to i (resp., [0, +∞[ ) is a one-to-one dense mapping such that x and f(x) are homeomorphic. proposition 2.12. let x be a tychonoff space and f : x −→ i be a continuous map which is orthogonal to i. then f is an homeomorphism. proof. by proposition 2.11, it is enough to see that f is a surjective map. suppose that f(x) 6= i and let y be in i not in f(x). we have two cases to discuss. first case: 0 < y < 1. let us denote by: x < = {x ∈ x : 0 ≤ f(x) < y} and x> = {x ∈ x : y < f(x) ≤ 1}. so, one can check easily that x< and x> are a disjoint union of x. then f(x) = f(x<) ∪ f(x>) which implies that i = f(x) = f(x>) ∪ f(x>). now since f(x<) (resp., f(x>)) is closed containing [0, y[ (resp., ]y, 1] ), then f(x<) = [0, y] (resp., f(x>) = [y, 1]). let consider the map g from x to i by g(x<) = {y 2 } and g(x>) = {y+1 2 }. it is clear that g is continuous and thus by orthogonality of f to i, let g̃ be the continuous map from i to itself such that g̃ ◦ f = g. by density of f(x<) (resp.,f(x>)) in [0, y] (resp., [y, 1] ), consider a sequence (xn) (resp., (zn) ) in x < (resp., x> ) with (f(xn)) (resp., (f(zn))) in [0, y] (resp., [y, 1] ) converges to y. by preserving continuity under continuous maps, the constant sequences (g(xn) = y 2 ) and (g(zn) = y+1 2 ) must both converge to g̃(y) which is impossible. second case: y ∈ {0, 1}. in this case f(x) ∈ {]0, 1], [0, 1[, ]0, 1[}. without loss in generality we can suppose that f(x) =]0, 1]. now, consider the map g from x to i defined by g(x) =| sin 1 f(x) |. clearly g is a continuous map and since f is orthogonal to i, there exists a unique map g̃ from i to itself such that g̃ ◦ f = g. so that for any y ∈]0, 1], g̃(y) =| sin 1 y | which leads to a contradiction since g̃ is continuous in 0. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 248 the hull orthogonal of the unit interval [0,1] 3. hcomp ={i}⊥⊥ before giving the main result of our paper, let us recall two important results introduced in chapter 6 in [5]. theorem 3.1 ([5, theorem 6.4]). let x be dense in t . the following statements are equivalent. (1) every continuous mapping τ from x into any compact space y has an extension to a continuous mapping from t into y . (2) x is c⋆-embedded in t . (3) any two disjoint zero-sets in x have disjoint closures in t . (4) for any two zero-sets z1 and z2 in x, clt (z1 ∩ z2) = clt z1 ∩ clt z2. (5) every point of t is the limit of a unique z-ultrafilter on x. theorem 3.2 (compactification theorem, [5, theorem 6.5]). every completely regular space x has a compactification βx, with the following equivalent properties. (1) (stone) every continuous mapping τ from x into any compact space y has a continuous extension τ from βx into y . (2) (stone-cech) every function f in c⋆(x) has an extension to a function fβ in c(βx). (3) (cech) any two disjoint zero-sets in x have disjoint closures in βx. (4) for any two zero-sets z1 and z2 in x, clβx(z1 ∩ z2) = clβxz1 ∩ clβxz2. (5) distinct z-ultrafilter on x have distinct limits in βx. remark 3.3. the compactification βx in theorem 3.2 is unique, in the following sense: if a compactification t of x satisfies anyone of the listed previous conditions, then there exists a homeomorphism from βx onto t that leaves x pointwise fixed. now, we are in a position to give our main result. theorem 3.4. hcomp ⊥ = {i}⊥. proof. clearly, hcomp⊥ ⊂ {i}⊥. conversely, let f : x −→ t be a continuous map orthogonal to i, y a hausdorff compact space and g a continuous map from x to y . by remark 2.4, we may assume x and t are completely regular spaces. now, using proposition 2.11, we may assume x as a dense subset of the completely regular space t and replace f by the canonical injection from x to t. now (2) =⇒ (1) of theorem 3.1 applies, and thus g has a continuous extension g̃ from t into y . furthermore, this extension is unique, since any two such continuous extensions must coincides on the dense subset x of the hausdorff space t , and thus must be equal. � c© agt, upv, 2018 appl. gen. topol. 19, no. 2 249 s. lazaar and s. nacib the following corollaries are immediate. corollary 3.5. hcomp = {i}⊥⊥. corollary 3.6. let f be a continuous map. then β(ρ(f)) is a homeomorphism if and only if f is orthogonal to i. in particular, for a continuous map f between two tychonoff spaces, β(f) is an homeomorphism if and only if f is orthogonal to i. proof. since the family of all morphisms rendered invertible by the reflector β ◦ ρ is exactly hcopm⊥, an application of theorem 3.4 gives the result. � let us recall the definition introduced by echi and lazaar in [3]. definition 3.7 ([3, definition 3.2]). let x be a topological space and h a subset of c(x). we say that h has the finite intersection property (fip, for short) if for each finite subset j of h we have ∩[f−1({0}, f ∈ j] 6= ∅. theorem 3.8. let f : x −→ y be a continuous map which is orthogonal to i. then the following statements are equivalent. (1) for each subset h of c(y ) satisfying the fip, ∩[f−1({0}) : f ∈ h] 6= ∅; (2) β(ρ(x)) = ρ(y ). proof. (1) =⇒ (2) by [3, proposition 3.6] ρ(y ) is a completely regular compact space. then ρ(f) is a continuous map from the tychonoff space ρ(x) to the compact tychonoff space ρ(y ). using the previous results, one can see that ρ(f)(ρ(x)) is a dense subset of the compact hausdorff space ρ(y ) = β(ρ(y )) which is c⋆-embedding. hence by the theorem 3.2 (2) and the remark 3.3, β(ρ(f)(ρ(x)) = ρ(y ). finally, since ρ(f)(ρ(x)) and ρ(x) are homeomorphic, (2) is satisfied. (2) =⇒ (1) is an immediate consequence of [3, proposition 3.6]. � corollary 3.9. let f : x −→ y be a continuous map between tychonoff spaces, with f⊥ i. then the following statements are equivalent. (1) y is compact; (2) β(x) = y (up to homeomorphism). examples 3.10. (1) let i :]0, 1] −→ [0, 1] be the canonical injection. clearly i is a dense mapping between two tychonoff spaces and the second space is compact. so, β(]0, 1]) 6= [0, 1] because i is not orthogonal to i. indeed the continuous mapping g :]0, 1] → [0, 1] defined by g(x) = | sin( 1 x )| can not be extended to x = 0. (2) let i :]0, 1] −→]0, 1] the canonical injection. clearly i is a dense mapping between two tychonoff spaces. since i is an isomorphism it is orthogonal to i, but β(]0, 1]) 6=]0, 1] because ]0, 1] is not compact. remark 3.11. regarding [1], the authors in [1, proposition 4.11], proved that, for any continuous map f : x −→ y between two tychonoff spaces, β(f) is c© agt, upv, 2018 appl. gen. topol. 19, no. 2 250 the hull orthogonal of the unit interval [0,1] a homeomorphism if and only if β(f(x)) = β(y ), so in our case this result becomes trivial because β(f) : β(x) −→ β(y ) is a homeomorphism if and only if f is orthogonal to i and in this situation, by proposition 2.9, x is homoeomorphic to f(x) and consequently β(f(x)) is homeomorphic to β(x). finally, β(f(x)) = β(y ). to finish this paper, we shield some light on the hull orthogonal of a given topological space. by [6], hewitt = {r}⊥⊥ and it is clear that any homeomorphic topological space to r satisfies also this property. the following example shows that the topological space [0, +∞[, which is not homeomorphic to r, satisfies also hewitt = {[0, +∞[}⊥⊥. proposition 3.12. hewitt = {[0, +∞[}⊥⊥. proof. since {r}⊥ = hewitt ⊥ and {[0, +∞[} ⊂ hewitt, then {r}⊥ ⊂ {[0, +∞[}⊥. conversely, let f : x −→ y be a continuous map which is orthogonal to [0, +∞[ and let us show that it is orthogonal to r. by proposition 2.3, we can suppose that x and y are tychonoff spaces. now according to proposition 2.11, we can suppose that x is a dense subset of a tychonoff space y and f is the canonical injection from x to y . for this, let g be a continuous map from x to r. then g+ = max(g, 0) ( resp., −g− = − min{g, 0} ) is a continuous map from x to [0, +∞[. by orthogonality of f to [0, +∞[, there exists a continuous map g̃+ (resp.,−̃g−) from y to [0, +∞[ such that g̃+ ◦ f = g+ (resp., (−̃g−) ◦ f = (−g−) ). hence (g̃+ − (−̃g−)) ◦ f = (g̃+ ◦ f) − ((−̃g−) ◦ f) = (g+) − (−g−) = g+ + g− = g. so the existence of a continuous map g̃ = g̃+ − −̃g− from y to r such that g̃ ◦ g = f. the uniqueness of a such function follows immediately from the density of x in y and the fact that r is hausdorff. � acknowledgements. the authors thank the referee for his/her comments, corrections and suggestions improving both the presentation and the mathematical content of this paper. lazaar would like to thank the laboratory of research latao (faculty of sciences of tunis, university tunis el manar, tunisia) for its support (lr11es12). c© agt, upv, 2018 appl. gen. topol. 19, no. 2 251 s. lazaar and s. nacib references [1] a. ayech and o. echi, the envelope of a subcategory in topology and group theory, int. j. math. sci. 2005, no. 21, 3387–3404. [2] c. cassidy, m. hebert and g. m. kelly, reflective subcategories, localizations and factorization systems, j. austral. math. soc. (series a) 41 (1986), 286. [3] o. echi and s. lazaar, universal spaces, tychonoff and spectral spaces, math. proc. r. ir. acad. 109, no. 1(2009), 35–48. [4] p. j. freyd and g. m. kelly, categories of continuous functors (i), j. pure appl. algebra 2 (1972), 169–191. [5] l. gillman and m. jerison, rings of continuous functions, springer-verlag (1976). [6] a. haouati and s. lazaar, real-compact spaces and the real line orthogonal, topology appl. 209 (2016), 30–32. [7] e. hewitt, rings of real-valued continuous function, i, trans. amer. math. soc. 64 (1948), 54–99. [8] d. holgate, linking the closure and orthogonality properties of perfect morphisms in a category, comment. math. univ. carolin. 39, no. 3 (1998), 587–607. [9] s. maclane, categories for the working mathematician, graduate texts in math. vol. 5, springer-verlag, new york, (1971). [10] m. h. stone, on the compactification of topological spaces, ann. soc. polon. math. 21 (1948), 153–160. [11] w. tholen, reflective subcategories, topology appl. 27 (1987), 201–212. [12] r. c. walker, the stone-čech compactification, springer-verlag: berlin, 1974. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 252 () @ appl. gen. topol. 18, no. 2 (2017), 289-299doi:10.4995/agt.2017.6797 c© agt, upv, 2017 generalized open sets in grill n-topology m. lellis thivagar a, i. l.reilly b, m. arockia dasan a and v. ramesh a a school of mathematics, madurai kamaraj university, madurai-625 021, tamil nadu, india. (mlthivagar@yahoo.co.in, dassfredy@gmail.com, kabilanchelian@gmail.com) b department of mathematics, the university of auckland, new zealand. (i.reilly@auckland.ac.nz) communicated by h.-p. a. künzi abstract the aim of this paper is to give a systematic development of grill ntopological spaces and discuss a few properties of local function. we build a topology for the corresponding grill by using the local function. furthermore, we investigate the properties of weak forms of open sets in the grill n-topological spaces and discuss the relationships between them. 2010 msc: 54a05; 54a99; 54c10. keywords: grill n-topological space; gnτ-α open sets; gnτ-semi open sets; gnτ-pre open sets; gnτ-β open sets. 1. introduction the grill concept proved to be an important and useful tool like nets and filters, for studying some topological concepts such as proximity spaces, closure spaces, the theory of compactifications and other similar extension problems. the idea of grill on a topological space was first introduced by choquet [4]. later chattopadhyay and thorn [3] proved that grills are always unions of ultra filters. further roy and mukherjee [13] defined and studied the typical topology associated with grill on a given topological space. recently, hatir and jafari [6] and al-omari and noiri [1] investigated new classes of generalized open sets and the relevant generalizations of continuity in grill topological spaces. many more researchers [5, 7, 9, 10, 11, 12] defined and established the received 03 november 2016 – accepted 09 march 2017 http://dx.doi.org/10.4995/agt.2017.6797 m. lellis thivagar, i. l. reilly, m. arockia dasan and v. ramesh properties of generalized open sets in classical topology. we note that corson and michael [5] used the term locally dense for pre open sets. lellis thivagar et al. [8] introduced the concept of n-topological space that is a set equipped with τ1, τ2, ..., τn, and also established its open sets. in this paper, we extend the notion of grill topological spaces into the grill n-topological spaces and we obtain a kind of topology by an operator which satisfies kuratowski’s closure axioms for the corresponding grill. we also investigate the properties of some generalized open sets in grill n-topological spaces. 2. preliminaries in this section we recall some known results of n-topological spaces and grill topological spaces which are used in the following sections. by a space x, we mean a grill n-topological space (x, nτ, g) with n-topology nτ and grill g on x on which no separation axioms are assumed unless explicitly stated. definition 2.1 ([8]). let x be a non empty set, τ1, τ2, ... , τn be n-arbitrary topologies defined on x and let the collection nτ be defined by nτ = {s ⊆ x : s = ( ⋃ n i=1 ai) ∪ ( ⋂ n i=1 bi), ai, bi ∈ τi}, satisfying the following axioms: (i) x, ∅ ∈ nτ (ii) ⋃ ∞ i=1 si ∈ nτ for all {si} ∞ i=1 ∈ nτ (iii) ⋂ n i=1 si ∈ nτ for all {si} n i=1 ∈ nτ. then the pair (x, nτ) is called a n-topological space on x and the elements of the collection nτ are known as nτ-open sets on x. a subset a of x is said to be nτ-closed on x if the complement of a is nτ-open on x. the set of all nτ-open sets on x and the set of all nτ-closed sets on x are respectively denoted by nτo(x) and nτc(x). definition 2.2 ([8]). let (x, nτ) be a n-topological space and s be a subset of x. then (i) the nτ-interior of s, denoted by nτ-int(s), and is defined by nτ-int(s) = ∪{g : g ⊆ s and g is nτ-open}. (ii) the nτ-closure of s, denoted by nτ-cl(s), and is defined by nτ-cl(s) = ∩{f : s ⊆ f and f is nτ-closed}. theorem 2.3 ([8]). let (x, nτ) be a n-topological space on x and a ⊆ x. then x ∈ nτ-cl(a) if and only if o ∩ a 6= ∅, for every nτ-open set o containing x. definition 2.4 ([4]). a non empty collection g of non empty subsets of a topological space (x, τ) is called a grill on x if (i) a ∈ g and a ⊂ b ⇒ b ∈ g and (ii) a, b ⊂ x and a ∪ b ∈ g ⇒ a ∈ g or b ∈ g. a topological space (x, τ) together with a grill g on x is called a grill topological space and is denoted by (x, τ, g). for any point x of a topological space (x, τ), τ(x) means the collection of all open sets containing x. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 290 generalized open sets in grill n-topology definition 2.5 ([13]). let (x, τ, g) be a grill topological space and for every a ⊆ x, the operator φg(a, τ) = {x ∈ x : a ∩ u ∈ g, ∀u ∈ τ(x)} is called the local function associated with the grill g and the topology τ. definition 2.6 ([13]). corresponding to a grill g on a topological space (x, τ), then the operator τg-cl : p(x) → p(x) defined by τg-cl(a) = a ∪ φ(a) ∀a ⊆ x, satisfies kuratowski’s closure axioms and also there exists a unique topology τg = {u ⊆ x : τg-cl(u c) = uc} which is finer than τ. 3. closure operator in grill n-topological spaces in this section we introduce grill n-topological spaces and investigate the properties of the local function φg(a, nτ). further we derive a topology by the closure operator τg-cl and we discuss some of its properties. definition 3.1. a non empty collection g of non empty subsets of a ntopological space (x, nτ) is called a grill on x if (i) a ∈ g and a ⊂ b ⇒ b ∈ g and (ii) a, b ⊂ x and a ∪ b ∈ g ⇒ a ∈ g or b ∈ g. then a n-topological space (x, nτ) together with a grill g is called a grill n-topological space and is denoted by (x, nτ, g). particularly, if n = 1, then (x, 1τ = τ, g) is called the grill topological space, if n = 2, then (x, 2τ, g) is called the grill bitopological space, if n = 3, then (x, 3τ, g) is called the grill tritopological space defined on x and so on. remark 3.2. (i) the grill g = p(x) − {∅} is the maximal grill in any n-topological space (x, nτ). (ii) the grill g = {x} is the minimal grill in any n-topological space (x, nτ). definition 3.3. let (x, nτ, g) be a grill n-topological space and for each a ⊆ x, the operator φg(a, nτ) = {x ∈ x : a ∩ u ∈ g, ∀u ∈ nτ(x)}, is called the local function associated with the grill g and the n-topology nτ. it is denoted as φg(a). for any point x of a n-topological space (x, nτ), nτ(x) means the collection of all nτ-open sets containing x. theorem 3.4. let (x, nτ) be a n-topological space. then the following are true: (i) if g is any grill on x, then φg is an increasing function in the sense that a ⊆ b implies φg(a, nτ) ⊆ φg(b, nτ). (ii) if g1 and g2 are two grills on x with g1 ⊆ g2, then φg1(a, nτ) ⊆ φg2(a, nτ), for all a ⊆ x. (iii) for any grill g on x and if a /∈ g, then φg(a, nτ) = ∅. proof. it trivially follows from the definition 3.3. � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 291 m. lellis thivagar, i. l. reilly, m. arockia dasan and v. ramesh theorem 3.5. let (x, nτ, g) be a grill n-topological space. then for all a, b ⊆ x, (i) φg(a ∪ b) ⊇ φg(a) ∪ φg(b). (ii) φg(φg(a)) ⊆ φg(a) = nτ-cl(φg(a)) ⊆ nτ-cl(a). proof. we prove the part (ii) only and part(i) is trivial. (ii). if x /∈ nτ-cl(a), then there exists u ∈ nτ(x) such that u ∩ a = ∅ /∈ g implies x /∈ φg(a). thus φg(a) ⊆ nτ-cl(a). now we shall show that nτcl(φg(a)) ⊆ φg(a). suppose that x ∈ nτ-cl(φg(a)), then there exists a u ∈ nτ(x) such that u ∩ φg(a) 6= ∅. let y ∈ u ∩ φg(a). then u ∩ a ∈ g and so x ∈ φg(a). thus nτ-cl(φg(a)) = φg(a). hence φg(φg(a)) ⊆ nτcl(φg(a)) = φg(a) ⊆ nτ-cl(a). � remark 3.6. equality does not always hold in (i) of theorem 3.5. let n = 2 and x = {a, b, c, d}, and consider τ1o(x) = {∅, x, {a}}, τ2o(x) = {∅, x, {a, b}}. then 2τo(x) = {∅, x, {a}, {a, b}} is a bitopology and consider the grill g = {{a, b}, {a, b, c}, {a, b, d}, x}. thus (x, 2τ, g) is a grill bitopological space on x. if a = {a} and b = {b, c}, then φg(a)∪φg(b) = ∅ ⊂ {b, c, d} = φg(a ∪ b). definition 3.7. corresponding to a grill g on a n-topological space (x, nτ), the operator nτg-cl : p(x) → p(x) defined by nτg-cl(a) = a ∪ φg(a) ∀a ⊆ x, satisfies kuratowski’s closure axioms and also there exists a unique topology nτg = {u ⊆ x : nτg-cl(u c) = uc} which is finer than nτ. example 3.8. let n = 3 and x = {a, b, c} and consider τ1o(x) = {∅, x, {a}}, τ2o(x) = {∅, x, {b}} and τ3o(x) = {∅, x, {a, b}}. then 3τo(x) = {∅, x, {a}, {b}, {a, b}} is a tritopology and consider the grill g = {{a}, {a, b}, {a, c}, x}. thus (x, 3τ, g) is a grill tritopological space on x and 3τg = {u ⊆ x : 3τgcl(uc) = uc} = {∅, {a}, {b}, {a, b}, {a, c}, x} which is finer than 3τo(x). theorem 3.9. (i) if g1 and g2 are two grills on a n-topological space (x, nτ) with g1 ⊆ g2, then nτg2 ⊆ nτg1. (ii) if g is a grill on a n-topological space (x, nτ) and b /∈ g, then b is nτg-closed set in (x, nτg). (iii) for any subset a of a n-topological space (x, nτ) and any grill g on x, φg(a) is nτg-closed set in (x, nτg). (iv) if a is a nτg-closed, then φg(a) ⊆ a. proof. (i) u ∈ nτg2 ⇒ nτg2-cl(u c) = uc ⇒ φg2(u c) ⊆ uc ⇒ φg1(u c) ⊆ uc ⇒ nτg1-cl(u c) = uc ⇒ u ∈ nτg1. (ii) if b /∈ g, then φg(b) = ∅ and nτg-cl(b) = b. (iii) we have, nτg-cl(φg(a)) = φg(a) ∪ φg(φg(a)) = φg(a) ⇒ φg(a) is nτg-closed. (iv) assume that x /∈ a = nτg-cl(a) ⇒ x /∈ φg(a). thus φg(a) ⊆ a. ✷ c© agt, upv, 2017 appl. gen. topol. 18, no. 2 292 generalized open sets in grill n-topology theorem 3.10. let (x, nτ, g) be a grill n-topological space. then the collection β(g, nτ) = {v − a : v ∈ nτ and a /∈ g} is an open basis for nτg. proof. let (x, nτ, g) be a grill n-topological space and u ∈ nτg and x ∈ u ⇒ x − u is nτg-closed ⇒ φg(x − u) ⊆ x − u ⇒ u ⊆ x − φg(x − u). therefore, x ∈ u which implies that x /∈ φg(x − u). then there exists a v ∈ nτ(x) such that v ∩ (x − u) /∈ g. let us take a = v ∩ (x − u) /∈ g and we have x ∈ v − a ⊆ u where v is nτ-open set and a /∈ g. thus u is the union of sets in β(g, nτ). clearly, β(g, nτ) is closed under finite intersections, that is if v1 − a, v2 − b ∈ β(g, nτ), then v1, v2 ∈ nτ and a, b /∈ g and also v1 ∩v2 ∈ nτ and a∪b /∈ g. now, (v1 −a)∩(v2 −b) = (v1 ∩v2)−(a∪b) ∈ β(g, nτ), and hence β(g, nτ) = {v − a : v ∈ nτ and a /∈ g} is an open base for nτg. � theorem 3.11. in a grill n-topological space (x, nτ, g), nτ ⊆ β(g, nτ) ⊆ nτg and in particular if g = p(x) − {∅}, then nτ = β(g, nτ) = nτg. proof. let v ∈ nτ. then v = v − ∅ ∈ β(g, nτ). hence nτ ⊆ β(g, nτ). now, let a ∈ β(g, nτ), then there exists v ∈ nτ and h /∈ g such that a = v − h. then, nτg-cl(a c) = nτg-cl((v − h) c) = (v − h)c ∪ φg((v − h)c) = (v c ∪h)∪(φg(v c)∪φg(h)). but, h /∈ g, then, by theorem 3.4(iii), φg(v c) ∪ φg(h) = φg(v c). since v c is nτ-closed and by theorem 3.9(iv), φg(v c) ⊆ v c. thus, nτg-cl(a c) ⊆ ac and hence a ∈ nτg. in particular, if g = p(x) − {∅}, then nτg = nτ. now v ∈ β(g, nτ) ⇒ v = u − a with u ∈ nτ and a /∈ g, we have a = ∅, so that v = u ∈ nτ and so nτ = β(g, nτ) = nτg. � corollary 3.12. let (x, nτ, g) be a grill n-topological space. if u ∈ nτ, then u ∩ φg(a) = u ∩ φg(u ∩ a), for any a ⊆ x. proof. clearly, u ∩ φg(a) ⊇ u ∩ φg(u ∩ a). on the other hand, let x ∈ u ∩ φg(a) and v ∈ nτ(x). then u ∩v ∈ nτ(x) and x ∈ φg(a) ⇒ (u ∩v )∩a ∈ g, that is, (u ∩ a) ∩ v ∈ g ⇒ x ∈ φg(u ∩ a) ⇒ x ∈ u ∩ φg(u ∩ a). thus u ∩ φg(a) = u ∩ φg(u ∩ a). � corollary 3.13. let (x, nτ, g) be a grill n-topological space. if nτ −{∅} ⊆ g, then u ⊆ φg(u) for all u ∈ nτ. proof. if u = ∅, then φg(u) = ∅ = u and if nτ−{∅} ⊆ g, then φg(x) = x. by corollary 3.12, we have for any u ∈ nτ −{∅}, u ∩φg(x) = u ∩φg(u ∩x) and implies u = u ∩ φg(u). thus, φg(u) ⊇ u. � corollary 3.14. let a be a subset of a grill n-topological space (x, nτ, g). if u ∈ nτ, then u ∩ nτg-cl(a) ⊆ nτg-cl(u ∩ a). proof. since u ∈ nτ and by corollary 3.12, we obtain u ∩ nτg-cl(a) = (u ∩ a) ∪ (u ∩ φg(a)) ⊆ (u ∩ a) ∪ φg(u ∩ a) = nτg-cl(u ∩ a). � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 293 m. lellis thivagar, i. l. reilly, m. arockia dasan and v. ramesh 4. generalized open sets in grill n-topological spaces in this section we introduce some weak forms of open sets in grill n-topological spaces and also we discuss the relationships between them. definition 4.1. let (x, nτ, g) be a grill n-topological space and a ⊆ x. then a is said to be (i) gnτ-open if a ⊆ nτ-int(φg(a)). (ii) gnτ-α open if a ⊆ nτ-int(nτg-cl(nτ-int(a))). (iii) gnτ-semi open if a ⊆ nτg-cl(nτ-int(a)). (iv) gnτ-pre open if a ⊆ nτ-int(nτg-cl(a)). (v) gnτ-β open if a ⊆ nτ-cl(nτ-int(nτg-cl(a))). the set of all gnτ-open (resp. gnτ-α open, gnτ-semi open, gnτ-pre open, gnτ-β open) sets in a grill n-topological space (x, nτ, g) is denoted by gnτo(x) (resp. gnταo(x), gnτso(x), gnτpo(x), gnτβo(x)). the complements of gnτ-open ( resp. gnτ-α open, gnτ-semi open, gnτpre open, gnτ-β open) sets in a grill n-topological space (x, nτ, g) are called their respective closed sets and the set of all gnτ-closed (resp. gnτ-α closed, gnτ-semi closed, gnτ-pre closed, gnτ-β closed) sets in a grill n-topological space (x, nτ, g) is denoted by gnτc(x) (resp. gnταc(x), gnτsc(x), gnτpc(x), gnτβc(x)). for n = 1, then we take g1τo(x) (resp. gαo(x), gso(x), gpo(x), gβo(x)). for n = 2, then we take g2τo(x) (resp. g2ταo(x), g2τso(x), g2τpo(x), g2τβo(x)) and so on. we observe that part (iii) of the next theorem is analogous to the 1985 topological space result of reilly and vamanamurthy [12]. theorem 4.2. let a be a subset of a grill n-topological space (x, nτ, g). (i) if a is nτ-open, then a is gnτ-α open. (ii) if a is gnτ-open, then a is gnτ-pre open. (iii) a is gnτ-α open if and only if it is gnτ-semi open and gnτ-pre open. (iv) if a is gnτ-semi open, then a is gnτ-β open. (v) if a is gnτ-pre open, then a is gnτ-β open. proof. here we prove part (iii) only, and note that the remaining parts have similar proofs. (iii). since a is gnτ-α-open, then a ⊆ nτ-int(nτg-cl(nτ-int(a))) ⊆ nτint(nτg-cl(a)) and a ⊆ nτ-int(nτg-cl(nτ-int(a))) ⊆ nτg-cl(nτ-int(a)). on the other hand, since a is gnτ-semi open and gnτ-pre open, then a ⊆ nτ-int(nτg-cl(a)) ⊆ nτ-int(nτg-cl(nτg-cl(nτ-int(a)))) ⊆ nτ-int(nτgcl(nτ-int(a))). � the following examples show that the converse of the above theorem need not be true, that gnτ-open sets and nτ-open sets are independent, and that gnτ-semi open sets and gnτ-pre open sets are independent. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 294 generalized open sets in grill n-topology example 4.3. let n = 5 and x = {a, b, c} and consider τ1o(x) = {∅, x, {a}}, τ2o(x) = {∅, x, {b}}, τ3o(x) = {∅, x, {a, b}}, τ4o(x) = {∅, x, {a}, {a, b}} and τ5o(x) = {∅, x, {b}, {a, b}}. then 5τo(x) = {∅, x, {a}, {b}, {a, b}} is a 5τ-topology and consider the grill g = {{a}, {a, b}, {a, c}, x}. thus (x, 5τ, g) is a grill 5-topological space. here the set {b} is a 5τ-open but not a g5τ-open set and the set {b, c} is g5τ-β-open but not g5τ-semi open, not g5τ-pre open and not g5τ-α open. also the set {a, c} is g5τ-semi open but not g5τ-pre open and not g5τ-α open and the set {a, b} is g5τ-pre open but not g5τ-open. example 4.4. let n = 3 and x = {a, b, c, d} and consider τ1o(x) = {∅, x}, τ2o(x) = {∅, x, {a}} and τ3o(x) = {∅, x, {a, b}}. then 3τo(x) = {∅, x, {a}, {a, b}} is a 3τ-topology and consider the grill g = {{a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}, x}. thus (x, 3τ, g) is a grill tritopological space. here the set {a, b, c} is g3τ-open and g3τ-α open but not a 3τ-open set. also if n = 2 and x = {a, b, c}, and consider τ1o(x) = {∅, x, {a}} and τ2o(x) = {∅, x}. then 2τo(x) = {∅, x, {a}} is a 2τ-topology and consider the grill g = {{a, b}, x}. thus (x, 2τ, g) is a grill bitopological space. here the set {a, b} is g2τ-pre open but not g2τ-semi open and not g2τ-α open. theorem 4.5. in a grill n-topological space (x, nτ, g), the following are true: (i) if nτ − {∅} ⊆ g, then every nτ-open set is gnτ-open. (ii) if a ⊆ x is gnτ-open and nτg-closed, then a is nτ-open. (iii) if a ⊆ x is gnτ-closed, then φg(nτ-int(a)) ⊆ nτ-cl(nτ-int(a)) ⊆ a. proof. (i) let a be a nτ-open set and by corollary 3.13, a = nτ-int(a) ⊆ nτint(φg(a)). (ii) since a is nτg-closed, a = nτg-cl(a) = a ∪ φg(a) ⇒ φg(a) ⊆ a and since a is gnτ-open, a ⊆ nτ-int(φg(a)) ⊆ nτ-int(a). thus a = nτ-int(a). (iii) assume a is gnτ-closed, x − a is gnτ-open and x − a ⊆ nτint(φg(x − a)) ⊆ φg(x − a). then by theorem 3.5, φg(x − a) = nτ-cl(x − a) = x − nτ-int(a) and now x − a ⊆ nτ-int(φg(x − a)) ⊆ nτ-int(x − nτ-int(a)) ⊆ x − nτ-cl(nτ-int(a)). thus nτcl(nτ-int(a)) ⊆ a and also φg(nτ-int(a)) ⊆ nτg-cl(nτ-int(a)) ⊆ nτ-cl(nτ-int(a)) ⊆ a. � theorem 4.6. let (x, nτ, g) be a grill n-topological space and ω be an index set. (i) if {ai}i∈ω ∈ gnτo(x), then ⋃ i∈ω ai ∈ gnτo(x). (ii) if {ai}i∈ω ∈ gnταo(x), then ⋃ i∈ω ai ∈ gnταo(x). (iii) if {ai}i∈ω ∈ gnτso(x), then ⋃ i∈ω ai ∈ gnτso(x). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 295 m. lellis thivagar, i. l. reilly, m. arockia dasan and v. ramesh (iv) if {ai}i∈ω ∈ gnτpo(x), then ⋃ i∈ω ai ∈ gnτpo(x). (v) if {ai}i∈ω ∈ gnτβo(x), then ⋃ i∈ω ai ∈ gnτβo(x). proof. we prove part (v) only, and note that the remaining parts have similar proofs. (v). assume {ai}i∈ω ∈ gnτβo(x), then for each i ∈ ω, ai ⊆ nτ-cl(nτint(nτg-cl(ai))) ⇒ ⋃ i∈ω ai ⊆ ⋃ i∈ω (nτ-cl(nτint(nτg-cl(ai)))) = nτ-cl( ⋃ i∈ω (nτ-int(nτg-cl(ai)))) ⊆ nτ-cl(nτ-int( ⋃ i∈ω (nτg-cl(ai)))) ⊆ nτ-cl(nτ-int (nτg-cl( ⋃ i∈ω ai))). this shows that ⋃ i∈ω ai ∈ gnτβo(x). � theorem 4.7. let (x, nτ, g) be a grill n-topological space and a, b ⊆ x, then the following statements are true: (i) if a ∈ gnτso(x) and b ∈ gnταo(x), then a ∩ b ∈ gnτso(x). (ii) if a ∈ gnτpo(x) and b ∈ gnταo(x), then a ∩ b ∈ gnτpo(x). proof. here we prove part (i) only, and note that part (ii) has a similar proof. (i) since a ⊆ nτg-cl(nτ-int(a)), b ⊆ nτ-int(nτg-cl(nτ-int(a))) and by corollary 3.14, a ∩ b ⊆ nτg-cl(nτ-int(a)) ∩ nτ-int(nτg-cl(nτ-int(b))) ⊆ nτg-cl(nτ-int(a)∩nτ-int(nτg-cl(nτ-int(b)))) ⊆ nτg-cl(nτ-int(a)∩nτgcl(nτ-int(b))) ⊆ nτg-cl(nτg-cl(nτ-int(a ∩ b))). this shows that a ∩ b ∈ gnτso(x). � lemma 4.8. let (x, nτ, g) be a grill n-topological space and a, b ⊆ x, then the following statements are true: (i) if a ∈ gnτso(x) and b ∈ nτo(x), then a ∩ b ∈ gnτso(x). (ii) if a ∈ gnτpo(x) and b ∈ nτo(x), then a ∩ b ∈ gnτpo(x). example 4.9. let n = 5 and x = {a, b, c} and consider τ1o(x) = {∅, x, {a}}, τ2o(x) = {∅, x, {b}}, τ3o(x) = {∅, x, {a, b}}, τ4o(x) = {∅, x, {a}, {a, b}} and τ5o(x) = {∅, x, {b}, {a, b}}. then 5τo(x) = {∅, x, {a}, {b}, {a, b}} is a 5τ-topology and consider the grill g = {{a}, {b}, {a, b}, {a, c}, {b, c}, x}. thus (x, 5τ, g) is a grill 5-topological space. the sets {a, c} and {b, c} are g5τ-open ( resp. g5τ-pre open, g5τ-β open) sets but their intersection {c} is not a g5τopen ( resp. g5τ-pre open, g5τ-β open) set. in the same 5τ-topology, consider the maximal grill g = p(x) − {∅}. thus (x, 5τ, g) is a grill 5-topological space. the set {a, c} and {b, c} are g5τ-semi open sets but their intersection {c} is not a g5τ-semi-open set. theorem 4.10. let (x, nτ, g) be a grill n-topological space and a, b ∈ gnταo(x), then a ∩ b ∈ gnταo(x). proof. since a, b ∈ gnταo(x), then by using theorem 4.2 and theorem 4.7 we get a ∩ b ∈ gnτso(x), a ∩ b ∈ gnτpo(x), and therefore a ∩ b ∈ gnταo(x). � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 296 generalized open sets in grill n-topology theorem 4.11. in a grill n-topological space (x, nτ, g), the family gnταo(x) is a topology and nτo(x) ⊆ gnταo(x). proof. clearly, ∅, x ∈ gnταo(x). the desired result follows from the theorem 4.2, theorem 4.6 and theorem 4.10. � theorem 4.12. let (x, nτ, g) be a grill n-topological space. then a ⊆ x is (i) gnτ-semi open if and only if nτg-cl(a) = nτg-cl(nτ-int(a)). (ii) gnτ-pre open, then nτ-cl(nτ-int(nτg-cl(a))) = nτ-cl(a). proof. (i) assume that a is gnτ-semi open, then a ⊆ nτg-cl(nτ-int(a)) ⇒ nτg-cl(a) ⊆ nτg-cl(nτ-int(a)) ⊆ nτg-cl(a). thus nτg-cl(a) = nτg-cl(nτ-int(a)). converse is obvious, since a ⊆ nτg-cl(a). (ii) assume that a is gnτ-pre open, then a ⊆ nτ-int(nτg-cl(a)) ⇒ nτ-cl(a) ⊆ nτ-cl(nτ-int(nτg-cl(a))) ⊆ nτ-cl(nτg-cl(a)) = nτcl(a∪φg(a)) = nτ-cl(a)∪nτ-cl(φg(a)) = nτ-cl(a)∪φg(a) ⊆ nτcl(a). thus nτ-cl(nτ-int(nτg-cl(a))) = nτ-cl(a). � theorem 4.13. let (x, nτ, g) be a grill n-topological space and a ⊆ x. (i) then a is gnτ-semi open if and only if there exists a u ∈ nτ such that u ⊆ a ⊆ nτg-cl(u). (ii) if a is a gnτ-semi open and a ⊆ b ⊆ nτg-cl(a), then b is gnτsemi open. proof. (i) since a is gnτ-semi open, then a ⊆ nτg-cl(nτ-int(a)). take u = nτ-int(a). then we have u ⊆ a ⊆ nτg-cl(u). on the other hand, assume u ⊆ a ⊆ nτg-cl(u) for some u ∈ nτ. since u ⊆ a, then u ⊆ nτ-int(a) ⇒ nτg-cl(u) ⊆ nτg-cl(nτ-int(a)). thus a ⊆ nτgcl(nτ-int(a)). (ii) since a is gnτ-semi open, then there exists a u ∈ nτ such that u ⊆ a ⊆ nτg-cl(u). then u ⊆ a ⊆ b ⊆ nτg-cl(a) ⊆ nτg-cl(nτgcl(u)) = nτg-cl(u). by part(i), we have b is gnτ-semi open. � theorem 4.14. let (x, nτ, g) be a grill n-topological space and a ⊆ x. (i) if a is gnτ-α closed, then nτ-cl(nτ-int(nτg-cl(a))) ⊆ a. (ii) if a is gnτ-semi closed, then nτ-int(nτg-cl(a)) ⊆ a. (iii) if a is gnτ-pre closed, then nτg-cl(nτ-int(a)) ⊆ a. (iv) if a is gnτ-β-closed, then nτ-int(nτg-cl(nτ-int(a))) ⊆ a. proof. (i). assume a is gnτ-α closed, then x − a is gnτ-α open and implies x−a ⊆ nτ-int(nτg-cl(nτ-int(x−a))) ⊆ nτ-int(nτg-cl(x−nτ-cl(a))) ⊆ nτ-int(nτ-cl(x−nτg-cl(a))) ⊆ nτ-int(x−nτ-int(nτg-cl(a))) ⊆ x−nτcl(nτ-int(nτg-cl(a))). thus nτ-cl(nτ-int(nτg-cl(a))) ⊆ a. similarly we can prove the remaining parts. � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 297 m. lellis thivagar, i. l. reilly, m. arockia dasan and v. ramesh the proof of the next theorem is straightforward. theorem 4.15. let a be a subset of a grill n-topological space (x, nτ, g). (i) if a is nτ-closed, then a is gnτ-α closed. (ii) if a is gnτ-closed, then a is gnτ-pre closed. (iii) a is gnτ-α closed if and only if it is gnτ-semi closed and gnτ-pre closed. (iv) if a is gnτ-semi closed, then a is gnτ-β closed. (v) if a is gnτ-pre closed, then a is gnτ-β closed. remark 4.16. from the above theorems, lemmas and examples we have the following diagram. we depict by arrow the implications between the classes of generalized open sets. (1) nτ-open, (2) gnτ-α open, (3) gnτ-semi open, (4) gnτ-open, (5) gnτ-pre open, (6) gnτ-β open. conclusion a set is merely an amorphous collection of elements, without coherence or form. when some kind of algebraic or geometric structure is imposed on a set, so that its elements are organized into a systematic whole, then it becomes a space. this paper is an attempt to provide a rigorous definition of generalized open sets of grill topologies on a non empty set, and to establish their properties with suitable examples. the grill n-topological concepts can be extended to other applicable research areas of topology such as nano topology, fuzzy topology, intuitionistic topology, digital topology and so on. references [1] a. al-omari and t. noiri, decompositions of continuity via grills, jordan j. math. stat. 4, no. 1 (2011), 33–46. [2] d. andrijevic, semi-preopen sets, mat. vesnik. 38, no. 1(1986), 24–32. [3] k. c. chattopadhyay, o. njastad and w. j. thron, merotopic spaces and extensions of closure spaces, can. j. math. 35, no. 4 (1983), 613–629. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 298 generalized open sets in grill n-topology [4] g. choquet, sur les notions de filter et grill, comptes rendus acad. sci. paris. 224 (1947), 171–173. [5] h. corson and e. michael, metrizability of certain countable unions, illinois j. math. 8 (1964), 351–360. [6] e. hatir and s. jafari, on some new classes of sets and a new decomposition of continuity via grills, j. adv. math. studies. 3, no. 1 (2010), 33–40. [7] n. levine, semi-open sets and semi-continuity in topological spaces, amer math. monthly. 70 (1963), 36–41. [8] m. lellis thivagar, v. ramesh and m. arockia dasan, on new structure of n-topology, cogent math. 3 (2016), 1204104, 10 pp. [9] a. s. mashhour, i. a. hasanien and s. n. el-deeb, on pre continuous and weak pre continuous mappings, proc. math. phys. soc. egypt. 53 (1982), 47–53. [10] a. s. mashhour, i. a. hasanien and s. n. el-deeb, α-continuous and α-open mappings, acta. math. hungar. 41, no. 3-4(1983), 213–218. [11] o. njastad, on some classes of nearly open sets, pacific j.math. 15 (1965), 961–970. [12] i. l reilly and m. k. vamanamurthy, on α-continuity in topological spaces, acta math. hungar. 45 (1985), 27–32. [13] b. roy and m. n. mukherjee, on a typical topology induced by a grill, soochow j. math. 33, no. 4 (2007), 771–786. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 299 () @ appl. gen. topol. 18, no. 2 (2017), 401-427doi:10.4995/agt.2017.7798 c© agt, upv, 2017 generalized normal product adjacency in digital topology laurence boxer department of computer and information sciences, niagara university, niagara university, ny 14109, usa; and department of computer science and engineering, state university of new york at buffalo. (boxer@niagara.edu) communicated by s. romaguera abstract we study properties of cartesian products of digital images for which adjacencies based on the normal product adjacency are used. we show that the use of such adjacencies lets us obtain many “product properties” for which the analogous statement is either unknown or invalid if, instead, we were to use cu-adjacencies. 2010 msc: 54h99. keywords: digital topology; digital image; continuous multivalued function; shy map; retraction. 1. introduction we study adjacency relations based on the normal product adjacency for cartesian products of multiple digital images. most of the literature of digital topology focuses on images that use a cu-adjacency; however, the results of this paper seem to indicate that for cartesian products of digital images, the natural adjacencies to use are based on the normal product adjacency of the factor adjacencies, in the sense of preservation of many properties in cartesian products. received 27 june 2017 – accepted 25 july 2017 http://dx.doi.org/10.4995/agt.2017.7798 2. preliminaries we use n, z, and r to represent the sets of natural numbers, integers, and real numbers, respectively. much of the material that appears in this section is quoted or paraphrased from [15, 17], and other papers cited in this section. we will assume familiarity with the topological theory of digital images. see, e.g., [3] for many of the standard definitions. all digital images x are assumed to carry their own adjacency relations (which may differ from one image to another). when we wish to emphasize the particular adjacency relation we write the image as (x, κ), where κ represents the adjacency relation. 2.1. common adjacencies. among the commonly used adjacencies are the cu-adjacencies. let x, y ∈ z n, x 6= y. let u be an integer, 1 ≤ u ≤ n. we say x and y are cu-adjacent if • there are at most u indices i for which |xi − yi| = 1, and • for all indices j such that |xj − yj| 6= 1 we have xj = yj. we often label a cu-adjacency by the number of points adjacent to a given point in zn using this adjacency. e.g., • in z1, c1-adjacency is 2-adjacency. • in z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency. • in z3, c1-adjacency is 6-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. given digital images or graphs (x, κ) and (y, λ), the normal product adjacency np(κ, λ) (also called the strong adjacency [37] and denoted κ∗(κ, λ) in [13]) generated by κ and λ on the cartesian product x × y is defined as follows. definition 2.1 ([1]). let x, x′ ∈ x, y, y′ ∈ y . then (x, y) and (x′, y′) are np(κ, λ)-adjacent in x × y if and only if • x = x′ and y and y′ are λ-adjacent; or • x and x′ are κ-adjacent and y = y′; or • x and x′ are κ-adjacent and y and y′ are λ-adjacent. � 2.2. connectedness. a subset y of a digital image (x, κ) is κ-connected [32], or connected when κ is understood, if for every pair of points a, b ∈ y there exists a sequence {yi} m i=0 ⊂ y such that a = y0, b = ym, and yi and yi+1 are κ-adjacent for 0 ≤ i < m. for two subsets a, b ⊂ x, we will say that a and b are adjacent when there exist points a ∈ a and b ∈ b such that a and b are equal or adjacent. thus sets with nonempty intersection are automatically adjacent, while disjoint sets may or may not be adjacent. it is easy to see that a finite union of connected adjacent sets is connected. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 402 generalized normal product adjacency in digital topology 2.3. continuous functions. the following generalizes a definition of [32]. definition 2.2 ([4]). let (x, κ) and (y, λ) be digital images. a function f : x → y is (κ, λ)-continuous if for every κ-connected a ⊂ x we have that f(a) is a λ-connected subset of y . when the adjacency relations are understood, we will simply say that f is continuous. continuity can be reformulated in terms of adjacency of points: theorem 2.3 ([32, 4]). a function f : x → y is continuous if and only if, for any adjacent points x, x′ ∈ x, the points f(x) and f(x′) are equal or adjacent. � note that similar notions appear in [18, 19] under the names immersion, gradually varied operator, and gradually varied mapping. theorem 2.4 ([3, 4]). if f : (a, κ) → (b, λ) and g : (b, λ) → (c, µ) are continuous, then g ◦ f : (a, κ) → (c, µ) is continuous. � example 2.5 ([32]). a constant function between digital images is continuous. � example 2.6. the identity function 1x : (x, κ) → (x, κ) is continuous. proof. this is an immediate consequence of theorem 2.3. � definition 2.7. let (x, κ) be a digital image in zn. let x, y ∈ x. a κ-path of length m from x to y is a set {xi} m i=0 ⊂ x such that x = x0, xm = y, and xi−1 and xi are equal or κ-adjacent for 1 ≤ i ≤ m. if x = y, we say {x} is a path of length 0 from x to x. notice that for a path from x to y as described above, the function f : [0, m]z → x defined by f(i) = xi is (c1, κ)-continuous. such a function is also called a κ-path of length m from x to y. 2.4. digital homotopy. a homotopy between continuous functions may be thought of as a continuous deformation of one of the functions into the other over a finite time period. definition 2.8 ([4]; see also [26]). let x and y be digital images. let f, g : x → y be (κ, κ′)-continuous functions. suppose there is a positive integer m and a function f : x × [0, m]z → y such that • for all x ∈ x, f(x, 0) = f(x) and f(x, m) = g(x); • for all x ∈ x, the induced function fx : [0, m]z → y defined by fx(t) = f(x, t) for all t ∈ [0, m]z is (2, κ′)−continuous. that is, fx(t) is a path in y . • for all t ∈ [0, m]z, the induced function ft : x → y defined by ft(x) = f(x, t) for all x ∈ x is (κ, κ′)−continuous. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 403 then f is a digital (κ, κ′)−homotopy between f and g, and f and g are digitally (κ, κ′)−homotopic in y . if for some x ∈ x we have f(x, t) = f(x, 0) for all t ∈ [0, m]z, we say f holds x fixed, and f is a pointed homotopy. ✷ we denote a pair of homotopic functions as described above by f ≃κ,κ′ g. when the adjacency relations κ and κ′ are understood in context, we say f and g are digitally homotopic (or just homotopic) to abbreviate “digitally (κ, κ′)−homotopic in y ,” and write f ≃ g. proposition 2.9 ([26, 4]). digital homotopy is an equivalence relation among digitally continuous functions f : x → y . ✷ definition 2.10 ([5]). let f : x → y be a (κ, κ′)-continuous function and let g : y → x be a (κ′, κ)-continuous function such that f ◦ g ≃κ′,κ′ 1x and g ◦ f ≃κ,κ 1y . then we say x and y have the same (κ, κ′)-homotopy type and that x and y are (κ, κ′)-homotopy equivalent, denoted x ≃κ,κ′ y or as x ≃ y when κ and κ′ are understood. if for some x0 ∈ x and y0 ∈ y we have f(x0) = y0, g(y0) = x0, and there exists a homotopy between f ◦ g and 1x that holds x0 fixed, and a homotopy between g ◦ f and 1y that holds y0 fixed, we say (x, x0, κ) and (y, y0, κ ′) are pointed homotopy equivalent and that (x, x0) and (y, y0) have the same pointed homotopy type, denoted (x, x0) ≃κ,κ′ (y, y0) or as (x, x0) ≃ (y, y0) when κ and κ ′ are understood. ✷ it is easily seen, from proposition 2.9, that having the same homotopy type (respectively, the same pointed homotopy type) is an equivalence relation among digital images (respectively, among pointed digital images). 2.5. connectivity preserving and continuous multivalued functions. a multivalued function f : x → y assigns a subset of y to each point of x. we will write f : x ⊸ y . for a ⊂ x and a multivalued function f : x ⊸ y , let f(a) = ⋃ x∈a f(x). definition 2.11 ([30]). a multivalued function f : x ⊸ y is connectivity preserving if f(a) ⊂ y is connected whenever a ⊂ x is connected. as is the case with definition 2.2, we can reformulate connectivity preservation in terms of adjacencies. theorem 2.12 ([15]). a multivalued function f : x ⊸ y is connectivity preserving if and only if the following are satisfied: • for every x ∈ x, f(x) is a connected subset of y . • for any adjacent points x, x′ ∈ x, the sets f(x) and f(x′) are adjacent. � definition 2.11 is related to a definition of multivalued continuity for subsets of zn given and explored by escribano, giraldo, and sastre in [20, 21] based on subdivisions. (these papers make a small error with respect to compositions, that is corrected in [22].) their definitions are as follows: c© agt, upv, 2017 appl. gen. topol. 18, no. 2 404 generalized normal product adjacency in digital topology definition 2.13. for any positive integer r, the r-th subdivision of zn is z n r = {(z1/r, . . . , zn/r) | zi ∈ z}. an adjacency relation κ on zn naturally induces an adjacency relation (which we also call κ) on znr as follows: (z1/r, . . . , zn/r), (z ′ 1/r, . . . , z ′ n/r) are adjacent in znr if and only if (z1, . . . , zn) and (z1, . . . , zn) are adjacent in z n. given a digital image (x, κ) ⊂ (zn, κ), the r-th subdivision of x is s(x, r) = {(x1, . . . , xn) ∈ z n r | (⌊x1⌋, . . . , ⌊xn⌋) ∈ x}. let er : s(x, r) → x be the natural map sending (x1, . . . , xn) ∈ s(x, r) to (⌊x1⌋, . . . , ⌊xn⌋). � definition 2.14. for a digital image (x, κ) ⊂ (zn, κ), a function f : s(x, r) → y induces a multivalued function f : x ⊸ y if x ∈ x implies f(x) = ⋃ x′∈e −1 r (x) {f(x′)}. � definition 2.15. a multivalued function f : x ⊸ y is called continuous when there is some r such that f is induced by some single valued continuous function f : s(x, r) → y . � figure 1. [15] two images x and y with their second subdivisions. note [15] that the subdivision construction (and thus the notion of continuity) depends on the particular embedding of x as a subset of zn. in particular we may have x, y ⊂ zn with x isomorphic to y but s(x, r) not isomorphic to s(y, r). this in fact is the case for the two images in figure 1, when we use 8-adjacency for all images. then the spaces x and y in the figure are isomorphic, each being a set of two adjacent points. but s(x, 2) and s(y, 2) are not isomorphic since s(x, 2) can be disconnected by removing a single point, while this is impossible in s(y, 2). the definition of connectivity preservation makes no reference to x as being embedded inside of any particular integer lattice zn. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 405 proposition 2.16 ([20, 21]). let f : x ⊸ y be a continuous multivalued function between digital images. then • for all x ∈ x, f(x) is connected; and • for all connected subsets a of x, f(a) is connected. � theorem 2.17 ([15]). for (x, κ) ⊂ (zn, κ), if f : x ⊸ y is a continuous multivalued function, then f is connectivity preserving. � the subdivision machinery often makes it difficult to prove that a given multivalued function is continuous. by contrast, many maps can easily be shown to be connectivity preserving. proposition 2.18 ([15]). let x and y be digital images. suppose y is connected. then the multivalued function f : x ⊸ y defined by f(x) = y for all x ∈ x is connectivity preserving. � proposition 2.19 ([15]). let f : (x, κ) ⊸ (y, λ) be a multivalued surjection between digital images (x, κ), (y, κ) ⊂ (zn, κ). if x is finite and y is infinite, then f is not continuous. � corollary 2.20 ([15]). let f : x ⊸ y be the multivalued function between digital images defined by f(x) = y for all x ∈ x. if x is finite and y is infinite and connected, then f is connectivity preserving but not continuous. � examples of connectivity preserving but not continuous multivalued functions on finite spaces are given in [15]. 2.6. other notions of multivalued continuity. other notions of continuity have been given for multivalued functions between graphs (equivalently, between digital images). we have the following. definition 2.21 ([36]). let f : x ⊸ y be a multivalued function between digital images. • f has weak continuity if for each pair of adjacent x, y ∈ x, f(x) and f(y) are adjacent subsets of y . • f has strong continuity if for each pair of adjacent x, y ∈ x, every point of f(x) is adjacent or equal to some point of f(y) and every point of f(y) is adjacent or equal to some point of f(x). � proposition 2.22 ([15]). let f : x ⊸ y be a multivalued function between digital images. then f is connectivity preserving if and only if f has weak continuity and for all x ∈ x, f(x) is connected. � example 2.23 ([15]). if f : [0, 1]z ⊸ [0, 2]z is defined by f(0) = {0, 2}, f(1) = {1}, then f has both weak and strong continuity. thus a multivalued function between digital images that has weak or strong continuity need not have connected point-images. by theorem 2.12 and proposition 2.16 it follows that neither having weak continuity nor having strong continuity implies that a multivalued function is connectivity preserving or continuous. ✷ c© agt, upv, 2017 appl. gen. topol. 18, no. 2 406 generalized normal product adjacency in digital topology example 2.24 ([15]). let f : [0, 1]z ⊸ [0, 2]z be defined by f(0) = {0, 1}, f(1) = {2}. then f is continuous and has weak continuity but does not have strong continuity. ✷ proposition 2.25 ([15]). let f : x ⊸ y be a multivalued function between digital images. if f has strong continuity and for each x ∈ x, f(x) is connected, then f is connectivity preserving. � the following shows that not requiring the images of points to be connected yields topologically unsatisfying consequences for weak and strong continuity. example 2.26 ([15]). let x and y be nonempty digital images. let the multivalued function f : x ⊸ y be defined by f(x) = y for all x ∈ x. • f has both weak and strong continuity. • f is connectivity preserving if and only if y is connected. � as a specific example [15] consider x = {0} ⊂ z and y = {0, 2}, all with c1 adjacency. then the function f : x ⊸ y with f(0) = y has both weak and strong continuity, even though it maps a connected image surjectively onto a disconnected image. 2.7. shy maps and their inverses. definition 2.27 ([5]). let f : x → y be a continuous surjection of digital images. we say f is shy if • for each y ∈ y , f−1(y) is connected, and • for every y0, y1 ∈ y such that y0 and y1 are adjacent, f −1({y0, y1}) is connected. � shy maps induce surjections on fundamental groups [5]. some relationships between shy maps f and their inverses f−1 as multivalued functions were studied in [8, 15, 9]. we have the following. theorem 2.28 ([15, 9]). let f : x → y be a continuous surjection between digital images. then the following are equivalent. • f is a shy map. • for every connected y0 ⊂ y , f −1(y0) is a connected subset of x. • f−1 : y ⊸ x is a connectivity preserving multi-valued function. • f−1 : y ⊸ x is a multi-valued function with weak continuity such that for all y ∈ y , f−1(y) is a connected subset of x. � 2.8. other tools. other terminology we use includes the following. given a digital image (x, κ) ⊂ zn and x ∈ x, the set of points adjacent to x ∈ zn and the neighborhood of x in zn are, respectively, nκ(x) = {y ∈ z n | y is κ-adjacent to x}, n∗κ(x) = nκ(x) ∪ {x}. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 407 3. extensions of normal product adjacency in this section, we define extensions of the normal product adjacency, as follows. definition 3.1. let u and v be positive integers, 1 ≤ u ≤ v. let {(xi, κi)} v i=1 be digital images. let npu(κ1, . . . , κv) be the adjacency defined on the cartesian product πvi=1xi as follows. for xi, x ′ i ∈ xi, p = (x1, . . . , xv) and q = (x′1, . . . , x ′ v) are npu(κ1, . . . , κv)-adjacent if and only if • for at least 1 and at most u indices i, xi and x ′ i are κi-adjacent, and • for all other indices i, xi = x ′ i. � throughout this paper, the reader should be careful to note that some of our results for npu(κ1, . . . , κv) are stated for all u ∈ {1, . . . , v} and others are stated only for u = 1 or u = 2 or u = v. proposition 3.2. np(κ, λ) = np2(κ, λ). i.e., given x, x ′ ∈ (x, κ) and y, y′ ∈ (y, λ), p = (x, y) and p′ = (x′, y′) are np(κ, λ)-adjacent in x × y if and only if p and p′ are np2(κ, λ)-adjacent. proof. this follows immediately from definitions 2.1 and 3.1. � theorem 3.3 ([13]). for x ∈ zm, y ∈ zn, np2(cm, cn) = cm+n, i.e., the normal product adjacency for (x, cm) × (y, cn) coincides with the cm+nadjacency for x × y . � examples are also given in [13] that show that if x ∈ zm, y ∈ zn, and a < m or b < n, then np2(ca, cb) 6= ca+b. the following shows that npv obeys a recursive property. proposition 3.4. let v > 2. then npv(κ1, . . . , κv) = np2(npv−1(κ1, . . . , κv−1), κv). proof. let xi, x ′ i ∈ xi for 1 ≤ i ≤ v. then p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) are npv(κ1, . . . , κv)-adjacent if and only if for at least 1 and at most v indices i, xi and x ′ i are κi-adjacent and for all other indices i, xi = x ′ i. hence p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) are npv(κ1, . . . , κu)-adjacent if and only if either • xi and x ′ i are κi-adjacent for from 1 to v−1 indices among {1, . . . , v−1}, xi = x ′ i for all other indices among {1, . . . , v − 1}, and xv = x ′ v; or • xi and x ′ i are κi-adjacent for from 1 to v−1 indices among {1, . . . , v−1}, xi = x ′ i for all other indices among {1, . . . , v − 1}, and xv and x ′ v are κv-adjacent. thus, p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) are npv(κ1, . . . , κv)-adjacent if and only if p and p′ are np2(npu−1(κ1, . . . , κv−1), κv)-adjacent. � notice proposition 3.4 may fail to extend to npu(κ1, . . . , κv) if u < v, as shown in the following (suggested by an example in [13]). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 408 generalized normal product adjacency in digital topology example 3.5. let xi, x ′ i ∈ (xi, κi), i ∈ {1, 2, 3}. suppose x1 and x ′ 1 are κ1-adjacent, x2 and x ′ 2 are κ2-adjacent, and x3 = x ′ 3. then (x1, x2, x3) and (x′1, x ′ 2, x ′ 3) are np2(κ1, κ2, κ3)-adjacent in x1 × x2 × x3, but (x1, x2) and (x′1, x ′ 2) are not np1(κ1, κ2)-adjacent in x1 × x2. thus, np2(κ1, κ2, κ3) 6= np2(np1(κ1, κ2), κ3). � theorem 3.6. let f, g : (x, κ) → (y, λ) be functions. let h : x×[0, m]z → y be a function such that h(x, 0) = f(x) and h(x, m) = g(x) for all x ∈ x. then h is a homotopy if and only if h is (np1(κ, c1), λ)-continuous. proof. in the following, we consider arbitrary (np1(κ, c1), λ)-adjacent (x, t) and (x′, t′) in x × [0, m]z with x, x ′ ∈ x and t, t′ ∈ [0, m]z. such points offer the following cases. (1) x and x′ are κ-adjacent and t = t′; or (2) x = x′ and t and t′ are c1-adjacent, i.e., |t − t ′| = 1. let h be a homotopy. then f and g are continuous, and given (np1(κ, c1), λ)adjacent (x, t) and (x′, t′) in x × [0, m]z, we consider the cases listed above. • in case 1, since h is a homotopy, h(x, t) and h(x′, t) = h(x′, t′) are equal or λ-adjacent. • in case 2, since h is a homotopy, h(x, t) and h(x′, t′) = h(x, t′) are equal or λ-adjacent. therefore, h is (np1(κ, c1), λ)-continuous. suppose h is (np1(κ, c1), λ)-continuous. then for κ-adjacent x, x ′, f(x) = h(x, 0) and h(x′, 0) = f(x′) are equal or λ-adjacent, so f is continuous. similarly, g(x) = h(x, m) and g(x′) = h(x′, m) are equal or λ-adjacent, so g is continuous. also, the continuity of h implies that h(x, t) and h(x′, t) must be equal or λ-adjacent, so the induced function ht is (κ, λ)-continuous. for c1-adjacent t, t ′, the continuity of h implies that h(x, t) and h(x, t′) are equal or λ-adjacent, so the induced function hx is continuous. by definition 2.8, h is a homotopy. � 4. npu and maps on products given functions fi : (xi, κi) → (yi, λi), 1 ≤ i ≤ v, the function πvi=1fi : (π v i=1xi, npu(κ1, . . . , κv)) → (π v i=1yi, npu(λ1, . . . , λv)) is defined by πvi=1fi(x1, . . . , xv) = (f(x1), . . . , f(xv)), where xi ∈ xi. the following generalizes a result in [9, 13]. theorem 4.1. let fi : (xi, κi) → (yi, λi), 1 ≤ i ≤ v. then the product map f = πvi=1fi : (π v i=1xi, npv(κ1, . . . , κv)) → (π v i=1yi, npv(λ1, . . . , λv)) is continuous if and only if each fi is continuous. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 409 proof. in the following, we let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v), where xi, x ′ i ∈ xi. suppose each fi is continuous and p and p ′ are npv(κ1, . . . , κv)-adjacent. then for all indices i, xi and x ′ i are equal or κi-adjacent, so fi(xi) and fi(x ′ i) are equal or λi-adjacent. therefore, f(p) and f(p ′) are equal or npv(λ1, . . . , λv)adjacent. thus, f is continuous. suppose f is continuous and for all indices i, xi and x ′ i are κi-adjacent. then f(p) and f(p′) are equal or npv(λ1, . . . , λi)-adjacent. therefore, for each index i, fi(xi) and fi(x ′ i) are equal or λi-adjacent. thus, each fi is continuous. � the statement analogous to theorem 4.1 is not generally true if cu-adjacencies are used instead of normal product adjacencies, as shown in the following. example 4.2. let x = {(0, 0), (1, 0)} ⊂ z2. let y = {(0, 0), (1, 1)} ⊂ z2. clearly, there is an isomorphism f : (x, c2) → (y, c2). consider x ′ = x × {0} ⊂ z3 and y ′ = y × {0} ⊂ z3. note that the product map f × 1{0} is not (c1, c1)-continuous, since x ′ is c1-connected and y ′ = (f × 1{0})(x ′) is not c1-connected. � the following is a generalization of a result of [25]. theorem 4.3. the projection maps pi : (π v i=1xi, npu(κ1, . . . , κv)) → (xi, κi) defined by pi(x1, . . . , xv) = xi for xi ∈ (xi, κi), are all continuous, for 1 ≤ u ≤ v. proof. let p = (x1, . . . , xv) and p ′ = (x′1, . . . , x ′ v) be npu(κ1, . . . , κv)-adjacent in (πvi=1xi, npu(κ1, . . . , κv)), where xi, x ′ i ∈ xi. then for all indices i, xi = pi(p) and x ′ i = pi(p ′) are equal or κi-adjacent. thus, pi is continuous. � the statement analogous to theorem 4.3 is not generally true if a cuadjacency is used instead of a normal product adjacency, as shown in the following. example 4.4 ([13]). let x = [0, 1]z ⊂ z. let y = {(0, 0), (1, 1)} ⊂ z 2. then the projection map p2 : (x × y, c3) → (y, c1) is not continuous, since x × y is c3-connected and y is not c1-connected. � we see in the next result that isomorphism is preserved by taking cartesian products with a normal product adjacency. theorem 4.5. let x = πvi=1xi. let fi : (xi, κi) → (yi λi), 1 ≤ i ≤ v. • for 1 ≤ u ≤ v, if the product map f = πvi=1fi : (x, npu(κ1, . . . , κv)) → (πvi=1yi, npu(λ1, . . . , κv)) is an isomorphism, then for 1 ≤ i ≤ v, fi is an isomorphism. • if fi is an isomorphism for all i, then the product map f = π v i=1fi : (x, npv(κ1, . . . , κv)) → (π v i=1yi, npv(λ1, . . . , κv)) is an isomorphism. proof. let f be an isomorphism. then each fi must be one-to-one and onto. let xi ∈ xi. let ii : xi → x be defined by ii(x) = (x1, . . . , xi−1, x, xi+1, . . . , xv). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 410 generalized normal product adjacency in digital topology define i′i : yi → y similarly. clearly, ii is (κi, npu(κ1, . . . , κv)-continuous and i′i is (λi, npu(λ1, . . . , λv)-continuous. let pi : x → xi and p ′ i : y → yi be the projections to the i-th coordinate. by theorems 2.4 and 4.3, fi = p ′ i ◦ f ◦ ii and f−1i = pi ◦ f −1 ◦ i′i are continuous. hence, fi is an isomorphism. let fi : xi → yi be an isomorphism. one sees easily that f is one-to-one and onto, and by theorem 4.1, f is continuous. the inverse function f−1 is the product function of the f−1i , hence is continuous by theorem 4.1. thus, f is an isomorphism. � the statement analogous to theorem 4.5 is not generally true for all cuadjacencies, as shown by the following. example 4.6. let x = {(0, 0), (1, 1)} ⊂ z2. let y = {(0, 0), (1, 0)} ⊂ z2. clearly, (x, c2) and (y, c2) are isomorphic. consider x ′ = x × {0} ⊂ z3 and y ′ = y × {0} ⊂ z3. note (x′, c1) and (y ′, c1) are not isomorphic, since the former is c1-disconnected and the latter is c1-connected. � 5. npv and connectedness theorem 5.1. let (xi, κi) be digital images, i ∈ {1, 2, . . . , v}. then (xi, κi) is connected for all i if and only (πvi=1xi, npu(κ1, . . . , κv)) is connected. proof. suppose (xi, κi) is connected for all i. let xi, x ′ i ∈ xi. then there are paths pi in xi from xi to x ′ i. let p = (x1, . . . , xv), p ′ = (x′1, . . . , x ′ v) ∈ π v i=1xi. then ⋃v i=1 p ′ i , where p ′1 = p1 × {(x2, . . . , xv)}, p ′i = {(x ′ 1, . . . , x ′ i−1)} × pi × {(xi+1, . . . , xv)} for 2 ≤ i < v, p ′v = {(x ′ 1, . . . , x ′ v−1)} × pv, is a path in πvi=1xi from p to p ′. since p and p′ are arbitrarily selected points in πvi=1xi, it follows that (π v i=1xi, npu(κ1, . . . , κv)) is connected. if (πvi=1xi, npu(κ1, . . . , κv)) is connected, then (xi, κi) = pi(π v i=1xi) is connected, by definition 2.2 and theorem 4.3. � the statement analogous to theorem 5.1 is not generally true if a cuadjacency is used instead of npu(κ1, . . . , κv) for x × y , as shown by the following. example 5.2 ([13]). let x = [0, 1]z, y = {(0, 0), (1, 1)} ⊂ z 2. then x × y is c2-connected, but y is not c1-connected. also, x is c1-connected and y is c2-connected, but x × y is not c1-connected. � 6. npv and homotopy relations in this section, we show that normal products preserve a variety of digital homotopy relations. these include homotopy type and several generalizations introduced in [17]. these generalizations homotopic similarity, long homotopy type, and real homotopy type all coincide with homotopy type on pairs of finite digital images; however, for each of these relationships, an example is given c© agt, upv, 2017 appl. gen. topol. 18, no. 2 411 in [17] of a pair of digital images, at least one member of which is infinite, such that the two images have the given relation but are not homotopy equivalent. by contrast with euclidean topology, in which a bounded space such as a single point and an unbounded space such as rn with euclidean topology can have the same homotopy type, a finite digital image and an image with infinite diameter e.g., a single point and (zn, c1) cannot share the same homotopy type. however, examples in [17] show that a finite digital image and an image with infinite diameter can share homotopic similarity, long homotopy type, or real homotopy type. 6.1. homotopic maps and homotopy type. theorem 6.1. let (xi, κi) and (yi, λi) be digital images, 1 ≤ i ≤ v. let x = (πvi=1xi, npv(κ1, . . . , κi)). let y = (π v i=1yi, npv(λ1, . . . , λi)). let fi, gi : xi → yi be continuous and let hi : xi ×[0, mi]z → yi be a homotopy from fi to gi. then there is a homotopy h between the product maps f = π v i=1fi : x → y and g = πvi=1gi : x → y . if the homotopies hi are pointed, then h is pointed. proof. let m = max{mi} v i=1. let h ′ i : xi × [0, m]z → yi be defined by h′i(x, t) = { hi(x, t) for 0 ≤ t ≤ mi; hi(x, mi) for mi ≤ t ≤ m. clearly, h′i is a homotopy from fi to gi. let h : x × m → y be defined by h((x1, . . . , xv), t) = (h ′ 1(x1, t), . . . , h ′ v(xv, t)). it is easily seen that h is a homotopy from f to g, and that if each hi is pointed, then h is pointed. � the following theorem generalizes results of [17]. theorem 6.2. suppose (6.1) xi ≃κi,λi yi for 1 ≤ i ≤ v. then (6.2) x = πvi=1xi ≃npv(κ1,...,κv),npv(λ1,...,λv) y = π v i=1yi. further, if the homotopy equivalences (6.1) are all pointed with respect to xi ∈ xi and yi ∈ yi, then the homotopy equivalence (6.2) is pointed with respect to (x1, . . . , xv) ∈ x and (y1, . . . , yv) ∈ y . proof. we give a proof for the unpointed assertion. with minor modifications, the pointed assertion is proven similarly. by hypothesis, there exist continuous functions fi : xi → yi and gi : yi → xi and homotopies hi : xi × [0, mi]z → xi from gi ◦ fi to 1xi and ki : yi × [0, ni]z → yi from fi ◦ gi to 1yi. let m = max{mi} v i=1. then h ′ i : xi × [0, m]z → xi, defined by h′i(x, t) = { hi(x, t) for 0 ≤ t ≤ mi; hi(x, mi) for mi ≤ t ≤ m, c© agt, upv, 2017 appl. gen. topol. 18, no. 2 412 generalized normal product adjacency in digital topology is clearly a homotopy from gi ◦ fi to 1xi. let f = πvi=1fi : x → y . let g = π v i=1gi : y → x. by theorem 4.1, f and g are continuous. let h : x × [0, m]z → x be defined by h(x1, . . . , xv, t) = (h ′ 1(x1, t), . . . , h ′ v(xv, t)). then h is easily seen to be a homotopy from g ◦ f to πvi=11xi = 1x. we can similarly show that f ◦ g ≃ 1y . therefore, x ≃ y . � the statements analogous to theorems 6.1 and 6.2 are not generally true if a cu-adjacency is used instead of a normal product adjacency for the cartesian product. consider, e.g., x and y as in example 4.6. let f : y → x be a (c1, c2)-isomorphism. then f is (c1, c2)-homotopic to the constant map (0, 0) of y to (0, 0). however, f × 1{0} is not even (c1, c1)-continuous, hence is not (c1, c1)-homotopic to (0, 0) × 1{0}. although x and y are (c2, c1)-homotopy equivalent, x′ = x × {0} and y ′ = y × {0} are not (c1, c1)-homotopy equivalent, since x′ is not c1-connected and y ′ is c1-connected. 6.2. homotopic similarity. definition 6.3 ([17]). let x and y be digital images. we say (x, κ) and (y, λ) are homotopically similar, denoted x ≃sκ,λ y , if there exist subsets {xj} ∞ j=1 of x and {yj} ∞ j=1 of y such that: • x = ⋃∞ j=1 xj, y = ⋃∞ j=1 yj, and, for all j, xj ⊂ xj+1, yj ⊂ yj+1. • there are continuous functions fj : xj → yj, gj : yj → xj such that gj ◦ fj ≃κ,κ 1xj and fj ◦ gj ≃λ,λ 1yj . • for m ≤ n, fn|xm ≃κ,λ fm in ym and gn|ym ≃λ,κ gm in xm. if all of these homotopies are pointed with respect to some x1 ∈ x1 and y1 ∈ y1, we say (x, x1) and (y, y1) are pointed homotopically similar, denoted (x, x1) ≃ s κ,λ (y, y1) or (x, x1) ≃ s (y, y1) when κ and λ are understood. ✷ theorem 6.4. let xi ≃ s κi,λi yi, 1 ≤ i ≤ v. let x = π v i=1xi, x = π v i=1xi. then x ≃s npv(κ1,...,κv),npv(λ1,...,λv) y . if the similarities xi ≃ s κi,λi yi are pointed at xi ∈ xi, yi ∈ yi, then the similarity x ≃ s npv(κ1,...,κv),npv(λ1,...,λv) y is pointed at x0 = (x1, . . . , xv) ∈ x, y0 = (y1, . . . , yv) ∈ y . proof. we give a proof for the unpointed assertion. a virtually identical argument can be given for the pointed assertion. by hypothesis, for j ∈ n there exist digital images xi,j ⊂ xi, yi,j ⊂ yi such that xi,j ⊂ xi,j+1, xi = ⋃∞ j=1 xi,j, yi,j ⊂ yi,j+1, yi = ⋃∞ j=1 yi,j, and continuous functions fi,j : xi,j → yi,j, gi,j : yi,j → xi,j, such that gi,j ◦ fi,j ≃κi,κi 1xi,j , fi,j ◦ gi,j ≃λi,λi 1yi,j , and m ≤ n implies fi,n|xi,m ≃κi,λi fi,m in yi,m and gi,n|xi,m ≃λi,κi gi,m in xi,m. let xj = π v i=1xi,j, yj = π v i=1yi,j. clearly we have x = ⋃∞ j=1 xj, y = ⋃∞ j=1 yj, xj ⊂ xj+1, yj ⊂ yj+1. let fj = π v i=1fi,j : xj → yj, gj = c© agt, upv, 2017 appl. gen. topol. 18, no. 2 413 πvi=1gi,j : yj → xj. by theorem 4.1, fj is (npv(κ1, . . . , κv), npv(λ1, . . . , λv))continuous and gj is (npv(λ1, . . . , λv), npv(κ1, . . . , κv))-continuous. by theorem 6.1, gj◦fj ≃npv(κ1,...,κv),npv(κ1,...,κv) 1xj and fj◦gj ≃npv(λ1,...,λv),npv(λ1,...,λv) 1yj . also by theorem 6.1, m ≤ n implies fn|xm ≃npv(κ1,...,κv),npv(λ1,...,λv) fm in ym and gn|ym ≃npv(λ1,...,λv),npv(κ1,...,κv) gm in xm. this completes the proof. � 6.3. long homotopy type. definition 6.5 ([17]). let (x, κ) and (y, λ) be digital images. let f, g : x → y be continuous. let f : x × z → y be a function such that • for all x ∈ x, there exists nf,x ∈ n such that t ≤ −nf,x implies f(x, t) = f(x) and t ≥ nf,x implies f(x, t) = g(x). • for all x ∈ x, the induced function fx : z → y defined by fx(t) = f(x, t) for all t ∈ z is (c1, λ)-continuous. • for all t ∈ z, the induced function ft : x → y defined by ft(x) = f(x, t) for all x ∈ x is (κ, λ)-continuous. then f is a long homotopy from f to g. if for some x0 ∈ x and y0 ∈ y we have f(x0, t) = y0 for all t ∈ n ∗, we say f is a pointed long homotopy. we write f ≃l κ,λ g, or f ≃l g when the adjacencies κ and λ are understood, to indicate that f and g are long homotopic functions. ✷ we have the following. theorem 6.6. let fi, gi : (xi, κi) → (yi, λi) be continuous functions that are long homotopic, 1 ≤ i ≤ v. then f = πvi=1fi and g = π v i=1gi are long homotopic maps from (πvi=1xi, npv(κ1, . . . , κv)) to (π v i=1yi, npv(λ1, . . . , λv)). if the long homotopies fi ≃ l gi are pointed with respect to xi ∈ xi and yi ∈ yi, then the long homotopy f ≃l g is pointed with respect to (x1, . . . , xv) ∈ π v i=1xi and (y1, . . . , yv) ∈ π v i=1yi. proof. we give a proof for the unpointed assertion. minor modifications yield a proof for the pointed assertion. let hi : xi × z → yi be a long homotopy from fi to gi. for all xi ∈ xi, there exists nfi,xi ∈ n such that t ≤ −nfi,xi implies hi(xi, t) = fi(xi) and t ≥ nfi,xi implies hi(xi, t) = gi(xi). for all x = (x1, . . . , xv) ∈ π v i=1xi, let nx = max{nfi,xi | 1 ≤ i ≤ v}. let h = π v i=1hi : π v i=1xi × z → π v i=1yi. then t ≤ −nx implies h(x, t) = f(x) and t ≥ nx implies h(x, t) = g(x). for all x ∈ πvi=1xi, the induced function hx(t) = (hi(x1, t), . . . , hv(xv, t)) is (c1, npv(λ1, . . . , λv))-continuous, by an argument similar to that given in the proof of theorem 6.1. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 414 generalized normal product adjacency in digital topology for all t ∈ z, the induced function ht(x) = (hi(x1, t), . . . , hv(xv, t)) is (npv(κ1, . . . , κv), npv(λ1, . . . , λv))-continuous, by an argument similar to that given in the proof of theorem 6.1. the assertion follows. � definition 6.7 ([17]). let f : (x, κ) → (y, λ) and g : (y, λ) → (x, κ) be continuous functions. suppose g◦f ≃l 1x and f◦g ≃ l 1y . then we say (x, κ) and (y, λ) have the same long homotopy type, denoted x ≃l κ,λ y or simply x ≃l y . if there exist x0 ∈ x and y0 ∈ y such that f(x0) = y0, g(y0) = x0, the long homotopy g ◦ f ≃l 1x holds x0 fixed, and the long homotopy f ◦ g ≃ l 1y holds y0 fixed, then (x, x0, κ) and (y, y0, λ) have the same pointed long homotopy type, denoted (x, x0) ≃ l κ,λ (y, y0) or (x, x0) ≃ l (y, y0). ✷ theorem 6.8. let xi ≃ l κi,λi yi, 1 ≤ i ≤ v. let x = π v i=1xi, y = π v i=1yi. then x ≃l npv(κ1,...,κv),npv(λ1,...,λv) y . if for each i the long homotopy equivalence xi ≃ l κi,λi yi is pointed with respect to xi ∈ xi and yi ∈ yi, then the long homotopy equivalence x ≃l npv(κ1,...,κv),npv(λ1,...,λv) y is pointed with respect to x0 = (x1, . . . , xv) ∈ x and y0 = (y1, . . . , yv) ∈ y . proof. this follows easily from definition 6.7 and theorem 6.6. � 6.4. real homotopy type. definition 6.9 ([17]). let (x, κ) be a digital image, and [0, 1] ⊂ r be the unit interval. a function f : [0, 1] → x is a real [digital] [κ-]path in x if: • there exists ǫ0 > 0 such that f is constant on (0, ǫ0) with constant value equal or κ-adjacent to f(0), and • for each t ∈ (0, 1) there exists ǫt > 0 such that f is constant on each of the intervals (t − ǫt, t) and (t, t + ǫt), and these two constant values are equal or κ-adjacent, with at least one of them equal to f(t), and • there exists ǫ1 > 0 such that f is constant on (1 − ǫ1, 1) with constant value equal or κ-adjacent to f(1). if t = 0 and f(0) 6= f((0, ǫ0)), or 0 < t < 1 and the two constant values f((t − ǫt, t)) and f((t, t + ǫt)) are not equal, or t = 1 and f(1) 6= f((1 − ǫ1, 1)), we say t is a jump of f. proposition 6.10 ([17]). let p, q ∈ (x, κ). let f : [a, b] → x be a real κ-path from p to q. then the number of jumps of f is finite. definition 6.11 ([17]). let (x, κ) and (y, κ′) be digital images, and let f, g : x → y be (κ, κ′) continuous. then a real [digital] homotopy of f and g is a function f : x × [0, 1] → y such that: • for all x ∈ x, f(x, 0) = f(x) and f(x, 1) = g(x) • for all x ∈ x, the induced function fx : [0, 1] → y defined by fx(t) = f(x, t) for all t ∈ [0, 1] is a real κ-path in x. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 415 • for all t ∈ [0, 1], the induced function ft : x → y defined by ft(x) = f(x, t) for all x ∈ x is (κ, κ′)–continuous. if such a function exists we say f and g are real homotopic and write f ≃r g. if there are points x0 ∈ x and y0 ∈ y such that f(x0, t) = y0 for all t ∈ [0, 1], we say f and g are pointed real homotopic. definition 6.12 ([17]). we say digital images (x, κ) and (y, κ′) have the same real homotopy type, denoted x ≃rκ,κ′ y or x ≃ r y when κ and κ′ are understood, if there are continuous functions f : x → y and g : y → x such that g ◦ f ≃r 1x and f ◦ g ≃ r 1y . if there exist x0 ∈ x and y0 ∈ y such that f(x0) = y0, g(y0) = x0, and the real homotopies above are pointed with respect to x0 and y0, we say x and y have the same pointed real homotopy type, denoted (x, x0) ≃ r κ,κ′ (y, y0) or (x, x0) ≃ r (y, y0). theorem 6.13. suppose (6.3) xi ≃ r κi,λi yi for 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi. then (6.4) x ≃rnpv(κ1,...,κv),npv(λ1,...,λv) y. if the equivalences (6.3) are all pointed with respect to xi ∈ xi and yi ∈ yi, then the equivalence (6.4) is pointed with respect to x0 = (x1, . . . , xv) ∈ x and y0 = (y1, . . . , yv) ∈ y . proof. we give a proof for the unpointed assertion. with minor modifications, the same argument yields the pointed assertion. by hypothesis, there exist continuous functions fi : xi → yi, gi : yi → xi and real homotopies hi : xi×[0, 1] → xi from gi ◦fi to 1xi, ki : yi ×[0, 1] → xi from fi ◦ gi to 1yi. let f = πvi=1fi : x → y . let g = π v i=1gi : y → x. for x = (x1, . . . , xv) ∈ x with xi ∈ xi, let h : x × [0, 1] → x be defined by h(x, t) = (h1(x1, t), . . . , hv(xv, t)). then h(x, 0) = g ◦ f(x) and h(x, 1) = x. for x ∈ x, the induced function hx has jumps only at the finitely many (by proposition 6.10) jumps of the functions hi. it follows that hx is a real npv(λ1, . . . , λv)-path in y . let x′ = (x′1, . . . , x ′ v) be npv(κ1, . . . , κv)-adjacent to x in x. then, for any t ∈ [0, 1], ht(x ′) = (h1(x ′ 1, t), . . . , hv(x ′ v, t)) is npv(λ1, . . . , λv)-adjacent to ht(x) = (h1(x1, t), . . . , hv(xv, t)), since each hi is a real homotopy. hence ht is continuous. thus, h is a real homotopy from g ◦ f to 1x. a similar argument lets us conclude that f ◦ g ≃r 1y . therefore, x ≃ r y . � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 416 generalized normal product adjacency in digital topology 7. npv and retractions definition 7.1 ([2, 3]). let y ⊂ (x, κ). a (κ, κ)-continuous function r : x → y is a retraction, and a is a retract of x, if r(y) = y for all y ∈ y . � theorem 7.2. let ai ⊂ (xi, κi), i ∈ {1, . . . , v}. then ai is a retract of xi for all i if and only if πvi=1ai is a retract of (π v i=1xi, npv(κ1, . . . , κv)). proof. suppose, for all i, ai is a retract of xi. let ri : xi → ai be a retraction. then, by theorem 4.1, πvi=1ri : π v i=1xi → π v i=1ai is continuous, and therefore is easily seen to be a retraction. suppose there is a retraction r : πvi=1xi → π v i=1ai. we construct retractions rj : xj → aj as follows. let ai ∈ ai. define ij : xj → π v i=1xi by ij(x) = (a1, . . . , aj−1, x, aj+1, . . . , av). clearly, ij is continuous. then rj = pj ◦ r ◦ ij is continuous, by theorem 2.4 and corollary 4.3, and is easily seen to be a retraction. � let a ⊂ (x, κ). we say a is a deformation retract of x if there is a κ-homotopy h : x × [0, m]z → x from 1x to a retraction of x to a. if h(a, t) = a for all (a, t) ∈ y × [0, m]z, we say h is a strong deformation and a is a strong deformation retract of x. we have the following. theorem 7.3. let ai ⊂ (xi, κi), i ∈ {1, . . . , v}. then ai is a (strong) deformation retract of xi for all i if and only if a = π v i=1ai is a (strong) deformation retract of x = (πui=1xi, npv(κ1, . . . , κv)). proof. suppose ai is a deformation retract of xi, 1 ≤ i ≤ v. it follows from theorems 6.1 and 7.2 and that a is a deformation retract of x. if each ai is a strong deformation retract of xi, then by using the argument in the proof of theorem 6.1 we can construct a homotopy from 1x to a retraction of x to a that holds every point of a fixed, so a is a strong deformation retract of x. suppose a is a (strong) deformation retract of x. this means there is a homotopy h : x×[0, m]z → x from 1x to a retraction r of x onto a (such that h(a, t) = a for all (a, t) ∈ a × [0, m]z). let ii : xi → x be as in theorem 7.2. let hi : xi × [0, m]z → xi be defined by hi(x, t) = pi(h(ii(x), t)). then hi is a homotopy between pi ◦ ii = 1xi and pi ◦ r ◦ ii (such that hi(ai, t) = ai for all ai ∈ ai). since pi ◦ r ◦ ii(xj) ⊂ pi ◦ r(x) = pi(a) = ai and a ∈ ai implies pi ◦ r ◦ ii(a) = a, pi ◦ r ◦ ii is a retraction. thus, ai is a (strong) deformation retract of xi. � 8. npv and the digital borsuk-ulam theorem the borsuk-ulam theorem of euclidean topology states that if f : sn → rn is a continuous function, where rn is n-dimensional euclidean space and sn is c© agt, upv, 2017 appl. gen. topol. 18, no. 2 417 the unit sphere in rn+1, i.e., sn = {(x1, . . . , xn+1) ∈ r n+1 | n+1 ∑ i=1 x2i = 1}, then there exists x ∈ sn such that f(−x) = f(x). a “layman’s example” of this theorem for n = 2 is that there are opposite points x, −x on the earth’s surface with the same temperature and the same barometric pressure. we say a set x ⊂ zn is symmetric with respect to the origin if for every x ∈ x, −x ∈ x. the assertion analogous to the borsuk-ulam theorem is not generally true in digital topology. an example is given in [7] of a continuous function f : s → z from a simple closed curve s ⊂ (z2, c2), a digital analog of s 1, into the digital line z, with s symmetric with respect to the origin, such that f(x) 6= f(−x) for all x ∈ s. however, the papers [7, 34] give conditions under which a continuous function f from a digital version sn of s n to zn must have a point x ∈ sn for which f(x) and f(−x) are equal or adjacent. in particular, [34] uses the “boundary” of a digital box as a digital model of a euclidean sphere. let bn = π n i=1[−ai, ai]z, for ai ∈ n. let δbn = n ⋃ i=1 {(x1, . . . , xn) ∈ bn | xi ∈ {−ai, ai}}. theorem 8.1. we have the following. • [7] let (s, κ) be a digital simple closed curve in zn such that s is symmetric with respect to the origin. let f : s → z be a (κ, c1)continuous function. then for some x ∈ s, f(x) and f(−x) are equal or c1-adjacent, i.e., |f(x) − f(−x)| ≤ 1. • [34] let u ∈ {1, n − 1} and letf : δbn → z n−1 be a (cn, cu)-continuous function. then for some x ∈ δbn, f(x) and f(−x) are equal or cuadjacent. � notice that for xi ⊂ z ni, πvi=1xi is symmetric with respect to the origin of z ∑ v i=1 ni if and only if xi is symmetric with respect to the origin of z ni for all indices i. suppose m, n ∈ n, 1 ≤ m ≤ n. let’s say a digital image s ⊂ zn+1 that is symmetric with respect to the origin has the (m, κ, λ)-borsuk-ulam property if for every (κ, λ)-continuous function f : s → zm there exists x ∈ x such that f(x) and f(−x) are equal or λ-adjacent in zn. we have the following. theorem 8.2. suppose • v > 1, • si ⊂ z ni+1 is symmetric with respect to the origin of zni+1 for 1 ≤ i ≤ v, and c© agt, upv, 2017 appl. gen. topol. 18, no. 2 418 generalized normal product adjacency in digital topology • πvi=1si has the (m, npv(κ1, . . . , κv), npv(λ1, . . . , λv))-borsuk-ulam property for some adjacencies κi for z ni+1 and λi for z ni, where m = ∑v i=1 ni. then, for all i, si has the (ni, κi, λi)-borsuk-ulam property. proof. notice πvi=1si ⊂ z m+v. let f : πvi=1si → z m be any function that is (npv(κ1, . . . , κv), npv(λ1, . . . , λv))-continuous. by hypothesis, there exists x ∈ πvi=1si such that f(x) and f(−x) are equal or npv(λ1, . . . , λv)-adjacent. in particular, we can let f be the product of arbitrary continuous functions fi : si → z ni, since if fi : si → z ni is (κi, λi)-continuous, then by theorem 4.1, f = πvi=1fi is (npv(κ1, . . . , κv), npv(λ1, . . . , λv))-continuous. therefore, there exists x = (x1, . . . , xv) ∈ x where xi ∈ si such that f(x) = (f1(x1), . . . , fv(xv)) and f(−x) = (f1(−x1), . . . , fv(−xv)) are equal or are npv(λ1, . . . , λv)-adjacent. hence, for all indices i, fi(xi) and fi(−xi) are equal or λi-adjacent. since the fi were arbitrarily chosen, the assertion follows. � 9. npu and the approximate fixed point property in both topology and digital topology, • a fixed point of a continuous function f : x → x is a point x ∈ x satisfying f(x) = x; • if every continuous f : x → x has a fixed point, then x has the fixed point property (fpp). however, a digital image x has the fpp if and only if x has a single point [10]. therefore, it turns out that the approximate fixed point property is more interesting for digital images. definition 9.1 ([10]). a digital image (x, κ) has the approximate fixed point property (afpp) if every continuous f : x → x has an approximate fixed point, i.e., a point x ∈ x such that f(x) is equal or κ-adjacent to x. � a number of results concerning the afpp were presented in [10], including the following. theorem 9.2 ([10]). suppose (x, κ) has the afpp. let h : x → y be a (κ, λ)-isomorphism. then (y, λ) has the afpp. � theorem 9.3 ([10]). suppose y is a retract of (x, κ). if (x, κ) has the afpp, then (y, κ) has the afpp. � the following is a generalization of theorem 5.10 of [10]. theorem 9.4. let (xi, κi) be digital images, 1 ≤ i ≤ v. then for any u ∈ z such that 1 ≤ u ≤ v, if (πvi=1xi, npu(κ1, . . . , κv)) has the afpp then (xi, κi) has the afpp for all i. proof. let x = (πvi=1xi, npu(κ1, . . . , κv)). suppose x has the afpp. let xi ∈ xi. let x′1 = x1 × {(x2, . . . , xv)}, c© agt, upv, 2017 appl. gen. topol. 18, no. 2 419 x′i = {(x1, . . . , xi−1)} × xi × {(xi+1, . . . , xv)} for 2 ≤ i < v, x′v = {(x1, . . . , xv−1)} × xv . clearly, each x′i is a retract of x and is isomorphic to xi. by theorems 9.2 and 9.3, xi has the afpp. � 10. npv and fundamental groups several versions of the fundamental group for digital images exist in the literature, including those of [35, 27, 4, 16]. in this paper, we use the version of [4], which was shown in [16] to be equivalent to the version developed in the latter paper. other papers cited in this section use the version of the digital fundamental group presented in [4]. the author of [25] attempted to study the fundamental group of a cartesian product of digital simple closed curves. errors of [25] were corrected in [13]. the notion of a covering map [25] is often useful in computing the fundamental group. the following is a somewhat simpler characterization of a covering map than that given in [25]. theorem 10.1 ([6]). let (e, κ) and (b, λ) be digital images. let g : e → b be a (κ, λ)-continuous surjection. then g is a (κ, λ)-covering map if and only if for each b ∈ b, there is an index set m such that • g−1(n∗λ(b, 1, b)) = ⋃ i∈m n ∗ κ(ei, 1, e) where ei ∈ g −1(b); • if i, j ∈ m and i 6= j then n∗κ(ei, 1, e) ∩ n ∗ κ(ej, 1, e) = ∅; and • the restriction map g|n∗ κ (ei,1,e) : n ∗ κ(ei, 1, e) → n ∗ λ (b, 1, b) is a (κ, λ)isomorphism for all i ∈ m. � example 10.2 ([25]). let c ⊂ zn be a simple closed κ-curve, as realized by a (c1, κ)-continuous surjection f : [0, m − 1]z → c such that f(0) and f(m − 1) are κ-adjacent. define g : z → c by g(z) = f(z mod m). then g is a covering map. � proposition 10.3 ([13]). suppose for i ∈ {1, 2}, gi : ei → bi is a (κi, λi)covering map. then g1×g2 : e1×e2 → b1×b2 is a (np2(κ1, κ2), np2(λ1, λ2))covering map. � corollary 10.4. suppose for i ∈ {1, . . . , v}, gi : ei → bi is a (κi, λi)-covering map. then πvi=1gi : π v i=1ei → π v i=1bi is a (npv(κ1, . . . , κv), npv(λ1, . . . , λv))covering map. � proof. this follows from propositions 10.3 and 3.4, and theorem 10.1. � a digital image with the homotopy type of a single point is called contractible. for the following theorem, it is useful to know that a digital simple closed curve s is not contractible if and only if |s| > 4 [4, 7]. theorem 10.5 ([29, 4, 25]). let s ⊂ (zn, κ) be a digital simple closed κcurve that is not contractible. let s0 ∈ s. then the fundamental group of s is πκ1(s, s0) ≈ z. � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 420 generalized normal product adjacency in digital topology the following theorem was discussed in [25], but the argument given for it in [25] had errors. a correct proof was given in [13]. theorem 10.6 ([13]). let si ⊂ (z ni, cni), for i ∈ {1, 2}, be a noncontractible digital simple closed curve. let si ∈ si. then the fundamental group π cn1+n2 1 (s1 × s2, (s1, s2)) ≈ z 2. � � the significance of the adjacency cn1+n2 in the proof of theorem 10.6 is that, per theorem 3.3, np(cn1, cn2) = cn1+n2. thus, trivial modifications of the proof given in [13] for theorem 10.6 yield the following generalization. theorem 10.7. for i ∈ {1, . . . , v}, let si ⊂ (z ni, κi) be a noncontractible digital simple closed curve. let si ∈ si. then the fundamental group π npv(κ1,...,κv) 1 (π v i=1si, (s1, . . . , sv)) ≈ z v. � many results concerning digital covering maps depend on the radius 2 local isomorphism property (e.g., [24, 6, 11, 12, 13, 7, 14]). we have the following. definition 10.8 ([24]). let n ∈ n. a (κ, λ)-covering (e, p, b) is a radius n local isomorphism if, for all i ∈ m, the restriction map p|n∗κ(ei,n) : n ∗ κ(ei, n) → n∗λ(bi, n) is an isomorphism, where ei, bi, m are as in theorem 10.1. lemma 10.9. let xi ∈ (xi, κi). then n∗npv(κ1,...,κv)((x1, . . . , xn), n) = π v i=1n ∗ κi (xi, n). proof. let x = (x1, . . . , xv). let y ∈ n ∗ npv(κ1,...,κv) (x, n). for some m ≤ n, there is a path {yi} m i=0 from x to y. let yi = (yi,1, . . . , yi,v) where yi,j ∈ xi. since yi and yi+1 are npv(κ1, . . . , κv)-adjacent, yi,j and yi,j+1 are equal or κi-adjacent. therefore, {yi,j} m j=1 is a κi path in xi from yi,0 to yi,m. hence n∗ npv(κ1,...,κv) ((x1, . . . , xn), n) ⊂ π v i=1n ∗ κ1 (xi, n). let y = (y1, . . . , yv) ∈ π v i=1n ∗ κ1 (xi, n). for each i and for some mi ≤ n, there is a κi-path pi = {yi,j} mi j=1 from xi to yi. there is no loss of generality in assuming mi = n, since we can take pi = {yi,j} n j=1 where yi,j = yi,mi for mi ≤ j ≤ n. then for each i < n, y′i = (yi,1, . . . , yi,v) and y ′ i+1 = (yi+1,1, . . . , yi+1,v) are equal or npv(κ1, . . . , κv)-adjacent. then {y ′ i} n i=1 is an npv(κ1, . . . , κv)path from x to y. thus, πvi=1nκ1(xi, n) ⊂ nnpv(κ1,...,κv)(x, n). the assertion follows. � theorem 10.10. for 1 ≤ i ≤ v, let pi : (ei, κi) → (bi, λi) be continuous and let n ∈ n. if (ei, pi, bi) is a covering and a radius n local isomorphism for all i, then the product function πvi=1pi : π v i=1ei → π v i=1bi is a (npv(κ1, . . . , κv), npv(λ1, . . . , λv)) covering map that is a radius n local isomorphism. proof. this follows from corollary 10.4 and lemma 10.9. � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 421 11. npv and multivalued functions we study properties of multivalued functions that are preserved by npv. 11.1. weak and strong continuity. theorem 11.1. let fi : (xi, κi) ⊸ (yi, λi) be multivalued functions for 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, and f = π v i=1fi : (x, npv(κ1, . . . , κv)) ⊸ (y, npv(λ1, . . . , λv)). then f has weak continuity if and only if each fi has weak continuity. proof. let xi and x ′ i be κi-adjacent or equal in xi. then x = (x1, . . . , xv) and x′ = (x′1, . . . , x ′ v) are npv(κ1, . . . , κv)-adjacent or equal in x. the multivalued function f has weak continuity ⇔ for x, x′ as above, f(x) and f(x′) are npv(λ1, . . . , λv)-adjacent subsets of y ⇔ for each i and for all xi, x ′ i as above, fi(xi) and fi(x ′ i) are λi-adjacent subsets of yi ⇔ for each i, fi has weak continuity. � theorem 11.2. let fi : (xi, κi) ⊸ (yi, λi) be multivalued functions for 1 ≤ i ≤ v. let x = πvi=1xi, y = π v i=1yi, and f = π v i=1fi : (x, npv(κ1, . . . , κv)) ⊸ (y, npv(λ1, . . . , λv)). then f has strong continuity if and only if each fi has strong continuity. proof. let xi and x ′ i be κi-adjacent or equal in xi. then x = (x1, . . . , xv) and x′ = (x′1, . . . , x ′ v) are npv(κ1, . . . , κv)-adjacent or equal in x. the multivalued function f has strong continuity ⇔ for x, x′ as above, every point of f(x) is npv(λ1, . . . , λv)-adjacent or equal to a point of f(x ′) and every point of f(x′) is npv(λ1, . . . , λv)-adjacent or equal to a point of f(x) ⇔ for each i and for all xi, x ′ i as above, every point of fi(xi) is λi-adjacent or equal to a point of fi(x ′ i) and every point of fi(x ′ i) is λi-adjacent or equal to a point of fi(xi) ⇔ for each i, fi has strong continuity. � 11.2. continuous multifunctions. lemma 11.3. let x ⊂ zm, y ⊂ zn. let f : (x, ca) ⊸ (y, cb) be a continuous multivalued function. let f : (s(x, r), ca) → (y, cb) be a continuous function that induces f. let s ∈ n. then there is a continuous function fs : (s(x, rs), ca) → (y, cb) that induces f. proof. given a point x = (x1, . . . , xm) ∈ s(x, rs), there is a unique point i(x) = x′ = (x′1, . . . , x ′ m) ∈ s(x, r) such that x ′ “contains” x in the sense that the fractional part of each component of x, xi − ⌊xi⌋, “truncates” to the fractional part of the corresponding component of x′, x′i − ⌊x ′ i⌋, i.e., x′i − ⌊x ′ i⌋ ≤ xi − ⌊xi⌋ < x ′ i − ⌊x ′ i⌋ + 1/r. (see figure 2.) define fs(x) = f(i(x)). we must show fs is a continuous multivalued function that induces f . if x, x′ are ca-adjacent in s(x, rs), then one sees easily that i(x) and i(x ′) are c© agt, upv, 2017 appl. gen. topol. 18, no. 2 422 generalized normal product adjacency in digital topology figure 2. the digital image x = [0, 2]z × [0, 1]z with its partitions s(x, 2) with member coordinates on heavy lines, and s(x, 6) with member coordinates on both heavy and light lines. in the notation used in the proof of lemma 11.3, we have, e.g., i(7/6, 2/3) = (1, 1/2). equal or ca-adjacent in s(x, r). hence fs(x) = f(i(x)) and fs(x ′) = f(i(x′)) are equal or cb-adjacent in y . thus, fs is continuous. for w ∈ x we have f(w) = ⋃ y∈e −1 r (w) f(y) = ⋃ u∈e −1 rs (w) fs(u). therefore, f induces f . � for multivalued functions fi : (xi, κi) ⊸ (yi, λi), 1 ≤ i ≤ v, define the product multivalued function πvi=1fi : (π v i=1xi, npv(κ1, . . . , κv)) ⊸ (π v i=1yi, npv(λ1, . . . , λv)) by (πvi=1fi)(x1, . . . , xv) = π v i=1fi(xi). theorem 11.4. given multivalued functions fi : (xi, cai) ⊸ (yi, cbi), 1 ≤ i ≤ v, if each fi is continuous then the product multivalued function πvi=1fi : (π v i=1xi, npv(ca1, . . . , cav)) ⊸ (π v i=1yi, npv(cb1, . . . , cbv )) is continuous. proof. if each fi is continuous, there exists a continuous fi : (s(xi, ri), cai) → (yi, cbi) that generates fi. by lemma 11.3, we may assume that all the ri are c© agt, upv, 2017 appl. gen. topol. 18, no. 2 423 equal. thus, for some positive integer r, we have fi : (s(xi, r), cai) → (yi, cbi) generating fi. by theorem 4.1, the product multivalued function πvi=1fi : (π v i=1s(xi, r), npv(ca1, . . . , cav )) → (π v i=1(yi, npv(cb1, . . . , cbv )) is continuous. it is clear that this function generates the multivalued function πvi=1fi. � the paper [20] has several results concerning the following notions. definition 11.5 ([20]). let (x, κ) ⊂ zn be a digital image and y ⊂ x. we say that y is a κ-retract of x if there exists a κ-continuous multivalued function f : x ⊸ y (a multivalued κ-retraction) such that f(y) = {y} if y ∈ y . if moreover f(x) ⊂ n∗cn(x) for every x ∈ x \ y , we say that f is a multivalued (n, κ)-retraction, and y is a multivalued (n, κ)-retract of x. we generalize theorem 7.2 as follows. theorem 11.6. for 1 ≤ i ≤ v, let ai ⊂ (xi, κi) ⊂ z ni. suppose fi : xi ⊸ ai is a continuous multivalued function for all i. then fi is a multivalued retraction for all i if and only if f = πvi=1fi : π v i=1xi ⊸ π v i=1ai is a multivalued npv(κ1, . . . , κv)-retraction. further, fi is an (n, κi)-retraction for all i if and only if f is a multivalued (n, npv(κ1, . . . , κv))-retraction. proof. let x = πvi=1xi, a = π v i=1ai. suppose each fi is a multivalued retraction. by theorem 11.4, the product multivalued function f is continuous. clearly, f(x) ⊂ a. also, given a = (a1, . . . , av) ∈ a, we have f(a) = πvi=1fi(ai) = π v i=1{ai} = {a}. therefore, f(x) = a, and f is a multivalued retraction. conversely, suppose f is a multivalued retraction. by theorem 11.4, each fi is continuous. also, since f(x) = a, we must have, for each i, fi(xi) = ai, and since f is a retraction, fi(a) = {a} for a ∈ ai. therefore, fi is a multivalued retraction. further, from lemma 10.9, for x = (x1, . . . , xv) ∈ x, n ∗ npv(cn1 ,...,cnv ) (x) = πvi=1ncni (xi). it follows that fi is an (n, κi)-retraction for all i if and only if f is a multivalued (n, npv(κ1, . . . , κv))-retraction. � 11.3. connectivity preserving multifunctions. theorem 11.7. let fi : (xi, κi) ⊸ (yi, λi) be a multivalued function between digital images, 1 ≤ i ≤ v. then the product map πvi=1fi : (π v i=1xi, npv(κ1, . . . , κv)) ⊸ (π v i=1yi, npv(λ1, . . . , λv)) is a connectivity preserving multifunction if and only if each fi is a connectivity preserving multifunction. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 424 generalized normal product adjacency in digital topology proof. let x = πvi=1xi, y = π v i=1yi, f = π v i=1fi : x ⊸ y . assume x = (x1, . . . , xv), x ′ = (x′1, . . . , x ′ v) with xi, x ′ i ∈ xi. using theorem 2.12, we argue as follows. f is connectivity preserving ⇔ • for every x ∈ x, f(x) = πvi=1fi(xi) is a connected subset of y , and • for adjacent x, x′ ∈ x, f(x) = πvi=1fi(xi) and f(x ′) = πvi=1fi(x ′ i) are adjacent subsets of y . ⇔ • for every xi ∈ xi, fi(x) is a connected subset of yi, and • for adjacent xi, x ′ i ∈ xi, fi(xi) and fi(x ′ i) are adjacent subsets of yi. ⇔ each fi is connectivity preserving. � 12. npv and shy maps the following generalizes a result of [9]. theorem 12.1. let fi : (xi, κi) → (yi, λi) be a continuous surjection between digital images, 1 ≤ i ≤ v. then the product map f = πvi=1fi : (π v i=1xi, npv(κ1, . . . , κv)) → (π v i=1yi, npv(λ1, . . . , λv)) is shy if and only if each fi is a shy map. proof. suppose the product map is shy. since fi = pi ◦ f ◦ ii, where ii is the continuous injection of the proof of theorem 7.2, it follows from theorems 2.4 and 4.3 that fi is continuous. also, since f is surjective, fi must be surjective. let y ′i be a λi-connected subset of yi. by theorem 5.1, π v i=1y ′ i is connected in πvi=1yi. since the product map is shy, we have from theorem 2.28 that x′ = f−1(πvi=1y ′ i ) = π v i=1f −1 i (y ′ i ) is npv(κ1, . . . , κv)-connected. then f −1 i (y ′ i ) = pi(x ′) is κi-connected. from theorem 2.28, it follows that fi is shy. conversely, suppose each fi is shy. by theorem 4.1, the product map π v i=1fi is continuous, and it is easily seen to be surjective. let yi ∈ yi. then (π v i=1fi) −1(y1, . . . , yv) = π v i=1f −1 i (yi) is connected, by definition 2.27 and theorem 5.1. let yi, y ′ i be λi-adjacent in yi, and let y = (y1, . . . , yv), y ′ = (y′1, . . . , y ′ v). then y and y′ are adjacent in y , and (πvi=1fi) −1({y, y′}) = πvi=1f −1 i ({yi, y ′ i}) is connected, by definition 2.27 and theorem 5.1. thus, by definition 2.27, πvi=1fi is shy. � the statement analogous to theorem 12.1 is not generally true if cu-adjacencies are used instead of normal product adjacencies, as shown in the following. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 425 example 12.2. recall example 4.2, in which x = {(0, 0), (1, 0)} ⊂ z2, y = {(0, 0), (1, 1)} ⊂ z2. there is a (c1, c2)-isomorphism f : x → y . consider x′ = x × {0} ⊂ z3, y ′ = y × {0} ⊂ z3. although the maps f and 1{0} are, respectively, (c1, c2)and (c1, c1)-isomorphisms and therefore are, respectively, (c1, c2)and (c1, c1)-shy, the product map f ×1{0} : x ′ → y ′ is not (c1, c1)-shy, by theorem 2.28, since, as observed in example 4.2, x′ is c1-connected and y ′ is not c1-connected. � 13. further remarks we have studied adjacencies that are extensions of the normal product adjacency for finite cartesian products of digital images. we have shown that such adjacencies preserve many properties for finite cartesian products of digital images that, in some cases, are not preserved by the use of the cu-adjacencies most commonly used in the literature of 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[37] j. h. van lint and r. m. wilson, a course in combinatorics, cambridge university press, new york, 1992 c© agt, upv, 2017 appl. gen. topol. 18, no. 2 427 () @ appl. gen. topol. 19, no. 1 (2018), 9-20doi:10.4995/agt.2018.7061 c© agt, upv, 2018 topological n-cells and hilbert cubes in inverse limits leonard r. rubin department of mathematics, university of oklahoma, norman, oklahoma 73019, usa (lrubin@ou.edu) communicated by e. a. sánchez-pérez abstract it has been shown by s. mardešić that if a compact metrizable space x has dim x ≥ 1 and x is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then x must contain an arc. we are going to prove that if x = (|ka|, p b a, (a, �)) is an inverse system in set theory of triangulated polyhedra |ka| with simplicial bonding functions pba and x = lim x, then there exists a uniquely determined sub-inverse system xx = (|la|, p b a ∣ ∣|lb|, (a, �)) of x where for each a, la is a subcomplex of ka, each p b a ∣ ∣|lb| : |lb| → |la| is surjective, and lim xx = x. we shall use this to generalize the mardešić result by characterizing when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain a topological n-cell and do the same in the case of an inverse system of finite triangulated polyhedra with simplicial bonding maps. we shall also characterize when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain an embedded copy of the hilbert cube. in each of the above settings, all the polyhedra have the weak topology or all have the metric topology (these topologies being identical when the polyhedra are finite). 2010 msc: 54b35; 54c25; 54f45. keywords: hilbert cube; inverse limit; inverse sequence; inverse system; polyhedron; simplicial inverse system; simplicial map; topological n-cell; triangulation. received 29 december 2016 – accepted 01 september 2017 http://dx.doi.org/10.4995/agt.2018.7061 l. r. rubin 1. introduction theorem 4.10.10 of [10] reads as follows. theorem 1.1. every completely metrizable space x is homeomorphic to the inverse limit of an inverse sequence (|ki|m, p i+1 i ) of metric polyhedra and pl maps such that each ki is locally finite-dimensional, card ki ≤ wt x, and each bonding map pi+1i : |ki+1|m → |ki|m is simplicial for some admissible subdivision k′i of ki, where admissibility guarantees the continuity of p i+1 i : |ki+1|m → |ki|m. the notion of locally finite-dimensional used in theorem 1.1 goes this way. let k be a simplicial complex. whenever v is a vertex of k, then st(v, k) will be the closed star of v in k, which is the subcomplex of k consisting of the simplexes of k having v as a vertex and all faces of such simplexes. then k is called locally finite-dimensional if dim(st(v, k)) < ∞ for each v ∈ k(0). one might wonder if an inverse sequence such as that in theorem 1.1 could be designed so that all the bonding maps1 are simplicial with respect to the given triangulations; unfortunately this is not the case. it was shown by s. mardešić in theorem 2.1 of [7], that if a compact metrizable space x has dim x ≥ 1 and x is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then x must contain an arc. since pseudo-arcs (see [8]) are metrizable compacta with dim ≥ 1 that contain no arcs, then he was able to obtain corollary 2.2 of [7], which says that there exist metrizable compacta that cannot be written as the limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps. the proof of theorem 2.1 of [7] is given without the assumption that the bonding maps are surjective, but if they were, then by an observation of m. levin, its proof would be trivial. the question of whether a given metrizable compactum could be written as the limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps arose from our research in [9]. there we were able to find, for the sake of extension theory, a “substitute” z for any given compact metrizable space x. this metrizable compactum z is represented as the limit of an inverse sequence of finite triangulated polyhedra in such a manner that all the bonding maps are simplicial with respect to these triangulations. since the process of determining such a z was complex, we were concerned to know if it was necessary, that is, could we represent the given x “simplicially” from the outset; the result of [7] made it apparent that we could not escape such a complication. we shall demonstrate, proposition 2.7, that if x = (|ka|, p b a, (a, �)) is an inverse system in set theory of triangulated polyhedra |ka| with simplicial bonding functions pba, and x = lim x, then there exists a uniquely determined sub-inverse system xx = (|la|, p b a ∣ ∣|lb|, (a, �)) of x where for each a, la is a subcomplex of ka, each p b a ∣ ∣|lb| : |lb| → |la| is surjective, and lim xx = x. 1in this paper map means continuous function. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 10 topological n-cells and hilbert cubes in inverse limits hence for such a “simplicial” inverse system in which the polyhedra |ka| are given either the cw (weak) topology or the metric topology m, one may as well assume for topological purposes that the bonding functions are surjective maps. in corollary 3.3 we will characterize when the limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain a topological n-cell. in proposition 3.5, we display a similar characterization in case we are dealing with an inverse system of finite polyhedra and simplicial bonding maps. our theorem 4.13 characterizes when the limit of an inverse sequence of triangulated polyhedra and simplicial bonding maps must contain a copy of the hilbert cube i∞. we were not successful in obtaining such a result for inverse systems even in the case that the coordinate spaces are finite polyhedra. in section 5 we shall provide what we could do for such systems. 2. simplicial inverse systems let k be a simplicial complex. then by |k|cw we mean the polyhedron |k| with the cw-topology (sometimes called the weak topology) and by |k|m we mean |k| with the metric topology m.2 if k is finite, then the cw-topology is the same as the metric topology m, so we usually just write |k| with no subscript. in case l is a simplicial complex and f : k → l is a simplicial function, then f induces a function |f| : |k| → |l| which we say is simplicial from |k| to |l|. in this setting we usually just write f instead of |f|; moreover, one has that both f : |k|cw → |l|cw and f : |k|m → |l|m are maps. we shall be concerned with inverse systems x = (xa, p b a, (a, �)) with a directed set (a, �) as indexing set. if x = lim x, then pa : x → xa will denote the a-coordinate projection. for x ∈ x, we shall typically write pa(x) = xa, and denote x = (xa)a∈a or just x = (xa). if for each a ∈ a, ya ⊂ xa and whenever a � b, pba(yb) ⊂ ya, then we call y = (ya, p b a|yb, (a, �)) a sub-inverse system of x. clearly lim y ⊂ lim x. in case (a, �) is (n, ≤), we simply denote the inverse system x = (xi, p i+1 i ) and call it an inverse sequence. the main result of this section is proposition 2.7. it shows that if x is the inverse limit of an inverse system in set theory of triangulated polyhedra and simplicial maps, then there is a sub-inverse system consisting of subpolyhedra determined by subcomplexes of the given triangulations such that the limit of this sub-inverse system is x and that the restricted, and hence simplicial, maps are surjective. definition 2.1. let x = (|ka|, p b a, (a, �)) be an inverse system in set theory of triangulated polyhedra and simplicial bonding functions pba. we shall refer to x as a simplicial inverse system. in case all |ka| have the topology cw or all have the topology m, then we shall denote all |ka| respectively as |ka|cw or |ka|m, and understand that all the functions p b a in set theory are simultaneously maps. if all the functions pba are surjective, then we shall call 2one may consult [10] for more information about polyhedra. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 11 l. r. rubin x a surjective inverse system. let x = lim x, x ∈ x, and for each a ∈ a, denote by σx,a the unique simplex of ka with xa ∈ int σx,a. lemma 2.2. let x = (|ka|, p b a, (a, �)) be a simplicial inverse system, x = lim x, and x ∈ x. then the trace {σx,a | a ∈ a} of x in x has the property that whenever a � b, pba(σx,b) = σx,a. hence xx = (σx,a, p b a|σx,b, (a, �)) is a surjective simplicial sub-inverse system of x with bonding functions that are simultaneously maps. moreover, x ∈ lim xx ⊂ x. definition 2.3. we shall refer to the uniquely determined inverse system xx = (σx,a, p b a|σx,b, (a, �)) of lemma 2.2 as the trace of x in x. definition 2.4. let x = (|ka|, p b a, (a, �)) be a simplicial inverse system, x = lim x, q ⊂ x, for each a ∈ a denote mq,a = {σy,a | y ∈ q}, and define lq,a to be the collection of faces of elements of mq,a. lemma 2.5. let x = (|ka|, p b a, (a, �)) be a simplicial inverse system, x = lim x, and q ⊂ x. then for each a ∈ a: (1) lq,a is a uniquely determined subcomplex of ka, (2) if n ∈ n, and for all y ∈ q, dim σy,a ≤ n, then dim lq,a ≤ n, and (3) if b ∈ a and a � b, pba(|lq,b|) = |lq,a|. hence xq = (|lq,a|, p b a ∣ ∣|lq,b|, (a, �)), which is uniquely determined by q, is a surjective simplicial sub-inverse system of x. moreover, for each x ∈ q, xx (see lemma 2.2) is a sub-inverse system of xq with x ∈ lim xx, so q ⊂ lim xq. proof. parts (1) and (2) are obviously true. to obtain (3), suppose that a � b. first we show that pba(|lq,b|) ⊂ |lq,a|. suppose that τ ∈ lq,b, i.e., τ is a face of an element σy,b ∈ mq,b. then lemma 2.2 shows that p b a(σy,b) = σy,a ∈ mq,a. since pba(τ) is a face of σy,a, then p b a(τ) ∈ lq,a, so p b a(τ) ⊂ |lq,a|. now we show the opposite inclusion, |lq,a| ⊂ p b a(|lq,b|). suppose that τ ∈ lq,a. then there exists y ∈ q such that τ is a face of σy,a. as before, we know that pba(σy,b) = σy,a; hence τ ⊂ σy,a = p b a(σy,b) ⊂ p b a(|lq,b|), which proves the desired inclusion. � definition 2.6. let x = (|ka|, p b a, (a, �)) be a simplicial inverse system, x = lim x, and q ⊂ x. then we shall refer to the uniquely determined inverse system xq of lemma 2.5 as the trace of q in x. applying lemmas 2.5 and 2.2, one arrives at the next result. proposition 2.7. if x = (|ka|, p b a, (a, �)) is a simplicial inverse system and x = lim x, then xx, the trace of x in x, is a surjective simplicial sub-inverse system of x with lim xx = x. this shows that x can be represented as the limit of a surjective simplicial inverse system. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 12 topological n-cells and hilbert cubes in inverse limits 3. topological cells in inverse limits in corollary 3.3 we shall characterize the conditions under which the inverse limit of a simplicial inverse sequence contains a topological n-cell. the same will be done in corollary 3.5 for a simplicial inverse system in which the coordinate spaces are finite polyhedra. for inverse sequences, we will make use of the class of stratifiable spaces; such spaces are convenient for applications when considering limits of inverse sequences. an exposition of generalized metrizable spaces, including stratifiable spaces, is given by g. gruenhage [3] in the handbook of set-theoretic topology. in that work, it is assumed that all spaces under consideration are t1 and regular. but for our purposes, we will only require that they be t1. we note that stratifiable spaces were first called m3-spaces, but the term stratifiable was introduced in [1] and this nomenclature became standard thenceforward. lemma 3.1 contains a list of properties of stratifiable spaces. let us first see which ones can be verified by reference to page numbers from [3]. the definition is given on page 426; we shall not repeat it here. using that definition and the t1 property, it is easy to prove that stratifiable spaces are regular; hence they are hausdorff. theorem 5.7 on page 457 gives us paracompactness, and theorem 5.10 on page 458 shows that they are hereditarily stratifiable and countably productive. hence the limit of an inverse sequence of stratifiable spaces is stratifiable. corollary 5.12(ii) on page 459 gives us that metrizable spaces are stratifiable. so the only statements in lemma 3.1 yet to be verified are the one in (4) concerning |k|cw, (5), and (7). we need to get at these from other references. every polyhedron |k|cw has the structure of a cw-complex. if one views the introduction of [2] (see corollary 8.6), one can see that all cw-complexes and hence all polyhedra are stratifiable spaces. this gives us the first part of (4). the main result of [5] (see also [6]) shows that covering dimension dim is preserved in the inverse limit of an inverse sequence of stratifiable spaces, so (7) is established. we get (5) from theorem 3.6 of [4]. lemma 3.1. the following are some facts about stratifiable spaces. (1) every stratifiable space is paracompact and hausdorff. (2) every subspace of a stratifiable space is stratifiable. (3) all metrizable spaces are stratifiable. (4) for each simplicial complex k, both |k|cw and |k|m are stratifiable. (5) if y ⊂ x and x is a stratifiable space, then dim y ≤ dim x. (6) the limit of an inverse sequence of stratifiable spaces is stratifiable. (7) if x = (xi, p i+1 i ) is an inverse sequence of stratifiable spaces, x = lim x, n ≥ 0, and for each i, dim xi ≤ n, then dim x ≤ n. proposition 3.2. let x = (|ki|cw, p i+1 i ) be a simplicial inverse sequence, x = lim x, and n ∈ n. if dim x ≥ n, then there exist i0 ∈ n and a sequence (τi)i≥i0 such that for each i ≥ i0, τi is an n-simplex of ki and p i+1 i carries τi+1 c© agt, upv, 2018 appl. gen. topol. 19, no. 1 13 l. r. rubin topologically onto τi. the same is true if we replace the topology cw, where it appears above, by the metric topology m. proof. applying proposition 2.7, there is no loss of generality in assuming that pi+1i is surjective for all i. using lemma 3.1(4,6), one sees that x is stratifiable. an application of lemma 3.1(7) shows this: if it is true that for all i, dim |ki| < n, then one would have that dim x < n. so there is a first i0 ∈ n with dim ki0 ≥ n. let τi0 be a simplex of ki0 with dim τi0 = n. using the fact that for each i ≥ i0, p i+1 i is simplicial and surjective, one can choose a sequence (τi)i≥i0 as requested. the same argument can be applied if we replace the topology cw, where it appears, by the metric topology m. � we obtain a corollary to lemma 3.1(4,5) and proposition 3.2. corollary 3.3. let x = (|ki|cw, p i+1 i ) be a simplicial inverse sequence, x = lim x, and n ∈ n. then x contains a topological n-cell if and only if dim x ≥ n. the same is true if we replace the topology cw, where it appears, by the metric topology m. proposition 3.4. let x = (|ka|, p b a, (a, �)) be a simplicial inverse system where all the |ka| are finite polyhedra, x = lim x, and n ∈ n. if dim x ≥ n, then there exists d ∈ a such that for each a ∈ a with d � a, there is an nsimplex τa of ka such that if b ∈ a with a � b, then p b a carries τb topologically onto τa. thus, x contains a topological n-cell. proof. we may assume that (a, �) has no upper bound. applying proposition 2.7, there is no loss of generality in assuming that pba is surjective for all a � b. it is moreover true that x is a compact hausdorff space. since dim x ≥ n, there has to be a cofinal subset a0 of a such that dim ka ≥ n for all a ∈ a0. we may as well require that a has this property from the outset. fix d ∈ a. then the set of a ∈ a with d � a is cofinal in a, so we shall assume that for all a ∈ a, d � a. now fix an n-simplex τd in kd, let xτd ∈ int τd, and hd = {xτd}. for each a ∈ a, there is at least one n-simplex τ ∈ ka such that p a d (τ) = τd. let fa be the collection of such n-simplexes, and for each τ ∈ fa, select the unique element xτ ∈ int τ with p a d (xτ ) = xτd. denote ha = {xτ | τ ∈ fa}. then for all a ∈ a, ha is a finite, nonempty subset of |ka|, and if u ∈ ha, then p a d(u) = xτd. we claim that if a � b, then pba(hb) ⊂ ha. for let τ ∈ fb; we must show that pba(xτ ) ∈ ha. now p a d ◦ p b a(xτ ) = p b d(xτ ) = xτd. also, p b d(τ) = τd. it follows that τ∗ = pba(τ) is an n-simplex of ka and p a d (τ∗) = τd. thus, τ ∗ ∈ fa and pba(xτ ) = xτ∗ ∈ ha as required. from this we get a sub-inverse system h = (ha, p b a|hb, (a �)) of x consisting of nonempty discrete finite sets ha. thus lim h 6= ∅. select y ∈ lim h ⊂ lim x. from lemma 2.2, the trace of y in x, xy = (σy,a, p b a|σy,b, (a, �)) is a surjective simplicial sub-inverse system of x. since dim σy,a = n for all a, then each p b a|σy,b : σy,b → σy,a is a homeomorphism. clearly, lim xy ⊂ lim x is a topological n-cell. � c© agt, upv, 2018 appl. gen. topol. 19, no. 1 14 topological n-cells and hilbert cubes in inverse limits corollary 3.5. let x = (|ka|, p b a, (a, �)) be a simplicial inverse system where all the |ka| are finite polyhedra, x = lim x, and n ∈ n. then x contains a topological n-cell if and only if dim x ≥ n. 4. hilbert cubes in limits of inverse sequences the main result of this section is theorem 4.13. it characterizes when the limit of a simplicial inverse sequence must contain a copy of the hilbert cube. first, let us review some concepts from dimension theory. recall that an infinite-dimensional space is called countable-dimensional if it can be written as the union of subspaces xn, n ∈ n, each xn having dimension ≤ n. it is called strongly countable-dimensional if it can be written as the union of closed subspaces xn, n ∈ n, each xn having dimension ≤ n. of course, strongly countable-dimensional spaces are countable-dimensional. from corollaries 3.3 and 3.5, respectively, we get propositions 4.1 and 4.2. proposition 4.1. let x = (|ki|cw, p i+1 i ) be a simplicial inverse sequence and x = lim x. if dim x = ∞, then x contains a strongly countable dimensional subspace y = ⋃ {yi | i ∈ n} such that for each i, yi is a topological i-cell. the same is true if we replace the topology cw, where it appears above, by the metric topology m. proposition 4.2. let x = (|ka|cw, p b a, (a, �)) be a simplicial inverse system and x = lim x. if all the |ka| are finite polyhedra and dim x = ∞, then x contains a strongly countable dimensional subspace y = ⋃ {yi | i ∈ n} such that for each i, yi is a topological i-cell. as usual, i = [0, 1], the unit interval. we shall denote the hilbert cube as i∞, that is, i∞ = ∏ {ii | i ∈ n} where for each i, ii = i. for each i ∈ n, let pi+1i : i i+1 → ii be the i-coordinate projection. remember that strongly infinite-dimensional spaces are not countable-dimensional. since i∞ is strongly infinite-dimensional, it is not countable-dimensional. one may consult [10] for more information on this subject. lemma 4.3. let g = (ii, pi+1i ) be the inverse sequence having the property that for each i, pi+1 i : ii+1 → ii is the coordinate projection. then lim g ∼= i∞. proof. since both i∞ and lim g are compact metrizable spaces, it is sufficient to find a bijective map from i∞ to lim g. define a map h : i∞ → lim g by setting h(x1, x2, x3, . . . ) = (x1, (x1, x2), (x1, x2, x3), . . . ). surely h is a map; we leave it to the reader to show that h is a bijection. � whenever v is the vertex set of a simplex σ, then an arbitrary element x of σ will be written x = ∑ {xvv | v ∈ v}, where for each v ∈ v, xv is the v-barycentric coordinate of x. lemma 4.4. let n ∈ n, σ be an n-simplex with vertex set v, τ0 an (n−1)-face of σ, w the vertex set of τ0, v ∈ v \ w, and µ : σ → τ0 a simplicial retraction. then µ(u) = u for each u ∈ w, and there is a unique w ∈ w with µ(v) = w. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 15 l. r. rubin indeed, if x = ∑ {xvv | v ∈ v} ∈ σ, then µ(x) = ∑ {buu | u ∈ w} ∈ τ0 where bu = xv + xw if u = w, and bu = xu otherwise. lemma 4.5. let n ∈ n, σ be an n-simplex, τ an (n − 1)-simplex, and q : σ → τ a simplicial surjection. then there exist a unique (n − 1)-face τ0 of σ, a unique simplicial retraction µ : σ → τ0, and a unique simplicial isomorphism q0 : τ0 → τ, such that q = q0 ◦ µ. lemma 4.6. let n ∈ n, σ be an n-simplex, τ0 an (n − 1)-face of σ, and µ : σ → τ0 a simplicial retraction of σ to τ0. suppose that d ⊂ int τ0 is nonempty and compact. let v, w, v, and w come from lemma 4.4. we claim that for any neighborhood u of ∂σ in σ, there is an embedding h : d × i → u ∩ int σ such that (µ| im(h)) ◦ h = p : d × i → d, where p : d × i → d is the coordinate projection. proof. let (x, t) ∈ d × i, x = ∑ {xuu | u ∈ w} ∈ d ⊂ int τ0. define h(x, t) ∈ σ so that its v-barycentric coordinate is (1 − t)xw, its w-barycentric coordinate is txw, and for any u ∈ v \ {v, w}, its u-barycentric coordinate is xu. then clearly h : d×i → σ is a map. to show that h is injective, let y = ∑ {yuu | u ∈ w} ∈ d, {t, s} ⊂ i, and (x, t) 6= (y, s). if u ∈ w \ {w}, and xu 6= yu, then h(x, t) 6= h(y, s) independently of t and s. hence we may as well assume that xu = yu for all u ∈ w \ {w}. suppose that h(x, t) = h(y, s). if t = s, then x 6= y, that is, xw 6= yw. by the definition of h, (1 − t)xw = (1 − t)yw and txw = tyw. since one of {1 − t, t} does not equal 0, then xw = yw, a contradiction. hence t 6= s. since d ⊂ int τ0, then xw 6= 0, so txw 6= sxw. this implies that t = s, another contradiction. therefore h(x, t) 6= h(y, s). we have demonstrated that h is injective which shows that h is an embedding because of compactness. one easily checks that (µ| im(h))◦h = p : d×i → d. notice that for x ∈ d ⊂ int τ0 as above, for all u ∈ w, xu > 0. this is true in particular if u = w. if t /∈ {0, 1}, both (1 − t)xw > 0 and txw > 0. hence for all u ∈ w, the u-barycentric coordinates of h(x, t) are > 0. therefore if 0 < a < b < 1 and we restrict h to d × [a, b], we get an embedding of d × [a, b] into int σ. but, the v-barycentric coordinate of h(x, 1) equals 0. so h(d × {1}) ⊂ ∂σ. taking a sufficiently close to 1, we get that h(d × [a, b]) ⊂ u ∩ int σ. it is now simply a matter of reparameterizing [a, b] so that it is replaced by [0, 1], and we have our proof. � lemma 4.7. let n ∈ n, σ be an n-simplex, τ an (n−1)-simplex, and q : σ → τ a simplicial surjection. suppose that e is a nonempty compact subset of int τ. then for any neighborhood u of ∂σ in σ, there is an embedding h∗ : e × i → u ∩ int σ such that (q| im(h∗)) ◦ h∗ = p : e × i → e, where p : e × i → e is the coordinate projection. proof. apply lemma 4.5 to q : σ → τ. let τ0 be the unique (n − 1)-face of σ, µ : σ → τ0 the unique simplicial retraction, and q0 : τ0 → τ the unique simplicial isomorphism such that q = q0 ◦ µ. put d = q −1 0 (e) ⊂ int τ0. apply lemma 4.6 to get an embedding h : d × i → u ∩ int σ having the property that (µ| im(h)) ◦ h = p : d × i → d, where p : d × i → d is the coordinate c© agt, upv, 2018 appl. gen. topol. 19, no. 1 16 topological n-cells and hilbert cubes in inverse limits projection. define h∗ : e × i → int σ by h∗(e, t) = h(q−10 (e), t). surely h ∗ is an embedding of e × i into u ∩ int σ. suppose that (e, t) ∈ e × i. then q ◦ h∗(e, t) = q0 ◦ µ ◦ h ∗(e, t) = q0 ◦ µ ◦ h(q −1 0 (e), t) = q0 ◦ p(q −1 0 (e), t) = q0 ◦ q −1 0 (e) = e. � lemma 4.8. let m < n ∈ n and {σi | 0 ≤ i ≤ n − m} be a set such that for each 0 ≤ i ≤ n − m, σi is an (m + i)-simplex. for each 1 ≤ i ≤ n − m, let qi : σi → σi−1 be a simplicial surjection and put q = q1◦· · ·◦qn−m : σn−m → σ0. let e be a nonempty compact subset of int σ0 and u a neighborhood of ∂σn−m in σn−m. then there is an embedding h ∗ : e × in−m → u ∩ int σn−m such that (q| im(h∗)) ◦ h∗ = p : e × in−m → e, where p : e × in−m → e is the coordinate projection. proof. an application of lemma 4.7 shows that this result is true in every case where n − m = 1. suppose that k ∈ n, and the lemma is true in every case where n − m = k. now assume that n − m = k + 1 and we are given the above data, only this time with one more map in the composition. note that in this setting, q = q′ ◦ qk+1 where qk+1 : σk+1 → σk, dim σk+1 = dim σk + 1, q′ = q1 ◦· · ·◦qk : σk → σ0, and k = n−(m+1) > 0. also, u is a neighborhood of ∂σk+1 in σk+1. thus, m + 1 < n, so we may apply the inductive hypothesis to the map q′. this gives us an embedding h : e × ik → int σk such that (q′| im(h)) ◦ h = p′ : e × ik → e, where p′ : e × ik → e is the coordinate projection. we now have the nonempty compact subset im h ⊂ int σk and of course k + 1 − k = 1. so we may apply the fact that our result is true for n = k + 1, m = k. this gives us an embedding h′ : (im h)× i into u ∩ int σk+1 such that (qk+1| im(h ′)) ◦ h′ = p∗ : (im h) × i → im h, where p∗ : (im h) × i → im h is the coordinate projection. define h∗ : e × ik × i → u ∩ int σk+1 by h∗(e, s, t) = h′(h(e, s), t). it follows that h∗ is an embedding. we must prove that (q| im(h∗)) ◦ h∗ = p : e × ik+1 → e, where p : e × ik+1 → e is the coordinate projection. let (e, s, t) ∈ e × ik × i. then q◦h∗(e, s, t) = q◦h′(h(e, s), t) = q′◦qk+1◦h ′(h(e, s), t) = q′◦p∗(h(e, s), t) = q′ ◦ h(e, s) = p′(e, s) = e. our proof is complete. � applying lemmas 4.5 and 4.8, one obtains a corollary. corollary 4.9. let σ and τ be simplexes such that dim τ = m < dim σ = n, suppose that p : σ → τ is a simplicial surjection, e is a compact subset of int τ and u is a neighborhood of bd σ in σ. then there exists an embedding h : e × in−m → u ∩ int σ such that p ◦ h : e × in−m → τ is the coordinate projection e × in−m to e. proposition 4.10. suppose that s = (σi, q i+1 i ) is a surjective simplicial inverse sequence such that for each i, σi is an i-simplex. then lim s contains an embedded copy of i∞. proof. let e ⊂ int σ1 be a closed interval, and identify e with i. apply lemma 4.7 in such a way that i × i ⊂ int σ2 and q 2 1|i × i : i × i → i ⊂ int σ1 c© agt, upv, 2018 appl. gen. topol. 19, no. 1 17 l. r. rubin is the coordinate projection (t1, t2) 7→ t1. next apply lemma 4.7 again in such a way that i2 ×i ⊂ int σ3 and q 3 2|i 2 ×i : i2 ×i → i2 ⊂ int σ2 is the coordinate projection (t1, t2, t3) 7→ (t1, t2). continuing recursively in this manner, we land up with a sub-inverse sequence of s of the form g = (ii, pi+1i ) from lemma 4.3. therefore lim g ∼= i∞ ⊂ lim s as requested. � corollary 4.11. suppose that s = (σi, q i+1 i ) is a surjective simplicial inverse sequence such that for each i, σi is a simplex, and there exists an increasing sequence (ni) in n such that for each i, dim σni < dim σni+1. then lim s contains an embedded copy of i∞. proof. since the sequence (ni) is increasing, we may replace s with the inverse sequence (σni, q ni+1 ni ) whose inverse limit is homeomorphic to lim s. to conserve notation, let us assume that the given inverse sequence s = (σi, q i+1 i ) already has the property that dim σi < dim σi+1 for all i. one may also assume that 1 ≤ dim σ1. select a 1-face τ1 of σ1. choose a 2-face τ2 of σ2 with q 2 1(τ2) = τ1. similarly, choose a 3-face τ3 of σ3 with q 3 2(τ3) = τ2. this process can be continued recursively so that we end up with a sequence (τi) having the property that for each i, dim τi = i, τi is a face of σi, and q i+1 i |τi+1 : τi+1 → τi is a simplicial surjection. the surjective simplicial sub-inverse sequence s0 = (τi, q i+1 i |τi+1) of s replicates the inverse sequence in proposition 4.10, so i ∞ embeds in lim s0 which in turn embeds in lim s. � lemma 4.12. let x = (|ki|cw, p i+1 i ) be a simplicial inverse sequence, and put x = lim x. suppose that x contains a strongly infinite-dimensional subspace q. then there exist x ∈ q and an increasing sequence (ni) in n, so that the trace xx of x in x has the property that for each i, dim σx,ni < dim σx,ni+1. the same is true if we replace the topology cw, where it appears above, by the metric topology m. proof. for each x ∈ q ⊂ x, let xx be the trace of x in x. then for all i, σx,i ∈ ki and p i+1 i (σx,i+1) = σx,i, so dim σx,i ≤ dim σx,i+1; moreover, x ∈ lim xx. let us suppose, for the purpose of reaching a contradiction, that for all x ∈ q, there exists nx ∈ n such that dim σx,i ≤ nx for all i. for each n ∈ n, let qn = {x ∈ q | nx ≤ n}. then q = ⋃ {qn | n ∈ n}. fix n ∈ n, and for each i ∈ n, let mqn,i be as in definition 2.4. then all the simplexes in mqn,i have dimension ≤ n. so by lemma 2.5(2), dim lqn,i ≤ n. applying proposition 2.7, we get the sub-inverse sequence xqn = (|lqn,i|, pi+1i ∣ ∣|lqn,i+1|) of x, with qn ⊂ xn = lim xqn. surely xn is a stratifiable space and dim xn ≤ n. thus, dim(qn ∩xn) ≤ n. hence q = ⋃ {qn ∩xn | n ∈ n} is countable-dimensional, which is false. this same argument works if we replace the topology cw, where it appears, by the metric topology m. our proof is complete. � putting together corollary 4.11 and lemma 4.12, we obtain a theorem. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 18 topological n-cells and hilbert cubes in inverse limits theorem 4.13. let x = (|ki|cw, p i+1 i ) be a simplicial inverse sequence, and put x = lim x. then x contains an embedded copy of i∞ if and only if there is a collection {σi | i ∈ n} and an increasing sequence (ni) in n, such that for each i, (1) σi is a simplex of ki, (2) pi+1i (σi+1) = σi, and (3) dim σni < dim σni+1. the same is true if we replace the topology cw, where it appears above, by the metric topology m. 5. strongly infinite dimensional sets in limits of inverse systems of finite polyhedra we present a result for inverse systems of finite polyhedra that is parallel to lemma 4.12. we however do not have a result that is similar to that of theorem 4.13. proposition 5.1. let x = (|ka|, p b a, (a, �)) be a simplicial inverse system where all the |ka| are finite polyhedra, and let x = lim x. suppose that x contains a strongly infinite-dimensional closed subspace q. then there exists x ∈ x (indeed, x ∈ q) so that the trace xx of x in x satisfies the property that for each a ∈ a and n ∈ n, there exists a � b such that dim σx,b ≥ n. hence there exists a sequence (ai) in a such that for each i, ai � ai+1, ai 6= ai+1, and dim σai < dim σai+1. proof. since x contains a strongly infinite-dimensional closed subspace, then (a, �) has no upper bound. for each x ∈ q ⊂ x, let xx be the trace of x in x. let us suppose, for the purpose of reaching a contradiction, that for all x ∈ q, there exist ax ∈ a and nx ∈ n such that for all ax � b, dim σx,b ≤ nx. for each n ∈ n, let qn = {x ∈ q | nx ≤ n}. then q = ⋃ {qn | n ∈ n}. fix n ∈ n, and for each a ∈ a, let mqn,a be as in definition 2.4. then whenever ax � b, by lemma 2.5(2), dim lqn,b ≤ n. one should note that {b ∈ a | ax � b} is cofinal in a. applying definition 2.6, we get the subinverse system xqn = (|lqn,a|, p b a ∣ ∣|lqn,b|, (a �)) of x, with qn ⊂ xn = lim xqn. surely, xn is a compact hausdorff space and dim xn ≤ n. hence q = ⋃ {qn ∩xn | n ∈ n} is strongly countable-dimensional, which is false since q is strongly infinite-dimensional. our proof is complete. � references 1. c. j. r. borges, on stratifiable spaces, pacific j. math. 17 (1966), 1–16. 2. j. g. ceder, some generalizations of metric spaces, pacific j. math. 11 (1961), 105–125. 3. g. gruenhage, generalized metric spaces, handbook of set-theoretic topology (edited by k. kunen and j. e. vaughan), elsevier science publishers, amsterdam 1984., 423-501. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 19 l. r. rubin 4. i. ivanšić and l. rubin, the extension dimension of universal spaces, glas. mat. ser. iii 38 (58) (2003), 121–127. 5. s. mardešić, extension dimension of inverse limits, glas. mat. ser. iii 35 (55) (2000), 339–354. 6. s. mardešić, extension dimension of inverse limits. correction of a proof, glas. mat. ser. iii 39 (59) (2004), 337–339. 7. s. mardešić, not every metrizable compactum is the limit of an inverse sequence with simplicial bonding maps, topology appl., to appear. 8. s. nadler, continuum theory, marcel dekker, inc., new york, basel, hong kong, 1992. 9. l. rubin and v. tonić, simplicial inverse sequences in extension theory, preprint. 10. k. sakai, geometric aspects of general topology, springer monographs in mathematics, tokyo, heidelberg, new york, dordrecht, london, 2013. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 20 () @ appl. gen. topol. 19, no. 2 (2018), 239-244doi:10.4995/agt.2018.7962 c© agt, upv, 2018 dynamics of real projective transformations sharan gopal a and srikanth ravulapalli b a bits-pilani, hyderabad campus, hyderabad, india. (sharanraghu@gmail.com) b school of mathematics and statistics, university of hyderabad, hyderabad, india. (srikanth.hcu@gmail.com) communicated by f. balibrea abstract the dynamics of a projective transformation on a real projective space are studied in this paper. the two main aspects of these transformations that are studied here are the topological entropy and the zeta function. topological entropy is an inherent property of a dynamical system whereas the zeta function is a useful tool for the study of periodic points. we find the zeta function for a general projective transformation but entropy only for certain transformations on the real projective line. 2010 msc: 54h20; 37b40. keywords: topological entropy; zeta function; projective transformation. 1. introduction an n−dimensional real projective space, denoted by pn(r) is the quotient space sn/∼, where the antipodal points are identified under the relation, ∼. p1(r) is also called the real projective line. let π : s n → pn(r) be the quotient map. a projective transformation on pn(r) associated to a matrix a ∈ gln+1(r), denoted by ā is defined as ā(π(x)) = π(ax), for every x ∈ s n. a discrete dynamical system is, by definition, a pair (x, f), where x is a topological space and f is a self map on x i.e., f : x → x. though f can be any map in a general setting, we need it to be a continuous map in many cases. so, unless otherwise mentioned, we assume the map to be continuous. since we consider only discrete dynamical systems in this paper, received 26 august 2017 – accepted 05 february 2018 http://dx.doi.org/10.4995/agt.2018.7962 s. gopal and s. ravulapalli hereafter, we refer to them simply as dynamical systems. given x ∈ x, the sequence (x, f(x), f2(x), f3(x), ...) is called the trajectory of x, where fk(x) = f ◦ f ◦ ... ◦ f(x) (k times) for k ∈ n and f0(x) = x. the set {fk(x) : k is a non-negative integer} is called the orbit of x. the study of dynamics is mainly about the eventual behavior of trajectories. a point x ∈ x is said to be periodic if there is a k ∈ n such that fk(x) = x; any such k is called a period of x and the least among them is called the least period of x. a periodic point x of period 1 is called a fixed point i.e., f(x) = x and the set of fixed points of f is denoted by fix(f). we also use the notation |y | to denote the cardinality of any set y . in this paper, the typical dynamical system that we are going to consider is (pn(r), ā). the periodic points of this system can be found very easily. if v ∈ sn is an eigenvector of a with eigenvalue λ, then ā(π(v)) = π(av) = π(λv) = π(v); hence π(v) is a fixed point. conversely, if π(v) is a periodic point with period k, then it is a fixed point of āk and thus π(akv) = π(v) i.e., akv = µv for some scalar µ. hence, v is an eigenvector of ak. to sum up, we have shown that π(v) is periodic if and only if v is an eigenvector of ak for some k ∈ n. the dynamics of projective transformations are well studied in the literature. see for instance [4] and [6]. in this paper, we study the topological entropy and the zeta function of projective transformations. one of the best ways of measuring the complexity of a dynamical system is finding its topological entropy. as stated in [3], topological entropy measures the exponential growth rate of the number of essentially different orbit segments of length n. on the other hand, the zeta function collects combinatorial information about the periodic points. in the next section, we calculate the entropy of certain projective transformations on the real projective line, followed by a section on finding the zeta function of a projective transformation on a projective space of any dimension. 2. topological entropy topological entropy was introduced by adler, konheim and mcandrew [1] and here, we will use an equivalent definition for maps on compact metric spaces given by bowen [2]. most of the basic facts about entropy, that we mention here can be found in [3]. given a compact metric space (x, d) and a continuous map f : x → x, we define a new metric dn, for every n ∈ n as dn(x, y) = max {d(f i(x), fi(y)) : 0 ≤ i < n}. it can be shown that each of these metrics induces the same topology on x as induced by d. a subset e ⊂ x is called an (n, ǫ)−separated set if for any two distinct points x, y ∈ e, dn(x, y) ≥ ǫ. since x is compact, every (n, ǫ)-separated set is a finite set; otherwise, there will be a sequence (xk) in e with no convergent subsequence, as d(xk, xk+1) ≥ ǫ for every k ∈ n, thus contradicting the compactness of x. now, let sep(n, ǫ, f) be the cardinality of an (n, ǫ)−separated set with maximum cardinality. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 240 dynamics of real projective transformations definition 2.1 (see [3]). the entropy h(f) of a system (x, f) is defined as (2.1) h(f) = lim ǫ→0+ lim sup n→∞ 1 n log(sep(n, ǫ, f)). similar to sep(n, ǫ, f), two more numbers, namely span(n, ǫ, f) and cov(n, ǫ, f) can be defined for every ǫ > 0 and n ∈ n. here, cov(n, ǫ, f) is the cardinality of a covering of x by least number of sets of dn-diameter less than ǫ. it is well defined because, there does exist a finite cover of x by sets of dn-diameter less than ǫ, as any cover of x with open sets of dn-diameter less than ǫ will have a finite subcover for x. finally, a subset a ⊂ x is called an (n, ǫ)-spanning set in x, if for every x ∈ x, there is y ∈ a such that dn(x, y) < ǫ. as x is compact, the open cover {bdn(x, ǫ) : x ∈ x} (where bdn(x, ǫ) is the open ball centered at x and has radius ǫ with respect to the dn-diameter) has a finite subcover, say {bdn(x1, ǫ), bdn(x2, ǫ), . . ., bdn(xk, ǫ)}. then, the set {x1, x2, . . , xk} is an (n, ǫ)-spanning set. since finite (n, ǫ)-spanning sets exist in a compact space, we can find an (n, ǫ)-spanning set with minimum cardinality. this minimum cardinality is called span(n, ǫ, f). lemma 2.2 (see [3]). (2.2) cov(n, 2ǫ, f) ≤ span(n, ǫ, f) ≤ sep(n, ǫ, f) ≤ cov(n, ǫ, f). using this lemma, it follows easily that h(f) = lim ǫ→0+ lim sup n→∞ 1 n log(sep(n, ǫ, f))(2.3) = lim ǫ→0+ lim sup n→∞ 1 n log(span(n, ǫ, f))(2.4) = lim ǫ→0+ lim sup n→∞ 1 n log(cov(n, ǫ, f))(2.5) proposition 2.3 (proposition 2.5.3 in [3]). the topological entropy of a continuous map f : x → x does not depend on the choice of a particular metric generating the topology of x. proposition 2.4 ([3]). the topological entropy of an isometry is zero. in the following proposition, t 2 denotes the torus, r2/z2. any automorphism of this topological group, r2/z2 which will be called a toral automorphism, is of the form π′(x) 7→ π′(mx), where π′ : r2 → t 2 is the canonical projection and m ∈ gl2(z). we say that the automorphism is induced by the matrix m and denote it by tm. if no eigenvalue of m has modulus 1, then tm is called a hyperbolic toral automorphism. proposition 2.5 (proposition 2.6.1 in [3]). the topological entropy of a hyperbolic toral automorphism tm : t 2 → t 2, with det(m) = 1 is equal to log |λ|, where λ is the eigenvalue of m such that |λ| > 1. all the above propositions can be found in [3]. our proof of theorem 2.6 relies mostly on the proof of proposition 2.5, as given in [3]. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 241 s. gopal and s. ravulapalli theorem 2.6. let a be a 2 × 2 matrix of determinant 1 with eigenvalues λ and 1 λ , where |λ| > 1. then the entropy of the corresponding projective transformation ā is log |λ|. proof. let v1 and v2 be unit eigenvectors of a corresponding to λ and 1 λ respectively. now, for u, v ∈ s1, define d̃(u, v) = max(|a1|, |a2|), where u − v = a1v1 + a2v2. d̃ is a metric on s 1 and it induces the metric d on p1(r), where d(x, y) is the d̃-distance between the sets π−1(x) and π−1(y). note that the definition of d̃ can be extended to a metric on r2 and an open ball of radius ǫ centered at (u0, v0) ∈ r 2 under d̃ is a parallelogram centered at (u0, v0) with its sides parallel to v1 and v2 and each having a length 2ǫ. then an open ball of radius ǫ in s1 centered at (u0, v0) ∈ s 1 is an arc centered at (u0, v0), which is formed by the intersection of s 1 with the above parallelogram. further, an ǫ d̃n-ball in r 2, with respect to the map induced by a is again a parallelogram with sides of lengths 2ǫ and 2ǫ |λ|n which are parallel to v1 and v2 respectively. now, an ǫ d̃n-ball in s 1 is thus an arc passing through the center of a parallelogram with the above dimensions. so, its length is at least the smaller side of the parallelogram i.e., 2ǫ |λ|n . on the other hand, its length is at most the perimeter of the parallelogram, which is equal to 4ǫ + 4ǫ |λ|n (see the figure). it follows from archimedean property of real numbers that, if a and b are any two positive real numbers, then there is a positive integer k such that (a + b) ≤ kab. thus, we can find a positive integer depending on ǫ, say k(ǫ), such that 4ǫ + 4ǫ |λ|n ≤ k(ǫ)ǫ2 |λ|n . since π is a local isometry, for sufficiently small ǫ, we can assume that ǫ dn-balls in p1(r) have the same dimensions. since the diameter of an ǫ dn-ball in p1(r) is at most k(ǫ)ǫ2 |λ|n , the minimum number of such balls that are required to cover p1(r) is π k(ǫ)ǫ2 |λ|n , as the euclidean length of p1(r) is π. since a set of diameter 2ǫ is contained in an open ball of radius ǫ, we have, cov(n, 2ǫ, ā) ≥ π|λ|n k(ǫ)ǫ2 . thus, we have h(ā) ≥ log |λ|. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 242 dynamics of real projective transformations similarly, since the diameter of an ǫ dn-ball in p1(r) is at least 2ǫ |λ|n , p1(r) can be covered by π2ǫ |λ|n number of arcs. hence, cov(n, 2ǫ, ā) ≤ π|λ|n 2ǫ . so, h(ā) ≤ log |λ|. thus, we conclude that h(ā) = log |λ|. � in [3], the authors have also given a proof of the fact that the entropy of a hyperbolic toral automorphism induced by a matrix a on an n−dimensional torus, t n is equal to m∑ i=1 log|αi|, where α1, α2, . . . , αm are those eigenvalues of a that have modulus strictly larger than 1 (see proposition 2.6.4 in [3]). the proof relies on the idea of decomposing rn in to generalised eigenspaces of a and is similar to the proof of the corresponding result on 2−dimensional torus, mentioned above (proposition 2.5). on the same lines, it is hoped that a result similar to theorem 2.6 can be obtained for projective transformations on higher dimensional projective spaces also. 3. zeta function the zeta function collects combinatorial information about the periodic points. we follow [3] for the definition and other basic facts of the zeta function. for a dynamical system (x, f), if |fix(fk)| is finite for every k, we define the zeta function ζf (z) of f to be the formal power series ζf (z) = exp( ∑∞ k=1 1 k |fix(fk)|zk). the zeta function can also be expressed by a product formula. let p(f) denote the collection of all periodic orbits of f i.e., a typical element of p(f) will be {x0, f(x0), . . ., f k−1(x0)}, where k is the least period of x0. now, the zeta function of f can be written as ζf (z) = ∏ γ∈p(f) (1 − z|γ|)−1 where |γ| is the number of elements in γ. we use the following lemma in proving theorem 3.2. lemma 3.1. if µ is a non-zero eigenvalue of ak for some k ∈ n such that there is a unique eigenvalue λ of a with λk = µ, then the eigenspaces of ak and a corresponding to µ and λ respectively, are same. the lemma follows easily from the facts that, under the assumptions of the hypothesis, the number of jordan blocks in the jordan normal form of a corresponding to λ is same as the number of jordan blocks in the jordan normal form of ak corresponding to µ. theorem 3.2. let ā be a projective transformation on pn(r) induced by a matrix a ∈ gln+1(r). ā possesses zeta function if and only if each eigenspace of a is one-dimensional and no two eigenvalues have same absolute value. in such case, the zeta function is given by ζf (z) = 1 (1−z)l . proof. suppose that each eigenspace of a is one-dimensional and no two eigenvalues of a have same absolute value. if π(v) is a periodic point of ā where v ∈ s1, then v is an eigenvector of ak for some k ∈ n, say akv = µv, µ ∈ r. then there is an eigenvalue λ of a such that λk = µ. if λ1 and λ2 are two c© agt, upv, 2018 appl. gen. topol. 19, no. 2 243 s. gopal and s. ravulapalli different eigenvalues of a such that λk1 = λ k 2 = µ, then |λ1| = |λ2|, contrary to the hypothesis. thus, λ is unique. hence, by the above lemma, v lies in the eigenspace of a corresponding to λ. thus v̄ is a fixed point of ā i.e., fixed points are the only periodic points. in other words, fix(āk) = fix(ā) for any k. further, since each eigenspace of a is one-dimensional, there are as many fixed points as the eigenvalues. thus, ζf (z) = exp(l ∑∞ k=1 z k k ) = 1 (1−z)l , where l is the number of eigenvalues of a. conversely, suppose ā possesses zeta function. then there should be finitely many fixed points and thus each eigenspace should be one dimensional. if possible, suppose there are two different eigenvalues λ1 and λ2 such that |λ1| = |λ2|. since λ1 and λ2 are real, λ 2 1 = λ 2 2; say µ = λ 2 1. then µ is an eigenvalue of a2. if v1 and v2 are eigenvectors of a corresponding to λ1 and λ2 respectively, then v1 and v2 are eigenvectors of a 2 corresponding to the same eigenvalue µ. so, the dimension of eigenspace of a2 corresponding to µ is greater than 1 and thus there are infinitely many periodic points of ā with period 2, implying that the zeta function doesn’t exist, contradicting the hypothesis. � acknowledgements. the authors thank the referee for his suggestions. the first author acknowledges the financial support received under the research initiation grant provided by bits-pilani. the second author thanks ugc, india for receiving the financial support as a ugc senior research fellow. references [1] r. l. adler, a. g. konheim and m. h. mcandrew, topological entropy, transactions of the american mathematical society 114 (1965), 309–319. [2] r. bowen, entropy for group endomorphisms and homogeneous spaces, transactions of the american mathematical society 153 (1971), 401–414. [3] m. brin and g. stuck, introduction to dynamical systems, cambridge university press (2004). [4] s. g. dani, dynamical properties of linear and projective transformations and their applications, indian j. pure appl. math. 35 (2004), 1365–1394. [5] r. devaney, an introduction to chaotic dynamical systems, second edition, addisonwesley publishing company advanced book program, redwood city, ca, 1989. [6] n. h. kuiper, topological conjugacy of real projective transformations, topology 15 (1976), 13–22. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 244 () @ appl. gen. topol. 18, no. 2 (2017), 255-275doi:10.4995/agt.2017.5868 c© agt, upv, 2017 oriented components and their separations bärbel m. r. stadler a and peter f. stadler a−d a max planck institute for mathematics in the sciences, inselstraße 22, d-04103 leipzig, germany. (baer@bioinf.uni-leipzig.de) b bioinformatics group, department of computer science; interdisciplinary center for bioinformatics; german centre for integrative biodiversity research (idiv) halle-jena-leipzig; competence center for scalable data services and solutions; and leipzig research center for civilization diseases, leipzig university, härtelstraße 16-18, d-04107 leipzig, germany. (studla@bioinf.uni-leipzig.de) c institute for theoretical chemistry, university of vienna, währingerstraße 17, a-1090 wien, austria. d santa fe institute, 1399 hyde park rd., santa fe, nm 87501 communicated by e. induráin abstract there is a tight connection between connectedness, connected components, and certain types of separation spaces. recently, axiom systems for oriented connectedness were proposed leading to the notion of reaches. here, we introduce production relations as a further generalization of connectivity spaces and reaches and derive associated systems of oriented components that generalize connected components in a natural manner. the main result is a characterization of generalized reaches in terms of equivalent separation spaces. 2010 msc: 54d05; 54e05. keywords: separation spaces; pre-proximities; generalized topological spaces. 1. introduction already in the 1940’s, a. d. wallace asked whether topological concepts can be axiomatized starting from a notion of connectivity or an associated received 02 june 2016 – accepted 21 march 2017 http://dx.doi.org/10.4995/agt.2017.5868 concept of separation [29], the latter being the complement of a generalization of proximity [24]. ch. ronse characterized the separations that are equivalent to abstract connectivities [20]. these results were generalized further in [26]. since connectedness is an intrinsically symmetric notion, the theory requires that separations satisfy a symmetry axiom. tankyevych et al. [28] recently introduced so-called semi-connections as a generalized model of connected components in directed graphs. ronse explored this concept in more detail [22] as a generalization of connectivity openings [20, 23] and discussed some fundamental properties of the equivalent notion of oriented components. the latter form a system of pairs (p, q) with the interpretation that “every point of q is reachable from p within q”. this construction is insufficient, however, to capture the natural connectivity structure of chemical reaction networks and directed hypergraphs in general. the key point is that in a chemical reaction, the “output molecules” depend on a set of “input molecules” rather than on a single input molecule. this suggests to generalize the work of [28, 22] to pairs (p, q) such that every point of q can be reached from the start set p within q. this formalization of reachability does not appear to be connected to concepts familiar from topology in an obvious manner. on the other hand, reaches are rather natural generalizations of connectivity structures, and the latter have been shown to be equivalent to certain generalized proximities in [20, 26]. it is the purpose of this contribution to show that there is also a 1-1 correspondence of reachability structures general enough to encompass chemical reaction networks, and thus directed hypergraphs, and a suitable class of generalized separations or, equivalently, proximities. to this end we consider two binary relations ≻− and | on 2x × 2x that we interpret as follows: p ≻− q means that p can produce q, i.e., q are the points eventually reachable from p . production relations capture the salient structure of (directed) hypergraphs, which to our knowledge rarely have been considered with regard of their topological properties [10]. on the other hand, we think of a | b as “a is separated from b”. the negation of separation relations are proximity relations, aδ b, which are well studied in the literature as a starting point for constructing topological theories [24, 13, 14, 18]. before we turn to establishing the formal connection between production relations and separations, we briefly argue that production relations are an interesting notion to study in their own right. 2. production relations in the real world the paradigmatic examples of production relations derive from chemical reactions. given a set x of molecular types, usually called compounds or reactants, chemical reactions describe the transformation of subsets into each other. an example is the burning of methane in oxygen, which produces carbon dioxide and water ch4 + 2o2 → co2+ 2h2o. an important property of systems of chemical reactions is that the products can undergo further reactions, for instance with each other, e.g. co2 + h2o → h2co3, or with additional c© agt, upv, 2017 appl. gen. topol. 18, no. 2 256 oriented components and their separations reactants, say co2 + h2 → co + h2o. chemical reactions thus can be concatenated into overall reactions such as ch4 + 2o2 → h2co3 + h2o. as long as we are only interested in which molecules can be transformed into each other, we can neglect the multiplicities (called stoichiometric coefficients) and think of reactions as a relation on 2x. in our example {ch4, o2} → {co2, h2o}, and so on. an important question in the analysis of reaction networks is to understand which compounds can eventually be generated from which initial conditions. in particular in the field of biochemistry, there are also large databases of the chemical reactions that together describe the metabolic network of a cell [8]. in a more general setting rules that describe which reactions are possible and which are not can be specified in the form of graph grammars. these can then be used to computationally generate large chemical networks [1]. reachability in such a network amounts to asking whether, given a set q of compounds of interest and set p of starting molecules, it is possible to produce q by a sequence of consecutive reactions p ′ → q′ where initially p ′ ⊆ p and always q′ ⊆ q must hold. in subsequent steps we also allow the educts p ′ to be contained in the union of the products of earlier steps. this construction naturally leads to the notion of what we call here the production relation p ≻− q. since chemical reactions may run in parallel, reachability must be “additive” in the sense that pi ≻− qi for all i ∈ i should imply ⋃ i pi ≻− ⋃ qi. the re-use of products for consecutive reactions leads us to conclude that p ≻− q and s ≻− t for some s ⊆ q must imply p ≻− q ∪ t . in fact, we will be content with essentially these two properties when we formally define production relations in the next section. a somewhat different motivation for the same formal construction comes from evolutionary algorithms. here, the problem is to optimize a cost function f : x → r over some set x using a search operator that produces from a set of parents a ⊆ x a set of offsprings c(a). the same framework of course also models real evolution in biology [27, 25]. in the simplest case, mutations transform individual genotypes into their mutant offsprings, thus imposing a graph structure on x. in the usual setup of evolutionary strategies [19] and genetic algorithms [11], however, recombinants of two parents are used, i.e., the structure of the search space is determined implicitly by the relations of the form {x, y} → rxy. such relations have been investigated extensively under the name transit functions [16, 5], which serve as generalizations of betweenness relations and abstract convexities. the cost function f : x → r together with the topological structure on x implied by the search spaces give rise to “fitness landscapes” [30], in which concepts such as local minima, saddle points, and basins of attraction are well defined. these notions are inherently topological in nature and require only the definition of connected sets on x [9, 12, 26]. search operators are not necessarily symmetric, however. most recombination operators used in evolutionary computation tend to reduce diversity in the “search population” and thus are inherently directional. it is of interest, therefore, to consider reachability as a generalization of connectedness. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 257 in general, production relations naturally appear in all types of generative systems. when generation rules apply to individual elements, as in the case of mutations of gene sequences, reaches in the sense of [28] and [22] are obtained. generative grammars and (term) rewriting systems could also be viewed in this way. alternatively, generation rules apply to sets or multisets of elements rather than individual elements. a proper description then requires production relations in the sense of this contribution. within computer science this pertains in particular to large classes of “chemistry-like” formal systems such as berry’s chemical abstract machine [3]. certain constructions in natural languages, such the well known property of the german language to produce new nouns by concatenations of nouns, or the self-assembly of macromolecular complexes may serve as further examples. 3. separations, productions, and oriented components 3.1. galois correspondence. to relate the production relation ≻− and a separation relation | with each other, we introduce another relation ≬ on (2x × 2x) × (2x × 2x) that expresses when a pair (a, b) splits a pair (p, q): (3.1) (a, b) ≬ (p, q) ⇐⇒ p ∪ q ⊆ a ∪ b, p ⊆ b, and q ∩ a 6= ∅ the negation of ≬ will be written as ≬−. thus we have (a, b) ≬− (p, q) if and only if p ∪ q 6⊆ a ∪ b, or p /∈ b, or a ∩ q = ∅. the relation ≬− is a very natural way of connecting ≻− and |: if p ≻− q there is no a | b that splits p from q, and a | b means that there is no p ≻− q that “reaches” from b to a. fig. 1 illustrates the interesting case. p p q q b a b a (a, b) ≬ ({p}, q) (a, b) ≬− ({p}, q) figure 1. illustration of the splitting relation ≬ between two pairs of subsets. more formally, we have the galois connection comprising the two maps φ : 22 x ×2x → 22 x ×2x , {(p, q)|p ≻− q} 7→ {(a, b)|a |̇ b} θ : 22 x ×2x → 22 x ×2x , {(a, b)|a | b} 7→ {(p, q)|a ≻̇− b}. (3.2) where the induced relations |̇ and ≻̇−, resp., are defined by a |̇ b ⇐⇒ (a, b) ≬− (p, q) ∀(p, q) ∈ 2x with p ≻− q p ≻̇− q ⇐⇒ (a, b) ≬− (p, q) ∀(a, b) ∈ 2x with a | b (3.3) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 258 oriented components and their separations the theory of galois connections implies that θ(φ) and φ(θ) are closure operations on 22 x ×2x defining from a | and ≻− relations |̈ := φ(≻̇−) = φ(θ(|)) and ≻̇−:= θ(|̇) = θ(φ(≻−)), resp., that are in 1-1 correspondence. in other words, θ and φ induce a bijection between img θ and img φ. we are interested, therefore, in the properties of relations | and ≻− that correspond to elements of img θ and img φ, respectively. 3.2. basic properties. throughout this section we write |̇:= φ(≻−) for the separation relation induced by a given production relation ≻−; conversely ≻̇−:= θ(|) denotes the production relation induced by a given separation relation |. we start from an arbitrary production relation ≻− and strive to identify the properties of |̇ that are necessary for membership in img φ. theorem 3.1. given an arbitrary production relation ≻−, the corresponding relation |̇ satisfies for all a, b ∈ 2x: (s0) ∅ |̇ b for all b. (s1) a′ ⊆ a, b′ ⊆ b, and a |̇ b implies a′ |̇ b′. (sr1) a |̇ c and b |̇ a ∪ c implies a ∪ b |̇ c. (sr2) ai ∪ bi = z and ai |̇ bi for all i ∈ i implies ⋃ i∈i ai |̇ ⋂ i∈i bi proof. (s0) follows immediately from the definition of ≬−. (s1) suppose a′ ⊆ a and b′ ⊆ b. from a |̇ b we know that for all p ≻− q with p ∪ q ⊆ a ∪ b and p ⊆ b holds q ∩ a = ∅. the pairs p ≻− q satisfying p ∪ q ⊆ a′ ∪ b′ and p ⊆ b′ are a subset of the latter. furthermore, they also satisfy q ∩ a′ = ∅. thus a′ |̇ b′ holds. (sr1) suppose a |̇ c and b |̇ a ∪ c are satisfied but a ∪ b |̇ c does not hold, i.e., there is a production p ≻− q with p ⊆ c, q ⊆ a ∪ b ∪ c, and (a∪b)∩q 6= ∅. suppose first that q∩b = ∅; then a |̇ c implies q∩a = ∅. the desired production therefore must satisfy q∩b 6= ∅. since p ⊆ c implies p ⊆ a ∪ c we infer from b |̇ a ∪ c that q ∩ b = ∅, a contradiction. thus a ∪ b |̇ c cannot be violated. (sr2) suppose ai ∪ bi = z and ai |̇ bi holds for all i ∈ i. consider all p ≻− q with p∪q ⊆ z and p ⊆ bi for all i ∈ i, i.e., p ⊆ ⋂ i bi. by assumption, q ∩ ai = ∅ for all i and thus q ∩ ⋃ i ai = ∅. therefore ⋃ i ai |̇ ⋂ bi. � a space satisfying (s0) and (s1) can be seen as the most general form of a separation space generalizing even further the setting of wallace [29]. the axioms (sr1) and (sr2), on the other hand, appeared in ronse’s characterization of separation spaces that are defined by connectedness [20]. now we take the converse point of view. starting from an arbitrary “separation” relation | we determine properties of the production relation ≻̇− that is defined by the map θ to obtain necessary conditions for membership in img θ. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 259 lemma 3.2. given an arbitrary separation relation |, the corresponding relation ≻̇− satisfies (o) p ≻̇− ∅. (u) if pi ≻̇− qi for all i ∈ i then ⋃ pi ≻̇− ⋃ qi. ( union property) (t) if p ≻̇− q, s ⊆ q, and s ≻− t then p ≻− q ∪ t . ( transitivity) (t+) if p ≻̇− q, s ⊆ q ⊆ t and p ∪ s ≻− t then p ≻̇− t . ( source closure) proof. (o) given any collection of a|b we always have a ∩ q = ∅ if q = ∅, i.e, (a, b) ≬− (p, ∅) holds for all p ∈ 2x. (u) suppose pi ≻̇− qi for some family i ∈ i. thus, for all a | b with pi ⊆ b and pi ∪ qi ⊆ a ∪ b holds a ∩ qi = ∅. in particular, therefore, every a | b satisfying ⋃ i∈i pi ∈ b and ⋃ i∈i pi ∪ ⋃ i∈i qi ⊆ a ∪ b therefore also satisfies ⋃ i∈i qi ∩ a = ∅, and therefore ⋃ i∈i pi ≻̇− ⋃ i∈i qi holds. (t) suppose p ≻̇− q, s ⊆ q and s ≻̇− t . suppose p ≻̇− q ∪ t does not hold, i.e., there is a separation a | b with p ∪ q ∪ t ⊆ a ∪ b, p ⊆ b, and (q ∪ t ) ∩ a 6= ∅. we observe that q ∩ a 6= ∅ contradicts p ≻̇− q, hence q ∩ a = ∅, which implies q ⊆ b and therefore also s ⊆ b. therefore a ∩ t 6= ∅ contradicts s ≻̇− t , i.e., no such separation a | b can exist, and p ≻̇− q ∪ t must be true. (t+) suppose p ≻̇− q and p ∪ s ≻− t holds for s ⊆ q ⊆ t . note that p ∪s ≻− t implies by (u) also p ∪s ≻− t ∪q, hence we may choose t such that q ⊆ t . now suppose for contradiction that p ≻̇− t does not hold. then there is a a | b with p ∪ t = a ∪ b such that p ⊆ b and t ∩ a 6= ∅. since p ≻̇− q we have a ∩ q = ∅ and thus q ⊆ b, which further implies s ⊆ b. from p ∪ s ≻̇− t we know that a′ ∩ t = ∅ holds for all a′ | b′ with s ∪ p ⊆ b′ and a′ ∪ b′ = p ∪ q. this is in particular also true for a | b, a contradiction. � axiom (u) generalizes the “union property” of [22]. it allows multiple production to be “applied” at the same time. the transitivity axiom (t) encapsulates the idea that q is an “attractor” that is reached eventually. lemma 3.3. if (o) and (u) holds, then (t+) implies (t). proof. suppose p ≻− q and s ⊆ q and s ≻− t . by (u) also have p ∪s ≻− q∪t . setting t ′ = q ∪ t we have s ⊆ q ⊆ t ′ and p ∪ s ≻− t ′. now (t+) implies p ≻− t ′, i.e., t ≻− q ∪ t , and thus (t) holds. � we next observe that it is sufficient to consider the relationship of ≻− and | on a given subset y ∈ 2x. to this end we define for a given separation relation | and all pairs (p, q) ∈ 2x × 2x the collections (3.4) s(p, q) := { (a, b) ∈ 2x × 2x ∣ ∣a ∪ b = p ∪ q, p ⊆ b , and a | b } . of separated pairs on y = p ∪ q. the production relation ≻̇− can be specified completely by the sets s(p, q) by virtue of the following simple condition: lemma 3.4. p ≻̇− q if and only if q ∩ a = ∅ for all (a, b) ∈ s(p, q). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 260 oriented components and their separations proof. by definition, we have p ≻̇− q if and only if (a, b) ≬− (p, q) for all (a, b) ∈ 2x with a | b. if p 6⊆ b or p ∪q := y 6⊆ a∪b then (a, b) ≬− (p, q) always holds, i.e., these pairs never impose a condition and therefore they can be ignored. now consider the remaining case p ⊆ b and y ⊆ a ∪ b. thus (a, b) ≬− (p, q) holds if and only if a∩q = ∅. set a′ = a∩y and b′ = b ∩y and observe that p ⊆ b ∩ y iff p ⊆ b′ and q ∩ a = q ∩ (a ∩ y ) = q ∩ a′. therefore, (a, b) ≬− (p, q) if and only if (a′, b′) ≬− (p, q) and a ∩ q = ∅ if and only if a′ ∩ q = ∅. since a | b implies a′ | b′ by (s1) we conclude that p ≻̇− q holds if and only if (a′, b′) ≬− (p, q) and a′ | b′. the latter statement is equivalent expressed as q ∩ a′′ = ∅ for all (a′′, b′′) ∈ s(p, q). � now let us fix z and p ⊆ z and suppose q ∪ p = z. as a consequence of (sr2) there is a unique “extremal” separation a∗|b∗ with a∗ ∪ b∗ = z and p ⊆ b∗, which is defined by (3.5) a∗ = ⋃ (a,b)∈s(p,q) a and b∗ = ⋂ (a,b)∈s(p,q) b since a ∩ q = ∅ and p ⊆ b for all (a, b) ∈ s(p, q). corollary 3.5. suppose a∗ | b∗ as defined by s(p, q) as in equ.(3.5). then we have p ≻̇− q if and only if q ∩ a∗ = ∅. proof. by lemma 3.4, p ≻̇− q if and only if q∩a = ∅ for all (a, b) ∈ s(p, q). by equ.(3.5) this is equivalent to a∗ ∩ q = ∅. � the key observation here is that a∗|b∗ depends only on p and z but not on the exact choice of q as long as z \ p ⊆ q ⊆ z. thus p ≻̇− q implies a∗ ⊆ p \ q. furthermore, q ∩ a∗ = ∅ implies the same condition also for all subsets q′ of q. since the corollary holds as long as q′ ∪ p = q ∪ p = z, we have in particular the following implication: (a) p ≻̇− q implies p ≻̇− q′ for all q′ satisfying q \ p ⊆ q′ ⊆ q. we will give a more intuitive interpretation of property (a) in the following section. 3.3. oriented components. for every pair (p, q) ∈ 2x × 2x we define the set (3.6) q[p ] = ⋃ { q′ ∈ 2x ∣ ∣ q′ ⊆ q, p ′ ⊆ p and p ′ ≻− q′ } the map γ : 2x × 2x → 2x : (p, q) 7→ q[p ] generalizes the openings that play a central role e.g. in topological approaches to image analysis [23, 20, 15]. the sets q[p ] will be referred to as (generalized) oriented components. lemma 3.6. suppose ≻− is a production relation satisfying (u). then (o1) q[p ] ⊆ q. (contraction) (o2) p ′ ⊆ p and q′ ⊆ q implies q′[p ′] ⊆ q[p ]. (isotony) (o3) (q[p ])[p ] = q[p ]. (idempotency) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 261 proof. property (o1) is an immediate consequence of the definition. if p shrinks in equ.(3.6) then the union runs over fewer productions, and thus q[p ] cannot increase, i.e., p ′ ⊆ p implies q[p ′] ⊆ q[p ]. the same argument can be applied if q is reduced, hence q′ ⊆ q implies q′[p ] ⊆ q[p ]. now (o2) follows by combining the two inclusions. fix p and q. by definition every q′ ⊆ q with p ′ ⊆ p and p ′ ≻− q′ implies q′ ⊆ q[p ]. thus replacing q in the r.h.s. of equ.(3.6) by q[p ] does not change the collection of sets. since this substitution turns the definition of q[p ] into the definition of (q[p ])[p ], property (o3) holds. � we call a map 2x × 2x → 2x : (p, q) 7→ q[p ] that satisfies (o1), (o2), and (o3) a generalized opening. it defines a production relation by means of (3.7) p ≻− q if and only if q[p ] = q. an immediate consequene is the following fact 3.7. if ≻− satisfies (o) and (u) then p ≻− q[p ] for all p, q ∈ 2x. lemma 3.8. if (p, q) 7→ q[p ] is a generalized opening, then the corresponding production relation ≻− satisfies (o) and (u). proof. setting q = ∅ we have ∅[p ] = ∅, i.e., p ≻− q. suppose q[p ] = q and p ⊆ p ′. then isotony w.r.t. p implies q = q[p ] ⊆ q[p ′] ⊆ q, and thus q[p ′] = q. now consider an arbitrary family f of pairs (p, q) satisfying q[p ] = q and let p ∗ = ⋃ {p |(p, q) ∈ f}. we have q[p ] = q[p ∗] for all (p, q) ∈ f. isotony w.r.t. to q and condition (o1) now imply ⋃ q:(p,q)∈f q = ⋃ q:(p,q)∈f q[p ∗] ⊆   ⋃ q:(p,q)∈f q   [p ∗] ⊆ ⋃ q:(p,q)∈f q . with the abbreviation q∗ := ⋃ (p,q)∈f q we therefore have q ∗[p ∗] = q∗. in other words, p ≻− q for all (p, q) ∈ f implies p ∗ ≻− q∗, i.e., (u) holds. � theorem 3.9. eqns.(3.6) and (3.7) define a bijection between relations ≻− satisfying (o) and (u) on 2x and maps 2x × 2x → 2x satisfying (o1), (o2), and (o3). proof. it is shown e.g. in [22] that there is a bijection between the openings γp : (p, q) 7→ q[p ] and the set systems with the union property fp := {q|p ≻− q} for fixed p . the theorem now follows directly from lemma 3.6 and lemma 3.8, whose proofs also establish the correspondence p ≻− q =⇒ p ′ ≻− q for all p ⊆ p ′ and q[p ] = q =⇒ q[p ′] = q for all p ⊆ p ′ � let us now turn to additional properties of production relations that derive from separations. property (t) translates into a simple transitivity condition for generalized openings. the following lemma parallels prop. 2 of [22]: c© agt, upv, 2017 appl. gen. topol. 18, no. 2 262 oriented components and their separations lemma 3.10. let ≻− be a production relation satisfying (o) and (u) and let (q, p) 7→ q[p ] be the corresponding generalized opening. then axiom (t) is equivalent to (o4) s ⊆ q[p ] implies q[s] ⊆ q[p ]. proof. suppose (t) holds. from p ≻− q[p ], s ≻− q[s], and s ⊆ q[p ] we conclude p ≻− q[p ] ∪ q[s] ⊆ q. by maximality of the oriented connected components we therefore have q[p ] ∪ q[s] ⊆ q[p ], i.e., q[s] ⊆ q[p ], i.e., (o4) holds. conversely suppose (o4) is satisfied. assume p ≻− q, s ≻− t and s ⊆ q. thus we have q[p ] = q and t [s] = t and further q[p ] ⊆ (q ∪ t )[p ], s ⊆ (q∪t )[p ], and t ⊆ (q∪t )[s]. now (o4) implies (q∪t )[s] ⊆ (q∪t )[p ] and therefore t ⊆ (q∪t )[p ]. taken together we have q∪t ⊆ (q∪t )[p ] and thus q ∪ t = (q ∪ t )[p ]. this translates to p ≻− q ∪ t , i.e., (t) holds. � property (t+) becomes a transitivity condition in the arguments: lemma 3.11. if ≻− satisfies (o) and (u) and (q, p) 7→ q[p ] is the corresponding generalized opening, then (t+) is equivalent to (t+) if r ⊆ q[p ] then q[p ∪ r] = q[p ]. proof. suppose (t+) holds. substituting q by q[p ] and t by t [p ∪ s] transforms (t+) to: s ⊆ q[p ] ⊆ t [p ∪ s] and p ∪ s ≻− t [p ∪ s] implies p ≻− t [p ∪ s]. for the first precondition observe that q[p ] ⊆ t [p ∪ s] implies q[p ] = (q[p ])[p ] ⊆ (t [p ∪ s])[p ] ⊆ t [p ], and hence in particular s ⊆ q[p ] ⊆ t [p ]. the second precondition is always true and thus can be omitted. the definition of generalized oriented components, finally, implies t [p ∪ s] ⊆ t [p ] because t [p ∪ s] ⊆ t . on the other hand t [p ] ⊆ t [p ∪ s] by isotony. thus s ⊆ t [p ] implies t [p ] = t [p ∪ s]. conversely, assume p ≻− q, s ⊆ q ⊆ t , and p ∪ s ≻− t . therefore q = q[p ] ⊆ t [p ] and thus s ⊆ t [p ] and t [p ∪ s] = t . by (t+) we therefore have t [p ∪s] = t [p ]. by definition p ≻− t [p ], and thus t = t [p ], i.e., p ≻− t , i.e., (t+) holds. � in analogy to lemma 3.3 we have the following fact 3.12. if (o2) holds, then (t+) implies (o4). proof. assume (t+) and suppose s ∈ q[p ]. then q[s] ⊆ q[p ∪ s] = q[p ], i.e., (o4) holds � condition (a) is also easily translated to the language of generalized oriented components: lemma 3.13. if ≻− satisfies (o) and (u) and (q, p) 7→ q[p ] is the corresponding generalized opening, then (a) is equivalent to (a) q[p ] = (p ∪ q)[p ] ∩ q. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 263 proof. suppose ≻− satisfies (a). the set z[p ] is uniquely defined and by (o2) q[p ] ⊆ z[p ] for all q ⊆ z and by (o1) we have q[p ] ⊆ z[p ], and thus q[p ] ⊆ z[p ] ∩ q. if q ∪ p = z then (a) implies p ≻− z[p ] ∩ q which in turn guarantees z[p ] ∩ q ⊆ q[p ]; hence q[p ] = z[p ] ∩ q whenever p ∪ q = z. substituting z = p ∪ q establishes (a). to see the converse set z = p ∪q and assume (a), i.e., q[p ] = z[p ]∩q for all q satisfying z \ p ⊆ q ⊆ z. by construction of the generalized oriented components we have p ≻̇− q[p ] and thus p ≻̇− z[p ] ∩ q for z \ p ⊆ q ⊆ z. in particular for q ⊆ z[p ] this implies p ≻̇− q. on the other hand, p ≻̇− q and q ⊆ z implies q ⊆ z[p ]. therefore p ≻− q implies p ≻− q′ for all q′ with z \ p ⊆ q′ ⊆ q ⊆ z[p ], i.e., condition (a) holds. � in this form, property (a) lends itself to a simple intuitive interpretation. it states that the generalized opening is completely defined by the (q∪p)[p ], i.e., the oriented components q[p ] with p ⊆ q. this matches, of course, with the observation in the previous section that separations are defined by z = p ∪ q and p independent of the exact choice of q. for later reference we finally record a simple consequence of property (a): lemma 3.14. if (p, q) → q[p ] satisfies (o1) through (o4) and (a) then p \ q[p ] = p \ q′[p ] for q \ p ⊆ q′ ⊆ q ∪ p. proof. we set z = p ∪ q and hence z \ p ⊆ q ⊆ z. by isotony we have p \ z[p ] ⊆ p \ q[p ] ⊆ p \ (z \ p)[p ]) = p \ (z[p ] ∩ (z \ p)) = p \ z[p ] ∪ p \ (z \ p) = p \ z[p ]; here the first equality uses (a). thus p \ q′[p ] does not depend on the choice of q′ between z \ p = q \ p ⊆ q′ ⊆ p ∪ q. � 3.4. bijection. in order to show that there is a bijection between the production relations satisfying (o), (u), (t+), and (a) and the separation relations satisfying (s0), (s1), (sr1), and (sr2) it suffices to show that |̈ = φ(θ(|)) =| or ≻̈− = θ(φ(≻−)) =|. consider the map 2x × 2x → 2x : (p, q) 7→ q(p) such that (3.8) q(p) = q \ { a ∈ 2x ∣ ∣(a, b) ∈ s(p, q) } = q \ a∗ lemma 3.15. the map (p, q) 7→ q(p) satisfies (o1), (o2), (o3), (a), (t+), and thus also (o4). proof. (o1). q(p) ⊆ q follows directly from the definition of q(p). (o2). suppose p ′ ⊆ p , a ∪ b = p ∪ q, p ⊆ b, and a | b. thus (a, b) ∈ s(p, q). set y ′ = p ′ ∪ q. by isotony, a ∩ y ′ | b ∩ y ′ and thus (a ∩ y ′, b ∩ y ′) ∈ s(p ′, q). therefore y ′ ∩ a∗ := y ′ ∩ ⋃ {a|(a, b) ∈ s(p, q)} = ⋃ {y ′ ∩a|(a, b) ∈ s(p, q)} ⊆ ⋃ {y ′ ∩a|(a∩y ′, b∩y ′) ∈ s(p ′, q) =: y ′ ∩ã. by definition, q(p) = q \ a∗ = q \ (q ∩ a∗) = q \ (y ′ ∩ a∗) where we have used q ⊆ y ′. analogously we have q(p ′) = q \ ã = q \ (y ′ ∩ ã). therefore q(p ′) ⊆ q(p). next consider q′ ⊆ q and set y ′ = p ∪ q′. from a ∪ b = p ∪ q and p ⊆ b we infer (a ∩ y ′) ∪ b = p ∪ q′ and a ∩ q′ | b. therefore q′ ∩ {a|(a, b) ∈ s(p, q)} ⊆ {a ∩ y ′|(a ∩ y ′, b) ∈ s(p, q′)} = q′ ∩ c© agt, upv, 2017 appl. gen. topol. 18, no. 2 264 oriented components and their separations {a|(a ∩ y ′, b) ∈ s(p, q′)} = q′ ∩ {a|(a, b) ∈ s(p, q′)}, which in turn implies q′(p) ⊆ q(p). therefore (o2) holds. (a). we have q(p) := q \ a∗ = (z \ a∗) ∩ q = z(p) ∩ q = (p ∪ q)(p) because z(p) = z \ a∗, q ⊆ z and p ∪ q = z, and thus (a) is satisfied. before we proceed, we show that the sets q(p) are separated from their complements in q. from a∗ | b∗, z \ a∗ ⊆ b∗ and p ⊂ b∗ we obtain p ∪z \a∗ ⊆ b∗, which by isotony implies a∗ | p ∪z \a∗. from z(p) = z \a∗ we have a∗ = z \ z(p) and thus z \ z(p) | z(p) ∪ p . isotony immediately yields q ∩ (z \ z(p)) | (q ∩ z(p)) ∪ p , which by (a) reduces to (3.9) q \ q(p) | q(p) ∪ p for all p, q ∈ 2x. (t+) let r ⊆ q(p) and set a := q\q(p) and b := q(p)∪p . by equ.(3.9), a | b is a separation with r ∪ p ⊆ b. therefore q(r ∪ p) ⊆ q \ a = q(p). by (o2) we have q(p) ⊆ q(r ∪ p); therefore q(r ∪ p) = q(p). (o3) suppose that there is a separation a | c with a ∪ c = z(p) ∪ p and p ⊆ c. with b := z \ z(p) we have b | a ∪ c and thus by (sr1) also a∪b | c, i.e., a∪(z\z(p)) | c. since this separation by definition is contained in s(p, q), we have a∪(z \z(p)) ⊆ a∗ = z \z(p), i.e., a ⊆ z \z(p). thus a∩z(p) = ∅. we conclude: (*) all separations a | b with a∪b = z(p)∪p and p ⊆ b satisfy a ∩ z(p) = ∅. the definition of (z(p))(p) and (*) together imply z(p) ⊆ (z(p))(p). thus (o1) implies (z(p))(p) = z(p). now consider q(p) with p ∪ q = z. from (a) we have q(p) = q ∩ z(p) ⊆ z(p) and thus q(p) ∪ p = z. now (a) implies with q′(p) = q′ ∩ z(p) in particular also for q′ = q(p), i.e., (q(p))(p) = q(p) ∩ z(p). since q(p) ⊆ z(p) by (o2), we arrive at (q(p))(p) = q(p). (o4). property (t+) with r = q(p) implies q(q(p) ∪ p) ⊆ q(p) and thus by isotony of q( . ) we have q(q(p)) ⊆ q(q(p) ∪ p) ⊆ q(p). thus for any s ⊆ q(p), isotony also implies q(s) ⊆ q(q(p)) ⊆ q(p). � lemma 3.16. let | satisfy (s0), (s1), (sr1), and (sr2), let ≻̇− be the derived production relation with generalized oriented connected components (p, q) 7→ q[p ] and let (p, q) 7→ q(p) be the map defined in equ.(3.8). then for all p, q ∈ 2x holds q(p) = q[p ]. proof. by definition of q(p) we have q(p) = q if and only if a∗ = ∅, i.e., iff there is no separation a | b with p ∈ b and a 6= ∅, i.e., if and only if p ≻̇− q. by thm. 3.9 (p, q) 7→ q(p) bijectively maps to a unique production relation, which we have just seen is the same as p ≻̇− q. the bijection between the production relation ≻̇− and the generalized opening (p, q) 7→ q[p ] completes the proof. � theorem 3.17. there is a bijection between production relations ≻− satisfying (o), (u), (t+), and (a) and separation relations satisfying (s0), (s1), (sr1), and (sr2). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 265 proof. it suffices to show that a |̈ b if and only if a | b. the fact that | and ≻̇− are related by the galois connection defined in equ.(3.3) then immediately implies the theorem. suppose a |̈ b. for a = ∅ we trivially have a | b. thus assume a 6= ∅ and set y = a ∪ b. from a |̈ b we know that p ≻̇− q with p ⊆ b and p ∪ q ⊆ y must satisfy q ∩ a = ∅. in particular, the corresponding generalized oriented components satisfy q(p) ∩ a = ∅ for all p ⊆ b and q ⊆ y in particular y (b) ∩ a = ∅. by equ.(3.9) we have y \ y (b) | y (b) ∪ b. from a ⊆ y and y (b) ∩ a = ∅ we conclude a ⊆ y \ y (b) and thus, by (s1) a | y (b) ∪ b and finally a | b. conversely, assume a | b. lemma 3.16 implies that the generalized connected components w.r.t. ≻̇− are given by equ.(3.8). now suppose a |̈ b does not hold. then there is a p ≻̇− q with p ∪q ⊆ a∪b, p ⊆ b, and q∩a 6= ∅. this is impossible, however, since by construction a ∩ y (b) = ∅ and q ⊆ y (b). thus a |̈ b. � we recall, finally, that axiom (a) means that (p, q) → q[p ] and hence the relation ≻− is uniquely defined by the p ≻− q or q[p ] with p ⊆ q. in other words, we might want to restrict the definition of production relations ≻− and of generalized oriented components q[p ] to the domain {(p, q)|p ⊆ q ⊆ x}. in this condition axiom (a) becomes void and thus can be omitted. 4. properties of separation and production relations throughout this section we assume | and ≻− are connected by their natural bijection (3.3) and that the q[p ] are the equivalent generalized oriented components. in particular, | satisfies (s0), (s1), (sr1), and (sr2), ≻− satisfies (o), (u), (a), (t+), and (p, q) 7→ q[p ] satisfies (o1), (o2), (o3), (t+), and (a). recall that axioms (o4) and (t), resp., also hold by lemma 3.3 and fact 3.12. 4.1. the membership property. theorem 4.1. the following three conditions are equivalent (sr0) a | b implies a ∪ b | a ∩ b. (m) if p ⊆ q then p ′ ⊆ p \ q[p ] implies q[p ′] = ∅. (m) p ≻− q, p ′ ≻− q′, p ′ ⊆ p \ q, and q′ ⊆ q implies q′ = ∅. proof. suppose (sr0) holds, a | b and z = a ∪ b. we have b ≻− z[b] and thus z[b]∩a = ∅, i.e., z[b] ⊆ b \ a = z \ a. furthermore, let {ai|i ∈ i|} be the family of sets with ai ∪ b = z and ai | b and set â = ⋃ i∈i ai. by (sr2) we have â | b. from z[b] ∩ ai = ∅ for all i and z \ z[b] | z[b], which is a special case of equ.(3.9), we conclude â = z \ z[b]. by (sr0) a ∪ b | a ∩ b, i.e., for every p ′ ⊆ a∩b holds z[p ′]∩(a∪b) = z[p ′]∩z = ∅, i.e., z[p ′] = ∅. this is true in particular also for all p ′ ⊆ â ∩ b = b ∩ (z \ z[b]) = b \ z[b]. by lemma 3.14 we have b \ z[b] = b \ q[b], i.e., p ′ ∈ b \ q[b] implies q[p ′] = z[p ′] = ∅. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 266 oriented components and their separations conversely, suppose (m) holds but there is a pair a | b such that a ∪ b | a∩b does not hold, i.e., there is a point y ∈ a∪b = z and a p ′ ∈ a∩b with y ∈ q[p ′] for some q ⊆ z. from a | b we have q[b] ⊆ b \ a. since a | b we have a ⊆ z \ z[b] and thus p ′ ⊆ a ∩ b ⊆ b ∩ (z \ z[b]) = b ⊆ z[b]. now property (m) implies z[p ′] = ∅. since q[p ′] ⊆ z[p ′] we arrive at the desired contradiction. suppose (m) holds, p ≻− q, p ′ ≻− q′, p ′ ⊆ p \ q[p ], and q ⊆ q′. then by definition, q = q[p ] and q′ = q′[p ′]. isotony implies q′[p ′] ⊆ q[p ′]. by (m) q[p ′] = ∅ and thus q′ = ∅, i.e., (m) holds. finally, suppose (m) holds. consider p ′ ⊆ p \ q[p ] and suppose p ′ ≻− q′ for some q′ ⊆ q. by definition of generalized connected components, q′ ⊆ q[p ′] ⊆ q. from p ≻− q[p ′] and (m) we conclude q[p ′] = ∅, i.e., (m) holds. � we shall see below that (m) corresponds to the “membership condition” employed in [22] in the context of oriented components: “p /∈ q[{p}] implies q[{p}] = ∅” for p ∈ q. property (sr0), on the other hand, appeared in [26] as a condition for the existence of a bijection between connected components and symmetric separations. 4.2. the point source property. ronse’s [22] work on oriented components can be embedded into the current framework by assuming that reachability from individual points completely specifies the production map. this property, which we call here the point source property is most easily expressed in terms of generalized oriented components as q[p ] = ⋃ p∈p q[{p}]. theorem 4.2. the following three conditions are equivalent (sr2+) let ai ∪ bi = z and ai | bi for all i ∈ i. then ⋂ ai | ⋃ bi. (d) if p ≻− q and q is maximal in p ∪ q then there are {p} ≻− qp for p ∈ p such that ⋃ p∈p qp = q. (d) if p ⊆ q then q[p ] = ⋃ p∈p q[{p}]. proof. (d)=⇒(sr2+). suppose (d) holds, ai ∪ bi = z and ai | bi for all i ∈ i. from ai | bi we known that for every p ⊆ bi we have, for every q ⊆ z, q[p ] ∩ ai = ∅ and thus in particular also q[{p}] ∩ ai = ∅. for every p ∈ ⋃ i∈i bi we therefore have q[{p}] ∩ ⋂ i∈i ai = ∅ and thus by (d) z[p ]∩ ⋂ i∈i ai = ∅ and therefore also q[p ]∩ ⋂ i∈i ai = ∅ for all p ⊆ ⋃ i∈i bi, and thus ⋂ i∈i ai | ⋃ i∈i bi. (sr2+)=⇒(d). fix z and p ⊆ z. choose i so that bi = {p} and ai = z\z[{p}] for some p ∈ p . we have shown in the proof of lemma (3.15) that z\ z[p ] | z[p ]∪p . therefore ai | bi holds, and hence by (sr2+), also ⋂ p∈p (z \ z[p ]) | ⋃ p∈p z[{p}], i.e., z \ ⋃ p∈p z[{p}] | ⋃ p∈p z[{p}]. furthermore we know that p ≻− ⋃ p∈p z[{p}] by (u). by equ.(3.9) we have z \ ⋃ p∈p z[p] | p ∪ ⋃ p∈p z[{p}]. now p ≻− z[p ] implies z[p ] ∩ z \ ⋃ p∈p z[{p}] = ∅, i.e., z[p ] ⊆ ⋃ p∈p z[{p}], i.e., z[p ] = ⋃ p∈p z[{p}]. since z = q for p ⊆ q, (d) follows. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 267 (d)=⇒(d). suppose p ≻− z and assume that z is maximal in p ∪ z. thus z = (p ∪ z)[p ]. set q = p ∪ z. by construction z = q[p ]. property (d) implies that there are qp with {p} ≻− qp and qp ⊆ q such that ⋃ p∈p qp = q[p]. we have qp ⊆ q[{p}] ⊆ q and thus q[p ] = ⋃ p∈p qp ⊆ ⋃ p∈p q[p] ⊆ q[p ], and hence q[p ] = ⋃ p∈p q[p] whenever p ⊆ q, i.e., (d) holds. (d)=⇒(d) we may assume p ⊆ q, thus q[p ] is the maximal set in q∪p = q satisfying p ≻− q. we simply choose qp = q[{p}]. by construction {p} ≻− qp and q[p ] = ⋃ p∈p q[{p}] = ⋃ p∈p qp. � property (a) now implies q[p ] = ⋃ p∈p (q ∪ p)[{p}] ∩ q whenever p 6⊆ q. in general we only have (4.1) ⋃ p∈p q[{p}] ⊆ q[p ] equality is guaranteed only if p ⊆ q. lemma 4.3. the following three axioms are equivalent for separation relations, production relations and generalized connected components satisfying the conditions of thm. 3.17. (s0+) a | ∅ for all a ∈ 2x. (g) ∅ ≻− q implies q = ∅. (g) q[∅] = ∅. proof. suppose (s0+) holds, i.e., a ∩ q = ∅ for all a ∈ 2x and all q such that ∅ ≻− q. thus q = ∅, i.e. (g) follows. conversely suppose g holds. then a | ∅ follows since a ∩ q = ∅ for q = ∅. � fact 4.4. axiom (d) implies axiom (g). proof. q[∅] = ⋃ p∈∅ q[{p}] = ∅. � lemma 4.5. if (d) holds, then (t) implies (t+). proof. suppose r ∈ q[p ]. then q[p ∪ r] = ⋃ p∈p ∪r (p ∪ q)[{p}] ∩ q ⊆ ⋃ p∈p ((p ∪ q)[{p}] ∩ q) ∪ ⋃ r∈r ((p ∪ q)[{r}] ∩ q) = (p ∪ q)[p ] ∩ q = q[p ] because, by (t), (p ∪ q)[{r}] ⊆ (p ∩ q)[p ] for all r ∈ q[p ]. thus (t+) holds. � 4.3. reaches. a reach [22] is a family r ⊆ 2x × x satisfying the following three axioms: (r1) let f ⊆ 2x and (f, p) ∈ r for all f ∈ f. then ( ⋃ f ∈f f, p) ∈ r. (union property) (r2) if (q, p) ∈ r, s ∈ q, and (t, s) ∈ r then (q ∪ t, p) ∈ r. (transitivity) (r3) if (p, f) ∈ r then f = ∅ or p ∈ f . (membership property) as shown in [22] there is a bijection of set systems satisfying (r1) and systems of point openings, that is, maps 2x × x → 2x, (f, p) 7→ f(p) satisfying the following three axioms for all p ∈ x and all f, f ′ ∈ 2x: (p1) f(p) ⊆ f , c© agt, upv, 2017 appl. gen. topol. 18, no. 2 268 oriented components and their separations (p2) f ′ ⊆ f implies f ′(p) ⊆ f(p), (p3) (f(p))(p) = f(p), the bijection is established by virtue of f(p) := ⋃ {s|(s, p) ∈ r and s ⊆ f} r = {(f, p)|f = f(p)} . (4.2) now consider a production relation ≻− satisfying (o), (u), (d), and (a) and the corresponding generalized opening. in this case q[p ] is completely determined by the q[{p}] with p ∈ q. we can think of the q[{p}] as a point opening. axiom (a) specializes to (pa) if p /∈ f then f(p) = (f ∪ {p})(p) \ {p}. and uniquely defines the sets f(p) whenever p /∈ f . an equivalent condition is (ra) if p ∈ f and (f, p) ∈ r then (f \ {p}, p) ∈ r. we call a set system r ⊆ 2x × x a pre-reach if it satisfies (r1) and (ra). we note here that this construction differs from the definition in [22], where f(p) = ∅ is stipulated for p /∈ f . in either case, however, (f, p) 7→ f(p) is uniquely determined by the subset {(f, p)|p ∈ f} on which [22] and our definition does agree. therefore there is an obvious 1-1 correspondence. lemma 4.6. there is a bijection between pre-reaches, i.e., set systems r on 2x × x satisfying (r1) and (ra) and production relations ≻− on 2x satisfying (o), (u), (a), and (d), by virtue of (f, p) ∈ r if and only if {p} ≻− f. proof. it is easier to argue in terms of the openings f(p) and the generalized oriented components q[p ], respectively. first we recall that q[p ] is uniquely determined by (q ∪ p)[p ] through property (a), i.e., the q[p ] with p 6⊆ q are determined by those with p ⊆ q. for the latter, however, property (d) implies that they are uniquely determined by the q[{p}] with p ∈ q. if p /∈ q, property (a) again determines q[{p}]. since (a) coincides with (ra) for |p | = 1, it extends to equality f(p) = f [{p}] to all p ∈ x. it follows that f(p) = f [{p}] induces a bijection between openings satisfying (ra) and generalized openings satisfying (a) and (d). indeed (o1), (o2), and (o3) trivially specialize to (p1), (p2), and (p3). it remains to show that (p1), (p2), and (p3) together with (d) implies (o1), (o2), and (o3). for p ⊆ q we have q[p ] = ⋃ p∈p q[{p}] ⊆ q and otherwise q[p ] = (q ∪ p)[p ] ∩ q ⊆ q, i.e., (o1) holds. next suppose p ′ ⊆ p and q′ ⊆ q. by (ii) we have (q′ ∪ p ′)[p] ⊆ (q ∪ p)[p]. therefore q′[p ′] = (q′ ∪p ′)[p ′]∩q′ = q′ ∩ ⋃ p∈p ′ (q′ ∪p ′)[p] ⊆ q∩ ⋃ p∈p (q∪ p)[p] = q ∩ (q ∪ p)[p ] = q[p ], i.e. (o2) holds. in particular, we have (q[{p}])[{p}] ⊆ (q[p ])[{p}] ⊆ (q[p ])[p ] for all p ∈ p and therefore also ⋃ p∈p (q[p])[{p}] ⊆ (q[p ])[p ]. now suppose p ⊆ q. then (d) implies q[p ] = ⋃ p∈p q[{p}] and (p3) implies q[p ] = ⋃ p∈p (q[{p}])[{p}] and therefore q[p ] ⊆ (q[p ])[p ]. now (o2) implies the desired equality q[p ] = (q[p ])[p ]. finally, if p 6⊆ q we have (q[p ])[p ] = (q[p ] ∪ p)[p ] ∩ q[p ] = c© agt, upv, 2017 appl. gen. topol. 18, no. 2 269 ((q∪p)[p ]∩q)[p ]∩q[p ] by (o2) ((q∪p)[p ])[p ]∩q[p ] ⊆ ((q∪p)[p ]∩q)[p ]. on the other hand ((q ∪ p)[p ])[p ] ∩ q[p ] = (q ∪ p)[p ] ∩ q[p ] = q[p ], and thus q[p ] ⊆ (q[p ])[p ]. thus (o3) holds. � a family r ⊆ 2x × x is transitive if (rt) (f, p) ∈ r, q ∈ f and (g, q) ∈ r implies (f ∪ g, p) ∈ r. it is shown in [22] that if r satisfies (r1), then (rt) is equivalent to (pt) q ∈ f(p) implies f(q) ⊆ f(p) for the corresponding system of point openings. in our setting we have the following, analogous result: lemma 4.7. let ≻− be a production relation satisfying (o), (u), and (d) and let r be the corresponding pre-reach. then (r2) is equivalent to (t) and (t+). proof. specializing (t) to single point sources yields {p} ≻− q, s ∈ q, and {s} ≻− t implies {p} ≻− q ∪ t , which translates to (q, p) ∈ r, s ∈ q, and (t, s) ∈ r implies (q ∪ t, p) ∈ r, i.e, (r2). now suppose {p} ≻− qp for all p ∈ p , s ⊆ q = ∪p∈p qp and s ≻− ts for s ∈ s. then p ≻− q, and for every s ∈ s there is p such that s ∈ qp and therefore {p} ≻− qp ∪ ts. using t = ⋃ s∈s ts and thus by (u) s ≻− t , and applying (u) again yields p ≻− q, s ⊆ q and s ≻− t implies p → q ∪ t , i.e., (t) holds. finally, by lemma 4.5 (d) and (t) imply (t+). � in the presence of (r1), it is shown in [22] that (r3) is equivalent to p ∈ q(p) or q(p) = ∅. this condition obviously clashes with (pa). the reason is that for p /∈ q, [22] stipulates q(p) = ∅, while we have made another choice with condition (pa), which in turn is motivated by (a). instead, we use the specialization of (m) to singleton sets p : (pm) if p ∈ q then p ∈ q(p) or q(p) = ∅ to see that (pm) is the proper specialization of (m) we simply note that for p ′ = p = {p} we have p ′ ⊆ p \ q[p ] = {p} ⊆ {p} \ q[{p}] which translates to p /∈ q[{p}]. lemma 4.8. let ≻− be a production relation satisfying (o), (u), (a) and (d) and let r be the corresponding pre-reach. then (m) and (pm) are equivalent. proof. we first show that (m) reduces to (pm) for p = {p}. assume p ∈ q. if p ∈ q[{p}] then p ′ = ∅; otherwise p ′ = {p} = p and thus p /∈ q[{p}]. now (m) implies q[p ′] = q[{p}] = ∅, and hence (pm) follows. conversely, suppose (pm) holds and suppose p ⊆ q. then by (d) q[p ] = ⋃ p∈p q[{p}]. let p̄ = p \ q[p ]. then for p′ ∈ p̄ holds p′ /∈ q[p ] and thus also p′ /∈ q[{p′}], hence (pm) implies q[{p′}] = ∅. by (d) we have q[p̄] = ⋃ p′∈p̄ q[{p′}] = ∅. isotony now implies q[p ′] = ∅ for all p ′ ⊆ p \ q[p ], i.e., (m) holds. � we can summarize this discussion in this section in the following form: c© agt, upv, 2017 appl. gen. topol. 18, no. 2 270 oriented components and their separations corollary 4.9. there is a bijection between the generalized oriented components satisfying (o1), (o2), (o3), (o4), (m), (a), and (d) and the system of point openings satisfying (p1), (p2), (p3), (pt), (pm), and (pa). the latter coincides for p ∈ q with the system of point openings that is equivalent to reaches in the sense of ronse [22]. proof. the first statement is a direct consequence of lemmas 4.6, 4.7, and 4.8. by virtue of (pa) and the choice made in [22] to set q(p) = ∅ whenever p /∈ q, the system of point openings is completely defined by (q, p) 7→ q(p) for p ∈ q. here q(p) = q[{p}], i.e., the bijection is just the identity on this subset. � finally, we can relate reaches to separation spaces: corollary 4.10. there is a bijection between the generalized oriented components satisfying (o1), (o2), (o3), (o4), (m), (a), and (d) and separation spaces satisfying (s0), (s0+), (s1), (sr0), (sr1), (sr2), and (sr2+). the same is true for the reaches as defined in [22]. this result directly generalizes the 1-1 correspondence between connectivity spaces and symmetric separations satisfying the same axioms [26]. we will return to this point in section 4.5. 4.4. disjunctiveness. consider the following properties: (s3) a | b implies a ∩ b = ∅. (p) p ≻− p for all p ∈ 2x. (p) p ⊆ q implies p ⊆ q[p ]. lemma 4.11. suppose | and ≻− are corresponding separation and production relations, and let {q[p ]} be the corresponding system of generalized oriented components. then (s3), (p), and (p) are equivalent. proof. suppose a | b. by assumption, we have in particular b ≻− b. it follows immediately that b ∩ a = ∅. conversely, assume a | b implies a ∩ b = ∅ and suppose p ≻− p does not hold for some p ∈ 2x. then there is a | b such that p ⊆ b and p ∩ a 6= ∅, whence a ∩ b 6= ∅, a contradiction. suppose p ⊆ q. from p ≻− p we immediately conclude p ⊆ q[p ]. conversely, consider p ⊆ p [p ]. by isotony we have p [p ] ⊆ p and therefore p = p [p ] and thus p ≻− p . � lemma 4.12. suppose (p) holds. then (m) is equivalent to (g) and, equivalently, (sr0) reduces to (s0+). proof. since p ⊆ q[p ] for all p ⊆ q, we have p ′ = p \ q[p ] = ∅. thus (m) reduces to q[∅] = ∅, i.e., axiom (g). analogously, we can argue that (s3) simplifies (sr0) to a | b implies a ∪ b | ∅. by (s0) we have ∅ | b and thus b | ∅ for all b ∈ 2x. � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 271 4.5. symmetry axioms. the natural symmetry axiom for separations is (s2) a | b implies b | a. lemma 4.13. suppose | satisfies (s2). then (s0) and (s0+), (sr2) and (sr2+) are equivalent. furthermore, (s2) and (sr2) implies (sr0). proof. ∅ | a implies a | ∅ for all a ∈ 2x and vice versa. suppose ai | bi for all ∈ i and (sr2) holds. then by (s2) bi | ai. (sr2) ensures ⋃ i∈i bi | ⋂ i∈i ai. using (s2) again we also have ⋂ i∈i ai | ⋃ i∈i bi, i.e., (sr2+) holds. the implication in the other direction is shown analogously. finally, if a | b then by (s2) also b | a. applying (sr2) with z = a ∪ b, a1 = a, b1 = b, a2 = b, and b2 = a results in a ∩ b | b ∩ a, i.e., (sr0) holds. � as shown in [26], there is a bijection between the “partial connections” of [21] and separation spaces satisfying (s0), (s1), (s2), (sr0), (sr1), and (sr2). equivalent constructions have been considered e.g. in [23, 7, 4, 17, 6], see [26] for a detailed overview. lemma 4.13 above shows that the (sr0) axiom thus is redundant in [26] and can simply be omitted. the (s2) axiom, via (sr0) and (sr2+) implies both the membership property (m) and point definedness (d) for the corresponding production relation and the generalized oriented components, i.e., the (s2) axiom takes us automatically to the reaches of [22]. the symmetry condition for the corresponding oriented components can be paraphrased in our notation as (s) if p ∈ q and r ∈ q[{p}] then q[{p}] ⊆ q[{r}]. by (o4) we have q[{r}] ⊆ q[{p}] for r ∈ q[{p}], and thus under our general assumptions axiom (s) is equivalent to q[{p}] = q[{r}]. lemma 4.14. (s2) is equivalent to (m), (d), and (s). proof. suppose (s2) holds. by lemma 4.13 we have (sr2+) and (sr0), and thus (m) and (d). in particular either x ∈ q[{x}] or q[{x}] = ∅ for all x ∈ x. suppose p ∈ q and r ∈ q[{p}]; by (o4) this implies q[{r}] ⊆ q[{p}]. now there are two cases to consider. (1) if p ∈ q[{r}] then (o4) implies q[{p}] ⊆ q[{r}], and thus q[{r}] = q[{p}]. (2) otherwise p /∈ q[{r}]. by equ.(3.9) we have q \ q[{r}] | q[{r}] and by (s2) also q[{r}] | q \ q[{r}]. since p ∈ q \ q[{r}], the correspondence of separation and generalized oriented components implies q[{p}] ∩ q[{r}] = ∅. the assumption r ∈ q[{p}] thus implies r /∈ q[{r}] and thus by (m) q[{r}] = ∅ and therefore q | {r}. using (s2) again implies {r} | q and thus p ∈ q implies q[{p}] ∩ {r} = ∅, i.e., r /∈ q[{p}], a contradiction. thus r ∈ q[{p}] implies p ∈ q[{r}]. conversely, suppose (m), (d), and (s) hold. suppose (s2) does not hold, i.e., there is a | b with a ∪ b = q but b | a does not hold. then for all p ∈ b holds q[{p}] ∩ a = ∅ and there is some y ∈ a such that q[{y}] ∩ b 6= ∅. consider x ∈ b ∩ q[{y}]. from x ∈ b we infer q[{x}] ∩ a = ∅. on the other hand, axiom (s) implies q[{x}] = q[{y}] and thus q[{y}] ∩ a = ∅, whence c© agt, upv, 2017 appl. gen. topol. 18, no. 2 272 oriented components and their separations y /∈ q[{y}]. now (m) implies q[{y}] = ∅, a contradiction. thus (s2) must hold. � the axiom for production relations corresponding to (s) is (s) if {p} ≻− s then q ≻− s for all q ∈ s. if ≻− corresponds to a reach, i.e., in addition to (o), (u), (t), and (a) we also have (d) and (m), then (s) is equivalent to (s) for the corresponding system of generalized oriented components [22]. concluding remarks we have shown here that production relations are in 1-1 correspondence with a class of proximity spaces satisfying simple, rather natural axioms. this establishes a natural topological interpretation of directed hypergraphs in general and chemical reaction networks in particular. the usefulness of this correspondence is further supported by the equivalence of simple properties in both axiom systems. many open questions remain for future research. the natural morphisms between proximity spaces, proximal maps f : (x, δ ) → (y, δ ), preserve proximity, i.e., a δ b implies f(a) δ f(b). it will be interesting to investigate the properties of the corresponding maps for production relations. these should be related to the “catenous functions” introduced in [17]. on this basis one can hope to investigate the natural product structures. this might be interesting for production relations defined on cartesian products such as the sequence spaces (hamming graphs) typically used to model molecular evolution. the wallace closure w : 2x → 2x associated with a proximity is usually defined as x ∈ w(b) iff {x} δ b. the wallace closure can be viewed as a generalization of kuratowski’s closure function for topologies. we will explore this connection to generalized topologies in forthcoming work. such closure functions have been used as an alternative way of associating a topological structure with chemical reaction networks [2]. exactly how this approach is connected with the production relations used here remains to be explored in the future. acknowledgements. bmrs gratefully acknowledges the hospitality of the santa fe institute, where much of this work has been performed. references [1] j. l. andersen, c. flamm, d. merkle and p. f. stadler, generic strategies for chemical space exploration, int. j. comp. biol. drug design 7 (2014), 225–258. 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[30] s. wright, the roles of mutation, inbreeding, crossbreeeding and selection in evolution, in: d. f. jones (ed.), proceedings of the sixth international congress on genetics, vol. 1, 1932, pp. 356–366. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 275 () @ appl. gen. topol. 19, no. 1 (2018), 55-64doi:10.4995/agt.2018.7630 c© agt, upv, 2018 set-open topologies on function spaces wafa khalaf alqurashi a, liaqat ali khan b and alexander v. osipov c a department of mathematical science, college of applied sciences, umm al-qura university, p.o. box 7944, makkah-24381, saudi arabia. (wafa-math@hotmail.com) b department of mathematics, quaid-i-azam university, islamabad, pakistan. (lkhan@kau.edu.sa) c krasovskii institute of mathematics and mechanics, ural federal university, ural state university of economics, p.o. box 620219, yekaterinburg, russia. (oab@list.ru) communicated by a. tamariz-mascarúa abstract let x and y be topological spaces, f(x, y ) the set of all functions from x into y and c(x, y ) the set of all continuous functions in f(x, y ). we study various set-open topologies tλ (λ ⊆ p(x)) on f(x, y ) and consider their existence, comparison and coincidence in the setting of y a general topological space as well as for y = r. further, we consider the parallel notion of quasi-uniform convergence topologies uλ (λ ⊆ p(x)) on f(x, y ) to discuss uλ-closedness and right uλ-k-completeness properties of a certain subspace of f(x, y ) in the case of y a locally symmetric quasi-uniform space. we include some counter-examples to justify our comments. 2010 msc: 54c35; 46a16; 54e15; 54c08. keywords: set-open topology; pseudocompact-open topology; c-compactopen topology; quasi-uniform convergence topology; right kcompleteness, α-continuous function. 1. introduction one of the original set-open topologies on c(x, y ) is the compact-open topology tk, which was introduced by fox [7] and further developed by arens [2], gale [8], myers [24], arens-dugundji [3] and jackson [12]. later, many received 28 april 2017 – accepted 05 february 2018 http://dx.doi.org/10.4995/agt.2018.7630 w. k. alqurashi, l. a. khan and a. v. osipov other set-open topologies tλ (λ ⊆ p(x)) were investigated that lie between tk and tw (the largest set-open topology) (see, e.g., [9, 10, 15, 17, 5, 27, 29]). apart from the set-open topologies, there is also a parallel notion of ”uniform convergence topologies” uλ (λ ⊆ p(x)) on f(x, y ) which were discussed in datail by kelley [13] in the case of y = (y, u) a uniform space and by naimpally [25] in the case of y = (y, u) a quasi-uniform space. these have been further investigated by several authors, including papadopoulos [34], kunzi and romaguera [20] and more recently in [1]. these topologies uλ (λ ⊆ p(x)) are, in general, different from their corresponding set-open topologies tλ (λ ⊆ p(x)) even in the case of y a metric space, but the two notions coincide in some particular cases (see [4, 13, 17, 23, 28, 29, 33]). in this paper, we study various set-open topologies on f(x, y ) in the setting of x and y arbitrary topological spaces. in section 2, we study their comparability and also coincidence of such topologies; we also discuss their existence and their relationship with some uniform convergence topologies. in section 3, we establish some results on closedness and completeness of the space cα(x, y ) of all α-continuous functions, from x into y [16, 21]. here, we shall need to assume that y is a regular topological space, which is equivalent to y being a locally symmetric quasi-uniform space [6, 25]. 2. set-open topologies on f(x, y ) recall that the space x is said to be pseudocompact if every f ∈ c(x) is bounded on x. a subset a of x is called c-compact (resp. bounded) if the set f(a) is compact (resp. bounded) in r for every f ∈ c(x). if a = x, the property of the set a to be c-compact (bounded) coincides with the pseudocompactness of x. notations. for any topological space x, let p(x) denote the power set of x, and let f(x) = {a ⊆ x : a is finite}, k(x) = {a ⊆ x : a is compact}, ps(x) = {a ⊆ x : a is pseudocompact}, rc(x) = {a ⊆ x : a is c-compact}. clearly, f(x) ⊆ k(x) ⊆ ps(x) ⊆ rc(x). recall that a collection λ ⊆ p(x) is called a network on x if, for each x ∈ x and each open neighborhood u of x, there exists an a ∈ λ such that x ∈ a ⊆ u. a network λ on x is called a closed network on x if each a ∈ λ is closed. since, for each x ∈ x and each open neighborhood u of x, x ∈ {x} ⊆ u, it is clear that each of the collections f(x), k(x), ps(x), rc(x), p(x) is a network on x. if x is a hausdorff space, then f(x), k(x) are closed networks. definition 2.1 (cf. [22, 5]). let x and y be topological spaces, and let λ ⊆ p(x) be a network which covers x. for any a ∈ λ and open g ⊆ y , let n(a, g) = {f ∈ f(x, y ) : f(a) ⊆ g}. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 56 set-open topologies on function spaces then the collection {n(a, g) : a ∈ λ, open g ⊆ y } forms a subbase for a ”set-open” topology on f(x, y ), called the λ-open topology (cf. [3], p. 13) and denoted by tλ; see also [7, 2, 13, 23]). using this terminology, we can consider the following topologies on f(x, y ): (1) f(x)-open topology, the usual point-open topology tp. (2) k(x)-open topology, the usual compact-open topology tk. (3) ps(x)-open topology, the pseudocompact-open topology tps [18]. (4) rc(x)-open topology, the c-compact-open topology trc [29, 30, 31, 32, 33]. if x and y are any two spaces with the same underlying set, then we use x = y, x ≤ y , and x < y to indicate, respectively, that x and y have the same topology, that the topology on y is finer than or equal to the topology on x, and that the topology on y is strictly finer than the topology on x. the symbols r and n denote the spaces of real numbers and natural numbers, respectively. for convenience, we shall some times denote (f(x, y ), tλ) by fλ(x, y ). if (x, τ) is a topological space and a ⊆ x, the closure of a is denoted by a or τ-cl(a); the interior of a is denoted by int(a) or τ-int(a). lemma 2.2. let x and y be non-empty sets. suppose that a, b ⊆ x, and let g, h ⊆ y be non-empty sets such that n(b, h) ⊆ n(a, g). then: (i) if a 6= ∅, then h ⊆ g. (ii) if g 6= y, then a ⊆ b. proof. (i) suppose a 6= ∅, but h * g, and let c ∈ h\g. consider the constant function fc : x → y defined by fc(x) = c (x ∈ x). if b = ∅, then fc(b) = ∅ ⊆ h; if b 6= ∅, then fc(b) = {c} ⊆ h. hence fc ∈ n(b, h). since a 6= ∅, fc(a) = {c} * g and so fc /∈ n(a, g). this contradicts n(b, h) ⊆ n(a, g). therefore h ⊆ g. (ii) suppose a * b, and let x0 ∈ a\b. since g 6= y , choose a ∈ y \g. let p ∈ h. define g = gb,p : x → y by g(x) = p if x ∈ b, g(x) = a if x ∈ x\b. then g ∈ f(x, y ). if b = ∅, then g(b) = ∅ ⊆ h; if b 6= ∅, then g(b) = {p} ⊆ h. hence g ∈ n(b, h). since x0 ∈ a\b ⊆ x\b, g(x0) = a /∈ g, g /∈ n(a, g). this contradicts n(b, h) ⊆ n(a, g). therefore a ⊆ b. � theorem 2.3. let x be a hausdorff topological space and y any topological space. then: (a) fp(x, y ) ≤ fk(x, y ) ≤ fps(x, y ) ≤ frc(x, y ). (b) fk(x, y ) = fps(x, y ) iff every closed pseudocompact subset of x is compact. (c) fk(x, y ) = frc(x, y ) iff every closed c-compact subset of x is compact. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 57 w. k. alqurashi, l. a. khan and a. v. osipov (d) fp(x, y ) = frc(x, y ) iff every c-compact subset of x is finite. in particular, if x is discrete, then fp(x, y ) = fk(x, y ) = fps(x, y ) = frc(x, y ). proof. (a) since f(x) ⊆ k(x) ⊆ ps(x) ⊆ rc(x), it follows that fp(x, y ) ≤ fk(x, y ) ≤ fps(x, y ) ≤ frc(x, y ). (b) suppose fps(x, y ) ≤ fk(x, y ), and let h ⊆ y be open and f be any closed pseudocompact subset of x, with h 6= y . let fc(x) = c (x ∈ x) where c ∈ h. then there exist a compact subsets k1, ..., kn of x and open subsets g1, ..., gn of y such that fc ∈ n⋂ i=1 n(ki, gi) ⊆ n(f, h). consider k = n⋃ i=1 ki and g = n⋂ i=1 gi, then n(k, g) ⊆ n(f, h). by lemma 2.2(ii), f ⊆ k. thus, f is compact. conversely, suppose that every closed pseudocompact subset of x is compact. note that for a subset a of x, n(a, g) ⊆ n(a, g). it follows that fps(x, y ) ≤ fk(x, y ). (c) this can be proved in a manner similar to (b). (d) suppose frc(x, y ) ≤ fp(x, y ), and let k ⊆ x be a c-compact subset of x and h ⊆ y be open, with h 6= y . let fc(x) = c (x ∈ x) where c ∈ h. then there exist a finite subsets f1, ..., fn of x and open subsets g1, ..., gn of y such that fc ∈ n⋂ i=1 n(fi, gi) ⊆ n(k, h). consider f = n⋃ i=1 ki and g = n⋂ i=1 gi, then n(f, g) ⊆ n(k, h). by lemma 2.2(ii), k ⊆ f ; hence k is finite. conversely, suppose that every c-compact subset of x is finite. to show frc(x, y ) ≤ fp(x, y ), take arbitrary n(k, g) ∈ frc(x, y ) with k ⊆ x a c-compact subset of x and g ⊆ y an open set. then k is finite. taking f = k, n(f, g) ∈ fp(x, y ) and n(f, g) ⊆ n(k, g). hence n(k, g) ∈ fp(x, y ) and so frc(x, y ) ≤ fp(x, y ). in particular, if x is discrete, then every c-compact subset of x is finite and hence fp(x, y ) = frc(x, y ). � remark 2.4. we can also define the tλ-open topologies on c(x, y ) by taking the collection {nc(a, g) : a ∈ λ, open g ⊆ y } as its subbase, where nc(a, g) = {f ∈ c(x, y ) : f(a) ⊆ g}. in this case, lemma 2.2 need not hold. example 2.5. let x = r (with the usual topology), y = {0, 1} (with the discrete topology) and λ be a family of connected subsets of x. then nc([0, 1], {0}) ⊆ nc(r, {0}), but r *[0, 1]. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 58 set-open topologies on function spaces however, under some additional hypotheses, an analogue of lemma 2.2 for c(x, y ) may be obtained as follows: proposition 2.6. let x be a tychonoff (completely regular hausdorff) space, and y a hausdorff topological space containing a non-trivial path p : [0, 1] 7→ y such that p(0) 6= p(1), and let g and h be open sets in y such that p(0) ∈ h and p(1) /∈ g. suppose that a, b ⊆ x are non-empty sets such that nc(b, h) ⊆ nc(a, g). then a ⊆ b. proof. (cf. [23], p. 5) let p : [0, 1] → y be a path (continuous function) in y such that p(0) 6= p(1). suppose nc(b, h) ⊆ nc(a, g), but a * b. then there exists some x0 ∈ a\b. since x is a tychonoff space, there exists a ϕ ∈ c(x, [0, 1]) such that ϕ(b) = {0} and ϕ(x0) = 1. then p◦ϕ ∈ c(x, y ) with p◦ϕ ∈ nc(b, h). indeed, for any x ∈ b, (p◦ϕ)(x) = p(ϕ(x)) = p(0) ∈ h, hence (p◦ϕ)(b) ⊆ h. but p◦ϕ /∈ nc(a, g), since x0 ∈ a and (p ◦ ϕ)(x0) = p(ϕ(x0)) = p(1) /∈ g, which is a contradiction. � remark 2.7. let x and y be topological spaces. if λ = {x}, we may consider the notion of {x}-open topology on f(x, y ), denoted by tx. however, this topology would not be of much use in most settings. in fact, if x is a compact space, then tx 6= tk, in general. example 2.8. let x = [0, 1] and y = r. then tx 6= tk on f([0, 1], r). the set n({0}, (0, 1)) ⋂ n({1}, (2, 3)) ∈ tk, but is not in tx. remark 2.9. if λ = σ(x) is a family of all σ-compact subsets of x or λ = σ0(x) is a family of all countable subsets of x, we may consider the notions of σ(x)open and σ0(x)-open topologies on f(x, y ), denoted by tσ and tσ0, respectively. however, these topologies may not have ”good” topological-algebraic properties. a space fλ(x, y ) may not be a topological vector space or a topological group, as is shown by the following example. example 2.10. let x = y = r. we consider f(x, y ) and λ = {n and all finite subsets of x}. then the set w = n(n, (−π/2, π/2)) is a tσ0-open set in f(x, y ). consider the function f(x) = arctan(x), x ∈ r. clearly, that f ∈ w . but we do not find a basis neighborhood b of 0 ∈ f(x, y ) such that f + b ⊆ w . it follows that the space fλ(x, y ) is not a topological vector space. remark 2.11. we mention that the situation is more useful and interesting if these topologies are considered on c(x, y ) with y = (y, ρ) a metrizable topological vector space and in particulat on c(x) = c(x, r). let x be a tychonoff space and λ ⊆ p(x). in addition to the tλ-topology on c(x, y ), we can define (following the terminology of [22, 17, 29]) the notion of tλ∗-topology on c(x, y ) which has a subbase as the collection {n∗c (a, g) : a ∈ λ, open g ⊆ y }, where n∗(a, g) = {f ∈ c(x, y ) : f(a) ⊆ g}. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 59 w. k. alqurashi, l. a. khan and a. v. osipov here the modification f(a) ⊆ g in place of f(a) ⊆ g is due to mccoy and ntantu ([22]) who used it in order to generalize the compact-open topology to real-valued noncontinuous functions, to balance the disadvantage a is compact but f(a) is not compact. in this regard, we can also consider the topology of uniform convergence on elements of λ (the λ-topology) on c(x, y ), denoted by cλ,u(x, y ), which has a base at each f ∈ c(x, y ) as the collection {< f, a, ε >: a ∈ λ, ε > 0}, where < f, a, ε >= {g ∈ c(x, y ) : sup x∈a ρ(f(x), g(x)) < ε}. theorem 2.12 ([29]). let x be a tychonoff space and y a metrizable topological vector space, and let λ ⊆ p(x). (a) if cλ(x, y ) = cλ,u(x, y ), then the family λ consists of c-compact sets. conversely, if λ consists of c-compact sets, then cλ(x, y ) ≤ cλ,u(x, y ). (b) if cλ∗(x, y ) = cλ,u(x, y ), then the family λ consists of bounded sets. conversely, if λ is a family consisting of bounded sets such that a ∩ w ∈ λ for every functionally open (co-zero) set w with a∩w 6= ∅, then cλ∗(x, y ) ≤ cλ,u(x, y ). we next give a brief account of the σ-compact-open and some related topologies on c(x) = c(x, r) ([17]). let λ ⊆ p(x) be any family satisfying the condition: if a, b ∈ λ, then there exists a c ∈ λ such that a ∪ b ⊆ c. the σ(x)-open topology (the usual σ-compact-open topology) on c(x) has a subbase as the family {n∗σ(a, g) : a ∈ σ(x), g ∈ b},where b is the set of bounded open intervals in r and we denote this space by cσ(x). since, for each f ∈ c(x), f(a) ⊆ f(a); so that f(a) = f(a). hence the same topology is obtained by using n∗c (a, g), where a ∈ σ(x) and b ∈ b. when λ = σ0(x), we get the countable-open topology, denoted by cσ0(x). for reader’s convenience, we summarize some important known properties of these topologies (without proof), as follows: theorem 2.13 ([17]). let x be a tychonoff space. then: (a) ck(x) ≤ cσ(x) ≤ cσ,u(x) ≤ cu(x). (b) ck(x) = cσ(x) iff every σ-compact subset of x has compact closure. (c) cσ(x) = cσ,u(x) iff x is pseudocompact. (d) cσ,u(x) = cu(x) iff x contains a dense σ-compact subset. (e) if x is separable, then cσ,u(x) = cu(x). (f) if every countable subset of x has compact closure, then cσ0,u(x) ≤ cu(x). 3. closedness and completeness in f(x, y ) the results of this section are motivated by those given in [13, 25, 20] regarding the closedness and completeness of certain function subspaces in (f(x, y ), ux). it is well-known (e.g., [13], p. 227-229) that, for y a uniform space, c(x, y ) is ux-closed in f(x, y ) but not necessarily up-closed. later, c© agt, upv, 2018 appl. gen. topol. 19, no. 1 60 set-open topologies on function spaces some authors also obtained variants of these results for some other classes of functions, not necessarily continuous [11, 16, 34]. in this section, we establish some results for the class cα(x, y ) of all ”α-continuous” functions from x into y [21, 16]. we first recall necessary background for quasi-uniform spaces. let y be a non-empty set. a filter u on y ×y is called a quasi-uniformity on y [6] if it satisfies the following conditions: (qu1) △(y ) = {(y, y) : y ∈ y } ⊆ u for all u ∈ u. (qu2) if u ∈ u, there is some v ∈ u such that v 2 ⊆ u. (here v 2 = v ◦ v = {(x, y) ∈ y × y : ∃ z ∈ y such that (x, z) ∈ v and (z, y) ∈ v }.) in this case, the pair (y, u) is called a quasi-uniform space. if, in addition, u satisfies the symmetry condition: (u3) u ∈ u implies u −1 := {(y, x) : (x, y) ∈ u} ∈ u, then u is called a uniformity on y and the pair (y, u) is called a uniform space. a quasi-uniform space (y, u) is called locally symmetric if, for each y ∈ y and each u ∈ u, there is a symmetric v ∈ u such that v 2[y] ⊆ u[y] [6, 25], where u[y] = {z ∈ y : (y, z) ∈ u}. if (y, u) is a quasi-uniform space, then the collection t (u) = {h ⊆ y : for each y ∈ h, there is u ∈ u such that u[y] ⊆ h} is a topology on y , called the topology induced by u. it is well-known that every topological space is quasi-uniformizable [35, 6] and every regular topological space is a locally symmetric quasi-uniform space [6, 25]. in view of this, for any topological space y , we may assume, without loss of generality, that y = (y, u) with u a quasi-uniformity. main advantage of this assumption is that one can introduce various notions of cauchy nets and completeness. in contrast to the case of uniform spaces, the formulation of the notion of ”cauchy net” or ”cauchy filter” in quasi-uniform spaces has been fairly complicated, and has been approached by several authors (see, e.g., the survey paper by kunzi [19]). we shall find it convenient to restrict ourselves to the notions of ”right k-cauchy net” and ”right k-complete”, as in [20, 36]. recall that a net {yα : α ∈ d} in a quasi-uniform space (y, u) is said to be t (u)-convergent to y ∈ y if, for each u ∈ u, there exists an α0 ∈ d such that yα ∈ u[y] for all α ≥ α0. a net {yα : α ∈ d} in y is called a right k-cauchy net provided that, for each u ∈ u, there exists some α0 ∈ d such that (yα, yβ) ∈ u for all α, β ∈ d with α ≥ β ≥ α0. (y, u) is called right k-complete if each right k-cauchy net is t (u)-convergent in y (cf. [20], lemma 1, p. 289). we now consider the notions of quasi-uniform convergence topologies on f(x, y ), which are parallel to those of the set-open topologies. let x be a topological space and (y, u) a quasi-uniform space, and let λ ⊆ p(x) be a collection which covers x. for any a ∈ λ and u ∈ u, let û|a = {(f, g) ∈ f(x, y ) × f(x, y ) : (f(x), g(x)) ∈ u for all x ∈ a}. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 61 w. k. alqurashi, l. a. khan and a. v. osipov then the collection {û|a : a ∈ λ and u ∈ u} forms a subbase for a quasiuniformity, called the quasi-uniformity of quasi-uniform convergence on the sets in λ induced by u. the resultant topology on f(x, y ) is called the topology of quasi-uniform convergence on the sets in λ [20] and is denoted uλ (or by fλ,u(x, y ) by some authors). (1) if λ = {x}, uλ is called the quasi-uniform topology of uniform convergence on f(x, y ) and denoted by ux. (2) if λ = k(x), uλ is called the quasi-uniform topology of compact convergence on f(x, y ) and denoted by uk. (3) if λ = f(x), uλ is called the quasi-uniform topology of pointwise convergence on f(x, y ) and denoted by up. clearly, up ≤ uk ≤ ux. we shall require the following result which extends ([13], theorem 8, p. 226-227) from uniform to quasi-uniform spaces. lemma 3.1 ([1]). let x be a topological space and (y, u) a quasi-uniform space. let {fα : α ∈ d} be a net in f(x, y ) such that: (i) {fα : α ∈ d} is a right k-cauchy net in (f(x, y ), ux), (ii) fα up −→ f on x. then fα ux −→ f. recall that a subset a of (x, τ) is called: α-open [26] if a ⊆ int(cl(inta))). a function f : x → y is said to be α-continuous [21] if f−1(h) is α-open in x for each open set h in y ; equivalently, if, for each point x of x and for each neighborhood h of f(x), there exists an α-neighborhood g of x such that f(g) ⊂ h. let cα(x, y ) denote the set of all α-continuous functions from x into y . it is easy to see that c(x, y ) ⊆ cα(x, y ). theorem 3.2. let x be a topological space and (y, u) a locally symmetric quasi-uniform space. then: (a) cα(x, y ) is ux-closed in f(x, y ). (b) if y is right k-complete, then cα(x, y ) is right ux − k-complete. proof. (a) let f ∈ f(x, y ) with f ∈ ux − cl[c α(x, y )]. let x0 ∈ x and h any open neighborhood of f(x0) in y . choose u ∈ u such that u[f(x0)] ⊆ h. choose a v ∈ u with v = v −1 and such that v 2[f(x0)] ⊆ u[f(x0)]. choose w ∈ u such that w 2 ⊆ v . there exists g ∈ cα(x, y ) such that g ∈ ŵ [f]. then we have (f(y), g(y)) ∈ w ⊆ w 2 ⊆ v for all y ∈ x. by α-continuity of g at x0, there exists an α-neighborhood g containing x0 in x such that g(g) ⊆ v [f(x0)]. then, for any y ∈ g, f(y) ∈ v −1[g(y)] = v [g(y)] ⊆ v [v [f(x0)]] ⊆ u[f(x0)] ⊆ h. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 62 set-open topologies on function spaces thus f ∈ cα(x, y ). (b) suppose y is right k-complete, and let {fα : α ∈ i} be a right ux − kcauchy net in cα(x, y ). let u ∈ u and let x ∈ x be fixed. there exists α0 ∈ i such that (fα(y), fβ(y)) ∈ u for all α ≥ β ≥ α0 and y ∈ x. in particular, {fα(x) : α ∈ i} is a right k-cauchy net in y . consequently, we have an f ∈ f(x, y ) such that fα up −→ f. then, by lemma 3.1, fα ux −→ f, and, by part (a), f ∈ cα(x, y ). thus (cα(x, y ), ux) is right k-complete. � corollary 3.3 ([16], theorem 3.2, p. 950). let xbe a topological space and (y, u) a uniform space. then: (a) cα(x, y ) is ux-closed in f(x, y ). (b) if y is complete then cα(x, y ) is ux-complete. finally, we mention that the quasi-uniform topologies uλ (λ ⊆ p(x)) are, in general, different from their corresponding tλ-open topologies (λ ⊆ p(x)) even in the case of y = r. example 3.4. let x = r = y , and let λ = {n} and g = (−1, 1), an open set in y = r. then nc(a, g) = {f ∈ c(r, r) : f(n) ⊆ (−1, 1)} is a tλ-open set in c(r, r), but it is not an open set in the topology of uniform convergence uλ on c(r, r). for more recent contribution on the coincidence of set-open and uniform convergence topologies, the interested reader is refered to the papers [4, 29, 33]. acknowledgements. the authors wish to thank professors h. p. a. künzi and r. a. mccoy for communicating to us useful information of various concepts used in this paper and also the anonymous referee for his/her comments that helped us to correct some errors and improve the presentation. references [1] w. k. alqurashi and l. a. khan, quasi-uniform convergence topologies on function spacesrevisited, appl. gen. top. 18, no. 2, (2017), 301–316. 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[36] s. romaguera, on hereditary precompactness and completeness in quasi-uniform spaces, acta math. hungar. 73 (1996), 159–178. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 64 () @ appl. gen. topol. 19, no. 2 (2018), 253-260doi:10.4995/agt.2018.9009 c© agt, upv, 2018 τ-metrizable spaces a. c. megaritis technological educational institute of peloponnese, department of computer engineering, 23100, sparta, greece (thanasismeg13@gmail.com) communicated by d. georgiou abstract in [1], a. a. borubaev introduced the concept of τ-metric space, where τ is an arbitrary cardinal number. the class of τ-metric spaces as τ runs through the cardinal numbers contains all ordinary metric spaces (for τ = 1) and thus these spaces are a generalization of metric spaces. in this paper the notion of τ-metrizable space is considered. 2010 msc: 54a05; 54e35. keywords: τ-metric space; τ-metrizable space; τ-metrization theorem. 1. preliminaries and notations our notation and terminology is standard and generally follows [2]. the cardinality of a set x is denoted by |x|. throughout, we denote by τ an arbitrary nonzero cardinal number. the cardinalities of the natural numbers and of the real numbers are denoted by ℵ0 and c, respectively. the character, the weight and the density of a topological space x are denoted by χ(x), w(x) and d(x), respectively. as usual i denotes the closed unit interval [0, 1] with the euclidean metric topology. by rτ+ we denote the topological product of τ copies of the space r+ = [0, +∞) (with the natural topology). on the space rτ+, the operations of addition, multiplication, and multiplication by a scalar, as well as a partial ordering, are defined in a natural way (coordinatewise). now, we present the notion of τ-metric space [1]. let x be a nonempty set. a mapping ρτ : x × x → r τ + is called a τ-metric on x if the following axioms hold: received 28 november 2017 – accepted 02 april 2018 http://dx.doi.org/10.4995/agt.2018.9009 a. c. megaritis (1) ρτ(x, y) = θ if and only if x = y, where θ is the point of the space r τ + whose all coordinates are zeros. (2) ρτ(x, y) = ρτ(y, x) for all x, y ∈ x. (3) ρτ(x, z) 6 ρτ (x, y) + ρτ(y, z) for all x, y, z ∈ x. the pair (x, ρτ) is called a τ-metric space and the elements of x are called points. every τ-metric space (x, ρτ ) generates a tychonoff (that is, completely regular and hausdorff) topological space (x, tρτ ). the topology tρτ on x defined by the local basis consisting of the sets of the form g(x) = {y ∈ x : ρτ (x, y) ∈ o(θ)}, where o(θ) runs through all open neighbourhoods of the point θ in the space r τ +, of each point x ∈ x is called the topology induced by the τ-metric ρτ. in this paper the notion of τ-metrizable space is introduced. the paper is organized as follows. section 2 contains the basic concepts of τ-metrizable spaces. generally, τ-metrizable spaces may be not metrizable. we prove that if τ 6 ℵ0, then every τ-metrizable space is metrizable. in section 3 we obtain a generalization of the classical metrization theorem of urysohn. more precisely, we prove that every tychonoff space of weight τ > ℵ0 is τ-metrizable. finally, in section 4 we prove that every compact τ-metrizable space has density less than or equal to τ. 2. basic concepts the notion of a τ-metric space leads to the notion of a τ-metrizable space which is inserted in the following definition. definition 2.1. a topological space (x, t ) is called τ-metrizable if there exists a τ-metric ρτ on the set x such that the topology tρτ induced by the τ-metric ρτ coincides with the original topology t of x. τ-metrics on the set x which induce the original topology of x will be called τ-metrics on the space x. note that τ-metrizable spaces are useful because only such spaces can be presented as limits of τ-long projective systems of metric spaces [1, theorem 3]. proposition 2.2. a metric space is τ-metrizable. proof. let (x, ρ) be a metric space, tρ be the topology induced by the metric ρ, and let τ be a cardinal number. consider a set λ such that |λ| = τ and set ρλ = ρ for each λ ∈ λ. the mapping ρτ : x × x → r τ + defined by ρτ(x, y) = {ρλ(x, y)}λ∈λ for every x, y ∈ x is a τ-metric on x. it is easy to see that tρ = tρτ . � proposition 2.3. a τ-metrizable space is τ′-metrizable for every cardinal number τ′ > τ. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 254 τ-metrizable spaces proof. let x be a τ-metrizable space, ρτ be a τ-metric on the space x and τ′ be a cardinal number such that τ′ > τ. consider two sets k and λ such that k ⊂ λ, |k| = τ and |λ| = τ′, and set ρτ(x, y) = {ρ k τ(x, y)}k∈k for every x, y ∈ x. let k0 be one fixed element of k. the mapping ρτ′ : x × x → r τ′ + defined by ρτ′(x, y) = {ρ λ τ′(x, y)}λ∈λ for every x, y ∈ x, where ρλτ′(x, y) = { ρλτ (x, y), if λ ∈ k ρk0τ (x, y), if λ ∈ λ \ k, is a τ′-metric on x such that tρ τ ′ = tρτ . � the following examples show that τ-metrizable spaces may be not metrizable. example 2.4. the product rc = ∏ λ∈λ xλ, where xλ = r for every λ ∈ λ and |λ| = c, of uncountably many copies of the real line r is not metrizable, since it is not first-countable. however, the space rc is c-metrizable. assuming each copy xλ of r has its usual metric dλ, the mapping ρc : r c×rc → rc+ defined by ρc(x, y) = {dλ(xλ, yλ)}λ∈λ for every x = {xλ}λ∈λ ∈ r c and y = {yλ}λ∈λ ∈ r c is a c-metric on rc and the topology induced by ρc coincides with the product topology. example 2.5. let r be the set of real numbers with the discrete topology d and (r∞, d∞) be the alexandroff’s one-point compactification of the space (r, d), that is r∞ = r∪{∞} and d∞ = d∪{r∞\k : k is a finite subset of r}. the space (r∞, d∞) is not metrizable (because it is not separable). we prove that the space (r∞, d∞) is c-metrizable. let fin(r) be the collection of all the nonempty finite subsets of r with |fin(r)| = c. for every f ∈ fin(r) we define: (1) ρf (x, x) = 0 for each x ∈ r∞. (2) ρf (x, ∞) = ρf (∞, x) = { 0, if x /∈ f 1, otherwise for each x ∈ r. (3) ρf (x, y) = { 0, if x /∈ f and y /∈ f 1, otherwise for each x, y ∈ r with x 6= y. the mapping ρc : r∞ ×r∞ → r c + defined by ρc(x, y) = {ρf (x, y)}f ∈f in(r) for every x, y ∈ r∞ is a c-metric on r∞. we prove that the topology tρ c induced by the c-metric ρc coincides with the topology d∞. let x ∈ r. if g(x) = {y ∈ r∞ : ρc(x, y) ∈ o(θ)}, where o(θ) is an open neighbourhood of the point θ in the space rc+, then {x} ∈ d∞ and {x} ⊆ g(x). moreover, for the open neighbourhood ∏ f ∈f in(r) wf of the point θ, where wf = { [0, 1 2 ), if f = {x} r+, otherwise we have g(x) = {y ∈ r∞ : ρc(x, y) ∈ ∏ f ∈f in(r) wf } ⊆ {x}. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 255 a. c. megaritis now, we consider the point ∞ of r∞. if {∞} ∪ (r \ k), where k ∈ fin(r) is an open neighbourhood of the point ∞ in the space r∞, then for the open neighbourhood ∏ f ∈f in(r) wf of the point θ, where wf = { [0, 1 2 ), if f = k r+, otherwise we have g(∞) = {y ∈ r∞ : ρc(∞, y) ∈ ∏ f ∈f in(r) wf } ⊆ {∞} ∪ (r \ k). finally, let ∏ f ∈f in(r) uf be an open neighbourhood of the point θ in the space rc+ and suppose that {f ∈ fin(r) : uf 6= r+} = {k1, . . . , km}. then, {∞} ∪ (r \ (k1 ∪ . . . ∪ km)) ⊆ g(∞) = {y ∈ r∞ : ρc(∞, y) ∈ ∏ f ∈f in(r) uf }. however, a τ-metrizable space may be metrizable considering addition conditions as the following assertions show. proposition 2.6. a n-metric space is metrizable for every finite cardinal number n. proof. let (x, ρn) be a n-metric space and tρn be the topology induced by ρn. consider a vector expression of the form ρn(x, y) = (ρ 1 n(x, y), . . . , ρ n n(x, y)) for every x, y ∈ x. the mapping ρ : x × x → r+ defined by ρ(x, y) = max{ρ1n(x, y), . . . , ρ n n(x, y)} for every x, y ∈ x is a metric on x. it is easy to see that the metric topology is the same as tρn. � definition 2.7. two τ-metrics ρ1τ and ρ2τ on a set x are called equivalent if they induce the same topology on x, that is tρ1τ = tρ2τ . example 2.8. let ρτ be a τ-metric on x. consider a set λ such that |λ| = τ and let us set ρτ(x, y) = {ρ λ τ (x, y)}λ∈λ for every x, y ∈ x. the mapping ρ∗τ : x × x → r τ + defined by ρ ∗ τ (x, y) = { min{1, ρλτ (x, y)} } λ∈λ for every x, y ∈ x is a τ-metric on x equivalent to ρτ. proposition 2.9. an ℵ0-metric space is metrizable. proof. let (x, ρ ℵ0 ) be an ℵ0-metric space. consider the equivalent ℵ0-metric ρ∗ ℵ0 to ρ ℵ0 of example 2.8. let ρ∗ ℵ0 (x, y) = (ρ∗1 ℵ0 (x, y), ρ∗2 ℵ0 (x, y), . . .) for every x, y ∈ x. the mapping ρ : x × x → r+ defined by ρ(x, y) = ∞ ∑ i=1 1 2i ρ∗i ℵ0 (x, y) for every x, y ∈ x is a metric on x. the process of proving that the topology induced by the metric ρ coincides with the topology tρ ℵ0 is similar to the proof of the theorem 4.2.2 of [2]. � corollary 2.10. if τ 6 ℵ0, then every τ-metrizable space is metrizable. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 256 τ-metrizable spaces proof. follows directly from propositions 2.6 and 2.9. � proposition 2.11. for each τ > ℵ0 there is a τ-metrizable space xτ with w(xτ ) = τ, which is not metrizable. proof. let xτ be the alexandroff’s one-point compactification of a discrete space x of cardinality τ, where τ > ℵ0. the space xτ is not metrizable (because it is not separable). it is known that |fin(x)| = |x| = τ. therefore, in the same manner as in example 2.5, we can prove that the space xτ is τ-metrizable. let us note that w(xτ ) = τ. � proposition 2.12. for every τ > ℵ0 and every τ-metrizable space x, we have χ(x) 6 τ. proof. let x be a τ-metrizable space and ρτ be a τ-metric on the space x with τ > ℵ0. consider a set λ such that |λ| = τ. the family bθ of all products ∏ λ∈λ wλ, where finitely many wλ are intervals of the form [0, b) with rational b and the remaining wλ = r+, form a local basis of the point θ in the space r τ +. hence, for every x ∈ x, the family b(x) = {g(x) = {y ∈ x : ρτ(x, y) ∈ b} : b ∈ bθ} is a local basis of the point x in the space x. since |bθ| = τ, we have |b(x)| 6 τ. � 3. a τ-metrization theorem metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable (see [2,5]). in this section we obtain a generalization of the classical metrization theorem of urysohn. lemma 3.1. if (x, ρτ) is a τ-metric space and a is a subspace of x, then the topology induced by the restriction of the τ-metric ρτ to a × a is the same as the subspace topology of a in x. theorem 3.2. every tychonoff space of weight τ > ℵ0 is τ-metrizable. proof. let x be a tychonoff space such that w(x) = τ > ℵ0. the space iτ = ∏ λ∈λ xλ, where xλ = i for every λ ∈ λ and |λ| = τ is τ-metrizable (see example 2.4). assuming each copy xλ of i has its usual metric dλ, the mapping dτ : i τ × iτ → rτ+ defined by dτ(x, y) = {dλ(xλ, yλ)}λ∈λ for every x = {xλ}λ∈λ ∈ i τ and y = {yλ}λ∈λ ∈ i τ is a τ-metric on iτ. we shall prove that x is τ-metrizable by imbedding x into the τ-metrizable space iτ, i.e. by showing that x is homeomorphic with a subspace of iτ . but this follows immediately from the fact that the tychonoff cube iτ is universal for all tychonoff spaces of weight τ (see [2, theorem 2.3.23]). by lemma 3.1, the space x is τ-metrizable. � as every τ-metrizable space is tychonoff (see [1]), we get the following result. corollary 3.3. a space of weight τ > ℵ0 is τ-metrizable if and only if it is tychonoff. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 257 a. c. megaritis remark 3.4. we can use theorem 3.2 to find τ-metrizable spaces, where τ > ℵ0, that are not metrizable. below we consider some examples. example 3.5 is a c-metrizable space which is not second-countable, example 3.6 is a cmetrizable space which is not normal and example 3.7 is a 2τ-metrizable space, where τ > c, which is not metrizable. example 3.5. let s be the sorgenfrey line, that is the real line with the topology in which local basis of x are the sets [x, y) for y > x. since x is separable but not second-countable, it cannot be metrizable. furthermore, s is tychonoff and w(s) = c. from theorem 3.2 it follows that the sorgenfrey line is a c-metrizable space. example 3.6. let p = {(α, β) ∈ r2 : β > 0} be the open upper half-plane with the euclidean topology and l = {(α, β) ∈ r2 : β = 0}. we set x = p ∪l. for every x ∈ p let b(x) be the family of all open discs in p centered at x. for every x ∈ l let b(x) be the family of all sets of the form {x}∪d, where d is an open disc in p which is tangent to l at the point x. the family t of all subsets of x that are unions of subfamilies of ∪{b(x) : x ∈ x} is a topology on x and the family {b(x) : x ∈ x} is a neighbourhood system for the topological space (x, t ). the space x is called the niemytzki plane (see, for example, [2, 4]). x is a tychonoff space with w(x) = c, which is not normal. therefore, by theorem 3.2, x is a c-metrizable space, but not metrizable. example 3.7. let βd(τ) be the čech-stone compactification of the discrete space d(τ) of cardinality τ > c. then, w(βd(τ)) = 2τ (see [2, theorem 3.6.11]). since βd(τ) is zero-dimensional (see [2, theorem 3.6.13]), it is tychonoff. the space d(τ) is not compact. therefore, βd(τ) is not metrizable (see [3, exercise 9, §38, ch.5]). from theorem 3.2 it follows that βd(τ) is 2τmetrizable. particularly, if one assumes the continuum hypothesis, the čechstone compactification βω of the discrete space of the non-negative integers ω = {0, 1, 2, . . .} is c-metrizable. remark 3.8. a space x may be τ-metrizable for some infinite cardinal number τ < w(x), as shown in the following example. example 3.9. let λ be a set of cardinality τ > ℵ0, d(κ) the discrete space of cardinality κ > τ, and f = ∏ λ∈λ xλ, where xλ = d(κ) for every λ ∈ λ, with the tychonoff product topology. we note that the points of f are functions from λ to d(κ). the space f is not metrizable for χ(f) = τ (see [2, exercise 2.3.f(b)]). moreover, w(f) = κ (see [2, exercise 2.3.f(a)]). we prove that the space f is τ-metrizable. for every λ ∈ λ we define: (1) ρλ(f, f) = 0 for each f ∈ f . (2) ρλ(f, g) = { 0, if f(λ) = g(λ) 1, otherwise for each f, g ∈ f with f 6= g. the mapping ρτ : f × f → r τ + defined by ρτ (f, g) = {ρλ(f, g)}λ∈λ for every f, g ∈ f is a τ-metric on f . we prove that the topology tρ τ induced by the τ-metric ρτ coincides with the tychonoff product topology. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 258 τ-metrizable spaces let f ∈ f , ∏ λ∈λ uλ be an open neighbourhood of the point θ in the space r τ +, and suppose that {λ ∈ λ : uλ 6= r+} = {λ1, . . . , λm}. for the open neighbourhood ∏ λ∈λ wλ of the point f, where wλ = { {f(λ)}, if λ ∈ {λ1, . . . , λm} d(κ), otherwise we have ∏ λ∈λ wλ ⊆ g(f) = {g ∈ f : ρτ(f, g) ∈ ∏ λ∈λ uλ}. now, let f ∈ f and ∏ λ∈λ wλ be an open neighbourhood of the point f in the space f , and suppose that {λ ∈ λ : wλ 6= d(κ)} = {λ1, . . . , λm}. for the open neighbourhood ∏ λ∈λ uλ of the point θ, where uλ = { [0, 1 2 ), if λ ∈ {λ1, . . . , λm} r+, otherwise we have g(f) = {g ∈ f : ρτ(f, g) ∈ ∏ λ∈λ uλ} ⊆ ∏ λ∈λ wλ. 4. compact τ-metrizable spaces it is well known that every compact metrizable space is separable. an analogous result for τ-metrizable spaces is stated in this section. let us consider a set λ such that |λ| = τ > ℵ0 and let bε be the family of all open subsets ∏ λ∈λ wλ of the product r τ +, where finitely many wλ are intervals of the form [0, ε) and the remaining wλ = r+. definition 4.1. let (x, ρτ ) be a τ-metric space. a subset a of x is called oε-dense in (x, ρτ), where oε ∈ bε, if for every x ∈ x there exists a ∈ a such that ρτ (x, a) ∈ oε. definition 4.2. a τ-metric space (x, ρτ) is called ε-totally bounded if for every oε ∈ bε there exists a finite subset a of x which is oε-dense in (x, ρτ). the τ-metric space (x, ρτ ) is called totally bounded if it is ε-totally bounded for every ε > 0. recall that the density d(x) of a topological space x, is defined to be d(x) = min{|d| : d is a dense subset of x}. proposition 4.3. for every totally bounded τ-metric space x, the inequality d(x) 6 τ holds. proof. let n ∈ {1, 2, . . .}. for each o1/n ∈ b1/n, let a(o1/n) be a finite o1/ndense subset of x and consider the subset an = ∪{a(o1/n) : o1/n ∈ b1/n} of x with |an| 6 τ. the subset a = ∪ ∞ n=1an of x is dense and |a| 6 τ. � proposition 4.4. every compact τ-metric space x is totally bounded. proof. let ε > 0. for every oε ∈ bε the family {g(x) = {y ∈ x : ρτ (x, y) ∈ oε} : x ∈ x} forms an open cover of x. by compactness of x, there exists a finite subset a of x such that ⋃ a∈a g(a) = x. for every x ∈ x there exists a ∈ a with c© agt, upv, 2018 appl. gen. topol. 19, no. 2 259 a. c. megaritis x ∈ g(a). therefore, ρτ(x, a) ∈ oε and the subset a of x is oε-dense in (x, ρτ). � theorem 4.5. for every compact τ-metrizable space x we have d(x) 6 τ. proof. let x be a compact τ-metrizable space. according to proposition 4.4, the space x is totally bounded. therefore, by virtue of proposition 4.3, d(x) 6 τ. � acknowledgements. the author would like to thank both referees for their valuable comments and suggestions. references [1] a. a. borubaev, on some generalizations of metric, normed, and unitary spaces, topology and its applications 201 (2016), 344–349. [2] r. engelking, general topology, sigma series in pure mathematics, 6. heldermann verlag, berlin, 1989. [3] j. r. munkres, topology: a first course, prentice-hall, inc., englewood cliffs, n.j., 1975. [4] l. a. steen and j. a. jr. seebach, counterexamples in topology, dover publications, inc., mineola, ny, 1995. [5] s. willard, general topology, dover publications, inc., mineola, ny, 2004. c© agt, upv, 2018 appl. gen. topol. 19, no. 2 260 @ �� !#"%$'&'� ( )*(,+�-*. � /10" � � � � )2( "�34)*. 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¦/��kczb1�2� ��y�~�y�í �xyðs"! q2w:~�w$#&%&%(' mon2p�q2rts:qvu�nxw8w8y�z|{}q2p�~ �n2r ) 7moih$nks�h �c7�4is�2p5e7+*[bd³ 2�l�21h-,. h / 4ib0%$7mnklkb�5eh�510,3(s2,2$4!2 �ch4397�s¬n�2ko�2�³ bds�h$46h$q e65 s�h�oih$³´h�oih 0 %$w , 46w87?h1hmhe7�³20,s³ b f�µ�2:9j2ew+;�<>= 0 x?)a@ w x 2$lks�h$³cb�b�d>b�e�f(q+gtw x wih�j�ekf�h ) wl*�w qi3m,7�f�b�µk2 n .��ch$bd³qhe5e5�n�7ml�l3�io�p�qsr�t&u�v>w�xzy[w-u xzy\x]t () @ appl. gen. topol. 18, no. 2 (2017), 331-344doi:10.4995/agt.2017.7149 c© agt, upv, 2017 a study of function space topologies for multifunctions ankit gupta a,∗ and ratna dev sarma b, † a department of mathematics, university of delhi, delhi 110007, india (ankitsince1988@yahoo.co.in) b department of mathematics, rajdhani college, university of delhi, delhi 110015, india. (ratna sarma@yahoo.com) communicated by a. tamariz-mascarúa abstract function space topologies are investigated for the class of continuous multifunctions. using the notion of continuous convergence, splittingness and admissibility are discussed for the topologies on continuous multifunctions. the theory of net of sets is further developed for this purpose. the (τ, µ)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. the point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting. 2010 msc: 54c35; 54a05; 54c60. keywords: multifunction; function space topology; continuous convergence; splittingness; admissibility. 1. introduction the interplay of properties of the topological spaces x and y and those of the function space c(x, y ) of continuous functions from x to y has been an area of active research in topology. several different sets of conditions under which the compact-open, isbell or natural topologies on the set of continuous ∗corresponding author. †this paper was prepared when the second author was on sabbatical leave. received 24 january 2017 – accepted 14 august 2017 http://dx.doi.org/10.4995/agt.2017.7149 a. gupta and r. d. sarma real-valued functions on a space may coincide, have been studied in [13]. a unified theory of function spaces and hyperspaces has been developed in [3]. in [4], it is shown that the intersection of all admissible topologies on c(x, y ) is admissible under certain conditions. these and many other research papers published in the recent years are the testimony to the keen interest of the researchers in the study of function spaces. in [13], while discussing coincidence of the function space topologies, a natural topology on the set of upper semi continuous set-valued functions has been constructed. apart from this, the continuous multifunctions in the study of function spaces have been investigated by several researchers [9, 10, 12, 15, 20, 21, 22]. at the same time, the multifunctions are being extensively used now-a-days in several areas of mathematics such as optimization theory, frame theory, approximation theory etc., to name a few. in this paper, we further develop the topological aspects of the function spaces for multifunctions. starting from the basic level, we provide discussions for several new as well as already existing topologies for continuous multifunctions. we have adopted the net theoretic approach to discuss continuous convergence for the topology of multifunctions. the net theory for sets is further developed for the same purpose. here it may be mentioned that characterizations of upper semi-continuity and lower semi-continuity are provided in [14] using net convergence. in our paper, we show that under certain conditions, continuity of multifunctions implies a net-theoretic result which is similar to its counterpart for single-valued functions. our study here is purely topological, unlike [11], where metric spaces and normed spaces are considered for similar results. similarly, the continuous convergence introduced in our paper is different from that of [2] and [18]. in [2] and [18], upper and lower topologies, defined on the second space, are used for defining continuous convergence. however our definition is more straight forward and appears similar to its counterpart of single-valued functions. conditions for splittingness (resp. upper and lower splittingness) and admissibility (resp. upper and lower admissibility) are obtained by using the concept of continuous convergence. the characterizations of admissibility and splittingness using net theory as shown in arens and dugundji [1] do not hold for multifunctions. their variants are investigated in our paper. several examples are provided to explain the intrinsic differences between the topologies of continuous functions and topologies of continuous multifunctions. in the last section, several topologies on cm(y, z), the class of continuous multifunctions are studied in the light of splittingness and admissibility. while the (τ, µ)-topology is found to be upper admissible, the compact-open topology on cm(y, z) is upper splitting. the point-open topology is found to be the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting. during our investigation, we have also noticed that for a multifunction f : (x, τ) → (y, µ) and u ∈ µ, the two different types of inverse images, that is, f +(u) and f −(u) types give rise to two families of open sets of τ. this leads to the possibility of having more than one dual topology for a given function c© agt, upv, 2017 appl. gen. topol. 18, no. 2 332 a study of function space topologies for multifunctions space topology on cm(y, z). however, further in depth research is required in this regard, which is beyond the scope of this paper. 2. preliminaries definition 2.1. a multifunction f : x → y is a point-to-set correspondence from x to y . we always assume that f(x) 6= ∅ for all x ∈ x. for each b ⊆ y , f +(b) = {x ∈ x | f(x) ⊆ b} and f −(b) = {x ∈ x | f(x) ∩ b 6= ∅}. for each a ⊆ x, f(a) = ⋃ x∈a f(x). the collection of all the multifunctions from x to y is denoted by y xm. the following definitions and results are taken from the available literature. definition 2.2. let (x, τ) and (y, µ) be two topological spaces. then f : x → y is called (i) upper semi continuous (or u.s.c., in brief) at x ∈ x if for each open set v ⊆ y with f(x) ⊆ v , there exists an open set u of x such that x ∈ u and f(u) ⊆ v ; (ii) lower semi continuous (or l.s.c, in brief) at x ∈ x if for each open set v ⊆ y with f(x) ∩ v 6= ∅, there exists an open set u of x such that x ∈ u and f(u) ∩ v 6= ∅ for every u ∈ u; (iii) continuous at x ∈ x, if it is both u.s.c. and l.s.c. at x; (iv) continuous (resp. u.s.c., l.s.c.) if it is continuous (resp. u.s.c., l.s.c.) at each point of x. if (x, τ) and (y, µ) are two topological spaces and f : x → y is a multifunction, then the following conditions are equivalent: (i) f is l.s.c. (resp. u.s.c.); (ii) f −(u) (resp. f +(u)) is open in x for each open subset u of y ; (iii) f +(a) (resp. f −(a)) is closed in x for each closed subset a of y . definition 2.3. a multifunction f : x → y is called a closed map if f(a) is closed in y whenever a is closed in x. 3. a topology on cm(y, z) now we proceed to define a topology on zym in the following way: let (y, τ) and (z, µ) be two topological spaces. for u ∈ τ and v ∈ µ, we define (u, v ) = {f ∈ zym | f(u) ⊆ v } let smτ,µ = {(u, v ) | u ∈ τ, v ∈ µ}. lemma 3.1. smτ,µ is a subbasis for a topology on z y m. proof. for f ∈ zym, we have f(∅) ⊆ v , for each v ∈ µ. hence, we get f ∈ (∅, v ) for each v ∈ µ. therefore ⋃ smτ,µ = z y m. � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 333 a. gupta and r. d. sarma in our discussion, we take f = cm(y, z), the collection of all continuous multifunctions from y to z. the topology on f obtained in the above manner is denoted by tmτ,µ, and is called the (τ, µ)-topology on cm(y, z). this topology reduces to open-open topology τoo[17] if the multifunctions are replaced by functions. in fact, τoo turns out to be the relative topology of t m τ,µ, when we consider the subspace c(y, z) of cm(y, z). lemma 3.1 ensures the existence of a topology on cm(y, z). in fact, several other interesting topologies exist on cm(y, z). we will discuss some of them in the last section. definition 3.2. let (y, τ) and (z, µ) be two topological spaces. let (x, λ) be another topological space. for a multifunction g : x × y → z, we define a map g∗ : x → cm(y, z) by g ∗(x)(y) = g(x, y). the mappings g and g∗ related in this way are called associated maps. definition 3.3. let (y, τ) and (z, µ) be two topological spaces. a topology t on cm(y, z) is called (i) admissible (resp. upper admissible, lower admissible) if the evaluation mapping e : cm(y, z)× y → z defined by e(f, y) = f(y) is continuous (resp. u.s.c., l.s.c.). (ii) splitting (resp. upper splitting, lower splitting) if for each topological space x, continuity (resp. u.s.c., l.s.c.) of g : x × y → z implies the continuity of g∗ : x → cm(y, z), where g ∗ is the associated map of g. in [5], georgiou, iliadis and papadopoulos have introduced one more variation of admissibility and splittingness for function spaces. we extend those definitions for multifunctions as follow: definition 3.4. let (y, τ) and (z, µ) be two topological spaces. a topology t on cm(y, z) is called (i) coordinately admissible (resp. upper coordinately admissible, lower coordinately admissible) if the evaluation mapping e : cm(y, z) × y → z defined by e(f, y) = f(y) is coordinately continuous (resp. coordinately u.s.c., coordinately l.s.c.). (ii) coordinately splitting (resp. coordinately upper splitting, coordinately lower splitting) if for each topological space x, coordinately continuity (resp. coordinately u.s.c., coordinately l.s.c.) of g : x × y → z implies the continuity of g∗ : x → cm(y, z), where g ∗ is the associated map of g. here, a map f : x × y → z is said to be coordinately continuous (resp. coordinately u.s.c., coordinately l.s.c.) if the maps fx : y → z and fy : x → z defined by fx(y) = f(x, y) and fy(x) = f(x, y) are continuous (resp. u.s.c., l.s.c.) for every x ∈ x and for every y ∈ y . the following two observations will be used at several places in the paper: (i) let f : x → y and g : y → z be a continuous function and a continuous (resp. u.s.c., l.s.c.) multifunction respectively. then gof is continuous (resp. u.s.c., l.s.c.). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 334 a study of function space topologies for multifunctions (ii) let (y, τ) and (z, µ) be two topological spaces. a topology t on cm(y, z) is admissible (resp. upper admissible, lower admissible) if and only if for each topological space x, continuity of g∗ : x → cm(y, z) implies continuity (resp. u.s.c., l.s.c.) of g : x×y → z, where g is the associated map of g∗. 4. continuous convergence of multifunctions in this section, we first investigate the relationships between the net convergence criteria and the continuity of multifunctions. convergence of net of sets is required for this purpose. definition 4.1 ([2, 16]). let s = {an}n∈∆ be a net of sets in a topological space (x, τ). then for any x ∈ x, we say (i) x ∈ lim inf(an) (or, x ∈ li(an), in brief) if s eventually intersects every open neighbourhood of x, that is, given an open neighbourhood u of x, there exists m ∈ ∆, such that an ∩ u 6= ∅ for all n ≥ m. (ii) x ∈ lim sup(an) (or, x ∈ ls(an), in brief) if s frequently intersects every open neighbourhood of x, that is, given an open neighbourhood u of x and any m ∈ ∆, there exists an n ≥ m such that an ∩ u 6= ∅. (iii) the net of sets s = {an}n∈∆ is said to converge to a and we write lim(an) = a if li(an) = ls(an) = a. lemma 4.2. for any net of sets, we have li(an) ⊆ ls(an). theorem 4.3. let f be a multifunction from a topological space (x, τ) to a regular topological space (y, µ). let {xn}n∈∆ be a net in x, which converges to x in x. then the net {f(xn)}n∈∆ converges to f(x), if f is continuous at x ∈ x and f(x) is closed in (y, µ). proof. let y ∈ f(x) and v be any open neighbourhood of y in y . then v ∩ f(x) 6= ∅. thus x ∈ f −(v ). as f is continuous, f −(v ) is open. therefore, there exists an open neighbourhood u of x such that x ∈ u ⊆ f −(v ). since {xn}n∈∆ converges to x, we have xn ∈ u ⊆ f −(v ) eventually. therefore, f(xn) ∩ v 6= ∅ eventually and hence y ∈ li(f(xn)n∈∆). thus, we have f(x) ⊆ li (f(xn)n∈∆). now we claim that ls(f(xn)n∈∆) ⊆ f(x). let y /∈ f(x). since (y, µ) is regular and f(x) is a closed set not containing y, therefore, there exist disjoint open sets u and v such that y ∈ u and f(x) ⊆ v . as multifunction f is given to be continuous at x ∈ x, there exists an open neighbourhood w of x such that f(w) ⊆ v . again, since the given net {xn}n∈∆ converges to x, therefore xn ∈ w eventually. then we have f(xn) ⊆ v for all n ≥ n0, for some n0 ∈ ∆. this implies that f(xn) ∩ u = ∅ for all n ≥ n0. hence, y /∈ ls(f(xn)n∈∆). thus, we have ls(f(xn)n∈∆) ⊆ f(x). hence, f(x) = li(f(xn)n∈∆) = ls(f(xn)n∈∆). therefore {f(xn)}n∈∆ converges to f(x). � c© agt, upv, 2017 appl. gen. topol. 18, no. 2 335 a. gupta and r. d. sarma in the above theorem, the condition of regularity of the space (y, µ) and closedness of f(x) can not be relaxed. here we provide examples to demonstrate this. example 4.4. let x = r be the set of real numbers with the usual topology u and µ be the irrational slope topology defined on y = {(x, y) | y ≥ 0, x, y ∈ q}. we fix some irrational number θ. the irrational slope topology[23] µ on y is generated by neighbourhoods of the form nǫ((x, y)) = {(x, y)} ∪ bǫ(x + y/θ) ∪ bǫ(x − y/θ) where bǫ(a) = {(r, 0) ∈ y | |r − a| < ǫ} is the collection of all rationals in (a − ǫ, a + ǫ). this irrational slope topology is hausdorff but not regular[23]. let f : (x, u) → (y, µ) be a multifunction defined by f(x) =        {(1, 5)}, x = 1 bǫ(1 − 5/θ), for all x ∈ (0, 1) bǫ(1 + 5/θ), for all x ∈ (1, 2) {(5, 10)} otherwise. we claim that (i) f(1) = {(1, 5)} is closed in (y, µ); (ii) f is continuous at x = 1; (iii) {f(1 − 1/n)}n∈n does not converge to f(1) in (y, µ), while {1 − 1/n}n∈n converges to 1 in (x, u). below we provide justifications for claim (i), (ii) and (iii): (i) since (y, τ) is a hausdorff space, there does not exist any y ∈ y distinct from the point (1, 5) such that every neighbourhood u of y intersects (1, 5). thus cl{(1, 5)} = {(1, 5)}. thus f(1) = {(1, 5)} is closed; (ii) first we show that f is u.s.c. at x = 1. let v be any open set in y such that f(1) ⊆ v . since v is open in y , we have {(1, 5)} ∪ bǫ(1 + 5/θ) ∪ bǫ(1 − 5/θ) ⊆ v , for some ǫ > 0. therefore f ((0, 2)) ⊆ v , where (0, 2) is neighbourhood of 1. hence f is u.s.c. at x = 1. now, we show that f is l.s.c. at x = 1. let v be any open set in y such that f(1) ∩ v 6= ∅. any open set containing (1, 5) must contain bǫ(1 + 5/θ) ∪ bǫ(1 − 5/θ) also. thus, again we have f(x) ∩ v 6= ∅ for all x ∈ (0, 2). therefore f is l.s.c. at x = 1. hence f is continuous at x = 1; (iii) next, we prove that {f(1 − 1/n)}n∈n does not converge to f(1), that is f(1) 6= li(f(xn)), where xn = 1 − 1/n. let y ∈ bǫ(1−5/θ). then every neighbourhood of y intersects f(1−1/n) for all n. thus y ∈ li(f(xn)). but y /∈ f(1). hence {f(1 − 1/n)}n∈n does not converge to f(1) in (y, µ) while {1 − 1/n}n∈n converges to 1 in (x, u). in the next example, we show that the condition of closedness can not be relaxed. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 336 a study of function space topologies for multifunctions example 4.5. let x = r be the set of real numbers with the usual topology u and let y = {a, b, c, d} with topology τ = {∅, y, {a, b}, {c, d}}. let f : (x, u) → (y, τ) be a multifunction defined by f(x) =    {a} for x = 1, {a, b} for all x ∈ (1 − 1/n, 1) ∪ (1, 1 + 1/n) for all n ≥ m, for some m ∈ n, {c} otherwise. here (y, τ) is regular and f(1) = {a} is not closed in y . now, we claim that (i) f is continuous at x = 1; (ii) {f(1-1/n)}n∈n does not converge to f(1) in (y, τ), while {1-1/n}n∈n converges to 1 in (x, u). we have (i) first we show that f is u.s.c. at x = 1. let v be any open set in y such that f(1) ⊆ v . since v is open in y , we have {a, b} ⊆ v . therefore there exists (1-1/m, 1 + 1/m), an open neighbourhood of 1 in (x, u) such that f((1-1/m, 1 + 1/m)) ⊆ v . hence f is u.s.c. at x = 1. now, we show that f is l.s.c. at x = 1. let v be any open set in y such that f(1) ∩ v 6= ∅. any open set containing {a} must contains {a, b} also. thus, again we have f(x)∩v 6= ∅ for all x ∈ (1 − 1/m, 1 + 1/m). therefore f is l.s.c. at x = 1. hence f is continuous at x = 1. (ii) now, we show that {f(1-1/n)}n∈n does not converge to f(1), that is f(1) 6= li(f(xn)), where xn = 1-1/n. let y = b, then every neighbourhood of y intersects f(1 − 1/n) for all n ≥ m. thus y ∈ li(f(xn)). but y /∈ f(1). hence {f(1 − 1/n)}n∈n does not converge to f(1) in (y, τ), while {1 − 1/n}n∈n converges to 1 in (x, u). the condition which we imposed on f in theorem 4.3, that is, f(x) is closed for every netwise limit x ∈ x can not be replaced by the condition that f is a closed map. (here, x is a netwise limit means x is a limit of some non-trivial net in x.) for this, first we define a topology on r, which is the countable complement extension topology [23], as given below. definition 4.6. let x = r be the set of real numbers and τ1 be the euclidean topology and τ2 be the topology of countable complements on x. we define τ to be the smallest topology generated by τ1 ∪ τ2. here a set u is open in τ if and only if u = o�a, where o ∈ τ1 and a is countable. then τ is the countable complement extension topology on r. example 4.7. let x and y be the set of real numbers having the co-countable topology and the countable complement extension topology respectively. let f : x → y be a multifunction defined as f(x) = (−∞, −2 − 1/x] ∪ {x} for all c© agt, upv, 2017 appl. gen. topol. 18, no. 2 337 a. gupta and r. d. sarma x ∈ x. here f(n) = (−∞, −2) ∪ n, which is not a closed set in y . hence f is not a closed map, although f(x) is closed for all x ∈ x. the following result available in [14], may be treated as a partial converse of theorem 4.3. theorem 4.8. let (x, τ) and (y, µ) be two topological spaces. let f : x → y be a multifunction. then f is lower semi continuous at x ∈ x if for any net {xn}n∈∆ in x converging to x ∈ x, the image net {f(xn)}n∈∆ converges to f(x). here we also mention to our readers that in [14], it is proved that if (x, τ) is a compact hausdroff space and {an}n∈∆ is a net of sets, then lim(an) = a if and only if {an}n∈∆ converges to a under vietoris topology τv of x. generalized nets, defined in [19], were used in [7] to introduce the notion of continuous convergence for function spaces on generalized topologies. here, we use net theory to the introduce the concept of continuous convergence for multifunctions. we shall use this concept extensively in our paper for classifying various topologies on cm(y, z). however, before coming to that, below we provide a small discussion related to directed sets and convergence of nets. let ∆ be a directed set. we add a point ∞ to ∆ satisfying ∞ ≥ n for all n ∈ ∆ and write ∆0 = ∆ ∪ {∞}. then a topology τ0 may be generated on ∆0 by declaring every singleton of ∆ is open and neighbourhood of ∞ to be all the sets of the form un0 = {n : n ≥ n0}, where n0 is any arbitrary member of ∆. lemma 4.9 ([1]). let (y, µ) be a topological space and {yn}n∈∆ be a net in y . then {yn}n∈∆ converges to y if and only if the function s : ∆0 → y defined by s(n) = yn for n ∈ ∆ and s(∞) = y is continuous at ∞. in the next set of theorems, we will provide some characterizations of splittingness and admissibility of topologies on cm(y, z). first we define continuous convergence for multifunctions. definition 4.10. let {fn}n∈∆ be a net in cm(y, z). then {fn}n∈∆ is said to continuously converge to f if for each net {ym}m∈σ in y converging to y, {fn(ym)}(n,m)∈∆×σ converges to f(y) in z. if we take functions in place of multifunctions, the above definition coincides with that of continuous convergence of functions defined in [1]. theorem 4.11. let (y, τ) and (z, µ) be two topological spaces in which (z, µ) is regular. let t be a topology on cm(y, z) such that for each net {fn}n∈∆ in cm(y, z), continuous convergence of {fn}n∈∆ to f implies {fn}n∈∆ converges to f under t, provided g(x, y) is closed for every continuous map g : x ×y → z and for every netwise limit (x, y) ∈ x × y . then t is splitting. proof. suppose, continuous convergence implies convergence. let g : x ×y → z be continuous. let {xn}n∈∆ be a convergent net which converges to x in x. we need to show that {g∗(xn)}n∈∆ converges to g ∗(x) in cm(y, z). c© agt, upv, 2017 appl. gen. topol. 18, no. 2 338 a study of function space topologies for multifunctions let {ym}m∈σ be a net in y converging to y. then {(xn, ym)}(n,m)∈∆×σ converges to (x, y) in x × y . hence {g(xn, ym)} converges to g(x, y), in view of theorem 4.3 because g is continuous and z is regular. let us define that g∗(xn) = fn and g ∗(x) = f . then g(xn, ym) = g ∗(xn)(ym) = fn(ym) and g(x, y) = g∗(x)(y) = f(y). that is, {fn(ym)}(n,m)∈∆×σ converges to f(y). thus {fn}n∈∆ continuously converges to f . therefore by the given condition {fn}n∈∆ converges to f in cm(y, z). that is, {g ∗(xn)}n∈∆ converges to g∗(x). hence g∗ is continuous, that is, t is splitting. � our next result provides a partial converse of the above theorem. theorem 4.12. let (y, τ) and (z, µ) be two topological spaces such that t on cm(y, z) is lower splitting. then for each net {fn}n∈∆ in cm(y, z), continuous convergence of {fn}n∈∆ to f implies {fn}n∈∆ converges to f under t. proof. let t be lower splitting and {fn}n∈∆ converge continuously to f . let ∆0 = ∆ ∪ {∞} be the topological space generated from ∆. we define g : ∆0 × y → z by g(n, y) = fn(y) and g(∞, y) = f(y). now, we claim that g is l.s.c.. let w be an open set in z such that g(n, y) ∩ w 6= ∅, that is, fn(y) ∩ w 6= ∅. since fn ∈ cm(y, z), therefore there exists an open neighbourhood v of y such that fn(v)∩w 6= ∅ for each v ∈ v . thus, we have g(n, v) ∩ w 6= ∅ for all (n, v) ∈ n × v . therefore g is l.s.c. at (n, y). now, we show that g is l.s.c. at (∞, y). it is clear that the only non-constant net in ∆0 is {n}, which converges to ∞. hence if s is a convergent net in ∆0 × y , then we have, s = s1 × s2, where s1 = {n} and s2 = {ym}m∈σ is some convergent net in y . then s converges to {∞}×{y}, for some y ∈ y such that {ym}m∈σ converges to y. then we have, g(s) = {fn(ym)}(n,m)∈∆×σ. by continuous convergence of {fn}, g(s) converges to f(y) = g(∞, y). hence g is lower semi continuous at (∞, y). thus g is l.s.c. on ∆0 × y . as t is lower splitting, this implies that g∗ is continuous. as {n} converges to ∞ in ∆0, we have , {g∗(n)}n∈∆ converges to g ∗(∞). now, g∗(n)(y) = g(n, y) = fn(y) and g∗(∞)(y) = g(∞, y) = f(y). that is, g∗(n) = fn and g ∗(∞) = f , hence {fn} converges to f in cm(y, z). � remark 4.13. if we take the subfamily c(y, z), instead of cm(y, z), then theorem 4.11 and theorem 4.12 reduce to the following, which was proved in [1]. corollary 4.14. let (y, µ) and (z, τ) be two topological spaces. a topology t in c(y, z) is splitting if and only if for any net {fn}n∈∆ in c(y, z), continuous convergence of {fn}n∈∆ to f implies convergence of {fn}n∈∆ to f in t. proof. it holds in view of the fact that net-theoretic characterization of continuity of functions does not need regularity, nor closedness of f(x). � in the following two results, we investigate the relationship between admissibility and continuous convergence. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 339 a. gupta and r. d. sarma theorem 4.15. let (y, τ) and (z, µ) be two topological spaces. a topology t on cm(y, z) is lower admissible if for any net {fn}n∈∆ in cm(y, z), {fn}n∈∆ converges to f in t implies continuous convergence of {fn}n∈∆ to f. proof. let g∗ : x → cm(y, z) be continuous. we have to show that the associated map g is lower semi continuous. let {xn, yn}n∈∆ be a convergent net which converges to (x, y) ∈ x × y . then {xn}n∈∆ converges to x in x and {yn}n∈∆ converges to y in y . since {xn}n∈∆ converges to x and g ∗ is continuous, therefore {g∗(xn)}n∈∆ converges to g ∗(x), that is, {fxn}n∈∆ converges to fx in cm(y, z), where fxn = g ∗(xn) and fx = g ∗(x) respectively. then, by the given hypothesis, {fxn}n∈∆ continuously converges to fx. hence for the convergent net {yn}n∈∆ which converges to y in y , we have {fxn(yn)}n∈∆ converges to fx(y), that is, {g(xn, yn)}n∈∆ converges to g(x, y). hence g is lower semi continuous. therefore t is lower admissible. � for the converse part, we have the following result: theorem 4.16. let (y, τ) and (z, µ) be two topological spaces in which (z, µ) is regular. suppose a topology t on cm(y, z) is admissible. then for each net {fn}n∈∆ in cm(y, z), convergence of {fn}n∈∆ to f implies {fn}n∈∆ continuously converges to f under t, provided g(x, y) is closed for every netwise limit (x, y) ∈ x × y , for every map g : x × y → z and for any topological space (x, λ). proof. let t be admissible and {ym}m∈σ be any net in y such that {ym}m∈σ converges to y in y . let {fn}n∈∆ be any net in cm(y, z) such that {fn}n∈∆ converges to f . let us define g∗ : ∆0 → cm(y, z) as g ∗(n) = fn and g∗(∞) = f , where ∆0 is generated by ∆. now the only non-constant net in ∆0 is {n} which converges to ∞. also, {g ∗(n)}n∈∆ = {fn}n∈∆ converges to f = g∗(∞). hence, g∗ is continuous. therefore g : ∆0 ×y → z is continuous as t is admissible. now {n, ym}(n,m)∈∆×σ is a convergent net in ∆0 × y which converges to (∞, y). therefore, in view of theorem 4.3, {g(n, ym)}(n,m)∈∆×σ converges to g(∞, y), that is, {g∗(n)(ym)}(n,m)∈∆×σ converges to g ∗(∞)(y). this implies {fn(ym)}(n,m)∈∆×σ converges to f(y). hence {fn} continuously converges to f . � as a corollary of theorem 4.15 and theorem 4.16, we get the following result for c(y, z) [1]: corollary 4.17. let (y, τ) and (z, µ) be two topological spaces. a topology t on c(y, z) is admissible if and only if for any net {fn}n∈∆ in c(y, z), {fn}n∈∆ converges to f in t implies continuous convergence of {fn}n∈∆ to f. in [1], arens and dugundji provided few lemmas for function spaces which are valid for function spaces for multifunctions as well. below we mention them without proof. here µ ≥ τ, means τ ⊆ µ. lemma 4.18. let τ and µ be two topologies on cm(y, z). if τ is admissible (resp. upper admissible, lower admissible) and µ ≥ τ, then µ is admissible c© agt, upv, 2017 appl. gen. topol. 18, no. 2 340 a study of function space topologies for multifunctions (resp. upper admissible, lower admissible). if µ is splitting (resp. upper splitting, lower splitting) and µ ≥ τ, then τ is splitting (resp. upper splitting, lower splitting). lemma 4.19. if µ is splitting (resp. upper splitting, lower splitting) and τ is admissible (resp. upper admissible, lower admissible) on cm(y, z), then µ ≤ τ. the following theorem, provided by georgiou, iliadis and papadopoulos in [5] for function space is also valid for the function spaces for multifunctions. that is, theorem 4.20. the following hold good: (i) every coordinately splitting (resp. coordinately upper splitting, coordinately lower splitting) topology on cm(y, z) is splitting (resp. upper splitting, lower splitting). (ii) every admissible (resp. upper admissible, lower admissible) topology on cm(y, z) is coordinately admissible (resp. coordinately upper admissible, coordinately lower admissible). 5. various topologies over cm(y, z) in section 3, we introduce the (τ, µ)-topology, denoted by tmτ,µ, on cm(y, z). in this section, we study several other topologies over cm(y, z) in the light of splittingness and admissibility. we also study the interrelationship between these topologies. first, we come back to the topology tmτ,µ on cm(y, z) defined in section 3. we show that tmτ,µ is upper admissible on cm(y, z). theorem 5.1. let (y, τ) and (z, µ) be two topological spaces. then the topology tmτ,µ is upper admissible for cm(y, z). proof. let (f, y) ∈ cm(y, z) × y and let v ∈ µ such that e(f, y) ⊆ v . this implies f(y) ⊆ v . since f is continuous, there exists an open set w containing y such that f(w) ⊆ v , that is, f ∈ (w, v ). thus e((w, v ) × w) ⊆ v . thus the evaluation map e is u.s.c.. hence cm(y, z) is upper admissible. � below we provide definitions for the compact-open topology and the pointopen topology for multifunctions, and investigate some of their properties. let (y, τ) and (z, µ) be two topological spaces. then we define (c, v ) = {f ∈ cm(y, z) | f(c) ⊆ v } (y, v ) = {f ∈ cm(y, z) | f(y) ⊆ v } where c is a compact subset of y , y ∈ y and v ∈ µ. let smco = {(c, v ) | c is compact in y and v ∈ µ} and s m po = {(y, v ) | y ∈ y, v ∈ µ}. it can be shown that smco and s m po form subbasis for two topologies on cm(y, z). these topologies are called the compact-open topology and the point-open topology respectively. they are denoted by tmco and t m po respectively. clearly t m co is finer than tmpo . for further study on compact-open topology and compact convergence, one may refer to [9, 10, 12]. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 341 a. gupta and r. d. sarma theorem 5.2. the compact-open topology over cm(y, z) is upper splitting. proof. left for the readers. � in our next result, we prove that the point-open topology for multifunctions is coordinately admissible. theorem 5.3. let (y, τ) and (z, µ) be two topological spaces. then the topology tmpo on cm(y, z) is coordinately admissible. also, this is the coarsest topology on cm(y, z) which is coordinately admissible. proof. first we show that tmpo is coordinately upper admissible. for this, let f ∈ cm(y, z) and v ∈ µ such that ey(f) ⊆ v . this implies f(y) ⊆ v . consider, (y, v ) ∈ tmpo . we have ey((y, v )) ⊆ v . hence the evaluation map ey is coordinately u.s.c. now, let y ∈ y and v ∈ µ such that ef (y) ⊆ v . this implies f(y) ⊆ v . since f ∈ cm(y, z), then there exists an open neighbourhood w of y such that f(w) ⊆ v , then we have ef (w) ⊆ v for every w ∈ w . hence the evaluation map ef is coordinately u.s.c. therefore, t m po is coordinately upper admissible. now, consider f ∈ cm(y, z) and v ∈ µ such that ey(f) ∩ v 6= ∅, that is, f(y) ∩ v 6= ∅. then again, we have (y, v ) ∈ tmpo such that ey(f) ∩ v 6= ∅ for each f ∈ (y, v ). hence the evaluation map ey is coordinately l.s.c. similarly, we can show that ef is coordinately l.s.c.. thus t m po is coordinately lower admissible as well. accordingly, tmpo is coordinately admissible. now, we show that it is the coarsest topology on cm(y, z) having this property. let (y, u) be a subbasic open set in tmpo and u be another topology which is coordinately admissible. we show that (y, u) is open in u. consider f ∈ (y, u), that is, f(y) ⊆ u. therefore, ey(f) ⊆ u. since the topology u is coordinately admissible, the evaluation map ey is continuous. therefore there exists an open set v in u containing f such that ey(v ) ⊆ u. that is, f ∈ v ⊆ (y, u). hence (y, u) is open in u. � our next result shows that tmpo is coordinately upper splitting and hence upper splitting. theorem 5.4. let (y, τ) and (z, µ) be two topological spaces. then the topology tmpo is coordinately upper splitting. proof. let (x, λ) be any topological space such that g : x × y → z is coordinately upper semi continuous. we have to show that its associated mapping g∗ : x → cm(y, z) is continuous. let x ∈ x and (y, v ) be a subbasic open set in tmpo with g ∗(x) ∈ (y, v ). that is, g∗(x)(y) ⊆ v . then, g(x, y) ⊆ v . since the map g : x × y → z is coordinately upper semi continuous, therefore gy : x → z is upper semi continuous for each y. hence there exists an open neighbourhood ox of x such that gy(ox) ⊆ v . consequently, we have c© agt, upv, 2017 appl. gen. topol. 18, no. 2 342 a study of function space topologies for multifunctions g∗(ox) ⊆ (y, v ). hence g ∗ is continuous. accordingly tmpo is coordinately upper splitting. � in view of lemma 4.18 and 4.19, the above two results lead us to the following interesting result: theorem 5.5. the point-open topology tmpo over cm(y, z) is the unique topology which is coordinately upper splitting as well as coordinately upper admissible. it is also the finest coordinately upper splitting as well as the coarsest coordinately upper admissible topology on cm(y, z). the corollary 3.6 of [5] on c(y, z) can be viewed as a particular case of the above result. remark 5.6. the open sets of the domain space which can be realized as preimages of continuous functions have been used to define the dual topologies for a given function space topology in [6] and [8]. several interesting relationships are established between a function space topology and its dual topology in these studies. investigations in this regard may also be carried out for multifunction as well. however, in case of multifunctions, the development is not straightforward. it is due to the fact that, for a continuous multifunction f : (x, τ) → (y, µ), the inverse images of open sets form two different classes in (x, τ) formed by f +(u) and f −(u) types of sets where u ∈ µ. a detailed discussion about the same is beyond the scope of the present paper and needs further investigation. references [1] r. arens and j. dugundji, topologies for function spaces, pacific j. math. 1 (1951), 5–31. [2] j. cao, i. l. reilly and m. v. vamanamurthy, comparison of convergences for multifunctions, demonstratio math. 30 (1997), 171–182. [3] s. dolecki and f. mynard, a unified theory of function spaces and hyperspaces: local properties, houston j. math. 40, no. 1 (2014), 285–318. [4] d. n. georgiou and s. d. iliadis, on the greatest splitting topology, topology appl. 156 (2008), 70–75. [5] d. n. georgiou, s. d. iliadis and b. k. papadopoulos, topology on function spaces and the coordinate continuity, topology proc. 25 (2000), 507–517. [6] d. n. georgiou, s. d. iliadis and b. k. papadopoulos, on dual topologies, topology appl. 140 (2004), 57–68. [7] a. gupta and r. d. sarma, function space topologies for generalized topological spaces, j. adv. res. pure math. 7, no. 4 (2015), 103–112. [8] a. gupta and r. d. sarma, on dual topologies concerning function spaces over cµ,ν(y, z), preprint. [9] v. g. gupta, compact convergence for multifunctions, pure appl. math. sci. 17 (1983), 35–40. [10] v. g. gupta, compact convergence topology for multi-valued functions, proc. nat. acad. sci. india sect. a 53 (1983), 164–167. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 343 a. gupta and r. d. sarma [11] s. hu and n. s. papageorgiou, handbook of multivalued analysis, vol. i theory, kluwer academic publishers, dordrecht, 1997. [12] p. jain and s. p. arya, some function space topologies for multifunctions, india j. pure appl. math. 6 (1975), 1488–1506. [13] f. jordan, coincidence of function space topologies, topology appl. 157 (2010), 336– 351. [14] e. klein and a. thompson, theory of correspondences: including applications to mathematical economics, canadian mathematical society series of monographs and advanced texts. j. wiley & sons, 1984. [15] v. j. mancuso, an ascoli theorem for multi-valued functions, j. austral. math. soc. 12 (1971), 466–472. [16] s. mrowka, on convergence of nets of sets, fund. math. 45 (1958) 237–246. 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[23] l. a. steen and j. a. seebach, counterexamples in topology, springer, new york 1978. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 344 () @ appl. gen. topol. 19, no. 1 (2018), 65-84doi:10.4995/agt.2018.7677 c© agt, upv, 2018 relation-theoretic metrical coincidence and common fixed point theorems under nonlinear contractions md ahmadullah, mohammad imdad and mohammad arif department of mathematics, aligarh muslim university, aligarh,-202002, u.p., india (ahmadullah2201@gmail.com, mhimdad@gmail.com, mohdarif154c@gmail.com) communicated by s. romaguera abstract in this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. our results extend, generalize, modify and unify several known results especially those are contained in berzig [j. fixed point theory appl. 12, 221-238 (2012))] and alam and imdad [to appear in filomat (arxiv:1603.09159 (2016))]. interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to berzig. finally, we use examples to highlight the accomplished improvements in the results of this paper. 2010 msc: 47h10; 54h25. keywords: complete metric spaces; binary relations; contraction mappings; fixed point. 1. introduction banach contraction principle (see [8]) continues to be one of the most inspiring and core result of metric fixed point theory which also has various applications in classical functional analysis besides several other domains especially in mathematical economics and psychology. in the course of last several received 14 may 2017 – accepted 18 september 2017 http://dx.doi.org/10.4995/agt.2018.7677 m. ahmadullah, m. imdad and m. arif years, numerous authors have extended this result by weakening the contraction conditions besides enlarging the class of underlying metric space. in recent years such type of results are also established employing order-theoretic notions. historically speaking, the idea of order-theoretic fixed points was initiated by turinici [23] in 1986. in 2004, ran and reurings [21] formulated a relatively more natural order-theoretic version of classical banach contraction principle. the existing literature contains several relation-theoretic results on fixed, coincidence and common fixed point (e.g., partial order: ran and reurings [21] and nieto and rodŕıguez-lópez [20], tolerance: turinici [25, 26], strict order: ghods et al. [12], transitive: ben-el-mechaiekh [10], preorder: turinici [24] etc). berzig [9] established the common fixed point theorem for nonlinear contraction under symmetric closure of a arbitrary relation. most recently, alam and imdad [5] proved a relation-theoretic version of banach contraction principle employing amorphous relation which in turn unify the several well known relevant order-theoretic fixed point theorems. moreover, for further details one can consults [1, 2, 4, 5, 6, 10, 9, 11, 14, 21, 20, 22, 25, 26, 13]. our aim in this work is to proved some coincidence and common fixed point theorems for nonlinear contraction on metric space endowed with amorphous relation. the results proved herein generalize and unify main results of berzig [9], alam and imdad [5] and several others. to demonstrate the validity of the hypotheses and degree of generality of our results, we also furnish some examples. 2. preliminaries for the sake of simplicity to have possibly self-contained presentation, we require some basic definitions, lemmas and propositions for our subsequent discussion. definition 2.1 ([15, 16]). let (f, g) be a pair of self-mappings defined on a non-empty set x. then (i) a point u ∈ x is said to be a coincidence point of the pair (f, g) if fu = gu, (ii) a point v ∈ x is said to be a point of coincidence of the pair (f, g) if there exists u ∈ x such that v = fu = gu, (iii) a coincidence point u ∈ x of the pair (f, g) is said to be a common fixed point if u = fu = gu, (iv) a pair (f, g) is called commuting if f(gu) = g(fu), ∀ u ∈ x. (v) a pair (f, g) is said to be weakly compatible if f and g commutes at their coincidence points i.e., f(gu) = g(fu) whenever f(u) = g(u) for any u ∈ x. definition 2.2 ([17, 28, 27]). let (f, g) be a pair of self-mappings defined on a metric space (x, d). then (i) (f, g) is said to be weakly commuting if for all u ∈ x, d(f(gu), g(fu)) ≤ d(fu, gu), c© agt, upv, 2018 appl. gen. topol. 19, no. 1 66 relation-theoretic metrical coincidence and common fixed point theorems (ii) (f, g) is said to be compatible if limn→∞ d(f(gun), g(fun)) = 0 whenever {un} ⊂ x is a sequence such that limn→∞ gun = limn→∞ fun, (iii) f is said to be a g-continuous at u ∈ x if gun d−→ gu, for all sequence {un} ⊂ x, we have fun d−→ fu. moreover, f is said to be a g-continuous if it is continuous at every point of x. definition 2.3 ([18]). a subset r of x × x is called a binary relation on x. we say that “u relates v under r” if and only if (u, v) ∈ r. throughout this paper, r stands for a ‘non-empty binary relation’ (i.e., r 6= ∅) instead of ‘binary relation’ and rs := r ∪ r−1, while n0, q and qc stand the set of whole numbers (n0 = n ∪ {0}), the set of rational numbers and the set of irrational numbers respectively. definition 2.4 ([19]). a binary relation r defined on a non-empty set x is called complete if every pair of elements of x are comparable under that relation i.e., for all u, v in x, either (u, v) ∈ r or (v, u) ∈ r which is denoted by [u, v] ∈ r. proposition 2.5 ([4]). let r be a binary relation defined on a non-empty set x. then (u, v) ∈ rs if and only if [u, v] ∈ r. definition 2.6 ([4]). let f be a self-mapping defined on a non-empty set x. then a binary relation r on x is called f-closed if for all u, v ∈ x (u, v) ∈ r ⇒ (fu, fv) ∈ r. definition 2.7 ([5]). let (f, g) be a pair of self-mappings defined on a nonempty set x. then a binary relation r on x is called (f, g)-closed if for all u, v ∈ x, (gu, gv) ∈ r ⇒ (fu, fv) ∈ r. notice that on setting g = i, (the identity mapping on x) definition 2.7 reduces to definition 2.6. definition 2.8 ([4]). let r be a binary relation defined on a non-empty set x. then a sequence {un} ⊂ x is said to be an r-preserving if (un, un+1) ∈ r, ∀ n ∈ n0. definition 2.9 ([5]). let (x, d) be a metric space equipped with a binary relation r. then (x, d) is said to be an r-complete if every r-preserving cauchy sequence in x converges to a point in x. remark 2.10 ([5]). every complete metric space is r-complete, where r denotes a binary relation. moreover, if r is universal relation, then notions of completeness and r-completeness are same. definition 2.11 ([5]). let (x, d) be a metric space equipped with a binary relation r. then a mappings f : x → x is said to be an r-continuous at u if un d−→ u, for any r-preserving sequence {un} ⊂ x, we have fun d−→ fu. moreover, f is said to be an r-continuous if it is r-continuous at every point of x. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 67 m. ahmadullah, m. imdad and m. arif definition 2.12 ([5]). let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r. then f is said to be a (g, r)continuous at u if gun d−→ gu, for any r-preserving sequence {un} ⊂ x, we have fun d−→ fu. moreover, f is called a (g, r)-continuous if it is (g, r)continuous at every point of x. notice that on setting g = i (the identity mapping on x), definition 2.12 reduces to definition 2.11. remark 2.13. every continuous mapping is r-continuous, where r denotes a binary relation. moreover, if r is universal relation, then notions of rcontinuity and continuity are same. definition 2.14. let (x, d) be a metric space equipped with a binary relation r and g a self-mapping on x. then (x, d) is said to be (g, r)-regular if for any r-preserving sequence {un} with un → u such that [gun, gu] ∈ r ∀ n ∈ n. definition 2.15 ([4]). let (x, d) be a metric space. then a binary relation r on x is said to be d-self-closed if for any r-preserving sequence {un} with un d−→ u, there is a subsequence {unk} of {un} such that [unk, u] ∈ r, for all k ∈ n. definition 2.16 ([5]). let g be a self-mapping on a metric space (x, d). then a binary relation r on x is said to be (g, d)-self-closed if for any r-preserving sequence {un} with un d−→ u, there is a subsequence {unk} of {un} such that [gunk, gu] ∈ r, for all k ∈ n. notice that under the consideration g = i (the identity mapping on x), definition 2.16 turn out to be definition 2.15. definition 2.17 ([22]). let (x, d) be a metric space endowed with an arbitrary binary relation r. then a subset d of x is said to be an r-directed if for every pair of points u, v in d, there is w in x such that (u, w) ∈ r and (v, w) ∈ r. definition 2.18 ([5]). let g be a self-mapping on a metric space (x, d) endowed with a binary relation r. then a subset d of x is said to be a (g, r)-directed if for every pair of points u, v in d, there is w in x such that (u, gw) ∈ r and (v, gw) ∈ r. notice that on setting g = i (the identity mapping on x), definition 2.18 turn out to be definition 2.17. definition 2.19 ([5]). let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r. then the pair (f, g) is said to be an r-compatible if lim n→∞ d(g(fun), f(gun)) = 0, whenever lim n→∞ g(un) = lim n→∞ f(un), for any sequence {un} ⊂ x such that {fun} and {gun} are rpreserving. for a given non-empty set x, together with a binary relation r on x and a pair of self-mappings (f, g) on x, we use the following notations: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 68 relation-theoretic metrical coincidence and common fixed point theorems • c(f, g): the collection of all coincidence points of (f, g); • mf (gu, gv) := max { d(gu, gv), d(gu, fu), d(gv, fv), 1 2 [d(gu, fv) + d(gv, fu)] } ; and • nf (gu, gv) := max { d(gu, gv), 1 2 [d(gu, fu) + d(gv, fv)], 1 2 [d(gu, fv) + d(gv, fu)] } . remark 2.20. observe that, nf (gu, gv) ≤ mf (gu, gv), for all u, v ∈ x. let φ be the family of all mappings ϕ : [0, ∞) → [0, ∞) satisfying the following properties: (φ1): ϕ is increasing; (φ2): ∞ ∑ n=1 ϕn(t) < ∞ for each t > 0, where ϕn is the n-th iterate of ϕ. lemma 2.21 ([22]). let ϕ ∈ φ. then for all s > 0, we have ϕ(s) < s. proposition 2.22. let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r and ϕ ∈ φ. then the following conditions are equivalent: (i): d(fu, fv) ≤ ϕ(mf (gu, gv)) with (gu, gv) ∈ r; (ii): d(fu, fv) ≤ ϕ(mf (gu, gv)) with [gu, gv] ∈ r. proof. the implication (ii) ⇒ (i) is straightforward. to show that (i) ⇒ (ii), choose u, v ∈ x such that [gu, gv] ∈ r. if (gu, gv) ∈ r, then (ii) immediately follows from (i). otherwise, if (gv, gu) ∈ r, then by (i) and the symmetry of metric d, we obtained the conclusion. � for the sake of completeness, we state the following theorems: theorem 2.23 ([9, theorems 3.2]). let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a symmetric closure s := r∪r−1 of any binary relation r. suppose the following conditions hold: (a) (x, d) is complete; (b) there exists w0 ∈ x such that (gw0, fw0) ∈ s; (c) s is (f, g) closed; (d) (x, d, s) is regular; (e) there exists ϕ ∈ φ such that d(fu, fv) ≤ ϕ(nf (gu, gv)) for all u, v ∈ x with (gu, gv) ∈ s. then (f, g) has a unique coincidence point. moreover, if c(f, g) is (g, s)directed and (f, g) is weakly compatible, then (f, g) has a unique common fixed point. theorem 2.24 ([5, theorem 2]). let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r and y a subspace of x. assume that the following conditions hold: (f) (y, d) is r-complete subspace of x; (g) f(x) ⊆ y ∩ g(x); (h) ∃ w0 ∈ x such that (gw0, fw0) ∈ r; c© agt, upv, 2018 appl. gen. topol. 19, no. 1 69 m. ahmadullah, m. imdad and m. arif (i) r is (f, g)-closed; (j) there exists α ∈ [0, 1) such that d(fu, fv) ≤ αd(gu, gv) for all u, v ∈ x with (gu, gv) ∈ r; (k) (k1) y ⊆ g(x); (k2) either f is (g, r)-continuous or f and g are continuous or r|y is d-self-closed; or, alternatively (l) (l1) (f, g) is r-compatible; (l2) g is r-continuous; (l3) f is r-continuous or r is (g, d)-self-closed. then (f, g) has a coincidence point. indeed, the main results of this paper are based on the following points: • theorem 2.23 is improved by replacing symmetric closure s of any binary relation with arbitrary binary relation r, • theorems 2.23 (upto coincidence point) and 2.24 are unified by replacing more general contraction condition, • theorem 2.23 is generalized by replacing comparatively weaker notions namely r-completeness of any subspace y ⊆ x, with fx ⊆ y ∩ gx rather than completeness of whole space x, • theorem 2.23 is improved by replacing d-self-closedness or (g, d)-selfclosedness of r instead of regularity of the whole space, • some examples are addopted to demonstrate the realized improvement in the results proved in this article. 3. main results now, we are equipped to prove our main result as follows: theorem 3.1. let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r. assume that the conditions (f), (g), (h), (i) and together with the following conditions hold: (m) there exists ϕ ∈ φ such that d(fu, fv) ≤ ϕ(mf (gu, gv)) (for all u, v ∈ x with (gu, gv) ∈ r); (k′) (k1) y ⊆ g(x); (k′2) either f is (g, r)-continuous, or (f is continuous and g is bicontinuous), or r|y is d-self-closed; or, alternatively (l′) (l′1) (f, g) is r-compatible; (l′2) g is r-continuous and (either f is r-continuous or (x, d) is (g, r)regular); or, alternatively (l′′2 ) g is continuous and (either f is r-continuous or r is (g, d)-selfclosed). then (f, g) has a coincidence point. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 70 relation-theoretic metrical coincidence and common fixed point theorems proof. let w0 ∈ x such that (gw0, fw0) ∈ r. construct a picard jungck sequence {gwn}, with the initial point w0, i.e., (3.1) g(wn+1) = f(wn), for all n ∈ n0. also as (gw0, fw0) ∈ r and r is (f, g)-closed, we have (fw0, fw1), (fw2, fw3), · · · , (fwn, fwn+1), · · · ∈ r. thus, (3.2) (gwn, gwn+1) ∈ r, for all n ∈ n0, therefore {gwn} is r-preserving. from condition (m), we have (for all n ∈ n) (3.3) d(gwn, gwn+1) = d(fwn−1, fwn) ≤ ϕ(mf (gwn−1, gwn)) where, mf (gwn−1, gwn) ≤ max { d(gwn−1, gwn), d(gwn−1, fwn−1), d(gwn, fwn), 1 2 [ d(gwn, fwn−1) + d(gwn−1, fwn) ]} on using (3.1) and tringular inequality, we have (for all n ∈ n) (3.4) mf(gwn−1, gwn) ≤ max { d(gwn−1, gwn), d(gwn, gwn+1) } . on using (3.3), (3.4) and the property (φ1), we obtain (for all n ∈ n) (3.5) d(gwn, gwn+1) ≤ ϕ ( max { d(gwn−1, gwn), d(gwn, gwn+1) }) . now, we show that the sequence {gwn} is cauchy in (x, d). in case gwn0 = gwn0+1 for some n0 ∈ n0, then the result is follows. otherwise, gwn 6= gwn+1 for all n ∈ n0. suppose that d(gwn1−1, gwn1) ≤ d(gwn1, gwn1+1), for some n1 ∈ n. on using (3.5) and lemma 2.21, we get d(gwn1, gwn1+1) ≤ ϕ(d(gwn1 , gwn1+1)) < d(gwn1, gwn1+1), which is a contradiction. thus d(gwn, gwn+1) < d(gwn−1, gwn) (for all n ∈ n), so that d(gwn, gwn+1) ≤ ϕ(d(gwn−1, gwn)), for all n ∈ n. employing induction on n and the property (φ1), we get d(gwn, gwn+1) ≤ ϕn(d(gw0, gw1)), for all n ∈ n0. now, for all m, n ∈ n0 with m ≥ n, we have d(gwn, gwm) ≤ d(gwn, gwn+1) + d(gwn+1, gwn+2) + · · · + d(gwm−1, gwm) ≤ ϕn(d(gw0, gw1)) + ϕn+1(d(gw0, gw1)) + · · · + ϕm−1(d(gw0, gw1)) = m−1 ∑ k=n ϕk(d(gw0, gw1)) ≤ ∑ k≥n ϕk(d(gw0, gw1)) → 0 as n → ∞. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 71 m. ahmadullah, m. imdad and m. arif therefore, {gwn} is r-preserving cauchy sequence in x. as {gwn} ⊆ g(x) and {gwn} ⊆ y ⊆ g(x) (due to (3.1) and (k1)), therefore {gwn} is rpreserving cauchy sequence in y. since (y, d) is r-complete, there exists y ∈ y such that gwn d−→ y. as y ⊆ g(x), there exists x ∈ x such that (3.6) lim n→∞ gwn = y = gx. since f is (g, r)-continuous, and on using (3.1) and (3.6), we have (3.7) lim n→∞ gwn+1 = lim n→∞ fwn = fx. due to uniqueness of the limit, we have fx = gx. hence x is a coincidence point of (f, g). next, we assume that f is continuous and g is bi-continuous. then on using (3.1) and (3.6), we get fx = fg−1(gx) = fg−1( lim n→∞ gwn) = lim n→∞ fg−1(gwn) = lim n→∞ fwn = gx. hence x is a coincidence point of (f, g). finally, if r|y is d-self-closed, then for any r-preserving sequence {gwn} in y with gwn d−→ gx, there is a subsequence {gwnk} of {gwn} such that [gwnk , gx] ∈ r|y ⊆ r, for all k ∈ n0. set δ := d(fx, gx) ≥ 0. suppose on contrary that δ > 0. on using condition (m), proposition 2.22 and [gwnk , gx] ∈ r, for all k ∈ n0, we have (3.8) d(gwnk+1, fx) = d(fwnk , fx) ≤ ϕ(mf (gwnk , gx)), where, mf (gwnk , gx) = max { d(gwnk , gx), d(gwnk , gwnk+1), d(gx, fx), 1 2 [d(gwnk , fx) + d(gx, gwnk+1)] } . if mf(gwnk , gx) = d(gx, fx) = δ, then (3.8) reduces to d(gwnk+1, fx) ≤ ϕ(δ), which on making k → ∞, gives arise δ ≤ ϕ(δ), which is a contradiction. otherwise, if mf (gwnk, gx) = max { d(gwnk , gx), d(gwnk , gwnk+1), 1 2 [d(gwnk , fx) + d(gx, gwnk+1)] } , then due to the fact that gwn d−→ gx, there exists a positive integer n = n(δ) such that mf (gwnk , gx) ≤ 4 5 δ, for all k ≥ n. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 72 relation-theoretic metrical coincidence and common fixed point theorems as ϕ is increasing, we have (3.9) ϕ(mf (gwnk, gx)) ≤ ϕ( 4 5 δ), for all k ≥ n. on using (3.8) and (3.9), we get d(gwnk+1, fx) = d(fwnk , fx) ≤ ϕ( 4 5 δ), for all k ≥ n. letting k → ∞ and using lemma 2.21, we get δ ≤ ϕ(4 5 δ) < 4 5 δ < δ, which is again a contradiction. hence, δ = 0, so that d(fx, gx) = δ = 0 ⇒ fx = gx. thus, x is a coincidence point of (f, g). alternatively, we suppose that (l′) holds. firstly, assume that (l′2) holds. as {gwn} ⊂ f(x) ⊆ y (in view (3.1)) we notice that {gwn} is r-preserving cauchy sequence in y. since y is r-complete, there exists y ∈ y such that (3.10) lim n→∞ gwn = y and lim n→∞ fwn = y. as {fwn} and {gwn} are r-preserving sequences (due to (3.1) and (3.2)), utilizing the condition (l′1) and (3.10), we obtain (3.11) lim n→∞ d(gfwn, fgwn) = 0. using (3.2), (3.10), and due to r-continuity of f and g, we have (3.12) lim n→∞ g(fwn) = g( lim n→∞ fwn) = gy, and (3.13) lim n→∞ f(gwn) = f( lim n→∞ gwn) = fy. on using (3.11)–(3.13) and continuity of d, we have fy = gy. hence y is a coincidence point of (f, g). next, assume that (x, d) is (g, r)-regular. as {gwn} is r-preserving and gwn d−→ y (due to (g, r)-regularity of (x, d)), we have [ggwn, gy] ∈ r, ∀ n ∈ n0. set η := d(fy, gy) ≥ 0. suppose on contrary that η > 0. on utilizing the condition (m), proposition 2.22 and [ggwn, gy] ∈ r, for all n ∈ n0, we have (3.14) d(fgwn+1, fy) ≤ ϕ(mf (ggwn+1, gy)), where, mf(ggwn+1, gy) = max { d(ggwn+1, gy), d(ggwn+1, fgwn+1), d(gy, fy), 1 2 [d(ggwn+1, fy) + d(gy, fgwn+1)] } . c© agt, upv, 2018 appl. gen. topol. 19, no. 1 73 m. ahmadullah, m. imdad and m. arif if mf(ggwn, gy) = d(gy, fy) = η, then (3.14) yields |d(fgwn+1, gfwn+1) − d(gfwn+1, fy)| ≤ d(fgwn+1, fy) ≤ ϕ(η), on making n → ∞; using (3.10), (3.11), continuity of d and r-continuity of g, we get η ≤ ϕ(η), which is a contradiction. therefore, η = 0, so that d(fy, gy) = η = 0 ⇒ fy = gy. hence y is a coincidence point of (f, g). otherwise, let mf (ggwn+1, gy) = max { d(ggwn+1, gy), d(ggwn+1, fgwn+1), 1 2 [d(ggwn+1, fy) + d(gy, fgwn+1)] } . now, on using triangular inequality, we have mf (ggwn+1, gy) ≤ max { d(ggwn+1, gy), d(ggwn+1, ggwn+2) + d(ggwn+2, fgwn+1), 1 2 [d(ggwn+1, fy) + d(gy, gfwn+1) + d(gfwn+1, fgwn+1)] } . on making n → ∞, on using (3.1), (3.10), (3.11), continuity of d and rcontinuity of g, we get lim n→∞ mf(ggwn+1, gy) = 1 2 η. since η > 0. by definition, there exists a positive integer n = n(η) such that mf (ggwn+1, gy) ≤ 4 5 η, for all n ≥ n. as ϕ is increasing, we have ϕ(mf (ggwn+1, gy)) ≤ ϕ( 4 5 η), for all n ≥ n, again (3.14) yields that ( for all n ≥ n) (3.15) |d(fgwn+1, gfwn+1) − d(gfwn+1, fy)| ≤ d(fgwn+1, fy) ≤ ϕ( 4 5 η). hence, |d(fgwn+1, gfwn+1) − d(gfwn+1, fy)| ≤ ( 4 5 η), for all n ≥ n. letting n → ∞, on using (3.10), (3.11), continuity of d and r-continuity of g, we get η ≤ ϕ(4 5 η) < 4 5 η < η, which is again a contradiction. hence, η = 0, so that d(fy, gy) = η = 0 ⇒ fy = gy. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 74 relation-theoretic metrical coincidence and common fixed point theorems hence, y is a coincidence point of (f, g). alternatively, suppose that (l′′2 ) holds. firstly, we assume that f is rcontinuous. since continuity of g implies r-continuity of g. thus on the similar lines of proof of (l′2) conclusion follows. secondly, we assume that r is (g, d)-self-closed. as {gwn} is r-preserving and gwn d−→ y (due to (g, d)-selfclosedness of r), there exists a subsequence {gwnk} of {gwn} such that [ggwnk, gy] ∈ r, ∀ k ∈ n0. since gwn d−→ y, therefore gwnk d−→ y for any subsequence {gwnk} of {gwn}. thus on the similar lines of above proof, we obtain (for all k ≥ n) (3.16) |d(fgwnk+1, gfwnk+1) − d(gfwnk+1, fy)| ≤ d(fgwnk+1, fy) ≤ ϕ( 4 5 η). hence, |d(fgwnk+1, gfwnk+1) − d(gfwnk+1, fy)| ≤ ( 4 5 η), for all k ≥ n. letting k → ∞, on using (3.10), (3.11), continuity of d and g, we get η ≤ ϕ(4 5 η) < 4 5 η < η, which is again a contradiction. hence, η = 0, so that d(fy, gy) = η = 0 ⇒ fy = gy. hence, y is a coincidence point of (f, g). this completes the proof. � on account taking y = x in theorem 3.1, we deduce a corollary which is sharpened version of theorem 2.23 up to coincidence point in view of comparatively weaker notions in the considerations of completeness, regularity and contraction condition. corollary 3.2. let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r. suppose that the conditions (h), (i), (m) together with the following conditions hold: (n) (x, d) is r-complete; (o) f(x) ⊆ g(x); (p) g is onto with together the condition (k′2) [or, alternatively condition (l′)]. then (f, g) has a coincidence point. in lieu of remarks 2.20, theorem 3.1 reduces to the following corollary. corollary 3.3. let (f, g) be a pair of self-mappings defined on a metric space (x, d) endowed with a binary relation r and y be a subspace of x. assume the conditions (f), (g), (h), (i) and (k′) (or (l′)) together with the following condition holds: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 75 m. ahmadullah, m. imdad and m. arif (q) there exists ϕ ∈ φ such that (for all u, v ∈ x with (gu, gv) ∈ r) d(fu, fv) ≤ ϕ ( max { d(gu, gv), 1 2 [d(gu, fu)+d(gv, fv)], 1 2 [d(gu, fv)+d(gv, fu)] ) . then (f, g) has a coincidence point. now, we establish the following results for the uniqueness of common fixed point (corresponding to corollary 3.3): theorem 3.4. in addition to the hypotheses of corollary 3.3, suppose that the following condition holds: (r) f(x) is (g, rs)-directed. then (f, g) has a unique point of coincidence. moreover, if (f, g) is weakly compatible, then (f, g) has a unique common fixed point. proof. we prove the result in three steps. step 1: by corollary 3.3, c(f, g) is non-empty. if c(f, g) is singleton, then there is nothing to prove. otherwise, to substantiate the proof, take two arbitrary elements u, v in c(f, g), so that fu = gu = x and fv = gv = y. now, we are required to show that x = y. since x, y ∈ fx and fx is (g, rs)directed, there exists u0 ∈ x such that [x, gu0] ∈ r and [y, gu0] ∈ r. now, we construct a sequence {gun} corresponding to u0, so that gun+1 = fun for all n ∈ n0. we claim that lim n→∞ d(x, gun) = 0. if d(x, gun0) = 0, for some n0 ∈ n0, then there is nothing to prove. otherwise, d(x, gun) > 0, for all n ∈ n0. as [x, gun] ∈ r, for all n ∈ n0 (due to the fact that (f, g)-closedness of r and [x, gu0] ∈ r), by proposition 2.22 and hypothesis (q), we get (3.17) d(x, gun+1) = d(fu, fun) ≤ ϕ(nf (gu, gun)), where, nf (gu, gun) = max { d(gu, gun), 1 2 [d(gu, fu) + d(gun, fun)], 1 2 [d(gu, fun) + d(gun, fu) ]} ≤ max { d(gu, gun), 1 2 [d(gun, gu) + d(fun, gu)], 1 2 [d(gu, fun) +d(gun, fu)] } ≤ max { d(gu, gun), 1 2 [d(gun, gu) + d(fun, gu)] } ≤ max { d(gu, gun), d(gu, fun) } = max { d(gu, gun), d(gu, gun+1) } , c© agt, upv, 2018 appl. gen. topol. 19, no. 1 76 relation-theoretic metrical coincidence and common fixed point theorems on using this and property (φ1) (3.17) yields (for all n ∈ n0) d(gu, gun+1) ≤ ϕ ( max { d(gu, gun), d(gu, gun+1) }) = ϕ ( d(gu, gun) ) , otherwise, we get a contradiction. so, by induction on n, we get d(gu, gun) ≤ ϕn(d(gu, gu0)), for all n ∈ n0, which on making n → ∞ and using the property (φ2), we get (3.18) lim n→∞ d(gu, gun) = 0. similarly, we can obtain (3.19) lim n→∞ d(gv, gun) = 0. using (3.18) and (3.19) we have d(x, y) ≤ d(gu, gun) + d(gun, gv) → 0, as n → ∞ ⇒ x = y i.e., (f, g) has a unique point of coincidence . step 2: now, we claim that the pair (f, g) has a common fixed point, let x ∈ c(f, g), i.e., fx = gx. due to weakly compatibility of the pair (f, g), we have (3.20) f(gx) = g(fx) = g(gx). put gx = y. then from (3.20), fy = gy. hence y is also a coincidence point of f and g. in view of step 1, we have y = gx = gy = fy, so that y is a common fixed point (f, g). step 3: to prove the uniqueness of common fixed point of (f, g), let us assume that w is another common fixed point of (f, g). then w ∈ c(f, g), by step 1, w = gw = gy = y. hence (f, g) has a unique common fixed point. � remark 3.5. in view of theorem 3.4, we have used comparatively more natural condition “(g, rs)-directedness of f(x)” instead of “(g, rs)-directedness of c(f, g)” which is too restrictive. our proof carry on even if we take “c(f, g) is (g, rs)-directed”. since point of coincidence implies that coincidence point due to weakly compatible of (f, g), as in our theorem 3.4 we want to find unique common fixed point of f and g which is the point in c(f, g). theorem 3.6. in addition to the hypotheses of theorem 3.1, assume the condition (r) together with the following condition holds: (s) one of f and g is one to one . then (f, g) has a unique coincidence point. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 77 m. ahmadullah, m. imdad and m. arif proof. due to theorem 3.1, c(f, g) 6= ∅. let u, v ∈ c(f, g), and hence in similar lines of the proof of theorem 3.4, we have gu = fu = fv = gv. since either f or g is one-one, we have u = v. � notice that theorem 3.6 is a natural improved version of theorem 4 due to alam and imdad [5]. theorem 3.7. in addition to the hypotheses of theorem 3.1, assume the following condition holds: (t) r|fx is complete. then (f, g) has a unique point of coincidence. moreover, if (f, g) is weakly compatible, then (f, g) has a unique common fixed point. proof. from theorem 3.1, we have c(f, g) 6= ∅. if c(f, g) is singleton, then proof is over. otherwise, choose any two elements x 6= y in c(f, g), so that fx = gx = x and fy = gy = y. as r|fx is complete, [x, y] ∈ r. using proposition 2.22 and condition (m), we get d(x, y) = d(fx, fy) ≤ ϕ ( max { d(gx, gy), d(gx, fx), d(gy, fy), 1 2 [d(gx, fy) + d(gy, fx)] }) = ϕ ( d(gx, gy) ) < d(gx, gy) = d(x, y), which is a contradiction, hence d(x, y) = 0, therefore x = y. thus (f, g) has a unique point of coincidence. thus the remaining part of the proof can be obtained from theorem 3.4. � remark 3.8. indeed, theorem 3.7 is more general as compared to corollary 5.1 of berzig [9] and corollary 3 due to alam and imdad [5]. in regard of remark 3.5, on considering symmetric closure s of any binary relation r in theorem 3.4, we obtain the following sharpened version of theorem 2.23. corollary 3.9. let (f, g) be a pair of self-mappings defined on a metric space (x, d) endowed with symmetric closure s of any arbitrary binary relation defined on x and y be a subspace of x. assume that the conditions (b), (c), (e) and (g) together with the following conditions hold: (u) (y, d) is s-complete subspace of x; (v) (v1) y ⊆ g(x); c© agt, upv, 2018 appl. gen. topol. 19, no. 1 78 relation-theoretic metrical coincidence and common fixed point theorems (v2) either f is (g, s)-continuous or (f is continuous and g is bicontinuous) or s|y is d-self-closed; or, alternatively (w) (w1) (f, g) is s-compatible; (w2) g is s-continuous and (either f is s-continuous or (x, d) is (g, s)regular); or, alternatively (w′2) g is continuous and (either f is s-continuous or s is (g, d)-selfclosed). then (f, g) has a coincidence point. moreover, if c(f, g) is (g, rs)-directed and (f, g) is weakly compatible, then (f, g) has a unique common fixed point. notice that the hypotheses ‘s is (f, g)-closed’ is equivalent to f is a ‘gcomparative’ and ‘s|y is d-self-closed’ is more natural ‘the regular property of (y, d, s)’. further ‘s is (g, d)-self-closed’ is more natural the ‘s is d-self-closed’. 4. consequences as consequences of our former proved results, we deduce several well known results of the existing literature. on the setting ϕ(t) = kt, with k ∈ [0, 1), we obtain the following corollaries which are immediate consequences of theorem 3.4. corollary 4.1. let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r and y a subspace of x. suppose that the conditions (f), (g), (h), (i) and (k′) (or (l′)) together with the following condition holds: (q1) there exists k ∈ [0, 1] such that ( for all u, v ∈ x with (gu, gv) ∈ r) d(fu, fv) ≤ k(max { d(gu, gv), 1 2 [d(gu, fu) + d(gv, fv)], 1 2 [d(gu, fv) + d(gv, fu)] }) . then (f, g) has a coincidence point. moreover, if f(x) is (g, rs)-directed and (f, g) is weakly compatible, then (f, g) has a unique common fixed point. remark 4.2. corollary 4.1 is a sharpened version of corollary 5.10 of berzig [9] and corollary 3.3 (corresponding to condition (20)) due to ahmadullah et al. [2]. corollary 4.3. let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r. suppose that the conditions (f), (g), (h), (i) and (k′) (or (l′)) together with the following condition holds: (q2) there exist a, b, c ≥ 0 with a+2b+2c < 1 such that (for all u, v ∈ x with (gu, gv) ∈ r) d(fu, fv) ≤ ad(gu, gv) + b[d(gu, fu) + d(gv, fv)] + c[d(gu, fv) + d(gv, fu)]. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 79 m. ahmadullah, m. imdad and m. arif then (f, g) has a coincidence point. moreover, if f(x) is (g, rs)-directed and (f, g) is weakly compatible, then (f, g) has a unique common fixed point. remark 4.4. corollary 4.3 remains a sharpened version of corollary 5.11 due to berzig [9] and corollary 3.3, (corresponding to condition (22)) in view of ahmadullah et al. [2]. remark 4.5. if b = 0 and c = 0 in corollary 4.3, then we deduces theorem 2.24 (see alam and imdad [5]). corollary 4.6. let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped a binary relation r and y a subspace of x. assume that the conditions (f), (g), (h), (i) and (k′) (or (l′)) together with the following condition holds: (q3) there exists k ∈ [0, 1/2) such that (for all u, v ∈ x with (gu, gv) ∈ r) d(fu, fv) ≤ k[d(gu, fu) + d(gv, fv)]. then (f, g) has a coincidence point. moreover, if f(x) is (g, rs)-directed and (f, g) is weakly compatible, then (f, g) has a unique common fixed point. remark 4.7. corollary 4.6 remains a improved version of corollary 5.13 established in berzig [9] and corollary 3.3 (corresponding to condition (18)) in ahmadullah et al. [2]. corollary 4.8. let (f, g) be a pair of self-mappings defined on a metric space (x, d) equipped with a binary relation r and y a subspace of x. assume that the conditions (f), (g), (h), (i) and (k′) (or (l′)) togetrher with the following condition holds: (q4) there exists k ∈ [0, 1/2) such that (for all u, v ∈ x with (gu, gv) ∈ r) d(fu, fv) ≤ k [ d(gu, fv) + d(gv, fu) ] . then (f, g) has a coincidence point. moreover, if f(x) is (g, rs)-directed and (f, g) is weakly compatible, then (f, g) has a unique common fixed point. remark 4.9. corollary 4.8 is an improved version of corollary 5.14 of berzig [9] and corollary 3.3 (corresponding to condition (19)) due to ahmadullah et al. [2]. remark 4.10. under the consideration g = i (identity mapping on x), theorems 3.1 and 3.4 deduce the fixed point results of ahmadullah et al. [3, theorem 2.1 and 2.5]. remark 4.11. on setting g = i (identity mapping on x), in corollaries 3.2-4.8, we deduce the fixed point results which are the sharpened version of several results in the existing literature. under the universal relation (i.e., r = x × x), theorems 3.4 and 3.7 unify to the following lone corollary: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 80 relation-theoretic metrical coincidence and common fixed point theorems corollary 4.12. let (x, d) be a metric space and (f, g) a pair of self-mappings on x. suppose that the following conditions hold: (a) there exists y ⊆ x, f(x) ⊆ y ⊆ g(x) such that (y, d) is complete; (b) there exists ϕ ∈ φ such that (for all u, v ∈ x) d(fu, fv) ≤ ϕ ( max { d(gu, gv), 1 2 [d(gu, fu) + d(gv, fv)], 1 2 [d(gu, fv) + d(gv, fu)] }) . then (f, g) has a unique common fixed point. 5. illustrative examples in this section, we furnish some examples to demonstrate the realized improvement of our proved results. example 5.1. let (x, d) be a metric space, where x = (−2, 4) and d(x, y) = |x − y|, ∀ x y ∈ x. now, define a binary relation r = {(x, y) ∈ x2 | x ≥ y, x, y ≥ 0} ∪ {(x, y) ∈ x2 | x ≤ y, x, y ≤ 0}, an increasing mapping ϕ : [0, ∞) → [0, ∞) by ϕ(s) = 1 3 s and two self-mappings f, g : x → x by f(x) = 0, ∀ x ∈ (−2, 4) and g(x) = { x 3 , x ∈ (−2, 3]; 1, x ∈ (3, 4). let y = [−1 2 , 1), so that f(x) = {0} ⊂ y ⊂ gx = (−2 3 , 1] and y is rcomplete but x is not r-complete. indeed, r is (f, g)-closed, f and g are r-continuous and (f, g) is r-compatible. by straightforward calculations, one can easily verify hypothesis (m) of theorems 3.1 thus in all by theorem 3.1 we obtain, (f, g) has a coincidence point (observe that, c(f, g) = {0}). moreover, as fx is (g, rs)-directed, r|fx is complete and (f, g) commute at their coincidence point i.e., x = 0 therefore, all the hypotheses of theorems 3.4 and 3.7 are satisfied, ensuring the uniqueness of the common fixed point. notice that, x = 0 is the only common fixed point of (f, g). with a view to show the genuineness of our results, notice that r is not symmetric and r can not be a symmetric closure of any binary relation. also (x, d) is not complete and even not r-complete which shows that theorems 3.1, 3.4 and 3.7 are applicable to the present example, while theorem 2.23 and even corollary 3.2 are not, which substantiates the utility of theorems 3.1, 3.4 and 3.7. example 5.2. let x = [0, 4) with usual metric d and r = {(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 3)} be a binary relation whose symmetric closure s = {(0, 0), (0, 1), (1, 0), (2, 3), (1, 1), (1, 2), (2, 1), (3, 2)} and (f, g) a pair of self-mappings on x defined by f(x) = { 0, x ∈ [0, 4) ∩ q; 1, x ∈ [0, 4) ∩ qc, and g(x) = { x, x ∈ {0, 1, 2}; 3, otherwise. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 81 m. ahmadullah, m. imdad and m. arif let y = {0, 1} which is r-complete and fx = {0, 1} ⊆ y ⊆ gx = {0, 1, 2, 3}. define an increasing function ϕ : [0, ∞) → [0, ∞) by ϕ(s) = 5 6 s. clearly, ϕ ∈ φ and both f and g are not continuous. also, r is (f, g)-closed. take any r-preserving sequence {xn} in y, i.e., (xn, xn+1) ∈ r|y , for all n ∈ n with xn d−→ x. here, one can notice that if (xn, xn+1) ∈ r|y , for all n ∈ n then there exists n ∈ n such that xn = x ∈ {0, 1}, for all n ≥ n. so, we choose a subsequence {xnk} of the sequence {xn} such that xnk = x, for all k ∈ n, which amounts to saying that [xnk , x] ∈ r|y , for all k ∈ n. therefore, r|y is d-self-closed. now, to substantiate the contraction condition (m) of theorems 3.1. for this, we need to verify for (gx, gy) ∈ {(2, 3)}, otherwise, d(fx, fy) = 0. if (gx, gy) ∈ {(2, 3)} ⇒ x = 2, y ∈ [0, 4) − {0, 1, 2}, then there are two cases arises: case (1): if x = 2, y ∈ ([0, 4) − {0, 1, 2}) ∩ q, then condition (m) is obvious. case (2): if x = 2, y ∈ ([0, 4) − {0, 1, 2, 3}) ∩ qc, then we have d(f2, fy) = 1 ≤ ϕ(max{d(g2, gy), d(g2, f2), d(gy, fy), 1 2 [d(g2, fy) + d(gy, f2)]}) = ϕ(max{d(2, 3), d(2, 0), d(3, 1), 1 2 [d(2, 1) + d(3, 0)]} = ϕ(2). thus all the conditions of theorem 3.1 are satisfied, hence (f, g) has a coincidence point (namely c(f, g) = {0}). also fx is (g, rs)-directed, (f, g) commutes at their coincidence point i.e., at x = 0 and condition (m) of theorem 3.4 holds. therefore all the hypotheses of theorem 3.4 are satisfied. notice that, x = 0 is the only common fixed point of (f, g). now, since (gx, gy) = (2, 3) ∈ r, clearly x = 2, we choose y = √ 2 but 1 = d(f2, f √ 2) ≤ αd(g2, g √ 2) = α, which shows that contraction condition of theorem 2.24 (due to alam and imdad [5]) is not satisfied. further, theorem 2.23 is not applicable to the present example as underlying metric space (x, d) is not complete and r is not symmetric closure of any binary relation. thus, our results are an improvement over theorem 2.23 (due to berzig [9]) and theorem 2.24 (alam and imdad [5]). acknowledgements. all the authors are thankful to an anonymous referee for his/her valuable comments and suggestions. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 82 relation-theoretic metrical coincidence and common fixed point theorems references [1] m. ahmadullah, j. ali and m. imdad, unified relation-theoretic metrical fixed point theorems under an implicit contractive condition with an application, fixed point theory appl. 2016:42 (2016), 1–15 . 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[26] m. turinici, nieto-lopez theorems in ordered metric spaces, math. student 81, no. 1-4 (2012), 219–229. [27] s. sessa, on a weak commutativity condition of mappings in fixed point considerations, pub. inst. math. soc. 32 (1982),149–153. [28] k. p. r. sastry and i. s. r. krishna murthy, common fixed points of two partially commuting tangential selfmaps on a metric space, j. math. anal. appl. 250, no. 2 (2000), 731–734. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 84 @ appl. gen. topol. 20, no. 2 (2019), 307-324 doi:10.4995/agt.2019.7910 c© agt, upv, 2019 simple dynamical systems k. ali akbar a, v. kannan b and i. subramania pillai c a k. ali akbar, department of mathematics, central university of kerala, kasaragod 671320, kerala, india. (aliakbar.pkd@gmail.com, aliakbar@cukerala.ac.in) b v. kannan, school of mathematics and statistics, university of hyderabad, hyderabad 500 046, telangana, india. (vksm@uohyd.ernet.in) c i. subramania pillai, department of mathematics, pondicherry university, puducherry-605014, india. (ispillai@gmail.com) communicated by m. sanchis abstract in this paper, we study the class of simple systems on r induced by homeomorphisms having finitely many non-ordinary points. we characterize the family of homeomorphisms on r having finitely many nonordinary points upto (order) conjugacy. for x, y ∈ r, we say x ∼ y on a dynamical system (r, f) if x and y have same dynamical properties, which is an equivalence relation. said precisely, x ∼ y if there exists an increasing homeomorphism h : r → r such that h ◦ f = f ◦ h and h(x) = y. an element x ∈ r is ordinary in (r, f) if its equivalence class [x] is a neighbourhood of it. 2010 msc: 54h20; 26a21; 26a48. keywords: special points; non-ordinary points; critical points; order conjugacy. 1. introduction a dynamical system is a pair (x,f) where x is a metric space and f is a continuous self map on x. two dynamical systems (x,f), (y,g) are said to be topological conjugate if there exists a homeomorphism h : x → y (called topological conjugacy) such that h◦f = g◦h. the properties of dynamical systems which are preserved by topological conjugacies are called dynamical properties. received 26 july 2017 – accepted 30 july 2019 http://dx.doi.org/10.4995/agt.2019.7910 k. ali akbar, v. kannan and i. subramania pillai the points which are unique upto some dynamical property are called dynamically special points. said differently, a special point has a dynamical property which no other point has. the idea of special points is relatively new to the literature (see [7]). in this paper, we introduce the notion of non-ordinary point. throughout this paper we will be working with continuous self homeomorphisms of the real line. since r has order structure, we would like to consider the topological conjugacies (simply we call conjugacies) preserving the order. hence the conjugacies which we mainly consider in this paper are order preserving conjugacies (increasing conjugacies). the increasing conjugacies are usually called order conjugacies. when we are working with a single system, any self conjugacy can utmost shuffle points with same dynamical behavior. therefore a point which is unique upto its behavior must be fixed by every self conjugacy. on the other hand, if a point is fixed by all self conjugacies then it must have a special property (some times it may not be known explicitly). these ideas motivated us to call the set of all points fixed by all self conjugacies as set of special points. for x,y ∈ r, we write x ∼ y if x and y have the same dynamical properties in the dynamical system (r,f). said precisely, x ∼ y if there exists an increasing homeomorphism h : r → r such that h◦f = f ◦h and h(x) = y. it is easy to see that ∼ is an equivalence relation. since the equivalence relation is coming from self conjugacy it is important in the field of topological dynamics. let [x] denote the equivalence class of x ∈ r. in a dynamical system (x,f), we say that a point x is ordinary if its “like” points are near to it. that is, an element x ∈ r is ordinary in (r,f) if its equivalence class [x] is a neighbourhood of it, i.e., the equivalence class of x contains an open interval around x. a point which is not ordinary is called non-ordinary . let n(f) be the set of all non-ordinary points of f. we call a point to be special if [x] = {x}. let s(f) be the set of all special points of f. a point x in a topological space x is said to be rigid if it is fixed by every self homeomorphism of x. for example, the point 1 is rigid in (0, 1]. according to the above definition all rigid points are special even though there is no role for the map f. we make this as a convention. by definition, the points of [x] are dynamically same. we consider systems for which there are only finitely many equivalence classes. this means there are only finitely many kinds of orbits upto conjugacy. in particular, their sets of periods per(f) are contained in {1, 2, 22, ...}. if f : r → r is continuous and per(f) properly contained in {1, 2, 22, ...} then f is not li-yorke chaotic (see [1]). also note that if f : r → r is devaney chaotic then 6 ∈ per(f) (see [2]). therefore, if f : r → r is a continuous map having finitely many non-ordinary points then it is neither li-yorke chaotic nor devaney chaotic due to sharkovskii’s theorem. for these reasons, we call such systems as simple systems. these are the system in which the phase portrait can be drawn. phase portraits (see [5]) are frequently used to graphically represent the dynamics of a system. a phase portrait consists of a diagram representing possible beginning positions in the system and arrows that indicate the change in these positions under iteration c© agt, upv, 2019 appl. gen. topol. 20, no. 2 308 simple dynamical systems of the function. the drawable systems are interesting to physicists and for this reason the study of the class of simple dynamical systems can be useful. our main results characterize the family of homeomorphisms on r having finitely many non-ordinary points upto (order) conjugacy. in particular, we prove that: (i) the number of all increasing continuous bijections (upto order conjugacy) on r with exactly n non-ordinary points is equal to an = c1(1 + √ 3)n +c2(1−√ 3)n, where c1 = (5+3 √ 3) 2 √ 3 and c2 = (3 √ 3−5) 2 √ 3 . (ii) if tn denotes the number of increasing homeomorphisms (upto conjugacy) on r with exactly n non-ordinary points, then t0 = 2, t1 = 5 and t2 = 12, and we have tn =   an+2a n−4 2 2 if n is even an+2a n−3 2 2 if n is odd for n ≥ 3. (iii) if sn denotes the number of decreasing homeomorphisms (upto order conjugacy) on r with exactly n non-ordinary points, then sn = { 0 if n is even an−1 2 if n is odd for all n. (iv) if kn denotes the number of decreasing homeomorphisms (upto conjugacy) on r with exactly n non-ordinary points, then kn = { 0 if n is even tn−1 2 if n is odd for all n. 2. basic results let (x,f) be a dynamical system. we denote the full orbit of a point x ∈ x by the set õ(x) = {y ∈ x : fn(x) = fm(y) for some m,n ∈ n}. for any subset a ⊂ r, let õ(a) = ⋃ x∈a õ(x) = ⋃ x∈a {y ∈ r : fn(y) = fm(x) for some m, n ∈ n}. a point x in a dynamical system (x,f) is said to be a critical point if f fails to be one-one in every neighbourhood of x. the set of all critical points of f is denoted by c(f), and by p(f) we denote the set of all periodic points of f. recall that a point x in a dynamical system (x,f) is said to be periodic if fn(x) = x for some n ∈ n. let f : r → r and d(f) be the set õ(c(f) ∪ p(f)∪{f(∞),f(−∞)}), where f(∞) and f(−∞) are the limits of f at ∞ and −∞ respectively, provide they are finite. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 309 k. ali akbar, v. kannan and i. subramania pillai we will prove below (see proposition 2.13) that if the map has only finitely many non-ordinary points then n(f) = s(f). hence the following characterization theorem holds for the set n(f) since a similar type of characterization holds for s(f). for a continuous map f : r → r, we define sp(r,f) = ⋂ h{x ∈ r : h(x) = x, h : r → r is a homeomorphism such that h◦f = f ◦h}. theorem 2.1. for continuous self maps of the real line r, the set of all special points is contained in the closure of the union of full orbits of critical points, periodic points and the limits at infinity (if they exist and finite). that is, n(f) is a subset of the closure of d(f). proof. for a continuous map f : r → r, first observe that s(f) ⊂ sp(r,f). by theorem 1 of [7], we have sp(r,f) ⊂ d(f). hence the proof follows. � remark 2.2. consider the map f(x) = x + sin(x) for all x ∈ r. observe that all integral multiples of π are fixed points for f but the increasing bijection x 7→ x + 2π commutes with f and fixes none of them. hence in this case n(f) is properly contained in the closure of d(f). now consider the following theorems. theorem 2.3. for polynomials of even degree the equality d(f) = s(f) holds. proof. let f : r → r be a polynomial map of even degree. by theorem 2 of [7], we have sp(r,f) = s(f) = d(f). � theorem 2.4. for polynomial maps f of r, s(f) has to be either empty or a singleton or the closure of d(f). proof. the ideas involved in the proof of theorem 3 as in [7] can be adapted to order conjugacies. hence the proof follows. � from the definition of special points, it is clear that the set of special points s(f) is always closed. the following theorem is about the converse. theorem 2.5. given any closed subset f of r, there exists a continuous map f : r → r such that s(f) = f . proof. the ideas involved in the proof of lemma 2, lemma 3, and theorem 4 can be adapted to order conjugacies. hence the proof follows. � the following total order on n is called the sharkovskii’s ordering: 3 � 5 � 7 � 9 � ... � 2 × 3 � 2 × 5 � 2 × 7 � ... � 2n × 3 � 2n × 5 � 2n × 7 � ... ... � 2n � .... � 22 � 2 � 1. we write m � n if m precedes n (not necessarily immediately) in this order. an n-cycle means a cycle of length n. theorem 2.6 (sharkovskii’s theorem, see[8]). let m � n in the sharkovskii’s ordering. for every continuous self map of r, if there is an m-cycle, then there is an n-cycle. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 310 simple dynamical systems for any continuous map f : r → r, topological conjugacy (respectively order conjugacy) of f is a homeomorphism (respectively increasing homeomorphism) h : r → r such that h◦f = f ◦h. for any continuous map f : r → r, let gf be the set of all topological conjugacies of f and let gf↑ be the set of all order conjugacies of f. proposition 2.7. if x is an ordinary point of f and if h is a self conjugacy of f, then h(x) is ordinary. proof. since x is ordinary there exists an open interval v contained in [x]. we prove that the open interval (since h is a homeomorphism) h(v ) is contained in [h(x)]. take s ∈ h(v ). then s = h(t) for some t ∈ v. since v ⊂ [x], there exists ϕ ∈ gf↑ such that ϕ(t) = x. then the increasing homeomorphism ψ = hϕh−1 carries s to h(x) and commutes with f. � proposition 2.8. if x is a non-ordinary point of f and if h is a self conjugacy of f, then h(x) is non-ordinary. proof. note that if h is a self conjugacy of f then h−1 is also a self conjugacy of f. now, the proof follows from proposition 2.7. � for any subset a of r, we write ∂a = a ∩ (x −a) for the boundary of a, where a denotes the closure of a in r. recall that the properties which are preserved under topological conjugacies are called dynamical properties. hence, if two points x,y in the dynamical system (x,f) differ by a dynamical property, then no conjugacy can map one to the other. hence the following proposition follows. proposition 2.9. the points of ∂sp are non-ordinary for any dynamical property p , where sp denotes the set of all points in (x,f) having the dynamical property p . corollary 2.10. let f : r → r be constant in a neighbourhood of a point x0. then the end points of the maximal interval around x0 on which f is constant are non-ordinary. remark 2.11. note that, being a point in a particular equivalence class [x] is a dynamical property. therefore when there are n non-ordinary points then there are n + 1 equivalence classes. but the converse is not true. consider the map x 7→ x + sin(x) on r. there are two equivalence classes but infinitely many non-ordinary points. now we ask: for a continuous map f : r → r, how the equivalence classes look like? the following lemma answers this question. lemma 2.12. let f : r → r be continuous. suppose a < b and (a,b)∩n(f) = ∅. then x ∼ y for all x,y ∈ (a,b). proof. assume without loss of generality that x < y. suppose x � y, so z = sup([x] ∩ (−∞,y]) exists. clearly z ∈ [x]. if z = y then z ∈ [y] ⊂ r\ [x]. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 311 k. ali akbar, v. kannan and i. subramania pillai otherwise z < y and [z,y) ∩ (r\ [x]) 6= ∅ for every y −x > � > 0 which again shows z ∈ r\ [x]. then z ∈ ∂([x]) and hence z ∈ n(f) by proposition 2.9. but a < x ≤ z ≤ y < b. hence z ∈ (a,b) ∩n(f) contradicting our hypothesis. � proposition 2.13. if f : r → r has only finitely many non-ordinary points then every non-ordinary point is special. proof. since the set of all non-ordinary points n(f) is finite, it follows from proposition 2.7 and proposition 2.8 that h(n(f)) = n(f) for all h ∈ gf↑. then we must have h(x) = x for all x ∈ n(f) because of the order preserving nature of h. hence all points of n(f) are special. � thus by above proposition the idea of special points and the idea of non-ordinary points coincide in the class of maps with finitely many non-ordinary points. for a set a, we denote |a| for the cardinality of a. now we consider the following theorem. theorem 2.14. let f : r → r be continuous. if |n(f)| = n then |{[x] : x ∈ r}| = 2n + 1. proof. let n(f) = {x1,x2, · · · ,xn} where x1 < x2 < · · · < xn. by proposition 2.13, each {xi} is an equivalence class. then each of these intervals (−∞,x1), (x1,x2), · · · , (xn−1,xn), (xn,∞) is invariant under every element of gf↑. hence all the remaining equivalence classes are contained in one of these intervals. lemma 2.12 above now shows that each of these interval is an equivalence class, giving |{[x] : x ∈ r}| = 2n + 1. � remark 2.15. note that, being a point in a particular equivalence class [x] is a dynamical property. remark 2.16. if f : r → r has a unique fixed point then it is non-ordinary and vice-versa. proof. since the topological conjugacies carry fixed points to fixed points, the unique fixed point must be fixed by every self conjugacy and hence special. next suppose x0 ∈ r is the unique non-ordinary point of f. then h(x0) = x0 for all h ∈ gf↑. now, for any h ∈ gf↑ we have h(f(x0)) = f(h(x0)) = f(x0). that is, the point f(x0) is special. then we have f(x0) = x0 since x0 is the only special point. � remark 2.17. if f : r → r has finitely many fixed points (critical points) then all fixed (critical) points are special and hence non-ordinary. proof. this remark follows from the fact that under a topological conjugacy fixed points will be mapped to fixed points and critical points will be mapped to critical points and the fact that it takes the finite set f (of fixed points) to f bijectively preserving the order. � remark 2.18. if there are only finitely many periodic cycles then all periodic points are special. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 312 simple dynamical systems the following remark says that in general non-ordinary points need not be special. remark 2.19. it is immediate from the definition that every special point is non-ordinary. but every non-ordinary point may not be special. for example, consider the map x 7→ x + sin x on r which has countably many fixed points. note that all the fixed points are non-ordinary and they all together form a single equivalence class. hence they are not special. proposition 2.20. for maps with finitely many non-ordinary points, f(x) is non-ordinary whenever x is non-ordinary. proof. since x is non-ordinary and since there are only finitely many nonordinary points, we have h(x) = x for all h ∈ gf↑. now for any h ∈ gf↑, we have h(f(x)) = f(h(x)) = f(x). hence f(x) is non-ordinary. � for a map f : a → r, a ⊂ r, we denote sup f and inf f for the supremum of f(a) and infimum of f(a) respectively. recall that, if f : r → r has a unique non-ordinary point then it must be a fixed point. proposition 2.21. let f : r → r be a continuous function and let x ∈ r. then (i) if x ∈ r is both critical and ordinary then f is locally constant at x. (ii) if x is ordinary then so is f(x) unless f is constant in a neighbourhood of x. proof. (i) let x0 ∈ r be both critical and ordinary. claim: f is constant in some neighbourhood of x0. since x0 is ordinary, there exists η > 0 such that all points in (x0−η,x0 +η) will look alike. so it is enough to prove that f is somewhere constant in (x0 −η,x0 + η). case 1: suppose some point in (x0 − η, x0 + η) is point of local maximum of f. then we can prove easily that every point of (x0 − η, x0 + η) is a point of local maximum. that is there exists � > 0 such that f(x0) ≥ f(t) ∀ t ∈ (x0 − �, x0 + �). next choose δ < �, η. then there exists y ∈ [x0 − δ, x0 + δ] such that (2.1) f(y) ≤ f(t)∀ t ∈ [x0 −δ,x0 + δ]. but y is a point of local maximum (since δ < η). that is there exists α > 0 such that (2.2) f(y) ≥ f(s) ∀ s ∈ (y −α, y + α) from (1) and (2), it follows that f is constant in some neighbourhood y and hence constant in some neighbourhood of x0. case 2: no point is a point of local maximum of f. that is, f attains its maximum at one of the endpoints in every subinterval. if f assumes supremum always on c© agt, upv, 2019 appl. gen. topol. 20, no. 2 313 k. ali akbar, v. kannan and i. subramania pillai the right end (or always on the left end) then f is strictly monotone. note that, it is enough if we prove monotone somewhere. take a neighbourhood (α,β) of x0 such that (α,β) ⊂ (x0 − η, x0 + η) and let sup f on (α,β) be attained at the right end point β. suppose sup f is attained at the right end point in every subinterval of (α,β) containing x0. then f is increasing in (x0, β). we are done. suppose there is a subinterval say (x0 − �1, x0 + �2) of (α,β) on which f attains its supremum at the left end point. then f attains its infimum on (x0 − �1,β) at some interior point. we now argue as in case 1. this ends the proof of part (i). (ii) we make use of (i). assume that f is not constant on any neighbourhood of x. because x is ordinary, there exists an open interval j around x, in which all points are equivalent to x, such that f is not constant on j. it follows that f is not constant on any non-trivial subinterval of j, because the endpoints of intervals of constancy are non-ordinary. from (i), it follows that j has no critical point. therefore f(j) is an open interval. we claim that any two elements of f(j) are equivalent. let f(y) be a general element of f(j), where y ∈ j, y 6= x. by choice of j, there exists a self conjugacy h of f such that h(y) = x, which implies h◦f(y) = f ◦h(y) = f(x). therefore f(y) is equivalent to f(x). this proves f(x) is ordinary. � remark 2.22. let f : r → r be continuous. then sup f(r), inf f(r), limx→∞f(x) and limx→−∞f(x) are non-ordinary provided they are finite. note that, for maps with finitely many non-ordinary points both limx→∞f(x) and limx→−∞f(x) always exist in r∪{∞,−∞}. proof. for any h ∈ gf↑, h(f(r)) = f(h(r)) = f(r). that is, h takes the range of f to itself. since h is increasing, h(sup f) = sup f and h(inf f) = inf f. now we prove that for maps with finitely many non-ordinary points, limx→∞f(x) always exists in r ∪ {∞,−∞}. for this, let t0 be the largest non-ordinary point and let a be the set of all critical points > t0. suppose a is empty. then f is monotone on [t0,∞) and hence limx→∞f(x) exists. suppose a is nonempty. then ∂a is nonempty. but every element of ∂a is non-ordinary. hence ∂a = {t0}. therefore a = (t0,∞). therefore f is constant on a (we argue as in the proof of case 2 of (i) in proposition 2.21). hence limx→∞f(x) exists. next we will prove limx→∞f(x) is special. we denote limx→∞f(x) by l. let h ∈ gf↑. note that for any sequence (xn) → ∞, we have f(xn) → l and h(xn) → ∞. hence h(f(xn)) = f(h(xn)) → h(l). being h(xn) → ∞, by the definition of l we find f(h(xn)) = h(f(xn)) → l. hence h(l) = l. this completes the proof. � proposition 2.23. the maps x+ 1 and x−1 on r are topologically conjugate; but not order conjugate. proof. the maps x+ 1 and x−1 are conjugate to each other through the order conjugacy −x + 1 2 . if possible, let h be an order conjugacy from f(x) = x + 1 to g(x) = x − 1. then h(x + 1) = h(f(x)) = g(h(x)) = h(x) − 1. i.e., c© agt, upv, 2019 appl. gen. topol. 20, no. 2 314 simple dynamical systems h(x + 1) −h(x) = −1 < 0, which is a contradiction to the assumption that h is increasing. � remark 2.24. note that for the map x + 1 on r, all points are ordinary. this is because, if a,b ∈ r then the map x + b−a is the order conjugacy of x + 1 which maps a to b. the following proposition is proved in [6]. for the sake of completeness, we included its proof. proposition 2.25. let f : r → r be a homeomorphism without fixed points. then (i) if f(0) > 0 then f is order conjugate to x + 1. (ii) if f(0) < 0 then f is order conjugate to x− 1. proof. define h : r → r as follows. assume f(0) > 0. define h(t) = t f(0) , 0 ≤ t < f(0). we know that (fn(0)) increases and diverges to ∞ and (f−n(0)) decreases and diverges to −∞ for all n ∈ n. moreover for t ∈ r there exists unique n ∈ z such that, fn(0) ≤ t < fn+1(0). define h(t) = h(f−n(t)) + n. then h◦f(t) = h(t) + 1 for all t ∈ r. this h gives a conjugacy from f to x + 1. if f(0) < 0 then we can give a similar proof. � for a map f : a → r, a ⊂ r, we define graph(f) := {(x,f(x)) : x ∈ a}. for continuous maps f,g : a → r, we say that graph(f) and graph(g) are above the diagonal if f(x) > x and g(x) > x for all x ∈ a. similarly, graph(f) and graph(g) are said to be below the diagonal if f(x) < x and g(x) < x for all x ∈ a, and graph(f) and graph(g) are said to be on the diagonal if f(x) = g(x) = x for all x ∈ a. we say that graph(f) and graph(g) are on the same side of the diagonal if it is either above the diagonal or below the diagonal or on the diagonal. corollary 2.26. let f,g : (a,b) → (a,b) be homeomorphisms without fixed points. then f is order conjugate to g if and only if both graph(f) and graph(g) are on the same side of the diagonal. in particular: (i) if f(x) > x for all x ∈ (a,b) then f is order conjugate to x + 1. (ii) if f(x) < x for all x ∈ (a,b) then f is order conjugate to x− 1. remark 2.27. in fact, the interval (a,b) involved in corollary 2.26 can be replaced by any open ray in r. remark 2.28. if f : r → r is an increasing bijection with finitely many nonordinary points then all non-ordinary points are fixed points. proof. we know that for maps with finitely many non-ordinary points all nonordinary points are fixed by every order conjugacy. here f itself is a self conjugacy. � c© agt, upv, 2019 appl. gen. topol. 20, no. 2 315 k. ali akbar, v. kannan and i. subramania pillai for a continuous map f : r → r, let fix(f) denote the set of all fixed points of f. it follows from the continuity of f that fix(f) is closed. for any subset a of a metric space x, we denote ac for the complement of a and int(a) for the interior of a. recall that (2.3) (∂a)c = int(a) ∪ int(ac). the following proposition provides a characterization for the non-ordinary points of increasing homeomorphisms. proposition 2.29. let f : r → r be an increasing bijection and let x ∈ r. then x is non-ordinary if and only if x ∈ ∂(fix(f)). proof. let x ∈ ∂fix(f). then x is non-ordinary since every open interval around x contains fixed and non-fixed points. now suppose x /∈ ∂fix(f). we shall prove that x is ordinary. now, x /∈ ∂fix(f) implies x ∈ (∂fix(f))c = int(fix(f)) ∪ int((fix(f))c) by equation (1). hence x ∈ int(fix(f)) or x ∈ int(fix(f)c). case 1: x ∈ int(fix(f)) suppose x ∈ int(fix(f)). then choose a,b ∈ r such that x ∈ (a, b) ⊂ fix(f). let y ∈ (a,b) be such that y 6= x. then define an increasing continuous bijection φy : r → r such that φy(t) =   t if t /∈ (a,b) y if t = x piecewise linear otherwise. this φy maps x to y. both [a,b] and its complement are invariant under both φy and f. note that f is identity on [a,b] and φy is identity on the complement of [a,b]. hence φy commutes with f on [a,b]. this proves x is an ordinary point. case 2: x ∈ int(fix(f)c) suppose x ∈ int(fix(f)c). let (a,b) be the component interval (open) of (fix(f))c containing x. then f(a) = a and f(b) = b, and the map f|(a,b) is a fixed point free self map of (a,b) since f is increasing. hence by corollary 2.26, the map f|(a,b) is order conjugate to either x + 1 or x− 1, for which all points are ordinary. this completes the proof. � remark 2.30. there are continuous maps f : r → r having finitely many equivalence classes (hence only finitely many special points) but infinitely many non-ordinary points. for example, consider the map f(x) = x + sin x on r. there are two classes of fixed points. since increasing orbits must map to increasing orbits under increasing conjugacies, points like π 2 (increasing orbit) and 3π 2 (decreasing orbit) cannot be equivalent. hence there must be at least four equivalence classes. to see that there are exactly four equivalence classes, let ik = (2kπ, (2k + 1)π),dk = ((2k + 1)π, 2(k + 1)π) and observe that ik ∩ n(f) = ∅ = dk ∩n(f) for each k ∈ z by proposition 2.29. hence by lemma 2.12, each ik and dk is contained in a single equivalence class. conjugacies of the form x 7→ x + 2kπ complete the argument. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 316 simple dynamical systems 3. main results: class of homeomorphisms note that, under a topological conjugacy a point can be mapped to a point with similar dynamics. by definition, the points of [x] are dynamically the same, i.e., all have the dynamics similar to that of x. we now consider the systems for which there are only finitely many equivalence classes. this means there are only finitely many kinds of orbits upto conjugacy. for this reason, we call such systems as simple systems. in this paper, we try to understand some simple systems on r. recall that, if sp denote the set of all points having the dynamical property p then the points of ∂sp (the boundary of sp ) are non-ordinary. in particular, being a point in a particular equivalence class is a dynamical property of the point. hence, by the very nature of the order conjugacies, it follows that when there are finitely many non-ordinary points (therefore special points) there are only finitely many equivalence classes. these are the simple systems we study in this paper. we describe completely, the homeomorphisms on r, having finitely many non-ordinary points and give a general formula for counting. by remark 2.11, for systems with finitely many non-ordinary points there are only finitely many equivalence classes. we now study, in the next subsections, the class of simple systems induced by homeomorphisms having finitely many non-ordinary points. 3.1. class of increasing homeomorphisms. note that the complement of fix(f) is a countable union of open intervals (including rays) whose end points are fixed points. since f is increasing and the end points are fixed, no point in a component interval can be mapped to a point in any other component interval by f. hence, it is observed that, for an increasing bijection f on r, if fix(f) c = tin then f|in is a self map of in, where t denotes the disjoint union. proposition 3.1. let f,g be two increasing bijections such that fix(f) = fix(g) and let fix(f) c = tin. if f|in is order conjugate to g|in for every n, then f is order conjugate to g. proof. for each n ∈ n, let hn : in → in be an order conjugacy from f|in to g|in . define h : r → r by h(x) = { hn(x) if x ∈ in x otherwise. then h is an increasing bijection such that h◦f = g ◦h. � an alphabet is a finite set of letters with at least two elements. a finite sequence of letters from an alphabet is often referred to as a word. for example, if σ = {a,b} be an alphabet then abab, aaabbbab are words over σ. number of letters (may not be distinct) in a word is called its length. any word of consecutive characters in a word w is said to be a subword of w. throughout this section we will be working with the alphabet {a, b, o}. let ã = b, c© agt, upv, 2019 appl. gen. topol. 20, no. 2 317 k. ali akbar, v. kannan and i. subramania pillai b̃ = a and õ = o. if w = w1w2...wn then the dual of w is defined as w̃ = w̃nw̃n−1...w̃1. if w̃ = w then the word w is said to be self conjugate. here a stands for “above the diagonal” and b stands for “below the diagonal” and o stands for “on the diagonal”. let f : r → r be an increasing homeomorphism with finitely many non-ordinary (hence special) x1, ...,xn for some n ∈ n. without loss of generality, assume that x1 < x2 < ... < xn for some n ∈ n. this finite set of points gives rise to an ordered partition {(−∞,x1), (x1,x2), ..., (xn,∞)} of r\{x1,x2, ...,xn}. note that, on each component interval exactly one of the following holds by proposition 2.29 (since the only subsets of r with empty boundary are the empty set and r): (i) f(t) > t ∀ t (ii) f(t) < t ∀ t (iii) f(t) = t ∀ t. this gives rise to a word w(f) over {a, b, o} of length n + 1 by associating a to (i), b to (ii) and o to (iii). note that the subword oo is forbidden. for this, suppose o is occurring at ith and (i+ 1)th place then in both(xi,xi+1) and (xi+1,xi+2) all points are fixed. then xi+1 becomes ordinary, a contradiction to the assumption that xi+1 is a non-ordinary point. conversely, suppose a word w of length n + 1 in which oo is forbidden is given. then we can construct an increasing bijection on r such that its associated word is w, as follows: take the points 0, 1, 2, ...,n and consider {(−∞, 0), (0, 1), (1, 2), ....., (n,∞)}, a partition of r. if w = w1w2.....wn+1 then associate w1 to (−∞, 0), w2 to (0, 1), ......, and wn+1 to (n,∞). now it is easy to construct an increasing bijection f : r → r such that w(f) = w. firstly, define f(t) on (−∞, 0) according to the value of w1 as follows: f(t) =   1 2 t if w1 = a 3 2 t if w1 = b t if w1 = o. secondly, define f(t) on the remaining subintervals (i−2, i−1) for i = 2, ...,n as follows: to be precise, if i− 2 < t < i− 1, i = 2, 3, ...,n, then consider f(t) =   i− 2 + (t− i + 2)2 if wi = b i− 2 + √ t− i + 2 if wi = a t if wi = o. and, finally, define f(t) on (n,∞) according to the value of wn+1 as follows: f(t) =   1 2 (t−n) + n if wn+1 = b 3 2 (t−n) + n if wn+1 = a t if wn+1 = o. now by the following proposition the increasing bijection constructed above is unique upto order conjugacy. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 318 simple dynamical systems proposition 3.2. let f, g be two increasing bijection on r with finitely many (same number of ) non-ordinary points. then f and g are order conjugate if and only if w(f) = w(g). proof. suppose w(f) = w(g) = w1w2...wn. let x1 < x2 < ... < xn and y1 < y2 < ... < yn be the non-ordinary points f and g respectively. the former gives the ordered partition {(−∞,x1), (x1,x2), ...(xn,∞)} of r\{x1,x2, ...,xn} and the later gives the ordered partition {(−∞,y1), (y1,y2), ..., (yn,∞)} of r\{y1,y2, ...,yn}. now by proposition 2.29, it follows that both f|(xi,xi+1) and g|(yi,yi+1) are fixed point free self maps (homeomorphisms) for each i and hence by corollary 2.26, both are order conjugate to x + 1 if wi+1 = a, and order conjugate to x − 1 if wi+1 = b. hence, by proposition 3.1, f is order conjugate to g. converse follows from corollary 2.26. � thus we have proved: proposition 3.3. there is a one to one correspondence between the set of all increasing continuous bijections (upto order conjugacy) on r with exactly n non-ordinary points and the set of all words of length n + 1 on three symbols a, b, o such that oo is forbidden. now we consider the following proposition. proposition 3.4. let an be the number of words of length n + 1 over {a, b, o}, where oo is forbidden. then an = c1(1 + √ 3)n + c2(1 − √ 3)n, where c1 = (5+3 √ 3) 2 √ 3 and c2 = (3 √ 3−5) 2 √ 3 . proof. let an be the set of all words of length n + 1 over {a, b, o} in which oo is forbidden. a general element in an+2 is of the form (i) aw or bw for some w ∈ an+1 or (ii) oav or obv for some v ∈ an. therefore an+2 = an+1 + an+1 + an + an since an+2 is the disjoint union of four types of the elements described above. hence an+2 = 2(an + an+1). this is a linear homogeneous recurrence relation with constant coefficients. the corresponding characteristic equation is α2 − 2α − 2 = 0 which has the two distinct roots α1 = 1 + √ 3 and α2 = 1 − √ 3. it follows that an = c1(1 + √ 3)n + c2(1− √ 3)n, where the constants c1 and c2 can be determined by using the boundary conditions a0 = 3 and a1 = 8. here c1 = (5+3 √ 3) 2 √ 3 and c2 = (3 √ 3−5) 2 √ 3 . � the following result is one of our principal theorems. it follows from propositions 3.3 and 3.4. theorem 3.5. the number of all increasing continuous bijections (upto order conjugacy) on r with exactly n non-ordinary points is an = c1(1 + √ 3)n + c2(1 − √ 3)n, where c1 = (5+3 √ 3) 2 √ 3 and c2 = (3 √ 3−5) 2 √ 3 . for any two continuous map f, g : r → r, we say that f is decreasingly conjugate to g if there is a decreasing homeomorphism h : r → r such that c© agt, upv, 2019 appl. gen. topol. 20, no. 2 319 k. ali akbar, v. kannan and i. subramania pillai h◦f = f ◦h. the following proposition is an analogue of proposition 3.2, and its proof will be omitted. proposition 3.6. let f, g be two increasing bijections on r with finitely many (same number of ) non-ordinary points. then f and g are decreasingly conjugate if and only if w(g) = w(f). let tn denote the number of increasing homeomorphisms on r with exactly n non-ordinary points upto topological conjugacy, an denote the number of increasing homeomorphisms on r with exactly n non-ordinary points upto order conjugacy, and νn+1 denote the number of self conjugate words of length n + 1 over {a, b, o}. if f, g : r → r are two increasing homeomorphisms such that they are topologically conjugate then either w(g) = w(f) or w(g) = w(f) by proposition 3.2 and proposition 3.6. first observe that t0 = 2, t1 = 5, and t2 = 12. in general, we have tn = 1 2 (an −νn+1) + νn+1 for all n ∈ n. case 1: when n is even. say n = 2m. a self conjugate word w of length 2m + 1 (oo is forbidden) is of the form w1w2...wmwm+1wm+2...w2m+1 such that wm+1 = o and wm,wm+2 ∈ {a, b} such that wm 6= wm+2. therefore the number of self conjugate words is 2am−2. hence t2m = a2m+2am−2 2 for all m ≥ 2. case 2: when n is odd. say n = 2m+1. in this case any self conjugate word of length 2m+2 (oo is forbidden) is of the form w1w2...wmwm+1wm+2...w2m+2 such that wm+1,wm+2 ∈{a, b} and wm+1 6= wm+2. hence the number of self conjugate words of length 2m + 2 is 2am−1. therefore t2m+1 = a2m+1+2am−1 2 for all m ≥ 1. thus we have proved: theorem 3.7. if tn denotes the number of increasing homeomorphisms upto topological conjugacy. then t0 = 2, t1 = 5 and t2 = 12 by direct computation and for all n ≥ 3 we have: tn =   an+2a n−4 2 2 if n is even an+2a n−3 2 2 if n is odd 3.2. class of decreasing homeomorphisms. we now ask: given a whole number n, how many decreasing bijections are there on r upto order conjugacy having exactly n non-ordinary points? for a map f : r → r, we denote the composition of f with itself by f2. proposition 3.8. two decreasing bijections f and g are order conjugate (respectively topologically conjugate) if and only if f2|[a,∞) and g2|[b,∞) are order conjugate (respectively topologically conjugate), where a and b are the fixed points of f and g respectively. note that every decreasing homeomorphism has a unique fixed point. proof. suppose f and g are order conjugate (respectively topologically conjugate). then the same conjugacy between f and g when we restrict forms an order conjugacy (respectively topological conjugacy) between f2|[a,∞) and c© agt, upv, 2019 appl. gen. topol. 20, no. 2 320 simple dynamical systems g2|[b,∞). conversely, suppose f2|[a,∞) and g2|[b,∞) are order conjugate through the increasing homeomorphism h1. then h1([a,∞)) = [b,∞) and h1(a) = b. also note that f((−∞,a]) = [a,∞) and g((−∞,b]) = [b,∞). that is, f−1([a,∞)) = (−∞,a] and g−1([b,∞)) = (−∞,b]. define h : r → r such that h(x) = { h1(x) if x ∈ [a,∞) g−1hf(x) if x < a. if t < a then by definition h◦f(t) = g◦h(t). if t > a then f(t) < a. therefore h(f(t)) = g−1(h1(f(f(t)))) = g −1(h1(f 2(t))) = g−1(g2(h1(t))) = g(h1(t)) = g(h(t)). hence h forms an order conjugacy from f to g. � the following proposition is analogous to proposition 3.8, and its proof will be omitted. proposition 3.9. two decreasing bijections f and g are order conjugate (respectively topologically conjugate) if and only if f2|(−∞,a] and g2|(−∞,b] are order conjugate (respectively topologically conjugate), where a and b are the fixed points of f and g respectively. a map f : r → r is said to be odd if f(−x) = −f(x) for all x ∈ r. proposition 3.10. let f : r → r be an odd increasing bijection. then, there exist a decreasing homeomorphism fr such that f 2 r = f. such an fr is called a decreasing square root of f. proof. note that f(0) = 0. define fr such that fr(x) = { −f(x) if x ≥ 0 −x if x < 0. clearly, fr is a decreasing bijection. then fr(x) ≤ 0 for all x ≥ 0. therefore fr(fr(x)) = −fr(x) = f(x). also we have fr(fr(x)) = fr(−x) = −f(−x) = f(x) for all x < 0. � remark 3.11. the conclusion of the above proposition is not true in general. for this, let h(x) = { x 2 if x ≥ 0 x if x < 0. clearly, h is an increasing bijection from r to r. there is no decreasing bijection f : r → r such that f ◦f = h. let if possible f be one such function. then we have f(f(x)) = h(x) = x for all x < 0. choose y > 0 such that f(y) < 0. therefore f2(f(y)) = f(y) = f(f2(y)). then f2(y) = y since f is one-one. therefore h(y) = y, a contradiction since h(y) = y 2 . proposition 3.12. let f : (0,∞) → (0,∞) be an increasing bijection. then there exists a unique decreasing bijection fr : r → r upto order conjugacy such that f2r |(0,∞) = f. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 321 k. ali akbar, v. kannan and i. subramania pillai proof. let f : (0,∞) → (0,∞) be an increasing bijection. this forces that f(0) := limx→0+f(x) = 0. any map f : (0,∞) → (0,∞) can be extended uniquely to an odd function f̃ : r → r. then there exists fr : r → r such that f2r |(0,∞) = f, by proposition 3.10. this f is unique upto order conjugacy by proposition 3.9. � proposition 3.13. let f : r → r be a decreasing bijection. then the nonordinary points of f2 are precisely the non-ordinary points of f. proof. suppose x is an ordinary point for f. then the resut follows from the fact that if h commutes with f then it commutes with f2 also. conversely, suppose x is an ordinary point of f2. let the unique fixed point of f be zero, i.e., f(x) = 0 if and only if x = 0. now let x > 0. then there exists a neighbourhood (x−δ,x+δ) such that for y in (x−δ,x+δ) there is an h ∈ gf◦f for which h(x) = y. then h|(0,∞) is a topological conjugacy between f◦f|(0,∞) and f ◦f|(0,∞). then h induces h̃ a conjugacy between f and f by proposition 3.9. by the way h̃ is defined, we have h̃(x) = h(x) = y. therefore x is an ordinary point of f. � proposition 3.14. let f be a decreasing bijection from r to r with fixed point a. then, f has 2n+1 non-ordinary points if and only if f2|(a,∞) : (a,∞) → (a,∞) has n non-ordinary points. proof. suppose that f has 2n + 1 non-ordinary points. let them be x1 < x2 < ... < xn < xn+1 < xn+2 < ... < x2n+1. let n = {x1,x2, ...,x2n+1}. then f(n) ⊂ n by proposition 2.20. since f is a decreasing bijection, we have f(n) = n and a = xn. hence f 2|(a,∞) : (a,∞) → (a,∞) has n nonordinary points. conversely, suppose f2|(a,∞) : (a,∞) → (a,∞) has n nonordinary points. then by proposition 3.14, we have n(f) = n(f2|(a,∞)) ∪ f(n(f2|(a,∞))) ∪{a}. thus f has 2n + 1 non-ordinary points. � remark 3.15. from the above proposition it follows that there does not exist a decreasing homeomorphism with even number of non-ordinary points. theorem 3.16. if sn denotes the number of decreasing homeomorphisms upto order conjugacy, then sn = { 0 if n is even an−1 2 if n is odd for all n. proof. by proposition 3.14, we have s2n = 0 for each n ∈ n. now we will prove that s2n+1 = an for all n ∈ n. let f : r → r be a decreasing bijection with 2n + 1 non-ordinary points. without loss of generality we can assume that 0 is the unique fixed point. then g = f2|(0,∞) : (0,∞) → (0,∞) is an increasing bijection with n non-ordinary points. since (0,∞) is homeomorphic to r, we get an increasing homeomorphism g′ : r → r (unique upto order conjugacy) with n non-ordinary points and which is order conjugate to g. on the other c© agt, upv, 2019 appl. gen. topol. 20, no. 2 322 simple dynamical systems hand, suppose h : r → r is an increasing bijection with n non-ordinary points. since (0,∞) is homeomorphic to r, corresponding to each h we have a unique (upto order conjugacy) increasing bijection h′ : (0,∞) → (0,∞) with n nonordinary points. then by proposition 3.10, there exists unique (upto order conjugacy) decreasing square root f : r → r for h′ such that f ◦f|(0,∞) = h′. by proposition 3.14, f has 2n+ 1 non-ordinary points. thus, there is a one-one correspondence between the set of all increasing bijections with n non-ordinary points (upto order conjugacy) and the set of all decreasing bijections with 2n+1 non-ordinary points (upto order conjugacy). hence s2n+1 = an. � theorem 3.17. if kn denotes the number of decreasing homeomorphisms having n non-ordinary points upto topological conjugacy, then kn = { 0 if n is even tn−1 2 if n is odd for all n. proof. if n is even then we have kn = 0 since sn = 0. if n is odd then we will argue as in theorem 3.16 to prove that there is a one-one correspondence between the set of all increasing bijections (upto topological conjugacy) on r having n non-ordinary points and the set of all decreasing bijections (upto topological conjugacy) on r with 2n + 1 non-ordinary points. thus k2n+1 = tn. � 4. summary we conclude this paper with the following table: n an sn tn kn 0 3 0 2 0 1 8 0 5 2 2 22 0 12 0 3 60 8 33 5 4 164 0 85 0 5 448 22 232 12 where an be the number of increasing bijections on r with exactly n nonordinary points upto order conjugacy, tn be the number of increasing bijections on r with exactly n non-ordinary points upto topological conjugacy, sn be the number of decreasing bijections on r with exactly n non-ordinary points upto order conjugacy, and kn be the number of decreasing bijections on r with exactly n non-ordinary points upto topological conjugacy. acknowledgements. the authors are thankful to the referee for his/her valuable suggestions. the first author acknowledges ugc, india for financial support. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 323 k. ali akbar, v. kannan and i. subramania pillai references [1] l. s. block and w. a. coppel, dynamics in one dimension, volume 1513 of lecture notes in mathematics, springer-verlag, berline, 1992. [2] l. block and e. coven, topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, trans. amer. math. soc. 300 (1987), 297–306. [3] m. brin and g. stuck, introduction to dynamical systems, cambridge university press, 2002. [4] r. l. devaney, an introduction to chaotic dynamical systems, addison-wesley publishing company advanced book program, redwood city, ca, second edition, 1989. [5] r. a. holmgren, a first course in discrete dynamical systems, springer-verlag, new york, 1996. [6] s. sai, symbolic dynamics for complete classification, ph.d thesis, university of hyderabad, 2000. [7] b. sankara rao, i. subramania pillai and v. kannan, the set of dynamically special points, aequationes mathematicae 82, no. 1-2 (2011), 81–90. [8] a. n. sharkovskii, coexistence of cycles of a continuous map of a line into itself, ukr. math. z. 16 (1964), 61–71. c© agt, upv, 2019 appl. gen. topol. 20, no. 2 324 @ appl. gen. topol. 20, no. 1 (2019), 81-95doi:10.4995/agt.2019.9949 c© agt, upv, 2019 fixed point results concerning α-f-contraction mappings in metric spaces lakshmi kanta dey a, poom kumam b and tanusri senapati c a department of mathematics, national institute of technology durgapur, india (lakshmikdey@yahoo.co.in) b department of mathematics, king mongkut’s university of technology thonburi (kmutt), 126 pracha uthit rd., bang mod, thung khru, bangkok 10140, thailand. (poom.kum@kmutt.ac.th) c department of mathematics, indian institute of technology guwahati, assam, india (senapati.tanusri@gmail.com) communicated by i. altun abstract in this paper, we introduce the notions of generalized α-f-contraction and modified generalized α-f-contraction. then, we present sufficient conditions for existence and uniqueness of fixed points for the above kind of contractions. necessarily, our results generalize and unify several results of the existing literature. some examples are presented to substantiate the usability of our obtained results. 2010 msc: 47h10; 54h25. keywords: metric space; fixed point; generalized α-f-contraction; modified generalized α-f-contraction. 1. introduction and preliminaries throughout this paper we denote by r+, r, n and n0, the set of positive real numbers, set of real numbers, set of natural numbers and set of nonnegative integers respectively. it is widely known that the banach contraction principle [1] is the first metric fixed point theorem and one of the most powerful and versatile results in the field of functional analysis. due to its significance and several applications, over the years, it has been generalized in different directions by received 12 april 2018 – accepted 11 november 2018 http://dx.doi.org/10.4995/agt.2019.9949 l. k. dey, p. kumam, t. senapati several mathematicians (for example, see ([2, 3, 4, 5, 7, 10, 17, 18, 15, 16, 19]) and references therein). before stating our main results, at first we recollect some useful definitions and results in the comparable literature which will be needed throughout the study. so, we start by presenting the concept of α-admissible mappings and triangular α-admissible mappings as follows: definition 1.1 ([14]). a mapping g : x → x is said to be an α-admissible mapping if there exists a function α : x × x → r+ such that for all x, y ∈ x α(x, y) ≥ 1 ⇒ α(gx, gy) ≥ 1. definition 1.2 ([11]). a mapping g : x → x is said to be a triangular α-admissible mapping if there exists a function α : x × x → r+ such that (1) for all x, y ∈ x, α(x, y) ≥ 1 ⇒ α(gx, gy) ≥ 1, (2) for all x, y, z ∈ x, α(x, y) ≥ 1, α(y, z) ≥ 1 ⇒ α(x, z) ≥ 1. note 1.3. [11] let g be a triangular α-admissible mapping. if (xn) is any sequence defined by xn+1 = gxn and α(xn, xn+1) ≥ 1, then for all n, m ∈ n, we get α(xn, xm) ≥ 1. in 2012, wardowski [19] introduced the concept of f-contractions which plays a crucial part in the recent trend of research in fixed point theory. after that, wardowski and dung [20] and dung and hang [6] extended the concept of f-contractions to f-weak contractions and generalized f-contractions respectively. by mixing up the concept of α-admissible mappings with f-contractions [19] and f-weak contractions [20], gopal et al. [8] introduced the concept of α-type f-contractions and α-type f-weak contractions as follows: definition 1.4 ([8]). let (x, d) be a metric space and g : x → x be a mapping. suppose α : x × x → {−∞} ∪ (0, ∞) be a function. the function g is said to be an α-type f-contraction if there exists τ > 0 such that for all x, y ∈ x, d(gx, gy) > 0 ⇒ τ + α(x, y)f(d(gx, gy)) ≤ f(d(x, y)). definition 1.5 ([8]). let (x, d) be a metric space and g : x → x be a selfmapping. let α : x × x → {−∞} ∪ (0, ∞) be a function. the function g is said to be an α-type f-weak contraction if there exists τ > 0 such that for all x, y ∈ x, d(gx, gy) > 0 implies that τ +α(x, y)f(d(gx, gy)) ≤ f ( max { d(x, y), d(x, gx), d(y, gy), d(x,gy)+d(y,gx) 2 }) . in the above definitions, the function f belongs to the family f of mappings from (0, ∞) → r satisfying the following conditions: (f1) f is a strictly increasing function, i.e., for all x, y ∈ r+ with x < y, f(x) < f(y); (f2) for each sequence (αn) of positive numbers, lim n→∞ αn = 0 ⇐⇒ lim n→∞ f(αn) = −∞; c© agt, upv, 2019 appl. gen. topol. 20, no. 1 82 fixed point theorems for α-f -contraction mappings (f3) there exists a k ∈ (0, 1) such that lim α→0+ αkf(α) = 0. in this sequel, the authors of [8] established some fixed point results and finally they presented an application to nonlinear fractional differential equations. subsequently, piri and kumam [13] established some new fixed point results by taking a weaker family of functions as well as by weakening the contraction condition given by: definition 1.6 ([13]). let (x, d) be a metric space and let g : x → x be a mapping. the function g is said to be a modified generalized f-contraction of type (a) if there exists τ > 0 such that for all x, y ∈ x, d(gx, gy) > 0 ⇒ τ + f(d(gx, gy)) ≤ f(ng(x, y)), where, ng(x, y) = max { d(x, y), d(x, gy) + d(y, gx) 2 , d(g2x, x) + d(g2x, gy) 2 , d(g2x, gx), d(g2x, y), d(gx, y) + d(y, gy), d(g2x, gy) + d(x, gx) } and f satisfies the following conditions: (1) f is strictly increasing, (2) f is continuous. in a similar fashion, they also defined modified generalized f-contraction of type (b) by considering different class of functions satisfying the above contractive condition along with the following properties: (1) f is strictly increasing; (2) there exists a k ∈ (0, 1) such that lim α→0+ αkf(α) = 0. using the notions of modified generalized f-contraction of type (a) and type (b), the authors presented some new fixed point results which generalized and extended several related results discussed in wardowski [19], piri and kumam [12], dung and hang [6] and wardowski and dung [20]. for the sake of completeness of our paper, we need to recall the definition of α-complete metric spaces and α-continuous mappings. definition 1.7 ([9]). let (x, d) be a metric space and α : x × x → [0, ∞) be a function. the metric space (x, d) is said to be an α-complete metric space if and only if every cauchy sequence with α(xn, xn+1) ≥ 1, for all n ∈ n0, converges in x. definition 1.8 ([9]). let (x, d) be a metric space. let g be a self-map defined on x and α : x × x → [0, ∞) be a function. then g is said to be an αcontinuous mapping if for every x ∈ x and sequence (xn) ∈ x with (xn) converging to x, α(xn, xn+1) ≥ 1, for all n ∈ n0 ⇒ gxn → gx. here, we provide an example of an α-continuous mapping which is not continuous. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 83 l. k. dey, p. kumam, t. senapati example 1.9. let x = [0, ∞) and d(x, y) = |x − y|, for all x, y ∈ x. we define α : x × x → [0, ∞) by α(x, y) = { 1, for all x, y ∈ [0, 1]; 1 2 , otherwise and the mapping g : x → x by gx =    x 2 , for all x ∈ [0, 1]; 2x, 1 < x ≤ 3; x2, otherwise clearly, g is not continuous as x = 1 and x = 3 are points of discontinuity but g is an α-continuous map. remark 1.10. every complete metric space is α-complete and every continuous map is α-continuous but in both the cases, the converse does not hold in general. in this article, by f, we denote the following family of functions given by f = {f/f : (0, ∞) → r} satisfying the following conditions: (f ′) f is a strictly increasing function, i.e., for all x, y ∈ r+ with x < y, f(x) < f(y); (f ′′) there exists a k ∈ (0, 1) such that lim α→0+ αkf(α) = 0. the aim of this article is to present some new fixed point results in αcomplete metric spaces and show that our obtained results generalize several existing results in the literature. for this, we introduce the concept of generalized α-type f-contractions and modified generalized α-type f-contractions. for simplicity, we call these contractions as generalized α-f-contractions and modified generalized α-f-contractions respectively. finally, we construct some non-trivial examples to validate the potential of our results. 2. main results we begin with this section by presenting the new concept of generalized α-f-contractions and modified generalized α-f-contractions respectively. definition 2.1. let (x, d) be a metric space and g : x → x be a mapping. let α : x × x → [0, ∞) be a function and f ∈ f. the function g is said to be a generalized α-f-contraction mapping if there exists τ > 0 such that for all x, y ∈ x, d(gx, gy) > 0 ⇒ τ + α(x, y)f(d(gx, gy)) ≤ f(mg(x, y)) where, mg(x, y) = max { d(x, y), d(x, gx), d(y, gy), d(x, gy) + d(y, gx) 2 , d(g2x, x) + d(g2x, gy) 2 , d(g2x, gx), d(g2x, y), d(g2x, gy) } . c© agt, upv, 2019 appl. gen. topol. 20, no. 1 84 fixed point theorems for α-f -contraction mappings definition 2.2. let (x, d) be a metric space and g : x → x be a mapping. let α : x × x → [0, ∞) be a function and f ∈ f. the function g is said to be a modified generalized α-f-contraction mapping if there exists τ > 0 such that for all x, y ∈ x, d(gx, gy) > 0 ⇒ τ + α(x, y)f(d(gx, gy)) ≤ f(ng(x, y)) where, ng(x, y) = max { d(x, y), d(x, gy) + d(y, gx) 2 , d(g2x, x) + d(g2x, gy) 2 , d(g2x, gx), d(g2x, y), d(gx, y) + d(y, gy), d(g2x, gy) + d(x, gx) } . remark 2.3. every modified generalized f-contraction (respectively, generalized f-contraction) is a modified generalized α-f-contraction (respectively, generalized α-f-contraction). the reverse implications do not hold. we illustrate this by presenting an example. example 2.4. let x = {0, 1, 2, 3, 4} and we define the distance function d as follows d(x, y) =    0, iff x = y; 5 2 , (x, y) ∈ {(0, 3), (3, 0)}; 3 2 , otherwise. also, we define a mapping g : x → x by g(0) = g(3) = 1; g(1) = g(4) = 3; g(2) = 0. therefore, we get d(gx, gy) > 0 ⇐⇒ [x ∈ {0, 3}∧y ∈ {1, 4}; x ∈ {0, 3}∧y = 2; x ∈ {1, 4}∧y = 2]. now, we are interested to find ng(x, y). for this purpose, we consider the following cases: case-i. let x ∈ {0, 3} and y ∈ {1, 4}. then for any (x, y) ∈ {(0, 1), (0, 4), (3, 1), (3, 4)}, we get d(gx, gy) = d(1, 3) = 3 2 . let (x, y) = (0, 1). then, we have ng(0, 1) = max{d(0, 1), d(0, g1) + d(1, g0) 2 , d(g20, 0) + d(g20, g1) 2 , d(g20, g0), d(g20, 1), d(g0, 1) + d(1, g1), d(g20, g1) + d(0, g0)} = max { 3 2 , 5 4 } = 3 2 . c© agt, upv, 2019 appl. gen. topol. 20, no. 1 85 l. k. dey, p. kumam, t. senapati for (x, y) = (0, 4), we get ng(0, 4) = max { d(0, 4), d(0, g4) + d(4, g0) 2 , d(g20, 0) + d(g20, g4) 2 , d(g20, g0), d(g20, 4), d(g0, 4) + d(4, g4), d(g20, g4) + d(0, g0) } = max { 3 2 , 5 4 , 2 } = 2. for (x, y) = (3, 1), we obtain ng(3, 1) = max { d(3, 1), d(3, g1) + d(1, g3) 2 , d(g23, 3) + d(g23, g1) 2 , d(g23, g3), d(g23, 1), d(g3, 1) + d(1, g1), d(g23, g1) + d(3, g3) } = max { 3 2 , 0 } = 3 2 and for (x, y) = (3, 4), also have ng(3, 4) = max { d(3, 4), d(3, g4) + d(4, g3) 2 , d(g23, 3) + d(g23, g4) 2 , d(g23, g3), d(g23, 4), d(g3, 4) + d(4, g4), d(g23, g4) + d(3, g3) } = max { 3 2 , 3 4 , 0, 3 } = 3. case-ii. let x ∈ {0, 3} and y = 2. then for (x, y) ∈ {(0, 2), (3, 2)}, d(gx, gy) = d(1, 0) = 3 2 . then, for (x, y) = (0, 2), we have ng(0, 2) = max { d(0, 2), d(0, g2) + d(2, g0) 2 , d(g20, 0) + d(g20, g2) 2 , d(g20, g0), d(g20, 2), d(g0, 2) + d(2, g2), d(g20, g2) + d(0, g0) } = max { 3 2 , 3 4 , 5 2 , 3, 4 } = 4. for (x, y) = (3, 2), we get ng(3, 2) = max { d(3, 2), d(3, g2) + d(2, g3) 2 , d(g23, 3) + d(g23, g2) 2 , d(g23, g3), d(g23, 2), d(g3, 2) + d(2, g2), d(g23, g2) + d(3, g3) } = max { 3 2 , 3 4 , 2, 3, 4 } = 4. case-iii. let x ∈ {1, 4} and y = 2 . then (x, y) ∈ {(1, 2), (4, 2)}, d(gx, gy) = d(3, 0) = 5 2 . c© agt, upv, 2019 appl. gen. topol. 20, no. 1 86 fixed point theorems for α-f -contraction mappings let (x, y) = (1, 2). then ng(1, 2) = max { d(1, 2), d(1, g2) + d(2, g1) 2 , d(g21, 1) + d(g21, g2) 2 , d(g21, g1), d(g21, 2), d(g1, 2) + d(2, g2), d(g21, g2) + d(1, g1) } = max { 3 2 , 3 4 , 3 } = 3. for (x, y) = (4, 2), we have ng(4, 2) = max { d(4, 2), d(4, g2) + d(2, g4) 2 , d(g24, 4) + d(g24, g2) 2 , d(g24, g4), d(g24, 2), d(g4, 2) + d(2, g2), d(g24, g2) + d(4, g4) } = max { 3 2 , 3 } = 3. from the above cases, we observe that whenever (x, y) ∈ {(0, 1), (3, 1)}, d(gx, gy) = ng(x, y). since f is increasing, we can’t find any τ > 0 such that τ + f(d(gx, gy)) ≤ f(ng(x, y)). this shows that g is not a modified generalized f-contraction. hence, g can not be an f-contraction, f-weak contraction and generalized f-contraction. let us consider f(x) = ln x for all x ∈ (0, ∞). clearly, f ∈ f. now, we define a function α : x × x → [0, ∞) by α(x, y) = { 1 2 , (x, y) ∈ {(0, 1), (3, 1)}; 1 otherwise. then, we can find τ > 0 such that τ + α(x, y)f(d(gx, gy)) ≤ f(ng(x, y)), whenever d(gx, gy) > 0. in particular, when α(x, y) = 1 2 one can choose τ ∈ (0, 1 5 ). therefore g is a modified generalized α-f-contraction. in the following, we present an example to show that the class of modified generalized α-f-contraction mappings is larger than that of generalized α-fcontraction mappings. example 2.5. let x = {−1, 0, 1} and g be a self-mapping on x defined by g(−1) = g(0) = 0, g(1) = −1. we define a distance function d on x by d(x, y) =    0, x = y; 1 2 , (x, y) ∈ {(1, −1), (−1, 1)}; 1 otherwise. so, (x, d) is a complete metric space. now d(gx, gy) > 0 for (x, y) = (0, 1) and (x, y) = (−1, 1). therefore we consider the following two cases. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 87 l. k. dey, p. kumam, t. senapati case-i. let (x, y) = (0, 1). then, d(g0, g1) = d(0, −1) = 1 mg(0, 1) = max { d(0, 1), d(0, g0), d(1, g1), d(0, g1) + d(1, g0) 2 , d(g20, 0) + d(g20, g1) 2 , d(g20, g0), d(g20, 1), d(g20, g1) } = max { 1, 0, 1 2 } = 1. and ng(0, 1) = max { d(0, 1), d(0, g1) + d(1, g0) 2 , d(g20, 0) + d(g20, g1) 2 , d(g20, g0), d(g20, 1), d(g0, 1) + d(1, g1), d(g20, g1) + d(0, g0) } = max { 1, 0, 1 2 , 3 2 } = 3 2 . case-ii. let (x, y) = (−1, 1). then d(g(−1), g1) = d(0, −1) = 1 and mg(−1, 1) = max {d(−1, 1), d(−1, g(−1)), d(1, g1), d(−1, g1) + d(1, g(−1)) 2 , d(g2(−1), −1) + d(g2(−1), g1) 2 , d(g2(−1), g(−1)), d(g2(−1), 1), d(g2(−1), g1)} = max { 1, 1 2 , 0 } = 1. ng(−1, 1) = max { d(−1, 1), d(−1, g1) + d(1, g(−1)) 2 , d(g2(−1), −1) + d(g2(−1), g1) 2 , d(g2(−1), g(−1)), d(g2(−1), 1), d(g(−1), 1) + d(1, g1), d(g2(−1), g1) + d(−1, g(−1)) } = max { 1, 0, 1 2 , 2, 3 2 } = 2. if we choose f(x) = ln(x) for all x ∈ (0, ∞) and α(x, y) ≥ 0, then g can not be a generalized α-f-contraction, since τ + α(0, 1)f(d(g0, g1)) ≤ f(mg(0, 1)) ⇒ τ + α(0, 1) ln(1) ≤ ln(1) ⇒ τ ≤ 0. if we choose ng(0, 1) instead of mg(0, 1), one can check that g is a modified generalized f-contraction and hence modified generalized α-f-contraction. in a similar fashion, for case-ii, it can be shown that g is a modified generalized α-f-contraction but not generalized α-f-contraction. now, we are in a position to state our main results. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 88 fixed point theorems for α-f -contraction mappings theorem 2.6. let (x, d) be an α-complete metric space and g : x → x be a modified generalized α-f-contraction where f ∈ f. assume that the following conditions hold: (1) g is α-admissible, α-continuous mapping; (2) there exists x0 ∈ x such that α(x0, gx0) ≥ 1. then g has a fixed point. proof. by the hypothesis, there exists a point x0 ∈ x such that α(x0, gx0) ≥ 1. now we define a sequence (xn) by xn+1 = gxn, for all n ∈ n0. if for some n ∈ n, xn = gxn, then xn is a fixed point of g and the proof is complete. so we assume that there exists no such integer n for which xn = gxn. now α(x0, gx0) ≥ 1 ⇒ α(x0, x1) ≥ 1. since g is an α-admissible mapping, for all n ∈ n0, we get α(xn, xn+1) ≥ 1. as d(gxn−1, gxn) > 0 and g is a modified generalized α-f-contraction, for some τ > 0, we have f(d(xn, xn+1)) = f(d(gxn−1, gxn)) ≤ τ + α(xn−1, xn)f(d(gxn−1, gxn)) ≤ f(ng(xn−1, xn)).(2.1) now, by simple computations, we have ng(xn−1, xn) = max { d(xn−1, xn), d(xn−1, gxn) + d(xn, gxn−1) 2 , d(g2xn−1, xn−1) + d(g 2xn−1, gxn) 2 , d(g2xn−1, gxn−1), d(g 2xn−1, xn), d(gxn−1, xn) + d(xn, gxn), d(g2xn−1, gxn) + d(xn−1, gxn−1) } = max { d(xn−1, xn), d(xn, xn+1), d(xn−1, xn+1) 2 } . if max{d(xn−1, xn), d(xn, xn+1)} = d(xn, xn+1), then (2.1) shows that τ + α(xn−1, xn)f(d(xn, xn+1)) ≤ f(d(xn, xn+1)) which is impossible. we must have max{d(xn−1, xn), d(xn, xn+1)} = d(xn−1, xn). therefore, (2.1) implies that f(d(xn, xn+1)) ≤ α(xn−1, xn)f(d(xn, xn+1)) ≤ f(d(xn−1, xn)) − τ(2.2) ⇒ f(d(xn, xn+1)) < f(d(xn−1, xn)) as τ > 0 ⇒ d(xn, xn+1) < d(xn−1, xn). this shows that (xn) is a decreasing sequence of nonnegative real numbers. we claim that lim n→∞ d(xn+1, xn) = 0. if possible, let lim n→∞ d(xn+1, xn) = δ for some c© agt, upv, 2019 appl. gen. topol. 20, no. 1 89 l. k. dey, p. kumam, t. senapati δ > 0. therefore, for every n ∈ n, we have d(xn, xn+1) ≥ δ. by (f ′) and (2.3), we have f(δ) ≤ f(d(xn, xn+1)) ≤ α(xn+1, xn)f(d(xn, xn+1)) < f(d(xn−1, xn)) − τ < f(d(xn−2, xn−1)) − 2τ ... < f(d(x0, x1)) − nτ.(2.3) as lim n→∞ (f(d(x0, x1)) − nτ) = −∞, so we can find some m ∈ n such that f(d(x0, x1))−nτ < f(δ) for all n > m, which contradicts the above equation. therefore, we must have lim n→∞ d(xn, xn+1) = 0. next, we claim that (xn) is a cauchy sequence. by (f ′′), there exists k ∈ (0, 1) such that (2.4) lim n→∞ (αkn)f(αn) = 0, where lim n→∞ αn = lim n→∞ d(xn, xn+1) = 0. again, from (2.3) and (2.4), we can obtain lim n→∞ (αkn)(f(αn) − f(α0)) ≤ lim n→∞ − (αkn)nτ ≤ 0 ⇒ lim n→∞ {nαkn} = 0 as τ > 0.(2.5) so, we can find some n0 ∈ n such that n(αn) k ≤ 1, for all n ≥ n0 ⇒ αn ≤ 1 n 1 k , for all n ≥ n0.(2.6) in view of (2.6), for all m > n > n0, we have d(xn, xm) ≤ d(xn, xn+1) + d(xn+1, xn+2) + . . . . + d(xm−1, xm) < σ∞j=1αj ≤ σ ∞ j=1 1 j 1 k . as 1 k > 1, the above series is convergent. this implies that lim n,m→∞ d(xn, xm) = 0, i.e., (xn) is a cauchy sequence. since, (x, d) is an α-complete metric space and (xn) is a cauchy sequence with α(xn, xn+1) ≥ 1 for all n ∈ n, we can find some x ∈ x such that xn → x whenever n → ∞. now, we claim that x is a fixed point of g. since xn → x as n → ∞ and α(xn, xn+1) ≥ 1, for all n ∈ n0, the α-continuity property of g implies that gxn → gx as n → ∞. finally, we have xn+1 = gxn ⇒ lim n→∞ xn+1 = lim n→∞ gxn ⇒ x = gx. hence x is a fixed point of g. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 90 fixed point theorems for α-f -contraction mappings notice that the condition of α-continuity of g in theorem 2.6 can actually be replaced by another weaker condition. in the sequel, we present the following result. theorem 2.7. let (x, d) be an α-complete metric space and let g : x → x be a modified generalized α-f-contraction, where f ∈ f. assume that the following conditions hold: (1) g is α-admissible; (2) there exists x0 ∈ x such that α(x0, gx0) ≥ 1; (3) if (xn) is a sequence in x with α(xn, xn+1) ≥ 1, for all n ∈ n0 and xn → x as n → ∞, we have α(xn, x) ≥ 1, for all n ∈ n0. then g has a fixed point. proof. following the proof of theorem 2.6, we know that (xn) is a cauchy sequence with α(xn, xn+1) ≥ 1, for all n ∈ n0 and it converges to some point x ∈ (x, d). by the hypothesis (3), we have α(xn, x) ≥ 1, for all n ∈ n0. we claim that x is a fixed point of g. on the contrary, suppose that gx 6= x ⇒ d(x, gx) > 0. we can find a number n ∈ n such that d(xm, gx) > 0, for all m ≥ n ⇒ d(gxm−1, gx) > 0. so by the condition of the theorem and by the property of f , we can find some τ > 0 such that τ + α(xm−1, x)f(d(gxm−1, gx)) ≤ f(ng(xm−1, x)) ⇒ f(d(gxm−1, gx)) < f(ng(xm−1, x)), [as α(xm−1, x) ≥ 1; τ > 0] ⇒ d(gxm−1, gx) < ng(xm−1, x) ⇒ lim m→∞ d(xm, gx) < lim m→∞ ng(xm−1, x).(2.7) now, we compute ng(xm−1, x) = max { d(xm−1, x), d(xm−1, gx) + d(x, gxm−1) 2 , d(g2xm−1, x) + d(g 2x, gx) 2 , d(g2xm−1, gxm−1), d(g 2xm−1, gx), d(g2xm−1, gx) + d(xm−1, gxm−1), d(gxm−1, x) + d(x, gx) } . using this in the above inequality, we get lim m→∞ d(xm, gx) < max{d(x, x), d(x, gx)} which leads to a contradiction. hence, our assumption was wrong. we must have d(x, gx) = 0, i.e., x is a fixed point of g. � in the following theorem, we present a fixed point result for a modified generalized α-f-contraction where the function f satisfies only (f ′) property. theorem 2.8. let (x, d) be an α-complete metric space and let g : x → x be a modified generalized α-f-contraction where f is strictly increasing function on (0, ∞). assume that the following conditions hold: c© agt, upv, 2019 appl. gen. topol. 20, no. 1 91 l. k. dey, p. kumam, t. senapati (1) g is triangular α-admissible; (2) there exists x0 ∈ x such that α(x0, gx0) ≥ 1; (3) if (xn) is a sequence in x with α(xn, xn+1) ≥ 1, for all n ∈ n0 and xn → x as n → ∞, we have α(xn, x) ≥ 1, for all n ∈ n0. then g has a fixed point. proof. following the proof of theorem 2.6, we have lim n→∞ d(xn, xn+1) = 0. now, we prove that (xn) is a cauchy sequence. if possible, suppose by contradiction that (xn) is not a cauchy sequence. then for some ǫ > 0, we can find sequences p(n) and q(n) of natural numbers such that p(n) > q(n) > n, d(xp(n), xq(n)) ≥ ǫ and d(xp(n)−1, xq(n)) < ǫ,(2.8) for all n ∈ n. therefore, we have ǫ ≤ d(xp(n), xq(n)) ≤ d(xp(n), xp(n)−1) + d(xp(n)−1, xq(n)) < d(xp(n), xp(n)−1) + ǫ which implies that lim n→∞ d(xp(n), xq(n)) = ǫ.(2.9) again, from (2.8), we can find n0 ∈ n such that d(xp(n), gxp(n)) < ǫ 4 and d(xq(n), gxq(n)) < ǫ 4 , for all n ≥ n0 ∈ n.(2.10) now, we claim that d(gxp(n), gxq(n)) > 0. indeed, if no, then there exists m ≥ n0 such that d(gxp(m), gxq(m)) = d(xp(m)+1, xq(m)+1) = 0. from (2.10), it follows that ǫ ≤ d(xp(m), xq(m)) ≤ d(xp(m), xp(m)+1) + d(xp(m)+1, xq(m)) ≤ d(xp(m), xp(m)+1) + d(xp(m)+1, xq(m)+1) + d(xq(m)+1, xq(m)) = d(xp(m), gxp(m)) + d(xp(m)+1, xq(m)+1) + d(xq(m), gxq(m)) ≤ ǫ 4 + 0 + ǫ 4 = ǫ 2 which is a contradiction. therefore, we get d(gxp(m), gxq(m)) > 0, for all m ∈ n. from (2.9), we get lim m→∞ d(gxp(m), gxq(m)) = lim m→∞ d(xp(m)+1, xq(m)+1) = ǫ. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 92 fixed point theorems for α-f -contraction mappings since g is a modified generalized α-f-contraction, we can find τ > 0 such that τ + α(xp(n), xq(n))f(d(gxp(n), gxq(n))) ≤f(ng(xp(n), xq(n))), for all n ≥ n0 ⇒ α(xp(n), xq(n))f(d(gxp(n), gxq(n))) ≤f(ng(xp(n), xq(n))) − τ. since, α(xp(n), xq(n)) ≥ 1, for all n ∈ n0; τ > 0 and f is strictly increasing, we have f(d(gxp(n), gxq(n))) < f(ng(xp(n), xq(n))) ⇒ d(gxp(n), gxq(n)) < ng(xp(n), xq(n)), ∀n ∈ n ⇒ lim n→∞ d(gxp(n), gxq(n)) < lim n→∞ ng(xp(n), xq(n)).(2.11) now, we observe that lim n→∞ ng(xp(n), xq(n)) = max{ lim n→∞ {d(xp(n), xq(n)), d(xp(n), xq(n)+1) + d(xq(n), xp(n)+1) 2 , d(xp(n)+2, xp(n)) + d(xp(n)+2, xq(n)+1) 2 , d(xp(n)+2, xp(n)+1), d(xp(n)+2, xq(n)), d(xp(n)+2, xq(n)+1) + d(xp(n), xp(n)+1), d(xp(n)+1, xq(n)) + d(xq(n), xq(n)+1)}}. using the triangle inequality and by some simple computations, one can easily check that lim n→∞ ng(xp(n), xq(n)) = ǫ. using this in (2.11), we have ǫ = lim n→∞ d(gxp(n), gxq(n)) < ǫ which implies that our assumption was wrong. so (xn) must be a cauchy sequence with the property α(xn, xn+1) ≥ 1, hence it converges to some point x̃ in x as (x, d) is an α-complete metric space. next, we show that x̃ is a fixed point of g. by the hypothesis of the theorem, we have α(xn, x̃) ≥ 1. again, by the property of f , we obtain f(d(xn, gx̃)) ≤ τ + α(xn−1, x)f(d(gxn−1, gx̃)) ≤ f(ng(xn−1, x̃)) ⇒ d(xn, gx̃) ≤ ng(xn−1, gx̃) ⇒ lim n→∞ d(xn, gx̃) ≤ lim n→∞ ng(xn−1, x̃) ⇒ d(x̃, gx̃)) = 0. this shows that x̃ is a fixed point of g. � now, we present an additional condition to ensure the uniqueness of fixed point. theorem 2.9. let g be a modified generalized α-f-contraction. if g has two fixed points x, y ∈ x with α(x, y) ≥ 1, then we must have x = y. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 93 l. k. dey, p. kumam, t. senapati proof. given x, y ∈ fix(g) with x 6= y ⇒ gx 6= gy ⇒ d(gx, gy) > 0. for any n ∈ n, we have gnx = x and gny = y. as g is an α-f-contraction with d(gx, gy) > 0, there exists some τ > 0 such that τ + α(x, y)f(d(gx, gy)) ≤ f(ng(x, y)) ⇒ τ + α(x, y)f(d(gx, gy)) < f(d(x, y)) ⇒ f(d(gx, gy)) < f(d(x, y)), [as α(x, y) ≥ 1; τ > 0] ⇒ f(d(x, y)) < f(d(x, y)). this contradiction shows that x = y. � remark 2.10. notice that the above theorems establish the existence and then uniqueness of fixed point of the function g without assuming the continuity property of f as well as the continuity property of g. remark 2.11. our results generalize several fixed point results in the existing literature. for instance, taking α(x, y) = 1, we can obtain the main results of piri and kumam [13] and dung and hang [6] as a corollary of our main results. most importantly, our results are the generalized versions of the fixed point results given by gopal et al. 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[20] d. wardowski and n. v. dung, fixed points of f-weak contractions on complete metric spaces, demonstr. math. 47, no. 1 (2014), 146–155. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 95 @ appl. gen. topol. 20, no. 1 (2019), 57-73doi:10.4995/agt.2019.9679 c© agt, upv, 2019 infinite games and quasi-uniform box products hope sabao a and olivier olela otafudu b a department of mathematics and statistics, university of zambia, great east rd campus, lusaka, zambia (hope@aims.edu.gh) b school of mathematics, university of the witwatersrand, johannesburg 2050, south africa (olivier.olelaotafudu@wits.ac.za) communicated by h.-p. a. künzi abstract we introduce new infinite games, played in a quasi-uniform space, that generalise the proximal game to the framework of quasi-uniform spaces. we then introduce bi-proximal spaces, a concept that generalises proximal spaces to the quasi-uniform setting. we show that every bi-proximal space is a w -space and as consequence of this, the bi-proximal property is preserved under σ-products and closed subsets. it is known that the sorgenfrey line is almost proximal but not proximal. however, in this paper we show that the sorgenfrey line is biproximal, which shows that our concept of bi-proximal spaces is more general than that of proximal spaces. we then present separation properties of certain bi-proximal spaces and apply them to quasi-uniform box products. 2010 msc: 54e15; 54b10; 54e35; 54d15. keywords: infinite games; w -spaces; σ-products; quasi-uniform spaces; quasi-uniform box products. 1. introduction jocelyn r. bell, in [2], introduced an infinite game played in a uniform space which she called the proximal game. she showed that every proximal space is a w-space and the proximal property is preserved under σ-products, countable products and closed subsets. also, every proximal space is collectionwise normal, countably paracompact and collectionwise hausdorff. she then used this received 16 february 2018 – accepted 11 november 2018 http://dx.doi.org/10.4995/agt.2019.9679 h. sabao and o. olela otafudu game to show that the uniform box product of countably many copies of a fortspace is collectionwise normal, countably paracompact and collectionwise hausdorff. during the 29th summer conference on topology and its applications held in new york 2014, ralph kopperman asked whether the proximal game, played in a uniform space, can be extended to generalised uniform spaces, for example, quasi-uniform spaces. in this article we answer ralph kopperman’s question for the class of quasi-uniform spaces. in particular, we introduce infinite games played in a quasi-uniform space which generalise the proximal game. since for any quasi-uniform space (x, u), u−1 is a quasi-uniformity on x and us is a uniformity on x, there are, atleast, three types of infinite games played in a quasi-uniform space that generalise the proximal game. we call the game played in (x, u), the left-proximal game; the game played in (x, u−1), the right proximal game; and the game played in (x, us), the us-proximal game. we show that a quasi-uniform space is left proximal if and only if it is right proximal. since for any quasi-uniformity u on x, us is a uniformity on x, we observe that the us-proximal game corresponds to the proximal game. furthermore, we say that a space is bi-proximal provided it is left and right proximal, and thereafter, show that a space is bi-proximal if and only if it is us-proximal. in [2, example 5], bell showed that the sorgenfrey line is almost proximal but it is not proximal. however, in this paper we show that the sorgenfrey line is bi-proximal, which proves that the bi-proximal property is more general than the proximal property. also, we show that every bi-proximal space is a w-space, and as the consequence of this, the bi-proximal property is closed under closed subsets, σ-products and countable products. this implies that the sorgenfrey plane is bi-proximal. however, it is known that the sorgenfey plane is not normal and collectionwise hausdorff. therefore, unlike proximal spaces, bi-proximal spaces are not, in general, collectionwise normal and collectionwise hausdorff. furthermore, in [9], kunz̈i and watson showed that there exists a quasi-metric space which is not countably metacompact. therefore, bi-proximal spaces are not, in general, countably metacompact. if we restrict ourselves to the quasi-uniform spaces (x, u) which satisfy the property that u and us are both compatible with the topology on x, then most topological properties satisfied by proximal spaces are also satisfied by bi-proximal spaces. we then use this fact to show that the quasi-uniform box product of countably many copies of a fort-space is bi-proximal, and as a consequence of this, it is collectionwise normal, countably paracompact and collectionwise hausdorff. we point out that our work is in parallel with bell [2]. in fact, we shall adapt some ideas and techniques of [2], which will be appropriately mentioned. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 58 infinite games and quasi-uniform box products 2. quasi-uniform spaces definition 2.1 ([8]). a quasi-uniformity on a set x is a filter u on x × x such that (i) each member u of u contains the diagonal △ = {(x, x) : x ∈ x} of x, (ii) for each u ∈ u there is v ∈ u such that 2v ⊆ u where 2v = v ◦ v = {(x, z) ∈ x ×x : there is y ∈ x such that (x, y) ∈ v and (y, z) ∈ v }. the members u ∈ u are called entourages of u and the elements of x are called points. the pair (x, u) is called a quasi-uniform space if u is a quasi-uniformity on a set x, then the filter u−1 = {u−1 : u ∈ u} on x × x is also a quasi-uniformity on x. the quasi-uniformity u−1 is called the conjugate of u. a quasi-uniformity that is equal to its conjugate is called a uniformity. the union of a quasi-uniformity u and its conjugate u−1 yields a subbase of the coarsest uniformity, denoted us, finer than u. if u ∈ u, the elements of us are of the form u ∩u−1 and are denoted by us. for u ∈ u, x ∈ x and z ⊂ x, put u(x) = {y ∈ x : (x, y) ∈ u} and u(z) = ⋃ {u(z) : z ∈ z}. a quasi-uniformity u generates a topology τ(u) on x for which the family of sets {u(x) : u ∈ u} is a base of neighbourhoods of any point x ∈ x. a subset a of x belongs to τ(u) if and only if for each x ∈ a, there is an entourage u ∈ u such that u(x) ⊂ a. thus for each x ∈ x and u ∈ u, u(x) is a τ(u)-neighborhood of x. note that u(x) need not be τ(u)-open in general. however, there is always a base b for u such that for each b ∈ b and x ∈ x, b(x) ∈ τ(u). proposition 2.2 ([4]). let u and v be quasi-uniformities on x. let u ∈ u, v ∈ v and m ⊂ x ×x. then u ◦m ◦v is a neighborhood of m in the topology of u−1 × v. corollary 2.3 ([4]). let (x, u) be a quasi-uniform space. then {u : u ∈ u and u is τ(u−1) × τ(u) open in x × x} is a base for u. definition 2.4 ([12]). let (x, τ) be a topological space. then the family of subsets b = {[a × a] ∪ [(x \ a) × x] : a ∈ τ} is a subbase of a quasi-uniformity on x that generates the topology on x. the quasi-uniformity generated by this subbase is called the pervin quasiuniformity. pervin remarked in [12] that the quasi-uniformity generating the topology is not unique. example 2.5. a fort-space is the one point compactification of a discrete space. if w is a discrete space, we will denote the one point compactification x of w by x = w ∪ {∞}. if w is uncountable, we say x is an uncountable fort-space. if we equip the fort-space x = w ∪{∞} with the quasi-uniformity u which has subbase {uf : f ⊆ w is finite }, where uf = △ ∪ [(x \ f) × x], c© agt, upv, 2019 appl. gen. topol. 20, no. 1 59 h. sabao and o. olela otafudu then u−1 has subbase {u−1f : f ⊆ w is finite}, where uf = △ ∪ [x × x \ f ]. thus uf ∩ u −1 f = △ ∪ (x \ f) × (x \ f). it follows that if x ∈ f , then u−1f (x) = {x} ∪ (x \ f) and uf (x) = {x}. if x /∈ f , then we have u−1f (x) = x \ f and uf (x) = x. moreover, it follows that for any x ∈ f, uf (x) ∩ u −1 f (x) = [{x} ∪ (x \ f)] ∩ {x} = {x}. also, for any x /∈ f , we have uf (x) ∩ u −1 f (x) = x \ f ∩ x = x \ f. definition 2.6. let x be a set and d : x ×x → [0, ∞) be a function mapping into the set [0, ∞) of nonnegative real numbers. then d is called a quasipseudometric on x if d(x, x) = 0 for all x ∈ x, and d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ x. the pair (x, d) is called a quasi-pseudometric space. if in addition, for any x, y ∈ x, d(x, y) = 0 = d(y, x) =⇒ x = y, then d is called a t0-quasi-metric and the pair (x, d) is called a t0-quasi-metric space. example 2.7 ([8]). let d be a quasi-pseudometric on a set x. for each ǫ > 0, set uǫ = {(x, y) ∈ x × x : d(x, y) < ǫ}. since for each ǫ > 0, 2uǫ/2 ⊆ uǫ, the filter generated by the base {uǫ : ǫ > 0} is a quasi-uniformity on x and is called the quasi-pseudometric quasi-uniformity ud induced by d on x. 3. the proximal game played on a quasi-uniform space this section presents infinite games, played in a quasi-uniform space, that generalise the proximal game to the quasi-uniform setting. since for any quasiuniform space (x, u), u−1 is a quasi-uniformity and us is a uniformity on x, there are, atleast, three types of infinite games played in a quasi-uniform space namely; the left proximal game, the right proximal game and the us-proximal game, as defined in section 1. similarly to [2, section 3], the left-proximal game is a game of perfect information that is played in a quasi-uniform space (x, u). following [2, section 3], in this game, player a, the entourage picking player, chooses elements of the quasi-uniformity u while player b, the point picking player, chooses elements of x. the first two rounds of the game are as follows: (i) player a chooses u1 ∈ u player b chooses x1 ∈ x (ii) player a chooses u2 ∈ u with u2 ⊆ u1 player b chooses x2 ∈ u1(x1) ∩ u −1 1 (x1) in general, if x1, x2, · · · , xn are the n choices of player b, player a chooses un+1 ⊆ un and then player b must choose xn+1 ∈ un(xn) ∩ u −1 n (xn). then player a wins the game if (i) there exists z ∈ x such that x1, x2, · · · τ(u s)-converges to z or (ii) ⋂ i∈n ui(xi) ∩ u −1 i (xi) = ∅. remark 3.1. note that player b chooses a point xn+1 in u(xn)∩u −1 n (xn) and not in un(xn) or u −1 n (xn) only. this is because the space in which player b chooses only elements in un(xn) or u −1 n (xn) does not generalise the proximal game to the framework of quasi-uniform spaces. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 60 infinite games and quasi-uniform box products a left-play of the game is a sequence (u1, x1, u2, x2, · · · ), where ui ∈ u and xi ∈ x are chosen according to the rules of the game. a finite sequence of points x1, x2, · · · , xn is left admissible if for some sequences of entourages u1 ⊇ u2, · · · ⊇ un from u, (u1, x1, u2, x2, · · · , un, xn) is a left-partial play of the game. a left strategy, in the left-proximal game on (x, u), is a recursively defined map w from the set of left admissible finite sequences a of x to u, that is, w : a → u such that (i) w(∅) = x × x, (ii) xn+1 ∈ w(x1, x2, · · · , xn−1)(xn) ∩ w(x1, x2, · · · , xn−1) −1(xn), and (iii) w(x1, x2, · · · , xn−1) ⊇ w(x1, x2, · · · , xn). therefore, w(x1, x2, · · · , xn) would be an element of u chosen by player a if x1, x2, · · · , xn are the n choices of player b and w(x1, x2, · · · , xn) −1 is the conjugate of w(x1, x2, · · · , xn). a sequence of points x1, x2, · · · resulting from a left-play of a left strategy is a left-proximal sequence. a left-strategy is winning if (i) every left-proximal sequence τ(us)-converges to a point z ∈ x or (ii) ⋂ i∈n w(x1, x2, · · · , xi−1)(xi) ∩ w(x1, x2, · · · , xi−1) −1(xi) = ∅. definition 3.2 (compare [2, definition 3]). a quasi-uniform space (x, u) is left-proximal provided player a has a winning strategy in the left-proximal game on (x, u). if a quasi-uniformizable space x has a compatible quasiuniformity u for which x is left-proximal, we say x is left-proximal. remark 3.3. if u is a uniformity on x, that is, u = u−1, then the left-proximal game is exactly the proximal game in the sense of [2]. in this paper, we will work with a filter base, consisting of τ(u−1) × τ(u)open entourages, rather than the whole quasi-uniformity. lemma 3.4. suppose (x, e) is a left-proximal quasi-uniform space and u is the filter base for e. then (x, u) is left-proximal. proof. suppose w : a → e is a left-winning strategy in the left-proximal game on (x, e), where a is the set of left-admissible finite sequences. then the construction of the left winning strategy v : a′ → u, where a′ is the set of leftadmissible finite sequences in the left-proximal game on (x, u) follows exactly that of [2, lemma 3]. � corollary 3.5. suppose (x, e) is a left-proximal quasi-uniform space and u is the subbase for e. then (x, u) is left-proximal. lemma 3.6 (compare [2, lemma 4]). every t0-quasi-metric space (x, d) is left-proximal with respect to the quasi-uniformity generated by the filter base u = {un : n ∈ n}, where un = {(x, y) ∈ x × x : d(x, y) < 2 −n}. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 61 h. sabao and o. olela otafudu proof. suppose xn, x2, · · · xn are the n choices for player b. then player a chooses the entourage w(x1, x2, · · · , xn) = un+1, where un+1 = {(x, y) : d(x, y) < 2−(n+1)}. then x1, x2, · · · is a d s-cauchy sequence. therefore, x1, x2, · · · τ(u s)-converges to z ∈ x or ⋂ i∈n ui(xi) ∩ u −1 i (xi) = ∅. hence w is a markov winning strategy (depending only on the opponent’s last choice and the round number) for player a in the left-proximal game on (x, u). � example 3.7. let r be the set of reals equipped with the t0-quasi-metric d, defined by d(x, y) = x − y if x ≥ y and d(x, y) = 1 otherwise. consider the quasi-uniformity generated by the base u = {(x, y) ∈ r × r : d(x, y) < 2−n}, where n ∈ n. then one can easily check that τ(u) is the sorgenfrey topology on r, which is generated by the base {(a, b] : a, b ∈ r, a < b} on r. similarly, τ(u−1) is a topology on r generated by the base {[a, b) : a, b ∈ r, a < b}. then r equipped with the sorgenfrey t0-quasi metric is left-proximal by lemma 3.6. example 3.8. the sorgenfrey plane has a topology generated by clopen boxes (a, b]×(c, d]. if p = (x, y) ∈ r×r and ǫ > 0 we write b(p, ǫ) = (x, x+ǫ]×(y, y+ǫ] and call it the clopen square cornered at p with side ǫ. this topology is quasimetrizable by the quasi-metric defined as follows: for any points p1 = (x1, y1) and p2 = (x2, y2) in r × r, ρ(p1, p2) = max{d(x1, x2), d(y1, y2)}, where d is the t0-quasi-metric defined in example 3.7. consider the quasi-uniformity generated by the base u = {(p1, p2) ∈ r × r : ρ(p1, p2) < 2 −n}, where n ∈ n. then one can easily check that τ(u) is the basis for the topology on the sorgenfrey plane r×r which is generated by the base {(a, b]×(c, d] : a, b, c, d ∈ r, a < b, c < d}. since (r × r, ρ) is a quasi-metric space, it is left proximal by lemma 3.6. a t0-quasi-metric space may not be left-proximal with respect to a quasiuniformity other than the one inherited from the t0-quasi-metric as the next example shows. example 3.9 (compare [2, example 1]). let w be uncountable set with the discrete topology. then w is quasi-metrizable with the sorgenfrey t0-quasimetric. consider the quasi-uniformity generated by the subbase u = {uf : f ⊆ w is finite}, where uf = [f × f ] ∪ [(w \ f) × w ]. then u −1 = {u−1f : f ⊆ w is finite}, where u−1f = [f × f ] ∪ [w × (w \ f)], is also a subbase generating a quasi-uniformity on w . then w is not left proximal. to see this, suppose w : a → u is any strategy for player a and w(∅) = uf1. then player b chooses x1 so that x1 /∈ f1. then uf1(x1) = w and u −1 f1 (x1) = w \ f1 and so uf1(x1)∩u −1 f1 (x1) = w \f1. in general, if for all k ≤ n, w(x1, x2, · · · , xk) = ufk, player b chooses xn /∈ f1 ∪ f2, · · · ∪ fn, so that xn is distinct from all previous choices of player b. then ⋂ n∈n ufn(xn) ∩ u −1 fn (xn) = w \ fn 6= ∅ and x1, x2, · · · is not τ(u s)-convergent to any point in w . we now give an example of a left-proximal space which is not quasi-metrizable. example 3.10 (compare [2, example 2]). an uncountable fort-space x = w ∪ {∞} is a non quasi-metrizable space which is left proximal. to see c© agt, upv, 2019 appl. gen. topol. 20, no. 1 62 infinite games and quasi-uniform box products this, consider the quasi-uniformity generated by the subbase u = {uf : f ⊆ w is finite}, where uf = [f × f ] ∪ [(x \ f) × x]. then u −1 = {u−1f : f ⊆ w is finite}, where u−1f = [f ×f ]∪[x ×(x \f)], is also a subbase for a quasiuniformity on x. suppose player b chooses x1 ∈ x. if x1 6= ∞, player a lets w(x1) = uf1, where f1 = {x1}. then player b chooses x2. if x2 6= x1, then uf1(x2) = x and u −1 f1 = x \ f1 and so uf1(x2) ∩ u −1 f1 (x2) = x ∩ (x \ f1) = x \ f1. since f1 = {x1}, then uf1(x2) ∩ u −1 f1 (x2) = x \ {x1} and player b cannot choose the point x1 in the future rounds of the game. if x2 = x1, then uf1(x2) = {x1} and u −1 f1 (x2) = x. therefore, uf1(x2) ∩ u −1 f1 (x2) = {x1}. this means that player b is forced to pick x1 at every rounds of the game. the left-winning strategy for player a is to add player b’s last choice (as long as it is not ∞) to the finite set which determines the element of the quasi-uniformity. precisely, if x1, x2, · · · xn are the first n choices of player b, then w(x1, x2, · · · , xn) = ufn, where fn = {x1, x2, · · · , xn} \ {∞}. then any left proximal sequence x1, x2, · · · either τ(u s)-converges to ∞ or is eventually constant. the right proximal game proceeds as the left proximal game except that player a, picks elements in the conjugate quasi-uniformity u−1, that is, elements of the form u−1, where u ∈ u. the winning criteria for player a, in the right proximal game, is the same as the winning criteria for player a in the left proximal. definition 3.11. a quasi-uniform space (x, u) is right-proximal provided player a has a winning strategy in the right-proximal game on (x, u). if a quasi-uniformizable space x has a compatible quasi-uniformity u for which x is right-proximal, we say x is right-proximal. lemma 3.12. a quasi-uniform space (x, u) is left-proximal if and only if it is right proximal. proof. suppose x is left-proximal, a is the set of left admissible finite sequences and b is the set of right admissible finite sequences. suppose w : a → u is a left winning strategy and m : b → u−1 is a right strategy. let w(∅) = x × x and m(∅) = x × x. suppose player b, in the left-proximal game, chooses x1 ∈ x. then player a chooses w(x1). suppose x1 is a choice for player b in the right proximal game. then player a, in the right proximal game, chooses m(x1), where m(x1) is a conjugate of w(x1). in general, if x1, x2, · · · , xn are the n choices for player b in the left proximal game and x1, x2, · · · , xn is a right admissible sequence, then player a, in the left-proximal game, chooses w(x1, x2, · · · , xn) ⊆ w(x1, x2, · · · , xn−1). also, player a, in the right proximal game, chooses m(x1, x2, · · · , xn) ⊆ m(x1, x2, · · · , xn−1), where m(x1, x2, · · · , xn) is a conjugate of w(x1, x2, · · · , xn). now suppose player b, in the left-proximal game, chooses xn+1 ∈ w(x1, x2, · · · , xn−1)(xn) ∩ w(x1, x2, · · · , xn−1) −1(xn). c© agt, upv, 2019 appl. gen. topol. 20, no. 1 63 h. sabao and o. olela otafudu then player b, in the right proximal game, can choose xn+1 ∈ m(x1, x2, · · · , xn−1)(xn) ∩ m(x1, x2, · · · , xn−1) −1(xn) since m(x1, x2, · · · , xn−1) is a conjugate of w(x1, x2, · · · , xn−1). therefore, x1, x2, · · · , xn+1 is a right admissible sequence. since (x, u) is left-proximal, x1, x2, · · · τ(u s)-converges to z ∈ x. if this does not hold, then ⋂ i∈n m(x1, x2, · · · , xi−1)(xi) ∩ m(x1, x2, · · · , xi−1) −1(xi) = ⋂ i∈n w(x1, x2, · · · , xi−1)(xi) ∩ w(x1, x2, · · · , xi−1) −1(xi) = ∅. therefore, m is a right winning strategy. the converse follows the same argument. � remark 3.13. since any left-proximal quasi-uniform space (x, u) is right proximal by lemma 3.12, we have that a = b, where a is a set of left-admissible finite sequences and b is a set of right-admissible finite sequences. moreover, for any choices x1, x2 · · · , xn for player b in the left and right proximal games, player b can choose xn+1 ∈ w(x1, x2 · · · , xn−1)(xn) ∩ m(x1, x2 · · · , xn−1)(xn) since w(x1, x2 · · · , xn−1) is a conjugate of m(x1, x2 · · · , xn−1). remark 3.14. one can easily show that the sorgenfrey line and the fort-space are right proximal. however, the uncountable space w is not right proximal with respect to the pervin quasi-uniformity. the us-proximal game proceeds as the left proximal game except that player a, picks elements in the symmetrised uniformity us, that is, elements of the for u ∩ u−1, where u ∈ u. the winning criteria for player a, in the us-proximal game, is the same as the winning criteria for player a in the left proximal. definition 3.15. a quasi-uniform space (x, u) is us-proximal provided player a has a winning strategy in the us-proximal game on (x, u). if a quasiunifomizable space x has a compatible quasi-uniformity u for which x is us-proximal, we will say x is us-proximal. remark 3.16. since for any quasi-uniform space (x, u), us is a uniformity on x, then the us-proximal game corresponds to the proximal game. we now present bi-proximal spaces, a concept that generalises proximal spaces to the framework of quasi-uniform spaces. these are spaces that posses a winning strategy for player a in the left and right proximal games. definition 3.17. let (x, u) be a quasi-uniform space. we say (x, u) is biproximal provided it is left and right-proximal. if a quasi-uniformizable topological space x has a compatible quasi-uniformity u for which x is bi-proximal, we will say x is bi-proximal. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 64 infinite games and quasi-uniform box products remark 3.18. one can easily show that the sorgenfrey line and the fort-space are bi-proximal. however, the uncountable space w is not bi-proximal with respect to the pervin quasi-uniformity. moreover, the sorgenfrey line is an example of a space which is bi-proximal but not proximal in the sense of bell [2]. therefore, bi-proximal spaces are more general than proximal spaces. also, the sorgenfrey line is first countable, hence, it is a w-space in the sense of [5]. as we will show later, all bi-proximal spaces are, in fact, w-spaces. theorem 3.19. a quasi-uniform space (x, u) is us-proximal if and only if it is bi-proximal. proof. suppose (x, u) is us-proximal, m is a set of us-admissible finite sequences and β : m → us is the us-winning strategy. then we define the winning strategy w : a → u for player a in the left-proximal game, where a be a set of left admissible finite sequences. suppose player b, in the us-proximal game, chooses x1 and player a chooses β(x1). suppose x1 is a choice for player a in the left proximal game. then player a in the left proximal game chooses w(x1), where β(x1) = w(x1) ∩ w(x1) −1. suppose player b, in the us-proximal game, has chosen x1, x2, · · · xn according to the rules of the us-proximal game and player a chooses β(x1, x2, · · · xn) ⊆ β(x1, x2, · · · xn−1). suppose x1, x2, · · · xn is a left admissible finite sequence. then player a, in the left-proximal game, chooses w(x1, x2, · · · xn) ⊆ w(x1, x2, · · · xn−1), where β(x1, x2, · · · xn) = w(x1, x2, · · · xn) ∩ w(x1, x2, · · · xn) −1. suppose player b, in the us-proximal game, chooses xn+1 ∈ β(x1, x2, · · · , xn−1)(xn). then xn+1 ∈ w(x1, x2, · · · , xn−1)(xn) ∩ w(x1, x2, · · · , xn−1) −1(xn) and player b, in the left-proximal game can choose xn+1. thus x1, x2, · · · , xn, xn+1 is a left admissible finite sequence. since β is a u s winning strategy, then x1, x2, · · · τ(u s)-converges to z ∈ x. if this does not hold, then ⋂ i∈n β(x1, x2, · · · , xi−1)(xi) = ⋂ i∈n (w(x1, x2, · · · , xi−1)(xi) ∩ w(x1, x2, · · · , xi−1) −1(xi)) = ∅. therefore, (x, u) is left-proximal and by lemma 3.12, (x, u) is right proximal. conversely, suppose (x, u) is a bi-proximal quasi-uniform space. suppose w : a → u and m : a → u−1 are the left and right winning strategies respectively, where a is a set of left and right admissible finite sequences. then we need to define the us-winning strategy β : m → us for player a in the us-proximal game, where m is a set of us-admissible finite sequences. suppose player b, in the left and right-proximal games, chooses x1 ∈ x. then player a, in the left proximal game chooses w(x1), whereas player a, in c© agt, upv, 2019 appl. gen. topol. 20, no. 1 65 h. sabao and o. olela otafudu the right proximal game, chooses m(x1), where m(x1) is a conjugate of w(x1). then player a, in the us-proximal game chooses β(x1), where β(x1) = w(x1)∩ m(x1). in general, if player b, at stage n, in the left and right proximal games, has chosen x1, x2, · · · , xn and that x1, x2, · · · , xn is a u s-admissible finite sequence. then player a, in the left-proximal game, chooses w(x1, x2, · · · , xn) ⊆ w(x1, x2, · · · , xn−1), whereas player a, in the right-proximal game, chooses m(x1, x2, · · · , xn) ⊆ m(x1, x2, · · · , xn−1), where m(x1, x2, · · · , xn) is a conjugate of w(x1, x2, · · · , xn). then player a in the u s-proximal game chooses β(x1, x2, · · · , xn) = w(x1, x2, · · · , xn) ∩ m(x1, x2, · · · , xn). now, player b, in the left and right proximal games chooses xn+1 ∈ w(x1, x2, · · · , xn−1)(xn) ∩ m(x1, x2, · · · , xn−1)(xn). then player b, in the u s-proximal game, can choose xn+1 ∈ β(x1, x2, · · · , xn)(xn) and x1, x2, · · · , xn, xn+1 is a u s-admissible finite sequence. since (x, u) is bi-proximal, we have ⋂ i∈n β(x1, x2, · · · , xi−1)(xi) = ⋂ i∈n w(x1, x2, · · · , xi−1)(xi) ∩ m(x1, x2, · · · , xi−1)(xi) = ∅. if not, then x1, x2, · · · τ(u s)-converges to z ∈ x. therefore, (x, u) is usproximal. � remark 3.20. in a bi-proximal space, the set of left admissible finite sequences, right-admissible finite sequences and us-admissible finite is the same, that is m = a. therefore, we simply call the set a, the set of admissible finite sequences. the same is true for proximal sequences. if we restrict the winning strategy for player a in the left-proximal game to only require convergence, then we say player a absolutely wins the left-proximal game. also, if we restrict the winning strategy for player a in the rightproximal game to only require convergence, then we say player a absolutely wins the right-proximal game. thus a space is absolutely bi-proximal provided player a has absolutely left and right winning strategies in the left and right proximal games. for t0-quasi-metric spaces, it turns out that an absolutely bi-proximal space is bicomplete. a t0-quasi-metric space (x, d) is said to be bicomplete if the metric space (x, ds) is complete. lemma 3.21. a t0-quasi-metric space (x, d) is bi-proximal if and only if is bicomplete. proof. suppose (x, d) is bicomplete. then (x, ds) is complete and by [2, lemma 5] (x, d) is us-proximal. conversely, suppose (x, d) is bi-proximal with respect to the quasi-uniformity generated by the filter base having elements of the form un = {(x, y) ∈ x × x : d(x, y) < 2 −n}. then (x, d) is us-proximal by theorem 3.19 and (x, ds) is complete by [2, lemma 5]. therefore, (x, d) is bicomplete. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 66 infinite games and quasi-uniform box products we now present the weakening of the left proximal game, the right proximal game and the us-proximal game. in these games, player a wins the game if he forces player b′s choices to cluster. definition 3.22. let (x, u) be a quasi-uniform space and w : a → u be a left-strategy for player a in the left proximal game. player a almost wins the left-proximal game if for every left-proximal sequence x1, x2, · · · , either (i) x1, x2, · · · has a τ(u s)-accumulation point or (ii) ⋂ i∈n w(x1, x2, · · · , xi−1)(xi) ∩ w(x1, x2, · · · , xi−1) −1(xi) = ∅. if player a has an almost left winning strategy, we say the space is almostleft proximal. if x is a quasi-uniformizable space and there exists a quasiuniformity u for which x is almost left proximal, we say x almost left-proximal. definition 3.23. let (x, u) be a quasi-uniform space and m : a → u−1 be a right-strategy for player a in the right proximal game. player a almost wins the right-proximal game if for every right-proximal sequence x1, x2, · · · , either (i) x1, x2, · · · has a τ(u s)-accumulation point or (ii) ⋂ i∈n m(x1, x2, · · · , xi−1)(xi) ∩ m(x1, x2, · · · , xi−1) −1(xi) = ∅. if player a has an almost right winning strategy, we say the space is almostright proximal. if x is a quasi-uniformizable space and there exists a quasiuniformity u for which x is almost right proximal, we say x almost rightproximal. definition 3.24. let (x, u) be a quasi-uniform space and β : a → u be a us-strategy for player a in the us-proximal game. player a almost wins the us-proximal game if for every us-proximal sequence x1, x2, · · · , either (i) x1, x2, · · · has a τ(u s)-accumulation point or (ii) ⋂ i∈n β(x1, x2, · · · , xi−1)(xi) = ∅. if player a has an almost us-winning strategy, we say the space is almost us-proximal. if x is a quasi-uniformizable space and there exists a quasiuniformity u for which x is almost us-proximal, we say x almost us-proximal. definition 3.25. a quasi-uniform space is almost bi-proximal provided it is almost left and almost right proximal. a quasi-uniformizable space x is almost bi-proximal provided there is a quasi-uniformity for which x is almost biproximal. remark 3.26. it is clear that every bi-proximal space is almost bi-proximal. however, the converse is not necessarily true. also, one can use the arguments from lemma 3.12 to show that a space is almost left-proximal if and only if it is almost right proximal. in addition, one can use arguments from theorem 3.19 to show that a space is almost bi-proximal if and only if it is almost us-proximal. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 67 h. sabao and o. olela otafudu 4. basic properties of bi-proximal spaces we begin this section by showing that every bi-proximal space is a wspace. w-spaces, introduced by gruenhage in [5], are generalized first countable spaces. they have many nice properties, for example, every w-space is a frechet-urysohn space. also, subspaces and σ-products of w-spaces are again w-spaces. definition 4.1 ([5]). a w-space is a topological space (x, τ) in which player 1 has a winning strategy in the following infinite game: (i) player 1 chooses v1 ∈ τ with x ∈ v1 player 2 chooses x1 ∈ v1 (ii) player 1 chooses v2 ∈ τ with x ∈ v2 player 2 chooses x2 ∈ v2 a strategy for player a is winning if the sequence x1, x2, · · · converges to x. this game is called the gruenhage game at x. lemma 4.2 (compare [2, lemma 6]). every bi-proximal space is a w-space. proof. suppose u is filter base generating the quasi-uniformity on x, consisting of τ(u−1)×τ(u) open entourages in x ×x, witnessing that x is bi-proximal. suppose player a, in the left-proximal game, picks elements of u and player b, the point picking player, picks points in x. similarly, suppose player a, in the right proximal game, picks elements of u−1 while player b picks elements of x. let player 1, in the gruenhage game, pick τ(u)-open sets while player 2 pick points in x. suppose w : a → u and m : a → u−1 are left and right winning strategies for player a in the left and right proximal games on (x, u), where a is the set of admissible finite sequences. fix a point x ∈ x and let nx denote the τ(u)-open neighbourhoods of x. then following the proof of [2, lemma 6] with modifications, one can use w and m to inductively construct a winning strategy δ : a′ → nx in the gruenhage game at x, where a ′ is a set of admissible finite sequences in the gruenhage game at x. � lemma 4.3 (compare [2, lemma 7]). suppose (x, u) is a bi-proximal space. then a τ(us)-closed subspace of x is bi-proximal. proof. suppose (x, u) is a bi-proximal quasi-uniform space, β : a → u is a us-winning strategy and k ⊆ x, where k is τ(us)-closed. define a filter base generating a quasi-uniformity on k by uk = {u ∩ (k × k) : u ∈ u}. then usk = {v ∩v −1 : v ∈ uk} is a filter base generating a uniformity on k. define βk : f(k) → u s k by βk(x1, x2, · · · , xn) = β(x1, x2, · · · , xn) ∩ (k × k). then βk is a u s k-winning strategy in (k, uk) since if ⋂ i∈n β(x1, x2, · · · xi)(xi+1) 6= ∅, then there is z ∈ x such that x1, x2, · · · τ(u s)-converges to z and z ∈ k since k is τ(us)-closed. by theorem 3.19, (k, uk) is bi-proximal. � remark 4.4. a τ(us)-open subspace of a quasi-uniform space (x, u) need not have a winning strategy in the bi-proximal game played with the subspace of quasi-uniformity generated by u and a subspace of a bi-proximal space need not be bi-proximal. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 68 infinite games and quasi-uniform box products definition 4.5. suppose (xn, un) is a quasi-uniform space for each n ∈ n. then the product quasi-uniformity on ∏ n∈n xn is generated by the subbase ǔ = {ǔn : un ∈ un}, where for each n ∈ n and each un ∈ un, ǔn = { (x, y) ∈ ∏ n∈n xn × ∏ n∈n xn : (x(n), y(n)) ∈ un } remark 4.6. it was observed by stoltenberg in [14] that ǔ induces the tychonov product topology on ∏ n∈n xn. for a fixed z ∈ ∏ n∈n xn, a σ-product with base point z is the set z = {x ∈ ∏ n∈n xn : |{β : x(β) 6= z(β)}| ≤ ω}. theorem 4.7. a σ-product of bi-proximal spaces is bi-proximal. proof. suppose (xn, un) is a quasi-uniform space for each n ∈ n, z is the σproduct with base point z and βn : an → un is a u s n-winning on (xn, un), where an is a set of finite admissible sequences in xn. then (z, ǔ) is ǔ s-proximal by [2, theorem 8]. therefore, (z, ǔ) is bi-proximal by theorem 3.19. � corollary 4.8. let (xn, un) be a bi-proximal quasi-uniform space for each n ∈ n. for each n ∈ n and each un ∈ un, let ûn = { (x, y) ∈ ∏ n∈n xn × ∏ n∈n xn : (x(n), y(n)) ∈ un } and define û = {ûn : un ∈ un}. then ( ∏ n∈n xn, û ) is bi-proximal. proof. since (x, un) is bi-proximal for each n ∈ n, then (x, un) is u s n-proximal by theorem 3.19 for each n ∈ n. therefore, ( ∏ n∈n xn, û ) is ûs-proximal by [2, corollary 1]. hence ( ∏ n∈n xn, û ) is bi-proximal by theorem 3.19. � 5. separation and covering properties a topological space (x, τ) is said to be collectionwise normal (respectively, collectionwise hausdorff) provided that each discrete collection of closed sets (respectively, each closed discrete point set) can be simultaneously separated by a pairwise disjoint collection of open sets [15]. bell [2] showed that proximal spaces are collectionwise normal and collectionwise hausdorff. however, this does not hold, in general, for bi-proximal spaces. in example 3.8, we showed that the sorgenfrey plane is bi-proximal. however, the sorgenfrey plane is not collectionwise normal and collectionwise hausdorff. for this reason, we restrict our discussion to only those bi-proximal spaces x for which u is a compatible quasi-uniformity for which x is bi-proximal and us is a compatible uniformity for which x is proximal. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 69 h. sabao and o. olela otafudu theorem 5.1. a bi-proximal space x for which for which u is a compatible quasi-uniformity for which x is bi-proximal and us is a compatible uniformity for which x is us-proximal is collectionwise normal. proof. since x is bi-proximal, then it is us-proximal by theorem 3.19. this implies that x is collectionwise normal by [2, theorem 10]. � theorem 5.2. an almost bi-proximal space x for which u is a compatible quasi-uniformity for which x is almost bi-proximal and us is a compatible uniformity for which x is almost us-proximal is collectionwise hausdorff. proof. since x is almost bi-proximal, then it is almost us-proximal by remark 3.26. this implies that x is collectionwise hausdorff by [2, theorem 12]. � a topological space (x, τ) is countably paracompact if every countable open cover has a locally finite open refinement and countably metacompact if every countable open cover has a point finite open refinement [15]. we use the following definition for a countably metacompact space. definition 5.3 ([6]). a topological space (x, τ) is countably metacompact if and only if for any descending sequence {fi}i∈n of nonempty closed sets such that ⋂ i∈n fi = ∅, there exists a descending sequence {gi}i∈n of nonempty open sets such that fi ⊂ gi for each i ∈ n and ⋂ i∈n cl(gi) = ∅. in theorem 11, bell [2] showed that all proximal spaces are countably metacompact. this implies that all metric spaces are countably metacompact since they are proximal. however, in [9], künzi, and watson showed that there exists a quasi-uniformity which is not countably metacompact. this shows that biproximal spaces are not, in general, countably metacompact. for this reason, in the next result, we restrict our discussion to only those bi-proximal spaces x for which u is a compatible quasi-uniformity for which x is bi-proximal and us is a compatible uniformity for which x is proximal. theorem 5.4. an almost bi-proximal space x for which u is a compatible quasi-uniformity for which x is almost bi-proximal and us is a compatible uniformity for which x is almost us-proximal is countably metacompact. proof. since x is almost bi-proximal, then it is almost us-proximal by theorem 3.26. this implies that x is countably metcompact by [2, theorem 11]. � 6. application to quasi-uniform box products the quasi-uniform box product, introduced in [11], is a topology, on the product of countably many copies of a quasi-uniform space, which is finer than the tychonov product topology but coarser than the uniform box product. it is generated by a quasi-uniformity, called the constant quasi-uniformity, whose symmetrised uniformity coincides with the constant uniformity in the sense of bell [1]. in this section, we discuss some separation and covering properties of the quasi-uniform box product of countably many copies of the uncountable fort-space. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 70 infinite games and quasi-uniform box products definition 6.1 ([11]). let (x, u) be a quasi-uniform space and ∏ n∈n x be the product of countably many copies of x. for each u ∈ u, let u = { (x, y) ∈ ∏ n∈n x × ∏ n∈n x : for all n ∈ n, (x(n), y(n)) ∈ u } . then u = {u : u ∈ u} is a filter base generating a quasi-uniformity on ∏ n∈n x. the quasi-uniformity u is called the constant quasi-uniformity on the product ∏ n∈n x, the topology τ(u) is called the constant quasi-uniform topology on ∏ n∈n x and the pair ( ∏ n∈nx, u ) is called the quasi-uniform box product. definition 6.2 (compare [2, definition 7]). let x = w ∪{∞} be uncountable fort-space and xω denote the tychonov product of countably many copies of x. for each finite set f ⊆ w , let uf = △ ∪ [(x \ f) × x] and for each k ∈ n, define uf,k = {(x, y) ∈ x ω × xω : ∀n < k, (x(n), y(n)) ∈ uf }. then u = {uf,k : f ⊆ w is finite and k ∈ n} is a subbase for a quasi uniformity on xω compatible with the topology. remark 6.3. note that u−1 = {u−1f,k : f ⊆ w is finite and k ∈ n}, where u−1f,k = {(x, y) ∈ x ω × xω : ∀n < k, (x(n), y(n)) ∈ u−1f } and u−1f = △ ∪ [x × (x \ f)], is a subbase for a quasi-uniformity on x ω. furthermore, us = {uf,k ∩ u −1 f,k : f ⊆ w is finite and k ∈ n}, where usf,k = {(x, y) ∈ x ω × xω : ∀n < k, (x(n), y(n)) ∈ usf } and usf = △ ∪ [(x \ f) × (x \ f)], is a uniformity base compatible with the topology on xω [2]. for any x ∈ xω, we have uf,k(x) = ∏ x(n)∈f,n sup{|xn| : n ∈ n}. � c© agt, upv, 2019 appl. gen. topol. 20, no. 1 72 infinite games and quasi-uniform box products since ( ∏ n∈n x ω, u ) , where x is a fort-space and u = {uf : f ⊂ w is finite}, where uf = △ ∪ [(x \ f) × x], contains the quasi-uniform box product ( ∏ n∈n x, u ) as a τ(u s )-closed subspace [2], we have the following: theorem 6.6. the quasi-uniform box product ( ∏ n∈n x, u ) , where x = w ∪ {∞} and u = {uf : f ⊂ wis finite}, where uf = △ ∪ [(x \ f) × x], is collectionwise normal, countably paracompact and collectionwise hausdorff. acknowledgements. the authors would like to thank the reviewer for the insightful comments that have greatly improved the presentation of this paper. references [1] j. r. bell, the uniform box product, proc. amer. soc. 142 (2014), 2161–2171. 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[10] k. morita, paracompactness and product spaces, fundam. math. 50, no. 3 (1962), 223– 236. [11] o. olela otafudu and h. sabao, on quasi-uniform box products, appl. gen. topol. 18, no. 1 (2017), 61–74. [12] w. j. pervin, quasi-uniformization of topological spaces, math. ann. 147 (1962), 316– 317. [13] j. roitman, paracompactness and normality in box products: old and new, set theory and its applications, contemp. math. 533 (2011), 157–181. [14] r. stoltenberg, some properties of quasi-uniform spaces, proc. london math. soc. 17 (1967), 226–240. [15] s. willard, general topology, dover publications, inc. mineols, new york, 2004. c© agt, upv, 2019 appl. gen. topol. 20, no. 1 73 () @ appl. gen. topol. 18, no. 2 (2017), 345-360doi:10.4995/agt.2017.7257 c© agt, upv, 2017 convergence theorems for finding the split common null point in banach spaces suthep suantai a, kittipong srisap b, natthapong naprang b, manatsawin mamat b, vithoon yundon b and prasit cholamjiak b a department of mathematics, faculty of science, chiang mai university, chiang mai 50200, thailand (suthep.s.cmu.ac.th) b school of science, university of phayao, phayao 56000, thailand. (prasitch2008@yahoo.com) communicated by e. a. sánchez-pérez abstract in this paper, we introduce a new iterative scheme for solving the split common null point problem. we then prove the strong convergence theorem under suitable conditions. finally, we give some numerical examples for supporting our main results. 2010 msc: 47h04; 47h10; 54h25. keywords: convergence theorem; split common null point problem; banach space; bounded linear operator. 1. introduction let h1 and h2 be real hilbert spaces and t : h1 → h2 a bounded linear operator (we denote a∗ by its adjoint) . let c and q be nonempty, closed and convex subsets of h1 and h2, respectively. the split feasibility problem is to find x ∈ c such that t x ∈ q. in order to solve the split feasibility problem (sfp), byrne [5] proposed the following iterative algorithm in the framework of hilbert spaces: x1 ∈ c and (1.1) xn+1 = pc(xn − λt ∗(i − pq)t xn), n ≥ 1, which is often called the cq algorithm, where λ > 0, pc and pq are the metric projections on c and q, respectively. it was shown that the sequence received 09 february 2017 – accepted 04 may 2017 http://dx.doi.org/10.4995/agt.2017.7257 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak {xn} converges weakly to a solution of sfp. since then several iterations have been invented for solving the sfp (see, for example, [2, 11, 13, 17]). let a : h1 → 2 h1 and b : h2 → 2 h2 be set-valued mappings. byrne et al. [6] considered the problem of finding a point z in h1 such that (1.2) z ∈ a−10 ∩ t −1(b−10), where the set of null points of a is defined by a−10 = {z ∈ h1 : 0 ∈ az}. we know that a−10 is closed and convex. this problem is called the split common null point problem and includes the spit feasibility problem as special cases; see also [8]. in 1953, mann [10] introduced the following iteration process. let c be a nonempty , closed and convex subset of a banach space e. a mapping t : c → c is called nonexpansive if (1.3) ‖t x − t y‖ ≤ ‖x − y‖ for all x, y ∈ c. we denote by f(t ) the fixed point set of t . for an initial point x1 ∈ c, an iteration process {xn} is defined recursively by (1.4) xn+1 = αnxn + (1 − αn)t xn, n ∈ n, where {αn} is a sequence in [0,1] and t is a nonexpansive mapping on c. in 1967, halpern [7] defined an iteration process as follows: take x0, x1 ∈ c arbitrarily and define {xn} recursively by (1.5) xn+1 = αnx0 + (1 − αn)t xn, n ∈ n, where {αn} is a sequence in [0, 1] and t is a nonexpansive mapping on c. a mapping f : c → c is said to be a contraction if there exists α ∈ (0, 1) such that (1.6) ‖f(x) − f(y)‖ ≤ α‖x − y‖, ∀x, y ∈ c. in 2000, moudafi [12] introduced the following algorithm: for x1 ∈ c, define the sequence {xn} by (1.7) xn+1 = αnf(xn) + (1 − αn)t xn, n ∈ n, where {αn} ⊂ (0, 1) and t is a nonexpansive mapping. this method is called the viscosity approximation method. let h be a hilbert space and let f be a strictly convex, reflexive and smooth banach space. let jf be the duality mapping on f . let c and d be nonempty, closed and convex subsets of h and f , respectively. let pc and pd be the metric projections of h onto c and f onto d, respectively. let t : h → f be a bounded linear operator such that t 6= 0 and let t ∗ be the adjoint operator of t . suppose that c ∩ a−1d 6= ∅. in 2015, alsulami and takahashi [2] defined the following algorithm: for any x1 ∈ h, (1.8) xn+1 = βnxn + (1 − βn)pc(i − rt ∗ jf (t − pdt ))xn, n ∈ n, where {βn} ⊂ [0, 1] and r ∈ (0, ∞). it was proved that if (1.9) 0 < a ≤ βn ≤ b < 1 and 0 < r‖t ‖ 2 < 2 c© agt, upv, 2017 appl. gen. topol. 18, no. 2 346 convergence theorems for finding the split common null point for some a, b ∈ r, then {xn} converges weakly to z0 ∈ c ∩ t −1d, where z0 = limn→∞ pc∩t −1dxn. they introduced the following halpern’s type iteration: for any x1 ∈ h, (1.10) xn+1 = βnxn + (1 − βn)(αnun + (1 − αn)pc(i − rt ∗ jf (i − pd)t )xn), n ∈ n, where r ∈ (0, ∞), {αn} ⊂ (0, 1) and {βn} ⊂ (0, 1). it was proved that if (1.11) 0 < r‖t ‖2 < 2, lim n→∞ αn = 0, (1.12) ∞ ∑ n=1 αn = ∞ and 0 < a ≤ βn ≤ b < 1 where a, b ∈ r. then {xn} converges strongly to a point z0 = c ∩ a −1d, for some z0 = pc∩a−1du. recently, using the idea of halpern’s iteration, alofi et al. [1] proved the following strong convergence theorem for finding a solution of the split common null point problem in banach spaces. theorem 1.1. let h be a hilbert space and let f be a uniformly convex and smooth banach space. let jf be the duality mapping on f. let a and b be maximal monotone operators of h into 2h and f into 2f ∗ such that a−10 6= ∅ and b−10 6= ∅, respectively. let jλ be the resolvent of a for λ > 0 and let qµ be the metric resolvent of b for µ > 0. let t : h → f be a bounded linear operator such that t 6= 0 and let t ∗ be the adjoint operator of t . suppose that a−10 ∩ t −1(b−10) 6= ∅. let {un} be a sequence in h such that un → u. let x1 = x ∈ h and let {xn} ⊂ h be a sequence generated by xn+1 = βnxn + (1 − βn)(αnun + (1 − αn)jλn (i − λnt ∗ jf (i − qµn)t )xn) (1.13) for all n ∈ n, where {λn}, {µn} ⊂ (0, ∞), {αn} ⊂ (0, 1) and {βn} ⊂ (0, 1) satisfy the following conditions (1.14) 0 < a ≤ λn‖t ‖ 2 ≤ b < 2, 0 < k ≤ µn, 0 < c ≤ βn ≤ d < 1, (1.15) lim n→∞ αn = 0 and ∞ ∑ n=1 αn = ∞ for some a, b, c, d, k ∈ r. then {xn} converges strongly to z0 ∈ a −10 ∩ t −1(b−10), where z0 = pa−10∩t −1(b−10)u. motivated by the previous works, we introduce a new iterative scheme for solving the split common null point problem. we then prove the strong convergence theorem under suitable conditions. finally, we give some numerical examples for supporting our main results. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 347 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak 2. preliminaries and lemmas let h be a real hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖, respectively. for x, y ∈ h and λ ∈ r, we know from [15] that (2.1) ‖x + y‖2 ≤ ‖x‖2 + 2〈y, x + y〉; (2.2) ‖λx + (1 − λ)y‖2 = λ‖x‖2 + (1 − λ)‖y‖2 − λ(1 − λ)‖x − y‖2. furthermore, for x, y, u, v ∈ h, (2.3) 2〈x − y, u − v〉 = ‖x − v‖2 + ‖y − u‖2 − ‖x − u‖2 − ‖y − v‖2. the nearest point projection of a nonempty, closed and convex set c is denoted by pc, that is, ‖x − pcx‖ ≤ ‖x − y‖ for all x ∈ h and y ∈ c. such pc is called the metric projection of h onto c. we know the metric projection pc is firmly nonexpansive, i.e., (2.4) ‖pcx − pcy‖ 2 ≤ 〈pcx − pcy, x − y〉 for all x, y ∈ h. moreover 〈x − pcx, y − pcx〉 ≤ 0 holds for all x ∈ h and y ∈ c; see [15]. let e be a real banach space with norm ‖ · ‖ and let e∗ be the dual space of e. we denote the value of y∗ ∈ e∗ at x ∈ e by 〈x, y∗〉. when {xn} is a sequence in e, we denote the strong convergence of {xn} to x ∈ e by xn → x and the weak convergence by xn ⇀ x. the modulus δ of convexity of e is defined by δ(ǫ) = inf { 1 − ‖x + y‖ 2 : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ǫ } (2.5) for every ǫ with 0 ≤ ǫ ≤ 2. a banach space e is said to be uniformly convex if δ(ǫ) > 0. it is known that a banach space e is uniformly convex if and only if for any two sequences {xn} and {yn} in e such that lim n→∞ ‖xn‖ = lim n→∞ ‖yn‖ = 1 and lim n→∞ ‖xn − yn‖ = 2,(2.6) limn→∞ ‖xn − yn‖ = 0 holds. a uniformly convex banach space is strictly convex and reflexive. the duality mapping j from e into 2e ∗ is defined by j(x) = {x∗ ∈ e∗ : 〈x, x∗〉 = ‖x‖2 = ‖x∗‖2}(2.7) for every x ∈ e. let u = {x ∈ e : ‖x‖ = 1}. the norm of e is said to be gâteaux differentiable if for each x, y ∈ u, the limit (2.8) lim t→0 ‖x + ty‖ − ‖x‖ t exists. in this case, e is called smooth. we know that e is smooth if and only if j is a single-valued mapping of e into e∗. we also know that e is reflexive if and only if j is surjective, and e is strictly convex if and only if j is one-to-one therefore, if e is a smooth, strictly convex and reflexive banach space, then j c© agt, upv, 2017 appl. gen. topol. 18, no. 2 348 convergence theorems for finding the split common null point is a single-valued bijection and in this case, the inverse mapping j−1 coincides with the duality mapping j∗ on e∗. for more details, see [14, 16]. let c be a nonempty, closed and convex subset of a strictly convex and reflexive banach space e. then we know that for any x ∈ e, there exists a unique element z ∈ c such that ‖x − z‖ ≤ ‖x − y‖ for all y ∈ c. putting z = pcx, we call pc the metric projection of e onto c. lemma 2.1 ([16]). let e be a smooth, strictly convex and reflexive banach space. let c be a nonempty, closed and convex subset of e, and let x1 ∈ e and z ∈ c then, the following conditions are equivalent: (1) z = pcx1; (2) 〈z − y, j(x1 − z)〉 ≥ 0, ∀ y ∈ c. let e be a banach space and let a be a mapping of e into 2e ∗ . the effective domain of a is denoted by dom(a), that is, dom(a) = {x ∈ e : ax 6= ∅}. a multi-valued mapping a on e is said to be monotone if 〈x − y, u∗ − v∗〉 ≥ 0 for all x, y ∈ dom(a), u∗ ∈ ax, and v∗ ∈ ay. a monotone operator a on e is said to be maximal if its graph is not properly contained in the graph of any other monotone operator on e. the following theorem is due to browder [4]; see also [14]. lemma 2.2 ([4]). let e be a uniformly convex and smooth banach space and let j be the duality mapping on e into e∗. let a be a monotone operator of e into 2e ∗ . then a is maximal if and only for any r > 0, (2.9) r(j + ra) = e∗, where r(j + ra) is the range of j + ra. let e be a uniformly convex and smooth banach space with a gâteaux differentiable norm and let a be a monotone operator of e into 2e ∗ . for all x ∈ e and r > 0, we consider the following equation (2.10) 0 ∈ j(xr − x) + raxr. this equation has a unique solution xr. we define jr by xr = jrx. such jr where r > 0 are called the metric resolvent of a. in a hilbert space h, the metric resolvent jr of a is simply called the resolvent of a. we also know the following lemmas: lemma 2.3 ([3, 18]). let {sn} be a sequence of nonnegative real numbers, let {αn} be a sequence in [0, 1] with ∑ ∞ n=1 αn = ∞, let {βn} be a sequence of nonnegative real numbers with ∑ ∞ n=1 βn < ∞ and {γn} be a sequence of real numbers with lim supn→∞ γn ≤ 0. suppose that (2.11) sn+1 = (1 − αn)sn + αnγn + βn for all n = 1, 2, .... then limn→∞ sn = 0. lemma 2.4 ([9]). let {γn} be a sequence of real numbers that does not decrease at infinity in the sense the there exists a subsequence {γni} of {γn} which c© agt, upv, 2017 appl. gen. topol. 18, no. 2 349 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak satisfies γni < γni+1 for all i ∈ n. define the sequence {τ(n)}n>n0 of integers as follows: (2.12) τ(n) = max {k ≤ n : γk < γk+1} , where n0 ∈ n such that {k ≤ n0 : γk < γk+1} 6= ∅. then, the following hold: (i) τ(n0) ≤ τ(n0 + 1) ≤ ... and τ(n) → ∞; (ii) γτ(n) ≤ γτ(n)+1 and γn ≤ γτ(n)+1, ∀n ≥ n0. 3. main results in this section, we prove strong convergence theorems for finding a solution of the split common null point problem in banach spaces. theorem 3.1. let h be a hilbert space and let f be a uniformly convex and smooth banach space. let jf be the duality mapping on f. let f : h → h be a contraction. let a and b be maximal monotone operators of h into 2h and f into 2f ∗ , respectively. let jλ be the resolvent of a for λ > 0 and let qµ be the metric resolvent of b for µ > 0. let t : h → f be a bounded linear operator such that t 6= 0 and let t ∗ be the adjoint operator of t . suppose that a−10∩t −1(b−10) 6= ∅. let x1 ∈ h and let {xn} ⊂ h be a sequence generated by xn+1 = αnf(xn) + βnxn + γnjλn(i − λnt ∗jf (i − qµn)t )xn(3.1) for all n ∈ n, where {µn}, {λn} ⊂ (0, ∞), {αn} ⊂ (0, 1), {βn} ⊂ (0, 1) and {γn} ⊂ (0, 1) satisfy the following conditions: (3.2) 0 < a ≤ λn‖t 2‖ ≤ b < 2, 0 < k ≤ µn, 0 < c ≤ γn ≤ d < 1, (3.3) lim n→∞ αn = 0 and ∞ ∑ n=1 αn = ∞ for some a, b, c, d, k ∈ r. then {xn} converges strongly to z0 ∈ a −10 ∩ t −1(b−10), where z0 = pa−10∩t −1(b−10)f(z0). proof. put zn = jλn(i − λnt ∗jf (i − qµn)t )xn for all n ∈ n and let z ∈ a−10 ∩ t −1(b−10). we have that z = jλnz and t z = qµnt z for all n ∈ n. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 350 convergence theorems for finding the split common null point since jλn is nonexpansive, we have ‖zn − z‖ 2 = ‖jλn(i − λnt ∗jf (i − qµn)t )xn − jλnz‖ 2 ≤ ‖xn − λnt ∗jf (i − qµn)t )xn − z‖ 2 = ‖xn − z‖ 2 − 2λn〈xn − z, t ∗jf (i − qµn)t xn〉 +λ2n‖t ∗jf (i − qµn)t xn‖ 2 ≤ ‖xn − z‖ 2 − 2λn〈t xn − t z, jf (i − qµn)t xn〉 + λ2n‖t ‖ 2‖(i − qµn)t xn‖ 2 = ‖xn − z‖ 2 − 2λn〈t xn − qµnt xn, jf (i − qµn)t xn〉 −2λn〈qµnt xn − t z, jf (i − qµn)t xn〉 + λ2n‖t ‖ 2‖(i − qµn)t xn‖ 2 = ‖xn − z‖ 2 − 2λn‖t xn − qµnt xn‖ 2 −2λn〈qµnt xn − t z, jf (i − qµn)t xn〉 + λ2n‖t ‖ 2‖(i − qµn)t xn‖ 2 ≤ ‖xn − z‖ 2 − 2λn‖t xn − qµnt xn‖ 2 + λ2n‖t ‖ 2‖(i − qµn)t xn‖ 2 = ‖xn − z‖ 2 + λn(λn‖t ‖ 2 − 2)‖(i − qµn)t xn‖ 2.(3.4) since 0 < λn‖t ‖ 2 < 2, it follows that ‖zn − z‖ ≤ ‖xn − z‖ for all n ∈ n. so we obtain ‖xn+1 − z‖ = ‖αnf(xn) + βnxn + γnzn − z‖ ≤ αn‖f(xn) − z‖ + βn‖xn − z‖ + γn‖xn − z‖ ≤ αnα‖xn − z‖ + αn‖f(z) − z‖ + (1 − αn)‖xn − z‖ = (1 − αn(1 − α))‖xn − z‖ + αn‖f(z) − z‖.(3.5) by induction, we conclude that {xn} is bounded. so are {t xn}, {zn} and {yn}. put z0 = pa−10∩t −1(b−10)f(z0). we see that xn+1 − xn = αn(f(xn) − xn) + γn(zn − xn),(3.6) which implies that xn+1 − xn − αn(f(xn) − xn) = γn(zn − xn).(3.7) it follows that 〈xn+1 − xn − αn(f(xn) − xn), xn − z0〉 = γn〈zn − xn, xn − z0〉 = −γn〈xn − zn, xn − z0〉.(3.8) from (2.3), we obtain 2〈xn − zn, xn − z0〉 = ‖xn − z0‖ 2 + ‖zn − xn‖ 2 − ‖zn − z0‖ 2 ≥ ‖xn − z0‖ 2 + ‖zn − xn‖ 2 − ‖xn − z0‖ 2 = ‖zn − xn‖ 2.(3.9) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 351 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak from (3.8) and (3.9), we obtain 2〈xn+1 − xn, xn − z0〉 = 2αn〈f(xn) − xn, xn − z0〉 − 2γn〈xn − zn, xn − z0〉 ≤ 2αn〈f(xn) − xn, xn − z0〉 − γn‖zn − xn‖ 2.(3.10) using (2.3) and (3.10), we have (3.11) ‖xn+1−z0‖ 2−‖xn−xn+1‖ 2−‖xn−z0‖ 2 ≤ 2αn〈f(xn)−xn, xn−z0〉−γn‖zn−xn‖ 2. putting γn = ‖xn − z0‖ 2 for all n ∈ n, we see that (3.12) γn+1 − γn − ‖xn − xn+1‖ 2 ≤ 2αn〈f(xn) − xn, xn − z0〉 − γn‖zn − xn‖ 2 . we note that ‖xn+1 − xn‖ = ‖αnf(xn) + βnxn + γnzn − xn‖ ≤ αn‖f(xn) − xn‖ + γn‖zn − xn‖.(3.13) this shows that ‖xn+1 − xn‖ 2 ≤ (αn‖f(xn) − xn‖ + γn‖zn − xn‖) 2 = α2n‖f(xn) − xn‖ 2 + 2αnγn‖f(xn) − xn‖‖zn − xn‖ + γ2n‖zn − xn‖ 2 .(3.14) hence by (3.12) and (3.14), we have γn+1 − γn ≤ αn(αn‖f(xn) − xn‖ 2 + 2γn‖f(xn) − xn‖‖zn − xn‖) + γ2n‖zn − xn‖ 2 + 2αn〈f(xn) − xn, xn − z0〉 − γn‖zn − xn‖ 2 = αn(αn‖f(xn) − xn‖ 2 + 2γn‖f(xn) − xn‖‖zn − xn‖) + γn(γn − 1)‖zn − xn‖ 2 + 2αn〈f(xn) − z0, xn − z0〉 − 2αn‖xn − z0‖ 2.(3.15) so we obtain γn+1 − γn + γn(1 − γn)‖zn − xn‖ 2 ≤ αn(αn‖f(xn) − xn‖ 2 + 2γn‖f(xn) − xn‖‖zn − xn‖) + 2αn〈f(xn) − z0, xn − z0〉 − 2αn‖xn − z0‖ 2.(3.16) we next split the proof into two cases. case 1: suppose that there exists a natural number n such that γn+1 ≤ γn for all n ≥ n. in this case, limn→∞ γn exists and then limn→∞(γn+1−γn) = 0. since limn→∞ αn = 0 and 0 < c ≤ γn ≤ d < 1, by (3.16), we have lim n→∞ ‖zn − xn‖ = 0.(3.17) from (3.13) we have lim n→∞ ‖xn+1 − xn‖ = 0.(3.18) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 352 convergence theorems for finding the split common null point we next show that lim supn→∞〈f(z0) − z0, zn − z0〉 ≤ 0. put (3.19) l = lim sup n→∞ 〈f(z0) − z0, zn − z0〉. then without loss of generality, there exists a subsequence {zni} of {zn} such that l = limi→∞〈f(z0)−z0, zni −z0〉 and {zni} converges weakly to some point w ∈ h. since ‖xn − zn‖ → 0, we also have that {xni} converges weakly to w ∈ h. on the other hand, from (3.4) we have λn(2 − λn‖t ‖ 2)‖(i − qµn)t xn‖ 2 ≤ ‖xn − zn‖ 2 − ‖zn − z‖ 2 ≤ ‖xn − zn‖(‖xn − z‖ + ‖zn − z‖).(3.20) then since ‖xn − zn‖ → 0 and 0 < a ≤ λn‖t ‖ 2 ≤ b < 2, (3.21) lim n→∞ ‖t xn − qµnt xn‖ = 0. since {xni} converges weakly to w ∈ h and t is bounded and linear, we also have {t xni} converges weakly to t w. using this and limn→∞ ‖t xn − qµnt xn‖ = 0, we have that qµni t xni ⇀ t w. since qµn is the metric resolvent of b for µn > 0, we have that jf (t xn−qµn t xn) µn ∈ bqµnt xn for all n ∈ n. by the monotonicity of b we obtain 0 ≤ 〈 u − qµni t xni, v ∗ − jf (t xni − qµni t xni) µni 〉 (3.22) for all (u, v∗) ∈ b. we observe that ‖jf (t xni − qµni t xni)‖ = ‖t xni − qµni t xni‖ → 0 as i → ∞. since 0 < k ≤ µni, it follows that 0 ≤ 〈u−t w, v ∗ − 0〉 for all (u, v∗) ∈ b. because b is maximal monotone, we have t w ∈ b−10. this implies that w ∈ t −1(b−10). using zn = jλn(xn − λnt ∗jf (t xn − qλnt xn)), we obtain zn = jλn (xn − λnt ∗ jf (t xn − qµnt xn))(3.23) ⇔ xn − λnt ∗jf (t xn − qµnt xn) ∈ zn + λnaz ⇔ xn − zn − λnt ∗ jf (t xn − qµnt xn) ∈ λnazn ⇔ 1 λn (xn − zn − λnt ∗jf (t xn − qµnt xn)) ∈ azn. since a is monotone, we have that for (u, v) ∈ a, 〈 zn − u, 1 λn (xn − zn − λnt ∗jf (t xn − qµnt xn)) − v 〉 ≥ 0(3.24) which implies that 〈 zn − u, xn − zn λn − t ∗jf (t xn − qµnt xn)) − v 〉 ≥ 0.(3.25) replacing n by ni, we have 〈 zni − u, xni − zni λni − t ∗jf (t xni − qµni t xni) − v 〉 ≥ 0.(3.26) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 353 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak since xni −zni → 0, 0 < a ≤ λni‖t ‖ 2, zni ⇀ w and t ∗jf (t xn −qµni t xni) → 0, we get that 〈w − u, −v〉 ≥ 0. since a is maximal, we have 0 ∈ aw. therefore, w ∈ a−10 ∩ t −1(b−10). since {zni} converges weakly to w ∈ a−10 ∩ t −1(b−10), it follows that (3.27) l = lim i→∞ 〈f(z0) − z0, zni − z0〉 = 〈f(z0) − z0, w − z0〉 ≤ 0. on the other hand, we see that ‖xn+1 − z0‖ 2 = 〈xn+1 − z0, xn+1 − z0〉 = 〈αnf(xn) + βnxn + γnzn − z0, xn+1 − z0〉 = 〈αn(f(xn) − z0) + βn(xn − z0) + γn(zn − z0), xn+1 − z0〉 = αn〈f(xn) − f(z0) + f(z0) − z0, xn+1 − z0〉 + βn〈xn − z0, xn+1 − z0〉 + γn〈zn − z0, xn+1 − z0〉 = αn〈f(xn) − f(z0), xn+1 − z0〉 + αn〈f(z0) − z0, xn+1 − z0〉 + βn〈xn − z0, xn+1 − z0〉 + γn〈zn − z0, xn+1 − z0〉 ≤ αnα‖xn − z0‖‖xn+1 − z0‖ + βn‖xn − z0‖‖xn+1 − z0‖ + γn‖xn − z0‖‖xn+1 − z0‖ + αn〈f(z0) − z0, xn+1 − z0〉 = (αnα + βn + γn)‖|xn − z0‖‖xn+1 − z0‖ + αn〈f(z0) − z0, xn+1 − z0〉 ≤ (αnα + 1 − αn) 1 2 (‖xn − z0‖ 2 + ‖xn+1 − z0‖ 2) + αn〈f(z0) − z0, xn+1 − z0〉 = ( ααn + 1 − αn 2 ) ‖xn − z0‖ 2 + ( ααn + 1 − αn 2 ) ‖xn+1 − z0‖ 2 + αn〈f(z0) − z0, xn+1 − z0〉 = ( 1 − 2(1 − α)αn 1 + (1 − α)αn ) ‖xn − z0‖ 2 + ( 2(1 − α)αn 1 + (1 − α)αn ) ( 1 1 − α ) 〈f(z0) − z0, xn+1 − z0〉.(3.28) also, we have lim n→∞ ‖zn − xn+1‖ ≤ lim n→∞ (‖zn − xn‖ + ‖xn+1 − xn‖) = 0.(3.29) then lim sup n→∞ 〈f(z0) − z0, xn+1 − z0〉 ≤ 0.(3.30) since ∑ ∞ n=1 αn = ∞, by lemma 2.3 we conclude that xn → z0 as n → ∞. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 354 convergence theorems for finding the split common null point case 2. suppose that there exists a subsequence {γni} of the sequence {γni} such that γni ≤ γni+1 for all i ∈ n. in this case, we define τ : n → n by τ(n) = max{k ≤ n : γk < γk+1}.(3.31) then by lemma 2.6 we have γτ(n) < γτ(n)+1. thus by (3.16) we have for all n ∈ n, γτ(n)(1 − γτ(n))‖zτ(n) − xτ(n)‖ 2 ≤ α2τ(n)‖f(xτ(n)) − xτ(n)‖ 2 + 2ατ(n)γτ(n)(‖f(xτ(n)) − xτ(n)‖ × ‖zτ(n) − xτ(n)‖) + 2ατ(n)〈f(xτ(n)) − z0, xτ(n) − z0〉 − 2ατ(n)‖xτ(n) − z0‖ 2 .(3.32) using limn→∞ αn = 0 and 0 < c ≤ γn ≤ d < 1, we have lim n→∞ ‖zτ(n) − xτ(n)‖ = 0.(3.33) as in the proof of case 1, we can show that lim n→∞ ‖xτ(n)+1 − xτ(n)‖ = 0.(3.34) this gives lim n→∞ ‖zτ(n) − xτ(n)+1‖ = 0.(3.35) we next show that lim supn→∞〈f(z0) − z0, xτ(n)+1 − z0〉 ≤ 0. put l = lim sup n→∞ 〈f(z0) − z0, xτ(n)+1 − z0〉.(3.36) so we have l = lim sup n→∞ 〈f(z0) − z0, zτ(n) − z0〉.(3.37) without loss of generality, there exists a subsequence {zτ(ni)} of {zτ(n)} such that l = lim i→∞ 〈f(z0) − z0, zτ(ni) − z0〉(3.38) and {zτ(ni)} converges weakly to some point w ∈ h. as in the proof of case 1, we can show that w ∈ a−10 ∩ t −1(b−10). then it follows that l = lim i→∞ 〈f(z0) − z0, zτ(ni) − z0〉 = 〈f(z0) − z0, w − z0〉 ≤ 0.(3.39) as in the proof of case 1, we also obtain ‖xτ(n)+1 − z0‖ 2 ≤ ( 1 − 2(1 − α)ατ(n) 1 + (1 − α)ατ(n) ) ‖xτ(n) − z0‖ 2 + ( 2(1 − α)αn 1 + (1 − α)αn ) ( 1 1 − α ) 〈f(z0) − z0, xτ(n)+1 − z0〉.(3.40) c© agt, upv, 2017 appl. gen. topol. 18, no. 2 355 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak since γτ(n) ≤ γτ(n)+1, ( 2(1 − α)ατ(n) 1 + (1 − α)ατ(n) ) ‖xτ(n) − z0‖ 2 ≤ ( 2(1 − α)αn 1 + (1 − α)αn ) × ( 1 1 − α ) 〈f(z0) − z0, xτ(n)+1 − z0〉.(3.41) it is easily seen that ( 2(1−α)ατ(n) 1+(1−α)ατ(n) ) > 0. then we have ‖xτ(n) − z0‖ 2 ≤ ( 1 1 − α ) 〈f(z0) − z0, xτ(n)+1 − z0〉.(3.42) this shows that lim sup n→∞ ‖xτ(n) − z0‖ 2 ≤ 0(3.43) and hence ‖xτ(n) − z0‖ → 0 as n → ∞. thus ‖xτ(n)+1 − z0‖ → 0 as n → ∞. by lemma 2.6, we obtain ‖xn − z0‖ ≤ ‖xτ(n)+1 − z0‖ → 0(3.44) as n → ∞. this completes the proof. � 4. examples and numerical results in this section, we give examples including its numerical results for supporting our main theorem. example 4.1. let h = r. for x ∈ r, we define g : r → r by g(x) = { ωx if x ≥ 0, +∞ otherwise. let f : r → r be defined by f(x) = ω|x| − ln(1 + ω|x|). choose x1 = 2, ω = 1, αn = 1 2n+1 , βn = n 2n+1 , γn = n 2n+1 for all n ∈ n. let f(x) = x 2 and t x = x . we aim to find the minimizers of f and g. using algorithm (3.1), we have the following numerical results: c© agt, upv, 2017 appl. gen. topol. 18, no. 2 356 convergence theorems for finding the split common null point n xn | xn+1 − xn | 1 1.471404251 5.2859547 × 10−1 2 1.265727840 7.8045565 × 10−1 3 0.527004926 7.8045565 × 10−1 4 0.250611252 1.3092799 × 10−1 5 0.160205803 7.9460312 × 10−2 6 0.114183169 5.3450423 × 10−2 7 0.083166153 3.7671482 × 10−2 8 0.061057394 2.7173989 × 10−2 9 0.045012810 1.9840027 × 10−2 10 0.033265124 1.4579710 × 10−2 11 0.024621382 1.0752746 × 10−2 12 0.018243176 7.9469695 × 10−3 ... ... ... 50 0.000000260 1.1017075 × 10−7 table 1 numerical results of example 4.1 for iteration process (3.1) 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 number of iterations x n figure 1: convergence behavior of {xn} in table 1. 5 10 15 20 25 30 35 40 45 50 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 number of iterations e rr o r figure 2: error plots for all sequences {xn} in table 1. example 4.2. let h = r3. for x ∈ r3, define g : r3 → r by g(x) = ‖lx − y‖2, where l =   1 2 2 3 0 1 2 1 −1  , y =   2 −3 1   and f : r3 → r by f(x) = 5‖x‖2 + (15, 6, −7)x + 10. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 357 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak let t =   1 0 1 0 −1 2 1 2 3   . find x ∈ r3 such that x is a minimizer of f and t x also is a minimizer of g. choose x1=   1 −1 −1  , αn = 1 2n+1 , βn = n 2n+1 , γn = n 2n+1 for all n ∈ n and let f(x) = x 2 . using algorithm (3.1), we have the following numerical results: n xn ‖xn+1 − xn‖ 1 (0.1527,-0.6014,0.8512) 20.745517681470 ×10−1 2 (-0.5968,-0.6646,-0.0706) 11.896884171498 ×10−1 3 (-0.8523,-0.5312,0.7977) 9.149067074771 ×10−1 4 (-1.1354,-0.6011,0.3222) 5.577239966457 ×10−1 5 (-1.2027,-0.5415,0.7430) 4.302752764182 ×10−1 6 (-1.3156,-0.5838,0.4941) 2.766128927383 ×10−1 7 (-1.3323,-0.5554,0.7039) 2.124210254520 ×10−1 8 (-1.3822,-0.5792,0.5747) 1.406007674047 ×10−1 9 (-1.3869,-0.5656,0.6815) 1.078492388367 ×10−1 10 (-1.4115,-0.5788,0.6150) 7.214423578793 ×10−2 11 (-1.4139,-0.5724,0.6703) 5.574083184995 ×10−2 12 (-1.4273,-0.5798,0.6365) 3.720017649222 ×10−2 13 (-1.4295,-0.5769,0.6655) 2.926886211227 ×10−2 14 (-1.4375,-0.5811,0.6485) 1.930348053252 ×10−2 15 (-1.4397,-0.5800.0.6640) 1.568794063595 ×10−2 ... ... ... 100 (-1.4790,-0.5913,0.6743) 8.809288634531 ×10−5 table 2 numerical results of example 4.2 for iteration process (3.1) 0 10 20 30 40 50 60 70 80 90 100 −1.5 −1 −0.5 0 0.5 1 number of iterations x n figure 3: convergence behavior of {xn} in table 2 . c© agt, upv, 2017 appl. gen. topol. 18, no. 2 358 convergence theorems for finding the split common null point 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 number of iterations e rr o r figure 4: error plots for all sequences {xn} in table 2 . from table 2, we see that   −1.5 −0.6 0.7   is a minimizer of f such that t   −1.5 −0.6 0.7   =   −0.8 2 −0.6   is a minimizer of g. acknowledgements. s. suantai wish to thank chiang mai university for financial supports. p. cholamjiak was supported by the thailand research fund and the commission on higher education under grant mrg5980248. references [1] a. s. alofi, m. alsulami and w. takahashi, strongly convergent iterative method for the split common null point problem in banach spaces, j. nonlinear convex anal. 2 (2016), 311–324. [2] s. m. alsulami and w. takahashi, iterative methods for the split feasibility problem in banach spaces, j. nonlinear convex anal. 16 (2015), 585–596. [3] k. aoyama, y. yasunori, w. takahashi and m. toyoda, on a strongly nonexpansive sequence in a hilbert space, j. nonlinear convex anal. 8 (2007), 471–489. [4] f. e. browder, nonlinear maximal monotone operators in banach spaces, math. ann. 175 (1968), 89–113. [5] c. byrne, iterative oblique projection onto convex sets and the split feasibility problem, inverse prob. 18 (2002), 441–453. [6] c. byrne, y. censor, a. gibili and s. reich, the split common null point problem, j. nonlinear convex anal. 13 (2012), 759–775. [7] b. halpern, fixed point of nonexpanding maps, bull. amer. math. soc. 73 (1967), 506-9-61. [8] y. censor and t. elfving, a multiprojection algorithm using bregman projections in product space, numer. algor. 8 (1994), 221–239 [9] p. e. maingé, strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, set-valued anal. 16 (2008), 899–912. [10] w. r. mann, mean value methods in it iteration, proc. amer. math. soc. 4 (1953), 506–510. [11] a. moudafi, split monotone variational inclusions, j. optim. theory appl. 150 (2011), 275–283. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 359 s. suantai, k. srisap, n. naprang, m. mamat, y. yundon and p. cholamjiak [12] a. moudafi, viscosity approximation method for fixed-points problems, j. math. anal. appl. 241 (2000), 46–55. [13] a. moudafi and b. s. thakur, solving proximal split feasibility problems without prior knowledge of operator norms, optim. lett. 8 (2014), 2099–2110. [14] w. takahashi, convex analysis and approximation of fixed point, yokohama publishers, yokohama, 2009. [15] w. takahashi, introduction to nonlinear and convex analysis, yokohama publishers, yokohama, 2009. [16] w. takahashi, nonlinear functional analysis, yokohama publishers, yokohama, 2000. [17] f. wang, a new algorithm for solving the multiple-sets split feasibility problem in certain banach spaces, numer. funct. anal. optim. 35 (2014), 99–110. [18] h. k. xu, another control condition in an iterative method for nonexpansive mappings, bull. austral. math. soc. 65 (2002), 109–113. c© agt, upv, 2017 appl. gen. topol. 18, no. 2 360 () @ appl. gen. topol. 19, no. 1 (2018), 101-127doi:10.4995/agt.2018.7812 c© agt, upv, 2018 some categorical aspects of the inverse limits in ditopological context fi̇li̇z yildiz department of mathematics, hacettepe university, ankara, turkey (yfiliz@hacettepe.edu.tr) communicated by s. romaguera abstract this paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifpditop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. in this context, the category invifpditop consisting of the inverse systems constructed by the objects and morphisms of ifpditop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical functorial setting. in conclusion, an identity natural transformation is obtained in the context of inverse systems limits constructed in ifpditop and the ditopological infinite products are characterized by the finite products via inverse limits. 2010 msc: primary: 18a30; 46m40; 46a13; 54b10; 54b30; secondary: 06b23; 18b35; 54e55. keywords: inverse limit; natural transformation; co-adjoint functor; ditopology; concrete isomorphism; joint topology. 1. introduction and preliminaries just as the methods used to derive a new space from two or more spaces are the products, subtextures and quotients of that spaces, so the another effective method is the theory of inverse systems ( projective spectrums) and inverse limits (projective limits). received 01 july 2017 – accepted 03 august 2017 http://dx.doi.org/10.4995/agt.2018.7812 f. yıldız the origins of the study of inverse limits date back to the 1920 ’s. classical theory of inverse systems and inverse limits are important in the extension of homology and cohomology theory. an exhaustive discussion of inverse systems which are in the some classical categories such as set, top, grp and rng defined in [1], was presented by the paper [5] which is a milestone in the development of that theory. as is the case with products, the inverse limit might not exist in any category in general whereas inverse systems exist in every category. note from that [5] inverse limits exist in any category when that category has products of objects and the equalizers [1] of pairs of morphisms, in other words, the inverse limits exist in any category if the category is complete, in the sense of [1]. additionally, an inverse system has at most one limit. that is, if an inverse limit of any inverse system exists in any category c, this limit is unique up to c-isomorphism. incidentally, inverse limits always exist in the categories set, top, grp and rng. note also that inverse limits are generally restricted to diagrams over directed sets. similarly, a suitable theory of inverse systems and inverse limits for the categories consisting of textures and ditopological spaces is handled first-time in [17] and [18]. incidentally, let ’s recall the notions of texture and ditopology introduced in 1993, by lawrence m. brown : for a nonempty set s, the family s ⊆ p(s) is called a texturing on s if (s,⊆) is a point-separating, complete, completely distributive lattice containing s and ∅, with meet coinciding with intersection and finite joins with union. the pair (s,s) is then called a texture. if s is closed under arbitrary unions, it is called plain texturing and (s,s) is called plain texture. since a texturing s need not be closed under the operation of taking the set-complement, the notion of topology is replaced by that of dichotomous topology or ditopology, namely a pair (τ,κ) of subsets of s, where the set of open sets τ and the set of closed sets κ, satisfy the some dual conditions. hence a ditopology is essentially a “topology” for which there is no a priori relation between the open and closed sets. in addition, a ditopological texture space or shortly ditopological space with respect to a ditopology (τ,κ) on the texture (s,s) is denoted by (s,s,τ,κ). there is now a considerable literature on the theory of ditopological spaces. an adequate introduction to this theory and the motivation for its study may be obtained from [2, 3, 4, 8, 9, 10, 13]. as will be clear from these general references, it is shown that ditopological spaces provide a unified setting for the study of topology, bitopology and fuzzy topology on hutton algebras. some of the links with hutton spaces and fuzzy topologies are expressed in a categorical setting in [14]. in addition, there are close and deep relationships between the bitopological and ditopological spaces as shown in [11, 12] and [15, 16]. in this study, we will use those close relationships insofar as the inverse systems and their inverse limits are concerned in a categorical view. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 102 some categorical aspects of the inverse limits in ditopological context as it is stated before, in [2, 3, 4] we have a few methods, such as product space, subtexture space and quotient space, to derive a new ditopological space from two or more ditopological spaces just like classical case. recently, it is seen in [17, 18] that the another method used to construct a new ditopological space is the theory of ditopological inverse systems and their limit spaces under the name ditopological inverse limits as the subspaces of ditopological product spaces described in [3, 4, 18]. there are considerable difficulties involved in constructing a suitable theory of inverse systems for general ditopological spaces. hence, in [17] we confined our attention to a special category whose objects are plain textures, and the basic properties of inverse systems and their inverse limits are investigated in the first-time for texture theory in the context of that category. accordingly, the various aspects of the inverse systems limits for texture theory are investigated for plain case and placed them in a categorical functorial setting. later, in [18], the theory of inverse systems and inverse limits is handled first-time in the ditopological textural context and we gave a detailed analysis of the theory of ditopological inverse systems and inverse limits insofar as the category ifpditop whose objects are the ditopological texture spaces which have plain texturing and morphisms are the bicontinuous, w-preserving point functions, is concerned. (for a detailed information and some basic facts about the point-functions between texture spaces, see [3, 10, 11]). by the way, no attempt isn ’t made at the direct systems of ditopological spaces even plain ones, and their (direct) limits as the dual notions of inverse limits. returning to work at the moment, our main aim in the present paper is to give some further results on the theory of inverse systems and their inverse limits in the context of category ifpditop. especially, this paper will present some intriguing connections between the bitopological inverse systems limit spaces and their ditopological counterparts, in a categorical functorial setting. here we will continue to work within the same framework given in [17, 18] that are the major sources of the topic on which we study. according to that, frequent reference will be made to the author ’s papers [17] and [18] which present all details related to the subjects inverse system and inverse limit constructed in the textural context for the plain case, besides providing some useful historical information located in the literature about inverse systems. otherwise, this paper is largely self-contained although the reader may wish to refer to the literature cited in these papers, for motivation and additional background material specific to the main topic of this paper. especially, the significant reference in the general field of inverse system theory is [5] and in addition, the reader is referred to [6] for the information about the inverse systems consisting of topological spaces. specifically, the reader may consult [7] for terms from lattice theory not mentioned here. in addition, we follow the terminology of [1] for all the general c© agt, upv, 2018 appl. gen. topol. 19, no. 1 103 f. yıldız concepts relating to category theory. thus, if a is a category, ob a will denote the class of objects and mor a the class of morphisms of a. in this paper, generally we have tried to give enough details of the proofs to make it clear where various of the conditions imposed are needed, but at the same time to avoid boring the reader with routine verifications. accordingly, this paper consists of six sections and the layout of paper is as follows: after presenting some background information via the references mentioned in the first section, we introduce and study the category invifpditop in section 2, mainly. for the paper, it will denote the category whose objects are the inverse systems constructed by the objects of ifpditop and morphisms are the inverse systems of mappings in the sense of mappings defined between inverse systems. following that, by describing another related categories and the required functors between the corresponding categories which have some useful properties, we continued to discuss various aspects of the inverse systems and their limits in ifpditop. in addition, there is a close relationship between ditopological spaces restricted to plain textures and bitopological spaces, as exemplified by a special functor isomorphism given in that section. hence, we are interested in the connections between bitopological and ditopological inverse systems together with their limits, via that isomorphism. in the end of this section, as one of the principal aims of paper, we obtained an identity natural transformation constructed between the related appropriate functors, described via those connections just mentioned. specifically, this section contains some examples and other results that are important in their own right and which will also be needed later on. in a similar way, in section 3 we presented a few connections between the category of topological spaces and the category ifpditop insofar as the inverse systems and their inverse limits are concerned in a categorical setting. besides these, in section 4 we investigated the effect of closure operators on inverse systems and limits in ifpditop, with respect to the joint topologies correspond to the ditopologies located on those inverse systems and limits. a significant characterization theorem which says that by applying the inverse limit operation, any cartesian products of ditopological plain spaces which are the objects of ifpditop can be expressed in terms of the finite cartesian products of those spaces, is proved in section 5. following that, this section ends with two principal corollaries of that characterization. as the last part of paper, section 6 gives a conclusion about the whole of this study. 2. relationships between the inverse systems-limits in the categories of bitopological and ditopological spaces in this section, firstly, let ’s recall all the considerations presented in [12, section 2] as follows: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 104 some categorical aspects of the inverse limits in ditopological context let bitop be the category whose objects are bitopological spaces and morphisms are pairwise continuous functions, and the category ifpditop, introduced in [18], is known from the previous section. accordingly, consider the mapping u from ifpditop to bitop by u((s,s,τs,κs) ϕ −→ (t,t,τt ,κt )) = (s,τs,κ c s) ϕ −→ (t,τt ,κ c t ). it is trivial to verify that this is indeed a functor and we omit the details. when applied to many important ditopological spaces, such as the unit interval and real space, the corresponding ditopological t0 axiom as a separation axiom is described as qs 6⊆ qt =⇒ ∃c ∈ τ ∪ κ with ps 6⊆ c 6⊆ qt and it behaves more like the bitopological weak pairwise t0 axiom, x ∈ yu ∩ yv and y ∈ xu ∩ xv =⇒ x = y. why this is so, at least in the case of plain textures, we now see by setting up a new functor in the opposite direction of u. to define the suitable functor such that preserves t0 axiom, we restrict ourselves to weakly pairwise t0 bitopological spaces (x,u,v), and consider the smallest subset kuv of p(x) which contains u∪v c and is closed under arbitrary intersections and unions. clearly the elements of kuv have the form (2.1) a = ⋂ j∈j aj, where aj = uj ∪ ⋃ i∈ij {(v j i ) c | v j i ∈ v}, uj ∈ u, j ∈ j. in summary, for a weakly pairwise t0 bitopological space (x,u,v), the set u ∪ vc generates a texturing, denoted by kuv on x. moreover, it is easy to verify that kuv is a plain texturing of x since it separates points, by using the property “weakly pairwise t0” of the space (x,u,v). finally, we have the plain ditopological space (x,kuv,u,v c) ∈ ob ifpditop satisfying the ditopological t0 separation axiom. specifically, for a space (s,s,τ,κ) ∈ ob ifpditop the equality kτκc = s is known from [12, corollary 3.8]. with all these considerations, this process gives a mapping between the subcategory bitopw0 of bitop, consisting of weakly pairwise t0 bitopological spaces pairwise continuous functions and the subcategory ifpditop0 of ifpditop, consisting of t0 ditopological spaces and bicontinuous, w-preserving point functions, as follows: h((x,ux,vx) ϕ −→ (y,uy ,vy )) = (x,kuxvx ,ux,v c x) ϕ −→ (y,kuy vy ,uy ,v c y ) clearly, it defines a functor h : bitopw0 → ifpditop0 as mentioned in [9]. note that this concrete functor is a variant of the functor with the same name considered in [12, 15] in connection with real dicompactness. we are now in a position to give two examples denote the importance of the functor h. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 105 f. yıldız example 2.1. (1) the unit interval ditopological space (i,i,τi,κi) ∈ ob ifpditop0 is the image of the bitopological space (i,ui,vi) ∈ ob bitopw0 under h, where τi = ui = {[0,r) | r ∈ i} ∪ {i} and κ c i = vi = {(r,1] | r ∈ i} ∪ {i}. (2) the real ditopological space (r,r,τr,κr) ∈ ob ifpditop0 is the image of the bitopological space (r,ur,vr) ∈ ob bitopw0 under h, where τr = ur = {(−∞,r) | r ∈ r} ∪ {r,∅} and κ c r = vr = {(r,∞) | r ∈ r} ∪ {r,∅}. it may be verified that h preserves the other basic ditopological separation axioms, besides t0 axiom. consequently, we have the following fact from [9, 12]: theorem 2.2. h is a concrete isomorphism between the constructs bitopw0 and ifpditop0. remark 2.3. in view of the above statements, the equalities u ◦ h = 1bitopw0 and h ◦ u = 1ifpditop0 are trivial for the functor u : ifpditop0 → bitopw0 defined as above. hence, u is the inverse of h as an isomorphism functor. incidentally, it is concrete isomorphism since u is identity carried, as well. now, we can turn our attention to the inverse systems and their inverse limits constructed in ifpditop, in the light of [18]. before everything, note that: remark 2.4. the inverse systems constructed by the objects and morphisms of the category ifpditop, which are the bonding maps satisfying some conditions given in [18, definition 3.1], have an inverse limit space described as in [18, definition 4.1], since ifpditop has products and equalizers as stated in [18, corollary 2.6]. also, the uniqueness of the limit space in the category ifpditop was mentioned just before [18, examples 4.5]. hence, the operation lim ← will be meaningful for the inverse systems given in the context of that category. notation: according to the major theorem given as [18, theorem 4.6], if take the inverse system {(sα,sα,τα,κα),ϕαβ}α≥β constructed in ifpditop, over a directed set λ, then the notations (τ∞,κ∞) and (s∞,s∞,τ∞,κ∞) will be used as inverse limit ditopology and (ditopological) inverse limit space, respectively, where s∞ = lim ← {sα}, in the remainder of paper. according to let ’s take a glimpse of the mappings between inverse systems: consider two inverse systems a = {(sα,sα,τα,κα),ϕαβ}α≥β and b = {(tα,tα,τ ′ α,κ ′ α),ψαβ}α≥β over λ described in ifpditop, as in [18, definition 3.1]. take into consideration [17, definition 3.9] which introduces the notion inverse system of mappings or mapping of inverse systems denoted by {tα} : a → b, consisting of the components tα ∈ mor ifpditop, satisfying the c© agt, upv, 2018 appl. gen. topol. 19, no. 1 106 some categorical aspects of the inverse limits in ditopological context equality ψβα ◦ tβ = tα ◦ ϕβα, that is, the commutativity of diagram sβ tβ // ϕβα �� tβ ψβα �� sα tα // tα which associates the bonding maps with the components tα. hence, by recalling the notion inverse limit space with the notation s∞ defined as in [18, definition 4.1] and the map t∞ = lim ← {tα}α∈λ defined in [17, theorem 4.14], called inverse limit map of the inverse system {tα} of mappings, now let ’s focus on the following crucial theorem proved in [18, theorem 4.24]: theorem 2.5. let {tα} : {(sα,sα,τα,κα),ϕβα}β≥α → {(tα,tα,τ ′ α,κ ′ α),ψβα}β≥α be an inverse system of mappings in ifpditop, over a directed set λ. then there exists a unique map t∞ ∈ mor ifpditop between the spaces (s∞,s∞,τ∞,κ∞) and (t∞,t∞,τ ′ ∞,κ ′ ∞) having the property that for each α ∈ λ ,the diagram s∞ t∞ // µα �� t∞ ηα �� sα tα // tα is commutative, that is tα ◦ µα = ηα ◦ t∞. in this case, i) if each tα is an ifpditop-isomorphism, t∞ is an ifpditop-isomorphism. ii) if each tα ◦ µα is surjective, t∞(s∞) is jointly dense in t∞. notations: in this study, invc denotes the category whose objects are the inverse systems constructed by the objects of category c and morphisms are the mappings of inverse systems, described as just before theorem 2.5, namely, the inverse systems of c-morphisms defined between the objects of c. particulary, the following notation will be required for the remainder of paper, mostly: invifpditop0 will denote the category consisting of inverse systems constructed by t0 ditopological plain texture spaces as objects of ifpditop0, and by the mappings between inverse systems, namely, the inverse systems of mappings defined as in theorem 2.5. incidentally, we have the following categorical fact about the inverse systems due to [18, remark 3.2]: remark 2.6. an inverse system in any category admits an alternative description in terms of functors. a directed set λ becomes a category if each relation α ≤ β is regarded as a map α → β, that is the morphisms consist of arrows α → β if and only if α ≤ β. then, c© agt, upv, 2018 appl. gen. topol. 19, no. 1 107 f. yıldız any inverse system in the category ifpditop over the directed set λ is actually a contravariant functor from λ to ifpditop. in the light of remark 2.6, note that the objects and morphisms of invifpditop may be regarded as the functors and natural transformations, respectively. example 2.7. if {(sα,uα,vα),fαβ}α≥β ∈ ob invbitopw0 then the system {(sα,kuαvα,uα,v c α),ϕαβ}α≥β consisting of the spaces h(sα,uα,vα) = (sα,kuαvα,uα,v c α) ∈ ob ifpditop0, corresponding to the bitopological spaces (sα,uα,vα) ∈ ob bitopw0, describes an inverse system via the isomorphism functor h given in theorem 2.2 and all the above considerations. trivially, this system is an object of invifpditop0. now, by taking into account example 2.7, immediately we have the following: example 2.8. if (s∞,u∞,v∞) ∈ ob bitopw0 is the inverse limit of the inverse system {(sα,uα,vα),fαβ}α≥β ∈ ob invbitopw0 then the corresponding plain space (s∞,ku∞v∞,u∞,v c ∞) ∈ ob ifpditop0 is the inverse limit of corresponding inverse system {(sα,kuαvα,uα,v c α),ϕαβ}α≥β ∈ ob invifpditop0, where ϕαβ = fαβ for α ≥ β. let ’s prove it: first of all, recall the fact s∞ ⊆ ∏ α sα. thus, it is clear that u∞ = ( ∏ α uα)|s∞ = ( ⊗ α uα)|s∞ since the textural and classical products of topologies are coincide by the plainness property. on the other hand, similar to the explanations given in [15, section 3] we have vc∞ = ( ⊗ α vcα)|s∞ since ( ⊗ α vα)|s∞ = ( ∏ α vα)|s∞ = v∞ and by [3, lemma 2.7] which is peculiar to the theory of product ditopologies. hence, it remains to prove the equality ku∞v∞ = ( ⊗ α kuαvα)|s∞. for it, we can show that the types of elements of these two families are absolutely the same: if a ∈ ku∞v∞ then let ’s recall the form of a as follows: a = ⋂ j∈j aj, where aj = uj ∪ ⋃ i∈ij {(v j i ) c | v j i ∈ v∞}, uj ∈ u∞, j ∈ j here, v j i ∈ v∞ = ( ∏ α vα)|s∞ and so v j i = c j i ∩ s∞, where c j i ∈ ∏ α vα. in this case, for tjαi ∈ vαi c j i = ⋃ ⋂ π−1αi [t j αi ] and so (v j i ) c = ⋂ ⋃ (π−1αi [sαi \ t j αi ]). similarly, uj = bj ∩ s∞ where bj ∈ ⊗ α uα, and so uj = ( ⋃ ⋂ π−1αi [g j αi ]) ∩ s∞ where g j αi ∈ uαi, by the definition of product topology. hence, aj = ( ⋃ ⋂ (παi|s∞) −1[gjαi])∪( ⋃ ⋂ (παi|s∞) −1[sαi \t j αi ]) and finally, by the fact that a = ⋂ j∈j aj we have a = ⋂ j∈j[( ⋃ ⋂ (παi|s∞) −1[gjαi])∪( ⋃ ⋂ (παi|s∞) −1[sαi \ tjαi])]. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 108 some categorical aspects of the inverse limits in ditopological context on the other hand, if b ∈ ( ⊗ α kuαvα)|s∞ then b = m ∩ s∞ where m ∈ ⊗ α∈i kuαvα. in this case, m = ⋂ ⋃ α∈i π−1α [kα], where kα ∈ kuαvα. thus, we have the form kα = ⋂ j∈j dαj , where d α j = w α j ∪ ⋃ i∈ij {sα \ (z j i ) α | (z j i ) α ∈ vα}, wj α ∈ uα, j ∈ j. hence kα = ⋂ (uj α ∪ ( ⋃ sα \ (z j i ) α)), and so m = ⋂ ( ⋃ (π−1j [ ⋂ (wαj ∪ ⋃ sα \ (z j i ) α])). in this case, with b = m ∩s∞ we have b = ⋂ [ ⋃ (((πj|s∞) −1[ ⋂ wαj ]∪(πj|s∞) −1[ ⋂ ⋃ (sα\(z j i ) α)])] = ⋂ [ ⋃ ⋂ (πj|s∞) −1[wαj ] ∪ ⋂ ⋃ (πj|s∞) −1[sα \ (z j i ) α]]. consequently, it is easy to check that the sets a and b have the same type if consider gjαi as w α j and t j αi as z j i by neglecting the details of indices, as well as by leaving the other details of required equality to the interested reader. now, let ’s recall the notion of inverse limit map introduced in [17, theorem 4.14] as a notion of peculiar to the texture theory, as well as mentioned in section 1. accordingly, in order to prove the next theorem, we need a special property of inverse limits maps, which is proved in the following: proposition 2.9. consider {hα} : {(sα,sα),ϕβα}α≤β → {(tα,tα),ψβα}α≤β and {gα} : {(tα,tα),ψβα}α≤β → {(zα,zα),φβα}α≤β between the inverse systems of textures then {gα ◦ hα} : {(sα,sα),ϕβα}α≤β → {(zα,zα),φβα}α≤β is also a mapping of inverse system and lim ← {gα ◦ hα}α∈λ = lim ← {gα}α∈λ ◦ lim ← {hα}α∈λ proof. at first, we define the composition operation for the mappings of inverse systems as follows : {gα} ◦ {hα} = {gα ◦ hα} by using the composition operation on the morphisms of ifpditop. on the other hand, because of the first inverse system, we have the equality ψβα ◦ hβ = haα ◦ ϕβα by the commutativity of related diagram constructed between the sets sα,tα,s∞ and t∞. similarly, from the second inverse system, we have the equality φβα ◦ gβ = gaα ◦ ψβα by the commutativity of related diagram constructed between the sets tα,zα,t∞ and z∞. hence, by considering the above two equalities, we have the result: φβα ◦ (gβ ◦ hβ) = (gα ◦ hα) ◦ ϕβα in fact, it says that {gα ◦ hα} becomes an inverse system of mappings by [17, definition 3.9]. therefore, now we can look at the commutativity of diagram. firstly, recall µα ◦ h∞ = hα ◦ λα and ηα ◦ g∞ = gα ◦ µα by [17, theorem 4.14]. thus, due to these equalities, we have ηα◦(g∞◦h∞) = (ηα◦g∞)◦h∞ = (gα◦µα)◦h∞ = gα◦(µα◦h∞) = (gα◦hα)◦λα and so the related diagram is commutative. finally, from the uniqueness of inverse limit maps, mentioned in theorem 2.5, the required result lim ← {gα ◦ c© agt, upv, 2018 appl. gen. topol. 19, no. 1 109 f. yıldız hα}α∈λ = g∞ ◦ h∞ is proved. that is, lim ← {gα ◦ hα}α∈λ = lim ← {gα}α∈λ ◦ lim ← {hα}α∈λ. � remark 2.10. for the remainder of paper, we will use the above final equality under the name transitivity property of inverse limit maps. from remark 2.4, the inverse systems which are the objects of invifpditop have a unique inverse limit space as an object of ifpditop. with the reference to this fact, we have the following immediately; theorem 2.11. the limit operation lim ← of assigning an inverse limit in ifpditop to each object in invifpditop and an inverse limit map t∞ ∈ mor ifpditop to each inverse system {tα}α ∈ mor invifpditop of maps tα ∈ mor ifpditop, forms the covariant functor lim ← : invifpditop → ifpditop. proof. let ’s recall that for each inverse system which is an object of invifpditop0 we can obtain an inverse limit space in ifpditop and moreover, it is unique by remark 2.4. now, according to theorem 2.5, if take the morphism {tα}α : {(sα,sα,τα,κα),ϕβα}β≥α → {(tα,tα,τ ′ α,κ ′ α),ψβα}β≥α in invifpditop then there exists a unique map t∞ = lim ← {tα}α∈λ ∈ mor ifpditop between the corresponding inverse limit spaces (s∞,s∞,τ∞,κ∞) and (t∞,t∞,τ ′ ∞,κ ′ ∞) which are the objects of ifpditop, having the property that for each α ∈ λ the diagram s∞ t∞ // µα �� t∞ ηα �� sα tα // tα is commutative, that is tα◦µα = ηα◦t∞. also, t∞ is the identity id(s∞,s∞,τ∞,κ∞) if suppose that the mapping {tα}α of inverse systems is identity, that is each map tα : sα → tα, α ∈ λ is the identity id(sα,sα,τα,κα) on sα. additionally, as it is stated in proposition 2.9, the inverse limit maps have the transitivity property and so the limit operation lim ← satisfies the composition rule lim ← {tα ◦hα} = lim ← {tα} ◦ lim ← {hα}. hence, the mapping lim ← : invifpditop → ifpditop is a covariant functor. � notation: the covariant functor lim ← described in theorem 2.11, as the limit operation in the context of ifpditop, will be used under the notation e for the remainder of paper. actually, note that we can always define a covariant functor between the categories c and invc, for any category c which has the equalizers and products. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 110 some categorical aspects of the inverse limits in ditopological context remark 2.12. (1) by virtue of the fact that any inverse system consisting of the objects of bitop has an inverse limit since bitop has equalizers and products, we can describe covariant functor, under the name b between the categories bitop and invbitop. (2) the above functor b introduced in (1) may be considered as the restricted mapping between the full subcategory bitopw0 of bitop and the full subcategory invbitopw0 of invbitop. obviously, that restriction is a covariant functor, as well. (3) furthermore, if we recall that the categories bitopw0 and ifpditop0 are isomorphic via the functor h constructed by using the fact that weakly pairwise t0 bitopology generates the smallest plain texturing and t0 ditopology, as mentioned in theorem 2.2, then we may describe a functor between the categories invbitopw0 and invifpditop0 in a natural way. according to the statement (3), we are now in a position to give a next isomorphism functor as follows: theorem 2.13. the categories invbitopw0 and invifpditop0 are concretely isomorphic. proof. first of all, if consider the isomorphism functor h given in theorem 2.2, between the categories bitopw0 and ifpditop0, clearly the mapping x : invbitop w0 → invifpditop0 may be defined by using h: taking into account the ideas given in example 2.7, then we may define the map x({(sα,uα,vα),fαβ}α≥β) = {(sα,kuαvα,uα,v c α),fαβ}α≥β where h(sα,uα,vα) = (sα,kuαvα,uα,v c α), h(fαβ) = fαβ, and if take the inverse system {tα} of mappings as the morphism between two inverse systems which are objects of invbitopw0 then it is easy to show that it is also a morphism in invbitop. indeed, if take tα ∈ mor bitop, for each α, that is, tα is pairwise continuous then it is w-preserving and bicontinuous between the corresponding ditopological plain spaces and finally, the equality x({tα}) = {tα} is meaningful, as well. in this case, for the inverse system mappings, the equality x({tα} ◦ {hα}) = x({tα ◦ hα}) = {tα ◦ hα} = {tα} ◦ {hα} is trivial. also, from x(id{(sα,uα,vα),fαβ}α≥β ) = idx({(sα,uα,vα),fαβ}α≥β), the map x describes a functor, naturally. now we will turn our attention to the isomorphism conditions for x. it is easy to show that x is full and faithful, since it is bijective between hom-set restrictions by the fact that the functor h given in theorem 2.2 is full and faithful. as the final step, it remains to prove that the bijectivity of x on objects of invbitop w0 and invifpditop0 , and it is clear from the bijectivity of the functor h. � in the light of considerations presented in remark 2.12 and theorem 2.13, now we can start to construct a major part in that theory, consisting of the c© agt, upv, 2018 appl. gen. topol. 19, no. 1 111 f. yıldız useful implications and an identity natural transformation which arises from those implications: a natural transformation in the context of inverse systems and limits located inside the categories bitopw0 and ifpditop0 : as we promised in section 1, firstly a natural transformation will be described between the corresponding functors, and later, that the natural transformation is identity will be proved, thoroughly. let ’s start by recalling the corresponding required functors as follows: invbitop w0 b −→ bitopw0 h −→ ifpditop0 {(xα,uα,vα),ϕαβ}α≥β 7→ (x∞,u∞,v∞) 7→ (x∞,ku∞v∞,u∞,(v∞) c) invbitop w0 x −→ invifpditop 0 e −→ ifpditop0 {(xα,uα,vα),ϕαβ}α≥β 7→ {(xα,kuαvα,uα,(vα) c),ϕαβ}α≥β 7→ (x∞,z,t,k) where z = ( ⊗ α kuαvα)|x∞, t = ( ⊗ α uα)|x∞ and k = ( ⊗ α vcα)|x∞ now, with the previous considerations, if take the equalities f = h ◦ b : invbitop w0 → ifpditop0 g = e ◦ x : invbitop w0 → ifpditop0 then it is clear that f and g are functors as compositions of the functors h,b and e,x, respectively. consider a mapping τ : f → g. in particular; theorem 2.14. τ is an identity natural transformation between the functors f and g. proof. let the inverse system a = {(xα,uα,vα),ϕαβ}α≥β ∈ ob invbitop w0 over λ and the mapping τa : fa → ga. firstly, it is easy to verify that fa = ga by the considerations mentioned in example 2.8 and thus, the mapping τa is an ifpditop0-identity morphism. on the other hand, for the inverse system a′ = {(x′α,u ′ α,v ′ α),ϕ ′ αβ}α≥β ∈ ob invbitop w0 over λ, take the inverse system {kα} : a → a ′ ∈ mor invbitop w0 of mappings kα : xα → x ′ α, α ∈ λ, as in described in theorem 2.5. also, assume that lim ← a = lim ← {xα}α∈λ = x∞ and lim ← a′ = lim ← {x′α}α∈λ = x ′ ∞. let ξ : {(xα,uα,vα),ϕαβ}α≥β → {(x ′ α,u ′ α,v ′ α),ϕ ′ αβ}α≥β be the mapping {kα}α∈λ of inverse systems, with the components kα : xα → x ′ α ∈ mor bitopw0, α ∈ λ. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 112 some categorical aspects of the inverse limits in ditopological context with all the above notations, now we may construct the following diagram: (x′∞,ku′∞v′∞,u ′ ∞,(v ′ ∞) c) τa′ // (x′∞,( ⊗ α ku′αv ′ α )|x′∞,( ⊗ α u′α)|x′∞,( ⊗ α (v′α) c)|x′∞) (x∞,ku∞v∞,u∞,(v∞) c) fξ oo τa // (x∞,( ⊗ α kuαvα)|x∞,( ⊗ α uα)|x∞,( ⊗ α (vα) c)|x∞ gξ oo )) in order to see that this diagram is commutative, we need to show the equality fξ = gξ for all ξ ∈ mor invbitopw0 : clearly, each kα : xα → x ′ α is pairwise continuous and by f = h ◦ b we have f({kα}α∈λ) = h(b{kα}α∈λ) = h(k∞) where k∞ = lim ← {kα}α∈λ ∈ mor bitopw0 and by applying the isomorphism h : bitopw0 → ifpditop0 to the limit map k∞ ∈ mor invbitopw0, we obtained h(k∞) = k∞ since h is identity on morphisms. finally, fξ = f({kα}α∈λ) = k∞. on the other hand, now let ’s turn our attention to g(ξ) and recall the equality g = e◦x. according to that, we have g({kα}α∈λ) = e(x{kα}α∈λ) = e({kα}α∈λ) since the isomorphism x described in theorem 2.13 is the identity on morphisms of invbitopw0 and invifpditop0. hence, by applying the functor e : invifpditop0 → ifpditop0 to the mapping {kα}α∈λ, we describe the map e({kα}) = h∞, where h∞ = lim ← {kα}α∈λ ∈ mor bitopw0. hence gξ = g({kα}α∈λ) = h∞. now, let ’s see that t∞ = h∞: the inverse systems considered above are exactly same since the spaces and bonding maps are the same. also, the property of commutativity ηα ◦t∞ = tα ◦µα, α ∈ λ is satisfied for the map h∞, as well. in this case, by virtue of the fact that the inverse limit of the mappings of inverse systems is unique by [17, theorem 4.14], we have fξ = gξ. thus, the equality f = g is verified and τ is identity. moreover, we have gξ◦τa = τa′◦fξ since τa,τa′ are identities and so, the diagram is commutative. � as a result of the above considerations, τ : f → g is the identity natural transformation. in a similar way to the considerations given in section 2, next section will discuss the relations between the topological inverse systems limits and ditopological inverse systems limits insofar as the theory of plain textures are concerned. 3. relationships between the inverse systems-limits in the categories of topological and ditopological spaces now we will show that we may associate with the ditopology (τ,κ) on a plain texture (s,s) a topology jτκ on s, by adapting the notion of appropriate joint topology for a ditopology described in [11], to the plain case: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 113 f. yıldız definition 3.1. let (s,s,τ,κ) ∈ ob ifpditop. we define the joint topology on s in terms of its family jcτκ of closed sets by the condition w ∈ jcτκ ⇐⇒ (s ∈ s, g ∈ η(s),k ∈ µ(s) =⇒ g ∩ w 6⊆ k) =⇒ s ∈ w. here η(s) = {n ∈ s | ps ⊆ g ⊆ n 6⊆ qs for some g ∈ τ} and µ(s) = {m ∈ s | ps 6⊆ m ⊆ k ⊆ qs for some k ∈ κ}. for the details about filter η(s) and cofilter µ(s) for s ∈ s, see [8, 11, 16]. the verification of that jcτκ satisfies the closed-set axioms is straightforward and on passing to the complement this reveals that (i) {g ⊆ s | g ∈ τ} ∪ {s \ k ⊆ s | k ∈ κ} is a subbase, and (ii) {g ∩ (s \ k) ⊆ s | g ∈ τ,k ∈ κ} a base of open sets for the topology jτκ on s. in case (x,u,v) is an object of bitop, we have the space (x,p(x),u,vc) ∈ ob ifpditop, and clearly obtain jτκ = u ∨ v as the joint topology of (u,v), where τ = u and κ = vc. hence we will refer to jτκ as the joint topology of (τ,κ) on s. remark 3.2. (1) for (s,s,τ,κ) ∈ ob ifpditop, it is trivial to see that κ ⊆ jcτκ and τ ⊆ jτκ. in addition, the family τ ∪ κ c is the subbase for the joint topology jτκ. (2) from now on, in this work we will use the terms jointly closed (open, dense) for the set which is closed (open, dense) with respect to the appropriate joint topology of the ditopology on space. note that the following statements are adapted forms of general cases given in [11] to the category ifpditop. here top will denote the category of topological spaces and continuous functions. theorem 3.3. the mapping j : ifpditop → top defined by j : ((s,s,τs,κs) ϕ −→ (t,t,τt ,κt )) = (s,jτsκs ) ϕ −→ (t,jτt κt ) is an adjoint functor. it is clear that j is full, faithful and isomorphism-dense functor although it is not a functor isomorphism since it is not one-to-one on the objects. corollary 3.4. the functor t : top → ifpditop given by t(x,t) = (x,p(x),t,tc), t(ϕ) = ϕ is the co-adjoint of j. here note also that t is not a functor isomorphism. in this section, we will be interested in the category invtop whose objects are the inverse systems constructed by the objects of top and morphisms are the inverse systems constructed by the morphisms of top, as well as the mappings between the inverse systems constructed in top. naturally, a covariant functor may be established between the categories top and invtop since any c© agt, upv, 2018 appl. gen. topol. 19, no. 1 114 some categorical aspects of the inverse limits in ditopological context inverse system constructed in top has an inverse limit by the fact that top has equalizers and products as mentioned in [5]. obviously, we can’t expect to find an isomorphism between the categories invtop and invifpditop and now, we may turn our attention to the relationships between the objects of categories invtop and invifpditop: it is known that an object of invifpditop can be obtained as the natural counterpart of an object of invtop by [18, example 3.4]. thus, by applying the similar considerations to corollary 3.4 we can describe a co-adjoint functor from invtop to invifpditop. conversely, in order to construct an opposite functor from invifpditop to invtop, let’s consider the reciprocal objects, and the adjoint functor j firstly. that is, take {(sα,sα,τα,κα),ϕαβ}α≥β ∈ ob invifpditop, and construct the image j(sα,sα,τα,κα) = (sα,jτακα) ∈ ob top. in this case, for the bonding map ϕαβ : sα → sβ ∈ mor ifpditop, we have j(ϕαβ) = ϕαβ : (sα,jτακα) → (sβ,jτβκβ ) as a morphism of top since j is a functor. in fact, j is the identity on morphisms. hence, we construct the inverse system {(sα,jτακα),ϕβα}β≥α ∈ ob invtop and so a mapping which is described as follows : theorem 3.5. the mapping jinv : invifpditop → invtop defined by jinv : ({(sα,sα,τα,κα),ϕαβ}α≥β {tα} −→ {(tα,tα,τ ′ α,κ ′ α),ψαβ}α≥β) = {(sα,jτακα),ϕαβ}α≥β {tα} −→ {(tα,jτ′ α κ′ α ),ψαβ}α≥β is an adjoint functor. proof. firstly, we need to check that jinv is a functor. assume that {tα}α ∈ mor invifpditop. in this case, the maps tα : sα → tα for each α, are bicontinuous and w-preserving as the morphisms in ifpditop. by the definition of joint topology, it is easy to show that tα is continuous for each α, as the morphism of top between the joint topological spaces (sα,jτακα) and (tα,j ′ τ′ακ ′ α ). to show jinv is an adjoint, now take {(xα,tα),φαβ}α≥β ∈ ob invtop. then ({idxα},{(xα,p(xα),tα,t c α),φαβ}α≥β) is a jinv-structured arrow by jinv({(xα,p(xα),tα,t c α),φαβ}α≥β) = {(xα,jtαtcα),φαβ}α≥β) and by the fact that {idxα} : {(xα,jtακα),φαβ}α≥β → {(xα,jtακα),φαβ}α≥β is an invtopmorphism. to show ({idxα},{(xα,p(xα),tα,t c α),φαβ}α≥β) has the universal property, take {(sα,sα,τ ∗ α,κ ∗ α),θαβ}α≥β ∈ ob invifpditop and let {ϕα} : {(xα,tα),φαβ}α≥β → jinv{(sα,sα,τ ∗ α,κ ∗ α),θαβ}α≥β = {(sα,jτ∗ακ∗α),θαβ}α≥β be an invtop-morphism. (x,t) idx // ϕ **❚❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ jinv(x,p(x),t,t c) = (x,t) jinv(ϕ̄) �� ✤ ✤ ✤ ✤ jinv(s,s,τ,κ) = (s,jτκ) since ϕ maps into s the only point function ϕ̄ : x → s making the above diagram commutative is ϕ, so it remains only to verify that ϕ : (x,p(x),t,tc) → c© agt, upv, 2018 appl. gen. topol. 19, no. 1 115 f. yıldız (s,s,τ,κ) is a morphism in ifpditop. certainly ϕ is ω-preserving, due to the fact that ϕ(x)ωs ϕ(x) for all x ∈ x. moreover, ϕ is bicontinuous. to see this, note that we have ϕ←a = ϕ−1a = ϕ−1(a ∩ sp) for all a ∈ s. hence, g ∈ τ =⇒ g ∩ sp ∈ jτκ =⇒ ϕ ←g = ϕ−1(g ∩ sp) ∈ t, and k ∈ κ =⇒ ϕ←k ∈ tc likewise. � particularly, by virtue of theorem 3.3 and theorem 3.5 we have the following: remark 3.6. let ’s take an inverse system a = {(sα,sα,τα,κα),ϕαβ}α≥β ∈ ob invifpditop. in this case, we construct the system jinv(a) = {(sα,jτακα),ϕαβ}α≥β ∈ ob invtop, and have the inverse limit space (s∞,s∞,τ∞,κ∞) ∈ ob ifpditop by theorem 2.11. thus, j(s∞,s∞,τ∞,κ∞) = (s∞,jτ∞κ∞) ∈ ob top. in addition, we have an inverse limit lim ← jinv(a) = (y,v) ∈ ob top due to the fact that top has equalizers and products as mentioned in [5]. now let ’s turn our attention to the main question; is the space (y,v) same with the space (s∞,jτ∞κ∞) in top ? firstly note that the systems {(sα,sα,τα,κα),ϕαβ}α≥β and {(sα,jτακα), ϕαβ}α≥β have the same bonding maps, so y = s∞ as a subset of ∏ α sα, trivially. as a next step, in order to prove the equality jτ∞κ∞ = v, note that the facts ( ∏ α jτακα)|s∞ = v, ( ⊗ α sα)|s∞ = s∞, ( ⊗ α τα)|s∞ = τ∞ and ( ⊗ α κα)|s∞ = κ∞. accordingly, let a ∈ jτ∞κ∞, so a = ⋃ δ (gδ ∩(s∞\kδ)) where gδ ∈ ( ⊗ α τα)|s∞, kδ ∈ ( ⊗ α κα)|s∞ for each δ. note here that gδ = ( ⋃ ( ⋂ finite π−1αi [g δ αi ]))∩s∞ and similarly, kδ = ( ⋂ ( ⋃ finite π−1αi [k δ αi ])) ∩ s∞, where g δ αi ∈ ταi and k δ αi ∈ καi. hence, a = ( ⋃ ⋂ finite π−1αi [ ⋃ gδαi]) ∩ ( ⋃ ⋂ finite π−1αi [ ⋃ (sαi \ k δ αi )]) ∩ s∞. on the other hand, let the set b ∈ ( ∏ α jτακα)|s∞ = v, where v denotes the product topology on s∞. in this case, b can be written as ( ⋃ ⋂ finite π−1αj [gαj ])∩ s∞, gαj ∈ jτακα. thus, b = ( ⋃ ⋂ finite π−1αj [ ⋃ (cαj ∩ (sαj \ dαj ))]) ∩ s∞ where cαj ∈ ταj , dαj ∈ καj and so, b = ( ⋃ ⋂ finite π−1αj [ ⋃ cαj ]) ∩ ( ⋃ ⋂ finite π−1αj [ ⋃ (sαj \ dαj )]) ∩ s∞. consequently, it is easy to check that the types of sets a and b are the same. it means that the topologies v and jτ∞κ∞ coincides. 4. effect of the closure operator on inverse systems and limits in the category ifpditop by recalling the notion of appropriate joint topology described for a ditopology, as presented in the previous section, we have the following significant theorem, immediately: c© agt, upv, 2018 appl. gen. topol. 19, no. 1 116 some categorical aspects of the inverse limits in ditopological context theorem 4.1. let λ be a directed set. for subspace (u,su,τu,κu) ∈ ob ifpditop of the inverse limit space (s∞,s∞,τ∞,κ∞) ∈ ob ifpditop of the inverse system {(sα,sα,τα,κα),ϕαβ}α≥β ∈ ob invifpditop, the families {(uα,sα|uα,τα|uα,κα|uα),ϕαβ|uα }α≥β and {(uα,sα|uα,τα|uα,κα|uα),ϕαβ}α≥β describe two objects in invifpditop as the inverse systems, where uα = µα(u) = πα|s∞(u), ϕαβ = ϕαβ|uα and uα denotes the closure in sα of the subset uα ⊆ sα with respect to the joint topology of the ditopology (τα,κα), α ∈ λ. proof. firstly, let us prove that {(uα,sα|uα,τα|uα,κα|uα),ϕαβ}α≥β is an object of invifpditop: note that we have µβ(s) = ϕαβ(µα(s)) for s ∈ u and β ≤ α. indeed, if s ∈ u then µα(s) ∈ µα(u) and so µα(s) ∈ uα. in this case, ϕαβ(µα(s)) = ϕαβ(µα(s)). also the equality ϕαβ(µα(s)) = µβ(s) for α ≥ β is known by [17, lemma 4.3], thus we have ϕαβ(µα(s)) = µβ(s) for s ∈ u and α ≥ β, as required. on the other hand, with the continuity of bonding map ϕαβ we have ϕαβ(uα) = ϕαβ(µα(u)) ⊆ ϕαβ(µα(u)) = µβ(u) = uβ and then, it is clear that the point function ϕαβ is defined from uα onto uβ. following that, ϕαβ is a morphism of ifpditop since it is a restriction of ϕαβ ∈ mor ifpditop to the subset uα ⊆ sα. incidentally, the equality ϕβγ ◦ ϕαβ = ϕαγ may be easily proved for the elements of uα via the equality ϕβγ ◦ ϕαβ = ϕαγ. as a next step, we have the equality ϕαα(s) = ϕαα(s) = s for s ∈ uα, as ϕαα is the identity idsα on sα. that is, ϕαα = iduα = idsα|uα. consequently, the family {(uα,sα|uα,τα|uα,κα|uα),ϕαβ}α≥β forms an object in invifpditop. furthermore, in a similar way to the above proof, it is easy to check that the family {(uα,sα|uα,τα|uα,κα|uα),ϕαβ|uα }α≥β describes an inverse system in ifpditop, and so an object in invifpditop. � according to remark 2.4, we have the following, right away. proposition 4.2. u∞ = lim ← {uα} ⊆ lim ← {sα} = s∞ proof. conversely, assume that u∞ = lim ← {uα} 6⊆ lim ← {sα} = s∞, so there exists s = {sα} ∈ ∏ α∈λ sα such that u∞ 6⊆ qs and ps 6⊆ s∞. in this case, s ∈ ∏ α∈λ uα and ϕαβ|uα(sα) = sβ for every sα ∈ uα, α,β ∈ λ such that α ≥ β. moreover, we have the equality ϕαβ|uα(sα) = ϕαβ(sα) for sα ∈ uα. thus, because of the facts sα ∈ sα, α ∈ λ and ϕαβ(sα) = sβ for α ≥ β, the point s = {sα} becomes an element of s∞, obviously and this gives a contradiction. � proposition 4.3. let {(sα,sα,τα,κα),ϕαβ}α≥β ∈ ob invifpditop be an inverse system over a directed set λ and (s∞,s∞,τ∞,κ∞) ∈ ob ifpditop be the c© agt, upv, 2018 appl. gen. topol. 19, no. 1 117 f. yıldız inverse limit of that system. if uα ∈ j c τακα , α ∈ λ and lim ← {uα} = u∞ for the inverse subsystem {(uα,sα|uα,τα|uα,κα|uα),ϕαβ|uα}α≥β ∈ ob invifpditop, then u∞ ∈ j c τ∞κ∞ . proof. by the definition of inverse limit and the equality ϕαβ|uα(sα) = ϕαβ(sα) for sα ∈ uα, α ≥ β, the inclusion u∞ = lim ← {uα} ⊆ lim ← {sα} = s∞ is immediate, as mentioned in proposition 4.2 as well. now, let us prove u∞ ∈ j c τ∞κ∞ : if pa 6⊆ u∞, that is a /∈ u∞ for a = {aα} ∈ s∞, then a /∈ ∏ α∈λ uα due to the equality ϕαβ|uα(sα) = ϕαβ(sα) for sα ∈ uα, α ≥ β. in this case, there exists α0 ∈ λ such that aα0 /∈ uα0, that is paα0 6⊆ uα0. additionally, the subset µ −1 α0 [uα0] ⊆ s∞ is an element of j c τ∞κ∞ since the limiting projection map µα0 : s∞ → sα0 is continuous between the corresponding joint topological spaces and uα0 ∈ j c τα0κα0 . on the other hand, the statements pa 6⊆ µ −1 α0 [uα0] and u∞ ⊆ µ −1 α0 [uα0] may be showed as follows: conversely, if pa ⊆ µ −1 α0 [uα0] then we have µα0(a) = aα0 ∈ uα0 which is a contradiction. also, assume that u∞ 6⊆ µ −1 α0 [uα0]. thus there exists a point z ∈ s∞ such that u∞ 6⊆ qz and pz 6⊆ µ −1 α0 [uα0]. hence, µα0(z) = zα0 /∈ uα0 and z = {zα} /∈ ∏ α∈λ uα gives the fact that z /∈ u∞ which is a contradiction. � from now on, in the remainder of this section we will use all of the above notations, in exactly the same form. by virtue of theorem 4.1 and the last proposition, now we have the next: theorem 4.4. if u denotes the closure of the subset u ⊆ s∞ with respect to the joint topology of the limit ditopology (τ∞,κ∞) then (1) lim ← {uα} is jointly closed subspace of s∞ (2) lim ← {uα} = u ⊆ s (3) u = ⋂ α∈λ µ−1α [uα] proof. (1) before everything, let ’s see that lim ← {uα} ⊆ s∞, where s∞ = lim ← {sα}: conversely, if the inclusion is not true, then there exists a point s = {sα} ∈∏ α∈λ sα such that lim ← {uα} 6⊆ qs and ps 6⊆ s∞. hence, by the facts uσ 6⊆ qsσ and sσ ∈ uσ for every σ ∈ λ, we have ϕαβ(sα) = sβ for α ≥ β. on the other hand, it is easy to see that psσ ⊆ sσ since the set uσ is a subset of sσ for every σ ∈ λ, and so ps = ∏ σ∈λ psσ ⊆ ∏ σ∈λ sσ. also, if recall the equality ϕαβ(sα) = ϕαβ(sα) for sα ∈ uα and α ≥ β, then we have ϕαβ(sα) = sβ due to c© agt, upv, 2018 appl. gen. topol. 19, no. 1 118 some categorical aspects of the inverse limits in ditopological context the fact that ϕαβ(sα) = sβ for sα ∈ uα and α ≥ β. thus, by the definition of inverse limit, s = {sα} ∈ s∞ and it is a contradiction. accordingly, now let us show that lim ← {uα} is a jointly closed subspace of s∞: take a point s = {sα} ∈ s∞ such that s /∈ lim ← {uα}. in this case, because of the fact that s /∈ ∏ α∈λ uα there exists an element σ ∈ λ such that sσ /∈ uσ. thus, s /∈ µ−1 σ∈λ[uσ] by the equality µσ(s) = sσ and in view of the fact that uσ is jointly closed in sσ, the subset µ −1 σ [uσ] ⊆ s∞ is jointly closed in s∞ due to the continuity of limiting projection µσ : s∞ → sσ as given in [18, proposition 4.4]. now, we can prove that lim ← {uσ} ⊆ µ −1 σ [uσ]: if there exists a point a = {aα} ∈ s∞ such that lim ← {uσ} 6⊆ qa and pa 6⊆ µ −1 σ [uσ] then a ∈ lim ← {uσ} and so a ∈ ∏ σ∈λ uσ. but also, the fact aσ = µσ(a) /∈ uσ gives a contradiction. as a result of the above considerations lim ← {uα} is a jointly closed subspace of s∞. in addition, now we will show that u = lim ← {uα}: (2) first of all, let ’s prove the inclusion u ⊆ lim ← {uα}. conversely, if u 6⊆ lim ← {uα}, then there exists b ∈ s∞ = lim ← {sα} such that u 6⊆ qb and pb 6⊆ lim ← {uα}. in this case, pbα ⊆ uα because of µα(b) ∈ µα(u). thus, pb = ∏ α∈λ pbα ⊆ ∏ α uα∈λ. on the other hand, b ∈ ∏ α sα∈λ and ϕαβ(bα) = bβ for α ≥ β, α,β ∈ λ. also, by the definition of ϕαβ for α ≥ β and the fact bα ∈ uα for every α ∈ λ, the equality ϕαβ(bα) = ϕαβ(bα) is satisfied. hence, ϕαβ(bα) = bβ for α ≥ β. that is, we obtained b ∈ lim ← {uα} which is a contradiction. therefore, from (1) if recall the fact that lim ← {uα} is jointly closed with respect to the limit ditopology (τ∞,κ∞) on (s∞,s∞), then the inclusion u ⊆ lim ← {uα} is immediate. for the other direction, assume lim ← {uα} 6⊆ u. thus, there exists a point a = {aα} ∈ s∞ such that lim ← {uα} 6⊆ qa and pa 6⊆ u. by the definition of joint topology, there exist m ∈ µ(a) and n ∈ η(a) such that u ⊆ n ∩ (s∞ \ m) and so we have the sets g ∈ τ∞ and k ∈ κ∞ such that pa ⊆ g ⊆ m, n ⊆ k ⊆ qa and u ⊆ k ∩ (s∞ \ g). hence, by [18, theorem 4.6], there exist α0,α1 ∈ λ and aα0 ∈ τα0, bα1 ∈ κα1 such that the conditions pa ⊆ µ−1α0 [aα0] ⊆ g and k ⊆ µ −1 α1 [bα1] ⊆ qa are satisfied. in this case, the inclusion u ⊆ (s∞\µ −1 α0 [aα0])∩µ −1 α1 [bα1] is trivial. finally, we obtained α1 ∈ λ satisfying the conditions u ⊆ u ⊆ µ−1α1 [bα1] and pa 6⊆ µ −1 α1 [bα1]. thus uα1 ⊆ bα1 for α1 ∈ λ, because of the inclusions µα1(u) ⊆ µα1(µα1 −1[bα1]) ⊆ bα1. if c© agt, upv, 2018 appl. gen. topol. 19, no. 1 119 f. yıldız we consider the closure operator on these sets, it is clear that uα1 ⊆ bα1 and so µα1(pa) 6⊆ uα1 by µα1(a) /∈ bα1. moreover, it is easy to verify that µα1(pa) = paα1 : µα1(pa) = {µα1(x) | x ∈ pa} = {xα1 | x ∈ pa} = {xα1 | x ∈ ∏ α∈λ paα} = {xα1 | xα ∈ paα, ∀α} = paα1 . as a result of these facts, we have paα1 6⊆ uα1 and so aα1 /∈ uα1 for α1 ∈ λ. this argument gives a /∈ ∏ α uα∈λ, clearly. it means that a /∈ lim ← {uα} and so, a contradiction. (3) note that the closure set uα is jointly closed in the space sα for each α. thus, the sets µ−1α [uα], α ∈ λ are jointly closed in the limit space s∞ since the limiting projection µα is continuous for α ∈ λ, between the corresponding joint topological spaces (s∞,jτ∞κ∞), (sα,jτακα) of the spaces (s∞,s∞,τ∞,κ∞),(sα,sα,τα,κα) ∈ ob ifpditop, respectively. in addition, with the equality µα(u) = uα , α ∈ λ given in the hypothesis, it is clear that u ⊆ µ−1α [uα] and so u ⊆ ⋂ α∈λ µ−1α [uα]. hence we have u ⊆ ⋂ α∈λ µ−1α [uα] since ⋂ α∈λ µ−1α [uα] is jointly closed in s∞. for the converse, suppose that ⋂ α∈λ µ−1α [uα] 6⊆ u. in this case, there exists a point a = {aα} ∈ s∞ such that ⋂ α∈λ µα −1[uα] 6⊆ qa and pa 6⊆ u. thus, a ∈ µα −1[uα] and µα(a) = aα ∈ uα for every α ∈ λ. on the other hand, if pa 6⊆ u and u is closed in s∞ with respect to the joint topology of the ditopology on (s∞,s∞) ∈ob ifptex, then there exist m ∈ µ(a) and n ∈ η(a) such that u ⊆ n ∩ (s∞\m). so we have the sets g ∈ τ∞, k ∈ κ∞ such that g ⊆ m, n ⊆ k and u ⊆ k ∩ (s∞\g). therefore, by [18, theorem 4.6] there exist α0,α1 ∈ λ and aα0 ∈ τα0, bα1 ∈ κα1 satisfying the conditions µ−1α0 [aα0 ] ⊆ g, µ −1 α0 [aα0] 6⊆ qa and k ⊆ µ −1 α1 [bα1], pa 6⊆ µ −1 α1 [bα1]. in this case, the inclusion u ⊆ (s∞\µ −1 α0 [aα0]) ∩ µ −1 α1 [bα1] is trivial and so, we have uα1 ⊆ bα1 for α1 ∈ λ by u ⊆ µ −1 α1 [bα1]. consequently, uα1 ⊆ bα1 and the fact that µα1(a) = aα1 /∈ bα1 means that aα1 /∈ uα1 which is a contradiction. � with the above notations, we have also the next result: corollary 4.5. i) u ⊆ lim ← {uα} ii) lim ← {uα} ⊆ lim ← {uα} proof. i) if the inclusion is not true, there exists a point a = {aα} ∈ s∞ such that u 6⊆ qa and pa 6⊆ lim ← {uα}. in this case, by the fact µα(a) ∈ µα(u) = uα we have aα ∈ uα for every α ∈ λ and so a ∈ ∏ α∈λ uα, obviously. also, we have ϕαβ|uα(aα) = ϕαβ(aα) = aβ since aα ∈ uα, α ∈ λ. as a result c© agt, upv, 2018 appl. gen. topol. 19, no. 1 120 some categorical aspects of the inverse limits in ditopological context of these considerations, we get a = {aα} ∈ lim ← {uα} which contradicts with pa 6⊆ lim ← {uα}. ii) firstly, note that the limit sets lim ← {uα} and lim ← {uα} are subsets of s∞, due to proposition 4.2 and theorem 4.4. now assume the converse of required inclusion. thus, there exists a point s = {sα} ∈ s∞ such that lim ← {uα} 6⊆ qs and ps 6⊆ lim ← {uα}. in this case, s = {sα} ∈ ∏ α∈λ uα and so ϕαβ|uα(sα) = sβ for α ≥ β, α,β ∈ λ because of sα ∈ uα. hence, s = {sα} ∈ ∏ α∈λ uα by uα ⊆ uα. also, for α ≥ β, we have the equalities ϕαβ(sα) = ϕαβ|uα(sα) = ϕαβ(sα) and ϕαβ(sα) = ϕαβ|uα(sα) due to sα ∈ uα. consequently, the point s = {sα} ∈∏ α∈λ uα is also an element of the inverse limit set lim ← {uα} since we have the equality ϕαβ(sα) = ϕαβ|uα(sα) = sβ for α ≥ β, and it is a contradiction. � according to all considerations presented above, we can mention a further result as the final stage of this section, besides the fact that it will be considered as the converse of proposition 4.3. corollary 4.6. let the system {(sα,sα,τα,κα),ϕαβ}α≥β ∈ ob invifpditop over a directed set λ. if take the ditopological subtexture space (u,su,τu,κu) ∈ ob ifpditop of the inverse limit space (s∞,s∞,τ∞,κ∞) where u ∈ j c τ∞κ∞ , then (u,su,τu,κu) ∈ ob ifpditop is the inverse limit space of the inverse system {(uα,suα,τuα,κuα),ϕαβ}α≥β ∈ ob invifpditop consisting of jointly closed subspaces (uα,suα,τuα,κuα) of the spaces (sα,sα,τα,κα) ∈ ob ifpditop, where πα|s∞(u) = µα(u) = uα, suα = sα|uα, τuα = τα|uα, κuα = κα|uα, α ∈ λ and ϕαβ = ϕαβ|uα, for α,β ∈ λ such that α ≥ β. in other words, if u ∈ jcτ∞κ∞ then we have the equality u = lim ← {uα} = lim ← {uα}. proof. if choose the set u as an element of jcτ∞κ∞, that is a closed set with respect to the joint topology of the limit ditopology (τ∞,κ∞) defined on the inverse limit texture, then by theorem 4.4 (2) and the two inclusions presented in corollary 4.5, the required equalities are straightforward. � 5. identification of the ditopological products as an inverse limit in ifpditop take into account all the previous considerations, it can be mentioned that the notion of inverse limit as an object of ifpditop for any inverse system which is the object of invifpditop is derived from the products as the objects of ifpditop. conversely, by applying the limit operation lim ← located in the theory of inverse systems, to the objects of invifpditop, one can express infinite ditopological cartesian products [3, 4, 18] of the spaces which are the objects of c© agt, upv, 2018 appl. gen. topol. 19, no. 1 121 f. yıldız ifpditop in terms of the finite cartesian products of those spaces belong to ob ifpditop. now, let ’s mention and prove this significant characterization as a theorem: theorem 5.1. for a directed set λ and any family {(xs,ss,τs,κs)}s∈λ of the objects in ifpditop, the product space ( ∏ s∈λ xs, ⊗ s∈λ ss, ⊗ s∈λ τs, ⊗ s∈λ κs) ∈ ob ifpditop may be expressed as the inverse limit of an inverse system over γ, which is the object of invifpditop and constructed by the finite cartesian product spaces ( ∏ s∈i xs, ⊗ s∈i ss, ⊗ s∈i τs, ⊗ s∈i κs) ∈ ob ifpditop for i ∈ γ, where the set γ = {i ⊆ λ | i is finite} is directed by the set inclusion. in other words, any arbitrary textural product of the objects in ifpditop is exactly the inverse limit space of the inverse system consisting of finite products of those objects. proof. let (xs,ss,τs,κs) ∈ ob ifpditop, s ∈ λ and γ be directed by the set inclusion, that is j ≤ i ⇐⇒ j ⊆ i for every i,j ∈ γ. now assume j ≤ i for any j ∈ γ. if x = {xs}s∈i ∈ ∏ s∈i xs = xi then xs ∈ xs for all s ∈ i. in this case, {xs}s∈j ∈ ∏ s∈j xs = xj by the facts that if s ∈ j then s ∈ i and xs ∈ xs for all s ∈ i. therefore, for j ≤ i, describe the mapping ϕij : xi → xj {xs}s∈i 7→ {xs}s∈j. now let us prove that ϕij is ω-preserving and bicontinuous for j ≤ i : assume that p{xs}s∈i 6⊆ q{x′s}s∈i for {xs},{x ′ s} ∈ xi. if s0 ∈ j, then s0 ∈ i by j ≤ i. thus, pxs0 6⊆ qx′s0 and p{xs}s∈j 6⊆ q{x′s}s∈j by [18, corollary 1.2], since pxs 6⊆ qx′s for all s ∈ j. hence pϕij (x) 6⊆ qϕij (x′) and ϕij is ω-preserving. for the second part, we prove that ϕij is bicontinuous between the product ditopological spaces (xi,si,τi,κi) and (xj,sj,τj,κj) as follows: suppose that j = {1,2, ...,m}, i = {1,2, ..., t} and j ⊆ i. in this case, m < t. now let g ∈ ⊗ s∈j τs = τj and ϕ −1 ij [g] 6⊆ qx for x = {xs}s∈i ∈ xi. in this case, ϕij(x) = {xs}s∈j ∈ g, that is g 6⊆ q{xs}s∈j . thus, there exists b ∈ bτj , where bτj denotes the base for τj, such that b 6⊆ q{xs}s∈j and b ⊆ g, so there exists finite set j0 = {1,2, ...,n} ≤ j (n < m) such that b = ⋂ j∈j0 (πjj ) −1[gj], where gj ∈ τj, j ∈ j0. thus, xj ∈ gj for j ∈ j0. also, ϕ−1ij [b] = ϕ −1 ij ( ⋂ j∈j0 (πj j)−1[gj]) ⊆ ϕ −1 ij [g] because of b ⊆ g. thus, with all the arrangements, we have b′ = ⋂ j∈j0 (πjj ◦ϕij) −1[gj] = ⋂ j∈j0 (πij ) −1[gj] ⊆ ϕ−1 ij [g] and so b′ ∈ bτi where bτi denotes the base for τi. c© agt, upv, 2018 appl. gen. topol. 19, no. 1 122 some categorical aspects of the inverse limits in ditopological context on the other hand, we have {xs}s∈j ∈ (π j j ) −1[gj] since {xs}s∈j ∈ b and πjj ({xs}s∈j) ∈ gj for every j ∈ j0 . thus x1 ∈ g1, x2 ∈ g2,...,xn ∈ gn. in this case, by the fact n < t, πij ({xs}) ∈ g for j ∈ j0 and so {xs}s∈i ∈ b ′. since, ϕ−1 ij [g] ∈ τi for g ∈ τj, ϕij is continuous. dually, by using closed sets as the elements of κj, it is proved that ϕij is cocontinuous and so bicontinuous. furthermore, note that the mappings ϕij for j ≤ i are the bonding maps: indeed, for the mapping ϕii : xi → xi, the equality ϕii({xs}s∈i) = {xs}s∈i is clear and so ϕii is the identity idxi . in addition, for k ≥ i ≥ j, let ’s prove ϕij ◦ ϕki = ϕkj. if {xs}s∈k ∈ xk then (ϕij ◦ ϕki)({xs}s∈k) = ϕij(ϕki({xs}s∈k)) = ϕij({xs}s∈i) = {xs}s∈j = ϕkj({xs}s∈k). consequently, thanks to the above expressions, the fact {(xi,si,τi,κi),ϕij}i≥j ∈ ob invifpditop is trivial. now let us turn to our main aim: the inverse limit space of inverse system {(xi,si,τi,κi),ϕij}i≥j ∈ ob invifpditop over γ is ifpditop-isomorphic to the arbitrary ditopological product space constructed on the set ∏ s∈λ xs. for proof, first of all we define a mapping between lim ← {xi}i∈γ and ∏ s∈λ xs: if {xi} ∈ lim ← {xi}i∈γ then {xi} ∈ ∏ i∈γ xi and so xi ∈ xi for every i ∈ γ. now, for any s ∈ λ let is = {s} ∈ γ, so by the fact xis = ∏ z∈is xz = ∏ z∈{s} xz = xs, we have x{s} = xis ∈ xs, s ∈ λ. thus {xis} ∈ ∏ s∈λ xs and finally, we can define the mapping ψ : lim ← {xi}i∈γ → ∏ s∈λ xs {xi}i∈γ 7→ {xis}s∈λ it is easy to verify that ψ is well-defined. now let us show that ψ is an ifpditop-isomorphism: ψ is ω-preserving: let {xi},{x ′ i} ∈ lim ← {xi}i∈γ such that p{xi} 6⊆ q{x′i}. in this case, pxi 6⊆ qx′i for all i ∈ γ, by [18, corollary 1.2]. take s ∈ λ, so is = {s} ⊆ λ, that is is ∈ γ. thus, pxis 6⊆ qx′is by the fact that pxi 6⊆ qx′i for all i ∈ γ. it means that pxis 6⊆ qx′is for all s ∈ λ. hence p{xis}s∈λ 6⊆ q{x′ is } s∈λ , that is pψ{xi} 6⊆ qψ{x′i}. in addition, the bijectivity of ψ is straightforward. now, if consider the product ditopological spaces (xi,si,τi,κi) for i ∈ γ, with the plain texturings then the product texturing ⊗ i∈γ si and product ditopology ( ⊗ i∈γ τi, ⊗ i∈γ κi) can be constructed over the product set ∏ i∈γ xi in a suitable way. therefore, the restricted texturing and ditopology will be taken over the subset lim ← {xi}i∈γ of ∏ i∈γ xi. shortly, if we use the notations t = c© agt, upv, 2018 appl. gen. topol. 19, no. 1 123 f. yıldız ( ⊗ i∈γ si)|lim ← {xi}i∈γ, v = ( ⊗ i∈γ τi)|lim ← {xi}i∈γ and z = ( ⊗ i∈γ κi)|lim ← {xi}i∈γ for the induced texturing, topology and cotopology, respectively, then now we will prove that ψ is bicontinuous with respect to the ditopologies ( ⊗ s∈λ τs, ⊗ s∈λ κs) and (v,z): let g ∈ ⊗ s∈λ τs = τλ and ψ −1[g] 6⊆ q{xi}i∈γ. in this case, g 6⊆ qψ({xi}i∈γ) and so g 6⊆ q{xis}s∈λ. thus, there exists b ∈ bτλ which is the base for the product topology τλ, such that b ⊆ g and b 6⊆ qψ({xi}i∈γ). note here that b = ⋂ j∈j0⊆λ π−1j [gj], where gj ∈ τj and j ∈ j0 for the finite set j0 ⊆ λ. thus, we have ψ−1( ⋂ j∈j0⊆λ π−1j [gj]) ⊆ ψ −1[g] and so ⋂ j∈j0 (πj ◦ ψ) −1[gj] ⊆ ψ −1[g]. on the other hand, the equality πj ◦ ψ = πij |lim ← {xi}i∈γ is obvious by the definition of projection map πij : ∏ i∈γ xi → xij = xj and by the facts j ∈ λ and ij = {j} ⊆ λ which means that ij ∈ γ for j ∈ j0. additionally, if take ϕ as the inverse of ψ, then we have πij |lim ← {xi}i∈γ ◦ ϕ = πj. here, the restriction πij |lim ← {xi}i∈γ is bicontinuous since ij. projection map πij is bicontinuous. hence, if a = ⋂ j∈j0 (πj ◦ ψ) −1[gj] = ⋂ j∈j0 (πij |lim ← {xi}i∈γ) −1[gj] then a ∈ bv. here, bv denotes the base for topology v. in this case, the fact a ⊆ ψ −1[g] is clear. now let us prove a 6⊆ q{xi}i∈γ: firstly, recall b 6⊆ q{xis}s∈λ and so π−1j [gj] 6⊆ q{xis}s∈λ for all j ∈ j0. that is, πj({xis}) ∈ gj and (πj ◦ ψ)({xi}i∈γ) ∈ gj for all j ∈ j0. therefore, {xi}i∈γ ∈ (πij |lim ← {xi}i∈γ) −1[gj] is clear for all j ∈ j0. finally, {xi}i∈γ ∈ ⋂ j∈j0 (πij |lim ← {xi}i∈γ) −1[gj] = a, and so a 6⊆ q{xi}i∈γ since the related texturings are plain. hence ψ −1[g] ∈ v and ψ is continuous. dually, it is easy to verify that ψ is cocontinuous by dealing with the closed sets. then ψ is bicontinuous. as the final step, that the map ϕ as the inverse of ψ is bicontinuous can be shown in a like manner. � the above theorem could be also summarized for the subcategory ifpdicomp2 consisting of dicompact [11] and bi-t2 (bi-hausdorff) [4] objects of the category ifpditop. hence, with the above arguments, note that: corollary 5.2. the infinite ditopological products of the objects which belong to ifpdicomp2 can be expressed via inverse limits, in terms of the finite ditopological products in ifpdicomp2 of those objects. proof. for all the details about category of dicompact spaces see [11], and from [4], note that (s,s,τ,κ) is bi-t2 if and only if for s,t ∈ s, qs 6⊆ qt =⇒ ∃h ∈ τ, k ∈ κ with h ⊆ k, ps 6⊆ k and h 6⊆ qt. thus, the required c© agt, upv, 2018 appl. gen. topol. 19, no. 1 124 some categorical aspects of the inverse limits in ditopological context characaterization is seen as a result of theorem 5.1. indeed, by the facts that the jointly closed subtexture spaces and the product spaces of dicompact, bit2 ditopological spaces are dicompact and bi-t2 from [18, theorem 4.16] and tychonoff property, respectively, and from [18, theorem 4.17 a)], the proof is completed. � definition 5.3. a property p is called ditopological property if it is a property defined for ditopological texture spaces, as a natural counterpart of the classical notion, named topological property. according to this, we have the following as a final result, as well. corollary 5.4. let p be a ditopological property which is hereditary with respect to the jointly closed subsets of a ditopological space and finitely multiplicative (that is, p is preserved under the finite multiplications of ditopological spaces). in this case, (s,s,τ,κ) ∈ ob ifpditop is ifpditop-isomorphic to the inverse limit of an inverse system constructed over a directed set λ, via bi-t2 spaces (sα,sα,τα,κα) ∈ ob ifpditop, α ∈ λ, which have the property p if and only if (s,s,τ,κ) is ifpditop-isomorphic to a jointly closed subspace of the product space ( ∏ α∈λ sα, ⊗ α∈λ sα, ⊗ α∈λ τα, ⊗ α∈λ κα). proof. necessity. suppose that (s,s,τ,κ) ∈ ob ifpditop is isomorphic to the inverse limit space (s∞,s∞,τ∞,κ∞) ∈ ob ifpditop of the inverse system {(sα,sα,τα,κα),ϕαβ}α≥β ∈ ob invifpditop over a directed set λ, where s∞ = lim ← {sα}. also, if recall that the inverse limit space s∞ is jointly closed in the product ∏ α∈λ sα by [18, theorem 4.17 a)], then the required assertion is proved. sufficiency. let {(sα,sα,τα,κα)}α∈λ be a family consisting of the objects in ifpditop, which have the properties bi-t2 and p . assume that (s,s,τ,κ) is ifpditop-isomorphic to a jointly closed subspace (u,( ⊗ sα)|u,( ⊗ τα)|u,( ⊗ κα)|u) of the product space ( ∏ sα, ⊗ sα, ⊗ τα, ⊗ κα). by theorem 5.1, it is known that the product ∏ sα can be expressed as the inverse limit of an inverse system consisting of finite cartesian product spaces n∏ i=1 si for n ∈ n. hence, with the same notations used in corollary 4.6, (u,( ⊗ sα)|u,( ⊗ τα)|u,( ⊗ κα)|u)) becomes the inverse limit of inverse system a = {(un,( n⊗ i=1 si)|un,( n⊗ i=1 τi)|un,( n⊗ i=1 κi)|un),ϕnm}n≥m constructed by the bonding maps ϕnm : un → um for n ≥ m, as well as the jointly closed subspaces (un,( n⊗ i=1 si)|un,( n⊗ i=1 τi)|un,( n⊗ i=1 κi)|un) of finite cartesian product spaces ( n∏ i=1 si, n⊗ i=1 si, n⊗ i=1 τi, n⊗ i=1 κi) for every n ∈ n. here, un denotes the closure of un for each n, with respect to the joint topology appropriate for the finite product space of the spaces (si,si,τi,κi), i = 1,2, ...,n. on the other hand, since each space sα, α ∈ λ has the property bi-t2 from [4], the product space ∏ sα has the property bi-t2 and so the ditopologies on c© agt, upv, 2018 appl. gen. topol. 19, no. 1 125 f. yıldız subsets un have the property bi-t2 as well. furthermore, each finite product space n∏ i=1 si has the property p since each space sα, α ∈ λ has the property p by hypothesis. thus, the jointly closed subspaces un, n ∈ n have the common property p as p is hereditary with respect to the jointly closed subspaces. consequently, a is the required inverse system in ifpditop and by the fact lim ← a = u, the proof is concluded. � 6. conclusion this paper studied some further categorical aspects of the inverse systems (projective spectrums) and inverse limits constructed in the subcategory ifpditop of ditopological plain spaces. as one of the investigations here, an identity natural transformation which is peculiar to the theory of inverse systems and inverse limits, as well as consisting of the adjoint and isomorphism functors introduced between the suitable related main subcategories of bitop and ifpditop, consisting of the spaces which satisfy a special separation axiom, is established. as another one, we proved a representation theorem which shows any infinite textural product of the objects in category ifpditop can be expressed as the inverse limit of the inverse system in invifpditop, constructed by the finite products of those objects in ifpditop. besides that, the textural products of dicompact bi-t2 ditopological spaces are characterized in terms of finite products, via inverse limits. there are considerable difficulties involved in constructing a suitable theory of inverse systems for general ditopological spaces. hence, we confined our attention to the inverse systems limits constructed in the special category ifpditop and we leaved as an open problem the task of extending the further results obtained here to more general categories established in the theory of ditopological spaces. acknowledgements. the author would like to thank the referees and editors for their constructive comments which have helped improve the exposition and the readability of the paper. references 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³!��¼ ½ ` f�4 g ra4�+�+n),-.0�0��¾�¿�à:á�âtãºä�ådàt¿dæ�ç�èdé�êºë;á'ìhçhë applied general topology c© universidad politécnica de valencia volume 3, no. 1, 2002 pp. 1–11 hausdorff compactifications and zero-one measures ii georgi d. dimov ∗ and gino tironi † abstract. the notion of pbs-sublattice is introduced and, using it, a simplification of the results of [6] and of some results of [5] is obtained. two propositions concerning wallman-type compactifications are presented as well. 2000 ams classification: primary 54d35, 28a60, 06d99; secondary 54d80, 54e05, 06b99. keywords: hausdorff compactifications, zero-one measures on boolean algebras, maximal spectrum of distributive lattices, efremovič proximities, wallman-type compactifications. 1. introduction in 1977, v. m. ul′janov ([15]) obtained a negative answer to the famous frink’s question, posed in [8], whether each hausdorff compactification of a tychonoff space x is a wallman-type compactification (we shall use from now on the term “wallman compactification” instead of “wallman-type compactification”). o. frink introduced the wallman compactifications of a space x as spaces of all c-ultrafilters, where c is a ring of subsets of x and a special closed base of x (called normal base) (we will denote such compactifications by ω(x,c)). passing to the complements in x of all elements of a normal base c, one obtains a special open base b = c′ of x (which is again a ring of sets), called normal wallman base. this leads to a dual description of the wallman compactifications of x as spaces of the type max(b) (= maximal spectrum of b), where b is a normal wallman base of x (see, e.g., [9]). hence, in general, not every hausdorff compactification of a tychonoff space x can be obtained as a maximal spectrum of a normal wallman base of x. in our paper [6], using ∗the first author was partially supported by a fellowship for mathematics of the natocnr outreach fellowships programme 1999, bando 219.32/16.07.1999. †the second author was supported by the national group “analisi reale” of the italian ministry of the university and scientific research at the university of trieste. 2 g. d. dimov and g. tironi the notion of pb-sublattice introduced in [5], we answered affirmatively two natural questions. the first one was: problem 1.1. is it possible to correlate (in a canonical way) to each tychonoff space x a boolean algebra bx and a set lx of sublattices of bx in order to obtain that the set of all, up to equivalence, hausdorff compactifications of x is represented by the set {max(l) : l ∈lx}? this question was motivated also by some measure-theoretic constructions of hausdorff compactifications. it was well known (see [1, 3, 4, 14]) that, when c is a normal base of x, then the space ir(c) (of all regular zero-one measures on the boolean subalgebra b(c) of the boolean algebra exp(x) (of all subsets of x, with the natural operations), generated by the sublattice c of exp(x)) is a hausdorff compactification of x equivalent to ω(x,c) and max(c′). the second problem was: problem 1.2. is it possible to construct in a similar way (by means of zero-one measures) every hausdorff compactification of x? in this paper we introduce the notion of pbs-sublattice and, using it, we obtain a simplification of the results of [6] and of some results of [5]. we also present the notion of pb-sublattice in a simpler but equivalent form. finally, a necessary and sufficient condition and a sufficient condition, as well, which a lattice l ∈ lx has to satisfy in order to obtain that max(l) is a wallman compactification of x, are stated and proved. 2. preliminaries we first fix some notations. note 2.1. we denote by ω the set of all positive natural numbers. all lattices will be with top (= unit) and bottom (= zero) elements, denoted respectively by 1 and 0 and all sublattices of a lattice l are assumed to contain the top and the bottom elements of l. we don’t require the elements 0 and 1 to be distinct. let a be a distributive lattice. the set of all ideals of a will be denoted by idl(a) and the set of all maximal ideals of a (which will be, as usual, always proper) — by max(a). put ta = {oi = {j ∈ max(a) : i 6⊆ j} : i ∈ idl(a)}. the space (max(a),ta) is called maximal spectrum of a and the topology ta is called spectral topology on the set max(a). (max(a),ta) is always a compact t1-space (see, e.g., [9]). if the lattice a is normal (i.e., for each pair a,b ∈ a with a∨ b = 1, there exist u,v ∈ a such that a∨u = 1 = b∨v and u∧v = 0) then (max(a),ta) is a compact t2-space. if l is a sublattice of a boolean algebra b then we will denote by b(l) the boolean subalgebra of b generated by l. by exp(x) we denote the set of all subsets of the set x. the ordered set of all, up to equivalence, hausdorff compactifications of a tychonoff space x will be denoted by (k(x),≤). hausdorff compactifications and zero-one measures ii 3 if (x,t ) is a topological space then we write coz(x,t ) or, simply, coz(x) for the set of all cozero-subsets of x; the closure of a subset m of (x,t ) will be denoted by clx m; a dense embedding will mean an embedding with dense image. by a proximity we shall always mean an efremovič proximity. if δ is a proximity on a set x, then δ will be the complement of the relation δ. if (x,t ) is a topological space and δ is a proximity on the set x, we say that δ is a proximity on the space (x,t ) if the topology tδ, generated by δ on the set x, coincides with t . the ordered set of all proximities δ on a topological space (x,t ) will be denoted by (pt (x),≤). for all undefined terms and notations see [7], [9] and [11]. we shall recall the smirnoff compactification theorem: theorem 2.2 ([13]). let (x,t ) be a tychonoff space. if (cx,c) is a hausdorff compactification of x, then putting, for every a,b ⊆ x, aδcb iff clcx c(a) ∩ clcx c(b) = ∅, we obtain a proximity δc on (x,t ). the correspondence s: (k(x),≤) −→ (pt (x),≤), defined by s(cx,c) = δc, is an isomorphism. if δ ∈ pt (x) then the compactification s−1(δ) of x, which will be denoted by (cδx,cδ), is called smirnoff compactification of (x,t ). definition 2.3 ([9, 8]). let (x,t ) be a topological space. a sublattice b of t is called a wallman base for x if b is a base of t and satisfies the following condition: (w) whenever u ∈b and x ∈ u, there exists v ∈b with u ∪v = x and x 6∈ v . if b is a wallman base for a t0-space x, then the map ηb : x −→ max(b), x 7→ ηb(x) = {u ∈b : x 6∈ u}, is a dense embedding. hence, for every t1-space x, (max(b),ηb) is a t1compactification of x. if b is a normal wallman base, then (max(b),ηb) is a t2-compactification of x, called wallman compactification. a family c of closed subsets of x, such that the family b = c′ = {x \f : f ∈ c} is a normal wallman base of x, is called a normal base of x. let ω(x,c) denote the set of all c-ultrafilters. topologize this set by using as a base for the closed sets all sets of the form a− = {f ∈ ω(x,c) : a ∈ f}, where a ∈ c. then the map ωc : x −→ ω(x,c), defined by the formula ωc(x) = {f ∈c : x ∈ f}, where x ∈ x, is a dense embedding of x in ω(x,c) and (ω(x,c),ωc) is a compactification of x equivalent to (max(b),ηb). we will need the following theorem of o. nj̊astad: theorem 2.4 ([12]). let (x,t ) be a tychonoff space. a compactification (cx,c) of x is a wallman compactification if and only if there exists a subfamily b of t which is closed under finite unions and satisfies the following two conditions: 4 g. d. dimov and g. tironi (b1) if u,v ∈b and u ∪v = x then (x \u)δc(x \v ); (b2) if a,b ⊆ x and aδcb then there exist u,v ∈b such that a ⊆ x \u, b ⊆ x \v and u ∪v = x. recall that (see, e.g., [1, 3, 4, 14]) a measure on a boolean algebra a is a non-negative real-valued function µ on a such that µ(a∨ b) = µ(a) + µ(b) for all a,b ∈ a with a∧b = 0; in the case when µ(a) = {0, 1}, µ is called a zero-one measure. let b be a boolean algebra and l be a sublattice of b. a measure µ, defined on the boolean algebra b(l), is called l-regular measure (or, simply, regular measure) if µ(x) = sup{µ(a) : a ∈ l,x ≥ a} for any x ∈ b(l). the set of all l-regular zero-one measures on the boolean algebra b(l) will be denoted by ir(l). the topology dw on ir(l) is defined as follows: a base for the closed sets of dw consists of all sets of the form w(a) = {µ ∈ ir(l) : µ(a) = 1}, where a ∈ l. the space (ir(l),dw) is a compact t1-space. if x is a tychonoff space and c is a normal base of x then (ir(c),dw) is a compact hausdorff space. the map mc : x −→ (ir(c),dw), defined by the formula mc(x) = µx, where x ∈ x and, for every element f of the boolean subalgebra b(c) of exp(x), µx(f) = 1 if x ∈ f, and mux(f) = 0 if x 6∈ f, is a dense embedding. ((ir(c),dw),mc) is a compactification of x equivalent to (ω(x,c),ωc) and (max(c′),ηc′). we will recall a theorem of j. kerstan. definition 2.5 ([10, 2]). a family u of open subsets of a topological space space x is called completely regular if for every u ∈u there exist two sequences (ui)i∈ω and (v i)i∈ω in u such that u = ⋃ {ui : i ∈ ω} and ui ⊆ x \v i ⊆ u for each i ∈ ω. theorem 2.6 ([10, 2]). a subset of a topological space is a cozero-set if and only if it belongs to a completely regular family. 3. the results definition 3.1. let (x,t ) be a space and u be an open subset of x. if there is a sequence (ui,uci)i∈ω in t ×t with u = ⋃ i∈ω u i, ui ⊆ x\uci ⊆ ui+1, for every i ∈ ω, then such a sequence (ui,uci)i∈ω will be called ur−representation of u. we put tur = {u ∈t : u has an ur-representation}. definition 3.2. let (x,t ) be a space. denote by l(x) the set of all ur-representations of the elements of tur. the elements of l(x) will be written in the following way: ū = (ui,uci)i∈ω, where (ui,uci)i∈ω is a ur-representation of u0 = ⋃ {ui : i ∈ ω}; two elements ū = (ui,uci)i∈ω and v̄ = (v i,v ci)i∈ω of l(x) are equal if ui = v i, uci = v ci, for every i ∈ ω. define two operations ∧ and ∨ in l(x) by ū ∨ v̄ = (ui ∪v i,uci ∩v ci)i∈ω hausdorff compactifications and zero-one measures ii 5 and ū ∧ v̄ = (ui ∩v i,uci ∪v ci)i∈ω, where ū = (ui,uci)i∈ω and v̄ = (v i,v ci)i∈ω, and let 0̄ = (0i, 0ci)i∈ω, 1̄ = (1i, 1ci)i∈ω, where ∅ = 0i = 1ci, x = 1i = 0ci, i ∈ ω. fact 3.3. (l(x),∨,∧) is a distributive lattice and 0̄, 1̄ are its zero and one. definition 3.4 (see also [5]). let x be a tychonoff space. a sublattice l of l(x) is said to be a pb-sublattice if (l1) the set l0 = {u0 = ⋃ {ui : i ∈ ω} : (ui,uci)i∈ω ∈ l} is an open base of the space x; (l2) for every ū = (ui,uci)i∈ω ∈ l and every j ∈ ω, there exist k ∈ ω and v̄ = (v i,v ci)i∈ω,w̄ = (wi,wci)i∈ω ∈ l (which depend on the choice of ū and j) such that uc(j+1) ⊆ wk ⊆ w 0 = ucj, uj−1 ⊆ v k ⊆ v 0 = uj (for j > 1), and v 0 = uj (for j = 1). proposition 3.5. let l be a pb-sublattice of l(x). then, for every element ū = (ui,uci)i∈ω of l and for every i ∈ ω, we have that ui,uci ∈ coz(x). hence, l0 ⊆ coz(x). proof. for every ū = (ui,uci)i∈ω ∈ l and every j ∈ ω, we have, by (l2), that there exist v̄ = (v i,v ci)i∈ω ∈ l and w̄ = (wi,wci)i∈ω ∈ l such that uj = v 0 and ucj = w 0. hence, in order to prove our proposition, we need only to show, according to kerstan theorem (see 2.6), that l0 is a completely regular family (see 2.5). so, let ū = (ui,uci)i∈ω ∈ l. then {ui : i ∈ ω}⊆ l0 and u0 = ⋃ {ui : i ∈ ω}. we let (ui)i∈ω to be the first required sequence. as it follows from 3.1, (uci)i∈ω can serve as the second required sequence. therefore, l0 is a completely regular family. � definition 3.6 ([5]). let (x,τ) be a space. denote by l(coz(x)) the set of all ur-representations of all elements of coz(x) by elements of coz(x). we will regard l(coz(x)) as a sublattice of the lattice l(x). remark 3.7. let us remark that in [5] the notion of “pb-sublattice” was introduced with the redundant (as proposition 3.5 shows now) requirement that a pb-sublattice is (by definition) a sublattice of l(coz(x)). proposition 3.8 ([5]). (l(coz(x)),∨,∧) is the greatest (with respect to the inclusion) pb-sublattice of (l(x),∨,∧). note 3.9. let x be a set. we will denote by s(x) the complete boolean algebra (exp(x))ℵ0 . definition 3.10. let (x,t ) be a topological space. we put ois(x,t ) = {ū = (ui)i∈ω : ui ∈t ,ui ⊆ ui+1,∀i ∈ ω}. instead of ois(x,t ), we shall often write simply ois(x). for every (ui)i∈ω ∈ ois(x), we put u0 = ⋃ {ui : i ∈ ω}. we will regard ois(x) as a sublattice of s(x). 6 g. d. dimov and g. tironi definition 3.11. define a relation ∼ in s(x) putting: for every ū = (ui)i∈ω, v̄ = (v i)i∈ω ∈ ois(x), ū ∼ v̄ if and only if there exists an i0 ∈ ω such that ui = v i, for every i ≥ i0. then ∼ is a congruence relation on the boolean algebra s(x). so, a quotient boolean algebra s(x)/∼, which will be denoted by [s(x)], is defined. the natural mapping between s(x) and [s(x)] will be denoted by π. we put, for every ū ∈ s(x), π(ū) = [ū]. definition 3.12. let (x,t ) be a tychonoff space. a sublattice l of the lattice ois(x) is said to be a pbs-sublattice in x, if (ls1) the set l0 = {u0 : (ui)i∈ω ∈ l} is a base of the space x; (ls2) for every ū = (ui)i∈ω ∈ l and for every j ∈ ω, there exist v̄ = (v i)i∈ω,w̄ = (wi)i∈ω ∈ l and k ∈ ω (which depend on the choice of ū and j) such that x\uj+1 ⊆ v k ⊆ v 0 ⊆ x\uj, and uj−1 ⊆ wk ⊆ w 0 = uj (for j > 1), uj = w 0 (for j = 1). fact 3.13. the restriction of the relation ∼ (defined in 3.11) to any pbssublattice l in x is a congruence relation in l. so, a quotient lattice [l] = l/∼ is defined. lemma 3.14. let l′ be a pb-sublattice of l(x). then l = {ū = (ui)i∈ω : there exists an ū′ ∈ l′ such that ū′ = (ui,uci)i∈ω} is a pbs-sublattice in x. for every ū′ = (ui,uci)i∈ω ∈ l′ put p(ū′) = (ui)i∈ω. then p: l′ −→ l is a lattice homomorphism, l = p(l′) and the correspondence [p] : [l′] −→ [l], [ū′] −→ [p(ū′)] is a lattice isomorphism. proof. for proving that l is a pbs-sublattice in x, we need only to check that the first part in the condition (ls2) (see 3.12) is satisfied. let ū = (ui)i∈ω ∈ l and j ∈ ω. there exists an ū′ ∈ l′ such that ū′ = (ui,uci)i∈ω. by (l2) of 3.4, there exist w̄ ′ = (wi,wci)i∈ω ∈ l′ and l ∈ ω such that uj ⊆ w l ⊆ w 0 = uj+1. then uj ⊆ w l ⊆ w l+2 ⊆ uj+1 and hence x\uj ⊇ x\w l ⊇ x\w l+2 ⊇ x\uj+1. we have that x\w l+2 ⊆ wc(l+1) ⊆ x\w l+1 ⊆ wcl ⊆ x\w l. using again (l2) of 3.4, we obtain that there exist v̄ ′ = (v i,v ci)i∈ω ∈ l′ and k ∈ ω such that wc(l+1) ⊆ v k ⊆ v 0 = wcl. therefore v̄ = (v i)i∈ω ∈ l and x \uj+1 ⊆ v k ⊆ v 0 ⊆ x \uj. it is easy to see that [p] is a lattice isomorphism. � lemma 3.15. for every pbs-sublattice l in x there exists a pb-sublattice l′ of l(x) such that p(l′) = l and [p] : [l′] −→ [l] is a lattice isomorphism (see 3.14 for the notations). proof. let ū = (ui)i∈ω ∈ l. then, by (ls2) (see 3.12), for every j ∈ ω there exist v̄j = (v ij )i∈ω ∈ l and kj ∈ ω such that u j ⊆ x \v 0j ⊆ x \v kj j ⊆ u j+1. put ucj = v 0j , for every j ∈ ω. then u j ⊆ x \ ucj ⊆ uj+1, for every j ∈ ω, and hence ū′ = (ui,uci)i∈ω ∈ l(x). put l′′ = {ū′ : ū ∈ l}. then hausdorff compactifications and zero-one measures ii 7 l′′ ⊆ l(x). let l′ be the sublattice of l(x) generated by l′′. in order to show that l′ is a pb-sublattice of l(x), we need only to check that the first part of the condition (l2) (see 3.4) is satisfied. let ū = (ui)i∈ω ∈ l. then ū′ = (ui,uci)i∈ω ∈ l′′. let j ∈ ω. by the construction of ū′, we have that uc(j+1) ⊆ x \uj+1 ⊆ v kjj ⊆ v 0 j = u cj. since (v̄j)′ ∈ l′′ ⊆ l′ and kj ∈ ω, we obtain that (l2) is satisfied by the elements of l′′. from the facts that l is a lattice and l′′ generates l′, we obtain that (l2) is satisfied also by all elements of l′. so, l′ is a pb-sublattice of l(x). the construction of l′ shows that p(l′) = l. the rest follows from 3.14. � corollary 3.16. let l be a pbs-sublattice in (x,t ). then, for every element ū = (ui)i∈ω of l and for every i ∈ ω, we have that ui ∈ coz(x). hence, l0 ⊆ coz(x). proof. it follows from 3.15 and 3.5. � theorem 3.17. let (x,t ) be a tychonoff space and l be a pbs-sublattice in x. define for a,b ⊆ x: aδlb iff there exist ū = (ui)i∈ω ∈ l and k ∈ ω such that a ⊆ uk ⊆ u0 ⊆ x \b. then δl is an efremovič proximity on the topological space (x,t ). (we will say that the proximity δl is generated by the pbs-sublattice l in x.) moreover, for any proximity δ on (x,t ) there exists a pbs-sublattice l in x such that δ = δl. the set of all pbs-sublattices in x generating a proximity δ on (x,t ) has a greatest element (with respect to the inclusion), which will be denoted by lδ. proof. by lemma 3.15, there exists a pb-sublattice l′ of l(x) such that p(l′) = l. in proposition 2.12 of [5] we show that the relation δl′ generated by l′, defined in the same way as we define here the relation δl, is a proximity on the space (x,t ). hence, δl is such one, as well. this fact can be also obtained directly, modifying the proof of proposition 2.12 of [5]. if (cx,c) is a compactification of x then the family f = {f : x −→ [0, 1] : f has a continuous extension to cx} generates (cx,c). the pb-sublattice lf of l(x), constructed in example 2.4 of [5], has the property that δlf = δc (see theorem 3.1(a) of [5]). by lemma 3.14, the lattice l = p(lf) is a pbs-sublattice in x. obviously, δlf = δl. hence, by theorem 2.2, for any proximity δ on (x,t ) there exists a pbs-sublattice l in x such that δ = δl. finally, one easily infer from proposition 2.11 of [5] and our lemmas 3.14 and 3.15 that the set of all pbs-sublattices in x generating a proximity δ on (x,t ) has a greatest element (with respect to the inclusion). � theorem 3.18. let (x,t ) be a tychonoff space and l be a pbs-sublattice in x. put, for every x ∈ x, ix = {ū ∈ l : x 6∈ u0}. then: (a) π(ix) = {[ū] : ū ∈ ix} ∈ max([l]) and the map el : (x,t ) −→ max([l]), defined by the formula el(x) = π(ix), is a dense embedding; (b) (max([l]),el) is a hausdorff compactification of x, equivalent to the smirnoff compactification (cδlx,cδl) (see 3.17 for δl and 2.2 for (cδlx,cδl)). 8 g. d. dimov and g. tironi hence, the set k(x) of all, up to equivalence, hausdorff compactifications of x is represented by the set {(max([lδ]),elδ ) : δ ∈ pt (x)}. moreover, the following is true: (cδ1x,cδ1 ) ≤ (cδ2x,cδ2 ) iff lδ1 ⊆ lδ2 (see 3.17 for lδ). therefore, putting bx = [s(x)] and lx = {[lδ] : δ ∈ pt (x)}, we obtain a new (simpler) solution of our problem 1. proof. in [6] the pb-sublattice version of this theorem (i.e., the version obtained by substituting everywhere in the theorem “pbs-” with “pb-”) was proved (see theorem 3.8 there). now our result follows from it, from lemmas 3.14, 3.15 and theorem 3.17 proved above, and from 2.17, 2.13 of [5]. � definition 3.19. let b be a boolean algebra and l be a sublattice of b. a measure µ, defined on the boolean algebra b(l), is called u-regular measure (or u-l-regular measure) if µ(x) = inf{µ(a) : a ∈ l,x ≤ a} for any x ∈ b(l). the set of all u-l-regular zero-one measures on the boolean algebra b(l) will be denoted by iur(l). the following lemma is essentialy known (see [1], theorem 2.1): lemma 3.20. let b be a boolean algebra and l be a sublattice of b. then there exists a bijection between the sets max(l) and iur(l). lemma 3.21. let (x,t ) be a tychonoff space and l be a pbs-sublattice in x. then [l] is a sublattice of [s(x)] (see 3.11 and 3.9 for the notations). for every [ū] ∈ [l] put [ū]∗ = {µ ∈ iur([l]) : µ([ū]) = 1}. then the family b∗ = {[ū]∗ : [ū] ∈ [l]} is a base of a topology t ∗ on the set iur([l]). if δ is the proximity on (x,t ) generated by l (see 3.17), then for every x ∈ x and every [ū] = [(ui)i∈ω] ∈ b([l]) put: µx([ū]) = { 0 if there exists an i0 ∈ ω such that xδui for every i ≥ i0, and 1 if there exists an j0 ∈ ω such that xδ(x \uj) for every j ≥ j0. then, for every x ∈ x, µx is a well-defined zero-one u-[l]-regular measure on the boolean subalgebra b([l]) of the complete boolean algebra [s(x)] and the mapping ml : (x,t ) −→ (iur([l]),t ∗), defined by the formula ml(x) = µx, is a dense embedding. ((iur([l]),t ∗),ml) is a hausdorff compactification of (x,t ) equivalent to the compactification (max([l]),el) of (x,t ) (and, hence, to the smirnoff compactification (cδx,cδ)). the map φ : (iur([l]),t ∗) −→ max([l]), defined by the formula φ(µ) = µ−1(0) ∩ [l], carries out this equivalence. proof. in [6] the pb-sublattice version of this lemma was proved (see lemma 3.16 there). our result follows from it and from lemma 3.15 proved above. � theorem 3.22 (the main theorem). let (x,t ) be a tychonoff space. then for every hausdorff compactification (cx,c) of x there exists a sublattice [l] of the complete boolean algebra [s(x)] (where l is a pbs-sublattice in x) such that (max([l]),el) (see 3.18 for the definition of the map el) and hausdorff compactifications and zero-one measures ii 9 ((iur([l]),t ∗),ml) (see 3.19 and 3.21 for the notations) are hausdorff compactification of x equivalent to the compactification (cx,c) of x. proof. let (cx,c) be a hausdorff compactification of (x,t ). then, by theorem 3.17, there exists a pbs-sublattice l in x such that δl = δc (see 2.2 and 3.17 for the notations). now, theorem 3.18, lemma 3.21 and theorem 2.2 imply that (max([l]),el) and ((iur([l]),t ∗),ml) are hausdorff compactification of x equivalent to the compactification (cx,c) of x. � corollary 3.23. let (x,t ) be a tychonoff space. put ma(x) = {(max([lδ]),elδ ) : δ ∈ pt (x)} and me(x) = {((iur([lδ]),t ∗),mlδ ) : δ ∈ pt (x)}. order these sets putting for every δ1,δ2 ∈ pt (x), max([lδ1 ]) ≤ max([lδ2 ]) iff δ1 ≤ δ2, and iur([lδ1 ]) ≤ iur([lδ2 ]) iff δ1 ≤ δ2. then the ordered sets (ma(x),≤) and (me(x),≤) are isomorphic to the ordered set (k(x),≤) of all, up to equivalence, hausdorff compactifications of x. in the next proposition, the o. nj̊astad’s characterization of wallman compactifications by means of proximities (see 2.4) is restated in the language of pbs-sublattices. proposition 3.24. let (x,t ) be a tychonoff space and l be a pbs-sublattice in x. then (max([l]),el) is a wallman compactification of x iff there exists a family b, consisting of open subsets of x, such that (i) b is closed under finite unions; (ii) if a,b ∈ b and a ∪ b = x then there exist ū = (ui)i∈ω ∈ l and j ∈ ω with x \a ⊆ uj ⊆ u0 ⊆ b; (iii) if ū = (ui)i∈ω ∈ l and j ∈ ω then there exist a,b ∈ b such that uj ⊆ x \a ⊆ b ⊆ u0. proof. the proximity generated by the compactification (max([l]),el) is exactly the proximity δl (see theorem 3.18(b)). hence, by theorem 2.4, (max([l]),el) is a wallman compactification of x if and only if there exists a subfamily b of t which is closed under finite unions and satisfies the conditions (b1) and (b2). since our proximity δl is generated by l, these conditions can be rewritten now as follows: (b1l) if u,v ∈ b and u ∪ v = x then there exist ū = (ui)i∈ω ∈ l and j ∈ ω such that x \u ⊆ uj ⊆ u0 ⊆ v ; (b2l) if a,b ⊆ x and there exist ū = (ui)i∈ω ∈ l and j ∈ ω such that a ⊆ uj ⊆ u0 ⊆ x \ b then there are v,w ∈ b with a ⊆ x \ v , b ⊆ x \w and v ∪w = x. 10 g. d. dimov and g. tironi obviously, condition (b1l) coincides with condition (ii) of our proposition and condition (i) is also satisfied. since for every ū = (ui)i∈ω ∈ l and j ∈ ω we have that ujδl(x \ u0), condition (b2l) is equivalent to the condition (iii). this completes the proof. � now we will give a sufficient condition for (max([l]),el) to be a wallman compactification: proposition 3.25. let (x,t ) be a tychonoff space and l be a pbs-sublattice in x. if l satisfies the following condition: (wa) if ū, v̄ ∈ l and u0 ∪v 0 = x then there exist w̄ = (wi)i∈ω ∈ l and j ∈ ω such that x \u0 ⊆ wj ⊆ w 0 ⊆ v 0, then (max([l]),el) is a wallman compactification of x. in fact, we have that (max([l]),el) is equivalent to the wallman compactification (max(l0),ηl0 ) (see 2.3 and 3.12 for the notations). proof. let us recall that o. nj̊astad ([12]) proved that if (x,δ) is a proximity space then a subfamily b of the topology tδ, generated by the proximity δ, is a normal wallman base of (x,tδ) if it is a ring of sets and satisfies the conditions (b1) and (b2) from 2.4; moreover, he showed that (max(b),ηb) and (cδx,cδ) are equivalent compactifications of (x,tδ) (see 2.2 and 2.3 for the notations). by theorem 3.18(b), we have that (max([l]),el) and (cδlx,cδl) are equivalent compactifications of (x,t ). obviously, l0(= {u0 = ⋃ {ui : i ∈ ω} : (ui)i∈ω ∈ l}) is a ring of open sets in (x,t ) and t = tδl (see theorem 3.17). so, in order to prove our proposition, it is enough to show that the family l0 satisfies the conditions (b1) and (b2) from 2.4. the condition (wa) says that if u0,v 0 ∈ l0 and u0 ∪ v 0 = x then (x \ u0)δl(x \ v 0). hence (b1) is satisfied. for proving (b2), let a,b ⊆ x and aδlb. then, by the definition of δl, there exist ū = (ui)i∈ω ∈ l and j ∈ ω such that a ⊆ uj ⊆ u0 ⊆ x \ b. since l is a pbs-sublattice in x, we obtain (by the condition (ls2) from definition 3.12) that there exist v̄ = (v i)i∈ω ∈ l and k ∈ ω with uj ⊆ x \ v 0 ⊆ x \ v k ⊆ uj+1 ⊆ u0. hence a ⊆ x \ v 0, b ⊆ x \u0, u0 ∪v 0 = x and u0,v 0 ∈ l0. therefore, l0 satisfies (b2). the proof of our proposition is completed. � references [1] g. bachman and r. cohen, regular lattice measures and repleteness, comm. on pure and applied math. 26 (1973), 587–599. [2] h. brandenburg and a. mysior, short proof of an internal characterization of complete regularity, canad. math. bull. 27 (4) (1984), 461–462. [3] g. bachman and p. stratigos, on the relation of lattice repleteness and c-real compactness, indian j. pure and applied math. 20 (11) (1989), 1043–1066. [4] r. cohen, lattice measures and topologies, ann. mat. pura appl. 109 (1976), 147–164. [5] g. dimov and g. tironi, compactifications, a-compactifications and proximities, ann. mat. pura appl. 169 (1995), 87–108. [6] g. dimov and g. tironi, hausdorff compactifications and zero-one measures, math. proc. camb. phil. soc. (2001) (to appear). [7] r. engelking, general topology, (pwn, warszawa, 1977). hausdorff compactifications and zero-one measures ii 11 [8] o. frink, compactifications and semi-normal spaces, amer. j. math. 86 (1964), 602– 607. [9] p. t. johnstone, stone spaces, cambridge studies in advanced math. no. 3 (cambridge university press, 1982). [10] j. kerstan, eine charakterisierung der vollständig regulären räume, math. nachrichten 17 (1958), 27–46. [11] s. a. naimpally and b. d. warrack, proximity spaces, (cambridge univ. press, 1970). [12] o. nj̊astad, on wallman-type compactifications, math. zeitschr 91 (1966), 267-276. [13] y. m. smirnoff, on proximity spaces, mat. sb. (russian) 31 (1952), 543-574; am. math. soc. transl. (english translation) 38 ser. 2, 5–35. [14] a. sultan, measure, compactification and representation, can. j. math. 30 (1) (1978), 54–65. [15] v. m. ul′janov, solution of a basic problem on compactifications of wallman type, soviet math. dokl. 18 (1977), 567–571. [16] h. wallman, lattices and topological spaces, annals of math. 39 (1938), 112–126. received february 2001 revised may 2001 georgi d. dimov department of mathematics and computer science university of sofia blvd. j. bourchier 5 1126 sofia, bulgaria e-mail address : gdimov@fmi.uni-sofia.bg gino tironi department of mathematical sciences university of trieste via a. valerio 12/1 34127 trieste, italy e-mail address : tironi@univ.trieste.it @ �� !#"%$'&'� ( )*(,+�-*. � /10" � � � � )2( "�34)*. 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